\(\int \frac {1}{(-3-4 x^2-2 x^4)^{3/2}} \, dx\) [274]

Optimal result

Integrand size = 16, antiderivative size = 247 \[ \int \frac {1}{\left (-3-4 x^2-2 x^4\right )^{3/2}} \, dx=\frac {x \left (1+2 x^2\right )}{6 \sqrt {-3-4 x^2-2 x^4}}+\frac {x \sqrt {-3-4 x^2-2 x^4}}{3 \left (\sqrt {6}+2 x^2\right )}+\frac {\left (3+\sqrt {6} x^2\right ) \sqrt {\frac {3+4 x^2+2 x^4}{\left (3+\sqrt {6} x^2\right )^2}} E\left (2 \arctan \left (\sqrt [4]{\frac {2}{3}} x\right )|\frac {1}{2}-\frac {1}{\sqrt {6}}\right )}{6^{3/4} \sqrt {-3-4 x^2-2 x^4}}-\frac {\left (2+\sqrt {6}\right ) \left (3+\sqrt {6} x^2\right ) \sqrt {\frac {3+4 x^2+2 x^4}{\left (3+\sqrt {6} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{\frac {2}{3}} x\right ),\frac {1}{2}-\frac {1}{\sqrt {6}}\right )}{4\ 6^{3/4} \sqrt {-3-4 x^2-2 x^4}} \] Output:

1/6*x*(2*x^2+1)/(-2*x^4-4*x^2-3)^(1/2)+x*(-2*x^4-4*x^2-3)^(1/2)/(3*6^(1/2) 
+6*x^2)+1/6*(3+6^(1/2)*x^2)*((2*x^4+4*x^2+3)/(3+6^(1/2)*x^2)^2)^(1/2)*Elli 
pticE(sin(2*arctan(1/3*2^(1/4)*3^(3/4)*x)),1/6*(18-6*6^(1/2))^(1/2))*6^(1/ 
4)/(-2*x^4-4*x^2-3)^(1/2)-1/24*(2+6^(1/2))*(3+6^(1/2)*x^2)*((2*x^4+4*x^2+3 
)/(3+6^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(1/3*2^(1/4)*3^(3/4)*x) 
,1/6*(18-6*6^(1/2))^(1/2))*6^(1/4)/(-2*x^4-4*x^2-3)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.50 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.31 \[ \int \frac {1}{\left (-3-4 x^2-2 x^4\right )^{3/2}} \, dx=\frac {2 \sqrt {-\frac {i}{-2 i+\sqrt {2}}} x \left (1+2 x^2\right )+2 \left (1+i \sqrt {2}\right ) \sqrt {\frac {-2 i+\sqrt {2}-2 i x^2}{-2 i+\sqrt {2}}} \sqrt {\frac {2 i+\sqrt {2}+2 i x^2}{2 i+\sqrt {2}}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {2 i}{-2 i+\sqrt {2}}} x\right )|\frac {2 i-\sqrt {2}}{2 i+\sqrt {2}}\right )+i \left (2 i+\sqrt {2}\right ) \sqrt {\frac {-2 i+\sqrt {2}-2 i x^2}{-2 i+\sqrt {2}}} \sqrt {\frac {2 i+\sqrt {2}+2 i x^2}{2 i+\sqrt {2}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {2 i}{-2 i+\sqrt {2}}} x\right ),\frac {2 i-\sqrt {2}}{2 i+\sqrt {2}}\right )}{12 \sqrt {-\frac {i}{-2 i+\sqrt {2}}} \sqrt {-3-4 x^2-2 x^4}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1405, 27, 1511, 27, 1416, 1509}

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

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1511

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1416

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

\(\Big \downarrow \) 1509

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

Input:

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

Output:

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

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 1416

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

rule 1509

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q 
^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* 
x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 
/(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 
- 4*a*c, 0] && PosQ[c/a]
 

rule 1511

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q   Int[1/Sqrt[a + b*x^2 + c*x^ 
4], x], x] - Simp[e/q   Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; 
NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos 
Q[c/a]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 2.55 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.96

