3.2462 \(\int \frac{(5-x) (2+5 x+3 x^2)^{7/2}}{(3+2 x)^{11}} \, dx\)
Optimal. Leaf size=209 \[ -\frac{29 \left (3 x^2+5 x+2\right )^{9/2}}{125 (2 x+3)^9}-\frac{13 \left (3 x^2+5 x+2\right )^{9/2}}{50 (2 x+3)^{10}}+\frac{1893 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{40000 (2 x+3)^8}-\frac{4417 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{1600000 (2 x+3)^6}+\frac{4417 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{25600000 (2 x+3)^4}-\frac{13251 (8 x+7) \sqrt{3 x^2+5 x+2}}{1024000000 (2 x+3)^2}+\frac{13251 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{2048000000 \sqrt{5}} \]
[Out]
(-13251*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(1024000000*(3 + 2*x)^2) + (4417*(7 + 8*x)*(2 + 5*x + 3*x^2)^(3/2))/(
25600000*(3 + 2*x)^4) - (4417*(7 + 8*x)*(2 + 5*x + 3*x^2)^(5/2))/(1600000*(3 + 2*x)^6) + (1893*(7 + 8*x)*(2 +
5*x + 3*x^2)^(7/2))/(40000*(3 + 2*x)^8) - (13*(2 + 5*x + 3*x^2)^(9/2))/(50*(3 + 2*x)^10) - (29*(2 + 5*x + 3*x^
2)^(9/2))/(125*(3 + 2*x)^9) + (13251*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(2048000000*Sqrt[5]
)
________________________________________________________________________________________
Rubi [A] time = 0.120977, antiderivative size = 209, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used =
{834, 806, 720, 724, 206} \[ -\frac{29 \left (3 x^2+5 x+2\right )^{9/2}}{125 (2 x+3)^9}-\frac{13 \left (3 x^2+5 x+2\right )^{9/2}}{50 (2 x+3)^{10}}+\frac{1893 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{40000 (2 x+3)^8}-\frac{4417 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{1600000 (2 x+3)^6}+\frac{4417 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{25600000 (2 x+3)^4}-\frac{13251 (8 x+7) \sqrt{3 x^2+5 x+2}}{1024000000 (2 x+3)^2}+\frac{13251 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{2048000000 \sqrt{5}} \]
Antiderivative was successfully verified.
[In]
Int[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^11,x]
[Out]
(-13251*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(1024000000*(3 + 2*x)^2) + (4417*(7 + 8*x)*(2 + 5*x + 3*x^2)^(3/2))/(
25600000*(3 + 2*x)^4) - (4417*(7 + 8*x)*(2 + 5*x + 3*x^2)^(5/2))/(1600000*(3 + 2*x)^6) + (1893*(7 + 8*x)*(2 +
5*x + 3*x^2)^(7/2))/(40000*(3 + 2*x)^8) - (13*(2 + 5*x + 3*x^2)^(9/2))/(50*(3 + 2*x)^10) - (29*(2 + 5*x + 3*x^
2)^(9/2))/(125*(3 + 2*x)^9) + (13251*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(2048000000*Sqrt[5]
)
Rule 834
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])
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)^{11}} \, dx &=-\frac{13 \left (2+5 x+3 x^2\right )^{9/2}}{50 (3+2 x)^{10}}-\frac{1}{50} \int \frac{\left (-\frac{405}{2}+39 x\right ) \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}}{50 (3+2 x)^{10}}-\frac{29 \left (2+5 x+3 x^2\right )^{9/2}}{125 (3+2 x)^9}+\frac{1893}{500} \int \frac{\left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^9} \, dx\\ &=\frac{1893 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{40000 (3+2 x)^8}-\frac{13 \left (2+5 x+3 x^2\right )^{9/2}}{50 (3+2 x)^{10}}-\frac{29 \left (2+5 x+3 x^2\right )^{9/2}}{125 (3+2 x)^9}-\frac{13251 \int \frac{\left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx}{80000}\\ &=-\frac{4417 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{1600000 (3+2 x)^6}+\frac{1893 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{40000 (3+2 x)^8}-\frac{13 \left (2+5 x+3 x^2\right )^{9/2}}{50 (3+2 x)^{10}}-\frac{29 \left (2+5 x+3 x^2\right )^{9/2}}{125 (3+2 x)^9}+\frac{4417 \int \frac{\left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx}{640000}\\ &=\frac{4417 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600000 (3+2 x)^4}-\frac{4417 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{1600000 (3+2 x)^6}+\frac{1893 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{40000 (3+2 x)^8}-\frac{13 \left (2+5 x+3 x^2\right )^{9/2}}{50 (3+2 x)^{10}}-\frac{29 \left (2+5 x+3 x^2\right )^{9/2}}{125 (3+2 x)^9}-\frac{13251 \int \frac{\sqrt{2+5 x+3 x^2}}{(3+2 x)^3} \, dx}{51200000}\\ &=-\frac{13251 (7+8 x) \sqrt{2+5 x+3 x^2}}{1024000000 (3+2 x)^2}+\frac{4417 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600000 (3+2 x)^4}-\frac{4417 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{1600000 (3+2 