Constitutive equations are derived for the elastic response of nanocomposite hydrogels subjected to swelling under an arbitrary deformation with finite strains. Under rapid deformation of as-prepared specimens, the stress-strain relations involve four adjustable parameters that are found by fitting observations on poly(N-isopropylacrylamide), poly(N,N-dimethylacrylamide), and polyacrylamide hydrogels reinforced with Laponite nanoclay and graphene oxide. Good agreement is demonstrated between the observations and the results of numerical simulation at elongations up to 3000%. A pronounced difference is revealed between the effect of filler content on material parameters of chemical and physical nanocomposite gels.