# Maple integration test file: "1 Algebraic functions\1.1 Binomial products\1.1.1 Linear\1.1.1.5 P(x) (a+b x)^m (c+d x)^n.txt"

lst:=[

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

# Integrands of the form P[x] (a+b x)^m (c+d x)^(n/2)

# n>0

# n<0
[(a+b*x)^3*(A+B*x+C*x^2+D*x^3)/sqrt(c+d*x),x,2,-2/3*(b*c-a*d)^2*(a*d*(2*c*C*d-B*d^2-3*c^2*D)-b*(5*c^2*C*d-4*B*c*d^2+3*A*d^3-6*c^3*D))*(c+d*x)^(3/2)/d^7-2/5*(b*c-a*d)*(a^2*d^2*(C*d-3*c*D)-a*b*d*(8*c*C*d-3*B*d^2-15*c^2*D)+b^2*(10*c^2*C*d-6*B*c*d^2+3*A*d^3-15*c^3*D))*(c+d*x)^(5/2)/d^7+2/7*(a^3*d^3*D+3*a^2*b*d^2*(C*d-4*c*D)-3*a*b^2*d*(4*c*C*d-B*d^2-10*c^2*D)+b^3*(10*c^2*C*d-4*B*c*d^2+A*d^3-20*c^3*D))*(c+d*x)^(7/2)/d^7+2/9*b*(3*a^2*d^2*D+3*a*b*d*(C*d-5*c*D)-b^2*(5*c*C*d-B*d^2-15*c^2*D))*(c+d*x)^(9/2)/d^7+2/11*b^2*(b*C*d-6*b*c*D+3*a*d*D)*(c+d*x)^(11/2)/d^7+2/13*b^3*D*(c+d*x)^(13/2)/d^7-2*(b*c-a*d)^3*(c^2*C*d-B*c*d^2+A*d^3-c^3*D)*sqrt(c+d*x)/d^7],
[(a+b*x)^2*(A+B*x+C*x^2+D*x^3)/sqrt(c+d*x),x,2,2/3*(b*c-a*d)*(a*d*(2*c*C*d-B*d^2-3*c^2*D)-b*(4*c^2*C*d-3*B*c*d^2+2*A*d^3-5*c^3*D))*(c+d*x)^(3/2)/d^6+2/5*(a^2*d^2*(C*d-3*c*D)-2*a*b*d*(3*c*C*d-B*d^2-6*c^2*D)+b^2*(6*c^2*C*d-3*B*c*d^2+A*d^3-10*c^3*D))*(c+d*x)^(5/2)/d^6+2/7*(a^2*d^2*D+2*a*b*d*(C*d-4*c*D)-b^2*(4*c*C*d-B*d^2-10*c^2*D))*(c+d*x)^(7/2)/d^6+2/9*b*(b*C*d-5*b*c*D+2*a*d*D)*(c+d*x)^(9/2)/d^6+2/11*b^2*D*(c+d*x)^(11/2)/d^6+2*(b*c-a*d)^2*(c^2*C*d-B*c*d^2+A*d^3-c^3*D)*sqrt(c+d*x)/d^6],
[(a+b*x)*(A+B*x+C*x^2+D*x^3)/sqrt(c+d*x),x,2,-2/3*(a*d*(2*c*C*d-B*d^2-3*c^2*D)-b*(3*c^2*C*d-2*B*c*d^2+A*d^3-4*c^3*D))*(c+d*x)^(3/2)/d^5+2/5*(a*d*(C*d-3*c*D)-b*(3*c*C*d-B*d^2-6*c^2*D))*(c+d*x)^(5/2)/d^5+2/7*(b*C*d-4*b*c*D+a*d*D)*(c+d*x)^(7/2)/d^5+2/9*b*D*(c+d*x)^(9/2)/d^5-2*(b*c-a*d)*(c^2*C*d-B*c*d^2+A*d^3-c^3*D)*sqrt(c+d*x)/d^5],
[(A+B*x+C*x^2+D*x^3)/sqrt(c+d*x),x,2,-2/3*(2*c*C*d-B*d^2-3*c^2*D)*(c+d*x)^(3/2)/d^4+2/5*(C*d-3*c*D)*(c+d*x)^(5/2)/d^4+2/7*D*(c+d*x)^(7/2)/d^4+2*(c^2*C*d-B*c*d^2+A*d^3-c^3*D)*sqrt(c+d*x)/d^4],
[(A+B*x+C*x^2+D*x^3)/((a+b*x)*sqrt(c+d*x)),x,4,2/3*(b*C*d-2*b*c*D-a*d*D)*(c+d*x)^(3/2)/(b^2*d^3)+2/5*D*(c+d*x)^(5/2)/(b*d^3)-2*(A*b^3-a*(b^2*B-a*b*C+a^2*D))*arctanh(sqrt(b)*sqrt(c+d*x)/sqrt(b*c-a*d))/(b^(7/2)*sqrt(b*c-a*d))+2*(a^2*d^2*D-a*b*d*(C*d-c*D)-b^2*(c*C*d-B*d^2-c^2*D))*sqrt(c+d*x)/(b^3*d^3)],
