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[Axiom-developer] 20080430.01.tpd.patch (CATS schaum13 post-mortem fixup
|
From: |
daly |
|
Subject: |
[Axiom-developer] 20080430.01.tpd.patch (CATS schaum13 post-mortem fixup) |
|
Date: |
Fri, 2 May 2008 01:36:13 -0500 |
Thanks to Barry Trager's help, I can show that this set of expressions
(at least the closed forms) all result in differences by a constant.
14:280 Schaums and Axiom differ by a constant
14:281 Schaums and Axiom differ by a constant
14:282 Schaums and Axiom differ by a constant
14:283 Schaums and Axiom differ by a constant
14:284 Schaums and Axiom differ by a constant
14:285 Schaums and Axiom differ by a constant
14:286 Schaums and Axiom differ by a constant
14:287 Schaums and Axiom differ by a constant
14:288 Schaums and Axiom differ by a constant
14:289 Schaums and Axiom differ by a constant
14:290 Schaums and Axiom differ by a constant
14:291 Schaums and Axiom differ by a constant
14:292 Schaums and Axiom differ by a constant
14:293 Schaums and Axiom differ by a constant
14:294 Schaums and Axiom differ by a constant
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 0086e15..7587673 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,4 @@
+20080430 tpd src/input/schaum13.input post-mortem fixes
20080429 tpd src/input/schaum12.input post-mortem fixes
20080428 tpd src/input/schaum19.input post-mortem fixes
20080428 tpd src/input/schaum23.input post-mortem fixes
diff --git a/src/input/schaum13.input.pamphlet
b/src/input/schaum13.input.pamphlet
index c4a0351..70acff6 100644
--- a/src/input/schaum13.input.pamphlet
+++ b/src/input/schaum13.input.pamphlet
@@ -31,7 +31,6 @@ $$
--S 1
aa:=integrate(1/sqrt(a*x^2+b*x+c),x)
---R
--R
--R (1)
--R [
@@ -62,14 +61,14 @@ aa:=integrate(1/sqrt(a*x^2+b*x+c),x)
--E
--S 2
-bb1:=1/sqrt(a)*log(2*sqrt(a)*sqrt(a*x^2+b*x+c)*2*a*x+b)
+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 +--------------+
+--R +-+ | 2
+--R log(2\|a \|a x + b x + c + 2a x + b)
+--R (2) --------------------------------------
+--R +-+
+--R \|a
--R Type: Expression
Integer
--E
@@ -105,9 +104,9 @@ bb3:=1/sqrt(a)*asinh((2*a*x+b)/sqrt(4*a*c-b^2))
cc1:=bb1-aa.1
--R
--R (5)
---R +--------------+
---R +-+ | 2
---R log(4a x\|a \|a x + b x + c + b)
+--R +--------------+
+--R +-+ | 2
+--R log(2\|a \|a x + b x + c + 2a x + b)
--R +
--R -
--R log
@@ -131,9 +130,9 @@ cc1:=bb1-aa.1
cc2:=bb1-aa.2
--R
--R (6)
---R +--------------+
---R +---+ +-+ | 2
---R \|- a log(4a x\|a \|a x + b x + c + b)
+--R +--------------+
+--R +---+ +-+ | 2
+--R \|- a log(2\|a \|a x + b x + c + 2a x + b)
--R +
--R +--------------+
--R +---+ | 2 +---+ +-+
@@ -221,7 +220,7 @@ cc5:=bb3-aa.1
--R Type: Expression
Integer
--E
---S 10 14:280 Axiom cannot simplify these expressions
+--S 10
cc6:=bb3-aa.2
--R
--R (10)
@@ -237,6 +236,42 @@ cc6:=bb3-aa.2
--R \|- a \|a
--R Type: Expression
Integer
--E
+
+--S 11
+dd1:=simplifyLog cc1
+--R
+--R (11)
+--R log
+--R +--------------+
+--R +-+ +-+ | 2
+--R ((4a x + 2b)\|c + (- 2b x - 4c)\|a )\|a x + b x + c
+--R +
+--R 2 +-+ +-+ 2 2
+--R (4a x + 4b x + 4c)\|a \|c - 2a b x + (- 4a c - b )x - 2b c
+--R /
+--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 \|a
+--R Type: Expression
Integer
+--E
+
+--S 12 14:280 Schaums and Axiom differ by a constant
+ee1:=ratDenom dd1
+--R
+--R +-+ +-+
+--R +-+ - 2a\|c + b\|a
+--R \|a log(----------------)
+--R a
+--R (12) -------------------------
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.281~~~~~$\displaystyle
@@ -499,7 +534,7 @@ cc3:=bb2-aa.1
--R Type: Expression
Integer
--E
---S 18 14:281 Axiom cannot simplify these expressions
+--S 18
cc4:=bb2-aa.2
--R
--R +--------------+
@@ -512,6 +547,15 @@ cc4:=bb2-aa.2
--R Type: Expression
Integer
--E
+--S 19 14:281 Schaums and Axiom differ by a constant
+dd1:=ratDenom cc4
+--R
+--R +-+
+--R \|c
+--R (9) - ----
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.282~~~~~$\displaystyle
@@ -524,48 +568,85 @@ $$
)clear all
--S 19
-aa:=integrate(x^2/(a*x^2+b*x+c),x)
---R
+aa:=integrate(x^2/sqrt(a*x^2+b*x+c),x)
--R
--R (1)
--R [
---R 2
---R (2a c - b )
+--R +--------------+
+--R 3 2 2 +-+ | 2
+--R ((16a b c - 12b )x + 32a c - 24b c)\|c \|a x + b x + c
+--R +
+--R 2 2 2 4 2 2 3 3
2 2
+--R (- 16a c + 8a b c + 3b )x + (- 32a b c + 24b c)x - 32a c +
24b c
--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 +-+ +-+ | 2 +-+
+--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c
--R +
---R 3
---R - 4a b c + b
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
--R /
---R 2
---R a x + b x + c
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
--R +
---R +-----------+
---R 2 | 2
---R (- b log(a x + b x + c) + 2a x)\|- 4a c + b
+--R 2 2 3 3 2 2 2
+-+
+--R ((- 16a c - 4a b )x + (- 8a b c + 6b )x + (- 32a c + 24b
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 - 8a b )x + (24a b c - 18b )x
+--R +
+--R 2 2
+--R (32a c - 24b c)x
+--R *
+--R +-+ +-+
+--R \|a \|c
--R /
---R +-----------+
---R 2 | 2
---R 2a \|- 4a c + b
+--R +--------------+
+--R 2 2 +-+ +-+ | 2
