\[ \int \frac {\left (-1+6 x^4\right ) \sqrt {x+2 x^5}}{\left (1+2 x^4\right ) \left (1-x^2+4 x^4+4 x^8\right )} \, dx \]
Optimal antiderivative \[ \arctan \left (\frac {\sqrt {2 x^{5}+x}}{2 x^{4}+1}\right )-\arctanh \left (\frac {\sqrt {2 x^{5}+x}}{2 x^{4}+1}\right ) \]
command
Integrate[((-1 + 6*x^4)*Sqrt[x + 2*x^5])/((1 + 2*x^4)*(1 - x^2 + 4*x^4 + 4*x^8)),x]
Mathematica 13.1 output
\[ \frac {\sqrt {x+2 x^5} \left (\text {ArcTan}\left (\frac {\sqrt {x}}{\sqrt {1+2 x^4}}\right )-\tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+2 x^4}}\right )\right )}{\sqrt {x} \sqrt {1+2 x^4}} \]
Mathematica 12.3 output
\[ \int \frac {\left (-1+6 x^4\right ) \sqrt {x+2 x^5}}{\left (1+2 x^4\right ) \left (1-x^2+4 x^4+4 x^8\right )} \, dx \]________________________________________________________________________________________