Optimal. Leaf size=25 \[ -\frac {\left (a \left (b x^n\right )^p\right )^q}{(1-n p q) x} \]
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Rubi [A]
time = 0.01, 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 = {1971, 30}
\begin {gather*} -\frac {\left (a \left (b x^n\right )^p\right )^q}{x (1-n p q)} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 1971
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 = 0.00, size = 23, normalized size = 0.92 \begin {gather*} \frac {\left (a \left (b x^n\right )^p\right )^q}{(-1+n p q) x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 24, normalized size = 0.96
method | result | size |
gosper | \(\frac {\left (a \left (b \,x^{n}\right )^{p}\right )^{q}}{x \left (n p q -1\right )}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.39, size = 27, normalized size = 1.08 \begin {gather*} \frac {a^{q} b^{p q} {\left ({\left (x^{n}\right )}^{p}\right )}^{q}}{{\left (n p q - 1\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 29, normalized size = 1.16 \begin {gather*} \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 {gather*}
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 \begin {gather*} \int \frac {\left (a \left (b x^{n}\right )^{p}\right )^{q}}{x^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (a\,{\left (b\,x^n\right )}^p\right )}^q}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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