Mathematica gives an unexpected answer for Integrate

$\int\frac{\sqrt{C + (1 - C) x^3}}{x}dx=\int\frac{\left(\sqrt{C + (1 - C) x^3}\right)x^2}{x^3}dx$

Let $C + (1 - C) x^3=u^2$

$3(1 - C) x^2dx=2udu$

$\frac{2}{3}\int\frac{u^2du}{u^2-C}du$

$\frac{2}{3}\int\left(1+\frac{c}{u^2-C}\right)du$

$\frac{2}{3}\left(u+C\int\frac{du}{u^2-\sqrt{C}^2}\right)$

$\frac{2}{3}\left(u-C\frac{Arctanh(\frac{u}{\sqrt{C}})}{\sqrt{C}}\right)$

Substituting back gives

\begin{equation}\tag{2} \frac{2}{3} \sqrt{C + (1 - C)\, x^3} - \frac{2}{3} \sqrt{C} \, \mathrm{argtanh} \Big( \frac{\sqrt{C + (1 - C) \, x^3}}{\sqrt{C}} \Big). \end{equation}

FunctionDomain[ArcTanh[Sqrt[C + (1 - C) x^3]/Sqrt[C]], x]

-1 < Sqrt[C + x^3 - C x^3]/Sqrt[C] < 1 && -C - x^3 + C x^3 <= 0

Or

 ArcTanh[Sqrt[C + (1 - C) x^3]/Sqrt[C]] // TrigToExp

-(1/2) Log[1 - Sqrt[C + (1 - C) x^3]/Sqrt[C]] + 1/2 Log[1 + Sqrt[C + (1 - C) x^3]/Sqrt[C]]

Here are four five six forms for an antiderivative, the first ad1 being the same as the OP's and complex-valued over the intended domain. The other three four five are real-valued.

ad1 = Integrate[Sqrt[c + (1 - c) x^3]/x, x, Assumptions -> 0 < c < 1]
(*  2/3 Sqrt[c + x^3 - c x^3] - 2/3 Sqrt[c] ArcTanh[Sqrt[c + (1 - c) x^3]/Sqrt[c]]  *)

The imaginary part of ad is constant so we can subtract it off (which is the point of @JimB's comment under the OP):

ad2 = ad1 /. {{x -> 1}, {x -> x}} // Differences // First;

Assuming[0 < c < 1 && x > 0,
 TrigToExp@ad2 /. {Log[z_] /; Simplify[z < 0] :> Log[-1] + Log[-z]} // 
   Expand // FullSimplify
 ]
(*
2/3 (-1 + Sqrt[c + x^3 - c x^3] + 
   Sqrt[c] ArcTanh[(Sqrt[c] (x^3 - Sqrt[c + x^3 - c x^3]))/(c + x^3)])
*)

Applying the identity, $$\tanh ^{-1}\left(\frac{1}{z}\right) - \tanh ^{-1}(z)=\frac{i \pi}{2} \,,$$ changes ad1 by a constant, which results in another antiderivative:

ad3 = ad1 /. ArcTanh[z_] :> ArcTanh[1/z]
(*  2/3 Sqrt[c + x^3 - c x^3] - 2/3 Sqrt[c] ArcTanh[Sqrt[c]/Sqrt[c + (1 - c) x^3]] *)

Rewrite ad3 in terms of logarithms:

ad4 = TrigToExp@ad3 /. 
  a__ Log[u_] + b__ Log[v_] + w__ /; Times@a == -Times@b :> a Log@Simplify[u/v] + w
(*
  2/3 Sqrt[c + x^3 - c x^3] + 
   1/3 Sqrt[c] Log[(-Sqrt[c] + Sqrt[c + x^3 - c x^3]) /
     (Sqrt[c] + Sqrt[c + x^3 - c x^3])]
*)

Check:

D[{ad1, ad2, ad3, ad4}, x] // Simplify
(*
  {Sqrt[c + x^3 - c x^3]/x, Sqrt[c + x^3 - c x^3]/x,
   Sqrt[c + x^3 - c x^3]/x, Sqrt[c + x^3 - c x^3]/x}
*)

Addendum: Here is a fifth, the real part of ad1:

ad5 = ComplexExpand[Re[ad1], TargetFunctions -> {Re, Im}] // 
 Simplify[#, 0 < c < 1 && x > (c/(1 - c))^(1/3)] &
(*
  1/6 (4 Sqrt[c + x^3 - c x^3] + 
     Sqrt[c] Log[(x^3 - c (-2 + x^3) - 2 Sqrt[c (c + x^3 - c x^3)])/c] -
     Sqrt[c] Log[(x^3 - c (-2 + x^3) + 2 Sqrt[c (c + x^3 - c x^3)])/c])
*)

Addendum 2: Doing a line integral within the domain of interest will often give the desired result; however, this time it ran forever, until I turned off GenerateConditions. The antiderivative comes back in terms of HypergeometricPFQ which can be simplified with the aid of FunctionExpand.

Assuming[0 < c < 1 && x > 0 && t > 0, 
 ad6 = Simplify@ FunctionExpand@
    Integrate[Sqrt[c + (1 - c) t^3]/t, {t, 1, x}, 
     Assumptions -> 0 < c < 1 && x > 0, GenerateConditions -> False]
 ]
(*
  1/3 (-2 + 2 Sqrt[c + x^3 - c x^3] + 2 Sqrt[c] Log[1 + 1/Sqrt[c]] + 
     3 Sqrt[c] Log[x] - 2 Sqrt[c] Log[1 + Sqrt[1 + (-1 + 1/c) x^3]])
*)

(* Check *)
Simplify[D[ad6, x] - Sqrt[c + x^3 - c x^3]/x, 0 < c < 1 && x > 0]