∫ x9∕2 ____________________

3.9.50 $\int \frac {x^{9/2}}{(a+b x^2+c x^4)^3} \, dx$

Optimal. Leaf size=533 \[ -\frac {3 x^{3/2} \left (-4 a c+5 b^2+8 b c x^2\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {x^{3/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {3 \sqrt [4]{c} \left (4 b \sqrt {b^2-4 a c}+20 a c+11 b^2\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{16\ 2^{3/4} \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}+\frac {3 \sqrt [4]{c} \left (-4 b \sqrt {b^2-4 a c}+20 a c+11 b^2\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{16\ 2^{3/4} \left (b^2-4 a c\right )^{5/2} \sqrt [4]{\sqrt {b^2-4 a c}-b}}+\frac {3 \sqrt [4]{c} \left (4 b \sqrt {b^2-4 a c}+20 a c+11 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{16\ 2^{3/4} \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}-\frac {3 \sqrt [4]{c} \left (-4 b \sqrt {b^2-4 a c}+20 a c+11 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{16\ 2^{3/4} \left (b^2-4 a c\right )^{5/2} \sqrt [4]{\sqrt {b^2-4 a c}-b}} \]

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Rubi [A] time = 1.45, antiderivative size = 533, 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, 1500, 1510, 298, 205, 208} \begin {gather*} -\frac {3 x^{3/2} \left (-4 a c+5 b^2+8 b c x^2\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {x^{3/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {3 \sqrt [4]{c} \left (4 b \sqrt {b^2-4 a c}+20 a c+11 b^2\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{16\ 2^{3/4} \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}+\frac {3 \sqrt [4]{c} \left (-4 b \sqrt {b^2-4 a c}+20 a c+11 b^2\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{16\ 2^{3/4} \left (b^2-4 a c\right )^{5/2} \sqrt [4]{\sqrt {b^2-4 a c}-b}}+\frac {3 \sqrt [4]{c} \left (4 b \sqrt {b^2-4 a c}+20 a c+11 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{16\ 2^{3/4} \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}-\frac {3 \sqrt [4]{c} \left (-4 b \sqrt {b^2-4 a c}+20 a c+11 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{16\ 2^{3/4} \left (b^2-4 a c\right )^{5/2} \sqrt [4]{\sqrt {b^2-4 a c}-b}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[x^(9/2)/(a + b*x^2 + c*x^4)^3,x]

[Out]

(x^(3/2)*(2*a + b*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (3*x^(3/2)*(5*b^2 - 4*a*c + 8*b*c*x^2))/(16*

(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - (3*c^(1/4)*(11*b^2 + 20*a*c + 4*b*Sqrt[b^2 - 4*a*c])*ArcTan[(2^(1/4)*c^

(1/4)*Sqrt[x])/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(16*2^(3/4)*(b^2 - 4*a*c)^(5/2)*(-b - Sqrt[b^2 - 4*a*c])^(1/4)

) + (3*c^(1/4)*(11*b^2 + 20*a*c - 4*b*Sqrt[b^2 - 4*a*c])*ArcTan[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b + Sqrt[b^2 - 4*a

*c])^(1/4)])/(16*2^(3/4)*(b^2 - 4*a*c)^(5/2)*(-b + Sqrt[b^2 - 4*a*c])^(1/4)) + (3*c^(1/4)*(11*b^2 + 20*a*c + 4

*b*Sqrt[b^2 - 4*a*c])*ArcTanh[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(16*2^(3/4)*(b^2 - 4*

a*c)^(5/2)*(-b - Sqrt[b^2 - 4*a*c])^(1/4)) - (3*c^(1/4)*(11*b^2 + 20*a*c - 4*b*Sqrt[b^2 - 4*a*c])*ArcTanh[(2^(

1/4)*c^(1/4)*Sqrt[x])/(-b + Sqrt[b^2 - 4*a*c])^(1/4)])/(16*2^(3/4)*(b^2 - 4*a*c)^(5/2)*(-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 1500

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

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

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

imp[d*(b^2*(m + n*(p + 1) + 1) - 2*a*c*(m + 2*n*(p + 1) + 1)) - a*b*e*(m + 1) + c*(m + n*(2*p + 3) + 1)*(b*d -

2*a*e)*x^n, x], 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

] && LtQ[p, -1] && 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^{9/2}}{\left (a+b x^2+c x^4\right )^3} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^{10}}{\left (a+b x^4+c x^8\right )^3} \, dx,x,\sqrt {x}\right )\\ &=\frac {x^{3/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (6 a-9 b x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx,x,\sqrt {x}\right )}{4 \left (b^2-4 a c\right )}\\ &=\frac {x^{3/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {3 x^{3/2} \left (5 b^2-4 a c+8 b c x^2\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (3 a \left (7 b^2+20 a c\right )-24 a b c x^4\right )}{a+b x^4+c x^8} \, dx,x,\sqrt {x}\right )}{16 a \left (b^2-4 a c\right )^2}\\ &=\frac {x^{3/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {3 x^{3/2} \left (5 b^2-4 a c+8 b c x^2\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\left (3 c \left (11 b^2+20 a c-4 b \sqrt {b^2-4 a c}\right )\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 )}{16 \left (b^2-4 a c\right )^{5/2}}-\frac {\left (3 c \left (11 b^2+20 a c+4 b \sqrt {b^2-4 a c}\right )\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 )}{16 \left (b^2-4 a c\right )^{5/2}}\\ &=\frac {x^{3/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {3 x^{3/2} \left (5 b^2-4 a c+8 b c x^2\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {\left (3 \sqrt {c} \left (11 b^2+20 a c-4 b \sqrt {b^2-4 a c}\right )\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 )}{16 \sqrt {2} \left (b^2-4 a c\right )^{5/2}}+\frac {\left (3 \sqrt {c} \left (11 b^2+20 a c-4 b \sqrt {b^2-4 a c}\right )\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 )}{16 \sqrt {2} \left (b^2-4 a c\right )^{5/2}}+\frac {\left (3 \sqrt {c} \left (11 b^2+20 a c+4 b \sqrt {b^2-4 a c}\right )\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 )}{16 \sqrt {2} \left (b^2-4 a c\right )^{5/2}}-\frac {\left (3 \sqrt {c} \left (11 b^2+20 a c+4 b \sqrt {b^2-4 a c}\right )\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 )}{16 \sqrt {2} \left (b^2-4 a c\right )^{5/2}}\\ &=\frac {x^{3/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {3 x^{3/2} \left (5 b^2-4 a c+8 b c x^2\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {3 \sqrt [4]{c} \left (11 b^2+20 a c+4 b \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 )}{16\ 2^{3/4} \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-b-\sqrt {b^2-4 a c}}}+\frac {3 \sqrt [4]{c} \left (11 b^2+20 a c-4 b \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 )}{16\ 2^{3/4} \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-b+\sqrt {b^2-4 a c}}}+\frac {3 \sqrt [4]{c} \left (11 b^2+20 a c+4 b \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 )}{16\ 2^{3/4} \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-b-\sqrt {b^2-4 a c}}}-\frac {3 \sqrt [4]{c} \left (11 b^2+20 a c-4 b \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 )}{16\ 2^{3/4} \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-b+\sqrt {b^2-4 a c}}}\\ \end {align*}

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Mathematica [C] time = 0.38, size = 176, normalized size = 0.33 \begin {gather*} \frac {-3 \text {RootSum}\left [\text {$\#$1}^8 c+\text {$\#$1}^4 b+a\&,\frac {8 \text {$\#$1}^4 b c \log \left (\sqrt {x}-\text {$\#$1}\right )-20 a c \log \left (\sqrt {x}-\text {$\#$1}\right )-7 b^2 \log \left (\sqrt {x}-\text {$\#$1}\right )}{2 \text {$\#$1}^5 c+\text {$\#$1} b}\&\right ]-\frac {12 x^{3/2} \left (-4 a c+5 b^2+8 b c x^2\right )}{a+b x^2+c x^4}+\frac {16 x^{3/2} \left (b^2-4 a c\right ) \left (2 a+b x^2\right )}{\left (a+b x^2+c x^4\right )^2}}{64 \left (b^2-4 a c\right )^2} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^(9/2)/(a + b*x^2 + c*x^4)^3,x]

[Out]

((16*(b^2 - 4*a*c)*x^(3/2)*(2*a + b*x^2))/(a + b*x^2 + c*x^4)^2 - (12*x^(3/2)*(5*b^2 - 4*a*c + 8*b*c*x^2))/(a

+ b*x^2 + c*x^4) - 3*RootSum[a + b*#1^4 + c*#1^8 & , (-7*b^2*Log[Sqrt[x] - #1] - 20*a*c*Log[Sqrt[x] - #1] + 8*

b*c*Log[Sqrt[x] - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ])/(64*(b^2 - 4*a*c)^2)

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IntegrateAlgebraic [C] time = 0.97, size = 344, normalized size = 0.65 \begin {gather*} -\frac {\text {RootSum}\left [\text {$\#$1}^8 c+\text {$\#$1}^4 b+a\&,\frac {-8 \text {$\#$1}^4 a b c^2 \log \left (\sqrt {x}-\text {$\#$1}\right )+8 \text {$\#$1}^4 b^3 c \log \left (\sqrt {x}-\text {$\#$1}\right )+260 a^2 c^2 \log \left (\sqrt {x}-\text {$\#$1}\right )-133 a b^2 c \log \left (\sqrt {x}-\text {$\#$1}\right )+8 b^4 \log \left (\sqrt {x}-\text {$\#$1}\right )}{2 \text {$\#$1}^5 c+\text {$\#$1} b}\&\right ]}{64 a c \left (4 a c-b^2\right )^2}-\frac {\text {RootSum}\left [\text {$\#$1}^8 c+\text {$\#$1}^4 b+a\&,\frac {\text {$\#$1}^4 b c \log \left (\sqrt {x}-\text {$\#$1}\right )-10 a c \log \left (\sqrt {x}-\text {$\#$1}\right )+b^2 \log \left (\sqrt {x}-\text {$\#$1}\right )}{2 \text {$\#$1}^5 c+\text {$\#$1} b}\&\right ]}{8 a c \left (4 a c-b^2\right )}-\frac {x^{3/2} \left (20 a^2 c+7 a b^2+28 a b c x^2-12 a c^2 x^4+11 b^3 x^2+39 b^2 c x^4+24 b c^2 x^6\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )^2} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

