Using Langevin dynamics simulations, we investigate the dynamics of polymer translocation into a circular nanocontainer through a nanopore under a driving force $F$. We observe that the translocation probability initially increases and then saturates with increasing $F$, independent of $phi$, which is the average density of the whole chain in the nanocontainer. The translocation time distribution undergoes a transition from a Gaussian distribution to an asymmetric distribution with increasing $phi$. Moreover, we find a nonuniversal scaling exponent of the translocation time as chain length, depending on $phi$ and $F$. These results are interpreted by the conformation of the translocated chain in the nanocontainer and the time of an individual segment passing through the pore during translocation.