\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (-d g +e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}{3 e^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 g \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {5}{2}}}{5 c \,e^{2} \left (e x +d \right )^{\frac {5}{2}}}-\frac {2 \left (-b e +2 c d \right )^{\frac {3}{2}} \left (-d g +e f \right ) \arctanh \left (\frac {\sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{\sqrt {-b e +2 c d}\, \sqrt {e x +d}}\right )}{e^{2}}+\frac {2 \left (-b e +2 c d \right ) \left (-d g +e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{e^{2} \sqrt {e x +d}} \]
command
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+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}{15} \, {\left (\frac {15 \, {\left (4 \, c^{2} d^{3} g - 4 \, c^{2} d^{2} f e - 4 \, b c d^{2} g e + 4 \, b c d f e^{2} + b^{2} d g e^{2} - b^{2} f e^{3}\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 {30 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{6} d^{2} g - 30 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{6} d f e - 15 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{5} d g e + 5 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{5} d g + 15 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{5} f e^{2} - 5 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{5} f e + 3 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{4} g}{c^{5}}\right )} e^{\left (-2\right )} + \frac {2 \, {\left (60 \, c^{3} d^{3} g \arctan \left (\frac {\sqrt {2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right ) - 60 \, c^{3} d^{2} f \arctan \left (\frac {\sqrt {2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right ) e - 60 \, b c^{2} d^{2} g \arctan \left (\frac {\sqrt {2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right ) e + 60 \, b c^{2} d f \arctan \left (\frac {\sqrt {2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right ) e^{2} + 15 \, b^{2} c d g \arctan \left (\frac {\sqrt {2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right ) e^{2} + 52 \, \sqrt {2 \, c d - b e} \sqrt {-2 \, c d + b e} c^{2} d^{2} g - 15 \, b^{2} c f \arctan \left (\frac {\sqrt {2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right ) e^{3} - 40 \, \sqrt {2 \, c d - b e} \sqrt {-2 \, c d + b e} c^{2} d f e - 32 \, \sqrt {2 \, c d - b e} \sqrt {-2 \, c d + b e} b c d g e + 20 \, \sqrt {2 \, c d - b e} \sqrt {-2 \, c d + b e} b c f e^{2} + 3 \, \sqrt {2 \, c d - b e} \sqrt {-2 \, c d + b e} b^{2} g e^{2}\right )} e^{\left (-2\right )}}{15 \, \sqrt {-2 \, c d + b e} c} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________