icon loop, damaged battery?

Hello,

recently the battery of my phone fell out when I dropped it (there is no back case to it) and after putting it back in the phone wouldn't turn back on. When I plugged it into a computer and tried turning it on and a few things can happen.

1) the apple logo turns on, stays for a bit, then turns off again, and does this continuously. In this scenario it doesn't come up on itunes.

2) i'll press the homescreen after i try and restart it and occasionally it'll say plug into itunes (which it already is plugged into) and then when I try and restore it on the computer it doesn't finish, saying there's a problem after the extracting software part.

I've tried the homescreen+power button, but that doesn't work to get things started. And whenever I disconnect it from the computer it stops working all together, and the apple symbol won't even show up.

Is there anything I can do?