Integrating an exponential with upper incomplete gamma functions

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.

• What happened when you executed the code you give? Have you tried NIntegrate with specific values of $L$, $t$, $z$, $w$, $a$, and $n$? – JimB Jun 13 at 23:25
• You should explain where this integral comes from and why it is related to probability-or-statistics. More details regarding constants is crucial. – Artes Jun 14 at 0:25
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• Do you have good reason to believe that a closed-form solution actually exists? – bbgodfrey Jun 14 at 0:52
• Crossposted here. – Rohit Namjoshi Jun 14 at 2:18

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 $$t$$.

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.