Optimal. Leaf size=26 \[ \frac{x^{m+1} \left (a \left (b x^n\right )^p\right )^q}{m+n p q+1} \]
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Rubi [A] time = 0.0451479, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {6679, 30} \[ \frac{x^{m+1} \left (a \left (b x^n\right )^p\right )^q}{m+n p q+1} \]
Antiderivative was successfully verified.
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Rule 6679
Rule 30
Rubi steps
\begin{align*} \int x^m \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^{m+n p q} \, dx\\ &=\frac{x^{1+m} \left (a \left (b x^n\right )^p\right )^q}{1+m+n p q}\\ \end{align*}
Mathematica [A] time = 0.0078345, size = 26, normalized size = 1. \[ \frac{x^{m+1} \left (a \left (b x^n\right )^p\right )^q}{m+n p q+1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 27, normalized size = 1. \begin{align*}{\frac{{x}^{1+m} \left ( a \left ( b{x}^{n} \right ) ^{p} \right ) ^{q}}{npq+m+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.42313, size = 45, normalized size = 1.73 \begin{align*} \frac{a^{q}{\left (b^{p}\right )}^{q} x e^{\left (m \log \left (x\right ) + q \log \left ({\left (x^{n}\right )}^{p}\right )\right )}}{n p q + m + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77252, size = 86, normalized size = 3.31 \begin{align*} \frac{x x^{m} e^{\left (n p q \log \left (x\right ) + p q \log \left (b\right ) + q \log \left (a\right )\right )}}{n p q + m + 1} \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^{m} \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 = 1.12154, size = 42, normalized size = 1.62 \begin{align*} \frac{x x^{m} e^{\left (n p q \log \left (x\right ) + p q \log \left (b\right ) + q \log \left (a\right )\right )}}{n p q + m + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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