Integrand size = 24, antiderivative size = 131 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^7} \, dx=-\frac {9 (4-9 x) \sqrt {2+3 x^2}}{17500 (3+2 x)^2}-\frac {(4-9 x) \left (2+3 x^2\right )^{3/2}}{500 (3+2 x)^4}-\frac {13 \left (2+3 x^2\right )^{5/2}}{210 (3+2 x)^6}-\frac {29 \left (2+3 x^2\right )^{5/2}}{1750 (3+2 x)^5}-\frac {27 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{8750 \sqrt {35}} \]
-1/500*(4-9*x)*(3*x^2+2)^(3/2)/(3+2*x)^4-13/210*(3*x^2+2)^(5/2)/(3+2*x)^6- 29/1750*(3*x^2+2)^(5/2)/(3+2*x)^5-27/306250*arctanh(1/35*(4-9*x)*35^(1/2)/ (3*x^2+2)^(1/2))*35^(1/2)-9/17500*(4-9*x)*(3*x^2+2)^(1/2)/(3+2*x)^2
Time = 1.48 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.71 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^7} \, dx=-\frac {\sqrt {2+3 x^2} \left (39748+3675 x+33180 x^2-39195 x^3+2160 x^4+432 x^5\right )}{52500 (3+2 x)^6}+\frac {27 \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )}{4375 \sqrt {35}} \]
-1/52500*(Sqrt[2 + 3*x^2]*(39748 + 3675*x + 33180*x^2 - 39195*x^3 + 2160*x ^4 + 432*x^5))/(3 + 2*x)^6 + (27*ArcTanh[(3*Sqrt[3] + 2*Sqrt[3]*x - 2*Sqrt [2 + 3*x^2])/Sqrt[35]])/(4375*Sqrt[35])
Time = 0.26 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {688, 27, 679, 486, 486, 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.
| \(\displaystyle \int \frac {(5-x) \left (3 x^2+2\right )^{3/2}}{(2 x+3)^7} \, dx\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle -\frac {1}{210} \int -\frac {3 (82-13 x) \left (3 x^2+2\right )^{3/2}}{(2 x+3)^6}dx-\frac {13 \left (3 x^2+2\right )^{5/2}}{210 (2 x+3)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{70} \int \frac {(82-13 x) \left (3 x^2+2\right )^{3/2}}{(2 x+3)^6}dx-\frac {13 \left (3 x^2+2\right )^{5/2}}{210 (2 x+3)^6}\) |
| \(\Big \downarrow \) 679 |
| \(\displaystyle \frac {1}{70} \left (\frac {98}{5} \int \frac {\left (3 x^2+2\right )^{3/2}}{(2 x+3)^5}dx-\frac {29 \left (3 x^2+2\right )^{5/2}}{25 (2 x+3)^5}\right )-\frac {13 \left (3 x^2+2\right )^{5/2}}{210 (2 x+3)^6}\) |
| \(\Big \downarrow \) 486 |
| \(\displaystyle \frac {1}{70} \left (\frac {98}{5} \left (\frac {9}{70} \int \frac {\sqrt {3 x^2+2}}{(2 x+3)^3}dx-\frac {(4-9 x) \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\right )-\frac {29 \left (3 x^2+2\right )^{5/2}}{25 (2 x+3)^5}\right )-\frac {13 \left (3 x^2+2\right )^{5/2}}{210 (2 x+3)^6}\) |
| \(\Big \downarrow \) 486 |
| \(\displaystyle \frac {1}{70} \left (\frac {98}{5} \left (\frac {9}{70} \left (\frac {3}{35} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {(4-9 x) \sqrt {3 x^2+2}}{70 (2 x+3)^2}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\right )-\frac {29 \left (3 x^2+2\right )^{5/2}}{25 (2 x+3)^5}\right )-\frac {13 \left (3 x^2+2\right )^{5/2}}{210 (2 x+3)^6}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {1}{70} \left (\frac {98}{5} \left (\frac {9}{70} \left (-\frac {3}{35} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}-\frac {\sqrt {3 x^2+2} (4-9 x)}{70 (2 x+3)^2}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\right )-\frac {29 \left (3 x^2+2\right )^{5/2}}{25 (2 x+3)^5}\right )-\frac {13 \left (3 x^2+2\right )^{5/2}}{210 (2 x+3)^6}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{70} \left (\frac {98}{5} \left (\frac {9}{70} \left (-\frac {3 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{35 \sqrt {35}}-\frac {\sqrt {3 x^2+2} (4-9 x)}{70 (2 x+3)^2}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\right )-\frac {29 \left (3 x^2+2\right )^{5/2}}{25 (2 x+3)^5}\right )-\frac {13 \left (3 x^2+2\right )^{5/2}}{210 (2 x+3)^6}\) |
