Get either an IP boost or RP if you want to get a champ as soon as possible. I know a while back RIOT implemented a change that made it so the longer your games were the more IP you would get. I also hear that a good option for farming IP is playing Dominion but I myself am not sure if that's true.
The most reliable way to play is in a PvP on summoner's rift, as it gives the best rewards out of all the game types, and to win your first win of the day. That will give you the maximum amount of IP with the least amount of effort. If you want, you can continue playing, but as long as you get the first win of the day IP (+150) every day, you can get a lot of IP.
10 days gives you an extra 1500 IP.
20 days gives you an extra 3000 IP.
40 days gives you an extra 6000 IP.
As you can see, it still takes a while, but that's assuming you play one game a day (and you win), and doesn't take into account the normal IP recieved from playing matches. If you were to win 10 matches a day, you'd probably get somewhere around 800 - 900 IP or more.
Also, another tip. Don't play Co-op vs. AI unless you are having trouble with a champion and need to learn them better. Or if you want to practice. Riot pretty much drained Co-op vs. AI of all the rewards, and basically made it a learning experience. Where you would recieve 76 IP for a bot game, you would get like 120~ for a normal game.
If you really want to get IP quickly, buy like a 10 day IP boost or something. Each normal game on summoner's rift gives like 170~ extra IP per win.
10 days gives you an extra 1500 IP.
20 days gives you an extra 3000 IP.
40 days gives you an extra 6000 IP.
As you can see, it still takes a while, but that's assuming you play one game a day (and you win), and doesn't take into account the normal IP recieved from playing matches. If you were to win 10 matches a day, you'd probably get somewhere around 800 - 900 IP or more.
Also, another tip. Don't play Co-op vs. AI unless you are having trouble with a champion and need to learn them better. Or if you want to practice. Riot pretty much drained Co-op vs. AI of all the rewards, and basically made it a learning experience. Where you would recieve 76 IP for a bot game, you would get like 120~ for a normal game.
If you really want to get IP quickly, buy like a 10 day IP boost or something. Each normal game on summoner's rift gives like 170~ extra IP per win.

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As you have your IP boost you should reap around 200+ IP from a resonably long Rift PVP. So as long as you play a few games a day you should get a good income.
IP boost resolves this issue.
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LOL, this may seem a bit off but it works. Don't even bother with your IP and just play. Soon, after a surprisingly short time-frame, you'll have a ton-load of IP. I did a 42 hour gaming spree with my friend and made SOOOO much IP.
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Normal 5v5 has the richest gameplay and highest IP rewards. Play it a lot and win as much as possible. That's still the fastest way to gain IP as far as I know. It's also a lot more rewarding than just spamming AI games since you're pitted against real players. You really don't learn much from playing against bots. If you want to improve as a player, you have to play PvP.

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I know they recently changed the way the rewards are given and I just played a 15 minute game solo vs ai and only got 75 ip meaning i'd have to play another 46 games doing the same thing which is 11 hours and 40 minutes ish of actually attempting to farm games.
I was just curious so then that way I can just get one of these champs asap, preferably Ez or Graves because they seem a much more high aggressive lane, which is the style I enjoy.
Thanks. :]