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

Optimal. Leaf size=211 \[ -\frac {3}{2} \text {RootSum}\left [\text {$\#$1}^8-3 \text {$\#$1}^4+3\& ,\frac {-\text {$\#$1}^4 \log \left (\sqrt [4]{x^4+x^3}-\text {$\#$1} x\right )+\text {$\#$1}^4 \log (x)+\log \left (\sqrt [4]{x^4+x^3}-\text {$\#$1} x\right )-\log (x)}{2 \text {$\#$1}^7-3 \text {$\#$1}^3}\& \right ]+\frac {1}{2} \text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^4+1\& ,\frac {-\text {$\#$1}^4 \log \left (\sqrt [4]{x^4+x^3}-\text {$\#$1} x\right )+\text {$\#$1}^4 \log (x)-\log \left (\sqrt [4]{x^4+x^3}-\text {$\#$1} x\right )+\log (x)}{2 \text {$\#$1}^7-\text {$\#$1}^3}\& \right ]-2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+x^3}}\right )+2 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+x^3}}\right ) \]

________________________________________________________________________________________

Rubi [C]  time = 3.16, antiderivative size = 1271, normalized size of antiderivative = 6.02, number of steps used = 35, number of rules used = 12, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {2056, 6728, 906, 63, 331, 298, 203, 206, 6725, 93, 205, 208} \begin {gather*} -\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{x^4+x^3} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[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 63

