Optimal. Leaf size=26 \[ \frac {x \left (a x^m\right )^r \left (b x^n\right )^s}{m r+n s+1} \]
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Rubi [A] time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {15, 30} \[ \frac {x \left (a x^m\right )^r \left (b x^n\right )^s}{m r+n s+1} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin {align*} \int \left (a x^m\right )^r \left (b x^n\right )^s \, dx &=\left (x^{-m r} \left (a x^m\right )^r\right ) \int x^{m r} \left (b x^n\right )^s \, dx\\ &=\left (x^{-m r-n s} \left (a x^m\right )^r \left (b x^n\right )^s\right ) \int x^{m r+n s} \, dx\\ &=\frac {x \left (a x^m\right )^r \left (b x^n\right )^s}{1+m r+n s}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 1.00 \[ \frac {x \left (a x^m\right )^r \left (b x^n\right )^s}{m r+n s+1} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 32, normalized size = 1.23 \[ \frac {x e^{\left (m r \log \relax (x) + n s \log \relax (x) + r \log \relax (a) + s \log \relax (b)\right )}}{m r + n s + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 32, normalized size = 1.23 \[ \frac {x e^{\left (m r \log \relax (x) + n s \log \relax (x) + r \log \relax (a) + s \log \relax (b)\right )}}{m r + n s + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 27, normalized size = 1.04 \[ \frac {x \left (a \,x^{m}\right )^{r} \left (b \,x^{n}\right )^{s}}{m r +n s +1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 32, normalized size = 1.23 \[ \frac {a^{r} b^{s} x e^{\left (r \log \left (x^{m}\right ) + s \log \left (x^{n}\right )\right )}}{m r + n s + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.92, size = 26, normalized size = 1.00 \[ \frac {x\,{\left (a\,x^m\right )}^r\,{\left (b\,x^n\right )}^s}{m\,r+n\,s+1} \]
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 \[ \begin {cases} \frac {a^{r} b^{s} x \left (x^{m}\right )^{r} \left (x^{n}\right )^{s}}{m r + n s + 1} & \text {for}\: m \neq - \frac {n s + 1}{r} \\\int \left (b x^{n}\right )^{s} \left (a x^{- \frac {1}{r}} x^{- \frac {n s}{r}}\right )^{r}\, dx & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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