Iphone5 battery life drains extremely fast?

So i purchased a broken iPhone, and fixed it myself. It needed a new screen so i replaced that, and the battery was punctured my a piece of glass so i replaced that to, the battery was brand new. It charges super slow, to the point where if i use it while its charging it drains the battery at about the rate the iPhone 4 would, but it should just take longer to charge not drain it. From 100%, i get about 5 hours of use and 3 hours of standby (I use the phone quite often). Even my iPhone 4 i could use it just as much from 7 in the morning till around 10 at night when i put it on the charger. This can't be right. Is there anyways I can make the battery last like its supposed to?

You shouldnt use it while its chargeing that seems to be the main problem. Because it gets uses to having the battery constantly going down abit then chargeing all while its on the charger

I think the battery may have some problem. You should go to a special store to ask them test it for you.