Oops! I just noticed that in my previous post I wrote the wrong denominator in the sqrt for the p = 5/2 case. It should be 6 not 5. IOW
R_avg = √[(r_2^2 + r_2^(8/5)*r_1^(2/5) + r_2^(6/5)*r_1^(4/5) + r_2^(4/5)*r_1^(6/5) + r_2^(2/5)*r_1^(8/5 ) + r_1^2))/6] when p = 5/2.
Whenever there is a finite sum replacing the quotient of differences of powers the denominator, i.e. 2*p +1, is always equal to the number of terms in the finite sum.
Also Tom van Baak's great post mentioning of Gauss' AGM brings up a nice relationship between elliptic integrals and the AGM which, among other things, gives a quite useful in formula the numerical evaluation of elliptic integrals in that one can use it to get a recursive formula to evaluate a complete elliptic integral of a large modulus (where the Taylor power series in the modulus converges slowly) in terms of another complete elliptic integral with a significantly smaller modulus which is much easier to evaluate. The formula can even be recursively used all the way down to a modulus indistinguishable from zero that needs no evaluation other than the π/2 coefficient in front. In particular, note for any k such that 0 ≤ k < 1 the value of K(k) can be recursively calculated using K(k) = (2/(1 + k'))*K((k/(1 + k'))^2) where k' ≡ √[1 - k^2]. The cool thing here is that the new modulus, i.e. (k/(1 + k'))^2 , is significantly smaller than the original one, k. Just a few iterations of this recursion knocks the modulus down so far that it becomes insignificant, and the complete elliptic integral can then be easily evaluated. This is useful in finding such things as the period of a simple pendulum whose maximum swing amplitude is an obtuse angle.