\(\int e^{a+b x} x^2 \operatorname {ExpIntegralEi}(c+d x) \, dx\) [56]

Optimal result

Integrand size = 17, antiderivative size = 238 \[ \int e^{a+b x} x^2 \operatorname {ExpIntegralEi}(c+d x) \, dx=\frac {e^{a+c+(b+d) x}}{b (b+d)^2}+\frac {2 e^{a+c+(b+d) x}}{b^2 (b+d)}+\frac {c e^{a+c+(b+d) x}}{b d (b+d)}-\frac {e^{a+c+(b+d) x} x}{b (b+d)}+\frac {2 e^{a+b x} \operatorname {ExpIntegralEi}(c+d x)}{b^3}-\frac {2 e^{a+b x} x \operatorname {ExpIntegralEi}(c+d x)}{b^2}+\frac {e^{a+b x} x^2 \operatorname {ExpIntegralEi}(c+d x)}{b}-\frac {2 e^{a-\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (\frac {(b+d) (c+d x)}{d}\right )}{b^3}-\frac {c^2 e^{a-\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (\frac {(b+d) (c+d x)}{d}\right )}{b d^2}-\frac {2 c e^{a-\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (\frac {(b+d) (c+d x)}{d}\right )}{b^2 d} \] Output:

exp(a+c+(b+d)*x)/b/(b+d)^2+2*exp(a+c+(b+d)*x)/b^2/(b+d)+c*exp(a+c+(b+d)*x) 
/b/d/(b+d)-exp(a+c+(b+d)*x)*x/b/(b+d)+2*exp(b*x+a)*Ei(d*x+c)/b^3-2*exp(b*x 
+a)*x*Ei(d*x+c)/b^2+exp(b*x+a)*x^2*Ei(d*x+c)/b-2*exp(a-b*c/d)*Ei((b+d)*(d* 
x+c)/d)/b^3-c^2*exp(a-b*c/d)*Ei((b+d)*(d*x+c)/d)/b/d^2-2*c*exp(a-b*c/d)*Ei 
((b+d)*(d*x+c)/d)/b^2/d
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.56 \[ \int e^{a+b x} x^2 \operatorname {ExpIntegralEi}(c+d x) \, dx=\frac {e^a \left (e^{b x} \left (2-2 b x+b^2 x^2\right ) \operatorname {ExpIntegralEi}(c+d x)+\frac {e^{-\frac {b c}{d}} \left (b d e^{\frac {(b+d) (c+d x)}{d}} \left (2 d^2+b^2 (c-d x)+b d (3+c-d x)\right )-(b+d)^2 \left (b^2 c^2+2 b c d+2 d^2\right ) \operatorname {ExpIntegralEi}\left (\frac {(b+d) (c+d x)}{d}\right )\right )}{d^2 (b+d)^2}\right )}{b^3} \] Input:

Integrate[E^(a + b*x)*x^2*ExpIntegralEi[c + d*x],x]
 

Output:

(E^a*(E^(b*x)*(2 - 2*b*x + b^2*x^2)*ExpIntegralEi[c + d*x] + (b*d*E^(((b + 
 d)*(c + d*x))/d)*(2*d^2 + b^2*(c - d*x) + b*d*(3 + c - d*x)) - (b + d)^2* 
(b^2*c^2 + 2*b*c*d + 2*d^2)*ExpIntegralEi[((b + d)*(c + d*x))/d])/(d^2*(b 
+ d)^2*E^((b*c)/d))))/b^3
 

Rubi [A] (verified)

Time = 1.38 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {7044, 2629, 2009, 7044, 2629, 2009, 7043, 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.

\(\displaystyle \int x^2 e^{a+b x} \operatorname {ExpIntegralEi}(c+d x) \, dx\)

\(\Big \downarrow \) 7044

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

\(\Big \downarrow \) 2629

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 7044

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

\(\Big \downarrow \) 2629

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 7043

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

\(\Big \downarrow \) 2609

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

Input:

Int[E^(a + b*x)*x^2*ExpIntegralEi[c + d*x],x]
 

Output:

(E^(a + b*x)*x^2*ExpIntegralEi[c + d*x])/b - (d*(-(E^(a + c + (b + d)*x)/( 
d*(b + d)^2)) - (c*E^(a + c + (b + d)*x))/(d^2*(b + d)) + (E^(a + c + (b + 
 d)*x)*x)/(d*(b + d)) + (c^2*E^(a - (b*c)/d)*ExpIntegralEi[((b + d)*(c + d 
*x))/d])/d^3))/b - (2*((E^(a + b*x)*x*ExpIntegralEi[c + d*x])/b - ((E^(a + 
 b*x)*ExpIntegralEi[c + d*x])/b - (E^(a - (b*c)/d)*ExpIntegralEi[((b + d)* 
(c + d*x))/d])/b)/b - (d*(E^(a + c + (b + d)*x)/(d*(b + d)) - (c*E^(a - (b 
*c)/d)*ExpIntegralEi[((b + d)*(c + d*x))/d])/d^2))/b))/b
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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 2629

