Optimal. Leaf size=228 \[ -\frac{3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{2 e^7 (d+e x)^2}+\frac{(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{3 e^7 (d+e x)^3}-\frac{3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{4 e^7 (d+e x)^4}+\frac{3 c^2 (2 c d-b e)}{e^7 (d+e x)}+\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{5 e^7 (d+e x)^5}-\frac{d^3 (c d-b e)^3}{6 e^7 (d+e x)^6}+\frac{c^3 \log (d+e x)}{e^7} \]
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Rubi [A] time = 0.174597, antiderivative size = 228, 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} \[ -\frac{3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{2 e^7 (d+e x)^2}+\frac{(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{3 e^7 (d+e x)^3}-\frac{3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{4 e^7 (d+e x)^4}+\frac{3 c^2 (2 c d-b e)}{e^7 (d+e x)}+\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{5 e^7 (d+e x)^5}-\frac{d^3 (c d-b e)^3}{6 e^7 (d+e x)^6}+\frac{c^3 \log (d+e x)}{e^7} \]
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)^7} \, dx &=\int \left (\frac{d^3 (c d-b e)^3}{e^6 (d+e x)^7}-\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{e^6 (d+e x)^6}+\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)^5}+\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)^4}+\frac{3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^6 (d+e x)^3}-\frac{3 c^2 (2 c d-b e)}{e^6 (d+e x)^2}+\frac{c^3}{e^6 (d+e x)}\right ) \, dx\\ &=-\frac{d^3 (c d-b e)^3}{6 e^7 (d+e x)^6}+\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{5 e^7 (d+e x)^5}-\frac{3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{4 e^7 (d+e x)^4}+\frac{(2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right )}{3 e^7 (d+e x)^3}-\frac{3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{2 e^7 (d+e x)^2}+\frac{3 c^2 (2 c d-b e)}{e^7 (d+e x)}+\frac{c^3 \log (d+e x)}{e^7}\\ \end{align*}
Mathematica [A] time = 0.0813611, size = 231, normalized size = 1.01 \[ \frac{-6 b^2 c e^2 \left (15 d^2 e^2 x^2+6 d^3 e x+d^4+20 d e^3 x^3+15 e^4 x^4\right )-b^3 e^3 \left (6 d^2 e x+d^3+15 d e^2 x^2+20 e^3 x^3\right )-30 b c^2 e \left (15 d^3 e^2 x^2+20 d^2 e^3 x^3+6 d^4 e x+d^5+15 d e^4 x^4+6 e^5 x^5\right )+c^3 d \left (1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+822 d^4 e x+147 d^5+1350 d e^4 x^4+360 e^5 x^5\right )+60 c^3 (d+e x)^6 \log (d+e x)}{60 e^7 (d+e x)^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 387, normalized size = 1.7 \begin{align*} -{\frac{3\,{b}^{2}c}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}+{\frac{15\,b{c}^{2}d}{2\,{e}^{6} \left ( ex+d \right ) ^{2}}}-{\frac{15\,{c}^{3}{d}^{2}}{2\,{e}^{7} \left ( ex+d \right ) ^{2}}}-{\frac{3\,{b}^{3}{d}^{2}}{5\,{e}^{4} \left ( ex+d \right ) ^{5}}}+{\frac{12\,{b}^{2}c{d}^{3}}{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}-3\,{\frac{b{c}^{2}{d}^{4}}{{e}^{6} \left ( ex+d \right ) ^{5}}}+{\frac{6\,{c}^{3}{d}^{5}}{5\,{e}^{7} \left ( ex+d \right ) ^{5}}}+{\frac{{c}^{3}\ln \left ( ex+d \right ) }{{e}^{7}}}-{\frac{{b}^{3}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}+4\,{\frac{{b}^{2}cd}{{e}^{5} \left ( ex+d \right ) ^{3}}}-10\,{\frac{b{c}^{2}{d}^{2}}{{e}^{6} \left ( ex+d \right ) ^{3}}}+{\frac{20\,{c}^{3}{d}^{3}}{3\,{e}^{7} \left ( ex+d \right ) ^{3}}}+{\frac{{b}^{3}{d}^{3}}{6\,{e}^{4} \left ( ex+d \right ) ^{6}}}-{\frac{{b}^{2}c{d}^{4}}{2\,{e}^{5} \left ( ex+d \right ) ^{6}}}+{\frac{b{c}^{2}{d}^{5}}{2\,{e}^{6} \left ( ex+d \right ) ^{6}}}-{\frac{{c}^{3}{d}^{6}}{6\,{e}^{7} \left ( ex+d \right ) ^{6}}}-3\,{\frac{b{c}^{2}}{{e}^{6} \left ( ex+d \right ) }}+6\,{\frac{{c}^{3}d}{{e}^{7} \left ( ex+d \right ) }}+{\frac{3\,{b}^{3}d}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{9\,{b}^{2}c{d}^{2}}{2\,{e}^{5} \left ( ex+d \right ) ^{4}}}+{\frac{15\,b{c}^{2}{d}^{3}}{2\,{e}^{6} \left ( ex+d \right ) ^{4}}}-{\frac{15\,{c}^{3}{d}^{4}}{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 = 1.20406, size = 447, normalized size = 1.96 \begin{align*} \frac{147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} + 180 \,{\left (2 \, c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} + 90 \,{\left (15 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} - b^{2} c e^{6}\right )} x^{4} + 20 \,{\left (110 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} - 6 \, b^{2} c d e^{5} - b^{3} e^{6}\right )} x^{3} + 15 \,{\left (125 \, c^{3} d^{4} e^{2} - 30 \, b c^{2} d^{3} e^{3} - 6 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} + 6 \,{\left (137 \, c^{3} d^{5} e - 30 \, b c^{2} d^{4} e^{2} - 6 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x}{60 \,{\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} + \frac{c^{3} \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 [A] time = 1.66275, size = 840, normalized size = 3.68 \begin{align*} \frac{147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} + 180 \,{\left (2 \, c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} + 90 \,{\left (15 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} - b^{2} c e^{6}\right )} x^{4} + 20 \,{\left (110 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} - 6 \, b^{2} c d e^{5} - b^{3} e^{6}\right )} x^{3} + 15 \,{\left (125 \, c^{3} d^{4} e^{2} - 30 \, b c^{2} d^{3} e^{3} - 6 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} + 6 \,{\left (137 \, c^{3} d^{5} e - 30 \, b c^{2} d^{4} e^{2} - 6 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x + 60 \,{\left (c^{3} e^{6} x^{6} + 6 \, c^{3} d e^{5} x^{5} + 15 \, c^{3} d^{2} e^{4} x^{4} + 20 \, c^{3} d^{3} e^{3} x^{3} + 15 \, c^{3} d^{4} e^{2} x^{2} + 6 \, c^{3} d^{5} e x + c^{3} d^{6}\right )} \log \left (e x + d\right )}{60 \,{\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 103.83, size = 343, normalized size = 1.5 \begin{align*} \frac{c^{3} \log{\left (d + e x \right )}}{e^{7}} - \frac{b^{3} d^{3} e^{3} + 6 b^{2} c d^{4} e^{2} + 30 b c^{2} d^{5} e - 147 c^{3} d^{6} + x^{5} \left (180 b c^{2} e^{6} - 360 c^{3} d e^{5}\right ) + x^{4} \left (90 b^{2} c e^{6} + 450 b c^{2} d e^{5} - 1350 c^{3} d^{2} e^{4}\right ) + x^{3} \left (20 b^{3} e^{6} + 120 b^{2} c d e^{5} + 600 b c^{2} d^{2} e^{4} - 2200 c^{3} d^{3} e^{3}\right ) + x^{2} \left (15 b^{3} d e^{5} + 90 b^{2} c d^{2} e^{4} + 450 b c^{2} d^{3} e^{3} - 1875 c^{3} d^{4} e^{2}\right ) + x \left (6 b^{3} d^{2} e^{4} + 36 b^{2} c d^{3} e^{3} + 180 b c^{2} d^{4} e^{2} - 822 c^{3} d^{5} e\right )}{60 d^{6} e^{7} + 360 d^{5} e^{8} x + 900 d^{4} e^{9} x^{2} + 1200 d^{3} e^{10} x^{3} + 900 d^{2} e^{11} x^{4} + 360 d e^{12} x^{5} + 60 e^{13} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26178, size = 351, normalized size = 1.54 \begin{align*} c^{3} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{{\left (180 \,{\left (2 \, c^{3} d e^{4} - b c^{2} e^{5}\right )} x^{5} + 90 \,{\left (15 \, c^{3} d^{2} e^{3} - 5 \, b c^{2} d e^{4} - b^{2} c e^{5}\right )} x^{4} + 20 \,{\left (110 \, c^{3} d^{3} e^{2} - 30 \, b c^{2} d^{2} e^{3} - 6 \, b^{2} c d e^{4} - b^{3} e^{5}\right )} x^{3} + 15 \,{\left (125 \, c^{3} d^{4} e - 30 \, b c^{2} d^{3} e^{2} - 6 \, b^{2} c d^{2} e^{3} - b^{3} d e^{4}\right )} x^{2} + 6 \,{\left (137 \, c^{3} d^{5} - 30 \, b c^{2} d^{4} e - 6 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )} x +{\left (147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} e^{\left (-1\right )}\right )} e^{\left (-6\right )}}{60 \,{\left (x e + d\right )}^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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