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 93

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 203

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

Rule 205

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

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 208

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

Rule 298

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 331

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 906

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 2056

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 6725

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 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 (-1+x^2\right ) \sqrt [4]{x^3+x^4}}{1+x^2+x^4} \, dx &=\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 (1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\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 ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1-i \sqrt {3}}-\sqrt {-\sqrt {2}+\sqrt {-1-i \sqrt {3}}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 \sqrt {-\sqrt {2}+\sqrt {-1-i \sqrt {3}}} 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 ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1-i \sqrt {3}}+\sqrt {-\sqrt {2}+\sqrt {-1-i \sqrt {3}}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 \sqrt {-\sqrt {2}+\sqrt {-1-i \sqrt {3}}} 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 ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1-i \sqrt {3}}-\sqrt {\sqrt {2}+\sqrt {-1-i \sqrt {3}}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 \sqrt {\sqrt {2}+\sqrt {-1-i \sqrt {3}}} 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 ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1-i \sqrt {3}}+\sqrt {\sqrt {2}+\sqrt {-1-i \sqrt {3}}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 \sqrt {\sqrt {2}+\sqrt {-1-i \sqrt {3}}} 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 ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1+i \sqrt {3}}-\sqrt {-\sqrt {2}+\sqrt {-1+i \sqrt {3}}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 \sqrt {-\sqrt {2}+\sqrt {-1+i \sqrt {3}}} 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 ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1+i \sqrt {3}}+\sqrt {-\sqrt {2}+\sqrt {-1+i \sqrt {3}}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 \sqrt {-\sqrt {2}+\sqrt {-1+i \sqrt {3}}} 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 ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1+i \sqrt {3}}-\sqrt {\sqrt {2}+\sqrt {-1+i \sqrt {3}}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 \sqrt {\sqrt {2}+\sqrt {-1+i \sqrt {3}}} 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 ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1+i \sqrt {3}}+\sqrt {\sqrt {2}+\sqrt {-1+i \sqrt {3}}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 \sqrt {\sqrt {2}+\sqrt {-1+i \sqrt {3}}} x^{3/4} \sqrt [4]{1+x}}\\ &=-\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{-\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1-i \sqrt {3}} \sqrt [4]{1+x}}\right )}{2 \sqrt [8]{-1-i \sqrt {3}} x^{3/4} \sqrt [4]{1+x}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1-i \sqrt {3}} \sqrt [4]{1+x}}\right )}{2 \sqrt [8]{-1-i \sqrt {3}} x^{3/4} \sqrt [4]{1+x}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{-\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1+i \sqrt {3}} \sqrt [4]{1+x}}\right )}{2 \sqrt [8]{-1+i \sqrt {3}} x^{3/4} \sqrt [4]{1+x}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1+i \sqrt {3}} \sqrt [4]{1+x}}\right )}{2 \sqrt [8]{-1+i \sqrt {3}} x^{3/4} \sqrt [4]{1+x}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{-\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1-i \sqrt {3}} \sqrt [4]{1+x}}\right )}{2 \sqrt [8]{-1-i \sqrt {3}} x^{3/4} \sqrt [4]{1+x}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1-i \sqrt {3}} \sqrt [4]{1+x}}\right )}{2 \sqrt [8]{-1-i \sqrt {3}} x^{3/4} \sqrt [4]{1+x}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{-\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1+i \sqrt {3}} \sqrt [4]{1+x}}\right )}{2 \sqrt [8]{-1+i \sqrt {3}} x^{3/4} \sqrt [4]{1+x}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1+i \sqrt {3}} \sqrt [4]{1+x}}\right )}{2 \sqrt [8]{-1+i \sqrt {3}} x^{3/4} \sqrt [4]{1+x}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 1.01, size = 489, normalized size = 2.32 \begin {gather*} \frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{x^3 (x+1)} \left (x^{3/4} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};-x\right )-\frac {\left (1-\frac {1}{\sqrt {\frac {1}{2} \left (-1-i \sqrt {3}\right )}}\right ) x^{3/4} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {\left (1-i \sqrt {3}\right ) x}{2 (x+1)}\right )}{(x+1)^{3/4}}\right )}{3 x^{3/4} \sqrt [4]{x+1}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{x^3 (x+1)} \left (x^{3/4} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};-x\right )-\frac {\left (1-\frac {1}{\sqrt {\frac {1}{2} i \left (\sqrt {3}+i\right )}}\right ) x^{3/4} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {\left (1+i \sqrt {3}\right ) x}{2 (x+1)}\right )}{(x+1)^{3/4}}\right )}{3 x^{3/4} \sqrt [4]{x+1}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{x^3 (x+1)} \left (2 x^{3/4} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};-x\right )-\frac {\left (3+i \sqrt {3}\right ) x^{3/4} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {\left (1+\frac {1}{\sqrt {\frac {1}{2} \left (-1-i \sqrt {3}\right )}}\right ) x}{x+1}\right )}{(x+1)^{3/4}}\right )}{6 x^{3/4} \sqrt [4]{x+1}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{x^3 (x+1)} \left (2 x^{3/4} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};-x\right )-\frac {\left (3-i \sqrt {3}\right ) x^{3/4} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {\left (1+\frac {1}{\sqrt {\frac {1}{2} i \left (i+\sqrt {3}\right )}}\right ) x}{x+1}\right )}{(x+1)^{3/4}}\right )}{6 x^{3/4} \sqrt [4]{x+1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((1 - I*Sqrt[3])*(x^3*(1 + x))^(1/4)*(x^(3/4)*Hypergeometric2F1[3/4, 3/4, 7/4, -x] - ((1 - 1/Sqrt[(-1 - I*Sqrt
[3])/2])*x^(3/4)*Hypergeometric2F1[3/4, 1, 7/4, ((1 - I*Sqrt[3])*x)/(2*(1 + x))])/(1 + x)^(3/4)))/(3*x^(3/4)*(
1 + x)^(1/4)) + ((1 + I*Sqrt[3])*(x^3*(1 + x))^(1/4)*(x^(3/4)*Hypergeometric2F1[3/4, 3/4, 7/4, -x] - ((1 - 1/S
qrt[(I/2)*(I + Sqrt[3])])*x^(3/4)*Hypergeometric2F1[3/4, 1, 7/4, ((1 + I*Sqrt[3])*x)/(2*(1 + x))])/(1 + x)^(3/
4)))/(3*x^(3/4)*(1 + x)^(1/4)) + ((1 - I*Sqrt[3])*(x^3*(1 + x))^(1/4)*(2*x^(3/4)*Hypergeometric2F1[3/4, 3/4, 7
/4, -x] - ((3 + I*Sqrt[3])*x^(3/4)*Hypergeometric2F1[3/4, 1, 7/4, ((1 + 1/Sqrt[(-1 - I*Sqrt[3])/2])*x)/(1 + x)
])/(1 + x)^(3/4)))/(6*x^(3/4)*(1 + x)^(1/4)) + ((1 + I*Sqrt[3])*(x^3*(1 + x))^(1/4)*(2*x^(3/4)*Hypergeometric2
F1[3/4, 3/4, 7/4, -x] - ((3 - I*Sqrt[3])*x^(3/4)*Hypergeometric2F1[3/4, 1, 7/4, ((1 + 1/Sqrt[(I/2)*(I + Sqrt[3
])])*x)/(1 + x)])/(1 + x)^(3/4)))/(6*x^(3/4)*(1 + x)^(1/4))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.41, size = 211, normalized size = 1.00 \begin {gather*} -2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )+2 \tanh ^{-1}\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 ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.92, size = 3011, normalized size = 14.27

