Carrying the Bat Around the Bases

 
No out and no one on base. The batter grounds deep to the shortstop, but the batter-runner is safe at first on a banger. However, the batter-runner carried his bat to and thru first base. Is there a penalty for this?
 
Is it a different call if the batter doubles and carries his bat all the way to second base?
 
Call
 
There is no penalty for the batter carrying his bat to any base -- provided he does not use it to interfere with any playing action.