IntegrateAlgebraic[x^(9/2)/(a + b*x^2 + c*x^4)^3,x]

[Out]

-1/16*(x^(3/2)*(7*a*b^2 + 20*a^2*c + 11*b^3*x^2 + 28*a*b*c*x^2 + 39*b^2*c*x^4 - 12*a*c^2*x^4 + 24*b*c^2*x^6))/

((b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)^2) - RootSum[a + b*#1^4 + c*#1^8 & , (b^2*Log[Sqrt[x] - #1] - 10*a*c*Log[

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

+ c*#1^8 & , (8*b^4*Log[Sqrt[x] - #1] - 133*a*b^2*c*Log[Sqrt[x] - #1] + 260*a^2*c^2*Log[Sqrt[x] - #1] + 8*b^3*

c*Log[Sqrt[x] - #1]*#1^4 - 8*a*b*c^2*Log[Sqrt[x] - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ]/(64*a*c*(-b^2 + 4*a*c)^2)

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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^(9/2)/(c*x^4+b*x^2+a)^3,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^(9/2)/(c*x^4+b*x^2+a)^3,x, algorithm="giac")

[Out]

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

uation time: 191.51Unable to convert to real 1/4 Error: Bad Argument Value

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maple [C] time = 0.04, size = 244, normalized size = 0.46 \begin {gather*} -\frac {3 \left (8 \RootOf \left (c \,\textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+a \right )^{6} b c +\left (-20 a c -7 b^{2}\right ) \RootOf \left (c \,\textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+a \right )^{2}\right ) \ln \left (-\RootOf \left (c \,\textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+a \right )+\sqrt {x}\right )}{64 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \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 {3 b \,c^{2} x^{\frac {15}{2}}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {3 \left (4 a c -13 b^{2}\right ) c \,x^{\frac {11}{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {\left (28 a c +11 b^{2}\right ) b \,x^{\frac {7}{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {\left (20 a c +7 b^{2}\right ) a \,x^{\frac {3}{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^(9/2)/(c*x^4+b*x^2+a)^3,x)

[Out]

2*(-1/32*a*(20*a*c+7*b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(3/2)-1/32*b*(28*a*c+11*b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)

*x^(7/2)+3/32*(4*a*c-13*b^2)*c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(11/2)-3/4*b*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(15/

2))/(c*x^4+b*x^2+a)^2-3/64/(16*a^2*c^2-8*a*b^2*c+b^4)*sum((8*_R^6*b*c+(-20*a*c-7*b^2)*_R^2)/(2*_R^7*c+_R^3*b)*

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

________________________________________________________________________________________

maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {24 \, b c^{2} x^{\frac {15}{2}} + 3 \, {\left (13 \, b^{2} c - 4 \, a c^{2}\right )} x^{\frac {11}{2}} + {\left (11 \, b^{3} + 28 \, a b c\right )} x^{\frac {7}{2}} + {\left (7 \, a b^{2} + 20 \, a^{2} c\right )} x^{\frac {3}{2}}}{16 \, {\left ({\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{8} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{6} + a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{4} + 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x^{2}\right )}} - \int \frac {3 \, {\left (8 \, b c x^{\frac {5}{2}} - {\left (7 \, b^{2} + 20 \, a c\right )} \sqrt {x}\right )}}{32 \, {\left (a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{4} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{2}\right )}}\,{d x} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(9/2)/(c*x^4+b*x^2+a)^3,x, algorithm="maxima")

[Out]

-1/16*(24*b*c^2*x^(15/2) + 3*(13*b^2*c - 4*a*c^2)*x^(11/2) + (11*b^3 + 28*a*b*c)*x^(7/2) + (7*a*b^2 + 20*a^2*c

)*x^(3/2))/((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^8 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^6 + a^2*b^4 -

8*a^3*b^2*c + 16*a^4*c^2 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^4 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x^2) -

integrate(3/32*(8*b*c*x^(5/2) - (7*b^2 + 20*a*c)*sqrt(x))/(a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2 + (b^4*c - 8*a*b^2

*c^2 + 16*a^2*c^3)*x^4 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*x^2), x)

________________________________________________________________________________________

mupad [B] time = 7.66, size = 37678, normalized size = 70.69

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^(9/2)/(a + b*x^2 + c*x^4)^3,x)

[Out]

- atan(((((27*(5754585088*a*b^27*c^4 + 309622474381721600*a^14*b*c^17 - 161128382464*a^2*b^25*c^5 + 1626181992

448*a^3*b^23*c^6 - 3983582167040*a^4*b^21*c^7 - 56328496087040*a^5*b^19*c^8 + 557813172535296*a^6*b^17*c^9 - 1

961803621859328*a^7*b^15*c^10 + 715782069682176*a^8*b^13*c^11 + 15816474765557760*a^9*b^11*c^12 - 392965455767

14240*a^10*b^9*c^13 - 32756650414702592*a^11*b^7*c^14 + 300756012615335936*a^12*b^5*c^15 - 517069532217475072*

a^13*b^3*c^16))/(268435456*(b^28 + 268435456*a^14*c^14 + 1456*a^2*b^24*c^2 - 23296*a^3*b^22*c^3 + 256256*a^4*b

^20*c^4 - 2050048*a^5*b^18*c^5 + 12300288*a^6*b^16*c^6 - 56229888*a^7*b^14*c^7 + 196804608*a^8*b^12*c^8 - 5248

12288*a^9*b^10*c^9 + 1049624576*a^10*b^8*c^10 - 1526726656*a^11*b^6*c^11 + 1526726656*a^12*b^4*c^12 - 93952409

6*a^13*b^2*c^13 - 56*a*b^26*c)) - (9*x^(1/2)*((81*(2401*b^4*(-(4*a*c - b^2)^25)^(1/2) - 2401*b^29 - 7046430720

00*a^14*b*c^14 + 1323600*a^2*b^25*c^2 - 28243200*a^3*b^23*c^3 + 271415040*a^4*b^21*c^4 - 1437284352*a^5*b^19*c

^5 + 3989852160*a^6*b^17*c^6 - 2793799680*a^7*b^15*c^7 - 13327073280*a^8*b^13*c^8 + 19977994240*a^9*b^11*c^9 +

66059239424*a^10*b^9*c^10 - 143696855040*a^11*b^7*c^11 - 230770606080*a^12*b^5*c^12 + 887850270720*a^13*b^3*c

^13 + 10000*a^2*c^2*(-(4*a*c - b^2)^25)^(1/2) + 9400*a*b^27*c + 9400*a*b^2*c*(-(4*a*c - b^2)^25)^(1/2)))/(3355

4432*(a*b^40 + 1099511627776*a^21*c^20 - 80*a^2*b^38*c + 3040*a^3*b^36*c^2 - 72960*a^4*b^34*c^3 + 1240320*a^5*

b^32*c^4 - 15876096*a^6*b^30*c^5 + 158760960*a^7*b^28*c^6 - 1270087680*a^8*b^26*c^7 + 8255569920*a^9*b^24*c^8

- 44029706240*a^10*b^22*c^9 + 193730707456*a^11*b^20*c^10 - 704475299840*a^12*b^18*c^11 + 2113425899520*a^13*b

^16*c^12 - 5202279137280*a^14*b^14*c^13 + 10404558274560*a^15*b^12*c^14 - 16647293239296*a^16*b^10*c^15 + 2080

9116549120*a^17*b^8*c^16 - 19585050869760*a^18*b^6*c^17 + 13056700579840*a^19*b^4*c^18 - 5497558138880*a^20*b^

2*c^19)))^(1/4)*(822083584*a*b^26*c^4 - 14073748835532800*a^14*c^17 - 27950841856*a^2*b^24*c^5 + 399431958528*

a^3*b^22*c^6 - 2968896143360*a^4*b^20*c^7 + 10329396346880*a^5*b^18*c^8 + 6262062317568*a^6*b^16*c^9 - 2028598

95324672*a^7*b^14*c^10 + 658057709223936*a^8*b^12*c^11 + 346346162749440*a^9*b^10*c^12 - 8653156510597120*a^10

*b^8*c^13 + 28569710136131584*a^11*b^6*c^14 - 47076689854857216*a^12*b^4*c^15 + 40250921669623808*a^13*b^2*c^1

6))/(4194304*(b^24 + 16777216*a^12*c^12 + 1056*a^2*b^20*c^2 - 14080*a^3*b^18*c^3 + 126720*a^4*b^16*c^4 - 81100

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

69206016*a^10*b^4*c^10 - 50331648*a^11*b^2*c^11 - 48*a*b^22*c)))*((81*(2401*b^4*(-(4*a*c - b^2)^25)^(1/2) - 24

01*b^29 - 704643072000*a^14*b*c^14 + 1323600*a^2*b^25*c^2 - 28243200*a^3*b^23*c^3 + 271415040*a^4*b^21*c^4 - 1

437284352*a^5*b^19*c^5 + 3989852160*a^6*b^17*c^6 - 2793799680*a^7*b^15*c^7 - 13327073280*a^8*b^13*c^8 + 199779

