ja[x_] = 
      Sqrt[2/Pi/
            x] (Cos[x - Pi/4] (1 - 9/128/x^2) + Sin[x - Pi/4]/8/
              x)

f[x_] = 
      x*BesselJ[0, x*Sqrt[2]]*BesselJ[0, x*Sqrt[3]]*BesselJ[0, x*Sqrt[5]]*
        BesselJ[0, x*Sqrt[7]]*BesselJ[0, x*Sqrt[11]]

fa[x_] = 
        x*ja[x*Sqrt[2]]*ja[x*Sqrt[3]]*ja[x*Sqrt[5]]*ja[x*Sqrt[7]]*
          ja[x*Sqrt[11]];

fa[x_] = TrigReduce[fa[x]];

fa[x_] = N[fa[x], 20];

fa[x_] = TrigExpand[fa[x]];

fb1[x_] = 
        Coefficient[PowerExpand[fa[x^2]], 1/x^3]/x^3 + 
          Coefficient[PowerExpand[fa[x^2]], 1/x^5]/x^5 + 
          Coefficient[PowerExpand[fa[x^2]], 1/x^7]/x^7;

fb2[x_] = PowerExpand[fb1[Sqrt[x]]];

p1 = 
      Sum[NIntegrate[f[x], {x, n - 1, n}, WorkingPrecision -> 20], {n, 
          1, 100}]

p2 = N[Integrate[fb2[x], {x, 100, Infinity}], 20]

p1 + p2

q1 = 
      Sum[NIntegrate[f[x], {x, n - 1, n}, WorkingPrecision -> 20], {n, 
          1, 200}]

q2 = N[Integrate[fb2[x], {x, 200, Infinity}], 20]

q1 + q2