\[ \int \frac {\sqrt {c d^2-c e^2 x^2}}{(d+e x)^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {\arctanh \left (\frac {\sqrt {-c \,e^{2} x^{2}+c \,d^{2}}\, \sqrt {2}}{2 \sqrt {c}\, \sqrt {d}\, \sqrt {e x +d}}\right ) \sqrt {c}\, \sqrt {2}}{16 d^{\frac {3}{2}} e}-\frac {\sqrt {-c \,e^{2} x^{2}+c \,d^{2}}}{2 e \left (e x +d \right )^{\frac {5}{2}}}+\frac {\sqrt {-c \,e^{2} x^{2}+c \,d^{2}}}{8 d e \left (e x +d \right )^{\frac {3}{2}}} \]
command
integrate((-c*e^2*x^2+c*d^2)^(1/2)/(e*x+d)^(7/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {1}{16} \, {\left (\frac {\sqrt {2} c \arctan \left (\frac {\sqrt {2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d}}{2 \, \sqrt {-c d}}\right )}{\sqrt {-c d} d} + \frac {2 \, {\left (2 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d} c^{2} d + {\left (-{\left (x e + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}} c\right )}}{{\left (x e + d\right )}^{2} c^{2} d}\right )} e^{\left (-1\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {\sqrt {-c e^{2} x^{2} + c d^{2}}}{{\left (e x + d\right )}^{\frac {7}{2}}}\,{d x} \]________________________________________________________________________________________