How much IP do you get for winning & losing games in the various modes/maps?

league-of-legends

My understanding is that you get a base number of Influence Points (IP) for losing each game and a little more for winning a game. In addition, the amount of time a games goes on seems to factor into the calculation as well. For the following modes, what is the base IP gain for wins and losses of the various game lengths? If there's a predictable calculation for IP per minute for wins/losses, that would be an acceptable answer.

I'm not interested in calculations including IP bonuses for your first game, bonuses for streaks of games where you didn't leave, or any other bonuses.

PvP
- Twisted Treeline (3v3)
- Summoner's Rift (5v5)
Co-op vs. AI
- Beginner, Summoner's Rift (5v5)
- Intermediate, Summoner's Rift (5v5)
Custom
- Twisted Treeline (3v3)
- Summoner's Rift (5v5)

Bonus points if you take the initiative to calculate the optimal game length for IP gain. By that, I mean "at what point does ending the current game one way or the other and being able to start a new game result in more IP gain over the course of a play session than if the current game continued".

Solution 1:

General

Custom games are only worth 75% of the IP you would have earned in a normal game.
Co-op vs. AI games incur penalties based on summoner level and difficulty:
- Level 10-19: Beginner games are worth 85%.
- Level 20-29
  - Beginner games are worth 70%.
  - Intermediate games are worth 90%.
- Level 30
  - Beginner games are worth 55%.
  - Intermediate games are worth 80%.

Win

2.305 IP per minute up until 55 minutes.
18 IP bonus for completing a match (~7.8 minutes worth of IP)

Loss

1.402 IP per minute up until 55 minutes.
16 IP bonus for completing a match (~11.4 minutes worth of IP)

Graph

IP Gain Graph

Optimum Game Length

Riot Games has designed their IP system to reward playing the game to completion and to reward longer games:

Q: Will I be getting more or less IP and XP as a result of this change?

A: While you will get more or less in some games, the average XP/IP gain will be much more consistent. If you typically played very long games, you will tend to get more IP and XP; however, if you typically play short games, you will get slightly less IP and XP.

While you do get a bonus beyond the "per minute IP", that bonus is designed to compensate players for the time between matches:

Q: It seems unfair that I spend 5 minutes milling around in queue and in champion select, and don't get an IP/XP reward for that. Does this system help with this at all?

A: Actually, it does! All game types give a slight 'completion bonus' that is equivalent to several minutes of play. This is embedded in IP you earn.

That being said: provided you could get into another game in substantially less that 7.8 minutes, you'd make out slightly better by surrendering and starting a new game. You'd have to do this a lot to see a substantial gain, however. If it took you longer to get into the next game, you'd actually incur a loss overall.

In short, it'll usually be best to play to 55 minutes, then bail out if you're very likely to lose. At that point, you're essentially playing to see if you'll get 145 IP vs 94 IP. If losing seems like a foregone conclusion, then you're mostly playing for the experience at that point.

Sources

Items in the general section: League of Legends @ Wikia
Win/Loss Accrual Rates: Based on recorded values after the completion of games. There are some instances where (rate * minutes) does not exactly equal what I recorded from the game client when you round up, but overall the above rates serve as a good enough rule of thumb for my purposes.
IP System FAQs

Reference

League of Legends Release Notes v1.0.0.111
League of Legends Release Notes v1.53