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

Optimal. Leaf size=111 \[ -\frac {3 \left (x^3-x^2\right )^{2/3}}{2 x^2}-\log \left (\sqrt [3]{x^3-x^2}-x\right )+\frac {1}{2} \log \left (x^2+\sqrt [3]{x^3-x^2} x+\left (x^3-x^2\right )^{2/3}\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-x^2}+x}\right ) \]

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Rubi [A]  time = 0.11, antiderivative size = 155, normalized size of antiderivative = 1.40, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2052, 2011, 59, 2014} \begin {gather*} -\frac {3 \left (x^3-x^2\right )^{2/3}}{2 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}}-\frac {\sqrt {3} \sqrt [3]{x-1} x^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{x-1}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{x^3-x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(-x^2 + x^3)^(2/3))/(2*x^2) - (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 59

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

Rule 2011

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(a*x^j + b*x^n)^FracPart[p]/(x^(j*FracPart[p
])*(a + b*x^(n - j))^FracPart[p]), Int[x^(j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, j, n, p}, x] &&  !I
ntegerQ[p] && NeQ[n, j] && PosQ[n - j]

Rule 2014

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

Rule 2052

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(c*x)
^m*Pq*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) &&  !In
tegerQ[p] && NeQ[n, j]

Rubi steps

\begin {align*} \int \frac {-1+x}{x \sqrt [3]{-x^2+x^3}} \, dx &=\int \left (\frac {1}{\sqrt [3]{-x^2+x^3}}-\frac {1}{x \sqrt [3]{-x^2+x^3}}\right ) \, dx\\ &=\int \frac {1}{\sqrt [3]{-x^2+x^3}} \, dx-\int \frac {1}{x \sqrt [3]{-x^2+x^3}} \, dx\\ &=-\frac {3 \left (-x^2+x^3\right )^{2/3}}{2 x^2}+\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 \left (-x^2+x^3\right )^{2/3}}{2 x^2}-\frac {\sqrt {3} \sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\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*}

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Mathematica [C]  time = 0.02, size = 35, normalized size = 0.32 \begin {gather*} \frac {3 \left ((x-1) x^2\right )^{5/3} \, _2F_1\left (\frac {5}{3},\frac {5}{3};\frac {8}{3};1-x\right )}{5 x^{10/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*((-1 + x)*x^2)^(5/3)*Hypergeometric2F1[5/3, 5/3, 8/3, 1 - x])/(5*x^(10/3))

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IntegrateAlgebraic [A]  time = 0.24, size = 111, normalized size = 1.00 \begin {gather*} -\frac {3 \left (-x^2+x^3\right )^{2/3}}{2 x^2}+\sqrt {3} \tan ^{-1}\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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.52, size = 119, normalized size = 1.07 \begin {gather*} -\frac {2 \, \sqrt {3} x^{2} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) + 2 \, x^{2} \log \left (-\frac {x - {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - x^{2} \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 ) + 3 \, {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.21, size = 74, normalized size = 0.67 \begin {gather*} -\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}{2} \, {\left (-\frac {1}{x} + 1\right )}^{\frac {2}{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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maple [C]  time = 0.58, size = 42, normalized size = 0.38

method	result	size
risch	\(-\frac {3 \left (-1+x \right )}{2 \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}+\frac {3 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {1}{3}} x^{\frac {1}{3}} \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x\right )}{\mathrm {signum}\left (-1+x \right )^{\frac {1}{3}}}\)	\(42\)
meijerg	\(\frac {3 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {1}{3}} \left (1-x \right )^{\frac {2}{3}}}{2 \mathrm {signum}\left (-1+x \right )^{\frac {1}{3}} x^{\frac {2}{3}}}+\frac {3 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {1}{3}} x^{\frac {1}{3}} \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x\right )}{\mathrm {signum}\left (-1+x \right )^{\frac {1}{3}}}\)	\(54\)
trager	\(-\frac {3 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}}{2 x^{2}}+6 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \ln \left (\frac {180 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{2}-360 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x +144 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+144 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x +114 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}-18 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x -9 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-9 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}-4 x^{2}+x}{x}\right )-6 \ln \left (\frac {180 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{2}-360 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x -144 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-144 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x -174 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}+138 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x +15 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+15 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}+20 x^{2}-12 x}{x}\right ) \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+\ln \left (\frac {180 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{2}-360 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x -144 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-144 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x -174 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}+138 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x +15 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+15 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}+20 x^{2}-12 x}{x}\right )\)	\(504\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x - 1}{{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.17, size = 44, normalized size = 0.40 \begin {gather*} \frac {3\,x\,{\left (1-x\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {1}{3};\ \frac {4}{3};\ x\right )}{{\left (x^3-x^2\right )}^{1/3}}-\frac {3\,{\left (x^3-x^2\right )}^{2/3}}{2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - 1)/(x*(x^3 - x^2)^(1/3)),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x - 1}{x \sqrt [3]{x^{2} \left (x - 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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