Optimal. Leaf size=20 \[ -\left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}} \]
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Rubi [A] time = 0.0442031, antiderivative size = 20, 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} \[ -\left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}} \]
Antiderivative was successfully verified.
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Rule 6679
Rule 30
Rubi steps
\begin{align*} \int \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*}
Mathematica [A] time = 0.0022073, size = 20, normalized size = 1. \[ -\left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 22, normalized size = 1.1 \begin{align*} - \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 = 1.53204, size = 55, normalized size = 2.75 \begin{align*} -\frac{1}{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 = 1.80817, size = 47, normalized size = 2.35 \begin{align*} -\frac{e^{\left (-\frac{n \log \left (b\right ) + \log \left (a\right )}{m n}\right )}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.77898, size = 153, normalized size = 7.65 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (\left (b x^{m}\right )^{n} a\right )^{\frac{1}{m n}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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