Optimal. Leaf size=36 \[ \frac{\left (a F^{c+d x}\right )^m \left (b F^{e+f x}\right )^n}{\log (F) (d m+f n)} \]
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Rubi [A] time = 0.0986903, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2281, 2227, 2194} \[ \frac{\left (a F^{c+d x}\right )^m \left (b F^{e+f x}\right )^n}{\log (F) (d m+f n)} \]
Antiderivative was successfully verified.
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Rule 2281
Rule 2227
Rule 2194
Rubi steps
\begin{align*} \int \left (a F^{c+d x}\right )^m \left (b F^{e+f x}\right )^n \, dx &=\left (F^{-m (c+d x)} \left (a F^{c+d x}\right )^m\right ) \int F^{m (c+d x)} \left (b F^{e+f x}\right )^n \, dx\\ &=\left (F^{-m (c+d x)-n (e+f x)} \left (a F^{c+d x}\right )^m \left (b F^{e+f x}\right )^n\right ) \int F^{m (c+d x)+n (e+f x)} \, dx\\ &=\left (F^{-m (c+d x)-n (e+f x)} \left (a F^{c+d x}\right )^m \left (b F^{e+f x}\right )^n\right ) \int F^{c m+e n+(d m+f n) x} \, dx\\ &=\frac{\left (a F^{c+d x}\right )^m \left (b F^{e+f x}\right )^n}{(d m+f n) \log (F)}\\ \end{align*}
Mathematica [A] time = 0.0464839, size = 36, normalized size = 1. \[ \frac{\left (a F^{c+d x}\right )^m \left (b F^{e+f x}\right )^n}{d m \log (F)+f n \log (F)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 37, normalized size = 1. \begin{align*}{\frac{ \left ( a{F}^{dx+c} \right ) ^{m} \left ( b{F}^{fx+e} \right ) ^{n}}{\ln \left ( F \right ) \left ( md+fn \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1545, size = 88, normalized size = 2.44 \begin{align*} \frac{{\left (F^{e}\right )}^{n} a^{m} b^{n} e^{\left (m \log \left (F^{d x + c}\right ) + n \log \left ({\left (F^{d x + c}\right )}^{\frac{f}{d}}\right )\right )}}{{\left (d m + f n\right )}{\left (F^{\frac{c f}{d}}\right )}^{n} \log \left (F\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.913505, size = 124, normalized size = 3.44 \begin{align*} \frac{e^{\left ({\left (d m x + c m\right )} \log \left (F\right ) +{\left (f n x + e n\right )} \log \left (F\right ) + m \log \left (a\right ) + n \log \left (b\right )\right )}}{{\left (d m + f n\right )} \log \left (F\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 65.8399, size = 143, normalized size = 3.97 \begin{align*} \begin{cases} a^{m} b^{n} x & \text{for}\: F = 1 \wedge \left (F = 1 \vee d = - \frac{f n}{m}\right ) \\a^{m} b^{n} x \left (F^{c}\right )^{m} \left (F^{e}\right )^{n} \left (F^{f x}\right )^{n} \left (F^{- \frac{f n x}{m}}\right )^{m} + \frac{a^{m} b^{n} \left (F^{c}\right )^{m} \left (F^{e}\right )^{n} \left (F^{f x}\right )^{n} \left (F^{- \frac{f n x}{m}}\right )^{m}}{f n \log{\left (F \right )}} & \text{for}\: d = - \frac{f n}{m} \\\frac{a^{m} b^{n} \left (F^{c}\right )^{m} \left (F^{e}\right )^{n} \left (F^{d x}\right )^{m} \left (F^{f x}\right )^{n}}{d m \log{\left (F \right )} + f n \log{\left (F \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.68147, size = 63, normalized size = 1.75 \begin{align*} \frac{e^{\left (d m x \log \left (F\right ) + f n x \log \left (F\right ) + c m \log \left (F\right ) + n e \log \left (F\right ) + m \log \left (a\right ) + n \log \left (b\right )\right )}}{d m \log \left (F\right ) + f n \log \left (F\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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