[Community Question] Calculus: Convergence of $\int_0^{+\infty}\frac{\ln(x^{3\over2})|\ln^{\alpha-{2\over3}}(x+x^2)|}{(x^{1\over3+1})^2x^{\alpha}}$

One of our user asked:

Firstly,if I evaluate behavior of function at $+\infty$: $$f(x)\sim\frac{\ln(x^{3\over2})|\ln^{\alpha-{2\over3}}(x^2)|}{x^{\alpha+{2\over3}}}$$ And I know that it is convergent when $\alpha+{2\over3}\gt1\implies\alpha\gt{1\over3}$.
As for $x\to0^+$ $f(x)\sim\frac{|\ln^{\alpha-{2\over3}}(x)|}{x^\alpha}$ and I know that when $\alpha={2\over3}$ we have ${1\over x^{2\over3}}$,which is convergent because ${2\over3}\lt1$.And I observe that $\forall\alpha\leq0$ limit of the quotient equals $0$ and $\forall\alpha\gt0$ to $\infty$.But I am not sure what it gives me and I can't think of any "standart conditions" I can use,like I did before in this problem.