The phrase "be taken up with something" refers to being absorbed or preoccupied with a specific task or activity. There are several synonyms which can be used to convey the same meaning, such as "engrossed", "immersed", "focused", or "captivated". These words all imply a deep level of involvement with a particular endeavor. Other possible synonyms include "consumed", "enthralled", "distracted", or "obsessed". Depending on the context, there are several different terms that could be used to describe being engrossed in a specific pursuit. Regardless of the specific word chosen, the concept remains the same: complete and total concentration on a particular activity.