3.1391 \(\int \frac{(5-x) (2+3 x^2)^{5/2}}{(3+2 x)^6} \, dx\)

Optimal. Leaf size=133 \[ \frac{(76 x+23) \left (3 x^2+2\right )^{5/2}}{140 (2 x+3)^5}+\frac{(8193 x+6637) \left (3 x^2+2\right )^{3/2}}{9800 (2 x+3)^3}-\frac{9 (2643 x+8575) \sqrt{3 x^2+2}}{19600 (2 x+3)}+\frac{789723 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{39200 \sqrt{35}}+\frac{63}{32} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

[Out]

(-9*(8575 + 2643*x)*Sqrt[2 + 3*x^2])/(19600*(3 + 2*x)) + ((6637 + 8193*x)*(2 + 3*x^2)^(3/2))/(9800*(3 + 2*x)^3
) + ((23 + 76*x)*(2 + 3*x^2)^(5/2))/(140*(3 + 2*x)^5) + (63*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/32 + (789723*ArcTanh
[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(39200*Sqrt[35])

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Rubi [A]  time = 0.0806538, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {811, 813, 844, 215, 725, 206} \[ \frac{(76 x+23) \left (3 x^2+2\right )^{5/2}}{140 (2 x+3)^5}+\frac{(8193 x+6637) \left (3 x^2+2\right )^{3/2}}{9800 (2 x+3)^3}-\frac{9 (2643 x+8575) \sqrt{3 x^2+2}}{19600 (2 x+3)}+\frac{789723 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{39200 \sqrt{35}}+\frac{63}{32} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^6,x]

[Out]

(-9*(8575 + 2643*x)*Sqrt[2 + 3*x^2])/(19600*(3 + 2*x)) + ((6637 + 8193*x)*(2 + 3*x^2)^(3/2))/(9800*(3 + 2*x)^3
) + ((23 + 76*x)*(2 + 3*x^2)^(5/2))/(140*(3 + 2*x)^5) + (63*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/32 + (789723*ArcTanh
[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(39200*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 813

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + c*x^2)^p)/(e^2*(m + 1)*(m + 2*p + 2)), x] + Di
st[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c
*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2,
0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

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)^6} \, dx &=\frac{(23+76 x) \left (2+3 x^2\right )^{5/2}}{140 (3+2 x)^5}-\frac{\int \frac{(-1248+1752 x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx}{1120}\\ &=\frac{(6637+8193 x) \left (2+3 x^2\right )^{3/2}}{9800 (3+2 x)^3}+\frac{(23+76 x) \left (2+3 x^2\right )^{5/2}}{140 (3+2 x)^5}+\frac{\int \frac{(372096-1522368 x) \sqrt{2+3 x^2}}{(3+2 x)^2} \, dx}{627200}\\ &=-\frac{9 (8575+2643 x) \sqrt{2+3 x^2}}{19600 (3+2 x)}+\frac{(6637+8193 x) \left (2+3 x^2\right )^{3/2}}{9800 (3+2 x)^3}+\frac{(23+76 x) \left (2+3 x^2\right )^{5/2}}{140 (3+2 x)^5}-\frac{\int \frac{12178944-59270400 x}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{5017600}\\ &=-\frac{9 (8575+2643 x) \sqrt{2+3 x^2}}{19600 (3+2 x)}+\frac{(6637+8193 x) \left (2+3 x^2\right )^{3/2}}{9800 (3+2 x)^3}+\frac{(23+76 x) \left (2+3 x^2\right )^{5/2}}{140 (3+2 x)^5}+\frac{189}{32} \int \frac{1}{\sqrt{2+3 x^2}} \, dx-\frac{789723 \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{39200}\\ &=-\frac{9 (8575+2643 x) \sqrt{2+3 x^2}}{19600 (3+2 x)}+\frac{(6637+8193 x) \left (2+3 x^2\right )^{3/2}}{9800 (3+2 x)^3}+\frac{(23+76 x) \left (2+3 x^2\right )^{5/2}}{140 (3+2 x)^5}+\frac{63}{32} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )+\frac{789723 \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )}{39200}\\ &=-\frac{9 (8575+2643 x) \sqrt{2+3 x^2}}{19600 (3+2 x)}+\frac{(6637+8193 x) \left (2+3 x^2\right )^{3/2}}{9800 (3+2 x)^3}+\frac{(23+76 x) \left (2+3 x^2\right )^{5/2}}{140 (3+2 x)^5}+\frac{63}{32} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )+\frac{789723 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )}{39200 \sqrt{35}}\\ \end{align*}

Mathematica [A]  time = 0.195889, size = 100, normalized size = 0.75 \[ \frac{789723 \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )-\frac{70 \sqrt{3 x^2+2} \left (88200 x^5+2740188 x^4+11367738 x^3+20911298 x^2+17940463 x+5999363\right )}{(2 x+3)^5}}{1372000}+\frac{63}{32} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^6,x]

[Out]

(63*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/32 + ((-70*Sqrt[2 + 3*x^2]*(5999363 + 17940463*x + 20911298*x^2 + 11367738*x
^3 + 2740188*x^4 + 88200*x^5))/(3 + 2*x)^5 + 789723*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/13
72000

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Maple [B]  time = 0.013, size = 248, normalized size = 1.9 \begin{align*} -{\frac{13}{5600} \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{11}{24500} \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{521}{857500} \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{2241}{30012500} \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{1131399\,x}{525218750} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{377133}{525218750} \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{267723\,x}{12005000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{248967\,x}{686000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}+{\frac{63\,\sqrt{3}}{32}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{789723\,\sqrt{35}}{1372000}{\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{789723}{262609375} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{789723}{1372000}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}-{\frac{263241}{6002500} \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)^6,x)

