\(\int \frac {(2+x^2) (-2-2 x+x^2) \sqrt {4-3 x^2+x^4}}{x^2 (-2+x^2) (-4+x+2 x^2)} \, dx\) [1789]

Optimal result

Integrand size = 48, antiderivative size = 121 \[ \int \frac {\left (2+x^2\right ) \left (-2-2 x+x^2\right ) \sqrt {4-3 x^2+x^4}}{x^2 \left (-2+x^2\right ) \left (-4+x+2 x^2\right )} \, dx=\frac {\sqrt {4-3 x^2+x^4}}{2 x}+4 \text {arctanh}\left (\frac {x}{-2+x^2+\sqrt {4-3 x^2+x^4}}\right )-\frac {5}{2} \sqrt {5} \text {arctanh}\left (\frac {\sqrt {5} x}{-4+x+2 x^2+2 \sqrt {4-3 x^2+x^4}}\right )+\frac {5 \log (x)}{4}-\frac {5}{4} \log \left (-2+x^2+\sqrt {4-3 x^2+x^4}\right ) \]

[Out]

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

Rubi [C] (verified)

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

Time = 2.27 (sec) , antiderivative size = 917, normalized size of antiderivative = 7.58, number of steps used = 53, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {6857, 1131, 1211, 1117, 1209, 1128, 748, 857, 633, 221, 738, 212, 1222, 1224, 1712, 213, 6860, 1742, 1230, 1720, 1261} \[ \int \frac {\left (2+x^2\right ) \left (-2-2 x+x^2\right ) \sqrt {4-3 x^2+x^4}}{x^2 \left (-2+x^2\right ) \left (-4+x+2 x^2\right )} \, dx=\frac {5}{64} \left (7+\sqrt {33}\right ) \text {arcsinh}\left (\frac {3-2 x^2}{\sqrt {7}}\right )+\frac {5}{64} \left (7-\sqrt {33}\right ) \text {arcsinh}\left (\frac {3-2 x^2}{\sqrt {7}}\right )-\frac {15}{32} \text {arcsinh}\left (\frac {3-2 x^2}{\sqrt {7}}\right )+2 \text {arctanh}\left (\frac {x}{\sqrt {x^4-3 x^2+4}}\right )-\frac {5}{4} \sqrt {5} \text {arctanh}\left (\frac {\sqrt {5} x}{2 \sqrt {x^4-3 x^2+4}}\right )+\frac {5}{8} \text {arctanh}\left (\frac {8-3 x^2}{4 \sqrt {x^4-3 x^2+4}}\right )-\frac {5}{8} \sqrt {5} \text {arctanh}\left (\frac {2 \left (5-\sqrt {33}\right ) x^2+3 \sqrt {33}+13}{2 \sqrt {10 \left (17-\sqrt {33}\right )} \sqrt {x^4-3 x^2+4}}\right )+\frac {5}{8} \sqrt {5} \text {arctanh}\left (\frac {2 \left (5+\sqrt {33}\right ) x^2-3 \sqrt {33}+13}{2 \sqrt {10 \left (17+\sqrt {33}\right )} \sqrt {x^4-3 x^2+4}}\right )-\frac {25 \left (17+\sqrt {33}\right ) \left (x^2+2\right ) \sqrt {\frac {x^4-3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {7}{8}\right )}{16 \sqrt {2} \left (33+\sqrt {33}\right ) \sqrt {x^4-3 x^2+4}}+\frac {5 \left (9+\sqrt {33}\right ) \left (x^2+2\right ) \sqrt {\frac {x^4-3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {7}{8}\right )}{32 \sqrt {2} \sqrt {x^4-3 x^2+4}}-\frac {25 \left (17-\sqrt {33}\right ) \left (x^2+2\right ) \sqrt {\frac {x^4-3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {7}{8}\right )}{16 \sqrt {2} \left (33-\sqrt {33}\right ) \sqrt {x^4-3 x^2+4}}+\frac {5 \left (9-\sqrt {33}\right ) \left (x^2+2\right ) \sqrt {\frac {x^4-3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {7}{8}\right )}{32 \sqrt {2} \sqrt {x^4-3 x^2+4}}-\frac {5 \left (x^2+2\right ) \sqrt {\frac {x^4-3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {7}{8}\right )}{4 \sqrt {2} \sqrt {x^4-3 x^2+4}}+\frac {25 \left (17+\sqrt {33}\right ) \left (x^2+2\right ) \sqrt {\frac {x^4-3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticPi}\left (\frac {33}{32},2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {7}{8}\right )}{32 \sqrt {2} \left (33+17 \sqrt {33}\right ) \sqrt {x^4-3 x^2+4}}+\frac {25 \left (17-\sqrt {33}\right ) \left (x^2+2\right ) \sqrt {\frac {x^4-3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticPi}\left (\frac {33}{32},2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {7}{8}\right )}{32 \sqrt {2} \left (33-17 \sqrt {33}\right ) \sqrt {x^4-3 x^2+4}}+\frac {\sqrt {x^4-3 x^2+4}}{2 x}+\frac {5}{32} \left (1+\sqrt {33}\right ) \sqrt {x^4-3 x^2+4}+\frac {5}{32} \left (1-\sqrt {33}\right ) \sqrt {x^4-3 x^2+4}-\frac {5}{16} \sqrt {x^4-3 x^2+4} \]

