∫ x11∕2 ____________________

3.9.49 $\int \frac {x^{11/2}}{(a+b x^2+c x^4)^3} \, dx$

Optimal. Leaf size=569 \[ \frac {\sqrt {x} \left (x^2 \left (20 a c+7 b^2\right )+24 a b\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {x^{5/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {3 \left (\sqrt {b^2-4 a c} \left (20 a c+7 b^2\right )+36 a b c+7 b^3\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} \sqrt [4]{c} \left (b^2-4 a c\right )^{5/2} \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {3 \left (-\frac {36 a b c+7 b^3}{\sqrt {b^2-4 a c}}+20 a c+7 b^2\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} \sqrt [4]{c} \left (b^2-4 a c\right )^2 \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {3 \left (\sqrt {b^2-4 a c} \left (20 a c+7 b^2\right )+36 a b c+7 b^3\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} \sqrt [4]{c} \left (b^2-4 a c\right )^{5/2} \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {3 \left (-\frac {36 a b c+7 b^3}{\sqrt {b^2-4 a c}}+20 a c+7 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} \sqrt [4]{c} \left (b^2-4 a c\right )^2 \left (\sqrt {b^2-4 a c}-b\right )^{3/4}} \]

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Rubi [A] time = 1.96, antiderivative size = 569, 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, 1498, 1422, 212, 208, 205} \begin {gather*} \frac {x^{5/2} \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 {\sqrt {x} \left (x^2 \left (20 a c+7 b^2\right )+24 a b\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {3 \left (\sqrt {b^2-4 a c} \left (20 a c+7 b^2\right )+36 a b c+7 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 )}{32 \sqrt [4]{2} \sqrt [4]{c} \left (b^2-4 a c\right )^{5/2} \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {3 \left (-\frac {36 a b c+7 b^3}{\sqrt {b^2-4 a c}}+20 a c+7 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 )}{32 \sqrt [4]{2} \sqrt [4]{c} \left (b^2-4 a c\right )^2 \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {3 \left (\sqrt {b^2-4 a c} \left (20 a c+7 b^2\right )+36 a b c+7 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 )}{32 \sqrt [4]{2} \sqrt [4]{c} \left (b^2-4 a c\right )^{5/2} \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {3 \left (-\frac {36 a b c+7 b^3}{\sqrt {b^2-4 a c}}+20 a c+7 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 )}{32 \sqrt [4]{2} \sqrt [4]{c} \left (b^2-4 a c\right )^2 \left (\sqrt {b^2-4 a c}-b\right )^{3/4}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[x^(11/2)/(a + b*x^2 + c*x^4)^3,x]

[Out]

(x^(5/2)*(2*a + b*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (Sqrt[x]*(24*a*b + (7*b^2 + 20*a*c)*x^2))/(1

6*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - (3*(7*b^3 + 36*a*b*c + Sqrt[b^2 - 4*a*c]*(7*b^2 + 20*a*c))*ArcTan[(2^

(1/4)*c^(1/4)*Sqrt[x])/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(32*2^(1/4)*c^(1/4)*(b^2 - 4*a*c)^(5/2)*(-b - Sqrt[b^2

- 4*a*c])^(3/4)) - (3*(7*b^2 + 20*a*c - (7*b^3 + 36*a*b*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)])/(32*2^(1/4)*c^(1/4)*(b^2 - 4*a*c)^2*(-b + Sqrt[b^2 - 4*a*c])^(3/4)) - (3*(7

*b^3 + 36*a*b*c + Sqrt[b^2 - 4*a*c]*(7*b^2 + 20*a*c))*ArcTanh[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b - Sqrt[b^2 - 4*a*c

])^(1/4)])/(32*2^(1/4)*c^(1/4)*(b^2 - 4*a*c)^(5/2)*(-b - Sqrt[b^2 - 4*a*c])^(3/4)) - (3*(7*b^2 + 20*a*c - (7*b

^3 + 36*a*b*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)])/(32*2^(1/

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

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 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]

]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), 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 1422

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*

c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^n), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), In

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

[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 - 4*a*c] || !IGtQ[n/2, 0])

