8.1 Problem number 424

\[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{15}} \, dx \]

Optimal antiderivative \[ -\frac {3 b^{3} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{128 x^{8}}-\frac {3 b^{2} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{80 x^{10}}-\frac {3 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{56 x^{12}}-\frac {\left (b \,x^{2}+a \right )^{\frac {9}{2}}}{14 x^{14}}-\frac {9 b^{7} \arctanh \! \left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {a}}\right )}{2048 a^{\frac {5}{2}}}-\frac {3 b^{4} \sqrt {b \,x^{2}+a}}{256 x^{6}}-\frac {3 b^{5} \sqrt {b \,x^{2}+a}}{1024 a \,x^{4}}+\frac {9 b^{6} \sqrt {b \,x^{2}+a}}{2048 a^{2} x^{2}} \]

command

integrate((b*x**2+a)**(9/2)/x**15,x)

Sympy 1.10.1 under Python 3.10.4 output

\[ \text {Timed out} \]

Sympy 1.8 under Python 3.8.8 output

\[ - \frac {a^{5}}{14 \sqrt {b} x^{15} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {23 a^{4} \sqrt {b}}{56 x^{13} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {541 a^{3} b^{\frac {3}{2}}}{560 x^{11} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {5249 a^{2} b^{\frac {5}{2}}}{4480 x^{9} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {6653 a b^{\frac {7}{2}}}{8960 x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {1027 b^{\frac {9}{2}}}{5120 x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {3 b^{\frac {11}{2}}}{2048 a x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {9 b^{\frac {13}{2}}}{2048 a^{2} x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {9 b^{7} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2048 a^{\frac {5}{2}}} \]