# Maple integration test file: "1 Algebraic functions\1.2 Trinomial products\1.2.1 Quadratic\1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.txt"

lst:=[

# Integrands of the form (g+h x) (a+b x+c x^2)^p (d+e x+f x^2)^q

# Integrands of the form (g+h x) (a+b x+c x^2)^p (d+e x+f x^2)^q with b^2-4 a c=0

# Integrands of the form (g+h x) (a+b x+c x^2)^p (d+e x+f x^2)^q with e=0

# Integrands of the form (g+h x) (a+b x+c x^2)^p (d+f x^2)^q

# p>0
[(A+B*x)*(a+b*x+c*x^2)/(d+f*x^2),x,5,(b*B+A*c)*x/f+1/2*B*c*x^2/f-1/2*(B*c*d-A*b*f-a*B*f)*log(d+f*x^2)/f^2-(b*B*d+A*c*d-a*A*f)*arctan(x*sqrt(f)/sqrt(d))/(f^(3/2)*sqrt(d))],
[(A+B*x)*(a+b*x+c*x^2)^2/(d+f*x^2),x,5,(A*b^2*f-A*c*(c*d-2*a*f)-b*B*(2*c*d-2*a*f))*x/f^2+1/2*(2*A*b*c*f-B*(c^2*d-b^2*f-2*a*c*f))*x^2/f^2+1/3*c*(2*b*B+A*c)*x^3/f+1/4*B*c^2*x^4/f-1/2*(2*A*b*f*(c*d-a*f)-B*(c^2*d^2-2*a*c*d*f-f*(b^2*d-a^2*f)))*log(d+f*x^2)/f^3-(A*b^2*d*f-2*b*B*d*(c*d-a*f)-A*(c*d-a*f)^2)*arctan(x*sqrt(f)/sqrt(d))/(f^(5/2)*sqrt(d))],
[(A+B*x)*(a+b*x+c*x^2)^3/(d+f*x^2),x,5,-(b^3*B*d*f+3*A*b^2*f*(c*d-a*f)-3*b*B*(c*d-a*f)^2-A*c*(c^2*d^2-3*a*c*d*f+3*a^2*f^2))*x/f^3-1/2*(A*b*f*(3*c^2*d-b^2*f-6*a*c*f)-B*(c^3*d^2-3*a*c^2*d*f+3*a*b^2*f^2-3*c*f*(b^2*d-a^2*f)))*x^2/f^3+1/3*(b^3*B*f+3*A*b^2*c*f-A*c^2*(c*d-3*a*f)-3*b*B*c*(c*d-2*a*f))*x^3/f^2+1/4*c*(3*A*b*c*f-B*(c^2*d-3*b^2*f-3*a*c*f))*x^4/f^2+1/5*c^2*(3*b*B+A*c)*x^5/f+1/6*B*c^3*x^6/f+1/2*(A*b*f*(3*c^2*d^2-6*a*c*d*f-f*(b^2*d-3*a^2*f))-B*(c*d-a*f)*(c^2*d^2-2*a*c*d*f-f*(3*b^2*d-a^2*f)))*log(d+f*x^2)/f^4+(b^3*B*d^2*f+3*A*b^2*d*f*(c*d-a*f)-3*b*B*d*(c*d-a*f)^2-A*(c*d-a*f)^3)*arctan(x*sqrt(f)/sqrt(d))/(f^(7/2)*sqrt(d))],

# p<0
[(A+B*x)/((a+b*x+c*x^2)*(d+f*x^2)),x,8,1/2*(B*c*d+A*b*f-a*B*f)*log(a+b*x+c*x^2)/(c^2*d^2-2*a*c*d*f+f*(b^2*d+a^2*f))-1/2*(B*c*d+A*b*f-a*B*f)*log(d+f*x^2)/(c^2*d^2-2*a*c*d*f+f*(b^2*d+a^2*f))-(A*b^2*f+2*A*c*(c*d-a*f)-b*B*(c*d+a*f))*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/((c^2*d^2-2*a*c*d*f+f*(b^2*d+a^2*f))*sqrt(b^2-4*a*c))+(b*B*d-A*c*d+a*A*f)*arctan(x*sqrt(f)/sqrt(d))*sqrt(f)/((c^2*d^2-2*a*c*d*f+f*(b^2*d+a^2*f))*sqrt(d))],
[(A+B*x)/((a+b*x+c*x^2)^2*(d+f*x^2)),x,9,(A*b*c*(c*d+a*f)-(A*b-a*B)*(2*c^2*d+b^2*f-2*a*c*f)-c*(A*b^2*f+2*A*c*(c*d-a*f)-b*B*(c*d+a*f))*x)/((b^2-4*a*c)*(b^2*d*f+(c*d-a*f)^2)*(a+b*x+c*x^2))-(b^5*B*d*f^2-2*A*b^4*f^2*(c*d-a*f)-4*A*c^2*(c*d-3*a*f)*(c*d-a*f)^2+b^3*B*f*(5*c^2*d^2-4*a*c*d*f-a^2*f^2)-4*A*b^2*c*f*(2*c^2*d^2-3*a*c*d*f+3*a^2*f^2)+2*b*B*c*(c^3*d^3-7*a*c^2*d^2*f+3*a^2*c*d*f^2+3*a^3*f^3))*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/((b^2-4*a*c)^(3/2)*(c^2*d^2-2*a*c*d*f+f*(b^2*d+a^2*f))^2)-1/2*f*(2*A*b*f*(c*d-a*f)+B*(c^2*d^2-2*a*c*d*f-f*(b^2*d-a^2*f)))*log(a+b*x+c*x^2)/(c^2*d^2-2*a*c*d*f+f*(b^2*d+a^2*f))^2+1/2*f*(2*A*b*f*(c*d-a*f)+B*(c^2*d^2-2*a*c*d*f-f*(b^2*d-a^2*f)))*log(d+f*x^2)/(c^2*d^2-2*a*c*d*f+f*(b^2*d+a^2*f))^2-f^(3/2)*(A*b^2*d*f+2*b*B*d*(c*d-a*f)-A*(c*d-a*f)^2)*arctan(x*sqrt(f)/sqrt(d))/((c^2*d^2-2*a*c*d*f+f*(b^2*d+a^2*f))^2*sqrt(d))],

# Integrands of the form (g+h x) (a+b x+c x^2)^(p/2) (d+f x^2)^q

# p>0
[(A+B*x)*(a+b*x+c*x^2)^(1/2)/(d-f*x^2),x,9,-1/2*(b*B+2*A*c)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(f*sqrt(c))-B*sqrt(a+b*x+c*x^2)/f-1/2*arctanh(1/2*(b*sqrt(d)-2*a*sqrt(f)+x*(2*c*sqrt(d)-b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))))*(B*sqrt(d)-A*sqrt(f))*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))/(f^(3/2)*sqrt(d))+1/2*arctanh(1/2*(b*sqrt(d)+2*a*sqrt(f)+x*(2*c*sqrt(d)+b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))))*(B*sqrt(d)+A*sqrt(f))*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))/(f^(3/2)*sqrt(d))],

#  {(A + B*x)*(a + b*x + c*x^2)^(3/2)/(d + f*x^2), x, 10, -(((B*c*d - A*b*f - a*B*f)*Sqrt[a + b*x + c*x^2])/f^2) + ((b*B + 2*A*c)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(8*c*f) + (B*(a + b*x + c*x^2)^(3/2))/(3*f) + ((b*f*(b*B*d + A*c*d - a*A*f) - (c*d - a*f)*(B*c*d - A*b*f - a*B*f) - (Sqrt[f]*(A*b^2*d*f - 2*b*B*d*(c*d - a*f) - A*(c*d - a*f)^2))/Sqrt[-d])*ArcTan[(b*Sqrt[-d] - 2*a*Sqrt[f] + (2*c*Sqrt[-d] - b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[-d]*Sqrt[f] - a*f]*Sqrt[a + b*x + c*x^2])])/(2*f^(5/2)*Sqrt[c*d + b*Sqrt[-d]*Sqrt[f] - a*f]) - ((b*f*(b*B*d + A*c*d - a*A*f) - (c*d - a*f)*(B*c*d - A*b*f - a*B*f) + (Sqrt[f]*(A*b^2*d*f - 2*b*B*d*(c*d - a*f) - A*(c*d - a*f)^2))/Sqrt[-d])*ArcTan[(b*Sqrt[-d] + 2*a*Sqrt[f] + (2*c*Sqrt[-d] + b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[-d]*Sqrt[f] - a*f]*Sqrt[a + b*x + c*x^2])])/(2*f^(5/2)*Sqrt[c*d - b*Sqrt[-d]*Sqrt[f] - a*f]) - ((b^2 - 4*a*c)*(b*B + 2*A*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(3/2)*f) + ((A*b^2*f - 2*A*c*(c*d - a*f) - b*B*(3*c*d - a*f))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[c]*f^2)} 

# p<0
[(A+B*x)/((a+b*x+c*x^2)^(1/2)*(d-f*x^2)),x,5,-1/2*arctanh(1/2*(b*sqrt(d)-2*a*sqrt(f)+x*(2*c*sqrt(d)-b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))))*(B-A*sqrt(f)/sqrt(d))/(sqrt(f)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f)))+1/2*arctanh(1/2*(b*sqrt(d)+2*a*sqrt(f)+x*(2*c*sqrt(d)+b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))))*(B+A*sqrt(f)/sqrt(d))/(sqrt(f)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f)))],
[(A+B*x)/((a+b*x+c*x^2)^(3/2)*(d-f*x^2)),x,6,-1/2*arctanh(1/2*(b*sqrt(d)-2*a*sqrt(f)+x*(2*c*sqrt(d)-b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))))*sqrt(f)*(B*sqrt(d)-A*sqrt(f))/(sqrt(d)*(c*d+a*f-b*sqrt(d)*sqrt(f))^(3/2))+1/2*arctanh(1/2*(b*sqrt(d)+2*a*sqrt(f)+x*(2*c*sqrt(d)+b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))))*sqrt(f)*(B*sqrt(d)+A*sqrt(f))/(sqrt(d)*(c*d+a*f+b*sqrt(d)*sqrt(f))^(3/2))-2*(a*B*(2*c^2*d-b^2*f+2*a*c*f)+A*(b^3*f-b*c*(c*d+3*a*f))+c*(A*b^2*f+b*B*(c*d-a*f)-2*A*c*(c*d+a*f))*x)/((b^2-4*a*c)*(b^2*d*f-(c*d+a*f)^2)*sqrt(a+b*x+c*x^2))],
[(A+B*x)/((a+b*x+c*x^2)^(5/2)*(d-f*x^2)),x,7,-2/3*(a*B*(2*c^2*d-b^2*f+2*a*c*f)+A*(b^3*f-b*c*(c*d+3*a*f))+c*(A*b^2*f+b*B*(c*d-a*f)-2*A*c*(c*d+a*f))*x)/((b^2-4*a*c)*(b^2*d*f-(c*d+a*f)^2)*(a+b*x+c*x^2)^(3/2))-1/2*f^(3/2)*arctanh(1/2*(b*sqrt(d)-2*a*sqrt(f)+x*(2*c*sqrt(d)-b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))))*(B*sqrt(d)-A*sqrt(f))/(sqrt(d)*(c*d+a*f-b*sqrt(d)*sqrt(f))^(5/2))+1/2*f^(3/2)*arctanh(1/2*(b*sqrt(d)+2*a*sqrt(f)+x*(2*c*sqrt(d)+b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))))*(B*sqrt(d)+A*sqrt(f))/(sqrt(d)*(c*d+a*f+b*sqrt(d)*sqrt(f))^(5/2))-2/3*(3*b^6*B*d*f^2+24*a^2*B*c^2*f*(c*d+a*f)^2-A*b^5*f^2*(7*c*d+6*a*f)-b^4*B*f*(7*c^2*d^2+14*a*c*d*f-3*a^2*f^2)+A*b^3*c*f*(15*c^2*d^2+46*a*c*d*f+43*a^2*f^2)+2*b^2*B*c*(2*c^3*d^3+5*a*c^2*d^2*f+4*a^2*c*d*f^2-11*a^3*f^3)-4*A*b*c^2*(2*c^3*d^3+9*a*c^2*d^2*f+24*a^2*c*d*f^2+17*a^3*f^3)+c*(3*b^5*B*d*f^2-2*A*b^4*f^2*(4*c*d+3*a*f)-8*A*c^2*(c*d+a*f)^2*(2*c*d+5*a*f)-b^3*B*f*(17*c^2*d^2+10*a*c*d*f-3*a^2*f^2)+2*A*b^2*c*f*(15*c^2*d^2+22*a*c*d*f+19*a^2*f^2)+4*b*B*c*(2*c^3*d^3+11*a*c^2*d^2*f+4*a^2*c*d*f^2-5*a^3*f^3))*x)/((b^2-4*a*c)^2*(c^2*d^2+2*a*c*d*f-f*(b^2*d-a^2*f))^2*sqrt(a+b*x+c*x^2))],
[(1+2*x)/((-1+x^2)*sqrt(-1+x+x^2)),x,5,-1/2*arctan(1/2*(3+x)/sqrt(-1+x+x^2))+3/2*arctanh(1/2*(1-3*x)/sqrt(-1+x+x^2))],
[(1+2*x)/((1+x^2)*sqrt(-1+x+x^2)),x,5,arctanh((5-2*sqrt(5)+x*sqrt(5))/(sqrt(-1+x+x^2)*sqrt(10*(-2+sqrt(5)))))*sqrt(1/2*(-2+sqrt(5)))-arctan((5+2*sqrt(5)-x*sqrt(5))/(sqrt(-1+x+x^2)*sqrt(10*(2+sqrt(5)))))*sqrt(1/2*(2+sqrt(5)))],
[(a-c+b*x)/((1+x^2)*sqrt(a+b*x+c*x^2)),x,5,-arctan((b*sqrt(a^2+b^2-2*a*c+c^2)-x*(b^2+(a-c)*(a-c+sqrt(a^2+b^2-2*a*c+c^2))))/((a^2+b^2-2*a*c+c^2)^(1/4)*sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(a^2+b^2+c*(c-sqrt(a^2+b^2-2*a*c+c^2))-a*(2*c-sqrt(a^2+b^2-2*a*c+c^2)))))*sqrt(a^2+b^2+c*(c-sqrt(a^2+b^2-2*a*c+c^2))-a*(2*c-sqrt(a^2+b^2-2*a*c+c^2)))/((a^2+b^2-2*a*c+c^2)^(1/4)*sqrt(2))-arctanh((x*(b^2+(a-c)*(a-c-sqrt(a^2+b^2-2*a*c+c^2)))+b*sqrt(a^2+b^2-2*a*c+c^2))/((a^2+b^2-2*a*c+c^2)^(1/4)*sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(a^2+b^2+c*(c+sqrt(a^2+b^2-2*a*c+c^2))-a*(2*c+sqrt(a^2+b^2-2*a*c+c^2)))))*sqrt(a^2+b^2+c*(c+sqrt(a^2+b^2-2*a*c+c^2))-a*(2*c+sqrt(a^2+b^2-2*a*c+c^2)))/((a^2+b^2-2*a*c+c^2)^(1/4)*sqrt(2))],

# Integrands of the form (g+h x) (a+b x+c x^2)^p (d+e x+f x^2)^q

# Integrands of the form (g+h x) (a+b x+c x^2)^p (d+e x+f x^2)^q

# p>0
[(A+B*x)*(a+b*x+c*x^2)/(d+e*x+f*x^2),x,6,-(B*c*e-b*B*f-A*c*f)*x/f^2+1/2*B*c*x^2/f-1/2*(A*f*(c*e-b*f)-B*(c*e^2-c*d*f-b*e*f+a*f^2))*log(d+e*x+f*x^2)/f^3-(A*f*(c*e^2-2*c*d*f-b*e*f+2*a*f^2)+B*(f*(b*e^2-2*b*d*f-a*e*f)-c*(e^3-3*d*e*f)))*arctanh((e+2*f*x)/sqrt(e^2-4*d*f))/(f^3*sqrt(e^2-4*d*f))],
[(A+B*x)*(a+b*x+c*x^2)^2/(d+e*x+f*x^2),x,6,(B*(c*e-b*f)*(f*(b*e-2*a*f)-c*(e^2-2*d*f))+A*f*(b^2*f^2-2*c*f*(b*e-a*f)+c^2*(e^2-d*f)))*x/f^4-1/2*(A*c*f*(c*e-2*b*f)-B*(b^2*f^2-2*c*f*(b*e-a*f)+c^2*(e^2-d*f)))*x^2/f^3-1/3*c*(B*c*e-2*b*B*f-A*c*f)*x^3/f^2+1/4*B*c^2*x^4/f+1/2*(A*f*(c*e-b*f)*(f*(b*e-2*a*f)-c*(e^2-2*d*f))+B*(c^2*(e^4-3*d*e^2*f+d^2*f^2)-f^2*(2*a*b*e*f-a^2*f^2-b^2*(e^2-d*f))+2*c*f*(a*f*(e^2-d*f)-b*(e^3-2*d*e*f))))*log(d+e*x+f*x^2)/f^5-(A*f*(c^2*(e^4-4*d*e^2*f+2*d^2*f^2)-f^2*(2*a*b*e*f-2*a^2*f^2-b^2*(e^2-2*d*f))+2*c*f*(a*f*(e^2-2*d*f)-b*(e^3-3*d*e*f)))-B*(c^2*(e^5-5*d*e^3*f+5*d^2*e*f^2)+f^2*(a^2*e*f^2-2*a*b*f*(e^2-2*d*f)+b^2*(e^3-3*d*e*f))+2*c*f*(a*e*f*(e^2-3*d*f)-b*(e^4-4*d*e^2*f+2*d^2*f^2))))*arctanh((e+2*f*x)/sqrt(e^2-4*d*f))/(f^5*sqrt(e^2-4*d*f))],

