1,1,33,33,0.0125132,"\int x^2 \left(a+b x^2\right) \left(A+B x^2\right) \, dx","Integrate[x^2*(a + b*x^2)*(A + B*x^2),x]","\frac{1}{5} x^5 (a B+A b)+\frac{1}{3} a A x^3+\frac{1}{7} b B x^7","\frac{1}{5} x^5 (a B+A b)+\frac{1}{3} a A x^3+\frac{1}{7} b B x^7",1,"(a*A*x^3)/3 + ((A*b + a*B)*x^5)/5 + (b*B*x^7)/7","A",1
2,1,33,33,0.0221558,"\int x \left(a+b x^2\right) \left(A+B x^2\right) \, dx","Integrate[x*(a + b*x^2)*(A + B*x^2),x]","\frac{1}{4} x^4 (a B+A b)+\frac{1}{2} a A x^2+\frac{1}{6} b B x^6","\frac{1}{4} x^4 (a B+A b)+\frac{1}{2} a A x^2+\frac{1}{6} b B x^6",1,"(a*A*x^2)/2 + ((A*b + a*B)*x^4)/4 + (b*B*x^6)/6","A",1
3,1,28,28,0.0047529,"\int \left(a+b x^2\right) \left(A+B x^2\right) \, dx","Integrate[(a + b*x^2)*(A + B*x^2),x]","\frac{1}{3} x^3 (a B+A b)+a A x+\frac{1}{5} b B x^5","\frac{1}{3} x^3 (a B+A b)+a A x+\frac{1}{5} b B x^5",1,"a*A*x + ((A*b + a*B)*x^3)/3 + (b*B*x^5)/5","A",1
4,1,29,29,0.0184082,"\int \frac{\left(a+b x^2\right) \left(A+B x^2\right)}{x} \, dx","Integrate[((a + b*x^2)*(A + B*x^2))/x,x]","\frac{1}{2} x^2 (a B+A b)+a A \log (x)+\frac{1}{4} b B x^4","\frac{1}{2} x^2 (a B+A b)+a A \log (x)+\frac{1}{4} b B x^4",1,"((A*b + a*B)*x^2)/2 + (b*B*x^4)/4 + a*A*Log[x]","A",1
5,1,26,26,0.0077587,"\int \frac{\left(a+b x^2\right) \left(A+B x^2\right)}{x^2} \, dx","Integrate[((a + b*x^2)*(A + B*x^2))/x^2,x]","x (a B+A b)-\frac{a A}{x}+\frac{1}{3} b B x^3","x (a B+A b)-\frac{a A}{x}+\frac{1}{3} b B x^3",1,"-((a*A)/x) + (A*b + a*B)*x + (b*B*x^3)/3","A",1
6,1,29,29,0.0098244,"\int \frac{\left(a+b x^2\right) \left(A+B x^2\right)}{x^3} \, dx","Integrate[((a + b*x^2)*(A + B*x^2))/x^3,x]","\log (x) (a B+A b)-\frac{a A}{2 x^2}+\frac{1}{2} b B x^2","\log (x) (a B+A b)-\frac{a A}{2 x^2}+\frac{1}{2} b B x^2",1,"-1/2*(a*A)/x^2 + (b*B*x^2)/2 + (A*b + a*B)*Log[x]","A",1
7,1,27,26,0.0313637,"\int \frac{\left(a+b x^2\right) \left(A+B x^2\right)}{x^4} \, dx","Integrate[((a + b*x^2)*(A + B*x^2))/x^4,x]","\frac{-a B-A b}{x}-\frac{a A}{3 x^3}+b B x","-\frac{a B+A b}{x}-\frac{a A}{3 x^3}+b B x",1,"-1/3*(a*A)/x^3 + (-(A*b) - a*B)/x + b*B*x","A",1
8,1,31,29,0.0157158,"\int \frac{\left(a+b x^2\right) \left(A+B x^2\right)}{x^5} \, dx","Integrate[((a + b*x^2)*(A + B*x^2))/x^5,x]","\frac{-a B-A b}{2 x^2}-\frac{a A}{4 x^4}+b B \log (x)","-\frac{a B+A b}{2 x^2}-\frac{a A}{4 x^4}+b B \log (x)",1,"-1/4*(a*A)/x^4 + (-(A*b) - a*B)/(2*x^2) + b*B*Log[x]","A",1
9,1,33,31,0.0101178,"\int \frac{\left(a+b x^2\right) \left(A+B x^2\right)}{x^6} \, dx","Integrate[((a + b*x^2)*(A + B*x^2))/x^6,x]","\frac{-a B-A b}{3 x^3}-\frac{a A}{5 x^5}-\frac{b B}{x}","-\frac{a B+A b}{3 x^3}-\frac{a A}{5 x^5}-\frac{b B}{x}",1,"-1/5*(a*A)/x^5 + (-(A*b) - a*B)/(3*x^3) - (b*B)/x","A",1
10,1,35,33,0.0171433,"\int \frac{\left(a+b x^2\right) \left(A+B x^2\right)}{x^7} \, dx","Integrate[((a + b*x^2)*(A + B*x^2))/x^7,x]","\frac{-a B-A b}{4 x^4}-\frac{a A}{6 x^6}-\frac{b B}{2 x^2}","-\frac{a B+A b}{4 x^4}-\frac{a A}{6 x^6}-\frac{b B}{2 x^2}",1,"-1/6*(a*A)/x^6 + (-(A*b) - a*B)/(4*x^4) - (b*B)/(2*x^2)","A",1
11,1,55,55,0.0070919,"\int x^2 \left(a+b x^2\right)^2 \left(A+B x^2\right) \, dx","Integrate[x^2*(a + b*x^2)^2*(A + B*x^2),x]","\frac{1}{3} a^2 A x^3+\frac{1}{7} b x^7 (2 a B+A b)+\frac{1}{5} a x^5 (a B+2 A b)+\frac{1}{9} b^2 B x^9","\frac{1}{3} a^2 A x^3+\frac{1}{7} b x^7 (2 a B+A b)+\frac{1}{5} a x^5 (a B+2 A b)+\frac{1}{9} b^2 B x^9",1,"(a^2*A*x^3)/3 + (a*(2*A*b + a*B)*x^5)/5 + (b*(A*b + 2*a*B)*x^7)/7 + (b^2*B*x^9)/9","A",1
12,1,51,42,0.0095166,"\int x \left(a+b x^2\right)^2 \left(A+B x^2\right) \, dx","Integrate[x*(a + b*x^2)^2*(A + B*x^2),x]","\frac{1}{24} x^2 \left(12 a^2 A+4 b x^4 (2 a B+A b)+6 a x^2 (a B+2 A b)+3 b^2 B x^6\right)","\frac{\left(a+b x^2\right)^3 (A b-a B)}{6 b^2}+\frac{B \left(a+b x^2\right)^4}{8 b^2}",1,"(x^2*(12*a^2*A + 6*a*(2*A*b + a*B)*x^2 + 4*b*(A*b + 2*a*B)*x^4 + 3*b^2*B*x^6))/24","A",1
13,1,50,50,0.0062549,"\int \left(a+b x^2\right)^2 \left(A+B x^2\right) \, dx","Integrate[(a + b*x^2)^2*(A + B*x^2),x]","a^2 A x+\frac{1}{5} b x^5 (2 a B+A b)+\frac{1}{3} a x^3 (a B+2 A b)+\frac{1}{7} b^2 B x^7","a^2 A x+\frac{1}{5} b x^5 (2 a B+A b)+\frac{1}{3} a x^3 (a B+2 A b)+\frac{1}{7} b^2 B x^7",1,"a^2*A*x + (a*(2*A*b + a*B)*x^3)/3 + (b*(A*b + 2*a*B)*x^5)/5 + (b^2*B*x^7)/7","A",1
14,1,51,43,0.0176744,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x^2\right)}{x} \, dx","Integrate[((a + b*x^2)^2*(A + B*x^2))/x,x]","a^2 A \log (x)+\frac{1}{4} b x^4 (2 a B+A b)+\frac{1}{2} a x^2 (a B+2 A b)+\frac{1}{6} b^2 B x^6","a^2 A \log (x)+a A b x^2+\frac{B \left(a+b x^2\right)^3}{6 b}+\frac{1}{4} A b^2 x^4",1,"(a*(2*A*b + a*B)*x^2)/2 + (b*(A*b + 2*a*B)*x^4)/4 + (b^2*B*x^6)/6 + a^2*A*Log[x]","A",1
15,1,48,48,0.0336008,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x^2\right)}{x^2} \, dx","Integrate[((a + b*x^2)^2*(A + B*x^2))/x^2,x]","-\frac{a^2 A}{x}+\frac{1}{3} b x^3 (2 a B+A b)+a x (a B+2 A b)+\frac{1}{5} b^2 B x^5","-\frac{a^2 A}{x}+\frac{1}{3} b x^3 (2 a B+A b)+a x (a B+2 A b)+\frac{1}{5} b^2 B x^5",1,"-((a^2*A)/x) + a*(2*A*b + a*B)*x + (b*(A*b + 2*a*B)*x^3)/3 + (b^2*B*x^5)/5","A",1
16,1,49,51,0.033196,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x^2\right)}{x^3} \, dx","Integrate[((a + b*x^2)^2*(A + B*x^2))/x^3,x]","\frac{1}{4} \left(-\frac{2 a^2 A}{x^2}+2 b x^2 (2 a B+A b)+4 a \log (x) (a B+2 A b)+b^2 B x^4\right)","-\frac{a^2 A}{2 x^2}+\frac{1}{2} b x^2 (2 a B+A b)+a \log (x) (a B+2 A b)+\frac{1}{4} b^2 B x^4",1,"((-2*a^2*A)/x^2 + 2*b*(A*b + 2*a*B)*x^2 + b^2*B*x^4 + 4*a*(2*A*b + a*B)*Log[x])/4","A",1
17,1,50,48,0.0175072,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x^2\right)}{x^4} \, dx","Integrate[((a + b*x^2)^2*(A + B*x^2))/x^4,x]","\frac{a^2 (-B)-2 a A b}{x}-\frac{a^2 A}{3 x^3}+b x (2 a B+A b)+\frac{1}{3} b^2 B x^3","-\frac{a^2 A}{3 x^3}+b x (2 a B+A b)-\frac{a (a B+2 A b)}{x}+\frac{1}{3} b^2 B x^3",1,"-1/3*(a^2*A)/x^3 + (-2*a*A*b - a^2*B)/x + b*(A*b + 2*a*B)*x + (b^2*B*x^3)/3","A",1
18,1,50,51,0.0220626,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x^2\right)}{x^5} \, dx","Integrate[((a + b*x^2)^2*(A + B*x^2))/x^5,x]","b \log (x) (2 a B+A b)-\frac{a^2 \left(A+2 B x^2\right)+4 a A b x^2-2 b^2 B x^6}{4 x^4}","-\frac{a^2 A}{4 x^4}-\frac{a (a B+2 A b)}{2 x^2}+b \log (x) (2 a B+A b)+\frac{1}{2} b^2 B x^2",1,"-1/4*(4*a*A*b*x^2 - 2*b^2*B*x^6 + a^2*(A + 2*B*x^2))/x^4 + b*(A*b + 2*a*B)*Log[x]","A",1
19,1,48,48,0.017524,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x^2\right)}{x^6} \, dx","Integrate[((a + b*x^2)^2*(A + B*x^2))/x^6,x]","-\frac{a^2 A}{5 x^5}-\frac{a (a B+2 A b)}{3 x^3}-\frac{b (2 a B+A b)}{x}+b^2 B x","-\frac{a^2 A}{5 x^5}-\frac{a (a B+2 A b)}{3 x^3}-\frac{b (2 a B+A b)}{x}+b^2 B x",1,"-1/5*(a^2*A)/x^5 - (a*(2*A*b + a*B))/(3*x^3) - (b*(A*b + 2*a*B))/x + b^2*B*x","A",1
20,1,54,51,0.0270472,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x^2\right)}{x^7} \, dx","Integrate[((a + b*x^2)^2*(A + B*x^2))/x^7,x]","b^2 B \log (x)-\frac{a^2 \left(2 A+3 B x^2\right)+6 a b x^2 \left(A+2 B x^2\right)+6 A b^2 x^4}{12 x^6}","-\frac{a^2 A}{6 x^6}-\frac{a (a B+2 A b)}{4 x^4}-\frac{b (2 a B+A b)}{2 x^2}+b^2 B \log (x)",1,"-1/12*(6*A*b^2*x^4 + 6*a*b*x^2*(A + 2*B*x^2) + a^2*(2*A + 3*B*x^2))/x^6 + b^2*B*Log[x]","A",1
21,1,56,53,0.0294174,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x^2\right)}{x^8} \, dx","Integrate[((a + b*x^2)^2*(A + B*x^2))/x^8,x]","-\frac{3 a^2 \left(5 A+7 B x^2\right)+14 a b x^2 \left(3 A+5 B x^2\right)+35 b^2 x^4 \left(A+3 B x^2\right)}{105 x^7}","-\frac{a^2 A}{7 x^7}-\frac{a (a B+2 A b)}{5 x^5}-\frac{b (2 a B+A b)}{3 x^3}-\frac{b^2 B}{x}",1,"-1/105*(35*b^2*x^4*(A + 3*B*x^2) + 14*a*b*x^2*(3*A + 5*B*x^2) + 3*a^2*(5*A + 7*B*x^2))/x^7","A",1
22,1,55,48,0.0320988,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x^2\right)}{x^9} \, dx","Integrate[((a + b*x^2)^2*(A + B*x^2))/x^9,x]","-\frac{a^2 \left(3 A+4 B x^2\right)+4 a b x^2 \left(2 A+3 B x^2\right)+6 b^2 x^4 \left(A+2 B x^2\right)}{24 x^8}","\frac{\left(a+b x^2\right)^3 (A b-4 a B)}{24 a^2 x^6}-\frac{A \left(a+b x^2\right)^3}{8 a x^8}",1,"-1/24*(6*b^2*x^4*(A + 2*B*x^2) + 4*a*b*x^2*(2*A + 3*B*x^2) + a^2*(3*A + 4*B*x^2))/x^8","A",1
23,1,117,117,0.0199675,"\int x^9 \left(a+b x^2\right)^5 \left(A+B x^2\right) \, dx","Integrate[x^9*(a + b*x^2)^5*(A + B*x^2),x]","\frac{1}{10} a^5 A x^{10}+\frac{1}{12} a^4 x^{12} (a B+5 A b)+\frac{5}{14} a^3 b x^{14} (a B+2 A b)+\frac{5}{8} a^2 b^2 x^{16} (a B+A b)+\frac{1}{20} b^4 x^{20} (5 a B+A b)+\frac{5}{18} a b^3 x^{18} (2 a B+A b)+\frac{1}{22} b^5 B x^{22}","\frac{1}{10} a^5 A x^{10}+\frac{1}{12} a^4 x^{12} (a B+5 A b)+\frac{5}{14} a^3 b x^{14} (a B+2 A b)+\frac{5}{8} a^2 b^2 x^{16} (a B+A b)+\frac{1}{20} b^4 x^{20} (5 a B+A b)+\frac{5}{18} a b^3 x^{18} (2 a B+A b)+\frac{1}{22} b^5 B x^{22}",1,"(a^5*A*x^10)/10 + (a^4*(5*A*b + a*B)*x^12)/12 + (5*a^3*b*(2*A*b + a*B)*x^14)/14 + (5*a^2*b^2*(A*b + a*B)*x^16)/8 + (5*a*b^3*(A*b + 2*a*B)*x^18)/18 + (b^4*(A*b + 5*a*B)*x^20)/20 + (b^5*B*x^22)/22","A",1
