Optimal. Leaf size=73 \[ \frac{\left ((d+e x)^3\right )^{-m} F^{c \left (a-\frac{b d}{e}\right )} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{3 m} \text{Gamma}\left (1-3 m,-\frac{b c \log (F) (d+e x)}{e}\right )}{b c \log (F)} \]
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Rubi [A] time = 0.0597445, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.049, Rules used = {2188, 2181} \[ \frac{\left ((d+e x)^3\right )^{-m} F^{c \left (a-\frac{b d}{e}\right )} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{3 m} \text{Gamma}\left (1-3 m,-\frac{b c \log (F) (d+e x)}{e}\right )}{b c \log (F)} \]
Antiderivative was successfully verified.
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Rule 2188
Rule 2181
Rubi steps
\begin{align*} \int F^{c (a+b x)} \left (d^3+3 d^2 e x+3 d e^2 x^2+e^3 x^3\right )^{-m} \, dx &=(d+e x)^{3 m} \left ((d+e x)^3\right )^{-m} \int F^{c (a+b x)} (d+e x)^{-3 m} \, dx\\ &=\frac{F^{c \left (a-\frac{b d}{e}\right )} \left ((d+e x)^3\right )^{-m} \Gamma \left (1-3 m,-\frac{b c (d+e x) \log (F)}{e}\right ) \left (-\frac{b c (d+e x) \log (F)}{e}\right )^{3 m}}{b c \log (F)}\\ \end{align*}
Mathematica [A] time = 0.0108804, size = 73, normalized size = 1. \[ \frac{\left ((d+e x)^3\right )^{-m} F^{c \left (a-\frac{b d}{e}\right )} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{3 m} \text{Gamma}\left (1-3 m,-\frac{b c \log (F) (d+e x)}{e}\right )}{b c \log (F)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.075, size = 0, normalized size = 0. \begin{align*} \int{\frac{{F}^{c \left ( bx+a \right ) }}{ \left ({e}^{3}{x}^{3}+3\,d{e}^{2}{x}^{2}+3\,{d}^{2}ex+{d}^{3} \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 \frac{F^{{\left (b x + a\right )} c}}{{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )}^{m}}\,{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 (\frac{F^{b c x + a c}}{{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )}^{m}}, 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 \frac{F^{{\left (b x + a\right )} c}}{{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )}^{m}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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