\(\int \frac {x^m}{\log ^{\frac {3}{2}}(a x^n)} \, dx\) [165]

Optimal result

Integrand size = 14, antiderivative size = 83 \[ \int \frac {x^m}{\log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=\frac {2 \sqrt {1+m} \sqrt {\pi } x^{1+m} \left (a x^n\right )^{-\frac {1+m}{n}} \text {erfi}\left (\frac {\sqrt {1+m} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{n^{3/2}}-\frac {2 x^{1+m}}{n \sqrt {\log \left (a x^n\right )}} \]

[Out]

2*x^(1+m)*erfi((1+m)^(1/2)*ln(a*x^n)^(1/2)/n^(1/2))*(1+m)^(1/2)*Pi^(1/2)/n^(3/2)/((a*x^n)^((1+m)/n))-2*x^(1+m)
/n/ln(a*x^n)^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2343, 2347, 2211, 2235} \[ \int \frac {x^m}{\log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=\frac {2 \sqrt {\pi } \sqrt {m+1} x^{m+1} \left (a x^n\right )^{-\frac {m+1}{n}} \text {erfi}\left (\frac {\sqrt {m+1} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{n^{3/2}}-\frac {2 x^{m+1}}{n \sqrt {\log \left (a x^n\right )}} \]

[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 2211

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] &&  !TrueQ[$UseGamma]

Rule 2235

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]

Rule 2343

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 2347

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)/n)*x)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rubi steps \begin{align*} \text {integral}& = -\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 ) \text {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 ) \text {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] (verified)

Time = 1.28 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.88 \[ \int \frac {x^m}{\log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=\frac {2 x^{1+m} \left (-1+\left (a x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1}{2},-\frac {(1+m) \log \left (a x^n\right )}{n}\right ) \sqrt {-\frac {(1+m) \log \left (a x^n\right )}{n}}\right )}{n \sqrt {\log \left (a x^n\right )}} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {x^{m}}{\ln \left (a \,x^{n}\right )^{\frac {3}{2}}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {x^m}{\log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=\int { \frac {x^{m}}{\log \left (a x^{n}\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

integral(x^m/log(a*x^n)^(3/2), x)

Sympy [F]

\[ \int \frac {x^m}{\log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=\int \frac {x^{m}}{\log {\left (a x^{n} \right )}^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

Integral(x**m/log(a*x**n)**(3/2), x)

Maxima [F]

\[ \int \frac {x^m}{\log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=\int { \frac {x^{m}}{\log \left (a x^{n}\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

integrate(x^m/log(a*x^n)^(3/2), x)

Giac [F]

\[ \int \frac {x^m}{\log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=\int { \frac {x^{m}}{\log \left (a x^{n}\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

integrate(x^m/log(a*x^n)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^m}{\log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=\int \frac {x^m}{{\ln \left (a\,x^n\right )}^{3/2}} \,d x \]

[In]

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

[Out]

int(x^m/log(a*x^n)^(3/2), x)