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

Optimal. Leaf size=79 \[ \frac {\sqrt {x^4-5 x^2+4}}{x}-4 \tanh ^{-1}\left (\frac {x^2+x-2}{\sqrt {x^4-5 x^2+4}}\right )+2 \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {x^4-5 x^2+4}}{\sqrt {3} \left (x^2+x-2\right )}\right ) \]

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Rubi [C]  time = 2.14, antiderivative size = 938, normalized size of antiderivative = 11.87, number of steps used = 57, number of rules used = 21, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {6728, 1117, 1183, 1096, 1182, 1114, 734, 843, 621, 206, 724, 1728, 1208, 1214, 1456, 540, 421, 420, 538, 537, 1247} \begin {gather*} \tanh ^{-1}\left (\frac {8-5 x^2}{4 \sqrt {x^4-5 x^2+4}}\right )+\frac {1}{8} \left (9+\sqrt {3}\right ) \tanh ^{-1}\left (\frac {5-2 x^2}{2 \sqrt {x^4-5 x^2+4}}\right )+\frac {1}{8} \left (9-\sqrt {3}\right ) \tanh ^{-1}\left (\frac {5-2 x^2}{2 \sqrt {x^4-5 x^2+4}}\right )-\frac {5}{4} \tanh ^{-1}\left (\frac {5-2 x^2}{2 \sqrt {x^4-5 x^2+4}}\right )+\frac {1}{2} \sqrt {3} \tanh ^{-1}\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} \tanh ^{-1}\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} F\left (\cos ^{-1}\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} F\left (\cos ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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} \Pi \left (\frac {1}{2} \left (2-\sqrt {3}\right );\sin ^{-1}(x)|\frac {1}{4}\right )}{2 \sqrt {x^4-5 x^2+4}}-\frac {3 \sqrt {1-x^2} \sqrt {4-x^2} \Pi \left (\frac {1}{2} \left (2+\sqrt {3}\right );\sin ^{-1}(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} \end {gather*}

Warning: Unable to verify antiderivative.

[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 206

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

Rule 420

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

Rule 421

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

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), 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 538

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

Rule 540

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 621

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 724

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 734

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 843

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 1096

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)]*EllipticF[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(2*q*Sqrt[a + b*x^2 + c
*x^4]), x]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] && GtQ[c/a, 0] && LtQ[b/a, 0]

Rule 1114

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 1117

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 1182

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)]*EllipticE[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(q*Sqrt[a + b*x^2 + c*x^4]), 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 1183

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 1208

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 1214

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 1247

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 1456

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*x^n)/e)^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 1728

