Dirichlet-Gamma-Poisson mixture distribution, or something like that?

[maxbiostat]

@martinmodrak

Anyway, I can recommend Wolfram alpha (or Wolfram Cloud for more advanced uses) for solving integrals. Even the free versions can definitely solve integrals I am completely unable to crack :-). I can however see how you could (if Q is large-ish) assume that Q is roughly gamma distributed and treat it as additional parameter in the model…

wxMaxima is your friend also. For a nice frontend, see this. I don’t do any integration/differentiation by hand if I can avoid it. This is not ~just~ out of lazyness: it also ensures fewer errors propagate through calculations.