3.17 \(\int x^2 (27+8 x^3)^{2/3} \, dx\)

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

[Out]

(27 + 8*x^3)^(5/3)/40

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Rubi [A]  time = 0.0030355, antiderivative size = 15, normalized size of antiderivative = 1., 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 = {261} \[ \frac{1}{40} \left (8 x^3+27\right )^{5/3} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(27 + 8*x^3)^(2/3),x]

[Out]

(27 + 8*x^3)^(5/3)/40

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

Mathematica [A]  time = 0.0033828, size = 15, normalized size = 1. \[ \frac{1}{40} \left (8 x^3+27\right )^{5/3} \]

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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Maple [B]  time = 0.005, size = 27, normalized size = 1.8 \begin{align*}{\frac{ \left ( 3+2\,x \right ) \left ( 4\,{x}^{2}-6\,x+9 \right ) }{40} \left ( 8\,{x}^{3}+27 \right ) ^{{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/40*(3+2*x)*(4*x^2-6*x+9)*(8*x^3+27)^(2/3)

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Maxima [A]  time = 0.950064, size = 15, normalized size = 1. \begin{align*} \frac{1}{40} \,{\left (8 \, x^{3} + 27\right )}^{\frac{5}{3}} \end{align*}

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 = 0.42443, size = 34, normalized size = 2.27 \begin{align*} \frac{1}{40} \,{\left (8 \, x^{3} + 27\right )}^{\frac{5}{3}} \end{align*}

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]  time = 0.349162, size = 27, normalized size = 1.8 \begin{align*} \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{align*}

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 = 1.09399, size = 15, normalized size = 1. \begin{align*} \frac{1}{40} \,{\left (8 \, x^{3} + 27\right )}^{\frac{5}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/40*(8*x^3 + 27)^(5/3)