Integrand size = 16, antiderivative size = 235 \[ \int \frac {1}{\left (-3+4 x^2+2 x^4\right )^{3/2}} \, dx=-\frac {x \left (7+2 x^2\right )}{30 \sqrt {-3+4 x^2+2 x^4}}+\frac {\sqrt {3-\left (2-\sqrt {10}\right ) x^2} \sqrt {3-\left (2+\sqrt {10}\right ) x^2} E\left (\arcsin \left (\sqrt {\frac {1}{3} \left (2+\sqrt {10}\right )} x\right )|\frac {1}{3} \left (-7+2 \sqrt {10}\right )\right )}{15 \sqrt {2 \left (-2+\sqrt {10}\right )} \sqrt {-3+4 x^2+2 x^4}}-\frac {\sqrt {3-\left (2-\sqrt {10}\right ) x^2} \sqrt {3-\left (2+\sqrt {10}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1}{3} \left (2+\sqrt {10}\right )} x\right ),\frac {1}{3} \left (-7+2 \sqrt {10}\right )\right )}{6 \sqrt {5 \left (-2+\sqrt {10}\right )} \sqrt {-3+4 x^2+2 x^4}} \] Output:
-1/30*x*(2*x^2+7)/(2*x^4+4*x^2-3)^(1/2)+1/15*(3-(2-10^(1/2))*x^2)^(1/2)*(3 -(2+10^(1/2))*x^2)^(1/2)*EllipticE(1/3*(6+3*10^(1/2))^(1/2)*x,1/3*I*15^(1/ 2)-1/3*I*6^(1/2))/(-4+2*10^(1/2))^(1/2)/(2*x^4+4*x^2-3)^(1/2)-1/6*(3-(2-10 ^(1/2))*x^2)^(1/2)*(3-(2+10^(1/2))*x^2)^(1/2)*EllipticF(1/3*(6+3*10^(1/2)) ^(1/2)*x,1/3*I*15^(1/2)-1/3*I*6^(1/2))/(-10+5*10^(1/2))^(1/2)/(2*x^4+4*x^2 -3)^(1/2)
Result contains complex when optimal does not.
Time = 6.92 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.73 \[ \int \frac {1}{\left (-3+4 x^2+2 x^4\right )^{3/2}} \, dx=\frac {-2 x \left (7+2 x^2\right )+2 i \sqrt {-2+\sqrt {10}} \sqrt {3-4 x^2-2 x^4} E\left (i \text {arcsinh}\left (\sqrt {\frac {2}{2+\sqrt {10}}} x\right )|-\frac {7}{3}-\frac {2 \sqrt {10}}{3}\right )-\frac {2 i \left (-5+\sqrt {10}\right ) \sqrt {3-4 x^2-2 x^4} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {2}{2+\sqrt {10}}} x\right ),-\frac {7}{3}-\frac {2 \sqrt {10}}{3}\right )}{\sqrt {-2+\sqrt {10}}}}{60 \sqrt {-3+4 x^2+2 x^4}} \] Input:
Integrate[(-3 + 4*x^2 + 2*x^4)^(-3/2),x]
Output:
(-2*x*(7 + 2*x^2) + (2*I)*Sqrt[-2 + Sqrt[10]]*Sqrt[3 - 4*x^2 - 2*x^4]*Elli pticE[I*ArcSinh[Sqrt[2/(2 + Sqrt[10])]*x], -7/3 - (2*Sqrt[10])/3] - ((2*I) *(-5 + Sqrt[10])*Sqrt[3 - 4*x^2 - 2*x^4]*EllipticF[I*ArcSinh[Sqrt[2/(2 + S qrt[10])]*x], -7/3 - (2*Sqrt[10])/3])/Sqrt[-2 + Sqrt[10]])/(60*Sqrt[-3 + 4 *x^2 + 2*x^4])
Time = 0.74 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.58, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1405, 27, 1501, 27, 1411, 1498}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{\left (2 x^4+4 x^2-3\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1405 |
| \(\displaystyle \frac {1}{120} \int -\frac {4 \left (3-2 x^2\right )}{\sqrt {2 x^4+4 x^2-3}}dx-\frac {x \left (2 x^2+7\right )}{30 \sqrt {2 x^4+4 x^2-3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{30} \int \frac {3-2 x^2}{\sqrt {2 x^4+4 x^2-3}}dx-\frac {x \left (2 x^2+7\right )}{30 \sqrt {2 x^4+4 x^2-3}}\) |
