\(\int x \sqrt {c W(a+b x)} \, dx\) [399]

Optimal result

Integrand size = 14, antiderivative size = 214 \[ \int x \sqrt {c W(a+b x)} \, dx=-\frac {a \sqrt {c} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c W(a+b x)}}{\sqrt {c}}\right )}{4 b^2}-\frac {3 \sqrt {c} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {c W(a+b x)}}{\sqrt {c}}\right )}{64 b^2}+\frac {3 c^2 (a+b x)^2}{32 b^2 (c W(a+b x))^{3/2}}+\frac {a c (a+b x)}{2 b^2 \sqrt {c W(a+b x)}}-\frac {c (a+b x)^2}{8 b^2 \sqrt {c W(a+b x)}}-\frac {a (a+b x) \sqrt {c W(a+b x)}}{b^2}+\frac {(a+b x)^2 \sqrt {c W(a+b x)}}{2 b^2} \] Output:

-1/4*a*c^(1/2)*Pi^(1/2)*erfi((c*LambertW(b*x+a))^(1/2)/c^(1/2))/b^2-3/128* 
c^(1/2)*2^(1/2)*Pi^(1/2)*erfi(2^(1/2)*(c*LambertW(b*x+a))^(1/2)/c^(1/2))/b 
^2+3/32*c^2*(b*x+a)^2/b^2/(c*LambertW(b*x+a))^(3/2)+1/2*a*c*(b*x+a)/b^2/(c 
*LambertW(b*x+a))^(1/2)-1/8*c*(b*x+a)^2/b^2/(c*LambertW(b*x+a))^(1/2)-a*(b 
*x+a)*(c*LambertW(b*x+a))^(1/2)/b^2+1/2*(b*x+a)^2*(c*LambertW(b*x+a))^(1/2 
)/b^2
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.64 \[ \int x \sqrt {c W(a+b x)} \, dx=\frac {c^2 \left (12 (a+b x)^2+16 \left (3 a^2+2 a b x-b^2 x^2\right ) W(a+b x)-\sqrt {\pi } \left (32 a \text {erfi}\left (\sqrt {W(a+b x)}\right )+3 \sqrt {2} \text {erfi}\left (\sqrt {2} \sqrt {W(a+b x)}\right )\right ) W(a+b x)^{3/2}-64 \left (a^2-b^2 x^2\right ) W(a+b x)^2\right )}{128 b^2 (c W(a+b x))^{3/2}} \] Input:

Integrate[x*Sqrt[c*ProductLog[a + b*x]],x]
 

Output:

(c^2*(12*(a + b*x)^2 + 16*(3*a^2 + 2*a*b*x - b^2*x^2)*ProductLog[a + b*x] 
- Sqrt[Pi]*(32*a*Erfi[Sqrt[ProductLog[a + b*x]]] + 3*Sqrt[2]*Erfi[Sqrt[2]* 
Sqrt[ProductLog[a + b*x]]])*ProductLog[a + b*x]^(3/2) - 64*(a^2 - b^2*x^2) 
*ProductLog[a + b*x]^2))/(128*b^2*(c*ProductLog[a + b*x])^(3/2))
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.92, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {7168, 2009}

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.

\(\displaystyle \int x \sqrt {c W(a+b x)} \, dx\)

\(\Big \downarrow \) 7168

\(\displaystyle \frac {\int \left ((a+b x) \sqrt {c W(a+b x)}-a \sqrt {c W(a+b x)}\right )d(a+b x)}{b^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {3 c^2 (a+b x)^2}{32 (c W(a+b x))^{3/2}}-\frac {1}{4} \sqrt {\pi } a \sqrt {c} \text {erfi}\left (\frac {\sqrt {c W(a+b x)}}{\sqrt {c}}\right )-\frac {3}{64} \sqrt {\frac {\pi }{2}} \sqrt {c} \text {erfi}\left (\frac {\sqrt {2} \sqrt {c W(a+b x)}}{\sqrt {c}}\right )+\frac {1}{2} (a+b x)^2 \sqrt {c W(a+b x)}-\frac {c (a+b x)^2}{8 \sqrt {c W(a+b x)}}-a (a+b x) \sqrt {c W(a+b x)}+\frac {a c (a+b x)}{2 \sqrt {c W(a+b x)}}}{b^2}\)

Input:

Int[x*Sqrt[c*ProductLog[a + b*x]],x]
 

Output:

(-1/4*(a*Sqrt[c]*Sqrt[Pi]*Erfi[Sqrt[c*ProductLog[a + b*x]]/Sqrt[c]]) - (3* 
Sqrt[c]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[c*ProductLog[a + b*x]])/Sqrt[c]])/64 
 + (3*c^2*(a + b*x)^2)/(32*(c*ProductLog[a + b*x])^(3/2)) + (a*c*(a + b*x) 
)/(2*Sqrt[c*ProductLog[a + b*x]]) - (c*(a + b*x)^2)/(8*Sqrt[c*ProductLog[a 
 + b*x]]) - a*(a + b*x)*Sqrt[c*ProductLog[a + b*x]] + ((a + b*x)^2*Sqrt[c* 
ProductLog[a + b*x]])/2)/b^2
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7168

