3.1393 \(\int \frac{(5-x) (2+3 x^2)^{5/2}}{(3+2 x)^8} \, dx\)

Optimal. Leaf size=136 \[ -\frac{13 \left (3 x^2+2\right )^{7/2}}{245 (2 x+3)^7}-\frac{41 (4-9 x) \left (3 x^2+2\right )^{5/2}}{7350 (2 x+3)^6}-\frac{41 (4-9 x) \left (3 x^2+2\right )^{3/2}}{34300 (2 x+3)^4}-\frac{369 (4-9 x) \sqrt{3 x^2+2}}{1200500 (2 x+3)^2}-\frac{1107 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{600250 \sqrt{35}} \]

[Out]

(-369*(4 - 9*x)*Sqrt[2 + 3*x^2])/(1200500*(3 + 2*x)^2) - (41*(4 - 9*x)*(2 + 3*x^2)^(3/2))/(34300*(3 + 2*x)^4)
- (41*(4 - 9*x)*(2 + 3*x^2)^(5/2))/(7350*(3 + 2*x)^6) - (13*(2 + 3*x^2)^(7/2))/(245*(3 + 2*x)^7) - (1107*ArcTa
nh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(600250*Sqrt[35])

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Rubi [A]  time = 0.0653373, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {807, 721, 725, 206} \[ -\frac{13 \left (3 x^2+2\right )^{7/2}}{245 (2 x+3)^7}-\frac{41 (4-9 x) \left (3 x^2+2\right )^{5/2}}{7350 (2 x+3)^6}-\frac{41 (4-9 x) \left (3 x^2+2\right )^{3/2}}{34300 (2 x+3)^4}-\frac{369 (4-9 x) \sqrt{3 x^2+2}}{1200500 (2 x+3)^2}-\frac{1107 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{600250 \sqrt{35}} \]

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^8,x]

[Out]

(-369*(4 - 9*x)*Sqrt[2 + 3*x^2])/(1200500*(3 + 2*x)^2) - (41*(4 - 9*x)*(2 + 3*x^2)^(3/2))/(34300*(3 + 2*x)^4)
- (41*(4 - 9*x)*(2 + 3*x^2)^(5/2))/(7350*(3 + 2*x)^6) - (13*(2 + 3*x^2)^(7/2))/(245*(3 + 2*x)^7) - (1107*ArcTa
nh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(600250*Sqrt[35])

Rule 807

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] + Dist[(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] && NeQ[c*d^2 + a*e^2, 0]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rule 721

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*(-2*a*e + (2*c*
d)*x)*(a + c*x^2)^p)/(2*(m + 1)*(c*d^2 + a*e^2)), x] - Dist[(4*a*c*p)/(2*(m + 1)*(c*d^2 + a*e^2)), Int[(d + e*
x)^(m + 2)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 2,
0] && GtQ[p, 0]

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)^8} \, dx &=-\frac{13 \left (2+3 x^2\right )^{7/2}}{245 (3+2 x)^7}+\frac{41}{35} \int \frac{\left (2+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx\\ &=-\frac{41 (4-9 x) \left (2+3 x^2\right )^{5/2}}{7350 (3+2 x)^6}-\frac{13 \left (2+3 x^2\right )^{7/2}}{245 (3+2 x)^7}+\frac{41}{245} \int \frac{\left (2+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx\\ &=-\frac{41 (4-9 x) \left (2+3 x^2\right )^{3/2}}{34300 (3+2 x)^4}-\frac{41 (4-9 x) \left (2+3 x^2\right )^{5/2}}{7350 (3+2 x)^6}-\frac{13 \left (2+3 x^2\right )^{7/2}}{245 (3+2 x)^7}+\frac{369 \int \frac{\sqrt{2+3 x^2}}{(3+2 x)^3} \, dx}{17150}\\ &=-\frac{369 (4-9 x) \sqrt{2+3 x^2}}{1200500 (3+2 x)^2}-\frac{41 (4-9 x) \left (2+3 x^2\right )^{3/2}}{34300 (3+2 x)^4}-\frac{41 (4-9 x) \left (2+3 x^2\right )^{5/2}}{7350 (3+2 x)^6}-\frac{13 \left (2+3 x^2\right )^{7/2}}{245 (3+2 x)^7}+\frac{1107 \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{600250}\\ &=-\frac{369 (4-9 x) \sqrt{2+3 x^2}}{1200500 (3+2 x)^2}-\frac{41 (4-9 x) \left (2+3 x^2\right )^{3/2}}{34300 (3+2 x)^4}-\frac{41 (4-9 x) \left (2+3 x^2\right )^{5/2}}{7350 (3+2 x)^6}-\frac{13 \left (2+3 x^2\right )^{7/2}}{245 (3+2 x)^7}-\frac{1107 \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )}{600250}\\ &=-\frac{369 (4-9 x) \sqrt{2+3 x^2}}{1200500 (3+2 x)^2}-\frac{41 (4-9 x) \left (2+3 x^2\right )^{3/2}}{34300 (3+2 x)^4}-\frac{41 (4-9 x) \left (2+3 x^2\right )^{5/2}}{7350 (3+2 x)^6}-\frac{13 \left (2+3 x^2\right )^{7/2}}{245 (3+2 x)^7}-\frac{1107 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )}{600250 \sqrt{35}}\\ \end{align*}

