Conjugate prior for the gamma distribution (with parameters $\displaystyle \alpha,\beta$) is supposed to be:

$\displaystyle \frac{1}{Z}\frac{p^{\alpha -1}e^{-\beta q}}{{\gamma(\alpha)}^r \beta^{-\alpha s}}$

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 $\displaystyle \alpha$ and $\displaystyle \beta$. And I have a conjugate prior with hyperparemeters p,q,r and s. Can anybody help me solving the integration over the parameters $\displaystyle \alpha,\beta$?

$\displaystyle \int_{a,b} \frac{1}{\gamma(\alpha)\beta^\alpha} x^{\alpha-1} e^{-x/\beta} \frac{1}{Z}\frac{p^{\alpha -1}e^{-\beta q}}{{\gamma(\alpha)}^r \beta^{-\alpha s}} d{a,b}$

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!