Does this $\int_{9}^{+\infty}\frac{\ln\left(x^5+1\right)+2x^3+1}{x^5\arctan\left(x^3\right)-2\sin\left(5x\right)} dx$ converge or diverge?

Question

$$\int_{9}^{+\infty}\frac{\ln\left(x^5+1\right)+2x^3+1}{x^5\arctan\left(x^3\right)-2\sin\left(5x\right)} dx$$ converge or diverge?

The answer of this question is converge and all the explanation given is

The function integrates is asymptotic to $x^{-2}$

but i can't find a way to get $x^{-2}$.

Answer 1

We have that as $x\to +\infty$, $$\frac{\ln\left(x^5+1\right)+2x^3+1}{x^5\arctan\left(x^3\right)-2\sin\left(5x\right)}=\frac{x^3}{x^5}\cdot\frac{2+\overbrace{\frac{\ln\left(x^5+1\right)+1}{x^3}}^{\to 0}}{\underbrace{\arctan\left(x^3\right)}_{\to \pi/2}-\underbrace{\frac{2\sin\left(5x\right)}{x^5}}_{\to 0}}\sim \frac{4/\pi}{x^2}.$$

Answer 2

For large $x$, the integrand's numerator and denominator respectively approximate $2x^3$ and $\pi x^5/2$, so the ratio approximates $\frac{4}{\pi}x^{-2}$.