\(\int \frac {x^2}{\sqrt {2+x^3}} \, dx\) [157]

Optimal result

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

[Out]

2/3*(x^3+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} \[ \int \frac {x^2}{\sqrt {2+x^3}} \, dx=\frac {2 \sqrt {x^3+2}}{3} \]

[In]

Int[x^2/Sqrt[2 + x^3],x]

[Out]

(2*Sqrt[2 + x^3])/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 {2 \sqrt {2+x^3}}{3} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

Integrate[x^2/Sqrt[2 + x^3],x]

[Out]

(2*Sqrt[2 + x^3])/3

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77

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

[In]

int(x^2/(x^3+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \frac {x^2}{\sqrt {2+x^3}} \, dx=\frac {2}{3} \, \sqrt {x^{3} + 2} \]

[In]

integrate(x^2/(x^3+2)^(1/2),x, algorithm="fricas")

[Out]

2/3*sqrt(x^3 + 2)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \frac {x^2}{\sqrt {2+x^3}} \, dx=\frac {2 \sqrt {x^{3} + 2}}{3} \]

[In]

integrate(x**2/(x**3+2)**(1/2),x)

[Out]

2*sqrt(x**3 + 2)/3

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \frac {x^2}{\sqrt {2+x^3}} \, dx=\frac {2}{3} \, \sqrt {x^{3} + 2} \]

[In]

integrate(x^2/(x^3+2)^(1/2),x, algorithm="maxima")

[Out]

2/3*sqrt(x^3 + 2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \frac {x^2}{\sqrt {2+x^3}} \, dx=\frac {2}{3} \, \sqrt {x^{3} + 2} \]

[In]

integrate(x^2/(x^3+2)^(1/2),x, algorithm="giac")

[Out]

2/3*sqrt(x^3 + 2)

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \frac {x^2}{\sqrt {2+x^3}} \, dx=\frac {2\,\sqrt {x^3+2}}{3} \]

[In]

int(x^2/(x^3 + 2)^(1/2),x)

[Out]

(2*(x^3 + 2)^(1/2))/3