1,1,74,74,0.015008,"\int x^2 \left(A+B x+C x^2\right) \left(a+b x^2+c x^4\right) \, dx","Integrate[x^2*(A + B*x + C*x^2)*(a + b*x^2 + c*x^4),x]","\frac{1}{5} x^5 (a C+A b)+\frac{1}{3} a A x^3+\frac{1}{4} a B x^4+\frac{1}{7} x^7 (A c+b C)+\frac{1}{6} b B x^6+\frac{1}{8} B c x^8+\frac{1}{9} c C x^9","\frac{1}{5} x^5 (a C+A b)+\frac{1}{3} a A x^3+\frac{1}{4} a B x^4+\frac{1}{7} x^7 (A c+b C)+\frac{1}{6} b B x^6+\frac{1}{8} B c x^8+\frac{1}{9} c C x^9",1,"(a*A*x^3)/3 + (a*B*x^4)/4 + ((A*b + a*C)*x^5)/5 + (b*B*x^6)/6 + ((A*c + b*C)*x^7)/7 + (B*c*x^8)/8 + (c*C*x^9)/9","A",1
2,1,74,74,0.0112315,"\int x \left(A+B x+C x^2\right) \left(a+b x^2+c x^4\right) \, dx","Integrate[x*(A + B*x + C*x^2)*(a + b*x^2 + c*x^4),x]","\frac{1}{4} x^4 (a C+A b)+\frac{1}{2} a A x^2+\frac{1}{3} a B x^3+\frac{1}{6} x^6 (A c+b C)+\frac{1}{5} b B x^5+\frac{1}{7} B c x^7+\frac{1}{8} c C x^8","\frac{1}{4} x^4 (a C+A b)+\frac{1}{2} a A x^2+\frac{1}{3} a B x^3+\frac{1}{6} x^6 (A c+b C)+\frac{1}{5} b B x^5+\frac{1}{7} B c x^7+\frac{1}{8} c C x^8",1,"(a*A*x^2)/2 + (a*B*x^3)/3 + ((A*b + a*C)*x^4)/4 + (b*B*x^5)/5 + ((A*c + b*C)*x^6)/6 + (B*c*x^7)/7 + (c*C*x^8)/8","A",1
3,1,69,69,0.0121768,"\int \left(A+B x+C x^2\right) \left(a+b x^2+c x^4\right) \, dx","Integrate[(A + B*x + C*x^2)*(a + b*x^2 + c*x^4),x]","\frac{1}{3} x^3 (a C+A b)+a A x+\frac{1}{2} a B x^2+\frac{1}{5} x^5 (A c+b C)+\frac{1}{4} b B x^4+\frac{1}{6} B c x^6+\frac{1}{7} c C x^7","\frac{1}{3} x^3 (a C+A b)+a A x+\frac{1}{2} a B x^2+\frac{1}{5} x^5 (A c+b C)+\frac{1}{4} b B x^4+\frac{1}{6} B c x^6+\frac{1}{7} c C x^7",1,"a*A*x + (a*B*x^2)/2 + ((A*b + a*C)*x^3)/3 + (b*B*x^4)/4 + ((A*c + b*C)*x^5)/5 + (B*c*x^6)/6 + (c*C*x^7)/7","A",1
4,1,65,65,0.0146151,"\int \frac{\left(A+B x+C x^2\right) \left(a+b x^2+c x^4\right)}{x} \, dx","Integrate[((A + B*x + C*x^2)*(a + b*x^2 + c*x^4))/x,x]","\frac{1}{2} x^2 (a C+A b)+a A \log (x)+a B x+\frac{1}{4} x^4 (A c+b C)+\frac{1}{3} b B x^3+\frac{1}{5} B c x^5+\frac{1}{6} c C x^6","\frac{1}{2} x^2 (a C+A b)+a A \log (x)+a B x+\frac{1}{4} x^4 (A c+b C)+\frac{1}{3} b B x^3+\frac{1}{5} B c x^5+\frac{1}{6} c C x^6",1,"a*B*x + ((A*b + a*C)*x^2)/2 + (b*B*x^3)/3 + ((A*c + b*C)*x^4)/4 + (B*c*x^5)/5 + (c*C*x^6)/6 + a*A*Log[x]","A",1
5,1,63,63,0.0224133,"\int \frac{\left(A+B x+C x^2\right) \left(a+b x^2+c x^4\right)}{x^2} \, dx","Integrate[((A + B*x + C*x^2)*(a + b*x^2 + c*x^4))/x^2,x]","x (a C+A b)-\frac{a A}{x}+a B \log (x)+\frac{1}{3} x^3 (A c+b C)+\frac{1}{2} b B x^2+\frac{1}{4} B c x^4+\frac{1}{5} c C x^5","x (a C+A b)-\frac{a A}{x}+a B \log (x)+\frac{1}{3} x^3 (A c+b C)+\frac{1}{2} b B x^2+\frac{1}{4} B c x^4+\frac{1}{5} c C x^5",1,"-((a*A)/x) + (A*b + a*C)*x + (b*B*x^2)/2 + ((A*c + b*C)*x^3)/3 + (B*c*x^4)/4 + (c*C*x^5)/5 + a*B*Log[x]","A",1
6,1,58,63,0.037558,"\int \frac{\left(A+B x+C x^2\right) \left(a+b x^2+c x^4\right)}{x^3} \, dx","Integrate[((A + B*x + C*x^2)*(a + b*x^2 + c*x^4))/x^3,x]","\log (x) (a C+A b)-\frac{a (A+2 B x)}{2 x^2}+\frac{1}{12} x \left(c x \left(6 A+4 B x+3 C x^2\right)+6 b (2 B+C x)\right)","\log (x) (a C+A b)-\frac{a A}{2 x^2}-\frac{a B}{x}+\frac{1}{2} x^2 (A c+b C)+b B x+\frac{1}{3} B c x^3+\frac{1}{4} c C x^4",1,"-1/2*(a*(A + 2*B*x))/x^2 + (x*(6*b*(2*B + C*x) + c*x*(6*A + 4*B*x + 3*C*x^2)))/12 + (A*b + a*C)*Log[x]","A",1
7,1,60,63,0.0446089,"\int \frac{\left(A+B x+C x^2\right) \left(a+b x^2+c x^4\right)}{x^4} \, dx","Integrate[((A + B*x + C*x^2)*(a + b*x^2 + c*x^4))/x^4,x]","-\frac{a (2 A+3 x (B+2 C x))}{6 x^3}-\frac{A b}{x}+A c x+b B \log (x)+b C x+\frac{1}{2} B c x^2+\frac{1}{3} c C x^3","-\frac{a C+A b}{x}-\frac{a A}{3 x^3}-\frac{a B}{2 x^2}+x (A c+b C)+b B \log (x)+\frac{1}{2} B c x^2+\frac{1}{3} c C x^3",1,"-((A*b)/x) + A*c*x + b*C*x + (B*c*x^2)/2 + (c*C*x^3)/3 - (a*(2*A + 3*x*(B + 2*C*x)))/(6*x^3) + b*B*Log[x]","A",1
8,1,62,63,0.0268259,"\int \frac{\left(A+B x+C x^2\right) \left(a+b x^2+c x^4\right)}{x^5} \, dx","Integrate[((A + B*x + C*x^2)*(a + b*x^2 + c*x^4))/x^5,x]","-\frac{a \left(3 A+4 B x+6 C x^2\right)}{12 x^4}+\frac{-A b-2 b B x+c x^3 (2 B+C x)}{2 x^2}+\log (x) (A c+b C)","-\frac{a C+A b}{2 x^2}-\frac{a A}{4 x^4}-\frac{a B}{3 x^3}+\log (x) (A c+b C)-\frac{b B}{x}+B c x+\frac{1}{2} c C x^2",1,"-1/12*(a*(3*A + 4*B*x + 6*C*x^2))/x^4 + (-(A*b) - 2*b*B*x + c*x^3*(2*B + C*x))/(2*x^2) + (A*c + b*C)*Log[x]","A",1
9,1,63,63,0.0536747,"\int \frac{\left(A+B x+C x^2\right) \left(a+b x^2+c x^4\right)}{x^6} \, dx","Integrate[((A + B*x + C*x^2)*(a + b*x^2 + c*x^4))/x^6,x]","B c \log (x)-\frac{12 a A+5 a x (3 B+4 C x)+20 A x^2 \left(b+3 c x^2\right)+30 b x^3 (B+2 C x)-60 c C x^6}{60 x^5}","-\frac{a C+A b}{3 x^3}-\frac{a A}{5 x^5}-\frac{a B}{4 x^4}-\frac{A c+b C}{x}-\frac{b B}{2 x^2}+B c \log (x)+c C x",1,"-1/60*(12*a*A - 60*c*C*x^6 + 30*b*x^3*(B + 2*C*x) + 5*a*x*(3*B + 4*C*x) + 20*A*x^2*(b + 3*c*x^2))/x^5 + B*c*Log[x]","A",1
10,1,68,68,0.0452748,"\int \frac{\left(A+B x+C x^2\right) \left(a+b x^2+c x^4\right)}{x^7} \, dx","Integrate[((A + B*x + C*x^2)*(a + b*x^2 + c*x^4))/x^7,x]","c C \log (x)-\frac{a (10 A+3 x (4 B+5 C x))+5 x^2 \left(3 A \left(b+2 c x^2\right)+2 x \left(2 b B+3 b C x+6 B c x^2\right)\right)}{60 x^6}","-\frac{a C+A b}{4 x^4}-\frac{a A}{6 x^6}-\frac{a B}{5 x^5}-\frac{A c+b C}{2 x^2}-\frac{b B}{3 x^3}-\frac{B c}{x}+c C \log (x)",1,"-1/60*(a*(10*A + 3*x*(4*B + 5*C*x)) + 5*x^2*(3*A*(b + 2*c*x^2) + 2*x*(2*b*B + 3*b*C*x + 6*B*c*x^2)))/x^6 + c*C*Log[x]","A",1
11,1,159,159,0.0437631,"\int x^2 \left(A+B x+C x^2\right) \left(a+b x^2+c x^4\right)^2 \, dx","Integrate[x^2*(A + B*x + C*x^2)*(a + b*x^2 + c*x^4)^2,x]","\frac{1}{3} a^2 A x^3+\frac{1}{4} a^2 B x^4+\frac{1}{9} x^9 \left(2 a c C+2 A b c+b^2 C\right)+\frac{1}{7} x^7 \left(2 a A c+2 a b C+A b^2\right)+\frac{1}{5} a x^5 (a C+2 A b)+\frac{1}{8} B x^8 \left(2 a c+b^2\right)+\frac{1}{3} a b B x^6+\frac{1}{11} c x^{11} (A c+2 b C)+\frac{1}{5} b B c x^{10}+\frac{1}{12} B c^2 x^{12}+\frac{1}{13} c^2 C x^{13}","\frac{1}{3} a^2 A x^3+\frac{1}{4} a^2 B x^4+\frac{1}{9} x^9 \left(C \left(2 a c+b^2\right)+2 A b c\right)+\frac{1}{7} x^7 \left(A \left(2 a c+b^2\right)+2 a b C\right)+\frac{1}{5} a x^5 (a C+2 A b)+\frac{1}{8} B x^8 \left(2 a c+b^2\right)+\frac{1}{3} a b B x^6+\frac{1}{11} c x^{11} (A c+2 b C)+\frac{1}{5} b B c x^{10}+\frac{1}{12} B c^2 x^{12}+\frac{1}{13} c^2 C x^{13}",1,"(a^2*A*x^3)/3 + (a^2*B*x^4)/4 + (a*(2*A*b + a*C)*x^5)/5 + (a*b*B*x^6)/3 + ((A*b^2 + 2*a*A*c + 2*a*b*C)*x^7)/7 + (B*(b^2 + 2*a*c)*x^8)/8 + ((2*A*b*c + b^2*C + 2*a*c*C)*x^9)/9 + (b*B*c*x^10)/5 + (c*(A*c + 2*b*C)*x^11)/11 + (B*c^2*x^12)/12 + (c^2*C*x^13)/13","A",1
12,1,159,159,0.0358781,"\int x \left(A+B x+C x^2\right) \left(a+b x^2+c x^4\right)^2 \, dx","Integrate[x*(A + B*x + C*x^2)*(a + b*x^2 + c*x^4)^2,x]","\frac{1}{2} a^2 A x^2+\frac{1}{3} a^2 B x^3+\frac{1}{8} x^8 \left(2 a c C+2 A b c+b^2 C\right)+\frac{1}{6} x^6 \left(2 a A c+2 a b C+A b^2\right)+\frac{1}{4} a x^4 (a C+2 A b)+\frac{1}{7} B x^7 \left(2 a c+b^2\right)+\frac{2}{5} a b B x^5+\frac{1}{10} c x^{10} (A c+2 b C)+\frac{2}{9} b B c x^9+\frac{1}{11} B c^2 x^{11}+\frac{1}{12} c^2 C x^{12}","\frac{1}{2} a^2 A x^2+\frac{1}{3} a^2 B x^3+\frac{1}{8} x^8 \left(C \left(2 a c+b^2\right)+2 A b c\right)+\frac{1}{6} x^6 \left(A \left(2 a c+b^2\right)+2 a b C\right)+\frac{1}{4} a x^4 (a C+2 A b)+\frac{1}{7} B x^7 \left(2 a c+b^2\right)+\frac{2}{5} a b B x^5+\frac{1}{10} c x^{10} (A c+2 b C)+\frac{2}{9} b B c x^9+\frac{1}{11} B c^2 x^{11}+\frac{1}{12} c^2 C x^{12}",1,"(a^2*A*x^2)/2 + (a^2*B*x^3)/3 + (a*(2*A*b + a*C)*x^4)/4 + (2*a*b*B*x^5)/5 + ((A*b^2 + 2*a*A*c + 2*a*b*C)*x^6)/6 + (B*(b^2 + 2*a*c)*x^7)/7 + ((2*A*b*c + b^2*C + 2*a*c*C)*x^8)/8 + (2*b*B*c*x^9)/9 + (c*(A*c + 2*b*C)*x^10)/10 + (B*c^2*x^11)/11 + (c^2*C*x^12)/12","A",1
13,1,154,154,0.028942,"\int \left(A+B x+C x^2\right) \left(a+b x^2+c x^4\right)^2 \, dx","Integrate[(A + B*x + C*x^2)*(a + b*x^2 + c*x^4)^2,x]","a^2 A x+\frac{1}{2} a^2 B x^2+\frac{1}{7} x^7 \left(2 a c C+2 A b c+b^2 C\right)+\frac{1}{5} x^5 \left(2 a A c+2 a b C+A b^2\right)+\frac{1}{3} a x^3 (a C+2 A b)+\frac{1}{6} B x^6 \left(2 a c+b^2\right)+\frac{1}{2} a b B x^4+\frac{1}{9} c x^9 (A c+2 b C)+\frac{1}{4} b B c x^8+\frac{1}{10} B c^2 x^{10}+\frac{1}{11} c^2 C x^{11}","a^2 A x+\frac{1}{2} a^2 B x^2+\frac{1}{7} x^7 \left(C \left(2 a c+b^2\right)+2 A b c\right)+\frac{1}{5} x^5 \left(A \left(2 a c+b^2\right)+2 a b C\right)+\frac{1}{3} a x^3 (a C+2 A b)+\frac{1}{6} B x^6 \left(2 a c+b^2\right)+\frac{1}{2} a b B x^4+\frac{1}{9} c x^9 (A c+2 b C)+\frac{1}{4} b B c x^8+\frac{1}{10} B c^2 x^{10}+\frac{1}{11} c^2 C x^{11}",1,"a^2*A*x + (a^2*B*x^2)/2 + (a*(2*A*b + a*C)*x^3)/3 + (a*b*B*x^4)/2 + ((A*b^2 + 2*a*A*c + 2*a*b*C)*x^5)/5 + (B*(b^2 + 2*a*c)*x^6)/6 + ((2*A*b*c + b^2*C + 2*a*c*C)*x^7)/7 + (b*B*c*x^8)/4 + (c*(A*c + 2*b*C)*x^9)/9 + (B*c^2*x^10)/10 + (c^2*C*x^11)/11","A",1
14,1,150,150,0.0381284,"\int \frac{\left(A+B x+C x^2\right) \left(a+b x^2+c x^4\right)^2}{x} \, dx","Integrate[((A + B*x + C*x^2)*(a + b*x^2 + c*x^4)^2)/x,x]","a^2 A \log (x)+a^2 B x+\frac{1}{6} x^6 \left(2 a c C+2 A b c+b^2 C\right)+\frac{1}{4} x^4 \left(2 a A c+2 a b C+A b^2\right)+\frac{1}{2} a x^2 (a C+2 A b)+\frac{1}{5} B x^5 \left(2 a c+b^2\right)+\frac{2}{3} a b B x^3+\frac{1}{8} c x^8 (A c+2 b C)+\frac{2}{7} b B c x^7+\frac{1}{9} B c^2 x^9+\frac{1}{10} c^2 C x^{10}","a^2 A \log (x)+a^2 B x+\frac{1}{6} x^6 \left(C \left(2 a c+b^2\right)+2 A b c\right)+\frac{1}{4} x^4 \left(A \left(2 a c+b^2\right)+2 a b C\right)+\frac{1}{2} a x^2 (a C+2 A b)+\frac{1}{5} B x^5 \left(2 a c+b^2\right)+\frac{2}{3} a b B x^3+\frac{1}{8} c x^8 (A c+2 b C)+\frac{2}{7} b B c x^7+\frac{1}{9} B c^2 x^9+\frac{1}{10} c^2 C x^{10}",1,"a^2*B*x + (a*(2*A*b + a*C)*x^2)/2 + (2*a*b*B*x^3)/3 + ((A*b^2 + 2*a*A*c + 2*a*b*C)*x^4)/4 + (B*(b^2 + 2*a*c)*x^5)/5 + ((2*A*b*c + b^2*C + 2*a*c*C)*x^6)/6 + (2*b*B*c*x^7)/7 + (c*(A*c + 2*b*C)*x^8)/8 + (B*c^2*x^9)/9 + (c^2*C*x^10)/10 + a^2*A*Log[x]","A",1
15,1,145,145,0.0914288,"\int \frac{\left(A+B x+C x^2\right) \left(a+b x^2+c x^4\right)^2}{x^2} \, dx","Integrate[((A + B*x + C*x^2)*(a + b*x^2 + c*x^4)^2)/x^2,x]","-\frac{a^2 A}{x}+a^2 B \log (x)+\frac{1}{5} x^5 \left(2 a c C+2 A b c+b^2 C\right)+\frac{1}{3} x^3 \left(2 a A c+2 a b C+A b^2\right)+a x (a C+2 A b)+\frac{1}{4} B x^4 \left(2 a c+b^2\right)+a b B x^2+\frac{1}{7} c x^7 (A c+2 b C)+\frac{1}{3} b B c x^6+\frac{1}{8} B c^2 x^8+\frac{1}{9} c^2 C x^9","-\frac{a^2 A}{x}+a^2 B \log (x)+\frac{1}{5} x^5 \left(C \left(2 a c+b^2\right)+2 A b c\right)+\frac{1}{3} x^3 \left(A \left(2 a c+b^2\right)+2 a b C\right)+a x (a C+2 A b)+\frac{1}{4} B x^4 \left(2 a c+b^2\right)+a b B x^2+\frac{1}{7} c x^7 (A c+2 b C)+\frac{1}{3} b B c x^6+\frac{1}{8} B c^2 x^8+\frac{1}{9} c^2 C x^9",1,"-((a^2*A)/x) + a*(2*A*b + a*C)*x + a*b*B*x^2 + ((A*b^2 + 2*a*A*c + 2*a*b*C)*x^3)/3 + (B*(b^2 + 2*a*c)*x^4)/4 + ((2*A*b*c + b^2*C + 2*a*c*C)*x^5)/5 + (b*B*c*x^6)/3 + (c*(A*c + 2*b*C)*x^7)/7 + (B*c^2*x^8)/8 + (c^2*C*x^9)/9 + a^2*B*Log[x]","A",1
16,1,139,149,0.0939864,"\int \frac{\left(A+B x+C x^2\right) \left(a+b x^2+c x^4\right)^2}{x^3} \, dx","Integrate[((A + B*x + C*x^2)*(a + b*x^2 + c*x^4)^2)/x^3,x]","-\frac{a^2 (A+2 B x)}{2 x^2}+\frac{1}{6} a x \left(c x \left(6 A+4 B x+3 C x^2\right)+6 b (2 B+C x)\right)+a \log (x) (a C+2 A b)+\frac{1}{840} x^2 \left(140 A \left(3 b^2+3 b c x^2+c^2 x^4\right)+70 b^2 x (4 B+3 C x)+56 b c x^3 (6 B+5 C x)+15 c^2 x^5 (8 B+7 C x)\right)","-\frac{a^2 A}{2 x^2}-\frac{a^2 B}{x}+\frac{1}{4} x^4 \left(C \left(2 a c+b^2\right)+2 A b c\right)+\frac{1}{2} x^2 \left(A \left(2 a c+b^2\right)+2 a b C\right)+a \log (x) (a C+2 A b)+\frac{1}{3} B x^3 \left(2 a c+b^2\right)+2 a b B x+\frac{1}{6} c x^6 (A c+2 b C)+\frac{2}{5} b B c x^5+\frac{1}{7} B c^2 x^7+\frac{1}{8} c^2 C x^8",1,"-1/2*(a^2*(A + 2*B*x))/x^2 + (a*x*(6*b*(2*B + C*x) + c*x*(6*A + 4*B*x + 3*C*x^2)))/6 + (x^2*(70*b^2*x*(4*B + 3*C*x) + 56*b*c*x^3*(6*B + 5*C*x) + 15*c^2*x^5*(8*B + 7*C*x) + 140*A*(3*b^2 + 3*b*c*x^2 + c^2*x^4)))/840 + a*(2*A*b + a*C)*Log[x]","A",1
17,1,151,149,0.0767649,"\int \frac{\left(A+B x+C x^2\right) \left(a+b x^2+c x^4\right)^2}{x^4} \, dx","Integrate[((A + B*x + C*x^2)*(a + b*x^2 + c*x^4)^2)/x^4,x]","\frac{a^2 (-C)-2 a A b}{x}-\frac{a^2 A}{3 x^3}-\frac{a^2 B}{2 x^2}+\frac{1}{3} x^3 \left(2 a c C+2 A b c+b^2 C\right)+x \left(2 a A c+2 a b C+A b^2\right)+\frac{1}{2} B x^2 \left(2 a c+b^2\right)+2 a b B \log (x)+\frac{1}{5} c x^5 (A c+2 b C)+\frac{1}{2} b B c x^4+\frac{1}{6} B c^2 x^6+\frac{1}{7} c^2 C x^7","-\frac{a^2 A}{3 x^3}-\frac{a^2 B}{2 x^2}+\frac{1}{3} x^3 \left(C \left(2 a c+b^2\right)+2 A b c\right)+x \left(A \left(2 a c+b^2\right)+2 a b C\right)-\frac{a (a C+2 A b)}{x}+\frac{1}{2} B x^2 \left(2 a c+b^2\right)+2 a b B \log (x)+\frac{1}{5} c x^5 (A c+2 b C)+\frac{1}{2} b B c x^4+\frac{1}{6} B c^2 x^6+\frac{1}{7} c^2 C x^7",1,"-1/3*(a^2*A)/x^3 - (a^2*B)/(2*x^2) + (-2*a*A*b - a^2*C)/x + (A*b^2 + 2*a*A*c + 2*a*b*C)*x + (B*(b^2 + 2*a*c)*x^2)/2 + ((2*A*b*c + b^2*C + 2*a*c*C)*x^3)/3 + (b*B*c*x^4)/2 + (c*(A*c + 2*b*C)*x^5)/5 + (B*c^2*x^6)/6 + (c^2*C*x^7)/7 + 2*a*b*B*Log[x]","A",1
18,1,130,148,0.0805895,"\int \frac{\left(A+B x+C x^2\right) \left(a+b x^2+c x^4\right)^2}{x^5} \, dx","Integrate[((A + B*x + C*x^2)*(a + b*x^2 + c*x^4)^2)/x^5,x]","-\frac{a^2 \left(3 A+4 B x+6 C x^2\right)}{12 x^4}+\log (x) \left(A \left(2 a c+b^2\right)+2 a b C\right)+\frac{a \left(-A b-2 b B x+c x^3 (2 B+C x)\right)}{x^2}+\frac{1}{60} x \left(10 b c x (6 A+x (4 B+3 C x))+c^2 x^3 (15 A+2 x (6 B+5 C x))+30 b^2 (2 B+C x)\right)","-\frac{a^2 A}{4 x^4}-\frac{a^2 B}{3 x^3}+\frac{1}{2} x^2 \left(C \left(2 a c+b^2\right)+2 A b c\right)+\log (x) \left(A \left(2 a c+b^2\right)+2 a b C\right)-\frac{a (a C+2 A b)}{2 x^2}+B x \left(2 a c+b^2\right)-\frac{2 a b B}{x}+\frac{1}{4} c x^4 (A c+2 b C)+\frac{2}{3} b B c x^3+\frac{1}{5} B c^2 x^5+\frac{1}{6} c^2 C x^6",1,"-1/12*(a^2*(3*A + 4*B*x + 6*C*x^2))/x^4 + (a*(-(A*b) - 2*b*B*x + c*x^3*(2*B + C*x)))/x^2 + (x*(30*b^2*(2*B + C*x) + 10*b*c*x*(6*A + x*(4*B + 3*C*x)) + c^2*x^3*(15*A + 2*x*(6*B + 5*C*x))))/60 + (A*(b^2 + 2*a*c) + 2*a*b*C)*Log[x]","A",1
19,1,142,143,0.0767171,"\int \frac{\left(A+B x+C x^2\right) \left(a+b x^2+c x^4\right)^2}{x^6} \, dx","Integrate[((A + B*x + C*x^2)*(a + b*x^2 + c*x^4)^2)/x^6,x]","-\frac{a^2 A}{5 x^5}-\frac{a^2 B}{4 x^4}-\frac{2 a A c+2 a b C+A b^2}{x}-\frac{a (a C+2 A b)}{3 x^3}+B \log (x) \left(2 a c+b^2\right)+C x \left(2 a c+b^2\right)-\frac{a b B}{x^2}+\frac{1}{3} c x^3 (A c+2 b C)+2 A b c x+b B c x^2+\frac{1}{4} B c^2 x^4+\frac{1}{5} c^2 C x^5","-\frac{a^2 A}{5 x^5}-\frac{a^2 B}{4 x^4}+x \left(C \left(2 a c+b^2\right)+2 A b c\right)-\frac{A \left(2 a c+b^2\right)+2 a b C}{x}-\frac{a (a C+2 A b)}{3 x^3}+B \log (x) \left(2 a c+b^2\right)-\frac{a b B}{x^2}+\frac{1}{3} c x^3 (A c+2 b C)+b B c x^2+\frac{1}{4} B c^2 x^4+\frac{1}{5} c^2 C x^5",1,"-1/5*(a^2*A)/x^5 - (a^2*B)/(4*x^4) - (a*(2*A*b + a*C))/(3*x^3) - (a*b*B)/x^2 - (A*b^2 + 2*a*A*c + 2*a*b*C)/x + 2*A*b*c*x + (b^2 + 2*a*c)*C*x + b*B*c*x^2 + (c*(A*c + 2*b*C)*x^3)/3 + (B*c^2*x^4)/4 + (c^2*C*x^5)/5 + B*(b^2 + 2*a*c)*Log[x]","A",1
20,1,144,149,0.0921903,"\int \frac{\left(A+B x+C x^2\right) \left(a+b x^2+c x^4\right)^2}{x^7} \, dx","Integrate[((A + B*x + C*x^2)*(a + b*x^2 + c*x^4)^2)/x^7,x]","-\frac{a^2 (10 A+3 x (4 B+5 C x))}{60 x^6}+\log (x) \left(C \left(2 a c+b^2\right)+2 A b c\right)-\frac{a \left(3 A \left(b+2 c x^2\right)+2 x \left(2 b B+3 b C x+6 B c x^2\right)\right)}{6 x^4}+\frac{A \left(c^2 x^4-b^2\right)}{2 x^2}-\frac{b^2 B}{x}+b c x (2 B+C x)+\frac{1}{12} c^2 x^3 (4 B+3 C x)","-\frac{a^2 A}{6 x^6}-\frac{a^2 B}{5 x^5}-\frac{A \left(2 a c+b^2\right)+2 a b C}{2 x^2}+\log (x) \left(C \left(2 a c+b^2\right)+2 A b c\right)-\frac{a (a C+2 A b)}{4 x^4}-\frac{B \left(2 a c+b^2\right)}{x}-\frac{2 a b B}{3 x^3}+\frac{1}{2} c x^2 (A c+2 b C)+2 b B c x+\frac{1}{3} B c^2 x^3+\frac{1}{4} c^2 C x^4",1,"-((b^2*B)/x) + b*c*x*(2*B + C*x) + (c^2*x^3*(4*B + 3*C*x))/12 + (A*(-b^2 + c^2*x^4))/(2*x^2) - (a^2*(10*A + 3*x*(4*B + 5*C*x)))/(60*x^6) - (a*(3*A*(b + 2*c*x^2) + 2*x*(2*b*B + 3*b*C*x + 6*B*c*x^2)))/(6*x^4) + (2*A*b*c + (b^2 + 2*a*c)*C)*Log[x]","A",1
21,1,460,339,0.5719036,"\int \frac{x^4 \left(A+B x+C x^2\right)}{a+b x^2+c x^4} \, dx","Integrate[(x^4*(A + B*x + C*x^2))/(a + b*x^2 + c*x^4),x]","\frac{\frac{6 \sqrt{2} \left(A c \left(-b \sqrt{b^2-4 a c}-2 a c+b^2\right)+C \left(b^2 \sqrt{b^2-4 a c}-a c \sqrt{b^2-4 a c}+3 a b c-b^3\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{6 \sqrt{2} \left(C \left(b^2 \sqrt{b^2-4 a c}-a c \sqrt{b^2-4 a c}-3 a b c+b^3\right)-A c \left(b \sqrt{b^2-4 a c}-2 a c+b^2\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{3 B \sqrt{c} \left(b \sqrt{b^2-4 a c}+2 a c-b^2\right) \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)}{\sqrt{b^2-4 a c}}-\frac{3 B \sqrt{c} \left(b \sqrt{b^2-4 a c}-2 a c+b^2\right) \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\sqrt{b^2-4 a c}}+12 \sqrt{c} x (A c-b C)+6 B c^{3/2} x^2+4 c^{3/2} C x^3}{12 c^{5/2}}","-\frac{\left(-\frac{A c \left(b^2-2 a c\right)-b C \left(b^2-3 a c\right)}{\sqrt{b^2-4 a c}}+a c C+A b c+b^2 (-C)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} c^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left(\frac{A c \left(b^2-2 a c\right)-b C \left(b^2-3 a c\right)}{\sqrt{b^2-4 a c}}+a c C+A b c+b^2 (-C)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} c^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{B \left(b^2-2 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{2 c^2 \sqrt{b^2-4 a c}}-\frac{b B \log \left(a+b x^2+c x^4\right)}{4 c^2}+\frac{x (A c-b C)}{c^2}+\frac{B x^2}{2 c}+\frac{C x^3}{3 c}",1,"(12*Sqrt[c]*(A*c - b*C)*x + 6*B*c^(3/2)*x^2 + 4*c^(3/2)*C*x^3 + (6*Sqrt[2]*(A*c*(b^2 - 2*a*c - b*Sqrt[b^2 - 4*a*c]) + (-b^3 + 3*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - a*c*Sqrt[b^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (6*Sqrt[2]*(-(A*c*(b^2 - 2*a*c + b*Sqrt[b^2 - 4*a*c])) + (b^3 - 3*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - a*c*Sqrt[b^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]) - (3*B*Sqrt[c]*(-b^2 + 2*a*c + b*Sqrt[b^2 - 4*a*c])*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/Sqrt[b^2 - 4*a*c] - (3*B*Sqrt[c]*(b^2 - 2*a*c + b*Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/Sqrt[b^2 - 4*a*c])/(12*c^(5/2))","A",1
22,1,377,278,0.4200462,"\int \frac{x^3 \left(A+B x+C x^2\right)}{a+b x^2+c x^4} \, dx","Integrate[(x^3*(A + B*x + C*x^2))/(a + b*x^2 + c*x^4),x]","\frac{\frac{\left(A c \left(\sqrt{b^2-4 a c}-b\right)+C \left(-b \sqrt{b^2-4 a c}-2 a c+b^2\right)\right) \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)}{\sqrt{b^2-4 a c}}-\frac{\left(C \left(b \sqrt{b^2-4 a c}-2 a c+b^2\right)-A c \left(\sqrt{b^2-4 a c}+b\right)\right) \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\sqrt{b^2-4 a c}}-\frac{2 \sqrt{2} B \sqrt{c} \left(b \sqrt{b^2-4 a c}+2 a c-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{2 \sqrt{2} B \sqrt{c} \left(b \sqrt{b^2-4 a c}-2 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+4 B c x+2 c C x^2}{4 c^2}","\frac{\left(2 a c C+A b c+b^2 (-C)\right) \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{2 c^2 \sqrt{b^2-4 a c}}+\frac{(A c-b C) \log \left(a+b x^2+c x^4\right)}{4 c^2}-\frac{B \left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} c^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{B \left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} c^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{B x}{c}+\frac{C x^2}{2 c}",1,"(4*B*c*x + 2*c*C*x^2 - (2*Sqrt[2]*B*Sqrt[c]*(-b^2 + 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (2*Sqrt[2]*B*Sqrt[c]*(b^2 - 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + ((A*c*(-b + Sqrt[b^2 - 4*a*c]) + (b^2 - 2*a*c - b*Sqrt[b^2 - 4*a*c])*C)*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/Sqrt[b^2 - 4*a*c] - ((-(A*c*(b + Sqrt[b^2 - 4*a*c])) + (b^2 - 2*a*c + b*Sqrt[b^2 - 4*a*c])*C)*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/Sqrt[b^2 - 4*a*c])/(4*c^2)","A",1
23,1,360,270,0.3657424,"\int \frac{x^2 \left(A+B x+C x^2\right)}{a+b x^2+c x^4} \, dx","Integrate[(x^2*(A + B*x + C*x^2))/(a + b*x^2 + c*x^4),x]","\frac{-\frac{2 \sqrt{2} \left(A c \left(b-\sqrt{b^2-4 a c}\right)+C \left(b \sqrt{b^2-4 a c}+2 a c-b^2\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{2 \sqrt{2} \left(C \left(b \sqrt{b^2-4 a c}-2 a c+b^2\right)-A c \left(\sqrt{b^2-4 a c}+b\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{B \sqrt{c} \left(\sqrt{b^2-4 a c}-b\right) \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)}{\sqrt{b^2-4 a c}}+\frac{B \sqrt{c} \left(\sqrt{b^2-4 a c}+b\right) \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\sqrt{b^2-4 a c}}+4 \sqrt{c} C x}{4 c^{3/2}}","\frac{\left(-\frac{A b c-C \left(b^2-2 a c\right)}{\sqrt{b^2-4 a c}}+A c-b C\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} c^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left(\frac{2 a c C+A b c+b^2 (-C)}{\sqrt{b^2-4 a c}}+A c-b C\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} c^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{b B \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{2 c \sqrt{b^2-4 a c}}+\frac{B \log \left(a+b x^2+c x^4\right)}{4 c}+\frac{C x}{c}",1,"(4*Sqrt[c]*C*x - (2*Sqrt[2]*(A*c*(b - Sqrt[b^2 - 4*a*c]) + (-b^2 + 2*a*c + b*Sqrt[b^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (2*Sqrt[2]*(-(A*c*(b + Sqrt[b^2 - 4*a*c])) + (b^2 - 2*a*c + b*Sqrt[b^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (B*Sqrt[c]*(-b + Sqrt[b^2 - 4*a*c])*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/Sqrt[b^2 - 4*a*c] + (B*Sqrt[c]*(b + Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/Sqrt[b^2 - 4*a*c])/(4*c^(3/2))","A",1
24,1,240,223,0.3589868,"\int \frac{x \left(A+B x+C x^2\right)}{a+b x^2+c x^4} \, dx","Integrate[(x*(A + B*x + C*x^2))/(a + b*x^2 + c*x^4),x]","\frac{\left(C \left(\sqrt{b^2-4 a c}-b\right)+2 A c\right) \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)-\left(2 A c-C \left(\sqrt{b^2-4 a c}+b\right)\right) \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)-2 \sqrt{2} B \sqrt{c} \sqrt{b-\sqrt{b^2-4 a c}} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)+2 \sqrt{2} B \sqrt{c} \sqrt{\sqrt{b^2-4 a c}+b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{4 c \sqrt{b^2-4 a c}}","-\frac{(2 A c-b C) \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{2 c \sqrt{b^2-4 a c}}-\frac{B \sqrt{b-\sqrt{b^2-4 a c}} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}+\frac{B \sqrt{\sqrt{b^2-4 a c}+b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}+\frac{C \log \left(a+b x^2+c x^4\right)}{4 c}",1,"(-2*Sqrt[2]*B*Sqrt[c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]] + 2*Sqrt[2]*B*Sqrt[c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]] + (2*A*c + (-b + Sqrt[b^2 - 4*a*c])*C)*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2] - (2*A*c - (b + Sqrt[b^2 - 4*a*c])*C)*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(4*c*Sqrt[b^2 - 4*a*c])","A",1
25,1,234,211,0.2086407,"\int \frac{A+B x+C x^2}{a+b x^2+c x^4} \, dx","Integrate[(A + B*x + C*x^2)/(a + b*x^2 + c*x^4),x]","\frac{\frac{\sqrt{2} \left(C \left(\sqrt{b^2-4 a c}-b\right)+2 A c\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{c} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \left(C \left(\sqrt{b^2-4 a c}+b\right)-2 A c\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{c} \sqrt{\sqrt{b^2-4 a c}+b}}+B \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)-B \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{2 \sqrt{b^2-4 a c}}","\frac{\left(\frac{2 A c-b C}{\sqrt{b^2-4 a c}}+C\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} \sqrt{c} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left(C-\frac{2 A c-b C}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} \sqrt{c} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{B \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c}}",1,"((Sqrt[2]*(2*A*c + (-b + Sqrt[b^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(-2*A*c + (b + Sqrt[b^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + B*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2] - B*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(2*Sqrt[b^2 - 4*a*c])","A",1
