Optimal. Leaf size=20 \[ -\left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}} \]
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Rubi [A] time = 0.04, antiderivative size = 20, 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} \[ -\left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}} \]
Antiderivative was successfully verified.
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Rule 30
Rule 6679
Rubi steps
\begin {align*} \int \frac {\left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}}}{x} \, dx &=\left (x \left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}}\right ) \int \frac {1}{x^2} \, dx\\ &=-\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 = 20, normalized size = 1.00 \[ -\left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 21, normalized size = 1.05 \[ -\frac {e^{\left (-\frac {n \log \relax (b) + \log \relax (a)}{m n}\right )}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 21, normalized size = 1.05 \[ -\frac {1}{\left (\left (b x^{m}\right )^{n} a\right )^{\frac {1}{m n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 22, normalized size = 1.10 \[ -\left (a \left (b \,x^{m}\right )^{n}\right )^{-\frac {1}{m n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.11, size = 35, normalized size = 1.75 \[ -\frac {1}{a^{\frac {1}{m n}} b^{\left (\frac {1}{m}\right )} {\left ({\left (x^{m}\right )}^{n}\right )}^{\frac {1}{m n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.97, size = 21, normalized size = 1.05 \[ -\frac {1}{{\left (a\,{\left (b\,x^m\right )}^n\right )}^{\frac {1}{m\,n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.88, size = 153, normalized size = 7.65 \[ \begin {cases} - \left (0^{m n}\right )^{- \frac {1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{- \frac {1}{m n}} \left (\left (\left (0^{m n}\right )^{\frac {1}{n}}\right )^{n}\right )^{- \frac {1}{m n}} & \text {for}\: a = 0^{m n} \wedge b = \left (0^{m n}\right )^{\frac {1}{n}} \\- a^{- \frac {1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{- \frac {1}{m n}} \left (\left (\left (0^{m n}\right )^{\frac {1}{n}}\right )^{n}\right )^{- \frac {1}{m n}} & \text {for}\: b = \left (0^{m n}\right )^{\frac {1}{n}} \\- \left (0^{m n}\right )^{- \frac {1}{m n}} \left (b^{n}\right )^{- \frac {1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{- \frac {1}{m n}} & \text {for}\: a = 0^{m n} \\- a^{- \frac {1}{m n}} \left (b^{n}\right )^{- \frac {1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{- \frac {1}{m n}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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