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[Axiom-developer] 20080420.01.tpd.patch (CATS Schaums-Axiom equivalence
|
From: |
daly |
|
Subject: |
[Axiom-developer] 20080420.01.tpd.patch (CATS Schaums-Axiom equivalence testing (13)) |
|
Date: |
Mon, 21 Apr 2008 02:21:56 -0500 |
Clearly I've exceeded my ability to simplify these expressions, or
Axiom gets the wrong answers, or Schaums has the wrong answers.
Personally, I'm betting on the first case.
schaum13.input.pamphlet
14:280 Axiom cannot simplify these expressions
14:281 Axiom cannot simplify these expressions
14:282 Axiom cannot simplify these expressions
14:283 Axiom cannot simplify these expressions
14:284 Axiom cannot simplify these expressions
14:285 Axiom cannot simplify these expressions
14:286 Axiom cannot simplify these expressions
14:287 Axiom cannot simplify these expressions
14:288 Axiom cannot simplify these expressions
14:289 Axiom cannot simplify these expressions
14:290 Axiom cannot simplify these expressions
14:291 Axiom cannot simplify these expressions
14:292 Axiom cannot simplify these expressions
14:293 Axiom cannot simplify these expressions
14:284 Axiom cannot simplify these expressions
14:295 Axiom cannot compute this integral
14:296 Axiom cannot compute this integral
14:297 Axiom cannot compute this integral
14:298 Axiom cannot compute this integral
========================================================================
diff --git a/changelog b/changelog
index d61af40..29750e1 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,4 @@
+20080420 tpd src/input/schaum13.input show Schaums-Axiom equivalence
20080419 tpd src/input/schaum12.input show Schaums-Axiom equivalence
20080418 tpd src/input/schaum11.input show Schaums-Axiom equivalence
20080418 tpd src/input/schaum10.input fix Schaums-Axiom equivalence
diff --git a/src/input/schaum13.input.pamphlet
b/src/input/schaum13.input.pamphlet
index eb1a6eb..c4a0351 100644
--- a/src/input/schaum13.input.pamphlet
+++ b/src/input/schaum13.input.pamphlet
@@ -29,7 +29,7 @@ $$
)set message auto off
)clear all
---S 1 of 19
+--S 1
aa:=integrate(1/sqrt(a*x^2+b*x+c),x)
--R
--R
@@ -60,6 +60,183 @@ aa:=integrate(1/sqrt(a*x^2+b*x+c),x)
--R \|- a
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 2
+bb1:=1/sqrt(a)*log(2*sqrt(a)*sqrt(a*x^2+b*x+c)*2*a*x+b)
+--R
+--R +--------------+
+--R +-+ | 2
+--R log(4a x\|a \|a x + b x + c + b)
+--R (2) ----------------------------------
+--R +-+
+--R \|a
+--R Type: Expression
Integer
+--E
+
+--S 3
+bb2:=-1/sqrt(-a)*asin((2*a*x+b)/sqrt(b^2-4*a*c))
+--R
+--R 2a x + b
+--R asin(--------------)
+--R +-----------+
+--R | 2
+--R \|- 4a c + b
+--R (3) - --------------------
+--R +---+
+--R \|- a
+--R Type: Expression
Integer
+--E
+
+--S 4
+bb3:=1/sqrt(a)*asinh((2*a*x+b)/sqrt(4*a*c-b^2))
+--R
+--R 2a x + b
+--R asinh(------------)
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R (4) -------------------
+--R +-+
+--R \|a
+--R Type: Expression
Integer
+--E
+
+--S 5
+cc1:=bb1-aa.1
+--R
+--R (5)
+--R +--------------+
+--R +-+ | 2
+--R log(4a x\|a \|a x + b x + c + b)
+--R +
+--R -
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R /
+--R +-+
+--R \|a
+--R Type: Expression
Integer
+--E
+
+--S 6
+cc2:=bb1-aa.2
+--R
+--R (6)
+--R +--------------+
+--R +---+ +-+ | 2
+--R \|- a log(4a x\|a \|a x + b x + c + b)
+--R +
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R +-+ \|- a \|a x + b x + c - \|- a \|c
+--R - 2\|a atan(------------------------------------)
+--R a x
+--R /
+--R +---+ +-+
+--R \|- a \|a
+--R Type: Expression
Integer
+--E
+
+--S 7
+cc3:=bb2-aa.1
+--R
+--R (7)
+--R -
+--R +---+
+--R \|- a
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +-+ 2a x + b
+--R - \|a asin(--------------)
+--R +-----------+
+--R | 2
+--R \|- 4a c + b
+--R /
+--R +---+ +-+
+--R \|- a \|a
+--R Type: Expression
Integer
+--E
+
+--S 8
+cc4:=bb2-aa.2
+--R
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c 2a x + b
+--R - 2atan(------------------------------------) - asin(--------------)
+--R a x +-----------+
+--R | 2
+--R \|- 4a c + b
+--R (8) --------------------------------------------------------------------
+--R +---+
+--R \|- a
+--R Type: Expression
Integer
+--E
+
+--S 9
+cc5:=bb3-aa.1
+--R
+--R (9)
+--R -
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R 2a x + b
+--R asinh(------------)
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R /
+--R +-+
+--R \|a
+--R Type: Expression
Integer
+--E
+
+--S 10 14:280 Axiom cannot simplify these expressions
+cc6:=bb3-aa.2
+--R
+--R (10)
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R +-+ \|- a \|a x + b x + c - \|- a \|c +---+ 2a x +
b
+--R - 2\|a atan(------------------------------------) + \|- a
asinh(------------)
+--R a x
+---------+
+--R |
2
+--R \|4a c -
b
+--R
-----------------------------------------------------------------------------
+--R +---+ +-+
+--R \|- a \|a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.281~~~~~$\displaystyle
@@ -70,7 +247,7 @@ $$
<<*>>=
)clear all
---S 2 of 19
+--S 11
aa:=integrate(x/sqrt(a*x^2+b*x+c),x)
--R
--R
@@ -121,6 +298,220 @@ aa:=integrate(x/sqrt(a*x^2+b*x+c),x)
--R ]
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 12
+t1:=integrate(1/sqrt(a*x^2+b*x+c),x)
+--R
+--R (2)
+--R [
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R /
+--R +-+
+--R \|a
+--R ,
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R 2atan(------------------------------------)
+--R a x
+--R -------------------------------------------]
+--R +---+
+--R \|- a
+--R Type: Union(List Expression
Integer,...)
+--E
+
+--S 13
+bb1:=sqrt(a*x^2+b*x+c)/a-b/(2*a)*t1.1
+--R
+--R (3)
+--R -
+--R b
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +--------------+
+--R +-+ | 2
+--R 2\|a \|a x + b x + c
+--R /
+--R +-+
+--R 2a\|a
+--R Type: Expression
Integer
+--E
+
+--S 14
+bb2:=sqrt(a*x^2+b*x+c)/a-b/(2*a)*t1.2
+--R
+--R (4)
+--R +--------------+
+--R +---+ | 2 +---+ +-+ +--------------+
+--R \|- a \|a x + b x + c - \|- a \|c +---+ | 2
+--R - b atan(------------------------------------) + \|- a \|a x + b x + c
+--R a x
+--R ------------------------------------------------------------------------
+--R +---+
+--R a\|- a
+--R Type: Expression
Integer
+--E
+
+--S 15
+cc1:=bb1-aa.1
+--R
+--R (5)
+--R +--------------+
+--R +-+ | 2 2
+--R (- 2b\|c \|a x + b x + c + b x + 2b c)
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +--------------+
+--R +-+ | 2 2
+--R (- 2b\|c \|a x + b x + c + b x + 2b c)
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +--------------+
+--R +-+ | 2 +-+ +-+
+--R - 4c\|a \|a x + b x + c + (2b x + 4c)\|a \|c
+--R /
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R 4a\|a \|c \|a x + b x + c + (- 2a b x - 4a c)\|a
+--R Type: Expression
Integer
+--E
+
+--S 16
+cc2:=bb1-aa.2
+--R
+--R (6)
+--R +--------------+
+--R +---+ +-+ | 2 2 +---+
+--R (- 2b\|- a \|c \|a x + b x + c + (b x + 2b c)\|- a )
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +--------------+
+--R +-+ +-+ | 2 2 +-+
+--R (4b\|a \|c \|a x + b x + c + (- 2b x - 4b c)\|a )
+--R *
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R atan(------------------------------------)
+--R a x
+--R +
+--R +--------------+
+--R +---+ +-+ | 2 +---+ +-+ +-+
+--R - 4c\|- a \|a \|a x + b x + c + (2b x + 4c)\|- a \|a \|c
+--R /
+--R +--------------+
+--R +---+ +-+ +-+ | 2 +---+ +-+
+--R 4a\|- a \|a \|c \|a x + b x + c + (- 2a b x - 4a c)\|- a \|a
+--R Type: Expression
Integer
+--E
+
+--S 17
+cc3:=bb2-aa.1
+--R
+--R (7)
+--R +--------------+
+--R +---+ +-+ | 2 2 +---+
+--R (- 2b\|- a \|c \|a x + b x + c + (b x + 2b c)\|- a )
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +--------------+
+--R +-+ +-+ | 2 2 +-+
+--R (- 4b\|a \|c \|a x + b x + c + (2b x + 4b c)\|a )
+--R *
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R atan(------------------------------------)
+--R a x
+--R +
+--R +--------------+
+--R +---+ +-+ | 2 +---+ +-+ +-+
+--R - 4c\|- a \|a \|a x + b x + c + (2b x + 4c)\|- a \|a \|c
+--R /
+--R +--------------+
+--R +---+ +-+ +-+ | 2 +---+ +-+
+--R 4a\|- a \|a \|c \|a x + b x + c + (- 2a b x - 4a c)\|- a \|a
+--R Type: Expression
Integer
+--E
+
+--S 18 14:281 Axiom cannot simplify these expressions
+cc4:=bb2-aa.2
+--R
+--R +--------------+
+--R | 2 +-+
+--R - 2c\|a x + b x + c + (b x + 2c)\|c
+--R (8) --------------------------------------
+--R +--------------+
+--R +-+ | 2
+--R 2a\|c \|a x + b x + c - a b x - 2a c
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.282~~~~~$\displaystyle
@@ -132,7 +523,7 @@ $$
<<*>>=
)clear all
---S 3 of 19
+--S 19
aa:=integrate(x^2/(a*x^2+b*x+c),x)
--R
--R
@@ -177,7 +568,247 @@ aa:=integrate(x^2/(a*x^2+b*x+c),x)
--R 2a \|4a c - b
--R ]
--R Type: Union(List Expression
Integer,...)
---E
+--E
+
+--S 20
+t1:=integrate(1/sqrt(a*x^2+b*x+c),x)
+--R
+--R (2)
+--R [
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R /
+--R +-+
+--R \|a
+--R ,
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R 2atan(------------------------------------)
+--R a x
+--R -------------------------------------------]
+--R +---+
+--R \|- a
+--R Type: Union(List Expression
Integer,...)
+--E
+
+--S 21
+bb1:=(2*a*x-3*b)/(4*a^2)*sqrt(a*x^2+b*x+c)+(3*b^2-4*a*c)/(8*a^2)*t1.1
+--R
+--R (3)
+--R 2
+--R (- 4a c + 3b )
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +--------------+
+--R +-+ | 2
+--R (4a x - 6b)\|a \|a x + b x + c
+--R /
+--R 2 +-+
+--R 8a \|a
+--R Type: Expression
Integer
+--E
+
+--S 22
+bb2:=(2*a*x-3*b)/(4*a^2)*sqrt(a*x^2+b*x+c)+(3*b^2-4*a*c)/(8*a^2)*t1.2
+--R
+--R (4)
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R 2 \|- a \|a x + b x + c - \|- a \|c
+--R (- 4a c + 3b )atan(------------------------------------)
+--R a x
+--R +
+--R +--------------+
+--R +---+ | 2
+--R (2a x - 3b)\|- a \|a x + b x + c
+--R /
+--R 2 +---+
+--R 4a \|- a
+--R Type: Expression
Integer
+--E
+
+--S 23
+cc1:=aa.1-bb1
+--R
+--R (5)
+--R +-----------+
+--R 2 | 2
+--R (4a c - 3b )\|- 4a c + b
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R 2 +-+
+--R (8a c - 4b )\|a
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b
)x
+--R +
+--R 3
+--R - 4a b c + b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +-----------+ +--------------+
+--R | 2 +-+ | 2
+--R (- 4a x + 6b)\|- 4a c + b \|a \|a x + b x + c
+--R +
+--R +-----------+
+--R 2 | 2 +-+
+--R (- 4b log(a x + b x + c) + 8a x)\|- 4a c + b \|a
+--R /
+--R +-----------+
+--R 2 | 2 +-+
+--R 8a \|- 4a c + b \|a
+--R Type: Expression
Integer
+--E
+
+--S 24
+cc2:=aa.2-bb1
+--R
+--R (6)
+--R +---------+
+--R 2 | 2
+--R (4a c - 3b )\|4a c - b
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +---------+
+--R | 2
+--R 2 +-+ (2a x + b)\|4a c - b
+--R (- 16a c + 8b )\|a atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R +---------+ +--------------+
+--R +-+ | 2 | 2
+--R (- 4a x + 6b)\|a \|4a c - b \|a x + b x + c
+--R +
+--R +---------+
+--R 2 +-+ | 2
+--R (- 4b log(a x + b x + c) + 8a x)\|a \|4a c - b
+--R /
+--R +---------+
+--R 2 +-+ | 2
+--R 8a \|a \|4a c - b
+--R Type: Expression
Integer
+--E
+
+--S 25
+cc3:=aa.2-bb1
+--R
+--R (7)
+--R +---------+
+--R 2 | 2
+--R (4a c - 3b )\|4a c - b
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +---------+
+--R | 2
+--R 2 +-+ (2a x + b)\|4a c - b
+--R (- 16a c + 8b )\|a atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R +---------+ +--------------+
+--R +-+ | 2 | 2
+--R (- 4a x + 6b)\|a \|4a c - b \|a x + b x + c
+--R +
+--R +---------+
+--R 2 +-+ | 2
+--R (- 4b log(a x + b x + c) + 8a x)\|a \|4a c - b
+--R /
+--R +---------+
+--R 2 +-+ | 2
+--R 8a \|a \|4a c - b
+--R Type: Expression
Integer
+--E
+
+--S 26 14:282 Axiom cannot simplify these expressions
+cc4:=aa.2-bb2
+--R
+--R (8)
+--R +--------------+
+--R +---------+ +---+ | 2 +---+ +-+
+--R 2 | 2 \|- a \|a x + b x + c - \|- a \|c
+--R (4a c - 3b )\|4a c - b atan(------------------------------------)
+--R a x
+--R +
+--R +---------+
+--R | 2
+--R 2 +---+ (2a x + b)\|4a c - b
+--R (- 8a c + 4b )\|- a atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R +---------+ +--------------+
+--R +---+ | 2 | 2
+--R (- 2a x + 3b)\|- a \|4a c - b \|a x + b x + c
+--R +
+--R +---------+
+--R 2 +---+ | 2
+--R (- 2b log(a x + b x + c) + 4a x)\|- a \|4a c - b
+--R /
+--R +---------+
+--R 2 +---+ | 2
+--R 4a \|- a \|4a c - b
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.283~~~~~$\displaystyle
@@ -200,7 +831,7 @@ $$
<<*>>=
)clear all
---S 4 of 19
+--S 27
aa:=integrate(1/x*(a*x^2+b*x+c),x)
--R
--R
@@ -210,6 +841,91 @@ aa:=integrate(1/x*(a*x^2+b*x+c),x)
--R 2
--R Type: Union(Expression
Integer,...)
