$\int_{\frac{1}{2}}^1\left(\left(1-x^2\right)^{\frac{3}{2}}x^2\right)dx$

$-\frac{3}{8}\cdot \frac{1}{2}\cdot \frac{\sqrt{3}}{2}-\frac{3}{8}\arcsin\left(\frac{1}{2}\right)+\frac{-\frac{1}{2}\cdot \frac{\sqrt{\left(3\right)^{3}}}{8}}{4}+\frac{3\pi }{16}+\frac{5}{16}\cdot \frac{1}{2}\cdot \frac{\sqrt{3}}{2}+\frac{5}{16}\arcsin\left(\frac{1}{2}\right)+\frac{\frac{5}{2}\cdot \frac{\sqrt{\left(3\right)^{3}}}{8}}{24}+\frac{\frac{\sqrt{\left(3\right)^{5}}}{32}\cdot \frac{1}{2}}{6}-\frac{5}{16}\cdot \frac{\pi }{2}$