\[ \int \frac {x^{3/2}}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {15 b^{2} \arctanh \left (\frac {\sqrt {a}\, \sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{2 a^{\frac {7}{2}}}-\frac {4 x^{\frac {3}{2}}}{a \sqrt {b \sqrt {x}+a x}}-\frac {15 b \sqrt {b \sqrt {x}+a x}}{2 a^{3}}+\frac {5 \sqrt {x}\, \sqrt {b \sqrt {x}+a x}}{a^{2}} \]
command
integrate(x^(3/2)/(b*x^(1/2)+a*x)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{2} \, \sqrt {a x + b \sqrt {x}} {\left (\frac {2 \, \sqrt {x}}{a^{2}} - \frac {7 \, b}{a^{3}}\right )} - \frac {15 \, b^{2} \log \left ({\left | -2 \, \sqrt {a} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} - b \right |}\right )}{4 \, a^{\frac {7}{2}}} - \frac {4 \, b^{3}}{{\left (a {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} + \sqrt {a} b\right )} a^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________