Rule 1498

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

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

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

*Simp[(n - m - 1)*(b*d - 2*a*e) + (2*n*p + 2*n + m + 1)*(b*e - 2*c*d)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e,

f}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m, n - 1] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {x^{11/2}}{\left (a+b x^2+c x^4\right )^3} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^{12}}{\left (a+b x^4+c x^8\right )^3} \, dx,x,\sqrt {x}\right )\\ &=\frac {x^{5/2} \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^4 \left (10 a-7 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^{5/2} \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 {\sqrt {x} \left (24 a b+\left (7 b^2+20 a c\right ) 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 {-24 a b+3 \left (7 b^2+20 a c\right ) x^4}{a+b x^4+c x^8} \, dx,x,\sqrt {x}\right )}{16 \left (b^2-4 a c\right )^2}\\ &=\frac {x^{5/2} \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 {\sqrt {x} \left (24 a b+\left (7 b^2+20 a c\right ) x^2\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\left (3 \left (7 b^3+36 a b c+\sqrt {b^2-4 a c} \left (7 b^2+20 a c\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^4} \, dx,x,\sqrt {x}\right )}{32 \left (b^2-4 a c\right )^{5/2}}+\frac {\left (3 \left (7 b^2+20 a c-\frac {7 b^3+36 a b c}{\sqrt {b^2-4 a c}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^4} \, dx,x,\sqrt {x}\right )}{32 \left (b^2-4 a c\right )^2}\\ &=\frac {x^{5/2} \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 {\sqrt {x} \left (24 a b+\left (7 b^2+20 a c\right ) x^2\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {\left (3 \left (7 b^3+36 a b c+\sqrt {b^2-4 a c} \left (7 b^2+20 a c\right )\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 )}{32 \left (b^2-4 a c\right )^{5/2} \sqrt {-b-\sqrt {b^2-4 a c}}}-\frac {\left (3 \left (7 b^3+36 a b c+\sqrt {b^2-4 a c} \left (7 b^2+20 a c\right )\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 )}{32 \left (b^2-4 a c\right )^{5/2} \sqrt {-b-\sqrt {b^2-4 a c}}}-\frac {\left (3 \left (7 b^2+20 a c-\frac {7 b^3+36 a b c}{\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 )}{32 \left (b^2-4 a c\right )^2 \sqrt {-b+\sqrt {b^2-4 a c}}}-\frac {\left (3 \left (7 b^2+20 a c-\frac {7 b^3+36 a b c}{\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 )}{32 \left (b^2-4 a c\right )^2 \sqrt {-b+\sqrt {b^2-4 a c}}}\\ &=\frac {x^{5/2} \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 {\sqrt {x} \left (24 a b+\left (7 b^2+20 a c\right ) x^2\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {3 \left (7 b^3+36 a b c+\sqrt {b^2-4 a c} \left (7 b^2+20 a c\right )\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{32 \sqrt [4]{2} \sqrt [4]{c} \left (b^2-4 a c\right )^{5/2} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {3 \left (7 b^2+20 a c-\frac {7 b^3+36 a b c}{\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 )}{32 \sqrt [4]{2} \sqrt [4]{c} \left (b^2-4 a c\right )^2 \left (-b+\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {3 \left (7 b^3+36 a b c+\sqrt {b^2-4 a c} \left (7 b^2+20 a c\right )\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{32 \sqrt [4]{2} \sqrt [4]{c} \left (b^2-4 a c\right )^{5/2} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {3 \left (7 b^2+20 a c-\frac {7 b^3+36 a b c}{\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 )}{32 \sqrt [4]{2} \sqrt [4]{c} \left (b^2-4 a c\right )^2 \left (-b+\sqrt {b^2-4 a c}\right )^{3/4}}\\ \end {align*}

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Mathematica [C] time = 0.39, size = 219, normalized size = 0.38 \begin {gather*} \frac {3 c \left (a+b x^2+c x^4\right )^2 \text {RootSum}\left [\text {$\#$1}^8 c+\text {$\#$1}^4 b+a\&,\frac {20 \text {$\#$1}^4 a c \log \left (\sqrt {x}-\text {$\#$1}\right )+7 \text {$\#$1}^4 b^2 \log \left (\sqrt {x}-\text {$\#$1}\right )-8 a b \log \left (\sqrt {x}-\text {$\#$1}\right )}{2 \text {$\#$1}^7 c+\text {$\#$1}^3 b}\&\right ]-16 \sqrt {x} \left (b^2-4 a c\right ) \left (a \left (b-2 c x^2\right )+b^2 x^2\right )+4 \sqrt {x} \left (8 a b c+20 a c^2 x^2+4 b^3+7 b^2 c x^2\right ) \left (a+b x^2+c x^4\right )}{64 c \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]

Integrate[x^(11/2)/(a + b*x^2 + c*x^4)^3,x]

[Out]

(4*Sqrt[x]*(4*b^3 + 8*a*b*c + 7*b^2*c*x^2 + 20*a*c^2*x^2)*(a + b*x^2 + c*x^4) - 16*(b^2 - 4*a*c)*Sqrt[x]*(b^2*

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

7*b^2*Log[Sqrt[x] - #1]*#1^4 + 20*a*c*Log[Sqrt[x] - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ])/(64*c*(b^2 - 4*a*c)^2*

