3.2373 \(\int \frac{(d+e x) (f+g x)}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=221 \[ \frac{(b+2 c x) \left (c (2 a e g-3 b (d g+e f))+b^2 e g+6 c^2 d f\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac{-x \left (-c (2 a e g+b d g+b e f)+b^2 e g+2 c^2 d f\right )-b (a e g+c d f)+2 a c (d g+e f)}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (c (2 a e g-3 b (d g+e f))+b^2 e g+6 c^2 d f\right )}{\left (b^2-4 a c\right )^{5/2}} \]

[Out]

(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c*(b*e*f + b*d*g
+ 2*a*e*g))*x)/(2*c*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + ((6*c^2*d*f + b^2*e*g
+ c*(2*a*e*g - 3*b*(e*f + d*g)))*(b + 2*c*x))/(2*c*(b^2 - 4*a*c)^2*(a + b*x + c*
x^2)) - (2*(6*c^2*d*f + b^2*e*g + c*(2*a*e*g - 3*b*(e*f + d*g)))*ArcTanh[(b + 2*
c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(5/2)

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Rubi [A]  time = 0.406561, antiderivative size = 219, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{(b+2 c x) \left (2 a c e g+b^2 e g-3 b c (d g+e f)+6 c^2 d f\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac{-x \left (-c (2 a e g+b d g+b e f)+b^2 e g+2 c^2 d f\right )-b (a e g+c d f)+2 a c (d g+e f)}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (2 a c e g+b^2 e g-3 b c (d g+e f)+6 c^2 d f\right )}{\left (b^2-4 a c\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)*(f + g*x))/(a + b*x + c*x^2)^3,x]

[Out]

(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c*(b*e*f + b*d*g
+ 2*a*e*g))*x)/(2*c*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + ((6*c^2*d*f + b^2*e*g
+ 2*a*c*e*g - 3*b*c*(e*f + d*g))*(b + 2*c*x))/(2*c*(b^2 - 4*a*c)^2*(a + b*x + c*
x^2)) - (2*(6*c^2*d*f + b^2*e*g + 2*a*c*e*g - 3*b*c*(e*f + d*g))*ArcTanh[(b + 2*
c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(5/2)

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Rubi in Sympy [A]  time = 38.5507, size = 211, normalized size = 0.95 \[ \frac{2 \left (- 2 a c e g - b^{2} e g + 3 c \left (b \left (d g + e f\right ) - 2 c d f\right )\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} - \frac{\left (b + 2 c x\right ) \left (- 2 a c e g - b^{2} e g + 3 c \left (b \left (d g + e f\right ) - 2 c d f\right )\right )}{2 c \left (- 4 a c + b^{2}\right )^{2} \left (a + b x + c x^{2}\right )} - \frac{- 2 a c \left (d g + e f\right ) + b \left (a e g + c d f\right ) - x \left (- b^{2} e g + b c \left (d g + e f\right ) + 2 c \left (a e g - c d f\right )\right )}{2 c \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)*(g*x+f)/(c*x**2+b*x+a)**3,x)

[Out]

2*(-2*a*c*e*g - b**2*e*g + 3*c*(b*(d*g + e*f) - 2*c*d*f))*atanh((b + 2*c*x)/sqrt
(-4*a*c + b**2))/(-4*a*c + b**2)**(5/2) - (b + 2*c*x)*(-2*a*c*e*g - b**2*e*g + 3
*c*(b*(d*g + e*f) - 2*c*d*f))/(2*c*(-4*a*c + b**2)**2*(a + b*x + c*x**2)) - (-2*
a*c*(d*g + e*f) + b*(a*e*g + c*d*f) - x*(-b**2*e*g + b*c*(d*g + e*f) + 2*c*(a*e*
g - c*d*f)))/(2*c*(-4*a*c + b**2)*(a + b*x + c*x**2)**2)

