\(\int \frac {(2-3 x+x^2) (d+e x+f x^2+g x^3+h x^4)}{(4-5 x^2+x^4)^2} \, dx\) [95]

Optimal result

Integrand size = 41, antiderivative size = 131 \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2+g x^3+h x^4\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=-\frac {d-e+f-g+h}{6 (1+x)}-\frac {d-2 e+4 f-8 g+16 h}{12 (2+x)}-\frac {1}{36} (d+e+f+g+h) \log (1-x)+\frac {1}{144} (d+2 e+4 f+8 g+16 h) \log (2-x)-\frac {1}{36} (7 d-13 e+19 f-25 g+31 h) \log (1+x)+\frac {1}{144} (31 d-50 e+76 f-104 g+112 h) \log (2+x) \]

[Out]

1/6*(-d+e-f+g-h)/(1+x)+1/12*(-d+2*e-4*f+8*g-16*h)/(2+x)-1/36*(d+e+f+g+h)*ln(1-x)+1/144*(d+2*e+4*f+8*g+16*h)*ln
(2-x)-1/36*(7*d-13*e+19*f-25*g+31*h)*ln(1+x)+1/144*(31*d-50*e+76*f-104*g+112*h)*ln(2+x)

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 131, 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, 6860} \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2+g x^3+h x^4\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=-\frac {d-e+f-g+h}{6 (x+1)}-\frac {d-2 e+4 f-8 g+16 h}{12 (x+2)}-\frac {1}{36} \log (1-x) (d+e+f+g+h)+\frac {1}{144} \log (2-x) (d+2 e+4 f+8 g+16 h)-\frac {1}{36} \log (x+1) (7 d-13 e+19 f-25 g+31 h)+\frac {1}{144} \log (x+2) (31 d-50 e+76 f-104 g+112 h) \]

[In]

Int[((2 - 3*x + x^2)*(d + e*x + f*x^2 + g*x^3 + h*x^4))/(4 - 5*x^2 + x^4)^2,x]

[Out]

-1/6*(d - e + f - g + h)/(1 + x) - (d - 2*e + 4*f - 8*g + 16*h)/(12*(2 + x)) - ((d + e + f + g + h)*Log[1 - x]
)/36 + ((d + 2*e + 4*f + 8*g + 16*h)*Log[2 - x])/144 - ((7*d - 13*e + 19*f - 25*g + 31*h)*Log[1 + x])/36 + ((3
1*d - 50*e + 76*f - 104*g + 112*h)*Log[2 + x])/144

Rule 1600

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]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \frac {d+e x+f x^2+g x^3+h x^4}{\left (2-3 x+x^2\right ) \left (2+3 x+x^2\right )^2} \, dx \\ & = \int \left (\frac {d+2 e+4 f+8 g+16 h}{144 (-2+x)}+\frac {-d-e-f-g-h}{36 (-1+x)}+\frac {d-e+f-g+h}{6 (1+x)^2}+\frac {-7 d+13 e-19 f+25 g-31 h}{36 (1+x)}+\frac {d-2 e+4 f-8 g+16 h}{12 (2+x)^2}+\frac {31 d-50 e+76 f-104 g+112 h}{144 (2+x)}\right ) \, dx \\ & = -\frac {d-e+f-g+h}{6 (1+x)}-\frac {d-2 e+4 f-8 g+16 h}{12 (2+x)}-\frac {1}{36} (d+e+f+g+h) \log (1-x)+\frac {1}{144} (d+2 e+4 f+8 g+16 h) \log (2-x)-\frac {1}{36} (7 d-13 e+19 f-25 g+31 h) \log (1+x)+\frac {1}{144} (31 d-50 e+76 f-104 g+112 h) \log (2+x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.04 \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2+g x^3+h x^4\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {1}{144} \left (-\frac {12 (d (5+3 x)+2 (4 f-6 g+10 h+3 f x-5 g x+9 h x-e (3+2 x)))}{2+3 x+x^2}-4 (d+e+f+g+h) \log (1-x)+(d+2 (e+2 f+4 g+8 h)) \log (2-x)-4 (7 d-13 e+19 f-25 g+31 h) \log (1+x)+(31 d-50 e+76 f-104 g+112 h) \log (2+x)\right ) \]

