Integrand size = 21, antiderivative size = 21 \[ \int x^{-1-n p q} \left (a \left (b x^n\right )^p\right )^q \, dx=x^{-n p q} \left (a \left (b x^n\right )^p\right )^q \log (x) \]
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Time = 0.01 (sec) , antiderivative size = 21, 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, 29} \[ \int x^{-1-n p q} \left (a \left (b x^n\right )^p\right )^q \, dx=\log (x) x^{-n p q} \left (a \left (b x^n\right )^p\right )^q \]
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Rule 29
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 \frac {1}{x} \, dx \\ & = x^{-n p q} \left (a \left (b x^n\right )^p\right )^q \log (x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int x^{-1-n p q} \left (a \left (b x^n\right )^p\right )^q \, dx=x^{-n p q} \left (a \left (b x^n\right )^p\right )^q \log (x) \]
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\[\int x^{-n p q -1} {\left (a \left (b \,x^{n}\right )^{p}\right )}^{q}d x\]
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none
Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67 \[ \int x^{-1-n p q} \left (a \left (b x^n\right )^p\right )^q \, dx=e^{\left (p q \log \left (b\right ) + q \log \left (a\right )\right )} \log \left (x\right ) \]
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Time = 0.74 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int x^{-1-n p q} \left (a \left (b x^n\right )^p\right )^q \, dx=x x^{- n p q - 1} \left (a \left (b x^{n}\right )^{p}\right )^{q} \log {\left (x \right )} \]
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\[ \int x^{-1-n p q} \left (a \left (b x^n\right )^p\right )^q \, dx=\int { \left (\left (b x^{n}\right )^{p} a\right )^{q} x^{-n p q - 1} \,d x } \]
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\[ \int x^{-1-n p q} \left (a \left (b x^n\right )^p\right )^q \, dx=\int { \left (\left (b x^{n}\right )^{p} a\right )^{q} x^{-n p q - 1} \,d x } \]
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Timed out. \[ \int x^{-1-n p q} \left (a \left (b x^n\right )^p\right )^q \, dx=\int \frac {{\left (a\,{\left (b\,x^n\right )}^p\right )}^q}{x^{n\,p\,q+1}} \,d x \]
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