Integrand size = 30, antiderivative size = 73 \[ \int F^{c (a+b x)} \left (d^2+2 d e x+e^2 x^2\right )^{-m} \, dx=\frac {F^{c \left (a-\frac {b d}{e}\right )} \left ((d+e x)^2\right )^{-m} \Gamma \left (1-2 m,-\frac {b c (d+e x) \log (F)}{e}\right ) \left (-\frac {b c (d+e x) \log (F)}{e}\right )^{2 m}}{b c \log (F)} \]
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Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2219, 2212} \[ \int F^{c (a+b x)} \left (d^2+2 d e x+e^2 x^2\right )^{-m} \, dx=\frac {\left ((d+e x)^2\right )^{-m} F^{c \left (a-\frac {b d}{e}\right )} \left (-\frac {b c \log (F) (d+e x)}{e}\right )^{2 m} \Gamma \left (1-2 m,-\frac {b c (d+e x) \log (F)}{e}\right )}{b c \log (F)} \]
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Rule 2212
Rule 2219
Rubi steps \begin{align*} \text {integral}& = (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*}
Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00 \[ \int F^{c (a+b x)} \left (d^2+2 d e x+e^2 x^2\right )^{-m} \, dx=\frac {F^{c \left (a-\frac {b d}{e}\right )} \left ((d+e x)^2\right )^{-m} \Gamma \left (1-2 m,-\frac {b c (d+e x) \log (F)}{e}\right ) \left (-\frac {b c (d+e x) \log (F)}{e}\right )^{2 m}}{b c \log (F)} \]
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\[\int F^{c \left (b x +a \right )} \left (e^{2} x^{2}+2 d e x +d^{2}\right )^{-m}d x\]
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\[ \int F^{c (a+b x)} \left (d^2+2 d e x+e^2 x^2\right )^{-m} \, dx=\int { \frac {F^{{\left (b x + a\right )} c}}{{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )}^{m}} \,d x } \]
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Exception generated. \[ \int F^{c (a+b x)} \left (d^2+2 d e x+e^2 x^2\right )^{-m} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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\[ \int F^{c (a+b x)} \left (d^2+2 d e x+e^2 x^2\right )^{-m} \, dx=\int { \frac {F^{{\left (b x + a\right )} c}}{{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )}^{m}} \,d x } \]
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\[ \int F^{c (a+b x)} \left (d^2+2 d e x+e^2 x^2\right )^{-m} \, dx=\int { \frac {F^{{\left (b x + a\right )} c}}{{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )}^{m}} \,d x } \]
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Timed out. \[ \int F^{c (a+b x)} \left (d^2+2 d e x+e^2 x^2\right )^{-m} \, dx=\int \frac {F^{c\,\left (a+b\,x\right )}}{{\left (d^2+2\,d\,e\,x+e^2\,x^2\right )}^m} \,d x \]
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