  We show that $U(N)$ $3d$ $\mathcal{N}=4$ supersymmetric gauge theories on
$S^{3}$ with $N_{f}$ massive fundamental hypermultiplets and with a
Fayet-Iliopoulos (FI) term are solvable in terms of generalized Selberg
integrals. Finite $N$ expressions for the partition function and Wilson loop in
arbitrary representations are given. We obtain explicit analytical expressions
for Wilson loops with symmetric, antisymmetric, rectangular and hook
representations, in terms of Gamma functions of complex argument. The free
energy for orthogonal and symplectic gauge group is also given. The asymptotic
expansion of the free energy is also presented, including a discussion of the
appearance of exponentially small contributions. Duality checks of the
analytical expressions for the partition functions are also carried out
explicitly.
