# Finding all values $p$ for which $\int_e^{+\infty} \frac{\ln(x)}{(1+x^3)^\frac{1}{p}}dx$ converges

I've been stuck for a while with this exercise. Find all positive real values $$p$$ for which the integral $$\int_e^{+\infty} \frac{\ln(x)}{(1+x^3)^\frac{1}{p}}dx$$ converges. So far I've came up with this: $$\int_e^{+\infty} \frac{\ln(x)}{(1+x^3)^\frac{1}{p}}dx\ge\int_e^{+\infty} \frac{1}{(1+x^3)^\frac{1}{p}}dx \\ \text{Take the limit of dividing the second function by }\frac{1}{x^\frac{3}{p}} \\ \lim_{x\to{+\infty}}\frac{x^\frac{3}{p}}{(1+x^3)^\frac{1}{p}}=1\ \\ \int_e^{+\infty}\frac{1}{x^\frac{3}{p}}\text{ Diverges } \leftrightarrow p\ge3$$ So, when $$p\ge3$$ my p-series diverges, which means my lower boundary for the main function diverges, which implies that the main function diverges. But I'm unable to prove any other implication. Any suggestions?

Hint. One may use that (with an integration by parts) $$\int_e^\infty \frac{\ln x}{x^a}\,dx \qquad \text{converges iff} \qquad a>1.$$ then one may observe that, as $$x \to \infty$$, $$\frac{\ln x}{(1+x^3)^\frac{1}{p}} \sim \frac{\ln x}{x^{3/p}}.$$