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/72*12^(3/4)*4^(1/4)*3^(7/8)*sqrt(12^(3/4)*3^(1/4)*(4*sqrt(3)*sqrt(2) - 7*sqrt(2))*sqrt(sqrt(3) + 2) - 24*sq
rt(3) + 48)*sqrt(sqrt(3) + 2)*arctan(1/432*(4^(3/4)*3^(3/8)*sqrt(2)*sqrt(12^(3/4)*3^(1/4)*(4*sqrt(3)*sqrt(2) -
7*sqrt(2))*sqrt(sqrt(3) + 2) - 24*sqrt(3) + 48)*(12^(1/4)*3^(3/4)*x*sqrt(sqrt(3) + 2) - sqrt(3)*(sqrt(3)*sqrt
(2)*x + 3*sqrt(2)*x))*sqrt((72*3^(1/4)*x^2 + 4^(1/4)*3^(1/8)*(12^(3/4)*3^(1/4)*(x^4 + x^3)^(1/4)*(sqrt(3)*x -
3*x)*sqrt(sqrt(3) + 2) + 6*sqrt(3)*sqrt(2)*(x^4 + x^3)^(1/4)*x)*sqrt(12^(3/4)*3^(1/4)*(4*sqrt(3)*sqrt(2) - 7*s
qrt(2))*sqrt(sqrt(3) + 2) - 24*sqrt(3) + 48) + 72*sqrt(x^4 + x^3))/x^2) - 72*12^(3/4)*3^(1/4)*sqrt(2)*x*sqrt(s
qrt(3) + 2) + 12*4^(3/4)*3^(3/8)*sqrt(12^(3/4)*3^(1/4)*(4*sqrt(3)*sqrt(2) - 7*sqrt(2))*sqrt(sqrt(3) + 2) - 24*
sqrt(3) + 48)*(sqrt(3)*(x^4 + x^3)^(1/4)*(sqrt(3)*sqrt(2) + 3*sqrt(2)) - 12^(1/4)*3^(3/4)*(x^4 + x^3)^(1/4)*sq
rt(sqrt(3) + 2)) + 432*sqrt(3)*x + 864*x)/x) - 1/72*12^(3/4)*4^(1/4)*3^(7/8)*sqrt(12^(3/4)*3^(1/4)*(4*sqrt(3)*
sqrt(2) - 7*sqrt(2))*sqrt(sqrt(3) + 2) - 24*sqrt(3) + 48)*sqrt(sqrt(3) + 2)*arctan(1/432*(4^(3/4)*3^(3/8)*sqrt
(2)*sqrt(12^(3/4)*3^(1/4)*(4*sqrt(3)*sqrt(2) - 7*sqrt(2))*sqrt(sqrt(3) + 2) - 24*sqrt(3) + 48)*(12^(1/4)*3^(3/
4)*x*sqrt(sqrt(3) + 2) - sqrt(3)*(sqrt(3)*sqrt(2)*x + 3*sqrt(2)*x))*sqrt((72*3^(1/4)*x^2 - 4^(1/4)*3^(1/8)*(12
^(3/4)*3^(1/4)*(x^4 + x^3)^(1/4)*(sqrt(3)*x - 3*x)*sqrt(sqrt(3) + 2) + 6*sqrt(3)*sqrt(2)*(x^4 + x^3)^(1/4)*x)*
sqrt(12^(3/4)*3^(1/4)*(4*sqrt(3)*sqrt(2) - 7*sqrt(2))*sqrt(sqrt(3) + 2) - 24*sqrt(3) + 48) + 72*sqrt(x^4 + x^3
))/x^2) + 72*12^(3/4)*3^(1/4)*sqrt(2)*x*sqrt(sqrt(3) + 2) + 12*4^(3/4)*3^(3/8)*sqrt(12^(3/4)*3^(1/4)*(4*sqrt(3
)*sqrt(2) - 7*sqrt(2))*sqrt(sqrt(3) + 2) - 24*sqrt(3) + 48)*(sqrt(3)*(x^4 + x^3)^(1/4)*(sqrt(3)*sqrt(2) + 3*sq
rt(2)) - 12^(1/4)*3^(3/4)*(x^4 + x^3)^(1/4)*sqrt(sqrt(3) + 2)) - 432*sqrt(3)*x - 864*x)/x) - 1/144*12^(3/4)*4^
