Optimal. Leaf size=200 \[ -\frac{x (c d-b e) \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )}{e^6}+\frac{3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right ) \log (d+e x)}{e^7}-\frac{c^2 x^3 (c d-b e)}{e^4}+\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{e^7 (d+e x)}-\frac{d^3 (c d-b e)^3}{2 e^7 (d+e x)^2}+\frac{3 c x^2 (c d-b e) (2 c d-b e)}{2 e^5}+\frac{c^3 x^4}{4 e^3} \]
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Rubi [A] time = 0.21873, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {698} \[ -\frac{x (c d-b e) \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )}{e^6}+\frac{3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right ) \log (d+e x)}{e^7}-\frac{c^2 x^3 (c d-b e)}{e^4}+\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{e^7 (d+e x)}-\frac{d^3 (c d-b e)^3}{2 e^7 (d+e x)^2}+\frac{3 c x^2 (c d-b e) (2 c d-b e)}{2 e^5}+\frac{c^3 x^4}{4 e^3} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int \frac{\left (b x+c x^2\right )^3}{(d+e x)^3} \, dx &=\int \left (\frac{(c d-b e) \left (-10 c^2 d^2+8 b c d e-b^2 e^2\right )}{e^6}+\frac{3 c (c d-b e) (2 c d-b e) x}{e^5}-\frac{3 c^2 (c d-b e) x^2}{e^4}+\frac{c^3 x^3}{e^3}+\frac{d^3 (c d-b e)^3}{e^6 (d+e x)^3}-\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{e^6 (d+e x)^2}+\frac{3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^6 (d+e x)}\right ) \, dx\\ &=-\frac{(c d-b e) \left (10 c^2 d^2-8 b c d e+b^2 e^2\right ) x}{e^6}+\frac{3 c (c d-b e) (2 c d-b e) x^2}{2 e^5}-\frac{c^2 (c d-b e) x^3}{e^4}+\frac{c^3 x^4}{4 e^3}-\frac{d^3 (c d-b e)^3}{2 e^7 (d+e x)^2}+\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{e^7 (d+e x)}+\frac{3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) \log (d+e x)}{e^7}\\ \end{align*}
Mathematica [A] time = 0.131485, size = 207, normalized size = 1.03 \[ \frac{6 c e^2 x^2 \left (b^2 e^2-3 b c d e+2 c^2 d^2\right )+4 e x \left (-9 b^2 c d e^2+b^3 e^3+18 b c^2 d^2 e-10 c^3 d^3\right )+12 d \left (6 b^2 c d e^2-b^3 e^3-10 b c^2 d^2 e+5 c^3 d^3\right ) \log (d+e x)-4 c^2 e^3 x^3 (c d-b e)+\frac{12 d^2 (c d-b e)^2 (2 c d-b e)}{d+e x}-\frac{2 d^3 (c d-b e)^3}{(d+e x)^2}+c^3 e^4 x^4}{4 e^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 335, normalized size = 1.7 \begin{align*}{\frac{{c}^{3}{x}^{4}}{4\,{e}^{3}}}+{\frac{b{x}^{3}{c}^{2}}{{e}^{3}}}-{\frac{{c}^{3}d{x}^{3}}{{e}^{4}}}+{\frac{3\,{b}^{2}{x}^{2}c}{2\,{e}^{3}}}-{\frac{9\,b{x}^{2}{c}^{2}d}{2\,{e}^{4}}}+3\,{\frac{{x}^{2}{c}^{3}{d}^{2}}{{e}^{5}}}+{\frac{{b}^{3}x}{{e}^{3}}}-9\,{\frac{{b}^{2}cdx}{{e}^{4}}}+18\,{\frac{b{c}^{2}{d}^{2}x}{{e}^{5}}}-10\,{\frac{{c}^{3}{d}^{3}x}{{e}^{6}}}+{\frac{{d}^{3}{b}^{3}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{3\,{d}^{4}{b}^{2}c}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}+{\frac{3\,{d}^{5}b{c}^{2}}{2\,{e}^{6} \left ( ex+d \right ) ^{2}}}-{\frac{{c}^{3}{d}^{6}}{2\,{e}^{7} \left ( ex+d \right ) ^{2}}}-3\,{\frac{d\ln \left ( ex+d \right ){b}^{3}}{{e}^{4}}}+18\,{\frac{{d}^{2}\ln \left ( ex+d \right ){b}^{2}c}{{e}^{5}}}-30\,{\frac{{d}^{3}\ln \left ( ex+d \right ) b{c}^{2}}{{e}^{6}}}+15\,{\frac{{d}^{4}\ln \left ( ex+d \right ){c}^{3}}{{e}^{7}}}-3\,{\frac{{b}^{3}{d}^{2}}{{e}^{4} \left ( ex+d \right ) }}+12\,{\frac{{b}^{2}c{d}^{3}}{{e}^{5} \left ( ex+d \right ) }}-15\,{\frac{b{c}^{2}{d}^{4}}{{e}^{6} \left ( ex+d \right ) }}+6\,{\frac{{c}^{3}{d}^{5}}{{e}^{7} \left ( ex+d \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.156, size = 378, normalized size = 1.89 \begin{align*} \frac{11 \, c^{3} d^{6} - 27 \, b c^{2} d^{5} e + 21 \, b^{2} c d^{4} e^{2} - 5 \, b^{3} d^{3} e^{3} + 6 \,{\left (2 \, c^{3} d^{5} e - 5 \, b c^{2} d^{4} e^{2} + 4 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x}{2 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} + \frac{c^{3} e^{3} x^{4} - 4 \,{\left (c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{3} + 6 \,{\left (2 \, c^{3} d^{2} e - 3 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} x^{2} - 4 \,{\left (10 \, c^{3} d^{3} - 18 \, b c^{2} d^{2} e + 9 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} x}{4 \, e^{6}} + \frac{3 \,{\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )} \log \left (e x + d\right )}{e^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.66007, size = 868, normalized size = 4.34 \begin{align*} \frac{c^{3} e^{6} x^{6} + 22 \, c^{3} d^{6} - 54 \, b c^{2} d^{5} e + 42 \, b^{2} c d^{4} e^{2} - 10 \, b^{3} d^{3} e^{3} - 2 \,{\left (c^{3} d e^{5} - 2 \, b c^{2} e^{6}\right )} x^{5} +{\left (5 \, c^{3} d^{2} e^{4} - 10 \, b c^{2} d e^{5} + 6 \, b^{2} c e^{6}\right )} x^{4} - 4 \,{\left (5 \, c^{3} d^{3} e^{3} - 10 \, b c^{2} d^{2} e^{4} + 6 \, b^{2} c d e^{5} - b^{3} e^{6}\right )} x^{3} - 2 \,{\left (34 \, c^{3} d^{4} e^{2} - 63 \, b c^{2} d^{3} e^{3} + 33 \, b^{2} c d^{2} e^{4} - 4 \, b^{3} d e^{5}\right )} x^{2} - 4 \,{\left (4 \, c^{3} d^{5} e - 3 \, b c^{2} d^{4} e^{2} - 3 \, b^{2} c d^{3} e^{3} + 2 \, b^{3} d^{2} e^{4}\right )} x + 12 \,{\left (5 \, c^{3} d^{6} - 10 \, b c^{2} d^{5} e + 6 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} +{\left (5 \, c^{3} d^{4} e^{2} - 10 \, b c^{2} d^{3} e^{3} + 6 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} + 2 \,{\left (5 \, c^{3} d^{5} e - 10 \, b c^{2} d^{4} e^{2} + 6 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x\right )} \log \left (e x + d\right )}{4 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.79209, size = 277, normalized size = 1.38 \begin{align*} \frac{c^{3} x^{4}}{4 e^{3}} - \frac{3 d \left (b e - c d\right ) \left (b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) \log{\left (d + e x \right )}}{e^{7}} - \frac{5 b^{3} d^{3} e^{3} - 21 b^{2} c d^{4} e^{2} + 27 b c^{2} d^{5} e - 11 c^{3} d^{6} + x \left (6 b^{3} d^{2} e^{4} - 24 b^{2} c d^{3} e^{3} + 30 b c^{2} d^{4} e^{2} - 12 c^{3} d^{5} e\right )}{2 d^{2} e^{7} + 4 d e^{8} x + 2 e^{9} x^{2}} + \frac{x^{3} \left (b c^{2} e - c^{3} d\right )}{e^{4}} + \frac{x^{2} \left (3 b^{2} c e^{2} - 9 b c^{2} d e + 6 c^{3} d^{2}\right )}{2 e^{5}} + \frac{x \left (b^{3} e^{3} - 9 b^{2} c d e^{2} + 18 b c^{2} d^{2} e - 10 c^{3} d^{3}\right )}{e^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3611, size = 356, normalized size = 1.78 \begin{align*} 3 \,{\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{4} \,{\left (c^{3} x^{4} e^{9} - 4 \, c^{3} d x^{3} e^{8} + 12 \, c^{3} d^{2} x^{2} e^{7} - 40 \, c^{3} d^{3} x e^{6} + 4 \, b c^{2} x^{3} e^{9} - 18 \, b c^{2} d x^{2} e^{8} + 72 \, b c^{2} d^{2} x e^{7} + 6 \, b^{2} c x^{2} e^{9} - 36 \, b^{2} c d x e^{8} + 4 \, b^{3} x e^{9}\right )} e^{\left (-12\right )} + \frac{{\left (11 \, c^{3} d^{6} - 27 \, b c^{2} d^{5} e + 21 \, b^{2} c d^{4} e^{2} - 5 \, b^{3} d^{3} e^{3} + 6 \,{\left (2 \, c^{3} d^{5} e - 5 \, b c^{2} d^{4} e^{2} + 4 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x\right )} e^{\left (-7\right )}}{2 \,{\left (x e + d\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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