+--R (32a b x + 64a c)\|a \|c \|a x + b x + c
+--R +
+--R 3 2 2 2 2 2 2 +-+
+--R ((- 32a c - 8a b )x - 64a b c x - 64a c )\|a
--R ,
--R
---R +---------+
---R | 2
---R 2 (2a x + b)\|4a c - b
---R (- 4a c + 2b )atan(----------------------)
---R 2
---R 4a c - b
+--R +--------------+
+--R 3 2 2 +-+ | 2
+--R ((- 16a b c + 12b )x - 32a c + 24b c)\|c \|a x + b x + c
+--R +
+--R 2 2 2 4 2 2 3 3
2 2
+--R (16a c - 8a b c - 3b )x + (32a b c - 24b c)x + 32a c - 24b
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 + (- 4a b c + 3b )x + (- 16a c + 12b
c)x)\|- a
+--R *
+--R +--------------+
+--R | 2
+--R \|a x + b x + c
--R +
---R +---------+
---R 2 | 2
---R (- b log(a x + b x + c) + 2a x)\|4a c - b
+--R 2 4 2 2 3 3 2 2 2
+--R (8a b x + (16a c - 4a b )x + (12a b c - 9b )x + (16a c - 12b
c)x)
+--R *
+--R +---+ +-+
+--R \|- a \|c
--R /
---R +---------+
---R 2 | 2
---R 2a \|4a c - b
+--R +--------------+
+--R 2 2 +---+ +-+ | 2
+--R (16a b x + 32a c)\|- a \|c \|a x + b x + c
+--R +
+--R 3 2 2 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
@@ -652,14 +733,17 @@
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
cc1:=aa.1-bb1
--R
--R (5)
---R +-----------+
---R 2 | 2
---R (4a c - 3b )\|- 4a c + b
+--R +--------------+
+--R 3 2 2 +-+ | 2
+--R ((16a b c - 12b )x + 32a c - 24b c)\|c \|a x + b x + c
+--R +
+--R 2 2 2 4 2 2 3 3
2 2
+--R (- 16a c + 8a b c + 3b )x + (- 32a b c + 24b c)x - 32a c + 24b
c
--R *
--R log
--R +--------------+
--R +-+ +-+ | 2 +-+
---R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c
--R +
--R 2 +-+
--R (- 2a x - b x - 2c)\|a
@@ -668,31 +752,38 @@ cc1:=aa.1-bb1
--R +-+ | 2
--R 2\|c \|a x + b x + c - b x - 2c
--R +
---R 2 +-+
---R (8a c - 4b )\|a
+--R +--------------+
+--R 3 2 2 +-+ | 2
+--R ((16a b c - 12b )x + 32a c - 24b c)\|c \|a x + b x + c
+--R +
+--R 2 2 2 4 2 2 3 3
2 2
+--R (- 16a c + 8a b c + 3b )x + (- 32a b c + 24b c)x - 32a c + 24b
c
--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 +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
--R +
---R 3
---R - 4a b c + b
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
--R /
---R 2
---R a x + b x + c
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
--R +
---R +-----------+ +--------------+
---R | 2 +-+ | 2
---R (- 4a x + 6b)\|- 4a c + b \|a \|a x + b x + c
+--R +--------------+
+--R 2 2 +-+ | 2
+--R (- 24b c x - 48b c )\|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 3 2 2 2 +-+ +-+
+--R ((24a b c + 6b )x + 48b c x + 48b c )\|a \|c
--R /
---R +-----------+
---R 2 | 2 +-+
---R 8a \|- 4a c + b \|a
+--R +--------------+
+--R 2 2 +-+ +-+ | 2
+--R (32a b x + 64a c)\|a \|c \|a x + b x + c
+--R +
+--R 3 2 2 2 2 2 2 +-+
+--R ((- 32a c - 8a b )x - 64a b c x - 64a c )\|a
--R Type: Expression
Integer
--E
@@ -700,9 +791,18 @@ cc1:=aa.1-bb1
cc2:=aa.2-bb1
--R
--R (6)
---R +---------+
---R 2 | 2
---R (4a c - 3b )\|4a c - b
+--R +--------------+
+--R 3 2 2 +---+ +-+ | 2
+--R ((16a b c - 12b )x + 32a c - 24b c)\|- a \|c \|a x + b x + c
+--R +
+--R 2 2 2 4 2 2 3 3
+--R (- 16a c + 8a b c + 3b )x + (- 32a b c + 24b c)x - 32a c
+--R +
+--R 2 2
+--R 24b c
+--R *
+--R +---+
+--R \|- a
--R *
--R log
--R +--------------+
@@ -716,24 +816,38 @@ cc2:=aa.2-bb1
--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 3 2 2 +-+ +-+ | 2
+--R ((- 32a b c + 24b )x - 64a c + 48b c)\|a \|c \|a x + b x + c
+--R +
+--R 2 2 2 4 2 2 3 3
+--R (32a c - 16a b c - 6b )x + (64a b c - 48b c)x + 64a c
+--R +
+--R 2 2
+--R - 48b 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 +---------+ +--------------+
---R +-+ | 2 | 2
---R (- 4a x + 6b)\|a \|4a c - b \|a x + b x + c
+--R +--------------+
+--R 2 2 +---+ +-+ | 2
+--R (- 24b c x - 48b c )\|- a \|a \|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 3 2 2 2 +---+ +-+ +-+
+--R ((24a b c + 6b )x + 48b c x + 48b c )\|- a \|a \|c
--R /
---R +---------+
---R 2 +-+ | 2
---R 8a \|a \|4a c - b
+--R +--------------+
+--R 2 2 +---+ +-+ +-+ | 2
+--R (32a b x + 64a c)\|- a \|a \|c \|a x + b x + c
+--R +
+--R 3 2 2 2 2 2 2 +---+ +-+
+--R ((- 32a c - 8a b )x - 64a b c x - 64a c )\|- a \|a
--R Type: Expression
Integer
--E
@@ -741,9 +855,18 @@ cc2:=aa.2-bb1
cc3:=aa.2-bb1
--R
--R (7)
---R +---------+
---R 2 | 2
---R (4a c - 3b )\|4a c - b
+--R +--------------+
+--R 3 2 2 +---+ +-+ | 2
+--R ((16a b c - 12b )x + 32a c - 24b c)\|- a \|c \|a x + b x + c
+--R +
+--R 2 2 2 4 2 2 3 3
+--R (- 16a c + 8a b c + 3b )x + (- 32a b c + 24b c)x - 32a c
+--R +
+--R 2 2
+--R 24b c
+--R *
+--R +---+
+--R \|- a
--R *
--R log
--R +--------------+
@@ -757,58 +880,71 @@ cc3:=aa.2-bb1
--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 3 2 2 +-+ +-+ | 2
+--R ((- 32a b c + 24b )x - 64a c + 48b c)\|a \|c \|a x + b x + c
+--R +
+--R 2 2 2 4 2 2 3 3
+--R (32a c - 16a b c - 6b )x + (64a b c - 48b c)x + 64a c
+--R +
+--R 2 2
+--R - 48b 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 +---------+ +--------------+
---R +-+ | 2 | 2
---R (- 4a x + 6b)\|a \|4a c - b \|a x + b x + c
+--R +--------------+
+--R 2 2 +---+ +-+ | 2