24,1,117,117,0.0340737,"\int x^8 \left(a+b x^2\right)^5 \left(A+B x^2\right) \, dx","Integrate[x^8*(a + b*x^2)^5*(A + B*x^2),x]","\frac{1}{9} a^5 A x^9+\frac{1}{11} a^4 x^{11} (a B+5 A b)+\frac{5}{13} a^3 b x^{13} (a B+2 A b)+\frac{2}{3} a^2 b^2 x^{15} (a B+A b)+\frac{1}{19} b^4 x^{19} (5 a B+A b)+\frac{5}{17} a b^3 x^{17} (2 a B+A b)+\frac{1}{21} b^5 B x^{21}","\frac{1}{9} a^5 A x^9+\frac{1}{11} a^4 x^{11} (a B+5 A b)+\frac{5}{13} a^3 b x^{13} (a B+2 A b)+\frac{2}{3} a^2 b^2 x^{15} (a B+A b)+\frac{1}{19} b^4 x^{19} (5 a B+A b)+\frac{5}{17} a b^3 x^{17} (2 a B+A b)+\frac{1}{21} b^5 B x^{21}",1,"(a^5*A*x^9)/9 + (a^4*(5*A*b + a*B)*x^11)/11 + (5*a^3*b*(2*A*b + a*B)*x^13)/13 + (2*a^2*b^2*(A*b + a*B)*x^15)/3 + (5*a*b^3*(A*b + 2*a*B)*x^17)/17 + (b^4*(A*b + 5*a*B)*x^19)/19 + (b^5*B*x^21)/21","A",1
25,1,117,122,0.0150211,"\int x^7 \left(a+b x^2\right)^5 \left(A+B x^2\right) \, dx","Integrate[x^7*(a + b*x^2)^5*(A + B*x^2),x]","\frac{1}{8} a^5 A x^8+\frac{1}{10} a^4 x^{10} (a B+5 A b)+\frac{5}{12} a^3 b x^{12} (a B+2 A b)+\frac{5}{7} a^2 b^2 x^{14} (a B+A b)+\frac{1}{18} b^4 x^{18} (5 a B+A b)+\frac{5}{16} a b^3 x^{16} (2 a B+A b)+\frac{1}{20} b^5 B x^{20}","-\frac{a^3 \left(a+b x^2\right)^6 (A b-a B)}{12 b^5}+\frac{a^2 \left(a+b x^2\right)^7 (3 A b-4 a B)}{14 b^5}+\frac{\left(a+b x^2\right)^9 (A b-4 a B)}{18 b^5}-\frac{3 a \left(a+b x^2\right)^8 (A b-2 a B)}{16 b^5}+\frac{B \left(a+b x^2\right)^{10}}{20 b^5}",1,"(a^5*A*x^8)/8 + (a^4*(5*A*b + a*B)*x^10)/10 + (5*a^3*b*(2*A*b + a*B)*x^12)/12 + (5*a^2*b^2*(A*b + a*B)*x^14)/7 + (5*a*b^3*(A*b + 2*a*B)*x^16)/16 + (b^4*(A*b + 5*a*B)*x^18)/18 + (b^5*B*x^20)/20","A",1
26,1,117,117,0.0311176,"\int x^6 \left(a+b x^2\right)^5 \left(A+B x^2\right) \, dx","Integrate[x^6*(a + b*x^2)^5*(A + B*x^2),x]","\frac{1}{7} a^5 A x^7+\frac{1}{9} a^4 x^9 (a B+5 A b)+\frac{5}{11} a^3 b x^{11} (a B+2 A b)+\frac{10}{13} a^2 b^2 x^{13} (a B+A b)+\frac{1}{17} b^4 x^{17} (5 a B+A b)+\frac{1}{3} a b^3 x^{15} (2 a B+A b)+\frac{1}{19} b^5 B x^{19}","\frac{1}{7} a^5 A x^7+\frac{1}{9} a^4 x^9 (a B+5 A b)+\frac{5}{11} a^3 b x^{11} (a B+2 A b)+\frac{10}{13} a^2 b^2 x^{13} (a B+A b)+\frac{1}{17} b^4 x^{17} (5 a B+A b)+\frac{1}{3} a b^3 x^{15} (2 a B+A b)+\frac{1}{19} b^5 B x^{19}",1,"(a^5*A*x^7)/7 + (a^4*(5*A*b + a*B)*x^9)/9 + (5*a^3*b*(2*A*b + a*B)*x^11)/11 + (10*a^2*b^2*(A*b + a*B)*x^13)/13 + (a*b^3*(A*b + 2*a*B)*x^15)/3 + (b^4*(A*b + 5*a*B)*x^17)/17 + (b^5*B*x^19)/19","A",1
27,1,107,95,0.0409688,"\int x^5 \left(a+b x^2\right)^5 \left(A+B x^2\right) \, dx","Integrate[x^5*(a + b*x^2)^5*(A + B*x^2),x]","\frac{x^6 \left(168 a^5 A+126 a^4 x^2 (a B+5 A b)+504 a^3 b x^4 (a B+2 A b)+840 a^2 b^2 x^6 (a B+A b)+63 b^4 x^{10} (5 a B+A b)+360 a b^3 x^8 (2 a B+A b)+56 b^5 B x^{12}\right)}{1008}","\frac{a^2 \left(a+b x^2\right)^6 (A b-a B)}{12 b^4}+\frac{\left(a+b x^2\right)^8 (A b-3 a B)}{16 b^4}-\frac{a \left(a+b x^2\right)^7 (2 A b-3 a B)}{14 b^4}+\frac{B \left(a+b x^2\right)^9}{18 b^4}",1,"(x^6*(168*a^5*A + 126*a^4*(5*A*b + a*B)*x^2 + 504*a^3*b*(2*A*b + a*B)*x^4 + 840*a^2*b^2*(A*b + a*B)*x^6 + 360*a*b^3*(A*b + 2*a*B)*x^8 + 63*b^4*(A*b + 5*a*B)*x^10 + 56*b^5*B*x^12))/1008","A",1
28,1,117,117,0.0140562,"\int x^4 \left(a+b x^2\right)^5 \left(A+B x^2\right) \, dx","Integrate[x^4*(a + b*x^2)^5*(A + B*x^2),x]","\frac{1}{5} a^5 A x^5+\frac{1}{7} a^4 x^7 (a B+5 A b)+\frac{5}{9} a^3 b x^9 (a B+2 A b)+\frac{10}{11} a^2 b^2 x^{11} (a B+A b)+\frac{1}{15} b^4 x^{15} (5 a B+A b)+\frac{5}{13} a b^3 x^{13} (2 a B+A b)+\frac{1}{17} b^5 B x^{17}","\frac{1}{5} a^5 A x^5+\frac{1}{7} a^4 x^7 (a B+5 A b)+\frac{5}{9} a^3 b x^9 (a B+2 A b)+\frac{10}{11} a^2 b^2 x^{11} (a B+A b)+\frac{1}{15} b^4 x^{15} (5 a B+A b)+\frac{5}{13} a b^3 x^{13} (2 a B+A b)+\frac{1}{17} b^5 B x^{17}",1,"(a^5*A*x^5)/5 + (a^4*(5*A*b + a*B)*x^7)/7 + (5*a^3*b*(2*A*b + a*B)*x^9)/9 + (10*a^2*b^2*(A*b + a*B)*x^11)/11 + (5*a*b^3*(A*b + 2*a*B)*x^13)/13 + (b^4*(A*b + 5*a*B)*x^15)/15 + (b^5*B*x^17)/17","A",1
29,1,114,67,0.0332962,"\int x^3 \left(a+b x^2\right)^5 \left(A+B x^2\right) \, dx","Integrate[x^3*(a + b*x^2)^5*(A + B*x^2),x]","\frac{1}{4} a^5 A x^4+\frac{1}{6} a^4 x^6 (a B+5 A b)+\frac{5}{8} a^3 b x^8 (a B+2 A b)+a^2 b^2 x^{10} (a B+A b)+\frac{1}{14} b^4 x^{14} (5 a B+A b)+\frac{5}{12} a b^3 x^{12} (2 a B+A b)+\frac{1}{16} b^5 B x^{16}","\frac{\left(a+b x^2\right)^7 (A b-2 a B)}{14 b^3}-\frac{a \left(a+b x^2\right)^6 (A b-a B)}{12 b^3}+\frac{B \left(a+b x^2\right)^8}{16 b^3}",1,"(a^5*A*x^4)/4 + (a^4*(5*A*b + a*B)*x^6)/6 + (5*a^3*b*(2*A*b + a*B)*x^8)/8 + a^2*b^2*(A*b + a*B)*x^10 + (5*a*b^3*(A*b + 2*a*B)*x^12)/12 + (b^4*(A*b + 5*a*B)*x^14)/14 + (b^5*B*x^16)/16","A",1
30,1,117,117,0.0139911,"\int x^2 \left(a+b x^2\right)^5 \left(A+B x^2\right) \, dx","Integrate[x^2*(a + b*x^2)^5*(A + B*x^2),x]","\frac{1}{3} a^5 A x^3+\frac{1}{5} a^4 x^5 (a B+5 A b)+\frac{5}{7} a^3 b x^7 (a B+2 A b)+\frac{10}{9} a^2 b^2 x^9 (a B+A b)+\frac{1}{13} b^4 x^{13} (5 a B+A b)+\frac{5}{11} a b^3 x^{11} (2 a B+A b)+\frac{1}{15} b^5 B x^{15}","\frac{1}{3} a^5 A x^3+\frac{1}{5} a^4 x^5 (a B+5 A b)+\frac{5}{7} a^3 b x^7 (a B+2 A b)+\frac{10}{9} a^2 b^2 x^9 (a B+A b)+\frac{1}{13} b^4 x^{13} (5 a B+A b)+\frac{5}{11} a b^3 x^{11} (2 a B+A b)+\frac{1}{15} b^5 B x^{15}",1,"(a^5*A*x^3)/3 + (a^4*(5*A*b + a*B)*x^5)/5 + (5*a^3*b*(2*A*b + a*B)*x^7)/7 + (10*a^2*b^2*(A*b + a*B)*x^9)/9 + (5*a*b^3*(A*b + 2*a*B)*x^11)/11 + (b^4*(A*b + 5*a*B)*x^13)/13 + (b^5*B*x^15)/15","A",1
31,1,107,42,0.0388507,"\int x \left(a+b x^2\right)^5 \left(A+B x^2\right) \, dx","Integrate[x*(a + b*x^2)^5*(A + B*x^2),x]","\frac{1}{84} x^2 \left(42 a^5 A+21 a^4 x^2 (a B+5 A b)+70 a^3 b x^4 (a B+2 A b)+105 a^2 b^2 x^6 (a B+A b)+7 b^4 x^{10} (5 a B+A b)+42 a b^3 x^8 (2 a B+A b)+6 b^5 B x^{12}\right)","\frac{\left(a+b x^2\right)^6 (A b-a B)}{12 b^2}+\frac{B \left(a+b x^2\right)^7}{14 b^2}",1,"(x^2*(42*a^5*A + 21*a^4*(5*A*b + a*B)*x^2 + 70*a^3*b*(2*A*b + a*B)*x^4 + 105*a^2*b^2*(A*b + a*B)*x^6 + 42*a*b^3*(A*b + 2*a*B)*x^8 + 7*b^4*(A*b + 5*a*B)*x^10 + 6*b^5*B*x^12))/84","B",1
32,1,109,109,0.0133127,"\int \left(a+b x^2\right)^5 \left(A+B x^2\right) \, dx","Integrate[(a + b*x^2)^5*(A + B*x^2),x]","a^5 A x+\frac{1}{3} a^4 x^3 (a B+5 A b)+a^3 b x^5 (a B+2 A b)+\frac{10}{7} a^2 b^2 x^7 (a B+A b)+\frac{1}{11} b^4 x^{11} (5 a B+A b)+\frac{5}{9} a b^3 x^9 (2 a B+A b)+\frac{1}{13} b^5 B x^{13}","a^5 A x+\frac{1}{3} a^4 x^3 (a B+5 A b)+a^3 b x^5 (a B+2 A b)+\frac{10}{7} a^2 b^2 x^7 (a B+A b)+\frac{1}{11} b^4 x^{11} (5 a B+A b)+\frac{5}{9} a b^3 x^9 (2 a B+A b)+\frac{1}{13} b^5 B x^{13}",1,"a^5*A*x + (a^4*(5*A*b + a*B)*x^3)/3 + a^3*b*(2*A*b + a*B)*x^5 + (10*a^2*b^2*(A*b + a*B)*x^7)/7 + (5*a*b^3*(A*b + 2*a*B)*x^9)/9 + (b^4*(A*b + 5*a*B)*x^11)/11 + (b^5*B*x^13)/13","A",1
33,1,113,88,0.0440601,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x} \, dx","Integrate[((a + b*x^2)^5*(A + B*x^2))/x,x]","a^5 A \log (x)+\frac{1}{2} a^4 x^2 (a B+5 A b)+\frac{5}{4} a^3 b x^4 (a B+2 A b)+\frac{5}{3} a^2 b^2 x^6 (a B+A b)+\frac{1}{10} b^4 x^{10} (5 a B+A b)+\frac{5}{8} a b^3 x^8 (2 a B+A b)+\frac{1}{12} b^5 B x^{12}","a^5 A \log (x)+\frac{5}{2} a^4 A b x^2+\frac{5}{2} a^3 A b^2 x^4+\frac{5}{3} a^2 A b^3 x^6+\frac{5}{8} a A b^4 x^8+\frac{B \left(a+b x^2\right)^6}{12 b}+\frac{1}{10} A b^5 x^{10}",1,"(a^4*(5*A*b + a*B)*x^2)/2 + (5*a^3*b*(2*A*b + a*B)*x^4)/4 + (5*a^2*b^2*(A*b + a*B)*x^6)/3 + (5*a*b^3*(A*b + 2*a*B)*x^8)/8 + (b^4*(A*b + 5*a*B)*x^10)/10 + (b^5*B*x^12)/12 + a^5*A*Log[x]","A",1
34,1,108,108,0.0270567,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^2} \, dx","Integrate[((a + b*x^2)^5*(A + B*x^2))/x^2,x]","-\frac{a^5 A}{x}+a^4 x (a B+5 A b)+\frac{5}{3} a^3 b x^3 (a B+2 A b)+2 a^2 b^2 x^5 (a B+A b)+\frac{1}{9} b^4 x^9 (5 a B+A b)+\frac{5}{7} a b^3 x^7 (2 a B+A b)+\frac{1}{11} b^5 B x^{11}","-\frac{a^5 A}{x}+a^4 x (a B+5 A b)+\frac{5}{3} a^3 b x^3 (a B+2 A b)+2 a^2 b^2 x^5 (a B+A b)+\frac{1}{9} b^4 x^9 (5 a B+A b)+\frac{5}{7} a b^3 x^7 (2 a B+A b)+\frac{1}{11} b^5 B x^{11}",1,"-((a^5*A)/x) + a^4*(5*A*b + a*B)*x + (5*a^3*b*(2*A*b + a*B)*x^3)/3 + 2*a^2*b^2*(A*b + a*B)*x^5 + (5*a*b^3*(A*b + 2*a*B)*x^7)/7 + (b^4*(A*b + 5*a*B)*x^9)/9 + (b^5*B*x^11)/11","A",1
35,1,115,113,0.0508889,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^3} \, dx","Integrate[((a + b*x^2)^5*(A + B*x^2))/x^3,x]","-\frac{a^5 A}{2 x^2}+\frac{5}{2} a^3 b x^2 (a B+2 A b)+\frac{5}{2} a^2 b^2 x^4 (a B+A b)+\log (x) \left(a^5 B+5 a^4 A b\right)+\frac{1}{8} b^4 x^8 (5 a B+A b)+\frac{5}{6} a b^3 x^6 (2 a B+A b)+\frac{1}{10} b^5 B x^{10}","-\frac{a^5 A}{2 x^2}+a^4 \log (x) (a B+5 A b)+\frac{5}{2} a^3 b x^2 (a B+2 A b)+\frac{5}{2} a^2 b^2 x^4 (a B+A b)+\frac{1}{8} b^4 x^8 (5 a B+A b)+\frac{5}{6} a b^3 x^6 (2 a B+A b)+\frac{1}{10} b^5 B x^{10}",1,"-1/2*(a^5*A)/x^2 + (5*a^3*b*(2*A*b + a*B)*x^2)/2 + (5*a^2*b^2*(A*b + a*B)*x^4)/2 + (5*a*b^3*(A*b + 2*a*B)*x^6)/6 + (b^4*(A*b + 5*a*B)*x^8)/8 + (b^5*B*x^10)/10 + (5*a^4*A*b + a^5*B)*Log[x]","A",1
