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