[(A+B*x+C*x^2+D*x^3)/((a+b*x)^2*sqrt(c+d*x)),x,5,2/3*D*(c+d*x)^(3/2)/(b^2*d^2)-(b^3*(2*B*c-A*d)-a*b^2*(4*c*C+B*d)-5*a^3*d*D+3*a^2*b*(C*d+2*c*D))*arctanh(sqrt(b)*sqrt(c+d*x)/sqrt(b*c-a*d))/(b^(7/2)*(b*c-a*d)^(3/2))+2*(b*C*d-b*c*D-2*a*d*D)*sqrt(c+d*x)/(b^3*d^2)-(A-a*(b^2*B-a*b*C+a^2*D)/b^3)*sqrt(c+d*x)/((b*c-a*d)*(a+b*x))],
[(A+B*x+C*x^2+D*x^3)/((a+b*x)^3*sqrt(c+d*x)),x,5,-1/4*(b^3*(8*c^2*C-4*B*c*d+3*A*d^2)-15*a^3*d^2*D+3*a^2*b*d*(C*d+12*c*D)-a*b^2*(8*c*C*d-B*d^2+24*c^2*D))*arctanh(sqrt(b)*sqrt(c+d*x)/sqrt(b*c-a*d))/(b^(7/2)*(b*c-a*d)^(5/2))+2*D*sqrt(c+d*x)/(b^3*d)-1/2*(A*b^3-a*(b^2*B-a*b*C+a^2*D))*sqrt(c+d*x)/(b^3*(b*c-a*d)*(a+b*x)^2)-1/4*(b^3*(4*B*c-3*A*d)-a*b^2*(8*c*C+B*d)-9*a^3*d*D+a^2*b*(5*C*d+12*c*D))*sqrt(c+d*x)/(b^3*(b*c-a*d)^2*(a+b*x))],
[(A+B*x+C*x^2+D*x^3)/((a+b*x)^4*sqrt(c+d*x)),x,5,1/8*(5*a^3*d^3*D+a^2*b*d^2*(C*d-18*c*D)-a*b^2*d*(4*c*C*d-B*d^2-24*c^2*D)+b^3*(8*c^2*C*d-6*B*c*d^2+5*A*d^3-16*c^3*D))*arctanh(sqrt(b)*sqrt(c+d*x)/sqrt(b*c-a*d))/(b^(7/2)*(b*c-a*d)^(7/2))-1/3*(A*b^3-a*(b^2*B-a*b*C+a^2*D))*sqrt(c+d*x)/(b^3*(b*c-a*d)*(a+b*x)^3)-1/12*(b^3*(6*B*c-5*A*d)-a*b^2*(12*c*C+B*d)-13*a^3*d*D+a^2*b*(7*C*d+18*c*D))*sqrt(c+d*x)/(b^3*(b*c-a*d)^2*(a+b*x)^2)-1/8*(b^3*(8*c^2*C-6*B*c*d+5*A*d^2)-11*a^3*d^2*D+a^2*b*d*(C*d+30*c*D)-a*b^2*(4*c*C*d-B*d^2+24*c^2*D))*sqrt(c+d*x)/(b^3*(b*c-a*d)^3*(a+b*x))],
[(A+B*x+C*x^2+D*x^3)/((a+b*x)^5*sqrt(c+d*x)),x,6,-1/64*d*(5*a^3*d^3*D+3*a^2*b*d^2*(C*d-8*c*D)-a*b^2*d*(16*c*C*d-5*B*d^2-48*c^2*D)+b^3*(48*c^2*C*d-40*B*c*d^2+35*A*d^3-64*c^3*D))*arctanh(sqrt(b)*sqrt(c+d*x)/sqrt(b*c-a*d))/(b^(7/2)*(b*c-a*d)^(9/2))-1/4*(A*b^3-a*(b^2*B-a*b*C+a^2*D))*sqrt(c+d*x)/(b^3*(b*c-a*d)*(a+b*x)^4)-1/24*(b^3*(8*B*c-7*A*d)-a*b^2*(16*c*C+B*d)-17*a^3*d*D+3*a^2*b*(3*C*d+8*c*D))*sqrt(c+d*x)/(b^3*(b*c-a*d)^2*(a+b*x)^3)-1/96*(b^3*(48*c^2*C-40*B*c*d+35*A*d^2)-59*a^3*d^2*D+3*a^2*b*d*(C*d+56*c*D)-a*b^2*(16*c*C*d-5*B*d^2+144*c^2*D))*sqrt(c+d*x)/(b^3*(b*c-a*d)^3*(a+b*x)^2)+1/64*(5*a^3*d^3*D+3*a^2*b*d^2*(C*d-8*c*D)-a*b^2*d*(16*c*C*d-5*B*d^2-48*c^2*D)+b^3*(48*c^2*C*d-40*B*c*d^2+35*A*d^3-64*c^3*D))*sqrt(c+d*x)/(b^3*(b*c-a*d)^4*(a+b*x))],
[(a+b*x)^3*(A+B*x+C*x^2+D*x^3)/(c+d*x)^(3/2),x,2,-2/3*(b*c-a*d)*(a^2*d^2*(C*d-3*c*D)-a*b*d*(8*c*C*d-3*B*d^2-15*c^2*D)+b^2*(10*c^2*C*d-6*B*c*d^2+3*A*d^3-15*c^3*D))*(c+d*x)^(3/2)/d^7+2/5*(a^3*d^3*D+3*a^2*b*d^2*(C*d-4*c*D)-3*a*b^2*d*(4*c*C*d-B*d^2-10*c^2*D)+b^3*(10*c^2*C*d-4*B*c*d^2+A*d^3-20*c^3*D))*(c+d*x)^(5/2)/d^7+2/7*b*(3*a^2*d^2*D+3*a*b*d*(C*d-5*c*D)-b^2*(5*c*C*d-B*d^2-15*c^2*D))*(c+d*x)^(7/2)/d^7+2/9*b^2*(b*C*d-6*b*c*D+3*a*d*D)*(c+d*x)^(9/2)/d^7+2/11*b^3*D*(c+d*x)^(11/2)/d^7+2*(b*c-a*d)^3*(c^2*C*d-B*c*d^2+A*d^3-c^3*D)/(d^7*sqrt(c+d*x))-2*(b*c-a*d)^2*(a*d*(2*c*C*d-B*d^2-3*c^2*D)-b*(5*c^2*C*d-4*B*c*d^2+3*A*d^3-6*c^3*D))*sqrt(c+d*x)/d^7],
