Optimal. Leaf size=109 \[ \frac{\log \left (a+c x^2\right ) (B d-A e)}{2 \left (a e^2+c d^2\right )}-\frac{(B d-A e) \log (d+e x)}{a e^2+c d^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) (a B e+A c d)}{\sqrt{a} \sqrt{c} \left (a e^2+c d^2\right )} \]
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Rubi [A] time = 0.10618, antiderivative size = 109, 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} \[ \frac{\log \left (a+c x^2\right ) (B d-A e)}{2 \left (a e^2+c d^2\right )}-\frac{(B d-A e) \log (d+e x)}{a e^2+c d^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) (a B e+A c d)}{\sqrt{a} \sqrt{c} \left (a e^2+c d^2\right )} \]
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) \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)}+\frac{A c d+a B e+c (B d-A e) x}{\left (c d^2+a e^2\right ) \left (a+c x^2\right )}\right ) \, dx\\ &=-\frac{(B d-A e) \log (d+e x)}{c d^2+a e^2}+\frac{\int \frac{A c d+a B e+c (B d-A e) x}{a+c x^2} \, dx}{c d^2+a e^2}\\ &=-\frac{(B d-A e) \log (d+e x)}{c d^2+a e^2}+\frac{(c (B d-A e)) \int \frac{x}{a+c x^2} \, dx}{c d^2+a e^2}+\frac{(A c d+a B e) \int \frac{1}{a+c x^2} \, dx}{c d^2+a e^2}\\ &=\frac{(A c d+a B e) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c} \left (c d^2+a e^2\right )}-\frac{(B d-A e) \log (d+e x)}{c d^2+a e^2}+\frac{(B d-A e) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )}\\ \end{align*}
Mathematica [A] time = 0.0674397, size = 91, normalized size = 0.83 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) (a B e+A c d)-\sqrt{a} \sqrt{c} (B d-A e) \left (2 \log (d+e x)-\log \left (a+c x^2\right )\right )}{2 \sqrt{a} \sqrt{c} \left (a e^2+c d^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 159, normalized size = 1.5 \begin{align*}{\frac{\ln \left ( ex+d \right ) Ae}{a{e}^{2}+c{d}^{2}}}-{\frac{\ln \left ( ex+d \right ) Bd}{a{e}^{2}+c{d}^{2}}}-{\frac{\ln \left ( c{x}^{2}+a \right ) Ae}{2\,a{e}^{2}+2\,c{d}^{2}}}+{\frac{\ln \left ( c{x}^{2}+a \right ) Bd}{2\,a{e}^{2}+2\,c{d}^{2}}}+{\frac{Acd}{a{e}^{2}+c{d}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{aBe}{a{e}^{2}+c{d}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \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 [A] time = 8.56581, size = 456, normalized size = 4.18 \begin{align*} \left [-\frac{{\left (A c d + B a e\right )} \sqrt{-a c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right ) -{\left (B a c d - A a c e\right )} \log \left (c x^{2} + a\right ) + 2 \,{\left (B a c d - A a c e\right )} \log \left (e x + d\right )}{2 \,{\left (a c^{2} d^{2} + a^{2} c e^{2}\right )}}, \frac{2 \,{\left (A c d + B a e\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) +{\left (B a c d - A a c e\right )} \log \left (c x^{2} + a\right ) - 2 \,{\left (B a c d - A a c e\right )} \log \left (e x + d\right )}{2 \,{\left (a c^{2} d^{2} + a^{2} c e^{2}\right )}}\right ] \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 = 1.13114, size = 140, normalized size = 1.28 \begin{align*} \frac{{\left (B d - A e\right )} \log \left (c x^{2} + a\right )}{2 \,{\left (c d^{2} + a e^{2}\right )}} - \frac{{\left (B d e - A e^{2}\right )} \log \left ({\left | x e + d \right |}\right )}{c d^{2} e + a e^{3}} + \frac{{\left (A c d + B a e\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{{\left (c d^{2} + a e^{2}\right )} \sqrt{a c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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