3.1392 \(\int \frac{(5-x) (2+3 x^2)^{5/2}}{(3+2 x)^7} \, dx\)

Optimal. Leaf size=133 \[ \frac{(159 x+11) \left (3 x^2+2\right )^{5/2}}{420 (2 x+3)^6}+\frac{(403 x+202) \left (3 x^2+2\right )^{3/2}}{1568 (2 x+3)^4}+\frac{9 (5167 x+4373) \sqrt{3 x^2+2}}{109760 (2 x+3)^2}-\frac{159759 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{219520 \sqrt{35}}-\frac{9}{128} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

[Out]

(9*(4373 + 5167*x)*Sqrt[2 + 3*x^2])/(109760*(3 + 2*x)^2) + ((202 + 403*x)*(2 + 3*x^2)^(3/2))/(1568*(3 + 2*x)^4
) + ((11 + 159*x)*(2 + 3*x^2)^(5/2))/(420*(3 + 2*x)^6) - (9*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/128 - (159759*ArcTan
h[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(219520*Sqrt[35])

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Rubi [A]  time = 0.0778891, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {811, 844, 215, 725, 206} \[ \frac{(159 x+11) \left (3 x^2+2\right )^{5/2}}{420 (2 x+3)^6}+\frac{(403 x+202) \left (3 x^2+2\right )^{3/2}}{1568 (2 x+3)^4}+\frac{9 (5167 x+4373) \sqrt{3 x^2+2}}{109760 (2 x+3)^2}-\frac{159759 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{219520 \sqrt{35}}-\frac{9}{128} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^7,x]

[Out]

(9*(4373 + 5167*x)*Sqrt[2 + 3*x^2])/(109760*(3 + 2*x)^2) + ((202 + 403*x)*(2 + 3*x^2)^(3/2))/(1568*(3 + 2*x)^4
) + ((11 + 159*x)*(2 + 3*x^2)^(5/2))/(420*(3 + 2*x)^6) - (9*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/128 - (159759*ArcTan
h[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(219520*Sqrt[35])

Rule 811

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((d + e*x)^
(m + 1)*(a + c*x^2)^p*((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*p*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e
^2) + 2*c*d*p*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2
+ a*e^2)), Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p + 1
) - e*f*(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2
, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx &=\frac{(11+159 x) \left (2+3 x^2\right )^{5/2}}{420 (3+2 x)^6}-\frac{\int \frac{(-1560+1260 x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx}{1680}\\ &=\frac{(202+403 x) \left (2+3 x^2\right )^{3/2}}{1568 (3+2 x)^4}+\frac{(11+159 x) \left (2+3 x^2\right )^{5/2}}{420 (3+2 x)^6}+\frac{\int \frac{(496800-1058400 x) \sqrt{2+3 x^2}}{(3+2 x)^3} \, dx}{1881600}\\ &=\frac{9 (4373+5167 x) \sqrt{2+3 x^2}}{109760 (3+2 x)^2}+\frac{(202+403 x) \left (2+3 x^2\right )^{3/2}}{1568 (3+2 x)^4}+\frac{(11+159 x) \left (2+3 x^2\right )^{5/2}}{420 (3+2 x)^6}-\frac{\int \frac{-100051200+444528000 x}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{1053696000}\\ &=\frac{9 (4373+5167 x) \sqrt{2+3 x^2}}{109760 (3+2 x)^2}+\frac{(202+403 x) \left (2+3 x^2\right )^{3/2}}{1568 (3+2 x)^4}+\frac{(11+159 x) \left (2+3 x^2\right )^{5/2}}{420 (3+2 x)^6}-\frac{27}{128} \int \frac{1}{\sqrt{2+3 x^2}} \, dx+\frac{159759 \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{219520}\\ &=\frac{9 (4373+5167 x) \sqrt{2+3 x^2}}{109760 (3+2 x)^2}+\frac{(202+403 x) \left (2+3 x^2\right )^{3/2}}{1568 (3+2 x)^4}+\frac{(11+159 x) \left (2+3 x^2\right )^{5/2}}{420 (3+2 x)^6}-\frac{9}{128} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\frac{159759 \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )}{219520}\\ &=\frac{9 (4373+5167 x) \sqrt{2+3 x^2}}{109760 (3+2 x)^2}+\frac{(202+403 x) \left (2+3 x^2\right )^{3/2}}{1568 (3+2 x)^4}+\frac{(11+159 x) \left (2+3 x^2\right )^{5/2}}{420 (3+2 x)^6}-\frac{9}{128} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\frac{159759 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )}{219520 \sqrt{35}}\\ \end{align*}

Mathematica [A]  time = 0.252436, size = 100, normalized size = 0.75 \[ \frac{\frac{70 \sqrt{3 x^2+2} \left (4369608 x^5+18915336 x^4+47453802 x^3+59256588 x^2+39843609 x+10361807\right )}{(2 x+3)^6}-479277 \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{23049600}-\frac{9}{128} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^7,x]

[Out]

(-9*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/128 + ((70*Sqrt[2 + 3*x^2]*(10361807 + 39843609*x + 59256588*x^2 + 47453802*
x^3 + 18915336*x^4 + 4369608*x^5))/(3 + 2*x)^6 - 479277*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])]
)/23049600

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Maple [B]  time = 0.014, size = 269, normalized size = 2. \begin{align*} -{\frac{13}{13440} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-6}}-{\frac{1}{3136} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-5}}-{\frac{113}{548800} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{1039}{9604000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{6561}{84035000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}+{\frac{123129\,x}{1470612500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{41043}{1470612500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-{\frac{27009\,x}{67228000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{45711\,x}{3841600}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}-{\frac{9\,\sqrt{3}}{128}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }-{\frac{159759\,\sqrt{35}}{7683200}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) }+{\frac{159759}{1470612500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}+{\frac{159759}{7683200}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}+{\frac{53253}{33614000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+2)^(5/2)/(3+2*x)^7,x)