(a + b*x^2 + c*x^4)^2)

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IntegrateAlgebraic [C] time = 0.93, size = 413, normalized size = 0.73 \begin {gather*} \frac {3 \text {RootSum}\left [\text {$\#$1}^8 c+\text {$\#$1}^4 b+a\&,\frac {-44 \text {$\#$1}^4 a^2 c^3 \log \left (\sqrt {x}-\text {$\#$1}\right )-9 \text {$\#$1}^4 a b^2 c^2 \log \left (\sqrt {x}-\text {$\#$1}\right )+8 \text {$\#$1}^4 b^4 c \log \left (\sqrt {x}-\text {$\#$1}\right )+152 a^2 b c^2 \log \left (\sqrt {x}-\text {$\#$1}\right )-72 a b^3 c \log \left (\sqrt {x}-\text {$\#$1}\right )+8 b^5 \log \left (\sqrt {x}-\text {$\#$1}\right )}{2 \text {$\#$1}^7 c+\text {$\#$1}^3 b}\&\right ]}{64 a c^2 \left (4 a c-b^2\right )^2}+\frac {3 \text {RootSum}\left [\text {$\#$1}^8 c+\text {$\#$1}^4 b+a\&,\frac {2 \text {$\#$1}^4 a c^2 \log \left (\sqrt {x}-\text {$\#$1}\right )+\text {$\#$1}^4 b^2 c \log \left (\sqrt {x}-\text {$\#$1}\right )-5 a b c \log \left (\sqrt {x}-\text {$\#$1}\right )+b^3 \log \left (\sqrt {x}-\text {$\#$1}\right )}{2 \text {$\#$1}^7 c+\text {$\#$1}^3 b}\&\right ]}{8 a c^2 \left (4 a c-b^2\right )}+\frac {24 a^2 b \sqrt {x}-12 a^2 c x^{5/2}+39 a b^2 x^{5/2}+28 a b c x^{9/2}+20 a c^2 x^{13/2}+11 b^3 x^{9/2}+7 b^2 c x^{13/2}}{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^(11/2)/(a + b*x^2 + c*x^4)^3,x]

[Out]

(24*a^2*b*Sqrt[x] + 39*a*b^2*x^(5/2) - 12*a^2*c*x^(5/2) + 11*b^3*x^(9/2) + 28*a*b*c*x^(9/2) + 7*b^2*c*x^(13/2)

+ 20*a*c^2*x^(13/2))/(16*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)^2) + (3*RootSum[a + b*#1^4 + c*#1^8 & , (b^3*Log

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

*#1^3 + 2*c*#1^7) & ])/(8*a*c^2*(-b^2 + 4*a*c)) + (3*RootSum[a + b*#1^4 + c*#1^8 & , (8*b^5*Log[Sqrt[x] - #1]

- 72*a*b^3*c*Log[Sqrt[x] - #1] + 152*a^2*b*c^2*Log[Sqrt[x] - #1] + 8*b^4*c*Log[Sqrt[x] - #1]*#1^4 - 9*a*b^2*c^

2*Log[Sqrt[x] - #1]*#1^4 - 44*a^2*c^3*Log[Sqrt[x] - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ])/(64*a*c^2*(-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^(11/2)/(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^(11/2)/(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: 191.55Unable to convert to real 1/4 Error: Bad Argument Value

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maple [C] time = 0.04, size = 241, normalized size = 0.42 \begin {gather*} \frac {3 \left (\left (20 a c +7 b^{2}\right ) \RootOf \left (c \,\textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+a \right )^{4}-8 a b \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 {2 \left (20 a c +7 b^{2}\right ) c \,x^{\frac {13}{2}}}{512 a^{2} c^{2}-256 a \,b^{2} c +32 b^{4}}+\frac {2 \left (28 a c +11 b^{2}\right ) b \,x^{\frac {9}{2}}}{512 a^{2} c^{2}-256 a \,b^{2} c +32 b^{4}}+\frac {3 a^{2} b \sqrt {x}}{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 ) a \,x^{\frac {5}{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^(11/2)/(c*x^4+b*x^2+a)^3,x)

[Out]

2*(3/4/(16*a^2*c^2-8*a*b^2*c+b^4)*a^2*b*x^(1/2)-3/32*(4*a*c-13*b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*a*x^(5/2)+1/32*

b*(28*a*c+11*b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(9/2)+1/32*c*(20*a*c+7*b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(13/2)

