3.296 \(\int (d+e x)^2 (b x+c x^2)^{3/2} \, dx\)

Optimal. Leaf size=214 \[ -\frac{b^2 (b+2 c x) \sqrt{b x+c x^2} \left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right )}{512 c^4}+\frac{(b+2 c x) \left (b x+c x^2\right )^{3/2} \left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right )}{192 c^3}+\frac{b^4 \left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{512 c^{9/2}}+\frac{7 e \left (b x+c x^2\right )^{5/2} (2 c d-b e)}{60 c^2}+\frac{e \left (b x+c x^2\right )^{5/2} (d+e x)}{6 c} \]

[Out]

-(b^2*(24*c^2*d^2 - 24*b*c*d*e + 7*b^2*e^2)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(512*c^4) + ((24*c^2*d^2 - 24*b*c*d
*e + 7*b^2*e^2)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(192*c^3) + (7*e*(2*c*d - b*e)*(b*x + c*x^2)^(5/2))/(60*c^2)
+ (e*(d + e*x)*(b*x + c*x^2)^(5/2))/(6*c) + (b^4*(24*c^2*d^2 - 24*b*c*d*e + 7*b^2*e^2)*ArcTanh[(Sqrt[c]*x)/Sqr
t[b*x + c*x^2]])/(512*c^(9/2))

________________________________________________________________________________________

Rubi [A]  time = 0.179299, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {742, 640, 612, 620, 206} \[ -\frac{b^2 (b+2 c x) \sqrt{b x+c x^2} \left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right )}{512 c^4}+\frac{(b+2 c x) \left (b x+c x^2\right )^{3/2} \left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right )}{192 c^3}+\frac{b^4 \left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{512 c^{9/2}}+\frac{7 e \left (b x+c x^2\right )^{5/2} (2 c d-b e)}{60 c^2}+\frac{e \left (b x+c x^2\right )^{5/2} (d+e x)}{6 c} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2*(b*x + c*x^2)^(3/2),x]

[Out]

-(b^2*(24*c^2*d^2 - 24*b*c*d*e + 7*b^2*e^2)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(512*c^4) + ((24*c^2*d^2 - 24*b*c*d
*e + 7*b^2*e^2)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(192*c^3) + (7*e*(2*c*d - b*e)*(b*x + c*x^2)^(5/2))/(60*c^2)
+ (e*(d + e*x)*(b*x + c*x^2)^(5/2))/(6*c) + (b^4*(24*c^2*d^2 - 24*b*c*d*e + 7*b^2*e^2)*ArcTanh[(Sqrt[c]*x)/Sqr
t[b*x + c*x^2]])/(512*c^(9/2))

Rule 742

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2
*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]*(a + b*x + c*x^2)^p, x], x] /;
FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]
&& If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m,
p, x]

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 612

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

Rule 620

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

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])