--E
+
+--S 28
+bb1:=-1/sqrt(c)*log((2*sqrt(c)*sqrt(a*x^2+b*x+c)+b*x+2*c)/x)
+--R
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c + b x + 2c
+--R log(---------------------------------)
+--R x
+--R (2) - --------------------------------------
+--R +-+
+--R \|c
+--R Type: Expression
Integer
+--E
+
+--S 29
+bb2:=1/sqrt(-c)*asin((b*x+2*c)/(x*sqrt(b^2-4*a*c)))
+--R
+--R b x + 2c
+--R asin(---------------)
+--R +-----------+
+--R | 2
+--R x\|- 4a c + b
+--R (3) ---------------------
+--R +---+
+--R \|- c
+--R Type: Expression
Integer
+--E
+
+--S 30
+bb3:=-1/sqrt(c)*asinh((b*x+2*c)/(x*sqrt(b^2-4*a*c)))
+--R
+--R b x + 2c
+--R asinh(---------------)
+--R +-----------+
+--R | 2
+--R x\|- 4a c + b
+--R (4) - ----------------------
+--R +-+
+--R \|c
+--R Type: Expression
Integer
+--E
+
+--S 31
+cc1:=aa-bb1
+--R
+--R (5)
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c + b x + 2c 2 +-+
+--R 2log(---------------------------------) + (2c log(x) + a x + 2b x)\|c
+--R x
+--R -----------------------------------------------------------------------
+--R +-+
+--R 2\|c
+--R Type: Expression
Integer
+--E
+
+--S 32
+cc2:=aa-bb2
+--R
+--R b x + 2c 2 +---+
+--R - 2asin(---------------) + (2c log(x) + a x + 2b x)\|- c
+--R +-----------+
+--R | 2
+--R x\|- 4a c + b
+--R (6) ----------------------------------------------------------
+--R +---+
+--R 2\|- c
+--R Type: Expression
Integer
+--E
+
+--S 33 14:283 Axiom cannot simplify these expressions
+cc3:=aa-bb3
+--R
+--R b x + 2c 2 +-+
+--R 2asinh(---------------) + (2c log(x) + a x + 2b x)\|c
+--R +-----------+
+--R | 2
+--R x\|- 4a c + b
+--R (7) -------------------------------------------------------
+--R +-+
+--R 2\|c
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.284~~~~~$\displaystyle
@@ -220,7 +936,7 @@ $$
<<*>>=
)clear all
---S 5 of 19
+--S 34
aa:=integrate(1/(x^2*(a*x^2+b*x+c)),x)
--R
--R
@@ -266,6 +982,93 @@ aa:=integrate(1/(x^2*(a*x^2+b*x+c)),x)
--R ]
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 35
+t1:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x)
+--R
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R log(---------------------------------)
+--R x
+--R (2) --------------------------------------
+--R +-+
+--R \|c
+--R Type: Union(Expression
Integer,...)
+--E
+
+--S 36
+bb:=-sqrt(a*x^2+b*x+c)/(c*x)-b/(2*x)*t1
+--R
+--R +--------------+
+--R +-+ | 2
+--------------+
+--R 2\|c \|a x + b x + c - b x - 2c +-+ | 2
+--R - b c log(---------------------------------) - 2\|c \|a x + b x + c
+--R x
+--R (3)
---------------------------------------------------------------------
+--R +-+
+--R 2c x\|c
+--R Type: Expression
Integer
+--E
+
+--S 37
+cc1:=aa.1-bb1
+--R
+--R (4)
+--R 2
+--R (2a c - b )x
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b
)x
+--R +
+--R 3
+--R - 4a b c + b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +-----------+
+--R 2 2 | 2
+--R (b x log(a x + b x + c) - 2b x log(x) - 2bb1 c x - 2c)\|- 4a c + b
+--R /
+--R +-----------+
+--R 2 | 2
+--R 2c x\|- 4a c + b
+--R Type: Expression
Integer
+--E
+
+--S 38 14:284 Axiom cannot simplify these expressions
+cc2:=aa.2-bb
+--R
+--R (5)
+--R +--------------+
+--R +---------+ +-+ | 2
+--R 2 | 2 2\|c \|a x + b x + c - b x - 2c
+--R b c \|4a c - b log(---------------------------------)
+--R x
+--R +
+--R +---------+
+--R | 2
+--R 2 +-+ (2a x + b)\|4a c - b
+--R (- 4a c + 2b )x\|c atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R +---------+ +--------------+
+--R +-+ | 2 | 2
+--R 2c\|c \|4a c - b \|a x + b x + c
+--R +
+--R +---------+
+--R 2 +-+ | 2
+--R (b x log(a x + b x + c) - 2b x log(x) - 2c)\|c \|4a c - b
+--R /
+--R +---------+
+--R 2 +-+ | 2
+--R 2c x\|c \|4a c - b
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.285~~~~~$\displaystyle
@@ -277,7 +1080,7 @@ $$
<<*>>=
)clear all
---S 6 of 19
+--S 39
aa:=integrate(sqrt(a*x^2+b*x+c),x)
--R
--R
@@ -361,6 +1164,228 @@ aa:=integrate(sqrt(a*x^2+b*x+c),x)
--R ]
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 40
+t1:=integrate(1/sqrt(a*x^2+b*x+c),x)
+--R
+--R (2)
+--R [
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R /
+--R +-+
+--R \|a
+--R ,
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R 2atan(------------------------------------)
+--R a x
+--R -------------------------------------------]
+--R +---+
+--R \|- a
+--R Type: Union(List Expression
Integer,...)
+--E
+
+--S 41
+bb1:=((2*a*x+b)*sqrt(a*x^2+b*x+c))/(4*a)+(4*a*c-b^2)/(8*a)*t1.1
+--R
+--R (3)
+--R 2
+--R (4a c - b )
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +--------------+
+--R +-+ | 2
+--R (4a x + 2b)\|a \|a x + b x + c
+--R /
+--R +-+
+--R 8a\|a
+--R Type: Expression
Integer
+--E
+
+--S 42
+bb2:=((2*a*x+b)*sqrt(a*x^2+b*x+c))/(4*a)+(4*a*c-b^2)/(8*a)*t1.2
+--R
+--R (4)
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R 2 \|- a \|a x + b x + c - \|- a \|c
+--R (4a c - b )atan(------------------------------------)
+--R a x
+--R +
+--R +--------------+
+--R +---+ | 2
+--R (2a x + b)\|- a \|a x + b x + c
+--R /
+--R +---+
+--R 4a\|- a
+--R Type: Expression
Integer
+--E
+
+--S 43
+cc1:=aa.1-bb1
+--R
+--R (5)
+--R +--------------+
+--R 2 2 | 2
+--R (4b c x + 8b c )\|a x + b x + c
+--R +
+--R 3 2 2 2 +-+
+--R ((- 4a b c - b )x - 8b c x - 8b c )\|c
+--R /
+--R +--------------+
+--R +-+ | 2 2 2 2
+--R (16a b x + 32a c)\|c \|a x + b x + c + (- 16a c - 4a b )x - 32a b
c x
+--R +
+--R 2
+--R - 32a c
+--R Type: Expression
Integer
+--E
+
+--S 44
+cc2:=aa.2-bb1
+--R
+--R (6)
+--R +--------------+
+--R 3 2 2 +---+ +-+ | 2
+--R ((- 16a b c + 4b )x - 32a c + 8b c)\|- a \|c \|a x + b x + c
+--R +
+--R 2 2 4 2 2 3 3 2 2 +---+
+--R ((16a c - b )x + (32a b c - 8b c)x + 32a c - 8b c )\|- a
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +--------------+
+--R 3 2 2 +-+ +-+ | 2
+--R ((32a b c - 8b )x + 64a c - 16b c)\|a \|c \|a x + b x + c
+--R +
+--R 2 2 4 2 2 3 3 2 2
+-+
+--R ((- 32a c + 2b )x + (- 64a b c + 16b c)x - 64a c + 16b c )\|a
+--R *
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R atan(------------------------------------)
+--R a x
+--R +
+--R +--------------+
+--R 2 2 +---+ +-+ | 2
+--R (8b c x + 16b c )\|- a \|a \|a x + b x + c
+--R +
+--R 3 2 2 2 +---+ +-+ +-+
+--R ((- 8a b c - 2b )x - 16b c x - 16b c )\|- a \|a \|c
+--R /
+--R +--------------+
+--R +---+ +-+ +-+ | 2
+--R (32a b x + 64a c)\|- a \|a \|c \|a x + b x + c
+--R +
+--R 2 2 2 2 +---+ +-+
+--R ((- 32a c - 8a b )x - 64a b c x - 64a c )\|- a \|a
+--R Type: Expression
Integer
+--E
+
+--S 45
+cc3:=aa.1-bb2
+--R
+--R (7)
+--R +--------------+
+--R 3 2 2 +---+ +-+ | 2
+--R ((16a b c - 4b )x + 32a c - 8b c)\|- a \|c \|a x + b x + c
+--R +
+--R 2 2 4 2 2 3 3 2 2 +---+
+--R ((- 16a c + b )x + (- 32a b c + 8b c)x - 32a c + 8b c )\|- a
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +--------------+
+--R 3 2 2 +-+ +-+ | 2
+--R ((- 32a b c + 8b )x - 64a c + 16b c)\|a \|c \|a x + b x + c
+--R +
+--R 2 2 4 2 2 3 3 2 2 +-+
+--R ((32a c - 2b )x + (64a b c - 16b c)x + 64a c - 16b c )\|a
+--R *
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R atan(------------------------------------)
+--R a x
+--R +
+--R +--------------+
+--R 2 2 +---+ +-+ | 2
+--R (8b c x + 16b c )\|- a \|a \|a x + b x + c
+--R +
+--R 3 2 2 2 +---+ +-+ +-+
+--R ((- 8a b c - 2b )x - 16b c x - 16b c )\|- a \|a \|c
+--R /
+--R +--------------+
+--R +---+ +-+ +-+ | 2
+--R (32a b x + 64a c)\|- a \|a \|c \|a x + b x + c
+--R +
+--R 2 2 2 2 +---+ +-+
+--R ((- 32a c - 8a b )x - 64a b c x - 64a c )\|- a \|a
+--R Type: Expression
Integer
+--E
+
+--S 46 14:285 Axiom cannot simplify these expressions
+cc4:=aa.2-bb2
+--R
+--R (8)
+--R +--------------+
+--R 2 2 | 2
+--R (4b c x + 8b c )\|a x + b x + c
+--R +
+--R 3 2 2 2 +-+
+--R ((- 4a b c - b )x - 8b c x - 8b c )\|c
+--R /
+--R +--------------+
+--R +-+ | 2 2 2 2
+--R (16a b x + 32a c)\|c \|a x + b x + c + (- 16a c - 4a b )x - 32a b
c x
+--R +
+--R 2
+--R - 32a c
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.286~~~~~$\displaystyle
@@ -378,7 +1403,7 @@ $$
<<*>>=
)clear all
---S 7 of 19
+--S 47
aa:=integrate(x*sqrt(a*x^2+b*x+c),x)
--R
--R
@@ -522,6 +1547,416 @@ aa:=integrate(x*sqrt(a*x^2+b*x+c),x)
--R ]
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 48
+t1:=integrate(1/sqrt(a*x^2+b*x+c),x)
+--R
+--R (2)
+--R [
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R /
+--R +-+
+--R \|a
+--R ,
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R 2atan(------------------------------------)
+--R a x
+--R -------------------------------------------]
+--R +---+
+--R \|- a
+--R Type: Union(List Expression
Integer,...)
