Optimal. Leaf size=24 \[ \frac{1}{2} x^{2-n p q} \left (a \left (b x^n\right )^p\right )^q \]
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Rubi [A] time = 0.0421745, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {6679, 30} \[ \frac{1}{2} x^{2-n p q} \left (a \left (b x^n\right )^p\right )^q \]
Antiderivative was successfully verified.
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Rule 6679
Rule 30
Rubi steps
\begin{align*} \int x^{1-n p q} \left (a \left (b x^n\right )^p\right )^q \, dx &=\left (x^{-n p q} \left (a \left (b x^n\right )^p\right )^q\right ) \int x \, dx\\ &=\frac{1}{2} x^{2-n p q} \left (a \left (b x^n\right )^p\right )^q\\ \end{align*}
Mathematica [A] time = 0.0037881, size = 24, normalized size = 1. \[ \frac{1}{2} x^{2-n p q} \left (a \left (b x^n\right )^p\right )^q \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 23, normalized size = 1. \begin{align*}{\frac{{x}^{-npq+2} \left ( a \left ( b{x}^{n} \right ) ^{p} \right ) ^{q}}{2}} \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 \left (\left (b x^{n}\right )^{p} a\right )^{q} x^{-n p q + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71207, size = 47, normalized size = 1.96 \begin{align*} \frac{1}{2} \, x^{2} e^{\left (p q \log \left (b\right ) + q \log \left (a\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{- n p q + 1} \left (a \left (b x^{n}\right )^{p}\right )^{q}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15266, size = 22, normalized size = 0.92 \begin{align*} \frac{1}{2} \, x e^{\left (p q \log \left (b\right ) + q \log \left (a\right ) + \log \left (x\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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