[(a+b*x)^2*(A+B*x+C*x^2+D*x^3)/(c+d*x)^(3/2),x,2,2/3*(a^2*d^2*(C*d-3*c*D)-2*a*b*d*(3*c*C*d-B*d^2-6*c^2*D)+b^2*(6*c^2*C*d-3*B*c*d^2+A*d^3-10*c^3*D))*(c+d*x)^(3/2)/d^6+2/5*(a^2*d^2*D+2*a*b*d*(C*d-4*c*D)-b^2*(4*c*C*d-B*d^2-10*c^2*D))*(c+d*x)^(5/2)/d^6+2/7*b*(b*C*d-5*b*c*D+2*a*d*D)*(c+d*x)^(7/2)/d^6+2/9*b^2*D*(c+d*x)^(9/2)/d^6-2*(b*c-a*d)^2*(c^2*C*d-B*c*d^2+A*d^3-c^3*D)/(d^6*sqrt(c+d*x))+2*(b*c-a*d)*(a*d*(2*c*C*d-B*d^2-3*c^2*D)-b*(4*c^2*C*d-3*B*c*d^2+2*A*d^3-5*c^3*D))*sqrt(c+d*x)/d^6],
[(a+b*x)*(A+B*x+C*x^2+D*x^3)/(c+d*x)^(3/2),x,2,2/3*(a*d*(C*d-3*c*D)-b*(3*c*C*d-B*d^2-6*c^2*D))*(c+d*x)^(3/2)/d^5+2/5*(b*C*d-4*b*c*D+a*d*D)*(c+d*x)^(5/2)/d^5+2/7*b*D*(c+d*x)^(7/2)/d^5+2*(b*c-a*d)*(c^2*C*d-B*c*d^2+A*d^3-c^3*D)/(d^5*sqrt(c+d*x))-2*(a*d*(2*c*C*d-B*d^2-3*c^2*D)-b*(3*c^2*C*d-2*B*c*d^2+A*d^3-4*c^3*D))*sqrt(c+d*x)/d^5],
[(A+B*x+C*x^2+D*x^3)/(c+d*x)^(3/2),x,2,2/3*(C*d-3*c*D)*(c+d*x)^(3/2)/d^4+2/5*D*(c+d*x)^(5/2)/d^4-2*(c^2*C*d-B*c*d^2+A*d^3-c^3*D)/(d^4*sqrt(c+d*x))-2*(2*c*C*d-B*d^2-3*c^2*D)*sqrt(c+d*x)/d^4],
[(A+B*x+C*x^2+D*x^3)/((a+b*x)*(c+d*x)^(3/2)),x,6,2/3*D*(c+d*x)^(3/2)/(b*d^3)-2*(A*b^3-a*(b^2*B-a*b*C+a^2*D))*arctanh(sqrt(b)*sqrt(c+d*x)/sqrt(b*c-a*d))/(b^(5/2)*(b*c-a*d)^(3/2))+2*(c^2*C*d-B*c*d^2+A*d^3-c^3*D)/(d^3*(b*c-a*d)*sqrt(c+d*x))-2*c*D*sqrt(c+d*x)/(b*d^3)+2*(b*C*d-b*c*D-a*d*D)*sqrt(c+d*x)/(b^2*d^3)],
[(A+B*x+C*x^2+D*x^3)/((a+b*x)^2*(c+d*x)^(3/2)),x,5,-(b^3*(2*B*c-3*A*d)-a*b^2*(4*c*C-B*d)-3*a^3*d*D+a^2*b*(C*d+6*c*D))*arctanh(sqrt(b)*sqrt(c+d*x)/sqrt(b*c-a*d))/(b^(5/2)*(b*c-a*d)^(5/2))+(a*b^2*B*d^3-a^2*b*C*d^3+a^3*d^3*D-b^3*(2*c^2*C*d-2*B*c*d^2+3*A*d^3-2*c^3*D))/(b^3*d^2*(b*c-a*d)^2*sqrt(c+d*x))+(-A+a*(b^2*B-a*b*C+a^2*D)/b^3)/((b*c-a*d)*(a+b*x)*sqrt(c+d*x))+2*D*sqrt(c+d*x)/(b^2*d^2)],
[(A+B*x+C*x^2+D*x^3)/((a+b*x)^3*(c+d*x)^(3/2)),x,5,-1/4*(b^3*(8*c^2*C-12*B*c*d+15*A*d^2)-3*a^3*d^2*D-a^2*b*d*(C*d-12*c*D)+a*b^2*(8*c*C*d-3*B*d^2-24*c^2*D))*arctanh(sqrt(b)*sqrt(c+d*x)/sqrt(b*c-a*d))/(b^(5/2)*(b*c-a*d)^(7/2))+1/2*(-a*b^2*B*d^3+a^2*b*C*d^3-a^3*d^3*D+b^3*(4*c^2*C*d-4*B*c*d^2+5*A*d^3-4*c^3*D))/(b^3*d*(b*c-a*d)^3*sqrt(c+d*x))+1/2*(-A*b^3+a*(b^2*B-a*b*C+a^2*D))/(b^3*(b*c-a*d)*(a+b*x)^2*sqrt(c+d*x))-1/4*(b^3*(4*B*c-5*A*d)-a*b^2*(8*c*C-B*d)-7*a^3*d*D+3*a^2*b*(C*d+4*c*D))*sqrt(c+d*x)/(b^2*(b*c-a*d)^3*(a+b*x))],
