Optimal. Leaf size=36 \[ \frac{x \left (a x^m\right )^r \left (b x^n\right )^s \left (c x^p\right )^t}{m r+n s+p t+1} \]
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Rubi [A] time = 0.0155787, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {15, 30} \[ \frac{x \left (a x^m\right )^r \left (b x^n\right )^s \left (c x^p\right )^t}{m r+n s+p t+1} \]
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 \left (c x^p\right )^t \, dx &=\left (x^{-m r} \left (a x^m\right )^r\right ) \int x^{m r} \left (b x^n\right )^s \left (c x^p\right )^t \, 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} \left (c x^p\right )^t \, dx\\ &=\left (x^{-m r-n s-p t} \left (a x^m\right )^r \left (b x^n\right )^s \left (c x^p\right )^t\right ) \int x^{m r+n s+p t} \, dx\\ &=\frac{x \left (a x^m\right )^r \left (b x^n\right )^s \left (c x^p\right )^t}{1+m r+n s+p t}\\ \end{align*}
Mathematica [A] time = 0.012172, size = 36, normalized size = 1. \[ \frac{x \left (a x^m\right )^r \left (b x^n\right )^s \left (c x^p\right )^t}{m r+n s+p t+1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 37, normalized size = 1. \begin{align*}{\frac{x \left ( a{x}^{m} \right ) ^{r} \left ( b{x}^{n} \right ) ^{s} \left ( c{x}^{p} \right ) ^{t}}{mr+ns+pt+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.44166, size = 59, normalized size = 1.64 \begin{align*} \frac{a^{r} b^{s} c^{t} x e^{\left (r \log \left (x^{m}\right ) + s \log \left (x^{n}\right ) + t \log \left (x^{p}\right )\right )}}{m r + n s + p t + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.989281, size = 134, normalized size = 3.72 \begin{align*} \frac{x e^{\left (m r \log \left (x\right ) + n s \log \left (x\right ) + p t \log \left (x\right ) + r \log \left (a\right ) + s \log \left (b\right ) + t \log \left (c\right )\right )}}{m r + n s + p t + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15558, size = 59, normalized size = 1.64 \begin{align*} \frac{x e^{\left (m r \log \left (x\right ) + n s \log \left (x\right ) + p t \log \left (x\right ) + r \log \left (a\right ) + s \log \left (b\right ) + t \log \left (c\right )\right )}}{m r + n s + p t + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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