I want to calculate the following integral $$ \int_0^{\infty}\int_0^{\infty}\frac{f(r_1,r_2)}{\lvert r_1-r_2\rvert}\; r_1^2 r_2^2\; dr_1 dr_2 \tag{1} $$ also I intend to use the following change variables $$ r=\lvert r_1-r_2\rvert \tag{2} $$ and $$ R=\frac{m_1r_1+m_2r_2}{m_1+m_2} \tag{3} $$
to transform $(1)$ using $(2)$ and $(3)$, assuming we know the form of $f(r,R)$ as $$ f(r,R)=\text{exp}(-ar-br^2) $$ where $a$ and $b$ are positive parameters. Well we know from $(2)$ and $(3)$ that $$ r_1=R-\frac{m_2 r}{(m_1+m_2)} \tag{4} $$ and $$ r_2=R-\frac{m_1 r}{(m_1+m_2)} \tag{5} $$ Should I simply replace $r_1^2$ and $r_2^2$ by $(4)$ and $(5)$? and what about differentials $dr_1$ and $dr_2$?