We rewrite it in a more compact form as $$\begin{equation*} \int f(x_1,\dots,x_d)F(x_1,\dots,x_d)\prod_{i=1}^6dx_i, \end{equation*}$$ where $f$ is the above normal distribution and $$\begin{equation*} F(x_1,\dots,x_6)=F(\mathbf{x, y})=(\mathbf{x}-\mathbf{y})^2, \end{equation*}$$