Closure of Bessel Functions of the first kind

I need to use the Bessel functions of the first kind to solve some initial value problem. For this I need the closure equation

$$
int_0^infty J_m(au)J_m(bu)u,text{d}u = frac{delta(a-b)}{a} quadquad text{for}quad a, b, m in mathbb{R},
$$

which can also be found on the Wolfram functions webpage.
However Mathematica (12.3.1.0) does some weird things:

In(12):= Refine(Integrate(u*BesselJ(1, b u)* BesselJ(1, a u), {u, 0, Infinity}), a > 0 && b > 0 && a != b)

Out(12)= ConditionalExpression(0, a > b)
In(13):= Integrate(u*BesselJ(1, 2 u)* BesselJ(1,  3 u), {u, 0, Infinity})

During evaluation of In(13):= Integrate::idiv: Integral of u BesselJ(1,2 u) BesselJ(1,3 u) does not converge on {0,(Infinity)}.

So according to the first expression AND the closure equation the second integral should evaluate to 0. However the integral diverges.

Some colleagues tried evaluating the expression in Mathematica 12.0 and get the correct result.

Is there something wrong in my code?