∫ x3∕2 ____________________

3.1086 $\int \frac {x^{3/2}}{(a+b x^2+c x^4)^3} \, dx$

Optimal. Leaf size=594 \[ -\frac {\sqrt {x} \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {\sqrt {x} \left (c x^2 \left (44 a c+b^2\right )+b \left (20 a c+b^2\right )\right )}{16 a \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {3 c^{3/4} \left (\frac {68 a b c}{\sqrt {b^2-4 a c}}-\frac {b^3}{\sqrt {b^2-4 a c}}+44 a c+b^2\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} a \left (b^2-4 a c\right )^2 \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {3 c^{3/4} \left (\sqrt {b^2-4 a c} \left (44 a c+b^2\right )-68 a b c+b^3\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} a \left (b^2-4 a c\right )^{5/2} \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {3 c^{3/4} \left (\frac {68 a b c}{\sqrt {b^2-4 a c}}-\frac {b^3}{\sqrt {b^2-4 a c}}+44 a c+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} a \left (b^2-4 a c\right )^2 \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {3 c^{3/4} \left (\sqrt {b^2-4 a c} \left (44 a c+b^2\right )-68 a b c+b^3\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} a \left (b^2-4 a c\right )^{5/2} \left (\sqrt {b^2-4 a c}-b\right )^{3/4}} \]

[Out]

-3/64*c^(3/4)*arctan(2^(1/4)*c^(1/4)*x^(1/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4))*(b^2+44*a*c-b^3/(-4*a*c+b^2)^(1/2)

+68*a*b*c/(-4*a*c+b^2)^(1/2))*2^(3/4)/a/(-4*a*c+b^2)^2/(-b-(-4*a*c+b^2)^(1/2))^(3/4)-3/64*c^(3/4)*arctanh(2^(1

/4)*c^(1/4)*x^(1/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4))*(b^2+44*a*c-b^3/(-4*a*c+b^2)^(1/2)+68*a*b*c/(-4*a*c+b^2)^(1

/2))*2^(3/4)/a/(-4*a*c+b^2)^2/(-b-(-4*a*c+b^2)^(1/2))^(3/4)-3/64*c^(3/4)*arctan(2^(1/4)*c^(1/4)*x^(1/2)/(-b+(-

4*a*c+b^2)^(1/2))^(1/4))*(b^3-68*a*b*c+(44*a*c+b^2)*(-4*a*c+b^2)^(1/2))*2^(3/4)/a/(-4*a*c+b^2)^(5/2)/(-b+(-4*a

*c+b^2)^(1/2))^(3/4)-3/64*c^(3/4)*arctanh(2^(1/4)*c^(1/4)*x^(1/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/4))*(b^3-68*a*b*c

+(44*a*c+b^2)*(-4*a*c+b^2)^(1/2))*2^(3/4)/a/(-4*a*c+b^2)^(5/2)/(-b+(-4*a*c+b^2)^(1/2))^(3/4)-1/4*(2*c*x^2+b)*x

^(1/2)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^2+1/16*(b*(20*a*c+b^2)+c*(44*a*c+b^2)*x^2)*x^(1/2)/a/(-4*a*c+b^2)^2/(c*x^4

+b*x^2+a)

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Rubi [A] time = 2.37, antiderivative size = 594, 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, 1364, 1430, 1422, 212, 208, 205} \[ -\frac {3 c^{3/4} \left (-\frac {b^3}{\sqrt {b^2-4 a c}}+\frac {68 a b c}{\sqrt {b^2-4 a c}}+44 a c+b^2\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} a \left (b^2-4 a c\right )^2 \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {3 c^{3/4} \left (\sqrt {b^2-4 a c} \left (44 a c+b^2\right )-68 a b c+b^3\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} a \left (b^2-4 a c\right )^{5/2} \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {3 c^{3/4} \left (-\frac {b^3}{\sqrt {b^2-4 a c}}+\frac {68 a b c}{\sqrt {b^2-4 a c}}+44 a c+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} a \left (b^2-4 a c\right )^2 \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {3 c^{3/4} \left (\sqrt {b^2-4 a c} \left (44 a c+b^2\right )-68 a b c+b^3\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} a \left (b^2-4 a c\right )^{5/2} \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {\sqrt {x} \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {\sqrt {x} \left (c x^2 \left (44 a c+b^2\right )+b \left (20 a c+b^2\right )\right )}{16 a \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(Sqrt[x]*(b + 2*c*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (Sqrt[x]*(b*(b^2 + 20*a*c) + c*(b^2 + 44*a*

c)*x^2))/(16*a*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - (3*c^(3/4)*(b^2 + 44*a*c - b^3/Sqrt[b^2 - 4*a*c] + (68*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)*a*(b^2

- 4*a*c)^2*(-b - Sqrt[b^2 - 4*a*c])^(3/4)) - (3*c^(3/4)*(b^3 - 68*a*b*c + Sqrt[b^2 - 4*a*c]*(b^2 + 44*a*c))*Ar

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

^2 - 4*a*c])^(3/4)) - (3*c^(3/4)*(b^2 + 44*a*c - b^3/Sqrt[b^2 - 4*a*c] + (68*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)*a*(b^2 - 4*a*c)^2*(-b - Sqrt[b^2 - 4*a

*c])^(3/4)) - (3*c^(3/4)*(b^3 - 68*a*b*c + Sqrt[b^2 - 4*a*c]*(b^2 + 44*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*(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 1364

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

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

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

tQ[m, n - 1] && LeQ[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 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 {x^{3/2}}{\left (a+b x^2+c x^4\right )^3} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^4}{\left (a+b x^4+c x^8\right )^3} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\sqrt {x} \left (b+2 c 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 {b-22 c x^4}{\left (a+b x^4+c x^8\right )^2} \, dx,x,\sqrt {x}\right )}{4 \left (b^2-4 a c\right )}\\ &=-\frac {\sqrt {x} \left (b+2 c 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 (b \left (b^2+20 a c\right )+c \left (b^2+44 a c\right ) x^2\right )}{16 a \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {\operatorname {Subst}\left (\int \frac {-3 b \left (b^2-12 a c\right )-3 c \left (b^2+44 a c\right ) x^4}{a+b x^4+c x^8} \, dx,x,\sqrt {x}\right )}{16 a \left (b^2-4 a c\right )^2}\\ &=-\frac {\sqrt {x} \left (b+2 c 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 (b \left (b^2+20 a c\right )+c \left (b^2+44 a c\right ) x^2\right )}{16 a \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\left (3 c \left (b^2+44 a c+\frac {b^3}{\sqrt {b^2-4 a c}}-\frac {68 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 a \left (b^2-4 a c\right )^2}+\frac {\left (3 c \left (b^2+44 a c-\frac {b^3}{\sqrt {b^2-4 a c}}+\frac {68 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 a \left (b^2-4 a c\right )^2}\\ &=-\frac {\sqrt {x} \left (b+2 c 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 (b \left (b^2+20 a c\right )+c \left (b^2+44 a c\right ) x^2\right )}{16 a \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {\left (3 c \left (b^2+44 a c-\frac {b^3}{\sqrt {b^2-4 a c}}+\frac {68 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 a \left (b^2-4 a c\right )^2 \sqrt {-b-\sqrt {b^2-4 a c}}}-\frac {\left (3 c \left (b^2+44 a c-\frac {b^3}{\sqrt {b^2-4 a c}}+\frac {68 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 a \left (b^2-4 a c\right )^2 \sqrt {-b-\sqrt {b^2-4 a c}}}-\frac {\left (3 c \left (b^2+44 a c+\frac {b^3}{\sqrt {b^2-4 a c}}-\frac {68 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 a \left (b^2-4 a c\right )^2 \sqrt {-b+\sqrt {b^2-4 a c}}}-\frac {\left (3 c \left (b^2+44 a c+\frac {b^3}{\sqrt {b^2-4 a c}}-\frac {68 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 a \left (b^2-4 a c\right )^2 \sqrt {-b+\sqrt {b^2-4 a c}}}\\ &=-\frac {\sqrt {x} \left (b+2 c 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 (b \left (b^2+20 a c\right )+c \left (b^2+44 a c\right ) x^2\right )}{16 a \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {3 c^{3/4} \left (b^2+44 a c-\frac {b^3}{\sqrt {b^2-4 a c}}+\frac {68 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} a \left (b^2-4 a c\right )^2 \left (-b-\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {3 c^{3/4} \left (b^2+44 a c+\frac {b^3}{\sqrt {b^2-4 a c}}-\frac {68 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} a \left (b^2-4 a c\right )^2 \left (-b+\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {3 c^{3/4} \left (b^2+44 a c-\frac {b^3}{\sqrt {b^2-4 a c}}+\frac {68 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} a \left (b^2-4 a c\right )^2 \left (-b-\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {3 c^{3/4} \left (b^2+44 a c+\frac {b^3}{\sqrt {b^2-4 a c}}-\frac {68 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} a \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 = 0.41, size = 224, normalized size = 0.38 \[ \frac {3 \left (a+b x^2+c x^4\right )^2 \text {RootSum}\left [\text {$\#$1}^8 c+\text {$\#$1}^4 b+a\& ,\frac {44 \text {$\#$1}^4 a c^2 \log \left (\sqrt {x}-\text {$\#$1}\right )+\text {$\#$1}^4 b^2 c \log \left (\sqrt {x}-\text {$\#$1}\right )-12 a b c \log \left (\sqrt {x}-\text {$\#$1}\right )+b^3 \log \left (\sqrt {x}-\text {$\#$1}\right )}{2 \text {$\#$1}^7 c+\text {$\#$1}^3 b}\& \right ]-16 a \sqrt {x} \left (b^2-4 a c\right ) \left (b+2 c x^2\right )+4 \sqrt {x} \left (20 a b c+44 a c^2 x^2+b^3+b^2 c x^2\right ) \left (a+b x^2+c x^4\right )}{64 a \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )^2} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-16*a*(b^2 - 4*a*c)*Sqrt[x]*(b + 2*c*x^2) + 4*Sqrt[x]*(b^3 + 20*a*b*c + b^2*c*x^2 + 44*a*c^2*x^2)*(a + b*x^2

