# Maple integration test file: "8 Special functions\8.3 Exponential integral functions.txt"

lst:=[

# Integration Problems Involving Exponential Integral Functions

# Integrands of the form (d x)^m ExpIntegralE[n, b x]

# Integrands of the form (d x)^m ExpIntegralE[n, b x]

# n>0
[x^2*Ei(1,b*x),x,1,-1/3*x^3*Ei(-2,b*x)+1/3*x^3*Ei(1,b*x)],
[x*Ei(1,b*x),x,1,-1/2*x^2*Ei(-1,b*x)+1/2*x^2*Ei(1,b*x)],
[Ei(1,b*x),x,1,-Ei(2,b*x)/b],
[Ei(1,b*x)/x,x,1,b*x*HypergeometricPFQ([1,1,1],[2,2,2],-b*x)-EulerGamma*log(x)-1/2*log(b*x)^2],
[Ei(1,b*x)/x^2,x,1,-Ei(1,b*x)/x+Ei(2,b*x)/x],
[Ei(1,b*x)/x^3,x,1,-1/2*Ei(1,b*x)/x^2+1/2*Ei(3,b*x)/x^2],
[Ei(1,b*x)/x^4,x,1,-1/3*Ei(1,b*x)/x^3+1/3*Ei(4,b*x)/x^3],
[x^2*Ei(2,b*x),x,1,-1/4*x^3*Ei(-2,b*x)+1/4*x^3*Ei(2,b*x)],
[x*Ei(2,b*x),x,1,-1/3*x^2*Ei(-1,b*x)+1/3*x^2*Ei(2,b*x)],
[Ei(2,b*x),x,1,-Ei(3,b*x)/b],
[Ei(2,b*x)/x,x,1,-Ei(1,b*x)+Ei(2,b*x)],
[Ei(2,b*x)/x^2,x,2,-Ei(2,b*x)/x-b^2*x*HypergeometricPFQ([1,1,1],[2,2,2],-b*x)+b*EulerGamma*log(x)+1/2*b*log(b*x)^2],
[Ei(2,b*x)/x^3,x,1,-Ei(2,b*x)/x^2+Ei(3,b*x)/x^2],
[Ei(2,b*x)/x^4,x,1,-1/2*Ei(2,b*x)/x^3+1/2*Ei(4,b*x)/x^3],
[Ei(2,b*x)/x^5,x,1,-1/3*Ei(2,b*x)/x^4+1/3*Ei(5,b*x)/x^4],
[x^2*Ei(3,b*x),x,1,-1/5*x^3*Ei(-2,b*x)+1/5*x^3*Ei(3,b*x)],
[x*Ei(3,b*x),x,1,-1/4*x^2*Ei(-1,b*x)+1/4*x^2*Ei(3,b*x)],
[Ei(3,b*x),x,1,-Ei(4,b*x)/b],
[Ei(3,b*x)/x,x,1,-1/2*Ei(1,b*x)+1/2*Ei(3,b*x)],
[Ei(3,b*x)/x^2,x,1,-Ei(2,b*x)/x+Ei(3,b*x)/x],
[Ei(3,b*x)/x^3,x,3,1/2*b*Ei(2,b*x)/x-1/2*Ei(3,b*x)/x^2+1/2*b^3*x*HypergeometricPFQ([1,1,1],[2,2,2],-b*x)-1/2*b^2*EulerGamma*log(x)-1/4*b^2*log(b*x)^2],
[Ei(3,b*x)/x^4,x,1,-Ei(3,b*x)/x^3+Ei(4,b*x)/x^3],
[Ei(3,b*x)/x^5,x,1,-1/2*Ei(3,b*x)/x^4+1/2*Ei(5,b*x)/x^4],
[Ei(3,b*x)/x^6,x,1,-1/3*Ei(3,b*x)/x^5+1/3*Ei(6,b*x)/x^5],

# n<0
[x^3*Ei(-1,b*x),x,1,-1/2*x^4*Ei(-3,b*x)+1/2*x^4*Ei(-1,b*x)],
[x^2*Ei(-1,b*x),x,1,-x^3*Ei(-2,b*x)+x^3*Ei(-1,b*x)],
[x*Ei(-1,b*x),x,3,(-1)/(b^2*exp(b*x))+Ei(-b*x)/b^2],
[Ei(-1,b*x),x,1,(-1)/(b^2*exp(b*x)*x)],
[Ei(-1,b*x)/x,x,1,-1/2*Ei(-1,b*x)+1/2*Ei(1,b*x)],
[Ei(-1,b*x)/x^2,x,1,-1/3*Ei(-1,b*x)/x+1/3*Ei(2,b*x)/x],
[Ei(-1,b*x)/x^3,x,1,-1/4*Ei(-1,b*x)/x^2+1/4*Ei(3,b*x)/x^2],
[x^4*Ei(-2,b*x),x,1,-1/2*x^5*Ei(-4,b*x)+1/2*x^5*Ei(-2,b*x)],
[x^3*Ei(-2,b*x),x,1,-x^4*Ei(-3,b*x)+x^4*Ei(-2,b*x)],
[x^2*Ei(-2,b*x),x,4,(-2)/(b^3*exp(b*x))+2*Ei(-b*x)/b^3-x^2*Ei(-1,b*x)/b],
[x*Ei(-2,b*x),x,1,-x^2*Ei(-2,b*x)+x^2*Ei(-1,b*x)],
[Ei(-1,b*x),x,1,(-1)/(b^2*exp(b*x)*x)],
[Ei(-2,b*x)/x,x,1,-1/3*Ei(-2,b*x)+1/3*Ei(1,b*x)],
[Ei(-2,b*x)/x^2,x,1,-1/4*Ei(-2,b*x)/x+1/4*Ei(2,b*x)/x],
[Ei(-2,b*x)/x^3,x,1,-1/5*Ei(-2,b*x)/x^2+1/5*Ei(3,b*x)/x^2],
[x^5*Ei(-3,b*x),x,1,-1/2*x^6*Ei(-5,b*x)+1/2*x^6*Ei(-3,b*x)],
[x^4*Ei(-3,b*x),x,1,-x^5*Ei(-4,b*x)+x^5*Ei(-3,b*x)],
[x^3*Ei(-3,b*x),x,5,(-6)/(b^4*exp(b*x))+6*Ei(-b*x)/b^4-x^3*Ei(-2,b*x)/b-3*x^2*Ei(-1,b*x)/b^2],
[x^2*Ei(-3,b*x),x,1,-x^3*Ei(-3,b*x)+x^3*Ei(-2,b*x)],
[x*Ei(-3,b*x),x,1,-1/2*x^2*Ei(-3,b*x)+1/2*x^2*Ei(-1,b*x)],
[Ei(-1,b*x),x,1,(-1)/(b^2*exp(b*x)*x)],
[Ei(-3,b*x)/x,x,1,-1/4*Ei(-3,b*x)+1/4*Ei(1,b*x)],
[Ei(-3,b*x)/x^2,x,1,-1/5*Ei(-3,b*x)/x+1/5*Ei(2,b*x)/x],
[Ei(-3,b*x)/x^3,x,1,-1/6*Ei(-3,b*x)/x^2+1/6*Ei(3,b*x)/x^2],
[x^3*Ei(-3,b*x),x,5,(-6)/(b^4*exp(b*x))+6*Ei(-b*x)/b^4-x^3*Ei(-2,b*x)/b-3*x^2*Ei(-1,b*x)/b^2],
[x^2*Ei(-2,b*x),x,4,(-2)/(b^3*exp(b*x))+2*Ei(-b*x)/b^3-x^2*Ei(-1,b*x)/b],
[x*Ei(-1,b*x),x,3,(-1)/(b^2*exp(b*x))+Ei(-b*x)/b^2],
[Ei(0,b*x),x,2,Ei(-b*x)/b],
[Ei(1,b*x)/x,x,1,b*x*HypergeometricPFQ([1,1,1],[2,2,2],-b*x)-EulerGamma*log(x)-1/2*log(b*x)^2],
[Ei(2,b*x)/x^2,x,2,-Ei(2,b*x)/x-b^2*x*HypergeometricPFQ([1,1,1],[2,2,2],-b*x)+b*EulerGamma*log(x)+1/2*b*log(b*x)^2],
[Ei(3,b*x)/x^3,x,3,1/2*b*Ei(2,b*x)/x-1/2*Ei(3,b*x)/x^2+1/2*b^3*x*HypergeometricPFQ([1,1,1],[2,2,2],-b*x)-1/2*b^2*EulerGamma*log(x)-1/4*b^2*log(b*x)^2],

