GDP of a country gives some idea about how well it is doing in terms of production. In the last two centuries, Western economies overwhelmed others by far.
In the 1950s, US and Europe combined made up 60% of world economy. That’s not counting Canada, Mexico and Australia.
If we look at the fractional share of US+major European countries over the last four decades:

GDP of the US and Europe as fraction of world GDP
We see that their share has been falling over the years. From 62% in the early ’70s, to about 47% now.
So who’s gaining?
There’s a lot of talk about the so-called BRIC (Brazil, Russia, India and China) countries. These countries are supposedly rising stars in the world economy- and have seen unprecedented economic rise. Let’s see if that’s true.
Here is BRIC’s share of world economy over the years:

Fraction of world GDP controlled by the BRIC countries
So it did indeed go up from about 8% of world economy to about 16% now. Impressive.
However, it’s not the whole picture. Let’s examine each of the countries separately. First, Brazil-

Brazil’s GDP divided by the GDP of the world
Brazil has seen a steady rise in the last decade in it’s share of world economy. It now commands about 2.7% of world economy, not very unlike it’s share in 1996 (2.9%), 1982 (2.7%), and 1976 (2.6%). So its position now is by no means “unprecedented.”
Next, Russia:

Russia’s share of world GDP
Russia saw a steady decline in it’s share of world economy through the ’90s. It has recovered it’s strength of the early ’90s now. It had 2.5% of world economy in 1990, now it has about 2.7% too.
What about India?

Indian economy as a Fraction of world GDP
We can see that India has seen a steady rise in it’s share of world economy from the ’90s from about 1% to 2.2% now. But it’s nowhere near it’s best days in the early ’60s share of 3.5%. (Note that we fought two major wars right around that time- with China and Pakistan, and focus shifted from development to armament.)
So, as you’ve guessed, among the BRIC countries only one that has seen real unprecedented rise is China. Here is how it’s share of world economy have changed over the years from a mere 2% to about 7% over the last two decades. It is interesting to note that it had a share of world economy similar to India in the ’60s. Now China’s economy is four times that of India’s.

China’s GDP/world GDP.
But that’s not all. Here is the total share of the major Muslim economies in the world (Turkey + Indonesia + Malaysia + Saudi Arabia + UAE + Qatar + Pakistan + Bangladesh + Libya + Algeria + Kazakhstan).

Fractional share of Muslim economies in the world (total gdp of Muslim countries/world gdp)
That goes up from a baseline of approximately 2.7% in the ’90s to 5% (and rising) now.
This list includes Kazakhstan (became free only in 1991) and Bangladesh (created in 1971). Excluding Kazakhstan, and Bangladesh we see the following:

Fractional share of Muslim economies in the world (Excl. Kazakhstan and Bangladesh)
Therefore, Muslim economies have yet to reach the share they had in the ’80s, in spite of their rise in the last decade. I can only hope and pray that their growth will not be stifled this time. Ameen.
It would be good to know the causes and be able to make some predictions.
Data is from Wolfram Alpha.