(1/4)*3^(7/8)*sqrt(-4*12^(3/4)*3^(1/4)*(4*sqrt(3)*sqrt(2) - 7*sqrt(2))*sqrt(sqrt(3) + 2) - 96*sqrt(3) + 192)*s
qrt(sqrt(3) + 2)*arctan(1/864*(4^(3/4)*3^(3/8)*sqrt(-4*12^(3/4)*3^(1/4)*(4*sqrt(3)*sqrt(2) - 7*sqrt(2))*sqrt(s
qrt(3) + 2) - 96*sqrt(3) + 192)*(12^(1/4)*3^(3/4)*x*sqrt(sqrt(3) + 2) + sqrt(3)*(sqrt(3)*sqrt(2)*x + 3*sqrt(2)
*x))*sqrt((144*3^(1/4)*x^2 + 4^(1/4)*3^(1/8)*(12^(3/4)*3^(1/4)*(x^4 + x^3)^(1/4)*(sqrt(3)*x - 3*x)*sqrt(sqrt(3
) + 2) - 6*sqrt(3)*sqrt(2)*(x^4 + x^3)^(1/4)*x)*sqrt(-4*12^(3/4)*3^(1/4)*(4*sqrt(3)*sqrt(2) - 7*sqrt(2))*sqrt(
sqrt(3) + 2) - 96*sqrt(3) + 192) + 144*sqrt(x^4 + x^3))/x^2) + 144*12^(3/4)*3^(1/4)*sqrt(2)*x*sqrt(sqrt(3) + 2
) - 12*4^(3/4)*3^(3/8)*sqrt(-4*12^(3/4)*3^(1/4)*(4*sqrt(3)*sqrt(2) - 7*sqrt(2))*sqrt(sqrt(3) + 2) - 96*sqrt(3)
+ 192)*(sqrt(3)*(x^4 + x^3)^(1/4)*(sqrt(3)*sqrt(2) + 3*sqrt(2)) + 12^(1/4)*3^(3/4)*(x^4 + x^3)^(1/4)*sqrt(sqr
t(3) + 2)) + 864*sqrt(3)*x + 1728*x)/x) - 1/144*12^(3/4)*4^(1/4)*3^(7/8)*sqrt(-4*12^(3/4)*3^(1/4)*(4*sqrt(3)*s
qrt(2) - 7*sqrt(2))*sqrt(sqrt(3) + 2) - 96*sqrt(3) + 192)*sqrt(sqrt(3) + 2)*arctan(1/864*(4^(3/4)*3^(3/8)*sqrt
(-4*12^(3/4)*3^(1/4)*(4*sqrt(3)*sqrt(2) - 7*sqrt(2))*sqrt(sqrt(3) + 2) - 96*sqrt(3) + 192)*(12^(1/4)*3^(3/4)*x
*sqrt(sqrt(3) + 2) + sqrt(3)*(sqrt(3)*sqrt(2)*x + 3*sqrt(2)*x))*sqrt((144*3^(1/4)*x^2 - 4^(1/4)*3^(1/8)*(12^(3
/4)*3^(1/4)*(x^4 + x^3)^(1/4)*(sqrt(3)*x - 3*x)*sqrt(sqrt(3) + 2) - 6*sqrt(3)*sqrt(2)*(x^4 + x^3)^(1/4)*x)*sqr
t(-4*12^(3/4)*3^(1/4)*(4*sqrt(3)*sqrt(2) - 7*sqrt(2))*sqrt(sqrt(3) + 2) - 96*sqrt(3) + 192) + 144*sqrt(x^4 + x
^3))/x^2) - 144*12^(3/4)*3^(1/4)*sqrt(2)*x*sqrt(sqrt(3) + 2) - 12*4^(3/4)*3^(3/8)*sqrt(-4*12^(3/4)*3^(1/4)*(4*
sqrt(3)*sqrt(2) - 7*sqrt(2))*sqrt(sqrt(3) + 2) - 96*sqrt(3) + 192)*(sqrt(3)*(x^4 + x^3)^(1/4)*(sqrt(3)*sqrt(2)
+ 3*sqrt(2)) + 12^(1/4)*3^(3/4)*(x^4 + x^3)^(1/4)*sqrt(sqrt(3) + 2)) - 864*sqrt(3)*x - 1728*x)/x) - 1/576*4^(
1/4)*3^(1/8)*sqrt(-4*12^(3/4)*3^(1/4)*(4*sqrt(3)*sqrt(2) - 7*sqrt(2))*sqrt(sqrt(3) + 2) - 96*sqrt(3) + 192)*(1
2^(3/4)*3^(1/4)*(2*sqrt(3) - 3)*sqrt(sqrt(3) + 2) - 12*sqrt(3)*sqrt(2))*log(1/9*(144*3^(1/4)*x^2 + 4^(1/4)*3^(
1/8)*(12^(3/4)*3^(1/4)*(x^4 + x^3)^(1/4)*(sqrt(3)*x - 3*x)*sqrt(sqrt(3) + 2) - 6*sqrt(3)*sqrt(2)*(x^4 + x^3)^(
1/4)*x)*sqrt(-4*12^(3/4)*3^(1/4)*(4*sqrt(3)*sqrt(2) - 7*sqrt(2))*sqrt(sqrt(3) + 2) - 96*sqrt(3) + 192) + 144*s
qrt(x^4 + x^3))/x^2) + 1/576*4^(1/4)*3^(1/8)*sqrt(-4*12^(3/4)*3^(1/4)*(4*sqrt(3)*sqrt(2) - 7*sqrt(2))*sqrt(sqr
t(3) + 2) - 96*sqrt(3) + 192)*(12^(3/4)*3^(1/4)*(2*sqrt(3) - 3)*sqrt(sqrt(3) + 2) - 12*sqrt(3)*sqrt(2))*log(1/
9*(144*3^(1/4)*x^2 - 4^(1/4)*3^(1/8)*(12^(3/4)*3^(1/4)*(x^4 + x^3)^(1/4)*(sqrt(3)*x - 3*x)*sqrt(sqrt(3) + 2) -
6*sqrt(3)*sqrt(2)*(x^4 + x^3)^(1/4)*x)*sqrt(-4*12^(3/4)*3^(1/4)*(4*sqrt(3)*sqrt(2) - 7*sqrt(2))*sqrt(sqrt(3)
+ 2) - 96*sqrt(3) + 192) + 144*sqrt(x^4 + x^3))/x^2) - 1/288*4^(1/4)*3^(1/8)*sqrt(12^(3/4)*3^(1/4)*(4*sqrt(3)*
sqrt(2) - 7*sqrt(2))*sqrt(sqrt(3) + 2) - 24*sqrt(3) + 48)*(12^(3/4)*3^(1/4)*(2*sqrt(3) - 3)*sqrt(sqrt(3) + 2)
+ 12*sqrt(3)*sqrt(2))*log(2/9*(72*3^(1/4)*x^2 + 4^(1/4)*3^(1/8)*(12^(3/4)*3^(1/4)*(x^4 + x^3)^(1/4)*(sqrt(3)*x
- 3*x)*sqrt(sqrt(3) + 2) + 6*sqrt(3)*sqrt(2)*(x^4 + x^3)^(1/4)*x)*sqrt(12^(3/4)*3^(1/4)*(4*sqrt(3)*sqrt(2) -
7*sqrt(2))*sqrt(sqrt(3) + 2) - 24*sqrt(3) + 48) + 72*sqrt(x^4 + x^3))/x^2) + 1/288*4^(1/4)*3^(1/8)*sqrt(12^(3/
4)*3^(1/4)*(4*sqrt(3)*sqrt(2) - 7*sqrt(2))*sqrt(sqrt(3) + 2) - 24*sqrt(3) + 48)*(12^(3/4)*3^(1/4)*(2*sqrt(3) -
3)*sqrt(sqrt(3) + 2) + 12*sqrt(3)*sqrt(2))*log(2/9*(72*3^(1/4)*x^2 - 4^(1/4)*3^(1/8)*(12^(3/4)*3^(1/4)*(x^4 +
x^3)^(1/4)*(sqrt(3)*x - 3*x)*sqrt(sqrt(3) + 2) + 6*sqrt(3)*sqrt(2)*(x^4 + x^3)^(1/4)*x)*sqrt(12^(3/4)*3^(1/4)
*(4*sqrt(3)*sqrt(2) - 7*sqrt(2))*sqrt(sqrt(3) + 2) - 24*sqrt(3) + 48) + 72*sqrt(x^4 + x^3))/x^2) - 1/8*(sqrt(3
) + 2)*sqrt(-4*sqrt(3) + 8)*log(8*(2*x^2 + (x^4 + x^3)^(1/4)*x*sqrt(-4*sqrt(3) + 8) + 2*sqrt(x^4 + x^3))/x^2)
+ 1/8*(sqrt(3) + 2)*sqrt(-4*sqrt(3) + 8)*log(8*(2*x^2 - (x^4 + x^3)^(1/4)*x*sqrt(-4*sqrt(3) + 8) + 2*sqrt(x^4
+ x^3))/x^2) + 1/4*sqrt(sqrt(3) + 2)*(sqrt(3) - 2)*log(16*(x^2 + (x^4 + x^3)^(1/4)*x*sqrt(sqrt(3) + 2) + sqrt(
x^4 + x^3))/x^2) - 1/4*sqrt(sqrt(3) + 2)*(sqrt(3) - 2)*log(16*(x^2 - (x^4 + x^3)^(1/4)*x*sqrt(sqrt(3) + 2) + s
qrt(x^4 + x^3))/x^2) + 1/2*sqrt(-4*sqrt(3) + 8)*arctan(1/2*(sqrt(2)*x*sqrt(-4*sqrt(3) + 8)*sqrt((2*x^2 + (x^4
+ x^3)^(1/4)*x*sqrt(-4*sqrt(3) + 8) + 2*sqrt(x^4 + x^3))/x^2) + 2*sqrt(3)*x - 4*x - 2*(x^4 + x^3)^(1/4)*sqrt(-
4*sqrt(3) + 8))/x) + 1/2*sqrt(-4*sqrt(3) + 8)*arctan(1/2*(sqrt(2)*x*sqrt(-4*sqrt(3) + 8)*sqrt((2*x^2 - (x^4 +
x^3)^(1/4)*x*sqrt(-4*sqrt(3) + 8) + 2*sqrt(x^4 + x^3))/x^2) - 2*sqrt(3)*x + 4*x - 2*(x^4 + x^3)^(1/4)*sqrt(-4*
sqrt(3) + 8))/x) + sqrt(sqrt(3) + 2)*arctan((2*x*sqrt(sqrt(3) + 2)*sqrt((x^2 + (x^4 + x^3)^(1/4)*x*sqrt(sqrt(3
) + 2) + sqrt(x^4 + x^3))/x^2) - sqrt(3)*x - 2*x - 2*(x^4 + x^3)^(1/4)*sqrt(sqrt(3) + 2))/x) + sqrt(sqrt(3) +
2)*arctan((2*x*sqrt(sqrt(3) + 2)*sqrt((x^2 - (x^4 + x^3)^(1/4)*x*sqrt(sqrt(3) + 2) + sqrt(x^4 + x^3))/x^2) + s
qrt(3)*x + 2*x - 2*(x^4 + x^3)^(1/4)*sqrt(sqrt(3) + 2))/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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{2} - 1\right )}}{x^{4} + x^{2} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [B]  time = 206.98, size = 14154, normalized size = 67.08

method	result	size
trager	\(\text {Expression too large to display}\)	\(14154\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{2} - 1\right )}}{x^{4} + x^{2} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (x^4+x^3\right )}^{1/4}\,\left (x^2-1\right )}{x^4+x^2+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________