# Integrands of the form (d x)^(m/2) ExpIntegralE[n/2, b x]
[(d*x)^(3/2)*Ei(-3/2,b*x),x,1,-4/25*(d*x)^(5/2)*HypergeometricPFQ([5/2,5/2],[7/2,7/2],-b*x)/d+3/4*(d*x)^(3/2)*log(x)*sqrt(Pi)/(b*(b*x)^(3/2))],
[(d*x)^(1/2)*Ei(-1/2,b*x),x,1,-4/9*(d*x)^(3/2)*HypergeometricPFQ([3/2,3/2],[5/2,5/2],-b*x)/d+1/2*log(x)*sqrt(Pi)*sqrt(d*x)/(b*sqrt(b*x))],
[Ei(1/2,b*x)/(d*x)^(1/2),x,1,log(x)*sqrt(Pi)*sqrt(b*x)/(b*sqrt(d*x))-4*HypergeometricPFQ([1/2,1/2],[3/2,3/2],-b*x)*sqrt(d*x)/d],
[Ei(3/2,b*x)/(d*x)^(3/2),x,1,-2*(b*x)^(3/2)*log(x)*sqrt(Pi)/(b*(d*x)^(3/2))-4*HypergeometricPFQ([-1/2,-1/2],[1/2,1/2],-b*x)/(d*sqrt(d*x))],
[Ei(5/2,b*x)/(d*x)^(5/2),x,1,-4/9*HypergeometricPFQ([-3/2,-3/2],[-1/2,-1/2],-b*x)/(d*(d*x)^(3/2))+4/3*(b*x)^(5/2)*log(x)*sqrt(Pi)/(b*(d*x)^(5/2))],

# Integrands of the form (d x)^m ExpIntegralE[n, b x] with n symbolic
[x^m*Ei(n,x),x,1,-x^(1+m)*Ei(-m,x)/(m+n)+x^(1+m)*Ei(n,x)/(m+n)],
[x^m*Ei(n,b*x),x,1,-x^(1+m)*Ei(-m,b*x)/(m+n)+x^(1+m)*Ei(n,b*x)/(m+n)],
[(d*x)^m*Ei(n,x),x,1,-(d*x)^(1+m)*Ei(-m,x)/(d*(m+n))+(d*x)^(1+m)*Ei(n,x)/(d*(m+n))],
[(d*x)^m*Ei(n,b*x),x,1,-(d*x)^(1+m)*Ei(-m,b*x)/(d*(m+n))+(d*x)^(1+m)*Ei(n,b*x)/(d*(m+n))],
[Ei(n,x)/x^n,x,1,-x^(1-n)*HypergeometricPFQ([1-n,1-n],[2-n,2-n],-x)/(1-n)^2+GAMMA(1-n)*log(x)],
[Ei(n,b*x)/x^n,x,1,-x^(1-n)*HypergeometricPFQ([1-n,1-n],[2-n,2-n],-b*x)/(1-n)^2+(b*x)^n*GAMMA(1-n)*log(x)/(b*x^n)],
[Ei(n,x)/(d*x)^n,x,1,-(d*x)^(1-n)*HypergeometricPFQ([1-n,1-n],[2-n,2-n],-x)/(d*(1-n)^2)+x^n*GAMMA(1-n)*log(x)/(d*x)^n],
[Ei(n,b*x)/(d*x)^n,x,1,-(d*x)^(1-n)*HypergeometricPFQ([1-n,1-n],[2-n,2-n],-b*x)/(d*(1-n)^2)+(b*x)^n*GAMMA(1-n)*log(x)/(b*(d*x)^n)],
[x^2*Ei(n,b*x),x,1,-x^3*Ei(-2,b*x)/(2+n)+x^3*Ei(n,b*x)/(2+n)],
[x*Ei(n,b*x),x,1,-x^2*Ei(-1,b*x)/(1+n)+x^2*Ei(n,b*x)/(1+n)],
[Ei(n,b*x),x,1,-Ei(1+n,b*x)/b],
[Ei(n,b*x)/x,x,1,Ei(1,b*x)/(1-n)-Ei(n,b*x)/(1-n)],
[Ei(n,b*x)/x^2,x,1,Ei(2,b*x)/((2-n)*x)-Ei(n,b*x)/((2-n)*x)],
[Ei(n,b*x)/x^3,x,1,Ei(3,b*x)/((3-n)*x^2)-Ei(n,b*x)/((3-n)*x^2)],

