\(\int \frac {(2+x^2) \sqrt {4-5 x^2+x^4}}{x^2 (-2+2 x+x^2)} \, dx\) [1048]

Optimal result

Integrand size = 33, antiderivative size = 79 \[ \int \frac {\left (2+x^2\right ) \sqrt {4-5 x^2+x^4}}{x^2 \left (-2+2 x+x^2\right )} \, dx=\frac {\sqrt {4-5 x^2+x^4}}{x}-4 \text {arctanh}\left (\frac {-2+x+x^2}{\sqrt {4-5 x^2+x^4}}\right )+2 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {4-5 x^2+x^4}}{\sqrt {3} \left (-2+x+x^2\right )}\right ) \]

[Out]

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

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 2.07 (sec) , antiderivative size = 938, normalized size of antiderivative = 11.87, number of steps used = 57, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {6860, 1131, 1197, 1110, 1196, 1128, 748, 857, 635, 212, 738, 1742, 1222, 1228, 1470, 554, 432, 431, 552, 551, 1261} \[ \int \frac {\left (2+x^2\right ) \sqrt {4-5 x^2+x^4}}{x^2 \left (-2+2 x+x^2\right )} \, dx=\text {arctanh}\left (\frac {8-5 x^2}{4 \sqrt {x^4-5 x^2+4}}\right )+\frac {1}{8} \left (9+\sqrt {3}\right ) \text {arctanh}\left (\frac {5-2 x^2}{2 \sqrt {x^4-5 x^2+4}}\right )+\frac {1}{8} \left (9-\sqrt {3}\right ) \text {arctanh}\left (\frac {5-2 x^2}{2 \sqrt {x^4-5 x^2+4}}\right )-\frac {5}{4} \text {arctanh}\left (\frac {5-2 x^2}{2 \sqrt {x^4-5 x^2+4}}\right )+\frac {1}{2} \sqrt {3} \text {arctanh}\left (\frac {2 \left (6-5 \sqrt {3}\right )-\left (3-4 \sqrt {3}\right ) x^2}{2 \sqrt {6 \left (2-\sqrt {3}\right )} \sqrt {x^4-5 x^2+4}}\right )-\frac {1}{2} \sqrt {3} \text {arctanh}\left (\frac {2 \left (6+5 \sqrt {3}\right )-\left (3+4 \sqrt {3}\right ) x^2}{2 \sqrt {6 \left (2+\sqrt {3}\right )} \sqrt {x^4-5 x^2+4}}\right )-\frac {\left (2+\sqrt {3}\right ) \sqrt {4-x^2} \sqrt {x^2-1} \operatorname {EllipticF}\left (\arccos \left (\frac {x}{2}\right ),\frac {4}{3}\right )}{\sqrt {x^4-5 x^2+4}}+\frac {\left (2-\sqrt {3}\right ) \sqrt {4-x^2} \sqrt {x^2-1} \operatorname {EllipticF}\left (\arccos \left (\frac {x}{2}\right ),\frac {4}{3}\right )}{\sqrt {x^4-5 x^2+4}}+\frac {\left (1+2 \sqrt {3}\right ) \left (x^2+2\right ) \sqrt {\frac {x^4-5 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {9}{8}\right )}{2 \sqrt {2} \sqrt {x^4-5 x^2+4}}-\frac {\sqrt {\frac {3}{2}} \left (2+\sqrt {3}\right ) \left (x^2+2\right ) \sqrt {\frac {x^4-5 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {9}{8}\right )}{2 \sqrt {x^4-5 x^2+4}}+\frac {\sqrt {\frac {3}{2}} \left (2-\sqrt {3}\right ) \left (x^2+2\right ) \sqrt {\frac {x^4-5 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {9}{8}\right )}{2 \sqrt {x^4-5 x^2+4}}+\frac {\left (1-2 \sqrt {3}\right ) \left (x^2+2\right ) \sqrt {\frac {x^4-5 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {9}{8}\right )}{2 \sqrt {2} \sqrt {x^4-5 x^2+4}}+\frac {\left (x^2+2\right ) \sqrt {\frac {x^4-5 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {9}{8}\right )}{2 \sqrt {2} \sqrt {x^4-5 x^2+4}}-\frac {3 \sqrt {1-x^2} \sqrt {4-x^2} \operatorname {EllipticPi}\left (\frac {1}{2} \left (2-\sqrt {3}\right ),\arcsin (x),\frac {1}{4}\right )}{2 \sqrt {x^4-5 x^2+4}}-\frac {3 \sqrt {1-x^2} \sqrt {4-x^2} \operatorname {EllipticPi}\left (\frac {1}{2} \left (2+\sqrt {3}\right ),\arcsin (x),\frac {1}{4}\right )}{2 \sqrt {x^4-5 x^2+4}}+\frac {\sqrt {x^4-5 x^2+4}}{x}+\frac {1}{4} \left (1+\sqrt {3}\right ) \sqrt {x^4-5 x^2+4}+\frac {1}{4} \left (1-\sqrt {3}\right ) \sqrt {x^4-5 x^2+4}-\frac {1}{2} \sqrt {x^4-5 x^2+4} \]

