Integrand size = 24, antiderivative size = 148 \[ \int \frac {5-x}{(3+2 x)^5 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {26+41 x}{70 (3+2 x)^4 \sqrt {2+3 x^2}}+\frac {58 \sqrt {2+3 x^2}}{1225 (3+2 x)^4}-\frac {298 \sqrt {2+3 x^2}}{18375 (3+2 x)^3}-\frac {708 \sqrt {2+3 x^2}}{42875 (3+2 x)^2}-\frac {14944 \sqrt {2+3 x^2}}{1500625 (3+2 x)}-\frac {30078 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{1500625 \sqrt {35}} \]
-30078/52521875*arctanh(1/35*(4-9*x)*35^(1/2)/(3*x^2+2)^(1/2))*35^(1/2)+1/ 70*(26+41*x)/(3+2*x)^4/(3*x^2+2)^(1/2)+58/1225*(3*x^2+2)^(1/2)/(3+2*x)^4-2 98/18375*(3*x^2+2)^(1/2)/(3+2*x)^3-708/42875*(3*x^2+2)^(1/2)/(3+2*x)^2-149 44/1500625*(3*x^2+2)^(1/2)/(3+2*x)
Time = 0.87 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.63 \[ \int \frac {5-x}{(3+2 x)^5 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {-\frac {35 \left (4197366+8562487 x+18957672 x^2+22188792 x^3+11467872 x^4+2151936 x^5\right )}{(3+2 x)^4 \sqrt {2+3 x^2}}+360936 \sqrt {35} \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )}{315131250} \]
((-35*(4197366 + 8562487*x + 18957672*x^2 + 22188792*x^3 + 11467872*x^4 + 2151936*x^5))/((3 + 2*x)^4*Sqrt[2 + 3*x^2]) + 360936*Sqrt[35]*ArcTanh[(3*S qrt[3] + 2*Sqrt[3]*x - 2*Sqrt[2 + 3*x^2])/Sqrt[35]])/315131250
Time = 0.33 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.14, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {686, 27, 688, 27, 688, 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.
| \(\displaystyle \int \frac {5-x}{(2 x+3)^5 \left (3 x^2+2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {41 x+26}{70 (2 x+3)^4 \sqrt {3 x^2+2}}-\frac {1}{210} \int -\frac {12 (82 x+65)}{(2 x+3)^5 \sqrt {3 x^2+2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{35} \int \frac {82 x+65}{(2 x+3)^5 \sqrt {3 x^2+2}}dx+\frac {41 x+26}{70 (2 x+3)^4 \sqrt {3 x^2+2}}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle \frac {2}{35} \left (\frac {29 \sqrt {3 x^2+2}}{35 (2 x+3)^4}-\frac {1}{140} \int -\frac {4 (261 x+913)}{(2 x+3)^4 \sqrt {3 x^2+2}}dx\right )+\frac {41 x+26}{70 (2 x+3)^4 \sqrt {3 x^2+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{35} \left (\frac {1}{35} \int \frac {261 x+913}{(2 x+3)^4 \sqrt {3 x^2+2}}dx+\frac {29 \sqrt {3 x^2+2}}{35 (2 x+3)^4}\right )+\frac {41 x+26}{70 (2 x+3)^4 \sqrt {3 x^2+2}}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle \frac {2}{35} \left (\frac {1}{35} \left (-\frac {1}{105} \int -\frac {21 (1323-298 x)}{(2 x+3)^3 \sqrt {3 x^2+2}}dx-\frac {149 \sqrt {3 x^2+2}}{15 (2 x+3)^3}\right )+\frac {29 \sqrt {3 x^2+2}}{35 (2 x+3)^4}\right )+\frac {41 x+26}{70 (2 x+3)^4 \sqrt {3 x^2+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{35} \left (\frac {1}{35} \left (\frac {1}{5} \int \frac {1323-298 x}{(2 x+3)^3 \sqrt {3 x^2+2}}dx-\frac {149 \sqrt {3 x^2+2}}{15 (2 x+3)^3}\right )+\frac {29 \sqrt {3 x^2+2}}{35 (2 x+3)^4}\right )+\frac {41 x+26}{70 (2 x+3)^4 \sqrt {3 x^2+2}}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle \frac {2}{35} \left (\frac {1}{35} \left (\frac {1}{5} \left (-\frac {1}{70} \int -\frac {10 (2143-1062 x)}{(2 x+3)^2 \sqrt {3 x^2+2}}dx-\frac {354 \sqrt {3 x^2+2}}{7 (2 x+3)^2}\right )-\frac {149 \sqrt {3 x^2+2}}{15 (2 x+3)^3}\right )+\frac {29 \sqrt {3 x^2+2}}{35 (2 x+3)^4}\right )+\frac {41 x+26}{70 (2 x+3)^4 \sqrt {3 x^2+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{35} \left (\frac {1}{35} \left (\frac {1}{5} \left (\frac {1}{7} \int \frac {2143-1062 x}{(2 x+3)^2 \sqrt {3 x^2+2}}dx-\frac {354 \sqrt {3 x^2+2}}{7 (2 x+3)^2}\right )-\frac {149 \sqrt {3 x^2+2}}{15 (2 x+3)^3}\right )+\frac {29 \sqrt {3 x^2+2}}{35 (2 x+3)^4}\right )+\frac {41 x+26}{70 (2 x+3)^4 \sqrt {3 x^2+2}}\) |
