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