  We study cosmological solutions of type II string theory with a metric of the
Kaluza--Klein type and nontrivial Ramond--Ramond forms. It is shown that models
with only one form excited can be integrated in general. Moreover, some
interesting cases with two nontrivial forms can be solved completely since they
correspond to Toda models. We find two types of solutions corresponding to a
negative time superinflating phase and a positive time subluminal expanding
phase. The two branches are separated by a curvature singularity. Within each
branch the effect of the forms is to interpolate between different solutions of
pure Kaluza--Klein theory.
