\(\int x^2 \operatorname {ExpIntegralEi}(a+b x)^2 \, dx\) [26]

Optimal result

Integrand size = 12, antiderivative size = 286 \[ \int x^2 \operatorname {ExpIntegralEi}(a+b x)^2 \, dx=-\frac {5 e^{2 a+2 b x}}{6 b^3}-\frac {2 a e^{2 a+2 b x}}{3 b^3}+\frac {e^{2 a+2 b x} x}{3 b^2}-\frac {4 e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{3 b^3}-\frac {2 a e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{3 b^3}-\frac {2 a^2 e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{3 b^3}+\frac {4 e^{a+b x} x \operatorname {ExpIntegralEi}(a+b x)}{3 b^2}+\frac {2 a e^{a+b x} x \operatorname {ExpIntegralEi}(a+b x)}{3 b^2}-\frac {2 e^{a+b x} x^2 \operatorname {ExpIntegralEi}(a+b x)}{3 b}-\frac {a^2 x \operatorname {ExpIntegralEi}(a+b x)^2}{3 b^2}+\frac {1}{3} x^3 \operatorname {ExpIntegralEi}(a+b x)^2+\frac {a^2 (a+b x) \operatorname {ExpIntegralEi}(a+b x)^2}{3 b^3}+\frac {4 \operatorname {ExpIntegralEi}(2 (a+b x))}{3 b^3}+\frac {2 a \operatorname {ExpIntegralEi}(2 (a+b x))}{b^3}+\frac {2 a^2 \operatorname {ExpIntegralEi}(2 (a+b x))}{b^3} \] Output:

-5/6*exp(2*b*x+2*a)/b^3-2/3*a*exp(2*b*x+2*a)/b^3+1/3*exp(2*b*x+2*a)*x/b^2- 
4/3*exp(b*x+a)*Ei(b*x+a)/b^3-2/3*a*exp(b*x+a)*Ei(b*x+a)/b^3-2/3*a^2*exp(b* 
x+a)*Ei(b*x+a)/b^3+4/3*exp(b*x+a)*x*Ei(b*x+a)/b^2+2/3*a*exp(b*x+a)*x*Ei(b* 
x+a)/b^2-2/3*exp(b*x+a)*x^2*Ei(b*x+a)/b-1/3*a^2*x*Ei(b*x+a)^2/b^2+1/3*x^3* 
Ei(b*x+a)^2+1/3*a^2*(b*x+a)*Ei(b*x+a)^2/b^3+4/3*Ei(2*b*x+2*a)/b^3+2*a*Ei(2 
*b*x+2*a)/b^3+2*a^2*Ei(2*b*x+2*a)/b^3
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.37 \[ \int x^2 \operatorname {ExpIntegralEi}(a+b x)^2 \, dx=\frac {e^{2 (a+b x)} (-5-4 a+2 b x)-4 e^{a+b x} \left (2+a+a^2-2 b x-a b x+b^2 x^2\right ) \operatorname {ExpIntegralEi}(a+b x)+2 \left (a^3+b^3 x^3\right ) \operatorname {ExpIntegralEi}(a+b x)^2+4 \left (2+3 a+3 a^2\right ) \operatorname {ExpIntegralEi}(2 (a+b x))}{6 b^3} \] Input:

Integrate[x^2*ExpIntegralEi[a + b*x]^2,x]
 

Output:

(E^(2*(a + b*x))*(-5 - 4*a + 2*b*x) - 4*E^(a + b*x)*(2 + a + a^2 - 2*b*x - 
 a*b*x + b^2*x^2)*ExpIntegralEi[a + b*x] + 2*(a^3 + b^3*x^3)*ExpIntegralEi 
[a + b*x]^2 + 4*(2 + 3*a + 3*a^2)*ExpIntegralEi[2*(a + b*x)])/(6*b^3)
 

Rubi [A] (verified)

Time = 2.29 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.41, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.250, Rules used = {7042, 7042, 7040, 7043, 2609, 7044, 2629, 2009, 7043, 2609, 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.

