3.197 \(\int \frac{\text{b1}+\text{c1} x}{\left (a+2 b x+c x^2\right )^4} \, dx\)

Optimal. Leaf size=173 \[ \frac{5 c^2 (\text{b1} c-b \text{c1}) \tanh ^{-1}\left (\frac{b+c x}{\sqrt{b^2-a c}}\right )}{16 \left (b^2-a c\right )^{7/2}}-\frac{5 c (b+c x) (\text{b1} c-b \text{c1})}{16 \left (b^2-a c\right )^3 \left (a+2 b x+c x^2\right )}+\frac{5 (b+c x) (\text{b1} c-b \text{c1})}{24 \left (b^2-a c\right )^2 \left (a+2 b x+c x^2\right )^2}-\frac{-a \text{c1}+x (\text{b1} c-b \text{c1})+b \text{b1}}{6 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )^3} \]

[Out]

-(b*b1 - a*c1 + (b1*c - b*c1)*x)/(6*(b^2 - a*c)*(a + 2*b*x + c*x^2)^3) + (5*(b1*
c - b*c1)*(b + c*x))/(24*(b^2 - a*c)^2*(a + 2*b*x + c*x^2)^2) - (5*c*(b1*c - b*c
1)*(b + c*x))/(16*(b^2 - a*c)^3*(a + 2*b*x + c*x^2)) + (5*c^2*(b1*c - b*c1)*ArcT
anh[(b + c*x)/Sqrt[b^2 - a*c]])/(16*(b^2 - a*c)^(7/2))

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Rubi [A]  time = 0.254358, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{5 c^2 (\text{b1} c-b \text{c1}) \tanh ^{-1}\left (\frac{b+c x}{\sqrt{b^2-a c}}\right )}{16 \left (b^2-a c\right )^{7/2}}-\frac{5 c (b+c x) (\text{b1} c-b \text{c1})}{16 \left (b^2-a c\right )^3 \left (a+2 b x+c x^2\right )}+\frac{5 (b+c x) (\text{b1} c-b \text{c1})}{24 \left (b^2-a c\right )^2 \left (a+2 b x+c x^2\right )^2}-\frac{-a \text{c1}+x (\text{b1} c-b \text{c1})+b \text{b1}}{6 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )^3} \]

Antiderivative was successfully verified.

[In]  Int[(b1 + c1*x)/(a + 2*b*x + c*x^2)^4,x]

[Out]

-(b*b1 - a*c1 + (b1*c - b*c1)*x)/(6*(b^2 - a*c)*(a + 2*b*x + c*x^2)^3) + (5*(b1*
c - b*c1)*(b + c*x))/(24*(b^2 - a*c)^2*(a + 2*b*x + c*x^2)^2) - (5*c*(b1*c - b*c
1)*(b + c*x))/(16*(b^2 - a*c)^3*(a + 2*b*x + c*x^2)) + (5*c^2*(b1*c - b*c1)*ArcT
anh[(b + c*x)/Sqrt[b^2 - a*c]])/(16*(b^2 - a*c)^(7/2))

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Rubi in Sympy [A]  time = 13.2722, size = 165, normalized size = 0.95 \[ - \frac{5 c^{2} \left (b c_{1} - b_{1} c\right ) \operatorname{atanh}{\left (\frac{b + c x}{\sqrt{- a c + b^{2}}} \right )}}{16 \left (- a c + b^{2}\right )^{\frac{7}{2}}} + \frac{5 c \left (2 b + 2 c x\right ) \left (b c_{1} - b_{1} c\right )}{32 \left (- a c + b^{2}\right )^{3} \left (a + 2 b x + c x^{2}\right )} - \frac{5 \left (2 b + 2 c x\right ) \left (b c_{1} - b_{1} c\right )}{48 \left (- a c + b^{2}\right )^{2} \left (a + 2 b x + c x^{2}\right )^{2}} + \frac{2 a c_{1} - 2 b b_{1} + x \left (2 b c_{1} - 2 b_{1} c\right )}{12 \left (- a c + b^{2}\right ) \left (a + 2 b x + c x^{2}\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c1*x+b1)/(c*x**2+2*b*x+a)**4,x)

