3.224 \(\int \frac{(2-x+3 x^2)^{5/2} (1+3 x+4 x^2)}{(1+2 x)^2} \, dx\)

Optimal. Leaf size=154 \[ -\frac{\left (3 x^2-x+2\right )^{7/2}}{13 (2 x+1)}-\frac{11 (37-60 x) \left (3 x^2-x+2\right )^{5/2}}{2340}-\frac{11}{864} (67-78 x) \left (3 x^2-x+2\right )^{3/2}-\frac{11 (4727-3090 x) \sqrt{3 x^2-x+2}}{6912}+\frac{429}{128} \sqrt{13} \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )-\frac{315623 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{13824 \sqrt{3}} \]

[Out]

(-11*(4727 - 3090*x)*Sqrt[2 - x + 3*x^2])/6912 - (11*(67 - 78*x)*(2 - x + 3*x^2)^(3/2))/864 - (11*(37 - 60*x)*
(2 - x + 3*x^2)^(5/2))/2340 - (2 - x + 3*x^2)^(7/2)/(13*(1 + 2*x)) - (315623*ArcSinh[(1 - 6*x)/Sqrt[23]])/(138
24*Sqrt[3]) + (429*Sqrt[13]*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/128

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Rubi [A]  time = 0.163071, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.219, Rules used = {1650, 814, 843, 619, 215, 724, 206} \[ -\frac{\left (3 x^2-x+2\right )^{7/2}}{13 (2 x+1)}-\frac{11 (37-60 x) \left (3 x^2-x+2\right )^{5/2}}{2340}-\frac{11}{864} (67-78 x) \left (3 x^2-x+2\right )^{3/2}-\frac{11 (4727-3090 x) \sqrt{3 x^2-x+2}}{6912}+\frac{429}{128} \sqrt{13} \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )-\frac{315623 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{13824 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[((2 - x + 3*x^2)^(5/2)*(1 + 3*x + 4*x^2))/(1 + 2*x)^2,x]

[Out]

(-11*(4727 - 3090*x)*Sqrt[2 - x + 3*x^2])/6912 - (11*(67 - 78*x)*(2 - x + 3*x^2)^(3/2))/864 - (11*(37 - 60*x)*
(2 - x + 3*x^2)^(5/2))/2340 - (2 - x + 3*x^2)^(7/2)/(13*(1 + 2*x)) - (315623*ArcSinh[(1 - 6*x)/Sqrt[23]])/(138
24*Sqrt[3]) + (429*Sqrt[13]*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/128

Rule 1650

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomia
lQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(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*ExpandToSum[(m + 1)*(c*d^2 - b*d*e + a*e^2)*Q + c*d*R*(m + 1) - b*e*R*(m + p + 2)
- c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]

