Test for convergence of improper integrals $\int_{0}^{1}\frac{\sqrt{x}}{(1+x)\ln^3(1+x)}dx$ and $\int_{1}^{\infty}\frac{\sqrt{x}}{(1+x)\ln^3(1+x)}dx$ I need to test if, integrals below, either converge or diverge:
1) $\displaystyle\int_{0}^{1}\frac{\sqrt{x}}{(1+x)\ln^3(1+x)}dx$
2) $\displaystyle\int_{1}^{\infty}\frac{\sqrt{x}}{(1+x)\ln^3(1+x)}dx$
I tried comparing with $\displaystyle\int_{0}^{1}\frac{1}{(1+x)\ln^3(1+x)}dx$, $\displaystyle\int_{0}^{1}\frac{\sqrt{x}}{(1+x)}dx$ but ended up with nothing. 
Do you have any suggestions? Thanks!

A related problem.
1) The integral diverges since as $x\sim 0$
$$\frac{\sqrt{x}}{(1+x)\ln^3(1+x)}\sim \frac{\sqrt{x}}{(1)(x^3) }\sim \frac{1}{x^{5/2}}.$$
Note:
$$ \ln(1+x) = x - \frac{x^2}{2} + \dots. $$
2) For the second integral, just replace $x \leftrightarrow 1/x $, so the integrand will behave as $x\sim 0$ as
$$ \frac{\sqrt{1/x}}{(1+1/x)\ln^3(1+1/x)}= \frac{\sqrt{x}}{(1+x)(\ln^3(1+1/x))}\sim \frac{\sqrt{x}}{(x)(\ln^3(1/x))} = -\frac{1}{\sqrt{x}\ln^3(x)}.$$
