3.118 \(\int \frac {\sqrt {\log (a x^n)}}{x} \, dx\)

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

[Out]

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

________________________________________________________________________________________

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} \[ \frac {2 \log ^{\frac {3}{2}}\left (a x^n\right )}{3 n} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[Log[a*x^n]]/x,x]

[Out]

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

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

________________________________________________________________________________________

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

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Log[a*x^n]]/x,x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.45, size = 14, normalized size = 0.82 \[ \frac {2 \, {\left (n \log \relax (x) + \log \relax (a)\right )}^{\frac {3}{2}}}{3 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(n*log(x) + log(a))^(3/2)/n

________________________________________________________________________________________

giac [A]  time = 0.36, size = 14, normalized size = 0.82 \[ \frac {2 \, {\left (n \log \relax (x) + \log \relax (a)\right )}^{\frac {3}{2}}}{3 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(a*x^n)^(1/2)/x,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 \[ \frac {2 \ln \left (a \,x^{n}\right )^{\frac {3}{2}}}{3 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.64, size = 13, normalized size = 0.76 \[ \frac {2 \, \log \left (a x^{n}\right )^{\frac {3}{2}}}{3 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 3.54, size = 13, normalized size = 0.76 \[ \frac {2\,{\ln \left (a\,x^n\right )}^{3/2}}{3\,n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 1.17, size = 29, normalized size = 1.71 \[ - \begin {cases} - \sqrt {\log {\relax (a )}} \log {\relax (x )} & \text {for}\: n = 0 \\- \frac {2 \log {\left (a x^{n} \right )}^{\frac {3}{2}}}{3 n} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Piecewise((-sqrt(log(a))*log(x), Eq(n, 0)), (-2*log(a*x**n)**(3/2)/(3*n), True))

________________________________________________________________________________________