+--R (- 24b c x - 48b c )\|- a \|a \|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 3 2 2 2 +---+ +-+ +-+
+--R ((24a b c + 6b )x + 48b c x + 48b c )\|- a \|a \|c
--R /
---R +---------+
---R 2 +-+ | 2
---R 8a \|a \|4a c - b
+--R +--------------+
+--R 2 2 +---+ +-+ +-+ | 2
+--R (32a b x + 64a c)\|- a \|a \|c \|a x + b x + c
+--R +
+--R 3 2 2 2 2 2 2 +---+ +-+
+--R ((- 32a c - 8a b )x - 64a b c x - 64a c )\|- a \|a
--R Type: Expression
Integer
--E
---S 26 14:282 Axiom cannot simplify these expressions
+--S 26
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 2 2 | 2
+--R (- 12b c x - 24b c )\|a x + b x + c
+--R +
+--R 3 2 2 2 +-+
+--R ((12a b c + 3b )x + 24b c x + 24b c )\|c
--R /
---R +---------+
---R 2 +---+ | 2
---R 4a \|- a \|4a c - b
+--R +--------------+
+--R 2 2 +-+ | 2 3 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 2
+--R - 32a c
--R Type: Expression
Integer
--E
+--S 27 14:282 Schaums and Axiom differ by a constant
+dd4:=ratDenom cc4
+--R
+--R +-+
+--R 3b\|c
+--R (9) - ------
+--R 2
+--R 4a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.283~~~~~$\displaystyle
@@ -832,13 +968,16 @@ $$
)clear all
--S 27
-aa:=integrate(1/x*(a*x^2+b*x+c),x)
---R
+aa:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x)
--R
---R 2
---R 2c log(x) + a x + 2b x
---R (1) -----------------------
---R 2
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R log(---------------------------------)
+--R x
+--R (1) --------------------------------------
+--R +-+
+--R \|c
--R Type: Union(Expression
Integer,...)
--E
@@ -871,16 +1010,16 @@ bb2:=1/sqrt(-c)*asin((b*x+2*c)/(x*sqrt(b^2-4*a*c)))
--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
+bb3:=-1/sqrt(c)*asinh((b*x+2*c)/(x*sqrt(4*a*c-b^2)))
+--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
@@ -888,42 +1027,96 @@ bb3:=-1/sqrt(c)*asinh((b*x+2*c)/(x*sqrt(b^2-4*a*c)))
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 +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c + b x + 2c
+--R log(---------------------------------)
+--R x
+--R +
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R log(---------------------------------)
+--R x
+--R /
+--R +-+
+--R \|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 (6)
+--R +--------------+
+--R +-+ | 2
+--R +---+ 2\|c \|a x + b x + c - b x - 2c +-+ b x + 2c
+--R \|- c log(---------------------------------) - \|c asin(---------------)
+--R x +-----------+
+--R | 2
+--R x\|- 4a c + b
+--R ------------------------------------------------------------------------
+--R +---+ +-+
+--R \|- c \|c
--R Type: Expression
Integer
--E
---S 33 14:283 Axiom cannot simplify these expressions
+--S 33
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 +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c b x + 2c
+--R log(---------------------------------) + asinh(-------------)
+--R x +---------+
+--R | 2
+--R x\|4a c - b
+--R (7) -------------------------------------------------------------
+--R +-+
+--R \|c
+--R Type: Expression
Integer
+--E
+
+--S 34
+dd1:=expandLog cc1
+--R
+--R (8)
+--R +--------------+
+--R +-+ | 2
+--R log(2\|c \|a x + b x + c + b x + 2c)
+--R +
+--R +--------------+
+--R +-+ | 2
+--R log(2\|c \|a x + b x + c - b x - 2c) - 2log(x)
+--R /
+--R +-+
+--R \|c
+--R Type: Expression
Integer
+--E
+
+--S 35
+ee1:=ratDenom dd1
+--R
+--R (9)
+--R +--------------+
+--R +-+ +-+ | 2
+--R \|c log(2\|c \|a x + b x + c + b x + 2c)
+--R +
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R \|c log(2\|c \|a x + b x + c - b x - 2c) - 2log(x)\|c
+--R /
+--R c
+--R Type: Expression
Integer
+--E
+
+--S 36 14:283 Schaums and Axiom differ by a constant
+ff1:=complexNormalize ee1
+--R
+--R 2 +-+
+--R log(4a c - b )\|c
+--R (10) ------------------
+--R c
--R Type: Expression
Integer
--E
@
@@ -936,54 +1129,31 @@ $$
<<*>>=
)clear all
---S 34
-aa:=integrate(1/(x^2*(a*x^2+b*x+c)),x)
---R
+--S 37
+aa:=integrate(1/(x^2*sqrt(a*x^2+b*x+c)),x)
--R
--R (1)
---R [
---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
---R (b x log(a x + b x + c) - 2b x log(x) - 2c)\|- 4a c + b
---R /
---R +-----------+
---R 2 | 2
---R 2c x\|- 4a c + b
---R ,
---R
---R +---------+
---R | 2
---R 2 (2a x + b)\|4a c - b
---R (- 4a c + 2b )x atan(----------------------)
---R 2
---R 4a c - b
---R +
---R +---------+
---R 2 | 2
---R (b x log(a x + b x + c) - 2b x log(x) - 2c)\|4a c - b
---R /
---R +---------+
---R 2 | 2
---R 2c x\|4a c - b
---R ]
---R Type: Union(List Expression
Integer,...)
+--R +--------------+
+--R +-+ | 2 2 2
+--R (- 4b x\|c \|a x + b x + c + 2b x + 4b c x)
+--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 2 2
+--R (2b x + 8c)\|c \|a x + b x + c + (- 8a c + b )x - 6b c x - 8c
+--R /
+--R +--------------+
+--R 2 | 2 2 2 +-+
+--R 8c x\|a x + b x + c + (- 4b c x - 8c x)\|c
+--R Type: Union(Expression
Integer,...)
--E
---S 35
+--S 38
t1:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x)
--R
--R +--------------+
@@ -997,13 +1167,13 @@ t1:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x)
--R Type: Union(Expression
Integer,...)
