Just as a curios observation, we could make integral "close" to elliptic with some Möbius transformation.
I've just played with it, like this:

Clear[u, t, x]; u = t/(1 + t);

FullSimplify[
 Integrate[FullSimplify[D[u, t]/Sqrt[1 + u^3]], t] /. 
  Solve[x == u, t]]

(*(2 (2 + (-1)^(2/3)) Sqrt[((1 + (-1)^(2/3) - x) (1 + x))/(2 + (-1)^(
   2/3))^2] Sqrt[(-1 + (-1)^(1/3) + x)/(-2 + (-1)^(1/3))]
  EllipticF[ArcSin[Sqrt[(1 + x)/(2 + (-1)^(2/3))]], 
  1/2 (1 + I Sqrt[3])])/Sqrt[1 + x^3]*)

I'm sure one can find a transformation that gives the simplest result, but I don’t know how (