[In]

Integrate[((2 - 3*x + x^2)*(d + e*x + f*x^2 + g*x^3 + h*x^4))/(4 - 5*x^2 + x^4)^2,x]

[Out]

((-12*(d*(5 + 3*x) + 2*(4*f - 6*g + 10*h + 3*f*x - 5*g*x + 9*h*x - e*(3 + 2*x))))/(2 + 3*x + x^2) - 4*(d + e +
f + g + h)*Log[1 - x] + (d + 2*(e + 2*f + 4*g + 8*h))*Log[2 - x] - 4*(7*d - 13*e + 19*f - 25*g + 31*h)*Log[1
+ x] + (31*d - 50*e + 76*f - 104*g + 112*h)*Log[2 + x])/144

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.01

method	result	size
default	\(-\frac {\frac {d}{12}-\frac {e}{6}+\frac {f}{3}-\frac {2 g}{3}+\frac {4 h}{3}}{x +2}+\left (\frac {31 d}{144}-\frac {25 e}{72}+\frac {19 f}{36}-\frac {13 g}{18}+\frac {7 h}{9}\right ) \ln \left (x +2\right )+\left (-\frac {7 d}{36}+\frac {13 e}{36}-\frac {19 f}{36}+\frac {25 g}{36}-\frac {31 h}{36}\right ) \ln \left (x +1\right )-\frac {\frac {d}{6}-\frac {e}{6}+\frac {f}{6}-\frac {g}{6}+\frac {h}{6}}{x +1}+\left (-\frac {d}{36}-\frac {e}{36}-\frac {f}{36}-\frac {g}{36}-\frac {h}{36}\right ) \ln \left (x -1\right )+\left (\frac {d}{144}+\frac {e}{72}+\frac {f}{36}+\frac {g}{18}+\frac {h}{9}\right ) \ln \left (x -2\right )\)	\(132\)
norman	\(\frac {\left (-\frac {d}{4}+\frac {e}{3}-\frac {f}{2}+\frac {5 g}{6}-\frac {3 h}{2}\right ) x^{3}+\left (\frac {3 d}{4}-\frac {5 e}{6}+f -\frac {4 g}{3}+2 h \right ) x +\left (\frac {d}{3}-\frac {e}{2}+\frac {5 f}{6}-\frac {3 g}{2}+\frac {17 h}{6}\right ) x^{2}-\frac {5 d}{6}+e +2 g -\frac {10 h}{3}-\frac {4 f}{3}}{x^{4}-5 x^{2}+4}+\left (-\frac {7 d}{36}+\frac {13 e}{36}-\frac {19 f}{36}+\frac {25 g}{36}-\frac {31 h}{36}\right ) \ln \left (x +1\right )+\left (-\frac {d}{36}-\frac {e}{36}-\frac {f}{36}-\frac {g}{36}-\frac {h}{36}\right ) \ln \left (x -1\right )+\left (\frac {d}{144}+\frac {e}{72}+\frac {f}{36}+\frac {g}{18}+\frac {h}{9}\right ) \ln \left (x -2\right )+\left (\frac {31 d}{144}-\frac {25 e}{72}+\frac {19 f}{36}-\frac {13 g}{18}+\frac {7 h}{9}\right ) \ln \left (x +2\right )\)	\(169\)
risch	\(\frac {\left (-\frac {d}{4}+\frac {e}{3}-\frac {f}{2}+\frac {5 g}{6}-\frac {3 h}{2}\right ) x -\frac {5 d}{12}+\frac {e}{2}-\frac {2 f}{3}+g -\frac {5 h}{3}}{x^{2}+3 x +2}+\frac {31 \ln \left (x +2\right ) d}{144}-\frac {25 \ln \left (x +2\right ) e}{72}+\frac {19 \ln \left (x +2\right ) f}{36}-\frac {13 \ln \left (x +2\right ) g}{18}+\frac {7 \ln \left (x +2\right ) h}{9}-\frac {\ln \left (x -1\right ) d}{36}-\frac {\ln \left (x -1\right ) e}{36}-\frac {\ln \left (x -1\right ) f}{36}-\frac {\ln \left (x -1\right ) g}{36}-\frac {\ln \left (x -1\right ) h}{36}+\frac {\ln \left (2-x \right ) d}{144}+\frac {\ln \left (2-x \right ) e}{72}+\frac {\ln \left (2-x \right ) f}{36}+\frac {\ln \left (2-x \right ) g}{18}+\frac {\ln \left (2-x \right ) h}{9}-\frac {7 \ln \left (-x -1\right ) d}{36}+\frac {13 \ln \left (-x -1\right ) e}{36}-\frac {19 \ln \left (-x -1\right ) f}{36}+\frac {25 \ln \left (-x -1\right ) g}{36}-\frac {31 \ln \left (-x -1\right ) h}{36}\)	\(205\)
