\(\int \frac {\sqrt {2-x+3 x^2} (1+3 x+4 x^2)}{1+2 x} \, dx\) [48]

Optimal result

Integrand size = 32, antiderivative size = 101 \[ \int \frac {\sqrt {2-x+3 x^2} \left (1+3 x+4 x^2\right )}{1+2 x} \, dx=\frac {1}{72} (13+30 x) \sqrt {2-x+3 x^2}+\frac {2}{9} \left (2-x+3 x^2\right )^{3/2}-\frac {43 \text {arcsinh}\left (\frac {1-6 x}{\sqrt {23}}\right )}{144 \sqrt {3}}-\frac {1}{8} \sqrt {13} \text {arctanh}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {2-x+3 x^2}}\right ) \] Output:

1/72*(13+30*x)*(3*x^2-x+2)^(1/2)+2/9*(3*x^2-x+2)^(3/2)-43/432*arcsinh(1/23 
*(1-6*x)*23^(1/2))*3^(1/2)-1/8*13^(1/2)*arctanh(1/26*(9-8*x)*13^(1/2)/(3*x 
^2-x+2)^(1/2))
 

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {2-x+3 x^2} \left (1+3 x+4 x^2\right )}{1+2 x} \, dx=\frac {1}{432} \left (6 \sqrt {2-x+3 x^2} \left (45+14 x+48 x^2\right )+108 \sqrt {13} \text {arctanh}\left (\frac {\sqrt {3}+2 \sqrt {3} x-2 \sqrt {2-x+3 x^2}}{\sqrt {13}}\right )-43 \sqrt {3} \log \left (1-6 x+2 \sqrt {6-3 x+9 x^2}\right )\right ) \] Input:

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

Output:

(6*Sqrt[2 - x + 3*x^2]*(45 + 14*x + 48*x^2) + 108*Sqrt[13]*ArcTanh[(Sqrt[3 
] + 2*Sqrt[3]*x - 2*Sqrt[2 - x + 3*x^2])/Sqrt[13]] - 43*Sqrt[3]*Log[1 - 6* 
x + 2*Sqrt[6 - 3*x + 9*x^2]])/432
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {2184, 27, 1231, 25, 1269, 1090, 222, 1154, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\sqrt {3 x^2-x+2} \left (4 x^2+3 x+1\right )}{2 x+1} \, dx\)

\(\Big \downarrow \) 2184

\(\displaystyle \frac {1}{36} \int \frac {12 (5 x+4) \sqrt {3 x^2-x+2}}{2 x+1}dx+\frac {2}{9} \left (3 x^2-x+2\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \int \frac {(5 x+4) \sqrt {3 x^2-x+2}}{2 x+1}dx+\frac {2}{9} \left (3 x^2-x+2\right )^{3/2}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {1}{3} \left (\frac {1}{24} (30 x+13) \sqrt {3 x^2-x+2}-\frac {1}{48} \int -\frac {86 x+277}{(2 x+1) \sqrt {3 x^2-x+2}}dx\right )+\frac {2}{9} \left (3 x^2-x+2\right )^{3/2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{3} \left (\frac {1}{48} \int \frac {86 x+277}{(2 x+1) \sqrt {3 x^2-x+2}}dx+\frac {1}{24} \sqrt {3 x^2-x+2} (30 x+13)\right )+\frac {2}{9} \left (3 x^2-x+2\right )^{3/2}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {1}{3} \left (\frac {1}{48} \left (43 \int \frac {1}{\sqrt {3 x^2-x+2}}dx+234 \int \frac {1}{(2 x+1) \sqrt {3 x^2-x+2}}dx\right )+\frac {1}{24} \sqrt {3 x^2-x+2} (30 x+13)\right )+\frac {2}{9} \left (3 x^2-x+2\right )^{3/2}\)

