\(\int \frac {x^3}{\sqrt {c W(a+b x)}} \, dx\) [403]

Optimal result

Integrand size = 16, antiderivative size = 449 \[ \int \frac {x^3}{\sqrt {c W(a+b x)}} \, dx=-\frac {a^3 \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c W(a+b x)}}{\sqrt {c}}\right )}{2 b^4 \sqrt {c}}-\frac {15 \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {c W(a+b x)}}{\sqrt {c}}\right )}{8192 b^4 \sqrt {c}}-\frac {3 a^2 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {c W(a+b x)}}{\sqrt {c}}\right )}{16 b^4 \sqrt {c}}-\frac {a \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {c W(a+b x)}}{\sqrt {c}}\right )}{24 b^4 \sqrt {c}}+\frac {15 c^3 (a+b x)^4}{2048 b^4 (c W(a+b x))^{7/2}}+\frac {a c^2 (a+b x)^3}{12 b^4 (c W(a+b x))^{5/2}}-\frac {5 c^2 (a+b x)^4}{256 b^4 (c W(a+b x))^{5/2}}+\frac {3 a^2 c (a+b x)^2}{8 b^4 (c W(a+b x))^{3/2}}-\frac {a c (a+b x)^3}{6 b^4 (c W(a+b x))^{3/2}}+\frac {c (a+b x)^4}{32 b^4 (c W(a+b x))^{3/2}}-\frac {a^3 (a+b x)}{b^4 \sqrt {c W(a+b x)}}+\frac {3 a^2 (a+b x)^2}{2 b^4 \sqrt {c W(a+b x)}}-\frac {a (a+b x)^3}{b^4 \sqrt {c W(a+b x)}}+\frac {(a+b x)^4}{4 b^4 \sqrt {c W(a+b x)}} \] Output:

-1/2*a^3*Pi^(1/2)*erfi((c*LambertW(b*x+a))^(1/2)/c^(1/2))/b^4/c^(1/2)-15/8 
192*Pi^(1/2)*erfi(2*(c*LambertW(b*x+a))^(1/2)/c^(1/2))/b^4/c^(1/2)-3/32*a^ 
2*2^(1/2)*Pi^(1/2)*erfi(2^(1/2)*(c*LambertW(b*x+a))^(1/2)/c^(1/2))/b^4/c^( 
1/2)-1/72*a*3^(1/2)*Pi^(1/2)*erfi(3^(1/2)*(c*LambertW(b*x+a))^(1/2)/c^(1/2 
))/b^4/c^(1/2)+15/2048*c^3*(b*x+a)^4/b^4/(c*LambertW(b*x+a))^(7/2)+1/12*a* 
c^2*(b*x+a)^3/b^4/(c*LambertW(b*x+a))^(5/2)-5/256*c^2*(b*x+a)^4/b^4/(c*Lam 
bertW(b*x+a))^(5/2)+3/8*a^2*c*(b*x+a)^2/b^4/(c*LambertW(b*x+a))^(3/2)-1/6* 
a*c*(b*x+a)^3/b^4/(c*LambertW(b*x+a))^(3/2)+1/32*c*(b*x+a)^4/b^4/(c*Lamber 
tW(b*x+a))^(3/2)-a^3*(b*x+a)/b^4/(c*LambertW(b*x+a))^(1/2)+3/2*a^2*(b*x+a) 
^2/b^4/(c*LambertW(b*x+a))^(1/2)-a*(b*x+a)^3/b^4/(c*LambertW(b*x+a))^(1/2) 
+1/4*(b*x+a)^4/b^4/(c*LambertW(b*x+a))^(1/2)
 

Mathematica [A] (verified)

Time = 0.96 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.49 \[ \int \frac {x^3}{\sqrt {c W(a+b x)}} \, dx=\frac {540 (a+b x)^4+96 (49 a-15 b x) (a+b x)^3 W(a+b x)+768 (a+b x)^2 \left (23 a^2-10 a b x+3 b^2 x^2\right ) W(a+b x)^2-18432 \left (a^4-b^4 x^4\right ) W(a+b x)^3-\sqrt {\pi } \left (36864 a^3 \text {erfi}\left (\sqrt {W(a+b x)}\right )+135 \text {erfi}\left (2 \sqrt {W(a+b x)}\right )+256 a \left (27 \sqrt {2} a \text {erfi}\left (\sqrt {2} \sqrt {W(a+b x)}\right )+4 \sqrt {3} \text {erfi}\left (\sqrt {3} \sqrt {W(a+b x)}\right )\right )\right ) W(a+b x)^{7/2}}{73728 b^4 W(a+b x)^3 \sqrt {c W(a+b x)}} \] Input:

Integrate[x^3/Sqrt[c*ProductLog[a + b*x]],x]
 

Output:

(540*(a + b*x)^4 + 96*(49*a - 15*b*x)*(a + b*x)^3*ProductLog[a + b*x] + 76 
8*(a + b*x)^2*(23*a^2 - 10*a*b*x + 3*b^2*x^2)*ProductLog[a + b*x]^2 - 1843 
2*(a^4 - b^4*x^4)*ProductLog[a + b*x]^3 - Sqrt[Pi]*(36864*a^3*Erfi[Sqrt[Pr 
oductLog[a + b*x]]] + 135*Erfi[2*Sqrt[ProductLog[a + b*x]]] + 256*a*(27*Sq 
rt[2]*a*Erfi[Sqrt[2]*Sqrt[ProductLog[a + b*x]]] + 4*Sqrt[3]*Erfi[Sqrt[3]*S 
qrt[ProductLog[a + b*x]]]))*ProductLog[a + b*x]^(7/2))/(73728*b^4*ProductL 
og[a + b*x]^3*Sqrt[c*ProductLog[a + b*x]])
 

Rubi [A] (verified)

Time = 1.08 (sec) , antiderivative size = 411, normalized size of antiderivative = 0.92, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, 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.

\(\displaystyle \int \frac {x^3}{\sqrt {c W(a+b x)}} \, dx\)

\(\Big \downarrow \) 7168

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {\sqrt {\pi } a^3 \text {erfi}\left (\frac {\sqrt {c W(a+b x)}}{\sqrt {c}}\right )}{2 \sqrt {c}}-\frac {a^3 (a+b x)}{\sqrt {c W(a+b x)}}-\frac {3 \sqrt {\frac {\pi }{2}} a^2 \text {erfi}\left (\frac {\sqrt {2} \sqrt {c W(a+b x)}}{\sqrt {c}}\right )}{16 \sqrt {c}}+\frac {3 a^2 (a+b x)^2}{2 \sqrt {c W(a+b x)}}+\frac {3 a^2 c (a+b x)^2}{8 (c W(a+b x))^{3/2}}+\frac {15 c^3 (a+b x)^4}{2048 (c W(a+b x))^{7/2}}-\frac {5 c^2 (a+b x)^4}{256 (c W(a+b x))^{5/2}}+\frac {a c^2 (a+b x)^3}{12 (c W(a+b x))^{5/2}}-\frac {15 \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {c W(a+b x)}}{\sqrt {c}}\right )}{8192 \sqrt {c}}-\frac {\sqrt {\frac {\pi }{3}} a \text {erfi}\left (\frac {\sqrt {3} \sqrt {c W(a+b x)}}{\sqrt {c}}\right )}{24 \sqrt {c}}+\frac {(a+b x)^4}{4 \sqrt {c W(a+b x)}}+\frac {c (a+b x)^4}{32 (c W(a+b x))^{3/2}}-\frac {a (a+b x)^3}{\sqrt {c W(a+b x)}}-\frac {a c (a+b x)^3}{6 (c W(a+b x))^{3/2}}}{b^4}\)

Input:

Int[x^3/Sqrt[c*ProductLog[a + b*x]],x]
 

Output:

(-1/2*(a^3*Sqrt[Pi]*Erfi[Sqrt[c*ProductLog[a + b*x]]/Sqrt[c]])/Sqrt[c] - ( 
15*Sqrt[Pi]*Erfi[(2*Sqrt[c*ProductLog[a + b*x]])/Sqrt[c]])/(8192*Sqrt[c]) 
- (3*a^2*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[c*ProductLog[a + b*x]])/Sqrt[c]])/( 
16*Sqrt[c]) - (a*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[c*ProductLog[a + b*x]])/Sqr 
t[c]])/(24*Sqrt[c]) + (15*c^3*(a + b*x)^4)/(2048*(c*ProductLog[a + b*x])^( 
7/2)) + (a*c^2*(a + b*x)^3)/(12*(c*ProductLog[a + b*x])^(5/2)) - (5*c^2*(a 
 + b*x)^4)/(256*(c*ProductLog[a + b*x])^(5/2)) + (3*a^2*c*(a + b*x)^2)/(8* 
(c*ProductLog[a + b*x])^(3/2)) - (a*c*(a + b*x)^3)/(6*(c*ProductLog[a + b* 
x])^(3/2)) + (c*(a + b*x)^4)/(32*(c*ProductLog[a + b*x])^(3/2)) - (a^3*(a 
+ b*x))/Sqrt[c*ProductLog[a + b*x]] + (3*a^2*(a + b*x)^2)/(2*Sqrt[c*Produc 
tLog[a + b*x]]) - (a*(a + b*x)^3)/Sqrt[c*ProductLog[a + b*x]] + (a + b*x)^ 
4/(4*Sqrt[c*ProductLog[a + b*x]]))/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 = 0.06 (sec) , antiderivative size = 597, normalized size of antiderivative = 1.33

