\[ \int \frac {(e+f x) \left (A+B x+C x^2\right )}{\sqrt {1-d x} \sqrt {1+d x}} \, dx \]
Optimal antiderivative \[ \frac {\left (2 A \,d^{2} e +B f +C e \right ) \arcsin \! \left (d x \right )}{2 d^{3}}-\frac {C \left (f x +e \right )^{2} \sqrt {-d^{2} x^{2}+1}}{3 d^{2} f}-\frac {\left (6 d^{2} f \left (A f +B e \right )-2 C \left (d^{2} e^{2}-2 f^{2}\right )-d^{2} f \left (-3 B f +C e \right ) x \right ) \sqrt {-d^{2} x^{2}+1}}{6 d^{4} f} \]
command
integrate((f*x+e)*(C*x**2+B*x+A)/(-d*x+1)**(1/2)/(d*x+1)**(1/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {output too large to display} \]