3.125 \(\int \frac{x^9 \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx\)

Optimal. Leaf size=254 \[ \frac{\left (-12 a^2 A c^3+30 a^2 b B c^2-10 a b^3 B c+b^5 B\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^3 \left (b^2-4 a c\right )^{5/2}}-\frac{x^2 \left (x^2 \left (16 a^2 B c^2+6 a A b c^2-15 a b^2 B c+2 b^4 B\right )+2 a \left (6 a A c^2-7 a b B c+b^3 B\right )\right )}{4 c^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{x^6 \left (x^2 \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{4 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{B \log \left (a+b x^2+c x^4\right )}{4 c^3} \]

[Out]

-(x^6*(a*(b*B - 2*A*c) + (b^2*B - A*b*c - 2*a*B*c)*x^2))/(4*c*(b^2 - 4*a*c)*(a +
b*x^2 + c*x^4)^2) - (x^2*(2*a*(b^3*B - 7*a*b*B*c + 6*a*A*c^2) + (2*b^4*B - 15*a
*b^2*B*c + 6*a*A*b*c^2 + 16*a^2*B*c^2)*x^2))/(4*c^2*(b^2 - 4*a*c)^2*(a + b*x^2 +
c*x^4)) + ((b^5*B - 10*a*b^3*B*c + 30*a^2*b*B*c^2 - 12*a^2*A*c^3)*ArcTanh[(b +
2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^3*(b^2 - 4*a*c)^(5/2)) + (B*Log[a + b*x^2 + c*
x^4])/(4*c^3)

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Rubi [A]  time = 0.802426, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{\left (-12 a^2 A c^3+30 a^2 b B c^2-10 a b^3 B c+b^5 B\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^3 \left (b^2-4 a c\right )^{5/2}}-\frac{x^2 \left (x^2 \left (16 a^2 B c^2+6 a A b c^2-15 a b^2 B c+2 b^4 B\right )+2 a \left (6 a A c^2-7 a b B c+b^3 B\right )\right )}{4 c^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{x^6 \left (x^2 \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{4 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{B \log \left (a+b x^2+c x^4\right )}{4 c^3} \]

Antiderivative was successfully verified.

[In]  Int[(x^9*(A + B*x^2))/(a + b*x^2 + c*x^4)^3,x]

[Out]

-(x^6*(a*(b*B - 2*A*c) + (b^2*B - A*b*c - 2*a*B*c)*x^2))/(4*c*(b^2 - 4*a*c)*(a +
b*x^2 + c*x^4)^2) - (x^2*(2*a*(b^3*B - 7*a*b*B*c + 6*a*A*c^2) + (2*b^4*B - 15*a
*b^2*B*c + 6*a*A*b*c^2 + 16*a^2*B*c^2)*x^2))/(4*c^2*(b^2 - 4*a*c)^2*(a + b*x^2 +
c*x^4)) + ((b^5*B - 10*a*b^3*B*c + 30*a^2*b*B*c^2 - 12*a^2*A*c^3)*ArcTanh[(b +
2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^3*(b^2 - 4*a*c)^(5/2)) + (B*Log[a + b*x^2 + c*
x^4])/(4*c^3)

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Rubi in Sympy [A]  time = 94.6042, size = 252, normalized size = 0.99 \[ \frac{B \log{\left (a + b x^{2} + c x^{4} \right )}}{4 c^{3}} + \frac{x^{6} \left (a \left (2 A c - B b\right ) - x^{2} \left (- A b c - 2 B a c + B b^{2}\right )\right )}{4 c \left (- 4 a c + b^{2}\right ) \left (a + b x^{2} + c x^{4}\right )^{2}} - \frac{x^{2} \left (2 a \left (6 A a c^{2} - 7 B a b c + B b^{3}\right ) + x^{2} \left (6 A a b c^{2} + 16 B a^{2} c^{2} - 15 B a b^{2} c + 2 B b^{4}\right )\right )}{4 c^{2} \left (- 4 a c + b^{2}\right )^{2} \left (a + b x^{2} + c x^{4}\right )} + \frac{\left (- 12 A a^{2} c^{3} + 30 B a^{2} b c^{2} - 10 B a b^{3} c + B b^{5}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 c^{3} \left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9*(B*x**2+A)/(c*x**4+b*x**2+a)**3,x)