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 6728

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*} \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 \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} \operatorname {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} \operatorname {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 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {9}{8}\right )}{2 \sqrt {2} \sqrt {4-5 x^2+x^4}}+\frac {5}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {4-5 x+x^2}} \, dx,x,x^2\right )-2 \operatorname {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 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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} \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {-5+2 x^2}{\sqrt {4-5 x^2+x^4}}\right )+4 \operatorname {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 ) \operatorname {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 ) \operatorname {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}+\tanh ^{-1}\left (\frac {8-5 x^2}{4 \sqrt {4-5 x^2+x^4}}\right )-\frac {5}{4} \tanh ^{-1}\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 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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 ) \operatorname {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 ) \operatorname {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}+\tanh ^{-1}\left (\frac {8-5 x^2}{4 \sqrt {4-5 x^2+x^4}}\right )-\frac {5}{4} \tanh ^{-1}\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 \tan ^{-1}\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 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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 ) \operatorname {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 ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {4-5 x+x^2}} \, dx,x,x^2\right )+\left (6 \left (1+\sqrt {3}\right )\right ) \operatorname {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 ) \operatorname {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}}\\ &=-\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}+\tanh ^{-1}\left (\frac {8-5 x^2}{4 \sqrt {4-5 x^2+x^4}}\right )-\frac {5}{4} \tanh ^{-1}\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 \tan ^{-1}\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 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {9}{8}\right )}{2 \sqrt {2} \sqrt {4-5 x^2+x^4}}-\left (12 \left (1-\sqrt {3}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{256-80 \left (2-2 \sqrt {3}\right )^2+4 \left (2-2 \sqrt {3}\right )^4-x^2} \, dx,x,\frac {-32+5 \left (2-2 \sqrt {3}\right )^2-4 \sqrt {3} \left (-4+\sqrt {3}\right ) x^2}{\sqrt {4-5 x^2+x^4}}\right )-\frac {1}{4} \left (9-\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {-5+2 x^2}{\sqrt {4-5 x^2+x^4}}\right )-\left (12 \left (1+\sqrt {3}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{256-80 \left (2+2 \sqrt {3}\right )^2+4 \left (2+2 \sqrt {3}\right )^4-x^2} \, dx,x,\frac {-32+5 \left (2+2 \sqrt {3}\right )^2-4 \sqrt {3} \left (4+\sqrt {3}\right ) x^2}{\sqrt {4-5 x^2+x^4}}\right )-\frac {1}{4} \left (9+\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {-5+2 x^2}{\sqrt {4-5 x^2+x^4}}\right )+\frac {\left (\left (3-2 \sqrt {3}\right ) \sqrt {-\frac {1}{2}+\frac {x^2}{2}} \sqrt {-8+2 x^2}\right ) \int \frac {1}{\sqrt {-\frac {1}{2}+\frac {x^2}{2}} \sqrt {-8+2 x^2}} \, dx}{\sqrt {4-5 x^2+x^4}}+\frac {\left (8 \sqrt {3} \left (3-2 \sqrt {3}\right ) \sqrt {-\frac {1}{2}+\frac {x^2}{2}} \sqrt {-8+2 x^2}\right ) \int \frac {1}{\left (\left (2-2 \sqrt {3}\right )^2-4 x^2\right ) \sqrt {-\frac {1}{2}+\frac {x^2}{2}} \sqrt {-8+2 x^2}} \, dx}{\sqrt {4-5 x^2+x^4}}+\frac {\left (\left (3+2 \sqrt {3}\right ) \sqrt {-\frac {1}{2}+\frac {x^2}{2}} \sqrt {-8+2 x^2}\right ) \int \frac {1}{\sqrt {-\frac {1}{2}+\frac {x^2}{2}} \sqrt {-8+2 x^2}} \, dx}{\sqrt {4-5 x^2+x^4}}-\frac {\left (8 \sqrt {3} \left (3+2 \sqrt {3}\right ) \sqrt {-\frac {1}{2}+\frac {x^2}{2}} \sqrt {-8+2 x^2}\right ) \int \frac {1}{\left (\left (2+2 \sqrt {3}\right )^2-4 x^2\right ) \sqrt {-\frac {1}{2}+\frac {x^2}{2}} \sqrt {-8+2 x^2}} \, dx}{\sqrt {4-5 x^2+x^4}}\\ &=-\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}+\tanh ^{-1}\left (\frac {8-5 x^2}{4 \sqrt {4-5 x^2+x^4}}\right )-\frac {5}{4} \tanh ^{-1}\left (\frac {5-2 x^2}{2 \sqrt {4-5 x^2+x^4}}\right )+\frac {1}{8} \left (9-\sqrt {3}\right ) \tanh ^{-1}\left (\frac {5-2 x^2}{2 \sqrt {4-5 x^2+x^4}}\right )+\frac {1}{8} \left (9+\sqrt {3}\right ) \tanh ^{-1}\left (\frac {5-2 