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*)