\(\int \frac {1}{(-c W(a+b x))^{7/2}} \, dx\) [383]

Optimal result

Integrand size = 13, antiderivative size = 119 \[ \int \frac {1}{(-c W(a+b x))^{7/2}} \, dx=\frac {28 \sqrt {\pi } \text {erf}\left (\frac {\sqrt {-c W(a+b x)}}{\sqrt {c}}\right )}{15 b c^{7/2}}-\frac {2 (a+b x)}{5 b (-c W(a+b x))^{7/2}}+\frac {14 (a+b x)}{15 b c (-c W(a+b x))^{5/2}}-\frac {28 (a+b x)}{15 b c^2 (-c W(a+b x))^{3/2}} \] Output:

28/15*Pi^(1/2)*erf((-c*LambertW(b*x+a))^(1/2)/c^(1/2))/b/c^(7/2)-2/5*(b*x+ 
a)/b/(-c*LambertW(b*x+a))^(7/2)+14/15*(b*x+a)/b/c/(-c*LambertW(b*x+a))^(5/ 
2)-28/15*(b*x+a)/b/c^2/(-c*LambertW(b*x+a))^(3/2)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.80 \[ \int \frac {1}{(-c W(a+b x))^{7/2}} \, dx=\frac {6 (a+b x)+14 (a+b x) W(a+b x)+28 (a+b x) W(a+b x)^2-28 \sqrt {\pi } \text {erfi}\left (\sqrt {W(a+b x)}\right ) W(a+b x)^{7/2}}{15 b c^3 W(a+b x)^3 \sqrt {-c W(a+b x)}} \] Input:

Integrate[(-(c*ProductLog[a + b*x]))^(-7/2),x]
 

Output:

(6*(a + b*x) + 14*(a + b*x)*ProductLog[a + b*x] + 28*(a + b*x)*ProductLog[ 
a + b*x]^2 - 28*Sqrt[Pi]*Erfi[Sqrt[ProductLog[a + b*x]]]*ProductLog[a + b* 
x]^(7/2))/(15*b*c^3*ProductLog[a + b*x]^3*Sqrt[-(c*ProductLog[a + b*x])])
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {7166, 7182, 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.

\(\displaystyle \int \frac {1}{(-c W(a+b x))^{7/2}} \, dx\)

\(\Big \downarrow \) 7166

\(\displaystyle -\frac {7 \int \frac {1}{(-c W(a+b x))^{5/2} (W(a+b x)+1)}dx}{5 c}-\frac {2 (a+b x)}{5 b (-c W(a+b x))^{7/2}}\)

\(\Big \downarrow \) 7182

\(\displaystyle -\frac {7 \left (-\frac {2 \int \frac {1}{(-c W(a+b x))^{3/2} (W(a+b x)+1)}dx}{3 c}-\frac {2 (a+b x)}{3 b (-c W(a+b x))^{5/2}}\right )}{5 c}-\frac {2 (a+b x)}{5 b (-c W(a+b x))^{7/2}}\)

\(\Big \downarrow \) 7182

\(\displaystyle -\frac {7 \left (-\frac {2 \left (-\frac {2 \int \frac {1}{\sqrt {-c W(a+b x)} (W(a+b x)+1)}dx}{c}-\frac {2 (a+b x)}{b (-c W(a+b x))^{3/2}}\right )}{3 c}-\frac {2 (a+b x)}{3 b (-c W(a+b x))^{5/2}}\right )}{5 c}-\frac {2 (a+b x)}{5 b (-c W(a+b x))^{7/2}}\)

\(\Big \downarrow \) 7181

\(\displaystyle -\frac {7 \left (-\frac {2 \left (\frac {2 \sqrt {\pi } \text {erf}\left (\frac {\sqrt {-c W(a+b x)}}{\sqrt {c}}\right )}{b c^{3/2}}-\frac {2 (a+b x)}{b (-c W(a+b x))^{3/2}}\right )}{3 c}-\frac {2 (a+b x)}{3 b (-c W(a+b x))^{5/2}}\right )}{5 c}-\frac {2 (a+b x)}{5 b (-c W(a+b x))^{7/2}}\)

Input:

Int[(-(c*ProductLog[a + b*x]))^(-7/2),x]
 

Output:

(-2*(a + b*x))/(5*b*(-(c*ProductLog[a + b*x]))^(7/2)) - (7*((-2*(a + b*x)) 
/(3*b*(-(c*ProductLog[a + b*x]))^(5/2)) - (2*((2*Sqrt[Pi]*Erf[Sqrt[-(c*Pro 
ductLog[a + b*x])]/Sqrt[c]])/(b*c^(3/2)) - (2*(a + b*x))/(b*(-(c*ProductLo 
g[a + b*x]))^(3/2))))/(3*c)))/(5*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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(209\) vs. \(2(97)=194\).

