Optimal. Leaf size=64 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) (A c d-a B e)}{\sqrt{a} c^{3/2}}+\frac{\log \left (a+c x^2\right ) (A e+B d)}{2 c}+\frac{B e x}{c} \]
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Rubi [A] time = 0.0429654, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {774, 635, 205, 260} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) (A c d-a B e)}{\sqrt{a} c^{3/2}}+\frac{\log \left (a+c x^2\right ) (A e+B d)}{2 c}+\frac{B e x}{c} \]
Antiderivative was successfully verified.
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Rule 774
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{(A+B x) (d+e x)}{a+c x^2} \, dx &=\frac{B e x}{c}+\frac{\int \frac{A c d-a B e+c (B d+A e) x}{a+c x^2} \, dx}{c}\\ &=\frac{B e x}{c}+(B d+A e) \int \frac{x}{a+c x^2} \, dx+\frac{(A c d-a B e) \int \frac{1}{a+c x^2} \, dx}{c}\\ &=\frac{B e x}{c}+\frac{(A c d-a B e) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} c^{3/2}}+\frac{(B d+A e) \log \left (a+c x^2\right )}{2 c}\\ \end{align*}
Mathematica [A] time = 0.05376, size = 65, normalized size = 1.02 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) (a B e-A c d)}{\sqrt{a} c^{3/2}}+\frac{\log \left (a+c x^2\right ) (A e+B d)}{2 c}+\frac{B e x}{c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 78, normalized size = 1.2 \begin{align*}{\frac{Bex}{c}}+{\frac{\ln \left ( c{x}^{2}+a \right ) Ae}{2\,c}}+{\frac{\ln \left ( c{x}^{2}+a \right ) Bd}{2\,c}}+{Ad\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-{\frac{aBe}{c}\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 = 1.85775, size = 350, normalized size = 5.47 \begin{align*} \left [\frac{2 \, B a c e x +{\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 \, a c^{2}}, \frac{2 \, B a c e x + 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 \, a c^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.65369, size = 212, normalized size = 3.31 \begin{align*} \frac{B e x}{c} + \left (\frac{A e + B d}{2 c} - \frac{\sqrt{- a c^{3}} \left (- A c d + B a e\right )}{2 a c^{3}}\right ) \log{\left (x + \frac{A a e + B a d - 2 a c \left (\frac{A e + B d}{2 c} - \frac{\sqrt{- a c^{3}} \left (- A c d + B a e\right )}{2 a c^{3}}\right )}{- A c d + B a e} \right )} + \left (\frac{A e + B d}{2 c} + \frac{\sqrt{- a c^{3}} \left (- A c d + B a e\right )}{2 a c^{3}}\right ) \log{\left (x + \frac{A a e + B a d - 2 a c \left (\frac{A e + B d}{2 c} + \frac{\sqrt{- a c^{3}} \left (- A c d + B a e\right )}{2 a c^{3}}\right )}{- A c d + B a e} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13072, size = 80, normalized size = 1.25 \begin{align*} \frac{B x e}{c} + \frac{{\left (B d + A e\right )} \log \left (c x^{2} + a\right )}{2 \, c} + \frac{{\left (A c d - B a e\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{\sqrt{a c} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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