How do I clear this cement pipe underground drainage system?
You could try hiring an electric eel



or get a plumber with a jetter



But either of these options may still not work it the pipe is broken or snapped and the two sections of pipe are now ofset from one another.
Even if these options do work to clear the pipe you still have to remember that the cause of the root problem is still there, cracked or broken joints or pipe, which will require ongoing treatment eg. running the eel through once a year or similar chemical treatment.

The other option is to get the shovel out or hire a mini digger and replace the pipe with a pvc drain