method	result	size
risch	\(\frac {x \left (2 x^{2}+1\right )}{6 \sqrt {-2 x^{4}-4 x^{2}-3}}-\frac {3 \sqrt {1-\left (-\frac {2}{3}-\frac {i \sqrt {2}}{3}\right ) x^{2}}\, \sqrt {1-\left (-\frac {2}{3}+\frac {i \sqrt {2}}{3}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-6-3 i \sqrt {2}}\, x}{3}, \frac {\sqrt {3-6 i \sqrt {2}}}{3}\right )}{2 \sqrt {-6-3 i \sqrt {2}}\, \sqrt {-2 x^{4}-4 x^{2}-3}}-\frac {6 \sqrt {1-\left (-\frac {2}{3}-\frac {i \sqrt {2}}{3}\right ) x^{2}}\, \sqrt {1-\left (-\frac {2}{3}+\frac {i \sqrt {2}}{3}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-6-3 i \sqrt {2}}\, x}{3}, \frac {\sqrt {3-6 i \sqrt {2}}}{3}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-6-3 i \sqrt {2}}\, x}{3}, \frac {\sqrt {3-6 i \sqrt {2}}}{3}\right )\right )}{\sqrt {-6-3 i \sqrt {2}}\, \sqrt {-2 x^{4}-4 x^{2}-3}\, \left (-4+2 i \sqrt {2}\right )}\)	\(237\)
default	\(\frac {\frac {1}{3} x^{3}+\frac {1}{6} x}{\sqrt {-2 x^{4}-4 x^{2}-3}}-\frac {3 \sqrt {1-\left (-\frac {2}{3}-\frac {i \sqrt {2}}{3}\right ) x^{2}}\, \sqrt {1-\left (-\frac {2}{3}+\frac {i \sqrt {2}}{3}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-6-3 i \sqrt {2}}\, x}{3}, \frac {\sqrt {3-6 i \sqrt {2}}}{3}\right )}{2 \sqrt {-6-3 i \sqrt {2}}\, \sqrt {-2 x^{4}-4 x^{2}-3}}-\frac {6 \sqrt {1-\left (-\frac {2}{3}-\frac {i \sqrt {2}}{3}\right ) x^{2}}\, \sqrt {1-\left (-\frac {2}{3}+\frac {i \sqrt {2}}{3}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-6-3 i \sqrt {2}}\, x}{3}, \frac {\sqrt {3-6 i \sqrt {2}}}{3}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-6-3 i \sqrt {2}}\, x}{3}, \frac {\sqrt {3-6 i \sqrt {2}}}{3}\right )\right )}{\sqrt {-6-3 i \sqrt {2}}\, \sqrt {-2 x^{4}-4 x^{2}-3}\, \left (-4+2 i \sqrt {2}\right )}\)	\(238\)
elliptic	\(\frac {\frac {1}{3} x^{3}+\frac {1}{6} x}{\sqrt {-2 x^{4}-4 x^{2}-3}}-\frac {3 \sqrt {1-\left (-\frac {2}{3}-\frac {i \sqrt {2}}{3}\right ) x^{2}}\, \sqrt {1-\left (-\frac {2}{3}+\frac {i \sqrt {2}}{3}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-6-3 i \sqrt {2}}\, x}{3}, \frac {\sqrt {3-6 i \sqrt {2}}}{3}\right )}{2 \sqrt {-6-3 i \sqrt {2}}\, \sqrt {-2 x^{4}-4 x^{2}-3}}-\frac {6 \sqrt {1-\left (-\frac {2}{3}-\frac {i \sqrt {2}}{3}\right ) x^{2}}\, \sqrt {1-\left (-\frac {2}{3}+\frac {i \sqrt {2}}{3}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-6-3 i \sqrt {2}}\, x}{3}, \frac {\sqrt {3-6 i \sqrt {2}}}{3}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-6-3 i \sqrt {2}}\, x}{3}, \frac {\sqrt {3-6 i \sqrt {2}}}{3}\right )\right )}{\sqrt {-6-3 i \sqrt {2}}\, \sqrt {-2 x^{4}-4 x^{2}-3}\, \left (-4+2 i \sqrt {2}\right )}\)	\(238\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.71 \[ \int \frac {1}{\left (-3-4 x^2-2 x^4\right )^{3/2}} \, dx=-\frac {2 \, {\left (\sqrt {-2} \sqrt {-3} {\left (2 \, x^{4} + 4 \, x^{2} + 3\right )} - 2 \, \sqrt {-3} {\left (2 \, x^{4} + 4 \, x^{2} + 3\right )}\right )} \sqrt {\frac {1}{3} \, \sqrt {-2} - \frac {2}{3}} E(\arcsin \left (x \sqrt {\frac {1}{3} \, \sqrt {-2} - \frac {2}{3}}\right )\,|\,\frac {2}{3} \, \sqrt {-2} + \frac {1}{3}) + {\left (\sqrt {-2} \sqrt {-3} {\left (2 \, x^{4} + 4 \, x^{2} + 3\right )} + 10 \, \sqrt {-3} {\left (2 \, x^{4} + 4 \, x^{2} + 3\right )}\right )} \sqrt {\frac {1}{3} \, \sqrt {-2} - \frac {2}{3}} F(\arcsin \left (x \sqrt {\frac {1}{3} \, \sqrt {-2} - \frac {2}{3}}\right )\,|\,\frac {2}{3} \, \sqrt {-2} + \frac {1}{3}) + 6 \, \sqrt {-2 \, x^{4} - 4 \, x^{2} - 3} {\left (2 \, x^{3} + x\right )}}{36 \, {\left (2 \, x^{4} + 4 \, x^{2} + 3\right )}} \] Input:

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

Output:

-1/36*(2*(sqrt(-2)*sqrt(-3)*(2*x^4 + 4*x^2 + 3) - 2*sqrt(-3)*(2*x^4 + 4*x^ 
2 + 3))*sqrt(1/3*sqrt(-2) - 2/3)*elliptic_e(arcsin(x*sqrt(1/3*sqrt(-2) - 2 
/3)), 2/3*sqrt(-2) + 1/3) + (sqrt(-2)*sqrt(-3)*(2*x^4 + 4*x^2 + 3) + 10*sq 
rt(-3)*(2*x^4 + 4*x^2 + 3))*sqrt(1/3*sqrt(-2) - 2/3)*elliptic_f(arcsin(x*s 
qrt(1/3*sqrt(-2) - 2/3)), 2/3*sqrt(-2) + 1/3) + 6*sqrt(-2*x^4 - 4*x^2 - 3) 
*(2*x^3 + x))/(2*x^4 + 4*x^2 + 3)
 

Sympy [F]

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

Maxima [F]

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

Giac [F]

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

Mupad [F(-1)]

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)
 

Reduce [F]

\[ \int \frac {1}{\left (-3-4 x^2-2 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {-2 x^{4}-4 x^{2}-3}}{4 x^{8}+16 x^{6}+28 x^{4}+24 x^{2}+9}d x \] 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 + 28*x**4 + 24*x**2 + 9 
),x)