[(A+B*x+C*x^2+D*x^3)/((a+b*x)^4*(c+d*x)^(3/2)),x,6,-1/8*(a^3*d^3*D+a^2*b*d^2*(C*d-6*c*D)-a*b^2*d*(12*c*C*d-5*B*d^2-24*c^2*D)-b^3*(24*c^2*C*d-30*B*c*d^2+35*A*d^3-16*c^3*D))*arctanh(sqrt(b)*sqrt(c+d*x)/sqrt(b*c-a*d))/(b^(5/2)*(b*c-a*d)^(9/2))+1/3*(a*b^2*B*d^3-a^2*b*C*d^3+a^3*d^3*D-b^3*(6*c^2*C*d-6*B*c*d^2+7*A*d^3-6*c^3*D))/(b^3*(b*c-a*d)^4*sqrt(c+d*x))+1/3*(-A*b^3+a*(b^2*B-a*b*C+a^2*D))/(b^3*(b*c-a*d)*(a+b*x)^3*sqrt(c+d*x))-1/12*(b^3*(6*B*c-7*A*d)-a*b^2*(12*c*C-B*d)-11*a^3*d*D+a^2*b*(5*C*d+18*c*D))*sqrt(c+d*x)/(b^2*(b*c-a*d)^3*(a+b*x)^2)-1/24*(b^3*(24*c^2*C-42*B*c*d+49*A*d^2)+5*a^3*d^2*D-a^2*b*d*(11*C*d-18*c*D)+a*b^2*(36*c*C*d-7*B*d^2-72*c^2*D))*sqrt(c+d*x)/(b^2*(b*c-a*d)^4*(a+b*x))],
[(a+b*x)^3*(A+B*x+C*x^2+D*x^3)/(c+d*x)^(5/2),x,2,2/3*(b*c-a*d)^3*(c^2*C*d-B*c*d^2+A*d^3-c^3*D)/(d^7*(c+d*x)^(3/2))+2/3*(a^3*d^3*D+3*a^2*b*d^2*(C*d-4*c*D)-3*a*b^2*d*(4*c*C*d-B*d^2-10*c^2*D)+b^3*(10*c^2*C*d-4*B*c*d^2+A*d^3-20*c^3*D))*(c+d*x)^(3/2)/d^7+2/5*b*(3*a^2*d^2*D+3*a*b*d*(C*d-5*c*D)-b^2*(5*c*C*d-B*d^2-15*c^2*D))*(c+d*x)^(5/2)/d^7+2/7*b^2*(b*C*d-6*b*c*D+3*a*d*D)*(c+d*x)^(7/2)/d^7+2/9*b^3*D*(c+d*x)^(9/2)/d^7+2*(b*c-a*d)^2*(a*d*(2*c*C*d-B*d^2-3*c^2*D)-b*(5*c^2*C*d-4*B*c*d^2+3*A*d^3-6*c^3*D))/(d^7*sqrt(c+d*x))-2*(b*c-a*d)*(a^2*d^2*(C*d-3*c*D)-a*b*d*(8*c*C*d-3*B*d^2-15*c^2*D)+b^2*(10*c^2*C*d-6*B*c*d^2+3*A*d^3-15*c^3*D))*sqrt(c+d*x)/d^7],
[(a+b*x)^2*(A+B*x+C*x^2+D*x^3)/(c+d*x)^(5/2),x,2,-2/3*(b*c-a*d)^2*(c^2*C*d-B*c*d^2+A*d^3-c^3*D)/(d^6*(c+d*x)^(3/2))+2/3*(a^2*d^2*D+2*a*b*d*(C*d-4*c*D)-b^2*(4*c*C*d-B*d^2-10*c^2*D))*(c+d*x)^(3/2)/d^6+2/5*b*(b*C*d-5*b*c*D+2*a*d*D)*(c+d*x)^(5/2)/d^6+2/7*b^2*D*(c+d*x)^(7/2)/d^6-2*(b*c-a*d)*(a*d*(2*c*C*d-B*d^2-3*c^2*D)-b*(4*c^2*C*d-3*B*c*d^2+2*A*d^3-5*c^3*D))/(d^6*sqrt(c+d*x))+2*(a^2*d^2*(C*d-3*c*D)-2*a*b*d*(3*c*C*d-B*d^2-6*c^2*D)+b^2*(6*c^2*C*d-3*B*c*d^2+A*d^3-10*c^3*D))*sqrt(c+d*x)/d^6],
[(a+b*x)*(A+B*x+C*x^2+D*x^3)/(c+d*x)^(5/2),x,2,2/3*(b*c-a*d)*(c^2*C*d-B*c*d^2+A*d^3-c^3*D)/(d^5*(c+d*x)^(3/2))+2/3*(b*C*d-4*b*c*D+a*d*D)*(c+d*x)^(3/2)/d^5+2/5*b*D*(c+d*x)^(5/2)/d^5+2*(a*d*(2*c*C*d-B*d^2-3*c^2*D)-b*(3*c^2*C*d-2*B*c*d^2+A*d^3-4*c^3*D))/(d^5*sqrt(c+d*x))+2*(a*d*(C*d-3*c*D)-b*(3*c*C*d-B*d^2-6*c^2*D))*sqrt(c+d*x)/d^5],
