The fifth and sixth virial coefficient for hard hyperspheres in four and five dimensions has been computed using Monte Carlo techniques. It is found that B-5/B-2(4) has values 0.035 63+/-0.000 07 and 0.012 87+/-0.000 06 and that B-6/B-2(5) has values 0.007 691+/-0.000 028 and 0.000 942+/-0.000 027 in four and five dimensions, respectively. These values are used to investigate the equation of state of hard and Weeks-Chandler-Anderson (WCA) hyperspheres in four and five dimensions. Molecular dynamics simulations are performed for WCA hyperspheres. When compared to the molecular dynamics calculations, it is found that both the hard hypersphere and WCA equations of state are well described by a variety of theoretical approaches as long as the density is in the low-to-moderate regime. At the highest fluid densities studied, the Luban-Michels procedure provides the best accuracy for hard hyperspheres. The WCA prescription for the scaling of the reference system to a hard hypersphere one is a very good approximation in the fluid region.