Integrand size = 41, antiderivative size = 96 \[ \int \frac {(2+x) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{4-5 x^2+x^4} \, dx=(g+2 h+5 i) x+\frac {1}{2} (h+2 i) x^2+\frac {i x^3}{3}-\frac {1}{2} (d+e+f+g+h+i) \log (1-x)+\frac {1}{3} (d+2 e+4 f+8 g+16 h+32 i) \log (2-x)+\frac {1}{6} (d-e+f-g+h-i) \log (1+x) \]
[Out]
Time = 0.09 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {1600, 2099} \[ \int \frac {(2+x) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{4-5 x^2+x^4} \, dx=-\frac {1}{2} \log (1-x) (d+e+f+g+h+i)+\frac {1}{3} \log (2-x) (d+2 e+4 f+8 g+16 h+32 i)+\frac {1}{6} \log (x+1) (d-e+f-g+h-i)+x (g+2 h+5 i)+\frac {1}{2} x^2 (h+2 i)+\frac {i x^3}{3} \]
[In]
[Out]
Rule 1600
Rule 2099
Rubi steps \begin{align*} \text {integral}& = \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{2-x-2 x^2+x^3} \, dx \\ & = \int \left (g \left (1+\frac {2 h+5 i}{g}\right )+\frac {d+2 e+4 f+8 g+16 h+32 i}{3 (-2+x)}+\frac {-d-e-f-g-h-i}{2 (-1+x)}+(h+2 i) x+i x^2+\frac {d-e+f-g+h-i}{6 (1+x)}\right ) \, dx \\ & = (g+2 h+5 i) x+\frac {1}{2} (h+2 i) x^2+\frac {i x^3}{3}-\frac {1}{2} (d+e+f+g+h+i) \log (1-x)+\frac {1}{3} (d+2 e+4 f+8 g+16 h+32 i) \log (2-x)+\frac {1}{6} (d-e+f-g+h-i) \log (1+x) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.95 \[ \int \frac {(2+x) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{4-5 x^2+x^4} \, dx=\frac {1}{6} \left (6 (g+2 h+5 i) x+3 (h+2 i) x^2+2 i x^3-3 (d+e+f+g+h+i) \log (1-x)+2 (d+2 e+4 (f+2 g+4 h+8 i)) \log (2-x)+(d-e+f-g+h-i) \log (1+x)\right ) \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.03
method | result | size |
norman | \(\left (\frac {h}{2}+i \right ) x^{2}+\left (g +2 h +5 i \right ) x +\frac {i \,x^{3}}{3}+\left (-\frac {d}{2}-\frac {e}{2}-\frac {f}{2}-\frac {g}{2}-\frac {h}{2}-\frac {i}{2}\right ) \ln \left (x -1\right )+\left (\frac {d}{3}+\frac {2 e}{3}+\frac {4 f}{3}+\frac {8 g}{3}+\frac {16 h}{3}+\frac {32 i}{3}\right ) \ln \left (x -2\right )+\left (\frac {d}{6}-\frac {e}{6}+\frac {f}{6}-\frac {g}{6}+\frac {h}{6}-\frac {i}{6}\right ) \ln \left (x +1\right )\) | \(99\) |
default | \(\frac {i \,x^{3}}{3}+\frac {h \,x^{2}}{2}+i \,x^{2}+g x +2 h x +5 i x +\left (\frac {d}{6}-\frac {e}{6}+\frac {f}{6}-\frac {g}{6}+\frac {h}{6}-\frac {i}{6}\right ) \ln \left (x +1\right )+\left (-\frac {d}{2}-\frac {e}{2}-\frac {f}{2}-\frac {g}{2}-\frac {h}{2}-\frac {i}{2}\right ) \ln \left (x -1\right )+\left (\frac {d}{3}+\frac {2 e}{3}+\frac {4 f}{3}+\frac {8 g}{3}+\frac {16 h}{3}+\frac {32 i}{3}\right ) \ln \left (x -2\right )\) | \(102\) |
parallelrisch | \(g x +i \,x^{2}+\frac {\ln \left (x -2\right ) d}{3}+\frac {2 \ln \left (x -2\right ) e}{3}-\frac {\ln \left (x -1\right ) d}{2}-\frac {\ln \left (x -1\right ) e}{2}+\frac {h \,x^{2}}{2}+\frac {i \,x^{3}}{3}-\frac {\ln \left (x +1\right ) i}{6}+\frac {\ln \left (x +1\right ) f}{6}+2 h x +\frac {\ln \left (x +1\right ) d}{6}-\frac {\ln \left (x +1\right ) e}{6}-\frac {\ln \left (x +1\right ) g}{6}+\frac {32 \ln \left (x -2\right ) i}{3}-\frac {\ln \left (x -1\right ) i}{2}+\frac {8 \ln \left (x -2\right ) g}{3}-\frac {\ln \left (x -1\right ) g}{2}+\frac {4 \ln \left (x -2\right ) f}{3}-\frac {\ln \left (x -1\right ) f}{2}+\frac {16 \ln \left (x -2\right ) h}{3}-\frac {\ln \left (x -1\right ) h}{2}+5 i x +\frac {\ln \left (x +1\right ) h}{6}\) | \(156\) |