[In]

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

[Out]

(-5*Sqrt[4 - 3*x^2 + x^4])/16 + (5*(1 - Sqrt[33])*Sqrt[4 - 3*x^2 + x^4])/32 + (5*(1 + Sqrt[33])*Sqrt[4 - 3*x^2
+ x^4])/32 + Sqrt[4 - 3*x^2 + x^4]/(2*x) - (15*ArcSinh[(3 - 2*x^2)/Sqrt[7]])/32 + (5*(7 - Sqrt[33])*ArcSinh[(
3 - 2*x^2)/Sqrt[7]])/64 + (5*(7 + Sqrt[33])*ArcSinh[(3 - 2*x^2)/Sqrt[7]])/64 + 2*ArcTanh[x/Sqrt[4 - 3*x^2 + x^
4]] - (5*Sqrt[5]*ArcTanh[(Sqrt[5]*x)/(2*Sqrt[4 - 3*x^2 + x^4])])/4 + (5*ArcTanh[(8 - 3*x^2)/(4*Sqrt[4 - 3*x^2
+ x^4])])/8 - (5*Sqrt[5]*ArcTanh[(13 + 3*Sqrt[33] + 2*(5 - Sqrt[33])*x^2)/(2*Sqrt[10*(17 - Sqrt[33])]*Sqrt[4 -
3*x^2 + x^4])])/8 + (5*Sqrt[5]*ArcTanh[(13 - 3*Sqrt[33] + 2*(5 + Sqrt[33])*x^2)/(2*Sqrt[10*(17 + Sqrt[33])]*S
qrt[4 - 3*x^2 + x^4])])/8 - (5*(2 + x^2)*Sqrt[(4 - 3*x^2 + x^4)/(2 + x^2)^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 7/
8])/(4*Sqrt[2]*Sqrt[4 - 3*x^2 + x^4]) + (5*(9 - Sqrt[33])*(2 + x^2)*Sqrt[(4 - 3*x^2 + x^4)/(2 + x^2)^2]*Ellipt
icF[2*ArcTan[x/Sqrt[2]], 7/8])/(32*Sqrt[2]*Sqrt[4 - 3*x^2 + x^4]) - (25*(17 - Sqrt[33])*(2 + x^2)*Sqrt[(4 - 3*
x^2 + x^4)/(2 + x^2)^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 7/8])/(16*Sqrt[2]*(33 - Sqrt[33])*Sqrt[4 - 3*x^2 + x^4]
) + (5*(9 + Sqrt[33])*(2 + x^2)*Sqrt[(4 - 3*x^2 + x^4)/(2 + x^2)^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 7/8])/(32*S
qrt[2]*Sqrt[4 - 3*x^2 + x^4]) - (25*(17 + Sqrt[33])*(2 + x^2)*Sqrt[(4 - 3*x^2 + x^4)/(2 + x^2)^2]*EllipticF[2*
ArcTan[x/Sqrt[2]], 7/8])/(16*Sqrt[2]*(33 + Sqrt[33])*Sqrt[4 - 3*x^2 + x^4]) + (25*(17 - Sqrt[33])*(2 + x^2)*Sq
rt[(4 - 3*x^2 + x^4)/(2 + x^2)^2]*EllipticPi[33/32, 2*ArcTan[x/Sqrt[2]], 7/8])/(32*Sqrt[2]*(33 - 17*Sqrt[33])*
Sqrt[4 - 3*x^2 + x^4]) + (25*(17 + Sqrt[33])*(2 + x^2)*Sqrt[(4 - 3*x^2 + x^4)/(2 + x^2)^2]*EllipticPi[33/32, 2
*ArcTan[x/Sqrt[2]], 7/8])/(32*Sqrt[2]*(33 + 17*Sqrt[33])*Sqrt[4 - 3*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 213

