∫ 1______√ x (a+bx2+cx4)3 dx

3.9.55 $\int \frac {1}{\sqrt {x} (a+b x^2+c x^4)^3} \, dx$

Optimal. Leaf size=658 \[ \frac {\sqrt {x} \left (60 a^2 c^2+b c x^2 \left (7 b^2-52 a c\right )-55 a b^2 c+7 b^4\right )}{16 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {3 c^{3/4} \left (280 a^2 c^2-66 a b^2 c-b \left (7 b^2-52 a c\right ) \sqrt {b^2-4 a c}+7 b^4\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} a^2 \left (b^2-4 a c\right )^{5/2} \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {3 c^{3/4} \left (280 a^2 c^2-66 a b^2 c+b \left (7 b^2-52 a c\right ) \sqrt {b^2-4 a c}+7 b^4\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} a^2 \left (b^2-4 a c\right )^{5/2} \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}+\frac {3 c^{3/4} \left (280 a^2 c^2-66 a b^2 c-b \left (7 b^2-52 a c\right ) \sqrt {b^2-4 a c}+7 b^4\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} a^2 \left (b^2-4 a c\right )^{5/2} \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {3 c^{3/4} \left (280 a^2 c^2-66 a b^2 c+b \left (7 b^2-52 a c\right ) \sqrt {b^2-4 a c}+7 b^4\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} a^2 \left (b^2-4 a c\right )^{5/2} \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}+\frac {\sqrt {x} \left (-2 a c+b^2+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

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Rubi [A] time = 5.79, antiderivative size = 658, 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, 1345, 1430, 1422, 212, 208, 205} \begin {gather*} \frac {\sqrt {x} \left (60 a^2 c^2+b c x^2 \left (7 b^2-52 a c\right )-55 a b^2 c+7 b^4\right )}{16 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {3 c^{3/4} \left (280 a^2 c^2-66 a b^2 c-b \left (7 b^2-52 a c\right ) \sqrt {b^2-4 a c}+7 b^4\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} a^2 \left (b^2-4 a c\right )^{5/2} \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {3 c^{3/4} \left (280 a^2 c^2-66 a b^2 c+b \left (7 b^2-52 a c\right ) \sqrt {b^2-4 a c}+7 b^4\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} a^2 \left (b^2-4 a c\right )^{5/2} \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}+\frac {3 c^{3/4} \left (280 a^2 c^2-66 a b^2 c-b \left (7 b^2-52 a c\right ) \sqrt {b^2-4 a c}+7 b^4\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} a^2 \left (b^2-4 a c\right )^{5/2} \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {3 c^{3/4} \left (280 a^2 c^2-66 a b^2 c+b \left (7 b^2-52 a c\right ) \sqrt {b^2-4 a c}+7 b^4\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} a^2 \left (b^2-4 a c\right )^{5/2} \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}+\frac {\sqrt {x} \left (-2 a c+b^2+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(Sqrt[x]*(b^2 - 2*a*c + b*c*x^2))/(4*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (Sqrt[x]*(7*b^4 - 55*a*b^2*c + 6

0*a^2*c^2 + b*c*(7*b^2 - 52*a*c)*x^2))/(16*a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (3*c^(3/4)*(7*b^4 - 66*a

*b^2*c + 280*a^2*c^2 - b*(7*b^2 - 52*a*c)*Sqrt[b^2 - 4*a*c])*ArcTan[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b - Sqrt[b^2 -

4*a*c])^(1/4)])/(32*2^(1/4)*a^2*(b^2 - 4*a*c)^(5/2)*(-b - Sqrt[b^2 - 4*a*c])^(3/4)) - (3*c^(3/4)*(7*b^4 - 66*

a*b^2*c + 280*a^2*c^2 + b*(7*b^2 - 52*a*c)*Sqrt[b^2 - 4*a*c])*ArcTan[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b + Sqrt[b^2

- 4*a*c])^(1/4)])/(32*2^(1/4)*a^2*(b^2 - 4*a*c)^(5/2)*(-b + Sqrt[b^2 - 4*a*c])^(3/4)) + (3*c^(3/4)*(7*b^4 - 66

*a*b^2*c + 280*a^2*c^2 - b*(7*b^2 - 52*a*c)*Sqrt[b^2 - 4*a*c])*ArcTanh[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b - Sqrt[b^

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

66*a*b^2*c + 280*a^2*c^2 + b*(7*b^2 - 52*a*c)*Sqrt[b^2 - 4*a*c])*ArcTanh[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b + Sqrt[

b^2 - 4*a*c])^(1/4)])/(32*2^(1/4)*a^2*(b^2 - 4*a*c)^(5/2)*(-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 1345

Int[((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(b^2 - 2*a*c + b*c*x^n)*(a + b*x^

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

+ n*(p + 1)*(b^2 - 4*a*c) + b*c*(n*(2*p + 3) + 1)*x^n)*(a + b*x^n + c*x^(2*n))^(p + 1), x], x] /; FreeQ[{a, b

, c, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && ILtQ[p, -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 1430

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

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

[1/(a*n*(p + 1)*(b^2 - 4*a*c)), Int[Simp[(n*p + n + 1)*d*b^2 - a*b*e - 2*a*c*d*(2*n*p + 2*n + 1) + (2*n*p + 3*

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

n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && ILtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {x} \left (a+b x^2+c x^4\right )^3} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{\left (a+b x^4+c x^8\right )^3} \, dx,x,\sqrt {x}\right )\\ &=\frac {\sqrt {x} \left (b^2-2 a c+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\operatorname {Subst}\left (\int \frac {b^2-2 a c-8 \left (b^2-4 a c\right )-11 b c x^4}{\left (a+b x^4+c x^8\right )^2} \, dx,x,\sqrt {x}\right )}{4 a \left (b^2-4 a c\right )}\\ &=\frac {\sqrt {x} \left (b^2-2 a c+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {\sqrt {x} \left (7 b^4-55 a b^2 c+60 a^2 c^2+b c \left (7 b^2-52 a c\right ) x^2\right )}{16 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\operatorname {Subst}\left (\int \frac {3 \left (7 b^4-59 a b^2 c+140 a^2 c^2\right )+3 b c \left (7 b^2-52 a c\right ) x^4}{a+b x^4+c x^8} \, dx,x,\sqrt {x}\right )}{16 a^2 \left (b^2-4 a c\right )^2}\\ &=\frac {\sqrt {x} \left (b^2-2 a c+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {\sqrt {x} \left (7 b^4-55 a b^2 c+60 a^2 c^2+b c \left (7 b^2-52 a c\right ) x^2\right )}{16 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {\left (3 c \left (7 b^4-66 a b^2 c+280 a^2 c^2-b \left (7 b^2-52 a c\right ) \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 a^2 \left (b^2-4 a c\right )^{5/2}}+\frac {\left (3 c \left (7 b^4-66 a b^2 c+280 a^2 c^2+b \left (7 b^2-52 a c\right ) \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 a^2 \left (b^2-4 a c\right )^{5/2}}\\ &=\frac {\sqrt {x} \left (b^2-2 a c+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {\sqrt {x} \left (7 b^4-55 a b^2 c+60 a^2 c^2+b c \left (7 b^2-52 a c\right ) x^2\right )}{16 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\left (3 c \left (7 b^4-66 a b^2 c+280 a^2 c^2-b \left (7 b^2-52 a c\right ) \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 a^2 \left (b^2-4 a c\right )^{5/2} \sqrt {-b-\sqrt {b^2-4 a c}}}+\frac {\left (3 c \left (7 b^4-66 a b^2 c+280 a^2 c^2-b \left (7 b^2-52 a c\right ) \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 a^2 \left (b^2-4 a c\right )^{5/2} \sqrt {-b-\sqrt {b^2-4 a c}}}-\frac {\left (3 c \left (7 b^4-66 a b^2 c+280 a^2 c^2+b \left (7 b^2-52 a c\right ) \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 a^2 \left (b^2-4 a c\right )^{5/2} \sqrt {-b+\sqrt {b^2-4 a c}}}-\frac {\left (3 c \left (7 b^4-66 a b^2 c+280 a^2 c^2+b \left (7 b^2-52 a c\right ) \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 a^2 \left (b^2-4 a c\right )^{5/2} \sqrt {-b+\sqrt {b^2-4 a c}}}\\ &=\frac {\sqrt {x} \left (b^2-2 a c+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {\sqrt {x} \left (7 b^4-55 a b^2 c+60 a^2 c^2+b c \left (7 b^2-52 a c\right ) x^2\right )}{16 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {3 c^{3/4} \left (7 b^4-66 a b^2 c+280 a^2 c^2-b \left (7 b^2-52 a c\right ) \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{32 \sqrt [4]{2} a^2 \left (b^2-4 a c\right )^{5/2} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {3 c^{3/4} \left (7 b^4-66 a b^2 c+280 a^2 c^2+b \left (7 b^2-52 a c\right ) \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{32 \sqrt [4]{2} a^2 \left (b^2-4 a c\right )^{5/2} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4}}+\frac {3 c^{3/4} \left (7 b^4-66 a b^2 c+280 a^2 c^2-b \left (7 b^2-52 a c\right ) \sqrt {b^2-4 a c}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{32 \sqrt [4]{2} a^2 \left (b^2-4 a c\right )^{5/2} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {3 c^{3/4} \left (7 b^4-66 a b^2 c+280 a^2 c^2+b \left (7 b^2-52 a c\right ) \sqrt {b^2-4 a c}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{32 \sqrt [4]{2} a^2 \left (b^2-4 a c\right )^{5/2} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4}}\\ \end {align*}