# Integrands of the form (c+d x)^m ExpIntegralE[n, a+b x]

# Integrands of the form (c+d x)^m ExpIntegralE[n, a+b x]

# n>0
[(c+d*x)^3*Ei(1,a+b*x),x,4,-(c+d*x)^3*Ei(2,a+b*x)/b-3*d*(c+d*x)^2*Ei(3,a+b*x)/b^2-6*d^2*(c+d*x)*Ei(4,a+b*x)/b^3-6*d^3*Ei(5,a+b*x)/b^4],
[(c+d*x)^2*Ei(1,a+b*x),x,3,-(c+d*x)^2*Ei(2,a+b*x)/b-2*d*(c+d*x)*Ei(3,a+b*x)/b^2-2*d^2*Ei(4,a+b*x)/b^3],
[(c+d*x)*Ei(1,a+b*x),x,2,-(c+d*x)*Ei(2,a+b*x)/b-d*Ei(3,a+b*x)/b^2],
[Ei(1,a+b*x),x,1,-Ei(2,a+b*x)/b],
[Ei(1,a+b*x)/(c+d*x),x,0,Unintegrable(Ei(1,a+b*x)/(c+d*x),x)],
[Ei(1,a+b*x)/(c+d*x)^2,x,5,-b*Ei(-a-b*x)/(d*(b*c-a*d))+b*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(d*(b*c-a*d))-Ei(1,a+b*x)/(d*(c+d*x))],
[Ei(1,a+b*x)/(c+d*x)^3,x,7,-1/2*b*exp(-a-b*x)/(d*(b*c-a*d)*(c+d*x))-1/2*b^2*Ei(-a-b*x)/(d*(b*c-a*d)^2)+1/2*b^2*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(d*(b*c-a*d)^2)-1/2*b^2*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(d^2*(b*c-a*d))-1/2*Ei(1,a+b*x)/(d*(c+d*x)^2)],
[Ei(1,a+b*x)/(c+d*x)^4,x,10,-1/6*b*exp(-a-b*x)/(d*(b*c-a*d)*(c+d*x)^2)-1/3*b^2*exp(-a-b*x)/(d*(b*c-a*d)^2*(c+d*x))+1/6*b^2*exp(-a-b*x)/(d^2*(b*c-a*d)*(c+d*x))-1/3*b^3*Ei(-a-b*x)/(d*(b*c-a*d)^3)+1/3*b^3*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(d*(b*c-a*d)^3)-1/3*b^3*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(d^2*(b*c-a*d)^2)+1/6*b^3*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(d^3*(b*c-a*d))-1/3*Ei(1,a+b*x)/(d*(c+d*x)^3)],
[(c+d*x)^3*Ei(2,a+b*x),x,4,-(c+d*x)^3*Ei(3,a+b*x)/b-3*d*(c+d*x)^2*Ei(4,a+b*x)/b^2-6*d^2*(c+d*x)*Ei(5,a+b*x)/b^3-6*d^3*Ei(6,a+b*x)/b^4],
[(c+d*x)^2*Ei(2,a+b*x),x,3,-(c+d*x)^2*Ei(3,a+b*x)/b-2*d*(c+d*x)*Ei(4,a+b*x)/b^2-2*d^2*Ei(5,a+b*x)/b^3],
[(c+d*x)*Ei(2,a+b*x),x,2,-(c+d*x)*Ei(3,a+b*x)/b-d*Ei(4,a+b*x)/b^2],
[Ei(2,a+b*x),x,1,-Ei(3,a+b*x)/b],
[Ei(2,a+b*x)/(c+d*x),x,0,Unintegrable(Ei(2,a+b*x)/(c+d*x),x)],
[Ei(2,a+b*x)/(c+d*x)^2,x,1,-Ei(2,a+b*x)/(d*(c+d*x))-b*Unintegrable(Ei(1,a+b*x)/(c+d*x),x)/d],
[Ei(2,a+b*x)/(c+d*x)^3,x,6,1/2*b^2*Ei(-a-b*x)/(d^2*(b*c-a*d))-1/2*b^2*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(d^2*(b*c-a*d))+1/2*b*Ei(1,a+b*x)/(d^2*(c+d*x))-1/2*Ei(2,a+b*x)/(d*(c+d*x)^2)],
[Ei(2,a+b*x)/(c+d*x)^4,x,8,1/6*b^2*exp(-a-b*x)/(d^2*(b*c-a*d)*(c+d*x))+1/6*b^3*Ei(-a-b*x)/(d^2*(b*c-a*d)^2)-1/6*b^3*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(d^2*(b*c-a*d)^2)+1/6*b^3*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(d^3*(b*c-a*d))+1/6*b*Ei(1,a+b*x)/(d^2*(c+d*x)^2)-1/3*Ei(2,a+b*x)/(d*(c+d*x)^3)],
[(c+d*x)^3*Ei(3,a+b*x),x,4,-(c+d*x)^3*Ei(4,a+b*x)/b-3*d*(c+d*x)^2*Ei(5,a+b*x)/b^2-6*d^2*(c+d*x)*Ei(6,a+b*x)/b^3-6*d^3*Ei(7,a+b*x)/b^4],
[(c+d*x)^2*Ei(3,a+b*x),x,3,-(c+d*x)^2*Ei(4,a+b*x)/b-2*d*(c+d*x)*Ei(5,a+b*x)/b^2-2*d^2*Ei(6,a+b*x)/b^3],
[(c+d*x)*Ei(3,a+b*x),x,2,-(c+d*x)*Ei(4,a+b*x)/b-d*Ei(5,a+b*x)/b^2],
[Ei(3,a+b*x),x,1,-Ei(4,a+b*x)/b],
[Ei(3,a+b*x)/(c+d*x),x,0,Unintegrable(Ei(3,a+b*x)/(c+d*x),x)],
[Ei(3,a+b*x)/(c+d*x)^2,x,1,-Ei(3,a+b*x)/(d*(c+d*x))-b*Unintegrable(Ei(2,a+b*x)/(c+d*x),x)/d],
[Ei(3,a+b*x)/(c+d*x)^3,x,2,1/2*b*Ei(2,a+b*x)/(d^2*(c+d*x))-1/2*Ei(3,a+b*x)/(d*(c+d*x)^2)+1/2*b^2*Unintegrable(Ei(1,a+b*x)/(c+d*x),x)/d^2],
[Ei(3,a+b*x)/(c+d*x)^4,x,7,-1/6*b^3*Ei(-a-b*x)/(d^3*(b*c-a*d))+1/6*b^3*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(d^3*(b*c-a*d))-1/6*b^2*Ei(1,a+b*x)/(d^3*(c+d*x))+1/6*b*Ei(2,a+b*x)/(d^2*(c+d*x)^2)-1/3*Ei(3,a+b*x)/(d*(c+d*x)^3)],

