\[ \int \frac {1}{\sqrt {1-d x} \sqrt {1+d x} \left (a+b x+c x^2\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {\left (b \left (b^{2} d^{2}-c \left (3 a \,d^{2}+c \right )\right )-c \left (2 a c \,d^{2}-b^{2} d^{2}+2 c^{2}\right ) x \right ) \sqrt {-d^{2} x^{2}+1}}{\left (-4 a c +b^{2}\right ) \left (b^{2} d^{2}-\left (a \,d^{2}+c \right )^{2}\right ) \left (c \,x^{2}+b x +a \right )}-\frac {c \arctanh \left (\frac {\left (2 c +d^{2} x \left (b -\sqrt {-4 a c +b^{2}}\right )\right ) \sqrt {2}}{2 \sqrt {-d^{2} x^{2}+1}\, \sqrt {2 c^{2}+2 a c \,d^{2}-b \,d^{2} \left (b -\sqrt {-4 a c +b^{2}}\right )}}\right ) \left (4 c^{3}+12 a \,c^{2} d^{2}-a b \,d^{4} \left (b +\sqrt {-4 a c +b^{2}}\right )-c \,d^{2} \left (5 b^{2}-8 a^{2} d^{2}-b \sqrt {-4 a c +b^{2}}\right )\right ) \sqrt {2}}{2 \left (-4 a c +b^{2}\right )^{\frac {3}{2}} \left (b^{2} d^{2}-\left (a \,d^{2}+c \right )^{2}\right ) \sqrt {2 c^{2}+2 a c \,d^{2}-b \,d^{2} \left (b -\sqrt {-4 a c +b^{2}}\right )}}+\frac {c \arctanh \left (\frac {\left (2 c +d^{2} x \left (b +\sqrt {-4 a c +b^{2}}\right )\right ) \sqrt {2}}{2 \sqrt {-d^{2} x^{2}+1}\, \sqrt {2 c^{2}+2 a c \,d^{2}-b \,d^{2} \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \left (4 c^{3}+12 a \,c^{2} d^{2}-2 a \,b^{2} d^{4}-4 c \,d^{2} \left (-2 a^{2} d^{2}+b^{2}\right )-b \,d^{2} \left (-a \,d^{2}+c \right ) \left (b +\sqrt {-4 a c +b^{2}}\right )\right ) \sqrt {2}}{2 \left (-4 a c +b^{2}\right )^{\frac {3}{2}} \left (b^{2} d^{2}-\left (a \,d^{2}+c \right )^{2}\right ) \sqrt {2 c^{2}+2 a c \,d^{2}-b \,d^{2} \left (b +\sqrt {-4 a c +b^{2}}\right )}} \]
command
integrate(1/(c*x^2+b*x+a)^2/(-d*x+1)^(1/2)/(d*x+1)^(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 \[ \text {Timed out} \]