All the attack speed, some AP and AD for his overpowered pew pew ult. Some results may vary.

This build lets you burst everything down almost instantly leading to lots of rage from the enemy team. Side effects may include: you getting fed, winning lots of games, making Thresh look like an idiot for taking your wife, and constantly pew pewing your super amazing ultimate.

Use at your own risk.

While Lucian is generally seen as an ADC, he is just as, if not more powerful, as a Hybrid Carry. Lucian has incredible burst and extremely low cooldowns. With this build, I will attempt to help you play Lucian as a Hybrid Mid Carry. You will watch your enemies Q-Q as you pew pew.

PROS
Lots of poke with his Q and W
Dash removes all slows and can dash over some walls
Amazing damage with his Ult
Passive double shot procs items/lifesteal

CONS
Has no CC
Can be squishy
Dash doesn't remove stuns, roots, snare, etc
Ult can be hard to aim

Flash: This is an absolute must. Flash can be used as a utility spell to get away or offensive to close the gap and get a kill.

Ignite: Ignite is a great spell to finish off any low healthed enemies. The true damage from this can be enough to kill someone and save your life.

Barrier is a good spell to have on an ADC. But generally, you will have Ignite for any given Mid champion. If you want to take Barrier instead of Ignite, by all means do so. I personally don't recommend it though.

Lucian has a very interesting kit. All his skills depend on each other for him to be successful. Once you get the hang of him, which doesn't take too long, you'll be able to burst anyone down incredibly fast.
PASSIVE
Lightslinger: After Lucian uses an ability, his next auto attack within 6 seconds will attack twice. The second shot does 50-75% of Lucian's AD.
- Lucian's passive is pretty over powered. Each shot from this will proc items, lifesteal and so on. Minions and monsters take 100% of his AD instead of the 50-75% and the second shot can critically hit as well

FIRST ABILITY (Q)
Piercing Light: Lucian shoots a laser in the targets direction after a short period of time and deals damage to anything it hits. The full range of the laser exceeds your target.
- once you get use to aiming this, you can snipe the enemy champions constantly and follow up by auto attacking with Lightslinger for even more damage. Max this first.

SECOND ABILITY (W)
Ardent Blaze: Lucian fires out a skill shot that explodes on enemy contact or when it reaches it's max range, then explodes into a cross pattern. The impact area is visible for a brief second upon exploding. Attacking an enemy that has been hit by Ardent Blaze increases your move speed for a short duration.
- This is where the AP portion of this build comes into play. This ability has a 90% AP ratio on it as well as a slight AD Ratio. This is used for clearing minion waves easy as well as checking brush.

THIRD ABILITY (E)
Relentless Pursuit: Lucian dashes a short distance and removed any slows that are on him.
- The mobility this gives is insane. Specially since it removes all slows and you can dash over some walls with it. Keep in mind, this does not remove any stuns, roots, etc.

ULTIMATE (R)
The Culling: Lucian fires a barrage of rounds in the chosen direction for 3 seconds and deals damage to the first enemy hit. Deals triple damage to minions but not monsters. If he kills an enemy, the cooldown on Relentless Pursuit is refreshed.
- This is Lucian's bread and butter ability. Full built, you can deal about 3600 damage with his ult. You are still able to use Relentless Pursuit and Flash while channeling this. He shoots more shots based on how much attack speed you have. More attack speed = more damage.

Greater Mark of Attack Speed - These runes are purely for maximizing the damage from your ult. Since his ult partially scales with attack speed, I believe these runes are a great choice. You can also use Greater Mark of Attack Damage if you want that small extra damage during early game. These fall off late game and the attack speed runes are more beneficial by my experiments.

Greater Seal of Armor - I know what you're thinking. "Why have armor runes if you're going mid?". These are so you can soak up a little more minion damage and the jungler is generally AD based so you can soak up some more damage from them as well

Greater Glyph of Scaling Ability Power - These runes will give your W some more damage early game before you start building your AP items. That's their only use.

Greater Quintessence of Attack Damage - These provide you with a little more damage to make it easier to farm in lane.

Offensive Tree
Fury - More attack speed means more damage with your ult
Deadliness - General Attack Damage
Blast - General Ability Power
Weapon Expertise - Cuts through enemy armor
Arcane Knowledge - Cuts through enemy magic resist
Brute Force - More Attack Damage
Spellsword - Combined with Nashor's Tooth, your auto attacks deal a bonus ~100 dmg once you get your Rabadon's Deathcap
Sunder - Most of Lucian's skills are physical damage, so the extra armor pen gives you a lot more damage
Executioner - 5% extra damage is always nice to have

Utility Tree
Summoner's Insight - Lowers the cooldown on Flash
Meditation - Provides Lucian with a small bit of Mana regen
Improved Recall - 1 second could mean life and death when recalling
Mastermind - An extra 10% cooldown reduction on Flash and Ignite. You'll be able to use them more often
Runic Affinity - Gotta have blue buff last longer. Don't cha?

Starting Items
Doran's Blade: This provides you with more health, slightly more damage and the 5 health per basic attack gives a good amount of sustain early game. Staying in lane longer means more gold for you.

First Recall
+ Fiendish Codex and Boots of Speed: If you can stay in lane, you should be able to get the 1155 gold to buy these in no time. the Fiendish Codex provides you with a nice AP boost as well as the 10% CDR knocks 10 seconds off your ult cooldown at rank 1. Boots are just for bonus move speed.

Early to Mid Game
+ Berserker's Greaves and Nashor's Tooth: Build your Boots into Berserker's Greaves right away. The attack speed from just this + Fiendish Codex will push your ult to do about 700 dmg rank 1.
Finish off your Nashor's Tooth. This is your main item. This gives you attack speed, AP, Cooldown Reduction, and you deal more damage every basic attack. Including damage from your passive. Easily 900 Damage with The Culling at this point.

Mid to Late Game
+ Blade of the Ruined King (BorK) and Rabadon's Deathcap: Bork is your second main item. This gives you your main Attack Damage bonus with it's attack speed, Damage, life steal, and it's passive.
Rabadon's Deathcap is what gives you all your damage on Ardent Blaze. That is it's soul purpose. It'll add about 40 bonus damage per shot onto your ult. Doesn't seem like much, but it stacks up quick.

Late Game
OR + Last Whisper or The Black Cleaver and Trinity Force: If the enemy team is stacking armor and has more than ~180 armor, then get Last Whisper. If not, Black Cleaver is the way to go.
Trinity Force is just an all round item. Bonus AD, AP, move speed, attack speed, and so on. This item benefits to every area that Lucian needs.

Simple. Just last hit minions and harass as you need to. Remember that Piercing Light can shoot well beyond the targeted enemy. Also Ardent Blaze has a fairly big explosion radius and can pick up some early game kills as well.

Thanks you for reading my guide for playing Lucian as a Hybrid Carry for Mid lane. I hope this helped you understand his play style a little better. If something was unclear or you have a suggestion, please leave a comment and I can get back to you right away. Thanks
-Tsugeta7