I am try to solve the following ODE by the method of separation of variables.
$$\frac{dF}{dx}=\frac{b}{x\left\{\ln^3 x+\ln x[c+d(ax+b)][2\ln x+c+d(ax+b)] \right\}},$$ where $a,b,c,d$ are constants. I think it could contain a special function, say, the logarithmic integral function $$li x=\int_0^x \frac{dt}{\ln t}.$$ After several days' effort, I still can not get the solution. Thus, I seek some help or hint here. There is a related topic posted by me previously.