\[ \int \frac {\sqrt {b+a x} \sqrt {1+\sqrt {b+a x}}}{x^2 \sqrt {1+\sqrt {1+\sqrt {b+a x}}}} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[(Sqrt[b + a*x]*Sqrt[1 + Sqrt[b + a*x]])/(x^2*Sqrt[1 + Sqrt[1 + Sqrt[b + a*x]]]),x]
Mathematica 13.1 output
\[ \frac {\sqrt {1+\sqrt {1+\sqrt {b+a x}}} \left (-1-\sqrt {b+a x}+\sqrt {1+\sqrt {b+a x}}\right )}{x}+a \text {RootSum}\left [b-4 \text {$\#$1}^4+4 \text {$\#$1}^6-\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {1+\sqrt {1+\sqrt {b+a x}}}-\text {$\#$1}\right )+\log \left (\sqrt {1+\sqrt {1+\sqrt {b+a x}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-2 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]-\frac {1}{8} a \text {RootSum}\left [b-4 \text {$\#$1}^4+4 \text {$\#$1}^6-\text {$\#$1}^8\&,\frac {6 \log \left (\sqrt {1+\sqrt {1+\sqrt {b+a x}}}-\text {$\#$1}\right )-7 \log \left (\sqrt {1+\sqrt {1+\sqrt {b+a x}}}-\text {$\#$1}\right ) \text {$\#$1}^2+3 \log \left (\sqrt {1+\sqrt {1+\sqrt {b+a x}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ] \]
Mathematica 12.3 output
\[ \int \frac {\sqrt {b+a x} \sqrt {1+\sqrt {b+a x}}}{x^2 \sqrt {1+\sqrt {1+\sqrt {b+a x}}}} \, dx \]________________________________________________________________________________________