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[Axiom-developer] 20080331.01.tpd.patch (CATS integration regression tes
|
From: |
daly |
|
Subject: |
[Axiom-developer] 20080331.01.tpd.patch (CATS integration regression testing) |
|
Date: |
Tue, 1 Apr 2008 00:36:30 -0600 |
More integrals tested.
======================================================================
diff --git a/changelog b/changelog
index 6b98774..a9a8d1b 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,5 @@
+20080331 tpd src/input/Makefile add integration regression testing
+20080331 tpd src/input/schaum12.input integrals of ax^2+bx+c
20080330 tpd src/input/Makefile add integration regression testing
20080330 tpd src/input/schaum11.input integrals of sqrt(a^2-x^2)
20080330 tpd src/input/schaum10.input integrals of sqrt(x^2-a^2)
diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet
index a1382e4..e42c5c0 100644
--- a/src/input/Makefile.pamphlet
+++ b/src/input/Makefile.pamphlet
@@ -357,7 +357,7 @@ REGRES= algaggr.regress algbrbf.regress algfacob.regress
alist.regress \
roman.regress roots.regress ruleset.regress rules.regress \
schaum1.regress schaum2.regress schaum3.regress schaum4.regress \
schaum5.regress schaum6.regress schaum7.regress schaum8.regress \
- schaum9.regress schaum10.regress schaum11.regress \
+ schaum9.regress schaum10.regress schaum11.regress schaum12.regress \
scherk.regress scope.regress seccsc.regress \
segbind.regress seg.regress \
series2.regress series.regress sersolve.regress set.regress \
@@ -633,7 +633,7 @@ FILES= ${OUT}/algaggr.input ${OUT}/algbrbf.input
${OUT}/algfacob.input \
${OUT}/schaum2.input ${OUT}/schaum3.input ${OUT}/schaum4.input \
${OUT}/schaum5.input ${OUT}/schaum6.input ${OUT}/schaum7.input \
${OUT}/schaum8.input ${OUT}/schaum9.input ${OUT}/schaum10.input \
- ${OUT}/schaum11.input \
+ ${OUT}/schaum11.input ${OUT}/schaum12.input \
${OUT}/saddle.input \
${OUT}/scherk.input ${OUT}/scope.input ${OUT}/seccsc.input \
${OUT}/segbind.input ${OUT}/seg.input ${OUT}/series2.input \
@@ -937,7 +937,7 @@ DOCFILES= \
${DOC}/schaum5.input.dvi ${DOC}/schaum6.input.dvi \
${DOC}/schaum7.input.dvi ${DOC}/schaum8.input.dvi \
${DOC}/schaum9.input.dvi ${DOC}/schaum10.input.dvi \
- ${DOC}/schaum11.input.dvi \
+ ${DOC}/schaum11.input.dvi ${DOC}/schaum12.input.dvi \
${DOC}/s01eaf.input.dvi ${DOC}/s13aaf.input.dvi \
${DOC}/s13acf.input.dvi ${DOC}/s13adf.input.dvi \
${DOC}/s14aaf.input.dvi ${DOC}/s14abf.input.dvi \
diff --git a/src/input/schaum12.input.pamphlet
b/src/input/schaum12.input.pamphlet
new file mode 100644
index 0000000..6027801
--- /dev/null
+++ b/src/input/schaum12.input.pamphlet
@@ -0,0 +1,780 @@
+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/input schaum12.input}
+\author{Timothy Daly}
+\maketitle
+\eject
+\tableofcontents
+\eject
+\section{\cite{1}:14.265~~~~~$\displaystyle
+\int{\frac{dx}{ax^2+bx+c}}$}
+$$\int{\frac{1}{ax^2+bx+c}}=
+\left\{
+\begin{array}{l}
+\displaystyle\frac{2}{\sqrt{4ac-b^2}}\tan^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}}\\
+\displaystyle\frac{1}{\sqrt{b^2-4ac}}\ln\left(
+\frac{2ax+b-\sqrt{b^2-4ac}}{2ax+b+\sqrt{b^2-4ac}}\right)
+\end{array}
+\right.