[Out]

B*log(a + b*x**2 + c*x**4)/(4*c**3) + x**6*(a*(2*A*c - B*b) - x**2*(-A*b*c - 2*B
*a*c + B*b**2))/(4*c*(-4*a*c + b**2)*(a + b*x**2 + c*x**4)**2) - x**2*(2*a*(6*A*
a*c**2 - 7*B*a*b*c + B*b**3) + x**2*(6*A*a*b*c**2 + 16*B*a**2*c**2 - 15*B*a*b**2
*c + 2*B*b**4))/(4*c**2*(-4*a*c + b**2)**2*(a + b*x**2 + c*x**4)) + (-12*A*a**2*
c**3 + 30*B*a**2*b*c**2 - 10*B*a*b**3*c + B*b**5)*atanh((b + 2*c*x**2)/sqrt(-4*a
*c + b**2))/(2*c**3*(-4*a*c + b**2)**(5/2))

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Mathematica [A]  time = 0.865887, size = 354, normalized size = 1.39 \[ \frac{-\frac{2 c \left (-12 a^2 A c^3+30 a^2 b B c^2-10 a b^3 B c+b^5 B\right ) \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}+\frac{2 a^2 b c^3 \left (11 A+25 B x^2\right )+4 a^2 c^3 \left (8 a B-5 A c x^2\right )+b^4 c \left (11 a B-2 A c x^2\right )-2 a b^3 c^2 \left (4 A+15 B x^2\right )+a b^2 c^2 \left (16 A c x^2-39 a B\right )+b^5 c \left (A+4 B x^2\right )+b^6 (-B)}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{2 a^3 B c^2+a^2 c \left (b c \left (3 A+5 B x^2\right )-2 A c^2 x^2-4 b^2 B\right )+a b^2 \left (-b c \left (A+5 B x^2\right )+4 A c^2 x^2+b^2 B\right )+b^4 x^2 (b B-A c)}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+B c \log \left (a+b x^2+c x^4\right )}{4 c^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^9*(A + B*x^2))/(a + b*x^2 + c*x^4)^3,x]

[Out]

((-(b^6*B) + b^5*c*(A + 4*B*x^2) - 2*a*b^3*c^2*(4*A + 15*B*x^2) + 2*a^2*b*c^3*(1
1*A + 25*B*x^2) + 4*a^2*c^3*(8*a*B - 5*A*c*x^2) + b^4*c*(11*a*B - 2*A*c*x^2) + a
*b^2*c^2*(-39*a*B + 16*A*c*x^2))/((b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (2*a^3*
B*c^2 + b^4*(b*B - A*c)*x^2 + a*b^2*(b^2*B + 4*A*c^2*x^2 - b*c*(A + 5*B*x^2)) +
a^2*c*(-4*b^2*B - 2*A*c^2*x^2 + b*c*(3*A + 5*B*x^2)))/((b^2 - 4*a*c)*(a + b*x^2
+ c*x^4)^2) - (2*c*(b^5*B - 10*a*b^3*B*c + 30*a^2*b*B*c^2 - 12*a^2*A*c^3)*ArcTan
[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2) + B*c*Log[a + b*x^2 + c
*x^4])/(4*c^4)

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Maple [B]  time = 0.034, size = 1274, normalized size = 5. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9*(B*x^2+A)/(c*x^4+b*x^2+a)^3,x)

[Out]

