3.1386 \(\int \frac{x^5}{\sqrt{2+x^6}} \, dx\)

Optimal. Leaf size=13 \[ \frac{\sqrt{x^6+2}}{3} \]

[Out]

Sqrt[2 + x^6]/3

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Rubi [A]  time = 0.0026282, 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} \[ \frac{\sqrt{x^6+2}}{3} \]

Antiderivative was successfully verified.

[In]

Int[x^5/Sqrt[2 + x^6],x]

[Out]

Sqrt[2 + x^6]/3

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^5}{\sqrt{2+x^6}} \, dx &=\frac{\sqrt{2+x^6}}{3}\\ \end{align*}

Mathematica [A]  time = 0.0018357, size = 13, normalized size = 1. \[ \frac{\sqrt{x^6+2}}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/Sqrt[2 + x^6],x]

[Out]

Sqrt[2 + x^6]/3

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Maple [A]  time = 0.003, size = 10, normalized size = 0.8 \begin{align*}{\frac{1}{3}\sqrt{{x}^{6}+2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(x^6+2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.990974, size = 12, normalized size = 0.92 \begin{align*} \frac{1}{3} \, \sqrt{x^{6} + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(x^6+2)^(1/2),x, algorithm="maxima")

[Out]

1/3*sqrt(x^6 + 2)

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Fricas [A]  time = 1.44953, size = 26, normalized size = 2. \begin{align*} \frac{1}{3} \, \sqrt{x^{6} + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(x^6+2)^(1/2),x, algorithm="fricas")

[Out]

1/3*sqrt(x^6 + 2)

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Sympy [A]  time = 0.252104, size = 8, normalized size = 0.62 \begin{align*} \frac{\sqrt{x^{6} + 2}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(x**6+2)**(1/2),x)

[Out]

sqrt(x**6 + 2)/3

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Giac [A]  time = 1.16945, size = 12, normalized size = 0.92 \begin{align*} \frac{1}{3} \, \sqrt{x^{6} + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(x^6+2)^(1/2),x, algorithm="giac")

[Out]

1/3*sqrt(x^6 + 2)