\[ \int \frac {\sqrt [3]{x}}{(a+b x)^3} \, dx \]
Optimal antiderivative \[ -\frac {x^{\frac {1}{3}}}{2 b \left (b x +a \right )^{2}}+\frac {x^{\frac {1}{3}}}{6 a b \left (b x +a \right )}+\frac {\ln \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x^{\frac {1}{3}}\right )}{6 a^{\frac {5}{3}} b^{\frac {4}{3}}}-\frac {\ln \left (b x +a \right )}{18 a^{\frac {5}{3}} b^{\frac {4}{3}}}-\frac {\arctan \left (\frac {\left (a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}}{9 a^{\frac {5}{3}} b^{\frac {4}{3}}} \]
command
integrate(x**(1/3)/(b*x+a)**3,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{3}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {3 x^{\frac {4}{3}}}{4 a^{3}} & \text {for}\: b = 0 \\- \frac {3}{5 b^{3} x^{\frac {5}{3}}} & \text {for}\: a = 0 \\- \frac {6 a^{2} \sqrt [3]{x}}{18 a^{4} b + 36 a^{3} b^{2} x + 18 a^{2} b^{3} x^{2}} - \frac {2 a^{2} \sqrt [3]{- \frac {a}{b}} \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{18 a^{4} b + 36 a^{3} b^{2} x + 18 a^{2} b^{3} x^{2}} + \frac {a^{2} \sqrt [3]{- \frac {a}{b}} \log {\left (4 x^{\frac {2}{3}} + 4 \sqrt [3]{x} \sqrt [3]{- \frac {a}{b}} + 4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} \right )}}{18 a^{4} b + 36 a^{3} b^{2} x + 18 a^{2} b^{3} x^{2}} + \frac {2 \sqrt {3} a^{2} \sqrt [3]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{18 a^{4} b + 36 a^{3} b^{2} x + 18 a^{2} b^{3} x^{2}} - \frac {2 a^{2} \sqrt [3]{- \frac {a}{b}} \log {\left (2 \right )}}{18 a^{4} b + 36 a^{3} b^{2} x + 18 a^{2} b^{3} x^{2}} + \frac {3 a b x^{\frac {4}{3}}}{18 a^{4} b + 36 a^{3} b^{2} x + 18 a^{2} b^{3} x^{2}} - \frac {4 a b x \sqrt [3]{- \frac {a}{b}} \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{18 a^{4} b + 36 a^{3} b^{2} x + 18 a^{2} b^{3} x^{2}} + \frac {2 a b x \sqrt [3]{- \frac {a}{b}} \log {\left (4 x^{\frac {2}{3}} + 4 \sqrt [3]{x} \sqrt [3]{- \frac {a}{b}} + 4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} \right )}}{18 a^{4} b + 36 a^{3} b^{2} x + 18 a^{2} b^{3} x^{2}} + \frac {4 \sqrt {3} a b x \sqrt [3]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{18 a^{4} b + 36 a^{3} b^{2} x + 18 a^{2} b^{3} x^{2}} - \frac {4 a b x \sqrt [3]{- \frac {a}{b}} \log {\left (2 \right )}}{18 a^{4} b + 36 a^{3} b^{2} x + 18 a^{2} b^{3} x^{2}} - \frac {2 b^{2} x^{2} \sqrt [3]{- \frac {a}{b}} \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{18 a^{4} b + 36 a^{3} b^{2} x + 18 a^{2} b^{3} x^{2}} + \frac {b^{2} x^{2} \sqrt [3]{- \frac {a}{b}} \log {\left (4 x^{\frac {2}{3}} + 4 \sqrt [3]{x} \sqrt [3]{- \frac {a}{b}} + 4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} \right )}}{18 a^{4} b + 36 a^{3} b^{2} x + 18 a^{2} b^{3} x^{2}} + \frac {2 \sqrt {3} b^{2} x^{2} \sqrt [3]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{18 a^{4} b + 36 a^{3} b^{2} x + 18 a^{2} b^{3} x^{2}} - \frac {2 b^{2} x^{2} \sqrt [3]{- \frac {a}{b}} \log {\left (2 \right )}}{18 a^{4} b + 36 a^{3} b^{2} x + 18 a^{2} b^{3} x^{2}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________