  This paper is a continuation of our earlier study on the integrability of the
Friedmann equations in the light of the Chebyshev theorem. Our main focus will
be on a series of important, yet not previously touched, problems when the
equation of state for the perfect-fluid universe is nonlinear. These include
the generalized Chaplygin gas, two-term energy density, trinomial Friedmann,
Born--Infeld, and two-fluid models. We show that some of these may be
integrated using Chebyshev's result while other are out of reach by the theorem
but may be integrated explicitly by other methods. With the explicit
integration, we are able to understand exactly the roles of the physical
parameters in various models play in the cosmological evolution. For example,
in the Chaplygin gas universe, it is seen that, as far as there is a tiny
presence of nonlinear matter, linear matter makes contribution to the dark
matter, which becomes significant near the phantom divide line. The Friedmann
equations also arise in areas of physics not directly related to cosmology. We
provide some examples ranging from geometric optics and central orbits to soap
films and the shape of glaciated valleys to which our results may be applied.