# p<0
[(A+B*x)/((a+b*x+c*x^2)*(d+e*x+f*x^2)),x,9,1/2*(B*c*d-A*c*e+A*b*f-a*B*f)*log(a+b*x+c*x^2)/(c^2*d^2+f*(b^2*d-a*b*e+a^2*f)-c*(b*d*e-a*(e^2-2*d*f)))-1/2*(B*c*d-A*c*e+A*b*f-a*B*f)*log(d+e*x+f*x^2)/(c^2*d^2+f*(b^2*d-a*b*e+a^2*f)-c*(b*d*e-a*(e^2-2*d*f)))-(A*b^2*f+2*c*(A*c*d+a*B*e-a*A*f)-b*(B*c*d+A*c*e+a*B*f))*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/((c^2*d^2+f*(b^2*d-a*b*e+a^2*f)-c*(b*d*e-a*(e^2-2*d*f)))*sqrt(b^2-4*a*c))+(B*(c*d*e-2*b*d*f+a*e*f)-A*(c*e^2-2*c*d*f-b*e*f+2*a*f^2))*arctanh((e+2*f*x)/sqrt(e^2-4*d*f))/((c^2*d^2+f*(b^2*d-a*b*e+a^2*f)-c*(b*d*e-a*(e^2-2*d*f)))*sqrt(e^2-4*d*f))],
[(A+B*x)/((a+b*x+c*x^2)^2*(d+e*x+f*x^2)),x,10,(-A*c*(2*a*c*e-b*(c*d+a*f))-(A*b-a*B)*(2*c^2*d+b^2*f-c*(b*e+2*a*f))-c*(A*b^2*f+2*c*(A*c*d+a*B*e-a*A*f)-b*(B*c*d+A*c*e+a*B*f))*x)/((b^2-4*a*c)*((c*d-a*f)^2-(b*d-a*e)*(c*e-b*f))*(a+b*x+c*x^2))-(b^5*(B*d-A*e)*f^2-2*b^4*f*(B*c*d*e-A*(c*e^2-c*d*f+a*f^2))-4*c^2*(A*(c^3*d^3-3*a^3*f^3-a^2*c*f*(e^2-7*d*f)+a*c^2*d*(3*e^2-5*d*f))-a*B*e*(c^2*d^2-3*a^2*f^2-a*c*(e^2-2*d*f)))-4*b^2*c*(B*c^2*d^2*e+A*f*(2*c^2*d^2+3*a^2*f^2+3*a*c*(e^2-d*f)))+2*b*c*(B*(c^3*d^3+3*a^3*f^3+a*c^2*d*(e^2-7*d*f)+3*a^2*c*f*(e^2+d*f))+A*c*e*(3*c^2*d^2+3*a^2*f^2+a*c*(3*e^2+2*d*f)))-b^3*(A*c*e*(c*e^2-2*c*d*f-4*a*f^2)+B*(4*a*c*d*f^2+a^2*f^3-c^2*d*(e^2+5*d*f))))*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/((b^2-4*a*c)^(3/2)*(c^2*d^2+f*(b^2*d-a*b*e+a^2*f)-c*(b*d*e-a*(e^2-2*d*f)))^2)+1/2*(A*(c*e-b*f)*(f*(b*e-2*a*f)-c*(e^2-2*d*f))-B*(2*c*d*f*(b*e-a*f)-f^2*(b^2*d-a^2*f)-c^2*d*(e^2-d*f)))*log(a+b*x+c*x^2)/(c^2*d^2+f*(b^2*d-a*b*e+a^2*f)-c*(b*d*e-a*(e^2-2*d*f)))^2-1/2*(A*(c*e-b*f)*(f*(b*e-2*a*f)-c*(e^2-2*d*f))-B*(2*c*d*f*(b*e-a*f)-f^2*(b^2*d-a^2*f)-c^2*d*(e^2-d*f)))*log(d+e*x+f*x^2)/(c^2*d^2+f*(b^2*d-a*b*e+a^2*f)-c*(b*d*e-a*(e^2-2*d*f)))^2+(B*(c^2*d*e*(e^2-3*d*f)-2*c*d*f*(b*e^2-2*b*d*f-a*e*f)+f^2*(b^2*d*e-4*a*b*d*f+a^2*e*f))-A*(c^2*(e^4-4*d*e^2*f+2*d^2*f^2)-f^2*(2*a*b*e*f-2*a^2*f^2-b^2*(e^2-2*d*f))+2*c*f*(a*f*(e^2-2*d*f)-b*(e^3-3*d*e*f))))*arctanh((e+2*f*x)/sqrt(e^2-4*d*f))/((c^2*d^2+f*(b^2*d-a*b*e+a^2*f)-c*(b*d*e-a*(e^2-2*d*f)))^2*sqrt(e^2-4*d*f))],
[(g+h*x)/((a+b*x+c*x^2)*(a*d+b*d*x+c*d*x^2)^2),x,5,1/2*(-b*g+2*a*h-(2*c*g-b*h)*x)/((b^2-4*a*c)*d^2*(a+b*x+c*x^2)^2)+3/2*(2*c*g-b*h)*(b+2*c*x)/((b^2-4*a*c)^2*d^2*(a+b*x+c*x^2))-6*c*(2*c*g-b*h)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/((b^2-4*a*c)^(5/2)*d^2)],
[(g+h*x)/((a+b*x+c*x^2)^2*(a*d+b*d*x+c*d*x^2)),x,5,1/2*(-b*g+2*a*h-(2*c*g-b*h)*x)/((b^2-4*a*c)*d*(a+b*x+c*x^2)^2)+3/2*(2*c*g-b*h)*(b+2*c*x)/((b^2-4*a*c)^2*d*(a+b*x+c*x^2))-6*c*(2*c*g-b*h)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/((b^2-4*a*c)^(5/2)*d)],

# Integrands of the form (g+h x) (a+b x+c x^2)^p (d+e x+f x^2)^(q/2)

# q>0
[(A+B*x)*(a+b*x+c*x^2)^(1/2)/(d+e*x+f*x^2),x,9,-1/2*(2*B*c*e-b*B*f-2*A*c*f)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(f^2*sqrt(c))+B*sqrt(a+b*x+c*x^2)/f+arctanh(1/2*(4*a*f+2*x*(b*f-c*(e-sqrt(e^2-4*d*f)))-b*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(c*e-b*f)*sqrt(e^2-4*d*f))))*(2*f*(A*f*(c*d-a*f)-B*d*(c*e-b*f))-(A*f*(c*e-b*f)+B*(f*(b*e-a*f)-c*(e^2-d*f)))*(e-sqrt(e^2-4*d*f)))/(f^2*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(c*e-b*f)*sqrt(e^2-4*d*f)))-arctanh(1/2*(4*a*f-b*(e+sqrt(e^2-4*d*f))+2*x*(b*f-c*(e+sqrt(e^2-4*d*f))))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(c*e-b*f)*sqrt(e^2-4*d*f))))*(2*f*(A*f*(c*d-a*f)-B*d*(c*e-b*f))-(A*f*(c*e-b*f)+B*(f*(b*e-a*f)-c*(e^2-d*f)))*(e+sqrt(e^2-4*d*f)))/(f^2*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(c*e-b*f)*sqrt(e^2-4*d*f)))],
[(A+B*x)*(a+b*x+c*x^2)^(3/2)/(d+e*x+f*x^2),x,10,1/3*B*(a+b*x+c*x^2)^(3/2)/f+1/16*(2*A*c*f*(3*b^2*f^2-12*c*f*(b*e-a*f)+8*c^2*(e^2-d*f))-B*(b^3*f^3+6*b*c*f^2*(b*e-2*a*f)-24*c^2*f*(b*e^2-b*d*f-a*e*f)+16*c^3*(e^3-2*d*e*f)))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(3/2)*f^4)-1/8*(2*A*c*f*(4*c*e-5*b*f)-B*(b^2*f^2-2*c*f*(5*b*e-4*a*f)+8*c^2*(e^2-d*f))+2*c*f*(2*B*c*e-b*B*f-2*A*c*f)*x)*sqrt(a+b*x+c*x^2)/(c*f^3)-arctanh(1/2*(4*a*f+2*x*(b*f-c*(e-sqrt(e^2-4*d*f)))-b*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(c*e-b*f)*sqrt(e^2-4*d*f))))*(2*c*f*(B*d*(c*e-b*f)*(c*e^2-2*c*d*f-b*e*f+2*a*f^2)+A*f*(2*c*d*f*(b*e-a*f)-f^2*(b^2*d-a^2*f)-c^2*d*(e^2-d*f)))-c*(A*f*(c*e-b*f)*(f*(b*e-2*a*f)-c*(e^2-2*d*f))+B*(c^2*(e^4-3*d*e^2*f+d^2*f^2)-f^2*(2*a*b*e*f-a^2*f^2-b^2*(e^2-d*f))+2*c*f*(a*f*(e^2-d*f)-b*(e^3-2*d*e*f))))*(e-sqrt(e^2-4*d*f)))/(c*f^4*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(c*e-b*f)*sqrt(e^2-4*d*f)))+arctanh(1/2*(4*a*f-b*(e+sqrt(e^2-4*d*f))+2*x*(b*f-c*(e+sqrt(e^2-4*d*f))))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(c*e-b*f)*sqrt(e^2-4*d*f))))*(2*f*(B*d*(c*e-b*f)*(c*e^2-2*c*d*f-b*e*f+2*a*f^2)+A*f*(2*c*d*f*(b*e-a*f)-f^2*(b^2*d-a^2*f)-c^2*d*(e^2-d*f)))-(A*f*(c*e-b*f)*(f*(b*e-2*a*f)-c*(e^2-2*d*f))+B*(c^2*(e^4-3*d*e^2*f+d^2*f^2)-f^2*(2*a*b*e*f-a^2*f^2-b^2*(e^2-d*f))+2*c*f*(a*f*(e^2-d*f)-b*(e^3-2*d*e*f))))*(e+sqrt(e^2-4*d*f)))/(f^4*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(c*e-b*f)*sqrt(e^2-4*d*f)))],

# q<0
[(A+B*x)/((a+b*x+c*x^2)*sqrt(d+e*x+f*x^2)),x,5,arctanh(1/2*(4*c*d-e*(b+sqrt(b^2-4*a*c))+2*x*(c*e-f*(b+sqrt(b^2-4*a*c))))/(sqrt(2)*sqrt(d+e*x+f*x^2)*sqrt(2*c^2*d-b*c*e+b^2*f-2*a*c*f-(c*e-b*f)*sqrt(b^2-4*a*c))))*(2*A*c-B*(b+sqrt(b^2-4*a*c)))/(sqrt(2)*sqrt(b^2-4*a*c)*sqrt(2*c^2*d-b*c*e+b^2*f-2*a*c*f-(c*e-b*f)*sqrt(b^2-4*a*c)))+arctanh(1/2*(4*c*d+2*x*(c*e-f*(b-sqrt(b^2-4*a*c)))-e*(b-sqrt(b^2-4*a*c)))/(sqrt(2)*sqrt(d+e*x+f*x^2)*sqrt(2*c^2*d-b*c*e+b^2*f-2*a*c*f+(c*e-b*f)*sqrt(b^2-4*a*c))))*(b*B-2*A*c-B*sqrt(b^2-4*a*c))/(sqrt(2)*sqrt(b^2-4*a*c)*sqrt(2*c^2*d-b*c*e+b^2*f-2*a*c*f+(c*e-b*f)*sqrt(b^2-4*a*c)))],
[(A+B*x)/((a+c*x^2)*sqrt(d+e*x+f*x^2)),x,5,-arctanh(sqrt(e)*(a*(A*c*e-B*(c*d-a*f-sqrt(c^2*d^2+a^2*f^2+a*c*(e^2-2*d*f))))-c*x*(a*B*e+A*(c*d-a*f+sqrt(c^2*d^2+a^2*f^2+a*c*(e^2-2*d*f)))))/(sqrt(2)*sqrt(a)*sqrt(c)*sqrt(d+e*x+f*x^2)*sqrt(-A*c*e+B*(c*d-a*f-sqrt(c^2*d^2+a^2*f^2+a*c*(e^2-2*d*f))))*sqrt(a*B*e+A*(c*d-a*f+sqrt(c^2*d^2+a^2*f^2+a*c*(e^2-2*d*f))))))*sqrt(-A*c*e+B*(c*d-a*f-sqrt(c^2*d^2+a^2*f^2+a*c*(e^2-2*d*f))))*sqrt(a*B*e+A*(c*d-a*f+sqrt(c^2*d^2+a^2*f^2+a*c*(e^2-2*d*f))))/(sqrt(2)*sqrt(a)*sqrt(c)*sqrt(e)*sqrt(c^2*d^2+a^2*f^2+a*c*(e^2-2*d*f)))+arctanh(sqrt(e)*(-c*x*(a*B*e+A*(c*d-a*f-sqrt(c^2*d^2+a^2*f^2+a*c*(e^2-2*d*f))))+a*(A*c*e-B*(c*d-a*f+sqrt(c^2*d^2+a^2*f^2+a*c*(e^2-2*d*f)))))/(sqrt(2)*sqrt(a)*sqrt(c)*sqrt(d+e*x+f*x^2)*sqrt(a*B*e+A*(c*d-a*f-sqrt(c^2*d^2+a^2*f^2+a*c*(e^2-2*d*f))))*sqrt(-A*c*e+B*(c*d-a*f+sqrt(c^2*d^2+a^2*f^2+a*c*(e^2-2*d*f))))))*sqrt(a*B*e+A*(c*d-a*f-sqrt(c^2*d^2+a^2*f^2+a*c*(e^2-2*d*f))))*sqrt(-A*c*e+B*(c*d-a*f+sqrt(c^2*d^2+a^2*f^2+a*c*(e^2-2*d*f))))/(sqrt(2)*sqrt(a)*sqrt(c)*sqrt(e)*sqrt(c^2*d^2+a^2*f^2+a*c*(e^2-2*d*f)))],
[(A+B*x)/((a+b*x+c*x^2)*sqrt(d+f*x^2)),x,5,arctanh((2*c*d-f*x*(b-sqrt(b^2-4*a*c)))/(sqrt(2)*sqrt(d+f*x^2)*sqrt(2*c^2*d-2*a*c*f+b*f*(b-sqrt(b^2-4*a*c)))))*(b*B-2*A*c-B*sqrt(b^2-4*a*c))/(sqrt(2)*sqrt(b^2-4*a*c)*sqrt(2*c^2*d-2*a*c*f+b*f*(b-sqrt(b^2-4*a*c))))+arctanh((2*c*d-f*x*(b+sqrt(b^2-4*a*c)))/(sqrt(2)*sqrt(d+f*x^2)*sqrt(2*c^2*d-2*a*c*f+b*f*(b+sqrt(b^2-4*a*c)))))*(2*A*c-B*(b+sqrt(b^2-4*a*c)))/(sqrt(2)*sqrt(b^2-4*a*c)*sqrt(2*c^2*d-2*a*c*f+b*f*(b+sqrt(b^2-4*a*c))))],
[(A+B*x)/((a+c*x^2)*sqrt(d+f*x^2)),x,6,A*arctan(x*sqrt(c*d-a*f)/(sqrt(a)*sqrt(d+f*x^2)))/(sqrt(a)*sqrt(c*d-a*f))-B*arctanh(sqrt(c)*sqrt(d+f*x^2)/sqrt(c*d-a*f))/(sqrt(c)*sqrt(c*d-a*f))],
[(2+x)/((2+4*x-3*x^2)*(1+3*x-2*x^2)^(1/2)),x,5,1/2*arctan(1/2*(3*(4-sqrt(10))+x*(1+4*sqrt(10)))/(sqrt(1+3*x-2*x^2)*sqrt(1+sqrt(10))))*sqrt(-13/5+sqrt(10))+1/2*arctanh(1/2*(x*(1-4*sqrt(10))+3*(4+sqrt(10)))/(sqrt(1+3*x-2*x^2)*sqrt(-1+sqrt(10))))*sqrt(13/5+sqrt(10))],
[(2+x)/((2+4*x-3*x^2)*(1+3*x-2*x^2)^(3/2)),x,7,-2/17*(15+14*x)/sqrt(1+3*x-2*x^2)-9/2*arctan(1/2*(3*(4-sqrt(10))+x*(1+4*sqrt(10)))/(sqrt(1+3*x-2*x^2)*sqrt(1+sqrt(10))))*sqrt(1/5*(-3+sqrt(10)))+9/2*arctanh(1/2*(x*(1-4*sqrt(10))+3*(4+sqrt(10)))/(sqrt(1+3*x-2*x^2)*sqrt(-1+sqrt(10))))*sqrt(1/5*(3+sqrt(10)))],
[(2+x)/((2+4*x-3*x^2)*(1+3*x-2*x^2)^(5/2)),x,7,-2/51*(15+14*x)/(1+3*x-2*x^2)^(3/2)-2/867*(291+4814*x)/sqrt(1+3*x-2*x^2)+9/2*arctan(1/2*(3*(4-sqrt(10))+x*(1+4*sqrt(10)))/(sqrt(1+3*x-2*x^2)*sqrt(1+sqrt(10))))*sqrt(1/5*(-53+17*sqrt(10)))+9/2*arctanh(1/2*(x*(1-4*sqrt(10))+3*(4+sqrt(10)))/(sqrt(1+3*x-2*x^2)*sqrt(-1+sqrt(10))))*sqrt(1/5*(53+17*sqrt(10)))],
[(2+x)/((2+4*x-3*x^2)*(1+3*x+2*x^2)^(1/2)),x,5,1/2*arctanh(1/2*(3*(4+sqrt(10))+x*(17+4*sqrt(10)))/(sqrt(1+3*x+2*x^2)*sqrt(55+17*sqrt(10))))*sqrt(1-7/5*sqrt(2/5))-1/2*arctanh(1/2*(x*(17-4*sqrt(10))+3*(4-sqrt(10)))/(sqrt(1+3*x+2*x^2)*sqrt(55-17*sqrt(10))))*sqrt(1+7/5*sqrt(2/5))],
[(2+x)/((2+4*x-3*x^2)*(1+3*x+2*x^2)^(3/2)),x,6,2/5*(21+22*x)/sqrt(1+3*x+2*x^2)+1/10*arctanh(1/2*(3*(4+sqrt(10))+x*(17+4*sqrt(10)))/(sqrt(1+3*x+2*x^2)*sqrt(55+17*sqrt(10))))*sqrt(3/5*(2065-653*sqrt(10)))-1/10*arctanh(1/2*(x*(17-4*sqrt(10))+3*(4-sqrt(10)))/(sqrt(1+3*x+2*x^2)*sqrt(55-17*sqrt(10))))*sqrt(3/5*(2065+653*sqrt(10)))],
[(2+x)/((2+4*x-3*x^2)*(1+3*x+2*x^2)^(5/2)),x,7,2/15*(21+22*x)/(1+3*x+2*x^2)^(3/2)+2/15*(273+230*x)/sqrt(1+3*x+2*x^2)+1/50*arctanh(1/2*(3*(4+sqrt(10))+x*(17+4*sqrt(10)))/(sqrt(1+3*x+2*x^2)*sqrt(55+17*sqrt(10))))*sqrt(1/3*(4885115-1544809*sqrt(10)))-1/50*arctanh(1/2*(x*(17-4*sqrt(10))+3*(4-sqrt(10)))/(sqrt(1+3*x+2*x^2)*sqrt(55-17*sqrt(10))))*sqrt(1/3*(4885115+1544809*sqrt(10)))],
[(1+x)/((4+2*x+x^2)*sqrt(5+2*x+x^2)),x,2,-arctanh(sqrt(5+2*x+x^2))],
[(4+x)/((4+2*x+x^2)*sqrt(5+2*x+x^2)),x,5,-arctanh(sqrt(5+2*x+x^2))+arctan((1+x)/(sqrt(3)*sqrt(5+2*x+x^2)))*sqrt(3)],
[(1+2*x)/((3+x+x^2)*sqrt(5+x+x^2)),x,2,-arctanh(sqrt(5+x+x^2)/sqrt(2))*sqrt(2)],
[x/((3+x+x^2)*sqrt(5+x+x^2)),x,5,-arctanh(sqrt(5+x+x^2)/sqrt(2))/sqrt(2)-arctan((1+2*x)*sqrt(2/11)/sqrt(5+x+x^2))/sqrt(22)],
[(A+B*x)/((a*e+b*e*x+b*f*x^2)^2*sqrt(d+e*x+f*x^2)),x,6,1/2*(B*e-2*A*f)*(8*a*e*f-b*(e^2+4*d*f))*arctanh((e+2*f*x)*sqrt(b*d-a*e)/(sqrt(e)*sqrt(b*e-4*a*f)*sqrt(d+e*x+f*x^2)))/(e^(3/2)*(b*d-a*e)^(3/2)*f*(b*e-4*a*f)^(3/2))+1/2*B*arctanh(sqrt(b)*sqrt(d+e*x+f*x^2)/sqrt(b*d-a*e))/((b*d-a*e)^(3/2)*f*sqrt(b))-((A*b-2*a*B)*e-b*(B*e-2*A*f)*x)*sqrt(d+e*x+f*x^2)/(e*(b*d-a*e)*(b*e-4*a*f)*(a*e+b*e*x+b*f*x^2))],
[(g+h*x)*sqrt(a+b*x+c*x^2)/(a*d+b*d*x+c*d*x^2)^2,x,2,-2*(b*g-2*a*h+(2*c*g-b*h)*x)/((b^2-4*a*c)*d^2*sqrt(a+b*x+c*x^2))],
[(3+2*x)/((3+4*x+2*x^2)*sqrt(-3-4*x-x^2)),x,2,arctanh(x/sqrt(-3-4*x-x^2))],
[(3+4*x)/((3+4*x+2*x^2)*sqrt(-3-4*x-x^2)),x,13,arctanh(x/sqrt(-3-4*x-x^2))+arctan((1+(-3-x)/sqrt(-3-4*x-x^2))/sqrt(2))*sqrt(2)-arctan((1+(3+x)/sqrt(-3-4*x-x^2))/sqrt(2))*sqrt(2)],