+$$
+<<*>>=
+)spool schaum12.output
+)set message test on
+)set message auto off
+)clear all
+
+--S 1 of 15
+aa:=integrate(1/sqrt(a*x^2+b*x+c),x)
+--R
+--R
+--R (1)
+--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
+@
+
+\section{\cite{1}:14.266~~~~~$\displaystyle
+\int{\frac{x~dx}{ax^2+bx+c}}$}
+$$\int{\frac{x}{ax^2+bx+c}}=
+\frac{1}{2a}\ln(ax^2+bx+c)-\frac{b}{2a}\int{\frac{1}{ax^2+bx+c}}
+$$
+<<*>>=
+)clear all
+
+--S 2 of 15
+aa:=integrate(x/sqrt(a*x^2+b*x+c),x)
+--R
+--R
+--R (1)
+--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 2 +-+ +-+
+--R - 2b x\|a \|a x + b x + c + (4a x + 2b x)\|a \|c
+--R /
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R 4a\|a \|c \|a x + b x + c + (- 2a b x - 4a c)\|a
+--R ,
+--R
+--R +--------------+
+--R +-+ | 2 2
+--R (- 2b\|c \|a x + b x + c + b x + 2b 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 - b x\|- a \|a x + b x + c + (2a x + b x)\|- a \|c
+--R /
+--R +--------------+
+--R +---+ +-+ | 2 +---+
+--R 2a\|- a \|c \|a x + b x + c + (- a b x - 2a c)\|- a
+--R ]
+--R Type: Union(List Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.267~~~~~$\displaystyle
+\int{\frac{x^2dx}{ax^2+bx+c}}$}
+$$\int{\frac{x^2}{ax^2+bx+c}}=
+\frac{x}{a}-\frac{b}{2a^2}\ln(ax^2+bx+c)+\frac{b^2-2ac}{2a^2}
+\int{\frac{1}{ax^2+bx+c}}
+$$
+<<*>>=
+)clear all
+
+--S 3 of 15
+aa:=integrate(x^2/(a*x^2+b*x+c),x)
+--R
+--R
+--R (1)
+--R [
+--R 2
+--R (2a c - b )
+--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 log(a x + b x + c) + 2a x)\|- 4a c + b
+--R /
+--R +-----------+
+--R 2 | 2
+--R 2a \|- 4a c + b
+--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 +---------+
+--R 2 | 2
+--R (- b log(a x + b x + c) + 2a x)\|4a c - b
+--R /
+--R +---------+
+--R 2 | 2
+--R 2a \|4a c - b
+--R ]
+--R Type: Union(List Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.268~~~~~$\displaystyle
+\int{\frac{x^m~dx}{ax^2+bx+c}}$}
+$$\int{\frac{x^m}{ax^2+bx+c}}=
+\frac{x^{m-1}}{(m-1)a}-\frac{c}{a}\int{\frac{x^{m-2}}{ax^2+bx+c}}-
+\frac{b}{a}\int{\frac{x^{m-1}}{ax^2+bx+c}}
+$$
+<<*>>=
+)clear all
+
+--S 4 of 15
+aa:=integrate(x^m/(a*x^2+b*x+c),x)
+--R
+--R
+--R x m
+--R ++ %N
+--R (1) | --------------- d%N
+--R ++ 2
+--R c + %N b + %N a
+--R Type: Union(Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.269~~~~~$\displaystyle
+\int{\frac{dx}{x(ax^2+bx+c)}}$}
+$$\int{\frac{1}{x(ax^2+bx+c)}}=
+\frac{1}{2c}\ln\left(\frac{x^2}{ax^2+bx+c}\right)-
+\frac{b}{2c}\int{\frac{1}{ax^2+bx+c}}
+$$
+<<*>>=
+)clear all
+
+--S 5 of 15
+aa:=integrate(1/(x*(a*x^2+b*x+c)),x)
