\[ \int \frac {e^{\frac {2 \left (4 x^2+x^3+\log \left (\frac {15}{40+3 e^x}\right )\right )}{4 x^2+x^3}} \left (e^x \left (-24 x-6 x^2\right )+\left (-640+e^x (-48-18 x)-240 x\right ) \log \left (\frac {15}{40+3 e^x}\right )\right )}{640 x^3+320 x^4+40 x^5+e^x \left (48 x^3+24 x^4+3 x^5\right )} \, dx \]
Optimal antiderivative \[ {\mathrm e}^{\frac {2 x +\frac {2 \ln \left (\frac {5}{{\mathrm e}^{x}+\frac {40}{3}}\right )}{\left (4+x \right ) x}}{x}} \]
command
Integrate[(E^((2*(4*x^2 + x^3 + Log[15/(40 + 3*E^x)]))/(4*x^2 + x^3))*(E^x*(-24*x - 6*x^2) + (-640 + E^x*(-48 - 18*x) - 240*x)*Log[15/(40 + 3*E^x)]))/(640*x^3 + 320*x^4 + 40*x^5 + E^x*(48*x^3 + 24*x^4 + 3*x^5)),x]
Mathematica 13.1 output
\[ \int \frac {e^{\frac {2 \left (4 x^2+x^3+\log \left (\frac {15}{40+3 e^x}\right )\right )}{4 x^2+x^3}} \left (e^x \left (-24 x-6 x^2\right )+\left (-640+e^x (-48-18 x)-240 x\right ) \log \left (\frac {15}{40+3 e^x}\right )\right )}{640 x^3+320 x^4+40 x^5+e^x \left (48 x^3+24 x^4+3 x^5\right )} \, dx \]
Mathematica 12.3 output
\[ 15^{\frac {2}{x^2 (4+x)}} e^2 \left (\frac {1}{40+3 e^x}\right )^{\frac {2}{x^2 (4+x)}} \]