# Integrands of the form (g+h x) (a+b x+c x^2)^(p/2) (d+e x+f x^2)^(q/2)

# q>0

# q<0
[(g+h*x)*sqrt(a+b*x+c*x^2)/(a*d+b*d*x+c*d*x^2)^(3/2),x,5,1/2*h*log(a+b*x+c*x^2)*sqrt(a+b*x+c*x^2)/(c*d*sqrt(a*d+b*d*x+c*d*x^2))-(2*c*g-b*h)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))*sqrt(a+b*x+c*x^2)/(c*d*sqrt(b^2-4*a*c)*sqrt(a*d+b*d*x+c*d*x^2))],

# Integrands of the form (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q

# Integrands of the form x^m (a+b x+c x^2)^p (d+e x+f x^2)^q

# Integrands of the form x^m (a^2+2 a b x+b^2 x^2)^(n/2) (d+f x^2)^(p/2)
[x^2*sqrt(a^2+2*a*b*x+b^2*x^2)*sqrt(c+d*x^2),x,6,1/5*b*x^2*(c+d*x^2)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(d*(a+b*x))-1/60*(8*b*c-15*a*d*x)*(c+d*x^2)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(d^2*(a+b*x))-1/8*a*c^2*arctanh(x*sqrt(d)/sqrt(c+d*x^2))*sqrt(a^2+2*a*b*x+b^2*x^2)/(d^(3/2)*(a+b*x))-1/8*a*c*x*sqrt(a^2+2*a*b*x+b^2*x^2)*sqrt(c+d*x^2)/(d*(a+b*x))],
[x*sqrt(a^2+2*a*b*x+b^2*x^2)*sqrt(c+d*x^2),x,5,1/12*(4*a+3*b*x)*(c+d*x^2)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(d*(a+b*x))-1/8*b*c^2*arctanh(x*sqrt(d)/sqrt(c+d*x^2))*sqrt(a^2+2*a*b*x+b^2*x^2)/(d^(3/2)*(a+b*x))-1/8*b*c*x*sqrt(a^2+2*a*b*x+b^2*x^2)*sqrt(c+d*x^2)/(d*(a+b*x))],
[sqrt(a^2+2*a*b*x+b^2*x^2)*sqrt(c+d*x^2),x,5,1/3*b*(c+d*x^2)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(d*(a+b*x))+1/2*a*c*arctanh(x*sqrt(d)/sqrt(c+d*x^2))*sqrt(a^2+2*a*b*x+b^2*x^2)/((a+b*x)*sqrt(d))+1/2*a*x*sqrt(a^2+2*a*b*x+b^2*x^2)*sqrt(c+d*x^2)/(a+b*x)],
[sqrt(a^2+2*a*b*x+b^2*x^2)*sqrt(c+d*x^2)/x,x,8,-a*arctanh(sqrt(c+d*x^2)/sqrt(c))*sqrt(c)*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+1/2*b*c*arctanh(x*sqrt(d)/sqrt(c+d*x^2))*sqrt(a^2+2*a*b*x+b^2*x^2)/((a+b*x)*sqrt(d))+1/2*(2*a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2)*sqrt(c+d*x^2)/(a+b*x)],
[sqrt(a^2+2*a*b*x+b^2*x^2)*sqrt(c+d*x^2)/x^2,x,8,-b*arctanh(sqrt(c+d*x^2)/sqrt(c))*sqrt(c)*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+a*arctanh(x*sqrt(d)/sqrt(c+d*x^2))*sqrt(d)*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)-(a-b*x)*sqrt(a^2+2*a*b*x+b^2*x^2)*sqrt(c+d*x^2)/(x*(a+b*x))],
[sqrt(a^2+2*a*b*x+b^2*x^2)*sqrt(c+d*x^2)/x^3,x,8,-1/2*a*d*arctanh(sqrt(c+d*x^2)/sqrt(c))*sqrt(a^2+2*a*b*x+b^2*x^2)/((a+b*x)*sqrt(c))+b*arctanh(x*sqrt(d)/sqrt(c+d*x^2))*sqrt(d)*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)-1/2*(a+2*b*x)*sqrt(a^2+2*a*b*x+b^2*x^2)*sqrt(c+d*x^2)/(x^2*(a+b*x))],

# Integrands of the form x^m (a^2+2 a b x+b^2 x^2)^(n/2) (d+e x+f x^2)^(p/2)
[x^2*sqrt(a^2+2*a*b*x+b^2*x^2)*sqrt(c+e*x+d*x^2),x,6,1/5*b*x^2*(c+e*x+d*x^2)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(d*(a+b*x))-1/240*(32*b*c*d+50*a*d*e-35*b*e^2-6*d*(10*a*d-7*b*e)*x)*(c+e*x+d*x^2)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(d^3*(a+b*x))-1/256*(4*c*d-e^2)*(8*a*c*d^2-12*b*c*d*e-10*a*d*e^2+7*b*e^3)*arctanh(1/2*(e+2*d*x)/(sqrt(d)*sqrt(c+e*x+d*x^2)))*sqrt(a^2+2*a*b*x+b^2*x^2)/(d^(9/2)*(a+b*x))-1/128*(2*a*d*(4*c*d-5*e^2)-b*(12*c*d*e-7*e^3))*(e+2*d*x)*sqrt(a^2+2*a*b*x+b^2*x^2)*sqrt(c+e*x+d*x^2)/(d^4*(a+b*x))],
[x*sqrt(a^2+2*a*b*x+b^2*x^2)*sqrt(c+e*x+d*x^2),x,5,1/24*(8*a*d-5*b*e+6*b*d*x)*(c+e*x+d*x^2)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(d^2*(a+b*x))-1/128*(4*c*d-e^2)*(4*b*c*d+8*a*d*e-5*b*e^2)*arctanh(1/2*(e+2*d*x)/(sqrt(d)*sqrt(c+e*x+d*x^2)))*sqrt(a^2+2*a*b*x+b^2*x^2)/(d^(7/2)*(a+b*x))-1/64*(4*b*c*d+8*a*d*e-5*b*e^2)*(e+2*d*x)*sqrt(a^2+2*a*b*x+b^2*x^2)*sqrt(c+e*x+d*x^2)/(d^3*(a+b*x))],
[sqrt(a^2+2*a*b*x+b^2*x^2)*sqrt(c+e*x+d*x^2),x,5,1/3*b*(c+e*x+d*x^2)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(d*(a+b*x))+1/16*(2*a*d-b*e)*(4*c*d-e^2)*arctanh(1/2*(e+2*d*x)/(sqrt(d)*sqrt(c+e*x+d*x^2)))*sqrt(a^2+2*a*b*x+b^2*x^2)/(d^(5/2)*(a+b*x))+1/8*(2*a*d-b*e)*(e+2*d*x)*sqrt(a^2+2*a*b*x+b^2*x^2)*sqrt(c+e*x+d*x^2)/(d^2*(a+b*x))],
[sqrt(a^2+2*a*b*x+b^2*x^2)*sqrt(c+e*x+d*x^2)/x,x,7,1/8*(4*b*c*d+4*a*d*e-b*e^2)*arctanh(1/2*(e+2*d*x)/(sqrt(d)*sqrt(c+e*x+d*x^2)))*sqrt(a^2+2*a*b*x+b^2*x^2)/(d^(3/2)*(a+b*x))-a*arctanh(1/2*(2*c+e*x)/(sqrt(c)*sqrt(c+e*x+d*x^2)))*sqrt(c)*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+1/4*(4*a*d+b*e+2*b*d*x)*sqrt(a^2+2*a*b*x+b^2*x^2)*sqrt(c+e*x+d*x^2)/(d*(a+b*x))],
[sqrt(a^2+2*a*b*x+b^2*x^2)*sqrt(c+e*x+d*x^2)/x^2,x,7,-1/2*(2*b*c+a*e)*arctanh(1/2*(2*c+e*x)/(sqrt(c)*sqrt(c+e*x+d*x^2)))*sqrt(a^2+2*a*b*x+b^2*x^2)/((a+b*x)*sqrt(c))+1/2*(2*a*d+b*e)*arctanh(1/2*(e+2*d*x)/(sqrt(d)*sqrt(c+e*x+d*x^2)))*sqrt(a^2+2*a*b*x+b^2*x^2)/((a+b*x)*sqrt(d))-(a-b*x)*sqrt(a^2+2*a*b*x+b^2*x^2)*sqrt(c+e*x+d*x^2)/(x*(a+b*x))],
[sqrt(a^2+2*a*b*x+b^2*x^2)*sqrt(c+e*x+d*x^2)/x^3,x,7,-1/8*(4*a*c*d+4*b*c*e-a*e^2)*arctanh(1/2*(2*c+e*x)/(sqrt(c)*sqrt(c+e*x+d*x^2)))*sqrt(a^2+2*a*b*x+b^2*x^2)/(c^(3/2)*(a+b*x))+b*arctanh(1/2*(e+2*d*x)/(sqrt(d)*sqrt(c+e*x+d*x^2)))*sqrt(d)*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)-1/4*(2*a*c+(4*b*c+a*e)*x)*sqrt(a^2+2*a*b*x+b^2*x^2)*sqrt(c+e*x+d*x^2)/(c*x^2*(a+b*x))],

# Integrands of the form x^m (a+b x+c x^2)^(p/2) / (d+e x+f x^2) when b=0

# p>0
[x^2*sqrt(a+c*x^2)/(d+e*x+f*x^2),x,9,1/2*(a*f^2+2*c*(e^2-d*f))*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/(f^3*sqrt(c))-1/2*(2*e-f*x)*sqrt(a+c*x^2)/f^2-arctanh((2*a*f-c*x*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f)))))*(-2*d*f*(a*f^2+c*(e^2-d*f))+e*(a*f^2+c*(e^2-2*d*f))*(e-sqrt(e^2-4*d*f)))/(f^3*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f))))+arctanh((2*a*f-c*x*(e+sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f)))))*(-2*d*f*(a*f^2+c*(e^2-d*f))+e*(a*f^2+c*(e^2-2*d*f))*(e+sqrt(e^2-4*d*f)))/(f^3*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f))))],
[x*sqrt(a+c*x^2)/(d+e*x+f*x^2),x,9,-e*arctanh(x*sqrt(c)/sqrt(a+c*x^2))*sqrt(c)/f^2+sqrt(a+c*x^2)/f-arctanh((2*a*f-c*x*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f)))))*(2*c*d*e*f-(a*f^2+c*(e^2-d*f))*(e-sqrt(e^2-4*d*f)))/(f^2*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f))))+arctanh((2*a*f-c*x*(e+sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f)))))*(2*c*d*e*f-(a*f^2+c*(e^2-d*f))*(e+sqrt(e^2-4*d*f)))/(f^2*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f))))],
[sqrt(a+c*x^2)/(d+e*x+f*x^2),x,8,arctanh(x*sqrt(c)/sqrt(a+c*x^2))*sqrt(c)/f-arctanh((2*a*f-c*x*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f)))))*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f)))/(f*sqrt(2)*sqrt(e^2-4*d*f))+arctanh((2*a*f-c*x*(e+sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f)))))*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f)))/(f*sqrt(2)*sqrt(e^2-4*d*f))],
[sqrt(a+c*x^2)/(x*(d+e*x+f*x^2)),x,12,-arctanh(sqrt(a+c*x^2)/sqrt(a))*sqrt(a)/d+arctanh((2*a*f-c*x*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f)))))*(2*a*e*f+(c*d-a*f)*(e-sqrt(e^2-4*d*f)))/(d*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f))))-arctanh((2*a*f-c*x*(e+sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f)))))*(2*a*e*f+(c*d-a*f)*(e+sqrt(e^2-4*d*f)))/(d*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f))))],
[sqrt(a+c*x^2)/(x^2*(d+e*x+f*x^2)),x,18,e*arctanh(sqrt(a+c*x^2)/sqrt(a))*sqrt(a)/d^2-sqrt(a+c*x^2)/(d*x)-f*arctanh((2*a*f-c*x*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f)))))*(2*c*d^2+a*(e^2-2*d*f+e*sqrt(e^2-4*d*f)))/(d^2*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f))))+f*arctanh((2*a*f-c*x*(e+sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f)))))*(2*c*d^2+a*(e^2-2*d*f-e*sqrt(e^2-4*d*f)))/(d^2*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f))))],
[sqrt(a+c*x^2)/(x^3*(d+e*x+f*x^2)),x,22,-1/2*c*arctanh(sqrt(a+c*x^2)/sqrt(a))/(d*sqrt(a))-(e^2-d*f)*arctanh(sqrt(a+c*x^2)/sqrt(a))*sqrt(a)/d^3-1/2*sqrt(a+c*x^2)/(d*x^2)+e*sqrt(a+c*x^2)/(d^2*x)+f*arctanh((2*a*f-c*x*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f)))))*(c*d^2*(e+sqrt(e^2-4*d*f))+a*(e^3-3*d*e*f+e^2*sqrt(e^2-4*d*f)-d*f*sqrt(e^2-4*d*f)))/(d^3*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f))))-f*arctanh((2*a*f-c*x*(e+sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f)))))*(c*d^2*(e-sqrt(e^2-4*d*f))+a*(e^3-3*d*e*f-e^2*sqrt(e^2-4*d*f)+d*f*sqrt(e^2-4*d*f)))/(d^3*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f))))],
[x^2*(a+c*x^2)^(3/2)/(d+e*x+f*x^2),x,10,-1/12*(4*e-3*f*x)*(a+c*x^2)^(3/2)/f^2+1/8*(3*a^2*f^4+12*a*c*f^2*(e^2-d*f)+8*c^2*(e^4-3*d*e^2*f+d^2*f^2))*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/(f^5*sqrt(c))-1/8*(8*e*(a*f^2+c*(e^2-2*d*f))-f*(3*a*f^2+4*c*(e^2-d*f))*x)*sqrt(a+c*x^2)/f^4-arctanh((2*a*f-c*x*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f)))))*(a^2*f^4*(e^2-2*d*f-e*sqrt(e^2-4*d*f))+2*a*c*f^2*(e^4-4*d*e^2*f+2*d^2*f^2-e^3*sqrt(e^2-4*d*f)+2*d*e*f*sqrt(e^2-4*d*f))+c^2*(e^6-6*d*e^4*f+9*d^2*e^2*f^2-2*d^3*f^3-e^5*sqrt(e^2-4*d*f)+4*d*e^3*f*sqrt(e^2-4*d*f)-3*d^2*e*f^2*sqrt(e^2-4*d*f)))/(f^5*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f))))+arctanh((2*a*f-c*x*(e+sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f)))))*(a^2*f^4*(e^2-2*d*f+e*sqrt(e^2-4*d*f))+2*a*c*f^2*(e^4-4*d*e^2*f+2*d^2*f^2+e^3*sqrt(e^2-4*d*f)-2*d*e*f*sqrt(e^2-4*d*f))+c^2*(e^6-6*d*e^4*f+9*d^2*e^2*f^2-2*d^3*f^3+e^5*sqrt(e^2-4*d*f)-4*d*e^3*f*sqrt(e^2-4*d*f)+3*d^2*e*f^2*sqrt(e^2-4*d*f)))/(f^5*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f))))],
[x*(a+c*x^2)^(3/2)/(d+e*x+f*x^2),x,10,1/3*(a+c*x^2)^(3/2)/f-1/2*e*(3*a*f^2+2*c*(e^2-2*d*f))*arctanh(x*sqrt(c)/sqrt(a+c*x^2))*sqrt(c)/f^4+1/2*(2*(a*f^2+c*(e^2-d*f))-c*e*f*x)*sqrt(a+c*x^2)/f^3-arctanh((2*a*f-c*x*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f)))))*(2*c*d*e*f*(2*a*f^2+c*(e^2-2*d*f))-(a^2*f^4+2*a*c*f^2*(e^2-d*f)+c^2*(e^4-3*d*e^2*f+d^2*f^2))*(e-sqrt(e^2-4*d*f)))/(f^4*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f))))+arctanh((2*a*f-c*x*(e+sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f)))))*(2*c*d*e*f*(2*a*f^2+c*(e^2-2*d*f))-(a^2*f^4+2*a*c*f^2*(e^2-d*f)+c^2*(e^4-3*d*e^2*f+d^2*f^2))*(e+sqrt(e^2-4*d*f)))/(f^4*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f))))],
[(a+c*x^2)^(3/2)/(d+e*x+f*x^2),x,9,1/2*(3*a*f^2+2*c*(e^2-d*f))*arctanh(x*sqrt(c)/sqrt(a+c*x^2))*sqrt(c)/f^3-1/2*c*(2*e-f*x)*sqrt(a+c*x^2)/f^2-arctanh((2*a*f-c*x*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f)))))*(-2*f*(2*a*c*d*f^2-a^2*f^3+c^2*d*(e^2-d*f))+c*e*(2*a*f^2+c*(e^2-2*d*f))*(e-sqrt(e^2-4*d*f)))/(f^3*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f))))+arctanh((2*a*f-c*x*(e+sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f)))))*(-2*f*(2*a*c*d*f^2-a^2*f^3+c^2*d*(e^2-d*f))+c*e*(2*a*f^2+c*(e^2-2*d*f))*(e+sqrt(e^2-4*d*f)))/(f^3*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f))))],
[(a+c*x^2)^(3/2)/(x*(d+e*x+f*x^2)),x,17,-c^(3/2)*e*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/f^2-a^(3/2)*arctanh(sqrt(a+c*x^2)/sqrt(a))/d+a*sqrt(a+c*x^2)/d+(c*d-a*f)*sqrt(a+c*x^2)/(d*f)-arctanh((2*a*f-c*x*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f)))))*(2*e*f*(c^2*d^2-a^2*f^2)-(c^2*d*e^2-f*(c*d-a*f)^2)*(e-sqrt(e^2-4*d*f)))/(d*f^2*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f))))+arctanh((2*a*f-c*x*(e+sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f)))))*(2*e*f*(c^2*d^2-a^2*f^2)-(c^2*d*e^2-f*(c*d-a*f)^2)*(e+sqrt(e^2-4*d*f)))/(d*f^2*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f))))],
[(a+c*x^2)^(3/2)/(x^2*(d+e*x+f*x^2)),x,21,-(a+c*x^2)^(3/2)/(d*x)+a^(3/2)*e*arctanh(sqrt(a+c*x^2)/sqrt(a))/d^2+3/2*a*arctanh(x*sqrt(c)/sqrt(a+c*x^2))*sqrt(c)/d+1/2*(2*c*d-3*a*f)*arctanh(x*sqrt(c)/sqrt(a+c*x^2))*sqrt(c)/(d*f)-a*e*sqrt(a+c*x^2)/d^2+3/2*c*x*sqrt(a+c*x^2)/d+1/2*(2*a*e-c*d*x)*sqrt(a+c*x^2)/d^2-arctanh((2*a*f-c*x*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f)))))*(4*a*c*d^2*f^2+c^2*d^2*(e^2-2*d*f-e*sqrt(e^2-4*d*f))+a^2*f^2*(e^2-2*d*f+e*sqrt(e^2-4*d*f)))/(d^2*f*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f))))+arctanh((2*a*f-c*x*(e+sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f)))))*(4*a*c*d^2*f^2+a^2*f^2*(e^2-2*d*f-e*sqrt(e^2-4*d*f))+c^2*d^2*(e^2-2*d*f+e*sqrt(e^2-4*d*f)))/(d^2*f*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f))))],
[(a+c*x^2)^(3/2)/(x^3*(d+e*x+f*x^2)),x,26,-1/2*(a+c*x^2)^(3/2)/(d*x^2)+e*(a+c*x^2)^(3/2)/(d^2*x)-a^(3/2)*(e^2-d*f)*arctanh(sqrt(a+c*x^2)/sqrt(a))/d^3-3/2*c*arctanh(sqrt(a+c*x^2)/sqrt(a))*sqrt(a)/d+3/2*c*sqrt(a+c*x^2)/d+a*(e^2-d*f)*sqrt(a+c*x^2)/d^3-3/2*c*e*x*sqrt(a+c*x^2)/d^2-1/2*(2*(c*d^2+a*(e^2-d*f))-c*d*e*x)*sqrt(a+c*x^2)/d^3+arctanh((2*a*f-c*x*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f)))))*(c^2*d^3*(e-sqrt(e^2-4*d*f))+2*a*c*d^2*f*(e+sqrt(e^2-4*d*f))+a^2*f*(e^3-3*d*e*f+e^2*sqrt(e^2-4*d*f)-d*f*sqrt(e^2-4*d*f)))/(d^3*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f))))-arctanh((2*a*f-c*x*(e+sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f)))))*(2*a*c*d^2*f*(e-sqrt(e^2-4*d*f))+c^2*d^3*(e+sqrt(e^2-4*d*f))+a^2*f*(e^3-3*d*e*f-e^2*sqrt(e^2-4*d*f)+d*f*sqrt(e^2-4*d*f)))/(d^3*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f))))],

