\(\int \frac {x^8}{a+b x^3+c x^6} \, dx\) [138]

Optimal result

Integrand size = 18, antiderivative size = 81 \[ \int \frac {x^8}{a+b x^3+c x^6} \, dx=\frac {x^3}{3 c}-\frac {\left (b^2-2 a c\right ) \text {arctanh}\left (\frac {b+2 c x^3}{\sqrt {b^2-4 a c}}\right )}{3 c^2 \sqrt {b^2-4 a c}}-\frac {b \log \left (a+b x^3+c x^6\right )}{6 c^2} \]

[Out]

1/3*x^3/c-1/6*b*ln(c*x^6+b*x^3+a)/c^2-1/3*(-2*a*c+b^2)*arctanh((2*c*x^3+b)/(-4*a*c+b^2)^(1/2))/c^2/(-4*a*c+b^2
)^(1/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1371, 717, 648, 632, 212, 642} \[ \int \frac {x^8}{a+b x^3+c x^6} \, dx=-\frac {\left (b^2-2 a c\right ) \text {arctanh}\left (\frac {b+2 c x^3}{\sqrt {b^2-4 a c}}\right )}{3 c^2 \sqrt {b^2-4 a c}}-\frac {b \log \left (a+b x^3+c x^6\right )}{6 c^2}+\frac {x^3}{3 c} \]

[In]

Int[x^8/(a + b*x^3 + c*x^6),x]

[Out]

x^3/(3*c) - ((b^2 - 2*a*c)*ArcTanh[(b + 2*c*x^3)/Sqrt[b^2 - 4*a*c]])/(3*c^2*Sqrt[b^2 - 4*a*c]) - (b*Log[a + b*
x^3 + c*x^6])/(6*c^2)

Rule 212

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

Rule 632

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

Rule 642

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

Rule 648

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

Rule 717

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

Rule 1371

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[
b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {x^2}{a+b x+c x^2} \, dx,x,x^3\right ) \\ & = \frac {x^3}{3 c}+\frac {\text {Subst}\left (\int \frac {-a-b x}{a+b x+c x^2} \, dx,x,x^3\right )}{3 c} \\ & = \frac {x^3}{3 c}-\frac {b \text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^3\right )}{6 c^2}+\frac {\left (b^2-2 a c\right ) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^3\right )}{6 c^2} \\ & = \frac {x^3}{3 c}-\frac {b \log \left (a+b x^3+c x^6\right )}{6 c^2}-\frac {\left (b^2-2 a c\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^3\right )}{3 c^2} \\ & = \frac {x^3}{3 c}-\frac {\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^3}{\sqrt {b^2-4 a c}}\right )}{3 c^2 \sqrt {b^2-4 a c}}-\frac {b \log \left (a+b x^3+c x^6\right )}{6 c^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.96 \[ \int \frac {x^8}{a+b x^3+c x^6} \, dx=\frac {2 c x^3+\frac {2 \left (b^2-2 a c\right ) \arctan \left (\frac {b+2 c x^3}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}-b \log \left (a+b x^3+c x^6\right )}{6 c^2} \]

[In]

Integrate[x^8/(a + b*x^3 + c*x^6),x]

[Out]

(2*c*x^3 + (2*(b^2 - 2*a*c)*ArcTan[(b + 2*c*x^3)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] - b*Log[a + b*x^3 + c
*x^6])/(6*c^2)

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.02

method	result	size
default	\(\frac {x^{3}}{3 c}+\frac {-\frac {b \ln \left (c \,x^{6}+b \,x^{3}+a \right )}{2 c}+\frac {2 \left (-a +\frac {b^{2}}{2 c}\right ) \arctan \left (\frac {2 c \,x^{3}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{3 c}\)	\(83\)
risch	\(\frac {x^{3}}{3 c}-\frac {2 \ln \left (\left (-8 a^{2} c^{2}+6 a \,b^{2} c -b^{4}+\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) x^{3}+2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, a \right ) a b}{3 c \left (4 a c -b^{2}\right )}+\frac {\ln \left (\left (-8 a^{2} c^{2}+6 a \,b^{2} c -b^{4}+\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) x^{3}+2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, a \right ) b^{3}}{6 c^{2} \left (4 a c -b^{2}\right )}+\frac {\ln \left (\left (-8 a^{2} c^{2}+6 a \,b^{2} c -b^{4}+\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) x^{3}+2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, a \right ) \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}}{6 c^{2} \left (4 a c -b^{2}\right )}-\frac {2 \ln \left (\left (-8 a^{2} c^{2}+6 a \,b^{2} c -b^{4}-\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) x^{3}-2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, a \right ) a b}{3 c \left (4 a c -b^{2}\right )}+\frac {\ln \left (\left (-8 a^{2} c^{2}+6 a \,b^{2} c -b^{4}-\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) x^{3}-2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, a \right ) b^{3}}{6 c^{2} \left (4 a c -b^{2}\right )}-\frac {\ln \left (\left (-8 a^{2} c^{2}+6 a \,b^{2} c -b^{4}-\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) x^{3}-2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, a \right ) \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}}{6 c^{2} \left (4 a c -b^{2}\right )}\)	\(681\)

[In]

int(x^8/(c*x^6+b*x^3+a),x,method=_RETURNVERBOSE)

[Out]

1/3*x^3/c+1/3/c*(-1/2*b/c*ln(c*x^6+b*x^3+a)+2*(-a+1/2/c*b^2)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^3+b)/(4*a*c-b^2)^
(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 254, normalized size of antiderivative = 3.14 \[ \int \frac {x^8}{a+b x^3+c x^6} \, dx=\left [\frac {2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} x^{3} - {\left (b^{2} - 2 \, a c\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{6} + 2 \, b c x^{3} + b^{2} - 2 \, a c + {\left (2 \, c x^{3} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{6} + b x^{3} + a}\right ) - {\left (b^{3} - 4 \, a b c\right )} \log \left (c x^{6} + b x^{3} + a\right )}{6 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}, \frac {2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} x^{3} - 2 \, {\left (b^{2} - 2 \, a c\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {{\left (2 \, c x^{3} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) - {\left (b^{3} - 4 \, a b c\right )} \log \left (c x^{6} + b x^{3} + a\right )}{6 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}\right ] \]

[In]

integrate(x^8/(c*x^6+b*x^3+a),x, algorithm="fricas")

[Out]

[1/6*(2*(b^2*c - 4*a*c^2)*x^3 - (b^2 - 2*a*c)*sqrt(b^2 - 4*a*c)*log((2*c^2*x^6 + 2*b*c*x^3 + b^2 - 2*a*c + (2*
c*x^3 + b)*sqrt(b^2 - 4*a*c))/(c*x^6 + b*x^3 + a)) - (b^3 - 4*a*b*c)*log(c*x^6 + b*x^3 + a))/(b^2*c^2 - 4*a*c^
3), 1/6*(2*(b^2*c - 4*a*c^2)*x^3 - 2*(b^2 - 2*a*c)*sqrt(-b^2 + 4*a*c)*arctan(-(2*c*x^3 + b)*sqrt(-b^2 + 4*a*c)
/(b^2 - 4*a*c)) - (b^3 - 4*a*b*c)*log(c*x^6 + b*x^3 + a))/(b^2*c^2 - 4*a*c^3)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (75) = 150\).

Time = 1.72 (sec) , antiderivative size = 316, normalized size of antiderivative = 3.90 \[ \int \frac {x^8}{a+b x^3+c x^6} \, dx=\left (- \frac {b}{6 c^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{6 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) \log {\left (x^{3} + \frac {- a b - 12 a c^{2} \left (- \frac {b}{6 c^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{6 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) + 3 b^{2} c \left (- \frac {b}{6 c^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{6 c^{2} \cdot \left (4 a c - b^{2}\right )}\right )}{2 a c - b^{2}} \right )} + \left (- \frac {b}{6 c^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{6 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) \log {\left (x^{3} + \frac {- a b - 12 a c^{2} \left (- \frac {b}{6 c^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{6 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) + 3 b^{2} c \left (- \frac {b}{6 c^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{6 c^{2} \cdot \left (4 a c - b^{2}\right )}\right )}{2 a c - b^{2}} \right )} + \frac {x^{3}}{3 c} \]

[In]

integrate(x**8/(c*x**6+b*x**3+a),x)

[Out]

