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

Optimal. Leaf size=147 \[ -\text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^4+1\& ,\frac {-\text {$\#$1}^4 \log \left (\sqrt [4]{x^4-x^3}-\text {$\#$1} x\right )+\text {$\#$1}^4 \log (x)+2 \log \left (\sqrt [4]{x^4-x^3}-\text {$\#$1} x\right )-2 \log (x)}{2 \text {$\#$1}^7-\text {$\#$1}^3}\& \right ]+\sqrt [4]{x^4-x^3}-\frac {7}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^3}}\right )+\frac {7}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^3}}\right ) \]

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Rubi [C]  time = 0.73, antiderivative size = 695, normalized size of antiderivative = 4.73, number of steps used = 25, number of rules used = 13, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.482, Rules used = {2056, 6728, 101, 157, 63, 240, 212, 206, 203, 93, 298, 205, 208} \begin {gather*} \frac {1}{2} \left (1+i \sqrt {3}\right ) \sqrt [4]{x^4-x^3}+\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt [4]{x^4-x^3}+\frac {\left (7+i \sqrt {3}\right ) \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{x-1} x^{3/4}}+\frac {\left (7-i \sqrt {3}\right ) \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{x-1} x^{3/4}}+\frac {\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \left (\sqrt {3}+i\right ) \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}+\frac {\left (-\frac {-\sqrt {3}+i}{\sqrt {3}+i}\right )^{3/4} \left (\sqrt {3}+i\right ) \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}+\frac {\left (7+i \sqrt {3}\right ) \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{x-1} x^{3/4}}+\frac {\left (7-i \sqrt {3}\right ) \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{x-1} x^{3/4}}-\frac {\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \left (\sqrt {3}+i\right ) \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}-\frac {\left (-\frac {-\sqrt {3}+i}{\sqrt {3}+i}\right )^{3/4} \left (\sqrt {3}+i\right ) \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((1 - I*Sqrt[3])*(-x^3 + x^4)^(1/4))/2 + ((1 + I*Sqrt[3])*(-x^3 + x^4)^(1/4))/2 + ((7 - I*Sqrt[3])*(-x^3 + x^4
)^(1/4)*ArcTan[(-1 + x)^(1/4)/x^(1/4)])/(4*(-1 + x)^(1/4)*x^(3/4)) + ((7 + I*Sqrt[3])*(-x^3 + x^4)^(1/4)*ArcTa
n[(-1 + x)^(1/4)/x^(1/4)])/(4*(-1 + x)^(1/4)*x^(3/4)) + ((-((I - Sqrt[3])/(I + Sqrt[3])))^(1/4)*(I + Sqrt[3])*
(-x^3 + x^4)^(1/4)*ArcTan[x^(1/4)/((-((I - Sqrt[3])/(I + Sqrt[3])))^(1/4)*(-1 + x)^(1/4))])/((-1 + x)^(1/4)*x^
(3/4)) + ((-((I - Sqrt[3])/(I + Sqrt[3])))^(3/4)*(I + Sqrt[3])*(-x^3 + x^4)^(1/4)*ArcTan[((-((I - Sqrt[3])/(I
+ Sqrt[3])))^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/((-1 + x)^(1/4)*x^(3/4)) + ((7 - I*Sqrt[3])*(-x^3 + x^4)^(1/4)*Ar
cTanh[(-1 + x)^(1/4)/x^(1/4)])/(4*(-1 + x)^(1/4)*x^(3/4)) + ((7 + I*Sqrt[3])*(-x^3 + x^4)^(1/4)*ArcTanh[(-1 +
x)^(1/4)/x^(1/4)])/(4*(-1 + x)^(1/4)*x^(3/4)) - ((-((I - Sqrt[3])/(I + Sqrt[3])))^(1/4)*(I + Sqrt[3])*(-x^3 +
x^4)^(1/4)*ArcTanh[x^(1/4)/((-((I - Sqrt[3])/(I + Sqrt[3])))^(1/4)*(-1 + x)^(1/4))])/((-1 + x)^(1/4)*x^(3/4))
- ((-((I - Sqrt[3])/(I + Sqrt[3])))^(3/4)*(I + Sqrt[3])*(-x^3 + x^4)^(1/4)*ArcTanh[((-((I - Sqrt[3])/(I + Sqrt
[3])))^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/((-1 + x)^(1/4)*x^(3/4))

