3.2461 \(\int \frac{(5-x) (2+5 x+3 x^2)^{7/2}}{(3+2 x)^{10}} \, dx\)

Optimal. Leaf size=184 \[ -\frac{13 \left (3 x^2+5 x+2\right )^{9/2}}{45 (2 x+3)^9}+\frac{47 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{800 (2 x+3)^8}-\frac{329 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{96000 (2 x+3)^6}+\frac{329 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{1536000 (2 x+3)^4}-\frac{329 (8 x+7) \sqrt{3 x^2+5 x+2}}{20480000 (2 x+3)^2}+\frac{329 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{40960000 \sqrt{5}} \]

[Out]

(-329*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(20480000*(3 + 2*x)^2) + (329*(7 + 8*x)*(2 + 5*x + 3*x^2)^(3/2))/(15360
00*(3 + 2*x)^4) - (329*(7 + 8*x)*(2 + 5*x + 3*x^2)^(5/2))/(96000*(3 + 2*x)^6) + (47*(7 + 8*x)*(2 + 5*x + 3*x^2
)^(7/2))/(800*(3 + 2*x)^8) - (13*(2 + 5*x + 3*x^2)^(9/2))/(45*(3 + 2*x)^9) + (329*ArcTanh[(7 + 8*x)/(2*Sqrt[5]
*Sqrt[2 + 5*x + 3*x^2])])/(40960000*Sqrt[5])

________________________________________________________________________________________

Rubi [A]  time = 0.0964, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {806, 720, 724, 206} \[ -\frac{13 \left (3 x^2+5 x+2\right )^{9/2}}{45 (2 x+3)^9}+\frac{47 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{800 (2 x+3)^8}-\frac{329 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{96000 (2 x+3)^6}+\frac{329 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{1536000 (2 x+3)^4}-\frac{329 (8 x+7) \sqrt{3 x^2+5 x+2}}{20480000 (2 x+3)^2}+\frac{329 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{40960000 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^10,x]

[Out]

(-329*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(20480000*(3 + 2*x)^2) + (329*(7 + 8*x)*(2 + 5*x + 3*x^2)^(3/2))/(15360
00*(3 + 2*x)^4) - (329*(7 + 8*x)*(2 + 5*x + 3*x^2)^(5/2))/(96000*(3 + 2*x)^6) + (47*(7 + 8*x)*(2 + 5*x + 3*x^2
)^(7/2))/(800*(3 + 2*x)^8) - (13*(2 + 5*x + 3*x^2)^(9/2))/(45*(3 + 2*x)^9) + (329*ArcTanh[(7 + 8*x)/(2*Sqrt[5]
*Sqrt[2 + 5*x + 3*x^2])])/(40960000*Sqrt[5])

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 720

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

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

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+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{10}} \, dx &=-\frac{13 \left (2+5 x+3 x^2\right )^{9/2}}{45 (3+2 x)^9}+\frac{47}{10} \int \frac{\left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^9} \, dx\\ &=\frac{47 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800 (3+2 x)^8}-\frac{13 \left (2+5 x+3 x^2\right )^{9/2}}{45 (3+2 x)^9}-\frac{329 \int \frac{\left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx}{1600}\\ &=-\frac{329 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{96000 (3+2 x)^6}+\frac{47 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800 (3+2 x)^8}-\frac{13 \left (2+5 x+3 x^2\right )^{9/2}}{45 (3+2 x)^9}+\frac{329 \int \frac{\left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx}{38400}\\ &=\frac{329 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{1536000 (3+2 x)^4}-\frac{329 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{96000 (3+2 x)^6}+\frac{47 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800 (3+2 x)^8}-\frac{13 \left (2+5 x+3 x^2\right )^{9/2}}{45 (3+2 x)^9}-\frac{329 \int \frac{\sqrt{2+5 x+3 x^2}}{(3+2 x)^3} \, dx}{1024000}\\ &=-\frac{329 (7+8 x) \sqrt{2+5 x+3 x^2}}{20480000 (3+2 x)^2}+\frac{329 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{1536000 (3+2 x)^4}-\frac{329 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{96000 (3+2 x)^6}+\frac{47 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800 (3+2 x)^8}-\frac{13 \left (2+5 x+3 x^2\right )^{9/2}}{45 (3+2 x)^9}+\frac{329 \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{40960000}\\ &=-\frac{329 (7+8 x) \sqrt{2+5 x+3 x^2}}{20480000 (3+2 x)^2}+\frac{329 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{1536000 (3+2 x)^4}-\frac{329 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{96000 (3+2 x)^6}+\frac{47 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800 (3+2 x)^8}-\frac{13 \left (2+5 x+3 x^2\right )^{9/2}}{45 (3+2 x)^9}-\frac{329 \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )}{20480000}\\ &=-\frac{329 (7+8 x) \sqrt{2+5 x+3 x^2}}{20480000 (3+2 x)^2}+\frac{329 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{1536000 (3+2 x)^4}-\frac{329 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{96000 (3+2 x)^6}+\frac{47 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800 (3+2 x)^8}-\frac{13 \left (2+5 x+3 x^2\right )^{9/2}}{45 (3+2 x)^9}+\frac{329 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{40960000 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.150289, size = 185, normalized size = 1.01 \[ -\frac{13 \left (3 x^2+5 x+2\right )^{9/2}}{45 (2 x+3)^9}+\frac{47 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{800 (2 x+3)^8}-\frac{329 \left (\frac{32 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^6}-\frac{2 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^4}+\frac{3 (8 x+7) \sqrt{3 x^2+5 x+2}}{20 (2 x+3)^2}+\frac{3 \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{40 \sqrt{5}}\right )}{3072000} \]