[Out]

-5*c**2*(b*c1 - b1*c)*atanh((b + c*x)/sqrt(-a*c + b**2))/(16*(-a*c + b**2)**(7/2
)) + 5*c*(2*b + 2*c*x)*(b*c1 - b1*c)/(32*(-a*c + b**2)**3*(a + 2*b*x + c*x**2))
- 5*(2*b + 2*c*x)*(b*c1 - b1*c)/(48*(-a*c + b**2)**2*(a + 2*b*x + c*x**2)**2) +
(2*a*c1 - 2*b*b1 + x*(2*b*c1 - 2*b1*c))/(12*(-a*c + b**2)*(a + 2*b*x + c*x**2)**
3)

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Mathematica [A]  time = 0.350171, size = 168, normalized size = 0.97 \[ \frac{\frac{15 c^2 (b \text{c1}-\text{b1} c) \tan ^{-1}\left (\frac{b+c x}{\sqrt{a c-b^2}}\right )}{\sqrt{a c-b^2}}-\frac{10 \left (b^2-a c\right ) (b+c x) (b \text{c1}-\text{b1} c)}{(a+x (2 b+c x))^2}+\frac{8 \left (b^2-a c\right )^2 (a \text{c1}-b \text{b1}+b \text{c1} x-\text{b1} c x)}{(a+x (2 b+c x))^3}+\frac{15 c (b+c x) (b \text{c1}-\text{b1} c)}{a+x (2 b+c x)}}{48 \left (b^2-a c\right )^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(b1 + c1*x)/(a + 2*b*x + c*x^2)^4,x]

[Out]

((8*(b^2 - a*c)^2*(-(b*b1) + a*c1 - b1*c*x + b*c1*x))/(a + x*(2*b + c*x))^3 - (1
0*(b^2 - a*c)*(-(b1*c) + b*c1)*(b + c*x))/(a + x*(2*b + c*x))^2 + (15*c*(-(b1*c)
+ b*c1)*(b + c*x))/(a + x*(2*b + c*x)) + (15*c^2*(-(b1*c) + b*c1)*ArcTan[(b + c
*x)/Sqrt[-b^2 + a*c]])/Sqrt[-b^2 + a*c])/(48*(b^2 - a*c)^3)

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Maple [B]  time = 0.008, size = 405, normalized size = 2.3 \[{\frac{ \left ( -2\,b{\it c1}+2\,{\it b1}\,c \right ) x+2\,b{\it b1}-2\,a{\it c1}}{ \left ( 12\,ac-12\,{b}^{2} \right ) \left ( c{x}^{2}+2\,bx+a \right ) ^{3}}}-{\frac{10\,cxb{\it c1}}{3\, \left ( 4\,ac-4\,{b}^{2} \right ) ^{2} \left ( c{x}^{2}+2\,bx+a \right ) ^{2}}}+{\frac{10\,x{c}^{2}{\it b1}}{3\, \left ( 4\,ac-4\,{b}^{2} \right ) ^{2} \left ( c{x}^{2}+2\,bx+a \right ) ^{2}}}-{\frac{10\,{b}^{2}{\it c1}}{3\, \left ( 4\,ac-4\,{b}^{2} \right ) ^{2} \left ( c{x}^{2}+2\,bx+a \right ) ^{2}}}+{\frac{10\,b{\it b1}\,c}{3\, \left ( 4\,ac-4\,{b}^{2} \right ) ^{2} \left ( c{x}^{2}+2\,bx+a \right ) ^{2}}}-20\,{\frac{x{c}^{2}b{\it c1}}{ \left ( 4\,ac-4\,{b}^{2} \right ) ^{3} \left ( c{x}^{2}+2\,bx+a \right ) }}+20\,{\frac{{c}^{3}x{\it b1}}{ \left ( 4\,ac-4\,{b}^{2} \right ) ^{3} \left ( c{x}^{2}+2\,bx+a \right ) }}-20\,{\frac{{b}^{2}c{\it c1}}{ \left ( 4\,ac-4\,{b}^{2} \right ) ^{3} \left ( c{x}^{2}+2\,bx+a \right ) }}+20\,{\frac{{\it b1}\,b{c}^{2}}{ \left ( 4\,ac-4\,{b}^{2} \right ) ^{3} \left ( c{x}^{2}+2\,bx+a \right ) }}-20\,{\frac{{\it c1}\,b{c}^{2}}{ \left ( 4\,ac-4\,{b}^{2} \right ) ^{3}\sqrt{ac-{b}^{2}}}\arctan \left ( 1/2\,{\frac{2\,cx+2\,b}{\sqrt{ac-{b}^{2}}}} \right ) }+20\,{\frac{{\it b1}\,{c}^{3}}{ \left ( 4\,ac-4\,{b}^{2} \right ) ^{3}\sqrt{ac-{b}^{2}}}\arctan \left ( 1/2\,{\frac{2\,cx+2\,b}{\sqrt{ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c1*x+b1)/(c*x^2+2*b*x+a)^4,x)

