\(\int e^{b x} x^3 \operatorname {ExpIntegralEi}(b x) \, dx\) [54]

Optimal result

Integrand size = 13, antiderivative size = 112 \[ \int e^{b x} x^3 \operatorname {ExpIntegralEi}(b x) \, dx=-\frac {4 e^{2 b x}}{b^4}+\frac {2 e^{2 b x} x}{b^3}-\frac {e^{2 b x} x^2}{2 b^2}-\frac {6 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b^4}+\frac {6 e^{b x} x \operatorname {ExpIntegralEi}(b x)}{b^3}-\frac {3 e^{b x} x^2 \operatorname {ExpIntegralEi}(b x)}{b^2}+\frac {e^{b x} x^3 \operatorname {ExpIntegralEi}(b x)}{b}+\frac {6 \operatorname {ExpIntegralEi}(2 b x)}{b^4} \] Output:

-4*exp(2*b*x)/b^4+2*exp(2*b*x)*x/b^3-1/2*exp(2*b*x)*x^2/b^2-6*exp(b*x)*Ei( 
b*x)/b^4+6*exp(b*x)*x*Ei(b*x)/b^3-3*exp(b*x)*x^2*Ei(b*x)/b^2+exp(b*x)*x^3* 
Ei(b*x)/b+6*Ei(2*b*x)/b^4
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.61 \[ \int e^{b x} x^3 \operatorname {ExpIntegralEi}(b x) \, dx=\frac {-e^{2 b x} \left (8-4 b x+b^2 x^2\right )+2 e^{b x} \left (-6+6 b x-3 b^2 x^2+b^3 x^3\right ) \operatorname {ExpIntegralEi}(b x)+12 \operatorname {ExpIntegralEi}(2 b x)}{2 b^4} \] Input:

Integrate[E^(b*x)*x^3*ExpIntegralEi[b*x],x]
 

Output:

(-(E^(2*b*x)*(8 - 4*b*x + b^2*x^2)) + 2*E^(b*x)*(-6 + 6*b*x - 3*b^2*x^2 + 
b^3*x^3)*ExpIntegralEi[b*x] + 12*ExpIntegralEi[2*b*x])/(2*b^4)
 

Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.69, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.154, Rules used = {7044, 27, 2607, 2607, 2624, 7044, 27, 2607, 2624, 7044, 27, 2624, 7043, 27, 2609}

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.

\(\displaystyle \int x^3 e^{b x} \operatorname {ExpIntegralEi}(b x) \, dx\)

\(\Big \downarrow \) 7044

\(\displaystyle -\frac {3 \int e^{b x} x^2 \operatorname {ExpIntegralEi}(b x)dx}{b}-\int \frac {e^{2 b x} x^2}{b}dx+\frac {x^3 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3 \int e^{b x} x^2 \operatorname {ExpIntegralEi}(b x)dx}{b}-\frac {\int e^{2 b x} x^2dx}{b}+\frac {x^3 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\)

\(\Big \downarrow \) 2607

\(\displaystyle -\frac {3 \int e^{b x} x^2 \operatorname {ExpIntegralEi}(b x)dx}{b}-\frac {\frac {x^2 e^{2 b x}}{2 b}-\frac {\int e^{2 b x} xdx}{b}}{b}+\frac {x^3 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\)

\(\Big \downarrow \) 2607

\(\displaystyle -\frac {3 \int e^{b x} x^2 \operatorname {ExpIntegralEi}(b x)dx}{b}-\frac {\frac {x^2 e^{2 b x}}{2 b}-\frac {\frac {x e^{2 b x}}{2 b}-\frac {\int e^{2 b x}dx}{2 b}}{b}}{b}+\frac {x^3 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\)

\(\Big \downarrow \) 2624

\(\displaystyle -\frac {3 \int e^{b x} x^2 \operatorname {ExpIntegralEi}(b x)dx}{b}-\frac {\frac {x^2 e^{2 b x}}{2 b}-\frac {\frac {x e^{2 b x}}{2 b}-\frac {e^{2 b x}}{4 b^2}}{b}}{b}+\frac {x^3 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\)

