1,1,33,33,0.0068698,"\int x^2 \left(A+B x^2\right) \left(b x^2+c x^4\right) \, dx","Integrate[x^2*(A + B*x^2)*(b*x^2 + c*x^4),x]","\frac{1}{7} x^7 (A c+b B)+\frac{1}{5} A b x^5+\frac{1}{9} B c x^9","\frac{1}{7} x^7 (A c+b B)+\frac{1}{5} A b x^5+\frac{1}{9} B c x^9",1,"(A*b*x^5)/5 + ((b*B + A*c)*x^7)/7 + (B*c*x^9)/9","A",1
2,1,33,33,0.010227,"\int x \left(A+B x^2\right) \left(b x^2+c x^4\right) \, dx","Integrate[x*(A + B*x^2)*(b*x^2 + c*x^4),x]","\frac{1}{6} x^6 (A c+b B)+\frac{1}{4} A b x^4+\frac{1}{8} B c x^8","\frac{1}{6} x^6 (A c+b B)+\frac{1}{4} A b x^4+\frac{1}{8} B c x^8",1,"(A*b*x^4)/4 + ((b*B + A*c)*x^6)/6 + (B*c*x^8)/8","A",1
3,1,33,33,0.0058404,"\int \left(A+B x^2\right) \left(b x^2+c x^4\right) \, dx","Integrate[(A + B*x^2)*(b*x^2 + c*x^4),x]","\frac{1}{5} x^5 (A c+b B)+\frac{1}{3} A b x^3+\frac{1}{7} B c x^7","\frac{1}{5} x^5 (A c+b B)+\frac{1}{3} A b x^3+\frac{1}{7} B c x^7",1,"(A*b*x^3)/3 + ((b*B + A*c)*x^5)/5 + (B*c*x^7)/7","A",1
4,1,33,33,0.0081648,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)}{x} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4))/x,x]","\frac{1}{4} x^4 (A c+b B)+\frac{1}{2} A b x^2+\frac{1}{6} B c x^6","\frac{1}{4} x^4 (A c+b B)+\frac{1}{2} A b x^2+\frac{1}{6} B c x^6",1,"(A*b*x^2)/2 + ((b*B + A*c)*x^4)/4 + (B*c*x^6)/6","A",1
5,1,28,28,0.0190273,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)}{x^2} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4))/x^2,x]","\frac{1}{3} x^3 (A c+b B)+A b x+\frac{1}{5} B c x^5","\frac{1}{3} x^3 (A c+b B)+A b x+\frac{1}{5} B c x^5",1,"A*b*x + ((b*B + A*c)*x^3)/3 + (B*c*x^5)/5","A",1
6,1,29,29,0.0094323,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)}{x^3} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4))/x^3,x]","\frac{1}{2} x^2 (A c+b B)+A b \log (x)+\frac{1}{4} B c x^4","\frac{1}{2} x^2 (A c+b B)+A b \log (x)+\frac{1}{4} B c x^4",1,"((b*B + A*c)*x^2)/2 + (B*c*x^4)/4 + A*b*Log[x]","A",1
7,1,26,26,0.0094189,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)}{x^4} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4))/x^4,x]","x (A c+b B)-\frac{A b}{x}+\frac{1}{3} B c x^3","x (A c+b B)-\frac{A b}{x}+\frac{1}{3} B c x^3",1,"-((A*b)/x) + (b*B + A*c)*x + (B*c*x^3)/3","A",1
8,1,29,29,0.014931,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)}{x^5} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4))/x^5,x]","\log (x) (A c+b B)-\frac{A b}{2 x^2}+\frac{1}{2} B c x^2","\log (x) (A c+b B)-\frac{A b}{2 x^2}+\frac{1}{2} B c x^2",1,"-1/2*(A*b)/x^2 + (B*c*x^2)/2 + (b*B + A*c)*Log[x]","A",1
9,1,27,26,0.0341037,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)}{x^6} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4))/x^6,x]","\frac{-A c-b B}{x}-\frac{A b}{3 x^3}+B c x","-\frac{A c+b B}{x}-\frac{A b}{3 x^3}+B c x",1,"-1/3*(A*b)/x^3 + (-(b*B) - A*c)/x + B*c*x","A",1
10,1,31,29,0.0266658,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)}{x^7} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4))/x^7,x]","\frac{-A c-b B}{2 x^2}-\frac{A b}{4 x^4}+B c \log (x)","-\frac{A c+b B}{2 x^2}-\frac{A b}{4 x^4}+B c \log (x)",1,"-1/4*(A*b)/x^4 + (-(b*B) - A*c)/(2*x^2) + B*c*Log[x]","A",1
11,1,33,31,0.0118246,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)}{x^8} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4))/x^8,x]","\frac{-A c-b B}{3 x^3}-\frac{A b}{5 x^5}-\frac{B c}{x}","-\frac{A c+b B}{3 x^3}-\frac{A b}{5 x^5}-\frac{B c}{x}",1,"-1/5*(A*b)/x^5 + (-(b*B) - A*c)/(3*x^3) - (B*c)/x","A",1
12,1,55,55,0.0103758,"\int \left(A+B x^2\right) \left(b x^2+c x^4\right)^2 \, dx","Integrate[(A + B*x^2)*(b*x^2 + c*x^4)^2,x]","\frac{1}{5} A b^2 x^5+\frac{1}{9} c x^9 (A c+2 b B)+\frac{1}{7} b x^7 (2 A c+b B)+\frac{1}{11} B c^2 x^{11}","\frac{1}{5} A b^2 x^5+\frac{1}{9} c x^9 (A c+2 b B)+\frac{1}{7} b x^7 (2 A c+b B)+\frac{1}{11} B c^2 x^{11}",1,"(A*b^2*x^5)/5 + (b*(b*B + 2*A*c)*x^7)/7 + (c*(2*b*B + A*c)*x^9)/9 + (B*c^2*x^11)/11","A",1
13,1,55,55,0.0088701,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^2}{x} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^2)/x,x]","\frac{1}{4} A b^2 x^4+\frac{1}{8} c x^8 (A c+2 b B)+\frac{1}{6} b x^6 (2 A c+b B)+\frac{1}{10} B c^2 x^{10}","\frac{1}{4} A b^2 x^4+\frac{1}{8} c x^8 (A c+2 b B)+\frac{1}{6} b x^6 (2 A c+b B)+\frac{1}{10} B c^2 x^{10}",1,"(A*b^2*x^4)/4 + (b*(b*B + 2*A*c)*x^6)/6 + (c*(2*b*B + A*c)*x^8)/8 + (B*c^2*x^10)/10","A",1
14,1,55,55,0.0094181,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^2}{x^2} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^2)/x^2,x]","\frac{1}{3} A b^2 x^3+\frac{1}{7} c x^7 (A c+2 b B)+\frac{1}{5} b x^5 (2 A c+b B)+\frac{1}{9} B c^2 x^9","\frac{1}{3} A b^2 x^3+\frac{1}{7} c x^7 (A c+2 b B)+\frac{1}{5} b x^5 (2 A c+b B)+\frac{1}{9} B c^2 x^9",1,"(A*b^2*x^3)/3 + (b*(b*B + 2*A*c)*x^5)/5 + (c*(2*b*B + A*c)*x^7)/7 + (B*c^2*x^9)/9","A",1
15,1,51,42,0.0161431,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^2}{x^3} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^2)/x^3,x]","\frac{1}{24} x^2 \left(12 A b^2+4 c x^4 (A c+2 b B)+6 b x^2 (2 A c+b B)+3 B c^2 x^6\right)","\frac{B \left(b+c x^2\right)^4}{8 c^2}-\frac{\left(b+c x^2\right)^3 (b B-A c)}{6 c^2}",1,"(x^2*(12*A*b^2 + 6*b*(b*B + 2*A*c)*x^2 + 4*c*(2*b*B + A*c)*x^4 + 3*B*c^2*x^6))/24","A",1
16,1,50,50,0.0094135,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^2}{x^4} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^2)/x^4,x]","A b^2 x+\frac{1}{5} c x^5 (A c+2 b B)+\frac{1}{3} b x^3 (2 A c+b B)+\frac{1}{7} B c^2 x^7","A b^2 x+\frac{1}{5} c x^5 (A c+2 b B)+\frac{1}{3} b x^3 (2 A c+b B)+\frac{1}{7} B c^2 x^7",1,"A*b^2*x + (b*(b*B + 2*A*c)*x^3)/3 + (c*(2*b*B + A*c)*x^5)/5 + (B*c^2*x^7)/7","A",1
17,1,51,43,0.0171974,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^2}{x^5} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^2)/x^5,x]","A b^2 \log (x)+\frac{1}{4} c x^4 (A c+2 b B)+\frac{1}{2} b x^2 (2 A c+b B)+\frac{1}{6} B c^2 x^6","A b^2 \log (x)+A b c x^2+\frac{1}{4} A c^2 x^4+\frac{B \left(b+c x^2\right)^3}{6 c}",1,"(b*(b*B + 2*A*c)*x^2)/2 + (c*(2*b*B + A*c)*x^4)/4 + (B*c^2*x^6)/6 + A*b^2*Log[x]","A",1
18,1,48,48,0.035475,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^2}{x^6} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^2)/x^6,x]","-\frac{A b^2}{x}+\frac{1}{3} c x^3 (A c+2 b B)+b x (2 A c+b B)+\frac{1}{5} B c^2 x^5","-\frac{A b^2}{x}+\frac{1}{3} c x^3 (A c+2 b B)+b x (2 A c+b B)+\frac{1}{5} B c^2 x^5",1,"-((A*b^2)/x) + b*(b*B + 2*A*c)*x + (c*(2*b*B + A*c)*x^3)/3 + (B*c^2*x^5)/5","A",1
19,1,49,51,0.0303624,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^2}{x^7} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^2)/x^7,x]","\frac{1}{4} \left(-\frac{2 A b^2}{x^2}+2 c x^2 (A c+2 b B)+4 b \log (x) (2 A c+b B)+B c^2 x^4\right)","-\frac{A b^2}{2 x^2}+\frac{1}{2} c x^2 (A c+2 b B)+b \log (x) (2 A c+b B)+\frac{1}{4} B c^2 x^4",1,"((-2*A*b^2)/x^2 + 2*c*(2*b*B + A*c)*x^2 + B*c^2*x^4 + 4*b*(b*B + 2*A*c)*Log[x])/4","A",1
20,1,50,48,0.0211723,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^2}{x^8} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^2)/x^8,x]","\frac{b^2 (-B)-2 A b c}{x}-\frac{A b^2}{3 x^3}+c x (A c+2 b B)+\frac{1}{3} B c^2 x^3","-\frac{A b^2}{3 x^3}+c x (A c+2 b B)-\frac{b (2 A c+b B)}{x}+\frac{1}{3} B c^2 x^3",1,"-1/3*(A*b^2)/x^3 + (-(b^2*B) - 2*A*b*c)/x + c*(2*b*B + A*c)*x + (B*c^2*x^3)/3","A",1
21,1,50,51,0.0406035,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^2}{x^9} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^2)/x^9,x]","c \log (x) (A c+2 b B)-\frac{A b \left(b+4 c x^2\right)+2 B x^2 \left(b^2-c^2 x^4\right)}{4 x^4}","-\frac{A b^2}{4 x^4}-\frac{b (2 A c+b B)}{2 x^2}+c \log (x) (A c+2 b B)+\frac{1}{2} B c^2 x^2",1,"-1/4*(A*b*(b + 4*c*x^2) + 2*B*x^2*(b^2 - c^2*x^4))/x^4 + c*(2*b*B + A*c)*Log[x]","A",1
22,1,48,48,0.0273741,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^2}{x^{10}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^2)/x^10,x]","-\frac{A b^2}{5 x^5}-\frac{b (2 A c+b B)}{3 x^3}-\frac{c (A c+2 b B)}{x}+B c^2 x","-\frac{A b^2}{5 x^5}-\frac{b (2 A c+b B)}{3 x^3}-\frac{c (A c+2 b B)}{x}+B c^2 x",1,"-1/5*(A*b^2)/x^5 - (b*(b*B + 2*A*c))/(3*x^3) - (c*(2*b*B + A*c))/x + B*c^2*x","A",1
23,1,53,51,0.0470734,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^2}{x^{11}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^2)/x^11,x]","B c^2 \log (x)-\frac{2 A \left(b^2+3 b c x^2+3 c^2 x^4\right)+3 b B x^2 \left(b+4 c x^2\right)}{12 x^6}","-\frac{A b^2}{6 x^6}-\frac{b (2 A c+b B)}{4 x^4}-\frac{c (A c+2 b B)}{2 x^2}+B c^2 \log (x)",1,"-1/12*(3*b*B*x^2*(b + 4*c*x^2) + 2*A*(b^2 + 3*b*c*x^2 + 3*c^2*x^4))/x^6 + B*c^2*Log[x]","A",1
24,1,59,53,0.0211957,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^2}{x^{12}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^2)/x^12,x]","\frac{b^2 (-B)-2 A b c}{5 x^5}-\frac{A b^2}{7 x^7}+\frac{-A c^2-2 b B c}{3 x^3}-\frac{B c^2}{x}","-\frac{A b^2}{7 x^7}-\frac{b (2 A c+b B)}{5 x^5}-\frac{c (A c+2 b B)}{3 x^3}-\frac{B c^2}{x}",1,"-1/7*(A*b^2)/x^7 + (-(b^2*B) - 2*A*b*c)/(5*x^5) + (-2*b*B*c - A*c^2)/(3*x^3) - (B*c^2)/x","A",1
25,1,75,75,0.0259478,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^3}{x^2} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^2,x]","\frac{1}{5} A b^3 x^5+\frac{1}{7} b^2 x^7 (3 A c+b B)+\frac{1}{11} c^2 x^{11} (A c+3 b B)+\frac{1}{3} b c x^9 (A c+b B)+\frac{1}{13} B c^3 x^{13}","\frac{1}{5} A b^3 x^5+\frac{1}{7} b^2 x^7 (3 A c+b B)+\frac{1}{11} c^2 x^{11} (A c+3 b B)+\frac{1}{3} b c x^9 (A c+b B)+\frac{1}{13} B c^3 x^{13}",1,"(A*b^3*x^5)/5 + (b^2*(b*B + 3*A*c)*x^7)/7 + (b*c*(b*B + A*c)*x^9)/3 + (c^2*(3*b*B + A*c)*x^11)/11 + (B*c^3*x^13)/13","A",1
26,1,69,68,0.0307112,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^3}{x^3} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^3,x]","\frac{1}{120} x^4 \left(30 A b^3+20 b^2 x^2 (3 A c+b B)+12 c^2 x^6 (A c+3 b B)+45 b c x^4 (A c+b B)+10 B c^3 x^8\right)","-\frac{\left(b+c x^2\right)^5 (2 b B-A c)}{10 c^3}+\frac{b \left(b+c x^2\right)^4 (b B-A c)}{8 c^3}+\frac{B \left(b+c x^2\right)^6}{12 c^3}",1,"(x^4*(30*A*b^3 + 20*b^2*(b*B + 3*A*c)*x^2 + 45*b*c*(b*B + A*c)*x^4 + 12*c^2*(3*b*B + A*c)*x^6 + 10*B*c^3*x^8))/120","A",1
27,1,75,75,0.0144436,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^3}{x^4} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^4,x]","\frac{1}{3} A b^3 x^3+\frac{1}{5} b^2 x^5 (3 A c+b B)+\frac{1}{9} c^2 x^9 (A c+3 b B)+\frac{3}{7} b c x^7 (A c+b B)+\frac{1}{11} B c^3 x^{11}","\frac{1}{3} A b^3 x^3+\frac{1}{5} b^2 x^5 (3 A c+b B)+\frac{1}{9} c^2 x^9 (A c+3 b B)+\frac{3}{7} b c x^7 (A c+b B)+\frac{1}{11} B c^3 x^{11}",1,"(A*b^3*x^3)/3 + (b^2*(b*B + 3*A*c)*x^5)/5 + (3*b*c*(b*B + A*c)*x^7)/7 + (c^2*(3*b*B + A*c)*x^9)/9 + (B*c^3*x^11)/11","A",1
28,1,69,42,0.0243083,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^3}{x^5} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^5,x]","\frac{1}{40} x^2 \left(20 A b^3+10 b^2 x^2 (3 A c+b B)+5 c^2 x^6 (A c+3 b B)+20 b c x^4 (A c+b B)+4 B c^3 x^8\right)","\frac{B \left(b+c x^2\right)^5}{10 c^2}-\frac{\left(b+c x^2\right)^4 (b B-A c)}{8 c^2}",1,"(x^2*(20*A*b^3 + 10*b^2*(b*B + 3*A*c)*x^2 + 20*b*c*(b*B + A*c)*x^4 + 5*c^2*(3*b*B + A*c)*x^6 + 4*B*c^3*x^8))/40","A",1
29,1,70,70,0.0123507,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^3}{x^6} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^6,x]","A b^3 x+\frac{1}{3} b^2 x^3 (3 A c+b B)+\frac{1}{7} c^2 x^7 (A c+3 b B)+\frac{3}{5} b c x^5 (A c+b B)+\frac{1}{9} B c^3 x^9","A b^3 x+\frac{1}{3} b^2 x^3 (3 A c+b B)+\frac{1}{7} c^2 x^7 (A c+3 b B)+\frac{3}{5} b c x^5 (A c+b B)+\frac{1}{9} B c^3 x^9",1,"A*b^3*x + (b^2*(b*B + 3*A*c)*x^3)/3 + (3*b*c*(b*B + A*c)*x^5)/5 + (c^2*(3*b*B + A*c)*x^7)/7 + (B*c^3*x^9)/9","A",1
30,1,71,60,0.0223813,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^3}{x^7} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^7,x]","A b^3 \log (x)+\frac{1}{2} b^2 x^2 (3 A c+b B)+\frac{1}{6} c^2 x^6 (A c+3 b B)+\frac{3}{4} b c x^4 (A c+b B)+\frac{1}{8} B c^3 x^8","A b^3 \log (x)+\frac{3}{2} A b^2 c x^2+\frac{3}{4} A b c^2 x^4+\frac{1}{6} A c^3 x^6+\frac{B \left(b+c x^2\right)^4}{8 c}",1,"(b^2*(b*B + 3*A*c)*x^2)/2 + (3*b*c*(b*B + A*c)*x^4)/4 + (c^2*(3*b*B + A*c)*x^6)/6 + (B*c^3*x^8)/8 + A*b^3*Log[x]","A",1
31,1,65,65,0.0286035,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^3}{x^8} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^8,x]","-\frac{A b^3}{x}+b^2 x (3 A c+b B)+\frac{1}{5} c^2 x^5 (A c+3 b B)+b c x^3 (A c+b B)+\frac{1}{7} B c^3 x^7","-\frac{A b^3}{x}+b^2 x (3 A c+b B)+\frac{1}{5} c^2 x^5 (A c+3 b B)+b c x^3 (A c+b B)+\frac{1}{7} B c^3 x^7",1,"-((A*b^3)/x) + b^2*(b*B + 3*A*c)*x + b*c*(b*B + A*c)*x^3 + (c^2*(3*b*B + A*c)*x^5)/5 + (B*c^3*x^7)/7","A",1
32,1,73,71,0.0437569,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^3}{x^9} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^9,x]","-\frac{A b^3}{2 x^2}+\log (x) \left(3 A b^2 c+b^3 B\right)+\frac{1}{4} c^2 x^4 (A c+3 b B)+\frac{3}{2} b c x^2 (A c+b B)+\frac{1}{6} B c^3 x^6","-\frac{A b^3}{2 x^2}+b^2 \log (x) (3 A c+b B)+\frac{1}{4} c^2 x^4 (A c+3 b B)+\frac{3}{2} b c x^2 (A c+b B)+\frac{1}{6} B c^3 x^6",1,"-1/2*(A*b^3)/x^2 + (3*b*c*(b*B + A*c)*x^2)/2 + (c^2*(3*b*B + A*c)*x^4)/4 + (B*c^3*x^6)/6 + (b^3*B + 3*A*b^2*c)*Log[x]","A",1
33,1,71,69,0.0354539,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^3}{x^{10}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^10,x]","-\frac{A b^3}{3 x^3}+\frac{b^3 (-B)-3 A b^2 c}{x}+\frac{1}{3} c^2 x^3 (A c+3 b B)+3 b c x (A c+b B)+\frac{1}{5} B c^3 x^5","-\frac{A b^3}{3 x^3}-\frac{b^2 (3 A c+b B)}{x}+\frac{1}{3} c^2 x^3 (A c+3 b B)+3 b c x (A c+b B)+\frac{1}{5} B c^3 x^5",1,"-1/3*(A*b^3)/x^3 + (-(b^3*B) - 3*A*b^2*c)/x + 3*b*c*(b*B + A*c)*x + (c^2*(3*b*B + A*c)*x^3)/3 + (B*c^3*x^5)/5","A",1
34,1,73,72,0.0339685,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^3}{x^{11}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^11,x]","\frac{B x^2 \left(-2 b^3+6 b c^2 x^4+c^3 x^6\right)-A \left(b^3+6 b^2 c x^2-2 c^3 x^6\right)}{4 x^4}+3 b c \log (x) (A c+b B)","-\frac{A b^3}{4 x^4}-\frac{b^2 (3 A c+b B)}{2 x^2}+\frac{1}{2} c^2 x^2 (A c+3 b B)+3 b c \log (x) (A c+b B)+\frac{1}{4} B c^3 x^4",1,"(-(A*(b^3 + 6*b^2*c*x^2 - 2*c^3*x^6)) + B*x^2*(-2*b^3 + 6*b*c^2*x^4 + c^3*x^6))/(4*x^4) + 3*b*c*(b*B + A*c)*Log[x]","A",1
35,1,68,68,0.0362389,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^3}{x^{12}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^12,x]","-\frac{A b^3}{5 x^5}-\frac{b^2 (3 A c+b B)}{3 x^3}+c^2 x (A c+3 b B)-\frac{3 b c (A c+b B)}{x}+\frac{1}{3} B c^3 x^3","-\frac{A b^3}{5 x^5}-\frac{b^2 (3 A c+b B)}{3 x^3}+c^2 x (A c+3 b B)-\frac{3 b c (A c+b B)}{x}+\frac{1}{3} B c^3 x^3",1,"-1/5*(A*b^3)/x^5 - (b^2*(b*B + 3*A*c))/(3*x^3) - (3*b*c*(b*B + A*c))/x + c^2*(3*b*B + A*c)*x + (B*c^3*x^3)/3","A",1
36,1,71,71,0.0472088,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^3}{x^{13}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^13,x]","-\frac{A b^3}{6 x^6}-\frac{b^2 (3 A c+b B)}{4 x^4}+c^2 \log (x) (A c+3 b B)-\frac{3 b c (A c+b B)}{2 x^2}+\frac{1}{2} B c^3 x^2","-\frac{A b^3}{6 x^6}-\frac{b^2 (3 A c+b B)}{4 x^4}+c^2 \log (x) (A c+3 b B)-\frac{3 b c (A c+b B)}{2 x^2}+\frac{1}{2} B c^3 x^2",1,"-1/6*(A*b^3)/x^6 - (b^2*(b*B + 3*A*c))/(4*x^4) - (3*b*c*(b*B + A*c))/(2*x^2) + (B*c^3*x^2)/2 + c^2*(3*b*B + A*c)*Log[x]","A",1
37,1,66,66,0.0357724,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^3}{x^{14}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^14,x]","-\frac{A b^3}{7 x^7}-\frac{b^2 (3 A c+b B)}{5 x^5}-\frac{c^2 (A c+3 b B)}{x}-\frac{b c (A c+b B)}{x^3}+B c^3 x","-\frac{A b^3}{7 x^7}-\frac{b^2 (3 A c+b B)}{5 x^5}-\frac{c^2 (A c+3 b B)}{x}-\frac{b c (A c+b B)}{x^3}+B c^3 x",1,"-1/7*(A*b^3)/x^7 - (b^2*(b*B + 3*A*c))/(5*x^5) - (b*c*(b*B + A*c))/x^3 - (c^2*(3*b*B + A*c))/x + B*c^3*x","A",1
38,1,77,63,0.0585843,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^3}{x^{15}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^15,x]","B c^3 \log (x)-\frac{3 A \left(b^3+4 b^2 c x^2+6 b c^2 x^4+4 c^3 x^6\right)+2 b B x^2 \left(2 b^2+9 b c x^2+18 c^2 x^4\right)}{24 x^8}","-\frac{A \left(b+c x^2\right)^4}{8 b x^8}-\frac{b^3 B}{6 x^6}-\frac{3 b^2 B c}{4 x^4}-\frac{3 b B c^2}{2 x^2}+B c^3 \log (x)",1,"-1/24*(2*b*B*x^2*(2*b^2 + 9*b*c*x^2 + 18*c^2*x^4) + 3*A*(b^3 + 4*b^2*c*x^2 + 6*b*c^2*x^4 + 4*c^3*x^6))/x^8 + B*c^3*Log[x]","A",1
39,1,73,73,0.0392782,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^3}{x^{16}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^16,x]","-\frac{A b^3}{9 x^9}-\frac{b^2 (3 A c+b B)}{7 x^7}-\frac{c^2 (A c+3 b B)}{3 x^3}-\frac{3 b c (A c+b B)}{5 x^5}-\frac{B c^3}{x}","-\frac{A b^3}{9 x^9}-\frac{b^2 (3 A c+b B)}{7 x^7}-\frac{c^2 (A c+3 b B)}{3 x^3}-\frac{3 b c (A c+b B)}{5 x^5}-\frac{B c^3}{x}",1,"-1/9*(A*b^3)/x^9 - (b^2*(b*B + 3*A*c))/(7*x^7) - (3*b*c*(b*B + A*c))/(5*x^5) - (c^2*(3*b*B + A*c))/(3*x^3) - (B*c^3)/x","A",1
40,1,78,49,0.0207133,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^3}{x^{17}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^17,x]","-\frac{A \left(4 b^3+15 b^2 c x^2+20 b c^2 x^4+10 c^3 x^6\right)+5 B x^2 \left(b^3+4 b^2 c x^2+6 b c^2 x^4+4 c^3 x^6\right)}{40 x^{10}}","-\frac{\left(b+c x^2\right)^4 (5 b B-A c)}{40 b^2 x^8}-\frac{A \left(b+c x^2\right)^4}{10 b x^{10}}",1,"-1/40*(5*B*x^2*(b^3 + 4*b^2*c*x^2 + 6*b*c^2*x^4 + 4*c^3*x^6) + A*(4*b^3 + 15*b^2*c*x^2 + 20*b*c^2*x^4 + 10*c^3*x^6))/x^10","A",1
41,1,119,119,0.085323,"\int \frac{x^{10} \left(A+B x^2\right)}{b x^2+c x^4} \, dx","Integrate[(x^10*(A + B*x^2))/(b*x^2 + c*x^4),x]","-\frac{b^{7/2} (b B-A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{c^{11/2}}+\frac{b^3 x (b B-A c)}{c^5}-\frac{b^2 x^3 (b B-A c)}{3 c^4}+\frac{b x^5 (b B-A c)}{5 c^3}+\frac{x^7 (A c-b B)}{7 c^2}+\frac{B x^9}{9 c}","-\frac{b^{7/2} (b B-A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{c^{11/2}}+\frac{b^3 x (b B-A c)}{c^5}-\frac{b^2 x^3 (b B-A c)}{3 c^4}+\frac{b x^5 (b B-A c)}{5 c^3}-\frac{x^7 (b B-A c)}{7 c^2}+\frac{B x^9}{9 c}",1,"(b^3*(b*B - A*c)*x)/c^5 - (b^2*(b*B - A*c)*x^3)/(3*c^4) + (b*(b*B - A*c)*x^5)/(5*c^3) + ((-(b*B) + A*c)*x^7)/(7*c^2) + (B*x^9)/(9*c) - (b^(7/2)*(b*B - A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/c^(11/2)","A",1
42,1,92,96,0.0408089,"\int \frac{x^9 \left(A+B x^2\right)}{b x^2+c x^4} \, dx","Integrate[(x^9*(A + B*x^2))/(b*x^2 + c*x^4),x]","\frac{12 b^3 (b B-A c) \log \left(b+c x^2\right)+c x^2 \left(6 b^2 c \left(2 A+B x^2\right)-2 b c^2 x^2 \left(3 A+2 B x^2\right)+c^3 x^4 \left(4 A+3 B x^2\right)-12 b^3 B\right)}{24 c^5}","\frac{b^3 (b B-A c) \log \left(b+c x^2\right)}{2 c^5}-\frac{b^2 x^2 (b B-A c)}{2 c^4}+\frac{b x^4 (b B-A c)}{4 c^3}-\frac{x^6 (b B-A c)}{6 c^2}+\frac{B x^8}{8 c}",1,"(c*x^2*(-12*b^3*B + 6*b^2*c*(2*A + B*x^2) - 2*b*c^2*x^2*(3*A + 2*B*x^2) + c^3*x^4*(4*A + 3*B*x^2)) + 12*b^3*(b*B - A*c)*Log[b + c*x^2])/(24*c^5)","A",1
43,1,98,98,0.0682188,"\int \frac{x^8 \left(A+B x^2\right)}{b x^2+c x^4} \, dx","Integrate[(x^8*(A + B*x^2))/(b*x^2 + c*x^4),x]","\frac{b^{5/2} (b B-A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{c^{9/2}}-\frac{b^2 x (b B-A c)}{c^4}+\frac{b x^3 (b B-A c)}{3 c^3}+\frac{x^5 (A c-b B)}{5 c^2}+\frac{B x^7}{7 c}","\frac{b^{5/2} (b B-A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{c^{9/2}}-\frac{b^2 x (b B-A c)}{c^4}+\frac{b x^3 (b B-A c)}{3 c^3}-\frac{x^5 (b B-A c)}{5 c^2}+\frac{B x^7}{7 c}",1,"-((b^2*(b*B - A*c)*x)/c^4) + (b*(b*B - A*c)*x^3)/(3*c^3) + ((-(b*B) + A*c)*x^5)/(5*c^2) + (B*x^7)/(7*c) + (b^(5/2)*(b*B - A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/c^(9/2)","A",1
44,1,71,75,0.0317649,"\int \frac{x^7 \left(A+B x^2\right)}{b x^2+c x^4} \, dx","Integrate[(x^7*(A + B*x^2))/(b*x^2 + c*x^4),x]","\frac{c x^2 \left(-3 b c \left(2 A+B x^2\right)+c^2 x^2 \left(3 A+2 B x^2\right)+6 b^2 B\right)+6 b^2 (A c-b B) \log \left(b+c x^2\right)}{12 c^4}","-\frac{b^2 (b B-A c) \log \left(b+c x^2\right)}{2 c^4}+\frac{b x^2 (b B-A c)}{2 c^3}-\frac{x^4 (b B-A c)}{4 c^2}+\frac{B x^6}{6 c}",1,"(c*x^2*(6*b^2*B - 3*b*c*(2*A + B*x^2) + c^2*x^2*(3*A + 2*B*x^2)) + 6*b^2*(-(b*B) + A*c)*Log[b + c*x^2])/(12*c^4)","A",1
45,1,77,77,0.0565585,"\int \frac{x^6 \left(A+B x^2\right)}{b x^2+c x^4} \, dx","Integrate[(x^6*(A + B*x^2))/(b*x^2 + c*x^4),x]","-\frac{b^{3/2} (b B-A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{c^{7/2}}+\frac{b x (b B-A c)}{c^3}+\frac{x^3 (A c-b B)}{3 c^2}+\frac{B x^5}{5 c}","-\frac{b^{3/2} (b B-A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{c^{7/2}}+\frac{b x (b B-A c)}{c^3}-\frac{x^3 (b B-A c)}{3 c^2}+\frac{B x^5}{5 c}",1,"(b*(b*B - A*c)*x)/c^3 + ((-(b*B) + A*c)*x^3)/(3*c^2) + (B*x^5)/(5*c) - (b^(3/2)*(b*B - A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/c^(7/2)","A",1
46,1,47,54,0.0213738,"\int \frac{x^5 \left(A+B x^2\right)}{b x^2+c x^4} \, dx","Integrate[(x^5*(A + B*x^2))/(b*x^2 + c*x^4),x]","\frac{c x^2 \left(2 A c-2 b B+B c x^2\right)+2 b (b B-A c) \log \left(b+c x^2\right)}{4 c^3}","\frac{b (b B-A c) \log \left(b+c x^2\right)}{2 c^3}-\frac{x^2 (b B-A c)}{2 c^2}+\frac{B x^4}{4 c}",1,"(c*x^2*(-2*b*B + 2*A*c + B*c*x^2) + 2*b*(b*B - A*c)*Log[b + c*x^2])/(4*c^3)","A",1
47,1,57,58,0.0428293,"\int \frac{x^4 \left(A+B x^2\right)}{b x^2+c x^4} \, dx","Integrate[(x^4*(A + B*x^2))/(b*x^2 + c*x^4),x]","\frac{\sqrt{b} (b B-A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{c^{5/2}}+\frac{x (A c-b B)}{c^2}+\frac{B x^3}{3 c}","\frac{\sqrt{b} (b B-A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{c^{5/2}}-\frac{x (b B-A c)}{c^2}+\frac{B x^3}{3 c}",1,"((-(b*B) + A*c)*x)/c^2 + (B*x^3)/(3*c) + (Sqrt[b]*(b*B - A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/c^(5/2)","A",1
48,1,31,35,0.0127218,"\int \frac{x^3 \left(A+B x^2\right)}{b x^2+c x^4} \, dx","Integrate[(x^3*(A + B*x^2))/(b*x^2 + c*x^4),x]","\frac{(A c-b B) \log \left(b+c x^2\right)+B c x^2}{2 c^2}","\frac{B x^2}{2 c}-\frac{(b B-A c) \log \left(b+c x^2\right)}{2 c^2}",1,"(B*c*x^2 + (-(b*B) + A*c)*Log[b + c*x^2])/(2*c^2)","A",1
49,1,40,40,0.0247636,"\int \frac{x^2 \left(A+B x^2\right)}{b x^2+c x^4} \, dx","Integrate[(x^2*(A + B*x^2))/(b*x^2 + c*x^4),x]","\frac{B x}{c}-\frac{(b B-A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{\sqrt{b} c^{3/2}}","\frac{B x}{c}-\frac{(b B-A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{\sqrt{b} c^{3/2}}",1,"(B*x)/c - ((b*B - A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(Sqrt[b]*c^(3/2))","A",1
50,1,34,34,0.0131654,"\int \frac{x \left(A+B x^2\right)}{b x^2+c x^4} \, dx","Integrate[(x*(A + B*x^2))/(b*x^2 + c*x^4),x]","\frac{(b B-A c) \log \left(b+c x^2\right)}{2 b c}+\frac{A \log (x)}{b}","\frac{(b B-A c) \log \left(b+c x^2\right)}{2 b c}+\frac{A \log (x)}{b}",1,"(A*Log[x])/b + ((b*B - A*c)*Log[b + c*x^2])/(2*b*c)","A",1
51,1,42,42,0.0489906,"\int \frac{A+B x^2}{b x^2+c x^4} \, dx","Integrate[(A + B*x^2)/(b*x^2 + c*x^4),x]","\frac{(b B-A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{b^{3/2} \sqrt{c}}-\frac{A}{b x}","\frac{(b B-A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{b^{3/2} \sqrt{c}}-\frac{A}{b x}",1,"-(A/(b*x)) + ((b*B - A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(b^(3/2)*Sqrt[c])","A",1
52,1,41,41,0.0263072,"\int \frac{A+B x^2}{b x^2-c x^4} \, dx","Integrate[(A + B*x^2)/(b*x^2 - c*x^4),x]","\frac{(A c+b B) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{b^{3/2} \sqrt{c}}-\frac{A}{b x}","\frac{(A c+b B) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{b^{3/2} \sqrt{c}}-\frac{A}{b x}",1,"-(A/(b*x)) + ((b*B + A*c)*ArcTanh[(Sqrt[c]*x)/Sqrt[b]])/(b^(3/2)*Sqrt[c])","A",1
53,1,49,49,0.0230294,"\int \frac{A+B x^2}{x \left(b x^2+c x^4\right)} \, dx","Integrate[(A + B*x^2)/(x*(b*x^2 + c*x^4)),x]","\frac{(A c-b B) \log \left(b+c x^2\right)}{2 b^2}+\frac{\log (x) (b B-A c)}{b^2}-\frac{A}{2 b x^2}","-\frac{(b B-A c) \log \left(b+c x^2\right)}{2 b^2}+\frac{\log (x) (b B-A c)}{b^2}-\frac{A}{2 b x^2}",1,"-1/2*A/(b*x^2) + ((b*B - A*c)*Log[x])/b^2 + ((-(b*B) + A*c)*Log[b + c*x^2])/(2*b^2)","A",1
54,1,60,61,0.0872825,"\int \frac{A+B x^2}{x^2 \left(b x^2+c x^4\right)} \, dx","Integrate[(A + B*x^2)/(x^2*(b*x^2 + c*x^4)),x]","-\frac{\sqrt{c} (b B-A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{b^{5/2}}+\frac{A c-b B}{b^2 x}-\frac{A}{3 b x^3}","-\frac{\sqrt{c} (b B-A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{b^{5/2}}-\frac{b B-A c}{b^2 x}-\frac{A}{3 b x^3}",1,"-1/3*A/(b*x^3) + (-(b*B) + A*c)/(b^2*x) - (Sqrt[c]*(b*B - A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/b^(5/2)","A",1
55,1,70,70,0.0383296,"\int \frac{A+B x^2}{x^3 \left(b x^2+c x^4\right)} \, dx","Integrate[(A + B*x^2)/(x^3*(b*x^2 + c*x^4)),x]","\frac{4 c x^4 \log (x) (A c-b B)-b \left(A b-2 A c x^2+2 b B x^2\right)+2 c x^4 (b B-A c) \log \left(b+c x^2\right)}{4 b^3 x^4}","\frac{c (b B-A c) \log \left(b+c x^2\right)}{2 b^3}-\frac{c \log (x) (b B-A c)}{b^3}-\frac{b B-A c}{2 b^2 x^2}-\frac{A}{4 b x^4}",1,"(-(b*(A*b + 2*b*B*x^2 - 2*A*c*x^2)) + 4*c*(-(b*B) + A*c)*x^4*Log[x] + 2*c*(b*B - A*c)*x^4*Log[b + c*x^2])/(4*b^3*x^4)","A",1
56,1,78,78,0.0891024,"\int \frac{A+B x^2}{x^4 \left(b x^2+c x^4\right)} \, dx","Integrate[(A + B*x^2)/(x^4*(b*x^2 + c*x^4)),x]","\frac{c^{3/2} (b B-A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{b^{7/2}}+\frac{c (b B-A c)}{b^3 x}+\frac{A c-b B}{3 b^2 x^3}-\frac{A}{5 b x^5}","\frac{c^{3/2} (b B-A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{b^{7/2}}+\frac{c (b B-A c)}{b^3 x}-\frac{b B-A c}{3 b^2 x^3}-\frac{A}{5 b x^5}",1,"-1/5*A/(b*x^5) + (-(b*B) + A*c)/(3*b^2*x^3) + (c*(b*B - A*c))/(b^3*x) + (c^(3/2)*(b*B - A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/b^(7/2)","A",1
57,1,96,92,0.0458051,"\int \frac{A+B x^2}{x^5 \left(b x^2+c x^4\right)} \, dx","Integrate[(A + B*x^2)/(x^5*(b*x^2 + c*x^4)),x]","\frac{\left(A c^3-b B c^2\right) \log \left(b+c x^2\right)}{2 b^4}+\frac{\log (x) \left(b B c^2-A c^3\right)}{b^4}+\frac{c (b B-A c)}{2 b^3 x^2}+\frac{A c-b B}{4 b^2 x^4}-\frac{A}{6 b x^6}","-\frac{c^2 (b B-A c) \log \left(b+c x^2\right)}{2 b^4}+\frac{c^2 \log (x) (b B-A c)}{b^4}+\frac{c (b B-A c)}{2 b^3 x^2}-\frac{b B-A c}{4 b^2 x^4}-\frac{A}{6 b x^6}",1,"-1/6*A/(b*x^6) + (-(b*B) + A*c)/(4*b^2*x^4) + (c*(b*B - A*c))/(2*b^3*x^2) + ((b*B*c^2 - A*c^3)*Log[x])/b^4 + ((-(b*B*c^2) + A*c^3)*Log[b + c*x^2])/(2*b^4)","A",1
58,1,134,133,0.1142422,"\int \frac{x^{12} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[(x^12*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]","\frac{b^{5/2} (9 b B-7 A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{2 c^{11/2}}-\frac{b^2 x (4 b B-3 A c)}{c^5}+\frac{x \left(A b^3 c-b^4 B\right)}{2 c^5 \left(b+c x^2\right)}+\frac{b x^3 (3 b B-2 A c)}{3 c^4}+\frac{x^5 (A c-2 b B)}{5 c^3}+\frac{B x^7}{7 c^2}","\frac{b^{5/2} (9 b B-7 A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{2 c^{11/2}}-\frac{b^3 x (b B-A c)}{2 c^5 \left(b+c x^2\right)}-\frac{b^2 x (4 b B-3 A c)}{c^5}+\frac{b x^3 (3 b B-2 A c)}{3 c^4}-\frac{x^5 (2 b B-A c)}{5 c^3}+\frac{B x^7}{7 c^2}",1,"-((b^2*(4*b*B - 3*A*c)*x)/c^5) + (b*(3*b*B - 2*A*c)*x^3)/(3*c^4) + ((-2*b*B + A*c)*x^5)/(5*c^3) + (B*x^7)/(7*c^2) + ((-(b^4*B) + A*b^3*c)*x)/(2*c^5*(b + c*x^2)) + (b^(5/2)*(9*b*B - 7*A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(2*c^(11/2))","A",1
59,1,93,105,0.0778097,"\int \frac{x^{11} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[(x^11*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]","\frac{\frac{6 b^3 (A c-b B)}{b+c x^2}+6 b^2 (3 A c-4 b B) \log \left(b+c x^2\right)+3 c^2 x^4 (A c-2 b B)+6 b c x^2 (3 b B-2 A c)+2 B c^3 x^6}{12 c^5}","-\frac{b^3 (b B-A c)}{2 c^5 \left(b+c x^2\right)}-\frac{b^2 (4 b B-3 A c) \log \left(b+c x^2\right)}{2 c^5}+\frac{b x^2 (3 b B-2 A c)}{2 c^4}-\frac{x^4 (2 b B-A c)}{4 c^3}+\frac{B x^6}{6 c^2}",1,"(6*b*c*(3*b*B - 2*A*c)*x^2 + 3*c^2*(-2*b*B + A*c)*x^4 + 2*B*c^3*x^6 + (6*b^3*(-(b*B) + A*c))/(b + c*x^2) + 6*b^2*(-4*b*B + 3*A*c)*Log[b + c*x^2])/(12*c^5)","A",1
60,1,111,110,0.090935,"\int \frac{x^{10} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[(x^10*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]","-\frac{b^{3/2} (7 b B-5 A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{2 c^{9/2}}-\frac{x \left(A b^2 c-b^3 B\right)}{2 c^4 \left(b+c x^2\right)}+\frac{b x (3 b B-2 A c)}{c^4}+\frac{x^3 (A c-2 b B)}{3 c^3}+\frac{B x^5}{5 c^2}","-\frac{b^{3/2} (7 b B-5 A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{2 c^{9/2}}+\frac{b^2 x (b B-A c)}{2 c^4 \left(b+c x^2\right)}+\frac{b x (3 b B-2 A c)}{c^4}-\frac{x^3 (2 b B-A c)}{3 c^3}+\frac{B x^5}{5 c^2}",1,"(b*(3*b*B - 2*A*c)*x)/c^4 + ((-2*b*B + A*c)*x^3)/(3*c^3) + (B*x^5)/(5*c^2) - ((-(b^3*B) + A*b^2*c)*x)/(2*c^4*(b + c*x^2)) - (b^(3/2)*(7*b*B - 5*A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(2*c^(9/2))","A",1
61,1,72,83,0.0576757,"\int \frac{x^9 \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[(x^9*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]","\frac{\frac{2 b^2 (b B-A c)}{b+c x^2}+2 c x^2 (A c-2 b B)+2 b (3 b B-2 A c) \log \left(b+c x^2\right)+B c^2 x^4}{4 c^4}","\frac{b^2 (b B-A c)}{2 c^4 \left(b+c x^2\right)}+\frac{b (3 b B-2 A c) \log \left(b+c x^2\right)}{2 c^4}-\frac{x^2 (2 b B-A c)}{2 c^3}+\frac{B x^4}{4 c^2}",1,"(2*c*(-2*b*B + A*c)*x^2 + B*c^2*x^4 + (2*b^2*(b*B - A*c))/(b + c*x^2) + 2*b*(3*b*B - 2*A*c)*Log[b + c*x^2])/(4*c^4)","A",1
62,1,89,89,0.0797871,"\int \frac{x^8 \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[(x^8*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]","\frac{x \left(A b c-b^2 B\right)}{2 c^3 \left(b+c x^2\right)}+\frac{\sqrt{b} (5 b B-3 A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{2 c^{7/2}}+\frac{x (A c-2 b B)}{c^3}+\frac{B x^3}{3 c^2}","\frac{\sqrt{b} (5 b B-3 A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{2 c^{7/2}}-\frac{b x (b B-A c)}{2 c^3 \left(b+c x^2\right)}-\frac{x (2 b B-A c)}{c^3}+\frac{B x^3}{3 c^2}",1,"((-2*b*B + A*c)*x)/c^3 + (B*x^3)/(3*c^2) + ((-(b^2*B) + A*b*c)*x)/(2*c^3*(b + c*x^2)) + (Sqrt[b]*(5*b*B - 3*A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(2*c^(7/2))","A",1