parallelrisch	\(\frac {-96 f +144 g +120 g x -60 d -240 h +72 e +48 \ln \left (x -2\right ) x h -12 \ln \left (x -1\right ) x h -372 \ln \left (x +1\right ) x h +336 \ln \left (x +2\right ) x h -36 d x +2 \ln \left (x -2\right ) d +4 \ln \left (x -2\right ) e -8 \ln \left (x -1\right ) d -8 \ln \left (x -1\right ) e +24 \ln \left (x -2\right ) x g -12 \ln \left (x -1\right ) x g +300 \ln \left (x +1\right ) x g -312 \ln \left (x +2\right ) x g +16 \ln \left (x -2\right ) x^{2} h -4 \ln \left (x -1\right ) x^{2} h -124 \ln \left (x +1\right ) x^{2} h +152 \ln \left (x +2\right ) f -152 \ln \left (x +1\right ) f +112 \ln \left (x +2\right ) x^{2} h -150 \ln \left (x +2\right ) x e +6 \ln \left (x -2\right ) x e -12 \ln \left (x -1\right ) x d -12 \ln \left (x -1\right ) x e -84 \ln \left (x +1\right ) x d +156 \ln \left (x +1\right ) x e +93 \ln \left (x +2\right ) x d +2 \ln \left (x -2\right ) x^{2} e -4 \ln \left (x -1\right ) x^{2} d -4 \ln \left (x -1\right ) x^{2} e -28 \ln \left (x +1\right ) x^{2} d +52 \ln \left (x +1\right ) x^{2} e +31 \ln \left (x +2\right ) x^{2} d -50 \ln \left (x +2\right ) x^{2} e +62 \ln \left (x +2\right ) d -216 h x +12 \ln \left (x -2\right ) x f -12 \ln \left (x -1\right ) x f -228 \ln \left (x +1\right ) x f +228 \ln \left (x +2\right ) x f -100 \ln \left (x +2\right ) e -56 \ln \left (x +1\right ) d +104 \ln \left (x +1\right ) e +3 \ln \left (x -2\right ) x d +200 \ln \left (x +1\right ) g -208 \ln \left (x +2\right ) g +8 \ln \left (x -2\right ) x^{2} g -4 \ln \left (x -1\right ) x^{2} g +100 \ln \left (x +1\right ) x^{2} g -104 \ln \left (x +2\right ) x^{2} g +48 e x +16 \ln \left (x -2\right ) g -8 \ln \left (x -1\right ) g +4 \ln \left (x -2\right ) x^{2} f -4 \ln \left (x -1\right ) x^{2} f -76 \ln \left (x +1\right ) x^{2} f +76 \ln \left (x +2\right ) x^{2} f +8 \ln \left (x -2\right ) f -8 \ln \left (x -1\right ) f +32 \ln \left (x -2\right ) h -8 \ln \left (x -1\right ) h +\ln \left (x -2\right ) x^{2} d -248 \ln \left (x +1\right ) h +224 \ln \left (x +2\right ) h -72 f x}{144 x^{2}+432 x +288}\)	\(548\)

[In]

int((x^2-3*x+2)*(h*x^4+g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4)^2,x,method=_RETURNVERBOSE)

[Out]

-(1/12*d-1/6*e+1/3*f-2/3*g+4/3*h)/(x+2)+(31/144*d-25/72*e+19/36*f-13/18*g+7/9*h)*ln(x+2)+(-7/36*d+13/36*e-19/3
6*f+25/36*g-31/36*h)*ln(x+1)-(1/6*d-1/6*e+1/6*f-1/6*g+1/6*h)/(x+1)+(-1/36*d-1/36*e-1/36*f-1/36*g-1/36*h)*ln(x-
1)+(1/144*d+1/72*e+1/36*f+1/18*g+1/9*h)*ln(x-2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 267 vs. \(2 (119) = 238\).