[(A+B*x+C*x^2+D*x^3)/(c+d*x)^(5/2),x,2,-2/3*(c^2*C*d-B*c*d^2+A*d^3-c^3*D)/(d^4*(c+d*x)^(3/2))+2/3*D*(c+d*x)^(3/2)/d^4+2*(2*c*C*d-B*d^2-3*c^2*D)/(d^4*sqrt(c+d*x))+2*(C*d-3*c*D)*sqrt(c+d*x)/d^4],
[(A+B*x+C*x^2+D*x^3)/((a+b*x)*(c+d*x)^(5/2)),x,4,2/3*(c^2*C*d-B*c*d^2+A*d^3-c^3*D)/(d^3*(b*c-a*d)*(c+d*x)^(3/2))-2*(A*b^3-a*(b^2*B-a*b*C+a^2*D))*arctanh(sqrt(b)*sqrt(c+d*x)/sqrt(b*c-a*d))/(b^(3/2)*(b*c-a*d)^(5/2))+2*(a*d*(2*c*C*d-B*d^2-3*c^2*D)-b*(c^2*C*d-A*d^3-2*c^3*D))/(d^3*(b*c-a*d)^2*sqrt(c+d*x))+2*D*sqrt(c+d*x)/(b*d^3)],
[(A+B*x+C*x^2+D*x^3)/((a+b*x)^2*(c+d*x)^(5/2)),x,5,1/3*(3*a*b^2*B*d^3-3*a^2*b*C*d^3+3*a^3*d^3*D-b^3*(2*c^2*C*d-2*B*c*d^2+5*A*d^3-2*c^3*D))/(b^3*d^2*(b*c-a*d)^2*(c+d*x)^(3/2))+(-A+a*(b^2*B-a*b*C+a^2*D)/b^3)/((b*c-a*d)*(a+b*x)*(c+d*x)^(3/2))-(b^3*(2*B*c-5*A*d)-a*b^2*(4*c*C-3*B*d)-a^3*d*D-a^2*b*(C*d-6*c*D))*arctanh(sqrt(b)*sqrt(c+d*x)/sqrt(b*c-a*d))/(b^(3/2)*(b*c-a*d)^(7/2))+(-a^2*b*C*d^3+a^3*d^3*D-a*b^2*d*(4*c*C*d-3*B*d^2-6*c^2*D)+b^3*(2*B*c*d^2-5*A*d^3-2*c^3*D))/(b^2*d^2*(b*c-a*d)^3*sqrt(c+d*x))],
[(A+B*x+C*x^2+D*x^3)/((a+b*x)^3*(c+d*x)^(5/2)),x,6,1/6*(-3*a*b^2*B*d^3+3*a^2*b*C*d^3-3*a^3*d^3*D+b^3*(4*c^2*C*d-4*B*c*d^2+7*A*d^3-4*c^3*D))/(b^3*d*(b*c-a*d)^3*(c+d*x)^(3/2))+1/2*(-A*b^3+a*(b^2*B-a*b*C+a^2*D))/(b^3*(b*c-a*d)*(a+b*x)^2*(c+d*x)^(3/2))-1/4*(b^3*(8*c^2*C-20*B*c*d+35*A*d^2)+a^3*d^2*D+3*a^2*b*d*(C*d-4*c*D)+3*a*b^2*(8*c*C*d-5*B*d^2-8*c^2*D))*arctanh(sqrt(b)*sqrt(c+d*x)/sqrt(b*c-a*d))/(b^(3/2)*(b*c-a*d)^(9/2))+(a^2*b*C*d^2+b^3*(2*c^2*C-4*B*c*d+7*A*d^2)-a^3*d^2*D+a*b^2*(4*c*C*d-3*B*d^2-6*c^2*D))/(b^2*(b*c-a*d)^4*sqrt(c+d*x))-1/4*(b^3*(4*B*c-7*A*d)-a*b^2*(8*c*C-3*B*d)-5*a^3*d*D+a^2*b*(C*d+12*c*D))*sqrt(c+d*x)/(b*(b*c-a*d)^4*(a+b*x))],

# Integrands of the form P[x] (a+b x)^m (c+d x)^n when n symbolic
[(a+b*x)^3*(c+d*x)^n*(A+B*x+C*x^2+D*x^3),x,2,-(b*c-a*d)^3*(c^2*C*d-B*c*d^2+A*d^3-c^3*D)*(c+d*x)^(1+n)/(d^7*(1+n))-(b*c-a*d)^2*(a*d*(2*c*C*d-B*d^2-3*c^2*D)-b*(5*c^2*C*d-4*B*c*d^2+3*A*d^3-6*c^3*D))*(c+d*x)^(2+n)/(d^7*(2+n))-(b*c-a*d)*(a^2*d^2*(C*d-3*c*D)-a*b*d*(8*c*C*d-3*B*d^2-15*c^2*D)+b^2*(10*c^2*C*d-6*B*c*d^2+3*A*d^3-15*c^3*D))*(c+d*x)^(3+n)/(d^7*(3+n))+(a^3*d^3*D+3*a^2*b*d^2*(C*d-4*c*D)-3*a*b^2*d*(4*c*C*d-B*d^2-10*c^2*D)+b^3*(10*c^2*C*d-4*B*c*d^2+A*d^3-20*c^3*D))*(c+d*x)^(4+n)/(d^7*(4+n))+b*(3*a^2*d^2*D+3*a*b*d*(C*d-5*c*D)-b^2*(5*c*C*d-B*d^2-15*c^2*D))*(c+d*x)^(5+n)/(d^7*(5+n))+b^2*(b*C*d-6*b*c*D+3*a*d*D)*(c+d*x)^(6+n)/(d^7*(6+n))+b^3*D*(c+d*x)^(7+n)/(d^7*(7+n))],
