I'm working on my poker math and had a situation come up that I would like some help with in figuring out my equity in the spot.

It's 1-2 with a $5 straddle and I'm in middle position with $280 and have everyone covered. UTG+1 limps. He's got a huge range here as he's been both limping and raising with all kinds of suited hands, any Ace, pairs, etc. He has about $200 left and always raises with hands like AA-QQ and AK that I've seen so I believe that they are out of his range.

I raise to $20 with AcKs. My image is a tight agressive one for the most part, but I have been a little more active the last couple of hands and not many people pay that close attention anyway in this group of players.

Player immediately behind me calls. He has about $200 and also a rather wide range here as I've played with him before. He can have a good Ace here, all kinds of suited cards, and QQ-22 here too. I believe he would three bet AA and KK here and just try to get it in with those hands now.

Button now three bets to $50. He's got $240. He sat down maybe about 20 hands ago. I've played with him before and he is an OK player who understands the game and I know he can make moves, however I believe his range here to be roughly AA-JJ, and AK. I don't think he gets overly crazy here with like QJs or TT or something when the pot is getting this big. The size of the raise almost seems like he wants to find out where his Queens or Jacks are here as I've seen him do that before, but he could be just milking his AA or KK here too or making some sort of move, althoughly that's much less likely I feel.

It gets back to UTG+1 and he calls $45 more, so now there is $140 in the pot. Now his range to me is pretty much hands like small pairs and suited connectors as I know him, and he would move in here with a strong hand but he just loves to see flops with his pairs and connectors, especially when the pot it getting large, so I'm pretty confident that that is his range.

Now it's back to me. I feel calling here is gross so that was out of my mind, and if I am against a hand like QQ I want the value of seeing the full five cards. I'm thinking with the ranges that I have on everyone and the fact that I do have fold equity by moving in that that would be the best play, but I would really like to have someone help me walk through the math to better understand a spot like this so I can run the calcuations myself next time to make sure my decision is mathmatically sound. Thanks for the help.

Shoving here is totally standard unless you are against the nittiest of 3bettors.

Board:
Dead:

equity win tie pots won pots tied
Hand 0: 42.819% 28.08% 14.74% 14424809 7571040.00 { AcKs }
Hand 1: 57.181% 42.44% 14.74% 21802231 7571040.00 { JJ+, AQs+, AKo }

Board:
Dead:

equity win tie pots won pots tied
Hand 0: 28.880% 21.62% 07.26% 1082239984 363494385.00 { AcKs }
Hand 1: 42.089% 34.87% 07.22% 1745731847 361228953.00 { JJ+, AQs+, AKo }
Hand 2: 29.031% 28.74% 00.29% 1438654378 14644061.00 { 99-22, ATs-A2s, KTs+, Q9s+, J9s+, T9s, 98s, 87s, 76s, 65s, 54s, 43s }

Raising AK and getting 3-bet is always a mess @LLS. The most important thing is the 3-bettor. Many people won't 3-bet anything but KK+, what indicates a clear fold. But if you can include JJ and QQ in his range, maybe some random light 3-bet (not likely), it is a shove.

We can expect that if the original 3-bettor calls your shove, the guy who limp/called $45 is entering the party too. His range is basically suited aces, big suited connectors ans pairs, so you're clearly good with AK on a 3-way AI pot.

However this is a spot of great variance cuz you'll have to hit A/K and hope to survive the river against 2 other guys. If you're certain that once you shove, other guy's entering too, the best play is to fold given you're equity is not great against both.

Just some math to help you decide:

equity win tie pots won pots tied
Hand 0: 45.478% 37.31% 08.16% 26791167228 5862194512.00 { JJ+, AKs, AKo }
Hand 1: 27.789% 19.14% 08.65% 13743826992 6208367146.00 { AKo }
Hand 2: 26.733% 25.72% 01.02% 18464001756 730536886.00 { TT-22, A2s+, KTs+, QTs+, J9s+, T8s+, 97s+, 86s+, 76s, 65s, 54s, AJo+, KQo }