Optimal. Leaf size=18 \[ \frac{1}{2} x^3 \left (a x^n\right )^{-1/n} \]
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Rubi [A] time = 0.001827, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {15, 30} \[ \frac{1}{2} x^3 \left (a x^n\right )^{-1/n} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin{align*} \int x^2 \left (a x^n\right )^{-1/n} \, dx &=\left (x \left (a x^n\right )^{-1/n}\right ) \int x \, dx\\ &=\frac{1}{2} x^3 \left (a x^n\right )^{-1/n}\\ \end{align*}
Mathematica [A] time = 0.001945, size = 18, normalized size = 1. \[ \frac{1}{2} x^3 \left (a x^n\right )^{-1/n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 17, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}}{2\,\sqrt [n]{a{x}^{n}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01288, size = 28, normalized size = 1.56 \begin{align*} \frac{x^{3}}{2 \, a^{\left (\frac{1}{n}\right )}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82491, size = 23, normalized size = 1.28 \begin{align*} \frac{x^{2}}{2 \, a^{\left (\frac{1}{n}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.24361, size = 58, normalized size = 3.22 \begin{align*} \begin{cases} \frac{a^{- \frac{1}{n}} x^{3} \left (x^{n}\right )^{- \frac{1}{n}}}{2} & \text{for}\: a \neq 0^{n} \\- \frac{x^{3}}{0^{n} \tilde{\infty }^{n} \left (0^{n}\right )^{\frac{1}{n}} \left (x^{n}\right )^{\frac{1}{n}} - 3 \left (0^{n}\right )^{\frac{1}{n}} \left (x^{n}\right )^{\frac{1}{n}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10506, size = 16, normalized size = 0.89 \begin{align*} \frac{x^{2}}{2 \, a^{\left (\frac{1}{n}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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