Optimal. Leaf size=128 \[ \frac{e x \left (A c e (3 c d-b e)+B \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )\right )}{c^3}+\frac{e^2 x^2 (A c e-b B e+3 B c d)}{2 c^2}+\frac{(b B-A c) (c d-b e)^3 \log (b+c x)}{b c^4}+\frac{A d^3 \log (x)}{b}+\frac{B e^3 x^3}{3 c} \]
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Rubi [A] time = 0.156239, 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} \[ \frac{e x \left (A c e (3 c d-b e)+B \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )\right )}{c^3}+\frac{e^2 x^2 (A c e-b B e+3 B c d)}{2 c^2}+\frac{(b B-A c) (c d-b e)^3 \log (b+c x)}{b c^4}+\frac{A d^3 \log (x)}{b}+\frac{B e^3 x^3}{3 c} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int \frac{(A+B x) (d+e x)^3}{b x+c x^2} \, dx &=\int \left (\frac{e \left (A c e (3 c d-b e)+B \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right )}{c^3}+\frac{A d^3}{b x}+\frac{e^2 (3 B c d-b B e+A c e) x}{c^2}+\frac{B e^3 x^2}{c}-\frac{(b B-A c) (-c d+b e)^3}{b c^3 (b+c x)}\right ) \, dx\\ &=\frac{e \left (A c e (3 c d-b e)+B \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right ) x}{c^3}+\frac{e^2 (3 B c d-b B e+A c e) x^2}{2 c^2}+\frac{B e^3 x^3}{3 c}+\frac{A d^3 \log (x)}{b}+\frac{(b B-A c) (c d-b e)^3 \log (b+c x)}{b c^4}\\ \end{align*}
Mathematica [A] time = 0.0874874, size = 118, normalized size = 0.92 \[ \frac{b c e x \left (3 A c e (-2 b e+6 c d+c e x)+B \left (6 b^2 e^2-3 b c e (6 d+e x)+c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )\right )-6 (b B-A c) (b e-c d)^3 \log (b+c x)+6 A c^4 d^3 \log (x)}{6 b c^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 252, normalized size = 2. \begin{align*}{\frac{B{e}^{3}{x}^{3}}{3\,c}}+{\frac{A{e}^{3}{x}^{2}}{2\,c}}-{\frac{B{e}^{3}{x}^{2}b}{2\,{c}^{2}}}+{\frac{3\,{e}^{2}B{x}^{2}d}{2\,c}}-{\frac{A{e}^{3}bx}{{c}^{2}}}+3\,{\frac{Ad{e}^{2}x}{c}}+{\frac{B{e}^{3}{b}^{2}x}{{c}^{3}}}-3\,{\frac{B{e}^{2}bdx}{{c}^{2}}}+3\,{\frac{B{d}^{2}ex}{c}}+{\frac{A{d}^{3}\ln \left ( x \right ) }{b}}+{\frac{{b}^{2}\ln \left ( cx+b \right ) A{e}^{3}}{{c}^{3}}}-3\,{\frac{b\ln \left ( cx+b \right ) Ad{e}^{2}}{{c}^{2}}}+3\,{\frac{\ln \left ( cx+b \right ) A{d}^{2}e}{c}}-{\frac{\ln \left ( cx+b \right ) A{d}^{3}}{b}}-{\frac{{b}^{3}\ln \left ( cx+b \right ) B{e}^{3}}{{c}^{4}}}+3\,{\frac{{b}^{2}\ln \left ( cx+b \right ) Bd{e}^{2}}{{c}^{3}}}-3\,{\frac{b\ln \left ( cx+b \right ) B{d}^{2}e}{{c}^{2}}}+{\frac{\ln \left ( cx+b \right ) B{d}^{3}}{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10675, size = 270, normalized size = 2.11 \begin{align*} \frac{A d^{3} \log \left (x\right )}{b} + \frac{2 \, B c^{2} e^{3} x^{3} + 3 \,{\left (3 \, B c^{2} d e^{2} -{\left (B b c - A c^{2}\right )} e^{3}\right )} x^{2} + 6 \,{\left (3 \, B c^{2} d^{2} e - 3 \,{\left (B b c - A c^{2}\right )} d e^{2} +{\left (B b^{2} - A b c\right )} e^{3}\right )} x}{6 \, c^{3}} + \frac{{\left ({\left (B b c^{3} - 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 )} \log \left (c x + b\right )}{b c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51282, size = 433, normalized size = 3.38 \begin{align*} \frac{2 \, B b c^{3} e^{3} x^{3} + 6 \, A c^{4} d^{3} \log \left (x\right ) + 3 \,{\left (3 \, B b c^{3} d e^{2} -{\left (B b^{2} c^{2} - A b c^{3}\right )} e^{3}\right )} x^{2} + 6 \,{\left (3 \, B b c^{3} d^{2} e - 3 \,{\left (B b^{2} c^{2} - A b c^{3}\right )} d e^{2} +{\left (B b^{3} c - A b^{2} c^{2}\right )} e^{3}\right )} x + 6 \,{\left ({\left (B b c^{3} - 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 )} \log \left (c x + b\right )}{6 \, b c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 5.62946, size = 260, normalized size = 2.03 \begin{align*} \frac{A d^{3} \log{\left (x \right )}}{b} + \frac{B e^{3} x^{3}}{3 c} - \frac{x^{2} \left (- A c e^{3} + B b e^{3} - 3 B c d e^{2}\right )}{2 c^{2}} + \frac{x \left (- A b c e^{3} + 3 A c^{2} d e^{2} + B b^{2} e^{3} - 3 B b c d e^{2} + 3 B c^{2} d^{2} e\right )}{c^{3}} - \frac{\left (- A c + B b\right ) \left (b e - c d\right )^{3} \log{\left (x + \frac{A b c^{3} d^{3} + \frac{b \left (- A c + B b\right ) \left (b e - c d\right )^{3}}{c}}{- 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 c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24461, size = 279, normalized size = 2.18 \begin{align*} \frac{A d^{3} \log \left ({\left | x \right |}\right )}{b} + \frac{2 \, B c^{2} x^{3} e^{3} + 9 \, B c^{2} d x^{2} e^{2} + 18 \, B c^{2} d^{2} x e - 3 \, B b c x^{2} e^{3} + 3 \, A c^{2} x^{2} e^{3} - 18 \, B b c d x e^{2} + 18 \, A c^{2} d x e^{2} + 6 \, B b^{2} x e^{3} - 6 \, A b c x e^{3}}{6 \, c^{3}} + \frac{{\left (B b c^{3} d^{3} - 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 )} \log \left ({\left | c x + b \right |}\right )}{b c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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