\(\int \frac {1}{(d+e x^2)^2 (a+c x^4)} \, dx\) [143]

Optimal result

Integrand size = 19, antiderivative size = 453 \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a+c x^4\right )} \, dx=\frac {e^2 x}{2 d \left (c d^2+a e^2\right ) \left (d+e x^2\right )}+\frac {2 c \sqrt {d} e^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\left (c d^2+a e^2\right )^2}+\frac {e^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \left (c d^2+a e^2\right )}-\frac {c^{3/4} \left (c d^2-2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^2}+\frac {c^{3/4} \left (c d^2-2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^2}-\frac {c^{3/4} \left (c d^2+2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^2}+\frac {c^{3/4} \left (c d^2+2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^2} \]

[Out]

1/2*e^2*x/d/(a*e^2+c*d^2)/(e*x^2+d)+1/2*e^(3/2)*arctan(x*e^(1/2)/d^(1/2))/d^(3/2)/(a*e^2+c*d^2)+1/4*c^(3/4)*ar
ctan(-1+c^(1/4)*x*2^(1/2)/a^(1/4))*(c*d^2-a*e^2-2*d*e*a^(1/2)*c^(1/2))/a^(3/4)/(a*e^2+c*d^2)^2*2^(1/2)+1/4*c^(
3/4)*arctan(1+c^(1/4)*x*2^(1/2)/a^(1/4))*(c*d^2-a*e^2-2*d*e*a^(1/2)*c^(1/2))/a^(3/4)/(a*e^2+c*d^2)^2*2^(1/2)-1
/8*c^(3/4)*ln(-a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(c*d^2-a*e^2+2*d*e*a^(1/2)*c^(1/2))/a^(3/4)/(a*e
^2+c*d^2)^2*2^(1/2)+1/8*c^(3/4)*ln(a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(c*d^2-a*e^2+2*d*e*a^(1/2)*c
^(1/2))/a^(3/4)/(a*e^2+c*d^2)^2*2^(1/2)+2*c*e^(3/2)*arctan(x*e^(1/2)/d^(1/2))*d^(1/2)/(a*e^2+c*d^2)^2

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {1185, 205, 211, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a+c x^4\right )} \, dx=-\frac {c^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) \left (-2 \sqrt {a} \sqrt {c} d e-a e^2+c d^2\right )}{2 \sqrt {2} a^{3/4} \left (a e^2+c d^2\right )^2}+\frac {c^{3/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) \left (-2 \sqrt {a} \sqrt {c} d e-a e^2+c d^2\right )}{2 \sqrt {2} a^{3/4} \left (a e^2+c d^2\right )^2}-\frac {c^{3/4} \left (2 \sqrt {a} \sqrt {c} d e-a e^2+c d^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (a e^2+c d^2\right )^2}+\frac {c^{3/4} \left (2 \sqrt {a} \sqrt {c} d e-a e^2+c d^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (a e^2+c d^2\right )^2}+\frac {2 c \sqrt {d} e^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\left (a e^2+c d^2\right )^2}+\frac {e^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \left (a e^2+c d^2\right )}+\frac {e^2 x}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )} \]

[In]

Int[1/((d + e*x^2)^2*(a + c*x^4)),x]

[Out]

(e^2*x)/(2*d*(c*d^2 + a*e^2)*(d + e*x^2)) + (2*c*Sqrt[d]*e^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(c*d^2 + a*e^2)^
2 + (e^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(3/2)*(c*d^2 + a*e^2)) - (c^(3/4)*(c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*
e - a*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(c*d^2 + a*e^2)^2) + (c^(3/4)*(c*d^2 -
2*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(c*d^2 + a*e^2)^2)
- (c^(3/4)*(c*d^2 + 2*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*
Sqrt[2]*a^(3/4)*(c*d^2 + a*e^2)^2) + (c^(3/4)*(c*d^2 + 2*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*Log[Sqrt[a] + Sqrt[2]*a^
(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(3/4)*(c*d^2 + a*e^2)^2)

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
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 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1182

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(-a)*c]

Rule 1185

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q/(a + c*x^
4), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[q]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {e^2}{\left (c d^2+a e^2\right ) \left (d+e x^2\right )^2}+\frac {2 c d e^2}{\left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}+\frac {c \left (c d^2-a e^2-2 c d e x^2\right )}{\left (c d^2+a e^2\right )^2 \left (a+c x^4\right )}\right ) \, dx \\ & = \frac {c \int \frac {c d^2-a e^2-2 c d e x^2}{a+c x^4} \, dx}{\left (c d^2+a e^2\right )^2}+\frac {\left (2 c d e^2\right ) \int \frac {1}{d+e x^2} \, dx}{\left (c d^2+a e^2\right )^2}+\frac {e^2 \int \frac {1}{\left (d+e x^2\right )^2} \, dx}{c d^2+a e^2} \\ & = \frac {e^2 x}{2 d \left (c d^2+a e^2\right ) \left (d+e x^2\right )}+\frac {2 c \sqrt {d} e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\left (c d^2+a e^2\right )^2}+\frac {\left (\sqrt {c} \left (c d^2-2 \sqrt {a} \sqrt {c} d e-a e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{2 \sqrt {a} \left (c d^2+a e^2\right )^2}+\frac {\left (\sqrt {c} \left (c d^2+2 \sqrt {a} \sqrt {c} d e-a e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{2 \sqrt {a} \left (c d^2+a e^2\right )^2}+\frac {e^2 \int \frac {1}{d+e x^2} \, dx}{2 d \left (c d^2+a e^2\right )} \\ & = \frac {e^2 x}{2 d \left (c d^2+a e^2\right ) \left (d+e x^2\right )}+\frac {2 c \sqrt {d} e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\left (c d^2+a e^2\right )^2}+\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \left (c d^2+a e^2\right )}+\frac {\left (\sqrt {c} \left (c d^2-2 \sqrt {a} \sqrt {c} d e-a e^2\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 \sqrt {a} \left (c d^2+a e^2\right )^2}+\frac {\left (\sqrt {c} \left (c d^2-2 \sqrt {a} \sqrt {c} d e-a e^2\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 \sqrt {a} \left (c d^2+a e^2\right )^2}-\frac {\left (c^{3/4} \left (c d^2+2 \sqrt {a} \sqrt {c} d e-a e^2\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^2}-\frac {\left (c^{3/4} \left (c d^2+2 \sqrt {a} \sqrt {c} d e-a e^2\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^2} \\ & = \frac {e^2 x}{2 d \left (c d^2+a e^2\right ) \left (d+e x^2\right )}+\frac {2 c \sqrt {d} e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\left (c d^2+a e^2\right )^2}+\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \left (c d^2+a e^2\right )}-\frac {c^{3/4} \left (c d^2+2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^2}+\frac {c^{3/4} \left (c d^2+2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^2}+\frac {\left (c^{3/4} \left (c d^2-2 \sqrt {a} \sqrt {c} d e-a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^2}-\frac {\left (c^{3/4} \left (c d^2-2 \sqrt {a} \sqrt {c} d e-a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^2} \\ & = \frac {e^2 x}{2 d \left (c d^2+a e^2\right ) \left (d+e x^2\right )}+\frac {2 c \sqrt {d} e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\left (c d^2+a e^2\right )^2}+\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \left (c d^2+a e^2\right )}-\frac {c^{3/4} \left (c d^2-2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^2}+\frac {c^{3/4} \left (c d^2-2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^2}-\frac {c^{3/4} \left (c d^2+2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^2}+\frac {c^{3/4} \left (c d^2+2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 362, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a+c x^4\right )} \, dx=\frac {\frac {4 e^2 \left (c d^2+a e^2\right ) x}{d \left (d+e x^2\right )}+\frac {4 e^{3/2} \left (5 c d^2+a e^2\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2}}+\frac {2 \sqrt {2} c^{3/4} \left (-c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{3/4}}-\frac {2 \sqrt {2} c^{3/4} \left (-c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{3/4}}+\frac {\sqrt {2} c^{3/4} \left (-c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{3/4}}+\frac {\sqrt {2} c^{3/4} \left (c d^2+2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{3/4}}}{8 \left (c d^2+a e^2\right )^2} \]

