Integrand size = 51, antiderivative size = 122 \[ \int \frac {\left (2-x-2 x^2+x^3\right ) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=i x+\frac {d-2 e+4 f-8 g+16 h-32 i}{12 (2+x)}-\frac {1}{18} (d+e+f+g+h+i) \log (1-x)+\frac {1}{48} (d+2 e+4 f+8 g+16 h+32 i) \log (2-x)+\frac {1}{6} (d-e+f-g+h-i) \log (1+x)-\frac {1}{144} (19 d-26 e+28 f-8 g-80 h+352 i) \log (2+x) \]
i*x+1/12*(d-2*e+4*f-8*g+16*h-32*i)/(2+x)-1/18*(d+e+f+g+h+i)*ln(1-x)+1/48*( d+2*e+4*f+8*g+16*h+32*i)*ln(2-x)+1/6*(d-e+f-g+h-i)*ln(1+x)-1/144*(19*d-26* e+28*f-8*g-80*h+352*i)*ln(2+x)
Time = 0.05 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.97 \[ \int \frac {\left (2-x-2 x^2+x^3\right ) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {1}{144} \left (144 i x+\frac {12 (d-2 (e-2 f+4 g-8 h+16 i))}{2+x}-8 (d+e+f+g+h+i) \log (1-x)+3 (d+2 e+4 (f+2 g+4 h+8 i)) \log (2-x)+24 (d-e+f-g+h-i) \log (1+x)+(-19 d+26 e-28 f+8 g+80 h-352 i) \log (2+x)\right ) \]
Integrate[((2 - x - 2*x^2 + x^3)*(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5) )/(4 - 5*x^2 + x^4)^2,x]
(144*i*x + (12*(d - 2*(e - 2*f + 4*g - 8*h + 16*i)))/(2 + x) - 8*(d + e + f + g + h + i)*Log[1 - x] + 3*(d + 2*e + 4*(f + 2*g + 4*h + 8*i))*Log[2 - x] + 24*(d - e + f - g + h - i)*Log[1 + x] + (-19*d + 26*e - 28*f + 8*g + 80*h - 352*i)*Log[2 + x])/144
Time = 0.44 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2019, 2462, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (x^3-2 x^2-x+2\right ) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{\left (x^4-5 x^2+4\right )^2} \, dx\) |
| \(\Big \downarrow \) 2019 |
| \(\displaystyle \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{(x+2)^2 \left (x^3-2 x^2-x+2\right )}dx\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {-19 d+26 e-28 f+8 g+80 h-352 i}{144 (x+2)}+\frac {d+2 e+4 f+8 g+16 h+32 i}{48 (x-2)}+\frac {-d-e-f-g-h-i}{18 (x-1)}+\frac {d-e+f-g+h-i}{6 (x+1)}+\frac {-d+2 e-4 f+8 g-16 h+32 i}{12 (x+2)^2}+i\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d-2 e+4 f-8 g+16 h-32 i}{12 (x+2)}-\frac {1}{18} \log (1-x) (d+e+f+g+h+i)+\frac {1}{48} \log (2-x) (d+2 e+4 f+8 g+16 h+32 i)+\frac {1}{6} \log (x+1) (d-e+f-g+h-i)-\frac {1}{144} \log (x+2) (19 d-26 e+28 f-8 g-80 h+352 i)+i x\) |
i*x + (d - 2*e + 4*f - 8*g + 16*h - 32*i)/(12*(2 + x)) - ((d + e + f + g + h + i)*Log[1 - x])/18 + ((d + 2*e + 4*f + 8*g + 16*h + 32*i)*Log[2 - x])/ 48 + ((d - e + f - g + h - i)*Log[1 + x])/6 - ((19*d - 26*e + 28*f - 8*g - 80*h + 352*i)*Log[2 + x])/144
3.1.90.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px , Qx, x]^p*Qx^(p + q), x] /; FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Time = 0.12 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.04
| method | result | size |
| default | \(i x +\left (\frac {5 h}{9}-\frac {22 i}{9}+\frac {g}{18}-\frac {7 f}{36}+\frac {13 e}{72}-\frac {19 d}{144}\right ) \ln \left (x +2\right )-\frac {-\frac {d}{12}+\frac {e}{6}-\frac {f}{3}+\frac {2 g}{3}-\frac {4 h}{3}+\frac {8 i}{3}}{x +2}+\left (\frac {d}{6}-\frac {e}{6}+\frac {f}{6}-\frac {g}{6}+\frac {h}{6}-\frac {i}{6}\right ) \ln \left (x +1\right )+\left (-\frac {d}{18}-\frac {e}{18}-\frac {f}{18}-\frac {g}{18}-\frac {h}{18}-\frac {i}{18}\right ) \ln \left (x -1\right )+\left (\frac {d}{48}+\frac {e}{24}+\frac {f}{12}+\frac {g}{6}+\frac {h}{3}+\frac {2 i}{3}\right ) \ln \left (x -2\right )\) | \(127\) |
