Integral of sequence of functions

Note that $$\int_0^1\frac{nx}{1+nx^3}\mathrm{d}x\gt\int_0^1\frac{nx^2}{1+nx^3}\mathrm{d}x=\left[\frac13\ln{\left|1+nx^3\right|}\right]_0^1=\frac13\ln{(1+n)}\to\infty$$

Or $$\int_{0}^{1}\frac{nx\,dx}{1+nx^3}\geq \int_{0}^{1}\frac{nx\,dx}{1+nx^2}=\frac{1}{2}\log(n+1). $$ The actual divergence speed is $\sim\frac{2\pi}{3\sqrt{3}}\sqrt[3]{n}$.