\(\int \frac {(1+x) \sqrt [4]{-x^3+x^4}}{1-x+x^2} \, dx\) [2051]

Optimal result

Integrand size = 27, antiderivative size = 147 \[ \int \frac {(1+x) \sqrt [4]{-x^3+x^4}}{1-x+x^2} \, dx=\sqrt [4]{-x^3+x^4}-\frac {7}{2} \arctan \left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )+\frac {7}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )-\text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 \log (x)+2 \log \left (\sqrt [4]{-x^3+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \]

[Out]

Unintegrable

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.55 (sec) , antiderivative size = 695, normalized size of antiderivative = 4.73, number of steps used = 25, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {2081, 6860, 103, 163, 65, 246, 218, 212, 209, 95, 304, 211, 214} \[ \int \frac {(1+x) \sqrt [4]{-x^3+x^4}}{1-x+x^2} \, dx=\frac {\left (7+i \sqrt {3}\right ) \sqrt [4]{x^4-x^3} \arctan \left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{x-1} x^{3/4}}+\frac {\left (7-i \sqrt {3}\right ) \sqrt [4]{x^4-x^3} \arctan \left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{x-1} x^{3/4}}+\frac {\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \left (\sqrt {3}+i\right ) \sqrt [4]{x^4-x^3} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}+\frac {\left (-\frac {-\sqrt {3}+i}{\sqrt {3}+i}\right )^{3/4} \left (\sqrt {3}+i\right ) \sqrt [4]{x^4-x^3} \arctan \left (\frac {\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}+\frac {\left (7+i \sqrt {3}\right ) \sqrt [4]{x^4-x^3} \text {arctanh}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{x-1} x^{3/4}}+\frac {\left (7-i \sqrt {3}\right ) \sqrt [4]{x^4-x^3} \text {arctanh}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{x-1} x^{3/4}}-\frac {\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \left (\sqrt {3}+i\right ) \sqrt [4]{x^4-x^3} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}-\frac {\left (-\frac {-\sqrt {3}+i}{\sqrt {3}+i}\right )^{3/4} \left (\sqrt {3}+i\right ) \sqrt [4]{x^4-x^3} \text {arctanh}\left (\frac {\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}+\frac {1}{2} \left (1+i \sqrt {3}\right ) \sqrt [4]{x^4-x^3}+\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt [4]{x^4-x^3} \]

[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 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 103

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 163

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 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 218

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] &&  !Gt
Q[a/b, 0]

Rule 246

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 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 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 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 {\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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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} \arctan \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} \arctan \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} \text {arctanh}\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} \text {arctanh}\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 ) \text {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 ) \text {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 ) \text {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 ) \text {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} \arctan \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} \arctan \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} \arctan \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} \arctan \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} \text {arctanh}\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} \text {arctanh}\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} \text {arctanh}\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} \text {arctanh}\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*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.08 \[ \int \frac {(1+x) \sqrt [4]{-x^3+x^4}}{1-x+x^2} \, dx=\frac {(-1+x)^{3/4} x^{9/4} \left (4 \sqrt [4]{-1+x} x^{3/4}-14 \arctan \left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )+14 \text {arctanh}\left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )-\text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 \log (x)+8 \log \left (\sqrt [4]{-1+x}-\sqrt [4]{x} \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-4 \log \left (\sqrt [4]{-1+x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ]\right )}{4 \left ((-1+x) x^3\right )^{3/4}} \]

[In]

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

[Out]

((-1 + x)^(3/4)*x^(9/4)*(4*(-1 + x)^(1/4)*x^(3/4) - 14*ArcTan[((-1 + x)/x)^(-1/4)] + 14*ArcTanh[((-1 + x)/x)^(
-1/4)] - RootSum[1 - #1^4 + #1^8 & , (-2*Log[x] + 8*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) & ]))/(4*((-1 + x)*x^3)^(3/4))

Maple [N/A] (verified)

Time = 7.82 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.79

method	result	size
pseudoelliptic	\(\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}+\frac {7 \ln \left (\frac {x +\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{4}-\frac {7 \ln \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}-x}{x}\right )}{4}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}-2\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (2 \textit {\_R}^{4}-1\right )}\right )+\frac {7 \arctan \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{2}\)	\(116\)
trager	\(\text {Expression too large to display}\)	\(1744\)
risch	\(\text {Expression too large to display}\)	\(3741\)

