Evaluate
$\displaystyle \int\int_S \left[\frac{1}{R}\nabla{\phi} - \phi\nabla{\left(\frac{1}{R}\right)}\right]\cdot d\bold{S}$
over the surface of the sphere $\displaystyle (x-3)^2 + y^2 + z^2 = 25$, where $\displaystyle \phi = xyz + 5$
I see a vector valued function dotted into the normal of a ball centered at (3,0,0). Can't you use the divergence theorem? So instead of evaluating
$\displaystyle
\vec f \cdot n dS
$
you instead evaluate
$\displaystyle
\nabla \cdot \vec f dV
$
It looks like the integrand was chosen so a couple of terms drop out. Then
$\displaystyle
\nabla \cdot \nabla \frac{1}{r}
$
gives you a constant because the origin is in your ball, and the other
$\displaystyle
\nabla \cdot \nabla (xyz+5)
$
should give you zero.