63,1,50,61,0.0382593,"\int \frac{x^7 \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[(x^7*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]","\frac{\frac{b (A c-b B)}{b+c x^2}+(A c-2 b B) \log \left(b+c x^2\right)+B c x^2}{2 c^3}","-\frac{b (b B-A c)}{2 c^3 \left(b+c x^2\right)}-\frac{(2 b B-A c) \log \left(b+c x^2\right)}{2 c^3}+\frac{B x^2}{2 c^2}",1,"(B*c*x^2 + (b*(-(b*B) + A*c))/(b + c*x^2) + (-2*b*B + A*c)*Log[b + c*x^2])/(2*c^3)","A",1
64,1,68,68,0.0556685,"\int \frac{x^6 \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[(x^6*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]","-\frac{(3 b B-A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{2 \sqrt{b} c^{5/2}}-\frac{x (A c-b B)}{2 c^2 \left(b+c x^2\right)}+\frac{B x}{c^2}","-\frac{(3 b B-A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{2 \sqrt{b} c^{5/2}}+\frac{x (b B-A c)}{2 c^2 \left(b+c x^2\right)}+\frac{B x}{c^2}",1,"(B*x)/c^2 - ((-(b*B) + A*c)*x)/(2*c^2*(b + c*x^2)) - ((3*b*B - A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(2*Sqrt[b]*c^(5/2))","A",1
65,1,41,41,0.0143745,"\int \frac{x^5 \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[(x^5*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]","\frac{b B-A c}{2 c^2 \left(b+c x^2\right)}+\frac{B \log \left(b+c x^2\right)}{2 c^2}","\frac{b B-A c}{2 c^2 \left(b+c x^2\right)}+\frac{B \log \left(b+c x^2\right)}{2 c^2}",1,"(b*B - A*c)/(2*c^2*(b + c*x^2)) + (B*Log[b + c*x^2])/(2*c^2)","A",1
66,1,63,63,0.0510277,"\int \frac{x^4 \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[(x^4*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]","\frac{(A c+b B) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{2 b^{3/2} c^{3/2}}-\frac{x (b B-A c)}{2 b c \left(b+c x^2\right)}","\frac{(A c+b B) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{2 b^{3/2} c^{3/2}}-\frac{x (b B-A c)}{2 b c \left(b+c x^2\right)}",1,"-1/2*((b*B - A*c)*x)/(b*c*(b + c*x^2)) + ((b*B + A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(2*b^(3/2)*c^(3/2))","A",1
67,1,46,51,0.0343394,"\int \frac{x^3 \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[(x^3*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]","\frac{\frac{b (A c-b B)}{c \left(b+c x^2\right)}-A \log \left(b+c x^2\right)+2 A \log (x)}{2 b^2}","-\frac{A \log \left(b+c x^2\right)}{2 b^2}+\frac{A \log (x)}{b^2}-\frac{b B-A c}{2 b c \left(b+c x^2\right)}",1,"((b*(-(b*B) + A*c))/(c*(b + c*x^2)) + 2*A*Log[x] - A*Log[b + c*x^2])/(2*b^2)","A",1
68,1,70,70,0.0359546,"\int \frac{x^2 \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[(x^2*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]","\frac{(b B-3 A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{2 b^{5/2} \sqrt{c}}+\frac{x (b B-A c)}{2 b^2 \left(b+c x^2\right)}-\frac{A}{b^2 x}","\frac{(b B-3 A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{2 b^{5/2} \sqrt{c}}+\frac{x (b B-A c)}{2 b^2 \left(b+c x^2\right)}-\frac{A}{b^2 x}",1,"-(A/(b^2*x)) + ((b*B - A*c)*x)/(2*b^2*(b + c*x^2)) + ((b*B - 3*A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(2*b^(5/2)*Sqrt[c])","A",1
69,1,64,73,0.0598139,"\int \frac{x \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[(x*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]","\frac{\frac{b (b B-A c)}{b+c x^2}+(2 A c-b B) \log \left(b+c x^2\right)+2 \log (x) (b B-2 A c)-\frac{A b}{x^2}}{2 b^3}","-\frac{(b B-2 A c) \log \left(b+c x^2\right)}{2 b^3}+\frac{\log (x) (b B-2 A c)}{b^3}+\frac{b B-A c}{2 b^2 \left(b+c x^2\right)}-\frac{A}{2 b^2 x^2}",1,"(-((A*b)/x^2) + (b*(b*B - A*c))/(b + c*x^2) + 2*(b*B - 2*A*c)*Log[x] + (-(b*B) + 2*A*c)*Log[b + c*x^2])/(2*b^3)","A",1
70,1,90,90,0.0817481,"\int \frac{A+B x^2}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[(A + B*x^2)/(b*x^2 + c*x^4)^2,x]","-\frac{\sqrt{c} (3 b B-5 A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{2 b^{7/2}}-\frac{c x (b B-A c)}{2 b^3 \left(b+c x^2\right)}+\frac{2 A c-b B}{b^3 x}-\frac{A}{3 b^2 x^3}","-\frac{\sqrt{c} (3 b B-5 A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{2 b^{7/2}}-\frac{c x (b B-A c)}{2 b^3 \left(b+c x^2\right)}-\frac{b B-2 A c}{b^3 x}-\frac{A}{3 b^2 x^3}",1,"-1/3*A/(b^2*x^3) + (-(b*B) + 2*A*c)/(b^3*x) - (c*(b*B - A*c)*x)/(2*b^3*(b + c*x^2)) - (Sqrt[c]*(3*b*B - 5*A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(2*b^(7/2))","A",1
71,1,85,97,0.1061995,"\int \frac{A+B x^2}{x \left(b x^2+c x^4\right)^2} \, dx","Integrate[(A + B*x^2)/(x*(b*x^2 + c*x^4)^2),x]","-\frac{\frac{A b^2}{x^4}+\frac{2 b c (b B-A c)}{b+c x^2}+\frac{2 b (b B-2 A c)}{x^2}+2 c (3 A c-2 b B) \log \left(b+c x^2\right)-4 c \log (x) (3 A c-2 b B)}{4 b^4}","\frac{c (2 b B-3 A c) \log \left(b+c x^2\right)}{2 b^4}-\frac{c \log (x) (2 b B-3 A c)}{b^4}-\frac{c (b B-A c)}{2 b^3 \left(b+c x^2\right)}-\frac{b B-2 A c}{2 b^3 x^2}-\frac{A}{4 b^2 x^4}",1,"-1/4*((A*b^2)/x^4 + (2*b*(b*B - 2*A*c))/x^2 + (2*b*c*(b*B - A*c))/(b + c*x^2) - 4*c*(-2*b*B + 3*A*c)*Log[x] + 2*c*(-2*b*B + 3*A*c)*Log[b + c*x^2])/b^4","A",1
72,1,112,111,0.0992909,"\int \frac{A+B x^2}{x^2 \left(b x^2+c x^4\right)^2} \, dx","Integrate[(A + B*x^2)/(x^2*(b*x^2 + c*x^4)^2),x]","\frac{c^{3/2} (5 b B-7 A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{2 b^{9/2}}+\frac{c^2 x (b B-A c)}{2 b^4 \left(b+c x^2\right)}+\frac{c (2 b B-3 A c)}{b^4 x}+\frac{2 A c-b B}{3 b^3 x^3}-\frac{A}{5 b^2 x^5}","\frac{c^{3/2} (5 b B-7 A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{2 b^{9/2}}+\frac{c^2 x (b B-A c)}{2 b^4 \left(b+c x^2\right)}+\frac{c (2 b B-3 A c)}{b^4 x}-\frac{b B-2 A c}{3 b^3 x^3}-\frac{A}{5 b^2 x^5}",1,"-1/5*A/(b^2*x^5) + (-(b*B) + 2*A*c)/(3*b^3*x^3) + (c*(2*b*B - 3*A*c))/(b^4*x) + (c^2*(b*B - A*c)*x)/(2*b^4*(b + c*x^2)) + (c^(3/2)*(5*b*B - 7*A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(2*b^(9/2))","A",1
73,1,133,140,0.1443568,"\int \frac{x^{14} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[(x^14*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]","\frac{x \left(-525 b^3 c \left(A-3 B x^2\right)+7 b^2 c^2 x^2 \left(72 B x^2-125 A\right)-8 b c^3 x^4 \left(35 A+9 B x^2\right)+8 c^4 x^6 \left(5 A+3 B x^2\right)+945 b^4 B\right)}{120 c^5 \left(b+c x^2\right)^2}-\frac{7 b^{3/2} (9 b B-5 A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{8 c^{11/2}}","-\frac{7 b^{3/2} (9 b B-5 A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{8 c^{11/2}}-\frac{b^3 x (b B-A c)}{4 c^5 \left(b+c x^2\right)^2}+\frac{b^2 x (17 b B-13 A c)}{8 c^5 \left(b+c x^2\right)}+\frac{3 b x (2 b B-A c)}{c^5}-\frac{x^3 (3 b B-A c)}{3 c^4}+\frac{B x^5}{5 c^3}",1,"(x*(945*b^4*B - 525*b^3*c*(A - 3*B*x^2) + 8*c^4*x^6*(5*A + 3*B*x^2) - 8*b*c^3*x^4*(35*A + 9*B*x^2) + 7*b^2*c^2*x^2*(-125*A + 72*B*x^2)))/(120*c^5*(b + c*x^2)^2) - (7*b^(3/2)*(9*b*B - 5*A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(8*c^(11/2))","A",1
74,1,94,111,0.0945087,"\int \frac{x^{13} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[(x^13*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]","\frac{\frac{b^3 (A c-b B)}{\left(b+c x^2\right)^2}+\frac{2 b^2 (4 b B-3 A c)}{b+c x^2}+2 c x^2 (A c-3 b B)+6 b (2 b B-A c) \log \left(b+c x^2\right)+B c^2 x^4}{4 c^5}","-\frac{b^3 (b B-A c)}{4 c^5 \left(b+c x^2\right)^2}+\frac{b^2 (4 b B-3 A c)}{2 c^5 \left(b+c x^2\right)}+\frac{3 b (2 b B-A c) \log \left(b+c x^2\right)}{2 c^5}-\frac{x^2 (3 b B-A c)}{2 c^4}+\frac{B x^4}{4 c^3}",1,"(2*c*(-3*b*B + A*c)*x^2 + B*c^2*x^4 + (b^3*(-(b*B) + A*c))/(b + c*x^2)^2 + (2*b^2*(4*b*B - 3*A*c))/(b + c*x^2) + 6*b*(2*b*B - A*c)*Log[b + c*x^2])/(4*c^5)","A",1
75,1,113,118,0.1338327,"\int \frac{x^{12} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[(x^12*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]","\frac{5 b^2 c x \left(9 A-35 B x^2\right)+b c^2 x^3 \left(75 A-56 B x^2\right)+8 c^3 x^5 \left(3 A+B x^2\right)-105 b^3 B x}{24 c^4 \left(b+c x^2\right)^2}+\frac{5 \sqrt{b} (7 b B-3 A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{8 c^{9/2}}","\frac{b^2 x (b B-A c)}{4 c^4 \left(b+c x^2\right)^2}+\frac{5 \sqrt{b} (7 b B-3 A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{8 c^{9/2}}-\frac{b x (13 b B-9 A c)}{8 c^4 \left(b+c x^2\right)}-\frac{x (3 b B-A c)}{c^4}+\frac{B x^3}{3 c^3}",1,"(-105*b^3*B*x + b*c^2*x^3*(75*A - 56*B*x^2) + 5*b^2*c*x*(9*A - 35*B*x^2) + 8*c^3*x^5*(3*A + B*x^2))/(24*c^4*(b + c*x^2)^2) + (5*Sqrt[b]*(7*b*B - 3*A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(8*c^(9/2))","A",1
76,1,92,89,0.0483252,"\int \frac{x^{11} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[(x^11*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]","\frac{2 A b c-3 b^2 B}{2 c^4 \left(b+c x^2\right)}+\frac{b^3 B-A b^2 c}{4 c^4 \left(b+c x^2\right)^2}+\frac{(A c-3 b B) \log \left(b+c x^2\right)}{2 c^4}+\frac{B x^2}{2 c^3}","\frac{b^2 (b B-A c)}{4 c^4 \left(b+c x^2\right)^2}-\frac{b (3 b B-2 A c)}{2 c^4 \left(b+c x^2\right)}-\frac{(3 b B-A c) \log \left(b+c x^2\right)}{2 c^4}+\frac{B x^2}{2 c^3}",1,"(B*x^2)/(2*c^3) + (b^3*B - A*b^2*c)/(4*c^4*(b + c*x^2)^2) + (-3*b^2*B + 2*A*b*c)/(2*c^4*(b + c*x^2)) + ((-3*b*B + A*c)*Log[b + c*x^2])/(2*c^4)","A",1
77,1,92,95,0.109305,"\int \frac{x^{10} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[(x^10*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]","\frac{x \left(b \left(25 B c x^2-3 A c\right)+c^2 x^2 \left(8 B x^2-5 A\right)+15 b^2 B\right)}{8 c^3 \left(b+c x^2\right)^2}-\frac{3 (5 b B-A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{8 \sqrt{b} c^{7/2}}","-\frac{3 (5 b B-A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{8 \sqrt{b} c^{7/2}}+\frac{x (9 b B-5 A c)}{8 c^3 \left(b+c x^2\right)}-\frac{b x (b B-A c)}{4 c^3 \left(b+c x^2\right)^2}+\frac{B x}{c^3}",1,"(x*(15*b^2*B + c^2*x^2*(-5*A + 8*B*x^2) + b*(-3*A*c + 25*B*c*x^2)))/(8*c^3*(b + c*x^2)^2) - (3*(5*b*B - A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(8*Sqrt[b]*c^(7/2))","A",1
78,1,64,67,0.0264455,"\int \frac{x^9 \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[(x^9*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]","\frac{-b c \left(A-4 B x^2\right)-2 A c^2 x^2+3 b^2 B+2 B \left(b+c x^2\right)^2 \log \left(b+c x^2\right)}{4 c^3 \left(b+c x^2\right)^2}","-\frac{b (b B-A c)}{4 c^3 \left(b+c x^2\right)^2}+\frac{2 b B-A c}{2 c^3 \left(b+c x^2\right)}+\frac{B \log \left(b+c x^2\right)}{2 c^3}",1,"(3*b^2*B - 2*A*c^2*x^2 - b*c*(A - 4*B*x^2) + 2*B*(b + c*x^2)^2*Log[b + c*x^2])/(4*c^3*(b + c*x^2)^2)","A",1
79,1,83,90,0.1222983,"\int \frac{x^8 \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[(x^8*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]","\frac{\frac{(A c+3 b B) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{b^{3/2}}+\frac{\sqrt{c} x \left(-b c \left(A+5 B x^2\right)+A c^2 x^2-3 b^2 B\right)}{b \left(b+c x^2\right)^2}}{8 c^{5/2}}","\frac{(A c+3 b B) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{8 b^{3/2} c^{5/2}}-\frac{x (5 b B-A c)}{8 b c^2 \left(b+c x^2\right)}+\frac{x (b B-A c)}{4 c^2 \left(b+c x^2\right)^2}",1,"((Sqrt[c]*x*(-3*b^2*B + A*c^2*x^2 - b*c*(A + 5*B*x^2)))/(b*(b + c*x^2)^2) + ((3*b*B + A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/b^(3/2))/(8*c^(5/2))","A",1
80,1,30,32,0.0144645,"\int \frac{x^7 \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[(x^7*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]","-\frac{c \left(A+2 B x^2\right)+b B}{4 c^2 \left(b+c x^2\right)^2}","\frac{\left(A+B x^2\right)^2}{4 \left(b+c x^2\right)^2 (b B-A c)}",1,"-1/4*(b*B + c*(A + 2*B*x^2))/(c^2*(b + c*x^2)^2)","A",1
81,1,84,92,0.0830374,"\int \frac{x^6 \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[(x^6*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]","\frac{(3 A c+b B) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{8 b^{5/2} c^{3/2}}+\frac{x \left(b c \left(5 A+B x^2\right)+3 A c^2 x^2+b^2 (-B)\right)}{8 b^2 c \left(b+c x^2\right)^2}","\frac{(3 A c+b B) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{8 b^{5/2} c^{3/2}}+\frac{x (3 A c+b B)}{8 b^2 c \left(b+c x^2\right)}-\frac{x (b B-A c)}{4 b c \left(b+c x^2\right)^2}",1,"(x*(-(b^2*B) + 3*A*c^2*x^2 + b*c*(5*A + B*x^2)))/(8*b^2*c*(b + c*x^2)^2) + ((b*B + 3*A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(8*b^(5/2)*c^(3/2))","A",1
82,1,59,68,0.0675582,"\int \frac{x^5 \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[(x^5*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]","\frac{\frac{b \left(3 A b c+2 A c^2 x^2+b^2 (-B)\right)}{c \left(b+c x^2\right)^2}-2 A \log \left(b+c x^2\right)+4 A \log (x)}{4 b^3}","-\frac{A \log \left(b+c x^2\right)}{2 b^3}+\frac{A \log (x)}{b^3}+\frac{A}{2 b^2 \left(b+c x^2\right)}-\frac{b B-A c}{4 b c \left(b+c x^2\right)^2}",1,"((b*(-(b^2*B) + 3*A*b*c + 2*A*c^2*x^2))/(c*(b + c*x^2)^2) + 4*A*Log[x] - 2*A*Log[b + c*x^2])/(4*b^3)","A",1
83,1,96,96,0.0973067,"\int \frac{x^4 \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[(x^4*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]","\frac{3 (b B-5 A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{8 b^{7/2} \sqrt{c}}+\frac{x (3 b B-7 A c)}{8 b^3 \left(b+c x^2\right)}-\frac{A}{b^3 x}+\frac{x (b B-A c)}{4 b^2 \left(b+c x^2\right)^2}","\frac{3 (b B-5 A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{8 b^{7/2} \sqrt{c}}+\frac{x (3 b B-7 A c)}{8 b^3 \left(b+c x^2\right)}-\frac{A}{b^3 x}+\frac{x (b B-A c)}{4 b^2 \left(b+c x^2\right)^2}",1,"-(A/(b^3*x)) + ((b*B - A*c)*x)/(4*b^2*(b + c*x^2)^2) + ((3*b*B - 7*A*c)*x)/(8*b^3*(b + c*x^2)) + (3*(b*B - 5*A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(8*b^(7/2)*Sqrt[c])","A",1
84,1,86,97,0.0999406,"\int \frac{x^3 \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[(x^3*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]","\frac{\frac{b^2 (b B-A c)}{\left(b+c x^2\right)^2}+\frac{2 b (b B-2 A c)}{b+c x^2}-2 (b B-3 A c) \log \left(b+c x^2\right)+4 \log (x) (b B-3 A c)-\frac{2 A b}{x^2}}{4 b^4}","-\frac{(b B-3 A c) \log \left(b+c x^2\right)}{2 b^4}+\frac{\log (x) (b B-3 A c)}{b^4}+\frac{b B-2 A c}{2 b^3 \left(b+c x^2\right)}-\frac{A}{2 b^3 x^2}+\frac{b B-A c}{4 b^2 \left(b+c x^2\right)^2}",1,"((-2*A*b)/x^2 + (b^2*(b*B - A*c))/(b + c*x^2)^2 + (2*b*(b*B - 2*A*c))/(b + c*x^2) + 4*(b*B - 3*A*c)*Log[x] - 2*(b*B - 3*A*c)*Log[b + c*x^2])/(4*b^4)","A",1
85,1,119,117,0.0768124,"\int \frac{x^2 \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[(x^2*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]","-\frac{5 \sqrt{c} (3 b B-7 A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{8 b^{9/2}}-\frac{x \left(7 b B c-11 A c^2\right)}{8 b^4 \left(b+c x^2\right)}+\frac{3 A c-b B}{b^4 x}-\frac{c x (b B-A c)}{4 b^3 \left(b+c x^2\right)^2}-\frac{A}{3 b^3 x^3}","-\frac{5 \sqrt{c} (3 b B-7 A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{8 b^{9/2}}-\frac{c x (7 b B-11 A c)}{8 b^4 \left(b+c x^2\right)}-\frac{b B-3 A c}{b^4 x}-\frac{c x (b B-A c)}{4 b^3 \left(b+c x^2\right)^2}-\frac{A}{3 b^3 x^3}",1,"-1/3*A/(b^3*x^3) + (-(b*B) + 3*A*c)/(b^4*x) - (c*(b*B - A*c)*x)/(4*b^3*(b + c*x^2)^2) - ((7*b*B*c - 11*A*c^2)*x)/(8*b^4*(b + c*x^2)) - (5*Sqrt[c]*(3*b*B - 7*A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(8*b^(9/2))","A",1
86,1,108,121,0.0879896,"\int \frac{x \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[(x*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]","\frac{\frac{b^2 c (A c-b B)}{\left(b+c x^2\right)^2}-\frac{A b^2}{x^4}+\frac{2 b c (3 A c-2 b B)}{b+c x^2}-\frac{2 b (b B-3 A c)}{x^2}+6 c (b B-2 A c) \log \left(b+c x^2\right)+12 c \log (x) (2 A c-b B)}{4 b^5}","\frac{3 c (b B-2 A c) \log \left(b+c x^2\right)}{2 b^5}-\frac{3 c \log (x) (b B-2 A c)}{b^5}-\frac{c (2 b B-3 A c)}{2 b^4 \left(b+c x^2\right)}-\frac{b B-3 A c}{2 b^4 x^2}-\frac{c (b B-A c)}{4 b^3 \left(b+c x^2\right)^2}-\frac{A}{4 b^3 x^4}",1,"(-((A*b^2)/x^4) - (2*b*(b*B - 3*A*c))/x^2 + (b^2*c*(-(b*B) + A*c))/(b + c*x^2)^2 + (2*b*c*(-2*b*B + 3*A*c))/(b + c*x^2) + 12*c*(-(b*B) + 2*A*c)*Log[x] + 6*c*(b*B - 2*A*c)*Log[b + c*x^2])/(4*b^5)","A",1
87,1,140,140,0.0845945,"\int \frac{A+B x^2}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[(A + B*x^2)/(b*x^2 + c*x^4)^3,x]","\frac{7 c^{3/2} (5 b B-9 A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{8 b^{11/2}}+\frac{c^2 x (11 b B-15 A c)}{8 b^5 \left(b+c x^2\right)}+\frac{3 c (b B-2 A c)}{b^5 x}+\frac{c^2 x (b B-A c)}{4 b^4 \left(b+c x^2\right)^2}-\frac{b B-3 A c}{3 b^4 x^3}-\frac{A}{5 b^3 x^5}","\frac{7 c^{3/2} (5 b B-9 A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{8 b^{11/2}}+\frac{c^2 x (11 b B-15 A c)}{8 b^5 \left(b+c x^2\right)}+\frac{3 c (b B-2 A c)}{b^5 x}+\frac{c^2 x (b B-A c)}{4 b^4 \left(b+c x^2\right)^2}-\frac{b B-3 A c}{3 b^4 x^3}-\frac{A}{5 b^3 x^5}",1,"-1/5*A/(b^3*x^5) - (b*B - 3*A*c)/(3*b^4*x^3) + (3*c*(b*B - 2*A*c))/(b^5*x) + (c^2*(b*B - A*c)*x)/(4*b^4*(b + c*x^2)^2) + (c^2*(11*b*B - 15*A*c)*x)/(8*b^5*(b + c*x^2)) + (7*c^(3/2)*(5*b*B - 9*A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(8*b^(11/2))","A",1
88,1,135,148,0.1362331,"\int \frac{A+B x^2}{x \left(b x^2+c x^4\right)^3} \, dx","Integrate[(A + B*x^2)/(x*(b*x^2 + c*x^4)^3),x]","\frac{-\frac{2 A b^3}{x^6}+\frac{3 b^2 c^2 (b B-A c)}{\left(b+c x^2\right)^2}-\frac{3 b^2 (b B-3 A c)}{x^4}+\frac{6 b c^2 (3 b B-4 A c)}{b+c x^2}+12 c^2 (5 A c-3 b B) \log \left(b+c x^2\right)+24 c^2 \log (x) (3 b B-5 A c)+\frac{18 b c (b B-2 A c)}{x^2}}{12 b^6}","-\frac{c^2 (3 b B-5 A c) \log \left(b+c x^2\right)}{b^6}+\frac{2 c^2 \log (x) (3 b B-5 A c)}{b^6}+\frac{c^2 (3 b B-4 A c)}{2 b^5 \left(b+c x^2\right)}+\frac{3 c (b B-2 A c)}{2 b^5 x^2}+\frac{c^2 (b B-A c)}{4 b^4 \left(b+c x^2\right)^2}-\frac{b B-3 A c}{4 b^4 x^4}-\frac{A}{6 b^3 x^6}",1,"((-2*A*b^3)/x^6 - (3*b^2*(b*B - 3*A*c))/x^4 + (18*b*c*(b*B - 2*A*c))/x^2 + (3*b^2*c^2*(b*B - A*c))/(b + c*x^2)^2 + (6*b*c^2*(3*b*B - 4*A*c))/(b + c*x^2) + 24*c^2*(3*b*B - 5*A*c)*Log[x] + 12*c^2*(-3*b*B + 5*A*c)*Log[b + c*x^2])/(12*b^6)","A",1
89,1,193,218,0.3201986,"\int x^7 \left(A+B x^2\right) \sqrt{b x^2+c x^4} \, dx","Integrate[x^7*(A + B*x^2)*Sqrt[b*x^2 + c*x^4],x]","\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(\sqrt{c} x \sqrt{\frac{c x^2}{b}+1} \left(-210 b^4 c \left(2 A+B x^2\right)+56 b^3 c^2 x^2 \left(5 A+3 B x^2\right)-16 b^2 c^3 x^4 \left(14 A+9 B x^2\right)+64 b c^4 x^6 \left(3 A+2 B x^2\right)+256 c^5 x^8 \left(6 A+5 B x^2\right)+315 b^5 B\right)-105 b^{9/2} (3 b B-4 A c) \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)\right)}{15360 c^{11/2} x \sqrt{\frac{c x^2}{b}+1}}","-\frac{7 b^5 (3 b B-4 A c) \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{1024 c^{11/2}}+\frac{7 b^3 \left(b+2 c x^2\right) \sqrt{b x^2+c x^4} (3 b B-4 A c)}{1024 c^5}-\frac{7 b^2 \left(b x^2+c x^4\right)^{3/2} (3 b B-4 A c)}{384 c^4}+\frac{7 b x^2 \left(b x^2+c x^4\right)^{3/2} (3 b B-4 A c)}{320 c^3}-\frac{x^4 \left(b x^2+c x^4\right)^{3/2} (3 b B-4 A c)}{40 c^2}+\frac{B x^6 \left(b x^2+c x^4\right)^{3/2}}{12 c}",1,"(Sqrt[x^2*(b + c*x^2)]*(Sqrt[c]*x*Sqrt[1 + (c*x^2)/b]*(315*b^5*B - 210*b^4*c*(2*A + B*x^2) + 64*b*c^4*x^6*(3*A + 2*B*x^2) + 56*b^3*c^2*x^2*(5*A + 3*B*x^2) + 256*c^5*x^8*(6*A + 5*B*x^2) - 16*b^2*c^3*x^4*(14*A + 9*B*x^2)) - 105*b^(9/2)*(3*b*B - 4*A*c)*ArcSinh[(Sqrt[c]*x)/Sqrt[b]]))/(15360*c^(11/2)*x*Sqrt[1 + (c*x^2)/b])","A",1
90,1,173,181,0.3023206,"\int x^5 \left(A+B x^2\right) \sqrt{b x^2+c x^4} \, dx","Integrate[x^5*(A + B*x^2)*Sqrt[b*x^2 + c*x^4],x]","\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(15 b^{7/2} (7 b B-10 A c) \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)+\sqrt{c} x \sqrt{\frac{c x^2}{b}+1} \left(10 b^3 c \left(15 A+7 B x^2\right)-4 b^2 c^2 x^2 \left(25 A+14 B x^2\right)+16 b c^3 x^4 \left(5 A+3 B x^2\right)+96 c^4 x^6 \left(5 A+4 B x^2\right)-105 b^4 B\right)\right)}{3840 c^{9/2} x \sqrt{\frac{c x^2}{b}+1}}","\frac{b^4 (7 b B-10 A c) \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{256 c^{9/2}}-\frac{b^2 \left(b+2 c x^2\right) \sqrt{b x^2+c x^4} (7 b B-10 A c)}{256 c^4}+\frac{b \left(b x^2+c x^4\right)^{3/2} (7 b B-10 A c)}{96 c^3}-\frac{x^2 \left(b x^2+c x^4\right)^{3/2} (7 b B-10 A c)}{80 c^2}+\frac{B x^4 \left(b x^2+c x^4\right)^{3/2}}{10 c}",1,"(Sqrt[x^2*(b + c*x^2)]*(Sqrt[c]*x*Sqrt[1 + (c*x^2)/b]*(-105*b^4*B + 16*b*c^3*x^4*(5*A + 3*B*x^2) + 96*c^4*x^6*(5*A + 4*B*x^2) + 10*b^3*c*(15*A + 7*B*x^2) - 4*b^2*c^2*x^2*(25*A + 14*B*x^2)) + 15*b^(7/2)*(7*b*B - 10*A*c)*ArcSinh[(Sqrt[c]*x)/Sqrt[b]]))/(3840*c^(9/2)*x*Sqrt[1 + (c*x^2)/b])","A",1
91,1,151,125,0.2271364,"\int x^3 \left(A+B x^2\right) \sqrt{b x^2+c x^4} \, dx","Integrate[x^3*(A + B*x^2)*Sqrt[b*x^2 + c*x^4],x]","\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(\sqrt{c} x \sqrt{\frac{c x^2}{b}+1} \left(-2 b^2 c \left(12 A+5 B x^2\right)+8 b c^2 x^2 \left(2 A+B x^2\right)+16 c^3 x^4 \left(4 A+3 B x^2\right)+15 b^3 B\right)-3 b^{5/2} (5 b B-8 A c) \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)\right)}{384 c^{7/2} x \sqrt{\frac{c x^2}{b}+1}}","-\frac{b^3 (5 b B-8 A c) \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{128 c^{7/2}}+\frac{b \left(b+2 c x^2\right) \sqrt{b x^2+c x^4} (5 b B-8 A c)}{128 c^3}-\frac{\left(b x^2+c x^4\right)^{3/2} \left(-8 A c+5 b B-6 B c x^2\right)}{48 c^2}",1,"(Sqrt[x^2*(b + c*x^2)]*(Sqrt[c]*x*Sqrt[1 + (c*x^2)/b]*(15*b^3*B + 8*b*c^2*x^2*(2*A + B*x^2) + 16*c^3*x^4*(4*A + 3*B*x^2) - 2*b^2*c*(12*A + 5*B*x^2)) - 3*b^(5/2)*(5*b*B - 8*A*c)*ArcSinh[(Sqrt[c]*x)/Sqrt[b]]))/(384*c^(7/2)*x*Sqrt[1 + (c*x^2)/b])","A",1
92,1,129,107,0.1992955,"\int x \left(A+B x^2\right) \sqrt{b x^2+c x^4} \, dx","Integrate[x*(A + B*x^2)*Sqrt[b*x^2 + c*x^4],x]","\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(3 b^{3/2} (b B-2 A c) \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)+\sqrt{c} x \sqrt{\frac{c x^2}{b}+1} \left(2 b c \left(3 A+B x^2\right)+4 c^2 x^2 \left(3 A+2 B x^2\right)-3 b^2 B\right)\right)}{48 c^{5/2} x \sqrt{\frac{c x^2}{b}+1}}","\frac{b^2 (b B-2 A c) \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{16 c^{5/2}}-\frac{\left(b+2 c x^2\right) \sqrt{b x^2+c x^4} (b B-2 A c)}{16 c^2}+\frac{B \left(b x^2+c x^4\right)^{3/2}}{6 c}",1,"(Sqrt[x^2*(b + c*x^2)]*(Sqrt[c]*x*Sqrt[1 + (c*x^2)/b]*(-3*b^2*B + 2*b*c*(3*A + B*x^2) + 4*c^2*x^2*(3*A + 2*B*x^2)) + 3*b^(3/2)*(b*B - 2*A*c)*ArcSinh[(Sqrt[c]*x)/Sqrt[b]]))/(48*c^(5/2)*x*Sqrt[1 + (c*x^2)/b])","A",1
93,1,91,100,0.1550486,"\int \frac{\left(A+B x^2\right) \sqrt{b x^2+c x^4}}{x} \, dx","Integrate[((A + B*x^2)*Sqrt[b*x^2 + c*x^4])/x,x]","\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(\sqrt{c} \left(4 A c+b B+2 B c x^2\right)-\frac{\sqrt{b} (b B-4 A c) \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{x \sqrt{\frac{c x^2}{b}+1}}\right)}{8 c^{3/2}}","-\frac{b (b B-4 A c) \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{8 c^{3/2}}-\frac{\sqrt{b x^2+c x^4} (b B-4 A c)}{8 c}+\frac{B \left(b x^2+c x^4\right)^{3/2}}{4 c x^2}",1,"(Sqrt[x^2*(b + c*x^2)]*(Sqrt[c]*(b*B + 4*A*c + 2*B*c*x^2) - (Sqrt[b]*(b*B - 4*A*c)*ArcSinh[(Sqrt[c]*x)/Sqrt[b]])/(x*Sqrt[1 + (c*x^2)/b])))/(8*c^(3/2))","A",1
94,1,78,97,0.1539813,"\int \frac{\left(A+B x^2\right) \sqrt{b x^2+c x^4}}{x^3} \, dx","Integrate[((A + B*x^2)*Sqrt[b*x^2 + c*x^4])/x^3,x]","\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(\frac{x (2 A c+b B) \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{\sqrt{b} \sqrt{c} \sqrt{\frac{c x^2}{b}+1}}-2 A+B x^2\right)}{2 x^2}","\frac{\sqrt{b x^2+c x^4} (2 A c+b B)}{2 b}+\frac{(2 A c+b B) \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{2 \sqrt{c}}-\frac{A \left(b x^2+c x^4\right)^{3/2}}{b x^4}",1,"(Sqrt[x^2*(b + c*x^2)]*(-2*A + B*x^2 + ((b*B + 2*A*c)*x*ArcSinh[(Sqrt[c]*x)/Sqrt[b]])/(Sqrt[b]*Sqrt[c]*Sqrt[1 + (c*x^2)/b])))/(2*x^2)","A",1
95,1,86,80,0.1298967,"\int \frac{\left(A+B x^2\right) \sqrt{b x^2+c x^4}}{x^5} \, dx","Integrate[((A + B*x^2)*Sqrt[b*x^2 + c*x^4])/x^5,x]","\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(-A \left(b+c x^2\right)+\frac{3 \sqrt{b} B \sqrt{c} x^3 \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{\sqrt{\frac{c x^2}{b}+1}}-3 b B x^2\right)}{3 b x^4}","-\frac{A \left(b x^2+c x^4\right)^{3/2}}{3 b x^6}-\frac{B \sqrt{b x^2+c x^4}}{x^2}+B \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)",1,"(Sqrt[x^2*(b + c*x^2)]*(-3*b*B*x^2 - A*(b + c*x^2) + (3*Sqrt[b]*B*Sqrt[c]*x^3*ArcSinh[(Sqrt[c]*x)/Sqrt[b]])/Sqrt[1 + (c*x^2)/b]))/(3*b*x^4)","A",1
96,1,44,61,0.018963,"\int \frac{\left(A+B x^2\right) \sqrt{b x^2+c x^4}}{x^7} \, dx","Integrate[((A + B*x^2)*Sqrt[b*x^2 + c*x^4])/x^7,x]","-\frac{\left(x^2 \left(b+c x^2\right)\right)^{3/2} \left(3 A b-2 A c x^2+5 b B x^2\right)}{15 b^2 x^8}","-\frac{\left(b x^2+c x^4\right)^{3/2} (5 b B-2 A c)}{15 b^2 x^6}-\frac{A \left(b x^2+c x^4\right)^{3/2}}{5 b x^8}",1,"-1/15*((x^2*(b + c*x^2))^(3/2)*(3*A*b + 5*b*B*x^2 - 2*A*c*x^2))/(b^2*x^8)","A",1
97,1,66,96,0.0268742,"\int \frac{\left(A+B x^2\right) \sqrt{b x^2+c x^4}}{x^9} \, dx","Integrate[((A + B*x^2)*Sqrt[b*x^2 + c*x^4])/x^9,x]","\frac{\left(x^2 \left(b+c x^2\right)\right)^{3/2} \left(A \left(-15 b^2+12 b c x^2-8 c^2 x^4\right)+7 b B x^2 \left(2 c x^2-3 b\right)\right)}{105 b^3 x^{10}}","\frac{2 c \left(b x^2+c x^4\right)^{3/2} (7 b B-4 A c)}{105 b^3 x^6}-\frac{\left(b x^2+c x^4\right)^{3/2} (7 b B-4 A c)}{35 b^2 x^8}-\frac{A \left(b x^2+c x^4\right)^{3/2}}{7 b x^{10}}",1,"((x^2*(b + c*x^2))^(3/2)*(7*b*B*x^2*(-3*b + 2*c*x^2) + A*(-15*b^2 + 12*b*c*x^2 - 8*c^2*x^4)))/(105*b^3*x^10)","A",1
98,1,88,133,0.0303804,"\int \frac{\left(A+B x^2\right) \sqrt{b x^2+c x^4}}{x^{11}} \, dx","Integrate[((A + B*x^2)*Sqrt[b*x^2 + c*x^4])/x^11,x]","-\frac{\left(x^2 \left(b+c x^2\right)\right)^{3/2} \left(A \left(35 b^3-30 b^2 c x^2+24 b c^2 x^4-16 c^3 x^6\right)+3 b B x^2 \left(15 b^2-12 b c x^2+8 c^2 x^4\right)\right)}{315 b^4 x^{12}}","-\frac{8 c^2 \left(b x^2+c x^4\right)^{3/2} (3 b B-2 A c)}{315 b^4 x^6}+\frac{4 c \left(b x^2+c x^4\right)^{3/2} (3 b B-2 A c)}{105 b^3 x^8}-\frac{\left(b x^2+c x^4\right)^{3/2} (3 b B-2 A c)}{21 b^2 x^{10}}-\frac{A \left(b x^2+c x^4\right)^{3/2}}{9 b x^{12}}",1,"-1/315*((x^2*(b + c*x^2))^(3/2)*(3*b*B*x^2*(15*b^2 - 12*b*c*x^2 + 8*c^2*x^4) + A*(35*b^3 - 30*b^2*c*x^2 + 24*b*c^2*x^4 - 16*c^3*x^6)))/(b^4*x^12)","A",1
99,1,94,170,0.0732104,"\int \frac{\left(A+B x^2\right) \sqrt{b x^2+c x^4}}{x^{13}} \, dx","Integrate[((A + B*x^2)*Sqrt[b*x^2 + c*x^4])/x^13,x]","\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(x^2 \left(\frac{c x^2}{b}+1\right) \left(35 b^3-30 b^2 c x^2+24 b c^2 x^4-16 c^3 x^6\right) (8 A c-11 b B)-315 A b^3 \left(b+c x^2\right)\right)}{3465 b^4 x^{12}}","\frac{16 c^3 \left(b x^2+c x^4\right)^{3/2} (11 b B-8 A c)}{3465 b^5 x^6}-\frac{8 c^2 \left(b x^2+c x^4\right)^{3/2} (11 b B-8 A c)}{1155 b^4 x^8}+\frac{2 c \left(b x^2+c x^4\right)^{3/2} (11 b B-8 A c)}{231 b^3 x^{10}}-\frac{\left(b x^2+c x^4\right)^{3/2} (11 b B-8 A c)}{99 b^2 x^{12}}-\frac{A \left(b x^2+c x^4\right)^{3/2}}{11 b x^{14}}",1,"(Sqrt[x^2*(b + c*x^2)]*(-315*A*b^3*(b + c*x^2) + (-11*b*B + 8*A*c)*x^2*(1 + (c*x^2)/b)*(35*b^3 - 30*b^2*c*x^2 + 24*b*c^2*x^4 - 16*c^3*x^6)))/(3465*b^4*x^12)","A",1
100,1,82,131,0.0637064,"\int x^4 \left(A+B x^2\right) \sqrt{b x^2+c x^4} \, dx","Integrate[x^4*(A + B*x^2)*Sqrt[b*x^2 + c*x^4],x]","\frac{\left(x^2 \left(b+c x^2\right)\right)^{3/2} \left(24 b^2 c \left(A+B x^2\right)-6 b c^2 x^2 \left(6 A+5 B x^2\right)+5 c^3 x^4 \left(9 A+7 B x^2\right)-16 b^3 B\right)}{315 c^4 x^3}","-\frac{8 b^2 \left(b x^2+c x^4\right)^{3/2} (2 b B-3 A c)}{315 c^4 x^3}+\frac{4 b \left(b x^2+c x^4\right)^{3/2} (2 b B-3 A c)}{105 c^3 x}-\frac{x \left(b x^2+c x^4\right)^{3/2} (2 b B-3 A c)}{21 c^2}+\frac{B x^3 \left(b x^2+c x^4\right)^{3/2}}{9 c}",1,"((x^2*(b + c*x^2))^(3/2)*(-16*b^3*B + 24*b^2*c*(A + B*x^2) - 6*b*c^2*x^2*(6*A + 5*B*x^2) + 5*c^3*x^4*(9*A + 7*B*x^2)))/(315*c^4*x^3)","A",1
101,1,64,94,0.0457976,"\int x^2 \left(A+B x^2\right) \sqrt{b x^2+c x^4} \, dx","Integrate[x^2*(A + B*x^2)*Sqrt[b*x^2 + c*x^4],x]","\frac{\left(x^2 \left(b+c x^2\right)\right)^{3/2} \left(-2 b c \left(7 A+6 B x^2\right)+3 c^2 x^2 \left(7 A+5 B x^2\right)+8 b^2 B\right)}{105 c^3 x^3}","\frac{2 b \left(b x^2+c x^4\right)^{3/2} (4 b B-7 A c)}{105 c^3 x^3}-\frac{\left(b x^2+c x^4\right)^{3/2} (4 b B-7 A c)}{35 c^2 x}+\frac{B x \left(b x^2+c x^4\right)^{3/2}}{7 c}",1,"((x^2*(b + c*x^2))^(3/2)*(8*b^2*B + 3*c^2*x^2*(7*A + 5*B*x^2) - 2*b*c*(7*A + 6*B*x^2)))/(105*c^3*x^3)","A",1
102,1,41,61,0.0281652,"\int \left(A+B x^2\right) \sqrt{b x^2+c x^4} \, dx","Integrate[(A + B*x^2)*Sqrt[b*x^2 + c*x^4],x]","\frac{\left(x^2 \left(b+c x^2\right)\right)^{3/2} \left(5 A c-2 b B+3 B c x^2\right)}{15 c^2 x^3}","\frac{B \left(b x^2+c x^4\right)^{3/2}}{5 c x}-\frac{\left(b x^2+c x^4\right)^{3/2} (2 b B-5 A c)}{15 c^2 x^3}",1,"((x^2*(b + c*x^2))^(3/2)*(-2*b*B + 5*A*c + 3*B*c*x^2))/(15*c^2*x^3)","A",1
103,1,84,78,0.0857497,"\int \frac{\left(A+B x^2\right) \sqrt{b x^2+c x^4}}{x^2} \, dx","Integrate[((A + B*x^2)*Sqrt[b*x^2 + c*x^4])/x^2,x]","\frac{x \left(\left(b+c x^2\right) \left(3 A c+b B+B c x^2\right)-3 A \sqrt{b} c \sqrt{b+c x^2} \tanh ^{-1}\left(\frac{\sqrt{b+c x^2}}{\sqrt{b}}\right)\right)}{3 c \sqrt{x^2 \left(b+c x^2\right)}}","\frac{A \sqrt{b x^2+c x^4}}{x}-A \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)+\frac{B \left(b x^2+c x^4\right)^{3/2}}{3 c x^3}",1,"(x*((b + c*x^2)*(b*B + 3*A*c + B*c*x^2) - 3*A*Sqrt[b]*c*Sqrt[b + c*x^2]*ArcTanh[Sqrt[b + c*x^2]/Sqrt[b]]))/(3*c*Sqrt[x^2*(b + c*x^2)])","A",1
104,1,94,100,0.0491453,"\int \frac{\left(A+B x^2\right) \sqrt{b x^2+c x^4}}{x^4} \, dx","Integrate[((A + B*x^2)*Sqrt[b*x^2 + c*x^4])/x^4,x]","-\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(\sqrt{b} \left(A-2 B x^2\right) \sqrt{b+c x^2}+x^2 (A c+2 b B) \tanh ^{-1}\left(\frac{\sqrt{b+c x^2}}{\sqrt{b}}\right)\right)}{2 \sqrt{b} x^3 \sqrt{b+c x^2}}","\frac{\sqrt{b x^2+c x^4} (A c+2 b B)}{2 b x}-\frac{(A c+2 b B) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)}{2 \sqrt{b}}-\frac{A \left(b x^2+c x^4\right)^{3/2}}{2 b x^5}",1,"-1/2*(Sqrt[x^2*(b + c*x^2)]*(Sqrt[b]*(A - 2*B*x^2)*Sqrt[b + c*x^2] + (2*b*B + A*c)*x^2*ArcTanh[Sqrt[b + c*x^2]/Sqrt[b]]))/(Sqrt[b]*x^3*Sqrt[b + c*x^2])","A",1
