\[ \int x^4 \left (d+c^2 d x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]
Optimal antiderivative \[ \frac {100976 b^{2} d^{3} x}{4002075 c^{4}}-\frac {50488 b^{2} d^{3} x^{3}}{12006225 c^{2}}+\frac {12622 b^{2} d^{3} x^{5}}{6670125}+\frac {9410 b^{2} c^{2} d^{3} x^{7}}{1120581}+\frac {182 b^{2} c^{4} d^{3} x^{9}}{29403}+\frac {2 b^{2} c^{6} d^{3} x^{11}}{1331}-\frac {16 b \,d^{3} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{693 c^{5}}+\frac {4 b \,d^{3} \left (c^{2} x^{2}+1\right )^{\frac {5}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{1155 c^{5}}-\frac {2 b \,d^{3} \left (c^{2} x^{2}+1\right )^{\frac {7}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{1617 c^{5}}+\frac {8 b \,d^{3} \left (c^{2} x^{2}+1\right )^{\frac {9}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{297 c^{5}}-\frac {2 b \,d^{3} \left (c^{2} x^{2}+1\right )^{\frac {11}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{121 c^{5}}+\frac {16 d^{3} x^{5} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{1155}+\frac {8 d^{3} x^{5} \left (c^{2} x^{2}+1\right ) \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{231}+\frac {2 d^{3} x^{5} \left (c^{2} x^{2}+1\right )^{2} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{33}+\frac {d^{3} x^{5} \left (c^{2} x^{2}+1\right )^{3} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{11}-\frac {256 b \,d^{3} \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{17325 c^{5}}+\frac {128 b \,d^{3} x^{2} \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{17325 c^{3}}-\frac {32 b \,d^{3} x^{4} \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{5775 c} \]
command
int(x^4*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2,x)
Maple 2022.1 output
\[\int x^{4} \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arcsinh \left (c x \right )\right )^{2}\, dx\]
Maple 2021.1 output
\[ \frac {d^{3} a^{2} \left (\frac {1}{11} c^{11} x^{11}+\frac {1}{3} c^{9} x^{9}+\frac {3}{7} c^{7} x^{7}+\frac {1}{5} c^{5} x^{5}\right )+d^{3} b^{2} \left (\frac {\arcsinh \left (c x \right )^{2} c^{3} x^{3} \left (c^{2} x^{2}+1\right )^{4}}{11}-\frac {\arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{4}}{33}+\frac {16 \arcsinh \left (c x \right )^{2} c x}{1155}+\frac {\arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{3}}{231}+\frac {2 \arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{2}}{385}+\frac {8 \arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{1155}-\frac {5487704 c x \left (c^{2} x^{2}+1\right )^{2}}{4622396625}-\frac {16 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{3465}-\frac {606416 c x \left (c^{2} x^{2}+1\right )}{13867189875}-\frac {428 c x \left (c^{2} x^{2}+1\right )^{4}}{323433}-\frac {2 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{1617}-\frac {4 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{1925}-\frac {148174 c x \left (c^{2} x^{2}+1\right )^{3}}{110937519}+\frac {34 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {9}{2}}}{3267}+\frac {2 c x \left (c^{2} x^{2}+1\right )^{5}}{1331}-\frac {32 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{1155}-\frac {2 \arcsinh \left (c x \right ) c^{2} x^{2} \left (c^{2} x^{2}+1\right )^{\frac {9}{2}}}{121}+\frac {382986368 c x}{13867189875}\right )+2 d^{3} a b \left (\frac {\arcsinh \left (c x \right ) c^{11} x^{11}}{11}+\frac {\arcsinh \left (c x \right ) c^{9} x^{9}}{3}+\frac {3 \arcsinh \left (c x \right ) c^{7} x^{7}}{7}+\frac {\arcsinh \left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{10} x^{10} \sqrt {c^{2} x^{2}+1}}{121}-\frac {91 c^{8} x^{8} \sqrt {c^{2} x^{2}+1}}{3267}-\frac {4705 c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{160083}-\frac {6311 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{1334025}+\frac {25244 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{4002075}-\frac {50488 \sqrt {c^{2} x^{2}+1}}{4002075}\right )}{c^{5}} \]