I know that if you do proper research on the internet you can find this too, but I thought I'd just post here in the guide section how much the gain is per Artifact Knowledge level. It can stack up to 25.
These levels apply to ALL of your artifacts, even the new ones you haven't obtained yet. The only exception is the Fishing Artifact, which cannot gain any Artifact Knowledge at all, and will continue to give 50 Artifact Power per rare fish. I hope this helps a few souls.
01 25 %02 50 %03 90 %04 140 %05 200 %06 275 %07 375 %08 500 %09 650 %10 850 %11 1,100 %12 1,400 %13 1,775 %14 2,250 %15 2,850 %16 3,600 %17 4,550 %18 5,700 %19 7,200 %20 9,000 %21 11,300 %22 14,200 %23 17,800 %24 22,300 %25 24,900 %
As for Artifact Power needed per next rank / trait / upgrade, please check out this spreadsheet:
Artifact trait level Power required to reach next level Total power required to reach this level % of total power to reach this level % of total power, not including paragon points to reach this level 1  100  -   0.0000% 0.00% 2  300  100 0.0002% 0.00% I use the term paragon points to be the points that incease max hp, damage, or healing depending on spec.  These are intended to be AP sinks generally for single spec play, or late expansion. 3  325  400 0.0006% 0.01% 4  350  725 0.0011% 0.01% 5  375  1,075 0.0016% 0.02% 6  400  1,450 0.0022...