3.2438 \(\int \frac{(5-x) (2+5 x+3 x^2)^{5/2}}{(3+2 x)^2} \, dx\)

Optimal. Leaf size=151 \[ -\frac{(x+34) \left (3 x^2+5 x+2\right )^{5/2}}{10 (2 x+3)}-\frac{1}{192} (65-1194 x) \left (3 x^2+5 x+2\right )^{3/2}-\frac{1}{512} (3865-8082 x) \sqrt{3 x^2+5 x+2}+\frac{41053 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{1024 \sqrt{3}}-\frac{1325}{128} \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right ) \]

[Out]

-((3865 - 8082*x)*Sqrt[2 + 5*x + 3*x^2])/512 - ((65 - 1194*x)*(2 + 5*x + 3*x^2)^(3/2))/192 - ((34 + x)*(2 + 5*
x + 3*x^2)^(5/2))/(10*(3 + 2*x)) + (41053*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(1024*Sqrt[3])
- (1325*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/128

________________________________________________________________________________________

Rubi [A]  time = 0.10216, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {812, 814, 843, 621, 206, 724} \[ -\frac{(x+34) \left (3 x^2+5 x+2\right )^{5/2}}{10 (2 x+3)}-\frac{1}{192} (65-1194 x) \left (3 x^2+5 x+2\right )^{3/2}-\frac{1}{512} (3865-8082 x) \sqrt{3 x^2+5 x+2}+\frac{41053 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{1024 \sqrt{3}}-\frac{1325}{128} \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((3865 - 8082*x)*Sqrt[2 + 5*x + 3*x^2])/512 - ((65 - 1194*x)*(2 + 5*x + 3*x^2)^(3/2))/192 - ((34 + x)*(2 + 5*
x + 3*x^2)^(5/2))/(10*(3 + 2*x)) + (41053*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(1024*Sqrt[3])
- (1325*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/128

Rule 812

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

Rule 814

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((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] - Dist[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] && NeQ[b^2 - 4*a*c, 0
] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

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]

Rubi steps

\begin{align*} \int \frac{(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^2} \, dx &=-\frac{(34+x) \left (2+5 x+3 x^2\right )^{5/2}}{10 (3+2 x)}-\frac{1}{8} \int \frac{(-332-398 x) \left (2+5 x+3 x^2\right )^{3/2}}{3+2 x} \, dx\\ &=-\frac{1}{192} (65-1194 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac{(34+x) \left (2+5 x+3 x^2\right )^{5/2}}{10 (3+2 x)}+\frac{1}{768} \int \frac{(40938+48492 x) \sqrt{2+5 x+3 x^2}}{3+2 x} \, dx\\ &=-\frac{1}{512} (3865-8082 x) \sqrt{2+5 x+3 x^2}-\frac{1}{192} (65-1194 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac{(34+x) \left (2+5 x+3 x^2\right )^{5/2}}{10 (3+2 x)}-\frac{\int \frac{-2525724-2955816 x}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{36864}\\ &=-\frac{1}{512} (3865-8082 x) \sqrt{2+5 x+3 x^2}-\frac{1}{192} (65-1194 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac{(34+x) \left (2+5 x+3 x^2\right )^{5/2}}{10 (3+2 x)}+\frac{41053 \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx}{1024}-\frac{6625}{128} \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{1}{512} (3865-8082 x) \sqrt{2+5 x+3 x^2}-\frac{1}{192} (65-1194 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac{(34+x) \left (2+5 x+3 x^2\right )^{5/2}}{10 (3+2 x)}+\frac{41053}{512} \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )+\frac{6625}{64} \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )\\ &=-\frac{1}{512} (3865-8082 x) \sqrt{2+5 x+3 x^2}-\frac{1}{192} (65-1194 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac{(34+x) \left (2+5 x+3 x^2\right )^{5/2}}{10 (3+2 x)}+\frac{41053 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{1024 \sqrt{3}}-\frac{1325}{128} \sqrt{5} \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0976969, size = 120, normalized size = 0.79 \[ \frac{-\frac{2 \sqrt{3 x^2+5 x+2} \left (6912 x^5-28512 x^4-80064 x^3-118996 x^2+40412 x+293973\right )}{2 x+3}+159000 \sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )+205265 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )}{15360} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*Sqrt[2 + 5*x + 3*x^2]*(293973 + 40412*x - 118996*x^2 - 80064*x^3 - 28512*x^4 + 6912*x^5))/(3 + 2*x) + 159
000*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])] + 205265*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6
+ 15*x + 9*x^2])])/15360

