\[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^8 (d+e x)^2} \, dx \]
Optimal antiderivative \[ -\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{7 x^{7}}+\frac {e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3 d \,x^{6}}-\frac {11 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{35 d^{2} x^{5}}+\frac {e^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4 d^{3} x^{4}}-\frac {22 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{105 d^{4} x^{3}}-\frac {e^{7} \arctanh \left (\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d}\right )}{8 d^{4}}+\frac {e^{5} \sqrt {-e^{2} x^{2}+d^{2}}}{8 d^{3} x^{2}} \]
command
integrate((-e^2*x^2+d^2)^(5/2)/x^8/(e*x+d)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {1}{53760} \, {\left (\frac {6720 \, e^{6} \log \left (\sqrt {\frac {2 \, d}{x e + d} - 1} + 1\right ) \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d^{4}} - \frac {6720 \, e^{6} \log \left ({\left | \sqrt {\frac {2 \, d}{x e + d} - 1} - 1 \right |}\right ) \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d^{4}} + \frac {32 \, {\left (105 \, e^{6} \log \left (2\right ) - 210 \, e^{6} \log \left (i + 1\right ) + 352 i \, e^{6}\right )} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d^{4}} - \frac {105 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {13}{2}} e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 3780 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {11}{2}} e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 189 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {9}{2}} e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 4992 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {7}{2}} e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 1981 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 700 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 105 \, \sqrt {\frac {2 \, d}{x e + d} - 1} e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d^{4} {\left (\frac {d}{x e + d} - 1\right )}^{7}}\right )} e \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________