Optimal. Leaf size=136 \[ \frac{(c d-b e)^3 (b e+3 c d)}{b^4 c^2 (b+c x)}+\frac{(c d-b e)^4}{2 b^3 c^2 (b+c x)^2}+\frac{d^3 (3 c d-4 b e)}{b^4 x}+\frac{6 d^2 \log (x) (c d-b e)^2}{b^5}-\frac{6 d^2 (c d-b e)^2 \log (b+c x)}{b^5}-\frac{d^4}{2 b^3 x^2} \]
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Rubi [A] time = 0.151547, antiderivative size = 136, 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} \[ \frac{(c d-b e)^3 (b e+3 c d)}{b^4 c^2 (b+c x)}+\frac{(c d-b e)^4}{2 b^3 c^2 (b+c x)^2}+\frac{d^3 (3 c d-4 b e)}{b^4 x}+\frac{6 d^2 \log (x) (c d-b e)^2}{b^5}-\frac{6 d^2 (c d-b e)^2 \log (b+c x)}{b^5}-\frac{d^4}{2 b^3 x^2} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int \frac{(d+e x)^4}{\left (b x+c x^2\right )^3} \, dx &=\int \left (\frac{d^4}{b^3 x^3}+\frac{d^3 (-3 c d+4 b e)}{b^4 x^2}+\frac{6 d^2 (-c d+b e)^2}{b^5 x}-\frac{(-c d+b e)^4}{b^3 c (b+c x)^3}+\frac{(-c d+b e)^3 (3 c d+b e)}{b^4 c (b+c x)^2}-\frac{6 c d^2 (-c d+b e)^2}{b^5 (b+c x)}\right ) \, dx\\ &=-\frac{d^4}{2 b^3 x^2}+\frac{d^3 (3 c d-4 b e)}{b^4 x}+\frac{(c d-b e)^4}{2 b^3 c^2 (b+c x)^2}+\frac{(c d-b e)^3 (3 c d+b e)}{b^4 c^2 (b+c x)}+\frac{6 d^2 (c d-b e)^2 \log (x)}{b^5}-\frac{6 d^2 (c d-b e)^2 \log (b+c x)}{b^5}\\ \end{align*}
Mathematica [A] time = 0.0835923, size = 130, normalized size = 0.96 \[ -\frac{-\frac{b^2 (c d-b e)^4}{c^2 (b+c x)^2}+\frac{b^2 d^4}{x^2}+\frac{2 b (b e-c d)^3 (b e+3 c d)}{c^2 (b+c x)}+\frac{2 b d^3 (4 b e-3 c d)}{x}-12 d^2 \log (x) (c d-b e)^2+12 d^2 (c d-b e)^2 \log (b+c x)}{2 b^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.062, size = 278, normalized size = 2. \begin{align*} -{\frac{{d}^{4}}{2\,{b}^{3}{x}^{2}}}-4\,{\frac{{d}^{3}e}{{b}^{3}x}}+3\,{\frac{{d}^{4}c}{{b}^{4}x}}+6\,{\frac{{d}^{2}\ln \left ( x \right ){e}^{2}}{{b}^{3}}}-12\,{\frac{{d}^{3}\ln \left ( x \right ) ce}{{b}^{4}}}+6\,{\frac{{d}^{4}\ln \left ( x \right ){c}^{2}}{{b}^{5}}}-{\frac{{e}^{4}}{{c}^{2} \left ( cx+b \right ) }}+6\,{\frac{{d}^{2}{e}^{2}}{{b}^{2} \left ( cx+b \right ) }}-8\,{\frac{{d}^{3}ec}{{b}^{3} \left ( cx+b \right ) }}+3\,{\frac{{c}^{2}{d}^{4}}{{b}^{4} \left ( cx+b \right ) }}+{\frac{{e}^{4}b}{2\,{c}^{2} \left ( cx+b \right ) ^{2}}}-2\,{\frac{d{e}^{3}}{c \left ( cx+b \right ) ^{2}}}+3\,{\frac{{d}^{2}{e}^{2}}{b \left ( cx+b \right ) ^{2}}}-2\,{\frac{{d}^{3}ec}{{b}^{2} \left ( cx+b \right ) ^{2}}}+{\frac{{c}^{2}{d}^{4}}{2\,{b}^{3} \left ( cx+b \right ) ^{2}}}-6\,{\frac{{d}^{2}\ln \left ( cx+b \right ){e}^{2}}{{b}^{3}}}+12\,{\frac{{d}^{3}\ln \left ( cx+b \right ) ce}{{b}^{4}}}-6\,{\frac{{d}^{4}\ln \left ( cx+b \right ){c}^{2}}{{b}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02357, size = 338, normalized size = 2.49 \begin{align*} -\frac{b^{3} c^{2} d^{4} - 2 \,{\left (6 \, c^{5} d^{4} - 12 \, b c^{4} d^{3} e + 6 \, b^{2} c^{3} d^{2} e^{2} - b^{4} c e^{4}\right )} x^{3} -{\left (18 \, b c^{4} d^{4} - 36 \, b^{2} c^{3} d^{3} e + 18 \, b^{3} c^{2} d^{2} e^{2} - 4 \, b^{4} c d e^{3} - b^{5} e^{4}\right )} x^{2} - 4 \,{\left (b^{2} c^{3} d^{4} - 2 \, b^{3} c^{2} d^{3} e\right )} x}{2 \,{\left (b^{4} c^{4} x^{4} + 2 \, b^{5} c^{3} x^{3} + b^{6} c^{2} x^{2}\right )}} - \frac{6 