An understanding of the dispersion of nanoparticles into polymer melts is needed in order to control material properties of polymer nanocomposites. Here we study the dispersion of polymer-grafted nanorods in homopolymer melts of the same chemistry, using both experiment and theory. The theoretical calculations are performed over the range of experimental system parameters. Polymer-grafted gold nanorods (Au NRs) were found to be dispersed when the matrix chain lengths were small relative to the brush chain lengths, and aggregated at higher matrix chain lengths. Both classical density functional theory (DFT) and self-consistent field theory (SCFT) are used to calculate the structure of a polymer brush around an isolated NR in a polymer melt Both theories predict a gradual transition from a "wet" to a "dry" brush as the grafting density, the NR radius, and/or the ratio of matrix to brush chain lengths is increased DFT calculations of the interaction free energy between two NRs find an attractive well at intermediate NR separations, with a repulsive barrier at closer NR separations The strength of the attraction increases as the brushes become more dry. Including the van der Waals attractions between the NRs gives an estimate of their total interaction free energy, which can be used to predict at which values of the system parameters the NRs are dispersed or aggregated A dispersion map shows good agreement between DFT calculations and experimental observations of dispersed and aggregated nanorods.