\[ \int \frac {1+x}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}} \, dx \]
Optimal antiderivative \[ \arctan \! \left (\frac {-4+4 x}{1-2 x +x^{2}-\sqrt {x^{4}-12 x^{3}+14 x^{2}+4 x -7}}\right )+\ln \! \left (-1+x \right )-\ln \! \left (-5+6 x -x^{2}+\sqrt {x^{4}-12 x^{3}+14 x^{2}+4 x -7}\right ) \]
command
integrate((1+x)/(x^4-12*x^3+14*x^2+4*x-7)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {Exception raised: NotImplementedError} \]
Giac 1.7.0 via sagemath 9.3 output
\[ -\frac {\arctan \left (\frac {1}{7} \, \sqrt {7} {\left (\sqrt {2} + \frac {3 \, {\left (\sqrt {7} \sqrt {-\frac {10}{x} - \frac {7}{x^{2}} + 1} - 4 \, \sqrt {2}\right )}}{\frac {7}{x} + 5}\right )}\right )}{\mathrm {sgn}\left (-\frac {1}{x^{2}} + \frac {1}{x^{3}}\right )} - \frac {\log \left ({\left | 10 \, \sqrt {7} + 40 \, \sqrt {2} + \frac {50 \, {\left (\sqrt {7} \sqrt {-\frac {10}{x} - \frac {7}{x^{2}} + 1} - 4 \, \sqrt {2}\right )}}{\frac {7}{x} + 5} \right |}\right )}{\mathrm {sgn}\left (-\frac {1}{x^{2}} + \frac {1}{x^{3}}\right )} + \frac {\log \left ({\left | -2 \, \sqrt {7} + 8 \, \sqrt {2} + \frac {10 \, {\left (\sqrt {7} \sqrt {-\frac {10}{x} - \frac {7}{x^{2}} + 1} - 4 \, \sqrt {2}\right )}}{\frac {7}{x} + 5} \right |}\right )}{\mathrm {sgn}\left (-\frac {1}{x^{2}} + \frac {1}{x^{3}}\right )} \]