3.125 \(\int \frac{\log ^{\frac{3}{2}}(a x^n)}{x} \, dx\)

Optimal. Leaf size=17 \[ \frac{2 \log ^{\frac{5}{2}}\left (a x^n\right )}{5 n} \]

[Out]

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

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Rubi [A]  time = 0.0136927, 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} \[ \frac{2 \log ^{\frac{5}{2}}\left (a x^n\right )}{5 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0017068, size = 17, normalized size = 1. \[ \frac{2 \log ^{\frac{5}{2}}\left (a x^n\right )}{5 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.039, size = 14, normalized size = 0.8 \begin{align*}{\frac{2}{5\,n} \left ( \ln \left ( a{x}^{n} \right ) \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.09763, size = 18, normalized size = 1.06 \begin{align*} \frac{2 \, \log \left (a x^{n}\right )^{\frac{5}{2}}}{5 \, n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(a*x^n)^(3/2)/x,x, algorithm="maxima")

[Out]

2/5*log(a*x^n)^(5/2)/n

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Fricas [B]  time = 0.993865, size = 104, normalized size = 6.12 \begin{align*} \frac{2 \,{\left (n^{2} \log \left (x\right )^{2} + 2 \, n \log \left (a\right ) \log \left (x\right ) + \log \left (a\right )^{2}\right )} \sqrt{n \log \left (x\right ) + \log \left (a\right )}}{5 \, n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(a*x^n)^(3/2)/x,x, algorithm="fricas")

[Out]

2/5*(n^2*log(x)^2 + 2*n*log(a)*log(x) + log(a)^2)*sqrt(n*log(x) + log(a))/n

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Sympy [A]  time = 37.1645, size = 75, normalized size = 4.41 \begin{align*} \begin{cases} \frac{2 n \sqrt{n \log{\left (x \right )} + \log{\left (a \right )}} \log{\left (x \right )}^{2}}{5} + \frac{4 \sqrt{n \log{\left (x \right )} + \log{\left (a \right )}} \log{\left (a \right )} \log{\left (x \right )}}{5} + \frac{2 \sqrt{n \log{\left (x \right )} + \log{\left (a \right )}} \log{\left (a \right )}^{2}}{5 n} & \text{for}\: n \neq 0 \\\log{\left (a \right )}^{\frac{3}{2}} \log{\left (x \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(a*x**n)**(3/2)/x,x)

[Out]

Piecewise((2*n*sqrt(n*log(x) + log(a))*log(x)**2/5 + 4*sqrt(n*log(x) + log(a))*log(a)*log(x)/5 + 2*sqrt(n*log(
x) + log(a))*log(a)**2/(5*n), Ne(n, 0)), (log(a)**(3/2)*log(x), True))

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Giac [A]  time = 1.33377, size = 19, normalized size = 1.12 \begin{align*} \frac{2 \,{\left (n \log \left (x\right ) + \log \left (a\right )\right )}^{\frac{5}{2}}}{5 \, n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(a*x^n)^(3/2)/x,x, algorithm="giac")

[Out]

2/5*(n*log(x) + log(a))^(5/2)/n