\(\int \frac {(c W(\frac {a}{x}))^p}{x^3} \, dx\) [227]

Optimal result

Integrand size = 14, antiderivative size = 128 \[ \int \frac {\left (c W\left (\frac {a}{x}\right )\right )^p}{x^3} \, dx=-\frac {2^{-2-p} e^{-W\left (\frac {a}{x}\right )} \Gamma \left (2+p,-2 W\left (\frac {a}{x}\right )\right ) \left (-W\left (\frac {a}{x}\right )\right )^{-1-p} \left (c W\left (\frac {a}{x}\right )\right )^p}{a x}-\frac {2^{-3-p} e^{-W\left (\frac {a}{x}\right )} \Gamma \left (3+p,-2 W\left (\frac {a}{x}\right )\right ) \left (-W\left (\frac {a}{x}\right )\right )^{-2-p} \left (c W\left (\frac {a}{x}\right )\right )^{1+p}}{a c x} \] Output:

-2^(-2-p)*GAMMA(2+p,-2*LambertW(a/x))*(-LambertW(a/x))^(-1-p)*(c*LambertW( 
a/x))^p/a/exp(LambertW(a/x))/x-2^(-3-p)*GAMMA(3+p,-2*LambertW(a/x))*(-Lamb 
ertW(a/x))^(-2-p)*(c*LambertW(a/x))^(p+1)/a/c/exp(LambertW(a/x))/x
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.48 \[ \int \frac {\left (c W\left (\frac {a}{x}\right )\right )^p}{x^3} \, dx=\frac {2^{-3-p} \left (2 \Gamma \left (2+p,-2 W\left (\frac {a}{x}\right )\right )-\Gamma \left (3+p,-2 W\left (\frac {a}{x}\right )\right )\right ) \left (-W\left (\frac {a}{x}\right )\right )^{-p} \left (c W\left (\frac {a}{x}\right )\right )^p}{a^2} \] Input:

Integrate[(c*ProductLog[a/x])^p/x^3,x]
 

Output:

(2^(-3 - p)*(2*Gamma[2 + p, -2*ProductLog[a/x]] - Gamma[3 + p, -2*ProductL 
og[a/x]])*(c*ProductLog[a/x])^p)/(a^2*(-ProductLog[a/x])^p)
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {7175, 7174, 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.

\(\displaystyle \int \frac {\left (c W\left (\frac {a}{x}\right )\right )^p}{x^3} \, dx\)

\(\Big \downarrow \) 7175

\(\displaystyle -\int \frac {\left (c W\left (\frac {a}{x}\right )\right )^p}{x}d\frac {1}{x}\)

\(\Big \downarrow \) 7174

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

\(\Big \downarrow \) 7207

\(\displaystyle -\frac {2^{-p-3} e^{-W\left (\frac {a}{x}\right )} \left (-W\left (\frac {a}{x}\right )\right )^{-p-2} \left (c W\left (\frac {a}{x}\right )\right )^{p+1} \Gamma \left (p+3,-2 W\left (\frac {a}{x}\right )\right )}{a c x}-\frac {2^{-p-2} e^{-W\left (\frac {a}{x}\right )} \left (-W\left (\frac {a}{x}\right )\right )^{-p-1} \left (c W\left (\frac {a}{x}\right )\right )^p \Gamma \left (p+2,-2 W\left (\frac {a}{x}\right )\right )}{a x}\)

Input:

Int[(c*ProductLog[a/x])^p/x^3,x]
 

Output:

-((2^(-2 - p)*Gamma[2 + p, -2*ProductLog[a/x]]*(-ProductLog[a/x])^(-1 - p) 
*(c*ProductLog[a/x])^p)/(a*E^ProductLog[a/x]*x)) - (2^(-3 - p)*Gamma[3 + p 
, -2*ProductLog[a/x]]*(-ProductLog[a/x])^(-2 - p)*(c*ProductLog[a/x])^(1 + 
 p))/(a*c*E^ProductLog[a/x]*x)
 

Defintions of rubi rules used

rule 7174

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]
 

rule 7175

Int[(x_)^(m_.)*((c_.)*ProductLog[(a_.)*(x_)^(n_)])^(p_.), x_Symbol] :> -Sub 
st[Int[(c*ProductLog[a/x^n])^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, c, p}, x 
] && ILtQ[n, 0] && IntegerQ[m] && NeQ[m, -1]
 

rule 7207

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]
 

Maple [F]

\[\int \frac {\left (c \operatorname {LambertW}\left (\frac {a}{x}\right )\right )^{p}}{x^{3}}d x\]

Input:

int((c*LambertW(a/x))^p/x^3,x)
 

Output:

int((c*LambertW(a/x))^p/x^3,x)
 

Fricas [F]

\[ \int \frac {\left (c W\left (\frac {a}{x}\right )\right )^p}{x^3} \, dx=\int { \frac {\left (c \operatorname {W}({\frac {a}{x}})\right )^{p}}{x^{3}} \,d x } \] Input:

integrate((c*lambert_w(a/x))^p/x^3,x, algorithm="fricas")
 

Output:

integral((c*lambert_w(a/x))^p/x^3, x)
 

Sympy [F]

\[ \int \frac {\left (c W\left (\frac {a}{x}\right )\right )^p}{x^3} \, dx=\int \frac {\left (c W\left (\frac {a}{x}\right )\right )^{p}}{x^{3}}\, dx \] Input:

integrate((c*LambertW(a/x))**p/x**3,x)
 

Output:

Integral((c*LambertW(a/x))**p/x**3, x)
 

Maxima [F]

\[ \int \frac {\left (c W\left (\frac {a}{x}\right )\right )^p}{x^3} \, dx=\int { \frac {\left (c \operatorname {W}({\frac {a}{x}})\right )^{p}}{x^{3}} \,d x } \] Input:

integrate((c*lambert_w(a/x))^p/x^3,x, algorithm="maxima")
 

Output:

integrate((c*lambert_w(a/x))^p/x^3, x)
 

Giac [F]

\[ \int \frac {\left (c W\left (\frac {a}{x}\right )\right )^p}{x^3} \, dx=\int { \frac {\left (c \operatorname {W}({\frac {a}{x}})\right )^{p}}{x^{3}} \,d x } \] Input:

integrate((c*lambert_w(a/x))^p/x^3,x, algorithm="giac")
 

Output:

integrate((c*lambert_w(a/x))^p/x^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c W\left (\frac {a}{x}\right )\right )^p}{x^3} \, dx=\int \frac {{\left (c\,\mathrm {LambertW}\left (\frac {a}{x}\right )\right )}^p}{x^3} \,d x \] Input:

int((c*LambertW(a/x))^p/x^3,x)
 

Output:

int((c*LambertW(a/x))^p/x^3, x)
 

Reduce [F]

\[ \int \frac {\left (c W\left (\frac {a}{x}\right )\right )^p}{x^3} \, dx=c^{p} \left (\int \frac {\textit {lambert\_w} \left (\frac {a}{x}\right )^{p}}{x^{3}}d x \right ) \] Input:

int((c*Lambert_W(a/x))^p/x^3,x)
 

Output:

c**p*int(lambert_w(a/x)**p/x**3,x)