Time = 0.05 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.76

method	result	size
default	\(\frac {-\frac {2 \left (b x +a \right )}{3 \left (-c \operatorname {LambertW}\left (b x +a \right )\right )^{\frac {3}{2}} \operatorname {LambertW}\left (b x +a \right )}-\frac {4 \left (-\frac {b x +a}{\sqrt {-c \operatorname {LambertW}\left (b x +a \right )}\, \operatorname {LambertW}\left (b x +a \right )}-\frac {\sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-c \operatorname {LambertW}\left (b x +a \right )}}{\sqrt {c}}\right )}{\sqrt {c}}\right )}{3 c}-2 c \left (-\frac {b x +a}{5 \left (-c \operatorname {LambertW}\left (b x +a \right )\right )^{\frac {5}{2}} \operatorname {LambertW}\left (b x +a \right )}-\frac {2 \left (-\frac {b x +a}{3 \left (-c \operatorname {LambertW}\left (b x +a \right )\right )^{\frac {3}{2}} \operatorname {LambertW}\left (b x +a \right )}-\frac {2 \left (-\frac {b x +a}{\sqrt {-c \operatorname {LambertW}\left (b x +a \right )}\, \operatorname {LambertW}\left (b x +a \right )}-\frac {\sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-c \operatorname {LambertW}\left (b x +a \right )}}{\sqrt {c}}\right )}{\sqrt {c}}\right )}{3 c}\right )}{5 c}\right )}{b \,c^{2}}\)	\(210\)

Input:

int(1/(-c*LambertW(b*x+a))^(7/2),x,method=_RETURNVERBOSE)
 

Output:

2/b/c^2*(-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)))-c*(-1/5/(-c*LambertW(b*x+a))^(5/2)*( 
b*x+a)/LambertW(b*x+a)-2/5/c*(-1/3/(-c*LambertW(b*x+a))^(3/2)*(b*x+a)/Lamb 
ertW(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]

\[ \int \frac {1}{(-c W(a+b x))^{7/2}} \, dx=\int { \frac {1}{\left (-c \operatorname {W}({b x + a})\right )^{\frac {7}{2}}} \,d x } \] Input:

integrate(1/(-c*lambert_w(b*x+a))^(7/2),x, algorithm="fricas")
 

Output:

integral(sqrt(-c*lambert_w(b*x + a))/(c^4*lambert_w(b*x + a)^4), x)
 

Sympy [F]

\[ \int \frac {1}{(-c W(a+b x))^{7/2}} \, dx=\int \frac {1}{\left (- c W\left (a + b x\right )\right )^{\frac {7}{2}}}\, dx \] Input:

integrate(1/(-c*LambertW(b*x+a))**(7/2),x)
 

Output:

Integral((-c*LambertW(a + b*x))**(-7/2), x)
 

Maxima [F]

\[ \int \frac {1}{(-c W(a+b x))^{7/2}} \, dx=\int { \frac {1}{\left (-c \operatorname {W}({b x + a})\right )^{\frac {7}{2}}} \,d x } \] Input:

integrate(1/(-c*lambert_w(b*x+a))^(7/2),x, algorithm="maxima")
 

Output:

integrate((-c*lambert_w(b*x + a))^(-7/2), x)
 

Giac [F]

\[ \int \frac {1}{(-c W(a+b x))^{7/2}} \, dx=\int { \frac {1}{\left (-c \operatorname {W}({b x + a})\right )^{\frac {7}{2}}} \,d x } \] Input:

integrate(1/(-c*lambert_w(b*x+a))^(7/2),x, algorithm="giac")
 

Output:

integrate((-c*lambert_w(b*x + a))^(-7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(-c W(a+b x))^{7/2}} \, dx=\int \frac {1}{{\left (-c\,\mathrm {LambertW}\left (a+b\,x\right )\right )}^{7/2}} \,d x \] Input:

int(1/(-c*LambertW(a + b*x))^(7/2),x)
 

Output:

int(1/(-c*LambertW(a + b*x))^(7/2), x)
 

Reduce [F]

\[ \int \frac {1}{(-c W(a+b x))^{7/2}} \, dx=\frac {\sqrt {c}\, \left (\int \frac {\sqrt {\textit {lambert\_w} \left (b x +a \right )}}{\textit {lambert\_w} \left (b x +a \right )^{4}}d x \right ) i}{c^{4}} \] Input:

int(1/(-c*Lambert_W(b*x+a))^(7/2),x)
 

Output:

(sqrt(c)*int(sqrt(lambert_w(a + b*x))/lambert_w(a + b*x)**4,x)*i)/c**4