\[ \int \frac {1}{x^{5/3} (a+b x)^2} \, dx \]
Optimal antiderivative \[ -\frac {5}{2 a^{2} x^{\frac {2}{3}}}+\frac {1}{a \,x^{\frac {2}{3}} \left (b x +a \right )}-\frac {5 b^{\frac {2}{3}} \ln \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x^{\frac {1}{3}}\right )}{2 a^{\frac {8}{3}}}+\frac {5 b^{\frac {2}{3}} \ln \left (b x +a \right )}{6 a^{\frac {8}{3}}}+\frac {5 b^{\frac {2}{3}} \arctan \left (\frac {\left (a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}}{3 a^{\frac {8}{3}}} \]
command
integrate(1/x**(5/3)/(b*x+a)**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {\tilde {\infty }}{x^{\frac {8}{3}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {3}{2 a^{2} x^{\frac {2}{3}}} & \text {for}\: b = 0 \\- \frac {3}{8 b^{2} x^{\frac {8}{3}}} & \text {for}\: a = 0 \\- \frac {9 a^{2}}{6 a^{4} x^{\frac {2}{3}} + 6 a^{3} b x^{\frac {5}{3}}} + \frac {10 a b x^{\frac {2}{3}} \sqrt [3]{- \frac {a}{b}} \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{6 a^{4} x^{\frac {2}{3}} + 6 a^{3} b x^{\frac {5}{3}}} - \frac {5 a b x^{\frac {2}{3}} \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 )}}{6 a^{4} x^{\frac {2}{3}} + 6 a^{3} b x^{\frac {5}{3}}} - \frac {10 \sqrt {3} a b x^{\frac {2}{3}} \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 )}}{6 a^{4} x^{\frac {2}{3}} + 6 a^{3} b x^{\frac {5}{3}}} + \frac {10 a b x^{\frac {2}{3}} \sqrt [3]{- \frac {a}{b}} \log {\left (2 \right )}}{6 a^{4} x^{\frac {2}{3}} + 6 a^{3} b x^{\frac {5}{3}}} - \frac {15 a b x}{6 a^{4} x^{\frac {2}{3}} + 6 a^{3} b x^{\frac {5}{3}}} + \frac {10 b^{2} x^{\frac {5}{3}} \sqrt [3]{- \frac {a}{b}} \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{6 a^{4} x^{\frac {2}{3}} + 6 a^{3} b x^{\frac {5}{3}}} - \frac {5 b^{2} x^{\frac {5}{3}} \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 )}}{6 a^{4} x^{\frac {2}{3}} + 6 a^{3} b x^{\frac {5}{3}}} - \frac {10 \sqrt {3} b^{2} x^{\frac {5}{3}} \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 )}}{6 a^{4} x^{\frac {2}{3}} + 6 a^{3} b x^{\frac {5}{3}}} + \frac {10 b^{2} x^{\frac {5}{3}} \sqrt [3]{- \frac {a}{b}} \log {\left (2 \right )}}{6 a^{4} x^{\frac {2}{3}} + 6 a^{3} b x^{\frac {5}{3}}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________