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

Optimal result

Integrand size = 21, antiderivative size = 107 \[ \int \frac {(1+x) \sqrt [3]{-1+x^3}}{(-1+x)^2 x} \, dx=-\frac {3 \sqrt [3]{-1+x^3}}{-1+x}+\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-1+x^3}}{-2+2 x+\sqrt [3]{-1+x^3}}\right )-\log \left (1-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{2} \log \left (1-2 x+x^2+(-1+x) \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \]

[Out] -3*(x^3-1)^(1/3)/(-1+x)+3^(1/2)*arctan(3^(1/2)*(x^3-1)^(1/3)/(-2+2*x+( 
x^3-1)^(1/3)))-ln(1-x+(x^3-1)^(1/3))+1/2*ln(1-2*x+x^2+(-1+x)*(x^3-1)^( 
1/3)+(x^3-1)^(2/3))
 

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.18 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.83, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {2582, 779, 778, 798, 61, 70, 1083, 217, 16, 796, 793, 889, 888, 817, 853} \[ \int \frac {(1+x) \sqrt [3]{-1+x^3}}{(-1+x)^2 x} \, dx=-\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\arctan \left (\frac {1-2 \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {3 \left (1-x^3\right )^{2/3} x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {5}{3},\frac {4}{3},x^3\right )}{\left (x^3-1\right )^{2/3}}-\frac {3 \left (1-x^3\right )^{2/3} x^4 \operatorname {Hypergeometric2F1}\left (\frac {4}{3},\frac {5}{3},\frac {7}{3},x^3\right )}{4 \left (x^3-1\right )^{2/3}}-\frac {3}{\left (x^3-1\right )^{2/3}}-\frac {1}{2} \log \left (x-\sqrt [3]{x^3-1}\right )-\frac {1}{2} \log \left (\sqrt [3]{x^3-1}+1\right )-\frac {3 x^2}{\left (x^3-1\right )^{2/3}}+\frac {\log (x)}{2} \]

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

[Out] -3/(-1 + x^3)^(2/3) - (3*x^2)/(-1 + x^3)^(2/3) - ArcTan[(1 + (2*x)/(-1 
 + x^3)^(1/3))/Sqrt[3]]/Sqrt[3] + ArcTan[(1 - 2*(-1 + x^3)^(1/3))/Sqrt 
[3]]/Sqrt[3] - (3*x*(1 - x^3)^(2/3)*Hypergeometric2F1[1/3, 5/3, 4/3, x 
^3])/(-1 + x^3)^(2/3) - (3*x^4*(1 - x^3)^(2/3)*Hypergeometric2F1[4/3, 
5/3, 7/3, x^3])/(4*(-1 + x^3)^(2/3)) + Log[x]/2 - Log[x - (-1 + x^3)^( 
1/3)]/2 - Log[1 + (-1 + x^3)^(1/3)]/2
 

Rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

Rule 61

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0 
] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d 
, m, n, x]
 

Rule 70

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[ 
{q = Rt[-(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2) 
, x] + (Simp[3/(2*b*q)   Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1 
/3)], x] + Simp[3/(2*b*q^2)   Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], 
 x])] /; FreeQ[{a, b, c, d}, x] && NegQ[(b*c - a*d)/b]
 

Rule 217

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

Rule 778

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F 
1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p 
, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p] || 
GtQ[a, 0])
 

Rule 779

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x 
^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p])   Int[(1 + b*(x^n/a))^p, x], x 
] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Si 
mplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])
 

Rule 793

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n) 
^(p + 1)/(b*n*(p + 1)), x] /; FreeQ[{a, b, m, n, p}, x] && EqQ[m, n - 1] && 
 NeQ[p, -1]
 

Rule 796

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* 
x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, 
 p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]
 

Rule 798

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n   Subst 
[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, 
b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
 

Rule 817

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^( 
n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[c^n 
*((m - n + 1)/(b*n*(p + 1)))   Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x 
] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  ! 
ILtQ[(m + n*(p + 1) + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]
                                                                                        
                                                                                        
 

