Optimal. Leaf size=108 \[ \frac{c^2 d^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^3}-\frac{c^2 d^2 \log (d+e x)}{\left (c d^2-a e^2\right )^3}+\frac{c d}{(d+e x) \left (c d^2-a e^2\right )^2}+\frac{1}{2 (d+e x)^2 \left (c d^2-a e^2\right )} \]
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Rubi [A] time = 0.0720214, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {626, 44} \[ \frac{c^2 d^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^3}-\frac{c^2 d^2 \log (d+e x)}{\left (c d^2-a e^2\right )^3}+\frac{c d}{(d+e x) \left (c d^2-a e^2\right )^2}+\frac{1}{2 (d+e x)^2 \left (c d^2-a e^2\right )} \]
Antiderivative was successfully verified.
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Rule 626
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx &=\int \frac{1}{(a e+c d x) (d+e x)^3} \, dx\\ &=\int \left (\frac{c^3 d^3}{\left (c d^2-a e^2\right )^3 (a e+c d x)}-\frac{e}{\left (c d^2-a e^2\right ) (d+e x)^3}-\frac{c d e}{\left (c d^2-a e^2\right )^2 (d+e x)^2}-\frac{c^2 d^2 e}{\left (c d^2-a e^2\right )^3 (d+e x)}\right ) \, dx\\ &=\frac{1}{2 \left (c d^2-a e^2\right ) (d+e x)^2}+\frac{c d}{\left (c d^2-a e^2\right )^2 (d+e x)}+\frac{c^2 d^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^3}-\frac{c^2 d^2 \log (d+e x)}{\left (c d^2-a e^2\right )^3}\\ \end{align*}
Mathematica [A] time = 0.0476664, size = 102, normalized size = 0.94 \[ \frac{2 c^2 d^2 (d+e x)^2 \log (a e+c d x)+\left (c d^2-a e^2\right ) \left (c d (3 d+2 e x)-a e^2\right )-2 c^2 d^2 (d+e x)^2 \log (d+e x)}{2 (d+e x)^2 \left (c d^2-a e^2\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 107, normalized size = 1. \begin{align*} -{\frac{1}{ \left ( 2\,a{e}^{2}-2\,c{d}^{2} \right ) \left ( ex+d \right ) ^{2}}}+{\frac{{c}^{2}{d}^{2}\ln \left ( ex+d \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{3}}}+{\frac{cd}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2} \left ( ex+d \right ) }}-{\frac{{c}^{2}{d}^{2}\ln \left ( cdx+ae \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.14348, size = 308, normalized size = 2.85 \begin{align*} \frac{c^{2} d^{2} \log \left (c d x + a e\right )}{c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}} - \frac{c^{2} d^{2} \log \left (e x + d\right )}{c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}} + \frac{2 \, c d e x + 3 \, c d^{2} - a e^{2}}{2 \,{\left (c^{2} d^{6} - 2 \, a c d^{4} e^{2} + a^{2} d^{2} e^{4} +{\left (c^{2} d^{4} e^{2} - 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} x^{2} + 2 \,{\left (c^{2} d^{5} e - 2 \, a c d^{3} e^{3} + a^{2} d e^{5}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.94723, size = 528, normalized size = 4.89 \begin{align*} \frac{3 \, c^{2} d^{4} - 4 \, a c d^{2} e^{2} + a^{2} e^{4} + 2 \,{\left (c^{2} d^{3} e - a c d e^{3}\right )} x + 2 \,{\left (c^{2} d^{2} e^{2} x^{2} + 2 \, c^{2} d^{3} e x + c^{2} d^{4}\right )} \log \left (c d x + a e\right ) - 2 \,{\left (c^{2} d^{2} e^{2} x^{2} + 2 \, c^{2} d^{3} e x + c^{2} d^{4}\right )} \log \left (e x + d\right )}{2 \,{\left (c^{3} d^{8} - 3 \, a c^{2} d^{6} e^{2} + 3 \, a^{2} c d^{4} e^{4} - a^{3} d^{2} e^{6} +{\left (c^{3} d^{6} e^{2} - 3 \, a c^{2} d^{4} e^{4} + 3 \, a^{2} c d^{2} e^{6} - a^{3} e^{8}\right )} x^{2} + 2 \,{\left (c^{3} d^{7} e - 3 \, a c^{2} d^{5} e^{3} + 3 \, a^{2} c d^{3} e^{5} - a^{3} d e^{7}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.52632, size = 471, normalized size = 4.36 \begin{align*} \frac{c^{2} d^{2} \log{\left (x + \frac{- \frac{a^{4} c^{2} d^{2} e^{8}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac{4 a^{3} c^{3} d^{4} e^{6}}{\left (a e^{2} - c d^{2}\right )^{3}} - \frac{6 a^{2} c^{4} d^{6} e^{4}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac{4 a c^{5} d^{8} e^{2}}{\left (a e^{2} - c d^{2}\right )^{3}} + a c^{2} d^{2} e^{2} - \frac{c^{6} d^{10}}{\left (a e^{2} - c d^{2}\right )^{3}} + c^{3} d^{4}}{2 c^{3} d^{3} e} \right )}}{\left (a e^{2} - c d^{2}\right )^{3}} - \frac{c^{2} d^{2} \log{\left (x + \frac{\frac{a^{4} c^{2} d^{2} e^{8}}{\left (a e^{2} - c d^{2}\right )^{3}} - \frac{4 a^{3} c^{3} d^{4} e^{6}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac{6 a^{2} c^{4} d^{6} e^{4}}{\left (a e^{2} - c d^{2}\right )^{3}} - \frac{4 a c^{5} d^{8} e^{2}}{\left (a e^{2} - c d^{2}\right )^{3}} + a c^{2} d^{2} e^{2} + \frac{c^{6} d^{10}}{\left (a e^{2} - c d^{2}\right )^{3}} + c^{3} d^{4}}{2 c^{3} d^{3} e} \right )}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac{- a e^{2} + 3 c d^{2} + 2 c d e x}{2 a^{2} d^{2} e^{4} - 4 a c d^{4} e^{2} + 2 c^{2} d^{6} + x^{2} \left (2 a^{2} e^{6} - 4 a c d^{2} e^{4} + 2 c^{2} d^{4} e^{2}\right ) + x \left (4 a^{2} d e^{5} - 8 a c d^{3} e^{3} + 4 c^{2} d^{5} e\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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