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