3.2443 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^8} \, dx\)

Optimal. Leaf size=154 \[ -\frac{13 \left (3 x^2+5 x+2\right )^{7/2}}{35 (2 x+3)^7}+\frac{47 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{600 (2 x+3)^6}-\frac{47 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{9600 (2 x+3)^4}+\frac{47 (8 x+7) \sqrt{3 x^2+5 x+2}}{128000 (2 x+3)^2}-\frac{47 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{256000 \sqrt{5}} \]

[Out]

(47*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(128000*(3 + 2*x)^2) - (47*(7 + 8*x)*(2 + 5
*x + 3*x^2)^(3/2))/(9600*(3 + 2*x)^4) + (47*(7 + 8*x)*(2 + 5*x + 3*x^2)^(5/2))/(
600*(3 + 2*x)^6) - (13*(2 + 5*x + 3*x^2)^(7/2))/(35*(3 + 2*x)^7) - (47*ArcTanh[(
7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(256000*Sqrt[5])

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Rubi [A]  time = 0.205421, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ -\frac{13 \left (3 x^2+5 x+2\right )^{7/2}}{35 (2 x+3)^7}+\frac{47 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{600 (2 x+3)^6}-\frac{47 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{9600 (2 x+3)^4}+\frac{47 (8 x+7) \sqrt{3 x^2+5 x+2}}{128000 (2 x+3)^2}-\frac{47 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{256000 \sqrt{5}} \]

Antiderivative was successfully verified.

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

[Out]

(47*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(128000*(3 + 2*x)^2) - (47*(7 + 8*x)*(2 + 5
*x + 3*x^2)^(3/2))/(9600*(3 + 2*x)^4) + (47*(7 + 8*x)*(2 + 5*x + 3*x^2)^(5/2))/(
600*(3 + 2*x)^6) - (13*(2 + 5*x + 3*x^2)^(7/2))/(35*(3 + 2*x)^7) - (47*ArcTanh[(
7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(256000*Sqrt[5])

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Rubi in Sympy [A]  time = 38.8906, size = 146, normalized size = 0.95 \[ \frac{47 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{1280000} + \frac{47 \left (8 x + 7\right ) \sqrt{3 x^{2} + 5 x + 2}}{128000 \left (2 x + 3\right )^{2}} - \frac{47 \left (8 x + 7\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{9600 \left (2 x + 3\right )^{4}} + \frac{47 \left (8 x + 7\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{600 \left (2 x + 3\right )^{6}} - \frac{13 \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{35 \left (2 x + 3\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3*x**2+5*x+2)**(5/2)/(3+2*x)**8,x)

[Out]

47*sqrt(5)*atanh(sqrt(5)*(-8*x - 7)/(10*sqrt(3*x**2 + 5*x + 2)))/1280000 + 47*(8
*x + 7)*sqrt(3*x**2 + 5*x + 2)/(128000*(2*x + 3)**2) - 47*(8*x + 7)*(3*x**2 + 5*
x + 2)**(3/2)/(9600*(2*x + 3)**4) + 47*(8*x + 7)*(3*x**2 + 5*x + 2)**(5/2)/(600*
(2*x + 3)**6) - 13*(3*x**2 + 5*x + 2)**(7/2)/(35*(2*x + 3)**7)

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Mathematica [A]  time = 0.204095, size = 105, normalized size = 0.68 \[ \frac{987 \sqrt{5} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )+\frac{10 \sqrt{3 x^2+5 x+2} \left (1089792 x^6+22620128 x^5+81951440 x^4+127557120 x^3+100711840 x^2+39981058 x+6404247\right )}{(2 x+3)^7}-987 \sqrt{5} \log (2 x+3)}{26880000} \]

Antiderivative was successfully verified.