36,1,110,108,0.044387,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^4} \, dx","Integrate[((a + b*x^2)^5*(A + B*x^2))/x^4,x]","-\frac{a^5 A}{3 x^3}+5 a^3 b x (a B+2 A b)+\frac{10}{3} a^2 b^2 x^3 (a B+A b)+\frac{a^5 (-B)-5 a^4 A b}{x}+\frac{1}{7} b^4 x^7 (5 a B+A b)+a b^3 x^5 (2 a B+A b)+\frac{1}{9} b^5 B x^9","-\frac{a^5 A}{3 x^3}-\frac{a^4 (a B+5 A b)}{x}+5 a^3 b x (a B+2 A b)+\frac{10}{3} a^2 b^2 x^3 (a B+A b)+\frac{1}{7} b^4 x^7 (5 a B+A b)+a b^3 x^5 (2 a B+A b)+\frac{1}{9} b^5 B x^9",1,"-1/3*(a^5*A)/x^3 + (-5*a^4*A*b - a^5*B)/x + 5*a^3*b*(2*A*b + a*B)*x + (10*a^2*b^2*(A*b + a*B)*x^3)/3 + a*b^3*(A*b + 2*a*B)*x^5 + (b^4*(A*b + 5*a*B)*x^7)/7 + (b^5*B*x^9)/9","A",1
37,1,112,112,0.0345632,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^5} \, dx","Integrate[((a + b*x^2)^5*(A + B*x^2))/x^5,x]","-\frac{a^5 A}{4 x^4}-\frac{a^4 (a B+5 A b)}{2 x^2}+5 a^3 b \log (x) (a B+2 A b)+5 a^2 b^2 x^2 (a B+A b)+\frac{1}{6} b^4 x^6 (5 a B+A b)+\frac{5}{4} a b^3 x^4 (2 a B+A b)+\frac{1}{8} b^5 B x^8","-\frac{a^5 A}{4 x^4}-\frac{a^4 (a B+5 A b)}{2 x^2}+5 a^3 b \log (x) (a B+2 A b)+5 a^2 b^2 x^2 (a B+A b)+\frac{1}{6} b^4 x^6 (5 a B+A b)+\frac{5}{4} a b^3 x^4 (2 a B+A b)+\frac{1}{8} b^5 B x^8",1,"-1/4*(a^5*A)/x^4 - (a^4*(5*A*b + a*B))/(2*x^2) + 5*a^2*b^2*(A*b + a*B)*x^2 + (5*a*b^3*(A*b + 2*a*B)*x^4)/4 + (b^4*(A*b + 5*a*B)*x^6)/6 + (b^5*B*x^8)/8 + 5*a^3*b*(2*A*b + a*B)*Log[x]","A",1
38,1,111,111,0.0336795,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^6} \, dx","Integrate[((a + b*x^2)^5*(A + B*x^2))/x^6,x]","-\frac{a^5 A}{5 x^5}-\frac{a^4 (a B+5 A b)}{3 x^3}-\frac{5 a^3 b (a B+2 A b)}{x}+10 a^2 b^2 x (a B+A b)+\frac{1}{5} b^4 x^5 (5 a B+A b)+\frac{5}{3} a b^3 x^3 (2 a B+A b)+\frac{1}{7} b^5 B x^7","-\frac{a^5 A}{5 x^5}-\frac{a^4 (a B+5 A b)}{3 x^3}-\frac{5 a^3 b (a B+2 A b)}{x}+10 a^2 b^2 x (a B+A b)+\frac{1}{5} b^4 x^5 (5 a B+A b)+\frac{5}{3} a b^3 x^3 (2 a B+A b)+\frac{1}{7} b^5 B x^7",1,"-1/5*(a^5*A)/x^5 - (a^4*(5*A*b + a*B))/(3*x^3) - (5*a^3*b*(2*A*b + a*B))/x + 10*a^2*b^2*(A*b + a*B)*x + (5*a*b^3*(A*b + 2*a*B)*x^3)/3 + (b^4*(A*b + 5*a*B)*x^5)/5 + (b^5*B*x^7)/7","A",1
39,1,116,114,0.0359581,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^7} \, dx","Integrate[((a + b*x^2)^5*(A + B*x^2))/x^7,x]","\frac{1}{12} \left(-\frac{a^5 \left(2 A+3 B x^2\right)}{x^6}-\frac{15 a^4 b \left(A+2 B x^2\right)}{x^4}-\frac{60 a^3 A b^2}{x^2}+120 a^2 b^2 \log (x) (a B+A b)+60 a^2 b^3 B x^2+15 a b^4 x^2 \left(2 A+B x^2\right)+b^5 x^4 \left(3 A+2 B x^2\right)\right)","-\frac{a^5 A}{6 x^6}-\frac{a^4 (a B+5 A b)}{4 x^4}-\frac{5 a^3 b (a B+2 A b)}{2 x^2}+10 a^2 b^2 \log (x) (a B+A b)+\frac{1}{4} b^4 x^4 (5 a B+A b)+\frac{5}{2} a b^3 x^2 (2 a B+A b)+\frac{1}{6} b^5 B x^6",1,"((-60*a^3*A*b^2)/x^2 + 60*a^2*b^3*B*x^2 + 15*a*b^4*x^2*(2*A + B*x^2) - (15*a^4*b*(A + 2*B*x^2))/x^4 + b^5*x^4*(3*A + 2*B*x^2) - (a^5*(2*A + 3*B*x^2))/x^6 + 120*a^2*b^2*(A*b + a*B)*Log[x])/12","A",1
40,1,111,111,0.0325981,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^8} \, dx","Integrate[((a + b*x^2)^5*(A + B*x^2))/x^8,x]","-\frac{a^5 A}{7 x^7}-\frac{a^4 (a B+5 A b)}{5 x^5}-\frac{5 a^3 b (a B+2 A b)}{3 x^3}-\frac{10 a^2 b^2 (a B+A b)}{x}+\frac{1}{3} b^4 x^3 (5 a B+A b)+5 a b^3 x (2 a B+A b)+\frac{1}{5} b^5 B x^5","-\frac{a^5 A}{7 x^7}-\frac{a^4 (a B+5 A b)}{5 x^5}-\frac{5 a^3 b (a B+2 A b)}{3 x^3}-\frac{10 a^2 b^2 (a B+A b)}{x}+\frac{1}{3} b^4 x^3 (5 a B+A b)+5 a b^3 x (2 a B+A b)+\frac{1}{5} b^5 B x^5",1,"-1/7*(a^5*A)/x^7 - (a^4*(5*A*b + a*B))/(5*x^5) - (5*a^3*b*(2*A*b + a*B))/(3*x^3) - (10*a^2*b^2*(A*b + a*B))/x + 5*a*b^3*(A*b + 2*a*B)*x + (b^4*(A*b + 5*a*B)*x^3)/3 + (b^5*B*x^5)/5","A",1
41,1,116,112,0.0544308,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^9} \, dx","Integrate[((a + b*x^2)^5*(A + B*x^2))/x^9,x]","5 a b^3 \log (x) (2 a B+A b)-\frac{a^5 \left(3 A+4 B x^2\right)+10 a^4 b x^2 \left(2 A+3 B x^2\right)+60 a^3 b^2 x^4 \left(A+2 B x^2\right)+120 a^2 A b^3 x^6-60 a b^4 B x^{10}-6 b^5 x^{10} \left(2 A+B x^2\right)}{24 x^8}","-\frac{a^5 A}{8 x^8}-\frac{a^4 (a B+5 A b)}{6 x^6}-\frac{5 a^3 b (a B+2 A b)}{4 x^4}-\frac{5 a^2 b^2 (a B+A b)}{x^2}+\frac{1}{2} b^4 x^2 (5 a B+A b)+5 a b^3 \log (x) (2 a B+A b)+\frac{1}{4} b^5 B x^4",1,"-1/24*(120*a^2*A*b^3*x^6 - 60*a*b^4*B*x^10 - 6*b^5*x^10*(2*A + B*x^2) + 60*a^3*b^2*x^4*(A + 2*B*x^2) + 10*a^4*b*x^2*(2*A + 3*B*x^2) + a^5*(3*A + 4*B*x^2))/x^8 + 5*a*b^3*(A*b + 2*a*B)*Log[x]","A",1
42,1,115,108,0.0293212,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^{10}} \, dx","Integrate[((a + b*x^2)^5*(A + B*x^2))/x^10,x]","-\frac{a^5 \left(7 A+9 B x^2\right)+9 a^4 b x^2 \left(5 A+7 B x^2\right)+42 a^3 b^2 x^4 \left(3 A+5 B x^2\right)+210 a^2 b^3 x^6 \left(A+3 B x^2\right)+315 a b^4 x^8 \left(A-B x^2\right)-21 b^5 x^{10} \left(3 A+B x^2\right)}{63 x^9}","-\frac{a^5 A}{9 x^9}-\frac{a^4 (a B+5 A b)}{7 x^7}-\frac{a^3 b (a B+2 A b)}{x^5}-\frac{10 a^2 b^2 (a B+A b)}{3 x^3}+b^4 x (5 a B+A b)-\frac{5 a b^3 (2 a B+A b)}{x}+\frac{1}{3} b^5 B x^3",1,"-1/63*(315*a*b^4*x^8*(A - B*x^2) - 21*b^5*x^10*(3*A + B*x^2) + 210*a^2*b^3*x^6*(A + 3*B*x^2) + 42*a^3*b^2*x^4*(3*A + 5*B*x^2) + 9*a^4*b*x^2*(5*A + 7*B*x^2) + a^5*(7*A + 9*B*x^2))/x^9","A",1
43,1,116,113,0.0528964,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^{11}} \, dx","Integrate[((a + b*x^2)^5*(A + B*x^2))/x^11,x]","b^4 \log (x) (5 a B+A b)-\frac{3 a^5 \left(4 A+5 B x^2\right)+25 a^4 b x^2 \left(3 A+4 B x^2\right)+100 a^3 b^2 x^4 \left(2 A+3 B x^2\right)+300 a^2 b^3 x^6 \left(A+2 B x^2\right)+300 a A b^4 x^8-60 b^5 B x^{12}}{120 x^{10}}","-\frac{a^5 A}{10 x^{10}}-\frac{a^4 (a B+5 A b)}{8 x^8}-\frac{5 a^3 b (a B+2 A b)}{6 x^6}-\frac{5 a^2 b^2 (a B+A b)}{2 x^4}+b^4 \log (x) (5 a B+A b)-\frac{5 a b^3 (2 a B+A b)}{2 x^2}+\frac{1}{2} b^5 B x^2",1,"-1/120*(300*a*A*b^4*x^8 - 60*b^5*B*x^12 + 300*a^2*b^3*x^6*(A + 2*B*x^2) + 100*a^3*b^2*x^4*(2*A + 3*B*x^2) + 25*a^4*b*x^2*(3*A + 4*B*x^2) + 3*a^5*(4*A + 5*B*x^2))/x^10 + b^4*(A*b + 5*a*B)*Log[x]","A",1
44,1,122,108,0.0386219,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^{12}} \, dx","Integrate[((a + b*x^2)^5*(A + B*x^2))/x^12,x]","-\frac{a^5 \left(9 A+11 B x^2\right)}{99 x^{11}}-\frac{5 a^4 b \left(7 A+9 B x^2\right)}{63 x^9}-\frac{2 a^3 b^2 \left(5 A+7 B x^2\right)}{7 x^7}-\frac{2 a^2 b^3 \left(3 A+5 B x^2\right)}{3 x^5}-\frac{5 a b^4 \left(A+3 B x^2\right)}{3 x^3}-\frac{A b^5}{x}+b^5 B x","-\frac{a^5 A}{11 x^{11}}-\frac{a^4 (a B+5 A b)}{9 x^9}-\frac{5 a^3 b (a B+2 A b)}{7 x^7}-\frac{2 a^2 b^2 (a B+A b)}{x^5}-\frac{b^4 (5 a B+A b)}{x}-\frac{5 a b^3 (2 a B+A b)}{3 x^3}+b^5 B x",1,"-((A*b^5)/x) + b^5*B*x - (5*a*b^4*(A + 3*B*x^2))/(3*x^3) - (2*a^2*b^3*(3*A + 5*B*x^2))/(3*x^5) - (2*a^3*b^2*(5*A + 7*B*x^2))/(7*x^7) - (5*a^4*b*(7*A + 9*B*x^2))/(63*x^9) - (a^5*(9*A + 11*B*x^2))/(99*x^11)","A",1
45,1,118,91,0.0534188,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^{13}} \, dx","Integrate[((a + b*x^2)^5*(A + B*x^2))/x^13,x]","b^5 B \log (x)-\frac{2 a^5 \left(5 A+6 B x^2\right)+15 a^4 b x^2 \left(4 A+5 B x^2\right)+50 a^3 b^2 x^4 \left(3 A+4 B x^2\right)+100 a^2 b^3 x^6 \left(2 A+3 B x^2\right)+150 a b^4 x^8 \left(A+2 B x^2\right)+60 A b^5 x^{10}}{120 x^{12}}","-\frac{a^5 B}{10 x^{10}}-\frac{5 a^4 b B}{8 x^8}-\frac{5 a^3 b^2 B}{3 x^6}-\frac{5 a^2 b^3 B}{2 x^4}-\frac{A \left(a+b x^2\right)^6}{12 a x^{12}}-\frac{5 a b^4 B}{2 x^2}+b^5 B \log (x)",1,"-1/120*(60*A*b^5*x^10 + 150*a*b^4*x^8*(A + 2*B*x^2) + 100*a^2*b^3*x^6*(2*A + 3*B*x^2) + 50*a^3*b^2*x^4*(3*A + 4*B*x^2) + 15*a^4*b*x^2*(4*A + 5*B*x^2) + 2*a^5*(5*A + 6*B*x^2))/x^12 + b^5*B*Log[x]","A",1
46,1,119,113,0.0294917,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^{14}} \, dx","Integrate[((a + b*x^2)^5*(A + B*x^2))/x^14,x]","-\frac{63 a^5 \left(11 A+13 B x^2\right)+455 a^4 b x^2 \left(9 A+11 B x^2\right)+1430 a^3 b^2 x^4 \left(7 A+9 B x^2\right)+2574 a^2 b^3 x^6 \left(5 A+7 B x^2\right)+3003 a b^4 x^8 \left(3 A+5 B x^2\right)+3003 b^5 x^{10} \left(A+3 B x^2\right)}{9009 x^{13}}","-\frac{a^5 A}{13 x^{13}}-\frac{a^4 (a B+5 A b)}{11 x^{11}}-\frac{5 a^3 b (a B+2 A b)}{9 x^9}-\frac{10 a^2 b^2 (a B+A b)}{7 x^7}-\frac{b^4 (5 a B+A b)}{3 x^3}-\frac{a b^3 (2 a B+A b)}{x^5}-\frac{b^5 B}{x}",1,"-1/9009*(3003*b^5*x^10*(A + 3*B*x^2) + 3003*a*b^4*x^8*(3*A + 5*B*x^2) + 2574*a^2*b^3*x^6*(5*A + 7*B*x^2) + 1430*a^3*b^2*x^4*(7*A + 9*B*x^2) + 455*a^4*b*x^2*(9*A + 11*B*x^2) + 63*a^5*(11*A + 13*B*x^2))/x^13","A",1