# n<0
[(c+d*x)^4*Ei(-1,a+b*x),x,10,-8*d^4*exp(-a-b*x)/b^5-4*d^3*(b*c-a*d)*exp(-a-b*x)/b^5-4*d^2*(b*c-a*d)^2*exp(-a-b*x)/b^5-8*d^3*exp(-a-b*x)*(c+d*x)/b^4-4*d^2*(b*c-a*d)*exp(-a-b*x)*(c+d*x)/b^4-4*d^2*exp(-a-b*x)*(c+d*x)^2/b^3-exp(-a-b*x)*(c+d*x)^4/(b*(a+b*x))+4*d*(b*c-a*d)^3*Ei(-a-b*x)/b^5],
[(c+d*x)^3*Ei(-1,a+b*x),x,7,-3*d^3*exp(-a-b*x)/b^4-3*d^2*(b*c-a*d)*exp(-a-b*x)/b^4-3*d^2*exp(-a-b*x)*(c+d*x)/b^3-exp(-a-b*x)*(c+d*x)^3/(b*(a+b*x))+3*d*(b*c-a*d)^2*Ei(-a-b*x)/b^4],
[(c+d*x)^2*Ei(-1,a+b*x),x,5,-2*d^2*exp(-a-b*x)/b^3-exp(-a-b*x)*(c+d*x)^2/(b*(a+b*x))+2*d*(b*c-a*d)*Ei(-a-b*x)/b^3],
[(c+d*x)*Ei(-1,a+b*x),x,2,-exp(-a-b*x)*(c+d*x)/(b*(a+b*x))+d*Ei(-a-b*x)/b^2],
[Ei(-1,a+b*x),x,1,-exp(-a-b*x)/(b*(a+b*x))],
[Ei(-1,a+b*x)/(c+d*x),x,7,-d*exp(-a-b*x)/(b*(b*c-a*d)*(c+d*x))-exp(-a-b*x)/(b*(a+b*x)*(c+d*x))-d*Ei(-a-b*x)/(b*c-a*d)^2+d*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(b*c-a*d)^2-exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(b*c-a*d)],
[Ei(-1,a+b*x)/(c+d*x)^2,x,10,-d*exp(-a-b*x)/(b*(b*c-a*d)*(c+d*x)^2)-exp(-a-b*x)/(b*(a+b*x)*(c+d*x)^2)-2*d*exp(-a-b*x)/((b*c-a*d)^2*(c+d*x))+exp(-a-b*x)/((b*c-a*d)*(c+d*x))-2*b*d*Ei(-a-b*x)/(b*c-a*d)^3+2*b*d*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(b*c-a*d)^3-2*b*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(b*c-a*d)^2+b*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(d*(b*c-a*d))],
[Ei(-1,a+b*x)/(c+d*x)^3,x,14,-d*exp(-a-b*x)/(b*(b*c-a*d)*(c+d*x)^3)-exp(-a-b*x)/(b*(a+b*x)*(c+d*x)^3)-3/2*d*exp(-a-b*x)/((b*c-a*d)^2*(c+d*x)^2)+1/2*exp(-a-b*x)/((b*c-a*d)*(c+d*x)^2)-3*b*d*exp(-a-b*x)/((b*c-a*d)^3*(c+d*x))+3/2*b*exp(-a-b*x)/((b*c-a*d)^2*(c+d*x))-1/2*b*exp(-a-b*x)/(d*(b*c-a*d)*(c+d*x))-3*b^2*d*Ei(-a-b*x)/(b*c-a*d)^4+3*b^2*d*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(b*c-a*d)^4-3*b^2*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(b*c-a*d)^3+3/2*b^2*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(d*(b*c-a*d)^2)-1/2*b^2*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(d^2*(b*c-a*d))],
[(c+d*x)^4*Ei(-2,a+b*x),x,8,-12*d^4*exp(-a-b*x)/b^5-12*d^3*(b*c-a*d)*exp(-a-b*x)/b^5-12*d^3*exp(-a-b*x)*(c+d*x)/b^4-4*d*exp(-a-b*x)*(c+d*x)^3/(b^2*(a+b*x))+12*d^2*(b*c-a*d)^2*Ei(-a-b*x)/b^5-(c+d*x)^4*Ei(-1,a+b*x)/b],
[(c+d*x)^3*Ei(-2,a+b*x),x,6,-6*d^3*exp(-a-b*x)/b^4-3*d*exp(-a-b*x)*(c+d*x)^2/(b^2*(a+b*x))+6*d^2*(b*c-a*d)*Ei(-a-b*x)/b^4-(c+d*x)^3*Ei(-1,a+b*x)/b],
[(c+d*x)^2*Ei(-2,a+b*x),x,3,-2*d*exp(-a-b*x)*(c+d*x)/(b^2*(a+b*x))+2*d^2*Ei(-a-b*x)/b^3-(c+d*x)^2*Ei(-1,a+b*x)/b],
[(c+d*x)*Ei(-2,a+b*x),x,2,-d*exp(-a-b*x)/(b^2*(a+b*x))-(c+d*x)*Ei(-1,a+b*x)/b],
[Ei(-2,a+b*x),x,1,-Ei(-1,a+b*x)/b],
[Ei(-2,a+b*x)/(c+d*x),x,11,d^2*exp(-a-b*x)/(b^2*(b*c-a*d)*(c+d*x)^2)+d*exp(-a-b*x)/(b^2*(a+b*x)*(c+d*x)^2)+2*d^2*exp(-a-b*x)/(b*(b*c-a*d)^2*(c+d*x))-d*exp(-a-b*x)/(b*(b*c-a*d)*(c+d*x))+2*d^2*Ei(-a-b*x)/(b*c-a*d)^3-2*d^2*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(b*c-a*d)^3+2*d*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(b*c-a*d)^2-exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(b*c-a*d)-Ei(-1,a+b*x)/(b*(c+d*x))],
[Ei(-2,a+b*x)/(c+d*x)^2,x,15,2*d^2*exp(-a-b*x)/(b^2*(b*c-a*d)*(c+d*x)^3)+2*d*exp(-a-b*x)/(b^2*(a+b*x)*(c+d*x)^3)+3*d^2*exp(-a-b*x)/(b*(b*c-a*d)^2*(c+d*x)^2)-d*exp(-a-b*x)/(b*(b*c-a*d)*(c+d*x)^2)+6*d^2*exp(-a-b*x)/((b*c-a*d)^3*(c+d*x))-3*d*exp(-a-b*x)/((b*c-a*d)^2*(c+d*x))+exp(-a-b*x)/((b*c-a*d)*(c+d*x))+6*b*d^2*Ei(-a-b*x)/(b*c-a*d)^4-6*b*d^2*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(b*c-a*d)^4+6*b*d*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(b*c-a*d)^3-3*b*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(b*c-a*d)^2+b*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(d*(b*c-a*d))-Ei(-1,a+b*x)/(b*(c+d*x)^2)],
[Ei(-2,a+b*x)/(c+d*x)^3,x,20,3*d^2*exp(-a-b*x)/(b^2*(b*c-a*d)*(c+d*x)^4)+3*d*exp(-a-b*x)/(b^2*(a+b*x)*(c+d*x)^4)+4*d^2*exp(-a-b*x)/(b*(b*c-a*d)^2*(c+d*x)^3)-d*exp(-a-b*x)/(b*(b*c-a*d)*(c+d*x)^3)+6*d^2*exp(-a-b*x)/((b*c-a*d)^3*(c+d*x)^2)-2*d*exp(-a-b*x)/((b*c-a*d)^2*(c+d*x)^2)+1/2*exp(-a-b*x)/((b*c-a*d)*(c+d*x)^2)+12*b*d^2*exp(-a-b*x)/((b*c-a*d)^4*(c+d*x))-6*b*d*exp(-a-b*x)/((b*c-a*d)^3*(c+d*x))+2*b*exp(-a-b*x)/((b*c-a*d)^2*(c+d*x))-1/2*b*exp(-a-b*x)/(d*(b*c-a*d)*(c+d*x))+12*b^2*d^2*Ei(-a-b*x)/(b*c-a*d)^5-12*b^2*d^2*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(b*c-a*d)^5+12*b^2*d*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(b*c-a*d)^4-6*b^2*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(b*c-a*d)^3+2*b^2*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(d*(b*c-a*d)^2)-1/2*b^2*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(d^2*(b*c-a*d))-Ei(-1,a+b*x)/(b*(c+d*x)^3)],
[(c+d*x)^4*Ei(-3,a+b*x),x,7,-24*d^4*exp(-a-b*x)/b^5-12*d^2*exp(-a-b*x)*(c+d*x)^2/(b^3*(a+b*x))+24*d^3*(b*c-a*d)*Ei(-a-b*x)/b^5-(c+d*x)^4*Ei(-2,a+b*x)/b-4*d*(c+d*x)^3*Ei(-1,a+b*x)/b^2],
[(c+d*x)^3*Ei(-3,a+b*x),x,4,-6*d^2*exp(-a-b*x)*(c+d*x)/(b^3*(a+b*x))+6*d^3*Ei(-a-b*x)/b^4-(c+d*x)^3*Ei(-2,a+b*x)/b-3*d*(c+d*x)^2*Ei(-1,a+b*x)/b^2],
[(c+d*x)^2*Ei(-3,a+b*x),x,3,-2*d^2*exp(-a-b*x)/(b^3*(a+b*x))-(c+d*x)^2*Ei(-2,a+b*x)/b-2*d*(c+d*x)*Ei(-1,a+b*x)/b^2],
[(c+d*x)*Ei(-3,a+b*x),x,2,-(c+d*x)*Ei(-2,a+b*x)/b-d*Ei(-1,a+b*x)/b^2],
[Ei(-3,a+b*x),x,1,-Ei(-2,a+b*x)/b],
[Ei(-3,a+b*x)/(c+d*x),x,16,-2*d^3*exp(-a-b*x)/(b^3*(b*c-a*d)*(c+d*x)^3)-2*d^2*exp(-a-b*x)/(b^3*(a+b*x)*(c+d*x)^3)-3*d^3*exp(-a-b*x)/(b^2*(b*c-a*d)^2*(c+d*x)^2)+d^2*exp(-a-b*x)/(b^2*(b*c-a*d)*(c+d*x)^2)-6*d^3*exp(-a-b*x)/(b*(b*c-a*d)^3*(c+d*x))+3*d^2*exp(-a-b*x)/(b*(b*c-a*d)^2*(c+d*x))-d*exp(-a-b*x)/(b*(b*c-a*d)*(c+d*x))-6*d^3*Ei(-a-b*x)/(b*c-a*d)^4+6*d^3*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(b*c-a*d)^4-6*d^2*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(b*c-a*d)^3+3*d*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(b*c-a*d)^2-exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(b*c-a*d)-Ei(-2,a+b*x)/(b*(c+d*x))+d*Ei(-1,a+b*x)/(b^2*(c+d*x)^2)],
[Ei(-3,a+b*x)/(c+d*x)^2,x,21,-6*d^3*exp(-a-b*x)/(b^3*(b*c-a*d)*(c+d*x)^4)-6*d^2*exp(-a-b*x)/(b^3*(a+b*x)*(c+d*x)^4)-8*d^3*exp(-a-b*x)/(b^2*(b*c-a*d)^2*(c+d*x)^3)+2*d^2*exp(-a-b*x)/(b^2*(b*c-a*d)*(c+d*x)^3)-12*d^3*exp(-a-b*x)/(b*(b*c-a*d)^3*(c+d*x)^2)+4*d^2*exp(-a-b*x)/(b*(b*c-a*d)^2*(c+d*x)^2)-d*exp(-a-b*x)/(b*(b*c-a*d)*(c+d*x)^2)-24*d^3*exp(-a-b*x)/((b*c-a*d)^4*(c+d*x))+12*d^2*exp(-a-b*x)/((b*c-a*d)^3*(c+d*x))-4*d*exp(-a-b*x)/((b*c-a*d)^2*(c+d*x))+exp(-a-b*x)/((b*c-a*d)*(c+d*x))-24*b*d^3*Ei(-a-b*x)/(b*c-a*d)^5+24*b*d^3*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(b*c-a*d)^5-24*b*d^2*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(b*c-a*d)^4+12*b*d*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(b*c-a*d)^3-4*b*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(b*c-a*d)^2+b*exp(-a+b*c/d)*Ei(-b*(c+d*x)/d)/(d*(b*c-a*d))-Ei(-2,a+b*x)/(b*(c+d*x)^2)+2*d*Ei(-1,a+b*x)/(b^2*(c+d*x)^3)],