94240*a^9*b^11*c^9 + 66059239424*a^10*b^9*c^10 - 143696855040*a^11*b^7*c^11 - 230770606080*a^12*b^5*c^12 + 887

850270720*a^13*b^3*c^13 + 10000*a^2*c^2*(-(4*a*c - b^2)^25)^(1/2) + 9400*a*b^27*c + 9400*a*b^2*c*(-(4*a*c - b^

2)^25)^(1/2)))/(33554432*(a*b^40 + 1099511627776*a^21*c^20 - 80*a^2*b^38*c + 3040*a^3*b^36*c^2 - 72960*a^4*b^3

4*c^3 + 1240320*a^5*b^32*c^4 - 15876096*a^6*b^30*c^5 + 158760960*a^7*b^28*c^6 - 1270087680*a^8*b^26*c^7 + 8255

569920*a^9*b^24*c^8 - 44029706240*a^10*b^22*c^9 + 193730707456*a^11*b^20*c^10 - 704475299840*a^12*b^18*c^11 +

2113425899520*a^13*b^16*c^12 - 5202279137280*a^14*b^14*c^13 + 10404558274560*a^15*b^12*c^14 - 16647293239296*a

^16*b^10*c^15 + 20809116549120*a^17*b^8*c^16 - 19585050869760*a^18*b^6*c^17 + 13056700579840*a^19*b^4*c^18 - 5

497558138880*a^20*b^2*c^19)))^(3/4) - (9*x^(1/2)*(200930625*a*b^13*c^5 - 3110400000*a^7*b*c^11 + 2093250600*a^

2*b^11*c^6 + 7523454960*a^3*b^9*c^7 + 10328580864*a^4*b^7*c^8 + 2354261760*a^5*b^5*c^9 - 5453568000*a^6*b^3*c^

10))/(4194304*(b^24 + 16777216*a^12*c^12 + 1056*a^2*b^20*c^2 - 14080*a^3*b^18*c^3 + 126720*a^4*b^16*c^4 - 8110

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

69206016*a^10*b^4*c^10 - 50331648*a^11*b^2*c^11 - 48*a*b^22*c)))*((81*(2401*b^4*(-(4*a*c - b^2)^25)^(1/2) - 2

401*b^29 - 704643072000*a^14*b*c^14 + 1323600*a^2*b^25*c^2 - 28243200*a^3*b^23*c^3 + 271415040*a^4*b^21*c^4 -

1437284352*a^5*b^19*c^5 + 3989852160*a^6*b^17*c^6 - 2793799680*a^7*b^15*c^7 - 13327073280*a^8*b^13*c^8 + 19977

994240*a^9*b^11*c^9 + 66059239424*a^10*b^9*c^10 - 143696855040*a^11*b^7*c^11 - 230770606080*a^12*b^5*c^12 + 88

7850270720*a^13*b^3*c^13 + 10000*a^2*c^2*(-(4*a*c - b^2)^25)^(1/2) + 9400*a*b^27*c + 9400*a*b^2*c*(-(4*a*c - b

^2)^25)^(1/2)))/(33554432*(a*b^40 + 1099511627776*a^21*c^20 - 80*a^2*b^38*c + 3040*a^3*b^36*c^2 - 72960*a^4*b^

34*c^3 + 1240320*a^5*b^32*c^4 - 15876096*a^6*b^30*c^5 + 158760960*a^7*b^28*c^6 - 1270087680*a^8*b^26*c^7 + 825

5569920*a^9*b^24*c^8 - 44029706240*a^10*b^22*c^9 + 193730707456*a^11*b^20*c^10 - 704475299840*a^12*b^18*c^11 +

2113425899520*a^13*b^16*c^12 - 5202279137280*a^14*b^14*c^13 + 10404558274560*a^15*b^12*c^14 - 16647293239296*

a^16*b^10*c^15 + 20809116549120*a^17*b^8*c^16 - 19585050869760*a^18*b^6*c^17 + 13056700579840*a^19*b^4*c^18 -

5497558138880*a^20*b^2*c^19)))^(1/4)*1i - (((27*(5754585088*a*b^27*c^4 + 309622474381721600*a^14*b*c^17 - 1611

28382464*a^2*b^25*c^5 + 1626181992448*a^3*b^23*c^6 - 3983582167040*a^4*b^21*c^7 - 56328496087040*a^5*b^19*c^8

+ 557813172535296*a^6*b^17*c^9 - 1961803621859328*a^7*b^15*c^10 + 715782069682176*a^8*b^13*c^11 + 158164747655

57760*a^9*b^11*c^12 - 39296545576714240*a^10*b^9*c^13 - 32756650414702592*a^11*b^7*c^14 + 300756012615335936*a

^12*b^5*c^15 - 517069532217475072*a^13*b^3*c^16))/(268435456*(b^28 + 268435456*a^14*c^14 + 1456*a^2*b^24*c^2 -

23296*a^3*b^22*c^3 + 256256*a^4*b^20*c^4 - 2050048*a^5*b^18*c^5 + 12300288*a^6*b^16*c^6 - 56229888*a^7*b^14*c

^7 + 196804608*a^8*b^12*c^8 - 524812288*a^9*b^10*c^9 + 1049624576*a^10*b^8*c^10 - 1526726656*a^11*b^6*c^11 + 1

526726656*a^12*b^4*c^12 - 939524096*a^13*b^2*c^13 - 56*a*b^26*c)) + (9*x^(1/2)*((81*(2401*b^4*(-(4*a*c - b^2)^

25)^(1/2) - 2401*b^29 - 704643072000*a^14*b*c^14 + 1323600*a^2*b^25*c^2 - 28243200*a^3*b^23*c^3 + 271415040*a^

4*b^21*c^4 - 1437284352*a^5*b^19*c^5 + 3989852160*a^6*b^17*c^6 - 2793799680*a^7*b^15*c^7 - 13327073280*a^8*b^1

3*c^8 + 19977994240*a^9*b^11*c^9 + 66059239424*a^10*b^9*c^10 - 143696855040*a^11*b^7*c^11 - 230770606080*a^12*

b^5*c^12 + 887850270720*a^13*b^3*c^13 + 10000*a^2*c^2*(-(4*a*c - b^2)^25)^(1/2) + 9400*a*b^27*c + 9400*a*b^2*c

*(-(4*a*c - b^2)^25)^(1/2)))/(33554432*(a*b^40 + 1099511627776*a^21*c^20 - 80*a^2*b^38*c + 3040*a^3*b^36*c^2 -

72960*a^4*b^34*c^3 + 1240320*a^5*b^32*c^4 - 15876096*a^6*b^30*c^5 + 158760960*a^7*b^28*c^6 - 1270087680*a^8*b

^26*c^7 + 8255569920*a^9*b^24*c^8 - 44029706240*a^10*b^22*c^9 + 193730707456*a^11*b^20*c^10 - 704475299840*a^1

2*b^18*c^11 + 2113425899520*a^13*b^16*c^12 - 5202279137280*a^14*b^14*c^13 + 10404558274560*a^15*b^12*c^14 - 16

647293239296*a^16*b^10*c^15 + 20809116549120*a^17*b^8*c^16 - 19585050869760*a^18*b^6*c^17 + 13056700579840*a^1

9*b^4*c^18 - 5497558138880*a^20*b^2*c^19)))^(1/4)*(822083584*a*b^26*c^4 - 14073748835532800*a^14*c^17 - 279508

41856*a^2*b^24*c^5 + 399431958528*a^3*b^22*c^6 - 2968896143360*a^4*b^20*c^7 + 10329396346880*a^5*b^18*c^8 + 62

62062317568*a^6*b^16*c^9 - 202859895324672*a^7*b^14*c^10 + 658057709223936*a^8*b^12*c^11 + 346346162749440*a^9

*b^10*c^12 - 8653156510597120*a^10*b^8*c^13 + 28569710136131584*a^11*b^6*c^14 - 47076689854857216*a^12*b^4*c^1

5 + 40250921669623808*a^13*b^2*c^16))/(4194304*(b^24 + 16777216*a^12*c^12 + 1056*a^2*b^20*c^2 - 14080*a^3*b^18

*c^3 + 126720*a^4*b^16*c^4 - 811008*a^5*b^14*c^5 + 3784704*a^6*b^12*c^6 - 12976128*a^7*b^10*c^7 + 32440320*a^8

*b^8*c^8 - 57671680*a^9*b^6*c^9 + 69206016*a^10*b^4*c^10 - 50331648*a^11*b^2*c^11 - 48*a*b^22*c)))*((81*(2401*

b^4*(-(4*a*c - b^2)^25)^(1/2) - 2401*b^29 - 704643072000*a^14*b*c^14 + 1323600*a^2*b^25*c^2 - 28243200*a^3*b^2

3*c^3 + 271415040*a^4*b^21*c^4 - 1437284352*a^5*b^19*c^5 + 3989852160*a^6*b^17*c^6 - 2793799680*a^7*b^15*c^7 -

13327073280*a^8*b^13*c^8 + 19977994240*a^9*b^11*c^9 + 66059239424*a^10*b^9*c^10 - 143696855040*a^11*b^7*c^11

- 230770606080*a^12*b^5*c^12 + 887850270720*a^13*b^3*c^13 + 10000*a^2*c^2*(-(4*a*c - b^2)^25)^(1/2) + 9400*a*b

^27*c + 9400*a*b^2*c*(-(4*a*c - b^2)^25)^(1/2)))/(33554432*(a*b^40 + 1099511627776*a^21*c^20 - 80*a^2*b^38*c +

3040*a^3*b^36*c^2 - 72960*a^4*b^34*c^3 + 1240320*a^5*b^32*c^4 - 15876096*a^6*b^30*c^5 + 158760960*a^7*b^28*c^

