3.2.1 \(\int \sqrt [3]{b x^n} \, dx\) [101]

Optimal. Leaf size=17 \[ \frac {3 x \sqrt [3]{b x^n}}{3+n} \]

[Out]

3*x*(b*x^n)^(1/3)/(3+n)

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Rubi [A]
time = 0.00, antiderivative size = 17, 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 {3 x \sqrt [3]{b x^n}}{n+3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x*(b*x^n)^(1/3))/(3 + n)

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^n} \, dx &=\left (x^{-n/3} \sqrt [3]{b x^n}\right ) \int x^{n/3} \, dx\\ &=\frac {3 x \sqrt [3]{b x^n}}{3+n}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {3 x \sqrt [3]{b x^n}}{3+n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x*(b*x^n)^(1/3))/(3 + n)

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Maple [A]
time = 0.01, size = 16, normalized size = 0.94

method	result	size
gosper	\(\frac {3 x \left (b \,x^{n}\right )^{\frac {1}{3}}}{3+n}\)	\(16\)
risch	\(\frac {3 x \left (b \,x^{n}\right )^{\frac {1}{3}}}{3+n}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x*(b*x^n)^(1/3)/(3+n)

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Maxima [A]
time = 0.30, size = 15, normalized size = 0.88 \begin {gather*} \frac {3 \, \left (b x^{n}\right )^{\frac {1}{3}} x}{n + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(b*x^n)^(1/3)*x/(n + 3)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} \frac {3 x \sqrt [3]{b x^{n}}}{n + 3} & \text {for}\: n \neq -3 \\\int \sqrt [3]{\frac {b}{x^{3}}}\, dx & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((3*x*(b*x**n)**(1/3)/(n + 3), Ne(n, -3)), (Integral((b/x**3)**(1/3), x), True))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^n)^(1/3), x)

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Mupad [B]
time = 1.00, size = 15, normalized size = 0.88 \begin {gather*} \frac {3\,x\,{\left (b\,x^n\right )}^{1/3}}{n+3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*x*(b*x^n)^(1/3))/(n + 3)

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