A probability vector $(X,Y)$ follows a distribution with PDF $$f(x,y)=C\exp(-\alpha \sqrt{x^2+y^2+2 \beta xy} )\quad -\infty \lt x,y \lt\infty$$
$\alpha, \beta, C$ are constants and $\alpha \gt0, -1\lt\beta\lt1$
I need to express $C$ using $\alpha, \beta$.
I set $$I = \displaystyle \int_{-\infty}^{ \infty } \displaystyle \int_{-\infty}^{ \infty } C\exp(-\alpha \sqrt{x^2+y^2+2 \beta xy} ) \,dx\,dy = 1$$
and but don't know how to tackle this integral. I first thought polar conversion may be a tool but could not deal with $2\beta xy$.