Integrand size = 40, antiderivative size = 137 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^2 (d+e x)} \, dx=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d x}-\frac {\left (c d^2-a e^2\right ) \text {arctanh}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{2 \sqrt {a} d^{3/2} \sqrt {e}} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {863, 820, 738, 212} \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^2 (d+e x)} \, dx=-\frac {\left (c d^2-a e^2\right ) \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \sqrt {a} d^{3/2} \sqrt {e}}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d x} \]
[In]
[Out]
Rule 212
Rule 738
Rule 820
Rule 863
Rubi steps \begin{align*} \text {integral}& = \int \frac {a e+c d x}{x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx \\ & = -\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d x}-\frac {\left (-2 a c d^2 e+a e \left (c d^2+a e^2\right )\right ) \int \frac {1}{x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2 a d e} \\ & = -\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d x}+\frac {\left (-2 a c d^2 e+a e \left (c d^2+a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{4 a d e-x^2} \, dx,x,\frac {2 a d e-\left (-c d^2-a e^2\right ) x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{a d e} \\ & = -\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d x}-\frac {\left (c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{2 \sqrt {a} d^{3/2} \sqrt {e}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(841\) vs. \(2(137)=274\).
Time = 9.25 (sec) , antiderivative size = 841, normalized size of antiderivative = 6.14 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^2 (d+e x)} \, dx=\frac {\sqrt {a e+c d x} \sqrt {d+e x} \left (-\frac {\sqrt {a e+c d x} \left (a^2 e^4 \sqrt {d+e x}+2 a c d e^2 \left (2 d \sqrt {d-\frac {a e^2}{c d}}+2 e \sqrt {d-\frac {a e^2}{c d}} x-4 d \sqrt {d+e x}-3 e x \sqrt {d+e x}\right )+c^2 \left (d^2 e^2 x^2 \left (-4 \sqrt {d-\frac {a e^2}{c d}}+\sqrt {d+e x}\right )+d^3 e x \left (-12 \sqrt {d-\frac {a e^2}{c d}}+8 \sqrt {d+e x}\right )+d^4 \left (-8 \sqrt {d-\frac {a e^2}{c d}}+8 \sqrt {d+e x}\right )\right )\right )}{a^2 d e^4 x-2 a c d^2 e^2 x \left (4 d+3 e x-2 \sqrt {d-\frac {a e^2}{c d}} \sqrt {d+e x}\right )+c^2 d^3 x \left (8 d^2+8 d e x+e^2 x^2-8 d \sqrt {d-\frac {a e^2}{c d}} \sqrt {d+e x}-4 e \sqrt {d-\frac {a e^2}{c d}} x \sqrt {d+e x}\right )}+\frac {\sqrt {c d^2-a e^2} \left (c d^2-a e^2+\sqrt {c} d \sqrt {c d^2-a e^2}\right ) \arctan \left (\frac {\sqrt {-2 c d^2+a e^2-2 \sqrt {c} d \sqrt {c d^2-a e^2}} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {c} \sqrt {d} \sqrt {e} \left (\sqrt {d-\frac {a e^2}{c d}}-\sqrt {d+e x}\right )}\right )}{\sqrt {a} d^{3/2} \sqrt {e} \sqrt {-2 c d^2+a e^2-2 \sqrt {c} d \sqrt {c d^2-a e^2}}}+\frac {\sqrt {c d^2-a e^2} \left (-c d^2+a e^2+\sqrt {c} d \sqrt {c d^2-a e^2}\right ) \arctan \left (\frac {\sqrt {-2 c d^2+a e^2+2 \sqrt {c} d \sqrt {c d^2-a e^2}} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {c} \sqrt {d} \sqrt {e} \left (\sqrt {d-\frac {a e^2}{c d}}-\sqrt {d+e x}\right )}\right )}{\sqrt {a} d^{3/2} \sqrt {e} \sqrt {-2 c d^2+a e^2+2 \sqrt {c} d \sqrt {c d^2-a e^2}}}\right )}{\sqrt {(a e+c d x) (d+e x)}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(707\) vs. \(2(117)=234\).
Time = 0.72 (sec) , antiderivative size = 708, normalized size of antiderivative = 5.17
method | result | size |
default | \(\frac {-\frac {{\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {3}{2}}}{a d e x}+\frac {\left (e^{2} a +c \,d^{2}\right ) \left (\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}+\frac {\left (e^{2} a +c \,d^{2}\right ) \ln \left (\frac {\frac {1}{2} e^{2} a +\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{2 \sqrt {c d e}}-\frac {a d e \ln \left (\frac {2 a d e +\left (e^{2} a +c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{x}\right )}{\sqrt {a d e}}\right )}{2 a d e}+\frac {2 c \left (\frac {\left (2 c d e x +e^{2} a +c \,d^{2}\right ) \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} e^{2} a +\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{8 c d e \sqrt {c d e}}\right )}{a}}{d}-\frac {e \left (\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}+\frac {\left (e^{2} a +c \,d^{2}\right ) \ln \left (\frac {\frac {1}{2} e^{2} a +\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{2 \sqrt {c d e}}-\frac {a d e \ln \left (\frac {2 a d e +\left (e^{2} a +c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{x}\right )}{\sqrt {a d e}}\right )}{d^{2}}+\frac {e \left (\sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}+\frac {\left (e^{2} a -c \,d^{2}\right ) \ln \left (\frac {\frac {e^{2} a}{2}-\frac {c \,d^{2}}{2}+c d e \left (x +\frac {d}{e}\right )}{\sqrt {c d e}}+\sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{2 \sqrt {c d e}}\right )}{d^{2}}\) | \(708\) |
[In]
[Out]
none
Time = 0.36 (sec) , antiderivative size = 355, normalized size of antiderivative = 2.59 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^2 (d+e x)} \, dx=\left [-\frac {4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} a d e + {\left (c d^{2} - a e^{2}\right )} \sqrt {a d e} x \log \left (\frac {8 \, a^{2} d^{2} e^{2} + {\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} + 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {a d e} + 8 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x}{x^{2}}\right )}{4 \, a d^{2} e x}, -\frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} a d e - {\left (c d^{2} - a e^{2}\right )} \sqrt {-a d e} x \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {-a d e}}{2 \, {\left (a c d^{2} e^{2} x^{2} + a^{2} d^{2} e^{2} + {\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )}}\right )}{2 \, a d^{2} e x}\right ] \]
[In]
[Out]
\[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^2 (d+e x)} \, dx=\int \frac {\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}{x^{2} \left (d + e x\right )}\, dx \]
[In]
[Out]
\[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^2 (d+e x)} \, dx=\int { \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{{\left (e x + d\right )} x^{2}} \,d x } \]
[In]
[Out]
none
Time = 0.33 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.61 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^2 (d+e x)} \, dx=\frac {{\left (c d^{2} - a e^{2}\right )} \arctan \left (-\frac {\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}}{\sqrt {-a d e}}\right )}{\sqrt {-a d e} d} - \frac {{\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} c d^{2} + {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} a e^{2} + 2 \, \sqrt {c d e} a d e}{{\left (a d e - {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )}^{2}\right )} d} \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^2 (d+e x)} \, dx=\int \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x^2\,\left (d+e\,x\right )} \,d x \]
[In]
[Out]