26,1,285,229,0.444546,"\int \frac{A+B x+C x^2}{x \left(a+b x^2+c x^4\right)} \, dx","Integrate[(A + B*x + C*x^2)/(x*(a + b*x^2 + c*x^4)),x]","-\frac{\left(A \left(\sqrt{b^2-4 a c}+b\right)-2 a C\right) \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)}{4 a \sqrt{b^2-4 a c}}-\frac{\left(A \left(\sqrt{b^2-4 a c}-b\right)+2 a C\right) \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{4 a \sqrt{b^2-4 a c}}+\frac{A \log (x)}{a}+\frac{\sqrt{2} B \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} B \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}","\frac{(A b-2 a C) \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{2 a \sqrt{b^2-4 a c}}-\frac{A \log \left(a+b x^2+c x^4\right)}{4 a}+\frac{A \log (x)}{a}+\frac{\sqrt{2} B \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} B \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}",1,"(Sqrt[2]*B*Sqrt[c]*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*B*Sqrt[c]*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (A*Log[x])/a - ((A*(b + Sqrt[b^2 - 4*a*c]) - 2*a*C)*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(4*a*Sqrt[b^2 - 4*a*c]) - ((A*(-b + Sqrt[b^2 - 4*a*c]) + 2*a*C)*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(4*a*Sqrt[b^2 - 4*a*c])","A",1
27,1,315,260,1.0798984,"\int \frac{A+B x+C x^2}{x^2 \left(a+b x^2+c x^4\right)} \, dx","Integrate[(A + B*x + C*x^2)/(x^2*(a + b*x^2 + c*x^4)),x]","-\frac{\frac{2 \sqrt{2} \sqrt{c} \left(A \left(\sqrt{b^2-4 a c}+b\right)-2 a C\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{2 \sqrt{2} \sqrt{c} \left(A \left(\sqrt{b^2-4 a c}-b\right)+2 a C\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{B \left(\sqrt{b^2-4 a c}+b\right) \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)}{\sqrt{b^2-4 a c}}+\frac{B \left(\sqrt{b^2-4 a c}-b\right) \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\sqrt{b^2-4 a c}}+\frac{4 A}{x}-4 B \log (x)}{4 a}","-\frac{\sqrt{c} \left(\frac{A b-2 a C}{\sqrt{b^2-4 a c}}+A\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} a \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left(A-\frac{A b-2 a C}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} a \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{A}{a x}+\frac{b B \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{2 a \sqrt{b^2-4 a c}}-\frac{B \log \left(a+b x^2+c x^4\right)}{4 a}+\frac{B \log (x)}{a}",1,"-1/4*((4*A)/x + (2*Sqrt[2]*Sqrt[c]*(A*(b + Sqrt[b^2 - 4*a*c]) - 2*a*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (2*Sqrt[2]*Sqrt[c]*(A*(-b + Sqrt[b^2 - 4*a*c]) + 2*a*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]) - 4*B*Log[x] + (B*(b + Sqrt[b^2 - 4*a*c])*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/Sqrt[b^2 - 4*a*c] + (B*(-b + Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/Sqrt[b^2 - 4*a*c])/a","A",1
28,1,377,288,0.8893621,"\int \frac{A+B x+C x^2}{x^3 \left(a+b x^2+c x^4\right)} \, dx","Integrate[(A + B*x + C*x^2)/(x^3*(a + b*x^2 + c*x^4)),x]","\frac{\frac{\left(A \left(b \sqrt{b^2-4 a c}-2 a c+b^2\right)-a C \left(\sqrt{b^2-4 a c}+b\right)\right) \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)}{\sqrt{b^2-4 a c}}+\frac{\left(A \left(b \sqrt{b^2-4 a c}+2 a c-b^2\right)+a C \left(b-\sqrt{b^2-4 a c}\right)\right) \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\sqrt{b^2-4 a c}}+4 \log (x) (a C-A b)-\frac{2 a A}{x^2}-\frac{2 \sqrt{2} a B \sqrt{c} \left(\sqrt{b^2-4 a c}+b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{2 \sqrt{2} a B \sqrt{c} \left(\sqrt{b^2-4 a c}-b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{4 a B}{x}}{4 a^2}","-\frac{\left(A \left(b^2-2 a c\right)-a b C\right) \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{2 a^2 \sqrt{b^2-4 a c}}+\frac{(A b-a C) \log \left(a+b x^2+c x^4\right)}{4 a^2}-\frac{\log (x) (A b-a C)}{a^2}-\frac{A}{2 a x^2}-\frac{B \sqrt{c} \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} a \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{B \sqrt{c} \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} a \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{B}{a x}",1,"((-2*a*A)/x^2 - (4*a*B)/x - (2*Sqrt[2]*a*B*Sqrt[c]*(b + Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (2*Sqrt[2]*a*B*Sqrt[c]*(-b + Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + 4*(-(A*b) + a*C)*Log[x] + ((A*(b^2 - 2*a*c + b*Sqrt[b^2 - 4*a*c]) - a*(b + Sqrt[b^2 - 4*a*c])*C)*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/Sqrt[b^2 - 4*a*c] + ((A*(-b^2 + 2*a*c + b*Sqrt[b^2 - 4*a*c]) + a*(b - Sqrt[b^2 - 4*a*c])*C)*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/Sqrt[b^2 - 4*a*c])/(4*a^2)","A",1
29,1,444,412,1.3815233,"\int \frac{x^4 \left(A+B x+C x^2\right)}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(x^4*(A + B*x + C*x^2))/(a + b*x^2 + c*x^4)^2,x]","\frac{1}{4} \left(\frac{2 \left(a (b (B+C x)-2 c x (A+x (B+C x)))+b x^2 (b (B+C x)-A c x)\right)}{c \left(4 a c-b^2\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \left(C \left(b^2 \sqrt{b^2-4 a c}-6 a c \sqrt{b^2-4 a c}+8 a b c-b^3\right)-A c \left(-b \sqrt{b^2-4 a c}+4 a c+b^2\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{c^{3/2} \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \left(A c \left(b \sqrt{b^2-4 a c}+4 a c+b^2\right)+C \left(b^2 \sqrt{b^2-4 a c}-6 a c \sqrt{b^2-4 a c}-8 a b c+b^3\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{c^{3/2} \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{4 a B \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}+\frac{4 a B \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}\right)","\frac{\left(-\frac{A c \left(4 a c+b^2\right)+b C \left(b^2-8 a c\right)}{\sqrt{b^2-4 a c}}+C \left(b^2-6 a c\right)+A b c\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} c^{3/2} \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left(\frac{A c \left(4 a c+b^2\right)+b C \left(b^2-8 a c\right)}{\sqrt{b^2-4 a c}}+C \left(b^2-6 a c\right)+A b c\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} c^{3/2} \left(b^2-4 a c\right) \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{x^3 \left(-2 a C+x^2 (2 A c-b C)+A b\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{x (2 A c-b C)}{2 c \left(b^2-4 a c\right)}+\frac{2 a B \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/2}}+\frac{B x^2 \left(2 a+b x^2\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"((2*(b*x^2*(-(A*c*x) + b*(B + C*x)) + a*(b*(B + C*x) - 2*c*x*(A + x*(B + C*x)))))/(c*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*(-(A*c*(b^2 + 4*a*c - b*Sqrt[b^2 - 4*a*c])) + (-b^3 + 8*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - 6*a*c*Sqrt[b^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(c^(3/2)*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(A*c*(b^2 + 4*a*c + b*Sqrt[b^2 - 4*a*c]) + (b^3 - 8*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - 6*a*c*Sqrt[b^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(c^(3/2)*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) - (4*a*B*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(3/2) + (4*a*B*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/4","A",1
30,1,358,347,0.8943242,"\int \frac{x^3 \left(A+B x+C x^2\right)}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(x^3*(A + B*x + C*x^2))/(a + b*x^2 + c*x^4)^2,x]","\frac{1}{4} \left(-\frac{2 \left(a (2 A c-b C+2 c x (B+C x))+b x^2 (A c-b C+B c x)\right)}{c \left(4 a c-b^2\right) \left(a+b x^2+c x^4\right)}+\frac{2 (A b-2 a C) \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}-\frac{2 (A b-2 a C) \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}+\frac{\sqrt{2} B \left(b \sqrt{b^2-4 a c}-4 a c-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{c} \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} B \left(b \sqrt{b^2-4 a c}+4 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{c} \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}\right)","-\frac{(A b-2 a C) \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/2}}+\frac{x^2 \left(2 a c C+A b c+b^2 (-C)\right)+a (2 A c-b C)}{2 c \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{B x \left(2 a+b x^2\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{B \left(b-\frac{4 a c+b^2}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} \sqrt{c} \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{B \left(b \sqrt{b^2-4 a c}+4 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} \sqrt{c} \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}",1,"((-2*(b*x^2*(A*c - b*C + B*c*x) + a*(2*A*c - b*C + 2*c*x*(B + C*x))))/(c*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*B*(-b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*B*(b^2 + 4*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (2*(A*b - 2*a*C)*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(3/2) - (2*(A*b - 2*a*C)*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/4","A",1
31,1,378,356,1.0389798,"\int \frac{x^2 \left(A+B x+C x^2\right)}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(x^2*(A + B*x + C*x^2))/(a + b*x^2 + c*x^4)^2,x]","\frac{1}{4} \left(\frac{4 a (B+C x)+2 x \left(b x (B+C x)-A \left(b+2 c x^2\right)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \left(C \left(b \sqrt{b^2-4 a c}-4 a c-b^2\right)-2 A c \left(\sqrt{b^2-4 a c}-2 b\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{c} \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \left(C \left(b \sqrt{b^2-4 a c}+4 a c+b^2\right)-2 A c \left(\sqrt{b^2-4 a c}+2 b\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{c} \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 b B \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}-\frac{2 b B \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}\right)","-\frac{x \left(-2 a C+x^2 (2 A c-b C)+A b\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-\frac{\left(-\frac{4 A b c-C \left(4 a c+b^2\right)}{\sqrt{b^2-4 a c}}+2 A c-b C\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} \sqrt{c} \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left(\frac{4 A b c-C \left(4 a c+b^2\right)}{\sqrt{b^2-4 a c}}+2 A c-b C\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} \sqrt{c} \left(b^2-4 a c\right) \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{b B \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/2}}+\frac{B \left(2 a+b x^2\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"((4*a*(B + C*x) + 2*x*(b*x*(B + C*x) - A*(b + 2*c*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*(-2*A*c*(-2*b + Sqrt[b^2 - 4*a*c]) + (-b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(-2*A*c*(2*b + Sqrt[b^2 - 4*a*c]) + (b^2 + 4*a*c + b*Sqrt[b^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (2*b*B*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(3/2) - (2*b*B*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/4","A",1
32,1,335,317,1.2494518,"\int \frac{x \left(A+B x+C x^2\right)}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(x*(A + B*x + C*x^2))/(a + b*x^2 + c*x^4)^2,x]","\frac{1}{2} \left(\frac{2 a C-A \left(b+2 c x^2\right)+x \left(-b B+b C x-2 B c x^2\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{(b C-2 A c) \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}+\frac{(2 A c-b C) \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}-\frac{\sqrt{2} B \sqrt{c} \left(\sqrt{b^2-4 a c}-2 b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} B \sqrt{c} \left(\sqrt{b^2-4 a c}+2 b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}\right)","\frac{(2 A c-b C) \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/2}}-\frac{-2 a C+x^2 (2 A c-b C)+A b}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-\frac{B x \left(b+2 c x^2\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{B \sqrt{c} \left(2 b-\sqrt{b^2-4 a c}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{B \sqrt{c} \left(\sqrt{b^2-4 a c}+2 b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}",1,"((2*a*C - A*(b + 2*c*x^2) + x*(-(b*B) + b*C*x - 2*B*c*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - (Sqrt[2]*B*Sqrt[c]*(-2*b + Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*B*Sqrt[c]*(2*b + Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + ((-2*A*c + b*C)*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(3/2) + ((2*A*c - b*C)*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/2","A",1
33,1,393,368,1.2238997,"\int \frac{A+B x+C x^2}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(A + B*x + C*x^2)/(a + b*x^2 + c*x^4)^2,x]","\frac{1}{4} \left(\frac{4 a c x (A+x (B+C x))+2 a b (B+C x)-2 A b x \left(b+c x^2\right)}{a \left(4 a c-b^2\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \sqrt{c} \left(A \left(b \sqrt{b^2-4 a c}-12 a c+b^2\right)-2 a C \left(\sqrt{b^2-4 a c}-2 b\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{a \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \sqrt{c} \left(A \left(-b \sqrt{b^2-4 a c}-12 a c+b^2\right)+2 a C \left(\sqrt{b^2-4 a c}+2 b\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{a \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{4 B c \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}+\frac{4 B c \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}\right)","\frac{x \left(c x^2 (A b-2 a C)-2 a A c-a b C+A b^2\right)}{2 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{c} \left(\frac{A \left(b^2-12 a c\right)+4 a b C}{\sqrt{b^2-4 a c}}-2 a C+A b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} a \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left(-\frac{-12 a A c+4 a b C+A b^2}{\sqrt{b^2-4 a c}}-2 a C+A b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} a \left(b^2-4 a c\right) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 B c \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/2}}-\frac{B \left(b+2 c x^2\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"((2*a*b*(B + C*x) - 2*A*b*x*(b + c*x^2) + 4*a*c*x*(A + x*(B + C*x)))/(a*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*Sqrt[c]*(A*(b^2 - 12*a*c + b*Sqrt[b^2 - 4*a*c]) - 2*a*(-2*b + Sqrt[b^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*Sqrt[c]*(A*(b^2 - 12*a*c - b*Sqrt[b^2 - 4*a*c]) + 2*a*(2*b + Sqrt[b^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) - (4*B*c*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(3/2) + (4*B*c*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/4","A",1
34,1,458,403,1.4733007,"\int \frac{A+B x+C x^2}{x \left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(A + B*x + C*x^2)/(x*(a + b*x^2 + c*x^4)^2),x]","\frac{-\frac{\left(4 a^2 c C+A \left(b^2 \sqrt{b^2-4 a c}-4 a c \sqrt{b^2-4 a c}-6 a b c+b^3\right)\right) \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}-\frac{\left(A \left(b^2 \sqrt{b^2-4 a c}-4 a c \sqrt{b^2-4 a c}+6 a b c-b^3\right)-4 a^2 c C\right) \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}-\frac{2 a \left(-A \left(-2 a c+b^2+b c x^2\right)+a b C+2 a c x (B+C x)-b B x \left(b+c x^2\right)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} a B \sqrt{c} \left(b \sqrt{b^2-4 a c}-12 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} a B \sqrt{c} \left(b \sqrt{b^2-4 a c}+12 a c-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+4 A \log (x)}{4 a^2}","\frac{\left(4 a^2 c C+A \left(b^3-6 a b c\right)\right) \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{2 a^2 \left(b^2-4 a c\right)^{3/2}}-\frac{A \log \left(a+b x^2+c x^4\right)}{4 a^2}+\frac{A \log (x)}{a^2}+\frac{A \left(b^2-2 a c\right)+c x^2 (A b-2 a C)-a b C}{2 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{B x \left(-2 a c+b^2+b c x^2\right)}{2 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{B \sqrt{c} \left(b \sqrt{b^2-4 a c}-12 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} a \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{B \sqrt{c} \left(-b \sqrt{b^2-4 a c}-12 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} a \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}",1,"((-2*a*(a*b*C + 2*a*c*x*(B + C*x) - b*B*x*(b + c*x^2) - A*(b^2 - 2*a*c + b*c*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*a*B*Sqrt[c]*(b^2 - 12*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*a*B*Sqrt[c]*(-b^2 + 12*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + 4*A*Log[x] - ((A*(b^3 - 6*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - 4*a*c*Sqrt[b^2 - 4*a*c]) + 4*a^2*c*C)*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(3/2) - ((A*(-b^3 + 6*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - 4*a*c*Sqrt[b^2 - 4*a*c]) - 4*a^2*c*C)*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/(4*a^2)","A",1
35,1,559,514,2.0271932,"\int \frac{A+B x+C x^2}{x^2 \left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(A + B*x + C*x^2)/(x^2*(a + b*x^2 + c*x^4)^2),x]","\frac{\frac{-4 a^2 c (B+C x)+2 a \left(b c x (3 A+x (B+C x))+2 A c^2 x^3+b^2 (B+C x)\right)-2 A b^2 x \left(b+c x^2\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \sqrt{c} \left(A \left(-3 b^2 \sqrt{b^2-4 a c}+10 a c \sqrt{b^2-4 a c}+16 a b c-3 b^3\right)+a C \left(b \sqrt{b^2-4 a c}-12 a c+b^2\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left(A \left(-3 b^2 \sqrt{b^2-4 a c}+10 a c \sqrt{b^2-4 a c}-16 a b c+3 b^3\right)+a C \left(b \sqrt{b^2-4 a c}+12 a c-b^2\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{B \left(b^2 \sqrt{b^2-4 a c}-4 a c \sqrt{b^2-4 a c}-6 a b c+b^3\right) \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}-\frac{B \left(b^2 \sqrt{b^2-4 a c}-4 a c \sqrt{b^2-4 a c}+6 a b c-b^3\right) \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}-\frac{4 A}{x}+4 B \log (x)}{4 a^2}","-\frac{-10 a A c-a b C+3 A b^2}{2 a^2 x \left(b^2-4 a c\right)}-\frac{\sqrt{c} \left(A \left(3 b^2 \sqrt{b^2-4 a c}-10 a c \sqrt{b^2-4 a c}-16 a b c+3 b^3\right)-a C \left(b \sqrt{b^2-4 a c}-12 a c+b^2\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} a^2 \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left(-\frac{A \left(3 b^3-16 a b c\right)-a C \left(b^2-12 a c\right)}{\sqrt{b^2-4 a c}}-10 a A c-a b C+3 A b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} a^2 \left(b^2-4 a c\right) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{b B \left(b^2-6 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{2 a^2 \left(b^2-4 a c\right)^{3/2}}-\frac{B \log \left(a+b x^2+c x^4\right)}{4 a^2}+\frac{B \log (x)}{a^2}+\frac{A \left(b^2-2 a c\right)+c x^2 (A b-2 a C)-a b C}{2 a x \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{B \left(-2 a c+b^2+b c x^2\right)}{2 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"((-4*A)/x + (-4*a^2*c*(B + C*x) - 2*A*b^2*x*(b + c*x^2) + 2*a*(2*A*c^2*x^3 + b^2*(B + C*x) + b*c*x*(3*A + x*(B + C*x))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*Sqrt[c]*(A*(-3*b^3 + 16*a*b*c - 3*b^2*Sqrt[b^2 - 4*a*c] + 10*a*c*Sqrt[b^2 - 4*a*c]) + a*(b^2 - 12*a*c + b*Sqrt[b^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(A*(3*b^3 - 16*a*b*c - 3*b^2*Sqrt[b^2 - 4*a*c] + 10*a*c*Sqrt[b^2 - 4*a*c]) + a*(-b^2 + 12*a*c + b*Sqrt[b^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + 4*B*Log[x] - (B*(b^3 - 6*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - 4*a*c*Sqrt[b^2 - 4*a*c])*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(3/2) - (B*(-b^3 + 6*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - 4*a*c*Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/(4*a^2)","A",1
36,1,655,534,2.4704375,"\int \frac{A+B x+C x^2}{x^3 \left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(A + B*x + C*x^2)/(x^3*(a + b*x^2 + c*x^4)^2),x]","\frac{-\frac{2 a \left(2 a^2 c C+A \left(-3 a b c-2 a c^2 x^2+b^3+b^2 c x^2\right)-a \left(b^2 C+b c x (3 B+C x)+2 B c^2 x^3\right)+b^2 B x \left(b+c x^2\right)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\left(2 A \left(6 a^2 c^2-6 a b^2 c-4 a b c \sqrt{b^2-4 a c}+b^3 \sqrt{b^2-4 a c}+b^4\right)+a C \left(-b^2 \sqrt{b^2-4 a c}+4 a c \sqrt{b^2-4 a c}+6 a b c-b^3\right)\right) \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}+\frac{\left(2 A \left(-6 a^2 c^2+6 a b^2 c-4 a b c \sqrt{b^2-4 a c}+b^3 \sqrt{b^2-4 a c}-b^4\right)+a C \left(-b^2 \sqrt{b^2-4 a c}+4 a c \sqrt{b^2-4 a c}-6 a b c+b^3\right)\right) \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}+4 \log (x) (a C-2 A b)-\frac{2 a A}{x^2}+\frac{\sqrt{2} a B \sqrt{c} \left(-3 b^2 \sqrt{b^2-4 a c}+10 a c \sqrt{b^2-4 a c}+16 a b c-3 b^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} a B \sqrt{c} \left(-3 b^2 \sqrt{b^2-4 a c}+10 a c \sqrt{b^2-4 a c}-16 a b c+3 b^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{4 a B}{x}}{4 a^3}","\frac{(2 A b-a C) \log \left(a+b x^2+c x^4\right)}{4 a^3}-\frac{\log (x) (2 A b-a C)}{a^3}-\frac{-6 a A c-a b C+2 A b^2}{2 a^2 x^2 \left(b^2-4 a c\right)}-\frac{B \left(3 b^2-10 a c\right)}{2 a^2 x \left(b^2-4 a c\right)}-\frac{B \sqrt{c} \left(\left(3 b^2-10 a c\right) \sqrt{b^2-4 a c}-16 a b c+3 b^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} a^2 \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{B \sqrt{c} \left(-\left(3 b^2-10 a c\right) \sqrt{b^2-4 a c}-16 a b c+3 b^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} a^2 \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{\left(2 A \left(6 a^2 c^2-6 a b^2 c+b^4\right)-a b C \left(b^2-6 a c\right)\right) \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{2 a^3 \left(b^2-4 a c\right)^{3/2}}+\frac{A \left(b^2-2 a c\right)+c x^2 (A b-2 a C)-a b C}{2 a x^2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{B \left(-2 a c+b^2+b c x^2\right)}{2 a x \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"((-2*a*A)/x^2 - (4*a*B)/x - (2*a*(2*a^2*c*C + b^2*B*x*(b + c*x^2) + A*(b^3 - 3*a*b*c + b^2*c*x^2 - 2*a*c^2*x^2) - a*(b^2*C + 2*B*c^2*x^3 + b*c*x*(3*B + C*x))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*a*B*Sqrt[c]*(-3*b^3 + 16*a*b*c - 3*b^2*Sqrt[b^2 - 4*a*c] + 10*a*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*a*B*Sqrt[c]*(3*b^3 - 16*a*b*c - 3*b^2*Sqrt[b^2 - 4*a*c] + 10*a*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + 4*(-2*A*b + a*C)*Log[x] + ((2*A*(b^4 - 6*a*b^2*c + 6*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 4*a*b*c*Sqrt[b^2 - 4*a*c]) + a*(-b^3 + 6*a*b*c - b^2*Sqrt[b^2 - 4*a*c] + 4*a*c*Sqrt[b^2 - 4*a*c])*C)*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(3/2) + ((2*A*(-b^4 + 6*a*b^2*c - 6*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 4*a*b*c*Sqrt[b^2 - 4*a*c]) + a*(b^3 - 6*a*b*c - b^2*Sqrt[b^2 - 4*a*c] + 4*a*c*Sqrt[b^2 - 4*a*c])*C)*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/(4*a^3)","A",1