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

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 633

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*(-4*(c/(b^2 - 4*a*c)))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/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 1117

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] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

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 1209

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] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1211

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] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

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 1224

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

Rule 1230

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

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 1712

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

Rule 1720

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[
{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(ArcTan[Rt[-b + c*(d/e) + a*(e/d), 2]*(x/Sqrt[a + b*x^2 + c*x^4])]/(2*d*
e*Rt[-b + c*(d/e) + a*(e/d), 2])), x] + Simp[(B*d + A*e)*(A + B*x^2)*(Sqrt[A^2*((a + b*x^2 + c*x^4)/(a*(A + B*
x^2)^2))]/(4*d*e*A*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticPi[Cancel[-(B*d - A*e)^2/(4*d*e*A*B)], 2*ArcTan[q*x], 1
/2 - b*(A/(4*a*B))], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^
2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0]

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 6857

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

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 {4-3 x^2+x^4}}{2 x^2}-\frac {5 \sqrt {4-3 x^2+x^4}}{8 x}-\frac {4 \sqrt {4-3 x^2+x^4}}{-2+x^2}+\frac {5 (17+2 x) \sqrt {4-3 x^2+x^4}}{8 \left (-4+x+2 x^2\right )}\right ) \, dx \\ & = -\left (\frac {1}{2} \int \frac {\sqrt {4-3 x^2+x^4}}{x^2} \, dx\right )-\frac {5}{8} \int \frac {\sqrt {4-3 x^2+x^4}}{x} \, dx+\frac {5}{8} \int \frac {(17+2 x) \sqrt {4-3 x^2+x^4}}{-4+x+2 x^2} \, dx-4 \int \frac {\sqrt {4-3 x^2+x^4}}{-2+x^2} \, dx \\ & = \frac {\sqrt {4-3 x^2+x^4}}{2 x}-\frac {5}{16} \text {Subst}\left (\int \frac {\sqrt {4-3 x+x^2}}{x} \, dx,x,x^2\right )-\frac {1}{2} \int \frac {-3+2 x^2}{\sqrt {4-3 x^2+x^4}} \, dx+\frac {5}{8} \int \left (\frac {\left (2+2 \sqrt {33}\right ) \sqrt {4-3 x^2+x^4}}{1-\sqrt {33}+4 x}+\frac {\left (2-2 \sqrt {33}\right ) \sqrt {4-3 x^2+x^4}}{1+\sqrt {33}+4 x}\right ) \, dx+4 \int \frac {1-x^2}{\sqrt {4-3 x^2+x^4}} \, dx-8 \int \frac {1}{\left (-2+x^2\right ) \sqrt {4-3 x^2+x^4}} \, dx \\ & = -\frac {5}{16} \sqrt {4-3 x^2+x^4}+\frac {\sqrt {4-3 x^2+x^4}}{2 x}+\frac {5}{32} \text {Subst}\left (\int \frac {-8+3 x}{x \sqrt {4-3 x+x^2}} \, dx,x,x^2\right )-\frac {1}{2} \int \frac {1}{\sqrt {4-3 x^2+x^4}} \, dx+2 \int \frac {1}{\sqrt {4-3 x^2+x^4}} \, dx+2 \int \frac {1-\frac {x^2}{2}}{\sqrt {4-3 x^2+x^4}} \, dx+2 \int \frac {-2-x^2}{\left (-2+x^2\right ) \sqrt {4-3 x^2+x^4}} \, dx-4 \int \frac {1}{\sqrt {4-3 x^2+x^4}} \, dx+8 \int \frac {1-\frac {x^2}{2}}{\sqrt {4-3 x^2+x^4}} \, dx+\frac {1}{4} \left (5 \left (1-\sqrt {33}\right )\right ) \int \frac {\sqrt {4-3 x^2+x^4}}{1+\sqrt {33}+4 x} \, dx+\frac {1}{4} \left (5 \left (1+\sqrt {33}\right )\right ) \int \frac {\sqrt {4-3 x^2+x^4}}{1-\sqrt {33}+4 x} \, dx \\ & = -\frac {5}{16} \sqrt {4-3 x^2+x^4}+\frac {\sqrt {4-3 x^2+x^4}}{2 x}-\frac {5 x \sqrt {4-3 x^2+x^4}}{2+x^2}+\frac {5 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{\sqrt {4-3 x^2+x^4}}+\frac {3 \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {7}{8}\right )}{4 \sqrt {2} \sqrt {4-3 x^2+x^4}}-\frac {\sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {7}{8}\right )}{\sqrt {4-3 x^2+x^4}}+\frac {15}{32} \text {Subst}\left (\int \frac {1}{\sqrt {4-3 x+x^2}} \, dx,x,x^2\right )-\frac {5}{4} \text {Subst}\left (\int \frac {1}{x \sqrt {4-3 x+x^2}} \, dx,x,x^2\right )-4 \text {Subst}\left (\int \frac {1}{-2+2 x^2} \, dx,x,\frac {x}{\sqrt {4-3 x^2+x^4}}\right )-40 \int \frac {\sqrt {4-3 x^2+x^4}}{\left (1-\sqrt {33}\right )^2-16 x^2} \, dx-40 \int \frac {\sqrt {4-3 x^2+x^4}}{\left (1+\sqrt {33}\right )^2-16 x^2} \, dx-\left (5 \left (1-\sqrt {33}\right )\right ) \int \frac {x \sqrt {4-3 x^2+x^4}}{\left (1+\sqrt {33}\right )^2-16 x^2} \, dx-\left (5 \left (1+\sqrt {33}\right )\right ) \int \frac {x \sqrt {4-3 x^2+x^4}}{\left (1-\sqrt {33}\right )^2-16 x^2} \, dx \\ & = -\frac {5}{16} \sqrt {4-3 x^2+x^4}+\frac {\sqrt {4-3 x^2+x^4}}{2 x}-\frac {5 x \sqrt {4-3 x^2+x^4}}{2+x^2}+2 \text {arctanh}\left (\frac {x}{\sqrt {4-3 x^2+x^4}}\right )+\frac {5 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{\sqrt {4-3 x^2+x^4}}+\frac {3 \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {7}{8}\right )}{4 \sqrt {2} \sqrt {4-3 x^2+x^4}}-\frac {\sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {7}{8}\right )}{\sqrt {4-3 x^2+x^4}}+\frac {5}{32} \int \frac {-48+\left (1-\sqrt {33}\right )^2+16 x^2}{\sqrt {4-3 x^2+x^4}} \, dx+\frac {5}{32} \int \frac {-48+\left (1+\sqrt {33}\right )^2+16 x^2}{\sqrt {4-3 x^2+x^4}} \, dx+\frac {5}{2} \text {Subst}\left (\int \frac {1}{16-x^2} \, dx,x,\frac {8-3 x^2}{\sqrt {4-3 x^2+x^4}}\right )+\frac {15 \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{7}}} \, dx,x,-3+2 x^2\right )}{32 \sqrt {7}}-\frac {1}{2} \left (5 \left (1-\sqrt {33}\right )\right ) \text {Subst}\left (\int \frac {\sqrt {4-3 x+x^2}}{\left (1+\sqrt {33}\right )^2-16 x} \, dx,x,x^2\right )-\frac {1}{4} \left (25 \left (17-\sqrt {33}\right )\right ) \int \frac {1}{\left (\left (1-\sqrt {33}\right )^2-16 x^2\right ) \sqrt {4-3 x^2+x^4}} \, dx-\frac {1}{2} \left (5 \left (1+\sqrt {33}\right )\right ) \text {Subst}\left (\int \frac {\sqrt {4-3 x+x^2}}{\left (1-\sqrt {33}\right )^2-16 x} \, dx,x,x^2\right )-\frac {1}{4} \left (25 \left (17+\sqrt {33}\right )\right ) \int \frac {1}{\left (\left (1+\sqrt {33}\right )^2-16 x^2\right ) \sqrt {4-3 x^2+x^4}} \, dx \\ & = -\frac {5}{16} \sqrt {4-3 x^2+x^4}+\frac {5}{32} \left (1-\sqrt {33}\right ) \sqrt {4-3 x^2+x^4}+\frac {5}{32} \left (1+\sqrt {33}\right ) \sqrt {4-3 x^2+x^4}+\frac {\sqrt {4-3 x^2+x^4}}{2 x}-\frac {5 x \sqrt {4-3 x^2+x^4}}{2+x^2}-\frac {15}{32} \text {arcsinh}\left (\frac {3-2 x^2}{\sqrt {7}}\right )+2 \text {arctanh}\left (\frac {x}{\sqrt {4-3 x^2+x^4}}\right )+\frac {5}{8} \text {arctanh}\left (\frac {8-3 x^2}{4 \sqrt {4-3 x^2+x^4}}\right )+\frac {5 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{\sqrt {4-3 x^2+x^4}}+\frac {3 \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {7}{8}\right )}{4 \sqrt {2} \sqrt {4-3 x^2+x^4}}-\frac {\sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {7}{8}\right )}{\sqrt {4-3 x^2+x^4}}-2 \left (5 \int \frac {1-\frac {x^2}{2}}{\sqrt {4-3 x^2+x^4}} \, dx\right )-\frac {1}{64} \left (5 \left (1-\sqrt {33}\right )\right ) \text {Subst}\left (\int \frac {2 \left (13-3 \sqrt {33}\right )+4 \left (5+\sqrt {33}\right ) x}{\left (\left (1+\sqrt {33}\right )^2-16 x\right ) \sqrt {4-3 x+x^2}} \, dx,x,x^2\right )+\frac {1}{16} \left (5 \left (9-\sqrt {33}\right )\right ) \int \frac {1}{\sqrt {4-3 x^2+x^4}} \, dx-\frac {1}{64} \left (5 \left (1+\sqrt {33}\right )\right ) \text {Subst}\left (\int \frac {2 \left (13+3 \sqrt {33}\right )+4 \left (5-\sqrt {33}\right ) x}{\left (\left (1-\sqrt {33}\right )^2-16 x\right ) \sqrt {4-3 x+x^2}} \, dx,x,x^2\right )+\frac {1}{16} \left (5 \left (9+\sqrt {33}\right )\right ) \int \frac {1}{\sqrt {4-3 x^2+x^4}} \, dx-\frac {\left (25 \left (17+\sqrt {33}\right )\right ) \int \frac {1}{\sqrt {4-3 x^2+x^4}} \, dx}{8 \left (33+\sqrt {33}\right )}-\frac {\left (100 \left (17+\sqrt {33}\right )\right ) \int \frac {1+\frac {x^2}{2}}{\left (\left (1+\sqrt {33}\right )^2-16 x^2\right ) \sqrt {4-3 x^2+x^4}} \, dx}{33+\sqrt {33}}-\frac {\left (400 \left (17-\sqrt {33}\right ) \left (-16+\frac {1}{2} \left (1-\sqrt {33}\right )^2\right )\right ) \int \frac {1+\frac {x^2}{2}}{\left (\left (1-\sqrt {33}\right )^2-16 x^2\right ) \sqrt {4-3 x^2+x^4}} \, dx}{-1024+\left (1-\sqrt {33}\right )^4}-\frac {\left (25 \left (17-\sqrt {33}\right ) \left (-32+\left (1-\sqrt {33}\right )^2\right )\right ) \int \frac {1}{\sqrt {4-3 x^2+x^4}} \, dx}{4 \left (-1024+\left (1-\sqrt {33}\right )^4\right )} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.01 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.97 \[ \int \frac {\left (2+x^2\right ) \left (-2-2 x+x^2\right ) \sqrt {4-3 x^2+x^4}}{x^2 \left (-2+x^2\right ) \left (-4+x+2 x^2\right )} \, dx=\frac {1}{4} \left (\frac {2 \sqrt {4-3 x^2+x^4}}{x}+16 \text {arctanh}\left (\frac {x}{-2+x^2+\sqrt {4-3 x^2+x^4}}\right )-10 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {5} x}{-4+x+2 x^2+2 \sqrt {4-3 x^2+x^4}}\right )+5 \log (x)-5 \log \left (-2+x^2+\sqrt {4-3 x^2+x^4}\right )\right ) \]

