What difficulty are you on?
I've done this on normal, and on hard. On normal I found it fairly straightforward. On hard it was a complete nightmare! At their main base there are a mass of anti-air units (~10), and a similar number of archons (who will kill you quickly and reveal your cloak). Then, when you finally get to the artifact, a bunch more of each pop out to polish you off. Nightmare. I bet it's fun on brutal...
Anyway... To start, build 5+ SCVs, and the gas extractors. Then make sure you build a second starport asap (with tech lab), and start pumping out banshees. When you have 6 or 7 (on normal... more on hard), send them in to the nearest enemy base, cloaked. You should be about depleting the first field when you move, assuming you built enough SCVs. Then move your base in sharpish, with SCVs, and start mining again. You'll need to build more refinaries. Also, you'll need a few anti-air units - wraiths are best since they can cloak, and you shouldn't be short of vespene. On normal, 4 should be enough.
Attack the next enemy base, to the eat of you, and send the base in afterwards. Go in with the banshees, and keep the wraiths just behind. Send them in only for the enemy air units, then retreat. Again, you'll need to start mining ASAP, and churning out more banshees and wraiths.
...then you attack the main base. On normal, ~20 craft total should be enough, and will be very easy to amass. On hard, the only way I could do it was to lift-off all my buildings and send them in first as a heavily armoured distraction
Then send my units in afterwards while at least some of the enemy is preoccupied.Anyway... try this. On normal, doing it this way with the buildings, you really shouldn't need very many forces at all.
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