105,1,95,103,0.1185905,"\int \frac{\left(A+B x^2\right) \sqrt{b x^2+c x^4}}{x^6} \, dx","Integrate[((A + B*x^2)*Sqrt[b*x^2 + c*x^4])/x^6,x]","-\frac{\left(b+c x^2\right) \left(2 A b+A c x^2+4 b B x^2\right)+c x^4 \sqrt{\frac{c x^2}{b}+1} (4 b B-A c) \tanh ^{-1}\left(\sqrt{\frac{c x^2}{b}+1}\right)}{8 b x^3 \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{c (4 b B-A c) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)}{8 b^{3/2}}-\frac{\sqrt{b x^2+c x^4} (4 b B-A c)}{8 b x^3}-\frac{A \left(b x^2+c x^4\right)^{3/2}}{4 b x^7}",1,"-1/8*((b + c*x^2)*(2*A*b + 4*b*B*x^2 + A*c*x^2) + c*(4*b*B - A*c)*x^4*Sqrt[1 + (c*x^2)/b]*ArcTanh[Sqrt[1 + (c*x^2)/b]])/(b*x^3*Sqrt[x^2*(b + c*x^2)])","A",1
106,1,215,223,0.3633861,"\int x^5 \left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2} \, dx","Integrate[x^5*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2),x]","\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(\sqrt{c} x \sqrt{\frac{c x^2}{b}+1} \left(-210 b^5 c \left(7 A+3 B x^2\right)+28 b^4 c^2 x^2 \left(35 A+18 B x^2\right)-16 b^3 c^3 x^4 \left(49 A+27 B x^2\right)+96 b^2 c^4 x^6 \left(7 A+4 B x^2\right)+256 b c^5 x^8 \left(91 A+75 B x^2\right)+2560 c^6 x^{10} \left(7 A+6 B x^2\right)+945 b^6 B\right)-105 b^{11/2} (9 b B-14 A c) \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)\right)}{215040 c^{11/2} x \sqrt{\frac{c x^2}{b}+1}}","-\frac{b^6 (9 b B-14 A c) \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{2048 c^{11/2}}+\frac{b^4 \left(b+2 c x^2\right) \sqrt{b x^2+c x^4} (9 b B-14 A c)}{2048 c^5}-\frac{b^2 \left(b+2 c x^2\right) \left(b x^2+c x^4\right)^{3/2} (9 b B-14 A c)}{768 c^4}+\frac{b \left(b x^2+c x^4\right)^{5/2} (9 b B-14 A c)}{240 c^3}-\frac{x^2 \left(b x^2+c x^4\right)^{5/2} (9 b B-14 A c)}{168 c^2}+\frac{B x^4 \left(b x^2+c x^4\right)^{5/2}}{14 c}",1,"(Sqrt[x^2*(b + c*x^2)]*(Sqrt[c]*x*Sqrt[1 + (c*x^2)/b]*(945*b^6*B - 210*b^5*c*(7*A + 3*B*x^2) + 96*b^2*c^4*x^6*(7*A + 4*B*x^2) + 2560*c^6*x^10*(7*A + 6*B*x^2) + 28*b^4*c^2*x^2*(35*A + 18*B*x^2) - 16*b^3*c^3*x^4*(49*A + 27*B*x^2) + 256*b*c^5*x^8*(91*A + 75*B*x^2)) - 105*b^(11/2)*(9*b*B - 14*A*c)*ArcSinh[(Sqrt[c]*x)/Sqrt[b]]))/(215040*c^(11/2)*x*Sqrt[1 + (c*x^2)/b])","A",1
107,1,193,167,0.3192709,"\int x^3 \left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2} \, dx","Integrate[x^3*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2),x]","\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(15 b^{9/2} (7 b B-12 A c) \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)+\sqrt{c} x \sqrt{\frac{c x^2}{b}+1} \left(10 b^4 c \left(18 A+7 B x^2\right)-8 b^3 c^2 x^2 \left(15 A+7 B x^2\right)+48 b^2 c^3 x^4 \left(2 A+B x^2\right)+64 b c^4 x^6 \left(33 A+26 B x^2\right)+256 c^5 x^8 \left(6 A+5 B x^2\right)-105 b^5 B\right)\right)}{15360 c^{9/2} x \sqrt{\frac{c x^2}{b}+1}}","\frac{b^5 (7 b B-12 A c) \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{1024 c^{9/2}}-\frac{b^3 \left(b+2 c x^2\right) \sqrt{b x^2+c x^4} (7 b B-12 A c)}{1024 c^4}+\frac{b \left(b+2 c x^2\right) \left(b x^2+c x^4\right)^{3/2} (7 b B-12 A c)}{384 c^3}-\frac{\left(b x^2+c x^4\right)^{5/2} \left(-12 A c+7 b B-10 B c x^2\right)}{120 c^2}",1,"(Sqrt[x^2*(b + c*x^2)]*(Sqrt[c]*x*Sqrt[1 + (c*x^2)/b]*(-105*b^5*B + 48*b^2*c^3*x^4*(2*A + B*x^2) + 256*c^5*x^8*(6*A + 5*B*x^2) - 8*b^3*c^2*x^2*(15*A + 7*B*x^2) + 10*b^4*c*(18*A + 7*B*x^2) + 64*b*c^4*x^6*(33*A + 26*B*x^2)) + 15*b^(9/2)*(7*b*B - 12*A*c)*ArcSinh[(Sqrt[c]*x)/Sqrt[b]]))/(15360*c^(9/2)*x*Sqrt[1 + (c*x^2)/b])","A",1
108,1,171,148,0.2674095,"\int x \left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2} \, dx","Integrate[x*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2),x]","\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(\sqrt{c} x \sqrt{\frac{c x^2}{b}+1} \left(-10 b^3 c \left(3 A+B x^2\right)+4 b^2 c^2 x^2 \left(5 A+2 B x^2\right)+16 b c^3 x^4 \left(15 A+11 B x^2\right)+32 c^4 x^6 \left(5 A+4 B x^2\right)+15 b^4 B\right)-15 b^{7/2} (b B-2 A c) \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)\right)}{1280 c^{7/2} x \sqrt{\frac{c x^2}{b}+1}}","-\frac{3 b^4 (b B-2 A c) \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{256 c^{7/2}}+\frac{3 b^2 \left(b+2 c x^2\right) \sqrt{b x^2+c x^4} (b B-2 A c)}{256 c^3}-\frac{\left(b+2 c x^2\right) \left(b x^2+c x^4\right)^{3/2} (b B-2 A c)}{32 c^2}+\frac{B \left(b x^2+c x^4\right)^{5/2}}{10 c}",1,"(Sqrt[x^2*(b + c*x^2)]*(Sqrt[c]*x*Sqrt[1 + (c*x^2)/b]*(15*b^4*B - 10*b^3*c*(3*A + B*x^2) + 4*b^2*c^2*x^2*(5*A + 2*B*x^2) + 32*c^4*x^6*(5*A + 4*B*x^2) + 16*b*c^3*x^4*(15*A + 11*B*x^2)) - 15*b^(7/2)*(b*B - 2*A*c)*ArcSinh[(Sqrt[c]*x)/Sqrt[b]]))/(1280*c^(7/2)*x*Sqrt[1 + (c*x^2)/b])","A",1
109,1,151,144,0.2425106,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2}}{x} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x,x]","\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(3 b^{5/2} (3 b B-8 A c) \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)+\sqrt{c} x \sqrt{\frac{c x^2}{b}+1} \left(6 b^2 c \left(4 A+B x^2\right)+8 b c^2 x^2 \left(14 A+9 B x^2\right)+16 c^3 x^4 \left(4 A+3 B x^2\right)-9 b^3 B\right)\right)}{384 c^{5/2} x \sqrt{\frac{c x^2}{b}+1}}","\frac{b^3 (3 b B-8 A c) \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{128 c^{5/2}}-\frac{b \left(b+2 c x^2\right) \sqrt{b x^2+c x^4} (3 b B-8 A c)}{128 c^2}-\frac{\left(b x^2+c x^4\right)^{3/2} (3 b B-8 A c)}{48 c}+\frac{B \left(b x^2+c x^4\right)^{5/2}}{8 c x^2}",1,"(Sqrt[x^2*(b + c*x^2)]*(Sqrt[c]*x*Sqrt[1 + (c*x^2)/b]*(-9*b^3*B + 6*b^2*c*(4*A + B*x^2) + 16*c^3*x^4*(4*A + 3*B*x^2) + 8*b*c^2*x^2*(14*A + 9*B*x^2)) + 3*b^(5/2)*(3*b*B - 8*A*c)*ArcSinh[(Sqrt[c]*x)/Sqrt[b]]))/(384*c^(5/2)*x*Sqrt[1 + (c*x^2)/b])","A",1
110,1,130,137,0.1255511,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2}}{x^3} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^3,x]","\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(\sqrt{c} x \sqrt{\frac{c x^2}{b}+1} \left(2 b c \left(15 A+7 B x^2\right)+4 c^2 x^2 \left(3 A+2 B x^2\right)+3 b^2 B\right)-3 b^{3/2} (b B-6 A c) \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)\right)}{48 c^{3/2} x \sqrt{\frac{c x^2}{b}+1}}","-\frac{b^2 (b B-6 A c) \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{16 c^{3/2}}+\frac{\left(b x^2+c x^4\right)^{3/2} (b B-6 A c)}{6 b}+\frac{\left(b+2 c x^2\right) \sqrt{b x^2+c x^4} (b B-6 A c)}{16 c}+\frac{A \left(b x^2+c x^4\right)^{5/2}}{b x^4}",1,"(Sqrt[x^2*(b + c*x^2)]*(Sqrt[c]*x*Sqrt[1 + (c*x^2)/b]*(3*b^2*B + 4*c^2*x^2*(3*A + 2*B*x^2) + 2*b*c*(15*A + 7*B*x^2)) - 3*b^(3/2)*(b*B - 6*A*c)*ArcSinh[(Sqrt[c]*x)/Sqrt[b]]))/(48*c^(3/2)*x*Sqrt[1 + (c*x^2)/b])","A",1
111,1,96,128,0.2045256,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2}}{x^5} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^5,x]","\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(\frac{3 \sqrt{b} x (4 A c+b B) \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{\sqrt{c} \sqrt{\frac{c x^2}{b}+1}}-8 A b+4 A c x^2+5 b B x^2+2 B c x^4\right)}{8 x^2}","\frac{\left(b x^2+c x^4\right)^{3/2} (4 A c+b B)}{4 b x^2}+\frac{3}{8} \sqrt{b x^2+c x^4} (4 A c+b B)+\frac{3 b (4 A c+b B) \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{8 \sqrt{c}}-\frac{A \left(b x^2+c x^4\right)^{5/2}}{b x^6}",1,"(Sqrt[x^2*(b + c*x^2)]*(-8*A*b + 5*b*B*x^2 + 4*A*c*x^2 + 2*B*c*x^4 + (3*Sqrt[b]*(b*B + 4*A*c)*x*ArcSinh[(Sqrt[c]*x)/Sqrt[b]])/(Sqrt[c]*Sqrt[1 + (c*x^2)/b])))/(8*x^2)","A",1
112,1,98,136,0.0541857,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2}}{x^7} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^7,x]","-\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(b x^2 (2 A c+3 b B) \, _2F_1\left(-\frac{3}{2},-\frac{1}{2};\frac{1}{2};-\frac{c x^2}{b}\right)+A \left(b+c x^2\right)^2 \sqrt{\frac{c x^2}{b}+1}\right)}{3 b x^4 \sqrt{\frac{c x^2}{b}+1}}","-\frac{\left(b x^2+c x^4\right)^{3/2} (2 A c+3 b B)}{3 b x^4}+\frac{c \sqrt{b x^2+c x^4} (2 A c+3 b B)}{2 b}+\frac{1}{2} \sqrt{c} (2 A c+3 b B) \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)-\frac{A \left(b x^2+c x^4\right)^{5/2}}{3 b x^8}",1,"-1/3*(Sqrt[x^2*(b + c*x^2)]*(A*(b + c*x^2)^2*Sqrt[1 + (c*x^2)/b] + b*(3*b*B + 2*A*c)*x^2*Hypergeometric2F1[-3/2, -1/2, 1/2, -((c*x^2)/b)]))/(b*x^4*Sqrt[1 + (c*x^2)/b])","C",1
113,1,94,104,0.0574648,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2}}{x^9} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^9,x]","-\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(3 A \left(b+c x^2\right)^2 \sqrt{\frac{c x^2}{b}+1}+5 b^2 B x^2 \, _2F_1\left(-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};-\frac{c x^2}{b}\right)\right)}{15 b x^6 \sqrt{\frac{c x^2}{b}+1}}","-\frac{A \left(b x^2+c x^4\right)^{5/2}}{5 b x^{10}}+B c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)-\frac{B c \sqrt{b x^2+c x^4}}{x^2}-\frac{B \left(b x^2+c x^4\right)^{3/2}}{3 x^6}",1,"-1/15*(Sqrt[x^2*(b + c*x^2)]*(3*A*(b + c*x^2)^2*Sqrt[1 + (c*x^2)/b] + 5*b^2*B*x^2*Hypergeometric2F1[-3/2, -3/2, -1/2, -((c*x^2)/b)]))/(b*x^6*Sqrt[1 + (c*x^2)/b])","C",1
114,1,44,61,0.0259178,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2}}{x^{11}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^11,x]","-\frac{\left(x^2 \left(b+c x^2\right)\right)^{5/2} \left(5 A b-2 A c x^2+7 b B x^2\right)}{35 b^2 x^{12}}","-\frac{\left(b x^2+c x^4\right)^{5/2} (7 b B-2 A c)}{35 b^2 x^{10}}-\frac{A \left(b x^2+c x^4\right)^{5/2}}{7 b x^{12}}",1,"-1/35*((x^2*(b + c*x^2))^(5/2)*(5*A*b + 7*b*B*x^2 - 2*A*c*x^2))/(b^2*x^12)","A",1
115,1,66,96,0.0321866,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2}}{x^{13}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^13,x]","\frac{\left(x^2 \left(b+c x^2\right)\right)^{5/2} \left(A \left(-35 b^2+20 b c x^2-8 c^2 x^4\right)+9 b B x^2 \left(2 c x^2-5 b\right)\right)}{315 b^3 x^{14}}","\frac{2 c \left(b x^2+c x^4\right)^{5/2} (9 b B-4 A c)}{315 b^3 x^{10}}-\frac{\left(b x^2+c x^4\right)^{5/2} (9 b B-4 A c)}{63 b^2 x^{12}}-\frac{A \left(b x^2+c x^4\right)^{5/2}}{9 b x^{14}}",1,"((x^2*(b + c*x^2))^(5/2)*(9*b*B*x^2*(-5*b + 2*c*x^2) + A*(-35*b^2 + 20*b*c*x^2 - 8*c^2*x^4)))/(315*b^3*x^14)","A",1
116,1,89,133,0.0379057,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2}}{x^{15}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^15,x]","-\frac{\left(x^2 \left(b+c x^2\right)\right)^{5/2} \left(3 A \left(105 b^3-70 b^2 c x^2+40 b c^2 x^4-16 c^3 x^6\right)+11 b B x^2 \left(35 b^2-20 b c x^2+8 c^2 x^4\right)\right)}{3465 b^4 x^{16}}","-\frac{8 c^2 \left(b x^2+c x^4\right)^{5/2} (11 b B-6 A c)}{3465 b^4 x^{10}}+\frac{4 c \left(b x^2+c x^4\right)^{5/2} (11 b B-6 A c)}{693 b^3 x^{12}}-\frac{\left(b x^2+c x^4\right)^{5/2} (11 b B-6 A c)}{99 b^2 x^{14}}-\frac{A \left(b x^2+c x^4\right)^{5/2}}{11 b x^{16}}",1,"-1/3465*((x^2*(b + c*x^2))^(5/2)*(11*b*B*x^2*(35*b^2 - 20*b*c*x^2 + 8*c^2*x^4) + 3*A*(105*b^3 - 70*b^2*c*x^2 + 40*b*c^2*x^4 - 16*c^3*x^6)))/(b^4*x^16)","A",1
117,1,95,170,0.0852624,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2}}{x^{17}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^17,x]","\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(\left(\frac{c x^3}{b}+x\right)^2 \left(105 b^3-70 b^2 c x^2+40 b c^2 x^4-16 c^3 x^6\right) (8 A c-13 b B)-1155 A b^2 \left(b+c x^2\right)^2\right)}{15015 b^3 x^{14}}","\frac{16 c^3 \left(b x^2+c x^4\right)^{5/2} (13 b B-8 A c)}{15015 b^5 x^{10}}-\frac{8 c^2 \left(b x^2+c x^4\right)^{5/2} (13 b B-8 A c)}{3003 b^4 x^{12}}+\frac{2 c \left(b x^2+c x^4\right)^{5/2} (13 b B-8 A c)}{429 b^3 x^{14}}-\frac{\left(b x^2+c x^4\right)^{5/2} (13 b B-8 A c)}{143 b^2 x^{16}}-\frac{A \left(b x^2+c x^4\right)^{5/2}}{13 b x^{18}}",1,"(Sqrt[x^2*(b + c*x^2)]*(-1155*A*b^2*(b + c*x^2)^2 + (-13*b*B + 8*A*c)*(x + (c*x^3)/b)^2*(105*b^3 - 70*b^2*c*x^2 + 40*b*c^2*x^4 - 16*c^3*x^6)))/(15015*b^3*x^14)","A",1
118,1,89,207,0.0768585,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2}}{x^{19}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^19,x]","\frac{\left(x^2 \left(b+c x^2\right)\right)^{5/2} \left(-3003 A b^5-x^2 \left(1155 b^4-840 b^3 c x^2+560 b^2 c^2 x^4-320 b c^3 x^6+128 c^4 x^8\right) (3 b B-2 A c)\right)}{45045 b^6 x^{20}}","-\frac{128 c^4 \left(b x^2+c x^4\right)^{5/2} (3 b B-2 A c)}{45045 b^6 x^{10}}+\frac{64 c^3 \left(b x^2+c x^4\right)^{5/2} (3 b B-2 A c)}{9009 b^5 x^{12}}-\frac{16 c^2 \left(b x^2+c x^4\right)^{5/2} (3 b B-2 A c)}{1287 b^4 x^{14}}+\frac{8 c \left(b x^2+c x^4\right)^{5/2} (3 b B-2 A c)}{429 b^3 x^{16}}-\frac{\left(b x^2+c x^4\right)^{5/2} (3 b B-2 A c)}{39 b^2 x^{18}}-\frac{A \left(b x^2+c x^4\right)^{5/2}}{15 b x^{20}}",1,"((x^2*(b + c*x^2))^(5/2)*(-3003*A*b^5 - (3*b*B - 2*A*c)*x^2*(1155*b^4 - 840*b^3*c*x^2 + 560*b^2*c^2*x^4 - 320*b*c^3*x^6 + 128*c^4*x^8)))/(45045*b^6*x^20)","A",1
119,1,113,168,0.0896249,"\int x^4 \left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2} \, dx","Integrate[x^4*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2),x]","\frac{x \left(b+c x^2\right)^3 \left(-16 b^3 c \left(13 A+20 B x^2\right)+40 b^2 c^2 x^2 \left(13 A+14 B x^2\right)-70 b c^3 x^4 \left(13 A+12 B x^2\right)+105 c^4 x^6 \left(13 A+11 B x^2\right)+128 b^4 B\right)}{15015 c^5 \sqrt{x^2 \left(b+c x^2\right)}}","\frac{16 b^3 \left(b x^2+c x^4\right)^{5/2} (8 b B-13 A c)}{15015 c^5 x^5}-\frac{8 b^2 \left(b x^2+c x^4\right)^{5/2} (8 b B-13 A c)}{3003 c^4 x^3}+\frac{2 b \left(b x^2+c x^4\right)^{5/2} (8 b B-13 A c)}{429 c^3 x}-\frac{x \left(b x^2+c x^4\right)^{5/2} (8 b B-13 A c)}{143 c^2}+\frac{B x^3 \left(b x^2+c x^4\right)^{5/2}}{13 c}",1,"(x*(b + c*x^2)^3*(128*b^4*B + 105*c^4*x^6*(13*A + 11*B*x^2) - 70*b*c^3*x^4*(13*A + 12*B*x^2) + 40*b^2*c^2*x^2*(13*A + 14*B*x^2) - 16*b^3*c*(13*A + 20*B*x^2)))/(15015*c^5*Sqrt[x^2*(b + c*x^2)])","A",1
120,1,92,131,0.0758462,"\int x^2 \left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2} \, dx","Integrate[x^2*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2),x]","\frac{x \left(b+c x^2\right)^3 \left(8 b^2 c \left(11 A+15 B x^2\right)-10 b c^2 x^2 \left(22 A+21 B x^2\right)+35 c^3 x^4 \left(11 A+9 B x^2\right)-48 b^3 B\right)}{3465 c^4 \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{8 b^2 \left(b x^2+c x^4\right)^{5/2} (6 b B-11 A c)}{3465 c^4 x^5}+\frac{4 b \left(b x^2+c x^4\right)^{5/2} (6 b B-11 A c)}{693 c^3 x^3}-\frac{\left(b x^2+c x^4\right)^{5/2} (6 b B-11 A c)}{99 c^2 x}+\frac{B x \left(b x^2+c x^4\right)^{5/2}}{11 c}",1,"(x*(b + c*x^2)^3*(-48*b^3*B + 35*c^3*x^4*(11*A + 9*B*x^2) + 8*b^2*c*(11*A + 15*B*x^2) - 10*b*c^2*x^2*(22*A + 21*B*x^2)))/(3465*c^4*Sqrt[x^2*(b + c*x^2)])","A",1
121,1,71,96,0.0535103,"\int \left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2} \, dx","Integrate[(A + B*x^2)*(b*x^2 + c*x^4)^(3/2),x]","\frac{x \left(b+c x^2\right)^3 \left(-2 b c \left(9 A+10 B x^2\right)+5 c^2 x^2 \left(9 A+7 B x^2\right)+8 b^2 B\right)}{315 c^3 \sqrt{x^2 \left(b+c x^2\right)}}","\frac{2 b \left(b x^2+c x^4\right)^{5/2} (4 b B-9 A c)}{315 c^3 x^5}-\frac{\left(b x^2+c x^4\right)^{5/2} (4 b B-9 A c)}{63 c^2 x^3}+\frac{B \left(b x^2+c x^4\right)^{5/2}}{9 c x}",1,"(x*(b + c*x^2)^3*(8*b^2*B + 5*c^2*x^2*(9*A + 7*B*x^2) - 2*b*c*(9*A + 10*B*x^2)))/(315*c^3*Sqrt[x^2*(b + c*x^2)])","A",1
122,1,48,61,0.0354962,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2}}{x^2} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^2,x]","\frac{x \left(b+c x^2\right)^3 \left(7 A c-2 b B+5 B c x^2\right)}{35 c^2 \sqrt{x^2 \left(b+c x^2\right)}}","\frac{B \left(b x^2+c x^4\right)^{5/2}}{7 c x^3}-\frac{\left(b x^2+c x^4\right)^{5/2} (2 b B-7 A c)}{35 c^2 x^5}",1,"(x*(b + c*x^2)^3*(-2*b*B + 7*A*c + 5*B*c*x^2))/(35*c^2*Sqrt[x^2*(b + c*x^2)])","A",1
123,1,109,102,0.0973072,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2}}{x^4} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^4,x]","\frac{\left(x^2 \left(b+c x^2\right)\right)^{3/2} \left(-15 A b^{3/2} c \tanh ^{-1}\left(\frac{\sqrt{b+c x^2}}{\sqrt{b}}\right)+5 A c \left(b+c x^2\right)^{3/2}+15 A b c \sqrt{b+c x^2}+3 B \left(b+c x^2\right)^{5/2}\right)}{15 c x^3 \left(b+c x^2\right)^{3/2}}","-A b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)+\frac{A b \sqrt{b x^2+c x^4}}{x}+\frac{A \left(b x^2+c x^4\right)^{3/2}}{3 x^3}+\frac{B \left(b x^2+c x^4\right)^{5/2}}{5 c x^5}",1,"((x^2*(b + c*x^2))^(3/2)*(15*A*b*c*Sqrt[b + c*x^2] + 5*A*c*(b + c*x^2)^(3/2) + 3*B*(b + c*x^2)^(5/2) - 15*A*b^(3/2)*c*ArcTanh[Sqrt[b + c*x^2]/Sqrt[b]]))/(15*c*x^3*(b + c*x^2)^(3/2))","A",1
124,1,109,133,0.0701544,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2}}{x^6} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^6,x]","\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(\sqrt{b+c x^2} \left(-3 A b+6 A c x^2+8 b B x^2+2 B c x^4\right)-3 \sqrt{b} x^2 (3 A c+2 b B) \tanh ^{-1}\left(\frac{\sqrt{b+c x^2}}{\sqrt{b}}\right)\right)}{6 x^3 \sqrt{b+c x^2}}","\frac{\sqrt{b x^2+c x^4} (3 A c+2 b B)}{2 x}-\frac{1}{2} \sqrt{b} (3 A c+2 b B) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)+\frac{\left(b x^2+c x^4\right)^{3/2} (3 A c+2 b B)}{6 b x^3}-\frac{A \left(b x^2+c x^4\right)^{5/2}}{2 b x^7}",1,"(Sqrt[x^2*(b + c*x^2)]*(Sqrt[b + c*x^2]*(-3*A*b + 8*b*B*x^2 + 6*A*c*x^2 + 2*B*c*x^4) - 3*Sqrt[b]*(2*b*B + 3*A*c)*x^2*ArcTanh[Sqrt[b + c*x^2]/Sqrt[b]]))/(6*x^3*Sqrt[b + c*x^2])","A",1
125,1,63,135,0.0395298,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2}}{x^8} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^8,x]","\frac{\left(x^2 \left(b+c x^2\right)\right)^{5/2} \left(c x^4 (A c+4 b B) \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};\frac{c x^2}{b}+1\right)-5 A b^2\right)}{20 b^3 x^9}","\frac{3 c \sqrt{b x^2+c x^4} (A c+4 b B)}{8 b x}-\frac{3 c (A c+4 b B) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)}{8 \sqrt{b}}-\frac{\left(b x^2+c x^4\right)^{3/2} (A c+4 b B)}{8 b x^5}-\frac{A \left(b x^2+c x^4\right)^{5/2}}{4 b x^9}",1,"((x^2*(b + c*x^2))^(5/2)*(-5*A*b^2 + c*(4*b*B + A*c)*x^4*Hypergeometric2F1[2, 5/2, 7/2, 1 + (c*x^2)/b]))/(20*b^3*x^9)","C",1
126,1,121,140,0.1432888,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2}}{x^{10}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^10,x]","-\frac{\left(b+c x^2\right) \left(A \left(8 b^2+14 b c x^2+3 c^2 x^4\right)+6 b B x^2 \left(2 b+5 c x^2\right)\right)+3 c^2 x^6 \sqrt{\frac{c x^2}{b}+1} (6 b B-A c) \tanh ^{-1}\left(\sqrt{\frac{c x^2}{b}+1}\right)}{48 b x^5 \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{c^2 (6 b B-A c) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)}{16 b^{3/2}}-\frac{\left(b x^2+c x^4\right)^{3/2} (6 b B-A c)}{24 b x^7}-\frac{c \sqrt{b x^2+c x^4} (6 b B-A c)}{16 b x^3}-\frac{A \left(b x^2+c x^4\right)^{5/2}}{6 b x^{11}}",1,"-1/48*((b + c*x^2)*(6*b*B*x^2*(2*b + 5*c*x^2) + A*(8*b^2 + 14*b*c*x^2 + 3*c^2*x^4)) + 3*c^2*(6*b*B - A*c)*x^6*Sqrt[1 + (c*x^2)/b]*ArcTanh[Sqrt[1 + (c*x^2)/b]])/(b*x^5*Sqrt[x^2*(b + c*x^2)])","A",1
127,1,66,177,0.0365035,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2}}{x^{12}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^12,x]","\frac{\left(x^2 \left(b+c x^2\right)\right)^{5/2} \left(c^3 x^8 (8 b B-3 A c) \, _2F_1\left(\frac{5}{2},4;\frac{7}{2};\frac{c x^2}{b}+1\right)-5 A b^4\right)}{40 b^5 x^{13}}","\frac{c^3 (8 b B-3 A c) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)}{128 b^{5/2}}-\frac{c^2 \sqrt{b x^2+c x^4} (8 b B-3 A c)}{128 b^2 x^3}-\frac{\left(b x^2+c x^4\right)^{3/2} (8 b B-3 A c)}{48 b x^9}-\frac{c \sqrt{b x^2+c x^4} (8 b B-3 A c)}{64 b x^5}-\frac{A \left(b x^2+c x^4\right)^{5/2}}{8 b x^{13}}",1,"((x^2*(b + c*x^2))^(5/2)*(-5*A*b^4 + c^3*(8*b*B - 3*A*c)*x^8*Hypergeometric2F1[5/2, 4, 7/2, 1 + (c*x^2)/b]))/(40*b^5*x^13)","C",1
128,1,65,214,0.0361988,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2}}{x^{14}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^14,x]","-\frac{\left(x^2 \left(b+c x^2\right)\right)^{5/2} \left(A b^5+c^4 x^{10} (2 b B-A c) \, _2F_1\left(\frac{5}{2},5;\frac{7}{2};\frac{c x^2}{b}+1\right)\right)}{10 b^6 x^{15}}","-\frac{3 c^4 (2 b B-A c) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)}{256 b^{7/2}}+\frac{3 c^3 \sqrt{b x^2+c x^4} (2 b B-A c)}{256 b^3 x^3}-\frac{c^2 \sqrt{b x^2+c x^4} (2 b B-A c)}{128 b^2 x^5}-\frac{\left(b x^2+c x^4\right)^{3/2} (2 b B-A c)}{16 b x^{11}}-\frac{c \sqrt{b x^2+c x^4} (2 b B-A c)}{32 b x^7}-\frac{A \left(b x^2+c x^4\right)^{5/2}}{10 b x^{15}}",1,"-1/10*((x^2*(b + c*x^2))^(5/2)*(A*b^5 + c^4*(2*b*B - A*c)*x^10*Hypergeometric2F1[5/2, 5, 7/2, 1 + (c*x^2)/b]))/(b^6*x^15)","C",1
129,1,66,251,0.0396367,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2}}{x^{16}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^16,x]","\frac{\left(x^2 \left(b+c x^2\right)\right)^{5/2} \left(c^5 x^{12} (12 b B-7 A c) \, _2F_1\left(\frac{5}{2},6;\frac{7}{2};\frac{c x^2}{b}+1\right)-5 A b^6\right)}{60 b^7 x^{17}}","\frac{c^5 (12 b B-7 A c) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)}{1024 b^{9/2}}-\frac{c^4 \sqrt{b x^2+c x^4} (12 b B-7 A c)}{1024 b^4 x^3}+\frac{c^3 \sqrt{b x^2+c x^4} (12 b B-7 A c)}{1536 b^3 x^5}-\frac{c^2 \sqrt{b x^2+c x^4} (12 b B-7 A c)}{1920 b^2 x^7}-\frac{\left(b x^2+c x^4\right)^{3/2} (12 b B-7 A c)}{120 b x^{13}}-\frac{c \sqrt{b x^2+c x^4} (12 b B-7 A c)}{320 b x^9}-\frac{A \left(b x^2+c x^4\right)^{5/2}}{12 b x^{17}}",1,"((x^2*(b + c*x^2))^(5/2)*(-5*A*b^6 + c^5*(12*b*B - 7*A*c)*x^12*Hypergeometric2F1[5/2, 6, 7/2, 1 + (c*x^2)/b]))/(60*b^7*x^17)","C",1
130,1,145,176,0.2619746,"\int \frac{x^7 \left(A+B x^2\right)}{\sqrt{b x^2+c x^4}} \, dx","Integrate[(x^7*(A + B*x^2))/Sqrt[b*x^2 + c*x^4],x]","\frac{x \left(15 b^3 \sqrt{b+c x^2} (7 b B-8 A c) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b+c x^2}}\right)-\sqrt{c} x \left(b+c x^2\right) \left(-10 b^2 c \left(12 A+7 B x^2\right)+8 b c^2 x^2 \left(10 A+7 B x^2\right)-16 c^3 x^4 \left(4 A+3 B x^2\right)+105 b^3 B\right)\right)}{384 c^{9/2} \sqrt{x^2 \left(b+c x^2\right)}}","\frac{5 b^3 (7 b B-8 A c) \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{128 c^{9/2}}-\frac{5 b^2 \sqrt{b x^2+c x^4} (7 b B-8 A c)}{128 c^4}+\frac{5 b x^2 \sqrt{b x^2+c x^4} (7 b B-8 A c)}{192 c^3}-\frac{x^4 \sqrt{b x^2+c x^4} (7 b B-8 A c)}{48 c^2}+\frac{B x^6 \sqrt{b x^2+c x^4}}{8 c}",1,"(x*(-(Sqrt[c]*x*(b + c*x^2)*(105*b^3*B - 16*c^3*x^4*(4*A + 3*B*x^2) + 8*b*c^2*x^2*(10*A + 7*B*x^2) - 10*b^2*c*(12*A + 7*B*x^2))) + 15*b^3*(7*b*B - 8*A*c)*Sqrt[b + c*x^2]*ArcTanh[(Sqrt[c]*x)/Sqrt[b + c*x^2]]))/(384*c^(9/2)*Sqrt[x^2*(b + c*x^2)])","A",1
131,1,123,139,0.1820895,"\int \frac{x^5 \left(A+B x^2\right)}{\sqrt{b x^2+c x^4}} \, dx","Integrate[(x^5*(A + B*x^2))/Sqrt[b*x^2 + c*x^4],x]","\frac{x \left(\sqrt{c} x \left(b+c x^2\right) \left(-2 b c \left(9 A+5 B x^2\right)+4 c^2 x^2 \left(3 A+2 B x^2\right)+15 b^2 B\right)-3 b^2 \sqrt{b+c x^2} (5 b B-6 A c) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b+c x^2}}\right)\right)}{48 c^{7/2} \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{b^2 (5 b B-6 A c) \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{16 c^{7/2}}+\frac{b \sqrt{b x^2+c x^4} (5 b B-6 A c)}{16 c^3}-\frac{x^2 \sqrt{b x^2+c x^4} (5 b B-6 A c)}{24 c^2}+\frac{B x^4 \sqrt{b x^2+c x^4}}{6 c}",1,"(x*(Sqrt[c]*x*(b + c*x^2)*(15*b^2*B + 4*c^2*x^2*(3*A + 2*B*x^2) - 2*b*c*(9*A + 5*B*x^2)) - 3*b^2*(5*b*B - 6*A*c)*Sqrt[b + c*x^2]*ArcTanh[(Sqrt[c]*x)/Sqrt[b + c*x^2]]))/(48*c^(7/2)*Sqrt[x^2*(b + c*x^2)])","A",1
132,1,97,83,0.1338178,"\int \frac{x^3 \left(A+B x^2\right)}{\sqrt{b x^2+c x^4}} \, dx","Integrate[(x^3*(A + B*x^2))/Sqrt[b*x^2 + c*x^4],x]","\frac{x \left(\sqrt{c} x \left(b+c x^2\right) \left(4 A c-3 b B+2 B c x^2\right)+b \sqrt{b+c x^2} (3 b B-4 A c) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b+c x^2}}\right)\right)}{8 c^{5/2} \sqrt{x^2 \left(b+c x^2\right)}}","\frac{b (3 b B-4 A c) \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{8 c^{5/2}}-\frac{\sqrt{b x^2+c x^4} \left(-4 A c+3 b B-2 B c x^2\right)}{8 c^2}",1,"(x*(Sqrt[c]*x*(b + c*x^2)*(-3*b*B + 4*A*c + 2*B*c*x^2) + b*(3*b*B - 4*A*c)*Sqrt[b + c*x^2]*ArcTanh[(Sqrt[c]*x)/Sqrt[b + c*x^2]]))/(8*c^(5/2)*Sqrt[x^2*(b + c*x^2)])","A",1
133,1,81,66,0.0508184,"\int \frac{x \left(A+B x^2\right)}{\sqrt{b x^2+c x^4}} \, dx","Integrate[(x*(A + B*x^2))/Sqrt[b*x^2 + c*x^4],x]","\frac{x \left(B \sqrt{c} x \left(b+c x^2\right)-\sqrt{b+c x^2} (b B-2 A c) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b+c x^2}}\right)\right)}{2 c^{3/2} \sqrt{x^2 \left(b+c x^2\right)}}","\frac{B \sqrt{b x^2+c x^4}}{2 c}-\frac{(b B-2 A c) \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{2 c^{3/2}}",1,"(x*(B*Sqrt[c]*x*(b + c*x^2) - (b*B - 2*A*c)*Sqrt[b + c*x^2]*ArcTanh[(Sqrt[c]*x)/Sqrt[b + c*x^2]]))/(2*c^(3/2)*Sqrt[x^2*(b + c*x^2)])","A",1
134,1,74,57,0.041447,"\int \frac{A+B x^2}{x \sqrt{b x^2+c x^4}} \, dx","Integrate[(A + B*x^2)/(x*Sqrt[b*x^2 + c*x^4]),x]","\frac{b B x \sqrt{b+c x^2} \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b+c x^2}}\right)-A \sqrt{c} \left(b+c x^2\right)}{b \sqrt{c} \sqrt{x^2 \left(b+c x^2\right)}}","\frac{B \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{\sqrt{c}}-\frac{A \sqrt{b x^2+c x^4}}{b x^2}",1,"(-(A*Sqrt[c]*(b + c*x^2)) + b*B*x*Sqrt[b + c*x^2]*ArcTanh[(Sqrt[c]*x)/Sqrt[b + c*x^2]])/(b*Sqrt[c]*Sqrt[x^2*(b + c*x^2)])","A",1
135,1,43,61,0.0408999,"\int \frac{A+B x^2}{x^3 \sqrt{b x^2+c x^4}} \, dx","Integrate[(A + B*x^2)/(x^3*Sqrt[b*x^2 + c*x^4]),x]","-\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(A \left(b-2 c x^2\right)+3 b B x^2\right)}{3 b^2 x^4}","-\frac{\sqrt{b x^2+c x^4} (3 b B-2 A c)}{3 b^2 x^2}-\frac{A \sqrt{b x^2+c x^4}}{3 b x^4}",1,"-1/3*(Sqrt[x^2*(b + c*x^2)]*(3*b*B*x^2 + A*(b - 2*c*x^2)))/(b^2*x^4)","A",1
136,1,64,96,0.0291489,"\int \frac{A+B x^2}{x^5 \sqrt{b x^2+c x^4}} \, dx","Integrate[(A + B*x^2)/(x^5*Sqrt[b*x^2 + c*x^4]),x]","\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(A \left(-3 b^2+4 b c x^2-8 c^2 x^4\right)-5 b B x^2 \left(b-2 c x^2\right)\right)}{15 b^3 x^6}","\frac{2 c \sqrt{b x^2+c x^4} (5 b B-4 A c)}{15 b^3 x^2}-\frac{\sqrt{b x^2+c x^4} (5 b B-4 A c)}{15 b^2 x^4}-\frac{A \sqrt{b x^2+c x^4}}{5 b x^6}",1,"(Sqrt[x^2*(b + c*x^2)]*(-5*b*B*x^2*(b - 2*c*x^2) + A*(-3*b^2 + 4*b*c*x^2 - 8*c^2*x^4)))/(15*b^3*x^6)","A",1
137,1,89,133,0.0510188,"\int \frac{A+B x^2}{x^7 \sqrt{b x^2+c x^4}} \, dx","Integrate[(A + B*x^2)/(x^7*Sqrt[b*x^2 + c*x^4]),x]","-\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(3 A \left(5 b^3-6 b^2 c x^2+8 b c^2 x^4-16 c^3 x^6\right)+7 b B x^2 \left(3 b^2-4 b c x^2+8 c^2 x^4\right)\right)}{105 b^4 x^8}","-\frac{8 c^2 \sqrt{b x^2+c x^4} (7 b B-6 A c)}{105 b^4 x^2}+\frac{4 c \sqrt{b x^2+c x^4} (7 b B-6 A c)}{105 b^3 x^4}-\frac{\sqrt{b x^2+c x^4} (7 b B-6 A c)}{35 b^2 x^6}-\frac{A \sqrt{b x^2+c x^4}}{7 b x^8}",1,"-1/105*(Sqrt[x^2*(b + c*x^2)]*(7*b*B*x^2*(3*b^2 - 4*b*c*x^2 + 8*c^2*x^4) + 3*A*(5*b^3 - 6*b^2*c*x^2 + 8*b*c^2*x^4 - 16*c^3*x^6)))/(b^4*x^8)","A",1
138,1,94,170,0.0965462,"\int \frac{A+B x^2}{x^9 \sqrt{b x^2+c x^4}} \, dx","Integrate[(A + B*x^2)/(x^9*Sqrt[b*x^2 + c*x^4]),x]","\frac{x^2 \left(\frac{c x^2}{b}+1\right) \left(5 b^3-6 b^2 c x^2+8 b c^2 x^4-16 c^3 x^6\right) (8 A c-9 b B)-35 A b^3 \left(b+c x^2\right)}{315 b^4 x^8 \sqrt{x^2 \left(b+c x^2\right)}}","\frac{16 c^3 \sqrt{b x^2+c x^4} (9 b B-8 A c)}{315 b^5 x^2}-\frac{8 c^2 \sqrt{b x^2+c x^4} (9 b B-8 A c)}{315 b^4 x^4}+\frac{2 c \sqrt{b x^2+c x^4} (9 b B-8 A c)}{105 b^3 x^6}-\frac{\sqrt{b x^2+c x^4} (9 b B-8 A c)}{63 b^2 x^8}-\frac{A \sqrt{b x^2+c x^4}}{9 b x^{10}}",1,"(-35*A*b^3*(b + c*x^2) + (-9*b*B + 8*A*c)*x^2*(1 + (c*x^2)/b)*(5*b^3 - 6*b^2*c*x^2 + 8*b*c^2*x^4 - 16*c^3*x^6))/(315*b^4*x^8*Sqrt[x^2*(b + c*x^2)])","A",1
139,1,85,131,0.1047169,"\int \frac{x^6 \left(A+B x^2\right)}{\sqrt{b x^2+c x^4}} \, dx","Integrate[(x^6*(A + B*x^2))/Sqrt[b*x^2 + c*x^4],x]","\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(8 b^2 c \left(7 A+3 B x^2\right)-2 b c^2 x^2 \left(14 A+9 B x^2\right)+3 c^3 x^4 \left(7 A+5 B x^2\right)-48 b^3 B\right)}{105 c^4 x}","-\frac{8 b^2 \sqrt{b x^2+c x^4} (6 b B-7 A c)}{105 c^4 x}+\frac{4 b x \sqrt{b x^2+c x^4} (6 b B-7 A c)}{105 c^3}-\frac{x^3 \sqrt{b x^2+c x^4} (6 b B-7 A c)}{35 c^2}+\frac{B x^5 \sqrt{b x^2+c x^4}}{7 c}",1,"(Sqrt[x^2*(b + c*x^2)]*(-48*b^3*B + 8*b^2*c*(7*A + 3*B*x^2) + 3*c^3*x^4*(7*A + 5*B*x^2) - 2*b*c^2*x^2*(14*A + 9*B*x^2)))/(105*c^4*x)","A",1
140,1,63,94,0.0711165,"\int \frac{x^4 \left(A+B x^2\right)}{\sqrt{b x^2+c x^4}} \, dx","Integrate[(x^4*(A + B*x^2))/Sqrt[b*x^2 + c*x^4],x]","\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(-2 b c \left(5 A+2 B x^2\right)+c^2 x^2 \left(5 A+3 B x^2\right)+8 b^2 B\right)}{15 c^3 x}","\frac{2 b \sqrt{b x^2+c x^4} (4 b B-5 A c)}{15 c^3 x}-\frac{x \sqrt{b x^2+c x^4} (4 b B-5 A c)}{15 c^2}+\frac{B x^3 \sqrt{b x^2+c x^4}}{5 c}",1,"(Sqrt[x^2*(b + c*x^2)]*(8*b^2*B - 2*b*c*(5*A + 2*B*x^2) + c^2*x^2*(5*A + 3*B*x^2)))/(15*c^3*x)","A",1
141,1,40,59,0.0521358,"\int \frac{x^2 \left(A+B x^2\right)}{\sqrt{b x^2+c x^4}} \, dx","Integrate[(x^2*(A + B*x^2))/Sqrt[b*x^2 + c*x^4],x]","\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(3 A c-2 b B+B c x^2\right)}{3 c^2 x}","\frac{B x \sqrt{b x^2+c x^4}}{3 c}-\frac{\sqrt{b x^2+c x^4} (2 b B-3 A c)}{3 c^2 x}",1,"(Sqrt[x^2*(b + c*x^2)]*(-2*b*B + 3*A*c + B*c*x^2))/(3*c^2*x)","A",1
142,1,73,55,0.0393757,"\int \frac{A+B x^2}{\sqrt{b x^2+c x^4}} \, dx","Integrate[(A + B*x^2)/Sqrt[b*x^2 + c*x^4],x]","\frac{x \left(\sqrt{b} B \left(b+c x^2\right)-A c \sqrt{b+c x^2} \tanh ^{-1}\left(\frac{\sqrt{b+c x^2}}{\sqrt{b}}\right)\right)}{\sqrt{b} c \sqrt{x^2 \left(b+c x^2\right)}}","\frac{B \sqrt{b x^2+c x^4}}{c x}-\frac{A \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)}{\sqrt{b}}",1,"(x*(Sqrt[b]*B*(b + c*x^2) - A*c*Sqrt[b + c*x^2]*ArcTanh[Sqrt[b + c*x^2]/Sqrt[b]]))/(Sqrt[b]*c*Sqrt[x^2*(b + c*x^2)])","A",1
143,1,87,68,0.0885183,"\int \frac{A+B x^2}{x^2 \sqrt{b x^2+c x^4}} \, dx","Integrate[(A + B*x^2)/(x^2*Sqrt[b*x^2 + c*x^4]),x]","\frac{x \sqrt{b+c x^2} \left(-\frac{2 \left(b B-\frac{A c}{2}\right) \tanh ^{-1}\left(\frac{\sqrt{b+c x^2}}{\sqrt{b}}\right)}{b^{3/2}}-\frac{A \sqrt{b+c x^2}}{b x^2}\right)}{2 \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{(2 b B-A c) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)}{2 b^{3/2}}-\frac{A \sqrt{b x^2+c x^4}}{2 b x^3}",1,"(x*Sqrt[b + c*x^2]*(-((A*Sqrt[b + c*x^2])/(b*x^2)) - (2*(b*B - (A*c)/2)*ArcTanh[Sqrt[b + c*x^2]/Sqrt[b]])/b^(3/2)))/(2*Sqrt[x^2*(b + c*x^2)])","A",1
