3.23.15 \(\int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=159 \[ -\frac {8}{27} \sqrt {1-2 x} (5 x+3)^{5/2}-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{3 (3 x+2)}+\frac {25}{12} \sqrt {1-2 x} (5 x+3)^{3/2}-\frac {3065 \sqrt {1-2 x} \sqrt {5 x+3}}{1296}-\frac {43 \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{3888}-\frac {181}{243} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {97, 154, 157, 54, 216, 93, 204} \begin {gather*} -\frac {8}{27} \sqrt {1-2 x} (5 x+3)^{5/2}-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{3 (3 x+2)}+\frac {25}{12} \sqrt {1-2 x} (5 x+3)^{3/2}-\frac {3065 \sqrt {1-2 x} \sqrt {5 x+3}}{1296}-\frac {43 \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{3888}-\frac {181}{243} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^2,x]

[Out]

(-3065*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1296 + (25*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/12 - (8*Sqrt[1 - 2*x]*(3 + 5*x)^
(5/2))/27 - ((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(3*(2 + 3*x)) - (43*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/
3888 - (181*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/243

Rule 54

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

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 97

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

Rule 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 204

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

Rule 216

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

Rubi steps

\begin {align*} \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^2} \, dx &=-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{3 (2+3 x)}+\frac {1}{3} \int \frac {\left (\frac {7}{2}-40 x\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{2+3 x} \, dx\\ &=-\frac {8}{27} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{3 (2+3 x)}+\frac {1}{135} \int \frac {\left (\frac {1835}{2}-3375 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {25}{12} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {8}{27} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{3 (2+3 x)}-\frac {\int \frac {\left (-2655-\frac {45975 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx}{1620}\\ &=-\frac {3065 \sqrt {1-2 x} \sqrt {3+5 x}}{1296}+\frac {25}{12} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {8}{27} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{3 (2+3 x)}+\frac {\int \frac {\frac {49605}{2}-\frac {3225 x}{4}}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{9720}\\ &=-\frac {3065 \sqrt {1-2 x} \sqrt {3+5 x}}{1296}+\frac {25}{12} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {8}{27} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{3 (2+3 x)}-\frac {215 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{7776}+\frac {1267}{486} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {3065 \sqrt {1-2 x} \sqrt {3+5 x}}{1296}+\frac {25}{12} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {8}{27} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{3 (2+3 x)}+\frac {1267}{243} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )-\frac {\left (43 \sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{3888}\\ &=-\frac {3065 \sqrt {1-2 x} \sqrt {3+5 x}}{1296}+\frac {25}{12} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {8}{27} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{3 (2+3 x)}-\frac {43 \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{3888}-\frac {181}{243} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.23, size = 135, normalized size = 0.85 \begin {gather*} \frac {-6 \sqrt {-(1-2 x)^2} \sqrt {5 x+3} \left (7200 x^3-1860 x^2-3513 x+730\right )-5792 (3 x+2) \sqrt {14 x-7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )+43 \sqrt {10-20 x} (3 x+2) \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{7776 \sqrt {2 x-1} (3 x+2)} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^2,x]

[Out]

(-6*Sqrt[-(1 - 2*x)^2]*Sqrt[3 + 5*x]*(730 - 3513*x - 1860*x^2 + 7200*x^3) + 43*Sqrt[10 - 20*x]*(2 + 3*x)*ArcSi
nh[Sqrt[5/11]*Sqrt[-1 + 2*x]] - 5792*(2 + 3*x)*Sqrt[-7 + 14*x]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/
(7776*Sqrt[-1 + 2*x]*(2 + 3*x))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.28, size = 178, normalized size = 1.12 \begin {gather*} -\frac {11 \sqrt {1-2 x} \left (\frac {76625 (1-2 x)^3}{(5 x+3)^3}+\frac {449175 (1-2 x)^2}{(5 x+3)^2}-\frac {36084 (1-2 x)}{5 x+3}-4732\right )}{1296 \sqrt {5 x+3} \left (\frac {1-2 x}{5 x+3}+7\right ) \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^3}+\frac {43 \sqrt {\frac {5}{2}} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{3888}-\frac {181}{243} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^2,x]