# p<0
[x^3/((d+e*x+f*x^2)*sqrt(a+c*x^2)),x,10,-e*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/(f^2*sqrt(c))+sqrt(a+c*x^2)/(c*f)-arctanh((2*a*f-c*x*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f)))))*(2*d*e*f-(e^2-d*f)*(e-sqrt(e^2-4*d*f)))/(f^2*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f))))+arctanh((2*a*f-c*x*(e+sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f)))))*(2*d*e*f-(e^2-d*f)*(e+sqrt(e^2-4*d*f)))/(f^2*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f))))],
[x^2/((d+e*x+f*x^2)*sqrt(a+c*x^2)),x,8,arctanh(x*sqrt(c)/sqrt(a+c*x^2))/(f*sqrt(c))-arctanh((2*a*f-c*x*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f)))))*(e^2-2*d*f-e*sqrt(e^2-4*d*f))/(f*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f))))-arctanh((2*a*f-c*x*(e+sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f)))))*(2*d*f-e*(e+sqrt(e^2-4*d*f)))/(f*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f))))],
[x/((d+e*x+f*x^2)*sqrt(a+c*x^2)),x,5,arctanh((2*a*f-c*x*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f)))))*(e-sqrt(e^2-4*d*f))/(sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f))))-arctanh((2*a*f-c*x*(e+sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f)))))*(e+sqrt(e^2-4*d*f))/(sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f))))],
[1/((d+e*x+f*x^2)*sqrt(a+c*x^2)),x,5,-f*arctanh((2*a*f-c*x*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f)))))*sqrt(2)/(sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f))))+f*arctanh((2*a*f-c*x*(e+sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f)))))*sqrt(2)/(sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f))))],
[1/(x*(d+e*x+f*x^2)*sqrt(a+c*x^2)),x,10,-arctanh(sqrt(a+c*x^2)/sqrt(a))/(d*sqrt(a))+f*arctanh((2*a*f-c*x*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f)))))*(e+sqrt(e^2-4*d*f))/(d*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f))))-f*arctanh((2*a*f-c*x*(e+sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f)))))*(e-sqrt(e^2-4*d*f))/(d*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f))))],
[1/(x^2*(d+e*x+f*x^2)*sqrt(a+c*x^2)),x,11,e*arctanh(sqrt(a+c*x^2)/sqrt(a))/(d^2*sqrt(a))-sqrt(a+c*x^2)/(a*d*x)-f*arctanh((2*a*f-c*x*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f)))))*(e^2-2*d*f+e*sqrt(e^2-4*d*f))/(d^2*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f))))+f*arctanh((2*a*f-c*x*(e+sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f)))))*(e^2-2*d*f-e*sqrt(e^2-4*d*f))/(d^2*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f))))],
[1/(x^3*(d+e*x+f*x^2)*sqrt(a+c*x^2)),x,15,1/2*c*arctanh(sqrt(a+c*x^2)/sqrt(a))/(a^(3/2)*d)-(e^2-d*f)*arctanh(sqrt(a+c*x^2)/sqrt(a))/(d^3*sqrt(a))-1/2*sqrt(a+c*x^2)/(a*d*x^2)+e*sqrt(a+c*x^2)/(a*d^2*x)+f*arctanh((2*a*f-c*x*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f)))))*(2*e^3-4*d*e*f-(e^2-d*f)*(e-sqrt(e^2-4*d*f)))/(d^3*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f))))-f*arctanh((2*a*f-c*x*(e+sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f)))))*(2*e^3-4*d*e*f-(e^2-d*f)*(e+sqrt(e^2-4*d*f)))/(d^3*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f))))],
[x^3/((a+c*x^2)^(3/2)*(d+e*x+f*x^2)),x,10,(-1)/(c*f*sqrt(a+c*x^2))-e*x/(a*f^2*sqrt(a+c*x^2))+(a*f*(c*d^2+a*(e^2-d*f))+c*e*(c*d^2+a*(e^2-2*d*f))*x)/(a*f^2*(a*c*e^2+(c*d-a*f)^2)*sqrt(a+c*x^2))-arctanh((2*a*f-c*x*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f)))))*(2*a*d*e*f-(c*d^2+a*(e^2-d*f))*(e-sqrt(e^2-4*d*f)))/((a*c*e^2+(c*d-a*f)^2)*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f))))+arctanh((2*a*f-c*x*(e+sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f)))))*(2*a*d*e*f-(c*d^2+a*(e^2-d*f))*(e+sqrt(e^2-4*d*f)))/((a*c*e^2+(c*d-a*f)^2)*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f))))],
[x^2/((a+c*x^2)^(3/2)*(d+e*x+f*x^2)),x,6,(-a*e-(c*d-a*f)*x)/((a*c*e^2+(c*d-a*f)^2)*sqrt(a+c*x^2))-f*arctanh((2*a*f-c*x*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f)))))*(2*d*(c*d-a*f)+a*e*(e-sqrt(e^2-4*d*f)))/((a*c*e^2+(c*d-a*f)^2)*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f))))+f*arctanh((2*a*f-c*x*(e+sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f)))))*(2*d*(c*d-a*f)+a*e*(e+sqrt(e^2-4*d*f)))/((a*c*e^2+(c*d-a*f)^2)*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f))))],
[x/((a+c*x^2)^(3/2)*(d+e*x+f*x^2)),x,6,(-c*d+a*f+c*e*x)/((a*c*e^2+(c*d-a*f)^2)*sqrt(a+c*x^2))+f*arctanh((2*a*f-c*x*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f)))))*(2*c*d*e-(c*d-a*f)*(e-sqrt(e^2-4*d*f)))/((a*c*e^2+(c*d-a*f)^2)*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f))))-f*arctanh((2*a*f-c*x*(e+sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f)))))*(2*c*d*e-(c*d-a*f)*(e+sqrt(e^2-4*d*f)))/((a*c*e^2+(c*d-a*f)^2)*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f))))],
[1/((a+c*x^2)^(3/2)*(d+e*x+f*x^2)),x,6,c*(a*e+(c*d-a*f)*x)/(a*(a*c*e^2+(c*d-a*f)^2)*sqrt(a+c*x^2))-f*arctanh((2*a*f-c*x*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f)))))*(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f)))/((a*c*e^2+(c*d-a*f)^2)*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f))))+f*arctanh((2*a*f-c*x*(e+sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f)))))*(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f)))/((a*c*e^2+(c*d-a*f)^2)*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f))))],
[1/(x*(a+c*x^2)^(3/2)*(d+e*x+f*x^2)),x,12,-arctanh(sqrt(a+c*x^2)/sqrt(a))/(a^(3/2)*d)+1/(a*d*sqrt(a+c*x^2))+(-a*(a*f^2+c*(e^2-d*f))-c^2*d*e*x)/(a*d*(a*c*e^2+(c*d-a*f)^2)*sqrt(a+c*x^2))+f*arctanh((2*a*f-c*x*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f)))))*(2*e*(a*f^2+c*(e^2-2*d*f))-(a*f^2+c*(e^2-d*f))*(e-sqrt(e^2-4*d*f)))/(d*(a*c*e^2+(c*d-a*f)^2)*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f))))-f*arctanh((2*a*f-c*x*(e+sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f)))))*(2*e*(a*f^2+c*(e^2-2*d*f))-(a*f^2+c*(e^2-d*f))*(e+sqrt(e^2-4*d*f)))/(d*(a*c*e^2+(c*d-a*f)^2)*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f))))],
[1/(x^2*(a+c*x^2)^(3/2)*(d+e*x+f*x^2)),x,14,e*arctanh(sqrt(a+c*x^2)/sqrt(a))/(a^(3/2)*d^2)-e/(a*d^2*sqrt(a+c*x^2))+(-1)/(a*d*x*sqrt(a+c*x^2))-2*c*x/(a^2*d*sqrt(a+c*x^2))+(a*e*(a*f^2+c*(e^2-2*d*f))+c*d*(a*f^2+c*(e^2-d*f))*x)/(a*d^2*(a*c*e^2+(c*d-a*f)^2)*sqrt(a+c*x^2))+f*arctanh((2*a*f-c*x*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f)))))*(-2*(a*f^2*(e^2-d*f)+c*(e^4-3*d*e^2*f+d^2*f^2))+e*(a*f^2+c*(e^2-2*d*f))*(e-sqrt(e^2-4*d*f)))/(d^2*(a*c*e^2+(c*d-a*f)^2)*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f))))-f*arctanh((2*a*f-c*x*(e+sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+c*x^2)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f)))))*(-2*(a*f^2*(e^2-d*f)+c*(e^4-3*d*e^2*f+d^2*f^2))+e*(a*f^2+c*(e^2-2*d*f))*(e+sqrt(e^2-4*d*f)))/(d^2*(a*c*e^2+(c*d-a*f)^2)*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(2*a*f^2+c*(e^2-2*d*f+e*sqrt(e^2-4*d*f))))],

