Answer
$\frac{2}{5}{x^2}{\left( {x - 2} \right)^{5/2}} - \frac{{8x}}{{35}}{\left( {x - 2} \right)^{7/2}} + \frac{{16}}{{315}}{\left( {x - 2} \right)^{9/2}} + C$
Work Step by Step
$$\eqalign{
& \int {{x^2}{{\left( {x - 2} \right)}^{3/2}}} dx \cr
& {\text{Integrate by tabulation }}\left( {{\text{See below}}} \right) \cr
& {\text{We can obtain the integral solution by adding the signed}} \cr
& {\text{the products of the diagonal entries:}} \cr
& = {x^2}\left( {\frac{2}{5}{{\left( {x - 2} \right)}^{5/2}}} \right) - 2x\left( {\frac{4}{{35}}{{\left( {x - 2} \right)}^{7/2}}} \right) + 2\left( {\frac{8}{{315}}{{\left( {x - 2} \right)}^{9/2}}} \right) + C \cr
& {\text{Simplifying}} \cr
& = \frac{2}{5}{x^2}{\left( {x - 2} \right)^{5/2}} - \frac{{8x}}{{35}}{\left( {x - 2} \right)^{7/2}} + \frac{{16}}{{315}}{\left( {x - 2} \right)^{9/2}} + C \cr} $$
