Optimal. Leaf size=77 \[ \frac{e x (A c e-b B e+2 B c d)}{c^2}+\frac{(b B-A c) (c d-b e)^2 \log (b+c x)}{b c^3}+\frac{A d^2 \log (x)}{b}+\frac{B e^2 x^2}{2 c} \]
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Rubi [A] time = 0.0863375, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {771} \[ \frac{e x (A c e-b B e+2 B c d)}{c^2}+\frac{(b B-A c) (c d-b e)^2 \log (b+c x)}{b c^3}+\frac{A d^2 \log (x)}{b}+\frac{B e^2 x^2}{2 c} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int \frac{(A+B x) (d+e x)^2}{b x+c x^2} \, dx &=\int \left (\frac{e (2 B c d-b B e+A c e)}{c^2}+\frac{A d^2}{b x}+\frac{B e^2 x}{c}+\frac{(b B-A c) (-c d+b e)^2}{b c^2 (b+c x)}\right ) \, dx\\ &=\frac{e (2 B c d-b B e+A c e) x}{c^2}+\frac{B e^2 x^2}{2 c}+\frac{A d^2 \log (x)}{b}+\frac{(b B-A c) (c d-b e)^2 \log (b+c x)}{b c^3}\\ \end{align*}
Mathematica [A] time = 0.0497905, size = 74, normalized size = 0.96 \[ \frac{b c e x (2 A c e+B (-2 b e+4 c d+c e x))+2 (b B-A c) (c d-b e)^2 \log (b+c x)+2 A c^3 d^2 \log (x)}{2 b c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 144, normalized size = 1.9 \begin{align*}{\frac{B{e}^{2}{x}^{2}}{2\,c}}+{\frac{{e}^{2}Ax}{c}}-{\frac{B{e}^{2}bx}{{c}^{2}}}+2\,{\frac{Bdex}{c}}+{\frac{A{d}^{2}\ln \left ( x \right ) }{b}}-{\frac{b\ln \left ( cx+b \right ) A{e}^{2}}{{c}^{2}}}+2\,{\frac{\ln \left ( cx+b \right ) Ade}{c}}-{\frac{\ln \left ( cx+b \right ) A{d}^{2}}{b}}+{\frac{{b}^{2}\ln \left ( cx+b \right ) B{e}^{2}}{{c}^{3}}}-2\,{\frac{b\ln \left ( cx+b \right ) Bde}{{c}^{2}}}+{\frac{\ln \left ( cx+b \right ) B{d}^{2}}{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16851, size = 155, normalized size = 2.01 \begin{align*} \frac{A d^{2} \log \left (x\right )}{b} + \frac{B c e^{2} x^{2} + 2 \,{\left (2 \, B c d e -{\left (B b - A c\right )} e^{2}\right )} x}{2 \, c^{2}} + \frac{{\left ({\left (B b c^{2} - A c^{3}\right )} d^{2} - 2 \,{\left (B b^{2} c - A b c^{2}\right )} d e +{\left (B b^{3} - A b^{2} c\right )} e^{2}\right )} \log \left (c x + b\right )}{b c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44331, size = 261, normalized size = 3.39 \begin{align*} \frac{B b c^{2} e^{2} x^{2} + 2 \, A c^{3} d^{2} \log \left (x\right ) + 2 \,{\left (2 \, B b c^{2} d e -{\left (B b^{2} c - A b c^{2}\right )} e^{2}\right )} x + 2 \,{\left ({\left (B b c^{2} - A c^{3}\right )} d^{2} - 2 \,{\left (B b^{2} c - A b c^{2}\right )} d e +{\left (B b^{3} - A b^{2} c\right )} e^{2}\right )} \log \left (c x + b\right )}{2 \, b c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.50883, size = 163, normalized size = 2.12 \begin{align*} \frac{A d^{2} \log{\left (x \right )}}{b} + \frac{B e^{2} x^{2}}{2 c} - \frac{x \left (- A c e^{2} + B b e^{2} - 2 B c d e\right )}{c^{2}} + \frac{\left (- A c + B b\right ) \left (b e - c d\right )^{2} \log{\left (x + \frac{- A b c^{2} d^{2} + \frac{b \left (- A c + B b\right ) \left (b e - c d\right )^{2}}{c}}{- A b^{2} c e^{2} + 2 A b c^{2} d e - 2 A c^{3} d^{2} + B b^{3} e^{2} - 2 B b^{2} c d e + B b c^{2} d^{2}} \right )}}{b c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18164, size = 158, normalized size = 2.05 \begin{align*} \frac{A d^{2} \log \left ({\left | x \right |}\right )}{b} + \frac{B c x^{2} e^{2} + 4 \, B c d x e - 2 \, B b x e^{2} + 2 \, A c x e^{2}}{2 \, c^{2}} + \frac{{\left (B b c^{2} d^{2} - A c^{3} d^{2} - 2 \, B b^{2} c d e + 2 \, A b c^{2} d e + B b^{3} e^{2} - A b^{2} c e^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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