\[ \int \frac {90 e^{53}-120 x^2}{9 e^{106}+100 x^2+4 e^2 x^2+80 x^3+16 x^4+e^{53} \left (60 x+12 e x+24 x^2\right )+e \left (40 x^2+16 x^3\right )} \, dx \]
Optimal antiderivative \[ \frac {15}{2 x +{\mathrm e}+5+\frac {3 \,{\mathrm e}^{3} {\mathrm e}^{50}}{2 x}} \]
command
integrate((90*exp(3)*exp(25)^2-120*x^2)/(9*exp(3)^2*exp(25)^4+(12*x*exp(1)+24*x^2+60*x)*exp(3)*exp(25)^2+4*x^2*exp(1)^2+(16*x^3+40*x^2)*exp(1)+16*x^4+80*x^3+100*x^2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {30 \, x}{4 \, x^{2} + 2 \, x e + 10 \, x + 3 \, e^{53}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int -\frac {30 \, {\left (4 \, x^{2} - 3 \, e^{53}\right )}}{16 \, x^{4} + 80 \, x^{3} + 4 \, x^{2} e^{2} + 100 \, x^{2} + 12 \, {\left (2 \, x^{2} + x e + 5 \, x\right )} e^{53} + 8 \, {\left (2 \, x^{3} + 5 \, x^{2}\right )} e + 9 \, e^{106}}\,{d x} \]________________________________________________________________________________________