[In]

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

[Out]

((4*e^2*(c*d^2 + a*e^2)*x)/(d*(d + e*x^2)) + (4*e^(3/2)*(5*c*d^2 + a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/d^(3/2)
+ (2*Sqrt[2]*c^(3/4)*(-(c*d^2) + 2*Sqrt[a]*Sqrt[c]*d*e + a*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(3
/4) - (2*Sqrt[2]*c^(3/4)*(-(c*d^2) + 2*Sqrt[a]*Sqrt[c]*d*e + a*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a
^(3/4) + (Sqrt[2]*c^(3/4)*(-(c*d^2) - 2*Sqrt[a]*Sqrt[c]*d*e + a*e^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x +
Sqrt[c]*x^2])/a^(3/4) + (Sqrt[2]*c^(3/4)*(c*d^2 + 2*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4
)*c^(1/4)*x + Sqrt[c]*x^2])/a^(3/4))/(8*(c*d^2 + a*e^2)^2)

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.68

method	result	size
default	\(-\frac {c \left (\frac {\left (a \,e^{2}-c \,d^{2}\right ) \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {d e \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{\left (a \,e^{2}+c \,d^{2}\right )^{2}}+\frac {e^{2} \left (\frac {\left (a \,e^{2}+c \,d^{2}\right ) x}{2 d \left (e \,x^{2}+d \right )}+\frac {\left (a \,e^{2}+5 c \,d^{2}\right ) \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{2 d \sqrt {e d}}\right )}{\left (a \,e^{2}+c \,d^{2}\right )^{2}}\)	\(307\)
risch	\(\text {Expression too large to display}\)	\(2526\)

[In]

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

[Out]

-c/(a*e^2+c*d^2)^2*(1/8*(a*e^2-c*d^2)*(a/c)^(1/4)/a*2^(1/2)*(ln((x^2+(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2))/(x^2-(
a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2)))+2*arctan(2^(1/2)/(a/c)^(1/4)*x+1)+2*arctan(2^(1/2)/(a/c)^(1/4)*x-1))+1/4*d*
e/(a/c)^(1/4)*2^(1/2)*(ln((x^2-(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2))/(x^2+(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2)))+2*a
rctan(2^(1/2)/(a/c)^(1/4)*x+1)+2*arctan(2^(1/2)/(a/c)^(1/4)*x-1)))+e^2/(a*e^2+c*d^2)^2*(1/2*(a*e^2+c*d^2)/d*x/
(e*x^2+d)+1/2*(a*e^2+5*c*d^2)/d/(e*d)^(1/2)*arctan(e*x/(e*d)^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4193 vs. \(2 (352) = 704\).

Time = 9.78 (sec) , antiderivative size = 8409, normalized size of antiderivative = 18.56 \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a+c x^4\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a+c x^4\right )} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a+c x^4\right )} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 531, normalized size of antiderivative = 1.17 \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a+c x^4\right )} \, dx=\frac {e^{2} x}{2 \, {\left (c d^{3} + a d e^{2}\right )} {\left (e x^{2} + d\right )}} + \frac {{\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} - \left (a c^{3}\right )^{\frac {1}{4}} a c e^{2} - 2 \, \left (a c^{3}\right )^{\frac {3}{4}} d e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{2 \, {\left (\sqrt {2} a c^{3} d^{4} + 2 \, \sqrt {2} a^{2} c^{2} d^{2} e^{2} + \sqrt {2} a^{3} c e^{4}\right )}} + \frac {{\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} - \left (a c^{3}\right )^{\frac {1}{4}} a c e^{2} - 2 \, \left (a c^{3}\right )^{\frac {3}{4}} d e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{2 \, {\left (\sqrt {2} a c^{3} d^{4} + 2 \, \sqrt {2} a^{2} c^{2} d^{2} e^{2} + \sqrt {2} a^{3} c e^{4}\right )}} + \frac {{\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} - \left (a c^{3}\right )^{\frac {1}{4}} a c e^{2} + 2 \, \left (a c^{3}\right )^{\frac {3}{4}} d e\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{4 \, {\left (\sqrt {2} a c^{3} d^{4} + 2 \, \sqrt {2} a^{2} c^{2} d^{2} e^{2} + \sqrt {2} a^{3} c e^{4}\right )}} - \frac {{\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} - \left (a c^{3}\right )^{\frac {1}{4}} a c e^{2} + 2 \, \left (a c^{3}\right )^{\frac {3}{4}} d e\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{4 \, {\left (\sqrt {2} a c^{3} d^{4} + 2 \, \sqrt {2} a^{2} c^{2} d^{2} e^{2} + \sqrt {2} a^{3} c e^{4}\right )}} + \frac {{\left (5 \, c d^{2} e^{2} + a e^{4}\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \, {\left (c^{2} d^{5} + 2 \, a c d^{3} e^{2} + a^{2} d e^{4}\right )} \sqrt {d e}} \]

[In]

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

[Out]