[(a+b*x)^2*(c+d*x)^n*(A+B*x+C*x^2+D*x^3),x,2,(b*c-a*d)^2*(c^2*C*d-B*c*d^2+A*d^3-c^3*D)*(c+d*x)^(1+n)/(d^6*(1+n))+(b*c-a*d)*(a*d*(2*c*C*d-B*d^2-3*c^2*D)-b*(4*c^2*C*d-3*B*c*d^2+2*A*d^3-5*c^3*D))*(c+d*x)^(2+n)/(d^6*(2+n))+(a^2*d^2*(C*d-3*c*D)-2*a*b*d*(3*c*C*d-B*d^2-6*c^2*D)+b^2*(6*c^2*C*d-3*B*c*d^2+A*d^3-10*c^3*D))*(c+d*x)^(3+n)/(d^6*(3+n))+(a^2*d^2*D+2*a*b*d*(C*d-4*c*D)-b^2*(4*c*C*d-B*d^2-10*c^2*D))*(c+d*x)^(4+n)/(d^6*(4+n))+b*(b*C*d-5*b*c*D+2*a*d*D)*(c+d*x)^(5+n)/(d^6*(5+n))+b^2*D*(c+d*x)^(6+n)/(d^6*(6+n))],
[(a+b*x)*(c+d*x)^n*(A+B*x+C*x^2+D*x^3),x,2,-(b*c-a*d)*(c^2*C*d-B*c*d^2+A*d^3-c^3*D)*(c+d*x)^(1+n)/(d^5*(1+n))-(a*d*(2*c*C*d-B*d^2-3*c^2*D)-b*(3*c^2*C*d-2*B*c*d^2+A*d^3-4*c^3*D))*(c+d*x)^(2+n)/(d^5*(2+n))+(a*d*(C*d-3*c*D)-b*(3*c*C*d-B*d^2-6*c^2*D))*(c+d*x)^(3+n)/(d^5*(3+n))+(b*C*d-4*b*c*D+a*d*D)*(c+d*x)^(4+n)/(d^5*(4+n))+b*D*(c+d*x)^(5+n)/(d^5*(5+n))],
[(c+d*x)^n*(A+B*x+C*x^2+D*x^3),x,2,(c^2*C*d-B*c*d^2+A*d^3-c^3*D)*(c+d*x)^(1+n)/(d^4*(1+n))-(2*c*C*d-B*d^2-3*c^2*D)*(c+d*x)^(2+n)/(d^4*(2+n))+(C*d-3*c*D)*(c+d*x)^(3+n)/(d^4*(3+n))+D*(c+d*x)^(4+n)/(d^4*(4+n))],
[(c+d*x)^n*(A+B*x+C*x^2+D*x^3)/(a+b*x),x,3,(a^2*d^2*D-a*b*d*(C*d-c*D)-b^2*(c*C*d-B*d^2-c^2*D))*(c+d*x)^(1+n)/(b^3*d^3*(1+n))+(b*C*d-2*b*c*D-a*d*D)*(c+d*x)^(2+n)/(b^2*d^3*(2+n))+D*(c+d*x)^(3+n)/(b*d^3*(3+n))-(A*b^3-a*(b^2*B-a*b*C+a^2*D))*(c+d*x)^(1+n)*hypergeom([1,1+n],[2+n],b*(c+d*x)/(b*c-a*d))/(b^3*(b*c-a*d)*(1+n))],
[(c+d*x)^n*(A+B*x+C*x^2+D*x^3)/(a+b*x)^2,x,4,(b*C*d-b*c*D-2*a*d*D)*(c+d*x)^(1+n)/(b^3*d^2*(1+n))-(A-a*(b^2*B-a*b*C+a^2*D)/b^3)*(c+d*x)^(1+n)/((b*c-a*d)*(a+b*x))+D*(c+d*x)^(2+n)/(b^2*d^2*(2+n))+(a^3*d*D*(3+n)-b^3*(B*c+A*d*n)+a*b^2*(2*c*C+B*d*(1+n))-a^2*b*(3*c*D+C*d*(2+n)))*(c+d*x)^(1+n)*hypergeom([1,1+n],[2+n],b*(c+d*x)/(b*c-a*d))/(b^3*(b*c-a*d)^2*(1+n))],
[(c+d*x)^n*(A+B*x+C*x^2+D*x^3)/(a+b*x)^3,x,4,D*(c+d*x)^(1+n)/(b^3*d*(1+n))-1/2*(A*b^3-a*(b^2*B-a*b*C+a^2*D))*(c+d*x)^(1+n)/(b^3*(b*c-a*d)*(a+b*x)^2)-1/2*(b^3*(2*B*c-A*d*(1-n))-a^3*d*D*(5+n)-a*b^2*(4*c*C+B*d*(1+n))+a^2*b*(6*c*D+C*d*(3+n)))*(c+d*x)^(1+n)/(b^3*(b*c-a*d)^2*(a+b*x))-1/2*(b^3*(2*c^2*C+2*B*c*d*n-A*d^2*(1-n)*n)-a^3*d^2*D*(6+5*n+n^2)+a^2*b*d*(2+n)*(6*c*D+C*d*(1+n))-a*b^2*(6*c^2*D+4*c*C*d*(1+n)+B*d^2*n*(1+n)))*(c+d*x)^(1+n)*hypergeom([1,1+n],[2+n],b*(c+d*x)/(b*c-a*d))/(b^3*(b*c-a*d)^3*(1+n))],

