38 HFOPDE, chapter 2.2.4

 38.1 problem number 1
 38.2 problem number 2
 38.3 problem number 3
 38.4 problem number 4
 38.5 problem number 5
 38.6 problem number 6
 38.7 problem number 7

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38.1 problem number 1

problem number 263

Added January 2, 2019.

Problem 2.2.4.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x +(a \sqrt{x} y) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (y e^{-\frac{2}{3} a x^{3/2}}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( y{{\rm e}^{-2/3\,{x}^{3/2}a}} \right ) \]

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38.2 problem number 2

problem number 264

Added January 2, 2019.

Problem 2.2.4.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x +(a \sqrt{x} y+ b \sqrt{y}) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{e^{-\frac{1}{3} a x^{3/2}} \left (3^{2/3} b e^{\frac{1}{3} a x^{3/2}} \text{Gamma}\left (\frac{2}{3},\frac{1}{3} a x^{3/2}\right )-3 a^{2/3} \sqrt{y}\right )}{3 a^{2/3}}\right )\right \},\left \{w(x,y)\to c_1\left (\frac{e^{-\frac{1}{3} a x^{3/2}} \left (3^{2/3} b e^{\frac{1}{3} a x^{3/2}} \text{Gamma}\left (\frac{2}{3},\frac{1}{3} a x^{3/2}\right )+3 a^{2/3} \sqrt{y}\right )}{3 a^{2/3}}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/10\,{\frac{ \left ( 3\,\sqrt [3]{3} \WhittakerM \left ( 1/3,5/6,1/3\,{x}^{3/2}a \right ){{\rm e}^{1/6\,{x}^{3/2}a}}bx+5\,bx\sqrt [3]{{x}^{3/2}a}-10\,\sqrt [3]{{x}^{3/2}a}\sqrt{y} \right ){{\rm e}^{-1/3\,{x}^{3/2}a}}}{\sqrt [3]{{x}^{3/2}a}}} \right ) \]

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38.3 problem number 3

problem number 265

Added January 2, 2019.

Problem 2.2.4.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x +(a \sqrt{x} y+ b x \sqrt{y}) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{e^{-\frac{1}{3} a x^{3/2}} \left (\sqrt [3]{3} b e^{\frac{1}{3} a x^{3/2}} \text{Gamma}\left (\frac{1}{3},\frac{1}{3} a x^{3/2}\right )-3 a^{4/3} \sqrt{y}+3 \sqrt [3]{a} b \sqrt{x}\right )}{3 a^{4/3}}\right )\right \},\left \{w(x,y)\to c_1\left (\frac{e^{-\frac{1}{3} a x^{3/2}} \left (\sqrt [3]{3} b e^{\frac{1}{3} a x^{3/2}} \text{Gamma}\left (\frac{1}{3},\frac{1}{3} a x^{3/2}\right )+3 a^{4/3} \sqrt{y}+3 \sqrt [3]{a} b \sqrt{x}\right )}{3 a^{4/3}}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/4\,{\frac{{{\rm e}^{-1/3\,{x}^{3/2}a}} \left ( 3\,\sqrt [6]{3} \WhittakerM \left ( 1/6,2/3,1/3\,{x}^{3/2}a \right ) \sqrt{x}{{\rm e}^{1/6\,{x}^{3/2}a}}b-4\,\sqrt{y}a\sqrt [6]{{x}^{3/2}a} \right ) }{\sqrt [6]{{x}^{3/2}a}a}} \right ) \]

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38.4 problem number 4

problem number 266

Added January 2, 2019.