+--E
+
+--S 49
+bb1:=(a*x^2+b*x+c)^(3/2)/(3*a)-(b*(2*a*x+b))/(8*a^2)*sqrt(a*x^2+b*x+c)-(b*(4*a*c-b^2))/(16*a^2)*t1.1
+--R
+--R (3)
+--R 3
+--R (- 12a b c + 3b )
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +--------------+
+--R 2 2 2 +-+ | 2
+--R (16a x + 4a b x + 16a c - 6b )\|a \|a x + b x + c
+--R /
+--R 2 +-+
+--R 48a \|a
+--R Type: Expression
Integer
+--E
+
+--S 50
+bb2:=(a*x^2+b*x+c)^(3/2)/(3*a)-(b*(2*a*x+b))/(8*a^2)*sqrt(a*x^2+b*x+c)-(b*(4*a*c-b^2))/(16*a^2)*t1.2
+--R
+--R (4)
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R 3 \|- a \|a x + b x + c - \|- a \|c
+--R (- 12a b c + 3b )atan(------------------------------------)
+--R a x
+--R +
+--R +--------------+
+--R 2 2 2 +---+ | 2
+--R (8a x + 2a b x + 8a c - 3b )\|- a \|a x + b x + c
+--R /
+--R 2 +---+
+--R 24a \|- a
+--R Type: Expression
Integer
+--E
+
+--S 51
+cc1:=aa.1-bb1
+--R
+--R (5)
+--R 2 2 3 5 2 2 2 4
3
+--R (96a b c + 48a b c - 18b )x + (384a b c - 96b c)x + 384a
b c
+--R +
+--R 3 2
+--R - 96b c
+--R *
+--R +--------------+
+--R +-+ | 2
+--R \|c \|a x + b x + c
+--R +
+--R 2 2 2 4 6 3 2 3 3 2
5 2
+--R (- 144a b c + 24a b c + 3b )x + (- 288a b c - 144a b c + 54b
c)x
+--R +
+--R 2 3 4 2 4 3 3
+--R (- 576a b c + 144b c )x - 384a b c + 96b c
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R 2 2 3 5 2 2 2 4
3
+--R (96a b c + 48a b c - 18b )x + (384a b c - 96b c)x + 384a
b c
+--R +
+--R 3 2
+--R - 96b c
+--R *
+--R +--------------+
+--R +-+ | 2
+--R \|c \|a x + b x + c
+--R +
+--R 2 2 2 4 6 3 2 3 3 2
5 2
+--R (- 144a b c + 24a b c + 3b )x + (- 288a b c - 144a b c + 54b
c)x
+--R +
+--R 2 3 4 2 4 3 3
+--R (- 576a b c + 144b c )x - 384a b c + 96b c
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R 2 3 2 2 4 2 3 3 2 4
+--R (128a c + 48a b c - 36b c)x + (512a b c - 192b c )x + 512a c
+--R +
+--R 2 3
+--R - 192b c
+--R *
+--R +--------------+
+--R +-+ | 2
+--R \|a \|a x + b x + c
+--R +
+--R 2 2 3 5 3 2 3 2 2 4
2
+--R (- 192a b c + 56a b c + 6b )x + (- 384a c - 144a b c + 108b
c)x
+--R +
+--R 3 3 2 4 2 3
+--R (- 768a b c + 288b c )x - 512a c + 192b c
+--R *
+--R +-+ +-+
+--R \|a \|c
+--R /
+--R
+--------------+
+--R 3 2 2 2 2 2 2 +-+ +-+ | 2
+--R ((384a c + 288a b )x + 1536a b c x + 1536a c )\|a \|c \|a x + b x
+ c
+--R +
+--R 3 2 3 3 3 2 2 2 2 2 2
+--R (- 576a b c - 48a b )x + (- 1152a c - 864a b c)x - 2304a b c x
+--R +
+--R 2 3
+--R - 1536a c
+--R *
+--R +-+
+--R \|a
+--R Type: Expression
Integer
+--E
+
+--S 52
+cc2:=aa.2-bb1
+--R
+--R (6)
+--R 2 2 3 5 2 2 2 4
3
+--R (96a b c + 48a b c - 18b )x + (384a b c - 96b c)x + 384a
b c
+--R +
+--R 3 2
+--R - 96b c
+--R *
+--R +--------------+
+--R +---+ +-+ | 2
+--R \|- a \|c \|a x + b x + c
+--R +
+--R 2 2 2 4 6 3
+--R (- 144a b c + 24a b c + 3b )x
+--R +
+--R 2 3 3 2 5 2 2 3 4 2
+--R (- 288a b c - 144a b c + 54b c)x + (- 576a b c + 144b c
)x
+--R +
+--R 4 3 3
+--R - 384a b c + 96b c
+--R *
+--R +---+
+--R \|- a
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R 2 2 3 5 2 2 2 4
+--R (- 192a b c - 96a b c + 36b )x + (- 768a b c + 192b c)x
+--R +
+--R 3 3 2
+--R - 768a b c + 192b c
+--R *
+--R +--------------+
+--R +-+ +-+ | 2
+--R \|a \|c \|a x + b x + c
+--R +
+--R 2 2 2 4 6 3
+--R (288a b c - 48a b c - 6b )x
+--R +
+--R 2 3 3 2 5 2 2 3 4 2
+--R (576a b c + 288a b c - 108b c)x + (1152a b c - 288b c )x
+--R +
+--R 4 3 3
+--R 768a b c - 192b c
+--R *
+--R +-+
+--R \|a
+--R *
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R atan(------------------------------------)
+--R a x
+--R +
+--R 2 3 2 2 4 2 3 3 2 4
+--R (128a c + 48a b c - 36b c)x + (512a b c - 192b c )x + 512a c
+--R +
+--R 2 3
+--R - 192b c
+--R *
+--R +--------------+
+--R +---+ +-+ | 2
+--R \|- a \|a \|a x + b x + c
+--R +
+--R 2 2 3 5 3 2 3 2 2 4
2
+--R (- 192a b c + 56a b c + 6b )x + (- 384a c - 144a b c + 108b
c)x
+--R +
+--R 3 3 2 4 2 3
+--R (- 768a b c + 288b c )x - 512a c + 192b c
+--R *
+--R +---+ +-+ +-+
+--R \|- a \|a \|c
+--R /
+--R 3 2 2 2 2 2 2 +---+ +-+ +-+
+--R ((384a c + 288a b )x + 1536a b c x + 1536a c )\|- a \|a \|c
+--R *
+--R +--------------+
+--R | 2
+--R \|a x + b x + c
+--R +
+--R 3 2 3 3 3 2 2 2 2 2 2
+--R (- 576a b c - 48a b )x + (- 1152a c - 864a b c)x - 2304a b c x
+--R +
+--R 2 3
+--R - 1536a c
+--R *
+--R +---+ +-+
+--R \|- a \|a
+--R Type: Expression
Integer
+--E
+
+--S 53
+cc3:=aa.1-bb2
+--R
+--R (7)
+--R 2 2 3 5 2 2 2 4
3
+--R (96a b c + 48a b c - 18b )x + (384a b c - 96b c)x + 384a
b c
+--R +
+--R 3 2
+--R - 96b c
+--R *
+--R +--------------+
+--R +---+ +-+ | 2
+--R \|- a \|c \|a x + b x + c
+--R +
+--R 2 2 2 4 6 3
+--R (- 144a b c + 24a b c + 3b )x
+--R +
+--R 2 3 3 2 5 2 2 3 4 2
+--R (- 288a b c - 144a b c + 54b c)x + (- 576a b c + 144b c
)x
+--R +
+--R 4 3 3
+--R - 384a b c + 96b c
+--R *
+--R +---+
+--R \|- a
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R 2 2 3 5 2 2 2 4
+--R (192a b c + 96a b c - 36b )x + (768a b c - 192b c)x
+--R +
+--R 3 3 2
+--R 768a b c - 192b c
+--R *
+--R +--------------+
+--R +-+ +-+ | 2
+--R \|a \|c \|a x + b x + c
+--R +
+--R 2 2 2 4 6 3
+--R (- 288a b c + 48a b c + 6b )x
+--R +
+--R 2 3 3 2 5 2 2 3 4
2
+--R (- 576a b c - 288a b c + 108b c)x + (- 1152a b c + 288b
c )x
+--R +
+--R 4 3 3
+--R - 768a b c + 192b c
+--R *
+--R +-+
+--R \|a
+--R *
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R atan(------------------------------------)
+--R a x
+--R +
+--R 2 3 2 2 4 2 3 3 2 4
+--R (128a c + 48a b c - 36b c)x + (512a b c - 192b c )x + 512a c
+--R +
+--R 2 3
+--R - 192b c
+--R *
+--R +--------------+
+--R +---+ +-+ | 2
+--R \|- a \|a \|a x + b x + c
+--R +
+--R 2 2 3 5 3 2 3 2 2 4
2
+--R (- 192a b c + 56a b c + 6b )x + (- 384a c - 144a b c + 108b
c)x
+--R +
+--R 3 3 2 4 2 3
+--R (- 768a b c + 288b c )x - 512a c + 192b c
+--R *
+--R +---+ +-+ +-+
+--R \|- a \|a \|c
+--R /
+--R 3 2 2 2 2 2 2 +---+ +-+ +-+
+--R ((384a c + 288a b )x + 1536a b c x + 1536a c )\|- a \|a \|c
+--R *
+--R +--------------+
+--R | 2
+--R \|a x + b x + c
+--R +
+--R 3 2 3 3 3 2 2 2 2 2 2
+--R (- 576a b c - 48a b )x + (- 1152a c - 864a b c)x - 2304a b c x
+--R +
+--R 2 3
+--R - 1536a c
+--R *
+--R +---+ +-+
+--R \|- a \|a
+--R Type: Expression
Integer
+--E
+
+--S 54 14:286 Axiom cannot simplify these expressions
+cc4:=aa.2-bb2
+--R
+--R (8)
+--R 2 3 2 2 4 2 3 3 2 4
+--R (64a c + 24a b c - 18b c)x + (256a b c - 96b c )x + 256a c
+--R +
+--R 2 3
+--R - 96b c
+--R *
+--R +--------------+
+--R | 2
+--R \|a x + b x + c
+--R +
+--R 2 2 3 5 3 2 3 2 2 4 2
+--R (- 96a b c + 28a b c + 3b )x + (- 192a c - 72a b c + 54b c)x
+--R +
+--R 3 3 2 4 2 3
+--R (- 384a b c + 144b c )x - 256a c + 96b c
+--R *
+--R +-+
+--R \|c
+--R /
+--R +--------------+
+--R 3 2 2 2 2 2 2 +-+ | 2
+--R ((192a c + 144a b )x + 768a b c x + 768a c )\|c \|a x + b x + c
+--R +
+--R 3 2 3 3 3 2 2 2 2 2 2
2 3
+--R (- 288a b c - 24a b )x + (- 576a c - 432a b c)x - 1152a b c x -
768a c
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.287~~~~~$\displaystyle
@@ -533,7 +1968,7 @@ $$
<<*>>=
)clear all
---S 8 of 19
+--S 55
aa:=integrate(x^2*sqrt(a*x^2+b*x+c),x)
--R
--R
@@ -763,7 +2198,891 @@ aa:=integrate(x^2*sqrt(a*x^2+b*x+c),x)
--R \|- a
--R ]
--R Type: Union(List Expression
Integer,...)
---E
+--E
+
+--S 56
+t1:=integrate(sqrt(a*x^2+b*x+c),x)
+--R
+--R (2)
+--R [
+--R +--------------+
+--R 3 2 2 +-+ | 2
+--R ((16a b c - 4b )x + 32a c - 8b c)\|c \|a x + b x + c
+--R +
+--R 2 2 4 2 2 3 3 2 2
+--R (- 16a c + b )x + (- 32a b c + 8b c)x - 32a c + 8b c
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R 2 2 3 3 2 2 2
+-+
+--R ((- 16a c - 4a b )x + (- 40a b c - 2b )x + (- 32a c - 8b
c)x)\|a
+--R *
+--R +--------------+
+--R | 2
+--R \|a x + b x + c
+--R +
+--R 2 4 2 2 3 3 2
+--R 16a b x + (32a c + 24a b )x + (56a b c + 6b )x
+--R +
+--R 2 2
+--R (32a c + 8b c)x
+--R *
+--R +-+ +-+
+--R \|a \|c
+--R /
+--R +--------------+
+--R +-+ +-+ | 2
+--R (32a b x + 64a c)\|a \|c \|a x + b x + c
+--R +
+--R 2 2 2 2 +-+
+--R ((- 32a c - 8a b )x - 64a b c x - 64a c )\|a
+--R ,
+--R
+--R +--------------+
+--R 3 2 2 +-+ | 2
+--R ((16a b c - 4b )x + 32a c - 8b c)\|c \|a x + b x + c
+--R +
+--R 2 2 4 2 2 3 3 2 2
+--R (- 16a c + b )x + (- 32a b c + 8b c)x - 32a c + 8b c
+--R *
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R atan(------------------------------------)
+--R a x
+--R +
+--R 2 2 3 3 2 2 2
+---+
+--R ((- 8a c - 2a b )x + (- 20a b c - b )x + (- 16a c - 4b
c)x)\|- a
+--R *
+--R +--------------+
+--R | 2
+--R \|a x + b x + c
+--R +
+--R 2 4 2 2 3 3 2 2 2
+--R (8a b x + (16a c + 12a b )x + (28a b c + 3b )x + (16a c + 4b
c)x)
+--R *
+--R +---+ +-+
+--R \|- a \|c
+--R /
+--R +--------------+
+--R +---+ +-+ | 2
+--R (16a b x + 32a c)\|- a \|c \|a x + b x + c
+--R +
+--R 2 2 2 2 +---+
+--R ((- 16a c - 4a b )x - 32a b c x - 32a c )\|- a
+--R ]
+--R Type: Union(List Expression
Integer,...)
