3.2453 \(\int \frac{(5-x) (2+5 x+3 x^2)^{7/2}}{(3+2 x)^2} \, dx\)

Optimal. Leaf size=174 \[ -\frac{(x+47) \left (3 x^2+5 x+2\right )^{7/2}}{14 (2 x+3)}+\frac{(8310 x+283) \left (3 x^2+5 x+2\right )^{5/2}}{1440}-\frac{(6925-151098 x) \left (3 x^2+5 x+2\right )^{3/2}}{13824}-\frac{(1454315-3037062 x) \sqrt{3 x^2+5 x+2}}{110592}+\frac{15434623 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{221184 \sqrt{3}}-\frac{9225}{512} \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right ) \]

[Out]

-((1454315 - 3037062*x)*Sqrt[2 + 5*x + 3*x^2])/110592 - ((6925 - 151098*x)*(2 + 5*x + 3*x^2)^(3/2))/13824 + ((
283 + 8310*x)*(2 + 5*x + 3*x^2)^(5/2))/1440 - ((47 + x)*(2 + 5*x + 3*x^2)^(7/2))/(14*(3 + 2*x)) + (15434623*Ar
cTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(221184*Sqrt[3]) - (9225*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt
[5]*Sqrt[2 + 5*x + 3*x^2])])/512

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Rubi [A]  time = 0.122866, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{(x+47) \left (3 x^2+5 x+2\right )^{7/2}}{14 (2 x+3)}+\frac{(8310 x+283) \left (3 x^2+5 x+2\right )^{5/2}}{1440}-\frac{(6925-151098 x) \left (3 x^2+5 x+2\right )^{3/2}}{13824}-\frac{(1454315-3037062 x) \sqrt{3 x^2+5 x+2}}{110592}+\frac{15434623 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{221184 \sqrt{3}}-\frac{9225}{512} \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^2,x]

[Out]

-((1454315 - 3037062*x)*Sqrt[2 + 5*x + 3*x^2])/110592 - ((6925 - 151098*x)*(2 + 5*x + 3*x^2)^(3/2))/13824 + ((
283 + 8310*x)*(2 + 5*x + 3*x^2)^(5/2))/1440 - ((47 + x)*(2 + 5*x + 3*x^2)^(7/2))/(14*(3 + 2*x)) + (15434623*Ar
cTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(221184*Sqrt[3]) - (9225*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt
[5]*Sqrt[2 + 5*x + 3*x^2])])/512

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 )^{7/2}}{(3+2 x)^2} \, dx &=-\frac{(47+x) \left (2+5 x+3 x^2\right )^{7/2}}{14 (3+2 x)}-\frac{1}{8} \int \frac{(-462-554 x) \left (2+5 x+3 x^2\right )^{5/2}}{3+2 x} \, dx\\ &=\frac{(283+8310 x) \left (2+5 x+3 x^2\right )^{5/2}}{1440}-\frac{(47+x) \left (2+5 x+3 x^2\right )^{7/2}}{14 (3+2 x)}+\frac{\int \frac{(84678+100732 x) \left (2+5 x+3 x^2\right )^{3/2}}{3+2 x} \, dx}{1152}\\ &=-\frac{(6925-151098 x) \left (2+5 x+3 x^2\right )^{3/2}}{13824}+\frac{(283+8310 x) \left (2+5 x+3 x^2\right )^{5/2}}{1440}-\frac{(47+x) \left (2+5 x+3 x^2\right )^{7/2}}{14 (3+2 x)}-\frac{\int \frac{(-10251972-12148248 x) \sqrt{2+5 x+3 x^2}}{3+2 x} \, dx}{110592}\\ &=-\frac{(1454315-3037062 x) \sqrt{2+5 x+3 x^2}}{110592}-\frac{(6925-151098 x) \left (2+5 x+3 x^2\right )^{3/2}}{13824}+\frac{(283+8310 x) \left (2+5 x+3 x^2\right )^{5/2}}{1440}-\frac{(47+x) \left (2+5 x+3 x^2\right )^{7/2}}{14 (3+2 x)}+\frac{\int \frac{633068856+740861904 x}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{5308416}\\ &=-\frac{(1454315-3037062 x) \sqrt{2+5 x+3 x^2}}{110592}-\frac{(6925-151098 x) \left (2+5 x+3 x^2\right )^{3/2}}{13824}+\frac{(283+8310 x) \left (2+5 x+3 x^2\right )^{5/2}}{1440}-\frac{(47+x) \left (2+5 x+3 x^2\right )^{7/2}}{14 (3+2 x)}+\frac{15434623 \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx}{221184}-\frac{46125}{512} \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{(1454315-3037062 x) \sqrt{2+5 x+3 x^2}}{110592}-\frac{(6925-151098 x) \left (2+5 x+3 x^2\right )^{3/2}}{13824}+\frac{(283+8310 x) \left (2+5 x+3 x^2\right )^{5/2}}{1440}-\frac{(47+x) \left (2+5 x+3 x^2\right )^{7/2}}{14 (3+2 x)}+\frac{15434623 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )}{110592}+\frac{46125}{256} \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )\\ &=-\frac{(1454315-3037062 x) \sqrt{2+5 x+3 x^2}}{110592}-\frac{(6925-151098 x) \left (2+5 x+3 x^2\right )^{3/2}}{13824}+\frac{(283+8310 x) \left (2+5 x+3 x^2\right )^{5/2}}{1440}-\frac{(47+x) \left (2+5 x+3 x^2\right )^{7/2}}{14 (3+2 x)}+\frac{15434623 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{221184 \sqrt{3}}-\frac{9225}{512} \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 = 0.117456, size = 130, normalized size = 0.75 \[ \frac{-\frac{6 \sqrt{3 x^2+5 x+2} \left (7464960 x^7-13893120 x^6-125632512 x^5-273531168 x^4-275126016 x^3-179819084 x^2+28017108 x+259165107\right )}{2 x+3}[phone] \sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )[phone] \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )}{23224320} \]

