\(\int \frac {(1+x) (-1+x^3)^{2/3}}{(-1+x)^3 x} \, dx\) [5]

Optimal result

Integrand size = 21, antiderivative size = 147 \[ \int \frac {(1+x) \left (-1+x^3\right )^{2/3}}{(-1+x)^3 x} \, dx=\frac {-3 \left (-1+x^3\right )^{2/3}-2 \sqrt {3} \left (1-2 x+x^2\right ) \arctan \left (\frac {\sqrt {3} \left (1+x+x^2\right )+4 \sqrt {3} (-1+x) \sqrt [3]{-1+x^3}-2 \sqrt {3} \left (-1+x^3\right )^{2/3}}{3 \left (3-5 x+3 x^2\right )}\right )-\left (1-2 x+x^2\right ) \log \left (\frac {x+(-1+x) \sqrt [3]{-1+x^3}-\left (-1+x^3\right )^{2/3}}{x}\right )}{2 \left (1-2 x+x^2\right )} \]

[Out]

1/2*(-2*3^(1/2)*(x^2-2*x+1)*arctan(1/3*(4*3^(1/2)*(x^3-1)^(1/3)*(-1+x)+3^(1/2)*(x^2+x+1)-2*3^(1/2)*(x^3-1)^(2/
3))/(3*x^2-5*x+3))-(x^2-2*x+1)*ln(((-1+x)*(x^3-1)^(1/3)+x-(x^3-1)^(2/3))/x)-3*(x^3-1)^(2/3))/(x^2-2*x+1)

Rubi [C] (verified)

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

Time = 0.15 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.63, number of steps used = 23, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.762, Rules used = {2180, 198, 197, 272, 53, 58, 632, 210, 31, 372, 371, 267, 270, 45, 294, 245} \[ \int \frac {(1+x) \left (-1+x^3\right )^{2/3}}{(-1+x)^3 x} \, dx=\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {1-2 \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {9 \sqrt [3]{1-x^3} x^5 \operatorname {Hypergeometric2F1}\left (\frac {5}{3},\frac {7}{3},\frac {8}{3},x^3\right )}{5 \sqrt [3]{x^3-1}}+\frac {9 \sqrt [3]{1-x^3} x^2 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {7}{3},\frac {5}{3},x^3\right )}{2 \sqrt [3]{x^3-1}}+\frac {2 x}{\sqrt [3]{x^3-1}}-\frac {x}{\left (x^3-1\right )^{4/3}}-\frac {3}{\sqrt [3]{x^3-1}}-\frac {9}{2 \left (x^3-1\right )^{4/3}}-\frac {1}{2} \log \left (\sqrt [3]{x^3-1}+1\right )-\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {7 x^4}{2 \left (x^3-1\right )^{4/3}}+\frac {\log (x)}{2} \]

[In]

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

[Out]

-9/(2*(-1 + x^3)^(4/3)) - x/(-1 + x^3)^(4/3) - (7*x^4)/(2*(-1 + x^3)^(4/3)) - 3/(-1 + x^3)^(1/3) + (2*x)/(-1 +
x^3)^(1/3) + 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] + (9*x^2*(1 - x^3)^(1/3)*Hypergeometric2F1[2/3, 7/3, 5/3, x^3])/(2*(-1 + x^3)^(1/3)) + (9*x^5*(1 - x^3
)^(1/3)*Hypergeometric2F1[5/3, 7/3, 8/3, x^3])/(5*(-1 + x^3)^(1/3)) + Log[x]/2 - Log[1 + (-1 + x^3)^(1/3)]/2 -
Log[-x + (-1 + x^3)^(1/3)]/2

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 53

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] - Dist[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] && NeQ[b*c - a*d, 0] && 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 58

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[-(b*c - a*d)/b, 3]}, Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), 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 197

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

Rule 198

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

Rule 210

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 245

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

Rule 267

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 270

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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[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 294

