3.1.17 \(\int x^2 (27+8 x^3)^{2/3} \, dx\) [17]

Optimal. Leaf size=15 \[ \frac {1}{40} \left (27+8 x^3\right )^{5/3} \]

[Out]

1/40*(8*x^3+27)^(5/3)

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Rubi [A]
time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {267} \begin {gather*} \frac {1}{40} \left (8 x^3+27\right )^{5/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(27 + 8*x^3)^(2/3),x]

[Out]

(27 + 8*x^3)^(5/3)/40

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 x^2 \left (27+8 x^3\right )^{2/3} \, dx &=\frac {1}{40} \left (27+8 x^3\right )^{5/3}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 15, normalized size = 1.00 \begin {gather*} \frac {1}{40} \left (27+8 x^3\right )^{5/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*(27 + 8*x^3)^(2/3),x]

[Out]

(27 + 8*x^3)^(5/3)/40

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Mathics [A]
time = 1.81, size = 11, normalized size = 0.73 \begin {gather*} \frac {{\left (27+8 x^3\right )}^{\frac {5}{3}}}{40} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x^2*(8*x^3 + 27)^(2/3),x]')

[Out]

(27 + 8 x ^ 3) ^ (5 / 3) / 40

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Maple [A]
time = 0.07, size = 12, normalized size = 0.80

method	result	size
derivativedivides	\(\frac {\left (8 x^{3}+27\right )^{\frac {5}{3}}}{40}\)	\(12\)
default	\(\frac {\left (8 x^{3}+27\right )^{\frac {5}{3}}}{40}\)	\(12\)
risch	\(\frac {\left (8 x^{3}+27\right )^{\frac {5}{3}}}{40}\)	\(12\)
meijerg	\(3 x^{3} \hypergeom \left (\left [-\frac {2}{3}, 1\right ], \left [2\right ], -\frac {8 x^{3}}{27}\right )\)	\(17\)
trager	\(\left (\frac {x^{3}}{5}+\frac {27}{40}\right ) \left (8 x^{3}+27\right )^{\frac {2}{3}}\)	\(18\)
gosper	\(\frac {\left (3+2 x \right ) \left (4 x^{2}-6 x +9\right ) \left (8 x^{3}+27\right )^{\frac {2}{3}}}{40}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(8*x^3+27)^(2/3),x,method=_RETURNVERBOSE)

[Out]

1/40*(8*x^3+27)^(5/3)

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Maxima [A]
time = 0.26, size = 11, normalized size = 0.73 \begin {gather*} \frac {1}{40} \, {\left (8 \, x^{3} + 27\right )}^{\frac {5}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(8*x^3+27)^(2/3),x, algorithm="maxima")

[Out]

1/40*(8*x^3 + 27)^(5/3)

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Fricas [A]
time = 0.32, size = 11, normalized size = 0.73 \begin {gather*} \frac {1}{40} \, {\left (8 \, x^{3} + 27\right )}^{\frac {5}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(8*x^3+27)^(2/3),x, algorithm="fricas")

[Out]

1/40*(8*x^3 + 27)^(5/3)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (10) = 20\)
time = 0.11, size = 27, normalized size = 1.80 \begin {gather*} \frac {x^{3} \left (8 x^{3} + 27\right )^{\frac {2}{3}}}{5} + \frac {27 \left (8 x^{3} + 27\right )^{\frac {2}{3}}}{40} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(8*x**3+27)**(2/3),x)

[Out]

x**3*(8*x**3 + 27)**(2/3)/5 + 27*(8*x**3 + 27)**(2/3)/40

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Giac [A]
time = 0.00, size = 23, normalized size = 1.53 \begin {gather*} \frac {1}{40} \left (\left (8 x^{3}+27\right )^{\frac {1}{3}}\right )^{2} \left (8 x^{3}+27\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(8*x^3+27)^(2/3),x)

[Out]

1/40*(8*x^3 + 27)^(5/3)

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Mupad [B]
time = 0.19, size = 11, normalized size = 0.73 \begin {gather*} \frac {{\left (8\,x^3+27\right )}^{5/3}}{40} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(8*x^3 + 27)^(2/3),x)

[Out]

(8*x^3 + 27)^(5/3)/40

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