3.2.9 \(\int (b x^n)^{2/3} \, dx\) [109]

Optimal. Leaf size=19 \[ \frac {3 x \left (b x^n\right )^{2/3}}{3+2 n} \]

[Out]

3*x*(b*x^n)^(2/3)/(3+2*n)

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Rubi [A]
time = 0.00, antiderivative size = 19, 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 \left (b x^n\right )^{2/3}}{2 n+3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x*(b*x^n)^(2/3))/(3 + 2*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 \left (b x^n\right )^{2/3} \, dx &=\left (x^{-2 n/3} \left (b x^n\right )^{2/3}\right ) \int x^{2 n/3} \, dx\\ &=\frac {3 x \left (b x^n\right )^{2/3}}{3+2 n}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 20, normalized size = 1.05 \begin {gather*} \frac {x \left (b x^n\right )^{2/3}}{1+\frac {2 n}{3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(b*x^n)^(2/3))/(1 + (2*n)/3)

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Maple [A]
time = 0.01, size = 18, normalized size = 0.95

method	result	size
gosper	\(\frac {3 x \left (b \,x^{n}\right )^{\frac {2}{3}}}{3+2 n}\)	\(18\)
risch	\(\frac {3 b x \,x^{n}}{\left (3+2 n \right ) \left (b \,x^{n}\right )^{\frac {1}{3}}}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^n)^(2/3),x,method=_RETURNVERBOSE)

[Out]

3*x*(b*x^n)^(2/3)/(3+2*n)

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Maxima [A]
time = 0.29, size = 17, normalized size = 0.89 \begin {gather*} \frac {3 \, \left (b x^{n}\right )^{\frac {2}{3}} x}{2 \, n + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(b*x^n)^(2/3)*x/(2*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)^(2/3),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 \left (b x^{n}\right )^{\frac {2}{3}}}{2 n + 3} & \text {for}\: n \neq - \frac {3}{2} \\\int \left (\frac {b}{x^{\frac {3}{2}}}\right )^{\frac {2}{3}}\, dx & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((3*x*(b*x**n)**(2/3)/(2*n + 3), Ne(n, -3/2)), (Integral((b/x**(3/2))**(2/3), 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)^(2/3),x, algorithm="giac")

[Out]

integrate((b*x^n)^(2/3), x)

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Mupad [B]
time = 0.98, size = 17, normalized size = 0.89 \begin {gather*} \frac {3\,x\,{\left (b\,x^n\right )}^{2/3}}{2\,n+3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*x*(b*x^n)^(2/3))/(2*n + 3)

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