\(\Big \downarrow \) 7044

\(\displaystyle -\frac {3 \left (-\frac {2 \int e^{b x} x \operatorname {ExpIntegralEi}(b x)dx}{b}-\int \frac {e^{2 b x} x}{b}dx+\frac {x^2 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\right )}{b}-\frac {\frac {x^2 e^{2 b x}}{2 b}-\frac {\frac {x e^{2 b x}}{2 b}-\frac {e^{2 b x}}{4 b^2}}{b}}{b}+\frac {x^3 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3 \left (-\frac {2 \int e^{b x} x \operatorname {ExpIntegralEi}(b x)dx}{b}-\frac {\int e^{2 b x} xdx}{b}+\frac {x^2 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\right )}{b}-\frac {\frac {x^2 e^{2 b x}}{2 b}-\frac {\frac {x e^{2 b x}}{2 b}-\frac {e^{2 b x}}{4 b^2}}{b}}{b}+\frac {x^3 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\)

\(\Big \downarrow \) 2607

\(\displaystyle -\frac {3 \left (-\frac {2 \int e^{b x} x \operatorname {ExpIntegralEi}(b x)dx}{b}-\frac {\frac {x e^{2 b x}}{2 b}-\frac {\int e^{2 b x}dx}{2 b}}{b}+\frac {x^2 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\right )}{b}-\frac {\frac {x^2 e^{2 b x}}{2 b}-\frac {\frac {x e^{2 b x}}{2 b}-\frac {e^{2 b x}}{4 b^2}}{b}}{b}+\frac {x^3 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\)

\(\Big \downarrow \) 2624

\(\displaystyle -\frac {3 \left (-\frac {2 \int e^{b x} x \operatorname {ExpIntegralEi}(b x)dx}{b}-\frac {\frac {x e^{2 b x}}{2 b}-\frac {e^{2 b x}}{4 b^2}}{b}+\frac {x^2 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\right )}{b}-\frac {\frac {x^2 e^{2 b x}}{2 b}-\frac {\frac {x e^{2 b x}}{2 b}-\frac {e^{2 b x}}{4 b^2}}{b}}{b}+\frac {x^3 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\)

\(\Big \downarrow \) 7044

\(\displaystyle -\frac {3 \left (-\frac {2 \left (-\frac {\int e^{b x} \operatorname {ExpIntegralEi}(b x)dx}{b}-\int \frac {e^{2 b x}}{b}dx+\frac {x e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\right )}{b}-\frac {\frac {x e^{2 b x}}{2 b}-\frac {e^{2 b x}}{4 b^2}}{b}+\frac {x^2 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\right )}{b}-\frac {\frac {x^2 e^{2 b x}}{2 b}-\frac {\frac {x e^{2 b x}}{2 b}-\frac {e^{2 b x}}{4 b^2}}{b}}{b}+\frac {x^3 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3 \left (-\frac {2 \left (-\frac {\int e^{b x} \operatorname {ExpIntegralEi}(b x)dx}{b}-\frac {\int e^{2 b x}dx}{b}+\frac {x e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\right )}{b}-\frac {\frac {x e^{2 b x}}{2 b}-\frac {e^{2 b x}}{4 b^2}}{b}+\frac {x^2 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\right )}{b}-\frac {\frac {x^2 e^{2 b x}}{2 b}-\frac {\frac {x e^{2 b x}}{2 b}-\frac {e^{2 b x}}{4 b^2}}{b}}{b}+\frac {x^3 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\)

\(\Big \downarrow \) 2624

\(\displaystyle -\frac {3 \left (-\frac {2 \left (-\frac {\int e^{b x} \operatorname {ExpIntegralEi}(b x)dx}{b}-\frac {e^{2 b x}}{2 b^2}+\frac {x e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\right )}{b}-\frac {\frac {x e^{2 b x}}{2 b}-\frac {e^{2 b x}}{4 b^2}}{b}+\frac {x^2 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\right )}{b}-\frac {\frac {x^2 e^{2 b x}}{2 b}-\frac {\frac {x e^{2 b x}}{2 b}-\frac {e^{2 b x}}{4 b^2}}{b}}{b}+\frac {x^3 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\)

\(\Big \downarrow \) 7043

\(\displaystyle -\frac {3 \left (-\frac {2 \left (-\frac {\frac {e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}-\int \frac {e^{2 b x}}{b x}dx}{b}-\frac {e^{2 b x}}{2 b^2}+\frac {x e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\right )}{b}-\frac {\frac {x e^{2 b x}}{2 b}-\frac {e^{2 b x}}{4 b^2}}{b}+\frac {x^2 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\right )}{b}-\frac {\frac {x^2 e^{2 b x}}{2 b}-\frac {\frac {x e^{2 b x}}{2 b}-\frac {e^{2 b x}}{4 b^2}}{b}}{b}+\frac {x^3 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3 \left (-\frac {2 \left (-\frac {\frac {e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}-\frac {\int \frac {e^{2 b x}}{x}dx}{b}}{b}-\frac {e^{2 b x}}{2 b^2}+\frac {x e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\right )}{b}-\frac {\frac {x e^{2 b x}}{2 b}-\frac {e^{2 b x}}{4 b^2}}{b}+\frac {x^2 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\right )}{b}-\frac {\frac {x^2 e^{2 b x}}{2 b}-\frac {\frac {x e^{2 b x}}{2 b}-\frac {e^{2 b x}}{4 b^2}}{b}}{b}+\frac {x^3 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\)

