Very nice guide, very experimental. I like it a lot!

Holy **** 0.o
Didn't mean for this to get so heated. I suggested BotRK simply because it offers vamp based on the target's health as well as attack speed, thus not requiring that my autoattacks or abilities themselves to a particularly high amount of damage to receive health. Also its active is way better than gunblade's in my opinion--as I stated before to each their own, I fancy my Jax with high attack speed to build passive AS faster and trigger more procs of Master-At-Arms.

Thank you for your advice, I greatly appreciate it. The idea of this build was to create something somewhat unique to Jax, yet still viable. So far this build has been viable in the ranked queue for me, so I seem to think it 'generally' isn't too bad of a build, and it's a great deal of fun ^.^

The whole point is that Eleisa's miracle costs you 1100g and will grant you barely anything.
It is just not worth it.

I am not saying this to argue with you, I am saying this to give you a tip.
Out of all the games I played with Jax, I really learned that he really needs this lifesteal/spellvamp, there is no exception.
And of course, if you play Jax AP, you need to get spellvamp, not bloody lifesteal :P
Do not recommend Blade of the Ruined King to me if the rest of your build is AP.
Take Hextech Gunblade instead, benefitting Jax in many many ways.

I gave you tips, do whatever you want with it.
I will not continue commenting about this.

The point of AP Jax (aggressive) is to burst down key targets quickly--he can do this, and with the items I have provided can do it relatively safely due to high Armor/MR/etc. Every item in the build is used as an efficient means of adding damage or utility to the end build.

Regarding the Eleisa's Miracle, it would be silly to expect that it pays entirely for itself. In my mind it pays for itself through:
1. The gold it very literally generates &
2. The gold that you receive through having longer sustain in the jungle/lane.

Also, I would argue that the regeneration is relatively significant. Especially when it takes up 0 slots.

This aside, if you want, you can take Zhonya's Hourglass out of the build and get a Blade of the Ruined King instead. Little more hybrid-ey, gives more AS which benefits Master-At-Arms's passive, and utilizes the already-high AS to proc a lot of life steal from the unique active, thus giving the build more sustain. I've run this version before, and it works. You just lose bonus armor from the build, a good deal of AP, and a killer unique passive. To each their own.

It can sound really appealing and all, but Eleisa's Miracle costs 1100g.
Eleisa's Miracle will ONLY grant health and mana regen for Jax, which will be strong early game, but late game, you will barely feel it.

If you were to buy Spellvamp / Lifesteal instead (which benefits Jax waaaaaaaaaaaaaaaaay more) you will AND deal more damage AND get more lategame sustain, which Jax really really really needs.

Sure, you could get Eleisa's miracle later on, saving your Philosopher's stone untill you basically get the item for free. But, that means the GP10 item must give you 1100g.

Since you get 5g/10seconds = 1g/2seconds, it will take you 1100 * 2 = 2200 seconds to finish Eleisa's without spending gold.
2200seconds / 60 = 36,7 minutes excluding the time you needed to get philosopher's stone. When you are that far in game, you should already be lvl 15+, which prevents the upgrade of Eleisa's.

So no my dear friend, I see no excuse at all to use Eleisa's Miracle :P
Just buy some spellvamp + lifesteal :P

Eleisa's Miracle is intended to be a support item, so has the reduced cooldowns for support spells--these spells are not particularly useful for jax. I listed heal as a potential alternative spell for those interested, if that counts for anything.

Mainly, though, I use Eleisa's Miracle because it gives you health and mana regen without taking up an item slot. Gives you some sustain without having to dedicate items to it, you can focus your 6 item slots more on burst at that point.

Guide is pretty good, build is a bit.... not so okay ;P

Especially having an item like eleisa's miracle, just seem like a waste of money for someone who does not use one of the named summoner spells in the unique active.

I will not vote you down, I will not vote you up,
but I will give you this:

CHEEEESE!! CHEESE FOR EVERYONE!