3.146 \(\int \frac{1}{x \log ^{\frac{5}{2}}(a x^n)} \, dx\)

Optimal. Leaf size=17 \[ -\frac{2}{3 n \log ^{\frac{3}{2}}\left (a x^n\right )} \]

[Out]

-2/(3*n*Log[a*x^n]^(3/2))

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Rubi [A]  time = 0.0139537, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2302, 30} \[ -\frac{2}{3 n \log ^{\frac{3}{2}}\left (a x^n\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Log[a*x^n]^(5/2)),x]

[Out]

-2/(3*n*Log[a*x^n]^(3/2))

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

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}{x \log ^{\frac{5}{2}}\left (a x^n\right )} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^{5/2}} \, dx,x,\log \left (a x^n\right )\right )}{n}\\ &=-\frac{2}{3 n \log ^{\frac{3}{2}}\left (a x^n\right )}\\ \end{align*}

Mathematica [A]  time = 0.001796, size = 17, normalized size = 1. \[ -\frac{2}{3 n \log ^{\frac{3}{2}}\left (a x^n\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Log[a*x^n]^(5/2)),x]

[Out]

-2/(3*n*Log[a*x^n]^(3/2))

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Maple [A]  time = 0.039, size = 14, normalized size = 0.8 \begin{align*} -{\frac{2}{3\,n} \left ( \ln \left ( a{x}^{n} \right ) \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/ln(a*x^n)^(5/2),x)

[Out]

-2/3/n/ln(a*x^n)^(3/2)

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Maxima [A]  time = 0.979796, size = 18, normalized size = 1.06 \begin{align*} -\frac{2}{3 \, n \log \left (a x^{n}\right )^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/log(a*x^n)^(5/2),x, algorithm="maxima")

[Out]

-2/3/(n*log(a*x^n)^(3/2))

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Fricas [B]  time = 0.984485, size = 108, normalized size = 6.35 \begin{align*} -\frac{2 \, \sqrt{n \log \left (x\right ) + \log \left (a\right )}}{3 \,{\left (n^{3} \log \left (x\right )^{2} + 2 \, n^{2} \log \left (a\right ) \log \left (x\right ) + n \log \left (a\right )^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/log(a*x^n)^(5/2),x, algorithm="fricas")

[Out]

-2/3*sqrt(n*log(x) + log(a))/(n^3*log(x)^2 + 2*n^2*log(a)*log(x) + n*log(a)^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/ln(a*x**n)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 1.27013, size = 19, normalized size = 1.12 \begin{align*} -\frac{2}{3 \,{\left (n \log \left (x\right ) + \log \left (a\right )\right )}^{\frac{3}{2}} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/log(a*x^n)^(5/2),x, algorithm="giac")

[Out]

-2/3/((n*log(x) + log(a))^(3/2)*n)