You're talking about an awful lot of river in 5-7 days. But I've learned never to say people shouldn't do things I think are impossible or impractical, so here's a little info.
Loma (the Ruby Horsethief put-in) to Moab is a nice leisurely five day float -- I've done it a couple of times. Lots of options for side canyon hikes in the upper half. But you have to have a Westwater permit for the 17-mile stretch in the middle, between the WW ranger station and Cisco. And those are obtainable in advance only from the BLM office in Moab -- no permits are issued at the ranger station itself. Below Dewey Bridge, the river parallels the road, but it's still a nice float.
The Moab (or more commonly Potash) to Hite run through Cataract Canyon is another easy five day possibility. Depending on how much wind and current you have, and how much side canyon hiking you want to do, it's very doable. And permits for Cat are pretty easy to get because the shuttle is not exactly convenient.
I think most people familiar with the vicinity would agree that to make the whole run in five days -- Loma to Hite -- would be pretty impractical. If you were serious, it would require a really fast run (most would say with a motor because of fairly predictable wind problems). And you'd still have to get the Westwater permit for the day you planned on going through there. And you'd have a much much longer shuttle, of course.
Better, I'd say, to pick one of the two and relax and enjoy the river and its canyons.