\[ \int \frac {1}{x^{4/3} (a+b x)^2} \, dx \]
Optimal antiderivative \[ -\frac {4}{a^{2} x^{\frac {1}{3}}}+\frac {1}{a \,x^{\frac {1}{3}} \left (b x +a \right )}+\frac {2 b^{\frac {1}{3}} \ln \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x^{\frac {1}{3}}\right )}{a^{\frac {7}{3}}}-\frac {2 b^{\frac {1}{3}} \ln \left (b x +a \right )}{3 a^{\frac {7}{3}}}+\frac {4 b^{\frac {1}{3}} \arctan \left (\frac {\left (a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}}{3 a^{\frac {7}{3}}} \]
command
integrate(1/x**(4/3)/(b*x+a)**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {\tilde {\infty }}{x^{\frac {7}{3}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {3}{a^{2} \sqrt [3]{x}} & \text {for}\: b = 0 \\- \frac {3}{7 b^{2} x^{\frac {7}{3}}} & \text {for}\: a = 0 \\- \frac {4 a \sqrt [3]{x} \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{3 a^{3} \sqrt [3]{x} \sqrt [3]{- \frac {a}{b}} + 3 a^{2} b x^{\frac {4}{3}} \sqrt [3]{- \frac {a}{b}}} + \frac {2 a \sqrt [3]{x} \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 )}}{3 a^{3} \sqrt [3]{x} \sqrt [3]{- \frac {a}{b}} + 3 a^{2} b x^{\frac {4}{3}} \sqrt [3]{- \frac {a}{b}}} - \frac {4 \sqrt {3} a \sqrt [3]{x} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{3 a^{3} \sqrt [3]{x} \sqrt [3]{- \frac {a}{b}} + 3 a^{2} b x^{\frac {4}{3}} \sqrt [3]{- \frac {a}{b}}} - \frac {4 a \sqrt [3]{x} \log {\left (2 \right )}}{3 a^{3} \sqrt [3]{x} \sqrt [3]{- \frac {a}{b}} + 3 a^{2} b x^{\frac {4}{3}} \sqrt [3]{- \frac {a}{b}}} - \frac {9 a \sqrt [3]{- \frac {a}{b}}}{3 a^{3} \sqrt [3]{x} \sqrt [3]{- \frac {a}{b}} + 3 a^{2} b x^{\frac {4}{3}} \sqrt [3]{- \frac {a}{b}}} - \frac {4 b x^{\frac {4}{3}} \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{3 a^{3} \sqrt [3]{x} \sqrt [3]{- \frac {a}{b}} + 3 a^{2} b x^{\frac {4}{3}} \sqrt [3]{- \frac {a}{b}}} + \frac {2 b x^{\frac {4}{3}} \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 )}}{3 a^{3} \sqrt [3]{x} \sqrt [3]{- \frac {a}{b}} + 3 a^{2} b x^{\frac {4}{3}} \sqrt [3]{- \frac {a}{b}}} - \frac {4 \sqrt {3} b x^{\frac {4}{3}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{3 a^{3} \sqrt [3]{x} \sqrt [3]{- \frac {a}{b}} + 3 a^{2} b x^{\frac {4}{3}} \sqrt [3]{- \frac {a}{b}}} - \frac {4 b x^{\frac {4}{3}} \log {\left (2 \right )}}{3 a^{3} \sqrt [3]{x} \sqrt [3]{- \frac {a}{b}} + 3 a^{2} b x^{\frac {4}{3}} \sqrt [3]{- \frac {a}{b}}} - \frac {12 b x \sqrt [3]{- \frac {a}{b}}}{3 a^{3} \sqrt [3]{x} \sqrt [3]{- \frac {a}{b}} + 3 a^{2} b x^{\frac {4}{3}} \sqrt [3]{- \frac {a}{b}}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________