Integrand size = 14, antiderivative size = 99 \[ \int \frac {x^2}{\sqrt {c W(a x)}} \, dx=\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {c W(a x)}}{\sqrt {c}}\right )}{72 a^3 \sqrt {c}}-\frac {c^2 x^3}{36 (c W(a x))^{5/2}}+\frac {c x^3}{18 (c W(a x))^{3/2}}+\frac {x^3}{3 \sqrt {c W(a x)}} \] Output:
1/216*3^(1/2)*Pi^(1/2)*erfi(3^(1/2)*(c*LambertW(a*x))^(1/2)/c^(1/2))/a^3/c ^(1/2)-1/36*c^2*x^3/(c*LambertW(a*x))^(5/2)+1/18*c*x^3/(c*LambertW(a*x))^( 3/2)+1/3*x^3/(c*LambertW(a*x))^(1/2)
Time = 0.02 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.90 \[ \int \frac {x^2}{\sqrt {c W(a x)}} \, dx=\frac {-6 a^3 x^3+12 a^3 x^3 W(a x)+72 a^3 x^3 W(a x)^2+\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {W(a x)}\right ) W(a x)^{5/2}}{216 a^3 W(a x)^2 \sqrt {c W(a x)}} \] Input:
Integrate[x^2/Sqrt[c*ProductLog[a*x]],x]
Output:
(-6*a^3*x^3 + 12*a^3*x^3*ProductLog[a*x] + 72*a^3*x^3*ProductLog[a*x]^2 + Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ProductLog[a*x]]]*ProductLog[a*x]^(5/2))/(216 *a^3*ProductLog[a*x]^2*Sqrt[c*ProductLog[a*x]])
Time = 1.08 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.09, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {7172, 7205, 7205, 7204}
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.
\(\displaystyle \int \frac {x^2}{\sqrt {c W(a x)}} \, dx\) |
\(\Big \downarrow \) 7172 |
\(\displaystyle \frac {1}{6} \int \frac {x^2}{\sqrt {c W(a x)} (W(a x)+1)}dx+\frac {x^3}{3 \sqrt {c W(a x)}}\) |
\(\Big \downarrow \) 7205 |
\(\displaystyle \frac {1}{6} \left (\frac {c x^3}{3 (c W(a x))^{3/2}}-\frac {1}{2} c \int \frac {x^2}{(c W(a x))^{3/2} (W(a x)+1)}dx\right )+\frac {x^3}{3 \sqrt {c W(a x)}}\) |
\(\Big \downarrow \) 7205 |
\(\displaystyle \frac {1}{6} \left (\frac {c x^3}{3 (c W(a x))^{3/2}}-\frac {1}{2} c \left (\frac {c x^3}{3 (c W(a x))^{5/2}}-\frac {1}{6} c \int \frac {x^2}{(c W(a x))^{5/2} (W(a x)+1)}dx\right )\right )+\frac {x^3}{3 \sqrt {c W(a x)}}\) |
\(\Big \downarrow \) 7204 |
\(\displaystyle \frac {1}{6} \left (\frac {c x^3}{3 (c W(a x))^{3/2}}-\frac {1}{2} c \left (\frac {c x^3}{3 (c W(a x))^{5/2}}-\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {c W(a x)}}{\sqrt {c}}\right )}{6 a^3 c^{3/2}}\right )\right )+\frac {x^3}{3 \sqrt {c W(a x)}}\) |
Input:
Int[x^2/Sqrt[c*ProductLog[a*x]],x]
Output:
x^3/(3*Sqrt[c*ProductLog[a*x]]) + ((c*x^3)/(3*(c*ProductLog[a*x])^(3/2)) - (c*(-1/6*(Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[c*ProductLog[a*x]])/Sqrt[c]])/(a^ 3*c^(3/2)) + (c*x^3)/(3*(c*ProductLog[a*x])^(5/2))))/2)/6
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_.)*((c_.)*ProductLog[(a_.)*(x_)^(n_.)])^(p_))/((d_) + (d_.)*Pr oductLog[(a_.)*(x_)^(n_.)]), x_Symbol] :> Simp[a^(p - 1/2)*c^(p - 1/2)*Rt[( -Pi)*(c/(p - 1/2)), 2]*(Erfi[Sqrt[c*ProductLog[a*x^n]]/Rt[-c/(p - 1/2), 2]] /(d*n)), x] /; FreeQ[{a, c, d, m, n}, x] && NeQ[m, -1] && IntegerQ[p - 1/2] && EqQ[m + n*(p - 1/2), -1] && NegQ[c/(p - 1/2)]