[Out]

1/3*((-2*b*c1+2*b1*c)*x+2*b*b1-2*a*c1)/(4*a*c-4*b^2)/(c*x^2+2*b*x+a)^3-10/3/(4*a
*c-4*b^2)^2/(c*x^2+2*b*x+a)^2*x*c*b*c1+10/3/(4*a*c-4*b^2)^2/(c*x^2+2*b*x+a)^2*x*
c^2*b1-10/3/(4*a*c-4*b^2)^2/(c*x^2+2*b*x+a)^2*b^2*c1+10/3/(4*a*c-4*b^2)^2/(c*x^2
+2*b*x+a)^2*b*b1*c-20/(4*a*c-4*b^2)^3*c^2/(c*x^2+2*b*x+a)*x*b*c1+20/(4*a*c-4*b^2
)^3*c^3/(c*x^2+2*b*x+a)*x*b1-20/(4*a*c-4*b^2)^3*c/(c*x^2+2*b*x+a)*b^2*c1+20/(4*a
*c-4*b^2)^3*c^2/(c*x^2+2*b*x+a)*b*b1-20/(4*a*c-4*b^2)^3*c^2/(a*c-b^2)^(1/2)*arct
an(1/2*(2*c*x+2*b)/(a*c-b^2)^(1/2))*b*c1+20/(4*a*c-4*b^2)^3*c^3/(a*c-b^2)^(1/2)*
arctan(1/2*(2*c*x+2*b)/(a*c-b^2)^(1/2))*b1

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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((c1*x + b1)/(c*x^2 + 2*b*x + a)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228335, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c1*x + b1)/(c*x^2 + 2*b*x + a)^4,x, algorithm="fricas")

[Out]

