3.25.6 \(\int (5-x) (3+2 x)^4 \sqrt {2+5 x+3 x^2} \, dx\) [2406]

3.25.6.1 Optimal result

Integrand size = 27, antiderivative size = 160 \[ \int (5-x) (3+2 x)^4 \sqrt {2+5 x+3 x^2} \, dx=\frac {25969 (5+6 x) \sqrt {2+5 x+3 x^2}}{15552}+\frac {478}{315} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}+\frac {229}{378} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{21} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2}+\frac {(874301+378774 x) \left (2+5 x+3 x^2\right )^{3/2}}{68040}-\frac {25969 \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{31104 \sqrt {3}} \]

output

478/315*(3+2*x)^2*(3*x^2+5*x+2)^(3/2)+229/378*(3+2*x)^3*(3*x^2+5*x+2)^(3/2 
)-1/21*(3+2*x)^4*(3*x^2+5*x+2)^(3/2)+1/68040*(874301+378774*x)*(3*x^2+5*x+ 
2)^(3/2)-25969/93312*arctanh(1/6*(5+6*x)*3^(1/2)/(3*x^2+5*x+2)^(1/2))*3^(1 
/2)+25969/15552*(5+6*x)*(3*x^2+5*x+2)^(1/2)
 

3.25.6.2 Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.51 \[ \int (5-x) (3+2 x)^4 \sqrt {2+5 x+3 x^2} \, dx=\frac {-3 \sqrt {2+5 x+3 x^2} \left (-47009103-161915450 x-208601544 x^2-123633360 x^3-28649088 x^4+1624320 x^5+1244160 x^6\right )-908915 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{1632960} \]

input

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

output

(-3*Sqrt[2 + 5*x + 3*x^2]*(-47009103 - 161915450*x - 208601544*x^2 - 12363 
3360*x^3 - 28649088*x^4 + 1624320*x^5 + 1244160*x^6) - 908915*Sqrt[3]*ArcT 
anh[Sqrt[2/3 + (5*x)/3 + x^2]/(1 + x)])/1632960
 

3.25.6.3 Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.12, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {1236, 27, 1236, 27, 1236, 27, 1225, 1087, 1092, 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.

\(\displaystyle \int (5-x) (2 x+3)^4 \sqrt {3 x^2+5 x+2} \, dx\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {1}{21} \int \frac {1}{2} (2 x+3)^3 (458 x+707) \sqrt {3 x^2+5 x+2}dx-\frac {1}{21} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{42} \int (2 x+3)^3 (458 x+707) \sqrt {3 x^2+5 x+2}dx-\frac {1}{21} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {1}{42} \left (\frac {1}{18} \int 3 (2 x+3)^2 (5736 x+7459) \sqrt {3 x^2+5 x+2}dx+\frac {229}{9} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^3\right )-\frac {1}{21} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{42} \left (\frac {1}{6} \int (2 x+3)^2 (5736 x+7459) \sqrt {3 x^2+5 x+2}dx+\frac {229}{9} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^3\right )-\frac {1}{21} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {1}{42} \left (\frac {1}{6} \left (\frac {1}{15} \int 3 (2 x+3) (42086 x+53569) \sqrt {3 x^2+5 x+2}dx+\frac {1912}{5} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^2\right )+\frac {229}{9} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^3\right )-\frac {1}{21} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{42} \left (\frac {1}{6} \left (\frac {1}{5} \int (2 x+3) (42086 x+53569) \sqrt {3 x^2+5 x+2}dx+\frac {1912}{5} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^2\right )+\frac {229}{9} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^3\right )-\frac {1}{21} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {1}{42} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {908915}{36} \int \sqrt {3 x^2+5 x+2}dx+\frac {1}{54} (378774 x+874301) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {1912}{5} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^2\right )+\frac {229}{9} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^3\right )-\frac {1}{21} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{42} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {908915}{36} \left (\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}-\frac {1}{24} \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx\right )+\frac {1}{54} (378774 x+874301) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {1912}{5} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^2\right )+\frac {229}{9} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^3\right )-\frac {1}{21} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {1}{42} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {908915}{36} \left (\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}-\frac {1}{12} \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}\right )+\frac {1}{54} (378774 x+874301) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {1912}{5} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^2\right )+\frac {229}{9} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^3\right )-\frac {1}{21} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{42} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {908915}{36} \left (\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}-\frac {\text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{24 \sqrt {3}}\right )+\frac {1}{54} (378774 x+874301) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {1912}{5} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^2\right )+\frac {229}{9} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^3\right )-\frac {1}{21} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{3/2}\)

