3.6.82 \(\int \frac {x^2}{\sqrt [4]{2+x^3}} \, dx\) [582]

Optimal. Leaf size=13 \[ \frac {4}{9} \left (2+x^3\right )^{3/4} \]

[Out]

4/9*(x^3+2)^(3/4)

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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 = {267} \begin {gather*} \frac {4}{9} \left (x^3+2\right )^{3/4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/(2 + x^3)^(1/4),x]

[Out]

(4*(2 + x^3)^(3/4))/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*} \int \frac {x^2}{\sqrt [4]{2+x^3}} \, dx &=\frac {4}{9} \left (2+x^3\right )^{3/4}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} \frac {4}{9} \left (2+x^3\right )^{3/4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/(2 + x^3)^(1/4),x]

[Out]

(4*(2 + x^3)^(3/4))/9

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Maple [A]
time = 0.14, size = 10, normalized size = 0.77

method	result	size
gosper	\(\frac {4 \left (x^{3}+2\right )^{\frac {3}{4}}}{9}\)	\(10\)
derivativedivides	\(\frac {4 \left (x^{3}+2\right )^{\frac {3}{4}}}{9}\)	\(10\)
default	\(\frac {4 \left (x^{3}+2\right )^{\frac {3}{4}}}{9}\)	\(10\)
trager	\(\frac {4 \left (x^{3}+2\right )^{\frac {3}{4}}}{9}\)	\(10\)
risch	\(\frac {4 \left (x^{3}+2\right )^{\frac {3}{4}}}{9}\)	\(10\)
meijerg	\(\frac {2^{\frac {3}{4}} x^{3} \hypergeom \left (\left [\frac {1}{4}, 1\right ], \left [2\right ], -\frac {x^{3}}{2}\right )}{6}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(x^3+2)^(1/4),x,method=_RETURNVERBOSE)

[Out]

4/9*(x^3+2)^(3/4)

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Maxima [A]
time = 0.30, size = 9, normalized size = 0.69 \begin {gather*} \frac {4}{9} \, {\left (x^{3} + 2\right )}^{\frac {3}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^3+2)^(1/4),x, algorithm="maxima")

[Out]

4/9*(x^3 + 2)^(3/4)

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Fricas [A]
time = 0.34, size = 9, normalized size = 0.69 \begin {gather*} \frac {4}{9} \, {\left (x^{3} + 2\right )}^{\frac {3}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^3+2)^(1/4),x, algorithm="fricas")

[Out]

4/9*(x^3 + 2)^(3/4)

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Sympy [A]
time = 0.05, size = 10, normalized size = 0.77 \begin {gather*} \frac {4 \left (x^{3} + 2\right )^{\frac {3}{4}}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(x**3+2)**(1/4),x)

[Out]

4*(x**3 + 2)**(3/4)/9

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Giac [A]
time = 1.69, size = 9, normalized size = 0.69 \begin {gather*} \frac {4}{9} \, {\left (x^{3} + 2\right )}^{\frac {3}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^3+2)^(1/4),x, algorithm="giac")

[Out]

4/9*(x^3 + 2)^(3/4)

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Mupad [B]
time = 1.13, size = 9, normalized size = 0.69 \begin {gather*} \frac {4\,{\left (x^3+2\right )}^{3/4}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(x^3 + 2)^(1/4),x)

[Out]

(4*(x^3 + 2)^(3/4))/9

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