+ c*x^4) + 3*(a + b*x^2 + c*x^4)^2*RootSum[a + b*#1^4 + c*#1^8 & , (b^3*Log[Sqrt[x] - #1] - 12*a*b*c*Log[Sqrt[

x] - #1] + b^2*c*Log[Sqrt[x] - #1]*#1^4 + 44*a*c^2*Log[Sqrt[x] - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ])/(64*a*(b^2

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)/(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 \[ \text {Exception raised: TypeError} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)/(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: 191.63Unable to convert to real 1/4 Error: Bad Argument Value

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maple [C] time = 0.04, size = 270, normalized size = 0.45 \[ \frac {3 \left (\left (44 a c +b^{2}\right ) \RootOf \left (c \,\textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+a \right )^{4} c -12 a b c +b^{3}\right ) \ln \left (-\RootOf \left (c \,\textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+a \right )+\sqrt {x}\right )}{64 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a \left (2 \RootOf \left (c \,\textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+a \right )^{7} c +\RootOf \left (c \,\textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+a \right )^{3} b \right )}+\frac {\frac {\left (44 a c +b^{2}\right ) c^{2} x^{\frac {13}{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {\left (32 a c +b^{2}\right ) b c \,x^{\frac {9}{2}}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {\left (76 a^{2} c^{2}+13 a \,b^{2} c +b^{4}\right ) x^{\frac {5}{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {3 \left (12 a c -b^{2}\right ) b \sqrt {x}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(3/32*b*(12*a*c-b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(1/2)+1/32*(76*a^2*c^2+13*a*b^2*c+b^4)/a/(16*a^2*c^2-8*a*b

^2*c+b^4)*x^(5/2)+1/16/a*c*b*(32*a*c+b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(9/2)+1/32*c^2*(44*a*c+b^2)/a/(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/(16*a^2*c^2-8*a*b^2*c+b^4)*sum((c*(44*a*c+b^2)*_R^4-12*a*

b*c+b^3)/(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 \[ \frac {3 \, {\left (b^{3} c^{2} - 12 \, a b c^{3}\right )} x^{\frac {17}{2}} + {\left (6 \, b^{4} c - 71 \, a b^{2} c^{2} + 44 \, a^{2} c^{3}\right )} x^{\frac {13}{2}} + {\left (3 \, b^{5} - 28 \, a b^{3} c - 8 \, a^{2} b c^{2}\right )} x^{\frac {9}{2}} + {\left (7 \, a b^{4} - 59 \, a^{2} b^{2} c + 76 \, a^{3} c^{2}\right )} x^{\frac {5}{2}}}{16 \, {\left ({\left (a^{2} b^{4} c^{2} - 8 \, a^{3} b^{2} c^{3} + 16 \, a^{4} c^{4}\right )} x^{8} + a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2} + 2 \, {\left (a^{2} b^{5} c - 8 \, a^{3} b^{3} c^{2} + 16 \, a^{4} b c^{3}\right )} x^{6} + {\left (a^{2} b^{6} - 6 \, a^{3} b^{4} c + 32 \, a^{5} c^{3}\right )} x^{4} + 2 \, {\left (a^{3} b^{5} - 8 \, a^{4} b^{3} c + 16 \, a^{5} b c^{2}\right )} x^{2}\right )}} + \int -\frac {3 \, {\left ({\left (b^{3} c - 12 \, a b c^{2}\right )} x^{\frac {7}{2}} + {\left (b^{4} - 13 \, a b^{2} c - 44 \, a^{2} c^{2}\right )} x^{\frac {3}{2}}\right )}}{32 \, {\left (a^{3} b^{4} - 8 \, a^{4} b^{2} c + 16 \, a^{5} c^{2} + {\left (a^{2} b^{4} c - 8 \, a^{3} b^{2} c^{2} + 16 \, a^{4} c^{3}\right )} x^{4} + {\left (a^{2} b^{5} - 8 \, a^{3} b^{3} c + 16 \, a^{4} b c^{2}\right )} x^{2}\right )}}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(3*(b^3*c^2 - 12*a*b*c^3)*x^(17/2) + (6*b^4*c - 71*a*b^2*c^2 + 44*a^2*c^3)*x^(13/2) + (3*b^5 - 28*a*b^3*c

- 8*a^2*b*c^2)*x^(9/2) + (7*a*b^4 - 59*a^2*b^2*c + 76*a^3*c^2)*x^(5/2))/((a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^

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

- 6*a^3*b^4*c + 32*a^5*c^3)*x^4 + 2*(a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*x^2) + integrate(-3/32*((b^3*c - 12

*a*b*c^2)*x^(7/2) + (b^4 - 13*a*b^2*c - 44*a^2*c^2)*x^(3/2))/(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2 + (a^2*b^4*c

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

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mupad [B] time = 9.35, size = 54027, normalized size = 90.95 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan(((((3*(230850*a*b^11*c^8 - 4455*b^13*c^7 + 24287662080*a^6*b*c^13 - 3679344*a^2*b^9*c^9 + 8309952*a^3*b^7

*c^10 - 548653824*a^4*b^5*c^11 + 9760227840*a^5*b^3*c^12))/(65536*(a^4*b^18 - 262144*a^13*c^9 - 36*a^5*b^16*c

+ 576*a^6*b^14*c^2 - 5376*a^7*b^12*c^3 + 32256*a^8*b^10*c^4 - 129024*a^9*b^8*c^5 + 344064*a^10*b^6*c^6 - 58982

