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 = 0.0091378, antiderivative size = 26, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0073769, size = 26, normalized size = 1. \[ \frac{x \left (a x^m\right )^r \left (b x^n\right )^s}{m r+n s+1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 27, normalized size = 1. \begin{align*}{\frac{x \left ( a{x}^{m} \right ) ^{r} \left ( b{x}^{n} \right ) ^{s}}{mr+ns+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50721, size = 43, normalized size = 1.65 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.01956, size = 93, normalized size = 3.58 \begin{align*} \frac{x e^{\left (m r \log \left (x\right ) + n s \log \left (x\right ) + r \log \left (a\right ) + s \log \left (b\right )\right )}}{m r + n s + 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 = 1.17513, size = 43, normalized size = 1.65 \begin{align*} \frac{x e^{\left (m r \log \left (x\right ) + n s \log \left (x\right ) + r \log \left (a\right ) + s \log \left (b\right )\right )}}{m r + n s + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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