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