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