method	result	size
default	\(\frac {\frac {c \left (c \operatorname {LambertW}\left (b x +a \right )\right )^{\frac {7}{2}} {\mathrm e}^{4 \operatorname {LambertW}\left (b x +a \right )}}{4}+\frac {c \left (\frac {c \left (c \operatorname {LambertW}\left (b x +a \right )\right )^{\frac {5}{2}} {\mathrm e}^{4 \operatorname {LambertW}\left (b x +a \right )}}{8}-\frac {5 c \left (\frac {c \left (c \operatorname {LambertW}\left (b x +a \right )\right )^{\frac {3}{2}} {\mathrm e}^{4 \operatorname {LambertW}\left (b x +a \right )}}{8}-\frac {3 c \left (\frac {c \sqrt {c \operatorname {LambertW}\left (b x +a \right )}\, {\mathrm e}^{4 \operatorname {LambertW}\left (b x +a \right )}}{8}-\frac {c \sqrt {\pi }\, \operatorname {erf}\left (2 \sqrt {-\frac {1}{c}}\, \sqrt {c \operatorname {LambertW}\left (b x +a \right )}\right )}{32 \sqrt {-\frac {1}{c}}}\right )}{8}\right )}{8}\right )}{4}-\frac {a^{3} c^{4} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-\frac {1}{c}}\, \sqrt {c \operatorname {LambertW}\left (b x +a \right )}\right )}{\sqrt {-\frac {1}{c}}}-2 a^{3} c^{3} \left (\frac {c \sqrt {c \operatorname {LambertW}\left (b x +a \right )}\, \left (b x +a \right )}{2 \operatorname {LambertW}\left (b x +a \right )}-\frac {c \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-\frac {1}{c}}\, \sqrt {c \operatorname {LambertW}\left (b x +a \right )}\right )}{4 \sqrt {-\frac {1}{c}}}\right )+6 a^{2} c^{3} \left (\frac {c \sqrt {c \operatorname {LambertW}\left (b x +a \right )}\, {\mathrm e}^{2 \operatorname {LambertW}\left (b x +a \right )}}{4}-\frac {c \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-\frac {2}{c}}\, \sqrt {c \operatorname {LambertW}\left (b x +a \right )}\right )}{8 \sqrt {-\frac {2}{c}}}\right )+6 a^{2} c^{2} \left (\frac {c \left (c \operatorname {LambertW}\left (b x +a \right )\right )^{\frac {3}{2}} {\mathrm e}^{2 \operatorname {LambertW}\left (b x +a \right )}}{4}-\frac {3 c \left (\frac {c \sqrt {c \operatorname {LambertW}\left (b x +a \right )}\, {\mathrm e}^{2 \operatorname {LambertW}\left (b x +a \right )}}{4}-\frac {c \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-\frac {2}{c}}\, \sqrt {c \operatorname {LambertW}\left (b x +a \right )}\right )}{8 \sqrt {-\frac {2}{c}}}\right )}{4}\right )-6 a \,c^{2} \left (\frac {c \left (c \operatorname {LambertW}\left (b x +a \right )\right )^{\frac {3}{2}} {\mathrm e}^{3 \operatorname {LambertW}\left (b x +a \right )}}{6}-\frac {c \left (\frac {c \sqrt {c \operatorname {LambertW}\left (b x +a \right )}\, {\mathrm e}^{3 \operatorname {LambertW}\left (b x +a \right )}}{6}-\frac {c \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-\frac {3}{c}}\, \sqrt {c \operatorname {LambertW}\left (b x +a \right )}\right )}{12 \sqrt {-\frac {3}{c}}}\right )}{2}\right )-6 a c \left (\frac {c \left (c \operatorname {LambertW}\left (b x +a \right )\right )^{\frac {5}{2}} {\mathrm e}^{3 \operatorname {LambertW}\left (b x +a \right )}}{6}-\frac {5 c \left (\frac {c \left (c \operatorname {LambertW}\left (b x +a \right )\right )^{\frac {3}{2}} {\mathrm e}^{3 \operatorname {LambertW}\left (b x +a \right )}}{6}-\frac {c \left (\frac {c \sqrt {c \operatorname {LambertW}\left (b x +a \right )}\, {\mathrm e}^{3 \operatorname {LambertW}\left (b x +a \right )}}{6}-\frac {c \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-\frac {3}{c}}\, \sqrt {c \operatorname {LambertW}\left (b x +a \right )}\right )}{12 \sqrt {-\frac {3}{c}}}\right )}{2}\right )}{6}\right )}{b^{4} c^{5}}\)	\(597\)

