\[ \int \frac {1}{\left (8 x-8 x^2+4 x^3-x^4\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {\left (5+\left (-1+x \right )^{2}\right ) \left (-1+x \right )}{72 \left (3-2 \left (-1+x \right )^{2}-\left (-1+x \right )^{4}\right )^{\frac {3}{2}}}-\frac {7 \EllipticE \left (-1+x , \frac {i \sqrt {3}}{3}\right ) \sqrt {3}}{432}+\frac {11 \EllipticF \left (-1+x , \frac {i \sqrt {3}}{3}\right ) \sqrt {3}}{432}+\frac {\left (26+7 \left (-1+x \right )^{2}\right ) \left (-1+x \right )}{432 \sqrt {3-2 \left (-1+x \right )^{2}-\left (-1+x \right )^{4}}} \]
command
integrate(1/(-x^4+4*x^3-8*x^2+8*x)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {43 \, \sqrt {2} {\left (x^{8} - 8 \, x^{7} + 32 \, x^{6} - 80 \, x^{5} + 128 \, x^{4} - 128 \, x^{3} + 64 \, x^{2}\right )} {\rm weierstrassPInverse}\left (-\frac {2}{3}, \frac {7}{54}, -\frac {x - 3}{3 \, x}\right ) - 84 \, \sqrt {2} {\left (x^{8} - 8 \, x^{7} + 32 \, x^{6} - 80 \, x^{5} + 128 \, x^{4} - 128 \, x^{3} + 64 \, x^{2}\right )} {\rm weierstrassZeta}\left (-\frac {2}{3}, \frac {7}{54}, {\rm weierstrassPInverse}\left (-\frac {2}{3}, \frac {7}{54}, -\frac {x - 3}{3 \, x}\right )\right ) + 6 \, {\left (7 \, x^{6} - 37 \, x^{5} + 115 \, x^{4} - 226 \, x^{3} + 274 \, x^{2} - 232 \, x + 36\right )} \sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x}}{2592 \, {\left (x^{8} - 8 \, x^{7} + 32 \, x^{6} - 80 \, x^{5} + 128 \, x^{4} - 128 \, x^{3} + 64 \, x^{2}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x}}{x^{12} - 12 \, x^{11} + 72 \, x^{10} - 280 \, x^{9} + 768 \, x^{8} - 1536 \, x^{7} + 2240 \, x^{6} - 2304 \, x^{5} + 1536 \, x^{4} - 512 \, x^{3}}, x\right ) \]