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)
\[ \int \left (c W\left (a x^n\right )\right )^p \, dx=\int \left (c W\left (a x^n\right )\right )^p \, dx \] Input:
Integrate[(c*ProductLog[a*x^n])^p,x]
Output:
Integrate[(c*ProductLog[a*x^n])^p, x]
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
\[\int {\left (c \operatorname {LambertW}\left (a \,x^{n}\right )\right )}^{p}d x\]
Input:
int((c*LambertW(a*x^n))^p,x)
Output:
int((c*LambertW(a*x^n))^p,x)
\[ \int \left (c W\left (a x^n\right )\right )^p \, dx=\int { \left (c \operatorname {W}({a x^{n}})\right )^{p} \,d x } \] Input:
integrate((c*lambert_w(a*x^n))^p,x, algorithm="fricas")
Output:
integral((c*lambert_w(a*x^n))^p, x)
\[ \int \left (c W\left (a x^n\right )\right )^p \, dx=\int \left (c W\left (a x^{n}\right )\right )^{p}\, dx \] Input:
integrate((c*LambertW(a*x**n))**p,x)
Output:
Integral((c*LambertW(a*x**n))**p, x)
\[ \int \left (c W\left (a x^n\right )\right )^p \, dx=\int { \left (c \operatorname {W}({a x^{n}})\right )^{p} \,d x } \] Input:
integrate((c*lambert_w(a*x^n))^p,x, algorithm="maxima")
Output:
integrate((c*lambert_w(a*x^n))^p, x)
\[ \int \left (c W\left (a x^n\right )\right )^p \, dx=\int { \left (c \operatorname {W}({a x^{n}})\right )^{p} \,d x } \] Input:
integrate((c*lambert_w(a*x^n))^p,x, algorithm="giac")
Output:
integrate((c*lambert_w(a*x^n))^p, x)
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)
\[ \int \left (c W\left (a x^n\right )\right )^p \, dx=c^{p} \left (\textit {lambert\_w} \left (x^{n} a \right )^{p} x -\left (\int \frac {x^{n} \textit {lambert\_w} \left (x^{n} a \right )^{p}}{e^{\textit {lambert\_w} \left (x^{n} a \right )} \textit {lambert\_w} \left (x^{n} a \right )^{2}+e^{\textit {lambert\_w} \left (x^{n} a \right )} \textit {lambert\_w} \left (x^{n} a \right )}d x \right ) a n p \right ) \] 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)