risch | \(\frac {i \,x^{3}}{3}+\frac {h \,x^{2}}{2}+i \,x^{2}+g x +2 h x +5 i x +\frac {\ln \left (x +1\right ) d}{6}-\frac {\ln \left (x +1\right ) e}{6}+\frac {\ln \left (x +1\right ) f}{6}-\frac {\ln \left (x +1\right ) g}{6}+\frac {\ln \left (x +1\right ) h}{6}-\frac {\ln \left (x +1\right ) i}{6}-\frac {\ln \left (1-x \right ) d}{2}-\frac {\ln \left (1-x \right ) e}{2}-\frac {\ln \left (1-x \right ) f}{2}-\frac {\ln \left (1-x \right ) g}{2}-\frac {\ln \left (1-x \right ) h}{2}-\frac {\ln \left (1-x \right ) i}{2}+\frac {\ln \left (2-x \right ) d}{3}+\frac {2 \ln \left (2-x \right ) e}{3}+\frac {4 \ln \left (2-x \right ) f}{3}+\frac {8 \ln \left (2-x \right ) g}{3}+\frac {16 \ln \left (2-x \right ) h}{3}+\frac {32 \ln \left (2-x \right ) i}{3}\) | \(180\) |
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.85 \[ \int \frac {(2+x) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{4-5 x^2+x^4} \, dx=\frac {1}{3} \, i x^{3} + \frac {1}{2} \, {\left (h + 2 \, i\right )} x^{2} + {\left (g + 2 \, h + 5 \, i\right )} x + \frac {1}{6} \, {\left (d - e + f - g + h - i\right )} \log \left (x + 1\right ) - \frac {1}{2} \, {\left (d + e + f + g + h + i\right )} \log \left (x - 1\right ) + \frac {1}{3} \, {\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h + 32 \, i\right )} \log \left (x - 2\right ) \]
[In]
[Out]
Timed out. \[ \int \frac {(2+x) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{4-5 x^2+x^4} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.18 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.85 \[ \int \frac {(2+x) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{4-5 x^2+x^4} \, dx=\frac {1}{3} \, i x^{3} + \frac {1}{2} \, {\left (h + 2 \, i\right )} x^{2} + {\left (g + 2 \, h + 5 \, i\right )} x + \frac {1}{6} \, {\left (d - e + f - g + h - i\right )} \log \left (x + 1\right ) - \frac {1}{2} \, {\left (d + e + f + g + h + i\right )} \log \left (x - 1\right ) + \frac {1}{3} \, {\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h + 32 \, i\right )} \log \left (x - 2\right ) \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.91 \[ \int \frac {(2+x) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{4-5 x^2+x^4} \, dx=\frac {1}{3} \, i x^{3} + \frac {1}{2} \, h x^{2} + i x^{2} + g x + 2 \, h x + 5 \, i x + \frac {1}{6} \, {\left (d - e + f - g + h - i\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{2} \, {\left (d + e + f + g + h + i\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac {1}{3} \, {\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h + 32 \, i\right )} \log \left ({\left | x - 2 \right |}\right ) \]
[In]
[Out]
Time = 7.93 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.03 \[ \int \frac {(2+x) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{4-5 x^2+x^4} \, dx=x\,\left (g+2\,h+5\,i\right )+\frac {i\,x^3}{3}-\ln \left (x-1\right )\,\left (\frac {d}{2}+\frac {e}{2}+\frac {f}{2}+\frac {g}{2}+\frac {h}{2}+\frac {i}{2}\right )+\ln \left (x+1\right )\,\left (\frac {d}{6}-\frac {e}{6}+\frac {f}{6}-\frac {g}{6}+\frac {h}{6}-\frac {i}{6}\right )+\ln \left (x-2\right )\,\left (\frac {d}{3}+\frac {2\,e}{3}+\frac {4\,f}{3}+\frac {8\,g}{3}+\frac {16\,h}{3}+\frac {32\,i}{3}\right )+x^2\,\left (\frac {h}{2}+i\right ) \]
[In]
[Out]