Answer
$$\frac{1}{8}\sqrt {{a^2} - {x^2}} \left( {5{a^2}x + \frac{{3a{{\sin }^{ - 1}}\left( {x/a} \right)}}{{\sqrt {{a^2} - {x^2}} }} - 2{x^3}} \right) + C$$
Work Step by Step
$$\eqalign{
& \int {{{\left( {{a^2} - {x^2}} \right)}^{3/2}}} dx \cr
& {\text{Integrate using a computer algebra system as the book says}} \cr
& {\text{Using wolfram alpha an simplifying we obtain:}} \cr} $$
