A snake did nuthin, hmmm
Try a better snake, those home cheapos don't do the job all the time. They are not usually big enough, and the blockage is out of reach. Don't do the acid with plastic pipe, not to good for the pipe.
You need to get a better, longer snake or call in the plumber.
Even if you have roots, the snake will work the best, then dump salt down there once and a while to kill em off for long term.