\(\int x^3 W(a+b x)^2 \, dx\) [391]

Optimal result

Integrand size = 12, antiderivative size = 456 \[ \int x^3 W(a+b x)^2 \, dx=-\frac {4 a^3 x}{b^3}+\frac {9 a^2 (a+b x)^2}{4 b^4}-\frac {8 a (a+b x)^3}{9 b^4}+\frac {5 (a+b x)^4}{32 b^4}+\frac {15 (a+b x)^4}{1024 b^4 W(a+b x)^4}+\frac {16 a (a+b x)^3}{81 b^4 W(a+b x)^3}-\frac {15 (a+b x)^4}{256 b^4 W(a+b x)^3}+\frac {9 a^2 (a+b x)^2}{8 b^4 W(a+b x)^2}-\frac {16 a (a+b x)^3}{27 b^4 W(a+b x)^2}+\frac {15 (a+b x)^4}{128 b^4 W(a+b x)^2}+\frac {4 a^3 (a+b x)}{b^4 W(a+b x)}-\frac {9 a^2 (a+b x)^2}{4 b^4 W(a+b x)}+\frac {8 a (a+b x)^3}{9 b^4 W(a+b x)}-\frac {5 (a+b x)^4}{32 b^4 W(a+b x)}+\frac {2 a^3 (a+b x) W(a+b x)}{b^4}-\frac {3 a^2 (a+b x)^2 W(a+b x)}{2 b^4}+\frac {2 a (a+b x)^3 W(a+b x)}{3 b^4}-\frac {(a+b x)^4 W(a+b x)}{8 b^4}-\frac {a^3 (a+b x) W(a+b x)^2}{b^4}+\frac {3 a^2 (a+b x)^2 W(a+b x)^2}{2 b^4}-\frac {a (a+b x)^3 W(a+b x)^2}{b^4}+\frac {(a+b x)^4 W(a+b x)^2}{4 b^4} \] Output:

-4*a^3*x/b^3+9/4*a^2*(b*x+a)^2/b^4-8/9*a*(b*x+a)^3/b^4+5/32*(b*x+a)^4/b^4+ 
15/1024*(b*x+a)^4/b^4/LambertW(b*x+a)^4+16/81*a*(b*x+a)^3/b^4/LambertW(b*x 
+a)^3-15/256*(b*x+a)^4/b^4/LambertW(b*x+a)^3+9/8*a^2*(b*x+a)^2/b^4/Lambert 
W(b*x+a)^2-16/27*a*(b*x+a)^3/b^4/LambertW(b*x+a)^2+15/128*(b*x+a)^4/b^4/La 
mbertW(b*x+a)^2+4*a^3*(b*x+a)/b^4/LambertW(b*x+a)-9/4*a^2*(b*x+a)^2/b^4/La 
mbertW(b*x+a)+8/9*a*(b*x+a)^3/b^4/LambertW(b*x+a)-5/32*(b*x+a)^4/b^4/Lambe 
rtW(b*x+a)+2*a^3*(b*x+a)*LambertW(b*x+a)/b^4-3/2*a^2*(b*x+a)^2*LambertW(b* 
x+a)/b^4+2/3*a*(b*x+a)^3*LambertW(b*x+a)/b^4-1/8*(b*x+a)^4*LambertW(b*x+a) 
/b^4-a^3*(b*x+a)*LambertW(b*x+a)^2/b^4+3/2*a^2*(b*x+a)^2*LambertW(b*x+a)^2 
/b^4-a*(b*x+a)^3*LambertW(b*x+a)^2/b^4+1/4*(b*x+a)^4*LambertW(b*x+a)^2/b^4
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.55 \[ \int x^3 W(a+b x)^2 \, dx=\frac {(a+b x) \left (1215 (a+b x)^3+4 (2881 a-1215 b x) (a+b x)^2 W(a+b x)+24 \left (2245 a^3+1007 a^2 b x-833 a b^2 x^2+405 b^3 x^3\right ) W(a+b x)^2+288 \left (715 a^3-271 a^2 b x+121 a b^2 x^2-45 b^3 x^3\right ) W(a+b x)^3-288 \left (715 a^3-271 a^2 b x+121 a b^2 x^2-45 b^3 x^3\right ) W(a+b x)^4+3456 \left (25 a^3-13 a^2 b x+7 a b^2 x^2-3 b^3 x^3\right ) W(a+b x)^5-20736 \left (a^3-a^2 b x+a b^2 x^2-b^3 x^3\right ) W(a+b x)^6\right )}{82944 b^4 W(a+b x)^4} \] Input:

Integrate[x^3*ProductLog[a + b*x]^2,x]
 

Output:

((a + b*x)*(1215*(a + b*x)^3 + 4*(2881*a - 1215*b*x)*(a + b*x)^2*ProductLo 
g[a + b*x] + 24*(2245*a^3 + 1007*a^2*b*x - 833*a*b^2*x^2 + 405*b^3*x^3)*Pr 
oductLog[a + b*x]^2 + 288*(715*a^3 - 271*a^2*b*x + 121*a*b^2*x^2 - 45*b^3* 
x^3)*ProductLog[a + b*x]^3 - 288*(715*a^3 - 271*a^2*b*x + 121*a*b^2*x^2 - 
45*b^3*x^3)*ProductLog[a + b*x]^4 + 3456*(25*a^3 - 13*a^2*b*x + 7*a*b^2*x^ 
2 - 3*b^3*x^3)*ProductLog[a + b*x]^5 - 20736*(a^3 - a^2*b*x + a*b^2*x^2 - 
b^3*x^3)*ProductLog[a + b*x]^6))/(82944*b^4*ProductLog[a + b*x]^4)
 

Rubi [A] (verified)

Time = 1.12 (sec) , antiderivative size = 398, normalized size of antiderivative = 0.87, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, 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.

\(\displaystyle \int x^3 W(a+b x)^2 \, dx\)

\(\Big \downarrow \) 7168

\(\displaystyle \frac {\int \left (-W(a+b x)^2 a^3+3 (a+b x) W(a+b x)^2 a^2-3 (a+b x)^2 W(a+b x)^2 a+(a+b x)^3 W(a+b x)^2\right )d(a+b x)}{b^4}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-a^3 (a+b x) W(a+b x)^2+2 a^3 (a+b x) W(a+b x)+\frac {4 a^3 (a+b x)}{W(a+b x)}-4 a^3 (a+b x)+\frac {3}{2} a^2 (a+b x)^2 W(a+b x)^2-\frac {3}{2} a^2 (a+b x)^2 W(a+b x)-\frac {9 a^2 (a+b x)^2}{4 W(a+b x)}+\frac {9 a^2 (a+b x)^2}{8 W(a+b x)^2}+\frac {9}{4} a^2 (a+b x)^2+\frac {1}{4} (a+b x)^4 W(a+b x)^2-\frac {1}{8} (a+b x)^4 W(a+b x)-\frac {5 (a+b x)^4}{32 W(a+b x)}+\frac {15 (a+b x)^4}{128 W(a+b x)^2}-\frac {15 (a+b x)^4}{256 W(a+b x)^3}+\frac {15 (a+b x)^4}{1024 W(a+b x)^4}-a (a+b x)^3 W(a+b x)^2+\frac {2}{3} a (a+b x)^3 W(a+b x)+\frac {8 a (a+b x)^3}{9 W(a+b x)}-\frac {16 a (a+b x)^3}{27 W(a+b x)^2}+\frac {16 a (a+b x)^3}{81 W(a+b x)^3}+\frac {5}{32} (a+b x)^4-\frac {8}{9} a (a+b x)^3}{b^4}\)

Input:

Int[x^3*ProductLog[a + b*x]^2,x]
 

Output:

(-4*a^3*(a + b*x) + (9*a^2*(a + b*x)^2)/4 - (8*a*(a + b*x)^3)/9 + (5*(a + 
b*x)^4)/32 + (15*(a + b*x)^4)/(1024*ProductLog[a + b*x]^4) + (16*a*(a + b* 
x)^3)/(81*ProductLog[a + b*x]^3) - (15*(a + b*x)^4)/(256*ProductLog[a + b* 
x]^3) + (9*a^2*(a + b*x)^2)/(8*ProductLog[a + b*x]^2) - (16*a*(a + b*x)^3) 
/(27*ProductLog[a + b*x]^2) + (15*(a + b*x)^4)/(128*ProductLog[a + b*x]^2) 
 + (4*a^3*(a + b*x))/ProductLog[a + b*x] - (9*a^2*(a + b*x)^2)/(4*ProductL 
og[a + b*x]) + (8*a*(a + b*x)^3)/(9*ProductLog[a + b*x]) - (5*(a + b*x)^4) 
/(32*ProductLog[a + b*x]) + 2*a^3*(a + b*x)*ProductLog[a + b*x] - (3*a^2*( 
a + b*x)^2*ProductLog[a + b*x])/2 + (2*a*(a + b*x)^3*ProductLog[a + b*x])/ 
3 - ((a + b*x)^4*ProductLog[a + b*x])/8 - a^3*(a + b*x)*ProductLog[a + b*x 
]^2 + (3*a^2*(a + b*x)^2*ProductLog[a + b*x]^2)/2 - a*(a + b*x)^3*ProductL 
og[a + b*x]^2 + ((a + b*x)^4*ProductLog[a + b*x]^2)/4)/b^4
 

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 = 0.16 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.16

