Optimal. Leaf size=193 \[ \frac{\log (x) \left (b^2 e^2+3 b c d e+6 c^2 d^2\right )}{b^5 d^3}-\frac{c^3 \left (10 b^2 e^2-15 b c d e+6 c^2 d^2\right ) \log (b+c x)}{b^5 (c d-b e)^3}+\frac{c^3 (3 c d-4 b e)}{b^4 (b+c x) (c d-b e)^2}+\frac{c^3}{2 b^3 (b+c x)^2 (c d-b e)}+\frac{b e+3 c d}{b^4 d^2 x}-\frac{1}{2 b^3 d x^2}+\frac{e^5 \log (d+e x)}{d^3 (c d-b e)^3} \]
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Rubi [A] time = 0.227096, antiderivative size = 193, 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} \[ \frac{\log (x) \left (b^2 e^2+3 b c d e+6 c^2 d^2\right )}{b^5 d^3}-\frac{c^3 \left (10 b^2 e^2-15 b c d e+6 c^2 d^2\right ) \log (b+c x)}{b^5 (c d-b e)^3}+\frac{c^3 (3 c d-4 b e)}{b^4 (b+c x) (c d-b e)^2}+\frac{c^3}{2 b^3 (b+c x)^2 (c d-b e)}+\frac{b e+3 c d}{b^4 d^2 x}-\frac{1}{2 b^3 d x^2}+\frac{e^5 \log (d+e x)}{d^3 (c d-b e)^3} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int \frac{1}{(d+e x) \left (b x+c x^2\right )^3} \, dx &=\int \left (\frac{1}{b^3 d x^3}+\frac{-3 c d-b e}{b^4 d^2 x^2}+\frac{6 c^2 d^2+3 b c d e+b^2 e^2}{b^5 d^3 x}+\frac{c^4}{b^3 (-c d+b e) (b+c x)^3}+\frac{c^4 (-3 c d+4 b e)}{b^4 (-c d+b e)^2 (b+c x)^2}+\frac{c^4 \left (6 c^2 d^2-15 b c d e+10 b^2 e^2\right )}{b^5 (-c d+b e)^3 (b+c x)}+\frac{e^6}{d^3 (c d-b e)^3 (d+e x)}\right ) \, dx\\ &=-\frac{1}{2 b^3 d x^2}+\frac{3 c d+b e}{b^4 d^2 x}+\frac{c^3}{2 b^3 (c d-b e) (b+c x)^2}+\frac{c^3 (3 c d-4 b e)}{b^4 (c d-b e)^2 (b+c x)}+\frac{\left (6 c^2 d^2+3 b c d e+b^2 e^2\right ) \log (x)}{b^5 d^3}-\frac{c^3 \left (6 c^2 d^2-15 b c d e+10 b^2 e^2\right ) \log (b+c x)}{b^5 (c d-b e)^3}+\frac{e^5 \log (d+e x)}{d^3 (c d-b e)^3}\\ \end{align*}
Mathematica [A] time = 0.196233, size = 192, normalized size = 0.99 \[ \frac{\log (x) \left (b^2 e^2+3 b c d e+6 c^2 d^2\right )}{b^5 d^3}+\frac{c^3 \left (10 b^2 e^2-15 b c d e+6 c^2 d^2\right ) \log (b+c x)}{b^5 (b e-c d)^3}+\frac{c^3 (3 c d-4 b e)}{b^4 (b+c x) (c d-b e)^2}-\frac{c^3}{2 b^3 (b+c x)^2 (b e-c d)}+\frac{b e+3 c d}{b^4 d^2 x}-\frac{1}{2 b^3 d x^2}+\frac{e^5 \log (d+e x)}{d^3 (c d-b e)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.061, size = 254, normalized size = 1.3 \begin{align*} -{\frac{1}{2\,d{b}^{3}{x}^{2}}}+{\frac{e}{{d}^{2}{b}^{3}x}}+3\,{\frac{c}{d{b}^{4}x}}+{\frac{\ln \left ( x \right ){e}^{2}}{{d}^{3}{b}^{3}}}+3\,{\frac{\ln \left ( x \right ) ce}{{d}^{2}{b}^{4}}}+6\,{\frac{\ln \left ( x \right ){c}^{2}}{d{b}^{5}}}-{\frac{{c}^{3}}{ \left ( 2\,be-2\,cd \right ){b}^{3} \left ( cx+b \right ) ^{2}}}-4\,{\frac{{c}^{3}e}{ \left ( be-cd \right ) ^{2}{b}^{3} \left ( cx+b \right ) }}+3\,{\frac{{c}^{4}d}{ \left ( be-cd \right ) ^{2}{b}^{4} \left ( cx+b \right ) }}+10\,{\frac{{c}^{3}\ln \left ( cx+b \right ){e}^{2}}{ \left ( be-cd \right ) ^{3}{b}^{3}}}-15\,{\frac{{c}^{4}\ln \left ( cx+b \right ) de}{ \left ( be-cd \right ) ^{3}{b}^{4}}}+6\,{\frac{{c}^{5}\ln \left ( cx+b \right ){d}^{2}}{ \left ( be-cd \right ) ^{3}{b}^{5}}}-{\frac{{e}^{5}\ln \left ( ex+d \right ) }{{d}^{3} \left ( be-cd \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.23846, size = 593, normalized size = 3.07 \begin{align*} \frac{e^{5} \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{{\left (6 \, c^{5} d^{2} - 15 \, b c^{4} d e + 10 \, b^{2} c^{3} e^{2}\right )} \log \left (c x + b\right )}{b^{5} c^{3} d^{3} - 3 \, b^{6} c^{2} d^{2} e + 3 \, b^{7} c d e^{2} - b^{8} e^{3}} - \frac{b^{3} c^{2} d^{3} - 2 \, b^{4} c d^{2} e + b^{5} d e^{2} - 2 \,{\left (6 \, c^{5} d^{3} - 9 \, b c^{4} d^{2} e + b^{2} c^{3} d e^{2} + b^{3} c^{2} e^{3}\right )} x^{3} -{\left (18 \, b c^{4} d^{3} - 27 \, b^{2} c^{3} d^{2} e + 3 \, b^{3} c^{2} d e^{2} + 4 \, b^{4} c e^{3}\right )} x^{2} - 2 \,{\left (2 \, b^{2} c^{3} d^{3} - 3 \, b^{3} c^{2} d^{2} e + b^{5} e^{3}\right )} x}{2 \,{\left ({\left (b^{4} c^{4} d^{4} - 2 \, b^{5} c^{3} d^{3} e + b^{6} c^{2} d^{2} e^{2}\right )} x^{4} + 2 \,{\left (b^{5} c^{3} d^{4} - 2 \, b^{6} c^{2} d^{3} e + b^{7} c d^{2} e^{2}\right )} x^{3} +{\left (b^{6} c^{2} d^{4} - 2 \, b^{7} c d^{3} e + b^{8} d^{2} e^{2}\right )} x^{2}\right )}} + \frac{{\left (6 \, c^{2} d^{2} + 3 \, b c d e + b^{2} e^{2}\right )} \log \left (x\right )}{b^{5} d^{3}} \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 = 1.3336, size = 559, normalized size = 2.9 \begin{align*} -\frac{{\left (6 \, c^{6} d^{2} - 15 \, b c^{5} d e + 10 \, b^{2} c^{4} e^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c^{4} d^{3} - 3 \, b^{6} c^{3} d^{2} e + 3 \, b^{7} c^{2} d e^{2} - b^{8} c e^{3}} + \frac{e^{6} \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{{\left (6 \, c^{2} d^{2} + 3 \, b c d e + b^{2} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5} d^{3}} - \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} - 2 \,{\left (6 \, c^{6} d^{5} - 15 \, b c^{5} d^{4} e + 10 \, b^{2} c^{4} d^{3} e^{2} - b^{4} c^{2} d e^{4}\right )} x^{3} -{\left (18 \, b c^{5} d^{5} - 45 \, b^{2} c^{4} d^{4} e + 30 \, b^{3} c^{3} d^{3} e^{2} + b^{4} c^{2} d^{2} e^{3} - 4 \, b^{5} c d e^{4}\right )} x^{2} - 2 \,{\left (2 \, b^{2} c^{4} d^{5} - 5 \, b^{3} c^{3} d^{4} e + 3 \, b^{4} c^{2} d^{3} e^{2} + b^{5} c d^{2} e^{3} - b^{6} d e^{4}\right )} x}{2 \,{\left (c d - b e\right )}^{3}{\left (c x + b\right )}^{2} b^{4} d^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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