Rout out is a phrasal verb that means to find out or discover something after extensive searching. The antonyms for rout out could be words like ignore, neglect, overlook, leave behind or miss. These words imply that the object or thing that one is looking for is either not found or intentionally left out of focus. Antonyms for rout out could also include terms like give up, abandon, surrender or relinquish, which indicate a lack of perseverance or the act of deliberately not pursuing the search further. In conclusion, for every action, there's an equal and opposite reaction, and antonyms for rout out are no exception.