Optimal. Leaf size=25 \[ -\frac{\left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}}}{2 x} \]
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Rubi [A] time = 0.0432139, 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} \[ -\frac{\left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}}}{2 x} \]
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^2} \, dx &=\left (x \left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}}\right ) \int \frac{1}{x^3} \, dx\\ &=-\frac{\left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}}}{2 x}\\ \end{align*}
Mathematica [A] time = 0.0032013, size = 25, normalized size = 1. \[ -\frac{\left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}}}{2 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 25, normalized size = 1. \begin{align*} -{\frac{1}{2\,x} \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 [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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82122, size = 55, normalized size = 2.2 \begin{align*} -\frac{e^{\left (-\frac{n \log \left (b\right ) + \log \left (a\right )}{m n}\right )}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.02892, size = 252, normalized size = 10.08 \begin{align*} \begin{cases} - \frac{1}{0^{m n} \tilde{\infty }^{m n} x \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}} + x \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}} \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 x} & \text{for}\: b = \left (0^{m n}\right )^{\frac{1}{n}} \\- \frac{1}{0^{m n} \tilde{\infty }^{m n} x \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}} + x \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}} \left (b^{n}\right )^{- \frac{1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{- \frac{1}{m n}}}{2 x} & \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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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