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