\[ \int \frac {1}{x^2 \sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx \]
Optimal antiderivative \[ \frac {b \arctanh \left (\frac {b x +2 a}{2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}\right )}{2 a^{\frac {3}{2}} d}-\frac {\sqrt {c \,x^{2}+b x +a}}{a d x}+\frac {f \arctanh \left (\frac {b \sqrt {d}-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )}{2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c d +a f -b \sqrt {d}\, \sqrt {f}}}\right )}{2 d^{\frac {3}{2}} \sqrt {c d +a f -b \sqrt {d}\, \sqrt {f}}}+\frac {f \arctanh \left (\frac {b \sqrt {d}+2 a \sqrt {f}+x \left (2 c \sqrt {d}+b \sqrt {f}\right )}{2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c d +a f +b \sqrt {d}\, \sqrt {f}}}\right )}{2 d^{\frac {3}{2}} \sqrt {c d +a f +b \sqrt {d}\, \sqrt {f}}} \]
command
integrate(1/x^2/(c*x^2+b*x+a)^(1/2)/(-f*x^2+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 \[ \text {Timed out} \]