\(\int \frac {1}{(3-2 x^2-x^4)^{5/2}} \, dx\) [341]

Optimal result

Integrand size = 16, antiderivative size = 85 \[ \int \frac {1}{\left (3-2 x^2-x^4\right )^{5/2}} \, dx=\frac {x \left (5+x^2\right )}{72 \left (3-2 x^2-x^4\right )^{3/2}}+\frac {x \left (26+7 x^2\right )}{432 \sqrt {3-2 x^2-x^4}}-\frac {7 E\left (\arcsin (x)\left |-\frac {1}{3}\right .\right )}{144 \sqrt {3}}+\frac {11 \operatorname {EllipticF}\left (\arcsin (x),-\frac {1}{3}\right )}{144 \sqrt {3}} \] Output:

1/72*x*(x^2+5)/(-x^4-2*x^2+3)^(3/2)+1/432*x*(7*x^2+26)/(-x^4-2*x^2+3)^(1/2 
)-7/432*EllipticE(x,1/3*I*3^(1/2))*3^(1/2)+11/432*EllipticF(x,1/3*I*3^(1/2 
))*3^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.07 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.26 \[ \int \frac {1}{\left (3-2 x^2-x^4\right )^{5/2}} \, dx=-\frac {-108 x+25 x^3+40 x^5+7 x^7+7 i \left (3-2 x^2-x^4\right )^{3/2} E\left (\left .i \text {arcsinh}\left (\frac {x}{\sqrt {3}}\right )\right |-3\right )+5 i \left (3-2 x^2-x^4\right )^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {x}{\sqrt {3}}\right ),-3\right )}{432 \left (3-2 x^2-x^4\right )^{3/2}} \] Input:

Integrate[(3 - 2*x^2 - x^4)^(-5/2),x]
 

Output:

-1/432*(-108*x + 25*x^3 + 40*x^5 + 7*x^7 + (7*I)*(3 - 2*x^2 - x^4)^(3/2)*E 
llipticE[I*ArcSinh[x/Sqrt[3]], -3] + (5*I)*(3 - 2*x^2 - x^4)^(3/2)*Ellipti 
cF[I*ArcSinh[x/Sqrt[3]], -3])/(3 - 2*x^2 - x^4)^(3/2)
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {1405, 27, 1492, 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.

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

\(\Big \downarrow \) 1405

\(\displaystyle \frac {x \left (x^2+5\right )}{72 \left (-x^4-2 x^2+3\right )^{3/2}}-\frac {1}{144} \int -\frac {2 \left (3 x^2+19\right )}{\left (-x^4-2 x^2+3\right )^{3/2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{72} \int \frac {3 x^2+19}{\left (-x^4-2 x^2+3\right )^{3/2}}dx+\frac {x \left (x^2+5\right )}{72 \left (-x^4-2 x^2+3\right )^{3/2}}\)

\(\Big \downarrow \) 1492

\(\displaystyle \frac {1}{72} \left (\frac {x \left (7 x^2+26\right )}{6 \sqrt {-x^4-2 x^2+3}}-\frac {1}{48} \int -\frac {8 \left (12-7 x^2\right )}{\sqrt {-x^4-2 x^2+3}}dx\right )+\frac {x \left (x^2+5\right )}{72 \left (-x^4-2 x^2+3\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{72} \left (\frac {1}{6} \int \frac {12-7 x^2}{\sqrt {-x^4-2 x^2+3}}dx+\frac {x \left (7 x^2+26\right )}{6 \sqrt {-x^4-2 x^2+3}}\right )+\frac {x \left (x^2+5\right )}{72 \left (-x^4-2 x^2+3\right )^{3/2}}\)

