Optimal. Leaf size=69 \[ \frac {(d+e x)^{-m} F^{c \left (a-\frac {b d}{e}\right )} \left (-\frac {b c \log (F) (d+e x)}{e}\right )^m \Gamma \left (1-m,-\frac {b c (d+e x) \log (F)}{e}\right )}{b c \log (F)} \]
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Rubi [A] time = 0.02, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2181} \[ \frac {(d+e x)^{-m} F^{c \left (a-\frac {b d}{e}\right )} \left (-\frac {b c \log (F) (d+e x)}{e}\right )^m \text {Gamma}\left (1-m,-\frac {b c \log (F) (d+e x)}{e}\right )}{b c \log (F)} \]
Antiderivative was successfully verified.
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Rule 2181
Rubi steps
\begin {align*} \int F^{c (a+b x)} (d+e x)^{-m} \, dx &=\frac {F^{c \left (a-\frac {b d}{e}\right )} (d+e x)^{-m} \Gamma \left (1-m,-\frac {b c (d+e x) \log (F)}{e}\right ) \left (-\frac {b c (d+e x) \log (F)}{e}\right )^m}{b c \log (F)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 69, normalized size = 1.00 \[ \frac {(d+e x)^{-m} F^{c \left (a-\frac {b d}{e}\right )} \left (-\frac {b c \log (F) (d+e x)}{e}\right )^m \Gamma \left (1-m,-\frac {b c (d+e x) \log (F)}{e}\right )}{b c \log (F)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 67, normalized size = 0.97 \[ \frac {e^{\left (\frac {e m \log \left (-\frac {b c \log \relax (F)}{e}\right ) - {\left (b c d - a c e\right )} \log \relax (F)}{e}\right )} \Gamma \left (-m + 1, -\frac {{\left (b c e x + b c d\right )} \log \relax (F)}{e}\right )}{b c \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{{\left (b x + a\right )} c}}{{\left (e x + d\right )}^{m}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int F^{\left (b x +a \right ) c} \left (e x +d \right )^{-m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{{\left (b x + a\right )} c}}{{\left (e x + d\right )}^{m}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {F^{c\,\left (a+b\,x\right )}}{{\left (d+e\,x\right )}^m} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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