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^10,x]

[Out]

(47*(7 + 8*x)*(2 + 5*x + 3*x^2)^(7/2))/(800*(3 + 2*x)^8) - (13*(2 + 5*x + 3*x^2)^(9/2))/(45*(3 + 2*x)^9) - (32
9*((3*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(20*(3 + 2*x)^2) - (2*(7 + 8*x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^4 +
(32*(7 + 8*x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^6 + (3*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])
/(40*Sqrt[5])))/3072000

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Maple [B]  time = 0.037, size = 369, normalized size = 2. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^10,x)

[Out]

-1457/400000/(x+3/2)^5*(3*(x+3/2)^2-4*x-19/4)^(9/2)-90287/16000000/(x+3/2)^4*(3*(x+3/2)^2-4*x-19/4)^(9/2)-2593
93/30000000/(x+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(9/2)-2621237/200000000/(x+3/2)^2*(3*(x+3/2)^2-4*x-19/4)^(9/2)+49
1479/50000000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(7/2)-191149/200000000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(5/2)-49147
9/25000000/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(9/2)+9541/96000000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(3/2)-329/2560000
0*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)-329/204800000*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x
-19)^(1/2))+47/200000000*(3*(x+3/2)^2-4*x-19/4)^(7/2)+329/800000000*(3*(x+3/2)^2-4*x-19/4)^(5/2)+329/384000000
*(3*(x+3/2)^2-4*x-19/4)^(3/2)+329/204800000*(12*(x+3/2)^2-16*x-19)^(1/2)-13/23040/(x+3/2)^9*(3*(x+3/2)^2-4*x-1
9/4)^(9/2)-47/51200/(x+3/2)^8*(3*(x+3/2)^2-4*x-19/4)^(9/2)-47/32000/(x+3/2)^7*(3*(x+3/2)^2-4*x-19/4)^(9/2)-893
/384000/(x+3/2)^6*(3*(x+3/2)^2-4*x-19/4)^(9/2)

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Maxima [B]  time = 2.12785, size = 693, normalized size = 3.77 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^10,x, algorithm="maxima")

[Out]

7863711/200000000*(3*x^2 + 5*x + 2)^(7/2) - 13/45*(3*x^2 + 5*x + 2)^(9/2)/(512*x^9 + 6912*x^8 + 41472*x^7 + 14
5152*x^6 + 326592*x^5 + 489888*x^4 + 489888*x^3 + 314928*x^2 + 118098*x + 19683) - 47/200*(3*x^2 + 5*x + 2)^(9
/2)/(256*x^8 + 3072*x^7 + 16128*x^6 + 48384*x^5 + 90720*x^4 + 108864*x^3 + 81648*x^2 + 34992*x + 6561) - 47/25
0*(3*x^2 + 5*x + 2)^(9/2)/(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187)
- 893/6000*(3*x^2 + 5*x + 2)^(9/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 1457/
12500*(3*x^2 + 5*x + 2)^(9/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 90287/1000000*(3*x^2 + 5
*x + 2)^(9/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 259393/3750000*(3*x^2 + 5*x + 2)^(9/2)/(8*x^3 + 36*x^
2 + 54*x + 27) - 2621237/50000000*(3*x^2 + 5*x + 2)^(9/2)/(4*x^2 + 12*x + 9) - 573447/100000000*(3*x^2 + 5*x +
2)^(5/2)*x - 3822651/800000000*(3*x^2 + 5*x + 2)^(5/2) - 491479/10000000*(3*x^2 + 5*x + 2)^(7/2)/(2*x + 3) +
9541/16000000*(3*x^2 + 5*x + 2)^(3/2)*x + 191149/384000000*(3*x^2 + 5*x + 2)^(3/2) - 987/12800000*sqrt(3*x^2 +
5*x + 2)*x - 329/204800000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 6
251/102400000*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 1.55295, size = 716, normalized size = 3.89 \begin{align*} \frac{2961 \, \sqrt{5}{\left (512 \, x^{9} + 6912 \, x^{8} + 41472 \, x^{7} + 145152 \, x^{6} + 326592 \, x^{5} + 489888 \, x^{4} + 489888 \, x^{3} + 314928 \, x^{2} + 118098 \, x + 19683\right )} \log \left (\frac{4 \, \sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 20 \,{\left (28394496 \, x^{8} + 2848109952 \, x^{7} + 15895201728 \, x^{6} + 38558367264 \, x^{5} + 51825176720 \, x^{4} + 41530110824 \, x^{3} + 19810691268 \, x^{2} + 5201574542 \, x + 578701331\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{3686400000 \,{\left (512 \, x^{9} + 6912 \, x^{8} + 41472 \, x^{7} + 145152 \, x^{6} + 326592 \, x^{5} + 489888 \, x^{4} + 489888 \, x^{3} + 314928 \, x^{2} + 118098 \, x + 19683\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^10,x, algorithm="fricas")

