3.51 \(\int \frac{d+e x+f x^2+g x^3+h x^4+i x^5}{\left (1+x^2+x^4\right )^3} \, dx\)

Optimal. Leaf size=269 \[ -\frac{1}{32} \log \left (x^2-x+1\right ) (9 d-4 f+3 h)+\frac{1}{32} \log \left (x^2+x+1\right ) (9 d-4 f+3 h)+\frac{x \left (x^2 (-(7 d-7 f+4 h))+2 d+3 f-h\right )}{24 \left (x^4+x^2+1\right )}+\frac{x \left (x^2 (-(d-2 f+h))+d+f-2 h\right )}{12 \left (x^4+x^2+1\right )^2}-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right ) (13 d+2 f+h)}{48 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) (13 d+2 f+h)}{48 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right ) (2 e-g+i)}{3 \sqrt{3}}+\frac{\left (2 x^2+1\right ) (2 e-g+i)}{12 \left (x^4+x^2+1\right )}+\frac{x^2 (2 e-g-i)+e-2 g+i}{12 \left (x^4+x^2+1\right )^2} \]

[Out]

(x*(d + f - 2*h - (d - 2*f + h)*x^2))/(12*(1 + x^2 + x^4)^2) + (e - 2*g + i + (2
*e - g - i)*x^2)/(12*(1 + x^2 + x^4)^2) + ((2*e - g + i)*(1 + 2*x^2))/(12*(1 + x
^2 + x^4)) + (x*(2*d + 3*f - h - (7*d - 7*f + 4*h)*x^2))/(24*(1 + x^2 + x^4)) -
((13*d + 2*f + h)*ArcTan[(1 - 2*x)/Sqrt[3]])/(48*Sqrt[3]) + ((13*d + 2*f + h)*Ar
cTan[(1 + 2*x)/Sqrt[3]])/(48*Sqrt[3]) + ((2*e - g + i)*ArcTan[(1 + 2*x^2)/Sqrt[3
]])/(3*Sqrt[3]) - ((9*d - 4*f + 3*h)*Log[1 - x + x^2])/32 + ((9*d - 4*f + 3*h)*L
og[1 + x + x^2])/32

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Rubi [A]  time = 0.629288, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 12, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{1}{32} \log \left (x^2-x+1\right ) (9 d-4 f+3 h)+\frac{1}{32} \log \left (x^2+x+1\right ) (9 d-4 f+3 h)+\frac{x \left (x^2 (-(7 d-7 f+4 h))+2 d+3 f-h\right )}{24 \left (x^4+x^2+1\right )}+\frac{x \left (x^2 (-(d-2 f+h))+d+f-2 h\right )}{12 \left (x^4+x^2+1\right )^2}-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right ) (13 d+2 f+h)}{48 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) (13 d+2 f+h)}{48 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right ) (2 e-g+i)}{3 \sqrt{3}}+\frac{\left (2 x^2+1\right ) (2 e-g+i)}{12 \left (x^4+x^2+1\right )}+\frac{x^2 (2 e-g-i)+e-2 g+i}{12 \left (x^4+x^2+1\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5)/(1 + x^2 + x^4)^3,x]

[Out]

(x*(d + f - 2*h - (d - 2*f + h)*x^2))/(12*(1 + x^2 + x^4)^2) + (e - 2*g + i + (2
*e - g - i)*x^2)/(12*(1 + x^2 + x^4)^2) + ((2*e - g + i)*(1 + 2*x^2))/(12*(1 + x
^2 + x^4)) + (x*(2*d + 3*f - h - (7*d - 7*f + 4*h)*x^2))/(24*(1 + x^2 + x^4)) -
((13*d + 2*f + h)*ArcTan[(1 - 2*x)/Sqrt[3]])/(48*Sqrt[3]) + ((13*d + 2*f + h)*Ar
cTan[(1 + 2*x)/Sqrt[3]])/(48*Sqrt[3]) + ((2*e - g + i)*ArcTan[(1 + 2*x^2)/Sqrt[3
]])/(3*Sqrt[3]) - ((9*d - 4*f + 3*h)*Log[1 - x + x^2])/32 + ((9*d - 4*f + 3*h)*L
og[1 + x + x^2])/32

