I don't think that's going to work.
What you have is a special case of the Beta function. Try looking at the Wiki page on that; it shows how the beta function relates to the gamma function.
I have a different method, but I don't know if it will help you.
Here we go
The convolution of two functions is defined to be
using this
now sub
Now taking the Laplace transform of both sides gives
Now solving for the integral gives
In...
Beta Function -- from Wolfram MathWorld
... You can verify that...
(1)
Merry Christmas from Italy