To solve the integral $\int x \sqrt{x + 6} \, dx$, we can use substitution to simplify it.

1. Substitute $u = x + 6$, so that $du = dx$ and $x = u - 6$.

   Now rewrite the integral in terms of $u$: $\int x \sqrt{x + 6} \, dx = \int (u - 6) \sqrt{u} \, du$

2. Expand the Integral by distributing $\sqrt{u}$: $= \int (u - 6) u^{1/2} \, du = \int \left( u^{3/2} - 6u^{1/2} \right) du$

3. Integrate Each Term:

   • For $\int u^{3/2} \, du$: $\int u^{3/2} \, du = \frac{2}{5} u^{5/2}$
   • For $\int 6u^{1/2} \, du$: $\int 6u^{1/2} \, du = 6 \cdot \frac{2}{3} u^{3/2} = 4 u^{3/2}$

   Putting these results together: $\int \left( u^{3/2} - 6u^{1/2} \right) du = \frac{2}{5} u^{5/2} - 4 u^{3/2} + C$

4. Substitute back $u = x + 6$: $= \frac{2}{5} (x + 6)^{5/2} - 4 (x + 6)^{3/2} + C$

5. Compare with the Options: None of the options match exactly in terms of this simplified form. However, we can simplify further if needed to see if any options might represent equivalent expressions.

The correct answer appears closest to one of the options if the simplified form matches an equivalent representation.