Integrand size = 28, antiderivative size = 99 \[ \int \frac {3-x+x^2}{\left (-2-2 x+x^2\right ) \sqrt {-1+x^3}} \, dx=-\frac {1}{6} \sqrt {15+14 \sqrt {3}} \arctan \left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt {-1+x^3}}{1+x+x^2}\right )-\frac {1}{6} \sqrt {-15+14 \sqrt {3}} \text {arctanh}\left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt {-1+x^3}}{1+x+x^2}\right ) \]
-1/6*(15+14*3^(1/2))^(1/2)*arctan((-3+2*3^(1/2))^(1/2)*(x^3-1)^(1/2)/(x^2+ x+1))-1/6*(-15+14*3^(1/2))^(1/2)*arctanh((3+2*3^(1/2))^(1/2)*(x^3-1)^(1/2) /(x^2+x+1))
Time = 1.59 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00 \[ \int \frac {3-x+x^2}{\left (-2-2 x+x^2\right ) \sqrt {-1+x^3}} \, dx=-\frac {1}{6} \sqrt {15+14 \sqrt {3}} \arctan \left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt {-1+x^3}}{1+x+x^2}\right )-\frac {1}{6} \sqrt {-15+14 \sqrt {3}} \text {arctanh}\left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt {-1+x^3}}{1+x+x^2}\right ) \]
-1/6*(Sqrt[15 + 14*Sqrt[3]]*ArcTan[(Sqrt[-3 + 2*Sqrt[3]]*Sqrt[-1 + x^3])/( 1 + x + x^2)]) - (Sqrt[-15 + 14*Sqrt[3]]*ArcTanh[(Sqrt[3 + 2*Sqrt[3]]*Sqrt [-1 + x^3])/(1 + x + x^2)])/6
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.99 (sec) , antiderivative size = 457, normalized size of antiderivative = 4.62, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {7279, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^2-x+3}{\left (x^2-2 x-2\right ) \sqrt {x^3-1}} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {1}{\sqrt {x^3-1}}+\frac {x+5}{\left (x^2-2 x-2\right ) \sqrt {x^3-1}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}+\frac {\sqrt {14-5 \sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{2\ 3^{3/4} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}+\frac {\sqrt {38-21 \sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{2\ 3^{3/4} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}+\frac {1}{6} \sqrt {15+14 \sqrt {3}} \arctan \left (\frac {\sqrt {2 \sqrt {3}-3} (1-x)}{\sqrt {x^3-1}}\right )+\frac {1}{6} \sqrt {14 \sqrt {3}-15} \text {arctanh}\left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {x^3-1}}\right )\) |
(Sqrt[15 + 14*Sqrt[3]]*ArcTan[(Sqrt[-3 + 2*Sqrt[3]]*(1 - x))/Sqrt[-1 + x^3 ]])/6 + (Sqrt[-15 + 14*Sqrt[3]]*ArcTanh[(Sqrt[3 + 2*Sqrt[3]]*(1 - x))/Sqrt [-1 + x^3]])/6 + (Sqrt[38 - 21*Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)/(1 - Sq rt[3] - x)^2]*EllipticF[ArcSin[(1 + Sqrt[3] - x)/(1 - Sqrt[3] - x)], -7 + 4*Sqrt[3]])/(2*3^(3/4)*Sqrt[-((1 - x)/(1 - Sqrt[3] - x)^2)]*Sqrt[-1 + x^3] ) + (Sqrt[14 - 5*Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)/(1 - Sqrt[3] - x)^2]* EllipticF[ArcSin[(1 + Sqrt[3] - x)/(1 - Sqrt[3] - x)], -7 + 4*Sqrt[3]])/(2 *3^(3/4)*Sqrt[-((1 - x)/(1 - Sqrt[3] - x)^2)]*Sqrt[-1 + x^3]) - (2*Sqrt[2 - Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)/(1 - Sqrt[3] - x)^2]*EllipticF[ArcSi n[(1 + Sqrt[3] - x)/(1 - Sqrt[3] - x)], -7 + 4*Sqrt[3]])/(3^(1/4)*Sqrt[-(( 1 - x)/(1 - Sqrt[3] - x)^2)]*Sqrt[-1 + x^3])
3.14.71.3.1 Defintions of rubi rules used
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 10.03 (sec) , antiderivative size = 591, normalized size of antiderivative = 5.97
| method | result | size |
