Integrate the following exponential function?

Questions
1

Consider the following code:

Integrate[
         Exp[-u]*Sqrt[
           u*(u - a)]*(-2*Power[b, 3]/(4*u*b + Power[a - b, 2])), {u, 0, 
          Infinity}]

I am a beginner. Since I want to solve the above integration, but I don not know the reason why Mathematica (ver.12) did not display the symbolic result which I want to do.
So I hope that have any idea to solve this integration.
Thank you very much.

2

I am not an expert in calculus, but I doubt if this integral can be expressed in any terms except of the input integral for arbitrary a and b.

For instance, if one substitutes a = - 1 and b = 1 he / she will get HypergeometricU in the result. Using a = 0 instead leads to ExpIntegralEi. And setting a to 1 results in unevaluated integral.

In any case, if a and b are not arbitrary I suggest using Assumptions option inside the Integrate function.

3

Yes, thank you very much for your helpful information. In my case, a and b are not arbitrary.