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