3.7.39 \(\int \frac {(d f+e f x)^3}{a+b (d+e x)^2+c (d+e x)^4} \, dx\) [639]

Optimal. Leaf size=87 \[ \frac {b f^3 \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 c \sqrt {b^2-4 a c} e}+\frac {f^3 \log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 c e} \]

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1156, 1128, 648, 632, 212, 642} \begin {gather*} \frac {b f^3 \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 c e \sqrt {b^2-4 a c}}+\frac {f^3 \log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 c e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*f + e*f*x)^3/(a + b*(d + e*x)^2 + c*(d + e*x)^4),x]

[Out]

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

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1128

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*(a +
b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]

Rule 1156

Int[(u_)^(m_.)*((a_.) + (b_.)*(v_)^2 + (c_.)*(v_)^4)^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m),
Subst[Int[x^m*(a + b*x^2 + c*x^(2*2))^p, x], x, v], x] /; FreeQ[{a, b, c, m, p}, x] && LinearPairQ[u, v, x]

Rubi steps

\begin {align*} \int \frac {(d f+e f x)^3}{a+b (d+e x)^2+c (d+e x)^4} \, dx &=\frac {f^3 \text {Subst}\left (\int \frac {x^3}{a+b x^2+c x^4} \, dx,x,d+e x\right )}{e}\\ &=\frac {f^3 \text {Subst}\left (\int \frac {x}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{2 e}\\ &=\frac {f^3 \text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{4 c e}-\frac {\left (b f^3\right ) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{4 c e}\\ &=\frac {f^3 \log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 c e}+\frac {\left (b f^3\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c (d+e x)^2\right )}{2 c e}\\ &=\frac {b f^3 \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 c \sqrt {b^2-4 a c} e}+\frac {f^3 \log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 c e}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 80, normalized size = 0.92 \begin {gather*} \frac {f^3 \left (-\frac {2 b \tan ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}+\log \left (a+b (d+e x)^2+c (d+e x)^4\right )\right )}{4 c e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d*f + e*f*x)^3/(a + b*(d + e*x)^2 + c*(d + e*x)^4),x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.27, size = 154, normalized size = 1.77

method	result	size
default	\(\frac {f^{3} \left (\munderset {\textit {\_R} =\RootOf \left (e^{4} c \,\textit {\_Z}^{4}+4 d \,e^{3} c \,\textit {\_Z}^{3}+\left (6 d^{2} e^{2} c +e^{2} b \right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 d e b \right ) \textit {\_Z} +d^{4} c +d^{2} b +a \right )}{\sum }\frac {\left (\textit {\_R}^{3} e^{3}+3 \textit {\_R}^{2} d \,e^{2}+3 \textit {\_R} \,d^{2} e +d^{3}\right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 d \,e^{2} c \,\textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 c \,d^{3}+e b \textit {\_R} +b d}\right )}{2 e}\)	\(154\)
risch	\(\frac {f^{3} \ln \left (\left (-4 a b c \,e^{2}+b^{3} e^{2}+\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \,e^{2}\right ) x^{2}+\left (-8 a b c d e +2 b^{3} d e +2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b d e \right ) x -4 a b c \,d^{2}+b^{3} d^{2}+\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \,d^{2}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a \right ) a}{\left (4 a c -b^{2}\right ) e}-\frac {f^{3} \ln \left (\left (-4 a b c \,e^{2}+b^{3} e^{2}+\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \,e^{2}\right ) x^{2}+\left (-8 a b c d e +2 b^{3} d e +2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b d e \right ) x -4 a b c \,d^{2}+b^{3} d^{2}+\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \,d^{2}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a \right ) b^{2}}{4 c \left (4 a c -b^{2}\right ) e}+\frac {f^{3} \ln \left (\left (-4 a b c \,e^{2}+b^{3} e^{2}+\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \,e^{2}\right ) x^{2}+\left (-8 a b c d e +2 b^{3} d e +2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b d e \right ) x -4 a b c \,d^{2}+b^{3} d^{2}+\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \,d^{2}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a \right ) \sqrt {-b^{2} \left (4 a c -b^{2}\right )}}{4 c \left (4 a c -b^{2}\right ) e}+\frac {f^{3} \ln \left (\left (-4 a b c \,e^{2}+b^{3} e^{2}-\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \,e^{2}\right ) x^{2}+\left (-8 a b c d e +2 b^{3} d e -2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b d e \right ) x -4 a b c \,d^{2}+b^{3} d^{2}-\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \,d^{2}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a \right ) a}{\left (4 a c -b^{2}\right ) e}-\frac {f^{3} \ln \left (\left (-4 a b c \,e^{2}+b^{3} e^{2}-\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \,e^{2}\right ) x^{2}+\left (-8 a b c d e +2 b^{3} d e -2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b d e \right ) x -4 a b c \,d^{2}+b^{3} d^{2}-\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \,d^{2}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a \right ) b^{2}}{4 c \left (4 a c -b^{2}\right ) e}-\frac {f^{3} \ln \left (\left (-4 a b c \,e^{2}+b^{3} e^{2}-\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \,e^{2}\right ) x^{2}+\left (-8 a b c d e +2 b^{3} d e -2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b d e \right ) x -4 a b c \,d^{2}+b^{3} d^{2}-\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \,d^{2}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a \right ) \sqrt {-b^{2} \left (4 a c -b^{2}\right )}}{4 c \left (4 a c -b^{2}\right ) e}\)	\(1020\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*f*x+d*f)^3/(a+b*(e*x+d)^2+c*(e*x+d)^4),x,method=_RETURNVERBOSE)

