Integrand size = 29, antiderivative size = 143 \[ \int \frac {(5-x) (3+2 x)^{3/2}}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {3+2 x} (121+139 x)}{3 \sqrt {2+5 x+3 x^2}}+\frac {274 \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{3 \sqrt {3} \sqrt {2+5 x+3 x^2}}-\frac {350 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{3 \sqrt {3} \sqrt {2+5 x+3 x^2}} \] Output:
-2/3*(3+2*x)^(1/2)*(121+139*x)/(3*x^2+5*x+2)^(1/2)+274/9*(-3*x^2-5*x-2)^(1 /2)*EllipticE((1+x)^(1/2)*3^(1/2),1/3*I*6^(1/2))*3^(1/2)/(3*x^2+5*x+2)^(1/ 2)-350/9*(-3*x^2-5*x-2)^(1/2)*EllipticF((1+x)^(1/2)*3^(1/2),1/3*I*6^(1/2)) *3^(1/2)/(3*x^2+5*x+2)^(1/2)
Time = 31.40 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.28 \[ \int \frac {(5-x) (3+2 x)^{3/2}}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {3+2 x} \left (541+607 x+12 x^2\right )-274 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )+64 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )}{9 (3+2 x) \sqrt {2+5 x+3 x^2}} \] Input:
Integrate[((5 - x)*(3 + 2*x)^(3/2))/(2 + 5*x + 3*x^2)^(3/2),x]
Output:
-1/9*(2*Sqrt[3 + 2*x]*(541 + 607*x + 12*x^2) - 274*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^2*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/ Sqrt[3 + 2*x]], 3/5] + 64*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^2*Sqrt [(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5])/((3 + 2*x)*Sqrt[2 + 5*x + 3*x^2])
Time = 0.47 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1233, 1269, 1172, 27, 321, 327}
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.
| \(\displaystyle \int \frac {(5-x) (2 x+3)^{3/2}}{\left (3 x^2+5 x+2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {2}{3} \int \frac {137 x+118}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {2 \sqrt {2 x+3} (139 x+121)}{3 \sqrt {3 x^2+5 x+2}}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {2}{3} \left (\frac {137}{2} \int \frac {\sqrt {2 x+3}}{\sqrt {3 x^2+5 x+2}}dx-\frac {175}{2} \int \frac {1}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )-\frac {2 \sqrt {2 x+3} (139 x+121)}{3 \sqrt {3 x^2+5 x+2}}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {2}{3} \left (\frac {137 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {3} \sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {175 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {3}}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {2 \sqrt {2 x+3} (139 x+121)}{3 \sqrt {3 x^2+5 x+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{3} \left (\frac {137 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {175 \sqrt {-3 x^2-5 x-2} \int \frac {1}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}\right )-\frac {2 \sqrt {2 x+3} (139 x+121)}{3 \sqrt {3 x^2+5 x+2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {2}{3} \left (\frac {137 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {175 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {2 \sqrt {2 x+3} (139 x+121)}{3 \sqrt {3 x^2+5 x+2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2}{3} \left (\frac {137 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {175 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {2 \sqrt {2 x+3} (139 x+121)}{3 \sqrt {3 x^2+5 x+2}}\) |
Input:
Int[((5 - x)*(3 + 2*x)^(3/2))/(2 + 5*x + 3*x^2)^(3/2),x]
Output:
(-2*Sqrt[3 + 2*x]*(121 + 139*x))/(3*Sqrt[2 + 5*x + 3*x^2]) + (2*((137*Sqrt [-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(Sqrt[3]* Sqrt[2 + 5*x + 3*x^2]) - (175*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt [3]*Sqrt[1 + x]], -2/3])/(Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])))/3
