\[ \int \frac {108 x+126 x^2+36 x^3+50 x^5+\left (100 x^3+150 x^4+50 x^5\right ) \log (3)+e^{3 x} \left (-50 x^2+\left (-100-150 x-50 x^2\right ) \log (3)\right )+e^{2 x} \left (150 x^3+\left (300 x+450 x^2+150 x^3\right ) \log (3)\right )+e^x \left (-36-126 x-90 x^2-18 x^3-150 x^4+\left (-300 x^2-450 x^3-150 x^4\right ) \log (3)\right )}{-200 x^5-300 x^6-150 x^7-25 x^8+e^{3 x} \left (200 x^2+300 x^3+150 x^4+25 x^5\right )+e^{2 x} \left (-600 x^3-900 x^4-450 x^5-75 x^6\right )+e^x \left (600 x^4+900 x^5+450 x^6+75 x^7\right )} \, dx \]
Optimal antiderivative \[ \frac {\frac {9}{\left (-5 \,{\mathrm e}^{x}+5 x \right )^{2}}+\frac {x}{2+x}+\ln \! \left (3\right )}{x \left (2+x \right )}+3 \]
command
Integrate[(108*x + 126*x^2 + 36*x^3 + 50*x^5 + (100*x^3 + 150*x^4 + 50*x^5)*Log[3] + E^(3*x)*(-50*x^2 + (-100 - 150*x - 50*x^2)*Log[3]) + E^(2*x)*(150*x^3 + (300*x + 450*x^2 + 150*x^3)*Log[3]) + E^x*(-36 - 126*x - 90*x^2 - 18*x^3 - 150*x^4 + (-300*x^2 - 450*x^3 - 150*x^4)*Log[3]))/(-200*x^5 - 300*x^6 - 150*x^7 - 25*x^8 + E^(3*x)*(200*x^2 + 300*x^3 + 150*x^4 + 25*x^5) + E^(2*x)*(-600*x^3 - 900*x^4 - 450*x^5 - 75*x^6) + E^x*(600*x^4 + 900*x^5 + 450*x^6 + 75*x^7)),x]
Mathematica 13.1 output
\[ \int \frac {108 x+126 x^2+36 x^3+50 x^5+\left (100 x^3+150 x^4+50 x^5\right ) \log (3)+e^{3 x} \left (-50 x^2+\left (-100-150 x-50 x^2\right ) \log (3)\right )+e^{2 x} \left (150 x^3+\left (300 x+450 x^2+150 x^3\right ) \log (3)\right )+e^x \left (-36-126 x-90 x^2-18 x^3-150 x^4+\left (-300 x^2-450 x^3-150 x^4\right ) \log (3)\right )}{-200 x^5-300 x^6-150 x^7-25 x^8+e^{3 x} \left (200 x^2+300 x^3+150 x^4+25 x^5\right )+e^{2 x} \left (-600 x^3-900 x^4-450 x^5-75 x^6\right )+e^x \left (600 x^4+900 x^5+450 x^6+75 x^7\right )} \, dx \]
Mathematica 12.3 output
\[ \frac {1}{100} \left (-\frac {50 (-2+x \log (3)+\log (9))}{(2+x)^2}+\frac {\frac {36}{\left (e^x-x\right )^2 (2+x)}+25 \log (9)}{x}\right ) \]