\(\int x^2 \sqrt {W(\frac {a}{x})} \, dx\) [208]

Optimal result

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)
 

Mathematica [A] (verified)

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
 

Rubi [A] (verified)

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
 

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 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 7206

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]
 

Maple [F]

\[\int x^{2} \sqrt {\operatorname {LambertW}\left (\frac {a}{x}\right )}d x\]

Input:

int(x^2*LambertW(a/x)^(1/2),x)
 

Output:

int(x^2*LambertW(a/x)^(1/2),x)
 

Fricas [F]

\[ \int x^2 \sqrt {W\left (\frac {a}{x}\right )} \, dx=\int { x^{2} \sqrt {\operatorname {W}({\frac {a}{x}})} \,d x } \] Input:

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

Output:

integral(x^2*sqrt(lambert_w(a/x)), x)
 

Sympy [F]

\[ \int x^2 \sqrt {W\left (\frac {a}{x}\right )} \, dx=\int x^{2} \sqrt {W\left (\frac {a}{x}\right )}\, dx \] Input:

integrate(x**2*LambertW(a/x)**(1/2),x)
 

Output:

Integral(x**2*sqrt(LambertW(a/x)), x)
 

Maxima [F]

\[ \int x^2 \sqrt {W\left (\frac {a}{x}\right )} \, dx=\int { x^{2} \sqrt {\operatorname {W}({\frac {a}{x}})} \,d x } \] Input:

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

Output:

integrate(x^2*sqrt(lambert_w(a/x)), x)
 

Giac [F]

\[ \int x^2 \sqrt {W\left (\frac {a}{x}\right )} \, dx=\int { x^{2} \sqrt {\operatorname {W}({\frac {a}{x}})} \,d x } \] Input:

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

Output:

integrate(x^2*sqrt(lambert_w(a/x)), x)
 

Mupad [F(-1)]

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)
 

Reduce [F]

\[ \int x^2 \sqrt {W\left (\frac {a}{x}\right )} \, dx=\frac {\sqrt {\textit {lambert\_w} \left (\frac {a}{x}\right )}\, x^{3}}{3}+\frac {\left (\int \frac {\sqrt {\textit {lambert\_w} \left (\frac {a}{x}\right )}\, x}{e^{\textit {lambert\_w} \left (\frac {a}{x}\right )} \textit {lambert\_w} \left (\frac {a}{x}\right )^{2}+e^{\textit {lambert\_w} \left (\frac {a}{x}\right )} \textit {lambert\_w} \left (\frac {a}{x}\right )}d x \right ) a}{6} \] 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