[Out]

-13/5600/(x+3/2)^5*(3*(x+3/2)^2-9*x-19/4)^(7/2)-11/24500/(x+3/2)^4*(3*(x+3/2)^2-9*x-19/4)^(7/2)-521/857500/(x+
3/2)^3*(3*(x+3/2)^2-9*x-19/4)^(7/2)-2241/30012500/(x+3/2)^2*(3*(x+3/2)^2-9*x-19/4)^(7/2)+1131399/525218750*x*(
3*(x+3/2)^2-9*x-19/4)^(5/2)-377133/525218750/(x+3/2)*(3*(x+3/2)^2-9*x-19/4)^(7/2)+267723/12005000*x*(3*(x+3/2)
^2-9*x-19/4)^(3/2)+248967/686000*x*(3*(x+3/2)^2-9*x-19/4)^(1/2)+63/32*arcsinh(1/2*x*6^(1/2))*3^(1/2)+789723/13
72000*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))-789723/262609375*(3*(x+3/2)^2-9*x-1
9/4)^(5/2)-789723/1372000*(12*(x+3/2)^2-36*x-19)^(1/2)-263241/6002500*(3*(x+3/2)^2-9*x-19/4)^(3/2)

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Maxima [B]  time = 1.55783, size = 329, normalized size = 2.47 \begin{align*} \frac{6723}{30012500} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} - \frac{13 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{175 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{44 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{6125 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{1042 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{214375 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{2241 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{7503125 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac{267723}{12005000} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x - \frac{263241}{6002500} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} - \frac{377133 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{30012500 \,{\left (2 \, x + 3\right )}} + \frac{248967}{686000} \, \sqrt{3 \, x^{2} + 2} x + \frac{63}{32} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) - \frac{789723}{1372000} \, \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{789723}{686000} \, \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)^6,x, algorithm="maxima")

[Out]

6723/30012500*(3*x^2 + 2)^(5/2) - 13/175*(3*x^2 + 2)^(7/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 24
3) - 44/6125*(3*x^2 + 2)^(7/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 1042/214375*(3*x^2 + 2)^(7/2)/(8*x^3
+ 36*x^2 + 54*x + 27) - 2241/7503125*(3*x^2 + 2)^(7/2)/(4*x^2 + 12*x + 9) + 267723/12005000*(3*x^2 + 2)^(3/2)
*x - 263241/6002500*(3*x^2 + 2)^(3/2) - 377133/30012500*(3*x^2 + 2)^(5/2)/(2*x + 3) + 248967/686000*sqrt(3*x^2
+ 2)*x + 63/32*sqrt(3)*arcsinh(1/2*sqrt(6)*x) - 789723/1372000*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) -
2/3*sqrt(6)/abs(2*x + 3)) - 789723/686000*sqrt(3*x^2 + 2)

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Fricas [A]  time = 1.94904, size = 589, normalized size = 4.43 \begin{align*} \frac{2701125 \, \sqrt{3}{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} \log \left (-\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 789723 \, \sqrt{35}{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\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 (88200 \, x^{5} + 2740188 \, x^{4} + 11367738 \, x^{3} + 20911298 \, x^{2} + 17940463 \, x + 5999363\right )} \sqrt{3 \, x^{2} + 2}}{2744000 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\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)^6,x, algorithm="fricas")

[Out]

1/2744000*(2701125*sqrt(3)*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)*log(-sqrt(3)*sqrt(3*x^2 + 2)*
x - 3*x^2 - 1) + 789723*sqrt(35)*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)*log((sqrt(35)*sqrt(3*x^
2 + 2)*(9*x - 4) - 93*x^2 + 36*x - 43)/(4*x^2 + 12*x + 9)) - 140*(88200*x^5 + 2740188*x^4 + 11367738*x^3 + 209
11298*x^2 + 17940463*x + 5999363)*sqrt(3*x^2 + 2))/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)

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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)**6,x)

[Out]

Timed out

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Giac [B]  time = 1.22076, size = 474, normalized size = 3.56 \begin{align*} -\frac{63}{32} \, \sqrt{3} \log \left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) - \frac{789723}{1372000} \, \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{9}{64} \, \sqrt{3 \, x^{2} + 2} - \frac{3 \,{\left (3103461 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{9} + 28143036 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{8} + 283092753 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{7} + 328235733 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{6} - 360132696 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{5} - 774358774 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{4} + 1736218428 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} - 495467552 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 199787184 \, \sqrt{3} x - 11086336 \, \sqrt{3} - 199787184 \, \sqrt{3 \, x^{2} + 2}\right )}}{156800 \,{\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 )}^{5}} \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)^6,x, algorithm="giac")

[Out]

-63/32*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) - 789723/1372000*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))) - 9/64*sqrt(3*x^2 + 2
) - 3/156800*(3103461*(sqrt(3)*x - sqrt(3*x^2 + 2))^9 + 28143036*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^8 + 283
092753*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 + 328235733*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^6 - 360132696*(sqrt(3
)*x - sqrt(3*x^2 + 2))^5 - 774358774*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 + 1736218428*(sqrt(3)*x - sqrt(3*
x^2 + 2))^3 - 495467552*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 199787184*sqrt(3)*x - 11086336*sqrt(3) - 199
787184*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)^5