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

\(\Big \downarrow \) 7042

\(\displaystyle -\frac {2}{3} \int e^{a+b x} x^2 \operatorname {ExpIntegralEi}(a+b x)dx-\frac {2 a \int x \operatorname {ExpIntegralEi}(a+b x)^2dx}{3 b}+\frac {1}{3} x^3 \operatorname {ExpIntegralEi}(a+b x)^2+\frac {a x^2 \operatorname {ExpIntegralEi}(a+b x)^2}{3 b}\)

\(\Big \downarrow \) 7042

\(\displaystyle -\frac {2}{3} \int e^{a+b x} x^2 \operatorname {ExpIntegralEi}(a+b x)dx-\frac {2 a \left (-\int e^{a+b x} x \operatorname {ExpIntegralEi}(a+b x)dx-\frac {a \int \operatorname {ExpIntegralEi}(a+b x)^2dx}{2 b}+\frac {1}{2} x^2 \operatorname {ExpIntegralEi}(a+b x)^2+\frac {a x \operatorname {ExpIntegralEi}(a+b x)^2}{2 b}\right )}{3 b}+\frac {1}{3} x^3 \operatorname {ExpIntegralEi}(a+b x)^2+\frac {a x^2 \operatorname {ExpIntegralEi}(a+b x)^2}{3 b}\)

\(\Big \downarrow \) 7040

\(\displaystyle -\frac {2 a \left (-\frac {a \left (\frac {(a+b x) \operatorname {ExpIntegralEi}(a+b x)^2}{b}-2 \int e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)dx\right )}{2 b}-\int e^{a+b x} x \operatorname {ExpIntegralEi}(a+b x)dx+\frac {1}{2} x^2 \operatorname {ExpIntegralEi}(a+b x)^2+\frac {a x \operatorname {ExpIntegralEi}(a+b x)^2}{2 b}\right )}{3 b}-\frac {2}{3} \int e^{a+b x} x^2 \operatorname {ExpIntegralEi}(a+b x)dx+\frac {1}{3} x^3 \operatorname {ExpIntegralEi}(a+b x)^2+\frac {a x^2 \operatorname {ExpIntegralEi}(a+b x)^2}{3 b}\)

\(\Big \downarrow \) 7043

\(\displaystyle -\frac {2 a \left (-\frac {a \left (\frac {(a+b x) \operatorname {ExpIntegralEi}(a+b x)^2}{b}-2 \left (\frac {e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}-\int \frac {e^{2 a+2 b x}}{a+b x}dx\right )\right )}{2 b}-\int e^{a+b x} x \operatorname {ExpIntegralEi}(a+b x)dx+\frac {1}{2} x^2 \operatorname {ExpIntegralEi}(a+b x)^2+\frac {a x \operatorname {ExpIntegralEi}(a+b x)^2}{2 b}\right )}{3 b}-\frac {2}{3} \int e^{a+b x} x^2 \operatorname {ExpIntegralEi}(a+b x)dx+\frac {1}{3} x^3 \operatorname {ExpIntegralEi}(a+b x)^2+\frac {a x^2 \operatorname {ExpIntegralEi}(a+b x)^2}{3 b}\)

\(\Big \downarrow \) 2609

\(\displaystyle -\frac {2 a \left (-\int e^{a+b x} x \operatorname {ExpIntegralEi}(a+b x)dx+\frac {1}{2} x^2 \operatorname {ExpIntegralEi}(a+b x)^2+\frac {a x \operatorname {ExpIntegralEi}(a+b x)^2}{2 b}-\frac {a \left (\frac {(a+b x) \operatorname {ExpIntegralEi}(a+b x)^2}{b}-2 \left (\frac {e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}-\frac {\operatorname {ExpIntegralEi}(2 (a+b x))}{b}\right )\right )}{2 b}\right )}{3 b}-\frac {2}{3} \int e^{a+b x} x^2 \operatorname {ExpIntegralEi}(a+b x)dx+\frac {1}{3} x^3 \operatorname {ExpIntegralEi}(a+b x)^2+\frac {a x^2 \operatorname {ExpIntegralEi}(a+b x)^2}{3 b}\)