37,1,296,399,0.9192659,"\int (d x)^m \left(A+B x+C x^2\right) \left(a+b x^2+c x^4\right)^3 \, dx","Integrate[(d*x)^m*(A + B*x + C*x^2)*(a + b*x^2 + c*x^4)^3,x]","x (d x)^m \left(\frac{a^3 A}{m+1}+\frac{a^3 B x}{m+2}+\frac{a^2 x^2 (a C+3 A b)}{m+3}+\frac{3 a^2 b B x^3}{m+4}+\frac{3 c x^{10} \left(C \left(a c+b^2\right)+A b c\right)}{m+11}+\frac{x^8 \left(3 A c \left(a c+b^2\right)+b C \left(6 a c+b^2\right)\right)}{m+9}+\frac{3 a x^4 \left(A \left(a c+b^2\right)+a b C\right)}{m+5}+\frac{x^6 \left(A \left(6 a b c+b^3\right)+3 a C \left(a c+b^2\right)\right)}{m+7}+\frac{3 B c x^9 \left(a c+b^2\right)}{m+10}+\frac{b B x^7 \left(6 a c+b^2\right)}{m+8}+\frac{3 a B x^5 \left(a c+b^2\right)}{m+6}+\frac{c^2 x^{12} (A c+3 b C)}{m+13}+\frac{3 b B c^2 x^{11}}{m+12}+\frac{B c^3 x^{13}}{m+14}+\frac{c^3 C x^{14}}{m+15}\right)","\frac{a^3 A (d x)^{m+1}}{d (m+1)}+\frac{a^3 B (d x)^{m+2}}{d^2 (m+2)}+\frac{a^2 (d x)^{m+3} (a C+3 A b)}{d^3 (m+3)}+\frac{3 a^2 b B (d x)^{m+4}}{d^4 (m+4)}+\frac{3 c (d x)^{m+11} \left(C \left(a c+b^2\right)+A b c\right)}{d^{11} (m+11)}+\frac{(d x)^{m+9} \left(3 A c \left(a c+b^2\right)+b C \left(6 a c+b^2\right)\right)}{d^9 (m+9)}+\frac{3 a (d x)^{m+5} \left(A \left(a c+b^2\right)+a b C\right)}{d^5 (m+5)}+\frac{(d x)^{m+7} \left(A \left(6 a b c+b^3\right)+3 a C \left(a c+b^2\right)\right)}{d^7 (m+7)}+\frac{3 B c \left(a c+b^2\right) (d x)^{m+10}}{d^{10} (m+10)}+\frac{b B \left(6 a c+b^2\right) (d x)^{m+8}}{d^8 (m+8)}+\frac{3 a B \left(a c+b^2\right) (d x)^{m+6}}{d^6 (m+6)}+\frac{c^2 (d x)^{m+13} (A c+3 b C)}{d^{13} (m+13)}+\frac{3 b B c^2 (d x)^{m+12}}{d^{12} (m+12)}+\frac{B c^3 (d x)^{m+14}}{d^{14} (m+14)}+\frac{c^3 C (d x)^{m+15}}{d^{15} (m+15)}",1,"x*(d*x)^m*((a^3*A)/(1 + m) + (a^3*B*x)/(2 + m) + (a^2*(3*A*b + a*C)*x^2)/(3 + m) + (3*a^2*b*B*x^3)/(4 + m) + (3*a*(A*(b^2 + a*c) + a*b*C)*x^4)/(5 + m) + (3*a*B*(b^2 + a*c)*x^5)/(6 + m) + ((A*(b^3 + 6*a*b*c) + 3*a*(b^2 + a*c)*C)*x^6)/(7 + m) + (b*B*(b^2 + 6*a*c)*x^7)/(8 + m) + ((3*A*c*(b^2 + a*c) + b*(b^2 + 6*a*c)*C)*x^8)/(9 + m) + (3*B*c*(b^2 + a*c)*x^9)/(10 + m) + (3*c*(A*b*c + (b^2 + a*c)*C)*x^10)/(11 + m) + (3*b*B*c^2*x^11)/(12 + m) + (c^2*(A*c + 3*b*C)*x^12)/(13 + m) + (B*c^3*x^13)/(14 + m) + (c^3*C*x^14)/(15 + m))","A",1
38,1,185,260,0.2767345,"\int (d x)^m \left(A+B x+C x^2\right) \left(a+b x^2+c x^4\right)^2 \, dx","Integrate[(d*x)^m*(A + B*x + C*x^2)*(a + b*x^2 + c*x^4)^2,x]","x (d x)^m \left(\frac{a^2 A}{m+1}+\frac{a^2 B x}{m+2}+\frac{x^6 \left(C \left(2 a c+b^2\right)+2 A b c\right)}{m+7}+\frac{x^4 \left(A \left(2 a c+b^2\right)+2 a b C\right)}{m+5}+\frac{a x^2 (a C+2 A b)}{m+3}+\frac{B x^5 \left(2 a c+b^2\right)}{m+6}+\frac{2 a b B x^3}{m+4}+\frac{c x^8 (A c+2 b C)}{m+9}+\frac{2 b B c x^7}{m+8}+\frac{B c^2 x^9}{m+10}+\frac{c^2 C x^{10}}{m+11}\right)","\frac{a^2 A (d x)^{m+1}}{d (m+1)}+\frac{a^2 B (d x)^{m+2}}{d^2 (m+2)}+\frac{(d x)^{m+7} \left(C \left(2 a c+b^2\right)+2 A b c\right)}{d^7 (m+7)}+\frac{(d x)^{m+5} \left(A \left(2 a c+b^2\right)+2 a b C\right)}{d^5 (m+5)}+\frac{a (d x)^{m+3} (a C+2 A b)}{d^3 (m+3)}+\frac{B \left(2 a c+b^2\right) (d x)^{m+6}}{d^6 (m+6)}+\frac{2 a b B (d x)^{m+4}}{d^4 (m+4)}+\frac{c (d x)^{m+9} (A c+2 b C)}{d^9 (m+9)}+\frac{2 b B c (d x)^{m+8}}{d^8 (m+8)}+\frac{B c^2 (d x)^{m+10}}{d^{10} (m+10)}+\frac{c^2 C (d x)^{m+11}}{d^{11} (m+11)}",1,"x*(d*x)^m*((a^2*A)/(1 + m) + (a^2*B*x)/(2 + m) + (a*(2*A*b + a*C)*x^2)/(3 + m) + (2*a*b*B*x^3)/(4 + m) + ((A*(b^2 + 2*a*c) + 2*a*b*C)*x^4)/(5 + m) + (B*(b^2 + 2*a*c)*x^5)/(6 + m) + ((2*A*b*c + (b^2 + 2*a*c)*C)*x^6)/(7 + m) + (2*b*B*c*x^7)/(8 + m) + (c*(A*c + 2*b*C)*x^8)/(9 + m) + (B*c^2*x^9)/(10 + m) + (c^2*C*x^10)/(11 + m))","A",1
39,1,90,137,0.1036623,"\int (d x)^m \left(A+B x+C x^2\right) \left(a+b x^2+c x^4\right) \, dx","Integrate[(d*x)^m*(A + B*x + C*x^2)*(a + b*x^2 + c*x^4),x]","x (d x)^m \left(\frac{x^2 (a C+A b)}{m+3}+\frac{a A}{m+1}+\frac{a B x}{m+2}+\frac{x^4 (A c+b C)}{m+5}+\frac{b B x^3}{m+4}+\frac{B c x^5}{m+6}+\frac{c C x^6}{m+7}\right)","\frac{(d x)^{m+3} (a C+A b)}{d^3 (m+3)}+\frac{a A (d x)^{m+1}}{d (m+1)}+\frac{a B (d x)^{m+2}}{d^2 (m+2)}+\frac{(d x)^{m+5} (A c+b C)}{d^5 (m+5)}+\frac{b B (d x)^{m+4}}{d^4 (m+4)}+\frac{B c (d x)^{m+6}}{d^6 (m+6)}+\frac{c C (d x)^{m+7}}{d^7 (m+7)}",1,"x*(d*x)^m*((a*A)/(1 + m) + (a*B*x)/(2 + m) + ((A*b + a*C)*x^2)/(3 + m) + (b*B*x^3)/(4 + m) + ((A*c + b*C)*x^4)/(5 + m) + (B*c*x^5)/(6 + m) + (c*C*x^6)/(7 + m))","A",1
40,1,438,368,0.4744012,"\int \frac{(d x)^m \left(A+B x+C x^2\right)}{a+b x^2+c x^4} \, dx","Integrate[((d*x)^m*(A + B*x + C*x^2))/(a + b*x^2 + c*x^4),x]","\frac{(d x)^m \left(A \left(m^2+3 m+2\right) \text{RootSum}\left[\text{$\#$1}^4 c+\text{$\#$1}^2 b+a\&,\frac{\left(\frac{x}{x-\text{$\#$1}}\right)^{-m} \, _2F_1\left(-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right)}{2 \text{$\#$1}^3 c+\text{$\#$1} b}\&\right]+B (m+2) \text{RootSum}\left[\text{$\#$1}^4 c+\text{$\#$1}^2 b+a\&,\frac{\text{$\#$1} m \left(\frac{x}{x-\text{$\#$1}}\right)^{-m} \, _2F_1\left(-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right)+\text{$\#$1} \left(\frac{x}{x-\text{$\#$1}}\right)^{-m} \, _2F_1\left(-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right)+m x}{2 \text{$\#$1}^3 c+\text{$\#$1} b}\&\right]+C \text{RootSum}\left[\text{$\#$1}^4 c+\text{$\#$1}^2 b+a\&,\frac{\text{$\#$1}^2 m^2 \left(\frac{x}{x-\text{$\#$1}}\right)^{-m} \, _2F_1\left(-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right)+3 \text{$\#$1}^2 m \left(\frac{x}{x-\text{$\#$1}}\right)^{-m} \, _2F_1\left(-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right)+2 \text{$\#$1}^2 \left(\frac{x}{x-\text{$\#$1}}\right)^{-m} \, _2F_1\left(-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right)+\text{$\#$1}^2 m \left(\frac{x}{\text{$\#$1}}\right)^{-m}+\text{$\#$1} m^2 x+2 \text{$\#$1} m x+m^2 x^2+m x^2}{2 \text{$\#$1}^3 c+\text{$\#$1} b}\&\right]\right)}{2 m (m+1) (m+2)}","\frac{(d x)^{m+1} \left(\frac{2 A c-b C}{\sqrt{b^2-4 a c}}+C\right) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right)}{d (m+1) \left(b-\sqrt{b^2-4 a c}\right)}+\frac{(d x)^{m+1} \left(C-\frac{2 A c-b C}{\sqrt{b^2-4 a c}}\right) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right)}{d (m+1) \left(\sqrt{b^2-4 a c}+b\right)}+\frac{2 B c (d x)^{m+2} \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right)}{d^2 (m+2) \sqrt{b^2-4 a c} \left(b-\sqrt{b^2-4 a c}\right)}-\frac{2 B c (d x)^{m+2} \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right)}{d^2 (m+2) \sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}+b\right)}",1,"((d*x)^m*(A*(2 + 3*m + m^2)*RootSum[a + b*#1^2 + c*#1^4 & , Hypergeometric2F1[-m, -m, 1 - m, -(#1/(x - #1))]/((x/(x - #1))^m*(b*#1 + 2*c*#1^3)) & ] + B*(2 + m)*RootSum[a + b*#1^2 + c*#1^4 & , (m*x + (Hypergeometric2F1[-m, -m, 1 - m, -(#1/(x - #1))]*#1)/(x/(x - #1))^m + (m*Hypergeometric2F1[-m, -m, 1 - m, -(#1/(x - #1))]*#1)/(x/(x - #1))^m)/(b*#1 + 2*c*#1^3) & ] + C*RootSum[a + b*#1^2 + c*#1^4 & , (m*x^2 + m^2*x^2 + 2*m*x*#1 + m^2*x*#1 + (2*Hypergeometric2F1[-m, -m, 1 - m, -(#1/(x - #1))]*#1^2)/(x/(x - #1))^m + (3*m*Hypergeometric2F1[-m, -m, 1 - m, -(#1/(x - #1))]*#1^2)/(x/(x - #1))^m + (m^2*Hypergeometric2F1[-m, -m, 1 - m, -(#1/(x - #1))]*#1^2)/(x/(x - #1))^m + (m*#1^2)/(x/#1)^m)/(b*#1 + 2*c*#1^3) & ]))/(2*m*(1 + m)*(2 + m))","C",0
41,1,242,685,0.3268035,"\int \frac{(d x)^m \left(A+B x+C x^2\right)}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[((d*x)^m*(A + B*x + C*x^2))/(a + b*x^2 + c*x^4)^2,x]","\frac{x (d x)^m \left(A \left(m^2+5 m+6\right) F_1\left(\frac{m+1}{2};2,2;\frac{m+3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right)+(m+1) x \left(B (m+3) F_1\left(\frac{m+2}{2};2,2;\frac{m+4}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right)+C (m+2) x F_1\left(\frac{m+3}{2};2,2;\frac{m+5}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right)\right)\right)}{a^2 (m+1) (m+2) (m+3)}","\frac{c (d x)^{m+1} \left(A \left(b (1-m) \sqrt{b^2-4 a c}-4 a c (3-m)+b^2 (1-m)\right)+2 a C \left(2 b-(1-m) \sqrt{b^2-4 a c}\right)\right) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right)}{2 a d (m+1) \left(b^2-4 a c\right)^{3/2} \left(b-\sqrt{b^2-4 a c}\right)}-\frac{c (d x)^{m+1} \left(A \left(-b (1-m) \sqrt{b^2-4 a c}-4 a c (3-m)+b^2 (1-m)\right)+2 a C \left((1-m) \sqrt{b^2-4 a c}+2 b\right)\right) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right)}{2 a d (m+1) \left(b^2-4 a c\right)^{3/2} \left(\sqrt{b^2-4 a c}+b\right)}+\frac{(d x)^{m+1} \left(A \left(b^2-2 a c\right)+c x^2 (A b-2 a C)-a b C\right)}{2 a d \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-\frac{B c (d x)^{m+2} \left(b m \left(\sqrt{b^2-4 a c}+b\right)+4 a c (2-m)\right) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right)}{2 a d^2 (m+2) \left(b^2-4 a c\right)^{3/2} \left(b-\sqrt{b^2-4 a c}\right)}+\frac{B c (d x)^{m+2} \left(b m \left(b-\sqrt{b^2-4 a c}\right)+4 a c (2-m)\right) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right)}{2 a d^2 (m+2) \left(b^2-4 a c\right)^{3/2} \left(\sqrt{b^2-4 a c}+b\right)}+\frac{B (d x)^{m+2} \left(-2 a c+b^2+b c x^2\right)}{2 a d^2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"(x*(d*x)^m*(A*(6 + 5*m + m^2)*AppellF1[(1 + m)/2, 2, 2, (3 + m)/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])] + (1 + m)*x*(B*(3 + m)*AppellF1[(2 + m)/2, 2, 2, (4 + m)/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])] + C*(2 + m)*x*AppellF1[(3 + m)/2, 2, 2, (5 + m)/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])))/(a^2*(1 + m)*(2 + m)*(3 + m))","C",0
42,1,378,356,1.0383579,"\int \frac{x^2 \left(A+B x+C x^2\right)}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(x^2*(A + B*x + C*x^2))/(a + b*x^2 + c*x^4)^2,x]","\frac{1}{4} \left(\frac{4 a (B+C x)+2 x \left(b x (B+C x)-A \left(b+2 c x^2\right)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \left(C \left(b \sqrt{b^2-4 a c}-4 a c-b^2\right)-2 A c \left(\sqrt{b^2-4 a c}-2 b\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{c} \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \left(C \left(b \sqrt{b^2-4 a c}+4 a c+b^2\right)-2 A c \left(\sqrt{b^2-4 a c}+2 b\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{c} \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 b B \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}-\frac{2 b B \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}\right)","-\frac{x \left(-2 a C+x^2 (2 A c-b C)+A b\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-\frac{\left(-\frac{4 A b c-C \left(4 a c+b^2\right)}{\sqrt{b^2-4 a c}}+2 A c-b C\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} \sqrt{c} \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left(\frac{4 A b c-C \left(4 a c+b^2\right)}{\sqrt{b^2-4 a c}}+2 A c-b C\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} \sqrt{c} \left(b^2-4 a c\right) \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{b B \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/2}}+\frac{B \left(2 a+b x^2\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"((4*a*(B + C*x) + 2*x*(b*x*(B + C*x) - A*(b + 2*c*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*(-2*A*c*(-2*b + Sqrt[b^2 - 4*a*c]) + (-b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(-2*A*c*(2*b + Sqrt[b^2 - 4*a*c]) + (b^2 + 4*a*c + b*Sqrt[b^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (2*b*B*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(3/2) - (2*b*B*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/4","A",1
43,1,378,356,0.4253254,"\int \frac{x \left(A x+B x^2+C x^3\right)}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(x*(A*x + B*x^2 + C*x^3))/(a + b*x^2 + c*x^4)^2,x]","\frac{1}{4} \left(\frac{4 a (B+C x)+2 x \left(b x (B+C x)-A \left(b+2 c x^2\right)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \left(C \left(b \sqrt{b^2-4 a c}-4 a c-b^2\right)-2 A c \left(\sqrt{b^2-4 a c}-2 b\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{c} \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \left(C \left(b \sqrt{b^2-4 a c}+4 a c+b^2\right)-2 A c \left(\sqrt{b^2-4 a c}+2 b\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{c} \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 b B \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}-\frac{2 b B \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}\right)","-\frac{x \left(-2 a C+x^2 (2 A c-b C)+A b\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-\frac{\left(-\frac{4 A b c-C \left(4 a c+b^2\right)}{\sqrt{b^2-4 a c}}+2 A c-b C\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} \sqrt{c} \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left(\frac{4 A b c-C \left(4 a c+b^2\right)}{\sqrt{b^2-4 a c}}+2 A c-b C\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} \sqrt{c} \left(b^2-4 a c\right) \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{b B \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/2}}+\frac{B \left(2 a+b x^2\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"((4*a*(B + C*x) + 2*x*(b*x*(B + C*x) - A*(b + 2*c*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*(-2*A*c*(-2*b + Sqrt[b^2 - 4*a*c]) + (-b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(-2*A*c*(2*b + Sqrt[b^2 - 4*a*c]) + (b^2 + 4*a*c + b*Sqrt[b^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (2*b*B*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(3/2) - (2*b*B*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/4","A",1
44,1,378,356,0.1544375,"\int \frac{A x^2+B x^3+C x^4}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(A*x^2 + B*x^3 + C*x^4)/(a + b*x^2 + c*x^4)^2,x]","\frac{1}{4} \left(\frac{4 a (B+C x)+2 x \left(b x (B+C x)-A \left(b+2 c x^2\right)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \left(C \left(b \sqrt{b^2-4 a c}-4 a c-b^2\right)-2 A c \left(\sqrt{b^2-4 a c}-2 b\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{c} \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \left(C \left(b \sqrt{b^2-4 a c}+4 a c+b^2\right)-2 A c \left(\sqrt{b^2-4 a c}+2 b\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{c} \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 b B \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}-\frac{2 b B \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}\right)","-\frac{x \left(-2 a C+x^2 (2 A c-b C)+A b\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-\frac{\left(-\frac{4 A b c-C \left(4 a c+b^2\right)}{\sqrt{b^2-4 a c}}+2 A c-b C\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} \sqrt{c} \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left(\frac{4 A b c-C \left(4 a c+b^2\right)}{\sqrt{b^2-4 a c}}+2 A c-b C\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} \sqrt{c} \left(b^2-4 a c\right) \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{b B \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/2}}+\frac{B \left(2 a+b x^2\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"((4*a*(B + C*x) + 2*x*(b*x*(B + C*x) - A*(b + 2*c*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*(-2*A*c*(-2*b + Sqrt[b^2 - 4*a*c]) + (-b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(-2*A*c*(2*b + Sqrt[b^2 - 4*a*c]) + (b^2 + 4*a*c + b*Sqrt[b^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (2*b*B*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(3/2) - (2*b*B*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/4","A",1
45,1,378,356,0.1539861,"\int \frac{A x^3+B x^4+C x^5}{x \left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(A*x^3 + B*x^4 + C*x^5)/(x*(a + b*x^2 + c*x^4)^2),x]","\frac{1}{4} \left(\frac{4 a (B+C x)+2 x \left(b x (B+C x)-A \left(b+2 c x^2\right)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \left(C \left(b \sqrt{b^2-4 a c}-4 a c-b^2\right)-2 A c \left(\sqrt{b^2-4 a c}-2 b\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{c} \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \left(C \left(b \sqrt{b^2-4 a c}+4 a c+b^2\right)-2 A c \left(\sqrt{b^2-4 a c}+2 b\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{c} \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 b B \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}-\frac{2 b B \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}\right)","-\frac{x \left(-2 a C+x^2 (2 A c-b C)+A b\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-\frac{\left(-\frac{4 A b c-C \left(4 a c+b^2\right)}{\sqrt{b^2-4 a c}}+2 A c-b C\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} \sqrt{c} \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left(\frac{4 A b c-C \left(4 a c+b^2\right)}{\sqrt{b^2-4 a c}}+2 A c-b C\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} \sqrt{c} \left(b^2-4 a c\right) \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{b B \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/2}}+\frac{B \left(2 a+b x^2\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"((4*a*(B + C*x) + 2*x*(b*x*(B + C*x) - A*(b + 2*c*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*(-2*A*c*(-2*b + Sqrt[b^2 - 4*a*c]) + (-b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(-2*A*c*(2*b + Sqrt[b^2 - 4*a*c]) + (b^2 + 4*a*c + b*Sqrt[b^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (2*b*B*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(3/2) - (2*b*B*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/4","A",1
46,1,378,356,0.1530057,"\int \frac{A x^4+B x^5+C x^6}{x^2 \left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(A*x^4 + B*x^5 + C*x^6)/(x^2*(a + b*x^2 + c*x^4)^2),x]","\frac{1}{4} \left(\frac{4 a (B+C x)+2 x \left(b x (B+C x)-A \left(b+2 c x^2\right)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \left(C \left(b \sqrt{b^2-4 a c}-4 a c-b^2\right)-2 A c \left(\sqrt{b^2-4 a c}-2 b\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{c} \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \left(C \left(b \sqrt{b^2-4 a c}+4 a c+b^2\right)-2 A c \left(\sqrt{b^2-4 a c}+2 b\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{c} \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 b B \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}-\frac{2 b B \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}\right)","-\frac{x \left(-2 a C+x^2 (2 A c-b C)+A b\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-\frac{\left(-\frac{4 A b c-C \left(4 a c+b^2\right)}{\sqrt{b^2-4 a c}}+2 A c-b C\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} \sqrt{c} \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left(\frac{4 A b c-C \left(4 a c+b^2\right)}{\sqrt{b^2-4 a c}}+2 A c-b C\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} \sqrt{c} \left(b^2-4 a c\right) \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{b B \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/2}}+\frac{B \left(2 a+b x^2\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"((4*a*(B + C*x) + 2*x*(b*x*(B + C*x) - A*(b + 2*c*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*(-2*A*c*(-2*b + Sqrt[b^2 - 4*a*c]) + (-b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(-2*A*c*(2*b + Sqrt[b^2 - 4*a*c]) + (b^2 + 4*a*c + b*Sqrt[b^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (2*b*B*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(3/2) - (2*b*B*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/4","A",1
47,1,260,273,0.1994124,"\int \frac{x^7 \left(d+e x^2+f x^4\right)}{a+b x^2+c x^4} \, dx","Integrate[(x^7*(d + e*x^2 + f*x^4))/(a + b*x^2 + c*x^4),x]","\frac{-\frac{12 \tan ^{-1}\left(\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right) \left(-2 a^2 c^3 e+b^3 c (c d-5 a f)+4 a b^2 c^2 e+a b c^2 (5 a f-3 c d)+b^5 f-b^4 c e\right)}{\sqrt{4 a c-b^2}}+6 c^2 x^4 \left(-c (a f+b e)+b^2 f+c^2 d\right)-12 c x^2 \left(b c (c d-2 a f)+a c^2 e+b^3 f-b^2 c e\right)+6 \log \left(a+b x^2+c x^4\right) \left(b^2 c (c d-3 a f)+2 a b c^2 e+a c^2 (a f-c d)+b^4 f-b^3 c e\right)+4 c^3 x^6 (c e-b f)+3 c^4 f x^8}{24 c^5}","-\frac{\tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right) \left(2 a^2 c^3 e-b^3 c (c d-5 a f)-4 a b^2 c^2 e+a b c^2 (3 c d-5 a f)+b^5 (-f)+b^4 c e\right)}{2 c^5 \sqrt{b^2-4 a c}}+\frac{x^4 \left(-c (a f+b e)+b^2 f+c^2 d\right)}{4 c^3}+\frac{x^2 \left(-b c (c d-2 a f)-a c^2 e+b^3 (-f)+b^2 c e\right)}{2 c^4}-\frac{\log \left(a+b x^2+c x^4\right) \left(-b^2 c (c d-3 a f)-2 a b c^2 e+a c^2 (c d-a f)+b^4 (-f)+b^3 c e\right)}{4 c^5}+\frac{x^6 (c e-b f)}{6 c^2}+\frac{f x^8}{8 c}",1,"(-12*c*(-(b^2*c*e) + a*c^2*e + b^3*f + b*c*(c*d - 2*a*f))*x^2 + 6*c^2*(c^2*d + b^2*f - c*(b*e + a*f))*x^4 + 4*c^3*(c*e - b*f)*x^6 + 3*c^4*f*x^8 - (12*(-(b^4*c*e) + 4*a*b^2*c^2*e - 2*a^2*c^3*e + b^5*f + b^3*c*(c*d - 5*a*f) + a*b*c^2*(-3*c*d + 5*a*f))*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + 6*(-(b^3*c*e) + 2*a*b*c^2*e + b^4*f + b^2*c*(c*d - 3*a*f) + a*c^2*(-(c*d) + a*f))*Log[a + b*x^2 + c*x^4])/(24*c^5)","A",1
48,1,193,203,0.1430639,"\int \frac{x^5 \left(d+e x^2+f x^4\right)}{a+b x^2+c x^4} \, dx","Integrate[(x^5*(d + e*x^2 + f*x^4))/(a + b*x^2 + c*x^4),x]","\frac{6 c x^2 \left(-c (a f+b e)+b^2 f+c^2 d\right)-3 \log \left(a+b x^2+c x^4\right) \left(b c (c d-2 a f)+a c^2 e+b^3 f-b^2 c e\right)+\frac{6 \tan ^{-1}\left(\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right) \left(b^2 c (c d-4 a f)+3 a b c^2 e+2 a c^2 (a f-c d)+b^4 f-b^3 c e\right)}{\sqrt{4 a c-b^2}}+3 c^2 x^4 (c e-b f)+2 c^3 f x^6}{12 c^4}","\frac{x^2 \left(-c (a f+b e)+b^2 f+c^2 d\right)}{2 c^3}+\frac{\log \left(a+b x^2+c x^4\right) \left(-b c (c d-2 a f)-a c^2 e+b^3 (-f)+b^2 c e\right)}{4 c^4}+\frac{\tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right) \left(-b^2 c (c d-4 a f)-3 a b c^2 e+2 a c^2 (c d-a f)+b^4 (-f)+b^3 c e\right)}{2 c^4 \sqrt{b^2-4 a c}}+\frac{x^4 (c e-b f)}{4 c^2}+\frac{f x^6}{6 c}",1,"(6*c*(c^2*d + b^2*f - c*(b*e + a*f))*x^2 + 3*c^2*(c*e - b*f)*x^4 + 2*c^3*f*x^6 + (6*(-(b^3*c*e) + 3*a*b*c^2*e + b^4*f + b^2*c*(c*d - 4*a*f) + 2*a*c^2*(-(c*d) + a*f))*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] - 3*(-(b^2*c*e) + a*c^2*e + b^3*f + b*c*(c*d - 2*a*f))*Log[a + b*x^2 + c*x^4])/(12*c^4)","A",1
49,1,136,144,0.0992839,"\int \frac{x^3 \left(d+e x^2+f x^4\right)}{a+b x^2+c x^4} \, dx","Integrate[(x^3*(d + e*x^2 + f*x^4))/(a + b*x^2 + c*x^4),x]","\frac{\log \left(a+b x^2+c x^4\right) \left(-c (a f+b e)+b^2 f+c^2 d\right)-\frac{2 \tan ^{-1}\left(\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right) \left(b c (c d-3 a f)+2 a c^2 e+b^3 f-b^2 c e\right)}{\sqrt{4 a c-b^2}}+2 c x^2 (c e-b f)+c^2 f x^4}{4 c^3}","\frac{\log \left(a+b x^2+c x^4\right) \left(-c (a f+b e)+b^2 f+c^2 d\right)}{4 c^3}-\frac{\tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right) \left(-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right)}{2 c^3 \sqrt{b^2-4 a c}}+\frac{x^2 (c e-b f)}{2 c^2}+\frac{f x^4}{4 c}",1,"(2*c*(c*e - b*f)*x^2 + c^2*f*x^4 - (2*(-(b^2*c*e) + 2*a*c^2*e + b^3*f + b*c*(c*d - 3*a*f))*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + (c^2*d + b^2*f - c*(b*e + a*f))*Log[a + b*x^2 + c*x^4])/(4*c^3)","A",1
50,1,100,103,0.0658989,"\int \frac{x \left(d+e x^2+f x^4\right)}{a+b x^2+c x^4} \, dx","Integrate[(x*(d + e*x^2 + f*x^4))/(a + b*x^2 + c*x^4),x]","\frac{\frac{2 \tan ^{-1}\left(\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right) \left(-c (2 a f+b e)+b^2 f+2 c^2 d\right)}{\sqrt{4 a c-b^2}}+(c e-b f) \log \left(a+b x^2+c x^4\right)+2 c f x^2}{4 c^2}","-\frac{\tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right) \left(-2 a c f+b^2 f-b c e+2 c^2 d\right)}{2 c^2 \sqrt{b^2-4 a c}}+\frac{(c e-b f) \log \left(a+b x^2+c x^4\right)}{4 c^2}+\frac{f x^2}{2 c}",1,"(2*c*f*x^2 + (2*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + (c*e - b*f)*Log[a + b*x^2 + c*x^4])/(4*c^2)","A",1
51,1,178,97,0.1370667,"\int \frac{d+e x^2+f x^4}{x \left(a+b x^2+c x^4\right)} \, dx","Integrate[(d + e*x^2 + f*x^4)/(x*(a + b*x^2 + c*x^4)),x]","\frac{-\log \left(-\sqrt{b^2-4 a c}+b+2 c x^2\right) \left(c d \sqrt{b^2-4 a c}-a f \sqrt{b^2-4 a c}+a b f-2 a c e+b c d\right)+\log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right) \left(-c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}+a b f-2 a c e+b c d\right)+4 c d \log (x) \sqrt{b^2-4 a c}}{4 a c \sqrt{b^2-4 a c}}","\frac{\tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right) (a b f-2 a c e+b c d)}{2 a c \sqrt{b^2-4 a c}}-\frac{(c d-a f) \log \left(a+b x^2+c x^4\right)}{4 a c}+\frac{d \log (x)}{a}",1,"(4*c*Sqrt[b^2 - 4*a*c]*d*Log[x] - (b*c*d + c*Sqrt[b^2 - 4*a*c]*d - 2*a*c*e + a*b*f - a*Sqrt[b^2 - 4*a*c]*f)*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2] + (b*c*d - c*Sqrt[b^2 - 4*a*c]*d - 2*a*c*e + a*b*f + a*Sqrt[b^2 - 4*a*c]*f)*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(4*a*c*Sqrt[b^2 - 4*a*c])","A",1
