Consider functions of the form
$$ f(x)= e^{- x^2/\sigma^2} \frac{a_0 + a_1 x + \cdots +a_n x^n }{b_0+ b_1 x + \cdots + b_m x^m } $$
Under which circumstances do these functions have closed forms for their Fourier transforms? Assume for simplicity that the polynomial $b_0+ b_1 x + \cdots + b_m x^m$ has no real roots.
Which forms are known?