\[ \int \frac {\sqrt {1+x^5} \left (-2+3 x^5\right )}{a-x^4+2 a x^5+a x^{10}} \, dx \]
Optimal antiderivative \[ -\frac {\arctan \left (\frac {x}{a^{\frac {1}{4}} \sqrt {x^{5}+1}}\right )}{a^{\frac {3}{4}}}-\frac {\arctanh \left (\frac {x}{a^{\frac {1}{4}} \sqrt {x^{5}+1}}\right )}{a^{\frac {3}{4}}} \]
command
Integrate[(Sqrt[1 + x^5]*(-2 + 3*x^5))/(a - x^4 + 2*a*x^5 + a*x^10),x]
Mathematica 13.1 output
\[ -\frac {\text {ArcTan}\left (\frac {x}{\sqrt [4]{a} \sqrt {1+x^5}}\right )+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{a} \sqrt {1+x^5}}\right )}{a^{3/4}} \]
Mathematica 12.3 output
\[ \int \frac {\sqrt {1+x^5} \left (-2+3 x^5\right )}{a-x^4+2 a x^5+a x^{10}} \, dx \]________________________________________________________________________________________