Optimal. Leaf size=89 \[ \frac{c d x \left (c d^2-a e^2\right )^2}{e^3}+\frac{1}{2} \left (a-\frac{c d^2}{e^2}\right ) (a e+c d x)^2-\frac{\left (c d^2-a e^2\right )^3 \log (d+e x)}{e^4}+\frac{(a e+c d x)^3}{3 e} \]
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Rubi [A] time = 0.0486573, antiderivative size = 89, 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} \[ \frac{c d x \left (c d^2-a e^2\right )^2}{e^3}+\frac{1}{2} \left (a-\frac{c d^2}{e^2}\right ) (a e+c d x)^2-\frac{\left (c d^2-a e^2\right )^3 \log (d+e x)}{e^4}+\frac{(a e+c d x)^3}{3 e} \]
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)^4} \, dx &=\int \frac{(a e+c d x)^3}{d+e x} \, dx\\ &=\int \left (\frac{c d \left (c d^2-a e^2\right )^2}{e^3}-\frac{c d \left (c d^2-a e^2\right ) (a e+c d x)}{e^2}+\frac{c d (a e+c d x)^2}{e}+\frac{\left (-c d^2+a e^2\right )^3}{e^3 (d+e x)}\right ) \, dx\\ &=\frac{c d \left (c d^2-a e^2\right )^2 x}{e^3}+\frac{1}{2} \left (a-\frac{c d^2}{e^2}\right ) (a e+c d x)^2+\frac{(a e+c d x)^3}{3 e}-\frac{\left (c d^2-a e^2\right )^3 \log (d+e x)}{e^4}\\ \end{align*}
Mathematica [A] time = 0.0358542, size = 85, normalized size = 0.96 \[ \frac{c d e x \left (18 a^2 e^4+9 a c d e^2 (e x-2 d)+c^2 d^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )-6 \left (c d^2-a e^2\right )^3 \log (d+e x)}{6 e^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 138, normalized size = 1.6 \begin{align*}{\frac{{x}^{3}{c}^{3}{d}^{3}}{3\,e}}+{\frac{3\,{c}^{2}{d}^{2}{x}^{2}a}{2}}-{\frac{{c}^{3}{d}^{4}{x}^{2}}{2\,{e}^{2}}}+3\,cde{a}^{2}x-3\,{\frac{{c}^{2}{d}^{3}ax}{e}}+{\frac{{c}^{3}{d}^{5}x}{{e}^{3}}}+{e}^{2}\ln \left ( ex+d \right ){a}^{3}-3\,\ln \left ( ex+d \right ){a}^{2}c{d}^{2}+3\,{\frac{\ln \left ( ex+d \right ) a{c}^{2}{d}^{4}}{{e}^{2}}}-{\frac{\ln \left ( ex+d \right ){c}^{3}{d}^{6}}{{e}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12567, size = 177, normalized size = 1.99 \begin{align*} \frac{2 \, c^{3} d^{3} e^{2} x^{3} - 3 \,{\left (c^{3} d^{4} e - 3 \, a c^{2} d^{2} e^{3}\right )} x^{2} + 6 \,{\left (c^{3} d^{5} - 3 \, a c^{2} d^{3} e^{2} + 3 \, a^{2} c d e^{4}\right )} x}{6 \, e^{3}} - \frac{{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\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 [A] time = 1.67437, size = 262, normalized size = 2.94 \begin{align*} \frac{2 \, c^{3} d^{3} e^{3} x^{3} - 3 \,{\left (c^{3} d^{4} e^{2} - 3 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 6 \,{\left (c^{3} d^{5} e - 3 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x - 6 \,{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \log \left (e x + d\right )}{6 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.651007, size = 100, normalized size = 1.12 \begin{align*} \frac{c^{3} d^{3} x^{3}}{3 e} + \frac{x^{2} \left (3 a c^{2} d^{2} e^{2} - c^{3} d^{4}\right )}{2 e^{2}} + \frac{x \left (3 a^{2} c d e^{4} - 3 a c^{2} d^{3} e^{2} + c^{3} d^{5}\right )}{e^{3}} + \frac{\left (a e^{2} - c d^{2}\right )^{3} \log{\left (d + e x \right )}}{e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.34463, size = 173, normalized size = 1.94 \begin{align*} -{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{6} \,{\left (2 \, c^{3} d^{3} x^{3} e^{11} - 3 \, c^{3} d^{4} x^{2} e^{10} + 6 \, c^{3} d^{5} x e^{9} + 9 \, a c^{2} d^{2} x^{2} e^{12} - 18 \, a c^{2} d^{3} x e^{11} + 18 \, a^{2} c d x e^{13}\right )} e^{\left (-12\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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