\[ \int \frac {A+B x}{\left (a+b x+c x^2\right ) \sqrt {d+e x+f x^2}} \, dx \]
Optimal antiderivative \[ \frac {\arctanh \left (\frac {\left (4 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )+2 x \left (c e -f \left (b +\sqrt {-4 a c +b^{2}}\right )\right )\right ) \sqrt {2}}{4 \sqrt {f \,x^{2}+e x +d}\, \sqrt {2 c^{2} d -b c e +b^{2} f -2 a c f -\left (-b f +c e \right ) \sqrt {-4 a c +b^{2}}}}\right ) \left (2 A c -B \left (b +\sqrt {-4 a c +b^{2}}\right )\right ) \sqrt {2}}{2 \sqrt {-4 a c +b^{2}}\, \sqrt {2 c^{2} d -b c e +b^{2} f -2 a c f -\left (-b f +c e \right ) \sqrt {-4 a c +b^{2}}}}+\frac {\arctanh \left (\frac {\left (4 c d +2 x \left (c e -f \left (b -\sqrt {-4 a c +b^{2}}\right )\right )-e \left (b -\sqrt {-4 a c +b^{2}}\right )\right ) \sqrt {2}}{4 \sqrt {f \,x^{2}+e x +d}\, \sqrt {2 c^{2} d -b c e +b^{2} f -2 a c f +\left (-b f +c e \right ) \sqrt {-4 a c +b^{2}}}}\right ) \left (b B -2 A c -B \sqrt {-4 a c +b^{2}}\right ) \sqrt {2}}{2 \sqrt {-4 a c +b^{2}}\, \sqrt {2 c^{2} d -b c e +b^{2} f -2 a c f +\left (-b f +c e \right ) \sqrt {-4 a c +b^{2}}}} \]
command
integrate((B*x+A)/(c*x^2+b*x+a)/(f*x^2+e*x+d)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ \text {Timed out} \]