1/2*e^2*x/((c*d^3 + a*d*e^2)*(e*x^2 + d)) + 1/2*((a*c^3)^(1/4)*c^2*d^2 - (a*c^3)^(1/4)*a*c*e^2 - 2*(a*c^3)^(3/
4)*d*e)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(sqrt(2)*a*c^3*d^4 + 2*sqrt(2)*a^2*c^2*d^2
*e^2 + sqrt(2)*a^3*c*e^4) + 1/2*((a*c^3)^(1/4)*c^2*d^2 - (a*c^3)^(1/4)*a*c*e^2 - 2*(a*c^3)^(3/4)*d*e)*arctan(1
/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(sqrt(2)*a*c^3*d^4 + 2*sqrt(2)*a^2*c^2*d^2*e^2 + sqrt(2)*a
^3*c*e^4) + 1/4*((a*c^3)^(1/4)*c^2*d^2 - (a*c^3)^(1/4)*a*c*e^2 + 2*(a*c^3)^(3/4)*d*e)*log(x^2 + sqrt(2)*x*(a/c
)^(1/4) + sqrt(a/c))/(sqrt(2)*a*c^3*d^4 + 2*sqrt(2)*a^2*c^2*d^2*e^2 + sqrt(2)*a^3*c*e^4) - 1/4*((a*c^3)^(1/4)*
c^2*d^2 - (a*c^3)^(1/4)*a*c*e^2 + 2*(a*c^3)^(3/4)*d*e)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(sqrt(2)*a
*c^3*d^4 + 2*sqrt(2)*a^2*c^2*d^2*e^2 + sqrt(2)*a^3*c*e^4) + 1/2*(5*c*d^2*e^2 + a*e^4)*arctan(e*x/sqrt(d*e))/((
c^2*d^5 + 2*a*c*d^3*e^2 + a^2*d*e^4)*sqrt(d*e))

Mupad [B] (verification not implemented)

Time = 16.80 (sec) , antiderivative size = 16369, normalized size of antiderivative = 36.13 \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a+c x^4\right )} \, dx=\text {Too large to display} \]

[In]

int(1/((a + c*x^4)*(d + e*x^2)^2),x)

[Out]

