\(\int \frac {1}{(-c W(a+b x))^{5/2}} \, dx\) [382]

Optimal result

Integrand size = 13, antiderivative size = 91 \[ \int \frac {1}{(-c W(a+b x))^{5/2}} \, dx=-\frac {10 \sqrt {\pi } \text {erf}\left (\frac {\sqrt {-c W(a+b x)}}{\sqrt {c}}\right )}{3 b c^{5/2}}-\frac {2 (a+b x)}{3 b (-c W(a+b x))^{5/2}}+\frac {10 (a+b x)}{3 b c (-c W(a+b x))^{3/2}} \] Output:

-10/3*Pi^(1/2)*erf((-c*LambertW(b*x+a))^(1/2)/c^(1/2))/b/c^(5/2)-2/3*(b*x+ 
a)/b/(-c*LambertW(b*x+a))^(5/2)+10/3*(b*x+a)/b/c/(-c*LambertW(b*x+a))^(3/2 
)
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(-c W(a+b x))^{5/2}} \, dx=\frac {2 \left (-a-b x-5 (a+b x) W(a+b x)+5 \sqrt {\pi } \text {erfi}\left (\sqrt {W(a+b x)}\right ) W(a+b x)^{5/2}\right )}{3 b (-c W(a+b x))^{5/2}} \] Input:

Integrate[(-(c*ProductLog[a + b*x]))^(-5/2),x]
 

Output:

(2*(-a - b*x - 5*(a + b*x)*ProductLog[a + b*x] + 5*Sqrt[Pi]*Erfi[Sqrt[Prod 
uctLog[a + b*x]]]*ProductLog[a + b*x]^(5/2)))/(3*b*(-(c*ProductLog[a + b*x 
]))^(5/2))
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {7166, 7182, 7181}

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.

\(\displaystyle \int \frac {1}{(-c W(a+b x))^{5/2}} \, dx\)

\(\Big \downarrow \) 7166

\(\displaystyle -\frac {5 \int \frac {1}{(-c W(a+b x))^{3/2} (W(a+b x)+1)}dx}{3 c}-\frac {2 (a+b x)}{3 b (-c W(a+b x))^{5/2}}\)

\(\Big \downarrow \) 7182

\(\displaystyle -\frac {5 \left (-\frac {2 \int \frac {1}{\sqrt {-c W(a+b x)} (W(a+b x)+1)}dx}{c}-\frac {2 (a+b x)}{b (-c W(a+b x))^{3/2}}\right )}{3 c}-\frac {2 (a+b x)}{3 b (-c W(a+b x))^{5/2}}\)

\(\Big \downarrow \) 7181

\(\displaystyle -\frac {5 \left (\frac {2 \sqrt {\pi } \text {erf}\left (\frac {\sqrt {-c W(a+b x)}}{\sqrt {c}}\right )}{b c^{3/2}}-\frac {2 (a+b x)}{b (-c W(a+b x))^{3/2}}\right )}{3 c}-\frac {2 (a+b x)}{3 b (-c W(a+b x))^{5/2}}\)

Input:

Int[(-(c*ProductLog[a + b*x]))^(-5/2),x]
 

Output:

(-2*(a + b*x))/(3*b*(-(c*ProductLog[a + b*x]))^(5/2)) - (5*((2*Sqrt[Pi]*Er 
f[Sqrt[-(c*ProductLog[a + b*x])]/Sqrt[c]])/(b*c^(3/2)) - (2*(a + b*x))/(b* 
(-(c*ProductLog[a + b*x]))^(3/2))))/(3*c)
 

Defintions of rubi rules used

rule 7166

Int[((c_.)*ProductLog[(a_.) + (b_.)*(x_)])^(p_), x_Symbol] :> Simp[(a + b*x 
)*((c*ProductLog[a + b*x])^p/(b*(p + 1))), x] + Simp[p/(c*(p + 1))   Int[(c 
*ProductLog[a + b*x])^(p + 1)/(1 + ProductLog[a + b*x]), x], x] /; FreeQ[{a 
, b, c}, x] && LtQ[p, -1]
 

rule 7181

Int[1/(Sqrt[(c_.)*ProductLog[(a_.) + (b_.)*(x_)]]*((d_) + (d_.)*ProductLog[ 
(a_.) + (b_.)*(x_)])), x_Symbol] :> Simp[Rt[(-Pi)*c, 2]*(Erf[Sqrt[c*Product 
Log[a + b*x]]/Rt[-c, 2]]/(b*c*d)), x] /; FreeQ[{a, b, c, d}, x] && NegQ[c]
 

rule 7182

Int[((c_.)*ProductLog[(a_.) + (b_.)*(x_)])^(p_)/((d_) + (d_.)*ProductLog[(a 
_.) + (b_.)*(x_)]), x_Symbol] :> Simp[(a + b*x)*((c*ProductLog[a + b*x])^p/ 
(b*d*(p + 1))), x] - Simp[1/(c*(p + 1))   Int[(c*ProductLog[a + b*x])^(p + 
1)/(d + d*ProductLog[a + b*x]), x], x] /; FreeQ[{a, b, c, d}, x] && LtQ[p, 
-1]
 