| \(\Big \downarrow \) 679 |
| \(\displaystyle \frac {2}{35} \left (\frac {1}{35} \left (\frac {1}{5} \left (\frac {1}{7} \left (\frac {15039}{35} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {7472 \sqrt {3 x^2+2}}{35 (2 x+3)}\right )-\frac {354 \sqrt {3 x^2+2}}{7 (2 x+3)^2}\right )-\frac {149 \sqrt {3 x^2+2}}{15 (2 x+3)^3}\right )+\frac {29 \sqrt {3 x^2+2}}{35 (2 x+3)^4}\right )+\frac {41 x+26}{70 (2 x+3)^4 \sqrt {3 x^2+2}}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {2}{35} \left (\frac {1}{35} \left (\frac {1}{5} \left (\frac {1}{7} \left (-\frac {15039}{35} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}-\frac {7472 \sqrt {3 x^2+2}}{35 (2 x+3)}\right )-\frac {354 \sqrt {3 x^2+2}}{7 (2 x+3)^2}\right )-\frac {149 \sqrt {3 x^2+2}}{15 (2 x+3)^3}\right )+\frac {29 \sqrt {3 x^2+2}}{35 (2 x+3)^4}\right )+\frac {41 x+26}{70 (2 x+3)^4 \sqrt {3 x^2+2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2}{35} \left (\frac {1}{35} \left (\frac {1}{5} \left (\frac {1}{7} \left (-\frac {15039 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{35 \sqrt {35}}-\frac {7472 \sqrt {3 x^2+2}}{35 (2 x+3)}\right )-\frac {354 \sqrt {3 x^2+2}}{7 (2 x+3)^2}\right )-\frac {149 \sqrt {3 x^2+2}}{15 (2 x+3)^3}\right )+\frac {29 \sqrt {3 x^2+2}}{35 (2 x+3)^4}\right )+\frac {41 x+26}{70 (2 x+3)^4 \sqrt {3 x^2+2}}\) |
(26 + 41*x)/(70*(3 + 2*x)^4*Sqrt[2 + 3*x^2]) + (2*((29*Sqrt[2 + 3*x^2])/(3 5*(3 + 2*x)^4) + ((-149*Sqrt[2 + 3*x^2])/(15*(3 + 2*x)^3) + ((-354*Sqrt[2 + 3*x^2])/(7*(3 + 2*x)^2) + ((-7472*Sqrt[2 + 3*x^2])/(35*(3 + 2*x)) - (150 39*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(35*Sqrt[35]))/7)/5)/35) )/35
3.15.17.3.1 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 = 0.32 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.51
| method | result | size |
| risch | \(-\frac {2151936 x^{5}+11467872 x^{4}+22188792 x^{3}+18957672 x^{2}+8562487 x +4197366}{9003750 \left (3+2 x \right )^{4} \sqrt {3 x^{2}+2}}-\frac {30078 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{52521875}\) | \(75\) |
| trager | \(-\frac {2151936 x^{5}+11467872 x^{4}+22188792 x^{3}+18957672 x^{2}+8562487 x +4197366}{9003750 \left (3+2 x \right )^{4} \sqrt {3 x^{2}+2}}-\frac {30078 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x +35 \sqrt {3 x^{2}+2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )}{3+2 x}\right )}{52521875}\) | \(91\) |
| default | \(-\frac {913}{117600 \left (x +\frac {3}{2}\right )^{3} \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}-\frac {9}{1000 \left (x +\frac {3}{2}\right )^{2} \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}-\frac {2143}{171500 \left (x +\frac {3}{2}\right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}+\frac {15039}{1500625 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}-\frac {22416 x}{1500625 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}-\frac {30078 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{52521875}-\frac {13}{2240 \left (x +\frac {3}{2}\right )^{4} \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}\) | \(149\) |
