\[ \int \frac {x^3}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {693 b^{5} \arctanh \left (\frac {\sqrt {a}\, \sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{64 a^{\frac {13}{2}}}-\frac {4 x^{3}}{a \sqrt {b \sqrt {x}+a x}}+\frac {693 b^{4} \sqrt {b \sqrt {x}+a x}}{64 a^{6}}+\frac {231 b^{2} x \sqrt {b \sqrt {x}+a x}}{40 a^{4}}-\frac {99 b \,x^{\frac {3}{2}} \sqrt {b \sqrt {x}+a x}}{20 a^{3}}+\frac {22 x^{2} \sqrt {b \sqrt {x}+a x}}{5 a^{2}}-\frac {231 b^{3} \sqrt {x}\, \sqrt {b \sqrt {x}+a x}}{32 a^{5}} \]
command
integrate(x^3/(b*x^(1/2)+a*x)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{320} \, \sqrt {a x + b \sqrt {x}} {\left (2 \, {\left (4 \, {\left (2 \, \sqrt {x} {\left (\frac {8 \, \sqrt {x}}{a^{2}} - \frac {19 \, b}{a^{3}}\right )} + \frac {71 \, b^{2}}{a^{4}}\right )} \sqrt {x} - \frac {515 \, b^{3}}{a^{5}}\right )} \sqrt {x} + \frac {2185 \, b^{4}}{a^{6}}\right )} + \frac {693 \, b^{5} \log \left ({\left | -2 \, \sqrt {a} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} - b \right |}\right )}{128 \, a^{\frac {13}{2}}} + \frac {4 \, b^{6}}{{\left (a {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} + \sqrt {a} b\right )} a^{6}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________