1/2*(-1/c^2*(10*A*a^2*c^3-8*A*a*b^2*c^2+A*b^4*c-25*B*a^2*b*c^2+15*B*a*b^3*c-2*B*
b^5)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6+1/2*(2*A*a^2*b*c^3+8*A*a*b^3*c^2-A*b^5*c+32*
B*a^3*c^3+11*B*a^2*b^2*c^2-19*B*a*b^4*c+3*B*b^6)/c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*
x^4-a*(6*A*a^2*c^3-10*A*a*b^2*c^2+A*b^4*c-31*B*a^2*b*c^2+22*B*a*b^3*c-3*B*b^5)/c
^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2+1/2*a^2*(10*A*a*b*c^2-A*b^3*c+24*B*a^2*c^2-21*
B*a*b^2*c+3*B*b^4)/c^3/(16*a^2*c^2-8*a*b^2*c+b^4))/(c*x^4+b*x^2+a)^2+4/c/(16*a^2
*c^2-8*a*b^2*c+b^4)*ln(c^2*(16*a^2*c^2-8*a*b^2*c+b^4)*(c*x^4+b*x^2+a))*a^2*B-2/c
^2/(16*a^2*c^2-8*a*b^2*c+b^4)*ln(c^2*(16*a^2*c^2-8*a*b^2*c+b^4)*(c*x^4+b*x^2+a))
*a*b^2*B+1/4/c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*ln(c^2*(16*a^2*c^2-8*a*b^2*c+b^4)*(c
*x^4+b*x^2+a))*b^4*B+6/(1024*a^5*c^9-1280*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^
6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2)*arctan((2*(16*a^2*c^2-8*a*b^2*c+b^4)*c^3*x^2+
c^2*(16*a^2*c^2-8*a*b^2*c+b^4)*b)/(1024*a^5*c^9-1280*a^4*b^2*c^8+640*a^3*b^4*c^7
-160*a^2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2))*A*a^2*c^2-15/(1024*a^5*c^9-1280*a
^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2)*arctan((
2*(16*a^2*c^2-8*a*b^2*c+b^4)*c^3*x^2+c^2*(16*a^2*c^2-8*a*b^2*c+b^4)*b)/(1024*a^5
*c^9-1280*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/
2))*a^2*b*B*c+5/(1024*a^5*c^9-1280*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+2
0*a*b^8*c^5-b^10*c^4)^(1/2)*arctan((2*(16*a^2*c^2-8*a*b^2*c+b^4)*c^3*x^2+c^2*(16
*a^2*c^2-8*a*b^2*c+b^4)*b)/(1024*a^5*c^9-1280*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^
2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2))*B*a*b^3-1/2/(1024*a^5*c^9-1280*a^4*b^2*c
^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2)*arctan((2*(16*a^
2*c^2-8*a*b^2*c+b^4)*c^3*x^2+c^2*(16*a^2*c^2-8*a*b^2*c+b^4)*b)/(1024*a^5*c^9-128
0*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2))*b^5/
c*B

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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((B*x^2 + A)*x^9/(c*x^4 + b*x^2 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.432666, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^9/(c*x^4 + b*x^2 + a)^3,x, algorithm="fricas")

[Out]