52,1,203,118,0.1498933,"\int \frac{d+e x^2+f x^4}{x^3 \left(a+b x^2+c x^4\right)} \, dx","Integrate[(d + e*x^2 + f*x^4)/(x^3*(a + b*x^2 + c*x^4)),x]","\frac{\frac{\log \left(-\sqrt{b^2-4 a c}+b+2 c x^2\right) \left(a \left(-e \sqrt{b^2-4 a c}+2 a f-2 c d\right)+b \left(d \sqrt{b^2-4 a c}-a e\right)+b^2 d\right)}{\sqrt{b^2-4 a c}}+\frac{\log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right) \left(-a \left(e \sqrt{b^2-4 a c}+2 a f-2 c d\right)+b \left(d \sqrt{b^2-4 a c}+a e\right)+b^2 (-d)\right)}{\sqrt{b^2-4 a c}}+4 \log (x) (a e-b d)-\frac{2 a d}{x^2}}{4 a^2}","-\frac{\tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right) \left(-a b e-2 a (c d-a f)+b^2 d\right)}{2 a^2 \sqrt{b^2-4 a c}}+\frac{(b d-a e) \log \left(a+b x^2+c x^4\right)}{4 a^2}-\frac{\log (x) (b d-a e)}{a^2}-\frac{d}{2 a x^2}",1,"((-2*a*d)/x^2 + 4*(-(b*d) + a*e)*Log[x] + ((b^2*d + b*(Sqrt[b^2 - 4*a*c]*d - a*e) + a*(-2*c*d - Sqrt[b^2 - 4*a*c]*e + 2*a*f))*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/Sqrt[b^2 - 4*a*c] + ((-(b^2*d) + b*(Sqrt[b^2 - 4*a*c]*d + a*e) - a*(-2*c*d + Sqrt[b^2 - 4*a*c]*e + 2*a*f))*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/Sqrt[b^2 - 4*a*c])/(4*a^2)","A",1
53,1,314,174,0.3451234,"\int \frac{d+e x^2+f x^4}{x^5 \left(a+b x^2+c x^4\right)} \, dx","Integrate[(d + e*x^2 + f*x^4)/(x^5*(a + b*x^2 + c*x^4)),x]","-\frac{\frac{a^2 d}{x^4}-4 \log (x) \left(-a b e+a (a f-c d)+b^2 d\right)+\frac{\log \left(-\sqrt{b^2-4 a c}+b+2 c x^2\right) \left(a b \left(-e \sqrt{b^2-4 a c}+a f-3 c d\right)+a \left(-c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}+2 a c e\right)+b^2 \left(d \sqrt{b^2-4 a c}-a e\right)+b^3 d\right)}{\sqrt{b^2-4 a c}}+\frac{\log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right) \left(-a b \left(e \sqrt{b^2-4 a c}+a f-3 c d\right)+a \left(a f \sqrt{b^2-4 a c}-c \left(d \sqrt{b^2-4 a c}+2 a e\right)\right)+b^2 \left(d \sqrt{b^2-4 a c}+a e\right)+b^3 (-d)\right)}{\sqrt{b^2-4 a c}}+\frac{2 a (a e-b d)}{x^2}}{4 a^3}","-\frac{\log \left(a+b x^2+c x^4\right) \left(-a b e-a (c d-a f)+b^2 d\right)}{4 a^3}+\frac{\log (x) \left(-a b e-a (c d-a f)+b^2 d\right)}{a^3}+\frac{b d-a e}{2 a^2 x^2}+\frac{\tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right) \left(2 a^2 c e-a b^2 e-a b (3 c d-a f)+b^3 d\right)}{2 a^3 \sqrt{b^2-4 a c}}-\frac{d}{4 a x^4}",1,"-1/4*((a^2*d)/x^4 + (2*a*(-(b*d) + a*e))/x^2 - 4*(b^2*d - a*b*e + a*(-(c*d) + a*f))*Log[x] + ((b^3*d + b^2*(Sqrt[b^2 - 4*a*c]*d - a*e) + a*b*(-3*c*d - Sqrt[b^2 - 4*a*c]*e + a*f) + a*(-(c*Sqrt[b^2 - 4*a*c]*d) + 2*a*c*e + a*Sqrt[b^2 - 4*a*c]*f))*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/Sqrt[b^2 - 4*a*c] + ((-(b^3*d) + b^2*(Sqrt[b^2 - 4*a*c]*d + a*e) - a*b*(-3*c*d + Sqrt[b^2 - 4*a*c]*e + a*f) + a*(-(c*(Sqrt[b^2 - 4*a*c]*d + 2*a*e)) + a*Sqrt[b^2 - 4*a*c]*f))*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/Sqrt[b^2 - 4*a*c])/a^3","A",1
54,1,416,244,0.3480663,"\int \frac{d+e x^2+f x^4}{x^7 \left(a+b x^2+c x^4\right)} \, dx","Integrate[(d + e*x^2 + f*x^4)/(x^7*(a + b*x^2 + c*x^4)),x]","\frac{-\frac{2 a^3 d}{x^6}-12 \log (x) \left(a^2 c e-a b^2 e+a b (a f-2 c d)+b^3 d\right)+\frac{3 \log \left(-\sqrt{b^2-4 a c}+b+2 c x^2\right) \left(a^2 c \left(e \sqrt{b^2-4 a c}-2 a f+2 c d\right)+a b^2 \left(-e \sqrt{b^2-4 a c}+a f-4 c d\right)+a b \left(-2 c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}+3 a c e\right)+b^3 \left(d \sqrt{b^2-4 a c}-a e\right)+b^4 d\right)}{\sqrt{b^2-4 a c}}+\frac{3 \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right) \left(a^2 c \left(e \sqrt{b^2-4 a c}+2 a f-2 c d\right)-a b^2 \left(e \sqrt{b^2-4 a c}+a f-4 c d\right)+a b \left(-2 c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}-3 a c e\right)+b^3 \left(d \sqrt{b^2-4 a c}+a e\right)+b^4 (-d)\right)}{\sqrt{b^2-4 a c}}+\frac{3 a^2 (b d-a e)}{x^4}+\frac{6 a \left(a b e+a (c d-a f)+b^2 (-d)\right)}{x^2}}{12 a^4}","-\frac{-a b e-a (c d-a f)+b^2 d}{2 a^3 x^2}+\frac{b d-a e}{4 a^2 x^4}+\frac{\log \left(a+b x^2+c x^4\right) \left(a^2 c e-a b^2 e-a b (2 c d-a f)+b^3 d\right)}{4 a^4}-\frac{\log (x) \left(a^2 c e-a b^2 e-a b (2 c d-a f)+b^3 d\right)}{a^4}-\frac{\tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right) \left(3 a^2 b c e+2 a^2 c (c d-a f)-a b^3 e-a b^2 (4 c d-a f)+b^4 d\right)}{2 a^4 \sqrt{b^2-4 a c}}-\frac{d}{6 a x^6}",1,"((-2*a^3*d)/x^6 + (3*a^2*(b*d - a*e))/x^4 + (6*a*(-(b^2*d) + a*b*e + a*(c*d - a*f)))/x^2 - 12*(b^3*d - a*b^2*e + a^2*c*e + a*b*(-2*c*d + a*f))*Log[x] + (3*(b^4*d + b^3*(Sqrt[b^2 - 4*a*c]*d - a*e) + a^2*c*(2*c*d + Sqrt[b^2 - 4*a*c]*e - 2*a*f) + a*b^2*(-4*c*d - Sqrt[b^2 - 4*a*c]*e + a*f) + a*b*(-2*c*Sqrt[b^2 - 4*a*c]*d + 3*a*c*e + a*Sqrt[b^2 - 4*a*c]*f))*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/Sqrt[b^2 - 4*a*c] + (3*(-(b^4*d) + b^3*(Sqrt[b^2 - 4*a*c]*d + a*e) - a*b^2*(-4*c*d + Sqrt[b^2 - 4*a*c]*e + a*f) + a^2*c*(-2*c*d + Sqrt[b^2 - 4*a*c]*e + 2*a*f) + a*b*(-2*c*Sqrt[b^2 - 4*a*c]*d - 3*a*c*e + a*Sqrt[b^2 - 4*a*c]*f))*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/Sqrt[b^2 - 4*a*c])/(12*a^4)","A",1
55,1,456,369,0.509149,"\int \frac{x^4 \left(d+e x^2+f x^4\right)}{a+b x^2+c x^4} \, dx","Integrate[(x^4*(d + e*x^2 + f*x^4))/(a + b*x^2 + c*x^4),x]","\frac{x \left(-c (a f+b e)+b^2 f+c^2 d\right)}{c^3}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(a c^2 \left(e \sqrt{b^2-4 a c}-2 a f+2 c d\right)-b^2 c \left(e \sqrt{b^2-4 a c}-4 a f+c d\right)+b c \left(c d \sqrt{b^2-4 a c}-2 a f \sqrt{b^2-4 a c}-3 a c e\right)+b^3 \left(f \sqrt{b^2-4 a c}+c e\right)+b^4 (-f)\right)}{\sqrt{2} c^{7/2} \sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(a c^2 \left(e \sqrt{b^2-4 a c}+2 a f-2 c d\right)+b^2 c \left(-e \sqrt{b^2-4 a c}-4 a f+c d\right)+b c \left(c d \sqrt{b^2-4 a c}-2 a f \sqrt{b^2-4 a c}+3 a c e\right)+b^3 \left(f \sqrt{b^2-4 a c}-c e\right)+b^4 f\right)}{\sqrt{2} c^{7/2} \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x^3 (c e-b f)}{3 c^2}+\frac{f x^5}{5 c}","\frac{x \left(-c (a f+b e)+b^2 f+c^2 d\right)}{c^3}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(-\frac{-b^2 c (c d-4 a f)-3 a b c^2 e+2 a c^2 (c d-a f)+b^4 (-f)+b^3 c e}{\sqrt{b^2-4 a c}}-b c (c d-2 a f)-a c^2 e+b^3 (-f)+b^2 c e\right)}{\sqrt{2} c^{7/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(\frac{-b^2 c (c d-4 a f)-3 a b c^2 e+2 a c^2 (c d-a f)+b^4 (-f)+b^3 c e}{\sqrt{b^2-4 a c}}-b c (c d-2 a f)-a c^2 e+b^3 (-f)+b^2 c e\right)}{\sqrt{2} c^{7/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x^3 (c e-b f)}{3 c^2}+\frac{f x^5}{5 c}",1,"((c^2*d + b^2*f - c*(b*e + a*f))*x)/c^3 + ((c*e - b*f)*x^3)/(3*c^2) + (f*x^5)/(5*c) - ((-(b^4*f) - b^2*c*(c*d + Sqrt[b^2 - 4*a*c]*e - 4*a*f) + a*c^2*(2*c*d + Sqrt[b^2 - 4*a*c]*e - 2*a*f) + b^3*(c*e + Sqrt[b^2 - 4*a*c]*f) + b*c*(c*Sqrt[b^2 - 4*a*c]*d - 3*a*c*e - 2*a*Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(7/2)*Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - ((b^4*f + b^2*c*(c*d - Sqrt[b^2 - 4*a*c]*e - 4*a*f) + a*c^2*(-2*c*d + Sqrt[b^2 - 4*a*c]*e + 2*a*f) + b^3*(-(c*e) + Sqrt[b^2 - 4*a*c]*f) + b*c*(c*Sqrt[b^2 - 4*a*c]*d + 3*a*c*e - 2*a*Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(7/2)*Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]])","A",1
56,1,365,282,0.4918759,"\int \frac{x^2 \left(d+e x^2+f x^4\right)}{a+b x^2+c x^4} \, dx","Integrate[(x^2*(d + e*x^2 + f*x^4))/(a + b*x^2 + c*x^4),x]","\frac{\frac{3 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(-b c \left(e \sqrt{b^2-4 a c}-3 a f+c d\right)+c \left(c d \sqrt{b^2-4 a c}-a f \sqrt{b^2-4 a c}-2 a c e\right)+b^2 \left(f \sqrt{b^2-4 a c}+c e\right)+b^3 (-f)\right)}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(b c \left(-e \sqrt{b^2-4 a c}-3 a f+c d\right)+c \left(c d \sqrt{b^2-4 a c}-a f \sqrt{b^2-4 a c}+2 a c e\right)+b^2 \left(f \sqrt{b^2-4 a c}-c e\right)+b^3 f\right)}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+6 \sqrt{c} x (c e-b f)+2 c^{3/2} f x^3}{6 c^{5/2}}","\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(\frac{-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e}{\sqrt{b^2-4 a c}}-a c f+b^2 f-b c e+c^2 d\right)}{\sqrt{2} c^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(-\frac{-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e}{\sqrt{b^2-4 a c}}-a c f+b^2 f-b c e+c^2 d\right)}{\sqrt{2} c^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x (c e-b f)}{c^2}+\frac{f x^3}{3 c}",1,"(6*Sqrt[c]*(c*e - b*f)*x + 2*c^(3/2)*f*x^3 + (3*Sqrt[2]*(-(b^3*f) - b*c*(c*d + Sqrt[b^2 - 4*a*c]*e - 3*a*f) + b^2*(c*e + Sqrt[b^2 - 4*a*c]*f) + c*(c*Sqrt[b^2 - 4*a*c]*d - 2*a*c*e - a*Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[2]*(b^3*f + b*c*(c*d - Sqrt[b^2 - 4*a*c]*e - 3*a*f) + b^2*(-(c*e) + Sqrt[b^2 - 4*a*c]*f) + c*(c*Sqrt[b^2 - 4*a*c]*d + 2*a*c*e - a*Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(6*c^(5/2))","A",1
57,1,258,219,0.3256333,"\int \frac{d+e x^2+f x^4}{a+b x^2+c x^4} \, dx","Integrate[(d + e*x^2 + f*x^4)/(a + b*x^2 + c*x^4),x]","\frac{\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(c \left(e \sqrt{b^2-4 a c}-2 a f-b e\right)+b f \left(b-\sqrt{b^2-4 a c}\right)+2 c^2 d\right)}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(-c \left(e \sqrt{b^2-4 a c}+2 a f+b e\right)+b f \left(\sqrt{b^2-4 a c}+b\right)+2 c^2 d\right)}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+2 \sqrt{c} f x}{2 c^{3/2}}","\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(\frac{-c (2 a f+b e)+b^2 f+2 c^2 d}{\sqrt{b^2-4 a c}}-b f+c e\right)}{\sqrt{2} c^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(-\frac{-2 a c f+b^2 f-b c e+2 c^2 d}{\sqrt{b^2-4 a c}}-b f+c e\right)}{\sqrt{2} c^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{f x}{c}",1,"(2*Sqrt[c]*f*x + (Sqrt[2]*(2*c^2*d + b*(b - Sqrt[b^2 - 4*a*c])*f + c*(-(b*e) + Sqrt[b^2 - 4*a*c]*e - 2*a*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*(2*c^2*d + b*(b + Sqrt[b^2 - 4*a*c])*f - c*(b*e + Sqrt[b^2 - 4*a*c]*e + 2*a*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(2*c^(3/2))","A",1
58,1,253,213,0.3023829,"\int \frac{d+e x^2+f x^4}{x^2 \left(a+b x^2+c x^4\right)} \, dx","Integrate[(d + e*x^2 + f*x^4)/(x^2*(a + b*x^2 + c*x^4)),x]","\frac{-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(c d \sqrt{b^2-4 a c}-a f \sqrt{b^2-4 a c}+a b f-2 a c e+b c d\right)}{\sqrt{c} \sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(-c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}+a b f-2 a c e+b c d\right)}{\sqrt{c} \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{2 d}{x}}{2 a}","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(\frac{a b f-2 a c e+b c d}{\sqrt{b^2-4 a c}}-a f+c d\right)}{\sqrt{2} a \sqrt{c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(-\frac{a b f-2 a c e+b c d}{\sqrt{b^2-4 a c}}-a f+c d\right)}{\sqrt{2} a \sqrt{c} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{d}{a x}",1,"((-2*d)/x - (Sqrt[2]*(b*c*d + c*Sqrt[b^2 - 4*a*c]*d - 2*a*c*e + a*b*f - a*Sqrt[b^2 - 4*a*c]*f)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(b*c*d - c*Sqrt[b^2 - 4*a*c]*d - 2*a*c*e + a*b*f + a*Sqrt[b^2 - 4*a*c]*f)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(2*a)","A",1
59,1,284,267,0.3390095,"\int \frac{d+e x^2+f x^4}{x^4 \left(a+b x^2+c x^4\right)} \, dx","Integrate[(d + e*x^2 + f*x^4)/(x^4*(a + b*x^2 + c*x^4)),x]","\frac{\frac{3 \sqrt{2} \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(a \left(-e \sqrt{b^2-4 a c}+2 a f-2 c d\right)+b \left(d \sqrt{b^2-4 a c}-a e\right)+b^2 d\right)}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{2} \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(-a \left(e \sqrt{b^2-4 a c}+2 a f-2 c d\right)+b \left(d \sqrt{b^2-4 a c}+a e\right)+b^2 (-d)\right)}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{6 b d-6 a e}{x}-\frac{2 a d}{x^3}}{6 a^2}","\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(\frac{-a b e-2 a (c d-a f)+b^2 d}{\sqrt{b^2-4 a c}}-a e+b d\right)}{\sqrt{2} a^2 \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(-a \left(-e \sqrt{b^2-4 a c}-2 a f+2 c d\right)-b \left(d \sqrt{b^2-4 a c}+a e\right)+b^2 d\right)}{\sqrt{2} a^2 \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{b d-a e}{a^2 x}-\frac{d}{3 a x^3}",1,"((-2*a*d)/x^3 + (6*b*d - 6*a*e)/x + (3*Sqrt[2]*Sqrt[c]*(b^2*d + b*(Sqrt[b^2 - 4*a*c]*d - a*e) + a*(-2*c*d - Sqrt[b^2 - 4*a*c]*e + 2*a*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[2]*Sqrt[c]*(-(b^2*d) + b*(Sqrt[b^2 - 4*a*c]*d + a*e) - a*(-2*c*d + Sqrt[b^2 - 4*a*c]*e + 2*a*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(6*a^2)","A",1
60,1,394,329,0.5512712,"\int \frac{d+e x^2+f x^4}{x^6 \left(a+b x^2+c x^4\right)} \, dx","Integrate[(d + e*x^2 + f*x^4)/(x^6*(a + b*x^2 + c*x^4)),x]","\frac{-\frac{6 a^2 d}{x^5}+\frac{30 \left(a b e+a (c d-a f)+b^2 (-d)\right)}{x}-\frac{15 \sqrt{2} \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(a b \left(-e \sqrt{b^2-4 a c}+a f-3 c d\right)+a \left(-c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}+2 a c e\right)+b^2 \left(d \sqrt{b^2-4 a c}-a e\right)+b^3 d\right)}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{15 \sqrt{2} \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(a b \left(e \sqrt{b^2-4 a c}+a f-3 c d\right)+a \left(c d \sqrt{b^2-4 a c}-a f \sqrt{b^2-4 a c}+2 a c e\right)-b^2 \left(d \sqrt{b^2-4 a c}+a e\right)+b^3 d\right)}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{10 a (b d-a e)}{x^3}}{30 a^3}","-\frac{-a b e-a (c d-a f)+b^2 d}{a^3 x}+\frac{b d-a e}{3 a^2 x^3}-\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(\frac{2 a^2 c e-a b^2 e-a b (3 c d-a f)+b^3 d}{\sqrt{b^2-4 a c}}-a b e-a (c d-a f)+b^2 d\right)}{\sqrt{2} a^3 \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(-\frac{2 a^2 c e-a b^2 e-a b (3 c d-a f)+b^3 d}{\sqrt{b^2-4 a c}}-a b e-a (c d-a f)+b^2 d\right)}{\sqrt{2} a^3 \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{d}{5 a x^5}",1,"((-6*a^2*d)/x^5 + (10*a*(b*d - a*e))/x^3 + (30*(-(b^2*d) + a*b*e + a*(c*d - a*f)))/x - (15*Sqrt[2]*Sqrt[c]*(b^3*d + b^2*(Sqrt[b^2 - 4*a*c]*d - a*e) + a*b*(-3*c*d - Sqrt[b^2 - 4*a*c]*e + a*f) + a*(-(c*Sqrt[b^2 - 4*a*c]*d) + 2*a*c*e + a*Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (15*Sqrt[2]*Sqrt[c]*(b^3*d - b^2*(Sqrt[b^2 - 4*a*c]*d + a*e) + a*b*(-3*c*d + Sqrt[b^2 - 4*a*c]*e + a*f) + a*(c*Sqrt[b^2 - 4*a*c]*d + 2*a*c*e - a*Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(30*a^3)","A",1
61,1,309,320,0.4965887,"\int \frac{x^7 \left(d+e x^2+f x^4\right)}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(x^7*(d + e*x^2 + f*x^4))/(a + b*x^2 + c*x^4)^2,x]","\frac{\frac{2 \tan ^{-1}\left(\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right) \left(-12 a^2 c^3 e+b^3 c (c d-20 a f)+12 a b^2 c^2 e+6 a b c^2 (5 a f-c d)+3 b^5 f-2 b^4 c e\right)}{\left(4 a c-b^2\right)^{3/2}}+\frac{2 \left(2 a^3 c^2 f+a^2 c \left(-4 b^2 f+b c \left(3 e+5 f x^2\right)-2 c^2 \left(d+e x^2\right)\right)+a b \left(b^3 f-b^2 c \left(e+5 f x^2\right)+b c^2 \left(d+4 e x^2\right)-3 c^3 d x^2\right)+b^3 x^2 \left(b^2 f-b c e+c^2 d\right)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\log \left(a+b x^2+c x^4\right) \left(-2 c (a f+b e)+3 b^2 f+c^2 d\right)+2 c x^2 (c e-2 b f)+c^2 f x^4}{4 c^4}","-\frac{\tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right) \left(12 a^2 c^3 e-b^3 c (c d-20 a f)-12 a b^2 c^2 e+6 a b c^2 (c d-5 a f)-3 b^5 f+2 b^4 c e\right)}{2 c^4 \left(b^2-4 a c\right)^{3/2}}+\frac{x^4 \left(-2 c (4 a f+b e)+3 b^2 f+4 c^2 d\right)}{4 c^2 \left(b^2-4 a c\right)}+\frac{x^6 \left(-\left(x^2 \left(-2 a c f+b^2 f-b c e+2 c^2 d\right)\right)-b (a f+c d)+2 a c e\right)}{2 c \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\log \left(a+b x^2+c x^4\right) \left(-2 c (a f+b e)+3 b^2 f+c^2 d\right)}{4 c^4}+\frac{x^2 \left(-b c (c d-11 a f)-6 a c^2 e-3 b^3 f+2 b^2 c e\right)}{2 c^3 \left(b^2-4 a c\right)}",1,"(2*c*(c*e - 2*b*f)*x^2 + c^2*f*x^4 + (2*(2*a^3*c^2*f + b^3*(c^2*d - b*c*e + b^2*f)*x^2 + a*b*(b^3*f - 3*c^3*d*x^2 + b*c^2*(d + 4*e*x^2) - b^2*c*(e + 5*f*x^2)) + a^2*c*(-4*b^2*f - 2*c^2*(d + e*x^2) + b*c*(3*e + 5*f*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (2*(-2*b^4*c*e + 12*a*b^2*c^2*e - 12*a^2*c^3*e + 3*b^5*f + b^3*c*(c*d - 20*a*f) + 6*a*b*c^2*(-(c*d) + 5*a*f))*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + (c^2*d + 3*b^2*f - 2*c*(b*e + a*f))*Log[a + b*x^2 + c*x^4])/(4*c^4)","A",1
62,1,236,236,0.3559635,"\int \frac{x^5 \left(d+e x^2+f x^4\right)}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(x^5*(d + e*x^2 + f*x^4))/(a + b*x^2 + c*x^4)^2,x]","\frac{-\frac{2 \tan ^{-1}\left(\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right) \left(12 a^2 c^2 f-2 a c \left(6 b^2 f-3 b c e+2 c^2 d\right)+b^3 (2 b f-c e)\right)}{\left(4 a c-b^2\right)^{3/2}}-\frac{2 \left(a^2 c \left(2 c \left(e+f x^2\right)-3 b f\right)+a \left(b^3 f-b^2 c \left(e+4 f x^2\right)+b c^2 \left(d+3 e x^2\right)-2 c^3 d x^2\right)+b^2 x^2 \left(b^2 f-b c e+c^2 d\right)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+(c e-2 b f) \log \left(a+b x^2+c x^4\right)+2 c f x^2}{4 c^3}","-\frac{\tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right) \left(12 a^2 c^2 f-2 a c \left(6 b^2 f-3 b c e+2 c^2 d\right)-\left(b^3 (c e-2 b f)\right)\right)}{2 c^3 \left(b^2-4 a c\right)^{3/2}}+\frac{x^2 \left(-c (6 a f+b e)+2 b^2 f+2 c^2 d\right)}{2 c^2 \left(b^2-4 a c\right)}+\frac{x^4 \left(-\left(x^2 \left(-2 a c f+b^2 f-b c e+2 c^2 d\right)\right)-b (a f+c d)+2 a c e\right)}{2 c \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{(c e-2 b f) \log \left(a+b x^2+c x^4\right)}{4 c^3}",1,"(2*c*f*x^2 - (2*(b^2*(c^2*d - b*c*e + b^2*f)*x^2 + a^2*c*(-3*b*f + 2*c*(e + f*x^2)) + a*(b^3*f - 2*c^3*d*x^2 + b*c^2*(d + 3*e*x^2) - b^2*c*(e + 4*f*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - (2*(12*a^2*c^2*f + b^3*(-(c*e) + 2*b*f) - 2*a*c*(2*c^2*d - 3*b*c*e + 6*b^2*f))*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + (c*e - 2*b*f)*Log[a + b*x^2 + c*x^4])/(4*c^3)","A",1
63,1,175,165,0.2490863,"\int \frac{x^3 \left(d+e x^2+f x^4\right)}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(x^3*(d + e*x^2 + f*x^4))/(a + b*x^2 + c*x^4)^2,x]","\frac{\frac{2 \left(-2 a^2 c f+a \left(b^2 f-b c \left(e+3 f x^2\right)+2 c^2 \left(d+e x^2\right)\right)+b x^2 \left(b^2 f-b c e+c^2 d\right)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{2 \tan ^{-1}\left(\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right) \left(-2 b c (3 a f+c d)+4 a c^2 e+b^3 f\right)}{\left(4 a c-b^2\right)^{3/2}}+f \log \left(a+b x^2+c x^4\right)}{4 c^2}","\frac{x^2 \left(-\left(x^2 \left(-2 a c f+b^2 f-b c e+2 c^2 d\right)\right)-b (a f+c d)+2 a c e\right)}{2 c \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right) \left(-2 b c (3 a f+c d)+4 a c^2 e+b^3 f\right)}{2 c^2 \left(b^2-4 a c\right)^{3/2}}+\frac{f \log \left(a+b x^2+c x^4\right)}{4 c^2}",1,"((2*(-2*a^2*c*f + b*(c^2*d - b*c*e + b^2*f)*x^2 + a*(b^2*f + 2*c^2*(d + e*x^2) - b*c*(e + 3*f*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (2*(4*a*c^2*e + b^3*f - 2*b*c*(c*d + 3*a*f))*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + f*Log[a + b*x^2 + c*x^4])/(4*c^2)","A",1
64,1,130,123,0.1024633,"\int \frac{x \left(d+e x^2+f x^4\right)}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(x*(d + e*x^2 + f*x^4))/(a + b*x^2 + c*x^4)^2,x]","\frac{a b f-2 a c \left(e+f x^2\right)+b^2 f x^2+b c \left(d-e x^2\right)+2 c^2 d x^2}{2 c \left(4 a c-b^2\right) \left(a+b x^2+c x^4\right)}-\frac{\tan ^{-1}\left(\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right) (-2 a f+b e-2 c d)}{\left(4 a c-b^2\right)^{3/2}}","\frac{-\left(x^2 \left(-2 a c f+b^2 f-b c e+2 c^2 d\right)\right)-b (a f+c d)+2 a c e}{2 c \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right) (2 a f-b e+2 c d)}{\left(b^2-4 a c\right)^{3/2}}",1,"(a*b*f + 2*c^2*d*x^2 + b^2*f*x^2 + b*c*(d - e*x^2) - 2*a*c*(e + f*x^2))/(2*c*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)) - ((-2*c*d + b*e - 2*a*f)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2)","A",1
65,1,268,166,0.4454919,"\int \frac{d+e x^2+f x^4}{x \left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(d + e*x^2 + f*x^4)/(x*(a + b*x^2 + c*x^4)^2),x]","-\frac{-\frac{2 a \left(b \left(-a e+a f x^2+c d x^2\right)+2 a \left(a f-c \left(d+e x^2\right)\right)+b^2 d\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\log \left(-\sqrt{b^2-4 a c}+b+2 c x^2\right) \left(4 a c \left(a e-d \sqrt{b^2-4 a c}\right)+b^2 d \sqrt{b^2-4 a c}-2 a b (a f+3 c d)+b^3 d\right)}{\left(b^2-4 a c\right)^{3/2}}+\frac{\log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right) \left(-4 a c \left(d \sqrt{b^2-4 a c}+a e\right)+b^2 d \sqrt{b^2-4 a c}+2 a b (a f+3 c d)+b^3 (-d)\right)}{\left(b^2-4 a c\right)^{3/2}}-4 d \log (x)}{4 a^2}","\frac{\tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right) \left(4 a^2 c e-2 a b (a f+3 c d)+b^3 d\right)}{2 a^2 \left(b^2-4 a c\right)^{3/2}}-\frac{d \log \left(a+b x^2+c x^4\right)}{4 a^2}+\frac{d \log (x)}{a^2}+\frac{x^2 (a b f-2 a c e+b c d)-a b e-2 a (c d-a f)+b^2 d}{2 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"-1/4*((-2*a*(b^2*d + b*(-(a*e) + c*d*x^2 + a*f*x^2) + 2*a*(a*f - c*(d + e*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - 4*d*Log[x] + ((b^3*d + b^2*Sqrt[b^2 - 4*a*c]*d + 4*a*c*(-(Sqrt[b^2 - 4*a*c]*d) + a*e) - 2*a*b*(3*c*d + a*f))*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2) + ((-(b^3*d) + b^2*Sqrt[b^2 - 4*a*c]*d - 4*a*c*(Sqrt[b^2 - 4*a*c]*d + a*e) + 2*a*b*(3*c*d + a*f))*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/a^2","A",1
66,1,403,234,0.658012,"\int \frac{d+e x^2+f x^4}{x^3 \left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(d + e*x^2 + f*x^4)/(x^3*(a + b*x^2 + c*x^4)^2),x]","\frac{\frac{\log \left(-\sqrt{b^2-4 a c}+b+2 c x^2\right) \left(4 a^2 c \left(e \sqrt{b^2-4 a c}-a f+3 c d\right)-a b^2 \left(e \sqrt{b^2-4 a c}+12 c d\right)+2 a b c \left(3 a e-4 d \sqrt{b^2-4 a c}\right)+b^3 \left(2 d \sqrt{b^2-4 a c}-a e\right)+2 b^4 d\right)}{\left(b^2-4 a c\right)^{3/2}}+\frac{\log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right) \left(4 a^2 c \left(e \sqrt{b^2-4 a c}+a f-3 c d\right)+a b^2 \left(12 c d-e \sqrt{b^2-4 a c}\right)-2 a b c \left(4 d \sqrt{b^2-4 a c}+3 a e\right)+b^3 \left(2 d \sqrt{b^2-4 a c}+a e\right)-2 b^4 d\right)}{\left(b^2-4 a c\right)^{3/2}}-\frac{2 a \left(b^2 \left(c d x^2-a e\right)+a b \left(a f-c \left(3 d+e x^2\right)\right)+2 a c \left(a \left(e+f x^2\right)-c d x^2\right)+b^3 d\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+4 \log (x) (a e-2 b d)-\frac{2 a d}{x^2}}{4 a^3}","\frac{(2 b d-a e) \log \left(a+b x^2+c x^4\right)}{4 a^3}-\frac{\log (x) (2 b d-a e)}{a^3}-\frac{2 a^2 c e+c x^2 \left(-a b e-2 a (c d-a f)+b^2 d\right)-a b^2 e-a b (3 c d-a f)+b^3 d}{2 a^2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-\frac{d}{2 a^2 x^2}-\frac{\tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right) \left(6 a^2 b c e+4 a^2 c (3 c d-a f)-a b^3 e-12 a b^2 c d+2 b^4 d\right)}{2 a^3 \left(b^2-4 a c\right)^{3/2}}",1,"((-2*a*d)/x^2 - (2*a*(b^3*d + b^2*(-(a*e) + c*d*x^2) + a*b*(a*f - c*(3*d + e*x^2)) + 2*a*c*(-(c*d*x^2) + a*(e + f*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + 4*(-2*b*d + a*e)*Log[x] + ((2*b^4*d + b^3*(2*Sqrt[b^2 - 4*a*c]*d - a*e) + 2*a*b*c*(-4*Sqrt[b^2 - 4*a*c]*d + 3*a*e) - a*b^2*(12*c*d + Sqrt[b^2 - 4*a*c]*e) + 4*a^2*c*(3*c*d + Sqrt[b^2 - 4*a*c]*e - a*f))*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2) + ((-2*b^4*d + b^3*(2*Sqrt[b^2 - 4*a*c]*d + a*e) - 2*a*b*c*(4*Sqrt[b^2 - 4*a*c]*d + 3*a*e) + a*b^2*(12*c*d - Sqrt[b^2 - 4*a*c]*e) + 4*a^2*c*(-3*c*d + Sqrt[b^2 - 4*a*c]*e + a*f))*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/(4*a^3)","A",1
