The simplest partition function, associated with homogeneous symmetric forms S of degree r in n variables, is integral discriminant Jn|r(S) = ∫e-S(x1,...,xn)dx1...dxn. Actually, S-dependence remains the same if e-S in the integrand is substituted by arbitrary function f(S), i.e. integral discriminant is a characteristic of the form S itself, and not of the averaging procedure. The aim of the present paper is to calculate Jn|r in a number of non-Gaussian cases. Using Ward identities — linear differential equations, satisfied by integral discriminants — we calculate J2|3,J2|4,J2|5 and J3|3. In all these examples, integral discriminant appears to be a generalized hypergeometric function. It depends on several SL(n) invariants of S, with essential singularities controlled by the ordinary algebraic discriminant of S.