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]
Time = 0.05 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.03
method | result | size |
default | \(\frac {\frac {c \left (c \operatorname {LambertW}\left (x a \right )\right )^{\frac {5}{2}} {\mathrm e}^{3 \operatorname {LambertW}\left (x a \right )}}{3}+\frac {c \left (\frac {c \left (c \operatorname {LambertW}\left (x a \right )\right )^{\frac {3}{2}} {\mathrm e}^{3 \operatorname {LambertW}\left (x a \right )}}{6}-\frac {c \left (\frac {c \sqrt {c \operatorname {LambertW}\left (x a \right )}\, {\mathrm e}^{3 \operatorname {LambertW}\left (x a \right )}}{6}-\frac {c \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-\frac {3}{c}}\, \sqrt {c \operatorname {LambertW}\left (x a \right )}\right )}{12 \sqrt {-\frac {3}{c}}}\right )}{2}\right )}{3}}{a^{3} c^{4}}\) | \(102\) |
Input:
int(x^2/(c*LambertW(x*a))^(1/2),x,method=_RETURNVERBOSE)
Output:
2/a^3/c^4*(1/6*c*(c*LambertW(x*a))^(5/2)*exp(3*LambertW(x*a))+1/6*c*(1/6*c *(c*LambertW(x*a))^(3/2)*exp(3*LambertW(x*a))-1/2*c*(1/6*c*(c*LambertW(x*a ))^(1/2)*exp(3*LambertW(x*a))-1/12*c*Pi^(1/2)/(-3/c)^(1/2)*erf((-3/c)^(1/2 )*(c*LambertW(x*a))^(1/2)))))
\[ \int \frac {x^2}{\sqrt {c W(a x)}} \, dx=\int { \frac {x^{2}}{\sqrt {c \operatorname {W}({a x})}} \,d x } \] Input:
integrate(x^2/(c*lambert_w(a*x))^(1/2),x, algorithm="fricas")
Output:
integral(sqrt(c*lambert_w(a*x))*x^2/(c*lambert_w(a*x)), x)
\[ \int \frac {x^2}{\sqrt {c W(a x)}} \, dx=\int \frac {x^{2}}{\sqrt {c W\left (a x\right )}}\, dx \] Input:
integrate(x**2/(c*LambertW(a*x))**(1/2),x)
Output:
Integral(x**2/sqrt(c*LambertW(a*x)), x)
\[ \int \frac {x^2}{\sqrt {c W(a x)}} \, dx=\int { \frac {x^{2}}{\sqrt {c \operatorname {W}({a x})}} \,d x } \] Input:
integrate(x^2/(c*lambert_w(a*x))^(1/2),x, algorithm="maxima")
Output:
integrate(x^2/sqrt(c*lambert_w(a*x)), x)
\[ \int \frac {x^2}{\sqrt {c W(a x)}} \, dx=\int { \frac {x^{2}}{\sqrt {c \operatorname {W}({a x})}} \,d x } \] Input:
integrate(x^2/(c*lambert_w(a*x))^(1/2),x, algorithm="giac")
Output:
integrate(x^2/sqrt(c*lambert_w(a*x)), x)
Timed out. \[ \int \frac {x^2}{\sqrt {c W(a x)}} \, dx=\int \frac {x^2}{\sqrt {c\,\mathrm {LambertW}\left (a\,x\right )}} \,d x \] Input:
int(x^2/(c*LambertW(a*x))^(1/2),x)
Output:
int(x^2/(c*LambertW(a*x))^(1/2), x)
\[ \int \frac {x^2}{\sqrt {c W(a x)}} \, dx=\frac {\sqrt {c}\, \left (\int \frac {\sqrt {\textit {lambert\_w} \left (a x \right )}\, x^{2}}{\textit {lambert\_w} \left (a x \right )}d x \right )}{c} \] Input:
int(x^2/(c*Lambert_W(a*x))^(1/2),x)
Output:
(sqrt(c)*int((sqrt(lambert_w(a*x))*x**2)/lambert_w(a*x),x))/c