\(\int \frac {1}{(1-5 \sqrt {5} x^2+x^4)^{3/2}} \, dx\) [262]

Optimal result

Integrand size = 19, antiderivative size = 263 \[ \int \frac {1}{\left (1-5 \sqrt {5} x^2+x^4\right )^{3/2}} \, dx=\frac {x \left (123-5 \sqrt {5} x^2\right )}{121 \sqrt {1-5 \sqrt {5} x^2+x^4}}-\frac {5 \sqrt {\frac {5}{2} \left (11+5 \sqrt {5}\right )} \sqrt {2+\left (11-5 \sqrt {5}\right ) x^2} \sqrt {2-\left (11+5 \sqrt {5}\right ) x^2} E\left (\arcsin \left (\sqrt {\frac {1}{2} \left (11+5 \sqrt {5}\right )} x\right )|\frac {1}{2} \left (123-55 \sqrt {5}\right )\right )}{242 \sqrt {1-5 \sqrt {5} x^2+x^4}}+\frac {\sqrt {\frac {1}{2} \left (11+5 \sqrt {5}\right )} \sqrt {2+\left (11-5 \sqrt {5}\right ) x^2} \sqrt {2-\left (11+5 \sqrt {5}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1}{2} \left (11+5 \sqrt {5}\right )} x\right ),\frac {1}{2} \left (123-55 \sqrt {5}\right )\right )}{22 \sqrt {1-5 \sqrt {5} x^2+x^4}} \] Output:

1/121*x*(123-5*5^(1/2)*x^2)/(1-5*5^(1/2)*x^2+x^4)^(1/2)-5/484*(110+50*5^(1 
/2))^(1/2)*(2+(11-5*5^(1/2))*x^2)^(1/2)*(2-(11+5*5^(1/2))*x^2)^(1/2)*Ellip 
ticE(1/2*(22+10*5^(1/2))^(1/2)*x,5/2*5^(1/2)-11/2)/(1-5*5^(1/2)*x^2+x^4)^( 
1/2)+1/44*(22+10*5^(1/2))^(1/2)*(2+(11-5*5^(1/2))*x^2)^(1/2)*(2-(11+5*5^(1 
/2))*x^2)^(1/2)*EllipticF(1/2*(22+10*5^(1/2))^(1/2)*x,5/2*5^(1/2)-11/2)/(1 
-5*5^(1/2)*x^2+x^4)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 7.94 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (1-5 \sqrt {5} x^2+x^4\right )^{3/2}} \, dx=\frac {246 x-10 \sqrt {5} x^3-5 \sqrt {5} \sqrt {11+5 \sqrt {5}-2 x^2} \sqrt {2-\left (11+5 \sqrt {5}\right ) x^2} E\left (\arcsin \left (\sqrt {\frac {1}{2} \left (11+5 \sqrt {5}\right )} x\right )|\frac {123}{2}-\frac {55 \sqrt {5}}{2}\right )+\sqrt {11+5 \sqrt {5}-2 x^2} \left (-2 \sqrt {\frac {2}{11+5 \sqrt {5}}} \sqrt {-11+5 \sqrt {5}-2 x^2}+5 \sqrt {10-5 \left (11+5 \sqrt {5}\right ) x^2}\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1}{2} \left (11+5 \sqrt {5}\right )} x\right ),\frac {123}{2}-\frac {55 \sqrt {5}}{2}\right )}{242 \sqrt {1-5 \sqrt {5} x^2+x^4}} \] Input:

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

Output:

(246*x - 10*Sqrt[5]*x^3 - 5*Sqrt[5]*Sqrt[11 + 5*Sqrt[5] - 2*x^2]*Sqrt[2 - 
(11 + 5*Sqrt[5])*x^2]*EllipticE[ArcSin[Sqrt[(11 + 5*Sqrt[5])/2]*x], 123/2 
- (55*Sqrt[5])/2] + Sqrt[11 + 5*Sqrt[5] - 2*x^2]*(-2*Sqrt[2/(11 + 5*Sqrt[5 
])]*Sqrt[-11 + 5*Sqrt[5] - 2*x^2] + 5*Sqrt[10 - 5*(11 + 5*Sqrt[5])*x^2])*E 
llipticF[ArcSin[Sqrt[(11 + 5*Sqrt[5])/2]*x], 123/2 - (55*Sqrt[5])/2])/(242 
*Sqrt[1 - 5*Sqrt[5]*x^2 + x^4])
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.88, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {1405, 1497, 1409, 1496}

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.

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

\(\Big \downarrow \) 1405

\(\displaystyle \frac {x \left (123-5 \sqrt {5} x^2\right )}{121 \sqrt {x^4-5 \sqrt {5} x^2+1}}-\frac {1}{121} \int \frac {2-5 \sqrt {5} x^2}{\sqrt {x^4-5 \sqrt {5} x^2+1}}dx\)

\(\Big \downarrow \) 1497

\(\displaystyle \frac {1}{121} \left (-\left (\left (2-5 \sqrt {5}\right ) \int \frac {1}{\sqrt {x^4-5 \sqrt {5} x^2+1}}dx\right )-5 \sqrt {5} \int \frac {1-x^2}{\sqrt {x^4-5 \sqrt {5} x^2+1}}dx\right )+\frac {x \left (123-5 \sqrt {5} x^2\right )}{121 \sqrt {x^4-5 \sqrt {5} x^2+1}}\)

\(\Big \downarrow \) 1409

\(\displaystyle \frac {1}{121} \left (-5 \sqrt {5} \int \frac {1-x^2}{\sqrt {x^4-5 \sqrt {5} x^2+1}}dx-\frac {\left (2-5 \sqrt {5}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4-5 \sqrt {5} x^2+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{4} \left (2+5 \sqrt {5}\right )\right )}{2 \sqrt {x^4-5 \sqrt {5} x^2+1}}\right )+\frac {x \left (123-5 \sqrt {5} x^2\right )}{121 \sqrt {x^4-5 \sqrt {5} x^2+1}}\)

\(\Big \downarrow \) 1496

\(\displaystyle \frac {1}{121} \left (-\frac {\left (2-5 \sqrt {5}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4-5 \sqrt {5} x^2+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{4} \left (2+5 \sqrt {5}\right )\right )}{2 \sqrt {x^4-5 \sqrt {5} x^2+1}}-5 \sqrt {5} \left (\frac {\left (x^2+1\right ) \sqrt {\frac {x^4-5 \sqrt {5} x^2+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4} \left (2+5 \sqrt {5}\right )\right .\right )}{\sqrt {x^4-5 \sqrt {5} x^2+1}}-\frac {x \sqrt {x^4-5 \sqrt {5} x^2+1}}{x^2+1}\right )\right )+\frac {x \left (123-5 \sqrt {5} x^2\right )}{121 \sqrt {x^4-5 \sqrt {5} x^2+1}}\)

Input:

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

Output:

(x*(123 - 5*Sqrt[5]*x^2))/(121*Sqrt[1 - 5*Sqrt[5]*x^2 + x^4]) + (-5*Sqrt[5 
]*(-((x*Sqrt[1 - 5*Sqrt[5]*x^2 + x^4])/(1 + x^2)) + ((1 + x^2)*Sqrt[(1 - 5 
*Sqrt[5]*x^2 + x^4)/(1 + x^2)^2]*EllipticE[2*ArcTan[x], (2 + 5*Sqrt[5])/4] 
)/Sqrt[1 - 5*Sqrt[5]*x^2 + x^4]) - ((2 - 5*Sqrt[5])*(1 + x^2)*Sqrt[(1 - 5* 
Sqrt[5]*x^2 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], (2 + 5*Sqrt[5])/4]) 
/(2*Sqrt[1 - 5*Sqrt[5]*x^2 + x^4]))/121
 