47,1,118,48,0.0277309,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^{15}} \, dx","Integrate[((a + b*x^2)^5*(A + B*x^2))/x^15,x]","-\frac{a^5 \left(6 A+7 B x^2\right)+7 a^4 b x^2 \left(5 A+6 B x^2\right)+21 a^3 b^2 x^4 \left(4 A+5 B x^2\right)+35 a^2 b^3 x^6 \left(3 A+4 B x^2\right)+35 a b^4 x^8 \left(2 A+3 B x^2\right)+21 b^5 x^{10} \left(A+2 B x^2\right)}{84 x^{14}}","\frac{\left(a+b x^2\right)^6 (A b-7 a B)}{84 a^2 x^{12}}-\frac{A \left(a+b x^2\right)^6}{14 a x^{14}}",1,"-1/84*(21*b^5*x^10*(A + 2*B*x^2) + 35*a*b^4*x^8*(2*A + 3*B*x^2) + 35*a^2*b^3*x^6*(3*A + 4*B*x^2) + 21*a^3*b^2*x^4*(4*A + 5*B*x^2) + 7*a^4*b*x^2*(5*A + 6*B*x^2) + a^5*(6*A + 7*B*x^2))/x^14","B",1
48,1,121,117,0.0293622,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^{16}} \, dx","Integrate[((a + b*x^2)^5*(A + B*x^2))/x^16,x]","-\frac{231 a^5 \left(13 A+15 B x^2\right)+1575 a^4 b x^2 \left(11 A+13 B x^2\right)+4550 a^3 b^2 x^4 \left(9 A+11 B x^2\right)+7150 a^2 b^3 x^6 \left(7 A+9 B x^2\right)+6435 a b^4 x^8 \left(5 A+7 B x^2\right)+3003 b^5 x^{10} \left(3 A+5 B x^2\right)}{45045 x^{15}}","-\frac{a^5 A}{15 x^{15}}-\frac{a^4 (a B+5 A b)}{13 x^{13}}-\frac{5 a^3 b (a B+2 A b)}{11 x^{11}}-\frac{10 a^2 b^2 (a B+A b)}{9 x^9}-\frac{b^4 (5 a B+A b)}{5 x^5}-\frac{5 a b^3 (2 a B+A b)}{7 x^7}-\frac{b^5 B}{3 x^3}",1,"-1/45045*(3003*b^5*x^10*(3*A + 5*B*x^2) + 6435*a*b^4*x^8*(5*A + 7*B*x^2) + 7150*a^2*b^3*x^6*(7*A + 9*B*x^2) + 4550*a^3*b^2*x^4*(9*A + 11*B*x^2) + 1575*a^4*b*x^2*(11*A + 13*B*x^2) + 231*a^5*(13*A + 15*B*x^2))/x^15","A",1
49,1,121,76,0.0284431,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^{17}} \, dx","Integrate[((a + b*x^2)^5*(A + B*x^2))/x^17,x]","-\frac{3 a^5 \left(7 A+8 B x^2\right)+20 a^4 b x^2 \left(6 A+7 B x^2\right)+56 a^3 b^2 x^4 \left(5 A+6 B x^2\right)+84 a^2 b^3 x^6 \left(4 A+5 B x^2\right)+70 a b^4 x^8 \left(3 A+4 B x^2\right)+28 b^5 x^{10} \left(2 A+3 B x^2\right)}{336 x^{16}}","-\frac{b \left(a+b x^2\right)^6 (A b-4 a B)}{336 a^3 x^{12}}+\frac{\left(a+b x^2\right)^6 (A b-4 a B)}{56 a^2 x^{14}}-\frac{A \left(a+b x^2\right)^6}{16 a x^{16}}",1,"-1/336*(28*b^5*x^10*(2*A + 3*B*x^2) + 70*a*b^4*x^8*(3*A + 4*B*x^2) + 84*a^2*b^3*x^6*(4*A + 5*B*x^2) + 56*a^3*b^2*x^4*(5*A + 6*B*x^2) + 20*a^4*b*x^2*(6*A + 7*B*x^2) + 3*a^5*(7*A + 8*B*x^2))/x^16","A",1
50,1,117,117,0.0396011,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^{18}} \, dx","Integrate[((a + b*x^2)^5*(A + B*x^2))/x^18,x]","-\frac{a^5 A}{17 x^{17}}-\frac{a^4 (a B+5 A b)}{15 x^{15}}-\frac{5 a^3 b (a B+2 A b)}{13 x^{13}}-\frac{10 a^2 b^2 (a B+A b)}{11 x^{11}}-\frac{b^4 (5 a B+A b)}{7 x^7}-\frac{5 a b^3 (2 a B+A b)}{9 x^9}-\frac{b^5 B}{5 x^5}","-\frac{a^5 A}{17 x^{17}}-\frac{a^4 (a B+5 A b)}{15 x^{15}}-\frac{5 a^3 b (a B+2 A b)}{13 x^{13}}-\frac{10 a^2 b^2 (a B+A b)}{11 x^{11}}-\frac{b^4 (5 a B+A b)}{7 x^7}-\frac{5 a b^3 (2 a B+A b)}{9 x^9}-\frac{b^5 B}{5 x^5}",1,"-1/17*(a^5*A)/x^17 - (a^4*(5*A*b + a*B))/(15*x^15) - (5*a^3*b*(2*A*b + a*B))/(13*x^13) - (10*a^2*b^2*(A*b + a*B))/(11*x^11) - (5*a*b^3*(A*b + 2*a*B))/(9*x^9) - (b^4*(A*b + 5*a*B))/(7*x^7) - (b^5*B)/(5*x^5)","A",1
51,1,121,117,0.0265097,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^{19}} \, dx","Integrate[((a + b*x^2)^5*(A + B*x^2))/x^19,x]","-\frac{7 a^5 \left(8 A+9 B x^2\right)+45 a^4 b x^2 \left(7 A+8 B x^2\right)+120 a^3 b^2 x^4 \left(6 A+7 B x^2\right)+168 a^2 b^3 x^6 \left(5 A+6 B x^2\right)+126 a b^4 x^8 \left(4 A+5 B x^2\right)+42 b^5 x^{10} \left(3 A+4 B x^2\right)}{1008 x^{18}}","-\frac{a^5 A}{18 x^{18}}-\frac{a^4 (a B+5 A b)}{16 x^{16}}-\frac{5 a^3 b (a B+2 A b)}{14 x^{14}}-\frac{5 a^2 b^2 (a B+A b)}{6 x^{12}}-\frac{b^4 (5 a B+A b)}{8 x^8}-\frac{a b^3 (2 a B+A b)}{2 x^{10}}-\frac{b^5 B}{6 x^6}",1,"-1/1008*(42*b^5*x^10*(3*A + 4*B*x^2) + 126*a*b^4*x^8*(4*A + 5*B*x^2) + 168*a^2*b^3*x^6*(5*A + 6*B*x^2) + 120*a^3*b^2*x^4*(6*A + 7*B*x^2) + 45*a^4*b*x^2*(7*A + 8*B*x^2) + 7*a^5*(8*A + 9*B*x^2))/x^18","A",1
52,1,117,117,0.0387371,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^{20}} \, dx","Integrate[((a + b*x^2)^5*(A + B*x^2))/x^20,x]","-\frac{a^5 A}{19 x^{19}}-\frac{a^4 (a B+5 A b)}{17 x^{17}}-\frac{a^3 b (a B+2 A b)}{3 x^{15}}-\frac{10 a^2 b^2 (a B+A b)}{13 x^{13}}-\frac{b^4 (5 a B+A b)}{9 x^9}-\frac{5 a b^3 (2 a B+A b)}{11 x^{11}}-\frac{b^5 B}{7 x^7}","-\frac{a^5 A}{19 x^{19}}-\frac{a^4 (a B+5 A b)}{17 x^{17}}-\frac{a^3 b (a B+2 A b)}{3 x^{15}}-\frac{10 a^2 b^2 (a B+A b)}{13 x^{13}}-\frac{b^4 (5 a B+A b)}{9 x^9}-\frac{5 a b^3 (2 a B+A b)}{11 x^{11}}-\frac{b^5 B}{7 x^7}",1,"-1/19*(a^5*A)/x^19 - (a^4*(5*A*b + a*B))/(17*x^17) - (a^3*b*(2*A*b + a*B))/(3*x^15) - (10*a^2*b^2*(A*b + a*B))/(13*x^13) - (5*a*b^3*(A*b + 2*a*B))/(11*x^11) - (b^4*(A*b + 5*a*B))/(9*x^9) - (b^5*B)/(7*x^7)","A",1
53,1,121,117,0.028888,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^{21}} \, dx","Integrate[((a + b*x^2)^5*(A + B*x^2))/x^21,x]","-\frac{28 a^5 \left(9 A+10 B x^2\right)+175 a^4 b x^2 \left(8 A+9 B x^2\right)+450 a^3 b^2 x^4 \left(7 A+8 B x^2\right)+600 a^2 b^3 x^6 \left(6 A+7 B x^2\right)+420 a b^4 x^8 \left(5 A+6 B x^2\right)+126 b^5 x^{10} \left(4 A+5 B x^2\right)}{5040 x^{20}}","-\frac{a^5 A}{20 x^{20}}-\frac{a^4 (a B+5 A b)}{18 x^{18}}-\frac{5 a^3 b (a B+2 A b)}{16 x^{16}}-\frac{5 a^2 b^2 (a B+A b)}{7 x^{14}}-\frac{b^4 (5 a B+A b)}{10 x^{10}}-\frac{5 a b^3 (2 a B+A b)}{12 x^{12}}-\frac{b^5 B}{8 x^8}",1,"-1/5040*(126*b^5*x^10*(4*A + 5*B*x^2) + 420*a*b^4*x^8*(5*A + 6*B*x^2) + 600*a^2*b^3*x^6*(6*A + 7*B*x^2) + 450*a^3*b^2*x^4*(7*A + 8*B*x^2) + 175*a^4*b*x^2*(8*A + 9*B*x^2) + 28*a^5*(9*A + 10*B*x^2))/x^20","A",1
54,1,117,117,0.0388844,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^{22}} \, dx","Integrate[((a + b*x^2)^5*(A + B*x^2))/x^22,x]","-\frac{a^5 A}{21 x^{21}}-\frac{a^4 (a B+5 A b)}{19 x^{19}}-\frac{5 a^3 b (a B+2 A b)}{17 x^{17}}-\frac{2 a^2 b^2 (a B+A b)}{3 x^{15}}-\frac{b^4 (5 a B+A b)}{11 x^{11}}-\frac{5 a b^3 (2 a B+A b)}{13 x^{13}}-\frac{b^5 B}{9 x^9}","-\frac{a^5 A}{21 x^{21}}-\frac{a^4 (a B+5 A b)}{19 x^{19}}-\frac{5 a^3 b (a B+2 A b)}{17 x^{17}}-\frac{2 a^2 b^2 (a B+A b)}{3 x^{15}}-\frac{b^4 (5 a B+A b)}{11 x^{11}}-\frac{5 a b^3 (2 a B+A b)}{13 x^{13}}-\frac{b^5 B}{9 x^9}",1,"-1/21*(a^5*A)/x^21 - (a^4*(5*A*b + a*B))/(19*x^19) - (5*a^3*b*(2*A*b + a*B))/(17*x^17) - (2*a^2*b^2*(A*b + a*B))/(3*x^15) - (5*a*b^3*(A*b + 2*a*B))/(13*x^13) - (b^4*(A*b + 5*a*B))/(11*x^11) - (b^5*B)/(9*x^9)","A",1
55,1,121,117,0.0297456,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^{23}} \, dx","Integrate[((a + b*x^2)^5*(A + B*x^2))/x^23,x]","-\frac{126 a^5 \left(10 A+11 B x^2\right)+770 a^4 b x^2 \left(9 A+10 B x^2\right)+1925 a^3 b^2 x^4 \left(8 A+9 B x^2\right)+2475 a^2 b^3 x^6 \left(7 A+8 B x^2\right)+1650 a b^4 x^8 \left(6 A+7 B x^2\right)+462 b^5 x^{10} \left(5 A+6 B x^2\right)}{27720 x^{22}}","-\frac{a^5 A}{22 x^{22}}-\frac{a^4 (a B+5 A b)}{20 x^{20}}-\frac{5 a^3 b (a B+2 A b)}{18 x^{18}}-\frac{5 a^2 b^2 (a B+A b)}{8 x^{16}}-\frac{b^4 (5 a B+A b)}{12 x^{12}}-\frac{5 a b^3 (2 a B+A b)}{14 x^{14}}-\frac{b^5 B}{10 x^{10}}",1,"-1/27720*(462*b^5*x^10*(5*A + 6*B*x^2) + 1650*a*b^4*x^8*(6*A + 7*B*x^2) + 2475*a^2*b^3*x^6*(7*A + 8*B*x^2) + 1925*a^3*b^2*x^4*(8*A + 9*B*x^2) + 770*a^4*b*x^2*(9*A + 10*B*x^2) + 126*a^5*(10*A + 11*B*x^2))/x^22","A",1
56,1,98,98,0.0797481,"\int \frac{x^6 \left(A+B x^2\right)}{a+b x^2} \, dx","Integrate[(x^6*(A + B*x^2))/(a + b*x^2),x]","\frac{a^{5/2} (a B-A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{9/2}}-\frac{a^2 x (a B-A b)}{b^4}+\frac{a x^3 (a B-A b)}{3 b^3}+\frac{x^5 (A b-a B)}{5 b^2}+\frac{B x^7}{7 b}","-\frac{a^{5/2} (A b-a B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{9/2}}+\frac{a^2 x (A b-a B)}{b^4}-\frac{a x^3 (A b-a B)}{3 b^3}+\frac{x^5 (A b-a B)}{5 b^2}+\frac{B x^7}{7 b}",1,"-((a^2*(-(A*b) + a*B)*x)/b^4) + (a*(-(A*b) + a*B)*x^3)/(3*b^3) + ((A*b - a*B)*x^5)/(5*b^2) + (B*x^7)/(7*b) + (a^(5/2)*(-(A*b) + a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(9/2)","A",1
57,1,71,75,0.0308208,"\int \frac{x^5 \left(A+B x^2\right)}{a+b x^2} \, dx","Integrate[(x^5*(A + B*x^2))/(a + b*x^2),x]","\frac{b x^2 \left(6 a^2 B-3 a b \left(2 A+B x^2\right)+b^2 x^2 \left(3 A+2 B x^2\right)\right)+6 a^2 (A b-a B) \log \left(a+b x^2\right)}{12 b^4}","\frac{a^2 (A b-a B) \log \left(a+b x^2\right)}{2 b^4}-\frac{a x^2 (A b-a B)}{2 b^3}+\frac{x^4 (A b-a B)}{4 b^2}+\frac{B x^6}{6 b}",1,"(b*x^2*(6*a^2*B - 3*a*b*(2*A + B*x^2) + b^2*x^2*(3*A + 2*B*x^2)) + 6*a^2*(A*b - a*B)*Log[a + b*x^2])/(12*b^4)","A",1