Rubi steps

\begin{align*} \int (d+e x)^2 \left (b x+c x^2\right )^{3/2} \, dx &=\frac{e (d+e x) \left (b x+c x^2\right )^{5/2}}{6 c}+\frac{\int \left (\frac{1}{2} d (12 c d-5 b e)+\frac{7}{2} e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2} \, dx}{6 c}\\ &=\frac{7 e (2 c d-b e) \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac{e (d+e x) \left (b x+c x^2\right )^{5/2}}{6 c}+\frac{\left (c d (12 c d-5 b e)-\frac{7}{2} b e (2 c d-b e)\right ) \int \left (b x+c x^2\right )^{3/2} \, dx}{12 c^2}\\ &=\frac{\left (24 c^2 d^2-24 b c d e+7 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}+\frac{7 e (2 c d-b e) \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac{e (d+e x) \left (b x+c x^2\right )^{5/2}}{6 c}-\frac{\left (b^2 \left (24 c^2 d^2-24 b c d e+7 b^2 e^2\right )\right ) \int \sqrt{b x+c x^2} \, dx}{128 c^3}\\ &=-\frac{b^2 \left (24 c^2 d^2-24 b c d e+7 b^2 e^2\right ) (b+2 c x) \sqrt{b x+c x^2}}{512 c^4}+\frac{\left (24 c^2 d^2-24 b c d e+7 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}+\frac{7 e (2 c d-b e) \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac{e (d+e x) \left (b x+c x^2\right )^{5/2}}{6 c}+\frac{\left (b^4 \left (24 c^2 d^2-24 b c d e+7 b^2 e^2\right )\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{1024 c^4}\\ &=-\frac{b^2 \left (24 c^2 d^2-24 b c d e+7 b^2 e^2\right ) (b+2 c x) \sqrt{b x+c x^2}}{512 c^4}+\frac{\left (24 c^2 d^2-24 b c d e+7 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}+\frac{7 e (2 c d-b e) \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac{e (d+e x) \left (b x+c x^2\right )^{5/2}}{6 c}+\frac{\left (b^4 \left (24 c^2 d^2-24 b c d e+7 b^2 e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{512 c^4}\\ &=-\frac{b^2 \left (24 c^2 d^2-24 b c d e+7 b^2 e^2\right ) (b+2 c x) \sqrt{b x+c x^2}}{512 c^4}+\frac{\left (24 c^2 d^2-24 b c d e+7 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}+\frac{7 e (2 c d-b e) \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac{e (d+e x) \left (b x+c x^2\right )^{5/2}}{6 c}+\frac{b^4 \left (24 c^2 d^2-24 b c d e+7 b^2 e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{512 c^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.464965, size = 197, normalized size = 0.92 \[ \frac{(x (b+c x))^{3/2} \left (\frac{\left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right ) \left (\sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \left (2 b^2 c x-3 b^3+24 b c^2 x^2+16 c^3 x^3\right )+3 b^{7/2} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )\right )}{256 c^{7/2} (b+c x) \sqrt{\frac{c x}{b}+1}}+\frac{7 e x^{5/2} (b+c x) (2 c d-b e)}{10 c}+e x^{5/2} (b+c x) (d+e x)\right )}{6 c x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^2*(b*x + c*x^2)^(3/2),x]

[Out]

((x*(b + c*x))^(3/2)*((7*e*(2*c*d - b*e)*x^(5/2)*(b + c*x))/(10*c) + e*x^(5/2)*(b + c*x)*(d + e*x) + ((24*c^2*
d^2 - 24*b*c*d*e + 7*b^2*e^2)*(Sqrt[c]*Sqrt[x]*Sqrt[1 + (c*x)/b]*(-3*b^3 + 2*b^2*c*x + 24*b*c^2*x^2 + 16*c^3*x
^3) + 3*b^(7/2)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]]))/(256*c^(7/2)*(b + c*x)*Sqrt[1 + (c*x)/b])))/(6*c*x^(3/2))

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Maple [B]  time = 0.055, size = 420, normalized size = 2. \begin{align*}{\frac{{e}^{2}x}{6\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{7\,{e}^{2}b}{60\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{7\,{b}^{2}{e}^{2}x}{96\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{b}^{3}{e}^{2}}{192\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{7\,{e}^{2}{b}^{4}x}{256\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{7\,{e}^{2}{b}^{5}}{512\,{c}^{4}}\sqrt{c{x}^{2}+bx}}+{\frac{7\,{e}^{2}{b}^{6}}{1024}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{9}{2}}}}+{\frac{2\,de}{5\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{bdex}{4\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{2}de}{8\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{3\,de{b}^{3}x}{32\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,de{b}^{4}}{64\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{3\,de{b}^{5}}{128}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}}+{\frac{{d}^{2}x}{4} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{{d}^{2}b}{8\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{b}^{2}{d}^{2}x}{32\,c}\sqrt{c{x}^{2}+bx}}-{\frac{3\,{d}^{2}{b}^{3}}{64\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,{d}^{2}{b}^{4}}{128}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2*(c*x^2+b*x)^(3/2),x)

