Answer
$$\frac{1}{2}{\sin ^{ - 1}}\left( {\frac{{{x^2} - 4}}{5}} \right) + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{x}{{\sqrt {9 + 8{x^2} - {x^4}} }}} dx \cr
& {\text{Completing the square}} \cr
& \int {\frac{x}{{\sqrt {9 + 8{x^2} - {x^4}} }}} dx = \int {\frac{x}{{\sqrt { - \left( {{x^4} - 8{x^2} + 16 - 25} \right)} }}} dx \cr
& = \int {\frac{x}{{\sqrt {25 - {{\left( {{x^2} - 4} \right)}^2}} }}} dx \cr
& {\text{Let }}u = {x^2} - 4,{\text{ }}du = 2xdx,{\text{ }}dx = \frac{1}{{2x}}du{\text{ }} \cr
& \int {\frac{x}{{\sqrt {25 - {{\left( {{x^2} - 4} \right)}^2}} }}} dx = \int {\frac{x}{{\sqrt {25 - {u^2}} }}} \left( {\frac{1}{{2x}}} \right)du \cr
& = \frac{1}{2}\int {\frac{1}{{\sqrt {25 - {u^2}} }}} du \cr
& {\text{Integrate using basic integration rules}} \cr
& = \frac{1}{2}{\sin ^{ - 1}}\left( {\frac{u}{5}} \right) + C \cr
& {\text{Write in terms of }}x \cr
& = \frac{1}{2}{\sin ^{ - 1}}\left( {\frac{{{x^2} - 4}}{5}} \right) + C \cr} $$
