\[ \int \frac {\sqrt {x}}{\left (a+b x^2\right )^3} \, dx \]
Optimal antiderivative \[ \frac {x^{\frac {3}{2}}}{4 a \left (b \,x^{2}+a \right )^{2}}+\frac {5 x^{\frac {3}{2}}}{16 a^{2} \left (b \,x^{2}+a \right )}-\frac {5 \arctan \left (1-\frac {b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{a^{\frac {1}{4}}}\right ) \sqrt {2}}{64 a^{\frac {9}{4}} b^{\frac {3}{4}}}+\frac {5 \arctan \left (1+\frac {b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{a^{\frac {1}{4}}}\right ) \sqrt {2}}{64 a^{\frac {9}{4}} b^{\frac {3}{4}}}+\frac {5 \ln \left (\sqrt {a}+x \sqrt {b}-a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{128 a^{\frac {9}{4}} b^{\frac {3}{4}}}-\frac {5 \ln \left (\sqrt {a}+x \sqrt {b}+a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{128 a^{\frac {9}{4}} b^{\frac {3}{4}}} \]
command
integrate(x**(1/2)/(b*x**2+a)**3,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {\tilde {\infty }}{x^{\frac {9}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{9 b^{3} x^{\frac {9}{2}}} & \text {for}\: a = 0 \\\frac {2 x^{\frac {3}{2}}}{3 a^{3}} & \text {for}\: b = 0 \\\frac {5 a^{2} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b \sqrt [4]{- \frac {a}{b}} + 128 a^{3} b^{2} x^{2} \sqrt [4]{- \frac {a}{b}} + 64 a^{2} b^{3} x^{4} \sqrt [4]{- \frac {a}{b}}} - \frac {5 a^{2} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b \sqrt [4]{- \frac {a}{b}} + 128 a^{3} b^{2} x^{2} \sqrt [4]{- \frac {a}{b}} + 64 a^{2} b^{3} x^{4} \sqrt [4]{- \frac {a}{b}}} + \frac {10 a^{2} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{64 a^{4} b \sqrt [4]{- \frac {a}{b}} + 128 a^{3} b^{2} x^{2} \sqrt [4]{- \frac {a}{b}} + 64 a^{2} b^{3} x^{4} \sqrt [4]{- \frac {a}{b}}} + \frac {36 a b x^{\frac {3}{2}} \sqrt [4]{- \frac {a}{b}}}{64 a^{4} b \sqrt [4]{- \frac {a}{b}} + 128 a^{3} b^{2} x^{2} \sqrt [4]{- \frac {a}{b}} + 64 a^{2} b^{3} x^{4} \sqrt [4]{- \frac {a}{b}}} + \frac {10 a b x^{2} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b \sqrt [4]{- \frac {a}{b}} + 128 a^{3} b^{2} x^{2} \sqrt [4]{- \frac {a}{b}} + 64 a^{2} b^{3} x^{4} \sqrt [4]{- \frac {a}{b}}} - \frac {10 a b x^{2} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b \sqrt [4]{- \frac {a}{b}} + 128 a^{3} b^{2} x^{2} \sqrt [4]{- \frac {a}{b}} + 64 a^{2} b^{3} x^{4} \sqrt [4]{- \frac {a}{b}}} + \frac {20 a b x^{2} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{64 a^{4} b \sqrt [4]{- \frac {a}{b}} + 128 a^{3} b^{2} x^{2} \sqrt [4]{- \frac {a}{b}} + 64 a^{2} b^{3} x^{4} \sqrt [4]{- \frac {a}{b}}} + \frac {20 b^{2} x^{\frac {7}{2}} \sqrt [4]{- \frac {a}{b}}}{64 a^{4} b \sqrt [4]{- \frac {a}{b}} + 128 a^{3} b^{2} x^{2} \sqrt [4]{- \frac {a}{b}} + 64 a^{2} b^{3} x^{4} \sqrt [4]{- \frac {a}{b}}} + \frac {5 b^{2} x^{4} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b \sqrt [4]{- \frac {a}{b}} + 128 a^{3} b^{2} x^{2} \sqrt [4]{- \frac {a}{b}} + 64 a^{2} b^{3} x^{4} \sqrt [4]{- \frac {a}{b}}} - \frac {5 b^{2} x^{4} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b \sqrt [4]{- \frac {a}{b}} + 128 a^{3} b^{2} x^{2} \sqrt [4]{- \frac {a}{b}} + 64 a^{2} b^{3} x^{4} \sqrt [4]{- \frac {a}{b}}} + \frac {10 b^{2} x^{4} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{64 a^{4} b \sqrt [4]{- \frac {a}{b}} + 128 a^{3} b^{2} x^{2} \sqrt [4]{- \frac {a}{b}} + 64 a^{2} b^{3} x^{4} \sqrt [4]{- \frac {a}{b}}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________