\[ \int \frac {1}{x^{5/2} \left (a+b x^2\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {7}{6 a^{2} x^{\frac {3}{2}}}+\frac {1}{2 a \,x^{\frac {3}{2}} \left (b \,x^{2}+a \right )}+\frac {7 b^{\frac {3}{4}} \arctan \left (1-\frac {b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{a^{\frac {1}{4}}}\right ) \sqrt {2}}{8 a^{\frac {11}{4}}}-\frac {7 b^{\frac {3}{4}} \arctan \left (1+\frac {b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{a^{\frac {1}{4}}}\right ) \sqrt {2}}{8 a^{\frac {11}{4}}}+\frac {7 b^{\frac {3}{4}} \ln \left (\sqrt {a}+x \sqrt {b}-a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{16 a^{\frac {11}{4}}}-\frac {7 b^{\frac {3}{4}} \ln \left (\sqrt {a}+x \sqrt {b}+a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{16 a^{\frac {11}{4}}} \]
command
integrate(1/x**(5/2)/(b*x**2+a)**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {\tilde {\infty }}{x^{\frac {11}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{11 b^{2} x^{\frac {11}{2}}} & \text {for}\: a = 0 \\- \frac {2}{3 a^{2} x^{\frac {3}{2}}} & \text {for}\: b = 0 \\- \frac {16 a^{2}}{24 a^{4} x^{\frac {3}{2}} + 24 a^{3} b x^{\frac {7}{2}}} + \frac {21 a b x^{\frac {3}{2}} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{24 a^{4} x^{\frac {3}{2}} + 24 a^{3} b x^{\frac {7}{2}}} - \frac {21 a b x^{\frac {3}{2}} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{24 a^{4} x^{\frac {3}{2}} + 24 a^{3} b x^{\frac {7}{2}}} - \frac {42 a b x^{\frac {3}{2}} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{24 a^{4} x^{\frac {3}{2}} + 24 a^{3} b x^{\frac {7}{2}}} - \frac {28 a b x^{2}}{24 a^{4} x^{\frac {3}{2}} + 24 a^{3} b x^{\frac {7}{2}}} + \frac {21 b^{2} x^{\frac {7}{2}} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{24 a^{4} x^{\frac {3}{2}} + 24 a^{3} b x^{\frac {7}{2}}} - \frac {21 b^{2} x^{\frac {7}{2}} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{24 a^{4} x^{\frac {3}{2}} + 24 a^{3} b x^{\frac {7}{2}}} - \frac {42 b^{2} x^{\frac {7}{2}} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{24 a^{4} x^{\frac {3}{2}} + 24 a^{3} b x^{\frac {7}{2}}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________