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

[Out]

((10*Sqrt[2 + 5*x + 3*x^2]*(6404247 + 39981058*x + 100711840*x^2 + 127557120*x^3
+ 81951440*x^4 + 22620128*x^5 + 1089792*x^6))/(3 + 2*x)^7 - 987*Sqrt[5]*Log[3 +
2*x] + 987*Sqrt[5]*Log[-7 - 8*x + 2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]])/26880000

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Maple [B]  time = 0.027, size = 290, normalized size = 1.9 \[ -{\frac{47}{5000000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{47}{2400000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{47}{1280000}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}-{\frac{13}{4480} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-7}}-{\frac{47}{9600} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-6}}-{\frac{47}{6000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-5}}-{\frac{987}{80000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{2867}{150000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{87373}{3000000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}+{\frac{136535+163842\,x}{1250000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{27307}{625000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-{\frac{6815+8178\,x}{600000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{235+282\,x}{160000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}+{\frac{47\,\sqrt{5}}{1280000}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-47/5000000*(3*(x+3/2)^2-4*x-19/4)^(5/2)-47/2400000*(3*(x+3/2)^2-4*x-19/4)^(3/2)
-47/1280000*(12*(x+3/2)^2-16*x-19)^(1/2)-13/4480/(x+3/2)^7*(3*(x+3/2)^2-4*x-19/4
)^(7/2)-47/9600/(x+3/2)^6*(3*(x+3/2)^2-4*x-19/4)^(7/2)-47/6000/(x+3/2)^5*(3*(x+3
/2)^2-4*x-19/4)^(7/2)-987/80000/(x+3/2)^4*(3*(x+3/2)^2-4*x-19/4)^(7/2)-2867/1500
00/(x+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(7/2)-87373/3000000/(x+3/2)^2*(3*(x+3/2)^2-4
*x-19/4)^(7/2)+27307/1250000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(5/2)-27307/625000/(
x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(7/2)-1363/600000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(
3/2)+47/160000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)+47/1280000*5^(1/2)*arctanh(2
/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))

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Maxima [A]  time = 0.814191, size = 495, normalized size = 3.21 \[ \frac{87373}{1000000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{35 \,{\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac{47 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{150 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac{94 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{375 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{987 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{5000 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{2867 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{18750 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{87373 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{750000 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{1363}{100000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x - \frac{27307}{2400000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{27307 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{250000 \,{\left (2 \, x + 3\right )}} + \frac{141}{80000} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{47}{1280000} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) + \frac{893}{640000} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^8,x, algorithm="maxima")

[Out]

87373/1000000*(3*x^2 + 5*x + 2)^(5/2) - 13/35*(3*x^2 + 5*x + 2)^(7/2)/(128*x^7 +
1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187) - 47/
150*(3*x^2 + 5*x + 2)^(7/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 +
2916*x + 729) - 94/375*(3*x^2 + 5*x + 2)^(7/2)/(32*x^5 + 240*x^4 + 720*x^3 + 10
80*x^2 + 810*x + 243) - 987/5000*(3*x^2 + 5*x + 2)^(7/2)/(16*x^4 + 96*x^3 + 216*
x^2 + 216*x + 81) - 2867/18750*(3*x^2 + 5*x + 2)^(7/2)/(8*x^3 + 36*x^2 + 54*x +
27) - 87373/750000*(3*x^2 + 5*x + 2)^(7/2)/(4*x^2 + 12*x + 9) - 1363/100000*(3*x
^2 + 5*x + 2)^(3/2)*x - 27307/2400000*(3*x^2 + 5*x + 2)^(3/2) - 27307/250000*(3*
x^2 + 5*x + 2)^(5/2)/(2*x + 3) + 141/80000*sqrt(3*x^2 + 5*x + 2)*x + 47/1280000*
sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) +
893/640000*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 0.283169, size = 236, normalized size = 1.53 \[ \frac{\sqrt{5}{\left (4 \, \sqrt{5}{\left (1089792 \, x^{6} + 22620128 \, x^{5} + 81951440 \, x^{4} + 127557120 \, x^{3} + 100711840 \, x^{2} + 39981058 \, x + 6404247\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + 987 \,{\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )} \log \left (\frac{\sqrt{5}{\left (124 \, x^{2} + 212 \, x + 89\right )} - 20 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{53760000 \,{\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^8,x, algorithm="fricas")