--E
---S 36
-bb:=-sqrt(a*x^2+b*x+c)/(c*x)-b/(2*x)*t1
+--S 39
+bb:=-sqrt(a*x^2+b*x+c)/(c*x)-b/(2*c)*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 - b x log(---------------------------------) - 2\|c \|a x + b x + c
--R x
--R (3)
---------------------------------------------------------------------
--R +-+
@@ -1011,64 +1181,104 @@ bb:=-sqrt(a*x^2+b*x+c)/(c*x)-b/(2*x)*t1
--R Type: Expression
Integer
--E
---S 37
-cc1:=aa.1-bb1
+--S 40
+cc:=aa-bb
--R
--R (4)
---R 2
---R (2a c - b )x
+--R +--------------+
+--R | 2 2 +-+
+--R (4b c\|a x + b x + c + (- 2b x - 4b c)\|c )
--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 +-+ | 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 x + b x + c + (2b x + 4b c)\|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 | 2
---R (b x log(a x + b x + c) - 2b x log(x) - 2bb1 c x - 2c)\|- 4a c + b
+--R +--------------+
+--R | 2 2 +-+
+--R - 2b c\|a x + b x + c + (b x + 2b c)\|c
--R /
---R +-----------+
---R 2 | 2
---R 2c x\|- 4a c + b
+--R +--------------+
+--R 2 +-+ | 2 2 3
+--R 8c \|c \|a x + b x + c - 4b c x - 8c
--R Type: Expression
Integer
--E
---S 38 14:284 Axiom cannot simplify these expressions
-cc2:=aa.2-bb
+--S 41
+dd:=expandLog cc
--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 | 2 2 +-+
+--R (4b c\|a x + b x + c + (- 2b x - 4b c)\|c )
+--R *
+--R +--------------+
+--R +-+ | 2
+--R log(2\|c \|a x + b x + c - b x - 2c)
+--R +
+--R +--------------+
+--R | 2 2 +-+
+--R (- 4b c\|a x + b x + c + (2b x + 4b c)\|c )
+--R *
+--R +--------------+
+--R | 2 +-+
+--R log(2c\|a x + b x + c + (- b x - 2c)\|c )
+--R +
+--R +--------------+
+--R | 2
+--R (4b c log(c) + 4b c log(2) - 2b c)\|a x + b x + c
+--R +
+--R 2 2 2 +-+
+--R ((- 2b x - 4b c)log(c) + (- 2b x - 4b c)log(2) + b x + 2b c)\|c
--R /
---R +---------+
---R 2 +-+ | 2
---R 2c x\|c \|4a c - b
+--R +--------------+
+--R 2 +-+ | 2 2 3
+--R 8c \|c \|a x + b x + c - 4b c x - 8c
+--R Type: Expression
Integer
+--E
+
+--S 42
+ee:=ratDenom dd
+--R
+--R (6)
+--R +--------------+
+--R +-+ +-+ | 2
+--R 2b\|c log(2\|c \|a x + b x + c - b x - 2c)
+--R +
+--R +--------------+
+--R +-+ | 2 +-+
+--R - 2b\|c log(2c\|a x + b x + c + (- b x - 2c)\|c )
+--R +
+--R +-+
+--R (2b log(c) + 2b log(2) - b)\|c
+--R /
+--R 2
+--R 4c
+--R Type: Expression
Integer
+--E
+
+--S 43 14:284 Schaums and Axiom differ by a constant
+ff:=complexNormalize ee
+--R
+--R +-+
+--R (b log(c) + 2b log(2) - b)\|c
+--R (7) ------------------------------
+--R 2
+--R 4c
--R Type: Expression
Integer
--E
+
@
\section{\cite{1}:14.285~~~~~$\displaystyle
@@ -1080,7 +1290,7 @@ $$
<<*>>=
)clear all
---S 39
+--S 44
aa:=integrate(sqrt(a*x^2+b*x+c),x)
--R
--R
@@ -1165,7 +1375,7 @@ aa:=integrate(sqrt(a*x^2+b*x+c),x)
--R Type: Union(List Expression
Integer,...)
--E
---S 40
+--S 45
t1:=integrate(1/sqrt(a*x^2+b*x+c),x)
--R
--R (2)
@@ -1196,7 +1406,7 @@ t1:=integrate(1/sqrt(a*x^2+b*x+c),x)
--R Type: Union(List Expression
Integer,...)
--E
---S 41
+--S 46
bb1:=((2*a*x+b)*sqrt(a*x^2+b*x+c))/(4*a)+(4*a*c-b^2)/(8*a)*t1.1
--R
--R (3)
@@ -1224,7 +1434,7 @@
bb1:=((2*a*x+b)*sqrt(a*x^2+b*x+c))/(4*a)+(4*a*c-b^2)/(8*a)*t1.1
--R Type: Expression
Integer
--E
---S 42
+--S 47
bb2:=((2*a*x+b)*sqrt(a*x^2+b*x+c))/(4*a)+(4*a*c-b^2)/(8*a)*t1.2
--R
--R (4)
@@ -1243,7 +1453,7 @@
bb2:=((2*a*x+b)*sqrt(a*x^2+b*x+c))/(4*a)+(4*a*c-b^2)/(8*a)*t1.2
--R Type: Expression
Integer
--E
---S 43
+--S 48
cc1:=aa.1-bb1
--R
--R (5)
@@ -1263,7 +1473,7 @@ cc1:=aa.1-bb1
--R Type: Expression
Integer
--E
---S 44
+--S 49
cc2:=aa.2-bb1
--R
--R (6)
@@ -1315,7 +1525,7 @@ cc2:=aa.2-bb1
--R Type: Expression
Integer
--E
---S 45
+--S 50
cc3:=aa.1-bb2
--R
--R (7)
@@ -1367,7 +1577,7 @@ cc3:=aa.1-bb2
--R Type: Expression
Integer
--E
---S 46 14:285 Axiom cannot simplify these expressions
+--S 51
cc4:=aa.2-bb2
--R
--R (8)
@@ -1386,6 +1596,16 @@ cc4:=aa.2-bb2
--R - 32a c
--R Type: Expression
Integer
--E
+
+--S 52 14:285 Schaums and Axiom differ by a constant
+dd4:=ratDenom cc4
+--R
+--R +-+
+--R b\|c
+--R (9) -----
+--R 4a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.286~~~~~$\displaystyle
@@ -1403,7 +1623,7 @@ $$
<<*>>=
)clear all
---S 47
+--S 53
aa:=integrate(x*sqrt(a*x^2+b*x+c),x)
--R
--R
@@ -1548,7 +1768,7 @@ aa:=integrate(x*sqrt(a*x^2+b*x+c),x)
--R Type: Union(List Expression
Integer,...)
--E
---S 48
+--S 54
t1:=integrate(1/sqrt(a*x^2+b*x+c),x)
--R
--R (2)
@@ -1579,7 +1799,7 @@ t1:=integrate(1/sqrt(a*x^2+b*x+c),x)
--R Type: Union(List Expression
Integer,...)
