\(\int \frac {\log ^{\frac {3}{2}}(a x^n)}{x} \, dx\) [125]

Optimal result

Integrand size = 14, antiderivative size = 17 \[ \int \frac {\log ^{\frac {3}{2}}\left (a x^n\right )}{x} \, dx=\frac {2 \log ^{\frac {5}{2}}\left (a x^n\right )}{5 n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2339, 30} \[ \int \frac {\log ^{\frac {3}{2}}\left (a x^n\right )}{x} \, dx=\frac {2 \log ^{\frac {5}{2}}\left (a x^n\right )}{5 n} \]

[In]

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

[Out]

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

Rule 30

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

Rule 2339

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*} \text {integral}& = \frac {\text {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] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^{\frac {3}{2}}\left (a x^n\right )}{x} \, dx=\frac {2 \log ^{\frac {5}{2}}\left (a x^n\right )}{5 n} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82

method	result	size
derivativedivides	\(\frac {2 \ln \left (a \,x^{n}\right )^{\frac {5}{2}}}{5 n}\)	\(14\)
default	\(\frac {2 \ln \left (a \,x^{n}\right )^{\frac {5}{2}}}{5 n}\)	\(14\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (13) = 26\).

Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.00 \[ \int \frac {\log ^{\frac {3}{2}}\left (a x^n\right )}{x} \, dx=\frac {2 \, {\left (n^{2} \log \left (x\right )^{2} + 2 \, n \log \left (a\right ) \log \left (x\right ) + \log \left (a\right )^{2}\right )} \sqrt {n \log \left (x\right ) + \log \left (a\right )}}{5 \, n} \]

[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

Sympy [A] (verification not implemented)

Time = 1.46 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.41 \[ \int \frac {\log ^{\frac {3}{2}}\left (a x^n\right )}{x} \, dx=\begin {cases} \frac {2 \log {\left (a x^{n} \right )}^{\frac {5}{2}}}{5 n} & \text {for}\: n \neq 0 \\\log {\left (a \right )}^{\frac {3}{2}} \log {\left (x \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {\log ^{\frac {3}{2}}\left (a x^n\right )}{x} \, dx=\frac {2 \, \log \left (a x^{n}\right )^{\frac {5}{2}}}{5 \, n} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (13) = 26\).

Time = 0.30 (sec) , antiderivative size = 72, normalized size of antiderivative = 4.24 \[ \int \frac {\log ^{\frac {3}{2}}\left (a x^n\right )}{x} \, dx=\frac {2 \, {\left (3 \, {\left (n \log \left (x\right ) + \log \left (a\right )\right )}^{\frac {5}{2}} - 10 \, {\left (n \log \left (x\right ) + \log \left (a\right )\right )}^{\frac {3}{2}} \log \left (a\right ) + 30 \, \sqrt {n \log \left (x\right ) + \log \left (a\right )} \log \left (a\right )^{2} + 10 \, {\left ({\left (n \log \left (x\right ) + \log \left (a\right )\right )}^{\frac {3}{2}} - 3 \, \sqrt {n \log \left (x\right ) + \log \left (a\right )} \log \left (a\right )\right )} \log \left (a\right )\right )}}{15 \, n} \]

[In]

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

[Out]

2/15*(3*(n*log(x) + log(a))^(5/2) - 10*(n*log(x) + log(a))^(3/2)*log(a) + 30*sqrt(n*log(x) + log(a))*log(a)^2
+ 10*((n*log(x) + log(a))^(3/2) - 3*sqrt(n*log(x) + log(a))*log(a))*log(a))/n

Mupad [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {\log ^{\frac {3}{2}}\left (a x^n\right )}{x} \, dx=\frac {2\,{\ln \left (a\,x^n\right )}^{5/2}}{5\,n} \]

[In]

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

[Out]

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