\(\int (5-x) (3+2 x)^2 (2+5 x+3 x^2)^{5/2} \, dx\) [942]

Optimal result

Integrand size = 27, antiderivative size = 151 \[ \int (5-x) (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2} \, dx=\frac {22535 (5+6 x) \sqrt {2+5 x+3 x^2}}{5971968}-\frac {22535 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{746496}+\frac {4507 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{15552}-\frac {1}{27} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(10211+4298 x) \left (2+5 x+3 x^2\right )^{7/2}}{4536}-\frac {22535 \text {arctanh}\left (\frac {\sqrt {3} (1+x)}{\sqrt {2+5 x+3 x^2}}\right )}{5971968 \sqrt {3}} \] Output:

22535/5971968*(5+6*x)*(3*x^2+5*x+2)^(1/2)-22535/746496*(5+6*x)*(3*x^2+5*x+ 
2)^(3/2)+4507/15552*(5+6*x)*(3*x^2+5*x+2)^(5/2)-1/27*(3+2*x)^2*(3*x^2+5*x+ 
2)^(7/2)+1/4536*(10211+4298*x)*(3*x^2+5*x+2)^(7/2)-22535/17915904*arctanh( 
3^(1/2)*(1+x)/(3*x^2+5*x+2)^(1/2))*3^(1/2)
 

Mathematica [A] (verified)

Time = 0.90 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.60 \[ \int (5-x) (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2} \, dx=\frac {-3 \sqrt {2+5 x+3 x^2} \left (-871825317-6434937470 x-19762157208 x^2-32476001904 x^3-30355761024 x^4-15455860992 x^5-3275873280 x^6+268240896 x^7+167215104 x^8\right )-157745 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{125411328} \] Input:

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

Output:

(-3*Sqrt[2 + 5*x + 3*x^2]*(-871825317 - 6434937470*x - 19762157208*x^2 - 3 
2476001904*x^3 - 30355761024*x^4 - 15455860992*x^5 - 3275873280*x^6 + 2682 
40896*x^7 + 167215104*x^8) - 157745*Sqrt[3]*ArcTanh[Sqrt[2/3 + (5*x)/3 + x 
^2]/(1 + x)])/125411328
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.17, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {1236, 27, 1225, 1087, 1087, 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.

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

\(\Big \downarrow \) 1236

\(\displaystyle \frac {1}{27} \int \frac {1}{2} (2 x+3) (614 x+931) \left (3 x^2+5 x+2\right )^{5/2}dx-\frac {1}{27} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{54} \int (2 x+3) (614 x+931) \left (3 x^2+5 x+2\right )^{5/2}dx-\frac {1}{27} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {1}{54} \left (\frac {4507}{8} \int \left (3 x^2+5 x+2\right )^{5/2}dx+\frac {1}{84} (4298 x+10211) \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {1}{27} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{54} \left (\frac {4507}{8} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \int \left (3 x^2+5 x+2\right )^{3/2}dx\right )+\frac {1}{84} (4298 x+10211) \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {1}{27} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{54} \left (\frac {4507}{8} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}-\frac {1}{16} \int \sqrt {3 x^2+5 x+2}dx\right )\right )+\frac {1}{84} (4298 x+10211) \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {1}{27} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{54} \left (\frac {4507}{8} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{16} \left (\frac {1}{24} \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )+\frac {1}{84} (4298 x+10211) \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {1}{27} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {1}{54} \left (\frac {4507}{8} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{16} \left (\frac {1}{12} \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )+\frac {1}{84} (4298 x+10211) \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {1}{27} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{54} \left (\frac {4507}{8} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{16} \left (\frac {\text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{24 \sqrt {3}}-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )+\frac {1}{84} (4298 x+10211) \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {1}{27} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}\)

Input:

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

Output:

