\[ \int \frac {1}{\sqrt {x} \left (a+b x^2\right )^3} \, dx \]
Optimal antiderivative \[ -\frac {21 \arctan \left (1-\frac {b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{a^{\frac {1}{4}}}\right ) \sqrt {2}}{64 a^{\frac {11}{4}} b^{\frac {1}{4}}}+\frac {21 \arctan \left (1+\frac {b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{a^{\frac {1}{4}}}\right ) \sqrt {2}}{64 a^{\frac {11}{4}} b^{\frac {1}{4}}}-\frac {21 \ln \left (\sqrt {a}+x \sqrt {b}-a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{128 a^{\frac {11}{4}} b^{\frac {1}{4}}}+\frac {21 \ln \left (\sqrt {a}+x \sqrt {b}+a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{128 a^{\frac {11}{4}} b^{\frac {1}{4}}}+\frac {\sqrt {x}}{4 a \left (b \,x^{2}+a \right )^{2}}+\frac {7 \sqrt {x}}{16 a^{2} \left (b \,x^{2}+a \right )} \]
command
integrate(1/(b*x**2+a)**3/x**(1/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {\tilde {\infty }}{x^{\frac {11}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{11 b^{3} x^{\frac {11}{2}}} & \text {for}\: a = 0 \\\frac {2 \sqrt {x}}{a^{3}} & \text {for}\: b = 0 \\\frac {44 a^{2} \sqrt {x}}{64 a^{5} + 128 a^{4} b x^{2} + 64 a^{3} b^{2} x^{4}} - \frac {21 a^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{5} + 128 a^{4} b x^{2} + 64 a^{3} b^{2} x^{4}} + \frac {21 a^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{5} + 128 a^{4} b x^{2} + 64 a^{3} b^{2} x^{4}} + \frac {42 a^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{64 a^{5} + 128 a^{4} b x^{2} + 64 a^{3} b^{2} x^{4}} + \frac {28 a b x^{\frac {5}{2}}}{64 a^{5} + 128 a^{4} b x^{2} + 64 a^{3} b^{2} x^{4}} - \frac {42 a b x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{5} + 128 a^{4} b x^{2} + 64 a^{3} b^{2} x^{4}} + \frac {42 a b x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{5} + 128 a^{4} b x^{2} + 64 a^{3} b^{2} x^{4}} + \frac {84 a b x^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{64 a^{5} + 128 a^{4} b x^{2} + 64 a^{3} b^{2} x^{4}} - \frac {21 b^{2} x^{4} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{5} + 128 a^{4} b x^{2} + 64 a^{3} b^{2} x^{4}} + \frac {21 b^{2} x^{4} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{5} + 128 a^{4} b x^{2} + 64 a^{3} b^{2} x^{4}} + \frac {42 b^{2} x^{4} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{64 a^{5} + 128 a^{4} b x^{2} + 64 a^{3} b^{2} x^{4}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________