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.60

method	result	size
default	\(\frac {-\frac {2 \left (b x +a \right )}{\sqrt {-c \operatorname {LambertW}\left (b x +a \right )}\, \operatorname {LambertW}\left (b x +a \right )}-\frac {2 \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-c \operatorname {LambertW}\left (b x +a \right )}}{\sqrt {c}}\right )}{\sqrt {c}}-2 c \left (-\frac {b x +a}{3 \left (-c \operatorname {LambertW}\left (b x +a \right )\right )^{\frac {3}{2}} \operatorname {LambertW}\left (b x +a \right )}-\frac {2 \left (-\frac {b x +a}{\sqrt {-c \operatorname {LambertW}\left (b x +a \right )}\, \operatorname {LambertW}\left (b x +a \right )}-\frac {\sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-c \operatorname {LambertW}\left (b x +a \right )}}{\sqrt {c}}\right )}{\sqrt {c}}\right )}{3 c}\right )}{b \,c^{2}}\)	\(146\)

Input:

int(1/(-c*LambertW(b*x+a))^(5/2),x,method=_RETURNVERBOSE)
 

Output:

2/b/c^2*(-1/(-c*LambertW(b*x+a))^(1/2)*(b*x+a)/LambertW(b*x+a)-1/c^(1/2)*P 
i^(1/2)*erf((-c*LambertW(b*x+a))^(1/2)/c^(1/2))-c*(-1/3/(-c*LambertW(b*x+a 
))^(3/2)*(b*x+a)/LambertW(b*x+a)-2/3/c*(-1/(-c*LambertW(b*x+a))^(1/2)*(b*x 
+a)/LambertW(b*x+a)-1/c^(1/2)*Pi^(1/2)*erf((-c*LambertW(b*x+a))^(1/2)/c^(1 
/2)))))
 

Fricas [F]

\[ \int \frac {1}{(-c W(a+b x))^{5/2}} \, dx=\int { \frac {1}{\left (-c \operatorname {W}({b x + a})\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(-c*lambert_w(b*x+a))^(5/2),x, algorithm="fricas")
 

Output:

integral(-sqrt(-c*lambert_w(b*x + a))/(c^3*lambert_w(b*x + a)^3), x)
 

Sympy [F]

\[ \int \frac {1}{(-c W(a+b x))^{5/2}} \, dx=\int \frac {1}{\left (- c W\left (a + b x\right )\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(1/(-c*LambertW(b*x+a))**(5/2),x)
 

Output:

Integral((-c*LambertW(a + b*x))**(-5/2), x)
 

Maxima [F]

\[ \int \frac {1}{(-c W(a+b x))^{5/2}} \, dx=\int { \frac {1}{\left (-c \operatorname {W}({b x + a})\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(-c*lambert_w(b*x+a))^(5/2),x, algorithm="maxima")
 

Output:

integrate((-c*lambert_w(b*x + a))^(-5/2), x)
 

Giac [F]

\[ \int \frac {1}{(-c W(a+b x))^{5/2}} \, dx=\int { \frac {1}{\left (-c \operatorname {W}({b x + a})\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(-c*lambert_w(b*x+a))^(5/2),x, algorithm="giac")
 

Output:

integrate((-c*lambert_w(b*x + a))^(-5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(-c W(a+b x))^{5/2}} \, dx=\int \frac {1}{{\left (-c\,\mathrm {LambertW}\left (a+b\,x\right )\right )}^{5/2}} \,d x \] Input:

int(1/(-c*LambertW(a + b*x))^(5/2),x)
 

Output:

int(1/(-c*LambertW(a + b*x))^(5/2), x)
 

Reduce [F]

\[ \int \frac {1}{(-c W(a+b x))^{5/2}} \, dx=-\frac {\sqrt {c}\, \left (\int \frac {\sqrt {\textit {lambert\_w} \left (b x +a \right )}}{\textit {lambert\_w} \left (b x +a \right )^{3}}d x \right ) i}{c^{3}} \] Input:

int(1/(-c*Lambert_W(b*x+a))^(5/2),x)
 

Output:

( - sqrt(c)*int(sqrt(lambert_w(a + b*x))/lambert_w(a + b*x)**3,x)*i)/c**3