input

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

output

-1/21*((3 + 2*x)^4*(2 + 5*x + 3*x^2)^(3/2)) + ((229*(3 + 2*x)^3*(2 + 5*x + 
 3*x^2)^(3/2))/9 + ((1912*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^(3/2))/5 + (((8743 
01 + 378774*x)*(2 + 5*x + 3*x^2)^(3/2))/54 + (908915*(((5 + 6*x)*Sqrt[2 + 
5*x + 3*x^2])/12 - ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])]/(2 
4*Sqrt[3])))/36)/5)/6)/42
 

3.25.6.3.1 Defintions of rubi rules used

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 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1225

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( 
x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 
 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), 
 x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p 
+ 3))/(2*c^2*(2*p + 3))   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, p}, x] &&  !LeQ[p, -1]
 

rule 1236

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 
1)/(c*(m + 2*p + 2))), x] + Simp[1/(c*(m + 2*p + 2))   Int[(d + e*x)^(m - 1 
)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m 
*(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ 
{a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege 
rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 0])
 

3.25.6.4 Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.47

method	result	size
risch	\(-\frac {\left (1244160 x^{6}+1624320 x^{5}-28649088 x^{4}-123633360 x^{3}-208601544 x^{2}-161915450 x -47009103\right ) \sqrt {3 x^{2}+5 x +2}}{544320}-\frac {25969 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{93312}\)	\(75\)
trager	\(\left (-\frac {16}{7} x^{6}-\frac {188}{63} x^{5}+\frac {49738}{945} x^{4}+\frac {171713}{756} x^{3}+\frac {8691731}{22680} x^{2}+\frac {16191545}{54432} x +\frac {15669701}{181440}\right ) \sqrt {3 x^{2}+5 x +2}+\frac {25969 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}+5 x +2}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )\right )}{93312}\)	\(86\)
default	\(\frac {25969 \left (5+6 x \right ) \sqrt {3 x^{2}+5 x +2}}{15552}-\frac {25969 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{93312}+\frac {2654033 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{68040}-\frac {16 x^{4} \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{21}+\frac {52 x^{3} \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{189}+\frac {5542 x^{2} \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{315}+\frac {34931 x \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{756}\)	\(130\)

input

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

output

-1/544320*(1244160*x^6+1624320*x^5-28649088*x^4-123633360*x^3-208601544*x^ 
2-161915450*x-47009103)*(3*x^2+5*x+2)^(1/2)-25969/93312*ln(1/3*(5/2+3*x)*3 
^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)
 

3.25.6.5 Fricas [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.52 \[ \int (5-x) (3+2 x)^4 \sqrt {2+5 x+3 x^2} \, dx=-\frac {1}{544320} \, {\left (1244160 \, x^{6} + 1624320 \, x^{5} - 28649088 \, x^{4} - 123633360 \, x^{3} - 208601544 \, x^{2} - 161915450 \, x - 47009103\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {25969}{186624} \, \sqrt {3} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) \]

input

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

output

-1/544320*(1244160*x^6 + 1624320*x^5 - 28649088*x^4 - 123633360*x^3 - 2086 
01544*x^2 - 161915450*x - 47009103)*sqrt(3*x^2 + 5*x + 2) + 25969/186624*s 
qrt(3)*log(-4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 4 
9)
 

3.25.6.6 Sympy [A] (verification not implemented)

Time = 0.56 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.56 \[ \int (5-x) (3+2 x)^4 \sqrt {2+5 x+3 x^2} \, dx=\sqrt {3 x^{2} + 5 x + 2} \left (- \frac {16 x^{6}}{7} - \frac {188 x^{5}}{63} + \frac {49738 x^{4}}{945} + \frac {171713 x^{3}}{756} + \frac {8691731 x^{2}}{22680} + \frac {16191545 x}{54432} + \frac {15669701}{181440}\right ) - \frac {25969 \sqrt {3} \log {\left (6 x + 2 \sqrt {3} \sqrt {3 x^{2} + 5 x + 2} + 5 \right )}}{93312} \]

input

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

output

sqrt(3*x**2 + 5*x + 2)*(-16*x**6/7 - 188*x**5/63 + 49738*x**4/945 + 171713 
*x**3/756 + 8691731*x**2/22680 + 16191545*x/54432 + 15669701/181440) - 259 
69*sqrt(3)*log(6*x + 2*sqrt(3)*sqrt(3*x**2 + 5*x + 2) + 5)/93312
 