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Rubi in Sympy [A]  time = 115.85, size = 240, normalized size = 0.89 \[ \frac{x \left (6 d + 9 f - 3 h - x^{3} \left (18 e - 18 g + 12\right ) - x^{2} \left (21 d - 21 f + 12 h\right ) + x \left (6 e + 6 g\right )\right )}{72 \left (x^{4} + x^{2} + 1\right )} + \frac{x \left (d + f - 2 h - x^{3} \left (e - 2 g + 1\right ) - x^{2} \left (d - 2 f + h\right ) - x \left (- e - g + 2\right )\right )}{12 \left (x^{4} + x^{2} + 1\right )^{2}} - \left (\frac{9 d}{32} - \frac{f}{8} + \frac{3 h}{32}\right ) \log{\left (x^{2} - x + 1 \right )} + \left (\frac{9 d}{32} - \frac{f}{8} + \frac{3 h}{32}\right ) \log{\left (x^{2} + x + 1 \right )} + \frac{\sqrt{3} \left (13 d + 2 f + h\right ) \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{144} + \frac{\sqrt{3} \left (13 d + 2 f + h\right ) \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{144} + \frac{\sqrt{3} \left (2 e - g + 1\right ) \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{2}}{3} + \frac{1}{3}\right ) \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((i*x**5+h*x**4+g*x**3+f*x**2+e*x+d)/(x**4+x**2+1)**3,x)

[Out]

x*(6*d + 9*f - 3*h - x**3*(18*e - 18*g + 12) - x**2*(21*d - 21*f + 12*h) + x*(6*
e + 6*g))/(72*(x**4 + x**2 + 1)) + x*(d + f - 2*h - x**3*(e - 2*g + 1) - x**2*(d
- 2*f + h) - x*(-e - g + 2))/(12*(x**4 + x**2 + 1)**2) - (9*d/32 - f/8 + 3*h/32
)*log(x**2 - x + 1) + (9*d/32 - f/8 + 3*h/32)*log(x**2 + x + 1) + sqrt(3)*(13*d
+ 2*f + h)*atan(sqrt(3)*(2*x/3 - 1/3))/144 + sqrt(3)*(13*d + 2*f + h)*atan(sqrt(
3)*(2*x/3 + 1/3))/144 + sqrt(3)*(2*e - g + 1)*atan(sqrt(3)*(2*x**2/3 + 1/3))/9

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Mathematica [C]  time = 2.22492, size = 325, normalized size = 1.21 \[ \frac{1}{144} \left (\frac{12 \left (-d x^3+d x+2 e x^2+e+2 f x^3+f x-g \left (x^2+2\right )-h x^3-2 h x-i x^2+i\right )}{\left (x^4+x^2+1\right )^2}+\frac{6 \left (-7 d x^3+2 d x+e \left (8 x^2+4\right )+7 f x^3+3 f x-2 g \left (2 x^2+1\right )-4 h x^3-h x+4 i x^2+2 i\right )}{x^4+x^2+1}-\frac{\tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}-i\right ) x\right ) \left (\left (7 \sqrt{3}-47 i\right ) d+\left (-7 \sqrt{3}+17 i\right ) f+2 \left (2 \sqrt{3}-7 i\right ) h\right )}{\sqrt{\frac{1}{6} \left (1+i \sqrt{3}\right )}}-\frac{\tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}+i\right ) x\right ) \left (\left (7 \sqrt{3}+47 i\right ) d-\left (7 \sqrt{3}+17 i\right ) f+2 \left (2 \sqrt{3}+7 i\right ) h\right )}{\sqrt{\frac{1}{6} \left (1-i \sqrt{3}\right )}}-16 \sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3}}{2 x^2+1}\right ) (2 e-g+i)\right ) \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5)/(1 + x^2 + x^4)^3,x]

[Out]

((12*(e + i + d*x + f*x - 2*h*x + 2*e*x^2 - i*x^2 - d*x^3 + 2*f*x^3 - h*x^3 - g*
(2 + x^2)))/(1 + x^2 + x^4)^2 + (6*(2*i + 2*d*x + 3*f*x - h*x + 4*i*x^2 - 7*d*x^
3 + 7*f*x^3 - 4*h*x^3 - 2*g*(1 + 2*x^2) + e*(4 + 8*x^2)))/(1 + x^2 + x^4) - (((-
47*I + 7*Sqrt[3])*d + (17*I - 7*Sqrt[3])*f + 2*(-7*I + 2*Sqrt[3])*h)*ArcTan[((-I
+ Sqrt[3])*x)/2])/Sqrt[(1 + I*Sqrt[3])/6] - (((47*I + 7*Sqrt[3])*d - (17*I + 7*
Sqrt[3])*f + 2*(7*I + 2*Sqrt[3])*h)*ArcTan[((I + Sqrt[3])*x)/2])/Sqrt[(1 - I*Sqr
t[3])/6] - 16*Sqrt[3]*(2*e - g + i)*ArcTan[Sqrt[3]/(1 + 2*x^2)])/144

