8.55 Problem number 2830

\[ \int \frac {1}{\left (\frac {c}{(a+b x)^2}\right )^{5/2}} \, dx \]

Optimal antiderivative \[ \frac {\left (b x +a \right )^{5}}{6 b \,c^{2} \sqrt {\frac {c}{\left (b x +a \right )^{2}}}} \]

command

integrate(1/(c/(b*x+a)^2)^(5/2),x, algorithm="giac")

Giac 1.9.0-11 via sagemath 9.6 output

\[ \frac {b^{5} \sqrt {c} x^{6} + 6 \, a b^{4} \sqrt {c} x^{5} + 15 \, a^{2} b^{3} \sqrt {c} x^{4} + 20 \, a^{3} b^{2} \sqrt {c} x^{3} + 15 \, a^{4} b \sqrt {c} x^{2} + 6 \, a^{5} \sqrt {c} x}{6 \, c^{3} \mathrm {sgn}\left (b x + a\right )} \]

Giac 1.7.0 via sagemath 9.3 output

\[ \int \frac {1}{\left (\frac {c}{{\left (b x + a\right )}^{2}}\right )^{\frac {5}{2}}}\,{d x} \]________________________________________________________________________________________