Homebuilders don't put special supports under your bath tub. A jacuzzi tub can hold a hundred or more gallons of water and 2 full grown adults (or more, depending on what you're into. I don't judge) and nobody asks for advice on filling their bathtub. My very heavy electric reclining sectional sits in the middle of my living room and holds a family of 8. I don't see anybody posting "hey, my friends are coming over and they're kinda fat and will probably want to sit on my couch, should I add floor supports?" Remember waterbeds?? I had one in the early 90s. Pretty sure we didn't add support to the floor under that. I have solid oak dining table that weighs like 1,000 lbs. Should I add floor supports in time for Thanksgiving dinner? Just making the point that just because the live load rating of a floor is 40 lbs per square foot, doesn't mean that if I stand my 180lb body in one square foot of space(which I often do) I'm going to go crashing into the basement.
I'm not an engineer and I'm certainly not trying to dissuade you from consulting one, but I think you're over thinking it. I built a support structure under mine when I was finishing the basement because mine is parallel to the floor joists and only rested on a single joist. If my tank spanned 5 joists, I probably wouldn't have worried about it.
Sagging is the concern. A sagging floor can lead to sagging stand which leads to cracked tanks and failed seams. I would say that making sure your stand is sturdy and is a bigger concern.
If you have the space and you want to play it even safer, you can do like I did and build an 8.5 ft stand for your 6 ft tank and add some storage cabinets at the ends. That will spread the load over two more joists.