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 101

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a +
b*x)^m*(c + d*x)^n*(e + f*x)^(p + 1))/(f*(m + n + p + 1)), x] - Dist[1/(f*(m + n + p + 1)), Int[(a + b*x)^(m -
1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a*f) + b*n*(d*e - c*f))
*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (Integ
ersQ[2*m, 2*n, 2*p] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, 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 212

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

Rule 240

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 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)] && IntegerQ[p
+ 1/n]

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 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 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 {(1+x) \sqrt [4]{-x^3+x^4}}{1-x+x^2} \, dx &=\frac {\sqrt [4]{-x^3+x^4} \int \frac {\sqrt [4]{-1+x} x^{3/4} (1+x)}{1-x+x^2} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {\sqrt [4]{-x^3+x^4} \int \left (\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{-1+x} x^{3/4}}{-1-i \sqrt {3}+2 x}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{-1+x} x^{3/4}}{-1+i \sqrt {3}+2 x}\right ) \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {\sqrt [4]{-1+x} x^{3/4}}{-1-i \sqrt {3}+2 x} \, dx}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {\sqrt [4]{-1+x} x^{3/4}}{-1+i \sqrt {3}+2 x} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}-\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {\frac {3}{4} \left (1+i \sqrt {3}\right )+\frac {1}{2} \left (-1-2 i \sqrt {3}\right ) x}{(-1+x)^{3/4} \sqrt [4]{x} \left (-1-i \sqrt {3}+2 x\right )} \, dx}{2 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {\frac {3}{4} \left (1-i \sqrt {3}\right )+\frac {1}{2} \left (-1+2 i \sqrt {3}\right ) x}{(-1+x)^{3/4} \sqrt [4]{x} \left (-1+i \sqrt {3}+2 x\right )} \, dx}{2 \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}-\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x} \left (-1-i \sqrt {3}+2 x\right )} \, dx}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x} \left (-1+i \sqrt {3}+2 x\right )} \, dx}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1-i \sqrt {3}\right ) \left (-1-2 i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{8 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1+i \sqrt {3}\right ) \left (-1+2 i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{8 \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}-\frac {\left (4 \left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-1-i \sqrt {3}-\left (1-i \sqrt {3}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (4 \left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-1+i \sqrt {3}-\left (1+i \sqrt {3}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1-i \sqrt {3}\right ) \left (-1-2 i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4}} \, dx,x,\sqrt [4]{-1+x}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1+i \sqrt {3}\right ) \left (-1+2 i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4}} \, dx,x,\sqrt [4]{-1+x}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}-\frac {\left (\left (1-i \sqrt {3}\right ) \left (-1-2 i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1+i \sqrt {3}\right ) \left (-1+2 i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 i \left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {i+\sqrt {3}}-\sqrt {-i+\sqrt {3}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt {-i+\sqrt {3}} \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (2 i \left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {i+\sqrt {3}}+\sqrt {-i+\sqrt {3}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt {-i+\sqrt {3}} \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (2 i \left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-i+\sqrt {3}}-\sqrt {i+\sqrt {3}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt {i+\sqrt {3}} \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 i \left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-i+\sqrt {3}}+\sqrt {i+\sqrt {3}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt {i+\sqrt {3}} \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {2 \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-1+x}}\right )}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-1+x} x^{3/4}}+\frac {2 \sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {2 \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-1+x}}\right )}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-1+x} x^{3/4}}-\frac {2 \sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1-i \sqrt {3}\right ) \left (-1-2 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]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1-i \sqrt {3}\right ) \left (-1-2 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]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1+i \sqrt {3}\right ) \left (-1+2 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]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1+i \sqrt {3}\right ) \left (-1+2 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]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {\left (7-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (7+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}+\frac {2 \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-1+x}}\right )}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-1+x} x^{3/4}}+\frac {2 \sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (7-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (7+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}-\frac {2 \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-1+x}}\right )}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-1+x} x^{3/4}}-\frac {2 \sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ \end {align*}

