Optimal. Leaf size=23 \[ \frac{x^2 \left (a \left (b x^n\right )^p\right )^q}{n p q+2} \]
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Rubi [A] time = 0.028609, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {6679, 30} \[ \frac{x^2 \left (a \left (b x^n\right )^p\right )^q}{n p q+2} \]
Antiderivative was successfully verified.
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Rule 6679
Rule 30
Rubi steps
\begin{align*} \int x \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^{1+n p q} \, dx\\ &=\frac{x^2 \left (a \left (b x^n\right )^p\right )^q}{2+n p q}\\ \end{align*}
Mathematica [A] time = 0.0037556, size = 23, normalized size = 1. \[ \frac{x^2 \left (a \left (b x^n\right )^p\right )^q}{n p q+2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 24, normalized size = 1. \begin{align*}{\frac{{x}^{2} \left ( a \left ( b{x}^{n} \right ) ^{p} \right ) ^{q}}{npq+2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.39744, size = 36, normalized size = 1.57 \begin{align*} \frac{a^{q}{\left (b^{p}\right )}^{q} x^{2}{\left ({\left (x^{n}\right )}^{p}\right )}^{q}}{n p q + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77177, size = 78, normalized size = 3.39 \begin{align*} \frac{x^{2} e^{\left (n p q \log \left (x\right ) + p q \log \left (b\right ) + q \log \left (a\right )\right )}}{n p q + 2} \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 \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 = 1.16318, size = 39, normalized size = 1.7 \begin{align*} \frac{x^{2} e^{\left (n p q \log \left (x\right ) + p q \log \left (b\right ) + q \log \left (a\right )\right )}}{n p q + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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