[1/96*(15*(a^3*b1*c^3 - a^3*b*c^2*c1 + (b1*c^6 - b*c^5*c1)*x^6 + 6*(b*b1*c^5 - b
^2*c^4*c1)*x^5 + 3*(4*b^2*b1*c^4 + a*b1*c^5 - (4*b^3*c^3 + a*b*c^4)*c1)*x^4 + 4*
(2*b^3*b1*c^3 + 3*a*b*b1*c^4 - (2*b^4*c^2 + 3*a*b^2*c^3)*c1)*x^3 + 3*(4*a*b^2*b1
*c^3 + a^2*b1*c^4 - (4*a*b^3*c^2 + a^2*b*c^3)*c1)*x^2 + 6*(a^2*b*b1*c^3 - a^2*b^
2*c^2*c1)*x)*log((2*b^3 - 2*a*b*c + 2*(b^2*c - a*c^2)*x + (c^2*x^2 + 2*b*c*x + 2
*b^2 - a*c)*sqrt(b^2 - a*c))/(c*x^2 + 2*b*x + a)) - 2*(8*b^5*b1 - 26*a*b^3*b1*c
+ 33*a^2*b*b1*c^2 + 15*(b1*c^5 - b*c^4*c1)*x^5 + 75*(b*b1*c^4 - b^2*c^3*c1)*x^4
+ 10*(11*b^2*b1*c^3 + 4*a*b1*c^4 - (11*b^3*c^2 + 4*a*b*c^3)*c1)*x^3 + 30*(b^3*b1
*c^2 + 4*a*b*b1*c^3 - (b^4*c + 4*a*b^2*c^2)*c1)*x^2 + (2*a*b^4 - 9*a^2*b^2*c - 8
*a^3*c^2)*c1 - 3*(4*b^4*b1*c - 18*a*b^2*b1*c^2 - 11*a^2*b1*c^3 - (4*b^5 - 18*a*b
^3*c - 11*a^2*b*c^2)*c1)*x)*sqrt(b^2 - a*c))/((a^3*b^6 - 3*a^4*b^4*c + 3*a^5*b^2
*c^2 - a^6*c^3 + (b^6*c^3 - 3*a*b^4*c^4 + 3*a^2*b^2*c^5 - a^3*c^6)*x^6 + 6*(b^7*
c^2 - 3*a*b^5*c^3 + 3*a^2*b^3*c^4 - a^3*b*c^5)*x^5 + 3*(4*b^8*c - 11*a*b^6*c^2 +
9*a^2*b^4*c^3 - a^3*b^2*c^4 - a^4*c^5)*x^4 + 4*(2*b^9 - 3*a*b^7*c - 3*a^2*b^5*c
^2 + 7*a^3*b^3*c^3 - 3*a^4*b*c^4)*x^3 + 3*(4*a*b^8 - 11*a^2*b^6*c + 9*a^3*b^4*c^
2 - a^4*b^2*c^3 - a^5*c^4)*x^2 + 6*(a^2*b^7 - 3*a^3*b^5*c + 3*a^4*b^3*c^2 - a^5*
b*c^3)*x)*sqrt(b^2 - a*c)), -1/48*(15*(a^3*b1*c^3 - a^3*b*c^2*c1 + (b1*c^6 - b*c
^5*c1)*x^6 + 6*(b*b1*c^5 - b^2*c^4*c1)*x^5 + 3*(4*b^2*b1*c^4 + a*b1*c^5 - (4*b^3
*c^3 + a*b*c^4)*c1)*x^4 + 4*(2*b^3*b1*c^3 + 3*a*b*b1*c^4 - (2*b^4*c^2 + 3*a*b^2*
c^3)*c1)*x^3 + 3*(4*a*b^2*b1*c^3 + a^2*b1*c^4 - (4*a*b^3*c^2 + a^2*b*c^3)*c1)*x^
2 + 6*(a^2*b*b1*c^3 - a^2*b^2*c^2*c1)*x)*arctan(-sqrt(-b^2 + a*c)*(c*x + b)/(b^2
- a*c)) + (8*b^5*b1 - 26*a*b^3*b1*c + 33*a^2*b*b1*c^2 + 15*(b1*c^5 - b*c^4*c1)*
x^5 + 75*(b*b1*c^4 - b^2*c^3*c1)*x^4 + 10*(11*b^2*b1*c^3 + 4*a*b1*c^4 - (11*b^3*
c^2 + 4*a*b*c^3)*c1)*x^3 + 30*(b^3*b1*c^2 + 4*a*b*b1*c^3 - (b^4*c + 4*a*b^2*c^2)
*c1)*x^2 + (2*a*b^4 - 9*a^2*b^2*c - 8*a^3*c^2)*c1 - 3*(4*b^4*b1*c - 18*a*b^2*b1*
c^2 - 11*a^2*b1*c^3 - (4*b^5 - 18*a*b^3*c - 11*a^2*b*c^2)*c1)*x)*sqrt(-b^2 + a*c
))/((a^3*b^6 - 3*a^4*b^4*c + 3*a^5*b^2*c^2 - a^6*c^3 + (b^6*c^3 - 3*a*b^4*c^4 +
3*a^2*b^2*c^5 - a^3*c^6)*x^6 + 6*(b^7*c^2 - 3*a*b^5*c^3 + 3*a^2*b^3*c^4 - a^3*b*
c^5)*x^5 + 3*(4*b^8*c - 11*a*b^6*c^2 + 9*a^2*b^4*c^3 - a^3*b^2*c^4 - a^4*c^5)*x^
4 + 4*(2*b^9 - 3*a*b^7*c - 3*a^2*b^5*c^2 + 7*a^3*b^3*c^3 - 3*a^4*b*c^4)*x^3 + 3*
(4*a*b^8 - 11*a^2*b^6*c + 9*a^3*b^4*c^2 - a^4*b^2*c^3 - a^5*c^4)*x^2 + 6*(a^2*b^
7 - 3*a^3*b^5*c + 3*a^4*b^3*c^2 - a^5*b*c^3)*x)*sqrt(-b^2 + a*c))]