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Mathematica [C]  time = 0.08, size = 133, normalized size = 0.90 \begin {gather*} \frac {4 \sqrt [4]{(x-1) x^3} \left (\sqrt [4]{x} \, _2F_1\left (-\frac {3}{4},\frac {1}{4};\frac {5}{4};1-x\right )+\sqrt [4]{x} \, _2F_1\left (\frac {1}{4},\frac {1}{4};\frac {5}{4};1-x\right )-\, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {\left (-i+\sqrt {3}\right ) (x-1)}{\left (i+\sqrt {3}\right ) x}\right )-\, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {\left (i+\sqrt {3}\right ) (x-1)}{\left (-i+\sqrt {3}\right ) x}\right )\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*((-1 + x)*x^3)^(1/4)*(x^(1/4)*Hypergeometric2F1[-3/4, 1/4, 5/4, 1 - x] + x^(1/4)*Hypergeometric2F1[1/4, 1/4
, 5/4, 1 - x] - Hypergeometric2F1[1/4, 1, 5/4, ((-I + Sqrt[3])*(-1 + x))/((I + Sqrt[3])*x)] - Hypergeometric2F
1[1/4, 1, 5/4, ((I + Sqrt[3])*(-1 + x))/((-I + Sqrt[3])*x)]))/x

