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