3.2523 \(\int \frac {5-x}{(3+2 x)^3 (2+5 x+3 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=147 \[ -\frac {2 (47 x+37)}{5 (2 x+3)^2 \left (3 x^2+5 x+2\right )^{3/2}}+\frac {11808 \sqrt {3 x^2+5 x+2}}{125 (2 x+3)}+\frac {152 \sqrt {3 x^2+5 x+2}}{(2 x+3)^2}+\frac {4 (2112 x+1907)}{25 (2 x+3)^2 \sqrt {3 x^2+5 x+2}}+\frac {4884 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{125 \sqrt {5}} \]

[Out]

-2/5*(37+47*x)/(3+2*x)^2/(3*x^2+5*x+2)^(3/2)+4884/625*arctanh(1/10*(7+8*x)*5^(1/2)/(3*x^2+5*x+2)^(1/2))*5^(1/2
)+4/25*(1907+2112*x)/(3+2*x)^2/(3*x^2+5*x+2)^(1/2)+152*(3*x^2+5*x+2)^(1/2)/(3+2*x)^2+11808/125*(3*x^2+5*x+2)^(
1/2)/(3+2*x)

________________________________________________________________________________________

Rubi [A]  time = 0.09, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {822, 834, 806, 724, 206} \[ -\frac {2 (47 x+37)}{5 (2 x+3)^2 \left (3 x^2+5 x+2\right )^{3/2}}+\frac {11808 \sqrt {3 x^2+5 x+2}}{125 (2 x+3)}+\frac {152 \sqrt {3 x^2+5 x+2}}{(2 x+3)^2}+\frac {4 (2112 x+1907)}{25 (2 x+3)^2 \sqrt {3 x^2+5 x+2}}+\frac {4884 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{125 \sqrt {5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(37 + 47*x))/(5*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^(3/2)) + (4*(1907 + 2112*x))/(25*(3 + 2*x)^2*Sqrt[2 + 5*x +
3*x^2]) + (152*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x)^2 + (11808*Sqrt[2 + 5*x + 3*x^2])/(125*(3 + 2*x)) + (4884*ArcT
anh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(125*Sqrt[5])

Rule 206

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

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-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] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rubi steps

\begin {align*} \int \frac {5-x}{(3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}} \, dx &=-\frac {2 (37+47 x)}{5 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {2}{15} \int \frac {1251+1128 x}{(3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac {2 (37+47 x)}{5 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (1907+2112 x)}{25 (3+2 x)^2 \sqrt {2+5 x+3 x^2}}+\frac {4}{75} \int \frac {23766+25344 x}{(3+2 x)^3 \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 (37+47 x)}{5 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (1907+2112 x)}{25 (3+2 x)^2 \sqrt {2+5 x+3 x^2}}+\frac {152 \sqrt {2+5 x+3 x^2}}{(3+2 x)^2}-\frac {2}{375} \int \frac {-83970-85500 x}{(3+2 x)^2 \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 (37+47 x)}{5 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (1907+2112 x)}{25 (3+2 x)^2 \sqrt {2+5 x+3 x^2}}+\frac {152 \sqrt {2+5 x+3 x^2}}{(3+2 x)^2}+\frac {11808 \sqrt {2+5 x+3 x^2}}{125 (3+2 x)}+\frac {4884}{125} \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 (37+47 x)}{5 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (1907+2112 x)}{25 (3+2 x)^2 \sqrt {2+5 x+3 x^2}}+\frac {152 \sqrt {2+5 x+3 x^2}}{(3+2 x)^2}+\frac {11808 \sqrt {2+5 x+3 x^2}}{125 (3+2 x)}-\frac {9768}{125} \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=-\frac {2 (37+47 x)}{5 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (1907+2112 x)}{25 (3+2 x)^2 \sqrt {2+5 x+3 x^2}}+\frac {152 \sqrt {2+5 x+3 x^2}}{(3+2 x)^2}+\frac {11808 \sqrt {2+5 x+3 x^2}}{125 (3+2 x)}+\frac {4884 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{125 \sqrt {5}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.08, size = 143, normalized size = 0.97 \[ \frac {2 \left (142500 \left (3 x^2+5 x+2\right )^2+50 (6336 x+5721) \left (3 x^2+5 x+2\right )+18 (2 x+3) \left (3 x^2+5 x+2\right )^{3/2} \left (4920 \sqrt {3 x^2+5 x+2}-407 \sqrt {5} (2 x+3) \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )\right )-375 (47 x+37)\right )}{1875 (2 x+3)^2 \left (3 x^2+5 x+2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-375*(37 + 47*x) + 50*(5721 + 6336*x)*(2 + 5*x + 3*x^2) + 142500*(2 + 5*x + 3*x^2)^2 + 18*(3 + 2*x)*(2 + 5
*x + 3*x^2)^(3/2)*(4920*Sqrt[2 + 5*x + 3*x^2] - 407*Sqrt[5]*(3 + 2*x)*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5
*x + 3*x^2])])))/(1875*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^(3/2))

________________________________________________________________________________________

fricas [A]  time = 0.83, size = 155, normalized size = 1.05 \[ \frac {2 \, {\left (1221 \, \sqrt {5} {\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 5 \, {\left (106272 \, x^{5} + 599148 \, x^{4} + 1316616 \, x^{3} + 1405814 \, x^{2} + 727887 \, x + 146063\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}}{625 \, {\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/625*(1221*sqrt(5)*(36*x^6 + 228*x^5 + 589*x^4 + 794*x^3 + 589*x^2 + 228*x + 36)*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)) + 5*(106272*x^5 + 599148*x^4 + 1316616*x^3 + 14
05814*x^2 + 727887*x + 146063)*sqrt(3*x^2 + 5*x + 2))/(36*x^6 + 228*x^5 + 589*x^4 + 794*x^3 + 589*x^2 + 228*x
+ 36)

