\[ \int \frac {150+e (-6-4 x)+100 x+36 x^2+16 x^3}{5625 x^2+3750 x^3+1525 x^4+600 x^5+136 x^6+24 x^7+4 x^8+e^2 \left (9 x^2+6 x^3+x^4\right )+e \left (-450 x^2-300 x^3-86 x^4-24 x^5-4 x^6\right )} \, dx \]
Optimal antiderivative \[ -\frac {2}{x^{2} \left (2 x +\frac {25-{\mathrm e}}{x}\right ) \left (3+x \right )} \]
command
integrate(((-4*x-6)*exp(1)+16*x^3+36*x^2+100*x+150)/((x^4+6*x^3+9*x^2)*exp(1)^2+(-4*x^6-24*x^5-86*x^4-300*x^3-450*x^2)*exp(1)+4*x^8+24*x^7+136*x^6+600*x^5+1525*x^4+3750*x^3+5625*x^2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {2}{2 \, x^{4} + 6 \, x^{3} + 25 \, x^{2} - {\left (x^{2} + 3 \, x\right )} e + 75 \, x} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________