Rule 814

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

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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]
&&  !IGtQ[m, 0]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 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 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{\left (2-x+3 x^2\right )^{5/2} \left (1+3 x+4 x^2\right )}{(1+2 x)^2} \, dx &=-\frac{\left (2-x+3 x^2\right )^{7/2}}{13 (1+2 x)}-\frac{1}{13} \int \frac{\left (-\frac{11}{2}-44 x\right ) \left (2-x+3 x^2\right )^{5/2}}{1+2 x} \, dx\\ &=-\frac{11 (37-60 x) \left (2-x+3 x^2\right )^{5/2}}{2340}-\frac{\left (2-x+3 x^2\right )^{7/2}}{13 (1+2 x)}+\frac{\int \frac{(-286+14872 x) \left (2-x+3 x^2\right )^{3/2}}{1+2 x} \, dx}{1872}\\ &=-\frac{11}{864} (67-78 x) \left (2-x+3 x^2\right )^{3/2}-\frac{11 (37-60 x) \left (2-x+3 x^2\right )^{5/2}}{2340}-\frac{\left (2-x+3 x^2\right )^{7/2}}{13 (1+2 x)}-\frac{\int \frac{(641784-3534960 x) \sqrt{2-x+3 x^2}}{1+2 x} \, dx}{179712}\\ &=-\frac{11 (4727-3090 x) \sqrt{2-x+3 x^2}}{6912}-\frac{11}{864} (67-78 x) \left (2-x+3 x^2\right )^{3/2}-\frac{11 (37-60 x) \left (2-x+3 x^2\right )^{5/2}}{2340}-\frac{\left (2-x+3 x^2\right )^{7/2}}{13 (1+2 x)}+\frac{\int \frac{-178896432+393897504 x}{(1+2 x) \sqrt{2-x+3 x^2}} \, dx}{8626176}\\ &=-\frac{11 (4727-3090 x) \sqrt{2-x+3 x^2}}{6912}-\frac{11}{864} (67-78 x) \left (2-x+3 x^2\right )^{3/2}-\frac{11 (37-60 x) \left (2-x+3 x^2\right )^{5/2}}{2340}-\frac{\left (2-x+3 x^2\right )^{7/2}}{13 (1+2 x)}+\frac{315623 \int \frac{1}{\sqrt{2-x+3 x^2}} \, dx}{13824}-\frac{5577}{128} \int \frac{1}{(1+2 x) \sqrt{2-x+3 x^2}} \, dx\\ &=-\frac{11 (4727-3090 x) \sqrt{2-x+3 x^2}}{6912}-\frac{11}{864} (67-78 x) \left (2-x+3 x^2\right )^{3/2}-\frac{11 (37-60 x) \left (2-x+3 x^2\right )^{5/2}}{2340}-\frac{\left (2-x+3 x^2\right )^{7/2}}{13 (1+2 x)}+\frac{5577}{64} \operatorname{Subst}\left (\int \frac{1}{52-x^2} \, dx,x,\frac{9-8 x}{\sqrt{2-x+3 x^2}}\right )+\frac{315623 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+6 x\right )}{13824 \sqrt{69}}\\ &=-\frac{11 (4727-3090 x) \sqrt{2-x+3 x^2}}{6912}-\frac{11}{864} (67-78 x) \left (2-x+3 x^2\right )^{3/2}-\frac{11 (37-60 x) \left (2-x+3 x^2\right )^{5/2}}{2340}-\frac{\left (2-x+3 x^2\right )^{7/2}}{13 (1+2 x)}-\frac{315623 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{13824 \sqrt{3}}+\frac{429}{128} \sqrt{13} \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{2-x+3 x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.115087, size = 113, normalized size = 0.73 \[ \frac{\frac{6 \sqrt{3 x^2-x+2} \left (103680 x^6-65664 x^5+251424 x^4-115680 x^3+310660 x^2-322972 x-364257\right )}{2 x+1}+694980 \sqrt{13} \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )+1578115 \sqrt{3} \sinh ^{-1}\left (\frac{6 x-1}{\sqrt{23}}\right )}{207360} \]

Antiderivative was successfully verified.

[In]

Integrate[((2 - x + 3*x^2)^(5/2)*(1 + 3*x + 4*x^2))/(1 + 2*x)^2,x]

[Out]

((6*Sqrt[2 - x + 3*x^2]*(-364257 - 322972*x + 310660*x^2 - 115680*x^3 + 251424*x^4 - 65664*x^5 + 103680*x^6))/
(1 + 2*x) + 1578115*Sqrt[3]*ArcSinh[(-1 + 6*x)/Sqrt[23]] + 694980*Sqrt[13]*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[
2 - x + 3*x^2])])/207360

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Maple [A]  time = 0.061, size = 235, normalized size = 1.5 \begin{align*}{\frac{-1+6\,x}{36} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{-115+690\,x}{1728} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{-2645+15870\,x}{13824}\sqrt{3\,{x}^{2}-x+2}}+{\frac{315623\,\sqrt{3}}{41472}{\it Arcsinh} \left ({\frac{6\,\sqrt{23}}{23} \left ( x-{\frac{1}{6}} \right ) } \right ) }-{\frac{33}{260} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{{\frac{5}{2}}}}+{\frac{-19+114\,x}{192} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{-965+5790\,x}{1536}\sqrt{3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}}}}-{\frac{11}{16} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{429}{128}\sqrt{12\, \left ( x+1/2 \right ) ^{2}-16\,x+5}}+{\frac{429\,\sqrt{13}}{128}{\it Artanh} \left ({\frac{2\,\sqrt{13}}{13} \left ({\frac{9}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+1/2 \right ) ^{2}-16\,x+5}}}} \right ) }-{\frac{1}{26} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{1}{2}} \right ) ^{-1}}+{\frac{-1+6\,x}{52} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2-x+2)^(5/2)*(4*x^2+3*x+1)/(1+2*x)^2,x)

[Out]

