Optimal. Leaf size=213 \[ \frac{(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{e^7 (d+e x)}-\frac{3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{2 e^7 (d+e x)^2}+\frac{3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right ) \log (d+e x)}{e^7}-\frac{c^2 x (5 c d-3 b e)}{e^6}+\frac{d^2 (c d-b e)^2 (2 c d-b e)}{e^7 (d+e x)^3}-\frac{d^3 (c d-b e)^3}{4 e^7 (d+e x)^4}+\frac{c^3 x^2}{2 e^5} \]
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Rubi [A] time = 0.197327, antiderivative size = 213, 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} \[ \frac{(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{e^7 (d+e x)}-\frac{3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{2 e^7 (d+e x)^2}+\frac{3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right ) \log (d+e x)}{e^7}-\frac{c^2 x (5 c d-3 b e)}{e^6}+\frac{d^2 (c d-b e)^2 (2 c d-b e)}{e^7 (d+e x)^3}-\frac{d^3 (c d-b e)^3}{4 e^7 (d+e x)^4}+\frac{c^3 x^2}{2 e^5} \]
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)^5} \, dx &=\int \left (-\frac{c^2 (5 c d-3 b e)}{e^6}+\frac{c^3 x}{e^5}+\frac{d^3 (c d-b e)^3}{e^6 (d+e x)^5}-\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{e^6 (d+e x)^4}+\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)^3}+\frac{(2 c d-b e) \left (-10 c^2 d^2+10 b c d e-b^2 e^2\right )}{e^6 (d+e x)^2}+\frac{3 c \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^2 (5 c d-3 b e) x}{e^6}+\frac{c^3 x^2}{2 e^5}-\frac{d^3 (c d-b e)^3}{4 e^7 (d+e x)^4}+\frac{d^2 (c d-b e)^2 (2 c d-b e)}{e^7 (d+e x)^3}-\frac{3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{2 e^7 (d+e x)^2}+\frac{(2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right )}{e^7 (d+e x)}+\frac{3 c \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 = 0.114337, size = 210, normalized size = 0.99 \[ \frac{\frac{48 b^2 c d e^2-4 b^3 e^3-120 b c^2 d^2 e+80 c^3 d^3}{d+e x}+\frac{6 d \left (-6 b^2 c d e^2+b^3 e^3+10 b c^2 d^2 e-5 c^3 d^3\right )}{(d+e x)^2}+12 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right ) \log (d+e x)-4 c^2 e x (5 c d-3 b e)+\frac{4 d^2 (c d-b e)^2 (2 c d-b e)}{(d+e x)^3}-\frac{d^3 (c d-b e)^3}{(d+e x)^4}+2 c^3 e^2 x^2}{4 e^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 370, normalized size = 1.7 \begin{align*}{\frac{{c}^{3}{x}^{2}}{2\,{e}^{5}}}+3\,{\frac{{c}^{2}xb}{{e}^{5}}}-5\,{\frac{{c}^{3}dx}{{e}^{6}}}+{\frac{3\,d{b}^{3}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-9\,{\frac{{b}^{2}{d}^{2}c}{{e}^{5} \left ( ex+d \right ) ^{2}}}+15\,{\frac{b{c}^{2}{d}^{3}}{{e}^{6} \left ( ex+d \right ) ^{2}}}-{\frac{15\,{c}^{3}{d}^{4}}{2\,{e}^{7} \left ( ex+d \right ) ^{2}}}+3\,{\frac{c\ln \left ( ex+d \right ){b}^{2}}{{e}^{5}}}-15\,{\frac{{c}^{2}\ln \left ( ex+d \right ) bd}{{e}^{6}}}+15\,{\frac{{c}^{3}\ln \left ( ex+d \right ){d}^{2}}{{e}^{7}}}-{\frac{{d}^{2}{b}^{3}}{{e}^{4} \left ( ex+d \right ) ^{3}}}+4\,{\frac{{d}^{3}{b}^{2}c}{{e}^{5} \left ( ex+d \right ) ^{3}}}-5\,{\frac{b{c}^{2}{d}^{4}}{{e}^{6} \left ( ex+d \right ) ^{3}}}+2\,{\frac{{c}^{3}{d}^{5}}{{e}^{7} \left ( ex+d \right ) ^{3}}}-{\frac{{b}^{3}}{{e}^{4} \left ( ex+d \right ) }}+12\,{\frac{{b}^{2}cd}{{e}^{5} \left ( ex+d \right ) }}-30\,{\frac{b{c}^{2}{d}^{2}}{{e}^{6} \left ( ex+d \right ) }}+20\,{\frac{{c}^{3}{d}^{3}}{{e}^{7} \left ( ex+d \right ) }}+{\frac{{d}^{3}{b}^{3}}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{3\,{d}^{4}{b}^{2}c}{4\,{e}^{5} \left ( ex+d \right ) ^{4}}}+{\frac{3\,{d}^{5}b{c}^{2}}{4\,{e}^{6} \left ( ex+d \right ) ^{4}}}-{\frac{{c}^{3}{d}^{6}}{4\,{e}^{7} \left ( ex+d \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.2026, size = 409, normalized size = 1.92 \begin{align*} \frac{57 \, c^{3} d^{6} - 77 \, b c^{2} d^{5} e + 25 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} + 4 \,{\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 )} x^{3} + 6 \,{\left (35 \, c^{3} d^{4} e^{2} - 50 \, b c^{2} d^{3} e^{3} + 18 