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]

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

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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} \[ \frac {2 \log ^{\frac {5}{2}}\left (a x^n\right )}{5 n} \]

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 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 {\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*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \[ \frac {2 \log ^{\frac {5}{2}}\left (a x^n\right )}{5 n} \]

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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fricas [B]  time = 0.46, size = 34, normalized size = 2.00 \[ \frac {2 \, {\left (n^{2} \log \relax (x)^{2} + 2 \, n \log \relax (a) \log \relax (x) + \log \relax (a)^{2}\right )} \sqrt {n \log \relax (x) + \log \relax (a)}}{5 \, n} \]

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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giac [B]  time = 0.40, size = 72, normalized size = 4.24 \[ \frac {2 \, {\left (3 \, {\left (n \log \relax (x) + \log \relax (a)\right )}^{\frac {5}{2}} - 10 \, {\left (n \log \relax (x) + \log \relax (a)\right )}^{\frac {3}{2}} \log \relax (a) + 30 \, \sqrt {n \log \relax (x) + \log \relax (a)} \log \relax (a)^{2} + 10 \, {\left ({\left (n \log \relax (x) + \log \relax (a)\right )}^{\frac {3}{2}} - 3 \, \sqrt {n \log \relax (x) + \log \relax (a)} \log \relax (a)\right )} \log \relax (a)\right )}}{15 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [A]  time = 0.03, size = 14, normalized size = 0.82 \[ \frac {2 \ln \left (a \,x^{n}\right )^{\frac {5}{2}}}{5 n} \]

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 = 0.61, size = 13, normalized size = 0.76 \[ \frac {2 \, \log \left (a x^{n}\right )^{\frac {5}{2}}}{5 \, n} \]

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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mupad [B]  time = 3.53, size = 13, normalized size = 0.76 \[ \frac {2\,{\ln \left (a\,x^n\right )}^{5/2}}{5\,n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 21.92, size = 75, normalized size = 4.41 \[ \begin {cases} \frac {2 n \sqrt {n \log {\relax (x )} + \log {\relax (a )}} \log {\relax (x )}^{2}}{5} + \frac {4 \sqrt {n \log {\relax (x )} + \log {\relax (a )}} \log {\relax (a )} \log {\relax (x )}}{5} + \frac {2 \sqrt {n \log {\relax (x )} + \log {\relax (a )}} \log {\relax (a )}^{2}}{5 n} & \text {for}\: n \neq 0 \\\log {\relax (a )}^{\frac {3}{2}} \log {\relax (x )} & \text {otherwise} \end {cases} \]

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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