4*a^11*b^4*c^7 + 589824*a^12*b^2*c^8)) + ((3*(-(81*(b^35 + b^10*(-(4*a*c - b^2)^25)^(1/2) + 12505065717760*a^1

7*b*c^17 + 3910*a^2*b^31*c^2 - 91335*a^3*b^29*c^3 + 1329320*a^4*b^27*c^4 - 12356816*a^5*b^25*c^5 + 70316800*a^

6*b^23*c^6 - 181190400*a^7*b^21*c^7 - 668723200*a^8*b^19*c^8 + 10912870400*a^9*b^17*c^9 - 83490242560*a^10*b^1

5*c^10 + 502626713600*a^11*b^13*c^11 - 2379389337600*a^12*b^11*c^12 + 8291284418560*a^13*b^9*c^13 - 2011495923

7120*a^14*b^7*c^14 + 31974471237632*a^15*b^5*c^15 - 29919144837120*a^16*b^3*c^16 - 234256*a^5*c^5*(-(4*a*c - b

^2)^25)^(1/2) - 95*a*b^33*c + 510*a^2*b^6*c^2*(-(4*a*c - b^2)^25)^(1/2) + 2015*a^3*b^4*c^3*(-(4*a*c - b^2)^25)

^(1/2) - 33880*a^4*b^2*c^4*(-(4*a*c - b^2)^25)^(1/2) - 45*a*b^8*c*(-(4*a*c - b^2)^25)^(1/2)))/(33554432*(a^7*b

^40 + 1099511627776*a^27*c^20 - 80*a^8*b^38*c + 3040*a^9*b^36*c^2 - 72960*a^10*b^34*c^3 + 1240320*a^11*b^32*c^

4 - 15876096*a^12*b^30*c^5 + 158760960*a^13*b^28*c^6 - 1270087680*a^14*b^26*c^7 + 8255569920*a^15*b^24*c^8 - 4

4029706240*a^16*b^22*c^9 + 193730707456*a^17*b^20*c^10 - 704475299840*a^18*b^18*c^11 + 2113425899520*a^19*b^16

*c^12 - 5202279137280*a^20*b^14*c^13 + 10404558274560*a^21*b^12*c^14 - 16647293239296*a^22*b^10*c^15 + 2080911

6549120*a^23*b^8*c^16 - 19585050869760*a^24*b^6*c^17 + 13056700579840*a^25*b^4*c^18 - 5497558138880*a^26*b^2*c

^19)))^(1/4)*(774056185954304*a^16*c^16 - 16777216*a^4*b^24*c^4 + 889192448*a^5*b^22*c^5 - 20065550336*a^6*b^2

0*c^6 + 256355860480*a^7*b^18*c^7 - 2045478174720*a^8*b^16*c^8 + 10385230921728*a^9*b^14*c^9 - 31026843746304*

a^10*b^12*c^10 + 30099130810368*a^11*b^10*c^11 + 156680406958080*a^12*b^8*c^12 - 764160581304320*a^13*b^6*c^13

+ 1587694790508544*a^14*b^4*c^14 - 1706442046308352*a^15*b^2*c^15))/(65536*(a^4*b^18 - 262144*a^13*c^9 - 36*a

^5*b^16*c + 576*a^6*b^14*c^2 - 5376*a^7*b^12*c^3 + 32256*a^8*b^10*c^4 - 129024*a^9*b^8*c^5 + 344064*a^10*b^6*c

^6 - 589824*a^11*b^4*c^7 + 589824*a^12*b^2*c^8)) - (9*x^(1/2)*(3096224743817216*a^16*b*c^18 - 16777216*a^2*b^2

9*c^4 + 1157627904*a^3*b^27*c^5 - 34175188992*a^4*b^25*c^6 + 570425344000*a^5*b^23*c^7 - 5968393928704*a^6*b^2

1*c^8 + 40450001993728*a^7*b^19*c^9 - 171227461189632*a^8*b^17*c^10 + 350881648214016*a^9*b^15*c^11 + 52364241

2728320*a^10*b^13*c^12 - 6226534348095488*a^11*b^11*c^13 + 21186489555615744*a^12*b^9*c^14 - 39951854506868736

*a^13*b^7*c^15 + 42889749576286208*a^14*b^5*c^16 - 22517998136852480*a^15*b^3*c^17))/(4194304*(a^4*b^24 + 1677

7216*a^16*c^12 - 48*a^5*b^22*c + 1056*a^6*b^20*c^2 - 14080*a^7*b^18*c^3 + 126720*a^8*b^16*c^4 - 811008*a^9*b^1

4*c^5 + 3784704*a^10*b^12*c^6 - 12976128*a^11*b^10*c^7 + 32440320*a^12*b^8*c^8 - 57671680*a^13*b^6*c^9 + 69206

016*a^14*b^4*c^10 - 50331648*a^15*b^2*c^11)))*(-(81*(b^35 + b^10*(-(4*a*c - b^2)^25)^(1/2) + 12505065717760*a^

17*b*c^17 + 3910*a^2*b^31*c^2 - 91335*a^3*b^29*c^3 + 1329320*a^4*b^27*c^4 - 12356816*a^5*b^25*c^5 + 70316800*a

^6*b^23*c^6 - 181190400*a^7*b^21*c^7 - 668723200*a^8*b^19*c^8 + 10912870400*a^9*b^17*c^9 - 83490242560*a^10*b^

15*c^10 + 502626713600*a^11*b^13*c^11 - 2379389337600*a^12*b^11*c^12 + 8291284418560*a^13*b^9*c^13 - 201149592

37120*a^14*b^7*c^14 + 31974471237632*a^15*b^5*c^15 - 29919144837120*a^16*b^3*c^16 - 234256*a^5*c^5*(-(4*a*c -

b^2)^25)^(1/2) - 95*a*b^33*c + 510*a^2*b^6*c^2*(-(4*a*c - b^2)^25)^(1/2) + 2015*a^3*b^4*c^3*(-(4*a*c - b^2)^25

)^(1/2) - 33880*a^4*b^2*c^4*(-(4*a*c - b^2)^25)^(1/2) - 45*a*b^8*c*(-(4*a*c - b^2)^25)^(1/2)))/(33554432*(a^7*

b^40 + 1099511627776*a^27*c^20 - 80*a^8*b^38*c + 3040*a^9*b^36*c^2 - 72960*a^10*b^34*c^3 + 1240320*a^11*b^32*c

^4 - 15876096*a^12*b^30*c^5 + 158760960*a^13*b^28*c^6 - 1270087680*a^14*b^26*c^7 + 8255569920*a^15*b^24*c^8 -

44029706240*a^16*b^22*c^9 + 193730707456*a^17*b^20*c^10 - 704475299840*a^18*b^18*c^11 + 2113425899520*a^19*b^1

6*c^12 - 5202279137280*a^20*b^14*c^13 + 10404558274560*a^21*b^12*c^14 - 16647293239296*a^22*b^10*c^15 + 208091

16549120*a^23*b^8*c^16 - 19585050869760*a^24*b^6*c^17 + 13056700579840*a^25*b^4*c^18 - 5497558138880*a^26*b^2*

c^19)))^(3/4))*(-(81*(b^35 + b^10*(-(4*a*c - b^2)^25)^(1/2) + 12505065717760*a^17*b*c^17 + 3910*a^2*b^31*c^2 -

91335*a^3*b^29*c^3 + 1329320*a^4*b^27*c^4 - 12356816*a^5*b^25*c^5 + 70316800*a^6*b^23*c^6 - 181190400*a^7*b^2

1*c^7 - 668723200*a^8*b^19*c^8 + 10912870400*a^9*b^17*c^9 - 83490242560*a^10*b^15*c^10 + 502626713600*a^11*b^1

3*c^11 - 2379389337600*a^12*b^11*c^12 + 8291284418560*a^13*b^9*c^13 - 20114959237120*a^14*b^7*c^14 + 319744712

