:

## elliptic integrals

Maple
`J:=Int((3*v^2+4*v+2)*(-(3*v^2+2*v+1)/(3*v^2+8*v+4))^(1/2)/(1+2*v)^2,      v = -2 .. -2/3) Then J = -8/3*EllipticK(1/3) + 8/3*EllipticPi(-1/3,1/3) + 3*EllipticE(1/3)That integral caught my attention through a private discussion. Both MMA and Mapleonly succeded up to a limit in the lower boundary, which could not be determined.Re-writing the integrand as A/(polynomial of degree 4) using the command 'rationalize'and simplifying (assuming the boundaries for the variable) one can use a presentationas real partial fraction for 1/B by 'convert(1/B, parfrac, v)'. Then Maple finds the integrals for A*(each summand) and adding gives the above.After playing a bit more I recognized, that Maple sees Elliptics as integrals overthe unit interval (by definition). Which motivated me to transform the task to that.And - astonishingly - Maple finds the answer:  J;  `` = IntegrationTools[Change](%, x = 3/4*v+3/2, x); # over the unit interval  value(%);  evalf(%);      -8/3*EllipticK(1/3) + 8/3*EllipticPi(-1/3,1/3) + 3*EllipticE(1/3)Is there any reason, why the system does not try that (just forgotten to be done)?`

﻿