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 = 0.10, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, 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 2194
Rule 2227
Rule 2281
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*}
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Mathematica [A] time = 0.05, size = 36, normalized size = 1.00 \[ \frac {\left (a F^{c+d x}\right )^m \left (b F^{e+f x}\right )^n}{d m \log (F)+f n \log (F)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 46, normalized size = 1.28 \[ \frac {e^{\left ({\left (d m x + c m\right )} \log \relax (F) + {\left (f n x + e n\right )} \log \relax (F) + m \log \relax (a) + n \log \relax (b)\right )}}{{\left (d m + f n\right )} \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.73, size = 47, normalized size = 1.31 \[ \frac {e^{\left (d m x \log \relax (F) + f n x \log \relax (F) + c m \log \relax (F) + n e \log \relax (F) + m \log \relax (a) + n \log \relax (b)\right )}}{d m \log \relax (F) + f n \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 37, normalized size = 1.03 \[ \frac {\left (a \,F^{d x +c}\right )^{m} \left (b \,F^{f x +e}\right )^{n}}{\left (m d +f n \right ) \ln \relax (F )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 65, normalized size = 1.81 \[ \frac {{\left (F^{e}\right )}^{n} a^{m} b^{n} e^{\left (m \log \left (F^{d x + c}\right ) + n \log \left ({\left (F^{d x + c}\right )}^{\frac {f}{d}}\right )\right )}}{{\left (d m + f n\right )} {\left (F^{\frac {c f}{d}}\right )}^{n} \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int {\left (F^{c+d\,x}\,a\right )}^m\,{\left (F^{e+f\,x}\,b\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 43.31, size = 143, normalized size = 3.97 \[ \begin {cases} a^{m} b^{n} x & \text {for}\: F = 1 \wedge \left (F = 1 \vee d = - \frac {f n}{m}\right ) \\a^{m} b^{n} x \left (F^{c}\right )^{m} \left (F^{e}\right )^{n} \left (F^{f x}\right )^{n} \left (F^{- \frac {f n x}{m}}\right )^{m} + \frac {a^{m} b^{n} \left (F^{c}\right )^{m} \left (F^{e}\right )^{n} \left (F^{f x}\right )^{n} \left (F^{- \frac {f n x}{m}}\right )^{m}}{f n \log {\relax (F )}} & \text {for}\: d = - \frac {f n}{m} \\\frac {a^{m} b^{n} \left (F^{c}\right )^{m} \left (F^{e}\right )^{n} \left (F^{d x}\right )^{m} \left (F^{f x}\right )^{n}}{d m \log {\relax (F )} + f n \log {\relax (F )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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