\(\int \frac {1}{(-2+x^2+3 x^4)^{3/2}} \, dx\) [208]

Optimal result

Integrand size = 14, antiderivative size = 141 \[ \int \frac {1}{\left (-2+x^2+3 x^4\right )^{3/2}} \, dx=-\frac {x \left (13+3 x^2\right )}{50 \sqrt {-2+x^2+3 x^4}}+\frac {\sqrt {3} \sqrt {2-3 x^2} \sqrt {1+x^2} E\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right )|-\frac {2}{3}\right )}{50 \sqrt {-2+x^2+3 x^4}}-\frac {\sqrt {3} \sqrt {2-3 x^2} \sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right ),-\frac {2}{3}\right )}{10 \sqrt {-2+x^2+3 x^4}} \] Output:

-1/50*x*(3*x^2+13)/(3*x^4+x^2-2)^(1/2)+1/50*3^(1/2)*(-3*x^2+2)^(1/2)*(x^2+ 
1)^(1/2)*EllipticE(1/2*x*6^(1/2),1/3*I*6^(1/2))/(3*x^4+x^2-2)^(1/2)-1/10*( 
-3*x^2+2)^(1/2)*(x^2+1)^(1/2)*EllipticF(1/2*x*6^(1/2),1/3*I*6^(1/2))*3^(1/ 
2)/(3*x^4+x^2-2)^(1/2)
 

Mathematica [A] (verified)

Time = 6.74 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.70 \[ \int \frac {1}{\left (-2+x^2+3 x^4\right )^{3/2}} \, dx=\frac {-13 x-3 x^3+\sqrt {6-9 x^2} \sqrt {1+x^2} E\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right )|-\frac {2}{3}\right )-5 \sqrt {6-9 x^2} \sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right ),-\frac {2}{3}\right )}{50 \sqrt {-2+x^2+3 x^4}} \] Input:

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

Output:

(-13*x - 3*x^3 + Sqrt[6 - 9*x^2]*Sqrt[1 + x^2]*EllipticE[ArcSin[Sqrt[3/2]* 
x], -2/3] - 5*Sqrt[6 - 9*x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[Sqrt[3/2]*x], 
 -2/3])/(50*Sqrt[-2 + x^2 + 3*x^4])
 

Rubi [A] (warning: unable to verify)

Time = 0.55 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.50, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1405, 27, 1501, 27, 1410, 1498}

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.

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

\(\Big \downarrow \) 1405

\(\displaystyle \frac {1}{50} \int -\frac {3 \left (4-x^2\right )}{\sqrt {3 x^4+x^2-2}}dx-\frac {x \left (3 x^2+13\right )}{50 \sqrt {3 x^4+x^2-2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3}{50} \int \frac {4-x^2}{\sqrt {3 x^4+x^2-2}}dx-\frac {x \left (3 x^2+13\right )}{50 \sqrt {3 x^4+x^2-2}}\)

\(\Big \downarrow \) 1501

\(\displaystyle -\frac {3}{50} \left (\frac {10}{3} \int \frac {1}{\sqrt {3 x^4+x^2-2}}dx-\frac {1}{6} \int -\frac {2 \left (2-3 x^2\right )}{\sqrt {3 x^4+x^2-2}}dx\right )-\frac {x \left (3 x^2+13\right )}{50 \sqrt {3 x^4+x^2-2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3}{50} \left (\frac {10}{3} \int \frac {1}{\sqrt {3 x^4+x^2-2}}dx+\frac {1}{3} \int \frac {2-3 x^2}{\sqrt {3 x^4+x^2-2}}dx\right )-\frac {x \left (3 x^2+13\right )}{50 \sqrt {3 x^4+x^2-2}}\)

\(\Big \downarrow \) 1410

\(\displaystyle -\frac {3}{50} \left (\frac {1}{3} \int \frac {2-3 x^2}{\sqrt {3 x^4+x^2-2}}dx+\frac {2 \sqrt {5} \sqrt {x^2+1} \sqrt {3 x^2-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {5} x}{\sqrt {3 x^2-2}}\right ),\frac {3}{5}\right )}{3 \sqrt {3 x^4+x^2-2}}\right )-\frac {x \left (3 x^2+13\right )}{50 \sqrt {3 x^4+x^2-2}}\)

\(\Big \downarrow \) 1498

\(\displaystyle -\frac {3}{50} \left (\frac {2 \sqrt {5} \sqrt {x^2+1} \sqrt {3 x^2-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {5} x}{\sqrt {3 x^2-2}}\right ),\frac {3}{5}\right )}{3 \sqrt {3 x^4+x^2-2}}+\frac {1}{3} \left (\frac {\sqrt {5} \sqrt {\frac {x^2+1}{2-3 x^2}} \sqrt {3 x^2-2} E\left (\arcsin \left (\frac {\sqrt {5} x}{\sqrt {3 x^2-2}}\right )|\frac {3}{5}\right )}{\sqrt {\frac {1}{2-3 x^2}} \sqrt {3 x^4+x^2-2}}-\frac {3 x \left (x^2+1\right )}{\sqrt {3 x^4+x^2-2}}\right )\right )-\frac {x \left (3 x^2+13\right )}{50 \sqrt {3 x^4+x^2-2}}\)