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Mathematica [C] time = 0.46, size = 258, normalized size = 0.39 \begin {gather*} \frac {3 \text {RootSum}\left [\text {$\#$1}^8 c+\text {$\#$1}^4 b+a\&,\frac {-52 \text {$\#$1}^4 a b c^2 \log \left (\sqrt {x}-\text {$\#$1}\right )+7 \text {$\#$1}^4 b^3 c \log \left (\sqrt {x}-\text {$\#$1}\right )+140 a^2 c^2 \log \left (\sqrt {x}-\text {$\#$1}\right )-59 a b^2 c \log \left (\sqrt {x}-\text {$\#$1}\right )+7 b^4 \log \left (\sqrt {x}-\text {$\#$1}\right )}{2 \text {$\#$1}^7 c+\text {$\#$1}^3 b}\&\right ]+\frac {4 \sqrt {x} \left (60 a^2 c^2-55 a b^2 c-52 a b c^2 x^2+7 b^4+7 b^3 c x^2\right )}{a+b x^2+c x^4}-\frac {16 a \sqrt {x} \left (4 a c-b^2\right ) \left (-2 a c+b^2+b c x^2\right )}{\left (a+b x^2+c x^4\right )^2}}{64 a^2 \left (b^2-4 a c\right )^2} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((-16*a*(-b^2 + 4*a*c)*Sqrt[x]*(b^2 - 2*a*c + b*c*x^2))/(a + b*x^2 + c*x^4)^2 + (4*Sqrt[x]*(7*b^4 - 55*a*b^2*c

+ 60*a^2*c^2 + 7*b^3*c*x^2 - 52*a*b*c^2*x^2))/(a + b*x^2 + c*x^4) + 3*RootSum[a + b*#1^4 + c*#1^8 & , (7*b^4*

Log[Sqrt[x] - #1] - 59*a*b^2*c*Log[Sqrt[x] - #1] + 140*a^2*c^2*Log[Sqrt[x] - #1] + 7*b^3*c*Log[Sqrt[x] - #1]*#

1^4 - 52*a*b*c^2*Log[Sqrt[x] - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ])/(64*a^2*(b^2 - 4*a*c)^2)

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IntegrateAlgebraic [C] time = 0.59, size = 296, normalized size = 0.45 \begin {gather*} \frac {3 \text {RootSum}\left [\text {$\#$1}^8 c+\text {$\#$1}^4 b+a\&,\frac {-52 \text {$\#$1}^4 a b c^2 \log \left (\sqrt {x}-\text {$\#$1}\right )+7 \text {$\#$1}^4 b^3 c \log \left (\sqrt {x}-\text {$\#$1}\right )+140 a^2 c^2 \log \left (\sqrt {x}-\text {$\#$1}\right )-59 a b^2 c \log \left (\sqrt {x}-\text {$\#$1}\right )+7 b^4 \log \left (\sqrt {x}-\text {$\#$1}\right )}{2 \text {$\#$1}^7 c+\text {$\#$1}^3 b}\&\right ]}{64 a^2 \left (4 a c-b^2\right )^2}+\frac {\sqrt {x} \left (92 a^3 c^2-79 a^2 b^2 c-8 a^2 b c^2 x^2+60 a^2 c^3 x^4+11 a b^4-44 a b^3 c x^2-107 a b^2 c^2 x^4-52 a b c^3 x^6+7 b^5 x^2+14 b^4 c x^4+7 b^3 c^2 x^6\right )}{16 a^2 \left (4 a c-b^2\right )^2 \left (a+b x^2+c x^4\right )^2} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(Sqrt[x]*(11*a*b^4 - 79*a^2*b^2*c + 92*a^3*c^2 + 7*b^5*x^2 - 44*a*b^3*c*x^2 - 8*a^2*b*c^2*x^2 + 14*b^4*c*x^4 -

107*a*b^2*c^2*x^4 + 60*a^2*c^3*x^4 + 7*b^3*c^2*x^6 - 52*a*b*c^3*x^6))/(16*a^2*(-b^2 + 4*a*c)^2*(a + b*x^2 + c

*x^4)^2) + (3*RootSum[a + b*#1^4 + c*#1^8 & , (7*b^4*Log[Sqrt[x] - #1] - 59*a*b^2*c*Log[Sqrt[x] - #1] + 140*a^

2*c^2*Log[Sqrt[x] - #1] + 7*b^3*c*Log[Sqrt[x] - #1]*#1^4 - 52*a*b*c^2*Log[Sqrt[x] - #1]*#1^4)/(b*#1^3 + 2*c*#1

^7) & ])/(64*a^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(1/x^(1/2)/(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(1/x^(1/2)/(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: 191.11Unable to convert to real 1/4 Error: Bad Argument Value

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maple [C] time = 0.04, size = 316, normalized size = 0.48 \begin {gather*} \frac {3 \left (\left (-52 a c +7 b^{2}\right ) \RootOf \left (c \,\textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+a \right )^{4} b c +140 a^{2} c^{2}-59 a \,b^{2} c +7 b^{4}\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 ) a^{2} \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 {\left (52 a c -7 b^{2}\right ) b \,c^{2} x^{\frac {13}{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{2}}+\frac {\left (60 a^{2} c^{2}-107 a \,b^{2} c +14 b^{4}\right ) c \,x^{\frac {9}{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{2}}-\frac {\left (8 a^{2} c^{2}+44 a \,b^{2} c -7 b^{4}\right ) b \,x^{\frac {5}{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{2}}+\frac {\left (92 a^{2} c^{2}-79 a \,b^{2} c +11 b^{4}\right ) \sqrt {x}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(1/2)/(c*x^4+b*x^2+a)^3,x)

[Out]

2*(1/32*(92*a^2*c^2-79*a*b^2*c+11*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)/a*x^(1/2)-1/32*b*(8*a^2*c^2+44*a*b^2*c-7*b^4

)/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(5/2)+1/32/a^2*c*(60*a^2*c^2-107*a*b^2*c+14*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)

*x^(9/2)-1/32*c^2*b*(52*a*c-7*b^2)/a^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/a^2/(16*a^2

