\(\int \frac {1}{(3+x^2-2 x^4)^{3/2}} \, dx\) [190]

Optimal result

Integrand size = 14, antiderivative size = 73 \[ \int \frac {1}{\left (3+x^2-2 x^4\right )^{3/2}} \, dx=\frac {x \left (13-2 x^2\right )}{75 \sqrt {3+x^2-2 x^4}}+\frac {1}{75} \sqrt {2} E\left (\arcsin \left (\sqrt {\frac {2}{3}} x\right )|-\frac {3}{2}\right )+\frac {1}{15} \sqrt {2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{3}} x\right ),-\frac {3}{2}\right ) \] Output:

1/75*x*(-2*x^2+13)/(-2*x^4+x^2+3)^(1/2)+1/75*2^(1/2)*EllipticE(1/3*x*6^(1/ 
2),1/2*I*6^(1/2))+1/15*EllipticF(1/3*x*6^(1/2),1/2*I*6^(1/2))*2^(1/2)
 

Mathematica [A] (verified)

Time = 8.95 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.34 \[ \int \frac {1}{\left (3+x^2-2 x^4\right )^{3/2}} \, dx=\frac {13 x-2 x^3+\sqrt {6-4 x^2} \sqrt {1+x^2} E\left (\arcsin \left (\sqrt {\frac {2}{3}} x\right )|-\frac {3}{2}\right )+5 \sqrt {6-4 x^2} \sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{3}} x\right ),-\frac {3}{2}\right )}{75 \sqrt {3+x^2-2 x^4}} \] Input:

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

Output:

(13*x - 2*x^3 + Sqrt[6 - 4*x^2]*Sqrt[1 + x^2]*EllipticE[ArcSin[Sqrt[2/3]*x 
], -3/2] + 5*Sqrt[6 - 4*x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[Sqrt[2/3]*x], 
-3/2])/(75*Sqrt[3 + x^2 - 2*x^4])
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, 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.

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

\(\Big \downarrow \) 1405

\(\displaystyle \frac {x \left (13-2 x^2\right )}{75 \sqrt {-2 x^4+x^2+3}}-\frac {1}{75} \int -\frac {2 \left (x^2+6\right )}{\sqrt {-2 x^4+x^2+3}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{75} \int \frac {x^2+6}{\sqrt {-2 x^4+x^2+3}}dx+\frac {x \left (13-2 x^2\right )}{75 \sqrt {-2 x^4+x^2+3}}\)

\(\Big \downarrow \) 1494

\(\displaystyle \frac {4}{75} \sqrt {2} \int \frac {x^2+6}{2 \sqrt {2} \sqrt {3-2 x^2} \sqrt {x^2+1}}dx+\frac {x \left (13-2 x^2\right )}{75 \sqrt {-2 x^4+x^2+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{75} \int \frac {x^2+6}{\sqrt {3-2 x^2} \sqrt {x^2+1}}dx+\frac {x \left (13-2 x^2\right )}{75 \sqrt {-2 x^4+x^2+3}}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {2}{75} \left (5 \int \frac {1}{\sqrt {3-2 x^2} \sqrt {x^2+1}}dx+\int \frac {\sqrt {x^2+1}}{\sqrt {3-2 x^2}}dx\right )+\frac {x \left (13-2 x^2\right )}{75 \sqrt {-2 x^4+x^2+3}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {2}{75} \left (\int \frac {\sqrt {x^2+1}}{\sqrt {3-2 x^2}}dx+\frac {5 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{3}} x\right ),-\frac {3}{2}\right )}{\sqrt {2}}\right )+\frac {x \left (13-2 x^2\right )}{75 \sqrt {-2 x^4+x^2+3}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {2}{75} \left (\frac {5 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{3}} x\right ),-\frac {3}{2}\right )}{\sqrt {2}}+\frac {E\left (\arcsin \left (\sqrt {\frac {2}{3}} x\right )|-\frac {3}{2}\right )}{\sqrt {2}}\right )+\frac {x \left (13-2 x^2\right )}{75 \sqrt {-2 x^4+x^2+3}}\)

Input:

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

Output:

(x*(13 - 2*x^2))/(75*Sqrt[3 + x^2 - 2*x^4]) + (2*(EllipticE[ArcSin[Sqrt[2/ 
3]*x], -3/2]/Sqrt[2] + (5*EllipticF[ArcSin[Sqrt[2/3]*x], -3/2])/Sqrt[2]))/ 
75
 

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. 131 vs. \(2 (59 ) = 118\).

