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

Optimal. Leaf size=864 \[ \frac {x e^4}{2 d \left (c d^2+a e^2\right )^2 \left (e x^2+d\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) e^{7/2}}{2 d^{3/2} \left (c d^2+a e^2\right )^2}+\frac {4 c \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) e^{7/2}}{\left (c d^2+a e^2\right )^3}-\frac {c^{3/4} \left (3 c d^2-4 \sqrt {a} \sqrt {c} e d-a e^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) e^2}{2 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^3}+\frac {c^{3/4} \left (3 c d^2-4 \sqrt {a} \sqrt {c} e d-a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) e^2}{2 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^3}-\frac {c^{3/4} \left (3 c d^2+4 \sqrt {a} \sqrt {c} e d-a e^2\right ) \log \left (\sqrt {c} x^2-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right ) e^2}{4 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^3}+\frac {c^{3/4} \left (3 c d^2+4 \sqrt {a} \sqrt {c} e d-a e^2\right ) \log \left (\sqrt {c} x^2+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right ) e^2}{4 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^3}-\frac {c^{3/4} \left (3 c d^2-2 \sqrt {a} \sqrt {c} e d-3 a e^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac {c^{3/4} \left (3 c d^2-2 \sqrt {a} \sqrt {c} e d-3 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}-\frac {c^{3/4} \left (3 c d^2+2 \sqrt {a} \sqrt {c} e d-3 a e^2\right ) \log \left (\sqrt {c} x^2-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right )}{16 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac {c^{3/4} \left (3 c d^2+2 \sqrt {a} \sqrt {c} e d-3 a e^2\right ) \log \left (\sqrt {c} x^2+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right )}{16 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac {c x \left (c d^2-2 c e x^2 d-a e^2\right )}{4 a \left (c d^2+a e^2\right )^2 \left (c x^4+a\right )} \]

[Out]

1/2*e^4*x/d/(a*e^2+c*d^2)^2/(e*x^2+d)+1/4*c*x*(-2*c*d*e*x^2-a*e^2+c*d^2)/a/(a*e^2+c*d^2)^2/(c*x^4+a)+1/2*e^(7/
2)*arctan(x*e^(1/2)/d^(1/2))/d^(3/2)/(a*e^2+c*d^2)^2+1/4*c^(3/4)*e^2*arctan(-1+c^(1/4)*x*2^(1/2)/a^(1/4))*(3*c
*d^2-a*e^2-4*d*e*a^(1/2)*c^(1/2))/a^(3/4)/(a*e^2+c*d^2)^3*2^(1/2)+1/4*c^(3/4)*e^2*arctan(1+c^(1/4)*x*2^(1/2)/a
^(1/4))*(3*c*d^2-a*e^2-4*d*e*a^(1/2)*c^(1/2))/a^(3/4)/(a*e^2+c*d^2)^3*2^(1/2)+1/16*c^(3/4)*arctan(-1+c^(1/4)*x
*2^(1/2)/a^(1/4))*(3*c*d^2-3*a*e^2-2*d*e*a^(1/2)*c^(1/2))/a^(7/4)/(a*e^2+c*d^2)^2*2^(1/2)+1/16*c^(3/4)*arctan(
1+c^(1/4)*x*2^(1/2)/a^(1/4))*(3*c*d^2-3*a*e^2-2*d*e*a^(1/2)*c^(1/2))/a^(7/4)/(a*e^2+c*d^2)^2*2^(1/2)-1/32*c^(3
/4)*ln(-a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(3*c*d^2-3*a*e^2+2*d*e*a^(1/2)*c^(1/2))/a^(7/4)/(a*e^2+
c*d^2)^2*2^(1/2)+1/32*c^(3/4)*ln(a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(3*c*d^2-3*a*e^2+2*d*e*a^(1/2)
*c^(1/2))/a^(7/4)/(a*e^2+c*d^2)^2*2^(1/2)-1/8*c^(3/4)*e^2*ln(-a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(
3*c*d^2-a*e^2+4*d*e*a^(1/2)*c^(1/2))/a^(3/4)/(a*e^2+c*d^2)^3*2^(1/2)+1/8*c^(3/4)*e^2*ln(a^(1/4)*c^(1/4)*x*2^(1
/2)+a^(1/2)+x^2*c^(1/2))*(3*c*d^2-a*e^2+4*d*e*a^(1/2)*c^(1/2))/a^(3/4)/(a*e^2+c*d^2)^3*2^(1/2)+4*c*e^(7/2)*arc
tan(x*e^(1/2)/d^(1/2))*d^(1/2)/(a*e^2+c*d^2)^3

________________________________________________________________________________________

Rubi [A]  time = 0.91, antiderivative size = 864, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 10, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {1239, 199, 205, 1179, 1168, 1162, 617, 204, 1165, 628} \[ \frac {x e^4}{2 d \left (c d^2+a e^2\right )^2 \left (e x^2+d\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) e^{7/2}}{2 d^{3/2} \left (c d^2+a e^2\right )^2}+\frac {4 c \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) e^{7/2}}{\left (c d^2+a e^2\right )^3}-\frac {c^{3/4} \left (3 c d^2-4 \sqrt {a} \sqrt {c} e d-a e^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) e^2}{2 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^3}+\frac {c^{3/4} \left (3 c d^2-4 \sqrt {a} \sqrt {c} e d-a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) e^2}{2 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^3}-\frac {c^{3/4} \left (3 c d^2+4 \sqrt {a} \sqrt {c} e d-a e^2\right ) \log \left (\sqrt {c} x^2-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right ) e^2}{4 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^3}+\frac {c^{3/4} \left (3 c d^2+4 \sqrt {a} \sqrt {c} e d-a e^2\right ) \log \left (\sqrt {c} x^2+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right ) e^2}{4 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^3}-\frac {c^{3/4} \left (3 c d^2-2 \sqrt {a} \sqrt {c} e d-3 a e^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac {c^{3/4} \left (3 c d^2-2 \sqrt {a} \sqrt {c} e d-3 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}-\frac {c^{3/4} \left (3 c d^2+2 \sqrt {a} \sqrt {c} e d-3 a e^2\right ) \log \left (\sqrt {c} x^2-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right )}{16 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac {c^{3/4} \left (3 c d^2+2 \sqrt {a} \sqrt {c} e d-3 a e^2\right ) \log \left (\sqrt {c} x^2+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right )}{16 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac {c x \left (c d^2-2 c e x^2 d-a e^2\right )}{4 a \left (c d^2+a e^2\right )^2 \left (c x^4+a\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e^4*x)/(2*d*(c*d^2 + a*e^2)^2*(d + e*x^2)) + (c*x*(c*d^2 - a*e^2 - 2*c*d*e*x^2))/(4*a*(c*d^2 + a*e^2)^2*(a +
c*x^4)) + (4*c*Sqrt[d]*e^(7/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(c*d^2 + a*e^2)^3 + (e^(7/2)*ArcTan[(Sqrt[e]*x)/Sq
rt[d]])/(2*d^(3/2)*(c*d^2 + a*e^2)^2) - (c^(3/4)*e^2*(3*c*d^2 - 4*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTan[1 - (Sqr
t[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(c*d^2 + a*e^2)^3) - (c^(3/4)*(3*c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e -
3*a*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*(c*d^2 + a*e^2)^2) + (c^(3/4)*e^2*(3*c*d^
2 - 4*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)
^3) + (c^(3/4)*(3*c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e - 3*a*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]
*a^(7/4)*(c*d^2 + a*e^2)^2) - (c^(3/4)*e^2*(3*c*d^2 + 4*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)^3) - (c^(3/4)*(3*c*d^2 + 2*Sqrt[a]*Sqrt[c]*d
*e - 3*a*e^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(7/4)*(c*d^2 + a*e^2)^2) +
(c^(3/4)*e^2*(3*c*d^2 + 4*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)^3) + (c^(3/4)*(3*c*d^2 + 2*Sqrt[a]*Sqrt[c]*d*e - 3*a*e^2)*Log[Sqrt[a] + S
qrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(7/4)*(c*d^2 + a*e^2)^2)

Rule 199

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] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 204

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

Rule 205

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

Rule 617

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 628

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 1162

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 1165

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 1168

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 1179

Int[((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> -Simp[(x*(d + e*x^2)*(a + c*x^4)^(p + 1))/(
4*a*(p + 1)), x] + Dist[1/(4*a*(p + 1)), Int[Simp[d*(4*p + 5) + e*(4*p + 7)*x^2, x]*(a + c*x^4)^(p + 1), x], x
] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1239

