\[ \int \frac {(d+e x)^3 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (-b e g +c d g +c e f \right ) \left (e x +d \right )^{3}}{3 c \,e^{2} \left (-b e +2 c d \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}+\frac {g \arctan \! \left (\frac {e \left (2 c x +b \right )}{2 \sqrt {c}\, \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}\right )}{c^{\frac {5}{2}} e^{2}}-\frac {2 g \left (e x +d \right )}{c^{2} e^{2} \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}} \]
command
integrate((e*x+d)^3*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {Exception raised: TypeError} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \frac {\sqrt {-c e^{2}} g e^{\left (-3\right )} \log \left ({\left | -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 - \sqrt {-c e^{2}} b \right |}\right )}{c^{3}} + \frac {2 \, \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e} {\left ({\left ({\left (\frac {{\left (56 \, c^{5} d^{4} g e^{4} + 8 \, c^{5} d^{3} f e^{5} - 116 \, b c^{4} d^{3} g e^{5} - 12 \, b c^{4} d^{2} f e^{6} + 90 \, b^{2} c^{3} d^{2} g e^{6} + 6 \, b^{2} c^{3} d f e^{7} - 31 \, b^{3} c^{2} d g e^{7} - b^{3} c^{2} f e^{8} + 4 \, b^{4} c g e^{8}\right )} x}{16 \, c^{6} d^{4} e^{3} - 32 \, b c^{5} d^{3} e^{4} + 24 \, b^{2} c^{4} d^{2} e^{5} - 8 \, b^{3} c^{3} d e^{6} + b^{4} c^{2} e^{7}} + \frac {3 \, {\left (24 \, c^{5} d^{5} g e^{3} + 8 \, c^{5} d^{4} f e^{4} - 36 \, b c^{4} d^{4} g e^{4} - 12 \, b c^{4} d^{3} f e^{5} + 10 \, b^{2} c^{3} d^{3} g e^{5} + 6 \, b^{2} c^{3} d^{2} f e^{6} + 9 \, b^{3} c^{2} d^{2} g e^{6} - b^{3} c^{2} d f e^{7} - 6 \, b^{4} c d g e^{7} + b^{5} g e^{8}\right )}}{16 \, c^{6} d^{4} e^{3} - 32 \, b c^{5} d^{3} e^{4} + 24 \, b^{2} c^{4} d^{2} e^{5} - 8 \, b^{3} c^{3} d e^{6} + b^{4} c^{2} e^{7}}\right )} x - \frac {3 \, {\left (8 \, c^{5} d^{6} g e^{2} - 8 \, c^{5} d^{5} f e^{3} - 44 \, b c^{4} d^{5} g e^{3} + 12 \, b c^{4} d^{4} f e^{4} + 70 \, b^{2} c^{3} d^{4} g e^{4} - 6 \, b^{2} c^{3} d^{3} f e^{5} - 49 \, b^{3} c^{2} d^{3} g e^{5} + b^{3} c^{2} d^{2} f e^{6} + 16 \, b^{4} c d^{2} g e^{6} - 2 \, b^{5} d g e^{7}\right )}}{16 \, c^{6} d^{4} e^{3} - 32 \, b c^{5} d^{3} e^{4} + 24 \, b^{2} c^{4} d^{2} e^{5} - 8 \, b^{3} c^{3} d e^{6} + b^{4} c^{2} e^{7}}\right )} x - \frac {40 \, c^{5} d^{7} g e - 8 \, c^{5} d^{6} f e^{2} - 124 \, b c^{4} d^{6} g e^{2} + 12 \, b c^{4} d^{5} f e^{3} + 150 \, b^{2} c^{3} d^{5} g e^{3} - 6 \, b^{2} c^{3} d^{4} f e^{4} - 89 \, b^{3} c^{2} d^{4} g e^{4} + b^{3} c^{2} d^{3} f e^{5} + 26 \, b^{4} c d^{3} g e^{5} - 3 \, b^{5} d^{2} g e^{6}}{16 \, c^{6} d^{4} e^{3} - 32 \, b c^{5} d^{3} e^{4} + 24 \, b^{2} c^{4} d^{2} e^{5} - 8 \, b^{3} c^{3} d e^{6} + b^{4} c^{2} e^{7}}\right )}}{3 \, {\left (c x^{2} e^{2} - c d^{2} + b x e^{2} + b d e\right )}^{2}} \]