In the sense of "to rear," "to care for during childhood," this term is acceptable but is considered less refined and somewhat more informal than rear (See raise). Use bring up in ordinary conversation if it is a term widely used in your locality; use rear in writing and in all formal speaking situations. Bringing-up is acceptable in the sense of "upbringing" and "childhood training" but is less preferred than rearing and upbringing. As a verb meaning "to introduce" (Why bring up that subject?), the term is tiresomely overused. Possible substitutes for bring up in this meaning are introduce, inject, advance, and suggest.
Dictionary of problem words and expressions. Harry Shaw. 1975.