Optimal. Leaf size=16 \[ \frac {x \left (a x^m\right )^r}{m r+1} \]
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Rubi [A] time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.00, size = 16, normalized size = 1.00 \[ \frac {x \left (a x^m\right )^r}{m r+1} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 20, normalized size = 1.25 \[ \frac {x e^{\left (m r \log \relax (x) + r \log \relax (a)\right )}}{m r + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 20, normalized size = 1.25 \[ \frac {x e^{\left (m r \log \relax (x) + r \log \relax (a)\right )}}{m r + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 17, normalized size = 1.06 \[ \frac {x \left (a \,x^{m}\right )^{r}}{m r +1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.48, size = 17, normalized size = 1.06 \[ \frac {a^{r} x {\left (x^{m}\right )}^{r}}{m r + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.40, size = 16, normalized size = 1.00 \[ \frac {x\,{\left (a\,x^m\right )}^r}{m\,r+1} \]
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 \[ \begin {cases} \frac {a^{r} x \left (x^{m}\right )^{r}}{m r + 1} & \text {for}\: m \neq - \frac {1}{r} \\\int \left (a x^{- \frac {1}{r}}\right )^{r}\, dx & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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