Problem number 841

$\int \frac{d + e x}{{(d^{2} - e^{2} x^{2})}^{5 / 2}} d x$

Optimal antiderivative $\frac{e x + d}{3 d e {(- e^{2} x^{2} + d^{2})}^{\frac{3}{2}}} + \frac{2 x}{3 d^{3} \sqrt{- e^{2} x^{2} + d^{2}}}$

command

integrate((e*x+d)/(-e^2*x^2+d^2)^(5/2),x, algorithm="giac")

Giac 1.9.0-11 via sagemath 9.6 output

$could not integrate$

Giac 1.7.0 via sagemath 9.3 output

$- \frac{\sqrt{- x^{2} e^{2} + d^{2}} (x (\frac{2 x^{2} e^{2}}{d^{3}} - \frac{3}{d}) - e^{(- 1)})}{3 {(x^{2} e^{2} - d^{2})}^{2}}$