My iPhone 5 became slower after 1-2 years. If the battery was bad, I never noticed it (no shutdowns or anything like that). It coincided with a new iPhone and iOS version... I tried every trick, from formatting the phone to reducing animations. Then I found a thread on Apple forums about this and someone was suggesting getting a new phone!
I did buy a new phone (OnePlus One) and it's still working after ~5 years. The battery doesn't last as long, but it doesn't die when I receive a call or play a game.
Because of this, that iPhone 5 was my first and last iPhone.