37632*a^15*b^5*c^15 - 29919144837120*a^16*b^3*c^16 - 234256*a^5*c^5*(-(4*a*c - b^2)^25)^(1/2) - 95*a*b^33*c +

510*a^2*b^6*c^2*(-(4*a*c - b^2)^25)^(1/2) + 2015*a^3*b^4*c^3*(-(4*a*c - b^2)^25)^(1/2) - 33880*a^4*b^2*c^4*(-(

4*a*c - b^2)^25)^(1/2) - 45*a*b^8*c*(-(4*a*c - b^2)^25)^(1/2)))/(33554432*(a^7*b^40 + 1099511627776*a^27*c^20

- 80*a^8*b^38*c + 3040*a^9*b^36*c^2 - 72960*a^10*b^34*c^3 + 1240320*a^11*b^32*c^4 - 15876096*a^12*b^30*c^5 + 1

58760960*a^13*b^28*c^6 - 1270087680*a^14*b^26*c^7 + 8255569920*a^15*b^24*c^8 - 44029706240*a^16*b^22*c^9 + 193

730707456*a^17*b^20*c^10 - 704475299840*a^18*b^18*c^11 + 2113425899520*a^19*b^16*c^12 - 5202279137280*a^20*b^1

4*c^13 + 10404558274560*a^21*b^12*c^14 - 16647293239296*a^22*b^10*c^15 + 20809116549120*a^23*b^8*c^16 - 195850

50869760*a^24*b^6*c^17 + 13056700579840*a^25*b^4*c^18 - 5497558138880*a^26*b^2*c^19)))^(1/4) + (9*x^(1/2)*(245

025*b^14*c^9 - 1175522844672*a^7*c^16 - 13142250*a*b^12*c^10 + 966155040*a^2*b^10*c^11 - 22497354720*a^3*b^8*c

^12 + 112005110016*a^4*b^6*c^13 + 617614170624*a^5*b^4*c^14 + 19430129664*a^6*b^2*c^15))/(4194304*(a^4*b^24 +

16777216*a^16*c^12 - 48*a^5*b^22*c + 1056*a^6*b^20*c^2 - 14080*a^7*b^18*c^3 + 126720*a^8*b^16*c^4 - 811008*a^9

*b^14*c^5 + 3784704*a^10*b^12*c^6 - 12976128*a^11*b^10*c^7 + 32440320*a^12*b^8*c^8 - 57671680*a^13*b^6*c^9 + 6

9206016*a^14*b^4*c^10 - 50331648*a^15*b^2*c^11)))*(-(81*(b^35 + b^10*(-(4*a*c - b^2)^25)^(1/2) + 1250506571776

0*a^17*b*c^17 + 3910*a^2*b^31*c^2 - 91335*a^3*b^29*c^3 + 1329320*a^4*b^27*c^4 - 12356816*a^5*b^25*c^5 + 703168

00*a^6*b^23*c^6 - 181190400*a^7*b^21*c^7 - 668723200*a^8*b^19*c^8 + 10912870400*a^9*b^17*c^9 - 83490242560*a^1

0*b^15*c^10 + 502626713600*a^11*b^13*c^11 - 2379389337600*a^12*b^11*c^12 + 8291284418560*a^13*b^9*c^13 - 20114

959237120*a^14*b^7*c^14 + 31974471237632*a^15*b^5*c^15 - 29919144837120*a^16*b^3*c^16 - 234256*a^5*c^5*(-(4*a*

c - b^2)^25)^(1/2) - 95*a*b^33*c + 510*a^2*b^6*c^2*(-(4*a*c - b^2)^25)^(1/2) + 2015*a^3*b^4*c^3*(-(4*a*c - b^2

)^25)^(1/2) - 33880*a^4*b^2*c^4*(-(4*a*c - b^2)^25)^(1/2) - 45*a*b^8*c*(-(4*a*c - b^2)^25)^(1/2)))/(33554432*(

a^7*b^40 + 1099511627776*a^27*c^20 - 80*a^8*b^38*c + 3040*a^9*b^36*c^2 - 72960*a^10*b^34*c^3 + 1240320*a^11*b^

32*c^4 - 15876096*a^12*b^30*c^5 + 158760960*a^13*b^28*c^6 - 1270087680*a^14*b^26*c^7 + 8255569920*a^15*b^24*c^

8 - 44029706240*a^16*b^22*c^9 + 193730707456*a^17*b^20*c^10 - 704475299840*a^18*b^18*c^11 + 2113425899520*a^19

*b^16*c^12 - 5202279137280*a^20*b^14*c^13 + 10404558274560*a^21*b^12*c^14 - 16647293239296*a^22*b^10*c^15 + 20

809116549120*a^23*b^8*c^16 - 19585050869760*a^24*b^6*c^17 + 13056700579840*a^25*b^4*c^18 - 5497558138880*a^26*

b^2*c^19)))^(1/4)*1i - (((3*(230850*a*b^11*c^8 - 4455*b^13*c^7 + 24287662080*a^6*b*c^13 - 3679344*a^2*b^9*c^9

+ 8309952*a^3*b^7*c^10 - 548653824*a^4*b^5*c^11 + 9760227840*a^5*b^3*c^12))/(65536*(a^4*b^18 - 262144*a^13*c^9

- 36*a^5*b^16*c + 576*a^6*b^14*c^2 - 5376*a^7*b^12*c^3 + 32256*a^8*b^10*c^4 - 129024*a^9*b^8*c^5 + 344064*a^1

0*b^6*c^6 - 589824*a^11*b^4*c^7 + 589824*a^12*b^2*c^8)) + ((3*(-(81*(b^35 + b^10*(-(4*a*c - b^2)^25)^(1/2) + 1

2505065717760*a^17*b*c^17 + 3910*a^2*b^31*c^2 - 91335*a^3*b^29*c^3 + 1329320*a^4*b^27*c^4 - 12356816*a^5*b^25*

c^5 + 70316800*a^6*b^23*c^6 - 181190400*a^7*b^21*c^7 - 668723200*a^8*b^19*c^8 + 10912870400*a^9*b^17*c^9 - 834

90242560*a^10*b^15*c^10 + 502626713600*a^11*b^13*c^11 - 2379389337600*a^12*b^11*c^12 + 8291284418560*a^13*b^9*

c^13 - 20114959237120*a^14*b^7*c^14 + 31974471237632*a^15*b^5*c^15 - 29919144837120*a^16*b^3*c^16 - 234256*a^5

*c^5*(-(4*a*c - b^2)^25)^(1/2) - 95*a*b^33*c + 510*a^2*b^6*c^2*(-(4*a*c - b^2)^25)^(1/2) + 2015*a^3*b^4*c^3*(-

(4*a*c - b^2)^25)^(1/2) - 33880*a^4*b^2*c^4*(-(4*a*c - b^2)^25)^(1/2) - 45*a*b^8*c*(-(4*a*c - b^2)^25)^(1/2)))

/(33554432*(a^7*b^40 + 1099511627776*a^27*c^20 - 80*a^8*b^38*c + 3040*a^9*b^36*c^2 - 72960*a^10*b^34*c^3 + 124

0320*a^11*b^32*c^4 - 15876096*a^12*b^30*c^5 + 158760960*a^13*b^28*c^6 - 1270087680*a^14*b^26*c^7 + 8255569920*

a^15*b^24*c^8 - 44029706240*a^16*b^22*c^9 + 193730707456*a^17*b^20*c^10 - 704475299840*a^18*b^18*c^11 + 211342

5899520*a^19*b^16*c^12 - 5202279137280*a^20*b^14*c^13 + 10404558274560*a^21*b^12*c^14 - 16647293239296*a^22*b^

10*c^15 + 20809116549120*a^23*b^8*c^16 - 19585050869760*a^24*b^6*c^17 + 13056700579840*a^25*b^4*c^18 - 5497558

