Consider the following ode for $x\rightarrow\infty$
$$\left(x f(x)^3 \left(\frac{(x f^\prime)^\prime}{x}\right)^\prime\right)^\prime=0$$
Assume the function $f(x)>0$ with $x>0$ and introduce an asymptotic expansion for $x\rightarrow\infty$. However, a common power series expansion $$f(x)\sim\sum_{n=0}^\infty c_n x^n$$ for small $x$ may be not very useful.
It might also be easy to estimate the leading order term to be $\sim x^2$ without calculation since we have 4th-order derivative. However, how can I obtain some higher-order terms? In particular, can anyone please give me some hints about how to obtain the following result as $x\rightarrow\infty$:
$$f(x)\sim c x^2+\frac{3}{80c^2}\ln^2x+ O(\ln x),$$ where $c$ and $d$ are constants. I am wondering if it is possible to obtain such a result including $\ln x$-terms with an asymptotic expansion including power functions only?