Integrand size = 8, antiderivative size = 91 \[ \int W(a+b x)^4 \, dx=96 x-\frac {96 (a+b x)}{b W(a+b x)}-\frac {48 (a+b x) W(a+b x)}{b}+\frac {16 (a+b x) W(a+b x)^2}{b}-\frac {4 (a+b x) W(a+b x)^3}{b}+\frac {(a+b x) W(a+b x)^4}{b} \] Output:
96*x-96*(b*x+a)/b/LambertW(b*x+a)-48*(b*x+a)*LambertW(b*x+a)/b+16*(b*x+a)* LambertW(b*x+a)^2/b-4*(b*x+a)*LambertW(b*x+a)^3/b+(b*x+a)*LambertW(b*x+a)^ 4/b
Time = 0.04 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.69 \[ \int W(a+b x)^4 \, dx=\frac {(a+b x) (-2+W(a+b x)) \left (48-24 W(a+b x)+12 W(a+b x)^2-2 W(a+b x)^3+W(a+b x)^4\right )}{b W(a+b x)} \] Input:
Integrate[ProductLog[a + b*x]^4,x]
Output:
((a + b*x)*(-2 + ProductLog[a + b*x])*(48 - 24*ProductLog[a + b*x] + 12*Pr oductLog[a + b*x]^2 - 2*ProductLog[a + b*x]^3 + ProductLog[a + b*x]^4))/(b *ProductLog[a + b*x])
Time = 0.67 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {7167, 7178, 7178, 7178, 7177, 7176}
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.
\(\displaystyle \int W(a+b x)^4 \, dx\) |
\(\Big \downarrow \) 7167 |
\(\displaystyle \frac {(a+b x) W(a+b x)^4}{b}-4 \int \frac {W(a+b x)^4}{W(a+b x)+1}dx\) |
\(\Big \downarrow \) 7178 |
\(\displaystyle \frac {(a+b x) W(a+b x)^4}{b}-4 \left (\frac {(a+b x) W(a+b x)^3}{b}-4 \int \frac {W(a+b x)^3}{W(a+b x)+1}dx\right )\) |
\(\Big \downarrow \) 7178 |
\(\displaystyle \frac {(a+b x) W(a+b x)^4}{b}-4 \left (\frac {(a+b x) W(a+b x)^3}{b}-4 \left (\frac {(a+b x) W(a+b x)^2}{b}-3 \int \frac {W(a+b x)^2}{W(a+b x)+1}dx\right )\right )\) |
\(\Big \downarrow \) 7178 |
\(\displaystyle \frac {(a+b x) W(a+b x)^4}{b}-4 \left (\frac {(a+b x) W(a+b x)^3}{b}-4 \left (\frac {(a+b x) W(a+b x)^2}{b}-3 \left (\frac {(a+b x) W(a+b x)}{b}-2 \int \frac {W(a+b x)}{W(a+b x)+1}dx\right )\right )\right )\) |
\(\Big \downarrow \) 7177 |
\(\displaystyle \frac {(a+b x) W(a+b x)^4}{b}-4 \left (\frac {(a+b x) W(a+b x)^3}{b}-4 \left (\frac {(a+b x) W(a+b x)^2}{b}-3 \left (\frac {(a+b x) W(a+b x)}{b}-2 \left (x-\int \frac {1}{W(a+b x)+1}dx\right )\right )\right )\right )\) |
\(\Big \downarrow \) 7176 |
\(\displaystyle \frac {(a+b x) W(a+b x)^4}{b}-4 \left (\frac {(a+b x) W(a+b x)^3}{b}-4 \left (\frac {(a+b x) W(a+b x)^2}{b}-3 \left (\frac {(a+b x) W(a+b x)}{b}-2 \left (x-\frac {a+b x}{b W(a+b x)}\right )\right )\right )\right )\) |
Input:
Int[ProductLog[a + b*x]^4,x]
Output:
((a + b*x)*ProductLog[a + b*x]^4)/b - 4*(((a + b*x)*ProductLog[a + b*x]^3) /b - 4*(((a + b*x)*ProductLog[a + b*x]^2)/b - 3*(-2*(x - (a + b*x)/(b*Prod uctLog[a + b*x])) + ((a + b*x)*ProductLog[a + b*x])/b)))
Int[((c_.)*ProductLog[(a_.) + (b_.)*(x_)])^(p_.), x_Symbol] :> Simp[(a + b* x)*((c*ProductLog[a + b*x])^p/b), x] - Simp[p Int[(c*ProductLog[a + b*x]) ^p/(1 + ProductLog[a + b*x]), x], x] /; FreeQ[{a, b, c}, x] && !LtQ[p, -1]
