\(\int \frac {1}{(2+6 x^2+3 x^4)^{3/2}} \, dx\) [232]

Optimal result

Integrand size = 16, antiderivative size = 234 \[ \int \frac {1}{\left (2+6 x^2+3 x^4\right )^{3/2}} \, dx=\frac {\sqrt {3} x}{2 \left (3-\sqrt {3}\right ) \sqrt {2+6 x^2+3 x^4}}+\frac {3 \sqrt {2+6 x^2+3 x^4} E\left (\arctan \left (\sqrt {\frac {3}{3+\sqrt {3}}} x\right )|-1-\sqrt {3}\right )}{2 \sqrt {2 \left (3+\sqrt {3}\right )} \sqrt {3-\sqrt {3}+3 x^2} \sqrt {3+\sqrt {3}+3 x^2}}-\frac {\sqrt {2+\left (3-\sqrt {3}\right ) x^2} \sqrt {2+\left (3+\sqrt {3}\right ) x^2} \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{2} \left (3-\sqrt {3}\right )} x\right ),-1-\sqrt {3}\right )}{2 \sqrt {2 \left (3-\sqrt {3}\right )} \sqrt {2+6 x^2+3 x^4}} \] Output:

1/2*3^(1/2)*x/(3-3^(1/2))/(3*x^4+6*x^2+2)^(1/2)+3/2*(3*x^4+6*x^2+2)^(1/2)* 
EllipticE(3^(1/2)/(3+3^(1/2))^(1/2)*x/(1+3/(3+3^(1/2))*x^2)^(1/2),(-1-3^(1 
/2))^(1/2))/(6+2*3^(1/2))^(1/2)/(3-3^(1/2)+3*x^2)^(1/2)/(3+3^(1/2)+3*x^2)^ 
(1/2)-1/2*(2+(3-3^(1/2))*x^2)^(1/2)*(2+(3+3^(1/2))*x^2)^(1/2)*InverseJacob 
iAM(arctan(1/2*(6-2*3^(1/2))^(1/2)*x),(-1-3^(1/2))^(1/2))/(6-2*3^(1/2))^(1 
/2)/(3*x^4+6*x^2+2)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 6.28 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\left (2+6 x^2+3 x^4\right )^{3/2}} \, dx=\frac {12 x+9 x^3+3 i \left (-1+\sqrt {3}\right ) \sqrt {\frac {-3+\sqrt {3}-3 x^2}{-3+\sqrt {3}}} \sqrt {3+\sqrt {3}+3 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{3+\sqrt {3}}} x\right )|2+\sqrt {3}\right )+i \sqrt {\left (-3+\sqrt {3}\right ) \left (-3+\sqrt {3}-3 x^2\right )} \sqrt {3+\sqrt {3}+3 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{3+\sqrt {3}}} x\right ),2+\sqrt {3}\right )}{12 \sqrt {2+6 x^2+3 x^4}} \] Input:

Integrate[(2 + 6*x^2 + 3*x^4)^(-3/2),x]
 

Output:

(12*x + 9*x^3 + (3*I)*(-1 + Sqrt[3])*Sqrt[(-3 + Sqrt[3] - 3*x^2)/(-3 + Sqr 
t[3])]*Sqrt[3 + Sqrt[3] + 3*x^2]*EllipticE[I*ArcSinh[Sqrt[3/(3 + Sqrt[3])] 
*x], 2 + Sqrt[3]] + I*Sqrt[(-3 + Sqrt[3])*(-3 + Sqrt[3] - 3*x^2)]*Sqrt[3 + 
 Sqrt[3] + 3*x^2]*EllipticF[I*ArcSinh[Sqrt[3/(3 + Sqrt[3])]*x], 2 + Sqrt[3 
]])/(12*Sqrt[2 + 6*x^2 + 3*x^4])
 

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.22, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1405, 27, 1503, 1412, 1455}

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.

