Optimal. Leaf size=17 \[ \frac{c^2 (d+e x)^3}{3 e} \]
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Rubi [A] time = 0.0051889, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {27, 12, 32} \[ \frac{c^2 (d+e x)^3}{3 e} \]
Antiderivative was successfully verified.
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Rule 27
Rule 12
Rule 32
Rubi steps
\begin{align*} \int \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^2} \, dx &=\int c^2 (d+e x)^2 \, dx\\ &=c^2 \int (d+e x)^2 \, dx\\ &=\frac{c^2 (d+e x)^3}{3 e}\\ \end{align*}
Mathematica [A] time = 0.0014096, size = 17, normalized size = 1. \[ \frac{c^2 (d+e x)^3}{3 e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 16, normalized size = 0.9 \begin{align*}{\frac{{c}^{2} \left ( ex+d \right ) ^{3}}{3\,e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.22575, size = 39, normalized size = 2.29 \begin{align*} \frac{1}{3} \, c^{2} e^{2} x^{3} + c^{2} d e x^{2} + c^{2} d^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.95184, size = 58, normalized size = 3.41 \begin{align*} \frac{1}{3} \, c^{2} e^{2} x^{3} + c^{2} d e x^{2} + c^{2} d^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.115574, size = 29, normalized size = 1.71 \begin{align*} c^{2} d^{2} x + c^{2} d e x^{2} + \frac{c^{2} e^{2} x^{3}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1677, size = 20, normalized size = 1.18 \begin{align*} \frac{1}{3} \,{\left (x e + d\right )}^{3} c^{2} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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