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