Optimal. Leaf size=144 \[ -\frac{c^3}{b^2 (b+c x) (c d-b e)^2}+\frac{2 c^3 (c d-2 b e) \log (b+c x)}{b^3 (c d-b e)^3}-\frac{2 \log (x) (b e+c d)}{b^3 d^3}-\frac{1}{b^2 d^2 x}-\frac{e^3}{d^2 (d+e x) (c d-b e)^2}+\frac{2 e^3 (2 c d-b e) \log (d+e x)}{d^3 (c d-b e)^3} \]
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Rubi [A] time = 0.167268, antiderivative size = 144, 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} \[ -\frac{c^3}{b^2 (b+c x) (c d-b e)^2}+\frac{2 c^3 (c d-2 b e) \log (b+c x)}{b^3 (c d-b e)^3}-\frac{2 \log (x) (b e+c d)}{b^3 d^3}-\frac{1}{b^2 d^2 x}-\frac{e^3}{d^2 (d+e x) (c d-b e)^2}+\frac{2 e^3 (2 c d-b e) \log (d+e x)}{d^3 (c d-b e)^3} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^2 \left (b x+c x^2\right )^2} \, dx &=\int \left (\frac{1}{b^2 d^2 x^2}-\frac{2 (c d+b e)}{b^3 d^3 x}+\frac{c^4}{b^2 (-c d+b e)^2 (b+c x)^2}+\frac{2 c^4 (-c d+2 b e)}{b^3 (-c d+b e)^3 (b+c x)}+\frac{e^4}{d^2 (c d-b e)^2 (d+e x)^2}+\frac{2 e^4 (2 c d-b e)}{d^3 (c d-b e)^3 (d+e x)}\right ) \, dx\\ &=-\frac{1}{b^2 d^2 x}-\frac{c^3}{b^2 (c d-b e)^2 (b+c x)}-\frac{e^3}{d^2 (c d-b e)^2 (d+e x)}-\frac{2 (c d+b e) \log (x)}{b^3 d^3}+\frac{2 c^3 (c d-2 b e) \log (b+c x)}{b^3 (c d-b e)^3}+\frac{2 e^3 (2 c d-b e) \log (d+e x)}{d^3 (c d-b e)^3}\\ \end{align*}
Mathematica [A] time = 0.178529, size = 145, normalized size = 1.01 \[ -\frac{c^3}{b^2 (b+c x) (c d-b e)^2}+\frac{2 c^3 (2 b e-c d) \log (b+c x)}{b^3 (b e-c d)^3}-\frac{2 \log (x) (b e+c d)}{b^3 d^3}-\frac{1}{b^2 d^2 x}-\frac{e^3}{d^2 (d+e x) (c d-b e)^2}+\frac{2 e^3 (2 c d-b e) \log (d+e x)}{d^3 (c d-b e)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.065, size = 185, normalized size = 1.3 \begin{align*} -{\frac{1}{{b}^{2}{d}^{2}x}}-2\,{\frac{\ln \left ( x \right ) e}{{d}^{3}{b}^{2}}}-2\,{\frac{\ln \left ( x \right ) c}{{d}^{2}{b}^{3}}}-{\frac{{c}^{3}}{ \left ( be-cd \right ) ^{2}{b}^{2} \left ( cx+b \right ) }}+4\,{\frac{{c}^{3}\ln \left ( cx+b \right ) e}{ \left ( be-cd \right ) ^{3}{b}^{2}}}-2\,{\frac{{c}^{4}\ln \left ( cx+b \right ) d}{ \left ( be-cd \right ) ^{3}{b}^{3}}}-{\frac{{e}^{3}}{{d}^{2} \left ( be-cd \right ) ^{2} \left ( ex+d \right ) }}+2\,{\frac{{e}^{4}\ln \left ( ex+d \right ) b}{{d}^{3} \left ( be-cd \right ) ^{3}}}-4\,{\frac{{e}^{3}\ln \left ( ex+d \right ) c}{{d}^{2} \left ( be-cd \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.2827, size = 504, normalized size = 3.5 \begin{align*} \frac{2 \,{\left (c^{4} d - 2 \, b c^{3} e\right )} \log \left (c x + b\right )}{b^{3} c^{3} d^{3} - 3 \, b^{4} c^{2} d^{2} e + 3 \, b^{5} c d e^{2} - b^{6} e^{3}} + \frac{2 \,{\left (2 \, c d e^{3} - b e^{4}\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{b c^{2} d^{3} - 2 \, b^{2} c d^{2} e + b^{3} d e^{2} + 2 \,{\left (c^{3} d^{2} e - b c^{2} d e^{2} + b^{2} c e^{3}\right )} x^{2} +{\left (2 \, c^{3} d^{3} - b c^{2} d^{2} e - b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} x}{{\left (b^{2} c^{3} d^{4} e - 2 \, b^{3} c^{2} d^{3} e^{2} + b^{4} c d^{2} e^{3}\right )} x^{3} +{\left (b^{2} c^{3} d^{5} - b^{3} c^{2} d^{4} e - b^{4} c d^{3} e^{2} + b^{5} d^{2} e^{3}\right )} x^{2} +{\left (b^{3} c^{2} d^{5} - 2 \, b^{4} c d^{4} e + b^{5} d^{3} e^{2}\right )} x} - \frac{2 \,{\left (c d + b e\right )} \log \left (x\right )}{b^{3} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 85.7863, size = 1262, normalized size = 8.76 \begin{align*} -\frac{b^{2} c^{3} d^{5} - 3 \, b^{3} c^{2} d^{4} e + 3 \, b^{4} c d^{3} e^{2} - b^{5} d^{2} e^{3} + 2 \,{\left (b c^{4} d^{4} e - 2 \, b^{2} c^{3} d^{3} e^{2} + 2 \, b^{3} c^{2} d^{2} e^{3} - b^{4} c d e^{4}\right )} x^{2} +{\left (2 \, b c^{4} d^{5} - 3 \, b^{2} c^{3} d^{4} e + 3 \, b^{4} c d^{2} e^{3} - 2 \, b^{5} d e^{4}\right )} x - 2 \,{\left ({\left (c^{5} d^{4} e - 2 \, b c^{4} d^{3} e^{2}\right )} x^{3} +{\left (c^{5} d^{5} - b c^{4} d^{4} e - 2 \, b^{2} c^{3} d^{3} e^{2}\right )} x^{2} +{\left (b c^{4} d^{5} - 2 \, b^{2} c^{3} d^{4} e\right )} x\right )} \log \left (c x + b\right ) - 2 \,{\left ({\left (2 \, b^{3} c^{2} d e^{4} - b^{4} c e^{5}\right )} x^{3} +{\left (2 \, b^{3} c^{2} d^{2} e^{3} + b^{4} c d e^{4} - b^{5} e^{5}\right )} x^{2} +{\left (2 \, b^{4} c d^{2} e^{3} - b^{5} d e^{4}\right )} x\right )} \log \left (e x + d\right ) + 2 \,{\left ({\left (c^{5} d^{4} e - 2 \, b c^{4} d^{3} e^{2} + 2 \, b^{3} c^{2} d e^{4} - b^{4} c e^{5}\right )} x^{3} +{\left (c^{5} d^{5} - b c^{4} d^{4} e - 2 \, b^{2} c^{3} d^{3} e^{2} + 2 \, b^{3} c^{2} d^{2} e^{3} + b^{4} c d e^{4} - b^{5} e^{5}\right )} x^{2} +{\left (b c^{4} d^{5} - 2 \, b^{2} c^{3} d^{4} e + 2 \, b^{4} c d^{2} e^{3} - b^{5} d e^{4}\right )} x\right )} \log \left (x\right )}{{\left (b^{3} c^{4} d^{6} e - 3 \, b^{4} c^{3} d^{5} e^{2} + 3 \, b^{5} c^{2} d^{4} e^{3} - b^{6} c d^{3} e^{4}\right )} x^{3} +{\left (b^{3} c^{4} d^{7} - 2 \, b^{4} c^{3} d^{6} e + 2 \, b^{6} c d^{4} e^{3} - b^{7} d^{3} e^{4}\right )} x^{2} +{\left (b^{4} c^{3} d^{7} - 3 \, b^{5} c^{2} d^{6} e + 3 \, b^{6} c d^{5} e^{2} - b^{7} d^{4} e^{3}\right )} x} \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 = 1.84037, size = 749, normalized size = 5.2 \begin{align*} \frac{{\left (2 \, c^{4} d^{4} e^{2} - 4 \, b c^{3} d^{3} e^{3} + 2 \, b^{3} c d e^{5} - b^{4} e^{6}\right )} e^{\left (-2\right )} \log \left (\frac{{\left | -2 \, c d e + \frac{2 \, c d^{2} e}{x e + d} + b e^{2} - \frac{2 \, b d e^{2}}{x e + d} -{\left | b \right |} e^{2} \right |}}{{\left | -2 \, c d e + \frac{2 \, c d^{2} e}{x e + d} + b e^{2} - \frac{2 \, b d e^{2}}{x e + d} +{\left | b \right |} e^{2} \right |}}\right )}{{\left (b^{2} c^{3} d^{6} - 3 \, b^{3} c^{2} d^{5} e + 3 \, b^{4} c d^{4} e^{2} - b^{5} d^{3} e^{3}\right )}{\left | b \right |}} - \frac{{\left (2 \, c d e^{3} - b e^{4}\right )} \log \left ({\left | -c + \frac{2 \, c d}{x e + d} - \frac{c d^{2}}{{\left (x e + d\right )}^{2}} - \frac{b e}{x e + d} + \frac{b d e}{{\left (x e + d\right )}^{2}} \right |}\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{e^{7}}{{\left (c^{2} d^{4} e^{4} - 2 \, b c d^{3} e^{5} + b^{2} d^{2} e^{6}\right )}{\left (x e + d\right )}} - \frac{\frac{2 \, c^{4} d^{3} e - 3 \, b c^{3} d^{2} e^{2} + 3 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}}{c d^{2} - b d e} - \frac{{\left (2 \, c^{4} d^{4} e^{2} - 4 \, b c^{3} d^{3} e^{3} + 6 \, b^{2} c^{2} d^{2} e^{4} - 4 \, b^{3} c d e^{5} + b^{4} e^{6}\right )} e^{\left (-1\right )}}{{\left (c d^{2} - b d e\right )}{\left (x e + d\right )}}}{{\left (c d - b e\right )}^{2} b^{2}{\left (c - \frac{2 \, c d}{x e + d} + \frac{c d^{2}}{{\left (x e + d\right )}^{2}} + \frac{b e}{x e + d} - \frac{b d e}{{\left (x e + d\right )}^{2}}\right )} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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