*c^2-8*a*b^2*c+b^4)*sum((b*c*(-52*a*c+7*b^2)*_R^4+140*a^2*c^2-59*a*b^2*c+7*b^4)/(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 {3 \, {\left (7 \, b^{4} c^{2} - 59 \, a b^{2} c^{3} + 140 \, a^{2} c^{4}\right )} x^{\frac {17}{2}} + {\left (42 \, b^{5} c - 347 \, a b^{3} c^{2} + 788 \, a^{2} b c^{3}\right )} x^{\frac {13}{2}} + {\left (21 \, b^{6} - 121 \, a b^{4} c - 41 \, a^{2} b^{2} c^{2} + 900 \, a^{3} c^{3}\right )} x^{\frac {9}{2}} + {\left (49 \, a b^{5} - 398 \, a^{2} b^{3} c + 832 \, a^{3} b c^{2}\right )} x^{\frac {5}{2}} + 32 \, {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2}\right )} \sqrt {x}}{16 \, {\left (a^{5} b^{4} - 8 \, a^{6} b^{2} c + 16 \, a^{7} c^{2} + {\left (a^{3} b^{4} c^{2} - 8 \, a^{4} b^{2} c^{3} + 16 \, a^{5} c^{4}\right )} x^{8} + 2 \, {\left (a^{3} b^{5} c - 8 \, a^{4} b^{3} c^{2} + 16 \, a^{5} b c^{3}\right )} x^{6} + {\left (a^{3} b^{6} - 6 \, a^{4} b^{4} c + 32 \, a^{6} c^{3}\right )} x^{4} + 2 \, {\left (a^{4} b^{5} - 8 \, a^{5} b^{3} c + 16 \, a^{6} b c^{2}\right )} x^{2}\right )}} - \int \frac {3 \, {\left ({\left (7 \, b^{4} c - 59 \, a b^{2} c^{2} + 140 \, a^{2} c^{3}\right )} x^{\frac {7}{2}} + {\left (7 \, b^{5} - 66 \, a b^{3} c + 192 \, a^{2} b c^{2}\right )} x^{\frac {3}{2}}\right )}}{32 \, {\left (a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2} + {\left (a^{3} b^{4} c - 8 \, a^{4} b^{2} c^{2} + 16 \, a^{5} c^{3}\right )} x^{4} + {\left (a^{3} b^{5} - 8 \, a^{4} b^{3} c + 16 \, a^{5} b c^{2}\right )} x^{2}\right )}}\,{d x} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(1/2)/(c*x^4+b*x^2+a)^3,x, algorithm="maxima")

[Out]

1/16*(3*(7*b^4*c^2 - 59*a*b^2*c^3 + 140*a^2*c^4)*x^(17/2) + (42*b^5*c - 347*a*b^3*c^2 + 788*a^2*b*c^3)*x^(13/2

) + (21*b^6 - 121*a*b^4*c - 41*a^2*b^2*c^2 + 900*a^3*c^3)*x^(9/2) + (49*a*b^5 - 398*a^2*b^3*c + 832*a^3*b*c^2)

*x^(5/2) + 32*(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*sqrt(x))/(a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2 + (a^3*b^4*c^2

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

+ 32*a^6*c^3)*x^4 + 2*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*x^2) - integrate(3/32*((7*b^4*c - 59*a*b^2*c^2 +

140*a^2*c^3)*x^(7/2) + (7*b^5 - 66*a*b^3*c + 192*a^2*b*c^2)*x^(3/2))/(a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2 + (a

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

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mupad [B] time = 9.85, size = 60099, normalized size = 91.34

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(1/2)*(a + b*x^2 + c*x^4)^3),x)

[Out]

((x^(9/2)*(14*b^4*c + 60*a^2*c^3 - 107*a*b^2*c^2))/(16*a^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x^(1/2)*(11*b^4

+ 92*a^2*c^2 - 79*a*b^2*c))/(16*a*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) - (x^(5/2)*(8*a^2*b*c^2 - 7*b^5 + 44*a*b^3*c

))/(16*a^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) - (b*c^2*x^(13/2)*(52*a*c - 7*b^2))/(16*a^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((((((9*x^(1/2)*(15467049970250547

20*a^19*b*c^19 - 822083584*a^4*b^31*c^4 + 50851741696*a^5*b^29*c^5 - 1473677099008*a^6*b^27*c^6 + 265236879769

60*a^7*b^25*c^7 - 331351626612736*a^8*b^23*c^8 + 3041476258824192*a^9*b^21*c^9 - 21176692735213568*a^10*b^19*c

^10 + 113812892427485184*a^11*b^17*c^11 - 475720885626470400*a^12*b^15*c^12 + 1545406748670558208*a^13*b^13*c^

13 - 3867206695260258304*a^14*b^11*c^14 + 7315227880965799936*a^15*b^9*c^15 - 10117494892562219008*a^16*b^7*c^

16 + 9650897342106173440*a^17*b^5*c^17 - 5672002255696429056*a^18*b^3*c^18))/(4194304*(a^8*b^24 + 16777216*a^2

0*c^12 - 48*a^9*b^22*c + 1056*a^10*b^20*c^2 - 14080*a^11*b^18*c^3 + 126720*a^12*b^16*c^4 - 811008*a^13*b^14*c^

5 + 3784704*a^14*b^12*c^6 - 12976128*a^15*b^10*c^7 + 32440320*a^16*b^8*c^8 - 57671680*a^17*b^6*c^9 + 69206016*

a^18*b^4*c^10 - 50331648*a^19*b^2*c^11)) - (3*(-(81*(2401*b^39 - 2401*b^14*(-(4*a*c - b^2)^25)^(1/2) - 2405416

566784000*a^19*b*c^19 + 7445060*a^2*b^35*c^2 - 180851965*a^3*b^33*c^3 + 3112544495*a^4*b^31*c^4 - 40302663491*

a^5*b^29*c^5 + 406936342200*a^6*b^27*c^6 - 3276813600400*a^7*b^25*c^7 + 21341140889600*a^8*b^23*c^8 - 11333074

8025600*a^9*b^21*c^9 + 492398189373440*a^10*b^19*c^10 - 1748923551027200*a^11*b^17*c^11 + 5052644161945600*a^1

2*b^15*c^12 - 11756581147443200*a^13*b^13*c^13 + 21683350423470080*a^14*b^11*c^14 - 30929025701511168*a^15*b^9

*c^15 + 32836636093972480*a^16*b^7*c^16 - 24359874477424640*a^17*b^5*c^17 + 11224950044098560*a^18*b^3*c^18 +

24010000*a^7*c^7*(-(4*a*c - b^2)^25)^(1/2) - 193795*a*b^37*c - 996660*a^2*b^10*c^2*(-(4*a*c - b^2)^25)^(1/2) +

7556115*a^3*b^8*c^3*(-(4*a*c - b^2)^25)^(1/2) - 34052295*a^4*b^6*c^4*(-(4*a*c - b^2)^25)^(1/2) + 87808681*a^5

*b^4*c^5*(-(4*a*c - b^2)^25)^(1/2) - 108025400*a^6*b^2*c^6*(-(4*a*c - b^2)^25)^(1/2) + 73745*a*b^12*c*(-(4*a*c

- b^2)^25)^(1/2)))/(33554432*(a^11*b^40 + 1099511627776*a^31*c^20 - 80*a^12*b^38*c + 3040*a^13*b^36*c^2 - 729

60*a^14*b^34*c^3 + 1240320*a^15*b^32*c^4 - 15876096*a^16*b^30*c^5 + 158760960*a^17*b^28*c^6 - 1270087680*a^18*

b^26*c^7 + 8255569920*a^19*b^24*c^8 - 44029706240*a^20*b^22*c^9 + 193730707456*a^21*b^20*c^10 - 704475299840*a

