Integrand size = 25, antiderivative size = 228 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^m \, dx=\frac {a^2 (c+d x)^{1+m}}{d (1+m)}+\frac {2^{-1-m} b^2 F^{2 \left (e-\frac {c f}{d}\right ) g n-2 g n (e+f x)} \left (F^{e g+f g x}\right )^{2 n} (c+d x)^m \Gamma \left (1+m,-\frac {2 f g n (c+d x) \log (F)}{d}\right ) \left (-\frac {f g n (c+d x) \log (F)}{d}\right )^{-m}}{f g n \log (F)}+\frac {2 a b F^{\left (e-\frac {c f}{d}\right ) g n-g n (e+f x)} \left (F^{e g+f g x}\right )^n (c+d x)^m \Gamma \left (1+m,-\frac {f g n (c+d x) \log (F)}{d}\right ) \left (-\frac {f g n (c+d x) \log (F)}{d}\right )^{-m}}{f g n \log (F)} \]
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Time = 0.18 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2214, 2213, 2212} \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^m \, dx=\frac {a^2 (c+d x)^{m+1}}{d (m+1)}+\frac {2 a b (c+d x)^m \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac {c f}{d}\right )-g n (e+f x)} \left (-\frac {f g n \log (F) (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {f g n (c+d x) \log (F)}{d}\right )}{f g n \log (F)}+\frac {b^2 2^{-m-1} (c+d x)^m \left (F^{e g+f g x}\right )^{2 n} F^{2 g n \left (e-\frac {c f}{d}\right )-2 g n (e+f x)} \left (-\frac {f g n \log (F) (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {2 f g n (c+d x) \log (F)}{d}\right )}{f g n \log (F)} \]
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Rule 2212
Rule 2213
Rule 2214
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 (c+d x)^m+2 a b \left (F^{e g+f g x}\right )^n (c+d x)^m+b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^m\right ) \, dx \\ & = \frac {a^2 (c+d x)^{1+m}}{d (1+m)}+(2 a b) \int \left (F^{e g+f g x}\right )^n (c+d x)^m \, dx+b^2 \int \left (F^{e g+f g x}\right )^{2 n} (c+d x)^m \, dx \\ & = \frac {a^2 (c+d x)^{1+m}}{d (1+m)}+\left (2 a b F^{-n (e g+f g x)} \left (F^{e g+f g x}\right )^n\right ) \int F^{n (e g+f g x)} (c+d x)^m \, dx+\left (b^2 F^{-2 n (e g+f g x)} \left (F^{e g+f g x}\right )^{2 n}\right ) \int F^{2 n (e g+f g x)} (c+d x)^m \, dx \\ & = \frac {a^2 (c+d x)^{1+m}}{d (1+m)}+\frac {2^{-1-m} b^2 F^{2 \left (e-\frac {c f}{d}\right ) g n-2 g n (e+f x)} \left (F^{e g+f g x}\right )^{2 n} (c+d x)^m \Gamma \left (1+m,-\frac {2 f g n (c+d x) \log (F)}{d}\right ) \left (-\frac {f g n (c+d x) \log (F)}{d}\right )^{-m}}{f g n \log (F)}+\frac {2 a b F^{\left (e-\frac {c f}{d}\right ) g n-g n (e+f x)} \left (F^{e g+f g x}\right )^n (c+d x)^m \Gamma \left (1+m,-\frac {f g n (c+d x) \log (F)}{d}\right ) \left (-\frac {f g n (c+d x) \log (F)}{d}\right )^{-m}}{f g n \log (F)} \\ \end{align*}
\[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^m \, dx=\int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^m \, dx \]
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\[\int {\left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right )}^{2} \left (d x +c \right )^{m}d x\]
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Time = 0.08 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.84 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^m \, dx=\frac {4 \, {\left (a b d m + a b d\right )} e^{\left (\frac {{\left (d e - c f\right )} g n \log \left (F\right ) - d m \log \left (-\frac {f g n \log \left (F\right )}{d}\right )}{d}\right )} \Gamma \left (m + 1, -\frac {{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) + {\left (b^{2} d m + b^{2} d\right )} e^{\left (\frac {2 \, {\left (d e - c f\right )} g n \log \left (F\right ) - d m \log \left (-\frac {2 \, f g n \log \left (F\right )}{d}\right )}{d}\right )} \Gamma \left (m + 1, -\frac {2 \, {\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) + 2 \, {\left (a^{2} d f g n x + a^{2} c f g n\right )} {\left (d x + c\right )}^{m} \log \left (F\right )}{2 \, {\left (d f g m + d f g\right )} n \log \left (F\right )} \]
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\[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^m \, dx=\int \left (a + b \left (F^{e g + f g x}\right )^{n}\right )^{2} \left (c + d x\right )^{m}\, dx \]
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\[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^m \, dx=\int { {\left ({\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a\right )}^{2} {\left (d x + c\right )}^{m} \,d x } \]
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\[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^m \, dx=\int { {\left ({\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a\right )}^{2} {\left (d x + c\right )}^{m} \,d x } \]
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Timed out. \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^m \, dx=\int {\left (a+b\,{\left (F^{g\,\left (e+f\,x\right )}\right )}^n\right )}^2\,{\left (c+d\,x\right )}^m \,d x \]
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