Optimal. Leaf size=17 \[ \frac{(d+e x)^3}{3 c^3 e} \]
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Rubi [A] time = 0.0048614, 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} \[ \frac{(d+e x)^3}{3 c^3 e} \]
Antiderivative was successfully verified.
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Rule 27
Rule 12
Rule 32
Rubi steps
\begin{align*} \int \frac{(d+e x)^8}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx &=\int \frac{(d+e x)^2}{c^3} \, dx\\ &=\frac{\int (d+e x)^2 \, dx}{c^3}\\ &=\frac{(d+e x)^3}{3 c^3 e}\\ \end{align*}
Mathematica [A] time = 0.0009025, size = 17, normalized size = 1. \[ \frac{(d+e x)^3}{3 c^3 e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 16, normalized size = 0.9 \begin{align*}{\frac{ \left ( ex+d \right ) ^{3}}{3\,{c}^{3}e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07916, size = 35, normalized size = 2.06 \begin{align*} \frac{e^{2} x^{3} + 3 \, d e x^{2} + 3 \, d^{2} x}{3 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98393, size = 55, normalized size = 3.24 \begin{align*} \frac{e^{2} x^{3} + 3 \, d e x^{2} + 3 \, d^{2} x}{3 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.145524, size = 29, normalized size = 1.71 \begin{align*} \frac{d^{2} x}{c^{3}} + \frac{d e x^{2}}{c^{3}} + \frac{e^{2} x^{3}}{3 c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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