\[ \int \frac {e^{10} \left (5+e^4 \left (-3 x+4 x^2\right )\right )+e^5 \left (-5 x^2+e^4 \left (4 x^2-5 x^3\right )\right )}{3125 x^2-6250 x^3+3125 x^4+e^4 \left (3125 x^3-6250 x^4+3125 x^5\right )+e^8 \left (1250 x^4-2500 x^5+1250 x^6\right )+e^{12} \left (250 x^5-500 x^6+250 x^7\right )+e^{16} \left (25 x^6-50 x^7+25 x^8\right )+e^{20} \left (x^7-2 x^8+x^9\right )+e^{10} \left (3125-6250 x+3125 x^2+e^4 \left (3125 x-6250 x^2+3125 x^3\right )+e^8 \left (1250 x^2-2500 x^3+1250 x^4\right )+e^{12} \left (250 x^3-500 x^4+250 x^5\right )+e^{16} \left (25 x^4-50 x^5+25 x^6\right )+e^{20} \left (x^5-2 x^6+x^7\right )\right )+e^5 \left (-6250 x+12500 x^2-6250 x^3+e^4 \left (-6250 x^2+12500 x^3-6250 x^4\right )+e^8 \left (-2500 x^3+5000 x^4-2500 x^5\right )+e^{12} \left (-500 x^4+1000 x^5-500 x^6\right )+e^{16} \left (-50 x^5+100 x^6-50 x^7\right )+e^{20} \left (-2 x^6+4 x^7-2 x^8\right )\right )} \, dx \]
Optimal antiderivative \[ \frac {x}{\left (5+x \,{\mathrm e}^{4}\right )^{4} \left (1-x -\left (-x^{2}+x \right ) {\mathrm e}^{-5}\right )} \]
command
integrate((((4*x^2-3*x)*exp(4)+5)*exp(5)^2+((-5*x^3+4*x^2)*exp(4)-5*x^2)*exp(5))/(((x^7-2*x^6+x^5)*exp(4)^5+(25*x^6-50*x^5+25*x^4)*exp(4)^4+(250*x^5-500*x^4+250*x^3)*exp(4)^3+(1250*x^4-2500*x^3+1250*x^2)*exp(4)^2+(3125*x^3-6250*x^2+3125*x)*exp(4)+3125*x^2-6250*x+3125)*exp(5)^2+((-2*x^8+4*x^7-2*x^6)*exp(4)^5+(-50*x^7+100*x^6-50*x^5)*exp(4)^4+(-500*x^6+1000*x^5-500*x^4)*exp(4)^3+(-2500*x^5+5000*x^4-2500*x^3)*exp(4)^2+(-6250*x^4+12500*x^3-6250*x^2)*exp(4)-6250*x^3+12500*x^2-6250*x)*exp(5)+(x^9-2*x^8+x^7)*exp(4)^5+(25*x^8-50*x^7+25*x^6)*exp(4)^4+(250*x^7-500*x^6+250*x^5)*exp(4)^3+(1250*x^6-2500*x^5+1250*x^4)*exp(4)^2+(3125*x^5-6250*x^4+3125*x^3)*exp(4)+3125*x^4-6250*x^3+3125*x^2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {x e^{36} + x e^{31} + 20 \, x e^{27} + x e^{26} + 20 \, x e^{22} + 150 \, x e^{18} - 625 \, x e^{5} - e^{41} - e^{36} - 20 \, e^{32} - e^{31} - 20 \, e^{27} - e^{26} - 150 \, e^{23} - 20 \, e^{22} - 150 \, e^{18} - 500 \, e^{14}}{{\left (x^{2} - x e^{5} - x + e^{5}\right )} {\left (e^{52} + 20 \, e^{48} + 150 \, e^{44} + 20 \, e^{43} + 500 \, e^{40} + 400 \, e^{39} + 625 \, e^{36} + 3000 \, e^{35} + 150 \, e^{34} + 10000 \, e^{31} + 3000 \, e^{30} + 12500 \, e^{27} + 22500 \, e^{26} + 500 \, e^{25} + 75000 \, e^{22} + 10000 \, e^{21} + 93750 \, e^{18} + 75000 \, e^{17} + 625 \, e^{16} + 250000 \, e^{13} + 12500 \, e^{12} + 312500 \, e^{9} + 93750 \, e^{8} + 312500 \, e^{4} + 390625\right )}} + \frac {x^{3} e^{52} + x^{3} e^{47} + 20 \, x^{3} e^{43} + x^{3} e^{42} + 20 \, x^{3} e^{38} + 150 \, x^{3} e^{34} - 625 \, x^{3} e^{21} + x^{2} e^{52} + 20 \, x^{2} e^{48} + x^{2} e^{47} + 40 \, x^{2} e^{43} + 400 \, x^{2} e^{39} + 20 \, x^{2} e^{38} + 400 \, x^{2} e^{34} + 2500 \, x^{2} e^{30} - 625 \, x^{2} e^{26} - 625 \, x^{2} e^{21} - 12500 \, x^{2} e^{17} + x e^{52} + 20 \, x e^{48} + 150 \, x e^{44} + 20 \, x e^{43} + 400 \, x e^{39} + 2500 \, x e^{35} + 150 \, x e^{34} - 625 \, x e^{31} + 2500 \, x e^{30} + 11875 \, x e^{26} - 12500 \, x e^{22} - 625 \, x e^{21} - 12500 \, x e^{17} - 93750 \, x e^{13} - 625 \, e^{36} - 625 \, e^{31} - 12500 \, e^{27} - 625 \, e^{26} - 12500 \, e^{22} - 625 \, e^{21} - 93750 \, e^{18} - 12500 \, e^{17} - 93750 \, e^{13} - 312500 \, e^{9}}{{\left (x e^{4} + 5\right )}^{4} {\left (e^{52} + 20 \, e^{48} + 150 \, e^{44} + 20 \, e^{43} + 500 \, e^{40} + 400 \, e^{39} + 625 \, e^{36} + 3000 \, e^{35} + 150 \, e^{34} + 10000 \, e^{31} + 3000 \, e^{30} + 12500 \, e^{27} + 22500 \, e^{26} + 500 \, e^{25} + 75000 \, e^{22} + 10000 \, e^{21} + 93750 \, e^{18} + 75000 \, e^{17} + 625 \, e^{16} + 250000 \, e^{13} + 12500 \, e^{12} + 312500 \, e^{9} + 93750 \, e^{8} + 312500 \, e^{4} + 390625\right )}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________