[-1/4*(((B*b^5*c^2 - 10*B*a*b^3*c^3 + 30*B*a^2*b*c^4 - 12*A*a^2*c^5)*x^8 + B*a^2
*b^5 - 10*B*a^3*b^3*c + 30*B*a^4*b*c^2 - 12*A*a^4*c^3 + 2*(B*b^6*c - 10*B*a*b^4*
c^2 + 30*B*a^2*b^2*c^3 - 12*A*a^2*b*c^4)*x^6 + (B*b^7 - 8*B*a*b^5*c + 10*B*a^2*b
^3*c^2 - 24*A*a^3*c^4 + 12*(5*B*a^3*b - A*a^2*b^2)*c^3)*x^4 + 2*(B*a*b^6 - 10*B*
a^2*b^4*c + 30*B*a^3*b^2*c^2 - 12*A*a^3*b*c^3)*x^2)*log(-(b^3 - 4*a*b*c + 2*(b^2
*c - 4*a*c^2)*x^2 - (2*c^2*x^4 + 2*b*c*x^2 + b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(c*
x^4 + b*x^2 + a)) - (3*B*a^2*b^4 + 2*(2*B*b^5*c - 10*A*a^2*c^4 + (25*B*a^2*b + 8
*A*a*b^2)*c^3 - (15*B*a*b^3 + A*b^4)*c^2)*x^6 + (3*B*b^6 + 2*(16*B*a^3 + A*a^2*b
)*c^3 + (11*B*a^2*b^2 + 8*A*a*b^3)*c^2 - (19*B*a*b^4 + A*b^5)*c)*x^4 + 2*(12*B*a
^4 + 5*A*a^3*b)*c^2 + 2*(3*B*a*b^5 - 6*A*a^3*c^3 + (31*B*a^3*b + 10*A*a^2*b^2)*c
^2 - (22*B*a^2*b^3 + A*a*b^4)*c)*x^2 - (21*B*a^3*b^2 + A*a^2*b^3)*c + ((B*b^4*c^
2 - 8*B*a*b^2*c^3 + 16*B*a^2*c^4)*x^8 + B*a^2*b^4 - 8*B*a^3*b^2*c + 16*B*a^4*c^2
+ 2*(B*b^5*c - 8*B*a*b^3*c^2 + 16*B*a^2*b*c^3)*x^6 + (B*b^6 - 6*B*a*b^4*c + 32*
B*a^3*c^3)*x^4 + 2*(B*a*b^5 - 8*B*a^2*b^3*c + 16*B*a^3*b*c^2)*x^2)*log(c*x^4 + b
*x^2 + a))*sqrt(b^2 - 4*a*c))/((a^2*b^4*c^3 - 8*a^3*b^2*c^4 + 16*a^4*c^5 + (b^4*
c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*x^8 + 2*(b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*x
^6 + (b^6*c^3 - 6*a*b^4*c^4 + 32*a^3*c^6)*x^4 + 2*(a*b^5*c^3 - 8*a^2*b^3*c^4 + 1
6*a^3*b*c^5)*x^2)*sqrt(b^2 - 4*a*c)), -1/4*(2*((B*b^5*c^2 - 10*B*a*b^3*c^3 + 30*
B*a^2*b*c^4 - 12*A*a^2*c^5)*x^8 + B*a^2*b^5 - 10*B*a^3*b^3*c + 30*B*a^4*b*c^2 -
12*A*a^4*c^3 + 2*(B*b^6*c - 10*B*a*b^4*c^2 + 30*B*a^2*b^2*c^3 - 12*A*a^2*b*c^4)*
x^6 + (B*b^7 - 8*B*a*b^5*c + 10*B*a^2*b^3*c^2 - 24*A*a^3*c^4 + 12*(5*B*a^3*b - A
*a^2*b^2)*c^3)*x^4 + 2*(B*a*b^6 - 10*B*a^2*b^4*c + 30*B*a^3*b^2*c^2 - 12*A*a^3*b
*c^3)*x^2)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) - (3*B*a^2*b^
4 + 2*(2*B*b^5*c - 10*A*a^2*c^4 + (25*B*a^2*b + 8*A*a*b^2)*c^3 - (15*B*a*b^3 + A
*b^4)*c^2)*x^6 + (3*B*b^6 + 2*(16*B*a^3 + A*a^2*b)*c^3 + (11*B*a^2*b^2 + 8*A*a*b
^3)*c^2 - (19*B*a*b^4 + A*b^5)*c)*x^4 + 2*(12*B*a^4 + 5*A*a^3*b)*c^2 + 2*(3*B*a*
b^5 - 6*A*a^3*c^3 + (31*B*a^3*b + 10*A*a^2*b^2)*c^2 - (22*B*a^2*b^3 + A*a*b^4)*c
)*x^2 - (21*B*a^3*b^2 + A*a^2*b^3)*c + ((B*b^4*c^2 - 8*B*a*b^2*c^3 + 16*B*a^2*c^
4)*x^8 + B*a^2*b^4 - 8*B*a^3*b^2*c + 16*B*a^4*c^2 + 2*(B*b^5*c - 8*B*a*b^3*c^2 +
16*B*a^2*b*c^3)*x^6 + (B*b^6 - 6*B*a*b^4*c + 32*B*a^3*c^3)*x^4 + 2*(B*a*b^5 - 8
*B*a^2*b^3*c + 16*B*a^3*b*c^2)*x^2)*log(c*x^4 + b*x^2 + a))*sqrt(-b^2 + 4*a*c))/
((a^2*b^4*c^3 - 8*a^3*b^2*c^4 + 16*a^4*c^5 + (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7
)*x^8 + 2*(b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*x^6 + (b^6*c^3 - 6*a*b^4*c^4 +
32*a^3*c^6)*x^4 + 2*(a*b^5*c^3 - 8*a^2*b^3*c^4 + 16*a^3*b*c^5)*x^2)*sqrt(-b^2 +
4*a*c))]