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^2,x]

[Out]

((-6*Sqrt[2 + 5*x + 3*x^2]*(259165107 + 28017108*x - 179819084*x^2 - 275126016*x^3 - 273531168*x^4 - 125632512
*x^5 - 13893120*x^6 + 7464960*x^7))/(3 + 2*x) + 418446000*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x +
3*x^2])] + 540211805*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/23224320

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Maple [A]  time = 0.01, size = 232, normalized size = 1.3 \begin{align*} -{\frac{369}{140} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}}}+{\frac{1385+1662\,x}{288} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}+{\frac{125915+151098\,x}{13824} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{2530885+3037062\,x}{110592}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}+{\frac{15434623\,\sqrt{3}}{663552}\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{369}{80} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{615}{64} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{9225}{512}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}+{\frac{9225\,\sqrt{5}}{512}{\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{9}{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{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^2,x)

[Out]

-369/140*(3*(x+3/2)^2-4*x-19/4)^(7/2)+277/288*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(5/2)+25183/13824*(5+6*x)*(3*(x+3
/2)^2-4*x-19/4)^(3/2)+506177/110592*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)[phone]/663552*ln(1/3*(5/2+3*x)*3^(1
/2)+(3*(x+3/2)^2-4*x-19/4)^(1/2))*3^(1/2)-369/80*(3*(x+3/2)^2-4*x-19/4)^(5/2)-615/64*(3*(x+3/2)^2-4*x-19/4)^(3
/2)-9225/512*(12*(x+3/2)^2-16*x-19)^(1/2)+9225/512*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-1
9)^(1/2))-13/10/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(9/2)+13/20*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(7/2)

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Maxima [A]  time = 2.20814, size = 259, normalized size = 1.49 \begin{align*} -\frac{1}{28} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} + \frac{277}{48} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x + \frac{283}{1440} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{4 \,{\left (2 \, x + 3\right )}} + \frac{25183}{2304} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x - \frac{6925}{13824} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} + \frac{506177}{18432} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{15434623}{663552} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) + \frac{9225}{512} \, \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{1454315}{110592} \, \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)^(7/2)/(3+2*x)^2,x, algorithm="maxima")

[Out]

-1/28*(3*x^2 + 5*x + 2)^(7/2) + 277/48*(3*x^2 + 5*x + 2)^(5/2)*x + 283/1440*(3*x^2 + 5*x + 2)^(5/2) - 13/4*(3*
x^2 + 5*x + 2)^(7/2)/(2*x + 3) + 25183/2304*(3*x^2 + 5*x + 2)^(3/2)*x - 6925/13824*(3*x^2 + 5*x + 2)^(3/2) + 5
06177/18432*sqrt(3*x^2 + 5*x + 2)*x + 15434623/663552*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) +
9225/512*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 1454315/110592*sqrt
(3*x^2 + 5*x + 2)

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Fricas [A]  time = 1.38397, size = 524, normalized size = 3.01 \begin{align*} \frac{540211805 \, \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 ) + 418446000 \, \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 ) - 12 \,{\left (7464960 \, x^{7} - 13893120 \, x^{6} - 125632512 \, x^{5} - 273531168 \, x^{4} - 275126016 \, x^{3} - 179819084 \, x^{2} + 28017108 \, x + 259165107\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{46448640 \,{\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)^(7/2)/(3+2*x)^2,x, algorithm="fricas")

[Out]

1/46448640*(540211805*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) +
418446000*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)) - 12*(7464960*x^7 - 13893120*x^6 - 125632512*x^5 - 273531168*x^4 - 275126016*x^3 - 179819084*x^2 +
28017108*x + 259165107)*sqrt(3*x^2 + 5*x + 2))/(2*x + 3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x**2+5*x+2)**(7/2)/(3+2*x)**2,x)

[Out]

Timed out

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Giac [B]  time = 2.47552, size = 1162, normalized size = 6.68 \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)^(7/2)/(3+2*x)^2,x, algorithm="giac")

[Out]

-15434623/663552*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)) + 9225/512*s
qrt(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)) - 1
625/512*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3)*sgn(1/(2*x + 3)) + 1/3870720*(1702084195*(sqrt(-8/(2*x + 3) + 5
/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^13*sgn(1/(2*x + 3)) - 3595838400*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x +
3)^2 + 3) + sqrt(5)/(2*x + 3))^12*sgn(1/(2*x + 3)) - 462583100*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(
5)/(2*x + 3))^11*sgn(1/(2*x + 3)) + 12803555520*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x
+ 3))^10*sgn(1/(2*x + 3)) + 91554292599*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^9*sgn(1/
(2*x + 3)) - 132950643840*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^8*sgn(1/(2*x +
3)) - 221215739904*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^7*sgn(1/(2*x + 3)) + 432202780
800*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^6*sgn(1/(2*x + 3)) + 252015304401*(sq
rt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^5*sgn(1/(2*x + 3)) - 680038027200*sqrt(5)*(sqrt(-8/(
2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^4*sgn(1/(2*x + 3)) - 506502404100*(sqrt(-8/(2*x + 3) + 5/(2
*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^3*sgn(1/(2*x + 3)) + 786343723200*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)
^2 + 3) + sqrt(5)/(2*x + 3))^2*sgn(1/(2*x + 3)) + 178876045845*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(
5)/(2*x + 3))*sgn(1/(2*x + 3)) - 339366412800*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)^7