  Cosmological implications of a generalized Rastall theory where the
non-conservation equation is promoted to accepts arbitrary functions of the
Ricci scalar is considered. We have obtain the model parameters of the
power-law ansatz $f\propto R^\eta$, using $H_0$ and $f\sigma_8$ data sets and
show that the generalized Rastall theory could satisfy observational data. The
dynamical analysis of the model shows that for $\eta\in(0,1)$, the dust
dominated fixed point is lost and as a result the parameter range of $\eta$
should be restricted to $\eta\geq1$. We will also show that the genralized
Rastall theory will predict more accelerating universe with larger radius,
compared to the $\Lambda$CDM model.
