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

Optimal. Leaf size=25 \[ \frac {3 (3 x+2) \left (x^3-x^2\right )^{2/3}}{10 x^3} \]

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Rubi [A]  time = 0.05, antiderivative size = 41, normalized size of antiderivative = 1.64, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2016, 2014} \begin {gather*} \frac {9 \left (x^3-x^2\right )^{2/3}}{10 x^2}+\frac {3 \left (x^3-x^2\right )^{2/3}}{5 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(-x^2 + x^3)^(1/3)),x]

[Out]

(3*(-x^2 + x^3)^(2/3))/(5*x^3) + (9*(-x^2 + x^3)^(2/3))/(10*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])

Rule 2016

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*(m + j*p + 1)), x] - Dist[(b*(m + n*p + n - j + 1))/(a*c^(n - j)*(m + j*p + 1)
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c,
0])

Rubi steps

\begin {align*} \int \frac {1}{x^2 \sqrt [3]{-x^2+x^3}} \, dx &=\frac {3 \left (-x^2+x^3\right )^{2/3}}{5 x^3}+\frac {3}{5} \int \frac {1}{x \sqrt [3]{-x^2+x^3}} \, dx\\ &=\frac {3 \left (-x^2+x^3\right )^{2/3}}{5 x^3}+\frac {9 \left (-x^2+x^3\right )^{2/3}}{10 x^2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 0.92 \begin {gather*} \frac {3 \left ((x-1) x^2\right )^{2/3} (3 x+2)}{10 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*((-1 + x)*x^2)^(2/3)*(2 + 3*x))/(10*x^3)

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IntegrateAlgebraic [A]  time = 0.11, size = 25, normalized size = 1.00 \begin {gather*} \frac {3 (2+3 x) \left (-x^2+x^3\right )^{2/3}}{10 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(2 + 3*x)*(-x^2 + x^3)^(2/3))/(10*x^3)

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fricas [A]  time = 0.47, size = 21, normalized size = 0.84 \begin {gather*} \frac {3 \, {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}} {\left (3 \, x + 2\right )}}{10 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/10*(x^3 - x^2)^(2/3)*(3*x + 2)/x^3

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giac [A]  time = 0.47, size = 23, normalized size = 0.92 \begin {gather*} -\frac {3}{5} \, {\left (-\frac {1}{x} + 1\right )}^{\frac {5}{3}} + \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^2/(x^3-x^2)^(1/3),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 22, normalized size = 0.88

method	result	size
trager	\(\frac {3 \left (2+3 x \right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}}{10 x^{3}}\)	\(22\)
gosper	\(\frac {3 \left (-1+x \right ) \left (2+3 x \right )}{10 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}}\)	\(25\)
risch	\(\frac {-\frac {3}{5}-\frac {3}{10} x +\frac {9}{10} x^{2}}{x \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}\)	\(25\)
meijerg	\(-\frac {3 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {1}{3}} \left (1+\frac {3 x}{2}\right ) \left (1-x \right )^{\frac {2}{3}}}{5 \mathrm {signum}\left (-1+x \right )^{\frac {1}{3}} x^{\frac {5}{3}}}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/10*(2+3*x)*(x^3-x^2)^(2/3)/x^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^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.16, size = 33, normalized size = 1.32 \begin {gather*} \frac {9\,x\,{\left (x^3-x^2\right )}^{2/3}+6\,{\left (x^3-x^2\right )}^{2/3}}{10\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(x^3 - x^2)^(1/3)),x)

[Out]

(9*x*(x^3 - x^2)^(2/3) + 6*(x^3 - x^2)^(2/3))/(10*x^3)

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

[Out]

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

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