\[ \int \frac {(a-b x)^{5/2}}{x^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {15 a^{2} \arctan \left (\frac {\sqrt {b}\, \sqrt {x}}{\sqrt {-b x +a}}\right ) \sqrt {b}}{4}-\frac {2 \left (-b x +a \right )^{\frac {5}{2}}}{\sqrt {x}}-\frac {5 b \left (-b x +a \right )^{\frac {3}{2}} \sqrt {x}}{2}-\frac {15 a b \sqrt {x}\, \sqrt {-b x +a}}{4} \]
command
int((-b*x+a)^(5/2)/x^(3/2),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(-\frac {\sqrt {-b x +a}\, \left (-2 x^{2} b^{2}+9 a b x +8 a^{2}\right )}{4 \sqrt {x}}-\frac {15 a^{2} \sqrt {b}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-x^{2} b +a x}}\right ) \sqrt {x \left (-b x +a \right )}}{8 \sqrt {x}\, \sqrt {-b x +a}}\) | \(88\) |
Maple 2021.1 output
\[ \int \frac {\left (-b x +a \right )^{\frac {5}{2}}}{x^{\frac {3}{2}}}\, dx \]________________________________________________________________________________________