6 - 1270087680*a^8*b^26*c^7 + 8255569920*a^9*b^24*c^8 - 44029706240*a^10*b^22*c^9 + 193730707456*a^11*b^20*c^1

0 - 704475299840*a^12*b^18*c^11 + 2113425899520*a^13*b^16*c^12 - 5202279137280*a^14*b^14*c^13 + 10404558274560

*a^15*b^12*c^14 - 16647293239296*a^16*b^10*c^15 + 20809116549120*a^17*b^8*c^16 - 19585050869760*a^18*b^6*c^17

+ 13056700579840*a^19*b^4*c^18 - 5497558138880*a^20*b^2*c^19)))^(3/4) + (9*x^(1/2)*(200930625*a*b^13*c^5 - 311

0400000*a^7*b*c^11 + 2093250600*a^2*b^11*c^6 + 7523454960*a^3*b^9*c^7 + 10328580864*a^4*b^7*c^8 + 2354261760*a

^5*b^5*c^9 - 5453568000*a^6*b^3*c^10))/(4194304*(b^24 + 16777216*a^12*c^12 + 1056*a^2*b^20*c^2 - 14080*a^3*b^1

8*c^3 + 126720*a^4*b^16*c^4 - 811008*a^5*b^14*c^5 + 3784704*a^6*b^12*c^6 - 12976128*a^7*b^10*c^7 + 32440320*a^

8*b^8*c^8 - 57671680*a^9*b^6*c^9 + 69206016*a^10*b^4*c^10 - 50331648*a^11*b^2*c^11 - 48*a*b^22*c)))*((81*(2401

*b^4*(-(4*a*c - b^2)^25)^(1/2) - 2401*b^29 - 704643072000*a^14*b*c^14 + 1323600*a^2*b^25*c^2 - 28243200*a^3*b^

23*c^3 + 271415040*a^4*b^21*c^4 - 1437284352*a^5*b^19*c^5 + 3989852160*a^6*b^17*c^6 - 2793799680*a^7*b^15*c^7

- 13327073280*a^8*b^13*c^8 + 19977994240*a^9*b^11*c^9 + 66059239424*a^10*b^9*c^10 - 143696855040*a^11*b^7*c^11

- 230770606080*a^12*b^5*c^12 + 887850270720*a^13*b^3*c^13 + 10000*a^2*c^2*(-(4*a*c - b^2)^25)^(1/2) + 9400*a*

b^27*c + 9400*a*b^2*c*(-(4*a*c - b^2)^25)^(1/2)))/(33554432*(a*b^40 + 1099511627776*a^21*c^20 - 80*a^2*b^38*c

+ 3040*a^3*b^36*c^2 - 72960*a^4*b^34*c^3 + 1240320*a^5*b^32*c^4 - 15876096*a^6*b^30*c^5 + 158760960*a^7*b^28*c

^6 - 1270087680*a^8*b^26*c^7 + 8255569920*a^9*b^24*c^8 - 44029706240*a^10*b^22*c^9 + 193730707456*a^11*b^20*c^

10 - 704475299840*a^12*b^18*c^11 + 2113425899520*a^13*b^16*c^12 - 5202279137280*a^14*b^14*c^13 + 1040455827456

0*a^15*b^12*c^14 - 16647293239296*a^16*b^10*c^15 + 20809116549120*a^17*b^8*c^16 - 19585050869760*a^18*b^6*c^17

+ 13056700579840*a^19*b^4*c^18 - 5497558138880*a^20*b^2*c^19)))^(1/4)*1i)/((27*(103680000000*a^8*c^12 + 14065

14375*a*b^14*c^5 + 22129159500*a^2*b^12*c^6 + 140297799600*a^3*b^10*c^7 + 460920922560*a^4*b^8*c^8 + 844743271

680*a^5*b^6*c^9 + 869387904000*a^6*b^4*c^10 + 469670400000*a^7*b^2*c^11))/(134217728*(b^28 + 268435456*a^14*c^

14 + 1456*a^2*b^24*c^2 - 23296*a^3*b^22*c^3 + 256256*a^4*b^20*c^4 - 2050048*a^5*b^18*c^5 + 12300288*a^6*b^16*c

^6 - 56229888*a^7*b^14*c^7 + 196804608*a^8*b^12*c^8 - 524812288*a^9*b^10*c^9 + 1049624576*a^10*b^8*c^10 - 1526

726656*a^11*b^6*c^11 + 1526726656*a^12*b^4*c^12 - 939524096*a^13*b^2*c^13 - 56*a*b^26*c)) + (((27*(5754585088*

a*b^27*c^4 + 309622474381721600*a^14*b*c^17 - 161128382464*a^2*b^25*c^5 + 1626181992448*a^3*b^23*c^6 - 3983582

167040*a^4*b^21*c^7 - 56328496087040*a^5*b^19*c^8 + 557813172535296*a^6*b^17*c^9 - 1961803621859328*a^7*b^15*c

^10 + 715782069682176*a^8*b^13*c^11 + 15816474765557760*a^9*b^11*c^12 - 39296545576714240*a^10*b^9*c^13 - 3275

6650414702592*a^11*b^7*c^14 + 300756012615335936*a^12*b^5*c^15 - 517069532217475072*a^13*b^3*c^16))/(268435456

*(b^28 + 268435456*a^14*c^14 + 1456*a^2*b^24*c^2 - 23296*a^3*b^22*c^3 + 256256*a^4*b^20*c^4 - 2050048*a^5*b^18

*c^5 + 12300288*a^6*b^16*c^6 - 56229888*a^7*b^14*c^7 + 196804608*a^8*b^12*c^8 - 524812288*a^9*b^10*c^9 + 10496

24576*a^10*b^8*c^10 - 1526726656*a^11*b^6*c^11 + 1526726656*a^12*b^4*c^12 - 939524096*a^13*b^2*c^13 - 56*a*b^2

6*c)) - (9*x^(1/2)*((81*(2401*b^4*(-(4*a*c - b^2)^25)^(1/2) - 2401*b^29 - 704643072000*a^14*b*c^14 + 1323600*a

^2*b^25*c^2 - 28243200*a^3*b^23*c^3 + 271415040*a^4*b^21*c^4 - 1437284352*a^5*b^19*c^5 + 3989852160*a^6*b^17*c

^6 - 2793799680*a^7*b^15*c^7 - 13327073280*a^8*b^13*c^8 + 19977994240*a^9*b^11*c^9 + 66059239424*a^10*b^9*c^10

- 143696855040*a^11*b^7*c^11 - 230770606080*a^12*b^5*c^12 + 887850270720*a^13*b^3*c^13 + 10000*a^2*c^2*(-(4*a

*c - b^2)^25)^(1/2) + 9400*a*b^27*c + 9400*a*b^2*c*(-(4*a*c - b^2)^25)^(1/2)))/(33554432*(a*b^40 + 10995116277

76*a^21*c^20 - 80*a^2*b^38*c + 3040*a^3*b^36*c^2 - 72960*a^4*b^34*c^3 + 1240320*a^5*b^32*c^4 - 15876096*a^6*b^

30*c^5 + 158760960*a^7*b^28*c^6 - 1270087680*a^8*b^26*c^7 + 8255569920*a^9*b^24*c^8 - 44029706240*a^10*b^22*c^

9 + 193730707456*a^11*b^20*c^10 - 704475299840*a^12*b^18*c^11 + 2113425899520*a^13*b^16*c^12 - 5202279137280*a

^14*b^14*c^13 + 10404558274560*a^15*b^12*c^14 - 16647293239296*a^16*b^10*c^15 + 20809116549120*a^17*b^8*c^16 -

19585050869760*a^18*b^6*c^17 + 13056700579840*a^19*b^4*c^18 - 5497558138880*a^20*b^2*c^19)))^(1/4)*(822083584

*a*b^26*c^4 - 14073748835532800*a^14*c^17 - 27950841856*a^2*b^24*c^5 + 399431958528*a^3*b^22*c^6 - 29688961433

60*a^4*b^20*c^7 + 10329396346880*a^5*b^18*c^8 + 6262062317568*a^6*b^16*c^9 - 202859895324672*a^7*b^14*c^10 + 6

58057709223936*a^8*b^12*c^11 + 346346162749440*a^9*b^10*c^12 - 8653156510597120*a^10*b^8*c^13 + 28569710136131

584*a^11*b^6*c^14 - 47076689854857216*a^12*b^4*c^15 + 40250921669623808*a^13*b^2*c^16))/(4194304*(b^24 + 16777

216*a^12*c^12 + 1056*a^2*b^20*c^2 - 14080*a^3*b^18*c^3 + 126720*a^4*b^16*c^4 - 811008*a^5*b^14*c^5 + 3784704*a

^6*b^12*c^6 - 12976128*a^7*b^10*c^7 + 32440320*a^8*b^8*c^8 - 57671680*a^9*b^6*c^9 + 69206016*a^10*b^4*c^10 - 5

0331648*a^11*b^2*c^11 - 48*a*b^22*c)))*((81*(2401*b^4*(-(4*a*c - b^2)^25)^(1/2) - 2401*b^29 - 704643072000*a^1

4*b*c^14 + 1323600*a^2*b^25*c^2 - 28243200*a^3*b^23*c^3 + 271415040*a^4*b^21*c^4 - 1437284352*a^5*b^19*c^5 + 3

989852160*a^6*b^17*c^6 - 2793799680*a^7*b^15*c^7 - 13327073280*a^8*b^13*c^8 + 19977994240*a^9*b^11*c^9 + 66059

239424*a^10*b^9*c^10 - 143696855040*a^11*b^7*c^11 - 230770606080*a^12*b^5*c^12 + 887850270720*a^13*b^3*c^13 +

