∫ x13∕2 ____________________

3.9.38 $\int \frac {x^{13/2}}{(a+b x^2+c x^4)^2} \, dx$

Optimal. Leaf size=544 \[ -\frac {b x^{3/2}}{2 c \left (b^2-4 a c\right )}+\frac {x^{7/2} \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\left (\left (3 b^2-14 a c\right ) \sqrt {b^2-4 a c}-20 a b c+3 b^3\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{4\ 2^{3/4} c^{7/4} \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}-\frac {\left (-\left (3 b^2-14 a c\right ) \sqrt {b^2-4 a c}-20 a b c+3 b^3\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{4\ 2^{3/4} c^{7/4} \left (b^2-4 a c\right )^{3/2} \sqrt [4]{\sqrt {b^2-4 a c}-b}}-\frac {\left (\left (3 b^2-14 a c\right ) \sqrt {b^2-4 a c}-20 a b c+3 b^3\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{4\ 2^{3/4} c^{7/4} \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}+\frac {\left (-\left (3 b^2-14 a c\right ) \sqrt {b^2-4 a c}-20 a b c+3 b^3\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{4\ 2^{3/4} c^{7/4} \left (b^2-4 a c\right )^{3/2} \sqrt [4]{\sqrt {b^2-4 a c}-b}} \]

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Rubi [A] time = 2.58, antiderivative size = 544, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 20, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.350, Rules used = {1115, 1365, 1502, 1510, 298, 205, 208} \begin {gather*} \frac {\left (\left (3 b^2-14 a c\right ) \sqrt {b^2-4 a c}-20 a b c+3 b^3\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{4\ 2^{3/4} c^{7/4} \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}-\frac {\left (-\left (3 b^2-14 a c\right ) \sqrt {b^2-4 a c}-20 a b c+3 b^3\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{4\ 2^{3/4} c^{7/4} \left (b^2-4 a c\right )^{3/2} \sqrt [4]{\sqrt {b^2-4 a c}-b}}-\frac {\left (\left (3 b^2-14 a c\right ) \sqrt {b^2-4 a c}-20 a b c+3 b^3\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{4\ 2^{3/4} c^{7/4} \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}+\frac {\left (-\left (3 b^2-14 a c\right ) \sqrt {b^2-4 a c}-20 a b c+3 b^3\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{4\ 2^{3/4} c^{7/4} \left (b^2-4 a c\right )^{3/2} \sqrt [4]{\sqrt {b^2-4 a c}-b}}+\frac {x^{7/2} \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {b x^{3/2}}{2 c \left (b^2-4 a c\right )} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[x^(13/2)/(a + b*x^2 + c*x^4)^2,x]

[Out]

-(b*x^(3/2))/(2*c*(b^2 - 4*a*c)) + (x^(7/2)*(2*a + b*x^2))/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + ((3*b^3 - 2

0*a*b*c + (3*b^2 - 14*a*c)*Sqrt[b^2 - 4*a*c])*ArcTan[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b - Sqrt[b^2 - 4*a*c])^(1/4)]

)/(4*2^(3/4)*c^(7/4)*(b^2 - 4*a*c)^(3/2)*(-b - Sqrt[b^2 - 4*a*c])^(1/4)) - ((3*b^3 - 20*a*b*c - (3*b^2 - 14*a*

c)*Sqrt[b^2 - 4*a*c])*ArcTan[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b + Sqrt[b^2 - 4*a*c])^(1/4)])/(4*2^(3/4)*c^(7/4)*(b^

2 - 4*a*c)^(3/2)*(-b + Sqrt[b^2 - 4*a*c])^(1/4)) - ((3*b^3 - 20*a*b*c + (3*b^2 - 14*a*c)*Sqrt[b^2 - 4*a*c])*Ar

cTanh[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(4*2^(3/4)*c^(7/4)*(b^2 - 4*a*c)^(3/2)*(-b -

Sqrt[b^2 - 4*a*c])^(1/4)) + ((3*b^3 - 20*a*b*c - (3*b^2 - 14*a*c)*Sqrt[b^2 - 4*a*c])*ArcTanh[(2^(1/4)*c^(1/4)*

Sqrt[x])/(-b + Sqrt[b^2 - 4*a*c])^(1/4)])/(4*2^(3/4)*c^(7/4)*(b^2 - 4*a*c)^(3/2)*(-b + Sqrt[b^2 - 4*a*c])^(1/4

))

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 208

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

b}, x] && NegQ[a/b]

Rule 298

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),

2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &

& !GtQ[a/b, 0]

Rule 1115

Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[

k/d, Subst[Int[x^(k*(m + 1) - 1)*(a + (b*x^(2*k))/d^2 + (c*x^(4*k))/d^4)^p, x], x, (d*x)^(1/k)], x]] /; FreeQ[

{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && FractionQ[m] && IntegerQ[p]

Rule 1365

Int[((d_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(d^(2*n - 1)*(d*x

)^(m - 2*n + 1)*(2*a + b*x^n)*(a + b*x^n + c*x^(2*n))^(p + 1))/(n*(p + 1)*(b^2 - 4*a*c)), x] + Dist[d^(2*n)/(n

*(p + 1)*(b^2 - 4*a*c)), Int[(d*x)^(m - 2*n)*(2*a*(m - 2*n + 1) + b*(m + n*(2*p + 1) + 1)*x^n)*(a + b*x^n + c*

x^(2*n))^(p + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && ILt

Q[p, -1] && GtQ[m, 2*n - 1]

Rule 1502

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :>

Simp[(e*f^(n - 1)*(f*x)^(m - n + 1)*(a + b*x^n + c*x^(2*n))^(p + 1))/(c*(m + n*(2*p + 1) + 1)), x] - Dist[f^n

/(c*(m + n*(2*p + 1) + 1)), Int[(f*x)^(m - n)*(a + b*x^n + c*x^(2*n))^p*Simp[a*e*(m - n + 1) + (b*e*(m + n*p +

1) - c*d*(m + n*(2*p + 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[n2, 2*n] && NeQ[b^2

- 4*a*c, 0] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*(2*p + 1) + 1, 0] && IntegerQ[p]

Rule 1510

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_)))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> Wi

th[{q = Rt[b^2 - 4*a*c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[(f*x)^m/(b/2 - q/2 + c*x^n), x], x] + Dist[e/

2 - (2*c*d - b*e)/(2*q), Int[(f*x)^m/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[n2

, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {x^{13/2}}{\left (a+b x^2+c x^4\right )^2} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^{14}}{\left (a+b x^4+c x^8\right )^2} \, dx,x,\sqrt {x}\right )\\ &=\frac {x^{7/2} \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\operatorname {Subst}\left (\int \frac {x^6 \left (14 a+3 b x^4\right )}{a+b x^4+c x^8} \, dx,x,\sqrt {x}\right )}{2 \left (b^2-4 a c\right )}\\ &=-\frac {b x^{3/2}}{2 c \left (b^2-4 a c\right )}+\frac {x^{7/2} \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (9 a b+3 \left (3 b^2-14 a c\right ) x^4\right )}{a+b x^4+c x^8} \, dx,x,\sqrt {x}\right )}{6 c \left (b^2-4 a c\right )}\\ &=-\frac {b x^{3/2}}{2 c \left (b^2-4 a c\right )}+\frac {x^{7/2} \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\left (3 b^2-14 a c+\frac {3 b^3}{\sqrt {b^2-4 a c}}-\frac {20 a b c}{\sqrt {b^2-4 a c}}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^4} \, dx,x,\sqrt {x}\right )}{4 c \left (b^2-4 a c\right )}-\frac {\left (3 b^3-20 a b c-\left (3 b^2-14 a c\right ) \sqrt {b^2-4 a c}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^4} \, dx,x,\sqrt {x}\right )}{4 c \left (b^2-4 a c\right )^{3/2}}\\ &=-\frac {b x^{3/2}}{2 c \left (b^2-4 a c\right )}+\frac {x^{7/2} \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\left (3 b^2-14 a c+\frac {3 b^3}{\sqrt {b^2-4 a c}}-\frac {20 a b c}{\sqrt {b^2-4 a c}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x^2} \, dx,x,\sqrt {x}\right )}{4 \sqrt {2} c^{3/2} \left (b^2-4 a c\right )}+\frac {\left (3 b^2-14 a c+\frac {3 b^3}{\sqrt {b^2-4 a c}}-\frac {20 a b c}{\sqrt {b^2-4 a c}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}+\sqrt {2} \sqrt {c} x^2} \, dx,x,\sqrt {x}\right )}{4 \sqrt {2} c^{3/2} \left (b^2-4 a c\right )}+\frac {\left (3 b^3-20 a b c-\left (3 b^2-14 a c\right ) \sqrt {b^2-4 a c}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b+\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x^2} \, dx,x,\sqrt {x}\right )}{4 \sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{3/2}}-\frac {\left (3 b^3-20 a b c-\left (3 b^2-14 a c\right ) \sqrt {b^2-4 a c}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b+\sqrt {b^2-4 a c}}+\sqrt {2} \sqrt {c} x^2} \, dx,x,\sqrt {x}\right )}{4 \sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{3/2}}\\ &=-\frac {b x^{3/2}}{2 c \left (b^2-4 a c\right )}+\frac {x^{7/2} \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\left (3 b^3-20 a b c+\left (3 b^2-14 a c\right ) \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{4\ 2^{3/4} c^{7/4} \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-b-\sqrt {b^2-4 a c}}}-\frac {\left (3 b^3-20 a b c-\left (3 b^2-14 a c\right ) \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{4\ 2^{3/4} c^{7/4} \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-b+\sqrt {b^2-4 a c}}}-\frac {\left (3 b^3-20 a b c+\left (3 b^2-14 a c\right ) \sqrt {b^2-4 a c}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{4\ 2^{3/4} c^{7/4} \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-b-\sqrt {b^2-4 a c}}}+\frac {\left (3 b^3-20 a b c-\left (3 b^2-14 a c\right ) \sqrt {b^2-4 a c}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{4\ 2^{3/4} c^{7/4} \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-b+\sqrt {b^2-4 a c}}}\\ \end {align*}

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Mathematica [C] time = 0.28, size = 144, normalized size = 0.26 \begin {gather*} \frac {\text {RootSum}\left [\text {$\#$1}^8 c+\text {$\#$1}^4 b+a\&,\frac {-14 \text {$\#$1}^4 a c \log \left (\sqrt {x}-\text {$\#$1}\right )+3 \text {$\#$1}^4 b^2 \log \left (\sqrt {x}-\text {$\#$1}\right )+3 a b \log \left (\sqrt {x}-\text {$\#$1}\right )}{2 \text {$\#$1}^5 c+\text {$\#$1} b}\&\right ]-\frac {4 x^{3/2} \left (a \left (b-2 c x^2\right )+b^2 x^2\right )}{a+b x^2+c x^4}}{8 c \left (b^2-4 a c\right )} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^(13/2)/(a + b*x^2 + c*x^4)^2,x]