# Integrands of the form x^m (a+b x+c x^2)^(p/2) / (d+e x+f x^2) when e=0

# p>0
[x^3*(a+b*x+c*x^2)^(1/2)/(d-f*x^2),x,15,-1/3*(a+b*x+c*x^2)^(3/2)/(c*f)-1/16*b*(b^2-4*a*c)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(5/2)*f)-1/2*b*d*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(f^2*sqrt(c))-d*sqrt(a+b*x+c*x^2)/f^2+1/8*b*(b+2*c*x)*sqrt(a+b*x+c*x^2)/(c^2*f)-1/2*d*arctanh(1/2*(b*sqrt(d)-2*a*sqrt(f)+x*(2*c*sqrt(d)-b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))))*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))/f^(5/2)+1/2*d*arctanh(1/2*(b*sqrt(d)+2*a*sqrt(f)+x*(2*c*sqrt(d)+b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))))*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))/f^(5/2)],
[x^2*(a+b*x+c*x^2)^(1/2)/(d-f*x^2),x,9,-1/8*(8*c^2*d-b^2*f+4*a*c*f)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(3/2)*f^2)-1/4*(b+2*c*x)*sqrt(a+b*x+c*x^2)/(c*f)+1/2*arctanh(1/2*(b*sqrt(d)-2*a*sqrt(f)+x*(2*c*sqrt(d)-b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))))*sqrt(d)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))/f^2+1/2*arctanh(1/2*(b*sqrt(d)+2*a*sqrt(f)+x*(2*c*sqrt(d)+b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))))*sqrt(d)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))/f^2],
[x*(a+b*x+c*x^2)^(1/2)/(d-f*x^2),x,9,-1/2*b*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(f*sqrt(c))-sqrt(a+b*x+c*x^2)/f-1/2*arctanh(1/2*(b*sqrt(d)-2*a*sqrt(f)+x*(2*c*sqrt(d)-b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))))*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))/f^(3/2)+1/2*arctanh(1/2*(b*sqrt(d)+2*a*sqrt(f)+x*(2*c*sqrt(d)+b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))))*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))/f^(3/2)],
[(a+b*x+c*x^2)^(1/2)/(d-f*x^2),x,8,-arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))*sqrt(c)/f+1/2*arctanh(1/2*(b*sqrt(d)-2*a*sqrt(f)+x*(2*c*sqrt(d)-b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))))*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))/(f*sqrt(d))+1/2*arctanh(1/2*(b*sqrt(d)+2*a*sqrt(f)+x*(2*c*sqrt(d)+b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))))*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))/(f*sqrt(d))],
[(a+b*x+c*x^2)^(1/2)/(x*(d-f*x^2)),x,17,-arctanh(1/2*(2*a+b*x)/(sqrt(a)*sqrt(a+b*x+c*x^2)))*sqrt(a)/d-1/2*arctanh(1/2*(b*sqrt(d)-2*a*sqrt(f)+x*(2*c*sqrt(d)-b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))))*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))/(d*sqrt(f))+1/2*arctanh(1/2*(b*sqrt(d)+2*a*sqrt(f)+x*(2*c*sqrt(d)+b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))))*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))/(d*sqrt(f))],
[(a+b*x+c*x^2)^(1/2)/(x^2*(d-f*x^2)),x,16,-1/2*b*arctanh(1/2*(2*a+b*x)/(sqrt(a)*sqrt(a+b*x+c*x^2)))/(d*sqrt(a))-sqrt(a+b*x+c*x^2)/(d*x)+1/2*arctanh(1/2*(b*sqrt(d)-2*a*sqrt(f)+x*(2*c*sqrt(d)-b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))))*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))/d^(3/2)+1/2*arctanh(1/2*(b*sqrt(d)+2*a*sqrt(f)+x*(2*c*sqrt(d)+b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))))*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))/d^(3/2)],
[(a+b*x+c*x^2)^(1/2)/(x^3*(d-f*x^2)),x,20,1/8*(b^2-4*a*c)*arctanh(1/2*(2*a+b*x)/(sqrt(a)*sqrt(a+b*x+c*x^2)))/(a^(3/2)*d)-f*arctanh(1/2*(2*a+b*x)/(sqrt(a)*sqrt(a+b*x+c*x^2)))*sqrt(a)/d^2-1/4*(2*a+b*x)*sqrt(a+b*x+c*x^2)/(a*d*x^2)-1/2*arctanh(1/2*(b*sqrt(d)-2*a*sqrt(f)+x*(2*c*sqrt(d)-b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))))*sqrt(f)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))/d^2+1/2*arctanh(1/2*(b*sqrt(d)+2*a*sqrt(f)+x*(2*c*sqrt(d)+b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))))*sqrt(f)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))/d^2],
[x^3*(a+b*x+c*x^2)^(3/2)/(d-f*x^2),x,17,-1/3*d*(a+b*x+c*x^2)^(3/2)/f^2+1/16*b*(b+2*c*x)*(a+b*x+c*x^2)^(3/2)/(c^2*f)-1/5*(a+b*x+c*x^2)^(5/2)/(c*f)+3/256*b*(b^2-4*a*c)^2*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(7/2)*f)-1/16*b*d*(24*c^2*d-b^2*f+12*a*c*f)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(3/2)*f^3)-1/2*d*arctanh(1/2*(b*sqrt(d)-2*a*sqrt(f)+x*(2*c*sqrt(d)-b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))))*(c*d+a*f-b*sqrt(d)*sqrt(f))^(3/2)/f^(7/2)+1/2*d*arctanh(1/2*(b*sqrt(d)+2*a*sqrt(f)+x*(2*c*sqrt(d)+b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))))*(c*d+a*f+b*sqrt(d)*sqrt(f))^(3/2)/f^(7/2)-3/128*b*(b^2-4*a*c)*(b+2*c*x)*sqrt(a+b*x+c*x^2)/(c^3*f)-1/8*d*(8*c^2*d+b^2*f+8*a*c*f+2*b*c*f*x)*sqrt(a+b*x+c*x^2)/(c*f^3)],
[x^2*(a+b*x+c*x^2)^(3/2)/(d-f*x^2),x,10,-1/8*(b+2*c*x)*(a+b*x+c*x^2)^(3/2)/(c*f)-1/128*(128*c^4*d^2+192*a*c^3*d*f+3*b^4*f^2-24*a*b^2*c*f^2+48*c^2*f*(b^2*d+a^2*f))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(5/2)*f^3)+1/2*arctanh(1/2*(b*sqrt(d)-2*a*sqrt(f)+x*(2*c*sqrt(d)-b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))))*sqrt(d)*(c*d+a*f-b*sqrt(d)*sqrt(f))^(3/2)/f^3+1/2*arctanh(1/2*(b*sqrt(d)+2*a*sqrt(f)+x*(2*c*sqrt(d)+b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))))*sqrt(d)*(c*d+a*f+b*sqrt(d)*sqrt(f))^(3/2)/f^3-1/64*(b*(80*c^2*d-3*b^2*f+12*a*c*f)+2*c*(16*c^2*d-3*b^2*f+12*a*c*f)*x)*sqrt(a+b*x+c*x^2)/(c^2*f^2)],
[x*(a+b*x+c*x^2)^(3/2)/(d-f*x^2),x,10,-1/3*(a+b*x+c*x^2)^(3/2)/f-1/16*b*(24*c^2*d-b^2*f+12*a*c*f)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(3/2)*f^2)-1/2*arctanh(1/2*(b*sqrt(d)-2*a*sqrt(f)+x*(2*c*sqrt(d)-b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))))*(c*d+a*f-b*sqrt(d)*sqrt(f))^(3/2)/f^(5/2)+1/2*arctanh(1/2*(b*sqrt(d)+2*a*sqrt(f)+x*(2*c*sqrt(d)+b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))))*(c*d+a*f+b*sqrt(d)*sqrt(f))^(3/2)/f^(5/2)-1/8*(8*c^2*d+b^2*f+8*a*c*f+2*b*c*f*x)*sqrt(a+b*x+c*x^2)/(c*f^2)],
[(a+b*x+c*x^2)^(3/2)/(d-f*x^2),x,9,-1/8*(8*c^2*d+3*b^2*f+12*a*c*f)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(f^2*sqrt(c))+1/2*arctanh(1/2*(b*sqrt(d)-2*a*sqrt(f)+x*(2*c*sqrt(d)-b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))))*(c*d+a*f-b*sqrt(d)*sqrt(f))^(3/2)/(f^2*sqrt(d))+1/2*arctanh(1/2*(b*sqrt(d)+2*a*sqrt(f)+x*(2*c*sqrt(d)+b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))))*(c*d+a*f+b*sqrt(d)*sqrt(f))^(3/2)/(f^2*sqrt(d))-1/4*(5*b+2*c*x)*sqrt(a+b*x+c*x^2)/f],
[(a+b*x+c*x^2)^(3/2)/(x*(d-f*x^2)),x,19,-a^(3/2)*arctanh(1/2*(2*a+b*x)/(sqrt(a)*sqrt(a+b*x+c*x^2)))/d-1/16*b*(b^2-12*a*c)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(3/2)*d)-1/16*b*(24*c^2*d-b^2*f+12*a*c*f)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(3/2)*d*f)-1/2*arctanh(1/2*(b*sqrt(d)-2*a*sqrt(f)+x*(2*c*sqrt(d)-b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))))*(c*d+a*f-b*sqrt(d)*sqrt(f))^(3/2)/(d*f^(3/2))+1/2*arctanh(1/2*(b*sqrt(d)+2*a*sqrt(f)+x*(2*c*sqrt(d)+b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))))*(c*d+a*f+b*sqrt(d)*sqrt(f))^(3/2)/(d*f^(3/2))+1/8*(b^2+8*a*c+2*b*c*x)*sqrt(a+b*x+c*x^2)/(c*d)-1/8*(8*c^2*d+b^2*f+8*a*c*f+2*b*c*f*x)*sqrt(a+b*x+c*x^2)/(c*d*f)],
[(a+b*x+c*x^2)^(3/2)/(x^2*(d-f*x^2)),x,18,-(a+b*x+c*x^2)^(3/2)/(d*x)-3/2*b*arctanh(1/2*(2*a+b*x)/(sqrt(a)*sqrt(a+b*x+c*x^2)))*sqrt(a)/d+3/8*(b^2+4*a*c)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(d*sqrt(c))-1/8*(8*c^2*d+3*b^2*f+12*a*c*f)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(d*f*sqrt(c))+1/2*arctanh(1/2*(b*sqrt(d)-2*a*sqrt(f)+x*(2*c*sqrt(d)-b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))))*(c*d+a*f-b*sqrt(d)*sqrt(f))^(3/2)/(d^(3/2)*f)+1/2*arctanh(1/2*(b*sqrt(d)+2*a*sqrt(f)+x*(2*c*sqrt(d)+b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))))*(c*d+a*f+b*sqrt(d)*sqrt(f))^(3/2)/(d^(3/2)*f)+3/4*(3*b+2*c*x)*sqrt(a+b*x+c*x^2)/d-1/4*(5*b+2*c*x)*sqrt(a+b*x+c*x^2)/d],
[(a+b*x+c*x^2)^(3/2)/(x^3*(d-f*x^2)),x,26,-1/2*(a+b*x+c*x^2)^(3/2)/(d*x^2)-a^(3/2)*f*arctanh(1/2*(2*a+b*x)/(sqrt(a)*sqrt(a+b*x+c*x^2)))/d^2-1/16*b*(b^2-12*a*c)*f*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(3/2)*d^2)-1/16*b*(24*c^2*d-b^2*f+12*a*c*f)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(3/2)*d^2)-3/8*(b^2+4*a*c)*arctanh(1/2*(2*a+b*x)/(sqrt(a)*sqrt(a+b*x+c*x^2)))/(d*sqrt(a))+3/2*b*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))*sqrt(c)/d-1/2*arctanh(1/2*(b*sqrt(d)-2*a*sqrt(f)+x*(2*c*sqrt(d)-b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))))*(c*d+a*f-b*sqrt(d)*sqrt(f))^(3/2)/(d^2*sqrt(f))+1/2*arctanh(1/2*(b*sqrt(d)+2*a*sqrt(f)+x*(2*c*sqrt(d)+b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))))*(c*d+a*f+b*sqrt(d)*sqrt(f))^(3/2)/(d^2*sqrt(f))-3/4*(b-2*c*x)*sqrt(a+b*x+c*x^2)/(d*x)+1/8*f*(b^2+8*a*c+2*b*c*x)*sqrt(a+b*x+c*x^2)/(c*d^2)-1/8*(8*c^2*d+b^2*f+8*a*c*f+2*b*c*f*x)*sqrt(a+b*x+c*x^2)/(c*d^2)],
[(a+b*x+c*x^2)^(3/2)/(1-x^2),x,9,-1/2*(a-b+c)^(3/2)*arctanh(1/2*(2*a-b+(b-2*c)*x)/(sqrt(a-b+c)*sqrt(a+b*x+c*x^2)))+1/2*(a+b+c)^(3/2)*arctanh(1/2*(2*a+b+(b+2*c)*x)/(sqrt(a+b+c)*sqrt(a+b*x+c*x^2)))-1/8*(3*b^2+12*a*c+8*c^2)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/sqrt(c)-1/4*(5*b+2*c*x)*sqrt(a+b*x+c*x^2)],
[sqrt(-1-x+x^2)/(1-x^2),x,8,-1/2*arctan(1/2*(3-x)/sqrt(-1-x+x^2))+arctanh(1/2*(1-2*x)/sqrt(-1-x+x^2))+1/2*arctanh(1/2*(1+3*x)/sqrt(-1-x+x^2))],
[(x+x^2)^(3/2)/(1+x^2),x,10,-5/4*arctanh(x/sqrt(x+x^2))+1/4*(5+2*x)*sqrt(x+x^2)-arctanh((1-x-sqrt(2))/(sqrt(x+x^2)*sqrt(2*(-1+sqrt(2)))))*sqrt(-1+sqrt(2))+arctan((1-x+sqrt(2))/(sqrt(x+x^2)*sqrt(2*(1+sqrt(2)))))*sqrt(1+sqrt(2))],

# p<0
[x^4/((a+b*x+c*x^2)^(1/2)*(d-f*x^2)),x,13,-1/8*(3*b^2-4*a*c)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(5/2)*f)-d*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(f^2*sqrt(c))+3/4*b*sqrt(a+b*x+c*x^2)/(c^2*f)-1/2*x*sqrt(a+b*x+c*x^2)/(c*f)+1/2*d^(3/2)*arctanh(1/2*(b*sqrt(d)-2*a*sqrt(f)+x*(2*c*sqrt(d)-b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))))/(f^2*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f)))+1/2*d^(3/2)*arctanh(1/2*(b*sqrt(d)+2*a*sqrt(f)+x*(2*c*sqrt(d)+b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))))/(f^2*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f)))],
[x^3/((a+b*x+c*x^2)^(1/2)*(d-f*x^2)),x,10,1/2*b*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(3/2)*f)-sqrt(a+b*x+c*x^2)/(c*f)-1/2*d*arctanh(1/2*(b*sqrt(d)-2*a*sqrt(f)+x*(2*c*sqrt(d)-b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))))/(f^(3/2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f)))+1/2*d*arctanh(1/2*(b*sqrt(d)+2*a*sqrt(f)+x*(2*c*sqrt(d)+b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))))/(f^(3/2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f)))],
[x^2/((a+b*x+c*x^2)^(1/2)*(d-f*x^2)),x,8,-arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(f*sqrt(c))+1/2*arctanh(1/2*(b*sqrt(d)-2*a*sqrt(f)+x*(2*c*sqrt(d)-b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))))*sqrt(d)/(f*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f)))+1/2*arctanh(1/2*(b*sqrt(d)+2*a*sqrt(f)+x*(2*c*sqrt(d)+b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))))*sqrt(d)/(f*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f)))],
[x/((a+b*x+c*x^2)^(1/2)*(d-f*x^2)),x,5,-1/2*arctanh(1/2*(b*sqrt(d)-2*a*sqrt(f)+x*(2*c*sqrt(d)-b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))))/(sqrt(f)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f)))+1/2*arctanh(1/2*(b*sqrt(d)+2*a*sqrt(f)+x*(2*c*sqrt(d)+b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))))/(sqrt(f)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f)))],
[1/((a+b*x+c*x^2)^(1/2)*(d-f*x^2)),x,5,1/2*arctanh(1/2*(b*sqrt(d)-2*a*sqrt(f)+x*(2*c*sqrt(d)-b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))))/(sqrt(d)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f)))+1/2*arctanh(1/2*(b*sqrt(d)+2*a*sqrt(f)+x*(2*c*sqrt(d)+b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))))/(sqrt(d)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f)))],
[1/(x*(a+b*x+c*x^2)^(1/2)*(d-f*x^2)),x,9,-arctanh(1/2*(2*a+b*x)/(sqrt(a)*sqrt(a+b*x+c*x^2)))/(d*sqrt(a))-1/2*arctanh(1/2*(b*sqrt(d)-2*a*sqrt(f)+x*(2*c*sqrt(d)-b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))))*sqrt(f)/(d*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f)))+1/2*arctanh(1/2*(b*sqrt(d)+2*a*sqrt(f)+x*(2*c*sqrt(d)+b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))))*sqrt(f)/(d*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f)))],
[1/(x^2*(a+b*x+c*x^2)^(1/2)*(d-f*x^2)),x,10,1/2*b*arctanh(1/2*(2*a+b*x)/(sqrt(a)*sqrt(a+b*x+c*x^2)))/(a^(3/2)*d)-sqrt(a+b*x+c*x^2)/(a*d*x)+1/2*f*arctanh(1/2*(b*sqrt(d)-2*a*sqrt(f)+x*(2*c*sqrt(d)-b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))))/(d^(3/2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f)))+1/2*f*arctanh(1/2*(b*sqrt(d)+2*a*sqrt(f)+x*(2*c*sqrt(d)+b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))))/(d^(3/2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f)))],
[1/(x^3*(a+b*x+c*x^2)^(1/2)*(d-f*x^2)),x,13,-1/8*(3*b^2-4*a*c)*arctanh(1/2*(2*a+b*x)/(sqrt(a)*sqrt(a+b*x+c*x^2)))/(a^(5/2)*d)-f*arctanh(1/2*(2*a+b*x)/(sqrt(a)*sqrt(a+b*x+c*x^2)))/(d^2*sqrt(a))-1/2*sqrt(a+b*x+c*x^2)/(a*d*x^2)+3/4*b*sqrt(a+b*x+c*x^2)/(a^2*d*x)-1/2*f^(3/2)*arctanh(1/2*(b*sqrt(d)-2*a*sqrt(f)+x*(2*c*sqrt(d)-b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))))/(d^2*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f)))+1/2*f^(3/2)*arctanh(1/2*(b*sqrt(d)+2*a*sqrt(f)+x*(2*c*sqrt(d)+b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))))/(d^2*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f)))],
[x^4/((a+b*x+c*x^2)^(3/2)*(d-f*x^2)),x,13,-arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(3/2)*f)+1/2*d^(3/2)*arctanh(1/2*(b*sqrt(d)-2*a*sqrt(f)+x*(2*c*sqrt(d)-b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))))/(f*(c*d+a*f-b*sqrt(d)*sqrt(f))^(3/2))+1/2*d^(3/2)*arctanh(1/2*(b*sqrt(d)+2*a*sqrt(f)+x*(2*c*sqrt(d)+b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))))/(f*(c*d+a*f+b*sqrt(d)*sqrt(f))^(3/2))-2*x*(2*a+b*x)/((b^2-4*a*c)*f*sqrt(a+b*x+c*x^2))+2*d*(b+2*c*x)/((b^2-4*a*c)*f^2*sqrt(a+b*x+c*x^2))-2*d^2*(b*(b^2*f-c*(c*d+3*a*f))-c*(2*c^2*d-b^2*f+2*a*c*f)*x)/((b^2-4*a*c)*f^2*(b^2*d*f-(c*d+a*f)^2)*sqrt(a+b*x+c*x^2))+2*b*sqrt(a+b*x+c*x^2)/(c*(b^2-4*a*c)*f)],
[x^3/((a+b*x+c*x^2)^(3/2)*(d-f*x^2)),x,9,-1/2*d*arctanh(1/2*(b*sqrt(d)-2*a*sqrt(f)+x*(2*c*sqrt(d)-b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))))/(sqrt(f)*(c*d+a*f-b*sqrt(d)*sqrt(f))^(3/2))+1/2*d*arctanh(1/2*(b*sqrt(d)+2*a*sqrt(f)+x*(2*c*sqrt(d)+b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))))/(sqrt(f)*(c*d+a*f+b*sqrt(d)*sqrt(f))^(3/2))-2*(2*a+b*x)/((b^2-4*a*c)*f*sqrt(a+b*x+c*x^2))-2*d*(a*(2*c^2*d-b^2*f+2*a*c*f)+b*c*(c*d-a*f)*x)/((b^2-4*a*c)*f*(b^2*d*f-(c*d+a*f)^2)*sqrt(a+b*x+c*x^2))],
[x^2/((a+b*x+c*x^2)^(3/2)*(d-f*x^2)),x,6,1/2*arctanh(1/2*(b*sqrt(d)-2*a*sqrt(f)+x*(2*c*sqrt(d)-b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))))*sqrt(d)/(c*d+a*f-b*sqrt(d)*sqrt(f))^(3/2)+1/2*arctanh(1/2*(b*sqrt(d)+2*a*sqrt(f)+x*(2*c*sqrt(d)+b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))))*sqrt(d)/(c*d+a*f+b*sqrt(d)*sqrt(f))^(3/2)+2*(a*b*(c*d-a*f)+c*(b^2*d-2*a*(c*d+a*f))*x)/((b^2-4*a*c)*(b^2*d*f-(c*d+a*f)^2)*sqrt(a+b*x+c*x^2))],
[x/((a+b*x+c*x^2)^(3/2)*(d-f*x^2)),x,6,-1/2*arctanh(1/2*(b*sqrt(d)-2*a*sqrt(f)+x*(2*c*sqrt(d)-b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))))*sqrt(f)/(c*d+a*f-b*sqrt(d)*sqrt(f))^(3/2)+1/2*arctanh(1/2*(b*sqrt(d)+2*a*sqrt(f)+x*(2*c*sqrt(d)+b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))))*sqrt(f)/(c*d+a*f+b*sqrt(d)*sqrt(f))^(3/2)-2*(a*(2*c^2*d-b^2*f+2*a*c*f)+b*c*(c*d-a*f)*x)/((b^2-4*a*c)*(b^2*d*f-(c*d+a*f)^2)*sqrt(a+b*x+c*x^2))],
[1/((a+b*x+c*x^2)^(3/2)*(d-f*x^2)),x,6,1/2*f*arctanh(1/2*(b*sqrt(d)-2*a*sqrt(f)+x*(2*c*sqrt(d)-b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))))/(sqrt(d)*(c*d+a*f-b*sqrt(d)*sqrt(f))^(3/2))+1/2*f*arctanh(1/2*(b*sqrt(d)+2*a*sqrt(f)+x*(2*c*sqrt(d)+b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))))/(sqrt(d)*(c*d+a*f+b*sqrt(d)*sqrt(f))^(3/2))-2*(b*(b^2*f-c*(c*d+3*a*f))-c*(2*c^2*d-b^2*f+2*a*c*f)*x)/((b^2-4*a*c)*(b^2*d*f-(c*d+a*f)^2)*sqrt(a+b*x+c*x^2))],
[1/(x*(a+b*x+c*x^2)^(3/2)*(d-f*x^2)),x,12,-arctanh(1/2*(2*a+b*x)/(sqrt(a)*sqrt(a+b*x+c*x^2)))/(a^(3/2)*d)-1/2*f^(3/2)*arctanh(1/2*(b*sqrt(d)-2*a*sqrt(f)+x*(2*c*sqrt(d)-b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))))/(d*(c*d+a*f-b*sqrt(d)*sqrt(f))^(3/2))+1/2*f^(3/2)*arctanh(1/2*(b*sqrt(d)+2*a*sqrt(f)+x*(2*c*sqrt(d)+b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))))/(d*(c*d+a*f+b*sqrt(d)*sqrt(f))^(3/2))+2*(b^2-2*a*c+b*c*x)/(a*(b^2-4*a*c)*d*sqrt(a+b*x+c*x^2))-2*f*(a*(2*c^2*d-b^2*f+2*a*c*f)+b*c*(c*d-a*f)*x)/((b^2-4*a*c)*d*(b^2*d*f-(c*d+a*f)^2)*sqrt(a+b*x+c*x^2))],
[1/(x^2*(a+b*x+c*x^2)^(3/2)*(d-f*x^2)),x,12,3/2*b*arctanh(1/2*(2*a+b*x)/(sqrt(a)*sqrt(a+b*x+c*x^2)))/(a^(5/2)*d)+1/2*f^2*arctanh(1/2*(b*sqrt(d)-2*a*sqrt(f)+x*(2*c*sqrt(d)-b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f-b*sqrt(d)*sqrt(f))))/(d^(3/2)*(c*d+a*f-b*sqrt(d)*sqrt(f))^(3/2))+1/2*f^2*arctanh(1/2*(b*sqrt(d)+2*a*sqrt(f)+x*(2*c*sqrt(d)+b*sqrt(f)))/(sqrt(a+b*x+c*x^2)*sqrt(c*d+a*f+b*sqrt(d)*sqrt(f))))/(d^(3/2)*(c*d+a*f+b*sqrt(d)*sqrt(f))^(3/2))+2*(b^2-2*a*c+b*c*x)/(a*(b^2-4*a*c)*d*x*sqrt(a+b*x+c*x^2))-2*f*(b*(b^2*f-c*(c*d+3*a*f))-c*(2*c^2*d-b^2*f+2*a*c*f)*x)/((b^2-4*a*c)*d*(b^2*d*f-(c*d+a*f)^2)*sqrt(a+b*x+c*x^2))-(3*b^2-8*a*c)*sqrt(a+b*x+c*x^2)/(a^2*(b^2-4*a*c)*d*x)],

