Integrand size = 29, antiderivative size = 117 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {358+351 x}{7986 \left (2+3 x^2\right )^{3/2}}+\frac {1216+2133 x}{29282 \sqrt {2+3 x^2}}-\frac {8 \sqrt {2+3 x^2}}{1331 (1+2 x)^2}-\frac {8 \sqrt {2+3 x^2}}{1331 (1+2 x)}-\frac {1216 \text {arctanh}\left (\frac {4-3 x}{\sqrt {11} \sqrt {2+3 x^2}}\right )}{14641 \sqrt {11}} \]
1/7986*(358+351*x)/(3*x^2+2)^(3/2)-1216/161051*arctanh(1/11*(4-3*x)*11^(1/ 2)/(3*x^2+2)^(1/2))*11^(1/2)+1/29282*(1216+2133*x)/(3*x^2+2)^(1/2)-8/1331* (3*x^2+2)^(1/2)/(1+2*x)^2-8/1331*(3*x^2+2)^(1/2)/(1+2*x)
Time = 0.62 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.78 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {\frac {11 \left (7010+57371 x+109844 x^2+116937 x^3+111060 x^4+67284 x^5\right )}{(1+2 x)^2 \left (2+3 x^2\right )^{3/2}}+14592 \sqrt {11} \text {arctanh}\left (\frac {\sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {11}}\right )}{966306} \]
((11*(7010 + 57371*x + 109844*x^2 + 116937*x^3 + 111060*x^4 + 67284*x^5))/ ((1 + 2*x)^2*(2 + 3*x^2)^(3/2)) + 14592*Sqrt[11]*ArcTanh[(Sqrt[3] + 2*Sqrt [3]*x - 2*Sqrt[2 + 3*x^2])/Sqrt[11]])/966306
Time = 0.42 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2178, 27, 2178, 27, 2182, 27, 679, 488, 219}
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.
| \(\displaystyle \int \frac {4 x^2+3 x+1}{(2 x+1)^3 \left (3 x^2+2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2178 |
| \(\displaystyle \frac {351 x+358}{7986 \left (3 x^2+2\right )^{3/2}}-\frac {1}{18} \int -\frac {18 \left (936 x^3+2836 x^2+1914 x+607\right )}{1331 (2 x+1)^3 \left (3 x^2+2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {936 x^3+2836 x^2+1914 x+607}{(2 x+1)^3 \left (3 x^2+2\right )^{3/2}}dx}{1331}+\frac {351 x+358}{7986 \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2178 |
| \(\displaystyle \frac {\frac {2133 x+1216}{22 \sqrt {3 x^2+2}}-\frac {1}{6} \int -\frac {96 \left (304 x^2+315 x+142\right )}{11 (2 x+1)^3 \sqrt {3 x^2+2}}dx}{1331}+\frac {351 x+358}{7986 \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {16}{11} \int \frac {304 x^2+315 x+142}{(2 x+1)^3 \sqrt {3 x^2+2}}dx+\frac {2133 x+1216}{22 \sqrt {3 x^2+2}}}{1331}+\frac {351 x+358}{7986 \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2182 |
| \(\displaystyle \frac {\frac {16}{11} \left (-\frac {1}{22} \int -\frac {11 (271 x+196)}{(2 x+1)^2 \sqrt {3 x^2+2}}dx-\frac {11 \sqrt {3 x^2+2}}{2 (2 x+1)^2}\right )+\frac {2133 x+1216}{22 \sqrt {3 x^2+2}}}{1331}+\frac {351 x+358}{7986 \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {16}{11} \left (\frac {1}{2} \int \frac {271 x+196}{(2 x+1)^2 \sqrt {3 x^2+2}}dx-\frac {11 \sqrt {3 x^2+2}}{2 (2 x+1)^2}\right )+\frac {2133 x+1216}{22 \sqrt {3 x^2+2}}}{1331}+\frac {351 x+358}{7986 \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 679 |
