\(\int (c W(a x^n))^p \, dx\) [286]

Optimal result

Integrand size = 10, antiderivative size = 162 \[ \int \left (c W\left (a x^n\right )\right )^p \, dx=\frac {e^{-\frac {(1-n) W\left (a x^n\right )}{n}} x^{1-n} \Gamma \left (1+\frac {1}{n}+p,-\frac {W\left (a x^n\right )}{n}\right ) \left (c W\left (a x^n\right )\right )^{1+p} \left (-\frac {W\left (a x^n\right )}{n}\right )^{-\frac {1}{n}-p}}{a c}+\frac {e^{-\frac {(1-n) W\left (a x^n\right )}{n}} x^{1-n} \Gamma \left (\frac {1}{n}+p,-\frac {W\left (a x^n\right )}{n}\right ) \left (c W\left (a x^n\right )\right )^p \left (-\frac {W\left (a x^n\right )}{n}\right )^{1-\frac {1}{n}-p}}{a} \] Output:

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

Mathematica [F]

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

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

Output:

Integrate[(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.

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

\(\Big \downarrow \) 7271

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

\(\Big \downarrow \) 7299

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

Input:

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

Output:

$Aborted
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

c**p*(lambert_w(x**n*a)**p*x - int((x**n*lambert_w(x**n*a)**p)/(e**lambert 
_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)