(e^2*x)/(2*d*(d + e*x^2)*(a*e^2 + c*d^2)) - atan(((((((256*a^8*c^4*d*e^16 - 128*a*c^11*d^15*e^2 + 256*a^2*c^10
*d^13*e^4 + 3456*a^3*c^9*d^11*e^6 + 8960*a^4*c^8*d^9*e^8 + 10880*a^5*c^7*d^7*e^10 + 6912*a^6*c^6*d^5*e^12 + 21
76*a^7*c^5*d^3*e^14)/(2*(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4)) + (x
*((a^2*e^4*(-a^3*c^3)^(1/2) + c^2*d^4*(-a^3*c^3)^(1/2) + 4*a^2*c^3*d^3*e - 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a
^3*c^3)^(1/2))/(16*(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))^(1/2)*(
512*a^2*c^11*d^16*e^3 + 2560*a^3*c^10*d^14*e^5 + 4608*a^4*c^9*d^12*e^7 + 2560*a^5*c^8*d^10*e^9 - 2560*a^6*c^7*
d^8*e^11 - 4608*a^7*c^6*d^6*e^13 - 2560*a^8*c^5*d^4*e^15 - 512*a^9*c^4*d^2*e^17))/(c^4*d^10 + a^4*d^2*e^8 + 4*
a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4))*((a^2*e^4*(-a^3*c^3)^(1/2) + c^2*d^4*(-a^3*c^3)^(1/2) +
4*a^2*c^3*d^3*e - 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^(1/2))/(16*(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e
^6 + 4*a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))^(1/2) + (x*(32*a^6*c^5*d*e^14 - 48*a*c^10*d^11*e^4 - 16*c^11*d^1
3*e^2 + 1024*a^2*c^9*d^9*e^6 + 2208*a^3*c^8*d^7*e^8 + 1264*a^4*c^7*d^5*e^10 + 144*a^5*c^6*d^3*e^12))/(c^4*d^10
+ a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4))*((a^2*e^4*(-a^3*c^3)^(1/2) + c^2*d^4*
(-a^3*c^3)^(1/2) + 4*a^2*c^3*d^3*e - 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^(1/2))/(16*(a^7*e^8 + a^3*c^4*
d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))^(1/2) + (480*a^2*c^8*d^6*e^7 - 200*a*c^9*d^8*
e^5 - 8*a^5*c^5*e^13 + 784*a^3*c^7*d^4*e^9 + 96*a^4*c^6*d^2*e^11)/(2*(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2
+ 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4)))*((a^2*e^4*(-a^3*c^3)^(1/2) + c^2*d^4*(-a^3*c^3)^(1/2) + 4*a^2*c^3*d^
3*e - 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^(1/2))/(16*(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c
^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))^(1/2) + (x*(a^3*c^6*e^11 - 27*c^9*d^6*e^5 + 11*a*c^8*d^4*e^7 + 7*a^2*c^7*d^2
*e^9))/(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4))*((a^2*e^4*(-a^3*c^3)^
(1/2) + c^2*d^4*(-a^3*c^3)^(1/2) + 4*a^2*c^3*d^3*e - 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^(1/2))/(16*(a^
7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))^(1/2)*1i - (((((256*a^8*c^4*d
*e^16 - 128*a*c^11*d^15*e^2 + 256*a^2*c^10*d^13*e^4 + 3456*a^3*c^9*d^11*e^6 + 8960*a^4*c^8*d^9*e^8 + 10880*a^5
*c^7*d^7*e^10 + 6912*a^6*c^6*d^5*e^12 + 2176*a^7*c^5*d^3*e^14)/(2*(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 +
4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4)) - (x*((a^2*e^4*(-a^3*c^3)^(1/2) + c^2*d^4*(-a^3*c^3)^(1/2) + 4*a^2*c^3*d
^3*e - 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^(1/2))/(16*(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*
c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))^(1/2)*(512*a^2*c^11*d^16*e^3 + 2560*a^3*c^10*d^14*e^5 + 4608*a^4*c^9*d^12*e
^7 + 2560*a^5*c^8*d^10*e^9 - 2560*a^6*c^7*d^8*e^11 - 4608*a^7*c^6*d^6*e^13 - 2560*a^8*c^5*d^4*e^15 - 512*a^9*c
^4*d^2*e^17))/(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4))*((a^2*e^4*(-a^
3*c^3)^(1/2) + c^2*d^4*(-a^3*c^3)^(1/2) + 4*a^2*c^3*d^3*e - 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^(1/2))/
(16*(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))^(1/2) - (x*(32*a^6*c^5
*d*e^14 - 48*a*c^10*d^11*e^4 - 16*c^11*d^13*e^2 + 1024*a^2*c^9*d^9*e^6 + 2208*a^3*c^8*d^7*e^8 + 1264*a^4*c^7*d
^5*e^10 + 144*a^5*c^6*d^3*e^12))/(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e
^4))*((a^2*e^4*(-a^3*c^3)^(1/2) + c^2*d^4*(-a^3*c^3)^(1/2) + 4*a^2*c^3*d^3*e - 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2
*(-a^3*c^3)^(1/2))/(16*(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))^(1/
2) + (480*a^2*c^8*d^6*e^7 - 200*a*c^9*d^8*e^5 - 8*a^5*c^5*e^13 + 784*a^3*c^7*d^4*e^9 + 96*a^4*c^6*d^2*e^11)/(2
*(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4)))*((a^2*e^4*(-a^3*c^3)^(1/2)
+ c^2*d^4*(-a^3*c^3)^(1/2) + 4*a^2*c^3*d^3*e - 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^(1/2))/(16*(a^7*e^8
+ a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))^(1/2) - (x*(a^3*c^6*e^11 - 27*c^9*
d^6*e^5 + 11*a*c^8*d^4*e^7 + 7*a^2*c^7*d^2*e^9))/(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 +
6*a^2*c^2*d^6*e^4))*((a^2*e^4*(-a^3*c^3)^(1/2) + c^2*d^4*(-a^3*c^3)^(1/2) + 4*a^2*c^3*d^3*e - 4*a^3*c^2*d*e^3
- 6*a*c*d^2*e^2*(-a^3*c^3)^(1/2))/(16*(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2 + 6*a^5*c^
2*d^4*e^4)))^(1/2)*1i)/((((((256*a^8*c^4*d*e^16 - 128*a*c^11*d^15*e^2 + 256*a^2*c^10*d^13*e^4 + 3456*a^3*c^9*d
^11*e^6 + 8960*a^4*c^8*d^9*e^8 + 10880*a^5*c^7*d^7*e^10 + 6912*a^6*c^6*d^5*e^12 + 2176*a^7*c^5*d^3*e^14)/(2*(c
^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4)) + (x*((a^2*e^4*(-a^3*c^3)^(1/2
) + c^2*d^4*(-a^3*c^3)^(1/2) + 4*a^2*c^3*d^3*e - 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^(1/2))/(16*(a^7*e^
8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))^(1/2)*(512*a^2*c^11*d^16*e^3 + 25
