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