$\begin{aligned} & \text { Put } t=x^2+1 \Rightarrow d t=2 x d x \\ & \qquad \int_0^2 \frac{x^3}{\left(x^2+1\right)^{3 / 2}} d x=\frac{1}{2} \int_1^5 \frac{(t-1)}{t^{3 / 2}} d t=\frac{1}{2} \int_1^5\left[t^{-1 / 2}-t^{-3 / 2}\right] d t \\ & =\frac{1}{2}\left[2 \sqrt{t}+2 \frac{1}{\sqrt{t}}\right]_1^5=\frac{1}{2}\left[2 \sqrt{5}+\frac{2}{\sqrt{5}}-2-2\right] \\ & =\left[\sqrt{5}+\frac{1}{\sqrt{5}}-2\right]=\frac{6-2 \sqrt{5}}{\sqrt{5}}\end{aligned}$