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

Optimal. Leaf size=20 \[ \frac {3 \left (x^3-x^2\right )^{2/3}}{2 x^2} \]

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Rubi [A]  time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2014} \begin {gather*} \frac {3 \left (x^3-x^2\right )^{2/3}}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*(-x^2 + x^3)^(1/3)),x]

[Out]

(3*(-x^2 + x^3)^(2/3))/(2*x^2)

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])

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 18, normalized size = 0.90 \begin {gather*} \frac {3 (x-1)}{2 \sqrt [3]{(x-1) x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(-x^2 + x^3)^(1/3)),x]

[Out]

(3*(-1 + x))/(2*((-1 + x)*x^2)^(1/3))

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IntegrateAlgebraic [A]  time = 0.13, size = 20, normalized size = 1.00 \begin {gather*} \frac {3 \left (-x^2+x^3\right )^{2/3}}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(-x^2 + x^3)^(2/3))/(2*x^2)

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fricas [A]  time = 0.43, size = 16, normalized size = 0.80 \begin {gather*} \frac {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^3-x^2)^(1/3),x, algorithm="fricas")

[Out]

3/2*(x^3 - x^2)^(2/3)/x^2

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giac [A]  time = 0.28, size = 11, normalized size = 0.55 \begin {gather*} \frac {3}{2} \, {\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*(-1/x + 1)^(2/3)

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maple [A]  time = 0.07, size = 15, normalized size = 0.75

method	result	size
risch	\(\frac {-\frac {3}{2}+\frac {3 x}{2}}{\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}\)	\(15\)
gosper	\(\frac {-\frac {3}{2}+\frac {3 x}{2}}{\left (x^{3}-x^{2}\right )^{\frac {1}{3}}}\)	\(17\)
trager	\(\frac {3 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}}{2 x^{2}}\)	\(17\)
meijerg	\(-\frac {3 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {1}{3}} \left (1-x \right )^{\frac {2}{3}}}{2 \mathrm {signum}\left (-1+x \right )^{\frac {1}{3}} x^{\frac {2}{3}}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {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^3-x^2)^(1/3),x, algorithm="maxima")

[Out]

integrate(1/((x^3 - x^2)^(1/3)*x), x)

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mupad [B]  time = 0.20, size = 16, normalized size = 0.80 \begin {gather*} \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(1/(x*(x^3 - x^2)^(1/3)),x)

[Out]

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

[Out]

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

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