1/36*(-1+6*x)*(3*x^2-x+2)^(5/2)+115/1728*(-1+6*x)*(3*x^2-x+2)^(3/2)+2645/13824*(-1+6*x)*(3*x^2-x+2)^(1/2)+3156
23/41472*3^(1/2)*arcsinh(6/23*23^(1/2)*(x-1/6))-33/260*(3*(x+1/2)^2-4*x+5/4)^(5/2)+19/192*(-1+6*x)*(3*(x+1/2)^
2-4*x+5/4)^(3/2)+965/1536*(-1+6*x)*(3*(x+1/2)^2-4*x+5/4)^(1/2)-11/16*(3*(x+1/2)^2-4*x+5/4)^(3/2)-429/128*(12*(
x+1/2)^2-16*x+5)^(1/2)+429/128*13^(1/2)*arctanh(2/13*(9/2-4*x)*13^(1/2)/(12*(x+1/2)^2-16*x+5)^(1/2))-1/26/(x+1
/2)*(3*(x+1/2)^2-4*x+5/4)^(7/2)+1/52*(-1+6*x)*(3*(x+1/2)^2-4*x+5/4)^(5/2)

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Maxima [A]  time = 1.56935, size = 217, normalized size = 1.41 \begin{align*} \frac{1}{6} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} x - \frac{7}{90} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} + \frac{143}{144} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} x - \frac{737}{864} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} - \frac{{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}}}{4 \,{\left (2 \, x + 1\right )}} + \frac{5665}{1152} \, \sqrt{3 \, x^{2} - x + 2} x + \frac{315623}{41472} \, \sqrt{3} \operatorname{arsinh}\left (\frac{6}{23} \, \sqrt{23} x - \frac{1}{23} \, \sqrt{23}\right ) - \frac{429}{128} \, \sqrt{13} \operatorname{arsinh}\left (\frac{8 \, \sqrt{23} x}{23 \,{\left | 2 \, x + 1 \right |}} - \frac{9 \, \sqrt{23}}{23 \,{\left | 2 \, x + 1 \right |}}\right ) - \frac{51997}{6912} \, \sqrt{3 \, x^{2} - x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2-x+2)^(5/2)*(4*x^2+3*x+1)/(1+2*x)^2,x, algorithm="maxima")

[Out]

1/6*(3*x^2 - x + 2)^(5/2)*x - 7/90*(3*x^2 - x + 2)^(5/2) + 143/144*(3*x^2 - x + 2)^(3/2)*x - 737/864*(3*x^2 -
x + 2)^(3/2) - 1/4*(3*x^2 - x + 2)^(5/2)/(2*x + 1) + 5665/1152*sqrt(3*x^2 - x + 2)*x + 315623/41472*sqrt(3)*ar
csinh(6/23*sqrt(23)*x - 1/23*sqrt(23)) - 429/128*sqrt(13)*arcsinh(8/23*sqrt(23)*x/abs(2*x + 1) - 9/23*sqrt(23)
/abs(2*x + 1)) - 51997/6912*sqrt(3*x^2 - x + 2)

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Fricas [A]  time = 1.56708, size = 462, normalized size = 3. \begin{align*} \frac{1578115 \, \sqrt{3}{\left (2 \, x + 1\right )} \log \left (-4 \, \sqrt{3} \sqrt{3 \, x^{2} - x + 2}{\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) + 694980 \, \sqrt{13}{\left (2 \, x + 1\right )} \log \left (\frac{4 \, \sqrt{13} \sqrt{3 \, x^{2} - x + 2}{\left (8 \, x - 9\right )} - 220 \, x^{2} + 196 \, x - 185}{4 \, x^{2} + 4 \, x + 1}\right ) + 12 \,{\left (103680 \, x^{6} - 65664 \, x^{5} + 251424 \, x^{4} - 115680 \, x^{3} + 310660 \, x^{2} - 322972 \, x - 364257\right )} \sqrt{3 \, x^{2} - x + 2}}{414720 \,{\left (2 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2-x+2)^(5/2)*(4*x^2+3*x+1)/(1+2*x)^2,x, algorithm="fricas")

[Out]

