3.2388 \(\int \frac{(1+x^{2/3})^{3/2}}{\sqrt [3]{x}} \, dx\)

Optimal. Leaf size=15 \[ \frac{3}{5} \left (x^{2/3}+1\right )^{5/2} \]

[Out]

(3*(1 + x^(2/3))^(5/2))/5

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Rubi [A]  time = 0.0030703, 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} \[ \frac{3}{5} \left (x^{2/3}+1\right )^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^(2/3))^(3/2)/x^(1/3),x]

[Out]

(3*(1 + x^(2/3))^(5/2))/5

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

Mathematica [A]  time = 0.0050031, size = 15, normalized size = 1. \[ \frac{3}{5} \left (x^{2/3}+1\right )^{5/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(1 + x^(2/3))^(5/2))/5

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Maple [A]  time = 0.005, size = 10, normalized size = 0.7 \begin{align*}{\frac{3}{5} \left ( 1+{x}^{{\frac{2}{3}}} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.94605, size = 12, normalized size = 0.8 \begin{align*} \frac{3}{5} \,{\left (x^{\frac{2}{3}} + 1\right )}^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/5*(x^(2/3) + 1)^(5/2)

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Fricas [B]  time = 6.50195, size = 66, normalized size = 4.4 \begin{align*} \frac{3}{5} \,{\left (x^{\frac{4}{3}} + 2 \, x^{\frac{2}{3}} + 1\right )} \sqrt{x^{\frac{2}{3}} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/5*(x^(4/3) + 2*x^(2/3) + 1)*sqrt(x^(2/3) + 1)

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Sympy [B]  time = 1.18405, size = 49, normalized size = 3.27 \begin{align*} \frac{3 x^{\frac{4}{3}} \sqrt{x^{\frac{2}{3}} + 1}}{5} + \frac{6 x^{\frac{2}{3}} \sqrt{x^{\frac{2}{3}} + 1}}{5} + \frac{3 \sqrt{x^{\frac{2}{3}} + 1}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x**(4/3)*sqrt(x**(2/3) + 1)/5 + 6*x**(2/3)*sqrt(x**(2/3) + 1)/5 + 3*sqrt(x**(2/3) + 1)/5

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Giac [A]  time = 1.14004, size = 12, normalized size = 0.8 \begin{align*} \frac{3}{5} \,{\left (x^{\frac{2}{3}} + 1\right )}^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/5*(x^(2/3) + 1)^(5/2)