--E
---S 49
+--S 55
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)
@@ -1607,7 +1827,7 @@
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
--R Type: Expression
Integer
--E
---S 50
+--S 56
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)
@@ -1626,7 +1846,7 @@
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
--R Type: Expression
Integer
--E
---S 51
+--S 57
cc1:=aa.1-bb1
--R
--R (5)
@@ -1720,7 +1940,7 @@ cc1:=aa.1-bb1
--R Type: Expression
Integer
--E
---S 52
+--S 58
cc2:=aa.2-bb1
--R
--R (6)
@@ -1823,7 +2043,7 @@ cc2:=aa.2-bb1
--R Type: Expression
Integer
--E
---S 53
+--S 59
cc3:=aa.1-bb2
--R
--R (7)
@@ -1926,7 +2146,7 @@ cc3:=aa.1-bb2
--R Type: Expression
Integer
--E
---S 54 14:286 Axiom cannot simplify these expressions
+--S 60
cc4:=aa.2-bb2
--R
--R (8)
@@ -1957,6 +2177,17 @@ cc4:=aa.2-bb2
--R (- 288a b c - 24a b )x + (- 576a c - 432a b c)x - 1152a b c x -
768a c
--R Type: Expression
Integer
--E
+
+--S 61 14:286 Schaums and Axiom differ by a constant
+dd4:=ratDenom cc4
+--R
+--R 2 +-+
+--R (8a c - 3b )\|c
+--R (9) ----------------
+--R 2
+--R 24a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.287~~~~~$\displaystyle
@@ -1968,7 +2199,7 @@ $$
<<*>>=
)clear all
---S 55
+--S 62
aa:=integrate(x^2*sqrt(a*x^2+b*x+c),x)
--R
--R
@@ -2200,7 +2431,7 @@ aa:=integrate(x^2*sqrt(a*x^2+b*x+c),x)
--R Type: Union(List Expression
Integer,...)
--E
---S 56
+--S 63
t1:=integrate(sqrt(a*x^2+b*x+c),x)
--R
--R (2)
@@ -2284,7 +2515,7 @@ t1:=integrate(sqrt(a*x^2+b*x+c),x)
--R Type: Union(List Expression
Integer,...)
--E
---S 57
+--S 64
bb1:=(6*a*x-5*b)/(24*a^2)*(a*x^2+b*x+c)^(3/2)+(5*b^2-4*a*c)/(16*a^2)*t1.1
--R
--R (3)
@@ -2347,7 +2578,7 @@
bb1:=(6*a*x-5*b)/(24*a^2)*(a*x^2+b*x+c)^(3/2)+(5*b^2-4*a*c)/(16*a^2)*t1.1
--R Type: Expression
Integer
--E
---S 58
+--S 65
bb2:=(6*a*x-5*b)/(24*a^2)*(a*x^2+b*x+c)^(3/2)+(5*b^2-4*a*c)/(16*a^2)*t1.2
--R
--R (4)
@@ -2404,7 +2635,7 @@
bb2:=(6*a*x-5*b)/(24*a^2)*(a*x^2+b*x+c)^(3/2)+(5*b^2-4*a*c)/(16*a^2)*t1.2
--R Type: Expression
Integer
--E
---S 59
+--S 66
cc1:=aa.1-bb1
--R
--R (5)
@@ -2600,7 +2831,7 @@ cc1:=aa.1-bb1
--R Type: Expression
Integer
--E
---S 60
+--S 67
cc2:=aa.2-bb1
--R
--R (6)
@@ -2808,7 +3039,7 @@ cc2:=aa.2-bb1
--R Type: Expression
Integer
--E
---S 61
+--S 68
cc3:=aa.1-bb2
--R
--R (7)
@@ -3016,7 +3247,7 @@ cc3:=aa.1-bb2
--R Type: Expression
Integer
--E
---S 62 14:287 Axiom cannot simplify these expressions
+--S 69
cc4:=aa.2-bb2
--R
--R (8)
@@ -3083,6 +3314,17 @@ cc4:=aa.2-bb2
--R - 147456a b c x - 49152a c
--R Type: Expression
Integer
--E
+
+--S 70 14:287 Schaums and Axiom differ by a constant
+dd4:=ratDenom cc4
+--R
+--R +-+
+--R 5b c\|c
+--R (9) - --------
+--R 2
+--R 24a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.288~~~~~$\displaystyle
@@ -3094,7 +3336,7 @@ $$
<<*>>=
)clear all
---S 63
+--S 71
aa:=integrate(sqrt(a*x^2+b*x+c)/x,x)
--R
--R
@@ -3178,7 +3420,7 @@ aa:=integrate(sqrt(a*x^2+b*x+c)/x,x)
--R Type: Union(List Expression
Integer,...)
--E
---S 64
+--S 72
t1:=integrate(1/sqrt(a*x^2+b*x+c),x)
--R
--R (2)
@@ -3209,7 +3451,7 @@ t1:=integrate(1/sqrt(a*x^2+b*x+c),x)
--R Type: Union(List Expression
Integer,...)
--E
---S 65
+--S 73
t2:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x)
--R
--R +--------------+
@@ -3223,7 +3465,7 @@ t2:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x)
--R Type: Union(Expression
Integer,...)
--E
---S 66
+--S 74
bb1:=sqrt(a*x^2+b*x+c)+b/2*t1.1+c*t2
--R
--R (4)
@@ -3257,7 +3499,7 @@ bb1:=sqrt(a*x^2+b*x+c)+b/2*t1.1+c*t2
--R Type: Expression
Integer
--E
---S 67
+--S 75
bb2:=sqrt(a*x^2+b*x+c)+b/2*t1.2+c*t2
--R
--R (5)
@@ -3282,7 +3524,7 @@ bb2:=sqrt(a*x^2+b*x+c)+b/2*t1.2+c*t2
--R Type: Expression
Integer
--E
---S 68
+--S 76
cc1:=aa.1-bb1
--R
--R (6)
@@ -3347,7 +3589,7 @@ cc1:=aa.1-bb1
--R Type: Expression
Integer
--E
---S 69
+--S 77
cc2:=aa.2-bb1
--R
--R (7)
@@ -3392,7 +3634,7 @@ cc2:=aa.2-bb1
--R Type: Expression
Integer
--E
---S 70
+--S 78
cc3:=aa.1-bb2
--R
--R (8)
@@ -3444,7 +3686,7 @@ cc3:=aa.1-bb2
--R Type: Expression
Integer
--E
---S 71 14:288 Axiom cannot simplify these expressions
+--S 79
cc4:=aa.2-bb2
--R
--R (9)
@@ -3478,6 +3720,48 @@ cc4:=aa.2-bb2
--R \|- a \|c
--R Type: Expression
Integer
--E
+
+--S 80
+dd4:=ratDenom cc4
+--R
+--R (10)
+--R +--------------+
+--R +-+ | 2
+--R +-+ 2\|c \|a x + b x + c - b x - 2c
+--R - \|c log(---------------------------------)
+--R x
+--R +
+--R +--------------+
+--R | 2 +-+
+--R +-+ 2c\|a x + b x + c + (- b x - 2c)\|c +-+
+--R \|c log(--------------------------------------) + \|c
+--R 2c x
+--R Type: Expression
Integer
+--E
+
+--S 81
+ee4:=expandLog dd4
+--R
+--R (11)
+--R +--------------+
+--R +-+ +-+ | 2
+--R - \|c log(2\|c \|a x + b x + c - b x - 2c)
+--R +
+--R +--------------+
+--R +-+ | 2 +-+
+-+
+--R \|c log(2c\|a x + b x + c + (- b x - 2c)\|c ) + (- log(c) - log(2) +
1)\|c
+--R Type: Expression
Integer
+--E
+
+--S 82 14:288 Schaums and Axiom differ by a constant
+ff4:=complexNormalize ee4
+--R
+--R +-+
+--R (- log(c) - 2log(2) + 2)\|c
+--R (12) ----------------------------
+--R 2
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.289~~~~~$\displaystyle
@@ -3490,7 +3774,7 @@ $$
<<*>>=
)clear all
---S 72
+--S 83
aa:=integrate(sqrt(a*x^2+b*x+c)/x^2,x)
--R
--R
@@ -3559,7 +3843,7 @@ aa:=integrate(sqrt(a*x^2+b*x+c)/x^2,x)
--R Type: Union(List Expression
Integer,...)