\(\Big \downarrow \) 7044

\(\displaystyle -\frac {2 a \left (\frac {\int e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)dx}{b}+\int \frac {e^{2 a+2 b x} x}{a+b x}dx+\frac {1}{2} x^2 \operatorname {ExpIntegralEi}(a+b x)^2+\frac {a x \operatorname {ExpIntegralEi}(a+b x)^2}{2 b}-\frac {x e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}-\frac {a \left (\frac {(a+b x) \operatorname {ExpIntegralEi}(a+b x)^2}{b}-2 \left (\frac {e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}-\frac {\operatorname {ExpIntegralEi}(2 (a+b x))}{b}\right )\right )}{2 b}\right )}{3 b}-\frac {2}{3} \left (-\frac {2 \int e^{a+b x} x \operatorname {ExpIntegralEi}(a+b x)dx}{b}-\int \frac {e^{2 a+2 b x} x^2}{a+b x}dx+\frac {x^2 e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}\right )+\frac {1}{3} x^3 \operatorname {ExpIntegralEi}(a+b x)^2+\frac {a x^2 \operatorname {ExpIntegralEi}(a+b x)^2}{3 b}\)

\(\Big \downarrow \) 2629

\(\displaystyle -\frac {2}{3} \left (-\int \left (\frac {e^{2 a+2 b x} a^2}{b^2 (a+b x)}-\frac {e^{2 a+2 b x} a}{b^2}+\frac {e^{2 a+2 b x} x}{b}\right )dx-\frac {2 \int e^{a+b x} x \operatorname {ExpIntegralEi}(a+b x)dx}{b}+\frac {x^2 e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}\right )-\frac {2 a \left (\frac {\int e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)dx}{b}+\int \left (\frac {e^{2 a+2 b x}}{b}-\frac {a e^{2 a+2 b x}}{b (a+b x)}\right )dx+\frac {1}{2} x^2 \operatorname {ExpIntegralEi}(a+b x)^2+\frac {a x \operatorname {ExpIntegralEi}(a+b x)^2}{2 b}-\frac {x e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}-\frac {a \left (\frac {(a+b x) \operatorname {ExpIntegralEi}(a+b x)^2}{b}-2 \left (\frac {e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}-\frac {\operatorname {ExpIntegralEi}(2 (a+b x))}{b}\right )\right )}{2 b}\right )}{3 b}+\frac {1}{3} x^3 \operatorname {ExpIntegralEi}(a+b x)^2+\frac {a x^2 \operatorname {ExpIntegralEi}(a+b x)^2}{3 b}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2}{3} \left (-\frac {2 \int e^{a+b x} x \operatorname {ExpIntegralEi}(a+b x)dx}{b}-\frac {a^2 \operatorname {ExpIntegralEi}(2 (a+b x))}{b^3}+\frac {a e^{2 a+2 b x}}{2 b^3}+\frac {e^{2 a+2 b x}}{4 b^3}-\frac {x e^{2 a+2 b x}}{2 b^2}+\frac {x^2 e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}\right )-\frac {2 a \left (\frac {\int e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)dx}{b}-\frac {a \operatorname {ExpIntegralEi}(2 (a+b x))}{b^2}+\frac {e^{2 a+2 b x}}{2 b^2}+\frac {1}{2} x^2 \operatorname {ExpIntegralEi}(a+b x)^2+\frac {a x \operatorname {ExpIntegralEi}(a+b x)^2}{2 b}-\frac {x e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}-\frac {a \left (\frac {(a+b x) \operatorname {ExpIntegralEi}(a+b x)^2}{b}-2 \left (\frac {e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}-\frac {\operatorname {ExpIntegralEi}(2 (a+b x))}{b}\right )\right )}{2 b}\right )}{3 b}+\frac {1}{3} x^3 \operatorname {ExpIntegralEi}(a+b x)^2+\frac {a x^2 \operatorname {ExpIntegralEi}(a+b x)^2}{3 b}\)

