Definition of taken up with:
(v) :
To form a close relationship with (someone).
(v) :
To become interested in (something).
(v) :
To begin living together with; to lodge with.
(v) :
(archaic) To be contented to receive; to receive without opposition; to put up with
(v) :
Used other than figuratively or idiomatically: see take, up, take up, with.