I came across one such integral in my calculations, for which there is no analytical solution. But it exists numerical solution, so how can I derive analytical solution of this integral from numerical solution? Below is my attempt, it might take some time to run the calculation.
Clear["`*"]
$Assumptions = {{\[Alpha], \[Beta]} \[Element] Reals};
R[r_] = Exp[-(1/\[Alpha])*((r/\[Beta])^\[Alpha] - 3)];
RR[r_] = Integrate[R[r]*r^2, {r, 0, r}];
Table[{\[Alpha]1,
RR[r] /. \[Alpha] -> \[Alpha]1}, {\[Alpha]1, {0.13, 0.14, 0.15,
0.16, 0.17, 0.18, 0.19}}]
Thanks a lot for everyone's suggestions and help!