\[ \int \frac {\left (a+b \sqrt {c+d x}\right )^2}{x^3} \, dx \]
Optimal antiderivative \[ \frac {a b \,d^{2} \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 c^{\frac {3}{2}}}-\frac {b d \left (b c +a \sqrt {d x +c}\right )}{2 c x}-\frac {\left (a +b \sqrt {d x +c}\right )^{2}}{2 x^{2}} \]
command
integrate((a+b*(d*x+c)**(1/2))**2/x**3,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ - \frac {a^{2}}{2 x^{2}} - \frac {20 a b c^{2} d^{2} \sqrt {c + d x}}{- 8 c^{4} - 16 c^{3} d x + 8 c^{2} \left (c + d x\right )^{2}} + \frac {12 a b c d^{2} \left (c + d x\right )^{\frac {3}{2}}}{- 8 c^{4} - 16 c^{3} d x + 8 c^{2} \left (c + d x\right )^{2}} + \frac {3 a b c d^{2} \sqrt {\frac {1}{c^{5}}} \log {\left (- c^{3} \sqrt {\frac {1}{c^{5}}} + \sqrt {c + d x} \right )}}{4} - \frac {3 a b c d^{2} \sqrt {\frac {1}{c^{5}}} \log {\left (c^{3} \sqrt {\frac {1}{c^{5}}} + \sqrt {c + d x} \right )}}{4} - a b d^{2} \sqrt {\frac {1}{c^{3}}} \log {\left (- c^{2} \sqrt {\frac {1}{c^{3}}} + \sqrt {c + d x} \right )} + a b d^{2} \sqrt {\frac {1}{c^{3}}} \log {\left (c^{2} \sqrt {\frac {1}{c^{3}}} + \sqrt {c + d x} \right )} - \frac {2 a b d \sqrt {c + d x}}{c x} - \frac {b^{2} c}{2 x^{2}} - \frac {b^{2} d}{x} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________