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