\(\int e^{a+b x} x^3 \operatorname {ExpIntegralEi}(c+d x) \, dx\) [55]

Optimal result

Integrand size = 17, antiderivative size = 433 \[ \int e^{a+b x} x^3 \operatorname {ExpIntegralEi}(c+d x) \, dx=-\frac {2 e^{a+c+(b+d) x}}{b (b+d)^3}-\frac {3 e^{a+c+(b+d) x}}{b^2 (b+d)^2}-\frac {c e^{a+c+(b+d) x}}{b d (b+d)^2}-\frac {6 e^{a+c+(b+d) x}}{b^3 (b+d)}-\frac {c^2 e^{a+c+(b+d) x}}{b d^2 (b+d)}-\frac {3 c e^{a+c+(b+d) x}}{b^2 d (b+d)}+\frac {2 e^{a+c+(b+d) x} x}{b (b+d)^2}+\frac {3 e^{a+c+(b+d) x} x}{b^2 (b+d)}+\frac {c e^{a+c+(b+d) x} x}{b d (b+d)}-\frac {e^{a+c+(b+d) x} x^2}{b (b+d)}-\frac {6 e^{a+b x} \operatorname {ExpIntegralEi}(c+d x)}{b^4}+\frac {6 e^{a+b x} x \operatorname {ExpIntegralEi}(c+d x)}{b^3}-\frac {3 e^{a+b x} x^2 \operatorname {ExpIntegralEi}(c+d x)}{b^2}+\frac {e^{a+b x} x^3 \operatorname {ExpIntegralEi}(c+d x)}{b}+\frac {6 e^{a-\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (\frac {(b+d) (c+d x)}{d}\right )}{b^4}+\frac {c^3 e^{a-\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (\frac {(b+d) (c+d x)}{d}\right )}{b d^3}+\frac {3 c^2 e^{a-\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (\frac {(b+d) (c+d x)}{d}\right )}{b^2 d^2}+\frac {6 c e^{a-\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (\frac {(b+d) (c+d x)}{d}\right )}{b^3 d} \] Output:

-2*exp(a+c+(b+d)*x)/b/(b+d)^3-3*exp(a+c+(b+d)*x)/b^2/(b+d)^2-c*exp(a+c+(b+ 
d)*x)/b/d/(b+d)^2-6*exp(a+c+(b+d)*x)/b^3/(b+d)-c^2*exp(a+c+(b+d)*x)/b/d^2/ 
(b+d)-3*c*exp(a+c+(b+d)*x)/b^2/d/(b+d)+2*exp(a+c+(b+d)*x)*x/b/(b+d)^2+3*ex 
p(a+c+(b+d)*x)*x/b^2/(b+d)+c*exp(a+c+(b+d)*x)*x/b/d/(b+d)-exp(a+c+(b+d)*x) 
*x^2/b/(b+d)-6*exp(b*x+a)*Ei(d*x+c)/b^4+6*exp(b*x+a)*x*Ei(d*x+c)/b^3-3*exp 
(b*x+a)*x^2*Ei(d*x+c)/b^2+exp(b*x+a)*x^3*Ei(d*x+c)/b+6*exp(a-b*c/d)*Ei((b+ 
d)*(d*x+c)/d)/b^4+c^3*exp(a-b*c/d)*Ei((b+d)*(d*x+c)/d)/b/d^3+3*c^2*exp(a-b 
*c/d)*Ei((b+d)*(d*x+c)/d)/b^2/d^2+6*c*exp(a-b*c/d)*Ei((b+d)*(d*x+c)/d)/b^3 
/d
 

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.52 \[ \int e^{a+b x} x^3 \operatorname {ExpIntegralEi}(c+d x) \, dx=\frac {e^a \left (e^{b x} \left (-6+6 b x-3 b^2 x^2+b^3 x^3\right ) \operatorname {ExpIntegralEi}(c+d x)+\frac {e^{-\frac {b c}{d}} \left (-b d e^{\frac {(b+d) (c+d x)}{d}} \left (6 d^4+3 b d^3 (5+c-d x)+b^4 \left (c^2-c d x+d^2 x^2\right )+b^2 d^2 \left (11+c^2-8 d x+d^2 x^2+c (7-d x)\right )+b^3 d \left (2 c^2+c (4-2 d x)+d x (-5+2 d x)\right )\right )+(b+d)^3 \left (b^3 c^3+3 b^2 c^2 d+6 b c d^2+6 d^3\right ) \operatorname {ExpIntegralEi}\left (\frac {(b+d) (c+d x)}{d}\right )\right )}{d^3 (b+d)^3}\right )}{b^4} \] Input:

Integrate[E^(a + b*x)*x^3*ExpIntegralEi[c + d*x],x]
 

Output:

(E^a*(E^(b*x)*(-6 + 6*b*x - 3*b^2*x^2 + b^3*x^3)*ExpIntegralEi[c + d*x] + 
(-(b*d*E^(((b + d)*(c + d*x))/d)*(6*d^4 + 3*b*d^3*(5 + c - d*x) + b^4*(c^2 
 - c*d*x + d^2*x^2) + b^2*d^2*(11 + c^2 - 8*d*x + d^2*x^2 + c*(7 - d*x)) + 
 b^3*d*(2*c^2 + c*(4 - 2*d*x) + d*x*(-5 + 2*d*x)))) + (b + d)^3*(b^3*c^3 + 
 3*b^2*c^2*d + 6*b*c*d^2 + 6*d^3)*ExpIntegralEi[((b + d)*(c + d*x))/d])/(d 
^3*(b + d)^3*E^((b*c)/d))))/b^4
 

Rubi [A] (verified)

Time = 2.53 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.03, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.647, Rules used = {7044, 2629, 2009, 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.

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

\(\Big \downarrow \) 7044

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

\(\Big \downarrow \) 2629

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 7044

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

\(\Big \downarrow \) 2629

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 7044

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

\(\Big \downarrow \) 2629

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 7043

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

\(\Big \downarrow \) 2609

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

Input:

Int[E^(a + b*x)*x^3*ExpIntegralEi[c + d*x],x]
 

Output:

(E^(a + b*x)*x^3*ExpIntegralEi[c + d*x])/b - (d*((2*E^(a + c + (b + d)*x)) 
/(d*(b + d)^3) + (c*E^(a + c + (b + d)*x))/(d^2*(b + d)^2) + (c^2*E^(a + c 
 + (b + d)*x))/(d^3*(b + d)) - (2*E^(a + c + (b + d)*x)*x)/(d*(b + d)^2) - 
 (c*E^(a + c + (b + d)*x)*x)/(d^2*(b + d)) + (E^(a + c + (b + d)*x)*x^2)/( 
d*(b + d)) - (c^3*E^(a - (b*c)/d)*ExpIntegralEi[((b + d)*(c + d*x))/d])/d^ 
4))/b - (3*((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)*ExpIntegral 
Ei[((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))/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. \(964\) vs. \(2(415)=830\).

Time = 0.72 (sec) , antiderivative size = 965, normalized size of antiderivative = 2.23

method	result	size
default	\(\text {Expression too large to display}\)	\(965\)

Input:

int(exp(b*x+a)*x^3*Ei(d*x+c),x,method=_RETURNVERBOSE)
 

Output:

(-Ei(d*x+c)/d^2/b*(exp(1/d*b*(d*x+c)+(a*d-b*c)/d)/b^3*d^3*a^3-1/b^3*d^3*(e 
xp(1/d*b*(d*x+c)+(a*d-b*c)/d)*(1/d*b*(d*x+c)+(a*d-b*c)/d)^3-3*(1/d*b*(d*x+ 
c)+(a*d-b*c)/d)^2*exp(1/d*b*(d*x+c)+(a*d-b*c)/d)+6*(1/d*b*(d*x+c)+(a*d-b*c 
)/d)*exp(1/d*b*(d*x+c)+(a*d-b*c)/d)-6*exp(1/d*b*(d*x+c)+(a*d-b*c)/d))-3/b^ 
3*d^3*a^2*((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))+3/b^3*d^3*a*((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)))+1/d^2/b*(-exp(a)/exp(b*c 
/d)*c^3*Ei(1,-(1+b/d)*(d*x+c))-exp(a)/exp(b*c/d)/(1+b/d)^3*((1+b/d)^2*(d*x 
+c)^2*exp((1+b/d)*(d*x+c))-2*(1+b/d)*(d*x+c)*exp((1+b/d)*(d*x+c))+2*exp((1 
+b/d)*(d*x+c)))-3*exp(a)/exp(b*c/d)*c^2*exp((1+b/d)*(d*x+c))/(1+b/d)-6/b^2 
*d^2*exp(a)/exp(b*c/d)*exp((1+b/d)*(d*x+c))/(1+b/d)+3*exp(a)/exp(b*c/d)*c/ 
(1+b/d)^2*((1+b/d)*(d*x+c)*exp((1+b/d)*(d*x+c))-exp((1+b/d)*(d*x+c)))-6/b* 
d*exp(a)/exp(b*c/d)*c*exp((1+b/d)*(d*x+c))/(1+b/d)-6/b^3*d^3*exp(a)/exp(b* 
c/d)*Ei(1,-(1+b/d)*(d*x+c))+3/b*d*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)))-6/b^2*d^2*exp(a)/exp(b*c/d 
)*c*Ei(1,-(1+b/d)*(d*x+c))-3/b*d*exp(a)/exp(b*c/d)*c^2*Ei(1,-(1+b/d)*(d*x+ 
c))))/d
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.17 \[ \int e^{a+b x} x^3 \operatorname {ExpIntegralEi}(c+d x) \, dx=-\frac {{\left (6 \, b^{3} d^{3} + 18 \, b^{2} d^{4} + 18 \, b d^{5} + 6 \, d^{6} - {\left (b^{6} d^{3} + 3 \, b^{5} d^{4} + 3 \, b^{4} d^{5} + b^{3} d^{6}\right )} x^{3} + 3 \, {\left (b^{5} d^{3} + 3 \, b^{4} d^{4} + 3 \, b^{3} d^{5} + b^{2} d^{6}\right )} x^{2} - 6 \, {\left (b^{4} d^{3} + 3 \, b^{3} d^{4} + 3 \, b^{2} d^{5} + b d^{6}\right )} x\right )} {\rm Ei}\left (d x + c\right ) e^{\left (b x + a\right )} - {\left (b^{6} c^{3} + 6 \, {\left (b c + 3 \, b\right )} d^{5} + 6 \, d^{6} + 3 \, {\left (b^{2} c^{2} + 6 \, b^{2} c + 6 \, b^{2}\right )} d^{4} + {\left (b^{3} c^{3} + 9 \, b^{3} c^{2} + 18 \, b^{3} c + 6 \, b^{3}\right )} d^{3} + 3 \, {\left (b^{4} c^{3} + 3 \, b^{4} c^{2} + 2 \, b^{4} c\right )} d^{2} + 3 \, {\left (b^{5} c^{3} + b^{5} c^{2}\right )} d\right )} {\rm Ei}\left (\frac {b c + c d + {\left (b d + d^{2}\right )} x}{d}\right ) e^{\left (-\frac {b c - a d}{d}\right )} + {\left (b^{5} c^{2} d + 6 \, b d^{5} + 3 \, {\left (b^{2} c + 5 \, b^{2}\right )} d^{4} + {\left (b^{3} c^{2} + 7 \, b^{3} c + 11 \, b^{3}\right )} d^{3} + 2 \, {\left (b^{4} c^{2} + 2 \, b^{4} c\right )} d^{2} + {\left (b^{5} d^{3} + 2 \, b^{4} d^{4} + b^{3} d^{5}\right )} x^{2} - {\left (b^{5} c d^{2} + 3 \, b^{2} d^{5} + {\left (b^{3} c + 8 \, b^{3}\right )} d^{4} + {\left (2 \, b^{4} c + 5 \, b^{4}\right )} d^{3}\right )} x\right )} e^{\left (b x + d x + a + c\right )}}{b^{7} d^{3} + 3 \, b^{6} d^{4} + 3 \, b^{5} d^{5} + b^{4} d^{6}} \] Input:

integrate(exp(b*x+a)*x^3*Ei(d*x+c),x, algorithm="fricas")
 

Output:

-((6*b^3*d^3 + 18*b^2*d^4 + 18*b*d^5 + 6*d^6 - (b^6*d^3 + 3*b^5*d^4 + 3*b^ 
4*d^5 + b^3*d^6)*x^3 + 3*(b^5*d^3 + 3*b^4*d^4 + 3*b^3*d^5 + b^2*d^6)*x^2 - 
 6*(b^4*d^3 + 3*b^3*d^4 + 3*b^2*d^5 + b*d^6)*x)*Ei(d*x + c)*e^(b*x + a) - 
(b^6*c^3 + 6*(b*c + 3*b)*d^5 + 6*d^6 + 3*(b^2*c^2 + 6*b^2*c + 6*b^2)*d^4 + 
 (b^3*c^3 + 9*b^3*c^2 + 18*b^3*c + 6*b^3)*d^3 + 3*(b^4*c^3 + 3*b^4*c^2 + 2 
*b^4*c)*d^2 + 3*(b^5*c^3 + b^5*c^2)*d)*Ei((b*c + c*d + (b*d + d^2)*x)/d)*e 
^(-(b*c - a*d)/d) + (b^5*c^2*d + 6*b*d^5 + 3*(b^2*c + 5*b^2)*d^4 + (b^3*c^ 
2 + 7*b^3*c + 11*b^3)*d^3 + 2*(b^4*c^2 + 2*b^4*c)*d^2 + (b^5*d^3 + 2*b^4*d 
^4 + b^3*d^5)*x^2 - (b^5*c*d^2 + 3*b^2*d^5 + (b^3*c + 8*b^3)*d^4 + (2*b^4* 
c + 5*b^4)*d^3)*x)*e^(b*x + d*x + a + c))/(b^7*d^3 + 3*b^6*d^4 + 3*b^5*d^5 
 + b^4*d^6)
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int e^{a+b x} x^3 \operatorname {ExpIntegralEi}(c+d x) \, dx=\text {Timed out} \] Input:

integrate(exp(b*x+a)*x**3*Ei(d*x+c),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(exp(b*x+a)*x^3*Ei(d*x+c),x, algorithm="maxima")
 

