*outdated* Go here
I've been criticized on my insistence on going Brutalizer first on Ezreal. I've also been infuriated with Ezreals going Zerker Greaves. I've finally gotten my act together and made a spreadsheet to compute it, see if I was right. And I was.
In the sheet I'm assuming R>Q>E, that you're 21 in offense and that you have AD marks.
The enemy hit is assumed to have average armor plus flat seals plus the +5 mastery. I've added a Sunfire Cape at higher levels. R and Shiv only hits 2 targets, E hits a target half the time. All this can be modified in the sheet.
Brutalizer first item
The most cost efficient to go from the start is Brutalizer. Here's a comparison of Brutalizer vs some other typical items, showing loss of gold efficiency:
12% BF sword
47% Spirit of the Elder Lizard
67% Manamune (stack at 250)
Brutalizer -> Sheen -> CDR boots
People have then told me that it pays off when you get to Trinity Force. But that isn't true either. Going Brutalizer, Sheen and CDR boots is the most effective build at this price level (3-4k gold). Another comparison:
11% BT + CDR boots
16% BT+ Zerk
19% Brutalizer + Phage + CDR boots
27% Brutalizer + sheen + Zerk
32% Trinity Force
Here we can also see that CDR boots are better than Zerker Greaves. And it gets even worse at higher levels. Don't buy Zerker Greaves on Ezreal please.
Conclusion about early build paths:
The only other option, trading damage for lifesteal, is going BT. BF sword is only a little weaker than Brutalizer, and same goes for BT compared to Brutalizer+Sheen+CDR. It also has more burst.
The optimal path for the BT build is:
BT -> Brutalizer/CDR Boots -> Remaining -> LW -> Tri -> IE -> BC
If you're maxed and want a defensive item, BT is the weakest if the enemy has just a little bonus armor, followed by BC if they have a little more bonus armor, followed by LW if they have a lot of bonus armor.
Brutalizer -> Sheen -> CDR boots -> Trinity -> LW -> BoRK/IE/BT -> BC
If they have any bonus armor, LW is more cost efficient than Trinity
If they don't have much relevant bonus armor, while LW is still cost effective, other options provide more slot efficiency and more utility.
Going BT/BoRK means a loss in damage, but it's not terribly big.
If you're buying Tear anyway (I do), upgrading it after Sheen but before Trinity is more damage. Trinity is 27% less efficient than upgrading Tear, and that's without it transformed.
The single most damaging build with boots is BC+Tri+MM+LW+IE with 1000 DPS on an averagely armored target without items, and it includes both anti-armor items.
If you want a defensive item in there, the MM is the least slot effective unless the opponent has little armor, in which case LW is the least effective.
If you don't want boots, Static Shiv is the best output(1165). BoRK(1124) is close if you want the active and lifesteal.