\[ \int \frac {12 x^2+3 e^2 x^2-10 x^3+4 x^4+e^3 \left (8 x+2 e^2 x-5 x^2\right )}{16+e^4-40 x+57 x^2-40 x^3+16 x^4+e^2 \left (8-10 x+8 x^2\right )} \, dx \]
Optimal antiderivative \[ \frac {x^{2} \left ({\mathrm e}^{3}+x \right )}{4+4 x^{2}-5 x +{\mathrm e}^{2}} \]
command
integrate(((2*exp(2)*x-5*x^2+8*x)*exp(3)+3*x^2*exp(2)+4*x^4-10*x^3+12*x^2)/(exp(2)^2+(8*x^2-10*x+8)*exp(2)+16*x^4-40*x^3+57*x^2-40*x+16),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{4} \, x + \frac {20 \, x e^{3} - 4 \, x e^{2} + 9 \, x - 4 \, e^{5} - 16 \, e^{3} - 5 \, e^{2} - 20}{16 \, {\left (4 \, x^{2} - 5 \, x + e^{2} + 4\right )}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {4 \, x^{4} - 10 \, x^{3} + 3 \, x^{2} e^{2} + 12 \, x^{2} - {\left (5 \, x^{2} - 2 \, x e^{2} - 8 \, x\right )} e^{3}}{16 \, x^{4} - 40 \, x^{3} + 57 \, x^{2} + 2 \, {\left (4 \, x^{2} - 5 \, x + 4\right )} e^{2} - 40 \, x + e^{4} + 16}\,{d x} \]________________________________________________________________________________________