58,1,77,77,0.0509675,"\int \frac{x^4 \left(A+B x^2\right)}{a+b x^2} \, dx","Integrate[(x^4*(A + B*x^2))/(a + b*x^2),x]","-\frac{a^{3/2} (a B-A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{7/2}}+\frac{a x (a B-A b)}{b^3}+\frac{x^3 (A b-a B)}{3 b^2}+\frac{B x^5}{5 b}","\frac{a^{3/2} (A b-a B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{7/2}}-\frac{a x (A b-a B)}{b^3}+\frac{x^3 (A b-a B)}{3 b^2}+\frac{B x^5}{5 b}",1,"(a*(-(A*b) + a*B)*x)/b^3 + ((A*b - a*B)*x^3)/(3*b^2) + (B*x^5)/(5*b) - (a^(3/2)*(-(A*b) + a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(7/2)","A",1
59,1,47,54,0.0204729,"\int \frac{x^3 \left(A+B x^2\right)}{a+b x^2} \, dx","Integrate[(x^3*(A + B*x^2))/(a + b*x^2),x]","\frac{b x^2 \left(-2 a B+2 A b+b B x^2\right)+2 a (a B-A b) \log \left(a+b x^2\right)}{4 b^3}","-\frac{a (A b-a B) \log \left(a+b x^2\right)}{2 b^3}+\frac{x^2 (A b-a B)}{2 b^2}+\frac{B x^4}{4 b}",1,"(b*x^2*(2*A*b - 2*a*B + b*B*x^2) + 2*a*(-(A*b) + a*B)*Log[a + b*x^2])/(4*b^3)","A",1
60,1,57,58,0.0373411,"\int \frac{x^2 \left(A+B x^2\right)}{a+b x^2} \, dx","Integrate[(x^2*(A + B*x^2))/(a + b*x^2),x]","\frac{\sqrt{a} (a B-A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{5/2}}+\frac{x (A b-a B)}{b^2}+\frac{B x^3}{3 b}","-\frac{\sqrt{a} (A b-a B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{5/2}}+\frac{x (A b-a B)}{b^2}+\frac{B x^3}{3 b}",1,"((A*b - a*B)*x)/b^2 + (B*x^3)/(3*b) + (Sqrt[a]*(-(A*b) + a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(5/2)","A",1
61,1,31,35,0.0112271,"\int \frac{x \left(A+B x^2\right)}{a+b x^2} \, dx","Integrate[(x*(A + B*x^2))/(a + b*x^2),x]","\frac{(A b-a B) \log \left(a+b x^2\right)+b B x^2}{2 b^2}","\frac{(A b-a B) \log \left(a+b x^2\right)}{2 b^2}+\frac{B x^2}{2 b}",1,"(b*B*x^2 + (A*b - a*B)*Log[a + b*x^2])/(2*b^2)","A",1
62,1,40,39,0.0242105,"\int \frac{A+B x^2}{a+b x^2} \, dx","Integrate[(A + B*x^2)/(a + b*x^2),x]","\frac{B x}{b}-\frac{(a B-A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} b^{3/2}}","\frac{(A b-a B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} b^{3/2}}+\frac{B x}{b}",1,"(B*x)/b - ((-(A*b) + a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*b^(3/2))","A",1
63,1,34,34,0.0111428,"\int \frac{A+B x^2}{x \left(a+b x^2\right)} \, dx","Integrate[(A + B*x^2)/(x*(a + b*x^2)),x]","\frac{(a B-A b) \log \left(a+b x^2\right)}{2 a b}+\frac{A \log (x)}{a}","\frac{A \log (x)}{a}-\frac{(A b-a B) \log \left(a+b x^2\right)}{2 a b}",1,"(A*Log[x])/a + ((-(A*b) + a*B)*Log[a + b*x^2])/(2*a*b)","A",1
64,1,42,43,0.02543,"\int \frac{A+B x^2}{x^2 \left(a+b x^2\right)} \, dx","Integrate[(A + B*x^2)/(x^2*(a + b*x^2)),x]","\frac{(a B-A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2} \sqrt{b}}-\frac{A}{a x}","-\frac{(A b-a B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2} \sqrt{b}}-\frac{A}{a x}",1,"-(A/(a*x)) + ((-(A*b) + a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*Sqrt[b])","A",1
65,1,49,50,0.020722,"\int \frac{A+B x^2}{x^3 \left(a+b x^2\right)} \, dx","Integrate[(A + B*x^2)/(x^3*(a + b*x^2)),x]","\frac{(A b-a B) \log \left(a+b x^2\right)}{2 a^2}+\frac{\log (x) (a B-A b)}{a^2}-\frac{A}{2 a x^2}","\frac{(A b-a B) \log \left(a+b x^2\right)}{2 a^2}-\frac{\log (x) (A b-a B)}{a^2}-\frac{A}{2 a x^2}",1,"-1/2*A/(a*x^2) + ((-(A*b) + a*B)*Log[x])/a^2 + ((A*b - a*B)*Log[a + b*x^2])/(2*a^2)","A",1
66,1,60,59,0.0492264,"\int \frac{A+B x^2}{x^4 \left(a+b x^2\right)} \, dx","Integrate[(A + B*x^2)/(x^4*(a + b*x^2)),x]","-\frac{\sqrt{b} (a B-A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{5/2}}+\frac{A b-a B}{a^2 x}-\frac{A}{3 a x^3}","\frac{\sqrt{b} (A b-a B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{5/2}}+\frac{A b-a B}{a^2 x}-\frac{A}{3 a x^3}",1,"-1/3*A/(a*x^3) + (A*b - a*B)/(a^2*x) - (Sqrt[b]*(-(A*b) + a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(5/2)","A",1
67,1,70,69,0.0269023,"\int \frac{A+B x^2}{x^5 \left(a+b x^2\right)} \, dx","Integrate[(A + B*x^2)/(x^5*(a + b*x^2)),x]","\frac{4 b x^4 \log (x) (A b-a B)-a \left(a A+2 a B x^2-2 A b x^2\right)+2 b x^4 (a B-A b) \log \left(a+b x^2\right)}{4 a^3 x^4}","-\frac{b (A b-a B) \log \left(a+b x^2\right)}{2 a^3}+\frac{b \log (x) (A b-a B)}{a^3}+\frac{A b-a B}{2 a^2 x^2}-\frac{A}{4 a x^4}",1,"(-(a*(a*A - 2*A*b*x^2 + 2*a*B*x^2)) + 4*b*(A*b - a*B)*x^4*Log[x] + 2*b*(-(A*b) + a*B)*x^4*Log[a + b*x^2])/(4*a^3*x^4)","A",1
68,1,78,80,0.0503119,"\int \frac{A+B x^2}{x^6 \left(a+b x^2\right)} \, dx","Integrate[(A + B*x^2)/(x^6*(a + b*x^2)),x]","\frac{b^{3/2} (a B-A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{7/2}}+\frac{b (a B-A b)}{a^3 x}+\frac{A b-a B}{3 a^2 x^3}-\frac{A}{5 a x^5}","-\frac{b^{3/2} (A b-a B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{7/2}}-\frac{b (A b-a B)}{a^3 x}+\frac{A b-a B}{3 a^2 x^3}-\frac{A}{5 a x^5}",1,"-1/5*A/(a*x^5) + (A*b - a*B)/(3*a^2*x^3) + (b*(-(A*b) + a*B))/(a^3*x) + (b^(3/2)*(-(A*b) + a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(7/2)","A",1
69,1,96,93,0.0320483,"\int \frac{A+B x^2}{x^7 \left(a+b x^2\right)} \, dx","Integrate[(A + B*x^2)/(x^7*(a + b*x^2)),x]","\frac{\left(A b^3-a b^2 B\right) \log \left(a+b x^2\right)}{2 a^4}+\frac{\log (x) \left(a b^2 B-A b^3\right)}{a^4}+\frac{b (a B-A b)}{2 a^3 x^2}+\frac{A b-a B}{4 a^2 x^4}-\frac{A}{6 a x^6}","\frac{b^2 (A b-a B) \log \left(a+b x^2\right)}{2 a^4}-\frac{b^2 \log (x) (A b-a B)}{a^4}-\frac{b (A b-a B)}{2 a^3 x^2}+\frac{A b-a B}{4 a^2 x^4}-\frac{A}{6 a x^6}",1,"-1/6*A/(a*x^6) + (A*b - a*B)/(4*a^2*x^4) + (b*(-(A*b) + a*B))/(2*a^3*x^2) + ((-(A*b^3) + a*b^2*B)*Log[x])/a^4 + ((A*b^3 - a*b^2*B)*Log[a + b*x^2])/(2*a^4)","A",1
70,1,101,99,0.0657944,"\int \frac{A+B x^2}{x^8 \left(a+b x^2\right)} \, dx","Integrate[(A + B*x^2)/(x^8*(a + b*x^2)),x]","-\frac{b^{5/2} (a B-A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{9/2}}-\frac{b^2 (a B-A b)}{a^4 x}+\frac{b (a B-A b)}{3 a^3 x^3}+\frac{A b-a B}{5 a^2 x^5}-\frac{A}{7 a x^7}","\frac{b^{5/2} (A b-a B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{9/2}}+\frac{b^2 (A b-a B)}{a^4 x}-\frac{b (A b-a B)}{3 a^3 x^3}+\frac{A b-a B}{5 a^2 x^5}-\frac{A}{7 a x^7}",1,"-1/7*A/(a*x^7) + (A*b - a*B)/(5*a^2*x^5) + (b*(-(A*b) + a*B))/(3*a^3*x^3) - (b^2*(-(A*b) + a*B))/(a^4*x) - (b^(5/2)*(-(A*b) + a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(9/2)","A",1
71,1,113,126,0.0712718,"\int \frac{x^9 \left(A+B x^2\right)}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^9*(A + B*x^2))/(a + b*x^2)^2,x]","\frac{\frac{12 a^4 (a B-A b)}{a+b x^2}+12 a^3 (5 a B-4 A b) \log \left(a+b x^2\right)-12 a^2 b x^2 (4 a B-3 A b)+4 b^3 x^6 (A b-2 a B)+6 a b^2 x^4 (3 a B-2 A b)+3 b^4 B x^8}{24 b^6}","-\frac{a^4 (A b-a B)}{2 b^6 \left(a+b x^2\right)}-\frac{a^3 (4 A b-5 a B) \log \left(a+b x^2\right)}{2 b^6}+\frac{a^2 x^2 (3 A b-4 a B)}{2 b^5}-\frac{a x^4 (2 A b-3 a B)}{4 b^4}+\frac{x^6 (A b-2 a B)}{6 b^3}+\frac{B x^8}{8 b^2}",1,"(-12*a^2*b*(-3*A*b + 4*a*B)*x^2 + 6*a*b^2*(-2*A*b + 3*a*B)*x^4 + 4*b^3*(A*b - 2*a*B)*x^6 + 3*b^4*B*x^8 + (12*a^4*(-(A*b) + a*B))/(a + b*x^2) + 12*a^3*(-4*A*b + 5*a*B)*Log[a + b*x^2])/(24*b^6)","A",1
72,1,134,131,0.0993609,"\int \frac{x^8 \left(A+B x^2\right)}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^8*(A + B*x^2))/(a + b*x^2)^2,x]","\frac{a^{5/2} (9 a B-7 A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{11/2}}-\frac{a^2 x (4 a B-3 A b)}{b^5}+\frac{x \left(a^3 A b-a^4 B\right)}{2 b^5 \left(a+b x^2\right)}+\frac{a x^3 (3 a B-2 A b)}{3 b^4}+\frac{x^5 (A b-2 a B)}{5 b^3}+\frac{B x^7}{7 b^2}","-\frac{a^{5/2} (7 A b-9 a B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{11/2}}+\frac{a^3 x (A b-a B)}{2 b^5 \left(a+b x^2\right)}+\frac{a^2 x (3 A b-4 a B)}{b^5}-\frac{a x^3 (2 A b-3 a B)}{3 b^4}+\frac{x^5 (A b-2 a B)}{5 b^3}+\frac{B x^7}{7 b^2}",1,"-((a^2*(-3*A*b + 4*a*B)*x)/b^5) + (a*(-2*A*b + 3*a*B)*x^3)/(3*b^4) + ((A*b - 2*a*B)*x^5)/(5*b^3) + (B*x^7)/(7*b^2) + ((a^3*A*b - a^4*B)*x)/(2*b^5*(a + b*x^2)) + (a^(5/2)*(-7*A*b + 9*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(11/2))","A",1
73,1,93,104,0.0586446,"\int \frac{x^7 \left(A+B x^2\right)}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^7*(A + B*x^2))/(a + b*x^2)^2,x]","\frac{\frac{6 a^3 (A b-a B)}{a+b x^2}+6 a^2 (3 A b-4 a B) \log \left(a+b x^2\right)+3 b^2 x^4 (A b-2 a B)+6 a b x^2 (3 a B-2 A b)+2 b^3 B x^6}{12 b^5}","\frac{a^3 (A b-a B)}{2 b^5 \left(a+b x^2\right)}+\frac{a^2 (3 A b-4 a B) \log \left(a+b x^2\right)}{2 b^5}-\frac{a x^2 (2 A b-3 a B)}{2 b^4}+\frac{x^4 (A b-2 a B)}{4 b^3}+\frac{B x^6}{6 b^2}",1,"(6*a*b*(-2*A*b + 3*a*B)*x^2 + 3*b^2*(A*b - 2*a*B)*x^4 + 2*b^3*B*x^6 + (6*a^3*(A*b - a*B))/(a + b*x^2) + 6*a^2*(3*A*b - 4*a*B)*Log[a + b*x^2])/(12*b^5)","A",1
