3.7.19 \(\int \frac {1}{(d+e x)^3 (a+b (d+e x)^2+c (d+e x)^4)} \, dx\) [619]

Optimal. Leaf size=121 \[ -\frac {1}{2 a e (d+e x)^2}-\frac {\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 \sqrt {b^2-4 a c} e}-\frac {b \log (d+e x)}{a^2 e}+\frac {b \log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^2 e} \]

[Out]

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

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Rubi [A]
time = 0.13, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1156, 1128, 723, 814, 648, 632, 212, 642} \begin {gather*} -\frac {\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 e \sqrt {b^2-4 a c}}+\frac {b \log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^2 e}-\frac {b \log (d+e x)}{a^2 e}-\frac {1}{2 a e (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)),x]

[Out]

-1/2*1/(a*e*(d + e*x)^2) - ((b^2 - 2*a*c)*ArcTanh[(b + 2*c*(d + e*x)^2)/Sqrt[b^2 - 4*a*c]])/(2*a^2*Sqrt[b^2 -
4*a*c]*e) - (b*Log[d + e*x])/(a^2*e) + (b*Log[a + b*(d + e*x)^2 + c*(d + e*x)^4])/(4*a^2*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 723

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

Rule 814

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[(d + e*x)^m*((f + g*x)/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

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

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Mathematica [A]
time = 0.09, size = 154, normalized size = 1.27 \begin {gather*} \frac {-\frac {2 a}{(d+e x)^2}-4 b \log (d+e x)+\frac {\left (b^2-2 a c+b \sqrt {b^2-4 a c}\right ) \log \left (b-\sqrt {b^2-4 a c}+2 c (d+e x)^2\right )}{\sqrt {b^2-4 a c}}+\frac {\left (-b^2+2 a c+b \sqrt {b^2-4 a c}\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c (d+e x)^2\right )}{\sqrt {b^2-4 a c}}}{4 a^2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)),x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.24, size = 213, normalized size = 1.76

method	result	size
default	\(-\frac {1}{2 a e \left (e x +d \right )^{2}}-\frac {b \ln \left (e x +d \right )}{e \,a^{2}}+\frac {\munderset {\textit {\_R} =\RootOf \left (e^{4} c \,\textit {\_Z}^{4}+4 d \,e^{3} c \,\textit {\_Z}^{3}+\left (6 d^{2} e^{2} c +e^{2} b \right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 d e b \right ) \textit {\_Z} +d^{4} c +d^{2} b +a \right )}{\sum }\frac {\left (b c \,e^{3} \textit {\_R}^{3}+3 b c d \,e^{2} \textit {\_R}^{2}+e \left (3 b c \,d^{2}-a c +b^{2}\right ) \textit {\_R} +b c \,d^{3}-a c d +b^{2} d \right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 d \,e^{2} c \,\textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 c \,d^{3}+e b \textit {\_R} +b d}}{2 a^{2} e}\)	\(213\)
risch	\(-\frac {1}{2 a e \left (e x +d \right )^{2}}-\frac {b \ln \left (e x +d \right )}{e \,a^{2}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (4 a^{3} c \,e^{2}-a^{2} b^{2} e^{2}\right ) \textit {\_Z}^{2}+\left (-4 a b c e +b^{3} e \right ) \textit {\_Z} +c^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (10 a^{3} c \,e^{4}-3 a^{2} b^{2} e^{4}\right ) \textit {\_R}^{2}-4 a b c \,e^{3} \textit {\_R} +2 c^{2} e^{2}\right ) x^{2}+\left (\left (20 a^{3} c d \,e^{3}-6 a^{2} b^{2} d \,e^{3}\right ) \textit {\_R}^{2}-8 a b c d \,e^{2} \textit {\_R} +4 c^{2} d e \right ) x +\left (10 a^{3} c \,d^{2} e^{2}-3 a^{2} b^{2} d^{2} e^{2}-a^{3} b \,e^{2}\right ) \textit {\_R}^{2}+\left (-4 a b c \,d^{2} e +a^{2} c e -2 a \,b^{2} e \right ) \textit {\_R} +2 c^{2} d^{2}+2 b c \right )\right )}{2}\)	\(256\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^3/(a+b*(e*x+d)^2+c*(e*x+d)^4),x,method=_RETURNVERBOSE)