10000*a^2*c^2*(-(4*a*c - b^2)^25)^(1/2) + 9400*a*b^27*c + 9400*a*b^2*c*(-(4*a*c - b^2)^25)^(1/2)))/(33554432*(

a*b^40 + 1099511627776*a^21*c^20 - 80*a^2*b^38*c + 3040*a^3*b^36*c^2 - 72960*a^4*b^34*c^3 + 1240320*a^5*b^32*c

^4 - 15876096*a^6*b^30*c^5 + 158760960*a^7*b^28*c^6 - 1270087680*a^8*b^26*c^7 + 8255569920*a^9*b^24*c^8 - 4402

9706240*a^10*b^22*c^9 + 193730707456*a^11*b^20*c^10 - 704475299840*a^12*b^18*c^11 + 2113425899520*a^13*b^16*c^

12 - 5202279137280*a^14*b^14*c^13 + 10404558274560*a^15*b^12*c^14 - 16647293239296*a^16*b^10*c^15 + 2080911654

9120*a^17*b^8*c^16 - 19585050869760*a^18*b^6*c^17 + 13056700579840*a^19*b^4*c^18 - 5497558138880*a^20*b^2*c^19

)))^(3/4) - (9*x^(1/2)*(200930625*a*b^13*c^5 - 3110400000*a^7*b*c^11 + 2093250600*a^2*b^11*c^6 + 7523454960*a^

3*b^9*c^7 + 10328580864*a^4*b^7*c^8 + 2354261760*a^5*b^5*c^9 - 5453568000*a^6*b^3*c^10))/(4194304*(b^24 + 1677

7216*a^12*c^12 + 1056*a^2*b^20*c^2 - 14080*a^3*b^18*c^3 + 126720*a^4*b^16*c^4 - 811008*a^5*b^14*c^5 + 3784704*

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

50331648*a^11*b^2*c^11 - 48*a*b^22*c)))*((81*(2401*b^4*(-(4*a*c - b^2)^25)^(1/2) - 2401*b^29 - 704643072000*a^

14*b*c^14 + 1323600*a^2*b^25*c^2 - 28243200*a^3*b^23*c^3 + 271415040*a^4*b^21*c^4 - 1437284352*a^5*b^19*c^5 +

3989852160*a^6*b^17*c^6 - 2793799680*a^7*b^15*c^7 - 13327073280*a^8*b^13*c^8 + 19977994240*a^9*b^11*c^9 + 6605

9239424*a^10*b^9*c^10 - 143696855040*a^11*b^7*c^11 - 230770606080*a^12*b^5*c^12 + 887850270720*a^13*b^3*c^13 +

10000*a^2*c^2*(-(4*a*c - b^2)^25)^(1/2) + 9400*a*b^27*c + 9400*a*b^2*c*(-(4*a*c - b^2)^25)^(1/2)))/(33554432*

(a*b^40 + 1099511627776*a^21*c^20 - 80*a^2*b^38*c + 3040*a^3*b^36*c^2 - 72960*a^4*b^34*c^3 + 1240320*a^5*b^32*

c^4 - 15876096*a^6*b^30*c^5 + 158760960*a^7*b^28*c^6 - 1270087680*a^8*b^26*c^7 + 8255569920*a^9*b^24*c^8 - 440

29706240*a^10*b^22*c^9 + 193730707456*a^11*b^20*c^10 - 704475299840*a^12*b^18*c^11 + 2113425899520*a^13*b^16*c

^12 - 5202279137280*a^14*b^14*c^13 + 10404558274560*a^15*b^12*c^14 - 16647293239296*a^16*b^10*c^15 + 208091165

49120*a^17*b^8*c^16 - 19585050869760*a^18*b^6*c^17 + 13056700579840*a^19*b^4*c^18 - 5497558138880*a^20*b^2*c^1

9)))^(1/4) + (((27*(5754585088*a*b^27*c^4 + 309622474381721600*a^14*b*c^17 - 161128382464*a^2*b^25*c^5 + 16261

81992448*a^3*b^23*c^6 - 3983582167040*a^4*b^21*c^7 - 56328496087040*a^5*b^19*c^8 + 557813172535296*a^6*b^17*c^

9 - 1961803621859328*a^7*b^15*c^10 + 715782069682176*a^8*b^13*c^11 + 15816474765557760*a^9*b^11*c^12 - 3929654

5576714240*a^10*b^9*c^13 - 32756650414702592*a^11*b^7*c^14 + 300756012615335936*a^12*b^5*c^15 - 51706953221747

5072*a^13*b^3*c^16))/(268435456*(b^28 + 268435456*a^14*c^14 + 1456*a^2*b^24*c^2 - 23296*a^3*b^22*c^3 + 256256*

a^4*b^20*c^4 - 2050048*a^5*b^18*c^5 + 12300288*a^6*b^16*c^6 - 56229888*a^7*b^14*c^7 + 196804608*a^8*b^12*c^8 -

524812288*a^9*b^10*c^9 + 1049624576*a^10*b^8*c^10 - 1526726656*a^11*b^6*c^11 + 1526726656*a^12*b^4*c^12 - 939

524096*a^13*b^2*c^13 - 56*a*b^26*c)) + (9*x^(1/2)*((81*(2401*b^4*(-(4*a*c - b^2)^25)^(1/2) - 2401*b^29 - 70464

3072000*a^14*b*c^14 + 1323600*a^2*b^25*c^2 - 28243200*a^3*b^23*c^3 + 271415040*a^4*b^21*c^4 - 1437284352*a^5*b

^19*c^5 + 3989852160*a^6*b^17*c^6 - 2793799680*a^7*b^15*c^7 - 13327073280*a^8*b^13*c^8 + 19977994240*a^9*b^11*

c^9 + 66059239424*a^10*b^9*c^10 - 143696855040*a^11*b^7*c^11 - 230770606080*a^12*b^5*c^12 + 887850270720*a^13*

b^3*c^13 + 10000*a^2*c^2*(-(4*a*c - b^2)^25)^(1/2) + 9400*a*b^27*c + 9400*a*b^2*c*(-(4*a*c - b^2)^25)^(1/2)))/

(33554432*(a*b^40 + 1099511627776*a^21*c^20 - 80*a^2*b^38*c + 3040*a^3*b^36*c^2 - 72960*a^4*b^34*c^3 + 1240320

*a^5*b^32*c^4 - 15876096*a^6*b^30*c^5 + 158760960*a^7*b^28*c^6 - 1270087680*a^8*b^26*c^7 + 8255569920*a^9*b^24

*c^8 - 44029706240*a^10*b^22*c^9 + 193730707456*a^11*b^20*c^10 - 704475299840*a^12*b^18*c^11 + 2113425899520*a

^13*b^16*c^12 - 5202279137280*a^14*b^14*c^13 + 10404558274560*a^15*b^12*c^14 - 16647293239296*a^16*b^10*c^15 +

20809116549120*a^17*b^8*c^16 - 19585050869760*a^18*b^6*c^17 + 13056700579840*a^19*b^4*c^18 - 5497558138880*a^

20*b^2*c^19)))^(1/4)*(822083584*a*b^26*c^4 - 14073748835532800*a^14*c^17 - 27950841856*a^2*b^24*c^5 + 39943195

8528*a^3*b^22*c^6 - 2968896143360*a^4*b^20*c^7 + 10329396346880*a^5*b^18*c^8 + 6262062317568*a^6*b^16*c^9 - 20

2859895324672*a^7*b^14*c^10 + 658057709223936*a^8*b^12*c^11 + 346346162749440*a^9*b^10*c^12 - 8653156510597120

*a^10*b^8*c^13 + 28569710136131584*a^11*b^6*c^14 - 47076689854857216*a^12*b^4*c^15 + 40250921669623808*a^13*b^

2*c^16))/(4194304*(b^24 + 16777216*a^12*c^12 + 1056*a^2*b^20*c^2 - 14080*a^3*b^18*c^3 + 126720*a^4*b^16*c^4 -

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

^9 + 69206016*a^10*b^4*c^10 - 50331648*a^11*b^2*c^11 - 48*a*b^22*c)))*((81*(2401*b^4*(-(4*a*c - b^2)^25)^(1/2)

- 2401*b^29 - 704643072000*a^14*b*c^14 + 1323600*a^2*b^25*c^2 - 28243200*a^3*b^23*c^3 + 271415040*a^4*b^21*c^

4 - 1437284352*a^5*b^19*c^5 + 3989852160*a^6*b^17*c^6 - 2793799680*a^7*b^15*c^7 - 13327073280*a^8*b^13*c^8 + 1

9977994240*a^9*b^11*c^9 + 66059239424*a^10*b^9*c^10 - 143696855040*a^11*b^7*c^11 - 230770606080*a^12*b^5*c^12

+ 887850270720*a^13*b^3*c^13 + 10000*a^2*c^2*(-(4*a*c - b^2)^25)^(1/2) + 9400*a*b^27*c + 9400*a*b^2*c*(-(4*a*c

- b^2)^25)^(1/2)))/(33554432*(a*b^40 + 1099511627776*a^21*c^20 - 80*a^2*b^38*c + 3040*a^3*b^36*c^2 - 72960*a^

4*b^34*c^3 + 1240320*a^5*b^32*c^4 - 15876096*a^6*b^30*c^5 + 158760960*a^7*b^28*c^6 - 1270087680*a^8*b^26*c^7 +

8255569920*a^9*b^24*c^8 - 44029706240*a^10*b^22*c^9 + 193730707456*a^11*b^20*c^10 - 704475299840*a^12*b^18*c^

11 + 2113425899520*a^13*b^16*c^12 - 5202279137280*a^14*b^14*c^13 + 10404558274560*a^15*b^12*c^14 - 16647293239

296*a^16*b^10*c^15 + 20809116549120*a^17*b^8*c^16 - 19585050869760*a^18*b^6*c^17 + 13056700579840*a^19*b^4*c^1

8 - 5497558138880*a^20*b^2*c^19)))^(3/4) + (9*x^(1/2)*(200930625*a*b^13*c^5 - 3110400000*a^7*b*c^11 + 20932506

00*a^2*b^11*c^6 + 7523454960*a^3*b^9*c^7 + 10328580864*a^4*b^7*c^8 + 2354261760*a^5*b^5*c^9 - 5453568000*a^6*b

^3*c^10))/(4194304*(b^24 + 16777216*a^12*c^12 + 1056*a^2*b^20*c^2 - 14080*a^3*b^18*c^3 + 126720*a^4*b^16*c^4 -

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

c^9 + 69206016*a^10*b^4*c^10 - 50331648*a^11*b^2*c^11 - 48*a*b^22*c)))*((81*(2401*b^4*(-(4*a*c - b^2)^25)^(1/2

) - 2401*b^29 - 704643072000*a^14*b*c^14 + 1323600*a^2*b^25*c^2 - 28243200*a^3*b^23*c^3 + 271415040*a^4*b^21*c

^4 - 1437284352*a^5*b^19*c^5 + 3989852160*a^6*b^17*c^6 - 2793799680*a^7*b^15*c^7 - 13327073280*a^8*b^13*c^8 +

19977994240*a^9*b^11*c^9 + 66059239424*a^10*b^9*c^10 - 143696855040*a^11*b^7*c^11 - 230770606080*a^12*b^5*c^12

+ 887850270720*a^13*b^3*c^13 + 10000*a^2*c^2*(-(4*a*c - b^2)^25)^(1/2) + 9400*a*b^27*c + 9400*a*b^2*c*(-(4*a*

c - b^2)^25)^(1/2)))/(33554432*(a*b^40 + 1099511627776*a^21*c^20 - 80*a^2*b^38*c + 3040*a^3*b^36*c^2 - 72960*a

^4*b^34*c^3 + 1240320*a^5*b^32*c^4 - 15876096*a^6*b^30*c^5 + 158760960*a^7*b^28*c^6 - 1270087680*a^8*b^26*c^7

+ 8255569920*a^9*b^24*c^8 - 44029706240*a^10*b^22*c^9 + 193730707456*a^11*b^20*c^10 - 704475299840*a^12*b^18*c

^11 + 2113425899520*a^13*b^16*c^12 - 5202279137280*a^14*b^14*c^13 + 10404558274560*a^15*b^12*c^14 - 1664729323

9296*a^16*b^10*c^15 + 20809116549120*a^17*b^8*c^16 - 19585050869760*a^18*b^6*c^17 + 13056700579840*a^19*b^4*c^

18 - 5497558138880*a^20*b^2*c^19)))^(1/4)))*((81*(2401*b^4*(-(4*a*c - b^2)^25)^(1/2) - 2401*b^29 - 70464307200

0*a^14*b*c^14 + 1323600*a^2*b^25*c^2 - 28243200*a^3*b^23*c^3 + 271415040*a^4*b^21*c^4 - 1437284352*a^5*b^19*c^

5 + 3989852160*a^6*b^17*c^6 - 2793799680*a^7*b^15*c^7 - 13327073280*a^8*b^13*c^8 + 19977994240*a^9*b^11*c^9 +

66059239424*a^10*b^9*c^10 - 143696855040*a^11*b^7*c^11 - 230770606080*a^12*b^5*c^12 + 887850270720*a^13*b^3*c^

13 + 10000*a^2*c^2*(-(4*a*c - b^2)^25)^(1/2) + 9400*a*b^27*c + 9400*a*b^2*c*(-(4*a*c - b^2)^25)^(1/2)))/(33554

432*(a*b^40 + 1099511627776*a^21*c^20 - 80*a^2*b^38*c + 3040*a^3*b^36*c^2 - 72960*a^4*b^34*c^3 + 1240320*a^5*b

^32*c^4 - 15876096*a^6*b^30*c^5 + 158760960*a^7*b^28*c^6 - 1270087680*a^8*b^26*c^7 + 8255569920*a^9*b^24*c^8 -

44029706240*a^10*b^22*c^9 + 193730707456*a^11*b^20*c^10 - 704475299840*a^12*b^18*c^11 + 2113425899520*a^13*b^

16*c^12 - 5202279137280*a^14*b^14*c^13 + 10404558274560*a^15*b^12*c^14 - 16647293239296*a^16*b^10*c^15 + 20809

116549120*a^17*b^8*c^16 - 19585050869760*a^18*b^6*c^17 + 13056700579840*a^19*b^4*c^18 - 5497558138880*a^20*b^2

*c^19)))^(1/4)*2i - atan(((((27*(5754585088*a*b^27*c^4 + 309622474381721600*a^14*b*c^17 - 161128382464*a^2*b^2

5*c^5 + 1626181992448*a^3*b^23*c^6 - 3983582167040*a^4*b^21*c^7 - 56328496087040*a^5*b^19*c^8 + 55781317253529

6*a^6*b^17*c^9 - 1961803621859328*a^7*b^15*c^10 + 715782069682176*a^8*b^13*c^11 + 15816474765557760*a^9*b^11*c

^12 - 39296545576714240*a^10*b^9*c^13 - 32756650414702592*a^11*b^7*c^14 + 300756012615335936*a^12*b^5*c^15 - 5

17069532217475072*a^13*b^3*c^16))/(268435456*(b^28 + 268435456*a^14*c^14 + 1456*a^2*b^24*c^2 - 23296*a^3*b^22*

c^3 + 256256*a^4*b^20*c^4 - 2050048*a^5*b^18*c^5 + 12300288*a^6*b^16*c^6 - 56229888*a^7*b^14*c^7 + 196804608*a

^8*b^12*c^8 - 524812288*a^9*b^10*c^9 + 1049624576*a^10*b^8*c^10 - 1526726656*a^11*b^6*c^11 + 1526726656*a^12*b

^4*c^12 - 939524096*a^13*b^2*c^13 - 56*a*b^26*c)) - (9*x^(1/2)*(-(81*(2401*b^29 + 2401*b^4*(-(4*a*c - b^2)^25)

^(1/2) + 704643072000*a^14*b*c^14 - 1323600*a^2*b^25*c^2 + 28243200*a^3*b^23*c^3 - 271415040*a^4*b^21*c^4 + 14

37284352*a^5*b^19*c^5 - 3989852160*a^6*b^17*c^6 + 2793799680*a^7*b^15*c^7 + 13327073280*a^8*b^13*c^8 - 1997799

4240*a^9*b^11*c^9 - 66059239424*a^10*b^9*c^10 + 143696855040*a^11*b^7*c^11 + 230770606080*a^12*b^5*c^12 - 8878

50270720*a^13*b^3*c^13 + 10000*a^2*c^2*(-(4*a*c - b^2)^25)^(1/2) - 9400*a*b^27*c + 9400*a*b^2*c*(-(4*a*c - b^2

)^25)^(1/2)))/(33554432*(a*b^40 + 1099511627776*a^21*c^20 - 80*a^2*b^38*c + 3040*a^3*b^36*c^2 - 72960*a^4*b^34

*c^3 + 1240320*a^5*b^32*c^4 - 15876096*a^6*b^30*c^5 + 158760960*a^7*b^28*c^6 - 1270087680*a^8*b^26*c^7 + 82555

69920*a^9*b^24*c^8 - 44029706240*a^10*b^22*c^9 + 193730707456*a^11*b^20*c^10 - 704475299840*a^12*b^18*c^11 + 2

113425899520*a^13*b^16*c^12 - 5202279137280*a^14*b^14*c^13 + 10404558274560*a^15*b^12*c^14 - 16647293239296*a^

16*b^10*c^15 + 20809116549120*a^17*b^8*c^16 - 19585050869760*a^18*b^6*c^17 + 13056700579840*a^19*b^4*c^18 - 54

97558138880*a^20*b^2*c^19)))^(1/4)*(822083584*a*b^26*c^4 - 14073748835532800*a^14*c^17 - 27950841856*a^2*b^24*

c^5 + 399431958528*a^3*b^22*c^6 - 2968896143360*a^4*b^20*c^7 + 10329396346880*a^5*b^18*c^8 + 6262062317568*a^6

*b^16*c^9 - 202859895324672*a^7*b^14*c^10 + 658057709223936*a^8*b^12*c^11 + 346346162749440*a^9*b^10*c^12 - 86

53156510597120*a^10*b^8*c^13 + 28569710136131584*a^11*b^6*c^14 - 47076689854857216*a^12*b^4*c^15 + 40250921669

623808*a^13*b^2*c^16))/(4194304*(b^24 + 16777216*a^12*c^12 + 1056*a^2*b^20*c^2 - 14080*a^3*b^18*c^3 + 126720*a

^4*b^16*c^4 - 811008*a^5*b^14*c^5 + 3784704*a^6*b^12*c^6 - 12976128*a^7*b^10*c^7 + 32440320*a^8*b^8*c^8 - 5767

1680*a^9*b^6*c^9 + 69206016*a^10*b^4*c^10 - 50331648*a^11*b^2*c^11 - 48*a*b^22*c)))*(-(81*(2401*b^29 + 2401*b^

4*(-(4*a*c - b^2)^25)^(1/2) + 704643072000*a^14*b*c^14 - 1323600*a^2*b^25*c^2 + 28243200*a^3*b^23*c^3 - 271415

040*a^4*b^21*c^4 + 1437284352*a^5*b^19*c^5 - 3989852160*a^6*b^17*c^6 + 2793799680*a^7*b^15*c^7 + 13327073280*a

^8*b^13*c^8 - 19977994240*a^9*b^11*c^9 - 66059239424*a^10*b^9*c^10 + 143696855040*a^11*b^7*c^11 + 230770606080

