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