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 Maple
only 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 presentation
as 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 over
the 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)?