\(\int x^3 W(a x^2) \, dx\) [93]

Optimal result
Mathematica [A] (verified)
Rubi [A] (verified)
Maple [F]
Fricas [A] (verification not implemented)
Sympy [F]
Maxima [F]
Giac [F]
Mupad [F(-1)]
Reduce [F]

Optimal result

Integrand size = 10, antiderivative size = 51 \[ \int x^3 W\left (a x^2\right ) \, dx=-\frac {x^4}{8}-\frac {x^4}{16 W\left (a x^2\right )^2}+\frac {x^4}{8 W\left (a x^2\right )}+\frac {1}{4} x^4 W\left (a x^2\right ) \] Output:

-1/8*x^4-1/16*x^4/LambertW(a*x^2)^2+1/8*x^4/LambertW(a*x^2)+1/4*x^4*Lamber 
tW(a*x^2)
 

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00 \[ \int x^3 W\left (a x^2\right ) \, dx=-\frac {x^4}{8}-\frac {x^4}{16 W\left (a x^2\right )^2}+\frac {x^4}{8 W\left (a x^2\right )}+\frac {1}{4} x^4 W\left (a x^2\right ) \] Input:

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

Output:

-1/8*x^4 - x^4/(16*ProductLog[a*x^2]^2) + x^4/(8*ProductLog[a*x^2]) + (x^4 
*ProductLog[a*x^2])/4
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.20, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {7172, 7205, 7283, 7194, 7201}

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.

\(\displaystyle \int x^3 W\left (a x^2\right ) \, dx\)

\(\Big \downarrow \) 7172

\(\displaystyle \frac {1}{4} x^4 W\left (a x^2\right )-\frac {1}{2} \int \frac {x^3 W\left (a x^2\right )}{W\left (a x^2\right )+1}dx\)

\(\Big \downarrow \) 7205

\(\displaystyle \frac {1}{2} \left (\int \frac {x^3}{W\left (a x^2\right )+1}dx-\frac {x^4}{4}\right )+\frac {1}{4} x^4 W\left (a x^2\right )\)

\(\Big \downarrow \) 7283

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

\(\Big \downarrow \) 7194

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

\(\Big \downarrow \) 7201

\(\displaystyle \frac {1}{4} x^4 W\left (a x^2\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {x^4}{2 W\left (a x^2\right )}-\frac {x^4}{4 W\left (a x^2\right )^2}\right )-\frac {x^4}{4}\right )\)

Input:

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

Output:

(-1/4*x^4 + (-1/4*x^4/ProductLog[a*x^2]^2 + x^4/(2*ProductLog[a*x^2]))/2)/ 
2 + (x^4*ProductLog[a*x^2])/4
 

Defintions of rubi rules used

rule 7172
Int[(x_)^(m_.)*((c_.)*ProductLog[(a_.)*(x_)^(n_.)])^(p_.), x_Symbol] :> Sim 
p[x^(m + 1)*((c*ProductLog[a*x^n])^p/(m + 1)), x] - Simp[n*(p/(m + 1))   In 
t[x^m*((c*ProductLog[a*x^n])^p/(1 + ProductLog[a*x^n])), x], x] /; FreeQ[{a 
, c, m, n, p}, x] && NeQ[m, -1] && ((IntegerQ[p - 1/2] && IGtQ[2*Simplify[p 
 + (m + 1)/n], 0]) || ( !IntegerQ[p - 1/2] && IGtQ[Simplify[p + (m + 1)/n] 
+ 1, 0]))
 

rule 7194
Int[(x_)^(m_.)/((d_) + (d_.)*ProductLog[(a_.)*(x_)]), x_Symbol] :> Simp[x^( 
m + 1)/(d*(m + 1)*ProductLog[a*x]), x] - Simp[m/(m + 1)   Int[x^m/(ProductL 
og[a*x]*(d + d*ProductLog[a*x])), x], x] /; FreeQ[{a, d}, x] && GtQ[m, 0]
 

rule 7201
Int[((x_)^(m_.)*((c_.)*ProductLog[(a_.)*(x_)^(n_.)])^(p_.))/((d_) + (d_.)*P 
roductLog[(a_.)*(x_)^(n_.)]), x_Symbol] :> Simp[c*x^(m + 1)*((c*ProductLog[ 
a*x^n])^(p - 1)/(d*(m + 1))), x] /; FreeQ[{a, c, d, m, n, p}, x] && NeQ[m, 
-1] && EqQ[m + n*(p - 1), -1]
 

rule 7205
Int[((x_)^(m_.)*((c_.)*ProductLog[(a_.)*(x_)^(n_.)])^(p_.))/((d_) + (d_.)*P 
roductLog[(a_.)*(x_)^(n_.)]), x_Symbol] :> Simp[c*x^(m + 1)*((c*ProductLog[ 
a*x^n])^(p - 1)/(d*(m + 1))), x] - Simp[c*((m + n*(p - 1) + 1)/(m + 1))   I 
nt[x^m*((c*ProductLog[a*x^n])^(p - 1)/(d + d*ProductLog[a*x^n])), x], x] /; 
 FreeQ[{a, c, d, m, n, p}, x] && NeQ[m, -1] && GtQ[Simplify[p + (m + 1)/n], 
 1]
 

rule 7283
Int[(u_)*(x_)^(m_.), x_Symbol] :> With[{lst = PowerVariableExpn[u, m + 1, x 
]}, Simp[1/lst[[2]]   Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x], x] 
, x], x, (lst[[3]]*x)^lst[[2]]], x] /;  !FalseQ[lst] && NeQ[lst[[2]], m + 1 
]] /; IntegerQ[m] && NeQ[m, -1] && NonsumQ[u] && (GtQ[m, 0] ||  !AlgebraicF 
unctionQ[u, x])
 
Maple [F]

\[\int x^{3} \operatorname {LambertW}\left (a \,x^{2}\right )d x\]

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.12 \[ \int x^3 W\left (a x^2\right ) \, dx=\frac {4 \, x^{4} \operatorname {W}({a x^{2}})^{3} - 2 \, x^{4} \operatorname {W}({a x^{2}})^{2} + 2 \, x^{4} \operatorname {W}({a x^{2}}) - x^{4}}{16 \, \operatorname {W}({a x^{2}})^{2}} \] Input:

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

Output:

1/16*(4*x^4*lambert_w(a*x^2)^3 - 2*x^4*lambert_w(a*x^2)^2 + 2*x^4*lambert_ 
w(a*x^2) - x^4)/lambert_w(a*x^2)^2
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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