\(\int \frac {1}{(2+x^2-3 x^4)^{3/2}} \, dx\) [177]

Optimal result

Integrand size = 14, antiderivative size = 57 \[ \int \frac {1}{\left (2+x^2-3 x^4\right )^{3/2}} \, dx=\frac {x \left (13-3 x^2\right )}{50 \sqrt {2+x^2-3 x^4}}+\frac {E\left (\arcsin (x)\left |-\frac {3}{2}\right .\right )}{25 \sqrt {2}}+\frac {\operatorname {EllipticF}\left (\arcsin (x),-\frac {3}{2}\right )}{5 \sqrt {2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 8.98 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.67 \[ \int \frac {1}{\left (2+x^2-3 x^4\right )^{3/2}} \, dx=\frac {1}{50} \left (\frac {13 x}{\sqrt {2+x^2-3 x^4}}-\frac {3 x^3}{\sqrt {2+x^2-3 x^4}}+i \sqrt {3} E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {2}{3}\right )-5 i \sqrt {3} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),-\frac {2}{3}\right )\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1405, 27, 1494, 27, 399, 321, 327}

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

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1494

\(\displaystyle \frac {3}{25} \sqrt {3} \int \frac {x^2+4}{2 \sqrt {3} \sqrt {1-x^2} \sqrt {3 x^2+2}}dx+\frac {x \left (13-3 x^2\right )}{50 \sqrt {-3 x^4+x^2+2}}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 399

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

(x*(13 - 3*x^2))/(50*Sqrt[2 + x^2 - 3*x^4]) + (3*((Sqrt[2]*EllipticE[ArcSi 
n[x], -3/2])/3 + (5*Sqrt[2]*EllipticF[ArcSin[x], -3/2])/3))/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 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 399

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) 
^2]), x_Symbol] :> Simp[f/b   Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + 
 Simp[(b*e - a*f)/b   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr 
eeQ[{a, b, c, d, e, f}, x] &&  !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && 
(PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
 

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 [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (49 ) = 98\).

Time = 1.79 (sec) , antiderivative size = 115, normalized size of antiderivative = 2.02

method	result	size
risch	\(-\frac {x \left (3 x^{2}-13\right )}{50 \sqrt {-3 x^{4}+x^{2}+2}}+\frac {3 \sqrt {-x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (x , \frac {i \sqrt {6}}{2}\right )}{25 \sqrt {-3 x^{4}+x^{2}+2}}-\frac {\sqrt {-x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (x , \frac {i \sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (x , \frac {i \sqrt {6}}{2}\right )\right )}{50 \sqrt {-3 x^{4}+x^{2}+2}}\)	\(115\)
default	\(\frac {\frac {13}{50} x -\frac {3}{50} x^{3}}{\sqrt {-3 x^{4}+x^{2}+2}}+\frac {3 \sqrt {-x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (x , \frac {i \sqrt {6}}{2}\right )}{25 \sqrt {-3 x^{4}+x^{2}+2}}-\frac {\sqrt {-x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (x , \frac {i \sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (x , \frac {i \sqrt {6}}{2}\right )\right )}{50 \sqrt {-3 x^{4}+x^{2}+2}}\)	\(116\)
elliptic	\(\frac {\frac {13}{50} x -\frac {3}{50} x^{3}}{\sqrt {-3 x^{4}+x^{2}+2}}+\frac {3 \sqrt {-x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (x , \frac {i \sqrt {6}}{2}\right )}{25 \sqrt {-3 x^{4}+x^{2}+2}}-\frac {\sqrt {-x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (x , \frac {i \sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (x , \frac {i \sqrt {6}}{2}\right )\right )}{50 \sqrt {-3 x^{4}+x^{2}+2}}\)	\(116\)

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.40 \[ \int \frac {1}{\left (2+x^2-3 x^4\right )^{3/2}} \, dx=\frac {\sqrt {2} {\left (3 \, x^{4} - x^{2} - 2\right )} E(\arcsin \left (x\right )\,|\,-\frac {3}{2}) + 5 \, \sqrt {2} {\left (3 \, x^{4} - x^{2} - 2\right )} F(\arcsin \left (x\right )\,|\,-\frac {3}{2}) + \sqrt {-3 \, x^{4} + x^{2} + 2} {\left (3 \, x^{3} - 13 \, x\right )}}{50 \, {\left (3 \, x^{4} - x^{2} - 2\right )}} \] Input:

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

Output:

1/50*(sqrt(2)*(3*x^4 - x^2 - 2)*elliptic_e(arcsin(x), -3/2) + 5*sqrt(2)*(3 
*x^4 - x^2 - 2)*elliptic_f(arcsin(x), -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)