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

Optimal. Leaf size=90 \[ \frac{3 \sqrt{\frac{\pi }{2}} n^{3/2} \left (a x^n\right )^{2/n} \text{Erf}\left (\frac{\sqrt{2} \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{16 x^2}-\frac{\log ^{\frac{3}{2}}\left (a x^n\right )}{2 x^2}-\frac{3 n \sqrt{\log \left (a x^n\right )}}{8 x^2} \]

[Out]

(3*n^(3/2)*Sqrt[Pi/2]*(a*x^n)^(2/n)*Erf[(Sqrt[2]*Sqrt[Log[a*x^n]])/Sqrt[n]])/(16*x^2) - (3*n*Sqrt[Log[a*x^n]])
/(8*x^2) - Log[a*x^n]^(3/2)/(2*x^2)

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Rubi [A]  time = 0.0725493, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2305, 2310, 2180, 2205} \[ \frac{3 \sqrt{\frac{\pi }{2}} n^{3/2} \left (a x^n\right )^{2/n} \text{Erf}\left (\frac{\sqrt{2} \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{16 x^2}-\frac{\log ^{\frac{3}{2}}\left (a x^n\right )}{2 x^2}-\frac{3 n \sqrt{\log \left (a x^n\right )}}{8 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*n^(3/2)*Sqrt[Pi/2]*(a*x^n)^(2/n)*Erf[(Sqrt[2]*Sqrt[Log[a*x^n]])/Sqrt[n]])/(16*x^2) - (3*n*Sqrt[Log[a*x^n]])
/(8*x^2) - Log[a*x^n]^(3/2)/(2*x^2)

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2310

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*n*(c*x^n
)^((m + 1)/n)), Subst[Int[E^(((m + 1)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rule 2180

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - (c*
f)/d) + (f*g*x^2)/d), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2205

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]),
2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{\log ^{\frac{3}{2}}\left (a x^n\right )}{x^3} \, dx &=-\frac{\log ^{\frac{3}{2}}\left (a x^n\right )}{2 x^2}+\frac{1}{4} (3 n) \int \frac{\sqrt{\log \left (a x^n\right )}}{x^3} \, dx\\ &=-\frac{3 n \sqrt{\log \left (a x^n\right )}}{8 x^2}-\frac{\log ^{\frac{3}{2}}\left (a x^n\right )}{2 x^2}+\frac{1}{16} \left (3 n^2\right ) \int \frac{1}{x^3 \sqrt{\log \left (a x^n\right )}} \, dx\\ &=-\frac{3 n \sqrt{\log \left (a x^n\right )}}{8 x^2}-\frac{\log ^{\frac{3}{2}}\left (a x^n\right )}{2 x^2}+\frac{\left (3 n \left (a x^n\right )^{2/n}\right ) \operatorname{Subst}\left (\int \frac{e^{-\frac{2 x}{n}}}{\sqrt{x}} \, dx,x,\log \left (a x^n\right )\right )}{16 x^2}\\ &=-\frac{3 n \sqrt{\log \left (a x^n\right )}}{8 x^2}-\frac{\log ^{\frac{3}{2}}\left (a x^n\right )}{2 x^2}+\frac{\left (3 n \left (a x^n\right )^{2/n}\right ) \operatorname{Subst}\left (\int e^{-\frac{2 x^2}{n}} \, dx,x,\sqrt{\log \left (a x^n\right )}\right )}{8 x^2}\\ &=\frac{3 n^{3/2} \sqrt{\frac{\pi }{2}} \left (a x^n\right )^{2/n} \text{erf}\left (\frac{\sqrt{2} \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{16 x^2}-\frac{3 n \sqrt{\log \left (a x^n\right )}}{8 x^2}-\frac{\log ^{\frac{3}{2}}\left (a x^n\right )}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0638241, size = 88, normalized size = 0.98 \[ -\frac{3 \sqrt{2} n^2 \left (a x^n\right )^{2/n} \sqrt{\frac{\log \left (a x^n\right )}{n}} \text{Gamma}\left (\frac{1}{2},\frac{2 \log \left (a x^n\right )}{n}\right )+4 \log \left (a x^n\right ) \left (4 \log \left (a x^n\right )+3 n\right )}{32 x^2 \sqrt{\log \left (a x^n\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(3*Sqrt[2]*n^2*(a*x^n)^(2/n)*Gamma[1/2, (2*Log[a*x^n])/n]*Sqrt[Log[a*x^n]/n] + 4*Log[a*x^n]*(3*n + 4*Log[a*x^
n]))/(32*x^2*Sqrt[Log[a*x^n]])

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Maple [F]  time = 0.174, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \left ( \ln \left ( a{x}^{n} \right ) \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (a x^{n}\right )^{\frac{3}{2}}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log(a*x^n)^(3/2)/x^3, x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log{\left (a x^{n} \right )}^{\frac{3}{2}}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(log(a*x**n)**(3/2)/x**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (a x^{n}\right )^{\frac{3}{2}}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log(a*x^n)^(3/2)/x^3, x)