\[ \int \frac {1}{(-1+x) \left (-2 x^2-3 x^3+x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {\left (-453 x^{3}+1555 x^{2}+238 x -136\right ) \sqrt {x^{4}-3 x^{3}-2 x^{2}}}{544 x^{3} \left (x^{2}-3 x -2\right )}+\frac {\arctan \left (\frac {-\frac {x}{2}+\frac {x^{2}}{2}-\frac {\sqrt {x^{4}-3 x^{3}-2 x^{2}}}{2}}{x}\right )}{4}-\frac {119 \arctan \left (\frac {\frac {x^{2} \sqrt {2}}{2}-\frac {\sqrt {x^{4}-3 x^{3}-2 x^{2}}\, \sqrt {2}}{2}}{x}\right ) \sqrt {2}}{64} \]
command
integrate(1/(-1+x)/(x^4-3*x^3-2*x^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\frac {29 \, x}{\mathrm {sgn}\left (x\right )} - \frac {103}{\mathrm {sgn}\left (x\right )}}{68 \, \sqrt {x^{2} - 3 \, x - 2}} + \frac {119 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (x - \sqrt {x^{2} - 3 \, x - 2}\right )}\right )}{64 \, \mathrm {sgn}\left (x\right )} - \frac {\arctan \left (-\frac {1}{2} \, x + \frac {1}{2} \, \sqrt {x^{2} - 3 \, x - 2} + \frac {1}{2}\right )}{4 \, \mathrm {sgn}\left (x\right )} - \frac {47 \, {\left (x - \sqrt {x^{2} - 3 \, x - 2}\right )}^{3} + 16 \, {\left (x - \sqrt {x^{2} - 3 \, x - 2}\right )}^{2} + 98 \, x - 98 \, \sqrt {x^{2} - 3 \, x - 2} + 128}{32 \, {\left ({\left (x - \sqrt {x^{2} - 3 \, x - 2}\right )}^{2} + 2\right )}^{2} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________