\(\int \frac {1}{(-2-6 x^2-3 x^4)^{3/2}} \, dx\) [291]

Optimal result

Integrand size = 16, antiderivative size = 198 \[ \int \frac {1}{\left (-2-6 x^2-3 x^4\right )^{3/2}} \, dx=-\frac {\sqrt {3} x}{2 \left (3-\sqrt {3}\right ) \sqrt {-2-6 x^2-3 x^4}}+\frac {\sqrt {3-\sqrt {3}} \sqrt {-3+\sqrt {3}-3 x^2} E\left (\arctan \left (\sqrt {\frac {1}{2} \left (3-\sqrt {3}\right )} x\right )|-1-\sqrt {3}\right )}{4 \sqrt {3-\sqrt {3}+3 x^2}}+\frac {\sqrt {3-\sqrt {3}+3 x^2} \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {3}{3+\sqrt {3}}} x\right ),-1-\sqrt {3}\right )}{2 \sqrt {3-\sqrt {3}} \sqrt {-3+\sqrt {3}-3 x^2}} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 6.68 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\left (-2-6 x^2-3 x^4\right )^{3/2}} \, dx=\frac {-3 x \left (4+3 x^2\right )-3 i \left (-1+\sqrt {3}\right ) \sqrt {\frac {-3+\sqrt {3}-3 x^2}{-3+\sqrt {3}}} \sqrt {3+\sqrt {3}+3 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{3+\sqrt {3}}} x\right )|2+\sqrt {3}\right )-i \sqrt {\left (-3+\sqrt {3}\right ) \left (-3+\sqrt {3}-3 x^2\right )} \sqrt {3+\sqrt {3}+3 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{3+\sqrt {3}}} x\right ),2+\sqrt {3}\right )}{12 \sqrt {-2-6 x^2-3 x^4}} \] Input:

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

Output:

(-3*x*(4 + 3*x^2) - (3*I)*(-1 + Sqrt[3])*Sqrt[(-3 + Sqrt[3] - 3*x^2)/(-3 + 
 Sqrt[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 + Sq 
rt[3]])/(12*Sqrt[-2 - 6*x^2 - 3*x^4])
 

Rubi [A] (warning: unable to verify)

Time = 0.73 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.53, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1405, 27, 1494, 27, 406, 320, 388, 313}

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

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1494

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 406

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

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

Input:

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

Output:

-1/4*(x*(4 + 3*x^2))/Sqrt[-2 - 6*x^2 - 3*x^4] + (Sqrt[3]*(3*(-1/3*(x*Sqrt[ 
-3 + Sqrt[3] - 3*x^2])/Sqrt[3 + Sqrt[3] + 3*x^2] + (Sqrt[(3 - Sqrt[3])/3]* 
Sqrt[-3 + Sqrt[3] - 3*x^2]*EllipticE[ArcTan[Sqrt[3/(3 + Sqrt[3])]*x], -1 - 
 Sqrt[3]])/(3*Sqrt[(3 - Sqrt[3] + 3*x^2)/(3 + Sqrt[3] + 3*x^2)]*Sqrt[3 + S 
qrt[3] + 3*x^2])) - (2*Sqrt[-3 + Sqrt[3] - 3*x^2]*EllipticF[ArcTan[Sqrt[3/ 
(3 + Sqrt[3])]*x], -1 - Sqrt[3]])/(Sqrt[3*(3 - Sqrt[3])]*Sqrt[(3 - Sqrt[3] 
 + 3*x^2)/(3 + Sqrt[3] + 3*x^2)]*Sqrt[3 + Sqrt[3] + 3*x^2])))/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 313

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim 
p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c 
+ d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ 
[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
 

rule 320

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( 
c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre 
eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 388

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] 
 :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b   Int[Sqrt[ 
a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 406

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[e   Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim 
p[f   Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, 
f, p, q}, 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 1494

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

Maple [A] (verified)

Time = 2.10 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.13

method	result	size
risch	\(-\frac {x \left (3 x^{2}+4\right )}{4 \sqrt {-3 x^{4}-6 x^{2}-2}}+\frac {\sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}+\frac {\sqrt {3}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-6-2 \sqrt {3}}\, x}{2}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )}{\sqrt {-6-2 \sqrt {3}}\, \sqrt {-3 x^{4}-6 x^{2}-2}}+\frac {6 \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}+\frac {\sqrt {3}}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-6-2 \sqrt {3}}\, x}{2}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-6-2 \sqrt {3}}\, x}{2}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )\right )}{\sqrt {-6-2 \sqrt {3}}\, \sqrt {-3 x^{4}-6 x^{2}-2}\, \left (-6+2 \sqrt {3}\right )}\)	\(223\)
default	\(\frac {-\frac {3}{4} x^{3}-x}{\sqrt {-3 x^{4}-6 x^{2}-2}}+\frac {\sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}+\frac {\sqrt {3}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-6-2 \sqrt {3}}\, x}{2}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )}{\sqrt {-6-2 \sqrt {3}}\, \sqrt {-3 x^{4}-6 x^{2}-2}}+\frac {6 \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}+\frac {\sqrt {3}}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-6-2 \sqrt {3}}\, x}{2}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-6-2 \sqrt {3}}\, x}{2}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )\right )}{\sqrt {-6-2 \sqrt {3}}\, \sqrt {-3 x^{4}-6 x^{2}-2}\, \left (-6+2 \sqrt {3}\right )}\)	\(224\)
elliptic	\(\frac {-\frac {3}{4} x^{3}-x}{\sqrt {-3 x^{4}-6 x^{2}-2}}+\frac {\sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}+\frac {\sqrt {3}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-6-2 \sqrt {3}}\, x}{2}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )}{\sqrt {-6-2 \sqrt {3}}\, \sqrt {-3 x^{4}-6 x^{2}-2}}+\frac {6 \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}+\frac {\sqrt {3}}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-6-2 \sqrt {3}}\, x}{2}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-6-2 \sqrt {3}}\, x}{2}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )\right )}{\sqrt {-6-2 \sqrt {3}}\, \sqrt {-3 x^{4}-6 x^{2}-2}\, \left (-6+2 \sqrt {3}\right )}\)	\(224\)

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (-2-6 x^2-3 x^4\right )^{3/2}} \, dx=\frac {3 \, {\left (\sqrt {3} \sqrt {-2} {\left (3 \, x^{4} + 6 \, x^{2} + 2\right )} - 3 \, \sqrt {-2} {\left (3 \, x^{4} + 6 \, x^{2} + 2\right )}\right )} \sqrt {\frac {1}{2} \, \sqrt {3} - \frac {3}{2}} E(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {3} - \frac {3}{2}}\right )\,|\,\sqrt {3} + 2) - {\left (\sqrt {3} \sqrt {-2} {\left (3 \, x^{4} + 6 \, x^{2} + 2\right )} - 15 \, \sqrt {-2} {\left (3 \, x^{4} + 6 \, x^{2} + 2\right )}\right )} \sqrt {\frac {1}{2} \, \sqrt {3} - \frac {3}{2}} F(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {3} - \frac {3}{2}}\right )\,|\,\sqrt {3} + 2) + 6 \, \sqrt {-3 \, x^{4} - 6 \, x^{2} - 2} {\left (3 \, x^{3} + 4 \, x\right )}}{24 \, {\left (3 \, x^{4} + 6 \, x^{2} + 2\right )}} \] 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)