Simplifying an integral involving Gaussian PDF

Question

Let $\phi(x)$ be the standard Gaussian probability density function and $1<Y<2$.Consider the integral $$ \int_{x=0}^\infty \int_{y=0}^\infty \left(\phi(x+y)(x+y)-x\phi(x)-y\phi(y)\right)(xy)^{-Y} \, dy \, dx $$ Can it be simplified, i.e. either written as a single integral, or as the expected value of some function of Gaussian random variables?