Stats: 

+ 25 Attack Damage

+ 40% Attack Speed

+ 15% Life Steal

Unique Passive: Your attacks deal 5% of the target's current health in magic damage (60 max vs. minions) on hit. Unique Active: Deals 15% of the target champion's maximum Health (min. 100) as physical damage, heals for the same amount, and steals 30% of the target's Movement Speed for 3 seconds (90 second cooldown).

(Unique passives with the same name do not stack)

With the diversity of item choices for ad carries brought forth by season 3, which ad item is the most effective? I will assume that the player in my calculations has .625 attack speed, 60 attack damage and no other stats from other items, runes or masteries. I will also not consider armor because armor will act the same on all physical damage.

The Bloodthirster

The Bloodthirster gives 70 AD and 12% lifesteal. Additionaly it gains stacks on unit kills up to a cap of 30 with each stack giving 1 AD and .2% lifesteal. Fully stacked the Bloodthirster gives 100 AD and 18% lifesteal.

With no stacks this gives our player 130 AD and .625 attack speed. Not calculating the enemies armor this gives a DPS (damage per second) of 130*.625= 81.25

Fully stacked The Bloodthirster gives our player 160 and .625 attack speed. Again not considering armor this gives our player a DPS of 160*.625=100.

Of course the Bloodthirster also gives between 12 and 18% lifesteal. While this does not affect a players DPS, it will affect their survivability. This is why the Bloodthirster is a good choice for many short ranged AD carries like Vayne and Draven. Additionaly the Bloodthirster has the highest raw AD value of any item in the game so if your champion has a lot of AD scaling abilities it might be a good choice. However, for the purpose of these calculations I am not taking abilities into account when I calculate DPS.

Infinity Edge

Infinity Edge gives the player 70 AD, 25% crit chance and a unique passive that increases critical strike damage by 50%.

This gives our player 130 AD, .625 attack speed and 25% crit chance. Over the course of 100 basic attacks, the player is expected to crit 25 times. The 75 normal attacks of these 100 will deal 75*130=9750 damage. The 25 critical strikes will deal (130*2.5)*25=8125.

Over the course of 100 auto attacks, our player with an Infinity Edge will deal 8125+9750=17,875 Damage. Because these attacks are coming at  a speed of .625 attacks per second it will take 160 seconds to deliver 100 basic attacks. 17,875/160=111.71875 DPS.

This means that for pure auto attacking, Infinity Edge is better than a bloodthirster both with and without stacks. However it gives no lifesteal and less AD and is therefore more beneficial to long range champions that can stay our of danger like Caitlyn and Tristana.

  Blade of the Ruined king

BOTRK gives the player 25 AD, 40% attack speed and 15% lifesteal. Additionaly it has a unique passive that causes basic attacks to deal 4% of the opponents current health as physical damage on-hit. BOTRK also has an active effect that steals a percentage of the targets health and movement speed for a duration.

After buying this item our player has 85 AD, .625(1+.40)=.875 attack speed and 15% lifesteal.

This gives our player 85*.875=74.375 DPS.

However the unique passive of BOTRK gives you an additional 4 damage on hit for every 100 health the opponent has. This equates to 4*.875=3.5 bonus DPS per 100 hp of your opponent.

This means the opponent must have approximately 1300 hp (13*3.5=45.5 | 45.5+74.375=119.875) to be more effective than Infinity Edge.

If the opponent has 1300 hp BOTRK gives the player 119.875 DPS. I realize that because it is 4% of current health the DPS will decrease as the opponent gets lower, but this gives a pretty good image of how much damage you can deal.

This DPS is superior to both of the other items examined and this is without the active effect taken into account.

  Overall BOTRK is extremely effective if your champion is only auto-attacking, but it gives only 25 AD and is therefor not very effective when you are looking to improve the damage on spells with AD scaling.

Infinity Edge also gives very high DPS with pure auto-attacking, but provides no lifesteal or other survivability.