method	result	size
derivativedivides	\(\frac {-\frac {\operatorname {LambertW}\left (b x +a \right ) \left (b x +a \right )^{4}}{8}+\frac {5 \left (b x +a \right )^{4}}{32}-\frac {5 \left (b x +a \right )^{4}}{32 \operatorname {LambertW}\left (b x +a \right )}+\frac {15 \left (b x +a \right )^{4}}{128 \operatorname {LambertW}\left (b x +a \right )^{2}}-\frac {15 \left (b x +a \right )^{4}}{256 \operatorname {LambertW}\left (b x +a \right )^{3}}+\frac {15 \left (b x +a \right )^{4}}{1024 \operatorname {LambertW}\left (b x +a \right )^{4}}+\frac {\operatorname {LambertW}\left (b x +a \right )^{2} \left (b x +a \right )^{4}}{4}-a^{3} \left (\left (b x +a \right ) \operatorname {LambertW}\left (b x +a \right )-2 b x -2 a +\frac {2 b x +2 a}{\operatorname {LambertW}\left (b x +a \right )}\right )-a^{3} \left (\operatorname {LambertW}\left (b x +a \right )^{2} \left (b x +a \right )-3 \left (b x +a \right ) \operatorname {LambertW}\left (b x +a \right )+6 b x +6 a -\frac {6 \left (b x +a \right )}{\operatorname {LambertW}\left (b x +a \right )}\right )+3 a^{2} \left (\frac {\operatorname {LambertW}\left (b x +a \right ) \left (b x +a \right )^{2}}{2}-\frac {3 \left (b x +a \right )^{2}}{4}+\frac {3 \left (b x +a \right )^{2}}{4 \operatorname {LambertW}\left (b x +a \right )}-\frac {3 \left (b x +a \right )^{2}}{8 \operatorname {LambertW}\left (b x +a \right )^{2}}\right )+3 a^{2} \left (\frac {\operatorname {LambertW}\left (b x +a \right )^{2} \left (b x +a \right )^{2}}{2}-\operatorname {LambertW}\left (b x +a \right ) \left (b x +a \right )^{2}+\frac {3 \left (b x +a \right )^{2}}{2}-\frac {3 \left (b x +a \right )^{2}}{2 \operatorname {LambertW}\left (b x +a \right )}+\frac {3 \left (b x +a \right )^{2}}{4 \operatorname {LambertW}\left (b x +a \right )^{2}}\right )-3 a \left (\frac {\operatorname {LambertW}\left (b x +a \right ) \left (b x +a \right )^{3}}{3}-\frac {4 \left (b x +a \right )^{3}}{9}+\frac {4 \left (b x +a \right )^{3}}{9 \operatorname {LambertW}\left (b x +a \right )}-\frac {8 \left (b x +a \right )^{3}}{27 \operatorname {LambertW}\left (b x +a \right )^{2}}+\frac {8 \left (b x +a \right )^{3}}{81 \operatorname {LambertW}\left (b x +a \right )^{3}}\right )-3 a \left (\frac {\operatorname {LambertW}\left (b x +a \right )^{2} \left (b x +a \right )^{3}}{3}-\frac {5 \operatorname {LambertW}\left (b x +a \right ) \left (b x +a \right )^{3}}{9}+\frac {20 \left (b x +a \right )^{3}}{27}-\frac {20 \left (b x +a \right )^{3}}{27 \operatorname {LambertW}\left (b x +a \right )}+\frac {40 \left (b x +a \right )^{3}}{81 \operatorname {LambertW}\left (b x +a \right )^{2}}-\frac {40 \left (b x +a \right )^{3}}{243 \operatorname {LambertW}\left (b x +a \right )^{3}}\right )}{b^{4}}\)	\(530\)
default	\(\frac {-\frac {\operatorname {LambertW}\left (b x +a \right ) \left (b x +a \right )^{4}}{8}+\frac {5 \left (b x +a \right )^{4}}{32}-\frac {5 \left (b x +a \right )^{4}}{32 \operatorname {LambertW}\left (b x +a \right )}+\frac {15 \left (b x +a \right )^{4}}{128 \operatorname {LambertW}\left (b x +a \right )^{2}}-\frac {15 \left (b x +a \right )^{4}}{256 \operatorname {LambertW}\left (b x +a \right )^{3}}+\frac {15 \left (b x +a \right )^{4}}{1024 \operatorname {LambertW}\left (b x +a \right )^{4}}+\frac {\operatorname {LambertW}\left (b x +a \right )^{2} \left (b x +a \right )^{4}}{4}-a^{3} \left (\left (b x +a \right ) \operatorname {LambertW}\left (b x +a \right )-2 b x -2 a +\frac {2 b x +2 a}{\operatorname {LambertW}\left (b x +a \right )}\right )-a^{3} \left (\operatorname {LambertW}\left (b x +a \right )^{2} \left (b x +a \right )-3 \left (b x +a \right ) \operatorname {LambertW}\left (b x +a \right )+6 b x +6 a -\frac {6 \left (b x +a \right )}{\operatorname {LambertW}\left (b x +a \right )}\right )+3 a^{2} \left (\frac {\operatorname {LambertW}\left (b x +a \right ) \left (b x +a \right )^{2}}{2}-\frac {3 \left (b x +a \right )^{2}}{4}+\frac {3 \left (b x +a \right )^{2}}{4 \operatorname {LambertW}\left (b x +a \right )}-\frac {3 \left (b x +a \right )^{2}}{8 \operatorname {LambertW}\left (b x +a \right )^{2}}\right )+3 a^{2} \left (\frac {\operatorname {LambertW}\left (b x +a \right )^{2} \left (b x +a \right )^{2}}{2}-\operatorname {LambertW}\left (b x +a \right ) \left (b x +a \right )^{2}+\frac {3 \left (b x +a \right )^{2}}{2}-\frac {3 \left (b x +a \right )^{2}}{2 \operatorname {LambertW}\left (b x +a \right )}+\frac {3 \left (b x +a \right )^{2}}{4 \operatorname {LambertW}\left (b x +a \right )^{2}}\right )-3 a \left (\frac {\operatorname {LambertW}\left (b x +a \right ) \left (b x +a \right )^{3}}{3}-\frac {4 \left (b x +a \right )^{3}}{9}+\frac {4 \left (b x +a \right )^{3}}{9 \operatorname {LambertW}\left (b x +a \right )}-\frac {8 \left (b x +a \right )^{3}}{27 \operatorname {LambertW}\left (b x +a \right )^{2}}+\frac {8 \left (b x +a \right )^{3}}{81 \operatorname {LambertW}\left (b x +a \right )^{3}}\right )-3 a \left (\frac {\operatorname {LambertW}\left (b x +a \right )^{2} \left (b x +a \right )^{3}}{3}-\frac {5 \operatorname {LambertW}\left (b x +a \right ) \left (b x +a \right )^{3}}{9}+\frac {20 \left (b x +a \right )^{3}}{27}-\frac {20 \left (b x +a \right )^{3}}{27 \operatorname {LambertW}\left (b x +a \right )}+\frac {40 \left (b x +a \right )^{3}}{81 \operatorname {LambertW}\left (b x +a \right )^{2}}-\frac {40 \left (b x +a \right )^{3}}{243 \operatorname {LambertW}\left (b x +a \right )^{3}}\right )}{b^{4}}\)	\(530\)

