\(\int x^m W(a x^n) \, dx\) [244]

Optimal result

Integrand size = 10, antiderivative size = 181 \[ \int x^m W\left (a x^n\right ) \, dx=\frac {e^{\left (1-\frac {1+m}{n}\right ) W\left (a x^n\right )} x^{1+m-n} \Gamma \left (\frac {1+m+n}{n},-\frac {(1+m) W\left (a x^n\right )}{n}\right ) W\left (a x^n\right ) \left (-\frac {(1+m) W\left (a x^n\right )}{n}\right )^{-\frac {1+m}{n}}}{a (1+m)}+\frac {e^{\left (1-\frac {1+m}{n}\right ) W\left (a x^n\right )} x^{1+m-n} \Gamma \left (2+\frac {1+m}{n},-\frac {(1+m) W\left (a x^n\right )}{n}\right ) W\left (a x^n\right )^2 \left (-\frac {(1+m) W\left (a x^n\right )}{n}\right )^{-\frac {1+m+n}{n}}}{a (1+m)} \] Output:

exp((1-(1+m)/n)*LambertW(a*x^n))*x^(1+m-n)*GAMMA((1+m+n)/n,-(1+m)*LambertW 
(a*x^n)/n)*LambertW(a*x^n)/a/(1+m)/((-(1+m)*LambertW(a*x^n)/n)^((1+m)/n))+ 
exp((1-(1+m)/n)*LambertW(a*x^n))*x^(1+m-n)*GAMMA(2+(1+m)/n,-(1+m)*LambertW 
(a*x^n)/n)*LambertW(a*x^n)^2/a/(1+m)/((-(1+m)*LambertW(a*x^n)/n)^((1+m+n)/ 
n))
 

Mathematica [F]

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

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

Output:

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

Rubi [F]

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.

\(\displaystyle \int x^m W\left (a x^n\right ) \, dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int x^m W\left (a x^n\right )dx\)

Input:

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

Output:

$Aborted
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^m W\left (a x^n\right ) \, dx=\frac {x^{m} \textit {lambert\_w} \left (x^{n} a \right ) x -\left (\int \frac {x^{m +n}}{e^{\textit {lambert\_w} \left (x^{n} a \right )} \textit {lambert\_w} \left (x^{n} a \right ) m +e^{\textit {lambert\_w} \left (x^{n} a \right )} \textit {lambert\_w} \left (x^{n} a \right )+e^{\textit {lambert\_w} \left (x^{n} a \right )} m +e^{\textit {lambert\_w} \left (x^{n} a \right )}}d x \right ) a m n -\left (\int \frac {x^{m +n}}{e^{\textit {lambert\_w} \left (x^{n} a \right )} \textit {lambert\_w} \left (x^{n} a \right ) m +e^{\textit {lambert\_w} \left (x^{n} a \right )} \textit {lambert\_w} \left (x^{n} a \right )+e^{\textit {lambert\_w} \left (x^{n} a \right )} m +e^{\textit {lambert\_w} \left (x^{n} a \right )}}d x \right ) a n}{m +1} \] Input:

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

Output:

(x**m*lambert_w(x**n*a)*x - int(x**(m + n)/(e**lambert_w(x**n*a)*lambert_w 
(x**n*a)*m + e**lambert_w(x**n*a)*lambert_w(x**n*a) + e**lambert_w(x**n*a) 
*m + e**lambert_w(x**n*a)),x)*a*m*n - int(x**(m + n)/(e**lambert_w(x**n*a) 
*lambert_w(x**n*a)*m + e**lambert_w(x**n*a)*lambert_w(x**n*a) + e**lambert 
_w(x**n*a)*m + e**lambert_w(x**n*a)),x)*a*n)/(m + 1)