\[ \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (-c \,e^{2} x^{2}+c \,d^{2}\right )^{\frac {3}{2}}}{3 e \left (e x +d \right )^{\frac {3}{2}}}-\frac {4 c^{\frac {3}{2}} d^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {-c \,e^{2} x^{2}+c \,d^{2}}\, \sqrt {2}}{2 \sqrt {c}\, \sqrt {d}\, \sqrt {e x +d}}\right ) \sqrt {2}}{e}+\frac {4 c d \sqrt {-c \,e^{2} x^{2}+c \,d^{2}}}{e \sqrt {e x +d}} \]
command
integrate((-c*e^2*x^2+c*d^2)^(3/2)/(e*x+d)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2}{3} \, {\left (\frac {6 \, \sqrt {2} c d^{2} \arctan \left (\frac {\sqrt {2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d}}{2 \, \sqrt {-c d}}\right )}{\sqrt {-c d}} + \frac {6 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d} c^{3} d + {\left (-{\left (x e + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}} c^{2}}{c^{3}}\right )} c e^{\left (-1\right )} - \frac {4 \, \sqrt {2} {\left (3 \, c^{2} d^{2} \arctan \left (\frac {\sqrt {c d}}{\sqrt {-c d}}\right ) + 4 \, \sqrt {c d} \sqrt {-c d} c d\right )} e^{\left (-1\right )}}{3 \, \sqrt {-c d}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________