________________________________________________________________________________________

Maple [A]  time = 0.01, size = 195, normalized size = 1.3 \begin{align*} -{\frac{53}{20} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}+{\frac{995+1194\,x}{192} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{6735+8082\,x}{512}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}+{\frac{41053\,\sqrt{3}}{3072}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \right ) }-{\frac{265}{48} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{1325}{128}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}+{\frac{1325\,\sqrt{5}}{128}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) }-{\frac{13}{10} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{65+78\,x}{20} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-53/20*(3*(x+3/2)^2-4*x-19/4)^(5/2)+199/192*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(3/2)+1347/512*(5+6*x)*(3*(x+3/2)^2
-4*x-19/4)^(1/2)+41053/3072*ln(1/3*(5/2+3*x)*3^(1/2)+(3*(x+3/2)^2-4*x-19/4)^(1/2))*3^(1/2)-265/48*(3*(x+3/2)^2
-4*x-19/4)^(3/2)-1325/128*(12*(x+3/2)^2-16*x-19)^(1/2)+1325/128*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+
3/2)^2-16*x-19)^(1/2))-13/10/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(7/2)+13/20*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(5/2)

________________________________________________________________________________________

Maxima [A]  time = 1.53539, size = 220, normalized size = 1.46 \begin{align*} -\frac{1}{20} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} + \frac{199}{32} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x - \frac{65}{192} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{4 \,{\left (2 \, x + 3\right )}} + \frac{4041}{256} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{41053}{3072} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) + \frac{1325}{128} \, \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{3865}{512} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/20*(3*x^2 + 5*x + 2)^(5/2) + 199/32*(3*x^2 + 5*x + 2)^(3/2)*x - 65/192*(3*x^2 + 5*x + 2)^(3/2) - 13/4*(3*x^
2 + 5*x + 2)^(5/2)/(2*x + 3) + 4041/256*sqrt(3*x^2 + 5*x + 2)*x + 41053/3072*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 +
5*x + 2) + 3*x + 5/2) + 1325/128*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2
) - 3865/512*sqrt(3*x^2 + 5*x + 2)

________________________________________________________________________________________

Fricas [A]  time = 1.46098, size = 441, normalized size = 2.92 \begin{align*} \frac{205265 \, \sqrt{3}{\left (2 \, x + 3\right )} \log \left (4 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 159000 \, \sqrt{5}{\left (2 \, x + 3\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 ) - 4 \,{\left (6912 \, x^{5} - 28512 \, x^{4} - 80064 \, x^{3} - 118996 \, x^{2} + 40412 \, x + 293973\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{30720 \,{\left (2 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30720*(205265*sqrt(3)*(2*x + 3)*log(4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49) + 15900
0*sqrt(5)*(2*x + 3)*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)
) - 4*(6912*x^5 - 28512*x^4 - 80064*x^3 - 118996*x^2 + 40412*x + 293973)*sqrt(3*x^2 + 5*x + 2))/(2*x + 3)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{20 \sqrt{3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\, dx - \int - \frac{96 x \sqrt{3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\, dx - \int - \frac{165 x^{2} \sqrt{3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\, dx - \int - \frac{113 x^{3} \sqrt{3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\, dx - \int - \frac{15 x^{4} \sqrt{3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\, dx - \int \frac{9 x^{5} \sqrt{3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-20*sqrt(3*x**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x) - Integral(-96*x*sqrt(3*x**2 + 5*x + 2)/(4*x**2 +
12*x + 9), x) - Integral(-165*x**2*sqrt(3*x**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x) - Integral(-113*x**3*sqrt(3
*x**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x) - Integral(-15*x**4*sqrt(3*x**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x) -
Integral(9*x**5*sqrt(3*x**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x)

________________________________________________________________________________________

Giac [B]  time = 1.99492, size = 906, normalized size = 6. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-41053/3072*sqrt(3)*log(abs(-2*sqrt(3) + 2*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + 2*sqrt(5)/(2*x + 3))/abs(2
*sqrt(3) + 2*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + 2*sqrt(5)/(2*x + 3)))*sgn(1/(2*x + 3)) + 1325/128*sqrt(5
)*log(abs(sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3)) - 4))*sgn(1/(2*x + 3)) - 325/12
8*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3)*sgn(1/(2*x + 3)) + 1/7680*(1304805*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2
+ 3) + sqrt(5)/(2*x + 3))^9*sgn(1/(2*x + 3)) - 2064120*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt
(5)/(2*x + 3))^8*sgn(1/(2*x + 3)) - 4382950*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^7*sgn
(1/(2*x + 3)) + 10490640*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^6*sgn(1/(2*x + 3
)) + 19083456*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^5*sgn(1/(2*x + 3)) - 33372000*sqrt(
5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^4*sgn(1/(2*x + 3)) - 42760170*(sqrt(-8/(2*x +
3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^3*sgn(1/(2*x + 3)) + 60102000*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x
+ 3)^2 + 3) + sqrt(5)/(2*x + 3))^2*sgn(1/(2*x + 3)) + 21448395*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt
(5)/(2*x + 3))*sgn(1/(2*x + 3)) - 36498600*sqrt(5)*sgn(1/(2*x + 3)))/((sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3)
+ sqrt(5)/(2*x + 3))^2 - 3)^5