How long dose it take for a permanent tooth to grow out completely?

Depends: There is no true time to say like 3 mnths or 6 months or anything like that. If it appears to be slow maybe an x-ray is necessary to determine the reason why it is 'slow'. Roots do not completely form in the bone for 2-3 years after eruption starts.

Few to many: It can take a few to many months for full eruption depending on age, which tooth, available room for eruption, health of gum tissue, and so many other factors. If you have a concern, see an orthodontist, a member of the dental specialty trained to evaluate growth and development. Know that when there is one eruption anomaly there are often multiple eruption anomalies.