Optimal. Leaf size=133 \[ \frac{\log \left (a+c x^2\right ) (a C e-A c e+B c d)}{2 c \left (a e^2+c d^2\right )}+\frac{\log (d+e x) \left (A e^2-B d e+C d^2\right )}{e \left (a e^2+c d^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) (a B e-a C d+A c d)}{\sqrt{a} \sqrt{c} \left (a e^2+c d^2\right )} \]
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Rubi [A] time = 0.162396, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {1629, 635, 205, 260} \[ \frac{\log \left (a+c x^2\right ) (a C e-A c e+B c d)}{2 c \left (a e^2+c d^2\right )}+\frac{\log (d+e x) \left (A e^2-B d e+C d^2\right )}{e \left (a e^2+c d^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) (a B e-a C d+A c d)}{\sqrt{a} \sqrt{c} \left (a e^2+c d^2\right )} \]
Antiderivative was successfully verified.
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Rule 1629
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{A+B x+C x^2}{(d+e x) \left (a+c x^2\right )} \, dx &=\int \left (\frac{C d^2-B d e+A e^2}{\left (c d^2+a e^2\right ) (d+e x)}+\frac{A c d-a C d+a B e+(B c d-A c e+a C e) x}{\left (c d^2+a e^2\right ) \left (a+c x^2\right )}\right ) \, dx\\ &=\frac{\left (C d^2-B d e+A e^2\right ) \log (d+e x)}{e \left (c d^2+a e^2\right )}+\frac{\int \frac{A c d-a C d+a B e+(B c d-A c e+a C e) x}{a+c x^2} \, dx}{c d^2+a e^2}\\ &=\frac{\left (C d^2-B d e+A e^2\right ) \log (d+e x)}{e \left (c d^2+a e^2\right )}+\frac{(A c d-a C d+a B e) \int \frac{1}{a+c x^2} \, dx}{c d^2+a e^2}+\frac{(B c d-A c e+a C e) \int \frac{x}{a+c x^2} \, dx}{c d^2+a e^2}\\ &=\frac{(A c d-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{\left (C d^2-B d e+A e^2\right ) \log (d+e x)}{e \left (c d^2+a e^2\right )}+\frac{(B c d-A c e+a C e) \log \left (a+c x^2\right )}{2 c \left (c d^2+a e^2\right )}\\ \end{align*}
Mathematica [A] time = 0.115492, size = 120, normalized size = 0.9 \[ \frac{\sqrt{a} \left (e \log \left (a+c x^2\right ) (a C e-A c e+B c d)+2 c \log (d+e x) \left (A e^2-B d e+C d^2\right )\right )+2 \sqrt{c} e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) (a B e-a C d+A c d)}{2 \sqrt{a} c e \left (a e^2+c d^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 247, normalized size = 1.9 \begin{align*} -{\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{\ln \left ( c{x}^{2}+a \right ) aCe}{ \left ( 2\,a{e}^{2}+2\,c{d}^{2} \right ) c}}+{\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}}}}-{\frac{Cad}{a{e}^{2}+c{d}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{e\ln \left ( ex+d \right ) A}{a{e}^{2}+c{d}^{2}}}-{\frac{\ln \left ( ex+d \right ) Bd}{a{e}^{2}+c{d}^{2}}}+{\frac{\ln \left ( ex+d \right ) C{d}^{2}}{ \left ( a{e}^{2}+c{d}^{2} \right ) e}} \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 = 47.1002, size = 575, normalized size = 4.32 \begin{align*} \left [-\frac{{\left (B a e^{2} -{\left (C a - A c\right )} d 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 e +{\left (C a^{2} - A a c\right )} e^{2}\right )} \log \left (c x^{2} + a\right ) - 2 \,{\left (C a c d^{2} - B a c d e + A a c e^{2}\right )} \log \left (e x + d\right )}{2 \,{\left (a c^{2} d^{2} e + a^{2} c e^{3}\right )}}, \frac{2 \,{\left (B a e^{2} -{\left (C a - A c\right )} d e\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) +{\left (B a c d e +{\left (C a^{2} - A a c\right )} e^{2}\right )} \log \left (c x^{2} + a\right ) + 2 \,{\left (C a c d^{2} - B a c d e + A a c e^{2}\right )} \log \left (e x + d\right )}{2 \,{\left (a c^{2} d^{2} e + a^{2} c e^{3}\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 = 1.13254, size = 169, normalized size = 1.27 \begin{align*} \frac{{\left (B c d + C a e - A c e\right )} \log \left (c x^{2} + a\right )}{2 \,{\left (c^{2} d^{2} + a c e^{2}\right )}} + \frac{{\left (C d^{2} - B d e + A e^{2}\right )} \log \left ({\left | x e + d \right |}\right )}{c d^{2} e + a e^{3}} - \frac{{\left (C a d - 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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