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

Rubi steps

\begin {align*} \int \frac {1}{\left (d+e x^2\right )^2 \left (a+c x^4\right )^2} \, dx &=\int \left (\frac {e^4}{\left (c d^2+a e^2\right )^2 \left (d+e x^2\right )^2}+\frac {4 c d e^4}{\left (c d^2+a e^2\right )^3 \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 )^2}-\frac {c e^2 \left (-3 c d^2+a e^2+4 c d e x^2\right )}{\left (c d^2+a e^2\right )^3 \left (a+c x^4\right )}\right ) \, dx\\ &=-\frac {\left (c e^2\right ) \int \frac {-3 c d^2+a e^2+4 c d e x^2}{a+c x^4} \, dx}{\left (c d^2+a e^2\right )^3}+\frac {\left (4 c d e^4\right ) \int \frac {1}{d+e x^2} \, dx}{\left (c d^2+a e^2\right )^3}+\frac {c \int \frac {c d^2-a e^2-2 c d e x^2}{\left (a+c x^4\right )^2} \, dx}{\left (c d^2+a e^2\right )^2}+\frac {e^4 \int \frac {1}{\left (d+e x^2\right )^2} \, dx}{\left (c d^2+a e^2\right )^2}\\ &=\frac {e^4 x}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}+\frac {c x \left (c d^2-a e^2-2 c d e x^2\right )}{4 a \left (c d^2+a e^2\right )^2 \left (a+c x^4\right )}+\frac {4 c \sqrt {d} e^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\left (c d^2+a e^2\right )^3}+\frac {\left (\sqrt {c} e^2 \left (3 c d^2-4 \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 )^3}+\frac {\left (\sqrt {c} e^2 \left (3 c d^2+4 \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 )^3}-\frac {c \int \frac {-3 \left (c d^2-a e^2\right )+2 c d e x^2}{a+c x^4} \, dx}{4 a \left (c d^2+a e^2\right )^2}+\frac {e^4 \int \frac {1}{d+e x^2} \, dx}{2 d \left (c d^2+a e^2\right )^2}\\ &=\frac {e^4 x}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}+\frac {c x \left (c d^2-a e^2-2 c d e x^2\right )}{4 a \left (c d^2+a e^2\right )^2 \left (a+c x^4\right )}+\frac {4 c \sqrt {d} e^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\left (c d^2+a e^2\right )^3}+\frac {e^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \left (c d^2+a e^2\right )^2}+\frac {\left (\sqrt {c} e^2 \left (3 c d^2-4 \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 )^3}+\frac {\left (\sqrt {c} e^2 \left (3 c d^2-4 \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 )^3}-\frac {\left (c^{3/4} e^2 \left (3 c d^2+4 \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 )^3}-\frac {\left (c^{3/4} e^2 \left (3 c d^2+4 \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 )^3}+\frac {\left (\sqrt {c} \left (3 c d^2-2 \sqrt {a} \sqrt {c} d e-3 a e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{8 a^{3/2} \left (c d^2+a e^2\right )^2}+\frac {\left (\sqrt {c} \left (3 c d^2+2 \sqrt {a} \sqrt {c} d e-3 a e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{8 a^{3/2} \left (c d^2+a e^2\right )^2}\\ &=\frac {e^4 x}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}+\frac {c x \left (c d^2-a e^2-2 c d e x^2\right )}{4 a \left (c d^2+a e^2\right )^2 \left (a+c x^4\right )}+\frac {4 c \sqrt {d} e^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\left (c d^2+a e^2\right )^3}+\frac {e^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \left (c d^2+a e^2\right )^2}-\frac {c^{3/4} e^2 \left (3 c d^2+4 \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 )^3}+\frac {c^{3/4} e^2 \left (3 c d^2+4 \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 )^3}+\frac {\left (c^{3/4} e^2 \left (3 c d^2-4 \sqrt {a} \sqrt {c} d e-a e^2\right )\right ) \operatorname {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 )^3}-\frac {\left (c^{3/4} e^2 \left (3 c d^2-4 \sqrt {a} \sqrt {c} d e-a e^2\right )\right ) \operatorname {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 )^3}+\frac {\left (\sqrt {c} \left (3 c d^2-2 \sqrt {a} \sqrt {c} d e-3 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}{16 a^{3/2} \left (c d^2+a e^2\right )^2}+\frac {\left (\sqrt {c} \left (3 c d^2-2 \sqrt {a} \sqrt {c} d e-3 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}{16 a^{3/2} \left (c d^2+a e^2\right )^2}-\frac {\left (c^{3/4} \left (3 c d^2+2 \sqrt {a} \sqrt {c} d e-3 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}{16 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}-\frac {\left (c^{3/4} \left (3 c d^2+2 \sqrt {a} \sqrt {c} d e-3 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}{16 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}\\ &=\frac {e^4 x}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}+\frac {c x \left (c d^2-a e^2-2 c d e x^2\right )}{4 a \left (c d^2+a e^2\right )^2 \left (a+c x^4\right )}+\frac {4 c \sqrt {d} e^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\left (c d^2+a e^2\right )^3}+\frac {e^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \left (c d^2+a e^2\right )^2}-\frac {c^{3/4} e^2 \left (3 c d^2-4 \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 )^3}+\frac {c^{3/4} e^2 \left (3 c d^2-4 \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 )^3}-\frac {c^{3/4} e^2 \left (3 c d^2+4 \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 )^3}-\frac {c^{3/4} \left (3 c d^2+2 \sqrt {a} \sqrt {c} d e-3 a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac {c^{3/4} e^2 \left (3 c d^2+4 \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 )^3}+\frac {c^{3/4} \left (3 c d^2+2 \sqrt {a} \sqrt {c} d e-3 a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac {\left (c^{3/4} \left (3 c d^2-2 \sqrt {a} \sqrt {c} d e-3 a e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}-\frac {\left (c^{3/4} \left (3 c d^2-2 \sqrt {a} \sqrt {c} d e-3 a e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}\\ &=\frac {e^4 x}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}+\frac {c x \left (c d^2-a e^2-2 c d e x^2\right )}{4 a \left (c d^2+a e^2\right )^2 \left (a+c x^4\right )}+\frac {4 c \sqrt {d} e^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\left (c d^2+a e^2\right )^3}+\frac {e^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \left (c d^2+a e^2\right )^2}-\frac {c^{3/4} e^2 \left (3 c d^2-4 \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 )^3}-\frac {c^{3/4} \left (3 c d^2-2 \sqrt {a} \sqrt {c} d e-3 a e^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac {c^{3/4} e^2 \left (3 c d^2-4 \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 )^3}+\frac {c^{3/4} \left (3 c d^2-2 \sqrt {a} \sqrt {c} d e-3 a e^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}-\frac {c^{3/4} e^2 \left (3 c d^2+4 \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 )^3}-\frac {c^{3/4} \left (3 c d^2+2 \sqrt {a} \sqrt {c} d e-3 a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac {c^{3/4} e^2 \left (3 c d^2+4 \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 )^3}+\frac {c^{3/4} \left (3 c d^2+2 \sqrt {a} \sqrt {c} d e-3 a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}\\ \end {align*}

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Mathematica [A]  time = 0.58, size = 540, normalized size = 0.62 \[ \frac {-\frac {\sqrt {2} c^{3/4} \left (18 a^{3/2} \sqrt {c} d e^3-7 a^2 e^4+2 \sqrt {a} c^{3/2} d^3 e+12 a c d^2 e^2+3 c^2 d^4\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{a^{7/4}}+\frac {\sqrt {2} c^{3/4} \left (18 a^{3/2} \sqrt {c} d e^3-7 a^2 e^4+2 \sqrt {a} c^{3/2} d^3 e+12 a c d^2 e^2+3 c^2 d^4\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{a^{7/4}}+\frac {2 \sqrt {2} c^{3/4} \left (18 a^{3/2} \sqrt {c} d e^3+7 a^2 e^4+2 \sqrt {a} c^{3/2} d^3 e-12 a c d^2 e^2-3 c^2 d^4\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{7/4}}-\frac {2 \sqrt {2} c^{3/4} \left (18 a^{3/2} \sqrt {c} d e^3+7 a^2 e^4+2 \sqrt {a} c^{3/2} d^3 e-12 a c d^2 e^2-3 c^2 d^4\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{a^{7/4}}+\frac {8 c x \left (a e^2+c d^2\right ) \left (c d \left (d-2 e x^2\right )-a e^2\right )}{a \left (a+c x^4\right )}+\frac {16 e^4 x \left (a e^2+c d^2\right )}{d \left (d+e x^2\right )}+\frac {16 e^{7/2} \left (a e^2+9 c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2}}}{32 \left (a e^2+c d^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((16*e^4*(c*d^2 + a*e^2)*x)/(d*(d + e*x^2)) + (8*c*(c*d^2 + a*e^2)*x*(-(a*e^2) + c*d*(d - 2*e*x^2)))/(a*(a + c
*x^4)) + (16*e^(7/2)*(9*c*d^2 + a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/d^(3/2) + (2*Sqrt[2]*c^(3/4)*(-3*c^2*d^4 +
2*Sqrt[a]*c^(3/2)*d^3*e - 12*a*c*d^2*e^2 + 18*a^(3/2)*Sqrt[c]*d*e^3 + 7*a^2*e^4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*
x)/a^(1/4)])/a^(7/4) - (2*Sqrt[2]*c^(3/4)*(-3*c^2*d^4 + 2*Sqrt[a]*c^(3/2)*d^3*e - 12*a*c*d^2*e^2 + 18*a^(3/2)*
Sqrt[c]*d*e^3 + 7*a^2*e^4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(7/4) - (Sqrt[2]*c^(3/4)*(3*c^2*d^4 + 2*
Sqrt[a]*c^(3/2)*d^3*e + 12*a*c*d^2*e^2 + 18*a^(3/2)*Sqrt[c]*d*e^3 - 7*a^2*e^4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c
^(1/4)*x + Sqrt[c]*x^2])/a^(7/4) + (Sqrt[2]*c^(3/4)*(3*c^2*d^4 + 2*Sqrt[a]*c^(3/2)*d^3*e + 12*a*c*d^2*e^2 + 18
*a^(3/2)*Sqrt[c]*d*e^3 - 7*a^2*e^4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/a^(7/4))/(32*(c*d^
2 + a*e^2)^3)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [A]  time = 0.25, size = 855, normalized size = 0.99 \[ \frac {{\left (9 \, c d^{2} e^{4} + a e^{6}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {1}{2}\right )}}{2 \, {\left (c^{3} d^{7} + 3 \, a c^{2} d^{5} e^{2} + 3 \, a^{2} c d^{3} e^{4} + a^{3} d e^{6}\right )} \sqrt {d}} + \frac {{\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{4} + 12 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{2} e^{2} - 2 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{3} e - 7 \, \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c e^{4} - 18 \, \left (a c^{3}\right )^{\frac {3}{4}} a d e^{3}\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 )}{8 \, {\left (\sqrt {2} a^{2} c^{4} d^{6} + 3 \, \sqrt {2} a^{3} c^{3} d^{4} e^{2} + 3 \, \sqrt {2} a^{4} c^{2} d^{2} e^{4} + \sqrt {2} a^{5} c e^{6}\right )}} + \frac {{\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{4} + 12 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{2} e^{2} - 2 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{3} e - 7 \, \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c e^{4} - 18 \, \left (a c^{3}\right )^{\frac {3}{4}} a d e^{3}\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 )}{8 \, {\left (\sqrt {2} a^{2} c^{4} d^{6} + 3 \, \sqrt {2} a^{3} c^{3} d^{4} e^{2} + 3 \, \sqrt {2} a^{4} c^{2} d^{2} e^{4} + \sqrt {2} a^{5} c e^{6}\right )}} + \frac {{\left (3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{4} + 12 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{2} e^{2} + 2 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} c d^{3} e - 7 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c e^{4} + 18 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} a d e^{3}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, {\left (a^{2} c^{4} d^{6} + 3 \, a^{3} c^{3} d^{4} e^{2} + 3 \, a^{4} c^{2} d^{2} e^{4} + a^{5} c e^{6}\right )}} - \frac {{\left (3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{4} + 12 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{2} e^{2} + 2 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} c d^{3} e - 7 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c e^{4} + 18 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} a d e^{3}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, {\left (a^{2} c^{4} d^{6} + 3 \, a^{3} c^{3} d^{4} e^{2} + 3 \, a^{4} c^{2} d^{2} e^{4} + a^{5} c e^{6}\right )}} - \frac {2 \, c^{2} d^{2} x^{5} e^{2} + c^{2} d^{3} x^{3} e - 2 \, a c x^{5} e^{4} - c^{2} d^{4} x + a c d x^{3} e^{3} + a c d^{2} x e^{2} - 2 \, a^{2} x e^{4}}{4 \, {\left (a c^{2} d^{5} + 2 \, a^{2} c d^{3} e^{2} + a^{3} d e^{4}\right )} {\left (c x^{6} e + c d x^{4} + a x^{2} e + a d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(9*c*d^2*e^4 + a*e^6)*arctan(x*e^(1/2)/sqrt(d))*e^(-1/2)/((c^3*d^7 + 3*a*c^2*d^5*e^2 + 3*a^2*c*d^3*e^4 + a
^3*d*e^6)*sqrt(d)) + 1/8*(3*(a*c^3)^(1/4)*c^3*d^4 + 12*(a*c^3)^(1/4)*a*c^2*d^2*e^2 - 2*(a*c^3)^(3/4)*c*d^3*e -
7*(a*c^3)^(1/4)*a^2*c*e^4 - 18*(a*c^3)^(3/4)*a*d*e^3)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)^(1
/4))/(sqrt(2)*a^2*c^4*d^6 + 3*sqrt(2)*a^3*c^3*d^4*e^2 + 3*sqrt(2)*a^4*c^2*d^2*e^4 + sqrt(2)*a^5*c*e^6) + 1/8*(
3*(a*c^3)^(1/4)*c^3*d^4 + 12*(a*c^3)^(1/4)*a*c^2*d^2*e^2 - 2*(a*c^3)^(3/4)*c*d^3*e - 7*(a*c^3)^(1/4)*a^2*c*e^4
- 18*(a*c^3)^(3/4)*a*d*e^3)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(sqrt(2)*a^2*c^4*d^6
+ 3*sqrt(2)*a^3*c^3*d^4*e^2 + 3*sqrt(2)*a^4*c^2*d^2*e^4 + sqrt(2)*a^5*c*e^6) + 1/32*(3*sqrt(2)*(a*c^3)^(1/4)*c
^3*d^4 + 12*sqrt(2)*(a*c^3)^(1/4)*a*c^2*d^2*e^2 + 2*sqrt(2)*(a*c^3)^(3/4)*c*d^3*e - 7*sqrt(2)*(a*c^3)^(1/4)*a^
2*c*e^4 + 18*sqrt(2)*(a*c^3)^(3/4)*a*d*e^3)*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^2*c^4*d^6 + 3*a^3*
c^3*d^4*e^2 + 3*a^4*c^2*d^2*e^4 + a^5*c*e^6) - 1/32*(3*sqrt(2)*(a*c^3)^(1/4)*c^3*d^4 + 12*sqrt(2)*(a*c^3)^(1/4
)*a*c^2*d^2*e^2 + 2*sqrt(2)*(a*c^3)^(3/4)*c*d^3*e - 7*sqrt(2)*(a*c^3)^(1/4)*a^2*c*e^4 + 18*sqrt(2)*(a*c^3)^(3/
4)*a*d*e^3)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 + 3*a^4*c^2*d^2*e^4
+ a^5*c*e^6) - 1/4*(2*c^2*d^2*x^5*e^2 + c^2*d^3*x^3*e - 2*a*c*x^5*e^4 - c^2*d^4*x + a*c*d*x^3*e^3 + a*c*d^2*x*
e^2 - 2*a^2*x*e^4)/((a*c^2*d^5 + 2*a^2*c*d^3*e^2 + a^3*d*e^4)*(c*x^6*e + c*d*x^4 + a*x^2*e + a*d))