[Out]

-1/2/a/e/(e*x+d)^2-b*ln(e*x+d)/e/a^2+1/2/a^2/e*sum((b*c*e^3*_R^3+3*b*c*d*e^2*_R^2+e*(3*b*c*d^2-a*c+b^2)*_R+b*c
*d^3-a*c*d+b^2*d)/(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(1/(e*x+d)^3/(a+b*(e*x+d)^2+c*(e*x+d)^4),x, algorithm="maxima")

[Out]

-b*e^(-1)*log(x*e + d)/a^2 - 1/2/(a*x^2*e^3 + 2*a*d*x*e^2 + a*d^2*e) + integrate((b*c*x^3*e^3 + 3*b*c*d*x^2*e^
2 + b*c*d^3 + (b^2 - a*c)*d + (3*b*c*d^2*e + b^2*e - a*c*e)*x)/(c*x^4*e^4 + 4*c*d*x^3*e^3 + c*d^4 + b*d^2 + (6
*c*d^2*e^2 + b*e^2)*x^2 + 2*(2*c*d^3*e + b*d*e)*x + a), x)/a^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (112) = 224\).
time = 0.41, size = 802, normalized size = 6.63 \begin {gather*} \left [-\frac {2 \, a b^{2} - 8 \, a^{2} c + {\left ({\left (b^{2} - 2 \, a c\right )} x^{2} e^{2} + 2 \, {\left (b^{2} - 2 \, a c\right )} d x e + {\left (b^{2} - 2 \, a c\right )} d^{2}\right )} \sqrt {b^{2} - 4 \, a c} \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 ({\left (b^{3} - 4 \, a b c\right )} x^{2} e^{2} + 2 \, {\left (b^{3} - 4 \, a b c\right )} d x e + {\left (b^{3} - 4 \, a b c\right )} d^{2}\right )} \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 ) + 4 \, {\left ({\left (b^{3} - 4 \, a b c\right )} x^{2} e^{2} + 2 \, {\left (b^{3} - 4 \, a b c\right )} d x e + {\left (b^{3} - 4 \, a b c\right )} d^{2}\right )} \log \left (x e + d\right )}{4 \, {\left ({\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x^{2} e^{3} + 2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d x e^{2} + {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d^{2} e\right )}}, -\frac {2 \, a b^{2} - 8 \, a^{2} c + 2 \, {\left ({\left (b^{2} - 2 \, a c\right )} x^{2} e^{2} + 2 \, {\left (b^{2} - 2 \, a c\right )} d x e + {\left (b^{2} - 2 \, a c\right )} d^{2}\right )} \sqrt {-b^{2} + 4 \, a c} \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 ({\left (b^{3} - 4 \, a b c\right )} x^{2} e^{2} + 2 \, {\left (b^{3} - 4 \, a b c\right )} d x e + {\left (b^{3} - 4 \, a b c\right )} d^{2}\right )} \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 ) + 4 \, {\left ({\left (b^{3} - 4 \, a b c\right )} x^{2} e^{2} + 2 \, {\left (b^{3} - 4 \, a b c\right )} d x e + {\left (b^{3} - 4 \, a b c\right )} d^{2}\right )} \log \left (x e + d\right )}{4 \, {\left ({\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x^{2} e^{3} + 2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d x e^{2} + {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d^{2} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(a+b*(e*x+d)^2+c*(e*x+d)^4),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**3/(a+b*(e*x+d)**2+c*(e*x+d)**4),x)

[Out]

