I am trying to find the following integral
\begin{equation} \int_{0}^{\infty}\left(1+\kappa\text{e}^{\left(b-c\right)t}\right)^{-\frac{b}{b-c}}e^{-at}dt \end{equation}
I would like to know whether it is correct to start integrating the following term without integral bounds.
$$\int\left(1+\kappa\text{e}^{\left(b-c\right)t}\right)^{-\frac{b}{b-c}}e^{-at}dt$$
For ths last equation, I find that the solution is
$$-\frac{e^{-\rho t}{_{2}F_{1}}\left[\frac{b}{b-c},-\frac{\rho}{b-c},-\frac{c-b+\rho}{b-c},e^{\left(b-c\right)t}\kappa\right]}{\rho}$$
After finding this solution is it correct to put
$$\left[-\frac{e^{-\rho t}{_{2}F_{1}}\left[\frac{b}{b-c},-\frac{\rho}{b-c},-\frac{c-b+\rho}{b-c},e^{\left(b-c\right)t}\kappa\right]}{\rho}\right]_{t=0}^{t=\infty}=\frac{{_{2}F_{1}}\left[\frac{b}{b-c},-\frac{\rho}{b-c},-\frac{c-b+\rho}{b-c},\kappa\right]}{\rho}$$
All parameters $\rho$, $\kappa$, $a$, $b$, $c$ are constant and positive terms.
If not, how should I proceed?