x^2}{2 \sqrt {4-5 x^2+x^4}}\right )+\frac {1}{2} \sqrt {3} \tanh ^{-1}\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 {4-5 x^2+x^4}}\right )-\frac {1}{2} \sqrt {3} \tanh ^{-1}\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 {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 \tan ^{-1}\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 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {9}{8}\right )}{2 \sqrt {2} \sqrt {4-5 x^2+x^4}}+\frac {\left (\left (3-2 \sqrt {3}\right ) \sqrt {1-\frac {x^2}{4}} \sqrt {-\frac {1}{2}+\frac {x^2}{2}}\right ) \int \frac {1}{\sqrt {1-\frac {x^2}{4}} \sqrt {-\frac {1}{2}+\frac {x^2}{2}}} \, dx}{\sqrt {4-5 x^2+x^4}}+\frac {\left (\left (3+2 \sqrt {3}\right ) \sqrt {1-\frac {x^2}{4}} \sqrt {-\frac {1}{2}+\frac {x^2}{2}}\right ) \int \frac {1}{\sqrt {1-\frac {x^2}{4}} \sqrt {-\frac {1}{2}+\frac {x^2}{2}}} \, dx}{\sqrt {4-5 x^2+x^4}}+\frac {\left (8 \sqrt {3} \left (3-2 \sqrt {3}\right ) \sqrt {1-x^2} \sqrt {-8+2 x^2}\right ) \int \frac {1}{\left (\left (2-2 \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1-x^2} \sqrt {-8+2 x^2}} \, dx}{\sqrt {4-5 x^2+x^4}}-\frac {\left (8 \sqrt {3} \left (3+2 \sqrt {3}\right ) \sqrt {1-x^2} \sqrt {-8+2 x^2}\right ) \int \frac {1}{\left (\left (2+2 \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1-x^2} \sqrt {-8+2 x^2}} \, dx}{\sqrt {4-5 x^2+x^4}}\\ &=-\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}+\tanh ^{-1}\left (\frac {8-5 x^2}{4 \sqrt {4-5 x^2+x^4}}\right )-\frac {5}{4} \tanh ^{-1}\left (\frac {5-2 x^2}{2 \sqrt {4-5 x^2+x^4}}\right )+\frac {1}{8} \left (9-\sqrt {3}\right ) \tanh ^{-1}\left (\frac {5-2 x^2}{2 \sqrt {4-5 x^2+x^4}}\right )+\frac {1}{8} \left (9+\sqrt {3}\right ) \tanh ^{-1}\left (\frac {5-2 x^2}{2 \sqrt {4-5 x^2+x^4}}\right )+\frac {1}{2} \sqrt {3} \tanh ^{-1}\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 {4-5 x^2+x^4}}\right )-\frac {1}{2} \sqrt {3} \tanh ^{-1}\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 {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 \tan ^{-1}\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 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {9}{8}\right )}{\sqrt {4-5 x^2+x^4}}\right )+\frac {\left (2-\sqrt {3}\right ) \sqrt {4-x^2} \sqrt {-1+x^2} F\left (\cos ^{-1}\left (\frac {x}{2}\right )|\frac {4}{3}\right )}{\sqrt {4-5 x^2+x^4}}-\frac {\left (2+\sqrt {3}\right ) \sqrt {4-x^2} \sqrt {-1+x^2} F\left (\cos ^{-1}\left (\frac {x}{2}\right )|\frac {4}{3}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {9}{8}\right )}{2 \sqrt {2} \sqrt {4-5 x^2+x^4}}+\frac {\left (8 \sqrt {3} \left (3-2 \sqrt {3}\right ) \sqrt {1-x^2} \sqrt {1-\frac {x^2}{4}}\right ) \int \frac {1}{\left (\left (2-2 \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1-x^2} \sqrt {1-\frac {x^2}{4}}} \, dx}{\sqrt {4-5 x^2+x^4}}-\frac {\left (8 \sqrt {3} \left (3+2 \sqrt {3}\right ) \sqrt {1-x^2} \sqrt {1-\frac {x^2}{4}}\right ) \int \frac {1}{\left (\left (2+2 \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1-x^2} \sqrt {1-\frac {x^2}{4}}} \, dx}{\sqrt {4-5 x^2+x^4}}\\ &=-\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}+\tanh ^{-1}\left (\frac {8-5 x^2}{4 \sqrt {4-5 x^2+x^4}}\right )-\frac {5}{4} \tanh ^{-1}\left (\frac {5-2 x^2}{2 \sqrt {4-5 x^2+x^4}}\right )+\frac {1}{8} \left (9-\sqrt {3}\right ) \tanh ^{-1}\left (\frac {5-2 x^2}{2 \sqrt {4-5 x^2+x^4}}\right )+\frac {1}{8} \left (9+\sqrt {3}\right ) \tanh ^{-1}\left (\frac {5-2 x^2}{2 \sqrt {4-5 x^2+x^4}}\right )+\frac {1}{2} \sqrt {3} \tanh ^{-1}\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 {4-5 x^2+x^4}}\right )-\frac {1}{2} \sqrt {3} \tanh ^{-1}\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 {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 \tan ^{-1}\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 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {9}{8}\right )}{\sqrt {4-5 x^2+x^4}}\right )+\frac {\left (2-\sqrt {3}\right ) \sqrt {4-x^2} \sqrt {-1+x^2} F\left (\cos ^{-1}\left (\frac {x}{2}\right )|\frac {4}{3}\right )}{\sqrt {4-5 x^2+x^4}}-\frac {\left (2+\sqrt {3}\right ) \sqrt {4-x^2} \sqrt {-1+x^2} F\left (\cos ^{-1}\left (\frac {x}{2}\right )|\frac {4}{3}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {9}{8}\right )}{2 \sqrt {2} \sqrt {4-5 x^2+x^4}}-\frac {3 \sqrt {1-x^2} \sqrt {4-x^2} \Pi \left (\frac {1}{2} \left (2-\sqrt {3}\right );\sin ^{-1}(x)|\frac {1}{4}\right )}{2 \sqrt {4-5 x^2+x^4}}-\frac {3 \sqrt {1-x^2} \sqrt {4-x^2} \Pi \left (\frac {1}{2} \left (2+\sqrt {3}\right );\sin ^{-1}(x)|\frac {1}{4}\right )}{2 \sqrt {4-5 x^2+x^4}}\\ \end {align*}

