\[ \int \frac {(a-b x)^{5/2}}{x^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-b x +a \right )^{\frac {5}{2}}}{3 x^{\frac {3}{2}}}+5 a \,b^{\frac {3}{2}} \arctan \left (\frac {\sqrt {b}\, \sqrt {x}}{\sqrt {-b x +a}}\right )+\frac {10 b \left (-b x +a \right )^{\frac {3}{2}}}{3 \sqrt {x}}+5 b^{2} \sqrt {x}\, \sqrt {-b x +a} \]
command
int((-b*x+a)^(5/2)/x^(5/2),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(-\frac {\sqrt {-b x +a}\, \left (-3 x^{2} b^{2}-14 a b x +2 a^{2}\right )}{3 x^{\frac {3}{2}}}+\frac {5 a \,b^{\frac {3}{2}} \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-x^{2} b +a x}}\right ) \sqrt {x \left (-b x +a \right )}}{2 \sqrt {x}\, \sqrt {-b x +a}}\) | \(86\) |
Maple 2021.1 output
\[ \int \frac {\left (-b x +a \right )^{\frac {5}{2}}}{x^{\frac {5}{2}}}\, dx \]________________________________________________________________________________________