| \(\displaystyle \frac {\frac {16}{11} \left (\frac {1}{2} \left (152 \int \frac {1}{(2 x+1) \sqrt {3 x^2+2}}dx-\frac {11 \sqrt {3 x^2+2}}{2 x+1}\right )-\frac {11 \sqrt {3 x^2+2}}{2 (2 x+1)^2}\right )+\frac {2133 x+1216}{22 \sqrt {3 x^2+2}}}{1331}+\frac {351 x+358}{7986 \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {\frac {16}{11} \left (\frac {1}{2} \left (-152 \int \frac {1}{11-\frac {(4-3 x)^2}{3 x^2+2}}d\frac {4-3 x}{\sqrt {3 x^2+2}}-\frac {11 \sqrt {3 x^2+2}}{2 x+1}\right )-\frac {11 \sqrt {3 x^2+2}}{2 (2 x+1)^2}\right )+\frac {2133 x+1216}{22 \sqrt {3 x^2+2}}}{1331}+\frac {351 x+358}{7986 \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {16}{11} \left (\frac {1}{2} \left (-\frac {152 \text {arctanh}\left (\frac {4-3 x}{\sqrt {11} \sqrt {3 x^2+2}}\right )}{\sqrt {11}}-\frac {11 \sqrt {3 x^2+2}}{2 x+1}\right )-\frac {11 \sqrt {3 x^2+2}}{2 (2 x+1)^2}\right )+\frac {2133 x+1216}{22 \sqrt {3 x^2+2}}}{1331}+\frac {351 x+358}{7986 \left (3 x^2+2\right )^{3/2}}\) |
(358 + 351*x)/(7986*(2 + 3*x^2)^(3/2)) + ((1216 + 2133*x)/(22*Sqrt[2 + 3*x ^2]) + (16*((-11*Sqrt[2 + 3*x^2])/(2*(1 + 2*x)^2) + ((-11*Sqrt[2 + 3*x^2]) /(1 + 2*x) - (152*ArcTanh[(4 - 3*x)/(Sqrt[11]*Sqrt[2 + 3*x^2])])/Sqrt[11]) /2))/11)/1331
3.2.35.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 )/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[Simplify[m + 2*p + 3], 0]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{Qx = PolynomialQuotient[(d + e*x)^m*Pq, a + b*x^2, x], R = Coeff[Po lynomialRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 0], S = Coeff[Polynomia lRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*S - b*R*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*b*(p + 1)) Int[(d + e*x )^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*b*(p + 1)*Qx)/(d + e*x)^m + (b*R*( 2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, d, e}, x] && PolyQ[Pq, x ] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[e*R*(d + e*x)^(m + 1)*((a + b*x^2)^(p + 1)/((m + 1)*(b* d^2 + a*e^2))), x] + Simp[1/((m + 1)*(b*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[(m + 1)*(b*d^2 + a*e^2)*Qx + b*d*R*(m + 1) - b *e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[m, -1]
Time = 0.49 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.64
| method | result | size |
| risch | \(\frac {67284 x^{5}+111060 x^{4}+116937 x^{3}+109844 x^{2}+57371 x +7010}{87846 \left (3 x^{2}+2\right )^{\frac {3}{2}} \left (1+2 x \right )^{2}}-\frac {1216 \sqrt {11}\, \operatorname {arctanh}\left (\frac {2 \left (4-3 x \right ) \sqrt {11}}{11 \sqrt {12 \left (x +\frac {1}{2}\right )^{2}-12 x +5}}\right )}{161051}\) | \(75\) |
| trager | \(\frac {\left (67284 x^{5}+111060 x^{4}+116937 x^{3}+109844 x^{2}+57371 x +7010\right ) \sqrt {3 x^{2}+2}}{87846 \left (6 x^{3}+3 x^{2}+4 x +2\right )^{2}}+\frac {1216 \operatorname {RootOf}\left (\textit {\_Z}^{2}-11\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-11\right ) x +11 \sqrt {3 x^{2}+2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-11\right )}{1+2 x}\right )}{161051}\) | \(101\) |
