The difference between the highest exponents is 3 - 2 = 1. By the intergral p test (with p = 1) this integral diverges.
More precisely we estimate the integral with
![{\displaystyle \int _{1}^{\infty }{\frac {x^{2}}{x^{3}+2x+5}}dx>\int _{1}^{\infty }{\frac {x^{2}}{x^{3}+2x^{3}+5x^{3}}}dx={\frac {1}{8}}\int _{1}^{\infty }{\frac {1}{x}}dx}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/d5980dd2af2d3637c6f78d709e61934845dd04e2)
Here we see exactly how to use the integral p test. We also can continue with the anti-derivative
![{\displaystyle {\frac {1}{8}}\int _{1}^{\infty }{\frac {1}{x}}dx={\frac {1}{8}}\lim _{\alpha \rightarrow \infty }[{\text{ln}}(x)]_{1}^{\alpha }={\frac {1}{8}}\lim _{\alpha \rightarrow \infty }{\text{ln}}(\alpha )=\infty }](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/d1340b3171a65e8df0c79e790595463037c143ba)