-1/27*((3 + 2*x)^2*(2 + 5*x + 3*x^2)^(7/2)) + (((10211 + 4298*x)*(2 + 5*x 
+ 3*x^2)^(7/2))/84 + (4507*(((5 + 6*x)*(2 + 5*x + 3*x^2)^(5/2))/36 - (5*(( 
(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/24 + (-1/12*((5 + 6*x)*Sqrt[2 + 5*x + 3 
*x^2]) + ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])]/(24*Sqrt[3]) 
)/16))/72))/8)/54
 

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])
 

Maple [A] (verified)

Time = 1.62 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.56

method	result	size
risch	\(-\frac {\left (167215104 x^{8}[phone] x^{7}-3275873280 x^{6}-15455860992 x^{5}-30355761024 x^{4}-32476001904 x^{3}-19762157208 x^{2}-6434937470 x -871825317\right ) \sqrt {3 x^{2}+5 x +2}}{41803776}-\frac {22535 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{35831808}\)	\(85\)
trager	\(\left (-4 x^{8}-\frac {77}{12} x^{7}+\frac {13165}{168} x^{6}+\frac {2236091}{6048} x^{5}+\frac {26350487}{36288} x^{4}+\frac {225527791}{290304} x^{3}+\frac {823423217}{1741824} x^{2}+\frac {3217468735}{20901888} x +\frac {290608439}{13934592}\right ) \sqrt {3 x^{2}+5 x +2}+\frac {22535 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}+5 x +2}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )\right )}{35831808}\)	\(96\)
default	\(\frac {4507 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{15552}-\frac {22535 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{746496}+\frac {22535 \left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{5971968}-\frac {22535 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{35831808}+\frac {8699 \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{4536}+\frac {163 x \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{324}-\frac {4 x^{2} \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{27}\)	\(134\)

Input:

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

Output:

-1/41803776*(167215104*x^8+268240896*x^7-3275873280*x^6-15455860992*x^5-30 
355761024*x^4-32476001904*x^3-19762157208*x^2-6434937470*x-871825317)*(3*x 
^2+5*x+2)^(1/2)-22535/35831808*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2 
))*3^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.62 \[ \int (5-x) (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\frac {1}{41803776} \, {\left (167215104 \, x^{8} + 268240896 \, x^{7} - 3275873280 \, x^{6} - 15455860992 \, x^{5} - 30355761024 \, x^{4} - 32476001904 \, x^{3} - 19762157208 \, x^{2} - 6434937470 \, x - 871825317\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {22535}{71663616} \, \sqrt {3} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) \] Input:

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

Output:

-1/41803776*(167215104*x^8 + 268240896*x^7 - 3275873280*x^6 - 15455860992* 
x^5 - 30355761024*x^4 - 32476001904*x^3 - 19762157208*x^2 - 6434937470*x - 
 871825317)*sqrt(3*x^2 + 5*x + 2) + 22535/71663616*sqrt(3)*log(-4*sqrt(3)* 
sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49)
 

Sympy [A] (verification not implemented)

Time = 0.70 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.68 \[ \int (5-x) (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2} \, dx=\sqrt {3 x^{2} + 5 x + 2} \left (- 4 x^{8} - \frac {77 x^{7}}{12} + \frac {13165 x^{6}}{168} + \frac {2236091 x^{5}}{6048} + \frac {26350487 x^{4}}{36288} + \frac {225527791 x^{3}}{290304} + \frac {823423217 x^{2}}{1741824} + \frac {3217468735 x}{20901888} + \frac {290608439}{13934592}\right ) - \frac {22535 \sqrt {3} \log {\left (6 x + 2 \sqrt {3} \sqrt {3 x^{2} + 5 x + 2} + 5 \right )}}{35831808} \] Input:

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

Output:

sqrt(3*x**2 + 5*x + 2)*(-4*x**8 - 77*x**7/12 + 13165*x**6/168 + 2236091*x* 
*5/6048 + 26350487*x**4/36288 + 225527791*x**3/290304 + 823423217*x**2/174 
1824 + 3217468735*x/20901888 + 290608439/13934592) - 22535*sqrt(3)*log(6*x 
 + 2*sqrt(3)*sqrt(3*x**2 + 5*x + 2) + 5)/35831808
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.07 \[ \int (5-x) (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\frac {4}{27} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x^{2} + \frac {163}{324} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x + \frac {8699}{4536} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} + \frac {4507}{2592} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {22535}{15552} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {22535}{124416} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {112675}{746496} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {22535}{995328} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {22535}{35831808} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {112675}{5971968} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \] Input:

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

Output:

-4/27*(3*x^2 + 5*x + 2)^(7/2)*x^2 + 163/324*(3*x^2 + 5*x + 2)^(7/2)*x + 86 
99/4536*(3*x^2 + 5*x + 2)^(7/2) + 4507/2592*(3*x^2 + 5*x + 2)^(5/2)*x + 22 
535/15552*(3*x^2 + 5*x + 2)^(5/2) - 22535/124416*(3*x^2 + 5*x + 2)^(3/2)*x 
 - 112675/746496*(3*x^2 + 5*x + 2)^(3/2) + 22535/995328*sqrt(3*x^2 + 5*x + 
 2)*x - 22535/35831808*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 
 5) + 112675/5971968*sqrt(3*x^2 + 5*x + 2)
 

Giac [A] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.59 \[ \int (5-x) (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\frac {1}{41803776} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (6 \, {\left (36 \, {\left (14 \, {\left (48 \, x + 77\right )} x - 13165\right )} x - 2236091\right )} x - 26350487\right )} x - 225527791\right )} x - 823423217\right )} x - 3217468735\right )} x - 871825317\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {22535}{35831808} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \] Input:

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

Output:

-1/41803776*(2*(12*(6*(8*(6*(36*(14*(48*x + 77)*x - 13165)*x - 2236091)*x 
- 26350487)*x - 225527791)*x - 823423217)*x - 3217468735)*x - 871825317)*s 
qrt(3*x^2 + 5*x + 2) + 22535/35831808*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)* 
x - sqrt(3*x^2 + 5*x + 2)) - 5))
 

Mupad [F(-1)]

Timed out. \[ \int (5-x) (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\int {\left (2\,x+3\right )}^2\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.09 \[ \int (5-x) (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2} \, dx=-4 \sqrt {3 x^{2}+5 x +2}\, x^{8}-\frac {77 \sqrt {3 x^{2}+5 x +2}\, x^{7}}{12}+\frac {13165 \sqrt {3 x^{2}+5 x +2}\, x^{6}}{168}+\frac {2236091 \sqrt {3 x^{2}+5 x +2}\, x^{5}}{6048}+\frac {26350487 \sqrt {3 x^{2}+5 x +2}\, x^{4}}{36288}+\frac {225527791 \sqrt {3 x^{2}+5 x +2}\, x^{3}}{290304}+\frac {823423217 \sqrt {3 x^{2}+5 x +2}\, x^{2}}{1741824}+\frac {3217468735 \sqrt {3 x^{2}+5 x +2}\, x}{20901888}+\frac {290608439 \sqrt {3 x^{2}+5 x +2}}{13934592}-\frac {22535 \sqrt {3}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}+6 x +5\right )}{35831808} \] Input:

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

Output:

( - 1003290624*sqrt(3*x**2 + 5*x + 2)*x**8 - 1609445376*sqrt(3*x**2 + 5*x 
+ 2)*x**7 + 19655239680*sqrt(3*x**2 + 5*x + 2)*x**6 + 92735165952*sqrt(3*x 
**2 + 5*x + 2)*x**5 + 182134566144*sqrt(3*x**2 + 5*x + 2)*x**4 + 194856011 
424*sqrt(3*x**2 + 5*x + 2)*x**3 + 118572943248*sqrt(3*x**2 + 5*x + 2)*x**2 
 + 38609624820*sqrt(3*x**2 + 5*x + 2)*x + 5230951902*sqrt(3*x**2 + 5*x + 2 
) - 157745*sqrt(3)*log(2*sqrt(3*x**2 + 5*x + 2)*sqrt(3) + 6*x + 5))/250822 
656