\(\Big \downarrow \) 7043

\(\displaystyle -\frac {2}{3} \left (-\frac {2 \int e^{a+b x} x \operatorname {ExpIntegralEi}(a+b x)dx}{b}-\frac {a^2 \operatorname {ExpIntegralEi}(2 (a+b x))}{b^3}+\frac {a e^{2 a+2 b x}}{2 b^3}+\frac {e^{2 a+2 b x}}{4 b^3}-\frac {x e^{2 a+2 b x}}{2 b^2}+\frac {x^2 e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}\right )-\frac {2 a \left (\frac {\frac {e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}-\int \frac {e^{2 a+2 b x}}{a+b x}dx}{b}-\frac {a \operatorname {ExpIntegralEi}(2 (a+b x))}{b^2}+\frac {e^{2 a+2 b x}}{2 b^2}+\frac {1}{2} x^2 \operatorname {ExpIntegralEi}(a+b x)^2+\frac {a x \operatorname {ExpIntegralEi}(a+b x)^2}{2 b}-\frac {x e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}-\frac {a \left (\frac {(a+b x) \operatorname {ExpIntegralEi}(a+b x)^2}{b}-2 \left (\frac {e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}-\frac {\operatorname {ExpIntegralEi}(2 (a+b x))}{b}\right )\right )}{2 b}\right )}{3 b}+\frac {1}{3} x^3 \operatorname {ExpIntegralEi}(a+b x)^2+\frac {a x^2 \operatorname {ExpIntegralEi}(a+b x)^2}{3 b}\)

\(\Big \downarrow \) 2609

\(\displaystyle -\frac {2}{3} \left (-\frac {2 \int e^{a+b x} x \operatorname {ExpIntegralEi}(a+b x)dx}{b}-\frac {a^2 \operatorname {ExpIntegralEi}(2 (a+b x))}{b^3}+\frac {a e^{2 a+2 b x}}{2 b^3}+\frac {e^{2 a+2 b x}}{4 b^3}-\frac {x e^{2 a+2 b x}}{2 b^2}+\frac {x^2 e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}\right )-\frac {2 a \left (-\frac {a \operatorname {ExpIntegralEi}(2 (a+b x))}{b^2}+\frac {e^{2 a+2 b x}}{2 b^2}+\frac {1}{2} x^2 \operatorname {ExpIntegralEi}(a+b x)^2+\frac {a x \operatorname {ExpIntegralEi}(a+b x)^2}{2 b}-\frac {x e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}+\frac {\frac {e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}-\frac {\operatorname {ExpIntegralEi}(2 (a+b x))}{b}}{b}-\frac {a \left (\frac {(a+b x) \operatorname {ExpIntegralEi}(a+b x)^2}{b}-2 \left (\frac {e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}-\frac {\operatorname {ExpIntegralEi}(2 (a+b x))}{b}\right )\right )}{2 b}\right )}{3 b}+\frac {1}{3} x^3 \operatorname {ExpIntegralEi}(a+b x)^2+\frac {a x^2 \operatorname {ExpIntegralEi}(a+b x)^2}{3 b}\)

\(\Big \downarrow \) 7044

\(\displaystyle -\frac {2}{3} \left (-\frac {2 \left (-\frac {\int e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)dx}{b}-\int \frac {e^{2 a+2 b x} x}{a+b x}dx+\frac {x e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}\right )}{b}-\frac {a^2 \operatorname {ExpIntegralEi}(2 (a+b x))}{b^3}+\frac {a e^{2 a+2 b x}}{2 b^3}+\frac {e^{2 a+2 b x}}{4 b^3}-\frac {x e^{2 a+2 b x}}{2 b^2}+\frac {x^2 e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}\right )-\frac {2 a \left (-\frac {a \operatorname {ExpIntegralEi}(2 (a+b x))}{b^2}+\frac {e^{2 a+2 b x}}{2 b^2}+\frac {1}{2} x^2 \operatorname {ExpIntegralEi}(a+b x)^2+\frac {a x \operatorname {ExpIntegralEi}(a+b x)^2}{2 b}-\frac {x e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}+\frac {\frac {e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}-\frac {\operatorname {ExpIntegralEi}(2 (a+b x))}{b}}{b}-\frac {a \left (\frac {(a+b x) \operatorname {ExpIntegralEi}(a+b x)^2}{b}-2 \left (\frac {e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}-\frac {\operatorname {ExpIntegralEi}(2 (a+b x))}{b}\right )\right )}{2 b}\right )}{3 b}+\frac {1}{3} x^3 \operatorname {ExpIntegralEi}(a+b x)^2+\frac {a x^2 \operatorname {ExpIntegralEi}(a+b x)^2}{3 b}\)

