This integral was done in spherical coordinates
[itex]\int dxdydz = \int r^{2}sin(\theta)dr d\theta d\phi[/itex]
if what you're integrating over does not depend on [itex]\theta[/itex] or [itex]\phi[/itex], then you can integrate over those variables giving you an additional factor of [itex]4 \pi[/itex]. Thus
[itex]\int dxdydz = 4\pi \int r^{2}d r[/itex].
I expect d^3p is shorthand for dp_{x}dp_{y}dp_{z}.