\(\int \frac {x^2}{(2+x^6)^{3/2}} \, dx\) [1417]

Optimal result

Integrand size = 13, antiderivative size = 16 \[ \int \frac {x^2}{\left (2+x^6\right )^{3/2}} \, dx=\frac {x^3}{6 \sqrt {2+x^6}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, 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 = {270} \[ \int \frac {x^2}{\left (2+x^6\right )^{3/2}} \, dx=\frac {x^3}{6 \sqrt {x^6+2}} \]

[In]

Int[x^2/(2 + x^6)^(3/2),x]

[Out]

x^3/(6*Sqrt[2 + x^6])

Rule 270

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

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\left (2+x^6\right )^{3/2}} \, dx=\frac {x^3}{6 \sqrt {2+x^6}} \]

[In]

Integrate[x^2/(2 + x^6)^(3/2),x]

[Out]

x^3/(6*Sqrt[2 + x^6])

Maple [A] (verified)

Time = 4.46 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81

method	result	size
gosper	\(\frac {x^{3}}{6 \sqrt {x^{6}+2}}\)	\(13\)
trager	\(\frac {x^{3}}{6 \sqrt {x^{6}+2}}\)	\(13\)
risch	\(\frac {x^{3}}{6 \sqrt {x^{6}+2}}\)	\(13\)
pseudoelliptic	\(\frac {x^{3}}{6 \sqrt {x^{6}+2}}\)	\(13\)
meijerg	\(\frac {\sqrt {2}\, x^{3}}{12 \sqrt {1+\frac {x^{6}}{2}}}\)	\(18\)

[In]

int(x^2/(x^6+2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (12) = 24\).

Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.56 \[ \int \frac {x^2}{\left (2+x^6\right )^{3/2}} \, dx=\frac {x^{6} + \sqrt {x^{6} + 2} x^{3} + 2}{6 \, {\left (x^{6} + 2\right )}} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {x^2}{\left (2+x^6\right )^{3/2}} \, dx=\frac {x^{3}}{6 \sqrt {x^{6} + 2}} \]

[In]

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

[Out]

x**3/(6*sqrt(x**6 + 2))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {x^2}{\left (2+x^6\right )^{3/2}} \, dx=\frac {x^{3}}{6 \, \sqrt {x^{6} + 2}} \]

[In]

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

[Out]

1/6*x^3/sqrt(x^6 + 2)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {x^2}{\left (2+x^6\right )^{3/2}} \, dx=\frac {x^{3}}{6 \, \sqrt {x^{6} + 2}} \]

[In]

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

[Out]

1/6*x^3/sqrt(x^6 + 2)

Mupad [B] (verification not implemented)

Time = 5.56 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {x^2}{\left (2+x^6\right )^{3/2}} \, dx=\frac {x^3}{6\,\sqrt {x^6+2}} \]

[In]

int(x^2/(x^6 + 2)^(3/2),x)

[Out]

x^3/(6*(x^6 + 2)^(1/2))