^22*b^18*c^11 + 2113425899520*a^23*b^16*c^12 - 5202279137280*a^24*b^14*c^13 + 10404558274560*a^25*b^12*c^14 -

16647293239296*a^26*b^10*c^15 + 20809116549120*a^27*b^8*c^16 - 19585050869760*a^28*b^6*c^17 + 13056700579840*a

^29*b^4*c^18 - 5497558138880*a^30*b^2*c^19)))^(1/4)*(3377699720527872*a^19*b*c^16 + 117440512*a^7*b^25*c^4 - 5

804916736*a^8*b^23*c^5 + 132070244352*a^9*b^21*c^6 - 1828045455360*a^10*b^19*c^7 + 17136919511040*a^11*b^17*c^

8 - 114572547588096*a^12*b^15*c^9 + 559926296444928*a^13*b^13*c^10 - 2014580179992576*a^14*b^11*c^11 + 5294148

487741440*a^15*b^9*c^12 - 9906599766261760*a^16*b^7*c^13 + 12525636463624192*a^17*b^5*c^14 - 9605333580251136*

a^18*b^3*c^15))/(65536*(a^8*b^18 - 262144*a^17*c^9 - 36*a^9*b^16*c + 576*a^10*b^14*c^2 - 5376*a^11*b^12*c^3 +

32256*a^12*b^10*c^4 - 129024*a^13*b^8*c^5 + 344064*a^14*b^6*c^6 - 589824*a^15*b^4*c^7 + 589824*a^16*b^2*c^8)))

*(-(81*(2401*b^39 - 2401*b^14*(-(4*a*c - b^2)^25)^(1/2) - 2405416566784000*a^19*b*c^19 + 7445060*a^2*b^35*c^2

- 180851965*a^3*b^33*c^3 + 3112544495*a^4*b^31*c^4 - 40302663491*a^5*b^29*c^5 + 406936342200*a^6*b^27*c^6 - 32

76813600400*a^7*b^25*c^7 + 21341140889600*a^8*b^23*c^8 - 113330748025600*a^9*b^21*c^9 + 492398189373440*a^10*b

^19*c^10 - 1748923551027200*a^11*b^17*c^11 + 5052644161945600*a^12*b^15*c^12 - 11756581147443200*a^13*b^13*c^1

3 + 21683350423470080*a^14*b^11*c^14 - 30929025701511168*a^15*b^9*c^15 + 32836636093972480*a^16*b^7*c^16 - 243

59874477424640*a^17*b^5*c^17 + 11224950044098560*a^18*b^3*c^18 + 24010000*a^7*c^7*(-(4*a*c - b^2)^25)^(1/2) -

193795*a*b^37*c - 996660*a^2*b^10*c^2*(-(4*a*c - b^2)^25)^(1/2) + 7556115*a^3*b^8*c^3*(-(4*a*c - b^2)^25)^(1/2

) - 34052295*a^4*b^6*c^4*(-(4*a*c - b^2)^25)^(1/2) + 87808681*a^5*b^4*c^5*(-(4*a*c - b^2)^25)^(1/2) - 10802540

0*a^6*b^2*c^6*(-(4*a*c - b^2)^25)^(1/2) + 73745*a*b^12*c*(-(4*a*c - b^2)^25)^(1/2)))/(33554432*(a^11*b^40 + 10

99511627776*a^31*c^20 - 80*a^12*b^38*c + 3040*a^13*b^36*c^2 - 72960*a^14*b^34*c^3 + 1240320*a^15*b^32*c^4 - 15

876096*a^16*b^30*c^5 + 158760960*a^17*b^28*c^6 - 1270087680*a^18*b^26*c^7 + 8255569920*a^19*b^24*c^8 - 4402970

6240*a^20*b^22*c^9 + 193730707456*a^21*b^20*c^10 - 704475299840*a^22*b^18*c^11 + 2113425899520*a^23*b^16*c^12

- 5202279137280*a^24*b^14*c^13 + 10404558274560*a^25*b^12*c^14 - 16647293239296*a^26*b^10*c^15 + 2080911654912

0*a^27*b^8*c^16 - 19585050869760*a^28*b^6*c^17 + 13056700579840*a^29*b^4*c^18 - 5497558138880*a^30*b^2*c^19)))

^(3/4) + (3*(4356374400000*a^8*c^16 + 18475695*b^16*c^8 - 685712223*a*b^14*c^9 + 11424393414*a^2*b^12*c^10 - 1

10892005343*a^3*b^10*c^11 + 681741235260*a^4*b^8*c^12 - 2694857597280*a^5*b^6*c^13 + 6582295198080*a^6*b^4*c^1

4 - 8763424992000*a^7*b^2*c^15))/(65536*(a^8*b^18 - 262144*a^17*c^9 - 36*a^9*b^16*c + 576*a^10*b^14*c^2 - 5376

*a^11*b^12*c^3 + 32256*a^12*b^10*c^4 - 129024*a^13*b^8*c^5 + 344064*a^14*b^6*c^6 - 589824*a^15*b^4*c^7 + 58982

4*a^16*b^2*c^8)))*(-(81*(2401*b^39 - 2401*b^14*(-(4*a*c - b^2)^25)^(1/2) - 2405416566784000*a^19*b*c^19 + 7445

060*a^2*b^35*c^2 - 180851965*a^3*b^33*c^3 + 3112544495*a^4*b^31*c^4 - 40302663491*a^5*b^29*c^5 + 406936342200*

a^6*b^27*c^6 - 3276813600400*a^7*b^25*c^7 + 21341140889600*a^8*b^23*c^8 - 113330748025600*a^9*b^21*c^9 + 49239

8189373440*a^10*b^19*c^10 - 1748923551027200*a^11*b^17*c^11 + 5052644161945600*a^12*b^15*c^12 - 11756581147443

200*a^13*b^13*c^13 + 21683350423470080*a^14*b^11*c^14 - 30929025701511168*a^15*b^9*c^15 + 32836636093972480*a^

16*b^7*c^16 - 24359874477424640*a^17*b^5*c^17 + 11224950044098560*a^18*b^3*c^18 + 24010000*a^7*c^7*(-(4*a*c -

b^2)^25)^(1/2) - 193795*a*b^37*c - 996660*a^2*b^10*c^2*(-(4*a*c - b^2)^25)^(1/2) + 7556115*a^3*b^8*c^3*(-(4*a*

c - b^2)^25)^(1/2) - 34052295*a^4*b^6*c^4*(-(4*a*c - b^2)^25)^(1/2) + 87808681*a^5*b^4*c^5*(-(4*a*c - b^2)^25)

^(1/2) - 108025400*a^6*b^2*c^6*(-(4*a*c - b^2)^25)^(1/2) + 73745*a*b^12*c*(-(4*a*c - b^2)^25)^(1/2)))/(3355443

2*(a^11*b^40 + 1099511627776*a^31*c^20 - 80*a^12*b^38*c + 3040*a^13*b^36*c^2 - 72960*a^14*b^34*c^3 + 1240320*a

^15*b^32*c^4 - 15876096*a^16*b^30*c^5 + 158760960*a^17*b^28*c^6 - 1270087680*a^18*b^26*c^7 + 8255569920*a^19*b

^24*c^8 - 44029706240*a^20*b^22*c^9 + 193730707456*a^21*b^20*c^10 - 704475299840*a^22*b^18*c^11 + 211342589952

0*a^23*b^16*c^12 - 5202279137280*a^24*b^14*c^13 + 10404558274560*a^25*b^12*c^14 - 16647293239296*a^26*b^10*c^1

5 + 20809116549120*a^27*b^8*c^16 - 19585050869760*a^28*b^6*c^17 + 13056700579840*a^29*b^4*c^18 - 5497558138880

*a^30*b^2*c^19)))^(1/4) - (9*x^(1/2)*(1219784832000000*a^8*c^19 + 1755191025*b^16*c^11 - 67599928620*a*b^14*c^

