# Eulerian type of integral

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.

• Do you have any reason to think there is a closed form? Most integrals don't have one. Maybe the best you can do is numerical methods. – Gerry Myerson Aug 15 at 2:03
• Also $1-e^{-t}-2te^{-t} = -t+O(t^2)$ so it doesn't converge when $a \ne 0$ – reuns Aug 15 at 2:04
• No, I do not have any good reason for a closed form. Of course if there is one that will be great. Thank you. – George Aug 15 at 2:07
• @reuns thank you. Oh... – George Aug 15 at 2:41
• Even with $a=0$, I am quite skeptical about a possible closed form (even using special functions). By the way, Welcome to the site ! – Claude Leibovici Aug 15 at 3:00