  A generalized version for the Rastall theory is proposed showing the
agreement with the cosmic accelerating expansion. In this regard, a coupling
between geometry and the pressureless matter fields is derived which may play
the role of dark energy responsible for the current accelerating expansion
phase. Moreover, our study also shows that the radiation field may not be
coupled to the geometry in a non-minimal way which represents that the ordinary
energy-momentum conservation law is respected by the radiation source. It is
also shown that the primary inflationary era may be justified by the ability of
the geometry to couple to the energy-momentum source in an empty flat FRW
universe. In fact, this ability is independent of the existence of the
energy-momentum source and may compel the empty flat FRW universe to expand
exponentially. Finally, we consider a flat FRW universe field by a spatially
homogeneous scalar field evolving in potential $\mathcal{V}(\phi)$, and study
the results of applying the slow-roll approximation to the system which may
lead to an inflationary phase for the universe expansion.
