What bonuses?
It varies somewhat based on difficulty level. At Suicidal it's approximately 2.64 per 1 bonus research (or 1.64 bonus). So for instance taking the Technologists with a 20% racial research bonus, and grabbing Planetary Improvements on turn 1 (which is doable even in a Gigantic galaxy due to the invisible bonus granted) gives you not a 40% and then 50% bonus, but a 105.6% and then 132% bonus (or in other words 2.056x and 2.32x gross research output respectively).
I may be a little out of date, and it may only be 2.6 per 1, but in my most recent testing it came out to very close to 2.64, consistent across all testing factors, at Suicidal. Iconia's research bonus messes something up, but I couldn't be bothered to find out exactly where.
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However, economy, social, military, population, etc, are all untouched. As far as I know, research is the only thing that is changed, and only when a bonus is applied. Whether that bonus is selected (party/points), researched/obtained (planetary improvements/nano recorders/hyper computers), or run into via anomalies, it all appears to function the same. The one exception, as mentioned, is rings on a planet.
And, just to clarify, the hidden artificial bonus starts to accumulate beyond Tough (that'd be what, Crippling?) although it isn't particularly much at that point-maybe 1.05 to 1.1 per 1 bonus research if I recall correctly (which to be honest I probably don't).
But I don't have any idea what the chances on bonus tiles on the homeworlds are, at any difficulty level for that matter.
-Edit: For the record...I was originally under the impression that Suicidal had a factor of 2.5 rather than 2.6 (or 2.64, as the case may be), but I decided to have another go at it and REALLY crunch the numbers that time...but I don't quite remember my test parameters. I should really write this stuff down while I'm doing it...
Anyway. It's a very close guess.