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Mathematica [A]  time = 0.785193, size = 216, normalized size = 0.98 \[ \frac{1}{2} \left (\frac{(b+2 c x) \left (2 a c e g+b^2 e g-3 b c (d g+e f)+6 c^2 d f\right )}{c \left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac{a b e g-2 a c (d g+e (f+g x))+b^2 e g x+b c (d (f-g x)-e f x)+2 c^2 d f x}{c \left (4 a c-b^2\right ) (a+x (b+c x))^2}+\frac{4 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right ) \left (2 a c e g+b^2 e g-3 b c (d g+e f)+6 c^2 d f\right )}{\left (4 a c-b^2\right )^{5/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((d + e*x)*(f + g*x))/(a + b*x + c*x^2)^3,x]

[Out]

(((6*c^2*d*f + b^2*e*g + 2*a*c*e*g - 3*b*c*(e*f + d*g))*(b + 2*c*x))/(c*(b^2 - 4
*a*c)^2*(a + x*(b + c*x))) + (a*b*e*g + 2*c^2*d*f*x + b^2*e*g*x + b*c*(-(e*f*x)
+ d*(f - g*x)) - 2*a*c*(d*g + e*(f + g*x)))/(c*(-b^2 + 4*a*c)*(a + x*(b + c*x))^
2) + (4*(6*c^2*d*f + b^2*e*g + 2*a*c*e*g - 3*b*c*(e*f + d*g))*ArcTan[(b + 2*c*x)
/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2))/2

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Maple [B]  time = 0.015, size = 591, normalized size = 2.7 \[{\frac{1}{ \left ( c{x}^{2}+bx+a \right ) ^{2}} \left ({\frac{c \left ( 2\,aceg+{b}^{2}eg-3\,bcdg-3\,bcef+6\,{c}^{2}df \right ){x}^{3}}{16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4}}}+{\frac{3\,b \left ( 2\,aceg+{b}^{2}eg-3\,bcdg-3\,bcef+6\,{c}^{2}df \right ){x}^{2}}{32\,{a}^{2}{c}^{2}-16\,a{b}^{2}c+2\,{b}^{4}}}-{\frac{ \left ( 2\,{a}^{2}ceg-5\,a{b}^{2}eg+5\,abcdg+5\,abcef-10\,a{c}^{2}df+{b}^{3}dg+{b}^{3}ef-2\,{b}^{2}cdf \right ) x}{16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4}}}+{\frac{6\,{a}^{2}beg-8\,{a}^{2}cdg-8\,{a}^{2}cef-a{b}^{2}dg-a{b}^{2}ef+10\,abcdf-{b}^{3}df}{32\,{a}^{2}{c}^{2}-16\,a{b}^{2}c+2\,{b}^{4}}} \right ) }+4\,{\frac{aceg}{ \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+2\,{\frac{{b}^{2}eg}{ \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-6\,{\frac{bcdg}{ \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-6\,{\frac{bcef}{ \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+12\,{\frac{{c}^{2}df}{ \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(g*x+f)/(c*x^2+b*x+a)^3,x)

[Out]

(c*(2*a*c*e*g+b^2*e*g-3*b*c*d*g-3*b*c*e*f+6*c^2*d*f)/(16*a^2*c^2-8*a*b^2*c+b^4)*
x^3+3/2*b*(2*a*c*e*g+b^2*e*g-3*b*c*d*g-3*b*c*e*f+6*c^2*d*f)/(16*a^2*c^2-8*a*b^2*
c+b^4)*x^2-(2*a^2*c*e*g-5*a*b^2*e*g+5*a*b*c*d*g+5*a*b*c*e*f-10*a*c^2*d*f+b^3*d*g
+b^3*e*f-2*b^2*c*d*f)/(16*a^2*c^2-8*a*b^2*c+b^4)*x+1/2*(6*a^2*b*e*g-8*a^2*c*d*g-
8*a^2*c*e*f-a*b^2*d*g-a*b^2*e*f+10*a*b*c*d*f-b^3*d*f)/(16*a^2*c^2-8*a*b^2*c+b^4)
)/(c*x^2+b*x+a)^2+4/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b
)/(4*a*c-b^2)^(1/2))*a*c*e*g+2/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arct
an((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^2*e*g-6/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)
^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*c*d*g-6/(16*a^2*c^2-8*a*b^2*c+b^4)/
(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*c*e*f+12/(16*a^2*c^2-8*a
*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c^2*d*f

