\[ \int (b \sinh (c+d x))^{7/2} \, dx \]
Optimal antiderivative \[ \frac {2 b \cosh \left (d x +c \right ) \left (b \sinh \left (d x +c \right )\right )^{\frac {5}{2}}}{7 d}+\frac {10 i b^{4} \sqrt {\frac {1}{2}+\frac {\sin \left (i d x +i c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {1}{2} i c +\frac {1}{4} \pi +\frac {1}{2} i d x \right ), \sqrt {2}\right ) \sqrt {i \sinh \left (d x +c \right )}}{21 \sin \left (\frac {1}{2} i c +\frac {1}{4} \pi +\frac {1}{2} i d x \right ) d \sqrt {b \sinh \left (d x +c \right )}}-\frac {10 b^{3} \cosh \left (d x +c \right ) \sqrt {b \sinh \left (d x +c \right )}}{21 d} \]
command
integrate((b*sinh(d*x+c))^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {40 \, {\left (\sqrt {2} b^{3} \cosh \left (d x + c\right )^{3} + 3 \, \sqrt {2} b^{3} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, \sqrt {2} b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + \sqrt {2} b^{3} \sinh \left (d x + c\right )^{3}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (4, 0, \cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) + {\left (3 \, b^{3} \cosh \left (d x + c\right )^{6} + 18 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + 3 \, b^{3} \sinh \left (d x + c\right )^{6} - 23 \, b^{3} \cosh \left (d x + c\right )^{4} - 23 \, b^{3} \cosh \left (d x + c\right )^{2} + {\left (45 \, b^{3} \cosh \left (d x + c\right )^{2} - 23 \, b^{3}\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (15 \, b^{3} \cosh \left (d x + c\right )^{3} - 23 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \, b^{3} + {\left (45 \, b^{3} \cosh \left (d x + c\right )^{4} - 138 \, b^{3} \cosh \left (d x + c\right )^{2} - 23 \, b^{3}\right )} \sinh \left (d x + c\right )^{2} + 2 \, {\left (9 \, b^{3} \cosh \left (d x + c\right )^{5} - 46 \, b^{3} \cosh \left (d x + c\right )^{3} - 23 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \sqrt {b \sinh \left (d x + c\right )}}{84 \, {\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + d \sinh \left (d x + c\right )^{3}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {b \sinh \left (d x + c\right )} b^{3} \sinh \left (d x + c\right )^{3}, x\right ) \]