Optimal. Leaf size=22 \[ -x^{-n p q-1} \left (a \left (b x^n\right )^p\right )^q \]
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Rubi [A] time = 0.0420725, antiderivative size = 22, 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} \[ -x^{-n p q-1} \left (a \left (b x^n\right )^p\right )^q \]
Antiderivative was successfully verified.
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Rule 6679
Rule 30
Rubi steps
\begin{align*} \int x^{-2-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 \frac{1}{x^2} \, dx\\ &=-x^{-1-n p q} \left (a \left (b x^n\right )^p\right )^q\\ \end{align*}
Mathematica [A] time = 0.0050027, size = 22, normalized size = 1. \[ -x^{-n p q-1} \left (a \left (b x^n\right )^p\right )^q \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 23, normalized size = 1.1 \begin{align*} -{x}^{-npq-1} \left ( a \left ( b{x}^{n} \right ) ^{p} \right ) ^{q} \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 - 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81456, size = 41, normalized size = 1.86 \begin{align*} -\frac{e^{\left (p q \log \left (b\right ) + q \log \left (a\right )\right )}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 145.066, size = 24, normalized size = 1.09 \begin{align*} - \frac{a^{q} x^{- n p q} \left (b^{p}\right )^{q} \left (\left (x^{n}\right )^{p}\right )^{q}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17026, size = 24, normalized size = 1.09 \begin{align*} -x e^{\left (p q \log \left (b\right ) + q \log \left (a\right ) - 2 \, \log \left (x\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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