\[ \int \frac {x^2 \sqrt {b+a x}}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[(x^2*Sqrt[b + a*x])/(x^2 - Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]]),x]
Mathematica 13.1 output
\[ 4 a \sqrt {c+\sqrt {b+a x}}+\frac {2 \left (c^3+(b+a x)^{3/2}\right )}{3 a}-4 a \text {RootSum}\left [b^2-2 b c^2+c^4+a^2 c \text {$\#$1}+4 b c \text {$\#$1}^2-4 c^3 \text {$\#$1}^2-a^2 \text {$\#$1}^3-2 b \text {$\#$1}^4+6 c^2 \text {$\#$1}^4-4 c \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {b^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right )-2 b c^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right )+c^4 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right )+a^2 c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+4 b c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2-3 c^3 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2-a^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3-2 b \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^4+3 c^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^4-c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^6}{a^2 c+8 b c \text {$\#$1}-8 c^3 \text {$\#$1}-3 a^2 \text {$\#$1}^2-8 b \text {$\#$1}^3+24 c^2 \text {$\#$1}^3-24 c \text {$\#$1}^5+8 \text {$\#$1}^7}\&\right ] \]
Mathematica 12.3 output
\[ \int \frac {x^2 \sqrt {b+a x}}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx \]________________________________________________________________________________________