[Out]

((-4*x^(3/2)*(b^2*x^2 + a*(b - 2*c*x^2)))/(a + b*x^2 + c*x^4) + RootSum[a + b*#1^4 + c*#1^8 & , (3*a*b*Log[Sqr

t[x] - #1] + 3*b^2*Log[Sqrt[x] - #1]*#1^4 - 14*a*c*Log[Sqrt[x] - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ])/(8*c*(b^2 -

4*a*c))

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IntegrateAlgebraic [C] time = 0.65, size = 257, normalized size = 0.47 \begin {gather*} \frac {\text {RootSum}\left [\text {$\#$1}^8 c+\text {$\#$1}^4 b+a\&,\frac {-2 \text {$\#$1}^4 a c^2 \log \left (\sqrt {x}-\text {$\#$1}\right )+\text {$\#$1}^4 b^2 c \log \left (\sqrt {x}-\text {$\#$1}\right )+13 a b c \log \left (\sqrt {x}-\text {$\#$1}\right )-4 b^3 \log \left (\sqrt {x}-\text {$\#$1}\right )}{2 \text {$\#$1}^5 c+\text {$\#$1} b}\&\right ]}{8 c^2 \left (4 a c-b^2\right )}-\frac {\text {RootSum}\left [\text {$\#$1}^8 c+\text {$\#$1}^4 b+a\&,\frac {b \log \left (\sqrt {x}-\text {$\#$1}\right )-\text {$\#$1}^4 c \log \left (\sqrt {x}-\text {$\#$1}\right )}{2 \text {$\#$1}^5 c+\text {$\#$1} b}\&\right ]}{2 c^2}+\frac {a b x^{3/2}-2 a c x^{7/2}+b^2 x^{7/2}}{2 c \left (4 a c-b^2\right ) \left (a+b x^2+c x^4\right )} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

IntegrateAlgebraic[x^(13/2)/(a + b*x^2 + c*x^4)^2,x]

[Out]

(a*b*x^(3/2) + b^2*x^(7/2) - 2*a*c*x^(7/2))/(2*c*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)) - RootSum[a + b*#1^4 + c*

#1^8 & , (b*Log[Sqrt[x] - #1] - c*Log[Sqrt[x] - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ]/(2*c^2) + RootSum[a + b*#1^4 +

c*#1^8 & , (-4*b^3*Log[Sqrt[x] - #1] + 13*a*b*c*Log[Sqrt[x] - #1] + b^2*c*Log[Sqrt[x] - #1]*#1^4 - 2*a*c^2*Lo

g[Sqrt[x] - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ]/(8*c^2*(-b^2 + 4*a*c))

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(13/2)/(c*x^4+b*x^2+a)^2,x, algorithm="fricas")

[Out]

Timed out

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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(13/2)/(c*x^4+b*x^2+a)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval

uation time: 46.99Unable to convert to real 1/4 Error: Bad Argument Value

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maple [C] time = 0.02, size = 149, normalized size = 0.27 \begin {gather*} \frac {\left (\left (14 a c -3 b^{2}\right ) \RootOf \left (c \,\textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+a \right )^{6}-3 \RootOf \left (c \,\textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+a \right )^{2} a b \right ) \ln \left (-\RootOf \left (c \,\textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+a \right )+\sqrt {x}\right )}{8 \left (4 a c -b^{2}\right ) c \left (2 \RootOf \left (c \,\textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+a \right )^{7} c +\RootOf \left (c \,\textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+a \right )^{3} b \right )}+\frac {\frac {a b \,x^{\frac {3}{2}}}{2 \left (4 a c -b^{2}\right ) c}-\frac {\left (2 a c -b^{2}\right ) x^{\frac {7}{2}}}{2 \left (4 a c -b^{2}\right ) c}}{c \,x^{4}+b \,x^{2}+a} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^(13/2)/(c*x^4+b*x^2+a)^2,x)

[Out]

2*(-1/4*(2*a*c-b^2)/(4*a*c-b^2)/c*x^(7/2)+1/4/(4*a*c-b^2)*a*b/c*x^(3/2))/(c*x^4+b*x^2+a)+1/8/c/(4*a*c-b^2)*sum

(((14*a*c-3*b^2)*_R^6-3*_R^2*a*b)/(2*_R^7*c+_R^3*b)*ln(-_R+x^(1/2)),_R=RootOf(_Z^8*c+_Z^4*b+a))

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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {{\left (b^{2} - 2 \, a c\right )} x^{\frac {7}{2}} + a b x^{\frac {3}{2}}}{2 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + a b^{2} c - 4 \, a^{2} c^{2} + {\left (b^{3} c - 4 \, a b c^{2}\right )} x^{2}\right )}} + \int \frac {{\left (3 \, b^{2} - 14 \, a c\right )} x^{\frac {5}{2}} + 3 \, a b \sqrt {x}}{4 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + a b^{2} c - 4 \, a^{2} c^{2} + {\left (b^{3} c - 4 \, a b c^{2}\right )} x^{2}\right )}}\,{d x} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(13/2)/(c*x^4+b*x^2+a)^2,x, algorithm="maxima")

[Out]

-1/2*((b^2 - 2*a*c)*x^(7/2) + a*b*x^(3/2))/((b^2*c^2 - 4*a*c^3)*x^4 + a*b^2*c - 4*a^2*c^2 + (b^3*c - 4*a*b*c^2

)*x^2) + integrate(1/4*((3*b^2 - 14*a*c)*x^(5/2) + 3*a*b*sqrt(x))/((b^2*c^2 - 4*a*c^3)*x^4 + a*b^2*c - 4*a^2*c

^2 + (b^3*c - 4*a*b*c^2)*x^2), x)

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mupad [B] time = 7.01, size = 28774, normalized size = 52.89

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^(13/2)/(a + b*x^2 + c*x^4)^2,x)

[Out]

- ((x^(7/2)*(2*a*c - b^2))/(2*c*(4*a*c - b^2)) - (a*b*x^(3/2))/(2*c*(4*a*c - b^2)))/(a + b*x^2 + c*x^4) - atan

(((((46036680704*a^12*c^12 - 110592*a^3*b^18*c^3 + 4423680*a^4*b^16*c^4 - 77783040*a^5*b^14*c^5 + 788037632*a^

6*b^12*c^6 - 5065015296*a^7*b^10*c^7 + 21401960448*a^8*b^8*c^8 - 59401830400*a^9*b^6*c^9 + 104312340480*a^10*b

^4*c^10 - 104991817728*a^11*b^2*c^11)/(128*(16384*a^7*c^10 - b^14*c^3 + 28*a*b^12*c^4 - 336*a^2*b^10*c^5 + 224

0*a^3*b^8*c^6 - 8960*a^4*b^6*c^7 + 21504*a^5*b^4*c^8 - 28672*a^6*b^2*c^9)) - (x^(1/2)*((81*b^8*(-(4*a*c - b^2)

^15)^(1/2) - 81*b^23 + 741801984*a^11*b*c^11 - 90126*a^2*b^19*c^2 + 1201623*a^3*b^17*c^3 - 10588384*a^4*b^15*c

^4 + 64704576*a^5*b^13*c^5 - 279571968*a^6*b^11*c^6 + 853174784*a^7*b^9*c^7 - 1799626752*a^8*b^7*c^8 + 2494119

936*a^9*b^5*c^9 - 2038693888*a^10*b^3*c^10 + 9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) + 4023*a*b^21*c + 10746*a^

2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 26313*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593*a*b^6*c*(-(4*a*c - b^

2)^15)^(1/2))/(8192*(16777216*a^12*c^19 + b^24*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 - 14080*a^3*b^18*c^10 +

126720*a^4*b^16*c^11 - 811008*a^5*b^14*c^12 + 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10*c^14 + 32440320*a^8*b

^8*c^15 - 57671680*a^9*b^6*c^16 + 69206016*a^10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(1/4)*(6576668672*a^11*c^

13 + 36864*a^3*b^16*c^5 - 1302528*a^4*b^14*c^6 + 20480000*a^5*b^12*c^7 - 185991168*a^6*b^10*c^8 + 1061683200*a

^7*b^8*c^9 - 3886022656*a^8*b^6*c^10 + 8883535872*a^9*b^4*c^11 - 11576279040*a^10*b^2*c^12))/(16*(4096*a^6*c^9

+ b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))*((81

*b^8*(-(4*a*c - b^2)^15)^(1/2) - 81*b^23 + 741801984*a^11*b*c^11 - 90126*a^2*b^19*c^2 + 1201623*a^3*b^17*c^3 -

10588384*a^4*b^15*c^4 + 64704576*a^5*b^13*c^5 - 279571968*a^6*b^11*c^6 + 853174784*a^7*b^9*c^7 - 1799626752*a

^8*b^7*c^8 + 2494119936*a^9*b^5*c^9 - 2038693888*a^10*b^3*c^10 + 9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) + 4023

*a*b^21*c + 10746*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 26313*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593*a

*b^6*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^19 + b^24*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 - 1

4080*a^3*b^18*c^10 + 126720*a^4*b^16*c^11 - 811008*a^5*b^14*c^12 + 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10*c

^14 + 32440320*a^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 69206016*a^10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(3/4)

- (x^(1/2)*(9801*a^5*b^11 - 256905*a^6*b^9*c - 29042496*a^10*b*c^5 + 2642841*a^7*b^7*c^2 - 13243020*a^8*b^5*c

^3 + 31945648*a^9*b^3*c^4))/(16*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6

+ 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))*((81*b^8*(-(4*a*c - b^2)^15)^(1/2) - 81*b^23 + 741801984*a^11*b*c^11

- 90126*a^2*b^19*c^2 + 1201623*a^3*b^17*c^3 - 10588384*a^4*b^15*c^4 + 64704576*a^5*b^13*c^5 - 279571968*a^6*b^

11*c^6 + 853174784*a^7*b^9*c^7 - 1799626752*a^8*b^7*c^8 + 2494119936*a^9*b^5*c^9 - 2038693888*a^10*b^3*c^10 +

9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) + 4023*a*b^21*c + 10746*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 26313*a

^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593*a*b^6*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^19 + b^2

4*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 - 14080*a^3*b^18*c^10 + 126720*a^4*b^16*c^11 - 811008*a^5*b^14*c^12

+ 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10*c^14 + 32440320*a^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 69206016*a^

10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(1/4)*1i - (((46036680704*a^12*c^12 - 110592*a^3*b^18*c^3 + 4423680*a^

4*b^16*c^4 - 77783040*a^5*b^14*c^5 + 788037632*a^6*b^12*c^6 - 5065015296*a^7*b^10*c^7 + 21401960448*a^8*b^8*c^

