Optimal. Leaf size=25 \[ -\frac{\left (a \left (b x^n\right )^p\right )^q}{x (1-n p q)} \]
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Rubi [A] time = 0.0452928, antiderivative size = 25, 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} \[ -\frac{\left (a \left (b x^n\right )^p\right )^q}{x (1-n p q)} \]
Antiderivative was successfully verified.
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Rule 6679
Rule 30
Rubi steps
\begin{align*} \int \frac{\left (a \left (b x^n\right )^p\right )^q}{x^2} \, dx &=\left (x^{-n p q} \left (a \left (b x^n\right )^p\right )^q\right ) \int x^{-2+n p q} \, dx\\ &=-\frac{\left (a \left (b x^n\right )^p\right )^q}{(1-n p q) x}\\ \end{align*}
Mathematica [A] time = 0.0049682, size = 23, normalized size = 0.92 \[ \frac{\left (a \left (b x^n\right )^p\right )^q}{x (n p q-1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 24, normalized size = 1. \begin{align*}{\frac{ \left ( a \left ( b{x}^{n} \right ) ^{p} \right ) ^{q}}{x \left ( npq-1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.39295, size = 36, normalized size = 1.44 \begin{align*} \frac{a^{q}{\left (b^{p}\right )}^{q}{\left ({\left (x^{n}\right )}^{p}\right )}^{q}}{{\left (n p q - 1\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85115, size = 78, normalized size = 3.12 \begin{align*} \frac{e^{\left (n p q \log \left (x\right ) + p q \log \left (b\right ) + q \log \left (a\right )\right )}}{{\left (n p q - 1\right )} x} \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 \frac{\left (a \left (b x^{n}\right )^{p}\right )^{q}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\left (b x^{n}\right )^{p} a\right )^{q}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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