Problem 2.2.4.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x +A \sqrt{a x + b y+ c} w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{2 a^2 \log \left (-a e^{-\frac{\sqrt{a A^2 b^2 x+A^2 b^3 y+A^2 b^2 c}}{a}} \left (-\frac{\sqrt{a A^2 b^2 x+A^2 b^3 y+A^2 b^2 c}}{a}-1\right )\right )+a A^2 b^2 x+A^2 b^2 c}{a A^2 b^2}\right )\right \},\left \{w(x,y)\to c_1\left (\frac{2 a^2 \log \left (-a e^{\frac{\sqrt{a A^2 b^2 x+A^2 b^3 y+A^2 b^2 c}}{a}} \left (\frac{\sqrt{a A^2 b^2 x+A^2 b^3 y+A^2 b^2 c}}{a}-1\right )\right )+a A^2 b^2 x+A^2 b^2 c}{a A^2 b^2}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{x{A}^{2}{b}^{2}-2\,A\sqrt{ax+by+c}b+a\ln \left ( A\sqrt{ax+by+c}b+a \right ) -a\ln \left ( A\sqrt{ax+by+c}b-a \right ) +a\ln \left ({A}^{2}a{b}^{2}x+{A}^{2}{b}^{3}y+{A}^{2}{b}^{2}c-{a}^{2} \right ) }{{A}^{2}{b}^{2}}} \right ) \]

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38.5 problem number 5

problem number 267

Added January 2, 2019.

Problem 2.2.4.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left ( a y + b \sqrt{y^2+c x^2} \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (a y+b \sqrt{c x^2+y^2}\right )+x w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ \text{ sol=() } \] Timed out

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38.6 problem number 6

problem number 268

Added January 2, 2019.

Problem 2.2.4.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ \left (a x + b \sqrt{y} \right ) w_x - \left ( c \sqrt{x} + a y \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{9 a^3 x^3+16 b^2 c x^{3/2}}{24 b^2}\right )\right \},\left \{w(x,y)\to c_1\left (\frac{1}{3} \left (3 a x y-2 b y^{3/2}+2 c x^{3/2}\right )\right )\right \},\left \{w(x,y)\to c_1\left (\frac{1}{3} \left (3 a x y+2 b y^{3/2}+2 c x^{3/2}\right )\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( \RootOf \left ( 8\,a{y}^{5/2}b{c}^{2}+3\,{a}^{4}{y}^{4}-2\,\sqrt [3]{-4\,{y}^{3/2}b{c}^{2}-{a}^{3}{y}^{3}-6\,{c}^{2}{\it \_Z}+2\,\sqrt{4\,{y}^{3}{b}^{2}{c}^{2}+2\,{y}^{9/2}{a}^{3}b+12\,{y}^{3/2}{\it \_Z}\,b{c}^{2}+3\,{\it \_Z}\,{a}^{3}{y}^{3}+9\,{{\it \_Z}}^{2}{c}^{2}}c}{a}^{3}{y}^{3}+3\, \left ( -4\,{y}^{3/2}b{c}^{2}-{a}^{3}{y}^{3}-6\,{c}^{2}{\it \_Z}+2\,\sqrt{4\,{y}^{3}{b}^{2}{c}^{2}+2\,{y}^{9/2}{a}^{3}b+12\,{y}^{3/2}{\it \_Z}\,b{c}^{2}+3\,{\it \_Z}\,{a}^{3}{y}^{3}+9\,{{\it \_Z}}^{2}{c}^{2}}c \right ) ^{2/3}{a}^{2}{y}^{2}-4\,x{c}^{2} \left ( -4\,{y}^{3/2}b{c}^{2}-{a}^{3}{y}^{3}-6\,{c}^{2}{\it \_Z}+2\,\sqrt{4\,{y}^{3}{b}^{2}{c}^{2}+2\,{y}^{9/2}{a}^{3}b+12\,{y}^{3/2}{\it \_Z}\,b{c}^{2}+3\,{\it \_Z}\,{a}^{3}{y}^{3}+9\,{{\it \_Z}}^{2}{c}^{2}}c \right ) ^{2/3}+12\,a{c}^{2}y{\it \_Z}+ \left ( -4\,{y}^{3/2}b{c}^{2}-{a}^{3}{y}^{3}-6\,{c}^{2}{\it \_Z}+2\,\sqrt{4\,{y}^{3}{b}^{2}{c}^{2}+2\,{y}^{9/2}{a}^{3}b+12\,{y}^{3/2}{\it \_Z}\,b{c}^{2}+3\,{\it \_Z}\,{a}^{3}{y}^{3}+9\,{{\it \_Z}}^{2}{c}^{2}}c \right ) ^{4/3}-4\,\sqrt{4\,{y}^{3}{b}^{2}{c}^{2}+2\,{y}^{9/2}{a}^{3}b+12\,{y}^{3/2}{\it \_Z}\,b{c}^{2}+3\,{\it \_Z}\,{a}^{3}{y}^{3}+9\,{{\it \_Z}}^{2}{c}^{2}}acy \right ) \right ) \]

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38.7 problem number 7

problem number 269

Added January 2, 2019.

