Optimal. Leaf size=24 \[ \frac {1}{2} x^{2-n p q} \left (a \left (b x^n\right )^p\right )^q \]
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Rubi [A]
time = 0.01, antiderivative size = 24, 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 = {1971, 30}
\begin {gather*} \frac {1}{2} x^{2-n p q} \left (a \left (b x^n\right )^p\right )^q \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 1971
Rubi steps
\begin {align*} \int x^{1-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 x \, dx\\ &=\frac {1}{2} x^{2-n p q} \left (a \left (b x^n\right )^p\right )^q\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 24, normalized size = 1.00 \begin {gather*} \frac {1}{2} x^{2-n p q} \left (a \left (b x^n\right )^p\right )^q \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 23, normalized size = 0.96
method | result | size |
gosper | \(\frac {x^{-n p q +2} \left (a \left (b \,x^{n}\right )^{p}\right )^{q}}{2}\) | \(23\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 16, normalized size = 0.67 \begin {gather*} \frac {1}{2} \, x^{2} e^{\left (p q \log \left (b\right ) + q \log \left (a\right )\right )} \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 x^{- n p q + 1} \left (a \left (b x^{n}\right )^{p}\right )^{q}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.24, size = 16, normalized size = 0.67 \begin {gather*} \frac {1}{2} \, x e^{\left (p q \log \left (b\right ) + q \log \left (a\right ) + \log \left (x\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.00, size = 22, normalized size = 0.92 \begin {gather*} \frac {x^{2-n\,p\,q}\,{\left (a\,{\left (b\,x^n\right )}^p\right )}^q}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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