)/(c*x^4+b*x^2+a)^2+3/64/(16*a^2*c^2-8*a*b^2*c+b^4)*sum(((20*a*c+7*b^2)*_R^4-8*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 {24 \, b c^{2} x^{\frac {17}{2}} + {\left (41 \, b^{2} c - 20 \, a c^{2}\right )} x^{\frac {13}{2}} + {\left (13 \, b^{3} + 20 \, a b c\right )} x^{\frac {9}{2}} + 3 \, {\left (3 \, a b^{2} + 4 \, a^{2} c\right )} x^{\frac {5}{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 {7}{2}} + 5 \, {\left (3 \, b^{2} + 4 \, a c\right )} x^{\frac {3}{2}}\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^(11/2)/(c*x^4+b*x^2+a)^3,x, algorithm="maxima")

[Out]

-1/16*(24*b*c^2*x^(17/2) + (41*b^2*c - 20*a*c^2)*x^(13/2) + (13*b^3 + 20*a*b*c)*x^(9/2) + 3*(3*a*b^2 + 4*a^2*c

)*x^(5/2))/((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^(7/2) + 5*(3*b^2 + 4*a*c)*x^(3/2))/(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)

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mupad [B] time = 8.52, size = 45495, normalized size = 79.96

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^(11/2)/(a + b*x^2 + c*x^4)^3,x)

[Out]

((x^(9/2)*(11*b^3 + 28*a*b*c))/(16*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (3*x^(5/2)*(13*a*b^2 - 4*a^2*c))/(16*(b^4

+ 16*a^2*c^2 - 8*a*b^2*c)) + (c*x^(13/2)*(20*a*c + 7*b^2))/(16*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (3*a^2*b*x^(

1/2))/(2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))/(x^4*(2*a*c + b^2) + a^2 + c^2*x^8 + 2*a*b*x^2 + 2*b*c*x^6) - atan((

((((3*((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 - 27937996

80*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 - 1436968550

40*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*(b^40*c + 1099511627776*a^20*c^21

- 80*a*b^38*c^2 + 3040*a^2*b^36*c^3 - 72960*a^3*b^34*c^4 + 1240320*a^4*b^32*c^5 - 15876096*a^5*b^30*c^6 + 1587

60960*a^6*b^28*c^7 - 1270087680*a^7*b^26*c^8 + 8255569920*a^8*b^24*c^9 - 44029706240*a^9*b^22*c^10 + 193730707

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

+ 10404558274560*a^14*b^12*c^15 - 16647293239296*a^15*b^10*c^16 + 20809116549120*a^16*b^8*c^17 - 195850508697

60*a^17*b^6*c^18 + 13056700579840*a^18*b^4*c^19 - 5497558138880*a^19*b^2*c^20)))^(1/4)*(351843720888320*a^13*c

^15 + 251658240*a^2*b^22*c^4 - 9730785280*a^3*b^20*c^5 + 167772160000*a^4*b^18*c^6 - 1691143372800*a^5*b^16*c^

7 + 10952166604800*a^6*b^14*c^8 - 46901042872320*a^7*b^12*c^9 + 129879811031040*a^8*b^10*c^10 - 20615843020800

0*a^9*b^8*c^11 + 82463372083200*a^10*b^6*c^12 + 329853488332800*a^11*b^4*c^13 - 615726511554560*a^12*b^2*c^14)

)/(65536*(b^18 - 262144*a^9*c^9 + 576*a^2*b^14*c^2 - 5376*a^3*b^12*c^3 + 32256*a^4*b^10*c^4 - 129024*a^5*b^8*c

^5 + 344064*a^6*b^6*c^6 - 589824*a^7*b^4*c^7 + 589824*a^8*b^2*c^8 - 36*a*b^16*c)) - (9*x^(1/2)*(3774873600*a^2

*b^25*c^4 - 4222124650659840*a^14*b*c^16 - 147907936256*a^3*b^23*c^5 + 2590402150400*a^4*b^21*c^6 - 2660732239

8720*a^5*b^19*c^7 + 176329882337280*a^6*b^17*c^8 - 777217281884160*a^7*b^15*c^9 + 2233932749733888*a^8*b^13*c^

10 - 3727344418160640*a^9*b^11*c^11 + 1599789418414080*a^10*b^9*c^12 + 7124835347988480*a^11*b^7*c^13 - 160088

89300418560*a^12*b^5*c^14 + 13792273858822144*a^13*b^3*c^15))/(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^2

5*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 - 2

793799680*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 - 143

696855040*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*(b^40*c + 1099511627776*a^2

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

+ 158760960*a^6*b^28*c^7 - 1270087680*a^7*b^26*c^8 + 8255569920*a^8*b^24*c^9 - 44029706240*a^9*b^22*c^10 + 19

3730707456*a^10*b^20*c^11 - 704475299840*a^11*b^18*c^12 + 2113425899520*a^12*b^16*c^13 - 5202279137280*a^13*b^

