Optimal. Leaf size=105 \[ \frac{e^2 x \left (3 c d^2-2 a e^2\right )}{c^3 d^3}-\frac{\left (c d^2-a e^2\right )^3}{c^4 d^4 (a e+c d x)}+\frac{3 e \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^4 d^4}+\frac{e^3 x^2}{2 c^2 d^2} \]
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Rubi [A] time = 0.0928664, antiderivative size = 105, 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, 43} \[ \frac{e^2 x \left (3 c d^2-2 a e^2\right )}{c^3 d^3}-\frac{\left (c d^2-a e^2\right )^3}{c^4 d^4 (a e+c d x)}+\frac{3 e \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^4 d^4}+\frac{e^3 x^2}{2 c^2 d^2} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int \frac{(d+e x)^5}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx &=\int \frac{(d+e x)^3}{(a e+c d x)^2} \, dx\\ &=\int \left (\frac{3 c d^2 e^2-2 a e^4}{c^3 d^3}+\frac{e^3 x}{c^2 d^2}+\frac{\left (c d^2-a e^2\right )^3}{c^3 d^3 (a e+c d x)^2}+\frac{3 e \left (c d^2-a e^2\right )^2}{c^3 d^3 (a e+c d x)}\right ) \, dx\\ &=\frac{e^2 \left (3 c d^2-2 a e^2\right ) x}{c^3 d^3}+\frac{e^3 x^2}{2 c^2 d^2}-\frac{\left (c d^2-a e^2\right )^3}{c^4 d^4 (a e+c d x)}+\frac{3 e \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^4 d^4}\\ \end{align*}
Mathematica [A] time = 0.0475453, size = 142, normalized size = 1.35 \[ \frac{-3 a^2 c d^2 e^4+a^3 e^6+3 a c^2 d^4 e^2-c^3 d^6}{c^4 d^4 (a e+c d x)}+\frac{3 \left (a^2 e^5-2 a c d^2 e^3+c^2 d^4 e\right ) \log (a e+c d x)}{c^4 d^4}-\frac{e^2 x \left (2 a e^2-3 c d^2\right )}{c^3 d^3}+\frac{e^3 x^2}{2 c^2 d^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 184, normalized size = 1.8 \begin{align*}{\frac{{e}^{3}{x}^{2}}{2\,{c}^{2}{d}^{2}}}-2\,{\frac{{e}^{4}ax}{{c}^{3}{d}^{3}}}+3\,{\frac{{e}^{2}x}{{c}^{2}d}}+{\frac{{a}^{3}{e}^{6}}{{c}^{4}{d}^{4} \left ( cdx+ae \right ) }}-3\,{\frac{{a}^{2}{e}^{4}}{{c}^{3}{d}^{2} \left ( cdx+ae \right ) }}+3\,{\frac{a{e}^{2}}{{c}^{2} \left ( cdx+ae \right ) }}-{\frac{{d}^{2}}{c \left ( cdx+ae \right ) }}+3\,{\frac{{e}^{5}\ln \left ( cdx+ae \right ){a}^{2}}{{c}^{4}{d}^{4}}}-6\,{\frac{{e}^{3}\ln \left ( cdx+ae \right ) a}{{c}^{3}{d}^{2}}}+3\,{\frac{e\ln \left ( cdx+ae \right ) }{{c}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.033, size = 193, normalized size = 1.84 \begin{align*} -\frac{c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}}{c^{5} d^{5} x + a c^{4} d^{4} e} + \frac{c d e^{3} x^{2} + 2 \,{\left (3 \, c d^{2} e^{2} - 2 \, a e^{4}\right )} x}{2 \, c^{3} d^{3}} + \frac{3 \,{\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} \log \left (c d x + a e\right )}{c^{4} d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82145, size = 408, normalized size = 3.89 \begin{align*} \frac{c^{3} d^{3} e^{3} x^{3} - 2 \, c^{3} d^{6} + 6 \, a c^{2} d^{4} e^{2} - 6 \, a^{2} c d^{2} e^{4} + 2 \, a^{3} e^{6} + 3 \,{\left (2 \, c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (3 \, a c^{2} d^{3} e^{3} - 2 \, a^{2} c d e^{5}\right )} x + 6 \,{\left (a c^{2} d^{4} e^{2} - 2 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} +{\left (c^{3} d^{5} e - 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 )}{2 \,{\left (c^{5} d^{5} x + a c^{4} d^{4} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.08642, size = 131, normalized size = 1.25 \begin{align*} \frac{a^{3} e^{6} - 3 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} - c^{3} d^{6}}{a c^{4} d^{4} e + c^{5} d^{5} x} + \frac{e^{3} x^{2}}{2 c^{2} d^{2}} - \frac{x \left (2 a e^{4} - 3 c d^{2} e^{2}\right )}{c^{3} d^{3}} + \frac{3 e \left (a e^{2} - c d^{2}\right )^{2} \log{\left (a e + c d x \right )}}{c^{4} d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.37179, size = 629, normalized size = 5.99 \begin{align*} \frac{3 \,{\left (c^{5} d^{10} e - 5 \, a c^{4} d^{8} e^{3} + 10 \, a^{2} c^{3} d^{6} e^{5} - 10 \, a^{3} c^{2} d^{4} e^{7} + 5 \, a^{4} c d^{2} e^{9} - a^{5} e^{11}\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^{6} d^{8} - 2 \, a c^{5} d^{6} e^{2} + a^{2} c^{4} d^{4} e^{4}\right )} \sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} + \frac{{\left (c^{2} d^{2} x^{2} e^{7} + 6 \, c^{2} d^{3} x e^{6} - 4 \, a c d x e^{8}\right )} e^{\left (-4\right )}}{2 \, c^{4} d^{4}} + \frac{3 \,{\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} \log \left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}{2 \, c^{4} d^{4}} - \frac{c^{5} d^{11} - 5 \, a c^{4} d^{9} e^{2} + 10 \, a^{2} c^{3} d^{7} e^{4} - 10 \, a^{3} c^{2} d^{5} e^{6} + 5 \, a^{4} c d^{3} e^{8} - a^{5} d e^{10} +{\left (c^{5} d^{10} e - 5 \, a c^{4} d^{8} e^{3} + 10 \, a^{2} c^{3} d^{6} e^{5} - 10 \, a^{3} c^{2} d^{4} e^{7} + 5 \, a^{4} c d^{2} e^{9} - a^{5} e^{11}\right )} x}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}{\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )} c^{4} d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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