| norman | \(\frac {i \,x^{5}+\left (-\frac {4 h}{3}+\frac {20 i}{3}+\frac {2 g}{3}-\frac {f}{3}+\frac {e}{6}-\frac {d}{12}\right ) x +\left (-\frac {23 i}{3}+\frac {4 h}{3}-\frac {2 g}{3}+\frac {f}{3}-\frac {e}{6}+\frac {d}{12}\right ) x^{3}+\left (-\frac {d}{6}+\frac {e}{3}-\frac {2 f}{3}+\frac {4 g}{3}-\frac {8 h}{3}+\frac {16 i}{3}\right ) x^{2}+\frac {8 h}{3}-\frac {16 i}{3}+\frac {d}{6}-\frac {e}{3}+\frac {2 f}{3}-\frac {4 g}{3}}{x^{4}-5 x^{2}+4}+\left (-\frac {d}{18}-\frac {e}{18}-\frac {f}{18}-\frac {g}{18}-\frac {h}{18}-\frac {i}{18}\right ) \ln \left (x -1\right )+\left (\frac {d}{6}-\frac {e}{6}+\frac {f}{6}-\frac {g}{6}+\frac {h}{6}-\frac {i}{6}\right ) \ln \left (x +1\right )+\left (\frac {d}{48}+\frac {e}{24}+\frac {f}{12}+\frac {g}{6}+\frac {h}{3}+\frac {2 i}{3}\right ) \ln \left (x -2\right )+\left (\frac {5 h}{9}-\frac {22 i}{9}+\frac {g}{18}-\frac {7 f}{36}+\frac {13 e}{72}-\frac {19 d}{144}\right ) \ln \left (x +2\right )\) | \(202\) |
| risch | \(-\frac {\ln \left (x -1\right ) d}{18}-\frac {\ln \left (x -1\right ) e}{18}-\frac {\ln \left (x +1\right ) i}{6}+\frac {\ln \left (x +1\right ) f}{6}+\frac {\ln \left (2-x \right ) f}{12}+\frac {5 \ln \left (-x -2\right ) h}{9}-\frac {22 \ln \left (-x -2\right ) i}{9}+\frac {d}{12 x +24}+\frac {\ln \left (2-x \right ) d}{48}+\frac {\ln \left (2-x \right ) e}{24}+\frac {\ln \left (x +1\right ) d}{6}-\frac {\ln \left (x +1\right ) e}{6}+\frac {2 \ln \left (2-x \right ) i}{3}-\frac {\ln \left (x +1\right ) g}{6}+\frac {\ln \left (2-x \right ) g}{6}-\frac {\ln \left (x -1\right ) i}{18}-\frac {\ln \left (x -1\right ) g}{18}-\frac {19 \ln \left (-x -2\right ) d}{144}+\frac {13 \ln \left (-x -2\right ) e}{72}-\frac {e}{6 \left (x +2\right )}+\frac {f}{3 x +6}+\frac {4 h}{3 \left (x +2\right )}-\frac {8 i}{3 \left (x +2\right )}-\frac {\ln \left (x -1\right ) f}{18}-\frac {\ln \left (x -1\right ) h}{18}+i x -\frac {2 g}{3 \left (x +2\right )}+\frac {\ln \left (-x -2\right ) g}{18}+\frac {\ln \left (2-x \right ) h}{3}+\frac {\ln \left (x +1\right ) h}{6}-\frac {7 \ln \left (-x -2\right ) f}{36}\) | \(245\) |
| parallelrisch | \(\frac {-960 i +48 f -96 g -352 \ln \left (x +2\right ) x i +96 \ln \left (x -2\right ) x i +12 d +192 h -24 e +48 \ln \left (x -2\right ) x h -8 \ln \left (x -1\right ) x h +24 \ln \left (x +1\right ) x h +80 \ln \left (x +2\right ) x h +144 i \,x^{2}+6 \ln \left (x -2\right ) d +12 \ln \left (x -2\right ) e -16 \ln \left (x -1\right ) d -16 \ln \left (x -1\right ) e -48 \ln \left (x +1\right ) i +24 \ln \left (x -2\right ) x g -8 \ln \left (x -1\right ) x g -24 \ln \left (x +1\right ) x g +8 \ln \left (x +2\right ) x g -56 \ln \left (x +2\right ) f +48 \ln \left (x +1\right ) f +26 \ln \left (x +2\right ) x e +6 \ln \left (x -2\right ) x e -8 \ln \left (x -1\right ) x d -8 \ln \left (x -1\right ) x e +24 \ln \left (x +1\right ) x d -24 \ln \left (x +1\right ) x e -19 \ln \left (x +2\right ) x d -38 \ln \left (x +2\right ) d -8 \ln \left (x -1\right ) x i -24 \ln \left (x +1\right ) x i +12 \ln \left (x -2\right ) x f -8 \ln \left (x -1\right ) x f +24 \ln \left (x +1\right ) x f -28 \ln \left (x +2\right ) x f +52 \ln \left (x +2\right ) e +48 \ln \left (x +1\right ) d -48 \ln \left (x +1\right ) e +3 \ln \left (x -2\right ) x d -704 \ln \left (x +2\right ) i -48 \ln \left (x +1\right ) g +16 \ln \left (x +2\right ) g +192 \ln \left (x -2\right ) i -16 \ln \left (x -1\right ) i +48 \ln \left (x -2\right ) g -16 \ln \left (x -1\right ) g +24 \ln \left (x -2\right ) f -16 \ln \left (x -1\right ) f +96 \ln \left (x -2\right ) h -16 \ln \left (x -1\right ) h +48 \ln \left (x +1\right ) h +160 \ln \left (x +2\right ) h}{144 x +288}\) | \(393\) |
i*x+(5/9*h-22/9*i+1/18*g-7/36*f+13/72*e-19/144*d)*ln(x+2)-(-1/12*d+1/6*e-1 /3*f+2/3*g-4/3*h+8/3*i)/(x+2)+(1/6*d-1/6*e+1/6*f-1/6*g+1/6*h-1/6*i)*ln(x+1 )+(-1/18*d-1/18*e-1/18*f-1/18*g-1/18*h-1/18*i)*ln(x-1)+(1/48*d+1/24*e+1/12 *f+1/6*g+1/3*h+2/3*i)*ln(x-2)
Time = 19.69 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.64 \[ \int \frac {\left (2-x-2 x^2+x^3\right ) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {144 \, i x^{2} + 288 \, i x - {\left ({\left (19 \, d - 26 \, e + 28 \, f - 8 \, g - 80 \, h + 352 \, i\right )} x + 38 \, d - 52 \, e + 56 \, f - 16 \, g - 160 \, h + 704 \, i\right )} \log \left (x + 2\right ) + 24 \, {\left ({\left (d - e + f - g + h - i\right )} x + 2 \, d - 2 \, e + 2 \, f - 2 \, g + 2 \, h - 2 \, i\right )} \log \left (x + 1\right ) - 8 \, {\left ({\left (d + e + f + g + h + i\right )} x + 2 \, d + 2 \, e + 2 \, f + 2 \, g + 2 \, h + 2 \, i\right )} \log \left (x - 1\right ) + 3 \, {\left ({\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h + 32 \, i\right )} x + 2 \, d + 4 \, e + 8 \, f + 16 \, g + 32 \, h + 64 \, i\right )} \log \left (x - 2\right ) + 12 \, d - 24 \, e + 48 \, f - 96 \, g + 192 \, h - 384 \, i}{144 \, {\left (x + 2\right )}} \]
integrate((x^3-2*x^2-x+2)*(i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4)^2, x, algorithm="fricas")
1/144*(144*i*x^2 + 288*i*x - ((19*d - 26*e + 28*f - 8*g - 80*h + 352*i)*x + 38*d - 52*e + 56*f - 16*g - 160*h + 704*i)*log(x + 2) + 24*((d - e + f - g + h - i)*x + 2*d - 2*e + 2*f - 2*g + 2*h - 2*i)*log(x + 1) - 8*((d + e + f + g + h + i)*x + 2*d + 2*e + 2*f + 2*g + 2*h + 2*i)*log(x - 1) + 3*((d + 2*e + 4*f + 8*g + 16*h + 32*i)*x + 2*d + 4*e + 8*f + 16*g + 32*h + 64*i )*log(x - 2) + 12*d - 24*e + 48*f - 96*g + 192*h - 384*i)/(x + 2)
Timed out. \[ \int \frac {\left (2-x-2 x^2+x^3\right ) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=\text {Timed out} \]