\(\Big \downarrow \) 1090

\(\displaystyle \frac {1}{3} \left (\frac {1}{48} \left (234 \int \frac {1}{(2 x+1) \sqrt {3 x^2-x+2}}dx+\frac {43 \int \frac {1}{\sqrt {\frac {1}{23} (6 x-1)^2+1}}d(6 x-1)}{\sqrt {69}}\right )+\frac {1}{24} \sqrt {3 x^2-x+2} (30 x+13)\right )+\frac {2}{9} \left (3 x^2-x+2\right )^{3/2}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {1}{3} \left (\frac {1}{48} \left (234 \int \frac {1}{(2 x+1) \sqrt {3 x^2-x+2}}dx+\frac {43 \text {arcsinh}\left (\frac {6 x-1}{\sqrt {23}}\right )}{\sqrt {3}}\right )+\frac {1}{24} \sqrt {3 x^2-x+2} (30 x+13)\right )+\frac {2}{9} \left (3 x^2-x+2\right )^{3/2}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {1}{3} \left (\frac {1}{48} \left (\frac {43 \text {arcsinh}\left (\frac {6 x-1}{\sqrt {23}}\right )}{\sqrt {3}}-468 \int \frac {1}{52-\frac {(9-8 x)^2}{3 x^2-x+2}}d\frac {9-8 x}{\sqrt {3 x^2-x+2}}\right )+\frac {1}{24} \sqrt {3 x^2-x+2} (30 x+13)\right )+\frac {2}{9} \left (3 x^2-x+2\right )^{3/2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{3} \left (\frac {1}{48} \left (\frac {43 \text {arcsinh}\left (\frac {6 x-1}{\sqrt {23}}\right )}{\sqrt {3}}-18 \sqrt {13} \text {arctanh}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {3 x^2-x+2}}\right )\right )+\frac {1}{24} \sqrt {3 x^2-x+2} (30 x+13)\right )+\frac {2}{9} \left (3 x^2-x+2\right )^{3/2}\)

Input:

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

Output:

(2*(2 - x + 3*x^2)^(3/2))/9 + (((13 + 30*x)*Sqrt[2 - x + 3*x^2])/24 + ((43 
*ArcSinh[(-1 + 6*x)/Sqrt[23]])/Sqrt[3] - 18*Sqrt[13]*ArcTanh[(9 - 8*x)/(2* 
Sqrt[13]*Sqrt[2 - x + 3*x^2])])/48)/3
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 222

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 1090

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* 
(c/(b^2 - 4*a*c)))^p)   Subst[Int[Simp[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 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-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]
 

rule 1231

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(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] - Simp[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] && GtQ[p, 0] && (IntegerQ[p] ||  !R 
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Integer 
Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1269

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

rule 2184

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S 
imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q 
+ 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + 
b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 
1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q - 1) - c 
*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[ 
q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && Pol 
yQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !(IGt 
Q[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
 

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.69

method	result	size
risch	\(\frac {\left (48 x^{2}+14 x +45\right ) \sqrt {3 x^{2}-x +2}}{72}+\frac {43 \sqrt {3}\, \operatorname {arcsinh}\left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{432}-\frac {\sqrt {13}\, \operatorname {arctanh}\left (\frac {2 \left (\frac {9}{2}-4 x \right ) \sqrt {13}}{13 \sqrt {12 \left (\frac {1}{2}+x \right )^{2}+5-16 x}}\right )}{8}\)	\(70\)
default	\(\frac {5 \left (6 x -1\right ) \sqrt {3 x^{2}-x +2}}{72}+\frac {43 \sqrt {3}\, \operatorname {arcsinh}\left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{432}+\frac {2 \left (3 x^{2}-x +2\right )^{\frac {3}{2}}}{9}+\frac {\sqrt {12 \left (\frac {1}{2}+x \right )^{2}+5-16 x}}{8}-\frac {\sqrt {13}\, \operatorname {arctanh}\left (\frac {2 \left (\frac {9}{2}-4 x \right ) \sqrt {13}}{13 \sqrt {12 \left (\frac {1}{2}+x \right )^{2}+5-16 x}}\right )}{8}\)	\(95\)
trager	\(\left (\frac {2}{3} x^{2}+\frac {7}{36} x +\frac {5}{8}\right ) \sqrt {3 x^{2}-x +2}+\frac {43 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x -\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}-x +2}\right )}{432}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-13\right ) \ln \left (\frac {-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-13\right ) x +26 \sqrt {3 x^{2}-x +2}+9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-13\right )}{1+2 x}\right )}{8}\)	\(115\)

