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