\[ \int \frac {\sqrt {a+c x^2}}{d+e x+f x^2} \, dx \]
Optimal antiderivative \[ \frac {\arctanh \left (\frac {x \sqrt {c}}{\sqrt {c \,x^{2}+a}}\right ) \sqrt {c}}{f}-\frac {\arctanh \left (\frac {\left (2 a f -c x \left (e -\sqrt {-4 d f +e^{2}}\right )\right ) \sqrt {2}}{2 \sqrt {c \,x^{2}+a}\, \sqrt {2 a \,f^{2}+c \left (e^{2}-2 d f -e \sqrt {-4 d f +e^{2}}\right )}}\right ) \sqrt {2 a \,f^{2}+c \left (e^{2}-2 d f -e \sqrt {-4 d f +e^{2}}\right )}\, \sqrt {2}}{2 f \sqrt {-4 d f +e^{2}}}+\frac {\arctanh \left (\frac {\left (2 a f -c x \left (e +\sqrt {-4 d f +e^{2}}\right )\right ) \sqrt {2}}{2 \sqrt {c \,x^{2}+a}\, \sqrt {2 a \,f^{2}+c \left (e^{2}-2 d f +e \sqrt {-4 d f +e^{2}}\right )}}\right ) \sqrt {2 a \,f^{2}+c \left (e^{2}-2 d f +e \sqrt {-4 d f +e^{2}}\right )}\, \sqrt {2}}{2 f \sqrt {-4 d f +e^{2}}} \]
command
integrate((c*x^2+a)^(1/2)/(f*x^2+e*x+d),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 \[ \text {Timed out} \]