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