Optimal. Leaf size=142 \[ \frac{e^2}{(d+e x) \left (c d^2-a e^2\right )^3}+\frac{2 c d e}{\left (c d^2-a e^2\right )^3 (a e+c d x)}-\frac{c d}{2 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}+\frac{3 c d e^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}-\frac{3 c d e^2 \log (d+e x)}{\left (c d^2-a e^2\right )^4} \]
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Rubi [A] time = 0.112844, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {626, 44} \[ \frac{e^2}{(d+e x) \left (c d^2-a e^2\right )^3}+\frac{2 c d e}{\left (c d^2-a e^2\right )^3 (a e+c d x)}-\frac{c d}{2 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}+\frac{3 c d e^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}-\frac{3 c d e^2 \log (d+e x)}{\left (c d^2-a e^2\right )^4} \]
Antiderivative was successfully verified.
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Rule 626
Rule 44
Rubi steps
\begin{align*} \int \frac{d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx &=\int \frac{1}{(a e+c d x)^3 (d+e x)^2} \, dx\\ &=\int \left (\frac{c^2 d^2}{\left (c d^2-a e^2\right )^2 (a e+c d x)^3}-\frac{2 c^2 d^2 e}{\left (c d^2-a e^2\right )^3 (a e+c d x)^2}+\frac{3 c^2 d^2 e^2}{\left (c d^2-a e^2\right )^4 (a e+c d x)}-\frac{e^3}{\left (c d^2-a e^2\right )^3 (d+e x)^2}-\frac{3 c d e^3}{\left (c d^2-a e^2\right )^4 (d+e x)}\right ) \, dx\\ &=-\frac{c d}{2 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}+\frac{2 c d e}{\left (c d^2-a e^2\right )^3 (a e+c d x)}+\frac{e^2}{\left (c d^2-a e^2\right )^3 (d+e x)}+\frac{3 c d e^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}-\frac{3 c d e^2 \log (d+e x)}{\left (c d^2-a e^2\right )^4}\\ \end{align*}
Mathematica [A] time = 0.102824, size = 127, normalized size = 0.89 \[ \frac{\frac{4 c d e \left (c d^2-a e^2\right )}{a e+c d x}+\frac{2 c d^2 e^2-2 a e^4}{d+e x}-\frac{c d \left (c d^2-a e^2\right )^2}{(a e+c d x)^2}+6 c d e^2 \log (a e+c d x)-6 c d e^2 \log (d+e x)}{2 \left (c d^2-a e^2\right )^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 142, normalized size = 1. \begin{align*} -{\frac{{e}^{2}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{3} \left ( ex+d \right ) }}-3\,{\frac{{e}^{2}cd\ln \left ( ex+d \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{4}}}-{\frac{cd}{2\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{2} \left ( cdx+ae \right ) ^{2}}}+3\,{\frac{{e}^{2}cd\ln \left ( cdx+ae \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{4}}}-2\,{\frac{dec}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{3} \left ( cdx+ae \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.22685, size = 579, normalized size = 4.08 \begin{align*} \frac{3 \, c d e^{2} \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{3 \, c d e^{2} \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} - c^{2} d^{4} + 5 \, a c d^{2} e^{2} + 2 \, a^{2} e^{4} + 3 \,{\left (c^{2} d^{3} e + 3 \, a c d e^{3}\right )} x}{2 \,{\left (a^{2} c^{3} d^{7} e^{2} - 3 \, a^{3} c^{2} d^{5} e^{4} + 3 \, a^{4} c d^{3} e^{6} - a^{5} d e^{8} +{\left (c^{5} d^{8} e - 3 \, a c^{4} d^{6} e^{3} + 3 \, a^{2} c^{3} d^{4} e^{5} - a^{3} c^{2} d^{2} e^{7}\right )} x^{3} +{\left (c^{5} d^{9} - a c^{4} d^{7} e^{2} - 3 \, a^{2} c^{3} d^{5} e^{4} + 5 \, a^{3} c^{2} d^{3} e^{6} - 2 \, a^{4} c d e^{8}\right )} x^{2} +{\left (2 \, a c^{4} d^{8} e - 5 \, a^{2} c^{3} d^{6} e^{3} + 3 \, a^{3} c^{2} d^{4} e^{5} + a^{4} c d^{2} e^{7} - a^{5} e^{9}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.03729, size = 1099, normalized size = 7.74 \begin{align*} -\frac{c^{3} d^{6} - 6 \, a c^{2} d^{4} e^{2} + 3 \, 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 (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} - 3 \, a^{2} c d e^{5}\right )} x - 6 \,{\left (c^{3} d^{3} e^{3} x^{3} + a^{2} c d^{2} e^{4} +{\left (c^{3} d^{4} e^{2} + 