Transformations of random variables

I try to derive the probability density function (PDF) of the random variable $$S=\frac{ AX+BY+CZ+D }{ EX+FY+GZ+H }.$$ We know that the random variables $$X, Y, Z$$ are independent and each of them satisfies Gaussian distribution. I do not know how to derive the PDF of $$S$$. Any kind of help would be appreciated.

If $$X,Y,Z$$ are independent and Gaussian, then $$P = AX+BY+CZ+D$$ and $$Q = EX+FY+GZ+H$$ are both Gaussian, but not (necessarily) independent. $$(P,Q)$$ will jointly follow a bivariate normal distribution.