+--E
+
+--S 57
+bb1:=(6*a*x-5*b)/(24*a^2)*(a*x^2+b*x+c)^(3/2)+(5*b^2-4*a*c)/(16*a^2)*t1.1
+--R
+--R (3)
+--R 2 2 3 5 2 3 2 2 4
+--R ((- 192a b c + 288a b c - 60b )x - 384a c + 576a b c - 120b
c)
+--R *
+--R +--------------+
+--R +-+ | 2
+--R \|c \|a x + b x + c
+--R +
+--R 3 3 2 2 2 4 6 2
+--R (192a c - 240a b c - 12a b c + 15b )x
+--R +
+--R 2 3 3 2 5 2 4 2 3 4 2
+--R (384a b c - 576a b c + 120b c)x + 384a c - 576a b c + 120b c
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R 4 3 2 5 3 2 3 4
+--R (- 384a c - 96a b )x + (- 832a b c - 16a b )x
+--R +
+--R 3 2 2 2 4 3 2 2 3 5
2
+--R (- 960a c - 96a b c + 20a b )x + (- 96a b c + 144a b c - 30b
)x
+--R +
+--R 2 3 2 2 4 3
+--R (- 384a c + 896a b c - 120b c)x + 640a b c
+--R *
+--R +--------------+
+--R +-+ | 2
+--R \|a \|a x + b x + c
+--R +
+--R 4 6 4 3 2 5 3 2 3 4
+--R 384a b x + (768a c + 448a b )x + (1472a b c - 16a b )x
+--R +
+--R 3 2 2 2 4 3 2 2 3 5
2
+--R (1152a c - 192a b c + 40a b )x + (- 32a b c - 512a b c + 90b
)x
+--R +
+--R 2 3 2 2 4 3
+--R (384a c - 1216a b c + 120b c)x - 640a b c
+--R *
+--R +-+ +-+
+--R \|a \|c
+--R /
+--R +--------------+
+--R 3 3 +-+ +-+ | 2
+--R (1536a b x + 3072a c)\|a \|c \|a x + b x + c
+--R +
+--R 4 3 2 2 3 3 2 +-+
+--R ((- 1536a c - 384a b )x - 3072a b c x - 3072a c )\|a
+--R Type: Expression
Integer
+--E
+
+--S 58
+bb2:=(6*a*x-5*b)/(24*a^2)*(a*x^2+b*x+c)^(3/2)+(5*b^2-4*a*c)/(16*a^2)*t1.2
+--R
+--R (4)
+--R 2 2 3 5 2 3 2 2 4
+--R ((- 192a b c + 288a b c - 60b )x - 384a c + 576a b c - 120b
c)
+--R *
+--R +--------------+
+--R +-+ | 2
+--R \|c \|a x + b x + c
+--R +
+--R 3 3 2 2 2 4 6 2
+--R (192a c - 240a b c - 12a b c + 15b )x
+--R +
+--R 2 3 3 2 5 2 4 2 3 4 2
+--R (384a b c - 576a b c + 120b c)x + 384a c - 576a b c + 120b c
+--R *
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R atan(------------------------------------)
+--R a x
+--R +
+--R 4 3 2 5 3 2 3 4
+--R (- 192a c - 48a b )x + (- 416a b c - 8a b )x
+--R +
+--R 3 2 2 2 4 3 2 2 3 5
2
+--R (- 480a c - 48a b c + 10a b )x + (- 48a b c + 72a b c - 15b )x
+--R +
+--R 2 3 2 2 4 3
+--R (- 192a c + 448a b c - 60b c)x + 320a b c
+--R *
+--R +--------------+
+--R +---+ | 2
+--R \|- a \|a x + b x + c
+--R +
+--R 4 6 4 3 2 5 3 2 3 4
+--R 192a b x + (384a c + 224a b )x + (736a b c - 8a b )x
+--R +
+--R 3 2 2 2 4 3 2 2 3 5 2
+--R (576a c - 96a b c + 20a b )x + (- 16a b c - 256a b c + 45b )x
+--R +
+--R 2 3 2 2 4 3
+--R (192a c - 608a b c + 60b c)x - 320a b c
+--R *
+--R +---+ +-+
+--R \|- a \|c
+--R /
+--R +--------------+
+--R 3 3 +---+ +-+ | 2
+--R (768a b x + 1536a c)\|- a \|c \|a x + b x + c
+--R +
+--R 4 3 2 2 3 3 2 +---+
+--R ((- 768a c - 192a b )x - 1536a b c x - 1536a c )\|- a
+--R Type: Expression
Integer
+--E
+
+--S 59
+cc1:=aa.1-bb1
+--R
+--R (5)
+--R 4 4 3 3 3 2 5 2 7 9 5
+--R (9216a b c - 6144a b c - 8064a b c + 1536a b c + 180b )x
+--R +
+--R 4 5 3 2 4 2 4 3 8 4
+--R (18432a c + 36864a b c - 77568a b c + 4200b c)x
+--R +
+--R 3 5 2 3 4 5 3 7 2 3
+--R (147456a b c - 135168a b c - 82944a b c + 26880b c )x
+--R +
+--R 3 6 2 2 5 4 4 6 3 2
+--R (98304a c + 73728a b c - 301056a b c + 69120b c )x
+--R +
+--R 2 6 3 5 5 4 2 7
+--R (245760a b c - 368640a b c + 76800b c )x + 98304a c
+--R +
+--R 2 6 4 5
+--R - 147456a b c + 30720b c
+--R *
+--R +--------------+
+--R +-+ | 2
+--R \|c \|a x + b x + c
+--R +
+--R 5 5 4 2 4 3 4 3 2 6 2 8
+--R - 3072a c - 6912a b c + 13440a b c + 672a b c - 828a b c
+--R +
+--R 10
+--R - 15b
+--R *
+--R 6
+--R x
+--R +
+--R 4 5 3 3 4 2 5 3 7 2 9
5
+--R (- 55296a b c + 36864a b c + 48384a b c - 9216a b c - 1080b
c)x
+--R +
+--R 4 6 3 2 5 2 4 4 8 2 4
+--R (- 55296a c - 110592a b c + 232704a b c - 12600b c )x
+--R +
+--R 3 6 2 3 5 5 4 7 3 3
+--R (- 294912a b c + 270336a b c + 165888a b c - 53760b c )x
+--R +
+--R 3 7 2 2 6 4 5 6 4 2
+--R (- 147456a c - 110592a b c + 451584a b c - 103680b c )x
+--R +
+--R 2 7 3 6 5 5 2 8
+--R (- 294912a b c + 442368a b c - 92160b c )x - 98304a c
+--R +
+--R 2 7 4 6
+--R 147456a b c - 30720b c
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R 4 4 3 3 3 2 5 2 7 9 5
+--R (9216a b c - 6144a b c - 8064a b c + 1536a b c + 180b )x
+--R +
+--R 4 5 3 2 4 2 4 3 8 4
+--R (18432a c + 36864a b c - 77568a b c + 4200b c)x
+--R +
+--R 3 5 2 3 4 5 3 7 2 3
+--R (147456a b c - 135168a b c - 82944a b c + 26880b c )x
+--R +
+--R 3 6 2 2 5 4 4 6 3 2
+--R (98304a c + 73728a b c - 301056a b c + 69120b c )x
+--R +
+--R 2 6 3 5 5 4 2 7
+--R (245760a b c - 368640a b c + 76800b c )x + 98304a c
+--R +
+--R 2 6 4 5
+--R - 147456a b c + 30720b c
+--R *
+--R +--------------+
+--R +-+ | 2
+--R \|c \|a x + b x + c
+--R +
+--R 5 5 4 2 4 3 4 3 2 6 2 8
+--R - 3072a c - 6912a b c + 13440a b c + 672a b c - 828a b c
+--R +
+--R 10
+--R - 15b
+--R *
+--R 6
+--R x
+--R +
+--R 4 5 3 3 4 2 5 3 7 2 9
5
+--R (- 55296a b c + 36864a b c + 48384a b c - 9216a b c - 1080b
c)x
+--R +
+--R 4 6 3 2 5 2 4 4 8 2 4
+--R (- 55296a c - 110592a b c + 232704a b c - 12600b c )x
+--R +
+--R 3 6 2 3 5 5 4 7 3 3
+--R (- 294912a b c + 270336a b c + 165888a b c - 53760b c )x
+--R +
+--R 3 7 2 2 6 4 5 6 4 2
+--R (- 147456a c - 110592a b c + 451584a b c - 103680b c )x
+--R +
+--R 2 7 3 6 5 5 2 8
+--R (- 294912a b c + 442368a b c - 92160b c )x - 98304a c
+--R +
+--R 2 7 4 6
+--R 147456a b c - 30720b c
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R 3 2 4 2 4 3 6 2 5
+--R (- 15360a b c - 12800a b c - 960a b c )x
+--R +
+--R 3 5 2 3 4 5 3 4
+--R (- 30720a b c - 107520a b c - 22400a b c )x
+--R +
+--R 2 2 5 4 4 3 2 6 3
5 2
+--R (- 245760a b c - 143360a b c )x + (- 163840a b c - 368640a b
c )x
+--R +
+--R 2 6 7
+--R - 409600a b c x - 163840a b c
+--R *
+--R +--------------+
+--R +-+ | 2
+--R \|a \|a x + b x + c
+--R +
+--R 4 4 3 3 3 2 5 2 7 6
+--R (5120a b c + 19200a b c + 4800a b c + 80a b c)x
+--R +
+--R 3 2 4 2 4 3 6 2 5
+--R (92160a b c + 76800a b c + 5760a b c )x
+--R +
+--R 3 5 2 3 4 5 3 4
+--R (92160a b c + 322560a b c + 67200a b c )x
+--R +
+--R 2 2 5 4 4 3 2 6 3 5 2
+--R (491520a b c + 286720a b c )x + (245760a b c + 552960a b c )x
+--R +
+--R 2 6 7
+--R 491520a b c x + 163840a b c
+--R *
+--R +-+ +-+
+--R \|a \|c
+--R /
+--R 5 2 4 3 3 5 5
+--R (73728a b c + 61440a b c + 4608a b )x
+--R +
+--R 5 3 4 2 2 3 4 4
+--R (147456a c + 516096a b c + 107520a b c)x
+--R +
+--R 4 3 3 3 2 3 4 4 3 2 3 2
+--R (1179648a b c + 688128a b c )x + (786432a c + 1769472a b c )x
+--R +
+--R 3 4 3 5
+--R 1966080a b c x + 786432a c
+--R *
+--R +--------------+
+--R +-+ +-+ | 2
+--R \|a \|c \|a x + b x + c
+--R +
+--R 6 3 5 2 2 4 4 3 6 6
+--R (- 24576a c - 92160a b c - 23040a b c - 384a b )x
+--R +
+--R 5 3 4 3 2 3 5 5
+--R (- 442368a b c - 368640a b c - 27648a b c)x
+--R +
+--R 5 4 4 2 3 3 4 2 4
+--R (- 442368a c - 1548288a b c - 322560a b c )x
+--R +
+--R 4 4 3 3 3 3
+--R (- 2359296a b c - 1376256a b c )x
+--R +
+--R 4 5 3 2 4 2 3 5 3 6
+--R (- 1179648a c - 2654208a b c )x - 2359296a b c x - 786432a c
+--R *
+--R +-+
+--R \|a
+--R Type: Expression
Integer
+--E
+
+--S 60
+cc2:=aa.2-bb1
+--R
+--R (6)
+--R 4 4 3 3 3 2 5 2 7 9 5
+--R (9216a b c - 6144a b c - 8064a b c + 1536a b c + 180b )x
+--R +
+--R 4 5 3 2 4 2 4 3 8 4
+--R (18432a c + 36864a b c - 77568a b c + 4200b c)x
+--R +
+--R 3 5 2 3 4 5 3 7 2 3
+--R (147456a b c - 135168a b c - 82944a b c + 26880b c )x
+--R +
+--R 3 6 2 2 5 4 4 6 3 2
+--R (98304a c + 73728a b c - 301056a b c + 69120b c )x
+--R +
+--R 2 6 3 5 5 4 2 7
+--R (245760a b c - 368640a b c + 76800b c )x + 98304a c
+--R +
+--R 2 6 4 5
+--R - 147456a b c + 30720b c
+--R *
+--R +--------------+
+--R +---+ +-+ | 2
+--R \|- a \|c \|a x + b x + c
+--R +
+--R 5 5 4 2 4 3 4 3 2 6 2
8
+--R - 3072a c - 6912a b c + 13440a b c + 672a b c - 828a
b c
+--R +
+--R 10
+--R - 15b
+--R *
+--R 6
+--R x
+--R +
+--R 4 5 3 3 4 2 5 3 7 2
+--R - 55296a b c + 36864a b c + 48384a b c - 9216a b c
+--R +
+--R 9
+--R - 1080b c
+--R *
+--R 5
+--R x
+--R +
+--R 4 6 3 2 5 2 4 4 8 2 4
+--R (- 55296a c - 110592a b c + 232704a b c - 12600b c )x
+--R +
+--R 3 6 2 3 5 5 4 7 3 3
+--R (- 294912a b c + 270336a b c + 165888a b c - 53760b c )x
+--R +
+--R 3 7 2 2 6 4 5 6 4 2
+--R (- 147456a c - 110592a b c + 451584a b c - 103680b c )x
+--R +
+--R 2 7 3 6 5 5 2 8
+--R (- 294912a b c + 442368a b c - 92160b c )x - 98304a c
+--R +
+--R 2 7 4 6
+--R 147456a b c - 30720b c
+--R *
+--R +---+
+--R \|- a
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R 4 4 3 3 3 2 5 2 7
9 5
+--R (- 18432a b c + 12288a b c + 16128a b c - 3072a b c -
360b )x
+--R +
+--R 4 5 3 2 4 2 4 3 8 4
+--R (- 36864a c - 73728a b c + 155136a b c - 8400b c)x
+--R +
+--R 3 5 2 3 4 5 3 7 2 3
+--R (- 294912a b c + 270336a b c + 165888a b c - 53760b c )x
+--R +
+--R 3 6 2 2 5 4 4 6 3 2
+--R (- 196608a c - 147456a b c + 602112a b c - 138240b c )x
+--R +
+--R 2 6 3 5 5 4 2 7
+--R (- 491520a b c + 737280a b c - 153600b c )x - 196608a c
+--R +
+--R 2 6 4 5
+--R 294912a b c - 61440b c
+--R *
+--R +--------------+
+--R +-+ +-+ | 2
+--R \|a \|c \|a x + b x + c
+--R +
+--R 5 5 4 2 4 3 4 3 2 6 2
8
+--R 6144a c + 13824a b c - 26880a b c - 1344a b c +
1656a b c
+--R +
+--R 10
+--R 30b
+--R *
+--R 6
+--R x
+--R +
