  We apply the recently proposed perturbative technique to solve the
supergravity BPS equations of ${\cal N}=1^*$ theories put on $S^4$. In
particular, we have calculated the coefficients of the leading quartic terms
exactly, in the expression of the universal part for the holographic free
energy as a function of the mass parameters. We also report on the coefficients
of higher order terms upto 10th order, which are computed numerically.