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maple [A]  time = 0.02, size = 1169, normalized size = 1.35 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*e^6/(a*e^2+c*d^2)^3/d*x/(e*x^2+d)*a+1/2*e^4/(a*e^2+c*d^2)^3*d*x/(e*x^2+d)*c+1/2*e^6/(a*e^2+c*d^2)^3/d/(d*e
)^(1/2)*arctan(1/(d*e)^(1/2)*e*x)*a+9/2*e^4/(a*e^2+c*d^2)^3*d/(d*e)^(1/2)*arctan(1/(d*e)^(1/2)*e*x)*c-1/2*c^2/
(a*e^2+c*d^2)^3/(c*x^4+a)*d*e^3*x^3-1/2*c^3/(a*e^2+c*d^2)^3/(c*x^4+a)*d^3*e/a*x^3-1/4*c/(a*e^2+c*d^2)^3/(c*x^4
+a)*a*x*e^4+1/4*c^3/(a*e^2+c*d^2)^3/(c*x^4+a)/a*x*d^4-7/16*c/(a*e^2+c*d^2)^3*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2
)/(a/c)^(1/4)*x+1)*e^4+3/4*c^2/(a*e^2+c*d^2)^3/a*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)*d^2*e^2+3
/16*c^3/(a*e^2+c*d^2)^3/a^2*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)*d^4-7/16*c/(a*e^2+c*d^2)^3*(a/
c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)*e^4+3/4*c^2/(a*e^2+c*d^2)^3/a*(a/c)^(1/4)*2^(1/2)*arctan(2^(1
/2)/(a/c)^(1/4)*x-1)*d^2*e^2+3/16*c^3/(a*e^2+c*d^2)^3/a^2*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)*
d^4-7/32*c/(a*e^2+c*d^2)^3*(a/c)^(1/4)*2^(1/2)*ln((x^2+(a/c)^(1/4)*2^(1/2)*x+(a/c)^(1/2))/(x^2-(a/c)^(1/4)*2^(
1/2)*x+(a/c)^(1/2)))*e^4+3/8*c^2/(a*e^2+c*d^2)^3/a*(a/c)^(1/4)*2^(1/2)*ln((x^2+(a/c)^(1/4)*2^(1/2)*x+(a/c)^(1/
2))/(x^2-(a/c)^(1/4)*2^(1/2)*x+(a/c)^(1/2)))*d^2*e^2+3/32*c^3/(a*e^2+c*d^2)^3/a^2*(a/c)^(1/4)*2^(1/2)*ln((x^2+
(a/c)^(1/4)*2^(1/2)*x+(a/c)^(1/2))/(x^2-(a/c)^(1/4)*2^(1/2)*x+(a/c)^(1/2)))*d^4-9/16*c/(a*e^2+c*d^2)^3/(a/c)^(
1/4)*2^(1/2)*ln((x^2-(a/c)^(1/4)*2^(1/2)*x+(a/c)^(1/2))/(x^2+(a/c)^(1/4)*2^(1/2)*x+(a/c)^(1/2)))*d*e^3-1/16*c^
2/(a*e^2+c*d^2)^3/a/(a/c)^(1/4)*2^(1/2)*ln((x^2-(a/c)^(1/4)*2^(1/2)*x+(a/c)^(1/2))/(x^2+(a/c)^(1/4)*2^(1/2)*x+
(a/c)^(1/2)))*d^3*e-9/8*c/(a*e^2+c*d^2)^3/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)*d*e^3-1/8*c^2/(a
*e^2+c*d^2)^3/a/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)*d^3*e-9/8*c/(a*e^2+c*d^2)^3/(a/c)^(1/4)*2^
(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)*d*e^3-1/8*c^2/(a*e^2+c*d^2)^3/a/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)
^(1/4)*x-1)*d^3*e

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maxima [A]  time = 2.61, size = 732, normalized size = 0.85 \[ \frac {c {\left (\frac {2 \, \sqrt {2} {\left (3 \, c^{\frac {5}{2}} d^{4} - 2 \, \sqrt {a} c^{2} d^{3} e + 12 \, a c^{\frac {3}{2}} d^{2} e^{2} - 18 \, a^{\frac {3}{2}} c d e^{3} - 7 \, a^{2} \sqrt {c} e^{4}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (3 \, c^{\frac {5}{2}} d^{4} - 2 \, \sqrt {a} c^{2} d^{3} e + 12 \, a c^{\frac {3}{2}} d^{2} e^{2} - 18 \, a^{\frac {3}{2}} c d e^{3} - 7 \, a^{2} \sqrt {c} e^{4}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (3 \, c^{\frac {5}{2}} d^{4} + 2 \, \sqrt {a} c^{2} d^{3} e + 12 \, a c^{\frac {3}{2}} d^{2} e^{2} + 18 \, a^{\frac {3}{2}} c d e^{3} - 7 \, a^{2} \sqrt {c} e^{4}\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (3 \, c^{\frac {5}{2}} d^{4} + 2 \, \sqrt {a} c^{2} d^{3} e + 12 \, a c^{\frac {3}{2}} d^{2} e^{2} + 18 \, a^{\frac {3}{2}} c d e^{3} - 7 \, a^{2} \sqrt {c} e^{4}\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}\right )}}{32 \, {\left (a c^{3} d^{6} + 3 \, a^{2} c^{2} d^{4} e^{2} + 3 \, a^{3} c d^{2} e^{4} + a^{4} e^{6}\right )}} + \frac {{\left (9 \, c d^{2} e^{4} + a e^{6}\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \, {\left (c^{3} d^{7} + 3 \, a c^{2} d^{5} e^{2} + 3 \, a^{2} c d^{3} e^{4} + a^{3} d e^{6}\right )} \sqrt {d e}} - \frac {2 \, {\left (c^{2} d^{2} e^{2} - a c e^{4}\right )} x^{5} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x^{3} - {\left (c^{2} d^{4} - a c d^{2} e^{2} + 2 \, a^{2} e^{4}\right )} x}{4 \, {\left (a^{2} c^{2} d^{6} + 2 \, a^{3} c d^{4} e^{2} + a^{4} d^{2} e^{4} + {\left (a c^{3} d^{5} e + 2 \, a^{2} c^{2} d^{3} e^{3} + a^{3} c d e^{5}\right )} x^{6} + {\left (a c^{3} d^{6} + 2 \, a^{2} c^{2} d^{4} e^{2} + a^{3} c d^{2} e^{4}\right )} x^{4} + {\left (a^{2} c^{2} d^{5} e + 2 \, a^{3} c d^{3} e^{3} + a^{4} d e^{5}\right )} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*c*(2*sqrt(2)*(3*c^(5/2)*d^4 - 2*sqrt(a)*c^2*d^3*e + 12*a*c^(3/2)*d^2*e^2 - 18*a^(3/2)*c*d*e^3 - 7*a^2*sqr
t(c)*e^4)*arctan(1/2*sqrt(2)*(2*sqrt(c)*x + sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a)*sqrt(c)))/(sqrt(a)*sqrt(sqrt
(a)*sqrt(c))*sqrt(c)) + 2*sqrt(2)*(3*c^(5/2)*d^4 - 2*sqrt(a)*c^2*d^3*e + 12*a*c^(3/2)*d^2*e^2 - 18*a^(3/2)*c*d
*e^3 - 7*a^2*sqrt(c)*e^4)*arctan(1/2*sqrt(2)*(2*sqrt(c)*x - sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a)*sqrt(c)))/(s
qrt(a)*sqrt(sqrt(a)*sqrt(c))*sqrt(c)) + sqrt(2)*(3*c^(5/2)*d^4 + 2*sqrt(a)*c^2*d^3*e + 12*a*c^(3/2)*d^2*e^2 +
18*a^(3/2)*c*d*e^3 - 7*a^2*sqrt(c)*e^4)*log(sqrt(c)*x^2 + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(3/4
)) - sqrt(2)*(3*c^(5/2)*d^4 + 2*sqrt(a)*c^2*d^3*e + 12*a*c^(3/2)*d^2*e^2 + 18*a^(3/2)*c*d*e^3 - 7*a^2*sqrt(c)*
e^4)*log(sqrt(c)*x^2 - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(3/4)))/(a*c^3*d^6 + 3*a^2*c^2*d^4*e^2
+ 3*a^3*c*d^2*e^4 + a^4*e^6) + 1/2*(9*c*d^2*e^4 + a*e^6)*arctan(e*x/sqrt(d*e))/((c^3*d^7 + 3*a*c^2*d^5*e^2 + 3
*a^2*c*d^3*e^4 + a^3*d*e^6)*sqrt(d*e)) - 1/4*(2*(c^2*d^2*e^2 - a*c*e^4)*x^5 + (c^2*d^3*e + a*c*d*e^3)*x^3 - (c
^2*d^4 - a*c*d^2*e^2 + 2*a^2*e^4)*x)/(a^2*c^2*d^6 + 2*a^3*c*d^4*e^2 + a^4*d^2*e^4 + (a*c^3*d^5*e + 2*a^2*c^2*d
^3*e^3 + a^3*c*d*e^5)*x^6 + (a*c^3*d^6 + 2*a^2*c^2*d^4*e^2 + a^3*c*d^2*e^4)*x^4 + (a^2*c^2*d^5*e + 2*a^3*c*d^3
*e^3 + a^4*d*e^5)*x^2)

