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