[Out]

1/6*e^2*x*(c*x^2+b*x)^(5/2)/c-7/60*e^2*b/c^2*(c*x^2+b*x)^(5/2)+7/96*e^2*b^2/c^2*x*(c*x^2+b*x)^(3/2)+7/192*e^2*
b^3/c^3*(c*x^2+b*x)^(3/2)-7/256*e^2*b^4/c^3*(c*x^2+b*x)^(1/2)*x-7/512*e^2*b^5/c^4*(c*x^2+b*x)^(1/2)+7/1024*e^2
*b^6/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+2/5*d*e*(c*x^2+b*x)^(5/2)/c-1/4*d*e*b/c*x*(c*x^2+b*x)^(
3/2)-1/8*d*e*b^2/c^2*(c*x^2+b*x)^(3/2)+3/32*d*e*b^3/c^2*(c*x^2+b*x)^(1/2)*x+3/64*d*e*b^4/c^3*(c*x^2+b*x)^(1/2)
-3/128*d*e*b^5/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+1/4*d^2*x*(c*x^2+b*x)^(3/2)+1/8*d^2/c*(c*x^2+
b*x)^(3/2)*b-3/32*d^2*b^2/c*(c*x^2+b*x)^(1/2)*x-3/64*d^2*b^3/c^2*(c*x^2+b*x)^(1/2)+3/128*d^2*b^4/c^(5/2)*ln((1
/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(c*x^2+b*x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.32955, size = 1108, normalized size = 5.18 \begin{align*} \left [\frac{15 \,{\left (24 \, b^{4} c^{2} d^{2} - 24 \, b^{5} c d e + 7 \, b^{6} e^{2}\right )} \sqrt{c} \log \left (2 \, c x + b + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) + 2 \,{\left (1280 \, c^{6} e^{2} x^{5} - 360 \, b^{3} c^{3} d^{2} + 360 \, b^{4} c^{2} d e - 105 \, b^{5} c e^{2} + 128 \,{\left (24 \, c^{6} d e + 13 \, b c^{5} e^{2}\right )} x^{4} + 48 \,{\left (40 \, c^{6} d^{2} + 88 \, b c^{5} d e + b^{2} c^{4} e^{2}\right )} x^{3} + 8 \,{\left (360 \, b c^{5} d^{2} + 24 \, b^{2} c^{4} d e - 7 \, b^{3} c^{3} e^{2}\right )} x^{2} + 10 \,{\left (24 \, b^{2} c^{4} d^{2} - 24 \, b^{3} c^{3} d e + 7 \, b^{4} c^{2} e^{2}\right )} x\right )} \sqrt{c x^{2} + b x}}{15360 \, c^{5}}, -\frac{15 \,{\left (24 \, b^{4} c^{2} d^{2} - 24 \, b^{5} c d e + 7 \, b^{6} e^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (1280 \, c^{6} e^{2} x^{5} - 360 \, b^{3} c^{3} d^{2} + 360 \, b^{4} c^{2} d e - 105 \, b^{5} c e^{2} + 128 \,{\left (24 \, c^{6} d e + 13 \, b c^{5} e^{2}\right )} x^{4} + 48 \,{\left (40 \, c^{6} d^{2} + 88 \, b c^{5} d e + b^{2} c^{4} e^{2}\right )} x^{3} + 8 \,{\left (360 \, b c^{5} d^{2} + 24 \, b^{2} c^{4} d e - 7 \, b^{3} c^{3} e^{2}\right )} x^{2} + 10 \,{\left (24 \, b^{2} c^{4} d^{2} - 24 \, b^{3} c^{3} d e + 7 \, b^{4} c^{2} e^{2}\right )} x\right )} \sqrt{c x^{2} + b x}}{7680 \, c^{5}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(c*x^2+b*x)^(3/2),x, algorithm="fricas")

[Out]

