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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, 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 30

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

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]

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 = 0.00, size = 17, normalized size = 1.00 \[ -\frac {2}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )} \]

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))

________________________________________________________________________________________

fricas [B]  time = 0.45, size = 37, normalized size = 2.18 \[ -\frac {2 \, \sqrt {n \log \relax (x) + \log \relax (a)}}{3 \, {\left (n^{3} \log \relax (x)^{2} + 2 \, n^{2} \log \relax (a) \log \relax (x) + n \log \relax (a)^{2}\right )}} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.25, size = 14, normalized size = 0.82 \[ -\frac {2}{3 \, {\left (n \log \relax (x) + \log \relax (a)\right )}^{\frac {3}{2}} n} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.03, size = 14, normalized size = 0.82 \[ -\frac {2}{3 n \ln \left (a \,x^{n}\right )^{\frac {3}{2}}} \]

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)

________________________________________________________________________________________

maxima [A]  time = 0.81, size = 13, normalized size = 0.76 \[ -\frac {2}{3 \, n \log \left (a x^{n}\right )^{\frac {3}{2}}} \]

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))

________________________________________________________________________________________

mupad [B]  time = 3.43, size = 13, normalized size = 0.76 \[ -\frac {2}{3\,n\,{\ln \left (a\,x^n\right )}^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________