12 + 1172433971394*a^2*b^12*c^13 - 11911732472304*a^3*b^10*c^14 + 77626373024736*a^4*b^8*c^15 - 33360325130188

8*a^5*b^6*c^16 + 930302051212800*a^6*b^4*c^17 - 1556843742720000*a^7*b^2*c^18))/(4194304*(a^8*b^24 + 16777216*

a^20*c^12 - 48*a^9*b^22*c + 1056*a^10*b^20*c^2 - 14080*a^11*b^18*c^3 + 126720*a^12*b^16*c^4 - 811008*a^13*b^14

*c^5 + 3784704*a^14*b^12*c^6 - 12976128*a^15*b^10*c^7 + 32440320*a^16*b^8*c^8 - 57671680*a^17*b^6*c^9 + 692060

16*a^18*b^4*c^10 - 50331648*a^19*b^2*c^11)))*(-(81*(2401*b^39 - 2401*b^14*(-(4*a*c - b^2)^25)^(1/2) - 24054165

66784000*a^19*b*c^19 + 7445060*a^2*b^35*c^2 - 180851965*a^3*b^33*c^3 + 3112544495*a^4*b^31*c^4 - 40302663491*a

^5*b^29*c^5 + 406936342200*a^6*b^27*c^6 - 3276813600400*a^7*b^25*c^7 + 21341140889600*a^8*b^23*c^8 - 113330748

025600*a^9*b^21*c^9 + 492398189373440*a^10*b^19*c^10 - 1748923551027200*a^11*b^17*c^11 + 5052644161945600*a^12

*b^15*c^12 - 11756581147443200*a^13*b^13*c^13 + 21683350423470080*a^14*b^11*c^14 - 30929025701511168*a^15*b^9*

c^15 + 32836636093972480*a^16*b^7*c^16 - 24359874477424640*a^17*b^5*c^17 + 11224950044098560*a^18*b^3*c^18 + 2

4010000*a^7*c^7*(-(4*a*c - b^2)^25)^(1/2) - 193795*a*b^37*c - 996660*a^2*b^10*c^2*(-(4*a*c - b^2)^25)^(1/2) +

7556115*a^3*b^8*c^3*(-(4*a*c - b^2)^25)^(1/2) - 34052295*a^4*b^6*c^4*(-(4*a*c - b^2)^25)^(1/2) + 87808681*a^5*

b^4*c^5*(-(4*a*c - b^2)^25)^(1/2) - 108025400*a^6*b^2*c^6*(-(4*a*c - b^2)^25)^(1/2) + 73745*a*b^12*c*(-(4*a*c

- b^2)^25)^(1/2)))/(33554432*(a^11*b^40 + 1099511627776*a^31*c^20 - 80*a^12*b^38*c + 3040*a^13*b^36*c^2 - 7296

0*a^14*b^34*c^3 + 1240320*a^15*b^32*c^4 - 15876096*a^16*b^30*c^5 + 158760960*a^17*b^28*c^6 - 1270087680*a^18*b

^26*c^7 + 8255569920*a^19*b^24*c^8 - 44029706240*a^20*b^22*c^9 + 193730707456*a^21*b^20*c^10 - 704475299840*a^

22*b^18*c^11 + 2113425899520*a^23*b^16*c^12 - 5202279137280*a^24*b^14*c^13 + 10404558274560*a^25*b^12*c^14 - 1

6647293239296*a^26*b^10*c^15 + 20809116549120*a^27*b^8*c^16 - 19585050869760*a^28*b^6*c^17 + 13056700579840*a^

29*b^4*c^18 - 5497558138880*a^30*b^2*c^19)))^(1/4)*1i + ((((9*x^(1/2)*(1546704997025054720*a^19*b*c^19 - 82208

3584*a^4*b^31*c^4 + 50851741696*a^5*b^29*c^5 - 1473677099008*a^6*b^27*c^6 + 26523687976960*a^7*b^25*c^7 - 3313

51626612736*a^8*b^23*c^8 + 3041476258824192*a^9*b^21*c^9 - 21176692735213568*a^10*b^19*c^10 + 1138128924274851

84*a^11*b^17*c^11 - 475720885626470400*a^12*b^15*c^12 + 1545406748670558208*a^13*b^13*c^13 - 38672066952602583

04*a^14*b^11*c^14 + 7315227880965799936*a^15*b^9*c^15 - 10117494892562219008*a^16*b^7*c^16 + 96508973421061734

40*a^17*b^5*c^17 - 5672002255696429056*a^18*b^3*c^18))/(4194304*(a^8*b^24 + 16777216*a^20*c^12 - 48*a^9*b^22*c

+ 1056*a^10*b^20*c^2 - 14080*a^11*b^18*c^3 + 126720*a^12*b^16*c^4 - 811008*a^13*b^14*c^5 + 3784704*a^14*b^12*

c^6 - 12976128*a^15*b^10*c^7 + 32440320*a^16*b^8*c^8 - 57671680*a^17*b^6*c^9 + 69206016*a^18*b^4*c^10 - 503316

48*a^19*b^2*c^11)) + (3*(-(81*(2401*b^39 - 2401*b^14*(-(4*a*c - b^2)^25)^(1/2) - 2405416566784000*a^19*b*c^19

+ 7445060*a^2*b^35*c^2 - 180851965*a^3*b^33*c^3 + 3112544495*a^4*b^31*c^4 - 40302663491*a^5*b^29*c^5 + 4069363

42200*a^6*b^27*c^6 - 3276813600400*a^7*b^25*c^7 + 21341140889600*a^8*b^23*c^8 - 113330748025600*a^9*b^21*c^9 +

492398189373440*a^10*b^19*c^10 - 1748923551027200*a^11*b^17*c^11 + 5052644161945600*a^12*b^15*c^12 - 11756581

147443200*a^13*b^13*c^13 + 21683350423470080*a^14*b^11*c^14 - 30929025701511168*a^15*b^9*c^15 + 32836636093972

480*a^16*b^7*c^16 - 24359874477424640*a^17*b^5*c^17 + 11224950044098560*a^18*b^3*c^18 + 24010000*a^7*c^7*(-(4*

a*c - b^2)^25)^(1/2) - 193795*a*b^37*c - 996660*a^2*b^10*c^2*(-(4*a*c - b^2)^25)^(1/2) + 7556115*a^3*b^8*c^3*(

-(4*a*c - b^2)^25)^(1/2) - 34052295*a^4*b^6*c^4*(-(4*a*c - b^2)^25)^(1/2) + 87808681*a^5*b^4*c^5*(-(4*a*c - b^

2)^25)^(1/2) - 108025400*a^6*b^2*c^6*(-(4*a*c - b^2)^25)^(1/2) + 73745*a*b^12*c*(-(4*a*c - b^2)^25)^(1/2)))/(3

3554432*(a^11*b^40 + 1099511627776*a^31*c^20 - 80*a^12*b^38*c + 3040*a^13*b^36*c^2 - 72960*a^14*b^34*c^3 + 124

0320*a^15*b^32*c^4 - 15876096*a^16*b^30*c^5 + 158760960*a^17*b^28*c^6 - 1270087680*a^18*b^26*c^7 + 8255569920*

a^19*b^24*c^8 - 44029706240*a^20*b^22*c^9 + 193730707456*a^21*b^20*c^10 - 704475299840*a^22*b^18*c^11 + 211342

5899520*a^23*b^16*c^12 - 5202279137280*a^24*b^14*c^13 + 10404558274560*a^25*b^12*c^14 - 16647293239296*a^26*b^

10*c^15 + 20809116549120*a^27*b^8*c^16 - 19585050869760*a^28*b^6*c^17 + 13056700579840*a^29*b^4*c^18 - 5497558