[In]

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

[Out]

-1/2*Sqrt[4 - 5*x^2 + x^4] + ((1 - Sqrt[3])*Sqrt[4 - 5*x^2 + x^4])/4 + ((1 + Sqrt[3])*Sqrt[4 - 5*x^2 + x^4])/4
+ Sqrt[4 - 5*x^2 + x^4]/x + ArcTanh[(8 - 5*x^2)/(4*Sqrt[4 - 5*x^2 + x^4])] - (5*ArcTanh[(5 - 2*x^2)/(2*Sqrt[4
- 5*x^2 + x^4])])/4 + ((9 - Sqrt[3])*ArcTanh[(5 - 2*x^2)/(2*Sqrt[4 - 5*x^2 + x^4])])/8 + ((9 + Sqrt[3])*ArcTa
nh[(5 - 2*x^2)/(2*Sqrt[4 - 5*x^2 + x^4])])/8 + (Sqrt[3]*ArcTanh[(2*(6 - 5*Sqrt[3]) - (3 - 4*Sqrt[3])*x^2)/(2*S
qrt[6*(2 - Sqrt[3])]*Sqrt[4 - 5*x^2 + x^4])])/2 - (Sqrt[3]*ArcTanh[(2*(6 + 5*Sqrt[3]) - (3 + 4*Sqrt[3])*x^2)/(
2*Sqrt[6*(2 + Sqrt[3])]*Sqrt[4 - 5*x^2 + x^4])])/2 + ((2 - Sqrt[3])*Sqrt[4 - x^2]*Sqrt[-1 + x^2]*EllipticF[Arc
Cos[x/2], 4/3])/Sqrt[4 - 5*x^2 + x^4] - ((2 + Sqrt[3])*Sqrt[4 - x^2]*Sqrt[-1 + x^2]*EllipticF[ArcCos[x/2], 4/3
])/Sqrt[4 - 5*x^2 + x^4] + ((2 + x^2)*Sqrt[(4 - 5*x^2 + x^4)/(2 + x^2)^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 9/8])
/(2*Sqrt[2]*Sqrt[4 - 5*x^2 + x^4]) + ((1 - 2*Sqrt[3])*(2 + x^2)*Sqrt[(4 - 5*x^2 + x^4)/(2 + x^2)^2]*EllipticF[
2*ArcTan[x/Sqrt[2]], 9/8])/(2*Sqrt[2]*Sqrt[4 - 5*x^2 + x^4]) + (Sqrt[3/2]*(2 - Sqrt[3])*(2 + x^2)*Sqrt[(4 - 5*
x^2 + x^4)/(2 + x^2)^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 9/8])/(2*Sqrt[4 - 5*x^2 + x^4]) - (Sqrt[3/2]*(2 + Sqrt[
3])*(2 + x^2)*Sqrt[(4 - 5*x^2 + x^4)/(2 + x^2)^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 9/8])/(2*Sqrt[4 - 5*x^2 + x^4
]) + ((1 + 2*Sqrt[3])*(2 + x^2)*Sqrt[(4 - 5*x^2 + x^4)/(2 + x^2)^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 9/8])/(2*Sq
rt[2]*Sqrt[4 - 5*x^2 + x^4]) - (3*Sqrt[1 - x^2]*Sqrt[4 - x^2]*EllipticPi[(2 - Sqrt[3])/2, ArcSin[x], 1/4])/(2*
Sqrt[4 - 5*x^2 + x^4]) - (3*Sqrt[1 - x^2]*Sqrt[4 - x^2]*EllipticPi[(2 + Sqrt[3])/2, ArcSin[x], 1/4])/(2*Sqrt[4
- 5*x^2 + x^4])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 431