Mathematica [A]  time = 0.185729, size = 122, normalized size = 0.9 \[ \frac{1}{490} \left (-\frac{26 \left (3 x^2+2\right )^{7/2}}{(2 x+3)^7}+\frac{41 (9 x-4) \left (3 x^2+2\right )^{5/2}}{15 (2 x+3)^6}+\frac{41 \left (\frac{35 \sqrt{3 x^2+2} \left (1269 x^3+408 x^2+927 x-604\right )}{(2 x+3)^4}-54 \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )\right )}{85750}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^8,x]

[Out]

((41*(-4 + 9*x)*(2 + 3*x^2)^(5/2))/(15*(3 + 2*x)^6) - (26*(2 + 3*x^2)^(7/2))/(3 + 2*x)^7 + (41*((35*Sqrt[2 + 3
*x^2]*(-604 + 927*x + 408*x^2 + 1269*x^3))/(3 + 2*x)^4 - 54*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^
2])]))/85750)/490

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Maple [B]  time = 0.018, size = 278, normalized size = 2. \begin{align*} -{\frac{13}{31360} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-7}}-{\frac{41}{235200} \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{123}{1372000} \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{1189}{24010000} \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{12177}{420175000} \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{132471}{7353062500} \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{4612869\,x}{128678593750} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{1537623}{128678593750} \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{129519\,x}{1470612500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{9963\,x}{42017500}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}-{\frac{1107\,\sqrt{35}}{21008750}{\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{17712}{64339296875} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}+{\frac{1107}{21008750}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}+{\frac{1476}{367653125} \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)^8,x)

[Out]

-13/31360/(x+3/2)^7*(3*(x+3/2)^2-9*x-19/4)^(7/2)-41/235200/(x+3/2)^6*(3*(x+3/2)^2-9*x-19/4)^(7/2)-123/1372000/
(x+3/2)^5*(3*(x+3/2)^2-9*x-19/4)^(7/2)-1189/24010000/(x+3/2)^4*(3*(x+3/2)^2-9*x-19/4)^(7/2)-12177/420175000/(x
+3/2)^3*(3*(x+3/2)^2-9*x-19/4)^(7/2)-132471/7353062500/(x+3/2)^2*(3*(x+3/2)^2-9*x-19/4)^(7/2)+4612869/12867859
3750*x*(3*(x+3/2)^2-9*x-19/4)^(5/2)-1537623/128678593750/(x+3/2)*(3*(x+3/2)^2-9*x-19/4)^(7/2)+129519/147061250
0*x*(3*(x+3/2)^2-9*x-19/4)^(3/2)+9963/42017500*x*(3*(x+3/2)^2-9*x-19/4)^(1/2)-1107/21008750*35^(1/2)*arctanh(2
/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))+17712/64339296875*(3*(x+3/2)^2-9*x-19/4)^(5/2)+1107/2100875
0*(12*(x+3/2)^2-36*x-19)^(1/2)+1476/367653125*(3*(x+3/2)^2-9*x-19/4)^(3/2)

