Calculating the integral

The question is asked in the mathstackexchange. There is no answer. I can’t solve it by hand, also.
Maybe we can solve it with Mathematica.

Appreciate all your help, effort.

Best regards.

  [Psi][0][τ]:=  Piecewise[{{0,  t<0}, { 1-t/h}}];
      [Psi][n][τ]:=  Piecewise[{{0,  t < T-h, {1-(T-t)/h}}];
         [Psi][i][τ]:=   Piecewise[{(t-(i-h)*h)/h],(i-1)*h<=t<=i*h},{((i+1)*h-t)/h,i*h<=t<=(i+1)*h }}];
    
        ((ρ+1)^(1-α)/(Gamma[α]))*Integrate[(z^(ρ+1)-τ^(ρ+1))^(α-1)*τ^ρ*[Psi][i][τ], τ];