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Sympy [A]  time = 7.61434, size = 1027, normalized size = 5.94 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c1*x+b1)/(c*x**2+2*b*x+a)**4,x)

[Out]

5*c**2*sqrt(-1/(a*c - b**2)**7)*(b*c1 - b1*c)*log(x + (-5*a**4*c**6*sqrt(-1/(a*c
- b**2)**7)*(b*c1 - b1*c) + 20*a**3*b**2*c**5*sqrt(-1/(a*c - b**2)**7)*(b*c1 -
b1*c) - 30*a**2*b**4*c**4*sqrt(-1/(a*c - b**2)**7)*(b*c1 - b1*c) + 20*a*b**6*c**
3*sqrt(-1/(a*c - b**2)**7)*(b*c1 - b1*c) - 5*b**8*c**2*sqrt(-1/(a*c - b**2)**7)*
(b*c1 - b1*c) + 5*b**2*c**2*c1 - 5*b*b1*c**3)/(5*b*c**3*c1 - 5*b1*c**4))/32 - 5*
c**2*sqrt(-1/(a*c - b**2)**7)*(b*c1 - b1*c)*log(x + (5*a**4*c**6*sqrt(-1/(a*c -
b**2)**7)*(b*c1 - b1*c) - 20*a**3*b**2*c**5*sqrt(-1/(a*c - b**2)**7)*(b*c1 - b1*
c) + 30*a**2*b**4*c**4*sqrt(-1/(a*c - b**2)**7)*(b*c1 - b1*c) - 20*a*b**6*c**3*s
qrt(-1/(a*c - b**2)**7)*(b*c1 - b1*c) + 5*b**8*c**2*sqrt(-1/(a*c - b**2)**7)*(b*
c1 - b1*c) + 5*b**2*c**2*c1 - 5*b*b1*c**3)/(5*b*c**3*c1 - 5*b1*c**4))/32 - (8*a*
*3*c**2*c1 + 9*a**2*b**2*c*c1 - 33*a**2*b*b1*c**2 - 2*a*b**4*c1 + 26*a*b**3*b1*c
- 8*b**5*b1 + x**5*(15*b*c**4*c1 - 15*b1*c**5) + x**4*(75*b**2*c**3*c1 - 75*b*b
1*c**4) + x**3*(40*a*b*c**3*c1 - 40*a*b1*c**4 + 110*b**3*c**2*c1 - 110*b**2*b1*c
**3) + x**2*(120*a*b**2*c**2*c1 - 120*a*b*b1*c**3 + 30*b**4*c*c1 - 30*b**3*b1*c*
*2) + x*(33*a**2*b*c**2*c1 - 33*a**2*b1*c**3 + 54*a*b**3*c*c1 - 54*a*b**2*b1*c**
2 - 12*b**5*c1 + 12*b**4*b1*c))/(48*a**6*c**3 - 144*a**5*b**2*c**2 + 144*a**4*b*
*4*c - 48*a**3*b**6 + x**6*(48*a**3*c**6 - 144*a**2*b**2*c**5 + 144*a*b**4*c**4
- 48*b**6*c**3) + x**5*(288*a**3*b*c**5 - 864*a**2*b**3*c**4 + 864*a*b**5*c**3 -
288*b**7*c**2) + x**4*(144*a**4*c**5 + 144*a**3*b**2*c**4 - 1296*a**2*b**4*c**3
+ 1584*a*b**6*c**2 - 576*b**8*c) + x**3*(576*a**4*b*c**4 - 1344*a**3*b**3*c**3
+ 576*a**2*b**5*c**2 + 576*a*b**7*c - 384*b**9) + x**2*(144*a**5*c**4 + 144*a**4
*b**2*c**3 - 1296*a**3*b**4*c**2 + 1584*a**2*b**6*c - 576*a*b**8) + x*(288*a**5*
b*c**3 - 864*a**4*b**3*c**2 + 864*a**3*b**5*c - 288*a**2*b**7))

