\[ \int \frac {\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}}{x \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[(Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]])/(x*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]]),x]
Mathematica 13.1 output
\[ \frac {8}{15} \sqrt {d+\sqrt {c+\sqrt {b+a x}}} \left (3 c+8 d^2+3 \sqrt {b+a x}-4 d \sqrt {c+\sqrt {b+a x}}\right )-b \text {RootSum}\left [b-c^2+2 c d^2-d^4-4 c d \text {$\#$1}^2+4 d^3 \text {$\#$1}^2+2 c \text {$\#$1}^4-6 d^2 \text {$\#$1}^4+4 d \text {$\#$1}^6-\text {$\#$1}^8\&,\frac {d \log \left (\sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\text {$\#$1}\right )-\log \left (\sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-c \text {$\#$1}+d^2 \text {$\#$1}-2 d \text {$\#$1}^3+\text {$\#$1}^5}\&\right ] \]
Mathematica 12.3 output
\[ \int \frac {\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}}{x \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx \]________________________________________________________________________________________