67,1,592,329,1.216446,"\int \frac{d+e x^2+f x^4}{x^5 \left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(d + e*x^2 + f*x^4)/(x^5*(a + b*x^2 + c*x^4)^2),x]","-\frac{\frac{2 a \left(2 a^2 c \left(a f-c \left(d+e x^2\right)\right)+b^3 \left(a e-c d x^2\right)+a b^2 \left(-a f+4 c d+c e x^2\right)-a b c \left(3 a e+a f x^2-3 c d x^2\right)+b^4 (-d)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\log \left(-\sqrt{b^2-4 a c}+b+2 c x^2\right) \left(2 a^2 b c \left(4 e \sqrt{b^2-4 a c}-3 a f+15 c d\right)-4 a^2 c \left(-2 c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}+3 a c e\right)+a b^2 \left(-14 c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}+12 a c e\right)+b^4 \left(3 d \sqrt{b^2-4 a c}-2 a e\right)+a b^3 \left(-2 e \sqrt{b^2-4 a c}+a f-20 c d\right)+3 b^5 d\right)}{\left(b^2-4 a c\right)^{3/2}}+\frac{\log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right) \left(2 a^2 b c \left(4 e \sqrt{b^2-4 a c}+3 a f-15 c d\right)+4 a^2 c \left(2 c d \sqrt{b^2-4 a c}-a f \sqrt{b^2-4 a c}+3 a c e\right)+a b^2 \left(a f \sqrt{b^2-4 a c}-2 c \left(7 d \sqrt{b^2-4 a c}+6 a e\right)\right)+b^4 \left(3 d \sqrt{b^2-4 a c}+2 a e\right)-a b^3 \left(2 e \sqrt{b^2-4 a c}+a f-20 c d\right)-3 b^5 d\right)}{\left(b^2-4 a c\right)^{3/2}}+\frac{a^2 d}{x^4}-4 \log (x) \left(-2 a b e+a (a f-2 c d)+3 b^2 d\right)+\frac{2 a (a e-2 b d)}{x^2}}{4 a^4}","-\frac{\log \left(a+b x^2+c x^4\right) \left(-2 a b e-a (2 c d-a f)+3 b^2 d\right)}{4 a^4}+\frac{\log (x) \left(-2 a b e-a (2 c d-a f)+3 b^2 d\right)}{a^4}+\frac{2 b d-a e}{2 a^3 x^2}-\frac{d}{4 a^2 x^4}+\frac{c x^2 \left(2 a^2 c e-a b^2 e-a b (3 c d-a f)+b^3 d\right)+3 a^2 b c e+2 a^2 c (c d-a f)-a b^3 e-a b^2 (4 c d-a f)+b^4 d}{2 a^3 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right) \left(-12 a^3 c^2 e+12 a^2 b^2 c e+6 a^2 b c (5 c d-a f)-2 a b^4 e-a b^3 (20 c d-a f)+3 b^5 d\right)}{2 a^4 \left(b^2-4 a c\right)^{3/2}}",1,"-1/4*((a^2*d)/x^4 + (2*a*(-2*b*d + a*e))/x^2 + (2*a*(-(b^4*d) + b^3*(a*e - c*d*x^2) + a*b^2*(4*c*d - a*f + c*e*x^2) - a*b*c*(3*a*e - 3*c*d*x^2 + a*f*x^2) + 2*a^2*c*(a*f - c*(d + e*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - 4*(3*b^2*d - 2*a*b*e + a*(-2*c*d + a*f))*Log[x] + ((3*b^5*d + b^4*(3*Sqrt[b^2 - 4*a*c]*d - 2*a*e) + 2*a^2*b*c*(15*c*d + 4*Sqrt[b^2 - 4*a*c]*e - 3*a*f) + a*b^3*(-20*c*d - 2*Sqrt[b^2 - 4*a*c]*e + a*f) - 4*a^2*c*(-2*c*Sqrt[b^2 - 4*a*c]*d + 3*a*c*e + a*Sqrt[b^2 - 4*a*c]*f) + a*b^2*(-14*c*Sqrt[b^2 - 4*a*c]*d + 12*a*c*e + a*Sqrt[b^2 - 4*a*c]*f))*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2) + ((-3*b^5*d + b^4*(3*Sqrt[b^2 - 4*a*c]*d + 2*a*e) - a*b^3*(-20*c*d + 2*Sqrt[b^2 - 4*a*c]*e + a*f) + 2*a^2*b*c*(-15*c*d + 4*Sqrt[b^2 - 4*a*c]*e + 3*a*f) + 4*a^2*c*(2*c*Sqrt[b^2 - 4*a*c]*d + 3*a*c*e - a*Sqrt[b^2 - 4*a*c]*f) + a*b^2*(-2*c*(7*Sqrt[b^2 - 4*a*c]*d + 6*a*e) + a*Sqrt[b^2 - 4*a*c]*f))*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/a^4","A",1
68,1,648,550,2.1306957,"\int \frac{x^6 \left(d+e x^2+f x^4\right)}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(x^6*(d + e*x^2 + f*x^4))/(a + b*x^2 + c*x^4)^2,x]","\frac{-\frac{6 \sqrt{c} x \left(a^2 c \left(2 c \left(e+f x^2\right)-3 b f\right)+a \left(b^3 f-b^2 c \left(e+4 f x^2\right)+b c^2 \left(d+3 e x^2\right)-2 c^3 d x^2\right)+b^2 x^2 \left(b^2 f-b c e+c^2 d\right)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{3 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(a b c^2 \left(13 e \sqrt{b^2-4 a c}-52 a f+8 c d\right)+2 a c^2 \left(-3 c d \sqrt{b^2-4 a c}+7 a f \sqrt{b^2-4 a c}+10 a c e\right)+b^2 c \left(c d \sqrt{b^2-4 a c}-24 a f \sqrt{b^2-4 a c}-19 a c e\right)+b^4 \left(5 f \sqrt{b^2-4 a c}+3 c e\right)-b^3 c \left(3 e \sqrt{b^2-4 a c}-34 a f+c d\right)-5 b^5 f\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(a b c^2 \left(13 e \sqrt{b^2-4 a c}+52 a f-8 c d\right)-2 a c^2 \left(3 c d \sqrt{b^2-4 a c}-7 a f \sqrt{b^2-4 a c}+10 a c e\right)+b^2 c \left(c d \sqrt{b^2-4 a c}-24 a f \sqrt{b^2-4 a c}+19 a c e\right)+b^4 \left(5 f \sqrt{b^2-4 a c}-3 c e\right)+b^3 c \left(-3 e \sqrt{b^2-4 a c}-34 a f+c d\right)+5 b^5 f\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+12 \sqrt{c} x (c e-2 b f)+4 c^{3/2} f x^3}{12 c^{7/2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(-\frac{20 a^2 c^3 e-b^3 c (c d-34 a f)-19 a b^2 c^2 e+4 a b c^2 (2 c d-13 a f)-5 b^5 f+3 b^4 c e}{\sqrt{b^2-4 a c}}-b^2 c (c d-24 a f)-13 a b c^2 e+2 a c^2 (3 c d-7 a f)-5 b^4 f+3 b^3 c e\right)}{2 \sqrt{2} c^{7/2} \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(\frac{20 a^2 c^3 e-b^3 c (c d-34 a f)-19 a b^2 c^2 e+4 a b c^2 (2 c d-13 a f)-5 b^5 f+3 b^4 c e}{\sqrt{b^2-4 a c}}-b^2 c (c d-24 a f)-13 a b c^2 e+2 a c^2 (3 c d-7 a f)-5 b^4 f+3 b^3 c e\right)}{2 \sqrt{2} c^{7/2} \left(b^2-4 a c\right) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x \left(a \left(-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right)+x^2 \left(-b^2 c (c d-4 a f)-3 a b c^2 e+2 a c^2 (c d-a f)+b^4 (-f)+b^3 c e\right)\right)}{2 c^3 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{x (c e-2 b f)}{c^3}+\frac{f x^3}{3 c^2}",1,"(12*Sqrt[c]*(c*e - 2*b*f)*x + 4*c^(3/2)*f*x^3 - (6*Sqrt[c]*x*(b^2*(c^2*d - b*c*e + b^2*f)*x^2 + a^2*c*(-3*b*f + 2*c*(e + f*x^2)) + a*(b^3*f - 2*c^3*d*x^2 + b*c^2*(d + 3*e*x^2) - b^2*c*(e + 4*f*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (3*Sqrt[2]*(-5*b^5*f + a*b*c^2*(8*c*d + 13*Sqrt[b^2 - 4*a*c]*e - 52*a*f) - b^3*c*(c*d + 3*Sqrt[b^2 - 4*a*c]*e - 34*a*f) + b^4*(3*c*e + 5*Sqrt[b^2 - 4*a*c]*f) + b^2*c*(c*Sqrt[b^2 - 4*a*c]*d - 19*a*c*e - 24*a*Sqrt[b^2 - 4*a*c]*f) + 2*a*c^2*(-3*c*Sqrt[b^2 - 4*a*c]*d + 10*a*c*e + 7*a*Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[2]*(5*b^5*f + b^3*c*(c*d - 3*Sqrt[b^2 - 4*a*c]*e - 34*a*f) + a*b*c^2*(-8*c*d + 13*Sqrt[b^2 - 4*a*c]*e + 52*a*f) + b^4*(-3*c*e + 5*Sqrt[b^2 - 4*a*c]*f) + b^2*c*(c*Sqrt[b^2 - 4*a*c]*d + 19*a*c*e - 24*a*Sqrt[b^2 - 4*a*c]*f) - 2*a*c^2*(3*c*Sqrt[b^2 - 4*a*c]*d + 10*a*c*e - 7*a*Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(12*c^(7/2))","A",1
69,1,511,436,1.5415455,"\int \frac{x^4 \left(d+e x^2+f x^4\right)}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(x^4*(d + e*x^2 + f*x^4))/(a + b*x^2 + c*x^4)^2,x]","\frac{\frac{2 \sqrt{c} x \left(-2 a^2 c f+a \left(b^2 f-b c \left(e+3 f x^2\right)+2 c^2 \left(d+e x^2\right)\right)+b x^2 \left(b^2 f-b c e+c^2 d\right)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(2 a c^2 \left(3 e \sqrt{b^2-4 a c}-10 a f+2 c d\right)+b^2 c \left(-e \sqrt{b^2-4 a c}+19 a f+c d\right)-b c \left(c d \sqrt{b^2-4 a c}+13 a f \sqrt{b^2-4 a c}+8 a c e\right)+b^3 \left(3 f \sqrt{b^2-4 a c}+c e\right)-3 b^4 f\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(2 a c^2 \left(3 e \sqrt{b^2-4 a c}+10 a f-2 c d\right)-b^2 c \left(e \sqrt{b^2-4 a c}+19 a f+c d\right)-b c \left(c d \sqrt{b^2-4 a c}+13 a f \sqrt{b^2-4 a c}-8 a c e\right)+b^3 \left(3 f \sqrt{b^2-4 a c}-c e\right)+3 b^4 f\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+4 \sqrt{c} f x}{4 c^{5/2}}","\frac{x \left(a \left(-2 a c f+b^2 f-b c e+2 c^2 d\right)-x^2 \left(-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right)\right)}{2 c^2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(-\frac{b^2 c (19 a f+c d)-8 a b c^2 e+4 a c^2 (c d-5 a f)-3 b^4 f+b^3 c e}{\sqrt{b^2-4 a c}}+b c (13 a f+c d)-6 a c^2 e-3 b^3 f+b^2 c e\right)}{2 \sqrt{2} c^{5/2} \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(\frac{b^2 c (19 a f+c d)-8 a b c^2 e+4 a c^2 (c d-5 a f)-3 b^4 f+b^3 c e}{\sqrt{b^2-4 a c}}+b c (13 a f+c d)-6 a c^2 e-3 b^3 f+b^2 c e\right)}{2 \sqrt{2} c^{5/2} \left(b^2-4 a c\right) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{f x}{c^2}",1,"(4*Sqrt[c]*f*x + (2*Sqrt[c]*x*(-2*a^2*c*f + b*(c^2*d - b*c*e + b^2*f)*x^2 + a*(b^2*f + 2*c^2*(d + e*x^2) - b*c*(e + 3*f*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - (Sqrt[2]*(-3*b^4*f + 2*a*c^2*(2*c*d + 3*Sqrt[b^2 - 4*a*c]*e - 10*a*f) + b^2*c*(c*d - Sqrt[b^2 - 4*a*c]*e + 19*a*f) + b^3*(c*e + 3*Sqrt[b^2 - 4*a*c]*f) - b*c*(c*Sqrt[b^2 - 4*a*c]*d + 8*a*c*e + 13*a*Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*(3*b^4*f + 2*a*c^2*(-2*c*d + 3*Sqrt[b^2 - 4*a*c]*e + 10*a*f) - b^2*c*(c*d + Sqrt[b^2 - 4*a*c]*e + 19*a*f) + b^3*(-(c*e) + 3*Sqrt[b^2 - 4*a*c]*f) - b*c*(c*Sqrt[b^2 - 4*a*c]*d - 8*a*c*e + 13*a*Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(4*c^(5/2))","A",1
70,1,414,362,1.1027461,"\int \frac{x^2 \left(d+e x^2+f x^4\right)}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(x^2*(d + e*x^2 + f*x^4))/(a + b*x^2 + c*x^4)^2,x]","\frac{-\frac{2 \sqrt{c} x \left(a b f-2 a c \left(e+f x^2\right)+b^2 f x^2+b c \left(d-e x^2\right)+2 c^2 d x^2\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(b c \left(e \sqrt{b^2-4 a c}+8 a f+4 c d\right)-2 c \left(c d \sqrt{b^2-4 a c}+3 a f \sqrt{b^2-4 a c}+2 a c e\right)+b^2 \left(f \sqrt{b^2-4 a c}-c e\right)+b^3 (-f)\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(b c \left(e \sqrt{b^2-4 a c}-8 a f-4 c d\right)-2 c \left(c d \sqrt{b^2-4 a c}+3 a f \sqrt{b^2-4 a c}-2 a c e\right)+b^2 \left(f \sqrt{b^2-4 a c}+c e\right)+b^3 f\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}}{4 c^{3/2}}","-\frac{x \left(x^2 \left(-2 a c f+b^2 f-b c e+2 c^2 d\right)+a b f-2 a c e+b c d\right)}{2 c \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(\frac{-4 b c (2 a f+c d)+4 a c^2 e+b^3 f+b^2 c e}{c \sqrt{b^2-4 a c}}+6 a f-\frac{b^2 f}{c}-b e+2 c d\right)}{2 \sqrt{2} \sqrt{c} \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(-\frac{-4 b c (2 a f+c d)+4 a c^2 e+b^3 f+b^2 c e}{c \sqrt{b^2-4 a c}}+6 a f-\frac{b^2 f}{c}-b e+2 c d\right)}{2 \sqrt{2} \sqrt{c} \left(b^2-4 a c\right) \sqrt{\sqrt{b^2-4 a c}+b}}",1,"((-2*Sqrt[c]*x*(a*b*f + 2*c^2*d*x^2 + b^2*f*x^2 + b*c*(d - e*x^2) - 2*a*c*(e + f*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*(-(b^3*f) + b*c*(4*c*d + Sqrt[b^2 - 4*a*c]*e + 8*a*f) + b^2*(-(c*e) + Sqrt[b^2 - 4*a*c]*f) - 2*c*(c*Sqrt[b^2 - 4*a*c]*d + 2*a*c*e + 3*a*Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(b^3*f + b*c*(-4*c*d + Sqrt[b^2 - 4*a*c]*e - 8*a*f) + b^2*(c*e + Sqrt[b^2 - 4*a*c]*f) - 2*c*(c*Sqrt[b^2 - 4*a*c]*d - 2*a*c*e + 3*a*Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(4*c^(3/2))","A",1
71,1,382,346,1.0813824,"\int \frac{d+e x^2+f x^4}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(d + e*x^2 + f*x^4)/(a + b*x^2 + c*x^4)^2,x]","\frac{\frac{2 x \left(b \left(-a e+a f x^2+c d x^2\right)+2 a \left(a f-c \left(d+e x^2\right)\right)+b^2 d\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(b \left(c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}+4 a c e\right)-2 a c \left(e \sqrt{b^2-4 a c}+2 a f+6 c d\right)+b^2 (c d-a f)\right)}{\sqrt{c} \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(b \left(c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}-4 a c e\right)+2 a c \left(-e \sqrt{b^2-4 a c}+2 a f+6 c d\right)+b^2 (a f-c d)\right)}{\sqrt{c} \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}}{4 a}","\frac{x \left(x^2 (a b f-2 a c e+b c d)-a b e-2 a (c d-a f)+b^2 d\right)}{2 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(\frac{b^2 (c d-a f)+4 a b c e-4 a c (a f+3 c d)}{\sqrt{b^2-4 a c}}+a b f-2 a c e+b c d\right)}{2 \sqrt{2} a \sqrt{c} \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(-\frac{b^2 (c d-a f)+4 a b c e-4 a c (a f+3 c d)}{\sqrt{b^2-4 a c}}+a b f-2 a c e+b c d\right)}{2 \sqrt{2} a \sqrt{c} \left(b^2-4 a c\right) \sqrt{\sqrt{b^2-4 a c}+b}}",1,"((2*x*(b^2*d + b*(-(a*e) + c*d*x^2 + a*f*x^2) + 2*a*(a*f - c*(d + e*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*(b^2*(c*d - a*f) - 2*a*c*(6*c*d + Sqrt[b^2 - 4*a*c]*e + 2*a*f) + b*(c*Sqrt[b^2 - 4*a*c]*d + 4*a*c*e + a*Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(b^2*(-(c*d) + a*f) + 2*a*c*(6*c*d - Sqrt[b^2 - 4*a*c]*e + 2*a*f) + b*(c*Sqrt[b^2 - 4*a*c]*d - 4*a*c*e + a*Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(4*a)","A",1
72,1,444,399,1.3191724,"\int \frac{d+e x^2+f x^4}{x^2 \left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(d + e*x^2 + f*x^4)/(x^2*(a + b*x^2 + c*x^4)^2),x]","\frac{-\frac{2 x \left(b^2 \left(c d x^2-a e\right)+a b \left(a f-c \left(3 d+e x^2\right)\right)+2 a c \left(a \left(e+f x^2\right)-c d x^2\right)+b^3 d\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(a b \left(e \sqrt{b^2-4 a c}+4 a f+16 c d\right)-2 a \left(-5 c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}+6 a c e\right)+b^2 \left(a e-3 d \sqrt{b^2-4 a c}\right)-3 b^3 d\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(a b \left(e \sqrt{b^2-4 a c}-4 a f-16 c d\right)+2 a \left(5 c d \sqrt{b^2-4 a c}-a f \sqrt{b^2-4 a c}+6 a c e\right)-b^2 \left(3 d \sqrt{b^2-4 a c}+a e\right)+3 b^3 d\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{4 d}{x}}{4 a^2}","-\frac{x \left(a \left(\frac{b^3 d}{a}+a (b f+2 c e)-b (b e+3 c d)\right)+c x^2 \left(-a b e-2 a (c d-a f)+b^2 d\right)\right)}{2 a^2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(\frac{12 a^2 c e-a b^2 e-4 a b (a f+4 c d)+3 b^3 d}{\sqrt{b^2-4 a c}}-a b e-2 a (5 c d-a f)+3 b^2 d\right)}{2 \sqrt{2} a^2 \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(-\frac{12 a^2 c e-a b^2 e-4 a b (a f+4 c d)+3 b^3 d}{\sqrt{b^2-4 a c}}-a b e-2 a (5 c d-a f)+3 b^2 d\right)}{2 \sqrt{2} a^2 \left(b^2-4 a c\right) \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{d}{a^2 x}",1,"((-4*d)/x - (2*x*(b^3*d + b^2*(-(a*e) + c*d*x^2) + a*b*(a*f - c*(3*d + e*x^2)) + 2*a*c*(-(c*d*x^2) + a*(e + f*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*Sqrt[c]*(-3*b^3*d + b^2*(-3*Sqrt[b^2 - 4*a*c]*d + a*e) + a*b*(16*c*d + Sqrt[b^2 - 4*a*c]*e + 4*a*f) - 2*a*(-5*c*Sqrt[b^2 - 4*a*c]*d + 6*a*c*e + a*Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(3*b^3*d - b^2*(3*Sqrt[b^2 - 4*a*c]*d + a*e) + a*b*(-16*c*d + Sqrt[b^2 - 4*a*c]*e - 4*a*f) + 2*a*(5*c*Sqrt[b^2 - 4*a*c]*d + 6*a*c*e - a*Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(4*a^2)","A",1
73,1,548,575,1.7966739,"\int \frac{d+e x^2+f x^4}{x^4 \left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(d + e*x^2 + f*x^4)/(x^4*(a + b*x^2 + c*x^4)^2),x]","\frac{\frac{6 x \left(2 a^2 c \left(c \left(d+e x^2\right)-a f\right)+b^3 \left(c d x^2-a e\right)+a b^2 \left(a f-c \left(4 d+e x^2\right)\right)+a b c \left(3 a e+a f x^2-3 c d x^2\right)+b^4 d\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{3 \sqrt{2} \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(2 a^2 c \left(5 e \sqrt{b^2-4 a c}-6 a f+14 c d\right)+a b^2 \left(-3 e \sqrt{b^2-4 a c}+a f-29 c d\right)+a b \left(-19 c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}+16 a c e\right)+b^3 \left(5 d \sqrt{b^2-4 a c}-3 a e\right)+5 b^4 d\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{2} \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(2 a^2 c \left(5 e \sqrt{b^2-4 a c}+6 a f-14 c d\right)-a b^2 \left(3 e \sqrt{b^2-4 a c}+a f-29 c d\right)+a b \left(-19 c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}-16 a c e\right)+b^3 \left(5 d \sqrt{b^2-4 a c}+3 a e\right)-5 b^4 d\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{24 b d-12 a e}{x}-\frac{4 a d}{x^3}}{12 a^3}","\frac{2 b d-a e}{a^3 x}-\frac{d}{3 a^2 x^3}+\frac{x \left(a^2 \left(\frac{b^4 d}{a^2}-\frac{b^2 (b e+4 c d)}{a}-2 a c f+b^2 f+3 b c e+2 c^2 d\right)+c x^2 \left(2 a^2 c e-a b^2 e-a b (3 c d-a f)+b^3 d\right)\right)}{2 a^3 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(2 a^2 c \left(5 e \sqrt{b^2-4 a c}-6 a f+14 c d\right)-a b^2 \left(3 e \sqrt{b^2-4 a c}-a f+29 c d\right)-a b \left(19 c d \sqrt{b^2-4 a c}-a f \sqrt{b^2-4 a c}-16 a c e\right)+b^3 \left(5 d \sqrt{b^2-4 a c}-3 a e\right)+5 b^4 d\right)}{2 \sqrt{2} a^3 \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(2 a^2 c \left(-5 e \sqrt{b^2-4 a c}-6 a f+14 c d\right)-a b^2 \left(-3 e \sqrt{b^2-4 a c}-a f+29 c d\right)+a b \left(19 c d \sqrt{b^2-4 a c}-a f \sqrt{b^2-4 a c}+16 a c e\right)-b^3 \left(5 d \sqrt{b^2-4 a c}+3 a e\right)+5 b^4 d\right)}{2 \sqrt{2} a^3 \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}",1,"((-4*a*d)/x^3 + (24*b*d - 12*a*e)/x + (6*x*(b^4*d + b^3*(-(a*e) + c*d*x^2) + a*b*c*(3*a*e - 3*c*d*x^2 + a*f*x^2) + 2*a^2*c*(-(a*f) + c*(d + e*x^2)) + a*b^2*(a*f - c*(4*d + e*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (3*Sqrt[2]*Sqrt[c]*(5*b^4*d + b^3*(5*Sqrt[b^2 - 4*a*c]*d - 3*a*e) + 2*a^2*c*(14*c*d + 5*Sqrt[b^2 - 4*a*c]*e - 6*a*f) + a*b^2*(-29*c*d - 3*Sqrt[b^2 - 4*a*c]*e + a*f) + a*b*(-19*c*Sqrt[b^2 - 4*a*c]*d + 16*a*c*e + a*Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[2]*Sqrt[c]*(-5*b^4*d + b^3*(5*Sqrt[b^2 - 4*a*c]*d + 3*a*e) - a*b^2*(-29*c*d + 3*Sqrt[b^2 - 4*a*c]*e + a*f) + 2*a^2*c*(-14*c*d + 5*Sqrt[b^2 - 4*a*c]*e + 6*a*f) + a*b*(-19*c*Sqrt[b^2 - 4*a*c]*d - 16*a*c*e + a*Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(12*a^3)","A",1
74,1,62,68,0.0281317,"\int \frac{x^9 \left(4+x^2+3 x^4+5 x^6\right)}{\left(2+3 x^2+x^4\right)^2} \, dx","Integrate[(x^9*(4 + x^2 + 3*x^4 + 5*x^6))/(2 + 3*x^2 + x^4)^2,x]","\frac{1}{8} \left(5 x^8-36 x^6+196 x^4-1172 x^2+16 \log \left(x^2+1\right)+3136 \log \left(x^2+2\right)+\frac{4 \left(415 x^2+414\right)}{x^4+3 x^2+2}\right)","\frac{5 x^8}{8}-\frac{9 x^6}{2}+\frac{49 x^4}{2}-\frac{293 x^2}{2}+2 \log \left(x^2+1\right)+392 \log \left(x^2+2\right)+\frac{415 x^2+414}{2 \left(x^4+3 x^2+2\right)}",1,"(-1172*x^2 + 196*x^4 - 36*x^6 + 5*x^8 + (4*(414 + 415*x^2))/(2 + 3*x^2 + x^4) + 16*Log[1 + x^2] + 3136*Log[2 + x^2])/8","A",1
75,1,61,61,0.0271434,"\int \frac{x^7 \left(4+x^2+3 x^4+5 x^6\right)}{\left(2+3 x^2+x^4\right)^2} \, dx","Integrate[(x^7*(4 + x^2 + 3*x^4 + 5*x^6))/(2 + 3*x^2 + x^4)^2,x]","\frac{5 x^6}{6}-\frac{27 x^4}{4}+49 x^2-\frac{5}{2} \log \left(x^2+1\right)-144 \log \left(x^2+2\right)+\frac{-207 x^2-206}{2 \left(x^4+3 x^2+2\right)}","\frac{5 x^6}{6}-\frac{27 x^4}{4}+49 x^2-\frac{5}{2} \log \left(x^2+1\right)-144 \log \left(x^2+2\right)-\frac{207 x^2+206}{2 \left(x^4+3 x^2+2\right)}",1,"49*x^2 - (27*x^4)/4 + (5*x^6)/6 + (-206 - 207*x^2)/(2*(2 + 3*x^2 + x^4)) - (5*Log[1 + x^2])/2 - 144*Log[2 + x^2]","A",1
76,1,54,54,0.0240277,"\int \frac{x^5 \left(4+x^2+3 x^4+5 x^6\right)}{\left(2+3 x^2+x^4\right)^2} \, dx","Integrate[(x^5*(4 + x^2 + 3*x^4 + 5*x^6))/(2 + 3*x^2 + x^4)^2,x]","\frac{5 x^4}{4}-\frac{27 x^2}{2}+3 \log \left(x^2+1\right)+46 \log \left(x^2+2\right)+\frac{103 x^2+102}{2 \left(x^4+3 x^2+2\right)}","\frac{5 x^4}{4}-\frac{27 x^2}{2}+3 \log \left(x^2+1\right)+46 \log \left(x^2+2\right)+\frac{103 x^2+102}{2 \left(x^4+3 x^2+2\right)}",1,"(-27*x^2)/2 + (5*x^4)/4 + (102 + 103*x^2)/(2*(2 + 3*x^2 + x^4)) + 3*Log[1 + x^2] + 46*Log[2 + x^2]","A",1
77,1,49,49,0.0222216,"\int \frac{x^3 \left(4+x^2+3 x^4+5 x^6\right)}{\left(2+3 x^2+x^4\right)^2} \, dx","Integrate[(x^3*(4 + x^2 + 3*x^4 + 5*x^6))/(2 + 3*x^2 + x^4)^2,x]","\frac{5 x^2}{2}-\frac{7}{2} \log \left(x^2+1\right)-10 \log \left(x^2+2\right)+\frac{-51 x^2-50}{2 \left(x^4+3 x^2+2\right)}","\frac{5 x^2}{2}-\frac{7}{2} \log \left(x^2+1\right)-10 \log \left(x^2+2\right)-\frac{51 x^2+50}{2 \left(x^4+3 x^2+2\right)}",1,"(5*x^2)/2 + (-50 - 51*x^2)/(2*(2 + 3*x^2 + x^4)) - (7*Log[1 + x^2])/2 - 10*Log[2 + x^2]","A",1
78,1,42,42,0.0170409,"\int \frac{x \left(4+x^2+3 x^4+5 x^6\right)}{\left(2+3 x^2+x^4\right)^2} \, dx","Integrate[(x*(4 + x^2 + 3*x^4 + 5*x^6))/(2 + 3*x^2 + x^4)^2,x]","4 \log \left(x^2+1\right)-\frac{3}{2} \log \left(x^2+2\right)+\frac{25 x^2+24}{2 \left(x^4+3 x^2+2\right)}","4 \log \left(x^2+1\right)-\frac{3}{2} \log \left(x^2+2\right)+\frac{25 x^2+24}{2 \left(x^4+3 x^2+2\right)}",1,"(24 + 25*x^2)/(2*(2 + 3*x^2 + x^4)) + 4*Log[1 + x^2] - (3*Log[2 + x^2])/2","A",1
79,1,44,44,0.021686,"\int \frac{4+x^2+3 x^4+5 x^6}{x \left(2+3 x^2+x^4\right)^2} \, dx","Integrate[(4 + x^2 + 3*x^4 + 5*x^6)/(x*(2 + 3*x^2 + x^4)^2),x]","-\frac{9}{2} \log \left(x^2+1\right)+4 \log \left(x^2+2\right)+\frac{-12 x^2-11}{2 \left(x^4+3 x^2+2\right)}+\log (x)","-\frac{9}{2} \log \left(x^2+1\right)+4 \log \left(x^2+2\right)-\frac{12 x^2+11}{2 \left(x^4+3 x^2+2\right)}+\log (x)",1,"(-11 - 12*x^2)/(2*(2 + 3*x^2 + x^4)) + Log[x] - (9*Log[1 + x^2])/2 + 4*Log[2 + x^2]","A",1
80,1,50,55,0.0250996,"\int \frac{4+x^2+3 x^4+5 x^6}{x^3 \left(2+3 x^2+x^4\right)^2} \, dx","Integrate[(4 + x^2 + 3*x^4 + 5*x^6)/(x^3*(2 + 3*x^2 + x^4)^2),x]","\frac{1}{8} \left(-\frac{4}{x^2}+40 \log \left(x^2+1\right)-29 \log \left(x^2+2\right)+\frac{22 x^2+18}{x^4+3 x^2+2}-22 \log (x)\right)","-\frac{1}{2 x^2}+5 \log \left(x^2+1\right)-\frac{29}{8} \log \left(x^2+2\right)+\frac{11 x^2+9}{4 \left(x^4+3 x^2+2\right)}-\frac{11 \log (x)}{4}",1,"(-4/x^2 + (18 + 22*x^2)/(2 + 3*x^2 + x^4) - 22*Log[x] + 40*Log[1 + x^2] - 29*Log[2 + x^2])/8","A",1
81,1,56,64,0.0270839,"\int \frac{4+x^2+3 x^4+5 x^6}{x^5 \left(2+3 x^2+x^4\right)^2} \, dx","Integrate[(4 + x^2 + 3*x^4 + 5*x^6)/(x^5*(2 + 3*x^2 + x^4)^2),x]","\frac{1}{8} \left(-\frac{2}{x^4}+\frac{11}{x^2}-44 \log \left(x^2+1\right)+21 \log \left(x^2+2\right)-\frac{9 x^2+5}{x^4+3 x^2+2}+46 \log (x)\right)","-\frac{1}{4 x^4}+\frac{11}{8 x^2}-\frac{11}{2} \log \left(x^2+1\right)+\frac{21}{8} \log \left(x^2+2\right)-\frac{9 x^2+5}{8 \left(x^4+3 x^2+2\right)}+\frac{23 \log (x)}{4}",1,"(-2/x^4 + 11/x^2 - (5 + 9*x^2)/(2 + 3*x^2 + x^4) + 46*Log[x] - 44*Log[1 + x^2] + 21*Log[2 + x^2])/8","A",1
82,1,71,70,0.0435693,"\int \frac{x^8 \left(4+x^2+3 x^4+5 x^6\right)}{\left(2+3 x^2+x^4\right)^2} \, dx","Integrate[(x^8*(4 + x^2 + 3*x^4 + 5*x^6))/(2 + 3*x^2 + x^4)^2,x]","\frac{5 x^7}{7}-\frac{27 x^5}{5}+\frac{98 x^3}{3}+\frac{-207 x^3-206 x}{2 \left(x^4+3 x^2+2\right)}-293 x+\frac{9}{2} \tan ^{-1}(x)+340 \sqrt{2} \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)","\frac{5 x^7}{7}-\frac{27 x^5}{5}+\frac{98 x^3}{3}-\frac{\left(207 x^2+206\right) x}{2 \left(x^4+3 x^2+2\right)}-293 x+\frac{9}{2} \tan ^{-1}(x)+340 \sqrt{2} \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)",1,"-293*x + (98*x^3)/3 - (27*x^5)/5 + (5*x^7)/7 + (-206*x - 207*x^3)/(2*(2 + 3*x^2 + x^4)) + (9*ArcTan[x])/2 + 340*Sqrt[2]*ArcTan[x/Sqrt[2]]","A",1
83,1,58,57,0.0483803,"\int \frac{x^6 \left(4+x^2+3 x^4+5 x^6\right)}{\left(2+3 x^2+x^4\right)^2} \, dx","Integrate[(x^6*(4 + x^2 + 3*x^4 + 5*x^6))/(2 + 3*x^2 + x^4)^2,x]","x^5-9 x^3+\frac{103 x^3+102 x}{2 \left(x^4+3 x^2+2\right)}+98 x-\frac{11}{2} \tan ^{-1}(x)-118 \sqrt{2} \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)","x^5-9 x^3+\frac{\left(103 x^2+102\right) x}{2 \left(x^4+3 x^2+2\right)}+98 x-\frac{11}{2} \tan ^{-1}(x)-118 \sqrt{2} \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)",1,"98*x - 9*x^3 + x^5 + (102*x + 103*x^3)/(2*(2 + 3*x^2 + x^4)) - (11*ArcTan[x])/2 - 118*Sqrt[2]*ArcTan[x/Sqrt[2]]","A",1
84,1,57,56,0.0422797,"\int \frac{x^4 \left(4+x^2+3 x^4+5 x^6\right)}{\left(2+3 x^2+x^4\right)^2} \, dx","Integrate[(x^4*(4 + x^2 + 3*x^4 + 5*x^6))/(2 + 3*x^2 + x^4)^2,x]","\frac{5 x^3}{3}+\frac{-51 x^3-50 x}{2 \left(x^4+3 x^2+2\right)}-27 x+\frac{13}{2} \tan ^{-1}(x)+33 \sqrt{2} \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)","\frac{5 x^3}{3}-\frac{\left(51 x^2+50\right) x}{2 \left(x^4+3 x^2+2\right)}-27 x+\frac{13}{2} \tan ^{-1}(x)+33 \sqrt{2} \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)",1,"-27*x + (5*x^3)/3 + (-50*x - 51*x^3)/(2*(2 + 3*x^2 + x^4)) + (13*ArcTan[x])/2 + 33*Sqrt[2]*ArcTan[x/Sqrt[2]]","A",1
85,1,50,49,0.0377353,"\int \frac{x^2 \left(4+x^2+3 x^4+5 x^6\right)}{\left(2+3 x^2+x^4\right)^2} \, dx","Integrate[(x^2*(4 + x^2 + 3*x^4 + 5*x^6))/(2 + 3*x^2 + x^4)^2,x]","\frac{25 x^3+24 x}{2 \left(x^4+3 x^2+2\right)}+5 x-\frac{15}{2} \tan ^{-1}(x)-\frac{7 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)}{\sqrt{2}}","\frac{\left(25 x^2+24\right) x}{2 \left(x^4+3 x^2+2\right)}+5 x-\frac{15}{2} \tan ^{-1}(x)-\frac{7 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)}{\sqrt{2}}",1,"5*x + (24*x + 25*x^3)/(2*(2 + 3*x^2 + x^4)) - (15*ArcTan[x])/2 - (7*ArcTan[x/Sqrt[2]])/Sqrt[2]","A",1
