\[ \int \frac {1}{(c e+d e x)^{7/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx \]
Optimal antiderivative \[ -\frac {6 \EllipticE \left (\frac {\sqrt {d e x +c e}}{\sqrt {e}}, i\right )}{5 d \,e^{\frac {7}{2}}}+\frac {6 \EllipticF \left (\frac {\sqrt {d e x +c e}}{\sqrt {e}}, i\right )}{5 d \,e^{\frac {7}{2}}}-\frac {2 \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}}{5 d e \left (d e x +c e \right )^{\frac {5}{2}}}-\frac {6 \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}}{5 d \,e^{3} \sqrt {d e x +c e}} \]
command
integrate(1/(d*e*x+c*e)^(7/2)/(-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
\[ -\frac {2 \, {\left ({\left (3 \, d^{3} x^{2} + 6 \, c d^{2} x + {\left (3 \, c^{2} + 1\right )} d\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt {d x + c} e^{\frac {1}{2}} + 3 \, {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )} \sqrt {-d^{3} e} {\rm weierstrassZeta}\left (\frac {4}{d^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right )\right )} e^{\left (-4\right )}}{5 \, {\left (d^{5} x^{3} + 3 \, c d^{4} x^{2} + 3 \, c^{2} d^{3} x + c^{3} d^{2}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt {d e x + c e}}{d^{6} e^{4} x^{6} + 6 \, c d^{5} e^{4} x^{5} + {\left (15 \, c^{2} - 1\right )} d^{4} e^{4} x^{4} + 4 \, {\left (5 \, c^{3} - c\right )} d^{3} e^{4} x^{3} + 3 \, {\left (5 \, c^{4} - 2 \, c^{2}\right )} d^{2} e^{4} x^{2} + 2 \, {\left (3 \, c^{5} - 2 \, c^{3}\right )} d e^{4} x + {\left (c^{6} - c^{4}\right )} e^{4}}, x\right ) \]