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \log \left (c x + b\right )}{b^{5}} + \frac{6 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \log \left (x\right )}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.8601, size = 834, normalized size = 6.13 \begin{align*} -\frac{b^{4} c^{2} d^{4} - 2 \,{\left (6 \, b c^{5} d^{4} - 12 \, b^{2} c^{4} d^{3} e + 6 \, b^{3} c^{3} d^{2} e^{2} - b^{5} c e^{4}\right )} x^{3} -{\left (18 \, b^{2} c^{4} d^{4} - 36 \, b^{3} c^{3} d^{3} e + 18 \, b^{4} c^{2} d^{2} e^{2} - 4 \, b^{5} c d e^{3} - b^{6} e^{4}\right )} x^{2} - 4 \,{\left (b^{3} c^{3} d^{4} - 2 \, b^{4} c^{2} d^{3} e\right )} x + 12 \,{\left ({\left (c^{6} d^{4} - 2 \, b c^{5} d^{3} e + b^{2} c^{4} d^{2} e^{2}\right )} x^{4} + 2 \,{\left (b c^{5} d^{4} - 2 \, b^{2} c^{4} d^{3} e + b^{3} c^{3} d^{2} e^{2}\right )} x^{3} +{\left (b^{2} c^{4} d^{4} - 2 \, b^{3} c^{3} d^{3} e + b^{4} c^{2} d^{2} e^{2}\right )} x^{2}\right )} \log \left (c x + b\right ) - 12 \,{\left ({\left (c^{6} d^{4} - 2 \, b c^{5} d^{3} e + b^{2} c^{4} d^{2} e^{2}\right )} x^{4} + 2 \,{\left (b c^{5} d^{4} - 2 \, b^{2} c^{4} d^{3} e + b^{3} c^{3} d^{2} e^{2}\right )} x^{3} +{\left (b^{2} c^{4} d^{4} - 2 \, b^{3} c^{3} d^{3} e + b^{4} c^{2} d^{2} e^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (b^{5} c^{4} x^{4} + 2 \, b^{6} c^{3} x^{3} + b^{7} c^{2} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 8.51847, size = 389, normalized size = 2.86 \begin{align*} - \frac{b^{3} c^{2} d^{4} + x^{3} \left (2 b^{4} c e^{4} - 12 b^{2} c^{3} d^{2} e^{2} + 24 b c^{4} d^{3} e - 12 c^{5} d^{4}\right ) + x^{2} \left (b^{5} e^{4} + 4 b^{4} c d e^{3} - 18 b^{3} c^{2} d^{2} e^{2} + 36 b^{2} c^{3} d^{3} e - 18 b c^{4} d^{4}\right ) + x \left (8 b^{3} c^{2} d^{3} e - 4 b^{2} c^{3} d^{4}\right )}{2 b^{6} c^{2} x^{2} + 4 b^{5} c^{3} x^{3} + 2 b^{4} c^{4} x^{4}} + \frac{6 d^{2} \left (b e - c d\right )^{2} \log{\left (x + \frac{6 b^{3} d^{2} e^{2} - 12 b^{2} c d^{3} e + 6 b c^{2} d^{4} - 6 b d^{2} \left (b e - c d\right )^{2}}{12 b^{2} c d^{2} e^{2} - 24 b c^{2} d^{3} e + 12 c^{3} d^{4}} \right )}}{b^{5}} - \frac{6 d^{2} \left (b e - c d\right )^{2} \log{\left (x + \frac{6 b^{3} d^{2} e^{2} - 12 b^{2} c d^{3} e + 6 b c^{2} d^{4} + 6 b d^{2} \left (b e - c d\right )^{2}}{12 b^{2} c d^{2} e^{2} - 24 b c^{2} d^{3} e + 12 c^{3} d^{4}} \right )}}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2758, size = 343, normalized size = 2.52 \begin{align*} \frac{6 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} - \frac{6 \,{\left (c^{3} d^{4} - 2 \, b c^{2} d^{3} e + b^{2} c d^{2} e^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c} + \frac{12 \, c^{5} d^{4} x^{3} - 24 \, b c^{4} d^{3} x^{3} e + 18 \, b c^{4} d^{4} x^{2} + 12 \, b^{2} c^{3} d^{2} x^{3} e^{2} - 36 \, b^{2} c^{3} d^{3} x^{2} e + 4 \, b^{2} c^{3} d^{4} x + 18 \, b^{3} c^{2} d^{2} x^{2} e^{2} - 8 \, b^{3} c^{2} d^{3} x e - b^{3} c^{2} d^{4} - 2 \, b^{4} c x^{3} e^{4} - 4 \, b^{4} c d x^{2} e^{3} - b^{5} x^{2} e^{4}}{2 \,{\left (c x^{2} + b x\right )}^{2} b^{4} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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