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mupad [B]  time = 8.33, size = 28923, normalized size = 33.48 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*(2*a^2*e^4 + c^2*d^4 - a*c*d^2*e^2))/(4*a*d*(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2)) - (c*e*x^3)/(4*a*(a*e^2 +
c*d^2)) + (c*e^2*x^5*(a*e^2 - c*d^2))/(2*a*d*(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2)))/(a*d + a*e*x^2 + c*d*x^4 +
c*e*x^6) + atan(((((3584*a^10*c^5*e^21 + 1152*a*c^14*d^18*e^3 + 13184*a^2*c^13*d^16*e^5 + 54912*a^3*c^12*d^14
*e^7 + 296832*a^4*c^11*d^12*e^9 + 1282432*a^5*c^10*d^10*e^11 + 769152*a^6*c^9*d^8*e^13 - 1421440*a^7*c^8*d^6*e
^15 - 1254784*a^8*c^7*d^4*e^17 - 89088*a^9*c^6*d^2*e^19)/(512*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^1
4 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10
+ 28*a^10*c^2*d^6*e^12)) - (((65536*a^15*c^4*d*e^24 - 24576*a^4*c^15*d^23*e^2 - 212992*a^5*c^14*d^21*e^4 - 35
2256*a^6*c^13*d^19*e^6 + 1966080*a^7*c^12*d^17*e^8 + 10960896*a^8*c^11*d^15*e^10 + 25460736*a^9*c^10*d^13*e^12
+ 34750464*a^10*c^9*d^11*e^14 + 30081024*a^11*c^8*d^9*e^16 + 16588800*a^12*c^7*d^7*e^18 + 5554176*a^13*c^6*d^
5*e^20 + 991232*a^14*c^5*d^3*e^22)/(512*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2
+ 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^1
2)) - (x*(-(49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) - 12*a^4*c^5*d^7*e + 252*a^7*c^2*d*e^7 -
156*a^5*c^4*d^5*e^3 - 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2) - 492*a^3*c*d^2*e^6*(-a^7*c^3)^(
1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 + 6*a^12*c*d^2*e^10 + 6*a^8*c^5*d^1
0*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/2)*(65536*a^6*c^15*d^24*e^3 + 589
824*a^7*c^14*d^22*e^5 + 2293760*a^8*c^13*d^20*e^7 + 4915200*a^9*c^12*d^18*e^9 + 5898240*a^10*c^11*d^16*e^11 +
2752512*a^11*c^10*d^14*e^13 - 2752512*a^12*c^9*d^12*e^15 - 5898240*a^13*c^8*d^10*e^17 - 4915200*a^14*c^7*d^8*e
^19 - 2293760*a^15*c^6*d^6*e^21 - 589824*a^16*c^5*d^4*e^23 - 65536*a^17*c^4*d^2*e^25))/(128*(a^4*c^8*d^18 + a^
12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*
d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12)))*(-(49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)
^(1/2) - 12*a^4*c^5*d^7*e + 252*a^7*c^2*d*e^7 - 156*a^5*c^4*d^5*e^3 - 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(
-a^7*c^3)^(1/2) - 492*a^3*c*d^2*e^6*(-a^7*c^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 +
a^7*c^6*d^12 + 6*a^12*c*d^2*e^10 + 6*a^8*c^5*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2
*d^4*e^8)))^(1/2) - (x*(4096*a^12*c^5*d*e^22 - 1152*a^2*c^15*d^21*e^2 - 15232*a^3*c^14*d^19*e^4 - 78336*a^4*c^
13*d^17*e^6 - 140800*a^5*c^12*d^15*e^8 + 489728*a^6*c^11*d^13*e^10 + 2219776*a^7*c^10*d^11*e^12 + 3155456*a^8*
c^9*d^9*e^14 + 1901056*a^9*c^8*d^7*e^16 + 362368*a^10*c^7*d^5*e^18 - 32640*a^11*c^6*d^3*e^20))/(128*(a^4*c^8*d
^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*
a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12)))*(-(49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-
a^7*c^3)^(1/2) - 12*a^4*c^5*d^7*e + 252*a^7*c^2*d*e^7 - 156*a^5*c^4*d^5*e^3 - 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d
^6*e^2*(-a^7*c^3)^(1/2) - 492*a^3*c*d^2*e^6*(-a^7*c^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13
*e^12 + a^7*c^6*d^12 + 6*a^12*c*d^2*e^10 + 6*a^8*c^5*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*
a^11*c^2*d^4*e^8)))^(1/2))*(-(49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) - 12*a^4*c^5*d^7*e + 25
2*a^7*c^2*d*e^7 - 156*a^5*c^4*d^5*e^3 - 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2) - 492*a^3*c*d^
2*e^6*(-a^7*c^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 + 6*a^12*c*d^2*e^
10 + 6*a^8*c^5*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/2) - (x*(81*c^1
3*d^14*e^5 - 392*a^7*c^6*e^19 + 1206*a*c^12*d^12*e^7 + 12247*a^2*c^11*d^10*e^9 + 58636*a^3*c^10*d^8*e^11 + 114
927*a^4*c^9*d^6*e^13 - 1306*a^5*c^8*d^4*e^15 - 3575*a^6*c^7*d^2*e^17))/(128*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*
a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^
9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12)))*(-(49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) - 12*a^4*
c^5*d^7*e + 252*a^7*c^2*d*e^7 - 156*a^5*c^4*d^5*e^3 - 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2)
- 492*a^3*c*d^2*e^6*(-a^7*c^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 + 6
*a^12*c*d^2*e^10 + 6*a^8*c^5*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/2
)*1i - (((3584*a^10*c^5*e^21 + 1152*a*c^14*d^18*e^3 + 13184*a^2*c^13*d^16*e^5 + 54912*a^3*c^12*d^14*e^7 + 2968
32*a^4*c^11*d^12*e^9 + 1282432*a^5*c^10*d^10*e^11 + 769152*a^6*c^9*d^8*e^13 - 1421440*a^7*c^8*d^6*e^15 - 12547
84*a^8*c^7*d^4*e^17 - 89088*a^9*c^6*d^2*e^19)/(512*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c
^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*
c^2*d^6*e^12)) - (((65536*a^15*c^4*d*e^24 - 24576*a^4*c^15*d^23*e^2 - 212992*a^5*c^14*d^21*e^4 - 352256*a^6*c^
13*d^19*e^6 + 1966080*a^7*c^12*d^17*e^8 + 10960896*a^8*c^11*d^15*e^10 + 25460736*a^9*c^10*d^13*e^12 + 34750464
*a^10*c^9*d^11*e^14 + 30081024*a^11*c^8*d^9*e^16 + 16588800*a^12*c^7*d^7*e^18 + 5554176*a^13*c^6*d^5*e^20 + 99
1232*a^14*c^5*d^3*e^22)/(512*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c
^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12)) + (x*(-
(49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) - 12*a^4*c^5*d^7*e + 252*a^7*c^2*d*e^7 - 156*a^5*c^4
*d^5*e^3 - 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2) - 492*a^3*c*d^2*e^6*(-a^7*c^3)^(1/2) + 30*a
^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 + 6*a^12*c*d^2*e^10 + 6*a^8*c^5*d^10*e^2 + 15*
a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/2)*(65536*a^6*c^15*d^24*e^3 + 589824*a^7*c^1
4*d^22*e^5 + 2293760*a^8*c^13*d^20*e^7 + 4915200*a^9*c^12*d^18*e^9 + 5898240*a^10*c^11*d^16*e^11 + 2752512*a^1
1*c^10*d^14*e^13 - 2752512*a^12*c^9*d^12*e^15 - 5898240*a^13*c^8*d^10*e^17 - 4915200*a^14*c^7*d^8*e^19 - 22937
60*a^15*c^6*d^6*e^21 - 589824*a^16*c^5*d^4*e^23 - 65536*a^17*c^4*d^2*e^25))/(128*(a^4*c^8*d^18 + a^12*d^2*e^16
+ 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 +
56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12)))*(-(49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) - 12
*a^4*c^5*d^7*e + 252*a^7*c^2*d*e^7 - 156*a^5*c^4*d^5*e^3 - 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(
1/2) - 492*a^3*c*d^2*e^6*(-a^7*c^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^1
2 + 6*a^12*c*d^2*e^10 + 6*a^8*c^5*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))
^(1/2) + (x*(4096*a^12*c^5*d*e^22 - 1152*a^2*c^15*d^21*e^2 - 15232*a^3*c^14*d^19*e^4 - 78336*a^4*c^13*d^17*e^6
- 140800*a^5*c^12*d^15*e^8 + 489728*a^6*c^11*d^13*e^10 + 2219776*a^7*c^10*d^11*e^12 + 3155456*a^8*c^9*d^9*e^1
4 + 1901056*a^9*c^8*d^7*e^16 + 362368*a^10*c^7*d^5*e^18 - 32640*a^11*c^6*d^3*e^20))/(128*(a^4*c^8*d^18 + a^12*
d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^1
0*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12)))*(-(49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1
/2) - 12*a^4*c^5*d^7*e + 252*a^7*c^2*d*e^7 - 156*a^5*c^4*d^5*e^3 - 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^
7*c^3)^(1/2) - 492*a^3*c*d^2*e^6*(-a^7*c^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7
*c^6*d^12 + 6*a^12*c*d^2*e^10 + 6*a^8*c^5*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^
4*e^8)))^(1/2))*(-(49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) - 12*a^4*c^5*d^7*e + 252*a^7*c^2*d
*e^7 - 156*a^5*c^4*d^5*e^3 - 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2) - 492*a^3*c*d^2*e^6*(-a^7
*c^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 + 6*a^12*c*d^2*e^10 + 6*a^8*
c^5*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/2) + (x*(81*c^13*d^14*e^5
- 392*a^7*c^6*e^19 + 1206*a*c^12*d^12*e^7 + 12247*a^2*c^11*d^10*e^9 + 58636*a^3*c^10*d^8*e^11 + 114927*a^4*c^9
*d^6*e^13 - 1306*a^5*c^8*d^4*e^15 - 3575*a^6*c^7*d^2*e^17))/(128*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*
e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e
^10 + 28*a^10*c^2*d^6*e^12)))*(-(49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) - 12*a^4*c^5*d^7*e +
252*a^7*c^2*d*e^7 - 156*a^5*c^4*d^5*e^3 - 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2) - 492*a^3*c
*d^2*e^6*(-a^7*c^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 + 6*a^12*c*d^2
*e^10 + 6*a^8*c^5*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/2)*1i)/((((3
584*a^10*c^5*e^21 + 1152*a*c^14*d^18*e^3 + 13184*a^2*c^13*d^16*e^5 + 54912*a^3*c^12*d^14*e^7 + 296832*a^4*c^11
*d^12*e^9 + 1282432*a^5*c^10*d^10*e^11 + 769152*a^6*c^9*d^8*e^13 - 1421440*a^7*c^8*d^6*e^15 - 1254784*a^8*c^7*
d^4*e^17 - 89088*a^9*c^6*d^2*e^19)/(512*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2
+ 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^1
2)) - (((65536*a^15*c^4*d*e^24 - 24576*a^4*c^15*d^23*e^2 - 212992*a^5*c^14*d^21*e^4 - 352256*a^6*c^13*d^19*e^6
