Integrand size = 21, antiderivative size = 105 \[ \int \frac {(2+x) (d+e x)}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {d+e}{12 (1-x)}+\frac {d+2 e}{36 (2-x)}-\frac {d-e}{36 (1+x)}+\frac {1}{36} (2 d+5 e) \log (1-x)-\frac {1}{432} (35 d+58 e) \log (2-x)+\frac {1}{108} (2 d+e) \log (1+x)+\frac {1}{144} (d-2 e) \log (2+x) \]
[Out]
Time = 0.13 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1600, 6874} \[ \int \frac {(2+x) (d+e x)}{\left (4-5 x^2+x^4\right )^2} \, dx=-\frac {d-e}{36 (x+1)}+\frac {d+e}{12 (1-x)}+\frac {d+2 e}{36 (2-x)}+\frac {1}{36} (2 d+5 e) \log (1-x)-\frac {1}{432} (35 d+58 e) \log (2-x)+\frac {1}{108} (2 d+e) \log (x+1)+\frac {1}{144} (d-2 e) \log (x+2) \]
[In]
[Out]
Rule 1600
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {d+e x}{(2+x) \left (2-x-2 x^2+x^3\right )^2} \, dx \\ & = \int \left (\frac {d+2 e}{36 (-2+x)^2}+\frac {-35 d-58 e}{432 (-2+x)}+\frac {d+e}{12 (-1+x)^2}+\frac {2 d+5 e}{36 (-1+x)}+\frac {d-e}{36 (1+x)^2}+\frac {2 d+e}{108 (1+x)}+\frac {d-2 e}{144 (2+x)}\right ) \, dx \\ & = \frac {d+e}{12 (1-x)}+\frac {d+2 e}{36 (2-x)}-\frac {d-e}{36 (1+x)}+\frac {1}{36} (2 d+5 e) \log (1-x)-\frac {1}{432} (35 d+58 e) \log (2-x)+\frac {1}{108} (2 d+e) \log (1+x)+\frac {1}{144} (d-2 e) \log (2+x) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.92 \[ \int \frac {(2+x) (d+e x)}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {1}{432} \left (\frac {12 \left (d \left (5+6 x-5 x^2\right )+2 e \left (5-2 x^2\right )\right )}{2-x-2 x^2+x^3}+12 (2 d+5 e) \log (1-x)-(35 d+58 e) \log (2-x)+4 (2 d+e) \log (1+x)+3 (d-2 e) \log (2+x)\right ) \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.88
method | result | size |
default | \(\left (\frac {d}{144}-\frac {e}{72}\right ) \ln \left (x +2\right )-\frac {\frac {d}{36}-\frac {e}{36}}{x +1}+\left (\frac {d}{54}+\frac {e}{108}\right ) \ln \left (x +1\right )-\frac {\frac {d}{12}+\frac {e}{12}}{x -1}+\left (\frac {d}{18}+\frac {5 e}{36}\right ) \ln \left (x -1\right )+\left (-\frac {35 d}{432}-\frac {29 e}{216}\right ) \ln \left (x -2\right )-\frac {\frac {d}{36}+\frac {e}{18}}{x -2}\) | \(92\) |
norman | \(\frac {\left (-\frac {5 d}{36}-\frac {e}{9}\right ) x^{3}+\left (\frac {17 d}{36}+\frac {5 e}{18}\right ) x +\left (-\frac {d}{9}-\frac {2 e}{9}\right ) x^{2}+\frac {5 d}{18}+\frac {5 e}{9}}{x^{4}-5 x^{2}+4}+\left (-\frac {35 d}{432}-\frac {29 e}{216}\right ) \ln \left (x -2\right )+\left (\frac {d}{18}+\frac {5 e}{36}\right ) \ln \left (x -1\right )+\left (\frac {d}{54}+\frac {e}{108}\right ) \ln \left (x +1\right )+\left (\frac {d}{144}-\frac {e}{72}\right ) \ln \left (x +2\right )\) | \(101\) |
risch | \(\frac {\left (-\frac {5 d}{36}-\frac {e}{9}\right ) x^{2}+\frac {d x}{6}+\frac {5 d}{36}+\frac {5 e}{18}}{x^{3}-2 x^{2}-x +2}+\frac {\ln \left (x +2\right ) d}{144}-\frac {\ln \left (x +2\right ) e}{72}+\frac {\ln \left (x -1\right ) d}{18}+\frac {5 \ln \left (x -1\right ) e}{36}-\frac {35 \ln \left (2-x \right ) d}{432}-\frac {29 \ln \left (2-x \right ) e}{216}+\frac {\ln \left (-x -1\right ) d}{54}+\frac {\ln \left (-x -1\right ) e}{108}\) | \(104\) |
parallelrisch | \(-\frac {60 d \,x^{2}-60 d -120 e -72 d x -60 \ln \left (x -1\right ) x^{3} e -8 \ln \left (x +1\right ) x^{3} d -4 \ln \left (x +1\right ) x^{3} e -3 \ln \left (x +2\right ) x^{3} d +6 \ln \left (x +2\right ) x^{3} e +70 \ln \left (x -2\right ) d +116 \ln \left (x -2\right ) e -48 \ln \left (x -1\right ) d -120 \ln \left (x -1\right ) e +48 e \,x^{2}+58 \ln \left (x -2\right ) x^{3} e -24 \ln \left (x -1\right ) x^{3} d -6 \ln \left (x +2\right ) x e -58 \ln \left (x -2\right ) x e +24 \ln \left (x -1\right ) x d +60 \ln \left (x -1\right ) x e +8 \ln \left (x +1\right ) x d +4 \ln \left (x +1\right ) x e +3 \ln \left (x +2\right ) x d -116 \ln \left (x -2\right ) x^{2} e +48 \ln \left (x -1\right ) x^{2} d +120 \ln \left (x -1\right ) x^{2} e +16 \ln \left (x +1\right ) x^{2} d +8 \ln \left (x +1\right ) x^{2} e +6 \ln \left (x +2\right ) x^{2} d -12 \ln \left (x +2\right ) x^{2} e -6 \ln \left (x +2\right ) d +12 \ln \left (x +2\right ) e -16 \ln \left (x +1\right ) d -8 \ln \left (x +1\right ) e -35 \ln \left (x -2\right ) x d +35 \ln \left (x -2\right ) x^{3} d -70 \ln \left (x -2\right ) x^{2} d}{432 \left (x^{3}-2 x^{2}-x +2\right )}\) | \(321\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (87) = 174\).
