\[ \int \frac {-32 x^7-12 x^8+16 x^9+e^2 \left (144 x^3+576 x^5+576 x^7\right )+e \left (240 x^4+72 x^5+192 x^6+96 x^7-192 x^8\right )}{x^6+e^2 \left (9+36 x^2+36 x^4\right )+e \left (-6 x^3-12 x^5\right )} \, dx \]
Optimal antiderivative \[ 4 \left (x -\frac {4+x}{x -\left (6+\frac {3}{x^{2}}\right ) {\mathrm e}}\right ) x^{3} \]
command
integrate(((576*x^7+576*x^5+144*x^3)*exp(1)^2+(-192*x^8+96*x^7+192*x^6+72*x^5+240*x^4)*exp(1)+16*x^9-12*x^8-32*x^7)/((36*x^4+36*x^2+9)*exp(1)^2+(-12*x^5-6*x^3)*exp(1)+x^6),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ 4 \, x^{4} - 4 \, x^{3} - 24 \, x^{2} e - 16 \, x^{2} - 144 \, x e^{2} - 96 \, x e - \frac {12 \, {\left (432 \, x^{2} e^{4} + 288 \, x^{2} e^{3} + 12 \, x^{2} e^{2} + 4 \, x^{2} e + 36 \, x e^{3} + 24 \, x e^{2} + 216 \, e^{4} + 144 \, e^{3} + 3 \, e^{2}\right )}}{x^{3} - 6 \, x^{2} e - 3 \, e} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {4 \, {\left (4 \, x^{9} - 3 \, x^{8} - 8 \, x^{7} + 36 \, {\left (4 \, x^{7} + 4 \, x^{5} + x^{3}\right )} e^{2} - 6 \, {\left (8 \, x^{8} - 4 \, x^{7} - 8 \, x^{6} - 3 \, x^{5} - 10 \, x^{4}\right )} e\right )}}{x^{6} + 9 \, {\left (4 \, x^{4} + 4 \, x^{2} + 1\right )} e^{2} - 6 \, {\left (2 \, x^{5} + x^{3}\right )} e}\,{d x} \]________________________________________________________________________________________