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 = 0.04, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 30
Rule 6679
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*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 1.00 \[ -x^{-n p q-1} \left (a \left (b x^n\right )^p\right )^q \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 16, normalized size = 0.73 \[ -\frac {e^{\left (p q \log \relax (b) + q \log \relax (a)\right )}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 18, normalized size = 0.82 \[ -x e^{\left (p q \log \relax (b) + q \log \relax (a) - 2 \, \log \relax (x)\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 23, normalized size = 1.05 \[ -x^{-n p q -1} \left (a \left (b \,x^{n}\right )^{p}\right )^{q} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\left (b x^{n}\right )^{p} a\right )^{q} x^{-n p q - 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.01, size = 23, normalized size = 1.05 \[ -\frac {{\left (a\,{\left (b\,x^n\right )}^p\right )}^q}{x^{n\,p\,q+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 93.32, size = 24, normalized size = 1.09 \[ - \frac {a^{q} x^{- n p q} \left (b^{p}\right )^{q} \left (\left (x^{n}\right )^{p}\right )^{q}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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