In formal contexts, avoid splitting infinitives.
The rule against split infinitives is antiquated and perhaps even a little wrongheaded. It seems to stem from a belief that Latin is the purest language, and because Latin infinitives are one word and cannot be split, English infinitives should not be split, either.
The authoritative texts generally approve of split infinitives, and have done so for a while: the Chicago Manual has found split infinitives acceptable since the 13th edition in 1983, and Strunk and White provisionally endorsed the practice as early as 1959.
That said, our goal with all matters of style should be unobtrusiveness; we should call as little attention to these decisions as possible. Because split infinitives are jarring to many careful readers despite all the above, avoid them where there is a graceful alternative. Where a split is unavoidable (or even elegant), accept it.
This whole thing has been a labor of love.