3.102 \(\int \sqrt [3]{b x^3} \, dx\)

Optimal. Leaf size=14 \[ \frac{1}{2} x \sqrt [3]{b x^3} \]

[Out]

(x*(b*x^3)^(1/3))/2

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Rubi [A]  time = 0.0013163, 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} \[ \frac{1}{2} x \sqrt [3]{b x^3} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^3)^(1/3),x]

[Out]

(x*(b*x^3)^(1/3))/2

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 \sqrt [3]{b x^3} \, dx &=\frac{\sqrt [3]{b x^3} \int x \, dx}{x}\\ &=\frac{1}{2} x \sqrt [3]{b x^3}\\ \end{align*}

Mathematica [A]  time = 0.0010876, size = 14, normalized size = 1. \[ \frac{1}{2} x \sqrt [3]{b x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^3)^(1/3),x]

[Out]

(x*(b*x^3)^(1/3))/2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3)^(1/3),x)

[Out]

1/2*x*(b*x^3)^(1/3)

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Maxima [A]  time = 0.965428, size = 14, normalized size = 1. \begin{align*} \frac{1}{2} \, \left (b x^{3}\right )^{\frac{1}{3}} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3)^(1/3),x, algorithm="maxima")

[Out]

1/2*(b*x^3)^(1/3)*x

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Fricas [A]  time = 1.65456, size = 28, normalized size = 2. \begin{align*} \frac{1}{2} \, \left (b x^{3}\right )^{\frac{1}{3}} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3)^(1/3),x, algorithm="fricas")

[Out]

1/2*(b*x^3)^(1/3)*x

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Sympy [A]  time = 0.217048, size = 14, normalized size = 1. \begin{align*} \frac{\sqrt [3]{b} x \sqrt [3]{x^{3}}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3)**(1/3),x)

[Out]

b**(1/3)*x*(x**3)**(1/3)/2

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Giac [A]  time = 1.14929, size = 11, normalized size = 0.79 \begin{align*} \frac{1}{2} \, b^{\frac{1}{3}} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3)^(1/3),x, algorithm="giac")

[Out]

1/2*b^(1/3)*x^2