\(\int \frac {1}{(-1+5 x^2-x^4)^{3/2}} \, dx\) [293]

Optimal result

Integrand size = 16, antiderivative size = 118 \[ \int \frac {1}{\left (-1+5 x^2-x^4\right )^{3/2}} \, dx=-\frac {x \left (23-5 x^2\right )}{21 \sqrt {-1+5 x^2-x^4}}+\frac {5 E\left (\arccos \left (\sqrt {\frac {2}{5+\sqrt {21}}} x\right )|\frac {1}{42} \left (21+5 \sqrt {21}\right )\right )}{21^{3/4}}-\frac {\left (5-\sqrt {21}\right ) \operatorname {EllipticF}\left (\arccos \left (\sqrt {\frac {2}{5+\sqrt {21}}} x\right ),\frac {1}{42} \left (21+5 \sqrt {21}\right )\right )}{2\ 21^{3/4}} \] Output:

-1/21*x*(-5*x^2+23)/(-x^4+5*x^2-1)^(1/2)+5/21*EllipticE((1-2/(5+21^(1/2))* 
x^2)^(1/2),1/42*(882+210*21^(1/2))^(1/2))*21^(1/4)-1/42*(5-21^(1/2))*Inver 
seJacobiAM(arccos(2^(1/2)/(5+21^(1/2))^(1/2)*x),1/42*(882+210*21^(1/2))^(1 
/2))*21^(1/4)
 

Mathematica [A] (warning: unable to verify)

Time = 5.05 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.57 \[ \int \frac {1}{\left (-1+5 x^2-x^4\right )^{3/2}} \, dx=\frac {4 x \left (-23+5 x^2\right )+5 \left (5+\sqrt {21}\right ) \sqrt {5-\sqrt {21}-2 x^2} \sqrt {2+\left (-5+\sqrt {21}\right ) x^2} E\left (\arcsin \left (\sqrt {\frac {1}{2} \left (5+\sqrt {21}\right )} x\right )|\frac {23}{2}-\frac {5 \sqrt {21}}{2}\right )-\left (21+5 \sqrt {21}\right ) \sqrt {5-\sqrt {21}-2 x^2} \sqrt {2+\left (-5+\sqrt {21}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1}{2} \left (5+\sqrt {21}\right )} x\right ),\frac {23}{2}-\frac {5 \sqrt {21}}{2}\right )}{84 \sqrt {-1+5 x^2-x^4}} \] Input:

Integrate[(-1 + 5*x^2 - x^4)^(-3/2),x]
 

Output:

(4*x*(-23 + 5*x^2) + 5*(5 + Sqrt[21])*Sqrt[5 - Sqrt[21] - 2*x^2]*Sqrt[2 + 
(-5 + Sqrt[21])*x^2]*EllipticE[ArcSin[Sqrt[(5 + Sqrt[21])/2]*x], 23/2 - (5 
*Sqrt[21])/2] - (21 + 5*Sqrt[21])*Sqrt[5 - Sqrt[21] - 2*x^2]*Sqrt[2 + (-5 
+ Sqrt[21])*x^2]*EllipticF[ArcSin[Sqrt[(5 + Sqrt[21])/2]*x], 23/2 - (5*Sqr 
t[21])/2])/(84*Sqrt[-1 + 5*x^2 - x^4])
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1405, 1494, 399, 322, 328}

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.

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

\(\Big \downarrow \) 1405

\(\displaystyle \frac {1}{21} \int \frac {2-5 x^2}{\sqrt {-x^4+5 x^2-1}}dx-\frac {x \left (23-5 x^2\right )}{21 \sqrt {-x^4+5 x^2-1}}\)

\(\Big \downarrow \) 1494

\(\displaystyle \frac {2}{21} \int \frac {2-5 x^2}{\sqrt {-2 x^2+\sqrt {21}+5} \sqrt {2 x^2+\sqrt {21}-5}}dx-\frac {x \left (23-5 x^2\right )}{21 \sqrt {-x^4+5 x^2-1}}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {2}{21} \left (-\frac {1}{2} \left (21-5 \sqrt {21}\right ) \int \frac {1}{\sqrt {-2 x^2+\sqrt {21}+5} \sqrt {2 x^2+\sqrt {21}-5}}dx-\frac {5}{2} \int \frac {\sqrt {2 x^2+\sqrt {21}-5}}{\sqrt {-2 x^2+\sqrt {21}+5}}dx\right )-\frac {x \left (23-5 x^2\right )}{21 \sqrt {-x^4+5 x^2-1}}\)