--E
---S 73
+--S 84
t1:=integrate(1/sqrt(a*x^2+b*x+c),x)
--R
--R (2)
@@ -3590,7 +3874,7 @@ t1:=integrate(1/sqrt(a*x^2+b*x+c),x)
--R Type: Union(List Expression
Integer,...)
--E
---S 74
+--S 85
t2:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x)
--R
--R +--------------+
@@ -3604,7 +3888,7 @@ t2:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x)
--R Type: Union(Expression
Integer,...)
--E
---S 75
+--S 86
bb1:=-sqrt(a*x^2+b*x+c)/x+a*t1.1+b/2*t2
--R
--R (4)
@@ -3638,7 +3922,7 @@ bb1:=-sqrt(a*x^2+b*x+c)/x+a*t1.1+b/2*t2
--R Type: Expression
Integer
--E
---S 76
+--S 87
bb2:=-sqrt(a*x^2+b*x+c)/x+a*t1.2+b/2*t2
--R
--R (5)
@@ -3663,7 +3947,7 @@ bb2:=-sqrt(a*x^2+b*x+c)/x+a*t1.2+b/2*t2
--R Type: Expression
Integer
--E
---S 77
+--S 88
cc1:=aa.1-bb1
--R
--R (6)
@@ -3725,7 +4009,7 @@ cc1:=aa.1-bb1
--R Type: Expression
Integer
--E
---S 78
+--S 89
cc2:=aa.2-bb1
--R
--R (7)
@@ -3786,7 +4070,7 @@ cc2:=aa.2-bb1
--R Type: Expression
Integer
--E
---S 79
+--S 90
cc3:=aa.1-bb2
--R
--R (8)
@@ -3842,7 +4126,7 @@ cc3:=aa.1-bb2
--R Type: Expression
Integer
--E
---S 80 14:289 Axiom cannot simplify these expressions
+--S 91
cc4:=aa.2-bb2
--R
--R (9)
@@ -3896,6 +4180,55 @@ cc4:=aa.2-bb2
--R 8c\|- a \|c \|a x + b x + c + (- 4b c x - 8c )\|- a
--R Type: Expression
Integer
--E
+
+--S 92
+dd4:=ratDenom cc4
+--R
+--R (10)
+--R +--------------+
+--R +-+ | 2
+--R +-+ 2\|c \|a x + b x + c - b x - 2c
+--R - 2b\|c log(---------------------------------)
+--R x
+--R +
+--R +--------------+
+--R | 2 +-+
+--R +-+ 2c\|a x + b x + c + (- b x - 2c)\|c +-+
+--R 2b\|c log(--------------------------------------) - b\|c
+--R 2c x
+--R /
+--R 4c
+--R Type: Expression
Integer
+--E
+
+--S 93
+ee4:=expandLog dd4
+--R
+--R (11)
+--R +--------------+
+--R +-+ +-+ | 2
+--R - 2b\|c log(2\|c \|a x + b x + c - b x - 2c)
+--R +
+--R +--------------+
+--R +-+ | 2 +-+
+--R 2b\|c log(2c\|a x + b x + c + (- b x - 2c)\|c )
+--R +
+--R +-+
+--R (- 2b log(c) - 2b log(2) - b)\|c
+--R /
+--R 4c
+--R Type: Expression
Integer
+--E
+
+--S 94 14:289 Schaums and Axiom differ by a constant
+ff4:=complexNormalize ee4
+--R
+--R +-+
+--R (- b log(c) - 2b log(2) - b)\|c
+--R (12) --------------------------------
+--R 4c
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.290~~~~~$\displaystyle
@@ -3907,7 +4240,7 @@ $$
<<*>>=
)clear all
---S 81
+--S 95
aa:=integrate(1/(a*x^2+b*x+c)^(3/2),x)
--R
--R
@@ -3921,7 +4254,7 @@ aa:=integrate(1/(a*x^2+b*x+c)^(3/2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 82
+--S 96
bb:=(2*(2*a*x+b))/((4*a*c-b^2)*sqrt(a*x^2+b*x+c))
--R
--R 4a x + 2b
@@ -3932,7 +4265,7 @@ bb:=(2*(2*a*x+b))/((4*a*c-b^2)*sqrt(a*x^2+b*x+c))
--R Type: Expression
Integer
--E
---S 83 14:290 Axiom cannot simplify these expressions
+--S 97
cc:=aa-bb
--R
--R (3)
@@ -3945,6 +4278,17 @@ cc:=aa-bb
--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
+
+--S 98 14:290 Schaums and Axiom differ by a constant
+dd:=ratDenom cc
+--R
+--R +-+
+--R 2b\|c
+--R (4) -----------
+--R 2 2
+--R 4a c - b c
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.291~~~~~$\displaystyle
@@ -3955,7 +4299,7 @@ $$
<<*>>=
)clear all
---S 84
+--S 99
aa:=integrate(x/(a*x^2+b*x+c)^(3/2),x)
--R
--R
@@ -3968,7 +4312,7 @@ aa:=integrate(x/(a*x^2+b*x+c)^(3/2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 85
+--S 100
bb:=(2*(b*x+2*c))/((b^2-4*a*c)*sqrt(a*x^2+b*x+c))
--R
--R - 2b x - 4c
@@ -3979,7 +4323,7 @@ bb:=(2*(b*x+2*c))/((b^2-4*a*c)*sqrt(a*x^2+b*x+c))
--R Type: Expression
Integer
--E
---S 86 14:291 Axiom cannot simplify these expressions
+--S 101
cc:=aa-bb
--R
--R (3)
@@ -3992,6 +4336,17 @@ cc:=aa-bb
--R (8a c - 2b c)\|a x + b x + c + ((- 4a b c + b )x - 8a c + 2b c)\|c
--R Type: Expression
Integer
--E
+
+--S 102 14:291 Schaums and Axiom differ by a constant
+dd:=ratDenom cc
+--R
+--R +-+
+--R 4\|c
+--R (4) - ---------
+--R 2
+--R 4a c - b
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.292~~~~~$\displaystyle
@@ -4003,7 +4358,7 @@ $$
<<*>>=
)clear all
---S 87
+--S 103
aa:=integrate(x^2/(a*x^2+b*x+c)^(3/2),x)
--R
--R
@@ -4061,7 +4416,7 @@ aa:=integrate(x^2/(a*x^2+b*x+c)^(3/2),x)
--R Type: Union(List Expression
Integer,...)
