Optimal. Leaf size=232 \[ -\frac{x \left (A c e (3 c d-2 b e)-B \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )\right )}{e^5}+\frac{\log (d+e x) \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{e^6}+\frac{d^2 (B d-A e) (c d-b e)^2}{2 e^6 (d+e x)^2}-\frac{c x^2 (-A c e-2 b B e+3 B c d)}{2 e^4}-\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6 (d+e x)}+\frac{B c^2 x^3}{3 e^3} \]
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Rubi [A] time = 0.320576, antiderivative size = 232, 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} \[ -\frac{x \left (A c e (3 c d-2 b e)-B \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )\right )}{e^5}+\frac{\log (d+e x) \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{e^6}+\frac{d^2 (B d-A e) (c d-b e)^2}{2 e^6 (d+e x)^2}-\frac{c x^2 (-A c e-2 b B e+3 B c d)}{2 e^4}-\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6 (d+e x)}+\frac{B c^2 x^3}{3 e^3} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^3} \, dx &=\int \left (\frac{-A c e (3 c d-2 b e)+B \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )}{e^5}+\frac{c (-3 B c d+2 b B e+A c e) x}{e^4}+\frac{B c^2 x^2}{e^3}-\frac{d^2 (B d-A e) (c d-b e)^2}{e^5 (d+e x)^3}+\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^5 (d+e x)^2}+\frac{A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )}{e^5 (d+e x)}\right ) \, dx\\ &=-\frac{\left (A c e (3 c d-2 b e)-B \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )\right ) x}{e^5}-\frac{c (3 B c d-2 b B e-A c e) x^2}{2 e^4}+\frac{B c^2 x^3}{3 e^3}+\frac{d^2 (B d-A e) (c d-b e)^2}{2 e^6 (d+e x)^2}-\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6 (d+e x)}+\frac{\left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) \log (d+e x)}{e^6}\\ \end{align*}
Mathematica [A] time = 0.177888, size = 219, normalized size = 0.94 \[ \frac{6 e x \left (A c e (2 b e-3 c d)+B \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )\right )+6 \log (d+e x) \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )+B d \left (-3 b^2 e^2+12 b c d e-10 c^2 d^2\right )\right )+\frac{3 d^2 (B d-A e) (c d-b e)^2}{(d+e x)^2}+3 c e^2 x^2 (A c e+2 b B e-3 B c d)-\frac{6 d (c d-b e) (2 A e (b e-2 c d)+B d (5 c d-3 b e))}{d+e x}+2 B c^2 e^3 x^3}{6 e^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 420, normalized size = 1.8 \begin{align*}{\frac{B{c}^{2}{x}^{3}}{3\,{e}^{3}}}+{\frac{{b}^{2}Bx}{{e}^{3}}}+{\frac{A{d}^{3}bc}{{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{{d}^{4}Bbc}{{e}^{5} \left ( ex+d \right ) ^{2}}}-6\,{\frac{A{d}^{2}bc}{{e}^{4} \left ( ex+d \right ) }}+8\,{\frac{Bcb{d}^{3}}{{e}^{5} \left ( ex+d \right ) }}-6\,{\frac{\ln \left ( ex+d \right ) Abcd}{{e}^{4}}}+12\,{\frac{\ln \left ( ex+d \right ) Bbc{d}^{2}}{{e}^{5}}}-6\,{\frac{Bcbdx}{{e}^{4}}}+{\frac{\ln \left ( ex+d \right ) A{b}^{2}}{{e}^{3}}}+{\frac{A{c}^{2}{x}^{2}}{2\,{e}^{3}}}+2\,{\frac{Abcx}{{e}^{3}}}-3\,{\frac{A{c}^{2}dx}{{e}^{4}}}-{\frac{{d}^{2}A{b}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{{d}^{4}A{c}^{2}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}+6\,{\frac{\ln \left ( ex+d \right ) A{c}^{2}{d}^{2}}{{e}^{5}}}-3\,{\frac{\ln \left ( ex+d \right ) B{b}^{2}d}{{e}^{4}}}-10\,{\frac{\ln \left ( ex+d \right ) B{c}^{2}{d}^{3}}{{e}^{6}}}+4\,{\frac{A{d}^{3}{c}^{2}}{{e}^{5} \left ( ex+d \right ) }}-3\,{\frac{{b}^{2}B{d}^{2}}{{e}^{4} \left ( ex+d \right ) }}+6\,{\frac{B{c}^{2}{d}^{2}x}{{e}^{5}}}+{\frac{B{x}^{2}bc}{{e}^{3}}}-{\frac{3\,B{c}^{2}{x}^{2}d}{2\,{e}^{4}}}-5\,{\frac{B{c}^{2}{d}^{4}}{{e}^{6} \left ( ex+d \right ) }}+{\frac{B{d}^{3}{b}^{2}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}+{\frac{B{c}^{2}{d}^{5}}{2\,{e}^{6} \left ( ex+d \right ) ^{2}}}+2\,{\frac{dA{b}^{2}}{{e}^{3} \left ( ex+d \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0788, size = 408, normalized size = 1.76 \begin{align*} -\frac{9 \, B c^{2} d^{5} - 3 \, A b^{2} d^{2} e^{3} - 7 