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