\(\displaystyle \int \frac {1}{\left (3 x^4+6 x^2+2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 1405

\(\displaystyle \frac {x \left (3 x^2+4\right )}{4 \sqrt {3 x^4+6 x^2+2}}-\frac {1}{24} \int \frac {6 \left (3 x^2+2\right )}{\sqrt {3 x^4+6 x^2+2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x \left (3 x^2+4\right )}{4 \sqrt {3 x^4+6 x^2+2}}-\frac {1}{4} \int \frac {3 x^2+2}{\sqrt {3 x^4+6 x^2+2}}dx\)

\(\Big \downarrow \) 1503

\(\displaystyle \frac {1}{4} \left (-2 \int \frac {1}{\sqrt {3 x^4+6 x^2+2}}dx-3 \int \frac {x^2}{\sqrt {3 x^4+6 x^2+2}}dx\right )+\frac {x \left (3 x^2+4\right )}{4 \sqrt {3 x^4+6 x^2+2}}\)

\(\Big \downarrow \) 1412

\(\displaystyle \frac {1}{4} \left (-3 \int \frac {x^2}{\sqrt {3 x^4+6 x^2+2}}dx-\frac {\sqrt {\frac {2}{3+\sqrt {3}}} \sqrt {\frac {\left (3-\sqrt {3}\right ) x^2+2}{\left (3+\sqrt {3}\right ) x^2+2}} \left (\left (3+\sqrt {3}\right ) x^2+2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{2} \left (3+\sqrt {3}\right )} x\right ),-1+\sqrt {3}\right )}{\sqrt {3 x^4+6 x^2+2}}\right )+\frac {x \left (3 x^2+4\right )}{4 \sqrt {3 x^4+6 x^2+2}}\)

\(\Big \downarrow \) 1455

\(\displaystyle \frac {1}{4} \left (-\frac {\sqrt {\frac {2}{3+\sqrt {3}}} \sqrt {\frac {\left (3-\sqrt {3}\right ) x^2+2}{\left (3+\sqrt {3}\right ) x^2+2}} \left (\left (3+\sqrt {3}\right ) x^2+2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{2} \left (3+\sqrt {3}\right )} x\right ),-1+\sqrt {3}\right )}{\sqrt {3 x^4+6 x^2+2}}-3 \left (\frac {x \left (3 x^2+\sqrt {3}+3\right )}{3 \sqrt {3 x^4+6 x^2+2}}-\frac {\sqrt {\frac {1}{2} \left (3+\sqrt {3}\right )} \sqrt {\frac {\left (3-\sqrt {3}\right ) x^2+2}{\left (3+\sqrt {3}\right ) x^2+2}} \left (\left (3+\sqrt {3}\right ) x^2+2\right ) E\left (\arctan \left (\sqrt {\frac {1}{2} \left (3+\sqrt {3}\right )} x\right )|-1+\sqrt {3}\right )}{3 \sqrt {3 x^4+6 x^2+2}}\right )\right )+\frac {x \left (3 x^2+4\right )}{4 \sqrt {3 x^4+6 x^2+2}}\)

Input:

Int[(2 + 6*x^2 + 3*x^4)^(-3/2),x]
 

Output:

(x*(4 + 3*x^2))/(4*Sqrt[2 + 6*x^2 + 3*x^4]) + (-3*((x*(3 + Sqrt[3] + 3*x^2 
))/(3*Sqrt[2 + 6*x^2 + 3*x^4]) - (Sqrt[(3 + Sqrt[3])/2]*Sqrt[(2 + (3 - Sqr 
t[3])*x^2)/(2 + (3 + Sqrt[3])*x^2)]*(2 + (3 + Sqrt[3])*x^2)*EllipticE[ArcT 
an[Sqrt[(3 + Sqrt[3])/2]*x], -1 + Sqrt[3]])/(3*Sqrt[2 + 6*x^2 + 3*x^4])) - 
 (Sqrt[2/(3 + Sqrt[3])]*Sqrt[(2 + (3 - Sqrt[3])*x^2)/(2 + (3 + Sqrt[3])*x^ 
2)]*(2 + (3 + Sqrt[3])*x^2)*EllipticF[ArcTan[Sqrt[(3 + Sqrt[3])/2]*x], -1 
+ Sqrt[3]])/Sqrt[2 + 6*x^2 + 3*x^4])/4
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1405

