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