\[ \int x^2 \left (d+c^2 d x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]
Optimal antiderivative \[ -\frac {10516 b^{2} d^{3} x}{99225 c^{2}}+\frac {5258 b^{2} d^{3} x^{3}}{297675}+\frac {4198 b^{2} c^{2} d^{3} x^{5}}{165375}+\frac {374 b^{2} c^{4} d^{3} x^{7}}{27783}+\frac {2 b^{2} c^{6} d^{3} x^{9}}{729}+\frac {16 b \,d^{3} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{315 c^{3}}+\frac {4 b \,d^{3} \left (c^{2} x^{2}+1\right )^{\frac {5}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{525 c^{3}}+\frac {2 b \,d^{3} \left (c^{2} x^{2}+1\right )^{\frac {7}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{441 c^{3}}-\frac {2 b \,d^{3} \left (c^{2} x^{2}+1\right )^{\frac {9}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{81 c^{3}}+\frac {16 d^{3} x^{3} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{315}+\frac {8 d^{3} x^{3} \left (c^{2} x^{2}+1\right ) \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{105}+\frac {2 d^{3} x^{3} \left (c^{2} x^{2}+1\right )^{2} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{21}+\frac {d^{3} x^{3} \left (c^{2} x^{2}+1\right )^{3} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{9}+\frac {64 b \,d^{3} \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{945 c^{3}}-\frac {32 b \,d^{3} x^{2} \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{945 c} \]
command
int(x^2*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2,x)
Maple 2022.1 output
\[\int x^{2} \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arcsinh \left (c x \right )\right )^{2}\, dx\]
Maple 2021.1 output
\[ \frac {d^{3} a^{2} \left (\frac {1}{9} c^{9} x^{9}+\frac {3}{7} c^{7} x^{7}+\frac {3}{5} c^{5} x^{5}+\frac {1}{3} c^{3} x^{3}\right )+d^{3} b^{2} \left (\frac {\arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{4}}{9}-\frac {16 \arcsinh \left (c x \right )^{2} c x}{315}-\frac {\arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{3}}{63}-\frac {2 \arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{2}}{105}-\frac {8 \arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{315}-\frac {2 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {9}{2}}}{81}-\frac {3406208 c x}{31255875}+\frac {2 c x \left (c^{2} x^{2}+1\right )^{4}}{729}+\frac {622 c x \left (c^{2} x^{2}+1\right )^{3}}{250047}+\frac {15224 c x \left (c^{2} x^{2}+1\right )^{2}}{10418625}-\frac {115504 c x \left (c^{2} x^{2}+1\right )}{31255875}+\frac {32 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{315}+\frac {2 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{441}+\frac {4 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{525}+\frac {16 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{945}\right )+2 d^{3} a b \left (\frac {\arcsinh \left (c x \right ) c^{9} x^{9}}{9}+\frac {3 \arcsinh \left (c x \right ) c^{7} x^{7}}{7}+\frac {3 \arcsinh \left (c x \right ) c^{5} x^{5}}{5}+\frac {\arcsinh \left (c x \right ) c^{3} x^{3}}{3}-\frac {c^{8} x^{8} \sqrt {c^{2} x^{2}+1}}{81}-\frac {187 c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{3969}-\frac {2099 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{33075}-\frac {2629 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{99225}+\frac {5258 \sqrt {c^{2} x^{2}+1}}{99225}\right )}{c^{3}} \]