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 = 0.04, 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} \[ -\frac {\left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}}}{2 x} \]
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^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*}
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Mathematica [A] time = 0.00, size = 25, normalized size = 1.00 \[ -\frac {\left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}}}{2 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 21, normalized size = 0.84 \[ -\frac {e^{\left (-\frac {n \log \relax (b) + \log \relax (a)}{m n}\right )}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\left (b x^{m}\right )^{n} a\right )^{\frac {1}{m n}} x^{2}}\,{d x} \]
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 \[ -\frac {\left (a \left (b \,x^{m}\right )^{n}\right )^{-\frac {1}{m n}}}{2 x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\left (b x^{m}\right )^{n} a\right )^{\frac {1}{m n}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{x^2\,{\left (a\,{\left (b\,x^m\right )}^n\right )}^{\frac {1}{m\,n}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.06, size = 252, normalized size = 10.08 \[ \begin {cases} - \frac {1}{0^{m n} \tilde {\infty }^{m n} x \left (0^{m n}\right )^{\frac {1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{\frac {1}{m n}} \left (\left (\left (0^{m n}\right )^{\frac {1}{n}}\right )^{n}\right )^{\frac {1}{m n}} + x \left (0^{m n}\right )^{\frac {1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{\frac {1}{m n}} \left (\left (\left (0^{m n}\right )^{\frac {1}{n}}\right )^{n}\right )^{\frac {1}{m n}}} & \text {for}\: a = 0^{m n} \wedge b = \left (0^{m n}\right )^{\frac {1}{n}} \\- \frac {a^{- \frac {1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{- \frac {1}{m n}} \left (\left (\left (0^{m n}\right )^{\frac {1}{n}}\right )^{n}\right )^{- \frac {1}{m n}}}{2 x} & \text {for}\: b = \left (0^{m n}\right )^{\frac {1}{n}} \\- \frac {1}{0^{m n} \tilde {\infty }^{m n} x \left (0^{m n}\right )^{\frac {1}{m n}} \left (b^{n}\right )^{\frac {1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{\frac {1}{m n}} + x \left (0^{m n}\right )^{\frac {1}{m n}} \left (b^{n}\right )^{\frac {1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{\frac {1}{m n}}} & \text {for}\: a = 0^{m n} \\- \frac {a^{- \frac {1}{m n}} \left (b^{n}\right )^{- \frac {1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{- \frac {1}{m n}}}{2 x} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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