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 = 0.0050974, antiderivative size = 19, 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} \[ \frac{3 x \left (b x^n\right )^{2/3}}{2 n+3} \]

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*}

Mathematica [A]  time = 0.002495, 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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Maple [A]  time = 0.001, size = 18, normalized size = 1. \begin{align*} 3\,{\frac{x \left ( b{x}^{n} \right ) ^{2/3}}{3+2\,n}} \end{align*}

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Giac [A]  time = 1.20755, size = 24, normalized size = 1.26 \begin{align*} \frac{3 \, b^{\frac{2}{3}} x x^{\frac{2}{3} \, n}}{2 \, n + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*b^(2/3)*x*x^(2/3*n)/(2*n + 3)