Optimal. Leaf size=230 \[ \frac{3 \log (x) \left (b^2 e^2+2 b c d e+2 c^2 d^2\right )}{b^5 d^4}-\frac{3 c^4 \left (5 b^2 e^2-6 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 (c d-b e)^4}+\frac{c^4 (3 c d-5 b e)}{b^4 (b+c x) (c d-b e)^3}+\frac{c^4}{2 b^3 (b+c x)^2 (c d-b e)^2}+\frac{2 b e+3 c d}{b^4 d^3 x}-\frac{1}{2 b^3 d^2 x^2}-\frac{e^5}{d^3 (d+e x) (c d-b e)^3}+\frac{3 e^5 (2 c d-b e) \log (d+e x)}{d^4 (c d-b e)^4} \]
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Rubi [A] time = 0.33923, antiderivative size = 230, 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} \[ \frac{3 \log (x) \left (b^2 e^2+2 b c d e+2 c^2 d^2\right )}{b^5 d^4}-\frac{3 c^4 \left (5 b^2 e^2-6 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 (c d-b e)^4}+\frac{c^4 (3 c d-5 b e)}{b^4 (b+c x) (c d-b e)^3}+\frac{c^4}{2 b^3 (b+c x)^2 (c d-b e)^2}+\frac{2 b e+3 c d}{b^4 d^3 x}-\frac{1}{2 b^3 d^2 x^2}-\frac{e^5}{d^3 (d+e x) (c d-b e)^3}+\frac{3 e^5 (2 c d-b e) \log (d+e x)}{d^4 (c d-b e)^4} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^2 \left (b x+c x^2\right )^3} \, dx &=\int \left (\frac{1}{b^3 d^2 x^3}+\frac{-3 c d-2 b e}{b^4 d^3 x^2}+\frac{3 \left (2 c^2 d^2+2 b c d e+b^2 e^2\right )}{b^5 d^4 x}-\frac{c^5}{b^3 (-c d+b e)^2 (b+c x)^3}-\frac{c^5 (-3 c d+5 b e)}{b^4 (-c d+b e)^3 (b+c x)^2}-\frac{3 c^5 \left (2 c^2 d^2-6 b c d e+5 b^2 e^2\right )}{b^5 (-c d+b e)^4 (b+c x)}+\frac{e^6}{d^3 (c d-b e)^3 (d+e x)^2}+\frac{3 e^6 (2 c d-b e)}{d^4 (c d-b e)^4 (d+e x)}\right ) \, dx\\ &=-\frac{1}{2 b^3 d^2 x^2}+\frac{3 c d+2 b e}{b^4 d^3 x}+\frac{c^4}{2 b^3 (c d-b e)^2 (b+c x)^2}+\frac{c^4 (3 c d-5 b e)}{b^4 (c d-b e)^3 (b+c x)}-\frac{e^5}{d^3 (c d-b e)^3 (d+e x)}+\frac{3 \left (2 c^2 d^2+2 b c d e+b^2 e^2\right ) \log (x)}{b^5 d^4}-\frac{3 c^4 \left (2 c^2 d^2-6 b c d e+5 b^2 e^2\right ) \log (b+c x)}{b^5 (c d-b e)^4}+\frac{3 e^5 (2 c d-b e) \log (d+e x)}{d^4 (c d-b e)^4}\\ \end{align*}
Mathematica [A] time = 0.29037, size = 230, normalized size = 1. \[ \frac{3 \log (x) \left (b^2 e^2+2 b c d e+2 c^2 d^2\right )}{b^5 d^4}-\frac{3 c^4 \left (5 b^2 e^2-6 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 (c d-b e)^4}+\frac{c^4 (5 b e-3 c d)}{b^4 (b+c x) (b e-c d)^3}+\frac{c^4}{2 b^3 (b+c x)^2 (c d-b e)^2}+\frac{2 b e+3 c d}{b^4 d^3 x}-\frac{1}{2 b^3 d^2 x^2}-\frac{e^5}{d^3 (d+e x) (c d-b e)^3}+\frac{3 e^5 (2 c d-b e) \log (d+e x)}{d^4 (c d-b e)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.069, size = 306, normalized size = 1.3 \begin{align*} -{\frac{1}{2\,{d}^{2}{b}^{3}{x}^{2}}}+2\,{\frac{e}{{d}^{3}{b}^{3}x}}+3\,{\frac{c}{{d}^{2}{b}^{4}x}}+3\,{\frac{\ln \left ( x \right ){e}^{2}}{{d}^{4}{b}^{3}}}+6\,{\frac{\ln \left ( x \right ) ce}{{d}^{3}{b}^{4}}}+6\,{\frac{\ln \left ( x \right ){c}^{2}}{{d}^{2}{b}^{5}}}+{\frac{{c}^{4}}{2\, \left ( be-cd \right ) ^{2}{b}^{3} \left ( cx+b \right ) ^{2}}}+5\,{\frac{{c}^{4}e}{ \left ( be-cd \right ) ^{3}{b}^{3} \left ( cx+b \right ) }}-3\,{\frac{{c}^{5}d}{ \left ( be-cd \right ) ^{3}{b}^{4} \left ( cx+b \right ) }}-15\,{\frac{{c}^{4}\ln \left ( cx+b \right ){e}^{2}}{ \left ( be-cd \right ) ^{4}{b}^{3}}}+18\,{\frac{{c}^{5}\ln \left ( cx+b \right ) de}{ \left ( be-cd \right ) ^{4}{b}^{4}}}-6\,{\frac{{c}^{6}\ln \left ( cx+b \right ){d}^{2}}{ \left ( be-cd \right ) ^{4}{b}^{5}}}+{\frac{{e}^{5}}{{d}^{3} \left ( be-cd \right ) ^{3} \left ( ex+d \right ) }}-3\,{\frac{{e}^{6}\ln \left ( ex+d \right ) b}{{d}^{4} \left ( be-cd \right ) ^{4}}}+6\,{\frac{{e}^{5}\ln \left ( ex+d \right ) c}{{d}^{3} \left ( be-cd \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.29149, size = 1015, normalized size = 4.41 \begin{align*} -\frac{3 \,{\left (2 \, c^{6} d^{2} - 6 \, b c^{5} d e + 5 \, b^{2} c^{4} e^{2}\right )} \log \left (c x + b\right )}{b^{5} c^{4} d^{4} - 4 \, b^{6} c^{3} d^{3} e + 6 \, b^{7} c^{2} d^{2} e^{2} - 4 \, b^{8} c d e^{3} + b^{9} e^{4}} + \frac{3 \,{\left (2 \, c d e^{5} - b e^{6}\right )} \log \left (e x + d\right )}{c^{4} d^{8} - 4 \, b c^{3} d^{7} e + 6 \, b^{2} c^{2} d^{6} e^{2} - 4 \, b^{3} c d^{5} e^{3} + b^{4} d^{4} e^{4}} - \frac{b^{3} c^{3} d^{5} - 3 \, b^{4} c^{2} d^{4} e + 3 \, b^{5} c d^{3} e^{2} - b^{6} d^{2} e^{3} - 6 \,{\left (2 \, c^{6} d^{4} e - 4 \, b c^{5} d^{3} e^{2} + b^{2} c^{4} d^{2} e^{3} + b^{3} c^{3} d e^{4} - b^{4} c^{2} e^{5}\right )} x^{4} - 3 \,{\left (4 \, c^{6} d^{5} - 2 \, b c^{5} d^{4} e - 10 \, b^{2} c^{4} d^{3} e^{2} + 5 \, b^{3} c^{3} d^{2} e^{3} + 3 \, b^{4} c^{2} d e^{4} - 4 \, b^{5} c e^{5}\right )} x^{3} -{\left (18 \, b c^{5} d^{5} - 32 \, b^{2} c^{4} d^{4} e + b^{3} c^{3} d^{3} e^{2} + 13 \, b^{4} c^{2} d^{2} e^{3} - 6 \, b^{6} e^{5}\right )} x^{2} -{\left (4 \, b^{2} c^{4} d^{5} - 9 \, b^{3} c^{3} d^{4} e + 3 \, b^{4} c^{2} d^{3} e^{2} + 5 \, b^{5} c d^{2} e^{3} - 3 \, b^{6} d e^{4}\right )} x}{2 \,{\left ({\left (b^{4} c^{5} d^{6} e - 3 \, b^{5} c^{4} d^{5} e^{2} + 3 \, b^{6} c^{3} d^{4} e^{3} - b^{7} c^{2} d^{3} e^{4}\right )} x^{5} +{\left (b^{4} c^{5} d^{7} - b^{5} c^{4} d^{6} e - 3 \, b^{6} c^{3} d^{5} e^{2} + 5 \, b^{7} c^{2} d^{4} e^{3} - 2 \, b^{8} c d^{3} e^{4}\right )} x^{4} +{\left (2 \, b^{5} c^{4} d^{7} - 5 \, b^{6} c^{3} d^{6} e + 3 \, b^{7} c^{2} d^{5} e^{2} + b^{8} c d^{4} e^{3} - b^{9} d^{3} e^{4}\right )} x^{3} +{\left (b^{6} c^{3} d^{7} - 3 \, b^{7} c^{2} d^{6} e + 3 \, b^{8} c d^{5} e^{2} - b^{9} d^{4} e^{3}\right )} x^{2}\right )}} + \frac{3 \,{\left (2 \, c^{2} d^{2} + 2 \, b c d e + b^{2} e^{2}\right )} \log \left (x\right )}{b^{5} d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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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 [B] time = 1.35597, size = 1133, normalized size = 4.93 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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