Integrand size = 24, antiderivative size = 131 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {26+41 x}{210 (3+2 x)^2 \left (2+3 x^2\right )^{3/2}}+\frac {4+419 x}{1050 (3+2 x)^2 \sqrt {2+3 x^2}}+\frac {83 \sqrt {2+3 x^2}}{1225 (3+2 x)^2}+\frac {857 \sqrt {2+3 x^2}}{128625 (3+2 x)}-\frac {3072 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{42875 \sqrt {35}} \]
1/210*(26+41*x)/(3+2*x)^2/(3*x^2+2)^(3/2)-3072/1500625*arctanh(1/35*(4-9*x )*35^(1/2)/(3*x^2+2)^(1/2))*35^(1/2)+1/1050*(4+419*x)/(3+2*x)^2/(3*x^2+2)^ (1/2)+83/1225*(3*x^2+2)^(1/2)/(3+2*x)^2+857/128625*(3*x^2+2)^(1/2)/(3+2*x)
Time = 0.61 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.71 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {\frac {35 \left (41366+89749 x+91268 x^2+116367 x^3+67716 x^4+10284 x^5\right )}{(3+2 x)^2 \left (2+3 x^2\right )^{3/2}}+12288 \sqrt {35} \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )}{3001250} \]
((35*(41366 + 89749*x + 91268*x^2 + 116367*x^3 + 67716*x^4 + 10284*x^5))/( (3 + 2*x)^2*(2 + 3*x^2)^(3/2)) + 12288*Sqrt[35]*ArcTanh[(3*Sqrt[3] + 2*Sqr t[3]*x - 2*Sqrt[2 + 3*x^2])/Sqrt[35]])/3001250
Time = 0.28 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {686, 27, 686, 27, 688, 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.
| \(\displaystyle \int \frac {5-x}{(2 x+3)^3 \left (3 x^2+2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {41 x+26}{210 (2 x+3)^2 \left (3 x^2+2\right )^{3/2}}-\frac {1}{630} \int -\frac {6 (164 x+253)}{(2 x+3)^3 \left (3 x^2+2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{105} \int \frac {164 x+253}{(2 x+3)^3 \left (3 x^2+2\right )^{3/2}}dx+\frac {41 x+26}{210 (2 x+3)^2 \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {1}{105} \left (\frac {419 x+4}{10 (2 x+3)^2 \sqrt {3 x^2+2}}-\frac {1}{210} \int -\frac {84 (419 x+6)}{(2 x+3)^3 \sqrt {3 x^2+2}}dx\right )+\frac {41 x+26}{210 (2 x+3)^2 \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{105} \left (\frac {2}{5} \int \frac {419 x+6}{(2 x+3)^3 \sqrt {3 x^2+2}}dx+\frac {419 x+4}{10 (2 x+3)^2 \sqrt {3 x^2+2}}\right )+\frac {41 x+26}{210 (2 x+3)^2 \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle \frac {1}{105} \left (\frac {2}{5} \left (\frac {249 \sqrt {3 x^2+2}}{14 (2 x+3)^2}-\frac {1}{70} \int -\frac {5 (747 x+692)}{(2 x+3)^2 \sqrt {3 x^2+2}}dx\right )+\frac {419 x+4}{10 (2 x+3)^2 \sqrt {3 x^2+2}}\right )+\frac {41 x+26}{210 (2 x+3)^2 \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{105} \left (\frac {2}{5} \left (\frac {1}{14} \int \frac {747 x+692}{(2 x+3)^2 \sqrt {3 x^2+2}}dx+\frac {249 \sqrt {3 x^2+2}}{14 (2 x+3)^2}\right )+\frac {419 x+4}{10 (2 x+3)^2 \sqrt {3 x^2+2}}\right )+\frac {41 x+26}{210 (2 x+3)^2 \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 679 |
| \(\displaystyle \frac {1}{105} \left (\frac {2}{5} \left (\frac {1}{14} \left (\frac {9216}{35} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx+\frac {857 \sqrt {3 x^2+2}}{35 (2 x+3)}\right )+\frac {249 \sqrt {3 x^2+2}}{14 (2 x+3)^2}\right )+\frac {419 x+4}{10 (2 x+3)^2 \sqrt {3 x^2+2}}\right )+\frac {41 x+26}{210 (2 x+3)^2 \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {1}{105} \left (\frac {2}{5} \left (\frac {1}{14} \left (\frac {857 \sqrt {3 x^2+2}}{35 (2 x+3)}-\frac {9216}{35} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}\right )+\frac {249 \sqrt {3 x^2+2}}{14 (2 x+3)^2}\right )+\frac {419 x+4}{10 (2 x+3)^2 \sqrt {3 x^2+2}}\right )+\frac {41 x+26}{210 (2 x+3)^2 \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{105} \left (\frac {2}{5} \left (\frac {1}{14} \left (\frac {857 \sqrt {3 x^2+2}}{35 (2 x+3)}-\frac {9216 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{35 \sqrt {35}}\right )+\frac {249 \sqrt {3 x^2+2}}{14 (2 x+3)^2}\right )+\frac {419 x+4}{10 (2 x+3)^2 \sqrt {3 x^2+2}}\right )+\frac {41 x+26}{210 (2 x+3)^2 \left (3 x^2+2\right )^{3/2}}\) |
