Is it normal to get 3 hours usage time out of the iPhone 5? Not at all.
Is it normal to get significantly lower battery life when using resource hogging features on the iPhone 5? Yes.
Like others said above, if you're definition of entertainment means you're using Netflix over cellular and playing games on the iPhone (especially games that push the processor graphically), then yes, I could see a scenario where 3 hours battery life could be possible, but you'd really have to be pushing the iPhone to work to get that kind of result. 3 hours is seriously low, like bottom of the barrel performance, so either you're really REALLY using your iPhone 5 or there is a problem.
The only way we could really tell you what is up is if we had a base line for your battery life during a non-intensive usage cycle. Then we could compare that to your heavy usage cycle and tell you if it's an issue or if it's normal.
Also, just to put it out there, there are MANY battery life threads on this forum, including an official one for the iPhone 5. If you look in there, I think you'll get a good idea about what normal is vs. what you're experiencing...and i think you'll come to the same conclusion that i would have that there is probably something wrong with your iPhone.