(-13*(2 + 3*x^2)^(5/2))/(210*(3 + 2*x)^6) + ((-29*(2 + 3*x^2)^(5/2))/(25*( 3 + 2*x)^5) + (98*(-1/140*((4 - 9*x)*(2 + 3*x^2)^(3/2))/(3 + 2*x)^4 + (9*( -1/70*((4 - 9*x)*Sqrt[2 + 3*x^2])/(3 + 2*x)^2 - (3*ArcTanh[(4 - 9*x)/(Sqrt [35]*Sqrt[2 + 3*x^2])])/(35*Sqrt[35])))/70))/5)/70
3.14.79.3.1 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[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (c + d*x)^(n + 1)*(a*d - b*c*x)*((a + b*x^2)^p/((n + 1)*(b*c^2 + a*d^2))), x] - Simp[2*a*b*(p/((n + 1)*(b*c^2 + a*d^2))) Int[(c + d*x)^(n + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[n + 2*p + 2, 0] && GtQ[p, 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[(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 = 85, normalized size of antiderivative = 0.65
| method | result | size |
| risch | \(-\frac {1296 x^{7}+6480 x^{6}-116721 x^{5}+103860 x^{4}-67365 x^{3}+185604 x^{2}+7350 x +79496}{52500 \left (3+2 x \right )^{6} \sqrt {3 x^{2}+2}}-\frac {27 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{306250}\) | \(85\) |
| trager | \(-\frac {\left (432 x^{5}+2160 x^{4}-39195 x^{3}+33180 x^{2}+3675 x +39748\right ) \sqrt {3 x^{2}+2}}{52500 \left (3+2 x \right )^{6}}+\frac {27 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x +35 \sqrt {3 x^{2}+2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )}{3+2 x}\right )}{306250}\) | \(91\) |
| default | \(-\frac {29 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{56000 \left (x +\frac {3}{2}\right )^{5}}-\frac {\left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{4000 \left (x +\frac {3}{2}\right )^{4}}-\frac {9 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{70000 \left (x +\frac {3}{2}\right )^{3}}-\frac {93 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{1225000 \left (x +\frac {3}{2}\right )^{2}}-\frac {1053 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{21437500 \left (x +\frac {3}{2}\right )}+\frac {36 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{5359375}+\frac {243 x \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{612500}+\frac {27 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}{306250}-\frac {27 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{306250}+\frac {3159 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{21437500}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{13440 \left (x +\frac {3}{2}\right )^{6}}\) | \(224\) |
-1/52500*(1296*x^7+6480*x^6-116721*x^5+103860*x^4-67365*x^3+185604*x^2+735 0*x+79496)/(3+2*x)^6/(3*x^2+2)^(1/2)-27/306250*35^(1/2)*arctanh(2/35*(4-9* x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))
Time = 0.27 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.14 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^7} \, dx=\frac {81 \, \sqrt {35} {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 35 \, {\left (432 \, x^{5} + 2160 \, x^{4} - 39195 \, x^{3} + 33180 \, x^{2} + 3675 \, x + 39748\right )} \sqrt {3 \, x^{2} + 2}}{1837500 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} \]
1/1837500*(81*sqrt(35)*(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 35*(432*x^5 + 2160*x^4 - 39195*x^3 + 33180*x^2 + 3675*x + 39748)*sqrt(3*x^2 + 2))/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729)
Timed out. \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^7} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (108) = 216\).