| default | \(\frac {152}{3993 \left (3 \left (x +\frac {1}{2}\right )^{2}-3 x +\frac {5}{4}\right )^{\frac {3}{2}}}+\frac {87 x}{2662 \left (3 \left (x +\frac {1}{2}\right )^{2}-3 x +\frac {5}{4}\right )^{\frac {3}{2}}}+\frac {1869 x}{29282 \sqrt {3 \left (x +\frac {1}{2}\right )^{2}-3 x +\frac {5}{4}}}+\frac {608}{14641 \sqrt {3 \left (x +\frac {1}{2}\right )^{2}-3 x +\frac {5}{4}}}-\frac {1216 \sqrt {11}\, \operatorname {arctanh}\left (\frac {2 \left (4-3 x \right ) \sqrt {11}}{11 \sqrt {12 \left (x +\frac {1}{2}\right )^{2}-12 x +5}}\right )}{161051}+\frac {1}{484 \left (x +\frac {1}{2}\right ) \left (3 \left (x +\frac {1}{2}\right )^{2}-3 x +\frac {5}{4}\right )^{\frac {3}{2}}}-\frac {1}{88 \left (x +\frac {1}{2}\right )^{2} \left (3 \left (x +\frac {1}{2}\right )^{2}-3 x +\frac {5}{4}\right )^{\frac {3}{2}}}\) | \(140\) |
1/87846*(67284*x^5+111060*x^4+116937*x^3+109844*x^2+57371*x+7010)/(3*x^2+2 )^(3/2)/(1+2*x)^2-1216/161051*11^(1/2)*arctanh(2/11*(4-3*x)*11^(1/2)/(12*( x+1/2)^2-12*x+5)^(1/2))
Time = 0.28 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.27 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {3648 \, \sqrt {11} {\left (36 \, x^{6} + 36 \, x^{5} + 57 \, x^{4} + 48 \, x^{3} + 28 \, x^{2} + 16 \, x + 4\right )} \log \left (-\frac {\sqrt {11} \sqrt {3 \, x^{2} + 2} {\left (3 \, x - 4\right )} + 21 \, x^{2} - 12 \, x + 19}{4 \, x^{2} + 4 \, x + 1}\right ) + 11 \, {\left (67284 \, x^{5} + 111060 \, x^{4} + 116937 \, x^{3} + 109844 \, x^{2} + 57371 \, x + 7010\right )} \sqrt {3 \, x^{2} + 2}}{966306 \, {\left (36 \, x^{6} + 36 \, x^{5} + 57 \, x^{4} + 48 \, x^{3} + 28 \, x^{2} + 16 \, x + 4\right )}} \]
1/966306*(3648*sqrt(11)*(36*x^6 + 36*x^5 + 57*x^4 + 48*x^3 + 28*x^2 + 16*x + 4)*log(-(sqrt(11)*sqrt(3*x^2 + 2)*(3*x - 4) + 21*x^2 - 12*x + 19)/(4*x^ 2 + 4*x + 1)) + 11*(67284*x^5 + 111060*x^4 + 116937*x^3 + 109844*x^2 + 573 71*x + 7010)*sqrt(3*x^2 + 2))/(36*x^6 + 36*x^5 + 57*x^4 + 48*x^3 + 28*x^2 + 16*x + 4)
Timed out. \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \left (2+3 x^2\right )^{5/2}} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.26 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {1216}{161051} \, \sqrt {11} \operatorname {arsinh}\left (\frac {\sqrt {6} x}{2 \, {\left | 2 \, x + 1 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 1 \right |}}\right ) + \frac {1869 \, x}{29282 \, \sqrt {3 \, x^{2} + 2}} + \frac {608}{14641 \, \sqrt {3 \, x^{2} + 2}} + \frac {87 \, x}{2662 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {1}{22 \, {\left (4 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x^{2} + 4 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}\right )}} + \frac {1}{242 \, {\left (2 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}\right )}} + \frac {152}{3993 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \]
1216/161051*sqrt(11)*arcsinh(1/2*sqrt(6)*x/abs(2*x + 1) - 2/3*sqrt(6)/abs( 2*x + 1)) + 1869/29282*x/sqrt(3*x^2 + 2) + 608/14641/sqrt(3*x^2 + 2) + 87/ 2662*x/(3*x^2 + 2)^(3/2) - 1/22/(4*(3*x^2 + 2)^(3/2)*x^2 + 4*(3*x^2 + 2)^( 3/2)*x + (3*x^2 + 2)^(3/2)) + 1/242/(2*(3*x^2 + 2)^(3/2)*x + (3*x^2 + 2)^( 3/2)) + 152/3993/(3*x^2 + 2)^(3/2)