138880*a^26*b^2*c^19)))^(1/4)*(774056185954304*a^16*c^16 - 16777216*a^4*b^24*c^4 + 889192448*a^5*b^22*c^5 - 20

065550336*a^6*b^20*c^6 + 256355860480*a^7*b^18*c^7 - 2045478174720*a^8*b^16*c^8 + 10385230921728*a^9*b^14*c^9

- 31026843746304*a^10*b^12*c^10 + 30099130810368*a^11*b^10*c^11 + 156680406958080*a^12*b^8*c^12 - 764160581304

320*a^13*b^6*c^13 + 1587694790508544*a^14*b^4*c^14 - 1706442046308352*a^15*b^2*c^15))/(65536*(a^4*b^18 - 26214

4*a^13*c^9 - 36*a^5*b^16*c + 576*a^6*b^14*c^2 - 5376*a^7*b^12*c^3 + 32256*a^8*b^10*c^4 - 129024*a^9*b^8*c^5 +

344064*a^10*b^6*c^6 - 589824*a^11*b^4*c^7 + 589824*a^12*b^2*c^8)) + (9*x^(1/2)*(3096224743817216*a^16*b*c^18 -

16777216*a^2*b^29*c^4 + 1157627904*a^3*b^27*c^5 - 34175188992*a^4*b^25*c^6 + 570425344000*a^5*b^23*c^7 - 5968

393928704*a^6*b^21*c^8 + 40450001993728*a^7*b^19*c^9 - 171227461189632*a^8*b^17*c^10 + 350881648214016*a^9*b^1

5*c^11 + 523642412728320*a^10*b^13*c^12 - 6226534348095488*a^11*b^11*c^13 + 21186489555615744*a^12*b^9*c^14 -

39951854506868736*a^13*b^7*c^15 + 42889749576286208*a^14*b^5*c^16 - 22517998136852480*a^15*b^3*c^17))/(4194304

*(a^4*b^24 + 16777216*a^16*c^12 - 48*a^5*b^22*c + 1056*a^6*b^20*c^2 - 14080*a^7*b^18*c^3 + 126720*a^8*b^16*c^4

- 811008*a^9*b^14*c^5 + 3784704*a^10*b^12*c^6 - 12976128*a^11*b^10*c^7 + 32440320*a^12*b^8*c^8 - 57671680*a^1

3*b^6*c^9 + 69206016*a^14*b^4*c^10 - 50331648*a^15*b^2*c^11)))*(-(81*(b^35 + b^10*(-(4*a*c - b^2)^25)^(1/2) +

12505065717760*a^17*b*c^17 + 3910*a^2*b^31*c^2 - 91335*a^3*b^29*c^3 + 1329320*a^4*b^27*c^4 - 12356816*a^5*b^25

*c^5 + 70316800*a^6*b^23*c^6 - 181190400*a^7*b^21*c^7 - 668723200*a^8*b^19*c^8 + 10912870400*a^9*b^17*c^9 - 83

490242560*a^10*b^15*c^10 + 502626713600*a^11*b^13*c^11 - 2379389337600*a^12*b^11*c^12 + 8291284418560*a^13*b^9

*c^13 - 20114959237120*a^14*b^7*c^14 + 31974471237632*a^15*b^5*c^15 - 29919144837120*a^16*b^3*c^16 - 234256*a^

5*c^5*(-(4*a*c - b^2)^25)^(1/2) - 95*a*b^33*c + 510*a^2*b^6*c^2*(-(4*a*c - b^2)^25)^(1/2) + 2015*a^3*b^4*c^3*(

-(4*a*c - b^2)^25)^(1/2) - 33880*a^4*b^2*c^4*(-(4*a*c - b^2)^25)^(1/2) - 45*a*b^8*c*(-(4*a*c - b^2)^25)^(1/2))

)/(33554432*(a^7*b^40 + 1099511627776*a^27*c^20 - 80*a^8*b^38*c + 3040*a^9*b^36*c^2 - 72960*a^10*b^34*c^3 + 12

40320*a^11*b^32*c^4 - 15876096*a^12*b^30*c^5 + 158760960*a^13*b^28*c^6 - 1270087680*a^14*b^26*c^7 + 8255569920

*a^15*b^24*c^8 - 44029706240*a^16*b^22*c^9 + 193730707456*a^17*b^20*c^10 - 704475299840*a^18*b^18*c^11 + 21134

25899520*a^19*b^16*c^12 - 5202279137280*a^20*b^14*c^13 + 10404558274560*a^21*b^12*c^14 - 16647293239296*a^22*b

^10*c^15 + 20809116549120*a^23*b^8*c^16 - 19585050869760*a^24*b^6*c^17 + 13056700579840*a^25*b^4*c^18 - 549755

8138880*a^26*b^2*c^19)))^(3/4))*(-(81*(b^35 + b^10*(-(4*a*c - b^2)^25)^(1/2) + 12505065717760*a^17*b*c^17 + 39

10*a^2*b^31*c^2 - 91335*a^3*b^29*c^3 + 1329320*a^4*b^27*c^4 - 12356816*a^5*b^25*c^5 + 70316800*a^6*b^23*c^6 -

181190400*a^7*b^21*c^7 - 668723200*a^8*b^19*c^8 + 10912870400*a^9*b^17*c^9 - 83490242560*a^10*b^15*c^10 + 5026

26713600*a^11*b^13*c^11 - 2379389337600*a^12*b^11*c^12 + 8291284418560*a^13*b^9*c^13 - 20114959237120*a^14*b^7

*c^14 + 31974471237632*a^15*b^5*c^15 - 29919144837120*a^16*b^3*c^16 - 234256*a^5*c^5*(-(4*a*c - b^2)^25)^(1/2)

- 95*a*b^33*c + 510*a^2*b^6*c^2*(-(4*a*c - b^2)^25)^(1/2) + 2015*a^3*b^4*c^3*(-(4*a*c - b^2)^25)^(1/2) - 3388

0*a^4*b^2*c^4*(-(4*a*c - b^2)^25)^(1/2) - 45*a*b^8*c*(-(4*a*c - b^2)^25)^(1/2)))/(33554432*(a^7*b^40 + 1099511

627776*a^27*c^20 - 80*a^8*b^38*c + 3040*a^9*b^36*c^2 - 72960*a^10*b^34*c^3 + 1240320*a^11*b^32*c^4 - 15876096*

a^12*b^30*c^5 + 158760960*a^13*b^28*c^6 - 1270087680*a^14*b^26*c^7 + 8255569920*a^15*b^24*c^8 - 44029706240*a^

16*b^22*c^9 + 193730707456*a^17*b^20*c^10 - 704475299840*a^18*b^18*c^11 + 2113425899520*a^19*b^16*c^12 - 52022

79137280*a^20*b^14*c^13 + 10404558274560*a^21*b^12*c^14 - 16647293239296*a^22*b^10*c^15 + 20809116549120*a^23*

b^8*c^16 - 19585050869760*a^24*b^6*c^17 + 13056700579840*a^25*b^4*c^18 - 5497558138880*a^26*b^2*c^19)))^(1/4)

- (9*x^(1/2)*(245025*b^14*c^9 - 1175522844672*a^7*c^16 - 13142250*a*b^12*c^10 + 966155040*a^2*b^10*c^11 - 2249

7354720*a^3*b^8*c^12 + 112005110016*a^4*b^6*c^13 + 617614170624*a^5*b^4*c^14 + 19430129664*a^6*b^2*c^15))/(419

4304*(a^4*b^24 + 16777216*a^16*c^12 - 48*a^5*b^22*c + 1056*a^6*b^20*c^2 - 14080*a^7*b^18*c^3 + 126720*a^8*b^16

*c^4 - 811008*a^9*b^14*c^5 + 3784704*a^10*b^12*c^6 - 12976128*a^11*b^10*c^7 + 32440320*a^12*b^8*c^8 - 57671680