3.25.6.7 Maxima [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.86 \[ \int (5-x) (3+2 x)^4 \sqrt {2+5 x+3 x^2} \, dx=-\frac {16}{21} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x^{4} + \frac {52}{189} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x^{3} + \frac {5542}{315} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x^{2} + \frac {34931}{756} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {2654033}{68040} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {25969}{2592} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {25969}{93312} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {129845}{15552} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]

input

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

output

-16/21*(3*x^2 + 5*x + 2)^(3/2)*x^4 + 52/189*(3*x^2 + 5*x + 2)^(3/2)*x^3 + 
5542/315*(3*x^2 + 5*x + 2)^(3/2)*x^2 + 34931/756*(3*x^2 + 5*x + 2)^(3/2)*x 
 + 2654033/68040*(3*x^2 + 5*x + 2)^(3/2) + 25969/2592*sqrt(3*x^2 + 5*x + 2 
)*x - 25969/93312*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) + 
 129845/15552*sqrt(3*x^2 + 5*x + 2)
 

3.25.6.8 Giac [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.49 \[ \int (5-x) (3+2 x)^4 \sqrt {2+5 x+3 x^2} \, dx=-\frac {1}{544320} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (30 \, {\left (36 \, x + 47\right )} x - 24869\right )} x - 858565\right )} x - 8691731\right )} x - 80957725\right )} x - 47009103\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {25969}{93312} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]

input

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

output

-1/544320*(2*(12*(6*(8*(30*(36*x + 47)*x - 24869)*x - 858565)*x - 8691731) 
*x - 80957725)*x - 47009103)*sqrt(3*x^2 + 5*x + 2) + 25969/93312*sqrt(3)*l 
og(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))
 

3.25.6.9 Mupad [B] (verification not implemented)

Time = 12.63 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.06 \[ \int (5-x) (3+2 x)^4 \sqrt {2+5 x+3 x^2} \, dx=\frac {5542\,x^2\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{315}+\frac {52\,x^3\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{189}-\frac {16\,x^4\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{21}-\frac {118159\,\sqrt {3}\,\ln \left (\sqrt {3\,x^2+5\,x+2}+\frac {\sqrt {3}\,\left (3\,x+\frac {5}{2}\right )}{3}\right )}{27216}+\frac {118159\,\left (\frac {x}{2}+\frac {5}{12}\right )\,\sqrt {3\,x^2+5\,x+2}}{378}+\frac {2654033\,\sqrt {3\,x^2+5\,x+2}\,\left (72\,x^2+30\,x-27\right )}{1632960}+\frac {34931\,x\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{756}+\frac {2654033\,\sqrt {3}\,\ln \left (2\,\sqrt {3\,x^2+5\,x+2}+\frac {\sqrt {3}\,\left (6\,x+5\right )}{3}\right )}{653184} \]

input

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

output

(5542*x^2*(5*x + 3*x^2 + 2)^(3/2))/315 + (52*x^3*(5*x + 3*x^2 + 2)^(3/2))/ 
189 - (16*x^4*(5*x + 3*x^2 + 2)^(3/2))/21 - (118159*3^(1/2)*log((5*x + 3*x 
^2 + 2)^(1/2) + (3^(1/2)*(3*x + 5/2))/3))/27216 + (118159*(x/2 + 5/12)*(5* 
x + 3*x^2 + 2)^(1/2))/378 + (2654033*(5*x + 3*x^2 + 2)^(1/2)*(30*x + 72*x^ 
2 - 27))/1632960 + (34931*x*(5*x + 3*x^2 + 2)^(3/2))/756 + (2654033*3^(1/2 
)*log(2*(5*x + 3*x^2 + 2)^(1/2) + (3^(1/2)*(6*x + 5))/3))/653184