+--R 4 5 3 3 4 2 5 3 7 2
+--R 110592a b c - 73728a b c - 96768a b c + 18432a b c
+--R +
+--R 9
+--R 2160b c
+--R *
+--R 5
+--R x
+--R +
+--R 4 6 3 2 5 2 4 4 8 2 4
+--R (110592a c + 221184a b c - 465408a b c + 25200b c )x
+--R +
+--R 3 6 2 3 5 5 4 7 3 3
+--R (589824a b c - 540672a b c - 331776a b c + 107520b c )x
+--R +
+--R 3 7 2 2 6 4 5 6 4 2
+--R (294912a c + 221184a b c - 903168a b c + 207360b c )x
+--R +
+--R 2 7 3 6 5 5 2 8
+--R (589824a b c - 884736a b c + 184320b c )x + 196608a c
+--R +
+--R 2 7 4 6
+--R - 294912a b c + 61440b c
+--R *
+--R +-+
+--R \|a
+--R *
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R atan(------------------------------------)
+--R a x
+--R +
+--R 3 2 4 2 4 3 6 2 5
+--R (- 15360a b c - 12800a b c - 960a b c )x
+--R +
+--R 3 5 2 3 4 5 3 4
+--R (- 30720a b c - 107520a b c - 22400a b c )x
+--R +
+--R 2 2 5 4 4 3 2 6 3
5 2
+--R (- 245760a b c - 143360a b c )x + (- 163840a b c - 368640a b
c )x
+--R +
+--R 2 6 7
+--R - 409600a b c x - 163840a b c
+--R *
+--R +--------------+
+--R +---+ +-+ | 2
+--R \|- a \|a \|a x + b x + c
+--R +
+--R 4 4 3 3 3 2 5 2 7 6
+--R (5120a b c + 19200a b c + 4800a b c + 80a b c)x
+--R +
+--R 3 2 4 2 4 3 6 2 5
+--R (92160a b c + 76800a b c + 5760a b c )x
+--R +
+--R 3 5 2 3 4 5 3 4
+--R (92160a b c + 322560a b c + 67200a b c )x
+--R +
+--R 2 2 5 4 4 3 2 6 3 5 2
+--R (491520a b c + 286720a b c )x + (245760a b c + 552960a b c )x
+--R +
+--R 2 6 7
+--R 491520a b c x + 163840a b c
+--R *
+--R +---+ +-+ +-+
+--R \|- a \|a \|c
+--R /
+--R 5 2 4 3 3 5 5
+--R (73728a b c + 61440a b c + 4608a b )x
+--R +
+--R 5 3 4 2 2 3 4 4
+--R (147456a c + 516096a b c + 107520a b c)x
+--R +
+--R 4 3 3 3 2 3 4 4 3 2 3 2
+--R (1179648a b c + 688128a b c )x + (786432a c + 1769472a b c )x
+--R +
+--R 3 4 3 5
+--R 1966080a b c x + 786432a c
+--R *
+--R +--------------+
+--R +---+ +-+ +-+ | 2
+--R \|- a \|a \|c \|a x + b x + c
+--R +
+--R 6 3 5 2 2 4 4 3 6 6
+--R (- 24576a c - 92160a b c - 23040a b c - 384a b )x
+--R +
+--R 5 3 4 3 2 3 5 5
+--R (- 442368a b c - 368640a b c - 27648a b c)x
+--R +
+--R 5 4 4 2 3 3 4 2 4
+--R (- 442368a c - 1548288a b c - 322560a b c )x
+--R +
+--R 4 4 3 3 3 3
+--R (- 2359296a b c - 1376256a b c )x
+--R +
+--R 4 5 3 2 4 2 3 5 3 6
+--R (- 1179648a c - 2654208a b c )x - 2359296a b c x - 786432a c
+--R *
+--R +---+ +-+
+--R \|- a \|a
+--R Type: Expression
Integer
+--E
+
+--S 61
+cc3:=aa.1-bb2
+--R
+--R (7)
+--R 4 4 3 3 3 2 5 2 7 9 5
+--R (9216a b c - 6144a b c - 8064a b c + 1536a b c + 180b )x
+--R +
+--R 4 5 3 2 4 2 4 3 8 4
+--R (18432a c + 36864a b c - 77568a b c + 4200b c)x
+--R +
+--R 3 5 2 3 4 5 3 7 2 3
+--R (147456a b c - 135168a b c - 82944a b c + 26880b c )x
+--R +
+--R 3 6 2 2 5 4 4 6 3 2
+--R (98304a c + 73728a b c - 301056a b c + 69120b c )x
+--R +
+--R 2 6 3 5 5 4 2 7
+--R (245760a b c - 368640a b c + 76800b c )x + 98304a c
+--R +
+--R 2 6 4 5
+--R - 147456a b c + 30720b c
+--R *
+--R +--------------+
+--R +---+ +-+ | 2
+--R \|- a \|c \|a x + b x + c
+--R +
+--R 5 5 4 2 4 3 4 3 2 6 2
8
+--R - 3072a c - 6912a b c + 13440a b c + 672a b c - 828a
b c
+--R +
+--R 10
+--R - 15b
+--R *
+--R 6
+--R x
+--R +
+--R 4 5 3 3 4 2 5 3 7 2
+--R - 55296a b c + 36864a b c + 48384a b c - 9216a b c
+--R +
+--R 9
+--R - 1080b c
+--R *
+--R 5
+--R x
+--R +
+--R 4 6 3 2 5 2 4 4 8 2 4
+--R (- 55296a c - 110592a b c + 232704a b c - 12600b c )x
+--R +
+--R 3 6 2 3 5 5 4 7 3 3
+--R (- 294912a b c + 270336a b c + 165888a b c - 53760b c )x
+--R +
+--R 3 7 2 2 6 4 5 6 4 2
+--R (- 147456a c - 110592a b c + 451584a b c - 103680b c )x
+--R +
+--R 2 7 3 6 5 5 2 8
+--R (- 294912a b c + 442368a b c - 92160b c )x - 98304a c
+--R +
+--R 2 7 4 6
+--R 147456a b c - 30720b c
+--R *
+--R +---+
+--R \|- a
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R 4 4 3 3 3 2 5 2 7 9
5
+--R (18432a b c - 12288a b c - 16128a b c + 3072a b c + 360b
)x
+--R +
+--R 4 5 3 2 4 2 4 3 8 4
+--R (36864a c + 73728a b c - 155136a b c + 8400b c)x
+--R +
+--R 3 5 2 3 4 5 3 7 2 3
+--R (294912a b c - 270336a b c - 165888a b c + 53760b c )x
+--R +
+--R 3 6 2 2 5 4 4 6 3 2
+--R (196608a c + 147456a b c - 602112a b c + 138240b c )x
+--R +
+--R 2 6 3 5 5 4 2 7
+--R (491520a b c - 737280a b c + 153600b c )x + 196608a c
+--R +
+--R 2 6 4 5
+--R - 294912a b c + 61440b c
+--R *
+--R +--------------+
+--R +-+ +-+ | 2
+--R \|a \|c \|a x + b x + c
+--R +
+--R 5 5 4 2 4 3 4 3 2 6 2
+--R - 6144a c - 13824a b c + 26880a b c + 1344a b c
+--R +
+--R 8 10
+--R - 1656a b c - 30b
+--R *
+--R 6
+--R x
+--R +
+--R 4 5 3 3 4 2 5 3 7 2
+--R - 110592a b c + 73728a b c + 96768a b c - 18432a b c
+--R +
+--R 9
+--R - 2160b c
+--R *
+--R 5
+--R x
+--R +
+--R 4 6 3 2 5 2 4 4 8 2 4
+--R (- 110592a c - 221184a b c + 465408a b c - 25200b c )x
+--R +
+--R 3 6 2 3 5 5 4 7 3 3
+--R (- 589824a b c + 540672a b c + 331776a b c - 107520b c )x
+--R +
+--R 3 7 2 2 6 4 5 6 4 2
+--R (- 294912a c - 221184a b c + 903168a b c - 207360b c )x
+--R +
+--R 2 7 3 6 5 5 2 8
+--R (- 589824a b c + 884736a b c - 184320b c )x - 196608a c
+--R +
+--R 2 7 4 6
+--R 294912a b c - 61440b c
+--R *
+--R +-+
+--R \|a
+--R *
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R atan(------------------------------------)
+--R a x
+--R +
+--R 3 2 4 2 4 3 6 2 5
+--R (- 15360a b c - 12800a b c - 960a b c )x
+--R +
+--R 3 5 2 3 4 5 3 4
+--R (- 30720a b c - 107520a b c - 22400a b c )x
+--R +
+--R 2 2 5 4 4 3 2 6 3
5 2
+--R (- 245760a b c - 143360a b c )x + (- 163840a b c - 368640a b
c )x
+--R +
+--R 2 6 7
+--R - 409600a b c x - 163840a b c
+--R *
+--R +--------------+
+--R +---+ +-+ | 2
+--R \|- a \|a \|a x + b x + c
+--R +
+--R 4 4 3 3 3 2 5 2 7 6
+--R (5120a b c + 19200a b c + 4800a b c + 80a b c)x
+--R +
+--R 3 2 4 2 4 3 6 2 5
+--R (92160a b c + 76800a b c + 5760a b c )x
+--R +
+--R 3 5 2 3 4 5 3 4
+--R (92160a b c + 322560a b c + 67200a b c )x
+--R +
+--R 2 2 5 4 4 3 2 6 3 5 2
+--R (491520a b c + 286720a b c )x + (245760a b c + 552960a b c )x
+--R +
+--R 2 6 7
+--R 491520a b c x + 163840a b c
+--R *
+--R +---+ +-+ +-+
+--R \|- a \|a \|c
+--R /
+--R 5 2 4 3 3 5 5
+--R (73728a b c + 61440a b c + 4608a b )x
+--R +
+--R 5 3 4 2 2 3 4 4
+--R (147456a c + 516096a b c + 107520a b c)x
+--R +
+--R 4 3 3 3 2 3 4 4 3 2 3 2
+--R (1179648a b c + 688128a b c )x + (786432a c + 1769472a b c )x
+--R +
+--R 3 4 3 5
+--R 1966080a b c x + 786432a c
+--R *
+--R +--------------+
+--R +---+ +-+ +-+ | 2
+--R \|- a \|a \|c \|a x + b x + c
+--R +
+--R 6 3 5 2 2 4 4 3 6 6
+--R (- 24576a c - 92160a b c - 23040a b c - 384a b )x
+--R +
+--R 5 3 4 3 2 3 5 5
+--R (- 442368a b c - 368640a b c - 27648a b c)x
+--R +
+--R 5 4 4 2 3 3 4 2 4
+--R (- 442368a c - 1548288a b c - 322560a b c )x
+--R +
+--R 4 4 3 3 3 3
+--R (- 2359296a b c - 1376256a b c )x
+--R +
+--R 4 5 3 2 4 2 3 5 3 6
+--R (- 1179648a c - 2654208a b c )x - 2359296a b c x - 786432a c
+--R *
+--R +---+ +-+
+--R \|- a \|a
+--R Type: Expression
Integer
+--E
+
+--S 62 14:287 Axiom cannot simplify these expressions
+cc4:=aa.2-bb2
+--R
+--R (8)
+--R 2 2 4 4 3 6 2 5
+--R (- 960a b c - 800a b c - 60b c )x
+--R +
+--R 2 5 3 4 5 3 4
+--R (- 1920a b c - 6720a b c - 1400b c )x
+--R +
+--R 2 5 4 4 3 6 3 5 2
+--R (- 15360a b c - 8960b c )x + (- 10240a b c - 23040b c )x
+--R +
+--R 2 6 7
+--R - 25600b c x - 10240b c
+--R *
+--R +--------------+
+--R | 2
+--R \|a x + b x + c
+--R +
+--R 3 4 2 3 3 5 2 7 6
+--R (320a b c + 1200a b c + 300a b c + 5b c)x
+--R +
+--R 2 2 4 4 3 6 2 5
+--R (5760a b c + 4800a b c + 360b c )x
+--R +
+--R 2 5 3 4 5 3 4 2 5 4
4 3
+--R (5760a b c + 20160a b c + 4200b c )x + (30720a b c + 17920b
c )x
+--R +
+--R 6 3 5 2 2 6 7
+--R (15360a b c + 34560b c )x + 30720b c x + 10240b c
+--R *
+--R +-+
+--R \|c
+--R /
+--R 4 2 3 3 2 5 5
+--R (4608a b c + 3840a b c + 288a b )x
+--R +
+--R 4 3 3 2 2 2 4 4
+--R (9216a c + 32256a b c + 6720a b c)x
+--R +
+--R 3 3 2 3 2 3 3 4 2 2 3 2
+--R (73728a b c + 43008a b c )x + (49152a c + 110592a b c )x
+--R +
+--R 2 4 2 5
+--R 122880a b c x + 49152a c
+--R *
+--R +--------------+
+--R +-+ | 2
+--R \|c \|a x + b x + c
+--R +
+--R 5 3 4 2 2 3 4 2 6 6
+--R (- 1536a c - 5760a b c - 1440a b c - 24a b )x
+--R +
+--R 4 3 3 3 2 2 5 5
+--R (- 27648a b c - 23040a b c - 1728a b c)x
+--R +
+--R 4 4 3 2 3 2 4 2 4
+--R (- 27648a c - 96768a b c - 20160a b c )x
+--R +
+--R 3 4 2 3 3 3 3 5 2 2 4 2
+--R (- 147456a b c - 86016a b c )x + (- 73728a c - 165888a b c )x
+--R +
+--R 2 5 2 6
+--R - 147456a b c x - 49152a c
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.288~~~~~$\displaystyle
@@ -775,7 +3094,7 @@ $$
<<*>>=
)clear all
---S 9 of 19
+--S 63
aa:=integrate(sqrt(a*x^2+b*x+c)/x,x)
--R
--R
@@ -858,6 +3177,307 @@ aa:=integrate(sqrt(a*x^2+b*x+c)/x,x)
--R ]
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 64
+t1:=integrate(1/sqrt(a*x^2+b*x+c),x)
+--R
+--R (2)
+--R [
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R /
+--R +-+
+--R \|a
+--R ,
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R 2atan(------------------------------------)
+--R a x
+--R -------------------------------------------]
+--R +---+
+--R \|- a
+--R Type: Union(List Expression
Integer,...)