[1/15360*(15*(24*b^4*c^2*d^2 - 24*b^5*c*d*e + 7*b^6*e^2)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))
+ 2*(1280*c^6*e^2*x^5 - 360*b^3*c^3*d^2 + 360*b^4*c^2*d*e - 105*b^5*c*e^2 + 128*(24*c^6*d*e + 13*b*c^5*e^2)*x^
4 + 48*(40*c^6*d^2 + 88*b*c^5*d*e + b^2*c^4*e^2)*x^3 + 8*(360*b*c^5*d^2 + 24*b^2*c^4*d*e - 7*b^3*c^3*e^2)*x^2
+ 10*(24*b^2*c^4*d^2 - 24*b^3*c^3*d*e + 7*b^4*c^2*e^2)*x)*sqrt(c*x^2 + b*x))/c^5, -1/7680*(15*(24*b^4*c^2*d^2
- 24*b^5*c*d*e + 7*b^6*e^2)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) - (1280*c^6*e^2*x^5 - 360*b^3*c^
3*d^2 + 360*b^4*c^2*d*e - 105*b^5*c*e^2 + 128*(24*c^6*d*e + 13*b*c^5*e^2)*x^4 + 48*(40*c^6*d^2 + 88*b*c^5*d*e
+ b^2*c^4*e^2)*x^3 + 8*(360*b*c^5*d^2 + 24*b^2*c^4*d*e - 7*b^3*c^3*e^2)*x^2 + 10*(24*b^2*c^4*d^2 - 24*b^3*c^3*
d*e + 7*b^4*c^2*e^2)*x)*sqrt(c*x^2 + b*x))/c^5]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (x \left (b + c x\right )\right )^{\frac{3}{2}} \left (d + e x\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**2*(c*x**2+b*x)**(3/2),x)

[Out]

Integral((x*(b + c*x))**(3/2)*(d + e*x)**2, x)

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Giac [A]  time = 1.38887, size = 354, normalized size = 1.65 \begin{align*} \frac{1}{7680} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, c x e^{2} + \frac{24 \, c^{6} d e + 13 \, b c^{5} e^{2}}{c^{5}}\right )} x + \frac{3 \,{\left (40 \, c^{6} d^{2} + 88 \, b c^{5} d e + b^{2} c^{4} e^{2}\right )}}{c^{5}}\right )} x + \frac{360 \, b c^{5} d^{2} + 24 \, b^{2} c^{4} d e - 7 \, b^{3} c^{3} e^{2}}{c^{5}}\right )} x + \frac{5 \,{\left (24 \, b^{2} c^{4} d^{2} - 24 \, b^{3} c^{3} d e + 7 \, b^{4} c^{2} e^{2}\right )}}{c^{5}}\right )} x - \frac{15 \,{\left (24 \, b^{3} c^{3} d^{2} - 24 \, b^{4} c^{2} d e + 7 \, b^{5} c e^{2}\right )}}{c^{5}}\right )} - \frac{{\left (24 \, b^{4} c^{2} d^{2} - 24 \, b^{5} c d e + 7 \, b^{6} e^{2}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{1024 \, c^{\frac{9}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(c*x^2+b*x)^(3/2),x, algorithm="giac")

[Out]

1/7680*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(10*c*x*e^2 + (24*c^6*d*e + 13*b*c^5*e^2)/c^5)*x + 3*(40*c^6*d^2 + 88*b*c
^5*d*e + b^2*c^4*e^2)/c^5)*x + (360*b*c^5*d^2 + 24*b^2*c^4*d*e - 7*b^3*c^3*e^2)/c^5)*x + 5*(24*b^2*c^4*d^2 - 2
4*b^3*c^3*d*e + 7*b^4*c^2*e^2)/c^5)*x - 15*(24*b^3*c^3*d^2 - 24*b^4*c^2*d*e + 7*b^5*c*e^2)/c^5) - 1/1024*(24*b
^4*c^2*d^2 - 24*b^5*c*d*e + 7*b^6*e^2)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(9/2)