| trager | \(-\frac {\operatorname {RootOf}\left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right ) \ln \left (\frac {-3888 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{5} x^{2}-7776 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{5} x +1800 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{3} x^{2}-2448 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{3} x +7392 \sqrt {x^{3}-1}\, \operatorname {RootOf}\left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{2}+6048 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{3}+53 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right ) x^{2}+3074 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right ) x -1232 \sqrt {x^{3}-1}-2968 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )}{{\left (36 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{2} x +x -28\right )}^{2}}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{2}+30\right ) \ln \left (\frac {3888 \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{2}+30\right ) \operatorname {RootOf}\left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{4} x^{2}+7776 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{4} x \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{2}+30\right )+8280 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{2}+30\right ) x^{2}+10512 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{2}+30\right ) x +6048 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{2}+30\right )+4147 \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{2}+30\right ) x^{2}-44352 \sqrt {x^{3}-1}\, \operatorname {RootOf}\left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{2}+286 \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{2}+30\right ) x +8008 \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{2}+30\right )-44352 \sqrt {x^{3}-1}}{{\left (36 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}+360 \textit {\_Z}^{2}-121\right )^{2} x +29 x +28\right )}^{2}}\right )}{12}\) | \(591\) |
| default | \(\text {Expression too large to display}\) | \(1517\) |
| elliptic | \(\text {Expression too large to display}\) | \(1726\) |
-1/2*RootOf(432*_Z^4+360*_Z^2-121)*ln((-3888*RootOf(432*_Z^4+360*_Z^2-121) ^5*x^2-7776*RootOf(432*_Z^4+360*_Z^2-121)^5*x+1800*RootOf(432*_Z^4+360*_Z^ 2-121)^3*x^2-2448*RootOf(432*_Z^4+360*_Z^2-121)^3*x+7392*(x^3-1)^(1/2)*Roo tOf(432*_Z^4+360*_Z^2-121)^2+6048*RootOf(432*_Z^4+360*_Z^2-121)^3+53*RootO f(432*_Z^4+360*_Z^2-121)*x^2+3074*RootOf(432*_Z^4+360*_Z^2-121)*x-1232*(x^ 3-1)^(1/2)-2968*RootOf(432*_Z^4+360*_Z^2-121))/(36*RootOf(432*_Z^4+360*_Z^ 2-121)^2*x+x-28)^2)+1/12*RootOf(_Z^2+36*RootOf(432*_Z^4+360*_Z^2-121)^2+30 )*ln((3888*RootOf(_Z^2+36*RootOf(432*_Z^4+360*_Z^2-121)^2+30)*RootOf(432*_ Z^4+360*_Z^2-121)^4*x^2+7776*RootOf(432*_Z^4+360*_Z^2-121)^4*x*RootOf(_Z^2 +36*RootOf(432*_Z^4+360*_Z^2-121)^2+30)+8280*RootOf(432*_Z^4+360*_Z^2-121) ^2*RootOf(_Z^2+36*RootOf(432*_Z^4+360*_Z^2-121)^2+30)*x^2+10512*RootOf(432 *_Z^4+360*_Z^2-121)^2*RootOf(_Z^2+36*RootOf(432*_Z^4+360*_Z^2-121)^2+30)*x +6048*RootOf(432*_Z^4+360*_Z^2-121)^2*RootOf(_Z^2+36*RootOf(432*_Z^4+360*_ Z^2-121)^2+30)+4147*RootOf(_Z^2+36*RootOf(432*_Z^4+360*_Z^2-121)^2+30)*x^2 -44352*(x^3-1)^(1/2)*RootOf(432*_Z^4+360*_Z^2-121)^2+286*RootOf(_Z^2+36*Ro otOf(432*_Z^4+360*_Z^2-121)^2+30)*x+8008*RootOf(_Z^2+36*RootOf(432*_Z^4+36 0*_Z^2-121)^2+30)-44352*(x^3-1)^(1/2))/(36*RootOf(432*_Z^4+360*_Z^2-121)^2 *x+29*x+28)^2)
Leaf count of result is larger than twice the leaf count of optimal. 403 vs. \(2 (75) = 150\).