Defintions of rubi rules used

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 1409

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] && GtQ[c/a, 0] && LtQ[ 
b/a, 0]
 

rule 1496

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q 
^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* 
x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 
/(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 
- 4*a*c, 0] && GtQ[c/a, 0] && LtQ[b/a, 0]
 

rule 1497

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q   Int[1/Sqrt[a + b*x^2 + c*x^ 
4], x], x] - Simp[e/q   Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; 
NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0] && GtQ 
[c/a, 0] && LtQ[b/a, 0]
 

Maple [A] (verified)

Time = 1.14 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.95

method	result	size
default	\(-\frac {2 \left (\frac {5 \sqrt {5}\, x^{3}}{242}-\frac {123 x}{242}\right )}{\sqrt {1-5 \sqrt {5}\, x^{2}+x^{4}}}-\frac {4 \sqrt {1-\left (\frac {11}{2}+\frac {5 \sqrt {5}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {5 \sqrt {5}}{2}-\frac {11}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {22+10 \sqrt {5}}\, x}{2}, \sqrt {-1+5 \sqrt {5}\, \left (\frac {5 \sqrt {5}}{2}-\frac {11}{2}\right )}\right )}{121 \sqrt {22+10 \sqrt {5}}\, \sqrt {1-5 \sqrt {5}\, x^{2}+x^{4}}}-\frac {20 \sqrt {5}\, \sqrt {1-\left (\frac {11}{2}+\frac {5 \sqrt {5}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {5 \sqrt {5}}{2}-\frac {11}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {22+10 \sqrt {5}}\, x}{2}, \sqrt {-1+5 \sqrt {5}\, \left (\frac {5 \sqrt {5}}{2}-\frac {11}{2}\right )}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {22+10 \sqrt {5}}\, x}{2}, \sqrt {-1+5 \sqrt {5}\, \left (\frac {5 \sqrt {5}}{2}-\frac {11}{2}\right )}\right )\right )}{121 \sqrt {22+10 \sqrt {5}}\, \sqrt {1-5 \sqrt {5}\, x^{2}+x^{4}}\, \left (11-5 \sqrt {5}\right )}\)	\(249\)
elliptic	\(-\frac {2 \left (\frac {5 \sqrt {5}\, x^{3}}{242}-\frac {123 x}{242}\right )}{\sqrt {1-5 \sqrt {5}\, x^{2}+x^{4}}}-\frac {4 \sqrt {1-\left (\frac {11}{2}+\frac {5 \sqrt {5}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {5 \sqrt {5}}{2}-\frac {11}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {22+10 \sqrt {5}}\, x}{2}, \sqrt {-1+5 \sqrt {5}\, \left (\frac {5 \sqrt {5}}{2}-\frac {11}{2}\right )}\right )}{121 \sqrt {22+10 \sqrt {5}}\, \sqrt {1-5 \sqrt {5}\, x^{2}+x^{4}}}-\frac {20 \sqrt {5}\, \sqrt {1-\left (\frac {11}{2}+\frac {5 \sqrt {5}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {5 \sqrt {5}}{2}-\frac {11}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {22+10 \sqrt {5}}\, x}{2}, \sqrt {-1+5 \sqrt {5}\, \left (\frac {5 \sqrt {5}}{2}-\frac {11}{2}\right )}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {22+10 \sqrt {5}}\, x}{2}, \sqrt {-1+5 \sqrt {5}\, \left (\frac {5 \sqrt {5}}{2}-\frac {11}{2}\right )}\right )\right )}{121 \sqrt {22+10 \sqrt {5}}\, \sqrt {1-5 \sqrt {5}\, x^{2}+x^{4}}\, \left (11-5 \sqrt {5}\right )}\)	\(249\)
risch	\(-\frac {\sqrt {5}\, x \left (25 x^{2}-123 \sqrt {5}\right )}{605 \sqrt {1-5 \sqrt {5}\, x^{2}+x^{4}}}+\frac {\sqrt {5}\, \left (-\frac {4 \sqrt {5}\, \sqrt {1-\left (\frac {11}{2}+\frac {5 \sqrt {5}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {5 \sqrt {5}}{2}-\frac {11}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {22+10 \sqrt {5}}\, x}{2}, \sqrt {-1+5 \sqrt {5}\, \left (\frac {5 \sqrt {5}}{2}-\frac {11}{2}\right )}\right )}{\sqrt {22+10 \sqrt {5}}\, \sqrt {1-5 \sqrt {5}\, x^{2}+x^{4}}}-\frac {100 \sqrt {1-\left (\frac {11}{2}+\frac {5 \sqrt {5}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {5 \sqrt {5}}{2}-\frac {11}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {22+10 \sqrt {5}}\, x}{2}, \sqrt {-1+5 \sqrt {5}\, \left (\frac {5 \sqrt {5}}{2}-\frac {11}{2}\right )}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {22+10 \sqrt {5}}\, x}{2}, \sqrt {-1+5 \sqrt {5}\, \left (\frac {5 \sqrt {5}}{2}-\frac {11}{2}\right )}\right )\right )}{\sqrt {22+10 \sqrt {5}}\, \sqrt {1-5 \sqrt {5}\, x^{2}+x^{4}}\, \left (11-5 \sqrt {5}\right )}\right )}{605}\)	\(258\)

