\[ \int \frac {\left (-b+a^2 x^4\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx \]
Optimal antiderivative \[ \frac {3 \sqrt {a}\, b \,x^{2}+2 a^{\frac {5}{2}} x^{6}+2 a^{\frac {3}{2}} x^{4} \sqrt {a^{2} x^{4}+b}}{8 \sqrt {a}\, x \sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}}-\frac {11 b \ln \left (a \,x^{2}+\sqrt {a^{2} x^{4}+b}+\sqrt {2}\, \sqrt {a}\, x \sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}\right ) \sqrt {2}}{16 \sqrt {a}} \]
command
Integrate[((-b + a^2*x^4)*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[b + a^2*x^4],x]
Mathematica 13.1 output
\[ \frac {3 b x+2 a x^3 \left (a x^2+\sqrt {b+a^2 x^4}\right )}{8 \sqrt {a x^2+\sqrt {b+a^2 x^4}}}-\frac {11 b \tanh ^{-1}\left (\frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {2} \sqrt {a} x}\right )}{8 \sqrt {2} \sqrt {a}} \]
Mathematica 12.3 output
\[ \int \frac {\left (-b+a^2 x^4\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx \]________________________________________________________________________________________