\(\Big \downarrow \) 1494

\(\displaystyle \frac {1}{72} \left (\frac {1}{3} \int \frac {12-7 x^2}{2 \sqrt {1-x^2} \sqrt {x^2+3}}dx+\frac {x \left (7 x^2+26\right )}{6 \sqrt {-x^4-2 x^2+3}}\right )+\frac {x \left (x^2+5\right )}{72 \left (-x^4-2 x^2+3\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{72} \left (\frac {1}{6} \int \frac {12-7 x^2}{\sqrt {1-x^2} \sqrt {x^2+3}}dx+\frac {x \left (7 x^2+26\right )}{6 \sqrt {-x^4-2 x^2+3}}\right )+\frac {x \left (x^2+5\right )}{72 \left (-x^4-2 x^2+3\right )^{3/2}}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {1}{72} \left (\frac {1}{6} \left (33 \int \frac {1}{\sqrt {1-x^2} \sqrt {x^2+3}}dx-7 \int \frac {\sqrt {x^2+3}}{\sqrt {1-x^2}}dx\right )+\frac {x \left (7 x^2+26\right )}{6 \sqrt {-x^4-2 x^2+3}}\right )+\frac {x \left (x^2+5\right )}{72 \left (-x^4-2 x^2+3\right )^{3/2}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {1}{72} \left (\frac {1}{6} \left (11 \sqrt {3} \operatorname {EllipticF}\left (\arcsin (x),-\frac {1}{3}\right )-7 \int \frac {\sqrt {x^2+3}}{\sqrt {1-x^2}}dx\right )+\frac {x \left (7 x^2+26\right )}{6 \sqrt {-x^4-2 x^2+3}}\right )+\frac {x \left (x^2+5\right )}{72 \left (-x^4-2 x^2+3\right )^{3/2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {1}{72} \left (\frac {1}{6} \left (11 \sqrt {3} \operatorname {EllipticF}\left (\arcsin (x),-\frac {1}{3}\right )-7 \sqrt {3} E\left (\arcsin (x)\left |-\frac {1}{3}\right .\right )\right )+\frac {x \left (7 x^2+26\right )}{6 \sqrt {-x^4-2 x^2+3}}\right )+\frac {x \left (x^2+5\right )}{72 \left (-x^4-2 x^2+3\right )^{3/2}}\)

Input:

Int[(3 - 2*x^2 - x^4)^(-5/2),x]
 

Output:

(x*(5 + x^2))/(72*(3 - 2*x^2 - x^4)^(3/2)) + ((x*(26 + 7*x^2))/(6*Sqrt[3 - 
 2*x^2 - x^4]) + (-7*Sqrt[3]*EllipticE[ArcSin[x], -1/3] + 11*Sqrt[3]*Ellip 
ticF[ArcSin[x], -1/3])/6)/72
 

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 1492

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb 
ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*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[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 
7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, 
 b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 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 [A] (verified)

Time = 2.38 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.68

method	result	size
risch	\(\frac {x \left (7 x^{6}+40 x^{4}+25 x^{2}-108\right )}{432 \left (x^{4}+2 x^{2}-3\right ) \sqrt {-x^{4}-2 x^{2}+3}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \operatorname {EllipticF}\left (x , \frac {i \sqrt {3}}{3}\right )}{108 \sqrt {-x^{4}-2 x^{2}+3}}+\frac {7 \sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \left (\operatorname {EllipticF}\left (x , \frac {i \sqrt {3}}{3}\right )-\operatorname {EllipticE}\left (x , \frac {i \sqrt {3}}{3}\right )\right )}{432 \sqrt {-x^{4}-2 x^{2}+3}}\)	\(143\)
default	\(\frac {\left (\frac {5}{72} x +\frac {1}{72} x^{3}\right ) \sqrt {-x^{4}-2 x^{2}+3}}{\left (x^{4}+2 x^{2}-3\right )^{2}}+\frac {\frac {7}{432} x^{3}+\frac {13}{216} x}{\sqrt {-x^{4}-2 x^{2}+3}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \operatorname {EllipticF}\left (x , \frac {i \sqrt {3}}{3}\right )}{108 \sqrt {-x^{4}-2 x^{2}+3}}+\frac {7 \sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \left (\operatorname {EllipticF}\left (x , \frac {i \sqrt {3}}{3}\right )-\operatorname {EllipticE}\left (x , \frac {i \sqrt {3}}{3}\right )\right )}{432 \sqrt {-x^{4}-2 x^{2}+3}}\)	\(158\)
elliptic	\(\frac {\left (\frac {5}{72} x +\frac {1}{72} x^{3}\right ) \sqrt {-x^{4}-2 x^{2}+3}}{\left (x^{4}+2 x^{2}-3\right )^{2}}+\frac {\frac {7}{432} x^{3}+\frac {13}{216} x}{\sqrt {-x^{4}-2 x^{2}+3}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \operatorname {EllipticF}\left (x , \frac {i \sqrt {3}}{3}\right )}{108 \sqrt {-x^{4}-2 x^{2}+3}}+\frac {7 \sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \left (\operatorname {EllipticF}\left (x , \frac {i \sqrt {3}}{3}\right )-\operatorname {EllipticE}\left (x , \frac {i \sqrt {3}}{3}\right )\right )}{432 \sqrt {-x^{4}-2 x^{2}+3}}\)	\(158\)