86,1,46,48,0.0395998,"\int \frac{4+x^2+3 x^4+5 x^6}{\left(2+3 x^2+x^4\right)^2} \, dx","Integrate[(4 + x^2 + 3*x^4 + 5*x^6)/(2 + 3*x^2 + x^4)^2,x]","\frac{1}{4} \left(-\frac{2 x \left(12 x^2+11\right)}{x^4+3 x^2+2}+34 \tan ^{-1}(x)-19 \sqrt{2} \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right)","-\frac{x \left(12 x^2+11\right)}{2 \left(x^4+3 x^2+2\right)}+\frac{17}{2} \tan ^{-1}(x)-\frac{19 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)}{2 \sqrt{2}}",1,"((-2*x*(11 + 12*x^2))/(2 + 3*x^2 + x^4) + 34*ArcTan[x] - 19*Sqrt[2]*ArcTan[x/Sqrt[2]])/4","A",1
87,1,51,53,0.0486696,"\int \frac{4+x^2+3 x^4+5 x^6}{x^2 \left(2+3 x^2+x^4\right)^2} \, dx","Integrate[(4 + x^2 + 3*x^4 + 5*x^6)/(x^2*(2 + 3*x^2 + x^4)^2),x]","\frac{1}{8} \left(\frac{2 x \left(11 x^2+9\right)}{x^4+3 x^2+2}-\frac{8}{x}-76 \tan ^{-1}(x)+45 \sqrt{2} \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right)","\frac{x \left(11 x^2+9\right)}{4 \left(x^4+3 x^2+2\right)}-\frac{1}{x}-\frac{19}{2} \tan ^{-1}(x)+\frac{45 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)}{4 \sqrt{2}}",1,"(-8/x + (2*x*(9 + 11*x^2))/(2 + 3*x^2 + x^4) - 76*ArcTan[x] + 45*Sqrt[2]*ArcTan[x/Sqrt[2]])/8","A",1
88,1,56,62,0.0514504,"\int \frac{4+x^2+3 x^4+5 x^6}{x^4 \left(2+3 x^2+x^4\right)^2} \, dx","Integrate[(4 + x^2 + 3*x^4 + 5*x^6)/(x^4*(2 + 3*x^2 + x^4)^2),x]","\frac{1}{48} \left(-\frac{16}{x^3}-\frac{6 x \left(9 x^2+5\right)}{x^4+3 x^2+2}+\frac{132}{x}+504 \tan ^{-1}(x)-213 \sqrt{2} \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right)","-\frac{1}{3 x^3}-\frac{x \left(9 x^2+5\right)}{8 \left(x^4+3 x^2+2\right)}+\frac{11}{4 x}+\frac{21}{2} \tan ^{-1}(x)-\frac{71 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)}{8 \sqrt{2}}",1,"(-16/x^3 + 132/x - (6*x*(5 + 9*x^2))/(2 + 3*x^2 + x^4) + 504*ArcTan[x] - 213*Sqrt[2]*ArcTan[x/Sqrt[2]])/48","A",1
89,1,61,69,0.0579732,"\int \frac{4+x^2+3 x^4+5 x^6}{x^6 \left(2+3 x^2+x^4\right)^2} \, dx","Integrate[(4 + x^2 + 3*x^4 + 5*x^6)/(x^6*(2 + 3*x^2 + x^4)^2),x]","\frac{1}{480} \left(-\frac{96}{x^5}+\frac{440}{x^3}+\frac{30 x \left(5 x^2-3\right)}{x^4+3 x^2+2}-\frac{2760}{x}-5520 \tan ^{-1}(x)+1455 \sqrt{2} \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right)","-\frac{1}{5 x^5}+\frac{11}{12 x^3}-\frac{x \left(3-5 x^2\right)}{16 \left(x^4+3 x^2+2\right)}-\frac{23}{4 x}-\frac{23}{2} \tan ^{-1}(x)+\frac{97 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)}{16 \sqrt{2}}",1,"(-96/x^5 + 440/x^3 - 2760/x + (30*x*(-3 + 5*x^2))/(2 + 3*x^2 + x^4) - 5520*ArcTan[x] + 1455*Sqrt[2]*ArcTan[x/Sqrt[2]])/480","A",1
90,1,77,76,0.0549109,"\int \frac{4+x^2+3 x^4+5 x^6}{x^8 \left(2+3 x^2+x^4\right)^2} \, dx","Integrate[(4 + x^2 + 3*x^4 + 5*x^6)/(x^8*(2 + 3*x^2 + x^4)^2),x]","-\frac{1}{7 x^7}+\frac{11}{20 x^5}-\frac{23}{12 x^3}+\frac{3 x^3+19 x}{32 \left(x^4+3 x^2+2\right)}+\frac{137}{16 x}+\frac{25}{2} \tan ^{-1}(x)-\frac{123 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)}{32 \sqrt{2}}","-\frac{1}{7 x^7}+\frac{11}{20 x^5}-\frac{23}{12 x^3}+\frac{x \left(3 x^2+19\right)}{32 \left(x^4+3 x^2+2\right)}+\frac{137}{16 x}+\frac{25}{2} \tan ^{-1}(x)-\frac{123 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)}{32 \sqrt{2}}",1,"-1/7*1/x^7 + 11/(20*x^5) - 23/(12*x^3) + 137/(16*x) + (19*x + 3*x^3)/(32*(2 + 3*x^2 + x^4)) + (25*ArcTan[x])/2 - (123*ArcTan[x/Sqrt[2]])/(32*Sqrt[2])","A",1
91,1,71,81,0.056168,"\int \frac{x^{10} \left(4+x^2+3 x^4+5 x^6\right)}{\left(2+3 x^2+x^4\right)^3} \, dx","Integrate[(x^10*(4 + x^2 + 3*x^4 + 5*x^6))/(2 + 3*x^2 + x^4)^3,x]","\frac{x \left(8 x^{12}-64 x^{10}+1144 x^8+10581 x^6+26775 x^4+26736 x^2+9324\right)}{8 \left(x^4+3 x^2+2\right)^2}+\frac{477}{8} \tan ^{-1}(x)-351 \sqrt{2} \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)","x^5-14 x^3+\frac{\left(1669 x^2+824\right) x}{8 \left(x^4+3 x^2+2\right)}+\frac{\left(415 x^2+414\right) x}{4 \left(x^4+3 x^2+2\right)^2}+214 x+\frac{477}{8} \tan ^{-1}(x)-351 \sqrt{2} \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)",1,"(x*(9324 + 26736*x^2 + 26775*x^4 + 10581*x^6 + 1144*x^8 - 64*x^10 + 8*x^12))/(8*(2 + 3*x^2 + x^4)^2) + (477*ArcTan[x])/8 - 351*Sqrt[2]*ArcTan[x/Sqrt[2]]","A",1
92,1,66,80,0.0537246,"\int \frac{x^8 \left(4+x^2+3 x^4+5 x^6\right)}{\left(2+3 x^2+x^4\right)^3} \, dx","Integrate[(x^8*(4 + x^2 + 3*x^4 + 5*x^6))/(2 + 3*x^2 + x^4)^3,x]","\frac{x \left(40 x^{10}-768 x^8-6755 x^6-16233 x^4-15416 x^2-5124\right)}{24 \left(x^4+3 x^2+2\right)^2}-\frac{449}{8} \tan ^{-1}(x)+\frac{219 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)}{\sqrt{2}}","\frac{5 x^3}{3}+\frac{\left(24-409 x^2\right) x}{8 \left(x^4+3 x^2+2\right)}-\frac{\left(207 x^2+206\right) x}{4 \left(x^4+3 x^2+2\right)^2}-42 x-\frac{449}{8} \tan ^{-1}(x)+\frac{219 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)}{\sqrt{2}}",1,"(x*(-5124 - 15416*x^2 - 16233*x^4 - 6755*x^6 - 768*x^8 + 40*x^10))/(24*(2 + 3*x^2 + x^4)^2) - (449*ArcTan[x])/8 + (219*ArcTan[x/Sqrt[2]])/Sqrt[2]","A",1
93,1,60,75,0.0606619,"\int \frac{x^6 \left(4+x^2+3 x^4+5 x^6\right)}{\left(2+3 x^2+x^4\right)^3} \, dx","Integrate[(x^6*(4 + x^2 + 3*x^4 + 5*x^6))/(2 + 3*x^2 + x^4)^3,x]","\frac{1}{8} \left(\frac{x \left(40 x^8+225 x^6+231 x^4-76 x^2-124\right)}{\left(x^4+3 x^2+2\right)^2}+413 \tan ^{-1}(x)-382 \sqrt{2} \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right)","-\frac{\left(15 x^2+244\right) x}{8 \left(x^4+3 x^2+2\right)}+\frac{\left(103 x^2+102\right) x}{4 \left(x^4+3 x^2+2\right)^2}+5 x+\frac{413}{8} \tan ^{-1}(x)-\frac{191 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)}{2 \sqrt{2}}",1,"((x*(-124 - 76*x^2 + 231*x^4 + 225*x^6 + 40*x^8))/(2 + 3*x^2 + x^4)^2 + 413*ArcTan[x] - 382*Sqrt[2]*ArcTan[x/Sqrt[2]])/8","A",1
94,1,55,72,0.0576909,"\int \frac{x^4 \left(4+x^2+3 x^4+5 x^6\right)}{\left(2+3 x^2+x^4\right)^3} \, dx","Integrate[(x^4*(4 + x^2 + 3*x^4 + 5*x^6))/(2 + 3*x^2 + x^4)^3,x]","\frac{1}{8} \left(\frac{x \left(125 x^6+629 x^4+910 x^2+408\right)}{\left(x^4+3 x^2+2\right)^2}-369 \tan ^{-1}(x)+267 \sqrt{2} \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right)","-\frac{x \left(51 x^2+50\right)}{4 \left(x^4+3 x^2+2\right)^2}+\frac{x \left(125 x^2+254\right)}{8 \left(x^4+3 x^2+2\right)}-\frac{369}{8} \tan ^{-1}(x)+\frac{267 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)}{4 \sqrt{2}}",1,"((x*(408 + 910*x^2 + 629*x^4 + 125*x^6))/(2 + 3*x^2 + x^4)^2 - 369*ArcTan[x] + 267*Sqrt[2]*ArcTan[x/Sqrt[2]])/8","A",1
95,1,56,72,0.061063,"\int \frac{x^2 \left(4+x^2+3 x^4+5 x^6\right)}{\left(2+3 x^2+x^4\right)^3} \, dx","Integrate[(x^2*(4 + x^2 + 3*x^4 + 5*x^6))/(2 + 3*x^2 + x^4)^3,x]","\frac{1}{16} \left(-\frac{2 x \left(130 x^6+601 x^4+843 x^2+374\right)}{\left(x^4+3 x^2+2\right)^2}+634 \tan ^{-1}(x)-447 \sqrt{2} \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right)","\frac{x \left(25 x^2+24\right)}{4 \left(x^4+3 x^2+2\right)^2}-\frac{x \left(130 x^2+211\right)}{8 \left(x^4+3 x^2+2\right)}+\frac{317}{8} \tan ^{-1}(x)-\frac{447 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)}{8 \sqrt{2}}",1,"((-2*x*(374 + 843*x^2 + 601*x^4 + 130*x^6))/(2 + 3*x^2 + x^4)^2 + 634*ArcTan[x] - 447*Sqrt[2]*ArcTan[x/Sqrt[2]])/16","A",1
96,1,56,72,0.0585771,"\int \frac{4+x^2+3 x^4+5 x^6}{\left(2+3 x^2+x^4\right)^3} \, dx","Integrate[(4 + x^2 + 3*x^4 + 5*x^6)/(2 + 3*x^2 + x^4)^3,x]","\frac{1}{32} \left(\frac{2 x \left(217 x^6+986 x^4+1391 x^2+626\right)}{\left(x^4+3 x^2+2\right)^2}-1028 \tan ^{-1}(x)+731 \sqrt{2} \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right)","-\frac{x \left(12 x^2+11\right)}{4 \left(x^4+3 x^2+2\right)^2}+\frac{x \left(217 x^2+335\right)}{16 \left(x^4+3 x^2+2\right)}-\frac{257}{8} \tan ^{-1}(x)+\frac{731 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)}{16 \sqrt{2}}",1,"((2*x*(626 + 1391*x^2 + 986*x^4 + 217*x^6))/(2 + 3*x^2 + x^4)^2 - 1028*ArcTan[x] + 731*Sqrt[2]*ArcTan[x/Sqrt[2]])/32","A",1
97,1,63,79,0.0653168,"\int \frac{4+x^2+3 x^4+5 x^6}{x^2 \left(2+3 x^2+x^4\right)^3} \, dx","Integrate[(4 + x^2 + 3*x^4 + 5*x^6)/(x^2*(2 + 3*x^2 + x^4)^3),x]","\frac{1}{64} \left(-\frac{2 \left(363 x^8+1684 x^6+2499 x^4+1250 x^2+64\right)}{x \left(x^4+3 x^2+2\right)^2}+1512 \tan ^{-1}(x)-1119 \sqrt{2} \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right)","\frac{x \left(11 x^2+9\right)}{8 \left(x^4+3 x^2+2\right)^2}-\frac{x \left(347 x^2+547\right)}{32 \left(x^4+3 x^2+2\right)}-\frac{1}{2 x}+\frac{189}{8} \tan ^{-1}(x)-\frac{1119 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)}{32 \sqrt{2}}",1,"((-2*(64 + 1250*x^2 + 2499*x^4 + 1684*x^6 + 363*x^8))/(x*(2 + 3*x^2 + x^4)^2) + 1512*ArcTan[x] - 1119*Sqrt[2]*ArcTan[x/Sqrt[2]])/64","A",1
98,1,78,86,0.0585676,"\int \frac{4+x^2+3 x^4+5 x^6}{x^4 \left(2+3 x^2+x^4\right)^3} \, dx","Integrate[(4 + x^2 + 3*x^4 + 5*x^6)/(x^4*(2 + 3*x^2 + x^4)^3),x]","\frac{1}{384} \left(-\frac{64}{x^3}-\frac{24 x \left(9 x^2+5\right)}{\left(x^4+3 x^2+2\right)^2}+\frac{6 x \left(571 x^2+951\right)}{x^4+3 x^2+2}+\frac{816}{x}-5424 \tan ^{-1}(x)+4833 \sqrt{2} \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right)","-\frac{1}{6 x^3}-\frac{x \left(9 x^2+5\right)}{16 \left(x^4+3 x^2+2\right)^2}+\frac{x \left(571 x^2+951\right)}{64 \left(x^4+3 x^2+2\right)}+\frac{17}{8 x}-\frac{113}{8} \tan ^{-1}(x)+\frac{1611 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)}{64 \sqrt{2}}",1,"(-64/x^3 + 816/x - (24*x*(5 + 9*x^2))/(2 + 3*x^2 + x^4)^2 + (6*x*(951 + 571*x^2))/(2 + 3*x^2 + x^4) - 5424*ArcTan[x] + 4833*Sqrt[2]*ArcTan[x/Sqrt[2]])/384","A",1
99,1,73,93,0.0763364,"\int \frac{4+x^2+3 x^4+5 x^6}{x^6 \left(2+3 x^2+x^4\right)^3} \, dx","Integrate[(4 + x^2 + 3*x^4 + 5*x^6)/(x^6*(2 + 3*x^2 + x^4)^3),x]","\frac{-\frac{2 \left(26145 x^{12}+137120 x^{10}+246477 x^8+170702 x^6+30816 x^4-3136 x^2+768\right)}{x^5 \left(x^4+3 x^2+2\right)^2}+13920 \tan ^{-1}(x)-33105 \sqrt{2} \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)}{3840}","-\frac{1}{10 x^5}+\frac{17}{24 x^3}-\frac{x \left(3-5 x^2\right)}{32 \left(x^4+3 x^2+2\right)^2}-\frac{x \left(999 x^2+1771\right)}{128 \left(x^4+3 x^2+2\right)}-\frac{93}{16 x}+\frac{29}{8} \tan ^{-1}(x)-\frac{2207 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)}{128 \sqrt{2}}",1,"((-2*(768 - 3136*x^2 + 30816*x^4 + 170702*x^6 + 246477*x^8 + 137120*x^10 + 26145*x^12))/(x^5*(2 + 3*x^2 + x^4)^2) + 13920*ArcTan[x] - 33105*Sqrt[2]*ArcTan[x/Sqrt[2]])/3840","A",1
100,1,78,86,0.0448189,"\int \frac{x^9 \left(4+x^2+3 x^4+5 x^6\right)}{\left(3+2 x^2+x^4\right)^2} \, dx","Integrate[(x^9*(4 + x^2 + 3*x^4 + 5*x^6))/(3 + 2*x^2 + x^4)^2,x]","\frac{1}{48} \left(30 x^8-136 x^6+228 x^4+912 x^2+603 \sqrt{2} \tan ^{-1}\left(\frac{x^2+1}{\sqrt{2}}\right)-\frac{150 \left(7 x^2+15\right)}{x^4+2 x^2+3}-2196 \log \left(x^4+2 x^2+3\right)\right)","\frac{5 x^8}{8}-\frac{17 x^6}{6}+\frac{19 x^4}{4}+19 x^2+\frac{201 \tan ^{-1}\left(\frac{x^2+1}{\sqrt{2}}\right)}{8 \sqrt{2}}-\frac{25 \left(7 x^2+15\right)}{8 \left(x^4+2 x^2+3\right)}-\frac{183}{4} \log \left(x^4+2 x^2+3\right)",1,"(912*x^2 + 228*x^4 - 136*x^6 + 30*x^8 - (150*(15 + 7*x^2))/(3 + 2*x^2 + x^4) + 603*Sqrt[2]*ArcTan[(1 + x^2)/Sqrt[2]] - 2196*Log[3 + 2*x^2 + x^4])/48","A",1
101,1,73,81,0.0304004,"\int \frac{x^7 \left(4+x^2+3 x^4+5 x^6\right)}{\left(3+2 x^2+x^4\right)^2} \, dx","Integrate[(x^7*(4 + x^2 + 3*x^4 + 5*x^6))/(3 + 2*x^2 + x^4)^2,x]","\frac{1}{48} \left(40 x^6-204 x^4+456 x^2-1365 \sqrt{2} \tan ^{-1}\left(\frac{x^2+1}{\sqrt{2}}\right)+\frac{150 \left(5 x^2+3\right)}{x^4+2 x^2+3}+456 \log \left(x^4+2 x^2+3\right)\right)","\frac{5 x^6}{6}-\frac{17 x^4}{4}+\frac{19 x^2}{2}-\frac{455 \tan ^{-1}\left(\frac{x^2+1}{\sqrt{2}}\right)}{8 \sqrt{2}}+\frac{25 \left(5 x^2+3\right)}{8 \left(x^4+2 x^2+3\right)}+\frac{19}{2} \log \left(x^4+2 x^2+3\right)",1,"(456*x^2 - 204*x^4 + 40*x^6 + (150*(3 + 5*x^2))/(3 + 2*x^2 + x^4) - 1365*Sqrt[2]*ArcTan[(1 + x^2)/Sqrt[2]] + 456*Log[3 + 2*x^2 + x^4])/48","A",1
102,1,66,74,0.0292095,"\int \frac{x^5 \left(4+x^2+3 x^4+5 x^6\right)}{\left(3+2 x^2+x^4\right)^2} \, dx","Integrate[(x^5*(4 + x^2 + 3*x^4 + 5*x^6))/(3 + 2*x^2 + x^4)^2,x]","\frac{1}{16} \left(20 x^4-136 x^2+203 \sqrt{2} \tan ^{-1}\left(\frac{x^2+1}{\sqrt{2}}\right)-\frac{50 \left(x^2-3\right)}{x^4+2 x^2+3}+76 \log \left(x^4+2 x^2+3\right)\right)","\frac{5 x^4}{4}-\frac{17 x^2}{2}+\frac{203 \tan ^{-1}\left(\frac{x^2+1}{\sqrt{2}}\right)}{8 \sqrt{2}}+\frac{25 \left(3-x^2\right)}{8 \left(x^4+2 x^2+3\right)}+\frac{19}{4} \log \left(x^4+2 x^2+3\right)",1,"(-136*x^2 + 20*x^4 - (50*(-3 + x^2))/(3 + 2*x^2 + x^4) + 203*Sqrt[2]*ArcTan[(1 + x^2)/Sqrt[2]] + 76*Log[3 + 2*x^2 + x^4])/16","A",1
103,1,61,65,0.0255078,"\int \frac{x^3 \left(4+x^2+3 x^4+5 x^6\right)}{\left(3+2 x^2+x^4\right)^2} \, dx","Integrate[(x^3*(4 + x^2 + 3*x^4 + 5*x^6))/(3 + 2*x^2 + x^4)^2,x]","\frac{1}{16} \left(40 x^2-17 \sqrt{2} \tan ^{-1}\left(\frac{x^2+1}{\sqrt{2}}\right)-\frac{50 \left(x^2+3\right)}{x^4+2 x^2+3}-68 \log \left(x^4+2 x^2+3\right)\right)","\frac{5 x^2}{2}-\frac{17 \tan ^{-1}\left(\frac{x^2+1}{\sqrt{2}}\right)}{8 \sqrt{2}}-\frac{25 \left(x^2+3\right)}{8 \left(x^4+2 x^2+3\right)}-\frac{17}{4} \log \left(x^4+2 x^2+3\right)",1,"(40*x^2 - (50*(3 + x^2))/(3 + 2*x^2 + x^4) - 17*Sqrt[2]*ArcTan[(1 + x^2)/Sqrt[2]] - 68*Log[3 + 2*x^2 + x^4])/16","A",1
104,1,58,58,0.0219696,"\int \frac{x \left(4+x^2+3 x^4+5 x^6\right)}{\left(3+2 x^2+x^4\right)^2} \, dx","Integrate[(x*(4 + x^2 + 3*x^4 + 5*x^6))/(3 + 2*x^2 + x^4)^2,x]","-\frac{23 \tan ^{-1}\left(\frac{x^2+1}{\sqrt{2}}\right)}{8 \sqrt{2}}+\frac{25 \left(x^2+1\right)}{8 \left(x^4+2 x^2+3\right)}+\frac{5}{4} \log \left(x^4+2 x^2+3\right)","-\frac{23 \tan ^{-1}\left(\frac{x^2+1}{\sqrt{2}}\right)}{8 \sqrt{2}}+\frac{25 \left(x^2+1\right)}{8 \left(x^4+2 x^2+3\right)}+\frac{5}{4} \log \left(x^4+2 x^2+3\right)",1,"(25*(1 + x^2))/(8*(3 + 2*x^2 + x^4)) - (23*ArcTan[(1 + x^2)/Sqrt[2]])/(8*Sqrt[2]) + (5*Log[3 + 2*x^2 + x^4])/4","A",1
105,1,93,66,0.0623813,"\int \frac{4+x^2+3 x^4+5 x^6}{x \left(3+2 x^2+x^4\right)^2} \, dx","Integrate[(4 + x^2 + 3*x^4 + 5*x^6)/(x*(3 + 2*x^2 + x^4)^2),x]","\frac{1}{288} \left(-\sqrt{2} \left(16 \sqrt{2}+89 i\right) \log \left(x^2-i \sqrt{2}+1\right)+\sqrt{2} \left(-16 \sqrt{2}+89 i\right) \log \left(x^2+i \sqrt{2}+1\right)-\frac{300 \left(x^2-1\right)}{x^4+2 x^2+3}+128 \log (x)\right)","\frac{89 \tan ^{-1}\left(\frac{x^2+1}{\sqrt{2}}\right)}{72 \sqrt{2}}+\frac{25 \left(1-x^2\right)}{24 \left(x^4+2 x^2+3\right)}-\frac{1}{9} \log \left(x^4+2 x^2+3\right)+\frac{4 \log (x)}{9}",1,"((-300*(-1 + x^2))/(3 + 2*x^2 + x^4) + 128*Log[x] - Sqrt[2]*(89*I + 16*Sqrt[2])*Log[1 - I*Sqrt[2] + x^2] + Sqrt[2]*(89*I - 16*Sqrt[2])*Log[1 + I*Sqrt[2] + x^2])/288","C",1
106,1,97,71,0.0500943,"\int \frac{4+x^2+3 x^4+5 x^6}{x^3 \left(3+2 x^2+x^4\right)^2} \, dx","Integrate[(4 + x^2 + 3*x^4 + 5*x^6)/(x^3*(3 + 2*x^2 + x^4)^2),x]","\frac{1}{864} \left(-\frac{192}{x^2}+\sqrt{2} \left(52 \sqrt{2}+71 i\right) \log \left(x^2-i \sqrt{2}+1\right)+\sqrt{2} \left(52 \sqrt{2}-71 i\right) \log \left(x^2+i \sqrt{2}+1\right)-\frac{300 \left(x^2+5\right)}{x^4+2 x^2+3}-416 \log (x)\right)","-\frac{2}{9 x^2}-\frac{71 \tan ^{-1}\left(\frac{x^2+1}{\sqrt{2}}\right)}{216 \sqrt{2}}-\frac{25 \left(x^2+5\right)}{72 \left(x^4+2 x^2+3\right)}+\frac{13}{108} \log \left(x^4+2 x^2+3\right)-\frac{13 \log (x)}{27}",1,"(-192/x^2 - (300*(5 + x^2))/(3 + 2*x^2 + x^4) - 416*Log[x] + Sqrt[2]*(71*I + 52*Sqrt[2])*Log[1 - I*Sqrt[2] + x^2] + Sqrt[2]*(-71*I + 52*Sqrt[2])*Log[1 + I*Sqrt[2] + x^2])/864","C",1
107,1,105,80,0.0600714,"\int \frac{4+x^2+3 x^4+5 x^6}{x^5 \left(3+2 x^2+x^4\right)^2} \, dx","Integrate[(4 + x^2 + 3*x^4 + 5*x^6)/(x^5*(3 + 2*x^2 + x^4)^2),x]","\frac{1}{864} \left(-\frac{96}{x^4}+\frac{208}{x^2}-\sqrt{2} \left(52 \sqrt{2}+125 i\right) \log \left(x^2-i \sqrt{2}+1\right)+\sqrt{2} \left(-52 \sqrt{2}+125 i\right) \log \left(x^2+i \sqrt{2}+1\right)+\frac{100 \left(5 x^2+7\right)}{x^4+2 x^2+3}+416 \log (x)\right)","-\frac{1}{9 x^4}+\frac{13}{54 x^2}+\frac{125 \tan ^{-1}\left(\frac{x^2+1}{\sqrt{2}}\right)}{216 \sqrt{2}}+\frac{25 \left(5 x^2+7\right)}{216 \left(x^4+2 x^2+3\right)}-\frac{13}{108} \log \left(x^4+2 x^2+3\right)+\frac{13 \log (x)}{27}",1,"(-96/x^4 + 208/x^2 + (100*(7 + 5*x^2))/(3 + 2*x^2 + x^4) + 416*Log[x] - Sqrt[2]*(125*I + 52*Sqrt[2])*Log[1 - I*Sqrt[2] + x^2] + Sqrt[2]*(125*I - 52*Sqrt[2])*Log[1 + I*Sqrt[2] + x^2])/864","C",1
108,1,110,87,0.066489,"\int \frac{4+x^2+3 x^4+5 x^6}{x^7 \left(3+2 x^2+x^4\right)^2} \, dx","Integrate[(4 + x^2 + 3*x^4 + 5*x^6)/(x^7*(3 + 2*x^2 + x^4)^2),x]","\frac{-\frac{576}{x^6}+\frac{936}{x^4}-\frac{1872}{x^2}+\sqrt{2} \left(-244 \sqrt{2}+1237 i\right) \log \left(x^2-i \sqrt{2}+1\right)-\sqrt{2} \left(244 \sqrt{2}+1237 i\right) \log \left(x^2+i \sqrt{2}+1\right)-\frac{300 \left(7 x^2-1\right)}{x^4+2 x^2+3}+1952 \log (x)}{7776}","-\frac{2}{27 x^6}+\frac{13}{108 x^4}-\frac{13}{54 x^2}-\frac{1237 \tan ^{-1}\left(\frac{x^2+1}{\sqrt{2}}\right)}{1944 \sqrt{2}}+\frac{25 \left(1-7 x^2\right)}{648 \left(x^4+2 x^2+3\right)}-\frac{61}{972} \log \left(x^4+2 x^2+3\right)+\frac{61 \log (x)}{243}",1,"(-576/x^6 + 936/x^4 - 1872/x^2 - (300*(-1 + 7*x^2))/(3 + 2*x^2 + x^4) + 1952*Log[x] + Sqrt[2]*(1237*I - 244*Sqrt[2])*Log[1 - I*Sqrt[2] + x^2] - Sqrt[2]*(1237*I + 244*Sqrt[2])*Log[1 + I*Sqrt[2] + x^2])/7776","C",1
109,1,145,248,0.1719713,"\int \frac{x^8 \left(4+x^2+3 x^4+5 x^6\right)}{\left(3+2 x^2+x^4\right)^2} \, dx","Integrate[(x^8*(4 + x^2 + 3*x^4 + 5*x^6))/(3 + 2*x^2 + x^4)^2,x]","\frac{5 x^7}{7}-\frac{17 x^5}{5}+\frac{19 x^3}{3}+\frac{25 \left(5 x^2+3\right) x}{8 \left(x^4+2 x^2+3\right)}+38 x-\frac{\left(1339 \sqrt{2}+352 i\right) \tan ^{-1}\left(\frac{x}{\sqrt{1-i \sqrt{2}}}\right)}{16 \sqrt{2-2 i \sqrt{2}}}-\frac{\left(1339 \sqrt{2}-352 i\right) \tan ^{-1}\left(\frac{x}{\sqrt{1+i \sqrt{2}}}\right)}{16 \sqrt{2+2 i \sqrt{2}}}","\frac{5 x^7}{7}-\frac{17 x^5}{5}+\frac{19 x^3}{3}-\frac{1}{32} \sqrt{\frac{1}{2} \left(618291 \sqrt{3}-262771\right)} \log \left(x^2-\sqrt{2 \left(\sqrt{3}-1\right)} x+\sqrt{3}\right)+\frac{1}{32} \sqrt{\frac{1}{2} \left(618291 \sqrt{3}-262771\right)} \log \left(x^2+\sqrt{2 \left(\sqrt{3}-1\right)} x+\sqrt{3}\right)+\frac{25 \left(5 x^2+3\right) x}{8 \left(x^4+2 x^2+3\right)}+38 x+\frac{1}{16} \sqrt{\frac{1}{2} \left(262771+618291 \sqrt{3}\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(\sqrt{3}-1\right)}-2 x}{\sqrt{2 \left(1+\sqrt{3}\right)}}\right)-\frac{1}{16} \sqrt{\frac{1}{2} \left(262771+618291 \sqrt{3}\right)} \tan ^{-1}\left(\frac{2 x+\sqrt{2 \left(\sqrt{3}-1\right)}}{\sqrt{2 \left(1+\sqrt{3}\right)}}\right)",1,"38*x + (19*x^3)/3 - (17*x^5)/5 + (5*x^7)/7 + (25*x*(3 + 5*x^2))/(8*(3 + 2*x^2 + x^4)) - ((352*I + 1339*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/(16*Sqrt[2 - (2*I)*Sqrt[2]]) - ((-352*I + 1339*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/(16*Sqrt[2 + (2*I)*Sqrt[2]])","C",1
110,1,132,237,0.158331,"\int \frac{x^6 \left(4+x^2+3 x^4+5 x^6\right)}{\left(3+2 x^2+x^4\right)^2} \, dx","Integrate[(x^6*(4 + x^2 + 3*x^4 + 5*x^6))/(3 + 2*x^2 + x^4)^2,x]","x^5-\frac{17 x^3}{3}-\frac{25 \left(x^2-3\right) x}{8 \left(x^4+2 x^2+3\right)}+19 x+\frac{9 \left(31 \sqrt{2}+90 i\right) \tan ^{-1}\left(\frac{x}{\sqrt{1-i \sqrt{2}}}\right)}{16 \sqrt{2-2 i \sqrt{2}}}+\frac{9 \left(31 \sqrt{2}-90 i\right) \tan ^{-1}\left(\frac{x}{\sqrt{1+i \sqrt{2}}}\right)}{16 \sqrt{2+2 i \sqrt{2}}}","x^5-\frac{17 x^3}{3}+\frac{3}{32} \sqrt{\frac{3}{2} \left(8669+5011 \sqrt{3}\right)} \log \left(x^2-\sqrt{2 \left(\sqrt{3}-1\right)} x+\sqrt{3}\right)-\frac{3}{32} \sqrt{\frac{3}{2} \left(8669+5011 \sqrt{3}\right)} \log \left(x^2+\sqrt{2 \left(\sqrt{3}-1\right)} x+\sqrt{3}\right)+\frac{25 \left(3-x^2\right) x}{8 \left(x^4+2 x^2+3\right)}+19 x+\frac{3}{16} \sqrt{\frac{3}{2} \left(5011 \sqrt{3}-8669\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(\sqrt{3}-1\right)}-2 x}{\sqrt{2 \left(1+\sqrt{3}\right)}}\right)-\frac{3}{16} \sqrt{\frac{3}{2} \left(5011 \sqrt{3}-8669\right)} \tan ^{-1}\left(\frac{2 x+\sqrt{2 \left(\sqrt{3}-1\right)}}{\sqrt{2 \left(1+\sqrt{3}\right)}}\right)",1,"19*x - (17*x^3)/3 + x^5 - (25*x*(-3 + x^2))/(8*(3 + 2*x^2 + x^4)) + (9*(90*I + 31*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/(16*Sqrt[2 - (2*I)*Sqrt[2]]) + (9*(-90*I + 31*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/(16*Sqrt[2 + (2*I)*Sqrt[2]])","C",1
111,1,129,232,0.1592267,"\int \frac{x^4 \left(4+x^2+3 x^4+5 x^6\right)}{\left(3+2 x^2+x^4\right)^2} \, dx","Integrate[(x^4*(4 + x^2 + 3*x^4 + 5*x^6))/(3 + 2*x^2 + x^4)^2,x]","\frac{5 x^3}{3}-\frac{25 \left(x^2+3\right) x}{8 \left(x^4+2 x^2+3\right)}-17 x+\frac{\left(127 \sqrt{2}-356 i\right) \tan ^{-1}\left(\frac{x}{\sqrt{1-i \sqrt{2}}}\right)}{16 \sqrt{2-2 i \sqrt{2}}}+\frac{\left(127 \sqrt{2}+356 i\right) \tan ^{-1}\left(\frac{x}{\sqrt{1+i \sqrt{2}}}\right)}{16 \sqrt{2+2 i \sqrt{2}}}","\frac{5 x^3}{3}-\frac{1}{32} \sqrt{\frac{1}{2} \left(26499 \sqrt{3}-14395\right)} \log \left(x^2-\sqrt{2 \left(\sqrt{3}-1\right)} x+\sqrt{3}\right)+\frac{1}{32} \sqrt{\frac{1}{2} \left(26499 \sqrt{3}-14395\right)} \log \left(x^2+\sqrt{2 \left(\sqrt{3}-1\right)} x+\sqrt{3}\right)-\frac{25 \left(x^2+3\right) x}{8 \left(x^4+2 x^2+3\right)}-17 x-\frac{1}{16} \sqrt{\frac{1}{2} \left(14395+26499 \sqrt{3}\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(\sqrt{3}-1\right)}-2 x}{\sqrt{2 \left(1+\sqrt{3}\right)}}\right)+\frac{1}{16} \sqrt{\frac{1}{2} \left(14395+26499 \sqrt{3}\right)} \tan ^{-1}\left(\frac{2 x+\sqrt{2 \left(\sqrt{3}-1\right)}}{\sqrt{2 \left(1+\sqrt{3}\right)}}\right)",1,"-17*x + (5*x^3)/3 - (25*x*(3 + x^2))/(8*(3 + 2*x^2 + x^4)) + ((-356*I + 127*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/(16*Sqrt[2 - (2*I)*Sqrt[2]]) + ((356*I + 127*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/(16*Sqrt[2 + (2*I)*Sqrt[2]])","C",1
112,1,121,225,0.1635127,"\int \frac{x^2 \left(4+x^2+3 x^4+5 x^6\right)}{\left(3+2 x^2+x^4\right)^2} \, dx","Integrate[(x^2*(4 + x^2 + 3*x^4 + 5*x^6))/(3 + 2*x^2 + x^4)^2,x]","\frac{25 \left(x^3+x\right)}{8 \left(x^4+2 x^2+3\right)}+5 x-\frac{\left(111 \sqrt{2}-34 i\right) \tan ^{-1}\left(\frac{x}{\sqrt{1-i \sqrt{2}}}\right)}{16 \sqrt{2-2 i \sqrt{2}}}-\frac{\left(111 \sqrt{2}+34 i\right) \tan ^{-1}\left(\frac{x}{\sqrt{1+i \sqrt{2}}}\right)}{16 \sqrt{2+2 i \sqrt{2}}}","-\frac{1}{32} \sqrt{\frac{1}{6} \left(12899 \sqrt{3}-19291\right)} \log \left(x^2-\sqrt{2 \left(\sqrt{3}-1\right)} x+\sqrt{3}\right)+\frac{1}{32} \sqrt{\frac{1}{6} \left(12899 \sqrt{3}-19291\right)} \log \left(x^2+\sqrt{2 \left(\sqrt{3}-1\right)} x+\sqrt{3}\right)+\frac{25 \left(x^2+1\right) x}{8 \left(x^4+2 x^2+3\right)}+5 x+\frac{1}{16} \sqrt{\frac{1}{6} \left(19291+12899 \sqrt{3}\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(\sqrt{3}-1\right)}-2 x}{\sqrt{2 \left(1+\sqrt{3}\right)}}\right)-\frac{1}{16} \sqrt{\frac{1}{6} \left(19291+12899 \sqrt{3}\right)} \tan ^{-1}\left(\frac{2 x+\sqrt{2 \left(\sqrt{3}-1\right)}}{\sqrt{2 \left(1+\sqrt{3}\right)}}\right)",1,"5*x + (25*(x + x^3))/(8*(3 + 2*x^2 + x^4)) - ((-34*I + 111*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/(16*Sqrt[2 - (2*I)*Sqrt[2]]) - ((34*I + 111*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/(16*Sqrt[2 + (2*I)*Sqrt[2]])","C",1
113,1,115,224,0.2641039,"\int \frac{4+x^2+3 x^4+5 x^6}{\left(3+2 x^2+x^4\right)^2} \, dx","Integrate[(4 + x^2 + 3*x^4 + 5*x^6)/(3 + 2*x^2 + x^4)^2,x]","\frac{1}{48} \left(-\frac{50 x \left(x^2-1\right)}{x^4+2 x^2+3}+\frac{\left(95+44 i \sqrt{2}\right) \tan ^{-1}\left(\frac{x}{\sqrt{1-i \sqrt{2}}}\right)}{\sqrt{1-i \sqrt{2}}}+\frac{\left(95-44 i \sqrt{2}\right) \tan ^{-1}\left(\frac{x}{\sqrt{1+i \sqrt{2}}}\right)}{\sqrt{1+i \sqrt{2}}}\right)","\frac{1}{96} \sqrt{\frac{1}{6} \left(11567+12897 \sqrt{3}\right)} \log \left(x^2-\sqrt{2 \left(\sqrt{3}-1\right)} x+\sqrt{3}\right)-\frac{1}{96} \sqrt{\frac{1}{6} \left(11567+12897 \sqrt{3}\right)} \log \left(x^2+\sqrt{2 \left(\sqrt{3}-1\right)} x+\sqrt{3}\right)+\frac{25 x \left(1-x^2\right)}{24 \left(x^4+2 x^2+3\right)}-\frac{1}{48} \sqrt{\frac{1}{6} \left(12897 \sqrt{3}-11567\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(\sqrt{3}-1\right)}-2 x}{\sqrt{2 \left(1+\sqrt{3}\right)}}\right)+\frac{1}{48} \sqrt{\frac{1}{6} \left(12897 \sqrt{3}-11567\right)} \tan ^{-1}\left(\frac{2 x+\sqrt{2 \left(\sqrt{3}-1\right)}}{\sqrt{2 \left(1+\sqrt{3}\right)}}\right)",1,"((-50*x*(-1 + x^2))/(3 + 2*x^2 + x^4) + ((95 + (44*I)*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/Sqrt[1 - I*Sqrt[2]] + ((95 - (44*I)*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/Sqrt[1 + I*Sqrt[2]])/48","C",1
