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

Optimal. Leaf size=83 \[ \frac{2 \sqrt{\pi } \sqrt{m+1} x^{m+1} \left (a x^n\right )^{-\frac{m+1}{n}} \text{Erfi}\left (\frac{\sqrt{m+1} \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{n^{3/2}}-\frac{2 x^{m+1}}{n \sqrt{\log \left (a x^n\right )}} \]

[Out]

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

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Rubi [A]  time = 0.0679234, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2306, 2310, 2180, 2204} \[ \frac{2 \sqrt{\pi } \sqrt{m+1} x^{m+1} \left (a x^n\right )^{-\frac{m+1}{n}} \text{Erfi}\left (\frac{\sqrt{m+1} \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{n^{3/2}}-\frac{2 x^{m+1}}{n \sqrt{\log \left (a x^n\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2306

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log
[c*x^n])^(p + 1))/(b*d*n*(p + 1)), x] - Dist[(m + 1)/(b*n*(p + 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] && LtQ[p, -1]

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

Mathematica [A]  time = 0.193792, size = 86, normalized size = 1.04 \[ \frac{2 \sqrt{\pi } \sqrt{m+1} e^{-\frac{(m+1) \left (\log \left (a x^n\right )-n \log (x)\right )}{n}} \text{Erfi}\left (\frac{\sqrt{m+1} \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{n^{3/2}}-\frac{2 x^{m+1}}{n \sqrt{\log \left (a x^n\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.185, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m} \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(x^m/ln(a*x^n)^(3/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{\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(x^m/log(a*x^n)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^m/log(a*x^n)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^m/log(a*x^n)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**m/log(a*x**n)**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{\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(x^m/log(a*x^n)^(3/2),x, algorithm="giac")

[Out]

integrate(x^m/log(a*x^n)^(3/2), x)