x)^6}+\frac{1893 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{40000 (3+2 x)^8}-\frac{13 \left (2+5 x+3 x^2\right )^{9/2}}{50 (3+2 x)^{10}}-\frac{29 \left (2+5 x+3 x^2\right )^{9/2}}{125 (3+2 x)^9}+\frac{13251 \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{2048000000}\\ &=-\frac{13251 (7+8 x) \sqrt{2+5 x+3 x^2}}{1024000000 (3+2 x)^2}+\frac{4417 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600000 (3+2 x)^4}-\frac{4417 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{1600000 (3+2 x)^6}+\frac{1893 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{40000 (3+2 x)^8}-\frac{13 \left (2+5 x+3 x^2\right )^{9/2}}{50 (3+2 x)^{10}}-\frac{29 \left (2+5 x+3 x^2\right )^{9/2}}{125 (3+2 x)^9}-\frac{13251 \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )}{1024000000}\\ &=-\frac{13251 (7+8 x) \sqrt{2+5 x+3 x^2}}{1024000000 (3+2 x)^2}+\frac{4417 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600000 (3+2 x)^4}-\frac{4417 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{1600000 (3+2 x)^6}+\frac{1893 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{40000 (3+2 x)^8}-\frac{13 \left (2+5 x+3 x^2\right )^{9/2}}{50 (3+2 x)^{10}}-\frac{29 \left (2+5 x+3 x^2\right )^{9/2}}{125 (3+2 x)^9}+\frac{13251 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{2048000000 \sqrt{5}}\\ \end{align*}
Mathematica [A] time = 0.179438, size = 212, normalized size = 1.01 \[ \frac{1}{50} \left (-\frac{58 \left (3 x^2+5 x+2\right )^{9/2}}{5 (2 x+3)^9}-\frac{13 \left (3 x^2+5 x+2\right )^{9/2}}{(2 x+3)^{10}}+\frac{1893 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{800 (2 x+3)^8}-\frac{4417 \left (\frac{32 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^6}-\frac{2 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^4}+\frac{3 (8 x+7) \sqrt{3 x^2+5 x+2}}{20 (2 x+3)^2}+\frac{3 \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{40 \sqrt{5}}\right )}{1024000}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^11,x]
[Out]
((1893*(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))/(3 + 2*x)^10 - (58*
(2 + 5*x + 3*x^2)^(9/2))/(5*(3 + 2*x)^9) - (4417*((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])))/1024000)/50
________________________________________________________________________________________
Maple [B] time = 0.053, size = 390, normalized size = 1.9 \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)^11,x)
[Out]
-58683/20000000/(x+3/2)^5*(3*(x+3/2)^2-4*x-19/4)^(9/2)-3636453/800000000/(x+3/2)^4*(3*(x+3/2)^2-4*x-19/4)^(9/2
)-3482489/500000000/(x+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(9/2)-105574503/10000000000/(x+3/2)^2*(3*(x+3/2)^2-4*x-19
/4)^(9/2)+19795101/2500000000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(7/2)-7698831/10000000000*(5+6*x)*(3*(x+3/2)^2-4*
x-19/4)^(5/2)-19795101/1250000000/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(9/2)+128093/1600000000*(5+6*x)*(3*(x+3/2)^2-
4*x-19/4)^(3/2)-13251/1280000000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)-13251/10240000000*5^(1/2)*arctanh(2/5*(-
7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))+1893/10000000000*(3*(x+3/2)^2-4*x-19/4)^(7/2)+13251/40000000000
*(3*(x+3/2)^2-4*x-19/4)^(5/2)+4417/6400000000*(3*(x+3/2)^2-4*x-19/4)^(3/2)+13251/10240000000*(12*(x+3/2)^2-16*
x-19)^(1/2)-13/51200/(x+3/2)^10*(3*(x+3/2)^2-4*x-19/4)^(9/2)-29/64000/(x+3/2)^9*(3*(x+3/2)^2-4*x-19/4)^(9/2)-1
893/2560000/(x+3/2)^8*(3*(x+3/2)^2-4*x-19/4)^(9/2)-1893/1600000/(x+3/2)^7*(3*(x+3/2)^2-4*x-19/4)^(9/2)-11989/6
400000/(x+3/2)^6*(3*(x+3/2)^2-4*x-19/4)^(9/2)
________________________________________________________________________________________
Maxima [B] time = 2.05295, size = 782, normalized size = 3.74 \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)^11,x, algorithm="maxima")
[Out]
316723509/10000000000*(3*x^2 + 5*x + 2)^(7/2) - 13/50*(3*x^2 + 5*x + 2)^(9/2)/(1024*x^10 + 15360*x^9 + 103680*
x^8 + 414720*x^7 + 1088640*x^6 + 1959552*x^5 + 2449440*x^4 + 2099520*x^3 + 1180980*x^2 + 393660*x + 59049) - 2
9/125*(3*x^2 + 5*x + 2)^(9/2)/(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) - 1893/10000*(3*x^2 + 5*x + 2)^(9/2)/(256*x^8 + 3072*x^7 + 16128*x^6 + 48
