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