Guide 1 - AD Ezreal, I recently retried AD Ezreal and got the hang of it. I think this Guide is overall better as it is stronger early game.

Guide 2 = Hybrid/AP Ezreal which is attempting to get his Q to do a ton of damage

I can't stress this enough Ezreal is a champion best played with smart casting turned on because you can fire off all your skills almost simultaneously without needing to click at all.
(Update 2: switched around build 1 and 2, 1 is now AD (better build) also added alternative items chapter

I am just throwing this guide together pretty quick so I will give a short explanation. First off I do understand that manamune and archangels mana granting abilities do not stack they are in this build for the %mana to ap/ad ratios. Before creating this guide I have done the math as to this build compared to some other similar ideas I had and this one gave the most overall damage to all the abilities.(you can get more damage from guinsoo's rageblade instead of hextech gunblade but that sacrifices the lifesteal/spellvamp and doesn't do much if you dont have it at max stats often)

A quick note about Ezreal:
All his abilities are skill shots and as such if you are used to the range of his abilities consider turning on smart casting for all of his abilities, this will give you greater reaction speed and allow you to fire off abilities very fast.

Mystic Shot- Q
Ezreal's Q is by far his damage dealer and harass skill. It is important to note that the skill procs on hit effects ie: sheen and Lich bane. one of the main purposes of this item build is to make this skill hit really hard.
Lich Bane and Sheen both work uniquely with Mystic Shot. when Mystic shot is cast it triggers the effect on the item thus making the next attack deal the bonus damage but since it is a skill shot and applies on hit effects when it hits it will in essence effect itself. so You cast the spell and sheen/Lich bane takes effect and when the spell hits it will deal the bonus damage due to triggering on hit effects.
with this build all maxed with a lich bane trigger Mystic Shot can hit for 964 damage before resistances. and with it reducing its own and all other spells cooldowns by 1 second it is able to hit often.

As AD Ezreal (Build 1) you tend to be extremely mana hungry, so if you don't feel like getting blue buff all the time you may need to substitute an item, I would suggest changing out last whisper (if you don't need the armor pen) for manamune taking the tear before bloodthirster (or earlier if you feel you need it) I would be cautious putting it before trinity force as it takes up a lot of gold and won't really increase your damage output at all unless you get the full manamune. I have found that rushing trinity force does a lot for Ezreal.