Following Michael Renardy's suggestion, write $g=\phi\psi$, and write $$ g(x)=g(0)+x\cdot G(x) , $$ where $G$ is a smooth function. Then the integral becomes $$ \int fg = g(0)\int f + \sum_i\int \frac{x_i}{|x|}G_i(x)d^3x . $$ The first integral can be computed analytically, and the integrands in the latter integrals are bounded.

Following Michael Renardy's suggestion, write $g=\phi\psi$, and write $$ g(x)=g(0)+x\cdot G(x) , $$ where $G$ is a smooth function. Then the integral becomes $$ \int_\Omega fg = g(0)\int_\Omega f + \sum_i\int_\Omega \frac{x_i}{|x|}G_i(x)d^3x . $$ The integrands in the latter integrals are bounded, and the first integral can be written as $$ \int_\Omega f = \int_{B_r}f + \int_{\Omega\setminus B_r}f , $$ where the first can be computed analytically, and the second has a smooth integrand. Here $B_r$ is some ball centred at the origin.