  We apply the technique of supersymmetric localization to exactly compute the
$S^5$ partition function of several large $N$ superconformal field theories in
five dimensions that have $AdS_6$ duals in massive type IIA supergravity. The
localization computations are performed in the non-renormalizable effective
field theories obtained through relevant deformations of the UV superconformal
field theories. We compare the $S^5$ free energy to a holographic computation
of entanglement entropy in the $AdS_6$ duals and find perfect agreement. In
particular, we reproduce the $N^{5/2}$ scaling of the $S^5$ free energy that
was expected from supergravity.