[Out]

(-11*Sqrt[1 - 2*x]*(-4732 + (76625*(1 - 2*x)^3)/(3 + 5*x)^3 + (449175*(1 - 2*x)^2)/(3 + 5*x)^2 - (36084*(1 - 2
*x))/(3 + 5*x)))/(1296*Sqrt[3 + 5*x]*(7 + (1 - 2*x)/(3 + 5*x))*(2 + (5*(1 - 2*x))/(3 + 5*x))^3) + (43*Sqrt[5/2
]*ArcTan[(Sqrt[5/2]*Sqrt[1 - 2*x])/Sqrt[3 + 5*x]])/3888 - (181*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 +
5*x])])/243

________________________________________________________________________________________

fricas [A]  time = 1.64, size = 137, normalized size = 0.86 \begin {gather*} \frac {43 \, \sqrt {5} \sqrt {2} {\left (3 \, x + 2\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 5792 \, \sqrt {7} {\left (3 \, x + 2\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 12 \, {\left (7200 \, x^{3} - 1860 \, x^{2} - 3513 \, x + 730\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{15552 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^2,x, algorithm="fricas")

[Out]

1/15552*(43*sqrt(5)*sqrt(2)*(3*x + 2)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*
x^2 + x - 3)) - 5792*sqrt(7)*(3*x + 2)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 +
x - 3)) - 12*(7200*x^3 - 1860*x^2 - 3513*x + 730)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(3*x + 2)

________________________________________________________________________________________

giac [B]  time = 2.25, size = 305, normalized size = 1.92 \begin {gather*} \frac {181}{4860} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {1}{2160} \, {\left (4 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} - 85 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 835 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {43}{15552} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {154 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{81 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^2,x, algorithm="giac")

[Out]

181/4860*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^
2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 1/2160*(4*(8*sqrt(5)*(5*x + 3) - 85*sqrt(5))*(5*x +
3) + 835*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) - 43/15552*sqrt(10)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*((sqrt(2
)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 154/81*sqrt(10)*((sqrt
(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt(
2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)

________________________________________________________________________________________

maple [A]  time = 0.02, size = 180, normalized size = 1.13 \begin {gather*} -\frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (86400 \sqrt {-10 x^{2}-x +3}\, x^{3}-22320 \sqrt {-10 x^{2}-x +3}\, x^{2}+129 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-17376 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-42156 \sqrt {-10 x^{2}-x +3}\, x +86 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-11584 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+8760 \sqrt {-10 x^{2}-x +3}\right )}{15552 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x+1)^(3/2)*(5*x+3)^(5/2)/(3*x+2)^2,x)

[Out]

-1/15552*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(86400*(-10*x^2-x+3)^(1/2)*x^3+129*10^(1/2)*x*arcsin(20/11*x+1/11)-17376
*7^(1/2)*x*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-22320*(-10*x^2-x+3)^(1/2)*x^2+86*10^(1/2)*arcsin
(20/11*x+1/11)-11584*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-42156*(-10*x^2-x+3)^(1/2)*x+87
60*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(3*x+2)

________________________________________________________________________________________

maxima [A]  time = 1.36, size = 104, normalized size = 0.65 \begin {gather*} \frac {5}{27} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {245}{108} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {43}{15552} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {181}{486} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {1301}{1296} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{9 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^2,x, algorithm="maxima")

[Out]

5/27*(-10*x^2 - x + 3)^(3/2) + 245/108*sqrt(-10*x^2 - x + 3)*x - 43/15552*sqrt(10)*arcsin(20/11*x + 1/11) + 18
1/486*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 1301/1296*sqrt(-10*x^2 - x + 3) + 1/9*(-10*x
^2 - x + 3)^(3/2)/(3*x + 2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1 - 2*x)^(3/2)*(5*x + 3)^(5/2))/(3*x + 2)^2,x)

[Out]

int(((1 - 2*x)^(3/2)*(5*x + 3)^(5/2))/(3*x + 2)^2, x)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**(3/2)*(3+5*x)**(5/2)/(2+3*x)**2,x)

[Out]

Timed out

________________________________________________________________________________________