+ 1966080*a^7*c^12*d^17*e^8 + 10960896*a^8*c^11*d^15*e^10 + 25460736*a^9*c^10*d^13*e^12 + 34750464*a^10*c^9*d
^11*e^14 + 30081024*a^11*c^8*d^9*e^16 + 16588800*a^12*c^7*d^7*e^18 + 5554176*a^13*c^6*d^5*e^20 + 991232*a^14*c
^5*d^3*e^22)/(512*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4
+ 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12)) - (x*(-(49*a^4*e^8
*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) - 12*a^4*c^5*d^7*e + 252*a^7*c^2*d*e^7 - 156*a^5*c^4*d^5*e^3 -
404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2) - 492*a^3*c*d^2*e^6*(-a^7*c^3)^(1/2) + 30*a^2*c^2*d^4*
e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 + 6*a^12*c*d^2*e^10 + 6*a^8*c^5*d^10*e^2 + 15*a^9*c^4*d^8
*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/2)*(65536*a^6*c^15*d^24*e^3 + 589824*a^7*c^14*d^22*e^5
+ 2293760*a^8*c^13*d^20*e^7 + 4915200*a^9*c^12*d^18*e^9 + 5898240*a^10*c^11*d^16*e^11 + 2752512*a^11*c^10*d^14
*e^13 - 2752512*a^12*c^9*d^12*e^15 - 5898240*a^13*c^8*d^10*e^17 - 4915200*a^14*c^7*d^8*e^19 - 2293760*a^15*c^6
*d^6*e^21 - 589824*a^16*c^5*d^4*e^23 - 65536*a^17*c^4*d^2*e^25))/(128*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c
*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*
d^8*e^10 + 28*a^10*c^2*d^6*e^12)))*(-(49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) - 12*a^4*c^5*d^
7*e + 252*a^7*c^2*d*e^7 - 156*a^5*c^4*d^5*e^3 - 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2) - 492*
a^3*c*d^2*e^6*(-a^7*c^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 + 6*a^12*
c*d^2*e^10 + 6*a^8*c^5*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/2) - (x
*(4096*a^12*c^5*d*e^22 - 1152*a^2*c^15*d^21*e^2 - 15232*a^3*c^14*d^19*e^4 - 78336*a^4*c^13*d^17*e^6 - 140800*a
^5*c^12*d^15*e^8 + 489728*a^6*c^11*d^13*e^10 + 2219776*a^7*c^10*d^11*e^12 + 3155456*a^8*c^9*d^9*e^14 + 1901056
*a^9*c^8*d^7*e^16 + 362368*a^10*c^7*d^5*e^18 - 32640*a^11*c^6*d^3*e^20))/(128*(a^4*c^8*d^18 + a^12*d^2*e^16 +
8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*
a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12)))*(-(49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) - 12*a^
4*c^5*d^7*e + 252*a^7*c^2*d*e^7 - 156*a^5*c^4*d^5*e^3 - 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2
) - 492*a^3*c*d^2*e^6*(-a^7*c^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 +
6*a^12*c*d^2*e^10 + 6*a^8*c^5*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1
/2))*(-(49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) - 12*a^4*c^5*d^7*e + 252*a^7*c^2*d*e^7 - 156*
a^5*c^4*d^5*e^3 - 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2) - 492*a^3*c*d^2*e^6*(-a^7*c^3)^(1/2)
+ 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 + 6*a^12*c*d^2*e^10 + 6*a^8*c^5*d^10*e^
2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/2) - (x*(81*c^13*d^14*e^5 - 392*a^7*c
^6*e^19 + 1206*a*c^12*d^12*e^7 + 12247*a^2*c^11*d^10*e^9 + 58636*a^3*c^10*d^8*e^11 + 114927*a^4*c^9*d^6*e^13 -
1306*a^5*c^8*d^4*e^15 - 3575*a^6*c^7*d^2*e^17))/(128*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^
5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^
10*c^2*d^6*e^12)))*(-(49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) - 12*a^4*c^5*d^7*e + 252*a^7*c^
2*d*e^7 - 156*a^5*c^4*d^5*e^3 - 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2) - 492*a^3*c*d^2*e^6*(-
a^7*c^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 + 6*a^12*c*d^2*e^10 + 6*a
^8*c^5*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/2) + (((3584*a^10*c^5*e
^21 + 1152*a*c^14*d^18*e^3 + 13184*a^2*c^13*d^16*e^5 + 54912*a^3*c^12*d^14*e^7 + 296832*a^4*c^11*d^12*e^9 + 12
82432*a^5*c^10*d^10*e^11 + 769152*a^6*c^9*d^8*e^13 - 1421440*a^7*c^8*d^6*e^15 - 1254784*a^8*c^7*d^4*e^17 - 890
88*a^9*c^6*d^2*e^19)/(512*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*
d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12)) - (((65536
*a^15*c^4*d*e^24 - 24576*a^4*c^15*d^23*e^2 - 212992*a^5*c^14*d^21*e^4 - 352256*a^6*c^13*d^19*e^6 + 1966080*a^7
*c^12*d^17*e^8 + 10960896*a^8*c^11*d^15*e^10 + 25460736*a^9*c^10*d^13*e^12 + 34750464*a^10*c^9*d^11*e^14 + 300
81024*a^11*c^8*d^9*e^16 + 16588800*a^12*c^7*d^7*e^18 + 5554176*a^13*c^6*d^5*e^20 + 991232*a^14*c^5*d^3*e^22)/(
512*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*
d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12)) + (x*(-(49*a^4*e^8*(-a^7*c^3)^(1
/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) - 12*a^4*c^5*d^7*e + 252*a^7*c^2*d*e^7 - 156*a^5*c^4*d^5*e^3 - 404*a^6*c^3*d^
3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2) - 492*a^3*c*d^2*e^6*(-a^7*c^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)
^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 + 6*a^12*c*d^2*e^10 + 6*a^8*c^5*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10
*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/2)*(65536*a^6*c^15*d^24*e^3 + 589824*a^7*c^14*d^22*e^5 + 2293760*a^8*
c^13*d^20*e^7 + 4915200*a^9*c^12*d^18*e^9 + 5898240*a^10*c^11*d^16*e^11 + 2752512*a^11*c^10*d^14*e^13 - 275251
2*a^12*c^9*d^12*e^15 - 5898240*a^13*c^8*d^10*e^17 - 4915200*a^14*c^7*d^8*e^19 - 2293760*a^15*c^6*d^6*e^21 - 58
9824*a^16*c^5*d^4*e^23 - 65536*a^17*c^4*d^2*e^25))/(128*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*
a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*
a^10*c^2*d^6*e^12)))*(-(49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) - 12*a^4*c^5*d^7*e + 252*a^7*
c^2*d*e^7 - 156*a^5*c^4*d^5*e^3 - 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2) - 492*a^3*c*d^2*e^6*
(-a^7*c^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 + 6*a^12*c*d^2*e^10 + 6
*a^8*c^5*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/2) + (x*(4096*a^12*c^
5*d*e^22 - 1152*a^2*c^15*d^21*e^2 - 15232*a^3*c^14*d^19*e^4 - 78336*a^4*c^13*d^17*e^6 - 140800*a^5*c^12*d^15*e
^8 + 489728*a^6*c^11*d^13*e^10 + 2219776*a^7*c^10*d^11*e^12 + 3155456*a^8*c^9*d^9*e^14 + 1901056*a^9*c^8*d^7*e
^16 + 362368*a^10*c^7*d^5*e^18 - 32640*a^11*c^6*d^3*e^20))/(128*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e
^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^
10 + 28*a^10*c^2*d^6*e^12)))*(-(49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) - 12*a^4*c^5*d^7*e +
252*a^7*c^2*d*e^7 - 156*a^5*c^4*d^5*e^3 - 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2) - 492*a^3*c*
d^2*e^6*(-a^7*c^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 + 6*a^12*c*d^2*
e^10 + 6*a^8*c^5*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/2))*(-(49*a^4
*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) - 12*a^4*c^5*d^7*e + 252*a^7*c^2*d*e^7 - 156*a^5*c^4*d^5*e^
3 - 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2) - 492*a^3*c*d^2*e^6*(-a^7*c^3)^(1/2) + 30*a^2*c^2*
d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 + 6*a^12*c*d^2*e^10 + 6*a^8*c^5*d^10*e^2 + 15*a^9*c^4
*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/2) + (x*(81*c^13*d^14*e^5 - 392*a^7*c^6*e^19 + 1206
*a*c^12*d^12*e^7 + 12247*a^2*c^11*d^10*e^9 + 58636*a^3*c^10*d^8*e^11 + 114927*a^4*c^9*d^6*e^13 - 1306*a^5*c^8*
d^4*e^15 - 3575*a^6*c^7*d^2*e^17))/(128*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2
+ 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^1
2)))*(-(49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) - 12*a^4*c^5*d^7*e + 252*a^7*c^2*d*e^7 - 156*
a^5*c^4*d^5*e^3 - 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2) - 492*a^3*c*d^2*e^6*(-a^7*c^3)^(1/2)
+ 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 + 6*a^12*c*d^2*e^10 + 6*a^8*c^5*d^10*e^
2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/2) - (729*c^11*d^9*e^8 + 2916*a*c^10*
d^7*e^10 + 2009*a^4*c^7*d*e^16 - 2538*a^2*c^9*d^5*e^12 + 17764*a^3*c^8*d^3*e^14)/(256*(a^4*c^8*d^18 + a^12*d^2
*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e
^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12))))*(-(49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2
) - 12*a^4*c^5*d^7*e + 252*a^7*c^2*d*e^7 - 156*a^5*c^4*d^5*e^3 - 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*
c^3)^(1/2) - 492*a^3*c*d^2*e^6*(-a^7*c^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c
^6*d^12 + 6*a^12*c*d^2*e^10 + 6*a^8*c^5*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*
e^8)))^(1/2)*2i + atan(((((3584*a^10*c^5*e^21 + 1152*a*c^14*d^18*e^3 + 13184*a^2*c^13*d^16*e^5 + 54912*a^3*c^1
2*d^14*e^7 + 296832*a^4*c^11*d^12*e^9 + 1282432*a^5*c^10*d^10*e^11 + 769152*a^6*c^9*d^8*e^13 - 1421440*a^7*c^8
*d^6*e^15 - 1254784*a^8*c^7*d^4*e^17 - 89088*a^9*c^6*d^2*e^19)/(512*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d
^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^
8*e^10 + 28*a^10*c^2*d^6*e^12)) - (((65536*a^15*c^4*d*e^24 - 24576*a^4*c^15*d^23*e^2 - 212992*a^5*c^14*d^21*e^
4 - 352256*a^6*c^13*d^19*e^6 + 1966080*a^7*c^12*d^17*e^8 + 10960896*a^8*c^11*d^15*e^10 + 25460736*a^9*c^10*d^1
3*e^12 + 34750464*a^10*c^9*d^11*e^14 + 30081024*a^11*c^8*d^9*e^16 + 16588800*a^12*c^7*d^7*e^18 + 5554176*a^13*