# Integrands of the form P[x] (a+b x)^m (c+d x)^n when m and n symbolic
[(a+b*x)^m*(A+B*x)*(c+d*x)^n,x,3,B*(a+b*x)^(1+m)*(c+d*x)^(1+n)/(b*d*(2+m+n))+(A*b*d*(2+m+n)-B*(b*c*(1+m)+a*d*(1+n)))*(a+b*x)^(1+m)*(c+d*x)^n*hypergeom([1+m,-n],[2+m],-d*(a+b*x)/(b*c-a*d))/(b^2*d*(1+m)*(2+m+n)*(b*(c+d*x)/(b*c-a*d))^n)],
[(a+b*x)^m*(c+d*x)^n*(A+B*x+C*x^2),x,4,-(a*C*d*(4+m+2*n)+b*(c*C*(2+m)-B*d*(3+m+n)))*(a+b*x)^(1+m)*(c+d*x)^(1+n)/(b^2*d^2*(2+m+n)*(3+m+n))+C*(a+b*x)^(2+m)*(c+d*x)^(1+n)/(b^2*d*(3+m+n))-(d*(2+m+n)*(a*b*c*C*(2+m)+a^2*C*d*(1+n)-A*b^2*d*(3+m+n))-(b*c*(1+m)+a*d*(1+n))*(a*C*d*(4+m+2*n)+b*(c*C*(2+m)-B*d*(3+m+n))))*(a+b*x)^(1+m)*(c+d*x)^n*hypergeom([1+m,-n],[2+m],-d*(a+b*x)/(b*c-a*d))/(b^3*d^2*(1+m)*(2+m+n)*(3+m+n)*(b*(c+d*x)/(b*c-a*d))^n)],
[(a+b*x)^m*(c+d*x)^n*(A+B*x+C*x^2+D*x^3),x,5,(a^2*d^2*D*(m^2+m*(8+3*n)+3*(6+5*n+n^2))+b^2*(c^2*D*(6+5*m+m^2)-c*C*d*(2+m)*(4+m+n)+B*d^2*(12+m^2+7*n+n^2+m*(7+2*n)))+a*b*d*(c*D*(2+m)*(6+m+3*n)-C*d*(m^2+m*(8+3*n)+2*(8+6*n+n^2))))*(a+b*x)^(1+m)*(c+d*x)^(1+n)/(b^3*d^3*(2+m+n)*(3+m+n)*(4+m+n))-(a*d*D*(9+2*m+3*n)+b*(c*D*(3+m)-C*d*(4+m+n)))*(a+b*x)^(2+m)*(c+d*x)^(1+n)/(b^3*d^2*(3+m+n)*(4+m+n))+D*(a+b*x)^(3+m)*(c+d*x)^(1+n)/(b^3*d*(4+m+n))+(d*(2+m+n)*(a^3*d^2*D*(1+n)*(6+m+2*n)+a*b^2*c*(2+m)*(c*D*(3+m)-C*d*(4+m+n))+A*b^3*d^2*(12+m^2+7*n+n^2+m*(7+2*n))-a^2*b*d*(C*d*(1+n)*(4+m+n)-c*D*(2+m)*(6+m+3*n)))-(b*c*(1+m)+a*d*(1+n))*(a^2*d^2*D*(m^2+m*(8+3*n)+3*(6+5*n+n^2))+b^2*(c^2*D*(6+5*m+m^2)-c*C*d*(2+m)*(4+m+n)+B*d^2*(12+m^2+7*n+n^2+m*(7+2*n)))+a*b*d*(c*D*(2+m)*(6+m+3*n)-C*d*(m^2+m*(8+3*n)+2*(8+6*n+n^2)))))*(a+b*x)^(1+m)*(c+d*x)^n*hypergeom([1+m,-n],[2+m],-d*(a+b*x)/(b*c-a*d))/(b^4*d^3*(1+m)*(2+m+n)*(3+m+n)*(4+m+n)*(b*(c+d*x)/(b*c-a*d))^n)]]:

#  {(a + b*x)^m*(c + d*x)^n*(A + B*x + C*x^2 + D*x^3 + F*x^4), x, 6, (1/(b^4*d^4*(2 + m + n)*(3 + m + n)*(4 + m + n)*(5 + m + n)))*((d*(3 + m + n)*(2*a*b^2*c*(3 + m)*(c*F*(4 + m) - d*D*(5 + m + n)) + b^3*B*d^2*(20 + m^2 + 9*n + n^2 + m*(9 + 2*n)) + 2*a^3*d^2*F*(m^2 + m*(11 + 4*n) + 4*(8 + 6*n + n^2)) + a^2*b*d*(4*c*F*(3 + m)*(6 + m + 2*n) - d*D*(m^2 + m*(11 + 4*n) + 3*(10 + 7*n + n^2)))) - b*c*(2 + m)*(a^2*d^2*F*(3*m^2 + m*(29 + 8*n) + 6*(12 + 7*n + n^2)) + b^2*(c*(3 + m)*(c*F*(4 + m) - d*D*(5 + m + n)) + C*d^2*(20 + m^2 + 9*n + n^2 + m*(9 + 2*n))) + a*b*d*(2*c*F*(3 + m)*(6 + m + 2*n) - d*D*(2*m^2 + m*(19 + 5*n) + 3*(15 + 8*n + n^2)))) - a*d*(2 + n)*(a^2*d^2*F*(3*m^2 + m*(29 + 8*n) + 6*(12 + 7*n + n^2)) + b^2*(c*(3 + m)*(c*F*(4 + m) - d*D*(5 + m + n)) + C*d^2*(20 + m^2 + 9*n + n^2 + m*(9 + 2*n))) + a*b*d*(2*c*F*(3 + m)*(6 + m + 2*n) - d*D*(2*m^2 + m*(19 + 5*n) + 3*(15 + 8*n + n^2)))))*(a + b*x)^(1 + m)*(c + d*x)^(1 + n)) + ((a^2*d^2*F*(3*m^2 + m*(29 + 8*n) + 6*(12 + 7*n + n^2)) + b^2*(c*(3 + m)*(c*F*(4 + m) - d*D*(5 + m + n)) + C*d^2*(20 + m^2 + 9*n + n^2 + m*(9 + 2*n))) + a*b*d*(2*c*F*(3 + m)*(6 + m + 2*n) - d*D*(2*m^2 + m*(19 + 5*n) + 3*(15 + 8*n + n^2))))*x*(a + b*x)^(1 + m)*(c + d*x)^(1 + n))/(b^3*d^3*(3 + m + n)*(4 + m + n)*(5 + m + n)) - ((a*d*F*(3*m + 4*(4 + n)) + b*(c*F*(4 + m) - d*D*(5 + m + n)))*(a + b*x)^(3 + m)*(c + d*x)^(1 + n))/(b^4*d^2*(4 + m + n)*(5 + m + n)) + (F*(a + b*x)^(4 + m)*(c + d*x)^(1 + n))/(b^4*d*(5 + m + n)) + (1/(b^5*d^4*(1 + m)*(2 + m + n)*(3 + m + n)*(4 + m + n)*(5 + m + n)))*(((d*(2 + m + n)*(d*(3 + m + n)*(A*b^4*d^2*(20 + m^2 + 9*n + n^2 + m*(9 + 2*n)) + a^2*(a^2*d^2*F*(1 + n)*(2*m + 3*(4 + n)) + b^2*c*(3 + m)*(c*F*(4 + m) - d*D*(5 + m + n)) - a*b*d*(d*D*(1 + n)*(5 + m + n) - 2*c*F*(3 + m)*(6 + m + 2*n)))) - a*b*c*(a^2*d^2*F*(3*m^2 + m*(29 + 8*n) + 6*(12 + 7*n + n^2)) + b^2*(c*(3 + m)*(c*F*(4 + m) - d*D*(5 + m + n)) + C*d^2*(20 + m^2 + 9*n + n^2 + m*(9 + 2*n))) + a*b*d*(2*c*F*(3 + m)*(6 + m + 2*n) - d*D*(2*m^2 + m*(19 + 5*n) + 3*(15 + 8*n + n^2))))) - (b*c*(1 + m) + a*d*(1 + n))*(d*(3 + m + n)*(2*a*b^2*c*(3 + m)*(c*F*(4 + m) - d*D*(5 + m + n)) + b^3*B*d^2*(20 + m^2 + 9*n + n^2 + m*(9 + 2*n)) + 2*a^3*d^2*F*(m^2 + m*(11 + 4*n) + 4*(8 + 6*n + n^2)) + a^2*b*d*(4*c*F*(3 + m)*(6 + m + 2*n) - d*D*(m^2 + m*(11 + 4*n) + 3*(10 + 7*n + n^2)))) - b*c*(2 + m)*(a^2*d^2*F*(3*m^2 + m*(29 + 8*n) + 6*(12 + 7*n + n^2)) + b^2*(c*(3 + m)*(c*F*(4 + m) - d*D*(5 + m + n)) + C*d^2*(20 + m^2 + 9*n + n^2 + m*(9 + 2*n))) + a*b*d*(2*c*F*(3 + m)*(6 + m + 2*n) - d*D*(2*m^2 + m*(19 + 5*n) + 3*(15 + 8*n + n^2)))) - a*d*(2 + n)*(a^2*d^2*F*(3*m^2 + m*(29 + 8*n) + 6*(12 + 7*n + n^2)) + b^2*(c*(3 + m)*(c*F*(4 + m) - d*D*(5 + m + n)) + C*d^2*(20 + m^2 + 9*n + n^2 + m*(9 + 2*n))) + a*b*d*(2*c*F*(3 + m)*(6 + m + 2*n) - d*D*(2*m^2 + m*(19 + 5*n) + 3*(15 + 8*n + n^2))))))*(a + b*x)^(1 + m)*(c + d*x)^n*Hypergeometric2F1[1 + m, -n, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/((b*(c + d*x))/(b*c - a*d))^n)} 