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(g*x + f)/(c*x^2 + b*x + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.302809, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(g*x + f)/(c*x^2 + b*x + a)^3,x, algorithm="fricas")

[Out]

[1/2*(2*((3*(2*c^4*d - b*c^3*e)*f - (3*b*c^3*d - (b^2*c^2 + 2*a*c^3)*e)*g)*x^4 +
2*(3*(2*b*c^3*d - b^2*c^2*e)*f - (3*b^2*c^2*d - (b^3*c + 2*a*b*c^2)*e)*g)*x^3 +
(3*(2*(b^2*c^2 + 2*a*c^3)*d - (b^3*c + 2*a*b*c^2)*e)*f - (3*(b^3*c + 2*a*b*c^2)
*d - (b^4 + 4*a*b^2*c + 4*a^2*c^2)*e)*g)*x^2 + 3*(2*a^2*c^2*d - a^2*b*c*e)*f - (
3*a^2*b*c*d - (a^2*b^2 + 2*a^3*c)*e)*g + 2*(3*(2*a*b*c^2*d - a*b^2*c*e)*f - (3*a
*b^2*c*d - (a*b^3 + 2*a^2*b*c)*e)*g)*x)*log(-(b^3 - 4*a*b*c + 2*(b^2*c - 4*a*c^2
)*x - (2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(c*x^2 + b*x + a))
+ (2*(3*(2*c^3*d - b*c^2*e)*f - (3*b*c^2*d - (b^2*c + 2*a*c^2)*e)*g)*x^3 + 3*(3*
(2*b*c^2*d - b^2*c*e)*f - (3*b^2*c*d - (b^3 + 2*a*b*c)*e)*g)*x^2 - ((b^3 - 10*a*
b*c)*d + (a*b^2 + 8*a^2*c)*e)*f + (6*a^2*b*e - (a*b^2 + 8*a^2*c)*d)*g + 2*((2*(b
^2*c + 5*a*c^2)*d - (b^3 + 5*a*b*c)*e)*f - ((b^3 + 5*a*b*c)*d - (5*a*b^2 - 2*a^2
*c)*e)*g)*x)*sqrt(b^2 - 4*a*c))/((a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^4*c^2
- 8*a*b^2*c^3 + 16*a^2*c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (
b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x)*sq
rt(b^2 - 4*a*c)), 1/2*(4*((3*(2*c^4*d - b*c^3*e)*f - (3*b*c^3*d - (b^2*c^2 + 2*a
*c^3)*e)*g)*x^4 + 2*(3*(2*b*c^3*d - b^2*c^2*e)*f - (3*b^2*c^2*d - (b^3*c + 2*a*b
*c^2)*e)*g)*x^3 + (3*(2*(b^2*c^2 + 2*a*c^3)*d - (b^3*c + 2*a*b*c^2)*e)*f - (3*(b
^3*c + 2*a*b*c^2)*d - (b^4 + 4*a*b^2*c + 4*a^2*c^2)*e)*g)*x^2 + 3*(2*a^2*c^2*d -
a^2*b*c*e)*f - (3*a^2*b*c*d - (a^2*b^2 + 2*a^3*c)*e)*g + 2*(3*(2*a*b*c^2*d - a*
b^2*c*e)*f - (3*a*b^2*c*d - (a*b^3 + 2*a^2*b*c)*e)*g)*x)*arctan(-sqrt(-b^2 + 4*a
*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (2*(3*(2*c^3*d - b*c^2*e)*f - (3*b*c^2*d - (b^2
*c + 2*a*c^2)*e)*g)*x^3 + 3*(3*(2*b*c^2*d - b^2*c*e)*f - (3*b^2*c*d - (b^3 + 2*a
*b*c)*e)*g)*x^2 - ((b^3 - 10*a*b*c)*d + (a*b^2 + 8*a^2*c)*e)*f + (6*a^2*b*e - (a
*b^2 + 8*a^2*c)*d)*g + 2*((2*(b^2*c + 5*a*c^2)*d - (b^3 + 5*a*b*c)*e)*f - ((b^3
+ 5*a*b*c)*d - (5*a*b^2 - 2*a^2*c)*e)*g)*x)*sqrt(-b^2 + 4*a*c))/((a^2*b^4 - 8*a^
3*b^2*c + 16*a^4*c^2 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^4 + 2*(b^5*c - 8*a
*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8
*a^2*b^3*c + 16*a^3*b*c^2)*x)*sqrt(-b^2 + 4*a*c))]