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GIAC/XCAS [A]  time = 0.210147, size = 490, normalized size = 2.83 \[ -\frac{5 \,{\left (b_{1} c^{3} - b c^{2} c_{1}\right )} \arctan \left (\frac{c x + b}{\sqrt{-b^{2} + a c}}\right )}{16 \,{\left (b^{6} - 3 \, a b^{4} c + 3 \, a^{2} b^{2} c^{2} - a^{3} c^{3}\right )} \sqrt{-b^{2} + a c}} - \frac{15 \, b_{1} c^{5} x^{5} - 15 \, b c^{4} c_{1} x^{5} + 75 \, b b_{1} c^{4} x^{4} - 75 \, b^{2} c^{3} c_{1} x^{4} + 110 \, b^{2} b_{1} c^{3} x^{3} + 40 \, a b_{1} c^{4} x^{3} - 110 \, b^{3} c^{2} c_{1} x^{3} - 40 \, a b c^{3} c_{1} x^{3} + 30 \, b^{3} b_{1} c^{2} x^{2} + 120 \, a b b_{1} c^{3} x^{2} - 30 \, b^{4} c c_{1} x^{2} - 120 \, a b^{2} c^{2} c_{1} x^{2} - 12 \, b^{4} b_{1} c x + 54 \, a b^{2} b_{1} c^{2} x + 33 \, a^{2} b_{1} c^{3} x + 12 \, b^{5} c_{1} x - 54 \, a b^{3} c c_{1} x - 33 \, a^{2} b c^{2} c_{1} x + 8 \, b^{5} b_{1} - 26 \, a b^{3} b_{1} c + 33 \, a^{2} b b_{1} c^{2} + 2 \, a b^{4} c_{1} - 9 \, a^{2} b^{2} c c_{1} - 8 \, a^{3} c^{2} c_{1}}{48 \,{\left (b^{6} - 3 \, a b^{4} c + 3 \, a^{2} b^{2} c^{2} - a^{3} c^{3}\right )}{\left (c x^{2} + 2 \, b x + a\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c1*x + b1)/(c*x^2 + 2*b*x + a)^4,x, algorithm="giac")

[Out]

-5/16*(b1*c^3 - b*c^2*c1)*arctan((c*x + b)/sqrt(-b^2 + a*c))/((b^6 - 3*a*b^4*c +
3*a^2*b^2*c^2 - a^3*c^3)*sqrt(-b^2 + a*c)) - 1/48*(15*b1*c^5*x^5 - 15*b*c^4*c1*
x^5 + 75*b*b1*c^4*x^4 - 75*b^2*c^3*c1*x^4 + 110*b^2*b1*c^3*x^3 + 40*a*b1*c^4*x^3
- 110*b^3*c^2*c1*x^3 - 40*a*b*c^3*c1*x^3 + 30*b^3*b1*c^2*x^2 + 120*a*b*b1*c^3*x
^2 - 30*b^4*c*c1*x^2 - 120*a*b^2*c^2*c1*x^2 - 12*b^4*b1*c*x + 54*a*b^2*b1*c^2*x
+ 33*a^2*b1*c^3*x + 12*b^5*c1*x - 54*a*b^3*c*c1*x - 33*a^2*b*c^2*c1*x + 8*b^5*b1
- 26*a*b^3*b1*c + 33*a^2*b*b1*c^2 + 2*a*b^4*c1 - 9*a^2*b^2*c*c1 - 8*a^3*c^2*c1)
/((b^6 - 3*a*b^4*c + 3*a^2*b^2*c^2 - a^3*c^3)*(c*x^2 + 2*b*x + a)^3)