[Out]

1/3686400000*(2961*sqrt(5)*(512*x^9 + 6912*x^8 + 41472*x^7 + 145152*x^6 + 326592*x^5 + 489888*x^4 + 489888*x^3
+ 314928*x^2 + 118098*x + 19683)*log((4*sqrt(5)*sqrt(3*x^2 + 5*x + 2)*(8*x + 7) + 124*x^2 + 212*x + 89)/(4*x^
2 + 12*x + 9)) + 20*(28394496*x^8 + 2848109952*x^7 + 15895201728*x^6 + 38558367264*x^5 + 51825176720*x^4 + 415
30110824*x^3 + 19810691268*x^2 + 5201574542*x + 578701331)*sqrt(3*x^2 + 5*x + 2))/(512*x^9 + 6912*x^8 + 41472*
x^7 + 145152*x^6 + 326592*x^5 + 489888*x^4 + 489888*x^3 + 314928*x^2 + 118098*x + 19683)

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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+5*x+2)**(7/2)/(3+2*x)**10,x)

[Out]

Timed out

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Giac [B]  time = 1.30421, size = 760, normalized size = 4.13 \begin{align*} \frac{329}{204800000} \, \sqrt{5} \log \left (\frac{{\left | -4 \, \sqrt{3} x - 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt{3} x + 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}\right ) - \frac{14930678016 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{17} + 204061569408 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{16} + 3866707486848 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{15} + 14840812733760 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{14} + 114102022608000 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{13} + 198779998219488 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{12} + 649357338634272 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 207317438979984 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{10} - 2217334591351040 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{9} - 5247913396815000 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{8} - 20151247122371016 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{7} - 17924557725783828 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{6} - 35125577732048328 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{5} - 16953161853593070 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{4} - 17752204726475250 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} - 4253745315948057 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} - 1882391465118753 \, \sqrt{3} x - 129047626217736 \, \sqrt{3} + 1882391465118753 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{184320000 \,{\left (2 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^10,x, algorithm="giac")

[Out]

329/204800000*sqrt(5)*log(abs(-4*sqrt(3)*x - 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))/abs(-4*sqrt(3)*x
+ 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))) - 1/184320000*(14930678016*(sqrt(3)*x - sqrt(3*x^2 + 5*x
+ 2))^17 + 204061569408*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^16 + 3866707486848*(sqrt(3)*x - sqrt(3*x^2
+ 5*x + 2))^15 + 14840812733760*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^14 + 114102022608000*(sqrt(3)*x -
sqrt(3*x^2 + 5*x + 2))^13 + 198779998219488*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^12 + 649357338634272*
(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^11 + 207317438979984*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 - 2217
334591351040*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9 - 5247913396815000*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x +
2))^8 - 20151247122371016*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 - 17924557725783828*sqrt(3)*(sqrt(3)*x - sqrt(
3*x^2 + 5*x + 2))^6 - 35125577732048328*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 - 16953161853593070*sqrt(3)*(sqr
t(3)*x - sqrt(3*x^2 + 5*x + 2))^4 - 17752204726475250*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 - 4253745315948057
*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 - 1882391465118753*sqrt(3)*x - 129047626217736*sqrt(3) + 188239
1465118753*sqrt(3*x^2 + 5*x + 2))/(2*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 6*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2
+ 5*x + 2)) + 11)^9