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]

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

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Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \[ \frac {9}{25} \left (x^{5/3}+1\right )^{5/3} \]

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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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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giac [A]  time = 0.19, size = 9, normalized size = 0.60 \[ \frac {9}{25} \, {\left (x^{\frac {5}{3}} + 1\right )}^{\frac {5}{3}} \]

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)

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maple [A]  time = 0.00, size = 10, normalized size = 0.67 \[ \frac {9 \left (x^{\frac {5}{3}}+1\right )^{\frac {5}{3}}}{25} \]

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 = 0.56, size = 9, normalized size = 0.60 \[ \frac {9}{25} \, {\left (x^{\frac {5}{3}} + 1\right )}^{\frac {5}{3}} \]

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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mupad [B]  time = 1.54, size = 9, normalized size = 0.60 \[ \frac {9\,{\left (x^{5/3}+1\right )}^{5/3}}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(2/3)*(x^(5/3) + 1)^(2/3),x)

[Out]

(9*(x^(5/3) + 1)^(5/3))/25

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sympy [B]  time = 1.90, size = 31, normalized size = 2.07 \[ \frac {9 x^{\frac {5}{3}} \left (x^{\frac {5}{3}} + 1\right )^{\frac {2}{3}}}{25} + \frac {9 \left (x^{\frac {5}{3}} + 1\right )^{\frac {2}{3}}}{25} \]

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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