Optimal. Leaf size=22 \[ x^2 \left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}} \]
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Rubi [A] time = 0.03, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6679, 8} \[ x^2 \left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 6679
Rubi steps
\begin {align*} \int x \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 1 \, dx\\ &=x^2 \left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 22, normalized size = 1.00 \[ x^2 \left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 18, normalized size = 0.82 \[ x e^{\left (-\frac {n \log \relax (b) + \log \relax (a)}{m n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 18, normalized size = 0.82 \[ x e^{\left (-\frac {n \log \relax (b) + \log \relax (a)}{m n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int x \left (a \left (b \,x^{m}\right )^{n}\right )^{-\frac {1}{m n}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.13, size = 37, normalized size = 1.68 \[ \frac {x^{2}}{a^{\frac {1}{m n}} b^{\left (\frac {1}{m}\right )} {\left ({\left (x^{m}\right )}^{n}\right )}^{\frac {1}{m n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.98, size = 23, normalized size = 1.05 \[ \frac {x^2}{{\left (a\,{\left (b\,x^m\right )}^n\right )}^{\frac {1}{m\,n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.35, size = 245, normalized size = 11.14 \[ \begin {cases} - \frac {x^{2}}{0^{m n} \tilde {\infty }^{m n} \left (0^{m n}\right )^{\frac {1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{\frac {1}{m n}} \left (\left (\left (0^{m n}\right )^{\frac {1}{n}}\right )^{n}\right )^{\frac {1}{m n}} - 2 \left (0^{m n}\right )^{\frac {1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{\frac {1}{m n}} \left (\left (\left (0^{m n}\right )^{\frac {1}{n}}\right )^{n}\right )^{\frac {1}{m n}}} & \text {for}\: a = 0^{m n} \wedge b = \left (0^{m n}\right )^{\frac {1}{n}} \\a^{- \frac {1}{m n}} x^{2} \left (\left (x^{m}\right )^{n}\right )^{- \frac {1}{m n}} \left (\left (\left (0^{m n}\right )^{\frac {1}{n}}\right )^{n}\right )^{- \frac {1}{m n}} & \text {for}\: b = \left (0^{m n}\right )^{\frac {1}{n}} \\- \frac {x^{2}}{0^{m n} \tilde {\infty }^{m n} \left (0^{m n}\right )^{\frac {1}{m n}} \left (b^{n}\right )^{\frac {1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{\frac {1}{m n}} - 2 \left (0^{m n}\right )^{\frac {1}{m n}} \left (b^{n}\right )^{\frac {1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{\frac {1}{m n}}} & \text {for}\: a = 0^{m n} \\a^{- \frac {1}{m n}} x^{2} \left (b^{n}\right )^{- \frac {1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{- \frac {1}{m n}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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