74,1,111,110,0.0814731,"\int \frac{x^6 \left(A+B x^2\right)}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^6*(A + B*x^2))/(a + b*x^2)^2,x]","-\frac{a^{3/2} (7 a B-5 A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{9/2}}-\frac{x \left(a^2 A b-a^3 B\right)}{2 b^4 \left(a+b x^2\right)}+\frac{a x (3 a B-2 A b)}{b^4}+\frac{x^3 (A b-2 a B)}{3 b^3}+\frac{B x^5}{5 b^2}","\frac{a^{3/2} (5 A b-7 a B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{9/2}}-\frac{a^2 x (A b-a B)}{2 b^4 \left(a+b x^2\right)}-\frac{a x (2 A b-3 a B)}{b^4}+\frac{x^3 (A b-2 a B)}{3 b^3}+\frac{B x^5}{5 b^2}",1,"(a*(-2*A*b + 3*a*B)*x)/b^4 + ((A*b - 2*a*B)*x^3)/(3*b^3) + (B*x^5)/(5*b^2) - ((a^2*A*b - a^3*B)*x)/(2*b^4*(a + b*x^2)) - (a^(3/2)*(-5*A*b + 7*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(9/2))","A",1
75,1,72,82,0.0667483,"\int \frac{x^5 \left(A+B x^2\right)}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^5*(A + B*x^2))/(a + b*x^2)^2,x]","\frac{\frac{2 a^2 (a B-A b)}{a+b x^2}+2 b x^2 (A b-2 a B)+2 a (3 a B-2 A b) \log \left(a+b x^2\right)+b^2 B x^4}{4 b^4}","-\frac{a^2 (A b-a B)}{2 b^4 \left(a+b x^2\right)}-\frac{a (2 A b-3 a B) \log \left(a+b x^2\right)}{2 b^4}+\frac{x^2 (A b-2 a B)}{2 b^3}+\frac{B x^4}{4 b^2}",1,"(2*b*(A*b - 2*a*B)*x^2 + b^2*B*x^4 + (2*a^2*(-(A*b) + a*B))/(a + b*x^2) + 2*a*(-2*A*b + 3*a*B)*Log[a + b*x^2])/(4*b^4)","A",1
76,1,89,87,0.0766873,"\int \frac{x^4 \left(A+B x^2\right)}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^4*(A + B*x^2))/(a + b*x^2)^2,x]","\frac{x \left(a A b-a^2 B\right)}{2 b^3 \left(a+b x^2\right)}+\frac{\sqrt{a} (5 a B-3 A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{7/2}}+\frac{x (A b-2 a B)}{b^3}+\frac{B x^3}{3 b^2}","-\frac{\sqrt{a} (3 A b-5 a B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{7/2}}+\frac{a x (A b-a B)}{2 b^3 \left(a+b x^2\right)}+\frac{x (A b-2 a B)}{b^3}+\frac{B x^3}{3 b^2}",1,"((A*b - 2*a*B)*x)/b^3 + (B*x^3)/(3*b^2) + ((a*A*b - a^2*B)*x)/(2*b^3*(a + b*x^2)) + (Sqrt[a]*(-3*A*b + 5*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(7/2))","A",1
77,1,50,60,0.032509,"\int \frac{x^3 \left(A+B x^2\right)}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^3*(A + B*x^2))/(a + b*x^2)^2,x]","\frac{\frac{a (A b-a B)}{a+b x^2}+(A b-2 a B) \log \left(a+b x^2\right)+b B x^2}{2 b^3}","\frac{a (A b-a B)}{2 b^3 \left(a+b x^2\right)}+\frac{(A b-2 a B) \log \left(a+b x^2\right)}{2 b^3}+\frac{B x^2}{2 b^2}",1,"(b*B*x^2 + (a*(A*b - a*B))/(a + b*x^2) + (A*b - 2*a*B)*Log[a + b*x^2])/(2*b^3)","A",1
78,1,68,67,0.065865,"\int \frac{x^2 \left(A+B x^2\right)}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^2*(A + B*x^2))/(a + b*x^2)^2,x]","-\frac{(3 a B-A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 \sqrt{a} b^{5/2}}-\frac{x (A b-a B)}{2 b^2 \left(a+b x^2\right)}+\frac{B x}{b^2}","\frac{(A b-3 a B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 \sqrt{a} b^{5/2}}-\frac{x (A b-a B)}{2 b^2 \left(a+b x^2\right)}+\frac{B x}{b^2}",1,"(B*x)/b^2 - ((A*b - a*B)*x)/(2*b^2*(a + b*x^2)) - ((-(A*b) + 3*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*Sqrt[a]*b^(5/2))","A",1
79,1,41,41,0.0121852,"\int \frac{x \left(A+B x^2\right)}{\left(a+b x^2\right)^2} \, dx","Integrate[(x*(A + B*x^2))/(a + b*x^2)^2,x]","\frac{a B-A b}{2 b^2 \left(a+b x^2\right)}+\frac{B \log \left(a+b x^2\right)}{2 b^2}","\frac{B \log \left(a+b x^2\right)}{2 b^2}-\frac{A b-a B}{2 b^2 \left(a+b x^2\right)}",1,"(-(A*b) + a*B)/(2*b^2*(a + b*x^2)) + (B*Log[a + b*x^2])/(2*b^2)","A",1
80,1,63,63,0.043561,"\int \frac{A+B x^2}{\left(a+b x^2\right)^2} \, dx","Integrate[(A + B*x^2)/(a + b*x^2)^2,x]","\frac{(a B+A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} b^{3/2}}-\frac{x (a B-A b)}{2 a b \left(a+b x^2\right)}","\frac{(a B+A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} b^{3/2}}+\frac{x (A b-a B)}{2 a b \left(a+b x^2\right)}",1,"-1/2*((-(A*b) + a*B)*x)/(a*b*(a + b*x^2)) + ((A*b + a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*b^(3/2))","A",1
81,1,46,51,0.0275972,"\int \frac{A+B x^2}{x \left(a+b x^2\right)^2} \, dx","Integrate[(A + B*x^2)/(x*(a + b*x^2)^2),x]","\frac{\frac{a (A b-a B)}{b \left(a+b x^2\right)}-A \log \left(a+b x^2\right)+2 A \log (x)}{2 a^2}","-\frac{A \log \left(a+b x^2\right)}{2 a^2}+\frac{A \log (x)}{a^2}+\frac{A b-a B}{2 a b \left(a+b x^2\right)}",1,"((a*(A*b - a*B))/(b*(a + b*x^2)) + 2*A*Log[x] - A*Log[a + b*x^2])/(2*a^2)","A",1
82,1,70,71,0.0300024,"\int \frac{A+B x^2}{x^2 \left(a+b x^2\right)^2} \, dx","Integrate[(A + B*x^2)/(x^2*(a + b*x^2)^2),x]","\frac{(a B-3 A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{5/2} \sqrt{b}}+\frac{x (a B-A b)}{2 a^2 \left(a+b x^2\right)}-\frac{A}{a^2 x}","-\frac{(3 A b-a B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{5/2} \sqrt{b}}-\frac{x (A b-a B)}{2 a^2 \left(a+b x^2\right)}-\frac{A}{a^2 x}",1,"-(A/(a^2*x)) + ((-(A*b) + a*B)*x)/(2*a^2*(a + b*x^2)) + ((-3*A*b + a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(5/2)*Sqrt[b])","A",1
83,1,64,76,0.0452002,"\int \frac{A+B x^2}{x^3 \left(a+b x^2\right)^2} \, dx","Integrate[(A + B*x^2)/(x^3*(a + b*x^2)^2),x]","\frac{\frac{a (a B-A b)}{a+b x^2}+(2 A b-a B) \log \left(a+b x^2\right)+2 \log (x) (a B-2 A b)-\frac{a A}{x^2}}{2 a^3}","\frac{(2 A b-a B) \log \left(a+b x^2\right)}{2 a^3}-\frac{\log (x) (2 A b-a B)}{a^3}-\frac{A b-a B}{2 a^2 \left(a+b x^2\right)}-\frac{A}{2 a^2 x^2}",1,"(-((a*A)/x^2) + (a*(-(A*b) + a*B))/(a + b*x^2) + 2*(-2*A*b + a*B)*Log[x] + (2*A*b - a*B)*Log[a + b*x^2])/(2*a^3)","A",1
84,1,90,90,0.0761129,"\int \frac{A+B x^2}{x^4 \left(a+b x^2\right)^2} \, dx","Integrate[(A + B*x^2)/(x^4*(a + b*x^2)^2),x]","-\frac{\sqrt{b} (3 a B-5 A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{7/2}}-\frac{b x (a B-A b)}{2 a^3 \left(a+b x^2\right)}+\frac{2 A b-a B}{a^3 x}-\frac{A}{3 a^2 x^3}","\frac{\sqrt{b} (5 A b-3 a B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{7/2}}+\frac{b x (A b-a B)}{2 a^3 \left(a+b x^2\right)}+\frac{2 A b-a B}{a^3 x}-\frac{A}{3 a^2 x^3}",1,"-1/3*A/(a^2*x^3) + (2*A*b - a*B)/(a^3*x) - (b*(-(A*b) + a*B)*x)/(2*a^3*(a + b*x^2)) - (Sqrt[b]*(-5*A*b + 3*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(7/2))","A",1
85,1,85,97,0.0906808,"\int \frac{A+B x^2}{x^5 \left(a+b x^2\right)^2} \, dx","Integrate[(A + B*x^2)/(x^5*(a + b*x^2)^2),x]","-\frac{\frac{a^2 A}{x^4}+\frac{2 a b (a B-A b)}{a+b x^2}+\frac{2 a (a B-2 A b)}{x^2}+2 b (3 A b-2 a B) \log \left(a+b x^2\right)-4 b \log (x) (3 A b-2 a B)}{4 a^4}","-\frac{b (3 A b-2 a B) \log \left(a+b x^2\right)}{2 a^4}+\frac{b \log (x) (3 A b-2 a B)}{a^4}+\frac{b (A b-a B)}{2 a^3 \left(a+b x^2\right)}+\frac{2 A b-a B}{2 a^3 x^2}-\frac{A}{4 a^2 x^4}",1,"-1/4*((a^2*A)/x^4 + (2*a*(-2*A*b + a*B))/x^2 + (2*a*b*(-(A*b) + a*B))/(a + b*x^2) - 4*b*(3*A*b - 2*a*B)*Log[x] + 2*b*(3*A*b - 2*a*B)*Log[a + b*x^2])/a^4","A",1
86,1,112,113,0.07325,"\int \frac{A+B x^2}{x^6 \left(a+b x^2\right)^2} \, dx","Integrate[(A + B*x^2)/(x^6*(a + b*x^2)^2),x]","\frac{b^{3/2} (5 a B-7 A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{9/2}}+\frac{b^2 x (a B-A b)}{2 a^4 \left(a+b x^2\right)}+\frac{b (2 a B-3 A b)}{a^4 x}+\frac{2 A b-a B}{3 a^3 x^3}-\frac{A}{5 a^2 x^5}","-\frac{b^{3/2} (7 A b-5 a B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{9/2}}-\frac{b^2 x (A b-a B)}{2 a^4 \left(a+b x^2\right)}-\frac{b (3 A b-2 a B)}{a^4 x}+\frac{2 A b-a B}{3 a^3 x^3}-\frac{A}{5 a^2 x^5}",1,"-1/5*A/(a^2*x^5) + (2*A*b - a*B)/(3*a^3*x^3) + (b*(-3*A*b + 2*a*B))/(a^4*x) + (b^2*(-(A*b) + a*B)*x)/(2*a^4*(a + b*x^2)) + (b^(3/2)*(-7*A*b + 5*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(9/2))","A",1
87,1,110,124,0.0917936,"\int \frac{A+B x^2}{x^7 \left(a+b x^2\right)^2} \, dx","Integrate[(A + B*x^2)/(x^7*(a + b*x^2)^2),x]","\frac{-\frac{2 a^3 A}{x^6}-\frac{3 a^2 (a B-2 A b)}{x^4}+\frac{6 a b^2 (a B-A b)}{a+b x^2}+6 b^2 (4 A b-3 a B) \log \left(a+b x^2\right)+12 b^2 \log (x) (3 a B-4 A b)+\frac{6 a b (2 a B-3 A b)}{x^2}}{12 a^5}","\frac{b^2 (4 A b-3 a B) \log \left(a+b x^2\right)}{2 a^5}-\frac{b^2 \log (x) (4 A b-3 a B)}{a^5}-\frac{b^2 (A b-a B)}{2 a^4 \left(a+b x^2\right)}-\frac{b (3 A b-2 a B)}{2 a^4 x^2}+\frac{2 A b-a B}{4 a^3 x^4}-\frac{A}{6 a^2 x^6}",1,"((-2*a^3*A)/x^6 - (3*a^2*(-2*A*b + a*B))/x^4 + (6*a*b*(-3*A*b + 2*a*B))/x^2 + (6*a*b^2*(-(A*b) + a*B))/(a + b*x^2) + 12*b^2*(-4*A*b + 3*a*B)*Log[x] + 6*b^2*(4*A*b - 3*a*B)*Log[a + b*x^2])/(12*a^5)","A",1
88,1,136,150,0.080023,"\int \frac{x^{11} \left(A+B x^2\right)}{\left(a+b x^2\right)^3} \, dx","Integrate[(x^11*(A + B*x^2))/(a + b*x^2)^3,x]","\frac{\frac{6 a^5 (A b-a B)}{\left(a+b x^2\right)^2}+\frac{12 a^4 (6 a B-5 A b)}{a+b x^2}+60 a^3 (3 a B-2 A b) \log \left(a+b x^2\right)-24 a^2 b x^2 (5 a B-3 A b)+4 b^3 x^6 (A b-3 a B)+18 a b^2 x^4 (2 a B-A b)+3 b^4 B x^8}{24 b^7}","\frac{a^5 (A b-a B)}{4 b^7 \left(a+b x^2\right)^2}-\frac{a^4 (5 A b-6 a B)}{2 b^7 \left(a+b x^2\right)}-\frac{5 a^3 (2 A b-3 a B) \log \left(a+b x^2\right)}{2 b^7}+\frac{a^2 x^2 (3 A b-5 a B)}{b^6}-\frac{3 a x^4 (A b-2 a B)}{4 b^5}+\frac{x^6 (A b-3 a B)}{6 b^4}+\frac{B x^8}{8 b^3}",1,"(-24*a^2*b*(-3*A*b + 5*a*B)*x^2 + 18*a*b^2*(-(A*b) + 2*a*B)*x^4 + 4*b^3*(A*b - 3*a*B)*x^6 + 3*b^4*B*x^8 + (6*a^5*(A*b - a*B))/(a + b*x^2)^2 + (12*a^4*(-5*A*b + 6*a*B))/(a + b*x^2) + 60*a^3*(-2*A*b + 3*a*B)*Log[a + b*x^2])/(24*b^7)","A",1
