\(\int \frac {1}{(-3-x^2+2 x^4)^{3/2}} \, dx\) [225]

Optimal result

Integrand size = 16, antiderivative size = 147 \[ \int \frac {1}{\left (-3-x^2+2 x^4\right )^{3/2}} \, dx=-\frac {x \left (13-2 x^2\right )}{75 \sqrt {-3-x^2+2 x^4}}-\frac {\sqrt {2} \sqrt {3-2 x^2} \sqrt {1+x^2} E\left (\arcsin \left (\sqrt {\frac {2}{3}} x\right )|-\frac {3}{2}\right )}{75 \sqrt {-3-x^2+2 x^4}}-\frac {\sqrt {2} \sqrt {3-2 x^2} \sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{3}} x\right ),-\frac {3}{2}\right )}{15 \sqrt {-3-x^2+2 x^4}} \] Output:

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

Mathematica [A] (verified)

Time = 6.64 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.69 \[ \int \frac {1}{\left (-3-x^2+2 x^4\right )^{3/2}} \, dx=\frac {-13 x+2 x^3-\sqrt {6-4 x^2} \sqrt {1+x^2} E\left (\arcsin \left (\sqrt {\frac {2}{3}} x\right )|-\frac {3}{2}\right )-5 \sqrt {6-4 x^2} \sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{3}} x\right ),-\frac {3}{2}\right )}{75 \sqrt {-3-x^2+2 x^4}} \] 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] (warning: unable to verify)

Time = 0.56 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.50, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, 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.

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

\(\Big \downarrow \) 1405

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1501

\(\displaystyle -\frac {2}{75} \left (\frac {15}{2} \int \frac {1}{\sqrt {2 x^4-x^2-3}}dx+\frac {1}{4} \int -\frac {2 \left (3-2 x^2\right )}{\sqrt {2 x^4-x^2-3}}dx\right )-\frac {x \left (13-2 x^2\right )}{75 \sqrt {2 x^4-x^2-3}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2}{75} \left (\frac {15}{2} \int \frac {1}{\sqrt {2 x^4-x^2-3}}dx-\frac {1}{2} \int \frac {3-2 x^2}{\sqrt {2 x^4-x^2-3}}dx\right )-\frac {x \left (13-2 x^2\right )}{75 \sqrt {2 x^4-x^2-3}}\)

\(\Big \downarrow \) 1410

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

\(\Big \downarrow \) 1498

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

Input:

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

Output:

-1/75*(x*(13 - 2*x^2))/Sqrt[-3 - x^2 + 2*x^4] - (2*(((2*x*(1 + x^2))/Sqrt[ 
-3 - x^2 + 2*x^4] - (Sqrt[5]*Sqrt[(1 + x^2)/(3 - 2*x^2)]*Sqrt[-3 + 2*x^2]* 
EllipticE[ArcSin[(Sqrt[5]*x)/Sqrt[-3 + 2*x^2]], 2/5])/(Sqrt[(3 - 2*x^2)^(- 
1)]*Sqrt[-3 - x^2 + 2*x^4]))/2 + (3*Sqrt[5]*Sqrt[1 + x^2]*Sqrt[-3 + 2*x^2] 
*EllipticF[ArcSin[(Sqrt[5]*x)/Sqrt[-3 + 2*x^2]], 2/5])/(2*Sqrt[-3 - x^2 + 
2*x^4])))/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 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 = 2.24 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.87

method	result	size
risch	\(\frac {x \left (2 x^{2}-13\right )}{75 \sqrt {2 x^{4}-x^{2}-3}}+\frac {4 i \sqrt {x^{2}+1}\, \sqrt {-6 x^{2}+9}\, \operatorname {EllipticF}\left (i x , \frac {i \sqrt {6}}{3}\right )}{75 \sqrt {2 x^{4}-x^{2}-3}}+\frac {i \sqrt {x^{2}+1}\, \sqrt {-6 x^{2}+9}\, \left (\operatorname {EllipticF}\left (i x , \frac {i \sqrt {6}}{3}\right )-\operatorname {EllipticE}\left (i x , \frac {i \sqrt {6}}{3}\right )\right )}{75 \sqrt {2 x^{4}-x^{2}-3}}\)	\(128\)
default	\(-\frac {4 \left (\frac {13}{300} x -\frac {1}{150} x^{3}\right )}{\sqrt {2 x^{4}-x^{2}-3}}+\frac {4 i \sqrt {x^{2}+1}\, \sqrt {-6 x^{2}+9}\, \operatorname {EllipticF}\left (i x , \frac {i \sqrt {6}}{3}\right )}{75 \sqrt {2 x^{4}-x^{2}-3}}+\frac {i \sqrt {x^{2}+1}\, \sqrt {-6 x^{2}+9}\, \left (\operatorname {EllipticF}\left (i x , \frac {i \sqrt {6}}{3}\right )-\operatorname {EllipticE}\left (i x , \frac {i \sqrt {6}}{3}\right )\right )}{75 \sqrt {2 x^{4}-x^{2}-3}}\)	\(129\)
elliptic	\(-\frac {4 \left (\frac {13}{300} x -\frac {1}{150} x^{3}\right )}{\sqrt {2 x^{4}-x^{2}-3}}+\frac {4 i \sqrt {x^{2}+1}\, \sqrt {-6 x^{2}+9}\, \operatorname {EllipticF}\left (i x , \frac {i \sqrt {6}}{3}\right )}{75 \sqrt {2 x^{4}-x^{2}-3}}+\frac {i \sqrt {x^{2}+1}\, \sqrt {-6 x^{2}+9}\, \left (\operatorname {EllipticF}\left (i x , \frac {i \sqrt {6}}{3}\right )-\operatorname {EllipticE}\left (i x , \frac {i \sqrt {6}}{3}\right )\right )}{75 \sqrt {2 x^{4}-x^{2}-3}}\)	\(129\)

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\left (-3-x^2+2 x^4\right )^{3/2}} \, dx=\frac {2 \, \sqrt {\frac {2}{3}} \sqrt {-3} {\left (2 \, x^{4} - x^{2} - 3\right )} E(\arcsin \left (\sqrt {\frac {2}{3}} x\right )\,|\,-\frac {3}{2}) + 16 \, \sqrt {\frac {2}{3}} \sqrt {-3} {\left (2 \, x^{4} - x^{2} - 3\right )} F(\arcsin \left (\sqrt {\frac {2}{3}} x\right )\,|\,-\frac {3}{2}) + 3 \, \sqrt {2 \, x^{4} - x^{2} - 3} {\left (2 \, x^{3} - 13 \, x\right )}}{225 \, {\left (2 \, x^{4} - x^{2} - 3\right )}} \] Input:

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

Output:

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

Output:

int(1/(2*x^4 - x^2 - 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)