\[ \int \frac {-18+\sqrt {e} (6-4 x)+12 x+4 x^2}{9+e-6 x+7 x^2-2 x^3+x^4+\sqrt {e} \left (-6+2 x-2 x^2\right )} \, dx \]
Optimal antiderivative \[ \frac {2 \left (-3+x \right ) x}{x^{2}-x +3-{\mathrm e}^{\frac {1}{2}}}-2 \]
command
integrate(((6-4*x)*exp(1/2)+4*x^2+12*x-18)/(exp(1/2)^2+(-2*x^2+2*x-6)*exp(1/2)+x^4-2*x^3+7*x^2-6*x+9),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {2 \, {\left (2 \, x - e^{\frac {1}{2}} + 3\right )}}{x^{2} - x - e^{\frac {1}{2}} + 3} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {2 \, {\left (2 \, x^{2} - {\left (2 \, x - 3\right )} e^{\frac {1}{2}} + 6 \, x - 9\right )}}{x^{4} - 2 \, x^{3} + 7 \, x^{2} - 2 \, {\left (x^{2} - x + 3\right )} e^{\frac {1}{2}} - 6 \, x + e + 9}\,{d x} \]________________________________________________________________________________________