Integrand size = 22, antiderivative size = 25 \[ \int x^2 \left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}} \, dx=\frac {1}{2} x^3 \left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}} \]
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Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1971, 30} \[ \int x^2 \left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}} \, dx=\frac {1}{2} x^3 \left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}} \]
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Rule 30
Rule 1971
Rubi steps \begin{align*} \text {integral}& = \left (x \left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}}\right ) \int x \, dx \\ & = \frac {1}{2} x^3 \left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int x^2 \left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}} \, dx=\frac {1}{2} x^3 \left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}} \]
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Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00
method | result | size |
gosper | \(\frac {x^{3} {\left (a \left (b \,x^{m}\right )^{n}\right )}^{-\frac {1}{m n}}}{2}\) | \(25\) |
parallelrisch | \(\frac {x^{3} {\left (a \left (b \,x^{m}\right )^{n}\right )}^{-\frac {1}{m n}}}{2}\) | \(25\) |
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Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int x^2 \left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}} \, dx=\frac {1}{2} \, x^{2} e^{\left (-\frac {n \log \left (b\right ) + \log \left (a\right )}{m n}\right )} \]
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Time = 1.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68 \[ \int x^2 \left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}} \, dx=\frac {x^{3} \left (a \left (b x^{m}\right )^{n}\right )^{- \frac {1}{m n}}}{2} \]
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Time = 0.31 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.52 \[ \int x^2 \left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}} \, dx=\frac {x^{3}}{2 \, a^{\frac {1}{m n}} b^{\left (\frac {1}{m}\right )} {\left ({\left (x^{m}\right )}^{n}\right )}^{\frac {1}{m n}}} \]
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Time = 0.32 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int x^2 \left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}} \, dx=\frac {1}{2} \, x^{2} e^{\left (-\frac {n \log \left (b\right ) + \log \left (a\right )}{m n}\right )} \]
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Time = 5.68 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int x^2 \left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}} \, dx=\frac {x^3}{2\,{\left (a\,{\left (b\,x^m\right )}^n\right )}^{\frac {1}{m\,n}}} \]
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