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Maxima [B]  time = 1.5703, size = 436, normalized size = 3.21 \begin{align*} \frac{397413}{7353062500} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} - \frac{13 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{245 \,{\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac{41 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{3675 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac{123 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{42875 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{1189 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{1500625 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{12177 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{52521875 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{132471 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{1838265625 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac{129519}{1470612500} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{1476}{367653125} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} - \frac{1537623 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{7353062500 \,{\left (2 \, x + 3\right )}} + \frac{9963}{42017500} \, \sqrt{3 \, x^{2} + 2} x + \frac{1107}{21008750} \, \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{1107}{10504375} \, \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)^8,x, algorithm="maxima")

[Out]

397413/7353062500*(3*x^2 + 2)^(5/2) - 13/245*(3*x^2 + 2)^(7/2)/(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22
680*x^3 + 20412*x^2 + 10206*x + 2187) - 41/3675*(3*x^2 + 2)^(7/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 48
60*x^2 + 2916*x + 729) - 123/42875*(3*x^2 + 2)^(7/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 1
189/1500625*(3*x^2 + 2)^(7/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 12177/52521875*(3*x^2 + 2)^(7/2)/(8*x
^3 + 36*x^2 + 54*x + 27) - 132471/1838265625*(3*x^2 + 2)^(7/2)/(4*x^2 + 12*x + 9) + 129519/1470612500*(3*x^2 +
2)^(3/2)*x + 1476/367653125*(3*x^2 + 2)^(3/2) - 1537623/7353062500*(3*x^2 + 2)^(5/2)/(2*x + 3) + 9963/4201750
0*sqrt(3*x^2 + 2)*x + 1107/21008750*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) +
1107/10504375*sqrt(3*x^2 + 2)

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Fricas [A]  time = 1.86882, size = 532, normalized size = 3.91 \begin{align*} \frac{3321 \, \sqrt{35}{\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\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 (656424 \, x^{6} - 9455994 \, x^{5} - 2997810 \, x^{4} - 15015225 \, x^{3} + 3488490 \, x^{2} - 593639 \, x + 4499004\right )} \sqrt{3 \, x^{2} + 2}}{126052500 \,{\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\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)^8,x, algorithm="fricas")

[Out]

1/126052500*(3321*sqrt(35)*(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187
)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 35*(656424*x^6 - 945599
4*x^5 - 2997810*x^4 - 15015225*x^3 + 3488490*x^2 - 593639*x + 4499004)*sqrt(3*x^2 + 2))/(128*x^7 + 1344*x^6 +
6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187)

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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)**8,x)

[Out]

Timed out

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Giac [B]  time = 1.32946, size = 551, normalized size = 4.05 \begin{align*} \frac{1107}{21008750} \, \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 \,{\left (908247 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{13} + 3755004 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{12} + 52905908 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{11} + 114259794 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{10} + 422075810 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{9} - 16674486 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{8} - 1093657086 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{7} - 205745364 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{6} + 1886581864 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{5} - 1023977040 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{4} + 660654976 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} - 94952448 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} - 9114816 \, \sqrt{3} x - 1555968 \, \sqrt{3} + 9114816 \, \sqrt{3 \, x^{2} + 2}\right )}}{38416000 \,{\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 )}^{7}} \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)^8,x, algorithm="giac")

[Out]

1107/21008750*sqrt(35)*log(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*sqrt(3*x^2 + 2))/(2*sqrt(3)*x - sqrt(3
5) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2))) - 9/38416000*(908247*(sqrt(3)*x - sqrt(3*x^2 + 2))^13 + 3755004*sqrt(3)*(
sqrt(3)*x - sqrt(3*x^2 + 2))^12 + 52905908*(sqrt(3)*x - sqrt(3*x^2 + 2))^11 + 114259794*sqrt(3)*(sqrt(3)*x - s
qrt(3*x^2 + 2))^10 + 422075810*(sqrt(3)*x - sqrt(3*x^2 + 2))^9 - 16674486*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)
)^8 - 1093657086*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 - 205745364*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^6 + 1886581
864*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 - 1023977040*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 + 660654976*(sqrt(3)*
x - sqrt(3*x^2 + 2))^3 - 94952448*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 9114816*sqrt(3)*x - 1555968*sqrt(3
) + 9114816*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)^7