\(\int \frac {(5-x) (2+5 x+3 x^2)^{3/2}}{3+2 x} \, dx\) [931]

Optimal result

Integrand size = 27, antiderivative size = 113 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{3+2 x} \, dx=\frac {1}{128} (175-414 x) \sqrt {2+5 x+3 x^2}+\frac {1}{48} (47-6 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {2011 \text {arctanh}\left (\frac {\sqrt {3} (1+x)}{\sqrt {2+5 x+3 x^2}}\right )}{128 \sqrt {3}}+\frac {65}{16} \sqrt {5} \text {arctanh}\left (\frac {\sqrt {5} (1+x)}{\sqrt {2+5 x+3 x^2}}\right ) \] Output:

1/128*(175-414*x)*(3*x^2+5*x+2)^(1/2)+1/48*(47-6*x)*(3*x^2+5*x+2)^(3/2)-20 
11/384*arctanh(3^(1/2)*(1+x)/(3*x^2+5*x+2)^(1/2))*3^(1/2)+65/16*5^(1/2)*ar 
ctanh(5^(1/2)*(1+x)/(3*x^2+5*x+2)^(1/2))
 

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.85 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{3+2 x} \, dx=\frac {1}{384} \left (-\sqrt {2+5 x+3 x^2} \left (-1277-542 x-888 x^2+144 x^3\right )+1560 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )-2011 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )\right ) \] Input:

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

Output:

(-(Sqrt[2 + 5*x + 3*x^2]*(-1277 - 542*x - 888*x^2 + 144*x^3)) + 1560*Sqrt[ 
5]*ArcTanh[Sqrt[2/5 + x + (3*x^2)/5]/(1 + x)] - 2011*Sqrt[3]*ArcTanh[Sqrt[ 
2/3 + (5*x)/3 + x^2]/(1 + x)])/384
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.14, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1231, 27, 1231, 27, 1269, 1092, 219, 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.

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

\(\Big \downarrow \) 1231

\(\displaystyle \frac {1}{48} (47-6 x) \left (3 x^2+5 x+2\right )^{3/2}-\frac {1}{96} \int \frac {3 (414 x+361) \sqrt {3 x^2+5 x+2}}{2 x+3}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{48} (47-6 x) \left (3 x^2+5 x+2\right )^{3/2}-\frac {1}{32} \int \frac {(414 x+361) \sqrt {3 x^2+5 x+2}}{2 x+3}dx\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {1}{32} \left (\frac {1}{48} \int -\frac {6 (4022 x+3433)}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx+\frac {1}{4} \sqrt {3 x^2+5 x+2} (175-414 x)\right )+\frac {1}{48} (47-6 x) \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{32} \left (\frac {1}{4} (175-414 x) \sqrt {3 x^2+5 x+2}-\frac {1}{8} \int \frac {4022 x+3433}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )+\frac {1}{48} (47-6 x) \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {1}{32} \left (\frac {1}{8} \left (2600 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-2011 \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx\right )+\frac {1}{4} \sqrt {3 x^2+5 x+2} (175-414 x)\right )+\frac {1}{48} (47-6 x) \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {1}{32} \left (\frac {1}{8} \left (2600 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-4022 \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}\right )+\frac {1}{4} \sqrt {3 x^2+5 x+2} (175-414 x)\right )+\frac {1}{48} (47-6 x) \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{32} \left (\frac {1}{8} \left (2600 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-\frac {2011 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {3}}\right )+\frac {1}{4} \sqrt {3 x^2+5 x+2} (175-414 x)\right )+\frac {1}{48} (47-6 x) \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {1}{32} \left (\frac {1}{8} \left (-5200 \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )-\frac {2011 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {3}}\right )+\frac {1}{4} \sqrt {3 x^2+5 x+2} (175-414 x)\right )+\frac {1}{48} (47-6 x) \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{32} \left (\frac {1}{8} \left (520 \sqrt {5} \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )-\frac {2011 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {3}}\right )+\frac {1}{4} \sqrt {3 x^2+5 x+2} (175-414 x)\right )+\frac {1}{48} (47-6 x) \left (3 x^2+5 x+2\right )^{3/2}\)

Input:

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

Output:

((47 - 6*x)*(2 + 5*x + 3*x^2)^(3/2))/48 + (((175 - 414*x)*Sqrt[2 + 5*x + 3 
*x^2])/4 + ((-2011*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/S 
qrt[3] + 520*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])]) 
/8)/32
 

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 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 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]
 

Maple [A] (verified)

