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

Optimal. Leaf size=72 \[ \frac{3}{4} \sqrt{\pi } n^{3/2} x \left (a x^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )+x \log ^{\frac{3}{2}}\left (a x^n\right )-\frac{3}{2} n x \sqrt{\log \left (a x^n\right )} \]

[Out]

(3*n^(3/2)*Sqrt[Pi]*x*Erfi[Sqrt[Log[a*x^n]]/Sqrt[n]])/(4*(a*x^n)^n^(-1)) - (3*n*x*Sqrt[Log[a*x^n]])/2 + x*Log[
a*x^n]^(3/2)

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Rubi [A]  time = 0.0347186, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {2296, 2300, 2180, 2204} \[ \frac{3}{4} \sqrt{\pi } n^{3/2} x \left (a x^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )+x \log ^{\frac{3}{2}}\left (a x^n\right )-\frac{3}{2} n x \sqrt{\log \left (a x^n\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*n^(3/2)*Sqrt[Pi]*x*Erfi[Sqrt[Log[a*x^n]]/Sqrt[n]])/(4*(a*x^n)^n^(-1)) - (3*n*x*Sqrt[Log[a*x^n]])/2 + x*Log[
a*x^n]^(3/2)

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2300

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[E^(x/n)*(a +
b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, 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 2204

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

Rubi steps

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

Mathematica [A]  time = 0.0380405, size = 72, normalized size = 1. \[ \frac{3}{4} \sqrt{\pi } n^{3/2} x \left (a x^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )+x \log ^{\frac{3}{2}}\left (a x^n\right )-\frac{3}{2} n x \sqrt{\log \left (a x^n\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*n^(3/2)*Sqrt[Pi]*x*Erfi[Sqrt[Log[a*x^n]]/Sqrt[n]])/(4*(a*x^n)^n^(-1)) - (3*n*x*Sqrt[Log[a*x^n]])/2 + x*Log[
a*x^n]^(3/2)

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Maple [F]  time = 0.17, size = 0, normalized size = 0. \begin{align*} \int \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)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(log(a*x**n)**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log(a*x^n)^(3/2), x)