Which items reasonably increase or have an active effect on AD?

With active you have the Hydras (Titanic (+50dmg) and Ravenous (+75dmg)), both of them have a damage active AoE. The main difference is that Titanic is for tanking and Ravenous damage, the active of the tanky deals damage scaling with HP and the damage one, scales with AD. You also have BOTRK (+25 dmg), which it's acctive slows the target and steals some HP from him.

With no active, just damage stats, you have the three items made with B.F. Sword:

1. Infinity Edge: Good damage (+65), crit. chance, and crit. damage. Rushed in lot of ranged AD champs.
2. Bloodthirster: Great damage (+75), life steal, and gives a shield when you overheal.
3. Essence Reaver: Decent damage (+65), crit. chance, AA regenerates mana, gives you CD reduction based on crit. chance.

Also you have the Mercurial Scimitar which gives you great damage (+75), life steal, magic ressitance and the active removes a debuff on you (stun, snare, etc) and increase mov. speed for a second.

If you want deal damage you will probably also need armor pen. And Last Whisper gives you that and 40 AD also (and any of the 2 items you built with it, which are Mortal Reminder and Lord Dominik's Regards, check both because they have differenet passives). There are others items with armor pen, but I think that this one is the more viable. Maybe Youmuu's Ghostblade could be a good choice too since it has more damage (+65), CD reduction and also some armor pen. and an active that gives movement and attack speed.

Edit As @Probs and @user1337 said: Death's Dance has plenty of AD (+75) with CD reduction and some lifesteal, and a passive that makes you take damager over time insead of instant damage. Also added the Yomumu's Ghostblade to the list, thanks @Probst. And the links to both Last Whispers items, thanks @user1337.

Hope this help you. If anyone gives any idea on other item to add would be nice.

For pure burst

• Duskblade of Draktharr +75 Attack Damage +5% Movement Speed

UNIQUE Passive: +10 Armor Penetration UNIQUE Passive: Basic Attacks on an enemy champion apply Nightfall (120 second cooldown).

Nightfall: After 2 seconds, deal physical damage equal to 90 plus 25% of the target's missing health. If you get a kill or assist on the target before Nightfall ends, the cooldown is refunded.

• Infinity Edge +65 Attack Damage +20% Critical Strike Chance

UNIQUE Passive: Critical strike bonus damage is increased by 50%.

• Essence Reaver +65 Attack Damage +20% Critical Strike Chance

UNIQUE Passive: +10% Cooldown Reduction. UNIQUE Passive: Gain increasingly more Cooldown Reduction from Critical Strike Chance provided by other sources (maximum +20% additional Cooldown Reduction at 30% Critical Strike Chance). UNIQUE Passive: Critical strikes restore 3% of your maximum Mana pool.

• Yomuu's Ghostblade +65 Attack Damage +10% Cooldown Reduction

UNIQUE Passive: +20 Armor Penetration UNIQUE Active: Grants +20% Movement Speed and +40% Attack Speed for 6 seconds (45 second cooldown).

• Lord Dominik's Regards +40 Attack Damage

UNIQUE Passive - Giant Slayer: Grants up to +15% physical damage against enemy champions with greater maximum Health than you (+1.5% damage per 50 Health difference, maxing at 500 Health difference). UNIQUE Passive - Last Whisper: +45% Bonus Armor Penetration

For Burst and Sustain/Survivability

• Bloodthirster +75 Attack Damage

UNIQUE Passive: +20% Life Steal UNIQUE Passive: Your basic attacks can now overheal you. Excess life is stored as a shield that can block 50-350 damage, based on champion level.

This shield decays slowly if you haven't dealt or taken damage in the last 25 seconds.

• Blade of the Ruined King +25 Attack Damage +40% Attack Speed +10% Life Steal

UNIQUE Passive: Basic attacks deal 6% of the target's current Health in bonus physical damage (max 60 vs. monsters and minions) on hit. Life Steal applies to this damage. UNIQUE Active: Deals 10% of target champion's maximum Health (min. 100) as physical damage, heals for the same amount, and steals 25% of the target's Movement Speed for 3 seconds (90 second cooldown).

