3.582 \(\int \frac{x^2}{\sqrt [4]{2+x^3}} \, dx\)

Optimal. Leaf size=13 \[ \frac{4}{9} \left (x^3+2\right )^{3/4} \]

[Out]

(4*(2 + x^3)^(3/4))/9

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Rubi [A]  time = 0.0026733, 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} \[ \frac{4}{9} \left (x^3+2\right )^{3/4} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(2 + x^3)^(1/4),x]

[Out]

(4*(2 + x^3)^(3/4))/9

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^2}{\sqrt [4]{2+x^3}} \, dx &=\frac{4}{9} \left (2+x^3\right )^{3/4}\\ \end{align*}

Mathematica [A]  time = 0.0024442, size = 13, normalized size = 1. \[ \frac{4}{9} \left (x^3+2\right )^{3/4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(2 + x^3)^(1/4),x]

[Out]

(4*(2 + x^3)^(3/4))/9

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Maple [A]  time = 0.002, size = 10, normalized size = 0.8 \begin{align*}{\frac{4}{9} \left ({x}^{3}+2 \right ) ^{{\frac{3}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(x^3+2)^(1/4),x)

[Out]

4/9*(x^3+2)^(3/4)

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Maxima [A]  time = 0.974676, size = 12, normalized size = 0.92 \begin{align*} \frac{4}{9} \,{\left (x^{3} + 2\right )}^{\frac{3}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^3+2)^(1/4),x, algorithm="maxima")

[Out]

4/9*(x^3 + 2)^(3/4)

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Fricas [A]  time = 1.42635, size = 28, normalized size = 2.15 \begin{align*} \frac{4}{9} \,{\left (x^{3} + 2\right )}^{\frac{3}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^3+2)^(1/4),x, algorithm="fricas")

[Out]

4/9*(x^3 + 2)^(3/4)

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Sympy [A]  time = 0.220696, size = 10, normalized size = 0.77 \begin{align*} \frac{4 \left (x^{3} + 2\right )^{\frac{3}{4}}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(x**3+2)**(1/4),x)

[Out]

4*(x**3 + 2)**(3/4)/9

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Giac [A]  time = 1.08938, size = 12, normalized size = 0.92 \begin{align*} \frac{4}{9} \,{\left (x^{3} + 2\right )}^{\frac{3}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^3+2)^(1/4),x, algorithm="giac")

[Out]

4/9*(x^3 + 2)^(3/4)