60*a^3*c^10*d^14*e^5 + 4608*a^4*c^9*d^12*e^7 + 2560*a^5*c^8*d^10*e^9 - 2560*a^6*c^7*d^8*e^11 - 4608*a^7*c^6*d^
6*e^13 - 2560*a^8*c^5*d^4*e^15 - 512*a^9*c^4*d^2*e^17))/(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^
4*e^6 + 6*a^2*c^2*d^6*e^4))*((a^2*e^4*(-a^3*c^3)^(1/2) + c^2*d^4*(-a^3*c^3)^(1/2) + 4*a^2*c^3*d^3*e - 4*a^3*c^
2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^(1/2))/(16*(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2 + 6
*a^5*c^2*d^4*e^4)))^(1/2) + (x*(32*a^6*c^5*d*e^14 - 48*a*c^10*d^11*e^4 - 16*c^11*d^13*e^2 + 1024*a^2*c^9*d^9*e
^6 + 2208*a^3*c^8*d^7*e^8 + 1264*a^4*c^7*d^5*e^10 + 144*a^5*c^6*d^3*e^12))/(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d
^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4))*((a^2*e^4*(-a^3*c^3)^(1/2) + c^2*d^4*(-a^3*c^3)^(1/2) + 4*a^2*c
^3*d^3*e - 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^(1/2))/(16*(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*
a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))^(1/2) + (480*a^2*c^8*d^6*e^7 - 200*a*c^9*d^8*e^5 - 8*a^5*c^5*e^13 + 784
*a^3*c^7*d^4*e^9 + 96*a^4*c^6*d^2*e^11)/(2*(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2
*c^2*d^6*e^4)))*((a^2*e^4*(-a^3*c^3)^(1/2) + c^2*d^4*(-a^3*c^3)^(1/2) + 4*a^2*c^3*d^3*e - 4*a^3*c^2*d*e^3 - 6*
a*c*d^2*e^2*(-a^3*c^3)^(1/2))/(16*(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4
*e^4)))^(1/2) + (x*(a^3*c^6*e^11 - 27*c^9*d^6*e^5 + 11*a*c^8*d^4*e^7 + 7*a^2*c^7*d^2*e^9))/(c^4*d^10 + a^4*d^2
*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4))*((a^2*e^4*(-a^3*c^3)^(1/2) + c^2*d^4*(-a^3*c^3)
^(1/2) + 4*a^2*c^3*d^3*e - 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^(1/2))/(16*(a^7*e^8 + a^3*c^4*d^8 + 4*a^
6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))^(1/2) + (((((256*a^8*c^4*d*e^16 - 128*a*c^11*d^15*e^2 +
256*a^2*c^10*d^13*e^4 + 3456*a^3*c^9*d^11*e^6 + 8960*a^4*c^8*d^9*e^8 + 10880*a^5*c^7*d^7*e^10 + 6912*a^6*c^6*
d^5*e^12 + 2176*a^7*c^5*d^3*e^14)/(2*(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d
^6*e^4)) - (x*((a^2*e^4*(-a^3*c^3)^(1/2) + c^2*d^4*(-a^3*c^3)^(1/2) + 4*a^2*c^3*d^3*e - 4*a^3*c^2*d*e^3 - 6*a*
c*d^2*e^2*(-a^3*c^3)^(1/2))/(16*(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e
^4)))^(1/2)*(512*a^2*c^11*d^16*e^3 + 2560*a^3*c^10*d^14*e^5 + 4608*a^4*c^9*d^12*e^7 + 2560*a^5*c^8*d^10*e^9 -
2560*a^6*c^7*d^8*e^11 - 4608*a^7*c^6*d^6*e^13 - 2560*a^8*c^5*d^4*e^15 - 512*a^9*c^4*d^2*e^17))/(c^4*d^10 + a^4
*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4))*((a^2*e^4*(-a^3*c^3)^(1/2) + c^2*d^4*(-a^3*
c^3)^(1/2) + 4*a^2*c^3*d^3*e - 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^(1/2))/(16*(a^7*e^8 + a^3*c^4*d^8 +
4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))^(1/2) - (x*(32*a^6*c^5*d*e^14 - 48*a*c^10*d^11*e^4
- 16*c^11*d^13*e^2 + 1024*a^2*c^9*d^9*e^6 + 2208*a^3*c^8*d^7*e^8 + 1264*a^4*c^7*d^5*e^10 + 144*a^5*c^6*d^3*e^1
2))/(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4))*((a^2*e^4*(-a^3*c^3)^(1/
2) + c^2*d^4*(-a^3*c^3)^(1/2) + 4*a^2*c^3*d^3*e - 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^(1/2))/(16*(a^7*e
^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))^(1/2) + (480*a^2*c^8*d^6*e^7 - 2
00*a*c^9*d^8*e^5 - 8*a^5*c^5*e^13 + 784*a^3*c^7*d^4*e^9 + 96*a^4*c^6*d^2*e^11)/(2*(c^4*d^10 + a^4*d^2*e^8 + 4*
a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4)))*((a^2*e^4*(-a^3*c^3)^(1/2) + c^2*d^4*(-a^3*c^3)^(1/2) +
4*a^2*c^3*d^3*e - 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^(1/2))/(16*(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*
e^6 + 4*a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))^(1/2) - (x*(a^3*c^6*e^11 - 27*c^9*d^6*e^5 + 11*a*c^8*d^4*e^7 +
7*a^2*c^7*d^2*e^9))/(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4))*((a^2*e^
4*(-a^3*c^3)^(1/2) + c^2*d^4*(-a^3*c^3)^(1/2) + 4*a^2*c^3*d^3*e - 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^(
1/2))/(16*(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))^(1/2) + (5*c^8*d
^3*e^6 + a*c^7*d*e^8)/(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4)))*((a^2
*e^4*(-a^3*c^3)^(1/2) + c^2*d^4*(-a^3*c^3)^(1/2) + 4*a^2*c^3*d^3*e - 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3
)^(1/2))/(16*(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))^(1/2)*2i - (a
tan(((((x*(a^3*c^6*e^11 - 27*c^9*d^6*e^5 + 11*a*c^8*d^4*e^7 + 7*a^2*c^7*d^2*e^9))/(c^4*d^10 + a^4*d^2*e^8 + 4*
a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4) - (((240*a^2*c^8*d^6*e^7 - 100*a*c^9*d^8*e^5 - 4*a^5*c^5*
e^13 + 392*a^3*c^7*d^4*e^9 + 48*a^4*c^6*d^2*e^11)/(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6
+ 6*a^2*c^2*d^6*e^4) - (((x*(32*a^6*c^5*d*e^14 - 48*a*c^10*d^11*e^4 - 16*c^11*d^13*e^2 + 1024*a^2*c^9*d^9*e^6
+ 2208*a^3*c^8*d^7*e^8 + 1264*a^4*c^7*d^5*e^10 + 144*a^5*c^6*d^3*e^12))/(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*
e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4) - ((a*e^2 + 5*c*d^2)*((128*a^8*c^4*d*e^16 - 64*a*c^11*d^15*e^2 + 12
8*a^2*c^10*d^13*e^4 + 1728*a^3*c^9*d^11*e^6 + 4480*a^4*c^8*d^9*e^8 + 5440*a^5*c^7*d^7*e^10 + 3456*a^6*c^6*d^5*
