Optimal. Leaf size=73 \[ \frac {F^{c \left (a-\frac {b d}{e}\right )} \left ((d+e x)^2\right )^{-m} \Gamma \left (1-2 m,-\frac {b c (d+e x) \log (F)}{e}\right ) \left (-\frac {b c (d+e x) \log (F)}{e}\right )^{2 m}}{b c \log (F)} \]
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Rubi [A]
time = 0.04, antiderivative size = 73, normalized size of antiderivative = 1.00, 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 = {2219, 2212}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2212
Rule 2219
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*}
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Mathematica [A]
time = 0.03, size = 73, normalized size = 1.00 \begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int F^{c \left (b x +a \right )} \left (e^{2} x^{2}+2 d e x +d^{2}\right )^{-m}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {F^{c\,\left (a+b\,x\right )}}{{\left (d^2+2\,d\,e\,x+e^2\,x^2\right )}^m} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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