Integral of BesselJ function doesn't calculate in my version?

Questions
,
1

Why the integral of BesselJ function multiplied by the power function does not calculated in Wolfram Mathematica?

Open in Wolfram Cloud ↗
2

$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

3

Thank you. I know the answer. It can be found in INTEGRALS AND SERIES. Volume 2. Special Functions. By A.P. Prudnikov. Yu. A. Brychkov and O.I. Marichev.
But still it is not calculated in my version 13.0 and in Wolfram Alpha also.

4

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}]
5

Yes, it is integrated with Assumptions. But usually these Assumptions are not known in advance. Thank you.