Optimal. Leaf size=16 \[ \frac {x \left (a x^m\right )^r}{1+m r} \]
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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}
\begin {gather*} \frac {x \left (a x^m\right )^r}{m r+1} \end {gather*}
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 \begin {gather*} \frac {x \left (a x^m\right )^r}{1+m r} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 17, normalized size = 1.06
method | result | size |
gosper | \(\frac {x \left (a \,x^{m}\right )^{r}}{m r +1}\) | \(17\) |
norman | \(\frac {x \,{\mathrm e}^{r \ln \left (a \,{\mathrm e}^{m \ln \left (x \right )}\right )}}{m r +1}\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 17, normalized size = 1.06 \begin {gather*} \frac {a^{r} x {\left (x^{m}\right )}^{r}}{m r + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 20, normalized size = 1.25 \begin {gather*} \frac {x e^{\left (m r \log \left (x\right ) + r \log \left (a\right )\right )}}{m r + 1} \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*} \begin {cases} \frac {x \left (a 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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.70, size = 20, normalized size = 1.25 \begin {gather*} \frac {x e^{\left (m r \log \left (x\right ) + r \log \left (a\right )\right )}}{m r + 1} \end {gather*}
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 \begin {gather*} \frac {x\,{\left (a\,x^m\right )}^r}{m\,r+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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