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

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

[Out]

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

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Rubi [A]  time = 0.01, 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} \[ \frac {3 x \left (b x^n\right )^{2/3}}{2 n+3} \]

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 \[ \frac {x \left (b x^n\right )^{2/3}}{\frac {2 n}{3}+1} \]

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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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b x^{n}\right )^{\frac {2}{3}}\,{d x} \]

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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maple [A]  time = 0.00, size = 18, normalized size = 0.95 \[ \frac {3 \left (b \,x^{n}\right )^{\frac {2}{3}} x}{2 n +3} \]

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)/(3+2*n)

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maxima [A]  time = 1.34, size = 17, normalized size = 0.89 \[ \frac {3 \, \left (b x^{n}\right )^{\frac {2}{3}} x}{2 \, n + 3} \]

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

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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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} \frac {3 b^{\frac {2}{3}} x \left (x^{n}\right )^{\frac {2}{3}}}{2 n + 3} & \text {for}\: n \neq - \frac {3}{2} \\\int \left (\frac {b}{x^{\frac {3}{2}}}\right )^{\frac {2}{3}}\, dx & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((3*b**(2/3)*x*(x**n)**(2/3)/(2*n + 3), Ne(n, -3/2)), (Integral((b/x**(3/2))**(2/3), x), True))

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