+--E
+
+--S 65
+t2:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x)
+--R
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R log(---------------------------------)
+--R x
+--R (3) --------------------------------------
+--R +-+
+--R \|c
+--R Type: Union(Expression
Integer,...)
+--E
+
+--S 66
+bb1:=sqrt(a*x^2+b*x+c)+b/2*t1.1+c*t2
+--R
+--R (4)
+--R +--------------+
+--R +-+ | 2
+--R +-+ 2\|c \|a x + b x + c - b x - 2c
+--R 2c\|a log(---------------------------------)
+--R x
+--R +
+--R +-+
+--R b\|c
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +--------------+
+--R +-+ +-+ | 2
+--R 2\|a \|c \|a x + b x + c
+--R /
+--R +-+ +-+
+--R 2\|a \|c
+--R Type: Expression
Integer
+--E
+
+--S 67
+bb2:=sqrt(a*x^2+b*x+c)+b/2*t1.2+c*t2
+--R
+--R (5)
+--R +--------------+
+--R +-+ | 2
+--R +---+ 2\|c \|a x + b x + c - b x - 2c
+--R c\|- a log(---------------------------------)
+--R x
+--R +
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R +-+ \|- a \|a x + b x + c - \|- a \|c
+--R b\|c atan(------------------------------------)
+--R a x
+--R +
+--R +--------------+
+--R +---+ +-+ | 2
+--R \|- a \|c \|a x + b x + c
+--R /
+--R +---+ +-+
+--R \|- a \|c
+--R Type: Expression
Integer
+--E
+
+--S 68
+cc1:=aa.1-bb1
+--R
+--R (6)
+--R +--------------+
+--R +-+ | 2
+--R +-+ 2\|c \|a x + b x + c - b x - 2c
+--R - 2c\|a log(---------------------------------)
+--R x
+--R +
+--R +--------------+
+--R +-+ | 2
+--R +-+ 2\|c \|a x + b x + c - b x - 2c
+--R 2c\|a log(---------------------------------)
+--R +-+
+--R 2x\|c
+--R +
+--R -
+--R +-+
+--R b\|c
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +-+
+--R b\|c
+--R *
+--R log
+--R 2 +-+ 2 2 +-+
+--R ((- 2a b x - 8a c x)\|c + (4a c x + 4b c x + 8c )\|a )
+--R *
+--R +--------------+
+--R | 2
+--R \|a x + b x + c
+--R +
+--R 3 2 2 2 +-+ +-+ 2
3
+--R (- 2a b x + (- 8a c - b )x - 8b c x - 8c )\|a \|c + 4a c
x
+--R +
+--R 2 2
+--R 6a b c x + 8a c x
+--R /
+--R +--------------+
+--R 2 | 2
+--R (4b c x + 8c )\|a x + b x + c
+--R +
+--R 2 2 2 +-+
+--R ((- 4a c - b )x - 8b c x - 8c )\|c
+--R +
+--R +-+
+--R 2c\|a
+--R /
+--R +-+ +-+
+--R 2\|a \|c
+--R Type: Expression
Integer
+--E
+
+--S 69
+cc2:=aa.2-bb1
+--R
+--R (7)
+--R +--------------+
+--R +-+ | 2
+--R +---+ +-+ 2\|c \|a x + b x + c - b x - 2c
+--R - 2c\|- a \|a log(---------------------------------)
+--R x
+--R +
+--R +--------------+
+--R +-+ | 2
+--R +---+ +-+ 2\|c \|a x + b x + c - b x - 2c
+--R 2c\|- a \|a log(---------------------------------)
+--R +-+
+--R 2x\|c
+--R +
+--R -
+--R +---+ +-+
+--R b\|- a \|c
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +--------------+
+--R +---+ +-+ | 2 +---+
+--R +-+ +-+ \|- a \|c \|a x + b x + c - c\|- a +---+ +-+
+--R 2b\|a \|c atan(-------------------------------------) + 2c\|- a \|a
+--R +-+
+--R a x\|c
+--R /
+--R +---+ +-+ +-+
+--R 2\|- a \|a \|c
+--R Type: Expression
Integer
+--E
+
+--S 70
+cc3:=aa.1-bb2
+--R
+--R (8)
+--R +--------------+
+--R +-+ | 2
+--R +---+ +-+ 2\|c \|a x + b x + c - b x - 2c
+--R - 2c\|- a \|a log(---------------------------------)
+--R x
+--R +
+--R +--------------+
+--R +-+ | 2
+--R +---+ +-+ 2\|c \|a x + b x + c - b x - 2c
+--R 2c\|- a \|a log(---------------------------------)
+--R +-+
+--R 2x\|c
+--R +
+--R +---+ +-+
+--R b\|- a \|c
+--R *
+--R log
+--R 2 +-+ 2 2 +-+
+--R ((- 2a b x - 8a c x)\|c + (4a c x + 4b c x + 8c )\|a )
+--R *
+--R +--------------+
+--R | 2
+--R \|a x + b x + c
+--R +
+--R 3 2 2 2 +-+ +-+ 2
3
+--R (- 2a b x + (- 8a c - b )x - 8b c x - 8c )\|a \|c + 4a c
x
+--R +
+--R 2 2
+--R 6a b c x + 8a c x
+--R /
+--R +--------------+
+--R 2 | 2
+--R (4b c x + 8c )\|a x + b x + c
+--R +
+--R 2 2 2 +-+
+--R ((- 4a c - b )x - 8b c x - 8c )\|c
+--R +
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R +-+ +-+ \|- a \|a x + b x + c - \|- a \|c +---+ +-+
+--R - 2b\|a \|c atan(------------------------------------) + 2c\|- a \|a
+--R a x
+--R /
+--R +---+ +-+ +-+
+--R 2\|- a \|a \|c
+--R Type: Expression
Integer
+--E
+
+--S 71 14:288 Axiom cannot simplify these expressions
+cc4:=aa.2-bb2
+--R
+--R (9)
+--R +--------------+
+--R +-+ | 2
+--R +---+ 2\|c \|a x + b x + c - b x - 2c
+--R - c\|- a log(---------------------------------)
+--R x
+--R +
+--R +--------------+
+--R +-+ | 2
+--R +---+ 2\|c \|a x + b x + c - b x - 2c
+--R c\|- a log(---------------------------------)
+--R +-+
+--R 2x\|c
+--R +
+--R +--------------+
+--R +---+ +-+ | 2 +---+
+--R +-+ \|- a \|c \|a x + b x + c - c\|- a
+--R b\|c atan(-------------------------------------)
+--R +-+
+--R a x\|c
+--R +
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R +-+ \|- a \|a x + b x + c - \|- a \|c +---+
+--R - b\|c atan(------------------------------------) + c\|- a
+--R a x
+--R /
+--R +---+ +-+
+--R \|- a \|c
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.289~~~~~$\displaystyle
@@ -870,7 +3490,7 @@ $$
<<*>>=
)clear all
---S 10 of 19
+--S 72
aa:=integrate(sqrt(a*x^2+b*x+c)/x^2,x)
--R
--R
@@ -937,7 +3557,345 @@ aa:=integrate(sqrt(a*x^2+b*x+c)/x^2,x)
--R 8c x\|a x + b x + c + (- 4b x - 8c x)\|c
--R ]
--R Type: Union(List Expression
Integer,...)
---E
+--E
+
+--S 73
+t1:=integrate(1/sqrt(a*x^2+b*x+c),x)
+--R
+--R (2)
+--R [
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R /
+--R +-+
+--R \|a
+--R ,
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R 2atan(------------------------------------)
+--R a x
+--R -------------------------------------------]
+--R +---+
+--R \|- a
+--R Type: Union(List Expression
Integer,...)
+--E
+
+--S 74
+t2:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x)
+--R
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R log(---------------------------------)
+--R x
+--R (3) --------------------------------------
+--R +-+
+--R \|c
+--R Type: Union(Expression
Integer,...)
