I updated from 5.01 to 5.1.1 with my AT&T Note 4. I did a factory reset and then restored all my apps from Google. My battery life is definitely better. In fact, the battery life is very close to what I was getting on Kit-Kat when the phone was brand new. I am very happy. Off the charger at 7am and still have anywhere between 40% - 60% when I plug back in at 10pm depending on use thru the day. On 5.0.1, I was barely making it to 10pm with any battery left.
Apps running constantly in the background:
Aqua Mail with 4 accounts, Textra, Facebook, Snapchat, Twitter, ESPN. Instagram, Weather Bug. Wi-fi and Bluetooth always on.
If your battery life is now truly horrible on 5.1.1, you should do a factory reset as you must have a runaway app eating your battery.