\[ \int (d+e x)^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]
Optimal antiderivative \[ 2 b^{2} d^{3} x -\frac {4 b^{2} d \,e^{2} x}{3 c^{2}}+\frac {3 b^{2} d^{2} e \,x^{2}}{4}-\frac {3 b^{2} e^{3} x^{2}}{32 c^{2}}+\frac {2 b^{2} d \,e^{2} x^{3}}{9}+\frac {b^{2} e^{3} x^{4}}{32}-\frac {d^{4} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{4 e}+\frac {3 d^{2} e \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{4 c^{2}}-\frac {3 e^{3} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{32 c^{4}}+\frac {\left (e x +d \right )^{4} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{4 e}-\frac {2 b \,d^{3} \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{c}+\frac {4 b d \,e^{2} \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{3 c^{3}}-\frac {3 b \,d^{2} e x \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{2 c}+\frac {3 b \,e^{3} x \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{16 c^{3}}-\frac {2 b d \,e^{2} x^{2} \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{3 c}-\frac {b \,e^{3} x^{3} \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{8 c} \]
command
int((e*x+d)^3*(a+b*arcsinh(c*x))^2,x)
Maple 2022.1 output
\[\int \left (e x +d \right )^{3} \left (a +b \arcsinh \left (c x \right )\right )^{2}\, dx\]
Maple 2021.1 output
\[ \frac {\frac {\left (c e x +c d \right )^{4} a^{2}}{4 c^{3} e}+\frac {b^{2} \left (c^{3} d^{3} \left (\arcsinh \left (c x \right )^{2} c x -2 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )+\frac {3 c^{2} d^{2} e \left (2 \arcsinh \left (c x \right )^{2} c^{2} x^{2}-2 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x +\arcsinh \left (c x \right )^{2}+c^{2} x^{2}+1\right )}{4}+\frac {c d \,e^{2} \left (9 \arcsinh \left (c x \right )^{2} c^{3} x^{3}-6 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+27 \arcsinh \left (c x \right )^{2} c x +2 c^{3} x^{3}-42 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}+42 c x \right )}{9}+\frac {e^{3} \left (8 \arcsinh \left (c x \right )^{2} c^{4} x^{4}-4 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+16 \arcsinh \left (c x \right )^{2} c^{2} x^{2}+c^{4} x^{4}-10 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x +5 \arcsinh \left (c x \right )^{2}+5 c^{2} x^{2}+4\right )}{32}-3 c d \,e^{2} \left (\arcsinh \left (c x \right )^{2} c x -2 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )-\frac {e^{3} \left (2 \arcsinh \left (c x \right )^{2} c^{2} x^{2}-2 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x +\arcsinh \left (c x \right )^{2}+c^{2} x^{2}+1\right )}{4}\right )}{c^{3}}+\frac {2 a b \left (\frac {e^{3} \arcsinh \left (c x \right ) c^{4} x^{4}}{4}+e^{2} \arcsinh \left (c x \right ) c^{4} x^{3} d +\frac {3 e \arcsinh \left (c x \right ) c^{4} x^{2} d^{2}}{2}+\arcsinh \left (c x \right ) c^{4} x \,d^{3}+\frac {\arcsinh \left (c x \right ) c^{4} d^{4}}{4 e}-\frac {e^{4} \left (\frac {c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {c^{2} x^{2}+1}}{8}+\frac {3 \arcsinh \left (c x \right )}{8}\right )+4 c d \,e^{3} \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )+6 c^{2} d^{2} e^{2} \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\arcsinh \left (c x \right )}{2}\right )+4 c^{3} d^{3} e \sqrt {c^{2} x^{2}+1}+c^{4} d^{4} \arcsinh \left (c x \right )}{4 e}\right )}{c^{3}}}{c} \]