I've seen a lot of posts talking about your expansion in relation to admin cap, generally along the lines of "Keep to empire size to X multiple of your cap" or "spam cap at first and then forget about it." Which is fair, because the game is trying to give you the impression that Admin Cap is useful for expansionist empires. But it's lying, and we should be doing the opposite.

Short version: because sprawl increases linearly its maluses get proportionately less important, so admin cap bonuses are great for small empires but borderline useless for wide ones. Also you can trivially calculate your 'ideal' empire size.

Long version:

Let's Talk Incentives

Regardless of your cap, each point of sprawl gives the exact same maluses: 0.3% tech cost, 0.5% influence cost, and 1% leader cost and upkeep. Admin cap 30? 300? Doesn't matter, each sprawl's maluse is linearly identical. How much this malus hurts you overall depends on your play style and your game stage, so for sake of easy math, let's call this malus your 'Sprawl Burden' – a measure of how much each point of sprawl makes your life harder. How you weigh your Burden is up to you, but personally as a rule of thumb I like to say it's 0.4% Burden per sprawl for pure tech games on the one end (Since 0.3 Research is the most important malus, 0.5 influence is of secondary importance, and leader cost is trivial(1) ), and 0.1% Burden per sprawl for superwide games on the other end (Where large fleets, which are only slightly and indirectly harmed by sprawl, are the name of the game). This decreases once I've got all the techs/traditions I care for of course. You can assign whatever weights you like, it doesn't really change the below math.

Let's Talk Numbers

Let's say that Alice has weighed her Sprawl Burden at 0.4. If she has an admin cap of 30, and a 280 admin-point-sized empire, then her Burden is 250 x .04=100%. However, her base potential power(2) is 933% larger than if she stayed within her cap. 933% more potential power divided by double the work required means 933÷2=467% more potential power overall. If she increases her cap to 50, however, then her burden decreases to 230 x 0.4=92%, while her potential power is 560% of what she'd get if she stayed within her cap, 560÷1.92=291% more potential power. You'll notice that as admin cap increases, sprawl becomes LESS proportionately profitable, not more.

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She can actually calculate at what point her admin cap is high enough that she shouldn't expand into sprawl at ALL, by dividing 1 by her burden. So for Alice, it's 1/0.004=250. Well technically point 251 is where she breaks even, but why would she want to spend resources on breaking even?

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Now let's look at Baljit, who's playing a wider game. He has an empire admin size of 1,001, a cap of 30, and a Burden of 0.1 per sprawl. Another point would increase his Burden by 0.1% and his potential power by only 0.099%, but of course he's already sitting at 97% Burden, so the additional admin point would only proportionately increase his existing Burden by 0.1÷1.97=.051%. Basically Baljit can expand as much as he wants and the sprawl will never decrease his effective power.

Math Is Tricky

I'm no mathematician (I didn't do this because I like math, I did it because I suck at math but I REALLY like optimising), but playing with the calculator seems to imply that up until your perfect zen balance size of 1÷burden, no matter your cap:size ratio each system will make you more powerful. After that though, you enter for some time into a 'valley of suck' wherein a high enough admin cap means you actually get less powerful with each size increase, because the Burden applied to all of your previously perfectly-efficient empire outweighs the benefit gained from the new system or district or whatever(2). That's not to say that you're not more powerful overall than if your admin cap was lower, just that new systems are kind of crap for awhile. You do eventually expand out of the valley, but it seems to require at least doubling the size of your empire… And the valley seems to get wider the less your Sprawl Burden is weighted per point.

Everything Is Upside-Down

What this means is that admin cap bonuses are for small empires, period. Their target empire size is smaller so it's an easier target to reach, they're much better equipped to use repeatable tech to get to that target (I had 585 cap by 2397 in my standard-tech-rate research game), and the valley of suck is much shorter to cross if and when they decide "That's enough science for now, let's conquer the galaxy." Large empires do get some benefit from admin cap, but it's so minor that spending civics or ascension perks on it is a waste – it might be moderately useful in the early game, but there are other options that help with your early game and also don't do something silly late game like "Reduce my unity costs from from 500% to 490%."

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(1)Unless you're doing Something Weird. I can't give advice that will fit into all Something Weird scenarios so don't whinge about how leader cost is important for your game with 500 simultaneous planetary invasions or something

(2)Assuming all admin points are equal, which of course they're not. This is a rule of thumb, so don't whinge about me 'telling you' not to expand into a chokepoint dark matter-generating system or something