Time = 1.80 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.80

method	result	size
risch	\(-\frac {\left (144 x^{3}-888 x^{2}-542 x -1277\right ) \sqrt {3 x^{2}+5 x +2}}{384}-\frac {2011 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{768}-\frac {65 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{32}\)	\(90\)
trager	\(\left (-\frac {3}{8} x^{3}+\frac {37}{16} x^{2}+\frac {271}{192} x +\frac {1277}{384}\right ) \sqrt {3 x^{2}+5 x +2}+\frac {65 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{2 x +3}\right )}{32}-\frac {2011 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}+5 x +2}\right )}{768}\)	\(120\)
default	\(-\frac {\left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{48}+\frac {\left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{384}-\frac {\ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{2304}+\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{12}-\frac {13 \left (6 x +5\right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{24}-\frac {377 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}\right ) \sqrt {3}}{144}+\frac {65 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{32}-\frac {65 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{32}\)	\(183\)

Input:

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

Output:

-1/384*(144*x^3-888*x^2-542*x-1277)*(3*x^2+5*x+2)^(1/2)-2011/768*ln(1/3*(5 
/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)-65/32*5^(1/2)*arctanh(2/5*(-7 
/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.05 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{3+2 x} \, dx=-\frac {1}{384} \, {\left (144 \, x^{3} - 888 \, x^{2} - 542 \, x - 1277\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {2011}{1536} \, \sqrt {3} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + \frac {65}{64} \, \sqrt {5} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) \] Input:

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

Output:

-1/384*(144*x^3 - 888*x^2 - 542*x - 1277)*sqrt(3*x^2 + 5*x + 2) + 2011/153 
6*sqrt(3)*log(-4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x 
+ 49) + 65/64*sqrt(5)*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))
 

Sympy [F]

\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{3+2 x} \, dx=- \int \left (- \frac {10 \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \left (- \frac {23 x \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \left (- \frac {10 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\, dx \] Input:

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

Output:

-Integral(-10*sqrt(3*x**2 + 5*x + 2)/(2*x + 3), x) - Integral(-23*x*sqrt(3 
*x**2 + 5*x + 2)/(2*x + 3), x) - Integral(-10*x**2*sqrt(3*x**2 + 5*x + 2)/ 
(2*x + 3), x) - Integral(3*x**3*sqrt(3*x**2 + 5*x + 2)/(2*x + 3), x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.13 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{3+2 x} \, dx=-\frac {1}{8} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {47}{48} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {207}{64} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {2011}{768} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) - \frac {65}{32} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) + \frac {175}{128} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \] Input:

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

Output:

-1/8*(3*x^2 + 5*x + 2)^(3/2)*x + 47/48*(3*x^2 + 5*x + 2)^(3/2) - 207/64*sq 
rt(3*x^2 + 5*x + 2)*x - 2011/768*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) 
 + 3*x + 5/2) - 65/32*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 
3) + 5/2/abs(2*x + 3) - 2) + 175/128*sqrt(3*x^2 + 5*x + 2)
 

Giac [A] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.20 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{3+2 x} \, dx=-\frac {1}{384} \, {\left (2 \, {\left (12 \, {\left (6 \, x - 37\right )} x - 271\right )} x - 1277\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {65}{32} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac {2011}{768} \, \sqrt {3} \log \left ({\left | -6 \, \sqrt {3} x - 5 \, \sqrt {3} + 6 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}\right ) \] Input:

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

Output:

-1/384*(2*(12*(6*x - 37)*x - 271)*x - 1277)*sqrt(3*x^2 + 5*x + 2) + 65/32* 
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))) 
 + 2011/768*sqrt(3)*log(abs(-6*sqrt(3)*x - 5*sqrt(3) + 6*sqrt(3*x^2 + 5*x 
+ 2)))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.25 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{3+2 x} \, dx=-\frac {3 \sqrt {3 x^{2}+5 x +2}\, x^{3}}{8}+\frac {37 \sqrt {3 x^{2}+5 x +2}\, x^{2}}{16}+\frac {271 \sqrt {3 x^{2}+5 x +2}\, x}{192}+\frac {1277 \sqrt {3 x^{2}+5 x +2}}{384}+\frac {65 \sqrt {5}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}-\sqrt {15}+6 x +9\right )}{32}-\frac {65 \sqrt {5}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}+\sqrt {15}+6 x +9\right )}{32}-\frac {2011 \sqrt {3}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}+6 x +5\right )}{768} \] Input:

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

Output:

( - 288*sqrt(3*x**2 + 5*x + 2)*x**3 + 1776*sqrt(3*x**2 + 5*x + 2)*x**2 + 1 
084*sqrt(3*x**2 + 5*x + 2)*x + 2554*sqrt(3*x**2 + 5*x + 2) + 1560*sqrt(5)* 
log(2*sqrt(3*x**2 + 5*x + 2)*sqrt(3) - sqrt(15) + 6*x + 9) - 1560*sqrt(5)* 
log(2*sqrt(3*x**2 + 5*x + 2)*sqrt(3) + sqrt(15) + 6*x + 9) - 2011*sqrt(3)* 
log(2*sqrt(3*x**2 + 5*x + 2)*sqrt(3) + 6*x + 5))/768