\(\int \frac {(-1+x^2) \sqrt [4]{x^3+x^4}}{1+x^2+x^4} \, dx\) [2524]

Optimal result

Integrand size = 27, antiderivative size = 211 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^4}}{1+x^2+x^4} \, dx=-2 \arctan \left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )+2 \text {arctanh}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )-\frac {3}{2} \text {RootSum}\left [3-3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-3 \text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ]+\frac {1}{2} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x)-\log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \]

[Out]

Unintegrable

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.62 (sec) , antiderivative size = 1271, normalized size of antiderivative = 6.02, number of steps used = 35, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {2081, 6860, 920, 65, 338, 304, 209, 212, 6857, 95, 211, 214} \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^4}}{1+x^2+x^4} \, dx=-\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{x^{3/4} \sqrt [4]{x+1}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{x^{3/4} \sqrt [4]{x+1}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{-\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{-\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1-i \sqrt {3}} \sqrt [4]{x+1}}\right )}{2 \sqrt [8]{-1-i \sqrt {3}} x^{3/4} \sqrt [4]{x+1}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1-i \sqrt {3}} \sqrt [4]{x+1}}\right )}{2 \sqrt [8]{-1-i \sqrt {3}} x^{3/4} \sqrt [4]{x+1}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{-\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{-\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1+i \sqrt {3}} \sqrt [4]{x+1}}\right )}{2 \sqrt [8]{-1+i \sqrt {3}} x^{3/4} \sqrt [4]{x+1}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1+i \sqrt {3}} \sqrt [4]{x+1}}\right )}{2 \sqrt [8]{-1+i \sqrt {3}} x^{3/4} \sqrt [4]{x+1}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{x^{3/4} \sqrt [4]{x+1}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{x^{3/4} \sqrt [4]{x+1}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{-\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{-\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1-i \sqrt {3}} \sqrt [4]{x+1}}\right )}{2 \sqrt [8]{-1-i \sqrt {3}} x^{3/4} \sqrt [4]{x+1}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1-i \sqrt {3}} \sqrt [4]{x+1}}\right )}{2 \sqrt [8]{-1-i \sqrt {3}} x^{3/4} \sqrt [4]{x+1}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{-\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{-\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1+i \sqrt {3}} \sqrt [4]{x+1}}\right )}{2 \sqrt [8]{-1+i \sqrt {3}} x^{3/4} \sqrt [4]{x+1}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1+i \sqrt {3}} \sqrt [4]{x+1}}\right )}{2 \sqrt [8]{-1+i \sqrt {3}} x^{3/4} \sqrt [4]{x+1}} \]

[In]

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

[Out]

