Integrand size = 12, antiderivative size = 101 \[ \int \frac {1}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {b+2 c x}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 c (b+2 c x)}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {12 c^2 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \] Output:
-1/2*(2*c*x+b)/(-4*a*c+b^2)/(c*x^2+b*x+a)^2+3*c*(2*c*x+b)/(-4*a*c+b^2)^2/( c*x^2+b*x+a)-12*c^2*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(5/ 2)
Time = 0.09 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\left (a+b x+c x^2\right )^3} \, dx=\frac {-\frac {(b+2 c x) \left (b^2-6 b c x-2 c \left (5 a+3 c x^2\right )\right )}{(a+x (b+c x))^2}+\frac {24 c^2 \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}}{2 \left (b^2-4 a c\right )^2} \] Input:
Integrate[(a + b*x + c*x^2)^(-3),x]
Output:
(-(((b + 2*c*x)*(b^2 - 6*b*c*x - 2*c*(5*a + 3*c*x^2)))/(a + x*(b + c*x))^2 ) + (24*c^2*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(2 *(b^2 - 4*a*c)^2)
Time = 0.42 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.11, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1086, 1086, 1083, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{\left (a+b x+c x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 1086 |
\(\displaystyle -\frac {3 c \int \frac {1}{\left (c x^2+b x+a\right )^2}dx}{b^2-4 a c}-\frac {b+2 c x}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
\(\Big \downarrow \) 1086 |
\(\displaystyle -\frac {3 c \left (-\frac {2 c \int \frac {1}{c x^2+b x+a}dx}{b^2-4 a c}-\frac {b+2 c x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{b^2-4 a c}-\frac {b+2 c x}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle -\frac {3 c \left (\frac {4 c \int \frac {1}{b^2-(b+2 c x)^2-4 a c}d(b+2 c x)}{b^2-4 a c}-\frac {b+2 c x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{b^2-4 a c}-\frac {b+2 c x}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {3 c \left (\frac {4 c \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {b+2 c x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{b^2-4 a c}-\frac {b+2 c x}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
Input:
Int[(a + b*x + c*x^2)^(-3),x]
Output:
-1/2*(b + 2*c*x)/((b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - (3*c*(-((b + 2*c*x) /((b^2 - 4*a*c)*(a + b*x + c*x^2))) + (4*c*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(3/2)))/(b^2 - 4*a*c)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fre eQ[{a, b, c}, x] && ILtQ[p, -1]
Time = 0.71 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.15
method | result | size |
default | \(\frac {2 c x +b}{2 \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{2}}+\frac {3 c \left (\frac {2 c x +b}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )}+\frac {4 c \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}\right )}{4 a c -b^{2}}\) | \(116\) |
risch | \(\frac {\frac {6 c^{3} x^{3}}{16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}}+\frac {9 b \,c^{2} x^{2}}{16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}}+\frac {2 \left (5 a c +b^{2}\right ) c x}{16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}}+\frac {b \left (10 a c -b^{2}\right )}{32 a^{2} c^{2}-16 c a \,b^{2}+2 b^{4}}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {6 c^{2} \ln \left (\left (32 a^{2} c^{3}-16 a \,c^{2} b^{2}+2 b^{4} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {5}{2}}+16 a^{2} b \,c^{2}-8 a \,b^{3} c +b^{5}\right )}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}+\frac {6 c^{2} \ln \left (\left (-32 a^{2} c^{3}+16 a \,c^{2} b^{2}-2 b^{4} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {5}{2}}-16 a^{2} b \,c^{2}+8 a \,b^{3} c -b^{5}\right )}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}\) | \(288\) |
Input:
int(1/(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
1/2*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^2+3*c/(4*a*c-b^2)*((2*c*x+b)/(4*a* c-b^2)/(c*x^2+b*x+a)+4*c/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1 /2)))
Leaf count of result is larger than twice the leaf count of optimal. 382 vs. \(2 (95) = 190\).
Time = 0.10 (sec) , antiderivative size = 785, normalized size of antiderivative = 7.77 \[ \int \frac {1}{\left (a+b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:
integrate(1/(c*x^2+b*x+a)^3,x, algorithm="fricas")
Output:
[-1/2*(b^5 - 14*a*b^3*c + 40*a^2*b*c^2 - 12*(b^2*c^3 - 4*a*c^4)*x^3 - 18*( b^3*c^2 - 4*a*b*c^3)*x^2 - 12*(c^4*x^4 + 2*b*c^3*x^3 + 2*a*b*c^2*x + a^2*c ^2 + (b^2*c^2 + 2*a*c^3)*x^2)*sqrt(b^2 - 4*a*c)*log((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c - sqrt(b^2 - 4*a*c)*(2*c*x + b))/(c*x^2 + b*x + a)) - 4*(b^4* c + a*b^2*c^2 - 20*a^2*c^3)*x)/(a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3 + (b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*x^4 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*x^3 + (b^8 - 10*a *b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*x^2 + 2*(a*b^7 - 1 2*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*x), -1/2*(b^5 - 14*a*b^3*c + 40*a^2*b*c^2 - 12*(b^2*c^3 - 4*a*c^4)*x^3 - 18*(b^3*c^2 - 4*a*b*c^3)*x^2 + 24*(c^4*x^4 + 2*b*c^3*x^3 + 2*a*b*c^2*x + a^2*c^2 + (b^2*c^2 + 2*a*c^3)*x ^2)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c )) - 4*(b^4*c + a*b^2*c^2 - 20*a^2*c^3)*x)/(a^2*b^6 - 12*a^3*b^4*c + 48*a^ 4*b^2*c^2 - 64*a^5*c^3 + (b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3 *c^5)*x^4 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*x^3 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*x^2 + 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*x)]
Leaf count of result is larger than twice the leaf count of optimal. 474 vs. \(2 (95) = 190\).