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Mathematica [C]  time = 0.91, size = 390, normalized size = 4.94 \begin {gather*} \frac {2 x^4-10 x^2+3 \sqrt {1-x^2} \sqrt {4-x^2} x F\left (\sin ^{-1}(x)|\frac {1}{4}\right )-3 \sqrt {1-x^2} \sqrt {4-x^2} x \Pi \left (1-\frac {\sqrt {3}}{2};\sin ^{-1}(x)|\frac {1}{4}\right )-3 \sqrt {1-x^2} \sqrt {4-x^2} x \Pi \left (1+\frac {\sqrt {3}}{2};\sin ^{-1}(x)|\frac {1}{4}\right )+2 \sqrt {x^4-5 x^2+4} x \tanh ^{-1}\left (\frac {8-5 x^2}{4 \sqrt {x^4-5 x^2+4}}\right )-2 \sqrt {x^4-5 x^2+4} x \tanh ^{-1}\left (\frac {2 x^2-5}{2 \sqrt {x^4-5 x^2+4}}\right )-\sqrt {3} \sqrt {x^4-5 x^2+4} x \tanh ^{-1}\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 )+\sqrt {3} \sqrt {x^4-5 x^2+4} x \tanh ^{-1}\left (\frac {\left (4 \sqrt {3}-3\right ) x^2-10 \sqrt {3}+12}{2 \sqrt {6} \sqrt {-\left (\left (\sqrt {3}-2\right ) \left (x^4-5 x^2+4\right )\right )}}\right )+8}{2 x \sqrt {x^4-5 x^2+4}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(8 - 10*x^2 + 2*x^4 + 2*x*Sqrt[4 - 5*x^2 + x^4]*ArcTanh[(8 - 5*x^2)/(4*Sqrt[4 - 5*x^2 + x^4])] - 2*x*Sqrt[4 -
5*x^2 + x^4]*ArcTanh[(-5 + 2*x^2)/(2*Sqrt[4 - 5*x^2 + x^4])] - Sqrt[3]*x*Sqrt[4 - 5*x^2 + x^4]*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])] + Sqrt[3]*x*Sqrt[4 - 5*x^2
+ x^4]*ArcTanh[(12 - 10*Sqrt[3] + (-3 + 4*Sqrt[3])*x^2)/(2*Sqrt[6]*Sqrt[-((-2 + Sqrt[3])*(4 - 5*x^2 + x^4))])
] + 3*x*Sqrt[1 - x^2]*Sqrt[4 - x^2]*EllipticF[ArcSin[x], 1/4] - 3*x*Sqrt[1 - x^2]*Sqrt[4 - x^2]*EllipticPi[1 -
Sqrt[3]/2, ArcSin[x], 1/4] - 3*x*Sqrt[1 - x^2]*Sqrt[4 - x^2]*EllipticPi[1 + Sqrt[3]/2, ArcSin[x], 1/4])/(2*x*
Sqrt[4 - 5*x^2 + x^4])