-(((1 - I*Sqrt[3])*(x^3 + x^4)^(1/4)*ArcTan[x^(1/4)/(1 + x)^(1/4)])/(x^(3/4)*(1 + x)^(1/4))) - ((1 + I*Sqrt[3]
)*(x^3 + x^4)^(1/4)*ArcTan[x^(1/4)/(1 + x)^(1/4)])/(x^(3/4)*(1 + x)^(1/4)) + ((1 - I*Sqrt[3])*(-Sqrt[2] + Sqrt
[-1 - I*Sqrt[3]])^(1/4)*(x^3 + x^4)^(1/4)*ArcTan[((-Sqrt[2] + Sqrt[-1 - I*Sqrt[3]])^(1/4)*x^(1/4))/((-1 - I*Sq
rt[3])^(1/8)*(1 + x)^(1/4))])/(2*(-1 - I*Sqrt[3])^(1/8)*x^(3/4)*(1 + x)^(1/4)) + ((1 - I*Sqrt[3])*(Sqrt[2] + S
qrt[-1 - I*Sqrt[3]])^(1/4)*(x^3 + x^4)^(1/4)*ArcTan[((Sqrt[2] + Sqrt[-1 - I*Sqrt[3]])^(1/4)*x^(1/4))/((-1 - I*
Sqrt[3])^(1/8)*(1 + x)^(1/4))])/(2*(-1 - I*Sqrt[3])^(1/8)*x^(3/4)*(1 + x)^(1/4)) + ((1 + I*Sqrt[3])*(-Sqrt[2]
+ Sqrt[-1 + I*Sqrt[3]])^(1/4)*(x^3 + x^4)^(1/4)*ArcTan[((-Sqrt[2] + Sqrt[-1 + I*Sqrt[3]])^(1/4)*x^(1/4))/((-1
+ I*Sqrt[3])^(1/8)*(1 + x)^(1/4))])/(2*(-1 + I*Sqrt[3])^(1/8)*x^(3/4)*(1 + x)^(1/4)) + ((1 + I*Sqrt[3])*(Sqrt[
2] + Sqrt[-1 + I*Sqrt[3]])^(1/4)*(x^3 + x^4)^(1/4)*ArcTan[((Sqrt[2] + Sqrt[-1 + I*Sqrt[3]])^(1/4)*x^(1/4))/((-
1 + I*Sqrt[3])^(1/8)*(1 + x)^(1/4))])/(2*(-1 + I*Sqrt[3])^(1/8)*x^(3/4)*(1 + x)^(1/4)) + ((1 - I*Sqrt[3])*(x^3
+ x^4)^(1/4)*ArcTanh[x^(1/4)/(1 + x)^(1/4)])/(x^(3/4)*(1 + x)^(1/4)) + ((1 + I*Sqrt[3])*(x^3 + x^4)^(1/4)*Arc
Tanh[x^(1/4)/(1 + x)^(1/4)])/(x^(3/4)*(1 + x)^(1/4)) - ((1 - I*Sqrt[3])*(-Sqrt[2] + Sqrt[-1 - I*Sqrt[3]])^(1/4
)*(x^3 + x^4)^(1/4)*ArcTanh[((-Sqrt[2] + Sqrt[-1 - I*Sqrt[3]])^(1/4)*x^(1/4))/((-1 - I*Sqrt[3])^(1/8)*(1 + x)^
(1/4))])/(2*(-1 - I*Sqrt[3])^(1/8)*x^(3/4)*(1 + x)^(1/4)) - ((1 - I*Sqrt[3])*(Sqrt[2] + Sqrt[-1 - I*Sqrt[3]])^
(1/4)*(x^3 + x^4)^(1/4)*ArcTanh[((Sqrt[2] + Sqrt[-1 - I*Sqrt[3]])^(1/4)*x^(1/4))/((-1 - I*Sqrt[3])^(1/8)*(1 +
x)^(1/4))])/(2*(-1 - I*Sqrt[3])^(1/8)*x^(3/4)*(1 + x)^(1/4)) - ((1 + I*Sqrt[3])*(-Sqrt[2] + Sqrt[-1 + I*Sqrt[3
]])^(1/4)*(x^3 + x^4)^(1/4)*ArcTanh[((-Sqrt[2] + Sqrt[-1 + I*Sqrt[3]])^(1/4)*x^(1/4))/((-1 + I*Sqrt[3])^(1/8)*
(1 + x)^(1/4))])/(2*(-1 + I*Sqrt[3])^(1/8)*x^(3/4)*(1 + x)^(1/4)) - ((1 + I*Sqrt[3])*(Sqrt[2] + Sqrt[-1 + I*Sq
rt[3]])^(1/4)*(x^3 + x^4)^(1/4)*ArcTanh[((Sqrt[2] + Sqrt[-1 + I*Sqrt[3]])^(1/4)*x^(1/4))/((-1 + I*Sqrt[3])^(1/
8)*(1 + x)^(1/4))])/(2*(-1 + I*Sqrt[3])^(1/8)*x^(3/4)*(1 + x)^(1/4))

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 209

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

Rule 211

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

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 214

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

Rule 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rule 338

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]

