Optimal. Leaf size=71 \[ \frac{\left ((d+e x)^4\right )^m F^{c \left (a-\frac{b d}{e}\right )} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{-4 m} \text{Gamma}\left (4 m+1,-\frac{b c \log (F) (d+e x)}{e}\right )}{b c \log (F)} \]
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Rubi [A] time = 0.0699975, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 50, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {2188, 2181} \[ \frac{\left ((d+e x)^4\right )^m F^{c \left (a-\frac{b d}{e}\right )} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{-4 m} \text{Gamma}\left (4 m+1,-\frac{b c \log (F) (d+e x)}{e}\right )}{b c \log (F)} \]
Antiderivative was successfully verified.
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Rule 2188
Rule 2181
Rubi steps
\begin{align*} \int F^{c (a+b x)} \left (d^4+4 d^3 e x+6 d^2 e^2 x^2+4 d e^3 x^3+e^4 x^4\right )^m \, dx &=(d+e x)^{-4 m} \left ((d+e x)^4\right )^m \int F^{c (a+b x)} (d+e x)^{4 m} \, dx\\ &=\frac{F^{c \left (a-\frac{b d}{e}\right )} \left ((d+e x)^4\right )^m \Gamma \left (1+4 m,-\frac{b c (d+e x) \log (F)}{e}\right ) \left (-\frac{b c (d+e x) \log (F)}{e}\right )^{-4 m}}{b c \log (F)}\\ \end{align*}
Mathematica [A] time = 0.0147353, size = 71, normalized size = 1. \[ \frac{\left ((d+e x)^4\right )^m F^{c \left (a-\frac{b d}{e}\right )} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{-4 m} \text{Gamma}\left (4 m+1,-\frac{b c \log (F) (d+e x)}{e}\right )}{b c \log (F)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.158, size = 0, normalized size = 0. \begin{align*} \int{F}^{c \left ( bx+a \right ) } \left ({e}^{4}{x}^{4}+4\,d{e}^{3}{x}^{3}+6\,{d}^{2}{e}^{2}{x}^{2}+4\,{d}^{3}ex+{d}^{4} \right ) ^{m}\, 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 (e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}\right )}^{m} F^{{\left (b x + a\right )} c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}\right )}^{m} F^{b c x + a c}, x\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 (e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}\right )}^{m} F^{{\left (b x + a\right )} c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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