# Integrands of the form (c+d x)^m ExpIntegralE[n, a+b x] when m symbolic
[(c+d*x)^m*Ei(3,a+b*x),x,3,b^2*(c+d*x)^(3+m)*Ei(1,a+b*x)/(d^3*(3+m)*(2+3*m+m^2))+b*(c+d*x)^(2+m)*Ei(2,a+b*x)/(d^2*(1+m)*(2+m))+(c+d*x)^(1+m)*Ei(3,a+b*x)/(d*(1+m))+b^3*integrate(exp(-a-b*x)*(c+d*x)^(3+m)/(a+b*x),x)/(d^3*(3+m)*(2+3*m+m^2))],
[(c+d*x)^m*Ei(2,a+b*x),x,2,b^2*CannotIntegrate(exp(-a-b*x)*(c+d*x)^(2+m)/(a+b*x),x)/(d^2*(1+m)*(2+m))+b*(c+d*x)^(2+m)*Ei(1,a+b*x)/(d^2*(1+m)*(2+m))+(c+d*x)^(1+m)*Ei(2,a+b*x)/(d*(1+m))],
[(c+d*x)^m*Ei(1,a+b*x),x,1,b*CannotIntegrate(exp(-a-b*x)*(c+d*x)^(1+m)/(a+b*x),x)/(d*(1+m))+(c+d*x)^(1+m)*Ei(1,a+b*x)/(d*(1+m))],
[(c+d*x)^m*Ei(-1,a+b*x),x,1,-exp(-a-b*x)*(c+d*x)^m/(b*(a+b*x))+d*m*CannotIntegrate(exp(-a-b*x)*(c+d*x)^(-1+m)/(a+b*x),x)/b],
[(c+d*x)^m*Ei(-2,a+b*x),x,2,-d*exp(-a-b*x)*m*(c+d*x)^(-1+m)/(b^2*(a+b*x))-d^2*(1-m)*m*CannotIntegrate(exp(-a-b*x)*(c+d*x)^(-2+m)/(a+b*x),x)/b^2-(c+d*x)^m*Ei(-1,a+b*x)/b],
[(c+d*x)^m*Ei(-3,a+b*x),x,3,d^2*exp(-a-b*x)*(1-m)*m*(c+d*x)^(-2+m)/(b^3*(a+b*x))+d^3*(1-m)*(2-m)*m*CannotIntegrate(exp(-a-b*x)*(c+d*x)^(-3+m)/(a+b*x),x)/b^3-(c+d*x)^m*Ei(-2,a+b*x)/b-d*m*(c+d*x)^(-1+m)*Ei(-1,a+b*x)/b^2],

# Integrands of the form (c+d x)^m ExpIntegralE[n, a+b x] when n symbolic
[(c+d*x)^m*Ei(n,a+b*x),x,0,Unintegrable((c+d*x)^m*Ei(n,a+b*x),x)],
[(c+d*x)^3*Ei(n,a+b*x),x,4,-(c+d*x)^3*Ei(1+n,a+b*x)/b-3*d*(c+d*x)^2*Ei(2+n,a+b*x)/b^2-6*d^2*(c+d*x)*Ei(3+n,a+b*x)/b^3-6*d^3*Ei(4+n,a+b*x)/b^4],
[(c+d*x)^2*Ei(n,a+b*x),x,3,-(c+d*x)^2*Ei(1+n,a+b*x)/b-2*d*(c+d*x)*Ei(2+n,a+b*x)/b^2-2*d^2*Ei(3+n,a+b*x)/b^3],
[(c+d*x)*Ei(n,a+b*x),x,2,-(c+d*x)*Ei(1+n,a+b*x)/b-d*Ei(2+n,a+b*x)/b^2],
[Ei(n,a+b*x),x,1,-Ei(1+n,a+b*x)/b],
[Ei(n,a+b*x)/(c+d*x),x,0,Unintegrable(Ei(n,a+b*x)/(c+d*x),x)],
[Ei(n,a+b*x)/(c+d*x)^2,x,0,Unintegrable(Ei(n,a+b*x)/(c+d*x)^2,x)],

