Find the following integral: $\int \sqrt{x}(3x^{2} + 2x + 3) dx$

To find the integral $\int \sqrt{x}(3x^{2} + 2x + 3) dx$,we first distribute $\sqrt{x}$ (which is $x^{1/2}$) into the parenthesis:
$= \int (3x^{2} \cdot x^{1/2} + 2x \cdot x^{1/2} + 3 \cdot x^{1/2}) dx$
$= \int (3x^{5/2} + 2x^{3/2} + 3x^{1/2}) dx$
Now,apply the power rule for integration $\int x^{n} dx = \frac{x^{n+1}}{n+1} + C$:
$= 3 \int x^{5/2} dx + 2 \int x^{3/2} dx + 3 \int x^{1/2} dx$
$= 3 \left( \frac{x^{7/2}}{7/2} \right) + 2 \left( \frac{x^{5/2}}{5/2} \right) + 3 \left( \frac{x^{3/2}}{3/2} \right) + C$
$= 3 \cdot \frac{2}{7} x^{7/2} + 2 \cdot \frac{2}{5} x^{5/2} + 3 \cdot \frac{2}{3} x^{3/2} + C$
$= \frac{6}{7} x^{7/2} + \frac{4}{5} x^{5/2} + 2x^{3/2} + C$
where $C$ is the constant of integration.