\[ \int (c e+d e x)^{7/2} \left (a+b \sinh ^{-1}(c+d x)\right ) \, dx \]
Optimal antiderivative \[ \frac {2 \left (e \left (d x +c \right )\right )^{\frac {9}{2}} \left (a +b \arcsinh \left (d x +c \right )\right )}{9 d e}+\frac {28 b \,e^{2} \left (e \left (d x +c \right )\right )^{\frac {3}{2}} \sqrt {1+\left (d x +c \right )^{2}}}{405 d}-\frac {4 b \left (e \left (d x +c \right )\right )^{\frac {7}{2}} \sqrt {1+\left (d x +c \right )^{2}}}{81 d}-\frac {28 b \,e^{3} \sqrt {e \left (d x +c \right )}\, \sqrt {1+\left (d x +c \right )^{2}}}{135 d \left (d x +c +1\right )}+\frac {28 b \,e^{\frac {7}{2}} \left (d x +c +1\right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {\sqrt {e \left (d x +c \right )}}{\sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {\sqrt {e \left (d x +c \right )}}{\sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {1+\left (d x +c \right )^{2}}{\left (d x +c +1\right )^{2}}}}{135 \cos \left (2 \arctan \left (\frac {\sqrt {e \left (d x +c \right )}}{\sqrt {e}}\right )\right ) d \sqrt {1+\left (d x +c \right )^{2}}}-\frac {14 b \,e^{\frac {7}{2}} \left (d x +c +1\right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {\sqrt {e \left (d x +c \right )}}{\sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {\sqrt {e \left (d x +c \right )}}{\sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {1+\left (d x +c \right )^{2}}{\left (d x +c +1\right )^{2}}}}{135 \cos \left (2 \arctan \left (\frac {\sqrt {e \left (d x +c \right )}}{\sqrt {e}}\right )\right ) d \sqrt {1+\left (d x +c \right )^{2}}} \]
command
integrate((d*e*x+c*e)^(7/2)*(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 (45 \, {\left ({\left (b d^{5} x^{4} + 4 \, b c d^{4} x^{3} + 6 \, b c^{2} d^{3} x^{2} + 4 \, b c^{3} d^{2} x + b c^{4} d\right )} \cosh \left (1\right )^{3} + 3 \, {\left (b d^{5} x^{4} + 4 \, b c d^{4} x^{3} + 6 \, b c^{2} d^{3} x^{2} + 4 \, b c^{3} d^{2} x + b c^{4} d\right )} \cosh \left (1\right )^{2} \sinh \left (1\right ) + 3 \, {\left (b d^{5} x^{4} + 4 \, b c d^{4} x^{3} + 6 \, b c^{2} d^{3} x^{2} + 4 \, b c^{3} d^{2} x + b c^{4} d\right )} \cosh \left (1\right ) \sinh \left (1\right )^{2} + {\left (b d^{5} x^{4} + 4 \, b c d^{4} x^{3} + 6 \, b c^{2} d^{3} x^{2} + 4 \, b c^{3} d^{2} x + b c^{4} d\right )} \sinh \left (1\right )^{3}\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 ) + 42 \, \sqrt {d^{3} \cosh \left (1\right ) + d^{3} \sinh \left (1\right )} {\left (b \cosh \left (1\right )^{3} + 3 \, b \cosh \left (1\right )^{2} \sinh \left (1\right ) + 3 \, b \cosh \left (1\right ) \sinh \left (1\right )^{2} + b \sinh \left (1\right )^{3}\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 (45 \, {\left (a d^{5} x^{4} + 4 \, a c d^{4} x^{3} + 6 \, a c^{2} d^{3} x^{2} + 4 \, a c^{3} d^{2} x + a c^{4} d\right )} \cosh \left (1\right )^{3} + 135 \, {\left (a d^{5} x^{4} + 4 \, a c d^{4} x^{3} + 6 \, a c^{2} d^{3} x^{2} + 4 \, a c^{3} d^{2} x + a c^{4} d\right )} \cosh \left (1\right )^{2} \sinh \left (1\right ) + 135 \, {\left (a d^{5} x^{4} + 4 \, a c d^{4} x^{3} + 6 \, a c^{2} d^{3} x^{2} + 4 \, a c^{3} d^{2} x + a c^{4} d\right )} \cosh \left (1\right ) \sinh \left (1\right )^{2} + 45 \, {\left (a d^{5} x^{4} + 4 \, a c d^{4} x^{3} + 6 \, a c^{2} d^{3} x^{2} + 4 \, a c^{3} d^{2} x + a c^{4} d\right )} \sinh \left (1\right )^{3} - 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} {\left ({\left (5 \, b d^{4} x^{3} + 15 \, b c d^{3} x^{2} + {\left (15 \, b c^{2} - 7 \, b\right )} d^{2} x + {\left (5 \, b c^{3} - 7 \, b c\right )} d\right )} \cosh \left (1\right )^{3} + 3 \, {\left (5 \, b d^{4} x^{3} + 15 \, b c d^{3} x^{2} + {\left (15 \, b c^{2} - 7 \, b\right )} d^{2} x + {\left (5 \, b c^{3} - 7 \, b c\right )} d\right )} \cosh \left (1\right )^{2} \sinh \left (1\right ) + 3 \, {\left (5 \, b d^{4} x^{3} + 15 \, b c d^{3} x^{2} + {\left (15 \, b c^{2} - 7 \, b\right )} d^{2} x + {\left (5 \, b c^{3} - 7 \, b c\right )} d\right )} \cosh \left (1\right ) \sinh \left (1\right )^{2} + {\left (5 \, b d^{4} x^{3} + 15 \, b c d^{3} x^{2} + {\left (15 \, b c^{2} - 7 \, b\right )} d^{2} x + {\left (5 \, b c^{3} - 7 \, b c\right )} d\right )} \sinh \left (1\right )^{3}\right )}\right )} \sqrt {{\left (d x + c\right )} \cosh \left (1\right ) + {\left (d x + c\right )} \sinh \left (1\right )}\right )}}{405 \, d^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (a d^{3} e^{3} x^{3} + 3 \, a c d^{2} e^{3} x^{2} + 3 \, a c^{2} d e^{3} x + a c^{3} e^{3} + {\left (b d^{3} e^{3} x^{3} + 3 \, b c d^{2} e^{3} x^{2} + 3 \, b c^{2} d e^{3} x + b c^{3} e^{3}\right )} \operatorname {arsinh}\left (d x + c\right )\right )} \sqrt {d e x + c e}, x\right ) \]