Time = 0.19 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.89 \[ \int \frac {\left (2-x-2 x^2+x^3\right ) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=i x - \frac {1}{144} \, {\left (19 \, d - 26 \, e + 28 \, f - 8 \, g - 80 \, h + 352 \, i\right )} \log \left (x + 2\right ) + \frac {1}{6} \, {\left (d - e + f - g + h - i\right )} \log \left (x + 1\right ) - \frac {1}{18} \, {\left (d + e + f + g + h + i\right )} \log \left (x - 1\right ) + \frac {1}{48} \, {\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h + 32 \, i\right )} \log \left (x - 2\right ) + \frac {d - 2 \, e + 4 \, f - 8 \, g + 16 \, h - 32 \, i}{12 \, {\left (x + 2\right )}} \]
integrate((x^3-2*x^2-x+2)*(i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4)^2, x, algorithm="maxima")
i*x - 1/144*(19*d - 26*e + 28*f - 8*g - 80*h + 352*i)*log(x + 2) + 1/6*(d - e + f - g + h - i)*log(x + 1) - 1/18*(d + e + f + g + h + i)*log(x - 1) + 1/48*(d + 2*e + 4*f + 8*g + 16*h + 32*i)*log(x - 2) + 1/12*(d - 2*e + 4* f - 8*g + 16*h - 32*i)/(x + 2)
Time = 0.30 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.92 \[ \int \frac {\left (2-x-2 x^2+x^3\right ) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=i x - \frac {1}{144} \, {\left (19 \, d - 26 \, e + 28 \, f - 8 \, g - 80 \, h + 352 \, i\right )} \log \left ({\left | x + 2 \right |}\right ) + \frac {1}{6} \, {\left (d - e + f - g + h - i\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{18} \, {\left (d + e + f + g + h + i\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac {1}{48} \, {\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h + 32 \, i\right )} \log \left ({\left | x - 2 \right |}\right ) + \frac {d - 2 \, e + 4 \, f - 8 \, g + 16 \, h - 32 \, i}{12 \, {\left (x + 2\right )}} \]
integrate((x^3-2*x^2-x+2)*(i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4)^2, x, algorithm="giac")
i*x - 1/144*(19*d - 26*e + 28*f - 8*g - 80*h + 352*i)*log(abs(x + 2)) + 1/ 6*(d - e + f - g + h - i)*log(abs(x + 1)) - 1/18*(d + e + f + g + h + i)*l og(abs(x - 1)) + 1/48*(d + 2*e + 4*f + 8*g + 16*h + 32*i)*log(abs(x - 2)) + 1/12*(d - 2*e + 4*f - 8*g + 16*h - 32*i)/(x + 2)
Time = 8.37 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.04 \[ \int \frac {\left (2-x-2 x^2+x^3\right ) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=i\,x+\frac {\frac {d}{12}-\frac {e}{6}+\frac {f}{3}-\frac {2\,g}{3}+\frac {4\,h}{3}-\frac {8\,i}{3}}{x+2}+\ln \left (x+1\right )\,\left (\frac {d}{6}-\frac {e}{6}+\frac {f}{6}-\frac {g}{6}+\frac {h}{6}-\frac {i}{6}\right )+\ln \left (x-2\right )\,\left (\frac {d}{48}+\frac {e}{24}+\frac {f}{12}+\frac {g}{6}+\frac {h}{3}+\frac {2\,i}{3}\right )-\ln \left (x-1\right )\,\left (\frac {d}{18}+\frac {e}{18}+\frac {f}{18}+\frac {g}{18}+\frac {h}{18}+\frac {i}{18}\right )-\ln \left (x+2\right )\,\left (\frac {19\,d}{144}-\frac {13\,e}{72}+\frac {7\,f}{36}-\frac {g}{18}-\frac {5\,h}{9}+\frac {22\,i}{9}\right ) \]
int(-((x + 2*x^2 - x^3 - 2)*(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5))/(x^ 4 - 5*x^2 + 4)^2,x)
i*x + (d/12 - e/6 + f/3 - (2*g)/3 + (4*h)/3 - (8*i)/3)/(x + 2) + log(x + 1 )*(d/6 - e/6 + f/6 - g/6 + h/6 - i/6) + log(x - 2)*(d/48 + e/24 + f/12 + g /6 + h/3 + (2*i)/3) - log(x - 1)*(d/18 + e/18 + f/18 + g/18 + h/18 + i/18) - log(x + 2)*((19*d)/144 - (13*e)/72 + (7*f)/36 - g/18 - (5*h)/9 + (22*i) /9)