(-b/(6*c**2) - sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(6*c**2*(4*a*c - b**2)))*log(x**3 + (-a*b - 12*a*c**2*(-b/(6
*c**2) - sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(6*c**2*(4*a*c - b**2))) + 3*b**2*c*(-b/(6*c**2) - sqrt(-4*a*c + b
**2)*(2*a*c - b**2)/(6*c**2*(4*a*c - b**2))))/(2*a*c - b**2)) + (-b/(6*c**2) + sqrt(-4*a*c + b**2)*(2*a*c - b*
*2)/(6*c**2*(4*a*c - b**2)))*log(x**3 + (-a*b - 12*a*c**2*(-b/(6*c**2) + sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(6
*c**2*(4*a*c - b**2))) + 3*b**2*c*(-b/(6*c**2) + sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(6*c**2*(4*a*c - b**2))))/
(2*a*c - b**2)) + x**3/(3*c)

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^8}{a+b x^3+c x^6} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(x^8/(c*x^6+b*x^3+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

Giac [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.93 \[ \int \frac {x^8}{a+b x^3+c x^6} \, dx=\frac {x^{3}}{3 \, c} - \frac {b \log \left (c x^{6} + b x^{3} + a\right )}{6 \, c^{2}} + \frac {{\left (b^{2} - 2 \, a c\right )} \arctan \left (\frac {2 \, c x^{3} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{3 \, \sqrt {-b^{2} + 4 \, a c} c^{2}} \]

[In]

integrate(x^8/(c*x^6+b*x^3+a),x, algorithm="giac")

[Out]

1/3*x^3/c - 1/6*b*log(c*x^6 + b*x^3 + a)/c^2 + 1/3*(b^2 - 2*a*c)*arctan((2*c*x^3 + b)/sqrt(-b^2 + 4*a*c))/(sqr
t(-b^2 + 4*a*c)*c^2)

Mupad [B] (verification not implemented)

Time = 8.89 (sec) , antiderivative size = 1758, normalized size of antiderivative = 21.70 \[ \int \frac {x^8}{a+b x^3+c x^6} \, dx=\text {Too large to display} \]

[In]

int(x^8/(a + b*x^3 + c*x^6),x)

[Out]

