Could anyone give me some light on how to evaluate $$ \int_{0}^{1} \frac{\sqrt{1-t}\sqrt{a+t}}{t^2}\left(1-e^{-t}-2te^{-t} \right)dt$$ where $a$ is a positive real constant.
I tried several different things including contour integral, integration by parts, and expanding the square root $\sqrt{a+t}$ with the hope that the resultant series may eventually add up to some closed form functions. None of these efforts worked. Thank you for your inputs.