Loop spaces of finite complexes at large primes

Let $X$ be a finite, simply connected CW complex with only a finite number of nonzero rational homotopy groups. Localized away from a certain finite set of primes, the loop space of $X$ is shown to be homotopy equivalent to a product of spheres and loop spaces of spheres. Applications to the homotopy groups of $X$ and the homological properties of $\Omega X$ are given.