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(-(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 432

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*
x^2], Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + (d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 0]

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-f/e, -d/c])

Rule 552

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 +
(d/c)*x^2]/Sqrt[c + d*x^2], Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[c, 0]

Rule 554

Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[d/b, Int[1/
(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x] + Dist[(b*c - a*d)/b, Int[1/((a + b*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*
x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NegQ[d/c]

Rule 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 748

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((
a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x] - Dist[p/(e*(m + 2*p + 1)), Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*
d - b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ
[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || Lt
Q[m, 1]) &&  !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 1110

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 1128

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*(a +
b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]

Rule 1131

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*x^2
+ c*x^4)^p/(d*(m + 1))), x] - Dist[2*(p/(d^2*(m + 1))), Int[(d*x)^(m + 2)*(b + 2*c*x^2)*(a + b*x^2 + c*x^4)^(p
- 1), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && LtQ[m, -1] && IntegerQ[2*p] &&
(IntegerQ[p] || IntegerQ[m])

Rule 1196

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> 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 1197

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 2]}, Dist[(
e + d*q)/q, Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Dist[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]

Rule 1222

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[-(e^2)^(-1), Int[(c*d -
b*e - c*e*x^2)*(a + b*x^2 + c*x^4)^(p - 1), x], x] + Dist[(c*d^2 - b*d*e + a*e^2)/e^2, Int[(a + b*x^2 + c*x^4
)^(p - 1)/(d + e*x^2), 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] && IGtQ[p + 1/2, 0]

Rule 1228

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

Rule 1261

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 1470

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((f_) + (g_.)*(x_)^(n_))^(r_.)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^
(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + (c*x^n)/e)^FracPar
t[p]), Int[(d + e*x^n)^(p + q)*(f + g*x^n)^r*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p,
q, r}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p]

