\(\int \frac {x^5}{\sqrt {2+x^6}} \, dx\) [1386]

Optimal result

Integrand size = 13, antiderivative size = 13 \[ \int \frac {x^5}{\sqrt {2+x^6}} \, dx=\frac {\sqrt {2+x^6}}{3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {267} \[ \int \frac {x^5}{\sqrt {2+x^6}} \, dx=\frac {\sqrt {x^6+2}}{3} \]

[In]

Int[x^5/Sqrt[2 + x^6],x]

[Out]

Sqrt[2 + x^6]/3

Rule 267

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*} \text {integral}& = \frac {\sqrt {2+x^6}}{3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {x^5}{\sqrt {2+x^6}} \, dx=\frac {\sqrt {2+x^6}}{3} \]

[In]

Integrate[x^5/Sqrt[2 + x^6],x]

[Out]

Sqrt[2 + x^6]/3

Maple [A] (verified)

Time = 4.59 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77

method	result	size
gosper	\(\frac {\sqrt {x^{6}+2}}{3}\)	\(10\)
derivativedivides	\(\frac {\sqrt {x^{6}+2}}{3}\)	\(10\)
default	\(\frac {\sqrt {x^{6}+2}}{3}\)	\(10\)
trager	\(\frac {\sqrt {x^{6}+2}}{3}\)	\(10\)
risch	\(\frac {\sqrt {x^{6}+2}}{3}\)	\(10\)
pseudoelliptic	\(\frac {\sqrt {x^{6}+2}}{3}\)	\(10\)
meijerg	\(\frac {\sqrt {2}\, \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {1+\frac {x^{6}}{2}}\right )}{6 \sqrt {\pi }}\)	\(29\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \frac {x^5}{\sqrt {2+x^6}} \, dx=\frac {1}{3} \, \sqrt {x^{6} + 2} \]

[In]

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

[Out]

1/3*sqrt(x^6 + 2)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.62 \[ \int \frac {x^5}{\sqrt {2+x^6}} \, dx=\frac {\sqrt {x^{6} + 2}}{3} \]

[In]

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

[Out]

sqrt(x**6 + 2)/3

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \frac {x^5}{\sqrt {2+x^6}} \, dx=\frac {1}{3} \, \sqrt {x^{6} + 2} \]

[In]

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

[Out]

1/3*sqrt(x^6 + 2)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \frac {x^5}{\sqrt {2+x^6}} \, dx=\frac {1}{3} \, \sqrt {x^{6} + 2} \]

[In]

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

[Out]

1/3*sqrt(x^6 + 2)

Mupad [B] (verification not implemented)

Time = 5.63 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \frac {x^5}{\sqrt {2+x^6}} \, dx=\frac {\sqrt {x^6+2}}{3} \]

[In]

int(x^5/(x^6 + 2)^(1/2),x)

[Out]

(x^6 + 2)^(1/2)/3