13.1 Problem number 1309

\[ \int \frac {x^{13/2}}{\sqrt {a+b x^5}} \, dx \]

Optimal antiderivative \[ -\frac {a \arctanh \! \left (\frac {x^{\frac {5}{2}} \sqrt {b}}{\sqrt {b \,x^{5}+a}}\right )}{5 b^{\frac {3}{2}}}+\frac {x^{\frac {5}{2}} \sqrt {b \,x^{5}+a}}{5 b} \]

command

integrate(x**(13/2)/(b*x**5+a)**(1/2),x)

Sympy 1.10.1 under Python 3.10.4 output

\[ \text {Exception raised: SystemError} \]

Sympy 1.8 under Python 3.8.8 output

\[ \frac {\sqrt {a} x^{\frac {5}{2}} \sqrt {1 + \frac {b x^{5}}{a}}}{5 b} - \frac {a \operatorname {asinh}{\left (\frac {\sqrt {b} x^{\frac {5}{2}}}{\sqrt {a}} \right )}}{5 b^{\frac {3}{2}}} \]