Optimal. Leaf size=128 \[ \frac{(b B-A c) (c d-b e)^3}{b^2 c^3 (b+c x)}+\frac{(c d-b e)^2 \log (b+c x) \left (-b c (B d-A e)+2 A c^2 d-2 b^2 B e\right )}{b^3 c^3}+\frac{d^2 \log (x) (3 A b e-2 A c d+b B d)}{b^3}-\frac{A d^3}{b^2 x}+\frac{B e^3 x}{c^2} \]
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Rubi [A] time = 0.158671, antiderivative size = 128, 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} \[ \frac{(b B-A c) (c d-b e)^3}{b^2 c^3 (b+c x)}+\frac{(c d-b e)^2 \log (b+c x) \left (-b c (B d-A e)+2 A c^2 d-2 b^2 B e\right )}{b^3 c^3}+\frac{d^2 \log (x) (3 A b e-2 A c d+b B d)}{b^3}-\frac{A d^3}{b^2 x}+\frac{B e^3 x}{c^2} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int \frac{(A+B x) (d+e x)^3}{\left (b x+c x^2\right )^2} \, dx &=\int \left (\frac{B e^3}{c^2}+\frac{A d^3}{b^2 x^2}+\frac{d^2 (b B d-2 A c d+3 A b e)}{b^3 x}+\frac{(b B-A c) (-c d+b e)^3}{b^2 c^2 (b+c x)^2}+\frac{(c d-b e)^2 \left (2 A c^2 d-2 b^2 B e-b c (B d-A e)\right )}{b^3 c^2 (b+c x)}\right ) \, dx\\ &=-\frac{A d^3}{b^2 x}+\frac{B e^3 x}{c^2}+\frac{(b B-A c) (c d-b e)^3}{b^2 c^3 (b+c x)}+\frac{d^2 (b B d-2 A c d+3 A b e) \log (x)}{b^3}+\frac{(c d-b e)^2 \left (2 A c^2 d-2 b^2 B e-b c (B d-A e)\right ) \log (b+c x)}{b^3 c^3}\\ \end{align*}
Mathematica [A] time = 0.0778754, size = 128, normalized size = 1. \[ -\frac{(b B-A c) (b e-c d)^3}{b^2 c^3 (b+c x)}+\frac{(c d-b e)^2 \log (b+c x) \left (A b c e+2 A c^2 d-2 b^2 B e-b B c d\right )}{b^3 c^3}+\frac{d^2 \log (x) (3 A b e-2 A c d+b B d)}{b^3}-\frac{A d^3}{b^2 x}+\frac{B e^3 x}{c^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 286, normalized size = 2.2 \begin{align*}{\frac{B{e}^{3}x}{{c}^{2}}}-{\frac{A{d}^{3}}{{b}^{2}x}}+3\,{\frac{{d}^{2}\ln \left ( x \right ) Ae}{{b}^{2}}}-2\,{\frac{{d}^{3}\ln \left ( x \right ) Ac}{{b}^{3}}}+{\frac{{d}^{3}\ln \left ( x \right ) B}{{b}^{2}}}+{\frac{\ln \left ( cx+b \right ) A{e}^{3}}{{c}^{2}}}-3\,{\frac{\ln \left ( cx+b \right ) A{d}^{2}e}{{b}^{2}}}+2\,{\frac{c\ln \left ( cx+b \right ) A{d}^{3}}{{b}^{3}}}-2\,{\frac{b\ln \left ( cx+b \right ) B{e}^{3}}{{c}^{3}}}+3\,{\frac{\ln \left ( cx+b \right ) Bd{e}^{2}}{{c}^{2}}}-{\frac{\ln \left ( cx+b \right ) B{d}^{3}}{{b}^{2}}}+{\frac{Ab{e}^{3}}{{c}^{2} \left ( cx+b \right ) }}-3\,{\frac{Ad{e}^{2}}{c \left ( cx+b \right ) }}+3\,{\frac{A{d}^{2}e}{b \left ( cx+b \right ) }}-{\frac{A{d}^{3}c}{{b}^{2} \left ( cx+b \right ) }}-{\frac{B{e}^{3}{b}^{2}}{{c}^{3} \left ( cx+b \right ) }}+3\,{\frac{Bbd{e}^{2}}{{c}^{2} \left ( cx+b \right ) }}-3\,{\frac{B{d}^{2}e}{c \left ( cx+b \right ) }}+{\frac{B{d}^{3}}{b \left ( cx+b \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09645, size = 304, normalized size = 2.38 \begin{align*} \frac{B e^{3} x}{c^{2}} - \frac{A b c^{3} d^{3} -{\left ({\left (B b c^{3} - 2 \, A c^{4}\right )} d^{3} - 3 \,{\left (B b^{2} c^{2} - A b c^{3}\right )} d^{2} e + 3 \,{\left (B b^{3} c - A b^{2} c^{2}\right )} d e^{2} -{\left (B b^{4} - A b^{3} c\right )} e^{3}\right )} x}{b^{2} c^{4} x^{2} + b^{3} c^{3} x} + \frac{{\left (3 \, A b d^{2} e +{\left (B b - 2 \, A c\right )} d^{3}\right )} \log \left (x\right )}{b^{3}} - \frac{{\left (3 \, A b c^{3} d^{2} e - 3 \, B b^{3} c d e^{2} +{\left (B b c^{3} - 2 \, A