14*c^14 + 10404558274560*a^14*b^12*c^15 - 16647293239296*a^15*b^10*c^16 + 20809116549120*a^16*b^8*c^17 - 19585

050869760*a^17*b^6*c^18 + 13056700579840*a^18*b^4*c^19 - 5497558138880*a^19*b^2*c^20)))^(3/4) + (3*(570240000*

a^7*b*c^8 + 2917215*a^2*b^11*c^3 + 49009212*a^3*b^9*c^4 + 303385824*a^4*b^7*c^5 + 879403392*a^5*b^5*c^6 + 1191

801600*a^6*b^3*c^7))/(65536*(b^18 - 262144*a^9*c^9 + 576*a^2*b^14*c^2 - 5376*a^3*b^12*c^3 + 32256*a^4*b^10*c^4

- 129024*a^5*b^8*c^5 + 344064*a^6*b^6*c^6 - 589824*a^7*b^4*c^7 + 589824*a^8*b^2*c^8 - 36*a*b^16*c)))*((81*(24

01*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*(b^40*c + 1099511627776*a^20*c^21 - 80*a*b^38*c^

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

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

c^11 - 704475299840*a^11*b^18*c^12 + 2113425899520*a^12*b^16*c^13 - 5202279137280*a^13*b^14*c^14 + 10404558274

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

18 + 13056700579840*a^18*b^4*c^19 - 5497558138880*a^19*b^2*c^20)))^(1/4) - (9*x^(1/2)*(43758225*a^2*b^14*c^3 -

10368000000*a^9*c^10 + 682628310*a^3*b^12*c^4 + 4119250464*a^4*b^10*c^5 + 11404429344*a^5*b^8*c^6 + 112636500

48*a^6*b^6*c^7 - 8687347200*a^7*b^4*c^8 - 22394880000*a^8*b^2*c^9))/(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 - 129761

28*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^1

1 - 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*(b^40*c + 10995116277

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

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

0 + 193730707456*a^10*b^20*c^11 - 704475299840*a^11*b^18*c^12 + 2113425899520*a^12*b^16*c^13 - 5202279137280*a

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

19585050869760*a^17*b^6*c^18 + 13056700579840*a^18*b^4*c^19 - 5497558138880*a^19*b^2*c^20)))^(1/4)*1i - ((((3

*((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 - 2824

3200*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*(b^40*c + 1099511627776*a^20*c^21 - 80*

a*b^38*c^2 + 3040*a^2*b^36*c^3 - 72960*a^3*b^34*c^4 + 1240320*a^4*b^32*c^5 - 15876096*a^5*b^30*c^6 + 158760960

*a^6*b^28*c^7 - 1270087680*a^7*b^26*c^8 + 8255569920*a^8*b^24*c^9 - 44029706240*a^9*b^22*c^10 + 193730707456*a

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

404558274560*a^14*b^12*c^15 - 16647293239296*a^15*b^10*c^16 + 20809116549120*a^16*b^8*c^17 - 19585050869760*a^

17*b^6*c^18 + 13056700579840*a^18*b^4*c^19 - 5497558138880*a^19*b^2*c^20)))^(1/4)*(351843720888320*a^13*c^15 +

251658240*a^2*b^22*c^4 - 9730785280*a^3*b^20*c^5 + 167772160000*a^4*b^18*c^6 - 1691143372800*a^5*b^16*c^7 + 1

0952166604800*a^6*b^14*c^8 - 46901042872320*a^7*b^12*c^9 + 129879811031040*a^8*b^10*c^10 - 206158430208000*a^9

*b^8*c^11 + 82463372083200*a^10*b^6*c^12 + 329853488332800*a^11*b^4*c^13 - 615726511554560*a^12*b^2*c^14))/(65

536*(b^18 - 262144*a^9*c^9 + 576*a^2*b^14*c^2 - 5376*a^3*b^12*c^3 + 32256*a^4*b^10*c^4 - 129024*a^5*b^8*c^5 +

344064*a^6*b^6*c^6 - 589824*a^7*b^4*c^7 + 589824*a^8*b^2*c^8 - 36*a*b^16*c)) + (9*x^(1/2)*(3774873600*a^2*b^25

*c^4 - 4222124650659840*a^14*b*c^16 - 147907936256*a^3*b^23*c^5 + 2590402150400*a^4*b^21*c^6 - 26607322398720*

a^5*b^19*c^7 + 176329882337280*a^6*b^17*c^8 - 777217281884160*a^7*b^15*c^9 + 2233932749733888*a^8*b^13*c^10 -

3727344418160640*a^9*b^11*c^11 + 1599789418414080*a^10*b^9*c^12 + 7124835347988480*a^11*b^7*c^13 - 16008889300

