\(\int x^{-1-n} (c W(a x^n))^{5/2} \, dx\) [255]

Optimal result

Integrand size = 20, antiderivative size = 85 \[ \int x^{-1-n} \left (c W\left (a x^n\right )\right )^{5/2} \, dx=\frac {5 a c^{5/2} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {c W\left (a x^n\right )}}{\sqrt {c}}\right )}{4 n}-\frac {5 c x^{-n} \left (c W\left (a x^n\right )\right )^{3/2}}{2 n}-\frac {x^{-n} \left (c W\left (a x^n\right )\right )^{5/2}}{n} \] Output:

5/4*a*c^(5/2)*Pi^(1/2)*erf((c*LambertW(a*x^n))^(1/2)/c^(1/2))/n-5/2*c*(c*L 
ambertW(a*x^n))^(3/2)/n/(x^n)-(c*LambertW(a*x^n))^(5/2)/n/(x^n)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.95 \[ \int x^{-1-n} \left (c W\left (a x^n\right )\right )^{5/2} \, dx=-\frac {x^{-n} \left (c W\left (a x^n\right )\right )^{5/2} \left (-5 a \sqrt {\pi } x^n \text {erf}\left (\sqrt {W\left (a x^n\right )}\right )+10 W\left (a x^n\right )^{3/2}+4 W\left (a x^n\right )^{5/2}\right )}{4 n W\left (a x^n\right )^{5/2}} \] Input:

Integrate[x^(-1 - n)*(c*ProductLog[a*x^n])^(5/2),x]
 

Output:

-1/4*((c*ProductLog[a*x^n])^(5/2)*(-5*a*Sqrt[Pi]*x^n*Erf[Sqrt[ProductLog[a 
*x^n]]] + 10*ProductLog[a*x^n]^(3/2) + 4*ProductLog[a*x^n]^(5/2)))/(n*x^n* 
ProductLog[a*x^n]^(5/2))
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {7172, 7205, 7203}

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.

\(\displaystyle \int x^{-n-1} \left (c W\left (a x^n\right )\right )^{5/2} \, dx\)

\(\Big \downarrow \) 7172

\(\displaystyle \frac {5}{2} \int \frac {x^{-n-1} \left (c W\left (a x^n\right )\right )^{5/2}}{W\left (a x^n\right )+1}dx-\frac {x^{-n} \left (c W\left (a x^n\right )\right )^{5/2}}{n}\)

\(\Big \downarrow \) 7205

\(\displaystyle \frac {5}{2} \left (\frac {1}{2} c \int \frac {x^{-n-1} \left (c W\left (a x^n\right )\right )^{3/2}}{W\left (a x^n\right )+1}dx-\frac {c x^{-n} \left (c W\left (a x^n\right )\right )^{3/2}}{n}\right )-\frac {x^{-n} \left (c W\left (a x^n\right )\right )^{5/2}}{n}\)

\(\Big \downarrow \) 7203

\(\displaystyle \frac {5}{2} \left (\frac {\sqrt {\pi } a c^{5/2} \text {erf}\left (\frac {\sqrt {c W\left (a x^n\right )}}{\sqrt {c}}\right )}{2 n}-\frac {c x^{-n} \left (c W\left (a x^n\right )\right )^{3/2}}{n}\right )-\frac {x^{-n} \left (c W\left (a x^n\right )\right )^{5/2}}{n}\)

Input:

Int[x^(-1 - n)*(c*ProductLog[a*x^n])^(5/2),x]
 

Output:

-((c*ProductLog[a*x^n])^(5/2)/(n*x^n)) + (5*((a*c^(5/2)*Sqrt[Pi]*Erf[Sqrt[ 
c*ProductLog[a*x^n]]/Sqrt[c]])/(2*n) - (c*(c*ProductLog[a*x^n])^(3/2))/(n* 
x^n)))/2
 

Defintions of rubi rules used

rule 7172

Int[(x_)^(m_.)*((c_.)*ProductLog[(a_.)*(x_)^(n_.)])^(p_.), x_Symbol] :> Sim 
p[x^(m + 1)*((c*ProductLog[a*x^n])^p/(m + 1)), x] - Simp[n*(p/(m + 1))   In 
t[x^m*((c*ProductLog[a*x^n])^p/(1 + ProductLog[a*x^n])), x], x] /; FreeQ[{a 
, c, m, n, p}, x] && NeQ[m, -1] && ((IntegerQ[p - 1/2] && IGtQ[2*Simplify[p 
 + (m + 1)/n], 0]) || ( !IntegerQ[p - 1/2] && IGtQ[Simplify[p + (m + 1)/n] 
+ 1, 0]))
 

rule 7203

Int[((x_)^(m_.)*((c_.)*ProductLog[(a_.)*(x_)^(n_.)])^(p_))/((d_) + (d_.)*Pr 
oductLog[(a_.)*(x_)^(n_.)]), x_Symbol] :> Simp[a^(p - 1/2)*c^(p - 1/2)*Rt[P 
i*(c/(p - 1/2)), 2]*(Erf[Sqrt[c*ProductLog[a*x^n]]/Rt[c/(p - 1/2), 2]]/(d*n 
)), x] /; FreeQ[{a, c, d, m, n}, x] && NeQ[m, -1] && IntegerQ[p - 1/2] && E 
qQ[m + n*(p - 1/2), -1] && PosQ[c/(p - 1/2)]
 