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IntegrateAlgebraic [A]  time = 0.60, size = 79, normalized size = 1.00 \begin {gather*} \frac {\sqrt {4-5 x^2+x^4}}{x}-4 \tanh ^{-1}\left (\frac {-2+x+x^2}{\sqrt {4-5 x^2+x^4}}\right )+2 \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {4-5 x^2+x^4}}{\sqrt {3} \left (-2+x+x^2\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((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))]

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fricas [A]  time = 0.81, size = 115, normalized size = 1.46 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [C]  time = 0.67, size = 107, normalized size = 1.35

method	result	size
trager	\(\frac {\sqrt {x^{4}-5 x^{2}+4}}{x}+\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{2}+\RootOf \left (\textit {\_Z}^{2}-3\right ) x +3 \sqrt {x^{4}-5 x^{2}+4}-4 \RootOf \left (\textit {\_Z}^{2}-3\right )}{x^{2}+2 x -2}\right )+2 \ln \left (-\frac {-x^{2}+\sqrt {x^{4}-5 x^{2}+4}+2}{x}\right )\)	\(107\)
elliptic	\(-\ln \left (-\frac {5}{2}+x^{2}+\sqrt {x^{4}-5 x^{2}+4}\right )+\frac {9 \sqrt {3}\, \arctanh \left (\frac {24-12 \sqrt {3}+\left (3-4 \sqrt {3}\right ) \left (x^{2}-4+2 \sqrt {3}\right )}{2 \left (3-\sqrt {3}\right ) \sqrt {\left (x^{2}-4+2 \sqrt {3}\right )^{2}+\left (3-4 \sqrt {3}\right ) \left (x^{2}-4+2 \sqrt {3}\right )+12-6 \sqrt {3}}}\right )}{2 \left (\sqrt {3}-2\right ) \left (3-\sqrt {3}\right )}-\frac {15 \arctanh \left (\frac {24-12 \sqrt {3}+\left (3-4 \sqrt {3}\right ) \left (x^{2}-4+2 \sqrt {3}\right )}{2 \left (3-\sqrt {3}\right ) \sqrt {\left (x^{2}-4+2 \sqrt {3}\right )^{2}+\left (3-4 \sqrt {3}\right ) \left (x^{2}-4+2 \sqrt {3}\right )+12-6 \sqrt {3}}}\right )}{2 \left (\sqrt {3}-2\right ) \left (3-\sqrt {3}\right )}-\frac {\arctanh \left (\frac {-5 x^{2}+8}{4 \sqrt {x^{4}-5 x^{2}+4}}\right )}{\left (2+\sqrt {3}\right ) \left (\sqrt {3}-2\right )}+\frac {9 \sqrt {3}\, \arctanh \left (\frac {24+12 \sqrt {3}+\left (3+4 \sqrt {3}\right ) \left (x^{2}-4-2 \sqrt {3}\right )}{2 \left (3+\sqrt {3}\right ) \sqrt {\left (x^{2}-4-2 \sqrt {3}\right )^{2}+\left (3+4 \sqrt {3}\right ) \left (x^{2}-4-2 \sqrt {3}\right )+12+6 \sqrt {3}}}\right )}{2 \left (2+\sqrt {3}\right ) \left (3+\sqrt {3}\right )}+\frac {15 \arctanh \left (\frac {24+12 \sqrt {3}+\left (3+4 \sqrt {3}\right ) \left (x^{2}-4-2 \sqrt {3}\right )}{2 \left (3+\sqrt {3}\right ) \sqrt {\left (x^{2}-4-2 \sqrt {3}\right )^{2}+\left (3+4 \sqrt {3}\right ) \left (x^{2}-4-2 \sqrt {3}\right )+12+6 \sqrt {3}}}\right )}{2 \left (2+\sqrt {3}\right ) \left (3+\sqrt {3}\right )}+\frac {\left (\frac {\sqrt {x^{4}-5 x^{2}+4}\, \sqrt {2}}{x}-\sqrt {6}\, \arctanh \left (\frac {\sqrt {6}\, \sqrt {x^{4}-5 x^{2}+4}\, \sqrt {2}}{6 x}\right )\right ) \sqrt {2}}{2}\)	\(488\)