Input:

int(x^3*LambertW(b*x+a)^2,x,method=_RETURNVERBOSE)
 

Output:

1/b^4*(-1/8*LambertW(b*x+a)*(b*x+a)^4+5/32*(b*x+a)^4-5/32/LambertW(b*x+a)* 
(b*x+a)^4+15/128*(b*x+a)^4/LambertW(b*x+a)^2-15/256/LambertW(b*x+a)^3*(b*x 
+a)^4+15/1024*(b*x+a)^4/LambertW(b*x+a)^4+1/4*LambertW(b*x+a)^2*(b*x+a)^4- 
a^3*((b*x+a)*LambertW(b*x+a)-2*b*x-2*a+2*(b*x+a)/LambertW(b*x+a))-a^3*(Lam 
bertW(b*x+a)^2*(b*x+a)-3*(b*x+a)*LambertW(b*x+a)+6*b*x+6*a-6*(b*x+a)/Lambe 
rtW(b*x+a))+3*a^2*(1/2*LambertW(b*x+a)*(b*x+a)^2-3/4*(b*x+a)^2+3/4/Lambert 
W(b*x+a)*(b*x+a)^2-3/8*(b*x+a)^2/LambertW(b*x+a)^2)+3*a^2*(1/2*LambertW(b* 
x+a)^2*(b*x+a)^2-LambertW(b*x+a)*(b*x+a)^2+3/2*(b*x+a)^2-3/2/LambertW(b*x+ 
a)*(b*x+a)^2+3/4*(b*x+a)^2/LambertW(b*x+a)^2)-3*a*(1/3*LambertW(b*x+a)*(b* 
x+a)^3-4/9*(b*x+a)^3+4/9/LambertW(b*x+a)*(b*x+a)^3-8/27/LambertW(b*x+a)^2* 
(b*x+a)^3+8/81*(b*x+a)^3/LambertW(b*x+a)^3)-3*a*(1/3*LambertW(b*x+a)^2*(b* 
x+a)^3-5/9*LambertW(b*x+a)*(b*x+a)^3+20/27*(b*x+a)^3-20/27/LambertW(b*x+a) 
*(b*x+a)^3+40/81/LambertW(b*x+a)^2*(b*x+a)^3-40/243*(b*x+a)^3/LambertW(b*x 
+a)^3))
 

Fricas [F]

\[ \int x^3 W(a+b x)^2 \, dx=\int { x^{3} \operatorname {W}({b x + a})^{2} \,d x } \] Input:

integrate(x^3*lambert_w(b*x+a)^2,x, algorithm="fricas")
 

Output:

integral(x^3*lambert_w(b*x + a)^2, x)
 

Sympy [F]

\[ \int x^3 W(a+b x)^2 \, dx=\int x^{3} W^{2}\left (a + b x\right )\, dx \] Input:

integrate(x**3*LambertW(b*x+a)**2,x)
 

Output:

Integral(x**3*LambertW(a + b*x)**2, x)
 

Maxima [F]

\[ \int x^3 W(a+b x)^2 \, dx=\int { x^{3} \operatorname {W}({b x + a})^{2} \,d x } \] Input:

integrate(x^3*lambert_w(b*x+a)^2,x, algorithm="maxima")
 

Output:

integrate(x^3*lambert_w(b*x + a)^2, x)
 

Giac [F]

\[ \int x^3 W(a+b x)^2 \, dx=\int { x^{3} \operatorname {W}({b x + a})^{2} \,d x } \] Input:

integrate(x^3*lambert_w(b*x+a)^2,x, algorithm="giac")
 

Output:

integrate(x^3*lambert_w(b*x + a)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^3 W(a+b x)^2 \, dx=\int x^3\,{\mathrm {LambertW}\left (a+b\,x\right )}^2 \,d x \] Input:

int(x^3*LambertW(a + b*x)^2,x)
 

Output:

int(x^3*LambertW(a + b*x)^2, x)
 

Reduce [F]

\[ \int x^3 W(a+b x)^2 \, dx=\int \textit {lambert\_w} \left (b x +a \right )^{2} x^{3}d x \] Input:

int(x^3*Lambert_W(b*x+a)^2,x)
 

Output:

int(lambert_w(a + b*x)**2*x**3,x)