*a^13*b^6*c^9 + 69206016*a^14*b^4*c^10 - 50331648*a^15*b^2*c^11)))*(-(81*(b^35 + b^10*(-(4*a*c - b^2)^25)^(1/2

) + 12505065717760*a^17*b*c^17 + 3910*a^2*b^31*c^2 - 91335*a^3*b^29*c^3 + 1329320*a^4*b^27*c^4 - 12356816*a^5*

b^25*c^5 + 70316800*a^6*b^23*c^6 - 181190400*a^7*b^21*c^7 - 668723200*a^8*b^19*c^8 + 10912870400*a^9*b^17*c^9

- 83490242560*a^10*b^15*c^10 + 502626713600*a^11*b^13*c^11 - 2379389337600*a^12*b^11*c^12 + 8291284418560*a^13

*b^9*c^13 - 20114959237120*a^14*b^7*c^14 + 31974471237632*a^15*b^5*c^15 - 29919144837120*a^16*b^3*c^16 - 23425

6*a^5*c^5*(-(4*a*c - b^2)^25)^(1/2) - 95*a*b^33*c + 510*a^2*b^6*c^2*(-(4*a*c - b^2)^25)^(1/2) + 2015*a^3*b^4*c

^3*(-(4*a*c - b^2)^25)^(1/2) - 33880*a^4*b^2*c^4*(-(4*a*c - b^2)^25)^(1/2) - 45*a*b^8*c*(-(4*a*c - b^2)^25)^(1

/2)))/(33554432*(a^7*b^40 + 1099511627776*a^27*c^20 - 80*a^8*b^38*c + 3040*a^9*b^36*c^2 - 72960*a^10*b^34*c^3

+ 1240320*a^11*b^32*c^4 - 15876096*a^12*b^30*c^5 + 158760960*a^13*b^28*c^6 - 1270087680*a^14*b^26*c^7 + 825556

9920*a^15*b^24*c^8 - 44029706240*a^16*b^22*c^9 + 193730707456*a^17*b^20*c^10 - 704475299840*a^18*b^18*c^11 + 2

113425899520*a^19*b^16*c^12 - 5202279137280*a^20*b^14*c^13 + 10404558274560*a^21*b^12*c^14 - 16647293239296*a^

22*b^10*c^15 + 20809116549120*a^23*b^8*c^16 - 19585050869760*a^24*b^6*c^17 + 13056700579840*a^25*b^4*c^18 - 54

97558138880*a^26*b^2*c^19)))^(1/4)*1i)/((((3*(230850*a*b^11*c^8 - 4455*b^13*c^7 + 24287662080*a^6*b*c^13 - 367

9344*a^2*b^9*c^9 + 8309952*a^3*b^7*c^10 - 548653824*a^4*b^5*c^11 + 9760227840*a^5*b^3*c^12))/(65536*(a^4*b^18

- 262144*a^13*c^9 - 36*a^5*b^16*c + 576*a^6*b^14*c^2 - 5376*a^7*b^12*c^3 + 32256*a^8*b^10*c^4 - 129024*a^9*b^8

*c^5 + 344064*a^10*b^6*c^6 - 589824*a^11*b^4*c^7 + 589824*a^12*b^2*c^8)) + ((3*(-(81*(b^35 + b^10*(-(4*a*c - b

^2)^25)^(1/2) + 12505065717760*a^17*b*c^17 + 3910*a^2*b^31*c^2 - 91335*a^3*b^29*c^3 + 1329320*a^4*b^27*c^4 - 1

2356816*a^5*b^25*c^5 + 70316800*a^6*b^23*c^6 - 181190400*a^7*b^21*c^7 - 668723200*a^8*b^19*c^8 + 10912870400*a

^9*b^17*c^9 - 83490242560*a^10*b^15*c^10 + 502626713600*a^11*b^13*c^11 - 2379389337600*a^12*b^11*c^12 + 829128

4418560*a^13*b^9*c^13 - 20114959237120*a^14*b^7*c^14 + 31974471237632*a^15*b^5*c^15 - 29919144837120*a^16*b^3*

c^16 - 234256*a^5*c^5*(-(4*a*c - b^2)^25)^(1/2) - 95*a*b^33*c + 510*a^2*b^6*c^2*(-(4*a*c - b^2)^25)^(1/2) + 20

15*a^3*b^4*c^3*(-(4*a*c - b^2)^25)^(1/2) - 33880*a^4*b^2*c^4*(-(4*a*c - b^2)^25)^(1/2) - 45*a*b^8*c*(-(4*a*c -

b^2)^25)^(1/2)))/(33554432*(a^7*b^40 + 1099511627776*a^27*c^20 - 80*a^8*b^38*c + 3040*a^9*b^36*c^2 - 72960*a^

10*b^34*c^3 + 1240320*a^11*b^32*c^4 - 15876096*a^12*b^30*c^5 + 158760960*a^13*b^28*c^6 - 1270087680*a^14*b^26*

c^7 + 8255569920*a^15*b^24*c^8 - 44029706240*a^16*b^22*c^9 + 193730707456*a^17*b^20*c^10 - 704475299840*a^18*b

^18*c^11 + 2113425899520*a^19*b^16*c^12 - 5202279137280*a^20*b^14*c^13 + 10404558274560*a^21*b^12*c^14 - 16647

293239296*a^22*b^10*c^15 + 20809116549120*a^23*b^8*c^16 - 19585050869760*a^24*b^6*c^17 + 13056700579840*a^25*b

^4*c^18 - 5497558138880*a^26*b^2*c^19)))^(1/4)*(774056185954304*a^16*c^16 - 16777216*a^4*b^24*c^4 + 889192448*

a^5*b^22*c^5 - 20065550336*a^6*b^20*c^6 + 256355860480*a^7*b^18*c^7 - 2045478174720*a^8*b^16*c^8 + 10385230921

728*a^9*b^14*c^9 - 31026843746304*a^10*b^12*c^10 + 30099130810368*a^11*b^10*c^11 + 156680406958080*a^12*b^8*c^

12 - 764160581304320*a^13*b^6*c^13 + 1587694790508544*a^14*b^4*c^14 - 1706442046308352*a^15*b^2*c^15))/(65536*

(a^4*b^18 - 262144*a^13*c^9 - 36*a^5*b^16*c + 576*a^6*b^14*c^2 - 5376*a^7*b^12*c^3 + 32256*a^8*b^10*c^4 - 1290

24*a^9*b^8*c^5 + 344064*a^10*b^6*c^6 - 589824*a^11*b^4*c^7 + 589824*a^12*b^2*c^8)) - (9*x^(1/2)*(3096224743817

216*a^16*b*c^18 - 16777216*a^2*b^29*c^4 + 1157627904*a^3*b^27*c^5 - 34175188992*a^4*b^25*c^6 + 570425344000*a^

5*b^23*c^7 - 5968393928704*a^6*b^21*c^8 + 40450001993728*a^7*b^19*c^9 - 171227461189632*a^8*b^17*c^10 + 350881

648214016*a^9*b^15*c^11 + 523642412728320*a^10*b^13*c^12 - 6226534348095488*a^11*b^11*c^13 + 21186489555615744

*a^12*b^9*c^14 - 39951854506868736*a^13*b^7*c^15 + 42889749576286208*a^14*b^5*c^16 - 22517998136852480*a^15*b^

3*c^17))/(4194304*(a^4*b^24 + 16777216*a^16*c^12 - 48*a^5*b^22*c + 1056*a^6*b^20*c^2 - 14080*a^7*b^18*c^3 + 12

6720*a^8*b^16*c^4 - 811008*a^9*b^14*c^5 + 3784704*a^10*b^12*c^6 - 12976128*a^11*b^10*c^7 + 32440320*a^12*b^8*c