# Integrands of the form (d x)^m ExpIntegralEi[b x]^n

# Integrands of the form x^m ExpIntegralEi[b x]^n

# n>0
[x^3*Ei(b*x),x,6,3/2*exp(b*x)/b^4-3/2*exp(b*x)*x/b^3+3/4*exp(b*x)*x^2/b^2-1/4*exp(b*x)*x^3/b+1/4*x^4*Ei(b*x)],
[x^2*Ei(b*x),x,5,-2/3*exp(b*x)/b^3+2/3*exp(b*x)*x/b^2-1/3*exp(b*x)*x^2/b+1/3*x^3*Ei(b*x)],
[x*Ei(b*x),x,4,1/2*exp(b*x)/b^2-1/2*exp(b*x)*x/b+1/2*x^2*Ei(b*x)],
[Ei(b*x),x,1,-exp(b*x)/b+x*Ei(b*x)],
[Ei(b*x)/x,x,2,b*x*HypergeometricPFQ([1,1,1],[2,2,2],b*x)+EulerGamma*log(x)+(Ei(b*x)+Ei(1,-b*x))*log(x)+1/2*log(-b*x)^2],
[Ei(b*x)/x^2,x,4,-exp(b*x)/x+b*Ei(b*x)-Ei(b*x)/x],
[Ei(b*x)/x^3,x,5,-1/4*exp(b*x)/x^2-1/4*b*exp(b*x)/x+1/4*b^2*Ei(b*x)-1/2*Ei(b*x)/x^2],
[Ei(b*x)/x^4,x,6,-1/9*exp(b*x)/x^3-1/18*b*exp(b*x)/x^2-1/18*b^2*exp(b*x)/x+1/18*b^3*Ei(b*x)-1/3*Ei(b*x)/x^3],
[x^2*Ei(b*x)^2,x,11,-5/6*exp(2*b*x)/b^3+1/3*exp(2*b*x)*x/b^2-4/3*exp(b*x)*Ei(b*x)/b^3+4/3*exp(b*x)*x*Ei(b*x)/b^2-2/3*exp(b*x)*x^2*Ei(b*x)/b+1/3*x^3*Ei(b*x)^2+4/3*Ei(2*b*x)/b^3],
[x*Ei(b*x)^2,x,7,1/2*exp(2*b*x)/b^2+exp(b*x)*Ei(b*x)/b^2-exp(b*x)*x*Ei(b*x)/b+1/2*x^2*Ei(b*x)^2-Ei(2*b*x)/b^2],
[Ei(b*x)^2,x,4,-2*exp(b*x)*Ei(b*x)/b+x*Ei(b*x)^2+2*Ei(2*b*x)/b],
[Ei(b*x)^2/x,x,0,CannotIntegrate(Ei(b*x)^2/x,x)],
[Ei(b*x)^2/x^2,x,0,CannotIntegrate(Ei(b*x)^2/x^2,x)],

# n<0

# Integrands of the form (d x)^m ExpIntegralEi[b x]^n with m symbolic
[(d*x)^m*Ei(b*x)^3,x,0,CannotIntegrate((d*x)^m*Ei(b*x)^3,x)],
[(d*x)^m*Ei(b*x)^2,x,0,CannotIntegrate((d*x)^m*Ei(b*x)^2,x)],
[(d*x)^m*Ei(b*x),x,4,(d*x)^(1+m)*Ei(b*x)/(d*(1+m))-(d*x)^m*GAMMA(1+m,-b*x)/(b*(1+m)*(-b*x)^m)],
[(d*x)^m/Ei(b*x),x,0,CannotIntegrate((d*x)^m/Ei(b*x),x)],
[(d*x)^m/Ei(b*x)^2,x,0,CannotIntegrate((d*x)^m/Ei(b*x)^2,x)],

# Integrands of the form (c+d x)^m ExpIntegralEi[a+b x]^n

# Integrands of the form (c+d x)^m ExpIntegralEi[a+b x]^n

# n>0
[x^3*Ei(a+b*x),x,14,3/2*exp(a+b*x)/b^4+1/2*a*exp(a+b*x)/b^4+1/4*a^2*exp(a+b*x)/b^4+1/4*a^3*exp(a+b*x)/b^4-3/2*exp(a+b*x)*x/b^3-1/2*a*exp(a+b*x)*x/b^3-1/4*a^2*exp(a+b*x)*x/b^3+3/4*exp(a+b*x)*x^2/b^2+1/4*a*exp(a+b*x)*x^2/b^2-1/4*exp(a+b*x)*x^3/b-1/4*a^4*Ei(a+b*x)/b^4+1/4*x^4*Ei(a+b*x)],
[x^2*Ei(a+b*x),x,10,-2/3*exp(a+b*x)/b^3-1/3*a*exp(a+b*x)/b^3-1/3*a^2*exp(a+b*x)/b^3+2/3*exp(a+b*x)*x/b^2+1/3*a*exp(a+b*x)*x/b^2-1/3*exp(a+b*x)*x^2/b+1/3*a^3*Ei(a+b*x)/b^3+1/3*x^3*Ei(a+b*x)],
[x*Ei(a+b*x),x,7,1/2*exp(a+b*x)/b^2+1/2*a*exp(a+b*x)/b^2-1/2*exp(a+b*x)*x/b-1/2*a^2*Ei(a+b*x)/b^2+1/2*x^2*Ei(a+b*x)],
[Ei(a+b*x),x,1,-exp(a+b*x)/b+(a+b*x)*Ei(a+b*x)/b],
[Ei(a+b*x)/x,x,0,Unintegrable(Ei(a+b*x)/x,x)],
[Ei(a+b*x)/x^2,x,5,b*exp(a)*Ei(b*x)/a-b*Ei(a+b*x)/a-Ei(a+b*x)/x],
[Ei(a+b*x)/x^3,x,7,-1/2*b*exp(a+b*x)/(a*x)-1/2*b^2*exp(a)*Ei(b*x)/a^2+1/2*b^2*exp(a)*Ei(b*x)/a+1/2*b^2*Ei(a+b*x)/a^2-1/2*Ei(a+b*x)/x^2],