Int[((e_.) + (f_.)*(x_))^(m_.)*((c_.)*ProductLog[(a_) + (b_.)*(x_)])^(p_.), 
 x_Symbol] :> Simp[1/b^(m + 1)   Subst[Int[ExpandIntegrand[(c*ProductLog[x] 
)^p, (b*e - a*f + f*x)^m, x], x], x, a + b*x], x] /; FreeQ[{a, b, c, e, f, 
p}, x] && IGtQ[m, 0]
 

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.28

method	result	size
default	\(\frac {\frac {c \left (c \operatorname {LambertW}\left (b x +a \right )\right )^{\frac {5}{2}} {\mathrm e}^{2 \operatorname {LambertW}\left (b x +a \right )}}{2}-\frac {c \left (\frac {c \left (c \operatorname {LambertW}\left (b x +a \right )\right )^{\frac {3}{2}} {\mathrm e}^{2 \operatorname {LambertW}\left (b x +a \right )}}{4}-\frac {3 c \left (\frac {c \sqrt {c \operatorname {LambertW}\left (b x +a \right )}\, {\mathrm e}^{2 \operatorname {LambertW}\left (b x +a \right )}}{4}-\frac {c \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-\frac {2}{c}}\, \sqrt {c \operatorname {LambertW}\left (b x +a \right )}\right )}{8 \sqrt {-\frac {2}{c}}}\right )}{4}\right )}{2}-2 a \,c^{2} \left (\frac {c \sqrt {c \operatorname {LambertW}\left (b x +a \right )}\, \left (b x +a \right )}{2 \operatorname {LambertW}\left (b x +a \right )}-\frac {c \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-\frac {1}{c}}\, \sqrt {c \operatorname {LambertW}\left (b x +a \right )}\right )}{4 \sqrt {-\frac {1}{c}}}\right )-2 a c \left (\frac {\left (c \operatorname {LambertW}\left (b x +a \right )\right )^{\frac {3}{2}} \left (b x +a \right ) c}{2 \operatorname {LambertW}\left (b x +a \right )}-\frac {3 c \left (\frac {c \sqrt {c \operatorname {LambertW}\left (b x +a \right )}\, \left (b x +a \right )}{2 \operatorname {LambertW}\left (b x +a \right )}-\frac {c \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-\frac {1}{c}}\, \sqrt {c \operatorname {LambertW}\left (b x +a \right )}\right )}{4 \sqrt {-\frac {1}{c}}}\right )}{2}\right )}{b^{2} c^{3}}\)	\(274\)

Input:

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

Output:

2/b^2/c^3*(1/4*c*(c*LambertW(b*x+a))^(5/2)*exp(2*LambertW(b*x+a))-1/4*c*(1 
/4*c*(c*LambertW(b*x+a))^(3/2)*exp(2*LambertW(b*x+a))-3/4*c*(1/4*c*(c*Lamb 
ertW(b*x+a))^(1/2)*exp(2*LambertW(b*x+a))-1/8*c*Pi^(1/2)/(-2/c)^(1/2)*erf( 
(-2/c)^(1/2)*(c*LambertW(b*x+a))^(1/2))))-a*c^2*(1/2*c*(c*LambertW(b*x+a)) 
^(1/2)*(b*x+a)/LambertW(b*x+a)-1/4*c*Pi^(1/2)/(-1/c)^(1/2)*erf((-1/c)^(1/2 
)*(c*LambertW(b*x+a))^(1/2)))-a*c*(1/2*(c*LambertW(b*x+a))^(3/2)*(b*x+a)/L 
ambertW(b*x+a)*c-3/2*c*(1/2*c*(c*LambertW(b*x+a))^(1/2)*(b*x+a)/LambertW(b 
*x+a)-1/4*c*Pi^(1/2)/(-1/c)^(1/2)*erf((-1/c)^(1/2)*(c*LambertW(b*x+a))^(1/ 
2)))))
 

Fricas [F]

\[ \int x \sqrt {c W(a+b x)} \, dx=\int { \sqrt {c \operatorname {W}({b x + a})} x \,d x } \] Input:

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

Output:

integral(sqrt(c*lambert_w(b*x + a))*x, x)
 

Sympy [F]

\[ \int x \sqrt {c W(a+b x)} \, dx=\int x \sqrt {c W\left (a + b x\right )}\, dx \] Input:

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

Output:

Integral(x*sqrt(c*LambertW(a + b*x)), x)
 

Maxima [F]

\[ \int x \sqrt {c W(a+b x)} \, dx=\int { \sqrt {c \operatorname {W}({b x + a})} x \,d x } \] Input:

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

Output:

integrate(sqrt(c*lambert_w(b*x + a))*x, x)
 

Giac [F]

\[ \int x \sqrt {c W(a+b x)} \, dx=\int { \sqrt {c \operatorname {W}({b x + a})} x \,d x } \] Input:

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

Output:

integrate(sqrt(c*lambert_w(b*x + a))*x, x)
 

Mupad [F(-1)]

Timed out. \[ \int x \sqrt {c W(a+b x)} \, dx=\int x\,\sqrt {c\,\mathrm {LambertW}\left (a+b\,x\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int x \sqrt {c W(a+b x)} \, dx=\sqrt {c}\, \left (\int \sqrt {\textit {lambert\_w} \left (b x +a \right )}\, x d x \right ) \] Input:

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

Output:

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