Conjugate prior for the gamma distribution (with parameters ) is supposed to be:

with the hyperparameters, p, q, r, s when both shape parameter and the scale parameter are known.

Here I have a gamma distributed variable x, with parameters and . And I have a conjugate prior with hyperparemeters p,q,r and s. Can anybody help me solving the integration over the parameters ?

It has been a long time since I took a calculus class in the university, so I don't remember much about integration solving. Any assistance will be appreciated!