$Version
(* "13.0.1 for Mac OS X x86 (64-bit) (January 28, 2022)" *)
Clear["Global`*"]
int = Integrate[x/Sqrt[x^4 + 10*x^2 - 96*x - 71], x]
(* -(1/8) Log[-10001 - 3124 x^2 + 1408 x^3 - 54 x^4 + 128 x^5 - 20 x^6 - x^8 +
Sqrt[-71 - 96 x + 10 x^2 +
x^4] (781 - 528 x + 27 x^2 - 80 x^3 + 15 x^4 + x^6)] *)
intR = Integrate[x/Sqrt[x^4 + 10*x^2 - 96*x - 71], x,
Assumptions -> x ∈ Reals];
The results are identical with or without the Assumptions
int === intR
(* True *)
The result that you report from Maple
intMaple = 1/8*
Log[(x^6 + 15 x^4 - 80 x^3 + 27 x^2 - 528 x + 781)*
Sqrt[x^4 + 10 x^2 - 96 x - 71] +
x^8 + 20 x^6 - 128 x^5 + 54 x^4 - 1408 x^3 + 3124 x^2 + 10001];
There is little difference in their complexity
LeafCount /@ {int, intMaple}
(* {78, 76} *)
They are equivalent anti-derivatives
D[int, x] == D[intMaple, x] // Simplify
(* True *)
Since you are taking the indefinite integral (anti-derivative), results are equivalent if they differ by only a constant (which can be complex).
The two results differ by a piecewise complex constant
ReImPlot[int - intMaple, {x, -10, 10},
PlotLegends -> Placed["ReIm", {.8, .4}]]

int - intMaple /. x -> 5/2 // FullSimplify
(* (I π)/8 + Log[1/(4 3^(7/8))] *)
% // N
(* -2.34758 + 0.392699 I *)
int - intMaple /. x -> 5 // FullSimplify
(* -((I π)/8) + Log[1/(4 3^(7/8))] *)
% // N
(* -2.34758 - 0.392699 I *)
$Version
"13.0.0 for Microsoft Windows (64-bit) (December 3, 2021)"
$\endgroup$SetSystemOptions[ "IntegrateOptions" -> {"IntegrateAlgebraicTimeConstraint" -> 100}]
and see if that improves things. $\endgroup$