^8 - 57671680*a^13*b^6*c^9 + 69206016*a^14*b^4*c^10 - 50331648*a^15*b^2*c^11)))*(-(81*(b^35 + b^10*(-(4*a*c -

b^2)^25)^(1/2) + 12505065717760*a^17*b*c^17 + 3910*a^2*b^31*c^2 - 91335*a^3*b^29*c^3 + 1329320*a^4*b^27*c^4 -

12356816*a^5*b^25*c^5 + 70316800*a^6*b^23*c^6 - 181190400*a^7*b^21*c^7 - 668723200*a^8*b^19*c^8 + 10912870400*

a^9*b^17*c^9 - 83490242560*a^10*b^15*c^10 + 502626713600*a^11*b^13*c^11 - 2379389337600*a^12*b^11*c^12 + 82912

84418560*a^13*b^9*c^13 - 20114959237120*a^14*b^7*c^14 + 31974471237632*a^15*b^5*c^15 - 29919144837120*a^16*b^3

*c^16 - 234256*a^5*c^5*(-(4*a*c - b^2)^25)^(1/2) - 95*a*b^33*c + 510*a^2*b^6*c^2*(-(4*a*c - b^2)^25)^(1/2) + 2

015*a^3*b^4*c^3*(-(4*a*c - b^2)^25)^(1/2) - 33880*a^4*b^2*c^4*(-(4*a*c - b^2)^25)^(1/2) - 45*a*b^8*c*(-(4*a*c

- b^2)^25)^(1/2)))/(33554432*(a^7*b^40 + 1099511627776*a^27*c^20 - 80*a^8*b^38*c + 3040*a^9*b^36*c^2 - 72960*a

^10*b^34*c^3 + 1240320*a^11*b^32*c^4 - 15876096*a^12*b^30*c^5 + 158760960*a^13*b^28*c^6 - 1270087680*a^14*b^26

*c^7 + 8255569920*a^15*b^24*c^8 - 44029706240*a^16*b^22*c^9 + 193730707456*a^17*b^20*c^10 - 704475299840*a^18*

b^18*c^11 + 2113425899520*a^19*b^16*c^12 - 5202279137280*a^20*b^14*c^13 + 10404558274560*a^21*b^12*c^14 - 1664

7293239296*a^22*b^10*c^15 + 20809116549120*a^23*b^8*c^16 - 19585050869760*a^24*b^6*c^17 + 13056700579840*a^25*

b^4*c^18 - 5497558138880*a^26*b^2*c^19)))^(3/4))*(-(81*(b^35 + b^10*(-(4*a*c - b^2)^25)^(1/2) + 12505065717760

*a^17*b*c^17 + 3910*a^2*b^31*c^2 - 91335*a^3*b^29*c^3 + 1329320*a^4*b^27*c^4 - 12356816*a^5*b^25*c^5 + 7031680

0*a^6*b^23*c^6 - 181190400*a^7*b^21*c^7 - 668723200*a^8*b^19*c^8 + 10912870400*a^9*b^17*c^9 - 83490242560*a^10

*b^15*c^10 + 502626713600*a^11*b^13*c^11 - 2379389337600*a^12*b^11*c^12 + 8291284418560*a^13*b^9*c^13 - 201149

59237120*a^14*b^7*c^14 + 31974471237632*a^15*b^5*c^15 - 29919144837120*a^16*b^3*c^16 - 234256*a^5*c^5*(-(4*a*c

- b^2)^25)^(1/2) - 95*a*b^33*c + 510*a^2*b^6*c^2*(-(4*a*c - b^2)^25)^(1/2) + 2015*a^3*b^4*c^3*(-(4*a*c - b^2)

^25)^(1/2) - 33880*a^4*b^2*c^4*(-(4*a*c - b^2)^25)^(1/2) - 45*a*b^8*c*(-(4*a*c - b^2)^25)^(1/2)))/(33554432*(a

^7*b^40 + 1099511627776*a^27*c^20 - 80*a^8*b^38*c + 3040*a^9*b^36*c^2 - 72960*a^10*b^34*c^3 + 1240320*a^11*b^3

2*c^4 - 15876096*a^12*b^30*c^5 + 158760960*a^13*b^28*c^6 - 1270087680*a^14*b^26*c^7 + 8255569920*a^15*b^24*c^8

- 44029706240*a^16*b^22*c^9 + 193730707456*a^17*b^20*c^10 - 704475299840*a^18*b^18*c^11 + 2113425899520*a^19*

b^16*c^12 - 5202279137280*a^20*b^14*c^13 + 10404558274560*a^21*b^12*c^14 - 16647293239296*a^22*b^10*c^15 + 208

09116549120*a^23*b^8*c^16 - 19585050869760*a^24*b^6*c^17 + 13056700579840*a^25*b^4*c^18 - 5497558138880*a^26*b

^2*c^19)))^(1/4) + (9*x^(1/2)*(245025*b^14*c^9 - 1175522844672*a^7*c^16 - 13142250*a*b^12*c^10 + 966155040*a^2

*b^10*c^11 - 22497354720*a^3*b^8*c^12 + 112005110016*a^4*b^6*c^13 + 617614170624*a^5*b^4*c^14 + 19430129664*a^

6*b^2*c^15))/(4194304*(a^4*b^24 + 16777216*a^16*c^12 - 48*a^5*b^22*c + 1056*a^6*b^20*c^2 - 14080*a^7*b^18*c^3

+ 126720*a^8*b^16*c^4 - 811008*a^9*b^14*c^5 + 3784704*a^10*b^12*c^6 - 12976128*a^11*b^10*c^7 + 32440320*a^12*b

^8*c^8 - 57671680*a^13*b^6*c^9 + 69206016*a^14*b^4*c^10 - 50331648*a^15*b^2*c^11)))*(-(81*(b^35 + b^10*(-(4*a*

c - b^2)^25)^(1/2) + 12505065717760*a^17*b*c^17 + 3910*a^2*b^31*c^2 - 91335*a^3*b^29*c^3 + 1329320*a^4*b^27*c^

4 - 12356816*a^5*b^25*c^5 + 70316800*a^6*b^23*c^6 - 181190400*a^7*b^21*c^7 - 668723200*a^8*b^19*c^8 + 10912870

400*a^9*b^17*c^9 - 83490242560*a^10*b^15*c^10 + 502626713600*a^11*b^13*c^11 - 2379389337600*a^12*b^11*c^12 + 8

291284418560*a^13*b^9*c^13 - 20114959237120*a^14*b^7*c^14 + 31974471237632*a^15*b^5*c^15 - 29919144837120*a^16

*b^3*c^16 - 234256*a^5*c^5*(-(4*a*c - b^2)^25)^(1/2) - 95*a*b^33*c + 510*a^2*b^6*c^2*(-(4*a*c - b^2)^25)^(1/2)

+ 2015*a^3*b^4*c^3*(-(4*a*c - b^2)^25)^(1/2) - 33880*a^4*b^2*c^4*(-(4*a*c - b^2)^25)^(1/2) - 45*a*b^8*c*(-(4*

a*c - b^2)^25)^(1/2)))/(33554432*(a^7*b^40 + 1099511627776*a^27*c^20 - 80*a^8*b^38*c + 3040*a^9*b^36*c^2 - 729

60*a^10*b^34*c^3 + 1240320*a^11*b^32*c^4 - 15876096*a^12*b^30*c^5 + 158760960*a^13*b^28*c^6 - 1270087680*a^14*

b^26*c^7 + 8255569920*a^15*b^24*c^8 - 44029706240*a^16*b^22*c^9 + 193730707456*a^17*b^20*c^10 - 704475299840*a

^18*b^18*c^11 + 2113425899520*a^19*b^16*c^12 - 5202279137280*a^20*b^14*c^13 + 10404558274560*a^21*b^12*c^14 -

16647293239296*a^22*b^10*c^15 + 20809116549120*a^23*b^8*c^16 - 19585050869760*a^24*b^6*c^17 + 13056700579840*a

