\(\int \frac {(d+e x)^2}{(a+b x+c x^2)^{3/2}} \, dx\) [618]

Optimal result

Integrand size = 22, antiderivative size = 117 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \left (b c d^2-4 a c d e+a b e^2+\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {e^2 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{3/2}} \] Output:

(-2*b*c*d^2+8*a*c*d*e-2*a*b*e^2-2*(2*c^2*d^2+b^2*e^2-2*c*e*(a*e+b*d))*x)/c 
/(-4*a*c+b^2)/(c*x^2+b*x+a)^(1/2)+e^2*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2 
+b*x+a)^(1/2))/c^(3/2)
 

Mathematica [A] (verified)

Time = 0.71 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.01 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \left (a b e^2+2 c^2 d^2 x+b^2 e^2 x+b c d (d-2 e x)-2 a c e (2 d+e x)\right )}{c \left (-b^2+4 a c\right ) \sqrt {a+x (b+c x)}}+\frac {2 e^2 \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{c^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.20, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1164, 25, 27, 1160, 1092, 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.

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

\(\Big \downarrow \) 1164

\(\displaystyle -\frac {2 \int -\frac {e (b d-2 a e+(2 c d-b e) x)}{\sqrt {c x^2+b x+a}}dx}{b^2-4 a c}-\frac {2 (d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \int \frac {e (b d-2 a e+(2 c d-b e) x)}{\sqrt {c x^2+b x+a}}dx}{b^2-4 a c}-\frac {2 (d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 e \int \frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}dx}{b^2-4 a c}-\frac {2 (d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {2 e \left (\frac {e \left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{2 c}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{c}\right )}{b^2-4 a c}-\frac {2 (d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {2 e \left (\frac {e \left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{c}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{c}\right )}{b^2-4 a c}-\frac {2 (d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {2 e \left (\frac {e \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{3/2}}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{c}\right )}{b^2-4 a c}-\frac {2 (d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1160

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol 
] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b 
*e)/(2*c)   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] 
 && NeQ[p, -1]
 

rule 1164

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x 
+ c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a* 
c))   Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2* 
c*d^2*(2*p + 3) + e*(b*e - 2*d*c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p 
+ 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 1] && Int 
QuadraticQ[a, b, c, d, e, m, p, x]
 

Maple [A] (verified)

Time = 1.02 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.77

method	result	size
default	\(\frac {2 d^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+e^{2} \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )+2 d e \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )\)	\(207\)

Input:

int((e*x+d)^2/(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

2*d^2*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+e^2*(-x/c/(c*x^2+b*x+a)^(1 
/2)-1/2*b/c*(-1/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x 
+a)^(1/2))+1/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))+2*d*e*(- 
1/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 229 vs. \(2 (107) = 214\).

Time = 0.19 (sec) , antiderivative size = 461, normalized size of antiderivative = 3.94 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\left [\frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} e^{2} x^{2} + {\left (b^{3} - 4 \, a b c\right )} e^{2} x + {\left (a b^{2} - 4 \, a^{2} c\right )} e^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (b c^{2} d^{2} - 4 \, a c^{2} d e + a b c e^{2} + {\left (2 \, c^{3} d^{2} - 2 \, b c^{2} d e + {\left (b^{2} c - 2 \, a c^{2}\right )} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{2 \, {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3} + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{2} + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x\right )}}, -\frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} e^{2} x^{2} + {\left (b^{3} - 4 \, a b c\right )} e^{2} x + {\left (a b^{2} - 4 \, a^{2} c\right )} e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (b c^{2} d^{2} - 4 \, a c^{2} d e + a b c e^{2} + {\left (2 \, c^{3} d^{2} - 2 \, b c^{2} d e + {\left (b^{2} c - 2 \, a c^{2}\right )} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{a b^{2} c^{2} - 4 \, a^{2} c^{3} + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{2} + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x}\right ] \] Input:

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

Output:

[1/2*(((b^2*c - 4*a*c^2)*e^2*x^2 + (b^3 - 4*a*b*c)*e^2*x + (a*b^2 - 4*a^2* 
c)*e^2)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*( 
2*c*x + b)*sqrt(c) - 4*a*c) - 4*(b*c^2*d^2 - 4*a*c^2*d*e + a*b*c*e^2 + (2* 
c^3*d^2 - 2*b*c^2*d*e + (b^2*c - 2*a*c^2)*e^2)*x)*sqrt(c*x^2 + b*x + a))/( 
a*b^2*c^2 - 4*a^2*c^3 + (b^2*c^3 - 4*a*c^4)*x^2 + (b^3*c^2 - 4*a*b*c^3)*x) 
, -(((b^2*c - 4*a*c^2)*e^2*x^2 + (b^3 - 4*a*b*c)*e^2*x + (a*b^2 - 4*a^2*c) 
*e^2)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2* 
x^2 + b*c*x + a*c)) + 2*(b*c^2*d^2 - 4*a*c^2*d*e + a*b*c*e^2 + (2*c^3*d^2 
- 2*b*c^2*d*e + (b^2*c - 2*a*c^2)*e^2)*x)*sqrt(c*x^2 + b*x + a))/(a*b^2*c^ 
2 - 4*a^2*c^3 + (b^2*c^3 - 4*a*c^4)*x^2 + (b^3*c^2 - 4*a*b*c^3)*x)]
 

Sympy [F]

\[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (d + e x\right )^{2}}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((e*x+d)**2/(c*x**2+b*x+a)**(3/2),x)
 

Output:

Integral((d + e*x)**2/(a + b*x + c*x**2)**(3/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x+d)^2/(c*x^2+b*x+a)^(3/2),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
 

Giac [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.13 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {e^{2} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{c^{\frac {3}{2}}} - \frac {2 \, {\left (\frac {{\left (2 \, c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2} - 2 \, a c e^{2}\right )} x}{b^{2} c - 4 \, a c^{2}} + \frac {b c d^{2} - 4 \, a c d e + a b e^{2}}{b^{2} c - 4 \, a c^{2}}\right )}}{\sqrt {c x^{2} + b x + a}} \] Input:

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

Output:

-e^2*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(3/2) - 
 2*((2*c^2*d^2 - 2*b*c*d*e + b^2*e^2 - 2*a*c*e^2)*x/(b^2*c - 4*a*c^2) + (b 
*c*d^2 - 4*a*c*d*e + a*b*e^2)/(b^2*c - 4*a*c^2))/sqrt(c*x^2 + b*x + a)
 

Mupad [B] (verification not implemented)

Time = 6.37 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.28 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {e^2\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )}{c^{3/2}}+\frac {d^2\,\left (\frac {b}{2}+c\,x\right )}{\left (a\,c-\frac {b^2}{4}\right )\,\sqrt {c\,x^2+b\,x+a}}-\frac {2\,d\,e\,\left (4\,a+2\,b\,x\right )}{\left (4\,a\,c-b^2\right )\,\sqrt {c\,x^2+b\,x+a}}+\frac {e^2\,\left (\frac {a\,b}{2}-x\,\left (a\,c-\frac {b^2}{2}\right )\right )}{c\,\left (a\,c-\frac {b^2}{4}\right )\,\sqrt {c\,x^2+b\,x+a}} \] Input:

int((d + e*x)^2/(a + b*x + c*x^2)^(3/2),x)
 

Output:

(e^2*log((b/2 + c*x)/c^(1/2) + (a + b*x + c*x^2)^(1/2)))/c^(3/2) + (d^2*(b 
/2 + c*x))/((a*c - b^2/4)*(a + b*x + c*x^2)^(1/2)) - (2*d*e*(4*a + 2*b*x)) 
/((4*a*c - b^2)*(a + b*x + c*x^2)^(1/2)) + (e^2*((a*b)/2 - x*(a*c - b^2/2) 
))/(c*(a*c - b^2/4)*(a + b*x + c*x^2)^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.58 (sec) , antiderivative size = 619, normalized size of antiderivative = 5.29 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {-\sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{2} e^{2}-\sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{3} e^{2} x -\sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{2} c \,e^{2} x^{2}+2 \sqrt {c \,x^{2}+b x +a}\, a b c \,e^{2}-8 \sqrt {c \,x^{2}+b x +a}\, a \,c^{2} d e -4 \sqrt {c \,x^{2}+b x +a}\, a \,c^{2} e^{2} x +2 \sqrt {c \,x^{2}+b x +a}\, b^{2} c \,e^{2} x +4 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a^{2} c \,e^{2}-4 \sqrt {c}\, a \,c^{2} e^{2} x^{2}+2 \sqrt {c}\, b^{2} c \,e^{2} x^{2}+4 \sqrt {c}\, b \,c^{2} d^{2} x -4 \sqrt {c \,x^{2}+b x +a}\, b \,c^{2} d e x +4 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a \,c^{2} e^{2} x^{2}-4 \sqrt {c}\, a b c d e -4 \sqrt {c}\, a b c \,e^{2} x -4 \sqrt {c}\, b^{2} c d e x -4 \sqrt {c}\, b \,c^{2} d e \,x^{2}+2 \sqrt {c \,x^{2}+b x +a}\, b \,c^{2} d^{2}+4 \sqrt {c \,x^{2}+b x +a}\, c^{3} d^{2} x -4 \sqrt {c}\, a^{2} c \,e^{2}+2 \sqrt {c}\, a \,b^{2} e^{2}+4 \sqrt {c}\, a \,c^{2} d^{2}+2 \sqrt {c}\, b^{3} e^{2} x +4 \sqrt {c}\, c^{3} d^{2} x^{2}+4 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a b c \,e^{2} x}{c^{2} \left (4 a \,c^{2} x^{2}-b^{2} c \,x^{2}+4 a b c x -b^{3} x +4 a^{2} c -a \,b^{2}\right )} \] Input:

int((e*x+d)^2/(c*x^2+b*x+a)^(3/2),x)
 

Output:

(2*sqrt(a + b*x + c*x**2)*a*b*c*e**2 - 8*sqrt(a + b*x + c*x**2)*a*c**2*d*e 
 - 4*sqrt(a + b*x + c*x**2)*a*c**2*e**2*x + 2*sqrt(a + b*x + c*x**2)*b**2* 
c*e**2*x + 2*sqrt(a + b*x + c*x**2)*b*c**2*d**2 - 4*sqrt(a + b*x + c*x**2) 
*b*c**2*d*e*x + 4*sqrt(a + b*x + c*x**2)*c**3*d**2*x + 4*sqrt(c)*log((2*sq 
rt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*c*e**2 
- sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - 
b**2))*a*b**2*e**2 + 4*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 
 2*c*x)/sqrt(4*a*c - b**2))*a*b*c*e**2*x + 4*sqrt(c)*log((2*sqrt(c)*sqrt(a 
 + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*a*c**2*e**2*x**2 - sqrt( 
c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))* 
b**3*e**2*x - sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/s 
qrt(4*a*c - b**2))*b**2*c*e**2*x**2 - 4*sqrt(c)*a**2*c*e**2 + 2*sqrt(c)*a* 
b**2*e**2 - 4*sqrt(c)*a*b*c*d*e - 4*sqrt(c)*a*b*c*e**2*x + 4*sqrt(c)*a*c** 
2*d**2 - 4*sqrt(c)*a*c**2*e**2*x**2 + 2*sqrt(c)*b**3*e**2*x - 4*sqrt(c)*b* 
*2*c*d*e*x + 2*sqrt(c)*b**2*c*e**2*x**2 + 4*sqrt(c)*b*c**2*d**2*x - 4*sqrt 
(c)*b*c**2*d*e*x**2 + 4*sqrt(c)*c**3*d**2*x**2)/(c**2*(4*a**2*c - a*b**2 + 
 4*a*b*c*x + 4*a*c**2*x**2 - b**3*x - b**2*c*x**2))