(26 + 41*x)/(210*(3 + 2*x)^2*(2 + 3*x^2)^(3/2)) + ((4 + 419*x)/(10*(3 + 2* x)^2*Sqrt[2 + 3*x^2]) + (2*((249*Sqrt[2 + 3*x^2])/(14*(3 + 2*x)^2) + ((857 *Sqrt[2 + 3*x^2])/(35*(3 + 2*x)) - (9216*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[ 2 + 3*x^2])])/(35*Sqrt[35]))/14))/5)/105
3.15.27.3.1 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[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ [p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
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)/( (m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Time = 0.30 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.57
| method | result | size |
| risch | \(\frac {10284 x^{5}+67716 x^{4}+116367 x^{3}+91268 x^{2}+89749 x +41366}{85750 \left (3+2 x \right )^{2} \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {3072 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{1500625}\) | \(75\) |
| trager | \(\frac {\left (10284 x^{5}+67716 x^{4}+116367 x^{3}+91268 x^{2}+89749 x +41366\right ) \sqrt {3 x^{2}+2}}{85750 \left (6 x^{3}+9 x^{2}+4 x +6\right )^{2}}-\frac {3072 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x +35 \sqrt {3 x^{2}+2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )}{3+2 x}\right )}{1500625}\) | \(101\) |
| default | \(-\frac {107}{700 \left (x +\frac {3}{2}\right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}+\frac {128}{1225 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}-\frac {173 x}{2450 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}+\frac {857 x}{85750 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}+\frac {1536}{42875 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}-\frac {3072 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{1500625}-\frac {13}{280 \left (x +\frac {3}{2}\right )^{2} \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}\) | \(140\) |
1/85750*(10284*x^5+67716*x^4+116367*x^3+91268*x^2+89749*x+41366)/(3+2*x)^2 /(3*x^2+2)^(3/2)-3072/1500625*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*( x+3/2)^2-36*x-19)^(1/2))
Time = 0.37 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.14 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {3072 \, \sqrt {35} {\left (36 \, x^{6} + 108 \, x^{5} + 129 \, x^{4} + 144 \, x^{3} + 124 \, x^{2} + 48 \, x + 36\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) + 35 \, {\left (10284 \, x^{5} + 67716 \, x^{4} + 116367 \, x^{3} + 91268 \, x^{2} + 89749 \, x + 41366\right )} \sqrt {3 \, x^{2} + 2}}{3001250 \, {\left (36 \, x^{6} + 108 \, x^{5} + 129 \, x^{4} + 144 \, x^{3} + 124 \, x^{2} + 48 \, x + 36\right )}} \]
1/3001250*(3072*sqrt(35)*(36*x^6 + 108*x^5 + 129*x^4 + 144*x^3 + 124*x^2 + 48*x + 36)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43) /(4*x^2 + 12*x + 9)) + 35*(10284*x^5 + 67716*x^4 + 116367*x^3 + 91268*x^2 + 89749*x + 41366)*sqrt(3*x^2 + 2))/(36*x^6 + 108*x^5 + 129*x^4 + 144*x^3 + 124*x^2 + 48*x + 36)
Timed out. \[ \int \frac {5-x}{(3+2 x)^3 \left (2+3 x^2\right )^{5/2}} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.15 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {3072}{1500625} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {857 \, x}{85750 \, \sqrt {3 \, x^{2} + 2}} + \frac {1536}{42875 \, \sqrt {3 \, x^{2} + 2}} - \frac {173 \, x}{2450 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {13}{70 \, {\left (4 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + 9 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}\right )}} - \frac {107}{350 \, {\left (2 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + 3 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}\right )}} + \frac {128}{1225 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \]