144,1,104,103,0.2229481,"\int \frac{A+B x^2}{x^4 \sqrt{b x^2+c x^4}} \, dx","Integrate[(A + B*x^2)/(x^4*Sqrt[b*x^2 + c*x^4]),x]","-\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(b \sqrt{\frac{c x^2}{b}+1} \left(2 A b-3 A c x^2+4 b B x^2\right)+c x^4 (3 A c-4 b B) \tanh ^{-1}\left(\sqrt{\frac{c x^2}{b}+1}\right)\right)}{8 b^3 x^5 \sqrt{\frac{c x^2}{b}+1}}","\frac{c (4 b B-3 A c) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)}{8 b^{5/2}}-\frac{\sqrt{b x^2+c x^4} (4 b B-3 A c)}{8 b^2 x^3}-\frac{A \sqrt{b x^2+c x^4}}{4 b x^5}",1,"-1/8*(Sqrt[x^2*(b + c*x^2)]*(b*(2*A*b + 4*b*B*x^2 - 3*A*c*x^2)*Sqrt[1 + (c*x^2)/b] + c*(-4*b*B + 3*A*c)*x^4*ArcTanh[Sqrt[1 + (c*x^2)/b]]))/(b^3*x^5*Sqrt[1 + (c*x^2)/b])","A",1
145,1,136,184,0.2584073,"\int \frac{x^9 \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(x^9*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2),x]","\frac{x \left(\sqrt{c} x \left(b^2 \left(35 B c x^2-90 A c\right)-2 b c^2 x^2 \left(15 A+7 B x^2\right)+4 c^3 x^4 \left(3 A+2 B x^2\right)+105 b^3 B\right)-15 b^{5/2} \sqrt{\frac{c x^2}{b}+1} (7 b B-6 A c) \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)\right)}{48 c^{9/2} \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{5 b^2 (7 b B-6 A c) \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{16 c^{9/2}}+\frac{5 b \sqrt{b x^2+c x^4} (7 b B-6 A c)}{16 c^4}-\frac{5 x^2 \sqrt{b x^2+c x^4} (7 b B-6 A c)}{24 c^3}+\frac{x^4 \sqrt{b x^2+c x^4} (7 b B-6 A c)}{6 b c^2}-\frac{x^8 (b B-A c)}{b c \sqrt{b x^2+c x^4}}",1,"(x*(Sqrt[c]*x*(105*b^3*B + 4*c^3*x^4*(3*A + 2*B*x^2) - 2*b*c^2*x^2*(15*A + 7*B*x^2) + b^2*(-90*A*c + 35*B*c*x^2)) - 15*b^(5/2)*(7*b*B - 6*A*c)*Sqrt[1 + (c*x^2)/b]*ArcSinh[(Sqrt[c]*x)/Sqrt[b]]))/(48*c^(9/2)*Sqrt[x^2*(b + c*x^2)])","A",1
146,1,113,147,0.2092417,"\int \frac{x^7 \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(x^7*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2),x]","\frac{x \left(3 b^{3/2} \sqrt{\frac{c x^2}{b}+1} (5 b B-4 A c) \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)+\sqrt{c} x \left(b c \left(12 A-5 B x^2\right)+2 c^2 x^2 \left(2 A+B x^2\right)-15 b^2 B\right)\right)}{8 c^{7/2} \sqrt{x^2 \left(b+c x^2\right)}}","\frac{3 b (5 b B-4 A c) \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{8 c^{7/2}}-\frac{3 \sqrt{b x^2+c x^4} (5 b B-4 A c)}{8 c^3}+\frac{x^2 \sqrt{b x^2+c x^4} (5 b B-4 A c)}{4 b c^2}-\frac{x^6 (b B-A c)}{b c \sqrt{b x^2+c x^4}}",1,"(x*(Sqrt[c]*x*(-15*b^2*B + b*c*(12*A - 5*B*x^2) + 2*c^2*x^2*(2*A + B*x^2)) + 3*b^(3/2)*(5*b*B - 4*A*c)*Sqrt[1 + (c*x^2)/b]*ArcSinh[(Sqrt[c]*x)/Sqrt[b]]))/(8*c^(7/2)*Sqrt[x^2*(b + c*x^2)])","A",1
147,1,91,112,0.1651849,"\int \frac{x^5 \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(x^5*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2),x]","\frac{x \left(\sqrt{c} x \left(-2 A c+3 b B+B c x^2\right)-\sqrt{b} \sqrt{\frac{c x^2}{b}+1} (3 b B-2 A c) \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)\right)}{2 c^{5/2} \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{(3 b B-2 A c) \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{2 c^{5/2}}+\frac{\sqrt{b x^2+c x^4} (3 b B-2 A c)}{2 b c^2}-\frac{x^4 (b B-A c)}{b c \sqrt{b x^2+c x^4}}",1,"(x*(Sqrt[c]*x*(3*b*B - 2*A*c + B*c*x^2) - Sqrt[b]*(3*b*B - 2*A*c)*Sqrt[1 + (c*x^2)/b]*ArcSinh[(Sqrt[c]*x)/Sqrt[b]]))/(2*c^(5/2)*Sqrt[x^2*(b + c*x^2)])","A",1
148,1,75,67,0.1063327,"\int \frac{x^3 \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(x^3*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2),x]","\frac{x \left(\sqrt{c} x (A c-b B)+b^{3/2} B \sqrt{\frac{c x^2}{b}+1} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)\right)}{b c^{3/2} \sqrt{x^2 \left(b+c x^2\right)}}","\frac{B \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{c^{3/2}}-\frac{x^2 (b B-A c)}{b c \sqrt{b x^2+c x^4}}",1,"(x*(Sqrt[c]*(-(b*B) + A*c)*x + b^(3/2)*B*Sqrt[1 + (c*x^2)/b]*ArcSinh[(Sqrt[c]*x)/Sqrt[b]]))/(b*c^(3/2)*Sqrt[x^2*(b + c*x^2)])","A",1
149,1,37,37,0.0198601,"\int \frac{x \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(x*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2),x]","\frac{b B x^2-A \left(b+2 c x^2\right)}{b^2 \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{A b-x^2 (b B-2 A c)}{b^2 \sqrt{b x^2+c x^4}}",1,"(b*B*x^2 - A*(b + 2*c*x^2))/(b^2*Sqrt[x^2*(b + c*x^2)])","A",1
150,1,64,66,0.0426275,"\int \frac{A+B x^2}{x \left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(A + B*x^2)/(x*(b*x^2 + c*x^4)^(3/2)),x]","\frac{A \left(-b^2+4 b c x^2+8 c^2 x^4\right)-3 b B x^2 \left(b+2 c x^2\right)}{3 b^3 x^2 \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{\left(b+2 c x^2\right) (3 b B-4 A c)}{3 b^3 \sqrt{b x^2+c x^4}}-\frac{A}{3 b x^2 \sqrt{b x^2+c x^4}}",1,"(-3*b*B*x^2*(b + 2*c*x^2) + A*(-b^2 + 4*b*c*x^2 + 8*c^2*x^4))/(3*b^3*x^2*Sqrt[x^2*(b + c*x^2)])","A",1
151,1,85,101,0.0374361,"\int \frac{A+B x^2}{x^3 \left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(A + B*x^2)/(x^3*(b*x^2 + c*x^4)^(3/2)),x]","\frac{-3 A \left(b^3-2 b^2 c x^2+8 b c^2 x^4+16 c^3 x^6\right)-5 b B x^2 \left(b^2-4 b c x^2-8 c^2 x^4\right)}{15 b^4 x^4 \sqrt{x^2 \left(b+c x^2\right)}}","\frac{4 c \left(b+2 c x^2\right) (5 b B-6 A c)}{15 b^4 \sqrt{b x^2+c x^4}}-\frac{5 b B-6 A c}{15 b^2 x^2 \sqrt{b x^2+c x^4}}-\frac{A}{5 b x^4 \sqrt{b x^2+c x^4}}",1,"(-5*b*B*x^2*(b^2 - 4*b*c*x^2 - 8*c^2*x^4) - 3*A*(b^3 - 2*b^2*c*x^2 + 8*b*c^2*x^4 + 16*c^3*x^6))/(15*b^4*x^4*Sqrt[x^2*(b + c*x^2)])","A",1
152,1,75,138,0.0544188,"\int \frac{A+B x^2}{x^5 \left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(A + B*x^2)/(x^5*(b*x^2 + c*x^4)^(3/2)),x]","\frac{x^2 \left(b^3-2 b^2 c x^2+8 b c^2 x^4+16 c^3 x^6\right) (8 A c-7 b B)-5 A b^4}{35 b^5 x^6 \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{8 c^2 \left(b+2 c x^2\right) (7 b B-8 A c)}{35 b^5 \sqrt{b x^2+c x^4}}+\frac{2 c (7 b B-8 A c)}{35 b^3 x^2 \sqrt{b x^2+c x^4}}-\frac{7 b B-8 A c}{35 b^2 x^4 \sqrt{b x^2+c x^4}}-\frac{A}{7 b x^6 \sqrt{b x^2+c x^4}}",1,"(-5*A*b^4 + (-7*b*B + 8*A*c)*x^2*(b^3 - 2*b^2*c*x^2 + 8*b*c^2*x^4 + 16*c^3*x^6))/(35*b^5*x^6*Sqrt[x^2*(b + c*x^2)])","A",1
153,1,82,139,0.0763936,"\int \frac{x^8 \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(x^8*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2),x]","\frac{x \left(-8 b^2 c \left(5 A-3 B x^2\right)-2 b c^2 x^2 \left(10 A+3 B x^2\right)+c^3 x^4 \left(5 A+3 B x^2\right)+48 b^3 B\right)}{15 c^4 \sqrt{x^2 \left(b+c x^2\right)}}","\frac{8 b \sqrt{b x^2+c x^4} (6 b B-5 A c)}{15 c^4 x}-\frac{4 x \sqrt{b x^2+c x^4} (6 b B-5 A c)}{15 c^3}+\frac{x^3 \sqrt{b x^2+c x^4} (6 b B-5 A c)}{5 b c^2}-\frac{x^7 (b B-A c)}{b c \sqrt{b x^2+c x^4}}",1,"(x*(48*b^3*B - 8*b^2*c*(5*A - 3*B*x^2) + c^3*x^4*(5*A + 3*B*x^2) - 2*b*c^2*x^2*(10*A + 3*B*x^2)))/(15*c^4*Sqrt[x^2*(b + c*x^2)])","A",1
154,1,60,104,0.0414266,"\int \frac{x^6 \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(x^6*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2),x]","\frac{x \left(b \left(6 A c-4 B c x^2\right)+c^2 x^2 \left(3 A+B x^2\right)-8 b^2 B\right)}{3 c^3 \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{2 \sqrt{b x^2+c x^4} (4 b B-3 A c)}{3 c^3 x}+\frac{x \sqrt{b x^2+c x^4} (4 b B-3 A c)}{3 b c^2}-\frac{x^5 (b B-A c)}{b c \sqrt{b x^2+c x^4}}",1,"(x*(-8*b^2*B + c^2*x^2*(3*A + B*x^2) + b*(6*A*c - 4*B*c*x^2)))/(3*c^3*Sqrt[x^2*(b + c*x^2)])","A",1
155,1,35,69,0.0272725,"\int \frac{x^4 \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(x^4*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2),x]","\frac{x \left(-A c+2 b B+B c x^2\right)}{c^2 \sqrt{x^2 \left(b+c x^2\right)}}","\frac{\sqrt{b x^2+c x^4} (2 b B-A c)}{b c^2 x}-\frac{x^3 (b B-A c)}{b c \sqrt{b x^2+c x^4}}",1,"(x*(2*b*B - A*c + B*c*x^2))/(c^2*Sqrt[x^2*(b + c*x^2)])","A",1
156,1,73,64,0.034911,"\int \frac{x^2 \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(x^2*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2),x]","-\frac{x \left(\sqrt{b} (b B-A c)+A c \sqrt{b+c x^2} \tanh ^{-1}\left(\frac{\sqrt{b+c x^2}}{\sqrt{b}}\right)\right)}{b^{3/2} c \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{A \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)}{b^{3/2}}-\frac{x (b B-A c)}{b c \sqrt{b x^2+c x^4}}",1,"-((x*(Sqrt[b]*(b*B - A*c) + A*c*Sqrt[b + c*x^2]*ArcTanh[Sqrt[b + c*x^2]/Sqrt[b]]))/(b^(3/2)*c*Sqrt[x^2*(b + c*x^2)]))","A",1
157,1,61,142,0.0270904,"\int \frac{A+B x^2}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(A + B*x^2)/(b*x^2 + c*x^4)^(3/2),x]","\frac{x^2 (2 b B-3 A c) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{c x^2}{b}+1\right)-A b}{2 b^2 x \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{(2 b B-3 A c) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)}{2 b^{5/2}}+\frac{\sqrt{b x^2+c x^4} (2 b B-3 A c)}{2 b^2 c x^3}-\frac{2 b B-3 A c}{3 b c x \sqrt{b x^2+c x^4}}-\frac{B}{3 c x \sqrt{b x^2+c x^4}}",1,"(-(A*b) + (2*b*B - 3*A*c)*x^2*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (c*x^2)/b])/(2*b^2*x*Sqrt[x^2*(b + c*x^2)])","C",1
158,1,64,137,0.0483603,"\int \frac{A+B x^2}{x^2 \left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(A + B*x^2)/(x^2*(b*x^2 + c*x^4)^(3/2)),x]","\frac{c x^4 (5 A c-4 b B) \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};\frac{c x^2}{b}+1\right)-A b^2}{4 b^3 x^3 \sqrt{x^2 \left(b+c x^2\right)}}","\frac{3 c (4 b B-5 A c) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)}{8 b^{7/2}}-\frac{3 \sqrt{b x^2+c x^4} (4 b B-5 A c)}{8 b^3 x^3}+\frac{4 b B-5 A c}{4 b^2 x \sqrt{b x^2+c x^4}}-\frac{A}{4 b x^3 \sqrt{b x^2+c x^4}}",1,"(-(A*b^2) + c*(-4*b*B + 5*A*c)*x^4*Hypergeometric2F1[-1/2, 2, 1/2, 1 + (c*x^2)/b])/(4*b^3*x^3*Sqrt[x^2*(b + c*x^2)])","C",1
159,1,33,39,0.01759,"\int x^{7/2} \left(A+B x^2\right) \left(b x^2+c x^4\right) \, dx","Integrate[x^(7/2)*(A + B*x^2)*(b*x^2 + c*x^4),x]","\frac{2 x^{13/2} \left(273 x^2 (A c+b B)+357 A b+221 B c x^4\right)}{4641}","\frac{2}{17} x^{17/2} (A c+b B)+\frac{2}{13} A b x^{13/2}+\frac{2}{21} B c x^{21/2}",1,"(2*x^(13/2)*(357*A*b + 273*(b*B + A*c)*x^2 + 221*B*c*x^4))/4641","A",1
160,1,33,39,0.0179486,"\int x^{5/2} \left(A+B x^2\right) \left(b x^2+c x^4\right) \, dx","Integrate[x^(5/2)*(A + B*x^2)*(b*x^2 + c*x^4),x]","\frac{2 x^{11/2} \left(209 x^2 (A c+b B)+285 A b+165 B c x^4\right)}{3135}","\frac{2}{15} x^{15/2} (A c+b B)+\frac{2}{11} A b x^{11/2}+\frac{2}{19} B c x^{19/2}",1,"(2*x^(11/2)*(285*A*b + 209*(b*B + A*c)*x^2 + 165*B*c*x^4))/3135","A",1
161,1,33,39,0.0173757,"\int x^{3/2} \left(A+B x^2\right) \left(b x^2+c x^4\right) \, dx","Integrate[x^(3/2)*(A + B*x^2)*(b*x^2 + c*x^4),x]","\frac{2 x^{9/2} \left(153 x^2 (A c+b B)+221 A b+117 B c x^4\right)}{1989}","\frac{2}{13} x^{13/2} (A c+b B)+\frac{2}{9} A b x^{9/2}+\frac{2}{17} B c x^{17/2}",1,"(2*x^(9/2)*(221*A*b + 153*(b*B + A*c)*x^2 + 117*B*c*x^4))/1989","A",1
162,1,33,39,0.0166128,"\int \sqrt{x} \left(A+B x^2\right) \left(b x^2+c x^4\right) \, dx","Integrate[Sqrt[x]*(A + B*x^2)*(b*x^2 + c*x^4),x]","\frac{2 x^{7/2} \left(105 x^2 (A c+b B)+165 A b+77 B c x^4\right)}{1155}","\frac{2}{11} x^{11/2} (A c+b B)+\frac{2}{7} A b x^{7/2}+\frac{2}{15} B c x^{15/2}",1,"(2*x^(7/2)*(165*A*b + 105*(b*B + A*c)*x^2 + 77*B*c*x^4))/1155","A",1
163,1,33,39,0.0168145,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)}{\sqrt{x}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4))/Sqrt[x],x]","\frac{2}{585} x^{5/2} \left(65 x^2 (A c+b B)+117 A b+45 B c x^4\right)","\frac{2}{9} x^{9/2} (A c+b B)+\frac{2}{5} A b x^{5/2}+\frac{2}{13} B c x^{13/2}",1,"(2*x^(5/2)*(117*A*b + 65*(b*B + A*c)*x^2 + 45*B*c*x^4))/585","A",1
164,1,33,39,0.0160463,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)}{x^{3/2}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4))/x^(3/2),x]","\frac{2}{231} x^{3/2} \left(33 x^2 (A c+b B)+77 A b+21 B c x^4\right)","\frac{2}{7} x^{7/2} (A c+b B)+\frac{2}{3} A b x^{3/2}+\frac{2}{11} B c x^{11/2}",1,"(2*x^(3/2)*(77*A*b + 33*(b*B + A*c)*x^2 + 21*B*c*x^4))/231","A",1
165,1,33,37,0.0161304,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)}{x^{5/2}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4))/x^(5/2),x]","\frac{2}{45} \sqrt{x} \left(9 x^2 (A c+b B)+45 A b+5 B c x^4\right)","\frac{2}{5} x^{5/2} (A c+b B)+2 A b \sqrt{x}+\frac{2}{9} B c x^{9/2}",1,"(2*Sqrt[x]*(45*A*b + 9*(b*B + A*c)*x^2 + 5*B*c*x^4))/45","A",1
166,1,35,37,0.0166498,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)}{x^{7/2}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4))/x^(7/2),x]","\frac{2 \left(-21 A b+7 A c x^2+7 b B x^2+3 B c x^4\right)}{21 \sqrt{x}}","\frac{2}{3} x^{3/2} (A c+b B)-\frac{2 A b}{\sqrt{x}}+\frac{2}{7} B c x^{7/2}",1,"(2*(-21*A*b + 7*b*B*x^2 + 7*A*c*x^2 + 3*B*c*x^4))/(21*Sqrt[x])","A",1
167,1,63,63,0.0356564,"\int x^{7/2} \left(A+B x^2\right) \left(b x^2+c x^4\right)^2 \, dx","Integrate[x^(7/2)*(A + B*x^2)*(b*x^2 + c*x^4)^2,x]","\frac{2}{17} A b^2 x^{17/2}+\frac{2}{25} c x^{25/2} (A c+2 b B)+\frac{2}{21} b x^{21/2} (2 A c+b B)+\frac{2}{29} B c^2 x^{29/2}","\frac{2}{17} A b^2 x^{17/2}+\frac{2}{25} c x^{25/2} (A c+2 b B)+\frac{2}{21} b x^{21/2} (2 A c+b B)+\frac{2}{29} B c^2 x^{29/2}",1,"(2*A*b^2*x^(17/2))/17 + (2*b*(b*B + 2*A*c)*x^(21/2))/21 + (2*c*(2*b*B + A*c)*x^(25/2))/25 + (2*B*c^2*x^(29/2))/29","A",1
168,1,53,63,0.033724,"\int x^{5/2} \left(A+B x^2\right) \left(b x^2+c x^4\right)^2 \, dx","Integrate[x^(5/2)*(A + B*x^2)*(b*x^2 + c*x^4)^2,x]","\frac{2 x^{15/2} \left(3933 A b^2+2565 c x^4 (A c+2 b B)+3105 b x^2 (2 A c+b B)+2185 B c^2 x^6\right)}{58995}","\frac{2}{15} A b^2 x^{15/2}+\frac{2}{23} c x^{23/2} (A c+2 b B)+\frac{2}{19} b x^{19/2} (2 A c+b B)+\frac{2}{27} B c^2 x^{27/2}",1,"(2*x^(15/2)*(3933*A*b^2 + 3105*b*(b*B + 2*A*c)*x^2 + 2565*c*(2*b*B + A*c)*x^4 + 2185*B*c^2*x^6))/58995","A",1
169,1,63,63,0.0310342,"\int x^{3/2} \left(A+B x^2\right) \left(b x^2+c x^4\right)^2 \, dx","Integrate[x^(3/2)*(A + B*x^2)*(b*x^2 + c*x^4)^2,x]","\frac{2}{13} A b^2 x^{13/2}+\frac{2}{21} c x^{21/2} (A c+2 b B)+\frac{2}{17} b x^{17/2} (2 A c+b B)+\frac{2}{25} B c^2 x^{25/2}","\frac{2}{13} A b^2 x^{13/2}+\frac{2}{21} c x^{21/2} (A c+2 b B)+\frac{2}{17} b x^{17/2} (2 A c+b B)+\frac{2}{25} B c^2 x^{25/2}",1,"(2*A*b^2*x^(13/2))/13 + (2*b*(b*B + 2*A*c)*x^(17/2))/17 + (2*c*(2*b*B + A*c)*x^(21/2))/21 + (2*B*c^2*x^(25/2))/25","A",1
170,1,53,63,0.0324005,"\int \sqrt{x} \left(A+B x^2\right) \left(b x^2+c x^4\right)^2 \, dx","Integrate[Sqrt[x]*(A + B*x^2)*(b*x^2 + c*x^4)^2,x]","\frac{2 x^{11/2} \left(6555 A b^2+3795 c x^4 (A c+2 b B)+4807 b x^2 (2 A c+b B)+3135 B c^2 x^6\right)}{72105}","\frac{2}{11} A b^2 x^{11/2}+\frac{2}{19} c x^{19/2} (A c+2 b B)+\frac{2}{15} b x^{15/2} (2 A c+b B)+\frac{2}{23} B c^2 x^{23/2}",1,"(2*x^(11/2)*(6555*A*b^2 + 4807*b*(b*B + 2*A*c)*x^2 + 3795*c*(2*b*B + A*c)*x^4 + 3135*B*c^2*x^6))/72105","A",1
171,1,53,63,0.032376,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^2}{\sqrt{x}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^2)/Sqrt[x],x]","\frac{2 x^{9/2} \left(1547 A b^2+819 c x^4 (A c+2 b B)+1071 b x^2 (2 A c+b B)+663 B c^2 x^6\right)}{13923}","\frac{2}{9} A b^2 x^{9/2}+\frac{2}{17} c x^{17/2} (A c+2 b B)+\frac{2}{13} b x^{13/2} (2 A c+b B)+\frac{2}{21} B c^2 x^{21/2}",1,"(2*x^(9/2)*(1547*A*b^2 + 1071*b*(b*B + 2*A*c)*x^2 + 819*c*(2*b*B + A*c)*x^4 + 663*B*c^2*x^6))/13923","A",1
172,1,63,63,0.0310352,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^2}{x^{3/2}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^2)/x^(3/2),x]","\frac{2}{7} A b^2 x^{7/2}+\frac{2}{15} c x^{15/2} (A c+2 b B)+\frac{2}{11} b x^{11/2} (2 A c+b B)+\frac{2}{19} B c^2 x^{19/2}","\frac{2}{7} A b^2 x^{7/2}+\frac{2}{15} c x^{15/2} (A c+2 b B)+\frac{2}{11} b x^{11/2} (2 A c+b B)+\frac{2}{19} B c^2 x^{19/2}",1,"(2*A*b^2*x^(7/2))/7 + (2*b*(b*B + 2*A*c)*x^(11/2))/11 + (2*c*(2*b*B + A*c)*x^(15/2))/15 + (2*B*c^2*x^(19/2))/19","A",1
173,1,53,63,0.0321757,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^2}{x^{5/2}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^2)/x^(5/2),x]","\frac{2 x^{5/2} \left(1989 A b^2+765 c x^4 (A c+2 b B)+1105 b x^2 (2 A c+b B)+585 B c^2 x^6\right)}{9945}","\frac{2}{5} A b^2 x^{5/2}+\frac{2}{13} c x^{13/2} (A c+2 b B)+\frac{2}{9} b x^{9/2} (2 A c+b B)+\frac{2}{17} B c^2 x^{17/2}",1,"(2*x^(5/2)*(1989*A*b^2 + 1105*b*(b*B + 2*A*c)*x^2 + 765*c*(2*b*B + A*c)*x^4 + 585*B*c^2*x^6))/9945","A",1
174,1,53,63,0.0341907,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^2}{x^{7/2}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^2)/x^(7/2),x]","\frac{2 x^{3/2} \left(385 A b^2+105 c x^4 (A c+2 b B)+165 b x^2 (2 A c+b B)+77 B c^2 x^6\right)}{1155}","\frac{2}{3} A b^2 x^{3/2}+\frac{2}{11} c x^{11/2} (A c+2 b B)+\frac{2}{7} b x^{7/2} (2 A c+b B)+\frac{2}{15} B c^2 x^{15/2}",1,"(2*x^(3/2)*(385*A*b^2 + 165*b*(b*B + 2*A*c)*x^2 + 105*c*(2*b*B + A*c)*x^4 + 77*B*c^2*x^6))/1155","A",1
175,1,85,85,0.0522664,"\int x^{7/2} \left(A+B x^2\right) \left(b x^2+c x^4\right)^3 \, dx","Integrate[x^(7/2)*(A + B*x^2)*(b*x^2 + c*x^4)^3,x]","\frac{2}{21} A b^3 x^{21/2}+\frac{2}{25} b^2 x^{25/2} (3 A c+b B)+\frac{2}{33} c^2 x^{33/2} (A c+3 b B)+\frac{6}{29} b c x^{29/2} (A c+b B)+\frac{2}{37} B c^3 x^{37/2}","\frac{2}{21} A b^3 x^{21/2}+\frac{2}{25} b^2 x^{25/2} (3 A c+b B)+\frac{2}{33} c^2 x^{33/2} (A c+3 b B)+\frac{6}{29} b c x^{29/2} (A c+b B)+\frac{2}{37} B c^3 x^{37/2}",1,"(2*A*b^3*x^(21/2))/21 + (2*b^2*(b*B + 3*A*c)*x^(25/2))/25 + (6*b*c*(b*B + A*c)*x^(29/2))/29 + (2*c^2*(3*b*B + A*c)*x^(33/2))/33 + (2*B*c^3*x^(37/2))/37","A",1
176,1,85,85,0.0503565,"\int x^{5/2} \left(A+B x^2\right) \left(b x^2+c x^4\right)^3 \, dx","Integrate[x^(5/2)*(A + B*x^2)*(b*x^2 + c*x^4)^3,x]","\frac{2}{19} A b^3 x^{19/2}+\frac{2}{23} b^2 x^{23/2} (3 A c+b B)+\frac{2}{31} c^2 x^{31/2} (A c+3 b B)+\frac{2}{9} b c x^{27/2} (A c+b B)+\frac{2}{35} B c^3 x^{35/2}","\frac{2}{19} A b^3 x^{19/2}+\frac{2}{23} b^2 x^{23/2} (3 A c+b B)+\frac{2}{31} c^2 x^{31/2} (A c+3 b B)+\frac{2}{9} b c x^{27/2} (A c+b B)+\frac{2}{35} B c^3 x^{35/2}",1,"(2*A*b^3*x^(19/2))/19 + (2*b^2*(b*B + 3*A*c)*x^(23/2))/23 + (2*b*c*(b*B + A*c)*x^(27/2))/9 + (2*c^2*(3*b*B + A*c)*x^(31/2))/31 + (2*B*c^3*x^(35/2))/35","A",1
177,1,85,85,0.0464003,"\int x^{3/2} \left(A+B x^2\right) \left(b x^2+c x^4\right)^3 \, dx","Integrate[x^(3/2)*(A + B*x^2)*(b*x^2 + c*x^4)^3,x]","\frac{2}{17} A b^3 x^{17/2}+\frac{2}{21} b^2 x^{21/2} (3 A c+b B)+\frac{2}{29} c^2 x^{29/2} (A c+3 b B)+\frac{6}{25} b c x^{25/2} (A c+b B)+\frac{2}{33} B c^3 x^{33/2}","\frac{2}{17} A b^3 x^{17/2}+\frac{2}{21} b^2 x^{21/2} (3 A c+b B)+\frac{2}{29} c^2 x^{29/2} (A c+3 b B)+\frac{6}{25} b c x^{25/2} (A c+b B)+\frac{2}{33} B c^3 x^{33/2}",1,"(2*A*b^3*x^(17/2))/17 + (2*b^2*(b*B + 3*A*c)*x^(21/2))/21 + (6*b*c*(b*B + A*c)*x^(25/2))/25 + (2*c^2*(3*b*B + A*c)*x^(29/2))/29 + (2*B*c^3*x^(33/2))/33","A",1
178,1,85,85,0.0430127,"\int \sqrt{x} \left(A+B x^2\right) \left(b x^2+c x^4\right)^3 \, dx","Integrate[Sqrt[x]*(A + B*x^2)*(b*x^2 + c*x^4)^3,x]","\frac{2}{15} A b^3 x^{15/2}+\frac{2}{19} b^2 x^{19/2} (3 A c+b B)+\frac{2}{27} c^2 x^{27/2} (A c+3 b B)+\frac{6}{23} b c x^{23/2} (A c+b B)+\frac{2}{31} B c^3 x^{31/2}","\frac{2}{15} A b^3 x^{15/2}+\frac{2}{19} b^2 x^{19/2} (3 A c+b B)+\frac{2}{27} c^2 x^{27/2} (A c+3 b B)+\frac{6}{23} b c x^{23/2} (A c+b B)+\frac{2}{31} B c^3 x^{31/2}",1,"(2*A*b^3*x^(15/2))/15 + (2*b^2*(b*B + 3*A*c)*x^(19/2))/19 + (6*b*c*(b*B + A*c)*x^(23/2))/23 + (2*c^2*(3*b*B + A*c)*x^(27/2))/27 + (2*B*c^3*x^(31/2))/31","A",1
179,1,85,85,0.0426416,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^3}{\sqrt{x}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^3)/Sqrt[x],x]","\frac{2}{13} A b^3 x^{13/2}+\frac{2}{17} b^2 x^{17/2} (3 A c+b B)+\frac{2}{25} c^2 x^{25/2} (A c+3 b B)+\frac{2}{7} b c x^{21/2} (A c+b B)+\frac{2}{29} B c^3 x^{29/2}","\frac{2}{13} A b^3 x^{13/2}+\frac{2}{17} b^2 x^{17/2} (3 A c+b B)+\frac{2}{25} c^2 x^{25/2} (A c+3 b B)+\frac{2}{7} b c x^{21/2} (A c+b B)+\frac{2}{29} B c^3 x^{29/2}",1,"(2*A*b^3*x^(13/2))/13 + (2*b^2*(b*B + 3*A*c)*x^(17/2))/17 + (2*b*c*(b*B + A*c)*x^(21/2))/7 + (2*c^2*(3*b*B + A*c)*x^(25/2))/25 + (2*B*c^3*x^(29/2))/29","A",1
180,1,85,85,0.0425974,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^3}{x^{3/2}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^(3/2),x]","\frac{2}{11} A b^3 x^{11/2}+\frac{2}{15} b^2 x^{15/2} (3 A c+b B)+\frac{2}{23} c^2 x^{23/2} (A c+3 b B)+\frac{6}{19} b c x^{19/2} (A c+b B)+\frac{2}{27} B c^3 x^{27/2}","\frac{2}{11} A b^3 x^{11/2}+\frac{2}{15} b^2 x^{15/2} (3 A c+b B)+\frac{2}{23} c^2 x^{23/2} (A c+3 b B)+\frac{6}{19} b c x^{19/2} (A c+b B)+\frac{2}{27} B c^3 x^{27/2}",1,"(2*A*b^3*x^(11/2))/11 + (2*b^2*(b*B + 3*A*c)*x^(15/2))/15 + (6*b*c*(b*B + A*c)*x^(19/2))/19 + (2*c^2*(3*b*B + A*c)*x^(23/2))/23 + (2*B*c^3*x^(27/2))/27","A",1
181,1,85,85,0.0402071,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^3}{x^{5/2}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^(5/2),x]","\frac{2}{9} A b^3 x^{9/2}+\frac{2}{13} b^2 x^{13/2} (3 A c+b B)+\frac{2}{21} c^2 x^{21/2} (A c+3 b B)+\frac{6}{17} b c x^{17/2} (A c+b B)+\frac{2}{25} B c^3 x^{25/2}","\frac{2}{9} A b^3 x^{9/2}+\frac{2}{13} b^2 x^{13/2} (3 A c+b B)+\frac{2}{21} c^2 x^{21/2} (A c+3 b B)+\frac{6}{17} b c x^{17/2} (A c+b B)+\frac{2}{25} B c^3 x^{25/2}",1,"(2*A*b^3*x^(9/2))/9 + (2*b^2*(b*B + 3*A*c)*x^(13/2))/13 + (6*b*c*(b*B + A*c)*x^(17/2))/17 + (2*c^2*(3*b*B + A*c)*x^(21/2))/21 + (2*B*c^3*x^(25/2))/25","A",1
182,1,85,85,0.0505747,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^3}{x^{7/2}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^(7/2),x]","\frac{2}{7} A b^3 x^{7/2}+\frac{2}{11} b^2 x^{11/2} (3 A c+b B)+\frac{2}{19} c^2 x^{19/2} (A c+3 b B)+\frac{2}{5} b c x^{15/2} (A c+b B)+\frac{2}{23} B c^3 x^{23/2}","\frac{2}{7} A b^3 x^{7/2}+\frac{2}{11} b^2 x^{11/2} (3 A c+b B)+\frac{2}{19} c^2 x^{19/2} (A c+3 b B)+\frac{2}{5} b c x^{15/2} (A c+b B)+\frac{2}{23} B c^3 x^{23/2}",1,"(2*A*b^3*x^(7/2))/7 + (2*b^2*(b*B + 3*A*c)*x^(11/2))/11 + (2*b*c*(b*B + A*c)*x^(15/2))/5 + (2*c^2*(3*b*B + A*c)*x^(19/2))/19 + (2*B*c^3*x^(23/2))/23","A",1
183,1,133,278,0.2591679,"\int \frac{x^{13/2} \left(A+B x^2\right)}{b x^2+c x^4} \, dx","Integrate[(x^(13/2)*(A + B*x^2))/(b*x^2 + c*x^4),x]","\frac{2 x^{3/2} \left(-11 b c \left(7 A+3 B x^2\right)+3 c^2 x^2 \left(11 A+7 B x^2\right)+77 b^2 B\right)}{231 c^3}+\frac{b (-b)^{3/4} (b B-A c) \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-b}}\right)}{c^{15/4}}+\frac{(-b)^{7/4} (b B-A c) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-b}}\right)}{c^{15/4}}","-\frac{b^{7/4} (b B-A c) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} c^{15/4}}+\frac{b^{7/4} (b B-A c) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} c^{15/4}}+\frac{b^{7/4} (b B-A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{\sqrt{2} c^{15/4}}-\frac{b^{7/4} (b B-A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{\sqrt{2} c^{15/4}}+\frac{2 b x^{3/2} (b B-A c)}{3 c^3}-\frac{2 x^{7/2} (b B-A c)}{7 c^2}+\frac{2 B x^{11/2}}{11 c}",1,"(2*x^(3/2)*(77*b^2*B - 11*b*c*(7*A + 3*B*x^2) + 3*c^2*x^2*(11*A + 7*B*x^2)))/(231*c^3) + ((-b)^(3/4)*b*(b*B - A*c)*ArcTan[(c^(1/4)*Sqrt[x])/(-b)^(1/4)])/c^(15/4) + ((-b)^(7/4)*(b*B - A*c)*ArcTanh[(c^(1/4)*Sqrt[x])/(-b)^(1/4)])/c^(15/4)","A",1
184,1,227,276,0.3091192,"\int \frac{x^{11/2} \left(A+B x^2\right)}{b x^2+c x^4} \, dx","Integrate[(x^(11/2)*(A + B*x^2))/(b*x^2 + c*x^4),x]","\frac{\frac{45 \sqrt{2} b^{5/4} (b B-A c) \left(\log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)-\log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)\right)}{\sqrt[4]{c}}+\frac{90 \sqrt{2} b^{5/4} (b B-A c) \left(\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)-\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)\right)}{\sqrt[4]{c}}+72 c x^{5/2} (A c-b B)+360 b \sqrt{x} (b B-A c)+40 B c^2 x^{9/2}}{180 c^3}","\frac{b^{5/4} (b B-A c) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} c^{13/4}}-\frac{b^{5/4} (b B-A c) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} c^{13/4}}+\frac{b^{5/4} (b B-A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{\sqrt{2} c^{13/4}}-\frac{b^{5/4} (b B-A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{\sqrt{2} c^{13/4}}+\frac{2 b \sqrt{x} (b B-A c)}{c^3}-\frac{2 x^{5/2} (b B-A c)}{5 c^2}+\frac{2 B x^{9/2}}{9 c}",1,"(360*b*(b*B - A*c)*Sqrt[x] + 72*c*(-(b*B) + A*c)*x^(5/2) + 40*B*c^2*x^(9/2) + (90*Sqrt[2]*b^(5/4)*(b*B - A*c)*(ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] - ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)]))/c^(1/4) + (45*Sqrt[2]*b^(5/4)*(b*B - A*c)*(Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x]))/c^(1/4))/(180*c^3)","A",1
185,1,110,257,0.146078,"\int \frac{x^{9/2} \left(A+B x^2\right)}{b x^2+c x^4} \, dx","Integrate[(x^(9/2)*(A + B*x^2))/(b*x^2 + c*x^4),x]","\frac{2 c^{3/4} x^{3/2} \left(7 A c-7 b B+3 B c x^2\right)-21 (-b)^{3/4} (b B-A c) \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-b}}\right)+21 (-b)^{3/4} (b B-A c) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-b}}\right)}{21 c^{11/4}}","\frac{b^{3/4} (b B-A c) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} c^{11/4}}-\frac{b^{3/4} (b B-A c) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} c^{11/4}}-\frac{b^{3/4} (b B-A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{\sqrt{2} c^{11/4}}+\frac{b^{3/4} (b B-A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{\sqrt{2} c^{11/4}}-\frac{2 x^{3/2} (b B-A c)}{3 c^2}+\frac{2 B x^{7/2}}{7 c}",1,"(2*c^(3/4)*x^(3/2)*(-7*b*B + 7*A*c + 3*B*c*x^2) - 21*(-b)^(3/4)*(b*B - A*c)*ArcTan[(c^(1/4)*Sqrt[x])/(-b)^(1/4)] + 21*(-b)^(3/4)*(b*B - A*c)*ArcTanh[(c^(1/4)*Sqrt[x])/(-b)^(1/4)])/(21*c^(11/4))","A",1
186,1,208,255,0.2244491,"\int \frac{x^{7/2} \left(A+B x^2\right)}{b x^2+c x^4} \, dx","Integrate[(x^(7/2)*(A + B*x^2))/(b*x^2 + c*x^4),x]","\frac{-40 \sqrt{x} (b B-A c)+\frac{5 \sqrt{2} \sqrt[4]{b} (A c-b B) \left(\log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)-\log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)\right)}{\sqrt[4]{c}}-\frac{10 \sqrt{2} \sqrt[4]{b} (b B-A c) \left(\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)-\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)\right)}{\sqrt[4]{c}}+8 B c x^{5/2}}{20 c^2}","-\frac{\sqrt[4]{b} (b B-A c) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} c^{9/4}}+\frac{\sqrt[4]{b} (b B-A c) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} c^{9/4}}-\frac{\sqrt[4]{b} (b B-A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{\sqrt{2} c^{9/4}}+\frac{\sqrt[4]{b} (b B-A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{\sqrt{2} c^{9/4}}-\frac{2 \sqrt{x} (b B-A c)}{c^2}+\frac{2 B x^{5/2}}{5 c}",1,"(-40*(b*B - A*c)*Sqrt[x] + 8*B*c*x^(5/2) - (10*Sqrt[2]*b^(1/4)*(b*B - A*c)*(ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] - ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)]))/c^(1/4) + (5*Sqrt[2]*b^(1/4)*(-(b*B) + A*c)*(Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x]))/c^(1/4))/(20*c^2)","A",1
187,1,95,237,0.0804397,"\int \frac{x^{5/2} \left(A+B x^2\right)}{b x^2+c x^4} \, dx","Integrate[(x^(5/2)*(A + B*x^2))/(b*x^2 + c*x^4),x]","\frac{(3 A c-3 b B) \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-b}}\right)+3 (b B-A c) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-b}}\right)+2 \sqrt[4]{-b} B c^{3/4} x^{3/2}}{3 \sqrt[4]{-b} c^{7/4}}","-\frac{(b B-A c) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} \sqrt[4]{b} c^{7/4}}+\frac{(b B-A c) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} \sqrt[4]{b} c^{7/4}}+\frac{(b B-A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{\sqrt{2} \sqrt[4]{b} c^{7/4}}-\frac{(b B-A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{\sqrt{2} \sqrt[4]{b} c^{7/4}}+\frac{2 B x^{3/2}}{3 c}",1,"(2*(-b)^(1/4)*B*c^(3/4)*x^(3/2) + (-3*b*B + 3*A*c)*ArcTan[(c^(1/4)*Sqrt[x])/(-b)^(1/4)] + 3*(b*B - A*c)*ArcTanh[(c^(1/4)*Sqrt[x])/(-b)^(1/4)])/(3*(-b)^(1/4)*c^(7/4))","A",1
188,1,166,235,0.1460382,"\int \frac{x^{3/2} \left(A+B x^2\right)}{b x^2+c x^4} \, dx","Integrate[(x^(3/2)*(A + B*x^2))/(b*x^2 + c*x^4),x]","\frac{(b B-A c) \left(\log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)-\log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)+2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)\right)}{2 \sqrt{2} b^{3/4} c^{5/4}}+\frac{2 B \sqrt{x}}{c}","\frac{(b B-A c) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} b^{3/4} c^{5/4}}-\frac{(b B-A c) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} b^{3/4} c^{5/4}}+\frac{(b B-A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{\sqrt{2} b^{3/4} c^{5/4}}-\frac{(b B-A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{\sqrt{2} b^{3/4} c^{5/4}}+\frac{2 B \sqrt{x}}{c}",1,"(2*B*Sqrt[x])/c + ((b*B - A*c)*(2*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x]))/(2*Sqrt[2]*b^(3/4)*c^(5/4))","A",1
189,1,74,235,0.0968415,"\int \frac{\sqrt{x} \left(A+B x^2\right)}{b x^2+c x^4} \, dx","Integrate[(Sqrt[x]*(A + B*x^2))/(b*x^2 + c*x^4),x]","\frac{\frac{(b B-A c) \left(\tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-b}}\right)+\tanh ^{-1}\left(\frac{b \sqrt[4]{c} \sqrt{x}}{(-b)^{5/4}}\right)\right)}{\sqrt[4]{-b} c^{3/4}}-\frac{2 A}{\sqrt{x}}}{b}","\frac{(b B-A c) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} b^{5/4} c^{3/4}}-\frac{(b B-A c) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} b^{5/4} c^{3/4}}-\frac{(b B-A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{\sqrt{2} b^{5/4} c^{3/4}}+\frac{(b B-A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{\sqrt{2} b^{5/4} c^{3/4}}-\frac{2 A}{b \sqrt{x}}",1,"((-2*A)/Sqrt[x] + ((b*B - A*c)*(ArcTan[(c^(1/4)*Sqrt[x])/(-b)^(1/4)] + ArcTanh[(b*c^(1/4)*Sqrt[x])/(-b)^(5/4)]))/((-b)^(1/4)*c^(3/4)))/b","A",1
190,1,168,237,0.1536205,"\int \frac{A+B x^2}{\sqrt{x} \left(b x^2+c x^4\right)} \, dx","Integrate[(A + B*x^2)/(Sqrt[x]*(b*x^2 + c*x^4)),x]","-\frac{(b B-A c) \left(\log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)-\log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)+2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)\right)}{2 \sqrt{2} b^{7/4} \sqrt[4]{c}}-\frac{2 A}{3 b x^{3/2}}","-\frac{(b B-A c) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} b^{7/4} \sqrt[4]{c}}+\frac{(b B-A c) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} b^{7/4} \sqrt[4]{c}}-\frac{(b B-A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{\sqrt{2} b^{7/4} \sqrt[4]{c}}+\frac{(b B-A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{\sqrt{2} b^{7/4} \sqrt[4]{c}}-\frac{2 A}{3 b x^{3/2}}",1,"(-2*A)/(3*b*x^(3/2)) - ((b*B - A*c)*(2*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x]))/(2*Sqrt[2]*b^(7/4)*c^(1/4))","A",1