Timed out

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Giac [A]
time = 3.83, size = 102, normalized size = 0.84 \begin {gather*} \frac {b e^{\left (-1\right )} \log \left (c + \frac {b}{{\left (x e + d\right )}^{2}} + \frac {a}{{\left (x e + d\right )}^{4}}\right )}{4 \, a^{2}} + \frac {{\left (b^{2} - 2 \, a c\right )} \arctan \left (-\frac {b + \frac {2 \, a}{{\left (x e + d\right )}^{2}}}{\sqrt {-b^{2} + 4 \, a c}}\right ) e^{\left (-1\right )}}{2 \, \sqrt {-b^{2} + 4 \, a c} a^{2}} - \frac {e^{\left (-1\right )}}{2 \, {\left (x e + d\right )}^{2} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(a+b*(e*x+d)^2+c*(e*x+d)^4),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 5.86, size = 2500, normalized size = 20.66 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan((16*a^6*x^2*(4*a*c - b^2)^(3/2)*(((3*b^4 + a^2*c^2 - 9*a*b^2*c)*((((((20*a^3*c^4*e^18 + 2*a^2*b^2*c^3*e^
18)/a^3 + ((2*b^3*e - 8*a*b*c*e)*(40*a^4*b*c^3*e^19 - 12*a^3*b^3*c^2*e^19))/(2*a^3*(16*a^3*c*e^2 - 4*a^2*b^2*e
^2)))*(2*b^3*e - 8*a*b*c*e))/(2*(16*a^3*c*e^2 - 4*a^2*b^2*e^2)) + (6*b*c^4*e^17)/a^2)*(2*b^3*e - 8*a*b*c*e))/(
2*(16*a^3*c*e^2 - 4*a^2*b^2*e^2)) + (c^5*e^16)/a^3 - (((((20*a^3*c^4*e^18 + 2*a^2*b^2*c^3*e^18)/a^3 + ((2*b^3*
e - 8*a*b*c*e)*(40*a^4*b*c^3*e^19 - 12*a^3*b^3*c^2*e^19))/(2*a^3*(16*a^3*c*e^2 - 4*a^2*b^2*e^2)))*(2*a*c - b^2
))/(4*a^2*e*(4*a*c - b^2)^(1/2)) + ((2*a*c - b^2)*(2*b^3*e - 8*a*b*c*e)*(40*a^4*b*c^3*e^19 - 12*a^3*b^3*c^2*e^
19))/(8*a^5*e*(16*a^3*c*e^2 - 4*a^2*b^2*e^2)*(4*a*c - b^2)^(1/2)))*(2*a*c - b^2))/(4*a^2*e*(4*a*c - b^2)^(1/2)
) - ((2*a*c - b^2)^2*(2*b^3*e - 8*a*b*c*e)*(40*a^4*b*c^3*e^19 - 12*a^3*b^3*c^2*e^19))/(32*a^7*e^2*(16*a^3*c*e^
2 - 4*a^2*b^2*e^2)*(4*a*c - b^2))))/(8*a^3*c^2*(a^2*c^2 - 6*b^4 + 24*a*b^2*c)) + (((((((20*a^3*c^4*e^18 + 2*a^
2*b^2*c^3*e^18)/a^3 + ((2*b^3*e - 8*a*b*c*e)*(40*a^4*b*c^3*e^19 - 12*a^3*b^3*c^2*e^19))/(2*a^3*(16*a^3*c*e^2 -
4*a^2*b^2*e^2)))*(2*a*c - b^2))/(4*a^2*e*(4*a*c - b^2)^(1/2)) + ((2*a*c - b^2)*(2*b^3*e - 8*a*b*c*e)*(40*a^4*