Time = 0.30 (sec) , antiderivative size = 403, normalized size of antiderivative = 4.07 \[ \int \frac {3-x+x^2}{\left (-2-2 x+x^2\right ) \sqrt {-1+x^3}} \, dx=-\frac {1}{24} \, \sqrt {14 \, \sqrt {3} - 15} \log \left (\frac {11 \, x^{4} + 22 \, x^{3} + 66 \, x^{2} + 2 \, \sqrt {x^{3} - 1} {\left (4 \, x^{2} + \sqrt {3} {\left (3 \, x^{2} + 2 \, x + 4\right )} + 10 \, x - 2\right )} \sqrt {14 \, \sqrt {3} - 15} + 44 \, \sqrt {3} {\left (x^{3} - 1\right )} - 44 \, x + 44}{x^{4} - 4 \, x^{3} + 8 \, x + 4}\right ) + \frac {1}{24} \, \sqrt {14 \, \sqrt {3} - 15} \log \left (\frac {11 \, x^{4} + 22 \, x^{3} + 66 \, x^{2} - 2 \, \sqrt {x^{3} - 1} {\left (4 \, x^{2} + \sqrt {3} {\left (3 \, x^{2} + 2 \, x + 4\right )} + 10 \, x - 2\right )} \sqrt {14 \, \sqrt {3} - 15} + 44 \, \sqrt {3} {\left (x^{3} - 1\right )} - 44 \, x + 44}{x^{4} - 4 \, x^{3} + 8 \, x + 4}\right ) - \frac {1}{24} \, \sqrt {-14 \, \sqrt {3} - 15} \log \left (\frac {11 \, x^{4} + 22 \, x^{3} + 66 \, x^{2} + 2 \, \sqrt {x^{3} - 1} {\left (4 \, x^{2} - \sqrt {3} {\left (3 \, x^{2} + 2 \, x + 4\right )} + 10 \, x - 2\right )} \sqrt {-14 \, \sqrt {3} - 15} - 44 \, \sqrt {3} {\left (x^{3} - 1\right )} - 44 \, x + 44}{x^{4} - 4 \, x^{3} + 8 \, x + 4}\right ) + \frac {1}{24} \, \sqrt {-14 \, \sqrt {3} - 15} \log \left (\frac {11 \, x^{4} + 22 \, x^{3} + 66 \, x^{2} - 2 \, \sqrt {x^{3} - 1} {\left (4 \, x^{2} - \sqrt {3} {\left (3 \, x^{2} + 2 \, x + 4\right )} + 10 \, x - 2\right )} \sqrt {-14 \, \sqrt {3} - 15} - 44 \, \sqrt {3} {\left (x^{3} - 1\right )} - 44 \, x + 44}{x^{4} - 4 \, x^{3} + 8 \, x + 4}\right ) \]
-1/24*sqrt(14*sqrt(3) - 15)*log((11*x^4 + 22*x^3 + 66*x^2 + 2*sqrt(x^3 - 1 )*(4*x^2 + sqrt(3)*(3*x^2 + 2*x + 4) + 10*x - 2)*sqrt(14*sqrt(3) - 15) + 4 4*sqrt(3)*(x^3 - 1) - 44*x + 44)/(x^4 - 4*x^3 + 8*x + 4)) + 1/24*sqrt(14*s qrt(3) - 15)*log((11*x^4 + 22*x^3 + 66*x^2 - 2*sqrt(x^3 - 1)*(4*x^2 + sqrt (3)*(3*x^2 + 2*x + 4) + 10*x - 2)*sqrt(14*sqrt(3) - 15) + 44*sqrt(3)*(x^3 - 1) - 44*x + 44)/(x^4 - 4*x^3 + 8*x + 4)) - 1/24*sqrt(-14*sqrt(3) - 15)*l og((11*x^4 + 22*x^3 + 66*x^2 + 2*sqrt(x^3 - 1)*(4*x^2 - sqrt(3)*(3*x^2 + 2 *x + 4) + 10*x - 2)*sqrt(-14*sqrt(3) - 15) - 44*sqrt(3)*(x^3 - 1) - 44*x + 44)/(x^4 - 4*x^3 + 8*x + 4)) + 1/24*sqrt(-14*sqrt(3) - 15)*log((11*x^4 + 22*x^3 + 66*x^2 - 2*sqrt(x^3 - 1)*(4*x^2 - sqrt(3)*(3*x^2 + 2*x + 4) + 10* x - 2)*sqrt(-14*sqrt(3) - 15) - 44*sqrt(3)*(x^3 - 1) - 44*x + 44)/(x^4 - 4 *x^3 + 8*x + 4))
