Equations of state for the four- and five-dimensional hard hypersphere fluids have been obtained. Several procedures have been considered in each case: a) adapting the method of Carnahan and Starling to such dimensionalities; b) introducing Padé and Levin approximants suitable to the virial expansion and c) establishing a pole at a given density. In most cases, the results obtained agree satisfactorily with the available simulation data. The use of an additional fitting parameter furnishes nearly perfect agreement.