3.110 \(\int (b x^2)^{2/3} \, dx\)

Optimal. Leaf size=14 \[ \frac{3}{7} x \left (b x^2\right )^{2/3} \]

[Out]

(3*x*(b*x^2)^(2/3))/7

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Rubi [A]  time = 0.0014176, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {15, 30} \[ \frac{3}{7} x \left (b x^2\right )^{2/3} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2)^(2/3),x]

[Out]

(3*x*(b*x^2)^(2/3))/7

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \left (b x^2\right )^{2/3} \, dx &=\frac{\left (b x^2\right )^{2/3} \int x^{4/3} \, dx}{x^{4/3}}\\ &=\frac{3}{7} x \left (b x^2\right )^{2/3}\\ \end{align*}

Mathematica [A]  time = 0.0011185, size = 14, normalized size = 1. \[ \frac{3}{7} x \left (b x^2\right )^{2/3} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2)^(2/3),x]

[Out]

(3*x*(b*x^2)^(2/3))/7

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Maple [A]  time = 0.001, size = 11, normalized size = 0.8 \begin{align*}{\frac{3\,x}{7} \left ( b{x}^{2} \right ) ^{{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2)^(2/3),x)

[Out]

3/7*x*(b*x^2)^(2/3)

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Maxima [A]  time = 0.981949, size = 14, normalized size = 1. \begin{align*} \frac{3}{7} \, \left (b x^{2}\right )^{\frac{2}{3}} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^(2/3),x, algorithm="maxima")

[Out]

3/7*(b*x^2)^(2/3)*x

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Fricas [A]  time = 1.72182, size = 28, normalized size = 2. \begin{align*} \frac{3}{7} \, \left (b x^{2}\right )^{\frac{2}{3}} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^(2/3),x, algorithm="fricas")

[Out]

3/7*(b*x^2)^(2/3)*x

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Sympy [A]  time = 0.374301, size = 15, normalized size = 1.07 \begin{align*} \frac{3 b^{\frac{2}{3}} x \left (x^{2}\right )^{\frac{2}{3}}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2)**(2/3),x)

[Out]

3*b**(2/3)*x*(x**2)**(2/3)/7

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b x^{2}\right )^{\frac{2}{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^(2/3),x, algorithm="giac")

[Out]

integrate((b*x^2)^(2/3), x)