\[ \int \left (8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx \]
Optimal antiderivative \[ \frac {\left (3-2 \left (-1+x \right )^{2}-\left (-1+x \right )^{4}\right )^{\frac {3}{2}} \left (-1+x \right )}{7}-\frac {16 \EllipticE \left (-1+x , \frac {i \sqrt {3}}{3}\right ) \sqrt {3}}{5}+\frac {176 \EllipticF \left (-1+x , \frac {i \sqrt {3}}{3}\right ) \sqrt {3}}{35}+\frac {2 \left (13-3 \left (-1+x \right )^{2}\right ) \left (-1+x \right ) \sqrt {3-2 \left (-1+x \right )^{2}-\left (-1+x \right )^{4}}}{35} \]
command
integrate((-x^4+4*x^3-8*x^2+8*x)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {{\left (5 \, x^{6} - 30 \, x^{5} + 91 \, x^{4} - 164 \, x^{3} + 130 \, x^{2} - 12 \, x - 132\right )} \sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x}}{35 \, {\left (x - 1\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x\right )}^{\frac {3}{2}}, x\right ) \]