384*x^5 + 90720*x^4 + 108864*x^3 + 81648*x^2 + 34992*x + 6561) - 1893/12500*(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) - 11989/100000*(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) - 58683/625000*(3*x^2 + 5*x + 2)^(9/2)/
(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 3636453/50000000*(3*x^2 + 5*x + 2)^(9/2)/(16*x^4 + 96*
x^3 + 216*x^2 + 216*x + 81) - 3482489/62500000*(3*x^2 + 5*x + 2)^(9/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 10557450
3/2500000000*(3*x^2 + 5*x + 2)^(9/2)/(4*x^2 + 12*x + 9) - 23096493/5000000000*(3*x^2 + 5*x + 2)^(5/2)*x - 1539
63369/40000000000*(3*x^2 + 5*x + 2)^(5/2) - 19795101/500000000*(3*x^2 + 5*x + 2)^(7/2)/(2*x + 3) + 384279/8000
00000*(3*x^2 + 5*x + 2)^(3/2)*x + 2566277/6400000000*(3*x^2 + 5*x + 2)^(3/2) - 39753/640000000*sqrt(3*x^2 + 5*
x + 2)*x - 13251/10240000000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) -
251769/5120000000*sqrt(3*x^2 + 5*x + 2)
________________________________________________________________________________________
Fricas [A] time = 1.54026, size = 813, normalized size = 3.89 \begin{align*} \frac{13251 \, \sqrt{5}{\left (1024 \, x^{10} + 15360 \, x^{9} + 103680 \, x^{8} + 414720 \, x^{7} + 1088640 \, x^{6} + 1959552 \, x^{5} + 2449440 \, x^{4} + 2099520 \, x^{3} + 1180980 \, x^{2} + 393660 \, x + 59049\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 (371791872 \, x^{9} + 5268182272 \, x^{8} + 40186580992 \, x^{7} + 148740043392 \, x^{6} + 304078211712 \, x^{5} + 372602220928 \, x^{4} + 281702072128 \, x^{3} + 128970753208 \, x^{2} + 32786922608 \, x + 3544392763\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{20480000000 \,{\left (1024 \, x^{10} + 15360 \, x^{9} + 103680 \, x^{8} + 414720 \, x^{7} + 1088640 \, x^{6} + 1959552 \, x^{5} + 2449440 \, x^{4} + 2099520 \, x^{3} + 1180980 \, x^{2} + 393660 \, x + 59049\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)^11,x, algorithm="fricas")
[Out]
1/20480000000*(13251*sqrt(5)*(1024*x^10 + 15360*x^9 + 103680*x^8 + 414720*x^7 + 1088640*x^6 + 1959552*x^5 + 24
49440*x^4 + 2099520*x^3 + 1180980*x^2 + 393660*x + 59049)*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*(371791872*x^9 + 5268182272*x^8 + 40186580992*x^7 + 148740043392*x
^6 + 304078211712*x^5 + 372602220928*x^4 + 281702072128*x^3 + 128970753208*x^2 + 32786922608*x + 3544392763)*s
qrt(3*x^2 + 5*x + 2))/(1024*x^10 + 15360*x^9 + 103680*x^8 + 414720*x^7 + 1088640*x^6 + 1959552*x^5 + 2449440*x
^4 + 2099520*x^3 + 1180980*x^2 + 393660*x + 59049)
________________________________________________________________________________________
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)**11,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.27385, size = 829, normalized size = 3.97 \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)^11,x, algorithm="giac")
[Out]
13251/10240000000*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))) - 1/1024000000*(6784512*(sqrt(3)*x - sqrt(3*x^2 + 5*x
+ 2))^19 + 83137358592*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^18 + 2689605043456*(sqrt(3)*x - sqrt(3*x^2
+ 5*x + 2))^17 + 9174489217536*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^16 - 53080570863872*(sqrt(3)*x - s
qrt(3*x^2 + 5*x + 2))^15 - 898783135722624*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^14 - 13174687008250752*
(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^13 - 40507172795248512*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^12 - 27
0169596727110016*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^11 - 458790099197766656*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 +
5*x + 2))^10 - 1833183533173743552*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9 - 1939024456450048032*sqrt(3)*(sqrt(
3)*x - sqrt(3*x^2 + 5*x + 2))^8 - 4903074367120921776*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 - 3280073192617110
456*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 - 5164856211259534888*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5
- 2082844158764403144*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 - 1869656136275991262*(sqrt(3)*x - sqrt(3*
x^2 + 5*x + 2))^3 - 391066159205340747*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 - 153124376229353121*sqrt
(3)*x - 9387541838830536*sqrt(3) + 153124376229353121*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)^10