\(\Big \downarrow \) 322

\(\displaystyle \frac {2}{21} \left (\frac {\left (21-5 \sqrt {21}\right ) \operatorname {EllipticF}\left (\arccos \left (\sqrt {\frac {2}{5+\sqrt {21}}} x\right ),\frac {1}{42} \left (21+5 \sqrt {21}\right )\right )}{4 \sqrt [4]{21}}-\frac {5}{2} \int \frac {\sqrt {2 x^2+\sqrt {21}-5}}{\sqrt {-2 x^2+\sqrt {21}+5}}dx\right )-\frac {x \left (23-5 x^2\right )}{21 \sqrt {-x^4+5 x^2-1}}\)

\(\Big \downarrow \) 328

\(\displaystyle \frac {2}{21} \left (\frac {\left (21-5 \sqrt {21}\right ) \operatorname {EllipticF}\left (\arccos \left (\sqrt {\frac {2}{5+\sqrt {21}}} x\right ),\frac {1}{42} \left (21+5 \sqrt {21}\right )\right )}{4 \sqrt [4]{21}}+\frac {5}{2} \sqrt [4]{21} E\left (\arccos \left (\sqrt {\frac {2}{5+\sqrt {21}}} x\right )|\frac {1}{42} \left (21+5 \sqrt {21}\right )\right )\right )-\frac {x \left (23-5 x^2\right )}{21 \sqrt {-x^4+5 x^2-1}}\)

Input:

Int[(-1 + 5*x^2 - x^4)^(-3/2),x]
 

Output:

-1/21*(x*(23 - 5*x^2))/Sqrt[-1 + 5*x^2 - x^4] + (2*((5*21^(1/4)*EllipticE[ 
ArcCos[Sqrt[2/(5 + Sqrt[21])]*x], (21 + 5*Sqrt[21])/42])/2 + ((21 - 5*Sqrt 
[21])*EllipticF[ArcCos[Sqrt[2/(5 + Sqrt[21])]*x], (21 + 5*Sqrt[21])/42])/( 
4*21^(1/4))))/21
 

Defintions of rubi rules used

rule 322

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

rule 328

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

Leaf count of result is larger than twice the leaf count of optimal. \(220\) vs. \(2(95)=190\).

Time = 1.15 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.87

method	result	size
risch	\(\frac {x \left (5 x^{2}-23\right )}{21 \sqrt {-x^{4}+5 x^{2}-1}}+\frac {2 \sqrt {1-\left (\frac {5}{2}-\frac {\sqrt {21}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {5}{2}+\frac {\sqrt {21}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (x \left (\frac {\sqrt {7}}{2}-\frac {\sqrt {3}}{2}\right ), \frac {5}{2}+\frac {\sqrt {21}}{2}\right )}{21 \left (\frac {\sqrt {7}}{2}-\frac {\sqrt {3}}{2}\right ) \sqrt {-x^{4}+5 x^{2}-1}}-\frac {10 \sqrt {1-\left (\frac {5}{2}-\frac {\sqrt {21}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {5}{2}+\frac {\sqrt {21}}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (x \left (\frac {\sqrt {7}}{2}-\frac {\sqrt {3}}{2}\right ), \frac {5}{2}+\frac {\sqrt {21}}{2}\right )-\operatorname {EllipticE}\left (x \left (\frac {\sqrt {7}}{2}-\frac {\sqrt {3}}{2}\right ), \frac {5}{2}+\frac {\sqrt {21}}{2}\right )\right )}{21 \left (\frac {\sqrt {7}}{2}-\frac {\sqrt {3}}{2}\right ) \sqrt {-x^{4}+5 x^{2}-1}\, \left (5+\sqrt {21}\right )}\)	\(221\)
default	\(\frac {-\frac {23}{21} x +\frac {5}{21} x^{3}}{\sqrt {-x^{4}+5 x^{2}-1}}+\frac {2 \sqrt {1-\left (\frac {5}{2}-\frac {\sqrt {21}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {5}{2}+\frac {\sqrt {21}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (x \left (\frac {\sqrt {7}}{2}-\frac {\sqrt {3}}{2}\right ), \frac {5}{2}+\frac {\sqrt {21}}{2}\right )}{21 \left (\frac {\sqrt {7}}{2}-\frac {\sqrt {3}}{2}\right ) \sqrt {-x^{4}+5 x^{2}-1}}-\frac {10 \sqrt {1-\left (\frac {5}{2}-\frac {\sqrt {21}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {5}{2}+\frac {\sqrt {21}}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (x \left (\frac {\sqrt {7}}{2}-\frac {\sqrt {3}}{2}\right ), \frac {5}{2}+\frac {\sqrt {21}}{2}\right )-\operatorname {EllipticE}\left (x \left (\frac {\sqrt {7}}{2}-\frac {\sqrt {3}}{2}\right ), \frac {5}{2}+\frac {\sqrt {21}}{2}\right )\right )}{21 \left (\frac {\sqrt {7}}{2}-\frac {\sqrt {3}}{2}\right ) \sqrt {-x^{4}+5 x^{2}-1}\, \left (5+\sqrt {21}\right )}\)	\(222\)
elliptic	\(\frac {-\frac {23}{21} x +\frac {5}{21} x^{3}}{\sqrt {-x^{4}+5 x^{2}-1}}+\frac {2 \sqrt {1-\left (\frac {5}{2}-\frac {\sqrt {21}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {5}{2}+\frac {\sqrt {21}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (x \left (\frac {\sqrt {7}}{2}-\frac {\sqrt {3}}{2}\right ), \frac {5}{2}+\frac {\sqrt {21}}{2}\right )}{21 \left (\frac {\sqrt {7}}{2}-\frac {\sqrt {3}}{2}\right ) \sqrt {-x^{4}+5 x^{2}-1}}-\frac {10 \sqrt {1-\left (\frac {5}{2}-\frac {\sqrt {21}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {5}{2}+\frac {\sqrt {21}}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (x \left (\frac {\sqrt {7}}{2}-\frac {\sqrt {3}}{2}\right ), \frac {5}{2}+\frac {\sqrt {21}}{2}\right )-\operatorname {EllipticE}\left (x \left (\frac {\sqrt {7}}{2}-\frac {\sqrt {3}}{2}\right ), \frac {5}{2}+\frac {\sqrt {21}}{2}\right )\right )}{21 \left (\frac {\sqrt {7}}{2}-\frac {\sqrt {3}}{2}\right ) \sqrt {-x^{4}+5 x^{2}-1}\, \left (5+\sqrt {21}\right )}\)	\(222\)