Output:

integrate(x^3*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. 1154 vs. \(2 (415) = 830\).

Time = 0.13 (sec) , antiderivative size = 1154, normalized size of antiderivative = 2.67 \[ \int e^{a+b x} x^3 \operatorname {ExpIntegralEi}(c+d x) \, dx=\text {Too large to display} \] Input:

integrate(exp(b*x+a)*x^3*Ei(d*x+c),x, algorithm="giac")
 

Output:

-(b^5*d^3*x^2*e^(b*x + d*x + a + c) + 2*b^4*d^4*x^2*e^(b*x + d*x + a + c) 
+ b^3*d^5*x^2*e^(b*x + d*x + a + c) - b^6*c^3*Ei((b*d*x + d^2*x + b*c + c* 
d)/d)*e^(a + c - (b*c + c*d)/d) - 3*b^5*c^3*d*Ei((b*d*x + d^2*x + b*c + c* 
d)/d)*e^(a + c - (b*c + c*d)/d) - 3*b^4*c^3*d^2*Ei((b*d*x + d^2*x + b*c + 
c*d)/d)*e^(a + c - (b*c + c*d)/d) - b^3*c^3*d^3*Ei((b*d*x + d^2*x + b*c + 
c*d)/d)*e^(a + c - (b*c + c*d)/d) - b^5*c*d^2*x*e^(b*x + d*x + a + c) - 2* 
b^4*c*d^3*x*e^(b*x + d*x + a + c) - b^3*c*d^4*x*e^(b*x + d*x + a + c) - 3* 
b^5*c^2*d*Ei((b*d*x + d^2*x + b*c + c*d)/d)*e^(a + c - (b*c + c*d)/d) - 9* 
b^4*c^2*d^2*Ei((b*d*x + d^2*x + b*c + c*d)/d)*e^(a + c - (b*c + c*d)/d) - 
9*b^3*c^2*d^3*Ei((b*d*x + d^2*x + b*c + c*d)/d)*e^(a + c - (b*c + c*d)/d) 
- 3*b^2*c^2*d^4*Ei((b*d*x + d^2*x + b*c + c*d)/d)*e^(a + c - (b*c + c*d)/d 
) + b^5*c^2*d*e^(b*x + d*x + a + c) + 2*b^4*c^2*d^2*e^(b*x + d*x + a + c) 
+ b^3*c^2*d^3*e^(b*x + d*x + a + c) - 5*b^4*d^3*x*e^(b*x + d*x + a + c) - 
8*b^3*d^4*x*e^(b*x + d*x + a + c) - 3*b^2*d^5*x*e^(b*x + d*x + a + c) - 6* 
b^4*c*d^2*Ei((b*d*x + d^2*x + b*c + c*d)/d)*e^(a + c - (b*c + c*d)/d) - 18 
*b^3*c*d^3*Ei((b*d*x + d^2*x + b*c + c*d)/d)*e^(a + c - (b*c + c*d)/d) - 1 
8*b^2*c*d^4*Ei((b*d*x + d^2*x + b*c + c*d)/d)*e^(a + c - (b*c + c*d)/d) - 
6*b*c*d^5*Ei((b*d*x + d^2*x + b*c + c*d)/d)*e^(a + c - (b*c + c*d)/d) + 4* 
b^4*c*d^2*e^(b*x + d*x + a + c) + 7*b^3*c*d^3*e^(b*x + d*x + a + c) + 3*b^ 
2*c*d^4*e^(b*x + d*x + a + c) - 6*b^3*d^3*Ei((b*d*x + d^2*x + b*c + c*d...
 

Mupad [F(-1)]

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

int(x^3*ei(c + d*x)*exp(a + b*x),x)
 

Output:

int(x^3*ei(c + d*x)*exp(a + b*x), x)
 

Reduce [F]

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

int(exp(b*x+a)*x^3*Ei(d*x+c),x)
 

Output:

e**a*int(e**(b*x)*ei(c + d*x)*x**3,x)