Time = 0.68 (sec) , antiderivative size = 474, normalized size of antiderivative = 4.69 \[ \int \frac {1}{\left (a+b x+c x^2\right )^3} \, dx=- 6 c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \log {\left (x + \frac {- 384 a^{3} c^{5} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 288 a^{2} b^{2} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 72 a b^{4} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 6 b^{6} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 6 b c^{2}}{12 c^{3}} \right )} + 6 c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \log {\left (x + \frac {384 a^{3} c^{5} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 288 a^{2} b^{2} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 72 a b^{4} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 6 b^{6} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 6 b c^{2}}{12 c^{3}} \right )} + \frac {10 a b c - b^{3} + 18 b c^{2} x^{2} + 12 c^{3} x^{3} + x \left (20 a c^{2} + 4 b^{2} c\right )}{32 a^{4} c^{2} - 16 a^{3} b^{2} c + 2 a^{2} b^{4} + x^{4} \cdot \left (32 a^{2} c^{4} - 16 a b^{2} c^{3} + 2 b^{4} c^{2}\right ) + x^{3} \cdot \left (64 a^{2} b c^{3} - 32 a b^{3} c^{2} + 4 b^{5} c\right ) + x^{2} \cdot \left (64 a^{3} c^{3} - 12 a b^{4} c + 2 b^{6}\right ) + x \left (64 a^{3} b c^{2} - 32 a^{2} b^{3} c + 4 a b^{5}\right )} \] Input:
integrate(1/(c*x**2+b*x+a)**3,x)
Output:
-6*c**2*sqrt(-1/(4*a*c - b**2)**5)*log(x + (-384*a**3*c**5*sqrt(-1/(4*a*c - b**2)**5) + 288*a**2*b**2*c**4*sqrt(-1/(4*a*c - b**2)**5) - 72*a*b**4*c* *3*sqrt(-1/(4*a*c - b**2)**5) + 6*b**6*c**2*sqrt(-1/(4*a*c - b**2)**5) + 6 *b*c**2)/(12*c**3)) + 6*c**2*sqrt(-1/(4*a*c - b**2)**5)*log(x + (384*a**3* c**5*sqrt(-1/(4*a*c - b**2)**5) - 288*a**2*b**2*c**4*sqrt(-1/(4*a*c - b**2 )**5) + 72*a*b**4*c**3*sqrt(-1/(4*a*c - b**2)**5) - 6*b**6*c**2*sqrt(-1/(4 *a*c - b**2)**5) + 6*b*c**2)/(12*c**3)) + (10*a*b*c - b**3 + 18*b*c**2*x** 2 + 12*c**3*x**3 + x*(20*a*c**2 + 4*b**2*c))/(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))
Exception generated. \[ \int \frac {1}{\left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(c*x^2+b*x+a)^3,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Time = 0.16 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.35 \[ \int \frac {1}{\left (a+b x+c x^2\right )^3} \, dx=\frac {12 \, c^{2} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {12 \, c^{3} x^{3} + 18 \, b c^{2} x^{2} + 4 \, b^{2} c x + 20 \, a c^{2} x - b^{3} + 10 \, a b c}{2 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} {\left (c x^{2} + b x + a\right )}^{2}} \] Input:
integrate(1/(c*x^2+b*x+a)^3,x, algorithm="giac")
Output:
12*c^2*arctan((2*c*x + b)/sqrt(-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*x^3 + 18*b*c^2*x^2 + 4*b^2*c*x + 20* a*c^2*x - b^3 + 10*a*b*c)/((b^4 - 8*a*b^2*c + 16*a^2*c^2)*(c*x^2 + b*x + a )^2)
Time = 9.97 (sec) , antiderivative size = 285, normalized size of antiderivative = 2.82 \[ \int \frac {1}{\left (a+b x+c x^2\right )^3} \, dx=\frac {\frac {6\,c^3\,x^3}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}-\frac {b^3-10\,a\,b\,c}{2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {9\,b\,c^2\,x^2}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}+\frac {2\,c\,x\,\left (b^2+5\,a\,c\right )}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3}+\frac {12\,c^2\,\mathrm {atan}\left (\frac {\left (\frac {12\,c^3\,x}{{\left (4\,a\,c-b^2\right )}^{5/2}}+\frac {6\,c^2\,\left (16\,a^2\,b\,c^2-8\,a\,b^3\,c+b^5\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}{6\,c^2}\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}} \] Input:
int(1/(a + b*x + c*x^2)^3,x)
Output:
((6*c^3*x^3)/(b^4 + 16*a^2*c^2 - 8*a*b^2*c) - (b^3 - 10*a*b*c)/(2*(b^4 + 1 6*a^2*c^2 - 8*a*b^2*c)) + (9*b*c^2*x^2)/(b^4 + 16*a^2*c^2 - 8*a*b^2*c) + ( 2*c*x*(5*a*c + b^2))/(b^4 + 16*a^2*c^2 - 8*a*b^2*c))/(x^2*(2*a*c + b^2) + a^2 + c^2*x^4 + 2*a*b*x + 2*b*c*x^3) + (12*c^2*atan((((12*c^3*x)/(4*a*c - b^2)^(5/2) + (6*c^2*(b^5 + 16*a^2*b*c^2 - 8*a*b^3*c))/((4*a*c - b^2)^(5/2) *(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))*(b^4 + 16*a^2*c^2 - 8*a*b^2*c))/(6*c^2)) )/(4*a*c - b^2)^(5/2)
Time = 0.21 (sec) , antiderivative size = 598, normalized size of antiderivative = 5.92 \[ \int \frac {1}{\left (a+b x+c x^2\right )^3} \, dx=\frac {24 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a^{2} b \,c^{2}+48 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{2} c^{2} x +48 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a b \,c^{3} x^{2}+24 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{3} c^{2} x^{2}+48 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{2} c^{3} x^{3}+24 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b \,c^{4} x^{4}-24 a^{3} c^{3}+46 a^{2} b^{2} c^{2}+32 a^{2} b \,c^{3} x -48 a^{2} c^{4} x^{2}-14 a \,b^{4} c +8 a \,b^{3} c^{2} x +60 a \,b^{2} c^{3} x^{2}-24 a \,c^{5} x^{4}+b^{6}-4 b^{5} c x -12 b^{4} c^{2} x^{2}+6 b^{2} c^{4} x^{4}}{2 b \left (64 a^{3} c^{5} x^{4}-48 a^{2} b^{2} c^{4} x^{4}+12 a \,b^{4} c^{3} x^{4}-b^{6} c^{2} x^{4}+128 a^{3} b \,c^{4} x^{3}-96 a^{2} b^{3} c^{3} x^{3}+24 a \,b^{5} c^{2} x^{3}-2 b^{7} c \,x^{3}+128 a^{4} c^{4} x^{2}-32 a^{3} b^{2} c^{3} x^{2}-24 a^{2} b^{4} c^{2} x^{2}+10 a \,b^{6} c \,x^{2}-b^{8} x^{2}+128 a^{4} b \,c^{3} x -96 a^{3} b^{3} c^{2} x +24 a^{2} b^{5} c x -2 a \,b^{7} x +64 a^{5} c^{3}-48 a^{4} b^{2} c^{2}+12 a^{3} b^{4} c -a^{2} b^{6}\right )} \] Input:
int(1/(c*x^2+b*x+a)^3,x)
Output:
(24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b*c**2 + 48*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**2*c**2*x + 48*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b*c**3*x**2 + 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**3*c**2*x** 2 + 48*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**2*c**3*x **3 + 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b*c**4*x* *4 - 24*a**3*c**3 + 46*a**2*b**2*c**2 + 32*a**2*b*c**3*x - 48*a**2*c**4*x* *2 - 14*a*b**4*c + 8*a*b**3*c**2*x + 60*a*b**2*c**3*x**2 - 24*a*c**5*x**4 + b**6 - 4*b**5*c*x - 12*b**4*c**2*x**2 + 6*b**2*c**4*x**4)/(2*b*(64*a**5* c**3 - 48*a**4*b**2*c**2 + 128*a**4*b*c**3*x + 128*a**4*c**4*x**2 + 12*a** 3*b**4*c - 96*a**3*b**3*c**2*x - 32*a**3*b**2*c**3*x**2 + 128*a**3*b*c**4* x**3 + 64*a**3*c**5*x**4 - a**2*b**6 + 24*a**2*b**5*c*x - 24*a**2*b**4*c** 2*x**2 - 96*a**2*b**3*c**3*x**3 - 48*a**2*b**2*c**4*x**4 - 2*a*b**7*x + 10 *a*b**6*c*x**2 + 24*a*b**5*c**2*x**3 + 12*a*b**4*c**3*x**4 - b**8*x**2 - 2 *b**7*c*x**3 - b**6*c**2*x**4))