Rule 853

Int[(x_)/((a_) + (b_.)*(x_)^3)^(2/3), x_Symbol] :> With[{q = Rt[b, 3]}, Sim 
p[-ArcTan[(1 + 2*q*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*q^2), x] - Simp 
[Log[q*x - (a + b*x^3)^(1/3)]/(2*q^2), x]] /; FreeQ[{a, b}, x]
 

Rule 888

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p 
*((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 
, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] && (ILt 
Q[p, 0] || GtQ[a, 0])
 

Rule 889

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^I 
ntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p])   Int[(c*x) 
^m*(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0 
] &&  !(ILtQ[p, 0] || GtQ[a, 0])
 

Rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

Rule 2582

Int[(Px_)*(x_)^(m_.)*((c_) + (d_.)*(x_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)* 
(x_)^3)^(p_.), x_Symbol] :> Simp[1/c^q   Int[ExpandIntegrand[(c^3 - d^3*x^3 
)^q*(a + b*x^3)^p, x^m*(Px/(c - d*x)^q), x], x], x] /; FreeQ[{a, b, c, d, e 
, m, p}, x] && PolyQ[Px, x] && EqQ[d^2 - c*e, 0] && ILtQ[q, 0] && IntegerQ[ 
m] && RationalQ[p] && EqQ[Denominator[p], 3]
 

Rubi steps \begin{align*} \text {integral}= \int \left (\frac {3}{\left (-1+x^3\right )^{5/3}}+\frac {1}{x \left (-1+x^3\right )^{5/3}}+\frac {5 x}{\left (-1+x^3\right )^{5/3}}+\frac {5 x^2}{\left (-1+x^3\right )^{5/3}}+\frac {3 x^3}{\left (-1+x^3\right )^{5/3}}+\frac {x^4}{\left (-1+x^3\right )^{5/3}}\right ) \, dx \\ = 3 \int \frac {1}{\left (-1+x^3\right )^{5/3}} \, dx+3 \int \frac {x^3}{\left (-1+x^3\right )^{5/3}} \, dx+5 \int \frac {x}{\left (-1+x^3\right )^{5/3}} \, dx+5 \int \frac {x^2}{\left (-1+x^3\right )^{5/3}} \, dx+\int \frac {1}{x \left (-1+x^3\right )^{5/3}} \, dx+\int \frac {x^4}{\left (-1+x^3\right )^{5/3}} \, dx \\ = -\frac {5}{2 \left (-1+x^3\right )^{2/3}}-\frac {3 x^2}{\left (-1+x^3\right )^{2/3}}+\frac {1}{3} \text {Subst}\left (\int \frac {1}{(-1+x)^{5/3} x} \, dx,x,x^3\right )-\frac {3 \left (1-x^3\right )^{2/3}}{\left (-1+x^3\right )^{2/3}} \int \frac {1}{\left (1-x^3\right )^{5/3}} \, dx-\frac {3 \left (1-x^3\right )^{2/3}}{\left (-1+x^3\right )^{2/3}} \int \frac {x^3}{\left (1-x^3\right )^{5/3}} \, dx+\int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx \\ = -\frac {3}{\left (-1+x^3\right )^{2/3}}-\frac {3 x^2}{\left (-1+x^3\right )^{2/3}}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {3 x \left (1-x^3\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {5}{3};\frac {4}{3};x^3\right )}{\left (-1+x^3\right )^{2/3}}-\frac {3 x^4 \left (1-x^3\right )^{2/3} \, _2F_1\left (\frac {4}{3},\frac {5}{3};\frac {7}{3};x^3\right )}{4 \left (-1+x^3\right )^{2/3}}-\frac {1}{2} \log \left (x-\sqrt [3]{-1+x^3}\right )-\frac {1}{3} \text {Subst}\left (\int \frac {1}{(-1+x)^{2/3} x} \, dx,x,x^3\right ) \\ = -\frac {3}{\left (-1+x^3\right )^{2/3}}-\frac {3 x^2}{\left (-1+x^3\right )^{2/3}}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {3 x \left (1-x^3\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {5}{3};\frac {4}{3};x^3\right )}{\left (-1+x^3\right )^{2/3}}-\frac {3 x^4 \left (1-x^3\right )^{2/3} \, _2F_1\left (\frac {4}{3},\frac {5}{3};\frac {7}{3};x^3\right )}{4 \left (-1+x^3\right )^{2/3}}+\frac {\log (x)}{2}-\frac {1}{2} \log \left (x-\sqrt [3]{-1+x^3}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+x^3}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x^3}\right ) \\ = -\frac {3}{\left (-1+x^3\right )^{2/3}}-\frac {3 x^2}{\left (-1+x^3\right )^{2/3}}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {3 x \left (1-x^3\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {5}{3};\frac {4}{3};x^3\right )}{\left (-1+x^3\right )^{2/3}}-\frac {3 x^4 \left (1-x^3\right )^{2/3} \, _2F_1\left (\frac {4}{3},\frac {5}{3};\frac {7}{3};x^3\right )}{4 \left (-1+x^3\right )^{2/3}}+\frac {\log (x)}{2}-\frac {1}{2} \log \left (x-\sqrt [3]{-1+x^3}\right )-\frac {1}{2} \log \left (1+\sqrt [3]{-1+x^3}\right )+\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x^3}\right ) \\ = -\frac {3}{\left (-1+x^3\right )^{2/3}}-\frac {3 x^2}{\left (-1+x^3\right )^{2/3}}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\tan ^{-1}\left (\frac {1-2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {3 x \left (1-x^3\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {5}{3};\frac {4}{3};x^3\right )}{\left (-1+x^3\right )^{2/3}}-\frac {3 x^4 \left (1-x^3\right )^{2/3} \, _2F_1\left (\frac {4}{3},\frac {5}{3};\frac {7}{3};x^3\right )}{4 \left (-1+x^3\right )^{2/3}}+\frac {\log (x)}{2}-\frac {1}{2} \log \left (x-\sqrt [3]{-1+x^3}\right )-\frac {1}{2} \log \left (1+\sqrt [3]{-1+x^3}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 1.66 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00 \[ \int \frac {(1+x) \sqrt [3]{-1+x^3}}{(-1+x)^2 x} \, dx=-\frac {3 \sqrt [3]{-1+x^3}}{-1+x}+\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-1+x^3}}{-2+2 x+\sqrt [3]{-1+x^3}}\right )-\log \left (1-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{2} \log \left (1-2 x+x^2+(-1+x) \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \]

