\[ \int \sqrt {a+b \sinh (x)} (A+B \sinh (x)) \, dx \]
Optimal antiderivative \[ \frac {2 B \cosh \left (x \right ) \sqrt {a +b \sinh \left (x \right )}}{3}+\frac {2 i \left (3 A b +B a \right ) \sqrt {\frac {1}{2}+\frac {i \sinh \left (x \right )}{2}}\, \EllipticE \left (\cos \left (\frac {\pi }{4}+\frac {i x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{i a +b}}\right ) \sqrt {a +b \sinh \left (x \right )}}{3 \sin \left (\frac {\pi }{4}+\frac {i x}{2}\right ) b \sqrt {\frac {a +b \sinh \left (x \right )}{-i b +a}}}-\frac {2 i \left (a^{2}+b^{2}\right ) B \sqrt {\frac {1}{2}+\frac {i \sinh \left (x \right )}{2}}\, \EllipticF \left (\cos \left (\frac {\pi }{4}+\frac {i x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{i a +b}}\right ) \sqrt {\frac {a +b \sinh \left (x \right )}{-i b +a}}}{3 \sin \left (\frac {\pi }{4}+\frac {i x}{2}\right ) b \sqrt {a +b \sinh \left (x \right )}} \]
command
integrate((a+b*sinh(x))^(1/2)*(A+B*sinh(x)),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (\sqrt {2} {\left (2 \, B a^{2} - 3 \, A a b + 3 \, B b^{2}\right )} \cosh \left (x\right ) + \sqrt {2} {\left (2 \, B a^{2} - 3 \, A a b + 3 \, B b^{2}\right )} \sinh \left (x\right )\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right ) + 6 \, {\left (\sqrt {2} {\left (B a b + 3 \, A b^{2}\right )} \cosh \left (x\right ) + \sqrt {2} {\left (B a b + 3 \, A b^{2}\right )} \sinh \left (x\right )\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right )\right ) - 3 \, {\left (B b^{2} \cosh \left (x\right )^{2} + B b^{2} \sinh \left (x\right )^{2} + B b^{2} - 2 \, {\left (B a b + 3 \, A b^{2}\right )} \cosh \left (x\right ) + 2 \, {\left (B b^{2} \cosh \left (x\right ) - B a b - 3 \, A b^{2}\right )} \sinh \left (x\right )\right )} \sqrt {b \sinh \left (x\right ) + a}}{9 \, {\left (b^{2} \cosh \left (x\right ) + b^{2} \sinh \left (x\right )\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (B \sinh \left (x\right ) + A\right )} \sqrt {b \sinh \left (x\right ) + a}, x\right ) \]