\(\int \frac {1}{(3+6 x^2+2 x^4)^{3/2}} \, dx\) [248]

Optimal result

Integrand size = 16, antiderivative size = 234 \[ \int \frac {1}{\left (3+6 x^2+2 x^4\right )^{3/2}} \, dx=\frac {x}{\sqrt {3} \left (3-\sqrt {3}\right ) \sqrt {3+6 x^2+2 x^4}}+\frac {\sqrt {3-\sqrt {3}} \sqrt {3+6 x^2+2 x^4} E\left (\arctan \left (\sqrt {\frac {1}{3} \left (3-\sqrt {3}\right )} x\right )|-1-\sqrt {3}\right )}{6 \sqrt {\frac {3}{3+\sqrt {3}}+x^2} \sqrt {3+\sqrt {3}+2 x^2}}-\frac {\sqrt {\frac {1}{2} \left (3+\sqrt {3}\right )} \sqrt {3+\left (3-\sqrt {3}\right ) x^2} \sqrt {3+\left (3+\sqrt {3}\right ) x^2} \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{3} \left (3-\sqrt {3}\right )} x\right ),-1-\sqrt {3}\right )}{9 \sqrt {3+6 x^2+2 x^4}} \] Output:

1/3*x*3^(1/2)/(3-3^(1/2))/(2*x^4+6*x^2+3)^(1/2)+1/6*(3-3^(1/2))^(1/2)*(2*x 
^4+6*x^2+3)^(1/2)*EllipticE((9-3*3^(1/2))^(1/2)*x/(9+(9-3*3^(1/2))*x^2)^(1 
/2),(-1-3^(1/2))^(1/2))/(3/(3+3^(1/2))+x^2)^(1/2)/(3+3^(1/2)+2*x^2)^(1/2)- 
1/18*(6+2*3^(1/2))^(1/2)*(3+(3-3^(1/2))*x^2)^(1/2)*(3+(3+3^(1/2))*x^2)^(1/ 
2)*InverseJacobiAM(arctan(1/3*(9-3*3^(1/2))^(1/2)*x),(-1-3^(1/2))^(1/2))/( 
2*x^4+6*x^2+3)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 6.08 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\left (3+6 x^2+2 x^4\right )^{3/2}} \, dx=\frac {4 x \left (2+x^2\right )-i \sqrt {2} \left (-3+\sqrt {3}\right ) \sqrt {\frac {-3+\sqrt {3}-2 x^2}{-3+\sqrt {3}}} \sqrt {3+\sqrt {3}+2 x^2} E\left (i \text {arcsinh}\left (\sqrt {1-\frac {1}{\sqrt {3}}} x\right )|2+\sqrt {3}\right )+i \sqrt {2} \left (-1+\sqrt {3}\right ) \sqrt {\frac {-3+\sqrt {3}-2 x^2}{-3+\sqrt {3}}} \sqrt {3+\sqrt {3}+2 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {1-\frac {1}{\sqrt {3}}} x\right ),2+\sqrt {3}\right )}{12 \sqrt {3+6 x^2+2 x^4}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.19, 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.

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

\(\Big \downarrow \) 1405

\(\displaystyle \frac {x \left (x^2+2\right )}{3 \sqrt {2 x^4+6 x^2+3}}-\frac {1}{36} \int \frac {12 \left (x^2+1\right )}{\sqrt {2 x^4+6 x^2+3}}dx\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1503

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

\(\Big \downarrow \) 1412

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

\(\Big \downarrow \) 1455

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

Input:

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

Output:

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

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 = 1.98 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.95

