\[ \int \frac {16 x^5+4 x^6+e^{\frac {3 (-1+2 x)}{x}} \left (-1296 x-756 x^2\right )+e^{\frac {4 (-1+2 x)}{x}} \left (1296+324 x-108 x^2\right )+e^{\frac {2 (-1+2 x)}{x}} \left (432 x^2+540 x^3+72 x^4\right )+e^{\frac {-1+2 x}{x}} \left (-48 x^3-156 x^4-32 x^5\right )+\left (e^{\frac {4 (-1+2 x)}{x}} \left (-1728-432 x+108 x^2\right )+e^{\frac {3 (-1+2 x)}{x}} \left (1296 x+756 x^2+36 x^3\right )+e^{\frac {2 (-1+2 x)}{x}} \left (-288 x^2-360 x^3-60 x^4\right )+e^{\frac {-1+2 x}{x}} \left (16 x^3+52 x^4+12 x^5\right )\right ) \log (4+x)+\left (e^{\frac {4 (-1+2 x)}{x}} \left (864+216 x-36 x^2\right )+e^{\frac {3 (-1+2 x)}{x}} \left (-432 x-252 x^2-24 x^3\right )+e^{\frac {2 (-1+2 x)}{x}} \left (48 x^2+60 x^3+12 x^4\right )\right ) \log ^2(4+x)+\left (e^{\frac {4 (-1+2 x)}{x}} \left (-192-48 x+4 x^2\right )+e^{\frac {3 (-1+2 x)}{x}} \left (48 x+28 x^2+4 x^3\right )\right ) \log ^3(4+x)+e^{\frac {4 (-1+2 x)}{x}} (16+4 x) \log ^4(4+x)}{4 x^2+x^3} \, dx \]
Optimal antiderivative \[ \left ({\mathrm e}^{2-\frac {1}{x}} \left (3-\ln \! \left (4+x \right )\right )-x \right )^{4} \]
command
integrate(((4*x+16)*exp((-1+2*x)/x)**4*ln(4+x)**4+((4*x**2-48*x-192)*exp((-1+2*x)/x)**4+(4*x**3+28*x**2+48*x)*exp((-1+2*x)/x)**3)*ln(4+x)**3+((-36*x**2+216*x+864)*exp((-1+2*x)/x)**4+(-24*x**3-252*x**2-432*x)*exp((-1+2*x)/x)**3+(12*x**4+60*x**3+48*x**2)*exp((-1+2*x)/x)**2)*ln(4+x)**2+((108*x**2-432*x-1728)*exp((-1+2*x)/x)**4+(36*x**3+756*x**2+1296*x)*exp((-1+2*x)/x)**3+(-60*x**4-360*x**3-288*x**2)*exp((-1+2*x)/x)**2+(12*x**5+52*x**4+16*x**3)*exp((-1+2*x)/x))*ln(4+x)+(-108*x**2+324*x+1296)*exp((-1+2*x)/x)**4+(-756*x**2-1296*x)*exp((-1+2*x)/x)**3+(72*x**4+540*x**3+432*x**2)*exp((-1+2*x)/x)**2+(-32*x**5-156*x**4-48*x**3)*exp((-1+2*x)/x)+4*x**6+16*x**5)/(x**3+4*x**2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ x^{4} + \left (4 x^{3} \log {\left (x + 4 \right )} - 12 x^{3}\right ) e^{\frac {2 x - 1}{x}} + \left (6 x^{2} \log {\left (x + 4 \right )}^{2} - 36 x^{2} \log {\left (x + 4 \right )} + 54 x^{2}\right ) e^{\frac {2 \left (2 x - 1\right )}{x}} + \left (4 x \log {\left (x + 4 \right )}^{3} - 36 x \log {\left (x + 4 \right )}^{2} + 108 x \log {\left (x + 4 \right )} - 108 x\right ) e^{\frac {3 \left (2 x - 1\right )}{x}} + \left (\log {\left (x + 4 \right )}^{4} - 12 \log {\left (x + 4 \right )}^{3} + 54 \log {\left (x + 4 \right )}^{2} - 108 \log {\left (x + 4 \right )} + 81\right ) e^{\frac {4 \left (2 x - 1\right )}{x}} \]