+--R
+--R
+--R (1)
+--R [
+--R b
+--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 (- log(a x + b x + c) + 2log(x))\|- 4a c + b
+--R /
+--R +-----------+
+--R | 2
+--R 2c\|- 4a c + b
+--R ,
+--R
+--R +---------+
+--R | 2
+--R (2a x + b)\|4a c - b
+--R - 2b atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R +---------+
+--R 2 | 2
+--R (- log(a x + b x + c) + 2log(x))\|4a c - b
+--R /
+--R +---------+
+--R | 2
+--R 2c\|4a c - b
+--R ]
+--R Type: Union(List Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.270~~~~~$\displaystyle
+\int{\frac{dx}{x^2(ax^2+bx+c)}}$}
+$$\int{\frac{1}{x^2(ax^2+bx+c)}}=
+\frac{b}{2c^2}\ln\left(\frac{ax^2+bx+c}{x^2}\right)-\frac{1}{cx}+
+\frac{b^2-2ac}{2c^2}\int{\frac{1}{ax^2+bx+c}}
+$$
+<<*>>=
+)clear all
+
+--S 6 of 15
+aa:=integrate(1/(x^2*(a*x^2+b*x+c)),x)
+--R
+--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,...)
+--E
+@
+
+\section{\cite{1}:14.271~~~~~$\displaystyle
+\int{\frac{dx}{x^n(ax^2+bx+c)}}$}
+$$\int{\frac{1}{x^n(ax^2+bx+c)}}=
+-\frac{1}{(n-1)cx^{n-1}}-
+\frac{b}{c}\int{\frac{1}{x^{n-1}(ax^2+bx+c)}}-
+\frac{a}{c}\int{\frac{1}{x^{x-2}(ax^2+bx+c)}}
+$$
+<<*>>=
+)clear all
+
+--S 7 of 15
+aa:=integrate(1/(x^n*(ax^2+bx+c)),x)
+--R
+--R
+--R x
+--R (1) - ------------------------------------------
+--R 2 2 n log(x)
+--R ((c + bx + ax )n - c - bx - ax )%e
+--R Type: Union(Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.272~~~~~$\displaystyle
+\int{\frac{dx}{(ax^2+bx+c)^2}}$}
+$$\int{\frac{1}{(ax^2+bx+c)^2}}=
+\frac{2xa+b}{(4ac-b^2)(ax^2+bx+c)}+
+\frac{2a}{4ac-b^2}\int{\frac{1}{ax^2+bx+c}}
+$$
+<<*>>=
+)clear all
+
+--S 8 of 15
+aa:=integrate(1/(a*x^2+b*x+c)^2,x)
+--R
+--R
+--R (1)
+--R [
+--R 2 2
+--R (2a x + 2a b x + 2a 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 +-----------+
+--R | 2
+--R (2a x + b)\|- 4a c + b
+--R /
+--R +-----------+
+--R 2 2 2 3 2 2 | 2
+--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b
+--R ,
+--R +---------+
+--R | 2
+---------+
+--R 2 2 (2a x + b)\|4a c - b |
2
+--R (4a x + 4a b x + 4a c)atan(----------------------) + (2a x + b)\|4a c
- b
+--R 2
+--R 4a c - b
+--R
----------------------------------------------------------------------------
+--R +---------+
+--R 2 2 2 3 2 2 | 2
+--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|4a c - b
+--R ]
+--R Type: Union(List Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.273~~~~~$\displaystyle
+\int{\frac{x~dx}{(ax^2+bx+c)^2}}$}
+$$\int{\frac{x}{(ax^2+bx+c)^2}}=
+-\frac{bx+2c}{(4ac-b^2)(ax^2+bx+c)}-
