\[ \int \frac {x^{7/2} (A+B x)}{\sqrt {a+b x}} \, dx \]
Optimal antiderivative \[ \frac {7 a^{4} \left (10 A b -9 a B \right ) \arctanh \! \left (\frac {\sqrt {b}\, \sqrt {x}}{\sqrt {b x +a}}\right )}{128 b^{\frac {11}{2}}}+\frac {7 a^{2} \left (10 A b -9 a B \right ) x^{\frac {3}{2}} \sqrt {b x +a}}{192 b^{4}}-\frac {7 a \left (10 A b -9 a B \right ) x^{\frac {5}{2}} \sqrt {b x +a}}{240 b^{3}}+\frac {\left (10 A b -9 a B \right ) x^{\frac {7}{2}} \sqrt {b x +a}}{40 b^{2}}+\frac {B \,x^{\frac {9}{2}} \sqrt {b x +a}}{5 b}-\frac {7 a^{3} \left (10 A b -9 a B \right ) \sqrt {x}\, \sqrt {b x +a}}{128 b^{5}} \]
command
integrate(x**(7/2)*(B*x+A)/(b*x+a)**(1/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ - \frac {35 A a^{\frac {7}{2}} \sqrt {x}}{64 b^{4} \sqrt {1 + \frac {b x}{a}}} - \frac {35 A a^{\frac {5}{2}} x^{\frac {3}{2}}}{192 b^{3} \sqrt {1 + \frac {b x}{a}}} + \frac {7 A a^{\frac {3}{2}} x^{\frac {5}{2}}}{96 b^{2} \sqrt {1 + \frac {b x}{a}}} - \frac {A \sqrt {a} x^{\frac {7}{2}}}{24 b \sqrt {1 + \frac {b x}{a}}} + \frac {35 A a^{4} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{64 b^{\frac {9}{2}}} + \frac {A x^{\frac {9}{2}}}{4 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} + \frac {63 B a^{\frac {9}{2}} \sqrt {x}}{128 b^{5} \sqrt {1 + \frac {b x}{a}}} + \frac {21 B a^{\frac {7}{2}} x^{\frac {3}{2}}}{128 b^{4} \sqrt {1 + \frac {b x}{a}}} - \frac {21 B a^{\frac {5}{2}} x^{\frac {5}{2}}}{320 b^{3} \sqrt {1 + \frac {b x}{a}}} + \frac {3 B a^{\frac {3}{2}} x^{\frac {7}{2}}}{80 b^{2} \sqrt {1 + \frac {b x}{a}}} - \frac {B \sqrt {a} x^{\frac {9}{2}}}{40 b \sqrt {1 + \frac {b x}{a}}} - \frac {63 B a^{5} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{128 b^{\frac {11}{2}}} + \frac {B x^{\frac {11}{2}}}{5 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} \]