\[ \int \frac {e^{\frac {x}{1+x}} \left (x^2+\left (-5-5 x-6 x^2\right ) \log (2)\right )+e^{\frac {2 x}{1+x}} \left (-2-4 x-2 x^2-243 x^4-486 x^5-243 x^6+\left (2+4 x+2 x^2-1620 x^3-2997 x^4-1134 x^5+243 x^6\right ) \log (2)\right )}{\left (x^2+2 x^3+x^4\right ) \log (2)+e^{\frac {x}{1+x}} \left (-4 x-8 x^2-4 x^3+162 x^5+324 x^6+162 x^7\right ) \log (2)+e^{\frac {2 x}{1+x}} \left (4+8 x+4 x^2-324 x^4-648 x^5-324 x^6+6561 x^8+13122 x^9+6561 x^{10}\right ) \log (2)} \, dx \]
Optimal antiderivative \[ \frac {\frac {x}{\ln \left (2\right )}+5-x}{x \,{\mathrm e}^{-\frac {x}{1+x}}-2+81 x^{4}} \]
command
Integrate[(E^(x/(1 + x))*(x^2 + (-5 - 5*x - 6*x^2)*Log[2]) + E^((2*x)/(1 + x))*(-2 - 4*x - 2*x^2 - 243*x^4 - 486*x^5 - 243*x^6 + (2 + 4*x + 2*x^2 - 1620*x^3 - 2997*x^4 - 1134*x^5 + 243*x^6)*Log[2]))/((x^2 + 2*x^3 + x^4)*Log[2] + E^(x/(1 + x))*(-4*x - 8*x^2 - 4*x^3 + 162*x^5 + 324*x^6 + 162*x^7)*Log[2] + E^((2*x)/(1 + x))*(4 + 8*x + 4*x^2 - 324*x^4 - 648*x^5 - 324*x^6 + 6561*x^8 + 13122*x^9 + 6561*x^10)*Log[2]),x]
Mathematica 13.1 output
\[ \int \frac {e^{\frac {x}{1+x}} \left (x^2+\left (-5-5 x-6 x^2\right ) \log (2)\right )+e^{\frac {2 x}{1+x}} \left (-2-4 x-2 x^2-243 x^4-486 x^5-243 x^6+\left (2+4 x+2 x^2-1620 x^3-2997 x^4-1134 x^5+243 x^6\right ) \log (2)\right )}{\left (x^2+2 x^3+x^4\right ) \log (2)+e^{\frac {x}{1+x}} \left (-4 x-8 x^2-4 x^3+162 x^5+324 x^6+162 x^7\right ) \log (2)+e^{\frac {2 x}{1+x}} \left (4+8 x+4 x^2-324 x^4-648 x^5-324 x^6+6561 x^8+13122 x^9+6561 x^{10}\right ) \log (2)} \, dx \]
Mathematica 12.3 output
\[ \frac {3944312 x+19785288 \log (2)-1524858 x \log (2)+72868 \log (4)-595423 x \log (4)-36434 \log (16)+51192 x \log (16)-9104 \log (128)-204768 x \log (128)+\frac {3944312 e^{\frac {1}{1+x}} x \left (243 x^7 (-1+\log (2))-10 \log (2)-1215 x^4 \log (2)-81 x^5 (3+32 \log (2))+x^3 (-2+\log (4))-x (2+\log (256))-x^2 (2+\log (256))-81 x^6 (7+\log (256))\right )}{\left (2+2 x+2 x^2+243 x^4+567 x^5+243 x^6\right ) \left (e^{\frac {1}{1+x}} x+e \left (-2+81 x^4\right )\right )}}{3944312 \left (-2+81 x^4\right ) \log (2)} \]