The stock player won't play a lot of formats, so I don't even consider it a feasible option. MX has been my go-to video player for Android for some time now. It works so well, plays anything I throw at it, and has some nice options. So why bother using anything else?
Can't say I've compared the two, but don't think there would be much difference in battery life. I get close to 10 hours playing video on MX anyway. Not often I would even need that kind of playtime, except for long airplane flights.
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Phone: AT&T HTC One (M8)
Tablet: Samsung Galaxy Tab Pro 8.4 (WiFi only)