Integrand size = 21, antiderivative size = 24 \[ \int x^{2-n p q} \left (a \left (b x^n\right )^p\right )^q \, dx=\frac {1}{3} x^{3-n p q} \left (a \left (b x^n\right )^p\right )^q \]
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Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1971, 30} \[ \int x^{2-n p q} \left (a \left (b x^n\right )^p\right )^q \, dx=\frac {1}{3} x^{3-n p q} \left (a \left (b x^n\right )^p\right )^q \]
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Rule 30
Rule 1971
Rubi steps \begin{align*} \text {integral}& = \left (x^{-n p q} \left (a \left (b x^n\right )^p\right )^q\right ) \int x^2 \, dx \\ & = \frac {1}{3} x^{3-n p q} \left (a \left (b x^n\right )^p\right )^q \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int x^{2-n p q} \left (a \left (b x^n\right )^p\right )^q \, dx=\frac {1}{3} x^{3-n p q} \left (a \left (b x^n\right )^p\right )^q \]
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Time = 0.35 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96
method | result | size |
gosper | \(\frac {x^{-n p q +3} {\left (a \left (b \,x^{n}\right )^{p}\right )}^{q}}{3}\) | \(23\) |
parallelrisch | \(\frac {x \,x^{-n p q +2} {\left (a \left (b \,x^{n}\right )^{p}\right )}^{q}}{3}\) | \(24\) |
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Time = 0.30 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.67 \[ \int x^{2-n p q} \left (a \left (b x^n\right )^p\right )^q \, dx=\frac {1}{3} \, x^{3} e^{\left (p q \log \left (b\right ) + q \log \left (a\right )\right )} \]
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Time = 0.39 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int x^{2-n p q} \left (a \left (b x^n\right )^p\right )^q \, dx=\frac {x x^{- n p q + 2} \left (a \left (b x^{n}\right )^{p}\right )^{q}}{3} \]
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\[ \int x^{2-n p q} \left (a \left (b x^n\right )^p\right )^q \, dx=\int { \left (\left (b x^{n}\right )^{p} a\right )^{q} x^{-n p q + 2} \,d x } \]
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Time = 0.30 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.75 \[ \int x^{2-n p q} \left (a \left (b x^n\right )^p\right )^q \, dx=\frac {1}{3} \, x e^{\left (p q \log \left (b\right ) + q \log \left (a\right ) + 2 \, \log \left (x\right )\right )} \]
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Time = 5.66 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int x^{2-n p q} \left (a \left (b x^n\right )^p\right )^q \, dx=\frac {x^{3-n\,p\,q}\,{\left (a\,{\left (b\,x^n\right )}^p\right )}^q}{3} \]
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