I think (using Maple) that the integral equals Integrate[((2 + 4*v + 3*v^2)* Sqrt[-((1 + 2*v + 3*v^2)/(4 + 8*v + 3*v^2))])/(1 + 2*v)^2, {v, -2, -2/3}] = -8/3*EllipticK(1/3)+8/3*EllipticPi(-1/3,1/3)+3*EllipticE(1/3) I normalized it as one fraction as A/(polynomial of degree 4), did a presentation using partial fraction. Then Maple luckily succeeds for each summand and finally arrived after simplification with the above. A = ((v+2)*(3*v+2))^(1/2)*(3*v^2+4*v+2)*(3*v^2+2*v+1)^(1/2)/(v+2)/(3*v+2)/(1+2*v)^2*I