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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(x**9*(B*x**2+A)/(c*x**4+b*x**2+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 15.7499, size = 629, normalized size = 2.48 \[ -\frac{{\left (B b^{5} - 10 \, B a b^{3} c + 30 \, B a^{2} b c^{2} - 12 \, A a^{2} c^{3}\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \,{\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{B{\rm ln}\left (c x^{4} + b x^{2} + a\right )}{4 \, c^{3}} - \frac{3 \, B b^{4} c^{2} x^{8} - 24 \, B a b^{2} c^{3} x^{8} + 48 \, B a^{2} c^{4} x^{8} - 2 \, B b^{5} c x^{6} + 12 \, B a b^{3} c^{2} x^{6} + 4 \, A b^{4} c^{2} x^{6} - 4 \, B a^{2} b c^{3} x^{6} - 32 \, A a b^{2} c^{3} x^{6} + 40 \, A a^{2} c^{4} x^{6} - 3 \, B b^{6} x^{4} + 20 \, B a b^{4} c x^{4} + 2 \, A b^{5} c x^{4} - 22 \, B a^{2} b^{2} c^{2} x^{4} - 16 \, A a b^{3} c^{2} x^{4} + 32 \, B a^{3} c^{3} x^{4} - 4 \, A a^{2} b c^{3} x^{4} - 6 \, B a b^{5} x^{2} + 40 \, B a^{2} b^{3} c x^{2} + 4 \, A a b^{4} c x^{2} - 28 \, B a^{3} b c^{2} x^{2} - 40 \, A a^{2} b^{2} c^{2} x^{2} + 24 \, A a^{3} c^{3} x^{2} - 3 \, B a^{2} b^{4} + 18 \, B a^{3} b^{2} c + 2 \, A a^{2} b^{3} c - 20 \, A a^{3} b c^{2}}{8 \,{\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )}{\left (c x^{4} + b x^{2} + a\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^9/(c*x^4 + b*x^2 + a)^3,x, algorithm="giac")

[Out]

-1/2*(B*b^5 - 10*B*a*b^3*c + 30*B*a^2*b*c^2 - 12*A*a^2*c^3)*arctan((2*c*x^2 + b)
/sqrt(-b^2 + 4*a*c))/((b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*sqrt(-b^2 + 4*a*c)) +
1/4*B*ln(c*x^4 + b*x^2 + a)/c^3 - 1/8*(3*B*b^4*c^2*x^8 - 24*B*a*b^2*c^3*x^8 + 4
8*B*a^2*c^4*x^8 - 2*B*b^5*c*x^6 + 12*B*a*b^3*c^2*x^6 + 4*A*b^4*c^2*x^6 - 4*B*a^2
*b*c^3*x^6 - 32*A*a*b^2*c^3*x^6 + 40*A*a^2*c^4*x^6 - 3*B*b^6*x^4 + 20*B*a*b^4*c*
x^4 + 2*A*b^5*c*x^4 - 22*B*a^2*b^2*c^2*x^4 - 16*A*a*b^3*c^2*x^4 + 32*B*a^3*c^3*x
^4 - 4*A*a^2*b*c^3*x^4 - 6*B*a*b^5*x^2 + 40*B*a^2*b^3*c*x^2 + 4*A*a*b^4*c*x^2 -
28*B*a^3*b*c^2*x^2 - 40*A*a^2*b^2*c^2*x^2 + 24*A*a^3*c^3*x^2 - 3*B*a^2*b^4 + 18*
B*a^3*b^2*c + 2*A*a^2*b^3*c - 20*A*a^3*b*c^2)/((b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c
^5)*(c*x^4 + b*x^2 + a)^2)