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