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 = 0.05, antiderivative size = 25, normalized size of antiderivative = 1.00, 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 30
Rule 6679
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.56, size = 29, normalized size = 1.16 \[ \frac {e^{\left (n p q \log \relax (x) + p q \log \relax (b) + q \log \relax (a)\right )}}{{\left (n p q - 1\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\left (b x^{n}\right )^{p} a\right )^{q}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 24, normalized size = 0.96 \[ \frac {\left (a \left (b \,x^{n}\right )^{p}\right )^{q}}{\left (n p q -1\right ) x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.80, size = 27, normalized size = 1.08 \[ \frac {a^{q} b^{p q} {\left ({\left (x^{n}\right )}^{p}\right )}^{q}}{{\left (n p q - 1\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\left (a\,{\left (b\,x^n\right )}^p\right )}^q}{x^2} \,d x \]
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 \[ \int \frac {\left (a \left (b x^{n}\right )^{p}\right )^{q}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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