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Maple [A]  time = 0.025, size = 454, normalized size = 1.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(x^4+x^2+1)^3,x)

[Out]

1/16*((-7/3*d+7/3*f-4/3*h-4/3*e-1/3*g+1/3*i)*x^3+(-6*d+4*f-2*h-2*g+2*i)*x^2+(-20
/3*d+13/3*f-5/3*h+1/3*e-8/3*g+7/3*i)*x-4*d+4/3*f+2*e-2*g+4/3*i)/(x^2+x+1)^2+9/32
*d*ln(x^2+x+1)-1/8*ln(x^2+x+1)*f+3/32*ln(x^2+x+1)*h+13/144*d*arctan(1/3*(1+2*x)*
3^(1/2))*3^(1/2)-2/9*3^(1/2)*arctan(1/3*(1+2*x)*3^(1/2))*e+1/72*3^(1/2)*arctan(1
/3*(1+2*x)*3^(1/2))*f+1/9*3^(1/2)*arctan(1/3*(1+2*x)*3^(1/2))*g+1/144*3^(1/2)*ar
ctan(1/3*(1+2*x)*3^(1/2))*h-1/9*3^(1/2)*arctan(1/3*(1+2*x)*3^(1/2))*i-1/16*((7/3
*d-7/3*f+4/3*h-4/3*e-1/3*g+1/3*i)*x^3+(-6*d+4*f-2*h+2*g-2*i)*x^2+(20/3*d-13/3*f+
5/3*h+1/3*e-8/3*g+7/3*i)*x-4*d+4/3*f-2*e+2*g-4/3*i)/(x^2-x+1)^2-9/32*d*ln(x^2-x+
1)+1/8*ln(x^2-x+1)*f-3/32*ln(x^2-x+1)*h+13/144*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2
))*d+2/9*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))*e+1/72*3^(1/2)*arctan(1/3*(2*x-1)*3
^(1/2))*f-1/9*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))*g+1/144*3^(1/2)*arctan(1/3*(2*
x-1)*3^(1/2))*h+1/9*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))*i

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Maxima [A]  time = 0.778989, size = 309, normalized size = 1.15 \[ \frac{1}{144} \, \sqrt{3}{\left (13 \, d - 32 \, e + 2 \, f + 16 \, g + h - 16 \, i\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{144} \, \sqrt{3}{\left (13 \, d + 32 \, e + 2 \, f - 16 \, g + h + 16 \, i\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{32} \,{\left (9 \, d - 4 \, f + 3 \, h\right )} \log \left (x^{2} + x + 1\right ) - \frac{1}{32} \,{\left (9 \, d - 4 \, f + 3 \, h\right )} \log \left (x^{2} - x + 1\right ) - \frac{{\left (7 \, d - 7 \, f + 4 \, h\right )} x^{7} - 4 \,{\left (2 \, e - g + i\right )} x^{6} + 5 \,{\left (d - 2 \, f + h\right )} x^{5} - 6 \,{\left (2 \, e - g + i\right )} x^{4} + 7 \,{\left (d - 2 \, f + h\right )} x^{3} - 4 \,{\left (4 \, e - 2 \, g + i\right )} x^{2} -{\left (4 \, d + 5 \, f - 5 \, h\right )} x - 6 \, e + 6 \, g - 4 \, i}{24 \,{\left (x^{8} + 2 \, x^{6} + 3 \, x^{4} + 2 \, x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((i*x^5 + h*x^4 + g*x^3 + f*x^2 + e*x + d)/(x^4 + x^2 + 1)^3,x, algorithm="maxima")

[Out]