[Out]

1/2*f^3/e*sum((_R^3*e^3+3*_R^2*d*e^2+3*_R*d^2*e+d^3)/(2*_R^3*c*e^3+6*_R^2*c*d*e^2+6*_R*c*d^2*e+2*c*d^3+_R*b*e+
b*d)*ln(x-_R),_R=RootOf(e^4*c*_Z^4+4*d*e^3*c*_Z^3+(6*c*d^2*e^2+b*e^2)*_Z^2+(4*c*d^3*e+2*b*d*e)*_Z+d^4*c+d^2*b+
a))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*f*x+d*f)^3/(a+b*(e*x+d)^2+c*(e*x+d)^4),x, algorithm="maxima")

[Out]

integrate((f*x*e + d*f)^3/((x*e + d)^4*c + (x*e + d)^2*b + a), x)

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Fricas [A]
time = 0.36, size = 436, normalized size = 5.01 \begin {gather*} \left [\frac {{\left (\sqrt {b^{2} - 4 \, a c} b f^{3} \log \left (\frac {2 \, c^{2} x^{4} e^{4} + 8 \, c^{2} d x^{3} e^{3} + 2 \, c^{2} d^{4} + 2 \, b c d^{2} + 2 \, {\left (6 \, c^{2} d^{2} + b c\right )} x^{2} e^{2} + 4 \, {\left (2 \, c^{2} d^{3} + b c d\right )} x e + b^{2} - 2 \, a c + {\left (2 \, c x^{2} e^{2} + 4 \, c d x e + 2 \, c d^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} e^{4} + 4 \, c d x^{3} e^{3} + c d^{4} + {\left (6 \, c d^{2} + b\right )} x^{2} e^{2} + b d^{2} + 2 \, {\left (2 \, c d^{3} + b d\right )} x e + a}\right ) + {\left (b^{2} - 4 \, a c\right )} f^{3} \log \left (c x^{4} e^{4} + 4 \, c d x^{3} e^{3} + c d^{4} + {\left (6 \, c d^{2} + b\right )} x^{2} e^{2} + b d^{2} + 2 \, {\left (2 \, c d^{3} + b d\right )} x e + a\right )\right )} e^{\left (-1\right )}}{4 \, {\left (b^{2} c - 4 \, a c^{2}\right )}}, \frac {{\left (2 \, \sqrt {-b^{2} + 4 \, a c} b f^{3} \arctan \left (-\frac {{\left (2 \, c x^{2} e^{2} + 4 \, c d x e + 2 \, c d^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) + {\left (b^{2} - 4 \, a c\right )} f^{3} \log \left (c x^{4} e^{4} + 4 \, c d x^{3} e^{3} + c d^{4} + {\left (6 \, c d^{2} + b\right )} x^{2} e^{2} + b d^{2} + 2 \, {\left (2 \, c d^{3} + b d\right )} x e + a\right )\right )} e^{\left (-1\right )}}{4 \, {\left (b^{2} c - 4 \, a c^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*f*x+d*f)^3/(a+b*(e*x+d)^2+c*(e*x+d)^4),x, algorithm="fricas")