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Sympy [A]  time = 66.7002, size = 1234, normalized size = 5.58 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)*(g*x+f)/(c*x**2+b*x+a)**3,x)

[Out]

-sqrt(-1/(4*a*c - b**2)**5)*(2*a*c*e*g + b**2*e*g - 3*b*c*d*g - 3*b*c*e*f + 6*c*
*2*d*f)*log(x + (-64*a**3*c**3*sqrt(-1/(4*a*c - b**2)**5)*(2*a*c*e*g + b**2*e*g
- 3*b*c*d*g - 3*b*c*e*f + 6*c**2*d*f) + 48*a**2*b**2*c**2*sqrt(-1/(4*a*c - b**2)
**5)*(2*a*c*e*g + b**2*e*g - 3*b*c*d*g - 3*b*c*e*f + 6*c**2*d*f) - 12*a*b**4*c*s
qrt(-1/(4*a*c - b**2)**5)*(2*a*c*e*g + b**2*e*g - 3*b*c*d*g - 3*b*c*e*f + 6*c**2
*d*f) + 2*a*b*c*e*g + b**6*sqrt(-1/(4*a*c - b**2)**5)*(2*a*c*e*g + b**2*e*g - 3*
b*c*d*g - 3*b*c*e*f + 6*c**2*d*f) + b**3*e*g - 3*b**2*c*d*g - 3*b**2*c*e*f + 6*b
*c**2*d*f)/(4*a*c**2*e*g + 2*b**2*c*e*g - 6*b*c**2*d*g - 6*b*c**2*e*f + 12*c**3*
d*f)) + sqrt(-1/(4*a*c - b**2)**5)*(2*a*c*e*g + b**2*e*g - 3*b*c*d*g - 3*b*c*e*f
+ 6*c**2*d*f)*log(x + (64*a**3*c**3*sqrt(-1/(4*a*c - b**2)**5)*(2*a*c*e*g + b**
2*e*g - 3*b*c*d*g - 3*b*c*e*f + 6*c**2*d*f) - 48*a**2*b**2*c**2*sqrt(-1/(4*a*c -
b**2)**5)*(2*a*c*e*g + b**2*e*g - 3*b*c*d*g - 3*b*c*e*f + 6*c**2*d*f) + 12*a*b*
*4*c*sqrt(-1/(4*a*c - b**2)**5)*(2*a*c*e*g + b**2*e*g - 3*b*c*d*g - 3*b*c*e*f +
6*c**2*d*f) + 2*a*b*c*e*g - b**6*sqrt(-1/(4*a*c - b**2)**5)*(2*a*c*e*g + b**2*e*
g - 3*b*c*d*g - 3*b*c*e*f + 6*c**2*d*f) + b**3*e*g - 3*b**2*c*d*g - 3*b**2*c*e*f
+ 6*b*c**2*d*f)/(4*a*c**2*e*g + 2*b**2*c*e*g - 6*b*c**2*d*g - 6*b*c**2*e*f + 12
*c**3*d*f)) + (6*a**2*b*e*g - 8*a**2*c*d*g - 8*a**2*c*e*f - a*b**2*d*g - a*b**2*
e*f + 10*a*b*c*d*f - b**3*d*f + x**3*(4*a*c**2*e*g + 2*b**2*c*e*g - 6*b*c**2*d*g
- 6*b*c**2*e*f + 12*c**3*d*f) + x**2*(6*a*b*c*e*g + 3*b**3*e*g - 9*b**2*c*d*g -
9*b**2*c*e*f + 18*b*c**2*d*f) + x*(-4*a**2*c*e*g + 10*a*b**2*e*g - 10*a*b*c*d*g
- 10*a*b*c*e*f + 20*a*c**2*d*f - 2*b**3*d*g - 2*b**3*e*f + 4*b**2*c*d*f))/(32*a
**4*c**2 - 16*a**3*b**2*c + 2*a**2*b**4 + x**4*(32*a**2*c**4 - 16*a*b**2*c**3 +
2*b**4*c**2) + x**3*(64*a**2*b*c**3 - 32*a*b**3*c**2 + 4*b**5*c) + x**2*(64*a**3
*c**3 - 12*a*b**4*c + 2*b**6) + x*(64*a**3*b*c**2 - 32*a**2*b**3*c + 4*a*b**5))

