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

Notice that:
$$\frac{2x+3}{\sqrt{x^3+2x+5}}\approx x^{-0.5}$$ for large $x$ and we know that:
$$\int_0^\infty\frac{1}{x^n+c}dx$$
only converges for $c>0,n>1$
