3.1.27 \(\int \frac {x^3}{\sqrt [3]{-1+x^4}} \, dx\)

Optimal. Leaf size=13 \[ \frac {3}{8} \left (x^4-1\right )^{2/3} \]

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Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {261} \begin {gather*} \frac {3}{8} \left (x^4-1\right )^{2/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(-1 + x^4)^(1/3),x]

[Out]

(3*(-1 + x^4)^(2/3))/8

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {x^3}{\sqrt [3]{-1+x^4}} \, dx &=\frac {3}{8} \left (-1+x^4\right )^{2/3}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} \frac {3}{8} \left (x^4-1\right )^{2/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/(-1 + x^4)^(1/3),x]

[Out]

(3*(-1 + x^4)^(2/3))/8

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IntegrateAlgebraic [A]  time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} \frac {3}{8} \left (-1+x^4\right )^{2/3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(-1 + x^4)^(2/3))/8

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fricas [A]  time = 0.43, size = 9, normalized size = 0.69 \begin {gather*} \frac {3}{8} \, {\left (x^{4} - 1\right )}^{\frac {2}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*(x^4 - 1)^(2/3)

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giac [A]  time = 0.69, size = 9, normalized size = 0.69 \begin {gather*} \frac {3}{8} \, {\left (x^{4} - 1\right )}^{\frac {2}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*(x^4 - 1)^(2/3)

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maple [A]  time = 0.07, size = 10, normalized size = 0.77

method	result	size
derivativedivides	\(\frac {3 \left (x^{4}-1\right )^{\frac {2}{3}}}{8}\)	\(10\)
default	\(\frac {3 \left (x^{4}-1\right )^{\frac {2}{3}}}{8}\)	\(10\)
trager	\(\frac {3 \left (x^{4}-1\right )^{\frac {2}{3}}}{8}\)	\(10\)
risch	\(\frac {3 \left (x^{4}-1\right )^{\frac {2}{3}}}{8}\)	\(10\)
gosper	\(\frac {3 \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}{8 \left (x^{4}-1\right )^{\frac {1}{3}}}\)	\(21\)
meijerg	\(\frac {\left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{3}} \hypergeom \left (\left [\frac {1}{3}, 1\right ], \relax [2], x^{4}\right ) x^{4}}{4 \mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{3}}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*(x^4-1)^(2/3)

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maxima [A]  time = 0.42, size = 9, normalized size = 0.69 \begin {gather*} \frac {3}{8} \, {\left (x^{4} - 1\right )}^{\frac {2}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*(x^4 - 1)^(2/3)

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mupad [B]  time = 0.17, size = 9, normalized size = 0.69 \begin {gather*} \frac {3\,{\left (x^4-1\right )}^{2/3}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(x^4 - 1)^(1/3),x)

[Out]

(3*(x^4 - 1)^(2/3))/8

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sympy [A]  time = 0.15, size = 10, normalized size = 0.77 \begin {gather*} \frac {3 \left (x^{4} - 1\right )^{\frac {2}{3}}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(x**4 - 1)**(2/3)/8

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