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GIAC/XCAS [A]  time = 0.361359, size = 498, normalized size = 2.25 \[ \frac{2 \,{\left (6 \, c^{2} d f - 3 \, b c d g - 3 \, b c f e + b^{2} g e + 2 \, a c g e\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{12 \, c^{3} d f x^{3} - 6 \, b c^{2} d g x^{3} - 6 \, b c^{2} f x^{3} e + 2 \, b^{2} c g x^{3} e + 4 \, a c^{2} g x^{3} e + 18 \, b c^{2} d f x^{2} - 9 \, b^{2} c d g x^{2} - 9 \, b^{2} c f x^{2} e + 3 \, b^{3} g x^{2} e + 6 \, a b c g x^{2} e + 4 \, b^{2} c d f x + 20 \, a c^{2} d f x - 2 \, b^{3} d g x - 10 \, a b c d g x - 2 \, b^{3} f x e - 10 \, a b c f x e + 10 \, a b^{2} g x e - 4 \, a^{2} c g x e - b^{3} d f + 10 \, a b c d f - a b^{2} d g - 8 \, a^{2} c d g - a b^{2} f e - 8 \, a^{2} c f e + 6 \, a^{2} b g e}{2 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}{\left (c x^{2} + b x + a\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(g*x + f)/(c*x^2 + b*x + a)^3,x, algorithm="giac")

[Out]

2*(6*c^2*d*f - 3*b*c*d*g - 3*b*c*f*e + b^2*g*e + 2*a*c*g*e)*arctan((2*c*x + b)/s
qrt(-b^2 + 4*a*c))/((b^4 - 8*a*b^2*c + 16*a^2*c^2)*sqrt(-b^2 + 4*a*c)) + 1/2*(12
*c^3*d*f*x^3 - 6*b*c^2*d*g*x^3 - 6*b*c^2*f*x^3*e + 2*b^2*c*g*x^3*e + 4*a*c^2*g*x
^3*e + 18*b*c^2*d*f*x^2 - 9*b^2*c*d*g*x^2 - 9*b^2*c*f*x^2*e + 3*b^3*g*x^2*e + 6*
a*b*c*g*x^2*e + 4*b^2*c*d*f*x + 20*a*c^2*d*f*x - 2*b^3*d*g*x - 10*a*b*c*d*g*x -
2*b^3*f*x*e - 10*a*b*c*f*x*e + 10*a*b^2*g*x*e - 4*a^2*c*g*x*e - b^3*d*f + 10*a*b
*c*d*f - a*b^2*d*g - 8*a^2*c*d*g - a*b^2*f*e - 8*a^2*c*f*e + 6*a^2*b*g*e)/((b^4
- 8*a*b^2*c + 16*a^2*c^2)*(c*x^2 + b*x + a)^2)