418560*a^12*b^5*c^14 + 13792273858822144*a^13*b^3*c^15))/(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 - 279379

9680*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 - 14369685

5040*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)^2

5)^(1/2) + 9400*a*b^27*c + 9400*a*b^2*c*(-(4*a*c - b^2)^25)^(1/2)))/(33554432*(b^40*c + 1099511627776*a^20*c^2

1 - 80*a*b^38*c^2 + 3040*a^2*b^36*c^3 - 72960*a^3*b^34*c^4 + 1240320*a^4*b^32*c^5 - 15876096*a^5*b^30*c^6 + 15

8760960*a^6*b^28*c^7 - 1270087680*a^7*b^26*c^8 + 8255569920*a^8*b^24*c^9 - 44029706240*a^9*b^22*c^10 + 1937307

07456*a^10*b^20*c^11 - 704475299840*a^11*b^18*c^12 + 2113425899520*a^12*b^16*c^13 - 5202279137280*a^13*b^14*c^

14 + 10404558274560*a^14*b^12*c^15 - 16647293239296*a^15*b^10*c^16 + 20809116549120*a^16*b^8*c^17 - 1958505086

9760*a^17*b^6*c^18 + 13056700579840*a^18*b^4*c^19 - 5497558138880*a^19*b^2*c^20)))^(3/4) + (3*(570240000*a^7*b

*c^8 + 2917215*a^2*b^11*c^3 + 49009212*a^3*b^9*c^4 + 303385824*a^4*b^7*c^5 + 879403392*a^5*b^5*c^6 + 119180160

0*a^6*b^3*c^7))/(65536*(b^18 - 262144*a^9*c^9 + 576*a^2*b^14*c^2 - 5376*a^3*b^12*c^3 + 32256*a^4*b^10*c^4 - 12

9024*a^5*b^8*c^5 + 344064*a^6*b^6*c^6 - 589824*a^7*b^4*c^7 + 589824*a^8*b^2*c^8 - 36*a*b^16*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 - 1

3327073280*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^2

7*c + 9400*a*b^2*c*(-(4*a*c - b^2)^25)^(1/2)))/(33554432*(b^40*c + 1099511627776*a^20*c^21 - 80*a*b^38*c^2 + 3

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

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

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

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

13056700579840*a^18*b^4*c^19 - 5497558138880*a^19*b^2*c^20)))^(1/4) + (9*x^(1/2)*(43758225*a^2*b^14*c^3 - 1036

8000000*a^9*c^10 + 682628310*a^3*b^12*c^4 + 4119250464*a^4*b^10*c^5 + 11404429344*a^5*b^8*c^6 + 11263650048*a^

6*b^6*c^7 - 8687347200*a^7*b^4*c^8 - 22394880000*a^8*b^2*c^9))/(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 - 4

8*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 - 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*(b^40*c + 1099511627776*a^

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

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

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

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

5050869760*a^17*b^6*c^18 + 13056700579840*a^18*b^4*c^19 - 5497558138880*a^19*b^2*c^20)))^(1/4)*1i)/(((((3*((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^1

5*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) + 9

400*a*b^27*c + 9400*a*b^2*c*(-(4*a*c - b^2)^25)^(1/2)))/(33554432*(b^40*c + 1099511627776*a^20*c^21 - 80*a*b^3

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

b^28*c^7 - 1270087680*a^7*b^26*c^8 + 8255569920*a^8*b^24*c^9 - 44029706240*a^9*b^22*c^10 + 193730707456*a^10*b

^20*c^11 - 704475299840*a^11*b^18*c^12 + 2113425899520*a^12*b^16*c^13 - 5202279137280*a^13*b^14*c^14 + 1040455

8274560*a^14*b^12*c^15 - 16647293239296*a^15*b^10*c^16 + 20809116549120*a^16*b^8*c^17 - 19585050869760*a^17*b^

6*c^18 + 13056700579840*a^18*b^4*c^19 - 5497558138880*a^19*b^2*c^20)))^(1/4)*(351843720888320*a^13*c^15 + 2516

58240*a^2*b^22*c^4 - 9730785280*a^3*b^20*c^5 + 167772160000*a^4*b^18*c^6 - 1691143372800*a^5*b^16*c^7 + 109521

66604800*a^6*b^14*c^8 - 46901042872320*a^7*b^12*c^9 + 129879811031040*a^8*b^10*c^10 - 206158430208000*a^9*b^8*

c^11 + 82463372083200*a^10*b^6*c^12 + 329853488332800*a^11*b^4*c^13 - 615726511554560*a^12*b^2*c^14))/(65536*(

b^18 - 262144*a^9*c^9 + 576*a^2*b^14*c^2 - 5376*a^3*b^12*c^3 + 32256*a^4*b^10*c^4 - 129024*a^5*b^8*c^5 + 34406

