\[ \int \frac {\sqrt {x} \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx \]
Optimal antiderivative \[ \frac {\left (A b -B a \right ) x^{\frac {3}{2}}}{3 a b \left (b \,x^{3}+a \right )}+\frac {\left (A b +B a \right ) \arctan \left (\frac {x^{\frac {3}{2}} \sqrt {b}}{\sqrt {a}}\right )}{3 a^{\frac {3}{2}} b^{\frac {3}{2}}} \]
command
integrate((B*x**3+A)*x**(1/2)/(b*x**3+a)**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \tilde {\infty } \left (- \frac {2 A}{9 x^{\frac {9}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {\frac {2 A x^{\frac {3}{2}}}{3} + \frac {2 B x^{\frac {9}{2}}}{9}}{a^{2}} & \text {for}\: b = 0 \\\frac {- \frac {2 A}{9 x^{\frac {9}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}}{b^{2}} & \text {for}\: a = 0 \\\frac {2 A a b x^{\frac {3}{2}}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} - \frac {A a b \sqrt {- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [6]{- \frac {a}{b}} \right )}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} + \frac {A a b \sqrt {- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [6]{- \frac {a}{b}} \right )}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} + \frac {A a b \sqrt {- \frac {a}{b}} \log {\left (- 4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} - \frac {A a b \sqrt {- \frac {a}{b}} \log {\left (4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} - \frac {A b^{2} x^{3} \sqrt {- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [6]{- \frac {a}{b}} \right )}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} + \frac {A b^{2} x^{3} \sqrt {- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [6]{- \frac {a}{b}} \right )}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} + \frac {A b^{2} x^{3} \sqrt {- \frac {a}{b}} \log {\left (- 4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} - \frac {A b^{2} x^{3} \sqrt {- \frac {a}{b}} \log {\left (4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} - \frac {2 B a^{2} x^{\frac {3}{2}}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} - \frac {B a^{2} \sqrt {- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [6]{- \frac {a}{b}} \right )}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} + \frac {B a^{2} \sqrt {- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [6]{- \frac {a}{b}} \right )}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} + \frac {B a^{2} \sqrt {- \frac {a}{b}} \log {\left (- 4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} - \frac {B a^{2} \sqrt {- \frac {a}{b}} \log {\left (4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} - \frac {B a b x^{3} \sqrt {- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [6]{- \frac {a}{b}} \right )}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} + \frac {B a b x^{3} \sqrt {- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [6]{- \frac {a}{b}} \right )}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} + \frac {B a b x^{3} \sqrt {- \frac {a}{b}} \log {\left (- 4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} - \frac {B a b x^{3} \sqrt {- \frac {a}{b}} \log {\left (4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________