8 - 59401830400*a^9*b^6*c^9 + 104312340480*a^10*b^4*c^10 - 104991817728*a^11*b^2*c^11)/(128*(16384*a^7*c^10 -

b^14*c^3 + 28*a*b^12*c^4 - 336*a^2*b^10*c^5 + 2240*a^3*b^8*c^6 - 8960*a^4*b^6*c^7 + 21504*a^5*b^4*c^8 - 28672*

a^6*b^2*c^9)) + (x^(1/2)*((81*b^8*(-(4*a*c - b^2)^15)^(1/2) - 81*b^23 + 741801984*a^11*b*c^11 - 90126*a^2*b^19

*c^2 + 1201623*a^3*b^17*c^3 - 10588384*a^4*b^15*c^4 + 64704576*a^5*b^13*c^5 - 279571968*a^6*b^11*c^6 + 8531747

84*a^7*b^9*c^7 - 1799626752*a^8*b^7*c^8 + 2494119936*a^9*b^5*c^9 - 2038693888*a^10*b^3*c^10 + 9604*a^4*c^4*(-(

4*a*c - b^2)^15)^(1/2) + 4023*a*b^21*c + 10746*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 26313*a^3*b^2*c^3*(-(4*

a*c - b^2)^15)^(1/2) - 1593*a*b^6*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^19 + b^24*c^7 - 48*a*b^2

2*c^8 + 1056*a^2*b^20*c^9 - 14080*a^3*b^18*c^10 + 126720*a^4*b^16*c^11 - 811008*a^5*b^14*c^12 + 3784704*a^6*b^

12*c^13 - 12976128*a^7*b^10*c^14 + 32440320*a^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 69206016*a^10*b^4*c^17 - 50

331648*a^11*b^2*c^18)))^(1/4)*(6576668672*a^11*c^13 + 36864*a^3*b^16*c^5 - 1302528*a^4*b^14*c^6 + 20480000*a^5

*b^12*c^7 - 185991168*a^6*b^10*c^8 + 1061683200*a^7*b^8*c^9 - 3886022656*a^8*b^6*c^10 + 8883535872*a^9*b^4*c^1

1 - 11576279040*a^10*b^2*c^12))/(16*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*

c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))*((81*b^8*(-(4*a*c - b^2)^15)^(1/2) - 81*b^23 + 741801984*a^11*b*c

^11 - 90126*a^2*b^19*c^2 + 1201623*a^3*b^17*c^3 - 10588384*a^4*b^15*c^4 + 64704576*a^5*b^13*c^5 - 279571968*a^

6*b^11*c^6 + 853174784*a^7*b^9*c^7 - 1799626752*a^8*b^7*c^8 + 2494119936*a^9*b^5*c^9 - 2038693888*a^10*b^3*c^1

0 + 9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) + 4023*a*b^21*c + 10746*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 263

13*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593*a*b^6*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^19 +

b^24*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 - 14080*a^3*b^18*c^10 + 126720*a^4*b^16*c^11 - 811008*a^5*b^14*c

^12 + 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10*c^14 + 32440320*a^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 6920601

6*a^10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(3/4) + (x^(1/2)*(9801*a^5*b^11 - 256905*a^6*b^9*c - 29042496*a^10

*b*c^5 + 2642841*a^7*b^7*c^2 - 13243020*a^8*b^5*c^3 + 31945648*a^9*b^3*c^4))/(16*(4096*a^6*c^9 + b^12*c^3 - 24

*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))*((81*b^8*(-(4*a*c -

b^2)^15)^(1/2) - 81*b^23 + 741801984*a^11*b*c^11 - 90126*a^2*b^19*c^2 + 1201623*a^3*b^17*c^3 - 10588384*a^4*b^

15*c^4 + 64704576*a^5*b^13*c^5 - 279571968*a^6*b^11*c^6 + 853174784*a^7*b^9*c^7 - 1799626752*a^8*b^7*c^8 + 249

4119936*a^9*b^5*c^9 - 2038693888*a^10*b^3*c^10 + 9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) + 4023*a*b^21*c + 1074

6*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 26313*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593*a*b^6*c*(-(4*a*c

- b^2)^15)^(1/2))/(8192*(16777216*a^12*c^19 + b^24*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 - 14080*a^3*b^18*c^

10 + 126720*a^4*b^16*c^11 - 811008*a^5*b^14*c^12 + 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10*c^14 + 32440320*a

^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 69206016*a^10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(1/4)*1i)/((((4603668

0704*a^12*c^12 - 110592*a^3*b^18*c^3 + 4423680*a^4*b^16*c^4 - 77783040*a^5*b^14*c^5 + 788037632*a^6*b^12*c^6 -

5065015296*a^7*b^10*c^7 + 21401960448*a^8*b^8*c^8 - 59401830400*a^9*b^6*c^9 + 104312340480*a^10*b^4*c^10 - 10

4991817728*a^11*b^2*c^11)/(128*(16384*a^7*c^10 - b^14*c^3 + 28*a*b^12*c^4 - 336*a^2*b^10*c^5 + 2240*a^3*b^8*c^

6 - 8960*a^4*b^6*c^7 + 21504*a^5*b^4*c^8 - 28672*a^6*b^2*c^9)) - (x^(1/2)*((81*b^8*(-(4*a*c - b^2)^15)^(1/2) -

81*b^23 + 741801984*a^11*b*c^11 - 90126*a^2*b^19*c^2 + 1201623*a^3*b^17*c^3 - 10588384*a^4*b^15*c^4 + 6470457

6*a^5*b^13*c^5 - 279571968*a^6*b^11*c^6 + 853174784*a^7*b^9*c^7 - 1799626752*a^8*b^7*c^8 + 2494119936*a^9*b^5*

c^9 - 2038693888*a^10*b^3*c^10 + 9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) + 4023*a*b^21*c + 10746*a^2*b^4*c^2*(-

(4*a*c - b^2)^15)^(1/2) - 26313*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593*a*b^6*c*(-(4*a*c - b^2)^15)^(1/2)

)/(8192*(16777216*a^12*c^19 + b^24*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 - 14080*a^3*b^18*c^10 + 126720*a^4*

b^16*c^11 - 811008*a^5*b^14*c^12 + 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10*c^14 + 32440320*a^8*b^8*c^15 - 57

671680*a^9*b^6*c^16 + 69206016*a^10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(1/4)*(6576668672*a^11*c^13 + 36864*a

^3*b^16*c^5 - 1302528*a^4*b^14*c^6 + 20480000*a^5*b^12*c^7 - 185991168*a^6*b^10*c^8 + 1061683200*a^7*b^8*c^9 -

3886022656*a^8*b^6*c^10 + 8883535872*a^9*b^4*c^11 - 11576279040*a^10*b^2*c^12))/(16*(4096*a^6*c^9 + b^12*c^3

- 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))*((81*b^8*(-(4*a*

c - b^2)^15)^(1/2) - 81*b^23 + 741801984*a^11*b*c^11 - 90126*a^2*b^19*c^2 + 1201623*a^3*b^17*c^3 - 10588384*a^

4*b^15*c^4 + 64704576*a^5*b^13*c^5 - 279571968*a^6*b^11*c^6 + 853174784*a^7*b^9*c^7 - 1799626752*a^8*b^7*c^8 +

2494119936*a^9*b^5*c^9 - 2038693888*a^10*b^3*c^10 + 9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) + 4023*a*b^21*c +

10746*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 26313*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593*a*b^6*c*(-(4*

a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^19 + b^24*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 - 14080*a^3*b^1

8*c^10 + 126720*a^4*b^16*c^11 - 811008*a^5*b^14*c^12 + 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10*c^14 + 324403

20*a^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 69206016*a^10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(3/4) - (x^(1/2)*

(9801*a^5*b^11 - 256905*a^6*b^9*c - 29042496*a^10*b*c^5 + 2642841*a^7*b^7*c^2 - 13243020*a^8*b^5*c^3 + 3194564

8*a^9*b^3*c^4))/(16*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b

^4*c^7 - 6144*a^5*b^2*c^8)))*((81*b^8*(-(4*a*c - b^2)^15)^(1/2) - 81*b^23 + 741801984*a^11*b*c^11 - 90126*a^2*

b^19*c^2 + 1201623*a^3*b^17*c^3 - 10588384*a^4*b^15*c^4 + 64704576*a^5*b^13*c^5 - 279571968*a^6*b^11*c^6 + 853

174784*a^7*b^9*c^7 - 1799626752*a^8*b^7*c^8 + 2494119936*a^9*b^5*c^9 - 2038693888*a^10*b^3*c^10 + 9604*a^4*c^4

*(-(4*a*c - b^2)^15)^(1/2) + 4023*a*b^21*c + 10746*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 26313*a^3*b^2*c^3*(

-(4*a*c - b^2)^15)^(1/2) - 1593*a*b^6*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^19 + b^24*c^7 - 48*a

*b^22*c^8 + 1056*a^2*b^20*c^9 - 14080*a^3*b^18*c^10 + 126720*a^4*b^16*c^11 - 811008*a^5*b^14*c^12 + 3784704*a^

6*b^12*c^13 - 12976128*a^7*b^10*c^14 + 32440320*a^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 69206016*a^10*b^4*c^17

- 50331648*a^11*b^2*c^18)))^(1/4) + (((46036680704*a^12*c^12 - 110592*a^3*b^18*c^3 + 4423680*a^4*b^16*c^4 - 77

783040*a^5*b^14*c^5 + 788037632*a^6*b^12*c^6 - 5065015296*a^7*b^10*c^7 + 21401960448*a^8*b^8*c^8 - 59401830400

*a^9*b^6*c^9 + 104312340480*a^10*b^4*c^10 - 104991817728*a^11*b^2*c^11)/(128*(16384*a^7*c^10 - b^14*c^3 + 28*a

*b^12*c^4 - 336*a^2*b^10*c^5 + 2240*a^3*b^8*c^6 - 8960*a^4*b^6*c^7 + 21504*a^5*b^4*c^8 - 28672*a^6*b^2*c^9)) +

(x^(1/2)*((81*b^8*(-(4*a*c - b^2)^15)^(1/2) - 81*b^23 + 741801984*a^11*b*c^11 - 90126*a^2*b^19*c^2 + 1201623*

a^3*b^17*c^3 - 10588384*a^4*b^15*c^4 + 64704576*a^5*b^13*c^5 - 279571968*a^6*b^11*c^6 + 853174784*a^7*b^9*c^7

- 1799626752*a^8*b^7*c^8 + 2494119936*a^9*b^5*c^9 - 2038693888*a^10*b^3*c^10 + 9604*a^4*c^4*(-(4*a*c - b^2)^15

)^(1/2) + 4023*a*b^21*c + 10746*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 26313*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^

(1/2) - 1593*a*b^6*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^19 + b^24*c^7 - 48*a*b^22*c^8 + 1056*a^