\(\Big \downarrow \) 2629

\(\displaystyle -\frac {2}{3} \left (-\frac {2 \left (-\frac {\int e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)dx}{b}-\int \left (\frac {e^{2 a+2 b x}}{b}-\frac {a e^{2 a+2 b x}}{b (a+b x)}\right )dx+\frac {x e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}\right )}{b}-\frac {a^2 \operatorname {ExpIntegralEi}(2 (a+b x))}{b^3}+\frac {a e^{2 a+2 b x}}{2 b^3}+\frac {e^{2 a+2 b x}}{4 b^3}-\frac {x e^{2 a+2 b x}}{2 b^2}+\frac {x^2 e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}\right )-\frac {2 a \left (-\frac {a \operatorname {ExpIntegralEi}(2 (a+b x))}{b^2}+\frac {e^{2 a+2 b x}}{2 b^2}+\frac {1}{2} x^2 \operatorname {ExpIntegralEi}(a+b x)^2+\frac {a x \operatorname {ExpIntegralEi}(a+b x)^2}{2 b}-\frac {x e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}+\frac {\frac {e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}-\frac {\operatorname {ExpIntegralEi}(2 (a+b x))}{b}}{b}-\frac {a \left (\frac {(a+b x) \operatorname {ExpIntegralEi}(a+b x)^2}{b}-2 \left (\frac {e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}-\frac {\operatorname {ExpIntegralEi}(2 (a+b x))}{b}\right )\right )}{2 b}\right )}{3 b}+\frac {1}{3} x^3 \operatorname {ExpIntegralEi}(a+b x)^2+\frac {a x^2 \operatorname {ExpIntegralEi}(a+b x)^2}{3 b}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2}{3} \left (-\frac {2 \left (-\frac {\int e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)dx}{b}+\frac {a \operatorname {ExpIntegralEi}(2 (a+b x))}{b^2}-\frac {e^{2 a+2 b x}}{2 b^2}+\frac {x e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}\right )}{b}-\frac {a^2 \operatorname {ExpIntegralEi}(2 (a+b x))}{b^3}+\frac {a e^{2 a+2 b x}}{2 b^3}+\frac {e^{2 a+2 b x}}{4 b^3}-\frac {x e^{2 a+2 b x}}{2 b^2}+\frac {x^2 e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}\right )-\frac {2 a \left (-\frac {a \operatorname {ExpIntegralEi}(2 (a+b x))}{b^2}+\frac {e^{2 a+2 b x}}{2 b^2}+\frac {1}{2} x^2 \operatorname {ExpIntegralEi}(a+b x)^2+\frac {a x \operatorname {ExpIntegralEi}(a+b x)^2}{2 b}-\frac {x e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}+\frac {\frac {e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}-\frac {\operatorname {ExpIntegralEi}(2 (a+b x))}{b}}{b}-\frac {a \left (\frac {(a+b x) \operatorname {ExpIntegralEi}(a+b x)^2}{b}-2 \left (\frac {e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}-\frac {\operatorname {ExpIntegralEi}(2 (a+b x))}{b}\right )\right )}{2 b}\right )}{3 b}+\frac {1}{3} x^3 \operatorname {ExpIntegralEi}(a+b x)^2+\frac {a x^2 \operatorname {ExpIntegralEi}(a+b x)^2}{3 b}\)

