\(\int x (c W(a x^n))^p \, dx\) [285]

Optimal result

Integrand size = 12, antiderivative size = 189 \[ \int x \left (c W\left (a x^n\right )\right )^p \, dx=\frac {2^{-1-\frac {2}{n}-p} e^{-\frac {(2-n) W\left (a x^n\right )}{n}} x^{2-n} \Gamma \left (1+\frac {2}{n}+p,-\frac {2 W\left (a x^n\right )}{n}\right ) \left (c W\left (a x^n\right )\right )^{1+p} \left (-\frac {W\left (a x^n\right )}{n}\right )^{-\frac {2}{n}-p}}{a c}+\frac {2^{-\frac {2}{n}-p} e^{-\frac {(2-n) W\left (a x^n\right )}{n}} x^{2-n} \Gamma \left (\frac {2}{n}+p,-\frac {2 W\left (a x^n\right )}{n}\right ) \left (c W\left (a x^n\right )\right )^p \left (-\frac {W\left (a x^n\right )}{n}\right )^{1-\frac {2}{n}-p}}{a} \] Output:

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

Mathematica [F]

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

Integrate[x*(c*ProductLog[a*x^n])^p,x]
 

Output:

Integrate[x*(c*ProductLog[a*x^n])^p, 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.

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

\(\Big \downarrow \) 7271

\(\displaystyle W\left (a x^n\right )^{-p} \left (c W\left (a x^n\right )\right )^p \int x W\left (a x^n\right )^pdx\)

\(\Big \downarrow \) 7299

\(\displaystyle W\left (a x^n\right )^{-p} \left (c W\left (a x^n\right )\right )^p \int x W\left (a x^n\right )^pdx\)

Input:

Int[x*(c*ProductLog[a*x^n])^p,x]
 

Output:

$Aborted
 

Maple [F]

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

Input:

int(x*(c*LambertW(a*x^n))^p,x)
 

Output:

int(x*(c*LambertW(a*x^n))^p,x)
 

Fricas [F]

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

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

Output:

integral((c*lambert_w(a*x^n))^p*x, x)
 

Sympy [F]

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

integrate(x*(c*LambertW(a*x**n))**p,x)
 

Output:

Integral(x*(c*LambertW(a*x**n))**p, x)
 

Maxima [F]

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

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

Output:

integrate((c*lambert_w(a*x^n))^p*x, x)
 

Giac [F]

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

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

Output:

integrate((c*lambert_w(a*x^n))^p*x, x)
 

Mupad [F(-1)]

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

int(x*(c*LambertW(a*x^n))^p,x)
 

Output:

int(x*(c*LambertW(a*x^n))^p, x)
 

Reduce [F]

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

int(x*(c*Lambert_W(a*x^n))^p,x)
 

Output:

(c**p*(lambert_w(x**n*a)**p*x**2 - int((x**n*lambert_w(x**n*a)**p*x)/(e**l 
ambert_w(x**n*a)*lambert_w(x**n*a)**2 + e**lambert_w(x**n*a)*lambert_w(x** 
n*a)),x)*a*n*p))/2