<s>[INST] Does $\int_{0}^{\infty} \frac{2x +3}{\sqrt {x^3 + 2x + 5}} \,dx $ converge? Does the following integral converge? I will post my solution, but I am unsure if it is true.
$$\int_{0}^{\infty} \frac{2x +3}{\sqrt {x^3 + 2x + 5}} \,dx $$ 

My solution
Let 
$$ g(x) = \frac{2x}{\sqrt {x^3}}$$
$$ f(x) = \frac{2x +3}{\sqrt {x^3 + 2x + 5}} $$
Then $$\lim_{k \to \infty} \frac{f(x)}{g(x)} = 1$$ 
Therefore whatever one does, so does the other.
$$ \int_{0}^{\infty} g(x) = \int_{0}^{\infty} \frac{2}{\sqrt {x}} = +\infty $$
Therefore g(x) diverges, thus 
$$\int_{0}^{\infty} f(x) = \int_{0}^{\infty} \frac{2x +3}{\sqrt {x^3 + 2x + 5}} \,dx = + \infty$$ 
diverges too
 [/INST] Perhaps  redundant:
Let $x \ge 2$.
$\dfrac{x}{\sqrt{3x^3}}\lt \dfrac{2x+3}{\sqrt{x^3+2x+5}}$, or
$0<g(x):=(1/√3)x^{-1/2} <$
$\dfrac{2x+3}{\sqrt{x^3+2x+5}}=:f(x);$
$\int_{2}^{\infty} g(x)dx$ diverges, so does $\int_{2}^{\infty}f(x)dx$ (Monotonicity of integral).
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