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e + 5 \,{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} + 2 \,{\left (5 \, B c^{2} d^{4} e - 2 \, A b^{2} d e^{4} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x}{2 \,{\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} + \frac{2 \, B c^{2} e^{2} x^{3} - 3 \,{\left (3 \, B c^{2} d e -{\left (2 \, B b c + A c^{2}\right )} e^{2}\right )} x^{2} + 6 \,{\left (6 \, B c^{2} d^{2} - 3 \,{\left (2 \, B b c + A c^{2}\right )} d e +{\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )} x}{6 \, e^{5}} - \frac{{\left (10 \, B c^{2} d^{3} - A b^{2} e^{3} - 6 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )} \log \left (e x + d\right )}{e^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.41795, size = 1019, normalized size = 4.39 \begin{align*} \frac{2 \, B c^{2} e^{5} x^{5} - 27 \, B c^{2} d^{5} + 9 \, A b^{2} d^{2} e^{3} + 21 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e - 15 \,{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} -{\left (5 \, B c^{2} d e^{4} - 3 \,{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 2 \,{\left (10 \, B c^{2} d^{2} e^{3} - 6 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4} + 3 \,{\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} + 3 \,{\left (21 \, B c^{2} d^{3} e^{2} - 11 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 4 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 6 \,{\left (B c^{2} d^{4} e + 2 \, A b^{2} d e^{4} +{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} - 2 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x - 6 \,{\left (10 \, B c^{2} d^{5} - A b^{2} d^{2} e^{3} - 6 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} +{\left (10 \, B c^{2} d^{3} e^{2} - A b^{2} e^{5} - 6 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 2 \,{\left (10 \, B c^{2} d^{4} e - A b^{2} d e^{4} - 6 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x\right )} \log \left (e x + d\right )}{6 \,{\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.70473, size = 355, normalized size = 1.53 \begin{align*} \frac{B c^{2} x^{3}}{3 e^{3}} - \frac{- 3 A b^{2} d^{2} e^{3} + 10 A b c d^{3} e^{2} - 7 A c^{2} d^{4} e + 5 B b^{2} d^{3} e^{2} - 14 B b c d^{4} e + 9 B c^{2} d^{5} + x \left (- 4 A b^{2} d e^{4} + 12 A b c d^{2} e^{3} - 8 A c^{2} d^{3} e^{2} + 6 B b^{2} d^{2} e^{3} - 16 B b c d^{3} e^{2} + 10 B c^{2} d^{4} e\right )}{2 d^{2} e^{6} + 4 d e^{7} x + 2 e^{8} x^{2}} + \frac{x^{2} \left (A c^{2} e + 2 B b c e - 3 B c^{2} d\right )}{2 e^{4}} + \frac{x \left (2 A b c e^{2} - 3 A c^{2} d e + B b^{2} e^{2} - 6 B b c d e + 6 B c^{2} d^{2}\right )}{e^{5}} - \frac{\left (- A b^{2} e^{3} + 6 A b c d e^{2} - 6 A c^{2} d^{2} e + 3 B b^{2} d e^{2} - 12 B b c d^{2} e + 10 B c^{2} d^{3}\right ) \log{\left (d + e x \right )}}{e^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16148, size = 414, normalized size = 1.78 \begin{align*} -{\left (10 \, B c^{2} d^{3} - 12 \, B b c d^{2} e - 6 \, A c^{2} d^{2} e + 3 \, B b^{2} d e^{2} + 6 \, A b c d e^{2} - A b^{2} e^{3}\right )} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{6} \,{\left (2 \, B c^{2} x^{3} e^{6} - 9 \, B c^{2} d x^{2} e^{5} + 36 \, B c^{2} d^{2} x e^{4} + 6 \, B b c x^{2} e^{6} + 3 \, A c^{2} x^{2} e^{6} - 36 \, B b c d x e^{5} - 18 \, A c^{2} d x e^{5} + 6 \, B b^{2} x e^{6} + 12 \, A b c x e^{6}\right )} e^{\left (-9\right )} - \frac{{\left (9 \, B c^{2} d^{5} - 14 \, B b c d^{4} e - 7 \, A c^{2} d^{4} e + 5 \, B b^{2} d^{3} e^{2} + 10 \, A b c d^{3} e^{2} - 3 \, A b^{2} d^{2} e^{3} + 2 \,{\left (5 \, B c^{2} d^{4} e - 8 \, B b c d^{3} e^{2} - 4 \, A c^{2} d^{3} e^{2} + 3 \, B b^{2} d^{2} e^{3} + 6 \, A b c d^{2} e^{3} - 2 \, A b^{2} d e^{4}\right )} x\right )} e^{\left (-6\right )}}{2 \,{\left (x e + d\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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