1/414720*(1578115*sqrt(3)*(2*x + 1)*log(-4*sqrt(3)*sqrt(3*x^2 - x + 2)*(6*x - 1) - 72*x^2 + 24*x - 25) + 69498
0*sqrt(13)*(2*x + 1)*log((4*sqrt(13)*sqrt(3*x^2 - x + 2)*(8*x - 9) - 220*x^2 + 196*x - 185)/(4*x^2 + 4*x + 1))
+ 12*(103680*x^6 - 65664*x^5 + 251424*x^4 - 115680*x^3 + 310660*x^2 - 322972*x - 364257)*sqrt(3*x^2 - x + 2))
/(2*x + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (3 x^{2} - x + 2\right )^{\frac{5}{2}} \left (4 x^{2} + 3 x + 1\right )}{\left (2 x + 1\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**2-x+2)**(5/2)*(4*x**2+3*x+1)/(1+2*x)**2,x)

[Out]

Integral((3*x**2 - x + 2)**(5/2)*(4*x**2 + 3*x + 1)/(2*x + 1)**2, x)

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Giac [B]  time = 2.01138, size = 1026, normalized size = 6.66 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2-x+2)^(5/2)*(4*x^2+3*x+1)/(1+2*x)^2,x, algorithm="giac")

[Out]

429/128*sqrt(13)*log(sqrt(13)*(sqrt(-8/(2*x + 1) + 13/(2*x + 1)^2 + 3) + sqrt(13)/(2*x + 1)) - 4)*sgn(1/(2*x +
1)) - 315623/41472*sqrt(3)*log(1/2*abs(-2*sqrt(3) + 2*sqrt(-8/(2*x + 1) + 13/(2*x + 1)^2 + 3) + 2*sqrt(13)/(2
*x + 1))/(sqrt(3) + sqrt(-8/(2*x + 1) + 13/(2*x + 1)^2 + 3) + sqrt(13)/(2*x + 1)))*sgn(1/(2*x + 1)) - 169/128*
sqrt(-8/(2*x + 1) + 13/(2*x + 1)^2 + 3)*sgn(1/(2*x + 1)) + 1/34560*(5154065*(sqrt(-8/(2*x + 1) + 13/(2*x + 1)^
2 + 3) + sqrt(13)/(2*x + 1))^11*sgn(1/(2*x + 1)) - 7837020*sqrt(13)*(sqrt(-8/(2*x + 1) + 13/(2*x + 1)^2 + 3) +
sqrt(13)/(2*x + 1))^10*sgn(1/(2*x + 1)) + 39468815*(sqrt(-8/(2*x + 1) + 13/(2*x + 1)^2 + 3) + sqrt(13)/(2*x +
1))^9*sgn(1/(2*x + 1)) - 14445540*sqrt(13)*(sqrt(-8/(2*x + 1) + 13/(2*x + 1)^2 + 3) + sqrt(13)/(2*x + 1))^8*s
gn(1/(2*x + 1)) + 460893402*(sqrt(-8/(2*x + 1) + 13/(2*x + 1)^2 + 3) + sqrt(13)/(2*x + 1))^7*sgn(1/(2*x + 1))
- 343084680*sqrt(13)*(sqrt(-8/(2*x + 1) + 13/(2*x + 1)^2 + 3) + sqrt(13)/(2*x + 1))^6*sgn(1/(2*x + 1)) + 94415
0094*(sqrt(-8/(2*x + 1) + 13/(2*x + 1)^2 + 3) + sqrt(13)/(2*x + 1))^5*sgn(1/(2*x + 1)) - 22871160*sqrt(13)*(sq
rt(-8/(2*x + 1) + 13/(2*x + 1)^2 + 3) + sqrt(13)/(2*x + 1))^4*sgn(1/(2*x + 1)) + 1397032245*(sqrt(-8/(2*x + 1)
+ 13/(2*x + 1)^2 + 3) + sqrt(13)/(2*x + 1))^3*sgn(1/(2*x + 1)) - 683367516*sqrt(13)*(sqrt(-8/(2*x + 1) + 13/(
2*x + 1)^2 + 3) + sqrt(13)/(2*x + 1))^2*sgn(1/(2*x + 1)) + 392684355*(sqrt(-8/(2*x + 1) + 13/(2*x + 1)^2 + 3)
+ sqrt(13)/(2*x + 1))*sgn(1/(2*x + 1)) + 197538588*sqrt(13)*sgn(1/(2*x + 1)))/((sqrt(-8/(2*x + 1) + 13/(2*x +
1)^2 + 3) + sqrt(13)/(2*x + 1))^2 - 3)^6