[Out]

[1/4*(sqrt(b^2 - 4*a*c)*b*f^3*log((2*c^2*x^4*e^4 + 8*c^2*d*x^3*e^3 + 2*c^2*d^4 + 2*b*c*d^2 + 2*(6*c^2*d^2 + b*
c)*x^2*e^2 + 4*(2*c^2*d^3 + b*c*d)*x*e + b^2 - 2*a*c + (2*c*x^2*e^2 + 4*c*d*x*e + 2*c*d^2 + b)*sqrt(b^2 - 4*a*
c))/(c*x^4*e^4 + 4*c*d*x^3*e^3 + c*d^4 + (6*c*d^2 + b)*x^2*e^2 + b*d^2 + 2*(2*c*d^3 + b*d)*x*e + a)) + (b^2 -
4*a*c)*f^3*log(c*x^4*e^4 + 4*c*d*x^3*e^3 + c*d^4 + (6*c*d^2 + b)*x^2*e^2 + b*d^2 + 2*(2*c*d^3 + b*d)*x*e + a))
*e^(-1)/(b^2*c - 4*a*c^2), 1/4*(2*sqrt(-b^2 + 4*a*c)*b*f^3*arctan(-(2*c*x^2*e^2 + 4*c*d*x*e + 2*c*d^2 + b)*sqr
t(-b^2 + 4*a*c)/(b^2 - 4*a*c)) + (b^2 - 4*a*c)*f^3*log(c*x^4*e^4 + 4*c*d*x^3*e^3 + c*d^4 + (6*c*d^2 + b)*x^2*e
^2 + b*d^2 + 2*(2*c*d^3 + b*d)*x*e + a))*e^(-1)/(b^2*c - 4*a*c^2)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 332 vs. \(2 (75) = 150\).
time = 1.03, size = 332, normalized size = 3.82 \begin {gather*} \left (- \frac {b f^{3} \sqrt {- 4 a c + b^{2}}}{4 c e \left (4 a c - b^{2}\right )} + \frac {f^{3}}{4 c e}\right ) \log {\left (\frac {2 d x}{e} + x^{2} + \frac {- 8 a c e \left (- \frac {b f^{3} \sqrt {- 4 a c + b^{2}}}{4 c e \left (4 a c - b^{2}\right )} + \frac {f^{3}}{4 c e}\right ) + 2 a f^{3} + 2 b^{2} e \left (- \frac {b f^{3} \sqrt {- 4 a c + b^{2}}}{4 c e \left (4 a c - b^{2}\right )} + \frac {f^{3}}{4 c e}\right ) + b d^{2} f^{3}}{b e^{2} f^{3}} \right )} + \left (\frac {b f^{3} \sqrt {- 4 a c + b^{2}}}{4 c e \left (4 a c - b^{2}\right )} + \frac {f^{3}}{4 c e}\right ) \log {\left (\frac {2 d x}{e} + x^{2} + \frac {- 8 a c e \left (\frac {b f^{3} \sqrt {- 4 a c + b^{2}}}{4 c e \left (4 a c - b^{2}\right )} + \frac {f^{3}}{4 c e}\right ) + 2 a f^{3} + 2 b^{2} e \left (\frac {b f^{3} \sqrt {- 4 a c + b^{2}}}{4 c e \left (4 a c - b^{2}\right )} + \frac {f^{3}}{4 c e}\right ) + b d^{2} f^{3}}{b e^{2} f^{3}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*f*x+d*f)**3/(a+b*(e*x+d)**2+c*(e*x+d)**4),x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (80) = 160\).
