\(\int \frac {x^m}{d+d W(\frac {a}{x^2})} \, dx\) [340]

Optimal result

Integrand size = 16, antiderivative size = 84 \[ \int \frac {x^m}{d+d W\left (\frac {a}{x^2}\right )} \, dx=\frac {2^{-\frac {3}{2}-\frac {m}{2}} e^{\frac {1}{2} (3+m) W\left (\frac {a}{x^2}\right )} x^{3+m} \Gamma \left (\frac {1}{2} (-1-m),\frac {1}{2} (1+m) W\left (\frac {a}{x^2}\right )\right ) \left ((1+m) W\left (\frac {a}{x^2}\right )\right )^{\frac {3+m}{2}}}{a d (1+m)} \] Output:

2^(-3/2-1/2*m)*exp(1/2*(3+m)*LambertW(a/x^2))*x^(3+m)*GAMMA(-1/2-1/2*m,1/2 
*(1+m)*LambertW(a/x^2))*((1+m)*LambertW(a/x^2))^(3/2+1/2*m)/a/d/(1+m)
                                                                                    
                                                                                    
 

Mathematica [F]

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

Integrate[x^m/(d + d*ProductLog[a/x^2]),x]
 

Output:

Integrate[x^m/(d + d*ProductLog[a/x^2]), x]
 

Rubi [F]

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.

\(\displaystyle \int \frac {x^m}{d W\left (\frac {a}{x^2}\right )+d} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {x^m}{d \left (W\left (\frac {a}{x^2}\right )+1\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {x^m}{W\left (\frac {a}{x^2}\right )+1}dx}{d}\)

\(\Big \downarrow \) 7299

\(\displaystyle \frac {\int \frac {x^m}{W\left (\frac {a}{x^2}\right )+1}dx}{d}\)

Input:

Int[x^m/(d + d*ProductLog[a/x^2]),x]
 

Output:

$Aborted
 

Maple [F]

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

Input:

int(x^m/(d+d*LambertW(a/x^2)),x)
 

Output:

int(x^m/(d+d*LambertW(a/x^2)),x)
 

Fricas [F]

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

integrate(x^m/(d+d*lambert_w(a/x^2)),x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

integral(x^m/(d*lambert_w(a/x^2) + d), x)
 

Sympy [F]

\[ \int \frac {x^m}{d+d W\left (\frac {a}{x^2}\right )} \, dx=\frac {\int \frac {x^{m}}{W\left (\frac {a}{x^{2}}\right ) + 1}\, dx}{d} \] Input:

integrate(x**m/(d+d*LambertW(a/x**2)),x)
 

Output:

Integral(x**m/(LambertW(a/x**2) + 1), x)/d
 

Maxima [F]

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

integrate(x^m/(d+d*lambert_w(a/x^2)),x, algorithm="maxima")
 

Output:

integrate(x^m/(d*lambert_w(a/x^2) + d), x)
 

Giac [F]

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

integrate(x^m/(d+d*lambert_w(a/x^2)),x, algorithm="giac")
 

Output:

integrate(x^m/(d*lambert_w(a/x^2) + d), x)
 

Mupad [F(-1)]

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

int(x^m/(d + d*LambertW(a/x^2)),x)
 

Output:

int(x^m/(d + d*LambertW(a/x^2)), x)
 

Reduce [F]

\[ \int \frac {x^m}{d+d W\left (\frac {a}{x^2}\right )} \, dx=\frac {3 x^{m} x +\left (\int \frac {x^{m}}{\textit {lambert\_w} \left (\frac {a}{x^{2}}\right )+1}d x \right ) m +\int \frac {x^{m}}{\textit {lambert\_w} \left (\frac {a}{x^{2}}\right )+1}d x -3 \left (\int \frac {x^{m} \textit {lambert\_w} \left (\frac {a}{x^{2}}\right )}{\textit {lambert\_w} \left (\frac {a}{x^{2}}\right )+1}d x \right ) m -3 \left (\int \frac {x^{m} \textit {lambert\_w} \left (\frac {a}{x^{2}}\right )}{\textit {lambert\_w} \left (\frac {a}{x^{2}}\right )+1}d x \right )}{4 d \left (m +1\right )} \] Input:

int(x^m/(d+d*Lambert_W(a/x^2)),x)
 

Output:

(3*x**m*x + int(x**m/(lambert_w(a/x**2) + 1),x)*m + int(x**m/(lambert_w(a/ 
x**2) + 1),x) - 3*int((x**m*lambert_w(a/x**2))/(lambert_w(a/x**2) + 1),x)* 
m - 3*int((x**m*lambert_w(a/x**2))/(lambert_w(a/x**2) + 1),x))/(4*d*(m + 1 
))