\(\int \frac {x^{-1+n}}{(c W(a x^n))^{3/2}} \, dx\) [273]

Optimal result

Integrand size = 18, antiderivative size = 58 \[ \int \frac {x^{-1+n}}{\left (c W\left (a x^n\right )\right )^{3/2}} \, dx=\frac {3 \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c W\left (a x^n\right )}}{\sqrt {c}}\right )}{a c^{3/2} n}-\frac {2 x^n}{n \left (c W\left (a x^n\right )\right )^{3/2}} \] Output:

3*Pi^(1/2)*erfi((c*LambertW(a*x^n))^(1/2)/c^(1/2))/a/c^(3/2)/n-2*x^n/n/(c* 
LambertW(a*x^n))^(3/2)
 

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.93 \[ \int \frac {x^{-1+n}}{\left (c W\left (a x^n\right )\right )^{3/2}} \, dx=\frac {-2 a x^n+3 \sqrt {\pi } \text {erfi}\left (\sqrt {W\left (a x^n\right )}\right ) W\left (a x^n\right )^{3/2}}{a n \left (c W\left (a x^n\right )\right )^{3/2}} \] Input:

Integrate[x^(-1 + n)/(c*ProductLog[a*x^n])^(3/2),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {7173, 7204}

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.

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

\(\Big \downarrow \) 7173

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

\(\Big \downarrow \) 7204

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

Input:

Int[x^(-1 + n)/(c*ProductLog[a*x^n])^(3/2),x]
 

Output:

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

Defintions of rubi rules used

rule 7173

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

rule 7204

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[( 
-Pi)*(c/(p - 1/2)), 2]*(Erfi[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] 
 && EqQ[m + n*(p - 1/2), -1] && NegQ[c/(p - 1/2)]
 

Maple [F]

\[\int \frac {x^{-1+n}}{{\left (c \operatorname {LambertW}\left (a \,x^{n}\right )\right )}^{\frac {3}{2}}}d x\]

Input:

int(x^(-1+n)/(c*LambertW(a*x^n))^(3/2),x)
 

Output:

int(x^(-1+n)/(c*LambertW(a*x^n))^(3/2),x)
 

Fricas [F]

\[ \int \frac {x^{-1+n}}{\left (c W\left (a x^n\right )\right )^{3/2}} \, dx=\int { \frac {x^{n - 1}}{\left (c \operatorname {W}({a x^{n}})\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(x^(-1+n)/(c*lambert_w(a*x^n))^(3/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {x^{-1+n}}{\left (c W\left (a x^n\right )\right )^{3/2}} \, dx=\int \frac {x^{n - 1}}{\left (c W\left (a x^{n}\right )\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(x**(-1+n)/(c*LambertW(a*x**n))**(3/2),x)
 

Output:

Integral(x**(n - 1)/(c*LambertW(a*x**n))**(3/2), x)
 

Maxima [F]

\[ \int \frac {x^{-1+n}}{\left (c W\left (a x^n\right )\right )^{3/2}} \, dx=\int { \frac {x^{n - 1}}{\left (c \operatorname {W}({a x^{n}})\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(x^(-1+n)/(c*lambert_w(a*x^n))^(3/2),x, algorithm="maxima")
 

Output:

integrate(x^(n - 1)/(c*lambert_w(a*x^n))^(3/2), x)
 

Giac [F]

\[ \int \frac {x^{-1+n}}{\left (c W\left (a x^n\right )\right )^{3/2}} \, dx=\int { \frac {x^{n - 1}}{\left (c \operatorname {W}({a x^{n}})\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(x^(-1+n)/(c*lambert_w(a*x^n))^(3/2),x, algorithm="giac")
 

Output:

integrate(x^(n - 1)/(c*lambert_w(a*x^n))^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{-1+n}}{\left (c W\left (a x^n\right )\right )^{3/2}} \, dx=\int \frac {x^{n-1}}{{\left (c\,\mathrm {LambertW}\left (a\,x^n\right )\right )}^{3/2}} \,d x \] Input:

int(x^(n - 1)/(c*LambertW(a*x^n))^(3/2),x)
 

Output:

int(x^(n - 1)/(c*LambertW(a*x^n))^(3/2), x)
 

Reduce [F]

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

int(x^(-1+n)/(c*Lambert_W(a*x^n))^(3/2),x)
 

Output:

(sqrt(c)*int((x**n*sqrt(lambert_w(x**n*a)))/(lambert_w(x**n*a)**2*x),x))/c 
**2