Optimal. Leaf size=25 \[ \frac{1}{2} x^3 \left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}} \]
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Rubi [A] time = 0.0499666, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {6679, 30} \[ \frac{1}{2} x^3 \left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}} \]
Antiderivative was successfully verified.
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Rule 6679
Rule 30
Rubi steps
\begin{align*} \int x^2 \left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}} \, dx &=\left (x \left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}}\right ) \int x \, dx\\ &=\frac{1}{2} x^3 \left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}}\\ \end{align*}
Mathematica [A] time = 0.0095051, size = 25, normalized size = 1. \[ \frac{1}{2} x^3 \left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 25, normalized size = 1. \begin{align*}{\frac{{x}^{3}}{2} \left ( \left ( a \left ( b{x}^{m} \right ) ^{n} \right ) ^{{\frac{1}{mn}}} \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.54464, size = 59, normalized size = 2.36 \begin{align*} \frac{x^{3}}{2 \, a^{\frac{1}{m n}}{\left (b^{n}\right )}^{\frac{1}{m n}}{\left ({\left (x^{m}\right )}^{n}\right )}^{\frac{1}{m n}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68945, size = 54, normalized size = 2.16 \begin{align*} \frac{1}{2} \, x^{2} e^{\left (-\frac{n \log \left (b\right ) + \log \left (a\right )}{m n}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 16.5999, size = 248, normalized size = 9.92 \begin{align*} \begin{cases} - \frac{x^{3}}{0^{m n} \tilde{\infty }^{m n} \left (0^{m n}\right )^{\frac{1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{\frac{1}{m n}} \left (\left (\left (0^{m n}\right )^{\frac{1}{n}}\right )^{n}\right )^{\frac{1}{m n}} - 3 \left (0^{m n}\right )^{\frac{1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{\frac{1}{m n}} \left (\left (\left (0^{m n}\right )^{\frac{1}{n}}\right )^{n}\right )^{\frac{1}{m n}}} & \text{for}\: a = 0^{m n} \wedge b = \left (0^{m n}\right )^{\frac{1}{n}} \\\frac{a^{- \frac{1}{m n}} x^{3} \left (\left (x^{m}\right )^{n}\right )^{- \frac{1}{m n}} \left (\left (\left (0^{m n}\right )^{\frac{1}{n}}\right )^{n}\right )^{- \frac{1}{m n}}}{2} & \text{for}\: b = \left (0^{m n}\right )^{\frac{1}{n}} \\- \frac{x^{3}}{0^{m n} \tilde{\infty }^{m n} \left (0^{m n}\right )^{\frac{1}{m n}} \left (b^{n}\right )^{\frac{1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{\frac{1}{m n}} - 3 \left (0^{m n}\right )^{\frac{1}{m n}} \left (b^{n}\right )^{\frac{1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{\frac{1}{m n}}} & \text{for}\: a = 0^{m n} \\\frac{a^{- \frac{1}{m n}} x^{3} \left (b^{n}\right )^{- \frac{1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{- \frac{1}{m n}}}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13731, size = 28, normalized size = 1.12 \begin{align*} \frac{1}{2} \, x^{2} e^{\left (-\frac{n \log \left (b\right ) + \log \left (a\right )}{m n}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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