\[ \int (b \sinh (c+d x))^{5/2} \, dx \]
Optimal antiderivative \[ \frac {2 b \cosh \left (d x +c \right ) \left (b \sinh \left (d x +c \right )\right )^{\frac {3}{2}}}{5 d}-\frac {6 i b^{2} \sqrt {\frac {1}{2}+\frac {\sin \left (i d x +i c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {1}{2} i c +\frac {1}{4} \pi +\frac {1}{2} i d x \right ), \sqrt {2}\right ) \sqrt {b \sinh \left (d x +c \right )}}{5 \sin \left (\frac {1}{2} i c +\frac {1}{4} \pi +\frac {1}{2} i d x \right ) d \sqrt {i \sinh \left (d x +c \right )}} \]
command
integrate((b*sinh(d*x+c))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {12 \, {\left (\sqrt {2} b^{2} \cosh \left (d x + c\right )^{2} + 2 \, \sqrt {2} b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sqrt {2} b^{2} \sinh \left (d x + c\right )^{2}\right )} \sqrt {b} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )\right ) + {\left (b^{2} \cosh \left (d x + c\right )^{4} + 4 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b^{2} \sinh \left (d x + c\right )^{4} + 12 \, b^{2} \cosh \left (d x + c\right )^{2} + 6 \, {\left (b^{2} \cosh \left (d x + c\right )^{2} + 2 \, b^{2}\right )} \sinh \left (d x + c\right )^{2} - b^{2} + 4 \, {\left (b^{2} \cosh \left (d x + c\right )^{3} + 6 \, b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \sqrt {b \sinh \left (d x + c\right )}}{10 \, {\left (d \cosh \left (d x + c\right )^{2} + 2 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + d \sinh \left (d x + c\right )^{2}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {b \sinh \left (d x + c\right )} b^{2} \sinh \left (d x + c\right )^{2}, x\right ) \]