Answer
The integral $\int_{0}^{\infty} \dfrac{dx}{x^{3/2}(x+1)}$ diverges.
Work Step by Step
We are given the function
$f(x)=\int_{0}^{\infty} \dfrac{dx}{x^{3/2}(x+1)}$
Since, $x^{5/2}\leq x^{3/2}$
This yields:
$\dfrac{1}{x^{3/2}(x+1)} \geq \dfrac{1}{2x^{3/2}} $
Consider the integral $\int_{0}^{1} \dfrac{dx}{2x^{3/2}}=\dfrac{1}{2} [\dfrac{-2}{x^{1/2}}]_0^1 \\=\infty$
Thus, the integral $\int_{0}^{1} \dfrac{dx}{2x^{3/2}}$ diverges. Therefore, by the comparison test, the integral $\int_{0}^{\infty} \dfrac{dx}{x^{3/2}(x+1)}$ diverges as well.