[In]

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

[Out]

(x^3*(-1+x))^(1/4)+7/4*ln((x+(x^3*(-1+x))^(1/4))/x)-7/4*ln(((x^3*(-1+x))^(1/4)-x)/x)+sum((_R^4-2)*ln((-_R*x+(x
^3*(-1+x))^(1/4))/x)/_R^3/(2*_R^4-1),_R=RootOf(_Z^8-_Z^4+1))+7/2*arctan((x^3*(-1+x))^(1/4)/x)

Fricas [C] (verification not implemented)

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

Time = 0.27 (sec) , antiderivative size = 485, normalized size of antiderivative = 3.30 \[ \int \frac {(1+x) \sqrt [4]{-x^3+x^4}}{1-x+x^2} \, dx=-\frac {1}{2} \, \sqrt {2} \sqrt {-\sqrt {2 \, \sqrt {-3} + 2}} \log \left (\frac {\sqrt {2} x \sqrt {-\sqrt {2 \, \sqrt {-3} + 2}} + 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \sqrt {2} \sqrt {-\sqrt {2 \, \sqrt {-3} + 2}} \log \left (-\frac {\sqrt {2} x \sqrt {-\sqrt {2 \, \sqrt {-3} + 2}} - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \sqrt {2} \sqrt {-\sqrt {-2 \, \sqrt {-3} + 2}} \log \left (\frac {\sqrt {2} x \sqrt {-\sqrt {-2 \, \sqrt {-3} + 2}} + 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \sqrt {2} \sqrt {-\sqrt {-2 \, \sqrt {-3} + 2}} \log \left (-\frac {\sqrt {2} x \sqrt {-\sqrt {-2 \, \sqrt {-3} + 2}} - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \sqrt {2} {\left (2 \, \sqrt {-3} + 2\right )}^{\frac {1}{4}} \log \left (\frac {\sqrt {2} x {\left (2 \, \sqrt {-3} + 2\right )}^{\frac {1}{4}} + 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \sqrt {2} {\left (2 \, \sqrt {-3} + 2\right )}^{\frac {1}{4}} \log \left (-\frac {\sqrt {2} x {\left (2 \, \sqrt {-3} + 2\right )}^{\frac {1}{4}} - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \sqrt {2} {\left (-2 \, \sqrt {-3} + 2\right )}^{\frac {1}{4}} \log \left (\frac {\sqrt {2} x {\left (-2 \, \sqrt {-3} + 2\right )}^{\frac {1}{4}} + 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \sqrt {2} {\left (-2 \, \sqrt {-3} + 2\right )}^{\frac {1}{4}} \log \left (-\frac {\sqrt {2} x {\left (-2 \, \sqrt {-3} + 2\right )}^{\frac {1}{4}} - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{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 ) \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.14 \[ \int \frac {(1+x) \sqrt [4]{-x^3+x^4}}{1-x+x^2} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x - 1\right )} \left (x + 1\right )}{x^{2} - x + 1}\, dx \]

[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)

Maxima [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.18 \[ \int \frac {(1+x) \sqrt [4]{-x^3+x^4}}{1-x+x^2} \, dx=\int { \frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (x + 1\right )}}{x^{2} - x + 1} \,d x } \]

[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)

Giac [C] (verification not implemented)

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

Time = 0.32 (sec) , antiderivative size = 383, normalized size of antiderivative = 2.61 \[ \int \frac {(1+x) \sqrt [4]{-x^3+x^4}}{1-x+x^2} \, dx=-\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 ) \]

[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))*arct
an((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))) - 1/4*(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) + sq
rt(-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))

Mupad [N/A]

Not integrable

Time = 5.66 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.18 \[ \int \frac {(1+x) \sqrt [4]{-x^3+x^4}}{1-x+x^2} \, dx=\int \frac {{\left (x^4-x^3\right )}^{1/4}\,\left (x+1\right )}{x^2-x+1} \,d x \]

[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)