4*a^6*b^6*c^6 - 589824*a^7*b^4*c^7 + 589824*a^8*b^2*c^8 - 36*a*b^16*c)) - (9*x^(1/2)*(3774873600*a^2*b^25*c^4

- 4222124650659840*a^14*b*c^16 - 147907936256*a^3*b^23*c^5 + 2590402150400*a^4*b^21*c^6 - 26607322398720*a^5*b

^19*c^7 + 176329882337280*a^6*b^17*c^8 - 777217281884160*a^7*b^15*c^9 + 2233932749733888*a^8*b^13*c^10 - 37273

44418160640*a^9*b^11*c^11 + 1599789418414080*a^10*b^9*c^12 + 7124835347988480*a^11*b^7*c^13 - 1600888930041856

0*a^12*b^5*c^14 + 13792273858822144*a^13*b^3*c^15))/(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 - 28

243200*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*(b^40*c + 1099511627776*a^20*c^21 - 8

0*a*b^38*c^2 + 3040*a^2*b^36*c^3 - 72960*a^3*b^34*c^4 + 1240320*a^4*b^32*c^5 - 15876096*a^5*b^30*c^6 + 1587609

60*a^6*b^28*c^7 - 1270087680*a^7*b^26*c^8 + 8255569920*a^8*b^24*c^9 - 44029706240*a^9*b^22*c^10 + 193730707456

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

10404558274560*a^14*b^12*c^15 - 16647293239296*a^15*b^10*c^16 + 20809116549120*a^16*b^8*c^17 - 19585050869760*

a^17*b^6*c^18 + 13056700579840*a^18*b^4*c^19 - 5497558138880*a^19*b^2*c^20)))^(3/4) + (3*(570240000*a^7*b*c^8

+ 2917215*a^2*b^11*c^3 + 49009212*a^3*b^9*c^4 + 303385824*a^4*b^7*c^5 + 879403392*a^5*b^5*c^6 + 1191801600*a^6

*b^3*c^7))/(65536*(b^18 - 262144*a^9*c^9 + 576*a^2*b^14*c^2 - 5376*a^3*b^12*c^3 + 32256*a^4*b^10*c^4 - 129024*

a^5*b^8*c^5 + 344064*a^6*b^6*c^6 - 589824*a^7*b^4*c^7 + 589824*a^8*b^2*c^8 - 36*a*b^16*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 - 133270

73280*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 - 23077

0606080*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*(b^40*c + 1099511627776*a^20*c^21 - 80*a*b^38*c^2 + 3040*a

^2*b^36*c^3 - 72960*a^3*b^34*c^4 + 1240320*a^4*b^32*c^5 - 15876096*a^5*b^30*c^6 + 158760960*a^6*b^28*c^7 - 127

0087680*a^7*b^26*c^8 + 8255569920*a^8*b^24*c^9 - 44029706240*a^9*b^22*c^10 + 193730707456*a^10*b^20*c^11 - 704

475299840*a^11*b^18*c^12 + 2113425899520*a^12*b^16*c^13 - 5202279137280*a^13*b^14*c^14 + 10404558274560*a^14*b

^12*c^15 - 16647293239296*a^15*b^10*c^16 + 20809116549120*a^16*b^8*c^17 - 19585050869760*a^17*b^6*c^18 + 13056

700579840*a^18*b^4*c^19 - 5497558138880*a^19*b^2*c^20)))^(1/4) - (9*x^(1/2)*(43758225*a^2*b^14*c^3 - 103680000

00*a^9*c^10 + 682628310*a^3*b^12*c^4 + 4119250464*a^4*b^10*c^5 + 11404429344*a^5*b^8*c^6 + 11263650048*a^6*b^6

*c^7 - 8687347200*a^7*b^4*c^8 - 22394880000*a^8*b^2*c^9))/(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^1

0*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 - 27937

99680*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 - 1436968

55040*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*(b^40*c + 1099511627776*a^20*c^

21 - 80*a*b^38*c^2 + 3040*a^2*b^36*c^3 - 72960*a^3*b^34*c^4 + 1240320*a^4*b^32*c^5 - 15876096*a^5*b^30*c^6 + 1

58760960*a^6*b^28*c^7 - 1270087680*a^7*b^26*c^8 + 8255569920*a^8*b^24*c^9 - 44029706240*a^9*b^22*c^10 + 193730

707456*a^10*b^20*c^11 - 704475299840*a^11*b^18*c^12 + 2113425899520*a^12*b^16*c^13 - 5202279137280*a^13*b^14*c

^14 + 10404558274560*a^14*b^12*c^15 - 16647293239296*a^15*b^10*c^16 + 20809116549120*a^16*b^8*c^17 - 195850508