+\frac{b}{4ac-b^2}\int{\frac{1}{ax^2+bx+c}}
+$$
+<<*>>=
+)clear all
+
+--S 9 of 15
+aa:=integrate(x/(a*x^2+b*x+c)^2,x)
+--R
+--R
+--R (1)
+--R [
+--R 2 2
+--R (a b x + b x + b 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 +-----------+
+--R | 2
+--R (- b x - 2c)\|- 4a c + b
+--R /
+--R +-----------+
+--R 2 2 2 3 2 2 | 2
+--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b
+--R ,
+--R
+--R +---------+
+--R | 2
+--R 2 2 (2a x + b)\|4a c - b
+--R (- 2a b x - 2b x - 2b c)atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R +---------+
+--R | 2
+--R (- b x - 2c)\|4a c - b
+--R /
+--R +---------+
+--R 2 2 2 3 2 2 | 2
+--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|4a c - b
+--R ]
+--R Type: Union(List Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.274~~~~~$\displaystyle
+\int{\frac{x^2~dx}{(ax^2+bx+c)^2}}$}
+$$\int{\frac{x^2}{(ax^2+bx+c)^2}}=
+\frac{(b^2-2ac)x+bc}{a(4ac-b^2)(ax^2+bx+c)}+
+\frac{2c}{4ac-b^2}\int{\frac{1}{ax^2+bx+c}}
+$$
+<<*>>=
+)clear all
+
+--S 10 of 15
+aa:=integrate(x^2/(a*x^2+b*x+c)^2,x)
+--R
+--R
+--R (1)
+--R [
+--R 2 2 2
+--R (2a c x + 2a b c x + 2a 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 +-----------+
+--R 2 | 2
+--R ((- 2a c + b )x + b c)\|- 4a c + b
+--R /
+--R +-----------+
+--R 3 2 2 2 2 3 2 2 2 | 2
+--R ((4a c - a b )x + (4a b c - a b )x + 4a c - a b c)\|- 4a c + b
+--R ,
+--R
+--R +---------+
+--R | 2
+--R 2 2 2 (2a x + b)\|4a c - b
+--R (4a c x + 4a b c x + 4a c )atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R +---------+
+--R 2 | 2
+--R ((- 2a c + b )x + b c)\|4a c - b
+--R /
+--R +---------+
+--R 3 2 2 2 2 3 2 2 2 | 2
+--R ((4a c - a b )x + (4a b c - a b )x + 4a c - a b c)\|4a c - b
+--R ]
+--R Type: Union(List Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.275~~~~~$\displaystyle
+\int{\frac{x^m~dx}{(ax^2+bx+c)^n}}$}
+$$
+\begin{array}{lr}
+\displaystyle\int{\frac{x^m}{(ax^2+bx+c)^n}}=
+&\displaystyle-\frac{x^{m-1}}{(2n-m-1)a(ax^2+bx+c)^{n-1}}\\
+&\\
+&\displaystyle+\frac{(m-1)c}{(2n-m-1)a}\int{\frac{x^{m-2}}{(ax^2+bx+2)^n}}\\
+&\\
+&\displaystyle-\frac{(n-m)b}{(2n-m-1)a}\int{\frac{x^{m-1}}{(ax^2+bx+c)^n}}
+\end{array}
+$$
+<<*>>=
+)clear all
+
+--S 11 of 15
+aa:=integrate(x^m/(a*x^2+b*x+c)^n,x)
+--R
+--R
+--R x m
+--R ++ %N
+--R (1) | ------------------ d%N
+--R ++ 2 n
+--R (c + %N b + %N a)
+--R Type: Union(Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.276~~~~~$\displaystyle
+\int{\frac{x^{2n-1}~dx}{(ax^2+bx+c)^n}}$}
+$$\begin{array}{lr}
+\displaystyle\int{\frac{x^{2n-1}}{(ax^2+bx+c)^n}}=
