Integrand size = 18, antiderivative size = 59 \[ \int \frac {x^{-1+n}}{\sqrt {c W\left (a x^n\right )}} \, dx=\frac {\sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c W\left (a x^n\right )}}{\sqrt {c}}\right )}{2 a \sqrt {c} n}+\frac {x^n}{n \sqrt {c W\left (a x^n\right )}} \] Output:
1/2*Pi^(1/2)*erfi((c*LambertW(a*x^n))^(1/2)/c^(1/2))/a/c^(1/2)/n+x^n/n/(c* LambertW(a*x^n))^(1/2)
Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.95 \[ \int \frac {x^{-1+n}}{\sqrt {c W\left (a x^n\right )}} \, dx=\frac {2 a x^n+\sqrt {\pi } \text {erfi}\left (\sqrt {W\left (a x^n\right )}\right ) \sqrt {W\left (a x^n\right )}}{2 a n \sqrt {c W\left (a x^n\right )}} \] Input:
Integrate[x^(-1 + n)/Sqrt[c*ProductLog[a*x^n]],x]
Output:
(2*a*x^n + Sqrt[Pi]*Erfi[Sqrt[ProductLog[a*x^n]]]*Sqrt[ProductLog[a*x^n]]) /(2*a*n*Sqrt[c*ProductLog[a*x^n]])
Time = 0.44 (sec) , antiderivative size = 59, 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 = {7172, 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.
\(\displaystyle \int \frac {x^{n-1}}{\sqrt {c W\left (a x^n\right )}} \, dx\) |
\(\Big \downarrow \) 7172 |
\(\displaystyle \frac {1}{2} \int \frac {x^{n-1}}{\sqrt {c W\left (a x^n\right )} \left (W\left (a x^n\right )+1\right )}dx+\frac {x^n}{n \sqrt {c W\left (a x^n\right )}}\) |
\(\Big \downarrow \) 7204 |
\(\displaystyle \frac {\sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c W\left (a x^n\right )}}{\sqrt {c}}\right )}{2 a \sqrt {c} n}+\frac {x^n}{n \sqrt {c W\left (a x^n\right )}}\) |
Input:
Int[x^(-1 + n)/Sqrt[c*ProductLog[a*x^n]],x]
Output:
(Sqrt[Pi]*Erfi[Sqrt[c*ProductLog[a*x^n]]/Sqrt[c]])/(2*a*Sqrt[c]*n) + x^n/( n*Sqrt[c*ProductLog[a*x^n]])
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[( -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)]
\[\int \frac {x^{-1+n}}{\sqrt {c \operatorname {LambertW}\left (a \,x^{n}\right )}}d x\]
Input:
int(x^(-1+n)/(c*LambertW(a*x^n))^(1/2),x)
Output:
int(x^(-1+n)/(c*LambertW(a*x^n))^(1/2),x)
\[ \int \frac {x^{-1+n}}{\sqrt {c W\left (a x^n\right )}} \, dx=\int { \frac {x^{n - 1}}{\sqrt {c \operatorname {W}({a x^{n}})}} \,d x } \] Input:
integrate(x^(-1+n)/(c*lambert_w(a*x^n))^(1/2),x, algorithm="fricas")
Output:
integral(sqrt(c*lambert_w(a*x^n))*x^(n - 1)/(c*lambert_w(a*x^n)), x)
\[ \int \frac {x^{-1+n}}{\sqrt {c W\left (a x^n\right )}} \, dx=\int \frac {x^{n - 1}}{\sqrt {c W\left (a x^{n}\right )}}\, dx \] Input:
integrate(x**(-1+n)/(c*LambertW(a*x**n))**(1/2),x)
Output:
Integral(x**(n - 1)/sqrt(c*LambertW(a*x**n)), x)
\[ \int \frac {x^{-1+n}}{\sqrt {c W\left (a x^n\right )}} \, dx=\int { \frac {x^{n - 1}}{\sqrt {c \operatorname {W}({a x^{n}})}} \,d x } \] Input:
integrate(x^(-1+n)/(c*lambert_w(a*x^n))^(1/2),x, algorithm="maxima")
Output:
integrate(x^(n - 1)/sqrt(c*lambert_w(a*x^n)), x)
\[ \int \frac {x^{-1+n}}{\sqrt {c W\left (a x^n\right )}} \, dx=\int { \frac {x^{n - 1}}{\sqrt {c \operatorname {W}({a x^{n}})}} \,d x } \] Input:
integrate(x^(-1+n)/(c*lambert_w(a*x^n))^(1/2),x, algorithm="giac")
Output:
integrate(x^(n - 1)/sqrt(c*lambert_w(a*x^n)), x)
Timed out. \[ \int \frac {x^{-1+n}}{\sqrt {c W\left (a x^n\right )}} \, dx=\int \frac {x^{n-1}}{\sqrt {c\,\mathrm {LambertW}\left (a\,x^n\right )}} \,d x \] Input:
int(x^(n - 1)/(c*LambertW(a*x^n))^(1/2),x)
Output:
int(x^(n - 1)/(c*LambertW(a*x^n))^(1/2), x)
\[ \int \frac {x^{-1+n}}{\sqrt {c W\left (a x^n\right )}} \, dx=\frac {\sqrt {c}\, \left (\int \frac {x^{n} \sqrt {\textit {lambert\_w} \left (x^{n} a \right )}}{\textit {lambert\_w} \left (x^{n} a \right ) x}d x \right )}{c} \] Input:
int(x^(-1+n)/(c*Lambert_W(a*x^n))^(1/2),x)
Output:
(sqrt(c)*int((x**n*sqrt(lambert_w(x**n*a)))/(lambert_w(x**n*a)*x),x))/c