# Integrands of the form x^m (a+b x+c x^2)^(p/2) / (d+e x+f x^2)

# p>0
[x^2*(a+b*x+c*x^2)^(1/2)/(d+e*x+f*x^2),x,9,-1/8*(b^2*f^2+4*c*f*(b*e-a*f)-8*c^2*(e^2-d*f))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(3/2)*f^3)-1/4*(4*c*e-b*f-2*c*f*x)*sqrt(a+b*x+c*x^2)/(c*f^2)-arctanh(1/2*(4*a*f+2*x*(b*f-c*(e-sqrt(e^2-4*d*f)))-b*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(c*e-b*f)*sqrt(e^2-4*d*f))))*(c*(e^4-4*d*e^2*f+2*d^2*f^2-e^3*sqrt(e^2-4*d*f)+2*d*e*f*sqrt(e^2-4*d*f))+f*(a*f*(e^2-2*d*f-e*sqrt(e^2-4*d*f))-b*(e^3-3*d*e*f-e^2*sqrt(e^2-4*d*f)+d*f*sqrt(e^2-4*d*f))))/(f^3*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(f*(2*a*f-b*(e-sqrt(e^2-4*d*f)))+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f))))+arctanh(1/2*(4*a*f-b*(e+sqrt(e^2-4*d*f))+2*x*(b*f-c*(e+sqrt(e^2-4*d*f))))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(c*e-b*f)*sqrt(e^2-4*d*f))))*(c*(e^4-4*d*e^2*f+2*d^2*f^2+e^3*sqrt(e^2-4*d*f)-2*d*e*f*sqrt(e^2-4*d*f))+f*(a*f*(e^2-2*d*f+e*sqrt(e^2-4*d*f))-b*(e^3-3*d*e*f+e^2*sqrt(e^2-4*d*f)-d*f*sqrt(e^2-4*d*f))))/(f^3*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(c*(e^2-2*d*f+e*sqrt(e^2-4*d*f))+f*(2*a*f-b*(e+sqrt(e^2-4*d*f)))))],
[x*(a+b*x+c*x^2)^(1/2)/(d+e*x+f*x^2),x,9,-1/2*(2*c*e-b*f)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(f^2*sqrt(c))+sqrt(a+b*x+c*x^2)/f-arctanh(1/2*(4*a*f+2*x*(b*f-c*(e-sqrt(e^2-4*d*f)))-b*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(c*e-b*f)*sqrt(e^2-4*d*f))))*(2*d*f*(c*e-b*f)+(f*(b*e-a*f)-c*(e^2-d*f))*(e-sqrt(e^2-4*d*f)))/(f^2*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(c*e-b*f)*sqrt(e^2-4*d*f)))+arctanh(1/2*(4*a*f-b*(e+sqrt(e^2-4*d*f))+2*x*(b*f-c*(e+sqrt(e^2-4*d*f))))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(c*e-b*f)*sqrt(e^2-4*d*f))))*(2*d*f*(c*e-b*f)+(f*(b*e-a*f)-c*(e^2-d*f))*(e+sqrt(e^2-4*d*f)))/(f^2*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(c*e-b*f)*sqrt(e^2-4*d*f)))],
[(a+b*x+c*x^2)^(1/2)/(d+e*x+f*x^2),x,8,arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))*sqrt(c)/f-arctanh(1/2*(4*a*f+2*x*(b*f-c*(e-sqrt(e^2-4*d*f)))-b*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(c*e-b*f)*sqrt(e^2-4*d*f))))*sqrt(f*(2*a*f-b*(e-sqrt(e^2-4*d*f)))+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f)))/(f*sqrt(2)*sqrt(e^2-4*d*f))+arctanh(1/2*(4*a*f-b*(e+sqrt(e^2-4*d*f))+2*x*(b*f-c*(e+sqrt(e^2-4*d*f))))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(c*e-b*f)*sqrt(e^2-4*d*f))))*sqrt(c*(e^2-2*d*f+e*sqrt(e^2-4*d*f))+f*(2*a*f-b*(e+sqrt(e^2-4*d*f))))/(f*sqrt(2)*sqrt(e^2-4*d*f))],
[(a+b*x+c*x^2)^(1/2)/(x*(d+e*x+f*x^2)),x,17,-arctanh(1/2*(2*a+b*x)/(sqrt(a)*sqrt(a+b*x+c*x^2)))*sqrt(a)/d+arctanh(1/2*(4*a*f+2*x*(b*f-c*(e-sqrt(e^2-4*d*f)))-b*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(c*e-b*f)*sqrt(e^2-4*d*f))))*(c*d*(e-sqrt(e^2-4*d*f))-f*(2*b*d-a*(e+sqrt(e^2-4*d*f))))/(d*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(f*(2*a*f-b*(e-sqrt(e^2-4*d*f)))+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f))))-arctanh(1/2*(4*a*f-b*(e+sqrt(e^2-4*d*f))+2*x*(b*f-c*(e+sqrt(e^2-4*d*f))))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(c*e-b*f)*sqrt(e^2-4*d*f))))*(-f*(2*b*d-a*(e-sqrt(e^2-4*d*f)))+c*d*(e+sqrt(e^2-4*d*f)))/(d*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(c*(e^2-2*d*f+e*sqrt(e^2-4*d*f))+f*(2*a*f-b*(e+sqrt(e^2-4*d*f)))))],
[(a+b*x+c*x^2)^(1/2)/(x^2*(d+e*x+f*x^2)),x,23,-1/2*b*arctanh(1/2*(2*a+b*x)/(sqrt(a)*sqrt(a+b*x+c*x^2)))/(d*sqrt(a))+e*arctanh(1/2*(2*a+b*x)/(sqrt(a)*sqrt(a+b*x+c*x^2)))*sqrt(a)/d^2-1/2*b*e*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(d^2*sqrt(c))-1/2*(2*c*d-b*e)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(d^2*sqrt(c))+arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))*sqrt(c)/d-sqrt(a+b*x+c*x^2)/(d*x)-f*arctanh(1/2*(4*a*f+2*x*(b*f-c*(e-sqrt(e^2-4*d*f)))-b*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(c*e-b*f)*sqrt(e^2-4*d*f))))*(2*c*d^2-b*d*(e+sqrt(e^2-4*d*f))+a*(e^2-2*d*f+e*sqrt(e^2-4*d*f)))/(d^2*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(f*(2*a*f-b*(e-sqrt(e^2-4*d*f)))+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f))))+f*arctanh(1/2*(4*a*f-b*(e+sqrt(e^2-4*d*f))+2*x*(b*f-c*(e+sqrt(e^2-4*d*f))))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(c*e-b*f)*sqrt(e^2-4*d*f))))*(2*c*d^2-b*d*(e-sqrt(e^2-4*d*f))+a*(e^2-2*d*f-e*sqrt(e^2-4*d*f)))/(d^2*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(c*(e^2-2*d*f+e*sqrt(e^2-4*d*f))+f*(2*a*f-b*(e+sqrt(e^2-4*d*f)))))],
#  {x^2*(a + b*x + c*x^2)^(3/2)/(d + e*x + f*x^2), x, 15, ((d*f*(c*e - b*f) + e*(f*(c*d - a*f) - e*(c*e - b*f)))*Sqrt[a + b*x + c*x^2])/f^4 - (3*(b^2 - 4*a*c)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(64*c^2*f) + ((2*c*e^2 - 2*c*d*f - b*e*f)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(8*c*f^3) - (e*(a + b*x + c*x^2)^(3/2))/(3*f^2) + ((b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(8*c*f) + (3*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(128*c^(5/2)*f) - ((b^2 - 4*a*c)*(2*c*e^2 - 2*c*d*f - b*e*f)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(3/2)*f^3) - ((3*b*c*e^3*f - e*(6*b*c*d + (b^2 + 2*a*c)*e)*f^2 + (b^2*d + 2*a*c*d + a*b*e)*f^3 - 2*c^2*(e^4 - 3*d*e^2*f + d^2*f^2))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[c]*f^5) + (((f*(c*d - a*f) - e*(c*e - b*f))*(e - Sqrt[e^2 - 4*d*f])*(2*d*f*(c*e - b*f) + e*(f*(c*d - a*f) - e*(c*e - b*f))) - 2*d*f*(f*(c*d - a*f)*(f*(c*d - a*f) - e*(c*e - b*f)) - (c*e - b*f)*(d*f*(c*e - b*f) + e*(f*(c*d - a*f) - e*(c*e - b*f)))))*ArcTanh[(b*e - 4*a*f - b*Sqrt[e^2 - 4*d*f] + 2*(c*e - b*f - c*Sqrt[e^2 - 4*d*f])*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(f^5*Sqrt[e^2 - 4*d*f]*Sqrt[4*a*f^2 - 2*b*f*(e - Sqrt[e^2 - 4*d*f]) + c*(e - Sqrt[e^2 - 4*d*f])^2]) - (((f*(c*d - a*f) - e*(c*e - b*f))*(e + Sqrt[e^2 - 4*d*f])*(2*d*f*(c*e - b*f) + e*(f*(c*d - a*f) - e*(c*e - b*f))) - 2*d*f*(f*(c*d - a*f)*(f*(c*d - a*f) - e*(c*e - b*f)) - (c*e - b*f)*(d*f*(c*e - b*f) + e*(f*(c*d - a*f) - e*(c*e - b*f)))))*ArcTanh[(b*e - 4*a*f + b*Sqrt[e^2 - 4*d*f] + 2*(c*e - b*f + c*Sqrt[e^2 - 4*d*f])*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(f^5*Sqrt[e^2 - 4*d*f]*Sqrt[4*a*f^2 - 2*b*f*(e + Sqrt[e^2 - 4*d*f]) + c*(e + Sqrt[e^2 - 4*d*f])^2])}
# {x^1*(a + b*x + c*x^2)^(3/2)/(d + e*x + f*x^2), x, 10, ((c*e^2 - c*d*f - b*e*f + a*f^2)*Sqrt[a + b*x + c*x^2])/f^3 - ((2*c*e - b*f)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(8*c*f^2) + (a + b*x + c*x^2)^(3/2)/(3*f) + ((b^2 - 4*a*c)*(2*c*e - b*f)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(3/2)*f^2) + ((3*b*c*e^2*f - (3*b*c*d + b^2*e + 2*a*c*e)*f^2 + a*b*f^3 - 2*c^2*e*(e^2 - 2*d*f))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[c]*f^4) + ((2*d*f*(c*e - b*f)*(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2) + (e - Sqrt[e^2 - 4*d*f])*(2*b*c*e^3*f - e*(4*b*c*d + (b^2 + 2*a*c)*e)*f^2 + (b^2*d + 2*a*c*d + 2*a*b*e)*f^3 - a^2*f^4 - c^2*(e^4 - 3*d*e^2*f + d^2*f^2)))*ArcTanh[(b*e - 4*a*f - b*Sqrt[e^2 - 4*d*f] + 2*(c*e - b*f - c*Sqrt[e^2 - 4*d*f])*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(f^4*Sqrt[e^2 - 4*d*f]*Sqrt[4*a*f^2 - 2*b*f*(e - Sqrt[e^2 - 4*d*f]) + c*(e - Sqrt[e^2 - 4*d*f])^2]) - ((2*d*f*(c*e - b*f)*(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2) + (e + Sqrt[e^2 - 4*d*f])*(2*b*c*e^3*f - e*(4*b*c*d + (b^2 + 2*a*c)*e)*f^2 + (b^2*d + 2*a*c*d + 2*a*b*e)*f^3 - a^2*f^4 - c^2*(e^4 - 3*d*e^2*f + d^2*f^2)))*ArcTanh[(b*e - 4*a*f + b*Sqrt[e^2 - 4*d*f] + 2*(c*e - b*f + c*Sqrt[e^2 - 4*d*f])*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(f^4*Sqrt[e^2 - 4*d*f]*Sqrt[4*a*f^2 - 2*b*f*(e + Sqrt[e^2 - 4*d*f]) + c*(e + Sqrt[e^2 - 4*d*f])^2])}
# {x^0*(a + b*x + c*x^2)^(3/2)/(d + e*x + f*x^2), x, 9, -(((c*e - b*f)*Sqrt[a + b*x + c*x^2])/f^2) + ((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(4*f) - ((b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*Sqrt[c]*f) - ((3*b*c*e*f - (b^2 + 2*a*c)*f^2 - 2*c^2*(e^2 - d*f))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[c]*f^3) + (((c*e - b*f)*(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2)*(e - Sqrt[e^2 - 4*d*f]) + 2*f*(2*b*c*d*e*f - (b^2 + 2*a*c)*d*f^2 + a^2*f^3 - c^2*d*(e^2 - d*f)))*ArcTanh[(b*e - 4*a*f - b*Sqrt[e^2 - 4*d*f] + 2*(c*e - b*f - c*Sqrt[e^2 - 4*d*f])*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(f^3*Sqrt[e^2 - 4*d*f]*Sqrt[4*a*f^2 - 2*b*f*(e - Sqrt[e^2 - 4*d*f]) + c*(e - Sqrt[e^2 - 4*d*f])^2]) - (((c*e - b*f)*(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2)*(e + Sqrt[e^2 - 4*d*f]) + 2*f*(2*b*c*d*e*f - (b^2 + 2*a*c)*d*f^2 + a^2*f^3 - c^2*d*(e^2 - d*f)))*ArcTanh[(b*e - 4*a*f + b*Sqrt[e^2 - 4*d*f] + 2*(c*e - b*f + c*Sqrt[e^2 - 4*d*f])*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(f^3*Sqrt[e^2 - 4*d*f]*Sqrt[4*a*f^2 - 2*b*f*(e + Sqrt[e^2 - 4*d*f]) + c*(e + Sqrt[e^2 - 4*d*f])^2])}
# {1/x^1*(a + b*x + c*x^2)^(3/2)/(d + e*x + f*x^2), x, 17, ((c*d - a*f)*Sqrt[a + b*x + c*x^2])/(d*f) - (b*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(8*c*d) + ((b^2 + 8*a*c + 2*b*c*x)*Sqrt[a + b*x + c*x^2])/(8*c*d) - (a^(3/2)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/d - (b*(b^2 - 12*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(3/2)*d) + (b*(b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(3/2)*d) - ((2*c^2*d*e - 3*b*c*d*f + a*b*f^2)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[c]*d*f^2) + ((2*f*(c*d - a*f)*(c*d*e - 2*b*d*f + a*e*f) + (e - Sqrt[e^2 - 4*d*f])*(2*b*c*d*e*f - (b^2 + 2*a*c)*d*f^2 + a^2*f^3 - c^2*d*(e^2 - d*f)))*ArcTanh[(b*e - 4*a*f - b*Sqrt[e^2 - 4*d*f] + 2*(c*e - b*f - c*Sqrt[e^2 - 4*d*f])*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(d*f^2*Sqrt[e^2 - 4*d*f]*Sqrt[4*a*f^2 - 2*b*f*(e - Sqrt[e^2 - 4*d*f]) + c*(e - Sqrt[e^2 - 4*d*f])^2]) - ((2*f*(c*d - a*f)*(c*d*e - 2*b*d*f + a*e*f) + (e + Sqrt[e^2 - 4*d*f])*(2*b*c*d*e*f - (b^2 + 2*a*c)*d*f^2 + a^2*f^3 - c^2*d*(e^2 - d*f)))*ArcTanh[(b*e - 4*a*f + b*Sqrt[e^2 - 4*d*f] + 2*(c*e - b*f + c*Sqrt[e^2 - 4*d*f])*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(d*f^2*Sqrt[e^2 - 4*d*f]*Sqrt[4*a*f^2 - 2*b*f*(e + Sqrt[e^2 - 4*d*f]) + c*(e + Sqrt[e^2 - 4*d*f])^2])}
# {1/x^2*(a + b*x + c*x^2)^(3/2)/(d + e*x + f*x^2), x, 22, -(((b*d - a*e)*Sqrt[a + b*x + c*x^2])/d^2) - ((2*c*d - b*e)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(8*c*d^2) + (3*(3*b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(4*d) - (e*(b^2 + 8*a*c + 2*b*c*x)*Sqrt[a + b*x + c*x^2])/(8*c*d^2) - (a + b*x + c*x^2)^(3/2)/(d*x) - (3*Sqrt[a]*b*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(2*d) + (a^(3/2)*e*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/d^2 + (3*(b^2 + 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*Sqrt[c]*d) + (b*(b^2 - 12*a*c)*e*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(3/2)*d^2) + ((b^2 - 4*a*c)*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(3/2)*d^2) + ((2*c^2*d^2 - b^2*d*f - 2*a*c*d*f + a*b*e*f)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[c]*d^2*f) - ((2*a^2*d*f^3 + a*f^2*(2*b*d*e + 2*b*d*Sqrt[e^2 - 4*d*f] - a*e*(e + Sqrt[e^2 - 4*d*f])) + d^2*(2*f*(b*c*e - (b^2 + 2*a*c)*f - b*c*Sqrt[e^2 - 4*d*f]) + c^2*(2*d*f - e*(e - Sqrt[e^2 - 4*d*f]))))*ArcTanh[(b*e - 4*a*f - b*Sqrt[e^2 - 4*d*f] + 2*(c*e - b*f - c*Sqrt[e^2 - 4*d*f])*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(d^2*f*Sqrt[e^2 - 4*d*f]*Sqrt[4*a*f^2 - 2*b*f*(e - Sqrt[e^2 - 4*d*f]) + c*(e - Sqrt[e^2 - 4*d*f])^2]) + ((2*a^2*d*f^3 + a*f^2*(2*b*d*e - 2*b*d*Sqrt[e^2 - 4*d*f] - a*e*(e - Sqrt[e^2 - 4*d*f])) + d^2*(2*f*(b*c*e - (b^2 + 2*a*c)*f + b*c*Sqrt[e^2 - 4*d*f]) + c^2*(2*d*f - e*(e + Sqrt[e^2 - 4*d*f]))))*ArcTanh[(b*e - 4*a*f + b*Sqrt[e^2 - 4*d*f] + 2*(c*e - b*f + c*Sqrt[e^2 - 4*d*f])*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(d^2*f*Sqrt[e^2 - 4*d*f]*Sqrt[4*a*f^2 - 2*b*f*(e + Sqrt[e^2 - 4*d*f]) + c*(e + Sqrt[e^2 - 4*d*f])^2])}