\(\Big \downarrow \) 7043

\(\displaystyle -\frac {2}{3} \left (-\frac {2 \left (-\frac {\frac {e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}-\int \frac {e^{2 a+2 b x}}{a+b x}dx}{b}+\frac {a \operatorname {ExpIntegralEi}(2 (a+b x))}{b^2}-\frac {e^{2 a+2 b x}}{2 b^2}+\frac {x e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}\right )}{b}-\frac {a^2 \operatorname {ExpIntegralEi}(2 (a+b x))}{b^3}+\frac {a e^{2 a+2 b x}}{2 b^3}+\frac {e^{2 a+2 b x}}{4 b^3}-\frac {x e^{2 a+2 b x}}{2 b^2}+\frac {x^2 e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}\right )-\frac {2 a \left (-\frac {a \operatorname {ExpIntegralEi}(2 (a+b x))}{b^2}+\frac {e^{2 a+2 b x}}{2 b^2}+\frac {1}{2} x^2 \operatorname {ExpIntegralEi}(a+b x)^2+\frac {a x \operatorname {ExpIntegralEi}(a+b x)^2}{2 b}-\frac {x e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}+\frac {\frac {e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}-\frac {\operatorname {ExpIntegralEi}(2 (a+b x))}{b}}{b}-\frac {a \left (\frac {(a+b x) \operatorname {ExpIntegralEi}(a+b x)^2}{b}-2 \left (\frac {e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}-\frac {\operatorname {ExpIntegralEi}(2 (a+b x))}{b}\right )\right )}{2 b}\right )}{3 b}+\frac {1}{3} x^3 \operatorname {ExpIntegralEi}(a+b x)^2+\frac {a x^2 \operatorname {ExpIntegralEi}(a+b x)^2}{3 b}\)

\(\Big \downarrow \) 2609

\(\displaystyle -\frac {2}{3} \left (-\frac {a^2 \operatorname {ExpIntegralEi}(2 (a+b x))}{b^3}+\frac {a e^{2 a+2 b x}}{2 b^3}+\frac {e^{2 a+2 b x}}{4 b^3}-\frac {2 \left (\frac {a \operatorname {ExpIntegralEi}(2 (a+b x))}{b^2}-\frac {e^{2 a+2 b x}}{2 b^2}+\frac {x e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}-\frac {\frac {e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}-\frac {\operatorname {ExpIntegralEi}(2 (a+b x))}{b}}{b}\right )}{b}-\frac {x e^{2 a+2 b x}}{2 b^2}+\frac {x^2 e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}\right )-\frac {2 a \left (-\frac {a \operatorname {ExpIntegralEi}(2 (a+b x))}{b^2}+\frac {e^{2 a+2 b x}}{2 b^2}+\frac {1}{2} x^2 \operatorname {ExpIntegralEi}(a+b x)^2+\frac {a x \operatorname {ExpIntegralEi}(a+b x)^2}{2 b}-\frac {x e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}+\frac {\frac {e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}-\frac {\operatorname {ExpIntegralEi}(2 (a+b x))}{b}}{b}-\frac {a \left (\frac {(a+b x) \operatorname {ExpIntegralEi}(a+b x)^2}{b}-2 \left (\frac {e^{a+b x} \operatorname {ExpIntegralEi}(a+b x)}{b}-\frac {\operatorname {ExpIntegralEi}(2 (a+b x))}{b}\right )\right )}{2 b}\right )}{3 b}+\frac {1}{3} x^3 \operatorname {ExpIntegralEi}(a+b x)^2+\frac {a x^2 \operatorname {ExpIntegralEi}(a+b x)^2}{3 b}\)

Input:

Int[x^2*ExpIntegralEi[a + b*x]^2,x]
 

Output:

(a*x^2*ExpIntegralEi[a + b*x]^2)/(3*b) + (x^3*ExpIntegralEi[a + b*x]^2)/3 
- (2*a*(E^(2*a + 2*b*x)/(2*b^2) - (E^(a + b*x)*x*ExpIntegralEi[a + b*x])/b 
 + (a*x*ExpIntegralEi[a + b*x]^2)/(2*b) + (x^2*ExpIntegralEi[a + b*x]^2)/2 
 - (a*ExpIntegralEi[2*(a + b*x)])/b^2 + ((E^(a + b*x)*ExpIntegralEi[a + b* 
x])/b - ExpIntegralEi[2*(a + b*x)]/b)/b - (a*(((a + b*x)*ExpIntegralEi[a + 
 b*x]^2)/b - 2*((E^(a + b*x)*ExpIntegralEi[a + b*x])/b - ExpIntegralEi[2*( 
a + b*x)]/b)))/(2*b)))/(3*b) - (2*(E^(2*a + 2*b*x)/(4*b^3) + (a*E^(2*a + 2 
*b*x))/(2*b^3) - (E^(2*a + 2*b*x)*x)/(2*b^2) + (E^(a + b*x)*x^2*ExpIntegra 
lEi[a + b*x])/b - (a^2*ExpIntegralEi[2*(a + b*x)])/b^3 - (2*(-1/2*E^(2*a + 
 2*b*x)/b^2 + (E^(a + b*x)*x*ExpIntegralEi[a + b*x])/b + (a*ExpIntegralEi[ 
2*(a + b*x)])/b^2 - ((E^(a + b*x)*ExpIntegralEi[a + b*x])/b - ExpIntegralE 
i[2*(a + b*x)]/b)/b))/b))/3
 

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 7040

Int[ExpIntegralEi[(a_.) + (b_.)*(x_)]^2, x_Symbol] :> Simp[(a + b*x)*(ExpIn 
tegralEi[a + b*x]^2/b), x] - Simp[2   Int[E^(a + b*x)*ExpIntegralEi[a + b*x 
], x], x] /; FreeQ[{a, b}, x]
 

rule 7042

Int[ExpIntegralEi[(a_) + (b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 
 1)*(ExpIntegralEi[a + b*x]^2/(m + 1)), x] + (Simp[a*x^m*(ExpIntegralEi[a + 
 b*x]^2/(b*(m + 1))), x] - Simp[2/(m + 1)   Int[x^m*E^(a + b*x)*ExpIntegral 
Ei[a + b*x], x], x] - Simp[a*(m/(b*(m + 1)))   Int[x^(m - 1)*ExpIntegralEi[ 
a + b*x]^2, x], x]) /; FreeQ[{a, b}, x] && IGtQ[m, 0]
 

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 [F]

\[\int x^{2} \operatorname {expIntegral}\left (b x +a \right )^{2}d x\]

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.35 \[ \int x^2 \operatorname {ExpIntegralEi}(a+b x)^2 \, dx=\frac {2 \, {\left (b^{3} x^{3} + a^{3}\right )} {\rm Ei}\left (b x + a\right )^{2} - 4 \, {\left (b^{2} x^{2} - {\left (a + 2\right )} b x + a^{2} + a + 2\right )} {\rm Ei}\left (b x + a\right ) e^{\left (b x + a\right )} + 4 \, {\left (3 \, a^{2} + 3 \, a + 2\right )} {\rm Ei}\left (2 \, b x + 2 \, a\right ) + {\left (2 \, b x - 4 \, a - 5\right )} e^{\left (2 \, b x + 2 \, a\right )}}{6 \, b^{3}} \] Input:

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

Output:

1/6*(2*(b^3*x^3 + a^3)*Ei(b*x + a)^2 - 4*(b^2*x^2 - (a + 2)*b*x + a^2 + a 
+ 2)*Ei(b*x + a)*e^(b*x + a) + 4*(3*a^2 + 3*a + 2)*Ei(2*b*x + 2*a) + (2*b* 
x - 4*a - 5)*e^(2*b*x + 2*a))/b^3
 

Sympy [F]

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

integrate(x**2*Ei(b*x+a)**2,x)
 

Output:

Integral(x**2*Ei(a + b*x)**2, x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(ei(a + b*x)**2*x**2,x)