Way back in my younger days, I cleaned apartments that people had moved out of/been evicted from to get my rent for half-price. I cleaned EVERY SINGLE disgusting kind of thing you can imagine off of tubs, toilets, and (God help me) radiators.
For regular old tub dirt, I like Scrubbing Bubbles.
Bar Keeper's Friend, if you can find it, is an excellent cleanser. Old-fashioned, not super pleasant-smelling, but really good. Comet smells better, but this stuff works better, especially if you have to leave it set a while.
If all else fails, and I know this is going to sound weird, try Drano Max. I realize it isn't *designed* for this purpose, but it'll take off just about anything. Just be sure to keep a window open and/or leave the room when you get funky. Also wear rubber gloves.
And get a bunch of those green scrubby things (or steel wool, if you don't mind little scratches) instead of sponges or rags. Pay the extra quarter or whatever for the name-brand green scrubby thing. The others don't work well for long.
Good luck!