\[ \int x^2 \sinh ^{-1}(a+b x)^3 \, dx \]
Optimal antiderivative \[ -\frac {2 \left (1+\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{27 b^{3}}-\frac {3 a \arcsinh \! \left (b x +a \right )}{4 b^{3}}-\frac {4 \left (b x +a \right ) \arcsinh \! \left (b x +a \right )}{3 b^{3}}+\frac {6 a^{2} \left (b x +a \right ) \arcsinh \! \left (b x +a \right )}{b^{3}}-\frac {3 a \left (b x +a \right )^{2} \arcsinh \! \left (b x +a \right )}{2 b^{3}}+\frac {2 \left (b x +a \right )^{3} \arcsinh \! \left (b x +a \right )}{9 b^{3}}-\frac {a \arcsinh \! \left (b x +a \right )^{3}}{2 b^{3}}+\frac {a^{3} \arcsinh \! \left (b x +a \right )^{3}}{3 b^{3}}+\frac {x^{3} \arcsinh \! \left (b x +a \right )^{3}}{3}+\frac {14 \sqrt {1+\left (b x +a \right )^{2}}}{9 b^{3}}-\frac {6 a^{2} \sqrt {1+\left (b x +a \right )^{2}}}{b^{3}}+\frac {3 a \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}}{4 b^{3}}+\frac {2 \arcsinh \! \left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}}{3 b^{3}}-\frac {3 a^{2} \arcsinh \! \left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}}{b^{3}}+\frac {3 a \left (b x +a \right ) \arcsinh \! \left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}}{2 b^{3}}-\frac {\left (b x +a \right )^{2} \arcsinh \! \left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}}{3 b^{3}} \]
command
int(x^2*arcsinh(b*x+a)^3,x)
Maple 2022.1 output
\[\int x^{2} \arcsinh \left (b x +a \right )^{3}\, dx\]
Maple 2021.1 output
\[ \frac {-\frac {a \left (4 \arcsinh \left (b x +a \right )^{3} \left (b x +a \right )^{2}-6 \arcsinh \left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )+2 \arcsinh \left (b x +a \right )^{3}+6 \arcsinh \left (b x +a \right ) \left (b x +a \right )^{2}-3 \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}+3 \arcsinh \left (b x +a \right )\right )}{4}-\frac {\arcsinh \left (b x +a \right )^{3} \left (b x +a \right )}{3}+\frac {\arcsinh \left (b x +a \right )^{3} \left (b x +a \right ) \left (1+\left (b x +a \right )^{2}\right )}{3}+\arcsinh \left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}-\frac {14 \left (b x +a \right ) \arcsinh \left (b x +a \right )}{9}+\frac {14 \sqrt {1+\left (b x +a \right )^{2}}}{9}-\frac {\arcsinh \left (b x +a \right )^{2} \left (1+\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{3}+\frac {2 \left (b x +a \right ) \left (1+\left (b x +a \right )^{2}\right ) \arcsinh \left (b x +a \right )}{9}-\frac {2 \left (1+\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{27}+a^{2} \left (\arcsinh \left (b x +a \right )^{3} \left (b x +a \right )-3 \arcsinh \left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}+6 \left (b x +a \right ) \arcsinh \left (b x +a \right )-6 \sqrt {1+\left (b x +a \right )^{2}}\right )}{b^{3}} \]