[Out]

1/53760000*sqrt(5)*(4*sqrt(5)*(1089792*x^6 + 22620128*x^5 + 81951440*x^4 + 12755
7120*x^3 + 100711840*x^2 + 39981058*x + 6404247)*sqrt(3*x^2 + 5*x + 2) + 987*(12
8*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187
)*log((sqrt(5)*(124*x^2 + 212*x + 89) - 20*sqrt(3*x^2 + 5*x + 2)*(8*x + 7))/(4*x
^2 + 12*x + 9)))/(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]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{20 \sqrt{3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\right )\, dx - \int \left (- \frac{96 x \sqrt{3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\right )\, dx - \int \left (- \frac{165 x^{2} \sqrt{3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\right )\, dx - \int \left (- \frac{113 x^{3} \sqrt{3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\right )\, dx - \int \left (- \frac{15 x^{4} \sqrt{3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\right )\, dx - \int \frac{9 x^{5} \sqrt{3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-x)*(3*x**2+5*x+2)**(5/2)/(3+2*x)**8,x)

[Out]

-Integral(-20*sqrt(3*x**2 + 5*x + 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), x) - Integral(-9
6*x*sqrt(3*x**2 + 5*x + 2)/(256*x**8 + 3072*x**7 + 16128*x**6 + 48384*x**5 + 907
20*x**4 + 108864*x**3 + 81648*x**2 + 34992*x + 6561), x) - Integral(-165*x**2*sq
rt(3*x**2 + 5*x + 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), x) - Integral(-113*x**3*sqrt(3*x
**2 + 5*x + 2)/(256*x**8 + 3072*x**7 + 16128*x**6 + 48384*x**5 + 90720*x**4 + 10
8864*x**3 + 81648*x**2 + 34992*x + 6561), x) - Integral(-15*x**4*sqrt(3*x**2 + 5
*x + 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), x) - Integral(9*x**5*sqrt(3*x**2 + 5*x + 2)/(
256*x**8 + 3072*x**7 + 16128*x**6 + 48384*x**5 + 90720*x**4 + 108864*x**3 + 8164
8*x**2 + 34992*x + 6561), x)

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GIAC/XCAS [A]  time = 0.316284, size = 622, normalized size = 4.04 \[ -\frac{47}{1280000} \, \sqrt{5}{\rm ln}\left (\frac{{\left | -4 \, \sqrt{3} x - 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt{3} x + 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}\right ) - \frac{72512832 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{13} + 651952224 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{12} + 6898276448 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 8494566864 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{10} - 58878767920 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{9} - 326450774496 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{8} - 2207907445056 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{7} - 3147944405424 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{6} - 9314774279636 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{5} - 6492162811470 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{4} - 9472821206534 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} - 3070624865553 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} - 1792565462541 \, \sqrt{3} x - 158637115728 \, \sqrt{3} + 1792565462541 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{2688000 \,{\left (2 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^8,x, algorithm="giac")

[Out]

-47/1280000*sqrt(5)*ln(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)))
- 1/2688000*(72512832*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^13 + 651952224*sqrt(3)
*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^12 + 6898276448*(sqrt(3)*x - sqrt(3*x^2 + 5
*x + 2))^11 + 8494566864*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 - 588787
67920*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9 - 326450774496*sqrt(3)*(sqrt(3)*x -
sqrt(3*x^2 + 5*x + 2))^8 - 2207907445056*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 -
3147944405424*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 - 9314774279636*(sq
rt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 - 6492162811470*sqrt(3)*(sqrt(3)*x - sqrt(3*x
^2 + 5*x + 2))^4 - 9472821206534*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 - 3070624
865553*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 - 1792565462541*sqrt(3)*x -
158637115728*sqrt(3) + 1792565462541*sqrt(3*x^2 + 5*x + 2))/(2*(sqrt(3)*x - sqr
t(3*x^2 + 5*x + 2))^2 + 6*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) + 11)^7