$\int x^{3}\sqrt{3-x^{2}}dx$
$u=3-x^{2}\Rightarrow du=-2xdx$
$=\int \frac{(3-u)\sqrt{u}}{-2} du$
$=-u^{\frac{3}{2}}+\frac{u^{\frac{5}{2}}}{5}+c$
$=-{(3-x^{2})^{\frac{3}{2}}}+\frac{(3-x^{2})^{\frac{5}{2}}}{5}+c$
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