114,1,126,229,0.1760347,"\int \frac{4+x^2+3 x^4+5 x^6}{x^2 \left(3+2 x^2+x^4\right)^2} \, dx","Integrate[(4 + x^2 + 3*x^4 + 5*x^6)/(x^2*(3 + 2*x^2 + x^4)^2),x]","-\frac{25 x \left(x^2+5\right)}{72 \left(x^4+2 x^2+3\right)}-\frac{4}{9 x}-\frac{\left(19 \sqrt{2}+26 i\right) \tan ^{-1}\left(\frac{x}{\sqrt{1-i \sqrt{2}}}\right)}{48 \sqrt{2-2 i \sqrt{2}}}-\frac{\left(19 \sqrt{2}-26 i\right) \tan ^{-1}\left(\frac{x}{\sqrt{1+i \sqrt{2}}}\right)}{48 \sqrt{2+2 i \sqrt{2}}}","-\frac{1}{96} \sqrt{\frac{1}{6} \left(965+699 \sqrt{3}\right)} \log \left(x^2-\sqrt{2 \left(\sqrt{3}-1\right)} x+\sqrt{3}\right)+\frac{1}{96} \sqrt{\frac{1}{6} \left(965+699 \sqrt{3}\right)} \log \left(x^2+\sqrt{2 \left(\sqrt{3}-1\right)} x+\sqrt{3}\right)-\frac{25 x \left(x^2+5\right)}{72 \left(x^4+2 x^2+3\right)}-\frac{4}{9 x}+\frac{1}{48} \sqrt{\frac{1}{6} \left(699 \sqrt{3}-965\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(\sqrt{3}-1\right)}-2 x}{\sqrt{2 \left(1+\sqrt{3}\right)}}\right)-\frac{1}{48} \sqrt{\frac{1}{6} \left(699 \sqrt{3}-965\right)} \tan ^{-1}\left(\frac{2 x+\sqrt{2 \left(\sqrt{3}-1\right)}}{\sqrt{2 \left(1+\sqrt{3}\right)}}\right)",1,"-4/(9*x) - (25*x*(5 + x^2))/(72*(3 + 2*x^2 + x^4)) - ((26*I + 19*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/(48*Sqrt[2 - (2*I)*Sqrt[2]]) - ((-26*I + 19*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/(48*Sqrt[2 + (2*I)*Sqrt[2]])","C",1
115,1,131,238,0.2908431,"\int \frac{4+x^2+3 x^4+5 x^6}{x^4 \left(3+2 x^2+x^4\right)^2} \, dx","Integrate[(4 + x^2 + 3*x^4 + 5*x^6)/(x^4*(3 + 2*x^2 + x^4)^2),x]","\frac{1}{864} \left(\frac{4 \left(229 x^6+351 x^4+248 x^2-96\right)}{x^3 \left(x^4+2 x^2+3\right)}+\frac{2 \left(229+46 i \sqrt{2}\right) \tan ^{-1}\left(\frac{x}{\sqrt{1-i \sqrt{2}}}\right)}{\sqrt{1-i \sqrt{2}}}+\frac{2 \left(229-46 i \sqrt{2}\right) \tan ^{-1}\left(\frac{x}{\sqrt{1+i \sqrt{2}}}\right)}{\sqrt{1+i \sqrt{2}}}\right)","-\frac{4}{27 x^3}+\frac{1}{864} \sqrt{\frac{1}{6} \left(56673 \sqrt{3}-6073\right)} \log \left(x^2-\sqrt{2 \left(\sqrt{3}-1\right)} x+\sqrt{3}\right)-\frac{1}{864} \sqrt{\frac{1}{6} \left(56673 \sqrt{3}-6073\right)} \log \left(x^2+\sqrt{2 \left(\sqrt{3}-1\right)} x+\sqrt{3}\right)+\frac{25 x \left(5 x^2+7\right)}{216 \left(x^4+2 x^2+3\right)}+\frac{13}{27 x}-\frac{1}{432} \sqrt{\frac{1}{6} \left(6073+56673 \sqrt{3}\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(\sqrt{3}-1\right)}-2 x}{\sqrt{2 \left(1+\sqrt{3}\right)}}\right)+\frac{1}{432} \sqrt{\frac{1}{6} \left(6073+56673 \sqrt{3}\right)} \tan ^{-1}\left(\frac{2 x+\sqrt{2 \left(\sqrt{3}-1\right)}}{\sqrt{2 \left(1+\sqrt{3}\right)}}\right)",1,"((4*(-96 + 248*x^2 + 351*x^4 + 229*x^6))/(x^3*(3 + 2*x^2 + x^4)) + (2*(229 + (46*I)*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/Sqrt[1 - I*Sqrt[2]] + (2*(229 - (46*I)*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/Sqrt[1 + I*Sqrt[2]])/864","C",1
116,1,140,245,0.2901348,"\int \frac{4+x^2+3 x^4+5 x^6}{x^6 \left(3+2 x^2+x^4\right)^2} \, dx","Integrate[(4 + x^2 + 3*x^4 + 5*x^6)/(x^6*(3 + 2*x^2 + x^4)^2),x]","\frac{-\frac{4 \left(2435 x^8+2475 x^6+3928 x^4-984 x^2+864\right)}{x^5 \left(x^4+2 x^2+3\right)}-\frac{10 i \left(475 \sqrt{2}-487 i\right) \tan ^{-1}\left(\frac{x}{\sqrt{1-i \sqrt{2}}}\right)}{\sqrt{1-i \sqrt{2}}}+\frac{10 i \left(475 \sqrt{2}+487 i\right) \tan ^{-1}\left(\frac{x}{\sqrt{1+i \sqrt{2}}}\right)}{\sqrt{1+i \sqrt{2}}}}{12960}","-\frac{4}{45 x^5}+\frac{13}{81 x^3}-\frac{\sqrt{\frac{1}{6} \left(1139381+688419 \sqrt{3}\right)} \log \left(x^2-\sqrt{2 \left(\sqrt{3}-1\right)} x+\sqrt{3}\right)}{2592}+\frac{\sqrt{\frac{1}{6} \left(1139381+688419 \sqrt{3}\right)} \log \left(x^2+\sqrt{2 \left(\sqrt{3}-1\right)} x+\sqrt{3}\right)}{2592}+\frac{25 x \left(1-7 x^2\right)}{648 \left(x^4+2 x^2+3\right)}-\frac{13}{27 x}+\frac{\sqrt{\frac{1}{6} \left(688419 \sqrt{3}-1139381\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(\sqrt{3}-1\right)}-2 x}{\sqrt{2 \left(1+\sqrt{3}\right)}}\right)}{1296}-\frac{\sqrt{\frac{1}{6} \left(688419 \sqrt{3}-1139381\right)} \tan ^{-1}\left(\frac{2 x+\sqrt{2 \left(\sqrt{3}-1\right)}}{\sqrt{2 \left(1+\sqrt{3}\right)}}\right)}{1296}",1,"((-4*(864 - 984*x^2 + 3928*x^4 + 2475*x^6 + 2435*x^8))/(x^5*(3 + 2*x^2 + x^4)) - ((10*I)*(-487*I + 475*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/Sqrt[1 - I*Sqrt[2]] + ((10*I)*(487*I + 475*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/Sqrt[1 + I*Sqrt[2]])/12960","C",1
117,1,156,243,0.2155225,"\int \frac{x^{10} \left(4+x^2+3 x^4+5 x^6\right)}{\left(3+2 x^2+x^4\right)^3} \, dx","Integrate[(x^10*(4 + x^2 + 3*x^4 + 5*x^6))/(3 + 2*x^2 + x^4)^3,x]","x^5-9 x^3+\frac{\left(252 x^2+3305\right) x}{64 \left(x^4+2 x^2+3\right)}-\frac{25 \left(7 x^2+15\right) x}{16 \left(x^4+2 x^2+3\right)^2}+58 x+\frac{3 \left(148 \sqrt{2}+4795 i\right) \tan ^{-1}\left(\frac{x}{\sqrt{1-i \sqrt{2}}}\right)}{128 \sqrt{2-2 i \sqrt{2}}}+\frac{3 \left(148 \sqrt{2}-4795 i\right) \tan ^{-1}\left(\frac{x}{\sqrt{1+i \sqrt{2}}}\right)}{128 \sqrt{2+2 i \sqrt{2}}}","x^5-9 x^3+\frac{3}{512} \sqrt{8595619+7678611 \sqrt{3}} \log \left(x^2-\sqrt{2 \left(\sqrt{3}-1\right)} x+\sqrt{3}\right)-\frac{3}{512} \sqrt{8595619+7678611 \sqrt{3}} \log \left(x^2+\sqrt{2 \left(\sqrt{3}-1\right)} x+\sqrt{3}\right)+\frac{\left(252 x^2+3305\right) x}{64 \left(x^4+2 x^2+3\right)}-\frac{25 \left(7 x^2+15\right) x}{16 \left(x^4+2 x^2+3\right)^2}+58 x+\frac{3}{256} \sqrt{7678611 \sqrt{3}-8595619} \tan ^{-1}\left(\frac{\sqrt{2 \left(\sqrt{3}-1\right)}-2 x}{\sqrt{2 \left(1+\sqrt{3}\right)}}\right)-\frac{3}{256} \sqrt{7678611 \sqrt{3}-8595619} \tan ^{-1}\left(\frac{2 x+\sqrt{2 \left(\sqrt{3}-1\right)}}{\sqrt{2 \left(1+\sqrt{3}\right)}}\right)",1,"58*x - 9*x^3 + x^5 - (25*x*(15 + 7*x^2))/(16*(3 + 2*x^2 + x^4)^2) + (x*(3305 + 252*x^2))/(64*(3 + 2*x^2 + x^4)) + (3*(4795*I + 148*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/(128*Sqrt[2 - (2*I)*Sqrt[2]]) + (3*(-4795*I + 148*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/(128*Sqrt[2 + (2*I)*Sqrt[2]])","C",1
118,1,155,242,0.2035668,"\int \frac{x^8 \left(4+x^2+3 x^4+5 x^6\right)}{\left(3+2 x^2+x^4\right)^3} \, dx","Integrate[(x^8*(4 + x^2 + 3*x^4 + 5*x^6))/(3 + 2*x^2 + x^4)^3,x]","\frac{5 x^3}{3}-\frac{\left(835 x^2+1468\right) x}{64 \left(x^4+2 x^2+3\right)}+\frac{25 \left(5 x^2+3\right) x}{16 \left(x^4+2 x^2+3\right)^2}-27 x+\frac{21 \left(137 \sqrt{2}-175 i\right) \tan ^{-1}\left(\frac{x}{\sqrt{1-i \sqrt{2}}}\right)}{128 \sqrt{2-2 i \sqrt{2}}}+\frac{21 \left(137 \sqrt{2}+175 i\right) \tan ^{-1}\left(\frac{x}{\sqrt{1+i \sqrt{2}}}\right)}{128 \sqrt{2+2 i \sqrt{2}}}","\frac{5 x^3}{3}-\frac{21}{512} \sqrt{22721 \sqrt{3}-34271} \log \left(x^2-\sqrt{2 \left(\sqrt{3}-1\right)} x+\sqrt{3}\right)+\frac{21}{512} \sqrt{22721 \sqrt{3}-34271} \log \left(x^2+\sqrt{2 \left(\sqrt{3}-1\right)} x+\sqrt{3}\right)-\frac{\left(835 x^2+1468\right) x}{64 \left(x^4+2 x^2+3\right)}+\frac{25 \left(5 x^2+3\right) x}{16 \left(x^4+2 x^2+3\right)^2}-27 x-\frac{21}{256} \sqrt{34271+22721 \sqrt{3}} \tan ^{-1}\left(\frac{\sqrt{2 \left(\sqrt{3}-1\right)}-2 x}{\sqrt{2 \left(1+\sqrt{3}\right)}}\right)+\frac{21}{256} \sqrt{34271+22721 \sqrt{3}} \tan ^{-1}\left(\frac{2 x+\sqrt{2 \left(\sqrt{3}-1\right)}}{\sqrt{2 \left(1+\sqrt{3}\right)}}\right)",1,"-27*x + (5*x^3)/3 + (25*x*(3 + 5*x^2))/(16*(3 + 2*x^2 + x^4)^2) - (x*(1468 + 835*x^2))/(64*(3 + 2*x^2 + x^4)) + (21*(-175*I + 137*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/(128*Sqrt[2 - (2*I)*Sqrt[2]]) + (21*(175*I + 137*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/(128*Sqrt[2 + (2*I)*Sqrt[2]])","C",1
119,1,138,235,0.3171173,"\int \frac{x^6 \left(4+x^2+3 x^4+5 x^6\right)}{\left(3+2 x^2+x^4\right)^3} \, dx","Integrate[(x^6*(4 + x^2 + 3*x^4 + 5*x^6))/(3 + 2*x^2 + x^4)^3,x]","\frac{1}{256} \left(\frac{4 x \left(320 x^8+1686 x^6+4089 x^4+5112 x^2+3411\right)}{\left(x^4+2 x^2+3\right)^2}-\frac{i \left(185 \sqrt{2}-2644 i\right) \tan ^{-1}\left(\frac{x}{\sqrt{1-i \sqrt{2}}}\right)}{\sqrt{1-i \sqrt{2}}}+\frac{i \left(185 \sqrt{2}+2644 i\right) \tan ^{-1}\left(\frac{x}{\sqrt{1+i \sqrt{2}}}\right)}{\sqrt{1+i \sqrt{2}}}\right)","-\frac{1}{512} \sqrt{1176531 \sqrt{3}-827621} \log \left(x^2-\sqrt{2 \left(\sqrt{3}-1\right)} x+\sqrt{3}\right)+\frac{1}{512} \sqrt{1176531 \sqrt{3}-827621} \log \left(x^2+\sqrt{2 \left(\sqrt{3}-1\right)} x+\sqrt{3}\right)+\frac{7 \left(58 x^2+11\right) x}{64 \left(x^4+2 x^2+3\right)}+\frac{25 \left(3-x^2\right) x}{16 \left(x^4+2 x^2+3\right)^2}+5 x+\frac{1}{256} \sqrt{827621+1176531 \sqrt{3}} \tan ^{-1}\left(\frac{\sqrt{2 \left(\sqrt{3}-1\right)}-2 x}{\sqrt{2 \left(1+\sqrt{3}\right)}}\right)-\frac{1}{256} \sqrt{827621+1176531 \sqrt{3}} \tan ^{-1}\left(\frac{2 x+\sqrt{2 \left(\sqrt{3}-1\right)}}{\sqrt{2 \left(1+\sqrt{3}\right)}}\right)",1,"((4*x*(3411 + 5112*x^2 + 4089*x^4 + 1686*x^6 + 320*x^8))/(3 + 2*x^2 + x^4)^2 - (I*(-2644*I + 185*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/Sqrt[1 - I*Sqrt[2]] + (I*(2644*I + 185*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/Sqrt[1 + I*Sqrt[2]])/256","C",1
120,1,129,238,0.2969681,"\int \frac{x^4 \left(4+x^2+3 x^4+5 x^6\right)}{\left(3+2 x^2+x^4\right)^3} \, dx","Integrate[(x^4*(4 + x^2 + 3*x^4 + 5*x^6))/(3 + 2*x^2 + x^4)^3,x]","\frac{1}{256} \left(\frac{4 x \left(-59 x^6+120 x^4+199 x^2+414\right)}{\left(x^4+2 x^2+3\right)^2}+\frac{3 \left(174+133 i \sqrt{2}\right) \tan ^{-1}\left(\frac{x}{\sqrt{1-i \sqrt{2}}}\right)}{\sqrt{1-i \sqrt{2}}}+\frac{3 \left(174-133 i \sqrt{2}\right) \tan ^{-1}\left(\frac{x}{\sqrt{1+i \sqrt{2}}}\right)}{\sqrt{1+i \sqrt{2}}}\right)","\frac{1}{512} \sqrt{3 \left(48835+32827 \sqrt{3}\right)} \log \left(x^2-\sqrt{2 \left(\sqrt{3}-1\right)} x+\sqrt{3}\right)-\frac{1}{512} \sqrt{3 \left(48835+32827 \sqrt{3}\right)} \log \left(x^2+\sqrt{2 \left(\sqrt{3}-1\right)} x+\sqrt{3}\right)+\frac{x \left(238-59 x^2\right)}{64 \left(x^4+2 x^2+3\right)}-\frac{25 x \left(x^2+3\right)}{16 \left(x^4+2 x^2+3\right)^2}-\frac{1}{256} \sqrt{3 \left(32827 \sqrt{3}-48835\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(\sqrt{3}-1\right)}-2 x}{\sqrt{2 \left(1+\sqrt{3}\right)}}\right)+\frac{1}{256} \sqrt{3 \left(32827 \sqrt{3}-48835\right)} \tan ^{-1}\left(\frac{2 x+\sqrt{2 \left(\sqrt{3}-1\right)}}{\sqrt{2 \left(1+\sqrt{3}\right)}}\right)",1,"((4*x*(414 + 199*x^2 + 120*x^4 - 59*x^6))/(3 + 2*x^2 + x^4)^2 + (3*(174 + (133*I)*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/Sqrt[1 - I*Sqrt[2]] + (3*(174 - (133*I)*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/Sqrt[1 + I*Sqrt[2]])/256","C",1
121,1,133,246,0.2969648,"\int \frac{x^2 \left(4+x^2+3 x^4+5 x^6\right)}{\left(3+2 x^2+x^4\right)^3} \, dx","Integrate[(x^2*(4 + x^2 + 3*x^4 + 5*x^6))/(3 + 2*x^2 + x^4)^3,x]","\frac{1}{768} \left(-\frac{4 x \left(88 x^6+529 x^4+670 x^2+759\right)}{\left(x^4+2 x^2+3\right)^2}-\frac{11 i \left(31 \sqrt{2}-16 i\right) \tan ^{-1}\left(\frac{x}{\sqrt{1-i \sqrt{2}}}\right)}{\sqrt{1-i \sqrt{2}}}+\frac{11 i \left(31 \sqrt{2}+16 i\right) \tan ^{-1}\left(\frac{x}{\sqrt{1+i \sqrt{2}}}\right)}{\sqrt{1+i \sqrt{2}}}\right)","-\frac{11 \sqrt{\frac{1}{3} \left(1825+1089 \sqrt{3}\right)} \log \left(x^2-\sqrt{2 \left(\sqrt{3}-1\right)} x+\sqrt{3}\right)}{1536}+\frac{11 \sqrt{\frac{1}{3} \left(1825+1089 \sqrt{3}\right)} \log \left(x^2+\sqrt{2 \left(\sqrt{3}-1\right)} x+\sqrt{3}\right)}{1536}+\frac{25 x \left(x^2+1\right)}{16 \left(x^4+2 x^2+3\right)^2}-\frac{x \left(88 x^2+353\right)}{192 \left(x^4+2 x^2+3\right)}-\frac{11}{768} \sqrt{\frac{1}{3} \left(1089 \sqrt{3}-1825\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(\sqrt{3}-1\right)}-2 x}{\sqrt{2 \left(1+\sqrt{3}\right)}}\right)+\frac{11}{768} \sqrt{\frac{1}{3} \left(1089 \sqrt{3}-1825\right)} \tan ^{-1}\left(\frac{2 x+\sqrt{2 \left(\sqrt{3}-1\right)}}{\sqrt{2 \left(1+\sqrt{3}\right)}}\right)",1,"((-4*x*(759 + 670*x^2 + 529*x^4 + 88*x^6))/(3 + 2*x^2 + x^4)^2 - ((11*I)*(-16*I + 31*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/Sqrt[1 - I*Sqrt[2]] + ((11*I)*(16*I + 31*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/Sqrt[1 + I*Sqrt[2]])/768","C",1
122,1,129,248,0.2908445,"\int \frac{4+x^2+3 x^4+5 x^6}{\left(3+2 x^2+x^4\right)^3} \, dx","Integrate[(4 + x^2 + 3*x^4 + 5*x^6)/(3 + 2*x^2 + x^4)^3,x]","\frac{1}{768} \left(\frac{4 x \left(51 x^6+166 x^4+181 x^2+292\right)}{\left(x^4+2 x^2+3\right)^2}+\frac{3 \left(34+21 i \sqrt{2}\right) \tan ^{-1}\left(\frac{x}{\sqrt{1-i \sqrt{2}}}\right)}{\sqrt{1-i \sqrt{2}}}+\frac{3 \left(34-21 i \sqrt{2}\right) \tan ^{-1}\left(\frac{x}{\sqrt{1+i \sqrt{2}}}\right)}{\sqrt{1+i \sqrt{2}}}\right)","\frac{1}{512} \sqrt{\frac{1}{3} \left(1291+1019 \sqrt{3}\right)} \log \left(x^2-\sqrt{2 \left(\sqrt{3}-1\right)} x+\sqrt{3}\right)-\frac{1}{512} \sqrt{\frac{1}{3} \left(1291+1019 \sqrt{3}\right)} \log \left(x^2+\sqrt{2 \left(\sqrt{3}-1\right)} x+\sqrt{3}\right)+\frac{25 x \left(1-x^2\right)}{48 \left(x^4+2 x^2+3\right)^2}+\frac{x \left(51 x^2+64\right)}{192 \left(x^4+2 x^2+3\right)}-\frac{1}{256} \sqrt{\frac{1}{3} \left(1019 \sqrt{3}-1291\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(\sqrt{3}-1\right)}-2 x}{\sqrt{2 \left(1+\sqrt{3}\right)}}\right)+\frac{1}{256} \sqrt{\frac{1}{3} \left(1019 \sqrt{3}-1291\right)} \tan ^{-1}\left(\frac{2 x+\sqrt{2 \left(\sqrt{3}-1\right)}}{\sqrt{2 \left(1+\sqrt{3}\right)}}\right)",1,"((4*x*(292 + 181*x^2 + 166*x^4 + 51*x^6))/(3 + 2*x^2 + x^4)^2 + (3*(34 + (21*I)*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/Sqrt[1 - I*Sqrt[2]] + (3*(34 - (21*I)*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/Sqrt[1 + I*Sqrt[2]])/768","C",1
123,1,140,253,0.3657962,"\int \frac{4+x^2+3 x^4+5 x^6}{x^2 \left(3+2 x^2+x^4\right)^3} \, dx","Integrate[(4 + x^2 + 3*x^4 + 5*x^6)/(x^2*(3 + 2*x^2 + x^4)^3),x]","\frac{-\frac{12 \left(166 x^8+611 x^6+1412 x^4+1849 x^2+768\right)}{x \left(x^4+2 x^2+3\right)^2}+\frac{3 i \left(7 \sqrt{2}+332 i\right) \tan ^{-1}\left(\frac{x}{\sqrt{1-i \sqrt{2}}}\right)}{\sqrt{1-i \sqrt{2}}}-\frac{3 i \left(7 \sqrt{2}-332 i\right) \tan ^{-1}\left(\frac{x}{\sqrt{1+i \sqrt{2}}}\right)}{\sqrt{1+i \sqrt{2}}}}{6912}","-\frac{\sqrt{\frac{1}{3} \left(55161 \sqrt{3}-59711\right)} \log \left(x^2-\sqrt{2 \left(\sqrt{3}-1\right)} x+\sqrt{3}\right)}{4608}+\frac{\sqrt{\frac{1}{3} \left(55161 \sqrt{3}-59711\right)} \log \left(x^2+\sqrt{2 \left(\sqrt{3}-1\right)} x+\sqrt{3}\right)}{4608}-\frac{25 x \left(x^2+5\right)}{144 \left(x^4+2 x^2+3\right)^2}-\frac{x \left(242 x^2+325\right)}{1728 \left(x^4+2 x^2+3\right)}-\frac{4}{27 x}+\frac{\sqrt{\frac{1}{3} \left(59711+55161 \sqrt{3}\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(\sqrt{3}-1\right)}-2 x}{\sqrt{2 \left(1+\sqrt{3}\right)}}\right)}{2304}-\frac{\sqrt{\frac{1}{3} \left(59711+55161 \sqrt{3}\right)} \tan ^{-1}\left(\frac{2 x+\sqrt{2 \left(\sqrt{3}-1\right)}}{\sqrt{2 \left(1+\sqrt{3}\right)}}\right)}{2304}",1,"((-12*(768 + 1849*x^2 + 1412*x^4 + 611*x^6 + 166*x^8))/(x*(3 + 2*x^2 + x^4)^2) + ((3*I)*(332*I + 7*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/Sqrt[1 - I*Sqrt[2]] - ((3*I)*(-332*I + 7*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/Sqrt[1 + I*Sqrt[2]])/6912","C",1
124,1,139,262,0.3154617,"\int \frac{4+x^2+3 x^4+5 x^6}{x^4 \left(3+2 x^2+x^4\right)^3} \, dx","Integrate[(4 + x^2 + 3*x^4 + 5*x^6)/(x^4*(3 + 2*x^2 + x^4)^3),x]","\frac{\frac{4 \left(2369 x^{10}+8644 x^8+19939 x^6+20090 x^4+9024 x^2-2304\right)}{x^3 \left(x^4+2 x^2+3\right)^2}+\frac{\left(4738+127 i \sqrt{2}\right) \tan ^{-1}\left(\frac{x}{\sqrt{1-i \sqrt{2}}}\right)}{\sqrt{1-i \sqrt{2}}}+\frac{\left(4738-127 i \sqrt{2}\right) \tan ^{-1}\left(\frac{x}{\sqrt{1+i \sqrt{2}}}\right)}{\sqrt{1+i \sqrt{2}}}}{20736}","-\frac{4}{81 x^3}+\frac{\sqrt{\frac{1}{3} \left(11240451 \sqrt{3}-10004741\right)} \log \left(x^2-\sqrt{2 \left(\sqrt{3}-1\right)} x+\sqrt{3}\right)}{41472}-\frac{\sqrt{\frac{1}{3} \left(11240451 \sqrt{3}-10004741\right)} \log \left(x^2+\sqrt{2 \left(\sqrt{3}-1\right)} x+\sqrt{3}\right)}{41472}+\frac{25 x \left(5 x^2+7\right)}{432 \left(x^4+2 x^2+3\right)^2}+\frac{x \left(1025 x^2+1474\right)}{5184 \left(x^4+2 x^2+3\right)}+\frac{7}{27 x}-\frac{\sqrt{\frac{1}{3} \left(10004741+11240451 \sqrt{3}\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(\sqrt{3}-1\right)}-2 x}{\sqrt{2 \left(1+\sqrt{3}\right)}}\right)}{20736}+\frac{\sqrt{\frac{1}{3} \left(10004741+11240451 \sqrt{3}\right)} \tan ^{-1}\left(\frac{2 x+\sqrt{2 \left(\sqrt{3}-1\right)}}{\sqrt{2 \left(1+\sqrt{3}\right)}}\right)}{20736}",1,"((4*(-2304 + 9024*x^2 + 20090*x^4 + 19939*x^6 + 8644*x^8 + 2369*x^10))/(x^3*(3 + 2*x^2 + x^4)^2) + ((4738 + (127*I)*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/Sqrt[1 - I*Sqrt[2]] + ((4738 - (127*I)*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/Sqrt[1 + I*Sqrt[2]])/20736","C",1
125,1,142,149,0.1158888,"\int \frac{x \left(d+e x^2+f x^4+g x^6\right)}{a+b x^2+c x^4} \, dx","Integrate[(x*(d + e*x^2 + f*x^4 + g*x^6))/(a + b*x^2 + c*x^4),x]","\frac{\log \left(a+b x^2+c x^4\right) \left(-c (a g+b f)+b^2 g+c^2 e\right)+\frac{2 \tan ^{-1}\left(\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right) \left(-c^2 (2 a f+b e)+b c (3 a g+b f)+b^3 (-g)+2 c^3 d\right)}{\sqrt{4 a c-b^2}}+2 c x^2 (c f-b g)+c^2 g x^4}{4 c^3}","\frac{\log \left(a+b x^2+c x^4\right) \left(-c (a g+b f)+b^2 g+c^2 e\right)}{4 c^3}-\frac{\tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right) \left(-c^2 (2 a f+b e)+b c (3 a g+b f)+b^3 (-g)+2 c^3 d\right)}{2 c^3 \sqrt{b^2-4 a c}}+\frac{x^2 (c f-b g)}{2 c^2}+\frac{g x^4}{4 c}",1,"(2*c*(c*f - b*g)*x^2 + c^2*g*x^4 + (2*(2*c^3*d - c^2*(b*e + 2*a*f) - b^3*g + b*c*(b*f + 3*a*g))*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + (c^2*e + b^2*g - c*(b*f + a*g))*Log[a + b*x^2 + c*x^4])/(4*c^3)","A",1
126,1,721,594,2.6408829,"\int \frac{x^4 \left(d+e x^2+f x^4+g x^6\right)}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(x^4*(d + e*x^2 + f*x^4 + g*x^6))/(a + b*x^2 + c*x^4)^2,x]","\frac{\frac{6 \sqrt{c} x \left(a^2 c \left(3 b g-2 c \left(f+g x^2\right)\right)+a \left(b^3 (-g)+b^2 c \left(f+4 g x^2\right)-b c^2 \left(e+3 f x^2\right)+2 c^3 \left(d+e x^2\right)\right)+b x^2 \left(b^3 (-g)+b^2 c f-b c^2 e+c^3 d\right)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{3 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(-b^2 c \left(-c e \sqrt{b^2-4 a c}+24 a g \sqrt{b^2-4 a c}+19 a c f+c^2 d\right)+b c^2 \left(c \left(d \sqrt{b^2-4 a c}+8 a e\right)+13 a \left(f \sqrt{b^2-4 a c}-4 a g\right)\right)+2 a c^2 \left(-3 c e \sqrt{b^2-4 a c}+7 a g \sqrt{b^2-4 a c}+10 a c f-2 c^2 d\right)+b^4 \left(5 g \sqrt{b^2-4 a c}+3 c f\right)-b^3 c \left(3 f \sqrt{b^2-4 a c}-34 a g+c e\right)-5 b^5 g\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(b^2 c \left(c e \sqrt{b^2-4 a c}-24 a g \sqrt{b^2-4 a c}+19 a c f+c^2 d\right)+b c^2 \left(c \left(d \sqrt{b^2-4 a c}-8 a e\right)+13 a \left(f \sqrt{b^2-4 a c}+4 a g\right)\right)+2 a c^2 \left(-3 c e \sqrt{b^2-4 a c}+7 a g \sqrt{b^2-4 a c}-10 a c f+2 c^2 d\right)+b^4 \left(5 g \sqrt{b^2-4 a c}-3 c f\right)+b^3 c \left(-3 f \sqrt{b^2-4 a c}-34 a g+c e\right)+5 b^5 g\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+12 \sqrt{c} x (c f-2 b g)+4 c^{3/2} g x^3}{12 c^{7/2}}","\frac{x \left(a \left(-c^2 (2 a f+b e)+b c (3 a g+b f)+b^3 (-g)+2 c^3 d\right)+x^2 \left(-b^2 c (c e-4 a g)+b c^2 (c d-3 a f)+2 a c^2 (c e-a g)+b^4 (-g)+b^3 c f\right)\right)}{2 c^3 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(-b^2 c (c e-24 a g)-\frac{-b^3 c (c e-34 a g)-b^2 c^2 (19 a f+c d)+4 a b c^2 (2 c e-13 a g)-4 a c^3 (c d-5 a f)-5 b^5 g+3 b^4 c f}{\sqrt{b^2-4 a c}}-b c^2 (13 a f+c d)+2 a c^2 (3 c e-7 a g)-5 b^4 g+3 b^3 c f\right)}{2 \sqrt{2} c^{7/2} \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(-b^2 c (c e-24 a g)+\frac{-b^3 c (c e-34 a g)-b^2 c^2 (19 a f+c d)+4 a b c^2 (2 c e-13 a g)-4 a c^3 (c d-5 a f)-5 b^5 g+3 b^4 c f}{\sqrt{b^2-4 a c}}-b c^2 (13 a f+c d)+2 a c^2 (3 c e-7 a g)-5 b^4 g+3 b^3 c f\right)}{2 \sqrt{2} c^{7/2} \left(b^2-4 a c\right) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x (c f-2 b g)}{c^3}+\frac{g x^3}{3 c^2}",1,"(12*Sqrt[c]*(c*f - 2*b*g)*x + 4*c^(3/2)*g*x^3 + (6*Sqrt[c]*x*(b*(c^3*d - b*c^2*e + b^2*c*f - b^3*g)*x^2 + a^2*c*(3*b*g - 2*c*(f + g*x^2)) + a*(-(b^3*g) + 2*c^3*(d + e*x^2) - b*c^2*(e + 3*f*x^2) + b^2*c*(f + 4*g*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (3*Sqrt[2]*(-5*b^5*g - b^3*c*(c*e + 3*Sqrt[b^2 - 4*a*c]*f - 34*a*g) + b^4*(3*c*f + 5*Sqrt[b^2 - 4*a*c]*g) + 2*a*c^2*(-2*c^2*d - 3*c*Sqrt[b^2 - 4*a*c]*e + 10*a*c*f + 7*a*Sqrt[b^2 - 4*a*c]*g) - b^2*c*(c^2*d - c*Sqrt[b^2 - 4*a*c]*e + 19*a*c*f + 24*a*Sqrt[b^2 - 4*a*c]*g) + b*c^2*(c*(Sqrt[b^2 - 4*a*c]*d + 8*a*e) + 13*a*(Sqrt[b^2 - 4*a*c]*f - 4*a*g)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[2]*(5*b^5*g + b^3*c*(c*e - 3*Sqrt[b^2 - 4*a*c]*f - 34*a*g) + b^4*(-3*c*f + 5*Sqrt[b^2 - 4*a*c]*g) + b^2*c*(c^2*d + c*Sqrt[b^2 - 4*a*c]*e + 19*a*c*f - 24*a*Sqrt[b^2 - 4*a*c]*g) + 2*a*c^2*(2*c^2*d - 3*c*Sqrt[b^2 - 4*a*c]*e - 10*a*c*f + 7*a*Sqrt[b^2 - 4*a*c]*g) + b*c^2*(c*(Sqrt[b^2 - 4*a*c]*d - 8*a*e) + 13*a*(Sqrt[b^2 - 4*a*c]*f + 4*a*g)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(12*c^(7/2))","A",1
127,1,575,471,1.9462419,"\int \frac{x^2 \left(d+e x^2+f x^4+g x^6\right)}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(x^2*(d + e*x^2 + f*x^4 + g*x^6))/(a + b*x^2 + c*x^4)^2,x]","\frac{-\frac{2 \sqrt{c} x \left(2 c \left(a^2 g-a c \left(e+f x^2\right)+c^2 d x^2\right)+b^2 \left(c f x^2-a g\right)+b c \left(a \left(f+3 g x^2\right)+c \left(d-e x^2\right)\right)+b^3 (-g) x^2\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(2 c^2 \left(-10 a^2 g+c d \sqrt{b^2-4 a c}+3 a f \sqrt{b^2-4 a c}+2 a c e\right)-b c \left(c e \sqrt{b^2-4 a c}+13 a g \sqrt{b^2-4 a c}+8 a c f+4 c^2 d\right)+b^2 c \left(-f \sqrt{b^2-4 a c}+19 a g+c e\right)+b^3 \left(3 g \sqrt{b^2-4 a c}+c f\right)-3 b^4 g\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(2 c^2 \left(10 a^2 g+c d \sqrt{b^2-4 a c}+3 a f \sqrt{b^2-4 a c}-2 a c e\right)+b c \left(-c e \sqrt{b^2-4 a c}-13 a g \sqrt{b^2-4 a c}+8 a c f+4 c^2 d\right)-b^2 c \left(f \sqrt{b^2-4 a c}+19 a g+c e\right)+b^3 \left(3 g \sqrt{b^2-4 a c}-c f\right)+3 b^4 g\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+4 \sqrt{c} g x}{4 c^{5/2}}","-\frac{x \left(x^2 \left(-c^2 (2 a f+b e)+b c (3 a g+b f)+b^3 (-g)+2 c^3 d\right)-a b^2 g+b c (a f+c d)-2 a c (c e-a g)\right)}{2 c^2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(\frac{b^2 c (19 a g+c e)-4 b c^2 (2 a f+c d)+4 a c^2 (c e-5 a g)-3 b^4 g+b^3 c f}{\sqrt{b^2-4 a c}}-c^2 (b e-6 a f)-b c (13 a g+b f)+3 b^3 g+2 c^3 d\right)}{2 \sqrt{2} c^{5/2} \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(-\frac{b^2 c (19 a g+c e)-4 b c^2 (2 a f+c d)+4 a c^2 (c e-5 a g)-3 b^4 g+b^3 c f}{\sqrt{b^2-4 a c}}-c^2 (b e-6 a f)-b c (13 a g+b f)+3 b^3 g+2 c^3 d\right)}{2 \sqrt{2} c^{5/2} \left(b^2-4 a c\right) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{g x}{c^2}",1,"(4*Sqrt[c]*g*x - (2*Sqrt[c]*x*(-(b^3*g*x^2) + b^2*(-(a*g) + c*f*x^2) + 2*c*(a^2*g + c^2*d*x^2 - a*c*(e + f*x^2)) + b*c*(c*(d - e*x^2) + a*(f + 3*g*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - (Sqrt[2]*(-3*b^4*g + b^2*c*(c*e - Sqrt[b^2 - 4*a*c]*f + 19*a*g) + 2*c^2*(c*Sqrt[b^2 - 4*a*c]*d + 2*a*c*e + 3*a*Sqrt[b^2 - 4*a*c]*f - 10*a^2*g) + b^3*(c*f + 3*Sqrt[b^2 - 4*a*c]*g) - b*c*(4*c^2*d + c*Sqrt[b^2 - 4*a*c]*e + 8*a*c*f + 13*a*Sqrt[b^2 - 4*a*c]*g))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*(3*b^4*g - b^2*c*(c*e + Sqrt[b^2 - 4*a*c]*f + 19*a*g) + 2*c^2*(c*Sqrt[b^2 - 4*a*c]*d - 2*a*c*e + 3*a*Sqrt[b^2 - 4*a*c]*f + 10*a^2*g) + b^3*(-(c*f) + 3*Sqrt[b^2 - 4*a*c]*g) + b*c*(4*c^2*d - c*Sqrt[b^2 - 4*a*c]*e + 8*a*c*f - 13*a*Sqrt[b^2 - 4*a*c]*g))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(4*c^(5/2))","A",1