2*b^20*c^9 - 14080*a^3*b^18*c^10 + 126720*a^4*b^16*c^11 - 811008*a^5*b^14*c^12 + 3784704*a^6*b^12*c^13 - 12976

128*a^7*b^10*c^14 + 32440320*a^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 69206016*a^10*b^4*c^17 - 50331648*a^11*b^2

*c^18)))^(1/4)*(6576668672*a^11*c^13 + 36864*a^3*b^16*c^5 - 1302528*a^4*b^14*c^6 + 20480000*a^5*b^12*c^7 - 185

991168*a^6*b^10*c^8 + 1061683200*a^7*b^8*c^9 - 3886022656*a^8*b^6*c^10 + 8883535872*a^9*b^4*c^11 - 11576279040

*a^10*b^2*c^12))/(16*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*

b^4*c^7 - 6144*a^5*b^2*c^8)))*((81*b^8*(-(4*a*c - b^2)^15)^(1/2) - 81*b^23 + 741801984*a^11*b*c^11 - 90126*a^2

*b^19*c^2 + 1201623*a^3*b^17*c^3 - 10588384*a^4*b^15*c^4 + 64704576*a^5*b^13*c^5 - 279571968*a^6*b^11*c^6 + 85

3174784*a^7*b^9*c^7 - 1799626752*a^8*b^7*c^8 + 2494119936*a^9*b^5*c^9 - 2038693888*a^10*b^3*c^10 + 9604*a^4*c^

4*(-(4*a*c - b^2)^15)^(1/2) + 4023*a*b^21*c + 10746*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 26313*a^3*b^2*c^3*

(-(4*a*c - b^2)^15)^(1/2) - 1593*a*b^6*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^19 + b^24*c^7 - 48*

a*b^22*c^8 + 1056*a^2*b^20*c^9 - 14080*a^3*b^18*c^10 + 126720*a^4*b^16*c^11 - 811008*a^5*b^14*c^12 + 3784704*a

^6*b^12*c^13 - 12976128*a^7*b^10*c^14 + 32440320*a^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 69206016*a^10*b^4*c^17

- 50331648*a^11*b^2*c^18)))^(3/4) + (x^(1/2)*(9801*a^5*b^11 - 256905*a^6*b^9*c - 29042496*a^10*b*c^5 + 264284

1*a^7*b^7*c^2 - 13243020*a^8*b^5*c^3 + 31945648*a^9*b^3*c^4))/(16*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 2

40*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))*((81*b^8*(-(4*a*c - b^2)^15)^(1/2)

- 81*b^23 + 741801984*a^11*b*c^11 - 90126*a^2*b^19*c^2 + 1201623*a^3*b^17*c^3 - 10588384*a^4*b^15*c^4 + 647045

76*a^5*b^13*c^5 - 279571968*a^6*b^11*c^6 + 853174784*a^7*b^9*c^7 - 1799626752*a^8*b^7*c^8 + 2494119936*a^9*b^5

*c^9 - 2038693888*a^10*b^3*c^10 + 9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) + 4023*a*b^21*c + 10746*a^2*b^4*c^2*(

-(4*a*c - b^2)^15)^(1/2) - 26313*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593*a*b^6*c*(-(4*a*c - b^2)^15)^(1/2

))/(8192*(16777216*a^12*c^19 + b^24*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 - 14080*a^3*b^18*c^10 + 126720*a^4

*b^16*c^11 - 811008*a^5*b^14*c^12 + 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10*c^14 + 32440320*a^8*b^8*c^15 - 5

7671680*a^9*b^6*c^16 + 69206016*a^10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(1/4) - (107811*a^7*b^9 - 2531925*a^

8*b^7*c + 128002112*a^11*b*c^4 + 22295196*a^9*b^5*c^2 - 87242736*a^10*b^3*c^3)/(64*(16384*a^7*c^10 - b^14*c^3

+ 28*a*b^12*c^4 - 336*a^2*b^10*c^5 + 2240*a^3*b^8*c^6 - 8960*a^4*b^6*c^7 + 21504*a^5*b^4*c^8 - 28672*a^6*b^2*c

^9))))*((81*b^8*(-(4*a*c - b^2)^15)^(1/2) - 81*b^23 + 741801984*a^11*b*c^11 - 90126*a^2*b^19*c^2 + 1201623*a^3

*b^17*c^3 - 10588384*a^4*b^15*c^4 + 64704576*a^5*b^13*c^5 - 279571968*a^6*b^11*c^6 + 853174784*a^7*b^9*c^7 - 1

799626752*a^8*b^7*c^8 + 2494119936*a^9*b^5*c^9 - 2038693888*a^10*b^3*c^10 + 9604*a^4*c^4*(-(4*a*c - b^2)^15)^(

1/2) + 4023*a*b^21*c + 10746*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 26313*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/

2) - 1593*a*b^6*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^19 + b^24*c^7 - 48*a*b^22*c^8 + 1056*a^2*b

^20*c^9 - 14080*a^3*b^18*c^10 + 126720*a^4*b^16*c^11 - 811008*a^5*b^14*c^12 + 3784704*a^6*b^12*c^13 - 12976128

*a^7*b^10*c^14 + 32440320*a^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 69206016*a^10*b^4*c^17 - 50331648*a^11*b^2*c^

18)))^(1/4)*2i - atan(((((46036680704*a^12*c^12 - 110592*a^3*b^18*c^3 + 4423680*a^4*b^16*c^4 - 77783040*a^5*b^

14*c^5 + 788037632*a^6*b^12*c^6 - 5065015296*a^7*b^10*c^7 + 21401960448*a^8*b^8*c^8 - 59401830400*a^9*b^6*c^9

+ 104312340480*a^10*b^4*c^10 - 104991817728*a^11*b^2*c^11)/(128*(16384*a^7*c^10 - b^14*c^3 + 28*a*b^12*c^4 - 3

36*a^2*b^10*c^5 + 2240*a^3*b^8*c^6 - 8960*a^4*b^6*c^7 + 21504*a^5*b^4*c^8 - 28672*a^6*b^2*c^9)) - (x^(1/2)*(-(

81*b^23 + 81*b^8*(-(4*a*c - b^2)^15)^(1/2) - 741801984*a^11*b*c^11 + 90126*a^2*b^19*c^2 - 1201623*a^3*b^17*c^3

+ 10588384*a^4*b^15*c^4 - 64704576*a^5*b^13*c^5 + 279571968*a^6*b^11*c^6 - 853174784*a^7*b^9*c^7 + 1799626752

*a^8*b^7*c^8 - 2494119936*a^9*b^5*c^9 + 2038693888*a^10*b^3*c^10 + 9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) - 40

23*a*b^21*c + 10746*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 26313*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593

*a*b^6*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^19 + b^24*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 -

14080*a^3*b^18*c^10 + 126720*a^4*b^16*c^11 - 811008*a^5*b^14*c^12 + 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10

*c^14 + 32440320*a^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 69206016*a^10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(1/

4)*(6576668672*a^11*c^13 + 36864*a^3*b^16*c^5 - 1302528*a^4*b^14*c^6 + 20480000*a^5*b^12*c^7 - 185991168*a^6*b

^10*c^8 + 1061683200*a^7*b^8*c^9 - 3886022656*a^8*b^6*c^10 + 8883535872*a^9*b^4*c^11 - 11576279040*a^10*b^2*c^

12))/(16*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 61

44*a^5*b^2*c^8)))*(-(81*b^23 + 81*b^8*(-(4*a*c - b^2)^15)^(1/2) - 741801984*a^11*b*c^11 + 90126*a^2*b^19*c^2 -

1201623*a^3*b^17*c^3 + 10588384*a^4*b^15*c^4 - 64704576*a^5*b^13*c^5 + 279571968*a^6*b^11*c^6 - 853174784*a^7

*b^9*c^7 + 1799626752*a^8*b^7*c^8 - 2494119936*a^9*b^5*c^9 + 2038693888*a^10*b^3*c^10 + 9604*a^4*c^4*(-(4*a*c

- b^2)^15)^(1/2) - 4023*a*b^21*c + 10746*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 26313*a^3*b^2*c^3*(-(4*a*c -

b^2)^15)^(1/2) - 1593*a*b^6*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^19 + b^24*c^7 - 48*a*b^22*c^8

+ 1056*a^2*b^20*c^9 - 14080*a^3*b^18*c^10 + 126720*a^4*b^16*c^11 - 811008*a^5*b^14*c^12 + 3784704*a^6*b^12*c^1

3 - 12976128*a^7*b^10*c^14 + 32440320*a^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 69206016*a^10*b^4*c^17 - 50331648

*a^11*b^2*c^18)))^(3/4) - (x^(1/2)*(9801*a^5*b^11 - 256905*a^6*b^9*c - 29042496*a^10*b*c^5 + 2642841*a^7*b^7*c

^2 - 13243020*a^8*b^5*c^3 + 31945648*a^9*b^3*c^4))/(16*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*

c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))*(-(81*b^23 + 81*b^8*(-(4*a*c - b^2)^15)^(1/2)

- 741801984*a^11*b*c^11 + 90126*a^2*b^19*c^2 - 1201623*a^3*b^17*c^3 + 10588384*a^4*b^15*c^4 - 64704576*a^5*b^1

3*c^5 + 279571968*a^6*b^11*c^6 - 853174784*a^7*b^9*c^7 + 1799626752*a^8*b^7*c^8 - 2494119936*a^9*b^5*c^9 + 203

8693888*a^10*b^3*c^10 + 9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) - 4023*a*b^21*c + 10746*a^2*b^4*c^2*(-(4*a*c -

b^2)^15)^(1/2) - 26313*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593*a*b^6*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(

16777216*a^12*c^19 + b^24*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 - 14080*a^3*b^18*c^10 + 126720*a^4*b^16*c^11

- 811008*a^5*b^14*c^12 + 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10*c^14 + 32440320*a^8*b^8*c^15 - 57671680*a^

9*b^6*c^16 + 69206016*a^10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(1/4)*1i - (((46036680704*a^12*c^12 - 110592*a

^3*b^18*c^3 + 4423680*a^4*b^16*c^4 - 77783040*a^5*b^14*c^5 + 788037632*a^6*b^12*c^6 - 5065015296*a^7*b^10*c^7

+ 21401960448*a^8*b^8*c^8 - 59401830400*a^9*b^6*c^9 + 104312340480*a^10*b^4*c^10 - 104991817728*a^11*b^2*c^11)

/(128*(16384*a^7*c^10 - b^14*c^3 + 28*a*b^12*c^4 - 336*a^2*b^10*c^5 + 2240*a^3*b^8*c^6 - 8960*a^4*b^6*c^7 + 21

504*a^5*b^4*c^8 - 28672*a^6*b^2*c^9)) + (x^(1/2)*(-(81*b^23 + 81*b^8*(-(4*a*c - b^2)^15)^(1/2) - 741801984*a^1

1*b*c^11 + 90126*a^2*b^19*c^2 - 1201623*a^3*b^17*c^3 + 10588384*a^4*b^15*c^4 - 64704576*a^5*b^13*c^5 + 2795719

68*a^6*b^11*c^6 - 853174784*a^7*b^9*c^7 + 1799626752*a^8*b^7*c^8 - 2494119936*a^9*b^5*c^9 + 2038693888*a^10*b^

3*c^10 + 9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) - 4023*a*b^21*c + 10746*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2)

- 26313*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593*a*b^6*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c

^19 + b^24*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 - 14080*a^3*b^18*c^10 + 126720*a^4*b^16*c^11 - 811008*a^5*b

^14*c^12 + 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10*c^14 + 32440320*a^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 69

206016*a^10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(1/4)*(6576668672*a^11*c^13 + 36864*a^3*b^16*c^5 - 1302528*a^

4*b^14*c^6 + 20480000*a^5*b^12*c^7 - 185991168*a^6*b^10*c^8 + 1061683200*a^7*b^8*c^9 - 3886022656*a^8*b^6*c^10

+ 8883535872*a^9*b^4*c^11 - 11576279040*a^10*b^2*c^12))/(16*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^

2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))*(-(81*b^23 + 81*b^8*(-(4*a*c - b^2)^15)^

(1/2) - 741801984*a^11*b*c^11 + 90126*a^2*b^19*c^2 - 1201623*a^3*b^17*c^3 + 10588384*a^4*b^15*c^4 - 64704576*a

^5*b^13*c^5 + 279571968*a^6*b^11*c^6 - 853174784*a^7*b^9*c^7 + 1799626752*a^8*b^7*c^8 - 2494119936*a^9*b^5*c^9

+ 2038693888*a^10*b^3*c^10 + 9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) - 4023*a*b^21*c + 10746*a^2*b^4*c^2*(-(4*

a*c - b^2)^15)^(1/2) - 26313*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593*a*b^6*c*(-(4*a*c - b^2)^15)^(1/2))/(

8192*(16777216*a^12*c^19 + b^24*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 - 14080*a^3*b^18*c^10 + 126720*a^4*b^1

6*c^11 - 811008*a^5*b^14*c^12 + 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10*c^14 + 32440320*a^8*b^8*c^15 - 57671

680*a^9*b^6*c^16 + 69206016*a^10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(3/4) + (x^(1/2)*(9801*a^5*b^11 - 256905

*a^6*b^9*c - 29042496*a^10*b*c^5 + 2642841*a^7*b^7*c^2 - 13243020*a^8*b^5*c^3 + 31945648*a^9*b^3*c^4))/(16*(40

96*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c

^8)))*(-(81*b^23 + 81*b^8*(-(4*a*c - b^2)^15)^(1/2) - 741801984*a^11*b*c^11 + 90126*a^2*b^19*c^2 - 1201623*a^3

*b^17*c^3 + 10588384*a^4*b^15*c^4 - 64704576*a^5*b^13*c^5 + 279571968*a^6*b^11*c^6 - 853174784*a^7*b^9*c^7 + 1

799626752*a^8*b^7*c^8 - 2494119936*a^9*b^5*c^9 + 2038693888*a^10*b^3*c^10 + 9604*a^4*c^4*(-(4*a*c - b^2)^15)^(

1/2) - 4023*a*b^21*c + 10746*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 26313*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/

2) - 1593*a*b^6*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^19 + b^24*c^7 - 48*a*b^22*c^8 + 1056*a^2*b

^20*c^9 - 14080*a^3*b^18*c^10 + 126720*a^4*b^16*c^11 - 811008*a^5*b^14*c^12 + 3784704*a^6*b^12*c^13 - 12976128

*a^7*b^10*c^14 + 32440320*a^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 69206016*a^10*b^4*c^17 - 50331648*a^11*b^2*c^

18)))^(1/4)*1i)/((((46036680704*a^12*c^12 - 110592*a^3*b^18*c^3 + 4423680*a^4*b^16*c^4 - 77783040*a^5*b^14*c^5

+ 788037632*a^6*b^12*c^6 - 5065015296*a^7*b^10*c^7 + 21401960448*a^8*b^8*c^8 - 59401830400*a^9*b^6*c^9 + 1043

12340480*a^10*b^4*c^10 - 104991817728*a^11*b^2*c^11)/(128*(16384*a^7*c^10 - b^14*c^3 + 28*a*b^12*c^4 - 336*a^2

*b^10*c^5 + 2240*a^3*b^8*c^6 - 8960*a^4*b^6*c^7 + 21504*a^5*b^4*c^8 - 28672*a^6*b^2*c^9)) - (x^(1/2)*(-(81*b^2

3 + 81*b^8*(-(4*a*c - b^2)^15)^(1/2) - 741801984*a^11*b*c^11 + 90126*a^2*b^19*c^2 - 1201623*a^3*b^17*c^3 + 105

88384*a^4*b^15*c^4 - 64704576*a^5*b^13*c^5 + 279571968*a^6*b^11*c^6 - 853174784*a^7*b^9*c^7 + 1799626752*a^8*b

^7*c^8 - 2494119936*a^9*b^5*c^9 + 2038693888*a^10*b^3*c^10 + 9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) - 4023*a*b

^21*c + 10746*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 26313*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593*a*b^6

*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^19 + b^24*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 - 14080

*a^3*b^18*c^10 + 126720*a^4*b^16*c^11 - 811008*a^5*b^14*c^12 + 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10*c^14

+ 32440320*a^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 69206016*a^10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(1/4)*(65

76668672*a^11*c^13 + 36864*a^3*b^16*c^5 - 1302528*a^4*b^14*c^6 + 20480000*a^5*b^12*c^7 - 185991168*a^6*b^10*c^

8 + 1061683200*a^7*b^8*c^9 - 3886022656*a^8*b^6*c^10 + 8883535872*a^9*b^4*c^11 - 11576279040*a^10*b^2*c^12))/(

16*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5

*b^2*c^8)))*(-(81*b^23 + 81*b^8*(-(4*a*c - b^2)^15)^(1/2) - 741801984*a^11*b*c^11 + 90126*a^2*b^19*c^2 - 12016

23*a^3*b^17*c^3 + 10588384*a^4*b^15*c^4 - 64704576*a^5*b^13*c^5 + 279571968*a^6*b^11*c^6 - 853174784*a^7*b^9*c

^7 + 1799626752*a^8*b^7*c^8 - 2494119936*a^9*b^5*c^9 + 2038693888*a^10*b^3*c^10 + 9604*a^4*c^4*(-(4*a*c - b^2)

^15)^(1/2) - 4023*a*b^21*c + 10746*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 26313*a^3*b^2*c^3*(-(4*a*c - b^2)^1

5)^(1/2) - 1593*a*b^6*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^19 + b^24*c^7 - 48*a*b^22*c^8 + 1056

*a^2*b^20*c^9 - 14080*a^3*b^18*c^10 + 126720*a^4*b^16*c^11 - 811008*a^5*b^14*c^12 + 3784704*a^6*b^12*c^13 - 12

976128*a^7*b^10*c^14 + 32440320*a^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 69206016*a^10*b^4*c^17 - 50331648*a^11*

b^2*c^18)))^(3/4) - (x^(1/2)*(9801*a^5*b^11 - 256905*a^6*b^9*c - 29042496*a^10*b*c^5 + 2642841*a^7*b^7*c^2 - 1

3243020*a^8*b^5*c^3 + 31945648*a^9*b^3*c^4))/(16*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 -

1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))*(-(81*b^23 + 81*b^8*(-(4*a*c - b^2)^15)^(1/2) - 7418

01984*a^11*b*c^11 + 90126*a^2*b^19*c^2 - 1201623*a^3*b^17*c^3 + 10588384*a^4*b^15*c^4 - 64704576*a^5*b^13*c^5

+ 279571968*a^6*b^11*c^6 - 853174784*a^7*b^9*c^7 + 1799626752*a^8*b^7*c^8 - 2494119936*a^9*b^5*c^9 + 203869388

8*a^10*b^3*c^10 + 9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) - 4023*a*b^21*c + 10746*a^2*b^4*c^2*(-(4*a*c - b^2)^1

5)^(1/2) - 26313*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593*a*b^6*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(167772

16*a^12*c^19 + b^24*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 - 14080*a^3*b^18*c^10 + 126720*a^4*b^16*c^11 - 811

008*a^5*b^14*c^12 + 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10*c^14 + 32440320*a^8*b^8*c^15 - 57671680*a^9*b^6*

c^16 + 69206016*a^10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(1/4) + (((46036680704*a^12*c^12 - 110592*a^3*b^18*c

^3 + 4423680*a^4*b^16*c^4 - 77783040*a^5*b^14*c^5 + 788037632*a^6*b^12*c^6 - 5065015296*a^7*b^10*c^7 + 2140196

0448*a^8*b^8*c^8 - 59401830400*a^9*b^6*c^9 + 104312340480*a^10*b^4*c^10 - 104991817728*a^11*b^2*c^11)/(128*(16

384*a^7*c^10 - b^14*c^3 + 28*a*b^12*c^4 - 336*a^2*b^10*c^5 + 2240*a^3*b^8*c^6 - 8960*a^4*b^6*c^7 + 21504*a^5*b

^4*c^8 - 28672*a^6*b^2*c^9)) + (x^(1/2)*(-(81*b^23 + 81*b^8*(-(4*a*c - b^2)^15)^(1/2) - 741801984*a^11*b*c^11

+ 90126*a^2*b^19*c^2 - 1201623*a^3*b^17*c^3 + 10588384*a^4*b^15*c^4 - 64704576*a^5*b^13*c^5 + 279571968*a^6*b^

11*c^6 - 853174784*a^7*b^9*c^7 + 1799626752*a^8*b^7*c^8 - 2494119936*a^9*b^5*c^9 + 2038693888*a^10*b^3*c^10 +

9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) - 4023*a*b^21*c + 10746*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 26313*a

^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593*a*b^6*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^19 + b^2

4*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 - 14080*a^3*b^18*c^10 + 126720*a^4*b^16*c^11 - 811008*a^5*b^14*c^12

+ 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10*c^14 + 32440320*a^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 69206016*a^

10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(1/4)*(6576668672*a^11*c^13 + 36864*a^3*b^16*c^5 - 1302528*a^4*b^14*c^

6 + 20480000*a^5*b^12*c^7 - 185991168*a^6*b^10*c^8 + 1061683200*a^7*b^8*c^9 - 3886022656*a^8*b^6*c^10 + 888353

5872*a^9*b^4*c^11 - 11576279040*a^10*b^2*c^12))/(16*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5

