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

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Rubi [A]  time = 0.0135638, 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} \[ \frac{2 \log ^{\frac{3}{2}}\left (a x^n\right )}{3 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 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{\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 = 0.0015515, size = 17, normalized size = 1. \[ \frac{2 \log ^{\frac{3}{2}}\left (a x^n\right )}{3 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.038, 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(ln(a*x^n)^(1/2)/x,x)

[Out]

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

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Maxima [A]  time = 1.0185, size = 18, normalized size = 1.06 \begin{align*} \frac{2 \, \log \left (a x^{n}\right )^{\frac{3}{2}}}{3 \, n} \end{align*}

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

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Fricas [A]  time = 1.02915, size = 45, normalized size = 2.65 \begin{align*} \frac{2 \,{\left (n \log \left (x\right ) + \log \left (a\right )\right )}^{\frac{3}{2}}}{3 \, n} \end{align*}

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

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Sympy [A]  time = 1.92429, size = 29, normalized size = 1.71 \begin{align*} - \begin{cases} - \sqrt{\log{\left (a \right )}} \log{\left (x \right )} & \text{for}\: n = 0 \\- \frac{2 \log{\left (a x^{n} \right )}^{\frac{3}{2}}}{3 n} & \text{otherwise} \end{cases} \end{align*}

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))

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Giac [A]  time = 1.23235, size = 19, normalized size = 1.12 \begin{align*} \frac{2 \,{\left (n \log \left (x\right ) + \log \left (a\right )\right )}^{\frac{3}{2}}}{3 \, n} \end{align*}

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