-1/9003750*(2151936*x^5+11467872*x^4+22188792*x^3+18957672*x^2+8562487*x+4 197366)/(3+2*x)^4/(3*x^2+2)^(1/2)-30078/52521875*35^(1/2)*arctanh(2/35*(4- 9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))
Time = 0.33 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.01 \[ \int \frac {5-x}{(3+2 x)^5 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {90234 \, \sqrt {35} {\left (48 \, x^{6} + 288 \, x^{5} + 680 \, x^{4} + 840 \, x^{3} + 675 \, x^{2} + 432 \, x + 162\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 35 \, {\left (2151936 \, x^{5} + 11467872 \, x^{4} + 22188792 \, x^{3} + 18957672 \, x^{2} + 8562487 \, x + 4197366\right )} \sqrt {3 \, x^{2} + 2}}{315131250 \, {\left (48 \, x^{6} + 288 \, x^{5} + 680 \, x^{4} + 840 \, x^{3} + 675 \, x^{2} + 432 \, x + 162\right )}} \]
1/315131250*(90234*sqrt(35)*(48*x^6 + 288*x^5 + 680*x^4 + 840*x^3 + 675*x^ 2 + 432*x + 162)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 35*(2151936*x^5 + 11467872*x^4 + 22188792*x^3 + 18957672*x^2 + 8562487*x + 4197366)*sqrt(3*x^2 + 2))/(48*x^6 + 288*x^5 + 680*x^4 + 840*x^3 + 675*x^2 + 432*x + 162)
Timed out. \[ \int \frac {5-x}{(3+2 x)^5 \left (2+3 x^2\right )^{3/2}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (121) = 242\).
Time = 0.29 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.72 \[ \int \frac {5-x}{(3+2 x)^5 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {30078}{52521875} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) - \frac {22416 \, x}{1500625 \, \sqrt {3 \, x^{2} + 2}} + \frac {15039}{1500625 \, \sqrt {3 \, x^{2} + 2}} - \frac {13}{140 \, {\left (16 \, \sqrt {3 \, x^{2} + 2} x^{4} + 96 \, \sqrt {3 \, x^{2} + 2} x^{3} + 216 \, \sqrt {3 \, x^{2} + 2} x^{2} + 216 \, \sqrt {3 \, x^{2} + 2} x + 81 \, \sqrt {3 \, x^{2} + 2}\right )}} - \frac {913}{14700 \, {\left (8 \, \sqrt {3 \, x^{2} + 2} x^{3} + 36 \, \sqrt {3 \, x^{2} + 2} x^{2} + 54 \, \sqrt {3 \, x^{2} + 2} x + 27 \, \sqrt {3 \, x^{2} + 2}\right )}} - \frac {9}{250 \, {\left (4 \, \sqrt {3 \, x^{2} + 2} x^{2} + 12 \, \sqrt {3 \, x^{2} + 2} x + 9 \, \sqrt {3 \, x^{2} + 2}\right )}} - \frac {2143}{85750 \, {\left (2 \, \sqrt {3 \, x^{2} + 2} x + 3 \, \sqrt {3 \, x^{2} + 2}\right )}} \]
30078/52521875*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/a bs(2*x + 3)) - 22416/1500625*x/sqrt(3*x^2 + 2) + 15039/1500625/sqrt(3*x^2 + 2) - 13/140/(16*sqrt(3*x^2 + 2)*x^4 + 96*sqrt(3*x^2 + 2)*x^3 + 216*sqrt( 3*x^2 + 2)*x^2 + 216*sqrt(3*x^2 + 2)*x + 81*sqrt(3*x^2 + 2)) - 913/14700/( 8*sqrt(3*x^2 + 2)*x^3 + 36*sqrt(3*x^2 + 2)*x^2 + 54*sqrt(3*x^2 + 2)*x + 27 *sqrt(3*x^2 + 2)) - 9/250/(4*sqrt(3*x^2 + 2)*x^2 + 12*sqrt(3*x^2 + 2)*x + 9*sqrt(3*x^2 + 2)) - 2143/85750/(2*sqrt(3*x^2 + 2)*x + 3*sqrt(3*x^2 + 2))