Input:

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

Output:

-1/50*(x*(13 + 3*x^2))/Sqrt[-2 + x^2 + 3*x^4] - (3*(((-3*x*(1 + x^2))/Sqrt 
[-2 + x^2 + 3*x^4] + (Sqrt[5]*Sqrt[(1 + x^2)/(2 - 3*x^2)]*Sqrt[-2 + 3*x^2] 
*EllipticE[ArcSin[(Sqrt[5]*x)/Sqrt[-2 + 3*x^2]], 3/5])/(Sqrt[(2 - 3*x^2)^( 
-1)]*Sqrt[-2 + x^2 + 3*x^4]))/3 + (2*Sqrt[5]*Sqrt[1 + x^2]*Sqrt[-2 + 3*x^2 
]*EllipticF[ArcSin[(Sqrt[5]*x)/Sqrt[-2 + 3*x^2]], 3/5])/(3*Sqrt[-2 + x^2 + 
 3*x^4])))/50
 

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 1410

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

rule 1498

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

rule 1501

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

Maple [A] (verified)

Time = 1.77 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.87

method	result	size
risch	\(-\frac {x \left (3 x^{2}+13\right )}{50 \sqrt {3 x^{4}+x^{2}-2}}+\frac {3 i \sqrt {x^{2}+1}\, \sqrt {-6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {i \sqrt {6}}{2}\right )}{25 \sqrt {3 x^{4}+x^{2}-2}}-\frac {i \sqrt {x^{2}+1}\, \sqrt {-6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {i \sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {i \sqrt {6}}{2}\right )\right )}{50 \sqrt {3 x^{4}+x^{2}-2}}\)	\(122\)
default	\(-\frac {6 \left (\frac {13}{300} x +\frac {1}{100} x^{3}\right )}{\sqrt {3 x^{4}+x^{2}-2}}+\frac {3 i \sqrt {x^{2}+1}\, \sqrt {-6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {i \sqrt {6}}{2}\right )}{25 \sqrt {3 x^{4}+x^{2}-2}}-\frac {i \sqrt {x^{2}+1}\, \sqrt {-6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {i \sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {i \sqrt {6}}{2}\right )\right )}{50 \sqrt {3 x^{4}+x^{2}-2}}\)	\(123\)
elliptic	\(-\frac {6 \left (\frac {13}{300} x +\frac {1}{100} x^{3}\right )}{\sqrt {3 x^{4}+x^{2}-2}}+\frac {3 i \sqrt {x^{2}+1}\, \sqrt {-6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {i \sqrt {6}}{2}\right )}{25 \sqrt {3 x^{4}+x^{2}-2}}-\frac {i \sqrt {x^{2}+1}\, \sqrt {-6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {i \sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {i \sqrt {6}}{2}\right )\right )}{50 \sqrt {3 x^{4}+x^{2}-2}}\)	\(123\)

Input:

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

Output:

-1/50*x*(3*x^2+13)/(3*x^4+x^2-2)^(1/2)+3/25*I*(x^2+1)^(1/2)*(-6*x^2+4)^(1/ 
2)/(3*x^4+x^2-2)^(1/2)*EllipticF(I*x,1/2*I*6^(1/2))-1/50*I*(x^2+1)^(1/2)*( 
-6*x^2+4)^(1/2)/(3*x^4+x^2-2)^(1/2)*(EllipticF(I*x,1/2*I*6^(1/2))-Elliptic 
E(I*x,1/2*I*6^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.64 \[ \int \frac {1}{\left (-2+x^2+3 x^4\right )^{3/2}} \, dx=-\frac {3 \, \sqrt {\frac {3}{2}} \sqrt {-2} {\left (3 \, x^{4} + x^{2} - 2\right )} E(\arcsin \left (\sqrt {\frac {3}{2}} x\right )\,|\,-\frac {2}{3}) - 11 \, \sqrt {\frac {3}{2}} \sqrt {-2} {\left (3 \, x^{4} + x^{2} - 2\right )} F(\arcsin \left (\sqrt {\frac {3}{2}} x\right )\,|\,-\frac {2}{3}) + 2 \, \sqrt {3 \, x^{4} + x^{2} - 2} {\left (3 \, x^{3} + 13 \, x\right )}}{100 \, {\left (3 \, x^{4} + x^{2} - 2\right )}} \] Input:

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

Output:

-1/100*(3*sqrt(3/2)*sqrt(-2)*(3*x^4 + x^2 - 2)*elliptic_e(arcsin(sqrt(3/2) 
*x), -2/3) - 11*sqrt(3/2)*sqrt(-2)*(3*x^4 + x^2 - 2)*elliptic_f(arcsin(sqr 
t(3/2)*x), -2/3) + 2*sqrt(3*x^4 + x^2 - 2)*(3*x^3 + 13*x))/(3*x^4 + x^2 - 
2)
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (-2+x^2+3 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {3 x^{4}+x^{2}-2}}{9 x^{8}+6 x^{6}-11 x^{4}-4 x^{2}+4}d x \] Input:

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

Output:

int(sqrt(3*x**4 + x**2 - 2)/(9*x**8 + 6*x**6 - 11*x**4 - 4*x**2 + 4),x)