3.1.22 \(\int x^2 \sqrt {1+x^3} \, dx\)

Optimal. Leaf size=13 \[ \frac {2}{9} \left (x^3+1\right )^{3/2} \]

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

Antiderivative was successfully verified.

[In]

Int[x^2*Sqrt[1 + x^3],x]

[Out]

(2*(1 + x^3)^(3/2))/9

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 x^2 \sqrt {1+x^3} \, dx &=\frac {2}{9} \left (1+x^3\right )^{3/2}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} \frac {2}{9} \left (x^3+1\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*Sqrt[1 + x^3],x]

[Out]

(2*(1 + x^3)^(3/2))/9

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IntegrateAlgebraic [A]  time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} \frac {2}{9} \left (1+x^3\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^2*Sqrt[1 + x^3],x]

[Out]

(2*(1 + x^3)^(3/2))/9

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fricas [A]  time = 0.42, size = 9, normalized size = 0.69 \begin {gather*} \frac {2}{9} \, {\left (x^{3} + 1\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*(x^3 + 1)^(3/2)

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giac [A]  time = 0.44, size = 9, normalized size = 0.69 \begin {gather*} \frac {2}{9} \, {\left (x^{3} + 1\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*(x^3 + 1)^(3/2)

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maple [A]  time = 0.05, size = 10, normalized size = 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	\(\frac {2 \left (x^{3}+1\right )^{\frac {3}{2}}}{9}\)	\(10\)
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	\(-\frac {\frac {4 \sqrt {\pi }}{3}-\frac {2 \sqrt {\pi }\, \left (2 x^{3}+2\right ) \sqrt {x^{3}+1}}{3}}{6 \sqrt {\pi }}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.70, size = 9, normalized size = 0.69 \begin {gather*} \frac {2}{9} \, {\left (x^{3} + 1\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*(x^3 + 1)^(3/2)

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mupad [B]  time = 0.02, size = 9, normalized size = 0.69 \begin {gather*} \frac {2\,{\left (x^3+1\right )}^{3/2}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(x^3 + 1)^(1/2),x)

[Out]

(2*(x^3 + 1)^(3/2))/9

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sympy [B]  time = 0.14, size = 26, normalized size = 2.00 \begin {gather*} \frac {2 x^{3} \sqrt {x^{3} + 1}}{9} + \frac {2 \sqrt {x^{3} + 1}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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