\[ \int \sqrt {c e+d e x} \left (a+b \sinh ^{-1}(c+d x)\right ) \, dx \]
Optimal antiderivative \[ \frac {2 \left (e \left (d x +c \right )\right )^{\frac {3}{2}} \left (a +b \arcsinh \left (d x +c \right )\right )}{3 d e}-\frac {4 b \sqrt {e \left (d x +c \right )}\, \sqrt {1+\left (d x +c \right )^{2}}}{9 d}+\frac {2 b \left (d x +c +1\right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {\sqrt {e \left (d x +c \right )}}{\sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {\sqrt {e \left (d x +c \right )}}{\sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \sqrt {e}\, \sqrt {\frac {1+\left (d x +c \right )^{2}}{\left (d x +c +1\right )^{2}}}}{9 \cos \left (2 \arctan \left (\frac {\sqrt {e \left (d x +c \right )}}{\sqrt {e}}\right )\right ) d \sqrt {1+\left (d x +c \right )^{2}}} \]
command
integrate((a+b*arcsinh(d*x+c))*(d*e*x+c*e)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (3 \, {\left (b d^{3} x + b c d^{2}\right )} \sqrt {{\left (d x + c\right )} \cosh \left (1\right ) + {\left (d x + c\right )} \sinh \left (1\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) + 2 \, \sqrt {d^{3} \cosh \left (1\right ) + d^{3} \sinh \left (1\right )} b {\rm weierstrassPInverse}\left (-\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right ) + {\left (3 \, a d^{3} x + 3 \, a c d^{2} - 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} b d^{2}\right )} \sqrt {{\left (d x + c\right )} \cosh \left (1\right ) + {\left (d x + c\right )} \sinh \left (1\right )}\right )}}{9 \, d^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {d e x + c e} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}, x\right ) \]