\[ \int \frac {(a-b x)^{3/2}}{x^{3/2}} \, dx \]
Optimal antiderivative \[ -3 a \arctan \left (\frac {\sqrt {b}\, \sqrt {x}}{\sqrt {-b x +a}}\right ) \sqrt {b}-\frac {2 \left (-b x +a \right )^{\frac {3}{2}}}{\sqrt {x}}-3 b \sqrt {x}\, \sqrt {-b x +a} \]
command
int((-b*x+a)^(3/2)/x^(3/2),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(-\frac {\sqrt {-b x +a}\, \left (b x +2 a \right )}{\sqrt {x}}-\frac {3 a \sqrt {b}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-x^{2} b +a x}}\right ) \sqrt {x \left (-b x +a \right )}}{2 \sqrt {x}\, \sqrt {-b x +a}}\) | \(74\) |
Maple 2021.1 output
\[ \int \frac {\left (-b x +a \right )^{\frac {3}{2}}}{x^{\frac {3}{2}}}\, dx \]________________________________________________________________________________________