\[ \int \frac {3-x+x^2}{\left (-2-2 x+x^2\right ) \sqrt {-1+x^3}} \, dx=\int \frac {x^{2} - x + 3}{\sqrt {\left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} - 2 x - 2\right )}\, dx \]
\[ \int \frac {3-x+x^2}{\left (-2-2 x+x^2\right ) \sqrt {-1+x^3}} \, dx=\int { \frac {x^{2} - x + 3}{\sqrt {x^{3} - 1} {\left (x^{2} - 2 \, x - 2\right )}} \,d x } \]
\[ \int \frac {3-x+x^2}{\left (-2-2 x+x^2\right ) \sqrt {-1+x^3}} \, dx=\int { \frac {x^{2} - x + 3}{\sqrt {x^{3} - 1} {\left (x^{2} - 2 \, x - 2\right )}} \,d x } \]
Time = 6.22 (sec) , antiderivative size = 505, normalized size of antiderivative = 5.10 \[ \int \frac {3-x+x^2}{\left (-2-2 x+x^2\right ) \sqrt {-1+x^3}} \, dx=-\frac {2\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}}-\frac {\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (\sqrt {3}-6\right )\,\Pi \left (\frac {\sqrt {3}\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{3\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}}+\frac {\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (\sqrt {3}+6\right )\,\Pi \left (-\frac {\sqrt {3}\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{3\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
(((3^(1/2)*1i)/2 + 3/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 3/2 ))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(3^(1/2) + 6)*ellipticPi(-(3^(1/2)*((3^(1 /2)*1i)/2 + 3/2))/3, asin((-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^( 1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)))/(3*(((3^(1/2)*1i)/2 - 1/2)*((3^ (1/2)*1i)/2 + 1/2) - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) + x^3)^(1/2)) - (((3^(1/2)*1i)/2 + 3/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^ (1/2)*1i)/2 - 3/2))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 3/ 2))^(1/2)*(-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(3^(1/2) - 6)*ellipticPi ((3^(1/2)*((3^(1/2)*1i)/2 + 3/2))/3, asin((-(x - 1)/((3^(1/2)*1i)/2 + 3/2) )^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)))/(3*(((3^(1/2)*1 i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1 i)/2 + 1/2) + 1) + x^3)^(1/2)) - (2*((3^(1/2)*1i)/2 + 3/2)*(-(x - (3^(1/2) *1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/(( 3^(1/2)*1i)/2 + 3/2))^(1/2)*(-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipt icF(asin((-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2) /((3^(1/2)*1i)/2 - 3/2)))/(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) + x^3)^(1/2)