3.2424 \(\int \frac{(5-x) (2+5 x+3 x^2)^{3/2}}{(3+2 x)^2} \, dx\)

Optimal. Leaf size=128 \[ -\frac{(x+21) \left (3 x^2+5 x+2\right )^{3/2}}{6 (2 x+3)}-\frac{1}{96} (361-726 x) \sqrt{3 x^2+5 x+2}+\frac{3743 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{192 \sqrt{3}}-\frac{161}{32} \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right ) \]

[Out]

-((361 - 726*x)*Sqrt[2 + 5*x + 3*x^2])/96 - ((21 + x)*(2 + 5*x + 3*x^2)^(3/2))/(6*(3 + 2*x)) + (3743*ArcTanh[(
5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(192*Sqrt[3]) - (161*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2
+ 5*x + 3*x^2])])/32

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Rubi [A]  time = 0.0825067, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{(x+21) \left (3 x^2+5 x+2\right )^{3/2}}{6 (2 x+3)}-\frac{1}{96} (361-726 x) \sqrt{3 x^2+5 x+2}+\frac{3743 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{192 \sqrt{3}}-\frac{161}{32} \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((361 - 726*x)*Sqrt[2 + 5*x + 3*x^2])/96 - ((21 + x)*(2 + 5*x + 3*x^2)^(3/2))/(6*(3 + 2*x)) + (3743*ArcTanh[(
5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(192*Sqrt[3]) - (161*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2
+ 5*x + 3*x^2])])/32

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 )^{3/2}}{(3+2 x)^2} \, dx &=-\frac{(21+x) \left (2+5 x+3 x^2\right )^{3/2}}{6 (3+2 x)}-\frac{1}{8} \int \frac{(-202-242 x) \sqrt{2+5 x+3 x^2}}{3+2 x} \, dx\\ &=-\frac{1}{96} (361-726 x) \sqrt{2+5 x+3 x^2}-\frac{(21+x) \left (2+5 x+3 x^2\right )^{3/2}}{6 (3+2 x)}+\frac{1}{384} \int \frac{12798+14972 x}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{1}{96} (361-726 x) \sqrt{2+5 x+3 x^2}-\frac{(21+x) \left (2+5 x+3 x^2\right )^{3/2}}{6 (3+2 x)}+\frac{3743}{192} \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx-\frac{805}{32} \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{1}{96} (361-726 x) \sqrt{2+5 x+3 x^2}-\frac{(21+x) \left (2+5 x+3 x^2\right )^{3/2}}{6 (3+2 x)}+\frac{3743}{96} \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )+\frac{805}{16} \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}{96} (361-726 x) \sqrt{2+5 x+3 x^2}-\frac{(21+x) \left (2+5 x+3 x^2\right )^{3/2}}{6 (3+2 x)}+\frac{3743 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{192 \sqrt{3}}-\frac{161}{32} \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 = 0.0758001, size = 110, normalized size = 0.86 \[ \frac{1}{576} \left (-\frac{6 \sqrt{3 x^2+5 x+2} \left (48 x^3-364 x^2+256 x+1755\right )}{2 x+3}+2898 \sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )+3743 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-6*Sqrt[2 + 5*x + 3*x^2]*(1755 + 256*x - 364*x^2 + 48*x^3))/(3 + 2*x) + 2898*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*S
qrt[5]*Sqrt[2 + 5*x + 3*x^2])] + 3743*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/576

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Maple [A]  time = 0.01, size = 158, normalized size = 1.2 \begin{align*} -{\frac{161}{60} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{605+726\,x}{96}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}+{\frac{3743\,\sqrt{3}}{576}\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{161}{32}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}+{\frac{161\,\sqrt{5}}{32}{\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{5}{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{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-161/60*(3*(x+3/2)^2-4*x-19/4)^(3/2)+121/96*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)+3743/576*ln(1/3*(5/2+3*x)*3^(
1/2)+(3*(x+3/2)^2-4*x-19/4)^(1/2))*3^(1/2)-161/32*(12*(x+3/2)^2-16*x-19)^(1/2)+161/32*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)^(5/2)+13/20*(5+6*x)*(3*(x+3/
2)^2-4*x-19/4)^(3/2)

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Maxima [A]  time = 1.52465, size = 181, normalized size = 1.41 \begin{align*} -\frac{1}{12} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} + \frac{121}{16} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{3743}{576} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) + \frac{161}{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{361}{96} \, \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}{4 \,{\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)^(3/2)/(3+2*x)^2,x, algorithm="maxima")

