3.290 \(\int \frac{1-\sqrt{x}}{\sqrt [3]{x}} \, dx\)

Optimal. Leaf size=19 \[ \frac{3 x^{2/3}}{2}-\frac{6 x^{7/6}}{7} \]

[Out]

(3*x^(2/3))/2 - (6*x^(7/6))/7

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Rubi [A]  time = 0.003367, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {14} \[ \frac{3 x^{2/3}}{2}-\frac{6 x^{7/6}}{7} \]

Antiderivative was successfully verified.

[In]

Int[(1 - Sqrt[x])/x^(1/3),x]

[Out]

(3*x^(2/3))/2 - (6*x^(7/6))/7

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \frac{1-\sqrt{x}}{\sqrt [3]{x}} \, dx &=\int \left (\frac{1}{\sqrt [3]{x}}-\sqrt [6]{x}\right ) \, dx\\ &=\frac{3 x^{2/3}}{2}-\frac{6 x^{7/6}}{7}\\ \end{align*}

Mathematica [A]  time = 0.0034718, size = 19, normalized size = 1. \[ \frac{3 x^{2/3}}{2}-\frac{6 x^{7/6}}{7} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - Sqrt[x])/x^(1/3),x]

[Out]

(3*x^(2/3))/2 - (6*x^(7/6))/7

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Maple [A]  time = 0.002, size = 12, normalized size = 0.6 \begin{align*}{\frac{3}{2}{x}^{{\frac{2}{3}}}}-{\frac{6}{7}{x}^{{\frac{7}{6}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*x^(2/3)-6/7*x^(7/6)

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Maxima [A]  time = 0.939876, size = 15, normalized size = 0.79 \begin{align*} -\frac{6}{7} \, x^{\frac{7}{6}} + \frac{3}{2} \, x^{\frac{2}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6/7*x^(7/6) + 3/2*x^(2/3)

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Fricas [A]  time = 1.90134, size = 38, normalized size = 2. \begin{align*} -\frac{6}{7} \, x^{\frac{7}{6}} + \frac{3}{2} \, x^{\frac{2}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6/7*x^(7/6) + 3/2*x^(2/3)

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Sympy [A]  time = 1.85478, size = 15, normalized size = 0.79 \begin{align*} - \frac{6 x^{\frac{7}{6}}}{7} + \frac{3 x^{\frac{2}{3}}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6*x**(7/6)/7 + 3*x**(2/3)/2

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Giac [A]  time = 1.05121, size = 15, normalized size = 0.79 \begin{align*} -\frac{6}{7} \, x^{\frac{7}{6}} + \frac{3}{2} \, x^{\frac{2}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6/7*x^(7/6) + 3/2*x^(2/3)