| \(\Big \downarrow \) 1501 |
| \(\displaystyle \frac {1}{30} \left (\frac {1}{2} \int \frac {2 \left (2 x^2-\sqrt {10}+2\right )}{\sqrt {2 x^4+4 x^2-3}}dx-\left (5-\sqrt {10}\right ) \int \frac {1}{\sqrt {2 x^4+4 x^2-3}}dx\right )-\frac {x \left (2 x^2+7\right )}{30 \sqrt {2 x^4+4 x^2-3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{30} \left (\int \frac {2 x^2-\sqrt {10}+2}{\sqrt {2 x^4+4 x^2-3}}dx-\left (5-\sqrt {10}\right ) \int \frac {1}{\sqrt {2 x^4+4 x^2-3}}dx\right )-\frac {x \left (2 x^2+7\right )}{30 \sqrt {2 x^4+4 x^2-3}}\) |
| \(\Big \downarrow \) 1411 |
| \(\displaystyle \frac {1}{30} \left (\int \frac {2 x^2-\sqrt {10}+2}{\sqrt {2 x^4+4 x^2-3}}dx-\frac {\left (5-\sqrt {10}\right ) \sqrt {\frac {3-\left (2-\sqrt {10}\right ) x^2}{3-\left (2+\sqrt {10}\right ) x^2}} \sqrt {\left (2+\sqrt {10}\right ) x^2-3} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{3/4} \sqrt [4]{5} x}{\sqrt {\left (2+\sqrt {10}\right ) x^2-3}}\right ),\frac {1}{10} \left (5+\sqrt {10}\right )\right )}{2^{3/4} \sqrt {3} \sqrt [4]{5} \sqrt {\frac {1}{3-\left (2+\sqrt {10}\right ) x^2}} \sqrt {2 x^4+4 x^2-3}}\right )-\frac {x \left (2 x^2+7\right )}{30 \sqrt {2 x^4+4 x^2-3}}\) |
| \(\Big \downarrow \) 1498 |
| \(\displaystyle \frac {1}{30} \left (-\frac {\left (5-\sqrt {10}\right ) \sqrt {\frac {3-\left (2-\sqrt {10}\right ) x^2}{3-\left (2+\sqrt {10}\right ) x^2}} \sqrt {\left (2+\sqrt {10}\right ) x^2-3} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{3/4} \sqrt [4]{5} x}{\sqrt {\left (2+\sqrt {10}\right ) x^2-3}}\right ),\frac {1}{10} \left (5+\sqrt {10}\right )\right )}{2^{3/4} \sqrt {3} \sqrt [4]{5} \sqrt {\frac {1}{3-\left (2+\sqrt {10}\right ) x^2}} \sqrt {2 x^4+4 x^2-3}}-\frac {2^{3/4} \sqrt [4]{5} \sqrt {\frac {3-\left (2-\sqrt {10}\right ) x^2}{3-\left (2+\sqrt {10}\right ) x^2}} \sqrt {\left (2+\sqrt {10}\right ) x^2-3} E\left (\arcsin \left (\frac {2^{3/4} \sqrt [4]{5} x}{\sqrt {\left (2+\sqrt {10}\right ) x^2-3}}\right )|\frac {1}{10} \left (5+\sqrt {10}\right )\right )}{\sqrt {3} \sqrt {\frac {1}{3-\left (2+\sqrt {10}\right ) x^2}} \sqrt {2 x^4+4 x^2-3}}+\frac {x \left (2 x^2+\sqrt {10}+2\right )}{\sqrt {2 x^4+4 x^2-3}}\right )-\frac {x \left (2 x^2+7\right )}{30 \sqrt {2 x^4+4 x^2-3}}\) |
Input:
Int[(-3 + 4*x^2 + 2*x^4)^(-3/2),x]
Output:
-1/30*(x*(7 + 2*x^2))/Sqrt[-3 + 4*x^2 + 2*x^4] + ((x*(2 + Sqrt[10] + 2*x^2 ))/Sqrt[-3 + 4*x^2 + 2*x^4] - (2^(3/4)*5^(1/4)*Sqrt[(3 - (2 - Sqrt[10])*x^ 2)/(3 - (2 + Sqrt[10])*x^2)]*Sqrt[-3 + (2 + Sqrt[10])*x^2]*EllipticE[ArcSi n[(2^(3/4)*5^(1/4)*x)/Sqrt[-3 + (2 + Sqrt[10])*x^2]], (5 + Sqrt[10])/10])/ (Sqrt[3]*Sqrt[(3 - (2 + Sqrt[10])*x^2)^(-1)]*Sqrt[-3 + 4*x^2 + 2*x^4]) - ( (5 - Sqrt[10])*Sqrt[(3 - (2 - Sqrt[10])*x^2)/(3 - (2 + Sqrt[10])*x^2)]*Sqr t[-3 + (2 + Sqrt[10])*x^2]*EllipticF[ArcSin[(2^(3/4)*5^(1/4)*x)/Sqrt[-3 + (2 + Sqrt[10])*x^2]], (5 + Sqrt[10])/10])/(2^(3/4)*Sqrt[3]*5^(1/4)*Sqrt[(3 - (2 + Sqrt[10])*x^2)^(-1)]*Sqrt[-3 + 4*x^2 + 2*x^4]))/30
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(b^2 - 2*a*c + b*c*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c)) ), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(b^2 - 2*a*c + 2*(p + 1)*( b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; Fr eeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b ^2 - 4*a*c, 2]}, Simp[Sqrt[(2*a + (b - q)*x^2)/(2*a + (b + q)*x^2)]*(Sqrt[( 2*a + (b + q)*x^2)/q]/(2*Sqrt[a + b*x^2 + c*x^4]*Sqrt[a/(2*a + (b + q)*x^2) ]))*EllipticF[ArcSin[x/Sqrt[(2*a + (b + q)*x^2)/(2*q)]], (b + q)/(2*q)], x] ] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[a, 0] && GtQ[c, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[e*x*((b + q + 2*c*x^2)/(2*c*Sqrt[ a + b*x^2 + c*x^4])), x] - Simp[e*q*Sqrt[(2*a + (b - q)*x^2)/(2*a + (b + q) *x^2)]*(Sqrt[(2*a + (b + q)*x^2)/q]/(2*c*Sqrt[a + b*x^2 + c*x^4]*Sqrt[a/(2* a + (b + q)*x^2)]))*EllipticE[ArcSin[x/Sqrt[(2*a + (b + q)*x^2)/(2*q)]], (b + q)/(2*q)], x] /; EqQ[2*c*d - e*(b - q), 0]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[a, 0] && GtQ[c, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(2*c*d - e*(b - q))/(2*c) Int[1 /Sqrt[a + b*x^2 + c*x^4], x], x] + Simp[e/(2*c) Int[(b - q + 2*c*x^2)/Sqr t[a + b*x^2 + c*x^4], x], x] /; NeQ[2*c*d - e*(b - q), 0]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[a, 0] && GtQ[c, 0]
Time = 2.26 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.98
| method | result | size |
| risch | \(-\frac {x \left (2 x^{2}+7\right )}{30 \sqrt {2 x^{4}+4 x^{2}-3}}-\frac {3 \sqrt {1-\left (\frac {2}{3}-\frac {\sqrt {10}}{3}\right ) x^{2}}\, \sqrt {1-\left (\frac {2}{3}+\frac {\sqrt {10}}{3}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {6-3 \sqrt {10}}\, x}{3}, \frac {i \sqrt {6}}{3}+\frac {i \sqrt {15}}{3}\right )}{10 \sqrt {6-3 \sqrt {10}}\, \sqrt {2 x^{4}+4 x^{2}-3}}+\frac {6 \sqrt {1-\left (\frac {2}{3}-\frac {\sqrt {10}}{3}\right ) x^{2}}\, \sqrt {1-\left (\frac {2}{3}+\frac {\sqrt {10}}{3}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {6-3 \sqrt {10}}\, x}{3}, \frac {i \sqrt {6}}{3}+\frac {i \sqrt {15}}{3}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {6-3 \sqrt {10}}\, x}{3}, \frac {i \sqrt {6}}{3}+\frac {i \sqrt {15}}{3}\right )\right )}{5 \sqrt {6-3 \sqrt {10}}\, \sqrt {2 x^{4}+4 x^{2}-3}\, \left (4+2 \sqrt {10}\right )}\) | \(230\) |