191,1,46,255,0.021646,"\int \frac{A+B x^2}{x^{3/2} \left(b x^2+c x^4\right)} \, dx","Integrate[(A + B*x^2)/(x^(3/2)*(b*x^2 + c*x^4)),x]","-\frac{2 \left(5 x^2 (b B-A c) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\frac{c x^2}{b}\right)+A b\right)}{5 b^2 x^{5/2}}","-\frac{\sqrt[4]{c} (b B-A c) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} b^{9/4}}+\frac{\sqrt[4]{c} (b B-A c) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} b^{9/4}}+\frac{\sqrt[4]{c} (b B-A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{\sqrt{2} b^{9/4}}-\frac{\sqrt[4]{c} (b B-A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{\sqrt{2} b^{9/4}}-\frac{2 (b B-A c)}{b^2 \sqrt{x}}-\frac{2 A}{5 b x^{5/2}}",1,"(-2*(A*b + 5*(b*B - A*c)*x^2*Hypergeometric2F1[-1/4, 1, 3/4, -((c*x^2)/b)]))/(5*b^2*x^(5/2))","C",1
192,1,47,257,0.0196323,"\int \frac{A+B x^2}{x^{5/2} \left(b x^2+c x^4\right)} \, dx","Integrate[(A + B*x^2)/(x^(5/2)*(b*x^2 + c*x^4)),x]","\frac{14 x^2 (A c-b B) \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\frac{c x^2}{b}\right)-6 A b}{21 b^2 x^{7/2}}","\frac{c^{3/4} (b B-A c) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} b^{11/4}}-\frac{c^{3/4} (b B-A c) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} b^{11/4}}+\frac{c^{3/4} (b B-A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{\sqrt{2} b^{11/4}}-\frac{c^{3/4} (b B-A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{\sqrt{2} b^{11/4}}-\frac{2 (b B-A c)}{3 b^2 x^{3/2}}-\frac{2 A}{7 b x^{7/2}}",1,"(-6*A*b + 14*(-(b*B) + A*c)*x^2*Hypergeometric2F1[-3/4, 1, 1/4, -((c*x^2)/b)])/(21*b^2*x^(7/2))","C",1
193,1,47,276,0.0173722,"\int \frac{A+B x^2}{x^{7/2} \left(b x^2+c x^4\right)} \, dx","Integrate[(A + B*x^2)/(x^(7/2)*(b*x^2 + c*x^4)),x]","\frac{2 \left(9 x^2 (A c-b B) \, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};-\frac{c x^2}{b}\right)-5 A b\right)}{45 b^2 x^{9/2}}","\frac{c^{5/4} (b B-A c) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} b^{13/4}}-\frac{c^{5/4} (b B-A c) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} b^{13/4}}-\frac{c^{5/4} (b B-A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{\sqrt{2} b^{13/4}}+\frac{c^{5/4} (b B-A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{\sqrt{2} b^{13/4}}+\frac{2 c (b B-A c)}{b^3 \sqrt{x}}-\frac{2 (b B-A c)}{5 b^2 x^{5/2}}-\frac{2 A}{9 b x^{9/2}}",1,"(2*(-5*A*b + 9*(-(b*B) + A*c)*x^2*Hypergeometric2F1[-5/4, 1, -1/4, -((c*x^2)/b)]))/(45*b^2*x^(9/2))","C",1
194,1,47,278,0.0188116,"\int \frac{A+B x^2}{x^{9/2} \left(b x^2+c x^4\right)} \, dx","Integrate[(A + B*x^2)/(x^(9/2)*(b*x^2 + c*x^4)),x]","\frac{-22 x^2 (b B-A c) \, _2F_1\left(-\frac{7}{4},1;-\frac{3}{4};-\frac{c x^2}{b}\right)-14 A b}{77 b^2 x^{11/2}}","-\frac{c^{7/4} (b B-A c) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} b^{15/4}}+\frac{c^{7/4} (b B-A c) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} b^{15/4}}-\frac{c^{7/4} (b B-A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{\sqrt{2} b^{15/4}}+\frac{c^{7/4} (b B-A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{\sqrt{2} b^{15/4}}+\frac{2 c (b B-A c)}{3 b^3 x^{3/2}}-\frac{2 (b B-A c)}{7 b^2 x^{7/2}}-\frac{2 A}{11 b x^{11/2}}",1,"(-14*A*b - 22*(b*B - A*c)*x^2*Hypergeometric2F1[-7/4, 1, -3/4, -((c*x^2)/b)])/(77*b^2*x^(11/2))","C",1
195,1,417,332,0.6737165,"\int \frac{x^{19/2} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[(x^(19/2)*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]","\frac{90 \sqrt{2} b^{5/4} (13 b B-9 A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)-90 \sqrt{2} b^{5/4} (13 b B-9 A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)-405 \sqrt{2} A b^{5/4} c \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)+405 \sqrt{2} A b^{5/4} c \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)-\frac{360 A b^2 c^{5/4} \sqrt{x}}{b+c x^2}-2880 A b c^{5/4} \sqrt{x}+288 A c^{9/4} x^{5/2}+585 \sqrt{2} b^{9/4} B \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)-585 \sqrt{2} b^{9/4} B \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)+\frac{360 b^3 B \sqrt[4]{c} \sqrt{x}}{b+c x^2}+4320 b^2 B \sqrt[4]{c} \sqrt{x}-576 b B c^{5/4} x^{5/2}+160 B c^{9/4} x^{9/2}}{720 c^{17/4}}","\frac{b^{5/4} (13 b B-9 A c) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} c^{17/4}}-\frac{b^{5/4} (13 b B-9 A c) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} c^{17/4}}+\frac{b^{5/4} (13 b B-9 A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{4 \sqrt{2} c^{17/4}}-\frac{b^{5/4} (13 b B-9 A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{4 \sqrt{2} c^{17/4}}+\frac{b \sqrt{x} (13 b B-9 A c)}{2 c^4}-\frac{x^{5/2} (13 b B-9 A c)}{10 c^3}+\frac{x^{9/2} (13 b B-9 A c)}{18 b c^2}-\frac{x^{13/2} (b B-A c)}{2 b c \left(b+c x^2\right)}",1,"(4320*b^2*B*c^(1/4)*Sqrt[x] - 2880*A*b*c^(5/4)*Sqrt[x] - 576*b*B*c^(5/4)*x^(5/2) + 288*A*c^(9/4)*x^(5/2) + 160*B*c^(9/4)*x^(9/2) + (360*b^3*B*c^(1/4)*Sqrt[x])/(b + c*x^2) - (360*A*b^2*c^(5/4)*Sqrt[x])/(b + c*x^2) + 90*Sqrt[2]*b^(5/4)*(13*b*B - 9*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] - 90*Sqrt[2]*b^(5/4)*(13*b*B - 9*A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 585*Sqrt[2]*b^(9/4)*B*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - 405*Sqrt[2]*A*b^(5/4)*c*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - 585*Sqrt[2]*b^(9/4)*B*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + 405*Sqrt[2]*A*b^(5/4)*c*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(720*c^(17/4))","A",1
196,1,154,310,0.2966181,"\int \frac{x^{17/2} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[(x^(17/2)*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]","-\frac{(-b)^{3/4} (3 b B-2 A c) \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-b}}\right)}{c^{15/4}}+\frac{(-b)^{3/4} (3 b B-2 A c) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-b}}\right)}{c^{15/4}}+\frac{2 x^{3/2} (A c-b B) \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-\frac{c x^2}{b}\right)}{3 c^3}+\frac{2 x^{3/2} (A c-2 b B)}{3 c^3}+\frac{2 B x^{7/2}}{7 c^2}","\frac{b^{3/4} (11 b B-7 A c) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} c^{15/4}}-\frac{b^{3/4} (11 b B-7 A c) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} c^{15/4}}-\frac{b^{3/4} (11 b B-7 A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{4 \sqrt{2} c^{15/4}}+\frac{b^{3/4} (11 b B-7 A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{4 \sqrt{2} c^{15/4}}-\frac{x^{3/2} (11 b B-7 A c)}{6 c^3}+\frac{x^{7/2} (11 b B-7 A c)}{14 b c^2}-\frac{x^{11/2} (b B-A c)}{2 b c \left(b+c x^2\right)}",1,"(2*(-2*b*B + A*c)*x^(3/2))/(3*c^3) + (2*B*x^(7/2))/(7*c^2) - ((-b)^(3/4)*(3*b*B - 2*A*c)*ArcTan[(c^(1/4)*Sqrt[x])/(-b)^(1/4)])/c^(15/4) + ((-b)^(3/4)*(3*b*B - 2*A*c)*ArcTanh[(c^(1/4)*Sqrt[x])/(-b)^(1/4)])/c^(15/4) + (2*(-(b*B) + A*c)*x^(3/2)*Hypergeometric2F1[3/4, 2, 7/4, -((c*x^2)/b)])/(3*c^3)","C",1
197,1,385,310,0.5476362,"\int \frac{x^{15/2} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[(x^(15/2)*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]","\frac{-10 \sqrt{2} \sqrt[4]{b} (9 b B-5 A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)+10 \sqrt{2} \sqrt[4]{b} (9 b B-5 A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)+\frac{40 A b c^{5/4} \sqrt{x}}{b+c x^2}+25 \sqrt{2} A \sqrt[4]{b} c \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)-25 \sqrt{2} A \sqrt[4]{b} c \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)+160 A c^{5/4} \sqrt{x}-45 \sqrt{2} b^{5/4} B \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)+45 \sqrt{2} b^{5/4} B \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)-\frac{40 b^2 B \sqrt[4]{c} \sqrt{x}}{b+c x^2}-320 b B \sqrt[4]{c} \sqrt{x}+32 B c^{5/4} x^{5/2}}{80 c^{13/4}}","-\frac{\sqrt[4]{b} (9 b B-5 A c) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} c^{13/4}}+\frac{\sqrt[4]{b} (9 b B-5 A c) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} c^{13/4}}-\frac{\sqrt[4]{b} (9 b B-5 A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{4 \sqrt{2} c^{13/4}}+\frac{\sqrt[4]{b} (9 b B-5 A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{4 \sqrt{2} c^{13/4}}-\frac{\sqrt{x} (9 b B-5 A c)}{2 c^3}+\frac{x^{5/2} (9 b B-5 A c)}{10 b c^2}-\frac{x^{9/2} (b B-A c)}{2 b c \left(b+c x^2\right)}",1,"(-320*b*B*c^(1/4)*Sqrt[x] + 160*A*c^(5/4)*Sqrt[x] + 32*B*c^(5/4)*x^(5/2) - (40*b^2*B*c^(1/4)*Sqrt[x])/(b + c*x^2) + (40*A*b*c^(5/4)*Sqrt[x])/(b + c*x^2) - 10*Sqrt[2]*b^(1/4)*(9*b*B - 5*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 10*Sqrt[2]*b^(1/4)*(9*b*B - 5*A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] - 45*Sqrt[2]*b^(5/4)*B*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + 25*Sqrt[2]*A*b^(1/4)*c*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + 45*Sqrt[2]*b^(5/4)*B*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - 25*Sqrt[2]*A*b^(1/4)*c*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(80*c^(13/4))","A",1
198,1,136,289,0.2646127,"\int \frac{x^{13/2} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[(x^(13/2)*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]","\frac{(3 A c-6 b B) \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-b}}\right)+(6 b B-3 A c) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-b}}\right)+2 \sqrt[4]{-b} B c^{3/4} x^{3/2}}{3 \sqrt[4]{-b} c^{11/4}}+\frac{2 x^{3/2} (b B-A c) \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-\frac{c x^2}{b}\right)}{3 b c^2}","-\frac{(7 b B-3 A c) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} \sqrt[4]{b} c^{11/4}}+\frac{(7 b B-3 A c) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} \sqrt[4]{b} c^{11/4}}+\frac{(7 b B-3 A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{4 \sqrt{2} \sqrt[4]{b} c^{11/4}}-\frac{(7 b B-3 A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{4 \sqrt{2} \sqrt[4]{b} c^{11/4}}+\frac{x^{3/2} (7 b B-3 A c)}{6 b c^2}-\frac{x^{7/2} (b B-A c)}{2 b c \left(b+c x^2\right)}",1,"(2*(-b)^(1/4)*B*c^(3/4)*x^(3/2) + (-6*b*B + 3*A*c)*ArcTan[(c^(1/4)*Sqrt[x])/(-b)^(1/4)] + (6*b*B - 3*A*c)*ArcTanh[(c^(1/4)*Sqrt[x])/(-b)^(1/4)])/(3*(-b)^(1/4)*c^(11/4)) + (2*(b*B - A*c)*x^(3/2)*Hypergeometric2F1[3/4, 2, 7/4, -((c*x^2)/b)])/(3*b*c^2)","C",1
199,1,354,289,0.5281202,"\int \frac{x^{11/2} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[(x^(11/2)*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]","\frac{\frac{2 \sqrt{2} (5 b B-A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{b^{3/4}}-\frac{2 \sqrt{2} (5 b B-A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{b^{3/4}}-\frac{\sqrt{2} A c \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{b^{3/4}}+\frac{\sqrt{2} A c \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{b^{3/4}}-\frac{8 A c^{5/4} \sqrt{x}}{b+c x^2}+\frac{8 b B \sqrt[4]{c} \sqrt{x}}{b+c x^2}+5 \sqrt{2} \sqrt[4]{b} B \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)-5 \sqrt{2} \sqrt[4]{b} B \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)+32 B \sqrt[4]{c} \sqrt{x}}{16 c^{9/4}}","\frac{(5 b B-A c) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} b^{3/4} c^{9/4}}-\frac{(5 b B-A c) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} b^{3/4} c^{9/4}}+\frac{(5 b B-A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{4 \sqrt{2} b^{3/4} c^{9/4}}-\frac{(5 b B-A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{4 \sqrt{2} b^{3/4} c^{9/4}}+\frac{\sqrt{x} (5 b B-A c)}{2 b c^2}-\frac{x^{5/2} (b B-A c)}{2 b c \left(b+c x^2\right)}",1,"(32*B*c^(1/4)*Sqrt[x] + (8*b*B*c^(1/4)*Sqrt[x])/(b + c*x^2) - (8*A*c^(5/4)*Sqrt[x])/(b + c*x^2) + (2*Sqrt[2]*(5*b*B - A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/b^(3/4) - (2*Sqrt[2]*(5*b*B - A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/b^(3/4) + 5*Sqrt[2]*b^(1/4)*B*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - (Sqrt[2]*A*c*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/b^(3/4) - 5*Sqrt[2]*b^(1/4)*B*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + (Sqrt[2]*A*c*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/b^(3/4))/(16*c^(9/4))","A",1
200,1,95,261,0.1824834,"\int \frac{x^{9/2} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[(x^(9/2)*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]","\frac{2 x^{3/2} (A c-b B) \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-\frac{c x^2}{b}\right)}{3 b^2 c}+\frac{B \left(\tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-b}}\right)+\tanh ^{-1}\left(\frac{b \sqrt[4]{c} \sqrt{x}}{(-b)^{5/4}}\right)\right)}{\sqrt[4]{-b} c^{7/4}}","\frac{(A c+3 b B) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} b^{5/4} c^{7/4}}-\frac{(A c+3 b B) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} b^{5/4} c^{7/4}}-\frac{(A c+3 b B) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{4 \sqrt{2} b^{5/4} c^{7/4}}+\frac{(A c+3 b B) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{4 \sqrt{2} b^{5/4} c^{7/4}}-\frac{x^{3/2} (b B-A c)}{2 b c \left(b+c x^2\right)}",1,"(B*(ArcTan[(c^(1/4)*Sqrt[x])/(-b)^(1/4)] + ArcTanh[(b*c^(1/4)*Sqrt[x])/(-b)^(5/4)]))/((-b)^(1/4)*c^(7/4)) + (2*(-(b*B) + A*c)*x^(3/2)*Hypergeometric2F1[3/4, 2, 7/4, -((c*x^2)/b)])/(3*b^2*c)","C",1
201,1,203,261,0.366071,"\int \frac{x^{7/2} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[(x^(7/2)*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]","\frac{\frac{(3 A c+b B) \left(8 b^{3/4} \sqrt[4]{c} \sqrt{x}-3 \sqrt{2} \left(b+c x^2\right) \left(\log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)-\log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)+2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)\right)\right)}{b^{7/4} \sqrt[4]{c}}-32 B \sqrt{x}}{48 c \left(b+c x^2\right)}","-\frac{(3 A c+b B) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} b^{7/4} c^{5/4}}+\frac{(3 A c+b B) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} b^{7/4} c^{5/4}}-\frac{(3 A c+b B) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{4 \sqrt{2} b^{7/4} c^{5/4}}+\frac{(3 A c+b B) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{4 \sqrt{2} b^{7/4} c^{5/4}}-\frac{\sqrt{x} (b B-A c)}{2 b c \left(b+c x^2\right)}",1,"(-32*B*Sqrt[x] + ((b*B + 3*A*c)*(8*b^(3/4)*c^(1/4)*Sqrt[x] - 3*Sqrt[2]*(b + c*x^2)*(2*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])))/(b^(7/4)*c^(1/4)))/(48*c*(b + c*x^2))","A",1
202,1,117,284,0.3164249,"\int \frac{x^{5/2} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[(x^(5/2)*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]","\frac{2 x^{3/2} (b B-A c) \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-\frac{c x^2}{b}\right)+3 A \left((-b)^{3/4} \sqrt[4]{c} \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-b}}\right)-(-b)^{3/4} \sqrt[4]{c} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-b}}\right)-\frac{2 b}{\sqrt{x}}\right)}{3 b^3}","\frac{(b B-5 A c) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} b^{9/4} c^{3/4}}-\frac{(b B-5 A c) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} b^{9/4} c^{3/4}}-\frac{(b B-5 A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{4 \sqrt{2} b^{9/4} c^{3/4}}+\frac{(b B-5 A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{4 \sqrt{2} b^{9/4} c^{3/4}}+\frac{b B-5 A c}{2 b^2 c \sqrt{x}}-\frac{b B-A c}{2 b c \sqrt{x} \left(b+c x^2\right)}",1,"(3*A*((-2*b)/Sqrt[x] + (-b)^(3/4)*c^(1/4)*ArcTan[(c^(1/4)*Sqrt[x])/(-b)^(1/4)] - (-b)^(3/4)*c^(1/4)*ArcTanh[(c^(1/4)*Sqrt[x])/(-b)^(1/4)]) + 2*(b*B - A*c)*x^(3/2)*Hypergeometric2F1[3/4, 2, 7/4, -((c*x^2)/b)])/(3*b^3)","C",1
203,1,355,289,0.5824743,"\int \frac{x^{3/2} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[(x^(3/2)*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]","\frac{-\frac{24 A b^{3/4} c \sqrt{x}}{b+c x^2}-\frac{32 A b^{3/4}}{x^{3/2}}+\frac{6 \sqrt{2} (7 A c-3 b B) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{\sqrt[4]{c}}+\frac{6 \sqrt{2} (3 b B-7 A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{\sqrt[4]{c}}+21 \sqrt{2} A c^{3/4} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)-21 \sqrt{2} A c^{3/4} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)+\frac{24 b^{7/4} B \sqrt{x}}{b+c x^2}-\frac{9 \sqrt{2} b B \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{\sqrt[4]{c}}+\frac{9 \sqrt{2} b B \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{\sqrt[4]{c}}}{48 b^{11/4}}","-\frac{(3 b B-7 A c) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} b^{11/4} \sqrt[4]{c}}+\frac{(3 b B-7 A c) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} b^{11/4} \sqrt[4]{c}}-\frac{(3 b B-7 A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{4 \sqrt{2} b^{11/4} \sqrt[4]{c}}+\frac{(3 b B-7 A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{4 \sqrt{2} b^{11/4} \sqrt[4]{c}}+\frac{3 b B-7 A c}{6 b^2 c x^{3/2}}-\frac{b B-A c}{2 b c x^{3/2} \left(b+c x^2\right)}",1,"((-32*A*b^(3/4))/x^(3/2) + (24*b^(7/4)*B*Sqrt[x])/(b + c*x^2) - (24*A*b^(3/4)*c*Sqrt[x])/(b + c*x^2) + (6*Sqrt[2]*(-3*b*B + 7*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/c^(1/4) + (6*Sqrt[2]*(3*b*B - 7*A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/c^(1/4) - (9*Sqrt[2]*b*B*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/c^(1/4) + 21*Sqrt[2]*A*c^(3/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + (9*Sqrt[2]*b*B*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/c^(1/4) - 21*Sqrt[2]*A*c^(3/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(48*b^(11/4))","A",1
204,1,151,310,0.637205,"\int \frac{\sqrt{x} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[(Sqrt[x]*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]","\frac{2 c x^{3/2} (A c-b B) \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-\frac{c x^2}{b}\right)}{3 b^4}+\frac{4 A c-2 b B}{b^3 \sqrt{x}}-\frac{2 A}{5 b^2 x^{5/2}}+\frac{\sqrt[4]{c} (b B-2 A c) \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-b}}\right)}{(-b)^{13/4}}+\frac{b \sqrt[4]{c} (b B-2 A c) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-b}}\right)}{(-b)^{17/4}}","-\frac{\sqrt[4]{c} (5 b B-9 A c) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} b^{13/4}}+\frac{\sqrt[4]{c} (5 b B-9 A c) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} b^{13/4}}+\frac{\sqrt[4]{c} (5 b B-9 A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{4 \sqrt{2} b^{13/4}}-\frac{\sqrt[4]{c} (5 b B-9 A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{4 \sqrt{2} b^{13/4}}-\frac{5 b B-9 A c}{2 b^3 \sqrt{x}}+\frac{5 b B-9 A c}{10 b^2 c x^{5/2}}-\frac{b B-A c}{2 b c x^{5/2} \left(b+c x^2\right)}",1,"(-2*A)/(5*b^2*x^(5/2)) + (-2*b*B + 4*A*c)/(b^3*Sqrt[x]) + (c^(1/4)*(b*B - 2*A*c)*ArcTan[(c^(1/4)*Sqrt[x])/(-b)^(1/4)])/(-b)^(13/4) + (b*c^(1/4)*(b*B - 2*A*c)*ArcTanh[(c^(1/4)*Sqrt[x])/(-b)^(1/4)])/(-b)^(17/4) + (2*c*(-(b*B) + A*c)*x^(3/2)*Hypergeometric2F1[3/4, 2, 7/4, -((c*x^2)/b)])/(3*b^4)","C",1
205,1,385,310,0.732857,"\int \frac{A+B x^2}{\sqrt{x} \left(b x^2+c x^4\right)^2} \, dx","Integrate[(A + B*x^2)/(Sqrt[x]*(b*x^2 + c*x^4)^2),x]","\frac{\frac{168 A b^{3/4} c^2 \sqrt{x}}{b+c x^2}+\frac{448 A b^{3/4} c}{x^{3/2}}-\frac{96 A b^{7/4}}{x^{7/2}}+42 \sqrt{2} c^{3/4} (7 b B-11 A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)+42 \sqrt{2} c^{3/4} (11 A c-7 b B) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)-231 \sqrt{2} A c^{7/4} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)+231 \sqrt{2} A c^{7/4} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)-\frac{168 b^{7/4} B c \sqrt{x}}{b+c x^2}-\frac{224 b^{7/4} B}{x^{3/2}}+147 \sqrt{2} b B c^{3/4} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)-147 \sqrt{2} b B c^{3/4} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{336 b^{15/4}}","\frac{c^{3/4} (7 b B-11 A c) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} b^{15/4}}-\frac{c^{3/4} (7 b B-11 A c) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} b^{15/4}}+\frac{c^{3/4} (7 b B-11 A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{4 \sqrt{2} b^{15/4}}-\frac{c^{3/4} (7 b B-11 A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{4 \sqrt{2} b^{15/4}}-\frac{7 b B-11 A c}{6 b^3 x^{3/2}}+\frac{7 b B-11 A c}{14 b^2 c x^{7/2}}-\frac{b B-A c}{2 b c x^{7/2} \left(b+c x^2\right)}",1,"((-96*A*b^(7/4))/x^(7/2) - (224*b^(7/4)*B)/x^(3/2) + (448*A*b^(3/4)*c)/x^(3/2) - (168*b^(7/4)*B*c*Sqrt[x])/(b + c*x^2) + (168*A*b^(3/4)*c^2*Sqrt[x])/(b + c*x^2) + 42*Sqrt[2]*c^(3/4)*(7*b*B - 11*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 42*Sqrt[2]*c^(3/4)*(-7*b*B + 11*A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 147*Sqrt[2]*b*B*c^(3/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - 231*Sqrt[2]*A*c^(7/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - 147*Sqrt[2]*b*B*c^(3/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + 231*Sqrt[2]*A*c^(7/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(336*b^(15/4))","A",1
206,1,176,332,0.5933595,"\int \frac{A+B x^2}{x^{3/2} \left(b x^2+c x^4\right)^2} \, dx","Integrate[(A + B*x^2)/(x^(3/2)*(b*x^2 + c*x^4)^2),x]","\frac{2 c^2 x^{3/2} (b B-A c) \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-\frac{c x^2}{b}\right)}{3 b^5}+\frac{2 c (2 b B-3 A c)}{b^4 \sqrt{x}}-\frac{2 (b B-2 A c)}{5 b^3 x^{5/2}}-\frac{2 A}{9 b^2 x^{9/2}}+\frac{c^{5/4} (2 b B-3 A c) \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-b}}\right)}{(-b)^{17/4}}+\frac{c^{5/4} (3 A c-2 b B) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-b}}\right)}{(-b)^{17/4}}","\frac{c^{5/4} (9 b B-13 A c) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} b^{17/4}}-\frac{c^{5/4} (9 b B-13 A c) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} b^{17/4}}-\frac{c^{5/4} (9 b B-13 A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{4 \sqrt{2} b^{17/4}}+\frac{c^{5/4} (9 b B-13 A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{4 \sqrt{2} b^{17/4}}+\frac{c (9 b B-13 A c)}{2 b^4 \sqrt{x}}-\frac{9 b B-13 A c}{10 b^3 x^{5/2}}+\frac{9 b B-13 A c}{18 b^2 c x^{9/2}}-\frac{b B-A c}{2 b c x^{9/2} \left(b+c x^2\right)}",1,"(-2*A)/(9*b^2*x^(9/2)) - (2*(b*B - 2*A*c))/(5*b^3*x^(5/2)) + (2*c*(2*b*B - 3*A*c))/(b^4*Sqrt[x]) + (c^(5/4)*(2*b*B - 3*A*c)*ArcTan[(c^(1/4)*Sqrt[x])/(-b)^(1/4)])/(-b)^(17/4) + (c^(5/4)*(-2*b*B + 3*A*c)*ArcTanh[(c^(1/4)*Sqrt[x])/(-b)^(1/4)])/(-b)^(17/4) + (2*c^2*(b*B - A*c)*x^(3/2)*Hypergeometric2F1[3/4, 2, 7/4, -((c*x^2)/b)])/(3*b^5)","C",1
207,1,435,343,0.6282327,"\int \frac{x^{23/2} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[(x^(23/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]","\frac{-\frac{160 A b^2 c^{5/4} \sqrt{x}}{\left(b+c x^2\right)^2}-90 \sqrt{2} \sqrt[4]{b} (13 b B-5 A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)+90 \sqrt{2} \sqrt[4]{b} (13 b B-5 A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)+\frac{680 A b c^{5/4} \sqrt{x}}{b+c x^2}+225 \sqrt{2} A \sqrt[4]{b} c \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)-225 \sqrt{2} A \sqrt[4]{b} c \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)+1280 A c^{5/4} \sqrt{x}-585 \sqrt{2} b^{5/4} B \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)+585 \sqrt{2} b^{5/4} B \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)+\frac{160 b^3 B \sqrt[4]{c} \sqrt{x}}{\left(b+c x^2\right)^2}-\frac{1000 b^2 B \sqrt[4]{c} \sqrt{x}}{b+c x^2}-3840 b B \sqrt[4]{c} \sqrt{x}+256 B c^{5/4} x^{5/2}}{640 c^{17/4}}","-\frac{9 \sqrt[4]{b} (13 b B-5 A c) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} c^{17/4}}+\frac{9 \sqrt[4]{b} (13 b B-5 A c) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} c^{17/4}}-\frac{9 \sqrt[4]{b} (13 b B-5 A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{32 \sqrt{2} c^{17/4}}+\frac{9 \sqrt[4]{b} (13 b B-5 A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{32 \sqrt{2} c^{17/4}}-\frac{9 \sqrt{x} (13 b B-5 A c)}{16 c^4}+\frac{9 x^{5/2} (13 b B-5 A c)}{80 b c^3}-\frac{x^{9/2} (13 b B-5 A c)}{16 b c^2 \left(b+c x^2\right)}-\frac{x^{13/2} (b B-A c)}{4 b c \left(b+c x^2\right)^2}",1,"(-3840*b*B*c^(1/4)*Sqrt[x] + 1280*A*c^(5/4)*Sqrt[x] + 256*B*c^(5/4)*x^(5/2) + (160*b^3*B*c^(1/4)*Sqrt[x])/(b + c*x^2)^2 - (160*A*b^2*c^(5/4)*Sqrt[x])/(b + c*x^2)^2 - (1000*b^2*B*c^(1/4)*Sqrt[x])/(b + c*x^2) + (680*A*b*c^(5/4)*Sqrt[x])/(b + c*x^2) - 90*Sqrt[2]*b^(1/4)*(13*b*B - 5*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 90*Sqrt[2]*b^(1/4)*(13*b*B - 5*A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] - 585*Sqrt[2]*b^(5/4)*B*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + 225*Sqrt[2]*A*b^(1/4)*c*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + 585*Sqrt[2]*b^(5/4)*B*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - 225*Sqrt[2]*A*b^(1/4)*c*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(640*c^(17/4))","A",1
208,1,176,322,0.4692778,"\int \frac{x^{21/2} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[(x^(21/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]","\frac{\frac{2 c^{3/4} x^{3/2} (3 b B-2 A c) \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-\frac{c x^2}{b}\right)}{b}+\frac{2 c^{3/4} x^{3/2} (A c-b B) \, _2F_1\left(\frac{3}{4},3;\frac{7}{4};-\frac{c x^2}{b}\right)}{b}+\frac{(3 A c-9 b B) \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-b}}\right)}{\sqrt[4]{-b}}+\frac{(9 b B-3 A c) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-b}}\right)}{\sqrt[4]{-b}}+2 B c^{3/4} x^{3/2}}{3 c^{15/4}}","-\frac{7 (11 b B-3 A c) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} \sqrt[4]{b} c^{15/4}}+\frac{7 (11 b B-3 A c) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} \sqrt[4]{b} c^{15/4}}+\frac{7 (11 b B-3 A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{32 \sqrt{2} \sqrt[4]{b} c^{15/4}}-\frac{7 (11 b B-3 A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{32 \sqrt{2} \sqrt[4]{b} c^{15/4}}+\frac{7 x^{3/2} (11 b B-3 A c)}{48 b c^3}-\frac{x^{7/2} (11 b B-3 A c)}{16 b c^2 \left(b+c x^2\right)}-\frac{x^{11/2} (b B-A c)}{4 b c \left(b+c x^2\right)^2}",1,"(2*B*c^(3/4)*x^(3/2) + ((-9*b*B + 3*A*c)*ArcTan[(c^(1/4)*Sqrt[x])/(-b)^(1/4)])/(-b)^(1/4) + ((9*b*B - 3*A*c)*ArcTanh[(c^(1/4)*Sqrt[x])/(-b)^(1/4)])/(-b)^(1/4) + (2*c^(3/4)*(3*b*B - 2*A*c)*x^(3/2)*Hypergeometric2F1[3/4, 2, 7/4, -((c*x^2)/b)])/b + (2*c^(3/4)*(-(b*B) + A*c)*x^(3/2)*Hypergeometric2F1[3/4, 3, 7/4, -((c*x^2)/b)])/b)/(3*c^(15/4))","C",1
209,1,403,322,0.6267867,"\int \frac{x^{19/2} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[(x^(19/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]","\frac{\frac{10 \sqrt{2} (9 b B-A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{b^{3/4}}-\frac{10 \sqrt{2} (9 b B-A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{b^{3/4}}-\frac{5 \sqrt{2} A c \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{b^{3/4}}+\frac{5 \sqrt{2} A c \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{b^{3/4}}+\frac{32 A b c^{5/4} \sqrt{x}}{\left(b+c x^2\right)^2}-\frac{72 A c^{5/4} \sqrt{x}}{b+c x^2}-\frac{32 b^2 B \sqrt[4]{c} \sqrt{x}}{\left(b+c x^2\right)^2}+\frac{136 b B \sqrt[4]{c} \sqrt{x}}{b+c x^2}+45 \sqrt{2} \sqrt[4]{b} B \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)-45 \sqrt{2} \sqrt[4]{b} B \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)+256 B \sqrt[4]{c} \sqrt{x}}{128 c^{13/4}}","\frac{5 (9 b B-A c) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{3/4} c^{13/4}}-\frac{5 (9 b B-A c) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{3/4} c^{13/4}}+\frac{5 (9 b B-A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{32 \sqrt{2} b^{3/4} c^{13/4}}-\frac{5 (9 b B-A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{32 \sqrt{2} b^{3/4} c^{13/4}}+\frac{5 \sqrt{x} (9 b B-A c)}{16 b c^3}-\frac{x^{5/2} (9 b B-A c)}{16 b c^2 \left(b+c x^2\right)}-\frac{x^{9/2} (b B-A c)}{4 b c \left(b+c x^2\right)^2}",1,"(256*B*c^(1/4)*Sqrt[x] - (32*b^2*B*c^(1/4)*Sqrt[x])/(b + c*x^2)^2 + (32*A*b*c^(5/4)*Sqrt[x])/(b + c*x^2)^2 + (136*b*B*c^(1/4)*Sqrt[x])/(b + c*x^2) - (72*A*c^(5/4)*Sqrt[x])/(b + c*x^2) + (10*Sqrt[2]*(9*b*B - A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/b^(3/4) - (10*Sqrt[2]*(9*b*B - A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/b^(3/4) + 45*Sqrt[2]*b^(1/4)*B*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - (5*Sqrt[2]*A*c*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/b^(3/4) - 45*Sqrt[2]*b^(1/4)*B*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + (5*Sqrt[2]*A*c*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/b^(3/4))/(128*c^(13/4))","A",1
210,1,137,293,0.3190191,"\int \frac{x^{17/2} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[(x^(17/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]","\frac{2 c^{3/4} x^{3/2} (A c-2 b B) \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-\frac{c x^2}{b}\right)+2 c^{3/4} x^{3/2} (b B-A c) \, _2F_1\left(\frac{3}{4},3;\frac{7}{4};-\frac{c x^2}{b}\right)+3 (-b)^{7/4} B \left(\tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-b}}\right)+\tanh ^{-1}\left(\frac{b \sqrt[4]{c} \sqrt{x}}{(-b)^{5/4}}\right)\right)}{3 b^2 c^{11/4}}","\frac{3 (A c+7 b B) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{5/4} c^{11/4}}-\frac{3 (A c+7 b B) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{5/4} c^{11/4}}-\frac{3 (A c+7 b B) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{32 \sqrt{2} b^{5/4} c^{11/4}}+\frac{3 (A c+7 b B) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{32 \sqrt{2} b^{5/4} c^{11/4}}-\frac{x^{3/2} (A c+7 b B)}{16 b c^2 \left(b+c x^2\right)}-\frac{x^{7/2} (b B-A c)}{4 b c \left(b+c x^2\right)^2}",1,"(3*(-b)^(7/4)*B*(ArcTan[(c^(1/4)*Sqrt[x])/(-b)^(1/4)] + ArcTanh[(b*c^(1/4)*Sqrt[x])/(-b)^(5/4)]) + 2*c^(3/4)*(-2*b*B + A*c)*x^(3/2)*Hypergeometric2F1[3/4, 2, 7/4, -((c*x^2)/b)] + 2*c^(3/4)*(b*B - A*c)*x^(3/2)*Hypergeometric2F1[3/4, 3, 7/4, -((c*x^2)/b)])/(3*b^2*c^(11/4))","C",1