138880*a^30*b^2*c^19)))^(1/4)*(3377699720527872*a^19*b*c^16 + 117440512*a^7*b^25*c^4 - 5804916736*a^8*b^23*c^5

+ 132070244352*a^9*b^21*c^6 - 1828045455360*a^10*b^19*c^7 + 17136919511040*a^11*b^17*c^8 - 114572547588096*a^

12*b^15*c^9 + 559926296444928*a^13*b^13*c^10 - 2014580179992576*a^14*b^11*c^11 + 5294148487741440*a^15*b^9*c^1

2 - 9906599766261760*a^16*b^7*c^13 + 12525636463624192*a^17*b^5*c^14 - 9605333580251136*a^18*b^3*c^15))/(65536

*(a^8*b^18 - 262144*a^17*c^9 - 36*a^9*b^16*c + 576*a^10*b^14*c^2 - 5376*a^11*b^12*c^3 + 32256*a^12*b^10*c^4 -

129024*a^13*b^8*c^5 + 344064*a^14*b^6*c^6 - 589824*a^15*b^4*c^7 + 589824*a^16*b^2*c^8)))*(-(81*(2401*b^39 - 24

01*b^14*(-(4*a*c - b^2)^25)^(1/2) - 2405416566784000*a^19*b*c^19 + 7445060*a^2*b^35*c^2 - 180851965*a^3*b^33*c

^3 + 3112544495*a^4*b^31*c^4 - 40302663491*a^5*b^29*c^5 + 406936342200*a^6*b^27*c^6 - 3276813600400*a^7*b^25*c

^7 + 21341140889600*a^8*b^23*c^8 - 113330748025600*a^9*b^21*c^9 + 492398189373440*a^10*b^19*c^10 - 17489235510

27200*a^11*b^17*c^11 + 5052644161945600*a^12*b^15*c^12 - 11756581147443200*a^13*b^13*c^13 + 21683350423470080*

a^14*b^11*c^14 - 30929025701511168*a^15*b^9*c^15 + 32836636093972480*a^16*b^7*c^16 - 24359874477424640*a^17*b^

5*c^17 + 11224950044098560*a^18*b^3*c^18 + 24010000*a^7*c^7*(-(4*a*c - b^2)^25)^(1/2) - 193795*a*b^37*c - 9966

60*a^2*b^10*c^2*(-(4*a*c - b^2)^25)^(1/2) + 7556115*a^3*b^8*c^3*(-(4*a*c - b^2)^25)^(1/2) - 34052295*a^4*b^6*c

^4*(-(4*a*c - b^2)^25)^(1/2) + 87808681*a^5*b^4*c^5*(-(4*a*c - b^2)^25)^(1/2) - 108025400*a^6*b^2*c^6*(-(4*a*c

- b^2)^25)^(1/2) + 73745*a*b^12*c*(-(4*a*c - b^2)^25)^(1/2)))/(33554432*(a^11*b^40 + 1099511627776*a^31*c^20

- 80*a^12*b^38*c + 3040*a^13*b^36*c^2 - 72960*a^14*b^34*c^3 + 1240320*a^15*b^32*c^4 - 15876096*a^16*b^30*c^5 +

158760960*a^17*b^28*c^6 - 1270087680*a^18*b^26*c^7 + 8255569920*a^19*b^24*c^8 - 44029706240*a^20*b^22*c^9 + 1

93730707456*a^21*b^20*c^10 - 704475299840*a^22*b^18*c^11 + 2113425899520*a^23*b^16*c^12 - 5202279137280*a^24*b

^14*c^13 + 10404558274560*a^25*b^12*c^14 - 16647293239296*a^26*b^10*c^15 + 20809116549120*a^27*b^8*c^16 - 1958

5050869760*a^28*b^6*c^17 + 13056700579840*a^29*b^4*c^18 - 5497558138880*a^30*b^2*c^19)))^(3/4) - (3*(435637440

0000*a^8*c^16 + 18475695*b^16*c^8 - 685712223*a*b^14*c^9 + 11424393414*a^2*b^12*c^10 - 110892005343*a^3*b^10*c

^11 + 681741235260*a^4*b^8*c^12 - 2694857597280*a^5*b^6*c^13 + 6582295198080*a^6*b^4*c^14 - 8763424992000*a^7*

b^2*c^15))/(65536*(a^8*b^18 - 262144*a^17*c^9 - 36*a^9*b^16*c + 576*a^10*b^14*c^2 - 5376*a^11*b^12*c^3 + 32256

*a^12*b^10*c^4 - 129024*a^13*b^8*c^5 + 344064*a^14*b^6*c^6 - 589824*a^15*b^4*c^7 + 589824*a^16*b^2*c^8)))*(-(8

1*(2401*b^39 - 2401*b^14*(-(4*a*c - b^2)^25)^(1/2) - 2405416566784000*a^19*b*c^19 + 7445060*a^2*b^35*c^2 - 180

851965*a^3*b^33*c^3 + 3112544495*a^4*b^31*c^4 - 40302663491*a^5*b^29*c^5 + 406936342200*a^6*b^27*c^6 - 3276813

600400*a^7*b^25*c^7 + 21341140889600*a^8*b^23*c^8 - 113330748025600*a^9*b^21*c^9 + 492398189373440*a^10*b^19*c

^10 - 1748923551027200*a^11*b^17*c^11 + 5052644161945600*a^12*b^15*c^12 - 11756581147443200*a^13*b^13*c^13 + 2

1683350423470080*a^14*b^11*c^14 - 30929025701511168*a^15*b^9*c^15 + 32836636093972480*a^16*b^7*c^16 - 24359874

477424640*a^17*b^5*c^17 + 11224950044098560*a^18*b^3*c^18 + 24010000*a^7*c^7*(-(4*a*c - b^2)^25)^(1/2) - 19379

5*a*b^37*c - 996660*a^2*b^10*c^2*(-(4*a*c - b^2)^25)^(1/2) + 7556115*a^3*b^8*c^3*(-(4*a*c - b^2)^25)^(1/2) - 3

4052295*a^4*b^6*c^4*(-(4*a*c - b^2)^25)^(1/2) + 87808681*a^5*b^4*c^5*(-(4*a*c - b^2)^25)^(1/2) - 108025400*a^6

*b^2*c^6*(-(4*a*c - b^2)^25)^(1/2) + 73745*a*b^12*c*(-(4*a*c - b^2)^25)^(1/2)))/(33554432*(a^11*b^40 + 1099511

627776*a^31*c^20 - 80*a^12*b^38*c + 3040*a^13*b^36*c^2 - 72960*a^14*b^34*c^3 + 1240320*a^15*b^32*c^4 - 1587609

6*a^16*b^30*c^5 + 158760960*a^17*b^28*c^6 - 1270087680*a^18*b^26*c^7 + 8255569920*a^19*b^24*c^8 - 44029706240*

a^20*b^22*c^9 + 193730707456*a^21*b^20*c^10 - 704475299840*a^22*b^18*c^11 + 2113425899520*a^23*b^16*c^12 - 520

2279137280*a^24*b^14*c^13 + 10404558274560*a^25*b^12*c^14 - 16647293239296*a^26*b^10*c^15 + 20809116549120*a^2

7*b^8*c^16 - 19585050869760*a^28*b^6*c^17 + 13056700579840*a^29*b^4*c^18 - 5497558138880*a^30*b^2*c^19)))^(1/4

) - (9*x^(1/2)*(1219784832000000*a^8*c^19 + 1755191025*b^16*c^11 - 67599928620*a*b^14*c^12 + 1172433971394*a^2

*b^12*c^13 - 11911732472304*a^3*b^10*c^14 + 77626373024736*a^4*b^8*c^15 - 333603251301888*a^5*b^6*c^16 + 93030

2051212800*a^6*b^4*c^17 - 1556843742720000*a^7*b^2*c^18))/(4194304*(a^8*b^24 + 16777216*a^20*c^12 - 48*a^9*b^2

2*c + 1056*a^10*b^20*c^2 - 14080*a^11*b^18*c^3 + 126720*a^12*b^16*c^4 - 811008*a^13*b^14*c^5 + 3784704*a^14*b^

12*c^6 - 12976128*a^15*b^10*c^7 + 32440320*a^16*b^8*c^8 - 57671680*a^17*b^6*c^9 + 69206016*a^18*b^4*c^10 - 503

31648*a^19*b^2*c^11)))*(-(81*(2401*b^39 - 2401*b^14*(-(4*a*c - b^2)^25)^(1/2) - 2405416566784000*a^19*b*c^19 +

7445060*a^2*b^35*c^2 - 180851965*a^3*b^33*c^3 + 3112544495*a^4*b^31*c^4 - 40302663491*a^5*b^29*c^5 + 40693634

2200*a^6*b^27*c^6 - 3276813600400*a^7*b^25*c^7 + 21341140889600*a^8*b^23*c^8 - 113330748025600*a^9*b^21*c^9 +

492398189373440*a^10*b^19*c^10 - 1748923551027200*a^11*b^17*c^11 + 5052644161945600*a^12*b^15*c^12 - 117565811

47443200*a^13*b^13*c^13 + 21683350423470080*a^14*b^11*c^14 - 30929025701511168*a^15*b^9*c^15 + 328366360939724

80*a^16*b^7*c^16 - 24359874477424640*a^17*b^5*c^17 + 11224950044098560*a^18*b^3*c^18 + 24010000*a^7*c^7*(-(4*a

*c - b^2)^25)^(1/2) - 193795*a*b^37*c - 996660*a^2*b^10*c^2*(-(4*a*c - b^2)^25)^(1/2) + 7556115*a^3*b^8*c^3*(-

(4*a*c - b^2)^25)^(1/2) - 34052295*a^4*b^6*c^4*(-(4*a*c - b^2)^25)^(1/2) + 87808681*a^5*b^4*c^5*(-(4*a*c - b^2

)^25)^(1/2) - 108025400*a^6*b^2*c^6*(-(4*a*c - b^2)^25)^(1/2) + 73745*a*b^12*c*(-(4*a*c - b^2)^25)^(1/2)))/(33

554432*(a^11*b^40 + 1099511627776*a^31*c^20 - 80*a^12*b^38*c + 3040*a^13*b^36*c^2 - 72960*a^14*b^34*c^3 + 1240

320*a^15*b^32*c^4 - 15876096*a^16*b^30*c^5 + 158760960*a^17*b^28*c^6 - 1270087680*a^18*b^26*c^7 + 8255569920*a

^19*b^24*c^8 - 44029706240*a^20*b^22*c^9 + 193730707456*a^21*b^20*c^10 - 704475299840*a^22*b^18*c^11 + 2113425

899520*a^23*b^16*c^12 - 5202279137280*a^24*b^14*c^13 + 10404558274560*a^25*b^12*c^14 - 16647293239296*a^26*b^1

0*c^15 + 20809116549120*a^27*b^8*c^16 - 19585050869760*a^28*b^6*c^17 + 13056700579840*a^29*b^4*c^18 - 54975581

38880*a^30*b^2*c^19)))^(1/4)*1i)/(((((9*x^(1/2)*(1546704997025054720*a^19*b*c^19 - 822083584*a^4*b^31*c^4 + 50

851741696*a^5*b^29*c^5 - 1473677099008*a^6*b^27*c^6 + 26523687976960*a^7*b^25*c^7 - 331351626612736*a^8*b^23*c

^8 + 3041476258824192*a^9*b^21*c^9 - 21176692735213568*a^10*b^19*c^10 + 113812892427485184*a^11*b^17*c^11 - 47

5720885626470400*a^12*b^15*c^12 + 1545406748670558208*a^13*b^13*c^13 - 3867206695260258304*a^14*b^11*c^14 + 73

15227880965799936*a^15*b^9*c^15 - 10117494892562219008*a^16*b^7*c^16 + 9650897342106173440*a^17*b^5*c^17 - 567

2002255696429056*a^18*b^3*c^18))/(4194304*(a^8*b^24 + 16777216*a^20*c^12 - 48*a^9*b^22*c + 1056*a^10*b^20*c^2

- 14080*a^11*b^18*c^3 + 126720*a^12*b^16*c^4 - 811008*a^13*b^14*c^5 + 3784704*a^14*b^12*c^6 - 12976128*a^15*b^

10*c^7 + 32440320*a^16*b^8*c^8 - 57671680*a^17*b^6*c^9 + 69206016*a^18*b^4*c^10 - 50331648*a^19*b^2*c^11)) - (

3*(-(81*(2401*b^39 - 2401*b^14*(-(4*a*c - b^2)^25)^(1/2) - 2405416566784000*a^19*b*c^19 + 7445060*a^2*b^35*c^2

- 180851965*a^3*b^33*c^3 + 3112544495*a^4*b^31*c^4 - 40302663491*a^5*b^29*c^5 + 406936342200*a^6*b^27*c^6 - 3

276813600400*a^7*b^25*c^7 + 21341140889600*a^8*b^23*c^8 - 113330748025600*a^9*b^21*c^9 + 492398189373440*a^10*

b^19*c^10 - 1748923551027200*a^11*b^17*c^11 + 5052644161945600*a^12*b^15*c^12 - 11756581147443200*a^13*b^13*c^

13 + 21683350423470080*a^14*b^11*c^14 - 30929025701511168*a^15*b^9*c^15 + 32836636093972480*a^16*b^7*c^16 - 24

359874477424640*a^17*b^5*c^17 + 11224950044098560*a^18*b^3*c^18 + 24010000*a^7*c^7*(-(4*a*c - b^2)^25)^(1/2) -

193795*a*b^37*c - 996660*a^2*b^10*c^2*(-(4*a*c - b^2)^25)^(1/2) + 7556115*a^3*b^8*c^3*(-(4*a*c - b^2)^25)^(1/

2) - 34052295*a^4*b^6*c^4*(-(4*a*c - b^2)^25)^(1/2) + 87808681*a^5*b^4*c^5*(-(4*a*c - b^2)^25)^(1/2) - 1080254

00*a^6*b^2*c^6*(-(4*a*c - b^2)^25)^(1/2) + 73745*a*b^12*c*(-(4*a*c - b^2)^25)^(1/2)))/(33554432*(a^11*b^40 + 1

099511627776*a^31*c^20 - 80*a^12*b^38*c + 3040*a^13*b^36*c^2 - 72960*a^14*b^34*c^3 + 1240320*a^15*b^32*c^4 - 1

5876096*a^16*b^30*c^5 + 158760960*a^17*b^28*c^6 - 1270087680*a^18*b^26*c^7 + 8255569920*a^19*b^24*c^8 - 440297

06240*a^20*b^22*c^9 + 193730707456*a^21*b^20*c^10 - 704475299840*a^22*b^18*c^11 + 2113425899520*a^23*b^16*c^12

- 5202279137280*a^24*b^14*c^13 + 10404558274560*a^25*b^12*c^14 - 16647293239296*a^26*b^10*c^15 + 208091165491

20*a^27*b^8*c^16 - 19585050869760*a^28*b^6*c^17 + 13056700579840*a^29*b^4*c^18 - 5497558138880*a^30*b^2*c^19))

)^(1/4)*(3377699720527872*a^19*b*c^16 + 117440512*a^7*b^25*c^4 - 5804916736*a^8*b^23*c^5 + 132070244352*a^9*b^

21*c^6 - 1828045455360*a^10*b^19*c^7 + 17136919511040*a^11*b^17*c^8 - 114572547588096*a^12*b^15*c^9 + 55992629

6444928*a^13*b^13*c^10 - 2014580179992576*a^14*b^11*c^11 + 5294148487741440*a^15*b^9*c^12 - 9906599766261760*a

