\(\int \frac {x^m}{W(a x)} \, dx\) [82]

Optimal result

Integrand size = 10, antiderivative size = 105 \[ \int \frac {x^m}{W(a x)} \, dx=\frac {e^{-m W(a x)} x^m \Gamma (m,(-1-m) W(a x)) ((-1-m) W(a x))^{1-m}}{a (1+m) W(a x)}+\frac {e^{-m W(a x)} x^m \Gamma (1+m,(-1-m) W(a x)) ((-1-m) W(a x))^{-m}}{a (1+m)} \] Output:

x^m*GAMMA(m,(-1-m)*LambertW(a*x))*((-1-m)*LambertW(a*x))^(1-m)/a/exp(m*Lam 
bertW(a*x))/(1+m)/LambertW(a*x)+x^m*GAMMA(1+m,(-1-m)*LambertW(a*x))/a/exp( 
m*LambertW(a*x))/(1+m)/(((-1-m)*LambertW(a*x))^m)
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.61 \[ \int \frac {x^m}{W(a x)} \, dx=\frac {e^{-m W(a x)} x^m (-((1+m) \Gamma (m,-((1+m) W(a x))))+\Gamma (1+m,-((1+m) W(a x)))) (-((1+m) W(a x)))^{-m}}{a (1+m)} \] Input:

Integrate[x^m/ProductLog[a*x],x]
 

Output:

(x^m*(-((1 + m)*Gamma[m, -((1 + m)*ProductLog[a*x])]) + Gamma[1 + m, -((1 
+ m)*ProductLog[a*x])]))/(a*E^(m*ProductLog[a*x])*(1 + m)*(-((1 + m)*Produ 
ctLog[a*x]))^m)
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {7174, 7197, 7207}

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.

\(\displaystyle \int \frac {x^m}{W(a x)} \, dx\)

\(\Big \downarrow \) 7174

\(\displaystyle \int \frac {x^m}{W(a x)+1}dx+\int \frac {x^m}{W(a x) (W(a x)+1)}dx\)

\(\Big \downarrow \) 7197

\(\displaystyle \int \frac {x^m}{W(a x) (W(a x)+1)}dx+\frac {x^m e^{-m W(a x)} (-((m+1) W(a x)))^{-m} \Gamma (m+1,-((m+1) W(a x)))}{a (m+1)}\)

\(\Big \downarrow \) 7207

\(\displaystyle \frac {x^m e^{-m W(a x)} (-((m+1) W(a x)))^{1-m} \Gamma (m,-((m+1) W(a x)))}{a (m+1) W(a x)}+\frac {x^m e^{-m W(a x)} (-((m+1) W(a x)))^{-m} \Gamma (m+1,-((m+1) W(a x)))}{a (m+1)}\)

Input:

Int[x^m/ProductLog[a*x],x]
 

Output:

(x^m*Gamma[m, -((1 + m)*ProductLog[a*x])]*(-((1 + m)*ProductLog[a*x]))^(1 
- m))/(a*E^(m*ProductLog[a*x])*(1 + m)*ProductLog[a*x]) + (x^m*Gamma[1 + m 
, -((1 + m)*ProductLog[a*x])])/(a*E^(m*ProductLog[a*x])*(1 + m)*(-((1 + m) 
*ProductLog[a*x]))^m)
 

Defintions of rubi rules used

rule 7174

Int[(x_)^(m_.)*((c_.)*ProductLog[(a_.)*(x_)])^(p_.), x_Symbol] :> Int[x^m*( 
(c*ProductLog[a*x])^p/(1 + ProductLog[a*x])), x] + Simp[1/c   Int[x^m*((c*P 
roductLog[a*x])^(p + 1)/(1 + ProductLog[a*x])), x], x] /; FreeQ[{a, c, m}, 
x]
 

rule 7197

Int[(x_)^(m_.)/((d_) + (d_.)*ProductLog[(a_.)*(x_)]), x_Symbol] :> Simp[x^m 
*(Gamma[m + 1, (-(m + 1))*ProductLog[a*x]]/(a*d*(m + 1)*E^(m*ProductLog[a*x 
])*((-(m + 1))*ProductLog[a*x])^m)), x] /; FreeQ[{a, d, m}, x] &&  !Integer 
Q[m]
 

rule 7207

Int[((x_)^(m_.)*((c_.)*ProductLog[(a_.)*(x_)])^(p_.))/((d_) + (d_.)*Product 
Log[(a_.)*(x_)]), x_Symbol] :> Simp[x^m*Gamma[m + p + 1, (-(m + 1))*Product 
Log[a*x]]*((c*ProductLog[a*x])^p/(a*d*(m + 1)*E^(m*ProductLog[a*x])*((-(m + 
 1))*ProductLog[a*x])^(m + p))), x] /; FreeQ[{a, c, d, m, p}, x] && NeQ[m, 
-1]
 

Maple [F]

\[\int \frac {x^{m}}{\operatorname {LambertW}\left (x a \right )}d x\]

Input:

int(x^m/LambertW(x*a),x)
 

Output:

int(x^m/LambertW(x*a),x)
 

Fricas [F]

\[ \int \frac {x^m}{W(a x)} \, dx=\int { \frac {x^{m}}{\operatorname {W}({a x})} \,d x } \] Input:

integrate(x^m/lambert_w(a*x),x, algorithm="fricas")
 

Output:

integral(x^m/lambert_w(a*x), x)
 

Sympy [F]

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

integrate(x**m/LambertW(a*x),x)
 

Output:

Integral(x**m/LambertW(a*x), x)
 

Maxima [F]

\[ \int \frac {x^m}{W(a x)} \, dx=\int { \frac {x^{m}}{\operatorname {W}({a x})} \,d x } \] Input:

integrate(x^m/lambert_w(a*x),x, algorithm="maxima")
 

Output:

integrate(x^m/lambert_w(a*x), x)
 

Giac [F]

\[ \int \frac {x^m}{W(a x)} \, dx=\int { \frac {x^{m}}{\operatorname {W}({a x})} \,d x } \] Input:

integrate(x^m/lambert_w(a*x),x, algorithm="giac")
 

Output:

integrate(x^m/lambert_w(a*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^m}{W(a x)} \, dx=\int \frac {x^m}{\mathrm {LambertW}\left (a\,x\right )} \,d x \] Input:

int(x^m/LambertW(a*x),x)
 

Output:

int(x^m/LambertW(a*x), x)
 

Reduce [F]

\[ \int \frac {x^m}{W(a x)} \, dx=\frac {x^{m} x -\left (\int x^{m}d x \right ) m -\left (\int x^{m}d x \right )+2 \left (\int \frac {x^{m}}{\textit {lambert\_w} \left (a x \right )}d x \right ) m +2 \left (\int \frac {x^{m}}{\textit {lambert\_w} \left (a x \right )}d x \right )}{2 m +2} \] Input:

int(x^m/Lambert_W(a*x),x)
 

Output:

(x**m*x - int(x**m,x)*m - int(x**m,x) + 2*int(x**m/lambert_w(a*x),x)*m + 2 
*int(x**m/lambert_w(a*x),x))/(2*(m + 1))