# {1/x^3*(a + b*x + c*x^2)^(3/2)/(d + e*x + f*x^2), x, 27, ((e*(c*e - b*f)*(e^2 - 2*d*f) + (e^2 - d*f)*(f*(c*d - a*f) - e*(c*e - b*f)))*Sqrt[a + b*x + c*x^2])/(d^3*f^2) - (3*(b - 2*c*x)*Sqrt[a + b*x + c*x^2])/(4*d*x) + ((2*c*d*e - b*e^2 + b*d*f)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(8*c*d^3) - (3*e*(3*b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(4*d^2) + ((e^2 - d*f)*(b^2 + 8*a*c + 2*b*c*x)*Sqrt[a + b*x + c*x^2])/(8*c*d^3) - (a + b*x + c*x^2)^(3/2)/(2*d*x^2) + (e*(a + b*x + c*x^2)^(3/2))/(d^2*x) - (3*(b^2 + 4*a*c)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(8*Sqrt[a]*d) + (3*Sqrt[a]*b*e*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(2*d^2) - (a^(3/2)*(e^2 - d*f)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/d^3 + (3*b*Sqrt[c]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*d) - (3*(b^2 + 4*a*c)*e*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*Sqrt[c]*d^2) - (b*(b^2 - 12*a*c)*(e^2 - d*f)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(3/2)*d^3) - ((b^2 - 4*a*c)*(2*c*d*e - b*e^2 + b*d*f)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(3/2)*d^3) - ((3*b*c*d^2 - b^2*d*e - 2*a*c*d*e + a*b*e^2 - a*b*d*f)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[c]*d^3) - (1/(d^3*f^3*Sqrt[e^2 - 4*d*f]*Sqrt[4*a*f^2 - 2*b*f*(e - Sqrt[e^2 - 4*d*f]) + c*(e - Sqrt[e^2 - 4*d*f])^2]))*((2*f*(f*(c*d - a*f)*(e*(c*d - a*f)*(e^2 - 2*d*f) - d*(c*e - b*f)*(e^2 - d*f)) - d*(c*e - b*f)*(e*(c*e - b*f)*(e^2 - 2*d*f) + (e^2 - d*f)*(f*(c*d - a*f) - e*(c*e - b*f)))) - (e - Sqrt[e^2 - 4*d*f])*(f*(c*e - b*f)*(e*(c*d - a*f)*(e^2 - 2*d*f) - d*(c*e - b*f)*(e^2 - d*f)) + (f*(c*d - a*f) - e*(c*e - b*f))*(e*(c*e - b*f)*(e^2 - 2*d*f) + (e^2 - d*f)*(f*(c*d - a*f) - e*(c*e - b*f)))))*ArcTanh[(b*e - 4*a*f - b*Sqrt[e^2 - 4*d*f] + 2*(c*e - b*f - c*Sqrt[e^2 - 4*d*f])*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])]) + (1/(d^3*f^3*Sqrt[e^2 - 4*d*f]*Sqrt[4*a*f^2 - 2*b*f*(e + Sqrt[e^2 - 4*d*f]) + c*(e + Sqrt[e^2 - 4*d*f])^2]))*((2*f*(f*(c*d - a*f)*(e*(c*d - a*f)*(e^2 - 2*d*f) - d*(c*e - b*f)*(e^2 - d*f)) - d*(c*e - b*f)*(e*(c*e - b*f)*(e^2 - 2*d*f) + (e^2 - d*f)*(f*(c*d - a*f) - e*(c*e - b*f)))) - (e + Sqrt[e^2 - 4*d*f])*(f*(c*e - b*f)*(e*(c*d - a*f)*(e^2 - 2*d*f) - d*(c*e - b*f)*(e^2 - d*f)) + (f*(c*d - a*f) - e*(c*e - b*f))*(e*(c*e - b*f)*(e^2 - 2*d*f) + (e^2 - d*f)*(f*(c*d - a*f) - e*(c*e - b*f)))))*ArcTanh[(b*e - 4*a*f + b*Sqrt[e^2 - 4*d*f] + 2*(c*e - b*f + c*Sqrt[e^2 - 4*d*f])*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])} 

# p<0
[x^3/((a+b*x+c*x^2)^(1/2)*(d+e*x+f*x^2)),x,12,-1/2*b*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(3/2)*f)-e*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(f^2*sqrt(c))+sqrt(a+b*x+c*x^2)/(c*f)-arctanh(1/2*(4*a*f+2*x*(b*f-c*(e-sqrt(e^2-4*d*f)))-b*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(c*e-b*f)*sqrt(e^2-4*d*f))))*(2*d*e*f-(e^2-d*f)*(e-sqrt(e^2-4*d*f)))/(f^2*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(c*e-b*f)*sqrt(e^2-4*d*f)))+arctanh(1/2*(4*a*f-b*(e+sqrt(e^2-4*d*f))+2*x*(b*f-c*(e+sqrt(e^2-4*d*f))))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(c*e-b*f)*sqrt(e^2-4*d*f))))*(2*d*e*f-(e^2-d*f)*(e+sqrt(e^2-4*d*f)))/(f^2*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(c*e-b*f)*sqrt(e^2-4*d*f)))],
[x^2/((a+b*x+c*x^2)^(1/2)*(d+e*x+f*x^2)),x,8,arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(f*sqrt(c))-arctanh(1/2*(4*a*f+2*x*(b*f-c*(e-sqrt(e^2-4*d*f)))-b*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(c*e-b*f)*sqrt(e^2-4*d*f))))*(e^2-2*d*f-e*sqrt(e^2-4*d*f))/(f*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(c*e-b*f)*sqrt(e^2-4*d*f)))-arctanh(1/2*(4*a*f-b*(e+sqrt(e^2-4*d*f))+2*x*(b*f-c*(e+sqrt(e^2-4*d*f))))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(c*e-b*f)*sqrt(e^2-4*d*f))))*(2*d*f-e*(e+sqrt(e^2-4*d*f)))/(f*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(c*e-b*f)*sqrt(e^2-4*d*f)))],
[x/((a+b*x+c*x^2)^(1/2)*(d+e*x+f*x^2)),x,5,arctanh(1/2*(4*a*f+2*x*(b*f-c*(e-sqrt(e^2-4*d*f)))-b*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(c*e-b*f)*sqrt(e^2-4*d*f))))*(e-sqrt(e^2-4*d*f))/(sqrt(2)*sqrt(e^2-4*d*f)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(c*e-b*f)*sqrt(e^2-4*d*f)))-arctanh(1/2*(4*a*f-b*(e+sqrt(e^2-4*d*f))+2*x*(b*f-c*(e+sqrt(e^2-4*d*f))))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(c*e-b*f)*sqrt(e^2-4*d*f))))*(e+sqrt(e^2-4*d*f))/(sqrt(2)*sqrt(e^2-4*d*f)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(c*e-b*f)*sqrt(e^2-4*d*f)))],
[1/((a+b*x+c*x^2)^(1/2)*(d+e*x+f*x^2)),x,5,-f*arctanh(1/2*(4*a*f+2*x*(b*f-c*(e-sqrt(e^2-4*d*f)))-b*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(c*e-b*f)*sqrt(e^2-4*d*f))))*sqrt(2)/(sqrt(e^2-4*d*f)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(c*e-b*f)*sqrt(e^2-4*d*f)))+f*arctanh(1/2*(4*a*f-b*(e+sqrt(e^2-4*d*f))+2*x*(b*f-c*(e+sqrt(e^2-4*d*f))))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(c*e-b*f)*sqrt(e^2-4*d*f))))*sqrt(2)/(sqrt(e^2-4*d*f)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(c*e-b*f)*sqrt(e^2-4*d*f)))],
[1/(x*(a+b*x+c*x^2)^(1/2)*(d+e*x+f*x^2)),x,9,-arctanh(1/2*(2*a+b*x)/(sqrt(a)*sqrt(a+b*x+c*x^2)))/(d*sqrt(a))+f*arctanh(1/2*(4*a*f+2*x*(b*f-c*(e-sqrt(e^2-4*d*f)))-b*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(c*e-b*f)*sqrt(e^2-4*d*f))))*(e+sqrt(e^2-4*d*f))/(d*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(c*e-b*f)*sqrt(e^2-4*d*f)))-f*arctanh(1/2*(4*a*f-b*(e+sqrt(e^2-4*d*f))+2*x*(b*f-c*(e+sqrt(e^2-4*d*f))))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(c*e-b*f)*sqrt(e^2-4*d*f))))*(e-sqrt(e^2-4*d*f))/(d*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(c*e-b*f)*sqrt(e^2-4*d*f)))],
[1/(x^2*(a+b*x+c*x^2)^(1/2)*(d+e*x+f*x^2)),x,12,1/2*b*arctanh(1/2*(2*a+b*x)/(sqrt(a)*sqrt(a+b*x+c*x^2)))/(a^(3/2)*d)+e*arctanh(1/2*(2*a+b*x)/(sqrt(a)*sqrt(a+b*x+c*x^2)))/(d^2*sqrt(a))-sqrt(a+b*x+c*x^2)/(a*d*x)+f*arctanh(1/2*(4*a*f-b*(e+sqrt(e^2-4*d*f))+2*x*(b*f-c*(e+sqrt(e^2-4*d*f))))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(c*e-b*f)*sqrt(e^2-4*d*f))))*(e^2-2*d*f-e*sqrt(e^2-4*d*f))/(d^2*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(c*e-b*f)*sqrt(e^2-4*d*f)))-f*arctanh(1/2*(4*a*f+2*x*(b*f-c*(e-sqrt(e^2-4*d*f)))-b*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(c*e-b*f)*sqrt(e^2-4*d*f))))*(e^2-2*d*f+e*sqrt(e^2-4*d*f))/(d^2*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(f*(2*a*f-b*(e-sqrt(e^2-4*d*f)))+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f))))],
[1/(x^3*(a+b*x+c*x^2)^(1/2)*(d+e*x+f*x^2)),x,16,-1/8*(3*b^2-4*a*c)*arctanh(1/2*(2*a+b*x)/(sqrt(a)*sqrt(a+b*x+c*x^2)))/(a^(5/2)*d)-1/2*b*e*arctanh(1/2*(2*a+b*x)/(sqrt(a)*sqrt(a+b*x+c*x^2)))/(a^(3/2)*d^2)-(e^2-d*f)*arctanh(1/2*(2*a+b*x)/(sqrt(a)*sqrt(a+b*x+c*x^2)))/(d^3*sqrt(a))-1/2*sqrt(a+b*x+c*x^2)/(a*d*x^2)+3/4*b*sqrt(a+b*x+c*x^2)/(a^2*d*x)+e*sqrt(a+b*x+c*x^2)/(a*d^2*x)+f*arctanh(1/2*(4*a*f+2*x*(b*f-c*(e-sqrt(e^2-4*d*f)))-b*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(c*e-b*f)*sqrt(e^2-4*d*f))))*(2*e^3-4*d*e*f-(e^2-d*f)*(e-sqrt(e^2-4*d*f)))/(d^3*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(c*e-b*f)*sqrt(e^2-4*d*f)))-f*arctanh(1/2*(4*a*f-b*(e+sqrt(e^2-4*d*f))+2*x*(b*f-c*(e+sqrt(e^2-4*d*f))))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(c*e-b*f)*sqrt(e^2-4*d*f))))*(2*e^3-4*d*e*f-(e^2-d*f)*(e+sqrt(e^2-4*d*f)))/(d^3*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(c*e-b*f)*sqrt(e^2-4*d*f)))],
[x^3/((a+b*x+c*x^2)^(3/2)*(d+e*x+f*x^2)),x,10,2*(2*a+b*x)/((b^2-4*a*c)*f*sqrt(a+b*x+c*x^2))+2*e*(b+2*c*x)/((b^2-4*a*c)*f^2*sqrt(a+b*x+c*x^2))+2*(c*d*e*(b*c*d-2*a*c*e+a*b*f)-(b*d*e-a*e^2+a*d*f)*(2*c^2*d+b^2*f-c*(b*e+2*a*f))+c*((b*c*d-2*a*c*e+a*b*f)*(e^2-d*f)-d*e*(2*c^2*d+b^2*f-c*(b*e+2*a*f)))*x)/((b^2-4*a*c)*f^2*((c*d-a*f)^2-(b*d-a*e)*(c*e-b*f))*sqrt(a+b*x+c*x^2))+arctanh(1/2*(4*a*f+2*x*(b*f-c*(e-sqrt(e^2-4*d*f)))-b*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(c*e-b*f)*sqrt(e^2-4*d*f))))*(2*d*(b*d-a*e)*f+(c*d^2-b*d*e+a*(e^2-d*f))*(e-sqrt(e^2-4*d*f)))/(((c*d-a*f)^2-(b*d-a*e)*(c*e-b*f))*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(c*e-b*f)*sqrt(e^2-4*d*f)))-arctanh(1/2*(4*a*f-b*(e+sqrt(e^2-4*d*f))+2*x*(b*f-c*(e+sqrt(e^2-4*d*f))))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(c*e-b*f)*sqrt(e^2-4*d*f))))*(2*d*(b*d-a*e)*f+(c*d^2-b*d*e+a*(e^2-d*f))*(e+sqrt(e^2-4*d*f)))/(((c*d-a*f)^2-(b*d-a*e)*(c*e-b*f))*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(c*e-b*f)*sqrt(e^2-4*d*f)))],
[x^2/((a+b*x+c*x^2)^(3/2)*(d+e*x+f*x^2)),x,6,-2*(a*(b*c*d-2*a*c*e+a*b*f)+c*(b^2*d-a*b*e-2*a*(c*d-a*f))*x)/((b^2-4*a*c)*((c*d-a*f)^2-(b*d-a*e)*(c*e-b*f))*sqrt(a+b*x+c*x^2))-f*arctanh(1/2*(4*a*f+2*x*(b*f-c*(e-sqrt(e^2-4*d*f)))-b*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(c*e-b*f)*sqrt(e^2-4*d*f))))*(2*d*(c*d-a*f)-(b*d-a*e)*(e-sqrt(e^2-4*d*f)))/(((c*d-a*f)^2-(b*d-a*e)*(c*e-b*f))*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(c*e-b*f)*sqrt(e^2-4*d*f)))+f*arctanh(1/2*(4*a*f-b*(e+sqrt(e^2-4*d*f))+2*x*(b*f-c*(e+sqrt(e^2-4*d*f))))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(c*e-b*f)*sqrt(e^2-4*d*f))))*(2*d*(c*d-a*f)-(b*d-a*e)*(e+sqrt(e^2-4*d*f)))/(((c*d-a*f)^2-(b*d-a*e)*(c*e-b*f))*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(c*e-b*f)*sqrt(e^2-4*d*f)))],
[x/((a+b*x+c*x^2)^(3/2)*(d+e*x+f*x^2)),x,6,2*(a*(2*c^2*d-b*c*e+b^2*f-2*a*c*f)+c*(b*c*d-2*a*c*e+a*b*f)*x)/((b^2-4*a*c)*((c*d-a*f)^2-(b*d-a*e)*(c*e-b*f))*sqrt(a+b*x+c*x^2))+f*arctanh(1/2*(4*a*f+2*x*(b*f-c*(e-sqrt(e^2-4*d*f)))-b*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(c*e-b*f)*sqrt(e^2-4*d*f))))*(2*d*(c*e-b*f)-(c*d-a*f)*(e-sqrt(e^2-4*d*f)))/(((c*d-a*f)^2-(b*d-a*e)*(c*e-b*f))*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(c*e-b*f)*sqrt(e^2-4*d*f)))-f*arctanh(1/2*(4*a*f-b*(e+sqrt(e^2-4*d*f))+2*x*(b*f-c*(e+sqrt(e^2-4*d*f))))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(c*e-b*f)*sqrt(e^2-4*d*f))))*(2*d*(c*e-b*f)-(c*d-a*f)*(e+sqrt(e^2-4*d*f)))/(((c*d-a*f)^2-(b*d-a*e)*(c*e-b*f))*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(c*e-b*f)*sqrt(e^2-4*d*f)))],
[1/((a+b*x+c*x^2)^(3/2)*(d+e*x+f*x^2)),x,6,2*(b^2*c*e-2*a*c^2*e-b^3*f-b*c*(c*d-3*a*f)-c*(2*c^2*d-b*c*e+b^2*f-2*a*c*f)*x)/((b^2-4*a*c)*((c*d-a*f)^2-(b*d-a*e)*(c*e-b*f))*sqrt(a+b*x+c*x^2))-f*arctanh(1/2*(4*a*f+2*x*(b*f-c*(e-sqrt(e^2-4*d*f)))-b*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(c*e-b*f)*sqrt(e^2-4*d*f))))*(c*(e^2-2*d*f+e*sqrt(e^2-4*d*f))+f*(2*a*f-b*(e+sqrt(e^2-4*d*f))))/(((c*d-a*f)^2-(b*d-a*e)*(c*e-b*f))*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(f*(2*a*f-b*(e-sqrt(e^2-4*d*f)))+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f))))+f*arctanh(1/2*(4*a*f-b*(e+sqrt(e^2-4*d*f))+2*x*(b*f-c*(e+sqrt(e^2-4*d*f))))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(c*e-b*f)*sqrt(e^2-4*d*f))))*(f*(2*a*f-b*(e-sqrt(e^2-4*d*f)))+c*(e^2-2*d*f-e*sqrt(e^2-4*d*f)))/(((c*d-a*f)^2-(b*d-a*e)*(c*e-b*f))*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(c*(e^2-2*d*f+e*sqrt(e^2-4*d*f))+f*(2*a*f-b*(e+sqrt(e^2-4*d*f)))))],
[1/(x*(a+b*x+c*x^2)^(3/2)*(d+e*x+f*x^2)),x,12,-arctanh(1/2*(2*a+b*x)/(sqrt(a)*sqrt(a+b*x+c*x^2)))/(a^(3/2)*d)+2*(b^2-2*a*c+b*c*x)/(a*(b^2-4*a*c)*d*sqrt(a+b*x+c*x^2))+2*(c*e*(2*a*c*e-b*(c*d+a*f))+(b*e-a*f)*(2*c^2*d+b^2*f-c*(b*e+2*a*f))+c*(2*c^2*d*e+b*f*(b*e-a*f)-b*c*(e^2+d*f))*x)/((b^2-4*a*c)*d*((c*d-a*f)^2-(b*d-a*e)*(c*e-b*f))*sqrt(a+b*x+c*x^2))+f*arctanh(1/2*(4*a*f+2*x*(b*f-c*(e-sqrt(e^2-4*d*f)))-b*(e-sqrt(e^2-4*d*f)))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(c*e-b*f)*sqrt(e^2-4*d*f))))*(-2*(f*(b*e^2-b*d*f-a*e*f)-c*(e^3-2*d*e*f))+(f*(b*e-a*f)-c*(e^2-d*f))*(e-sqrt(e^2-4*d*f)))/(d*((c*d-a*f)^2-(b*d-a*e)*(c*e-b*f))*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(c*e-b*f)*sqrt(e^2-4*d*f)))-f*arctanh(1/2*(4*a*f-b*(e+sqrt(e^2-4*d*f))+2*x*(b*f-c*(e+sqrt(e^2-4*d*f))))/(sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(c*e-b*f)*sqrt(e^2-4*d*f))))*(-2*(f*(b*e^2-b*d*f-a*e*f)-c*(e^3-2*d*e*f))+(f*(b*e-a*f)-c*(e^2-d*f))*(e+sqrt(e^2-4*d*f)))/(d*((c*d-a*f)^2-(b*d-a*e)*(c*e-b*f))*sqrt(2)*sqrt(e^2-4*d*f)*sqrt(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(c*e-b*f)*sqrt(e^2-4*d*f)))],