*a^12*b^5*c^12 - 887850270720*a^13*b^3*c^13 + 10000*a^2*c^2*(-(4*a*c - b^2)^25)^(1/2) - 9400*a*b^27*c + 9400*a

*b^2*c*(-(4*a*c - b^2)^25)^(1/2)))/(33554432*(a*b^40 + 1099511627776*a^21*c^20 - 80*a^2*b^38*c + 3040*a^3*b^36

*c^2 - 72960*a^4*b^34*c^3 + 1240320*a^5*b^32*c^4 - 15876096*a^6*b^30*c^5 + 158760960*a^7*b^28*c^6 - 1270087680

*a^8*b^26*c^7 + 8255569920*a^9*b^24*c^8 - 44029706240*a^10*b^22*c^9 + 193730707456*a^11*b^20*c^10 - 7044752998

40*a^12*b^18*c^11 + 2113425899520*a^13*b^16*c^12 - 5202279137280*a^14*b^14*c^13 + 10404558274560*a^15*b^12*c^1

4 - 16647293239296*a^16*b^10*c^15 + 20809116549120*a^17*b^8*c^16 - 19585050869760*a^18*b^6*c^17 + 130567005798

40*a^19*b^4*c^18 - 5497558138880*a^20*b^2*c^19)))^(3/4) - (9*x^(1/2)*(200930625*a*b^13*c^5 - 3110400000*a^7*b*

c^11 + 2093250600*a^2*b^11*c^6 + 7523454960*a^3*b^9*c^7 + 10328580864*a^4*b^7*c^8 + 2354261760*a^5*b^5*c^9 - 5

453568000*a^6*b^3*c^10))/(4194304*(b^24 + 16777216*a^12*c^12 + 1056*a^2*b^20*c^2 - 14080*a^3*b^18*c^3 + 126720

*a^4*b^16*c^4 - 811008*a^5*b^14*c^5 + 3784704*a^6*b^12*c^6 - 12976128*a^7*b^10*c^7 + 32440320*a^8*b^8*c^8 - 57

671680*a^9*b^6*c^9 + 69206016*a^10*b^4*c^10 - 50331648*a^11*b^2*c^11 - 48*a*b^22*c)))*(-(81*(2401*b^29 + 2401*

b^4*(-(4*a*c - b^2)^25)^(1/2) + 704643072000*a^14*b*c^14 - 1323600*a^2*b^25*c^2 + 28243200*a^3*b^23*c^3 - 2714

15040*a^4*b^21*c^4 + 1437284352*a^5*b^19*c^5 - 3989852160*a^6*b^17*c^6 + 2793799680*a^7*b^15*c^7 + 13327073280

*a^8*b^13*c^8 - 19977994240*a^9*b^11*c^9 - 66059239424*a^10*b^9*c^10 + 143696855040*a^11*b^7*c^11 + 2307706060

80*a^12*b^5*c^12 - 887850270720*a^13*b^3*c^13 + 10000*a^2*c^2*(-(4*a*c - b^2)^25)^(1/2) - 9400*a*b^27*c + 9400

*a*b^2*c*(-(4*a*c - b^2)^25)^(1/2)))/(33554432*(a*b^40 + 1099511627776*a^21*c^20 - 80*a^2*b^38*c + 3040*a^3*b^

36*c^2 - 72960*a^4*b^34*c^3 + 1240320*a^5*b^32*c^4 - 15876096*a^6*b^30*c^5 + 158760960*a^7*b^28*c^6 - 12700876

80*a^8*b^26*c^7 + 8255569920*a^9*b^24*c^8 - 44029706240*a^10*b^22*c^9 + 193730707456*a^11*b^20*c^10 - 70447529

9840*a^12*b^18*c^11 + 2113425899520*a^13*b^16*c^12 - 5202279137280*a^14*b^14*c^13 + 10404558274560*a^15*b^12*c

^14 - 16647293239296*a^16*b^10*c^15 + 20809116549120*a^17*b^8*c^16 - 19585050869760*a^18*b^6*c^17 + 1305670057

9840*a^19*b^4*c^18 - 5497558138880*a^20*b^2*c^19)))^(1/4)*1i - (((27*(5754585088*a*b^27*c^4 + 3096224743817216

00*a^14*b*c^17 - 161128382464*a^2*b^25*c^5 + 1626181992448*a^3*b^23*c^6 - 3983582167040*a^4*b^21*c^7 - 5632849

6087040*a^5*b^19*c^8 + 557813172535296*a^6*b^17*c^9 - 1961803621859328*a^7*b^15*c^10 + 715782069682176*a^8*b^1

3*c^11 + 15816474765557760*a^9*b^11*c^12 - 39296545576714240*a^10*b^9*c^13 - 32756650414702592*a^11*b^7*c^14 +

300756012615335936*a^12*b^5*c^15 - 517069532217475072*a^13*b^3*c^16))/(268435456*(b^28 + 268435456*a^14*c^14

+ 1456*a^2*b^24*c^2 - 23296*a^3*b^22*c^3 + 256256*a^4*b^20*c^4 - 2050048*a^5*b^18*c^5 + 12300288*a^6*b^16*c^6

- 56229888*a^7*b^14*c^7 + 196804608*a^8*b^12*c^8 - 524812288*a^9*b^10*c^9 + 1049624576*a^10*b^8*c^10 - 1526726

656*a^11*b^6*c^11 + 1526726656*a^12*b^4*c^12 - 939524096*a^13*b^2*c^13 - 56*a*b^26*c)) + (9*x^(1/2)*(-(81*(240

1*b^29 + 2401*b^4*(-(4*a*c - b^2)^25)^(1/2) + 704643072000*a^14*b*c^14 - 1323600*a^2*b^25*c^2 + 28243200*a^3*b

^23*c^3 - 271415040*a^4*b^21*c^4 + 1437284352*a^5*b^19*c^5 - 3989852160*a^6*b^17*c^6 + 2793799680*a^7*b^15*c^7

+ 13327073280*a^8*b^13*c^8 - 19977994240*a^9*b^11*c^9 - 66059239424*a^10*b^9*c^10 + 143696855040*a^11*b^7*c^1

1 + 230770606080*a^12*b^5*c^12 - 887850270720*a^13*b^3*c^13 + 10000*a^2*c^2*(-(4*a*c - b^2)^25)^(1/2) - 9400*a

*b^27*c + 9400*a*b^2*c*(-(4*a*c - b^2)^25)^(1/2)))/(33554432*(a*b^40 + 1099511627776*a^21*c^20 - 80*a^2*b^38*c

+ 3040*a^3*b^36*c^2 - 72960*a^4*b^34*c^3 + 1240320*a^5*b^32*c^4 - 15876096*a^6*b^30*c^5 + 158760960*a^7*b^28*

c^6 - 1270087680*a^8*b^26*c^7 + 8255569920*a^9*b^24*c^8 - 44029706240*a^10*b^22*c^9 + 193730707456*a^11*b^20*c

^10 - 704475299840*a^12*b^18*c^11 + 2113425899520*a^13*b^16*c^12 - 5202279137280*a^14*b^14*c^13 + 104045582745

60*a^15*b^12*c^14 - 16647293239296*a^16*b^10*c^15 + 20809116549120*a^17*b^8*c^16 - 19585050869760*a^18*b^6*c^1

7 + 13056700579840*a^19*b^4*c^18 - 5497558138880*a^20*b^2*c^19)))^(1/4)*(822083584*a*b^26*c^4 - 14073748835532

800*a^14*c^17 - 27950841856*a^2*b^24*c^5 + 399431958528*a^3*b^22*c^6 - 2968896143360*a^4*b^20*c^7 + 1032939634

6880*a^5*b^18*c^8 + 6262062317568*a^6*b^16*c^9 - 202859895324672*a^7*b^14*c^10 + 658057709223936*a^8*b^12*c^11

+ 346346162749440*a^9*b^10*c^12 - 8653156510597120*a^10*b^8*c^13 + 28569710136131584*a^11*b^6*c^14 - 47076689

854857216*a^12*b^4*c^15 + 40250921669623808*a^13*b^2*c^16))/(4194304*(b^24 + 16777216*a^12*c^12 + 1056*a^2*b^2

0*c^2 - 14080*a^3*b^18*c^3 + 126720*a^4*b^16*c^4 - 811008*a^5*b^14*c^5 + 3784704*a^6*b^12*c^6 - 12976128*a^7*b

^10*c^7 + 32440320*a^8*b^8*c^8 - 57671680*a^9*b^6*c^9 + 69206016*a^10*b^4*c^10 - 50331648*a^11*b^2*c^11 - 48*a

*b^22*c)))*(-(81*(2401*b^29 + 2401*b^4*(-(4*a*c - b^2)^25)^(1/2) + 704643072000*a^14*b*c^14 - 1323600*a^2*b^25

*c^2 + 28243200*a^3*b^23*c^3 - 271415040*a^4*b^21*c^4 + 1437284352*a^5*b^19*c^5 - 3989852160*a^6*b^17*c^6 + 27

93799680*a^7*b^15*c^7 + 13327073280*a^8*b^13*c^8 - 19977994240*a^9*b^11*c^9 - 66059239424*a^10*b^9*c^10 + 1436

96855040*a^11*b^7*c^11 + 230770606080*a^12*b^5*c^12 - 887850270720*a^13*b^3*c^13 + 10000*a^2*c^2*(-(4*a*c - b^

2)^25)^(1/2) - 9400*a*b^27*c + 9400*a*b^2*c*(-(4*a*c - b^2)^25)^(1/2)))/(33554432*(a*b^40 + 1099511627776*a^21

*c^20 - 80*a^2*b^38*c + 3040*a^3*b^36*c^2 - 72960*a^4*b^34*c^3 + 1240320*a^5*b^32*c^4 - 15876096*a^6*b^30*c^5