Input:

int(1/(-x^4-2*x^2+3)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/432*x*(7*x^6+40*x^4+25*x^2-108)/(x^4+2*x^2-3)/(-x^4-2*x^2+3)^(1/2)+1/108 
*(-x^2+1)^(1/2)*(3*x^2+9)^(1/2)/(-x^4-2*x^2+3)^(1/2)*EllipticF(x,1/3*I*3^( 
1/2))+7/432*(-x^2+1)^(1/2)*(3*x^2+9)^(1/2)/(-x^4-2*x^2+3)^(1/2)*(EllipticF 
(x,1/3*I*3^(1/2))-EllipticE(x,1/3*I*3^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.38 \[ \int \frac {1}{\left (3-2 x^2-x^4\right )^{5/2}} \, dx=-\frac {7 \, \sqrt {3} {\left (x^{8} + 4 \, x^{6} - 2 \, x^{4} - 12 \, x^{2} + 9\right )} E(\arcsin \left (x\right )\,|\,-\frac {1}{3}) - 11 \, \sqrt {3} {\left (x^{8} + 4 \, x^{6} - 2 \, x^{4} - 12 \, x^{2} + 9\right )} F(\arcsin \left (x\right )\,|\,-\frac {1}{3}) + {\left (7 \, x^{7} + 40 \, x^{5} + 25 \, x^{3} - 108 \, x\right )} \sqrt {-x^{4} - 2 \, x^{2} + 3}}{432 \, {\left (x^{8} + 4 \, x^{6} - 2 \, x^{4} - 12 \, x^{2} + 9\right )}} \] Input:

integrate(1/(-x^4-2*x^2+3)^(5/2),x, algorithm="fricas")
 

Output:

-1/432*(7*sqrt(3)*(x^8 + 4*x^6 - 2*x^4 - 12*x^2 + 9)*elliptic_e(arcsin(x), 
 -1/3) - 11*sqrt(3)*(x^8 + 4*x^6 - 2*x^4 - 12*x^2 + 9)*elliptic_f(arcsin(x 
), -1/3) + (7*x^7 + 40*x^5 + 25*x^3 - 108*x)*sqrt(-x^4 - 2*x^2 + 3))/(x^8 
+ 4*x^6 - 2*x^4 - 12*x^2 + 9)
 

Sympy [F]

\[ \int \frac {1}{\left (3-2 x^2-x^4\right )^{5/2}} \, dx=\int \frac {1}{\left (- x^{4} - 2 x^{2} + 3\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(1/(-x**4-2*x**2+3)**(5/2),x)
 

Output:

Integral((-x**4 - 2*x**2 + 3)**(-5/2), x)
 

Maxima [F]

\[ \int \frac {1}{\left (3-2 x^2-x^4\right )^{5/2}} \, dx=\int { \frac {1}{{\left (-x^{4} - 2 \, x^{2} + 3\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(-x^4-2*x^2+3)^(5/2),x, algorithm="maxima")
 

Output:

integrate((-x^4 - 2*x^2 + 3)^(-5/2), x)
 

Giac [F]

\[ \int \frac {1}{\left (3-2 x^2-x^4\right )^{5/2}} \, dx=\int { \frac {1}{{\left (-x^{4} - 2 \, x^{2} + 3\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(-x^4-2*x^2+3)^(5/2),x, algorithm="giac")
 

Output:

integrate((-x^4 - 2*x^2 + 3)^(-5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (3-2 x^2-x^4\right )^{5/2}} \, dx=\int \frac {1}{{\left (-x^4-2\,x^2+3\right )}^{5/2}} \,d x \] Input:

int(1/(3 - x^4 - 2*x^2)^(5/2),x)
 

Output:

int(1/(3 - x^4 - 2*x^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {1}{\left (3-2 x^2-x^4\right )^{5/2}} \, dx=-\left (\int \frac {\sqrt {-x^{4}-2 x^{2}+3}}{x^{12}+6 x^{10}+3 x^{8}-28 x^{6}-9 x^{4}+54 x^{2}-27}d x \right ) \] Input:

int(1/(-x^4-2*x^2+3)^(5/2),x)
 

Output:

 - int(sqrt( - x**4 - 2*x**2 + 3)/(x**12 + 6*x**10 + 3*x**8 - 28*x**6 - 9* 
x**4 + 54*x**2 - 27),x)