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 )/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e *Rt[b^2 - 4*a*c, 2])))^m)) Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e }, x] && EqQ[m^2, 1/4]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m - 1))*(a + b*x + c*x^2) ^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c *(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[1/(c*( p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Sim p[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f *(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*( m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2* p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g]) | | !ILtQ[m + 2*p + 3, 0])
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]
Time = 1.65 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.92
| method | result | size |
| default | \(\frac {2 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, \left (57 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {2 x +3}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )-137 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {2 x +3}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )-12510 x^{2}-29655 x -16335\right )}{135 \left (6 x^{3}+19 x^{2}+19 x +6\right )}\) | \(131\) |
| elliptic | \(\frac {\sqrt {\left (3 x^{2}+5 x +2\right ) \left (2 x +3\right )}\, \left (-\frac {2 \left (6 x +9\right ) \left (\frac {121}{9}+\frac {139 x}{9}\right )}{\sqrt {\left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right ) \left (6 x +9\right )}}-\frac {236 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {30 x +45}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )}{45 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}-\frac {274 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {30 x +45}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-\operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{45 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}\) | \(198\) |
Input:
int((5-x)*(2*x+3)^(3/2)/(3*x^2+5*x+2)^(3/2),x,method=_RETURNVERBOSE)
Output:
2/135*(2*x+3)^(1/2)*(3*x^2+5*x+2)^(1/2)*(57*(-30*x-20)^(1/2)*(3+3*x)^(1/2) *15^(1/2)*(2*x+3)^(1/2)*EllipticF(1/5*(-30*x-20)^(1/2),1/2*10^(1/2))-137*( -30*x-20)^(1/2)*(3+3*x)^(1/2)*15^(1/2)*(2*x+3)^(1/2)*EllipticE(1/5*(-30*x- 20)^(1/2),1/2*10^(1/2))-12510*x^2-29655*x-16335)/(6*x^3+19*x^2+19*x+6)
Time = 0.12 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.60 \[ \int \frac {(5-x) (3+2 x)^{3/2}}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx=-\frac {479 \, \sqrt {6} {\left (3 \, x^{2} + 5 \, x + 2\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 2466 \, \sqrt {6} {\left (3 \, x^{2} + 5 \, x + 2\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) + 54 \, \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (139 \, x + 121\right )} \sqrt {2 \, x + 3}}{81 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}} \] Input:
integrate((5-x)*(3+2*x)^(3/2)/(3*x^2+5*x+2)^(3/2),x, algorithm="fricas")
Output:
-1/81*(479*sqrt(6)*(3*x^2 + 5*x + 2)*weierstrassPInverse(19/27, -28/729, x + 19/18) + 2466*sqrt(6)*(3*x^2 + 5*x + 2)*weierstrassZeta(19/27, -28/729, weierstrassPInverse(19/27, -28/729, x + 19/18)) + 54*sqrt(3*x^2 + 5*x + 2 )*(139*x + 121)*sqrt(2*x + 3))/(3*x^2 + 5*x + 2)
\[ \int \frac {(5-x) (3+2 x)^{3/2}}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx=- \int \left (- \frac {15 \sqrt {2 x + 3}}{3 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 5 x \sqrt {3 x^{2} + 5 x + 2} + 2 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac {7 x \sqrt {2 x + 3}}{3 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 5 x \sqrt {3 x^{2} + 5 x + 2} + 2 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac {2 x^{2} \sqrt {2 x + 3}}{3 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 5 x \sqrt {3 x^{2} + 5 x + 2} + 2 \sqrt {3 x^{2} + 5 x + 2}}\, dx \] Input:
integrate((5-x)*(3+2*x)**(3/2)/(3*x**2+5*x+2)**(3/2),x)
Output:
-Integral(-15*sqrt(2*x + 3)/(3*x**2*sqrt(3*x**2 + 5*x + 2) + 5*x*sqrt(3*x* *2 + 5*x + 2) + 2*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-7*x*sqrt(2*x + 3 )/(3*x**2*sqrt(3*x**2 + 5*x + 2) + 5*x*sqrt(3*x**2 + 5*x + 2) + 2*sqrt(3*x **2 + 5*x + 2)), x) - Integral(2*x**2*sqrt(2*x + 3)/(3*x**2*sqrt(3*x**2 + 5*x + 2) + 5*x*sqrt(3*x**2 + 5*x + 2) + 2*sqrt(3*x**2 + 5*x + 2)), x)
\[ \int \frac {(5-x) (3+2 x)^{3/2}}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx=\int { -\frac {{\left (2 \, x + 3\right )}^{\frac {3}{2}} {\left (x - 5\right )}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((5-x)*(3+2*x)^(3/2)/(3*x^2+5*x+2)^(3/2),x, algorithm="maxima")
Output:
-integrate((2*x + 3)^(3/2)*(x - 5)/(3*x^2 + 5*x + 2)^(3/2), x)
\[ \int \frac {(5-x) (3+2 x)^{3/2}}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx=\int { -\frac {{\left (2 \, x + 3\right )}^{\frac {3}{2}} {\left (x - 5\right )}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((5-x)*(3+2*x)^(3/2)/(3*x^2+5*x+2)^(3/2),x, algorithm="giac")
Output:
integrate(-(2*x + 3)^(3/2)*(x - 5)/(3*x^2 + 5*x + 2)^(3/2), x)
Timed out. \[ \int \frac {(5-x) (3+2 x)^{3/2}}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx=-\int \frac {{\left (2\,x+3\right )}^{3/2}\,\left (x-5\right )}{{\left (3\,x^2+5\,x+2\right )}^{3/2}} \,d x \] Input:
int(-((2*x + 3)^(3/2)*(x - 5))/(5*x + 3*x^2 + 2)^(3/2),x)
Output:
-int(((2*x + 3)^(3/2)*(x - 5))/(5*x + 3*x^2 + 2)^(3/2), x)
\[ \int \frac {(5-x) (3+2 x)^{3/2}}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx=\frac {-8 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x -46 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}-30 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{2}}{18 x^{5}+87 x^{4}+164 x^{3}+151 x^{2}+68 x +12}d x \right ) x^{2}-50 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{2}}{18 x^{5}+87 x^{4}+164 x^{3}+151 x^{2}+68 x +12}d x \right ) x -20 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{2}}{18 x^{5}+87 x^{4}+164 x^{3}+151 x^{2}+68 x +12}d x \right )+195 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}{18 x^{5}+87 x^{4}+164 x^{3}+151 x^{2}+68 x +12}d x \right ) x^{2}+325 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}{18 x^{5}+87 x^{4}+164 x^{3}+151 x^{2}+68 x +12}d x \right ) x +130 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}{18 x^{5}+87 x^{4}+164 x^{3}+151 x^{2}+68 x +12}d x \right )}{18 x^{2}+30 x +12} \] Input:
int((5-x)*(3+2*x)^(3/2)/(3*x^2+5*x+2)^(3/2),x)
Output:
( - 8*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x - 46*sqrt(2*x + 3)*sqrt(3*x** 2 + 5*x + 2) - 30*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**2)/(18*x**5 + 87*x**4 + 164*x**3 + 151*x**2 + 68*x + 12),x)*x**2 - 50*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**2)/(18*x**5 + 87*x**4 + 164*x**3 + 151*x**2 + 68*x + 12),x)*x - 20*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**2)/(18 *x**5 + 87*x**4 + 164*x**3 + 151*x**2 + 68*x + 12),x) + 195*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2))/(18*x**5 + 87*x**4 + 164*x**3 + 151*x**2 + 68 *x + 12),x)*x**2 + 325*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2))/(18*x**5 + 87*x**4 + 164*x**3 + 151*x**2 + 68*x + 12),x)*x + 130*int((sqrt(2*x + 3 )*sqrt(3*x**2 + 5*x + 2))/(18*x**5 + 87*x**4 + 164*x**3 + 151*x**2 + 68*x + 12),x))/(6*(3*x**2 + 5*x + 2))