[In]

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

[Out]

((2*Sqrt[4 - 3*x^2 + x^4])/x + 16*ArcTanh[x/(-2 + x^2 + Sqrt[4 - 3*x^2 + x^4])] - 10*Sqrt[5]*ArcTanh[(Sqrt[5]*
x)/(-4 + x + 2*x^2 + 2*Sqrt[4 - 3*x^2 + x^4])] + 5*Log[x] - 5*Log[-2 + x^2 + Sqrt[4 - 3*x^2 + x^4]])/4

Maple [A] (verified)

Time = 2.79 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.65

method	result	size
risch	\(\frac {\sqrt {x^{4}-3 x^{2}+4}}{2 x}+2 \,\operatorname {arctanh}\left (\frac {x}{\sqrt {x^{4}-3 x^{2}+4}}\right )-\frac {5 \,\operatorname {arcsinh}\left (\frac {x^{2}-2}{x}\right )}{4}+\frac {5 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (x^{2}-2 x -2\right ) \sqrt {5}}{5 \sqrt {x^{4}-3 x^{2}+4}}\right )}{4}\)	\(79\)
default	\(\frac {5 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (x^{2}-2 x -2\right ) \sqrt {5}}{5 \sqrt {x^{4}-3 x^{2}+4}}\right ) x +8 \,\operatorname {arctanh}\left (\frac {x}{\sqrt {x^{4}-3 x^{2}+4}}\right ) x -5 \,\operatorname {arcsinh}\left (\frac {x^{2}-2}{x}\right ) x +2 \sqrt {x^{4}-3 x^{2}+4}}{4 x}\)	\(84\)
pseudoelliptic	\(\frac {5 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (x^{2}-2 x -2\right ) \sqrt {5}}{5 \sqrt {x^{4}-3 x^{2}+4}}\right ) x +8 \,\operatorname {arctanh}\left (\frac {x}{\sqrt {x^{4}-3 x^{2}+4}}\right ) x -5 \,\operatorname {arcsinh}\left (\frac {x^{2}-2}{x}\right ) x +2 \sqrt {x^{4}-3 x^{2}+4}}{4 x}\)	\(84\)
elliptic	\(-\frac {5 \,\operatorname {arcsinh}\left (\frac {2 \sqrt {7}\, \left (x^{2}-\frac {3}{2}\right )}{7}\right )}{8}-\frac {160 \,\operatorname {arctanh}\left (\frac {-3 x^{2}+8}{4 \sqrt {x^{4}-3 x^{2}+4}}\right )}{\left (17+\sqrt {33}\right ) \left (-17+\sqrt {33}\right )}-\frac {25 \left (-297+25 \sqrt {33}\right ) \sqrt {33}\, \operatorname {arctanh}\left (\frac {\frac {85}{4}-\frac {5 \sqrt {33}}{4}+4 \left (\frac {5}{4}-\frac {\sqrt {33}}{4}\right ) \left (x^{2}-\frac {17}{8}+\frac {\sqrt {33}}{8}\right )}{\left (\frac {\sqrt {165}}{8}-\frac {\sqrt {5}}{8}\right ) \sqrt {64 \left (x^{2}-\frac {17}{8}+\frac {\sqrt {33}}{8}\right )^{2}+64 \left (\frac {5}{4}-\frac {\sqrt {33}}{4}\right ) \left (x^{2}-\frac {17}{8}+\frac {\sqrt {33}}{8}\right )+170-10 \sqrt {33}}}\right )}{16 \left (-1122+66 \sqrt {33}\right ) \left (\frac {\sqrt {165}}{8}-\frac {\sqrt {5}}{8}\right )}+\frac {25 \left (297+25 \sqrt {33}\right ) \sqrt {33}\, \operatorname {arctanh}\left (\frac {\frac {85}{4}+\frac {5 \sqrt {33}}{4}+4 \left (\frac {5}{4}+\frac {\sqrt {33}}{4}\right ) \left (x^{2}-\frac {17}{8}-\frac {\sqrt {33}}{8}\right )}{\left (\frac {\sqrt {165}}{8}+\frac {\sqrt {5}}{8}\right ) \sqrt {64 \left (x^{2}-\frac {17}{8}-\frac {\sqrt {33}}{8}\right )^{2}+64 \left (\frac {5}{4}+\frac {\sqrt {33}}{4}\right ) \left (x^{2}-\frac {17}{8}-\frac {\sqrt {33}}{8}\right )+170+10 \sqrt {33}}}\right )}{16 \left (1122+66 \sqrt {33}\right ) \left (\frac {\sqrt {165}}{8}+\frac {\sqrt {5}}{8}\right )}+\frac {\left (\frac {\sqrt {x^{4}-3 x^{2}+4}\, \sqrt {2}}{2 x}-\frac {5 \sqrt {10}\, \operatorname {arctanh}\left (\frac {\sqrt {10}\, \sqrt {x^{4}-3 x^{2}+4}\, \sqrt {2}}{5 x}\right )}{4}+2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{4}-3 x^{2}+4}}{x}\right )\right ) \sqrt {2}}{2}\)	\(364\)
trager	\(\frac {\sqrt {x^{4}-3 x^{2}+4}}{2 x}+\frac {\ln \left (\frac {131072+524288 x -30198 x^{20}+4373760 x^{10}+1296128 x^{9}-2095616 x^{7}+65536 \sqrt {x^{4}-3 x^{2}+4}+154096 x^{18}+1437984 x^{14}+9128 x^{21}+3507 x^{22}-32744 x^{19}-156064 x^{15}-1568 x^{23}+81008 x^{17}+2336768 x^{5}+3865344 x^{6}+237568 x^{2}-1605632 x^{3}-4931072 x^{8}-1795584 x^{4}-546720 x^{16}+292096 x^{13}-2875968 x^{12}-624256 x^{11}-16 x^{26}+128 x^{25}-116 x^{24}+16 \sqrt {x^{4}-3 x^{2}+4}\, x^{24}-128 \sqrt {x^{4}-3 x^{2}+4}\, x^{23}+140 \sqrt {x^{4}-3 x^{2}+4}\, x^{22}+1376 \sqrt {x^{4}-3 x^{2}+4}\, x^{21}-3311 \sqrt {x^{4}-3 x^{2}+4}\, x^{20}-6952 \sqrt {x^{4}-3 x^{2}+4}\, x^{19}+25088 \sqrt {x^{4}-3 