3.2446 \(\int x^{2/3} (1+x^{5/3})^{2/3} \, dx\)

Optimal. Leaf size=15 \[ \frac{9}{25} \left (x^{5/3}+1\right )^{5/3} \]

[Out]

(9*(1 + x^(5/3))^(5/3))/25

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Rubi [A]  time = 0.0031098, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {261} \[ \frac{9}{25} \left (x^{5/3}+1\right )^{5/3} \]

Antiderivative was successfully verified.

[In]

Int[x^(2/3)*(1 + x^(5/3))^(2/3),x]

[Out]

(9*(1 + x^(5/3))^(5/3))/25

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^{2/3} \left (1+x^{5/3}\right )^{2/3} \, dx &=\frac{9}{25} \left (1+x^{5/3}\right )^{5/3}\\ \end{align*}

Mathematica [A]  time = 0.0044365, size = 15, normalized size = 1. \[ \frac{9}{25} \left (x^{5/3}+1\right )^{5/3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(2/3)*(1 + x^(5/3))^(2/3),x]

[Out]

(9*(1 + x^(5/3))^(5/3))/25

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Maple [A]  time = 0.001, size = 10, normalized size = 0.7 \begin{align*}{\frac{9}{25} \left ( 1+{x}^{{\frac{5}{3}}} \right ) ^{{\frac{5}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(2/3)*(1+x^(5/3))^(2/3),x)

[Out]

9/25*(1+x^(5/3))^(5/3)

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Maxima [A]  time = 0.975876, size = 12, normalized size = 0.8 \begin{align*} \frac{9}{25} \,{\left (x^{\frac{5}{3}} + 1\right )}^{\frac{5}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(2/3)*(1+x^(5/3))^(2/3),x, algorithm="maxima")

[Out]

9/25*(x^(5/3) + 1)^(5/3)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(2/3)*(1+x^(5/3))^(2/3),x, algorithm="fricas")

[Out]

Timed out

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Sympy [B]  time = 2.11386, size = 31, normalized size = 2.07 \begin{align*} \frac{9 x^{\frac{5}{3}} \left (x^{\frac{5}{3}} + 1\right )^{\frac{2}{3}}}{25} + \frac{9 \left (x^{\frac{5}{3}} + 1\right )^{\frac{2}{3}}}{25} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(2/3)*(1+x**(5/3))**(2/3),x)

[Out]

9*x**(5/3)*(x**(5/3) + 1)**(2/3)/25 + 9*(x**(5/3) + 1)**(2/3)/25

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Giac [A]  time = 1.21286, size = 12, normalized size = 0.8 \begin{align*} \frac{9}{25} \,{\left (x^{\frac{5}{3}} + 1\right )}^{\frac{5}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(2/3)*(1+x^(5/3))^(2/3),x, algorithm="giac")

[Out]

9/25*(x^(5/3) + 1)^(5/3)