\(\int \frac {\sqrt {W(\frac {a}{x})}}{x^3} \, dx\) [213]

Optimal result

Integrand size = 14, antiderivative size = 85 \[ \int \frac {\sqrt {W\left (\frac {a}{x}\right )}}{x^3} \, dx=\frac {3 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {W\left (\frac {a}{x}\right )}\right )}{64 a^2}-\frac {3}{32 x^2 W\left (\frac {a}{x}\right )^{3/2}}+\frac {1}{8 x^2 \sqrt {W\left (\frac {a}{x}\right )}}-\frac {\sqrt {W\left (\frac {a}{x}\right )}}{2 x^2} \] Output:

3/128*2^(1/2)*Pi^(1/2)*erfi(2^(1/2)*LambertW(a/x)^(1/2))/a^2-3/32/x^2/Lamb 
ertW(a/x)^(3/2)+1/8/x^2/LambertW(a/x)^(1/2)-1/2*LambertW(a/x)^(1/2)/x^2
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {W\left (\frac {a}{x}\right )}}{x^3} \, dx=\frac {1}{128} \left (\frac {3 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {W\left (\frac {a}{x}\right )}\right )}{a^2}-\frac {12}{x^2 W\left (\frac {a}{x}\right )^{3/2}}+\frac {16}{x^2 \sqrt {W\left (\frac {a}{x}\right )}}-\frac {64 \sqrt {W\left (\frac {a}{x}\right )}}{x^2}\right ) \] Input:

Integrate[Sqrt[ProductLog[a/x]]/x^3,x]
 

Output:

((3*Sqrt[2*Pi]*Erfi[Sqrt[2]*Sqrt[ProductLog[a/x]]])/a^2 - 12/(x^2*ProductL 
og[a/x]^(3/2)) + 16/(x^2*Sqrt[ProductLog[a/x]]) - (64*Sqrt[ProductLog[a/x] 
])/x^2)/128
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {7172, 7205, 7205, 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.

\(\displaystyle \int \frac {\sqrt {W\left (\frac {a}{x}\right )}}{x^3} \, dx\)

\(\Big \downarrow \) 7172

\(\displaystyle -\frac {1}{4} \int \frac {\sqrt {W\left (\frac {a}{x}\right )}}{x^3 \left (W\left (\frac {a}{x}\right )+1\right )}dx-\frac {\sqrt {W\left (\frac {a}{x}\right )}}{2 x^2}\)

\(\Big \downarrow \) 7205

\(\displaystyle \frac {1}{4} \left (\frac {3}{4} \int \frac {1}{x^3 \sqrt {W\left (\frac {a}{x}\right )} \left (W\left (\frac {a}{x}\right )+1\right )}dx+\frac {1}{2 x^2 \sqrt {W\left (\frac {a}{x}\right )}}\right )-\frac {\sqrt {W\left (\frac {a}{x}\right )}}{2 x^2}\)

\(\Big \downarrow \) 7205

\(\displaystyle \frac {1}{4} \left (\frac {3}{4} \left (-\frac {1}{4} \int \frac {1}{x^3 W\left (\frac {a}{x}\right )^{3/2} \left (W\left (\frac {a}{x}\right )+1\right )}dx-\frac {1}{2 x^2 W\left (\frac {a}{x}\right )^{3/2}}\right )+\frac {1}{2 x^2 \sqrt {W\left (\frac {a}{x}\right )}}\right )-\frac {\sqrt {W\left (\frac {a}{x}\right )}}{2 x^2}\)

\(\Big \downarrow \) 7204

\(\displaystyle \frac {1}{4} \left (\frac {3}{4} \left (\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {W\left (\frac {a}{x}\right )}\right )}{4 a^2}-\frac {1}{2 x^2 W\left (\frac {a}{x}\right )^{3/2}}\right )+\frac {1}{2 x^2 \sqrt {W\left (\frac {a}{x}\right )}}\right )-\frac {\sqrt {W\left (\frac {a}{x}\right )}}{2 x^2}\)

Input:

Int[Sqrt[ProductLog[a/x]]/x^3,x]
 

Output:

((3*((Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ProductLog[a/x]]])/(4*a^2) - 1/(2*x^2*P 
roductLog[a/x]^(3/2))))/4 + 1/(2*x^2*Sqrt[ProductLog[a/x]]))/4 - Sqrt[Prod 
uctLog[a/x]]/(2*x^2)
 

Defintions of rubi rules used

rule 7172

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]))
 

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)]
 

rule 7205

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]
 

Maple [F]

\[\int \frac {\sqrt {\operatorname {LambertW}\left (\frac {a}{x}\right )}}{x^{3}}d x\]

Input:

int(LambertW(a/x)^(1/2)/x^3,x)
 

Output:

int(LambertW(a/x)^(1/2)/x^3,x)
 

Fricas [F]

\[ \int \frac {\sqrt {W\left (\frac {a}{x}\right )}}{x^3} \, dx=\int { \frac {\sqrt {\operatorname {W}({\frac {a}{x}})}}{x^{3}} \,d x } \] Input:

integrate(lambert_w(a/x)^(1/2)/x^3,x, algorithm="fricas")
 

Output:

integral(sqrt(lambert_w(a/x))/x^3, x)
 

Sympy [F]

\[ \int \frac {\sqrt {W\left (\frac {a}{x}\right )}}{x^3} \, dx=\int \frac {\sqrt {W\left (\frac {a}{x}\right )}}{x^{3}}\, dx \] Input:

integrate(LambertW(a/x)**(1/2)/x**3,x)
 

Output:

Integral(sqrt(LambertW(a/x))/x**3, x)
 

Maxima [F]

\[ \int \frac {\sqrt {W\left (\frac {a}{x}\right )}}{x^3} \, dx=\int { \frac {\sqrt {\operatorname {W}({\frac {a}{x}})}}{x^{3}} \,d x } \] Input:

integrate(lambert_w(a/x)^(1/2)/x^3,x, algorithm="maxima")
 

Output:

integrate(sqrt(lambert_w(a/x))/x^3, x)
 

Giac [F]

\[ \int \frac {\sqrt {W\left (\frac {a}{x}\right )}}{x^3} \, dx=\int { \frac {\sqrt {\operatorname {W}({\frac {a}{x}})}}{x^{3}} \,d x } \] Input:

integrate(lambert_w(a/x)^(1/2)/x^3,x, algorithm="giac")
 

Output:

integrate(sqrt(lambert_w(a/x))/x^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {W\left (\frac {a}{x}\right )}}{x^3} \, dx=\int \frac {\sqrt {\mathrm {LambertW}\left (\frac {a}{x}\right )}}{x^3} \,d x \] Input:

int(LambertW(a/x)^(1/2)/x^3,x)
 

Output:

int(LambertW(a/x)^(1/2)/x^3, x)
 

Reduce [F]

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

int(Lambert_W(a/x)^(1/2)/x^3,x)
 

Output:

int(sqrt(lambert_w(a/x))/x**3,x)