Optimal. Leaf size=16 \[ \frac{x \left (a x^m\right )^r}{m r+1} \]
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Rubi [A] time = 0.0046085, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {15, 30} \[ \frac{x \left (a x^m\right )^r}{m r+1} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin{align*} \int \left (a x^m\right )^r \, dx &=\left (x^{-m r} \left (a x^m\right )^r\right ) \int x^{m r} \, dx\\ &=\frac{x \left (a x^m\right )^r}{1+m r}\\ \end{align*}
Mathematica [A] time = 0.0031748, size = 16, normalized size = 1. \[ \frac{x \left (a x^m\right )^r}{m r+1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 17, normalized size = 1.1 \begin{align*}{\frac{x \left ( a{x}^{m} \right ) ^{r}}{mr+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.31621, size = 23, normalized size = 1.44 \begin{align*} \frac{a^{r} x{\left (x^{m}\right )}^{r}}{m r + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.00261, size = 53, normalized size = 3.31 \begin{align*} \frac{x e^{\left (m r \log \left (x\right ) + r \log \left (a\right )\right )}}{m r + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14182, size = 27, normalized size = 1.69 \begin{align*} \frac{x e^{\left (m r \log \left (x\right ) + r \log \left (a\right )\right )}}{m r + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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