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

[Out] (-3*(-1 + x^3)^(1/3))/(-1 + x) + Sqrt[3]*ArcTan[(Sqrt[3]*(-1 + x^3)^(1 
/3))/(-2 + 2*x + (-1 + x^3)^(1/3))] - Log[1 - x + (-1 + x^3)^(1/3)] + 
Log[1 - 2*x + x^2 + (-1 + x)*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)]/2
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.79 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.53

method	result	size
risch	\(-\frac {3 \left (x^{2}+x +1\right )}{\left (x^{3}-1\right )^{\frac {2}{3}}}+\frac {\left (\frac {\left (x^{3}-1\right )^{\frac {2}{3}} {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} x^{2} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{3}\right )}{2 {\left (\left (x^{3}-1\right )^{2}\right )}^{\frac {1}{3}} \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}-\frac {\left (x^{3}-1\right )^{\frac {2}{3}} {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} \left (\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )+\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], x^{3}\right )}{3}\right )}{3 {\left (\left (x^{3}-1\right )^{2}\right )}^{\frac {1}{3}} \Gamma \left (\frac {2}{3}\right ) \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}\right ) {\left (\left (x^{3}-1\right )^{2}\right )}^{\frac {1}{3}}}{\left (x^{3}-1\right )^{\frac {2}{3}}}\)	\(164\)
trager	\(-\frac {3 \left (x^{3}-1\right )^{\frac {1}{3}}}{-1+x}-\ln \left (\frac {12 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-30 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x -99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+127 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x -40 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+12 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-127 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}+157 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +127 \left (x^{3}-1\right )^{\frac {2}{3}}-28 \left (x^{3}-1\right )^{\frac {1}{3}} x -87 x^{2}-40 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+28 \left (x^{3}-1\right )^{\frac {1}{3}}-58 x -87}{x}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {46 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-115 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+28 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x -173 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+46 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-28 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}+214 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +28 \left (x^{3}-1\right )^{\frac {2}{3}}-127 \left (x^{3}-1\right )^{\frac {1}{3}} x +145 x^{2}-173 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+127 \left (x^{3}-1\right )^{\frac {1}{3}}-87 x +145}{x}\right )\)	\(384\)