#  {1/(x^2*(a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2)), x, 13, -((2*e*(b^2 - 2*a*c + b*c*x))/(a*(b^2 - 4*a*c)*d^2*Sqrt[a + b*x + c*x^2])) + (2*(b^2 - 2*a*c + b*c*x))/(a*(b^2 - 4*a*c)*d*x*Sqrt[a + b*x + c*x^2]) + (2*(2*a*c*(f*(b*e^2 - b*d*f - a*e*f) - c*(e^3 - 2*d*e*f)) - b*(a*e*f*(c*e - b*f) + (e^2 - d*f)*(c^2*d + b^2*f - c*(b*e + a*f))) + c*(b*c*e^3 - 2*a*c*d*f^2 - 2*c^2*d*(e^2 - d*f) - b*f*(b*e^2 - b*d*f - a*e*f))*x))/((b^2 - 4*a*c)*d^2*(c^2*d^2 + f*(b^2*d - a*b*e + a^2*f) - c*(b*d*e - a*(e^2 - 2*d*f)))*Sqrt[a + b*x + c*x^2]) - ((3*b^2 - 8*a*c)*Sqrt[a + b*x + c*x^2])/(a^2*(b^2 - 4*a*c)*d*x) + (3*b*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(2*a^(5/2)*d) + (e*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(a^(3/2)*d^2) - (f*((e - Sqrt[e^2 - 4*d*f])*(f*(b*e^2 - b*d*f - a*e*f) - c*(e^3 - 2*d*e*f)) - 2*(d*e*f*(c*e - b*f) + (e^2 - d*f)*(f*(b*e - a*f) - c*(e^2 - d*f))))*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*d^2*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*(c^2*d^2 + f*(b^2*d - a*b*e + a^2*f) - c*(b*d*e - a*(e^2 - 2*d*f)))) + (f*((e + Sqrt[e^2 - 4*d*f])*(f*(b*e^2 - b*d*f - a*e*f) - c*(e^3 - 2*d*e*f)) - 2*(d*e*f*(c*e - b*f) + (e^2 - d*f)*(f*(b*e - a*f) - c*(e^2 - d*f))))*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*d^2*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*(c^2*d^2 + f*(b^2*d - a*b*e + a^2*f) - c*(b*d*e - a*(e^2 - 2*d*f))))} 
[x^4/((3+4*x+2*x^2)*sqrt(-3-4*x-x^2)),x,24,11/2*arcsin(2+x)-5/4*arctanh(x/sqrt(-3-4*x-x^2))+1/2*arctan((1+(-3-x)/sqrt(-3-4*x-x^2))/sqrt(2))/sqrt(2)-1/2*arctan((1+(3+x)/sqrt(-3-4*x-x^2))/sqrt(2))/sqrt(2)+5/2*sqrt(-3-4*x-x^2)-1/4*x*sqrt(-3-4*x-x^2)],
[x^3/((3+4*x+2*x^2)*sqrt(-3-4*x-x^2)),x,20,-2*arcsin(2+x)+arctanh(x/sqrt(-3-4*x-x^2))+1/2*arctan((1+(-3-x)/sqrt(-3-4*x-x^2))/sqrt(2))/sqrt(2)-1/2*arctan((1+(3+x)/sqrt(-3-4*x-x^2))/sqrt(2))/sqrt(2)-1/2*sqrt(-3-4*x-x^2)],
[x^2/((3+4*x+2*x^2)*sqrt(-3-4*x-x^2)),x,16,1/2*arcsin(2+x)-1/2*arctanh(x/sqrt(-3-4*x-x^2))-arctan((1+(-3-x)/sqrt(-3-4*x-x^2))/sqrt(2))/sqrt(2)+arctan((1+(3+x)/sqrt(-3-4*x-x^2))/sqrt(2))/sqrt(2)],
[x/((3+4*x+2*x^2)*sqrt(-3-4*x-x^2)),x,6,-arctan((1-3*sqrt(-1-x)/sqrt(3+x))/sqrt(2))/sqrt(2)+arctan((1+3*sqrt(-1-x)/sqrt(3+x))/sqrt(2))/sqrt(2),arctan((1+(-3-x)/sqrt(-3-4*x-x^2))/sqrt(2))/sqrt(2)-arctan((1+(3+x)/sqrt(-3-4*x-x^2))/sqrt(2))/sqrt(2)],
[1/((3+4*x+2*x^2)*sqrt(-3-4*x-x^2)),x,10,1/3*arctanh(x/sqrt(-3-4*x-x^2))-1/3*arctan((1+(-3-x)/sqrt(-3-4*x-x^2))/sqrt(2))*sqrt(2)+1/3*arctan((1+(3+x)/sqrt(-3-4*x-x^2))/sqrt(2))*sqrt(2)],
[1/(x*(3+4*x+2*x^2)*sqrt(-3-4*x-x^2)),x,17,-4/9*arctanh(x/sqrt(-3-4*x-x^2))+1/9*arctan((1+(-3-x)/sqrt(-3-4*x-x^2))/sqrt(2))*sqrt(2)-1/9*arctan((1+(3+x)/sqrt(-3-4*x-x^2))/sqrt(2))*sqrt(2)-1/3*arctan((3+2*x)/(sqrt(3)*sqrt(-3-4*x-x^2)))/sqrt(3)],
[1/(x^2*(3+4*x+2*x^2)*sqrt(-3-4*x-x^2)),x,20,10/27*arctanh(x/sqrt(-3-4*x-x^2))+2/27*arctan((1+(-3-x)/sqrt(-3-4*x-x^2))/sqrt(2))*sqrt(2)-2/27*arctan((1+(3+x)/sqrt(-3-4*x-x^2))/sqrt(2))*sqrt(2)+2/3*arctan((3+2*x)/(sqrt(3)*sqrt(-3-4*x-x^2)))/sqrt(3)+1/9*sqrt(-3-4*x-x^2)/x],

# Integrands of the form (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q

# Integrands of the form (g+h x)^m (a+b x+c x^2)^(p/2) (d+e x+f x^2)^q
[(2+3*x)^2*(30+31*x-12*x^2)^2*(6+17*x+12*x^2)^(1/2),x,8,-125455/4718592*(17+24*x)*(6+17*x+12*x^2)^(3/2)+25091/24576*(17+24*x)*(6+17*x+12*x^2)^(5/2)-873/1792*(6+17*x+12*x^2)^(7/2)-1/32*(10-3*x)*(6+17*x+12*x^2)^(7/2)-125455/603979776*arctanh(1/4*(17+24*x)/(sqrt(3)*sqrt(6+17*x+12*x^2)))/sqrt(3)+125455/150994944*(17+24*x)*sqrt(6+17*x+12*x^2)],
[(2+3*x)*(30+31*x-12*x^2)*(6+17*x+12*x^2)^(1/2),x,6,97/768*(17+24*x)*(6+17*x+12*x^2)^(3/2)-1/20*(6+17*x+12*x^2)^(5/2)+97/98304*arctanh(1/4*(17+24*x)/(sqrt(3)*sqrt(6+17*x+12*x^2)))/sqrt(3)-97/24576*(17+24*x)*sqrt(6+17*x+12*x^2)],
[(6+17*x+12*x^2)^(1/2)/((2+3*x)*(30+31*x-12*x^2)),x,3,1/42*arctanh(1/84*(206+291*x)/sqrt(6+17*x+12*x^2))],
[(6+17*x+12*x^2)^(1/2)/((2+3*x)^2*(30+31*x-12*x^2)^2),x,5,97/3226944*arctanh(1/84*(206+291*x)/sqrt(6+17*x+12*x^2))+1/98*(-275-388*x)/((10-3*x)*sqrt(6+17*x+12*x^2))+3137/38416*sqrt(6+17*x+12*x^2)/(10-3*x)],
[(6+17*x+12*x^2)^(1/2)/((2+3*x)^3*(30+31*x-12*x^2)^3),x,7,1/294*(-275-388*x)/((10-3*x)^2*(6+17*x+12*x^2)^(3/2))+40325/637540872192*arctanh(1/84*(206+291*x)/sqrt(6+17*x+12*x^2))+1/8232*(738029+1042556*x)/((10-3*x)^2*sqrt(6+17*x+12*x^2))-50555899/19361664*sqrt(6+17*x+12*x^2)/(10-3*x)^2-1634466587/7589772288*sqrt(6+17*x+12*x^2)/(10-3*x)],

#  The following integrands are equal. 
[(-3+2*x)*(-3*x+x^2)^(2/3),x,1,3/5*(-3*x+x^2)^(5/3)],
[((-3+x)*x)^(2/3)*(-3+2*x),x,1,3/5*(-(3-x)*x)^(5/3)],
[x*(9-9*x+2*x^2)/(-3*x+x^2)^(1/3),x,2,3/5*(-3*x+x^2)^(5/3)],
[x*(9-9*x+2*x^2)/((-3+x)*x)^(1/3),x,3,3/5*(-3*x+x^2)^(5/3)],
#  {(A + B*x)^2*(a + ((B^2*c*d^2 + a*A^2*f^2)/(A*B*d*f))*x + c*x^2)^2*(d + ((B^2*d + A^2*f)/(A*B))*x + f*x^2)^(1/2), x, 8, (5*(B^2*d - A^2*f)^4*(9*B^4*c^2*d^4 + 2*A^2*B^2*c*d^2*f*(7*c*d - 16*a*f) + A^4*f^2*(9*c^2*d^2 - 32*a*c*d*f + 32*a^2*f^2))*((B*d)/A + (A*f)/B + 2*f*x)*Sqrt[d + ((B*d)/A + (A*f)/B)*x + f*x^2])/(16384*A^6*B^4*d^2*f^7) - (5*(B^2*d - A^2*f)^2*(9*B^4*c^2*d^4 + 2*A^2*B^2*c*d^2*f*(7*c*d - 16*a*f) + A^4*f^2*(9*c^2*d^2 - 32*a*c*d*f + 32*a^2*f^2))*((B*d)/A + (A*f)/B + 2*f*x)*(d + ((B*d)/A + (A*f)/B)*x + f*x^2)^(3/2))/(6144*A^4*B^2*d^2*f^6) + ((9*B^4*c^2*d^4 + 2*A^2*B^2*c*d^2*f*(7*c*d - 16*a*f) + A^4*f^2*(9*c^2*d^2 - 32*a*c*d*f + 32*a^2*f^2))*((B*d)/A + (A*f)/B + 2*f*x)*(d + ((B*d)/A + (A*f)/B)*x + f*x^2)^(5/2))/(384*A^2*d^2*f^5) - (9*B*c*((B^2*c*d)/(A*f) + A*(c - (2*a*f)/d))*(d + ((B*d)/A + (A*f)/B)*x + f*x^2)^(7/2))/(112*f^3) + (B*c*(a*A*f + B*c*d*x)*(d + ((B*d)/A + (A*f)/B)*x + f*x^2)^(7/2))/(8*d*f^3) - (5*(B^2*d - A^2*f)^6*(9*B^4*c^2*d^4 + 2*A^2*B^2*c*d^2*f*(7*c*d - 16*a*f) + A^4*f^2*(9*c^2*d^2 - 32*a*c*d*f + 32*a^2*f^2))*ArcTanh[((B*d)/A + (A*f)/B + 2*f*x)/(2*Sqrt[f]*Sqrt[d + ((B*d)/A + (A*f)/B)*x + f*x^2])])/(32768*A^8*B^6*d^2*f^(15/2))}
# {(A + B*x)^1*(a + ((B^2*c*d^2 + a*A^2*f^2)/(A*B*d*f))*x + c*x^2)^1*(d + ((B^2*d + A^2*f)/(A*B))*x + f*x^2)^(1/2), x, 6, (3*(B^2*d - A^2*f)^2*(B^2*c*d^2 + A^2*f*(c*d - 2*a*f))*((B*d)/A + (A*f)/B + 2*f*x)*Sqrt[d + ((B*d)/A + (A*f)/B)*x + f*x^2])/(128*A^3*B^2*d*f^4) - ((A*c + (B^2*c*d)/(A*f) - (2*a*A*f)/d)*((B*d)/A + (A*f)/B + 2*f*x)*(d + ((B*d)/A + (A*f)/B)*x + f*x^2)^(3/2))/(16*f^2) + (B*c*(d + ((B*d)/A + (A*f)/B)*x + f*x^2)^(5/2))/(5*f^2) - (3*(B^2*d - A^2*f)^4*(B^2*c*d^2 + A^2*f*(c*d - 2*a*f))*ArcTanh[((B*d)/A + (A*f)/B + 2*f*x)/(2*Sqrt[f]*Sqrt[d + ((B*d)/A + (A*f)/B)*x + f*x^2])])/(256*A^5*B^4*d*f^(9/2))}
# {(A + B*x)^(-1)*(a + ((B^2*c*d^2 + a*A^2*f^2)/(A*B*d*f))*x + c*x^2)^(-1)*(d + ((B^2*d + A^2*f)/(A*B))*x + f*x^2)^(1/2), x, 3, (d*f*ArcTanh[(d*f*(a*B - (2*B*c*d)/f + (a*A^2*f)/(B*d) - ((B^2*c*d)/(A*f) + A*(c - (2*a*f)/d))*x))/(2*Sqrt[c*d - a*f]*Sqrt[B^2*c*d^2 - a*A^2*f^2]*Sqrt[d + ((B*d)/A + (A*f)/B)*x + f*x^2])])/(Sqrt[c*d - a*f]*Sqrt[B^2*c*d^2 - a*A^2*f^2])}

# {(A + B*x)^(-2)*(a + ((B^2*c*d^2 + a*A^2*f^2)/(A*B*d*f))*x + c*x^2)^(-2)*(d + ((B^2*d + A^2*f)/(A*B))*x + f*x^2)^(1/2), x, 6, (2*A*B^2*d^2*f^3*(A*d*((B^3*c*d^2)/(A^2*f) - a*B*f + (A^2*f*(c*d - a*f))/(B*d)) + (B^2*c*d^2 + A^2*f*(c*d - 2*a*f))*x))/((c*d - a*f)*(B^2*d - A^2*f)^2*(B^2*c*d^2 - a*A^2*f^2)*(a*A*f + B*c*d*x)*Sqrt[d + ((B*d)/A + (A*f)/B)*x + f*x^2]) - (B^3*c*d^3*f^2*(3*B^4*c^2*d^4 - 2*A^2*B^2*c*d^2*f*(c*d + 2*a*f) + A^4*f^2*(3*c^2*d^2 - 4*a*c*d*f + 4*a^2*f^2))*Sqrt[d + ((B*d)/A + (A*f)/B)*x + f*x^2])/((c*d - a*f)^2*(B^2*d - A^2*f)^2*(B^2*c*d^2 - a*A^2*f^2)^2*(a*A*f + B*c*d*x)) + (3*B^2*c^2*d^4*f^2*(B^2*c*d^2 + A^2*f*(c*d - 2*a*f))*ArcTanh[(d*f*((2*B*c*d)/f - a*(B + (A^2*f)/(B*d)) + ((B^2*c*d)/(A*f) + A*(c - (2*a*f)/d))*x))/(2*Sqrt[c*d - a*f]*Sqrt[B^2*c*d^2 - a*A^2*f^2]*Sqrt[d + ((B*d)/A + (A*f)/B)*x + f*x^2])])/(2*A*(c*d - a*f)^(5/2)*(B^2*c*d^2 - a*A^2*f^2)^(5/2))} 
[(g+h*x)/((-c*g^2/h^2+9*c*x^2)^(1/3)*(g^2+3*h^2*x^2)),x,2,1/6*(1-9*h^2*x^2/g^2)^(1/3)*log(g^2+3*h^2*x^2)/(2^(2/3)*h*(-c*g^2/h^2+9*c*x^2)^(1/3))-1/2*(1-9*h^2*x^2/g^2)^(1/3)*log((1-3*h*x/g)^(2/3)+2^(1/3)*(1+3*h*x/g)^(1/3))/(2^(2/3)*h*(-c*g^2/h^2+9*c*x^2)^(1/3))+(1-9*h^2*x^2/g^2)^(1/3)*arctan(1/sqrt(3)-2^(2/3)*(1-3*h*x/g)^(2/3)/((1+3*h*x/g)^(1/3)*sqrt(3)))/(2^(2/3)*h*(-c*g^2/h^2+9*c*x^2)^(1/3)*sqrt(3))],
[(g+h*x)/((1/9*(-c^2*g^2+b*c*g*h+2*b^2*h^2)/(c*h^2)+b*x+c*x^2)^(1/3)*(f*(b^2+1/3*(c^2*g^2-b*c*g*h-2*b^2*h^2)/h^2)/c^2+b*f*x/c+f*x^2)),x,2,3*3^(1/6)*h*(c*h^2*((c*g-2*b*h)*(c*g+b*h)/(c*h^2)-9*b*x-9*c*x^2)/(2*c*g-b*h)^2)^(1/3)*arctan(1/sqrt(3)-2^(2/3)*(1-3*h*(b+2*c*x)/(2*c*g-b*h))^(2/3)/((1+3*h*(b+2*c*x)/(2*c*g-b*h))^(1/3)*sqrt(3)))/(f*(-(c*g-2*b*h)*(c*g+b*h)/(c*h^2)+9*b*x+9*c*x^2)^(1/3))+1/2*3^(2/3)*h*(c*h^2*((c*g-2*b*h)*(c*g+b*h)/(c*h^2)-9*b*x-9*c*x^2)/(2*c*g-b*h)^2)^(1/3)*log(1/3*f*(c^2*g^2-b*c*g*h+b^2*h^2)/(c^2*h^2)+b*f*x/c+f*x^2)/(f*(-(c*g-2*b*h)*(c*g+b*h)/(c*h^2)+9*b*x+9*c*x^2)^(1/3))-3/2*3^(2/3)*h*(c*h^2*((c*g-2*b*h)*(c*g+b*h)/(c*h^2)-9*b*x-9*c*x^2)/(2*c*g-b*h)^2)^(1/3)*log((1-3*h*(b+2*c*x)/(2*c*g-b*h))^(2/3)+2^(1/3)*(1+3*h*(b+2*c*x)/(2*c*g-b*h))^(1/3))/(f*(-(c*g-2*b*h)*(c*g+b*h)/(c*h^2)+9*b*x+9*c*x^2)^(1/3))]]:
