\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{d+e x} \, dx \]
Optimal antiderivative \[ -\frac {\left (-b e +2 c d \right ) \left (5 b e g -2 c \left (-d g +6 e f \right )\right ) \left (2 c x +b \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}{192 c^{2} e}+\frac {\left (-5 b e g -2 c d g +12 c e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {5}{2}}}{60 c \,e^{2}}-\frac {g \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {7}{2}}}{6 c \,e^{2} \left (e x +d \right )}-\frac {\left (-b e +2 c d \right )^{5} \left (5 b e g -2 c \left (-d g +6 e f \right )\right ) \arctan \left (\frac {e \left (2 c x +b \right )}{2 \sqrt {c}\, \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}\right )}{1024 c^{\frac {7}{2}} e^{2}}-\frac {\left (-b e +2 c d \right )^{3} \left (5 b e g -2 c \left (-d g +6 e f \right )\right ) \left (2 c x +b \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{512 c^{3} e} \]
command
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{7680} \, \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, c^{2} g x e^{3} - \frac {{\left (12 \, c^{7} d g e^{10} - 12 \, c^{7} f e^{11} - 25 \, b c^{6} g e^{11}\right )} e^{\left (-8\right )}}{c^{5}}\right )} x - \frac {{\left (140 \, c^{7} d^{2} g e^{9} + 120 \, c^{7} d f e^{10} - 8 \, b c^{6} d g e^{10} - 252 \, b c^{6} f e^{11} - 135 \, b^{2} c^{5} g e^{11}\right )} e^{\left (-8\right )}}{c^{5}}\right )} x + \frac {{\left (384 \, c^{7} d^{3} g e^{8} - 384 \, c^{7} d^{2} f e^{9} - 804 \, b c^{6} d^{2} g e^{9} + 24 \, b c^{6} d f e^{10} + 408 \, b^{2} c^{5} d g e^{10} + 372 \, b^{2} c^{5} f e^{11} + 5 \, b^{3} c^{4} g e^{11}\right )} e^{\left (-8\right )}}{c^{5}}\right )} x + \frac {{\left (240 \, c^{7} d^{4} g e^{7} + 2400 \, c^{7} d^{3} f e^{8} - 144 \, b c^{6} d^{3} g e^{8} - 5136 \, b c^{6} d^{2} f e^{9} - 216 \, b^{2} c^{5} d^{2} g e^{9} + 2616 \, b^{2} c^{5} d f e^{10} + 180 \, b^{3} c^{4} d g e^{10} + 60 \, b^{3} c^{4} f e^{11} - 25 \, b^{4} c^{3} g e^{11}\right )} e^{\left (-8\right )}}{c^{5}}\right )} x - \frac {{\left (1536 \, c^{7} d^{5} g e^{6} - 1536 \, c^{7} d^{4} f e^{7} - 3312 \, b c^{6} d^{4} g e^{7} + 672 \, b c^{6} d^{3} f e^{8} + 3216 \, b^{2} c^{5} d^{3} g e^{8} + 2064 \, b^{2} c^{5} d^{2} f e^{9} - 1880 \, b^{3} c^{4} d^{2} g e^{9} - 1560 \, b^{3} c^{4} d f e^{10} + 620 \, b^{4} c^{3} d g e^{10} + 180 \, b^{4} c^{3} f e^{11} - 75 \, b^{5} c^{2} g e^{11}\right )} e^{\left (-8\right )}}{c^{5}}\right )} - \frac {{\left (64 \, c^{6} d^{6} g - 384 \, c^{6} d^{5} f e + 960 \, b c^{5} d^{4} f e^{2} - 240 \, b^{2} c^{4} d^{4} g e^{2} - 960 \, b^{2} c^{4} d^{3} f e^{3} + 320 \, b^{3} c^{3} d^{3} g e^{3} + 480 \, b^{3} c^{3} d^{2} f e^{4} - 180 \, b^{4} c^{2} d^{2} g e^{4} - 120 \, b^{4} c^{2} d f e^{5} + 48 \, b^{5} c d g e^{5} + 12 \, b^{5} c f e^{6} - 5 \, b^{6} g e^{6}\right )} \sqrt {-c} e^{\left (-2\right )} \log \left ({\left | -b \sqrt {-c} e - 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} c \right |}\right )}{1024 \, c^{4}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________