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} + 4 \,{\left (47 \, c^{3} d^{5} e - 65 \, b c^{2} d^{4} e^{2} + 22 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x}{4 \,{\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )}} + \frac{c^{3} e x^{2} - 2 \,{\left (5 \, c^{3} d - 3 \, b c^{2} e\right )} x}{2 \, e^{6}} + \frac{3 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2}\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 = 1.58442, size = 976, normalized size = 4.58 \begin{align*} \frac{2 \, c^{3} e^{6} x^{6} + 57 \, c^{3} d^{6} - 77 \, b c^{2} d^{5} e + 25 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} - 12 \,{\left (c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} - 4 \,{\left (17 \, c^{3} d^{2} e^{4} - 12 \, b c^{2} d e^{5}\right )} x^{4} - 4 \,{\left (8 \, c^{3} d^{3} e^{3} + 12 \, b c^{2} d^{2} e^{4} - 12 \, b^{2} c d e^{5} + b^{3} e^{6}\right )} x^{3} + 6 \,{\left (22 \, c^{3} d^{4} e^{2} - 42 \, b c^{2} d^{3} e^{3} + 18 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} + 4 \,{\left (42 \, c^{3} d^{5} e - 62 \, b c^{2} d^{4} e^{2} + 22 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x + 12 \,{\left (5 \, c^{3} d^{6} - 5 \, b c^{2} d^{5} e + b^{2} c d^{4} e^{2} +{\left (5 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} + b^{2} c e^{6}\right )} x^{4} + 4 \,{\left (5 \, c^{3} d^{3} e^{3} - 5 \, b c^{2} d^{2} e^{4} + b^{2} c d e^{5}\right )} x^{3} + 6 \,{\left (5 \, c^{3} d^{4} e^{2} - 5 \, b c^{2} d^{3} e^{3} + b^{2} c d^{2} e^{4}\right )} x^{2} + 4 \,{\left (5 \, c^{3} d^{5} e - 5 \, b c^{2} d^{4} e^{2} + b^{2} c d^{3} e^{3}\right )} x\right )} \log \left (e x + d\right )}{4 \,{\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 14.4251, size = 314, normalized size = 1.47 \begin{align*} \frac{c^{3} x^{2}}{2 e^{5}} + \frac{3 c \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{b^{3} d^{3} e^{3} - 25 b^{2} c d^{4} e^{2} + 77 b c^{2} d^{5} e - 57 c^{3} d^{6} + x^{3} \left (4 b^{3} e^{6} - 48 b^{2} c d e^{5} + 120 b c^{2} d^{2} e^{4} - 80 c^{3} d^{3} e^{3}\right ) + x^{2} \left (6 b^{3} d e^{5} - 108 b^{2} c d^{2} e^{4} + 300 b c^{2} d^{3} e^{3} - 210 c^{3} d^{4} e^{2}\right ) + x \left (4 b^{3} d^{2} e^{4} - 88 b^{2} c d^{3} e^{3} + 260 b c^{2} d^{4} e^{2} - 188 c^{3} d^{5} e\right )}{4 d^{4} e^{7} + 16 d^{3} e^{8} x + 24 d^{2} e^{9} x^{2} + 16 d e^{10} x^{3} + 4 e^{11} x^{4}} + \frac{x \left (3 b c^{2} e - 5 c^{3} d\right )}{e^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27752, size = 522, normalized size = 2.45 \begin{align*} \frac{1}{2} \,{\left (c^{3} - \frac{6 \,{\left (2 \, c^{3} d e - b c^{2} e^{2}\right )} e^{\left (-1\right )}}{x e + d}\right )}{\left (x e + d\right )}^{2} e^{\left (-7\right )} - 3 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2}\right )} e^{\left (-7\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + \frac{1}{4} \,{\left (\frac{80 \, c^{3} d^{3} e^{29}}{x e + d} - \frac{30 \, c^{3} d^{4} e^{29}}{{\left (x e + d\right )}^{2}} + \frac{8 \, c^{3} d^{5} e^{29}}{{\left (x e + d\right )}^{3}} - \frac{c^{3} d^{6} e^{29}}{{\left (x e + d\right )}^{4}} - \frac{120 \, b c^{2} d^{2} e^{30}}{x e + d} + \frac{60 \, b c^{2} d^{3} e^{30}}{{\left (x e + d\right )}^{2}} - \frac{20 \, b c^{2} d^{4} e^{30}}{{\left (x e + d\right )}^{3}} + \frac{3 \, b c^{2} d^{5} e^{30}}{{\left (x e + d\right )}^{4}} + \frac{48 \, b^{2} c d e^{31}}{x e + d} - \frac{36 \, b^{2} c d^{2} e^{31}}{{\left (x e + d\right )}^{2}} + \frac{16 \, b^{2} c d^{3} e^{31}}{{\left (x e + d\right )}^{3}} - \frac{3 \, b^{2} c d^{4} e^{31}}{{\left (x e + d\right )}^{4}} - \frac{4 \, b^{3} e^{32}}{x e + d} + \frac{6 \, b^{3} d e^{32}}{{\left (x e + d\right )}^{2}} - \frac{4 \, b^{3} d^{2} e^{32}}{{\left (x e + d\right )}^{3}} + \frac{b^{3} d^{3} e^{32}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-36\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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