Input:

int((3*x^2-x+2)^(1/2)*(4*x^2+3*x+1)/(1+2*x),x,method=_RETURNVERBOSE)
 

Output:

1/72*(48*x^2+14*x+45)*(3*x^2-x+2)^(1/2)+43/432*3^(1/2)*arcsinh(6/23*23^(1/ 
2)*(x-1/6))-1/8*13^(1/2)*arctanh(2/13*(9/2-4*x)*13^(1/2)/(12*(1/2+x)^2+5-1 
6*x)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.14 \[ \int \frac {\sqrt {2-x+3 x^2} \left (1+3 x+4 x^2\right )}{1+2 x} \, dx=\frac {1}{72} \, {\left (48 \, x^{2} + 14 \, x + 45\right )} \sqrt {3 \, x^{2} - x + 2} + \frac {43}{864} \, \sqrt {3} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} - x + 2} {\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) + \frac {1}{16} \, \sqrt {13} \log \left (-\frac {4 \, \sqrt {13} \sqrt {3 \, x^{2} - x + 2} {\left (8 \, x - 9\right )} + 220 \, x^{2} - 196 \, x + 185}{4 \, x^{2} + 4 \, x + 1}\right ) \] Input:

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

Output:

1/72*(48*x^2 + 14*x + 45)*sqrt(3*x^2 - x + 2) + 43/864*sqrt(3)*log(-4*sqrt 
(3)*sqrt(3*x^2 - x + 2)*(6*x - 1) - 72*x^2 + 24*x - 25) + 1/16*sqrt(13)*lo 
g(-(4*sqrt(13)*sqrt(3*x^2 - x + 2)*(8*x - 9) + 220*x^2 - 196*x + 185)/(4*x 
^2 + 4*x + 1))
 

Sympy [F]

\[ \int \frac {\sqrt {2-x+3 x^2} \left (1+3 x+4 x^2\right )}{1+2 x} \, dx=\int \frac {\sqrt {3 x^{2} - x + 2} \cdot \left (4 x^{2} + 3 x + 1\right )}{2 x + 1}\, dx \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {2-x+3 x^2} \left (1+3 x+4 x^2\right )}{1+2 x} \, dx=\frac {2}{9} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} + \frac {5}{12} \, \sqrt {3 \, x^{2} - x + 2} x + \frac {43}{432} \, \sqrt {3} \operatorname {arsinh}\left (\frac {6}{23} \, \sqrt {23} x - \frac {1}{23} \, \sqrt {23}\right ) + \frac {1}{8} \, \sqrt {13} \operatorname {arsinh}\left (\frac {8 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 1 \right |}} - \frac {9 \, \sqrt {23}}{23 \, {\left | 2 \, x + 1 \right |}}\right ) + \frac {13}{72} \, \sqrt {3 \, x^{2} - x + 2} \] Input:

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

Output:

2/9*(3*x^2 - x + 2)^(3/2) + 5/12*sqrt(3*x^2 - x + 2)*x + 43/432*sqrt(3)*ar 
csinh(6/23*sqrt(23)*x - 1/23*sqrt(23)) + 1/8*sqrt(13)*arcsinh(8/23*sqrt(23 
)*x/abs(2*x + 1) - 9/23*sqrt(23)/abs(2*x + 1)) + 13/72*sqrt(3*x^2 - x + 2)
 

