Integrating an exponential with upper incomplete gamma functions

Question

I would greatly appreciate calculating an integral consisting of an upper incomplete gamma function and an exponential function.

Integrate[Exp[-L y] Gamma[t, (Sqrt[z (y + w)/a])/n], {y, 0, Infinity}]

I would like to find it in an exact non-numerical closed-form.

Answer 1

This is just an additional comment to @Artes answer. (All the real work was provided in that answer.)

The term

(Sqrt[z (y + w)/a])/n

can be simplified without loss of generality with

c Sqrt[y + w]

So instead of 3 parameters ($z$, $a$, and $n$), you only need 1. Doing so gets you explicit solutions for all positive integer values of $t$. For example:

t = 3;
Integrate[Exp[-L y] Gamma[t, c Sqrt[y + w]], {y, 0, Infinity}, 
  Assumptions -> L > 0 && w > 0 && c > 0] // FunctionExpand

One might see patterns to end up with a single function for any positive integer value of $t$.

Answer 2

Too many symbolic constants usually turn out to be obstructive to calculate integrals symbolically. Another problem comes up when no restriction for parameters is given. We can set specific values to a few constants using With and (or) restrict constants with an option Assumptions in Integrate e.g.

With[{n = 1, a = 1, z = 1}, 
     Table[{t, Integrate[ Exp[-L y] Gamma[t, (Sqrt[z (y + w)/a])/n],
                          {y, 0, Infinity}, 
                          Assumptions -> L > 0 && w > 0]},
            {t, 4}]]

We could also restrict appropriately all constant but one, however then Mathematica cannot provide symbolic results for certain values of t, e.g. for t = 3 the integral is not found

Table[{t, Integrate[ Exp[-L y] Gamma[t, (Sqrt[z (y + w)/a])/n],
                     {y, 0, Infinity}, 
                     Assumptions -> L > 0 && n > 0 && a > 0 && z > 0 && w > 0]},
      {t, 4}]

Perhaps next versions of the system should provide more general results.