What is the method for computing the double Fourier transform in 2 variables?

[zetafunction]

Homework Statement

compute the Fourier transform in 2 variables $$\iint_{R^{2}}dxdy\frac{x^{2}y}{1+x+y}exp(iax+iby)$$

Homework Equations

$$\iint_{R^{2}}\frac{x^{2}y}{1+x+y}exp(iax+iby)$$

The Attempt at a Solution

i have tried by FIRST substractin a polynomial on variable 'x' and considering 'y' to constant in order to get a finite expression for the integral over 'x' , then i do the same over 'y'

 

[lippyka]

by substracting a polynomial on variable 'y' and considering 'x' to constant but this method gives very long expression and i could not solve it....