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] - Dist[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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 372

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[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 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2180

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

Rubi steps \begin{align*} \text {integral}= \int \left (\frac {4}{\left (-1+x^3\right )^{7/3}}+\frac {1}{x \left (-1+x^3\right )^{7/3}}+\frac {9 x}{\left (-1+x^3\right )^{7/3}}+\frac {13 x^2}{\left (-1+x^3\right )^{7/3}}+\frac {13 x^3}{\left (-1+x^3\right )^{7/3}}+\frac {9 x^4}{\left (-1+x^3\right )^{7/3}}+\frac {4 x^5}{\left (-1+x^3\right )^{7/3}}+\frac {x^6}{\left (-1+x^3\right )^{7/3}}\right ) \, dx \\ = 4 \int \frac {1}{\left (-1+x^3\right )^{7/3}} \, dx+4 \int \frac {x^5}{\left (-1+x^3\right )^{7/3}} \, dx+9 \int \frac {x}{\left (-1+x^3\right )^{7/3}} \, dx+9 \int \frac {x^4}{\left (-1+x^3\right )^{7/3}} \, dx+13 \int \frac {x^2}{\left (-1+x^3\right )^{7/3}} \, dx+13 \int \frac {x^3}{\left (-1+x^3\right )^{7/3}} \, dx+\int \frac {1}{x \left (-1+x^3\right )^{7/3}} \, dx+\int \frac {x^6}{\left (-1+x^3\right )^{7/3}} \, dx \\ = -\frac {13}{4 \left (-1+x^3\right )^{4/3}}-\frac {x}{\left (-1+x^3\right )^{4/3}}-\frac {7 x^4}{2 \left (-1+x^3\right )^{4/3}}+\frac {1}{3} \text {Subst}\left (\int \frac {1}{(-1+x)^{7/3} x} \, dx,x,x^3\right )+\frac {4}{3} \text {Subst}\left (\int \frac {x}{(-1+x)^{7/3}} \, dx,x,x^3\right )-3 \int \frac {1}{\left (-1+x^3\right )^{4/3}} \, dx+\frac {\left (9 \sqrt [3]{1-x^3}\right ) \int \frac {x}{\left (1-x^3\right )^{7/3}} \, dx}{\sqrt [3]{-1+x^3}}+\frac {\left (9 \sqrt [3]{1-x^3}\right ) \int \frac {x^4}{\left (1-x^3\right )^{7/3}} \, dx}{\sqrt [3]{-1+x^3}}+\int \frac {x^3}{\left (-1+x^3\right )^{4/3}} \, dx \\ = -\frac {7}{2 \left (-1+x^3\right )^{4/3}}-\frac {x}{\left (-1+x^3\right )^{4/3}}-\frac {7 x^4}{2 \left (-1+x^3\right )^{4/3}}+\frac {2 x}{\sqrt [3]{-1+x^3}}+\frac {9 x^2 \sqrt [3]{1-x^3} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {7}{3},\frac {5}{3},x^3\right )}{2 \sqrt [3]{-1+x^3}}+\frac {9 x^5 \sqrt [3]{1-x^3} \operatorname {Hypergeometric2F1}\left (\frac {5}{3},\frac {7}{3},\frac {8}{3},x^3\right )}{5 \sqrt [3]{-1+x^3}}-\frac {1}{3} \text {Subst}\left (\int \frac {1}{(-1+x)^{4/3} x} \, dx,x,x^3\right )+\frac {4}{3} \text {Subst}\left (\int \left (\frac {1}{(-1+x)^{7/3}}+\frac {1}{(-1+x)^{4/3}}\right ) \, dx,x,x^3\right )+\int \frac {1}{\sqrt [3]{-1+x^3}} \, dx \\ = -\frac {9}{2 \left (-1+x^3\right )^{4/3}}-\frac {x}{\left (-1+x^3\right )^{4/3}}-\frac {7 x^4}{2 \left (-1+x^3\right )^{4/3}}-\frac {3}{\sqrt [3]{-1+x^3}}+\frac {2 x}{\sqrt [3]{-1+x^3}}+\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {9 x^2 \sqrt [3]{1-x^3} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {7}{3},\frac {5}{3},x^3\right )}{2 \sqrt [3]{-1+x^3}}+\frac {9 x^5 \sqrt [3]{1-x^3} \operatorname {Hypergeometric2F1}\left (\frac {5}{3},\frac {7}{3},\frac {8}{3},x^3\right )}{5 \sqrt [3]{-1+x^3}}-\frac {1}{2} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x} x} \, dx,x,x^3\right ) \\ = -\frac {9}{2 \left (-1+x^3\right )^{4/3}}-\frac {x}{\left (-1+x^3\right )^{4/3}}-\frac {7 x^4}{2 \left (-1+x^3\right )^{4/3}}-\frac {3}{\sqrt [3]{-1+x^3}}+\frac {2 x}{\sqrt [3]{-1+x^3}}+\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {9 x^2 \sqrt [3]{1-x^3} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {7}{3},\frac {5}{3},x^3\right )}{2 \sqrt [3]{-1+x^3}}+\frac {9 x^5 \sqrt [3]{1-x^3} \operatorname {Hypergeometric2F1}\left (\frac {5}{3},\frac {7}{3},\frac {8}{3},x^3\right )}{5 \sqrt [3]{-1+x^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 {9}{2 \left (-1+x^3\right )^{4/3}}-\frac {x}{\left (-1+x^3\right )^{4/3}}-\frac {7 x^4}{2 \left (-1+x^3\right )^{4/3}}-\frac {3}{\sqrt [3]{-1+x^3}}+\frac {2 x}{\sqrt [3]{-1+x^3}}+\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {9 x^2 \sqrt [3]{1-x^3} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {7}{3},\frac {5}{3},x^3\right )}{2 \sqrt [3]{-1+x^3}}+\frac {9 x^5 \sqrt [3]{1-x^3} \operatorname {Hypergeometric2F1}\left (\frac {5}{3},\frac {7}{3},\frac {8}{3},x^3\right )}{5 \sqrt [3]{-1+x^3}}+\frac {\log (x)}{2}-\frac {1}{2} \log \left (1+\sqrt [3]{-1+x^3}\right )-\frac {1}{2} \log \left (-x+\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 {9}{2 \left (-1+x^3\right )^{4/3}}-\frac {x}{\left (-1+x^3\right )^{4/3}}-\frac {7 x^4}{2 \left (-1+x^3\right )^{4/3}}-\frac {3}{\sqrt [3]{-1+x^3}}+\frac {2 x}{\sqrt [3]{-1+x^3}}+\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\arctan \left (\frac {-1+2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {9 x^2 \sqrt [3]{1-x^3} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {7}{3},\frac {5}{3},x^3\right )}{2 \sqrt [3]{-1+x^3}}+\frac {9 x^5 \sqrt [3]{1-x^3} \operatorname {Hypergeometric2F1}\left (\frac {5}{3},\frac {7}{3},\frac {8}{3},x^3\right )}{5 \sqrt [3]{-1+x^3}}+\frac {\log (x)}{2}-\frac {1}{2} \log \left (1+\sqrt [3]{-1+x^3}\right )-\frac {1}{2} \log \left (-x+\sqrt [3]{-1+x^3}\right ) \\ \end{align*}

