u[n_,l_,r_]=1/(n Sqrt[Pochhammer[(n-l), 2 l +1]]) (2 r/n)^(l+1) LaguerreL[n-l-1,2 l +1,2 r/n] Exp[-r/n]
{Integrate[u[5,3,r]^2,{r,0,Infinity}],Integrate[u[1,0,r]^2,{r,0,Infinity}],Integrate[u[9,7,r]^2,{r,0,Infinity}]}
Out[2]= {1, 1, 1}

ans[n_,l_]=1/n^2
{Integrate[u[5,3,r]^2/r,{r,0,Infinity}],Integrate[u[1,0,r]^2/r,{r,0,Infinity}],Integrate[u[9,7,r]^2/r,{r,0,Infinity}]}
{ans[5,3],ans[1,0],ans[9,7]}
         1      1
Out[6]= {--, 1, --}
         25     81

ans[n_,l_]=(3 n^2-l(l+1))/2
{Integrate[u[5,3,r]^2 r,{r,0,Infinity}],Integrate[u[1,0,r]^2 r,{r,0,Infinity}],Integrate[u[9,7,r]^2 r,{r,0,Infinity}]}
{ans[5,3],ans[1,0],ans[9,7]}
         63  3  187
Out[9]= {--, -, ---}
         2   2   2

ans[n_,l_]=n^2(5 n^2+1-3l(l+1))/2
{Integrate[u[5,3,r]^2 r^2,{r,0,Infinity}],Integrate[u[1,0,r]^2 r^2,{r,0,Infinity}],Integrate[u[9,7,r]^2 r^2,{r,0,Infinity}]}
{ans[5,3],ans[1,0],ans[9,7]}
Out[12]= {1125, 3, 9639}

ans[n_,l_]=1/((l+1/2)n^3)
{Integrate[u[5,3,r]^2 r^-2,{r,0,Infinity}],Integrate[u[1,0,r]^2 r^-2,{r,0,Infinity}],Integrate[u[9,7,r]^2 r^-2,{r,0,Infinity}]}
{ans[5,3],ans[1,0],ans[9,7]}
           2        2
Out[15]= {---, 2, -----}
          875     10935