b*c^3*e^19 - 12*a^3*b^3*c^2*e^19))/(8*a^5*e*(16*a^3*c*e^2 - 4*a^2*b^2*e^2)*(4*a*c - b^2)^(1/2)))*(2*b^3*e - 8*
a*b*c*e))/(2*(16*a^3*c*e^2 - 4*a^2*b^2*e^2)) + (((((20*a^3*c^4*e^18 + 2*a^2*b^2*c^3*e^18)/a^3 + ((2*b^3*e - 8*
a*b*c*e)*(40*a^4*b*c^3*e^19 - 12*a^3*b^3*c^2*e^19))/(2*a^3*(16*a^3*c*e^2 - 4*a^2*b^2*e^2)))*(2*b^3*e - 8*a*b*c
*e))/(2*(16*a^3*c*e^2 - 4*a^2*b^2*e^2)) + (6*b*c^4*e^17)/a^2)*(2*a*c - b^2))/(4*a^2*e*(4*a*c - b^2)^(1/2)) - (
(2*a*c - b^2)^3*(40*a^4*b*c^3*e^19 - 12*a^3*b^3*c^2*e^19))/(64*a^9*e^3*(4*a*c - b^2)^(3/2)))*(3*b^5 + 13*a^2*b
*c^2 - 15*a*b^3*c))/(8*a^3*c^2*(4*a*c - b^2)^(1/2)*(a^2*c^2 - 6*b^4 + 24*a*b^2*c))))/(4*a^2*c^4*e^14 + b^4*c^2
*e^14 - 4*a*b^2*c^3*e^14) + (16*a^6*x*(((3*b^4 + a^2*c^2 - 9*a*b^2*c)*(((((2*b^3*e - 8*a*b*c*e)*((2*(20*a^3*c^
4*d*e^17 + 2*a^2*b^2*c^3*d*e^17))/a^3 + ((40*a^4*b*c^3*d*e^18 - 12*a^3*b^3*c^2*d*e^18)*(2*b^3*e - 8*a*b*c*e))/
(a^3*(16*a^3*c*e^2 - 4*a^2*b^2*e^2))))/(2*(16*a^3*c*e^2 - 4*a^2*b^2*e^2)) + (12*b*c^4*d*e^16)/a^2)*(2*b^3*e -
8*a*b*c*e))/(2*(16*a^3*c*e^2 - 4*a^2*b^2*e^2)) + (2*c^5*d*e^15)/a^3 - ((((2*a*c - b^2)*((2*(20*a^3*c^4*d*e^17
+ 2*a^2*b^2*c^3*d*e^17))/a^3 + ((40*a^4*b*c^3*d*e^18 - 12*a^3*b^3*c^2*d*e^18)*(2*b^3*e - 8*a*b*c*e))/(a^3*(16*
a^3*c*e^2 - 4*a^2*b^2*e^2))))/(4*a^2*e*(4*a*c - b^2)^(1/2)) + ((40*a^4*b*c^3*d*e^18 - 12*a^3*b^3*c^2*d*e^18)*(
2*a*c - b^2)*(2*b^3*e - 8*a*b*c*e))/(4*a^5*e*(16*a^3*c*e^2 - 4*a^2*b^2*e^2)*(4*a*c - b^2)^(1/2)))*(2*a*c - b^2
))/(4*a^2*e*(4*a*c - b^2)^(1/2)) - ((40*a^4*b*c^3*d*e^18 - 12*a^3*b^3*c^2*d*e^18)*(2*a*c - b^2)^2*(2*b^3*e - 8
*a*b*c*e))/(16*a^7*e^2*(16*a^3*c*e^2 - 4*a^2*b^2*e^2)*(4*a*c - b^2))))/(8*a^3*c^2*(a^2*c^2 - 6*b^4 + 24*a*b^2*
c)) + ((((((2*a*c - b^2)*((2*(20*a^3*c^4*d*e^17 + 2*a^2*b^2*c^3*d*e^17))/a^3 + ((40*a^4*b*c^3*d*e^18 - 12*a^3*
b^3*c^2*d*e^18)*(2*b^3*e - 8*a*b*c*e))/(a^3*(16*a^3*c*e^2 - 4*a^2*b^2*e^2))))/(4*a^2*e*(4*a*c - b^2)^(1/2)) +
