\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{d+e x} \, dx \]
Optimal antiderivative \[ \frac {\left (-3 b e g -2 c d g +8 c e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}{24 c \,e^{2}}-\frac {g \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {5}{2}}}{4 c \,e^{2} \left (e x +d \right )}+\frac {\left (-b e +2 c d \right )^{3} \left (-3 b e g -2 c d g +8 c e f \right ) \arctan \left (\frac {e \left (2 c x +b \right )}{2 \sqrt {c}\, \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}\right )}{128 c^{\frac {5}{2}} e^{2}}+\frac {\left (-b e +2 c d \right ) \left (-3 b e g -2 c d g +8 c e f \right ) \left (2 c x +b \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{64 c^{2} e} \]
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),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {1}{192} \, \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e} {\left (2 \, {\left (4 \, {\left (6 \, c g x e - \frac {{\left (8 \, c^{4} d g e^{4} - 8 \, c^{4} f e^{5} - 9 \, b c^{3} g e^{5}\right )} e^{\left (-4\right )}}{c^{3}}\right )} x - \frac {{\left (12 \, c^{4} d^{2} g e^{3} + 48 \, c^{4} d f e^{4} - 4 \, b c^{3} d g e^{4} - 56 \, b c^{3} f e^{5} - 3 \, b^{2} c^{2} g e^{5}\right )} e^{\left (-4\right )}}{c^{3}}\right )} x + \frac {{\left (64 \, c^{4} d^{3} g e^{2} - 64 \, c^{4} d^{2} f e^{3} - 76 \, b c^{3} d^{2} g e^{3} + 16 \, b c^{3} d f e^{4} + 36 \, b^{2} c^{2} d g e^{4} + 24 \, b^{2} c^{2} f e^{5} - 9 \, b^{3} c g e^{5}\right )} e^{\left (-4\right )}}{c^{3}}\right )} - \frac {{\left (16 \, c^{4} d^{4} g - 64 \, c^{4} d^{3} f e + 96 \, b c^{3} d^{2} f e^{2} - 24 \, b^{2} c^{2} d^{2} g e^{2} - 48 \, b^{2} c^{2} d f e^{3} + 16 \, b^{3} c d g e^{3} + 8 \, b^{3} c f e^{4} - 3 \, b^{4} g e^{4}\right )} \sqrt {-c} e^{\left (-2\right )} \log \left ({\left | -b \sqrt {-c} e - 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} c \right |}\right )}{128 \, c^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________