128,1,512,449,1.651497,"\int \frac{d+e x^2+f x^4+g x^6}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(d + e*x^2 + f*x^4 + g*x^6)/(a + b*x^2 + c*x^4)^2,x]","\frac{\frac{2 \sqrt{c} x \left(b \left(a^2 (-g)-a c e+a c f x^2+c^2 d x^2\right)+b^2 \left(c d-a g x^2\right)+2 a c \left(a \left(f+g x^2\right)-c \left(d+e x^2\right)\right)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(b c \left(8 a^2 g+c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}+4 a c e\right)-a b^3 g-2 a c \left(c e \sqrt{b^2-4 a c}+3 a g \sqrt{b^2-4 a c}+2 a c f+6 c^2 d\right)+b^2 \left(a g \sqrt{b^2-4 a c}-a c f+c^2 d\right)\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(b c \left(-8 a^2 g+c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}-4 a c e\right)+a b^3 g+2 a c \left(-c e \sqrt{b^2-4 a c}-3 a g \sqrt{b^2-4 a c}+2 a c f+6 c^2 d\right)+b^2 \left(a g \sqrt{b^2-4 a c}+a c f+c^2 (-d)\right)\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}}{4 a c^{3/2}}","\frac{x \left(x^2 \left(-a b^2 g+b c (a f+c d)-2 a c (c e-a g)\right)+c \left(-\frac{a b (a g+c e)}{c}-2 a (c d-a f)+b^2 d\right)\right)}{2 a c \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(\frac{a b^2 g}{c}+\frac{-a b^3 g+b^2 c (c d-a f)+4 a b c (2 a g+c e)-4 a c^2 (a f+3 c d)}{c \sqrt{b^2-4 a c}}+b (a f+c d)-2 a (3 a g+c e)\right)}{2 \sqrt{2} a \sqrt{c} \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(\frac{a b^2 g}{c}-\frac{-a b^3 g+b^2 c (c d-a f)+4 a b c (2 a g+c e)-4 a c^2 (a f+3 c d)}{c \sqrt{b^2-4 a c}}+b (a f+c d)-2 a (3 a g+c e)\right)}{2 \sqrt{2} a \sqrt{c} \left(b^2-4 a c\right) \sqrt{\sqrt{b^2-4 a c}+b}}",1,"((2*Sqrt[c]*x*(b*(-(a*c*e) - a^2*g + c^2*d*x^2 + a*c*f*x^2) + b^2*(c*d - a*g*x^2) + 2*a*c*(-(c*(d + e*x^2)) + a*(f + g*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*(-(a*b^3*g) + b*c*(c*Sqrt[b^2 - 4*a*c]*d + 4*a*c*e + a*Sqrt[b^2 - 4*a*c]*f + 8*a^2*g) + b^2*(c^2*d - a*c*f + a*Sqrt[b^2 - 4*a*c]*g) - 2*a*c*(6*c^2*d + c*Sqrt[b^2 - 4*a*c]*e + 2*a*c*f + 3*a*Sqrt[b^2 - 4*a*c]*g))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(a*b^3*g + b*c*(c*Sqrt[b^2 - 4*a*c]*d - 4*a*c*e + a*Sqrt[b^2 - 4*a*c]*f - 8*a^2*g) + 2*a*c*(6*c^2*d - c*Sqrt[b^2 - 4*a*c]*e + 2*a*c*f - 3*a*Sqrt[b^2 - 4*a*c]*g) + b^2*(-(c^2*d) + a*c*f + a*Sqrt[b^2 - 4*a*c]*g))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(4*a*c^(3/2))","A",1
129,1,529,460,2.4344657,"\int \frac{d+e x^2+f x^4+g x^6}{x^2 \left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(d + e*x^2 + f*x^4 + g*x^6)/(x^2*(a + b*x^2 + c*x^4)^2),x]","-\frac{-\frac{2 x \left(2 a \left(a^2 g-a c \left(e+f x^2\right)+c^2 d x^2\right)+b^2 \left(a e-c d x^2\right)+a b \left(-a f+a g x^2+3 c d+c e x^2\right)+b^3 (-d)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(2 a c \left(2 a^2 g-5 c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}+6 a c e\right)+b^2 \left(a^2 g+3 c d \sqrt{b^2-4 a c}-a c e\right)-a b \left(c e \sqrt{b^2-4 a c}+a g \sqrt{b^2-4 a c}+4 a c f+16 c^2 d\right)+3 b^3 c d\right)}{\sqrt{c} \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(-2 a c \left(2 a^2 g+5 c d \sqrt{b^2-4 a c}-a f \sqrt{b^2-4 a c}+6 a c e\right)+b^2 \left(a^2 (-g)+3 c d \sqrt{b^2-4 a c}+a c e\right)+a b \left(-c e \sqrt{b^2-4 a c}-a g \sqrt{b^2-4 a c}+4 a c f+16 c^2 d\right)-3 b^3 c d\right)}{\sqrt{c} \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{4 d}{x}}{4 a^2}","-\frac{x \left(a \left(-2 a^2 g+\frac{b^3 d}{a}+a (b f+2 c e)-b (b e+3 c d)\right)+x^2 \left(-a b (a g+c e)-2 a c (c d-a f)+b^2 c d\right)\right)}{2 a^2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(\frac{4 a^2 c (a g+3 c e)-a b^2 (c e-a g)-4 a b c (a f+4 c d)+3 b^3 c d}{\sqrt{b^2-4 a c}}-a b (a g+c e)-2 a c (5 c d-a f)+3 b^2 c d\right)}{2 \sqrt{2} a^2 \sqrt{c} \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(-\frac{4 a^2 c (a g+3 c e)-a b^2 (c e-a g)-4 a b c (a f+4 c d)+3 b^3 c d}{\sqrt{b^2-4 a c}}-a b (a g+c e)-2 a c (5 c d-a f)+3 b^2 c d\right)}{2 \sqrt{2} a^2 \sqrt{c} \left(b^2-4 a c\right) \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{d}{a^2 x}",1,"-1/4*((4*d)/x - (2*x*(-(b^3*d) + b^2*(a*e - c*d*x^2) + a*b*(3*c*d - a*f + c*e*x^2 + a*g*x^2) + 2*a*(a^2*g + c^2*d*x^2 - a*c*(e + f*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*(3*b^3*c*d + b^2*(3*c*Sqrt[b^2 - 4*a*c]*d - a*c*e + a^2*g) + 2*a*c*(-5*c*Sqrt[b^2 - 4*a*c]*d + 6*a*c*e + a*Sqrt[b^2 - 4*a*c]*f + 2*a^2*g) - a*b*(16*c^2*d + c*Sqrt[b^2 - 4*a*c]*e + 4*a*c*f + a*Sqrt[b^2 - 4*a*c]*g))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(-3*b^3*c*d + b^2*(3*c*Sqrt[b^2 - 4*a*c]*d + a*c*e - a^2*g) - 2*a*c*(5*c*Sqrt[b^2 - 4*a*c]*d + 6*a*c*e - a*Sqrt[b^2 - 4*a*c]*f + 2*a^2*g) + a*b*(16*c^2*d - c*Sqrt[b^2 - 4*a*c]*e + 4*a*c*f - a*Sqrt[b^2 - 4*a*c]*g))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/a^2","A",1
130,1,612,542,2.1528463,"\int \frac{d+e x^2+f x^4+g x^6}{x^4 \left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(d + e*x^2 + f*x^4 + g*x^6)/(x^4*(a + b*x^2 + c*x^4)^2),x]","\frac{\frac{6 x \left(a b \left(a^2 (-g)+a c \left(3 e+f x^2\right)-3 c^2 d x^2\right)+2 a^2 c \left(c \left(d+e x^2\right)-a \left(f+g x^2\right)\right)+b^3 \left(c d x^2-a e\right)+a b^2 \left(a f-c \left(4 d+e x^2\right)\right)+b^4 d\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{3 \sqrt{2} \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(-2 a^2 \left(-5 c e \sqrt{b^2-4 a c}+a g \sqrt{b^2-4 a c}+6 a c f-14 c^2 d\right)+a b \left(4 a^2 g-19 c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}+16 a c e\right)+a b^2 \left(-3 e \sqrt{b^2-4 a c}+a f-29 c d\right)+b^3 \left(5 d \sqrt{b^2-4 a c}-3 a e\right)+5 b^4 d\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{3 \sqrt{2} \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(2 a^2 \left(-5 c e \sqrt{b^2-4 a c}+a g \sqrt{b^2-4 a c}-6 a c f+14 c^2 d\right)+a b \left(4 a^2 g+19 c d \sqrt{b^2-4 a c}-a f \sqrt{b^2-4 a c}+16 a c e\right)+a b^2 \left(3 e \sqrt{b^2-4 a c}+a f-29 c d\right)-b^3 \left(5 d \sqrt{b^2-4 a c}+3 a e\right)+5 b^4 d\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{24 b d-12 a e}{x}-\frac{4 a d}{x^3}}{12 a^3}","\frac{2 b d-a e}{a^3 x}-\frac{d}{3 a^2 x^3}+\frac{x \left(a^2 \left(\frac{b^4 d}{a^2}-\frac{b^2 (b e+4 c d)}{a}-a (b g+2 c f)+b^2 f+3 b c e+2 c^2 d\right)+c x^2 \left(2 a^2 (c e-a g)-a b^2 e-a b (3 c d-a f)+b^3 d\right)\right)}{2 a^3 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(\frac{4 a^2 b (a g+4 c e)+4 a^2 c (7 c d-3 a f)-3 a b^3 e-a b^2 (29 c d-a f)+5 b^4 d}{\sqrt{b^2-4 a c}}+2 a^2 (5 c e-a g)-3 a b^2 e-a b (19 c d-a f)+5 b^3 d\right)}{2 \sqrt{2} a^3 \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(-\frac{4 a^2 b (a g+4 c e)+4 a^2 c (7 c d-3 a f)-3 a b^3 e-a b^2 (29 c d-a f)+5 b^4 d}{\sqrt{b^2-4 a c}}+2 a^2 (5 c e-a g)-3 a b^2 e-a b (19 c d-a f)+5 b^3 d\right)}{2 \sqrt{2} a^3 \left(b^2-4 a c\right) \sqrt{\sqrt{b^2-4 a c}+b}}",1,"((-4*a*d)/x^3 + (24*b*d - 12*a*e)/x + (6*x*(b^4*d + b^3*(-(a*e) + c*d*x^2) + a*b^2*(a*f - c*(4*d + e*x^2)) + a*b*(-(a^2*g) - 3*c^2*d*x^2 + a*c*(3*e + f*x^2)) + 2*a^2*c*(c*(d + e*x^2) - a*(f + g*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (3*Sqrt[2]*Sqrt[c]*(5*b^4*d + b^3*(5*Sqrt[b^2 - 4*a*c]*d - 3*a*e) + a*b^2*(-29*c*d - 3*Sqrt[b^2 - 4*a*c]*e + a*f) + a*b*(-19*c*Sqrt[b^2 - 4*a*c]*d + 16*a*c*e + a*Sqrt[b^2 - 4*a*c]*f + 4*a^2*g) - 2*a^2*(-14*c^2*d - 5*c*Sqrt[b^2 - 4*a*c]*e + 6*a*c*f + a*Sqrt[b^2 - 4*a*c]*g))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (3*Sqrt[2]*Sqrt[c]*(5*b^4*d - b^3*(5*Sqrt[b^2 - 4*a*c]*d + 3*a*e) + a*b^2*(-29*c*d + 3*Sqrt[b^2 - 4*a*c]*e + a*f) + a*b*(19*c*Sqrt[b^2 - 4*a*c]*d + 16*a*c*e - a*Sqrt[b^2 - 4*a*c]*f + 4*a^2*g) + 2*a^2*(14*c^2*d - 5*c*Sqrt[b^2 - 4*a*c]*e - 6*a*c*f + a*Sqrt[b^2 - 4*a*c]*g))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(12*a^3)","A",1
131,1,20,20,0.1410599,"\int x^2 \left(a+b x^2+c x^4\right)^p \left(3 a+b (5+2 p) x^2+c (7+4 p) x^4\right) \, dx","Integrate[x^2*(a + b*x^2 + c*x^4)^p*(3*a + b*(5 + 2*p)*x^2 + c*(7 + 4*p)*x^4),x]","x^3 \left(a+b x^2+c x^4\right)^{p+1}","x^3 \left(a+b x^2+c x^4\right)^{p+1}",1,"x^3*(a + b*x^2 + c*x^4)^(1 + p)","A",1
132,1,265,210,1.3786281,"\int \frac{x^5 \left(a+b x^2+c x^4\right)}{\sqrt{d-e x} \sqrt{d+e x}} \, dx","Integrate[(x^5*(a + b*x^2 + c*x^4))/(Sqrt[d - e*x]*Sqrt[d + e*x]),x]","-\frac{\frac{630 d^{9/2} \sqrt{d+e x} \sin ^{-1}\left(\frac{\sqrt{d-e x}}{\sqrt{2} \sqrt{d}}\right) \left(a e^4+b d^2 e^2+c d^4\right)}{\sqrt{\frac{e x}{d}+1}}-630 d^5 \tan ^{-1}\left(\frac{\sqrt{d-e x}}{\sqrt{d+e x}}\right) \left(a e^4+b d^2 e^2+c d^4\right)+\sqrt{d-e x} \sqrt{d+e x} \left(21 a e^4 \left(8 d^4+4 d^2 e^2 x^2+3 e^4 x^4\right)+9 b \left(16 d^6 e^2+8 d^4 e^4 x^2+6 d^2 e^6 x^4+5 e^8 x^6\right)+c \left(128 d^8+64 d^6 e^2 x^2+48 d^4 e^4 x^4+40 d^2 e^6 x^6+35 e^8 x^8\right)\right)}{315 e^{10}}","-\frac{(d-e x)^{5/2} (d+e x)^{5/2} \left(a e^4+3 b d^2 e^2+6 c d^4\right)}{5 e^{10}}+\frac{d^2 (d-e x)^{3/2} (d+e x)^{3/2} \left(2 a e^4+3 b d^2 e^2+4 c d^4\right)}{3 e^{10}}-\frac{d^4 \sqrt{d-e x} \sqrt{d+e x} \left(a e^4+b d^2 e^2+c d^4\right)}{e^{10}}+\frac{(d-e x)^{7/2} (d+e x)^{7/2} \left(b e^2+4 c d^2\right)}{7 e^{10}}-\frac{c (d-e x)^{9/2} (d+e x)^{9/2}}{9 e^{10}}",1,"-1/315*(Sqrt[d - e*x]*Sqrt[d + e*x]*(21*a*e^4*(8*d^4 + 4*d^2*e^2*x^2 + 3*e^4*x^4) + 9*b*(16*d^6*e^2 + 8*d^4*e^4*x^2 + 6*d^2*e^6*x^4 + 5*e^8*x^6) + c*(128*d^8 + 64*d^6*e^2*x^2 + 48*d^4*e^4*x^4 + 40*d^2*e^6*x^6 + 35*e^8*x^8)) + (630*d^(9/2)*(c*d^4 + b*d^2*e^2 + a*e^4)*Sqrt[d + e*x]*ArcSin[Sqrt[d - e*x]/(Sqrt[2]*Sqrt[d])])/Sqrt[1 + (e*x)/d] - 630*d^5*(c*d^4 + b*d^2*e^2 + a*e^4)*ArcTan[Sqrt[d - e*x]/Sqrt[d + e*x]])/e^10","C",1
133,1,232,159,1.0693539,"\int \frac{x^3 \left(a+b x^2+c x^4\right)}{\sqrt{d-e x} \sqrt{d+e x}} \, dx","Integrate[(x^3*(a + b*x^2 + c*x^4))/(Sqrt[d - e*x]*Sqrt[d + e*x]),x]","-\frac{\frac{210 d^{5/2} \sqrt{d+e x} \sin ^{-1}\left(\frac{\sqrt{d-e x}}{\sqrt{2} \sqrt{d}}\right) \left(a e^4+b d^2 e^2+c d^4\right)}{\sqrt{\frac{e x}{d}+1}}+\sqrt{d-e x} \sqrt{d+e x} \left(35 a e^4 \left(2 d^2+e^2 x^2\right)+7 b \left(8 d^4 e^2+4 d^2 e^4 x^2+3 e^6 x^4\right)+3 c \left(16 d^6+8 d^4 e^2 x^2+6 d^2 e^4 x^4+5 e^6 x^6\right)\right)-210 d^3 \tan ^{-1}\left(\frac{\sqrt{d-e x}}{\sqrt{d+e x}}\right) \left(a e^4+b d^2 e^2+c d^4\right)}{105 e^8}","\frac{(d-e x)^{3/2} (d+e x)^{3/2} \left(a e^4+2 b d^2 e^2+3 c d^4\right)}{3 e^8}-\frac{d^2 \sqrt{d-e x} \sqrt{d+e x} \left(a e^4+b d^2 e^2+c d^4\right)}{e^8}-\frac{(d-e x)^{5/2} (d+e x)^{5/2} \left(b e^2+3 c d^2\right)}{5 e^8}+\frac{c (d-e x)^{7/2} (d+e x)^{7/2}}{7 e^8}",1,"-1/105*(Sqrt[d - e*x]*Sqrt[d + e*x]*(35*a*e^4*(2*d^2 + e^2*x^2) + 7*b*(8*d^4*e^2 + 4*d^2*e^4*x^2 + 3*e^6*x^4) + 3*c*(16*d^6 + 8*d^4*e^2*x^2 + 6*d^2*e^4*x^4 + 5*e^6*x^6)) + (210*d^(5/2)*(c*d^4 + b*d^2*e^2 + a*e^4)*Sqrt[d + e*x]*ArcSin[Sqrt[d - e*x]/(Sqrt[2]*Sqrt[d])])/Sqrt[1 + (e*x)/d] - 210*d^3*(c*d^4 + b*d^2*e^2 + a*e^4)*ArcTan[Sqrt[d - e*x]/Sqrt[d + e*x]])/e^8","C",1
134,1,194,109,0.6908735,"\int \frac{x \left(a+b x^2+c x^4\right)}{\sqrt{d-e x} \sqrt{d+e x}} \, dx","Integrate[(x*(a + b*x^2 + c*x^4))/(Sqrt[d - e*x]*Sqrt[d + e*x]),x]","-\frac{\sqrt{d-e x} \sqrt{d+e x} \left(5 \left(3 a e^4+2 b d^2 e^2+b e^4 x^2\right)+c \left(8 d^4+4 d^2 e^2 x^2+3 e^4 x^4\right)\right)+\frac{30 \sqrt{d} \sqrt{d+e x} \sin ^{-1}\left(\frac{\sqrt{d-e x}}{\sqrt{2} \sqrt{d}}\right) \left(a e^4+b d^2 e^2+c d^4\right)}{\sqrt{\frac{e x}{d}+1}}-30 d \tan ^{-1}\left(\frac{\sqrt{d-e x}}{\sqrt{d+e x}}\right) \left(a e^4+b d^2 e^2+c d^4\right)}{15 e^6}","-\frac{\sqrt{d-e x} \sqrt{d+e x} \left(a e^4+b d^2 e^2+c d^4\right)}{e^6}+\frac{(d-e x)^{3/2} (d+e x)^{3/2} \left(b e^2+2 c d^2\right)}{3 e^6}-\frac{c (d-e x)^{5/2} (d+e x)^{5/2}}{5 e^6}",1,"-1/15*(Sqrt[d - e*x]*Sqrt[d + e*x]*(5*(2*b*d^2*e^2 + 3*a*e^4 + b*e^4*x^2) + c*(8*d^4 + 4*d^2*e^2*x^2 + 3*e^4*x^4)) + (30*Sqrt[d]*(c*d^4 + b*d^2*e^2 + a*e^4)*Sqrt[d + e*x]*ArcSin[Sqrt[d - e*x]/(Sqrt[2]*Sqrt[d])])/Sqrt[1 + (e*x)/d] - 30*d*(c*d^4 + b*d^2*e^2 + a*e^4)*ArcTan[Sqrt[d - e*x]/Sqrt[d + e*x]])/e^6","C",1
135,1,217,93,0.880089,"\int \frac{a+b x^2+c x^4}{x \sqrt{d-e x} \sqrt{d+e x}} \, dx","Integrate[(a + b*x^2 + c*x^4)/(x*Sqrt[d - e*x]*Sqrt[d + e*x]),x]","\frac{-\frac{3 a \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{d \sqrt{d-e x}}+\frac{\left(e^2 x^2-d^2\right) \left(3 b e^2+2 c d^2+c e^2 x^2\right)}{e^4 \sqrt{d-e x}}+\frac{6 d \sqrt{d+e x} \left(b e^2+c d^2\right) \tan ^{-1}\left(\frac{\sqrt{d-e x}}{\sqrt{d+e x}}\right)}{e^4}-\frac{6 d^{3/2} \sqrt{\frac{e x}{d}+1} \left(b e^2+c d^2\right) \sin ^{-1}\left(\frac{\sqrt{d-e x}}{\sqrt{2} \sqrt{d}}\right)}{e^4}}{3 \sqrt{d+e x}}","-\frac{a \tanh ^{-1}\left(\frac{\sqrt{d-e x} \sqrt{d+e x}}{d}\right)}{d}-\frac{\sqrt{d-e x} \sqrt{d+e x} \left(b e^2+c d^2\right)}{e^4}+\frac{c (d-e x)^{3/2} (d+e x)^{3/2}}{3 e^4}",1,"(((-d^2 + e^2*x^2)*(2*c*d^2 + 3*b*e^2 + c*e^2*x^2))/(e^4*Sqrt[d - e*x]) - (6*d^(3/2)*(c*d^2 + b*e^2)*Sqrt[1 + (e*x)/d]*ArcSin[Sqrt[d - e*x]/(Sqrt[2]*Sqrt[d])])/e^4 + (6*d*(c*d^2 + b*e^2)*Sqrt[d + e*x]*ArcTan[Sqrt[d - e*x]/Sqrt[d + e*x]])/e^4 - (3*a*Sqrt[d^2 - e^2*x^2]*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(d*Sqrt[d - e*x]))/(3*Sqrt[d + e*x])","B",1
136,1,233,99,0.2131771,"\int \frac{a+b x^2+c x^4}{x^3 \sqrt{d-e x} \sqrt{d+e x}} \, dx","Integrate[(a + b*x^2 + c*x^4)/(x^3*Sqrt[d - e*x]*Sqrt[d + e*x]),x]","\frac{-e^2 x^2 \sqrt{d^2-e^2 x^2} \left(a e^2+2 b d^2\right) \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)-a d^3 e^2+a d e^4 x^2-4 c d^{9/2} x^2 \sqrt{d-e x} \sqrt{\frac{e x}{d}+1} \sin ^{-1}\left(\frac{\sqrt{d-e x}}{\sqrt{2} \sqrt{d}}\right)-2 c d^5 x^2+4 c d^4 x^2 \sqrt{d-e x} \sqrt{d+e x} \tan ^{-1}\left(\frac{\sqrt{d-e x}}{\sqrt{d+e x}}\right)+2 c d^3 e^2 x^4}{2 d^3 e^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}","-\frac{\left(a e^2+2 b d^2\right) \tanh ^{-1}\left(\frac{\sqrt{d-e x} \sqrt{d+e x}}{d}\right)}{2 d^3}-\frac{a \sqrt{d-e x} \sqrt{d+e x}}{2 d^2 x^2}-\frac{c \sqrt{d-e x} \sqrt{d+e x}}{e^2}",1,"(-(a*d^3*e^2) - 2*c*d^5*x^2 + a*d*e^4*x^2 + 2*c*d^3*e^2*x^4 - 4*c*d^(9/2)*x^2*Sqrt[d - e*x]*Sqrt[1 + (e*x)/d]*ArcSin[Sqrt[d - e*x]/(Sqrt[2]*Sqrt[d])] + 4*c*d^4*x^2*Sqrt[d - e*x]*Sqrt[d + e*x]*ArcTan[Sqrt[d - e*x]/Sqrt[d + e*x]] - e^2*(2*b*d^2 + a*e^2)*x^2*Sqrt[d^2 - e^2*x^2]*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(2*d^3*e^2*x^2*Sqrt[d - e*x]*Sqrt[d + e*x])","B",1
137,1,134,126,0.1605987,"\int \frac{a+b x^2+c x^4}{x^5 \sqrt{d-e x} \sqrt{d+e x}} \, dx","Integrate[(a + b*x^2 + c*x^4)/(x^5*Sqrt[d - e*x]*Sqrt[d + e*x]),x]","\frac{-\left(x^4 \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right) \left(3 a e^4+4 b d^2 e^2+8 c d^4\right)\right)-d \left(d^2-e^2 x^2\right) \left(2 a d^2+3 a e^2 x^2+4 b d^2 x^2\right)}{8 d^5 x^4 \sqrt{d-e x} \sqrt{d+e x}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{d-e x} \sqrt{d+e x}}{d}\right) \left(3 a e^4+4 b d^2 e^2+8 c d^4\right)}{8 d^5}-\frac{\sqrt{d-e x} \sqrt{d+e x} \left(3 a e^2+4 b d^2\right)}{8 d^4 x^2}-\frac{a \sqrt{d-e x} \sqrt{d+e x}}{4 d^2 x^4}",1,"(-(d*(d^2 - e^2*x^2)*(2*a*d^2 + 4*b*d^2*x^2 + 3*a*e^2*x^2)) - (8*c*d^4 + 4*b*d^2*e^2 + 3*a*e^4)*x^4*Sqrt[d^2 - e^2*x^2]*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(8*d^5*x^4*Sqrt[d - e*x]*Sqrt[d + e*x])","A",1
138,1,173,212,0.1923912,"\int \frac{a+b x^2+c x^4}{x^7 \sqrt{d-e x} \sqrt{d+e x}} \, dx","Integrate[(a + b*x^2 + c*x^4)/(x^7*Sqrt[d - e*x]*Sqrt[d + e*x]),x]","\frac{-3 e^2 x^6 \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right) \left(5 a e^4+6 b d^2 e^2+8 c d^4\right)-d \left(d^2-e^2 x^2\right) \left(a \left(8 d^4+10 d^2 e^2 x^2+15 e^4 x^4\right)+6 \left(2 b d^4 x^2+3 b d^2 e^2 x^4+4 c d^4 x^4\right)\right)}{48 d^7 x^6 \sqrt{d-e x} \sqrt{d+e x}}","-\frac{e^2 \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right) \left(5 a e^4+6 b d^2 e^2+8 c d^4\right)}{16 d^7 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\sqrt{d-e x} \sqrt{d+e x} \left(5 a e^4+6 b d^2 e^2+8 c d^4\right)}{16 d^6 x^2}-\frac{\sqrt{d-e x} \sqrt{d+e x} \left(5 a e^2+6 b d^2\right)}{24 d^4 x^4}-\frac{a \sqrt{d-e x} \sqrt{d+e x}}{6 d^2 x^6}",1,"(-(d*(d^2 - e^2*x^2)*(6*(2*b*d^4*x^2 + 4*c*d^4*x^4 + 3*b*d^2*e^2*x^4) + a*(8*d^4 + 10*d^2*e^2*x^2 + 15*e^4*x^4))) - 3*e^2*(8*c*d^4 + 6*b*d^2*e^2 + 5*a*e^4)*x^6*Sqrt[d^2 - e^2*x^2]*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(48*d^7*x^6*Sqrt[d - e*x]*Sqrt[d + e*x])","A",1
139,1,202,216,0.7906896,"\int \frac{x^2 \left(a+b x^2+c x^4\right)}{\sqrt{d-e x} \sqrt{d+e x}} \, dx","Integrate[(x^2*(a + b*x^2 + c*x^4))/(Sqrt[d - e*x]*Sqrt[d + e*x]),x]","-\frac{e x \sqrt{d-e x} \sqrt{d+e x} \left(6 \left(4 a e^4+3 b d^2 e^2+2 b e^4 x^2\right)+c \left(15 d^4+10 d^2 e^2 x^2+8 e^4 x^4\right)\right)+96 d^2 \tan ^{-1}\left(\frac{\sqrt{d-e x}}{\sqrt{d+e x}}\right) \left(a e^4+b d^2 e^2+c d^4\right)-\frac{6 d^{3/2} \sqrt{d+e x} \sin ^{-1}\left(\frac{\sqrt{d-e x}}{\sqrt{2} \sqrt{d}}\right) \left(8 a e^4+10 b d^2 e^2+11 c d^4\right)}{\sqrt{\frac{e x}{d}+1}}}{48 e^7}","\frac{d^2 \sqrt{d^2-e^2 x^2} \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right) \left(8 a e^4+6 b d^2 e^2+5 c d^4\right)}{16 e^7 \sqrt{d-e x} \sqrt{d+e x}}-\frac{x \sqrt{d-e x} \sqrt{d+e x} \left(8 a e^4+6 b d^2 e^2+5 c d^4\right)}{16 e^6}-\frac{x^3 \sqrt{d-e x} \sqrt{d+e x} \left(6 b e^2+5 c d^2\right)}{24 e^4}+\frac{c x^5 (e x-d) \sqrt{d+e x}}{6 e^2 \sqrt{d-e x}}",1,"-1/48*(e*x*Sqrt[d - e*x]*Sqrt[d + e*x]*(6*(3*b*d^2*e^2 + 4*a*e^4 + 2*b*e^4*x^2) + c*(15*d^4 + 10*d^2*e^2*x^2 + 8*e^4*x^4)) - (6*d^(3/2)*(11*c*d^4 + 10*b*d^2*e^2 + 8*a*e^4)*Sqrt[d + e*x]*ArcSin[Sqrt[d - e*x]/(Sqrt[2]*Sqrt[d])])/Sqrt[1 + (e*x)/d] + 96*d^2*(c*d^4 + b*d^2*e^2 + a*e^4)*ArcTan[Sqrt[d - e*x]/Sqrt[d + e*x]])/e^7","A",1
140,1,157,128,0.5623835,"\int \frac{a+b x^2+c x^4}{\sqrt{d-e x} \sqrt{d+e x}} \, dx","Integrate[(a + b*x^2 + c*x^4)/(Sqrt[d - e*x]*Sqrt[d + e*x]),x]","-\frac{16 \tan ^{-1}\left(\frac{\sqrt{d-e x}}{\sqrt{d+e x}}\right) \left(a e^4+b d^2 e^2+c d^4\right)+e x \sqrt{d-e x} \sqrt{d+e x} \left(4 b e^2+3 c d^2+2 c e^2 x^2\right)-\frac{2 d^{5/2} \sqrt{\frac{e x}{d}+1} \left(4 b e^2+5 c d^2\right) \sin ^{-1}\left(\frac{\sqrt{d-e x}}{\sqrt{2} \sqrt{d}}\right)}{\sqrt{d+e x}}}{8 e^5}","-\frac{\tan ^{-1}\left(\frac{\sqrt{d-e x}}{\sqrt{d+e x}}\right) \left(8 a e^4+4 b d^2 e^2+3 c d^4\right)}{4 e^5}-\frac{x \sqrt{d-e x} \sqrt{d+e x} \left(4 b e^2+3 c d^2\right)}{8 e^4}+\frac{c x^3 (e x-d) \sqrt{d+e x}}{4 e^2 \sqrt{d-e x}}",1,"-1/8*(e*x*Sqrt[d - e*x]*Sqrt[d + e*x]*(3*c*d^2 + 4*b*e^2 + 2*c*e^2*x^2) - (2*d^(5/2)*(5*c*d^2 + 4*b*e^2)*Sqrt[1 + (e*x)/d]*ArcSin[Sqrt[d - e*x]/(Sqrt[2]*Sqrt[d])])/Sqrt[d + e*x] + 16*(c*d^4 + b*d^2*e^2 + a*e^4)*ArcTan[Sqrt[d - e*x]/Sqrt[d + e*x]])/e^5","A",1
141,1,135,102,0.5550054,"\int \frac{a+b x^2+c x^4}{x^2 \sqrt{d-e x} \sqrt{d+e x}} \, dx","Integrate[(a + b*x^2 + c*x^4)/(x^2*Sqrt[d - e*x]*Sqrt[d + e*x]),x]","-\frac{\frac{e \sqrt{d-e x} \sqrt{d+e x} \left(2 a e^2+c d^2 x^2\right)}{d^2 x}+4 \left(b e^2+c d^2\right) \tan ^{-1}\left(\frac{\sqrt{d-e x}}{\sqrt{d+e x}}\right)-\frac{2 c d^{5/2} \sqrt{\frac{e x}{d}+1} \sin ^{-1}\left(\frac{\sqrt{d-e x}}{\sqrt{2} \sqrt{d}}\right)}{\sqrt{d+e x}}}{2 e^3}","-\frac{a \sqrt{d-e x} \sqrt{d+e x}}{d^2 x}-\frac{\left(2 b e^2+c d^2\right) \tan ^{-1}\left(\frac{\sqrt{d-e x}}{\sqrt{d+e x}}\right)}{e^3}+\frac{c x (e x-d) \sqrt{d+e x}}{2 e^2 \sqrt{d-e x}}",1,"-1/2*((e*Sqrt[d - e*x]*Sqrt[d + e*x]*(2*a*e^2 + c*d^2*x^2))/(d^2*x) - (2*c*d^(5/2)*Sqrt[1 + (e*x)/d]*ArcSin[Sqrt[d - e*x]/(Sqrt[2]*Sqrt[d])])/Sqrt[d + e*x] + 4*(c*d^2 + b*e^2)*ArcTan[Sqrt[d - e*x]/Sqrt[d + e*x]])/e^3","A",1
142,1,81,157,0.124023,"\int \frac{a+b x^2+c x^4}{x^4 \sqrt{d-e x} \sqrt{d+e x}} \, dx","Integrate[(a + b*x^2 + c*x^4)/(x^4*Sqrt[d - e*x]*Sqrt[d + e*x]),x]","-\frac{\sqrt{d-e x} \sqrt{d+e x} \left(a \left(d^2+2 e^2 x^2\right)+3 b d^2 x^2\right)}{3 d^4 x^3}-\frac{2 c \tan ^{-1}\left(\frac{\sqrt{d-e x}}{\sqrt{d+e x}}\right)}{e}","-\frac{\left(d^2-e^2 x^2\right) \left(2 a e^2+3 b d^2\right)}{3 d^4 x \sqrt{d-e x} \sqrt{d+e x}}-\frac{a \left(d^2-e^2 x^2\right)}{3 d^2 x^3 \sqrt{d-e x} \sqrt{d+e x}}+\frac{c \sqrt{d^2-e^2 x^2} \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e \sqrt{d-e x} \sqrt{d+e x}}",1,"-1/3*(Sqrt[d - e*x]*Sqrt[d + e*x]*(3*b*d^2*x^2 + a*(d^2 + 2*e^2*x^2)))/(d^4*x^3) - (2*c*ArcTan[Sqrt[d - e*x]/Sqrt[d + e*x]])/e","A",1
143,1,87,160,0.1183598,"\int \frac{a+b x^2+c x^4}{x^6 \sqrt{d-e x} \sqrt{d+e x}} \, dx","Integrate[(a + b*x^2 + c*x^4)/(x^6*Sqrt[d - e*x]*Sqrt[d + e*x]),x]","-\frac{\sqrt{d-e x} \sqrt{d+e x} \left(a \left(3 d^4+4 d^2 e^2 x^2+8 e^4 x^4\right)+5 b d^2 x^2 \left(d^2+2 e^2 x^2\right)+15 c d^4 x^4\right)}{15 d^6 x^5}","-\frac{\left(d^2-e^2 x^2\right) \left(8 a e^4+10 b d^2 e^2+15 c d^4\right)}{15 d^6 x \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left(d^2-e^2 x^2\right) \left(4 a e^2+5 b d^2\right)}{15 d^4 x^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{a \left(d^2-e^2 x^2\right)}{5 d^2 x^5 \sqrt{d-e x} \sqrt{d+e x}}",1,"-1/15*(Sqrt[d - e*x]*Sqrt[d + e*x]*(15*c*d^4*x^4 + 5*b*d^2*x^2*(d^2 + 2*e^2*x^2) + a*(3*d^4 + 4*d^2*e^2*x^2 + 8*e^4*x^4)))/(d^6*x^5)","A",1
144,1,124,226,0.1421111,"\int \frac{a+b x^2+c x^4}{x^8 \sqrt{d-e x} \sqrt{d+e x}} \, dx","Integrate[(a + b*x^2 + c*x^4)/(x^8*Sqrt[d - e*x]*Sqrt[d + e*x]),x]","-\frac{\sqrt{d-e x} \sqrt{d+e x} \left(3 a \left(5 d^6+6 d^4 e^2 x^2+8 d^2 e^4 x^4+16 e^6 x^6\right)+7 b \left(3 d^6 x^2+4 d^4 e^2 x^4+8 d^2 e^4 x^6\right)+35 c d^4 x^4 \left(d^2+2 e^2 x^2\right)\right)}{105 d^8 x^7}","-\frac{2 e^2 \left(d^2-e^2 x^2\right) \left(24 a e^4+28 b d^2 e^2+35 c d^4\right)}{105 d^8 x \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left(d^2-e^2 x^2\right) \left(24 a e^4+28 b d^2 e^2+35 c d^4\right)}{105 d^6 x^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left(d^2-e^2 x^2\right) \left(6 a e^2+7 b d^2\right)}{35 d^4 x^5 \sqrt{d-e x} \sqrt{d+e x}}-\frac{a \left(d^2-e^2 x^2\right)}{7 d^2 x^7 \sqrt{d-e x} \sqrt{d+e x}}",1,"-1/105*(Sqrt[d - e*x]*Sqrt[d + e*x]*(35*c*d^4*x^4*(d^2 + 2*e^2*x^2) + 7*b*(3*d^6*x^2 + 4*d^4*e^2*x^4 + 8*d^2*e^4*x^6) + 3*a*(5*d^6 + 6*d^4*e^2*x^2 + 8*d^2*e^4*x^4 + 16*e^6*x^6)))/(d^8*x^7)","A",1
145,1,158,292,0.1757747,"\int \frac{a+b x^2+c x^4}{x^{10} \sqrt{d-e x} \sqrt{d+e x}} \, dx","Integrate[(a + b*x^2 + c*x^4)/(x^10*Sqrt[d - e*x]*Sqrt[d + e*x]),x]","-\frac{\sqrt{d-e x} \sqrt{d+e x} \left(a \left(35 d^8+40 d^6 e^2 x^2+48 d^4 e^4 x^4+64 d^2 e^6 x^6+128 e^8 x^8\right)+9 b \left(5 d^8 x^2+6 d^6 e^2 x^4+8 d^4 e^4 x^6+16 d^2 e^6 x^8\right)+21 c d^4 x^4 \left(3 d^4+4 d^2 e^2 x^2+8 e^4 x^4\right)\right)}{315 d^{10} x^9}","-\frac{8 e^4 \left(d^2-e^2 x^2\right) \left(16 a e^4+18 b d^2 e^2+21 c d^4\right)}{315 d^{10} x \sqrt{d-e x} \sqrt{d+e x}}-\frac{4 e^2 \left(d^2-e^2 x^2\right) \left(16 a e^4+18 b d^2 e^2+21 c d^4\right)}{315 d^8 x^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left(d^2-e^2 x^2\right) \left(16 a e^4+18 b d^2 e^2+21 c d^4\right)}{105 d^6 x^5 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left(d^2-e^2 x^2\right) \left(8 a e^2+9 b d^2\right)}{63 d^4 x^7 \sqrt{d-e x} \sqrt{d+e x}}-\frac{a \left(d^2-e^2 x^2\right)}{9 d^2 x^9 \sqrt{d-e x} \sqrt{d+e x}}",1,"-1/315*(Sqrt[d - e*x]*Sqrt[d + e*x]*(21*c*d^4*x^4*(3*d^4 + 4*d^2*e^2*x^2 + 8*e^4*x^4) + 9*b*(5*d^8*x^2 + 6*d^6*e^2*x^4 + 8*d^4*e^4*x^6 + 16*d^2*e^6*x^8) + a*(35*d^8 + 40*d^6*e^2*x^2 + 48*d^4*e^4*x^4 + 64*d^2*e^6*x^6 + 128*e^8*x^8)))/(d^10*x^9)","A",1