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IntegrateAlgebraic [A]  time = 0.00, size = 147, normalized size = 1.00 \begin {gather*} \sqrt [4]{-x^3+x^4}-\frac {7}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )+\frac {7}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )-\text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 \log (x)+2 \log \left (\sqrt [4]{-x^3+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.54, size = 836, normalized size = 5.69 \begin {gather*} -\frac {1}{4} \, {\left (\sqrt {3} + 2\right )} \sqrt {-4 \, \sqrt {3} + 8} \log \left (\frac {2 \, {\left (2 \, x^{2} + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (\sqrt {3} x + 2 \, x\right )} \sqrt {-4 \, \sqrt {3} + 8} + 2 \, \sqrt {x^{4} - x^{3}}\right )}}{x^{2}}\right ) + \frac {1}{4} \, {\left (\sqrt {3} + 2\right )} \sqrt {-4 \, \sqrt {3} + 8} \log \left (\frac {2 \, {\left (2 \, x^{2} - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (\sqrt {3} x + 2 \, x\right )} \sqrt {-4 \, \sqrt {3} + 8} + 2 \, \sqrt {x^{4} - x^{3}}\right )}}{x^{2}}\right ) - \frac {1}{2} \, \sqrt {\sqrt {3} + 2} {\left (\sqrt {3} - 2\right )} \log \left (\frac {4 \, {\left (x^{2} + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (\sqrt {3} x - 2 \, x\right )} \sqrt {\sqrt {3} + 2} + \sqrt {x^{4} - x^{3}}\right )}}{x^{2}}\right ) + \frac {1}{2} \, \sqrt {\sqrt {3} + 2} {\left (\sqrt {3} - 2\right )} \log \left (\frac {4 \, {\left (x^{2} - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (\sqrt {3} x - 2 \, x\right )} \sqrt {\sqrt {3} + 2} + \sqrt {x^{4} - x^{3}}\right )}}{x^{2}}\right ) + \sqrt {-4 \, \sqrt {3} + 8} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {3} x + 2 \, x\right )} \sqrt {-4 \, \sqrt {3} + 8} \sqrt {\frac {2 \, x^{2} + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (\sqrt {3} x + 2 \, x\right )} \sqrt {-4 \, \sqrt {3} + 8} + 2 \, \sqrt {x^{4} - x^{3}}}{x^{2}}} - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (\sqrt {3} + 2\right )} \sqrt {-4 \, \sqrt {3} + 8} - 2 \, \sqrt {3} x - 4 \, x}{2 \, x}\right ) + \sqrt {-4 \, \sqrt {3} + 8} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {3} x + 2 \, x\right )} \sqrt {-4 \, \sqrt {3} + 8} \sqrt {\frac {2 \, x^{2} - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (\sqrt {3} x + 2 \, x\right )} \sqrt {-4 \, \sqrt {3} + 8} + 2 \, \sqrt {x^{4} - x^{3}}}{x^{2}}} - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (\sqrt {3} + 2\right )} \sqrt {-4 \, \sqrt {3} + 8} + 2 \, \sqrt {3} x + 4 \, x}{2 \, x}\right ) - 2 \, \sqrt {\sqrt {3} + 2} \arctan \left (\frac {2 \, {\left (\sqrt {3} x - 2 \, x\right )} \sqrt {\sqrt {3} + 2} \sqrt {\frac {x^{2} + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (\sqrt {3} x - 2 \, x\right )} \sqrt {\sqrt {3} + 2} + \sqrt {x^{4} - x^{3}}}{x^{2}}} - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} \sqrt {\sqrt {3} + 2} {\left (\sqrt {3} - 2\right )} + \sqrt {3} x - 2 \, x}{x}\right ) - 2 \, \sqrt {\sqrt {3} + 2} \arctan \left (\frac {2 \, {\left (\sqrt {3} x - 2 \, x\right )} \sqrt {\sqrt {3} + 2} \sqrt {\frac {x^{2} - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (\sqrt {3} x - 2 \, x\right )} \sqrt {\sqrt {3} + 2} + \sqrt {x^{4} - x^{3}}}{x^{2}}} - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} \sqrt {\sqrt {3} + 2} {\left (\sqrt {3} - 2\right )} - \sqrt {3} x + 2 \, x}{x}\right ) + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} + \frac {7}{2} \, \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {7}{4} \, \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {7}{4} \, \log \left (-\frac {x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(sqrt(3) + 2)*sqrt(-4*sqrt(3) + 8)*log(2*(2*x^2 + (x^4 - x^3)^(1/4)*(sqrt(3)*x + 2*x)*sqrt(-4*sqrt(3) + 8
) + 2*sqrt(x^4 - x^3))/x^2) + 1/4*(sqrt(3) + 2)*sqrt(-4*sqrt(3) + 8)*log(2*(2*x^2 - (x^4 - x^3)^(1/4)*(sqrt(3)
*x + 2*x)*sqrt(-4*sqrt(3) + 8) + 2*sqrt(x^4 - x^3))/x^2) - 1/2*sqrt(sqrt(3) + 2)*(sqrt(3) - 2)*log(4*(x^2 + (x
^4 - x^3)^(1/4)*(sqrt(3)*x - 2*x)*sqrt(sqrt(3) + 2) + sqrt(x^4 - x^3))/x^2) + 1/2*sqrt(sqrt(3) + 2)*(sqrt(3) -
2)*log(4*(x^2 - (x^4 - x^3)^(1/4)*(sqrt(3)*x - 2*x)*sqrt(sqrt(3) + 2) + sqrt(x^4 - x^3))/x^2) + sqrt(-4*sqrt(
3) + 8)*arctan(1/2*(sqrt(2)*(sqrt(3)*x + 2*x)*sqrt(-4*sqrt(3) + 8)*sqrt((2*x^2 + (x^4 - x^3)^(1/4)*(sqrt(3)*x
+ 2*x)*sqrt(-4*sqrt(3) + 8) + 2*sqrt(x^4 - x^3))/x^2) - 2*(x^4 - x^3)^(1/4)*(sqrt(3) + 2)*sqrt(-4*sqrt(3) + 8)
- 2*sqrt(3)*x - 4*x)/x) + sqrt(-4*sqrt(3) + 8)*arctan(1/2*(sqrt(2)*(sqrt(3)*x + 2*x)*sqrt(-4*sqrt(3) + 8)*sqr
t((2*x^2 - (x^4 - x^3)^(1/4)*(sqrt(3)*x + 2*x)*sqrt(-4*sqrt(3) + 8) + 2*sqrt(x^4 - x^3))/x^2) - 2*(x^4 - x^3)^
(1/4)*(sqrt(3) + 2)*sqrt(-4*sqrt(3) + 8) + 2*sqrt(3)*x + 4*x)/x) - 2*sqrt(sqrt(3) + 2)*arctan((2*(sqrt(3)*x -
2*x)*sqrt(sqrt(3) + 2)*sqrt((x^2 + (x^4 - x^3)^(1/4)*(sqrt(3)*x - 2*x)*sqrt(sqrt(3) + 2) + sqrt(x^4 - x^3))/x^
2) - 2*(x^4 - x^3)^(1/4)*sqrt(sqrt(3) + 2)*(sqrt(3) - 2) + sqrt(3)*x - 2*x)/x) - 2*sqrt(sqrt(3) + 2)*arctan((2
*(sqrt(3)*x - 2*x)*sqrt(sqrt(3) + 2)*sqrt((x^2 - (x^4 - x^3)^(1/4)*(sqrt(3)*x - 2*x)*sqrt(sqrt(3) + 2) + sqrt(
x^4 - x^3))/x^2) - 2*(x^4 - x^3)^(1/4)*sqrt(sqrt(3) + 2)*(sqrt(3) - 2) - sqrt(3)*x + 2*x)/x) + (x^4 - x^3)^(1/
4) + 7/2*arctan((x^4 - x^3)^(1/4)/x) + 7/4*log((x + (x^4 - x^3)^(1/4))/x) - 7/4*log(-(x - (x^4 - x^3)^(1/4))/x
)

