Optimal. Leaf size=166 \[ -\frac{c^2 x^4 (2 c d-3 b e)}{4 e^3}-\frac{d^3 (c d-b e)^3}{e^7 (d+e x)}-\frac{3 d^2 (c d-b e)^2 (2 c d-b e) \log (d+e x)}{e^7}+\frac{c x^3 (c d-b e)^2}{e^4}-\frac{x^2 (c d-b e)^2 (4 c d-b e)}{2 e^5}+\frac{d x (5 c d-2 b e) (c d-b e)^2}{e^6}+\frac{c^3 x^5}{5 e^2} \]
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Rubi [A] time = 0.186282, antiderivative size = 166, 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} \[ -\frac{c^2 x^4 (2 c d-3 b e)}{4 e^3}-\frac{d^3 (c d-b e)^3}{e^7 (d+e x)}-\frac{3 d^2 (c d-b e)^2 (2 c d-b e) \log (d+e x)}{e^7}+\frac{c x^3 (c d-b e)^2}{e^4}-\frac{x^2 (c d-b e)^2 (4 c d-b e)}{2 e^5}+\frac{d x (5 c d-2 b e) (c d-b e)^2}{e^6}+\frac{c^3 x^5}{5 e^2} \]
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)^2} \, dx &=\int \left (\frac{d (5 c d-2 b e) (c d-b e)^2}{e^6}+\frac{(-4 c d+b e) (-c d+b e)^2 x}{e^5}+\frac{3 c (c d-b e)^2 x^2}{e^4}-\frac{c^2 (2 c d-3 b e) x^3}{e^3}+\frac{c^3 x^4}{e^2}+\frac{d^3 (c d-b e)^3}{e^6 (d+e x)^2}-\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{e^6 (d+e x)}\right ) \, dx\\ &=\frac{d (5 c d-2 b e) (c d-b e)^2 x}{e^6}-\frac{(c d-b e)^2 (4 c d-b e) x^2}{2 e^5}+\frac{c (c d-b e)^2 x^3}{e^4}-\frac{c^2 (2 c d-3 b e) x^4}{4 e^3}+\frac{c^3 x^5}{5 e^2}-\frac{d^3 (c d-b e)^3}{e^7 (d+e x)}-\frac{3 d^2 (c d-b e)^2 (2 c d-b e) \log (d+e x)}{e^7}\\ \end{align*}
Mathematica [A] time = 0.0618319, size = 160, normalized size = 0.96 \[ \frac{-5 c^2 e^4 x^4 (2 c d-3 b e)-\frac{20 d^3 (c d-b e)^3}{d+e x}-60 d^2 (c d-b e)^2 (2 c d-b e) \log (d+e x)+20 c e^3 x^3 (c d-b e)^2+10 e^2 x^2 (c d-b e)^2 (b e-4 c d)+20 d e x (5 c d-2 b e) (c d-b e)^2+4 c^3 e^5 x^5}{20 e^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 318, normalized size = 1.9 \begin{align*}{\frac{{c}^{3}{x}^{5}}{5\,{e}^{2}}}+{\frac{3\,b{x}^{4}{c}^{2}}{4\,{e}^{2}}}-{\frac{{c}^{3}d{x}^{4}}{2\,{e}^{3}}}+{\frac{{b}^{2}c{x}^{3}}{{e}^{2}}}-2\,{\frac{b{x}^{3}{c}^{2}d}{{e}^{3}}}+{\frac{{x}^{3}{c}^{3}{d}^{2}}{{e}^{4}}}+{\frac{{x}^{2}{b}^{3}}{2\,{e}^{2}}}-3\,{\frac{{b}^{2}{x}^{2}cd}{{e}^{3}}}+{\frac{9\,b{x}^{2}{c}^{2}{d}^{2}}{2\,{e}^{4}}}-2\,{\frac{{x}^{2}{c}^{3}{d}^{3}}{{e}^{5}}}-2\,{\frac{{b}^{3}dx}{{e}^{3}}}+9\,{\frac{{b}^{2}c{d}^{2}x}{{e}^{4}}}-12\,{\frac{b{c}^{2}{d}^{3}x}{{e}^{5}}}+5\,{\frac{{c}^{3}{d}^{4}x}{{e}^{6}}}+3\,{\frac{{d}^{2}\ln \left ( ex+d \right ){b}^{3}}{{e}^{4}}}-12\,{\frac{{d}^{3}\ln \left ( ex+d \right ){b}^{2}c}{{e}^{5}}}+15\,{\frac{{d}^{4}\ln \left ( ex+d \right ) b{c}^{2}}{{e}^{6}}}-6\,{\frac{{d}^{5}\ln \left ( ex+d \right ){c}^{3}}{{e}^{7}}}+{\frac{{d}^{3}{b}^{3}}{{e}^{4} \left ( ex+d \right ) }}-3\,{\frac{{d}^{4}{b}^{2}c}{{e}^{5} \left ( ex+d \right ) }}+3\,{\frac{{d}^{5}b{c}^{2}}{{e}^{6} \left ( ex+d \right ) }}-{\frac{{c}^{3}{d}^{6}}{{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 = 1.0369, size = 369, normalized size = 2.22 \begin{align*} -\frac{c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}}{e^{8} x + d e^{7}} + \frac{4 \, c^{3} e^{4} x^{5} - 5 \,{\left (2 \, c^{3} d e^{3} - 3 \, b c^{2} e^{4}\right )} x^{4} + 20 \,{\left (c^{3} d^{2} e^{2} - 2 \, b c^{2} d e^{3} + b^{2} c e^{4}\right )} x^{3} - 10 \,{\left (4 \, c^{3} d^{3} e - 9 \, b c^{2} d^{2} e^{2} + 6 \, b^{2} c d e^{3} - b^{3} e^{4}\right )} x^{2} + 20 \,{\left (5 \, c^{3} d^{4} - 12 \, b c^{2} d^{3} e + 9 \, b^{2} c d^{2} e^{2} - 2 \, b^{3} d e^{3}\right )} x}{20 \, e^{6}} - \frac{3 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} 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 = 1.44909, size = 751, normalized size = 4.52 \begin{align*} \frac{4 \, c^{3} e^{6} x^{6} - 20 \, c^{3} d^{6} + 60 \, b c^{2} d^{5} e - 60 \, b^{2} c d^{4} e^{2} + 20 \, b^{3} d^{3} e^{3} - 3 \,{\left (2 \, c^{3} d e^{5} - 5 \, b c^{2} e^{6}\right )} x^{5} + 5 \,{\left (2 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} + 4 \, b^{2} c e^{6}\right )} x^{4} - 10 \,{\left (2 \, c^{3} d^{3} e^{3} - 5 \, b c^{2} d^{2} e^{4} + 4 \, b^{2} c d e^{5} - b^{3} e^{6}\right )} x^{3} + 30 \,{\left (2 \, c^{3} d^{4} e^{2} - 5 \, b c^{2} d^{3} e^{3} + 4 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} + 20 \,{\left (5 \, c^{3} d^{5} e - 12 \, b c^{2} d^{4} e^{2} + 9 \, b^{2} c d^{3} e^{3} - 2 \, b^{3} d^{2} e^{4}\right )} x - 60 \,{\left (2 \, c^{3} d^{6} - 5 \, b c^{2} d^{5} e + 4 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} +{\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 )} \log \left (e x + d\right )}{20 \,{\left (e^{8} x + d e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.46736, size = 248, normalized size = 1.49 \begin{align*} \frac{c^{3} x^{5}}{5 e^{2}} + \frac{3 d^{2} \left (b e - 2 c d\right ) \left (b e - c d\right )^{2} \log{\left (d + e x \right )}}{e^{7}} + \frac{b^{3} d^{3} e^{3} - 3 b^{2} c d^{4} e^{2} + 3 b c^{2} d^{5} e - c^{3} d^{6}}{d e^{7} + e^{8} x} + \frac{x^{4} \left (3 b c^{2} e - 2 c^{3} d\right )}{4 e^{3}} + \frac{x^{3} \left (b^{2} c e^{2} - 2 b c^{2} d e + c^{3} d^{2}\right )}{e^{4}} + \frac{x^{2} \left (b^{3} e^{3} - 6 b^{2} c d e^{2} + 9 b c^{2} d^{2} e - 4 c^{3} d^{3}\right )}{2 e^{5}} - \frac{x \left (2 b^{3} d e^{3} - 9 b^{2} c d^{2} e^{2} + 12 b c^{2} d^{3} e - 5 c^{3} d^{4}\right )}{e^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.43872, size = 448, normalized size = 2.7 \begin{align*} \frac{1}{20} \,{\left (4 \, c^{3} - \frac{15 \,{\left (2 \, c^{3} d e - b c^{2} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac{20 \,{\left (5 \, c^{3} d^{2} e^{2} - 5 \, b c^{2} d e^{3} + b^{2} c e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac{10 \,{\left (20 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} + 12 \, b^{2} c d e^{5} - b^{3} e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} + \frac{60 \,{\left (5 \, c^{3} d^{4} e^{4} - 10 \, b c^{2} d^{3} e^{5} + 6 \, b^{2} c d^{2} e^{6} - b^{3} d e^{7}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{4}}\right )}{\left (x e + d\right )}^{5} e^{\left (-7\right )} + 3 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )} e^{\left (-7\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) -{\left (\frac{c^{3} d^{6} e^{5}}{x e + d} - \frac{3 \, b c^{2} d^{5} e^{6}}{x e + d} + \frac{3 \, b^{2} c d^{4} e^{7}}{x e + d} - \frac{b^{3} d^{3} e^{8}}{x e + d}\right )} e^{\left (-12\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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