| default | \(-\frac {4 \left (\frac {7}{120} x +\frac {1}{60} x^{3}\right )}{\sqrt {2 x^{4}+4 x^{2}-3}}-\frac {3 \sqrt {1-\left (\frac {2}{3}-\frac {\sqrt {10}}{3}\right ) x^{2}}\, \sqrt {1-\left (\frac {2}{3}+\frac {\sqrt {10}}{3}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {6-3 \sqrt {10}}\, x}{3}, \frac {i \sqrt {6}}{3}+\frac {i \sqrt {15}}{3}\right )}{10 \sqrt {6-3 \sqrt {10}}\, \sqrt {2 x^{4}+4 x^{2}-3}}+\frac {6 \sqrt {1-\left (\frac {2}{3}-\frac {\sqrt {10}}{3}\right ) x^{2}}\, \sqrt {1-\left (\frac {2}{3}+\frac {\sqrt {10}}{3}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {6-3 \sqrt {10}}\, x}{3}, \frac {i \sqrt {6}}{3}+\frac {i \sqrt {15}}{3}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {6-3 \sqrt {10}}\, x}{3}, \frac {i \sqrt {6}}{3}+\frac {i \sqrt {15}}{3}\right )\right )}{5 \sqrt {6-3 \sqrt {10}}\, \sqrt {2 x^{4}+4 x^{2}-3}\, \left (4+2 \sqrt {10}\right )}\) | \(231\) |
| elliptic | \(-\frac {4 \left (\frac {7}{120} x +\frac {1}{60} x^{3}\right )}{\sqrt {2 x^{4}+4 x^{2}-3}}-\frac {3 \sqrt {1-\left (\frac {2}{3}-\frac {\sqrt {10}}{3}\right ) x^{2}}\, \sqrt {1-\left (\frac {2}{3}+\frac {\sqrt {10}}{3}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {6-3 \sqrt {10}}\, x}{3}, \frac {i \sqrt {6}}{3}+\frac {i \sqrt {15}}{3}\right )}{10 \sqrt {6-3 \sqrt {10}}\, \sqrt {2 x^{4}+4 x^{2}-3}}+\frac {6 \sqrt {1-\left (\frac {2}{3}-\frac {\sqrt {10}}{3}\right ) x^{2}}\, \sqrt {1-\left (\frac {2}{3}+\frac {\sqrt {10}}{3}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {6-3 \sqrt {10}}\, x}{3}, \frac {i \sqrt {6}}{3}+\frac {i \sqrt {15}}{3}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {6-3 \sqrt {10}}\, x}{3}, \frac {i \sqrt {6}}{3}+\frac {i \sqrt {15}}{3}\right )\right )}{5 \sqrt {6-3 \sqrt {10}}\, \sqrt {2 x^{4}+4 x^{2}-3}\, \left (4+2 \sqrt {10}\right )}\) | \(231\) |
Input:
int(1/(2*x^4+4*x^2-3)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/30*x*(2*x^2+7)/(2*x^4+4*x^2-3)^(1/2)-3/10/(6-3*10^(1/2))^(1/2)*(1-(2/3- 1/3*10^(1/2))*x^2)^(1/2)*(1-(2/3+1/3*10^(1/2))*x^2)^(1/2)/(2*x^4+4*x^2-3)^ (1/2)*EllipticF(1/3*(6-3*10^(1/2))^(1/2)*x,1/3*I*6^(1/2)+1/3*I*15^(1/2))+6 /5/(6-3*10^(1/2))^(1/2)*(1-(2/3-1/3*10^(1/2))*x^2)^(1/2)*(1-(2/3+1/3*10^(1 /2))*x^2)^(1/2)/(2*x^4+4*x^2-3)^(1/2)/(4+2*10^(1/2))*(EllipticF(1/3*(6-3*1 0^(1/2))^(1/2)*x,1/3*I*6^(1/2)+1/3*I*15^(1/2))-EllipticE(1/3*(6-3*10^(1/2) )^(1/2)*x,1/3*I*6^(1/2)+1/3*I*15^(1/2)))
