It'll go down for the inquiry, but depending on the increase, it'll go up due to more available credit and lower utilization.

Don't fret it. Most people on this board will go through a couple hard inquiries a year, and the majority of them see their credit scores remain roughly the same or even increase with more credit being allocated to them. As long as you continue to make on-time payments and apply for new cards sparingly, this really will not make a difference at all.

Edit: Plus, I see you said your score is 780. Truly, a drop in even 30 points (which won't happen from one hard inquiry) won't affect your credit worthiness in the slightest.