\(\Big \downarrow \) 2609

\(\displaystyle -\frac {3 \left (-\frac {2 \left (-\frac {e^{2 b x}}{2 b^2}+\frac {x e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}-\frac {\frac {e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}-\frac {\operatorname {ExpIntegralEi}(2 b x)}{b}}{b}\right )}{b}-\frac {\frac {x e^{2 b x}}{2 b}-\frac {e^{2 b x}}{4 b^2}}{b}+\frac {x^2 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\right )}{b}-\frac {\frac {x^2 e^{2 b x}}{2 b}-\frac {\frac {x e^{2 b x}}{2 b}-\frac {e^{2 b x}}{4 b^2}}{b}}{b}+\frac {x^3 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\)

Input:

Int[E^(b*x)*x^3*ExpIntegralEi[b*x],x]
 

Output:

-(((E^(2*b*x)*x^2)/(2*b) - (-1/4*E^(2*b*x)/b^2 + (E^(2*b*x)*x)/(2*b))/b)/b 
) + (E^(b*x)*x^3*ExpIntegralEi[b*x])/b - (3*(-((-1/4*E^(2*b*x)/b^2 + (E^(2 
*b*x)*x)/(2*b))/b) + (E^(b*x)*x^2*ExpIntegralEi[b*x])/b - (2*(-1/2*E^(2*b* 
x)/b^2 + (E^(b*x)*x*ExpIntegralEi[b*x])/b - ((E^(b*x)*ExpIntegralEi[b*x])/ 
b - ExpIntegralEi[2*b*x]/b)/b))/b))/b
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2607

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m 
_.), x_Symbol] :> Simp[(c + d*x)^m*((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), 
x] - Simp[d*(m/(f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x)))^ 
n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2* 
m] &&  !TrueQ[$UseGamma]
 

rule 2609

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Si 
mp[(F^(g*(e - c*(f/d)))/d)*ExpIntegralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; F 
reeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]
 

rule 2624

Int[((F_)^(v_))^(n_.), x_Symbol] :> Simp[(F^v)^n/(n*Log[F]*D[v, x]), x] /; 
FreeQ[{F, n}, x] && LinearQ[v, x]
 

rule 7043

Int[E^((a_.) + (b_.)*(x_))*ExpIntegralEi[(c_.) + (d_.)*(x_)], x_Symbol] :> 
Simp[E^(a + b*x)*(ExpIntegralEi[c + d*x]/b), x] - Simp[d/b   Int[E^(a + c + 
 (b + d)*x)/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x]
 

rule 7044

Int[E^((a_.) + (b_.)*(x_))*ExpIntegralEi[(c_.) + (d_.)*(x_)]*(x_)^(m_.), x_ 
Symbol] :> Simp[x^m*E^(a + b*x)*(ExpIntegralEi[c + d*x]/b), x] + (-Simp[d/b 
   Int[x^m*(E^(a + c + (b + d)*x)/(c + d*x)), x], x] - Simp[m/b   Int[x^(m 
- 1)*E^(a + b*x)*ExpIntegralEi[c + d*x], x], x]) /; FreeQ[{a, b, c, d}, x] 
&& IGtQ[m, 0]
 

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.79

method	result	size
derivativedivides	\(\frac {\operatorname {expIntegral}\left (b x \right ) \left ({\mathrm e}^{b x} b^{3} x^{3}-3 b^{2} x^{2} {\mathrm e}^{b x}+6 b x \,{\mathrm e}^{b x}-6 \,{\mathrm e}^{b x}\right )-\frac {{\mathrm e}^{2 b x} b^{2} x^{2}}{2}+2 \,{\mathrm e}^{2 b x} b x -4 \,{\mathrm e}^{2 b x}-6 \,\operatorname {expIntegral}_{1}\left (-2 b x \right )}{b^{4}}\)	\(89\)
default	\(\frac {\operatorname {expIntegral}\left (b x \right ) \left ({\mathrm e}^{b x} b^{3} x^{3}-3 b^{2} x^{2} {\mathrm e}^{b x}+6 b x \,{\mathrm e}^{b x}-6 \,{\mathrm e}^{b x}\right )-\frac {{\mathrm e}^{2 b x} b^{2} x^{2}}{2}+2 \,{\mathrm e}^{2 b x} b x -4 \,{\mathrm e}^{2 b x}-6 \,\operatorname {expIntegral}_{1}\left (-2 b x \right )}{b^{4}}\)	\(89\)

