Optimal. Leaf size=73 \[ \frac{\left ((d+e x)^2\right )^{-m} F^{c \left (a-\frac{b d}{e}\right )} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{2 m} \text{Gamma}\left (1-2 m,-\frac{b c \log (F) (d+e x)}{e}\right )}{b c \log (F)} \]
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Rubi [A] time = 0.0566247, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {2188, 2181} \[ \frac{\left ((d+e x)^2\right )^{-m} F^{c \left (a-\frac{b d}{e}\right )} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{2 m} \text{Gamma}\left (1-2 m,-\frac{b c \log (F) (d+e x)}{e}\right )}{b c \log (F)} \]
Antiderivative was successfully verified.
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Rule 2188
Rule 2181
Rubi steps
\begin{align*} \int F^{c (a+b x)} \left (d^2+2 d e x+e^2 x^2\right )^{-m} \, dx &=(d+e x)^{2 m} \left ((d+e x)^2\right )^{-m} \int F^{c (a+b x)} (d+e x)^{-2 m} \, dx\\ &=\frac{F^{c \left (a-\frac{b d}{e}\right )} \left ((d+e x)^2\right )^{-m} \Gamma \left (1-2 m,-\frac{b c (d+e x) \log (F)}{e}\right ) \left (-\frac{b c (d+e x) \log (F)}{e}\right )^{2 m}}{b c \log (F)}\\ \end{align*}
Mathematica [A] time = 0.0107577, size = 73, normalized size = 1. \[ \frac{\left ((d+e x)^2\right )^{-m} F^{c \left (a-\frac{b d}{e}\right )} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{2 m} \text{Gamma}\left (1-2 m,-\frac{b c \log (F) (d+e x)}{e}\right )}{b c \log (F)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.099, size = 0, normalized size = 0. \begin{align*} \int{\frac{{F}^{c \left ( bx+a \right ) }}{ \left ({e}^{2}{x}^{2}+2\,dex+{d}^{2} \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^{2} x^{2} + 2 \, d e x + d^{2}\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^{2} x^{2} + 2 \, d e x + d^{2}\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^{2} x^{2} + 2 \, d e x + d^{2}\right )}^{m}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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