2 \, a c^{2} d^{2} e^{4}\right )} x^{2} +{\left (2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} \log \left (c d x + a e\right ) + 6 \,{\left (c^{3} d^{3} e^{3} x^{3} + a^{2} c d^{2} e^{4} +{\left (c^{3} d^{4} e^{2} + 2 \, a c^{2} d^{2} e^{4}\right )} x^{2} +{\left (2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (a^{2} c^{4} d^{9} e^{2} - 4 \, a^{3} c^{3} d^{7} e^{4} + 6 \, a^{4} c^{2} d^{5} e^{6} - 4 \, a^{5} c d^{3} e^{8} + a^{6} d e^{10} +{\left (c^{6} d^{10} e - 4 \, a c^{5} d^{8} e^{3} + 6 \, a^{2} c^{4} d^{6} e^{5} - 4 \, a^{3} c^{3} d^{4} e^{7} + a^{4} c^{2} d^{2} e^{9}\right )} x^{3} +{\left (c^{6} d^{11} - 2 \, a c^{5} d^{9} e^{2} - 2 \, a^{2} c^{4} d^{7} e^{4} + 8 \, a^{3} c^{3} d^{5} e^{6} - 7 \, a^{4} c^{2} d^{3} e^{8} + 2 \, a^{5} c d e^{10}\right )} x^{2} +{\left (2 \, a c^{5} d^{10} e - 7 \, a^{2} c^{4} d^{8} e^{3} + 8 \, a^{3} c^{3} d^{6} e^{5} - 2 \, a^{4} c^{2} d^{4} e^{7} - 2 \, a^{5} c d^{2} e^{9} + a^{6} e^{11}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.62892, size = 734, normalized size = 5.17 \begin{align*} - \frac{3 c d e^{2} \log{\left (x + \frac{- \frac{3 a^{5} c d e^{12}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{15 a^{4} c^{2} d^{3} e^{10}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{30 a^{3} c^{3} d^{5} e^{8}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{30 a^{2} c^{4} d^{7} e^{6}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{15 a c^{5} d^{9} e^{4}}{\left (a e^{2} - c d^{2}\right )^{4}} + 3 a c d e^{4} + \frac{3 c^{6} d^{11} e^{2}}{\left (a e^{2} - c d^{2}\right )^{4}} + 3 c^{2} d^{3} e^{2}}{6 c^{2} d^{2} e^{3}} \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{3 c d e^{2} \log{\left (x + \frac{\frac{3 a^{5} c d e^{12}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{15 a^{4} c^{2} d^{3} e^{10}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{30 a^{3} c^{3} d^{5} e^{8}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{30 a^{2} c^{4} d^{7} e^{6}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{15 a c^{5} d^{9} e^{4}}{\left (a e^{2} - c d^{2}\right )^{4}} + 3 a c d e^{4} - \frac{3 c^{6} d^{11} e^{2}}{\left (a e^{2} - c d^{2}\right )^{4}} + 3 c^{2} d^{3} e^{2}}{6 c^{2} d^{2} e^{3}} \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{2 a^{2} e^{4} + 5 a c d^{2} e^{2} - c^{2} d^{4} + 6 c^{2} d^{2} e^{2} x^{2} + x \left (9 a c d e^{3} + 3 c^{2} d^{3} e\right )}{2 a^{5} d e^{8} - 6 a^{4} c d^{3} e^{6} + 6 a^{3} c^{2} d^{5} e^{4} - 2 a^{2} c^{3} d^{7} e^{2} + x^{3} \left (2 a^{3} c^{2} d^{2} e^{7} - 6 a^{2} c^{3} d^{4} e^{5} + 6 a c^{4} d^{6} e^{3} - 2 c^{5} d^{8} e\right ) + x^{2} \left (4 a^{4} c d e^{8} - 10 a^{3} c^{2} d^{3} e^{6} + 6 a^{2} c^{3} d^{5} e^{4} + 2 a c^{4} d^{7} e^{2} - 2 c^{5} d^{9}\right ) + x \left (2 a^{5} e^{9} - 2 a^{4} c d^{2} e^{7} - 6 a^{3} c^{2} d^{4} e^{5} + 10 a^{2} c^{3} d^{6} e^{3} - 4 a c^{4} d^{8} e\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.25553, size = 483, normalized size = 3.4 \begin{align*} \frac{6 \,{\left (c^{2} d^{3} e^{2} - a c d e^{4}\right )} \arctan \left (\frac{2 \, c d x e + c d^{2} + a e^{2}}{\sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right )}{{\left (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}\right )} \sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} + \frac{6 \, c^{3} d^{4} x^{3} e^{3} + 9 \, c^{3} d^{5} x^{2} e^{2} + 2 \, c^{3} d^{6} x e - c^{3} d^{7} - 6 \, a c^{2} d^{2} x^{3} e^{5} + 12 \, a c^{2} d^{4} x e^{3} + 6 \, a c^{2} d^{5} e^{2} - 9 \, a^{2} c d x^{2} e^{6} - 12 \, a^{2} c d^{2} x e^{5} - 3 \, a^{2} c d^{3} e^{4} - 2 \, a^{3} x e^{7} - 2 \, a^{3} d e^{6}}{2 \,{\left (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}\right )}{\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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