\(\int \frac {1}{(-3-6 x^2-2 x^4)^{3/2}} \, dx\) [276]

Optimal result

Integrand size = 16, antiderivative size = 200 \[ \int \frac {1}{\left (-3-6 x^2-2 x^4\right )^{3/2}} \, dx=-\frac {x}{\sqrt {3} \left (3-\sqrt {3}\right ) \sqrt {-3-6 x^2-2 x^4}}+\frac {\sqrt {3-\sqrt {3}} \sqrt {-3+\sqrt {3}-2 x^2} E\left (\arctan \left (\sqrt {\frac {1}{3} \left (3-\sqrt {3}\right )} x\right )|-1-\sqrt {3}\right )}{6 \sqrt {3-\sqrt {3}+2 x^2}}+\frac {\sqrt {\frac {1}{6} \left (3+\sqrt {3}\right )} \sqrt {3-\sqrt {3}+2 x^2} \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{3} \left (3-\sqrt {3}\right )} x\right ),-1-\sqrt {3}\right )}{3 \sqrt {-3+\sqrt {3}-2 x^2}} \] 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)*(- 
3+3^(1/2)-2*x^2)^(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^(1/2)+2*x^2)^(1/2)+1/18*(18+6*3^(1/2))^( 
1/2)*(3-3^(1/2)+2*x^2)^(1/2)*InverseJacobiAM(arctan(1/3*(9-3*3^(1/2))^(1/2 
)*x),(-1-3^(1/2))^(1/2))/(-3+3^(1/2)-2*x^2)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 6.56 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (-3-6 x^2-2 x^4\right )^{3/2}} \, dx=\frac {-4 x \left (2+x^2\right )+i \sqrt {2} \left (-3+\sqrt {3}\right ) \sqrt {\frac {-3+\sqrt {3}-2 x^2}{-3+\sqrt {3}}} \sqrt {3+\sqrt {3}+2 x^2} E\left (i \text {arcsinh}\left (\sqrt {1-\frac {1}{\sqrt {3}}} x\right )|2+\sqrt {3}\right )-i \sqrt {2} \left (-1+\sqrt {3}\right ) \sqrt {\frac {-3+\sqrt {3}-2 x^2}{-3+\sqrt {3}}} \sqrt {3+\sqrt {3}+2 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {1-\frac {1}{\sqrt {3}}} x\right ),2+\sqrt {3}\right )}{12 \sqrt {-3-6 x^2-2 x^4}} \] 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] (warning: unable to verify)

Time = 0.74 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.52, 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.

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

\(\Big \downarrow \) 1405

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1494

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 406

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

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

Input:

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

Output:

-1/3*(x*(2 + x^2))/Sqrt[-3 - 6*x^2 - 2*x^4] + (Sqrt[2]*((x*Sqrt[3 + Sqrt[3 
] + 2*x^2])/(2*Sqrt[-3 + Sqrt[3] - 2*x^2]) - (Sqrt[3/(3 - Sqrt[3])]*Sqrt[3 
 + Sqrt[3] + 2*x^2]*EllipticE[ArcTan[Sqrt[(3 + Sqrt[3])/3]*x], -1 + Sqrt[3 
]])/(2*Sqrt[-3 + Sqrt[3] - 2*x^2]*Sqrt[(3 + Sqrt[3] + 2*x^2)/(3 - Sqrt[3] 
+ 2*x^2)]) + (Sqrt[3/(3 - Sqrt[3])]*Sqrt[3 + Sqrt[3] + 2*x^2]*EllipticF[Ar 
cTan[Sqrt[(3 + Sqrt[3])/3]*x], -1 + Sqrt[3]])/((3 + Sqrt[3])*Sqrt[-3 + Sqr 
t[3] - 2*x^2]*Sqrt[(3 + Sqrt[3] + 2*x^2)/(3 - Sqrt[3] + 2*x^2)])))/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 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 = 1.50 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.10

method	result	size
risch	\(-\frac {\left (x^{2}+2\right ) x}{3 \sqrt {-2 x^{4}-6 x^{2}-3}}+\frac {\sqrt {1-\left (-1-\frac {\sqrt {3}}{3}\right ) x^{2}}\, \sqrt {1-\left (-1+\frac {\sqrt {3}}{3}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-9-3 \sqrt {3}}\, x}{3}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )}{\sqrt {-9-3 \sqrt {3}}\, \sqrt {-2 x^{4}-6 x^{2}-3}}+\frac {6 \sqrt {1-\left (-1-\frac {\sqrt {3}}{3}\right ) x^{2}}\, \sqrt {1-\left (-1+\frac {\sqrt {3}}{3}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-9-3 \sqrt {3}}\, x}{3}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-9-3 \sqrt {3}}\, x}{3}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )\right )}{\sqrt {-9-3 \sqrt {3}}\, \sqrt {-2 x^{4}-6 x^{2}-3}\, \left (-6+2 \sqrt {3}\right )}\)	\(221\)
default	\(\frac {-\frac {2}{3} x -\frac {1}{3} x^{3}}{\sqrt {-2 x^{4}-6 x^{2}-3}}+\frac {\sqrt {1-\left (-1-\frac {\sqrt {3}}{3}\right ) x^{2}}\, \sqrt {1-\left (-1+\frac {\sqrt {3}}{3}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-9-3 \sqrt {3}}\, x}{3}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )}{\sqrt {-9-3 \sqrt {3}}\, \sqrt {-2 x^{4}-6 x^{2}-3}}+\frac {6 \sqrt {1-\left (-1-\frac {\sqrt {3}}{3}\right ) x^{2}}\, \sqrt {1-\left (-1+\frac {\sqrt {3}}{3}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-9-3 \sqrt {3}}\, x}{3}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-9-3 \sqrt {3}}\, x}{3}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )\right )}{\sqrt {-9-3 \sqrt {3}}\, \sqrt {-2 x^{4}-6 x^{2}-3}\, \left (-6+2 \sqrt {3}\right )}\)	\(224\)
elliptic	\(\frac {-\frac {2}{3} x -\frac {1}{3} x^{3}}{\sqrt {-2 x^{4}-6 x^{2}-3}}+\frac {\sqrt {1-\left (-1-\frac {\sqrt {3}}{3}\right ) x^{2}}\, \sqrt {1-\left (-1+\frac {\sqrt {3}}{3}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-9-3 \sqrt {3}}\, x}{3}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )}{\sqrt {-9-3 \sqrt {3}}\, \sqrt {-2 x^{4}-6 x^{2}-3}}+\frac {6 \sqrt {1-\left (-1-\frac {\sqrt {3}}{3}\right ) x^{2}}\, \sqrt {1-\left (-1+\frac {\sqrt {3}}{3}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-9-3 \sqrt {3}}\, x}{3}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-9-3 \sqrt {3}}\, x}{3}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )\right )}{\sqrt {-9-3 \sqrt {3}}\, \sqrt {-2 x^{4}-6 x^{2}-3}\, \left (-6+2 \sqrt {3}\right )}\)	\(224\)

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)*El 
lipticF(1/3*(-9-3*3^(1/2))^(1/2)*x,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*(-9-3*3^(1/2))^(1/2)*x,1 
/2*6^(1/2)-1/2*2^(1/2))-EllipticE(1/3*(-9-3*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 = 149, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\left (-3-6 x^2-2 x^4\right )^{3/2}} \, dx=\frac {6 \, \sqrt {-3} {\left (2 \, x^{4} + 6 \, x^{2} + 3\right )} \sqrt {\frac {1}{3} \, \sqrt {3} - 1} F(\arcsin \left (x \sqrt {\frac {1}{3} \, \sqrt {3} - 1}\right )\,|\,\sqrt {3} + 2) + {\left (\sqrt {3} \sqrt {-3} {\left (2 \, x^{4} + 6 \, x^{2} + 3\right )} - 3 \, \sqrt {-3} {\left (2 \, x^{4} + 6 \, x^{2} + 3\right )}\right )} \sqrt {\frac {1}{3} \, \sqrt {3} - 1} E(\arcsin \left (x \sqrt {\frac {1}{3} \, \sqrt {3} - 1}\right )\,|\,\sqrt {3} + 2) + 6 \, \sqrt {-2 \, x^{4} - 6 \, x^{2} - 3} {\left (x^{3} + 2 \, x\right )}}{18 \, {\left (2 \, x^{4} + 6 \, x^{2} + 3\right )}} \] Input:

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

Output:

1/18*(6*sqrt(-3)*(2*x^4 + 6*x^2 + 3)*sqrt(1/3*sqrt(3) - 1)*elliptic_f(arcs 
in(x*sqrt(1/3*sqrt(3) - 1)), sqrt(3) + 2) + (sqrt(3)*sqrt(-3)*(2*x^4 + 6*x 
^2 + 3) - 3*sqrt(-3)*(2*x^4 + 6*x^2 + 3))*sqrt(1/3*sqrt(3) - 1)*elliptic_e 
(arcsin(x*sqrt(1/3*sqrt(3) - 1)), sqrt(3) + 2) + 6*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)