Problem 2.2.4.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ \sqrt{f(x)} w_x + \sqrt{f(y)} w_y = 0 \] Where \(f(t) = \sum _{n=0}^{4} a_n t^n \)

Mathematica

\[ \text{DSolve}\left [\sqrt{a(4) x^4+a(3) x^3+a(2) x^2+a(1) x} w^{(1,0)}(x,y)+\sqrt{a(4) y^4+a(3) y^3+a(2) y^2+a(1) y} w^{(0,1)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -2\,{1 \left ( 1/12\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}-1/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}+1/3\,{\frac{a_{{3}}}{a_{{4}}}}+i/2\sqrt{3} \left ( 1/6\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}} \right ) \right ) \sqrt{{x \left ( -1/4\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}-i/2\sqrt{3} \left ( 1/6\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}} \right ) \right ) \left ( -1/12\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+1/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}-1/3\,{\frac{a_{{3}}}{a_{{4}}}}-i/2\sqrt{3} \left ( 1/6\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}} \right ) \right ) ^{-1} \left ( x-1/6\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}+1/3\,{\frac{a_{{3}}}{a_{{4}}}} \right ) ^{-1}}} \left ( x-1/6\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}+1/3\,{\frac{a_{{3}}}{a_{{4}}}} \right ) ^{2}\sqrt{{1 \left ( 1/6\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}-2/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}-1/3\,{\frac{a_{{3}}}{a_{{4}}}} \right ) \left ( x+1/12\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}-1/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}+1/3\,{\frac{a_{{3}}}{a_{{4}}}}-i/2\sqrt{3} \left ( 1/6\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}} \right ) \right ) \left ( -1/12\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+1/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}-1/3\,{\frac{a_{{3}}}{a_{{4}}}}+i/2\sqrt{3} \left ( 1/6\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}} \right ) \right ) ^{-1} \left ( x-1/6\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}+1/3\,{\frac{a_{{3}}}{a_{{4}}}} \right ) ^{-1}}}\sqrt{{1 \left ( 1/6\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}-2/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}-1/3\,{\frac{a_{{3}}}{a_{{4}}}} \right ) \left ( x+1/12\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}-1/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}+1/3\,{\frac{a_{{3}}}{a_{{4}}}}+i/2\sqrt{3} \left ( 1/6\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}} \right ) \right ) \left ( -1/12\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+1/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}-1/3\,{\frac{a_{{3}}}{a_{{4}}}}-i/2\sqrt{3} \left ( 1/6\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}} \right ) \right ) ^{-1} \left ( x-1/6\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}+1/3\,{\frac{a_{{3}}}{a_{{4}}}} \right ) ^{-1}}}\EllipticF \left ( \sqrt{{x \left ( -1/4\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}-i/2\sqrt{3} \left ( 1/6\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}} \right ) \right ) \left ( -1/12\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+1/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}-1/3\,{\frac{a_{{3}}}{a_{{4}}}}-i/2\sqrt{3} \left ( 1/6\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}} \right ) \right ) ^{-1} \left ( x-1/6\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}+1/3\,{\frac{a_{{3}}}{a_{{4}}}} \right ) ^{-1}}},\sqrt{{1 \left ( 1/4\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}-{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}-i/2\sqrt{3} \left ( 1/6\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}} \right ) \right ) \left ( 1/12\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}-1/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}+1/3\,{\frac{a_{{3}}}{a_{{4}}}}+i/2\sqrt{3} \left ( 1/6\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}} \right ) \right ) \left ( 1/12\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}-1/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}+1/3\,{\frac{a_{{3}}}{a_{{4}}}}-i/2\sqrt{3} \left ( 1/6\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}} \right ) \right ) ^{-1} \left ( 