89,1,116,128,0.0681095,"\int \frac{x^9 \left(A+B x^2\right)}{\left(a+b x^2\right)^3} \, dx","Integrate[(x^9*(A + B*x^2))/(a + b*x^2)^3,x]","\frac{\frac{3 a^4 (a B-A b)}{\left(a+b x^2\right)^2}+\frac{6 a^3 (4 A b-5 a B)}{a+b x^2}+12 a^2 (3 A b-5 a B) \log \left(a+b x^2\right)+3 b^2 x^4 (A b-3 a B)+18 a b x^2 (2 a B-A b)+2 b^3 B x^6}{12 b^6}","-\frac{a^4 (A b-a B)}{4 b^6 \left(a+b x^2\right)^2}+\frac{a^3 (4 A b-5 a B)}{2 b^6 \left(a+b x^2\right)}+\frac{a^2 (3 A b-5 a B) \log \left(a+b x^2\right)}{b^6}-\frac{3 a x^2 (A b-2 a B)}{2 b^5}+\frac{x^4 (A b-3 a B)}{4 b^4}+\frac{B x^6}{6 b^3}",1,"(18*a*b*(-(A*b) + 2*a*B)*x^2 + 3*b^2*(A*b - 3*a*B)*x^4 + 2*b^3*B*x^6 + (3*a^4*(-(A*b) + a*B))/(a + b*x^2)^2 + (6*a^3*(4*A*b - 5*a*B))/(a + b*x^2) + 12*a^2*(3*A*b - 5*a*B)*Log[a + b*x^2])/(12*b^6)","A",1
90,1,94,109,0.0606482,"\int \frac{x^7 \left(A+B x^2\right)}{\left(a+b x^2\right)^3} \, dx","Integrate[(x^7*(A + B*x^2))/(a + b*x^2)^3,x]","\frac{\frac{a^3 (A b-a B)}{\left(a+b x^2\right)^2}+\frac{2 a^2 (4 a B-3 A b)}{a+b x^2}+2 b x^2 (A b-3 a B)+6 a (2 a B-A b) \log \left(a+b x^2\right)+b^2 B x^4}{4 b^5}","\frac{a^3 (A b-a B)}{4 b^5 \left(a+b x^2\right)^2}-\frac{a^2 (3 A b-4 a B)}{2 b^5 \left(a+b x^2\right)}-\frac{3 a (A b-2 a B) \log \left(a+b x^2\right)}{2 b^5}+\frac{x^2 (A b-3 a B)}{2 b^4}+\frac{B x^4}{4 b^3}",1,"(2*b*(A*b - 3*a*B)*x^2 + b^2*B*x^4 + (a^3*(A*b - a*B))/(a + b*x^2)^2 + (2*a^2*(-3*A*b + 4*a*B))/(a + b*x^2) + 6*a*(-(A*b) + 2*a*B)*Log[a + b*x^2])/(4*b^5)","A",1
91,1,92,88,0.035771,"\int \frac{x^5 \left(A+B x^2\right)}{\left(a+b x^2\right)^3} \, dx","Integrate[(x^5*(A + B*x^2))/(a + b*x^2)^3,x]","\frac{2 a A b-3 a^2 B}{2 b^4 \left(a+b x^2\right)}+\frac{a^3 B-a^2 A b}{4 b^4 \left(a+b x^2\right)^2}+\frac{(A b-3 a B) \log \left(a+b x^2\right)}{2 b^4}+\frac{B x^2}{2 b^3}","-\frac{a^2 (A b-a B)}{4 b^4 \left(a+b x^2\right)^2}+\frac{a (2 A b-3 a B)}{2 b^4 \left(a+b x^2\right)}+\frac{(A b-3 a B) \log \left(a+b x^2\right)}{2 b^4}+\frac{B x^2}{2 b^3}",1,"(B*x^2)/(2*b^3) + (-(a^2*A*b) + a^3*B)/(4*b^4*(a + b*x^2)^2) + (2*a*A*b - 3*a^2*B)/(2*b^4*(a + b*x^2)) + ((A*b - 3*a*B)*Log[a + b*x^2])/(2*b^4)","A",1
92,1,64,66,0.0233015,"\int \frac{x^3 \left(A+B x^2\right)}{\left(a+b x^2\right)^3} \, dx","Integrate[(x^3*(A + B*x^2))/(a + b*x^2)^3,x]","\frac{3 a^2 B-a b \left(A-4 B x^2\right)+2 B \left(a+b x^2\right)^2 \log \left(a+b x^2\right)-2 A b^2 x^2}{4 b^3 \left(a+b x^2\right)^2}","-\frac{A b-2 a B}{2 b^3 \left(a+b x^2\right)}+\frac{a (A b-a B)}{4 b^3 \left(a+b x^2\right)^2}+\frac{B \log \left(a+b x^2\right)}{2 b^3}",1,"(3*a^2*B - 2*A*b^2*x^2 - a*b*(A - 4*B*x^2) + 2*B*(a + b*x^2)^2*Log[a + b*x^2])/(4*b^3*(a + b*x^2)^2)","A",1
93,1,30,32,0.0122904,"\int \frac{x \left(A+B x^2\right)}{\left(a+b x^2\right)^3} \, dx","Integrate[(x*(A + B*x^2))/(a + b*x^2)^3,x]","-\frac{B \left(a+2 b x^2\right)+A b}{4 b^2 \left(a+b x^2\right)^2}","-\frac{\left(A+B x^2\right)^2}{4 \left(a+b x^2\right)^2 (A b-a B)}",1,"-1/4*(A*b + B*(a + 2*b*x^2))/(b^2*(a + b*x^2)^2)","A",1
94,1,59,68,0.044923,"\int \frac{A+B x^2}{x \left(a+b x^2\right)^3} \, dx","Integrate[(A + B*x^2)/(x*(a + b*x^2)^3),x]","\frac{\frac{a \left(a^2 (-B)+3 a A b+2 A b^2 x^2\right)}{b \left(a+b x^2\right)^2}-2 A \log \left(a+b x^2\right)+4 A \log (x)}{4 a^3}","-\frac{A \log \left(a+b x^2\right)}{2 a^3}+\frac{A \log (x)}{a^3}+\frac{A}{2 a^2 \left(a+b x^2\right)}+\frac{A b-a B}{4 a b \left(a+b x^2\right)^2}",1,"((a*(3*a*A*b - a^2*B + 2*A*b^2*x^2))/(b*(a + b*x^2)^2) + 4*A*Log[x] - 2*A*Log[a + b*x^2])/(4*a^3)","A",1
95,1,87,101,0.0612693,"\int \frac{A+B x^2}{x^3 \left(a+b x^2\right)^3} \, dx","Integrate[(A + B*x^2)/(x^3*(a + b*x^2)^3),x]","\frac{\frac{a^2 (a B-A b)}{\left(a+b x^2\right)^2}+\frac{2 a (a B-2 A b)}{a+b x^2}+2 (3 A b-a B) \log \left(a+b x^2\right)+4 \log (x) (a B-3 A b)-\frac{2 a A}{x^2}}{4 a^4}","\frac{(3 A b-a B) \log \left(a+b x^2\right)}{2 a^4}-\frac{\log (x) (3 A b-a B)}{a^4}-\frac{2 A b-a B}{2 a^3 \left(a+b x^2\right)}-\frac{A}{2 a^3 x^2}-\frac{A b-a B}{4 a^2 \left(a+b x^2\right)^2}",1,"((-2*a*A)/x^2 + (a^2*(-(A*b) + a*B))/(a + b*x^2)^2 + (2*a*(-2*A*b + a*B))/(a + b*x^2) + 4*(-3*A*b + a*B)*Log[x] + 2*(3*A*b - a*B)*Log[a + b*x^2])/(4*a^4)","A",1
96,1,108,124,0.0749751,"\int \frac{A+B x^2}{x^5 \left(a+b x^2\right)^3} \, dx","Integrate[(A + B*x^2)/(x^5*(a + b*x^2)^3),x]","\frac{\frac{a^2 b (A b-a B)}{\left(a+b x^2\right)^2}-\frac{a^2 A}{x^4}+\frac{2 a b (3 A b-2 a B)}{a+b x^2}-\frac{2 a (a B-3 A b)}{x^2}+6 b (a B-2 A b) \log \left(a+b x^2\right)+12 b \log (x) (2 A b-a B)}{4 a^5}","-\frac{3 b (2 A b-a B) \log \left(a+b x^2\right)}{2 a^5}+\frac{3 b \log (x) (2 A b-a B)}{a^5}+\frac{b (3 A b-2 a B)}{2 a^4 \left(a+b x^2\right)}+\frac{3 A b-a B}{2 a^4 x^2}+\frac{b (A b-a B)}{4 a^3 \left(a+b x^2\right)^2}-\frac{A}{4 a^3 x^4}",1,"(-((a^2*A)/x^4) - (2*a*(-3*A*b + a*B))/x^2 + (a^2*b*(A*b - a*B))/(a + b*x^2)^2 + (2*a*b*(3*A*b - 2*a*B))/(a + b*x^2) + 12*b*(2*A*b - a*B)*Log[x] + 6*b*(-2*A*b + a*B)*Log[a + b*x^2])/(4*a^5)","A",1
97,1,135,149,0.1147802,"\int \frac{A+B x^2}{x^7 \left(a+b x^2\right)^3} \, dx","Integrate[(A + B*x^2)/(x^7*(a + b*x^2)^3),x]","\frac{-\frac{2 a^3 A}{x^6}+\frac{3 a^2 b^2 (a B-A b)}{\left(a+b x^2\right)^2}-\frac{3 a^2 (a B-3 A b)}{x^4}+\frac{6 a b^2 (3 a B-4 A b)}{a+b x^2}+12 b^2 (5 A b-3 a B) \log \left(a+b x^2\right)+24 b^2 \log (x) (3 a B-5 A b)+\frac{18 a b (a B-2 A b)}{x^2}}{12 a^6}","\frac{b^2 (5 A b-3 a B) \log \left(a+b x^2\right)}{a^6}-\frac{2 b^2 \log (x) (5 A b-3 a B)}{a^6}-\frac{b^2 (4 A b-3 a B)}{2 a^5 \left(a+b x^2\right)}-\frac{3 b (2 A b-a B)}{2 a^5 x^2}-\frac{b^2 (A b-a B)}{4 a^4 \left(a+b x^2\right)^2}+\frac{3 A b-a B}{4 a^4 x^4}-\frac{A}{6 a^3 x^6}",1,"((-2*a^3*A)/x^6 - (3*a^2*(-3*A*b + a*B))/x^4 + (18*a*b*(-2*A*b + a*B))/x^2 + (3*a^2*b^2*(-(A*b) + a*B))/(a + b*x^2)^2 + (6*a*b^2*(-4*A*b + 3*a*B))/(a + b*x^2) + 24*b^2*(-5*A*b + 3*a*B)*Log[x] + 12*b^2*(5*A*b - 3*a*B)*Log[a + b*x^2])/(12*a^6)","A",1
98,1,158,158,0.0889239,"\int \frac{x^{10} \left(A+B x^2\right)}{\left(a+b x^2\right)^3} \, dx","Integrate[(x^10*(A + B*x^2))/(a + b*x^2)^3,x]","\frac{9 a^{5/2} (11 a B-7 A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 b^{13/2}}+\frac{a^4 x (a B-A b)}{4 b^6 \left(a+b x^2\right)^2}+\frac{a^3 x (17 A b-21 a B)}{8 b^6 \left(a+b x^2\right)}-\frac{2 a^2 x (5 a B-3 A b)}{b^6}+\frac{a x^3 (2 a B-A b)}{b^5}+\frac{x^5 (A b-3 a B)}{5 b^4}+\frac{B x^7}{7 b^3}","-\frac{9 a^{5/2} (7 A b-11 a B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 b^{13/2}}-\frac{a^4 x (A b-a B)}{4 b^6 \left(a+b x^2\right)^2}+\frac{a^3 x (17 A b-21 a B)}{8 b^6 \left(a+b x^2\right)}+\frac{2 a^2 x (3 A b-5 a B)}{b^6}-\frac{a x^3 (A b-2 a B)}{b^5}+\frac{x^5 (A b-3 a B)}{5 b^4}+\frac{B x^7}{7 b^3}",1,"(-2*a^2*(-3*A*b + 5*a*B)*x)/b^6 + (a*(-(A*b) + 2*a*B)*x^3)/b^5 + ((A*b - 3*a*B)*x^5)/(5*b^4) + (B*x^7)/(7*b^3) + (a^4*(-(A*b) + a*B)*x)/(4*b^6*(a + b*x^2)^2) + (a^3*(17*A*b - 21*a*B)*x)/(8*b^6*(a + b*x^2)) + (9*a^(5/2)*(-7*A*b + 11*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*b^(13/2))","A",1
99,1,133,138,0.0957491,"\int \frac{x^8 \left(A+B x^2\right)}{\left(a+b x^2\right)^3} \, dx","Integrate[(x^8*(A + B*x^2))/(a + b*x^2)^3,x]","\frac{x \left(945 a^4 B-525 a^3 b \left(A-3 B x^2\right)+7 a^2 b^2 x^2 \left(72 B x^2-125 A\right)-8 a b^3 x^4 \left(35 A+9 B x^2\right)+8 b^4 x^6 \left(5 A+3 B x^2\right)\right)}{120 b^5 \left(a+b x^2\right)^2}-\frac{7 a^{3/2} (9 a B-5 A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 b^{11/2}}","\frac{7 a^{3/2} (5 A b-9 a B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 b^{11/2}}+\frac{a^3 x (A b-a B)}{4 b^5 \left(a+b x^2\right)^2}-\frac{a^2 x (13 A b-17 a B)}{8 b^5 \left(a+b x^2\right)}-\frac{3 a x (A b-2 a B)}{b^5}+\frac{x^3 (A b-3 a B)}{3 b^4}+\frac{B x^5}{5 b^3}",1,"(x*(945*a^4*B - 525*a^3*b*(A - 3*B*x^2) + 8*b^4*x^6*(5*A + 3*B*x^2) - 8*a*b^3*x^4*(35*A + 9*B*x^2) + 7*a^2*b^2*x^2*(-125*A + 72*B*x^2)))/(120*b^5*(a + b*x^2)^2) - (7*a^(3/2)*(-5*A*b + 9*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*b^(11/2))","A",1
100,1,113,116,0.081501,"\int \frac{x^6 \left(A+B x^2\right)}{\left(a+b x^2\right)^3} \, dx","Integrate[(x^6*(A + B*x^2))/(a + b*x^2)^3,x]","\frac{-105 a^3 B x+5 a^2 b x \left(9 A-35 B x^2\right)+a b^2 x^3 \left(75 A-56 B x^2\right)+8 b^3 x^5 \left(3 A+B x^2\right)}{24 b^4 \left(a+b x^2\right)^2}+\frac{5 \sqrt{a} (7 a B-3 A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 b^{9/2}}","-\frac{a^2 x (A b-a B)}{4 b^4 \left(a+b x^2\right)^2}-\frac{5 \sqrt{a} (3 A b-7 a B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 b^{9/2}}+\frac{a x (9 A b-13 a B)}{8 b^4 \left(a+b x^2\right)}+\frac{x (A b-3 a B)}{b^4}+\frac{B x^3}{3 b^3}",1,"(-105*a^3*B*x + a*b^2*x^3*(75*A - 56*B*x^2) + 5*a^2*b*x*(9*A - 35*B*x^2) + 8*b^3*x^5*(3*A + B*x^2))/(24*b^4*(a + b*x^2)^2) + (5*Sqrt[a]*(-3*A*b + 7*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*b^(9/2))","A",1