c^6*d^5*e^20 + 991232*a^14*c^5*d^3*e^22)/(512*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^
16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d
^6*e^12)) - (x*((49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) + 12*a^4*c^5*d^7*e - 252*a^7*c^2*d*e
^7 + 156*a^5*c^4*d^5*e^3 + 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2) - 492*a^3*c*d^2*e^6*(-a^7*c
^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 + 6*a^12*c*d^2*e^10 + 6*a^8*c^
5*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/2)*(65536*a^6*c^15*d^24*e^3
+ 589824*a^7*c^14*d^22*e^5 + 2293760*a^8*c^13*d^20*e^7 + 4915200*a^9*c^12*d^18*e^9 + 5898240*a^10*c^11*d^16*e^
11 + 2752512*a^11*c^10*d^14*e^13 - 2752512*a^12*c^9*d^12*e^15 - 5898240*a^13*c^8*d^10*e^17 - 4915200*a^14*c^7*
d^8*e^19 - 2293760*a^15*c^6*d^6*e^21 - 589824*a^16*c^5*d^4*e^23 - 65536*a^17*c^4*d^2*e^25))/(128*(a^4*c^8*d^18
+ a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8
*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12)))*((49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*
c^3)^(1/2) + 12*a^4*c^5*d^7*e - 252*a^7*c^2*d*e^7 + 156*a^5*c^4*d^5*e^3 + 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e
^2*(-a^7*c^3)^(1/2) - 492*a^3*c*d^2*e^6*(-a^7*c^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^1
2 + a^7*c^6*d^12 + 6*a^12*c*d^2*e^10 + 6*a^8*c^5*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11
*c^2*d^4*e^8)))^(1/2) - (x*(4096*a^12*c^5*d*e^22 - 1152*a^2*c^15*d^21*e^2 - 15232*a^3*c^14*d^19*e^4 - 78336*a^
4*c^13*d^17*e^6 - 140800*a^5*c^12*d^15*e^8 + 489728*a^6*c^11*d^13*e^10 + 2219776*a^7*c^10*d^11*e^12 + 3155456*
a^8*c^9*d^9*e^14 + 1901056*a^9*c^8*d^7*e^16 + 362368*a^10*c^7*d^5*e^18 - 32640*a^11*c^6*d^3*e^20))/(128*(a^4*c
^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 +
70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12)))*((49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8
*(-a^7*c^3)^(1/2) + 12*a^4*c^5*d^7*e - 252*a^7*c^2*d*e^7 + 156*a^5*c^4*d^5*e^3 + 404*a^6*c^3*d^3*e^5 + 68*a*c^
3*d^6*e^2*(-a^7*c^3)^(1/2) - 492*a^3*c*d^2*e^6*(-a^7*c^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a
^13*e^12 + a^7*c^6*d^12 + 6*a^12*c*d^2*e^10 + 6*a^8*c^5*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 +
15*a^11*c^2*d^4*e^8)))^(1/2))*((49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) + 12*a^4*c^5*d^7*e -
252*a^7*c^2*d*e^7 + 156*a^5*c^4*d^5*e^3 + 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2) - 492*a^3*c*
d^2*e^6*(-a^7*c^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 + 6*a^12*c*d^2*
e^10 + 6*a^8*c^5*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/2) - (x*(81*c
^13*d^14*e^5 - 392*a^7*c^6*e^19 + 1206*a*c^12*d^12*e^7 + 12247*a^2*c^11*d^10*e^9 + 58636*a^3*c^10*d^8*e^11 + 1
14927*a^4*c^9*d^6*e^13 - 1306*a^5*c^8*d^4*e^15 - 3575*a^6*c^7*d^2*e^17))/(128*(a^4*c^8*d^18 + a^12*d^2*e^16 +
8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*
a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12)))*((49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) + 12*a^4
*c^5*d^7*e - 252*a^7*c^2*d*e^7 + 156*a^5*c^4*d^5*e^3 + 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2)
- 492*a^3*c*d^2*e^6*(-a^7*c^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 +
6*a^12*c*d^2*e^10 + 6*a^8*c^5*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/
2)*1i - (((3584*a^10*c^5*e^21 + 1152*a*c^14*d^18*e^3 + 13184*a^2*c^13*d^16*e^5 + 54912*a^3*c^12*d^14*e^7 + 296
832*a^4*c^11*d^12*e^9 + 1282432*a^5*c^10*d^10*e^11 + 769152*a^6*c^9*d^8*e^13 - 1421440*a^7*c^8*d^6*e^15 - 1254
784*a^8*c^7*d^4*e^17 - 89088*a^9*c^6*d^2*e^19)/(512*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*
c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10
*c^2*d^6*e^12)) - (((65536*a^15*c^4*d*e^24 - 24576*a^4*c^15*d^23*e^2 - 212992*a^5*c^14*d^21*e^4 - 352256*a^6*c
^13*d^19*e^6 + 1966080*a^7*c^12*d^17*e^8 + 10960896*a^8*c^11*d^15*e^10 + 25460736*a^9*c^10*d^13*e^12 + 3475046
4*a^10*c^9*d^11*e^14 + 30081024*a^11*c^8*d^9*e^16 + 16588800*a^12*c^7*d^7*e^18 + 5554176*a^13*c^6*d^5*e^20 + 9
91232*a^14*c^5*d^3*e^22)/(512*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*
c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12)) + (x*(
(49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) + 12*a^4*c^5*d^7*e - 252*a^7*c^2*d*e^7 + 156*a^5*c^4
*d^5*e^3 + 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2) - 492*a^3*c*d^2*e^6*(-a^7*c^3)^(1/2) + 30*a
^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 + 6*a^12*c*d^2*e^10 + 6*a^8*c^5*d^10*e^2 + 15*
a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/2)*(65536*a^6*c^15*d^24*e^3 + 589824*a^7*c^1
4*d^22*e^5 + 2293760*a^8*c^13*d^20*e^7 + 4915200*a^9*c^12*d^18*e^9 + 5898240*a^10*c^11*d^16*e^11 + 2752512*a^1
1*c^10*d^14*e^13 - 2752512*a^12*c^9*d^12*e^15 - 5898240*a^13*c^8*d^10*e^17 - 4915200*a^14*c^7*d^8*e^19 - 22937
60*a^15*c^6*d^6*e^21 - 589824*a^16*c^5*d^4*e^23 - 65536*a^17*c^4*d^2*e^25))/(128*(a^4*c^8*d^18 + a^12*d^2*e^16
+ 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 +
56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12)))*((49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) + 12*
a^4*c^5*d^7*e - 252*a^7*c^2*d*e^7 + 156*a^5*c^4*d^5*e^3 + 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1
/2) - 492*a^3*c*d^2*e^6*(-a^7*c^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12
+ 6*a^12*c*d^2*e^10 + 6*a^8*c^5*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^
(1/2) + (x*(4096*a^12*c^5*d*e^22 - 1152*a^2*c^15*d^21*e^2 - 15232*a^3*c^14*d^19*e^4 - 78336*a^4*c^13*d^17*e^6
- 140800*a^5*c^12*d^15*e^8 + 489728*a^6*c^11*d^13*e^10 + 2219776*a^7*c^10*d^11*e^12 + 3155456*a^8*c^9*d^9*e^14
+ 1901056*a^9*c^8*d^7*e^16 + 362368*a^10*c^7*d^5*e^18 - 32640*a^11*c^6*d^3*e^20))/(128*(a^4*c^8*d^18 + a^12*d
^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10
*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12)))*((49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2
) + 12*a^4*c^5*d^7*e - 252*a^7*c^2*d*e^7 + 156*a^5*c^4*d^5*e^3 + 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*
c^3)^(1/2) - 492*a^3*c*d^2*e^6*(-a^7*c^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c
^6*d^12 + 6*a^12*c*d^2*e^10 + 6*a^8*c^5*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*
e^8)))^(1/2))*((49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) + 12*a^4*c^5*d^7*e - 252*a^7*c^2*d*e^
7 + 156*a^5*c^4*d^5*e^3 + 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2) - 492*a^3*c*d^2*e^6*(-a^7*c^
3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 + 6*a^12*c*d^2*e^10 + 6*a^8*c^5
*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/2) + (x*(81*c^13*d^14*e^5 - 3
92*a^7*c^6*e^19 + 1206*a*c^12*d^12*e^7 + 12247*a^2*c^11*d^10*e^9 + 58636*a^3*c^10*d^8*e^11 + 114927*a^4*c^9*d^
6*e^13 - 1306*a^5*c^8*d^4*e^15 - 3575*a^6*c^7*d^2*e^17))/(128*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^1
4 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10
+ 28*a^10*c^2*d^6*e^12)))*((49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) + 12*a^4*c^5*d^7*e - 252
*a^7*c^2*d*e^7 + 156*a^5*c^4*d^5*e^3 + 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2) - 492*a^3*c*d^2
*e^6*(-a^7*c^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 + 6*a^12*c*d^2*e^1
0 + 6*a^8*c^5*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/2)*1i)/((((3584*
a^10*c^5*e^21 + 1152*a*c^14*d^18*e^3 + 13184*a^2*c^13*d^16*e^5 + 54912*a^3*c^12*d^14*e^7 + 296832*a^4*c^11*d^1
2*e^9 + 1282432*a^5*c^10*d^10*e^11 + 769152*a^6*c^9*d^8*e^13 - 1421440*a^7*c^8*d^6*e^15 - 1254784*a^8*c^7*d^4*
e^17 - 89088*a^9*c^6*d^2*e^19)/(512*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 2
8*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12))
- (((65536*a^15*c^4*d*e^24 - 24576*a^4*c^15*d^23*e^2 - 212992*a^5*c^14*d^21*e^4 - 352256*a^6*c^13*d^19*e^6 + 1
966080*a^7*c^12*d^17*e^8 + 10960896*a^8*c^11*d^15*e^10 + 25460736*a^9*c^10*d^13*e^12 + 34750464*a^10*c^9*d^11*
e^14 + 30081024*a^11*c^8*d^9*e^16 + 16588800*a^12*c^7*d^7*e^18 + 5554176*a^13*c^6*d^5*e^20 + 991232*a^14*c^5*d
^3*e^22)/(512*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 5
6*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12)) - (x*((49*a^4*e^8*(-a^
7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) + 12*a^4*c^5*d^7*e - 252*a^7*c^2*d*e^7 + 156*a^5*c^4*d^5*e^3 + 404*a
^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2) - 492*a^3*c*d^2*e^6*(-a^7*c^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(
-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 + 6*a^12*c*d^2*e^10 + 6*a^8*c^5*d^10*e^2 + 15*a^9*c^4*d^8*e^4
+ 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/2)*(65536*a^6*c^15*d^24*e^3 + 589824*a^7*c^14*d^22*e^5 + 229
3760*a^8*c^13*d^20*e^7 + 4915200*a^9*c^12*d^18*e^9 + 5898240*a^10*c^11*d^16*e^11 + 2752512*a^11*c^10*d^14*e^13
- 2752512*a^12*c^9*d^12*e^15 - 5898240*a^13*c^8*d^10*e^17 - 4915200*a^14*c^7*d^8*e^19 - 2293760*a^15*c^6*d^6*
e^21 - 589824*a^16*c^5*d^4*e^23 - 65536*a^17*c^4*d^2*e^25))/(128*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*
