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