\[ \int \frac {1}{(c e+d e x)^{13/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx \]
Optimal antiderivative \[ \frac {30 \EllipticF \left (\frac {\sqrt {d e x +c e}}{\sqrt {e}}, i\right )}{77 d \,e^{\frac {13}{2}}}-\frac {2 \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}}{11 d e \left (d e x +c e \right )^{\frac {11}{2}}}-\frac {18 \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}}{77 d \,e^{3} \left (d e x +c e \right )^{\frac {7}{2}}}-\frac {30 \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}}{77 d \,e^{5} \left (d e x +c e \right )^{\frac {3}{2}}} \]
command
integrate(1/(d*e*x+c*e)^(13/2)/(-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
\[ -\frac {2 \, {\left ({\left (15 \, d^{6} x^{4} + 60 \, c d^{5} x^{3} + 9 \, {\left (10 \, c^{2} + 1\right )} d^{4} x^{2} + 6 \, {\left (10 \, c^{3} + 3 \, c\right )} d^{3} x + {\left (15 \, c^{4} + 9 \, c^{2} + 7\right )} d^{2}\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt {d x + c} e^{\frac {1}{2}} + 15 \, {\left (d^{6} x^{6} + 6 \, c d^{5} x^{5} + 15 \, c^{2} d^{4} x^{4} + 20 \, c^{3} d^{3} x^{3} + 15 \, c^{4} d^{2} x^{2} + 6 \, c^{5} d x + c^{6}\right )} \sqrt {-d^{3} e} {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right )} e^{\left (-7\right )}}{77 \, {\left (d^{9} x^{6} + 6 \, c d^{8} x^{5} + 15 \, c^{2} d^{7} x^{4} + 20 \, c^{3} d^{6} x^{3} + 15 \, c^{4} d^{5} x^{2} + 6 \, c^{5} d^{4} x + c^{6} d^{3}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt {d e x + c e}}{d^{9} e^{7} x^{9} + 9 \, c d^{8} e^{7} x^{8} + {\left (36 \, c^{2} - 1\right )} d^{7} e^{7} x^{7} + 7 \, {\left (12 \, c^{3} - c\right )} d^{6} e^{7} x^{6} + 21 \, {\left (6 \, c^{4} - c^{2}\right )} d^{5} e^{7} x^{5} + 7 \, {\left (18 \, c^{5} - 5 \, c^{3}\right )} d^{4} e^{7} x^{4} + 7 \, {\left (12 \, c^{6} - 5 \, c^{4}\right )} d^{3} e^{7} x^{3} + 3 \, {\left (12 \, c^{7} - 7 \, c^{5}\right )} d^{2} e^{7} x^{2} + {\left (9 \, c^{8} - 7 \, c^{6}\right )} d e^{7} x + {\left (c^{9} - c^{7}\right )} e^{7}}, x\right ) \]