The damage bonus is already factored in. The first bow is normally only 38 DMG but got turned into a 48 DMG bow by the +10 bonus.
A 5-shot bow only deals maximum damage if all arrows actually hit their target, ~~which is difficult to achieve if you're not directly in front of the enemy anyway (like shooting a Lynel in the back of the head after the rodeo)~~ ((EDIT: I completely forgot about fused eyeballs, so please disregard that sentence.))
Personally, I use single-shot bows for sniping faraway targets (headshots deal extra damage) and multishot bows to carpet-bomb camps with explosive ammo, to get the most out of the bonuses.
PS: the second one is still better. If at least 2 arrows hit, it already deals 32*2 = 64 DMG whereas the Royal Bow is still only a 38 DMG bow. Plus, elemental affects apply per arrow, so if you for example fuse a shockfruit to an arrow and use a 5-shot bow during rain, you get five electric AoEs stunning and damaging foes nearby.
PS: all bow stats (and lots of other stuff) datamined from the game code: https://docs.google.com/spreadsheets/d/18pNtDx3z-8CwGJRmlW574xbQ6VphQOkvpZhClpOEVDA/edit#gid=0