3.198 \(\int (a (b x^m)^n)^{-\frac{1}{m n}} \, dx\)

Optimal. Leaf size=22 \[ x \log (x) \left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}} \]

[Out]

(x*Log[x])/(a*(b*x^m)^n)^(1/(m*n))

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Rubi [A]  time = 0.0109022, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6679, 29} \[ x \log (x) \left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}} \]

Antiderivative was successfully verified.

[In]

Int[(a*(b*x^m)^n)^(-(1/(m*n))),x]

[Out]

(x*Log[x])/(a*(b*x^m)^n)^(1/(m*n))

Rule 6679

Int[(u_.)*((c_.)*((d_.)*((a_.) + (b_.)*(x_))^(n_))^(p_))^(q_), x_Symbol] :> Dist[(c*(d*(a + b*x)^n)^p)^q/(a +
b*x)^(n*p*q), Int[u*(a + b*x)^(n*p*q), x], x] /; FreeQ[{a, b, c, d, n, p, q}, x] &&  !IntegerQ[p] &&  !Integer
Q[q]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

\begin{align*} \int \left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}} \, dx &=\left (x \left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}}\right ) \int \frac{1}{x} \, dx\\ &=x \left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}} \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0017707, size = 22, normalized size = 1. \[ x \log (x) \left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*(b*x^m)^n)^(-(1/(m*n))),x]

[Out]

(x*Log[x])/(a*(b*x^m)^n)^(1/(m*n))

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \begin{align*} \int \left ( \left ( a \left ( b{x}^{m} \right ) ^{n} \right ) ^{{\frac{1}{mn}}} \right ) ^{-1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*(b*x^m)^n)^(1/m/n)),x)

[Out]

int(1/((a*(b*x^m)^n)^(1/m/n)),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (\left (b x^{m}\right )^{n} a\right )^{\frac{1}{m n}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(((b*x^m)^n*a)^(1/(m*n))), x)

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Fricas [A]  time = 1.83432, size = 53, normalized size = 2.41 \begin{align*} e^{\left (-\frac{n \log \left (b\right ) + \log \left (a\right )}{m n}\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(-(n*log(b) + log(a))/(m*n))*log(x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \left (b x^{m}\right )^{n}\right )^{- \frac{1}{m n}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*(b*x**m)**n)**(1/m/n)),x)

[Out]

Integral((a*(b*x**m)**n)**(-1/(m*n)), x)

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Giac [A]  time = 1.16167, size = 26, normalized size = 1.18 \begin{align*} e^{\left (-\frac{n \log \left (b\right ) + \log \left (a\right )}{m n}\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(-(n*log(b) + log(a))/(m*n))*log(x)