risch	\(\frac {\sqrt {x^{4}-5 x^{2}+4}}{x}-\ln \left (-\frac {5}{2}+x^{2}+\sqrt {x^{4}-5 x^{2}+4}\right )+\frac {3 \sqrt {-x^{2}+1}\, \sqrt {-x^{2}+4}\, \EllipticF \left (x , \frac {1}{2}\right )}{2 \sqrt {x^{4}-5 x^{2}+4}}+\arctanh \left (\frac {-5 x^{2}+8}{4 \sqrt {x^{4}-5 x^{2}+4}}\right )-\frac {3 \arctanh \left (\frac {3}{2 \sqrt {-\frac {3 \sqrt {3}}{2}+3}\, \sqrt {\frac {1}{4} x^{4}-\frac {5}{4} x^{2}+1}}-\frac {5 \sqrt {3}}{4 \sqrt {-\frac {3 \sqrt {3}}{2}+3}\, \sqrt {\frac {1}{4} x^{4}-\frac {5}{4} x^{2}+1}}-\frac {3 x^{2}}{8 \sqrt {-\frac {3 \sqrt {3}}{2}+3}\, \sqrt {\frac {1}{4} x^{4}-\frac {5}{4} x^{2}+1}}+\frac {\sqrt {3}\, x^{2}}{2 \sqrt {-\frac {3 \sqrt {3}}{2}+3}\, \sqrt {\frac {1}{4} x^{4}-\frac {5}{4} x^{2}+1}}\right )}{2 \sqrt {-\frac {3 \sqrt {3}}{2}+3}}+\frac {3 \sqrt {-x^{2}+1}\, \sqrt {-x^{2}+4}\, \EllipticPi \left (x , \frac {1}{\left (\sqrt {3}-1\right )^{2}}, \frac {1}{2}\right )}{2 \sqrt {x^{4}-5 x^{2}+4}\, \left (\sqrt {3}-1\right )}+\frac {3 \sqrt {3}\, \arctanh \left (\frac {3}{2 \sqrt {-\frac {3 \sqrt {3}}{2}+3}\, \sqrt {\frac {1}{4} x^{4}-\frac {5}{4} x^{2}+1}}-\frac {5 \sqrt {3}}{4 \sqrt {-\frac {3 \sqrt {3}}{2}+3}\, \sqrt {\frac {1}{4} x^{4}-\frac {5}{4} x^{2}+1}}-\frac {3 x^{2}}{8 \sqrt {-\frac {3 \sqrt {3}}{2}+3}\, \sqrt {\frac {1}{4} x^{4}-\frac {5}{4} x^{2}+1}}+\frac {\sqrt {3}\, x^{2}}{2 \sqrt {-\frac {3 \sqrt {3}}{2}+3}\, \sqrt {\frac {1}{4} x^{4}-\frac {5}{4} x^{2}+1}}\right )}{2 \sqrt {-\frac {3 \sqrt {3}}{2}+3}}-\frac {3 \sqrt {3}\, \sqrt {-x^{2}+1}\, \sqrt {-x^{2}+4}\, \EllipticPi \left (x , \frac {1}{\left (\sqrt {3}-1\right )^{2}}, \frac {1}{2}\right )}{2 \sqrt {x^{4}-5 x^{2}+4}\, \left (\sqrt {3}-1\right )}+\frac {3 \sqrt {3}\, \arctanh \left (-\frac {3}{2 \sqrt {\frac {3 \sqrt {3}}{2}+3}\, \sqrt {\frac {1}{4} x^{4}-\frac {5}{4} x^{2}+1}}-\frac {5 \sqrt {3}}{4 \sqrt {\frac {3 \sqrt {3}}{2}+3}\, \sqrt {\frac {1}{4} x^{4}-\frac {5}{4} x^{2}+1}}+\frac {3 x^{2}}{8 \sqrt {\frac {3 \sqrt {3}}{2}+3}\, \sqrt {\frac {1}{4} x^{4}-\frac {5}{4} x^{2}+1}}+\frac {\sqrt {3}\, x^{2}}{2 \sqrt {\frac {3 \sqrt {3}}{2}+3}\, \sqrt {\frac {1}{4} x^{4}-\frac {5}{4} x^{2}+1}}\right )}{2 \sqrt {\frac {3 \sqrt {3}}{2}+3}}+\frac {3 \sqrt {3}\, \sqrt {-x^{2}+1}\, \sqrt {-x^{2}+4}\, \EllipticPi \left (x , \frac {1}{\left (-1-\sqrt {3}\right )^{2}}, \frac {1}{2}\right )}{2 \sqrt {x^{4}-5 x^{2}+4}\, \left (-1-\sqrt {3}\right )}+\frac {3 \arctanh \left (-\frac {3}{2 \sqrt {\frac {3 \sqrt {3}}{2}+3}\, \sqrt {\frac {1}{4} x^{4}-\frac {5}{4} x^{2}+1}}-\frac {5 \sqrt {3}}{4 \sqrt {\frac {3 \sqrt {3}}{2}+3}\, \sqrt {\frac {1}{4} x^{4}-\frac {5}{4} x^{2}+1}}+\frac {3 x^{2}}{8 \sqrt {\frac {3 \sqrt {3}}{2}+3}\, \sqrt {\frac {1}{4} x^{4}-\frac {5}{4} x^{2}+1}}+\frac {\sqrt {3}\, x^{2}}{2 \sqrt {\frac {3 \sqrt {3}}{2}+3}\, \sqrt {\frac {1}{4} x^{4}-\frac {5}{4} x^{2}+1}}\right )}{2 \sqrt {\frac {3 \sqrt {3}}{2}+3}}+\frac {3 \sqrt {-x^{2}+1}\, \sqrt {-x^{2}+4}\, \EllipticPi \left (x , \frac {1}{\left (-1-\sqrt {3}\right )^{2}}, \frac {1}{2}\right )}{2 \sqrt {x^{4}-5 x^{2}+4}\, \left (-1-\sqrt {3}\right )}\)	\(811\)
default	\(\frac {\sqrt {x^{4}-5 x^{2}+4}}{x}+\frac {11 \sqrt {-x^{2}+1}\, \sqrt {-x^{2}+4}\, \EllipticF \left (x , \frac {1}{2}\right )}{2 \sqrt {x^{4}-5 x^{2}+4}}-\frac {4 \sqrt {-x^{2}+1}\, \sqrt {-x^{2}+4}\, \left (\EllipticF \left (x , \frac {1}{2}\right )-\EllipticE \left (x , \frac {1}{2}\right )\right )}{\sqrt {x^{4}-5 x^{2}+4}}+\frac {5 \ln \left (-\frac {5}{2}+x^{2}+\sqrt {x^{4}-5 x^{2}+4}\right )}{4}+\arctanh \left (\frac {-5 x^{2}+8}{4 \sqrt {x^{4}-5 x^{2}+4}}\right )+9 \ln \relax (2)-\frac {9 \ln \left (2 x^{2}-5+2 \sqrt {x^{4}-5 x^{2}+4}\right )}{2}-\frac {4 \sqrt {-x^{2}+1}\, \sqrt {-x^{2}+4}\, \EllipticE \left (x , \frac {1}{2}\right )}{\sqrt {x^{4}-5 x^{2}+4}}-\frac {3 \arctanh \left (\frac {3}{2 \sqrt {-\frac {3 \sqrt {3}}{2}+3}\, \sqrt {\frac {1}{4} x^{4}-\frac {5}{4} x^{2}+1}}-\frac {5 \sqrt {3}}{4 \sqrt {-\frac {3 \sqrt {3}}{2}+3}\, \sqrt {\frac {1}{4} x^{4}-\frac {5}{4} x^{2}+1}}-\frac {3 x^{2}}{8 \sqrt {-\frac {3 \sqrt {3}}{2}+3}\, \sqrt {\frac {1}{4} x^{4}-\frac {5}{4} x^{2}+1}}+\frac {\sqrt {3}\, x^{2}}{2 \sqrt {-\frac {3 \sqrt {3}}{2}+3}\, \sqrt {\frac {1}{4} x^{4}-\frac {5}{4} x^{2}+1}}\right )}{2 \sqrt {-\frac {3 \sqrt {3}}{2}+3}}+\frac {3 \sqrt {-x^{2}+1}\, \sqrt {-x^{2}+4}\, \EllipticPi \left (x , \frac {1}{\left (\sqrt {3}-1\right )^{2}}, \frac {1}{2}\right )}{2 \sqrt {x^{4}-5 x^{2}+4}\, \left (\sqrt {3}-1\right )}+\frac {3 \sqrt {3}\, \arctanh \left (\frac {3}{2 \sqrt {-\frac {3 \sqrt {3}}{2}+3}\, \sqrt {\frac {1}{4} x^{4}-\frac {5}{4} x^{2}+1}}-\frac {5 \sqrt {3}}{4 \sqrt {-\frac {3 \sqrt {3}}{2}+3}\, \sqrt {\frac {1}{4} x^{4}-\frac {5}{4} x^{2}+1}}-\frac {3 x^{2}}{8 \sqrt {-\frac {3 \sqrt {3}}{2}+3}\, \sqrt {\frac {1}{4} x^{4}-\frac {5}{4} x^{2}+1}}+\frac {\sqrt {3}\, x^{2}}{2 \sqrt {-\frac {3 \sqrt {3}}{2}+3}\, \sqrt {\frac {1}{4} x^{4}-\frac {5}{4} x^{2}+1}}\right )}{2 \sqrt {-\frac {3 \sqrt {3}}{2}+3}}-\frac {3 \sqrt {3}\, \sqrt {-x^{2}+1}\, \sqrt {-x^{2}+4}\, \EllipticPi \left (x , \frac {1}{\left (\sqrt {3}-1\right )^{2}}, \frac {1}{2}\right )}{2 \sqrt {x^{4}-5 x^{2}+4}\, \left (\sqrt {3}-1\right )}+\frac {3 \sqrt {3}\, \arctanh \left (-\frac {3}{2 \sqrt {\frac {3 \sqrt {3}}{2}+3}\, \sqrt {\frac {1}{4} x^{4}-\frac {5}{4} x^{2}+1}}-\frac {5 \sqrt {3}}{4 \sqrt {\frac {3 \sqrt {3}}{2}+3}\, \sqrt {\frac {1}{4} x^{4}-\frac {5}{4} x^{2}+1}}+\frac {3 x^{2}}{8 \sqrt {\frac {3 \sqrt {3}}{2}+3}\, \sqrt {\frac {1}{4} x^{4}-\frac {5}{4} x^{2}+1}}+\frac {\sqrt {3}\, x^{2}}{2 \sqrt {\frac {3 \sqrt {3}}{2}+3}\, \sqrt {\frac {1}{4} x^{4}-\frac {5}{4} x^{2}+1}}\right )}{2 \sqrt {\frac {3 \sqrt {3}}{2}+3}}+\frac {3 \sqrt {3}\, \sqrt {-x^{2}+1}\, \sqrt {-x^{2}+4}\, \EllipticPi \left (x , \frac {1}{\left (-1-\sqrt {3}\right )^{2}}, \frac {1}{2}\right )}{2 \sqrt {x^{4}-5 x^{2}+4}\, \left (-1-\sqrt {3}\right )}+\frac {3 \arctanh \left (-\frac {3}{2 \sqrt {\frac {3 \sqrt {3}}{2}+3}\, \sqrt {\frac {1}{4} x^{4}-\frac {5}{4} x^{2}+1}}-\frac {5 \sqrt {3}}{4 \sqrt {\frac {3 \sqrt {3}}{2}+3}\, \sqrt {\frac {1}{4} x^{4}-\frac {5}{4} x^{2}+1}}+\frac {3 x^{2}}{8 \sqrt {\frac {3 \sqrt {3}}{2}+3}\, \sqrt {\frac {1}{4} x^{4}-\frac {5}{4} x^{2}+1}}+\frac {\sqrt {3}\, x^{2}}{2 \sqrt {\frac {3 \sqrt {3}}{2}+3}\, \sqrt {\frac {1}{4} x^{4}-\frac {5}{4} x^{2}+1}}\right )}{2 \sqrt {\frac {3 \sqrt {3}}{2}+3}}+\frac {3 \sqrt {-x^{2}+1}\, \sqrt {-x^{2}+4}\, \EllipticPi \left (x , \frac {1}{\left (-1-\sqrt {3}\right )^{2}}, \frac {1}{2}\right )}{2 \sqrt {x^{4}-5 x^{2}+4}\, \left (-1-\sqrt {3}\right )}\)	\(915\)

Verification of antiderivative is not currently implemented for this CAS.

[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+RootOf(_Z^2-3)*ln(-(2*RootOf(_Z^2-3)*x^2+RootOf(_Z^2-3)*x+3*(x^4-5*x^2+4)^(1/2)-4*RootOf
(_Z^2-3))/(x^2+2*x-2))+2*ln(-(-x^2+(x^4-5*x^2+4)^(1/2)+2)/x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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