Int[(F_)^(v_)*(Px_)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandInte 
grand[F^v, Px*(d + e*x)^m, x], x] /; FreeQ[{F, d, e, m}, x] && PolynomialQ[ 
Px, x] && LinearQ[v, x] &&  !TrueQ[$UseGamma]
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(522\) vs. \(2(228)=456\).

Time = 0.34 (sec) , antiderivative size = 523, normalized size of antiderivative = 2.20

method	result	size
default	\(\frac {\frac {\operatorname {expIntegral}\left (d x +c \right ) \left (\frac {{\mathrm e}^{\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}} d^{2} a^{2}}{b^{2}}+\frac {d^{2} \left (\left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )^{2} {\mathrm e}^{\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}}-2 \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right ) {\mathrm e}^{\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}}+2 \,{\mathrm e}^{\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}}\right )}{b^{2}}-\frac {2 d^{2} a \left (\left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right ) {\mathrm e}^{\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}}-{\mathrm e}^{\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}}\right )}{b^{2}}\right )}{d b}-\frac {\frac {{\mathrm e}^{a} {\mathrm e}^{-\frac {b c}{d}} \left (\left (1+\frac {b}{d}\right ) \left (d x +c \right ) {\mathrm e}^{\left (1+\frac {b}{d}\right ) \left (d x +c \right )}-{\mathrm e}^{\left (1+\frac {b}{d}\right ) \left (d x +c \right )}\right )}{\left (1+\frac {b}{d}\right )^{2}}-{\mathrm e}^{a} {\mathrm e}^{-\frac {b c}{d}} c^{2} \operatorname {expIntegral}_{1}\left (-\left (1+\frac {b}{d}\right ) \left (d x +c \right )\right )-\frac {2 \,{\mathrm e}^{a} {\mathrm e}^{-\frac {b c}{d}} c \,{\mathrm e}^{\left (1+\frac {b}{d}\right ) \left (d x +c \right )}}{1+\frac {b}{d}}-\frac {2 d \,{\mathrm e}^{a} {\mathrm e}^{-\frac {b c}{d}} {\mathrm e}^{\left (1+\frac {b}{d}\right ) \left (d x +c \right )}}{b \left (1+\frac {b}{d}\right )}-\frac {2 d^{2} {\mathrm e}^{a} {\mathrm e}^{-\frac {b c}{d}} \operatorname {expIntegral}_{1}\left (-\left (1+\frac {b}{d}\right ) \left (d x +c \right )\right )}{b^{2}}-\frac {2 d \,{\mathrm e}^{a} {\mathrm e}^{-\frac {b c}{d}} c \,\operatorname {expIntegral}_{1}\left (-\left (1+\frac {b}{d}\right ) \left (d x +c \right )\right )}{b}}{d b}}{d}\)	\(523\)

Input:

int(exp(b*x+a)*x^2*Ei(d*x+c),x,method=_RETURNVERBOSE)
 

Output:

(Ei(d*x+c)/d/b*(exp(1/d*b*(d*x+c)+(a*d-b*c)/d)/b^2*d^2*a^2+1/b^2*d^2*((1/d 
*b*(d*x+c)+(a*d-b*c)/d)^2*exp(1/d*b*(d*x+c)+(a*d-b*c)/d)-2*(1/d*b*(d*x+c)+ 
(a*d-b*c)/d)*exp(1/d*b*(d*x+c)+(a*d-b*c)/d)+2*exp(1/d*b*(d*x+c)+(a*d-b*c)/ 
d))-2/b^2*d^2*a*((1/d*b*(d*x+c)+(a*d-b*c)/d)*exp(1/d*b*(d*x+c)+(a*d-b*c)/d 
)-exp(1/d*b*(d*x+c)+(a*d-b*c)/d)))-1/d/b*(exp(a)/exp(b*c/d)/(1+b/d)^2*((1+ 
b/d)*(d*x+c)*exp((1+b/d)*(d*x+c))-exp((1+b/d)*(d*x+c)))-exp(a)/exp(b*c/d)* 
c^2*Ei(1,-(1+b/d)*(d*x+c))-2*exp(a)/exp(b*c/d)*c*exp((1+b/d)*(d*x+c))/(1+b 
/d)-2/b*d*exp(a)/exp(b*c/d)*exp((1+b/d)*(d*x+c))/(1+b/d)-2/b^2*d^2*exp(a)/ 
exp(b*c/d)*Ei(1,-(1+b/d)*(d*x+c))-2/b*d*exp(a)/exp(b*c/d)*c*Ei(1,-(1+b/d)* 
(d*x+c))))/d
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.13 \[ \int e^{a+b x} x^2 \operatorname {ExpIntegralEi}(c+d x) \, dx=\frac {{\left (2 \, b^{2} d^{2} + 4 \, b d^{3} + 2 \, d^{4} + {\left (b^{4} d^{2} + 2 \, b^{3} d^{3} + b^{2} d^{4}\right )} x^{2} - 2 \, {\left (b^{3} d^{2} + 2 \, b^{2} d^{3} + b d^{4}\right )} x\right )} {\rm Ei}\left (d x + c\right ) e^{\left (b x + a\right )} - {\left (b^{4} c^{2} + 2 \, {\left (b c + 2 \, b\right )} d^{3} + 2 \, d^{4} + {\left (b^{2} c^{2} + 4 \, b^{2} c + 2 \, b^{2}\right )} d^{2} + 2 \, {\left (b^{3} c^{2} + b^{3} c\right )} d\right )} {\rm Ei}\left (\frac {b c + c d + {\left (b d + d^{2}\right )} x}{d}\right ) e^{\left (-\frac {b c - a d}{d}\right )} + {\left (b^{3} c d + 2 \, b d^{3} + {\left (b^{2} c + 3 \, b^{2}\right )} d^{2} - {\left (b^{3} d^{2} + b^{2} d^{3}\right )} x\right )} e^{\left (b x + d x + a + c\right )}}{b^{5} d^{2} + 2 \, b^{4} d^{3} + b^{3} d^{4}} \] Input:

integrate(exp(b*x+a)*x^2*Ei(d*x+c),x, algorithm="fricas")
 

Output:

((2*b^2*d^2 + 4*b*d^3 + 2*d^4 + (b^4*d^2 + 2*b^3*d^3 + b^2*d^4)*x^2 - 2*(b 
^3*d^2 + 2*b^2*d^3 + b*d^4)*x)*Ei(d*x + c)*e^(b*x + a) - (b^4*c^2 + 2*(b*c 
 + 2*b)*d^3 + 2*d^4 + (b^2*c^2 + 4*b^2*c + 2*b^2)*d^2 + 2*(b^3*c^2 + b^3*c 
)*d)*Ei((b*c + c*d + (b*d + d^2)*x)/d)*e^(-(b*c - a*d)/d) + (b^3*c*d + 2*b 
*d^3 + (b^2*c + 3*b^2)*d^2 - (b^3*d^2 + b^2*d^3)*x)*e^(b*x + d*x + a + c)) 
/(b^5*d^2 + 2*b^4*d^3 + b^3*d^4)
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int e^{a+b x} x^2 \operatorname {ExpIntegralEi}(c+d x) \, dx=\text {Timed out} \] Input:

integrate(exp(b*x+a)*x**2*Ei(d*x+c),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int e^{a+b x} x^2 \operatorname {ExpIntegralEi}(c+d x) \, dx=\int { x^{2} {\rm Ei}\left (d x + c\right ) e^{\left (b x + a\right )} \,d x } \] Input:

integrate(exp(b*x+a)*x^2*Ei(d*x+c),x, algorithm="maxima")
 

Output:

integrate(x^2*Ei(d*x + c)*e^(b*x + a), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 565 vs. \(2 (228) = 456\).

Time = 0.13 (sec) , antiderivative size = 565, normalized size of antiderivative = 2.37 \[ \int e^{a+b x} x^2 \operatorname {ExpIntegralEi}(c+d x) \, dx=-\frac {b^{4} c^{2} {\rm Ei}\left (\frac {b d x + d^{2} x + b c + c d}{d}\right ) e^{\left (a + c - \frac {b c + c d}{d}\right )} + 2 \, b^{3} c^{2} d {\rm Ei}\left (\frac {b d x + d^{2} x + b c + c d}{d}\right ) e^{\left (a + c - \frac {b c + c d}{d}\right )} + b^{2} c^{2} d^{2} {\rm Ei}\left (\frac {b d x + d^{2} x + b c + c d}{d}\right ) e^{\left (a + c - \frac {b c + c d}{d}\right )} + b^{3} d^{2} x e^{\left (b x + d x + a + c\right )} + b^{2} d^{3} x e^{\left (b x + d x + a + c\right )} + 2 \, b^{3} c d {\rm Ei}\left (\frac {b d x + d^{2} x + b c + c d}{d}\right ) e^{\left (a + c - \frac {b c + c d}{d}\right )} + 4 \, b^{2} c d^{2} {\rm Ei}\left (\frac {b d x + d^{2} x + b c + c d}{d}\right ) e^{\left (a + c - \frac {b c + c d}{d}\right )} + 2 \, b c d^{3} {\rm Ei}\left (\frac {b d x + d^{2} x + b c + c d}{d}\right ) e^{\left (a + c - \frac {b c + c d}{d}\right )} - b^{3} c d e^{\left (b x + d x + a + c\right )} - b^{2} c d^{2} e^{\left (b x + d x + a + c\right )} + 2 \, b^{2} d^{2} {\rm Ei}\left (\frac {b d x + d^{2} x + b c + c d}{d}\right ) e^{\left (a + c - \frac {b c + c d}{d}\right )} + 4 \, b d^{3} {\rm Ei}\left (\frac {b d x + d^{2} x + b c + c d}{d}\right ) e^{\left (a + c - \frac {b c + c d}{d}\right )} + 2 \, d^{4} {\rm Ei}\left (\frac {b d x + d^{2} x + b c + c d}{d}\right ) e^{\left (a + c - \frac {b c + c d}{d}\right )} - 3 \, b^{2} d^{2} e^{\left (b x + d x + a + c\right )} - 2 \, b d^{3} e^{\left (b x + d x + a + c\right )}}{b^{5} d^{2} + 2 \, b^{4} d^{3} + b^{3} d^{4}} + \frac {{\left (b^{2} x^{2} - 2 \, b x + 2\right )} {\rm Ei}\left (d x + c\right ) e^{\left (b x + a\right )}}{b^{3}} \] Input:

integrate(exp(b*x+a)*x^2*Ei(d*x+c),x, algorithm="giac")
 

Output:

-(b^4*c^2*Ei((b*d*x + d^2*x + b*c + c*d)/d)*e^(a + c - (b*c + c*d)/d) + 2* 
b^3*c^2*d*Ei((b*d*x + d^2*x + b*c + c*d)/d)*e^(a + c - (b*c + c*d)/d) + b^ 
2*c^2*d^2*Ei((b*d*x + d^2*x + b*c + c*d)/d)*e^(a + c - (b*c + c*d)/d) + b^ 
3*d^2*x*e^(b*x + d*x + a + c) + b^2*d^3*x*e^(b*x + d*x + a + c) + 2*b^3*c* 
d*Ei((b*d*x + d^2*x + b*c + c*d)/d)*e^(a + c - (b*c + c*d)/d) + 4*b^2*c*d^ 
2*Ei((b*d*x + d^2*x + b*c + c*d)/d)*e^(a + c - (b*c + c*d)/d) + 2*b*c*d^3* 
Ei((b*d*x + d^2*x + b*c + c*d)/d)*e^(a + c - (b*c + c*d)/d) - b^3*c*d*e^(b 
*x + d*x + a + c) - b^2*c*d^2*e^(b*x + d*x + a + c) + 2*b^2*d^2*Ei((b*d*x 
+ d^2*x + b*c + c*d)/d)*e^(a + c - (b*c + c*d)/d) + 4*b*d^3*Ei((b*d*x + d^ 
2*x + b*c + c*d)/d)*e^(a + c - (b*c + c*d)/d) + 2*d^4*Ei((b*d*x + d^2*x + 
b*c + c*d)/d)*e^(a + c - (b*c + c*d)/d) - 3*b^2*d^2*e^(b*x + d*x + a + c) 
- 2*b*d^3*e^(b*x + d*x + a + c))/(b^5*d^2 + 2*b^4*d^3 + b^3*d^4) + (b^2*x^ 
2 - 2*b*x + 2)*Ei(d*x + c)*e^(b*x + a)/b^3
 

Mupad [F(-1)]

Timed out. \[ \int e^{a+b x} x^2 \operatorname {ExpIntegralEi}(c+d x) \, dx=\int x^2\,\mathrm {ei}\left (c+d\,x\right )\,{\mathrm {e}}^{a+b\,x} \,d x \] Input:

int(x^2*ei(c + d*x)*exp(a + b*x),x)
 

Output:

int(x^2*ei(c + d*x)*exp(a + b*x), x)
 

Reduce [F]

\[ \int e^{a+b x} x^2 \operatorname {ExpIntegralEi}(c+d x) \, dx=e^{a} \left (\int e^{b x} \mathit {ei} \left (d x +c \right ) x^{2}d x \right ) \] Input:

int(exp(b*x+a)*x^2*Ei(d*x+c),x)
 

Output:

e**a*int(e**(b*x)*ei(c + d*x)*x**2,x)