14.24 Problem number 618

\[ \int \frac {(d+e x)^{5/2}}{a+c x^2} \, dx \]

Optimal antiderivative \[ \frac {2 e \left (e x +d \right )^{\frac {3}{2}}}{3 c}+\frac {4 d e \sqrt {e x +d}}{c}-\frac {e \arctanh \left (\frac {-c^{\frac {1}{4}} \sqrt {2}\, \sqrt {e x +d}+\sqrt {d \sqrt {c}+\sqrt {a \,e^{2}+c \,d^{2}}}}{\sqrt {d \sqrt {c}-\sqrt {a \,e^{2}+c \,d^{2}}}}\right ) \left (2 c^{\frac {3}{2}} d^{3}+2 a d \,e^{2} \sqrt {c}-\left (-a \,e^{2}+3 c \,d^{2}\right ) \sqrt {a \,e^{2}+c \,d^{2}}\right ) \sqrt {2}}{2 c^{\frac {7}{4}} \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {d \sqrt {c}-\sqrt {a \,e^{2}+c \,d^{2}}}}+\frac {e \arctanh \left (\frac {c^{\frac {1}{4}} \sqrt {2}\, \sqrt {e x +d}+\sqrt {d \sqrt {c}+\sqrt {a \,e^{2}+c \,d^{2}}}}{\sqrt {d \sqrt {c}-\sqrt {a \,e^{2}+c \,d^{2}}}}\right ) \left (2 c^{\frac {3}{2}} d^{3}+2 a d \,e^{2} \sqrt {c}-\left (-a \,e^{2}+3 c \,d^{2}\right ) \sqrt {a \,e^{2}+c \,d^{2}}\right ) \sqrt {2}}{2 c^{\frac {7}{4}} \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {d \sqrt {c}-\sqrt {a \,e^{2}+c \,d^{2}}}}+\frac {e \ln \left (\left (e x +d \right ) \sqrt {c}+\sqrt {a \,e^{2}+c \,d^{2}}-c^{\frac {1}{4}} \sqrt {2}\, \sqrt {e x +d}\, \sqrt {d \sqrt {c}+\sqrt {a \,e^{2}+c \,d^{2}}}\right ) \left (2 c^{\frac {3}{2}} d^{3}+2 a d \,e^{2} \sqrt {c}+\left (-a \,e^{2}+3 c \,d^{2}\right ) \sqrt {a \,e^{2}+c \,d^{2}}\right ) \sqrt {2}}{4 c^{\frac {7}{4}} \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {d \sqrt {c}+\sqrt {a \,e^{2}+c \,d^{2}}}}-\frac {e \ln \left (\left (e x +d \right ) \sqrt {c}+\sqrt {a \,e^{2}+c \,d^{2}}+c^{\frac {1}{4}} \sqrt {2}\, \sqrt {e x +d}\, \sqrt {d \sqrt {c}+\sqrt {a \,e^{2}+c \,d^{2}}}\right ) \left (2 c^{\frac {3}{2}} d^{3}+2 a d \,e^{2} \sqrt {c}+\left (-a \,e^{2}+3 c \,d^{2}\right ) \sqrt {a \,e^{2}+c \,d^{2}}\right ) \sqrt {2}}{4 c^{\frac {7}{4}} \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {d \sqrt {c}+\sqrt {a \,e^{2}+c \,d^{2}}}} \]

command

integrate((e*x+d)^(5/2)/(c*x^2+a),x, algorithm="giac")

Giac 1.9.0-11 via sagemath 9.6 output

\[ \frac {{\left (c^{4} d^{4} - 3 \, a c^{3} d^{2} e^{2} + {\left (3 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} c^{2} + 2 \, {\left (\sqrt {-a c} c^{2} d^{3} e + \sqrt {-a c} a c d e^{3}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {c^{4} d + \sqrt {c^{8} d^{2} - {\left (c^{4} d^{2} + a c^{3} e^{2}\right )} c^{4}}}{c^{4}}}}\right )}{{\left (a c^{3} e + \sqrt {-a c} c^{3} d\right )} \sqrt {-c^{2} d - \sqrt {-a c} c e}} + \frac {{\left (c^{4} d^{4} - 3 \, a c^{3} d^{2} e^{2} + {\left (3 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} c^{2} - 2 \, {\left (\sqrt {-a c} c^{2} d^{3} e + \sqrt {-a c} a c d e^{3}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {c^{4} d - \sqrt {c^{8} d^{2} - {\left (c^{4} d^{2} + a c^{3} e^{2}\right )} c^{4}}}{c^{4}}}}\right )}{{\left (a c^{3} e - \sqrt {-a c} c^{3} d\right )} \sqrt {-c^{2} d + \sqrt {-a c} c e}} + \frac {2 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} c^{2} e + 6 \, \sqrt {x e + d} c^{2} d e\right )}}{3 \, c^{3}} \]

Giac 1.7.0 via sagemath 9.3 output

\[ \mathit {sage}_{0} x \]________________________________________________________________________________________