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

Optimal. Leaf size=20 \[ -\left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}} \]

[Out]

-1/((a*(b*x^m)^n)^(1/m/n))

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Rubi [A]  time = 0.04, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6679, 30} \[ -\left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*(b*x^m)^n)^(-(1/(m*n)))

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 20, normalized size = 1.00 \[ -\left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*(b*x^m)^n)^(-(1/(m*n)))

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fricas [A]  time = 0.73, size = 21, normalized size = 1.05 \[ -\frac {e^{\left (-\frac {n \log \relax (b) + \log \relax (a)}{m n}\right )}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^(-(n*log(b) + log(a))/(m*n))/x

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giac [A]  time = 0.17, size = 21, normalized size = 1.05 \[ -\frac {1}{\left (\left (b x^{m}\right )^{n} a\right )^{\frac {1}{m n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/((b*x^m)^n*a)^(1/(m*n))

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maple [A]  time = 0.00, size = 22, normalized size = 1.10 \[ -\left (a \left (b \,x^{m}\right )^{n}\right )^{-\frac {1}{m n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/((a*(b*x^m)^n)^(1/m/n))

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maxima [A]  time = 3.11, size = 35, normalized size = 1.75 \[ -\frac {1}{a^{\frac {1}{m n}} b^{\left (\frac {1}{m}\right )} {\left ({\left (x^{m}\right )}^{n}\right )}^{\frac {1}{m n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(a^(1/(m*n))*b^(1/m)*((x^m)^n)^(1/(m*n)))

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mupad [B]  time = 0.97, size = 21, normalized size = 1.05 \[ -\frac {1}{{\left (a\,{\left (b\,x^m\right )}^n\right )}^{\frac {1}{m\,n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(a*(b*x^m)^n)^(1/(m*n))

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sympy [A]  time = 5.88, size = 153, normalized size = 7.65 \[ \begin {cases} - \left (0^{m n}\right )^{- \frac {1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{- \frac {1}{m n}} \left (\left (\left (0^{m n}\right )^{\frac {1}{n}}\right )^{n}\right )^{- \frac {1}{m n}} & \text {for}\: a = 0^{m n} \wedge b = \left (0^{m n}\right )^{\frac {1}{n}} \\- a^{- \frac {1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{- \frac {1}{m n}} \left (\left (\left (0^{m n}\right )^{\frac {1}{n}}\right )^{n}\right )^{- \frac {1}{m n}} & \text {for}\: b = \left (0^{m n}\right )^{\frac {1}{n}} \\- \left (0^{m n}\right )^{- \frac {1}{m n}} \left (b^{n}\right )^{- \frac {1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{- \frac {1}{m n}} & \text {for}\: a = 0^{m n} \\- a^{- \frac {1}{m n}} \left (b^{n}\right )^{- \frac {1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{- \frac {1}{m n}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-(0**(m*n))**(-1/(m*n))*((x**m)**n)**(-1/(m*n))*(((0**(m*n))**(1/n))**n)**(-1/(m*n)), Eq(a, 0**(m*n
)) & Eq(b, (0**(m*n))**(1/n))), (-a**(-1/(m*n))*((x**m)**n)**(-1/(m*n))*(((0**(m*n))**(1/n))**n)**(-1/(m*n)),
Eq(b, (0**(m*n))**(1/n))), (-(0**(m*n))**(-1/(m*n))*(b**n)**(-1/(m*n))*((x**m)**n)**(-1/(m*n)), Eq(a, 0**(m*n)
)), (-a**(-1/(m*n))*(b**n)**(-1/(m*n))*((x**m)**n)**(-1/(m*n)), True))

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