+ 158760960*a^7*b^28*c^6 - 1270087680*a^8*b^26*c^7 + 8255569920*a^9*b^24*c^8 - 44029706240*a^10*b^22*c^9 + 193

730707456*a^11*b^20*c^10 - 704475299840*a^12*b^18*c^11 + 2113425899520*a^13*b^16*c^12 - 5202279137280*a^14*b^1

4*c^13 + 10404558274560*a^15*b^12*c^14 - 16647293239296*a^16*b^10*c^15 + 20809116549120*a^17*b^8*c^16 - 195850

50869760*a^18*b^6*c^17 + 13056700579840*a^19*b^4*c^18 - 5497558138880*a^20*b^2*c^19)))^(3/4) + (9*x^(1/2)*(200

930625*a*b^13*c^5 - 3110400000*a^7*b*c^11 + 2093250600*a^2*b^11*c^6 + 7523454960*a^3*b^9*c^7 + 10328580864*a^4

*b^7*c^8 + 2354261760*a^5*b^5*c^9 - 5453568000*a^6*b^3*c^10))/(4194304*(b^24 + 16777216*a^12*c^12 + 1056*a^2*b

^20*c^2 - 14080*a^3*b^18*c^3 + 126720*a^4*b^16*c^4 - 811008*a^5*b^14*c^5 + 3784704*a^6*b^12*c^6 - 12976128*a^7

*b^10*c^7 + 32440320*a^8*b^8*c^8 - 57671680*a^9*b^6*c^9 + 69206016*a^10*b^4*c^10 - 50331648*a^11*b^2*c^11 - 48

*a*b^22*c)))*(-(81*(2401*b^29 + 2401*b^4*(-(4*a*c - b^2)^25)^(1/2) + 704643072000*a^14*b*c^14 - 1323600*a^2*b^

25*c^2 + 28243200*a^3*b^23*c^3 - 271415040*a^4*b^21*c^4 + 1437284352*a^5*b^19*c^5 - 3989852160*a^6*b^17*c^6 +

2793799680*a^7*b^15*c^7 + 13327073280*a^8*b^13*c^8 - 19977994240*a^9*b^11*c^9 - 66059239424*a^10*b^9*c^10 + 14

3696855040*a^11*b^7*c^11 + 230770606080*a^12*b^5*c^12 - 887850270720*a^13*b^3*c^13 + 10000*a^2*c^2*(-(4*a*c -

b^2)^25)^(1/2) - 9400*a*b^27*c + 9400*a*b^2*c*(-(4*a*c - b^2)^25)^(1/2)))/(33554432*(a*b^40 + 1099511627776*a^

21*c^20 - 80*a^2*b^38*c + 3040*a^3*b^36*c^2 - 72960*a^4*b^34*c^3 + 1240320*a^5*b^32*c^4 - 15876096*a^6*b^30*c^

5 + 158760960*a^7*b^28*c^6 - 1270087680*a^8*b^26*c^7 + 8255569920*a^9*b^24*c^8 - 44029706240*a^10*b^22*c^9 + 1

93730707456*a^11*b^20*c^10 - 704475299840*a^12*b^18*c^11 + 2113425899520*a^13*b^16*c^12 - 5202279137280*a^14*b

^14*c^13 + 10404558274560*a^15*b^12*c^14 - 16647293239296*a^16*b^10*c^15 + 20809116549120*a^17*b^8*c^16 - 1958

5050869760*a^18*b^6*c^17 + 13056700579840*a^19*b^4*c^18 - 5497558138880*a^20*b^2*c^19)))^(1/4)*1i)/((27*(10368

0000000*a^8*c^12 + 1406514375*a*b^14*c^5 + 22129159500*a^2*b^12*c^6 + 140297799600*a^3*b^10*c^7 + 460920922560

*a^4*b^8*c^8 + 844743271680*a^5*b^6*c^9 + 869387904000*a^6*b^4*c^10 + 469670400000*a^7*b^2*c^11))/(134217728*(

b^28 + 268435456*a^14*c^14 + 1456*a^2*b^24*c^2 - 23296*a^3*b^22*c^3 + 256256*a^4*b^20*c^4 - 2050048*a^5*b^18*c

^5 + 12300288*a^6*b^16*c^6 - 56229888*a^7*b^14*c^7 + 196804608*a^8*b^12*c^8 - 524812288*a^9*b^10*c^9 + 1049624

576*a^10*b^8*c^10 - 1526726656*a^11*b^6*c^11 + 1526726656*a^12*b^4*c^12 - 939524096*a^13*b^2*c^13 - 56*a*b^26*

c)) + (((27*(5754585088*a*b^27*c^4 + 309622474381721600*a^14*b*c^17 - 161128382464*a^2*b^25*c^5 + 162618199244

8*a^3*b^23*c^6 - 3983582167040*a^4*b^21*c^7 - 56328496087040*a^5*b^19*c^8 + 557813172535296*a^6*b^17*c^9 - 196

1803621859328*a^7*b^15*c^10 + 715782069682176*a^8*b^13*c^11 + 15816474765557760*a^9*b^11*c^12 - 39296545576714

240*a^10*b^9*c^13 - 32756650414702592*a^11*b^7*c^14 + 300756012615335936*a^12*b^5*c^15 - 517069532217475072*a^

13*b^3*c^16))/(268435456*(b^28 + 268435456*a^14*c^14 + 1456*a^2*b^24*c^2 - 23296*a^3*b^22*c^3 + 256256*a^4*b^2

0*c^4 - 2050048*a^5*b^18*c^5 + 12300288*a^6*b^16*c^6 - 56229888*a^7*b^14*c^7 + 196804608*a^8*b^12*c^8 - 524812

288*a^9*b^10*c^9 + 1049624576*a^10*b^8*c^10 - 1526726656*a^11*b^6*c^11 + 1526726656*a^12*b^4*c^12 - 939524096*

a^13*b^2*c^13 - 56*a*b^26*c)) - (9*x^(1/2)*(-(81*(2401*b^29 + 2401*b^4*(-(4*a*c - b^2)^25)^(1/2) + 70464307200

0*a^14*b*c^14 - 1323600*a^2*b^25*c^2 + 28243200*a^3*b^23*c^3 - 271415040*a^4*b^21*c^4 + 1437284352*a^5*b^19*c^

5 - 3989852160*a^6*b^17*c^6 + 2793799680*a^7*b^15*c^7 + 13327073280*a^8*b^13*c^8 - 19977994240*a^9*b^11*c^9 -

66059239424*a^10*b^9*c^10 + 143696855040*a^11*b^7*c^11 + 230770606080*a^12*b^5*c^12 - 887850270720*a^13*b^3*c^

13 + 10000*a^2*c^2*(-(4*a*c - b^2)^25)^(1/2) - 9400*a*b^27*c + 9400*a*b^2*c*(-(4*a*c - b^2)^25)^(1/2)))/(33554

432*(a*b^40 + 1099511627776*a^21*c^20 - 80*a^2*b^38*c + 3040*a^3*b^36*c^2 - 72960*a^4*b^34*c^3 + 1240320*a^5*b

^32*c^4 - 15876096*a^6*b^30*c^5 + 158760960*a^7*b^28*c^6 - 1270087680*a^8*b^26*c^7 + 8255569920*a^9*b^24*c^8 -

44029706240*a^10*b^22*c^9 + 193730707456*a^11*b^20*c^10 - 704475299840*a^12*b^18*c^11 + 2113425899520*a^13*b^

16*c^12 - 5202279137280*a^14*b^14*c^13 + 10404558274560*a^15*b^12*c^14 - 16647293239296*a^16*b^10*c^15 + 20809

116549120*a^17*b^8*c^16 - 19585050869760*a^18*b^6*c^17 + 13056700579840*a^19*b^4*c^18 - 5497558138880*a^20*b^2

*c^19)))^(1/4)*(822083584*a*b^26*c^4 - 14073748835532800*a^14*c^17 - 27950841856*a^2*b^24*c^5 + 399431958528*a

^3*b^22*c^6 - 2968896143360*a^4*b^20*c^7 + 10329396346880*a^5*b^18*c^8 + 6262062317568*a^6*b^16*c^9 - 20285989

5324672*a^7*b^14*c^10 + 658057709223936*a^8*b^12*c^11 + 346346162749440*a^9*b^10*c^12 - 8653156510597120*a^10*

b^8*c^13 + 28569710136131584*a^11*b^6*c^14 - 47076689854857216*a^12*b^4*c^15 + 40250921669623808*a^13*b^2*c^16

))/(4194304*(b^24 + 16777216*a^12*c^12 + 1056*a^2*b^20*c^2 - 14080*a^3*b^18*c^3 + 126720*a^4*b^16*c^4 - 811008

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

9206016*a^10*b^4*c^10 - 50331648*a^11*b^2*c^11 - 48*a*b^22*c)))*(-(81*(2401*b^29 + 2401*b^4*(-(4*a*c - b^2)^25

)^(1/2) + 704643072000*a^14*b*c^14 - 1323600*a^2*b^25*c^2 + 28243200*a^3*b^23*c^3 - 271415040*a^4*b^21*c^4 + 1

437284352*a^5*b^19*c^5 - 3989852160*a^6*b^17*c^6 + 2793799680*a^7*b^15*c^7 + 13327073280*a^8*b^13*c^8 - 199779

94240*a^9*b^11*c^9 - 66059239424*a^10*b^9*c^10 + 143696855040*a^11*b^7*c^11 + 230770606080*a^12*b^5*c^12 - 887

850270720*a^13*b^3*c^13 + 10000*a^2*c^2*(-(4*a*c - b^2)^25)^(1/2) - 9400*a*b^27*c + 9400*a*b^2*c*(-(4*a*c - b^