Optimal. Leaf size=134 \[ \frac{e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right ) \log (d+e x)}{d^3 (c d-b e)^3}-\frac{c^3 \log (b+c x)}{b (c d-b e)^3}-\frac{e (2 c d-b e)}{d^2 (d+e x) (c d-b e)^2}-\frac{e}{2 d (d+e x)^2 (c d-b e)}+\frac{\log (x)}{b d^3} \]
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Rubi [A] time = 0.119824, antiderivative size = 134, 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} \[ \frac{e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right ) \log (d+e x)}{d^3 (c d-b e)^3}-\frac{c^3 \log (b+c x)}{b (c d-b e)^3}-\frac{e (2 c d-b e)}{d^2 (d+e x) (c d-b e)^2}-\frac{e}{2 d (d+e x)^2 (c d-b e)}+\frac{\log (x)}{b d^3} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^3 \left (b x+c x^2\right )} \, dx &=\int \left (\frac{1}{b d^3 x}+\frac{c^4}{b (-c d+b e)^3 (b+c x)}+\frac{e^2}{d (c d-b e) (d+e x)^3}+\frac{e^2 (2 c d-b e)}{d^2 (c d-b e)^2 (d+e x)^2}+\frac{e^2 \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )}{d^3 (c d-b e)^3 (d+e x)}\right ) \, dx\\ &=-\frac{e}{2 d (c d-b e) (d+e x)^2}-\frac{e (2 c d-b e)}{d^2 (c d-b e)^2 (d+e x)}+\frac{\log (x)}{b d^3}-\frac{c^3 \log (b+c x)}{b (c d-b e)^3}+\frac{e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right ) \log (d+e x)}{d^3 (c d-b e)^3}\\ \end{align*}
Mathematica [A] time = 0.257857, size = 116, normalized size = 0.87 \[ \frac{\frac{e \left (\frac{d (c d-b e) (c d (5 d+4 e x)-b e (3 d+2 e x))}{(d+e x)^2}-2 \left (b^2 e^2-3 b c d e+3 c^2 d^2\right ) \log (d+e x)\right )}{d^3}+\frac{2 c^3 \log (b+c x)}{b}}{2 (b e-c d)^3}+\frac{\log (x)}{b d^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 184, normalized size = 1.4 \begin{align*}{\frac{\ln \left ( x \right ) }{{d}^{3}b}}+{\frac{{c}^{3}\ln \left ( cx+b \right ) }{ \left ( be-cd \right ) ^{3}b}}+{\frac{e}{2\,d \left ( be-cd \right ) \left ( ex+d \right ) ^{2}}}+{\frac{{e}^{2}b}{{d}^{2} \left ( be-cd \right ) ^{2} \left ( ex+d \right ) }}-2\,{\frac{ce}{d \left ( be-cd \right ) ^{2} \left ( ex+d \right ) }}-{\frac{{e}^{3}\ln \left ( ex+d \right ){b}^{2}}{{d}^{3} \left ( be-cd \right ) ^{3}}}+3\,{\frac{{e}^{2}\ln \left ( ex+d \right ) bc}{{d}^{2} \left ( be-cd \right ) ^{3}}}-3\,{\frac{e\ln \left ( ex+d \right ){c}^{2}}{d \left ( be-cd \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.15808, size = 359, normalized size = 2.68 \begin{align*} -\frac{c^{3} \log \left (c x + b\right )}{b c^{3} d^{3} - 3 \, b^{2} c^{2} d^{2} e + 3 \, b^{3} c d e^{2} - b^{4} e^{3}} + \frac{{\left (3 \, c^{2} d^{2} e - 3 \, b c d e^{2} + b^{2} e^{3}\right )} \log \left (e x + d\right )}{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}} - \frac{5 \, c d^{2} e - 3 \, b d e^{2} + 2 \,{\left (2 \, c d e^{2} - b e^{3}\right )} x}{2 \,{\left (c^{2} d^{6} - 2 \, b c d^{5} e + b^{2} d^{4} e^{2} +{\left (c^{2} d^{4} e^{2} - 2 \, b c d^{3} e^{3} + b^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (c^{2} d^{5} e - 2 \, b c d^{4} e^{2} + b^{2} d^{3} e^{3}\right )} x\right )}} + \frac{\log \left (x\right )}{b d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 45.3123, size = 1008, normalized size = 7.52 \begin{align*} -\frac{5 \, b c^{2} d^{4} e - 8 \, b^{2} c d^{3} e^{2} + 3 \, b^{3} d^{2} e^{3} + 2 \,{\left (2 \, b c^{2} d^{3} e^{2} - 3 \, b^{2} c d^{2} e^{3} + b^{3} d e^{4}\right )} x + 2 \,{\left (c^{3} d^{3} e^{2} x^{2} + 2 \, c^{3} d^{4} e x + c^{3} d^{5}\right )} \log \left (c x + b\right ) - 2 \,{\left (3 \, b c^{2} d^{4} e - 3 \, b^{2} c d^{3} e^{2} + b^{3} d^{2} e^{3} +{\left (3 \, b c^{2} d^{2} e^{3} - 3 \, b^{2} c d e^{4} + b^{3} e^{5}\right )} x^{2} + 2 \,{\left (3 \, b c^{2} d^{3} e^{2} - 3 \, b^{2} c d^{2} e^{3} + b^{3} d e^{4}\right )} x\right )} \log \left (e x + d\right ) - 2 \,{\left (c^{3} d^{5} - 3 \, b c^{2} d^{4} e + 3 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3} +{\left (c^{3} d^{3} e^{2} - 3 \, b c^{2} d^{2} e^{3} + 3 \, b^{2} c d e^{4} - b^{3} e^{5}\right )} x^{2} + 2 \,{\left (c^{3} d^{4} e - 3 \, b c^{2} d^{3} e^{2} + 3 \, b^{2} c d^{2} e^{3} - b^{3} d e^{4}\right )} x\right )} \log \left (x\right )}{2 \,{\left (b c^{3} d^{8} - 3 \, b^{2} c^{2} d^{7} e + 3 \, b^{3} c d^{6} e^{2} - b^{4} d^{5} e^{3} +{\left (b c^{3} d^{6} e^{2} - 3 \, b^{2} c^{2} d^{5} e^{3} + 3 \, b^{3} c d^{4} e^{4} - b^{4} d^{3} e^{5}\right )} x^{2} + 2 \,{\left (b c^{3} d^{7} e - 3 \, b^{2} c^{2} d^{6} e^{2} + 3 \, b^{3} c d^{5} e^{3} - b^{4} d^{4} e^{4}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27047, size = 306, normalized size = 2.28 \begin{align*} -\frac{c^{4} \log \left ({\left | c x + b \right |}\right )}{b c^{4} d^{3} - 3 \, b^{2} c^{3} d^{2} e + 3 \, b^{3} c^{2} d e^{2} - b^{4} c e^{3}} + \frac{{\left (3 \, c^{2} d^{2} e^{2} - 3 \, b c d e^{3} + b^{2} e^{4}\right )} \log \left ({\left | x e + d \right |}\right )}{c^{3} d^{6} e - 3 \, b c^{2} d^{5} e^{2} + 3 \, b^{2} c d^{4} e^{3} - b^{3} d^{3} e^{4}} + \frac{\log \left ({\left | x \right |}\right )}{b d^{3}} - \frac{5 \, c^{2} d^{4} e - 8 \, b c d^{3} e^{2} + 3 \, b^{2} d^{2} e^{3} + 2 \,{\left (2 \, c^{2} d^{3} e^{2} - 3 \, b c d^{2} e^{3} + b^{2} d e^{4}\right )} x}{2 \,{\left (c d - b e\right )}^{3}{\left (x e + d\right )}^{2} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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