\[ \int \frac {x^{7/2} (A+B x)}{\left (b x+c x^2\right )^2} \, dx \]
Optimal antiderivative \[ \frac {\left (-3 A c +5 b B \right ) x^{\frac {3}{2}}}{3 b \,c^{2}}-\frac {\left (-A c +b B \right ) x^{\frac {5}{2}}}{b c \left (c x +b \right )}+\frac {\left (-3 A c +5 b B \right ) \arctan \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {b}}\right ) \sqrt {b}}{c^{\frac {7}{2}}}-\frac {\left (-3 A c +5 b B \right ) \sqrt {x}}{c^{3}} \]
command
integrate(x**(7/2)*(B*x+A)/(c*x**2+b*x)**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \tilde {\infty } \left (2 A \sqrt {x} + \frac {2 B x^{\frac {3}{2}}}{3}\right ) & \text {for}\: b = 0 \wedge c = 0 \\\frac {\frac {2 A x^{\frac {5}{2}}}{5} + \frac {2 B x^{\frac {7}{2}}}{7}}{b^{2}} & \text {for}\: c = 0 \\\frac {2 A \sqrt {x} + \frac {2 B x^{\frac {3}{2}}}{3}}{c^{2}} & \text {for}\: b = 0 \\- \frac {9 A b^{2} c \log {\left (\sqrt {x} - \sqrt {- \frac {b}{c}} \right )}}{6 b c^{4} \sqrt {- \frac {b}{c}} + 6 c^{5} x \sqrt {- \frac {b}{c}}} + \frac {9 A b^{2} c \log {\left (\sqrt {x} + \sqrt {- \frac {b}{c}} \right )}}{6 b c^{4} \sqrt {- \frac {b}{c}} + 6 c^{5} x \sqrt {- \frac {b}{c}}} + \frac {18 A b c^{2} \sqrt {x} \sqrt {- \frac {b}{c}}}{6 b c^{4} \sqrt {- \frac {b}{c}} + 6 c^{5} x \sqrt {- \frac {b}{c}}} - \frac {9 A b c^{2} x \log {\left (\sqrt {x} - \sqrt {- \frac {b}{c}} \right )}}{6 b c^{4} \sqrt {- \frac {b}{c}} + 6 c^{5} x \sqrt {- \frac {b}{c}}} + \frac {9 A b c^{2} x \log {\left (\sqrt {x} + \sqrt {- \frac {b}{c}} \right )}}{6 b c^{4} \sqrt {- \frac {b}{c}} + 6 c^{5} x \sqrt {- \frac {b}{c}}} + \frac {12 A c^{3} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}}}{6 b c^{4} \sqrt {- \frac {b}{c}} + 6 c^{5} x \sqrt {- \frac {b}{c}}} + \frac {15 B b^{3} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{c}} \right )}}{6 b c^{4} \sqrt {- \frac {b}{c}} + 6 c^{5} x \sqrt {- \frac {b}{c}}} - \frac {15 B b^{3} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{c}} \right )}}{6 b c^{4} \sqrt {- \frac {b}{c}} + 6 c^{5} x \sqrt {- \frac {b}{c}}} - \frac {30 B b^{2} c \sqrt {x} \sqrt {- \frac {b}{c}}}{6 b c^{4} \sqrt {- \frac {b}{c}} + 6 c^{5} x \sqrt {- \frac {b}{c}}} + \frac {15 B b^{2} c x \log {\left (\sqrt {x} - \sqrt {- \frac {b}{c}} \right )}}{6 b c^{4} \sqrt {- \frac {b}{c}} + 6 c^{5} x \sqrt {- \frac {b}{c}}} - \frac {15 B b^{2} c x \log {\left (\sqrt {x} + \sqrt {- \frac {b}{c}} \right )}}{6 b c^{4} \sqrt {- \frac {b}{c}} + 6 c^{5} x \sqrt {- \frac {b}{c}}} - \frac {20 B b c^{2} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}}}{6 b c^{4} \sqrt {- \frac {b}{c}} + 6 c^{5} x \sqrt {- \frac {b}{c}}} + \frac {4 B c^{3} x^{\frac {5}{2}} \sqrt {- \frac {b}{c}}}{6 b c^{4} \sqrt {- \frac {b}{c}} + 6 c^{5} x \sqrt {- \frac {b}{c}}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________