#  {x^3*ExpIntegralEi[a + b*x]^2, x, 51, (2*E^(2*a + 2*b*x))/b^4 + (3*a*E^(2*a + 2*b*x))/(2*b^4) + (3*a^2*E^(2*a + 2*b*x))/(4*b^4) - (E^(2*a + 2*b*x)*x)/b^3 - (a*E^(2*a + 2*b*x)*x)/(2*b^3) + (E^(2*a + 2*b*x)*x^2)/(4*b^2) + (3*E^(a + b*x)*ExpIntegralEi[a + b*x])/b^4 + (a*E^(a + b*x)*ExpIntegralEi[a + b*x])/b^4 + (a^2*E^(a + b*x)*ExpIntegralEi[a + b*x])/(2*b^4) + (a^3*E^(a + b*x)*ExpIntegralEi[a + b*x])/(2*b^4) - (3*E^(a + b*x)*x*ExpIntegralEi[a + b*x])/b^3 - (a*E^(a + b*x)*x*ExpIntegralEi[a + b*x])/b^3 - (a^2*E^(a + b*x)*x*ExpIntegralEi[a + b*x])/(2*b^3) + (3*E^(a + b*x)*x^2*ExpIntegralEi[a + b*x])/(2*b^2) + (a*E^(a + b*x)*x^2*ExpIntegralEi[a + b*x])/(2*b^2) - (E^(a + b*x)*x^3*ExpIntegralEi[a + b*x])/(2*b) + (a^3*x*ExpIntegralEi[a + b*x]^2)/(4*b^3) + (1/4)*x^4*ExpIntegralEi[a + b*x]^2 - (a^3*(a + b*x)*ExpIntegralEi[a + b*x]^2)/(4*b^4) - (3*ExpIntegralEi[2*a + 2*b*x])/b^4 - (4*a*ExpIntegralEi[2*a + 2*b*x])/b^4 - (3*a^2*ExpIntegralEi[2*a + 2*b*x])/b^4 - (2*a^3*ExpIntegralEi[2*a + 2*b*x])/b^4} 
[x^2*Ei(a+b*x)^2,x,26,-5/6*exp(2*a+2*b*x)/b^3-2/3*a*exp(2*a+2*b*x)/b^3+1/3*exp(2*a+2*b*x)*x/b^2-4/3*exp(a+b*x)*Ei(a+b*x)/b^3-2/3*a*exp(a+b*x)*Ei(a+b*x)/b^3-2/3*a^2*exp(a+b*x)*Ei(a+b*x)/b^3+4/3*exp(a+b*x)*x*Ei(a+b*x)/b^2+2/3*a*exp(a+b*x)*x*Ei(a+b*x)/b^2-2/3*exp(a+b*x)*x^2*Ei(a+b*x)/b-1/3*a^2*x*Ei(a+b*x)^2/b^2+1/3*x^3*Ei(a+b*x)^2+1/3*a^2*(a+b*x)*Ei(a+b*x)^2/b^3+4/3*Ei(2*(a+b*x))/b^3+2*a*Ei(2*(a+b*x))/b^3+2*a^2*Ei(2*(a+b*x))/b^3],
[x*Ei(a+b*x)^2,x,11,1/2*exp(2*a+2*b*x)/b^2+exp(a+b*x)*Ei(a+b*x)/b^2+a*exp(a+b*x)*Ei(a+b*x)/b^2-exp(a+b*x)*x*Ei(a+b*x)/b+1/2*a*x*Ei(a+b*x)^2/b+1/2*x^2*Ei(a+b*x)^2-1/2*a*(a+b*x)*Ei(a+b*x)^2/b^2-Ei(2*(a+b*x))/b^2-2*a*Ei(2*(a+b*x))/b^2],
[Ei(a+b*x)^2,x,3,-2*exp(a+b*x)*Ei(a+b*x)/b+(a+b*x)*Ei(a+b*x)^2/b+2*Ei(2*(a+b*x))/b],
[Ei(a+b*x)^2/x,x,0,CannotIntegrate(Ei(a+b*x)^2/x,x)],
[Ei(a+b*x)^2/x^2,x,0,CannotIntegrate(Ei(a+b*x)^2/x^2,x)],
[x^2*Ei(a+b*x)^3,x,0,CannotIntegrate(x^2*Ei(a+b*x)^3,x)],
[x*Ei(a+b*x)^3,x,0,CannotIntegrate(x*Ei(a+b*x)^3,x)],
[Ei(a+b*x)^3,x,1,CannotIntegrate(Ei(a+b*x)^3,x)],
[Ei(a+b*x)^3/x,x,0,CannotIntegrate(Ei(a+b*x)^3/x,x)],
[Ei(a+b*x)^3/x^2,x,0,CannotIntegrate(Ei(a+b*x)^3/x^2,x)],

# n<0

# Integrands of the form (c+d x)^m ExpIntegralEi[a+b x]^n with m symbolic
[(c+d*x)^m*Ei(a+b*x)^3,x,0,CannotIntegrate((c+d*x)^m*Ei(a+b*x)^3,x)],
[(c+d*x)^m*Ei(a+b*x)^2,x,0,CannotIntegrate((c+d*x)^m*Ei(a+b*x)^2,x)],
[(c+d*x)^m*Ei(a+b*x),x,1,-b*CannotIntegrate(exp(a+b*x)*(c+d*x)^(1+m)/(a+b*x),x)/(d*(1+m))+(c+d*x)^(1+m)*Ei(a+b*x)/(d*(1+m))],
[(c+d*x)^m/Ei(a+b*x),x,0,CannotIntegrate((c+d*x)^m/Ei(a+b*x),x)],
[(c+d*x)^m/Ei(a+b*x)^2,x,0,CannotIntegrate((c+d*x)^m/Ei(a+b*x)^2,x)],

# Integrands of the form (e x)^m ExpIntegralEi[d (a+b Log[c x^n])]
[x^2*Ei(d*(a+b*log(c*x^n))),x,3,1/3*x^3*Ei(d*(a+b*log(c*x^n)))-1/3*x^3*Ei((3+b*d*n)*(a+b*log(c*x^n))/(b*n))/(exp(3*a/(b*n))*(c*x^n)^(3/n))],
[x*Ei(d*(a+b*log(c*x^n))),x,3,1/2*x^2*Ei(d*(a+b*log(c*x^n)))-1/2*x^2*Ei((2+b*d*n)*(a+b*log(c*x^n))/(b*n))/(exp(2*a/(b*n))*(c*x^n)^(2/n))],
[Ei(d*(a+b*log(c*x^n))),x,4,x*Ei(d*(a+b*log(c*x^n)))-x*Ei((1+b*d*n)*(a+b*log(c*x^n))/(b*n))/(exp(a/(b*n))*(c*x^n)^(1/n))],
[Ei(d*(a+b*log(c*x^n)))/x,x,3,-exp(a*d)*(c*x^n)^(b*d)/(b*d*n)+Ei(d*(a+b*log(c*x^n)))*(a+b*log(c*x^n))/(b*n)],
[Ei(d*(a+b*log(c*x^n)))/x^2,x,3,-Ei(d*(a+b*log(c*x^n)))/x+exp(a/(b*n))*(c*x^n)^(1/n)*Ei(-(1-b*d*n)*(a+b*log(c*x^n))/(b*n))/x],
[Ei(d*(a+b*log(c*x^n)))/x^3,x,3,-1/2*Ei(d*(a+b*log(c*x^n)))/x^2+1/2*exp(2*a/(b*n))*(c*x^n)^(2/n)*Ei(-(2-b*d*n)*(a+b*log(c*x^n))/(b*n))/x^2],
[(e*x)^m*Ei(d*(a+b*log(c*x^n))),x,3,(e*x)^(1+m)*Ei(d*(a+b*log(c*x^n)))/(e*(1+m))-(e*x)^(1+m)*Ei((1+m+b*d*n)*(a+b*log(c*x^n))/(b*n))/(e*exp(a*(1+m)/(b*n))*(1+m)*(c*x^n)^((1+m)/n))],

