Optimal. Leaf size=139 \[ -\frac{e^{i d} F^{a c} (f x)^m (-x (b c \log (F)+i e))^{-m} \text{Gamma}(m+1,-x (b c \log (F)+i e))}{2 (e-i b c \log (F))}-\frac{e^{-i d} F^{a c} (f x)^m (x (-b c \log (F)+i e))^{-m} \text{Gamma}(m+1,x (-b c \log (F)+i e))}{2 (e+i b c \log (F))} \]
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Rubi [F] time = 0.490977, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int F^{c (a+b x)} (f x)^m \sin (d+e x) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int F^{c (a+b x)} (f x)^m \sin (d+e x) \, dx &=\int F^{a c+b c x} (f x)^m \sin (d+e x) \, dx\\ \end{align*}
Mathematica [A] time = 0.578081, size = 143, normalized size = 1.03 \[ \frac{1}{2} F^{a c} (f x)^m (-x (b c \log (F)+i e))^{-m} \left (-i x (\cos (d)-i \sin (d)) (-b c x \log (F)-i e x)^m (i x (e+i b c \log (F)))^{-m-1} \text{Gamma}(m+1,-b c x \log (F)+i e x)-\frac{(\cos (d)+i \sin (d)) \text{Gamma}(m+1,-x (b c \log (F)+i e))}{e-i b c \log (F)}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.366, size = 0, normalized size = 0. \begin{align*} \int{F}^{c \left ( bx+a \right ) } \left ( fx \right ) ^{m}\sin \left ( ex+d \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (f x\right )^{m} F^{{\left (b x + a\right )} c} \sin \left (e x + d\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.519904, size = 336, normalized size = 2.42 \begin{align*} \frac{{\left (i \, b c \log \left (F\right ) - e\right )} e^{\left (a c \log \left (F\right ) - m \log \left (-\frac{b c \log \left (F\right ) - i \, e}{f}\right ) - i \, d\right )} \Gamma \left (m + 1, -b c x \log \left (F\right ) + i \, e x\right ) +{\left (-i \, b c \log \left (F\right ) - e\right )} e^{\left (a c \log \left (F\right ) - m \log \left (-\frac{b c \log \left (F\right ) + i \, e}{f}\right ) + i \, d\right )} \Gamma \left (m + 1, -b c x \log \left (F\right ) - i \, e x\right )}{2 \,{\left (b^{2} c^{2} \log \left (F\right )^{2} + e^{2}\right )}} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (f x\right )^{m} F^{{\left (b x + a\right )} c} \sin \left (e x + d\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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