Mathematica high-dimensional indefinite integral

    int[var_] := Block[var, fun = Power[Plus @@ var, p]~Power~-1;
    con0 = <|1 -> (StringRiffle[# > 0 & /@ var, "&&"] // ToExpression), 2 -> Plus @@ var < 1|>;
    con = StringRiffle[Values@con0, "&&"] // ToExpression;
    Column[{fun, con}] // TraditionalForm // Print;
    (result = Integrate[fun, var ∈ ImplicitRegion[con, Evaluate@var], Assumptions -> p < Length@var] // Together) // TraditionalForm]
    int@{x, y, z}
    int@{x, y, z, u, v}