--E
---S 88
+--S 104
t1:=integrate(1/sqrt(a*x^2+b*x+c),x)
--R
--R (2)
@@ -4092,13 +4447,13 @@ t1:=integrate(1/sqrt(a*x^2+b*x+c),x)
--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
+--S 105
+bb1:=((2*b^2-4*a*c)*x+2*b*c)/(a*(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 2 | 2
+--R (4a c - b )\|a x + b x + c
--R *
--R log
--R +--------------+
@@ -4112,94 +4467,58 @@
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 +-+ | 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 2 +-+
+--R ((- 4a c + 2b )x + 2b c)\|a
--R /
---R +--------------+
---R 2 2 +-+ | 2
---R (8a c - 2a b )\|a \|a x + b x + c
+--R +--------------+
+--R 2 2 +-+ | 2
+--R (4a c - a 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
+--S 106
+bb2:=((2*b^2-4*a*c)*x+2*b*c)/(a*(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 +--------------+ +---+ | 2 +---+
+-+
+--R 2 | 2 \|- a \|a x + b x + c - \|- a \|c
+--R (8a c - 2b )\|a x + b x + c
atan(------------------------------------)
+--R a x
+--R +
+--R 2 +---+
+--R ((- 4a c + 2b )x + 2b c)\|- a
--R /
---R +--------------+
---R 2 2 +---+ | 2
---R (8a c - 2a b )\|- a \|a x + b x + c
+--R +--------------+
+--R 2 2 +---+ | 2
+--R (4a c - a b )\|- a \|a x + b x + c
--R Type: Expression
Integer
--E
---S 91
+--S 107
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 +--------------+
+--R +-+ | 2 2 2
+--R 4b c\|c \|a x + b x + c - 2b c x - 4b c
+--R
-----------------------------------------------------------------------------
+--R +--------------+
+--R 2 2 2 | 2 2 3 2 2 2
+-+
+--R (8a c - 2a b c)\|a x + b x + c + ((- 4a b c + a b )x - 8a c + 2a b
c)\|c
--R Type: Expression
Integer
--E
---S 92
+--S 108
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 2 +---+ | 2
+--R (- 8a c + 2b c)\|- 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 3 2 2 +---+ +-+
+--R ((4a b c - b )x + 8a c - 2b c)\|- a \|c
--R *
--R log
--R +--------------+
@@ -4213,21 +4532,12 @@ cc2:=aa.2-bb1
--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 2 +-+ | 2
+--R (16a c - 4b c)\|a \|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 3 2 2 +-+ +-+
+--R ((- 8a b c + 2b )x - 16a c + 4b c)\|a \|c
--R *
--R +--------------+
--R +---+ | 2 +---+ +-+
@@ -4235,65 +4545,29 @@ cc2:=aa.2-bb1
--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 +---+ +-+ +-+ | 2 2 2 +---+ +-+
+--R 4b c\|- a \|a \|c \|a x + b x + c + (- 2b c x - 4b c )\|- 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 2 2 2 +---+ +-+ | 2
+--R (8a c - 2a b c)\|- a \|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 2 3 2 2 2 +---+ +-+ +-+
+--R ((- 4a b c + a b )x - 8a c + 2a b c)\|- a \|a \|c
--R Type: Expression
Integer
--E
---S 93
+--S 109
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 2 +---+ | 2
+--R (8a c - 2b c)\|- 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 3 2 2 +---+ +-+
+--R ((- 4a b c + b )x - 8a c + 2b c)\|- a \|c
--R *
--R log
--R +--------------+
@@ -4307,21 +4581,12 @@ cc3:=aa.1-bb2
--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 2 +-+ | 2
+--R (- 16a c + 4b c)\|a \|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 3 2 2 +-+ +-+
+--R ((8a b c - 2b )x + 16a c - 4b c)\|a \|c
--R *
--R +--------------+
--R +---+ | 2 +---+ +-+
@@ -4329,87 +4594,43 @@ cc3:=aa.1-bb2
--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 +---+ +-+ +-+ | 2 2 2 +---+ +-+
+--R 4b c\|- a \|a \|c \|a x + b x + c + (- 2b c x - 4b c )\|- 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 2 2 2 +---+ +-+ | 2
+--R (8a c - 2a b c)\|- a \|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 2 3 2 2 2 +---+ +-+ +-+
+--R ((- 4a b c + a b )x - 8a c + 2a b c)\|- a \|a \|c
--R Type: Expression
Integer
--E
---S 94 14:292 Axiom cannot simplify these expressions
+--S 110
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 +--------------+
+--R +-+ | 2 2 2
+--R 4b c\|c \|a x + b x + c - 2b c x - 4b c
+--R
-----------------------------------------------------------------------------
+--R +--------------+
+--R 2 2 2 | 2 2 3 2 2 2
+-+
+--R (8a c - 2a b c)\|a x + b x + c + ((- 4a b c + a b )x - 8a c + 2a b
c)\|c
--R Type: Expression
Integer
--E
+--S 111 14:292 Schaums and Axiom differ by a constant
+dd4:=ratDenom cc4
+--R
+--R +-+
+--R 2b\|c
+--R (9) -----------
+--R 2 2
+--R 4a c - a b
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.293~~~~~$\displaystyle
@@ -4422,7 +4643,7 @@ $$
<<*>>=
)clear all
---S 95
+--S 112
aa:=integrate(1/(x*(a*x^2+b*x+c)^(3/2)),x)
--R
--R
@@ -4448,7 +4669,7 @@ aa:=integrate(1/(x*(a*x^2+b*x+c)^(3/2)),x)
--R Type: Union(Expression
Integer,...)
--E
---S 96
+--S 113
t1:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x)
--R
--R +--------------+
@@ -4462,7 +4683,7 @@ t1:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x)
--R Type: Union(Expression
Integer,...)