Rule 1742

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Dist[d, Int[(a + b*x^2 + c*x^
4)^p/(d^2 - e^2*x^2), x], x] - Dist[e, Int[x*((a + b*x^2 + c*x^4)^p/(d^2 - e^2*x^2)), x], x] /; FreeQ[{a, b, c
, d, e}, x] && IntegerQ[p + 1/2]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {\sqrt {4-5 x^2+x^4}}{x^2}-\frac {\sqrt {4-5 x^2+x^4}}{x}+\frac {(4+x) \sqrt {4-5 x^2+x^4}}{-2+2 x+x^2}\right ) \, dx \\ & = -\int \frac {\sqrt {4-5 x^2+x^4}}{x^2} \, dx-\int \frac {\sqrt {4-5 x^2+x^4}}{x} \, dx+\int \frac {(4+x) \sqrt {4-5 x^2+x^4}}{-2+2 x+x^2} \, dx \\ & = \frac {\sqrt {4-5 x^2+x^4}}{x}-\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {4-5 x+x^2}}{x} \, dx,x,x^2\right )-\int \frac {-5+2 x^2}{\sqrt {4-5 x^2+x^4}} \, dx+\int \left (\frac {\left (1+\sqrt {3}\right ) \sqrt {4-5 x^2+x^4}}{2-2 \sqrt {3}+2 x}+\frac {\left (1-\sqrt {3}\right ) \sqrt {4-5 x^2+x^4}}{2+2 \sqrt {3}+2 x}\right ) \, dx \\ & = -\frac {1}{2} \sqrt {4-5 x^2+x^4}+\frac {\sqrt {4-5 x^2+x^4}}{x}+\frac {1}{4} \text {Subst}\left (\int \frac {-8+5 x}{x \sqrt {4-5 x+x^2}} \, dx,x,x^2\right )+4 \int \frac {1-\frac {x^2}{2}}{\sqrt {4-5 x^2+x^4}} \, dx+\left (1-\sqrt {3}\right ) \int \frac {\sqrt {4-5 x^2+x^4}}{2+2 \sqrt {3}+2 x} \, dx+\left (1+\sqrt {3}\right ) \int \frac {\sqrt {4-5 x^2+x^4}}{2-2 \sqrt {3}+2 x} \, dx+\int \frac {1}{\sqrt {4-5 x^2+x^4}} \, dx \\ & = -\frac {1}{2} \sqrt {4-5 x^2+x^4}+\frac {\sqrt {4-5 x^2+x^4}}{x}-\frac {2 x \sqrt {4-5 x^2+x^4}}{2+x^2}+\frac {2 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-5 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {9}{8}\right )}{\sqrt {4-5 x^2+x^4}}+\frac {\left (2+x^2\right ) \sqrt {\frac {4-5 x^2+x^4}{\left (2+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {9}{8}\right )}{2 \sqrt {2} \sqrt {4-5 x^2+x^4}}+\frac {5}{4} \text {Subst}\left (\int \frac {1}{\sqrt {4-5 x+x^2}} \, dx,x,x^2\right )-2 \text {Subst}\left (\int \frac {1}{x \sqrt {4-5 x+x^2}} \, dx,x,x^2\right )-4 \int \frac {\sqrt {4-5 x^2+x^4}}{\left (2-2 \sqrt {3}\right )^2-4 x^2} \, dx-4 \int \frac {\sqrt {4-5 x^2+x^4}}{\left (2+2 \sqrt {3}\right )^2-4 x^2} \, dx-\left (2 \left (1-\sqrt {3}\right )\right ) \int \frac {x \sqrt {4-5 x^2+x^4}}{\left (2+2 \sqrt {3}\right )^2-4 x^2} \, dx-\left (2 \left (1+\sqrt {3}\right )\right ) \int \frac {x \sqrt {4-5 x^2+x^4}}{\left (2-2 \sqrt {3}\right )^2-4 x^2} \, dx \\ & = -\frac {1}{2} \sqrt {4-5 x^2+x^4}+\frac {\sqrt {4-5 x^2+x^4}}{x}-\frac {2 x \sqrt {4-5 x^2+x^4}}{2+x^2}+\frac {2 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-5 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {9}{8}\right )}{\sqrt {4-5 x^2+x^4}}+\frac {\left (2+x^2\right ) \sqrt {\frac {4-5 x^2+x^4}{\left (2+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {9}{8}\right )}{2 \sqrt {2} \sqrt {4-5 x^2+x^4}}+\frac {1}{4} \int \frac {-20+\left (2-2 \sqrt {3}\right )^2+4 x^2}{\sqrt {4-5 x^2+x^4}} \, dx+\frac {1}{4} \int \frac {-20+\left (2+2 \sqrt {3}\right )^2+4 x^2}{\sqrt {4-5 x^2+x^4}} \, dx+\frac {5}{2} \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {-5+2 x^2}{\sqrt {4-5 x^2+x^4}}\right )+4 \text {Subst}\left (\int \frac {1}{16-x^2} \, dx,x,\frac {8-5 x^2}{\sqrt {4-5 x^2+x^4}}\right )-\left (1-\sqrt {3}\right ) \text {Subst}\left (\int \frac {\sqrt {4-5 x+x^2}}{\left (2+2 \sqrt {3}\right )^2-4 x} \, dx,x,x^2\right )-\left (24 \left (2-\sqrt {3}\right )\right ) \int \frac {1}{\left (\left (2-2 \sqrt {3}\right )^2-4 x^2\right ) \sqrt {4-5 x^2+x^4}} \, dx-\left (1+\sqrt {3}\right ) \text {Subst}\left (\int \frac {\sqrt {4-5 x+x^2}}{\left (2-2 \sqrt {3}\right )^2-4 x} \, dx,x,x^2\right )-\left (24 \left (2+\sqrt {3}\right )\right ) \int \frac {1}{\left (\left (2+2 \sqrt {3}\right )^2-4 x^2\right ) \sqrt {4-5 x^2+x^4}} \, dx \\ & = -\frac {1}{2} \sqrt {4-5 x^2+x^4}+\frac {1}{4} \left (1-\sqrt {3}\right ) \sqrt {4-5 x^2+x^4}+\frac {1}{4} \left (1+\sqrt {3}\right ) \sqrt {4-5 x^2+x^4}+\frac {\sqrt {4-5 x^2+x^4}}{x}-\frac {2 x \sqrt {4-5 x^2+x^4}}{2+x^2}+\text {arctanh}\left (\frac {8-5 x^2}{4 \sqrt {4-5 x^2+x^4}}\right )-\frac {5}{4} \text {arctanh}\left (\frac {5-2 x^2}{2 \sqrt {4-5 x^2+x^4}}\right )+\frac {2 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-5 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {9}{8}\right )}{\sqrt {4-5 x^2+x^4}}+\frac {\left (2+x^2\right ) \sqrt {\frac {4-5 x^2+x^4}{\left (2+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {9}{8}\right )}{2 \sqrt {2} \sqrt {4-5 x^2+x^4}}-2 \left (2 \int \frac {1-\frac {x^2}{2}}{\sqrt {4-5 x^2+x^4}} \, dx\right )+\left (1-2 \sqrt {3}\right ) \int \frac {1}{\sqrt {4-5 x^2+x^4}} \, dx-\left (2 \left (3-2 \sqrt {3}\right )\right ) \int \frac {-8+2 x^2}{\left (\left (2-2 \sqrt {3}\right )^2-4 x^2\right ) \sqrt {4-5 x^2+x^4}} \, dx-\frac {1}{8} \left (1-\sqrt {3}\right ) \text {Subst}\left (\int \frac {-8 \left (6+5 \sqrt {3}\right )+4 \left (3+4 \sqrt {3}\right ) x}{\left (\left (2+2 \sqrt {3}\right )^2-4 x\right ) \sqrt {4-5 x+x^2}} \, dx,x,x^2\right )-\frac {1}{8} \left (1+\sqrt {3}\right ) \text {Subst}\left (\int \frac {-8 \left (6-5 \sqrt {3}\right )+4 \left (3-4 \sqrt {3}\right ) x}{\left (\left (2-2 \sqrt {3}\right )^2-4 x\right ) \sqrt {4-5 x+x^2}} \, dx,x,x^2\right )+\left (-3+2 \sqrt {3}\right ) \int \frac {1}{\sqrt {4-5 x^2+x^4}} \, dx+\left (1+2 \sqrt {3}\right ) \int \frac {1}{\sqrt {4-5 x^2+x^4}} \, dx-\left (3+2 \sqrt {3}\right ) \int \frac {1}{\sqrt {4-5 x^2+x^4}} \, dx-\left (2 \left (3+2 \sqrt {3}\right )\right ) \int \frac {-8+2 x^2}{\left (\left (2+2 \sqrt {3}\right )^2-4 x^2\right ) \sqrt {4-5 x^2+x^4}} \, dx \\ & = -\frac {1}{2} \sqrt {4-5 x^2+x^4}+\frac {1}{4} \left (1-\sqrt {3}\right ) \sqrt {4-5 x^2+x^4}+\frac {1}{4} \left (1+\sqrt {3}\right ) \sqrt {4-5 x^2+x^4}+\frac {\sqrt {4-5 x^2+x^4}}{x}-\frac {2 x \sqrt {4-5 x^2+x^4}}{2+x^2}+\text {arctanh}\left (\frac {8-5 x^2}{4 \sqrt {4-5 x^2+x^4}}\right )-\frac {5}{4} \text {arctanh}\left (\frac {5-2 x^2}{2 \sqrt {4-5 x^2+x^4}}\right )+\frac {2 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-5 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {9}{8}\right )}{\sqrt {4-5 x^2+x^4}}-2 \left (-\frac {x \sqrt {4-5 x^2+x^4}}{2+x^2}+\frac {\sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-5 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {9}{8}\right )}{\sqrt {4-5 x^2+x^4}}\right )+\frac {\left (2+x^2\right ) \sqrt {\frac {4-5 x^2+x^4}{\left (2+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {9}{8}\right )}{2 \sqrt {2} \sqrt {4-5 x^2+x^4}}+\frac {\left (1-2 \sqrt {3}\right ) \left (2+x^2\right ) \sqrt {\frac {4-5 x^2+x^4}{\left (2+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {9}{8}\right )}{2 \sqrt {2} \sqrt {4-5 x^2+x^4}}+\frac {\sqrt {\frac {3}{2}} \left (2-\sqrt {3}\right ) \left (2+x^2\right ) \sqrt {\frac {4-5 x^2+x^4}{\left (2+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {9}{8}\right )}{2 \sqrt {4-5 x^2+x^4}}-\frac {\sqrt {\frac {3}{2}} \left (2+\sqrt {3}\right ) \left (2+x^2\right ) \sqrt {\frac {4-5 x^2+x^4}{\left (2+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {9}{8}\right )}{2 \sqrt {4-5 x^2+x^4}}+\frac {\left (1+2 \sqrt {3}\right ) \left (2+x^2\right ) \sqrt {\frac {4-5 x^2+x^4}{\left (2+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {9}{8}\right )}{2 \sqrt {2} \sqrt {4-5 x^2+x^4}}+\left (6 \left (1-\sqrt {3}\right )\right ) \text {Subst}\left (\int \frac {1}{\left (\left (2-2 \sqrt {3}\right )^2-4 x\right ) \sqrt {4-5 x+x^2}} \, dx,x,x^2\right )-\frac {1}{8} \left (9-\sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {4-5 x+x^2}} \, dx,x,x^2\right )+\left (6 \left (1+\sqrt {3}\right )\right ) \text {Subst}\left (\int \frac {1}{\left (\left (2+2 \sqrt {3}\right )^2-4 x\right ) \sqrt {4-5 x+x^2}} \, dx,x,x^2\right )-\frac {1}{8} \left (9+\sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {4-5 x+x^2}} \, dx,x,x^2\right )-\frac {\left (2 \left (3-2 \sqrt {3}\right ) \sqrt {-\frac {1}{2}+\frac {x^2}{2}} \sqrt {-8+2 x^2}\right ) \int \frac {\sqrt {-8+2 x^2}}{\left (\left (2-2 \sqrt {3}\right )^2-4 x^2\right ) \sqrt {-\frac {1}{2}+\frac {x^2}{2}}} \, dx}{\sqrt {4-5 x^2+x^4}}-\frac {\left (2 \left (3+2 \sqrt {3}\right ) \sqrt {-\frac {1}{2}+\frac {x^2}{2}} \sqrt {-8+2 x^2}\right ) \int \frac {\sqrt {-8+2 x^2}}{\left (\left (2+2 \sqrt {3}\right )^2-4 x^2\right ) \sqrt {-\frac {1}{2}+\frac {x^2}{2}}} \, dx}{\sqrt {4-5 x^2+x^4}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00 \[ \int \frac {\left (2+x^2\right ) \sqrt {4-5 x^2+x^4}}{x^2 \left (-2+2 x+x^2\right )} \, dx=\frac {\sqrt {4-5 x^2+x^4}}{x}-4 \text {arctanh}\left (\frac {-2+x+x^2}{\sqrt {4-5 x^2+x^4}}\right )+2 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {4-5 x^2+x^4}}{\sqrt {3} \left (-2+x+x^2\right )}\right ) \]