211,1,389,298,0.7074279,"\int \frac{x^{15/2} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[(x^(15/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]","\frac{-\frac{2 \sqrt{2} (3 A c+5 b B) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{b^{7/4}}+\frac{2 \sqrt{2} (3 A c+5 b B) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{b^{7/4}}-\frac{3 \sqrt{2} A c \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{b^{7/4}}+\frac{3 \sqrt{2} A c \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{b^{7/4}}+\frac{8 A c^{5/4} \sqrt{x}}{b^2+b c x^2}-\frac{32 A c^{5/4} \sqrt{x}}{\left(b+c x^2\right)^2}-\frac{5 \sqrt{2} B \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{b^{3/4}}+\frac{5 \sqrt{2} B \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{b^{3/4}}-\frac{72 B \sqrt[4]{c} \sqrt{x}}{b+c x^2}+\frac{32 b B \sqrt[4]{c} \sqrt{x}}{\left(b+c x^2\right)^2}}{128 c^{9/4}}","-\frac{(3 A c+5 b B) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{7/4} c^{9/4}}+\frac{(3 A c+5 b B) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{7/4} c^{9/4}}-\frac{(3 A c+5 b B) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{32 \sqrt{2} b^{7/4} c^{9/4}}+\frac{(3 A c+5 b B) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{32 \sqrt{2} b^{7/4} c^{9/4}}-\frac{\sqrt{x} (3 A c+5 b B)}{16 b c^2 \left(b+c x^2\right)}-\frac{x^{5/2} (b B-A c)}{4 b c \left(b+c x^2\right)^2}",1,"((32*b*B*c^(1/4)*Sqrt[x])/(b + c*x^2)^2 - (32*A*c^(5/4)*Sqrt[x])/(b + c*x^2)^2 - (72*B*c^(1/4)*Sqrt[x])/(b + c*x^2) + (8*A*c^(5/4)*Sqrt[x])/(b^2 + b*c*x^2) - (2*Sqrt[2]*(5*b*B + 3*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/b^(7/4) + (2*Sqrt[2]*(5*b*B + 3*A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/b^(7/4) - (5*Sqrt[2]*B*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/b^(3/4) - (3*Sqrt[2]*A*c*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/b^(7/4) + (5*Sqrt[2]*B*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/b^(3/4) + (3*Sqrt[2]*A*c*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/b^(7/4))/(128*c^(9/4))","A",1
212,1,62,298,0.0871966,"\int \frac{x^{13/2} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[(x^(13/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]","\frac{2 x^{3/2} \left((A c-b B) \, _2F_1\left(\frac{3}{4},3;\frac{7}{4};-\frac{c x^2}{b}\right)+b B \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-\frac{c x^2}{b}\right)\right)}{3 b^3 c}","\frac{(5 A c+3 b B) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{9/4} c^{7/4}}-\frac{(5 A c+3 b B) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{9/4} c^{7/4}}-\frac{(5 A c+3 b B) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{32 \sqrt{2} b^{9/4} c^{7/4}}+\frac{(5 A c+3 b B) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{32 \sqrt{2} b^{9/4} c^{7/4}}+\frac{x^{3/2} (5 A c+3 b B)}{16 b^2 c \left(b+c x^2\right)}-\frac{x^{3/2} (b B-A c)}{4 b c \left(b+c x^2\right)^2}",1,"(2*x^(3/2)*(b*B*Hypergeometric2F1[3/4, 2, 7/4, -((c*x^2)/b)] + (-(b*B) + A*c)*Hypergeometric2F1[3/4, 3, 7/4, -((c*x^2)/b)]))/(3*b^3*c)","C",1
213,1,230,293,0.3781533,"\int \frac{x^{11/2} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[(x^(11/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]","\frac{\frac{(7 A c+b B) \left(7 \left(b+c x^2\right) \left(8 b^{3/4} \sqrt[4]{c} \sqrt{x}-3 \sqrt{2} \left(b+c x^2\right) \left(\log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)-\log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)+2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)\right)\right)+32 b^{7/4} \sqrt[4]{c} \sqrt{x}\right)}{b^{11/4} \sqrt[4]{c}}-256 B \sqrt{x}}{896 c \left(b+c x^2\right)^2}","-\frac{3 (7 A c+b B) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{11/4} c^{5/4}}+\frac{3 (7 A c+b B) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{11/4} c^{5/4}}-\frac{3 (7 A c+b B) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{32 \sqrt{2} b^{11/4} c^{5/4}}+\frac{3 (7 A c+b B) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{32 \sqrt{2} b^{11/4} c^{5/4}}+\frac{\sqrt{x} (7 A c+b B)}{16 b^2 c \left(b+c x^2\right)}-\frac{\sqrt{x} (b B-A c)}{4 b c \left(b+c x^2\right)^2}",1,"(-256*B*Sqrt[x] + ((b*B + 7*A*c)*(32*b^(7/4)*c^(1/4)*Sqrt[x] + 7*(b + c*x^2)*(8*b^(3/4)*c^(1/4)*Sqrt[x] - 3*Sqrt[2]*(b + c*x^2)*(2*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x]))))/(b^(11/4)*c^(1/4)))/(896*c*(b + c*x^2)^2)","A",1
214,1,147,316,0.2940896,"\int \frac{x^{9/2} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[(x^(9/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]","\frac{2 x^{3/2} (b B-A c) \, _2F_1\left(\frac{3}{4},3;\frac{7}{4};-\frac{c x^2}{b}\right)}{3 b^4}-\frac{2 A c x^{3/2} \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-\frac{c x^2}{b}\right)}{3 b^4}-\frac{2 A}{b^3 \sqrt{x}}+\frac{A \sqrt[4]{c} \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-b}}\right)}{(-b)^{13/4}}+\frac{A b \sqrt[4]{c} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-b}}\right)}{(-b)^{17/4}}","\frac{5 (b B-9 A c) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{13/4} c^{3/4}}-\frac{5 (b B-9 A c) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{13/4} c^{3/4}}-\frac{5 (b B-9 A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{32 \sqrt{2} b^{13/4} c^{3/4}}+\frac{5 (b B-9 A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{32 \sqrt{2} b^{13/4} c^{3/4}}+\frac{5 (b B-9 A c)}{16 b^3 c \sqrt{x}}-\frac{b B-9 A c}{16 b^2 c \sqrt{x} \left(b+c x^2\right)}-\frac{b B-A c}{4 b c \sqrt{x} \left(b+c x^2\right)^2}",1,"(-2*A)/(b^3*Sqrt[x]) + (A*c^(1/4)*ArcTan[(c^(1/4)*Sqrt[x])/(-b)^(1/4)])/(-b)^(13/4) + (A*b*c^(1/4)*ArcTanh[(c^(1/4)*Sqrt[x])/(-b)^(1/4)])/(-b)^(17/4) - (2*A*c*x^(3/2)*Hypergeometric2F1[3/4, 2, 7/4, -((c*x^2)/b)])/(3*b^4) + (2*(b*B - A*c)*x^(3/2)*Hypergeometric2F1[3/4, 3, 7/4, -((c*x^2)/b)])/(3*b^4)","C",1
215,1,400,322,0.4955706,"\int \frac{x^{7/2} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[(x^(7/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]","\frac{-\frac{96 A b^{7/4} c \sqrt{x}}{\left(b+c x^2\right)^2}-\frac{360 A b^{3/4} c \sqrt{x}}{b+c x^2}-\frac{256 A b^{3/4}}{x^{3/2}}+\frac{42 \sqrt{2} (11 A c-3 b B) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{\sqrt[4]{c}}+\frac{42 \sqrt{2} (3 b B-11 A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{\sqrt[4]{c}}+231 \sqrt{2} A c^{3/4} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)-231 \sqrt{2} A c^{3/4} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)+\frac{96 b^{11/4} B \sqrt{x}}{\left(b+c x^2\right)^2}+\frac{168 b^{7/4} B \sqrt{x}}{b+c x^2}-\frac{63 \sqrt{2} b B \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{\sqrt[4]{c}}+\frac{63 \sqrt{2} b B \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{\sqrt[4]{c}}}{384 b^{15/4}}","-\frac{7 (3 b B-11 A c) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{15/4} \sqrt[4]{c}}+\frac{7 (3 b B-11 A c) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{15/4} \sqrt[4]{c}}-\frac{7 (3 b B-11 A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{32 \sqrt{2} b^{15/4} \sqrt[4]{c}}+\frac{7 (3 b B-11 A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{32 \sqrt{2} b^{15/4} \sqrt[4]{c}}+\frac{7 (3 b B-11 A c)}{48 b^3 c x^{3/2}}-\frac{3 b B-11 A c}{16 b^2 c x^{3/2} \left(b+c x^2\right)}-\frac{b B-A c}{4 b c x^{3/2} \left(b+c x^2\right)^2}",1,"((-256*A*b^(3/4))/x^(3/2) + (96*b^(11/4)*B*Sqrt[x])/(b + c*x^2)^2 - (96*A*b^(7/4)*c*Sqrt[x])/(b + c*x^2)^2 + (168*b^(7/4)*B*Sqrt[x])/(b + c*x^2) - (360*A*b^(3/4)*c*Sqrt[x])/(b + c*x^2) + (42*Sqrt[2]*(-3*b*B + 11*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/c^(1/4) + (42*Sqrt[2]*(3*b*B - 11*A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/c^(1/4) - (63*Sqrt[2]*b*B*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/c^(1/4) + 231*Sqrt[2]*A*c^(3/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + (63*Sqrt[2]*b*B*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/c^(1/4) - 231*Sqrt[2]*A*c^(3/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(384*b^(15/4))","A",1
216,1,189,343,0.5174357,"\int \frac{x^{5/2} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[(x^(5/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]","-\frac{2 c x^{3/2} (b B-2 A c) \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-\frac{c x^2}{b}\right)}{3 b^5}+\frac{2 c x^{3/2} (A c-b B) \, _2F_1\left(\frac{3}{4},3;\frac{7}{4};-\frac{c x^2}{b}\right)}{3 b^5}+\frac{6 A c-2 b B}{b^4 \sqrt{x}}-\frac{2 A}{5 b^3 x^{5/2}}+\frac{\sqrt[4]{c} (3 A c-b B) \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-b}}\right)}{(-b)^{17/4}}+\frac{\sqrt[4]{c} (b B-3 A c) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-b}}\right)}{(-b)^{17/4}}","-\frac{9 \sqrt[4]{c} (5 b B-13 A c) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{17/4}}+\frac{9 \sqrt[4]{c} (5 b B-13 A c) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{17/4}}+\frac{9 \sqrt[4]{c} (5 b B-13 A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{32 \sqrt{2} b^{17/4}}-\frac{9 \sqrt[4]{c} (5 b B-13 A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{32 \sqrt{2} b^{17/4}}-\frac{9 (5 b B-13 A c)}{16 b^4 \sqrt{x}}+\frac{9 (5 b B-13 A c)}{80 b^3 c x^{5/2}}-\frac{5 b B-13 A c}{16 b^2 c x^{5/2} \left(b+c x^2\right)}-\frac{b B-A c}{4 b c x^{5/2} \left(b+c x^2\right)^2}",1,"(-2*A)/(5*b^3*x^(5/2)) + (-2*b*B + 6*A*c)/(b^4*Sqrt[x]) + (c^(1/4)*(-(b*B) + 3*A*c)*ArcTan[(c^(1/4)*Sqrt[x])/(-b)^(1/4)])/(-b)^(17/4) + (c^(1/4)*(b*B - 3*A*c)*ArcTanh[(c^(1/4)*Sqrt[x])/(-b)^(1/4)])/(-b)^(17/4) - (2*c*(b*B - 2*A*c)*x^(3/2)*Hypergeometric2F1[3/4, 2, 7/4, -((c*x^2)/b)])/(3*b^5) + (2*c*(-(b*B) + A*c)*x^(3/2)*Hypergeometric2F1[3/4, 3, 7/4, -((c*x^2)/b)])/(3*b^5)","C",1
217,1,433,343,0.5706694,"\int \frac{x^{3/2} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[(x^(3/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]","\frac{\frac{672 A b^{7/4} c^2 \sqrt{x}}{\left(b+c x^2\right)^2}+\frac{3864 A b^{3/4} c^2 \sqrt{x}}{b+c x^2}+\frac{5376 A b^{3/4} c}{x^{3/2}}-\frac{768 A b^{7/4}}{x^{7/2}}+462 \sqrt{2} c^{3/4} (7 b B-15 A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)+462 \sqrt{2} c^{3/4} (15 A c-7 b B) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)-3465 \sqrt{2} A c^{7/4} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)+3465 \sqrt{2} A c^{7/4} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)-\frac{672 b^{11/4} B c \sqrt{x}}{\left(b+c x^2\right)^2}-\frac{2520 b^{7/4} B c \sqrt{x}}{b+c x^2}-\frac{1792 b^{7/4} B}{x^{3/2}}+1617 \sqrt{2} b B c^{3/4} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)-1617 \sqrt{2} b B c^{3/4} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2688 b^{19/4}}","\frac{11 c^{3/4} (7 b B-15 A c) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{19/4}}-\frac{11 c^{3/4} (7 b B-15 A c) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{19/4}}+\frac{11 c^{3/4} (7 b B-15 A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{32 \sqrt{2} b^{19/4}}-\frac{11 c^{3/4} (7 b B-15 A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{32 \sqrt{2} b^{19/4}}-\frac{11 (7 b B-15 A c)}{48 b^4 x^{3/2}}+\frac{11 (7 b B-15 A c)}{112 b^3 c x^{7/2}}-\frac{7 b B-15 A c}{16 b^2 c x^{7/2} \left(b+c x^2\right)}-\frac{b B-A c}{4 b c x^{7/2} \left(b+c x^2\right)^2}",1,"((-768*A*b^(7/4))/x^(7/2) - (1792*b^(7/4)*B)/x^(3/2) + (5376*A*b^(3/4)*c)/x^(3/2) - (672*b^(11/4)*B*c*Sqrt[x])/(b + c*x^2)^2 + (672*A*b^(7/4)*c^2*Sqrt[x])/(b + c*x^2)^2 - (2520*b^(7/4)*B*c*Sqrt[x])/(b + c*x^2) + (3864*A*b^(3/4)*c^2*Sqrt[x])/(b + c*x^2) + 462*Sqrt[2]*c^(3/4)*(7*b*B - 15*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 462*Sqrt[2]*c^(3/4)*(-7*b*B + 15*A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 1617*Sqrt[2]*b*B*c^(3/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - 3465*Sqrt[2]*A*c^(7/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - 1617*Sqrt[2]*b*B*c^(3/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + 3465*Sqrt[2]*A*c^(7/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(2688*b^(19/4))","A",1
218,1,216,365,0.5519944,"\int \frac{\sqrt{x} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[(Sqrt[x]*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]","\frac{2 c^2 x^{3/2} (2 b B-3 A c) \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-\frac{c x^2}{b}\right)}{3 b^6}+\frac{2 c^2 x^{3/2} (b B-A c) \, _2F_1\left(\frac{3}{4},3;\frac{7}{4};-\frac{c x^2}{b}\right)}{3 b^6}+\frac{6 c (b B-2 A c)}{b^5 \sqrt{x}}-\frac{2 (b B-3 A c)}{5 b^4 x^{5/2}}-\frac{2 A}{9 b^3 x^{9/2}}-\frac{3 c^{5/4} (b B-2 A c) \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-b}}\right)}{(-b)^{21/4}}+\frac{3 c^{5/4} (b B-2 A c) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-b}}\right)}{(-b)^{21/4}}","\frac{13 c^{5/4} (9 b B-17 A c) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{21/4}}-\frac{13 c^{5/4} (9 b B-17 A c) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{21/4}}-\frac{13 c^{5/4} (9 b B-17 A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{32 \sqrt{2} b^{21/4}}+\frac{13 c^{5/4} (9 b B-17 A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{32 \sqrt{2} b^{21/4}}+\frac{13 c (9 b B-17 A c)}{16 b^5 \sqrt{x}}-\frac{13 (9 b B-17 A c)}{80 b^4 x^{5/2}}+\frac{13 (9 b B-17 A c)}{144 b^3 c x^{9/2}}-\frac{9 b B-17 A c}{16 b^2 c x^{9/2} \left(b+c x^2\right)}-\frac{b B-A c}{4 b c x^{9/2} \left(b+c x^2\right)^2}",1,"(-2*A)/(9*b^3*x^(9/2)) - (2*(b*B - 3*A*c))/(5*b^4*x^(5/2)) + (6*c*(b*B - 2*A*c))/(b^5*Sqrt[x]) - (3*c^(5/4)*(b*B - 2*A*c)*ArcTan[(c^(1/4)*Sqrt[x])/(-b)^(1/4)])/(-b)^(21/4) + (3*c^(5/4)*(b*B - 2*A*c)*ArcTanh[(c^(1/4)*Sqrt[x])/(-b)^(1/4)])/(-b)^(21/4) + (2*c^2*(2*b*B - 3*A*c)*x^(3/2)*Hypergeometric2F1[3/4, 2, 7/4, -((c*x^2)/b)])/(3*b^6) + (2*c^2*(b*B - A*c)*x^(3/2)*Hypergeometric2F1[3/4, 3, 7/4, -((c*x^2)/b)])/(3*b^6)","C",1
219,1,467,365,0.6275743,"\int \frac{A+B x^2}{\sqrt{x} \left(b x^2+c x^4\right)^3} \, dx","Integrate[(A + B*x^2)/(Sqrt[x]*(b*x^2 + c*x^4)^3),x]","\frac{-\frac{19096 A b^{3/4} c^3 \sqrt{x}}{b+c x^2}-\frac{2464 A b^{7/4} c^3 \sqrt{x}}{\left(b+c x^2\right)^2}-\frac{39424 A b^{3/4} c^2}{x^{3/2}}+\frac{8448 A b^{7/4} c}{x^{7/2}}-\frac{1792 A b^{11/4}}{x^{11/2}}+2310 \sqrt{2} c^{7/4} (19 A c-11 b B) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)+2310 \sqrt{2} c^{7/4} (11 b B-19 A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)+21945 \sqrt{2} A c^{11/4} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)-21945 \sqrt{2} A c^{11/4} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)+\frac{14168 b^{7/4} B c^2 \sqrt{x}}{b+c x^2}+\frac{2464 b^{11/4} B c^2 \sqrt{x}}{\left(b+c x^2\right)^2}+\frac{19712 b^{7/4} B c}{x^{3/2}}-\frac{2816 b^{11/4} B}{x^{7/2}}-12705 \sqrt{2} b B c^{7/4} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)+12705 \sqrt{2} b B c^{7/4} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{9856 b^{23/4}}","-\frac{15 c^{7/4} (11 b B-19 A c) \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{23/4}}+\frac{15 c^{7/4} (11 b B-19 A c) \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{23/4}}-\frac{15 c^{7/4} (11 b B-19 A c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{32 \sqrt{2} b^{23/4}}+\frac{15 c^{7/4} (11 b B-19 A c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{32 \sqrt{2} b^{23/4}}+\frac{5 c (11 b B-19 A c)}{16 b^5 x^{3/2}}-\frac{15 (11 b B-19 A c)}{112 b^4 x^{7/2}}+\frac{15 (11 b B-19 A c)}{176 b^3 c x^{11/2}}-\frac{11 b B-19 A c}{16 b^2 c x^{11/2} \left(b+c x^2\right)}-\frac{b B-A c}{4 b c x^{11/2} \left(b+c x^2\right)^2}",1,"((-1792*A*b^(11/4))/x^(11/2) - (2816*b^(11/4)*B)/x^(7/2) + (8448*A*b^(7/4)*c)/x^(7/2) + (19712*b^(7/4)*B*c)/x^(3/2) - (39424*A*b^(3/4)*c^2)/x^(3/2) + (2464*b^(11/4)*B*c^2*Sqrt[x])/(b + c*x^2)^2 - (2464*A*b^(7/4)*c^3*Sqrt[x])/(b + c*x^2)^2 + (14168*b^(7/4)*B*c^2*Sqrt[x])/(b + c*x^2) - (19096*A*b^(3/4)*c^3*Sqrt[x])/(b + c*x^2) + 2310*Sqrt[2]*c^(7/4)*(-11*b*B + 19*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 2310*Sqrt[2]*c^(7/4)*(11*b*B - 19*A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] - 12705*Sqrt[2]*b*B*c^(7/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + 21945*Sqrt[2]*A*c^(11/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + 12705*Sqrt[2]*b*B*c^(7/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - 21945*Sqrt[2]*A*c^(11/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(9856*b^(23/4))","A",1
220,1,136,243,0.1766385,"\int x^{5/2} \left(A+B x^2\right) \sqrt{b x^2+c x^4} \, dx","Integrate[x^(5/2)*(A + B*x^2)*Sqrt[b*x^2 + c*x^4],x]","\frac{2 \sqrt{x^2 \left(b+c x^2\right)} \left(\left(b+c x^2\right) \sqrt{\frac{c x^2}{b}+1} \left(-3 b c \left(25 A+21 B x^2\right)+7 c^2 x^2 \left(15 A+11 B x^2\right)+45 b^2 B\right)+15 b^2 (5 A c-3 b B) \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^2}{b}\right)\right)}{1155 c^3 \sqrt{x} \sqrt{\frac{c x^2}{b}+1}}","-\frac{2 b^{11/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (3 b B-5 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{231 c^{13/4} \sqrt{b x^2+c x^4}}+\frac{4 b^2 \sqrt{b x^2+c x^4} (3 b B-5 A c)}{231 c^3 \sqrt{x}}-\frac{4 b x^{3/2} \sqrt{b x^2+c x^4} (3 b B-5 A c)}{385 c^2}-\frac{2 x^{7/2} \sqrt{b x^2+c x^4} (3 b B-5 A c)}{55 c}+\frac{2 B x^{3/2} \left(b x^2+c x^4\right)^{3/2}}{15 c}",1,"(2*Sqrt[x^2*(b + c*x^2)]*((b + c*x^2)*Sqrt[1 + (c*x^2)/b]*(45*b^2*B + 7*c^2*x^2*(15*A + 11*B*x^2) - 3*b*c*(25*A + 21*B*x^2)) + 15*b^2*(-3*b*B + 5*A*c)*Hypergeometric2F1[-1/2, 1/4, 5/4, -((c*x^2)/b)]))/(1155*c^3*Sqrt[x]*Sqrt[1 + (c*x^2)/b])","C",1
221,1,111,369,0.1448656,"\int x^{3/2} \left(A+B x^2\right) \sqrt{b x^2+c x^4} \, dx","Integrate[x^(3/2)*(A + B*x^2)*Sqrt[b*x^2 + c*x^4],x]","\frac{2 \sqrt{x} \sqrt{x^2 \left(b+c x^2\right)} \left(b (7 b B-13 A c) \, _2F_1\left(-\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{b}\right)-\left(b+c x^2\right) \sqrt{\frac{c x^2}{b}+1} \left(-13 A c+7 b B-9 B c x^2\right)\right)}{117 c^2 \sqrt{\frac{c x^2}{b}+1}}","\frac{2 b^{9/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (7 b B-13 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{195 c^{11/4} \sqrt{b x^2+c x^4}}-\frac{4 b^{9/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (7 b B-13 A c) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{195 c^{11/4} \sqrt{b x^2+c x^4}}+\frac{4 b^2 x^{3/2} \left(b+c x^2\right) (7 b B-13 A c)}{195 c^{5/2} \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}-\frac{4 b \sqrt{x} \sqrt{b x^2+c x^4} (7 b B-13 A c)}{585 c^2}-\frac{2 x^{5/2} \sqrt{b x^2+c x^4} (7 b B-13 A c)}{117 c}+\frac{2 B \sqrt{x} \left(b x^2+c x^4\right)^{3/2}}{13 c}",1,"(2*Sqrt[x]*Sqrt[x^2*(b + c*x^2)]*(-((b + c*x^2)*Sqrt[1 + (c*x^2)/b]*(7*b*B - 13*A*c - 9*B*c*x^2)) + b*(7*b*B - 13*A*c)*Hypergeometric2F1[-1/2, 3/4, 7/4, -((c*x^2)/b)]))/(117*c^2*Sqrt[1 + (c*x^2)/b])","C",1
222,1,111,204,0.1388062,"\int \sqrt{x} \left(A+B x^2\right) \sqrt{b x^2+c x^4} \, dx","Integrate[Sqrt[x]*(A + B*x^2)*Sqrt[b*x^2 + c*x^4],x]","\frac{2 \sqrt{x^2 \left(b+c x^2\right)} \left(b (5 b B-11 A c) \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^2}{b}\right)-\left(b+c x^2\right) \sqrt{\frac{c x^2}{b}+1} \left(-11 A c+5 b B-7 B c x^2\right)\right)}{77 c^2 \sqrt{x} \sqrt{\frac{c x^2}{b}+1}}","\frac{2 b^{7/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (5 b B-11 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{231 c^{9/4} \sqrt{b x^2+c x^4}}-\frac{4 b \sqrt{b x^2+c x^4} (5 b B-11 A c)}{231 c^2 \sqrt{x}}-\frac{2 x^{3/2} \sqrt{b x^2+c x^4} (5 b B-11 A c)}{77 c}+\frac{2 B \left(b x^2+c x^4\right)^{3/2}}{11 c \sqrt{x}}",1,"(2*Sqrt[x^2*(b + c*x^2)]*(-((b + c*x^2)*Sqrt[1 + (c*x^2)/b]*(5*b*B - 11*A*c - 7*B*c*x^2)) + b*(5*b*B - 11*A*c)*Hypergeometric2F1[-1/2, 1/4, 5/4, -((c*x^2)/b)]))/(77*c^2*Sqrt[x]*Sqrt[1 + (c*x^2)/b])","C",1
223,1,94,326,0.0946064,"\int \frac{\left(A+B x^2\right) \sqrt{b x^2+c x^4}}{\sqrt{x}} \, dx","Integrate[((A + B*x^2)*Sqrt[b*x^2 + c*x^4])/Sqrt[x],x]","\frac{2 \sqrt{x} \sqrt{x^2 \left(b+c x^2\right)} \left((3 A c-b B) \, _2F_1\left(-\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{b}\right)+B \sqrt{\frac{c x^2}{b}+1} \left(b+c x^2\right)\right)}{9 c \sqrt{\frac{c x^2}{b}+1}}","-\frac{2 b^{5/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (b B-3 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{15 c^{7/4} \sqrt{b x^2+c x^4}}+\frac{4 b^{5/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (b B-3 A c) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{15 c^{7/4} \sqrt{b x^2+c x^4}}-\frac{4 b x^{3/2} \left(b+c x^2\right) (b B-3 A c)}{15 c^{3/2} \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}-\frac{2 \sqrt{x} \sqrt{b x^2+c x^4} (b B-3 A c)}{15 c}+\frac{2 B \left(b x^2+c x^4\right)^{3/2}}{9 c x^{3/2}}",1,"(2*Sqrt[x]*Sqrt[x^2*(b + c*x^2)]*(B*(b + c*x^2)*Sqrt[1 + (c*x^2)/b] + (-(b*B) + 3*A*c)*Hypergeometric2F1[-1/2, 3/4, 7/4, -((c*x^2)/b)]))/(9*c*Sqrt[1 + (c*x^2)/b])","C",1
224,1,94,165,0.0618901,"\int \frac{\left(A+B x^2\right) \sqrt{b x^2+c x^4}}{x^{3/2}} \, dx","Integrate[((A + B*x^2)*Sqrt[b*x^2 + c*x^4])/x^(3/2),x]","\frac{2 \sqrt{x^2 \left(b+c x^2\right)} \left((7 A c-b B) \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^2}{b}\right)+B \sqrt{\frac{c x^2}{b}+1} \left(b+c x^2\right)\right)}{7 c \sqrt{x} \sqrt{\frac{c x^2}{b}+1}}","-\frac{2 b^{3/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (b B-7 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{21 c^{5/4} \sqrt{b x^2+c x^4}}-\frac{2 \sqrt{b x^2+c x^4} (b B-7 A c)}{21 c \sqrt{x}}+\frac{2 B \left(b x^2+c x^4\right)^{3/2}}{7 c x^{5/2}}",1,"(2*Sqrt[x^2*(b + c*x^2)]*(B*(b + c*x^2)*Sqrt[1 + (c*x^2)/b] + (-(b*B) + 7*A*c)*Hypergeometric2F1[-1/2, 1/4, 5/4, -((c*x^2)/b)]))/(7*c*Sqrt[x]*Sqrt[1 + (c*x^2)/b])","C",1
225,1,97,323,0.0479658,"\int \frac{\left(A+B x^2\right) \sqrt{b x^2+c x^4}}{x^{5/2}} \, dx","Integrate[((A + B*x^2)*Sqrt[b*x^2 + c*x^4])/x^(5/2),x]","\frac{2 \sqrt{x^2 \left(b+c x^2\right)} \left(x^2 (5 A c+b B) \, _2F_1\left(-\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{b}\right)-3 A \left(b+c x^2\right) \sqrt{\frac{c x^2}{b}+1}\right)}{3 b x^{3/2} \sqrt{\frac{c x^2}{b}+1}}","\frac{2 \sqrt[4]{b} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (5 A c+b B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{5 c^{3/4} \sqrt{b x^2+c x^4}}-\frac{4 \sqrt[4]{b} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (5 A c+b B) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{5 c^{3/4} \sqrt{b x^2+c x^4}}+\frac{2 \sqrt{x} \sqrt{b x^2+c x^4} (5 A c+b B)}{5 b}+\frac{4 x^{3/2} \left(b+c x^2\right) (5 A c+b B)}{5 \sqrt{c} \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}-\frac{2 A \left(b x^2+c x^4\right)^{3/2}}{b x^{7/2}}",1,"(2*Sqrt[x^2*(b + c*x^2)]*(-3*A*(b + c*x^2)*Sqrt[1 + (c*x^2)/b] + (b*B + 5*A*c)*x^2*Hypergeometric2F1[-1/2, 3/4, 7/4, -((c*x^2)/b)]))/(3*b*x^(3/2)*Sqrt[1 + (c*x^2)/b])","C",1
226,1,97,163,0.0480802,"\int \frac{\left(A+B x^2\right) \sqrt{b x^2+c x^4}}{x^{7/2}} \, dx","Integrate[((A + B*x^2)*Sqrt[b*x^2 + c*x^4])/x^(7/2),x]","\frac{2 \sqrt{x^2 \left(b+c x^2\right)} \left(3 x^2 (A c+b B) \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^2}{b}\right)-A \left(b+c x^2\right) \sqrt{\frac{c x^2}{b}+1}\right)}{3 b x^{5/2} \sqrt{\frac{c x^2}{b}+1}}","\frac{2 \sqrt{b x^2+c x^4} (A c+b B)}{3 b \sqrt{x}}+\frac{2 x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (A c+b B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{3 \sqrt[4]{b} \sqrt[4]{c} \sqrt{b x^2+c x^4}}-\frac{2 A \left(b x^2+c x^4\right)^{3/2}}{3 b x^{9/2}}",1,"(2*Sqrt[x^2*(b + c*x^2)]*(-(A*(b + c*x^2)*Sqrt[1 + (c*x^2)/b]) + 3*(b*B + A*c)*x^2*Hypergeometric2F1[-1/2, 1/4, 5/4, -((c*x^2)/b)]))/(3*b*x^(5/2)*Sqrt[1 + (c*x^2)/b])","C",1
227,1,96,328,0.0523845,"\int \frac{\left(A+B x^2\right) \sqrt{b x^2+c x^4}}{x^{9/2}} \, dx","Integrate[((A + B*x^2)*Sqrt[b*x^2 + c*x^4])/x^(9/2),x]","-\frac{2 \sqrt{x^2 \left(b+c x^2\right)} \left(x^2 (A c+5 b B) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};-\frac{c x^2}{b}\right)+A \left(b+c x^2\right) \sqrt{\frac{c x^2}{b}+1}\right)}{5 b x^{7/2} \sqrt{\frac{c x^2}{b}+1}}","\frac{2 \sqrt[4]{c} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (A c+5 b B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{5 b^{3/4} \sqrt{b x^2+c x^4}}-\frac{4 \sqrt[4]{c} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (A c+5 b B) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{5 b^{3/4} \sqrt{b x^2+c x^4}}+\frac{4 \sqrt{c} x^{3/2} \left(b+c x^2\right) (A c+5 b B)}{5 b \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}-\frac{2 \sqrt{b x^2+c x^4} (A c+5 b B)}{5 b x^{3/2}}-\frac{2 A \left(b x^2+c x^4\right)^{3/2}}{5 b x^{11/2}}",1,"(-2*Sqrt[x^2*(b + c*x^2)]*(A*(b + c*x^2)*Sqrt[1 + (c*x^2)/b] + (5*b*B + A*c)*x^2*Hypergeometric2F1[-1/2, -1/4, 3/4, -((c*x^2)/b)]))/(5*b*x^(7/2)*Sqrt[1 + (c*x^2)/b])","C",1
228,1,98,167,0.0625055,"\int \frac{\left(A+B x^2\right) \sqrt{b x^2+c x^4}}{x^{11/2}} \, dx","Integrate[((A + B*x^2)*Sqrt[b*x^2 + c*x^4])/x^(11/2),x]","-\frac{2 \sqrt{x^2 \left(b+c x^2\right)} \left(x^2 (7 b B-A c) \, _2F_1\left(-\frac{3}{4},-\frac{1}{2};\frac{1}{4};-\frac{c x^2}{b}\right)+3 A \left(b+c x^2\right) \sqrt{\frac{c x^2}{b}+1}\right)}{21 b x^{9/2} \sqrt{\frac{c x^2}{b}+1}}","\frac{2 c^{3/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (7 b B-A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{21 b^{5/4} \sqrt{b x^2+c x^4}}-\frac{2 \sqrt{b x^2+c x^4} (7 b B-A c)}{21 b x^{5/2}}-\frac{2 A \left(b x^2+c x^4\right)^{3/2}}{7 b x^{13/2}}",1,"(-2*Sqrt[x^2*(b + c*x^2)]*(3*A*(b + c*x^2)*Sqrt[1 + (c*x^2)/b] + (7*b*B - A*c)*x^2*Hypergeometric2F1[-3/4, -1/2, 1/4, -((c*x^2)/b)]))/(21*b*x^(9/2)*Sqrt[1 + (c*x^2)/b])","C",1
229,1,99,369,0.0552885,"\int \frac{\left(A+B x^2\right) \sqrt{b x^2+c x^4}}{x^{13/2}} \, dx","Integrate[((A + B*x^2)*Sqrt[b*x^2 + c*x^4])/x^(13/2),x]","-\frac{2 \sqrt{x^2 \left(b+c x^2\right)} \left(3 x^2 (3 b B-A c) \, _2F_1\left(-\frac{5}{4},-\frac{1}{2};-\frac{1}{4};-\frac{c x^2}{b}\right)+5 A \left(b+c x^2\right) \sqrt{\frac{c x^2}{b}+1}\right)}{45 b x^{11/2} \sqrt{\frac{c x^2}{b}+1}}","\frac{2 c^{5/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (3 b B-A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{15 b^{7/4} \sqrt{b x^2+c x^4}}-\frac{4 c^{5/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (3 b B-A c) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{15 b^{7/4} \sqrt{b x^2+c x^4}}+\frac{4 c^{3/2} x^{3/2} \left(b+c x^2\right) (3 b B-A c)}{15 b^2 \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}-\frac{4 c \sqrt{b x^2+c x^4} (3 b B-A c)}{15 b^2 x^{3/2}}-\frac{2 \sqrt{b x^2+c x^4} (3 b B-A c)}{15 b x^{7/2}}-\frac{2 A \left(b x^2+c x^4\right)^{3/2}}{9 b x^{15/2}}",1,"(-2*Sqrt[x^2*(b + c*x^2)]*(5*A*(b + c*x^2)*Sqrt[1 + (c*x^2)/b] + 3*(3*b*B - A*c)*x^2*Hypergeometric2F1[-5/4, -1/2, -1/4, -((c*x^2)/b)]))/(45*b*x^(11/2)*Sqrt[1 + (c*x^2)/b])","C",1
230,1,98,204,0.0625041,"\int \frac{\left(A+B x^2\right) \sqrt{b x^2+c x^4}}{x^{15/2}} \, dx","Integrate[((A + B*x^2)*Sqrt[b*x^2 + c*x^4])/x^(15/2),x]","-\frac{2 \sqrt{x^2 \left(b+c x^2\right)} \left(x^2 (11 b B-5 A c) \, _2F_1\left(-\frac{7}{4},-\frac{1}{2};-\frac{3}{4};-\frac{c x^2}{b}\right)+7 A \left(b+c x^2\right) \sqrt{\frac{c x^2}{b}+1}\right)}{77 b x^{13/2} \sqrt{\frac{c x^2}{b}+1}}","-\frac{2 c^{7/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (11 b B-5 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{231 b^{9/4} \sqrt{b x^2+c x^4}}-\frac{4 c \sqrt{b x^2+c x^4} (11 b B-5 A c)}{231 b^2 x^{5/2}}-\frac{2 \sqrt{b x^2+c x^4} (11 b B-5 A c)}{77 b x^{9/2}}-\frac{2 A \left(b x^2+c x^4\right)^{3/2}}{11 b x^{17/2}}",1,"(-2*Sqrt[x^2*(b + c*x^2)]*(7*A*(b + c*x^2)*Sqrt[1 + (c*x^2)/b] + (11*b*B - 5*A*c)*x^2*Hypergeometric2F1[-7/4, -1/2, -3/4, -((c*x^2)/b)]))/(77*b*x^(13/2)*Sqrt[1 + (c*x^2)/b])","C",1
231,1,160,486,0.3045967,"\int x^{7/2} \left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2} \, dx","Integrate[x^(7/2)*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2),x]","\frac{2 \sqrt{x} \sqrt{x^2 \left(b+c x^2\right)} \left(385 b^4 (3 b B-5 A c) \, _2F_1\left(-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{b}\right)-\left(b+c x^2\right)^2 \sqrt{\frac{c x^2}{b}+1} \left(-55 b^2 c \left(35 A+39 B x^2\right)+65 b c^2 x^2 \left(55 A+51 B x^2\right)-221 c^3 x^4 \left(25 A+21 B x^2\right)+1155 b^3 B\right)\right)}{116025 c^4 \sqrt{\frac{c x^2}{b}+1}}","\frac{44 b^{21/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (3 b B-5 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{16575 c^{19/4} \sqrt{b x^2+c x^4}}-\frac{88 b^{21/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (3 b B-5 A c) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{16575 c^{19/4} \sqrt{b x^2+c x^4}}+\frac{88 b^5 x^{3/2} \left(b+c x^2\right) (3 b B-5 A c)}{16575 c^{9/2} \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}-\frac{88 b^4 \sqrt{x} \sqrt{b x^2+c x^4} (3 b B-5 A c)}{49725 c^4}+\frac{88 b^3 x^{5/2} \sqrt{b x^2+c x^4} (3 b B-5 A c)}{69615 c^3}-\frac{8 b^2 x^{9/2} \sqrt{b x^2+c x^4} (3 b B-5 A c)}{7735 c^2}-\frac{4 b x^{13/2} \sqrt{b x^2+c x^4} (3 b B-5 A c)}{595 c}-\frac{2 x^{9/2} \left(b x^2+c x^4\right)^{3/2} (3 b B-5 A c)}{105 c}+\frac{2 B x^{5/2} \left(b x^2+c x^4\right)^{5/2}}{25 c}",1,"(2*Sqrt[x]*Sqrt[x^2*(b + c*x^2)]*(-((b + c*x^2)^2*Sqrt[1 + (c*x^2)/b]*(1155*b^3*B - 221*c^3*x^4*(25*A + 21*B*x^2) - 55*b^2*c*(35*A + 39*B*x^2) + 65*b*c^2*x^2*(55*A + 51*B*x^2))) + 385*b^4*(3*b*B - 5*A*c)*Hypergeometric2F1[-3/2, 3/4, 7/4, -((c*x^2)/b)]))/(116025*c^4*Sqrt[1 + (c*x^2)/b])","C",1
232,1,160,321,0.2292287,"\int x^{5/2} \left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2} \, dx","Integrate[x^(5/2)*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2),x]","\frac{2 \sqrt{x^2 \left(b+c x^2\right)} \left(15 b^4 (13 b B-23 A c) \, _2F_1\left(-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^2}{b}\right)-\left(b+c x^2\right)^2 \sqrt{\frac{c x^2}{b}+1} \left(-3 b^2 c \left(115 A+143 B x^2\right)+11 b c^2 x^2 \left(69 A+65 B x^2\right)-55 c^3 x^4 \left(23 A+19 B x^2\right)+195 b^3 B\right)\right)}{24035 c^4 \sqrt{x} \sqrt{\frac{c x^2}{b}+1}}","\frac{12 b^{19/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (13 b B-23 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{33649 c^{17/4} \sqrt{b x^2+c x^4}}-\frac{24 b^4 \sqrt{b x^2+c x^4} (13 b B-23 A c)}{33649 c^4 \sqrt{x}}+\frac{72 b^3 x^{3/2} \sqrt{b x^2+c x^4} (13 b B-23 A c)}{168245 c^3}-\frac{8 b^2 x^{7/2} \sqrt{b x^2+c x^4} (13 b B-23 A c)}{24035 c^2}-\frac{4 b x^{11/2} \sqrt{b x^2+c x^4} (13 b B-23 A c)}{2185 c}-\frac{2 x^{7/2} \left(b x^2+c x^4\right)^{3/2} (13 b B-23 A c)}{437 c}+\frac{2 B x^{3/2} \left(b x^2+c x^4\right)^{5/2}}{23 c}",1,"(2*Sqrt[x^2*(b + c*x^2)]*(-((b + c*x^2)^2*Sqrt[1 + (c*x^2)/b]*(195*b^3*B - 55*c^3*x^4*(23*A + 19*B*x^2) + 11*b*c^2*x^2*(69*A + 65*B*x^2) - 3*b^2*c*(115*A + 143*B*x^2))) + 15*b^4*(13*b*B - 23*A*c)*Hypergeometric2F1[-3/2, 1/4, 5/4, -((c*x^2)/b)]))/(24035*c^4*Sqrt[x]*Sqrt[1 + (c*x^2)/b])","C",1
233,1,138,447,0.1915108,"\int x^{3/2} \left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2} \, dx","Integrate[x^(3/2)*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2),x]","\frac{2 \sqrt{x} \sqrt{x^2 \left(b+c x^2\right)} \left(7 b^3 (21 A c-11 b B) \, _2F_1\left(-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{b}\right)+\left(b+c x^2\right)^2 \sqrt{\frac{c x^2}{b}+1} \left(-b c \left(147 A+143 B x^2\right)+13 c^2 x^2 \left(21 A+17 B x^2\right)+77 b^2 B\right)\right)}{4641 c^3 \sqrt{\frac{c x^2}{b}+1}}","-\frac{4 b^{17/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (11 b B-21 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{3315 c^{15/4} \sqrt{b x^2+c x^4}}+\frac{8 b^{17/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (11 b B-21 A c) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{3315 c^{15/4} \sqrt{b x^2+c x^4}}-\frac{8 b^4 x^{3/2} \left(b+c x^2\right) (11 b B-21 A c)}{3315 c^{7/2} \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}+\frac{8 b^3 \sqrt{x} \sqrt{b x^2+c x^4} (11 b B-21 A c)}{9945 c^3}-\frac{8 b^2 x^{5/2} \sqrt{b x^2+c x^4} (11 b B-21 A c)}{13923 c^2}-\frac{4 b x^{9/2} \sqrt{b x^2+c x^4} (11 b B-21 A c)}{1547 c}-\frac{2 x^{5/2} \left(b x^2+c x^4\right)^{3/2} (11 b B-21 A c)}{357 c}+\frac{2 B \sqrt{x} \left(b x^2+c x^4\right)^{5/2}}{21 c}",1,"(2*Sqrt[x]*Sqrt[x^2*(b + c*x^2)]*((b + c*x^2)^2*Sqrt[1 + (c*x^2)/b]*(77*b^2*B + 13*c^2*x^2*(21*A + 17*B*x^2) - b*c*(147*A + 143*B*x^2)) + 7*b^3*(-11*b*B + 21*A*c)*Hypergeometric2F1[-3/2, 3/4, 7/4, -((c*x^2)/b)]))/(4641*c^3*Sqrt[1 + (c*x^2)/b])","C",1
234,1,138,282,0.1849098,"\int \sqrt{x} \left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2} \, dx","Integrate[Sqrt[x]*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2),x]","\frac{2 \sqrt{x^2 \left(b+c x^2\right)} \left(5 b^3 (19 A c-9 b B) \, _2F_1\left(-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^2}{b}\right)+\left(b+c x^2\right)^2 \sqrt{\frac{c x^2}{b}+1} \left(-b c \left(95 A+99 B x^2\right)+11 c^2 x^2 \left(19 A+15 B x^2\right)+45 b^2 B\right)\right)}{3135 c^3 \sqrt{x} \sqrt{\frac{c x^2}{b}+1}}","-\frac{4 b^{15/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (9 b B-19 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{4389 c^{13/4} \sqrt{b x^2+c x^4}}+\frac{8 b^3 \sqrt{b x^2+c x^4} (9 b B-19 A c)}{4389 c^3 \sqrt{x}}-\frac{8 b^2 x^{3/2} \sqrt{b x^2+c x^4} (9 b B-19 A c)}{7315 c^2}-\frac{4 b x^{7/2} \sqrt{b x^2+c x^4} (9 b B-19 A c)}{1045 c}-\frac{2 x^{3/2} \left(b x^2+c x^4\right)^{3/2} (9 b B-19 A c)}{285 c}+\frac{2 B \left(b x^2+c x^4\right)^{5/2}}{19 c \sqrt{x}}",1,"(2*Sqrt[x^2*(b + c*x^2)]*((b + c*x^2)^2*Sqrt[1 + (c*x^2)/b]*(45*b^2*B + 11*c^2*x^2*(19*A + 15*B*x^2) - b*c*(95*A + 99*B*x^2)) + 5*b^3*(-9*b*B + 19*A*c)*Hypergeometric2F1[-3/2, 1/4, 5/4, -((c*x^2)/b)]))/(3135*c^3*Sqrt[x]*Sqrt[1 + (c*x^2)/b])","C",1
235,1,115,408,0.1587859,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2}}{\sqrt{x}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/Sqrt[x],x]","\frac{2 \sqrt{x} \sqrt{x^2 \left(b+c x^2\right)} \left(b^2 (7 b B-17 A c) \, _2F_1\left(-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{b}\right)-\left(b+c x^2\right)^2 \sqrt{\frac{c x^2}{b}+1} \left(-17 A c+7 b B-13 B c x^2\right)\right)}{221 c^2 \sqrt{\frac{c x^2}{b}+1}}","\frac{4 b^{13/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (7 b B-17 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{1105 c^{11/4} \sqrt{b x^2+c x^4}}-\frac{8 b^{13/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (7 b B-17 A c) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{1105 c^{11/4} \sqrt{b x^2+c x^4}}+\frac{8 b^3 x^{3/2} \left(b+c x^2\right) (7 b B-17 A c)}{1105 c^{5/2} \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}-\frac{8 b^2 \sqrt{x} \sqrt{b x^2+c x^4} (7 b B-17 A c)}{3315 c^2}-\frac{2 \sqrt{x} \left(b x^2+c x^4\right)^{3/2} (7 b B-17 A c)}{221 c}-\frac{4 b x^{5/2} \sqrt{b x^2+c x^4} (7 b B-17 A c)}{663 c}+\frac{2 B \left(b x^2+c x^4\right)^{5/2}}{17 c x^{3/2}}",1,"(2*Sqrt[x]*Sqrt[x^2*(b + c*x^2)]*(-((b + c*x^2)^2*Sqrt[1 + (c*x^2)/b]*(7*b*B - 17*A*c - 13*B*c*x^2)) + b^2*(7*b*B - 17*A*c)*Hypergeometric2F1[-3/2, 3/4, 7/4, -((c*x^2)/b)]))/(221*c^2*Sqrt[1 + (c*x^2)/b])","C",1
236,1,115,239,0.1539415,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2}}{x^{3/2}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^(3/2),x]","\frac{2 \sqrt{x^2 \left(b+c x^2\right)} \left(5 b^2 (b B-3 A c) \, _2F_1\left(-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^2}{b}\right)-\left(b+c x^2\right)^2 \sqrt{\frac{c x^2}{b}+1} \left(-15 A c+5 b B-11 B c x^2\right)\right)}{165 c^2 \sqrt{x} \sqrt{\frac{c x^2}{b}+1}}","\frac{4 b^{11/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (b B-3 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{231 c^{9/4} \sqrt{b x^2+c x^4}}-\frac{8 b^2 \sqrt{b x^2+c x^4} (b B-3 A c)}{231 c^2 \sqrt{x}}-\frac{2 \left(b x^2+c x^4\right)^{3/2} (b B-3 A c)}{33 c \sqrt{x}}-\frac{4 b x^{3/2} \sqrt{b x^2+c x^4} (b B-3 A c)}{77 c}+\frac{2 B \left(b x^2+c x^4\right)^{5/2}}{15 c x^{5/2}}",1,"(2*Sqrt[x^2*(b + c*x^2)]*(-((b + c*x^2)^2*Sqrt[1 + (c*x^2)/b]*(5*b*B - 15*A*c - 11*B*c*x^2)) + 5*b^2*(b*B - 3*A*c)*Hypergeometric2F1[-3/2, 1/4, 5/4, -((c*x^2)/b)]))/(165*c^2*Sqrt[x]*Sqrt[1 + (c*x^2)/b])","C",1
237,1,98,369,0.107926,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2}}{x^{5/2}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^(5/2),x]","\frac{2 \sqrt{x} \sqrt{x^2 \left(b+c x^2\right)} \left(b (13 A c-3 b B) \, _2F_1\left(-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{b}\right)+3 B \sqrt{\frac{c x^2}{b}+1} \left(b+c x^2\right)^2\right)}{39 c \sqrt{\frac{c x^2}{b}+1}}","-\frac{4 b^{9/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (3 b B-13 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{195 c^{7/4} \sqrt{b x^2+c x^4}}+\frac{8 b^{9/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (3 b B-13 A c) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{195 c^{7/4} \sqrt{b x^2+c x^4}}-\frac{8 b^2 x^{3/2} \left(b+c x^2\right) (3 b B-13 A c)}{195 c^{3/2} \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}-\frac{4 b \sqrt{x} \sqrt{b x^2+c x^4} (3 b B-13 A c)}{195 c}-\frac{2 \left(b x^2+c x^4\right)^{3/2} (3 b B-13 A c)}{117 c x^{3/2}}+\frac{2 B \left(b x^2+c x^4\right)^{5/2}}{13 c x^{7/2}}",1,"(2*Sqrt[x]*Sqrt[x^2*(b + c*x^2)]*(3*B*(b + c*x^2)^2*Sqrt[1 + (c*x^2)/b] + b*(-3*b*B + 13*A*c)*Hypergeometric2F1[-3/2, 3/4, 7/4, -((c*x^2)/b)]))/(39*c*Sqrt[1 + (c*x^2)/b])","C",1
238,1,97,201,0.0730595,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2}}{x^{7/2}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^(7/2),x]","\frac{2 \sqrt{x^2 \left(b+c x^2\right)} \left(b (11 A c-b B) \, _2F_1\left(-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^2}{b}\right)+B \sqrt{\frac{c x^2}{b}+1} \left(b+c x^2\right)^2\right)}{11 c \sqrt{x} \sqrt{\frac{c x^2}{b}+1}}","-\frac{4 b^{7/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (b B-11 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{77 c^{5/4} \sqrt{b x^2+c x^4}}-\frac{4 b \sqrt{b x^2+c x^4} (b B-11 A c)}{77 c \sqrt{x}}-\frac{2 \left(b x^2+c x^4\right)^{3/2} (b B-11 A c)}{77 c x^{5/2}}+\frac{2 B \left(b x^2+c x^4\right)^{5/2}}{11 c x^{9/2}}",1,"(2*Sqrt[x^2*(b + c*x^2)]*(B*(b + c*x^2)^2*Sqrt[1 + (c*x^2)/b] + b*(-(b*B) + 11*A*c)*Hypergeometric2F1[-3/2, 1/4, 5/4, -((c*x^2)/b)]))/(11*c*Sqrt[x]*Sqrt[1 + (c*x^2)/b])","C",1
239,1,85,356,0.0801502,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2}}{x^{9/2}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^(9/2),x]","\frac{2 \sqrt{x^2 \left(b+c x^2\right)} \left(\frac{x^2 (9 A c+b B) \, _2F_1\left(-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{b}\right)}{\sqrt{\frac{c x^2}{b}+1}}-\frac{3 A \left(b+c x^2\right)^2}{b}\right)}{3 x^{3/2}}","\frac{4 b^{5/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (9 A c+b B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{15 c^{3/4} \sqrt{b x^2+c x^4}}-\frac{8 b^{5/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (9 A c+b B) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{15 c^{3/4} \sqrt{b x^2+c x^4}}+\frac{4}{15} \sqrt{x} \sqrt{b x^2+c x^4} (9 A c+b B)+\frac{2 \left(b x^2+c x^4\right)^{3/2} (9 A c+b B)}{9 b x^{3/2}}+\frac{8 b x^{3/2} \left(b+c x^2\right) (9 A c+b B)}{15 \sqrt{c} \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}-\frac{2 A \left(b x^2+c x^4\right)^{5/2}}{b x^{11/2}}",1,"(2*Sqrt[x^2*(b + c*x^2)]*((-3*A*(b + c*x^2)^2)/b + ((b*B + 9*A*c)*x^2*Hypergeometric2F1[-3/2, 3/4, 7/4, -((c*x^2)/b)])/Sqrt[1 + (c*x^2)/b]))/(3*x^(3/2))","C",1
240,1,101,200,0.0863135,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2}}{x^{11/2}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^(11/2),x]","-\frac{2 \sqrt{x^2 \left(b+c x^2\right)} \left(A \left(b+c x^2\right)^2 \sqrt{\frac{c x^2}{b}+1}-b x^2 (7 A c+3 b B) \, _2F_1\left(-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^2}{b}\right)\right)}{3 b x^{5/2} \sqrt{\frac{c x^2}{b}+1}}","\frac{4 b^{3/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (7 A c+3 b B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{21 \sqrt[4]{c} \sqrt{b x^2+c x^4}}+\frac{4 \sqrt{b x^2+c x^4} (7 A c+3 b B)}{21 \sqrt{x}}+\frac{2 \left(b x^2+c x^4\right)^{3/2} (7 A c+3 b B)}{21 b x^{5/2}}-\frac{2 A \left(b x^2+c x^4\right)^{5/2}}{3 b x^{13/2}}",1,"(-2*Sqrt[x^2*(b + c*x^2)]*(A*(b + c*x^2)^2*Sqrt[1 + (c*x^2)/b] - b*(3*b*B + 7*A*c)*x^2*Hypergeometric2F1[-3/2, 1/4, 5/4, -((c*x^2)/b)]))/(3*b*x^(5/2)*Sqrt[1 + (c*x^2)/b])","C",1
241,1,99,354,0.0676022,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2}}{x^{13/2}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^(13/2),x]","-\frac{2 \sqrt{x^2 \left(b+c x^2\right)} \left(5 b x^2 (A c+b B) \, _2F_1\left(-\frac{3}{2},-\frac{1}{4};\frac{3}{4};-\frac{c x^2}{b}\right)+A \left(b+c x^2\right)^2 \sqrt{\frac{c x^2}{b}+1}\right)}{5 b x^{7/2} \sqrt{\frac{c x^2}{b}+1}}","\frac{12 c \sqrt{x} \sqrt{b x^2+c x^4} (A c+b B)}{5 b}+\frac{12 \sqrt[4]{b} \sqrt[4]{c} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (A c+b B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{5 \sqrt{b x^2+c x^4}}-\frac{24 \sqrt[4]{b} \sqrt[4]{c} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (A c+b B) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{5 \sqrt{b x^2+c x^4}}-\frac{2 \left(b x^2+c x^4\right)^{3/2} (A c+b B)}{b x^{7/2}}+\frac{24 \sqrt{c} x^{3/2} \left(b+c x^2\right) (A c+b B)}{5 \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}-\frac{2 A \left(b x^2+c x^4\right)^{5/2}}{5 b x^{15/2}}",1,"(-2*Sqrt[x^2*(b + c*x^2)]*(A*(b + c*x^2)^2*Sqrt[1 + (c*x^2)/b] + 5*b*(b*B + A*c)*x^2*Hypergeometric2F1[-3/2, -1/4, 3/4, -((c*x^2)/b)]))/(5*b*x^(7/2)*Sqrt[1 + (c*x^2)/b])","C",1
242,1,101,204,0.057864,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2}}{x^{15/2}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^(15/2),x]","-\frac{2 \sqrt{x^2 \left(b+c x^2\right)} \left(b x^2 (3 A c+7 b B) \, _2F_1\left(-\frac{3}{2},-\frac{3}{4};\frac{1}{4};-\frac{c x^2}{b}\right)+3 A \left(b+c x^2\right)^2 \sqrt{\frac{c x^2}{b}+1}\right)}{21 b x^{9/2} \sqrt{\frac{c x^2}{b}+1}}","\frac{4 c^{3/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (3 A c+7 b B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{21 \sqrt[4]{b} \sqrt{b x^2+c x^4}}+\frac{4 c \sqrt{b x^2+c x^4} (3 A c+7 b B)}{21 b \sqrt{x}}-\frac{2 \left(b x^2+c x^4\right)^{3/2} (3 A c+7 b B)}{21 b x^{9/2}}-\frac{2 A \left(b x^2+c x^4\right)^{5/2}}{7 b x^{17/2}}",1,"(-2*Sqrt[x^2*(b + c*x^2)]*(3*A*(b + c*x^2)^2*Sqrt[1 + (c*x^2)/b] + b*(7*b*B + 3*A*c)*x^2*Hypergeometric2F1[-3/2, -3/4, 1/4, -((c*x^2)/b)]))/(21*b*x^(9/2)*Sqrt[1 + (c*x^2)/b])","C",1
243,1,100,364,0.0607142,"\int \frac{\left(A+B x^2\right) \left(b x^2+c x^4\right)^{3/2}}{x^{17/2}} \, dx","Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^(17/2),x]","-\frac{2 \sqrt{x^2 \left(b+c x^2\right)} \left(b x^2 (A c+9 b B) \, _2F_1\left(-\frac{3}{2},-\frac{5}{4};-\frac{1}{4};-\frac{c x^2}{b}\right)+5 A \left(b+c x^2\right)^2 \sqrt{\frac{c x^2}{b}+1}\right)}{45 b x^{11/2} \sqrt{\frac{c x^2}{b}+1}}","\frac{4 c^{5/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (A c+9 b B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{15 b^{3/4} \sqrt{b x^2+c x^4}}-\frac{8 c^{5/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (A c+9 b B) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{15 b^{3/4} \sqrt{b x^2+c x^4}}+\frac{8 c^{3/2} x^{3/2} \left(b+c x^2\right) (A c+9 b B)}{15 b \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}-\frac{2 \left(b x^2+c x^4\right)^{3/2} (A c+9 b B)}{45 b x^{11/2}}-\frac{4 c \sqrt{b x^2+c x^4} (A c+9 b B)}{15 b x^{3/2}}-\frac{2 A \left(b x^2+c x^4\right)^{5/2}}{9 b x^{19/2}}",1,"(-2*Sqrt[x^2*(b + c*x^2)]*(5*A*(b + c*x^2)^2*Sqrt[1 + (c*x^2)/b] + b*(9*b*B + A*c)*x^2*Hypergeometric2F1[-3/2, -5/4, -1/4, -((c*x^2)/b)]))/(45*b*x^(11/2)*Sqrt[1 + (c*x^2)/b])","C",1
244,1,143,243,0.1772328,"\int \frac{x^{13/2} \left(A+B x^2\right)}{\sqrt{b x^2+c x^4}} \, dx","Integrate[(x^(13/2)*(A + B*x^2))/Sqrt[b*x^2 + c*x^4],x]","\frac{2 x^{3/2} \left(15 b^3 \sqrt{\frac{c x^2}{b}+1} (13 b B-15 A c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^2}{b}\right)-\left(b+c x^2\right) \left(-9 b^2 c \left(25 A+13 B x^2\right)+b c^2 x^2 \left(135 A+91 B x^2\right)-7 c^3 x^4 \left(15 A+11 B x^2\right)+195 b^3 B\right)\right)}{1155 c^4 \sqrt{x^2 \left(b+c x^2\right)}}","\frac{b^{11/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (13 b B-15 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{77 c^{17/4} \sqrt{b x^2+c x^4}}-\frac{2 b^2 \sqrt{b x^2+c x^4} (13 b B-15 A c)}{77 c^4 \sqrt{x}}+\frac{6 b x^{3/2} \sqrt{b x^2+c x^4} (13 b B-15 A c)}{385 c^3}-\frac{2 x^{7/2} \sqrt{b x^2+c x^4} (13 b B-15 A c)}{165 c^2}+\frac{2 B x^{11/2} \sqrt{b x^2+c x^4}}{15 c}",1,"(2*x^(3/2)*(-((b + c*x^2)*(195*b^3*B - 7*c^3*x^4*(15*A + 11*B*x^2) - 9*b^2*c*(25*A + 13*B*x^2) + b*c^2*x^2*(135*A + 91*B*x^2))) + 15*b^3*(13*b*B - 15*A*c)*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^2)/b)]))/(1155*c^4*Sqrt[x^2*(b + c*x^2)])","C",1
245,1,122,369,0.1521632,"\int \frac{x^{11/2} \left(A+B x^2\right)}{\sqrt{b x^2+c x^4}} \, dx","Integrate[(x^(11/2)*(A + B*x^2))/Sqrt[b*x^2 + c*x^4],x]","\frac{2 x^{5/2} \left(\left(b+c x^2\right) \left(-b c \left(91 A+55 B x^2\right)+5 c^2 x^2 \left(13 A+9 B x^2\right)+77 b^2 B\right)+7 b^2 \sqrt{\frac{c x^2}{b}+1} (13 A c-11 b B) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{b}\right)\right)}{585 c^3 \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{7 b^{9/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (11 b B-13 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{195 c^{15/4} \sqrt{b x^2+c x^4}}+\frac{14 b^{9/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (11 b B-13 A c) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{195 c^{15/4} \sqrt{b x^2+c x^4}}-\frac{14 b^2 x^{3/2} \left(b+c x^2\right) (11 b B-13 A c)}{195 c^{7/2} \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}+\frac{14 b \sqrt{x} \sqrt{b x^2+c x^4} (11 b B-13 A c)}{585 c^3}-\frac{2 x^{5/2} \sqrt{b x^2+c x^4} (11 b B-13 A c)}{117 c^2}+\frac{2 B x^{9/2} \sqrt{b x^2+c x^4}}{13 c}",1,"(2*x^(5/2)*((b + c*x^2)*(77*b^2*B + 5*c^2*x^2*(13*A + 9*B*x^2) - b*c*(91*A + 55*B*x^2)) + 7*b^2*(-11*b*B + 13*A*c)*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[1/2, 3/4, 7/4, -((c*x^2)/b)]))/(585*c^3*Sqrt[x^2*(b + c*x^2)])","C",1
246,1,122,204,0.1463713,"\int \frac{x^{9/2} \left(A+B x^2\right)}{\sqrt{b x^2+c x^4}} \, dx","Integrate[(x^(9/2)*(A + B*x^2))/Sqrt[b*x^2 + c*x^4],x]","\frac{2 x^{3/2} \left(\left(b+c x^2\right) \left(-b c \left(55 A+27 B x^2\right)+3 c^2 x^2 \left(11 A+7 B x^2\right)+45 b^2 B\right)+5 b^2 \sqrt{\frac{c x^2}{b}+1} (11 A c-9 b B) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^2}{b}\right)\right)}{231 c^3 \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{5 b^{7/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (9 b B-11 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{231 c^{13/4} \sqrt{b x^2+c x^4}}+\frac{10 b \sqrt{b x^2+c x^4} (9 b B-11 A c)}{231 c^3 \sqrt{x}}-\frac{2 x^{3/2} \sqrt{b x^2+c x^4} (9 b B-11 A c)}{77 c^2}+\frac{2 B x^{7/2} \sqrt{b x^2+c x^4}}{11 c}",1,"(2*x^(3/2)*((b + c*x^2)*(45*b^2*B + 3*c^2*x^2*(11*A + 7*B*x^2) - b*c*(55*A + 27*B*x^2)) + 5*b^2*(-9*b*B + 11*A*c)*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^2)/b)]))/(231*c^3*Sqrt[x^2*(b + c*x^2)])","C",1
247,1,97,330,0.1218823,"\int \frac{x^{7/2} \left(A+B x^2\right)}{\sqrt{b x^2+c x^4}} \, dx","Integrate[(x^(7/2)*(A + B*x^2))/Sqrt[b*x^2 + c*x^4],x]","\frac{2 x^{5/2} \left(b \sqrt{\frac{c x^2}{b}+1} (7 b B-9 A c) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{b}\right)-\left(b+c x^2\right) \left(-9 A c+7 b B-5 B c x^2\right)\right)}{45 c^2 \sqrt{x^2 \left(b+c x^2\right)}}","\frac{b^{5/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (7 b B-9 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{15 c^{11/4} \sqrt{b x^2+c x^4}}-\frac{2 b^{5/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (7 b B-9 A c) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{15 c^{11/4} \sqrt{b x^2+c x^4}}+\frac{2 b x^{3/2} \left(b+c x^2\right) (7 b B-9 A c)}{15 c^{5/2} \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}-\frac{2 \sqrt{x} \sqrt{b x^2+c x^4} (7 b B-9 A c)}{45 c^2}+\frac{2 B x^{5/2} \sqrt{b x^2+c x^4}}{9 c}",1,"(2*x^(5/2)*(-((b + c*x^2)*(7*b*B - 9*A*c - 5*B*c*x^2)) + b*(7*b*B - 9*A*c)*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[1/2, 3/4, 7/4, -((c*x^2)/b)]))/(45*c^2*Sqrt[x^2*(b + c*x^2)])","C",1
248,1,97,167,0.1196656,"\int \frac{x^{5/2} \left(A+B x^2\right)}{\sqrt{b x^2+c x^4}} \, dx","Integrate[(x^(5/2)*(A + B*x^2))/Sqrt[b*x^2 + c*x^4],x]","\frac{2 x^{3/2} \left(b \sqrt{\frac{c x^2}{b}+1} (5 b B-7 A c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^2}{b}\right)-\left(b+c x^2\right) \left(-7 A c+5 b B-3 B c x^2\right)\right)}{21 c^2 \sqrt{x^2 \left(b+c x^2\right)}}","\frac{b^{3/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (5 b B-7 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{21 c^{9/4} \sqrt{b x^2+c x^4}}-\frac{2 \sqrt{b x^2+c x^4} (5 b B-7 A c)}{21 c^2 \sqrt{x}}+\frac{2 B x^{3/2} \sqrt{b x^2+c x^4}}{7 c}",1,"(2*x^(3/2)*(-((b + c*x^2)*(5*b*B - 7*A*c - 3*B*c*x^2)) + b*(5*b*B - 7*A*c)*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^2)/b)]))/(21*c^2*Sqrt[x^2*(b + c*x^2)])","C",1
249,1,81,293,0.1001446,"\int \frac{x^{3/2} \left(A+B x^2\right)}{\sqrt{b x^2+c x^4}} \, dx","Integrate[(x^(3/2)*(A + B*x^2))/Sqrt[b*x^2 + c*x^4],x]","\frac{2 x^{5/2} \left(\sqrt{\frac{c x^2}{b}+1} (5 A c-3 b B) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{b}\right)+3 B \left(b+c x^2\right)\right)}{15 c \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{\sqrt[4]{b} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (3 b B-5 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{5 c^{7/4} \sqrt{b x^2+c x^4}}+\frac{2 \sqrt[4]{b} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (3 b B-5 A c) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{5 c^{7/4} \sqrt{b x^2+c x^4}}-\frac{2 x^{3/2} \left(b+c x^2\right) (3 b B-5 A c)}{5 c^{3/2} \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}+\frac{2 B \sqrt{x} \sqrt{b x^2+c x^4}}{5 c}",1,"(2*x^(5/2)*(3*B*(b + c*x^2) + (-3*b*B + 5*A*c)*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[1/2, 3/4, 7/4, -((c*x^2)/b)]))/(15*c*Sqrt[x^2*(b + c*x^2)])","C",1
250,1,80,130,0.0667427,"\int \frac{\sqrt{x} \left(A+B x^2\right)}{\sqrt{b x^2+c x^4}} \, dx","Integrate[(Sqrt[x]*(A + B*x^2))/Sqrt[b*x^2 + c*x^4],x]","\frac{2 x^{3/2} \left(\sqrt{\frac{c x^2}{b}+1} (3 A c-b B) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^2}{b}\right)+B \left(b+c x^2\right)\right)}{3 c \sqrt{x^2 \left(b+c x^2\right)}}","\frac{2 B \sqrt{b x^2+c x^4}}{3 c \sqrt{x}}-\frac{x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (b B-3 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{3 \sqrt[4]{b} c^{5/4} \sqrt{b x^2+c x^4}}",1,"(2*x^(3/2)*(B*(b + c*x^2) + (-(b*B) + 3*A*c)*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^2)/b)]))/(3*c*Sqrt[x^2*(b + c*x^2)])","C",1
251,1,82,281,0.0521268,"\int \frac{A+B x^2}{\sqrt{x} \sqrt{b x^2+c x^4}} \, dx","Integrate[(A + B*x^2)/(Sqrt[x]*Sqrt[b*x^2 + c*x^4]),x]","\frac{2 \sqrt{x} \left(x^2 \sqrt{\frac{c x^2}{b}+1} (A c+b B) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{b}\right)-3 A \left(b+c x^2\right)\right)}{3 b \sqrt{x^2 \left(b+c x^2\right)}}","\frac{x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (A c+b B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{b^{3/4} c^{3/4} \sqrt{b x^2+c x^4}}-\frac{2 x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (A c+b B) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{b^{3/4} c^{3/4} \sqrt{b x^2+c x^4}}+\frac{2 x^{3/2} \left(b+c x^2\right) (A c+b B)}{b \sqrt{c} \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}-\frac{2 A \sqrt{b x^2+c x^4}}{b x^{3/2}}",1,"(2*Sqrt[x]*(-3*A*(b + c*x^2) + (b*B + A*c)*x^2*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[1/2, 3/4, 7/4, -((c*x^2)/b)]))/(3*b*Sqrt[x^2*(b + c*x^2)])","C",1
252,1,82,131,0.0703547,"\int \frac{A+B x^2}{x^{3/2} \sqrt{b x^2+c x^4}} \, dx","Integrate[(A + B*x^2)/(x^(3/2)*Sqrt[b*x^2 + c*x^4]),x]","-\frac{2 \left(x^2 \sqrt{\frac{c x^2}{b}+1} (A c-3 b B) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^2}{b}\right)+A \left(b+c x^2\right)\right)}{3 b \sqrt{x} \sqrt{x^2 \left(b+c x^2\right)}}","\frac{x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (3 b B-A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{3 b^{5/4} \sqrt[4]{c} \sqrt{b x^2+c x^4}}-\frac{2 A \sqrt{b x^2+c x^4}}{3 b x^{5/2}}",1,"(-2*(A*(b + c*x^2) + (-3*b*B + A*c)*x^2*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^2)/b)]))/(3*b*Sqrt[x]*Sqrt[x^2*(b + c*x^2)])","C",1
253,1,83,332,0.0680773,"\int \frac{A+B x^2}{x^{5/2} \sqrt{b x^2+c x^4}} \, dx","Integrate[(A + B*x^2)/(x^(5/2)*Sqrt[b*x^2 + c*x^4]),x]","-\frac{2 \left(x^2 \sqrt{\frac{c x^2}{b}+1} (5 b B-3 A c) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{c x^2}{b}\right)+A \left(b+c x^2\right)\right)}{5 b x^{3/2} \sqrt{x^2 \left(b+c x^2\right)}}","\frac{\sqrt[4]{c} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (5 b B-3 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{5 b^{7/4} \sqrt{b x^2+c x^4}}-\frac{2 \sqrt[4]{c} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (5 b B-3 A c) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{5 b^{7/4} \sqrt{b x^2+c x^4}}+\frac{2 \sqrt{c} x^{3/2} \left(b+c x^2\right) (5 b B-3 A c)}{5 b^2 \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}-\frac{2 \sqrt{b x^2+c x^4} (5 b B-3 A c)}{5 b^2 x^{3/2}}-\frac{2 A \sqrt{b x^2+c x^4}}{5 b x^{7/2}}",1,"(-2*(A*(b + c*x^2) + (5*b*B - 3*A*c)*x^2*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[-1/4, 1/2, 3/4, -((c*x^2)/b)]))/(5*b*x^(3/2)*Sqrt[x^2*(b + c*x^2)])","C",1
254,1,85,167,0.0521719,"\int \frac{A+B x^2}{x^{7/2} \sqrt{b x^2+c x^4}} \, dx","Integrate[(A + B*x^2)/(x^(7/2)*Sqrt[b*x^2 + c*x^4]),x]","\frac{2 x^2 \sqrt{\frac{c x^2}{b}+1} (5 A c-7 b B) \, _2F_1\left(-\frac{3}{4},\frac{1}{2};\frac{1}{4};-\frac{c x^2}{b}\right)-6 A \left(b+c x^2\right)}{21 b x^{5/2} \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{c^{3/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (7 b B-5 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{21 b^{9/4} \sqrt{b x^2+c x^4}}-\frac{2 \sqrt{b x^2+c x^4} (7 b B-5 A c)}{21 b^2 x^{5/2}}-\frac{2 A \sqrt{b x^2+c x^4}}{7 b x^{9/2}}",1,"(-6*A*(b + c*x^2) + 2*(-7*b*B + 5*A*c)*x^2*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[-3/4, 1/2, 1/4, -((c*x^2)/b)])/(21*b*x^(5/2)*Sqrt[x^2*(b + c*x^2)])","C",1
255,1,84,369,0.0554694,"\int \frac{A+B x^2}{x^{9/2} \sqrt{b x^2+c x^4}} \, dx","Integrate[(A + B*x^2)/(x^(9/2)*Sqrt[b*x^2 + c*x^4]),x]","-\frac{2 \left(x^2 \sqrt{\frac{c x^2}{b}+1} (9 b B-7 A c) \, _2F_1\left(-\frac{5}{4},\frac{1}{2};-\frac{1}{4};-\frac{c x^2}{b}\right)+5 A \left(b+c x^2\right)\right)}{45 b x^{7/2} \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{c^{5/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (9 b B-7 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{15 b^{11/4} \sqrt{b x^2+c x^4}}+\frac{2 c^{5/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (9 b B-7 A c) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{15 b^{11/4} \sqrt{b x^2+c x^4}}-\frac{2 c^{3/2} x^{3/2} \left(b+c x^2\right) (9 b B-7 A c)}{15 b^3 \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}+\frac{2 c \sqrt{b x^2+c x^4} (9 b B-7 A c)}{15 b^3 x^{3/2}}-\frac{2 \sqrt{b x^2+c x^4} (9 b B-7 A c)}{45 b^2 x^{7/2}}-\frac{2 A \sqrt{b x^2+c x^4}}{9 b x^{11/2}}",1,"(-2*(5*A*(b + c*x^2) + (9*b*B - 7*A*c)*x^2*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[-5/4, 1/2, -1/4, -((c*x^2)/b)]))/(45*b*x^(7/2)*Sqrt[x^2*(b + c*x^2)])","C",1
256,1,84,204,0.0657131,"\int \frac{A+B x^2}{x^{11/2} \sqrt{b x^2+c x^4}} \, dx","Integrate[(A + B*x^2)/(x^(11/2)*Sqrt[b*x^2 + c*x^4]),x]","-\frac{2 \left(x^2 \sqrt{\frac{c x^2}{b}+1} (11 b B-9 A c) \, _2F_1\left(-\frac{7}{4},\frac{1}{2};-\frac{3}{4};-\frac{c x^2}{b}\right)+7 A \left(b+c x^2\right)\right)}{77 b x^{9/2} \sqrt{x^2 \left(b+c x^2\right)}}","\frac{5 c^{7/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (11 b B-9 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{231 b^{13/4} \sqrt{b x^2+c x^4}}+\frac{10 c \sqrt{b x^2+c x^4} (11 b B-9 A c)}{231 b^3 x^{5/2}}-\frac{2 \sqrt{b x^2+c x^4} (11 b B-9 A c)}{77 b^2 x^{9/2}}-\frac{2 A \sqrt{b x^2+c x^4}}{11 b x^{13/2}}",1,"(-2*(7*A*(b + c*x^2) + (11*b*B - 9*A*c)*x^2*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[-7/4, 1/2, -3/4, -((c*x^2)/b)]))/(77*b*x^(9/2)*Sqrt[x^2*(b + c*x^2)])","C",1
257,1,134,251,0.2445325,"\int \frac{x^{17/2} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(x^(17/2)*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2),x]","\frac{x^{3/2} \left(15 b^2 \sqrt{\frac{c x^2}{b}+1} (11 A c-13 b B) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^2}{b}\right)+b^2 \left(78 B c x^2-165 A c\right)-2 b c^2 x^2 \left(33 A+13 B x^2\right)+2 c^3 x^4 \left(11 A+7 B x^2\right)+195 b^3 B\right)}{77 c^4 \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{15 b^{7/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (13 b B-11 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{154 c^{17/4} \sqrt{b x^2+c x^4}}+\frac{15 b \sqrt{b x^2+c x^4} (13 b B-11 A c)}{77 c^4 \sqrt{x}}-\frac{9 x^{3/2} \sqrt{b x^2+c x^4} (13 b B-11 A c)}{77 c^3}+\frac{x^{7/2} \sqrt{b x^2+c x^4} (13 b B-11 A c)}{11 b c^2}-\frac{x^{15/2} (b B-A c)}{b c \sqrt{b x^2+c x^4}}",1,"(x^(3/2)*(195*b^3*B + 2*c^3*x^4*(11*A + 7*B*x^2) - 2*b*c^2*x^2*(33*A + 13*B*x^2) + b^2*(-165*A*c + 78*B*c*x^2) + 15*b^2*(-13*b*B + 11*A*c)*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^2)/b)]))/(77*c^4*Sqrt[x^2*(b + c*x^2)])","C",1
258,1,110,377,0.1810165,"\int \frac{x^{15/2} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(x^(15/2)*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2),x]","\frac{2 x^{5/2} \left(7 b \sqrt{\frac{c x^2}{b}+1} (9 A c-11 b B) \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{c x^2}{b}\right)-b c \left(63 A+11 B x^2\right)+c^2 x^2 \left(9 A+5 B x^2\right)+77 b^2 B\right)}{45 c^3 \sqrt{x^2 \left(b+c x^2\right)}}","\frac{7 b^{5/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (11 b B-9 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{30 c^{15/4} \sqrt{b x^2+c x^4}}-\frac{7 b^{5/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (11 b B-9 A c) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{15 c^{15/4} \sqrt{b x^2+c x^4}}+\frac{7 b x^{3/2} \left(b+c x^2\right) (11 b B-9 A c)}{15 c^{7/2} \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}-\frac{7 \sqrt{x} \sqrt{b x^2+c x^4} (11 b B-9 A c)}{45 c^3}+\frac{x^{5/2} \sqrt{b x^2+c x^4} (11 b B-9 A c)}{9 b c^2}-\frac{x^{13/2} (b B-A c)}{b c \sqrt{b x^2+c x^4}}",1,"(2*x^(5/2)*(77*b^2*B + c^2*x^2*(9*A + 5*B*x^2) - b*c*(63*A + 11*B*x^2) + 7*b*(-11*b*B + 9*A*c)*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[3/4, 3/2, 7/4, -((c*x^2)/b)]))/(45*c^3*Sqrt[x^2*(b + c*x^2)])","C",1
259,1,110,214,0.174794,"\int \frac{x^{13/2} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(x^(13/2)*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2),x]","\frac{x^{3/2} \left(5 b \sqrt{\frac{c x^2}{b}+1} (9 b B-7 A c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^2}{b}\right)+b c \left(35 A-18 B x^2\right)+2 c^2 x^2 \left(7 A+3 B x^2\right)-45 b^2 B\right)}{21 c^3 \sqrt{x^2 \left(b+c x^2\right)}}","\frac{5 b^{3/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (9 b B-7 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{42 c^{13/4} \sqrt{b x^2+c x^4}}-\frac{5 \sqrt{b x^2+c x^4} (9 b B-7 A c)}{21 c^3 \sqrt{x}}+\frac{x^{3/2} \sqrt{b x^2+c x^4} (9 b B-7 A c)}{7 b c^2}-\frac{x^{11/2} (b B-A c)}{b c \sqrt{b x^2+c x^4}}",1,"(x^(3/2)*(-45*b^2*B + b*c*(35*A - 18*B*x^2) + 2*c^2*x^2*(7*A + 3*B*x^2) + 5*b*(9*b*B - 7*A*c)*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^2)/b)]))/(21*c^3*Sqrt[x^2*(b + c*x^2)])","C",1
