3.124 \(\int \frac {1}{(b x^n)^{2/3}} \, dx\)

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

[Out]

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

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

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x)/((3 - 2*n)*(b*x^n)^(2/3))

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 \frac {1}{\left (b x^n\right )^{2/3}} \, dx &=\frac {x^{2 n/3} \int x^{-2 n/3} \, dx}{\left (b x^n\right )^{2/3}}\\ &=\frac {3 x}{(3-2 n) \left (b x^n\right )^{2/3}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 1.05 \[ \frac {x}{\left (1-\frac {2 n}{3}\right ) \left (b x^n\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/((1 - (2*n)/3)*(b*x^n)^(2/3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^n)^(2/3),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^n)^(2/3),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(-(2*n)/3>0)', see `assume?` fo
r more details)Is -(2*n)/3 equal to -1?

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mupad [B]  time = 0.98, size = 26, normalized size = 1.37 \[ -\frac {3\,x^{1-n}\,{\left (b\,x^n\right )}^{1/3}}{b\,\left (2\,n-3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^n)^(2/3),x)

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**n)**(2/3),x)

[Out]

Timed out

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