Here is a problem that affects some HP printers. I can confirm that it affects the OfficeJet 3830 series. It appears that it only affects printers with two ink cartridges. The printer does a sweep, pauses for about 15 seconds, does the next sweep, pauses for about 15 seconds, and so on.

The problem with the printer is that one ink cartridge is bad. You need to remove one and test the printer in single cartridge mode. If the printer is fast again, you found your bad cartridge. If that doesn’t work, remove the other ink cartridge and return the first one. The printer should now be fast. If not, you will need to try again with a fresh cartridge, either color or black. If it still very slow, chances are good the problem is too expensive to fix.