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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}]

enter image description here

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

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  • $\begingroup$ What happened when you executed the code you give? Have you tried NIntegrate with specific values of $L$, $t$, $z$, $w$, $a$, and $n$? $\endgroup$
    – JimB
    Jun 13 at 23:25
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    $\begingroup$ You should explain where this integral comes from and why it is related to probability-or-statistics. More details regarding constants is crucial. $\endgroup$
    – Artes
    Jun 14 at 0:25
  • $\begingroup$ Welcome to Mathematica.SE! I hope you will become a regular contributor. To get started, 1) take the introductory tour now, 2) when you see good questions and answers, vote them up by clicking the gray triangles, because the credibility of the system is based on the reputation gained by users sharing their knowledge, 3) remember to accept the answer, if any, that solves your problem, by clicking the checkmark sign, and 4) give help too, by answering questions in your areas of expertise. $\endgroup$
    – bbgodfrey
    Jun 14 at 0:51
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    $\begingroup$ Do you have good reason to believe that a closed-form solution actually exists? $\endgroup$
    – bbgodfrey
    Jun 14 at 0:52
  • $\begingroup$ Crossposted here. $\endgroup$ Jun 14 at 2:18
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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

Integral for t = 3

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

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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}]]

enter image description here

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}]

enter image description here

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

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