\[ \int \frac {1}{(c e+d e x)^{9/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx \]
Optimal antiderivative \[ \frac {10 \EllipticF \left (\frac {\sqrt {d e x +c e}}{\sqrt {e}}, i\right )}{21 d \,e^{\frac {9}{2}}}-\frac {2 \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}}{7 d e \left (d e x +c e \right )^{\frac {7}{2}}}-\frac {10 \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}}{21 d \,e^{3} \left (d e x +c e \right )^{\frac {3}{2}}} \]
command
integrate(1/(d*e*x+c*e)^(9/2)/(-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
\[ -\frac {2 \, {\left ({\left (5 \, d^{4} x^{2} + 10 \, c d^{3} x + {\left (5 \, c^{2} + 3\right )} d^{2}\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt {d x + c} e^{\frac {1}{2}} + 5 \, {\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}\right )} \sqrt {-d^{3} e} {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right )} e^{\left (-5\right )}}{21 \, {\left (d^{7} x^{4} + 4 \, c d^{6} x^{3} + 6 \, c^{2} d^{5} x^{2} + 4 \, c^{3} d^{4} x + c^{4} d^{3}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt {d e x + c e}}{d^{7} e^{5} x^{7} + 7 \, c d^{6} e^{5} x^{6} + {\left (21 \, c^{2} - 1\right )} d^{5} e^{5} x^{5} + 5 \, {\left (7 \, c^{3} - c\right )} d^{4} e^{5} x^{4} + 5 \, {\left (7 \, c^{4} - 2 \, c^{2}\right )} d^{3} e^{5} x^{3} + {\left (21 \, c^{5} - 10 \, c^{3}\right )} d^{2} e^{5} x^{2} + {\left (7 \, c^{6} - 5 \, c^{4}\right )} d e^{5} x + {\left (c^{7} - c^{5}\right )} e^{5}}, x\right ) \]