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

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

[Out]

(3*(1 + x^4)^(2/3))/8

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Rubi [A]  time = 0.0026967, antiderivative size = 13, normalized size of antiderivative = 1., 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} \[ \frac{3}{8} \left (x^4+1\right )^{2/3} \]

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*}

Mathematica [A]  time = 0.0022577, size = 13, normalized size = 1. \[ \frac{3}{8} \left (x^4+1\right )^{2/3} \]

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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Maple [A]  time = 0.003, size = 10, normalized size = 0.8 \begin{align*}{\frac{3}{8} \left ({x}^{4}+1 \right ) ^{{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(x^4+1)^(1/3),x)

[Out]

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

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Maxima [A]  time = 0.961474, size = 12, normalized size = 0.92 \begin{align*} \frac{3}{8} \,{\left (x^{4} + 1\right )}^{\frac{2}{3}} \end{align*}

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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Fricas [A]  time = 1.44132, size = 28, normalized size = 2.15 \begin{align*} \frac{3}{8} \,{\left (x^{4} + 1\right )}^{\frac{2}{3}} \end{align*}

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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Sympy [A]  time = 0.197193, size = 10, normalized size = 0.77 \begin{align*} \frac{3 \left (x^{4} + 1\right )^{\frac{2}{3}}}{8} \end{align*}

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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Giac [A]  time = 1.16229, size = 12, normalized size = 0.92 \begin{align*} \frac{3}{8} \,{\left (x^{4} + 1\right )}^{\frac{2}{3}} \end{align*}

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)