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 = 0.03, antiderivative size = 23, normalized size of antiderivative = 1.00, 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 30
Rule 6679
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*}
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Mathematica [A] time = 0.00, size = 23, normalized size = 1.00 \[ \frac {x^2 \left (a \left (b x^n\right )^p\right )^q}{n p q+2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 29, normalized size = 1.26 \[ \frac {x^{2} e^{\left (n p q \log \relax (x) + p q \log \relax (b) + q \log \relax (a)\right )}}{n p q + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 29, normalized size = 1.26 \[ \frac {x^{2} e^{\left (n p q \log \relax (x) + p q \log \relax (b) + q \log \relax (a)\right )}}{n p q + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 24, normalized size = 1.04 \[ \frac {x^{2} \left (a \left (b \,x^{n}\right )^{p}\right )^{q}}{n p q +2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.70, size = 27, normalized size = 1.17 \[ \frac {a^{q} b^{p q} x^{2} {\left ({\left (x^{n}\right )}^{p}\right )}^{q}}{n p q + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.99, size = 23, normalized size = 1.00 \[ \frac {x^2\,{\left (a\,{\left (b\,x^n\right )}^p\right )}^q}{n\,p\,q+2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \left (a \left (b x^{n}\right )^{p}\right )^{q}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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