[Out]

-1/12*(3*x^2 + 5*x + 2)^(3/2) + 121/16*sqrt(3*x^2 + 5*x + 2)*x + 3743/576*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x
+ 2) + 3*x + 5/2) + 161/32*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 3
61/96*sqrt(3*x^2 + 5*x + 2) - 13/4*(3*x^2 + 5*x + 2)^(3/2)/(2*x + 3)

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Fricas [A]  time = 1.50778, size = 392, normalized size = 3.06 \begin{align*} \frac{3743 \, \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 ) + 2898 \, \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 (48 \, x^{3} - 364 \, x^{2} + 256 \, x + 1755\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{1152 \,{\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)^(3/2)/(3+2*x)^2,x, algorithm="fricas")

[Out]

1/1152*(3743*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) + 2898*sqr
t(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)) - 1
2*(48*x^3 - 364*x^2 + 256*x + 1755)*sqrt(3*x^2 + 5*x + 2))/(2*x + 3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{10 \sqrt{3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\, dx - \int - \frac{23 x \sqrt{3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\, dx - \int - \frac{10 x^{2} \sqrt{3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\, dx - \int \frac{3 x^{3} \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)**(3/2)/(3+2*x)**2,x)

[Out]

-Integral(-10*sqrt(3*x**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x) - Integral(-23*x*sqrt(3*x**2 + 5*x + 2)/(4*x**2 +
12*x + 9), x) - Integral(-10*x**2*sqrt(3*x**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x) - Integral(3*x**3*sqrt(3*x**
2 + 5*x + 2)/(4*x**2 + 12*x + 9), x)

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Giac [B]  time = 1.80065, size = 649, normalized size = 5.07 \begin{align*} -\frac{3743}{576} \, \sqrt{3} \log \left (\frac{{\left | -2 \, \sqrt{3} + 2 \, \sqrt{-\frac{8}{2 \, x + 3} + \frac{5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{2 \, \sqrt{5}}{2 \, x + 3} \right |}}{{\left | 2 \, \sqrt{3} + 2 \, \sqrt{-\frac{8}{2 \, x + 3} + \frac{5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{2 \, \sqrt{5}}{2 \, x + 3} \right |}}\right ) \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right ) + \frac{161}{32} \, \sqrt{5} \log \left ({\left | \sqrt{5}{\left (\sqrt{-\frac{8}{2 \, x + 3} + \frac{5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{5}}{2 \, x + 3}\right )} - 4 \right |}\right ) \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right ) - \frac{65}{32} \, \sqrt{-\frac{8}{2 \, x + 3} + \frac{5}{{\left (2 \, x + 3\right )}^{2}} + 3} \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right ) + \frac{4069 \,{\left (\sqrt{-\frac{8}{2 \, x + 3} + \frac{5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{5}}{2 \, x + 3}\right )}^{5} \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right ) - 4308 \, \sqrt{5}{\left (\sqrt{-\frac{8}{2 \, x + 3} + \frac{5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{5}}{2 \, x + 3}\right )}^{4} \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right ) - 14464 \,{\left (\sqrt{-\frac{8}{2 \, x + 3} + \frac{5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{5}}{2 \, x + 3}\right )}^{3} \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right ) + 17388 \, \sqrt{5}{\left (\sqrt{-\frac{8}{2 \, x + 3} + \frac{5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{5}}{2 \, x + 3}\right )}^{2} \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right ) + 12627 \,{\left (\sqrt{-\frac{8}{2 \, x + 3} + \frac{5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{5}}{2 \, x + 3}\right )} \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right ) - 17928 \, \sqrt{5} \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right )}{96 \,{\left ({\left (\sqrt{-\frac{8}{2 \, x + 3} + \frac{5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{5}}{2 \, x + 3}\right )}^{2} - 3\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3743/576*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*s
qrt(3) + 2*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + 2*sqrt(5)/(2*x + 3)))*sgn(1/(2*x + 3)) + 161/32*sqrt(5)*lo
g(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)) - 65/32*sqrt
(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3)*sgn(1/(2*x + 3)) + 1/96*(4069*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqr
t(5)/(2*x + 3))^5*sgn(1/(2*x + 3)) - 4308*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))
^4*sgn(1/(2*x + 3)) - 14464*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^3*sgn(1/(2*x + 3)) +
17388*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^2*sgn(1/(2*x + 3)) + 12627*(sqrt(-8
/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))*sgn(1/(2*x + 3)) - 17928*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)^3