Optimal. Leaf size=54 \[ \frac{\left (c d^2-a e^2\right ) (a e+c d x)^4}{4 c^2 d^2}+\frac{e (a e+c d x)^5}{5 c^2 d^2} \]
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Rubi [A] time = 0.0312005, antiderivative size = 54, 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} \[ \frac{\left (c d^2-a e^2\right ) (a e+c d x)^4}{4 c^2 d^2}+\frac{e (a e+c d x)^5}{5 c^2 d^2} \]
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)^2} \, dx &=\int (a e+c d x)^3 (d+e x) \, dx\\ &=\int \left (\frac{\left (c d^2-a e^2\right ) (a e+c d x)^3}{c d}+\frac{e (a e+c d x)^4}{c d}\right ) \, dx\\ &=\frac{\left (c d^2-a e^2\right ) (a e+c d x)^4}{4 c^2 d^2}+\frac{e (a e+c d x)^5}{5 c^2 d^2}\\ \end{align*}
Mathematica [A] time = 0.0259652, size = 79, normalized size = 1.46 \[ \frac{1}{20} x \left (10 a^2 c d e^2 x (3 d+2 e x)+10 a^3 e^3 (2 d+e x)+5 a c^2 d^2 e x^2 (4 d+3 e x)+c^3 d^3 x^3 (5 d+4 e x)\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.04, size = 136, normalized size = 2.5 \begin{align*}{\frac{{c}^{3}{d}^{3}e{x}^{5}}{5}}+{\frac{ \left ( 2\,a{c}^{2}{d}^{2}{e}^{2}+{c}^{2}{d}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ({a}^{2}{e}^{3}dc+2\,acde \left ( a{e}^{2}+c{d}^{2} \right ) +{c}^{2}{d}^{3}ae \right ){x}^{3}}{3}}+{\frac{ \left ({a}^{2}{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) +2\,{a}^{2}c{d}^{2}{e}^{2} \right ){x}^{2}}{2}}+{a}^{3}{e}^{3}dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08454, size = 128, normalized size = 2.37 \begin{align*} \frac{1}{5} \, c^{3} d^{3} e x^{5} + a^{3} d e^{3} x + \frac{1}{4} \,{\left (c^{3} d^{4} + 3 \, a c^{2} d^{2} e^{2}\right )} x^{4} +{\left (a c^{2} d^{3} e + a^{2} c d e^{3}\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37772, size = 193, normalized size = 3.57 \begin{align*} \frac{1}{5} \, c^{3} d^{3} e x^{5} + a^{3} d e^{3} x + \frac{1}{4} \,{\left (c^{3} d^{4} + 3 \, a c^{2} d^{2} e^{2}\right )} x^{4} +{\left (a c^{2} d^{3} e + a^{2} c d e^{3}\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.144167, size = 100, normalized size = 1.85 \begin{align*} a^{3} d e^{3} x + \frac{c^{3} d^{3} e x^{5}}{5} + x^{4} \left (\frac{3 a c^{2} d^{2} e^{2}}{4} + \frac{c^{3} d^{4}}{4}\right ) + x^{3} \left (a^{2} c d e^{3} + a c^{2} d^{3} e\right ) + x^{2} \left (\frac{a^{3} e^{4}}{2} + \frac{3 a^{2} c d^{2} e^{2}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19977, size = 197, normalized size = 3.65 \begin{align*} \frac{1}{20} \,{\left (4 \, c^{3} d^{3} - \frac{15 \,{\left (c^{3} d^{4} e - a c^{2} d^{2} e^{3}\right )} e^{\left (-1\right )}}{x e + d} + \frac{20 \,{\left (c^{3} d^{5} e^{2} - 2 \, a c^{2} d^{3} e^{4} + a^{2} c d e^{6}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac{10 \,{\left (c^{3} d^{6} e^{3} - 3 \, a c^{2} d^{4} e^{5} + 3 \, a^{2} c d^{2} e^{7} - a^{3} e^{9}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}}\right )}{\left (x e + d\right )}^{5} e^{\left (-4\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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