\[ \int (c e+d e x)^{3/2} \left (a+b \sinh ^{-1}(c+d x)\right ) \, dx \]
Optimal antiderivative \[ \frac {2 \left (e \left (d x +c \right )\right )^{\frac {5}{2}} \left (a +b \arcsinh \left (d x +c \right )\right )}{5 d e}-\frac {4 b \left (e \left (d x +c \right )\right )^{\frac {3}{2}} \sqrt {1+\left (d x +c \right )^{2}}}{25 d}+\frac {12 b e \sqrt {e \left (d x +c \right )}\, \sqrt {1+\left (d x +c \right )^{2}}}{25 d \left (d x +c +1\right )}-\frac {12 b \,e^{\frac {3}{2}} \left (d x +c +1\right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {\sqrt {e \left (d x +c \right )}}{\sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {\sqrt {e \left (d x +c \right )}}{\sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {1+\left (d x +c \right )^{2}}{\left (d x +c +1\right )^{2}}}}{25 \cos \left (2 \arctan \left (\frac {\sqrt {e \left (d x +c \right )}}{\sqrt {e}}\right )\right ) d \sqrt {1+\left (d x +c \right )^{2}}}+\frac {6 b \,e^{\frac {3}{2}} \left (d x +c +1\right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {\sqrt {e \left (d x +c \right )}}{\sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {\sqrt {e \left (d x +c \right )}}{\sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {1+\left (d x +c \right )^{2}}{\left (d x +c +1\right )^{2}}}}{25 \cos \left (2 \arctan \left (\frac {\sqrt {e \left (d x +c \right )}}{\sqrt {e}}\right )\right ) d \sqrt {1+\left (d x +c \right )^{2}}} \]
command
integrate((d*e*x+c*e)^(3/2)*(a+b*arcsinh(d*x+c)),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (5 \, {\left ({\left (b d^{3} x^{2} + 2 \, b c d^{2} x + b c^{2} d\right )} \cosh \left (1\right ) + {\left (b d^{3} x^{2} + 2 \, b c d^{2} x + b c^{2} d\right )} \sinh \left (1\right )\right )} \sqrt {{\left (d x + c\right )} \cosh \left (1\right ) + {\left (d x + c\right )} \sinh \left (1\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - 6 \, \sqrt {d^{3} \cosh \left (1\right ) + d^{3} \sinh \left (1\right )} {\left (b \cosh \left (1\right ) + b \sinh \left (1\right )\right )} {\rm weierstrassZeta}\left (-\frac {4}{d^{2}}, 0, {\rm weierstrassPInverse}\left (-\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right ) + {\left (5 \, {\left (a d^{3} x^{2} + 2 \, a c d^{2} x + a c^{2} d\right )} \cosh \left (1\right ) + 5 \, {\left (a d^{3} x^{2} + 2 \, a c d^{2} x + a c^{2} d\right )} \sinh \left (1\right ) - 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} {\left ({\left (b d^{2} x + b c d\right )} \cosh \left (1\right ) + {\left (b d^{2} x + b c d\right )} \sinh \left (1\right )\right )}\right )} \sqrt {{\left (d x + c\right )} \cosh \left (1\right ) + {\left (d x + c\right )} \sinh \left (1\right )}\right )}}{25 \, d^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (a d e x + a c e + {\left (b d e x + b c e\right )} \operatorname {arsinh}\left (d x + c\right )\right )} \sqrt {d e x + c e}, x\right ) \]