\[ \int (b \text {sech}(c+d x))^{5/2} \, dx \]
Optimal antiderivative \[ \frac {2 b \left (b \,\mathrm {sech}\left (d x +c \right )\right )^{\frac {3}{2}} \sinh \left (d x +c \right )}{3 d}-\frac {2 i b^{2} \sqrt {\frac {\cosh \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (i \sinh \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cosh }\left (d x +c \right )\right ) \sqrt {b \,\mathrm {sech}\left (d x +c \right )}}{3 \cosh \left (\frac {d x}{2}+\frac {c}{2}\right ) d} \]
command
integrate((b*sech(d*x+c))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (\sqrt {2} {\left (b^{2} \cosh \left (d x + c\right )^{2} + 2 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b^{2} \sinh \left (d x + c\right )^{2} + b^{2}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) + \sqrt {2} {\left (b^{2} \cosh \left (d x + c\right )^{2} + 2 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b^{2} \sinh \left (d x + c\right )^{2} - b^{2}\right )} \sqrt {\frac {b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )}{\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 1}}\right )}}{3 \, {\left (d \cosh \left (d x + c\right )^{2} + 2 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + d \sinh \left (d x + c\right )^{2} + d\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {b \operatorname {sech}\left (d x + c\right )} b^{2} \operatorname {sech}\left (d x + c\right )^{2}, x\right ) \]