[Out]

-13/13440/(x+3/2)^6*(3*(x+3/2)^2-9*x-19/4)^(7/2)-1/3136/(x+3/2)^5*(3*(x+3/2)^2-9*x-19/4)^(7/2)-113/548800/(x+3
/2)^4*(3*(x+3/2)^2-9*x-19/4)^(7/2)-1039/9604000/(x+3/2)^3*(3*(x+3/2)^2-9*x-19/4)^(7/2)-6561/84035000/(x+3/2)^2
*(3*(x+3/2)^2-9*x-19/4)^(7/2)+123129/1470612500*x*(3*(x+3/2)^2-9*x-19/4)^(5/2)-41043/1470612500/(x+3/2)*(3*(x+
3/2)^2-9*x-19/4)^(7/2)-27009/67228000*x*(3*(x+3/2)^2-9*x-19/4)^(3/2)-45711/3841600*x*(3*(x+3/2)^2-9*x-19/4)^(1
/2)-9/128*arcsinh(1/2*x*6^(1/2))*3^(1/2)-159759/7683200*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-3
6*x-19)^(1/2))+159759/1470612500*(3*(x+3/2)^2-9*x-19/4)^(5/2)+159759/7683200*(12*(x+3/2)^2-36*x-19)^(1/2)+5325
3/33614000*(3*(x+3/2)^2-9*x-19/4)^(3/2)

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Maxima [B]  time = 1.58342, size = 387, normalized size = 2.91 \begin{align*} \frac{19683}{84035000} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} - \frac{13 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{210 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac{{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{98 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{113 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{34300 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{1039 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{1200500 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{6561 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{21008750 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{27009}{67228000} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{53253}{33614000} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} - \frac{41043 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{84035000 \,{\left (2 \, x + 3\right )}} - \frac{45711}{3841600} \, \sqrt{3 \, x^{2} + 2} x - \frac{9}{128} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{159759}{7683200} \, \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{159759}{3841600} \, \sqrt{3 \, x^{2} + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(5/2)/(3+2*x)^7,x, algorithm="maxima")

[Out]

19683/84035000*(3*x^2 + 2)^(5/2) - 13/210*(3*x^2 + 2)^(7/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2
+ 2916*x + 729) - 1/98*(3*x^2 + 2)^(7/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 113/34300*(3
*x^2 + 2)^(7/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 1039/1200500*(3*x^2 + 2)^(7/2)/(8*x^3 + 36*x^2 + 54
*x + 27) - 6561/21008750*(3*x^2 + 2)^(7/2)/(4*x^2 + 12*x + 9) - 27009/67228000*(3*x^2 + 2)^(3/2)*x + 53253/336
14000*(3*x^2 + 2)^(3/2) - 41043/84035000*(3*x^2 + 2)^(5/2)/(2*x + 3) - 45711/3841600*sqrt(3*x^2 + 2)*x - 9/128
*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 159759/7683200*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs
(2*x + 3)) + 159759/3841600*sqrt(3*x^2 + 2)

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Fricas [A]  time = 1.97541, size = 648, normalized size = 4.87 \begin{align*} \frac{1620675 \, \sqrt{3}{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 479277 \, \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 ) + 140 \,{\left (4369608 \, x^{5} + 18915336 \, x^{4} + 47453802 \, x^{3} + 59256588 \, x^{2} + 39843609 \, x + 10361807\right )} \sqrt{3 \, x^{2} + 2}}{46099200 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(5/2)/(3+2*x)^7,x, algorithm="fricas")

[Out]

1/46099200*(1620675*sqrt(3)*(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729)*log(sqrt(3)*sqr
t(3*x^2 + 2)*x - 3*x^2 - 1) + 479277*sqrt(35)*(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 72
9)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) + 140*(4369608*x^5 + 189
15336*x^4 + 47453802*x^3 + 59256588*x^2 + 39843609*x + 10361807)*sqrt(3*x^2 + 2))/(64*x^6 + 576*x^5 + 2160*x^4
+ 4320*x^3 + 4860*x^2 + 2916*x + 729)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x**2+2)**(5/2)/(3+2*x)**7,x)

[Out]

Timed out

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Giac [B]  time = 1.32976, size = 520, normalized size = 3.91 \begin{align*} \frac{9}{128} \, \sqrt{3} \log \left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) + \frac{159759}{7683200} \, \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 (1700928 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{11} + 16427322 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{10} + 212377560 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{9} + 421378065 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{8} + 732041442 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{7} - 879808433 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{6} - 1537837812 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{5} + 2079633300 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{4} - 2495803200 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} + 500387712 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} - 155311488 \, \sqrt{3} x + 7768192 \, \sqrt{3} + 155311488 \, \sqrt{3 \, x^{2} + 2}\right )}}{878080 \,{\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(5/2)/(3+2*x)^7,x, algorithm="giac")

[Out]

9/128*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) + 159759/7683200*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))) + 3/878080*(1700928*(s
qrt(3)*x - sqrt(3*x^2 + 2))^11 + 16427322*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^10 + 212377560*(sqrt(3)*x - sq
rt(3*x^2 + 2))^9 + 421378065*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^8 + 732041442*(sqrt(3)*x - sqrt(3*x^2 + 2))
^7 - 879808433*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^6 - 1537837812*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 + 20796333
00*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 - 2495803200*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 + 500387712*sqrt(3)*(s
qrt(3)*x - sqrt(3*x^2 + 2))^2 - 155311488*sqrt(3)*x + 7768192*sqrt(3) + 155311488*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)^6