[In]

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

[Out]

Sqrt[4 - 5*x^2 + x^4]/x - 4*ArcTanh[(-2 + x + x^2)/Sqrt[4 - 5*x^2 + x^4]] + 2*Sqrt[3]*ArcTanh[Sqrt[4 - 5*x^2 +
x^4]/(Sqrt[3]*(-2 + x + x^2))]

Maple [A] (verified)

Time = 3.27 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.91

method	result	size
risch	\(\frac {\sqrt {x^{4}-5 x^{2}+4}}{x}-2 \ln \left (\frac {x^{2}+\sqrt {x^{4}-5 x^{2}+4}-2}{x}\right )+\sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (2 x^{2}+x -4\right ) \sqrt {3}}{3 \sqrt {x^{4}-5 x^{2}+4}}\right )\)	\(72\)
default	\(\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (2 x^{2}+x -4\right ) \sqrt {3}}{3 \sqrt {x^{4}-5 x^{2}+4}}\right ) x -2 \ln \left (\frac {x^{2}+\sqrt {x^{4}-5 x^{2}+4}-2}{x}\right ) x +\sqrt {x^{4}-5 x^{2}+4}}{x}\)	\(74\)
pseudoelliptic	\(\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (2 x^{2}+x -4\right ) \sqrt {3}}{3 \sqrt {x^{4}-5 x^{2}+4}}\right ) x -2 \ln \left (\frac {x^{2}+\sqrt {x^{4}-5 x^{2}+4}-2}{x}\right ) x +\sqrt {x^{4}-5 x^{2}+4}}{x}\)	\(74\)
trager	\(\frac {\sqrt {x^{4}-5 x^{2}+4}}{x}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+3 \sqrt {x^{4}-5 x^{2}+4}}{x^{2}+2 x -2}\right )-2 \ln \left (-\frac {x^{2}+\sqrt {x^{4}-5 x^{2}+4}-2}{x}\right )\)	\(105\)
elliptic	\(-\ln \left (-\frac {5}{2}+x^{2}+\sqrt {x^{4}-5 x^{2}+4}\right )-\frac {\operatorname {arctanh}\left (\frac {-5 x^{2}+8}{4 \sqrt {x^{4}-5 x^{2}+4}}\right )}{\left (2+\sqrt {3}\right ) \left (-2+\sqrt {3}\right )}+\frac {\left (9+5 \sqrt {3}\right ) \sqrt {3}\, \operatorname {arctanh}\left (\frac {24+12 \sqrt {3}+\left (3+4 \sqrt {3}\right ) \left (x^{2}-4-2 \sqrt {3}\right )}{2 \left (3+\sqrt {3}\right ) \sqrt {\left (x^{2}-4-2 \sqrt {3}\right )^{2}+\left (3+4 \sqrt {3}\right ) \left (x^{2}-4-2 \sqrt {3}\right )+12+6 \sqrt {3}}}\right )}{\left (4+2 \sqrt {3}\right ) \left (3+\sqrt {3}\right )}-\frac {\left (-9+5 \sqrt {3}\right ) \sqrt {3}\, \operatorname {arctanh}\left (\frac {24-12 \sqrt {3}+\left (3-4 \sqrt {3}\right ) \left (x^{2}-4+2 \sqrt {3}\right )}{2 \left (3-\sqrt {3}\right ) \sqrt {\left (x^{2}-4+2 \sqrt {3}\right )^{2}+\left (3-4 \sqrt {3}\right ) \left (x^{2}-4+2 \sqrt {3}\right )+12-6 \sqrt {3}}}\right )}{\left (-4+2 \sqrt {3}\right ) \left (3-\sqrt {3}\right )}+\frac {\left (\frac {\sqrt {x^{4}-5 x^{2}+4}\, \sqrt {2}}{x}-\sqrt {6}\, \operatorname {arctanh}\left (\frac {\sqrt {6}\, \sqrt {x^{4}-5 x^{2}+4}\, \sqrt {2}}{6 x}\right )\right ) \sqrt {2}}{2}\)	\(321\)

[In]

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

[Out]

(x^4-5*x^2+4)^(1/2)/x-2*ln((x^2+(x^4-5*x^2+4)^(1/2)-2)/x)+3^(1/2)*arctanh(1/3*(2*x^2+x-4)*3^(1/2)/(x^4-5*x^2+4
)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.46 \[ \int \frac {\left (2+x^2\right ) \sqrt {4-5 x^2+x^4}}{x^2 \left (-2+2 x+x^2\right )} \, dx=\frac {\sqrt {3} x \log \left (-\frac {7 \, x^{4} + 4 \, x^{3} + 2 \, \sqrt {3} \sqrt {x^{4} - 5 \, x^{2} + 4} {\left (2 \, x^{2} + x - 4\right )} - 30 \, x^{2} - 8 \, x + 28}{x^{4} + 4 \, x^{3} - 8 \, x + 4}\right ) + 4 \, x \log \left (\frac {x^{2} - \sqrt {x^{4} - 5 \, x^{2} + 4} - 2}{x}\right ) + 2 \, \sqrt {x^{4} - 5 \, x^{2} + 4}}{2 \, x} \]

[In]

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

[Out]

1/2*(sqrt(3)*x*log(-(7*x^4 + 4*x^3 + 2*sqrt(3)*sqrt(x^4 - 5*x^2 + 4)*(2*x^2 + x - 4) - 30*x^2 - 8*x + 28)/(x^4
+ 4*x^3 - 8*x + 4)) + 4*x*log((x^2 - sqrt(x^4 - 5*x^2 + 4) - 2)/x) + 2*sqrt(x^4 - 5*x^2 + 4))/x

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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