Input:

int(exp(b*x)*x^3*Ei(b*x),x,method=_RETURNVERBOSE)
 

Output:

1/b^4*(Ei(b*x)*(exp(b*x)*b^3*x^3-3*b^2*x^2*exp(b*x)+6*b*x*exp(b*x)-6*exp(b 
*x))-1/2*exp(b*x)^2*b^2*x^2+2*exp(b*x)^2*b*x-4*exp(b*x)^2-6*Ei(1,-2*b*x))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.57 \[ \int e^{b x} x^3 \operatorname {ExpIntegralEi}(b x) \, dx=\frac {2 \, {\left (b^{3} x^{3} - 3 \, b^{2} x^{2} + 6 \, b x - 6\right )} {\rm Ei}\left (b x\right ) e^{\left (b x\right )} - {\left (b^{2} x^{2} - 4 \, b x + 8\right )} e^{\left (2 \, b x\right )} + 12 \, {\rm Ei}\left (2 \, b x\right )}{2 \, b^{4}} \] Input:

integrate(exp(b*x)*x^3*Ei(b*x),x, algorithm="fricas")
 

Output:

1/2*(2*(b^3*x^3 - 3*b^2*x^2 + 6*b*x - 6)*Ei(b*x)*e^(b*x) - (b^2*x^2 - 4*b* 
x + 8)*e^(2*b*x) + 12*Ei(2*b*x))/b^4
 

Sympy [F]

\[ \int e^{b x} x^3 \operatorname {ExpIntegralEi}(b x) \, dx=\int x^{3} e^{b x} \operatorname {Ei}{\left (b x \right )}\, dx \] Input:

integrate(exp(b*x)*x**3*Ei(b*x),x)
 

Output:

Integral(x**3*exp(b*x)*Ei(b*x), x)
 

Maxima [F]

\[ \int e^{b x} x^3 \operatorname {ExpIntegralEi}(b x) \, dx=\int { x^{3} {\rm Ei}\left (b x\right ) e^{\left (b x\right )} \,d x } \] Input:

integrate(exp(b*x)*x^3*Ei(b*x),x, algorithm="maxima")
 

Output:

integrate(x^3*Ei(b*x)*e^(b*x), x)
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.67 \[ \int e^{b x} x^3 \operatorname {ExpIntegralEi}(b x) \, dx=\frac {{\left (b^{3} x^{3} - 3 \, b^{2} x^{2} + 6 \, b x - 6\right )} {\rm Ei}\left (b x\right ) e^{\left (b x\right )}}{b^{4}} - \frac {b^{2} x^{2} e^{\left (2 \, b x\right )} - 4 \, b x e^{\left (2 \, b x\right )} - 12 \, {\rm Ei}\left (2 \, b x\right ) + 8 \, e^{\left (2 \, b x\right )}}{2 \, b^{4}} \] Input:

integrate(exp(b*x)*x^3*Ei(b*x),x, algorithm="giac")
 

Output:

(b^3*x^3 - 3*b^2*x^2 + 6*b*x - 6)*Ei(b*x)*e^(b*x)/b^4 - 1/2*(b^2*x^2*e^(2* 
b*x) - 4*b*x*e^(2*b*x) - 12*Ei(2*b*x) + 8*e^(2*b*x))/b^4
 

Mupad [F(-1)]

Timed out. \[ \int e^{b x} x^3 \operatorname {ExpIntegralEi}(b x) \, dx=\int x^3\,{\mathrm {e}}^{b\,x}\,\mathrm {ei}\left (b\,x\right ) \,d x \] Input:

int(x^3*exp(b*x)*ei(b*x),x)
 

Output:

int(x^3*exp(b*x)*ei(b*x), x)
 

Reduce [F]

\[ \int e^{b x} x^3 \operatorname {ExpIntegralEi}(b x) \, dx=\int e^{b x} \mathit {ei} \left (b x \right ) x^{3}d x \] Input:

int(exp(b*x)*x^3*Ei(b*x),x)
 

Output:

int(e**(b*x)*ei(b*x)*x**3,x)