e^12 + 1088*a^7*c^5*d^3*e^14)/(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4)
- (x*(a*e^2 + 5*c*d^2)*(-d^3*e^3)^(1/2)*(512*a^2*c^11*d^16*e^3 + 2560*a^3*c^10*d^14*e^5 + 4608*a^4*c^9*d^12*e
^7 + 2560*a^5*c^8*d^10*e^9 - 2560*a^6*c^7*d^8*e^11 - 4608*a^7*c^6*d^6*e^13 - 2560*a^8*c^5*d^4*e^15 - 512*a^9*c
^4*d^2*e^17))/(4*(c^2*d^7 + a^2*d^3*e^4 + 2*a*c*d^5*e^2)*(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d
^4*e^6 + 6*a^2*c^2*d^6*e^4)))*(-d^3*e^3)^(1/2))/(4*(c^2*d^7 + a^2*d^3*e^4 + 2*a*c*d^5*e^2)))*(a*e^2 + 5*c*d^2)
*(-d^3*e^3)^(1/2))/(4*(c^2*d^7 + a^2*d^3*e^4 + 2*a*c*d^5*e^2)))*(a*e^2 + 5*c*d^2)*(-d^3*e^3)^(1/2))/(4*(c^2*d^
7 + a^2*d^3*e^4 + 2*a*c*d^5*e^2)))*(a*e^2 + 5*c*d^2)*(-d^3*e^3)^(1/2)*1i)/(4*(c^2*d^7 + a^2*d^3*e^4 + 2*a*c*d^
5*e^2)) + (((x*(a^3*c^6*e^11 - 27*c^9*d^6*e^5 + 11*a*c^8*d^4*e^7 + 7*a^2*c^7*d^2*e^9))/(c^4*d^10 + a^4*d^2*e^8
+ 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4) + (((240*a^2*c^8*d^6*e^7 - 100*a*c^9*d^8*e^5 - 4*a^5
*c^5*e^13 + 392*a^3*c^7*d^4*e^9 + 48*a^4*c^6*d^2*e^11)/(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4
*e^6 + 6*a^2*c^2*d^6*e^4) + (((x*(32*a^6*c^5*d*e^14 - 48*a*c^10*d^11*e^4 - 16*c^11*d^13*e^2 + 1024*a^2*c^9*d^9
*e^6 + 2208*a^3*c^8*d^7*e^8 + 1264*a^4*c^7*d^5*e^10 + 144*a^5*c^6*d^3*e^12))/(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3
*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4) + ((a*e^2 + 5*c*d^2)*((128*a^8*c^4*d*e^16 - 64*a*c^11*d^15*e^2
+ 128*a^2*c^10*d^13*e^4 + 1728*a^3*c^9*d^11*e^6 + 4480*a^4*c^8*d^9*e^8 + 5440*a^5*c^7*d^7*e^10 + 3456*a^6*c^6
*d^5*e^12 + 1088*a^7*c^5*d^3*e^14)/(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6
*e^4) + (x*(a*e^2 + 5*c*d^2)*(-d^3*e^3)^(1/2)*(512*a^2*c^11*d^16*e^3 + 2560*a^3*c^10*d^14*e^5 + 4608*a^4*c^9*d
^12*e^7 + 2560*a^5*c^8*d^10*e^9 - 2560*a^6*c^7*d^8*e^11 - 4608*a^7*c^6*d^6*e^13 - 2560*a^8*c^5*d^4*e^15 - 512*
a^9*c^4*d^2*e^17))/(4*(c^2*d^7 + a^2*d^3*e^4 + 2*a*c*d^5*e^2)*(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^
3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4)))*(-d^3*e^3)^(1/2))/(4*(c^2*d^7 + a^2*d^3*e^4 + 2*a*c*d^5*e^2)))*(a*e^2 + 5*c
*d^2)*(-d^3*e^3)^(1/2))/(4*(c^2*d^7 + a^2*d^3*e^4 + 2*a*c*d^5*e^2)))*(a*e^2 + 5*c*d^2)*(-d^3*e^3)^(1/2))/(4*(c
^2*d^7 + a^2*d^3*e^4 + 2*a*c*d^5*e^2)))*(a*e^2 + 5*c*d^2)*(-d^3*e^3)^(1/2)*1i)/(4*(c^2*d^7 + a^2*d^3*e^4 + 2*a
*c*d^5*e^2)))/((5*c^8*d^3*e^6 + a*c^7*d*e^8)/(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a
^2*c^2*d^6*e^4) - (((x*(a^3*c^6*e^11 - 27*c^9*d^6*e^5 + 11*a*c^8*d^4*e^7 + 7*a^2*c^7*d^2*e^9))/(c^4*d^10 + a^4
*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4) - (((240*a^2*c^8*d^6*e^7 - 100*a*c^9*d^8*e^5
- 4*a^5*c^5*e^13 + 392*a^3*c^7*d^4*e^9 + 48*a^4*c^6*d^2*e^11)/(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a
^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4) - (((x*(32*a^6*c^5*d*e^14 - 48*a*c^10*d^11*e^4 - 16*c^11*d^13*e^2 + 1024*a^2
*c^9*d^9*e^6 + 2208*a^3*c^8*d^7*e^8 + 1264*a^4*c^7*d^5*e^10 + 144*a^5*c^6*d^3*e^12))/(c^4*d^10 + a^4*d^2*e^8 +
4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4) - ((a*e^2 + 5*c*d^2)*((128*a^8*c^4*d*e^16 - 64*a*c^11*
d^15*e^2 + 128*a^2*c^10*d^13*e^4 + 1728*a^3*c^9*d^11*e^6 + 4480*a^4*c^8*d^9*e^8 + 5440*a^5*c^7*d^7*e^10 + 3456
*a^6*c^6*d^5*e^12 + 1088*a^7*c^5*d^3*e^14)/(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2
*c^2*d^6*e^4) - (x*(a*e^2 + 5*c*d^2)*(-d^3*e^3)^(1/2)*(512*a^2*c^11*d^16*e^3 + 2560*a^3*c^10*d^14*e^5 + 4608*a
^4*c^9*d^12*e^7 + 2560*a^5*c^8*d^10*e^9 - 2560*a^6*c^7*d^8*e^11 - 4608*a^7*c^6*d^6*e^13 - 2560*a^8*c^5*d^4*e^1
5 - 512*a^9*c^4*d^2*e^17))/(4*(c^2*d^7 + a^2*d^3*e^4 + 2*a*c*d^5*e^2)*(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^
2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4)))*(-d^3*e^3)^(1/2))/(4*(c^2*d^7 + a^2*d^3*e^4 + 2*a*c*d^5*e^2)))*(a*e
^2 + 5*c*d^2)*(-d^3*e^3)^(1/2))/(4*(c^2*d^7 + a^2*d^3*e^4 + 2*a*c*d^5*e^2)))*(a*e^2 + 5*c*d^2)*(-d^3*e^3)^(1/2
))/(4*(c^2*d^7 + a^2*d^3*e^4 + 2*a*c*d^5*e^2)))*(a*e^2 + 5*c*d^2)*(-d^3*e^3)^(1/2))/(4*(c^2*d^7 + a^2*d^3*e^4
+ 2*a*c*d^5*e^2)) + (((x*(a^3*c^6*e^11 - 27*c^9*d^6*e^5 + 11*a*c^8*d^4*e^7 + 7*a^2*c^7*d^2*e^9))/(c^4*d^10 + a
^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4) + (((240*a^2*c^8*d^6*e^7 - 100*a*c^9*d^8*e
^5 - 4*a^5*c^5*e^13 + 392*a^3*c^7*d^4*e^9 + 48*a^4*c^6*d^2*e^11)/(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4
*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4) + (((x*(32*a^6*c^5*d*e^14 - 48*a*c^10*d^11*e^4 - 16*c^11*d^13*e^2 + 1024*a
^2*c^9*d^9*e^6 + 2208*a^3*c^8*d^7*e^8 + 1264*a^4*c^7*d^5*e^10 + 144*a^5*c^6*d^3*e^12))/(c^4*d^10 + a^4*d^2*e^8
+ 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4) + ((a*e^2 + 5*c*d^2)*((128*a^8*c^4*d*e^16 - 64*a*c^1
1*d^15*e^2 + 128*a^2*c^10*d^13*e^4 + 1728*a^3*c^9*d^11*e^6 + 4480*a^4*c^8*d^9*e^8 + 5440*a^5*c^7*d^7*e^10 + 34
56*a^6*c^6*d^5*e^12 + 1088*a^7*c^5*d^3*e^14)/(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a
^2*c^2*d^6*e^4) + (x*(a*e^2 + 5*c*d^2)*(-d^3*e^3)^(1/2)*(512*a^2*c^11*d^16*e^3 + 2560*a^3*c^10*d^14*e^5 + 4608
*a^4*c^9*d^12*e^7 + 2560*a^5*c^8*d^10*e^9 - 2560*a^6*c^7*d^8*e^11 - 4608*a^7*c^6*d^6*e^13 - 2560*a^8*c^5*d^4*e
^15 - 512*a^9*c^4*d^2*e^17))/(4*(c^2*d^7 + a^2*d^3*e^4 + 2*a*c*d^5*e^2)*(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*
e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4)))*(-d^3*e^3)^(1/2))/(4*(c^2*d^7 + a^2*d^3*e^4 + 2*a*c*d^5*e^2)))*(a
*e^2 + 5*c*d^2)*(-d^3*e^3)^(1/2))/(4*(c^2*d^7 + a^2*d^3*e^4 + 2*a*c*d^5*e^2)))*(a*e^2 + 5*c*d^2)*(-d^3*e^3)^(1
/2))/(4*(c^2*d^7 + a^2*d^3*e^4 + 2*a*c*d^5*e^2)))*(a*e^2 + 5*c*d^2)*(-d^3*e^3)^(1/2))/(4*(c^2*d^7 + a^2*d^3*e^
4 + 2*a*c*d^5*e^2))))*(a*e^2 + 5*c*d^2)*(-d^3*e^3)^(1/2)*1i)/(2*(c^2*d^7 + a^2*d^3*e^4 + 2*a*c*d^5*e^2)) - ata
n(((((((256*a^8*c^4*d*e^16 - 128*a*c^11*d^15*e^2 + 256*a^2*c^10*d^13*e^4 + 3456*a^3*c^9*d^11*e^6 + 8960*a^4*c^
8*d^9*e^8 + 10880*a^5*c^7*d^7*e^10 + 6912*a^6*c^6*d^5*e^12 + 2176*a^7*c^5*d^3*e^14)/(2*(c^4*d^10 + a^4*d^2*e^8
+ 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4)) + (x*(-(a^2*e^4*(-a^3*c^3)^(1/2) + c^2*d^4*(-a^3*c^
3)^(1/2) - 4*a^2*c^3*d^3*e + 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^(1/2))/(16*(a^7*e^8 + a^3*c^4*d^8 + 4*
a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))^(1/2)*(512*a^2*c^11*d^16*e^3 + 2560*a^3*c^10*d^14*e^5
+ 4608*a^4*c^9*d^12*e^7 + 2560*a^5*c^8*d^10*e^9 - 2560*a^6*c^7*d^8*e^11 - 4608*a^7*c^6*d^6*e^13 - 2560*a^8*c^
5*d^4*e^15 - 512*a^9*c^4*d^2*e^17))/(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^
6*e^4))*(-(a^2*e^4*(-a^3*c^3)^(1/2) + c^2*d^4*(-a^3*c^3)^(1/2) - 4*a^2*c^3*d^3*e + 4*a^3*c^2*d*e^3 - 6*a*c*d^2
*e^2*(-a^3*c^3)^(1/2))/(16*(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))
^(1/2) + (x*(32*a^6*c^5*d*e^14 - 48*a*c^10*d^11*e^4 - 16*c^11*d^13*e^2 + 1024*a^2*c^9*d^9*e^6 + 2208*a^3*c^8*d
^7*e^8 + 1264*a^4*c^7*d^5*e^10 + 144*a^5*c^6*d^3*e^12))/(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^
4*e^6 + 6*a^2*c^2*d^6*e^4))*(-(a^2*e^4*(-a^3*c^3)^(1/2) + c^2*d^4*(-a^3*c^3)^(1/2) - 4*a^2*c^3*d^3*e + 4*a^3*c
^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^(1/2))/(16*(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2 +
6*a^5*c^2*d^4*e^4)))^(1/2) + (480*a^2*c^8*d^6*e^7 - 200*a*c^9*d^8*e^5 - 8*a^5*c^5*e^13 + 784*a^3*c^7*d^4*e^9 +
96*a^4*c^6*d^2*e^11)/(2*(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4)))*(-
(a^2*e^4*(-a^3*c^3)^(1/2) + c^2*d^4*(-a^3*c^3)^(1/2) - 4*a^2*c^3*d^3*e + 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3
*c^3)^(1/2))/(16*(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))^(1/2) + (
x*(a^3*c^6*e^11 - 27*c^9*d^6*e^5 + 11*a*c^8*d^4*e^7 + 7*a^2*c^7*d^2*e^9))/(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^
8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4))*(-(a^2*e^4*(-a^3*c^3)^(1/2) + c^2*d^4*(-a^3*c^3)^(1/2) - 4*a^2*c
^3*d^3*e + 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^(1/2))/(16*(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*
a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))^(1/2)*1i - (((((256*a^8*c^4*d*e^16 - 128*a*c^11*d^15*e^2 + 256*a^2*c^10
*d^13*e^4 + 3456*a^3*c^9*d^11*e^6 + 8960*a^4*c^8*d^9*e^8 + 10880*a^5*c^7*d^7*e^10 + 6912*a^6*c^6*d^5*e^12 + 21
76*a^7*c^5*d^3*e^14)/(2*(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4)) - (x
*(-(a^2*e^4*(-a^3*c^3)^(1/2) + c^2*d^4*(-a^3*c^3)^(1/2) - 4*a^2*c^3*d^3*e + 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-
a^3*c^3)^(1/2))/(16*(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))^(1/2)*
(512*a^2*c^11*d^16*e^3 + 2560*a^3*c^10*d^14*e^5 + 4608*a^4*c^9*d^12*e^7 + 2560*a^5*c^8*d^10*e^9 - 2560*a^6*c^7
*d^8*e^11 - 4608*a^7*c^6*d^6*e^13 - 2560*a^8*c^5*d^4*e^15 - 512*a^9*c^4*d^2*e^17))/(c^4*d^10 + a^4*d^2*e^8 + 4
*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4))*(-(a^2*e^4*(-a^3*c^3)^(1/2) + c^2*d^4*(-a^3*c^3)^(1/2)
- 4*a^2*c^3*d^3*e + 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^(1/2))/(16*(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2
*e^6 + 4*a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))^(1/2) - (x*(32*a^6*c^5*d*e^14 - 48*a*c^10*d^11*e^4 - 16*c^11*d
^13*e^2 + 1024*a^2*c^9*d^9*e^6 + 2208*a^3*c^8*d^7*e^8 + 1264*a^4*c^7*d^5*e^10 + 144*a^5*c^6*d^3*e^12))/(c^4*d^
10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4))*(-(a^2*e^4*(-a^3*c^3)^(1/2) + c^2*d
^4*(-a^3*c^3)^(1/2) - 4*a^2*c^3*d^3*e + 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^(1/2))/(16*(a^7*e^8 + a^3*c
^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))^(1/2) + (480*a^2*c^8*d^6*e^7 - 200*a*c^9*d
^8*e^5 - 8*a^5*c^5*e^13 + 784*a^3*c^7*d^4*e^9 + 96*a^4*c^6*d^2*e^11)/(2*(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*
e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4)))*(-(a^2*e^4*(-a^3*c^3)^(1/2) + c^2*d^4*(-a^3*c^3)^(1/2) - 4*a^2*c^
3*d^3*e + 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^(1/2))/(16*(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a
^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))^(1/2) - (x*(a^3*c^6*e^11 - 27*c^9*d^6*e^5 + 11*a*c^8*d^4*e^7 + 7*a^2*c^7
*d^2*e^9))/(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4))*(-(a^2*e^4*(-a^3*
