I've stripped my 2nd floor bathroom to the studs and joists, installed new plumbing and electricity, and moved a few things around. The tub I'm putting in is acrylic, and weighs less than the cast iron tub I removed. But it holds 25 gallons more water. I'm also installing a glass shower enclosure with a tiled shower pan. And that should add about 200lbs. to the floor as well.
My joists span at least 10' but not more than 12'. They are 16" apart, except for the one parallel to a load-bearing wall, which is 24" away, originally designed like that to make room for the waste plumbing of the 1930s. 
My question is: what kind of weight can I expect this thing to support? My real concern is the tub. It weighs in at only 105 lbs. But full of up to 60 gallons of water, it could weigh as much as 585 lbs. Do you think the floor can handle it?
I should also add that instead of using 1/4" cement board on the floors, I've poured self-leveling cement. At six bags, it adds at least another 360 lbs. to the floor.
Best Answer
If I assume you have 2x12 joists, then it's all good. If you want it better you can add full depth joist blocking perpendicular to the joists to try and help the floor out a little, but you don't have to.
If I assume you have 2x10 joists, then it's all good except for that joist under the shower/tub. It probably wouldn't catastrophically fail, but add a second joist next to it if possible. It'll also help prevent your new tile job from cracking by decreasing the live load deflection you'll get when filling the tub. If you absolutely can't add another joist there, you can add full depth joist blocking perpendicular to the joists to try and help the floor out a little.
If I assume you have 2x8 joists, that's bad. I hope you don't have 2x8 joists!