# Integrands of the form x^m E^(a+b x) ExpIntegralEi[c+d x]^n

# Integrands of the form x^m E^(b x) ExpIntegralEi[b x]^n

#  Integrands of x^m*E^(b*x)*ExpIntegralEi[b*x] where m is an integer 
[exp(b*x)*Ei(b*x)/x^3,x,10,-1/4*exp(2*b*x)/x^2-b*exp(2*b*x)/x-1/2*exp(b*x)*Ei(b*x)/x^2-1/2*b*exp(b*x)*Ei(b*x)/x+1/4*b^2*Ei(b*x)^2+2*b^2*Ei(2*b*x)],
[exp(b*x)*Ei(b*x)/x^2,x,5,-exp(2*b*x)/x-exp(b*x)*Ei(b*x)/x+1/2*b*Ei(b*x)^2+2*b*Ei(2*b*x)],
[exp(b*x)*Ei(b*x)/x,x,1,1/2*Ei(b*x)^2],
[exp(b*x)*Ei(b*x),x,3,exp(b*x)*Ei(b*x)/b-Ei(2*b*x)/b],
[exp(b*x)*x*Ei(b*x),x,6,-1/2*exp(2*b*x)/b^2-exp(b*x)*Ei(b*x)/b^2+exp(b*x)*x*Ei(b*x)/b+Ei(2*b*x)/b^2],
[exp(b*x)*x^2*Ei(b*x),x,10,5/4*exp(2*b*x)/b^3-1/2*exp(2*b*x)*x/b^2+2*exp(b*x)*Ei(b*x)/b^3-2*exp(b*x)*x*Ei(b*x)/b^2+exp(b*x)*x^2*Ei(b*x)/b-2*Ei(2*b*x)/b^3],
[exp(b*x)*x^3*Ei(b*x),x,15,-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],

# Integrands of the form x^m E^(a+b x) ExpIntegralEi[c+d x]^n
[exp(a+b*x)*x^3*Ei(c+d*x),x,24,-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*exp(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(a+b*x)*Ei(c+d*x)/b^4+6*exp(a+b*x)*x*Ei(c+d*x)/b^3-3*exp(a+b*x)*x^2*Ei(c+d*x)/b^2+exp(a+b*x)*x^3*Ei(c+d*x)/b+6*exp(a-b*c/d)*Ei((b+d)*(c+d*x)/d)/b^4+c^3*exp(a-b*c/d)*Ei((b+d)*(c+d*x)/d)/(b*d^3)+3*c^2*exp(a-b*c/d)*Ei((b+d)*(c+d*x)/d)/(b^2*d^2)+6*c*exp(a-b*c/d)*Ei((b+d)*(c+d*x)/d)/(b^3*d)],
[exp(a+b*x)*x^2*Ei(c+d*x),x,14,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(a+b*x)*Ei(c+d*x)/b^3-2*exp(a+b*x)*x*Ei(c+d*x)/b^2+exp(a+b*x)*x^2*Ei(c+d*x)/b-2*exp(a-b*c/d)*Ei((b+d)*(c+d*x)/d)/b^3-c^2*exp(a-b*c/d)*Ei((b+d)*(c+d*x)/d)/(b*d^2)-2*c*exp(a-b*c/d)*Ei((b+d)*(c+d*x)/d)/(b^2*d)],
[exp(a+b*x)*x*Ei(c+d*x),x,7,-exp(a+c+(b+d)*x)/(b*(b+d))-exp(a+b*x)*Ei(c+d*x)/b^2+exp(a+b*x)*x*Ei(c+d*x)/b+exp(a-b*c/d)*Ei((b+d)*(c+d*x)/d)/b^2+c*exp(a-b*c/d)*Ei((b+d)*(c+d*x)/d)/(b*d)],
[exp(a+b*x)*Ei(c+d*x),x,2,exp(a+b*x)*Ei(c+d*x)/b-exp(a-b*c/d)*Ei((b+d)*(c+d*x)/d)/b],
[exp(a+b*x)*Ei(c+d*x)/x,x,0,CannotIntegrate(exp(a+b*x)*Ei(c+d*x)/x,x)],
[exp(a+b*x)*Ei(c+d*x)/x^2,x,5,b*CannotIntegrate(exp(a+b*x)*Ei(c+d*x)/x,x)+d*exp(a+c)*Ei((b+d)*x)/c-exp(a+b*x)*Ei(c+d*x)/x-d*exp(a-b*c/d)*Ei((b+d)*(c+d*x)/d)/c],
[exp(a+b*x)*Ei(c+d*x)/x^3,x,12,-1/2*d*exp(a+c+(b+d)*x)/(c*x)+1/2*b^2*CannotIntegrate(exp(a+b*x)*Ei(c+d*x)/x,x)+1/2*b*d*exp(a+c)*Ei((b+d)*x)/c-1/2*d^2*exp(a+c)*Ei((b+d)*x)/c^2+1/2*d*(b+d)*exp(a+c)*Ei((b+d)*x)/c-1/2*exp(a+b*x)*Ei(c+d*x)/x^2-1/2*b*exp(a+b*x)*Ei(c+d*x)/x-1/2*b*d*exp(a-b*c/d)*Ei((b+d)*(c+d*x)/d)/c+1/2*d^2*exp(a-b*c/d)*Ei((b+d)*(c+d*x)/d)/c^2],

# Integrands of the form (d x)^m LogIntegral[b x]^n

# Integrands of the form x^m LogIntegral[b x]^n
[x^2*Li(b*x),x,3,-1/3*Ei(4*log(b*x))/b^3+1/3*x^3*Li(b*x)],
[x*Li(b*x),x,3,-1/2*Ei(3*log(b*x))/b^2+1/2*x^2*Li(b*x)],
[Li(b*x),x,1,-Ei(2*log(b*x))/b+x*Li(b*x)],
[Li(b*x)/x,x,1,-b*x+Li(b*x)*log(b*x)],
[Li(b*x)/x^2,x,3,-Li(b*x)/x+b*log(log(b*x))],
[Li(b*x)/x^3,x,3,1/2*b^2*Ei(-log(b*x))-1/2*Li(b*x)/x^2],

# Integrands of the form (d x)^m LogIntegral[b x]^n with m symbolic
[(d*x)^m*Li(b*x),x,3,-b*(b*x)^(-2-m)*(d*x)^(2+m)*Ei((2+m)*log(b*x))/(d^2*(1+m))+(d*x)^(1+m)*Li(b*x)/(d*(1+m))],

# Integrands of the form (c+d x)^m LogIntegral[a+b x]^n

# Integrands of the form x^m LogIntegral[a+b x]^n
[x^2*Li(a+b*x),x,14,-a^2*Ei(2*log(a+b*x))/b^3+a*Ei(3*log(a+b*x))/b^3-1/3*Ei(4*log(a+b*x))/b^3+1/3*a^3*Li(a+b*x)/b^3+1/3*x^3*Li(a+b*x)],
[x*Li(a+b*x),x,11,a*Ei(2*log(a+b*x))/b^2-1/2*Ei(3*log(a+b*x))/b^2-1/2*a^2*Li(a+b*x)/b^2+1/2*x^2*Li(a+b*x)],
[Li(a+b*x),x,1,-Ei(2*log(a+b*x))/b+(a+b*x)*Li(a+b*x)/b],
[Li(a+b*x)/x,x,0,Unintegrable(Li(a+b*x)/x,x)],
[Li(a+b*x)/x^2,x,1,-Li(a+b*x)/x+b*Unintegrable(1/(x*log(a+b*x)),x)],

# Integrands of the form (d x)^m LogIntegral[a+b x]^n with m symbolic
[(d*x)^m*Li(a+b*x),x,1,(d*x)^(1+m)*Li(a+b*x)/(d*(1+m))-b*Unintegrable((d*x)^(1+m)/log(a+b*x),x)/(d*(1+m))]]:
