Any hints on solving an integration of the following form,
$$\int_{x}^{+\infty}\left(1-\frac{1}{1+sy^{-1}}\right) \left(\text{exp}(-\sqrt{y})+ y^{-\frac{1}{2}}(1-\text{exp}(-\sqrt[4]y)\right)dy $$
This arises pretty much in Poisson point processes after using the probability generating function property.
I know the above looks pretty much hopeless, I tried using Mathematica and Wolfram Alpha, both don't really help much.
I also have $x>0$ and $s>0$ real numbers if that helps.
I tried the following Wolfram|Alpha input and it gives a result in terms of $i$ imaginary, I don't understand why.
Also is there any valid approximation for which I can say that this integral can be approximately upper and lower bound by? Thanks in advance for any help.