Optimal. Leaf size=17 \[ \frac{c^2 (d+e x)^5}{5 e} \]
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Rubi [A] time = 0.003937, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {27, 12, 32} \[ \frac{c^2 (d+e x)^5}{5 e} \]
Antiderivative was successfully verified.
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Rule 27
Rule 12
Rule 32
Rubi steps
\begin{align*} \int \left (c d^2+2 c d e x+c e^2 x^2\right )^2 \, dx &=\int c^2 (d+e x)^4 \, dx\\ &=c^2 \int (d+e x)^4 \, dx\\ &=\frac{c^2 (d+e x)^5}{5 e}\\ \end{align*}
Mathematica [A] time = 0.001093, size = 17, normalized size = 1. \[ \frac{c^2 (d+e x)^5}{5 e} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.039, size = 58, normalized size = 3.4 \begin{align*}{\frac{{c}^{2}{e}^{4}{x}^{5}}{5}}+{c}^{2}d{e}^{3}{x}^{4}+2\,{c}^{2}{d}^{2}{e}^{2}{x}^{3}+2\,{c}^{2}{d}^{3}e{x}^{2}+{c}^{2}{d}^{4}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.22374, size = 92, normalized size = 5.41 \begin{align*} \frac{1}{5} \, c^{2} e^{4} x^{5} + c^{2} d e^{3} x^{4} + \frac{4}{3} \, c^{2} d^{2} e^{2} x^{3} + c^{2} d^{4} x + \frac{2}{3} \,{\left (c e^{2} x^{3} + 3 \, c d e x^{2}\right )} c d^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.85975, size = 112, normalized size = 6.59 \begin{align*} \frac{1}{5} x^{5} e^{4} c^{2} + x^{4} e^{3} d c^{2} + 2 x^{3} e^{2} d^{2} c^{2} + 2 x^{2} e d^{3} c^{2} + x d^{4} c^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.074471, size = 60, normalized size = 3.53 \begin{align*} c^{2} d^{4} x + 2 c^{2} d^{3} e x^{2} + 2 c^{2} d^{2} e^{2} x^{3} + c^{2} d e^{3} x^{4} + \frac{c^{2} e^{4} x^{5}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19167, size = 74, normalized size = 4.35 \begin{align*} \frac{1}{5} \, c^{2} x^{5} e^{4} + c^{2} d x^{4} e^{3} + 2 \, c^{2} d^{2} x^{3} e^{2} + 2 \, c^{2} d^{3} x^{2} e + c^{2} d^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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