3.1417 \(\int \frac{x^2}{(2+x^6)^{3/2}} \, dx\)

Optimal. Leaf size=16 \[ \frac{x^3}{6 \sqrt{x^6+2}} \]

[Out]

x^3/(6*Sqrt[2 + x^6])

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Rubi [A]  time = 0.0030059, antiderivative size = 16, 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 = {264} \[ \frac{x^3}{6 \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(2 + x^6)^(3/2),x]

[Out]

x^3/(6*Sqrt[2 + x^6])

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{x^2}{\left (2+x^6\right )^{3/2}} \, dx &=\frac{x^3}{6 \sqrt{2+x^6}}\\ \end{align*}

Mathematica [A]  time = 0.0024186, size = 16, normalized size = 1. \[ \frac{x^3}{6 \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(2 + x^6)^(3/2),x]

[Out]

x^3/(6*Sqrt[2 + x^6])

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Maple [A]  time = 0.003, size = 13, normalized size = 0.8 \begin{align*}{\frac{{x}^{3}}{6}{\frac{1}{\sqrt{{x}^{6}+2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(x^6+2)^(3/2),x)

[Out]

1/6*x^3/(x^6+2)^(1/2)

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Maxima [A]  time = 1.0038, size = 16, normalized size = 1. \begin{align*} \frac{x^{3}}{6 \, \sqrt{x^{6} + 2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^6+2)^(3/2),x, algorithm="maxima")

[Out]

1/6*x^3/sqrt(x^6 + 2)

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Fricas [B]  time = 1.48822, size = 61, normalized size = 3.81 \begin{align*} \frac{x^{6} + \sqrt{x^{6} + 2} x^{3} + 2}{6 \,{\left (x^{6} + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^6+2)^(3/2),x, algorithm="fricas")

[Out]

1/6*(x^6 + sqrt(x^6 + 2)*x^3 + 2)/(x^6 + 2)

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Sympy [A]  time = 0.487208, size = 12, normalized size = 0.75 \begin{align*} \frac{x^{3}}{6 \sqrt{x^{6} + 2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(x**6+2)**(3/2),x)

[Out]

x**3/(6*sqrt(x**6 + 2))

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Giac [A]  time = 1.2325, size = 16, normalized size = 1. \begin{align*} \frac{x^{3}}{6 \, \sqrt{x^{6} + 2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^6+2)^(3/2),x, algorithm="giac")

[Out]

1/6*x^3/sqrt(x^6 + 2)