+&\displaystyle\frac{1}{a}\int{\frac{x^{2n-3}}{(ax^2+bx+c)^{n-1}}}\\
+&\\
+&\displaystyle-\frac{c}{a}\int{\frac{x^{2n-3}}{(ax^2+bx+c)^n}}\\
+&\\
+&\displaystyle-\frac{b}{a}\int{\frac{x^{2n-2}}{(ax^2+bx+c)^n}}
+\end{array}
+$$
+<<*>>=
+)clear all
+
+--S 12 of 15
+aa:=integrate(x^(2*n-1)/(a*x^2+b*x+c)^n,x)
+--R
+--R
+--R x 2n - 1
+--R ++ %N
+--R (1) | ------------------ d%N
+--R ++ 2 n
+--R (c + %N b + %N a)
+--R Type: Union(Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.277~~~~~$\displaystyle
+\int{\frac{dx}{x(ax^2+bx+c)^2}}$}
+$$\begin{array}{lr}
+\displaystyle\int{\frac{1}{x(ax^2+bx+c)^2}}=
+&\displaystyle\frac{1}{2c(ax^2+bx+2)}\\
+&\\
+&\displaystyle-\frac{b}{2c}\int{\frac{1}{(ax^2+bx+c)^2}}\\
+&\\
+&\displaystyle+\frac{1}{c}\int{\frac{1}{x(ax^2+bx+c)}}
+\end{array}
+$$
+<<*>>=
+)clear all
+
+--S 13 of 15
+aa:=integrate(1/(x*(a*x^2+b*x+c)^2),x)
+--R
+--R
+--R (1)
+--R [
+--R 2 3 2 2 4 2 3
+--R ((6a b c - a b )x + (6a b c - b )x + 6a b c - b 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 2 2 3 2 2
+--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c)
+--R *
+--R 2
+--R log(a x + b x + c)
+--R +
+--R 2 2 2 3 2 2
+--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x)
+--R +
+--R 2 2
+--R - 2a b c x + 4a c - 2b c
+--R *
+--R +-----------+
+--R | 2
+--R \|- 4a c + b
+--R /
+--R
+-----------+
+--R 2 3 2 2 2 3 3 2 4 2 3 |
2
+--R ((8a c - 2a b c )x + (8a b c - 2b c )x + 8a c - 2b c )\|- 4a c +
b
+--R ,
+--R
+--R 2 3 2 2 4 2 3
+--R ((- 12a b c + 2a b )x + (- 12a b c + 2b )x - 12a b c + 2b c)
+--R *
+--R +---------+
+--R | 2
+--R (2a x + b)\|4a c - b
+--R atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R 2 2 2 3 2 2
+--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c)
+--R *
+--R 2
+--R log(a x + b x + c)
+--R +
+--R 2 2 2 3 2 2
+--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x)
+--R +
+--R 2 2
+--R - 2a b c x + 4a c - 2b c
+--R *
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R /
+--R
+---------+
+--R 2 3 2 2 2 3 3 2 4 2 3 | 2
+--R ((8a c - 2a b c )x + (8a b c - 2b c )x + 8a c - 2b c )\|4a c - b
+--R ]
+--R Type: Union(List Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.278~~~~~$\displaystyle
+\int{\frac{dx}{x^2(ax^2+bx+c)^2}}$}
+$$\begin{array}{lr}
+\displaystyle\int{\frac{1}{x^2(ax^2+bx+c)^2}}=
+&\displaystyle-\frac{1}{cx(ax^2+bx+c)}\\
+&\\
+&\displaystyle-\frac{3a}{c}\int{\frac{1}{(ax^2+bx+c)^2}}\\
+&\\
+&\displaystyle-\frac{2b}{c}\int{\frac{1}{x(ax^2+bx+c)^2}}
+\end{array}
+$$
+<<*>>=
+)clear all
+
