Optimal. Leaf size=179 \[ \frac{3 d^4 \log (x) \left (5 b^2 e^2-6 b c d e+2 c^2 d^2\right )}{b^5}-\frac{3 (c d-b e)^4 \left (b^2 e^2+2 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 c^4}+\frac{3 (c d-b e)^5 (b e+c d)}{b^4 c^4 (b+c x)}+\frac{(c d-b e)^6}{2 b^3 c^4 (b+c x)^2}+\frac{3 d^5 (c d-2 b e)}{b^4 x}-\frac{d^6}{2 b^3 x^2}+\frac{e^6 x}{c^3} \]
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Rubi [A] time = 0.222377, antiderivative size = 179, 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} \[ \frac{3 d^4 \log (x) \left (5 b^2 e^2-6 b c d e+2 c^2 d^2\right )}{b^5}-\frac{3 (c d-b e)^4 \left (b^2 e^2+2 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 c^4}+\frac{3 (c d-b e)^5 (b e+c d)}{b^4 c^4 (b+c x)}+\frac{(c d-b e)^6}{2 b^3 c^4 (b+c x)^2}+\frac{3 d^5 (c d-2 b e)}{b^4 x}-\frac{d^6}{2 b^3 x^2}+\frac{e^6 x}{c^3} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int \frac{(d+e x)^6}{\left (b x+c x^2\right )^3} \, dx &=\int \left (\frac{e^6}{c^3}+\frac{d^6}{b^3 x^3}+\frac{3 d^5 (-c d+2 b e)}{b^4 x^2}+\frac{3 d^4 \left (2 c^2 d^2-6 b c d e+5 b^2 e^2\right )}{b^5 x}-\frac{(-c d+b e)^6}{b^3 c^3 (b+c x)^3}+\frac{3 (-c d+b e)^5 (c d+b e)}{b^4 c^3 (b+c x)^2}-\frac{3 (-c d+b e)^4 \left (2 c^2 d^2+2 b c d e+b^2 e^2\right )}{b^5 c^3 (b+c x)}\right ) \, dx\\ &=-\frac{d^6}{2 b^3 x^2}+\frac{3 d^5 (c d-2 b e)}{b^4 x}+\frac{e^6 x}{c^3}+\frac{(c d-b e)^6}{2 b^3 c^4 (b+c x)^2}+\frac{3 (c d-b e)^5 (c d+b e)}{b^4 c^4 (b+c x)}+\frac{3 d^4 \left (2 c^2 d^2-6 b c d e+5 b^2 e^2\right ) \log (x)}{b^5}-\frac{3 (c d-b e)^4 \left (2 c^2 d^2+2 b c d e+b^2 e^2\right ) \log (b+c x)}{b^5 c^4}\\ \end{align*}
Mathematica [A] time = 0.0874307, size = 179, normalized size = 1. \[ \frac{3 d^4 \log (x) \left (5 b^2 e^2-6 b c d e+2 c^2 d^2\right )}{b^5}-\frac{3 (c d-b e)^4 \left (b^2 e^2+2 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 c^4}+\frac{3 (c d-b e)^5 (b e+c d)}{b^4 c^4 (b+c x)}+\frac{(c d-b e)^6}{2 b^3 c^4 (b+c x)^2}+\frac{3 d^5 (c d-2 b e)}{b^4 x}-\frac{d^6}{2 b^3 x^2}+\frac{e^6 x}{c^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.062, size = 396, normalized size = 2.2 \begin{align*} -18\,{\frac{{d}^{5}\ln \left ( x \right ) ce}{{b}^{4}}}+18\,{\frac{c\ln \left ( cx+b \right ){d}^{5}e}{{b}^{4}}}-{\frac{{d}^{6}}{2\,{b}^{3}{x}^{2}}}+{\frac{{e}^{6}x}{{c}^{3}}}+15\,{\frac{{d}^{4}\ln \left ( x \right ){e}^{2}}{{b}^{3}}}+6\,{\frac{{d}^{6}\ln \left ( x \right ){c}^{2}}{{b}^{5}}}-6\,{\frac{{d}^{5}e}{{b}^{3}x}}+3\,{\frac{{d}^{6}c}{{b}^{4}x}}-3\,{\frac{b\ln \left ( cx+b \right ){e}^{6}}{{c}^{4}}}+6\,{\frac{\ln \left ( cx+b \right ) d{e}^{5}}{{c}^{3}}}-15\,{\frac{\ln \left ( cx+b \right ){d}^{4}{e}^{2}}{{b}^{3}}}-6\,{\frac{{c}^{2}\ln \left ( cx+b \right ){d}^{6}}{{b}^{5}}}-3\,{\frac{{b}^{2}{e}^{6}}{{c}^{4} \left ( cx+b \right ) }}+3\,{\frac{{c}^{2}{d}^{6}}{{b}^{4} \left ( cx+b \right ) }}+{\frac{{b}^{3}{e}^{6}}{2\,{c}^{4} \left ( cx+b \right ) ^{2}}}+{\frac{{c}^{2}{d}^{6}}{2\,{b}^{3} \left ( cx+b \right ) ^{2}}}-10\,{\frac{{d}^{3}{e}^{3}}{c \left ( cx+b \right ) ^{2}}}+{\frac{15\,{d}^{4}{e}^{2}}{2\,b \left ( cx+b \right ) ^{2}}}-15\,{\frac{{d}^{2}{e}^{4}}{{c}^{2} \left ( cx+b \right ) }}+15\,{\frac{{d}^{4}{e}^{2}}{{b}^{2} \left ( cx+b \right ) }}+12\,{\frac{bd{e}^{5}}{{c}^{3} \left ( cx+b \right ) }}-12\,{\frac{c{d}^{5}e}{{b}^{3} \left ( cx+b \right ) }}-3\,{\frac{{b}^{2}d{e}^{5}}{{c}^{3} \left ( cx+b \right ) ^{2}}}+{\frac{15\,b{d}^{2}{e}^{4}}{2\,{c}^{2} \left ( cx+b \right ) ^{2}}}-3\,{\frac{c{d}^{5}e}{{b}^{2} \left ( cx+b \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00397, size = 460, normalized size = 2.57 \begin{align*} \frac{e^{6} x}{c^{3}} - \frac{b^{3} c^{4} d^{6} - 6 \,{\left (2 \, c^{7} d^{6} - 6 \, b c^{6} d^{5} e + 5 \, b^{2} c^{5} d^{4} e^{2} - 5 \, b^{4} c^{3} d^{2} e^{4} + 4 \, b^{5} c^{2} d e^{5} - b^{6} c e^{6}\right )} x^{3} -{\left (18 \, b c^{6} d^{6} - 54 \, b^{2} c^{5} d^{5} e + 45 \, b^{3} c^{4} d^{4} e^{2} - 20 \, b^{4} c^{3} d^{3} e^{3} - 15 \, b^{5} c^{2} d^{2} e^{4} + 18 \, b^{6} c d e^{5} - 5 \, b^{7} e^{6}\right )} x^{2} - 4 \,{\left (b^{2} c^{5} d^{6} - 3 \, b^{3} c^{4} d^{5} e\right )} x}{2 \,{\left (b^{4} c^{6} x^{4} + 2 \, b^{5} c^{5} x^{3} + b^{6} c^{4} x^{2}\right )}} + \frac{3 \,{\left (2 \, c^{2} d^{6} - 6 \, b c d^{5} e + 5 \, b^{2} d^{4} e^{2}\right )} \log \left (x\right )}{b^{5}} - \frac{3 \,{\left (2 \, c^{6} d^{6} - 6 \, b c^{5} d^{5} e + 5 \, b^{2} c^{4} d^{4} e^{2} - 2 \, b^{5} c d e^{5} + b^{6} e^{6}\right )} \log \left (c x + b\right )}{b^{5} c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.98504, size = 1154, normalized size = 6.45 \begin{align*} \frac{2 \, b^{5} c^{3} e^{6} x^{5} + 4 \, b^{6} c^{2} e^{6} x^{4} - b^{4} c^{4} d^{6} + 2 \,{\left (6 \, b c^{7} d^{6} - 18 \, b^{2} c^{6} d^{5} e + 15 \, b^{3} c^{5} d^{4} e^{2} - 15 \, b^{5} c^{3} d^{2} e^{4} + 12 \, b^{6} c^{2} d e^{5} - 2 \, b^{7} c e^{6}\right )} x^{3} +{\left (18 \, b^{2} c^{6} d^{6} - 54 \, b^{3} c^{5} d^{5} e + 45 \, b^{4} c^{4} d^{4} e^{2} - 20 \, b^{5} c^{3} d^{3} e^{3} - 15 \, b^{6} c^{2} d^{2} e^{4} + 18 \, b^{7} c d e^{5} - 5 \, b^{8} e^{6}\right )} x^{2} + 4 \,{\left (b^{3} c^{5} d^{6} - 3 \, b^{4} c^{4} d^{5} e\right )} x - 6 \,{\left ({\left (2 \, c^{8} d^{6} - 6 \, b c^{7} d^{5} e + 5 \, b^{2} c^{6} d^{4} e^{2} - 2 \, b^{5} c^{3} d e^{5} + b^{6} c^{2} e^{6}\right )} x^{4} + 2 \,{\left (2 \, b c^{7} d^{6} - 6 \, b^{2} c^{6} d^{5} e + 5 \, b^{3} c^{5} d^{4} e^{2} - 2 \, b^{6} c^{2} d e^{5} + b^{7} c e^{6}\right )} x^{3} +{\left (2 \, b^{2} c^{6} d^{6} - 6 \, b^{3} c^{5} d^{5} e + 5 \, b^{4} c^{4} d^{4} e^{2} - 2 \, b^{7} c d e^{5} + b^{8} e^{6}\right )} x^{2}\right )} \log \left (c x + b\right ) + 6 \,{\left ({\left (2 \, c^{8} d^{6} - 6 \, b c^{7} d^{5} e + 5 \, b^{2} c^{6} d^{4} e^{2}\right )} x^{4} + 2 \,{\left (2 \, b c^{7} d^{6} - 6 \, b^{2} c^{6} d^{5} e + 5 \, b^{3} c^{5} d^{4} e^{2}\right )} x^{3} +{\left (2 \, b^{2} c^{6} d^{6} - 6 \, b^{3} c^{5} d^{5} e + 5 \, b^{4} c^{4} d^{4} e^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (b^{5} c^{6} x^{4} + 2 \, b^{6} c^{5} x^{3} + b^{7} c^{4} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 30.1229, size = 597, normalized size = 3.34 \begin{align*} - \frac{b^{3} c^{4} d^{6} + x^{3} \left (6 b^{6} c e^{6} - 24 b^{5} c^{2} d e^{5} + 30 b^{4} c^{3} d^{2} e^{4} - 30 b^{2} c^{5} d^{4} e^{2} + 36 b c^{6} d^{5} e - 12 c^{7} d^{6}\right ) + x^{2} \left (5 b^{7} e^{6} - 18 b^{6} c d e^{5} + 15 b^{5} c^{2} d^{2} e^{4} + 20 b^{4} c^{3} d^{3} e^{3} - 45 b^{3} c^{4} d^{4} e^{2} + 54 b^{2} c^{5} d^{5} e - 18 b c^{6} d^{6}\right ) + x \left (12 b^{3} c^{4} d^{5} e - 4 b^{2} c^{5} d^{6}\right )}{2 b^{6} c^{4} x^{2} + 4 b^{5} c^{5} x^{3} + 2 b^{4} c^{6} x^{4}} + \frac{e^{6} x}{c^{3}} + \frac{3 d^{4} \left (5 b^{2} e^{2} - 6 b c d e + 2 c^{2} d^{2}\right ) \log{\left (x + \frac{15 b^{3} c^{3} d^{4} e^{2} - 18 b^{2} c^{4} d^{5} e + 6 b c^{5} d^{6} - 3 b c^{3} d^{4} \left (5 b^{2} e^{2} - 6 b c d e + 2 c^{2} d^{2}\right )}{3 b^{6} e^{6} - 6 b^{5} c d e^{5} + 30 b^{2} c^{4} d^{4} e^{2} - 36 b c^{5} d^{5} e + 12 c^{6} d^{6}} \right )}}{b^{5}} - \frac{3 \left (b e - c d\right )^{4} \left (b^{2} e^{2} + 2 b c d e + 2 c^{2} d^{2}\right ) \log{\left (x + \frac{15 b^{3} c^{3} d^{4} e^{2} - 18 b^{2} c^{4} d^{5} e + 6 b c^{5} d^{6} + \frac{3 b \left (b e - c d\right )^{4} \left (b^{2} e^{2} + 2 b c d e + 2 c^{2} d^{2}\right )}{c}}{3 b^{6} e^{6} - 6 b^{5} c d e^{5} + 30 b^{2} c^{4} d^{4} e^{2} - 36 b c^{5} d^{5} e + 12 c^{6} d^{6}} \right )}}{b^{5} c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 3.19695, size = 427, normalized size = 2.39 \begin{align*} \frac{x e^{6}}{c^{3}} + \frac{3 \,{\left (2 \, c^{2} d^{6} - 6 \, b c d^{5} e + 5 \, b^{2} d^{4} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} - \frac{3 \,{\left (2 \, c^{6} d^{6} - 6 \, b c^{5} d^{5} e + 5 \, b^{2} c^{4} d^{4} e^{2} - 2 \, b^{5} c d e^{5} + b^{6} e^{6}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c^{4}} - \frac{b^{3} c^{4} d^{6} - 6 \,{\left (2 \, c^{7} d^{6} - 6 \, b c^{6} d^{5} e + 5 \, b^{2} c^{5} d^{4} e^{2} - 5 \, b^{4} c^{3} d^{2} e^{4} + 4 \, b^{5} c^{2} d e^{5} - b^{6} c e^{6}\right )} x^{3} -{\left (18 \, b c^{6} d^{6} - 54 \, b^{2} c^{5} d^{5} e + 45 \, b^{3} c^{4} d^{4} e^{2} - 20 \, b^{4} c^{3} d^{3} e^{3} - 15 \, b^{5} c^{2} d^{2} e^{4} + 18 \, b^{6} c d e^{5} - 5 \, b^{7} e^{6}\right )} x^{2} - 4 \,{\left (b^{2} c^{5} d^{6} - 3 \, b^{3} c^{4} d^{5} e\right )} x}{2 \,{\left (c x + b\right )}^{2} b^{4} c^{4} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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