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

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

[Out]

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

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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} \[ \frac {\sqrt {x^6+2}}{3} \]

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*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \[ \frac {\sqrt {x^6+2}}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/Sqrt[2 + x^6],x]

[Out]

Sqrt[2 + x^6]/3

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fricas [A]  time = 0.69, size = 9, normalized size = 0.69 \[ \frac {1}{3} \, \sqrt {x^{6} + 2} \]

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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giac [A]  time = 0.15, size = 9, normalized size = 0.69 \[ \frac {1}{3} \, \sqrt {x^{6} + 2} \]

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)

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maple [A]  time = 0.00, size = 10, normalized size = 0.77 \[ \frac {\sqrt {x^{6}+2}}{3} \]

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 = 1.11, size = 9, normalized size = 0.69 \[ \frac {1}{3} \, \sqrt {x^{6} + 2} \]

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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mupad [B]  time = 1.15, size = 9, normalized size = 0.69 \[ \frac {\sqrt {x^6+2}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(x^6 + 2)^(1/2),x)

[Out]

(x^6 + 2)^(1/2)/3

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sympy [A]  time = 0.53, size = 8, normalized size = 0.62 \[ \frac {\sqrt {x^{6} + 2}}{3} \]

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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