Integrand size = 10, antiderivative size = 105 \[ \int \frac {x^m}{W(a x)} \, dx=\frac {e^{-m W(a x)} x^m \Gamma (m,(-1-m) W(a x)) ((-1-m) W(a x))^{1-m}}{a (1+m) W(a x)}+\frac {e^{-m W(a x)} x^m \Gamma (1+m,(-1-m) W(a x)) ((-1-m) W(a x))^{-m}}{a (1+m)} \] Output:
x^m*GAMMA(m,(-1-m)*LambertW(a*x))*((-1-m)*LambertW(a*x))^(1-m)/a/exp(m*Lam bertW(a*x))/(1+m)/LambertW(a*x)+x^m*GAMMA(1+m,(-1-m)*LambertW(a*x))/a/exp( m*LambertW(a*x))/(1+m)/(((-1-m)*LambertW(a*x))^m)
Time = 0.09 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.61 \[ \int \frac {x^m}{W(a x)} \, dx=\frac {e^{-m W(a x)} x^m (-((1+m) \Gamma (m,-((1+m) W(a x))))+\Gamma (1+m,-((1+m) W(a x)))) (-((1+m) W(a x)))^{-m}}{a (1+m)} \] Input:
Integrate[x^m/ProductLog[a*x],x]
Output:
(x^m*(-((1 + m)*Gamma[m, -((1 + m)*ProductLog[a*x])]) + Gamma[1 + m, -((1 + m)*ProductLog[a*x])]))/(a*E^(m*ProductLog[a*x])*(1 + m)*(-((1 + m)*Produ ctLog[a*x]))^m)
Time = 0.75 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {7174, 7197, 7207}
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.
\(\displaystyle \int \frac {x^m}{W(a x)} \, dx\) |
\(\Big \downarrow \) 7174 |
\(\displaystyle \int \frac {x^m}{W(a x)+1}dx+\int \frac {x^m}{W(a x) (W(a x)+1)}dx\) |
\(\Big \downarrow \) 7197 |
\(\displaystyle \int \frac {x^m}{W(a x) (W(a x)+1)}dx+\frac {x^m e^{-m W(a x)} (-((m+1) W(a x)))^{-m} \Gamma (m+1,-((m+1) W(a x)))}{a (m+1)}\) |
\(\Big \downarrow \) 7207 |
\(\displaystyle \frac {x^m e^{-m W(a x)} (-((m+1) W(a x)))^{1-m} \Gamma (m,-((m+1) W(a x)))}{a (m+1) W(a x)}+\frac {x^m e^{-m W(a x)} (-((m+1) W(a x)))^{-m} \Gamma (m+1,-((m+1) W(a x)))}{a (m+1)}\) |
Input:
Int[x^m/ProductLog[a*x],x]
Output:
(x^m*Gamma[m, -((1 + m)*ProductLog[a*x])]*(-((1 + m)*ProductLog[a*x]))^(1 - m))/(a*E^(m*ProductLog[a*x])*(1 + m)*ProductLog[a*x]) + (x^m*Gamma[1 + m , -((1 + m)*ProductLog[a*x])])/(a*E^(m*ProductLog[a*x])*(1 + m)*(-((1 + m) *ProductLog[a*x]))^m)
Int[(x_)^(m_.)*((c_.)*ProductLog[(a_.)*(x_)])^(p_.), x_Symbol] :> Int[x^m*( (c*ProductLog[a*x])^p/(1 + ProductLog[a*x])), x] + Simp[1/c Int[x^m*((c*P roductLog[a*x])^(p + 1)/(1 + ProductLog[a*x])), x], x] /; FreeQ[{a, c, m}, x]
Int[(x_)^(m_.)/((d_) + (d_.)*ProductLog[(a_.)*(x_)]), x_Symbol] :> Simp[x^m *(Gamma[m + 1, (-(m + 1))*ProductLog[a*x]]/(a*d*(m + 1)*E^(m*ProductLog[a*x ])*((-(m + 1))*ProductLog[a*x])^m)), x] /; FreeQ[{a, d, m}, x] && !Integer Q[m]
Int[((x_)^(m_.)*((c_.)*ProductLog[(a_.)*(x_)])^(p_.))/((d_) + (d_.)*Product Log[(a_.)*(x_)]), x_Symbol] :> Simp[x^m*Gamma[m + p + 1, (-(m + 1))*Product Log[a*x]]*((c*ProductLog[a*x])^p/(a*d*(m + 1)*E^(m*ProductLog[a*x])*((-(m + 1))*ProductLog[a*x])^(m + p))), x] /; FreeQ[{a, c, d, m, p}, x] && NeQ[m, -1]
\[\int \frac {x^{m}}{\operatorname {LambertW}\left (x a \right )}d x\]
Input:
int(x^m/LambertW(x*a),x)
Output:
int(x^m/LambertW(x*a),x)
\[ \int \frac {x^m}{W(a x)} \, dx=\int { \frac {x^{m}}{\operatorname {W}({a x})} \,d x } \] Input:
integrate(x^m/lambert_w(a*x),x, algorithm="fricas")
Output:
integral(x^m/lambert_w(a*x), x)
\[ \int \frac {x^m}{W(a x)} \, dx=\int \frac {x^{m}}{W\left (a x\right )}\, dx \] Input:
integrate(x**m/LambertW(a*x),x)
Output:
Integral(x**m/LambertW(a*x), x)
\[ \int \frac {x^m}{W(a x)} \, dx=\int { \frac {x^{m}}{\operatorname {W}({a x})} \,d x } \] Input:
integrate(x^m/lambert_w(a*x),x, algorithm="maxima")
Output:
integrate(x^m/lambert_w(a*x), x)
\[ \int \frac {x^m}{W(a x)} \, dx=\int { \frac {x^{m}}{\operatorname {W}({a x})} \,d x } \] Input:
integrate(x^m/lambert_w(a*x),x, algorithm="giac")
Output:
integrate(x^m/lambert_w(a*x), x)
Timed out. \[ \int \frac {x^m}{W(a x)} \, dx=\int \frac {x^m}{\mathrm {LambertW}\left (a\,x\right )} \,d x \] Input:
int(x^m/LambertW(a*x),x)
Output:
int(x^m/LambertW(a*x), x)
\[ \int \frac {x^m}{W(a x)} \, dx=\frac {x^{m} x -\left (\int x^{m}d x \right ) m -\left (\int x^{m}d x \right )+2 \left (\int \frac {x^{m}}{\textit {lambert\_w} \left (a x \right )}d x \right ) m +2 \left (\int \frac {x^{m}}{\textit {lambert\_w} \left (a x \right )}d x \right )}{2 m +2} \] Input:
int(x^m/Lambert_W(a*x),x)
Output:
(x**m*x - int(x**m,x)*m - int(x**m,x) + 2*int(x**m/lambert_w(a*x),x)*m + 2 *int(x**m/lambert_w(a*x),x))/(2*(m + 1))