\(\int \frac {1}{x^4 \sqrt {c W(a x^2)}} \, dx\) [179]

Optimal result

Integrand size = 16, antiderivative size = 97 \[ \int \frac {1}{x^4 \sqrt {c W\left (a x^2\right )}} \, dx=-\frac {1}{4 x^3 \sqrt {c W\left (a x^2\right )}}-\frac {\sqrt {c W\left (a x^2\right )}}{8 c x^3}+\frac {3 e^{\frac {5}{2} W\left (a x^2\right )} \Gamma \left (0,\frac {3}{2} W\left (a x^2\right )\right ) W\left (a x^2\right ) \left (c W\left (a x^2\right )\right )^{3/2}}{16 a c^2 x^5} \] Output:

-1/4/x^3/(c*LambertW(a*x^2))^(1/2)-1/8*(c*LambertW(a*x^2))^(1/2)/c/x^3+3/1 
6*exp(5/2*LambertW(a*x^2))*Ei(1,3/2*LambertW(a*x^2))*LambertW(a*x^2)*(c*La 
mbertW(a*x^2))^(3/2)/a/c^2/x^5
 

Mathematica [F]

\[ \int \frac {1}{x^4 \sqrt {c W\left (a x^2\right )}} \, dx=\int \frac {1}{x^4 \sqrt {c W\left (a x^2\right )}} \, dx \] Input:

Integrate[1/(x^4*Sqrt[c*ProductLog[a*x^2]]),x]
 

Output:

Integrate[1/(x^4*Sqrt[c*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.

\(\displaystyle \int \frac {1}{x^4 \sqrt {c W\left (a x^2\right )}} \, dx\)

\(\Big \downarrow \) 7271

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

\(\Big \downarrow \) 7299

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

Input:

Int[1/(x^4*Sqrt[c*ProductLog[a*x^2]]),x]
 

Output:

$Aborted
 

Maple [F]

\[\int \frac {1}{x^{4} \sqrt {c \operatorname {LambertW}\left (a \,x^{2}\right )}}d x\]

Input:

int(1/x^4/(c*LambertW(a*x^2))^(1/2),x)
 

Output:

int(1/x^4/(c*LambertW(a*x^2))^(1/2),x)
 

Fricas [F]

\[ \int \frac {1}{x^4 \sqrt {c W\left (a x^2\right )}} \, dx=\int { \frac {1}{\sqrt {c \operatorname {W}({a x^{2}})} x^{4}} \,d x } \] Input:

integrate(1/x^4/(c*lambert_w(a*x^2))^(1/2),x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

integral(sqrt(c*lambert_w(a*x^2))/(c*x^4*lambert_w(a*x^2)), x)
 

Sympy [F]

\[ \int \frac {1}{x^4 \sqrt {c W\left (a x^2\right )}} \, dx=\int \frac {1}{x^{4} \sqrt {c W\left (a x^{2}\right )}}\, dx \] Input:

integrate(1/x**4/(c*LambertW(a*x**2))**(1/2),x)
 

Output:

Integral(1/(x**4*sqrt(c*LambertW(a*x**2))), x)
 

Maxima [F]

\[ \int \frac {1}{x^4 \sqrt {c W\left (a x^2\right )}} \, dx=\int { \frac {1}{\sqrt {c \operatorname {W}({a x^{2}})} x^{4}} \,d x } \] Input:

integrate(1/x^4/(c*lambert_w(a*x^2))^(1/2),x, algorithm="maxima")
 

Output:

integrate(1/(sqrt(c*lambert_w(a*x^2))*x^4), x)
 

Giac [F]

\[ \int \frac {1}{x^4 \sqrt {c W\left (a x^2\right )}} \, dx=\int { \frac {1}{\sqrt {c \operatorname {W}({a x^{2}})} x^{4}} \,d x } \] Input:

integrate(1/x^4/(c*lambert_w(a*x^2))^(1/2),x, algorithm="giac")
 

Output:

integrate(1/(sqrt(c*lambert_w(a*x^2))*x^4), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^4 \sqrt {c W\left (a x^2\right )}} \, dx=\int \frac {1}{x^4\,\sqrt {c\,\mathrm {LambertW}\left (a\,x^2\right )}} \,d x \] Input:

int(1/(x^4*(c*LambertW(a*x^2))^(1/2)),x)
 

Output:

int(1/(x^4*(c*LambertW(a*x^2))^(1/2)), x)
 

Reduce [F]

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

int(1/x^4/(c*Lambert_W(a*x^2))^(1/2),x)
 

Output:

(sqrt(c)*int(sqrt(lambert_w(a*x**2))/(lambert_w(a*x**2)*x**4),x))/c