rule 7205

Int[((x_)^(m_.)*((c_.)*ProductLog[(a_.)*(x_)^(n_.)])^(p_.))/((d_) + (d_.)*P 
roductLog[(a_.)*(x_)^(n_.)]), x_Symbol] :> Simp[c*x^(m + 1)*((c*ProductLog[ 
a*x^n])^(p - 1)/(d*(m + 1))), x] - Simp[c*((m + n*(p - 1) + 1)/(m + 1))   I 
nt[x^m*((c*ProductLog[a*x^n])^(p - 1)/(d + d*ProductLog[a*x^n])), x], x] /; 
 FreeQ[{a, c, d, m, n, p}, x] && NeQ[m, -1] && GtQ[Simplify[p + (m + 1)/n], 
 1]
 

Maple [F]

\[\int x^{-1-n} {\left (c \operatorname {LambertW}\left (a \,x^{n}\right )\right )}^{\frac {5}{2}}d x\]

Input:

int(x^(-1-n)*(c*LambertW(a*x^n))^(5/2),x)
 

Output:

int(x^(-1-n)*(c*LambertW(a*x^n))^(5/2),x)
 

Fricas [F]

\[ \int x^{-1-n} \left (c W\left (a x^n\right )\right )^{5/2} \, dx=\int { \left (c \operatorname {W}({a x^{n}})\right )^{\frac {5}{2}} x^{-n - 1} \,d x } \] Input:

integrate(x^(-1-n)*(c*lambert_w(a*x^n))^(5/2),x, algorithm="fricas")
 

Output:

integral(sqrt(c*lambert_w(a*x^n))*c^2*x^(-n - 1)*lambert_w(a*x^n)^2, x)
 

Sympy [F(-1)]

Timed out. \[ \int x^{-1-n} \left (c W\left (a x^n\right )\right )^{5/2} \, dx=\text {Timed out} \] Input:

integrate(x**(-1-n)*(c*LambertW(a*x**n))**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x^{-1-n} \left (c W\left (a x^n\right )\right )^{5/2} \, dx=\int { \left (c \operatorname {W}({a x^{n}})\right )^{\frac {5}{2}} x^{-n - 1} \,d x } \] Input:

integrate(x^(-1-n)*(c*lambert_w(a*x^n))^(5/2),x, algorithm="maxima")
 

Output:

integrate((c*lambert_w(a*x^n))^(5/2)*x^(-n - 1), x)
 

Giac [F]

\[ \int x^{-1-n} \left (c W\left (a x^n\right )\right )^{5/2} \, dx=\int { \left (c \operatorname {W}({a x^{n}})\right )^{\frac {5}{2}} x^{-n - 1} \,d x } \] Input:

integrate(x^(-1-n)*(c*lambert_w(a*x^n))^(5/2),x, algorithm="giac")
 

Output:

integrate((c*lambert_w(a*x^n))^(5/2)*x^(-n - 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int x^{-1-n} \left (c W\left (a x^n\right )\right )^{5/2} \, dx=\int \frac {{\left (c\,\mathrm {LambertW}\left (a\,x^n\right )\right )}^{5/2}}{x^{n+1}} \,d x \] Input:

int((c*LambertW(a*x^n))^(5/2)/x^(n + 1),x)
 

Output:

int((c*LambertW(a*x^n))^(5/2)/x^(n + 1), x)
 

Reduce [F]

\[ \int x^{-1-n} \left (c W\left (a x^n\right )\right )^{5/2} \, dx=\frac {\sqrt {c}\, c^{2} \left (-2 \sqrt {\textit {lambert\_w} \left (x^{n} a \right )}\, \textit {lambert\_w} \left (x^{n} a \right )^{2}+5 x^{n} \left (\int \frac {\sqrt {\textit {lambert\_w} \left (x^{n} a \right )}\, \textit {lambert\_w} \left (x^{n} a \right )}{e^{\textit {lambert\_w} \left (x^{n} a \right )} \textit {lambert\_w} \left (x^{n} a \right ) x +e^{\textit {lambert\_w} \left (x^{n} a \right )} x}d x \right ) a n \right )}{2 x^{n} n} \] Input:

int(x^(-1-n)*(c*Lambert_W(a*x^n))^(5/2),x)
 

Output:

(sqrt(c)*c**2*( - 2*sqrt(lambert_w(x**n*a))*lambert_w(x**n*a)**2 + 5*x**n* 
int((sqrt(lambert_w(x**n*a))*lambert_w(x**n*a))/(e**lambert_w(x**n*a)*lamb 
ert_w(x**n*a)*x + e**lambert_w(x**n*a)*x),x)*a*n))/(2*x**n*n)