e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e
^10 + 28*a^10*c^2*d^6*e^12)))*((49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) + 12*a^4*c^5*d^7*e -
252*a^7*c^2*d*e^7 + 156*a^5*c^4*d^5*e^3 + 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2) - 492*a^3*c*
d^2*e^6*(-a^7*c^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 + 6*a^12*c*d^2*
e^10 + 6*a^8*c^5*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/2) - (x*(4096
*a^12*c^5*d*e^22 - 1152*a^2*c^15*d^21*e^2 - 15232*a^3*c^14*d^19*e^4 - 78336*a^4*c^13*d^17*e^6 - 140800*a^5*c^1
2*d^15*e^8 + 489728*a^6*c^11*d^13*e^10 + 2219776*a^7*c^10*d^11*e^12 + 3155456*a^8*c^9*d^9*e^14 + 1901056*a^9*c
^8*d^7*e^16 + 362368*a^10*c^7*d^5*e^18 - 32640*a^11*c^6*d^3*e^20))/(128*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11
*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^
3*d^8*e^10 + 28*a^10*c^2*d^6*e^12)))*((49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) + 12*a^4*c^5*d
^7*e - 252*a^7*c^2*d*e^7 + 156*a^5*c^4*d^5*e^3 + 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2) - 492
*a^3*c*d^2*e^6*(-a^7*c^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 + 6*a^12
*c*d^2*e^10 + 6*a^8*c^5*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/2))*((
49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) + 12*a^4*c^5*d^7*e - 252*a^7*c^2*d*e^7 + 156*a^5*c^4*
d^5*e^3 + 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2) - 492*a^3*c*d^2*e^6*(-a^7*c^3)^(1/2) + 30*a^
2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 + 6*a^12*c*d^2*e^10 + 6*a^8*c^5*d^10*e^2 + 15*a
^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/2) - (x*(81*c^13*d^14*e^5 - 392*a^7*c^6*e^19
+ 1206*a*c^12*d^12*e^7 + 12247*a^2*c^11*d^10*e^9 + 58636*a^3*c^10*d^8*e^11 + 114927*a^4*c^9*d^6*e^13 - 1306*a^
5*c^8*d^4*e^15 - 3575*a^6*c^7*d^2*e^17))/(128*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^
16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d
^6*e^12)))*((49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) + 12*a^4*c^5*d^7*e - 252*a^7*c^2*d*e^7 +
156*a^5*c^4*d^5*e^3 + 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2) - 492*a^3*c*d^2*e^6*(-a^7*c^3)^
(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 + 6*a^12*c*d^2*e^10 + 6*a^8*c^5*d^
10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/2) + (((3584*a^10*c^5*e^21 + 115
2*a*c^14*d^18*e^3 + 13184*a^2*c^13*d^16*e^5 + 54912*a^3*c^12*d^14*e^7 + 296832*a^4*c^11*d^12*e^9 + 1282432*a^5
*c^10*d^10*e^11 + 769152*a^6*c^9*d^8*e^13 - 1421440*a^7*c^8*d^6*e^15 - 1254784*a^8*c^7*d^4*e^17 - 89088*a^9*c^
6*d^2*e^19)/(512*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4
+ 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12)) - (((65536*a^15*c^4
*d*e^24 - 24576*a^4*c^15*d^23*e^2 - 212992*a^5*c^14*d^21*e^4 - 352256*a^6*c^13*d^19*e^6 + 1966080*a^7*c^12*d^1
7*e^8 + 10960896*a^8*c^11*d^15*e^10 + 25460736*a^9*c^10*d^13*e^12 + 34750464*a^10*c^9*d^11*e^14 + 30081024*a^1
1*c^8*d^9*e^16 + 16588800*a^12*c^7*d^7*e^18 + 5554176*a^13*c^6*d^5*e^20 + 991232*a^14*c^5*d^3*e^22)/(512*(a^4*
c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6
+ 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12)) + (x*((49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^
4*d^8*(-a^7*c^3)^(1/2) + 12*a^4*c^5*d^7*e - 252*a^7*c^2*d*e^7 + 156*a^5*c^4*d^5*e^3 + 404*a^6*c^3*d^3*e^5 + 68
*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2) - 492*a^3*c*d^2*e^6*(-a^7*c^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(2
56*(a^13*e^12 + a^7*c^6*d^12 + 6*a^12*c*d^2*e^10 + 6*a^8*c^5*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e
^6 + 15*a^11*c^2*d^4*e^8)))^(1/2)*(65536*a^6*c^15*d^24*e^3 + 589824*a^7*c^14*d^22*e^5 + 2293760*a^8*c^13*d^20*
e^7 + 4915200*a^9*c^12*d^18*e^9 + 5898240*a^10*c^11*d^16*e^11 + 2752512*a^11*c^10*d^14*e^13 - 2752512*a^12*c^9
*d^12*e^15 - 5898240*a^13*c^8*d^10*e^17 - 4915200*a^14*c^7*d^8*e^19 - 2293760*a^15*c^6*d^6*e^21 - 589824*a^16*
c^5*d^4*e^23 - 65536*a^17*c^4*d^2*e^25))/(128*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^
16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d
^6*e^12)))*((49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) + 12*a^4*c^5*d^7*e - 252*a^7*c^2*d*e^7 +
156*a^5*c^4*d^5*e^3 + 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2) - 492*a^3*c*d^2*e^6*(-a^7*c^3)^
(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 + 6*a^12*c*d^2*e^10 + 6*a^8*c^5*d^
10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/2) + (x*(4096*a^12*c^5*d*e^22 -
1152*a^2*c^15*d^21*e^2 - 15232*a^3*c^14*d^19*e^4 - 78336*a^4*c^13*d^17*e^6 - 140800*a^5*c^12*d^15*e^8 + 489728
*a^6*c^11*d^13*e^10 + 2219776*a^7*c^10*d^11*e^12 + 3155456*a^8*c^9*d^9*e^14 + 1901056*a^9*c^8*d^7*e^16 + 36236
8*a^10*c^7*d^5*e^18 - 32640*a^11*c^6*d^3*e^20))/(128*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5
*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^1
0*c^2*d^6*e^12)))*((49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) + 12*a^4*c^5*d^7*e - 252*a^7*c^2*
d*e^7 + 156*a^5*c^4*d^5*e^3 + 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2) - 492*a^3*c*d^2*e^6*(-a^
7*c^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 + 6*a^12*c*d^2*e^10 + 6*a^8
*c^5*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/2))*((49*a^4*e^8*(-a^7*c^
3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) + 12*a^4*c^5*d^7*e - 252*a^7*c^2*d*e^7 + 156*a^5*c^4*d^5*e^3 + 404*a^6*c
^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2) - 492*a^3*c*d^2*e^6*(-a^7*c^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7
*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 + 6*a^12*c*d^2*e^10 + 6*a^8*c^5*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20
*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/2) + (x*(81*c^13*d^14*e^5 - 392*a^7*c^6*e^19 + 1206*a*c^12*d^12*
e^7 + 12247*a^2*c^11*d^10*e^9 + 58636*a^3*c^10*d^8*e^11 + 114927*a^4*c^9*d^6*e^13 - 1306*a^5*c^8*d^4*e^15 - 35
75*a^6*c^7*d^2*e^17))/(128*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6
*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12)))*((49*a^4
*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) + 12*a^4*c^5*d^7*e - 252*a^7*c^2*d*e^7 + 156*a^5*c^4*d^5*e^
3 + 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2) - 492*a^3*c*d^2*e^6*(-a^7*c^3)^(1/2) + 30*a^2*c^2*
d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 + 6*a^12*c*d^2*e^10 + 6*a^8*c^5*d^10*e^2 + 15*a^9*c^4
*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/2) - (729*c^11*d^9*e^8 + 2916*a*c^10*d^7*e^10 + 200
9*a^4*c^7*d*e^16 - 2538*a^2*c^9*d^5*e^12 + 17764*a^3*c^8*d^3*e^14)/(256*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11
*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^
3*d^8*e^10 + 28*a^10*c^2*d^6*e^12))))*((49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) + 12*a^4*c^5*
d^7*e - 252*a^7*c^2*d*e^7 + 156*a^5*c^4*d^5*e^3 + 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2) - 49
2*a^3*c*d^2*e^6*(-a^7*c^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 + 6*a^1
2*c*d^2*e^10 + 6*a^8*c^5*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/2)*2i
+ (atan(((((x*(81*c^13*d^14*e^5 - 392*a^7*c^6*e^19 + 1206*a*c^12*d^12*e^7 + 12247*a^2*c^11*d^10*e^9 + 58636*a
^3*c^10*d^8*e^11 + 114927*a^4*c^9*d^6*e^13 - 1306*a^5*c^8*d^4*e^15 - 3575*a^6*c^7*d^2*e^17))/(128*(a^4*c^8*d^1
8 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^
8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12)) - (((7*a^10*c^5*e^21 + (9*a*c^14*d^18*e^3)/4 + (
103*a^2*c^13*d^16*e^5)/4 + (429*a^3*c^12*d^14*e^7)/4 + (2319*a^4*c^11*d^12*e^9)/4 + (10019*a^5*c^10*d^10*e^11)
/4 + (6009*a^6*c^9*d^8*e^13)/4 - (11105*a^7*c^8*d^6*e^15)/4 - (9803*a^8*c^7*d^4*e^17)/4 - 174*a^9*c^6*d^2*e^19
)/(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^
12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12) + ((a*e^2 + 9*c*d^2)*(-d^3*e^7)^(1/
2)*((x*(4096*a^12*c^5*d*e^22 - 1152*a^2*c^15*d^21*e^2 - 15232*a^3*c^14*d^19*e^4 - 78336*a^4*c^13*d^17*e^6 - 14
0800*a^5*c^12*d^15*e^8 + 489728*a^6*c^11*d^13*e^10 + 2219776*a^7*c^10*d^11*e^12 + 3155456*a^8*c^9*d^9*e^14 + 1
901056*a^9*c^8*d^7*e^16 + 362368*a^10*c^7*d^5*e^18 - 32640*a^11*c^6*d^3*e^20))/(128*(a^4*c^8*d^18 + a^12*d^2*e
^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8
+ 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12)) - (((128*a^15*c^4*d*e^24 - 48*a^4*c^15*d^23*e^2 - 416*a^5*c^14
*d^21*e^4 - 688*a^6*c^13*d^19*e^6 + 3840*a^7*c^12*d^17*e^8 + 21408*a^8*c^11*d^15*e^10 + 49728*a^9*c^10*d^13*e^
12 + 67872*a^10*c^9*d^11*e^14 + 58752*a^11*c^8*d^9*e^16 + 32400*a^12*c^7*d^7*e^18 + 10848*a^13*c^6*d^5*e^20 +
1936*a^14*c^5*d^3*e^22)/(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^
14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12) - (x*(a*e^2 +
9*c*d^2)*(-d^3*e^7)^(1/2)*(65536*a^6*c^15*d^24*e^3 + 589824*a^7*c^14*d^22*e^5 + 2293760*a^8*c^13*d^20*e^7 + 4
915200*a^9*c^12*d^18*e^9 + 5898240*a^10*c^11*d^16*e^11 + 2752512*a^11*c^10*d^14*e^13 - 2752512*a^12*c^9*d^12*e
^15 - 5898240*a^13*c^8*d^10*e^17 - 4915200*a^14*c^7*d^8*e^19 - 2293760*a^15*c^6*d^6*e^21 - 589824*a^16*c^5*d^4
*e^23 - 65536*a^17*c^4*d^2*e^25))/(512*(c^3*d^9 + a^3*d^3*e^6 + 3*a*c^2*d^7*e^2 + 3*a^2*c*d^5*e^4)*(a^4*c^8*d^
18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a
^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12)))*(a*e^2 + 9*c*d^2)*(-d^3*e^7)^(1/2))/(4*(c^3*d^
9 + a^3*d^3*e^6 + 3*a*c^2*d^7*e^2 + 3*a^2*c*d^5*e^4))))/(4*(c^3*d^9 + a^3*d^3*e^6 + 3*a*c^2*d^7*e^2 + 3*a^2*c*
d^5*e^4)))*(a*e^2 + 9*c*d^2)*(-d^3*e^7)^(1/2))/(4*(c^3*d^9 + a^3*d^3*e^6 + 3*a*c^2*d^7*e^2 + 3*a^2*c*d^5*e^4))
)*(a*e^2 + 9*c*d^2)*(-d^3*e^7)^(1/2)*1i)/(4*(c^3*d^9 + a^3*d^3*e^6 + 3*a*c^2*d^7*e^2 + 3*a^2*c*d^5*e^4)) + (((
x*(81*c^13*d^14*e^5 - 392*a^7*c^6*e^19 + 1206*a*c^12*d^12*e^7 + 12247*a^2*c^11*d^10*e^9 + 58636*a^3*c^10*d^8*e
^11 + 114927*a^4*c^9*d^6*e^13 - 1306*a^5*c^8*d^4*e^15 - 3575*a^6*c^7*d^2*e^17))/(128*(a^4*c^8*d^18 + a^12*d^2*
e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^
8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12)) + (((7*a^10*c^5*e^21 + (9*a*c^14*d^18*e^3)/4 + (103*a^2*c^13*
d^16*e^5)/4 + (429*a^3*c^12*d^14*e^7)/4 + (2319*a^4*c^11*d^12*e^9)/4 + (10019*a^5*c^10*d^10*e^11)/4 + (6009*a^
6*c^9*d^8*e^13)/4 - (11105*a^7*c^8*d^6*e^15)/4 - (9803*a^8*c^7*d^4*e^17)/4 - 174*a^9*c^6*d^2*e^19)/(a^4*c^8*d^
18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a
^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12) - ((a*e^2 + 9*c*d^2)*(-d^3*e^7)^(1/2)*((x*(4096*
a^12*c^5*d*e^22 - 1152*a^2*c^15*d^21*e^2 - 15232*a^3*c^14*d^19*e^4 - 78336*a^4*c^13*d^17*e^6 - 140800*a^5*c^12
*d^15*e^8 + 489728*a^6*c^11*d^13*e^10 + 2219776*a^7*c^10*d^11*e^12 + 3155456*a^8*c^9*d^9*e^14 + 1901056*a^9*c^
8*d^7*e^16 + 362368*a^10*c^7*d^5*e^18 - 32640*a^11*c^6*d^3*e^20))/(128*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*
c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3
*d^8*e^10 + 28*a^10*c^2*d^6*e^12)) + (((128*a^15*c^4*d*e^24 - 48*a^4*c^15*d^23*e^2 - 416*a^5*c^14*d^21*e^4 - 6
88*a^6*c^13*d^19*e^6 + 3840*a^7*c^12*d^17*e^8 + 21408*a^8*c^11*d^15*e^10 + 49728*a^9*c^10*d^13*e^12 + 67872*a^
10*c^9*d^11*e^14 + 58752*a^11*c^8*d^9*e^16 + 32400*a^12*c^7*d^7*e^18 + 10848*a^13*c^6*d^5*e^20 + 1936*a^14*c^5
*d^3*e^22)/(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a
^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12) + (x*(a*e^2 + 9*c*d^2)*(-d
^3*e^7)^(1/2)*(65536*a^6*c^15*d^24*e^3 + 589824*a^7*c^14*d^22*e^5 + 2293760*a^8*c^13*d^20*e^7 + 4915200*a^9*c^
12*d^18*e^9 + 5898240*a^10*c^11*d^16*e^11 + 2752512*a^11*c^10*d^14*e^13 - 2752512*a^12*c^9*d^12*e^15 - 5898240
*a^13*c^8*d^10*e^17 - 4915200*a^14*c^7*d^8*e^19 - 2293760*a^15*c^6*d^6*e^21 - 589824*a^16*c^5*d^4*e^23 - 65536
*a^17*c^4*d^2*e^25))/(512*(c^3*d^9 + a^3*d^3*e^6 + 3*a*c^2*d^7*e^2 + 3*a^2*c*d^5*e^4)*(a^4*c^8*d^18 + a^12*d^2
*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e
^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12)))*(a*e^2 + 9*c*d^2)*(-d^3*e^7)^(1/2))/(4*(c^3*d^9 + a^3*d^3*e
^6 + 3*a*c^2*d^7*e^2 + 3*a^2*c*d^5*e^4))))/(4*(c^3*d^9 + a^3*d^3*e^6 + 3*a*c^2*d^7*e^2 + 3*a^2*c*d^5*e^4)))*(a
*e^2 + 9*c*d^2)*(-d^3*e^7)^(1/2))/(4*(c^3*d^9 + a^3*d^3*e^6 + 3*a*c^2*d^7*e^2 + 3*a^2*c*d^5*e^4)))*(a*e^2 + 9*
c*d^2)*(-d^3*e^7)^(1/2)*1i)/(4*(c^3*d^9 + a^3*d^3*e^6 + 3*a*c^2*d^7*e^2 + 3*a^2*c*d^5*e^4)))/(((729*c^11*d^9*e
^8)/256 + (729*a*c^10*d^7*e^10)/64 + (2009*a^4*c^7*d*e^16)/256 - (1269*a^2*c^9*d^5*e^12)/128 + (4441*a^3*c^8*d
^3*e^14)/64)/(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56
*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12) + (((x*(81*c^13*d^14*e^5
- 392*a^7*c^6*e^19 + 1206*a*c^12*d^12*e^7 + 12247*a^2*c^11*d^10*e^9 + 58636*a^3*c^10*d^8*e^11 + 114927*a^4*c^
9*d^6*e^13 - 1306*a^5*c^8*d^4*e^15 - 3575*a^6*c^7*d^2*e^17))/(128*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4
*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*
e^10 + 28*a^10*c^2*d^6*e^12)) - (((7*a^10*c^5*e^21 + (9*a*c^14*d^18*e^3)/4 + (103*a^2*c^13*d^16*e^5)/4 + (429*
a^3*c^12*d^14*e^7)/4 + (2319*a^4*c^11*d^12*e^9)/4 + (10019*a^5*c^10*d^10*e^11)/4 + (6009*a^6*c^9*d^8*e^13)/4 -
(11105*a^7*c^8*d^6*e^15)/4 - (9803*a^8*c^7*d^4*e^17)/4 - 174*a^9*c^6*d^2*e^19)/(a^4*c^8*d^18 + a^12*d^2*e^16
+ 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 5
6*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12) + ((a*e^2 + 9*c*d^2)*(-d^3*e^7)^(1/2)*((x*(4096*a^12*c^5*d*e^22 - 1
152*a^2*c^15*d^21*e^2 - 15232*a^3*c^14*d^19*e^4 - 78336*a^4*c^13*d^17*e^6 - 140800*a^5*c^12*d^15*e^8 + 489728*
a^6*c^11*d^13*e^10 + 2219776*a^7*c^10*d^11*e^12 + 3155456*a^8*c^9*d^9*e^14 + 1901056*a^9*c^8*d^7*e^16 + 362368
*a^10*c^7*d^5*e^18 - 32640*a^11*c^6*d^3*e^20))/(128*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*
c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10
*c^2*d^6*e^12)) - (((128*a^15*c^4*d*e^24 - 48*a^4*c^15*d^23*e^2 - 416*a^5*c^14*d^21*e^4 - 688*a^6*c^13*d^19*e^
6 + 3840*a^7*c^12*d^17*e^8 + 21408*a^8*c^11*d^15*e^10 + 49728*a^9*c^10*d^13*e^12 + 67872*a^10*c^9*d^11*e^14 +
58752*a^11*c^8*d^9*e^16 + 32400*a^12*c^7*d^7*e^18 + 10848*a^13*c^6*d^5*e^20 + 1936*a^14*c^5*d^3*e^22)/(a^4*c^8
*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 7
0*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12) - (x*(a*e^2 + 9*c*d^2)*(-d^3*e^7)^(1/2)*(6553
6*a^6*c^15*d^24*e^3 + 589824*a^7*c^14*d^22*e^5 + 2293760*a^8*c^13*d^20*e^7 + 4915200*a^9*c^12*d^18*e^9 + 58982
40*a^10*c^11*d^16*e^11 + 2752512*a^11*c^10*d^14*e^13 - 2752512*a^12*c^9*d^12*e^15 - 5898240*a^13*c^8*d^10*e^17
- 4915200*a^14*c^7*d^8*e^19 - 2293760*a^15*c^6*d^6*e^21 - 589824*a^16*c^5*d^4*e^23 - 65536*a^17*c^4*d^2*e^25)
)/(512*(c^3*d^9 + a^3*d^3*e^6 + 3*a*c^2*d^7*e^2 + 3*a^2*c*d^5*e^4)*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^
4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8
*e^10 + 28*a^10*c^2*d^6*e^12)))*(a*e^2 + 9*c*d^2)*(-d^3*e^7)^(1/2))/(4*(c^3*d^9 + a^3*d^3*e^6 + 3*a*c^2*d^7*e^
2 + 3*a^2*c*d^5*e^4))))/(4*(c^3*d^9 + a^3*d^3*e^6 + 3*a*c^2*d^7*e^2 + 3*a^2*c*d^5*e^4)))*(a*e^2 + 9*c*d^2)*(-d
^3*e^7)^(1/2))/(4*(c^3*d^9 + a^3*d^3*e^6 + 3*a*c^2*d^7*e^2 + 3*a^2*c*d^5*e^4)))*(a*e^2 + 9*c*d^2)*(-d^3*e^7)^(
1/2))/(4*(c^3*d^9 + a^3*d^3*e^6 + 3*a*c^2*d^7*e^2 + 3*a^2*c*d^5*e^4)) - (((x*(81*c^13*d^14*e^5 - 392*a^7*c^6*e
^19 + 1206*a*c^12*d^12*e^7 + 12247*a^2*c^11*d^10*e^9 + 58636*a^3*c^10*d^8*e^11 + 114927*a^4*c^9*d^6*e^13 - 130
6*a^5*c^8*d^4*e^15 - 3575*a^6*c^7*d^2*e^17))/(128*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^
7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c
^2*d^6*e^12)) + (((7*a^10*c^5*e^21 + (9*a*c^14*d^18*e^3)/4 + (103*a^2*c^13*d^16*e^5)/4 + (429*a^3*c^12*d^14*e^
7)/4 + (2319*a^4*c^11*d^12*e^9)/4 + (10019*a^5*c^10*d^10*e^11)/4 + (6009*a^6*c^9*d^8*e^13)/4 - (11105*a^7*c^8*
d^6*e^15)/4 - (9803*a^8*c^7*d^4*e^17)/4 - 174*a^9*c^6*d^2*e^19)/(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e
^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^
10 + 28*a^10*c^2*d^6*e^12) - ((a*e^2 + 9*c*d^2)*(-d^3*e^7)^(1/2)*((x*(4096*a^12*c^5*d*e^22 - 1152*a^2*c^15*d^2
1*e^2 - 15232*a^3*c^14*d^19*e^4 - 78336*a^4*c^13*d^17*e^6 - 140800*a^5*c^12*d^15*e^8 + 489728*a^6*c^11*d^13*e^
10 + 2219776*a^7*c^10*d^11*e^12 + 3155456*a^8*c^9*d^9*e^14 + 1901056*a^9*c^8*d^7*e^16 + 362368*a^10*c^7*d^5*e^
18 - 32640*a^11*c^6*d^3*e^20))/(128*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 2
8*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12))
+ (((128*a^15*c^4*d*e^24 - 48*a^4*c^15*d^23*e^2 - 416*a^5*c^14*d^21*e^4 - 688*a^6*c^13*d^19*e^6 + 3840*a^7*c^1
2*d^17*e^8 + 21408*a^8*c^11*d^15*e^10 + 49728*a^9*c^10*d^13*e^12 + 67872*a^10*c^9*d^11*e^14 + 58752*a^11*c^8*d
^9*e^16 + 32400*a^12*c^7*d^7*e^18 + 10848*a^13*c^6*d^5*e^20 + 1936*a^14*c^5*d^3*e^22)/(a^4*c^8*d^18 + a^12*d^2
*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e
^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12) + (x*(a*e^2 + 9*c*d^2)*(-d^3*e^7)^(1/2)*(65536*a^6*c^15*d^24*
e^3 + 589824*a^7*c^14*d^22*e^5 + 2293760*a^8*c^13*d^20*e^7 + 4915200*a^9*c^12*d^18*e^9 + 5898240*a^10*c^11*d^1
6*e^11 + 2752512*a^11*c^10*d^14*e^13 - 2752512*a^12*c^9*d^12*e^15 - 5898240*a^13*c^8*d^10*e^17 - 4915200*a^14*
c^7*d^8*e^19 - 2293760*a^15*c^6*d^6*e^21 - 589824*a^16*c^5*d^4*e^23 - 65536*a^17*c^4*d^2*e^25))/(512*(c^3*d^9
+ a^3*d^3*e^6 + 3*a*c^2*d^7*e^2 + 3*a^2*c*d^5*e^4)*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c
^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*
c^2*d^6*e^12)))*(a*e^2 + 9*c*d^2)*(-d^3*e^7)^(1/2))/(4*(c^3*d^9 + a^3*d^3*e^6 + 3*a*c^2*d^7*e^2 + 3*a^2*c*d^5*
e^4))))/(4*(c^3*d^9 + a^3*d^3*e^6 + 3*a*c^2*d^7*e^2 + 3*a^2*c*d^5*e^4)))*(a*e^2 + 9*c*d^2)*(-d^3*e^7)^(1/2))/(
4*(c^3*d^9 + a^3*d^3*e^6 + 3*a*c^2*d^7*e^2 + 3*a^2*c*d^5*e^4)))*(a*e^2 + 9*c*d^2)*(-d^3*e^7)^(1/2))/(4*(c^3*d^
9 + a^3*d^3*e^6 + 3*a*c^2*d^7*e^2 + 3*a^2*c*d^5*e^4))))*(a*e^2 + 9*c*d^2)*(-d^3*e^7)^(1/2)*1i)/(2*(c^3*d^9 + a
^3*d^3*e^6 + 3*a*c^2*d^7*e^2 + 3*a^2*c*d^5*e^4))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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