Optimal. Leaf size=107 \[ \frac{e}{\left (c d^2-a e^2\right )^2 (a e+c d x)}-\frac{1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2}+\frac{e^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^3}-\frac{e^2 \log (d+e x)}{\left (c d^2-a e^2\right )^3} \]
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Rubi [A] time = 0.0710984, antiderivative size = 107, 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} \[ \frac{e}{\left (c d^2-a e^2\right )^2 (a e+c d x)}-\frac{1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2}+\frac{e^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^3}-\frac{e^2 \log (d+e x)}{\left (c d^2-a e^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 626
Rule 44
Rubi steps
\begin{align*} \int \frac{(d+e x)^2}{\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)} \, dx\\ &=\int \left (\frac{c d}{\left (c d^2-a e^2\right ) (a e+c d x)^3}-\frac{c d e}{\left (c d^2-a e^2\right )^2 (a e+c d x)^2}+\frac{c d e^2}{\left (c d^2-a e^2\right )^3 (a e+c d x)}-\frac{e^3}{\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 ) (a e+c d x)^2}+\frac{e}{\left (c d^2-a e^2\right )^2 (a e+c d x)}+\frac{e^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^3}-\frac{e^2 \log (d+e x)}{\left (c d^2-a e^2\right )^3}\\ \end{align*}
Mathematica [A] time = 0.0770763, size = 83, normalized size = 0.78 \[ -\frac{\frac{\left (c d^2-a e^2\right ) \left (c d (d-2 e x)-3 a e^2\right )}{(a e+c d x)^2}-2 e^2 \log (a e+c d x)+2 e^2 \log (d+e x)}{2 \left (c d^2-a e^2\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 106, normalized size = 1. \begin{align*}{\frac{{e}^{2}\ln \left ( ex+d \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{3}}}+{\frac{1}{ \left ( 2\,a{e}^{2}-2\,c{d}^{2} \right ) \left ( cdx+ae \right ) ^{2}}}+{\frac{e}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2} \left ( cdx+ae \right ) }}-{\frac{{e}^{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 = 1.09754, size = 321, normalized size = 3. \begin{align*} \frac{e^{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{e^{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 - c d^{2} + 3 \, a e^{2}}{2 \,{\left (a^{2} c^{2} d^{4} e^{2} - 2 \, a^{3} c d^{2} e^{4} + a^{4} e^{6} +{\left (c^{4} d^{6} - 2 \, a c^{3} d^{4} e^{2} + a^{2} c^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (a c^{3} d^{5} e - 2 \, a^{2} c^{2} d^{3} e^{3} + a^{3} c 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 = 2.00002, size = 556, normalized size = 5.2 \begin{align*} -\frac{c^{2} d^{4} - 4 \, a c d^{2} e^{2} + 3 \, 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 \, a c d e^{3} x + a^{2} e^{4}\right )} \log \left (c d x + a e\right ) + 2 \,{\left (c^{2} d^{2} e^{2} x^{2} + 2 \, a c d e^{3} x + a^{2} e^{4}\right )} \log \left (e x + d\right )}{2 \,{\left (a^{2} c^{3} d^{6} e^{2} - 3 \, a^{3} c^{2} d^{4} e^{4} + 3 \, a^{4} c d^{2} e^{6} - a^{5} e^{8} +{\left (c^{5} d^{8} - 3 \, a c^{4} d^{6} e^{2} + 3 \, a^{2} c^{3} d^{4} e^{4} - a^{3} c^{2} d^{2} e^{6}\right )} x^{2} + 2 \,{\left (a c^{4} d^{7} e - 3 \, a^{2} c^{3} d^{5} e^{3} + 3 \, a^{3} c^{2} d^{3} e^{5} - a^{4} c 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 = 1.61083, size = 457, normalized size = 4.27 \begin{align*} \frac{e^{2} \log{\left (x + \frac{- \frac{a^{4} e^{10}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac{4 a^{3} c d^{2} e^{8}}{\left (a e^{2} - c d^{2}\right )^{3}} - \frac{6 a^{2} c^{2} d^{4} e^{6}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac{4 a c^{3} d^{6} e^{4}}{\left (a e^{2} - c d^{2}\right )^{3}} + a e^{4} - \frac{c^{4} d^{8} e^{2}}{\left (a e^{2} - c d^{2}\right )^{3}} + c d^{2} e^{2}}{2 c d e^{3}} \right )}}{\left (a e^{2} - c d^{2}\right )^{3}} - \frac{e^{2} \log{\left (x + \frac{\frac{a^{4} e^{10}}{\left (a e^{2} - c d^{2}\right )^{3}} - \frac{4 a^{3} c d^{2} e^{8}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac{6 a^{2} c^{2} d^{4} e^{6}}{\left (a e^{2} - c d^{2}\right )^{3}} - \frac{4 a c^{3} d^{6} e^{4}}{\left (a e^{2} - c d^{2}\right )^{3}} + a e^{4} + \frac{c^{4} d^{8} e^{2}}{\left (a e^{2} - c d^{2}\right )^{3}} + c d^{2} e^{2}}{2 c d e^{3}} \right )}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac{3 a e^{2} - c d^{2} + 2 c d e x}{2 a^{4} e^{6} - 4 a^{3} c d^{2} e^{4} + 2 a^{2} c^{2} d^{4} e^{2} + x^{2} \left (2 a^{2} c^{2} d^{2} e^{4} - 4 a c^{3} d^{4} e^{2} + 2 c^{4} d^{6}\right ) + x \left (4 a^{3} c d e^{5} - 8 a^{2} c^{2} d^{3} e^{3} + 4 a c^{3} d^{5} e\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.24947, size = 535, normalized size = 5. \begin{align*} \frac{2 \,{\left (c^{2} d^{4} e^{2} - 2 \, a c d^{2} e^{4} + a^{2} e^{6}\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{2 \, c^{3} d^{5} x^{3} e^{3} + 3 \, c^{3} d^{6} x^{2} e^{2} - c^{3} d^{8} - 4 \, a c^{2} d^{3} x^{3} e^{5} - 3 \, a c^{2} d^{4} x^{2} e^{4} + 6 \, a c^{2} d^{5} x e^{3} + 5 \, a c^{2} d^{6} e^{2} + 2 \, a^{2} c d x^{3} e^{7} - 3 \, a^{2} c d^{2} x^{2} e^{6} - 12 \, a^{2} c d^{3} x e^{5} - 7 \, a^{2} c d^{4} e^{4} + 3 \, a^{3} x^{2} e^{8} + 6 \, a^{3} d x e^{7} + 3 \, a^{3} d^{2} 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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