- 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))*(-(81*b^23 + 81*b^8*(-(4*a*c - b^2)^15)^(1/2) - 7

41801984*a^11*b*c^11 + 90126*a^2*b^19*c^2 - 1201623*a^3*b^17*c^3 + 10588384*a^4*b^15*c^4 - 64704576*a^5*b^13*c

^5 + 279571968*a^6*b^11*c^6 - 853174784*a^7*b^9*c^7 + 1799626752*a^8*b^7*c^8 - 2494119936*a^9*b^5*c^9 + 203869

3888*a^10*b^3*c^10 + 9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) - 4023*a*b^21*c + 10746*a^2*b^4*c^2*(-(4*a*c - b^2

)^15)^(1/2) - 26313*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593*a*b^6*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(167

77216*a^12*c^19 + b^24*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 - 14080*a^3*b^18*c^10 + 126720*a^4*b^16*c^11 -

811008*a^5*b^14*c^12 + 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10*c^14 + 32440320*a^8*b^8*c^15 - 57671680*a^9*b

^6*c^16 + 69206016*a^10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(3/4) + (x^(1/2)*(9801*a^5*b^11 - 256905*a^6*b^9*

c - 29042496*a^10*b*c^5 + 2642841*a^7*b^7*c^2 - 13243020*a^8*b^5*c^3 + 31945648*a^9*b^3*c^4))/(16*(4096*a^6*c^

9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))*(-(

81*b^23 + 81*b^8*(-(4*a*c - b^2)^15)^(1/2) - 741801984*a^11*b*c^11 + 90126*a^2*b^19*c^2 - 1201623*a^3*b^17*c^3

+ 10588384*a^4*b^15*c^4 - 64704576*a^5*b^13*c^5 + 279571968*a^6*b^11*c^6 - 853174784*a^7*b^9*c^7 + 1799626752

*a^8*b^7*c^8 - 2494119936*a^9*b^5*c^9 + 2038693888*a^10*b^3*c^10 + 9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) - 40

23*a*b^21*c + 10746*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 26313*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593

*a*b^6*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^19 + b^24*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 -

14080*a^3*b^18*c^10 + 126720*a^4*b^16*c^11 - 811008*a^5*b^14*c^12 + 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10

*c^14 + 32440320*a^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 69206016*a^10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(1/

4) - (107811*a^7*b^9 - 2531925*a^8*b^7*c + 128002112*a^11*b*c^4 + 22295196*a^9*b^5*c^2 - 87242736*a^10*b^3*c^3

)/(64*(16384*a^7*c^10 - b^14*c^3 + 28*a*b^12*c^4 - 336*a^2*b^10*c^5 + 2240*a^3*b^8*c^6 - 8960*a^4*b^6*c^7 + 21

504*a^5*b^4*c^8 - 28672*a^6*b^2*c^9))))*(-(81*b^23 + 81*b^8*(-(4*a*c - b^2)^15)^(1/2) - 741801984*a^11*b*c^11

+ 90126*a^2*b^19*c^2 - 1201623*a^3*b^17*c^3 + 10588384*a^4*b^15*c^4 - 64704576*a^5*b^13*c^5 + 279571968*a^6*b^

11*c^6 - 853174784*a^7*b^9*c^7 + 1799626752*a^8*b^7*c^8 - 2494119936*a^9*b^5*c^9 + 2038693888*a^10*b^3*c^10 +

9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) - 4023*a*b^21*c + 10746*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 26313*a

^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593*a*b^6*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^19 + b^2

4*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 - 14080*a^3*b^18*c^10 + 126720*a^4*b^16*c^11 - 811008*a^5*b^14*c^12

+ 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10*c^14 + 32440320*a^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 69206016*a^

10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(1/4)*2i - 2*atan(((((46036680704*a^12*c^12 - 110592*a^3*b^18*c^3 + 44

23680*a^4*b^16*c^4 - 77783040*a^5*b^14*c^5 + 788037632*a^6*b^12*c^6 - 5065015296*a^7*b^10*c^7 + 21401960448*a^

8*b^8*c^8 - 59401830400*a^9*b^6*c^9 + 104312340480*a^10*b^4*c^10 - 104991817728*a^11*b^2*c^11)/(128*(16384*a^7

*c^10 - b^14*c^3 + 28*a*b^12*c^4 - 336*a^2*b^10*c^5 + 2240*a^3*b^8*c^6 - 8960*a^4*b^6*c^7 + 21504*a^5*b^4*c^8

- 28672*a^6*b^2*c^9)) - (x^(1/2)*((81*b^8*(-(4*a*c - b^2)^15)^(1/2) - 81*b^23 + 741801984*a^11*b*c^11 - 90126*

a^2*b^19*c^2 + 1201623*a^3*b^17*c^3 - 10588384*a^4*b^15*c^4 + 64704576*a^5*b^13*c^5 - 279571968*a^6*b^11*c^6 +

853174784*a^7*b^9*c^7 - 1799626752*a^8*b^7*c^8 + 2494119936*a^9*b^5*c^9 - 2038693888*a^10*b^3*c^10 + 9604*a^4

*c^4*(-(4*a*c - b^2)^15)^(1/2) + 4023*a*b^21*c + 10746*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 26313*a^3*b^2*c

^3*(-(4*a*c - b^2)^15)^(1/2) - 1593*a*b^6*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^19 + b^24*c^7 -

48*a*b^22*c^8 + 1056*a^2*b^20*c^9 - 14080*a^3*b^18*c^10 + 126720*a^4*b^16*c^11 - 811008*a^5*b^14*c^12 + 378470

4*a^6*b^12*c^13 - 12976128*a^7*b^10*c^14 + 32440320*a^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 69206016*a^10*b^4*c

^17 - 50331648*a^11*b^2*c^18)))^(1/4)*(6576668672*a^11*c^13 + 36864*a^3*b^16*c^5 - 1302528*a^4*b^14*c^6 + 2048

0000*a^5*b^12*c^7 - 185991168*a^6*b^10*c^8 + 1061683200*a^7*b^8*c^9 - 3886022656*a^8*b^6*c^10 + 8883535872*a^9

*b^4*c^11 - 11576279040*a^10*b^2*c^12)*1i)/(16*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 12

80*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))*((81*b^8*(-(4*a*c - b^2)^15)^(1/2) - 81*b^23 + 7418019

84*a^11*b*c^11 - 90126*a^2*b^19*c^2 + 1201623*a^3*b^17*c^3 - 10588384*a^4*b^15*c^4 + 64704576*a^5*b^13*c^5 - 2

79571968*a^6*b^11*c^6 + 853174784*a^7*b^9*c^7 - 1799626752*a^8*b^7*c^8 + 2494119936*a^9*b^5*c^9 - 2038693888*a

^10*b^3*c^10 + 9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) + 4023*a*b^21*c + 10746*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^

(1/2) - 26313*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593*a*b^6*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*

a^12*c^19 + b^24*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 - 14080*a^3*b^18*c^10 + 126720*a^4*b^16*c^11 - 811008

*a^5*b^14*c^12 + 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10*c^14 + 32440320*a^8*b^8*c^15 - 57671680*a^9*b^6*c^1

6 + 69206016*a^10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(3/4)*1i + (x^(1/2)*(9801*a^5*b^11 - 256905*a^6*b^9*c -

29042496*a^10*b*c^5 + 2642841*a^7*b^7*c^2 - 13243020*a^8*b^5*c^3 + 31945648*a^9*b^3*c^4))/(16*(4096*a^6*c^9 +

b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))*((81*b

^8*(-(4*a*c - b^2)^15)^(1/2) - 81*b^23 + 741801984*a^11*b*c^11 - 90126*a^2*b^19*c^2 + 1201623*a^3*b^17*c^3 - 1

0588384*a^4*b^15*c^4 + 64704576*a^5*b^13*c^5 - 279571968*a^6*b^11*c^6 + 853174784*a^7*b^9*c^7 - 1799626752*a^8

*b^7*c^8 + 2494119936*a^9*b^5*c^9 - 2038693888*a^10*b^3*c^10 + 9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) + 4023*a

*b^21*c + 10746*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 26313*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593*a*b

^6*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^19 + b^24*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 - 140

80*a^3*b^18*c^10 + 126720*a^4*b^16*c^11 - 811008*a^5*b^14*c^12 + 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10*c^1

4 + 32440320*a^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 69206016*a^10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(1/4) -

(((46036680704*a^12*c^12 - 110592*a^3*b^18*c^3 + 4423680*a^4*b^16*c^4 - 77783040*a^5*b^14*c^5 + 788037632*a^6

*b^12*c^6 - 5065015296*a^7*b^10*c^7 + 21401960448*a^8*b^8*c^8 - 59401830400*a^9*b^6*c^9 + 104312340480*a^10*b^

4*c^10 - 104991817728*a^11*b^2*c^11)/(128*(16384*a^7*c^10 - b^14*c^3 + 28*a*b^12*c^4 - 336*a^2*b^10*c^5 + 2240

*a^3*b^8*c^6 - 8960*a^4*b^6*c^7 + 21504*a^5*b^4*c^8 - 28672*a^6*b^2*c^9)) + (x^(1/2)*((81*b^8*(-(4*a*c - b^2)^

15)^(1/2) - 81*b^23 + 741801984*a^11*b*c^11 - 90126*a^2*b^19*c^2 + 1201623*a^3*b^17*c^3 - 10588384*a^4*b^15*c^

4 + 64704576*a^5*b^13*c^5 - 279571968*a^6*b^11*c^6 + 853174784*a^7*b^9*c^7 - 1799626752*a^8*b^7*c^8 + 24941199

36*a^9*b^5*c^9 - 2038693888*a^10*b^3*c^10 + 9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) + 4023*a*b^21*c + 10746*a^2

*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 26313*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593*a*b^6*c*(-(4*a*c - b^2

)^15)^(1/2))/(8192*(16777216*a^12*c^19 + b^24*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 - 14080*a^3*b^18*c^10 +

126720*a^4*b^16*c^11 - 811008*a^5*b^14*c^12 + 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10*c^14 + 32440320*a^8*b^

8*c^15 - 57671680*a^9*b^6*c^16 + 69206016*a^10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(1/4)*(6576668672*a^11*c^1

3 + 36864*a^3*b^16*c^5 - 1302528*a^4*b^14*c^6 + 20480000*a^5*b^12*c^7 - 185991168*a^6*b^10*c^8 + 1061683200*a^

7*b^8*c^9 - 3886022656*a^8*b^6*c^10 + 8883535872*a^9*b^4*c^11 - 11576279040*a^10*b^2*c^12)*1i)/(16*(4096*a^6*c

^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))*((

81*b^8*(-(4*a*c - b^2)^15)^(1/2) - 81*b^23 + 741801984*a^11*b*c^11 - 90126*a^2*b^19*c^2 + 1201623*a^3*b^17*c^3