+--E
+
+--S 75
+bb1:=-sqrt(a*x^2+b*x+c)/x+a*t1.1+b/2*t2
+--R
+--R (4)
+--R +--------------+
+--R +-+ | 2
+--R +-+ 2\|c \|a x + b x + c - b x - 2c
+--R b x\|a log(---------------------------------)
+--R x
+--R +
+--R +-+
+--R 2a x\|c
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +--------------+
+--R +-+ +-+ | 2
+--R - 2\|a \|c \|a x + b x + c
+--R /
+--R +-+ +-+
+--R 2x\|a \|c
+--R Type: Expression
Integer
+--E
+
+--S 76
+bb2:=-sqrt(a*x^2+b*x+c)/x+a*t1.2+b/2*t2
+--R
+--R (5)
+--R +--------------+
+--R +-+ | 2
+--R +---+ 2\|c \|a x + b x + c - b x - 2c
+--R b x\|- a log(---------------------------------)
+--R x
+--R +
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R +-+ \|- a \|a x + b x + c - \|- a \|c
+--R 4a x\|c atan(------------------------------------)
+--R a x
+--R +
+--R +--------------+
+--R +---+ +-+ | 2
+--R - 2\|- a \|c \|a x + b x + c
+--R /
+--R +---+ +-+
+--R 2x\|- a \|c
+--R Type: Expression
Integer
+--E
+
+--S 77
+cc1:=aa.1-bb1
+--R
+--R (6)
+--R +--------------+
+--R +-+ | 2 2 +-+ +-+
+--R (- 4b c\|a \|a x + b x + c + (2b x + 4b c)\|a \|c )
+--R *
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R log(---------------------------------)
+--R x
+--R +
+--R +--------------+
+--R +-+ | 2 2 +-+ +-+
+--R (4b c\|a \|a x + b x + c + (- 2b x - 4b c)\|a \|c )
+--R *
+--R +--------------+
+--R | 2 +-+
+--R 2c\|a x + b x + c + (- b x - 2c)\|c
+--R log(--------------------------------------)
+--R 2c x
+--R +
+--R +--------------+
+--R +-+ | 2 2
+--R (- 8a c\|c \|a x + b x + c + 4a b c x + 8a c )
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +--------------+
+--R +-+ | 2 2
+--R (8a c\|c \|a x + b x + c - 4a b c x - 8a c )
+--R *
+--R +--------------+
+--R +-+ +-+ | 2 +-+ +-+ 2
+--R (2\|c - 2x\|a )\|a x + b x + c + 2x\|a \|c - 2a x - b x -
2c
+--R
log(-----------------------------------------------------------------)
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +--------------+
+--R +-+ | 2 2 +-+ +-+
+--R - 2b c\|a \|a x + b x + c + (b x + 2b c)\|a \|c
+--R /
+--R +--------------+
+--R +-+ +-+ | 2 2 +-+
+--R 8c\|a \|c \|a x + b x + c + (- 4b c x - 8c )\|a
+--R Type: Expression
Integer
+--E
+
+--S 78
+cc2:=aa.2-bb1
+--R
+--R (7)
+--R +--------------+
+--R +-+ | 2 2 +-+ +-+
+--R (- 4b c\|a \|a x + b x + c + (2b x + 4b c)\|a \|c )
+--R *
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R log(---------------------------------)
+--R x
+--R +
+--R +--------------+
+--R +-+ | 2 2 +-+ +-+
+--R (4b c\|a \|a x + b x + c + (- 2b x - 4b c)\|a \|c )
+--R *
+--R +--------------+
+--R | 2 +-+
+--R 2c\|a x + b x + c + (- b x - 2c)\|c
+--R log(--------------------------------------)
+--R 2c x
+--R +
+--R +--------------+
+--R +-+ | 2 2
+--R (- 8a c\|c \|a x + b x + c + 4a b c x + 8a c )
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +--------------+
+--R +---+ +-+ +-+ | 2 2 +---+ +-+
+--R (16c\|- a \|a \|c \|a x + b x + c + (- 8b c x - 16c )\|- a \|a )
+--R *
+--R +--------------+
+--R | 2 +-+
+--R \|a x + b x + c - \|c
+--R atan(------------------------)
+--R +---+
+--R x\|- a
+--R +
+--R +--------------+
+--R +-+ | 2 2 +-+ +-+
+--R - 2b c\|a \|a x + b x + c + (b x + 2b c)\|a \|c
+--R /
+--R +--------------+
+--R +-+ +-+ | 2 2 +-+
+--R 8c\|a \|c \|a x + b x + c + (- 4b c x - 8c )\|a
+--R Type: Expression
Integer
+--E
+
+--S 79
+cc3:=aa.1-bb2
+--R
+--R (8)
+--R +--------------+
+--R +---+ | 2 2 +---+ +-+
+--R (- 4b c\|- a \|a x + b x + c + (2b x + 4b c)\|- a \|c )
+--R *
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R log(---------------------------------)
+--R x
+--R +
+--R +--------------+
+--R +---+ | 2 2 +---+ +-+
+--R (4b c\|- a \|a x + b x + c + (- 2b x - 4b c)\|- a \|c )
+--R *
+--R +--------------+
+--R | 2 +-+
+--R 2c\|a x + b x + c + (- b x - 2c)\|c
+--R log(--------------------------------------)
+--R 2c x
+--R +
+--R +--------------+
+--R +---+ +-+ +-+ | 2 2 +---+ +-+
+--R (8c\|- a \|a \|c \|a x + b x + c + (- 4b c x - 8c )\|- a \|a )
+--R *
+--R +--------------+
+--R +-+ +-+ | 2 +-+ +-+ 2
+--R (2\|c - 2x\|a )\|a x + b x + c + 2x\|a \|c - 2a x - b x -
2c
+--R
log(-----------------------------------------------------------------)
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +--------------+
+--R +-+ | 2 2
+--R (- 16a c\|c \|a x + b x + c + 8a b c x + 16a c )
+--R *
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R atan(------------------------------------)
+--R a x
+--R +
+--R +--------------+
+--R +---+ | 2 2 +---+ +-+
+--R - 2b c\|- a \|a x + b x + c + (b x + 2b c)\|- a \|c
+--R /
+--R +--------------+
+--R +---+ +-+ | 2 2 +---+
+--R 8c\|- a \|c \|a x + b x + c + (- 4b c x - 8c )\|- a
+--R Type: Expression
Integer
+--E
+
+--S 80 14:289 Axiom cannot simplify these expressions
+cc4:=aa.2-bb2
+--R
+--R (9)
+--R +--------------+
+--R +---+ | 2 2 +---+ +-+
+--R (- 4b c\|- a \|a x + b x + c + (2b x + 4b c)\|- a \|c )
+--R *
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R log(---------------------------------)
+--R x
+--R +
+--R +--------------+
+--R +---+ | 2 2 +---+ +-+
+--R (4b c\|- a \|a x + b x + c + (- 2b x - 4b c)\|- a \|c )
+--R *
+--R +--------------+
+--R | 2 +-+
+--R 2c\|a x + b x + c + (- b x - 2c)\|c
+--R log(--------------------------------------)
+--R 2c x
+--R +
+--R +--------------+
+--R +-+ | 2 2
+--R (- 16a c\|c \|a x + b x + c + 8a b c x + 16a c )
+--R *
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R atan(------------------------------------)
+--R a x
+--R +
+--R +--------------+
+--R +-+ | 2 2
+--R (- 16a c\|c \|a x + b x + c + 8a b c x + 16a c )
+--R *
+--R +--------------+
+--R | 2 +-+
+--R \|a x + b x + c - \|c
+--R atan(------------------------)
+--R +---+
+--R x\|- a
+--R +
+--R +--------------+
+--R +---+ | 2 2 +---+ +-+
+--R - 2b c\|- a \|a x + b x + c + (b x + 2b c)\|- a \|c
+--R /
+--R +--------------+
+--R +---+ +-+ | 2 2 +---+
+--R 8c\|- a \|c \|a x + b x + c + (- 4b c x - 8c )\|- a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.290~~~~~$\displaystyle
@@ -949,7 +3907,7 @@ $$
<<*>>=
)clear all
---S 11 of 19
+--S 81
aa:=integrate(1/(a*x^2+b*x+c)^(3/2),x)
--R
--R
@@ -962,6 +3920,31 @@ aa:=integrate(1/(a*x^2+b*x+c)^(3/2),x)
--R (b x + 2c)\|c \|a x + b x + c - 2a c x - 2b c x - 2c
--R Type: Union(Expression
Integer,...)
--E
+
+--S 82
+bb:=(2*(2*a*x+b))/((4*a*c-b^2)*sqrt(a*x^2+b*x+c))
+--R
+--R 4a x + 2b
+--R (2) ----------------------------
+--R +--------------+
+--R 2 | 2
+--R (4a c - b )\|a x + b x + c
+--R Type: Expression
Integer
+--E
+
+--S 83 14:290 Axiom cannot simplify these expressions
+cc:=aa-bb
+--R
+--R (3)
+--R +--------------+
+--R +-+ | 2 2
+--R 4b\|c \|a x + b x + c - 2b x - 4b c
+--R -----------------------------------------------------------------------
+--R +--------------+
+--R 2 2 | 2 3 2 2 +-+
+--R (8a c - 2b c)\|a x + b x + c + ((- 4a b c + b )x - 8a c + 2b c)\|c
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.291~~~~~$\displaystyle
@@ -972,7 +3955,7 @@ $$
<<*>>=
)clear all
---S 12 of 19
+--S 84
aa:=integrate(x/(a*x^2+b*x+c)^(3/2),x)
--R
--R
@@ -984,6 +3967,31 @@ aa:=integrate(x/(a*x^2+b*x+c)^(3/2),x)
--R (b x + 2c)\|c \|a x + b x + c - 2a c x - 2b c x - 2c
--R Type: Union(Expression
Integer,...)
--E
+
+--S 85
+bb:=(2*(b*x+2*c))/((b^2-4*a*c)*sqrt(a*x^2+b*x+c))
+--R
+--R - 2b x - 4c
+--R (2) ----------------------------
+--R +--------------+
+--R 2 | 2
+--R (4a c - b )\|a x + b x + c
+--R Type: Expression
Integer
+--E
+
+--S 86 14:291 Axiom cannot simplify these expressions
+cc:=aa-bb
+--R
+--R (3)
+--R +--------------+
+--R +-+ | 2 2
+--R - 8c\|c \|a x + b x + c + 4b c x + 8c
+--R -----------------------------------------------------------------------
+--R +--------------+
+--R 2 2 | 2 3 2 2 +-+
+--R (8a c - 2b c)\|a x + b x + c + ((- 4a b c + b )x - 8a c + 2b c)\|c
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.292~~~~~$\displaystyle
@@ -995,7 +4003,7 @@ $$
<<*>>=
)clear all
---S 13 of 19
+--S 87
aa:=integrate(x^2/(a*x^2+b*x+c)^(3/2),x)
--R
--R
@@ -1052,6 +4060,356 @@ aa:=integrate(x^2/(a*x^2+b*x+c)^(3/2),x)
--R ]
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 88
+t1:=integrate(1/sqrt(a*x^2+b*x+c),x)
+--R
+--R (2)
+--R [
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R /
+--R +-+
+--R \|a
+--R ,
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R 2atan(------------------------------------)
+--R a x
+--R -------------------------------------------]
+--R +---+
+--R \|- a
+--R Type: Union(List Expression
Integer,...)
+--E
+
+--S 89
+bb1:=((2*b^2-4*a*c)*x+2*b*c)/(4*(4*a*c-b^2)*sqrt(a*x^2+b*x+c))+1/a*t1.1
+--R
+--R (3)
+--R +--------------+
+--R 2 | 2
+--R (8a c - 2b )\|a x + b x + c
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R 2 2 +-+
+--R ((- 2a c + a b )x + a b c)\|a
+--R /
+--R +--------------+
+--R 2 2 +-+ | 2
+--R (8a c - 2a b )\|a \|a x + b x + c
+--R Type: Expression
Integer
+--E
+
+--S 90
+bb2:=((2*b^2-4*a*c)*x+2*b*c)/(4*(4*a*c-b^2)*sqrt(a*x^2+b*x+c))+1/a*t1.2
+--R
+--R (4)
+--R +--------------+
+--R +--------------+ +---+ | 2 +---+
+-+
+--R 2 | 2 \|- a \|a x + b x + c - \|- a
\|c
+--R (16a c - 4b )\|a x + b x + c
atan(------------------------------------)
+--R a x
+--R +
+--R 2 2 +---+
+--R ((- 2a c + a b )x + a b c)\|- a
+--R /
+--R +--------------+
+--R 2 2 +---+ | 2
+--R (8a c - 2a b )\|- a \|a x + b x + c
+--R Type: Expression
Integer
+--E
+
+--S 91
+cc1:=aa.1-bb1
+--R
+--R (5)
+--R 2 3 2 2 2 2
+--R ((- 2a + 16a)b c + (a - 4)b )x + ((- 4a + 16a)c + (3a - 4)b
c)x
+--R +
+--R 2
+--R 2a b c
+--R *
+--R +--------------+
+--R +-+ | 2
+--R \|c \|a x + b x + c
+--R +
+--R 3 2 2 2 2 3
+--R ((4a - 16a )c + (- 2a + 4a)b c)x
+--R +
+--R 2 2 3 2 2 3 2 2
+--R ((2a - 16a)b c + (- 2a + 4)b c)x + ((4a - 16a)c + (- 4a + 4)b c
)x
+--R +
+--R 3
+--R - 2a b c
+--R /
+--R 3 2 2 2 2 2 2 3 2 3 2 2
+--R ((16a c - 4a b c)x + (16a b c - 4a b c)x + 16a c - 4a b c )
+--R *
+--R +--------------+
+--R | 2
+--R \|a x + b x + c
+--R +
+--R 3 2 3 3 3 2 2 2 4 2
+--R (- 8a b c + 2a b )x + (- 16a c - 4a b c + 2a b )x
+--R +
+--R 2 2 3 2 3 2 2
+--R (- 24a b c + 6a b c)x - 16a c + 4a b c
+--R *
+--R +-+
+--R \|c
+--R Type: Expression
Integer
+--E
+
+--S 92
+cc2:=aa.2-bb1
+--R
+--R (6)
+--R 2 2 2 2 2 3 3 2 2
+--R ((- 16a c + 4a b c)x + (- 16a b c + 4b c)x - 16a c + 4b c )
+--R *
+--R +--------------+
+--R +---+ | 2
+--R \|- a \|a x + b x + c
+--R +
+--R 2 3 3 2 2 2 4 2
+--R (8a b c - 2a b )x + (16a c + 4a b c - 2b )x
+--R +
+--R 2 3 3 2 2
+--R (24a b c - 6b c)x + 16a c - 4b c
+--R *
+--R +---+ +-+
+--R \|- a \|c
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R 2 2 2 2 2 3 3 2 2 +-+
+--R ((32a c - 8a b c)x + (32a b c - 8b c)x + 32a c - 8b c )\|a
