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