^16*b^7*c^13 + 12525636463624192*a^17*b^5*c^14 - 9605333580251136*a^18*b^3*c^15))/(65536*(a^8*b^18 - 262144*a^

17*c^9 - 36*a^9*b^16*c + 576*a^10*b^14*c^2 - 5376*a^11*b^12*c^3 + 32256*a^12*b^10*c^4 - 129024*a^13*b^8*c^5 +

344064*a^14*b^6*c^6 - 589824*a^15*b^4*c^7 + 589824*a^16*b^2*c^8)))*(-(81*(2401*b^39 - 2401*b^14*(-(4*a*c - b^2

)^25)^(1/2) - 2405416566784000*a^19*b*c^19 + 7445060*a^2*b^35*c^2 - 180851965*a^3*b^33*c^3 + 3112544495*a^4*b^

31*c^4 - 40302663491*a^5*b^29*c^5 + 406936342200*a^6*b^27*c^6 - 3276813600400*a^7*b^25*c^7 + 21341140889600*a^

8*b^23*c^8 - 113330748025600*a^9*b^21*c^9 + 492398189373440*a^10*b^19*c^10 - 1748923551027200*a^11*b^17*c^11 +

5052644161945600*a^12*b^15*c^12 - 11756581147443200*a^13*b^13*c^13 + 21683350423470080*a^14*b^11*c^14 - 30929

025701511168*a^15*b^9*c^15 + 32836636093972480*a^16*b^7*c^16 - 24359874477424640*a^17*b^5*c^17 + 1122495004409

8560*a^18*b^3*c^18 + 24010000*a^7*c^7*(-(4*a*c - b^2)^25)^(1/2) - 193795*a*b^37*c - 996660*a^2*b^10*c^2*(-(4*a

*c - b^2)^25)^(1/2) + 7556115*a^3*b^8*c^3*(-(4*a*c - b^2)^25)^(1/2) - 34052295*a^4*b^6*c^4*(-(4*a*c - b^2)^25)

^(1/2) + 87808681*a^5*b^4*c^5*(-(4*a*c - b^2)^25)^(1/2) - 108025400*a^6*b^2*c^6*(-(4*a*c - b^2)^25)^(1/2) + 73

745*a*b^12*c*(-(4*a*c - b^2)^25)^(1/2)))/(33554432*(a^11*b^40 + 1099511627776*a^31*c^20 - 80*a^12*b^38*c + 304

0*a^13*b^36*c^2 - 72960*a^14*b^34*c^3 + 1240320*a^15*b^32*c^4 - 15876096*a^16*b^30*c^5 + 158760960*a^17*b^28*c

^6 - 1270087680*a^18*b^26*c^7 + 8255569920*a^19*b^24*c^8 - 44029706240*a^20*b^22*c^9 + 193730707456*a^21*b^20*

c^10 - 704475299840*a^22*b^18*c^11 + 2113425899520*a^23*b^16*c^12 - 5202279137280*a^24*b^14*c^13 + 10404558274

560*a^25*b^12*c^14 - 16647293239296*a^26*b^10*c^15 + 20809116549120*a^27*b^8*c^16 - 19585050869760*a^28*b^6*c^

17 + 13056700579840*a^29*b^4*c^18 - 5497558138880*a^30*b^2*c^19)))^(3/4) + (3*(4356374400000*a^8*c^16 + 184756

95*b^16*c^8 - 685712223*a*b^14*c^9 + 11424393414*a^2*b^12*c^10 - 110892005343*a^3*b^10*c^11 + 681741235260*a^4

*b^8*c^12 - 2694857597280*a^5*b^6*c^13 + 6582295198080*a^6*b^4*c^14 - 8763424992000*a^7*b^2*c^15))/(65536*(a^8

*b^18 - 262144*a^17*c^9 - 36*a^9*b^16*c + 576*a^10*b^14*c^2 - 5376*a^11*b^12*c^3 + 32256*a^12*b^10*c^4 - 12902

4*a^13*b^8*c^5 + 344064*a^14*b^6*c^6 - 589824*a^15*b^4*c^7 + 589824*a^16*b^2*c^8)))*(-(81*(2401*b^39 - 2401*b^

14*(-(4*a*c - b^2)^25)^(1/2) - 2405416566784000*a^19*b*c^19 + 7445060*a^2*b^35*c^2 - 180851965*a^3*b^33*c^3 +

3112544495*a^4*b^31*c^4 - 40302663491*a^5*b^29*c^5 + 406936342200*a^6*b^27*c^6 - 3276813600400*a^7*b^25*c^7 +

21341140889600*a^8*b^23*c^8 - 113330748025600*a^9*b^21*c^9 + 492398189373440*a^10*b^19*c^10 - 1748923551027200

*a^11*b^17*c^11 + 5052644161945600*a^12*b^15*c^12 - 11756581147443200*a^13*b^13*c^13 + 21683350423470080*a^14*

b^11*c^14 - 30929025701511168*a^15*b^9*c^15 + 32836636093972480*a^16*b^7*c^16 - 24359874477424640*a^17*b^5*c^1

7 + 11224950044098560*a^18*b^3*c^18 + 24010000*a^7*c^7*(-(4*a*c - b^2)^25)^(1/2) - 193795*a*b^37*c - 996660*a^

2*b^10*c^2*(-(4*a*c - b^2)^25)^(1/2) + 7556115*a^3*b^8*c^3*(-(4*a*c - b^2)^25)^(1/2) - 34052295*a^4*b^6*c^4*(-

(4*a*c - b^2)^25)^(1/2) + 87808681*a^5*b^4*c^5*(-(4*a*c - b^2)^25)^(1/2) - 108025400*a^6*b^2*c^6*(-(4*a*c - b^

2)^25)^(1/2) + 73745*a*b^12*c*(-(4*a*c - b^2)^25)^(1/2)))/(33554432*(a^11*b^40 + 1099511627776*a^31*c^20 - 80*

a^12*b^38*c + 3040*a^13*b^36*c^2 - 72960*a^14*b^34*c^3 + 1240320*a^15*b^32*c^4 - 15876096*a^16*b^30*c^5 + 1587

60960*a^17*b^28*c^6 - 1270087680*a^18*b^26*c^7 + 8255569920*a^19*b^24*c^8 - 44029706240*a^20*b^22*c^9 + 193730

707456*a^21*b^20*c^10 - 704475299840*a^22*b^18*c^11 + 2113425899520*a^23*b^16*c^12 - 5202279137280*a^24*b^14*c

^13 + 10404558274560*a^25*b^12*c^14 - 16647293239296*a^26*b^10*c^15 + 20809116549120*a^27*b^8*c^16 - 195850508

69760*a^28*b^6*c^17 + 13056700579840*a^29*b^4*c^18 - 5497558138880*a^30*b^2*c^19)))^(1/4) - (9*x^(1/2)*(121978

4832000000*a^8*c^19 + 1755191025*b^16*c^11 - 67599928620*a*b^14*c^12 + 1172433971394*a^2*b^12*c^13 - 119117324

72304*a^3*b^10*c^14 + 77626373024736*a^4*b^8*c^15 - 333603251301888*a^5*b^6*c^16 + 930302051212800*a^6*b^4*c^1

7 - 1556843742720000*a^7*b^2*c^18))/(4194304*(a^8*b^24 + 16777216*a^20*c^12 - 48*a^9*b^22*c + 1056*a^10*b^20*c

^2 - 14080*a^11*b^18*c^3 + 126720*a^12*b^16*c^4 - 811008*a^13*b^14*c^5 + 3784704*a^14*b^12*c^6 - 12976128*a^15

*b^10*c^7 + 32440320*a^16*b^8*c^8 - 57671680*a^17*b^6*c^9 + 69206016*a^18*b^4*c^10 - 50331648*a^19*b^2*c^11)))