c^3)^(1/2) + c^2*d^4*(-a^3*c^3)^(1/2) - 4*a^2*c^3*d^3*e + 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^(1/2))/(1
6*(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))^(1/2)*1i)/((((((256*a^8*
c^4*d*e^16 - 128*a*c^11*d^15*e^2 + 256*a^2*c^10*d^13*e^4 + 3456*a^3*c^9*d^11*e^6 + 8960*a^4*c^8*d^9*e^8 + 1088
0*a^5*c^7*d^7*e^10 + 6912*a^6*c^6*d^5*e^12 + 2176*a^7*c^5*d^3*e^14)/(2*(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e
^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4)) + (x*(-(a^2*e^4*(-a^3*c^3)^(1/2) + c^2*d^4*(-a^3*c^3)^(1/2) - 4*a^2
*c^3*d^3*e + 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^(1/2))/(16*(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 +
4*a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))^(1/2)*(512*a^2*c^11*d^16*e^3 + 2560*a^3*c^10*d^14*e^5 + 4608*a^4*c^9*
d^12*e^7 + 2560*a^5*c^8*d^10*e^9 - 2560*a^6*c^7*d^8*e^11 - 4608*a^7*c^6*d^6*e^13 - 2560*a^8*c^5*d^4*e^15 - 512
*a^9*c^4*d^2*e^17))/(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4))*(-(a^2*e
^4*(-a^3*c^3)^(1/2) + c^2*d^4*(-a^3*c^3)^(1/2) - 4*a^2*c^3*d^3*e + 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^
(1/2))/(16*(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))^(1/2) + (x*(32*
a^6*c^5*d*e^14 - 48*a*c^10*d^11*e^4 - 16*c^11*d^13*e^2 + 1024*a^2*c^9*d^9*e^6 + 2208*a^3*c^8*d^7*e^8 + 1264*a^
4*c^7*d^5*e^10 + 144*a^5*c^6*d^3*e^12))/(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^
2*d^6*e^4))*(-(a^2*e^4*(-a^3*c^3)^(1/2) + c^2*d^4*(-a^3*c^3)^(1/2) - 4*a^2*c^3*d^3*e + 4*a^3*c^2*d*e^3 - 6*a*c
*d^2*e^2*(-a^3*c^3)^(1/2))/(16*(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^
4)))^(1/2) + (480*a^2*c^8*d^6*e^7 - 200*a*c^9*d^8*e^5 - 8*a^5*c^5*e^13 + 784*a^3*c^7*d^4*e^9 + 96*a^4*c^6*d^2*
e^11)/(2*(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4)))*(-(a^2*e^4*(-a^3*c
^3)^(1/2) + c^2*d^4*(-a^3*c^3)^(1/2) - 4*a^2*c^3*d^3*e + 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^(1/2))/(16
*(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))^(1/2) + (x*(a^3*c^6*e^11
- 27*c^9*d^6*e^5 + 11*a*c^8*d^4*e^7 + 7*a^2*c^7*d^2*e^9))/(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*
d^4*e^6 + 6*a^2*c^2*d^6*e^4))*(-(a^2*e^4*(-a^3*c^3)^(1/2) + c^2*d^4*(-a^3*c^3)^(1/2) - 4*a^2*c^3*d^3*e + 4*a^3
*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^(1/2))/(16*(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2
+ 6*a^5*c^2*d^4*e^4)))^(1/2) + (((((256*a^8*c^4*d*e^16 - 128*a*c^11*d^15*e^2 + 256*a^2*c^10*d^13*e^4 + 3456*a^
3*c^9*d^11*e^6 + 8960*a^4*c^8*d^9*e^8 + 10880*a^5*c^7*d^7*e^10 + 6912*a^6*c^6*d^5*e^12 + 2176*a^7*c^5*d^3*e^14
)/(2*(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4)) - (x*(-(a^2*e^4*(-a^3*c
^3)^(1/2) + c^2*d^4*(-a^3*c^3)^(1/2) - 4*a^2*c^3*d^3*e + 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^(1/2))/(16
*(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))^(1/2)*(512*a^2*c^11*d^16*
e^3 + 2560*a^3*c^10*d^14*e^5 + 4608*a^4*c^9*d^12*e^7 + 2560*a^5*c^8*d^10*e^9 - 2560*a^6*c^7*d^8*e^11 - 4608*a^
7*c^6*d^6*e^13 - 2560*a^8*c^5*d^4*e^15 - 512*a^9*c^4*d^2*e^17))/(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*
a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4))*(-(a^2*e^4*(-a^3*c^3)^(1/2) + c^2*d^4*(-a^3*c^3)^(1/2) - 4*a^2*c^3*d^3*e +
4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^(1/2))/(16*(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^
6*e^2 + 6*a^5*c^2*d^4*e^4)))^(1/2) - (x*(32*a^6*c^5*d*e^14 - 48*a*c^10*d^11*e^4 - 16*c^11*d^13*e^2 + 1024*a^2*
c^9*d^9*e^6 + 2208*a^3*c^8*d^7*e^8 + 1264*a^4*c^7*d^5*e^10 + 144*a^5*c^6*d^3*e^12))/(c^4*d^10 + a^4*d^2*e^8 +
4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4))*(-(a^2*e^4*(-a^3*c^3)^(1/2) + c^2*d^4*(-a^3*c^3)^(1/2)
- 4*a^2*c^3*d^3*e + 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^(1/2))/(16*(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^
2*e^6 + 4*a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))^(1/2) + (480*a^2*c^8*d^6*e^7 - 200*a*c^9*d^8*e^5 - 8*a^5*c^5*
e^13 + 784*a^3*c^7*d^4*e^9 + 96*a^4*c^6*d^2*e^11)/(2*(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e
^6 + 6*a^2*c^2*d^6*e^4)))*(-(a^2*e^4*(-a^3*c^3)^(1/2) + c^2*d^4*(-a^3*c^3)^(1/2) - 4*a^2*c^3*d^3*e + 4*a^3*c^2
*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^(1/2))/(16*(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2 + 6*
a^5*c^2*d^4*e^4)))^(1/2) - (x*(a^3*c^6*e^11 - 27*c^9*d^6*e^5 + 11*a*c^8*d^4*e^7 + 7*a^2*c^7*d^2*e^9))/(c^4*d^1
0 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4))*(-(a^2*e^4*(-a^3*c^3)^(1/2) + c^2*d^
4*(-a^3*c^3)^(1/2) - 4*a^2*c^3*d^3*e + 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^(1/2))/(16*(a^7*e^8 + a^3*c^
4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))^(1/2) + (5*c^8*d^3*e^6 + a*c^7*d*e^8)/(c^4*
d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2*d^6*e^4)))*(-(a^2*e^4*(-a^3*c^3)^(1/2) + c^
2*d^4*(-a^3*c^3)^(1/2) - 4*a^2*c^3*d^3*e + 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^(1/2))/(16*(a^7*e^8 + a^
3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))^(1/2)*2i