Are you asking about pictures or video? Battery life will depend on a number of variable such as the mode you shoot in.
Pictures:
"Energy-saving design that achieves approx. 500 shots* per charge (with EN-EL14, in single-frame mode). ...more
*Based on CIPA Standards."
Source: http

/imaging.nikon.com/lineup/dslr/d5200/
Video (not stated how long on a single charge):
Maximum length: 29 min. 59 s (3 min. in miniature effect mode)
Source:
http
/imaging.nikon.com/lineup/dslr/d5200/spec.htm