\[ \int \frac {1}{a b c-(c+a b x)^2 \sqrt {c+b x+a x^2}} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[(a*b*c - (c + a*b*x)^2*Sqrt[c + b*x + a*x^2])^(-1),x]
Mathematica 13.1 output
\[ -2 \text {RootSum}\left [a b^4 c+\sqrt {a} b^2 c^3-2 a^{3/2} b^2 c^3+a^{5/2} b^2 c^3-6 a^{3/2} b^3 c \text {$\#$1}-b^3 c^2 \text {$\#$1}+2 a b^3 c^2 \text {$\#$1}-a^2 b^3 c^2 \text {$\#$1}-4 a b c^3 \text {$\#$1}+4 a^2 b c^3 \text {$\#$1}+12 a^2 b^2 c \text {$\#$1}^2+5 \sqrt {a} b^2 c^2 \text {$\#$1}^2-4 a^{3/2} b^2 c^2 \text {$\#$1}^2-a^{5/2} b^2 c^2 \text {$\#$1}^2+4 a^{3/2} c^3 \text {$\#$1}^2-8 a^{5/2} b c \text {$\#$1}^3-2 a b^3 c \text {$\#$1}^3+2 a^2 b^3 c \text {$\#$1}^3-8 a b c^2 \text {$\#$1}^3+6 a^{3/2} b^2 c \text {$\#$1}^4-a^{5/2} b^2 c \text {$\#$1}^4+4 a^{3/2} c^2 \text {$\#$1}^4-a^2 b^3 \text {$\#$1}^5-4 a^2 b c \text {$\#$1}^5+a^{5/2} b^2 \text {$\#$1}^6\&,\frac {-\sqrt {a} b c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )+b^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 a c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}-3 \sqrt {a} b \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+2 a \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^3}{6 a^{3/2} b^3 c+b^3 c^2-2 a b^3 c^2+a^2 b^3 c^2+4 a b c^3-4 a^2 b c^3-24 a^2 b^2 c \text {$\#$1}-10 \sqrt {a} b^2 c^2 \text {$\#$1}+8 a^{3/2} b^2 c^2 \text {$\#$1}+2 a^{5/2} b^2 c^2 \text {$\#$1}-8 a^{3/2} c^3 \text {$\#$1}+24 a^{5/2} b c \text {$\#$1}^2+6 a b^3 c \text {$\#$1}^2-6 a^2 b^3 c \text {$\#$1}^2+24 a b c^2 \text {$\#$1}^2-24 a^{3/2} b^2 c \text {$\#$1}^3+4 a^{5/2} b^2 c \text {$\#$1}^3-16 a^{3/2} c^2 \text {$\#$1}^3+5 a^2 b^3 \text {$\#$1}^4+20 a^2 b c \text {$\#$1}^4-6 a^{5/2} b^2 \text {$\#$1}^5}\&\right ] \]
Mathematica 12.3 output
\[ \int \frac {1}{a b c-(c+a b x)^2 \sqrt {c+b x+a x^2}} \, dx \]________________________________________________________________________________________