Buy a new battery and dispose of the old one. Swollen batteries are dangerous to keep around. They can explode. Keep the psp though, that's still good.

I would take the battery out. Now you might be able to jump the positive and negative leads in the console to be able to play it plugged in. Or you could just buy a replacement battery.
Happened to my Wii U.

Get rid of that battery before it busts wide open and ruins everything with the corrosion...

Originally posted by Hex draught:

I found my old PSP
The battery is swollen like a massive buttocks, I don't think it will power on

Congrats in finding it! Also replace the battery before it causes damage to the device itself.

Fermented battery can sell for quite a bit