$Version
(*"13.2.0 for Microsoft Windows (64-bit) (November 18, 2022)"*)

Integrate[BesselJ[\[Alpha], x*y]*x^\[Beta], {x, 0, Infinity}]

(* ConditionalExpression[(
 2^\[Beta] y^\[Alpha] Abs[y]^(-1 - \[Alpha] - \[Beta])
   Gamma[1/2 (1 + \[Alpha] + \[Beta])])/
 Gamma[1/2 (1 + \[Alpha] - \[Beta])], 
 y \[Element] Reals && Re[\[Beta]] < 1/2 && 
  Re[\[Alpha] + \[Beta]] > -1]*)

$\fbox{$\frac{2^{\beta } y^{\alpha } \Gamma \left(\frac{1}{2} (\alpha +\beta +1)\right) | y| ^{-\alpha -\beta -1}}{\Gamma \left(\frac{1}{2} (\alpha -\beta +1)\right)}\text{ if }y\in \mathbb{R}\land \Re(\beta )<\frac{1}{2}\land \Re(\alpha +\beta )>-1$}$

Regards M.I

Probably Version issue. Maybe this helps:

Assuming[{Re[\[Beta]] < -1/2, Re[\[Alpha] + \[Beta]] > -1, Re[y] > 0}, Integrate[BesselJ[\[Alpha], x*y]*x^\[Beta], {x, 0, Infinity}]]
(* Or: *)
Integrate[BesselJ[\[Alpha], x*y]*x^\[Beta], {x, 0, Infinity}, Assumptions -> {Re[\[Beta]] < -1/2, Re[\[Alpha] + \[Beta]] > -1, Re[y] > 0}]