1/4\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}-{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}+i/2\sqrt{3} \left ( 1/6\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}} \right ) \right ) ^{-1}}} \right ) \left ( -1/4\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}-i/2\sqrt{3} \left ( 1/6\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}} \right ) \right ) ^{-1} \left ( 1/6\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}-2/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}-1/3\,{\frac{a_{{3}}}{a_{{4}}}} \right ) ^{-1}{\frac{1}{\sqrt{a_{{4}}x \left ( x-1/6\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}+1/3\,{\frac{a_{{3}}}{a_{{4}}}} \right ) \left ( x+1/12\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}-1/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}+1/3\,{\frac{a_{{3}}}{a_{{4}}}}-i/2\sqrt{3} \left ( 1/6\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}} \right ) \right ) \left ( x+1/12\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}-1/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}+1/3\,{\frac{a_{{3}}}{a_{{4}}}}+i/2\sqrt{3} \left ( 1/6\,{\frac{\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac{3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}} \right ) \right ) }}}}-48\,{\frac{ \left ( i \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}\sqrt{3}+12\,i\sqrt{3}a_{{2}}a_{{4}}-4\,i\sqrt{3}{a_{{3}}}^{2}+ \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}+4\,a_{{3}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}-12\,a_{{2}}a_{{4}}+4\,{a_{{3}}}^{2} \right ) \sqrt{6} \left ( 6\,ya_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}- \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}+2\,a_{{3}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}+12\,a_{{2}}a_{{4}}-4\,{a_{{3}}}^{2} \right ) ^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}} \left ( i \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}\sqrt{3}+12\,i\sqrt{3}a_{{2}}a_{{4}}-4\,i\sqrt{3}{a_{{3}}}^{2}+3\, \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}-36\,a_{{2}}a_{{4}}+12\,{a_{{3}}}^{2} \right ) \left ( \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}-2\,a_{{3}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}-12\,a_{{2}}a_{{4}}+4\,{a_{{3}}}^{2} \right ) }\sqrt{{\frac{ \left ( i \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}\sqrt{3}+12\,i\sqrt{3}a_{{2}}a_{{4}}-4\,i\sqrt{3}{a_{{3}}}^{2}+3\, \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}-36\,a_{{2}}a_{{4}}+12\,{a_{{3}}}^{2} \right ) ya_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{ \left ( i \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}\sqrt{3}+12\,i\sqrt{3}a_{{2}}a_{{4}}-4\,i\sqrt{3}{a_{{3}}}^{2}+ \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}+4\,a_{{3}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}-12\,a_{{2}}a_{{4}}+4\,{a_{{3}}}^{2} \right ) \left ( 6\,ya_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}- \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}+2\,a_{{3}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}+12\,a_{{2}}a_{{4}}-4\,{a_{{3}}}^{2} \right ) }}}\sqrt{-{\frac{ \left ( \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}-2\,a_{{3}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}-12\,a_{{2}}a_{{4}}+4\,{a_{{3}}}^{2} \right ) \left ( i \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}\sqrt{3}+12\,i\sqrt{3}a_{{2}}a_{{4}}-4\,i\sqrt{3}{a_{{3}}}^{2}-12\,ya_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}- \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}-4\,a_{{3}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}+12\,a_{{2}}a_{{4}}-4\,{a_{{3}}}^{2} \right ) }{ \left ( i \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}\sqrt{3}+12\,i\sqrt{3}a_{{2}}a_{{4}}-4\,i\sqrt{3}{a_{{3}}}^{2}- \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}-4\,a_{{3}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}+12\,a_{{2}}a_{{4}}-4\,{a_{{3}}}^{2} \right ) \left ( 6\,ya_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}- \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}+2\,a_{{3}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}+12\,a_{{2}}a_{{4}}-4\,{a_{{3}}}^{2} \right ) }}}\sqrt{-{\frac{ \left ( \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}-2\,a_{{3}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}-12\,a_{{2}}a_{{4}}+4\,{a_{{3}}}^{2} \right ) \left ( i \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}\sqrt{3}+12\,i\sqrt{3}a_{{2}}a_{{4}}-4\,i\sqrt{3}{a_{{3}}}^{2}+12\,ya_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}+ \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}+4\,a_{{3}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}-12\,a_{{2}}a_{{4}}+4\,{a_{{3}}}^{2} \right ) }{ \left ( i \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}\sqrt{3}+12\,i\sqrt{3}a_{{2}}a_{{4}}-4\,i\sqrt{3}{a_{{3}}}^{2}+ \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}+4\,a_{{3}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}-12\,a_{{2}}a_{{4}}+4\,{a_{{3}}}^{2} \right ) \left ( 6\,ya_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}- \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}+2\,a_{{3}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}+12\,a_{{2}}a_{{4}}-4\,{a_{{3}}}^{2} \right ) }}}\EllipticF \left ( \sqrt{6}\sqrt{{\frac{ \left ( i \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}\sqrt{3}+12\,i\sqrt{3}a_{{2}}a_{{4}}-4\,i\sqrt{3}{a_{{3}}}^{2}+3\, \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}-36\,a_{{2}}a_{{4}}+12\,{a_{{3}}}^{2} \right ) ya_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{ \left ( i \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}\sqrt{3}+12\,i\sqrt{3}a_{{2}}a_{{4}}-4\,i\sqrt{3}{a_{{3}}}^{2}+ \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}+4\,a_{{3}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}-12\,a_{{2}}a_{{4}}+4\,{a_{{3}}}^{2} \right ) \left ( 6\,ya_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}- \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}+2\,a_{{3}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}+12\,a_{{2}}a_{{4}}-4\,{a_{{3}}}^{2} \right ) }}},\sqrt{{\frac{ \left ( i \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}\sqrt{3}+12\,i\sqrt{3}a_{{2}}a_{{4}}-4\,i\sqrt{3}{a_{{3}}}^{2}-3\, \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}+36\,a_{{2}}a_{{4}}-12\,{a_{{3}}}^{2} \right ) \left ( i \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}\sqrt{3}+12\,i\sqrt{3}a_{{2}}a_{{4}}-4\,i\sqrt{3}{a_{{3}}}^{2}+ \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}+4\,a_{{3}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}-12\,a_{{2}}a_{{4}}+4\,{a_{{3}}}^{2} \right ) }{ \left ( i \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}\sqrt{3}+12\,i\sqrt{3}a_{{2}}a_{{4}}-4\,i\sqrt{3}{a_{{3}}}^{2}- \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}-4\,a_{{3}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}+12\,a_{{2}}a_{{4}}-4\,{a_{{3}}}^{2} \right ) \left ( i \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}\sqrt{3}+12\,i\sqrt{3}a_{{2}}a_{{4}}-4\,i\sqrt{3}{a_{{3}}}^{2}+3\, \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}-36\,a_{{2}}a_{{4}}+12\,{a_{{3}}}^{2} \right ) }}} \right ){\frac{1}{\sqrt{-6\,{\frac{y \left ( 6\,ya_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}- \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}+2\,a_{{3}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}+12\,a_{{2}}a_{{4}}-4\,{a_{{3}}}^{2} \right ) \left ( i \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}\sqrt{3}+12\,i\sqrt{3}a_{{2}}a_{{4}}-4\,i\sqrt{3}{a_{{3}}}^{2}-12\,ya_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}- \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}-4\,a_{{3}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}+12\,a_{{2}}a_{{4}}-4\,{a_{{3}}}^{2} \right ) \left ( i \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}\sqrt{3}+12\,i\sqrt{3}a_{{2}}a_{{4}}-4\,i\sqrt{3}{a_{{3}}}^{2}+12\,ya_{{4}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}+ \left ( 12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}+4\,a_{{3}}\sqrt [3]{12\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}-12\,a_{{2}}a_{{4}}+4\,{a_{{3}}}^{2} \right ) }{{a_{{4}}}^{2} \left ( 3\,\sqrt{3}\sqrt{27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-27\,a_{{1}}{a_{{4}}}^{2}+9\,a_{{2}}a_{{3}}a_{{4}}-2\,{a_{{3}}}^{3} \right ) }}}}}} \right ) \]