Answer
$$-3(9-x^2)^{3/2}+\frac{1}{5}(9-x^2)^{5/2}+C$$
Work Step by Step
\begin{aligned}
\int x^{3} \sqrt{9-x^{2}} d x &=\int(3 \sin \theta)^{3} \sqrt{9-(3 \sin \theta)^{2}} \cdot 3 \cos \theta d \theta \\
&=\int 27 \sin ^{3} \theta \sqrt{9-9 \sin ^{2} \theta} \cdot 3 \cos \theta d \theta \\
&=81 \int \sin ^{3} \theta \cos \theta \cdot 3 \sqrt{1-\sin ^{2} \theta} d \theta \\
&=243 \int \sin ^{3} \theta \cos \theta \sqrt{\cos ^{2} \theta} d \theta \\
&=243 \int \sin ^{3} \theta \cos ^{2} \theta d \theta\\
&= 243 \int (1- \cos ^{2} \theta ) \cos ^{2} \theta \sin \theta d \theta\\
&= 243 \int (\cos ^{2} \theta- \cos ^{4} \theta ) \sin \theta d \theta\\
&=-243\left(\frac{1}{3}\cos ^{3} \theta +\frac{1}{5} \cos ^{5} \theta\right) +C\\
&= -243 \left(\frac{1}{3}\left( \frac{\sqrt{9-x^{2}}}{3}\right)^3+\frac{1}{5}\left( \frac{\sqrt{9-x^{2}}}{3}\right)^5\right)+C \\
& = -3(9-x^2)^{3/2}+\frac{1}{5}(9-x^2)^{5/2}+C
\end{aligned}
