Optimal. Leaf size=94 \[ -\frac{c^2 d^2 x \left (2 c d^2-3 a e^2\right )}{e^3}+\frac{\left (c d^2-a e^2\right )^3}{e^4 (d+e x)}+\frac{3 c d \left (c d^2-a e^2\right )^2 \log (d+e x)}{e^4}+\frac{c^3 d^3 x^2}{2 e^2} \]
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Rubi [A] time = 0.0814272, antiderivative size = 94, 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} \[ -\frac{c^2 d^2 x \left (2 c d^2-3 a e^2\right )}{e^3}+\frac{\left (c d^2-a e^2\right )^3}{e^4 (d+e x)}+\frac{3 c d \left (c d^2-a e^2\right )^2 \log (d+e x)}{e^4}+\frac{c^3 d^3 x^2}{2 e^2} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^5} \, dx &=\int \frac{(a e+c d x)^3}{(d+e x)^2} \, dx\\ &=\int \left (-\frac{c^2 d^2 \left (2 c d^2-3 a e^2\right )}{e^3}+\frac{c^3 d^3 x}{e^2}+\frac{\left (-c d^2+a e^2\right )^3}{e^3 (d+e x)^2}+\frac{3 c d \left (c d^2-a e^2\right )^2}{e^3 (d+e x)}\right ) \, dx\\ &=-\frac{c^2 d^2 \left (2 c d^2-3 a e^2\right ) x}{e^3}+\frac{c^3 d^3 x^2}{2 e^2}+\frac{\left (c d^2-a e^2\right )^3}{e^4 (d+e x)}+\frac{3 c d \left (c d^2-a e^2\right )^2 \log (d+e x)}{e^4}\\ \end{align*}
Mathematica [A] time = 0.0412388, size = 129, normalized size = 1.37 \[ \frac{6 a^2 c d^2 e^4-2 a^3 e^6+6 a c^2 d^2 e^2 \left (-d^2+d e x+e^2 x^2\right )+6 c d (d+e x) \left (c d^2-a e^2\right )^2 \log (d+e x)+c^3 d^3 \left (-4 d^2 e x+2 d^3-3 d e^2 x^2+e^3 x^3\right )}{2 e^4 (d+e x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 156, normalized size = 1.7 \begin{align*}{\frac{{c}^{3}{d}^{3}{x}^{2}}{2\,{e}^{2}}}+3\,{\frac{a{c}^{2}{d}^{2}x}{e}}-2\,{\frac{{c}^{3}{d}^{4}x}{{e}^{3}}}+3\,dc\ln \left ( ex+d \right ){a}^{2}-6\,{\frac{{c}^{2}{d}^{3}\ln \left ( ex+d \right ) a}{{e}^{2}}}+3\,{\frac{{c}^{3}{d}^{5}\ln \left ( ex+d \right ) }{{e}^{4}}}-{\frac{{e}^{2}{a}^{3}}{ex+d}}+3\,{\frac{{a}^{2}c{d}^{2}}{ex+d}}-3\,{\frac{a{c}^{2}{d}^{4}}{{e}^{2} \left ( ex+d \right ) }}+{\frac{{c}^{3}{d}^{6}}{{e}^{4} \left ( ex+d \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06743, size = 184, normalized size = 1.96 \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}}{e^{5} x + d e^{4}} + \frac{c^{3} d^{3} e x^{2} - 2 \,{\left (2 \, c^{3} d^{4} - 3 \, a c^{2} d^{2} e^{2}\right )} x}{2 \, e^{3}} + \frac{3 \,{\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} \log \left (e x + d\right )}{e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.62843, size = 386, normalized size = 4.11 \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 (c^{3} d^{4} e^{2} - 2 \, a c^{2} d^{2} e^{4}\right )} x^{2} - 2 \,{\left (2 \, c^{3} d^{5} e - 3 \, a c^{2} d^{3} e^{3}\right )} x + 6 \,{\left (c^{3} d^{6} - 2 \, a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} +{\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 (e x + d\right )}{2 \,{\left (e^{5} x + d e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.09834, size = 119, normalized size = 1.27 \begin{align*} \frac{c^{3} d^{3} x^{2}}{2 e^{2}} + \frac{3 c d \left (a e^{2} - c d^{2}\right )^{2} \log{\left (d + e x \right )}}{e^{4}} - \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}}{d e^{4} + e^{5} x} + \frac{x \left (3 a c^{2} d^{2} e^{2} - 2 c^{3} d^{4}\right )}{e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2441, size = 240, normalized size = 2.55 \begin{align*} \frac{1}{2} \,{\left (c^{3} d^{3} - \frac{6 \,{\left (c^{3} d^{4} e - a c^{2} d^{2} e^{3}\right )} e^{\left (-1\right )}}{x e + d}\right )}{\left (x e + d\right )}^{2} e^{\left (-4\right )} - 3 \,{\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} e^{\left (-4\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) +{\left (\frac{c^{3} d^{6} e^{20}}{x e + d} - \frac{3 \, a c^{2} d^{4} e^{22}}{x e + d} + \frac{3 \, a^{2} c d^{2} e^{24}}{x e + d} - \frac{a^{3} e^{26}}{x e + d}\right )} e^{\left (-24\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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