Optimal. Leaf size=54 \[ \frac{\left (c d^2-a e^2\right ) (a e+c d x)^3}{3 c^2 d^2}+\frac{e (a e+c d x)^4}{4 c^2 d^2} \]
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Rubi [A] time = 0.0490563, 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} \[ \frac{\left (c d^2-a e^2\right ) (a e+c d x)^3}{3 c^2 d^2}+\frac{e (a e+c d x)^4}{4 c^2 d^2} \]
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 )^2}{d+e x} \, dx &=\int (a e+c d x)^2 (d+e x) \, dx\\ &=\int \left (\frac{\left (c d^2-a e^2\right ) (a e+c d x)^2}{c d}+\frac{e (a e+c d x)^3}{c d}\right ) \, dx\\ &=\frac{\left (c d^2-a e^2\right ) (a e+c d x)^3}{3 c^2 d^2}+\frac{e (a e+c d x)^4}{4 c^2 d^2}\\ \end{align*}
Mathematica [A] time = 0.0125745, size = 54, normalized size = 1. \[ \frac{1}{12} x \left (6 a^2 e^2 (2 d+e x)+4 a c d e x (3 d+2 e x)+c^2 d^2 x^2 (4 d+3 e x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 77, normalized size = 1.4 \begin{align*}{\frac{{c}^{2}{d}^{2}e{x}^{4}}{4}}+{\frac{ \left ( ad{e}^{2}c+cd \left ( a{e}^{2}+c{d}^{2} \right ) \right ){x}^{3}}{3}}+{\frac{ \left ( ae \left ( a{e}^{2}+c{d}^{2} \right ) +c{d}^{2}ae \right ){x}^{2}}{2}}+{a}^{2}{e}^{2}dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10627, size = 86, normalized size = 1.59 \begin{align*} \frac{1}{4} \, c^{2} d^{2} e x^{4} + a^{2} d e^{2} x + \frac{1}{3} \,{\left (c^{2} d^{3} + 2 \, a c d e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a c d^{2} e + a^{2} e^{3}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52867, size = 136, normalized size = 2.52 \begin{align*} \frac{1}{4} \, c^{2} d^{2} e x^{4} + a^{2} d e^{2} x + \frac{1}{3} \,{\left (c^{2} d^{3} + 2 \, a c d e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a c d^{2} e + a^{2} e^{3}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.168092, size = 66, normalized size = 1.22 \begin{align*} a^{2} d e^{2} x + \frac{c^{2} d^{2} e x^{4}}{4} + x^{3} \left (\frac{2 a c d e^{2}}{3} + \frac{c^{2} d^{3}}{3}\right ) + x^{2} \left (\frac{a^{2} e^{3}}{2} + a c d^{2} e\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1643, size = 97, normalized size = 1.8 \begin{align*} \frac{1}{12} \,{\left (3 \, c^{2} d^{2} x^{4} e^{5} + 4 \, c^{2} d^{3} x^{3} e^{4} + 8 \, a c d x^{3} e^{6} + 12 \, a c d^{2} x^{2} e^{5} + 6 \, a^{2} x^{2} e^{7} + 12 \, a^{2} d x e^{6}\right )} e^{\left (-4\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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