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