method	result	size
risch	\(\frac {\left (x^{2}+2\right ) x}{3 \sqrt {2 x^{4}+6 x^{2}+3}}-\frac {\sqrt {1-\left (-1+\frac {\sqrt {3}}{3}\right ) x^{2}}\, \sqrt {1-\left (-1-\frac {\sqrt {3}}{3}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-9+3 \sqrt {3}}}{3}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )}{\sqrt {-9+3 \sqrt {3}}\, \sqrt {2 x^{4}+6 x^{2}+3}}+\frac {6 \sqrt {1-\left (-1+\frac {\sqrt {3}}{3}\right ) x^{2}}\, \sqrt {1-\left (-1-\frac {\sqrt {3}}{3}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-9+3 \sqrt {3}}}{3}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-9+3 \sqrt {3}}}{3}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )\right )}{\sqrt {-9+3 \sqrt {3}}\, \sqrt {2 x^{4}+6 x^{2}+3}\, \left (6+2 \sqrt {3}\right )}\)	\(222\)
default	\(-\frac {4 \left (-\frac {1}{6} x -\frac {1}{12} x^{3}\right )}{\sqrt {2 x^{4}+6 x^{2}+3}}-\frac {\sqrt {1-\left (-1+\frac {\sqrt {3}}{3}\right ) x^{2}}\, \sqrt {1-\left (-1-\frac {\sqrt {3}}{3}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-9+3 \sqrt {3}}}{3}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )}{\sqrt {-9+3 \sqrt {3}}\, \sqrt {2 x^{4}+6 x^{2}+3}}+\frac {6 \sqrt {1-\left (-1+\frac {\sqrt {3}}{3}\right ) x^{2}}\, \sqrt {1-\left (-1-\frac {\sqrt {3}}{3}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-9+3 \sqrt {3}}}{3}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-9+3 \sqrt {3}}}{3}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )\right )}{\sqrt {-9+3 \sqrt {3}}\, \sqrt {2 x^{4}+6 x^{2}+3}\, \left (6+2 \sqrt {3}\right )}\)	\(225\)
elliptic	\(-\frac {4 \left (-\frac {1}{6} x -\frac {1}{12} x^{3}\right )}{\sqrt {2 x^{4}+6 x^{2}+3}}-\frac {\sqrt {1-\left (-1+\frac {\sqrt {3}}{3}\right ) x^{2}}\, \sqrt {1-\left (-1-\frac {\sqrt {3}}{3}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-9+3 \sqrt {3}}}{3}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )}{\sqrt {-9+3 \sqrt {3}}\, \sqrt {2 x^{4}+6 x^{2}+3}}+\frac {6 \sqrt {1-\left (-1+\frac {\sqrt {3}}{3}\right ) x^{2}}\, \sqrt {1-\left (-1-\frac {\sqrt {3}}{3}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-9+3 \sqrt {3}}}{3}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-9+3 \sqrt {3}}}{3}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )\right )}{\sqrt {-9+3 \sqrt {3}}\, \sqrt {2 x^{4}+6 x^{2}+3}\, \left (6+2 \sqrt {3}\right )}\)	\(225\)

Input:

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

Output:

1/3*(x^2+2)*x/(2*x^4+6*x^2+3)^(1/2)-1/(-9+3*3^(1/2))^(1/2)*(1-(-1+1/3*3^(1 
/2))*x^2)^(1/2)*(1-(-1-1/3*3^(1/2))*x^2)^(1/2)/(2*x^4+6*x^2+3)^(1/2)*Ellip 
ticF(1/3*x*(-9+3*3^(1/2))^(1/2),1/2*6^(1/2)+1/2*2^(1/2))+6/(-9+3*3^(1/2))^ 
(1/2)*(1-(-1+1/3*3^(1/2))*x^2)^(1/2)*(1-(-1-1/3*3^(1/2))*x^2)^(1/2)/(2*x^4 
+6*x^2+3)^(1/2)/(6+2*3^(1/2))*(EllipticF(1/3*x*(-9+3*3^(1/2))^(1/2),1/2*6^ 
(1/2)+1/2*2^(1/2))-EllipticE(1/3*x*(-9+3*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 = 141, normalized size of antiderivative = 0.60 \[ \int \frac {1}{\left (3+6 x^2+2 x^4\right )^{3/2}} \, dx=\frac {2 \, \sqrt {3} {\left (2 \, x^{4} + 6 \, x^{2} + 3\right )} \sqrt {\frac {1}{3} \, \sqrt {3} - 1} F(\arcsin \left (x \sqrt {\frac {1}{3} \, \sqrt {3} - 1}\right )\,|\,\sqrt {3} + 2) + {\left (2 \, x^{4} + 6 \, x^{2} - \sqrt {3} {\left (2 \, x^{4} + 6 \, x^{2} + 3\right )} + 3\right )} \sqrt {\frac {1}{3} \, \sqrt {3} - 1} E(\arcsin \left (x \sqrt {\frac {1}{3} \, \sqrt {3} - 1}\right )\,|\,\sqrt {3} + 2) + 2 \, \sqrt {2 \, x^{4} + 6 \, x^{2} + 3} {\left (x^{3} + 2 \, x\right )}}{6 \, {\left (2 \, x^{4} + 6 \, x^{2} + 3\right )}} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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