Hear me out first, I've been yelled at so many times for this build. But it works very well, you get wicked burst.
I start boots and pots, as per usual, and buy sheen (and hopefully berserker greaves) on my first trip back. Second trip is Tear of the Godess, followed by manamune. Time to start stacking!
Around Mid game, I end up with Berzerkers, Manamune, Sheen, Vampiric Scepter, and perhaps a BF sword or Zeal.
Regardless, next I get Trinity force, then IE, then finish with Last whisper and bloodthirster.
Lets do some math, to see how it compares to a regular ADC build.
Say a regular build is Berserkers, Bloodthirster, black cleaver, last whisper, infinity edge, and phantom dancer.
Both these builds are pretty expensive, trinity force is ~16.1k, and standerd ADC is 15.5k.
So lets take a look at some numbers. At level 18, draven has 110 damage and 0.979 AS. We'll assume all the items are fully stacked here (manamune usually transforms for me at about 30 min, maxes about 40.)
The only two item differences are Black Cleaver+Phantom Dancer or Trinity Force+Manamura. So lets compare!
With Black cleaver, you get 50 more damage, and wicked armour pen. This Pen alone may be a game changer. Phantom Dancer adds 50% AS, 30% critical and 5% MS.
Trinity force which is 30% AS, 10% critical strike, and 8% MS, and manamura is 20 damage + 2% of max mana (which is 44 more at full stacks), so 64 damage (more then black cleaver), but it also has the toggle to burn 3% of current mana and add double that as damage (works out to about 100 bonus).
The damage lays out as follows for each build:
Standerd ADC: 370 base + 315 (spinning axe bonus, 85% of base) to total 685 per hit.
Trinity+Manamura: 413 base + 165 (trinity force proc) + 350 (Spinning ax bonus, 85% of base) + 100 (manamura proc) to total at 1028 damage.
The trinity force only procs every 2 seconds, but you can make it proc every 2 second because W refreshes whenever you catch an axe, and with manamura you have the mana to sustain. So teamfights look like QWQWQWQWQWQWQWQWQ as you run around catching axes.
With both builds you get the criticals of IE, standerd ADc you have a 55% chance and 1.665 AS, with trinity force you have 35% chance with 1.469 AS.
What do you guys think? The Black cleaver armour pen may make up the difference here, but either way the trinity force has higher burst. Last game I played I two-hit the enemy ashe with it.