Int[((d_) + (d_.)*ProductLog[(a_.) + (b_.)*(x_)])^(-1), x_Symbol] :> Simp[( a + b*x)/(b*d*ProductLog[a + b*x]), x] /; FreeQ[{a, b, d}, x]
Int[ProductLog[(a_.) + (b_.)*(x_)]/((d_) + (d_.)*ProductLog[(a_.) + (b_.)*( x_)]), x_Symbol] :> Simp[d*x, x] - Int[1/(d + d*ProductLog[a + b*x]), x] /; FreeQ[{a, b, d}, x]
Int[((c_.)*ProductLog[(a_.) + (b_.)*(x_)])^(p_)/((d_) + (d_.)*ProductLog[(a _.) + (b_.)*(x_)]), x_Symbol] :> Simp[c*(a + b*x)*((c*ProductLog[a + b*x])^ (p - 1)/(b*d)), x] - Simp[c*p Int[(c*ProductLog[a + b*x])^(p - 1)/(d + d* ProductLog[a + b*x]), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[p, 0]
Time = 0.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(\frac {-4 \operatorname {LambertW}\left (b x +a \right )^{3} \left (b x +a \right )+16 \operatorname {LambertW}\left (b x +a \right )^{2} \left (b x +a \right )-48 \left (b x +a \right ) \operatorname {LambertW}\left (b x +a \right )+96 b x +96 a -\frac {96 \left (b x +a \right )}{\operatorname {LambertW}\left (b x +a \right )}+\operatorname {LambertW}\left (b x +a \right )^{4} \left (b x +a \right )}{b}\) | \(85\) |
default | \(\frac {-4 \operatorname {LambertW}\left (b x +a \right )^{3} \left (b x +a \right )+16 \operatorname {LambertW}\left (b x +a \right )^{2} \left (b x +a \right )-48 \left (b x +a \right ) \operatorname {LambertW}\left (b x +a \right )+96 b x +96 a -\frac {96 \left (b x +a \right )}{\operatorname {LambertW}\left (b x +a \right )}+\operatorname {LambertW}\left (b x +a \right )^{4} \left (b x +a \right )}{b}\) | \(85\) |
parallelrisch | \(-\frac {-x \operatorname {LambertW}\left (b x +a \right )^{5} b +4 x \operatorname {LambertW}\left (b x +a \right )^{4} b -\operatorname {LambertW}\left (b x +a \right )^{5} a -16 x \operatorname {LambertW}\left (b x +a \right )^{3} b +4 \operatorname {LambertW}\left (b x +a \right )^{4} a +48 \operatorname {LambertW}\left (b x +a \right )^{2} x b -16 \operatorname {LambertW}\left (b x +a \right )^{3} a -96 x \operatorname {LambertW}\left (b x +a \right ) b +48 a \operatorname {LambertW}\left (b x +a \right )^{2}+96 b x +96 a \operatorname {LambertW}\left (b x +a \right )+96 a}{b \operatorname {LambertW}\left (b x +a \right )}\) | \(133\) |
Input:
int(LambertW(b*x+a)^4,x,method=_RETURNVERBOSE)
Output:
1/b*(-4*LambertW(b*x+a)^3*(b*x+a)+16*LambertW(b*x+a)^2*(b*x+a)-48*(b*x+a)* LambertW(b*x+a)+96*b*x+96*a-96*(b*x+a)/LambertW(b*x+a)+LambertW(b*x+a)^4*( b*x+a))
Time = 0.08 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.38 \[ \int W(a+b x)^4 \, dx=\frac {{\left (b x + a\right )} \operatorname {W}({b x + a})^{5} - 4 \, {\left (b x + a\right )} \operatorname {W}({b x + a})^{4} - 48 \, b x \operatorname {W}({b x + a})^{2} + 16 \, {\left (b x + a\right )} \operatorname {W}({b x + a})^{3} + 96 \, b x \operatorname {W}({b x + a}) - 48 \, a \operatorname {W}({b x + a}) \log \left (b x + a\right ) + 48 \, a \operatorname {W}({b x + a}) \log \left (\operatorname {W}({b x + a})\right ) - 96 \, b x - 96 \, a}{b \operatorname {W}({b x + a})} \] Input:
integrate(lambert_w(b*x+a)^4,x, algorithm="fricas")
Output:
((b*x + a)*lambert_w(b*x + a)^5 - 4*(b*x + a)*lambert_w(b*x + a)^4 - 48*b* x*lambert_w(b*x + a)^2 + 16*(b*x + a)*lambert_w(b*x + a)^3 + 96*b*x*lamber t_w(b*x + a) - 48*a*lambert_w(b*x + a)*log(b*x + a) + 48*a*lambert_w(b*x + a)*log(lambert_w(b*x + a)) - 96*b*x - 96*a)/(b*lambert_w(b*x + a))
Time = 0.68 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.44 \[ \int W(a+b x)^4 \, dx=\begin {cases} 0 & \text {for}\: a = 0 \wedge b = 0 \\x W^{4}\left (a\right ) & \text {for}\: b = 0 \\0 & \text {for}\: a = - b x \\\frac {a W^{4}\left (a + b x\right )}{b} - \frac {4 a W^{3}\left (a + b x\right )}{b} + \frac {16 a W^{2}\left (a + b x\right )}{b} - \frac {48 a W\left (a + b x\right )}{b} - \frac {96 a}{b W\left (a + b x\right )} + x W^{4}\left (a + b x\right ) - 4 x W^{3}\left (a + b x\right ) + 16 x W^{2}\left (a + b x\right ) - 48 x W\left (a + b x\right ) + 96 x - \frac {96 x}{W\left (a + b x\right )} & \text {otherwise} \end {cases} \] Input:
integrate(LambertW(b*x+a)**4,x)
Output:
Piecewise((0, Eq(a, 0) & Eq(b, 0)), (x*LambertW(a)**4, Eq(b, 0)), (0, Eq(a , -b*x)), (a*LambertW(a + b*x)**4/b - 4*a*LambertW(a + b*x)**3/b + 16*a*La mbertW(a + b*x)**2/b - 48*a*LambertW(a + b*x)/b - 96*a/(b*LambertW(a + b*x )) + x*LambertW(a + b*x)**4 - 4*x*LambertW(a + b*x)**3 + 16*x*LambertW(a + b*x)**2 - 48*x*LambertW(a + b*x) + 96*x - 96*x/LambertW(a + b*x), True))
\[ \int W(a+b x)^4 \, dx=\int { \operatorname {W}({b x + a})^{4} \,d x } \] Input:
integrate(lambert_w(b*x+a)^4,x, algorithm="maxima")
Output:
integrate(lambert_w(b*x + a)^4, x)
\[ \int W(a+b x)^4 \, dx=\int { \operatorname {W}({b x + a})^{4} \,d x } \] Input:
integrate(lambert_w(b*x+a)^4,x, algorithm="giac")
Output:
integrate(lambert_w(b*x + a)^4, x)
Timed out. \[ \int W(a+b x)^4 \, dx=\int {\mathrm {LambertW}\left (a+b\,x\right )}^4 \,d x \] Input:
int(LambertW(a + b*x)^4,x)
Output:
int(LambertW(a + b*x)^4, x)
Time = 0.30 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.66 \[ \int W(a+b x)^4 \, dx=\frac {e^{\textit {lambert\_w} \left (b x +a \right )} \left (\textit {lambert\_w} \left (b x +a \right )^{5}-4 \textit {lambert\_w} \left (b x +a \right )^{4}+16 \textit {lambert\_w} \left (b x +a \right )^{3}-48 \textit {lambert\_w} \left (b x +a \right )^{2}+96 \textit {lambert\_w} \left (b x +a \right )-96\right )}{b} \] Input:
int(Lambert_W(b*x+a)^4,x)
Output:
(e**lambert_w(a + b*x)*(lambert_w(a + b*x)**5 - 4*lambert_w(a + b*x)**4 + 16*lambert_w(a + b*x)**3 - 48*lambert_w(a + b*x)**2 + 96*lambert_w(a + b*x ) - 96))/b