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)
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
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
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]))
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]
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]
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]
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])
\[\int x^{3} \operatorname {LambertW}\left (a \,x^{2}\right )d x\]
Input:
int(x^3*LambertW(a*x^2),x)
Output:
int(x^3*LambertW(a*x^2),x)
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
\[ \int x^3 W\left (a x^2\right ) \, dx=\int x^{3} W\left (a x^{2}\right )\, dx \] Input:
integrate(x**3*LambertW(a*x**2),x)
Output:
Integral(x**3*LambertW(a*x**2), x)
\[ \int x^3 W\left (a x^2\right ) \, dx=\int { x^{3} \operatorname {W}({a x^{2}}) \,d x } \] Input:
integrate(x^3*lambert_w(a*x^2),x, algorithm="maxima")
Output:
integrate(x^3*lambert_w(a*x^2), x)
\[ \int x^3 W\left (a x^2\right ) \, dx=\int { x^{3} \operatorname {W}({a x^{2}}) \,d x } \] Input:
integrate(x^3*lambert_w(a*x^2),x, algorithm="giac")
Output:
integrate(x^3*lambert_w(a*x^2), x)
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)
\[ \int x^3 W\left (a x^2\right ) \, dx=\int \textit {lambert\_w} \left (a \,x^{2}\right ) x^{3}d x \] Input:
int(x^3*Lambert_W(a*x^2),x)
Output:
int(lambert_w(a*x**2)*x**3,x)