- 10588384*a^4*b^15*c^4 + 64704576*a^5*b^13*c^5 - 279571968*a^6*b^11*c^6 + 853174784*a^7*b^9*c^7 - 1799626752

*a^8*b^7*c^8 + 2494119936*a^9*b^5*c^9 - 2038693888*a^10*b^3*c^10 + 9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) + 40

23*a*b^21*c + 10746*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 26313*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593

*a*b^6*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^19 + b^24*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 -

14080*a^3*b^18*c^10 + 126720*a^4*b^16*c^11 - 811008*a^5*b^14*c^12 + 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10

*c^14 + 32440320*a^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 69206016*a^10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(3/

4)*1i - (x^(1/2)*(9801*a^5*b^11 - 256905*a^6*b^9*c - 29042496*a^10*b*c^5 + 2642841*a^7*b^7*c^2 - 13243020*a^8*

b^5*c^3 + 31945648*a^9*b^3*c^4))/(16*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6

*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))*((81*b^8*(-(4*a*c - b^2)^15)^(1/2) - 81*b^23 + 741801984*a^11*b*

c^11 - 90126*a^2*b^19*c^2 + 1201623*a^3*b^17*c^3 - 10588384*a^4*b^15*c^4 + 64704576*a^5*b^13*c^5 - 279571968*a

^6*b^11*c^6 + 853174784*a^7*b^9*c^7 - 1799626752*a^8*b^7*c^8 + 2494119936*a^9*b^5*c^9 - 2038693888*a^10*b^3*c^

10 + 9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) + 4023*a*b^21*c + 10746*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 26

313*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593*a*b^6*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^19

+ b^24*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 - 14080*a^3*b^18*c^10 + 126720*a^4*b^16*c^11 - 811008*a^5*b^14*

c^12 + 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10*c^14 + 32440320*a^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 692060

16*a^10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(1/4))/((((46036680704*a^12*c^12 - 110592*a^3*b^18*c^3 + 4423680*

a^4*b^16*c^4 - 77783040*a^5*b^14*c^5 + 788037632*a^6*b^12*c^6 - 5065015296*a^7*b^10*c^7 + 21401960448*a^8*b^8*

c^8 - 59401830400*a^9*b^6*c^9 + 104312340480*a^10*b^4*c^10 - 104991817728*a^11*b^2*c^11)/(128*(16384*a^7*c^10

- b^14*c^3 + 28*a*b^12*c^4 - 336*a^2*b^10*c^5 + 2240*a^3*b^8*c^6 - 8960*a^4*b^6*c^7 + 21504*a^5*b^4*c^8 - 2867

2*a^6*b^2*c^9)) - (x^(1/2)*((81*b^8*(-(4*a*c - b^2)^15)^(1/2) - 81*b^23 + 741801984*a^11*b*c^11 - 90126*a^2*b^

19*c^2 + 1201623*a^3*b^17*c^3 - 10588384*a^4*b^15*c^4 + 64704576*a^5*b^13*c^5 - 279571968*a^6*b^11*c^6 + 85317

4784*a^7*b^9*c^7 - 1799626752*a^8*b^7*c^8 + 2494119936*a^9*b^5*c^9 - 2038693888*a^10*b^3*c^10 + 9604*a^4*c^4*(

-(4*a*c - b^2)^15)^(1/2) + 4023*a*b^21*c + 10746*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 26313*a^3*b^2*c^3*(-(

4*a*c - b^2)^15)^(1/2) - 1593*a*b^6*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^19 + b^24*c^7 - 48*a*b

^22*c^8 + 1056*a^2*b^20*c^9 - 14080*a^3*b^18*c^10 + 126720*a^4*b^16*c^11 - 811008*a^5*b^14*c^12 + 3784704*a^6*

b^12*c^13 - 12976128*a^7*b^10*c^14 + 32440320*a^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 69206016*a^10*b^4*c^17 -

50331648*a^11*b^2*c^18)))^(1/4)*(6576668672*a^11*c^13 + 36864*a^3*b^16*c^5 - 1302528*a^4*b^14*c^6 + 20480000*a

^5*b^12*c^7 - 185991168*a^6*b^10*c^8 + 1061683200*a^7*b^8*c^9 - 3886022656*a^8*b^6*c^10 + 8883535872*a^9*b^4*c

^11 - 11576279040*a^10*b^2*c^12)*1i)/(16*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3

*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))*((81*b^8*(-(4*a*c - b^2)^15)^(1/2) - 81*b^23 + 741801984*a^1

1*b*c^11 - 90126*a^2*b^19*c^2 + 1201623*a^3*b^17*c^3 - 10588384*a^4*b^15*c^4 + 64704576*a^5*b^13*c^5 - 2795719

68*a^6*b^11*c^6 + 853174784*a^7*b^9*c^7 - 1799626752*a^8*b^7*c^8 + 2494119936*a^9*b^5*c^9 - 2038693888*a^10*b^

3*c^10 + 9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) + 4023*a*b^21*c + 10746*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2)

- 26313*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593*a*b^6*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c

^19 + b^24*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 - 14080*a^3*b^18*c^10 + 126720*a^4*b^16*c^11 - 811008*a^5*b

^14*c^12 + 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10*c^14 + 32440320*a^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 69

206016*a^10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(3/4)*1i + (x^(1/2)*(9801*a^5*b^11 - 256905*a^6*b^9*c - 29042

496*a^10*b*c^5 + 2642841*a^7*b^7*c^2 - 13243020*a^8*b^5*c^3 + 31945648*a^9*b^3*c^4))/(16*(4096*a^6*c^9 + b^12*

c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))*((81*b^8*(-(

4*a*c - b^2)^15)^(1/2) - 81*b^23 + 741801984*a^11*b*c^11 - 90126*a^2*b^19*c^2 + 1201623*a^3*b^17*c^3 - 1058838

4*a^4*b^15*c^4 + 64704576*a^5*b^13*c^5 - 279571968*a^6*b^11*c^6 + 853174784*a^7*b^9*c^7 - 1799626752*a^8*b^7*c

^8 + 2494119936*a^9*b^5*c^9 - 2038693888*a^10*b^3*c^10 + 9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) + 4023*a*b^21*

c + 10746*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 26313*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593*a*b^6*c*(

-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^19 + b^24*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 - 14080*a^3

*b^18*c^10 + 126720*a^4*b^16*c^11 - 811008*a^5*b^14*c^12 + 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10*c^14 + 32

440320*a^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 69206016*a^10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(1/4)*1i + ((

(46036680704*a^12*c^12 - 110592*a^3*b^18*c^3 + 4423680*a^4*b^16*c^4 - 77783040*a^5*b^14*c^5 + 788037632*a^6*b^

12*c^6 - 5065015296*a^7*b^10*c^7 + 21401960448*a^8*b^8*c^8 - 59401830400*a^9*b^6*c^9 + 104312340480*a^10*b^4*c

^10 - 104991817728*a^11*b^2*c^11)/(128*(16384*a^7*c^10 - b^14*c^3 + 28*a*b^12*c^4 - 336*a^2*b^10*c^5 + 2240*a^

3*b^8*c^6 - 8960*a^4*b^6*c^7 + 21504*a^5*b^4*c^8 - 28672*a^6*b^2*c^9)) + (x^(1/2)*((81*b^8*(-(4*a*c - b^2)^15)

^(1/2) - 81*b^23 + 741801984*a^11*b*c^11 - 90126*a^2*b^19*c^2 + 1201623*a^3*b^17*c^3 - 10588384*a^4*b^15*c^4 +

64704576*a^5*b^13*c^5 - 279571968*a^6*b^11*c^6 + 853174784*a^7*b^9*c^7 - 1799626752*a^8*b^7*c^8 + 2494119936*

a^9*b^5*c^9 - 2038693888*a^10*b^3*c^10 + 9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) + 4023*a*b^21*c + 10746*a^2*b^

4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 26313*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593*a*b^6*c*(-(4*a*c - b^2)^1

5)^(1/2))/(8192*(16777216*a^12*c^19 + b^24*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 - 14080*a^3*b^18*c^10 + 126

720*a^4*b^16*c^11 - 811008*a^5*b^14*c^12 + 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10*c^14 + 32440320*a^8*b^8*c

^15 - 57671680*a^9*b^6*c^16 + 69206016*a^10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(1/4)*(6576668672*a^11*c^13 +

36864*a^3*b^16*c^5 - 1302528*a^4*b^14*c^6 + 20480000*a^5*b^12*c^7 - 185991168*a^6*b^10*c^8 + 1061683200*a^7*b

^8*c^9 - 3886022656*a^8*b^6*c^10 + 8883535872*a^9*b^4*c^11 - 11576279040*a^10*b^2*c^12)*1i)/(16*(4096*a^6*c^9

+ b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))*((81*

b^8*(-(4*a*c - b^2)^15)^(1/2) - 81*b^23 + 741801984*a^11*b*c^11 - 90126*a^2*b^19*c^2 + 1201623*a^3*b^17*c^3 -

10588384*a^4*b^15*c^4 + 64704576*a^5*b^13*c^5 - 279571968*a^6*b^11*c^6 + 853174784*a^7*b^9*c^7 - 1799626752*a^

8*b^7*c^8 + 2494119936*a^9*b^5*c^9 - 2038693888*a^10*b^3*c^10 + 9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) + 4023*

a*b^21*c + 10746*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 26313*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593*a*

b^6*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^19 + b^24*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 - 14

080*a^3*b^18*c^10 + 126720*a^4*b^16*c^11 - 811008*a^5*b^14*c^12 + 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10*c^

14 + 32440320*a^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 69206016*a^10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(3/4)*

1i - (x^(1/2)*(9801*a^5*b^11 - 256905*a^6*b^9*c - 29042496*a^10*b*c^5 + 2642841*a^7*b^7*c^2 - 13243020*a^8*b^5

*c^3 + 31945648*a^9*b^3*c^4))/(16*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^

6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))*((81*b^8*(-(4*a*c - b^2)^15)^(1/2) - 81*b^23 + 741801984*a^11*b*c^1

1 - 90126*a^2*b^19*c^2 + 1201623*a^3*b^17*c^3 - 10588384*a^4*b^15*c^4 + 64704576*a^5*b^13*c^5 - 279571968*a^6*

b^11*c^6 + 853174784*a^7*b^9*c^7 - 1799626752*a^8*b^7*c^8 + 2494119936*a^9*b^5*c^9 - 2038693888*a^10*b^3*c^10

+ 9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) + 4023*a*b^21*c + 10746*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 26313

*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593*a*b^6*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^19 + b

^24*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 - 14080*a^3*b^18*c^10 + 126720*a^4*b^16*c^11 - 811008*a^5*b^14*c^1

