Optimal. Leaf size=139 \[ \frac{c^2 d^2}{(d+e x) \left (c d^2-a e^2\right )^3}+\frac{c^3 d^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}-\frac{c^3 d^3 \log (d+e x)}{\left (c d^2-a e^2\right )^4}+\frac{c d}{2 (d+e x)^2 \left (c d^2-a e^2\right )^2}+\frac{1}{3 (d+e x)^3 \left (c d^2-a e^2\right )} \]
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Rubi [A] time = 0.0989909, antiderivative size = 139, 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} \[ \frac{c^2 d^2}{(d+e x) \left (c d^2-a e^2\right )^3}+\frac{c^3 d^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}-\frac{c^3 d^3 \log (d+e x)}{\left (c d^2-a e^2\right )^4}+\frac{c d}{2 (d+e x)^2 \left (c d^2-a e^2\right )^2}+\frac{1}{3 (d+e x)^3 \left (c d^2-a e^2\right )} \]
Antiderivative was successfully verified.
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Rule 626
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^3 \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)^4} \, dx\\ &=\int \left (\frac{c^4 d^4}{\left (c d^2-a e^2\right )^4 (a e+c d x)}-\frac{e}{\left (c d^2-a e^2\right ) (d+e x)^4}-\frac{c d e}{\left (c d^2-a e^2\right )^2 (d+e x)^3}-\frac{c^2 d^2 e}{\left (c d^2-a e^2\right )^3 (d+e x)^2}-\frac{c^3 d^3 e}{\left (c d^2-a e^2\right )^4 (d+e x)}\right ) \, dx\\ &=\frac{1}{3 \left (c d^2-a e^2\right ) (d+e x)^3}+\frac{c d}{2 \left (c d^2-a e^2\right )^2 (d+e x)^2}+\frac{c^2 d^2}{\left (c d^2-a e^2\right )^3 (d+e x)}+\frac{c^3 d^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}-\frac{c^3 d^3 \log (d+e x)}{\left (c d^2-a e^2\right )^4}\\ \end{align*}
Mathematica [A] time = 0.0684324, size = 135, normalized size = 0.97 \[ \frac{\left (c d^2-a e^2\right ) \left (2 a^2 e^4-a c d e^2 (7 d+3 e x)+c^2 d^2 \left (11 d^2+15 d e x+6 e^2 x^2\right )\right )+6 c^3 d^3 (d+e x)^3 \log (a e+c d x)-6 c^3 d^3 (d+e x)^3 \log (d+e x)}{6 (d+e x)^3 \left (c d^2-a e^2\right )^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 137, normalized size = 1. \begin{align*} -{\frac{1}{ \left ( 3\,a{e}^{2}-3\,c{d}^{2} \right ) \left ( ex+d \right ) ^{3}}}-{\frac{{c}^{2}{d}^{2}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{3} \left ( ex+d \right ) }}+{\frac{cd}{2\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{2} \left ( ex+d \right ) ^{2}}}-{\frac{{c}^{3}{d}^{3}\ln \left ( ex+d \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{4}}}+{\frac{{c}^{3}{d}^{3}\ln \left ( cdx+ae \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.15512, size = 531, normalized size = 3.82 \begin{align*} \frac{c^{3} d^{3} \log \left (c d x + a e\right )}{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} - \frac{c^{3} d^{3} \log \left (e x + d\right )}{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} + \frac{6 \, c^{2} d^{2} e^{2} x^{2} + 11 \, c^{2} d^{4} - 7 \, a c d^{2} e^{2} + 2 \, a^{2} e^{4} + 3 \,{\left (5 \, c^{2} d^{3} e - a c d e^{3}\right )} x}{6 \,{\left (c^{3} d^{9} - 3 \, a c^{2} d^{7} e^{2} + 3 \, a^{2} c d^{5} e^{4} - a^{3} d^{3} e^{6} +{\left (c^{3} d^{6} e^{3} - 3 \, a c^{2} d^{4} e^{5} + 3 \, a^{2} c d^{2} e^{7} - a^{3} e^{9}\right )} x^{3} + 3 \,{\left (c^{3} d^{7} e^{2} - 3 \, a c^{2} d^{5} e^{4} + 3 \, a^{2} c d^{3} e^{6} - a^{3} d e^{8}\right )} x^{2} + 3 \,{\left (c^{3} d^{8} e - 3 \, a c^{2} d^{6} e^{3} + 3 \, a^{2} c d^{4} e^{5} - a^{3} d^{2} e^{7}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.86856, size = 906, normalized size = 6.52 \begin{align*} \frac{11 \, c^{3} d^{6} - 18 \, a c^{2} d^{4} e^{2} + 9 \, a^{2} c d^{2} e^{4} - 2 \, a^{3} e^{6} + 6 \,{\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 3 \,{\left (5 \, c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x + 6 \,{\left (c^{3} d^{3} e^{3} x^{3} + 3 \, c^{3} d^{4} e^{2} x^{2} + 3 \, c^{3} d^{5} e x + c^{3} d^{6}\right )} \log \left (c d x + a e\right ) - 6 \,{\left (c^{3} d^{3} e^{3} x^{3} + 3 \, c^{3} d^{4} e^{2} x^{2} + 3 \, c^{3} d^{5} e x + c^{3} d^{6}\right )} \log \left (e x + d\right )}{6 \,{\left (c^{4} d^{11} - 4 \, a c^{3} d^{9} e^{2} + 6 \, a^{2} c^{2} d^{7} e^{4} - 4 \, a^{3} c d^{5} e^{6} + a^{4} d^{3} e^{8} +{\left (c^{4} d^{8} e^{3} - 4 \, a c^{3} d^{6} e^{5} + 6 \, a^{2} c^{2} d^{4} e^{7} - 4 \, a^{3} c d^{2} e^{9} + a^{4} e^{11}\right )} x^{3} + 3 \,{\left (c^{4} d^{9} e^{2} - 4 \, a c^{3} d^{7} e^{4} + 6 \, a^{2} c^{2} d^{5} e^{6} - 4 \, a^{3} c d^{3} e^{8} + a^{4} d e^{10}\right )} x^{2} + 3 \,{\left (c^{4} d^{10} e - 4 \, a c^{3} d^{8} e^{3} + 6 \, a^{2} c^{2} d^{6} e^{5} - 4 \, a^{3} c d^{4} e^{7} + a^{4} d^{2} e^{9}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.40193, size = 672, normalized size = 4.83 \begin{align*} - \frac{c^{3} d^{3} \log{\left (x + \frac{- \frac{a^{5} c^{3} d^{3} e^{10}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{5 a^{4} c^{4} d^{5} e^{8}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{10 a^{3} c^{5} d^{7} e^{6}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{10 a^{2} c^{6} d^{9} e^{4}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{5 a c^{7} d^{11} e^{2}}{\left (a e^{2} - c d^{2}\right )^{4}} + a c^{3} d^{3} e^{2} + \frac{c^{8} d^{13}}{\left (a e^{2} - c d^{2}\right )^{4}} + c^{4} d^{5}}{2 c^{4} d^{4} e} \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{c^{3} d^{3} \log{\left (x + \frac{\frac{a^{5} c^{3} d^{3} e^{10}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{5 a^{4} c^{4} d^{5} e^{8}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{10 a^{3} c^{5} d^{7} e^{6}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{10 a^{2} c^{6} d^{9} e^{4}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{5 a c^{7} d^{11} e^{2}}{\left (a e^{2} - c d^{2}\right )^{4}} + a c^{3} d^{3} e^{2} - \frac{c^{8} d^{13}}{\left (a e^{2} - c d^{2}\right )^{4}} + c^{4} d^{5}}{2 c^{4} d^{4} e} \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{2 a^{2} e^{4} - 7 a c d^{2} e^{2} + 11 c^{2} d^{4} + 6 c^{2} d^{2} e^{2} x^{2} + x \left (- 3 a c d e^{3} + 15 c^{2} d^{3} e\right )}{6 a^{3} d^{3} e^{6} - 18 a^{2} c d^{5} e^{4} + 18 a c^{2} d^{7} e^{2} - 6 c^{3} d^{9} + x^{3} \left (6 a^{3} e^{9} - 18 a^{2} c d^{2} e^{7} + 18 a c^{2} d^{4} e^{5} - 6 c^{3} d^{6} e^{3}\right ) + x^{2} \left (18 a^{3} d e^{8} - 54 a^{2} c d^{3} e^{6} + 54 a c^{2} d^{5} e^{4} - 18 c^{3} d^{7} e^{2}\right ) + x \left (18 a^{3} d^{2} e^{7} - 54 a^{2} c d^{4} e^{5} + 54 a c^{2} d^{6} e^{3} - 18 c^{3} d^{8} 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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