Input:

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

Output:

1/21*x*(5*x^2-23)/(-x^4+5*x^2-1)^(1/2)+2/21/(1/2*7^(1/2)-1/2*3^(1/2))*(1-( 
5/2-1/2*21^(1/2))*x^2)^(1/2)*(1-(5/2+1/2*21^(1/2))*x^2)^(1/2)/(-x^4+5*x^2- 
1)^(1/2)*EllipticF(x*(1/2*7^(1/2)-1/2*3^(1/2)),5/2+1/2*21^(1/2))-10/21/(1/ 
2*7^(1/2)-1/2*3^(1/2))*(1-(5/2-1/2*21^(1/2))*x^2)^(1/2)*(1-(5/2+1/2*21^(1/ 
2))*x^2)^(1/2)/(-x^4+5*x^2-1)^(1/2)/(5+21^(1/2))*(EllipticF(x*(1/2*7^(1/2) 
-1/2*3^(1/2)),5/2+1/2*21^(1/2))-EllipticE(x*(1/2*7^(1/2)-1/2*3^(1/2)),5/2+ 
1/2*21^(1/2)))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.08 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.35 \[ \int \frac {1}{\left (-1+5 x^2-x^4\right )^{3/2}} \, dx=-\frac {5 \, {\left (5 i \, x^{4} - 25 i \, x^{2} + \sqrt {21} {\left (i \, x^{4} - 5 i \, x^{2} + i\right )} + 5 i\right )} \sqrt {\frac {1}{2} \, \sqrt {21} + \frac {5}{2}} E(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {21} + \frac {5}{2}}\right )\,|\,-\frac {5}{2} \, \sqrt {21} + \frac {23}{2}) - {\left (15 i \, x^{4} - 75 i \, x^{2} - 7 \, \sqrt {21} {\left (-i \, x^{4} + 5 i \, x^{2} - i\right )} + 15 i\right )} \sqrt {\frac {1}{2} \, \sqrt {21} + \frac {5}{2}} F(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {21} + \frac {5}{2}}\right )\,|\,-\frac {5}{2} \, \sqrt {21} + \frac {23}{2}) + 2 \, \sqrt {-x^{4} + 5 \, x^{2} - 1} {\left (5 \, x^{3} - 23 \, x\right )}}{42 \, {\left (x^{4} - 5 \, x^{2} + 1\right )}} \] Input:

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

Output:

-1/42*(5*(5*I*x^4 - 25*I*x^2 + sqrt(21)*(I*x^4 - 5*I*x^2 + I) + 5*I)*sqrt( 
1/2*sqrt(21) + 5/2)*elliptic_e(arcsin(x*sqrt(1/2*sqrt(21) + 5/2)), -5/2*sq 
rt(21) + 23/2) - (15*I*x^4 - 75*I*x^2 - 7*sqrt(21)*(-I*x^4 + 5*I*x^2 - I) 
+ 15*I)*sqrt(1/2*sqrt(21) + 5/2)*elliptic_f(arcsin(x*sqrt(1/2*sqrt(21) + 5 
/2)), -5/2*sqrt(21) + 23/2) + 2*sqrt(-x^4 + 5*x^2 - 1)*(5*x^3 - 23*x))/(x^ 
4 - 5*x^2 + 1)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt( - x**4 + 5*x**2 - 1)/(x**8 - 10*x**6 + 27*x**4 - 10*x**2 + 1),x)