  We determine the critical equation of state of the three-dimensional O(N)
universality class, for N=4, 5, 6, 32, 64. The N=4 is relevant for the chiral
phase transition in QCD with two flavors, the N=5 model is relevant for the
SO(5) theory of high-T_c superconductivity, while the N=6 model is relevant for
the chiral phase transition in two-color QCD with two flavors. We first
consider the small-field expansion of the effective potential (Helmholtz free
energy). Then, we apply a systematic approximation scheme based on polynomial
parametric representations that are valid in the whole critical regime, satisfy
the correct analytic properties (Griffiths' analyticity), take into account the
Goldstone singularities at the coexistence curve, and match the small-field
expansion of the effective potential. From the approximate representations of
the equation of state, we obtain estimates of universal amplitude ratios. We
also compare our approximate solutions with those obtained in the large-N
expansion, up to order 1/N, finding good agreement for N \gtrsim 32.
