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