\[ \int \frac {1}{\sqrt {c e+d e x} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx \]
Optimal antiderivative \[ \frac {2 \EllipticF \left (\frac {\sqrt {d e x +c e}}{\sqrt {e}}, i\right )}{d \sqrt {e}} \]
command
integrate(1/(d*e*x+c*e)^(1/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, \sqrt {-d^{3} e} e^{\left (-1\right )} {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )}{d^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt {d e x + c e}}{d^{3} e x^{3} + 3 \, c d^{2} e x^{2} + {\left (3 \, c^{2} - 1\right )} d e x + {\left (c^{3} - c\right )} e}, x\right ) \]