It is included in Samuel Johnson's
A Dictionary of the English Language (1755), in which many vulgar words aren't, and there is no mark either about its register. But you might get a hint of it in the literary examples.
FART. n.s. [fert, Saxon.] Wind from behind.
Love is the fart
Of every heart;
It pains a man when ‘tis kept close;
And others doth offend, when ‘tis let loose.
-- Suckling.
TO FART. v. a. [from the noun.] To break wind behind.
As when we a gun discharge,
Although the bore be ne’er so large,
Before the flame from muzzle burst,
Just at the breech it flashes first;
So from my lord his passion broke,
He farted first and then he spoke.
-- Swift.