• Death's Dance +75 Attack Damage +10% Cooldown Reduction

UNIQUE Passive: Dealing physical damage heals for 15% of the damage dealt. This is 33% as effective for Area of Effect damage. UNIQUE Passive: 15% of damage taken is dealt as a Bleed effect over 3 seconds instead.

• Ravenous Hydra +75 Attack Damage +100% Base Health Regen +12% Life Steal

Passive: 50% of total Life Steal applies to damage dealt by this item. UNIQUE Passive - Cleave: Basic attacks deal 20% to 60% of total Attack Damage as bonus physical damage to enemies near the target on hit (enemies closest to the target take the most damage). UNIQUE Active - Crescent: Deals 60% to 100% of total Attack Damage as physical damage to nearby enemy units (closest enemies take the most damage) (10 second cooldown).

• Titanic Hydra +450 Health +50 Attack Damage +100% Base Health Regen

UNIQUE Passive - Cleave: Basic attacks deal 5 + 1% of your maximum health as bonus physical damage to your target and 40 + 2.5% of your maximum health as physical damage to other enemies in a cone on hit. UNIQUE Active - Crescent: Cleave damage to all targets is increased to 40 + 10% of your maximum health as bonus physical damage in a larger cone for your next basic attack (20 second cooldown).

• Mercurial Scimitar +75 Attack Damage +35 Magic Resist +10% Life Steal

UNIQUE Active - Quicksilver: Removes all debuffs and also grants +50% bonus Movement Speed for 1 second (90 second cooldown).

For countering specific champs/mechanics

• Mortal Reminder ((Healing Reduction)) +40 Attack Damage

UNIQUE Passive - Executioner: Physical damage inflicts Grievous Wounds on enemy champions for 5 seconds. UNIQUE Passive - Last Whisper: +45% Bonus Armor Penetration.

• Black Cleaver ((Armor Reduction)) +300 Health +55 Attack Damage +20% Cooldown Reduction

UNIQUE Passive: Dealing physical damage to an enemy champion Cleaves them, reducing their Armor by 5% for 6 seconds (stacks up to 6 times, up to 30%). UNIQUE Passive - Rage: Dealing physical damage grants 20 movement speed for 2 seconds. Assists on Cleaved enemy champions or kills on any unit grant 60 movement speed for 2 seconds instead. This Movement Speed is halved for ranged champions.

• Maw of Malmortius ((Magic Damage heavy team)) +55 Attack Damage +50 Magic Resist +10 Armor Penetration

UNIQUE Passive - Lifeline: Upon taking magic damage that would reduce Health below 30%, grants a shield that absorbs magic damage equal to 300 + 1 per bonus Magic Resistance for 5 seconds (90 second cooldown). UNIQUE Passive - Lifegrip: When Lifeline triggers, gain +25% Attack Speed, +10% Spell Vamp, and +10% Life Steal until out of combat.

I've left out Muramana and Gunblade as neither are very good at quickly bursting for AD Champs and have non-optimal build paths for burst in the current meta.

An interesting note is that Titanic Hydra's active acts as an Auto Attack reset, whereas Ravenous Hydra's does not. This is because Titanic Hydra's active is an On-Hit active, whereas Ravenous Hydra's active is a PBAoE Effect.

I'm going to assume if you're looking for pure burst, you're playing champs like Zed, Talon, and Kha'Zix. These guys are looking for pure AD and Armor Penetration. Their general playstyle is get in, murder someone, and get out alive. None of the typical AD Assassins make good use of crit, because they use abilities almost exclusively, so I would steer away from crit items. Duskblade is alright on Talon and Kha'Zix, but overkill for Zed, since he can 100-0 people very easily without it. Yomuu's is a very typical first item for Zed, and is a very good item on all three of these characters because of the flat AD and Armor Penetration, as well as the movespeed active. After either Yomuu's or Duskblade, you'll typically want some form of either sustain or survivability, unless you're Zed since he can get out of anything pretty easily. Both Maw and Black Cleaver offer good stats for their price, and their passives can be absolutely incredible when used optimally.

After you get some form of starting burst/damage item and then something for survivability, game flow will dictate the rest of your items. Crushing face? More damage. Winning but getting blown up? Pick up something with survivability. Can't blow anyone up? Get an Armor Pen item.

If you can provide some specific champs you're looking to pick up, I can give you further advice as well.