________________________________________________________________________________________

giac [A]  time = 0.32, size = 234, normalized size = 1.59 \[ \frac {4884}{625} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac {2 \, {\left ({\left (6 \, {\left (23826 \, x + 61591\right )} x + 309599\right )} x + 84259\right )}}{625 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {8 \, {\left (4106 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 16447 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 57729 \, \sqrt {3} x + 20987 \, \sqrt {3} - 57729 \, \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}}{625 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4884/625*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))) + 2/625*((6*(23826*x + 61591)*x + 309599)*x + 84259)/(3*x^2 +
5*x + 2)^(3/2) - 8/625*(4106*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 16447*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5
*x + 2))^2 + 57729*sqrt(3)*x + 20987*sqrt(3) - 57729*sqrt(3*x^2 + 5*x + 2))/(2*(sqrt(3)*x - sqrt(3*x^2 + 5*x +
2))^2 + 6*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) + 11)^2

________________________________________________________________________________________

maple [A]  time = 0.01, size = 148, normalized size = 1.01 \[ -\frac {4884 \sqrt {5}\, \arctanh \left (\frac {2 \left (-4 x -\frac {7}{2}\right ) \sqrt {5}}{5 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{625}-\frac {177}{50 \left (x +\frac {3}{2}\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}+\frac {407}{50 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}-\frac {106 \left (6 x +5\right )}{25 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}+\frac {\frac {17712 x}{125}+\frac {2952}{25}}{\sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}+\frac {2442}{125 \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}-\frac {13}{40 \left (x +\frac {3}{2}\right )^{2} \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-177/50/(x+3/2)/(-4*x+3*(x+3/2)^2-19/4)^(3/2)+407/50/(-4*x+3*(x+3/2)^2-19/4)^(3/2)-106/25*(6*x+5)/(-4*x+3*(x+3
/2)^2-19/4)^(3/2)+2952/125*(6*x+5)/(-4*x+3*(x+3/2)^2-19/4)^(1/2)+2442/125/(-4*x+3*(x+3/2)^2-19/4)^(1/2)-4884/6
25*5^(1/2)*arctanh(2/5*(-4*x-7/2)*5^(1/2)/(-16*x+12*(x+3/2)^2-19)^(1/2))-13/40/(x+3/2)^2/(-4*x+3*(x+3/2)^2-19/
4)^(3/2)

________________________________________________________________________________________

maxima [A]  time = 1.24, size = 186, normalized size = 1.27 \[ -\frac {4884}{625} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) + \frac {17712 \, x}{125 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} + \frac {17202}{125 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {636 \, x}{25 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {13}{10 \, {\left (4 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + 9 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}\right )}} - \frac {177}{25 \, {\left (2 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + 3 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}\right )}} - \frac {653}{50 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4884/625*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 17712/125*x/sqrt(3*
x^2 + 5*x + 2) + 17202/125/sqrt(3*x^2 + 5*x + 2) - 636/25*x/(3*x^2 + 5*x + 2)^(3/2) - 13/10/(4*(3*x^2 + 5*x +
2)^(3/2)*x^2 + 12*(3*x^2 + 5*x + 2)^(3/2)*x + 9*(3*x^2 + 5*x + 2)^(3/2)) - 177/25/(2*(3*x^2 + 5*x + 2)^(3/2)*x
+ 3*(3*x^2 + 5*x + 2)^(3/2)) - 653/50/(3*x^2 + 5*x + 2)^(3/2)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {x-5}{{\left (2\,x+3\right )}^3\,{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {x}{72 x^{7} \sqrt {3 x^{2} + 5 x + 2} + 564 x^{6} \sqrt {3 x^{2} + 5 x + 2} + 1862 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 3355 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 3560 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 2223 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 756 x \sqrt {3 x^{2} + 5 x + 2} + 108 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{72 x^{7} \sqrt {3 x^{2} + 5 x + 2} + 564 x^{6} \sqrt {3 x^{2} + 5 x + 2} + 1862 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 3355 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 3560 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 2223 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 756 x \sqrt {3 x^{2} + 5 x + 2} + 108 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(x/(72*x**7*sqrt(3*x**2 + 5*x + 2) + 564*x**6*sqrt(3*x**2 + 5*x + 2) + 1862*x**5*sqrt(3*x**2 + 5*x +
2) + 3355*x**4*sqrt(3*x**2 + 5*x + 2) + 3560*x**3*sqrt(3*x**2 + 5*x + 2) + 2223*x**2*sqrt(3*x**2 + 5*x + 2) +
756*x*sqrt(3*x**2 + 5*x + 2) + 108*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-5/(72*x**7*sqrt(3*x**2 + 5*x + 2) +
564*x**6*sqrt(3*x**2 + 5*x + 2) + 1862*x**5*sqrt(3*x**2 + 5*x + 2) + 3355*x**4*sqrt(3*x**2 + 5*x + 2) + 3560*
x**3*sqrt(3*x**2 + 5*x + 2) + 2223*x**2*sqrt(3*x**2 + 5*x + 2) + 756*x*sqrt(3*x**2 + 5*x + 2) + 108*sqrt(3*x**
2 + 5*x + 2)), x)

________________________________________________________________________________________