\(\int \frac {W(a x^n)^p}{d+d W(a x^n)} \, dx\) [357]

Optimal result

Integrand size = 21, antiderivative size = 79 \[ \int \frac {W\left (a x^n\right )^p}{d+d W\left (a x^n\right )} \, dx=\frac {e^{-\frac {(1-n) W\left (a x^n\right )}{n}} x^{1-n} \Gamma \left (\frac {1}{n}+p,-\frac {W\left (a x^n\right )}{n}\right ) W\left (a x^n\right )^p \left (-\frac {W\left (a x^n\right )}{n}\right )^{1-\frac {1}{n}-p}}{a d} \] Output:

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

Mathematica [F]

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

Integrate[ProductLog[a*x^n]^p/(d + d*ProductLog[a*x^n]),x]
 

Output:

Integrate[ProductLog[a*x^n]^p/(d + d*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.

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

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {W\left (a x^n\right )^p}{d \left (W\left (a x^n\right )+1\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {W\left (a x^n\right )^p}{W\left (a x^n\right )+1}dx}{d}\)

\(\Big \downarrow \) 7299

\(\displaystyle \frac {\int \frac {W\left (a x^n\right )^p}{W\left (a x^n\right )+1}dx}{d}\)

Input:

Int[ProductLog[a*x^n]^p/(d + d*ProductLog[a*x^n]),x]
 

Output:

$Aborted
 

Maple [F]

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

Input:

int(LambertW(a*x^n)^p/(d+d*LambertW(a*x^n)),x)
 

Output:

int(LambertW(a*x^n)^p/(d+d*LambertW(a*x^n)),x)
 

Fricas [F]

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

integrate(lambert_w(a*x^n)^p/(d+d*lambert_w(a*x^n)),x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

integral(lambert_w(a*x^n)^p/(d*lambert_w(a*x^n) + d), x)
 

Sympy [F]

\[ \int \frac {W\left (a x^n\right )^p}{d+d W\left (a x^n\right )} \, dx=\frac {\int \frac {W^{p}\left (a x^{n}\right )}{W\left (a x^{n}\right ) + 1}\, dx}{d} \] Input:

integrate(LambertW(a*x**n)**p/(d+d*LambertW(a*x**n)),x)
 

Output:

Integral(LambertW(a*x**n)**p/(LambertW(a*x**n) + 1), x)/d
 

Maxima [F]

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

integrate(lambert_w(a*x^n)^p/(d+d*lambert_w(a*x^n)),x, algorithm="maxima")
 

Output:

integrate(lambert_w(a*x^n)^p/(d*lambert_w(a*x^n) + d), x)
 

Giac [F]

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

integrate(lambert_w(a*x^n)^p/(d+d*lambert_w(a*x^n)),x, algorithm="giac")
 

Output:

integrate(lambert_w(a*x^n)^p/(d*lambert_w(a*x^n) + d), x)
 

Mupad [F(-1)]

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

int(LambertW(a*x^n)^p/(d + d*LambertW(a*x^n)),x)
 

Output:

int(LambertW(a*x^n)^p/(d + d*LambertW(a*x^n)), x)
 

Reduce [F]

\[ \int \frac {W\left (a x^n\right )^p}{d+d W\left (a x^n\right )} \, dx=\frac {\textit {lambert\_w} \left (x^{n} a \right )^{p} x +3 \left (\int \frac {\textit {lambert\_w} \left (x^{n} a \right )^{p}}{\textit {lambert\_w} \left (x^{n} a \right )+1}d x \right )-\left (\int \frac {\textit {lambert\_w} \left (x^{n} a \right )^{p} \textit {lambert\_w} \left (x^{n} a \right )}{\textit {lambert\_w} \left (x^{n} a \right )+1}d x \right )-\left (\int \frac {x^{n} \textit {lambert\_w} \left (x^{n} a \right )^{p}}{e^{\textit {lambert\_w} \left (x^{n} a \right )} \textit {lambert\_w} \left (x^{n} a \right )^{2}+e^{\textit {lambert\_w} \left (x^{n} a \right )} \textit {lambert\_w} \left (x^{n} a \right )}d x \right ) a n p}{4 d} \] Input:

int(Lambert_W(a*x^n)^p/(d+d*Lambert_W(a*x^n)),x)
 

Output:

(lambert_w(x**n*a)**p*x + 3*int(lambert_w(x**n*a)**p/(lambert_w(x**n*a) + 
1),x) - int((lambert_w(x**n*a)**p*lambert_w(x**n*a))/(lambert_w(x**n*a) + 
1),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)/(4*d)