69760*a^17*b^6*c^18 + 13056700579840*a^18*b^4*c^19 - 5497558138880*a^19*b^2*c^20)))^(1/4) + ((((3*((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*(b^40*c + 1099511627776*a^20*c^21 - 80*a*b^38*c^2 +

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

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

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

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

13056700579840*a^18*b^4*c^19 - 5497558138880*a^19*b^2*c^20)))^(1/4)*(351843720888320*a^13*c^15 + 251658240*a^

2*b^22*c^4 - 9730785280*a^3*b^20*c^5 + 167772160000*a^4*b^18*c^6 - 1691143372800*a^5*b^16*c^7 + 10952166604800

*a^6*b^14*c^8 - 46901042872320*a^7*b^12*c^9 + 129879811031040*a^8*b^10*c^10 - 206158430208000*a^9*b^8*c^11 + 8

2463372083200*a^10*b^6*c^12 + 329853488332800*a^11*b^4*c^13 - 615726511554560*a^12*b^2*c^14))/(65536*(b^18 - 2

62144*a^9*c^9 + 576*a^2*b^14*c^2 - 5376*a^3*b^12*c^3 + 32256*a^4*b^10*c^4 - 129024*a^5*b^8*c^5 + 344064*a^6*b^

6*c^6 - 589824*a^7*b^4*c^7 + 589824*a^8*b^2*c^8 - 36*a*b^16*c)) + (9*x^(1/2)*(3774873600*a^2*b^25*c^4 - 422212

4650659840*a^14*b*c^16 - 147907936256*a^3*b^23*c^5 + 2590402150400*a^4*b^21*c^6 - 26607322398720*a^5*b^19*c^7

+ 176329882337280*a^6*b^17*c^8 - 777217281884160*a^7*b^15*c^9 + 2233932749733888*a^8*b^13*c^10 - 3727344418160

640*a^9*b^11*c^11 + 1599789418414080*a^10*b^9*c^12 + 7124835347988480*a^11*b^7*c^13 - 16008889300418560*a^12*b

^5*c^14 + 13792273858822144*a^13*b^3*c^15))/(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 + 324403

20*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) + 94

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

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

^28*c^7 - 1270087680*a^7*b^26*c^8 + 8255569920*a^8*b^24*c^9 - 44029706240*a^9*b^22*c^10 + 193730707456*a^10*b^

20*c^11 - 704475299840*a^11*b^18*c^12 + 2113425899520*a^12*b^16*c^13 - 5202279137280*a^13*b^14*c^14 + 10404558

274560*a^14*b^12*c^15 - 16647293239296*a^15*b^10*c^16 + 20809116549120*a^16*b^8*c^17 - 19585050869760*a^17*b^6

*c^18 + 13056700579840*a^18*b^4*c^19 - 5497558138880*a^19*b^2*c^20)))^(3/4) + (3*(570240000*a^7*b*c^8 + 291721

5*a^2*b^11*c^3 + 49009212*a^3*b^9*c^4 + 303385824*a^4*b^7*c^5 + 879403392*a^5*b^5*c^6 + 1191801600*a^6*b^3*c^7

))/(65536*(b^18 - 262144*a^9*c^9 + 576*a^2*b^14*c^2 - 5376*a^3*b^12*c^3 + 32256*a^4*b^10*c^4 - 129024*a^5*b^8*

c^5 + 344064*a^6*b^6*c^6 - 589824*a^7*b^4*c^7 + 589824*a^8*b^2*c^8 - 36*a*b^16*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 + 2714150

40*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*(b^40*c + 1099511627776*a^20*c^21 - 80*a*b^38*c^2 + 3040*a^2*b^36*

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

a^7*b^26*c^8 + 8255569920*a^8*b^24*c^9 - 44029706240*a^9*b^22*c^10 + 193730707456*a^10*b^20*c^11 - 70447529984

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

- 16647293239296*a^15*b^10*c^16 + 20809116549120*a^16*b^8*c^17 - 19585050869760*a^17*b^6*c^18 + 1305670057984

0*a^18*b^4*c^19 - 5497558138880*a^19*b^2*c^20)))^(1/4) + (9*x^(1/2)*(43758225*a^2*b^14*c^3 - 10368000000*a^9*c

^10 + 682628310*a^3*b^12*c^4 + 4119250464*a^4*b^10*c^5 + 11404429344*a^5*b^8*c^6 + 11263650048*a^6*b^6*c^7 - 8

687347200*a^7*b^4*c^8 - 22394880000*a^8*b^2*c^9))/(4194304*(b^24 + 16777216*a^12*c^12 + 1056*a^2*b^20*c^2 - 14

080*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)))