x^3/(3*c) + (log(a + b*x^3 + c*x^6)*(3*b^3 - 12*a*b*c))/(2*(36*a*c^3 - 9*b^2*c^2)) + (atan((4*c^3*x^3*(4*a*c -
b^2)^(3/2)*((b*((b^5 + a^2*b*c^2 - 2*a*b^3*c)/c^3 + ((3*b^3 - 12*a*b*c)*((6*a^2*c^4 + 12*b^4*c^2 - 18*a*b^2*c
^3)/c^3 + ((3*b^3 - 12*a*b*c)*((45*b^3*c^4 - 36*a*b*c^5)/c^3 + (27*b^2*c^3*(3*b^3 - 12*a*b*c))/(36*a*c^3 - 9*b
^2*c^2)))/(2*(36*a*c^3 - 9*b^2*c^2))))/(2*(36*a*c^3 - 9*b^2*c^2)) - ((((2*a*c - b^2)*((45*b^3*c^4 - 36*a*b*c^5
)/c^3 + (27*b^2*c^3*(3*b^3 - 12*a*b*c))/(36*a*c^3 - 9*b^2*c^2)))/(6*c^2*(4*a*c - b^2)^(1/2)) + (9*b^2*c*(3*b^3
- 12*a*b*c)*(2*a*c - b^2))/(2*(4*a*c - b^2)^(1/2)*(36*a*c^3 - 9*b^2*c^2)))*(2*a*c - b^2))/(6*c^2*(4*a*c - b^2
)^(1/2)) - (3*b^2*(3*b^3 - 12*a*b*c)*(2*a*c - b^2)^2)/(4*c*(4*a*c - b^2)*(36*a*c^3 - 9*b^2*c^2))))/(4*a^2*c) +
((2*a*c - b^2)*(((3*b^3 - 12*a*b*c)*(((2*a*c - b^2)*((45*b^3*c^4 - 36*a*b*c^5)/c^3 + (27*b^2*c^3*(3*b^3 - 12*
a*b*c))/(36*a*c^3 - 9*b^2*c^2)))/(6*c^2*(4*a*c - b^2)^(1/2)) + (9*b^2*c*(3*b^3 - 12*a*b*c)*(2*a*c - b^2))/(2*(
4*a*c - b^2)^(1/2)*(36*a*c^3 - 9*b^2*c^2))))/(2*(36*a*c^3 - 9*b^2*c^2)) - (b^2*(2*a*c - b^2)^3)/(4*c^3*(4*a*c
- b^2)^(3/2)) + (((6*a^2*c^4 + 12*b^4*c^2 - 18*a*b^2*c^3)/c^3 + ((3*b^3 - 12*a*b*c)*((45*b^3*c^4 - 36*a*b*c^5)
/c^3 + (27*b^2*c^3*(3*b^3 - 12*a*b*c))/(36*a*c^3 - 9*b^2*c^2)))/(2*(36*a*c^3 - 9*b^2*c^2)))*(2*a*c - b^2))/(6*
c^2*(4*a*c - b^2)^(1/2))))/(4*a^2*c*(4*a*c - b^2)^(1/2))))/(b^6 - 8*a^3*c^3 + 12*a^2*b^2*c^2 - 6*a*b^4*c) - (c
^2*(2*a*c - b^2)*(4*a*c - b^2)*(((3*b^3 - 12*a*b*c)*((((36*a^2*c^5 - 72*a*b^2*c^4)/c^3 - (54*a*b*c^3*(3*b^3 -
12*a*b*c))/(36*a*c^3 - 9*b^2*c^2))*(2*a*c - b^2))/(6*c^2*(4*a*c - b^2)^(1/2)) - (9*a*b*c*(3*b^3 - 12*a*b*c)*(2
*a*c - b^2))/((4*a*c - b^2)^(1/2)*(36*a*c^3 - 9*b^2*c^2))))/(2*(36*a*c^3 - 9*b^2*c^2)) - (((15*a*b^3*c^2 - 12*
a^2*b*c^3)/c^3 - ((3*b^3 - 12*a*b*c)*((36*a^2*c^5 - 72*a*b^2*c^4)/c^3 - (54*a*b*c^3*(3*b^3 - 12*a*b*c))/(36*a*
c^3 - 9*b^2*c^2)))/(2*(36*a*c^3 - 9*b^2*c^2)))*(2*a*c - b^2))/(6*c^2*(4*a*c - b^2)^(1/2)) + (a*b*(2*a*c - b^2)
^3)/(2*c^3*(4*a*c - b^2)^(3/2))))/(a^2*(b^6 - 8*a^3*c^3 + 12*a^2*b^2*c^2 - 6*a*b^4*c)) + (b*c^2*(4*a*c - b^2)^
(3/2)*((a*b^4 - a^2*b^2*c)/c^3 + ((3*b^3 - 12*a*b*c)*((15*a*b^3*c^2 - 12*a^2*b*c^3)/c^3 - ((3*b^3 - 12*a*b*c)*
((36*a^2*c^5 - 72*a*b^2*c^4)/c^3 - (54*a*b*c^3*(3*b^3 - 12*a*b*c))/(36*a*c^3 - 9*b^2*c^2)))/(2*(36*a*c^3 - 9*b
^2*c^2))))/(2*(36*a*c^3 - 9*b^2*c^2)) + (((((36*a^2*c^5 - 72*a*b^2*c^4)/c^3 - (54*a*b*c^3*(3*b^3 - 12*a*b*c))/
(36*a*c^3 - 9*b^2*c^2))*(2*a*c - b^2))/(6*c^2*(4*a*c - b^2)^(1/2)) - (9*a*b*c*(3*b^3 - 12*a*b*c)*(2*a*c - b^2)
)/((4*a*c - b^2)^(1/2)*(36*a*c^3 - 9*b^2*c^2)))*(2*a*c - b^2))/(6*c^2*(4*a*c - b^2)^(1/2)) - (3*a*b*(3*b^3 - 1
2*a*b*c)*(2*a*c - b^2)^2)/(2*c*(4*a*c - b^2)*(36*a*c^3 - 9*b^2*c^2))))/(a^2*(b^6 - 8*a^3*c^3 + 12*a^2*b^2*c^2
- 6*a*b^4*c)))*(2*a*c - b^2))/(3*c^2*(4*a*c - b^2)^(1/2))