time = 4.13, size = 162, normalized size = 1.86 \begin {gather*} -\frac {b f^{3} \arctan \left (\frac {2 \, c d^{2} f + 2 \, {\left (f x^{2} e + 2 \, d f x\right )} c e + b f}{\sqrt {-b^{2} + 4 \, a c} f}\right ) e^{\left (-1\right )}}{2 \, \sqrt {-b^{2} + 4 \, a c} c} + \frac {f^{3} e^{\left (-1\right )} \log \left (c d^{4} f^{2} + 2 \, {\left (f x^{2} e + 2 \, d f x\right )} c d^{2} f e + b d^{2} f^{2} + {\left (f x^{2} e + 2 \, d f x\right )}^{2} c e^{2} + {\left (f x^{2} e + 2 \, d f x\right )} b f e + a f^{2}\right )}{4 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*f*x+d*f)^3/(a+b*(e*x+d)^2+c*(e*x+d)^4),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.44, size = 287, normalized size = 3.30 \begin {gather*} \frac {4\,a\,c\,e\,f^3\,\ln \left (c\,d^4+4\,c\,d^3\,e\,x+6\,c\,d^2\,e^2\,x^2+b\,d^2+4\,c\,d\,e^3\,x^3+2\,b\,d\,e\,x+c\,e^4\,x^4+b\,e^2\,x^2+a\right )}{16\,a\,c^2\,e^2-4\,b^2\,c\,e^2}-\frac {b\,f^3\,\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,d^2}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,e^2\,x^2}{\sqrt {4\,a\,c-b^2}}+\frac {4\,c\,d\,e\,x}{\sqrt {4\,a\,c-b^2}}\right )}{2\,c\,e\,\sqrt {4\,a\,c-b^2}}-\frac {b^2\,e\,f^3\,\ln \left (c\,d^4+4\,c\,d^3\,e\,x+6\,c\,d^2\,e^2\,x^2+b\,d^2+4\,c\,d\,e^3\,x^3+2\,b\,d\,e\,x+c\,e^4\,x^4+b\,e^2\,x^2+a\right )}{16\,a\,c^2\,e^2-4\,b^2\,c\,e^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*f + e*f*x)^3/(a + b*(d + e*x)^2 + c*(d + e*x)^4),x)

[Out]

(4*a*c*e*f^3*log(a + b*d^2 + c*d^4 + b*e^2*x^2 + c*e^4*x^4 + 2*b*d*e*x + 6*c*d^2*e^2*x^2 + 4*c*d^3*e*x + 4*c*d
*e^3*x^3))/(16*a*c^2*e^2 - 4*b^2*c*e^2) - (b*f^3*atan(b/(4*a*c - b^2)^(1/2) + (2*c*d^2)/(4*a*c - b^2)^(1/2) +
(2*c*e^2*x^2)/(4*a*c - b^2)^(1/2) + (4*c*d*e*x)/(4*a*c - b^2)^(1/2)))/(2*c*e*(4*a*c - b^2)^(1/2)) - (b^2*e*f^3
*log(a + b*d^2 + c*d^4 + b*e^2*x^2 + c*e^4*x^4 + 2*b*d*e*x + 6*c*d^2*e^2*x^2 + 4*c*d^3*e*x + 4*c*d*e^3*x^3))/(
16*a*c^2*e^2 - 4*b^2*c*e^2)

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