Time = 0.28 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.01 \[ \int \frac {(2+x) (d+e x)}{\left (4-5 x^2+x^4\right )^2} \, dx=-\frac {12 \, {\left (5 \, d + 4 \, e\right )} x^{2} - 72 \, d x - 3 \, {\left ({\left (d - 2 \, e\right )} x^{3} - 2 \, {\left (d - 2 \, e\right )} x^{2} - {\left (d - 2 \, e\right )} x + 2 \, d - 4 \, e\right )} \log \left (x + 2\right ) - 4 \, {\left ({\left (2 \, d + e\right )} x^{3} - 2 \, {\left (2 \, d + e\right )} x^{2} - {\left (2 \, d + e\right )} x + 4 \, d + 2 \, e\right )} \log \left (x + 1\right ) - 12 \, {\left ({\left (2 \, d + 5 \, e\right )} x^{3} - 2 \, {\left (2 \, d + 5 \, e\right )} x^{2} - {\left (2 \, d + 5 \, e\right )} x + 4 \, d + 10 \, e\right )} \log \left (x - 1\right ) + {\left ({\left (35 \, d + 58 \, e\right )} x^{3} - 2 \, {\left (35 \, d + 58 \, e\right )} x^{2} - {\left (35 \, d + 58 \, e\right )} x + 70 \, d + 116 \, e\right )} \log \left (x - 2\right ) - 60 \, d - 120 \, e}{432 \, {\left (x^{3} - 2 \, x^{2} - x + 2\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1034 vs. \(2 (82) = 164\).
Time = 5.46 (sec) , antiderivative size = 1034, normalized size of antiderivative = 9.85 \[ \int \frac {(2+x) (d+e x)}{\left (4-5 x^2+x^4\right )^2} \, dx=\text {Too large to display} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.84 \[ \int \frac {(2+x) (d+e x)}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {1}{144} \, {\left (d - 2 \, e\right )} \log \left (x + 2\right ) + \frac {1}{108} \, {\left (2 \, d + e\right )} \log \left (x + 1\right ) + \frac {1}{36} \, {\left (2 \, d + 5 \, e\right )} \log \left (x - 1\right ) - \frac {1}{432} \, {\left (35 \, d + 58 \, e\right )} \log \left (x - 2\right ) - \frac {{\left (5 \, d + 4 \, e\right )} x^{2} - 6 \, d x - 5 \, d - 10 \, e}{36 \, {\left (x^{3} - 2 \, x^{2} - x + 2\right )}} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.88 \[ \int \frac {(2+x) (d+e x)}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {1}{144} \, {\left (d - 2 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) + \frac {1}{108} \, {\left (2 \, d + e\right )} \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{36} \, {\left (2 \, d + 5 \, e\right )} \log \left ({\left | x - 1 \right |}\right ) - \frac {1}{432} \, {\left (35 \, d + 58 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) - \frac {{\left (5 \, d + 4 \, e\right )} x^{2} - 6 \, d x - 5 \, d - 10 \, e}{36 \, {\left (x + 1\right )} {\left (x - 1\right )} {\left (x - 2\right )}} \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.86 \[ \int \frac {(2+x) (d+e x)}{\left (4-5 x^2+x^4\right )^2} \, dx=\ln \left (x-1\right )\,\left (\frac {d}{18}+\frac {5\,e}{36}\right )-\frac {\left (-\frac {5\,d}{36}-\frac {e}{9}\right )\,x^2+\frac {d\,x}{6}+\frac {5\,d}{36}+\frac {5\,e}{18}}{-x^3+2\,x^2+x-2}+\ln \left (x+1\right )\,\left (\frac {d}{54}+\frac {e}{108}\right )+\ln \left (x+2\right )\,\left (\frac {d}{144}-\frac {e}{72}\right )-\ln \left (x-2\right )\,\left (\frac {35\,d}{432}+\frac {29\,e}{216}\right ) \]
[In]
[Out]