\[ \int \frac {\sqrt {b^2+a^2 x^2}}{d+c x^2+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[Sqrt[b^2 + a^2*x^2]/(d + c*x^2 + Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]),x]
Mathematica 13.1 output
\[ \frac {a \left (\log \left (a x+\sqrt {b^2+a^2 x^2}\right )-2 \text {RootSum}\left [b^4 c-2 b^2 c \text {$\#$1}^4+4 a^2 d \text {$\#$1}^4+4 a^2 \text {$\#$1}^5+c \text {$\#$1}^8\&,\frac {b^2 c \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )-a^2 d \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )-a^2 \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}}{2 b^2 c-4 a^2 d-5 a^2 \text {$\#$1}-2 c \text {$\#$1}^4}\&\right ]\right )}{c} \]
Mathematica 12.3 output
\[ \text {\$Aborted} \]________________________________________________________________________________________