c^{4}\right )} d^{3} +{\left (2 \, B b^{4} - A b^{3} c\right )} e^{3}\right )} \log \left (c x + b\right )}{b^{3} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.50293, size = 717, normalized size = 5.6 \begin{align*} \frac{B b^{3} c^{2} e^{3} x^{3} + B b^{4} c e^{3} x^{2} - A b^{2} c^{3} d^{3} +{\left ({\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{3} - 3 \,{\left (B b^{3} c^{2} - A b^{2} c^{3}\right )} d^{2} e + 3 \,{\left (B b^{4} c - A b^{3} c^{2}\right )} d e^{2} -{\left (B b^{5} - A b^{4} c\right )} e^{3}\right )} x -{\left ({\left (3 \, A b c^{4} d^{2} e - 3 \, B b^{3} c^{2} d e^{2} +{\left (B b c^{4} - 2 \, A c^{5}\right )} d^{3} +{\left (2 \, B b^{4} c - A b^{3} c^{2}\right )} e^{3}\right )} x^{2} +{\left (3 \, A b^{2} c^{3} d^{2} e - 3 \, B b^{4} c d e^{2} +{\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{3} +{\left (2 \, B b^{5} - A b^{4} c\right )} e^{3}\right )} x\right )} \log \left (c x + b\right ) +{\left ({\left (3 \, A b c^{4} d^{2} e +{\left (B b c^{4} - 2 \, A c^{5}\right )} d^{3}\right )} x^{2} +{\left (3 \, A b^{2} c^{3} d^{2} e +{\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{3}\right )} x\right )} \log \left (x\right )}{b^{3} c^{4} x^{2} + b^{4} c^{3} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 11.9035, size = 502, normalized size = 3.92 \begin{align*} \frac{B e^{3} x}{c^{2}} - \frac{A b c^{3} d^{3} + x \left (- A b^{3} c e^{3} + 3 A b^{2} c^{2} d e^{2} - 3 A b c^{3} d^{2} e + 2 A c^{4} d^{3} + B b^{4} e^{3} - 3 B b^{3} c d e^{2} + 3 B b^{2} c^{2} d^{2} e - B b c^{3} d^{3}\right )}{b^{3} c^{3} x + b^{2} c^{4} x^{2}} + \frac{d^{2} \left (3 A b e - 2 A c d + B b d\right ) \log{\left (x + \frac{3 A b^{2} c^{2} d^{2} e - 2 A b c^{3} d^{3} + B b^{2} c^{2} d^{3} - b c^{2} d^{2} \left (3 A b e - 2 A c d + B b d\right )}{- A b^{3} c e^{3} + 6 A b c^{3} d^{2} e - 4 A c^{4} d^{3} + 2 B b^{4} e^{3} - 3 B b^{3} c d e^{2} + 2 B b c^{3} d^{3}} \right )}}{b^{3}} - \frac{\left (b e - c d\right )^{2} \left (- A b c e - 2 A c^{2} d + 2 B b^{2} e + B b c d\right ) \log{\left (x + \frac{3 A b^{2} c^{2} d^{2} e - 2 A b c^{3} d^{3} + B b^{2} c^{2} d^{3} + \frac{b \left (b e - c d\right )^{2} \left (- A b c e - 2 A c^{2} d + 2 B b^{2} e + B b c d\right )}{c}}{- A b^{3} c e^{3} + 6 A b c^{3} d^{2} e - 4 A c^{4} d^{3} + 2 B b^{4} e^{3} - 3 B b^{3} c d e^{2} + 2 B b c^{3} d^{3}} \right )}}{b^{3} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31509, size = 309, normalized size = 2.41 \begin{align*} \frac{B x e^{3}}{c^{2}} + \frac{{\left (B b d^{3} - 2 \, A c d^{3} + 3 \, A b d^{2} e\right )} \log \left ({\left | x \right |}\right )}{b^{3}} - \frac{{\left (B b c^{3} d^{3} - 2 \, A c^{4} d^{3} + 3 \, A b c^{3} d^{2} e - 3 \, B b^{3} c d e^{2} + 2 \, B b^{4} e^{3} - A b^{3} c e^{3}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c^{3}} - \frac{A b c^{2} d^{3} - \frac{{\left (B b c^{3} d^{3} - 2 \, A c^{4} d^{3} - 3 \, B b^{2} c^{2} d^{2} e + 3 \, A b c^{3} d^{2} e + 3 \, B b^{3} c d e^{2} - 3 \, A b^{2} c^{2} d e^{2} - B b^{4} e^{3} + A b^{3} c e^{3}\right )} x}{c}}{{\left (c x + b\right )} b^{2} c^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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