For as long as I remember,Mathematica has always had trouble solving Elliptic integrals:
Using Rubi package:
<< Rubi`
F = Int[ Sqrt[t] /(Sqrt[(-1 + t) (-1 + 5 t)] (-3 + 5 t)), t];
Limit[F, t -> 7/5] - Limit[F, t -> 1] // FullSimplify
(*1/50 (-10 EllipticF[ArcTan[Sqrt[2/5]], -(1/4)] +
2 5^(3/4)
EllipticF[2 ArcTan[Sqrt[7]/5^(1/4)], 1/2 + 3/(2 Sqrt[5])] +
5 (-1 + Sqrt[5]) EllipticK[-(5/2) + (3 Sqrt[5])/2] -
4 5^(3/4) EllipticK[1/10 (5 + 3 Sqrt[5])] +
25 EllipticPi[-(3/2), ArcTan[Sqrt[2/5]], -(1/4)])*)
% // N
(*0.245649 + 0. I*)