260,1,85,340,0.132561,"\int \frac{x^{11/2} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(x^(11/2)*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2),x]","\frac{2 x^{5/2} \left(\sqrt{\frac{c x^2}{b}+1} (7 b B-5 A c) \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{c x^2}{b}\right)+5 A c-7 b B+B c x^2\right)}{5 c^2 \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{3 \sqrt[4]{b} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (7 b B-5 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{10 c^{11/4} \sqrt{b x^2+c x^4}}+\frac{3 \sqrt[4]{b} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (7 b B-5 A c) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{5 c^{11/4} \sqrt{b x^2+c x^4}}-\frac{3 x^{3/2} \left(b+c x^2\right) (7 b B-5 A c)}{5 c^{5/2} \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}+\frac{\sqrt{x} \sqrt{b x^2+c x^4} (7 b B-5 A c)}{5 b c^2}-\frac{x^{9/2} (b B-A c)}{b c \sqrt{b x^2+c x^4}}",1,"(2*x^(5/2)*(-7*b*B + 5*A*c + B*c*x^2 + (7*b*B - 5*A*c)*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[3/4, 3/2, 7/4, -((c*x^2)/b)]))/(5*c^2*Sqrt[x^2*(b + c*x^2)])","C",1
261,1,86,178,0.1301359,"\int \frac{x^{9/2} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(x^(9/2)*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2),x]","\frac{x^{3/2} \left(\sqrt{\frac{c x^2}{b}+1} (3 A c-5 b B) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^2}{b}\right)-3 A c+5 b B+2 B c x^2\right)}{3 c^2 \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (5 b B-3 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{6 \sqrt[4]{b} c^{9/4} \sqrt{b x^2+c x^4}}+\frac{\sqrt{b x^2+c x^4} (5 b B-3 A c)}{3 b c^2 \sqrt{x}}-\frac{x^{7/2} (b B-A c)}{b c \sqrt{b x^2+c x^4}}",1,"(x^(3/2)*(5*b*B - 3*A*c + 2*B*c*x^2 + (-5*b*B + 3*A*c)*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^2)/b)]))/(3*c^2*Sqrt[x^2*(b + c*x^2)])","C",1
262,1,78,299,0.1114809,"\int \frac{x^{7/2} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(x^(7/2)*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2),x]","-\frac{2 x^{5/2} \left(\sqrt{\frac{c x^2}{b}+1} (3 b B-A c) \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{c x^2}{b}\right)-3 b B\right)}{3 b c \sqrt{x^2 \left(b+c x^2\right)}}","\frac{x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (3 b B-A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{2 b^{3/4} c^{7/4} \sqrt{b x^2+c x^4}}-\frac{x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (3 b B-A c) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{b^{3/4} c^{7/4} \sqrt{b x^2+c x^4}}+\frac{x^{3/2} \left(b+c x^2\right) (3 b B-A c)}{b c^{3/2} \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}-\frac{x^{5/2} (b B-A c)}{b c \sqrt{b x^2+c x^4}}",1,"(-2*x^(5/2)*(-3*b*B + (3*b*B - A*c)*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[3/4, 3/2, 7/4, -((c*x^2)/b)]))/(3*b*c*Sqrt[x^2*(b + c*x^2)])","C",1
263,1,76,137,0.0788635,"\int \frac{x^{5/2} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(x^(5/2)*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2),x]","\frac{x^{3/2} \left(\sqrt{\frac{c x^2}{b}+1} (A c+b B) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^2}{b}\right)+A c-b B\right)}{b c \sqrt{x^2 \left(b+c x^2\right)}}","\frac{x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (A c+b B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{2 b^{5/4} c^{5/4} \sqrt{b x^2+c x^4}}-\frac{x^{3/2} (b B-A c)}{b c \sqrt{b x^2+c x^4}}",1,"(x^(3/2)*(-(b*B) + A*c + (b*B + A*c)*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^2)/b)]))/(b*c*Sqrt[x^2*(b + c*x^2)])","C",1
264,1,77,318,0.0820621,"\int \frac{x^{3/2} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(x^(3/2)*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2),x]","\frac{2 \sqrt{x} \left(x^2 \sqrt{\frac{c x^2}{b}+1} (b B-3 A c) \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{c x^2}{b}\right)-3 A b\right)}{3 b^2 \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (b B-3 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{2 b^{7/4} c^{3/4} \sqrt{b x^2+c x^4}}+\frac{x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (b B-3 A c) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{b^{7/4} c^{3/4} \sqrt{b x^2+c x^4}}+\frac{x^{5/2} (b B-3 A c)}{b^2 \sqrt{b x^2+c x^4}}-\frac{x^{3/2} \left(b+c x^2\right) (b B-3 A c)}{b^2 \sqrt{c} \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}-\frac{2 A \sqrt{x}}{b \sqrt{b x^2+c x^4}}",1,"(2*Sqrt[x]*(-3*A*b + (b*B - 3*A*c)*x^2*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[3/4, 3/2, 7/4, -((c*x^2)/b)]))/(3*b^2*Sqrt[x^2*(b + c*x^2)])","C",1
265,1,92,167,0.070662,"\int \frac{\sqrt{x} \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(Sqrt[x]*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2),x]","\frac{x^2 \sqrt{\frac{c x^2}{b}+1} (3 b B-5 A c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^2}{b}\right)-2 A b-5 A c x^2+3 b B x^2}{3 b^2 \sqrt{x} \sqrt{x^2 \left(b+c x^2\right)}}","\frac{x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (3 b B-5 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{6 b^{9/4} \sqrt[4]{c} \sqrt{b x^2+c x^4}}+\frac{x^{3/2} (3 b B-5 A c)}{3 b^2 \sqrt{b x^2+c x^4}}-\frac{2 A}{3 b \sqrt{x} \sqrt{b x^2+c x^4}}",1,"(-2*A*b + 3*b*B*x^2 - 5*A*c*x^2 + (3*b*B - 5*A*c)*x^2*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^2)/b)])/(3*b^2*Sqrt[x]*Sqrt[x^2*(b + c*x^2)])","C",1
266,1,79,368,0.0624707,"\int \frac{A+B x^2}{\sqrt{x} \left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(A + B*x^2)/(Sqrt[x]*(b*x^2 + c*x^4)^(3/2)),x]","\frac{2 x^2 \sqrt{\frac{c x^2}{b}+1} (7 A c-5 b B) \, _2F_1\left(-\frac{1}{4},\frac{3}{2};\frac{3}{4};-\frac{c x^2}{b}\right)-2 A b}{5 b^2 x^{3/2} \sqrt{x^2 \left(b+c x^2\right)}}","\frac{3 \sqrt[4]{c} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (5 b B-7 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{10 b^{11/4} \sqrt{b x^2+c x^4}}-\frac{3 \sqrt[4]{c} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (5 b B-7 A c) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{5 b^{11/4} \sqrt{b x^2+c x^4}}+\frac{3 \sqrt{c} x^{3/2} \left(b+c x^2\right) (5 b B-7 A c)}{5 b^3 \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}-\frac{3 \sqrt{b x^2+c x^4} (5 b B-7 A c)}{5 b^3 x^{3/2}}+\frac{\sqrt{x} (5 b B-7 A c)}{5 b^2 \sqrt{b x^2+c x^4}}-\frac{2 A}{5 b x^{3/2} \sqrt{b x^2+c x^4}}",1,"(-2*A*b + 2*(-5*b*B + 7*A*c)*x^2*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[-1/4, 3/2, 3/4, -((c*x^2)/b)])/(5*b^2*x^(3/2)*Sqrt[x^2*(b + c*x^2)])","C",1
267,1,79,203,0.0515091,"\int \frac{A+B x^2}{x^{3/2} \left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(A + B*x^2)/(x^(3/2)*(b*x^2 + c*x^4)^(3/2)),x]","\frac{2 x^2 \sqrt{\frac{c x^2}{b}+1} (9 A c-7 b B) \, _2F_1\left(-\frac{3}{4},\frac{3}{2};\frac{1}{4};-\frac{c x^2}{b}\right)-6 A b}{21 b^2 x^{5/2} \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{5 c^{3/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (7 b B-9 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{42 b^{13/4} \sqrt{b x^2+c x^4}}-\frac{5 \sqrt{b x^2+c x^4} (7 b B-9 A c)}{21 b^3 x^{5/2}}+\frac{7 b B-9 A c}{7 b^2 \sqrt{x} \sqrt{b x^2+c x^4}}-\frac{2 A}{7 b x^{5/2} \sqrt{b x^2+c x^4}}",1,"(-6*A*b + 2*(-7*b*B + 9*A*c)*x^2*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[-3/4, 3/2, 1/4, -((c*x^2)/b)])/(21*b^2*x^(5/2)*Sqrt[x^2*(b + c*x^2)])","C",1
268,1,79,405,0.0581425,"\int \frac{A+B x^2}{x^{5/2} \left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(A + B*x^2)/(x^(5/2)*(b*x^2 + c*x^4)^(3/2)),x]","\frac{2 x^2 \sqrt{\frac{c x^2}{b}+1} (11 A c-9 b B) \, _2F_1\left(-\frac{5}{4},\frac{3}{2};-\frac{1}{4};-\frac{c x^2}{b}\right)-10 A b}{45 b^2 x^{7/2} \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{7 c^{5/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (9 b B-11 A c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{30 b^{15/4} \sqrt{b x^2+c x^4}}+\frac{7 c^{5/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} (9 b B-11 A c) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{15 b^{15/4} \sqrt{b x^2+c x^4}}-\frac{7 c^{3/2} x^{3/2} \left(b+c x^2\right) (9 b B-11 A c)}{15 b^4 \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}+\frac{7 c \sqrt{b x^2+c x^4} (9 b B-11 A c)}{15 b^4 x^{3/2}}-\frac{7 \sqrt{b x^2+c x^4} (9 b B-11 A c)}{45 b^3 x^{7/2}}+\frac{9 b B-11 A c}{9 b^2 x^{3/2} \sqrt{b x^2+c x^4}}-\frac{2 A}{9 b x^{7/2} \sqrt{b x^2+c x^4}}",1,"(-10*A*b + 2*(-9*b*B + 11*A*c)*x^2*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[-5/4, 3/2, -1/4, -((c*x^2)/b)])/(45*b^2*x^(7/2)*Sqrt[x^2*(b + c*x^2)])","C",1
269,1,89,96,0.1424251,"\int x^m \left(A+B x^2\right) \left(b x^2+c x^4\right)^3 \, dx","Integrate[x^m*(A + B*x^2)*(b*x^2 + c*x^4)^3,x]","x^{m+7} \left(\frac{A b^3}{m+7}+\frac{b^2 x^2 (3 A c+b B)}{m+9}+\frac{c^2 x^6 (A c+3 b B)}{m+13}+\frac{3 b c x^4 (A c+b B)}{m+11}+\frac{B c^3 x^8}{m+15}\right)","\frac{A b^3 x^{m+7}}{m+7}+\frac{b^2 x^{m+9} (3 A c+b B)}{m+9}+\frac{c^2 x^{m+13} (A c+3 b B)}{m+13}+\frac{3 b c x^{m+11} (A c+b B)}{m+11}+\frac{B c^3 x^{m+15}}{m+15}",1,"x^(7 + m)*((A*b^3)/(7 + m) + (b^2*(b*B + 3*A*c)*x^2)/(9 + m) + (3*b*c*(b*B + A*c)*x^4)/(11 + m) + (c^2*(3*b*B + A*c)*x^6)/(13 + m) + (B*c^3*x^8)/(15 + m))","A",1
270,1,66,71,0.0710798,"\int x^m \left(A+B x^2\right) \left(b x^2+c x^4\right)^2 \, dx","Integrate[x^m*(A + B*x^2)*(b*x^2 + c*x^4)^2,x]","x^{m+5} \left(\frac{A b^2}{m+5}+\frac{c x^4 (A c+2 b B)}{m+9}+\frac{b x^2 (2 A c+b B)}{m+7}+\frac{B c^2 x^6}{m+11}\right)","\frac{A b^2 x^{m+5}}{m+5}+\frac{b x^{m+7} (2 A c+b B)}{m+7}+\frac{c x^{m+9} (A c+2 b B)}{m+9}+\frac{B c^2 x^{m+11}}{m+11}",1,"x^(5 + m)*((A*b^2)/(5 + m) + (b*(b*B + 2*A*c)*x^2)/(7 + m) + (c*(2*b*B + A*c)*x^4)/(9 + m) + (B*c^2*x^6)/(11 + m))","A",1
271,1,42,45,0.0537015,"\int x^m \left(A+B x^2\right) \left(b x^2+c x^4\right) \, dx","Integrate[x^m*(A + B*x^2)*(b*x^2 + c*x^4),x]","x^{m+3} \left(\frac{x^2 (A c+b B)}{m+5}+\frac{A b}{m+3}+\frac{B c x^4}{m+7}\right)","\frac{x^{m+5} (A c+b B)}{m+5}+\frac{A b x^{m+3}}{m+3}+\frac{B c x^{m+7}}{m+7}",1,"x^(3 + m)*((A*b)/(3 + m) + ((b*B + A*c)*x^2)/(5 + m) + (B*c*x^4)/(7 + m))","A",1
272,1,55,71,0.0760625,"\int \frac{x^m \left(A+B x^2\right)}{b x^2+c x^4} \, dx","Integrate[(x^m*(A + B*x^2))/(b*x^2 + c*x^4),x]","\frac{x^{m-1} \left((A c-b B) \, _2F_1\left(1,\frac{m-1}{2};\frac{m+1}{2};-\frac{c x^2}{b}\right)+b B\right)}{b c (m-1)}","\frac{x^{m-1} (b B-A c) \, _2F_1\left(1,\frac{m-1}{2};\frac{m+1}{2};-\frac{c x^2}{b}\right)}{b c (1-m)}-\frac{B x^{m-1}}{c (1-m)}",1,"(x^(-1 + m)*(b*B + (-(b*B) + A*c)*Hypergeometric2F1[1, (-1 + m)/2, (1 + m)/2, -((c*x^2)/b)]))/(b*c*(-1 + m))","A",1
273,1,80,98,0.1045669,"\int \frac{x^m \left(A+B x^2\right)}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[(x^m*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]","\frac{x^{m-3} \left((A c-b B) \, _2F_1\left(2,\frac{m-3}{2};\frac{m-1}{2};-\frac{c x^2}{b}\right)+b B \, _2F_1\left(1,\frac{m-3}{2};\frac{m-1}{2};-\frac{c x^2}{b}\right)\right)}{b^2 c (m-3)}","\frac{x^{m-3} (b B (3-m)-A c (5-m)) \, _2F_1\left(1,\frac{m-3}{2};\frac{m-1}{2};-\frac{c x^2}{b}\right)}{2 b^2 c (3-m)}-\frac{x^{m-3} (b B-A c)}{2 b c \left(b+c x^2\right)}",1,"(x^(-3 + m)*(b*B*Hypergeometric2F1[1, (-3 + m)/2, (-1 + m)/2, -((c*x^2)/b)] + (-(b*B) + A*c)*Hypergeometric2F1[2, (-3 + m)/2, (-1 + m)/2, -((c*x^2)/b)]))/(b^2*c*(-3 + m))","A",1
274,1,135,140,0.1224066,"\int x^m \left(A+B x^2\right) \left(b x^2+c x^4\right)^p \, dx","Integrate[x^m*(A + B*x^2)*(b*x^2 + c*x^4)^p,x]","\frac{x^{m+1} \left(x^2 \left(b+c x^2\right)\right)^p \left(\frac{c x^2}{b}+1\right)^{-p} \left(A (m+2 p+3) \, _2F_1\left(-p,\frac{1}{2} (m+2 p+1);\frac{1}{2} (m+2 p+3);-\frac{c x^2}{b}\right)+B x^2 (m+2 p+1) \, _2F_1\left(-p,\frac{1}{2} (m+2 p+3);\frac{1}{2} (m+2 p+5);-\frac{c x^2}{b}\right)\right)}{(m+2 p+1) (m+2 p+3)}","\frac{B x^{m-1} \left(b x^2+c x^4\right)^{p+1}}{c (m+4 p+3)}-\frac{x^{m+1} \left(\frac{c x^2}{b}+1\right)^{-p} \left(b x^2+c x^4\right)^p (b B (m+2 p+1)-A c (m+4 p+3)) \, _2F_1\left(-p,\frac{1}{2} (m+2 p+1);\frac{1}{2} (m+2 p+3);-\frac{c x^2}{b}\right)}{c (m+2 p+1) (m+4 p+3)}",1,"(x^(1 + m)*(x^2*(b + c*x^2))^p*(A*(3 + m + 2*p)*Hypergeometric2F1[-p, (1 + m + 2*p)/2, (3 + m + 2*p)/2, -((c*x^2)/b)] + B*(1 + m + 2*p)*x^2*Hypergeometric2F1[-p, (3 + m + 2*p)/2, (5 + m + 2*p)/2, -((c*x^2)/b)]))/((1 + m + 2*p)*(3 + m + 2*p)*(1 + (c*x^2)/b)^p)","A",1
275,1,63,95,0.0753364,"\int x^{-1+n-j p} \left(c+d x^n\right) \left(a x^j+b x^{j+n}\right)^p \, dx","Integrate[x^(-1 + n - j*p)*(c + d*x^n)*(a*x^j + b*x^(j + n))^p,x]","\frac{x^{-j p} \left(a+b x^n\right) \left(x^j \left(a+b x^n\right)\right)^p \left(-a d+b c (p+2)+b d (p+1) x^n\right)}{b^2 n (p+1) (p+2)}","\frac{d x^{n-j (p+1)} \left(a x^j+b x^{j+n}\right)^{p+1}}{b n (p+2)}-\frac{x^{-j (p+1)} (a d-b c (p+2)) \left(a x^j+b x^{j+n}\right)^{p+1}}{b^2 n (p+1) (p+2)}",1,"((a + b*x^n)*(x^j*(a + b*x^n))^p*(-(a*d) + b*c*(2 + p) + b*d*(1 + p)*x^n))/(b^2*n*(1 + p)*(2 + p)*x^(j*p))","A",1
276,1,111,113,0.1763048,"\int (e x)^m \left(c+d x^n\right)^q \left(a x^j+b x^{j+n}\right)^p \, dx","Integrate[(e*x)^m*(c + d*x^n)^q*(a*x^j + b*x^(j + n))^p,x]","\frac{x (e x)^m \left(\frac{b x^n}{a}+1\right)^{-p} \left(c+d x^n\right)^q \left(\frac{d x^n}{c}+1\right)^{-q} \left(x^j \left(a+b x^n\right)\right)^p F_1\left(\frac{m+j p+1}{n};-p,-q;\frac{m+n+j p+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)}{j p+m+1}","\frac{x (e x)^m \left(\frac{b x^n}{a}+1\right)^{-p} \left(c+d x^n\right)^q \left(\frac{d x^n}{c}+1\right)^{-q} \left(a x^j+b x^{j+n}\right)^p F_1\left(\frac{m+j p+1}{n};-p,-q;\frac{m+n+j p+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)}{j p+m+1}",1,"(x*(e*x)^m*(x^j*(a + b*x^n))^p*(c + d*x^n)^q*AppellF1[(1 + m + j*p)/n, -p, -q, (1 + m + n + j*p)/n, -((b*x^n)/a), -((d*x^n)/c)])/((1 + m + j*p)*(1 + (b*x^n)/a)^p*(1 + (d*x^n)/c)^q)","A",1
277,1,210,129,0.2463473,"\int (e x)^{7/4} \left(c+d x^n\right)^q \left(a x^j+b x^{j+n}\right)^{5/3} \, dx","Integrate[(e*x)^(7/4)*(c + d*x^n)^q*(a*x^j + b*x^(j + n))^(5/3),x]","\frac{12 (e x)^{7/4} x^{j+1} \left(x^j \left(a+b x^n\right)\right)^{2/3} \left(c+d x^n\right)^q \left(\frac{d x^n}{c}+1\right)^{-q} \left(b (20 j+33) x^n F_1\left(\frac{20 j+12 n+33}{12 n};-\frac{2}{3},-q;\frac{20 j+24 n+33}{12 n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)+a (20 j+12 n+33) F_1\left(\frac{20 j+33}{12 n};-\frac{2}{3},-q;\frac{\frac{5 j}{3}+n+\frac{11}{4}}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)\right)}{(20 j+33) (20 j+12 n+33) \left(\frac{b x^n}{a}+1\right)^{2/3}}","\frac{12 a e (e x)^{3/4} x^{j+2} \left(a x^j+b x^{j+n}\right)^{2/3} \left(c+d x^n\right)^q \left(\frac{d x^n}{c}+1\right)^{-q} F_1\left(\frac{20 j+33}{12 n};-\frac{5}{3},-q;\frac{20 j+12 n+33}{12 n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)}{(20 j+33) \left(\frac{b x^n}{a}+1\right)^{2/3}}",1,"(12*x^(1 + j)*(e*x)^(7/4)*(x^j*(a + b*x^n))^(2/3)*(c + d*x^n)^q*(a*(33 + 20*j + 12*n)*AppellF1[(33 + 20*j)/(12*n), -2/3, -q, (11/4 + (5*j)/3 + n)/n, -((b*x^n)/a), -((d*x^n)/c)] + b*(33 + 20*j)*x^n*AppellF1[(33 + 20*j + 12*n)/(12*n), -2/3, -q, (33 + 20*j + 24*n)/(12*n), -((b*x^n)/a), -((d*x^n)/c)]))/((33 + 20*j)*(33 + 20*j + 12*n)*(1 + (b*x^n)/a)^(2/3)*(1 + (d*x^n)/c)^q)","A",0
278,1,19,19,0.0055412,"\int \frac{4+3 x^4}{5 x+2 x^5} \, dx","Integrate[(4 + 3*x^4)/(5*x + 2*x^5),x]","\frac{7}{40} \log \left(2 x^4+5\right)+\frac{4 \log (x)}{5}","\frac{7}{40} \log \left(2 x^4+5\right)+\frac{4 \log (x)}{5}",1,"(4*Log[x])/5 + (7*Log[5 + 2*x^4])/40","A",1
279,1,15,15,0.0059067,"\int \frac{1+x^6}{x-x^7} \, dx","Integrate[(1 + x^6)/(x - x^7),x]","\log (x)-\frac{1}{3} \log \left(1-x^6\right)","\log (x)-\frac{1}{3} \log \left(1-x^6\right)",1,"Log[x] - Log[1 - x^6]/3","A",1
280,1,17,17,0.0058223,"\int \frac{8+5 x^{10}}{2 x-x^{11}} \, dx","Integrate[(8 + 5*x^10)/(2*x - x^11),x]","4 \log (x)-\frac{9}{10} \log \left(2-x^{10}\right)","4 \log (x)-\frac{9}{10} \log \left(2-x^{10}\right)",1,"4*Log[x] - (9*Log[2 - x^10])/10","A",1
281,1,16,16,0.0040418,"\int \frac{-3+2 x}{-x^2+x^3} \, dx","Integrate[(-3 + 2*x)/(-x^2 + x^3),x]","-\frac{3}{x}-\log (1-x)+\log (x)","-\frac{3}{x}-\log (1-x)+\log (x)",1,"-3/x - Log[1 - x] + Log[x]","A",1
282,1,52,54,0.0211757,"\int \frac{a x^m+b x^n}{c x^m+d x^n} \, dx","Integrate[(a*x^m + b*x^n)/(c*x^m + d*x^n),x]","\frac{x \left((b c-a d) \, _2F_1\left(1,\frac{1}{m-n};1+\frac{1}{m-n};-\frac{c x^{m-n}}{d}\right)+a d\right)}{c d}","\frac{x (b c-a d) \, _2F_1\left(1,\frac{1}{m-n};1+\frac{1}{m-n};-\frac{c x^{m-n}}{d}\right)}{c d}+\frac{a x}{c}",1,"(x*(a*d + (b*c - a*d)*Hypergeometric2F1[1, (m - n)^(-1), 1 + (m - n)^(-1), -((c*x^(m - n))/d)]))/(c*d)","A",1
283,1,116,18,0.2206419,"\int x^m \left(a+b x^n\right)^p \left(a (1+m+q) x^q+b (1+m+n (1+p)+q) x^{n+q}\right) \, dx","Integrate[x^m*(a + b*x^n)^p*(a*(1 + m + q)*x^q + b*(1 + m + n*(1 + p) + q)*x^(n + q)),x]","x^{m+q+1} \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} \left(\frac{b x^n (m+n p+n+q+1) \, _2F_1\left(-p,\frac{m+n+q+1}{n};\frac{m+2 n+q+1}{n};-\frac{b x^n}{a}\right)}{m+n+q+1}+a \, _2F_1\left(-p,\frac{m+q+1}{n};\frac{m+n+q+1}{n};-\frac{b x^n}{a}\right)\right)","x^{m+q+1} \left(a+b x^n\right)^{p+1}",1,"(x^(1 + m + q)*(a + b*x^n)^p*(a*Hypergeometric2F1[-p, (1 + m + q)/n, (1 + m + n + q)/n, -((b*x^n)/a)] + (b*(1 + m + n + n*p + q)*x^n*Hypergeometric2F1[-p, (1 + m + n + q)/n, (1 + m + 2*n + q)/n, -((b*x^n)/a)])/(1 + m + n + q)))/(1 + (b*x^n)/a)^p","C",1
284,0,0,64,0.1094687,"\int \frac{\left(a+\frac{b}{x}\right)^n x^m}{c+d x} \, dx","Integrate[((a + b/x)^n*x^m)/(c + d*x),x]","\int \frac{\left(a+\frac{b}{x}\right)^n x^m}{c+d x} \, dx","\frac{x^m \left(a+\frac{b}{x}\right)^n \left(\frac{b}{a x}+1\right)^{-n} F_1\left(-m;-n,1;1-m;-\frac{b}{a x},-\frac{c}{d x}\right)}{d m}",1,"Integrate[((a + b/x)^n*x^m)/(c + d*x), x]","F",-1
285,1,157,195,0.1744219,"\int \frac{\left(a+\frac{b}{x}\right)^n x^2}{c+d x} \, dx","Integrate[((a + b/x)^n*x^2)/(c + d*x),x]","\frac{(a x+b) \left(a+\frac{b}{x}\right)^n \left((a c-b d) \left(\left(2 a^2 c^2-2 a b c d n+b^2 d^2 (n-1) n\right) \, _2F_1\left(1,n+1;n+2;\frac{b}{a x}+1\right)+a d (n+1) x (a (d x-2 c)+b d (n-1))\right)-2 a^3 c^3 \, _2F_1\left(1,n+1;n+2;\frac{c \left(a+\frac{b}{x}\right)}{a c-b d}\right)\right)}{2 a^3 d^3 (n+1) x (a c-b d)}","-\frac{x \left(a+\frac{b}{x}\right)^{n+1} (2 a c+b d (1-n))}{2 a^2 d^2}+\frac{\left(a+\frac{b}{x}\right)^{n+1} \left(2 a^2 c^2-2 a b c d n-b^2 d^2 (1-n) n\right) \, _2F_1\left(1,n+1;n+2;\frac{b}{a x}+1\right)}{2 a^3 d^3 (n+1)}-\frac{c^3 \left(a+\frac{b}{x}\right)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{c \left(a+\frac{b}{x}\right)}{a c-b d}\right)}{d^3 (n+1) (a c-b d)}+\frac{x^2 \left(a+\frac{b}{x}\right)^{n+1}}{2 a d}",1,"((a + b/x)^n*(b + a*x)*(-2*a^3*c^3*Hypergeometric2F1[1, 1 + n, 2 + n, (c*(a + b/x))/(a*c - b*d)] + (a*c - b*d)*(a*d*(1 + n)*x*(b*d*(-1 + n) + a*(-2*c + d*x)) + (2*a^2*c^2 - 2*a*b*c*d*n + b^2*d^2*(-1 + n)*n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + b/(a*x)])))/(2*a^3*d^3*(a*c - b*d)*(1 + n)*x)","A",1
286,1,119,131,0.0791732,"\int \frac{\left(a+\frac{b}{x}\right)^n x}{c+d x} \, dx","Integrate[((a + b/x)^n*x)/(c + d*x),x]","\frac{(a x+b) \left(a+\frac{b}{x}\right)^n \left(a^2 c^2 \, _2F_1\left(1,n+1;n+2;\frac{c \left(a+\frac{b}{x}\right)}{a c-b d}\right)+(a c-b d) \left((b d n-a c) \, _2F_1\left(1,n+1;n+2;\frac{b}{a x}+1\right)+a d (n+1) x\right)\right)}{a^2 d^2 (n+1) x (a c-b d)}","-\frac{\left(a+\frac{b}{x}\right)^{n+1} (a c-b d n) \, _2F_1\left(1,n+1;n+2;\frac{b}{a x}+1\right)}{a^2 d^2 (n+1)}+\frac{c^2 \left(a+\frac{b}{x}\right)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{c \left(a+\frac{b}{x}\right)}{a c-b d}\right)}{d^2 (n+1) (a c-b d)}+\frac{x \left(a+\frac{b}{x}\right)^{n+1}}{a d}",1,"((a + b/x)^n*(b + a*x)*(a^2*c^2*Hypergeometric2F1[1, 1 + n, 2 + n, (c*(a + b/x))/(a*c - b*d)] + (a*c - b*d)*(a*d*(1 + n)*x + (-(a*c) + b*d*n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + b/(a*x)])))/(a^2*d^2*(a*c - b*d)*(1 + n)*x)","A",1
287,1,97,101,0.0364955,"\int \frac{\left(a+\frac{b}{x}\right)^n}{c+d x} \, dx","Integrate[(a + b/x)^n/(c + d*x),x]","\frac{(a x+b) \left(a+\frac{b}{x}\right)^n \left(a c \, _2F_1\left(1,n+1;n+2;\frac{c \left(a+\frac{b}{x}\right)}{a c-b d}\right)+(b d-a c) \, _2F_1\left(1,n+1;n+2;\frac{b}{a x}+1\right)\right)}{a d (n+1) x (b d-a c)}","\frac{\left(a+\frac{b}{x}\right)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{b}{a x}+1\right)}{a d (n+1)}-\frac{c \left(a+\frac{b}{x}\right)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{c \left(a+\frac{b}{x}\right)}{a c-b d}\right)}{d (n+1) (a c-b d)}",1,"((a + b/x)^n*(b + a*x)*(a*c*Hypergeometric2F1[1, 1 + n, 2 + n, (c*(a + b/x))/(a*c - b*d)] + (-(a*c) + b*d)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + b/(a*x)]))/(a*d*(-(a*c) + b*d)*(1 + n)*x)","A",1
288,1,54,54,0.0165777,"\int \frac{\left(a+\frac{b}{x}\right)^n}{x (c+d x)} \, dx","Integrate[(a + b/x)^n/(x*(c + d*x)),x]","\frac{\left(a+\frac{b}{x}\right)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{c \left(a+\frac{b}{x}\right)}{a c-b d}\right)}{(n+1) (a c-b d)}","\frac{\left(a+\frac{b}{x}\right)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{c \left(a+\frac{b}{x}\right)}{a c-b d}\right)}{(n+1) (a c-b d)}",1,"((a + b/x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (c*(a + b/x))/(a*c - b*d)])/((a*c - b*d)*(1 + n))","A",1
289,1,77,84,0.0264852,"\int \frac{\left(a+\frac{b}{x}\right)^n}{x^2 (c+d x)} \, dx","Integrate[(a + b/x)^n/(x^2*(c + d*x)),x]","\frac{(a x+b) \left(a+\frac{b}{x}\right)^n \left(b d \, _2F_1\left(1,n+1;n+2;\frac{c \left(a+\frac{b}{x}\right)}{a c-b d}\right)+a c-b d\right)}{b c (n+1) x (b d-a c)}","-\frac{d \left(a+\frac{b}{x}\right)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{c \left(a+\frac{b}{x}\right)}{a c-b d}\right)}{c (n+1) (a c-b d)}-\frac{\left(a+\frac{b}{x}\right)^{n+1}}{b c (n+1)}",1,"((a + b/x)^n*(b + a*x)*(a*c - b*d + b*d*Hypergeometric2F1[1, 1 + n, 2 + n, (c*(a + b/x))/(a*c - b*d)]))/(b*c*(-(a*c) + b*d)*(1 + n)*x)","A",1
290,1,112,115,0.1441555,"\int \frac{\left(a+\frac{b}{x}\right)^n}{x^3 (c+d x)} \, dx","Integrate[(a + b/x)^n/(x^3*(c + d*x)),x]","-\frac{(a x+b) \left(a+\frac{b}{x}\right)^n \left(b^2 d^2 (n+2) x \, _2F_1\left(1,n+1;n+2;\frac{c \left(a+\frac{b}{x}\right)}{a c-b d}\right)+(a c-b d) (a c x-b c (n+1)+b d (n+2) x)\right)}{b^2 c^2 (n+1) (n+2) x^2 (b d-a c)}","\frac{(a c+b d) \left(a+\frac{b}{x}\right)^{n+1}}{b^2 c^2 (n+1)}-\frac{\left(a+\frac{b}{x}\right)^{n+2}}{b^2 c (n+2)}+\frac{d^2 \left(a+\frac{b}{x}\right)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{c \left(a+\frac{b}{x}\right)}{a c-b d}\right)}{c^2 (n+1) (a c-b d)}",1,"-(((a + b/x)^n*(b + a*x)*((a*c - b*d)*(-(b*c*(1 + n)) + a*c*x + b*d*(2 + n)*x) + b^2*d^2*(2 + n)*x*Hypergeometric2F1[1, 1 + n, 2 + n, (c*(a + b/x))/(a*c - b*d)]))/(b^2*c^2*(-(a*c) + b*d)*(1 + n)*(2 + n)*x^2))","A",1
291,1,184,207,0.2390505,"\int \frac{\left(a+\frac{b}{x}\right)^n}{x^5 (c+d x)} \, dx","Integrate[(a + b/x)^n/(x^5*(c + d*x)),x]","\frac{\left(a+\frac{b}{x}\right)^{n+1} \left(-\frac{c \left(a+\frac{b}{x}\right) \left(3 a^2 c^2+2 a b c d+b^2 d^2\right)}{b^4 (n+2)}+\frac{(a c+b d) \left(a^2 c^2+b^2 d^2\right)}{b^4 (n+1)}-\frac{c^3 \left(a+\frac{b}{x}\right)^3}{b^4 (n+4)}+\frac{c^2 \left(a+\frac{b}{x}\right)^2 (3 a c+b d)}{b^4 (n+3)}+\frac{d^4 \, _2F_1\left(1,n+1;n+2;\frac{c \left(a+\frac{b}{x}\right)}{a c-b d}\right)}{(n+1) (a c-b d)}\right)}{c^4}","\frac{(a c+b d) \left(a^2 c^2+b^2 d^2\right) \left(a+\frac{b}{x}\right)^{n+1}}{b^4 c^4 (n+1)}-\frac{\left(3 a^2 c^2+2 a b c d+b^2 d^2\right) \left(a+\frac{b}{x}\right)^{n+2}}{b^4 c^3 (n+2)}+\frac{(3 a c+b d) \left(a+\frac{b}{x}\right)^{n+3}}{b^4 c^2 (n+3)}-\frac{\left(a+\frac{b}{x}\right)^{n+4}}{b^4 c (n+4)}+\frac{d^4 \left(a+\frac{b}{x}\right)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{c \left(a+\frac{b}{x}\right)}{a c-b d}\right)}{c^4 (n+1) (a c-b d)}",1,"((a + b/x)^(1 + n)*(((a*c + b*d)*(a^2*c^2 + b^2*d^2))/(b^4*(1 + n)) - (c*(3*a^2*c^2 + 2*a*b*c*d + b^2*d^2)*(a + b/x))/(b^4*(2 + n)) + (c^2*(3*a*c + b*d)*(a + b/x)^2)/(b^4*(3 + n)) - (c^3*(a + b/x)^3)/(b^4*(4 + n)) + (d^4*Hypergeometric2F1[1, 1 + n, 2 + n, (c*(a + b/x))/(a*c - b*d)])/((a*c - b*d)*(1 + n))))/c^4","A",1
292,0,0,73,0.1084143,"\int \frac{\left(a+\frac{b}{x}\right)^n x^m}{(c+d x)^2} \, dx","Integrate[((a + b/x)^n*x^m)/(c + d*x)^2,x]","\int \frac{\left(a+\frac{b}{x}\right)^n x^m}{(c+d x)^2} \, dx","-\frac{x^{m-1} \left(a+\frac{b}{x}\right)^n \left(\frac{b}{a x}+1\right)^{-n} F_1\left(1-m;-n,2;2-m;-\frac{b}{a x},-\frac{c}{d x}\right)}{d^2 (1-m)}",1,"Integrate[((a + b/x)^n*x^m)/(c + d*x)^2, x]","F",-1
293,1,179,202,0.2416211,"\int \frac{\left(a+\frac{b}{x}\right)^n x^2}{(c+d x)^2} \, dx","Integrate[((a + b/x)^n*x^2)/(c + d*x)^2,x]","\frac{\left(a+\frac{b}{x}\right)^{n+1} \left((c+d x) \left(a^2 c^2 (2 a c+b d (n-2)) \, _2F_1\left(1,n+1;n+2;\frac{c \left(a+\frac{b}{x}\right)}{a c-b d}\right)-(a c-b d)^2 (2 a c-b d n) \, _2F_1\left(1,n+1;n+2;\frac{b}{a x}+1\right)\right)+a d^2 (n+1) x^2 (a c-b d)^2+a c d (n+1) x (a c-b d) (2 a c-b d)\right)}{a^2 d^3 (n+1) (c+d x) (a c-b d)^2}","-\frac{\left(a+\frac{b}{x}\right)^{n+1} (2 a c-b d n) \, _2F_1\left(1,n+1;n+2;\frac{b}{a x}+1\right)}{a^2 d^3 (n+1)}+\frac{c^2 \left(a+\frac{b}{x}\right)^{n+1} (2 a c-b d (2-n)) \, _2F_1\left(1,n+1;n+2;\frac{c \left(a+\frac{b}{x}\right)}{a c-b d}\right)}{d^3 (n+1) (a c-b d)^2}+\frac{c (2 a c-b d) \left(a+\frac{b}{x}\right)^{n+1}}{a d^2 \left(\frac{c}{x}+d\right) (a c-b d)}+\frac{x \left(a+\frac{b}{x}\right)^{n+1}}{a d \left(\frac{c}{x}+d\right)}",1,"((a + b/x)^(1 + n)*(a*c*d*(a*c - b*d)*(2*a*c - b*d)*(1 + n)*x + a*d^2*(a*c - b*d)^2*(1 + n)*x^2 + (c + d*x)*(a^2*c^2*(2*a*c + b*d*(-2 + n))*Hypergeometric2F1[1, 1 + n, 2 + n, (c*(a + b/x))/(a*c - b*d)] - (a*c - b*d)^2*(2*a*c - b*d*n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + b/(a*x)])))/(a^2*d^3*(a*c - b*d)^2*(1 + n)*(c + d*x))","A",1
294,1,120,150,0.151386,"\int \frac{\left(a+\frac{b}{x}\right)^n x}{(c+d x)^2} \, dx","Integrate[((a + b/x)^n*x)/(c + d*x)^2,x]","\frac{\left(a+\frac{b}{x}\right)^{n+1} \left(-\frac{c (a c+b d (n-1)) \, _2F_1\left(1,n+1;n+2;\frac{c \left(a+\frac{b}{x}\right)}{a c-b d}\right)}{(n+1) (a c-b d)^2}-\frac{c d x}{(c+d x) (a c-b d)}+\frac{\, _2F_1\left(1,n+1;n+2;\frac{b}{a x}+1\right)}{a (n+1)}\right)}{d^2}","-\frac{c \left(a+\frac{b}{x}\right)^{n+1} (a c-b d (1-n)) \, _2F_1\left(1,n+1;n+2;\frac{c \left(a+\frac{b}{x}\right)}{a c-b d}\right)}{d^2 (n+1) (a c-b d)^2}-\frac{c \left(a+\frac{b}{x}\right)^{n+1}}{d \left(\frac{c}{x}+d\right) (a c-b d)}+\frac{\left(a+\frac{b}{x}\right)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{b}{a x}+1\right)}{a d^2 (n+1)}",1,"((a + b/x)^(1 + n)*(-((c*d*x)/((a*c - b*d)*(c + d*x))) - (c*(a*c + b*d*(-1 + n))*Hypergeometric2F1[1, 1 + n, 2 + n, (c*(a + b/x))/(a*c - b*d)])/((a*c - b*d)^2*(1 + n)) + Hypergeometric2F1[1, 1 + n, 2 + n, 1 + b/(a*x)]/(a*(1 + n))))/d^2","A",1
295,1,57,56,0.0404173,"\int \frac{\left(a+\frac{b}{x}\right)^n}{(c+d x)^2} \, dx","Integrate[(a + b/x)^n/(c + d*x)^2,x]","-\frac{b \left(a+\frac{b}{x}\right)^{n+1} \, _2F_1\left(2,n+1;n+2;-\frac{c \left(a+\frac{b}{x}\right)}{b d-a c}\right)}{(n+1) (b d-a c)^2}","-\frac{b \left(a+\frac{b}{x}\right)^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{c \left(a+\frac{b}{x}\right)}{a c-b d}\right)}{(n+1) (a c-b d)^2}",1,"-((b*(a + b/x)^(1 + n)*Hypergeometric2F1[2, 1 + n, 2 + n, -((c*(a + b/x))/(-(a*c) + b*d))])/((-(a*c) + b*d)^2*(1 + n)))","A",1
296,1,88,105,0.1237408,"\int \frac{\left(a+\frac{b}{x}\right)^n}{x (c+d x)^2} \, dx","Integrate[(a + b/x)^n/(x*(c + d*x)^2),x]","\frac{\left(a+\frac{b}{x}\right)^{n+1} \left(\frac{(a c-b d (n+1)) \, _2F_1\left(1,n+1;n+2;\frac{c \left(a+\frac{b}{x}\right)}{a c-b d}\right)}{n+1}+\frac{d x (b d-a c)}{c+d x}\right)}{c (a c-b d)^2}","\frac{\left(a+\frac{b}{x}\right)^{n+1} (a c-b d (n+1)) \, _2F_1\left(1,n+1;n+2;\frac{c \left(a+\frac{b}{x}\right)}{a c-b d}\right)}{c (n+1) (a c-b d)^2}-\frac{d \left(a+\frac{b}{x}\right)^{n+1}}{c \left(\frac{c}{x}+d\right) (a c-b d)}",1,"((a + b/x)^(1 + n)*((d*(-(a*c) + b*d)*x)/(c + d*x) + ((a*c - b*d*(1 + n))*Hypergeometric2F1[1, 1 + n, 2 + n, (c*(a + b/x))/(a*c - b*d)])/(1 + n)))/(c*(a*c - b*d)^2)","A",1
297,1,125,133,0.1107221,"\int \frac{\left(a+\frac{b}{x}\right)^n}{x^2 (c+d x)^2} \, dx","Integrate[(a + b/x)^n/(x^2*(c + d*x)^2),x]","-\frac{(a x+b) \left(a+\frac{b}{x}\right)^n \left(b d (c+d x) (2 a c-b d (n+2)) \, _2F_1\left(1,n+1;n+2;\frac{c \left(a+\frac{b}{x}\right)}{a c-b d}\right)+(a c-b d) (a c (c+d x)-b d (c+d (n+2) x))\right)}{b c^2 (n+1) x (c+d x) (a c-b d)^2}","\frac{d^2 \left(a+\frac{b}{x}\right)^{n+1}}{c^2 \left(\frac{c}{x}+d\right) (a c-b d)}-\frac{d \left(a+\frac{b}{x}\right)^{n+1} (2 a c-b d (n+2)) \, _2F_1\left(1,n+1;n+2;\frac{c \left(a+\frac{b}{x}\right)}{a c-b d}\right)}{c^2 (n+1) (a c-b d)^2}-\frac{\left(a+\frac{b}{x}\right)^{n+1}}{b c^2 (n+1)}",1,"-(((a + b/x)^n*(b + a*x)*((a*c - b*d)*(a*c*(c + d*x) - b*d*(c + d*(2 + n)*x)) + b*d*(2*a*c - b*d*(2 + n))*(c + d*x)*Hypergeometric2F1[1, 1 + n, 2 + n, (c*(a + b/x))/(a*c - b*d)]))/(b*c^2*(a*c - b*d)^2*(1 + n)*x*(c + d*x)))","A",1
298,1,182,217,0.3464993,"\int \frac{\left(a+\frac{b}{x}\right)^n}{x^3 (c+d x)^2} \, dx","Integrate[(a + b/x)^n/(x^3*(c + d*x)^2),x]","\frac{\left(a+\frac{b}{x}\right)^{n+1} \left(\frac{-a^2 c^2 (c+d x)-a b c d (c (n+2)+d (2 n+3) x)+b^2 d^2 (n+3) (c+d (n+2) x)}{b c^2 (n+1) (c+d x) (b d-a c)}-\frac{b d^2 (n+2) (b d (n+3)-3 a c) \, _2F_1\left(1,n+1;n+2;\frac{c \left(a+\frac{b}{x}\right)}{a c-b d}\right)}{c^2 (n+1) (a c-b d)^2}-\frac{1}{x (c+d x)}\right)}{b c (n+2)}","-\frac{\left(a+\frac{b}{x}\right)^{n+1} \left(d (b d (n+2) (a c+b d (n+3))-a c (a c+b d (3 n+5)))-\frac{c (a c-b d) (a c+b d (n+3))}{x}\right)}{b^2 c^3 (n+1) (n+2) \left(\frac{c}{x}+d\right) (a c-b d)}+\frac{d^2 \left(a+\frac{b}{x}\right)^{n+1} (3 a c-b d (n+3)) \, _2F_1\left(1,n+1;n+2;\frac{c \left(a+\frac{b}{x}\right)}{a c-b d}\right)}{c^3 (n+1) (a c-b d)^2}-\frac{\left(a+\frac{b}{x}\right)^{n+1}}{b c (n+2) x^2 \left(\frac{c}{x}+d\right)}",1,"((a + b/x)^(1 + n)*(-(1/(x*(c + d*x))) + (-(a^2*c^2*(c + d*x)) + b^2*d^2*(3 + n)*(c + d*(2 + n)*x) - a*b*c*d*(c*(2 + n) + d*(3 + 2*n)*x))/(b*c^2*(-(a*c) + b*d)*(1 + n)*(c + d*x)) - (b*d^2*(2 + n)*(-3*a*c + b*d*(3 + n))*Hypergeometric2F1[1, 1 + n, 2 + n, (c*(a + b/x))/(a*c - b*d)])/(c^2*(a*c - b*d)^2*(1 + n))))/(b*c*(2 + n))","A",1