+--S 14 of 15
+aa:=integrate(1/(x^2*(a*x^2+b*x+c)^2),x)
+--R
+--R
+--R (1)
+--R [
+--R 3 2 2 2 4 3 2 2 3 5 2
+--R (6a c - 6a b c + a b )x + (6a b c - 6a b c + b )x
+--R +
+--R 2 3 2 2 4
+--R (6a c - 6a b c + b c)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 2 3 3 2 4 2 2 3
+--R ((4a b c - a b )x + (4a b c - b )x + (4a b c - b c)x)
+--R *
+--R 2
+--R log(a x + b x + c)
+--R +
+--R 2 3 3 2 4 2 2 3
+--R ((- 8a b c + 2a b )x + (- 8a b c + 2b )x + (- 8a b c + 2b
c)x)
+--R *
+--R log(x)
+--R +
+--R 2 2 2 2 2 3 3 2 2
+--R (- 6a c + 2a b c)x + (- 7a b c + 2b c)x - 4a c + b c
+--R *
+--R +-----------+
+--R | 2
+--R \|- 4a c + b
+--R /
+--R
+-----------+
+--R 2 4 2 3 3 4 3 3 2 5 2 4 |
2
+--R ((4a c - a b c )x + (4a b c - b c )x + (4a c - b c )x)\|- 4a c
+ b
+--R ,
+--R
+--R 3 2 2 2 4 3 2 2 3 5 2
+--R (- 12a c + 12a b c - 2a b )x + (- 12a b c + 12a b c - 2b )x
+--R +
+--R 2 3 2 2 4
+--R (- 12a c + 12a b c - 2b c)x
+--R *
+--R +---------+
+--R | 2
+--R (2a x + b)\|4a c - b
+--R atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R 2 3 3 2 4 2 2 3
+--R ((4a b c - a b )x + (4a b c - b )x + (4a b c - b c)x)
+--R *
+--R 2
+--R log(a x + b x + c)
+--R +
+--R 2 3 3 2 4 2 2 3
+--R ((- 8a b c + 2a b )x + (- 8a b c + 2b )x + (- 8a b c + 2b
c)x)
+--R *
+--R log(x)
+--R +
+--R 2 2 2 2 2 3 3 2 2
+--R (- 6a c + 2a b c)x + (- 7a b c + 2b c)x - 4a c + b c
+--R *
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R /
+--R
+---------+
+--R 2 4 2 3 3 4 3 3 2 5 2 4 |
2
+--R ((4a c - a b c )x + (4a b c - b c )x + (4a c - b c )x)\|4a c - b
+--R ]
+--R Type: Union(List Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.279~~~~~$\displaystyle
+\int{\frac{dx}{x^m(ax^2+bx+c)^n}}$}
+$$\begin{array}{lr}
+\displaystyle\int{\frac{1}{x^m(ax^2+bx+c)^n}}=
+&\displaystyle-\frac{1}{(m-1)cx^{m-1}(ax^2+bx+c)^{n-1}}\\
+&\\
+&\displaystyle-\frac{(m+2n-3)a}{(m-1)c}\int{\frac{1}{x^{m-2}(ax^2+bx+c)^n}}\\
+&\\
+&\displaystyle-\frac{(m+n-2)b}{(m-1)c}\int{\frac{1}{x^{m-1}(ax^2+bx+c)^n}}
+\end{array}
+$$
+<<*>>=
+)clear all
+
+--S 15 of 15
+aa:=integrate(1/(x^m*(a*x^2+b*x+c)^n),x)
+--R
+--R
+--R x
+--R ++ 1
+--R (1) | --------------------- d%N
+--R ++ m 2 n
+--R %N (c + %N b + %N a)
+--R Type: Union(Expression
Integer,...)
+--E
+
+)spool
+)lisp (bye)
+@
+
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} Spiegel, Murray R.
+{\sl Mathematical Handbook of Formulas and Tables}\\
+Schaum's Outline Series McGraw-Hill 1968 p71
+\end{thebibliography}
+\end{document}
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