Rule 920

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[e*(g/c), In
t[(d + e*x)^(m - 1)*(f + g*x)^(n - 1), x], x] + Dist[1/c, Int[Simp[c*d*f - a*e*g + (c*e*f + c*d*g)*x, x]*(d +
e*x)^(m - 1)*((f + g*x)^(n - 1)/(a + c*x^2)), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0]
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[m, 0] && GtQ[n, 0]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 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}& = \frac {\sqrt [4]{x^3+x^4} \int \frac {x^{3/4} \sqrt [4]{1+x} \left (-1+x^2\right )}{1+x^2+x^4} \, dx}{x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {\sqrt [4]{x^3+x^4} \int \left (\frac {\left (1+i \sqrt {3}\right ) x^{3/4} \sqrt [4]{1+x}}{1-i \sqrt {3}+2 x^2}+\frac {\left (1-i \sqrt {3}\right ) x^{3/4} \sqrt [4]{1+x}}{1+i \sqrt {3}+2 x^2}\right ) \, dx}{x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{3/4} \sqrt [4]{1+x}}{1+i \sqrt {3}+2 x^2} \, dx}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{3/4} \sqrt [4]{1+x}}{1-i \sqrt {3}+2 x^2} \, dx}{x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4}} \, dx}{2 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {-1-i \sqrt {3}+2 x}{\sqrt [4]{x} (1+x)^{3/4} \left (1+i \sqrt {3}+2 x^2\right )} \, dx}{2 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4}} \, dx}{2 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {-1+i \sqrt {3}+2 x}{\sqrt [4]{x} (1+x)^{3/4} \left (1-i \sqrt {3}+2 x^2\right )} \, dx}{2 x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \int \left (\frac {\sqrt {2} \left (-1-i \sqrt {3}\right )+\left (-1-i \sqrt {3}\right )^{3/2}}{2 \left (1+i \sqrt {3}\right ) \sqrt [4]{x} (1+x)^{3/4} \left (\sqrt {-1-i \sqrt {3}}-\sqrt {2} x\right )}+\frac {-\sqrt {2} \left (-1-i \sqrt {3}\right )+\left (-1-i \sqrt {3}\right )^{3/2}}{2 \left (1+i \sqrt {3}\right ) \sqrt [4]{x} (1+x)^{3/4} \left (\sqrt {-1-i \sqrt {3}}+\sqrt {2} x\right )}\right ) \, dx}{2 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \left (1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \int \left (\frac {\sqrt {2} \left (-1+i \sqrt {3}\right )+\left (-1+i \sqrt {3}\right )^{3/2}}{2 \left (1-i \sqrt {3}\right ) \sqrt [4]{x} (1+x)^{3/4} \left (\sqrt {-1+i \sqrt {3}}-\sqrt {2} x\right )}+\frac {-\sqrt {2} \left (-1+i \sqrt {3}\right )+\left (-1+i \sqrt {3}\right )^{3/2}}{2 \left (1-i \sqrt {3}\right ) \sqrt [4]{x} (1+x)^{3/4} \left (\sqrt {-1+i \sqrt {3}}+\sqrt {2} x\right )}\right ) \, dx}{2 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {\left (2 \left (1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1-i \sqrt {3}\right ) \left (-\sqrt {2}-\sqrt {-1-i \sqrt {3}}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4} \left (\sqrt {-1-i \sqrt {3}}-\sqrt {2} x\right )} \, dx}{4 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1-i \sqrt {3}\right ) \left (\sqrt {2}-\sqrt {-1-i \sqrt {3}}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4} \left (\sqrt {-1-i \sqrt {3}}+\sqrt {2} x\right )} \, dx}{4 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1+i \sqrt {3}\right ) \left (-\sqrt {2}-\sqrt {-1+i \sqrt {3}}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4} \left (\sqrt {-1+i \sqrt {3}}-\sqrt {2} x\right )} \, dx}{4 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1+i \sqrt {3}\right ) \left (\sqrt {2}-\sqrt {-1+i \sqrt {3}}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4} \left (\sqrt {-1+i \sqrt {3}}+\sqrt {2} x\right )} \, dx}{4 x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1-i \sqrt {3}\right ) \left (-\sqrt {2}-\sqrt {-1-i \sqrt {3}}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {-1-i \sqrt {3}}-\left (\sqrt {2}+\sqrt {-1-i \sqrt {3}}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1-i \sqrt {3}\right ) \left (\sqrt {2}-\sqrt {-1-i \sqrt {3}}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {-1-i \sqrt {3}}-\left (-\sqrt {2}+\sqrt {-1-i \sqrt {3}}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1+i \sqrt {3}\right ) \left (-\sqrt {2}-\sqrt {-1+i \sqrt {3}}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {-1+i \sqrt {3}}-\left (\sqrt {2}+\sqrt {-1+i \sqrt {3}}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1+i \sqrt {3}\right ) \left (\sqrt {2}-\sqrt {-1+i \sqrt {3}}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {-1+i \sqrt {3}}-\left (-\sqrt {2}+\sqrt {-1+i \sqrt {3}}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.09 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^4}}{1+x^2+x^4} \, dx=\frac {x^{9/4} (1+x)^{3/4} \left (16 \left (-\arctan \left (\sqrt [4]{\frac {x}{1+x}}\right )+\text {arctanh}\left (\sqrt [4]{\frac {x}{1+x}}\right )\right )-3 \text {RootSum}\left [3-3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+4 \log \left (\sqrt [4]{1+x}-\sqrt [4]{x} \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-4 \log \left (\sqrt [4]{1+x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-3 \text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ]+\text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x)-4 \log \left (\sqrt [4]{1+x}-\sqrt [4]{x} \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-4 \log \left (\sqrt [4]{1+x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ]\right )}{8 \left (x^3 (1+x)\right )^{3/4}} \]

