La integral de

$$\begin{align}&\int_0^2 x^3 \sqrt{x+2}dx=\\ &\\ &\\ & x+2=t^2,\quad dx=2tdt,\quad x=t^2-2\\ &\\ &x=0 \implies t=\sqrt 2\\ &x=2 \implies t=\sqrt{2+2}=2\\ &\\ &=\int_{\sqrt 2}^2(t^2-2)^3t(2t)dt=\\ &\\ &\\ &2\int_{\sqrt 2}^2(t^6-6t^4+12t^2-8)t^2dt =\\ &\\ &\\ &2\int_{\sqrt 2}^2 (t^8-6t^6+12t^4-8t^2) dt=\\ &\\ &2\left[\frac{t^9}{9}-\frac{6t^7}{7}+\frac{12t^5}{5}-\frac{8t^3}{3} \right]_{\sqrt 2}^2=\\ &\\ &2\left(\frac{512}{9}-\frac{768}{7}+\frac{384}{5}-\frac{64}{3}-\frac{16 \sqrt 2}{9}+\frac{48 \sqrt 2}{7}-\frac{48 \sqrt 2}{5}+\frac{16 \sqrt 2}{3}   \right)=\\ &\\ &2\left(\frac{53760-103680+72576-20160-1680 \sqrt 2+6480 \sqrt 2-9072 \sqrt 2+5040 \sqrt 2}{945}  \right)=\\ &\\ &\\ &2\left(\frac{2496+768 \sqrt 2}{945}  \right)=\frac{4992+1536 \sqrt 2}{945}=\\ &\\ &\\ &\frac{1664+512 \sqrt 2}{315} \approx 7.581197917\end{align}$$

¡Uff!

Y eso es todo.