Battery swollen

See: https://www.apple.com/support/iphone5-battery/ for information on the iPhone 5 batteery replacement program.

Even if your battery on your iPhone 5 isn't eligible for a free replacement, I would turn off your iPhone and get the battery replaced as soon as possible. The swelling of the battery can put pressure on areas that shouldn't have that much pressure and can cause further damage.

So why are you telling us? Contact Apple and get it replaced.

Make an appointment at an Apple Store to have your device examined by a technician. Or contact Apple Support. Be sure to back up your device before taking or sending it to Apple.

DANGER!, DANGER! PU IT IN A LARGE, PADDED BOX, & RUSH IT TO THE FIRE DEPARTMENT. URGENT. VERY DANGEROUS!