1/144*sqrt(3)*(13*d - 32*e + 2*f + 16*g + h - 16*i)*arctan(1/3*sqrt(3)*(2*x + 1)
) + 1/144*sqrt(3)*(13*d + 32*e + 2*f - 16*g + h + 16*i)*arctan(1/3*sqrt(3)*(2*x
- 1)) + 1/32*(9*d - 4*f + 3*h)*log(x^2 + x + 1) - 1/32*(9*d - 4*f + 3*h)*log(x^2
- x + 1) - 1/24*((7*d - 7*f + 4*h)*x^7 - 4*(2*e - g + i)*x^6 + 5*(d - 2*f + h)*
x^5 - 6*(2*e - g + i)*x^4 + 7*(d - 2*f + h)*x^3 - 4*(4*e - 2*g + i)*x^2 - (4*d +
5*f - 5*h)*x - 6*e + 6*g - 4*i)/(x^8 + 2*x^6 + 3*x^4 + 2*x^2 + 1)

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Fricas [A]  time = 7.22812, size = 714, normalized size = 2.65 \[ \frac{\sqrt{3}{\left (3 \, \sqrt{3}{\left ({\left (9 \, d - 4 \, f + 3 \, h\right )} x^{8} + 2 \,{\left (9 \, d - 4 \, f + 3 \, h\right )} x^{6} + 3 \,{\left (9 \, d - 4 \, f + 3 \, h\right )} x^{4} + 2 \,{\left (9 \, d - 4 \, f + 3 \, h\right )} x^{2} + 9 \, d - 4 \, f + 3 \, h\right )} \log \left (x^{2} + x + 1\right ) - 3 \, \sqrt{3}{\left ({\left (9 \, d - 4 \, f + 3 \, h\right )} x^{8} + 2 \,{\left (9 \, d - 4 \, f + 3 \, h\right )} x^{6} + 3 \,{\left (9 \, d - 4 \, f + 3 \, h\right )} x^{4} + 2 \,{\left (9 \, d - 4 \, f + 3 \, h\right )} x^{2} + 9 \, d - 4 \, f + 3 \, h\right )} \log \left (x^{2} - x + 1\right ) + 2 \,{\left ({\left (13 \, d - 32 \, e + 2 \, f + 16 \, g + h - 16 \, i\right )} x^{8} + 2 \,{\left (13 \, d - 32 \, e + 2 \, f + 16 \, g + h - 16 \, i\right )} x^{6} + 3 \,{\left (13 \, d - 32 \, e + 2 \, f + 16 \, g + h - 16 \, i\right )} x^{4} + 2 \,{\left (13 \, d - 32 \, e + 2 \, f + 16 \, g + h - 16 \, i\right )} x^{2} + 13 \, d - 32 \, e + 2 \, f + 16 \, g + h - 16 \, i\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + 2 \,{\left ({\left (13 \, d + 32 \, e + 2 \, f - 16 \, g + h + 16 \, i\right )} x^{8} + 2 \,{\left (13 \, d + 32 \, e + 2 \, f - 16 \, g + h + 16 \, i\right )} x^{6} + 3 \,{\left (13 \, d + 32 \, e + 2 \, f - 16 \, g + h + 16 \, i\right )} x^{4} + 2 \,{\left (13 \, d + 32 \, e + 2 \, f - 16 \, g + h + 16 \, i\right )} x^{2} + 13 \, d + 32 \, e + 2 \, f - 16 \, g + h + 16 \, i\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - 4 \, \sqrt{3}{\left ({\left (7 \, d - 7 \, f + 4 \, h\right )} x^{7} - 4 \,{\left (2 \, e - g + i\right )} x^{6} + 5 \,{\left (d - 2 \, f + h\right )} x^{5} - 6 \,{\left (2 \, e - g + i\right )} x^{4} + 7 \,{\left (d - 2 \, f + h\right )} x^{3} - 4 \,{\left (4 \, e - 2 \, g + i\right )} x^{2} -{\left (4 \, d + 5 \, f - 5 \, h\right )} x - 6 \, e + 6 \, g - 4 \, i\right )}\right )}}{288 \,{\left (x^{8} + 2 \, x^{6} + 3 \, x^{4} + 2 \, x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((i*x^5 + h*x^4 + g*x^3 + f*x^2 + e*x + d)/(x^4 + x^2 + 1)^3,x, algorithm="fricas")

[Out]