[In]

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

[Out]

(x^(9/4)*(1 + x)^(3/4)*(16*(-ArcTan[(x/(1 + x))^(1/4)] + ArcTanh[(x/(1 + x))^(1/4)]) - 3*RootSum[3 - 3*#1^4 +
#1^8 & , (-Log[x] + 4*Log[(1 + x)^(1/4) - x^(1/4)*#1] + Log[x]*#1^4 - 4*Log[(1 + x)^(1/4) - x^(1/4)*#1]*#1^4)/
(-3*#1^3 + 2*#1^7) & ] + RootSum[1 - #1^4 + #1^8 & , (Log[x] - 4*Log[(1 + x)^(1/4) - x^(1/4)*#1] + Log[x]*#1^4
- 4*Log[(1 + x)^(1/4) - x^(1/4)*#1]*#1^4)/(-#1^3 + 2*#1^7) & ]))/(8*(x^3*(1 + x))^(3/4))

Maple [N/A] (verified)

Time = 32.94 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.76

method	result	size
pseudoelliptic	\(2 \arctan \left (\frac {\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )+\ln \left (\frac {\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}+x}{x}\right )-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}+1\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (2 \textit {\_R}^{4}-1\right )}\right )}{2}-\ln \left (\frac {-x +\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-3 \textit {\_Z}^{4}+3\right )}{\sum }\frac {\left (\textit {\_R}^{4}-1\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (2 \textit {\_R}^{4}-3\right )}\right )}{2}\)	\(160\)
trager	\(\text {Expression too large to display}\)	\(4123\)

[In]

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

[Out]

2*arctan((x^3*(1+x))^(1/4)/x)+ln(((x^3*(1+x))^(1/4)+x)/x)-1/2*sum((_R^4+1)*ln((-_R*x+(x^3*(1+x))^(1/4))/x)/_R^
3/(2*_R^4-1),_R=RootOf(_Z^8-_Z^4+1))-ln((-x+(x^3*(1+x))^(1/4))/x)+3/2*sum((_R^4-1)*ln((-_R*x+(x^3*(1+x))^(1/4)
)/x)/_R^3/(2*_R^4-3),_R=RootOf(_Z^8-3*_Z^4+3))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.30 (sec) , antiderivative size = 1070, normalized size of antiderivative = 5.07 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^4}}{1+x^2+x^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/4*sqrt(2)*sqrt(sqrt(2)*sqrt(sqrt(-3) + 1))*log((sqrt(2)*(sqrt(-3)*x + x)*sqrt(sqrt(2)*sqrt(sqrt(-3) + 1)) +
4*(x^4 + x^3)^(1/4))/x) + 1/4*sqrt(2)*sqrt(sqrt(2)*sqrt(sqrt(-3) + 1))*log(-(sqrt(2)*(sqrt(-3)*x + x)*sqrt(sq
rt(2)*sqrt(sqrt(-3) + 1)) - 4*(x^4 + x^3)^(1/4))/x) - 1/4*sqrt(2)*sqrt(-sqrt(2)*sqrt(sqrt(-3) + 1))*log((sqrt(
2)*(sqrt(-3)*x + x)*sqrt(-sqrt(2)*sqrt(sqrt(-3) + 1)) + 4*(x^4 + x^3)^(1/4))/x) + 1/4*sqrt(2)*sqrt(-sqrt(2)*sq
rt(sqrt(-3) + 1))*log(-(sqrt(2)*(sqrt(-3)*x + x)*sqrt(-sqrt(2)*sqrt(sqrt(-3) + 1)) - 4*(x^4 + x^3)^(1/4))/x) -
1/4*sqrt(2)*sqrt(sqrt(2)*sqrt(sqrt(-3) - 3))*log((sqrt(2)*(sqrt(-3)*x + x)*sqrt(sqrt(2)*sqrt(sqrt(-3) - 3)) +
4*(x^4 + x^3)^(1/4))/x) + 1/4*sqrt(2)*sqrt(sqrt(2)*sqrt(sqrt(-3) - 3))*log(-(sqrt(2)*(sqrt(-3)*x + x)*sqrt(sq
rt(2)*sqrt(sqrt(-3) - 3)) - 4*(x^4 + x^3)^(1/4))/x) - 1/4*sqrt(2)*sqrt(-sqrt(2)*sqrt(sqrt(-3) - 3))*log((sqrt(
2)*(sqrt(-3)*x + x)*sqrt(-sqrt(2)*sqrt(sqrt(-3) - 3)) + 4*(x^4 + x^3)^(1/4))/x) + 1/4*sqrt(2)*sqrt(-sqrt(2)*sq
rt(sqrt(-3) - 3))*log(-(sqrt(2)*(sqrt(-3)*x + x)*sqrt(-sqrt(2)*sqrt(sqrt(-3) - 3)) - 4*(x^4 + x^3)^(1/4))/x) +
1/4*sqrt(2)*sqrt(sqrt(2)*sqrt(-sqrt(-3) + 1))*log((sqrt(2)*(sqrt(-3)*x - x)*sqrt(sqrt(2)*sqrt(-sqrt(-3) + 1))
+ 4*(x^4 + x^3)^(1/4))/x) - 1/4*sqrt(2)*sqrt(sqrt(2)*sqrt(-sqrt(-3) + 1))*log(-(sqrt(2)*(sqrt(-3)*x - x)*sqrt
(sqrt(2)*sqrt(-sqrt(-3) + 1)) - 4*(x^4 + x^3)^(1/4))/x) + 1/4*sqrt(2)*sqrt(-sqrt(2)*sqrt(-sqrt(-3) + 1))*log((
sqrt(2)*(sqrt(-3)*x - x)*sqrt(-sqrt(2)*sqrt(-sqrt(-3) + 1)) + 4*(x^4 + x^3)^(1/4))/x) - 1/4*sqrt(2)*sqrt(-sqrt
(2)*sqrt(-sqrt(-3) + 1))*log(-(sqrt(2)*(sqrt(-3)*x - x)*sqrt(-sqrt(2)*sqrt(-sqrt(-3) + 1)) - 4*(x^4 + x^3)^(1/
4))/x) + 1/4*sqrt(2)*sqrt(sqrt(2)*sqrt(-sqrt(-3) - 3))*log((sqrt(2)*(sqrt(-3)*x - x)*sqrt(sqrt(2)*sqrt(-sqrt(-
3) - 3)) + 4*(x^4 + x^3)^(1/4))/x) - 1/4*sqrt(2)*sqrt(sqrt(2)*sqrt(-sqrt(-3) - 3))*log(-(sqrt(2)*(sqrt(-3)*x -
x)*sqrt(sqrt(2)*sqrt(-sqrt(-3) - 3)) - 4*(x^4 + x^3)^(1/4))/x) + 1/4*sqrt(2)*sqrt(-sqrt(2)*sqrt(-sqrt(-3) - 3
))*log((sqrt(2)*(sqrt(-3)*x - x)*sqrt(-sqrt(2)*sqrt(-sqrt(-3) - 3)) + 4*(x^4 + x^3)^(1/4))/x) - 1/4*sqrt(2)*sq
rt(-sqrt(2)*sqrt(-sqrt(-3) - 3))*log(-(sqrt(2)*(sqrt(-3)*x - x)*sqrt(-sqrt(2)*sqrt(-sqrt(-3) - 3)) - 4*(x^4 +
x^3)^(1/4))/x) + 2*arctan((x^4 + x^3)^(1/4)/x) + log((x + (x^4 + x^3)^(1/4))/x) - log(-(x - (x^4 + x^3)^(1/4))
/x)

Sympy [N/A]

Not integrable

Time = 1.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.15 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^4}}{1+x^2+x^4} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x + 1\right )} \left (x - 1\right ) \left (x + 1\right )}{\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.13 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^4}}{1+x^2+x^4} \, dx=\int { \frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{2} - 1\right )}}{x^{4} + x^{2} + 1} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^4}}{1+x^2+x^4} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:proot error [1,0,0,0,1,0,0,0,1]proot error [1,0,0,0,-1,0,0,0,1]proot error [1,0,-10,0,1]proot error [1,0,-1
0,0,1]proot

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.13 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^4}}{1+x^2+x^4} \, dx=\int \frac {{\left (x^4+x^3\right )}^{1/4}\,\left (x^2-1\right )}{x^4+x^2+1} \,d x \]

[In]

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

[Out]

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