Input:

int(x^3/(c*LambertW(b*x+a))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

2/b^4/c^5*(1/8*c*(c*LambertW(b*x+a))^(7/2)*exp(4*LambertW(b*x+a))+1/8*c*(1 
/8*c*(c*LambertW(b*x+a))^(5/2)*exp(4*LambertW(b*x+a))-5/8*c*(1/8*c*(c*Lamb 
ertW(b*x+a))^(3/2)*exp(4*LambertW(b*x+a))-3/8*c*(1/8*c*(c*LambertW(b*x+a)) 
^(1/2)*exp(4*LambertW(b*x+a))-1/32*c*Pi^(1/2)/(-1/c)^(1/2)*erf(2*(-1/c)^(1 
/2)*(c*LambertW(b*x+a))^(1/2)))))-1/2*a^3*c^4*Pi^(1/2)/(-1/c)^(1/2)*erf((- 
1/c)^(1/2)*(c*LambertW(b*x+a))^(1/2))-a^3*c^3*(1/2*c*(c*LambertW(b*x+a))^( 
1/2)*(b*x+a)/LambertW(b*x+a)-1/4*c*Pi^(1/2)/(-1/c)^(1/2)*erf((-1/c)^(1/2)* 
(c*LambertW(b*x+a))^(1/2)))+3*a^2*c^3*(1/4*c*(c*LambertW(b*x+a))^(1/2)*exp 
(2*LambertW(b*x+a))-1/8*c*Pi^(1/2)/(-2/c)^(1/2)*erf((-2/c)^(1/2)*(c*Lamber 
tW(b*x+a))^(1/2)))+3*a^2*c^2*(1/4*c*(c*LambertW(b*x+a))^(3/2)*exp(2*Lamber 
tW(b*x+a))-3/4*c*(1/4*c*(c*LambertW(b*x+a))^(1/2)*exp(2*LambertW(b*x+a))-1 
/8*c*Pi^(1/2)/(-2/c)^(1/2)*erf((-2/c)^(1/2)*(c*LambertW(b*x+a))^(1/2))))-3 
*a*c^2*(1/6*c*(c*LambertW(b*x+a))^(3/2)*exp(3*LambertW(b*x+a))-1/2*c*(1/6* 
c*(c*LambertW(b*x+a))^(1/2)*exp(3*LambertW(b*x+a))-1/12*c*Pi^(1/2)/(-3/c)^ 
(1/2)*erf((-3/c)^(1/2)*(c*LambertW(b*x+a))^(1/2))))-3*a*c*(1/6*c*(c*Lamber 
tW(b*x+a))^(5/2)*exp(3*LambertW(b*x+a))-5/6*c*(1/6*c*(c*LambertW(b*x+a))^( 
3/2)*exp(3*LambertW(b*x+a))-1/2*c*(1/6*c*(c*LambertW(b*x+a))^(1/2)*exp(3*L 
ambertW(b*x+a))-1/12*c*Pi^(1/2)/(-3/c)^(1/2)*erf((-3/c)^(1/2)*(c*LambertW( 
b*x+a))^(1/2))))))
 

Fricas [F]

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

integrate(x^3/(c*lambert_w(b*x+a))^(1/2),x, algorithm="fricas")
 

Output:

integral(sqrt(c*lambert_w(b*x + a))*x^3/(c*lambert_w(b*x + a)), x)
 

Sympy [F]

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

integrate(x**3/(c*LambertW(b*x+a))**(1/2),x)
 

Output:

Integral(x**3/sqrt(c*LambertW(a + b*x)), x)
 

Maxima [F]

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

integrate(x^3/(c*lambert_w(b*x+a))^(1/2),x, algorithm="maxima")
 

Output:

integrate(x^3/sqrt(c*lambert_w(b*x + a)), x)
 

Giac [F]

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

integrate(x^3/(c*lambert_w(b*x+a))^(1/2),x, algorithm="giac")
 

Output:

integrate(x^3/sqrt(c*lambert_w(b*x + a)), x)
 

Mupad [F(-1)]

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

int(x^3/(c*LambertW(a + b*x))^(1/2),x)
 

Output:

int(x^3/(c*LambertW(a + b*x))^(1/2), x)
 

Reduce [F]

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

int(x^3/(c*Lambert_W(b*x+a))^(1/2),x)
 

Output:

(sqrt(c)*int((sqrt(lambert_w(a + b*x))*x**3)/lambert_w(a + b*x),x))/c