3.2.2 \(\int \sqrt [3]{b x^3} \, dx\) [102]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {1}{2} x \sqrt [3]{b x^3} \end {gather*}

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*}

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Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {1}{2} x \sqrt [3]{b x^3} \end {gather*}

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 = 0.03, size = 11, normalized size = 0.79

method	result	size
gosper	\(\frac {x \left (b \,x^{3}\right )^{\frac {1}{3}}}{2}\)	\(11\)
risch	\(\frac {x \left (b \,x^{3}\right )^{\frac {1}{3}}}{2}\)	\(11\)
trager	\(\frac {\left (x -1\right ) \left (x +1\right ) \left (b \,x^{3}\right )^{\frac {1}{3}}}{2 x}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3)^(1/3),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.30, size = 10, normalized size = 0.71 \begin {gather*} \frac {1}{2} \, \left (b x^{3}\right )^{\frac {1}{3}} x \end {gather*}

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 = 0.36, size = 10, normalized size = 0.71 \begin {gather*} \frac {1}{2} \, \left (b x^{3}\right )^{\frac {1}{3}} x \end {gather*}

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 = 0.06, size = 10, normalized size = 0.71 \begin {gather*} \frac {x \sqrt [3]{b x^{3}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.85, size = 10, normalized size = 0.71 \begin {gather*} \frac {1}{2} \, \left (b x^{3}\right )^{\frac {1}{3}} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.07 \begin {gather*} \int {\left (b\,x^3\right )}^{1/3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((b*x^3)^(1/3), x)

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