3.2.24 \(\int \frac {\sqrt [3]{-x+x^3}}{x^4} \, dx\)

Optimal. Leaf size=18 \[ \frac {3 \left (x^3-x\right )^{4/3}}{8 x^4} \]

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Rubi [A]  time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2014} \begin {gather*} \frac {3 \left (x^3-x\right )^{4/3}}{8 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-x + x^3)^(1/3)/x^4,x]

[Out]

(3*(-x + x^3)^(4/3))/(8*x^4)

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 {\sqrt [3]{-x+x^3}}{x^4} \, dx &=\frac {3 \left (-x+x^3\right )^{4/3}}{8 x^4}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} \frac {3 \left (x \left (x^2-1\right )\right )^{4/3}}{8 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x + x^3)^(1/3)/x^4,x]

[Out]

(3*(x*(-1 + x^2))^(4/3))/(8*x^4)

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IntegrateAlgebraic [A]  time = 0.15, size = 18, normalized size = 1.00 \begin {gather*} \frac {3 \left (-x+x^3\right )^{4/3}}{8 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-x + x^3)^(1/3)/x^4,x]

[Out]

(3*(-x + x^3)^(4/3))/(8*x^4)

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fricas [A]  time = 0.47, size = 19, normalized size = 1.06 \begin {gather*} \frac {3 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (x^{2} - 1\right )}}{8 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-x)^(1/3)/x^4,x, algorithm="fricas")

[Out]

3/8*(x^3 - x)^(1/3)*(x^2 - 1)/x^3

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giac [A]  time = 0.33, size = 11, normalized size = 0.61 \begin {gather*} \frac {3}{8} \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {4}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-x)^(1/3)/x^4,x, algorithm="giac")

[Out]

3/8*(-1/x^2 + 1)^(4/3)

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maple [A]  time = 0.07, size = 20, normalized size = 1.11

method	result	size
trager	\(\frac {3 \left (x^{2}-1\right ) \left (x^{3}-x \right )^{\frac {1}{3}}}{8 x^{3}}\)	\(20\)
gosper	\(\frac {3 \left (-1+x \right ) \left (1+x \right ) \left (x^{3}-x \right )^{\frac {1}{3}}}{8 x^{3}}\)	\(21\)
risch	\(\frac {3 \left (x \left (x^{2}-1\right )\right )^{\frac {1}{3}} \left (x^{4}-2 x^{2}+1\right )}{8 x^{3} \left (x^{2}-1\right )}\)	\(32\)
meijerg	\(-\frac {3 \mathrm {signum}\left (x^{2}-1\right )^{\frac {1}{3}} \left (-x^{2}+1\right )^{\frac {4}{3}}}{8 \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {1}{3}} x^{\frac {8}{3}}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3-x)^(1/3)/x^4,x,method=_RETURNVERBOSE)

[Out]

3/8*(x^2-1)/x^3*(x^3-x)^(1/3)

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maxima [A]  time = 0.41, size = 22, normalized size = 1.22 \begin {gather*} \frac {3 \, {\left (x^{3} - x\right )} {\left (x + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )}^{\frac {1}{3}}}{8 \, x^{\frac {11}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-x)^(1/3)/x^4,x, algorithm="maxima")

[Out]

3/8*(x^3 - x)*(x + 1)^(1/3)*(x - 1)^(1/3)/x^(11/3)

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mupad [B]  time = 0.17, size = 31, normalized size = 1.72 \begin {gather*} -\frac {3\,{\left (x^3-x\right )}^{1/3}-3\,x^2\,{\left (x^3-x\right )}^{1/3}}{8\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3 - x)^(1/3)/x^4,x)

[Out]

-(3*(x^3 - x)^(1/3) - 3*x^2*(x^3 - x)^(1/3))/(8*x^3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )}}{x^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3-x)**(1/3)/x**4,x)

[Out]

Integral((x*(x - 1)*(x + 1))**(1/3)/x**4, x)

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