Mathematica [C] (verified)

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

Time = 10.16 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.84 \[ \int \frac {(1+x) \left (-1+x^3\right )^{2/3}}{(-1+x)^3 x} \, dx=-\frac {3 \left (-1+x^3\right )^{2/3}}{2 (-1+x)^2}+\frac {x \sqrt [3]{1-x^3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{3},\frac {4}{3},x^3\right )}{\sqrt [3]{-1+x^3}}+\frac {1}{6} \left (-2 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )-2 \log \left (1+\sqrt [3]{-1+x^3}\right )+\log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \]

[In]

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

[Out]

(-3*(-1 + x^3)^(2/3))/(2*(-1 + x)^2) + (x*(1 - x^3)^(1/3)*Hypergeometric2F1[1/3, 1/3, 4/3, x^3])/(-1 + x^3)^(1
/3) + (-2*Sqrt[3]*ArcTan[(1 - 2*(-1 + x^3)^(1/3))/Sqrt[3]] - 2*Log[1 + (-1 + x^3)^(1/3)] + Log[1 - (-1 + x^3)^
(1/3) + (-1 + x^3)^(2/3)])/6

Maple [C] (warning: unable to verify)

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

Time = 0.66 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.90

method	result	size
risch	\(-\frac {3 \left (x^{2}+x +1\right )}{2 \left (-1+x \right ) \left (x^{3}-1\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}} \left (\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \pi \sqrt {3}\, x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], x^{3}\right )}{9 \Gamma \left (\frac {2}{3}\right )}\right )}{6 \pi \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}}}+\frac {{\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}} x \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{3}\right )}{\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}}}\)	\(133\)
trager	\(-\frac {3 \left (x^{3}-1\right )^{\frac {2}{3}}}{2 \left (-1+x \right )^{2}}+\ln \left (-\frac {58 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-145 \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}}-99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x -157 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+58 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}+117 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -28 \left (x^{3}-1\right )^{\frac {2}{3}}-28 \left (x^{3}-1\right )^{\frac {1}{3}} x +30 x^{2}-157 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+28 \left (x^{3}-1\right )^{\frac {1}{3}}-18 x +30}{x}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {58 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-145 \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}}+99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x +41 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+58 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}+173 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -127 \left (x^{3}-1\right )^{\frac {2}{3}}-127 \left (x^{3}-1\right )^{\frac {1}{3}} x -69 x^{2}+41 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+127 \left (x^{3}-1\right )^{\frac {1}{3}}-46 x -69}{x}\right )+\ln \left (-\frac {58 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-145 \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}}+99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x +41 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+58 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}+173 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -127 \left (x^{3}-1\right )^{\frac {2}{3}}-127 \left (x^{3}-1\right )^{\frac {1}{3}} x -69 x^{2}+41 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+127 \left (x^{3}-1\right )^{\frac {1}{3}}-46 x -69}{x}\right )\)	\(574\)

[In]

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

[Out]

-3/2*(x^2+x+1)/(-1+x)/(x^3-1)^(1/3)+1/6/Pi*3^(1/2)*GAMMA(2/3)/signum(x^3-1)^(1/3)*(-signum(x^3-1))^(1/3)*(2/3*
(-1/6*Pi*3^(1/2)-3/2*ln(3)+3*ln(x)+I*Pi)*Pi*3^(1/2)/GAMMA(2/3)+2/9*Pi*3^(1/2)/GAMMA(2/3)*x^3*hypergeom([1,1,4/
3],[2,2],x^3))+1/signum(x^3-1)^(1/3)*(-signum(x^3-1))^(1/3)*x*hypergeom([1/3,1/3],[4/3],x^3)

Fricas [B] (verification not implemented)

Error detected during grading. Assigning place holder grade for now.

Time = 0.36 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.85 \[ \int \frac {(1+x) \left (-1+x^3\right )^{2/3}}{(-1+x)^3 x} \, dx=\frac {2 \, \sqrt {3} {\left (x^{2} - 2 \, x + 1\right )} \arctan \left (-\frac {4 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} + \sqrt {3} {\left (x^{2} + x + 1\right )} - 2 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{3 \, {\left (3 \, x^{2} - 5 \, x + 3\right )}}\right ) - {\left (x^{2} - 2 \, x + 1\right )} \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} + x - {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x}\right ) - 3 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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