Optimal. Leaf size=97 \[ -\frac{3 c^2 d^2 \left (c d^2-a e^2\right ) \log (d+e x)}{e^4}-\frac{3 c d \left (c d^2-a e^2\right )^2}{e^4 (d+e x)}+\frac{\left (c d^2-a e^2\right )^3}{2 e^4 (d+e x)^2}+\frac{c^3 d^3 x}{e^3} \]
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Rubi [A] time = 0.0737391, antiderivative size = 97, 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} \[ -\frac{3 c^2 d^2 \left (c d^2-a e^2\right ) \log (d+e x)}{e^4}-\frac{3 c d \left (c d^2-a e^2\right )^2}{e^4 (d+e x)}+\frac{\left (c d^2-a e^2\right )^3}{2 e^4 (d+e x)^2}+\frac{c^3 d^3 x}{e^3} \]
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)^6} \, dx &=\int \frac{(a e+c d x)^3}{(d+e x)^3} \, dx\\ &=\int \left (\frac{c^3 d^3}{e^3}+\frac{\left (-c d^2+a e^2\right )^3}{e^3 (d+e x)^3}+\frac{3 c d \left (c d^2-a e^2\right )^2}{e^3 (d+e x)^2}-\frac{3 c^2 d^2 \left (c d^2-a e^2\right )}{e^3 (d+e x)}\right ) \, dx\\ &=\frac{c^3 d^3 x}{e^3}+\frac{\left (c d^2-a e^2\right )^3}{2 e^4 (d+e x)^2}-\frac{3 c d \left (c d^2-a e^2\right )^2}{e^4 (d+e x)}-\frac{3 c^2 d^2 \left (c d^2-a e^2\right ) \log (d+e x)}{e^4}\\ \end{align*}
Mathematica [A] time = 0.0454638, size = 129, normalized size = 1.33 \[ \frac{-3 a^2 c d e^4 (d+2 e x)-a^3 e^6+3 a c^2 d^3 e^2 (3 d+4 e x)-6 c^2 d^2 (d+e x)^2 \left (c d^2-a e^2\right ) \log (d+e x)+c^3 d^3 \left (-4 d^2 e x-5 d^3+4 d e^2 x^2+2 e^3 x^3\right )}{2 e^4 (d+e x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 167, normalized size = 1.7 \begin{align*}{\frac{{c}^{3}{d}^{3}x}{{e}^{3}}}-{\frac{{e}^{2}{a}^{3}}{2\, \left ( ex+d \right ) ^{2}}}+{\frac{3\,{a}^{2}c{d}^{2}}{2\, \left ( ex+d \right ) ^{2}}}-{\frac{3\,a{c}^{2}{d}^{4}}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}+{\frac{{c}^{3}{d}^{6}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}+3\,{\frac{{c}^{2}{d}^{2}\ln \left ( ex+d \right ) a}{{e}^{2}}}-3\,{\frac{{c}^{3}{d}^{4}\ln \left ( ex+d \right ) }{{e}^{4}}}-3\,{\frac{cd{a}^{2}}{ex+d}}+6\,{\frac{{c}^{2}{d}^{3}a}{{e}^{2} \left ( ex+d \right ) }}-3\,{\frac{{c}^{3}{d}^{5}}{{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 = 1.09244, size = 192, normalized size = 1.98 \begin{align*} \frac{c^{3} d^{3} x}{e^{3}} - \frac{5 \, c^{3} d^{6} - 9 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} + 6 \,{\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{2 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} - \frac{3 \,{\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\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 = 1.5291, size = 408, normalized size = 4.21 \begin{align*} \frac{2 \, c^{3} d^{3} e^{3} x^{3} + 4 \, c^{3} d^{4} e^{2} x^{2} - 5 \, c^{3} d^{6} + 9 \, a c^{2} d^{4} e^{2} - 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} - 2 \,{\left (2 \, c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x - 6 \,{\left (c^{3} d^{6} - a c^{2} d^{4} e^{2} +{\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (c^{3} d^{5} e - a c^{2} d^{3} e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.92822, size = 144, normalized size = 1.48 \begin{align*} \frac{c^{3} d^{3} x}{e^{3}} + \frac{3 c^{2} d^{2} \left (a e^{2} - c d^{2}\right ) \log{\left (d + e x \right )}}{e^{4}} - \frac{a^{3} e^{6} + 3 a^{2} c d^{2} e^{4} - 9 a c^{2} d^{4} e^{2} + 5 c^{3} d^{6} + x \left (6 a^{2} c d e^{5} - 12 a c^{2} d^{3} e^{3} + 6 c^{3} d^{5} e\right )}{2 d^{2} e^{4} + 4 d e^{5} x + 2 e^{6} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.20684, size = 352, normalized size = 3.63 \begin{align*} c^{3} d^{3} x e^{\left (-3\right )} - 3 \,{\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) - \frac{{\left (5 \, c^{3} d^{9} - 9 \, a c^{2} d^{7} e^{2} + 3 \, a^{2} c d^{5} e^{4} + a^{3} d^{3} e^{6} + 6 \,{\left (c^{3} d^{5} e^{4} - 2 \, a c^{2} d^{3} e^{6} + a^{2} c d e^{8}\right )} x^{4} +{\left (23 \, c^{3} d^{6} e^{3} - 45 \, a c^{2} d^{4} e^{5} + 21 \, a^{2} c d^{2} e^{7} + a^{3} e^{9}\right )} x^{3} + 3 \,{\left (11 \, c^{3} d^{7} e^{2} - 21 \, a c^{2} d^{5} e^{4} + 9 \, a^{2} c d^{3} e^{6} + a^{3} d e^{8}\right )} x^{2} + 3 \,{\left (7 \, c^{3} d^{8} e - 13 \, a c^{2} d^{6} e^{3} + 5 \, a^{2} c d^{4} e^{5} + a^{3} d^{2} e^{7}\right )} x\right )} e^{\left (-4\right )}}{2 \,{\left (x e + d\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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