\[ \int \frac {e^{\frac {x}{4 \log (x)}} \left (-e^{\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}} x^5-x^6-x^7+\left (e^{\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}} x^5+x^6+x^7\right ) \log (x)+\left (4 x^5+8 x^6+e^{\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}} \left (-10000+12000 x-800 x^2-1760 x^3-352 x^5+32 x^6+96 x^7+16 x^8\right )\right ) \log ^2(x)\right )}{4 x^5 \log ^2(x)} \, dx \]
Optimal antiderivative \[ \left (x +{\mathrm e}^{\left (2+x -\frac {5}{x}\right )^{4}+3}+x^{2}\right ) {\mathrm e}^{\frac {x}{4 \ln \left (x \right )}} \]
command
integrate(1/4*(((16*x**8+96*x**7+32*x**6-352*x**5-1760*x**3-800*x**2+12000*x-10000)*exp((x**8+8*x**7+4*x**6-88*x**5-71*x**4+440*x**3+100*x**2-1000*x+625)/x**4)+8*x**6+4*x**5)*ln(x)**2+(x**5*exp((x**8+8*x**7+4*x**6-88*x**5-71*x**4+440*x**3+100*x**2-1000*x+625)/x**4)+x**7+x**6)*ln(x)-x**5*exp((x**8+8*x**7+4*x**6-88*x**5-71*x**4+440*x**3+100*x**2-1000*x+625)/x**4)-x**7-x**6)*exp(1/4*x/ln(x))/x**5/ln(x)**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Exception raised: TypeError} \]
Sympy 1.8 under Python 3.8.8 output
\[ \left (x^{2} + x\right ) e^{\frac {x}{4 \log {\left (x \right )}}} + e^{\frac {x^{8} + 8 x^{7} + 4 x^{6} - 88 x^{5} - 71 x^{4} + 440 x^{3} + 100 x^{2} - 1000 x + 625}{x^{4}}} e^{\frac {x}{4 \log {\left (x \right )}}} \]