((40*a^4*b*c^3*d*e^18 - 12*a^3*b^3*c^2*d*e^18)*(2*a*c - b^2)*(2*b^3*e - 8*a*b*c*e))/(4*a^5*e*(16*a^3*c*e^2 - 4
*a^2*b^2*e^2)*(4*a*c - b^2)^(1/2)))*(2*b^3*e - 8*a*b*c*e))/(2*(16*a^3*c*e^2 - 4*a^2*b^2*e^2)) + ((((2*b^3*e -
8*a*b*c*e)*((2*(20*a^3*c^4*d*e^17 + 2*a^2*b^2*c^3*d*e^17))/a^3 + ((40*a^4*b*c^3*d*e^18 - 12*a^3*b^3*c^2*d*e^18
)*(2*b^3*e - 8*a*b*c*e))/(a^3*(16*a^3*c*e^2 - 4*a^2*b^2*e^2))))/(2*(16*a^3*c*e^2 - 4*a^2*b^2*e^2)) + (12*b*c^4
*d*e^16)/a^2)*(2*a*c - b^2))/(4*a^2*e*(4*a*c - b^2)^(1/2)) - ((40*a^4*b*c^3*d*e^18 - 12*a^3*b^3*c^2*d*e^18)*(2
*a*c - b^2)^3)/(32*a^9*e^3*(4*a*c - b^2)^(3/2)))*(3*b^5 + 13*a^2*b*c^2 - 15*a*b^3*c))/(8*a^3*c^2*(4*a*c - b^2)
^(1/2)*(a^2*c^2 - 6*b^4 + 24*a*b^2*c)))*(4*a*c - b^2)^(3/2))/(4*a^2*c^4*e^14 + b^4*c^2*e^14 - 4*a*b^2*c^3*e^14
) + (2*a^3*(4*a*c - b^2)^(3/2)*(3*b^4 + a^2*c^2 - 9*a*b^2*c)*((b*c^4*e^14 + c^5*d^2*e^14)/a^3 + (((4*a*b^2*c^3
*e^15 - a^2*c^4*e^15 + 6*a*b*c^4*d^2*e^15)/a^3 + (((4*a^2*b^3*c^2*e^16 - 4*a^3*b*c^3*e^16 + 20*a^3*c^4*d^2*e^1
6 + 2*a^2*b^2*c^3*d^2*e^16)/a^3 - ((2*b^3*e - 8*a*b*c*e)*(4*a^4*b^2*c^2*e^17 + 12*a^3*b^3*c^2*d^2*e^17 - 40*a^
4*b*c^3*d^2*e^17))/(2*a^3*(16*a^3*c*e^2 - 4*a^2*b^2*e^2)))*(2*b^3*e - 8*a*b*c*e))/(2*(16*a^3*c*e^2 - 4*a^2*b^2
*e^2)))*(2*b^3*e - 8*a*b*c*e))/(2*(16*a^3*c*e^2 - 4*a^2*b^2*e^2)) - ((2*a*c - b^2)*((((4*a^2*b^3*c^2*e^16 - 4*
a^3*b*c^3*e^16 + 20*a^3*c^4*d^2*e^16 + 2*a^2*b^2*c^3*d^2*e^16)/a^3 - ((2*b^3*e - 8*a*b*c*e)*(4*a^4*b^2*c^2*e^1
7 + 12*a^3*b^3*c^2*d^2*e^17 - 40*a^4*b*c^3*d^2*e^17))/(2*a^3*(16*a^3*c*e^2 - 4*a^2*b^2*e^2)))*(2*a*c - b^2))/(
4*a^2*e*(4*a*c - b^2)^(1/2)) - ((2*a*c - b^2)*(2*b^3*e - 8*a*b*c*e)*(4*a^4*b^2*c^2*e^17 + 12*a^3*b^3*c^2*d^2*e
^17 - 40*a^4*b*c^3*d^2*e^17))/(8*a^5*e*(16*a^3*c*e^2 - 4*a^2*b^2*e^2)*(4*a*c - b^2)^(1/2))))/(4*a^2*e*(4*a*c -
b^2)^(1/2)) + ((2*a*c - b^2)^2*(2*b^3*e - 8*a*b*c*e)*(4*a^4*b^2*c^2*e^17 + 12*a^3*b^3*c^2*d^2*e^17 - 40*a^4*b
*c^3*d^2*e^17))/(32*a^7*e^2*(16*a^3*c*e^2 - 4*a...

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