\(\int \frac {x}{(-1+x) \sqrt [3]{-x^2+x^3}} \, dx\) [1696]

Optimal result

Integrand size = 20, antiderivative size = 114 \[ \int \frac {x}{(-1+x) \sqrt [3]{-x^2+x^3}} \, dx=-\frac {3 \left (-x^2+x^3\right )^{2/3}}{(-1+x) x}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x^2+x^3}}\right )-\log \left (-x+\sqrt [3]{-x^2+x^3}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{-x^2+x^3}+\left (-x^2+x^3\right )^{2/3}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.32, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2081, 49, 61} \[ \int \frac {x}{(-1+x) \sqrt [3]{-x^2+x^3}} \, dx=-\frac {\sqrt {3} \sqrt [3]{x-1} x^{2/3} \arctan \left (\frac {2 \sqrt [3]{x-1}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{x^3-x^2}}-\frac {3 x}{\sqrt [3]{x^3-x^2}}-\frac {3 \sqrt [3]{x-1} x^{2/3} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{x}}-1\right )}{2 \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log (x)}{2 \sqrt [3]{x^3-x^2}} \]

[In]

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

[Out]

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

Rule 49

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

Rule 61

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[d/b, 3]}, Simp[(-Sqrt
[3])*(q/d)*ArcTan[2*q*((a + b*x)^(1/3)/(Sqrt[3]*(c + d*x)^(1/3))) + 1/Sqrt[3]], x] + (-Simp[3*(q/(2*d))*Log[q*
((a + b*x)^(1/3)/(c + d*x)^(1/3)) - 1], x] - Simp[(q/(2*d))*Log[c + d*x], x])] /; FreeQ[{a, b, c, d}, x] && Ne
Q[b*c - a*d, 0] && PosQ[d/b]

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]

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.15 \[ \int \frac {x}{(-1+x) \sqrt [3]{-x^2+x^3}} \, dx=\frac {x^{2/3} \left (-6 \sqrt [3]{x}+2 \sqrt {3} \sqrt [3]{-1+x} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{2 \sqrt [3]{-1+x}+\sqrt [3]{x}}\right )-2 \sqrt [3]{-1+x} \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x}\right )+\sqrt [3]{-1+x} \log \left ((-1+x)^{2/3}+\sqrt [3]{-1+x} \sqrt [3]{x}+x^{2/3}\right )\right )}{2 \sqrt [3]{(-1+x) x^2}} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 0.38 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.24

method	result	size
meijerg	\(-\frac {3 \left (-\operatorname {signum}\left (-1+x \right )\right )^{\frac {1}{3}} x^{\frac {4}{3}} \operatorname {hypergeom}\left (\left [\frac {4}{3}, \frac {4}{3}\right ], \left [\frac {7}{3}\right ], x\right )}{4 \operatorname {signum}\left (-1+x \right )^{\frac {1}{3}}}\)	\(27\)
risch	\(-\frac {3 x}{\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}+\frac {3 \left (-\operatorname {signum}\left (-1+x \right )\right )^{\frac {1}{3}} x^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x\right )}{\operatorname {signum}\left (-1+x \right )^{\frac {1}{3}}}\)	\(40\)
pseudoelliptic	\(-\frac {\left (\sqrt {3}\, \arctan \left (\frac {\left (2 \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )+\ln \left (\frac {\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}-x}{x}\right )-\frac {\ln \left (\frac {\left (\left (-1+x \right ) x^{2}\right )^{\frac {2}{3}}+\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{2}\right ) \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+3 x}{\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}\)	\(102\)
trager	\(-\frac {3 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}}{\left (-1+x \right ) x}+\ln \left (-\frac {-45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )^{2} x^{2}+144 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+144 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x +90 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )^{2} x +174 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) x^{2}-60 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-60 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}-138 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) x -80 x^{2}+48 x}{x}\right )-\frac {3 \ln \left (-\frac {-45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )^{2} x^{2}+144 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+144 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x +90 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )^{2} x +174 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) x^{2}-60 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-60 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}-138 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) x -80 x^{2}+48 x}{x}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )}{2}+\frac {3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) \ln \left (\frac {45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )^{2} x^{2}+144 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+144 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x -90 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )^{2} x +114 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) x^{2}-36 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-36 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}-18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) x -16 x^{2}+4 x}{x}\right )}{2}\)	\(513\)

[In]

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

[Out]

-3/4/signum(-1+x)^(1/3)*(-signum(-1+x))^(1/3)*x^(4/3)*hypergeom([4/3,4/3],[7/3],x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.20 \[ \int \frac {x}{(-1+x) \sqrt [3]{-x^2+x^3}} \, dx=-\frac {2 \, \sqrt {3} {\left (x^{2} - x\right )} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) + 2 \, {\left (x^{2} - x\right )} \log \left (-\frac {x - {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - {\left (x^{2} - x\right )} \log \left (\frac {x^{2} + {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) + 6 \, {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{2 \, {\left (x^{2} - x\right )}} \]

[In]

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

[Out]

-1/2*(2*sqrt(3)*(x^2 - x)*arctan(1/3*(sqrt(3)*x + 2*sqrt(3)*(x^3 - x^2)^(1/3))/x) + 2*(x^2 - x)*log(-(x - (x^3
- x^2)^(1/3))/x) - (x^2 - x)*log((x^2 + (x^3 - x^2)^(1/3)*x + (x^3 - x^2)^(2/3))/x^2) + 6*(x^3 - x^2)^(2/3))/
(x^2 - x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.65 \[ \int \frac {x}{(-1+x) \sqrt [3]{-x^2+x^3}} \, dx=-\sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {3}{{\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}}} + \frac {1}{2} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) - \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(x/((x^3 - x^2)^(1/3)*(x - 1)),x)

[Out]

int(x/((x^3 - x^2)^(1/3)*(x - 1)), x)