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

[Out] -3*(x^2+x+1)/(x^3-1)^(2/3)+(1/2/((x^3-1)^2)^(1/3)*(x^3-1)^(2/3)/signum 
(x^3-1)^(2/3)*(-signum(x^3-1))^(2/3)*x^2*hypergeom([2/3,2/3],[5/3],x^3 
)-1/3/((x^3-1)^2)^(1/3)*(x^3-1)^(2/3)/GAMMA(2/3)/signum(x^3-1)^(2/3)*( 
-signum(x^3-1))^(2/3)*((1/6*Pi*3^(1/2)-3/2*ln(3)+3*ln(x)+I*Pi)*GAMMA(2 
/3)+2/3*GAMMA(2/3)*x^3*hypergeom([1,1,5/3],[2,2],x^3)))/(x^3-1)^(2/3)* 
((x^3-1)^2)^(1/3)
 

Fricas [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.02 \[ \int \frac {(1+x) \sqrt [3]{-1+x^3}}{(-1+x)^2 x} \, dx=-\frac {2 \, \sqrt {3} {\left (x - 1\right )} \arctan \left (-\frac {4 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} + \sqrt {3} {\left (x^{2} + x + 1\right )} - 2 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{3 \, {\left (3 \, x^{2} - 5 \, x + 3\right )}}\right ) + {\left (x - 1\right )} \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} + x - {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x}\right ) + 6 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{2 \, {\left (x - 1\right )}} \]

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

[Out] -1/2*(2*sqrt(3)*(x - 1)*arctan(-1/3*(4*sqrt(3)*(x^3 - 1)^(1/3)*(x - 1) 
 + sqrt(3)*(x^2 + x + 1) - 2*sqrt(3)*(x^3 - 1)^(2/3))/(3*x^2 - 5*x + 3 
)) + (x - 1)*log(((x^3 - 1)^(1/3)*(x - 1) + x - (x^3 - 1)^(2/3))/x) + 
6*(x^3 - 1)^(1/3))/(x - 1)
 

Sympy [F]

\[ \int \frac {(1+x) \sqrt [3]{-1+x^3}}{(-1+x)^2 x} \, dx=\int \frac {\sqrt [3]{\left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 1\right )}{x \left (x - 1\right )^{2}}\, dx \]

[In] integrate((1+x)*(x**3-1)**(1/3)/(-1+x)**2/x,x)
 

[Out] Integral(((x - 1)*(x**2 + x + 1))**(1/3)*(x + 1)/(x*(x - 1)**2), x)
 

Maxima [F]

\[ \int \frac {(1+x) \sqrt [3]{-1+x^3}}{(-1+x)^2 x} \, dx=\int { \frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}} {\left (x + 1\right )}}{{\left (x - 1\right )}^{2} x} \,d x } \]

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

[Out] integrate((x^3 - 1)^(1/3)*(x + 1)/((x - 1)^2*x), x)
 

Giac [F]

\[ \int \frac {(1+x) \sqrt [3]{-1+x^3}}{(-1+x)^2 x} \, dx=\int { \frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}} {\left (x + 1\right )}}{{\left (x - 1\right )}^{2} x} \,d x } \]

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

[Out] integrate((x^3 - 1)^(1/3)*(x + 1)/((x - 1)^2*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(1+x) \sqrt [3]{-1+x^3}}{(-1+x)^2 x} \, dx=\int \frac {{\left (x^3-1\right )}^{1/3}\,\left (x+1\right )}{x\,{\left (x-1\right )}^2} \,d x \]

[In] int(((x^3 - 1)^(1/3)*(x + 1))/(x*(x - 1)^2),x)
 

[Out] int(((x^3 - 1)^(1/3)*(x + 1))/(x*(x - 1)^2), x)