Answer
$$\frac{1}{3}\left( {{x^2} - 4{a^2}} \right)\sqrt {{x^2} - {a^2}} + {a^3}{\tan ^{ - 1}}\left( {\frac{{\sqrt {{x^2} - {a^2}} }}{a}} \right) + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{{{{\left( {{x^2} - {a^2}} \right)}^{3/2}}}}{x}} dx \cr
& {\text{Integrate using a computer algebra system as the book says}} \cr
& {\text{Using wolfram alpha we obtain:}} \cr} $$
