\(\int \frac {x^6}{W(a x^2)^2} \, dx\) [148]

Optimal result

Integrand size = 12, antiderivative size = 94 \[ \int \frac {x^6}{W\left (a x^2\right )^2} \, dx=\frac {4 x^7}{49 W\left (a x^2\right )^3}-\frac {2 \sqrt {\frac {2}{7}} e^{-\frac {5}{2} W\left (a x^2\right )} x^5 \Gamma \left (\frac {1}{2},-\frac {7}{2} W\left (a x^2\right )\right ) \sqrt {-W\left (a x^2\right )}}{49 a W\left (a x^2\right )^3}+\frac {x^7}{7 W\left (a x^2\right )^2} \] Output:

4/49*x^7/LambertW(a*x^2)^3-2/343*14^(1/2)*x^5*Pi^(1/2)*erfc(1/2*(-14*Lambe 
rtW(a*x^2))^(1/2))*(-LambertW(a*x^2))^(1/2)/a/exp(5/2*LambertW(a*x^2))/Lam 
bertW(a*x^2)^3+1/7*x^7/LambertW(a*x^2)^2
 

Mathematica [F]

\[ \int \frac {x^6}{W\left (a x^2\right )^2} \, dx=\int \frac {x^6}{W\left (a x^2\right )^2} \, dx \] Input:

Integrate[x^6/ProductLog[a*x^2]^2,x]
 

Output:

Integrate[x^6/ProductLog[a*x^2]^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.

\(\displaystyle \int \frac {x^6}{W\left (a x^2\right )^2} \, dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \frac {x^6}{W\left (a x^2\right )^2}dx\)

Input:

Int[x^6/ProductLog[a*x^2]^2,x]
 

Output:

$Aborted
 

Maple [F]

\[\int \frac {x^{6}}{\operatorname {LambertW}\left (a \,x^{2}\right )^{2}}d x\]

Input:

int(x^6/LambertW(a*x^2)^2,x)
 

Output:

int(x^6/LambertW(a*x^2)^2,x)
 

Fricas [F]

\[ \int \frac {x^6}{W\left (a x^2\right )^2} \, dx=\int { \frac {x^{6}}{\operatorname {W}({a x^{2}})^{2}} \,d x } \] Input:

integrate(x^6/lambert_w(a*x^2)^2,x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

integral(x^6/lambert_w(a*x^2)^2, x)
 

Sympy [F]

\[ \int \frac {x^6}{W\left (a x^2\right )^2} \, dx=\int \frac {x^{6}}{W^{2}\left (a x^{2}\right )}\, dx \] Input:

integrate(x**6/LambertW(a*x**2)**2,x)
 

Output:

Integral(x**6/LambertW(a*x**2)**2, x)
 

Maxima [F]

\[ \int \frac {x^6}{W\left (a x^2\right )^2} \, dx=\int { \frac {x^{6}}{\operatorname {W}({a x^{2}})^{2}} \,d x } \] Input:

integrate(x^6/lambert_w(a*x^2)^2,x, algorithm="maxima")
 

Output:

integrate(x^6/lambert_w(a*x^2)^2, x)
 

Giac [F]

\[ \int \frac {x^6}{W\left (a x^2\right )^2} \, dx=\int { \frac {x^{6}}{\operatorname {W}({a x^{2}})^{2}} \,d x } \] Input:

integrate(x^6/lambert_w(a*x^2)^2,x, algorithm="giac")
 

Output:

integrate(x^6/lambert_w(a*x^2)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^6}{W\left (a x^2\right )^2} \, dx=\int \frac {x^6}{{\mathrm {LambertW}\left (a\,x^2\right )}^2} \,d x \] Input:

int(x^6/LambertW(a*x^2)^2,x)
 

Output:

int(x^6/LambertW(a*x^2)^2, x)
 

Reduce [F]

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

int(x^6/Lambert_W(a*x^2)^2,x)
 

Output:

int(x**6/lambert_w(a*x**2)**2,x)