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

Optimal result

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

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

Mathematica [A] (verified)

Time = 2.28 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.04 \[ \int \frac {x^m}{\log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=\frac {2 e^{-\frac {(1+m) \left (-n \log (x)+\log \left (a x^n\right )\right )}{n}} \sqrt {1+m} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {1+m} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{n^{3/2}}-\frac {2 x^{1+m}}{n \sqrt {\log \left (a x^n\right )}} \] Input:

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

Output:

(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[L 
og[a*x^n]])
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2743, 2747, 2611, 2633}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2743

\(\displaystyle \frac {2 (m+1) \int \frac {x^m}{\sqrt {\log \left (a x^n\right )}}dx}{n}-\frac {2 x^{m+1}}{n \sqrt {\log \left (a x^n\right )}}\)

\(\Big \downarrow \) 2747

\(\displaystyle \frac {2 (m+1) x^{m+1} \left (a x^n\right )^{-\frac {m+1}{n}} \int \frac {\left (a x^n\right )^{\frac {m+1}{n}}}{\sqrt {\log \left (a x^n\right )}}d\log \left (a x^n\right )}{n^2}-\frac {2 x^{m+1}}{n \sqrt {\log \left (a x^n\right )}}\)

\(\Big \downarrow \) 2611

\(\displaystyle \frac {4 (m+1) x^{m+1} \left (a x^n\right )^{-\frac {m+1}{n}} \int \left (a x^n\right )^{\frac {m+1}{n}}d\sqrt {\log \left (a x^n\right )}}{n^2}-\frac {2 x^{m+1}}{n \sqrt {\log \left (a x^n\right )}}\)

\(\Big \downarrow \) 2633

\(\displaystyle \frac {2 \sqrt {\pi } \sqrt {m+1} x^{m+1} \left (a x^n\right )^{-\frac {m+1}{n}} \text {erfi}\left (\frac {\sqrt {m+1} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{n^{3/2}}-\frac {2 x^{m+1}}{n \sqrt {\log \left (a x^n\right )}}\)

Input:

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

Output:

(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]])
 

Defintions of rubi rules used

rule 2611

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : 
> Simp[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 2633

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 2743

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] - 
Simp[(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 2747

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol 
] :> Simp[(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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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