Time = 0.28 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.56 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {1216}{161051} \, \sqrt {11} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {11} - \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {11} + \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) + \frac {9 \, {\left ({\left (2133 \, x + 1216\right )} x + 1851\right )} x + 11234}{87846 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} + \frac {4 \, {\left (\sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 24 \, \sqrt {3} x - 8 \, \sqrt {3} - 24 \, \sqrt {3 \, x^{2} + 2}\right )}}{1331 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{2}} \]
1216/161051*sqrt(11)*log(-abs(-2*sqrt(3)*x - sqrt(11) - sqrt(3) + 2*sqrt(3 *x^2 + 2))/(2*sqrt(3)*x - sqrt(11) + sqrt(3) - 2*sqrt(3*x^2 + 2))) + 1/878 46*(9*((2133*x + 1216)*x + 1851)*x + 11234)/(3*x^2 + 2)^(3/2) + 4/1331*(sq rt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 24*sqrt(3)*x - 8*sqrt(3) - 24*sqrt (3*x^2 + 2))/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + sqrt(3)*(sqrt(3)*x - sqrt( 3*x^2 + 2)) - 2)^2
Time = 12.67 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.57 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {1216\,\sqrt {11}\,\ln \left (x+\frac {1}{2}\right )}{161051}-\frac {1216\,\sqrt {11}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {11}\,\sqrt {x^2+\frac {2}{3}}}{3}-\frac {4}{3}\right )}{161051}-\frac {179\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{95832\,\left (x^2+\frac {2{}\mathrm {i}\,\sqrt {6}\,x}{3}-\frac {2}{3}\right )}+\frac {711\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{58564\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {711\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{58564\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {2\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1331\,\left (x^2+x+\frac {1}{4}\right )}+\frac {179\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{95832\,\left (-x^2+\frac {2{}\mathrm {i}\,\sqrt {6}\,x}{3}+\frac {2}{3}\right )}-\frac {4\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1331\,\left (x+\frac {1}{2}\right )}+\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,13{}\mathrm {i}}{21296\,\left (x^2+\frac {2{}\mathrm {i}\,\sqrt {6}\,x}{3}-\frac {2}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,9265{}\mathrm {i}}{2108304\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,9265{}\mathrm {i}}{2108304\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,13{}\mathrm {i}}{21296\,\left (-x^2+\frac {2{}\mathrm {i}\,\sqrt {6}\,x}{3}+\frac {2}{3}\right )} \]
(1216*11^(1/2)*log(x + 1/2))/161051 - (1216*11^(1/2)*log(x - (3^(1/2)*11^( 1/2)*(x^2 + 2/3)^(1/2))/3 - 4/3))/161051 - (179*3^(1/2)*(x^2 + 2/3)^(1/2)) /(95832*((6^(1/2)*x*2i)/3 + x^2 - 2/3)) + (711*3^(1/2)*(x^2 + 2/3)^(1/2))/ (58564*(x - (6^(1/2)*1i)/3)) + (711*3^(1/2)*(x^2 + 2/3)^(1/2))/(58564*(x + (6^(1/2)*1i)/3)) - (2*3^(1/2)*(x^2 + 2/3)^(1/2))/(1331*(x + x^2 + 1/4)) + (179*3^(1/2)*(x^2 + 2/3)^(1/2))/(95832*((6^(1/2)*x*2i)/3 - x^2 + 2/3)) - (4*3^(1/2)*(x^2 + 2/3)^(1/2))/(1331*(x + 1/2)) + (3^(1/2)*6^(1/2)*(x^2 + 2 /3)^(1/2)*13i)/(21296*((6^(1/2)*x*2i)/3 + x^2 - 2/3)) - (3^(1/2)*6^(1/2)*( x^2 + 2/3)^(1/2)*9265i)/(2108304*(x - (6^(1/2)*1i)/3)) + (3^(1/2)*6^(1/2)* (x^2 + 2/3)^(1/2)*9265i)/(2108304*(x + (6^(1/2)*1i)/3)) + (3^(1/2)*6^(1/2) *(x^2 + 2/3)^(1/2)*13i)/(21296*((6^(1/2)*x*2i)/3 - x^2 + 2/3))