\[ \int \frac {\left (A+B x+C x^2\right ) \sqrt {d^2-e^2 x^2}}{(d+e x)^5} \, dx \]
Optimal antiderivative \[ -\frac {\left (A \,e^{2}-B d e +C \,d^{2}\right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{7 d \,e^{3} \left (e x +d \right )^{5}}+\frac {C \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{e^{3} \left (e x +d \right )^{4}}-\frac {\left (23 C \,d^{2}+e \left (2 A e +5 B d \right )\right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{35 d^{2} e^{3} \left (e x +d \right )^{4}}-\frac {\left (23 C \,d^{2}+e \left (2 A e +5 B d \right )\right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{105 d^{3} e^{3} \left (e x +d \right )^{3}} \]
command
integrate((C*x^2+B*x+A)*(-e^2*x^2+d^2)^(1/2)/(e*x+d)^5,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {1}{420} \, {\left (\frac {{\left (3 \, {\left (5 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {7}{2}} + 21 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} + 35 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} + 35 \, \sqrt {\frac {2 \, d}{x e + d} - 1}\right )} C \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 35 \, {\left (3 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} + 10 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} + 15 \, \sqrt {\frac {2 \, d}{x e + d} - 1}\right )} C \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 280 \, {\left ({\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} + 3 \, \sqrt {\frac {2 \, d}{x e + d} - 1}\right )} C \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - \frac {3 \, {\left (5 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {7}{2}} + 21 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} + 35 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} + 35 \, \sqrt {\frac {2 \, d}{x e + d} - 1}\right )} B e \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d} + \frac {21 \, {\left (3 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} + 10 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} + 15 \, \sqrt {\frac {2 \, d}{x e + d} - 1}\right )} B e \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d} - \frac {70 \, {\left ({\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} + 3 \, \sqrt {\frac {2 \, d}{x e + d} - 1}\right )} B e \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d} - 420 \, C \sqrt {\frac {2 \, d}{x e + d} - 1} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + \frac {3 \, {\left (5 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {7}{2}} + 21 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} + 35 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} + 35 \, \sqrt {\frac {2 \, d}{x e + d} - 1}\right )} A e^{2} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d^{2}} - \frac {7 \, {\left (3 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} + 10 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} + 15 \, \sqrt {\frac {2 \, d}{x e + d} - 1}\right )} A e^{2} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d^{2}}\right )} e^{\left (-4\right )}}{d} + \frac {4 \, {\left (23 i \, C d^{2} + 5 i \, B d e + 2 i \, A e^{2}\right )} e^{\left (-4\right )} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d^{3}}\right )} e \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________