1/288*sqrt(3)*(3*sqrt(3)*((9*d - 4*f + 3*h)*x^8 + 2*(9*d - 4*f + 3*h)*x^6 + 3*(9
*d - 4*f + 3*h)*x^4 + 2*(9*d - 4*f + 3*h)*x^2 + 9*d - 4*f + 3*h)*log(x^2 + x + 1
) - 3*sqrt(3)*((9*d - 4*f + 3*h)*x^8 + 2*(9*d - 4*f + 3*h)*x^6 + 3*(9*d - 4*f +
3*h)*x^4 + 2*(9*d - 4*f + 3*h)*x^2 + 9*d - 4*f + 3*h)*log(x^2 - x + 1) + 2*((13*
d - 32*e + 2*f + 16*g + h - 16*i)*x^8 + 2*(13*d - 32*e + 2*f + 16*g + h - 16*i)*
x^6 + 3*(13*d - 32*e + 2*f + 16*g + h - 16*i)*x^4 + 2*(13*d - 32*e + 2*f + 16*g
+ h - 16*i)*x^2 + 13*d - 32*e + 2*f + 16*g + h - 16*i)*arctan(1/3*sqrt(3)*(2*x +
1)) + 2*((13*d + 32*e + 2*f - 16*g + h + 16*i)*x^8 + 2*(13*d + 32*e + 2*f - 16*
g + h + 16*i)*x^6 + 3*(13*d + 32*e + 2*f - 16*g + h + 16*i)*x^4 + 2*(13*d + 32*e
+ 2*f - 16*g + h + 16*i)*x^2 + 13*d + 32*e + 2*f - 16*g + h + 16*i)*arctan(1/3*
sqrt(3)*(2*x - 1)) - 4*sqrt(3)*((7*d - 7*f + 4*h)*x^7 - 4*(2*e - g + i)*x^6 + 5*
(d - 2*f + h)*x^5 - 6*(2*e - g + i)*x^4 + 7*(d - 2*f + h)*x^3 - 4*(4*e - 2*g + i
)*x^2 - (4*d + 5*f - 5*h)*x - 6*e + 6*g - 4*i))/(x^8 + 2*x^6 + 3*x^4 + 2*x^2 + 1
)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((i*x**5+h*x**4+g*x**3+f*x**2+e*x+d)/(x**4+x**2+1)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.267752, size = 344, normalized size = 1.28 \[ \frac{1}{144} \, \sqrt{3}{\left (13 \, d + 2 \, f + 16 \, g + h - 16 \, i - 32 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{144} \, \sqrt{3}{\left (13 \, d + 2 \, f - 16 \, g + h + 16 \, i + 32 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{32} \,{\left (9 \, d - 4 \, f + 3 \, h\right )}{\rm ln}\left (x^{2} + x + 1\right ) - \frac{1}{32} \,{\left (9 \, d - 4 \, f + 3 \, h\right )}{\rm ln}\left (x^{2} - x + 1\right ) - \frac{7 \, d x^{7} - 7 \, f x^{7} + 4 \, h x^{7} + 4 \, g x^{6} - 4 \, i x^{6} - 8 \, x^{6} e + 5 \, d x^{5} - 10 \, f x^{5} + 5 \, h x^{5} + 6 \, g x^{4} - 6 \, i x^{4} - 12 \, x^{4} e + 7 \, d x^{3} - 14 \, f x^{3} + 7 \, h x^{3} + 8 \, g x^{2} - 4 \, i x^{2} - 16 \, x^{2} e - 4 \, d x - 5 \, f x + 5 \, h x + 6 \, g - 4 \, i - 6 \, e}{24 \,{\left (x^{4} + x^{2} + 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((i*x^5 + h*x^4 + g*x^3 + f*x^2 + e*x + d)/(x^4 + x^2 + 1)^3,x, algorithm="giac")

[Out]

1/144*sqrt(3)*(13*d + 2*f + 16*g + h - 16*i - 32*e)*arctan(1/3*sqrt(3)*(2*x + 1)
) + 1/144*sqrt(3)*(13*d + 2*f - 16*g + h + 16*i + 32*e)*arctan(1/3*sqrt(3)*(2*x
- 1)) + 1/32*(9*d - 4*f + 3*h)*ln(x^2 + x + 1) - 1/32*(9*d - 4*f + 3*h)*ln(x^2 -
x + 1) - 1/24*(7*d*x^7 - 7*f*x^7 + 4*h*x^7 + 4*g*x^6 - 4*i*x^6 - 8*x^6*e + 5*d*
x^5 - 10*f*x^5 + 5*h*x^5 + 6*g*x^4 - 6*i*x^4 - 12*x^4*e + 7*d*x^3 - 14*f*x^3 + 7
*h*x^3 + 8*g*x^2 - 4*i*x^2 - 16*x^2*e - 4*d*x - 5*f*x + 5*h*x + 6*g - 4*i - 6*e)
/(x^4 + x^2 + 1)^2