Integrand size = 14, antiderivative size = 83 \[ \int x^2 \sqrt {W\left (\frac {a}{x}\right )} \, dx=\frac {4}{5} a^3 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {W\left (\frac {a}{x}\right )}\right )+\frac {2}{5} x^3 \sqrt {W\left (\frac {a}{x}\right )}-\frac {2}{15} x^3 W\left (\frac {a}{x}\right )^{3/2}+\frac {4}{5} x^3 W\left (\frac {a}{x}\right )^{5/2} \] Output:
4/5*a^3*3^(1/2)*Pi^(1/2)*erf(3^(1/2)*LambertW(a/x)^(1/2))+2/5*x^3*LambertW (a/x)^(1/2)-2/15*x^3*LambertW(a/x)^(3/2)+4/5*x^3*LambertW(a/x)^(5/2)
Time = 0.01 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.95 \[ \int x^2 \sqrt {W\left (\frac {a}{x}\right )} \, dx=\frac {2}{15} \left (6 a^3 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {W\left (\frac {a}{x}\right )}\right )+3 x^3 \sqrt {W\left (\frac {a}{x}\right )}-x^3 W\left (\frac {a}{x}\right )^{3/2}+6 x^3 W\left (\frac {a}{x}\right )^{5/2}\right ) \] Input:
Integrate[x^2*Sqrt[ProductLog[a/x]],x]
Output:
(2*(6*a^3*Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ProductLog[a/x]]] + 3*x^3*Sqrt[Produ ctLog[a/x]] - x^3*ProductLog[a/x]^(3/2) + 6*x^3*ProductLog[a/x]^(5/2)))/15
Time = 0.83 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {7173, 7206, 7206, 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.
\(\displaystyle \int x^2 \sqrt {W\left (\frac {a}{x}\right )} \, dx\) |
\(\Big \downarrow \) 7173 |
\(\displaystyle \frac {2}{5} x^3 \sqrt {W\left (\frac {a}{x}\right )}-\frac {1}{5} \int \frac {x^2 W\left (\frac {a}{x}\right )^{3/2}}{W\left (\frac {a}{x}\right )+1}dx\) |
\(\Big \downarrow \) 7206 |
\(\displaystyle \frac {1}{5} \left (2 \int \frac {x^2 W\left (\frac {a}{x}\right )^{5/2}}{W\left (\frac {a}{x}\right )+1}dx-\frac {2}{3} x^3 W\left (\frac {a}{x}\right )^{3/2}\right )+\frac {2}{5} x^3 \sqrt {W\left (\frac {a}{x}\right )}\) |
\(\Big \downarrow \) 7206 |
\(\displaystyle \frac {1}{5} \left (2 \left (2 x^3 W\left (\frac {a}{x}\right )^{5/2}-6 \int \frac {x^2 W\left (\frac {a}{x}\right )^{7/2}}{W\left (\frac {a}{x}\right )+1}dx\right )-\frac {2}{3} x^3 W\left (\frac {a}{x}\right )^{3/2}\right )+\frac {2}{5} x^3 \sqrt {W\left (\frac {a}{x}\right )}\) |
\(\Big \downarrow \) 7203 |
\(\displaystyle \frac {1}{5} \left (2 \left (2 \sqrt {3 \pi } a^3 \text {erf}\left (\sqrt {3} \sqrt {W\left (\frac {a}{x}\right )}\right )+2 x^3 W\left (\frac {a}{x}\right )^{5/2}\right )-\frac {2}{3} x^3 W\left (\frac {a}{x}\right )^{3/2}\right )+\frac {2}{5} x^3 \sqrt {W\left (\frac {a}{x}\right )}\) |
Input:
Int[x^2*Sqrt[ProductLog[a/x]],x]
Output:
(2*x^3*Sqrt[ProductLog[a/x]])/5 + ((-2*x^3*ProductLog[a/x]^(3/2))/3 + 2*(2 *a^3*Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ProductLog[a/x]]] + 2*x^3*ProductLog[a/x] ^(5/2)))/5
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]))
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[x^(m + 1)*((c*ProductLog[a* x^n])^p/(d*(m + n*p + 1))), x] - Simp[(m + 1)/(c*(m + n*p + 1)) Int[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] && LtQ[Simplify[p + (m + 1)/n], 0]
\[\int x^{2} \sqrt {\operatorname {LambertW}\left (\frac {a}{x}\right )}d x\]
Input:
int(x^2*LambertW(a/x)^(1/2),x)
Output:
int(x^2*LambertW(a/x)^(1/2),x)
\[ \int x^2 \sqrt {W\left (\frac {a}{x}\right )} \, dx=\int { x^{2} \sqrt {\operatorname {W}({\frac {a}{x}})} \,d x } \] Input:
integrate(x^2*lambert_w(a/x)^(1/2),x, algorithm="fricas")
Output:
integral(x^2*sqrt(lambert_w(a/x)), x)
\[ \int x^2 \sqrt {W\left (\frac {a}{x}\right )} \, dx=\int x^{2} \sqrt {W\left (\frac {a}{x}\right )}\, dx \] Input:
integrate(x**2*LambertW(a/x)**(1/2),x)
Output:
Integral(x**2*sqrt(LambertW(a/x)), x)
\[ \int x^2 \sqrt {W\left (\frac {a}{x}\right )} \, dx=\int { x^{2} \sqrt {\operatorname {W}({\frac {a}{x}})} \,d x } \] Input:
integrate(x^2*lambert_w(a/x)^(1/2),x, algorithm="maxima")
Output:
integrate(x^2*sqrt(lambert_w(a/x)), x)
\[ \int x^2 \sqrt {W\left (\frac {a}{x}\right )} \, dx=\int { x^{2} \sqrt {\operatorname {W}({\frac {a}{x}})} \,d x } \] Input:
integrate(x^2*lambert_w(a/x)^(1/2),x, algorithm="giac")
Output:
integrate(x^2*sqrt(lambert_w(a/x)), x)
Timed out. \[ \int x^2 \sqrt {W\left (\frac {a}{x}\right )} \, dx=\int x^2\,\sqrt {\mathrm {LambertW}\left (\frac {a}{x}\right )} \,d x \] Input:
int(x^2*LambertW(a/x)^(1/2),x)
Output:
int(x^2*LambertW(a/x)^(1/2), x)
\[ \int x^2 \sqrt {W\left (\frac {a}{x}\right )} \, dx=\frac {\sqrt {\textit {lambert\_w} \left (\frac {a}{x}\right )}\, x^{3}}{3}+\frac {\left (\int \frac {\sqrt {\textit {lambert\_w} \left (\frac {a}{x}\right )}\, x}{e^{\textit {lambert\_w} \left (\frac {a}{x}\right )} \textit {lambert\_w} \left (\frac {a}{x}\right )^{2}+e^{\textit {lambert\_w} \left (\frac {a}{x}\right )} \textit {lambert\_w} \left (\frac {a}{x}\right )}d x \right ) a}{6} \] Input:
int(x^2*Lambert_W(a/x)^(1/2),x)
Output:
(2*sqrt(lambert_w(a/x))*x**3 + int((sqrt(lambert_w(a/x))*x)/(e**lambert_w( a/x)*lambert_w(a/x)**2 + e**lambert_w(a/x)*lambert_w(a/x)),x)*a)/6