Time = 1.80 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.81

method	result	size
risch	\(-\frac {x \left (2 x^{2}-13\right )}{75 \sqrt {-2 x^{4}+x^{2}+3}}+\frac {2 \sqrt {6}\, \sqrt {-6 x^{2}+9}\, \sqrt {x^{2}+1}\, \operatorname {EllipticF}\left (\frac {x \sqrt {6}}{3}, \frac {i \sqrt {6}}{2}\right )}{75 \sqrt {-2 x^{4}+x^{2}+3}}-\frac {\sqrt {6}\, \sqrt {-6 x^{2}+9}\, \sqrt {x^{2}+1}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {6}}{3}, \frac {i \sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {6}}{3}, \frac {i \sqrt {6}}{2}\right )\right )}{225 \sqrt {-2 x^{4}+x^{2}+3}}\)	\(132\)
default	\(\frac {\frac {13}{75} x -\frac {2}{75} x^{3}}{\sqrt {-2 x^{4}+x^{2}+3}}+\frac {2 \sqrt {6}\, \sqrt {-6 x^{2}+9}\, \sqrt {x^{2}+1}\, \operatorname {EllipticF}\left (\frac {x \sqrt {6}}{3}, \frac {i \sqrt {6}}{2}\right )}{75 \sqrt {-2 x^{4}+x^{2}+3}}-\frac {\sqrt {6}\, \sqrt {-6 x^{2}+9}\, \sqrt {x^{2}+1}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {6}}{3}, \frac {i \sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {6}}{3}, \frac {i \sqrt {6}}{2}\right )\right )}{225 \sqrt {-2 x^{4}+x^{2}+3}}\)	\(133\)
elliptic	\(\frac {\frac {13}{75} x -\frac {2}{75} x^{3}}{\sqrt {-2 x^{4}+x^{2}+3}}+\frac {2 \sqrt {6}\, \sqrt {-6 x^{2}+9}\, \sqrt {x^{2}+1}\, \operatorname {EllipticF}\left (\frac {x \sqrt {6}}{3}, \frac {i \sqrt {6}}{2}\right )}{75 \sqrt {-2 x^{4}+x^{2}+3}}-\frac {\sqrt {6}\, \sqrt {-6 x^{2}+9}\, \sqrt {x^{2}+1}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {6}}{3}, \frac {i \sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {6}}{3}, \frac {i \sqrt {6}}{2}\right )\right )}{225 \sqrt {-2 x^{4}+x^{2}+3}}\)	\(133\)

Input:

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

Output:

-1/75*x*(2*x^2-13)/(-2*x^4+x^2+3)^(1/2)+2/75*6^(1/2)*(-6*x^2+9)^(1/2)*(x^2 
+1)^(1/2)/(-2*x^4+x^2+3)^(1/2)*EllipticF(1/3*x*6^(1/2),1/2*I*6^(1/2))-1/22 
5*6^(1/2)*(-6*x^2+9)^(1/2)*(x^2+1)^(1/2)/(-2*x^4+x^2+3)^(1/2)*(EllipticF(1 
/3*x*6^(1/2),1/2*I*6^(1/2))-EllipticE(1/3*x*6^(1/2),1/2*I*6^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.32 \[ \int \frac {1}{\left (3+x^2-2 x^4\right )^{3/2}} \, dx=\frac {2 \, \sqrt {3} \sqrt {\frac {2}{3}} {\left (2 \, x^{4} - x^{2} - 3\right )} E(\arcsin \left (\sqrt {\frac {2}{3}} x\right )\,|\,-\frac {3}{2}) + 16 \, \sqrt {3} \sqrt {\frac {2}{3}} {\left (2 \, x^{4} - x^{2} - 3\right )} F(\arcsin \left (\sqrt {\frac {2}{3}} x\right )\,|\,-\frac {3}{2}) + 3 \, \sqrt {-2 \, x^{4} + x^{2} + 3} {\left (2 \, x^{3} - 13 \, x\right )}}{225 \, {\left (2 \, x^{4} - x^{2} - 3\right )}} \] Input:

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

Output:

1/225*(2*sqrt(3)*sqrt(2/3)*(2*x^4 - x^2 - 3)*elliptic_e(arcsin(sqrt(2/3)*x 
), -3/2) + 16*sqrt(3)*sqrt(2/3)*(2*x^4 - x^2 - 3)*elliptic_f(arcsin(sqrt(2 
/3)*x), -3/2) + 3*sqrt(-2*x^4 + x^2 + 3)*(2*x^3 - 13*x))/(2*x^4 - x^2 - 3)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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