We consider the hyperelastic response of semi-crystalline ethylene-co-butyl acrylate (EBA) samples filled with carbon black (CB) particles. Such material is structurally complex with its microstructure being characterized by many structural parameters including crosslink density, filler/matrix interfaces, crystallinity, filler network, and chain entanglement which have different degrees of influence on the effective mechanical properties. We evaluate the ability of a number of analytical models to correctly reproduce the non-linear elastic mechanical response of these samples. We do this by considering either dry samples, or samples which are swollen by a non-polar solvent (toluene) at equilibrium, and subjected to uniaxial tension at room temperature. As test cases, we focus on six physical models for the purpose of analyzing the stress-strain curves of samples with different cross-linking densities. Among these frameworks, we show that the Mooney-Rivlin (MR), Ogden, and eight-chain models accurately describe the stress-strain curves of both dry and swollen CB-EBA samples. These findings highlight the possibility of attaining a diverse set of mechanical properties of filled polymer samples by tailoring their structural parameters.