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giac [B]  time = 0.26, size = 384, normalized size = 2.61 \begin {gather*} \frac {1}{2} \, {\left (\sqrt {6} + \sqrt {2}\right )} \arctan \left (\frac {\sqrt {6} - \sqrt {2} + 4 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{2} \, {\left (\sqrt {6} + \sqrt {2}\right )} \arctan \left (-\frac {\sqrt {6} - \sqrt {2} - 4 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{2} \, {\left (\sqrt {6} - \sqrt {2}\right )} \arctan \left (\frac {\sqrt {6} + \sqrt {2} + 4 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{2} \, {\left (\sqrt {6} - \sqrt {2}\right )} \arctan \left (-\frac {\sqrt {6} + \sqrt {2} - 4 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{4} \, {\left (\sqrt {6} + \sqrt {2}\right )} \log \left (\frac {1}{2} \, {\left (\sqrt {6} + \sqrt {2}\right )} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) - \frac {1}{4} \, {\left (\sqrt {6} + \sqrt {2}\right )} \log \left (-\frac {1}{2} \, {\left (\sqrt {6} + \sqrt {2}\right )} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) + \frac {1}{4} \, {\left (\sqrt {6} - \sqrt {2}\right )} \log \left (\frac {1}{2} \, {\left (\sqrt {6} - \sqrt {2}\right )} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) - \frac {1}{4} \, {\left (\sqrt {6} - \sqrt {2}\right )} \log \left (-\frac {1}{2} \, {\left (\sqrt {6} - \sqrt {2}\right )} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) - x {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - \frac {7}{2} \, \arctan \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {7}{4} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {7}{4} \, \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(sqrt(6) + sqrt(2))*arctan((sqrt(6) - sqrt(2) + 4*(-1/x + 1)^(1/4))/(sqrt(6) + sqrt(2))) + 1/2*(sqrt(6) +
sqrt(2))*arctan(-(sqrt(6) - sqrt(2) - 4*(-1/x + 1)^(1/4))/(sqrt(6) + sqrt(2))) + 1/2*(sqrt(6) - sqrt(2))*arcta
n((sqrt(6) + sqrt(2) + 4*(-1/x + 1)^(1/4))/(sqrt(6) - sqrt(2))) + 1/2*(sqrt(6) - sqrt(2))*arctan(-(sqrt(6) + s
qrt(2) - 4*(-1/x + 1)^(1/4))/(sqrt(6) - sqrt(2))) + 1/4*(sqrt(6) + sqrt(2))*log(1/2*(sqrt(6) + sqrt(2))*(-1/x
+ 1)^(1/4) + sqrt(-1/x + 1) + 1) - 1/4*(sqrt(6) + sqrt(2))*log(-1/2*(sqrt(6) + sqrt(2))*(-1/x + 1)^(1/4) + sqr
t(-1/x + 1) + 1) + 1/4*(sqrt(6) - sqrt(2))*log(1/2*(sqrt(6) - sqrt(2))*(-1/x + 1)^(1/4) + sqrt(-1/x + 1) + 1)
- 1/4*(sqrt(6) - sqrt(2))*log(-1/2*(sqrt(6) - sqrt(2))*(-1/x + 1)^(1/4) + sqrt(-1/x + 1) + 1) - x*(-1/x + 1)^(
1/4) - 7/2*arctan((-1/x + 1)^(1/4)) - 7/4*log((-1/x + 1)^(1/4) + 1) + 7/4*log(abs((-1/x + 1)^(1/4) - 1))