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]
 

rule 1412

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b 
^2 - 4*a*c, 2]}, Simp[(2*a + (b + q)*x^2)*(Sqrt[(2*a + (b - q)*x^2)/(2*a + 
(b + q)*x^2)]/(2*a*Rt[(b + q)/(2*a), 2]*Sqrt[a + b*x^2 + c*x^4]))*EllipticF 
[ArcTan[Rt[(b + q)/(2*a), 2]*x], 2*(q/(b + q))], x] /; PosQ[(b + q)/a] && 
!(PosQ[(b - q)/a] && SimplerSqrtQ[(b - q)/(2*a), (b + q)/(2*a)])] /; FreeQ[ 
{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
 

rule 1455

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = 
 Rt[b^2 - 4*a*c, 2]}, Simp[x*((b + q + 2*c*x^2)/(2*c*Sqrt[a + b*x^2 + c*x^4 
])), x] - Simp[Rt[(b + q)/(2*a), 2]*(2*a + (b + q)*x^2)*(Sqrt[(2*a + (b - q 
)*x^2)/(2*a + (b + q)*x^2)]/(2*c*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[ArcTan 
[Rt[(b + q)/(2*a), 2]*x], 2*(q/(b + q))], x] /; PosQ[(b + q)/a] &&  !(PosQ[ 
(b - q)/a] && SimplerSqrtQ[(b - q)/(2*a), (b + q)/(2*a)])] /; FreeQ[{a, b, 
c}, x] && GtQ[b^2 - 4*a*c, 0]
 

rule 1503

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[d   Int[1/Sqrt[a + b*x^2 + c*x^4] 
, x], x] + Simp[e   Int[x^2/Sqrt[a + b*x^2 + c*x^4], x], x] /; PosQ[(b + q) 
/a] || PosQ[(b - q)/a]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0]
 

Maple [A] (verified)

Time = 2.60 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.96

method	result	size
risch	\(\frac {x \left (3 x^{2}+4\right )}{4 \sqrt {3 x^{4}+6 x^{2}+2}}-\frac {\sqrt {1-\left (-\frac {3}{2}+\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-6+2 \sqrt {3}}}{2}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )}{\sqrt {-6+2 \sqrt {3}}\, \sqrt {3 x^{4}+6 x^{2}+2}}+\frac {6 \sqrt {1-\left (-\frac {3}{2}+\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-6+2 \sqrt {3}}}{2}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-6+2 \sqrt {3}}}{2}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )\right )}{\sqrt {-6+2 \sqrt {3}}\, \sqrt {3 x^{4}+6 x^{2}+2}\, \left (6+2 \sqrt {3}\right )}\)	\(224\)
default	\(-\frac {6 \left (-\frac {1}{8} x^{3}-\frac {1}{6} x \right )}{\sqrt {3 x^{4}+6 x^{2}+2}}-\frac {\sqrt {1-\left (-\frac {3}{2}+\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-6+2 \sqrt {3}}}{2}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )}{\sqrt {-6+2 \sqrt {3}}\, \sqrt {3 x^{4}+6 x^{2}+2}}+\frac {6 \sqrt {1-\left (-\frac {3}{2}+\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-6+2 \sqrt {3}}}{2}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-6+2 \sqrt {3}}}{2}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )\right )}{\sqrt {-6+2 \sqrt {3}}\, \sqrt {3 x^{4}+6 x^{2}+2}\, \left (6+2 \sqrt {3}\right )}\)	\(225\)
elliptic	\(-\frac {6 \left (-\frac {1}{8} x^{3}-\frac {1}{6} x \right )}{\sqrt {3 x^{4}+6 x^{2}+2}}-\frac {\sqrt {1-\left (-\frac {3}{2}+\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-6+2 \sqrt {3}}}{2}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )}{\sqrt {-6+2 \sqrt {3}}\, \sqrt {3 x^{4}+6 x^{2}+2}}+\frac {6 \sqrt {1-\left (-\frac {3}{2}+\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-6+2 \sqrt {3}}}{2}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-6+2 \sqrt {3}}}{2}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )\right )}{\sqrt {-6+2 \sqrt {3}}\, \sqrt {3 x^{4}+6 x^{2}+2}\, \left (6+2 \sqrt {3}\right )}\)	\(225\)

Input:

int(1/(3*x^4+6*x^2+2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/4*x*(3*x^2+4)/(3*x^4+6*x^2+2)^(1/2)-1/(-6+2*3^(1/2))^(1/2)*(1-(-3/2+1/2* 
3^(1/2))*x^2)^(1/2)*(1-(-3/2-1/2*3^(1/2))*x^2)^(1/2)/(3*x^4+6*x^2+2)^(1/2) 
*EllipticF(1/2*x*(-6+2*3^(1/2))^(1/2),1/2*6^(1/2)+1/2*2^(1/2))+6/(-6+2*3^( 
1/2))^(1/2)*(1-(-3/2+1/2*3^(1/2))*x^2)^(1/2)*(1-(-3/2-1/2*3^(1/2))*x^2)^(1 
/2)/(3*x^4+6*x^2+2)^(1/2)/(6+2*3^(1/2))*(EllipticF(1/2*x*(-6+2*3^(1/2))^(1 
/2),1/2*6^(1/2)+1/2*2^(1/2))-EllipticE(1/2*x*(-6+2*3^(1/2))^(1/2),1/2*6^(1 
/2)+1/2*2^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\left (2+6 x^2+3 x^4\right )^{3/2}} \, dx=\frac {3 \, {\left (\sqrt {3} \sqrt {2} {\left (3 \, x^{4} + 6 \, x^{2} + 2\right )} - 3 \, \sqrt {2} {\left (3 \, x^{4} + 6 \, x^{2} + 2\right )}\right )} \sqrt {\frac {1}{2} \, \sqrt {3} - \frac {3}{2}} E(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {3} - \frac {3}{2}}\right )\,|\,\sqrt {3} + 2) - {\left (\sqrt {3} \sqrt {2} {\left (3 \, x^{4} + 6 \, x^{2} + 2\right )} - 15 \, \sqrt {2} {\left (3 \, x^{4} + 6 \, x^{2} + 2\right )}\right )} \sqrt {\frac {1}{2} \, \sqrt {3} - \frac {3}{2}} F(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {3} - \frac {3}{2}}\right )\,|\,\sqrt {3} + 2) + 6 \, \sqrt {3 \, x^{4} + 6 \, x^{2} + 2} {\left (3 \, x^{3} + 4 \, x\right )}}{24 \, {\left (3 \, x^{4} + 6 \, x^{2} + 2\right )}} \] Input:

integrate(1/(3*x^4+6*x^2+2)^(3/2),x, algorithm="fricas")
 

Output:

1/24*(3*(sqrt(3)*sqrt(2)*(3*x^4 + 6*x^2 + 2) - 3*sqrt(2)*(3*x^4 + 6*x^2 + 
2))*sqrt(1/2*sqrt(3) - 3/2)*elliptic_e(arcsin(x*sqrt(1/2*sqrt(3) - 3/2)), 
sqrt(3) + 2) - (sqrt(3)*sqrt(2)*(3*x^4 + 6*x^2 + 2) - 15*sqrt(2)*(3*x^4 + 
6*x^2 + 2))*sqrt(1/2*sqrt(3) - 3/2)*elliptic_f(arcsin(x*sqrt(1/2*sqrt(3) - 
 3/2)), sqrt(3) + 2) + 6*sqrt(3*x^4 + 6*x^2 + 2)*(3*x^3 + 4*x))/(3*x^4 + 6 
*x^2 + 2)
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {1}{\left (2+6 x^2+3 x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (3 x^{4} + 6 x^{2} + 2\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(3*x**4+6*x**2+2)**(3/2),x)
 

Output:

Integral((3*x**4 + 6*x**2 + 2)**(-3/2), x)
 

Maxima [F]

\[ \int \frac {1}{\left (2+6 x^2+3 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (3 \, x^{4} + 6 \, x^{2} + 2\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(3*x^4+6*x^2+2)^(3/2),x, algorithm="maxima")
 

Output:

integrate((3*x^4 + 6*x^2 + 2)^(-3/2), x)
 

Giac [F]

\[ \int \frac {1}{\left (2+6 x^2+3 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (3 \, x^{4} + 6 \, x^{2} + 2\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(3*x^4+6*x^2+2)^(3/2),x, algorithm="giac")
 

Output:

integrate((3*x^4 + 6*x^2 + 2)^(-3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (2+6 x^2+3 x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (3\,x^4+6\,x^2+2\right )}^{3/2}} \,d x \] Input:

int(1/(6*x^2 + 3*x^4 + 2)^(3/2),x)
 

Output:

int(1/(6*x^2 + 3*x^4 + 2)^(3/2), x)
 

Reduce [F]

\[ \int \frac {1}{\left (2+6 x^2+3 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {3 x^{4}+6 x^{2}+2}}{9 x^{8}+36 x^{6}+48 x^{4}+24 x^{2}+4}d x \] Input:

int(1/(3*x^4+6*x^2+2)^(3/2),x)
 

Output:

int(sqrt(3*x**4 + 6*x**2 + 2)/(9*x**8 + 36*x**6 + 48*x**4 + 24*x**2 + 4),x 
)