Giac [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.25 \[ \int \frac {\sqrt {2-x+3 x^2} \left (1+3 x+4 x^2\right )}{1+2 x} \, dx=\frac {1}{72} \, {\left (2 \, {\left (24 \, x + 7\right )} x + 45\right )} \sqrt {3 \, x^{2} - x + 2} - \frac {43}{432} \, \sqrt {3} \log \left (-6 \, \sqrt {3} x + \sqrt {3} + 6 \, \sqrt {3 \, x^{2} - x + 2}\right ) + \frac {1}{8} \, \sqrt {13} \log \left (-\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {13} - 2 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} - x + 2} \right |}}{2 \, {\left (2 \, \sqrt {3} x - \sqrt {13} + \sqrt {3} - 2 \, \sqrt {3 \, x^{2} - x + 2}\right )}}\right ) \] Input:

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

Output:

1/72*(2*(24*x + 7)*x + 45)*sqrt(3*x^2 - x + 2) - 43/432*sqrt(3)*log(-6*sqr 
t(3)*x + sqrt(3) + 6*sqrt(3*x^2 - x + 2)) + 1/8*sqrt(13)*log(-1/2*abs(-4*s 
qrt(3)*x - 2*sqrt(13) - 2*sqrt(3) + 4*sqrt(3*x^2 - x + 2))/(2*sqrt(3)*x - 
sqrt(13) + sqrt(3) - 2*sqrt(3*x^2 - x + 2)))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {2-x+3 x^2} \left (1+3 x+4 x^2\right )}{1+2 x} \, dx=\int \frac {\sqrt {3\,x^2-x+2}\,\left (4\,x^2+3\,x+1\right )}{2\,x+1} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.86 \[ \int \frac {\sqrt {2-x+3 x^2} \left (1+3 x+4 x^2\right )}{1+2 x} \, dx=\frac {\sqrt {13}\, \mathit {atan} \left (\frac {2 \sqrt {3 x^{2}-x +2}\, \sqrt {3}\, i +6 i x -i}{\sqrt {39}-4}\right ) i}{8}+\frac {2 \sqrt {3 x^{2}-x +2}\, x^{2}}{3}+\frac {7 \sqrt {3 x^{2}-x +2}\, x}{36}+\frac {5 \sqrt {3 x^{2}-x +2}}{8}+\frac {\sqrt {13}\, \mathrm {log}\left (24 \sqrt {3 x^{2}-x +2}\, \sqrt {3}\, x -4 \sqrt {3 x^{2}-x +2}\, \sqrt {3}+8 \sqrt {39}+72 x^{2}-24 x -30\right )}{16}-\frac {\sqrt {13}\, \mathrm {log}\left (2 \sqrt {3 x^{2}-x +2}\, \sqrt {3}+\sqrt {39}+6 x +3\right )}{8}+\frac {43 \sqrt {3}\, \mathrm {log}\left (\frac {2 \sqrt {3 x^{2}-x +2}\, \sqrt {3}+6 x -1}{\sqrt {23}}\right )}{432} \] Input:

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

Output:

(54*sqrt(13)*atan((2*sqrt(3*x**2 - x + 2)*sqrt(3)*i + 6*i*x - i)/(sqrt(39) 
 - 4))*i + 288*sqrt(3*x**2 - x + 2)*x**2 + 84*sqrt(3*x**2 - x + 2)*x + 270 
*sqrt(3*x**2 - x + 2) + 27*sqrt(13)*log(24*sqrt(3*x**2 - x + 2)*sqrt(3)*x 
- 4*sqrt(3*x**2 - x + 2)*sqrt(3) + 8*sqrt(39) + 72*x**2 - 24*x - 30) - 54* 
sqrt(13)*log(2*sqrt(3*x**2 - x + 2)*sqrt(3) + sqrt(39) + 6*x + 3) + 43*sqr 
t(3)*log((2*sqrt(3*x**2 - x + 2)*sqrt(3) + 6*x - 1)/sqrt(23)))/432