^25*b^4*c^18 - 5497558138880*a^26*b^2*c^19)))^(1/4) + (((3*(230850*a*b^11*c^8 - 4455*b^13*c^7 + 24287662080*a^

6*b*c^13 - 3679344*a^2*b^9*c^9 + 8309952*a^3*b^7*c^10 - 548653824*a^4*b^5*c^11 + 9760227840*a^5*b^3*c^12))/(65

536*(a^4*b^18 - 262144*a^13*c^9 - 36*a^5*b^16*c + 576*a^6*b^14*c^2 - 5376*a^7*b^12*c^3 + 32256*a^8*b^10*c^4 -

129024*a^9*b^8*c^5 + 344064*a^10*b^6*c^6 - 589824*a^11*b^4*c^7 + 589824*a^12*b^2*c^8)) + ((3*(-(81*(b^35 + b^1

0*(-(4*a*c - b^2)^25)^(1/2) + 12505065717760*a^17*b*c^17 + 3910*a^2*b^31*c^2 - 91335*a^3*b^29*c^3 + 1329320*a^

4*b^27*c^4 - 12356816*a^5*b^25*c^5 + 70316800*a^6*b^23*c^6 - 181190400*a^7*b^21*c^7 - 668723200*a^8*b^19*c^8 +

10912870400*a^9*b^17*c^9 - 83490242560*a^10*b^15*c^10 + 502626713600*a^11*b^13*c^11 - 2379389337600*a^12*b^11

*c^12 + 8291284418560*a^13*b^9*c^13 - 20114959237120*a^14*b^7*c^14 + 31974471237632*a^15*b^5*c^15 - 2991914483

7120*a^16*b^3*c^16 - 234256*a^5*c^5*(-(4*a*c - b^2)^25)^(1/2) - 95*a*b^33*c + 510*a^2*b^6*c^2*(-(4*a*c - b^2)^

25)^(1/2) + 2015*a^3*b^4*c^3*(-(4*a*c - b^2)^25)^(1/2) - 33880*a^4*b^2*c^4*(-(4*a*c - b^2)^25)^(1/2) - 45*a*b^

8*c*(-(4*a*c - b^2)^25)^(1/2)))/(33554432*(a^7*b^40 + 1099511627776*a^27*c^20 - 80*a^8*b^38*c + 3040*a^9*b^36*

c^2 - 72960*a^10*b^34*c^3 + 1240320*a^11*b^32*c^4 - 15876096*a^12*b^30*c^5 + 158760960*a^13*b^28*c^6 - 1270087

680*a^14*b^26*c^7 + 8255569920*a^15*b^24*c^8 - 44029706240*a^16*b^22*c^9 + 193730707456*a^17*b^20*c^10 - 70447

5299840*a^18*b^18*c^11 + 2113425899520*a^19*b^16*c^12 - 5202279137280*a^20*b^14*c^13 + 10404558274560*a^21*b^1

2*c^14 - 16647293239296*a^22*b^10*c^15 + 20809116549120*a^23*b^8*c^16 - 19585050869760*a^24*b^6*c^17 + 1305670

0579840*a^25*b^4*c^18 - 5497558138880*a^26*b^2*c^19)))^(1/4)*(774056185954304*a^16*c^16 - 16777216*a^4*b^24*c^

4 + 889192448*a^5*b^22*c^5 - 20065550336*a^6*b^20*c^6 + 256355860480*a^7*b^18*c^7 - 2045478174720*a^8*b^16*c^8

+ 10385230921728*a^9*b^14*c^9 - 31026843746304*a^10*b^12*c^10 + 30099130810368*a^11*b^10*c^11 + 1566804069580

80*a^12*b^8*c^12 - 764160581304320*a^13*b^6*c^13 + 1587694790508544*a^14*b^4*c^14 - 1706442046308352*a^15*b^2*

c^15))/(65536*(a^4*b^18 - 262144*a^13*c^9 - 36*a^5*b^16*c + 576*a^6*b^14*c^2 - 5376*a^7*b^12*c^3 + 32256*a^8*b

^10*c^4 - 129024*a^9*b^8*c^5 + 344064*a^10*b^6*c^6 - 589824*a^11*b^4*c^7 + 589824*a^12*b^2*c^8)) + (9*x^(1/2)*

(3096224743817216*a^16*b*c^18 - 16777216*a^2*b^29*c^4 + 1157627904*a^3*b^27*c^5 - 34175188992*a^4*b^25*c^6 + 5

70425344000*a^5*b^23*c^7 - 5968393928704*a^6*b^21*c^8 + 40450001993728*a^7*b^19*c^9 - 171227461189632*a^8*b^17

*c^10 + 350881648214016*a^9*b^15*c^11 + 523642412728320*a^10*b^13*c^12 - 6226534348095488*a^11*b^11*c^13 + 211

86489555615744*a^12*b^9*c^14 - 39951854506868736*a^13*b^7*c^15 + 42889749576286208*a^14*b^5*c^16 - 22517998136

852480*a^15*b^3*c^17))/(4194304*(a^4*b^24 + 16777216*a^16*c^12 - 48*a^5*b^22*c + 1056*a^6*b^20*c^2 - 14080*a^7

*b^18*c^3 + 126720*a^8*b^16*c^4 - 811008*a^9*b^14*c^5 + 3784704*a^10*b^12*c^6 - 12976128*a^11*b^10*c^7 + 32440

320*a^12*b^8*c^8 - 57671680*a^13*b^6*c^9 + 69206016*a^14*b^4*c^10 - 50331648*a^15*b^2*c^11)))*(-(81*(b^35 + b^

10*(-(4*a*c - b^2)^25)^(1/2) + 12505065717760*a^17*b*c^17 + 3910*a^2*b^31*c^2 - 91335*a^3*b^29*c^3 + 1329320*a

^4*b^27*c^4 - 12356816*a^5*b^25*c^5 + 70316800*a^6*b^23*c^6 - 181190400*a^7*b^21*c^7 - 668723200*a^8*b^19*c^8

+ 10912870400*a^9*b^17*c^9 - 83490242560*a^10*b^15*c^10 + 502626713600*a^11*b^13*c^11 - 2379389337600*a^12*b^1

1*c^12 + 8291284418560*a^13*b^9*c^13 - 20114959237120*a^14*b^7*c^14 + 31974471237632*a^15*b^5*c^15 - 299191448

37120*a^16*b^3*c^16 - 234256*a^5*c^5*(-(4*a*c - b^2)^25)^(1/2) - 95*a*b^33*c + 510*a^2*b^6*c^2*(-(4*a*c - b^2)

^25)^(1/2) + 2015*a^3*b^4*c^3*(-(4*a*c - b^2)^25)^(1/2) - 33880*a^4*b^2*c^4*(-(4*a*c - b^2)^25)^(1/2) - 45*a*b

^8*c*(-(4*a*c - b^2)^25)^(1/2)))/(33554432*(a^7*b^40 + 1099511627776*a^27*c^20 - 80*a^8*b^38*c + 3040*a^9*b^36

*c^2 - 72960*a^10*b^34*c^3 + 1240320*a^11*b^32*c^4 - 15876096*a^12*b^30*c^5 + 158760960*a^13*b^28*c^6 - 127008

7680*a^14*b^26*c^7 + 8255569920*a^15*b^24*c^8 - 44029706240*a^16*b^22*c^9 + 193730707456*a^17*b^20*c^10 - 7044

75299840*a^18*b^18*c^11 + 2113425899520*a^19*b^16*c^12 - 5202279137280*a^20*b^14*c^13 + 10404558274560*a^21*b^

12*c^14 - 16647293239296*a^22*b^10*c^15 + 20809116549120*a^23*b^8*c^16 - 19585050869760*a^24*b^6*c^17 + 130567

00579840*a^25*b^4*c^18 - 5497558138880*a^26*b^2*c^19)))^(3/4))*(-(81*(b^35 + b^10*(-(4*a*c - b^2)^25)^(1/2) +