2014 moto x battery life
Even though you already drained the battery and recharged it, I would recommend doing it once more (except let the phone die), just to recalibrate the battery. After that, check out what is eating up your battery in the settings. Try playing with the different radios (BT, WiFi) and seeing what effect each one has. If all of these don't work, try getting a new battery under warrenty or buying one and replacing it yourself (I would try Motorola Moto X 2nd Generation Battery Replacement - iFixit)

If ALL of these don't work, then 5.1 is messing with your phone (even though I have a Moto X running 5.1) and try installing a rom of 5.0.1 or even something else like cyanogenmod.

Good Luck!