--E
---S 97
+--S 114
t2:=integrate(1/(a*x^2+b*x+c)^(3/2),x)
--R
--R +--------------+
@@ -4475,7 +4696,7 @@ t2:=integrate(1/(a*x^2+b*x+c)^(3/2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 98
+--S 115
bb:=1/(c*sqrt(a*x^2+b*x+c))+1/c*t1-b/(2*c)*t2
--R
--R (4)
@@ -4505,7 +4726,7 @@ bb:=1/(c*sqrt(a*x^2+b*x+c))+1/c*t1-b/(2*c)*t2
--R Type: Expression
Integer
--E
---S 99 14:293 Axiom cannot simplify these expressions
+--S 116
cc:=aa-bb
--R
--R (5)
@@ -4551,6 +4772,58 @@ cc:=aa-bb
--R (- 8a c - 4b c )x - 12b c x - 8c
--R Type: Expression
Integer
--E
+
+--S 117
+dd:=ratDenom cc
+--R
+--R (6)
+--R +--------------+
+--R +-+ | 2
+--R +-+ 2\|c \|a x + b x + c - b x - 2c
+--R - \|c log(---------------------------------)
+--R x
+--R +
+--R +--------------+
+--R | 2 +-+
+--R +-+ 2c\|a x + b x + c + (- b x - 2c)\|c +-+
+--R \|c log(--------------------------------------) + \|c
+--R 2c x
+--R /
+--R 2
+--R c
+--R Type: Expression
Integer
+--E
+
+--S 118
+ee:=expandLog dd
+--R
+--R (7)
+--R +--------------+
+--R +-+ +-+ | 2
+--R - \|c log(2\|c \|a x + b x + c - b x - 2c)
+--R +
+--R +--------------+
+--R +-+ | 2 +-+
+--R \|c log(2c\|a x + b x + c + (- b x - 2c)\|c )
+--R +
+--R +-+
+--R (- log(c) - log(2) + 1)\|c
+--R /
+--R 2
+--R c
+--R Type: Expression
Integer
+--E
+
+--S 119 14:293 Schaums and Axiom differ by a constant
+ff:=complexNormalize ee
+--R
+--R +-+
+--R (- log(c) - 2log(2) + 2)\|c
+--R (8) ----------------------------
+--R 2
+--R 2c
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.294~~~~~$\displaystyle
@@ -4567,7 +4840,7 @@ $$
<<*>>=
)clear all
---S 100
+--S 120
aa:=integrate(1/(x^2*(a*x^2+b*x+c)^(3/2)),x)
--R
--R
@@ -4608,7 +4881,7 @@ aa:=integrate(1/(x^2*(a*x^2+b*x+c)^(3/2)),x)
--R Type: Union(Expression
Integer,...)
--E
---S 101
+--S 121
t1:=integrate(1/(a*x^2+b*x+c)^(3/2),x)
--R
--R +--------------+
@@ -4621,7 +4894,7 @@ t1:=integrate(1/(a*x^2+b*x+c)^(3/2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 102
+--S 122
t2:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x)
--R
--R +--------------+
@@ -4635,8 +4908,8 @@ t2:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x)
--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
+--S 123
+bb:=-(a*x^2+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 +--------------+
@@ -4652,15 +4925,15 @@
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
--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 3 2 2 2 2 3 | 2
+--R (2a b c x + (8a c + 2b c)x + 10b 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 (- 8a c + 2a b )x + (- 16a b c + 2b )x + (- 12a c - 6b c)x
--R +
---R 2 3
---R - 8b c x - 4c
+--R 2 3
+--R - 12b c x - 4c
--R *
--R +-+
--R \|c
@@ -4674,7 +4947,7 @@
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
--R Type: Expression
Integer
--E
---S 104 14:284 Axiom cannot simplify these expressions
+--S 124
cc:=aa-bb
--R
--R (5)
@@ -4720,18 +4993,18 @@ cc:=aa-bb
--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 2 4 3 2 3 2 2 2 3
+--R ((- 60a b c - 5b )x + (- 120a b c - 90b c)x - 240b c x - 160b c
)
--R *
---R +--------------+
---R | 2
---R \|a x + b x + c
+--R +--------------+
+--R +-+ | 2
+--R \|c \|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 (40a b c + 30a b c)x + (200a b c + 30b c)x + (200a b c + 190b c
)x
--R +
---R 2 3 4
---R 128b c x + 32b c
+--R 2 3 4
+--R 320b c x + 160b 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 )
@@ -4750,6 +5023,58 @@ cc:=aa-bb
--R \|c
--R Type: Expression
Integer
--E
+
+--S 125
+dd:=ratDenom cc
+--R
+--R (6)
+--R +--------------+
+--R +-+ | 2
+--R +-+ 2\|c \|a x + b x + c - b x - 2c
+--R 6b\|c log(---------------------------------)
+--R x
+--R +
+--R +--------------+
+--R | 2 +-+
+--R +-+ 2c\|a x + b x + c + (- b x - 2c)\|c +-+
+--R - 6b\|c log(--------------------------------------) - 5b\|c
+--R 2c x
+--R /
+--R 3
+--R 4c
+--R Type: Expression
Integer
+--E
+
+--S 126
+ee:=expandLog dd
+--R
+--R (7)
+--R +--------------+
+--R +-+ +-+ | 2
+--R 6b\|c log(2\|c \|a x + b x + c - b x - 2c)
+--R +
+--R +--------------+
+--R +-+ | 2 +-+
+--R - 6b\|c log(2c\|a x + b x + c + (- b x - 2c)\|c )
+--R +
+--R +-+
+--R (6b log(c) + 6b log(2) - 5b)\|c
+--R /
+--R 3
+--R 4c
+--R Type: Expression
Integer
+--E
+
+--S 127 14:284 Schaums and Axiom differ by a constant
+ff:=complexNormalize ee
+--R
+--R +-+
+--R (3b log(c) + 6b log(2) - 5b)\|c
+--R (8) --------------------------------
+--R 3
+--R 4c
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.295~~~~~$\displaystyle
@@ -4765,7 +5090,7 @@ $$
<<*>>=
)clear all
---S 105 14:295 Axiom cannot compute this integral
+--S 128 14:295 Axiom cannot compute this integral
aa:=integrate((a*x^2+b*x+c)^(n+1/2),x)
--R
--R
@@ -4787,7 +5112,7 @@ $$
<<*>>=
)clear all
---S 106 14:296 Axiom cannot compute this integral
+--S 129 14:296 Axiom cannot compute this integral
aa:=integrate(x*(a*x^2+b*x+c)^(n+1/2),x)
--R
--R
@@ -4813,7 +5138,7 @@ $$
<<*>>=
)clear all
---S 107 14:297 Axiom cannot compute this integral
+--S 130 14:297 Axiom cannot compute this integral
aa:=integrate(1/(a*x^2+b*x+c)^(n+1/2),x)
--R
--R
@@ -4844,7 +5169,7 @@ $$
<<*>>=
)clear all
---S 108 14:298 Axiom cannot compute this integral
+--S 131 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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- [Axiom-developer] 20080430.01.tpd.patch (CATS schaum13 post-mortem fixup),
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