\[ \int \frac {-x^6+120 x^7+1112 x^8+3210 x^9+3504 x^{10}+1300 x^{11}+150 x^{12}+e^9 \left (-1+50 x^2+780 x^3+2904 x^4+1300 x^5+150 x^6\right )+e^6 \left (-3 x^2+90 x^3+1374 x^4+6390 x^5+9612 x^6+3900 x^7+450 x^8\right )+e^3 \left (33 x^4+570 x^5+3336 x^6+8820 x^7+10212 x^8+3900 x^9+450 x^{10}\right )}{e^9 x+3 e^6 x^3+3 e^3 x^5+x^7} \, dx \]
Optimal antiderivative \[ \left (x +5+\frac {3}{x +\frac {{\mathrm e}^{3}}{x}}\right )^{2} \left (5 x^{2}+x \right )^{2}-\ln \left (-x \right ) \]
command
integrate(((150*x^6+1300*x^5+2904*x^4+780*x^3+50*x^2-1)*exp(3)^3+(450*x^8+3900*x^7+9612*x^6+6390*x^5+1374*x^4+90*x^3-3*x^2)*exp(3)^2+(450*x^10+3900*x^9+10212*x^8+8820*x^7+3336*x^6+570*x^5+33*x^4)*exp(3)+150*x^12+1300*x^11+3504*x^10+3210*x^9+1112*x^8+120*x^7-x^6)/(x*exp(3)^3+3*x^3*exp(3)^2+3*x^5*exp(3)+x^7),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ 25 \, x^{6} + 260 \, x^{5} + 876 \, x^{4} + 1070 \, x^{3} - 150 \, x^{2} e^{3} + 556 \, x^{2} - 810 \, x e^{3} + 120 \, x + \frac {3 \, {\left (270 \, x^{3} e^{6} - 70 \, x^{3} e^{3} - 50 \, x^{2} e^{9} + 327 \, x^{2} e^{6} - 6 \, x^{2} e^{3} + 270 \, x e^{9} - 40 \, x e^{6} - 50 \, e^{12} + 252 \, e^{9} - 3 \, e^{6}\right )}}{{\left (x^{2} + e^{3}\right )}^{2}} - \log \left ({\left | x \right |}\right ) \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________