Input:

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

Output:

-2*(5/242*5^(1/2)*x^3-123/242*x)/(1-5*5^(1/2)*x^2+x^4)^(1/2)-4/121/(22+10* 
5^(1/2))^(1/2)*(1-(11/2+5/2*5^(1/2))*x^2)^(1/2)*(1-(5/2*5^(1/2)-11/2)*x^2) 
^(1/2)/(1-5*5^(1/2)*x^2+x^4)^(1/2)*EllipticF(1/2*(22+10*5^(1/2))^(1/2)*x,( 
-1+5*5^(1/2)*(5/2*5^(1/2)-11/2))^(1/2))-20/121*5^(1/2)/(22+10*5^(1/2))^(1/ 
2)*(1-(11/2+5/2*5^(1/2))*x^2)^(1/2)*(1-(5/2*5^(1/2)-11/2)*x^2)^(1/2)/(1-5* 
5^(1/2)*x^2+x^4)^(1/2)/(11-5*5^(1/2))*(EllipticF(1/2*(22+10*5^(1/2))^(1/2) 
*x,(-1+5*5^(1/2)*(5/2*5^(1/2)-11/2))^(1/2))-EllipticE(1/2*(22+10*5^(1/2))^ 
(1/2)*x,(-1+5*5^(1/2)*(5/2*5^(1/2)-11/2))^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.65 \[ \int \frac {1}{\left (1-5 \sqrt {5} x^2+x^4\right )^{3/2}} \, dx=-\frac {5 \, {\left (25 \, x^{8} - 3075 \, x^{4} + 11 \, \sqrt {5} {\left (x^{8} - 123 \, x^{4} + 1\right )} + 25\right )} \sqrt {\frac {5}{2} \, \sqrt {5} + \frac {11}{2}} E(\arcsin \left (x \sqrt {\frac {5}{2} \, \sqrt {5} + \frac {11}{2}}\right )\,|\,-\frac {55}{2} \, \sqrt {5} + \frac {123}{2}) - 3 \, {\left (49 \, x^{8} - 6027 \, x^{4} + 15 \, \sqrt {5} {\left (x^{8} - 123 \, x^{4} + 1\right )} + 49\right )} \sqrt {\frac {5}{2} \, \sqrt {5} + \frac {11}{2}} F(\arcsin \left (x \sqrt {\frac {5}{2} \, \sqrt {5} + \frac {11}{2}}\right )\,|\,-\frac {55}{2} \, \sqrt {5} + \frac {123}{2}) + 2 \, {\left (2 \, x^{5} + 5 \, \sqrt {5} {\left (x^{7} - 122 \, x^{3}\right )} - 123 \, x\right )} \sqrt {x^{4} - 5 \, \sqrt {5} x^{2} + 1}}{242 \, {\left (x^{8} - 123 \, x^{4} + 1\right )}} \] Input:

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

Output:

-1/242*(5*(25*x^8 - 3075*x^4 + 11*sqrt(5)*(x^8 - 123*x^4 + 1) + 25)*sqrt(5 
/2*sqrt(5) + 11/2)*elliptic_e(arcsin(x*sqrt(5/2*sqrt(5) + 11/2)), -55/2*sq 
rt(5) + 123/2) - 3*(49*x^8 - 6027*x^4 + 15*sqrt(5)*(x^8 - 123*x^4 + 1) + 4 
9)*sqrt(5/2*sqrt(5) + 11/2)*elliptic_f(arcsin(x*sqrt(5/2*sqrt(5) + 11/2)), 
 -55/2*sqrt(5) + 123/2) + 2*(2*x^5 + 5*sqrt(5)*(x^7 - 122*x^3) - 123*x)*sq 
rt(x^4 - 5*sqrt(5)*x^2 + 1))/(x^8 - 123*x^4 + 1)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (1-5 \sqrt {5} x^2+x^4\right )^{3/2}} \, dx=\frac {10 \sqrt {-5 \sqrt {5}\, x^{2}+x^{4}+1}\, \sqrt {5}\, x +11 \sqrt {-5 \sqrt {5}\, x^{2}+x^{4}+1}\, x^{3}-5 \sqrt {5}\, \left (\int \frac {\sqrt {-5 \sqrt {5}\, x^{2}+x^{4}+1}}{x^{16}-246 x^{12}+15131 x^{8}-246 x^{4}+1}d x \right ) x^{8}+615 \sqrt {5}\, \left (\int \frac {\sqrt {-5 \sqrt {5}\, x^{2}+x^{4}+1}}{x^{16}-246 x^{12}+15131 x^{8}-246 x^{4}+1}d x \right ) x^{4}-5 \sqrt {5}\, \left (\int \frac {\sqrt {-5 \sqrt {5}\, x^{2}+x^{4}+1}}{x^{16}-246 x^{12}+15131 x^{8}-246 x^{4}+1}d x \right )-1770 \sqrt {5}\, \left (\int \frac {\sqrt {-5 \sqrt {5}\, x^{2}+x^{4}+1}\, x^{4}}{x^{16}-246 x^{12}+15131 x^{8}-246 x^{4}+1}d x \right ) x^{8}+217710 \sqrt {5}\, \left (\int \frac {\sqrt {-5 \sqrt {5}\, x^{2}+x^{4}+1}\, x^{4}}{x^{16}-246 x^{12}+15131 x^{8}-246 x^{4}+1}d x \right ) x^{4}-1770 \sqrt {5}\, \left (\int \frac {\sqrt {-5 \sqrt {5}\, x^{2}+x^{4}+1}\, x^{4}}{x^{16}-246 x^{12}+15131 x^{8}-246 x^{4}+1}d x \right )+33 \left (\int \frac {\sqrt {-5 \sqrt {5}\, x^{2}+x^{4}+1}\, x^{10}}{x^{16}-246 x^{12}+15131 x^{8}-246 x^{4}+1}d x \right ) x^{8}-4059 \left (\int \frac {\sqrt {-5 \sqrt {5}\, x^{2}+x^{4}+1}\, x^{10}}{x^{16}-246 x^{12}+15131 x^{8}-246 x^{4}+1}d x \right ) x^{4}+33 \left (\int \frac {\sqrt {-5 \sqrt {5}\, x^{2}+x^{4}+1}\, x^{10}}{x^{16}-246 x^{12}+15131 x^{8}-246 x^{4}+1}d x \right )+467 \left (\int \frac {\sqrt {-5 \sqrt {5}\, x^{2}+x^{4}+1}\, x^{2}}{x^{16}-246 x^{12}+15131 x^{8}-246 x^{4}+1}d x \right ) x^{8}-57441 \left (\int \frac {\sqrt {-5 \sqrt {5}\, x^{2}+x^{4}+1}\, x^{2}}{x^{16}-246 x^{12}+15131 x^{8}-246 x^{4}+1}d x \right ) x^{4}+467 \left (\int \frac {\sqrt {-5 \sqrt {5}\, x^{2}+x^{4}+1}\, x^{2}}{x^{16}-246 x^{12}+15131 x^{8}-246 x^{4}+1}d x \right )}{5 \sqrt {5}\, \left (x^{8}-123 x^{4}+1\right )} \] Input:

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

Output:

(10*sqrt( - 5*sqrt(5)*x**2 + x**4 + 1)*sqrt(5)*x + 11*sqrt( - 5*sqrt(5)*x* 
*2 + x**4 + 1)*x**3 - 5*sqrt(5)*int(sqrt( - 5*sqrt(5)*x**2 + x**4 + 1)/(x* 
*16 - 246*x**12 + 15131*x**8 - 246*x**4 + 1),x)*x**8 + 615*sqrt(5)*int(sqr 
t( - 5*sqrt(5)*x**2 + x**4 + 1)/(x**16 - 246*x**12 + 15131*x**8 - 246*x**4 
 + 1),x)*x**4 - 5*sqrt(5)*int(sqrt( - 5*sqrt(5)*x**2 + x**4 + 1)/(x**16 - 
246*x**12 + 15131*x**8 - 246*x**4 + 1),x) - 1770*sqrt(5)*int((sqrt( - 5*sq 
rt(5)*x**2 + x**4 + 1)*x**4)/(x**16 - 246*x**12 + 15131*x**8 - 246*x**4 + 
1),x)*x**8 + 217710*sqrt(5)*int((sqrt( - 5*sqrt(5)*x**2 + x**4 + 1)*x**4)/ 
(x**16 - 246*x**12 + 15131*x**8 - 246*x**4 + 1),x)*x**4 - 1770*sqrt(5)*int 
((sqrt( - 5*sqrt(5)*x**2 + x**4 + 1)*x**4)/(x**16 - 246*x**12 + 15131*x**8 
 - 246*x**4 + 1),x) + 33*int((sqrt( - 5*sqrt(5)*x**2 + x**4 + 1)*x**10)/(x 
**16 - 246*x**12 + 15131*x**8 - 246*x**4 + 1),x)*x**8 - 4059*int((sqrt( - 
5*sqrt(5)*x**2 + x**4 + 1)*x**10)/(x**16 - 246*x**12 + 15131*x**8 - 246*x* 
*4 + 1),x)*x**4 + 33*int((sqrt( - 5*sqrt(5)*x**2 + x**4 + 1)*x**10)/(x**16 
 - 246*x**12 + 15131*x**8 - 246*x**4 + 1),x) + 467*int((sqrt( - 5*sqrt(5)* 
x**2 + x**4 + 1)*x**2)/(x**16 - 246*x**12 + 15131*x**8 - 246*x**4 + 1),x)* 
x**8 - 57441*int((sqrt( - 5*sqrt(5)*x**2 + x**4 + 1)*x**2)/(x**16 - 246*x* 
*12 + 15131*x**8 - 246*x**4 + 1),x)*x**4 + 467*int((sqrt( - 5*sqrt(5)*x**2 
 + x**4 + 1)*x**2)/(x**16 - 246*x**12 + 15131*x**8 - 246*x**4 + 1),x))/(5* 
sqrt(5)*(x**8 - 123*x**4 + 1))