Optimal. Leaf size=21 \[ \frac{\left (a \left (b x^n\right )^p\right )^q}{n p q} \]
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Rubi [A] time = 0.044548, antiderivative size = 21, normalized size of antiderivative = 1., 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} \[ \frac{\left (a \left (b x^n\right )^p\right )^q}{n p q} \]
Antiderivative was successfully verified.
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Rule 6679
Rule 30
Rubi steps
\begin{align*} \int \frac{\left (a \left (b x^n\right )^p\right )^q}{x} \, dx &=\left (x^{-n p q} \left (a \left (b x^n\right )^p\right )^q\right ) \int x^{-1+n p q} \, dx\\ &=\frac{\left (a \left (b x^n\right )^p\right )^q}{n p q}\\ \end{align*}
Mathematica [A] time = 0.0021466, size = 21, normalized size = 1. \[ \frac{\left (a \left (b x^n\right )^p\right )^q}{n p q} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 22, normalized size = 1.1 \begin{align*}{\frac{ \left ( a \left ( b{x}^{n} \right ) ^{p} \right ) ^{q}}{npq}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.37418, size = 34, normalized size = 1.62 \begin{align*} \frac{a^{q}{\left (b^{p}\right )}^{q}{\left ({\left (x^{n}\right )}^{p}\right )}^{q}}{n p q} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79109, size = 68, normalized size = 3.24 \begin{align*} \frac{e^{\left (n p q \log \left (x\right ) + p q \log \left (b\right ) + q \log \left (a\right )\right )}}{n p q} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.771194, size = 42, normalized size = 2. \begin{align*} \begin{cases} \log{\left (x \right )} & \text{for}\: q = 0 \wedge \left (n = 0 \vee q = 0\right ) \wedge \left (p = 0 \vee q = 0\right ) \\\left (a b^{p}\right )^{q} \log{\left (x \right )} & \text{for}\: n = 0 \\a^{q} \log{\left (x \right )} & \text{for}\: p = 0 \\\frac{a^{q} \left (b^{p}\right )^{q} \left (\left (x^{n}\right )^{p}\right )^{q}}{n p q} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\left (b x^{n}\right )^{p} a\right )^{q}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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