Time = 0.29 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.92 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^7} \, dx=\frac {279}{1225000} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{210 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {29 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{1750 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {{\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{250 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {9 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{8750 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {93 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{306250 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {243}{612500} \, \sqrt {3 \, x^{2} + 2} x + \frac {27}{306250} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {27}{153125} \, \sqrt {3 \, x^{2} + 2} - \frac {1053 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{1225000 \, {\left (2 \, x + 3\right )}} \]
279/1225000*(3*x^2 + 2)^(3/2) - 13/210*(3*x^2 + 2)^(5/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 29/1750*(3*x^2 + 2)^(5 /2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 1/250*(3*x^2 + 2)^(5/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 9/8750*(3*x^2 + 2)^(5 /2)/(8*x^3 + 36*x^2 + 54*x + 27) - 93/306250*(3*x^2 + 2)^(5/2)/(4*x^2 + 12 *x + 9) + 243/612500*sqrt(3*x^2 + 2)*x + 27/306250*sqrt(35)*arcsinh(3/2*sq rt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 27/153125*sqrt(3*x^2 + 2) - 1053/1225000*(3*x^2 + 2)^(3/2)/(2*x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 367 vs. \(2 (108) = 216\).
Time = 0.30 (sec) , antiderivative size = 367, normalized size of antiderivative = 2.80 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^7} \, dx=\frac {27}{306250} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) - \frac {3 \, \sqrt {3} {\left (96 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{11} + 17877 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{10} - 4120 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{9} + 25860 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{8} - 225240 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{7} - 173964 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{6} - 648336 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} + 641040 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} - 309440 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} - 135120 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} - 10752 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} + 1536\right )}}{280000 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{6}} \]
27/306250*sqrt(35)*log(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*sqrt(3 *x^2 + 2))/(2*sqrt(3)*x - sqrt(35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2))) - 3/2 80000*sqrt(3)*(96*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^11 + 17877*(sqrt(3 )*x - sqrt(3*x^2 + 2))^10 - 4120*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^9 + 25860*(sqrt(3)*x - sqrt(3*x^2 + 2))^8 - 225240*sqrt(3)*(sqrt(3)*x - sqrt( 3*x^2 + 2))^7 - 173964*(sqrt(3)*x - sqrt(3*x^2 + 2))^6 - 648336*sqrt(3)*(s qrt(3)*x - sqrt(3*x^2 + 2))^5 + 641040*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 - 3 09440*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 - 135120*(sqrt(3)*x - sqrt(3 *x^2 + 2))^2 - 10752*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) + 1536)/((sqrt( 3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 2)^6
Time = 10.68 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.70 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^7} \, dx=\frac {27\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{306250}-\frac {27\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{306250}-\frac {5977\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{89600\,\left (x^4+6\,x^3+\frac {27\,x^2}{2}+\frac {27\,x}{2}+\frac {81}{16}\right )}+\frac {577\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{5120\,\left (x^5+\frac {15\,x^4}{2}+\frac {45\,x^3}{2}+\frac {135\,x^2}{4}+\frac {405\,x}{16}+\frac {243}{32}\right )}-\frac {9\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{70000\,\left (x+\frac {3}{2}\right )}-\frac {455\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{6144\,\left (x^6+9\,x^5+\frac {135\,x^4}{4}+\frac {135\,x^3}{2}+\frac {1215\,x^2}{16}+\frac {729\,x}{16}+\frac {729}{64}\right )}+\frac {9\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{28000\,\left (x^2+3\,x+\frac {9}{4}\right )}+\frac {2829\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{224000\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )} \]
(27*35^(1/2)*log(x + 3/2))/306250 - (27*35^(1/2)*log(x - (3^(1/2)*35^(1/2) *(x^2 + 2/3)^(1/2))/9 - 4/9))/306250 - (5977*3^(1/2)*(x^2 + 2/3)^(1/2))/(8 9600*((27*x)/2 + (27*x^2)/2 + 6*x^3 + x^4 + 81/16)) + (577*3^(1/2)*(x^2 + 2/3)^(1/2))/(5120*((405*x)/16 + (135*x^2)/4 + (45*x^3)/2 + (15*x^4)/2 + x^ 5 + 243/32)) - (9*3^(1/2)*(x^2 + 2/3)^(1/2))/(70000*(x + 3/2)) - (455*3^(1 /2)*(x^2 + 2/3)^(1/2))/(6144*((729*x)/16 + (1215*x^2)/16 + (135*x^3)/2 + ( 135*x^4)/4 + 9*x^5 + x^6 + 729/64)) + (9*3^(1/2)*(x^2 + 2/3)^(1/2))/(28000 *(3*x + x^2 + 9/4)) + (2829*3^(1/2)*(x^2 + 2/3)^(1/2))/(224000*((27*x)/4 + (9*x^2)/2 + x^3 + 27/8))