\[ \int \frac {25-20 x-20 x^2+4 x^4+\left (-25+10 x+20 x^2-4 x^3-4 x^4\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )}{1125 x^2-450 x^3-900 x^4+180 x^5+180 x^6+\left (-750 x+300 x^2+600 x^3-120 x^4-120 x^5\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )+\left (125-50 x-100 x^2+20 x^3+20 x^4\right ) \log ^2\left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )} \, dx \]
Optimal antiderivative \[ \frac {x}{15 x -5 \ln \! \left (\frac {2 x^{2}}{x^{2}-\frac {5}{2}}+2 x \right )} \]
command
Int[(25 - 20*x - 20*x^2 + 4*x^4 + (-25 + 10*x + 20*x^2 - 4*x^3 - 4*x^4)*Log[(-10*x + 4*x^2 + 4*x^3)/(-5 + 2*x^2)])/(1125*x^2 - 450*x^3 - 900*x^4 + 180*x^5 + 180*x^6 + (-750*x + 300*x^2 + 600*x^3 - 120*x^4 - 120*x^5)*Log[(-10*x + 4*x^2 + 4*x^3)/(-5 + 2*x^2)] + (125 - 50*x - 100*x^2 + 20*x^3 + 20*x^4)*Log[(-10*x + 4*x^2 + 4*x^3)/(-5 + 2*x^2)]^2),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {25-20 x-20 x^2+4 x^4+\left (-25+10 x+20 x^2-4 x^3-4 x^4\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )}{1125 x^2-450 x^3-900 x^4+180 x^5+180 x^6+\left (-750 x+300 x^2+600 x^3-120 x^4-120 x^5\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )+\left (125-50 x-100 x^2+20 x^3+20 x^4\right ) \log ^2\left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ -\frac {1}{15 \left (1-\frac {3 x}{\log \left (\frac {2 x \left (-2 x^2-2 x+5\right )}{5-2 x^2}\right )}\right )} \]