Time = 3.52 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.04 \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2+g x^3+h x^4\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=-\frac {12 \, {\left (3 \, d - 4 \, e + 6 \, f - 10 \, g + 18 \, h\right )} x - {\left ({\left (31 \, d - 50 \, e + 76 \, f - 104 \, g + 112 \, h\right )} x^{2} + 3 \, {\left (31 \, d - 50 \, e + 76 \, f - 104 \, g + 112 \, h\right )} x + 62 \, d - 100 \, e + 152 \, f - 208 \, g + 224 \, h\right )} \log \left (x + 2\right ) + 4 \, {\left ({\left (7 \, d - 13 \, e + 19 \, f - 25 \, g + 31 \, h\right )} x^{2} + 3 \, {\left (7 \, d - 13 \, e + 19 \, f - 25 \, g + 31 \, h\right )} x + 14 \, d - 26 \, e + 38 \, f - 50 \, g + 62 \, h\right )} \log \left (x + 1\right ) + 4 \, {\left ({\left (d + e + f + g + h\right )} x^{2} + 3 \, {\left (d + e + f + g + h\right )} x + 2 \, d + 2 \, e + 2 \, f + 2 \, g + 2 \, h\right )} \log \left (x - 1\right ) - {\left ({\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h\right )} x^{2} + 3 \, {\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h\right )} x + 2 \, d + 4 \, e + 8 \, f + 16 \, g + 32 \, h\right )} \log \left (x - 2\right ) + 60 \, d - 72 \, e + 96 \, f - 144 \, g + 240 \, h}{144 \, {\left (x^{2} + 3 \, x + 2\right )}} \]

[In]

integrate((x^2-3*x+2)*(h*x^4+g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4)^2,x, algorithm="fricas")

[Out]

-1/144*(12*(3*d - 4*e + 6*f - 10*g + 18*h)*x - ((31*d - 50*e + 76*f - 104*g + 112*h)*x^2 + 3*(31*d - 50*e + 76
*f - 104*g + 112*h)*x + 62*d - 100*e + 152*f - 208*g + 224*h)*log(x + 2) + 4*((7*d - 13*e + 19*f - 25*g + 31*h
)*x^2 + 3*(7*d - 13*e + 19*f - 25*g + 31*h)*x + 14*d - 26*e + 38*f - 50*g + 62*h)*log(x + 1) + 4*((d + e + f +
g + h)*x^2 + 3*(d + e + f + g + h)*x + 2*d + 2*e + 2*f + 2*g + 2*h)*log(x - 1) - ((d + 2*e + 4*f + 8*g + 16*h
)*x^2 + 3*(d + 2*e + 4*f + 8*g + 16*h)*x + 2*d + 4*e + 8*f + 16*g + 32*h)*log(x - 2) + 60*d - 72*e + 96*f - 14
4*g + 240*h)/(x^2 + 3*x + 2)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2+g x^3+h x^4\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=\text {Timed out} \]

[In]

integrate((x**2-3*x+2)*(h*x**4+g*x**3+f*x**2+e*x+d)/(x**4-5*x**2+4)**2,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.94 \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2+g x^3+h x^4\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {1}{144} \, {\left (31 \, d - 50 \, e + 76 \, f - 104 \, g + 112 \, h\right )} \log \left (x + 2\right ) - \frac {1}{36} \, {\left (7 \, d - 13 \, e + 19 \, f - 25 \, g + 31 \, h\right )} \log \left (x + 1\right ) - \frac {1}{36} \, {\left (d + e + f + g + h\right )} \log \left (x - 1\right ) + \frac {1}{144} \, {\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h\right )} \log \left (x - 2\right ) - \frac {{\left (3 \, d - 4 \, e + 6 \, f - 10 \, g + 18 \, h\right )} x + 5 \, d - 6 \, e + 8 \, f - 12 \, g + 20 \, h}{12 \, {\left (x^{2} + 3 \, x + 2\right )}} \]

