\(\int x^2 \sqrt {1+x^3} \, dx\) [22]

Optimal result

Integrand size = 13, antiderivative size = 13 \[ \int x^2 \sqrt {1+x^3} \, dx=\frac {2}{9} \left (1+x^3\right )^{3/2} \]

[Out]

2/9*(x^3+1)^(3/2)

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} \[ \int x^2 \sqrt {1+x^3} \, dx=\frac {2}{9} \left (x^3+1\right )^{3/2} \]

[In]

Int[x^2*Sqrt[1 + x^3],x]

[Out]

(2*(1 + x^3)^(3/2))/9

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}{9} \left (1+x^3\right )^{3/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int x^2 \sqrt {1+x^3} \, dx=\frac {2}{9} \left (1+x^3\right )^{3/2} \]

[In]

Integrate[x^2*Sqrt[1 + x^3],x]

[Out]

(2*(1 + x^3)^(3/2))/9

Maple [A] (verified)

Time = 0.75 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77

method	result	size
derivativedivides	\(\frac {2 \left (x^{3}+1\right )^{\frac {3}{2}}}{9}\)	\(10\)
default	\(\frac {2 \left (x^{3}+1\right )^{\frac {3}{2}}}{9}\)	\(10\)
risch	\(\frac {2 \left (x^{3}+1\right )^{\frac {3}{2}}}{9}\)	\(10\)
pseudoelliptic	\(\frac {2 \left (x^{3}+1\right )^{\frac {3}{2}}}{9}\)	\(10\)
trager	\(\left (\frac {2}{9}+\frac {2 x^{3}}{9}\right ) \sqrt {x^{3}+1}\)	\(16\)
gosper	\(\frac {2 \left (1+x \right ) \left (x^{2}-x +1\right ) \sqrt {x^{3}+1}}{9}\)	\(21\)
meijerg	\(-\frac {\frac {4 \sqrt {\pi }}{3}-\frac {2 \sqrt {\pi }\, \left (2 x^{3}+2\right ) \sqrt {x^{3}+1}}{3}}{6 \sqrt {\pi }}\)	\(31\)

[In]

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

[Out]

2/9*(x^3+1)^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x^2 \sqrt {1+x^3} \, dx=\frac {2}{9} \, {\left (x^{3} + 1\right )}^{\frac {3}{2}} \]

[In]

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

[Out]

2/9*(x^3 + 1)^(3/2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (10) = 20\).

Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 2.00 \[ \int x^2 \sqrt {1+x^3} \, dx=\frac {2 x^{3} \sqrt {x^{3} + 1}}{9} + \frac {2 \sqrt {x^{3} + 1}}{9} \]

[In]

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

[Out]

2*x**3*sqrt(x**3 + 1)/9 + 2*sqrt(x**3 + 1)/9

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x^2 \sqrt {1+x^3} \, dx=\frac {2}{9} \, {\left (x^{3} + 1\right )}^{\frac {3}{2}} \]

[In]

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

[Out]

2/9*(x^3 + 1)^(3/2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x^2 \sqrt {1+x^3} \, dx=\frac {2}{9} \, {\left (x^{3} + 1\right )}^{\frac {3}{2}} \]

[In]

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

[Out]

2/9*(x^3 + 1)^(3/2)

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x^2 \sqrt {1+x^3} \, dx=\frac {2\,{\left (x^3+1\right )}^{3/2}}{9} \]

[In]

int(x^2*(x^3 + 1)^(1/2),x)

[Out]

(2*(x^3 + 1)^(3/2))/9