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 = 0.05, antiderivative size = 25, normalized size of antiderivative = 1.00, 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 30
Rule 6679
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*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.00 \[ \frac {1}{2} x^3 \left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 21, normalized size = 0.84 \[ \frac {1}{2} \, x^{2} e^{\left (-\frac {n \log \relax (b) + \log \relax (a)}{m n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 21, normalized size = 0.84 \[ \frac {1}{2} \, x^{2} e^{\left (-\frac {n \log \relax (b) + \log \relax (a)}{m n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 25, normalized size = 1.00 \[ \frac {x^{3} \left (a \left (b \,x^{m}\right )^{n}\right )^{-\frac {1}{m n}}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.05, size = 38, normalized size = 1.52 \[ \frac {x^{3}}{2 \, a^{\frac {1}{m n}} b^{\left (\frac {1}{m}\right )} {\left ({\left (x^{m}\right )}^{n}\right )}^{\frac {1}{m n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.01, size = 24, normalized size = 0.96 \[ \frac {x^3}{2\,{\left (a\,{\left (b\,x^m\right )}^n\right )}^{\frac {1}{m\,n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 20.01, size = 248, normalized size = 9.92 \[ \begin {cases} - \frac {x^{3}}{0^{m n} \tilde {\infty }^{m n} \left (0^{m n}\right )^{\frac {1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{\frac {1}{m n}} \left (\left (\left (0^{m n}\right )^{\frac {1}{n}}\right )^{n}\right )^{\frac {1}{m n}} - 3 \left (0^{m n}\right )^{\frac {1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{\frac {1}{m n}} \left (\left (\left (0^{m n}\right )^{\frac {1}{n}}\right )^{n}\right )^{\frac {1}{m n}}} & \text {for}\: a = 0^{m n} \wedge b = \left (0^{m n}\right )^{\frac {1}{n}} \\\frac {a^{- \frac {1}{m n}} x^{3} \left (\left (x^{m}\right )^{n}\right )^{- \frac {1}{m n}} \left (\left (\left (0^{m n}\right )^{\frac {1}{n}}\right )^{n}\right )^{- \frac {1}{m n}}}{2} & \text {for}\: b = \left (0^{m n}\right )^{\frac {1}{n}} \\- \frac {x^{3}}{0^{m n} \tilde {\infty }^{m n} \left (0^{m n}\right )^{\frac {1}{m n}} \left (b^{n}\right )^{\frac {1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{\frac {1}{m n}} - 3 \left (0^{m n}\right )^{\frac {1}{m n}} \left (b^{n}\right )^{\frac {1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{\frac {1}{m n}}} & \text {for}\: a = 0^{m n} \\\frac {a^{- \frac {1}{m n}} x^{3} \left (b^{n}\right )^{- \frac {1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{- \frac {1}{m n}}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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