Optimal. Leaf size=17 \[ \frac{c^2 (d+e x)^6}{6 e} \]
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Rubi [A] time = 0.0043832, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {27, 12, 32} \[ \frac{c^2 (d+e x)^6}{6 e} \]
Antiderivative was successfully verified.
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Rule 27
Rule 12
Rule 32
Rubi steps
\begin{align*} \int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^2 \, dx &=\int c^2 (d+e x)^5 \, dx\\ &=c^2 \int (d+e x)^5 \, dx\\ &=\frac{c^2 (d+e x)^6}{6 e}\\ \end{align*}
Mathematica [A] time = 0.0020731, size = 17, normalized size = 1. \[ \frac{c^2 (d+e x)^6}{6 e} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.038, size = 72, normalized size = 4.2 \begin{align*}{\frac{{c}^{2}{e}^{5}{x}^{6}}{6}}+d{c}^{2}{e}^{4}{x}^{5}+{\frac{5\,{d}^{2}{c}^{2}{e}^{3}{x}^{4}}{2}}+{\frac{10\,{d}^{3}{c}^{2}{e}^{2}{x}^{3}}{3}}+{\frac{5\,{c}^{2}e{d}^{4}{x}^{2}}{2}}+{c}^{2}{d}^{5}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.22369, size = 41, normalized size = 2.41 \begin{align*} \frac{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{3}}{6 \, c e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.72845, size = 149, normalized size = 8.76 \begin{align*} \frac{1}{6} x^{6} e^{5} c^{2} + x^{5} e^{4} d c^{2} + \frac{5}{2} x^{4} e^{3} d^{2} c^{2} + \frac{10}{3} x^{3} e^{2} d^{3} c^{2} + \frac{5}{2} x^{2} e d^{4} c^{2} + x d^{5} c^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.084157, size = 80, normalized size = 4.71 \begin{align*} c^{2} d^{5} x + \frac{5 c^{2} d^{4} e x^{2}}{2} + \frac{10 c^{2} d^{3} e^{2} x^{3}}{3} + \frac{5 c^{2} d^{2} e^{3} x^{4}}{2} + c^{2} d e^{4} x^{5} + \frac{c^{2} e^{5} x^{6}}{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16845, size = 92, normalized size = 5.41 \begin{align*} \frac{1}{6} \, c^{2} x^{6} e^{5} + c^{2} d x^{5} e^{4} + \frac{5}{2} \, c^{2} d^{2} x^{4} e^{3} + \frac{10}{3} \, c^{2} d^{3} x^{3} e^{2} + \frac{5}{2} \, c^{2} d^{4} x^{2} e + c^{2} d^{5} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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