x^{2}+4}\, x^{18}+21280 \sqrt {x^{4}-3 x^{2}+4}\, x^{17}-113776 \sqrt {x^{4}-3 x^{2}+4}\, x^{16}-44608 \sqrt {x^{4}-3 x^{2}+4}\, x^{15}+358208 \sqrt {x^{4}-3 x^{2}+4}\, x^{14}+77632 \sqrt {x^{4}-3 x^{2}+4}\, x^{13}-828960 \sqrt {x^{4}-3 x^{2}+4}\, x^{12}-155264 \sqrt {x^{4}-3 x^{2}+4}\, x^{11}+1432832 \sqrt {x^{4}-3 x^{2}+4}\, x^{10}+356864 \sqrt {x^{4}-3 x^{2}+4}\, x^{9}-1820416 \sqrt {x^{4}-3 x^{2}+4}\, x^{8}-680960 \sqrt {x^{4}-3 x^{2}+4}\, x^{7}+1605632 \sqrt {x^{4}-3 x^{2}+4}\, x^{6}+889856 \sqrt {x^{4}-3 x^{2}+4}\, x^{5}-847616 \sqrt {x^{4}-3 x^{2}+4}\, x^{4}-704512 x^{3} \sqrt {x^{4}-3 x^{2}+4}+143360 \sqrt {x^{4}-3 x^{2}+4}\, x^{2}+262144 x \sqrt {x^{4}-3 x^{2}+4}}{\left (x^{2}-2\right )^{8} x^{5}}\right )}{4}-\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )-5 \sqrt {x^{4}-3 x^{2}+4}}{2 x^{2}+x -4}\right )}{4}\)	\(645\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.24 \[ \int \frac {\left (2+x^2\right ) \left (-2-2 x+x^2\right ) \sqrt {4-3 x^2+x^4}}{x^2 \left (-2+x^2\right ) \left (-4+x+2 x^2\right )} \, dx=\frac {5 \, \sqrt {5} x \log \left (\frac {6 \, x^{4} - 4 \, x^{3} + 2 \, \sqrt {5} \sqrt {x^{4} - 3 \, x^{2} + 4} {\left (x^{2} - 2 \, x - 2\right )} - 15 \, x^{2} + 8 \, x + 24}{4 \, x^{4} + 4 \, x^{3} - 15 \, x^{2} - 8 \, x + 16}\right ) + 16 \, x \log \left (-\frac {x + \sqrt {x^{4} - 3 \, x^{2} + 4}}{x^{2} - 2}\right ) + 10 \, x \log \left (-\frac {x^{2} - \sqrt {x^{4} - 3 \, x^{2} + 4} - 2}{x}\right ) + 4 \, \sqrt {x^{4} - 3 \, x^{2} + 4}}{8 \, x} \]

[In]

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

[Out]

1/8*(5*sqrt(5)*x*log((6*x^4 - 4*x^3 + 2*sqrt(5)*sqrt(x^4 - 3*x^2 + 4)*(x^2 - 2*x - 2) - 15*x^2 + 8*x + 24)/(4*
x^4 + 4*x^3 - 15*x^2 - 8*x + 16)) + 16*x*log(-(x + sqrt(x^4 - 3*x^2 + 4))/(x^2 - 2)) + 10*x*log(-(x^2 - sqrt(x
^4 - 3*x^2 + 4) - 2)/x) + 4*sqrt(x^4 - 3*x^2 + 4))/x

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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