\[ \int \frac {x^{7/2}}{\left (a+b x^2\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {x^{\frac {5}{2}}}{2 b \left (b \,x^{2}+a \right )}+\frac {5 a^{\frac {1}{4}} \arctan \left (1-\frac {b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{a^{\frac {1}{4}}}\right ) \sqrt {2}}{8 b^{\frac {9}{4}}}-\frac {5 a^{\frac {1}{4}} \arctan \left (1+\frac {b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{a^{\frac {1}{4}}}\right ) \sqrt {2}}{8 b^{\frac {9}{4}}}+\frac {5 a^{\frac {1}{4}} \ln \left (\sqrt {a}+x \sqrt {b}-a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{16 b^{\frac {9}{4}}}-\frac {5 a^{\frac {1}{4}} \ln \left (\sqrt {a}+x \sqrt {b}+a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{16 b^{\frac {9}{4}}}+\frac {5 \sqrt {x}}{2 b^{2}} \]
command
integrate(x**(7/2)/(b*x**2+a)**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \tilde {\infty } \sqrt {x} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 \sqrt {x}}{b^{2}} & \text {for}\: a = 0 \\\frac {2 x^{\frac {9}{2}}}{9 a^{2}} & \text {for}\: b = 0 \\\frac {20 a \sqrt {x}}{8 a b^{2} + 8 b^{3} x^{2}} + \frac {5 a \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{8 a b^{2} + 8 b^{3} x^{2}} - \frac {5 a \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{8 a b^{2} + 8 b^{3} x^{2}} - \frac {10 a \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{8 a b^{2} + 8 b^{3} x^{2}} + \frac {16 b x^{\frac {5}{2}}}{8 a b^{2} + 8 b^{3} x^{2}} + \frac {5 b x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{8 a b^{2} + 8 b^{3} x^{2}} - \frac {5 b x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{8 a b^{2} + 8 b^{3} x^{2}} - \frac {10 b x^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{8 a b^{2} + 8 b^{3} x^{2}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________