11.5 Problem number 22

\[ \int \frac {x^4 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx \]

Optimal antiderivative \[ \frac {x^{4} \left (e x +d \right )}{5 d e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {4 d^{2}}{15 e^{5} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {4}{5 e^{5} \sqrt {-e^{2} x^{2}+d^{2}}} \]

command

integrate(x^4*(e*x+d)/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

Giac 1.9.0-11 via sagemath 9.6 output

\[ \text {could not integrate} \]

Giac 1.7.0 via sagemath 9.3 output

\[ -\frac {{\left (8 \, d^{4} e^{\left (-5\right )} + {\left (3 \, x^{2} {\left (\frac {x}{d} + 5 \, e^{\left (-1\right )}\right )} - 20 \, d^{2} e^{\left (-3\right )}\right )} x^{2}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]