\[ \int \frac {270+90 e^{13/2}-90 x^2}{9+e^{13}+12 x-4 e^{39/4} x+10 x^2+4 x^3+x^4+e^{13/2} \left (6+4 x+6 x^2\right )+e^{13/4} \left (-12 x-8 x^2-4 x^3\right )} \, dx \]
Optimal antiderivative \[ \frac {90 x}{2 x +\left (x -{\mathrm e}^{\frac {13}{4}}\right )^{2}+3}+1 \]
command
integrate((90*exp(13/4)^2-90*x^2+270)/(exp(13/4)^4-4*x*exp(13/4)^3+(6*x^2+4*x+6)*exp(13/4)^2+(-4*x^3-8*x^2-12*x)*exp(13/4)+x^4+4*x^3+10*x^2+12*x+9),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {90 \, x}{x^{2} - 2 \, x e^{\frac {13}{4}} + 2 \, x + e^{\frac {13}{2}} + 3} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int -\frac {90 \, {\left (x^{2} - e^{\frac {13}{2}} - 3\right )}}{x^{4} + 4 \, x^{3} + 10 \, x^{2} - 4 \, x e^{\frac {39}{4}} + 2 \, {\left (3 \, x^{2} + 2 \, x + 3\right )} e^{\frac {13}{2}} - 4 \, {\left (x^{3} + 2 \, x^{2} + 3 \, x\right )} e^{\frac {13}{4}} + 12 \, x + e^{13} + 9}\,{d x} \]________________________________________________________________________________________