Obtaining general relations between macroscopic properties of random assemblies, such as density, and the microscopic properties of their constituent particles, such as shape, is a foundational challenge in the study of amorphous materials. By leveraging existing understanding of the random packing of spherical particles, we estimate the random packing density for all sufficiently spherical shapes. Our method uses the ensemble of random packing configurations of spheres as a reference point for a perturbative calculation, which we carry to linear order in the deformation. A fully analytic calculation shows that all sufficiently spherical shapes pack more densely than spheres. Additionally, we use simulation data for spheres to calculate numerical estimates for nonspherical particles and compare these estimates to simulations.