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