Optimal. Leaf size=173 \[ \frac{\log \left (a+c x^2\right ) \left (-a B e^2-2 A c d e+B c d^2\right )}{2 \left (a e^2+c d^2\right )^2}+\frac{B d-A e}{(d+e x) \left (a e^2+c d^2\right )}-\frac{\log (d+e x) \left (-a B e^2-2 A c d e+B c d^2\right )}{\left (a e^2+c d^2\right )^2}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (-a A e^2+2 a B d e+A c d^2\right )}{\sqrt{a} \left (a e^2+c d^2\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.169804, antiderivative size = 173, 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} \[ \frac{\log \left (a+c x^2\right ) \left (-a B e^2-2 A c d e+B c d^2\right )}{2 \left (a e^2+c d^2\right )^2}+\frac{B d-A e}{(d+e x) \left (a e^2+c d^2\right )}-\frac{\log (d+e x) \left (-a B e^2-2 A c d e+B c d^2\right )}{\left (a e^2+c d^2\right )^2}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (-a A e^2+2 a B d e+A c d^2\right )}{\sqrt{a} \left (a e^2+c d^2\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 801
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{A+B x}{(d+e x)^2 \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)^2}+\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)}+\frac{c \left (A c d^2+2 a B d e-a A e^2+\left (B c d^2-2 A c d e-a B e^2\right ) x\right )}{\left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}\right ) \, dx\\ &=\frac{B d-A e}{\left (c d^2+a e^2\right ) (d+e x)}-\frac{\left (B c d^2-2 A c d e-a B e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^2}+\frac{c \int \frac{A c d^2+2 a B d e-a A e^2+\left (B c d^2-2 A c d e-a B e^2\right ) x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^2}\\ &=\frac{B d-A e}{\left (c d^2+a e^2\right ) (d+e x)}-\frac{\left (B c d^2-2 A c d e-a B e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^2}+\frac{\left (c \left (A c d^2+2 a B d e-a A e^2\right )\right ) \int \frac{1}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^2}+\frac{\left (c \left (B c d^2-2 A c d e-a B e^2\right )\right ) \int \frac{x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^2}\\ &=\frac{B d-A e}{\left (c d^2+a e^2\right ) (d+e x)}+\frac{\sqrt{c} \left (A c d^2+2 a B d e-a A e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \left (c d^2+a e^2\right )^2}-\frac{\left (B c d^2-2 A c d e-a B e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^2}+\frac{\left (B c d^2-2 A c d e-a B e^2\right ) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^2}\\ \end{align*}
Mathematica [A] time = 0.196738, size = 148, normalized size = 0.86 \[ \frac{\log \left (a+c x^2\right ) \left (-a B e^2-2 A c d e+B c d^2\right )+\frac{2 \left (a e^2+c d^2\right ) (B d-A e)}{d+e x}+\log (d+e x) \left (2 a B e^2+4 A c d e-2 B c d^2\right )+\frac{2 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (-a A e^2+2 a B d e+A c d^2\right )}{\sqrt{a}}}{2 \left (a e^2+c d^2\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.01, size = 312, normalized size = 1.8 \begin{align*} -{\frac{Ae}{ \left ( a{e}^{2}+c{d}^{2} \right ) \left ( ex+d \right ) }}+{\frac{Bd}{ \left ( a{e}^{2}+c{d}^{2} \right ) \left ( ex+d \right ) }}+2\,{\frac{\ln \left ( ex+d \right ) Acde}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}}+{\frac{\ln \left ( ex+d \right ) aB{e}^{2}}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}}-{\frac{\ln \left ( ex+d \right ) Bc{d}^{2}}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}}-{\frac{c\ln \left ( c{x}^{2}+a \right ) Ade}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}}-{\frac{\ln \left ( c{x}^{2}+a \right ) aB{e}^{2}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}}+{\frac{c\ln \left ( c{x}^{2}+a \right ) B{d}^{2}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}}-{\frac{aAc{e}^{2}}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{A{c}^{2}{d}^{2}}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+2\,{\frac{aBcde}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 49.3874, size = 1211, normalized size = 7. \begin{align*} \left [\frac{2 \, B c d^{3} - 2 \, A c d^{2} e + 2 \, B a d e^{2} - 2 \, A a e^{3} -{\left (A c d^{3} + 2 \, B a d^{2} e - A a d e^{2} +{\left (A c d^{2} e + 2 \, B a d e^{2} - A a e^{3}\right )} x\right )} \sqrt{-\frac{c}{a}} \log \left (\frac{c x^{2} - 2 \, a x \sqrt{-\frac{c}{a}} - a}{c x^{2} + a}\right ) +{\left (B c d^{3} - 2 \, A c d^{2} e - B a d e^{2} +{\left (B c d^{2} e - 2 \, A c d e^{2} - B a e^{3}\right )} x\right )} \log \left (c x^{2} + a\right ) - 2 \,{\left (B c d^{3} - 2 \, A c d^{2} e - B a d e^{2} +{\left (B c d^{2} e - 2 \, A c d e^{2} - B a e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (c^{2} d^{5} + 2 \, a c d^{3} e^{2} + a^{2} d e^{4} +{\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x\right )}}, \frac{2 \, B c d^{3} - 2 \, A c d^{2} e + 2 \, B a d e^{2} - 2 \, A a e^{3} + 2 \,{\left (A c d^{3} + 2 \, B a d^{2} e - A a d e^{2} +{\left (A c d^{2} e + 2 \, B a d e^{2} - A a e^{3}\right )} x\right )} \sqrt{\frac{c}{a}} \arctan \left (x \sqrt{\frac{c}{a}}\right ) +{\left (B c d^{3} - 2 \, A c d^{2} e - B a d e^{2} +{\left (B c d^{2} e - 2 \, A c d e^{2} - B a e^{3}\right )} x\right )} \log \left (c x^{2} + a\right ) - 2 \,{\left (B c d^{3} - 2 \, A c d^{2} e - B a d e^{2} +{\left (B c d^{2} e - 2 \, A c d e^{2} - B a e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (c^{2} d^{5} + 2 \, a c d^{3} e^{2} + a^{2} d e^{4} +{\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.2111, size = 309, normalized size = 1.79 \begin{align*} \frac{{\left (A c^{2} d^{2} e^{2} + 2 \, B a c d e^{3} - A a c e^{4}\right )} \arctan \left (\frac{{\left (c d - \frac{c d^{2}}{x e + d} - \frac{a e^{2}}{x e + d}\right )} e^{\left (-1\right )}}{\sqrt{a c}}\right ) e^{\left (-2\right )}}{{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt{a c}} + \frac{{\left (B c d^{2} - 2 \, A c d e - B a e^{2}\right )} \log \left (c - \frac{2 \, c d}{x e + d} + \frac{c d^{2}}{{\left (x e + d\right )}^{2}} + \frac{a e^{2}}{{\left (x e + d\right )}^{2}}\right )}{2 \,{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} + \frac{\frac{B d e^{2}}{x e + d} - \frac{A e^{3}}{x e + d}}{c d^{2} e^{2} + a e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]