[In]

integrate((x^2-3*x+2)*(h*x^4+g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4)^2,x, algorithm="maxima")

[Out]

1/144*(31*d - 50*e + 76*f - 104*g + 112*h)*log(x + 2) - 1/36*(7*d - 13*e + 19*f - 25*g + 31*h)*log(x + 1) - 1/
36*(d + e + f + g + h)*log(x - 1) + 1/144*(d + 2*e + 4*f + 8*g + 16*h)*log(x - 2) - 1/12*((3*d - 4*e + 6*f - 1
0*g + 18*h)*x + 5*d - 6*e + 8*f - 12*g + 20*h)/(x^2 + 3*x + 2)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.97 \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2+g x^3+h x^4\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {1}{144} \, {\left (31 \, d - 50 \, e + 76 \, f - 104 \, g + 112 \, h\right )} \log \left ({\left | x + 2 \right |}\right ) - \frac {1}{36} \, {\left (7 \, d - 13 \, e + 19 \, f - 25 \, g + 31 \, h\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{36} \, {\left (d + e + f + g + h\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac {1}{144} \, {\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h\right )} \log \left ({\left | x - 2 \right |}\right ) - \frac {{\left (3 \, d - 4 \, e + 6 \, f - 10 \, g + 18 \, h\right )} x + 5 \, d - 6 \, e + 8 \, f - 12 \, g + 20 \, h}{12 \, {\left (x + 2\right )} {\left (x + 1\right )}} \]

[In]

integrate((x^2-3*x+2)*(h*x^4+g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4)^2,x, algorithm="giac")

[Out]

1/144*(31*d - 50*e + 76*f - 104*g + 112*h)*log(abs(x + 2)) - 1/36*(7*d - 13*e + 19*f - 25*g + 31*h)*log(abs(x
+ 1)) - 1/36*(d + e + f + g + h)*log(abs(x - 1)) + 1/144*(d + 2*e + 4*f + 8*g + 16*h)*log(abs(x - 2)) - 1/12*(
(3*d - 4*e + 6*f - 10*g + 18*h)*x + 5*d - 6*e + 8*f - 12*g + 20*h)/((x + 2)*(x + 1))

Mupad [B] (verification not implemented)

Time = 8.24 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.02 \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2+g x^3+h x^4\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=\ln \left (x-2\right )\,\left (\frac {d}{144}+\frac {e}{72}+\frac {f}{36}+\frac {g}{18}+\frac {h}{9}\right )-\ln \left (x-1\right )\,\left (\frac {d}{36}+\frac {e}{36}+\frac {f}{36}+\frac {g}{36}+\frac {h}{36}\right )-\ln \left (x+1\right )\,\left (\frac {7\,d}{36}-\frac {13\,e}{36}+\frac {19\,f}{36}-\frac {25\,g}{36}+\frac {31\,h}{36}\right )-\frac {\frac {5\,d}{12}-\frac {e}{2}+\frac {2\,f}{3}-g+\frac {5\,h}{3}+x\,\left (\frac {d}{4}-\frac {e}{3}+\frac {f}{2}-\frac {5\,g}{6}+\frac {3\,h}{2}\right )}{x^2+3\,x+2}+\ln \left (x+2\right )\,\left (\frac {31\,d}{144}-\frac {25\,e}{72}+\frac {19\,f}{36}-\frac {13\,g}{18}+\frac {7\,h}{9}\right ) \]

[In]

int(((x^2 - 3*x + 2)*(d + e*x + f*x^2 + g*x^3 + h*x^4))/(x^4 - 5*x^2 + 4)^2,x)

[Out]

log(x - 2)*(d/144 + e/72 + f/36 + g/18 + h/9) - log(x - 1)*(d/36 + e/36 + f/36 + g/36 + h/36) - log(x + 1)*((7
*d)/36 - (13*e)/36 + (19*f)/36 - (25*g)/36 + (31*h)/36) - ((5*d)/12 - e/2 + (2*f)/3 - g + (5*h)/3 + x*(d/4 - e
/3 + f/2 - (5*g)/6 + (3*h)/2))/(3*x + x^2 + 2) + log(x + 2)*((31*d)/144 - (25*e)/72 + (19*f)/36 - (13*g)/18 +
(7*h)/9)