Time = 0.31 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.58 \[ \int \frac {5-x}{(3+2 x)^5 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {2}{52521875} \, \sqrt {35} {\left (3736 \, \sqrt {35} \sqrt {3} + 15039 \, \log \left (\sqrt {35} \sqrt {3} - 9\right )\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {\frac {\frac {35 \, {\left (\frac {7 \, {\left (\frac {5 \, {\left (\frac {913}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} + \frac {1365}{{\left (2 \, x + 3\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}\right )}}{2 \, x + 3} + \frac {2646}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}\right )}}{2 \, x + 3} + \frac {12858}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}\right )}}{2 \, x + 3} - \frac {583956}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{2 \, x + 3} + \frac {134496}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{9003750 \, \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3}} - \frac {30078 \, \sqrt {35} \log \left (\sqrt {35} {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )} - 9\right )}{52521875 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} \]
2/52521875*sqrt(35)*(3736*sqrt(35)*sqrt(3) + 15039*log(sqrt(35)*sqrt(3) - 9))*sgn(1/(2*x + 3)) - 1/9003750*((35*(7*(5*(913/sgn(1/(2*x + 3)) + 1365/( (2*x + 3)*sgn(1/(2*x + 3))))/(2*x + 3) + 2646/sgn(1/(2*x + 3)))/(2*x + 3) + 12858/sgn(1/(2*x + 3)))/(2*x + 3) - 583956/sgn(1/(2*x + 3)))/(2*x + 3) + 134496/sgn(1/(2*x + 3)))/sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) - 30078 /52521875*sqrt(35)*log(sqrt(35)*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3)) - 9)/sgn(1/(2*x + 3))
Time = 0.17 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.65 \[ \int \frac {5-x}{(3+2 x)^5 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {30078\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{52521875}-\frac {30078\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{52521875}-\frac {13\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{19600\,\left (x^4+6\,x^3+\frac {27\,x^2}{2}+\frac {27\,x}{2}+\frac {81}{16}\right )}-\frac {168573\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{210087500\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {168573\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{210087500\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {354467\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{105043750\,\left (x+\frac {3}{2}\right )}-\frac {14499\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{6002500\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {323\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{205800\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,36471{}\mathrm {i}}{210087500\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,36471{}\mathrm {i}}{210087500\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \]
(30078*35^(1/2)*log(x + 3/2))/52521875 - (30078*35^(1/2)*log(x - (3^(1/2)* 35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/52521875 - (13*3^(1/2)*(x^2 + 2/3)^( 1/2))/(19600*((27*x)/2 + (27*x^2)/2 + 6*x^3 + x^4 + 81/16)) - (168573*3^(1 /2)*(x^2 + 2/3)^(1/2))/(210087500*(x - (6^(1/2)*1i)/3)) - (168573*3^(1/2)* (x^2 + 2/3)^(1/2))/(210087500*(x + (6^(1/2)*1i)/3)) - (354467*3^(1/2)*(x^2 + 2/3)^(1/2))/(105043750*(x + 3/2)) - (14499*3^(1/2)*(x^2 + 2/3)^(1/2))/( 6002500*(3*x + x^2 + 9/4)) - (323*3^(1/2)*(x^2 + 2/3)^(1/2))/(205800*((27* x)/4 + (9*x^2)/2 + x^3 + 27/8)) - (3^(1/2)*6^(1/2)*(x^2 + 2/3)^(1/2)*36471 i)/(210087500*(x - (6^(1/2)*1i)/3)) + (3^(1/2)*6^(1/2)*(x^2 + 2/3)^(1/2)*3 6471i)/(210087500*(x + (6^(1/2)*1i)/3))