rF =
  NDSolveValue[
    {r'[ϕ] == -(r[ϕ] - 1)^2/r[ϕ]/Sqrt[2] Sqrt[r[ϕ]^3 + r[ϕ]], r[0] == 100},
    r, {ϕ, 0, 8 π}]

PolarPlot[rF[ϕ], {ϕ, 0, 8 π}, PlotRange -> {{-2, 8}, {-1.5, 1.5}}]