Optimal. Leaf size=251 \[ \frac{c \log \left (a+c x^2\right ) \left (a A e^3-3 a B d e^2-3 A c d^2 e+B c d^3\right )}{2 \left (a e^2+c d^2\right )^3}+\frac{B d-A e}{2 (d+e x)^2 \left (a e^2+c d^2\right )}+\frac{-a B e^2-2 A c d e+B c d^2}{(d+e x) \left (a e^2+c d^2\right )^2}-\frac{c \log (d+e x) \left (a A e^3-3 a B d e^2-3 A c d^2 e+B c d^3\right )}{\left (a e^2+c d^2\right )^3}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (c d^2-3 a e^2\right )+a B e \left (3 c d^2-a e^2\right )\right )}{\sqrt{a} \left (a e^2+c d^2\right )^3} \]
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Rubi [A] time = 0.322575, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {801, 635, 205, 260} \[ \frac{c \log \left (a+c x^2\right ) \left (a A e^3-3 a B d e^2-3 A c d^2 e+B c d^3\right )}{2 \left (a e^2+c d^2\right )^3}+\frac{B d-A e}{2 (d+e x)^2 \left (a e^2+c d^2\right )}+\frac{-a B e^2-2 A c d e+B c d^2}{(d+e x) \left (a e^2+c d^2\right )^2}-\frac{c \log (d+e x) \left (a A e^3-3 a B d e^2-3 A c d^2 e+B c d^3\right )}{\left (a e^2+c d^2\right )^3}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (c d^2-3 a e^2\right )+a B e \left (3 c d^2-a e^2\right )\right )}{\sqrt{a} \left (a e^2+c d^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 801
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{A+B x}{(d+e x)^3 \left (a+c x^2\right )} \, dx &=\int \left (\frac{e (-B d+A e)}{\left (c d^2+a e^2\right ) (d+e x)^3}+\frac{e \left (-B c d^2+2 A c d e+a B e^2\right )}{\left (c d^2+a e^2\right )^2 (d+e x)^2}+\frac{c e \left (-B c d^3+3 A c d^2 e+3 a B d e^2-a A e^3\right )}{\left (c d^2+a e^2\right )^3 (d+e x)}+\frac{c \left (A c d \left (c d^2-3 a e^2\right )+a B e \left (3 c d^2-a e^2\right )+c \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\right ) x\right )}{\left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}\right ) \, dx\\ &=\frac{B d-A e}{2 \left (c d^2+a e^2\right ) (d+e x)^2}+\frac{B c d^2-2 A c d e-a B e^2}{\left (c d^2+a e^2\right )^2 (d+e x)}-\frac{c \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^3}+\frac{c \int \frac{A c d \left (c d^2-3 a e^2\right )+a B e \left (3 c d^2-a e^2\right )+c \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\right ) x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^3}\\ &=\frac{B d-A e}{2 \left (c d^2+a e^2\right ) (d+e x)^2}+\frac{B c d^2-2 A c d e-a B e^2}{\left (c d^2+a e^2\right )^2 (d+e x)}-\frac{c \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^3}+\frac{\left (c^2 \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\right )\right ) \int \frac{x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^3}+\frac{\left (c \left (A c d \left (c d^2-3 a e^2\right )+a B e \left (3 c d^2-a e^2\right )\right )\right ) \int \frac{1}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^3}\\ &=\frac{B d-A e}{2 \left (c d^2+a e^2\right ) (d+e x)^2}+\frac{B c d^2-2 A c d e-a B e^2}{\left (c d^2+a e^2\right )^2 (d+e x)}+\frac{\sqrt{c} \left (A c d \left (c d^2-3 a e^2\right )+a B e \left (3 c d^2-a e^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \left (c d^2+a e^2\right )^3}-\frac{c \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^3}+\frac{c \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\right ) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^3}\\ \end{align*}
Mathematica [A] time = 0.405469, size = 223, normalized size = 0.89 \[ \frac{c \log \left (a+c x^2\right ) \left (a A e^3-3 a B d e^2-3 A c d^2 e+B c d^3\right )+\frac{\left (a e^2+c d^2\right ) \left (B \left (c d^2 (3 d+2 e x)-a e^2 (d+2 e x)\right )-A e \left (a e^2+c d (5 d+4 e x)\right )\right )}{(d+e x)^2}-2 c \log (d+e x) \left (a A e^3-3 a B d e^2-3 A c d^2 e+B c d^3\right )+\frac{2 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (c d^2-3 a e^2\right )+a B e \left (3 c d^2-a e^2\right )\right )}{\sqrt{a}}}{2 \left (a e^2+c d^2\right )^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.011, size = 509, normalized size = 2. \begin{align*} -{\frac{Ae}{ \left ( 2\,a{e}^{2}+2\,c{d}^{2} \right ) \left ( ex+d \right ) ^{2}}}+{\frac{Bd}{ \left ( 2\,a{e}^{2}+2\,c{d}^{2} \right ) \left ( ex+d \right ) ^{2}}}-2\,{\frac{Acde}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( ex+d \right ) }}-{\frac{aB{e}^{2}}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( ex+d \right ) }}+{\frac{Bc{d}^{2}}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( ex+d \right ) }}-{\frac{c\ln \left ( ex+d \right ) aA{e}^{3}}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}}+3\,{\frac{{c}^{2}\ln \left ( ex+d \right ) A{d}^{2}e}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}}+3\,{\frac{c\ln \left ( ex+d \right ) aBd{e}^{2}}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}}-{\frac{{c}^{2}\ln \left ( ex+d \right ) B{d}^{3}}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}}+{\frac{c\ln \left ( c{x}^{2}+a \right ) Aa{e}^{3}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}}-{\frac{3\,{c}^{2}\ln \left ( c{x}^{2}+a \right ) A{d}^{2}e}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}}-{\frac{3\,c\ln \left ( c{x}^{2}+a \right ) Bad{e}^{2}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}}+{\frac{{c}^{2}\ln \left ( c{x}^{2}+a \right ) B{d}^{3}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}}-3\,{\frac{Ada{c}^{2}{e}^{2}}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) }+{\frac{A{c}^{3}{d}^{3}}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-{\frac{B{e}^{3}{a}^{2}c}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+3\,{\frac{B{c}^{2}a{d}^{2}e}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21858, size = 517, normalized size = 2.06 \begin{align*} \frac{{\left (B c^{2} d^{3} - 3 \, A c^{2} d^{2} e - 3 \, B a c d e^{2} + A a c e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \,{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )}} - \frac{{\left (B c^{2} d^{3} e - 3 \, A c^{2} d^{2} e^{2} - 3 \, B a c d e^{3} + A a c e^{4}\right )} \log \left ({\left | x e + d \right |}\right )}{c^{3} d^{6} e + 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} + a^{3} e^{7}} + \frac{{\left (A c^{3} d^{3} + 3 \, B a c^{2} d^{2} e - 3 \, A a c^{2} d e^{2} - B a^{2} c e^{3}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} \sqrt{a c}} + \frac{3 \, B c^{2} d^{5} - 5 \, A c^{2} d^{4} e + 2 \, B a c d^{3} e^{2} - 6 \, A a c d^{2} e^{3} - B a^{2} d e^{4} - A a^{2} e^{5} + 2 \,{\left (B c^{2} d^{4} e - 2 \, A c^{2} d^{3} e^{2} - 2 \, A a c d e^{4} - B a^{2} e^{5}\right )} x}{2 \,{\left (c d^{2} + a e^{2}\right )}^{3}{\left (x e + d\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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