\(\int \frac {x^m}{W(a x^n)} \, dx\) [245]

Optimal result

Integrand size = 12, antiderivative size = 177 \[ \int \frac {x^m}{W\left (a x^n\right )} \, dx=\frac {e^{\left (1-\frac {1+m}{n}\right ) W\left (a x^n\right )} x^{1+m-n} \Gamma \left (\frac {1+m}{n},-\frac {(1+m) W\left (a x^n\right )}{n}\right ) \left (-\frac {(1+m) W\left (a x^n\right )}{n}\right )^{1-\frac {1+m}{n}}}{a (1+m)}+\frac {e^{\left (1-\frac {1+m}{n}\right ) W\left (a x^n\right )} x^{1+m-n} \Gamma \left (-1+\frac {1+m}{n},-\frac {(1+m) W\left (a x^n\right )}{n}\right ) \left (-\frac {(1+m) W\left (a x^n\right )}{n}\right )^{2-\frac {1+m}{n}}}{a (1+m) W\left (a x^n\right )} \] Output:

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

Mathematica [F]

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

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

\(\Big \downarrow \) 7299

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

Input:

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

Output:

$Aborted
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Reduce [F]

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

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

Output:

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