2 + 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10*c^14 + 32440320*a^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 69206016*

a^10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(1/4)*1i + (107811*a^7*b^9 - 2531925*a^8*b^7*c + 128002112*a^11*b*c^

4 + 22295196*a^9*b^5*c^2 - 87242736*a^10*b^3*c^3)/(64*(16384*a^7*c^10 - b^14*c^3 + 28*a*b^12*c^4 - 336*a^2*b^1

0*c^5 + 2240*a^3*b^8*c^6 - 8960*a^4*b^6*c^7 + 21504*a^5*b^4*c^8 - 28672*a^6*b^2*c^9))))*((81*b^8*(-(4*a*c - b^

2)^15)^(1/2) - 81*b^23 + 741801984*a^11*b*c^11 - 90126*a^2*b^19*c^2 + 1201623*a^3*b^17*c^3 - 10588384*a^4*b^15

*c^4 + 64704576*a^5*b^13*c^5 - 279571968*a^6*b^11*c^6 + 853174784*a^7*b^9*c^7 - 1799626752*a^8*b^7*c^8 + 24941

19936*a^9*b^5*c^9 - 2038693888*a^10*b^3*c^10 + 9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) + 4023*a*b^21*c + 10746*

a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 26313*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593*a*b^6*c*(-(4*a*c -

b^2)^15)^(1/2))/(8192*(16777216*a^12*c^19 + b^24*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 - 14080*a^3*b^18*c^10

+ 126720*a^4*b^16*c^11 - 811008*a^5*b^14*c^12 + 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10*c^14 + 32440320*a^8

*b^8*c^15 - 57671680*a^9*b^6*c^16 + 69206016*a^10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(1/4) - 2*atan(((((4603

6680704*a^12*c^12 - 110592*a^3*b^18*c^3 + 4423680*a^4*b^16*c^4 - 77783040*a^5*b^14*c^5 + 788037632*a^6*b^12*c^

6 - 5065015296*a^7*b^10*c^7 + 21401960448*a^8*b^8*c^8 - 59401830400*a^9*b^6*c^9 + 104312340480*a^10*b^4*c^10 -

104991817728*a^11*b^2*c^11)/(128*(16384*a^7*c^10 - b^14*c^3 + 28*a*b^12*c^4 - 336*a^2*b^10*c^5 + 2240*a^3*b^8

*c^6 - 8960*a^4*b^6*c^7 + 21504*a^5*b^4*c^8 - 28672*a^6*b^2*c^9)) - (x^(1/2)*(-(81*b^23 + 81*b^8*(-(4*a*c - b^

2)^15)^(1/2) - 741801984*a^11*b*c^11 + 90126*a^2*b^19*c^2 - 1201623*a^3*b^17*c^3 + 10588384*a^4*b^15*c^4 - 647

04576*a^5*b^13*c^5 + 279571968*a^6*b^11*c^6 - 853174784*a^7*b^9*c^7 + 1799626752*a^8*b^7*c^8 - 2494119936*a^9*

b^5*c^9 + 2038693888*a^10*b^3*c^10 + 9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) - 4023*a*b^21*c + 10746*a^2*b^4*c^

2*(-(4*a*c - b^2)^15)^(1/2) - 26313*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593*a*b^6*c*(-(4*a*c - b^2)^15)^(

1/2))/(8192*(16777216*a^12*c^19 + b^24*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 - 14080*a^3*b^18*c^10 + 126720*

a^4*b^16*c^11 - 811008*a^5*b^14*c^12 + 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10*c^14 + 32440320*a^8*b^8*c^15

- 57671680*a^9*b^6*c^16 + 69206016*a^10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(1/4)*(6576668672*a^11*c^13 + 368

64*a^3*b^16*c^5 - 1302528*a^4*b^14*c^6 + 20480000*a^5*b^12*c^7 - 185991168*a^6*b^10*c^8 + 1061683200*a^7*b^8*c

^9 - 3886022656*a^8*b^6*c^10 + 8883535872*a^9*b^4*c^11 - 11576279040*a^10*b^2*c^12)*1i)/(16*(4096*a^6*c^9 + b^

12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))*(-(81*b^2

3 + 81*b^8*(-(4*a*c - b^2)^15)^(1/2) - 741801984*a^11*b*c^11 + 90126*a^2*b^19*c^2 - 1201623*a^3*b^17*c^3 + 105

88384*a^4*b^15*c^4 - 64704576*a^5*b^13*c^5 + 279571968*a^6*b^11*c^6 - 853174784*a^7*b^9*c^7 + 1799626752*a^8*b

^7*c^8 - 2494119936*a^9*b^5*c^9 + 2038693888*a^10*b^3*c^10 + 9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) - 4023*a*b

^21*c + 10746*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 26313*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593*a*b^6

*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^19 + b^24*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 - 14080

*a^3*b^18*c^10 + 126720*a^4*b^16*c^11 - 811008*a^5*b^14*c^12 + 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10*c^14

+ 32440320*a^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 69206016*a^10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(3/4)*1i

+ (x^(1/2)*(9801*a^5*b^11 - 256905*a^6*b^9*c - 29042496*a^10*b*c^5 + 2642841*a^7*b^7*c^2 - 13243020*a^8*b^5*c^

3 + 31945648*a^9*b^3*c^4))/(16*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 +

3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))*(-(81*b^23 + 81*b^8*(-(4*a*c - b^2)^15)^(1/2) - 741801984*a^11*b*c^11

+ 90126*a^2*b^19*c^2 - 1201623*a^3*b^17*c^3 + 10588384*a^4*b^15*c^4 - 64704576*a^5*b^13*c^5 + 279571968*a^6*b^

11*c^6 - 853174784*a^7*b^9*c^7 + 1799626752*a^8*b^7*c^8 - 2494119936*a^9*b^5*c^9 + 2038693888*a^10*b^3*c^10 +

9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) - 4023*a*b^21*c + 10746*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 26313*a

^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593*a*b^6*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^19 + b^2

4*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 - 14080*a^3*b^18*c^10 + 126720*a^4*b^16*c^11 - 811008*a^5*b^14*c^12

+ 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10*c^14 + 32440320*a^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 69206016*a^

10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(1/4) - (((46036680704*a^12*c^12 - 110592*a^3*b^18*c^3 + 4423680*a^4*b

^16*c^4 - 77783040*a^5*b^14*c^5 + 788037632*a^6*b^12*c^6 - 5065015296*a^7*b^10*c^7 + 21401960448*a^8*b^8*c^8 -

59401830400*a^9*b^6*c^9 + 104312340480*a^10*b^4*c^10 - 104991817728*a^11*b^2*c^11)/(128*(16384*a^7*c^10 - b^1

4*c^3 + 28*a*b^12*c^4 - 336*a^2*b^10*c^5 + 2240*a^3*b^8*c^6 - 8960*a^4*b^6*c^7 + 21504*a^5*b^4*c^8 - 28672*a^6

*b^2*c^9)) + (x^(1/2)*(-(81*b^23 + 81*b^8*(-(4*a*c - b^2)^15)^(1/2) - 741801984*a^11*b*c^11 + 90126*a^2*b^19*c

^2 - 1201623*a^3*b^17*c^3 + 10588384*a^4*b^15*c^4 - 64704576*a^5*b^13*c^5 + 279571968*a^6*b^11*c^6 - 853174784

*a^7*b^9*c^7 + 1799626752*a^8*b^7*c^8 - 2494119936*a^9*b^5*c^9 + 2038693888*a^10*b^3*c^10 + 9604*a^4*c^4*(-(4*

a*c - b^2)^15)^(1/2) - 4023*a*b^21*c + 10746*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 26313*a^3*b^2*c^3*(-(4*a*

c - b^2)^15)^(1/2) - 1593*a*b^6*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^19 + b^24*c^7 - 48*a*b^22*

c^8 + 1056*a^2*b^20*c^9 - 14080*a^3*b^18*c^10 + 126720*a^4*b^16*c^11 - 811008*a^5*b^14*c^12 + 3784704*a^6*b^12

*c^13 - 12976128*a^7*b^10*c^14 + 32440320*a^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 69206016*a^10*b^4*c^17 - 5033

1648*a^11*b^2*c^18)))^(1/4)*(6576668672*a^11*c^13 + 36864*a^3*b^16*c^5 - 1302528*a^4*b^14*c^6 + 20480000*a^5*b

^12*c^7 - 185991168*a^6*b^10*c^8 + 1061683200*a^7*b^8*c^9 - 3886022656*a^8*b^6*c^10 + 8883535872*a^9*b^4*c^11

- 11576279040*a^10*b^2*c^12)*1i)/(16*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6

*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))*(-(81*b^23 + 81*b^8*(-(4*a*c - b^2)^15)^(1/2) - 741801984*a^11*b

*c^11 + 90126*a^2*b^19*c^2 - 1201623*a^3*b^17*c^3 + 10588384*a^4*b^15*c^4 - 64704576*a^5*b^13*c^5 + 279571968*

a^6*b^11*c^6 - 853174784*a^7*b^9*c^7 + 1799626752*a^8*b^7*c^8 - 2494119936*a^9*b^5*c^9 + 2038693888*a^10*b^3*c

^10 + 9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) - 4023*a*b^21*c + 10746*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 2

6313*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593*a*b^6*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^19

+ b^24*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 - 14080*a^3*b^18*c^10 + 126720*a^4*b^16*c^11 - 811008*a^5*b^14

*c^12 + 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10*c^14 + 32440320*a^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 69206

016*a^10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(3/4)*1i - (x^(1/2)*(9801*a^5*b^11 - 256905*a^6*b^9*c - 29042496

*a^10*b*c^5 + 2642841*a^7*b^7*c^2 - 13243020*a^8*b^5*c^3 + 31945648*a^9*b^3*c^4))/(16*(4096*a^6*c^9 + b^12*c^3

- 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))*(-(81*b^23 + 81

*b^8*(-(4*a*c - b^2)^15)^(1/2) - 741801984*a^11*b*c^11 + 90126*a^2*b^19*c^2 - 1201623*a^3*b^17*c^3 + 10588384*

a^4*b^15*c^4 - 64704576*a^5*b^13*c^5 + 279571968*a^6*b^11*c^6 - 853174784*a^7*b^9*c^7 + 1799626752*a^8*b^7*c^8

- 2494119936*a^9*b^5*c^9 + 2038693888*a^10*b^3*c^10 + 9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) - 4023*a*b^21*c

+ 10746*a^2*b^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 26313*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593*a*b^6*c*(-(

4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^19 + b^24*c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 - 14080*a^3*b

^18*c^10 + 126720*a^4*b^16*c^11 - 811008*a^5*b^14*c^12 + 3784704*a^6*b^12*c^13 - 12976128*a^7*b^10*c^14 + 3244

0320*a^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 69206016*a^10*b^4*c^17 - 50331648*a^11*b^2*c^18)))^(1/4))/((((4603