\[ \int \frac {(d+e x)^{3/2}}{a+c x^2} \, dx \]
Optimal antiderivative \[ \frac {2 e \sqrt {e x +d}}{c}-\frac {e \arctanh \left (\frac {-c^{\frac {1}{4}} \sqrt {2}\, \sqrt {e x +d}+\sqrt {d \sqrt {c}+\sqrt {a \,e^{2}+c \,d^{2}}}}{\sqrt {d \sqrt {c}-\sqrt {a \,e^{2}+c \,d^{2}}}}\right ) \left (c \,d^{2}+a \,e^{2}-2 d \sqrt {c}\, \sqrt {a \,e^{2}+c \,d^{2}}\right ) \sqrt {2}}{2 c^{\frac {5}{4}} \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {d \sqrt {c}-\sqrt {a \,e^{2}+c \,d^{2}}}}+\frac {e \arctanh \left (\frac {c^{\frac {1}{4}} \sqrt {2}\, \sqrt {e x +d}+\sqrt {d \sqrt {c}+\sqrt {a \,e^{2}+c \,d^{2}}}}{\sqrt {d \sqrt {c}-\sqrt {a \,e^{2}+c \,d^{2}}}}\right ) \left (c \,d^{2}+a \,e^{2}-2 d \sqrt {c}\, \sqrt {a \,e^{2}+c \,d^{2}}\right ) \sqrt {2}}{2 c^{\frac {5}{4}} \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {d \sqrt {c}-\sqrt {a \,e^{2}+c \,d^{2}}}}+\frac {e \ln \left (\left (e x +d \right ) \sqrt {c}+\sqrt {a \,e^{2}+c \,d^{2}}-c^{\frac {1}{4}} \sqrt {2}\, \sqrt {e x +d}\, \sqrt {d \sqrt {c}+\sqrt {a \,e^{2}+c \,d^{2}}}\right ) \left (c \,d^{2}+a \,e^{2}+2 d \sqrt {c}\, \sqrt {a \,e^{2}+c \,d^{2}}\right ) \sqrt {2}}{4 c^{\frac {5}{4}} \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {d \sqrt {c}+\sqrt {a \,e^{2}+c \,d^{2}}}}-\frac {e \ln \left (\left (e x +d \right ) \sqrt {c}+\sqrt {a \,e^{2}+c \,d^{2}}+c^{\frac {1}{4}} \sqrt {2}\, \sqrt {e x +d}\, \sqrt {d \sqrt {c}+\sqrt {a \,e^{2}+c \,d^{2}}}\right ) \left (c \,d^{2}+a \,e^{2}+2 d \sqrt {c}\, \sqrt {a \,e^{2}+c \,d^{2}}\right ) \sqrt {2}}{4 c^{\frac {5}{4}} \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {d \sqrt {c}+\sqrt {a \,e^{2}+c \,d^{2}}}} \]
command
integrate((e*x+d)^(3/2)/(c*x^2+a),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (c^{3} d^{3} + a c^{2} d e^{2} + {\left (\sqrt {-a c} c d^{2} e + \sqrt {-a c} a e^{3}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {c^{2} d + \sqrt {c^{4} d^{2} - {\left (c^{2} d^{2} + a c e^{2}\right )} c^{2}}}{c^{2}}}}\right )}{{\left (a c^{2} e + \sqrt {-a c} c^{2} d\right )} \sqrt {-c^{2} d - \sqrt {-a c} c e}} + \frac {{\left (c^{3} d^{3} + a c^{2} d e^{2} - {\left (\sqrt {-a c} c d^{2} e + \sqrt {-a c} a e^{3}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {c^{2} d - \sqrt {c^{4} d^{2} - {\left (c^{2} d^{2} + a c e^{2}\right )} c^{2}}}{c^{2}}}}\right )}{{\left (a c^{2} e - \sqrt {-a c} c^{2} d\right )} \sqrt {-c^{2} d + \sqrt {-a c} c e}} + \frac {2 \, \sqrt {x e + d} e}{c} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________