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maple [B]  time = 48.67, size = 1746, normalized size = 11.88

method	result	size
trager	\(\text {Expression too large to display}\)	\(1746\)
risch	\(\text {Expression too large to display}\)	\(5889\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^4-x^3)^(1/4)+RootOf(_Z^8-_Z^4+1)*ln(-(RootOf(_Z^8-_Z^4+1)^11*x^3-2*RootOf(_Z^8-_Z^4+1)^11*x^2-3*RootOf(_Z^8
-_Z^4+1)^7*x^3-RootOf(_Z^8-_Z^4+1)^7*x^2+10*RootOf(_Z^8-_Z^4+1)^6*(x^4-x^3)^(1/4)*x^2-10*(x^4-x^3)^(1/2)*RootO
f(_Z^8-_Z^4+1)^5*x+10*(x^4-x^3)^(3/4)*RootOf(_Z^8-_Z^4+1)^4-10*RootOf(_Z^8-_Z^4+1)^3*x^3+6*RootOf(_Z^8-_Z^4+1)
^3*x^2+6*RootOf(_Z^8-_Z^4+1)^2*(x^4-x^3)^(1/4)*x^2-6*(x^4-x^3)^(1/2)*RootOf(_Z^8-_Z^4+1)*x+6*(x^4-x^3)^(3/4))/
(x*RootOf(_Z^8-_Z^4+1)^4-2*RootOf(_Z^8-_Z^4+1)^4+x+1)/x^2)-RootOf(_Z^8-_Z^4+1)^7*ln(-(3*RootOf(_Z^8-_Z^4+1)^9*
x^3-6*RootOf(_Z^8-_Z^4+1)^9*x^2+16*(x^4-x^3)^(1/2)*RootOf(_Z^8-_Z^4+1)^7*x-16*RootOf(_Z^8-_Z^4+1)^6*(x^4-x^3)^
(1/4)*x^2+10*RootOf(_Z^8-_Z^4+1)^5*x^3+RootOf(_Z^8-_Z^4+1)^5*x^2-10*(x^4-x^3)^(3/4)*RootOf(_Z^8-_Z^4+1)^4-6*(x
^4-x^3)^(1/2)*RootOf(_Z^8-_Z^4+1)^3*x+6*RootOf(_Z^8-_Z^4+1)^2*(x^4-x^3)^(1/4)*x^2-8*RootOf(_Z^8-_Z^4+1)*x^3+2*
RootOf(_Z^8-_Z^4+1)*x^2+16*(x^4-x^3)^(3/4))/(x*RootOf(_Z^8-_Z^4+1)^4-2*RootOf(_Z^8-_Z^4+1)^4-2*x+1)/x^2)+RootO
f(_Z^8-_Z^4+1)^3*ln(-(3*RootOf(_Z^8-_Z^4+1)^9*x^3-6*RootOf(_Z^8-_Z^4+1)^9*x^2+16*(x^4-x^3)^(1/2)*RootOf(_Z^8-_
Z^4+1)^7*x-16*RootOf(_Z^8-_Z^4+1)^6*(x^4-x^3)^(1/4)*x^2+10*RootOf(_Z^8-_Z^4+1)^5*x^3+RootOf(_Z^8-_Z^4+1)^5*x^2
-10*(x^4-x^3)^(3/4)*RootOf(_Z^8-_Z^4+1)^4-6*(x^4-x^3)^(1/2)*RootOf(_Z^8-_Z^4+1)^3*x+6*RootOf(_Z^8-_Z^4+1)^2*(x
^4-x^3)^(1/4)*x^2-8*RootOf(_Z^8-_Z^4+1)*x^3+2*RootOf(_Z^8-_Z^4+1)*x^2+16*(x^4-x^3)^(3/4))/(x*RootOf(_Z^8-_Z^4+
1)^4-2*RootOf(_Z^8-_Z^4+1)^4-2*x+1)/x^2)-RootOf(_Z^8-_Z^4+1)^7*ln(-(2*RootOf(_Z^8-_Z^4+1)^9*x^3-4*RootOf(_Z^8-
_Z^4+1)^9*x^2+16*(x^4-x^3)^(1/2)*RootOf(_Z^8-_Z^4+1)^7*x-10*RootOf(_Z^8-_Z^4+1)^6*(x^4-x^3)^(1/4)*x^2-13*RootO
f(_Z^8-_Z^4+1)^5*x^3+12*RootOf(_Z^8-_Z^4+1)^5*x^2+10*(x^4-x^3)^(3/4)*RootOf(_Z^8-_Z^4+1)^4-10*(x^4-x^3)^(1/2)*
RootOf(_Z^8-_Z^4+1)^3*x-6*RootOf(_Z^8-_Z^4+1)^2*(x^4-x^3)^(1/4)*x^2+15*RootOf(_Z^8-_Z^4+1)*x^3-9*RootOf(_Z^8-_
Z^4+1)*x^2+6*(x^4-x^3)^(3/4))/(x*RootOf(_Z^8-_Z^4+1)^4-2*RootOf(_Z^8-_Z^4+1)^4+x+1)/x^2)-RootOf(_Z^8-_Z^4+1)^5
*ln(-(RootOf(_Z^8-_Z^4+1)^11*x^3-2*RootOf(_Z^8-_Z^4+1)^11*x^2+RootOf(_Z^8-_Z^4+1)^7*x^3+5*RootOf(_Z^8-_Z^4+1)^
7*x^2+16*RootOf(_Z^8-_Z^4+1)^6*(x^4-x^3)^(1/4)*x^2+6*(x^4-x^3)^(1/2)*RootOf(_Z^8-_Z^4+1)^5*x-10*(x^4-x^3)^(3/4
)*RootOf(_Z^8-_Z^4+1)^4-12*RootOf(_Z^8-_Z^4+1)^3*x^3+3*RootOf(_Z^8-_Z^4+1)^3*x^2-6*RootOf(_Z^8-_Z^4+1)^2*(x^4-
x^3)^(1/4)*x^2+10*(x^4-x^3)^(1/2)*RootOf(_Z^8-_Z^4+1)*x+16*(x^4-x^3)^(3/4))/(x*RootOf(_Z^8-_Z^4+1)^4-2*RootOf(
_Z^8-_Z^4+1)^4-2*x+1)/x^2)-7/4*ln((2*(x^4-x^3)^(3/4)-2*(x^4-x^3)^(1/2)*x+2*x^2*(x^4-x^3)^(1/4)-2*x^3+x^2)/x^2)
-7/4*RootOf(_Z^8-_Z^4+1)^6*ln(-(2*RootOf(_Z^8-_Z^4+1)^6*(x^4-x^3)^(1/2)*x-2*RootOf(_Z^8-_Z^4+1)^6*x^3+RootOf(_
Z^8-_Z^4+1)^6*x^2-2*(x^4-x^3)^(3/4)+2*x^2*(x^4-x^3)^(1/4))/x^2)

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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 + 1\right )}}{x^{2} - x + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^4-x^3\right )}^{1/4}\,\left (x+1\right )}{x^2-x+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )}{x^{2} - x + 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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