Time = 0.07 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\left (-3+4 x^2+2 x^4\right )^{3/2}} \, dx=-\frac {2 \, {\left (\sqrt {10} \sqrt {-3} {\left (2 \, x^{4} + 4 \, x^{2} - 3\right )} + 2 \, \sqrt {-3} {\left (2 \, x^{4} + 4 \, x^{2} - 3\right )}\right )} \sqrt {\frac {1}{3} \, \sqrt {10} + \frac {2}{3}} E(\arcsin \left (x \sqrt {\frac {1}{3} \, \sqrt {10} + \frac {2}{3}}\right )\,|\,\frac {2}{3} \, \sqrt {10} - \frac {7}{3}) - {\left (5 \, \sqrt {10} \sqrt {-3} {\left (2 \, x^{4} + 4 \, x^{2} - 3\right )} - 2 \, \sqrt {-3} {\left (2 \, x^{4} + 4 \, x^{2} - 3\right )}\right )} \sqrt {\frac {1}{3} \, \sqrt {10} + \frac {2}{3}} F(\arcsin \left (x \sqrt {\frac {1}{3} \, \sqrt {10} + \frac {2}{3}}\right )\,|\,\frac {2}{3} \, \sqrt {10} - \frac {7}{3}) + 6 \, \sqrt {2 \, x^{4} + 4 \, x^{2} - 3} {\left (2 \, x^{3} + 7 \, x\right )}}{180 \, {\left (2 \, x^{4} + 4 \, x^{2} - 3\right )}} \] Input:
integrate(1/(2*x^4+4*x^2-3)^(3/2),x, algorithm="fricas")
Output:
-1/180*(2*(sqrt(10)*sqrt(-3)*(2*x^4 + 4*x^2 - 3) + 2*sqrt(-3)*(2*x^4 + 4*x ^2 - 3))*sqrt(1/3*sqrt(10) + 2/3)*elliptic_e(arcsin(x*sqrt(1/3*sqrt(10) + 2/3)), 2/3*sqrt(10) - 7/3) - (5*sqrt(10)*sqrt(-3)*(2*x^4 + 4*x^2 - 3) - 2* sqrt(-3)*(2*x^4 + 4*x^2 - 3))*sqrt(1/3*sqrt(10) + 2/3)*elliptic_f(arcsin(x *sqrt(1/3*sqrt(10) + 2/3)), 2/3*sqrt(10) - 7/3) + 6*sqrt(2*x^4 + 4*x^2 - 3 )*(2*x^3 + 7*x))/(2*x^4 + 4*x^2 - 3)
\[ \int \frac {1}{\left (-3+4 x^2+2 x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (2 x^{4} + 4 x^{2} - 3\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(2*x**4+4*x**2-3)**(3/2),x)
Output:
Integral((2*x**4 + 4*x**2 - 3)**(-3/2), x)
\[ \int \frac {1}{\left (-3+4 x^2+2 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (2 \, x^{4} + 4 \, x^{2} - 3\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(2*x^4+4*x^2-3)^(3/2),x, algorithm="maxima")
Output:
integrate((2*x^4 + 4*x^2 - 3)^(-3/2), x)
\[ \int \frac {1}{\left (-3+4 x^2+2 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (2 \, x^{4} + 4 \, x^{2} - 3\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(2*x^4+4*x^2-3)^(3/2),x, algorithm="giac")
Output:
integrate((2*x^4 + 4*x^2 - 3)^(-3/2), x)
Timed out. \[ \int \frac {1}{\left (-3+4 x^2+2 x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (2\,x^4+4\,x^2-3\right )}^{3/2}} \,d x \] Input:
int(1/(4*x^2 + 2*x^4 - 3)^(3/2),x)
Output:
int(1/(4*x^2 + 2*x^4 - 3)^(3/2), x)
\[ \int \frac {1}{\left (-3+4 x^2+2 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {2 x^{4}+4 x^{2}-3}}{4 x^{8}+16 x^{6}+4 x^{4}-24 x^{2}+9}d x \] Input:
int(1/(2*x^4+4*x^2-3)^(3/2),x)
Output:
int(sqrt(2*x**4 + 4*x**2 - 3)/(4*x**8 + 16*x**6 + 4*x**4 - 24*x**2 + 9),x)