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