13.5 Problem number 266

\[ \int \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \sinh ^{-1}(a+b x)^3 \, dx \]

Optimal antiderivative \[ -\frac {51 \left (b x +a \right )^{2}}{128 b}-\frac {3 \left (b x +a \right )^{4}}{128 b}+\frac {3 \left (b x +a \right ) \left (1+\left (b x +a \right )^{2}\right )^{\frac {3}{2}} \arcsinh \! \left (b x +a \right )}{32 b}-\frac {27 \arcsinh \! \left (b x +a \right )^{2}}{128 b}-\frac {9 \left (b x +a \right )^{2} \arcsinh \! \left (b x +a \right )^{2}}{16 b}-\frac {3 \left (1+\left (b x +a \right )^{2}\right )^{2} \arcsinh \! \left (b x +a \right )^{2}}{16 b}+\frac {\left (b x +a \right ) \left (1+\left (b x +a \right )^{2}\right )^{\frac {3}{2}} \arcsinh \! \left (b x +a \right )^{3}}{4 b}+\frac {3 \arcsinh \! \left (b x +a \right )^{4}}{32 b}+\frac {45 \left (b x +a \right ) \arcsinh \! \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}}{64 b}+\frac {3 \left (b x +a \right ) \arcsinh \! \left (b x +a \right )^{3} \sqrt {1+\left (b x +a \right )^{2}}}{8 b} \]

command

int((b^2*x^2+2*a*b*x+a^2+1)^(3/2)*arcsinh(b*x+a)^3,x)

Maple 2022.1 output

\[\int \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}} \arcsinh \left (b x +a \right )^{3}\, dx\]

Maple 2021.1 output

\[ \frac {-48-102 a b x +80 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \arcsinh \left (b x +a \right )^{3} x b -240 \arcsinh \left (b x +a \right )^{2} x a b +102 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \arcsinh \left (b x +a \right ) x b -51 b^{2} x^{2}-51 a^{2}-24 \arcsinh \left (b x +a \right )^{2} a^{4}-3 x^{4} b^{4}-120 \arcsinh \left (b x +a \right )^{2} a^{2}-96 \arcsinh \left (b x +a \right )^{2} x^{3} a \,b^{3}+32 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \arcsinh \left (b x +a \right )^{3} x^{3} b^{3}+12 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \arcsinh \left (b x +a \right ) x^{3} b^{3}-144 \arcsinh \left (b x +a \right )^{2} x^{2} a^{2} b^{2}-96 \arcsinh \left (b x +a \right )^{2} x \,a^{3} b -3 a^{4}-51 \arcsinh \left (b x +a \right )^{2}+32 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \arcsinh \left (b x +a \right )^{3} a^{3}+12 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \arcsinh \left (b x +a \right ) a^{3}-24 \arcsinh \left (b x +a \right )^{2} x^{4} b^{4}-12 x^{3} a \,b^{3}-18 x^{2} a^{2} b^{2}-12 x \,a^{3} b +12 \arcsinh \left (b x +a \right )^{4}-120 \arcsinh \left (b x +a \right )^{2} x^{2} b^{2}+80 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \arcsinh \left (b x +a \right )^{3} a +102 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \arcsinh \left (b x +a \right ) a +36 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \arcsinh \left (b x +a \right ) x \,a^{2} b +96 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \arcsinh \left (b x +a \right )^{3} x^{2} a \,b^{2}+96 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \arcsinh \left (b x +a \right )^{3} x \,a^{2} b +36 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \arcsinh \left (b x +a \right ) x^{2} a \,b^{2}}{128 b} \]