+--R *
+--R +--------------+
+--R | 2
+--R \|a x + b x + c
+--R +
+--R 2 3 3 2 2 2 4 2
+--R (- 16a b c + 4a b )x + (- 32a c - 8a b c + 4b )x
+--R +
+--R 2 3 3 2 2
+--R (- 48a b c + 12b c)x - 32a c + 8b c
+--R *
+--R +-+ +-+
+--R \|a \|c
+--R *
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R atan(------------------------------------)
+--R a x
+--R +
+--R 2 3 2 2 2 2
+--R ((- 2a + 16a)b c + (a - 4)b )x + ((- 4a + 16a)c + (3a - 4)b
c)x
+--R +
+--R 2
+--R 2a b c
+--R *
+--R +--------------+
+--R +---+ +-+ +-+ | 2
+--R \|- a \|a \|c \|a x + b x + c
+--R +
+--R 3 2 2 2 2 3
+--R ((4a - 16a )c + (- 2a + 4a)b c)x
+--R +
+--R 2 2 3 2
+--R ((2a - 16a)b c + (- 2a + 4)b c)x
+--R +
+--R 2 3 2 2 3
+--R ((4a - 16a)c + (- 4a + 4)b c )x - 2a b c
+--R *
+--R +---+ +-+
+--R \|- a \|a
+--R /
+--R 3 2 2 2 2 2 2 3 2 3 2 2
+---+
+--R ((16a c - 4a b c)x + (16a b c - 4a b c)x + 16a c - 4a b c )\|-
a
+--R *
+--R +--------------+
+--R +-+ | 2
+--R \|a \|a x + b x + c
+--R +
+--R 3 2 3 3 3 2 2 2 4 2
+--R (- 8a b c + 2a b )x + (- 16a c - 4a b c + 2a b )x
+--R +
+--R 2 2 3 2 3 2 2
+--R (- 24a b c + 6a b c)x - 16a c + 4a b c
+--R *
+--R +---+ +-+ +-+
+--R \|- a \|a \|c
+--R Type: Expression
Integer
+--E
+
+--S 93
+cc3:=aa.1-bb2
+--R
+--R (7)
+--R 2 2 2 2 2 3 3 2 2
+---+
+--R ((16a c - 4a b c)x + (16a b c - 4b c)x + 16a c - 4b c )\|-
a
+--R *
+--R +--------------+
+--R | 2
+--R \|a x + b x + c
+--R +
+--R 2 3 3 2 2 2 4 2
+--R (- 8a b c + 2a b )x + (- 16a c - 4a b c + 2b )x
+--R +
+--R 2 3 3 2 2
+--R (- 24a b c + 6b c)x - 16a c + 4b c
+--R *
+--R +---+ +-+
+--R \|- a \|c
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R 2 2 2 2 2 3 3 2 2
+-+
+--R ((- 32a c + 8a b c)x + (- 32a b c + 8b c)x - 32a c + 8b c
)\|a
+--R *
+--R +--------------+
+--R | 2
+--R \|a x + b x + c
+--R +
+--R 2 3 3 2 2 2 4 2
+--R (16a b c - 4a b )x + (32a c + 8a b c - 4b )x
+--R +
+--R 2 3 3 2 2
+--R (48a b c - 12b c)x + 32a c - 8b c
+--R *
+--R +-+ +-+
+--R \|a \|c
+--R *
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R atan(------------------------------------)
+--R a x
+--R +
+--R 2 3 2 2 2 2
+--R ((- 2a + 16a)b c + (a - 4)b )x + ((- 4a + 16a)c + (3a - 4)b
c)x
+--R +
+--R 2
+--R 2a b c
+--R *
+--R +--------------+
+--R +---+ +-+ +-+ | 2
+--R \|- a \|a \|c \|a x + b x + c
+--R +
+--R 3 2 2 2 2 3
+--R ((4a - 16a )c + (- 2a + 4a)b c)x
+--R +
+--R 2 2 3 2
+--R ((2a - 16a)b c + (- 2a + 4)b c)x
+--R +
+--R 2 3 2 2 3
+--R ((4a - 16a)c + (- 4a + 4)b c )x - 2a b c
+--R *
+--R +---+ +-+
+--R \|- a \|a
+--R /
+--R 3 2 2 2 2 2 2 3 2 3 2 2
+---+
+--R ((16a c - 4a b c)x + (16a b c - 4a b c)x + 16a c - 4a b c )\|-
a
+--R *
+--R +--------------+
+--R +-+ | 2
+--R \|a \|a x + b x + c
+--R +
+--R 3 2 3 3 3 2 2 2 4 2
+--R (- 8a b c + 2a b )x + (- 16a c - 4a b c + 2a b )x
+--R +
+--R 2 2 3 2 3 2 2
+--R (- 24a b c + 6a b c)x - 16a c + 4a b c
+--R *
+--R +---+ +-+ +-+
+--R \|- a \|a \|c
+--R Type: Expression
Integer
+--E
+
+--S 94 14:292 Axiom cannot simplify these expressions
+cc4:=aa.2-bb2
+--R
+--R (8)
+--R 2 3 2 2 2 2
+--R ((- 2a + 16a)b c + (a - 4)b )x + ((- 4a + 16a)c + (3a - 4)b
c)x
+--R +
+--R 2
+--R 2a b c
+--R *
+--R +--------------+
+--R +-+ | 2
+--R \|c \|a x + b x + c
+--R +
+--R 3 2 2 2 2 3
+--R ((4a - 16a )c + (- 2a + 4a)b c)x
+--R +
+--R 2 2 3 2 2 3 2 2
+--R ((2a - 16a)b c + (- 2a + 4)b c)x + ((4a - 16a)c + (- 4a + 4)b c
)x
+--R +
+--R 3
+--R - 2a b c
+--R /
+--R 3 2 2 2 2 2 2 3 2 3 2 2
+--R ((16a c - 4a b c)x + (16a b c - 4a b c)x + 16a c - 4a b c )
+--R *
+--R +--------------+
+--R | 2
+--R \|a x + b x + c
+--R +
+--R 3 2 3 3 3 2 2 2 4 2
+--R (- 8a b c + 2a b )x + (- 16a c - 4a b c + 2a b )x
+--R +
+--R 2 2 3 2 3 2 2
+--R (- 24a b c + 6a b c)x - 16a c + 4a b c
+--R *
+--R +-+
+--R \|c
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.293~~~~~$\displaystyle
@@ -1064,7 +4422,7 @@ $$
<<*>>=
)clear all
---S 14 of 19
+--S 95
aa:=integrate(1/(x*(a*x^2+b*x+c)^(3/2)),x)
--R
--R
@@ -1089,6 +4447,110 @@ aa:=integrate(1/(x*(a*x^2+b*x+c)^(3/2)),x)
--R (b c x + 2c )\|c \|a x + b x + c - 2a c x - 2b c x - 2c
--R Type: Union(Expression
Integer,...)
--E
+
+--S 96
+t1:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x)
+--R
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R log(---------------------------------)
+--R x
+--R (2) --------------------------------------
+--R +-+
+--R \|c
+--R Type: Union(Expression
Integer,...)
+--E
+
+--S 97
+t2:=integrate(1/(a*x^2+b*x+c)^(3/2),x)
+--R
+--R +--------------+
+--R | 2 +-+
+--R - 2x\|a x + b x + c + 2x\|c
+--R (3) --------------------------------------------------------
+--R +--------------+
+--R +-+ | 2 2 2
+--R (b x + 2c)\|c \|a x + b x + c - 2a c x - 2b c x - 2c
+--R Type: Union(Expression
Integer,...)
+--E
+
+--S 98
+bb:=1/(c*sqrt(a*x^2+b*x+c))+1/c*t1-b/(2*c)*t2
+--R
+--R (4)
+--R +--------------+
+--R 2 2 | 2
+--R (2a c x + 2b c x + 2c )\|a x + b x + c
+--R +
+--R 3 2 2 2 +-+
+--R (- a b x + (- 2a c - b )x - 3b c x - 2c )\|c
+--R *
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R log(---------------------------------)
+--R x
+--R +
+--R +--------------+
+--R 2 | 2 3 2 2 2 +-+
+--R - 2c \|a x + b x + c + (- a b x + (2a c - b )x + b c x + 2c )\|c
+--R /
+--R +--------------+
+--R 2 2 2 3 +-+ | 2 2 3
+--R (2a c x + 2b c x + 2c )\|c \|a x + b x + c - a b c x
+--R +
+--R 3 2 2 2 3 4
+--R (- 2a c - b c )x - 3b c x - 2c
+--R Type: Expression
Integer
+--E
+
+--S 99 14:293 Axiom cannot simplify these expressions
+cc:=aa-bb
+--R
+--R (5)
+--R +--------------+
+--R 2 2 2 | 2
+--R ((- 4a c - b )x - 8b c x - 8c )\|a x + b x + c
+--R +
+--R 3 2 2 2 +-+
+--R (4a b x + (8a c + 4b )x + 12b c x + 8c )\|c
+--R *
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R log(---------------------------------)
+--R x
+--R +
+--R +--------------+
+--R 2 2 2 | 2
+--R ((4a c + b )x + 8b c x + 8c )\|a x + b x + c
+--R +
+--R 3 2 2 2 +-+
+--R (- 4a b x + (- 8a c - 4b )x - 12b c x - 8c )\|c
+--R *
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R log(---------------------------------)
+--R +-+
+--R 2x\|c
+--R +
+--R +--------------+
+--R 2 2 2 | 2
+--R ((4a c + b )x + 8b c x + 8c )\|a x + b x + c
+--R +
+--R 3 2 2 2 +-+
+--R (- 4a b x + (- 8a c - 4b )x - 12b c x - 8c )\|c
+--R /
+--R +--------------+
+--R 2 2 2 2 3 +-+ | 2 2 3
+--R ((4a c + b c)x + 8b c x + 8c )\|c \|a x + b x + c - 4a b c x
+--R +
+--R 3 2 2 2 3 4
+--R (- 8a c - 4b c )x - 12b c x - 8c
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.294~~~~~$\displaystyle
@@ -1105,7 +4567,7 @@ $$
<<*>>=
)clear all
---S 15 of 19
+--S 100
aa:=integrate(1/(x^2*(a*x^2+b*x+c)^(3/2)),x)
--R
--R
@@ -1145,6 +4607,149 @@ aa:=integrate(1/(x^2*(a*x^2+b*x+c)^(3/2)),x)
--R (- 16a b c x + (- 32a c - 16b c )x - 48b c x - 32c x)\|c
--R Type: Union(Expression
Integer,...)
--E
+
+--S 101
+t1:=integrate(1/(a*x^2+b*x+c)^(3/2),x)
+--R
+--R +--------------+
+--R | 2 +-+
+--R - 2x\|a x + b x + c + 2x\|c
+--R (2) --------------------------------------------------------
+--R +--------------+
+--R +-+ | 2 2 2
+--R (b x + 2c)\|c \|a x + b x + c - 2a c x - 2b c x - 2c
+--R Type: Union(Expression
Integer,...)
+--E
+
+--S 102
+t2:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x)
+--R
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R log(---------------------------------)
+--R x
+--R (3) --------------------------------------
+--R +-+
+--R \|c
+--R Type: Union(Expression
Integer,...)
+--E
+
+--S 103
+bb:=-(a*x^2+b*x+c)/(c^2*x*sqrt(a*x^2+b*x+c))+(b^2-2*a*c)/(2*c^2)*t1-(3*b)/(2*c^2)*t2
+--R
+--R (4)
+--R +--------------+
+--R 3 2 2 2 | 2
+--R (- 6a b c x - 6b c x - 6b c x)\|a x + b x + c
+--R +
+--R 2 4 3 3 2 2 2 +-+
+--R (3a b x + (6a b c + 3b )x + 9b c x + 6b c x)\|c
+--R *
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R log(---------------------------------)
+--R x
+--R +
+--R +--------------+
+--R 3 2 2 2 3 | 2
+--R (2a b c x + 8a c x + 6b c x + 4c )\|a x + b x + c
+--R +
+--R 2 2 4 3 3 2 2 2
+--R (- 8a c + 2a b )x + (- 12a b c + 2b )x + (- 12a c - 2b c)x
+--R +
+--R 2 3
+--R - 8b c x - 4c
+--R *
+--R +-+
+--R \|c
+--R /
+--R +--------------+
+--R 3 3 3 2 4 +-+ | 2 3 4
+--R (4a c x + 4b c x + 4c x)\|c \|a x + b x + c - 2a b c x
+--R +
+--R 4 2 3 3 4 2 5
+--R (- 4a c - 2b c )x - 6b c x - 4c x
+--R Type: Expression
Integer
+--E
+
+--S 104 14:284 Axiom cannot simplify these expressions
+cc:=aa-bb
+--R
+--R (5)
+--R 2 4 3 2 3 2 2 2
3
+--R ((72a b c + 6b )x + (144a b c + 108b c)x + 288b c x + 192b
c )
+--R *
+--R +--------------+
+--R +-+ | 2
+--R \|c \|a x + b x + c
+--R +
+--R 2 2 3 4 2 2 4 3
+--R (- 48a b c - 36a b c)x + (- 240a b c - 36b c)x
+--R +
+--R 3 3 2 2 2 3 4
+--R (- 240a b c - 228b c )x - 384b c x - 192b c
+--R *
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R log(---------------------------------)
+--R x
+--R +
+--R 2 4 3 2 3 2 2 2
+--R (- 72a b c - 6b )x + (- 144a b c - 108b c)x - 288b c x
+--R +
+--R 3
+--R - 192b c
+--R *
+--R +--------------+
+--R +-+ | 2
+--R \|c \|a x + b x + c
+--R +
+--R 2 2 3 4 2 2 4 3
+--R (48a b c + 36a b c)x + (240a b c + 36b c)x
+--R +
+--R 3 3 2 2 2 3 4
+--R (240a b c + 228b c )x + 384b c x + 192b c
+--R *
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R log(---------------------------------)
+--R +-+
+--R 2x\|c
+--R +
+--R 2 4 3 2 3 2 2 2 3
+-+
+--R ((- 60a b c - 5b )x + (- 88a b c - 66b c)x - 112b c x - 32b c
)\|c
+--R *
+--R +--------------+
+--R | 2
+--R \|a x + b x + c
+--R +
+--R 2 2 3 4 2 2 4 3 3 3
2 2
+--R (40a b c + 30a b c)x + (152a b c + 26b c)x + (104a b c + 118b c
)x
+--R +
+--R 2 3 4
+--R 128b c x + 32b c
+--R /
+--R 4 3 3 3 5 2 4 2 5 6
+--R ((48a b c + 4b c )x + (96a c + 72b c )x + 192b c x + 128c )
+--R *
+--R +--------------+
+--R | 2
+--R \|a x + b x + c
+--R +
+--R 2 4 2 3 4 4 3 3 3
+--R (- 32a c - 24a b c )x + (- 160a b c - 24b c )x
+--R +
+--R 5 2 4 2 5 6
+--R (- 160a c - 152b c )x - 256b c x - 128c
+--R *
+--R +-+
+--R \|c
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.295~~~~~$\displaystyle
@@ -1160,7 +4765,7 @@ $$
<<*>>=
)clear all
---S 16 of 19
+--S 105 14:295 Axiom cannot compute this integral
aa:=integrate((a*x^2+b*x+c)^(n+1/2),x)
--R
--R
@@ -1182,7 +4787,7 @@ $$
<<*>>=
)clear all
---S 17 of 19
+--S 106 14:296 Axiom cannot compute this integral
aa:=integrate(x*(a*x^2+b*x+c)^(n+1/2),x)
--R
--R
@@ -1208,7 +4813,7 @@ $$
<<*>>=
)clear all
---S 18 of 19
+--S 107 14:297 Axiom cannot compute this integral
aa:=integrate(1/(a*x^2+b*x+c)^(n+1/2),x)
--R
--R
@@ -1239,7 +4844,7 @@ $$
<<*>>=
)clear all
---S 19 of 19
+--S 108 14:298 Axiom cannot compute this integral
aa:=integrate(1/(x*(a*x^2+b*x+c)^(n+1/2)),x)
--R
--R
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