\[ \int x \left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]
Optimal antiderivative \[ \frac {25 b^{2} d^{2} x^{2}}{288}+\frac {5 b^{2} c^{2} d^{2} x^{4}}{288}+\frac {b^{2} d^{2} \left (c^{2} x^{2}+1\right )^{3}}{108 c^{2}}-\frac {5 b \,d^{2} x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{72 c}-\frac {b \,d^{2} x \left (c^{2} x^{2}+1\right )^{\frac {5}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{18 c}-\frac {5 d^{2} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{96 c^{2}}+\frac {d^{2} \left (c^{2} x^{2}+1\right )^{3} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{6 c^{2}}-\frac {5 b \,d^{2} x \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{48 c} \]
command
int(x*(c^2*d*x^2+d)^2*(a+b*arcsinh(c*x))^2,x)
Maple 2022.1 output
\[\int x \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arcsinh \left (c x \right )\right )^{2}\, dx\]
Maple 2021.1 output
\[ \frac {d^{2} a^{2} \left (\frac {1}{6} c^{6} x^{6}+\frac {1}{2} c^{4} x^{4}+\frac {1}{2} c^{2} x^{2}\right )+d^{2} b^{2} \left (\frac {\arcsinh \left (c x \right )^{2} \left (c^{2} x^{2}+1\right )^{3}}{6}-\frac {\arcsinh \left (c x \right ) c x \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{18}-\frac {5 \arcsinh \left (c x \right ) c x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{72}-\frac {5 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x}{48}-\frac {5 \arcsinh \left (c x \right )^{2}}{96}+\frac {\left (c^{2} x^{2}+1\right )^{3}}{108}+\frac {5 \left (c^{2} x^{2}+1\right )^{2}}{288}+\frac {5 c^{2} x^{2}}{96}+\frac {5}{96}\right )+2 d^{2} a b \left (\frac {\arcsinh \left (c x \right ) c^{6} x^{6}}{6}+\frac {\arcsinh \left (c x \right ) c^{4} x^{4}}{2}+\frac {\arcsinh \left (c x \right ) c^{2} x^{2}}{2}-\frac {c^{5} x^{5} \sqrt {c^{2} x^{2}+1}}{36}-\frac {13 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{144}-\frac {11 c x \sqrt {c^{2} x^{2}+1}}{96}+\frac {11 \arcsinh \left (c x \right )}{96}\right )}{c^{2}} \]