3072/1500625*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs (2*x + 3)) + 857/85750*x/sqrt(3*x^2 + 2) + 1536/42875/sqrt(3*x^2 + 2) - 17 3/2450*x/(3*x^2 + 2)^(3/2) - 13/70/(4*(3*x^2 + 2)^(3/2)*x^2 + 12*(3*x^2 + 2)^(3/2)*x + 9*(3*x^2 + 2)^(3/2)) - 107/350/(2*(3*x^2 + 2)^(3/2)*x + 3*(3* x^2 + 2)^(3/2)) + 128/1225/(3*x^2 + 2)^(3/2)
Time = 0.30 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.59 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {3072}{1500625} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) + \frac {3 \, {\left ({\left (59203 \, x + 69168\right )} x + 37637\right )} x + 190066}{3001250 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {4 \, {\left (9588 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} + 27991 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} - 68448 \, \sqrt {3} x + 9736 \, \sqrt {3} + 68448 \, \sqrt {3 \, x^{2} + 2}\right )}}{1500625 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{2}} \]
3072/1500625*sqrt(35)*log(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*sqr t(3*x^2 + 2))/(2*sqrt(3)*x - sqrt(35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2))) + 1/3001250*(3*((59203*x + 69168)*x + 37637)*x + 190066)/(3*x^2 + 2)^(3/2) - 4/1500625*(9588*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 + 27991*sqrt(3)*(sqrt(3)* x - sqrt(3*x^2 + 2))^2 - 68448*sqrt(3)*x + 9736*sqrt(3) + 68448*sqrt(3*x^2 + 2))/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - sqrt(3*x^ 2 + 2)) - 2)^2
Time = 10.55 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.30 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {3072\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{1500625}-\frac {3072\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{1500625}-\frac {739\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1029000\,\left (x^2+\frac {2{}\mathrm {i}\,\sqrt {6}\,x}{3}-\frac {2}{3}\right )}+\frac {59203\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{18007500\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {59203\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{18007500\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {739\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1029000\,\left (-x^2+\frac {2{}\mathrm {i}\,\sqrt {6}\,x}{3}+\frac {2}{3}\right )}-\frac {4868\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1500625\,\left (x+\frac {3}{2}\right )}-\frac {26\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{42875\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,157{}\mathrm {i}}{6174000\,\left (x^2+\frac {2{}\mathrm {i}\,\sqrt {6}\,x}{3}-\frac {2}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,164201{}\mathrm {i}}{72030000\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,164201{}\mathrm {i}}{72030000\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,157{}\mathrm {i}}{6174000\,\left (-x^2+\frac {2{}\mathrm {i}\,\sqrt {6}\,x}{3}+\frac {2}{3}\right )} \]
(3072*35^(1/2)*log(x + 3/2))/1500625 - (3072*35^(1/2)*log(x - (3^(1/2)*35^ (1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/1500625 - (739*3^(1/2)*(x^2 + 2/3)^(1/2 ))/(1029000*((6^(1/2)*x*2i)/3 + x^2 - 2/3)) + (59203*3^(1/2)*(x^2 + 2/3)^( 1/2))/(18007500*(x - (6^(1/2)*1i)/3)) + (59203*3^(1/2)*(x^2 + 2/3)^(1/2))/ (18007500*(x + (6^(1/2)*1i)/3)) + (739*3^(1/2)*(x^2 + 2/3)^(1/2))/(1029000 *((6^(1/2)*x*2i)/3 - x^2 + 2/3)) - (4868*3^(1/2)*(x^2 + 2/3)^(1/2))/(15006 25*(x + 3/2)) - (26*3^(1/2)*(x^2 + 2/3)^(1/2))/(42875*(3*x + x^2 + 9/4)) - (3^(1/2)*6^(1/2)*(x^2 + 2/3)^(1/2)*157i)/(6174000*((6^(1/2)*x*2i)/3 + x^2 - 2/3)) - (3^(1/2)*6^(1/2)*(x^2 + 2/3)^(1/2)*164201i)/(72030000*(x - (6^( 1/2)*1i)/3)) + (3^(1/2)*6^(1/2)*(x^2 + 2/3)^(1/2)*164201i)/(72030000*(x + (6^(1/2)*1i)/3)) - (3^(1/2)*6^(1/2)*(x^2 + 2/3)^(1/2)*157i)/(6174000*((6^( 1/2)*x*2i)/3 - x^2 + 2/3))