A much better choice for the D5200 is the 18-300. It is made for DX cameras. The lower end wide angle means you'll almost never want to take it off.
It is a bit less expensive than the 28-300, and last spring Nikon had a sale on it for $300 off. I doubt they'll do that again but you might keep an eye out.
Thanks, what should I expect to pay? I moved up from the kit 55-200 to the 55-300 from a fellow photog for $265 and thought I did all right at that price, almost all of the American 5 year warranty was left on it.