101,1,91,94,0.0699724,"\int \frac{x^4 \left(A+B x^2\right)}{\left(a+b x^2\right)^3} \, dx","Integrate[(x^4*(A + B*x^2))/(a + b*x^2)^3,x]","\frac{x \left(15 a^2 B+a \left(25 b B x^2-3 A b\right)+b^2 x^2 \left(8 B x^2-5 A\right)\right)}{8 b^3 \left(a+b x^2\right)^2}+\frac{3 (A b-5 a B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 \sqrt{a} b^{7/2}}","\frac{3 (A b-5 a B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 \sqrt{a} b^{7/2}}-\frac{x (5 A b-9 a B)}{8 b^3 \left(a+b x^2\right)}+\frac{a x (A b-a B)}{4 b^3 \left(a+b x^2\right)^2}+\frac{B x}{b^3}",1,"(x*(15*a^2*B + b^2*x^2*(-5*A + 8*B*x^2) + a*(-3*A*b + 25*b*B*x^2)))/(8*b^3*(a + b*x^2)^2) + (3*(A*b - 5*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*Sqrt[a]*b^(7/2))","A",1
102,1,83,89,0.0805674,"\int \frac{x^2 \left(A+B x^2\right)}{\left(a+b x^2\right)^3} \, dx","Integrate[(x^2*(A + B*x^2))/(a + b*x^2)^3,x]","\frac{\frac{(3 a B+A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2}}+\frac{\sqrt{b} x \left(-3 a^2 B-a b \left(A+5 B x^2\right)+A b^2 x^2\right)}{a \left(a+b x^2\right)^2}}{8 b^{5/2}}","\frac{(3 a B+A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{3/2} b^{5/2}}+\frac{x (A b-5 a B)}{8 a b^2 \left(a+b x^2\right)}-\frac{x (A b-a B)}{4 b^2 \left(a+b x^2\right)^2}",1,"((Sqrt[b]*x*(-3*a^2*B + A*b^2*x^2 - a*b*(A + 5*B*x^2)))/(a*(a + b*x^2)^2) + ((A*b + 3*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(3/2))/(8*b^(5/2))","A",1
103,1,84,92,0.0605301,"\int \frac{A+B x^2}{\left(a+b x^2\right)^3} \, dx","Integrate[(A + B*x^2)/(a + b*x^2)^3,x]","\frac{(a B+3 A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} b^{3/2}}+\frac{x \left(a^2 (-B)+a b \left(5 A+B x^2\right)+3 A b^2 x^2\right)}{8 a^2 b \left(a+b x^2\right)^2}","\frac{(a B+3 A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} b^{3/2}}+\frac{x (a B+3 A b)}{8 a^2 b \left(a+b x^2\right)}+\frac{x (A b-a B)}{4 a b \left(a+b x^2\right)^2}",1,"(x*(-(a^2*B) + 3*A*b^2*x^2 + a*b*(5*A + B*x^2)))/(8*a^2*b*(a + b*x^2)^2) + ((3*A*b + a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*b^(3/2))","A",1
104,1,96,97,0.0577357,"\int \frac{A+B x^2}{x^2 \left(a+b x^2\right)^3} \, dx","Integrate[(A + B*x^2)/(x^2*(a + b*x^2)^3),x]","\frac{3 (a B-5 A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{7/2} \sqrt{b}}+\frac{x (3 a B-7 A b)}{8 a^3 \left(a+b x^2\right)}-\frac{A}{a^3 x}+\frac{x (a B-A b)}{4 a^2 \left(a+b x^2\right)^2}","-\frac{3 (5 A b-a B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{7/2} \sqrt{b}}-\frac{x (7 A b-3 a B)}{8 a^3 \left(a+b x^2\right)}-\frac{A}{a^3 x}-\frac{x (A b-a B)}{4 a^2 \left(a+b x^2\right)^2}",1,"-(A/(a^3*x)) + ((-(A*b) + a*B)*x)/(4*a^2*(a + b*x^2)^2) + ((-7*A*b + 3*a*B)*x)/(8*a^3*(a + b*x^2)) + (3*(-5*A*b + a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(7/2)*Sqrt[b])","A",1
105,1,116,117,0.0886194,"\int \frac{A+B x^2}{x^4 \left(a+b x^2\right)^3} \, dx","Integrate[(A + B*x^2)/(x^4*(a + b*x^2)^3),x]","\frac{5 \sqrt{b} (7 A b-3 a B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{9/2}}+\frac{-8 a^3 \left(A+3 B x^2\right)+a^2 b x^2 \left(56 A-75 B x^2\right)+5 a b^2 x^4 \left(35 A-9 B x^2\right)+105 A b^3 x^6}{24 a^4 x^3 \left(a+b x^2\right)^2}","\frac{5 \sqrt{b} (7 A b-3 a B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{9/2}}+\frac{b x (11 A b-7 a B)}{8 a^4 \left(a+b x^2\right)}+\frac{3 A b-a B}{a^4 x}+\frac{b x (A b-a B)}{4 a^3 \left(a+b x^2\right)^2}-\frac{A}{3 a^3 x^3}",1,"(105*A*b^3*x^6 + a^2*b*x^2*(56*A - 75*B*x^2) + 5*a*b^2*x^4*(35*A - 9*B*x^2) - 8*a^3*(A + 3*B*x^2))/(24*a^4*x^3*(a + b*x^2)^2) + (5*Sqrt[b]*(7*A*b - 3*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(9/2))","A",1
106,1,139,142,0.0966702,"\int \frac{A+B x^2}{x^6 \left(a+b x^2\right)^3} \, dx","Integrate[(A + B*x^2)/(x^6*(a + b*x^2)^3),x]","\frac{7 b^{3/2} (5 a B-9 A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{11/2}}+\frac{-8 a^4 \left(3 A+5 B x^2\right)+8 a^3 b x^2 \left(9 A+35 B x^2\right)+7 a^2 b^2 x^4 \left(125 B x^2-72 A\right)+525 a b^3 x^6 \left(B x^2-3 A\right)-945 A b^4 x^8}{120 a^5 x^5 \left(a+b x^2\right)^2}","-\frac{7 b^{3/2} (9 A b-5 a B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{11/2}}-\frac{b^2 x (15 A b-11 a B)}{8 a^5 \left(a+b x^2\right)}-\frac{3 b (2 A b-a B)}{a^5 x}-\frac{b^2 x (A b-a B)}{4 a^4 \left(a+b x^2\right)^2}+\frac{3 A b-a B}{3 a^4 x^3}-\frac{A}{5 a^3 x^5}",1,"(-945*A*b^4*x^8 + 525*a*b^3*x^6*(-3*A + B*x^2) - 8*a^4*(3*A + 5*B*x^2) + 8*a^3*b*x^2*(9*A + 35*B*x^2) + 7*a^2*b^2*x^4*(-72*A + 125*B*x^2))/(120*a^5*x^5*(a + b*x^2)^2) + (7*b^(3/2)*(-9*A*b + 5*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(11/2))","A",1
107,1,12,12,0.0057529,"\int \frac{a+b x^2}{1+x^2} \, dx","Integrate[(a + b*x^2)/(1 + x^2),x]","(a-b) \tan ^{-1}(x)+b x","(a-b) \tan ^{-1}(x)+b x",1,"b*x + (a - b)*ArcTan[x]","A",1
108,1,28,11,0.0098939,"\int \frac{a+b x^2}{1-x^2} \, dx","Integrate[(a + b*x^2)/(1 - x^2),x]","\frac{1}{2} (-(a+b) \log (1-x)+(a+b) \log (x+1)-2 b x)","(a+b) \tanh ^{-1}(x)-b x",1,"(-2*b*x - (a + b)*Log[1 - x] + (a + b)*Log[1 + x])/2","B",1
109,1,10,11,0.003923,"\int \frac{1+x^2}{\left(-1+x^2\right)^2} \, dx","Integrate[(1 + x^2)/(-1 + x^2)^2,x]","-\frac{x}{x^2-1}","\frac{x}{1-x^2}",1,"-(x/(-1 + x^2))","A",1
110,1,9,9,0.0039145,"\int \frac{1-x^2}{\left(1+x^2\right)^2} \, dx","Integrate[(1 - x^2)/(1 + x^2)^2,x]","\frac{x}{x^2+1}","\frac{x}{x^2+1}",1,"x/(1 + x^2)","A",1
111,1,19,19,0.061295,"\int \frac{3+2 x^2}{\left(1+x^2\right)^2} \, dx","Integrate[(3 + 2*x^2)/(1 + x^2)^2,x]","\frac{x}{2 \left(x^2+1\right)}+\frac{5}{2} \tan ^{-1}(x)","\frac{x}{2 \left(x^2+1\right)}+\frac{5}{2} \tan ^{-1}(x)",1,"x/(2*(1 + x^2)) + (5*ArcTan[x])/2","A",1
112,1,19,19,0.0066533,"\int \frac{-2+x^2}{\left(1+x^2\right)^2} \, dx","Integrate[(-2 + x^2)/(1 + x^2)^2,x]","-\frac{3 x}{2 \left(x^2+1\right)}-\frac{1}{2} \tan ^{-1}(x)","-\frac{3 x}{2 \left(x^2+1\right)}-\frac{1}{2} \tan ^{-1}(x)",1,"(-3*x)/(2*(1 + x^2)) - ArcTan[x]/2","A",1
113,1,14,14,0.005601,"\int \frac{3+x^2}{\left(1+x^2\right)^2} \, dx","Integrate[(3 + x^2)/(1 + x^2)^2,x]","\frac{x}{x^2+1}+2 \tan ^{-1}(x)","\frac{x}{x^2+1}+2 \tan ^{-1}(x)",1,"x/(1 + x^2) + 2*ArcTan[x]","A",1
114,1,14,12,0.0075676,"\int \frac{a+b x^2}{\left(-a+b x^2\right)^2} \, dx","Integrate[(a + b*x^2)/(-a + b*x^2)^2,x]","-\frac{x}{b x^2-a}","\frac{x}{a-b x^2}",1,"-(x/(-a + b*x^2))","A",1
115,1,14,12,0.0054565,"\int \frac{a+b x^2}{\left(a-b x^2\right)^2} \, dx","Integrate[(a + b*x^2)/(a - b*x^2)^2,x]","-\frac{x}{b x^2-a}","\frac{x}{a-b x^2}",1,"-(x/(-a + b*x^2))","A",1
116,1,39,39,0.0207787,"\int \frac{A+B x^2}{a-b x^2} \, dx","Integrate[(A + B*x^2)/(a - b*x^2),x]","\frac{(a B+A b) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} b^{3/2}}-\frac{B x}{b}","\frac{(a B+A b) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} b^{3/2}}-\frac{B x}{b}",1,"-((B*x)/b) + ((A*b + a*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*b^(3/2))","A",1
117,1,35,35,0.0088414,"\int \frac{1+x^2}{\left(16+x^2\right)^3} \, dx","Integrate[(1 + x^2)/(16 + x^2)^3,x]","\frac{19 x}{2048 \left(x^2+16\right)}-\frac{15 x}{64 \left(x^2+16\right)^2}+\frac{19 \tan ^{-1}\left(\frac{x}{4}\right)}{8192}","\frac{19 x}{2048 \left(x^2+16\right)}-\frac{15 x}{64 \left(x^2+16\right)^2}+\frac{19 \tan ^{-1}\left(\frac{x}{4}\right)}{8192}",1,"(-15*x)/(64*(16 + x^2)^2) + (19*x)/(2048*(16 + x^2)) + (19*ArcTan[x/4])/8192","A",1
118,1,14,14,0.0060706,"\int \frac{1+2 x^2}{x^5 \left(1+x^2\right)^3} \, dx","Integrate[(1 + 2*x^2)/(x^5*(1 + x^2)^3),x]","-\frac{1}{4 x^4 \left(x^2+1\right)^2}","-\frac{1}{4 x^4 \left(x^2+1\right)^2}",1,"-1/4*1/(x^4*(1 + x^2)^2)","A",1
119,1,1,1,0.0003098,"\int \frac{\left(1-x^2\right)^2}{\left(-1+x^2\right)^2} \, dx","Integrate[(1 - x^2)^2/(-1 + x^2)^2,x]","x","x",1,"x","A",1
120,1,8,8,0.0004802,"\int \frac{x^3 \left(a c+b c x^2\right)}{a+b x^2} \, dx","Integrate[(x^3*(a*c + b*c*x^2))/(a + b*x^2),x]","\frac{c x^4}{4}","\frac{c x^4}{4}",1,"(c*x^4)/4","A",1
121,1,8,8,0.0004376,"\int \frac{x^2 \left(a c+b c x^2\right)}{a+b x^2} \, dx","Integrate[(x^2*(a*c + b*c*x^2))/(a + b*x^2),x]","\frac{c x^3}{3}","\frac{c x^3}{3}",1,"(c*x^3)/3","A",1
122,1,8,8,0.0003079,"\int \frac{x \left(a c+b c x^2\right)}{a+b x^2} \, dx","Integrate[(x*(a*c + b*c*x^2))/(a + b*x^2),x]","\frac{c x^2}{2}","\frac{c x^2}{2}",1,"(c*x^2)/2","A",1
123,1,3,3,0.0002548,"\int \frac{a c+b c x^2}{a+b x^2} \, dx","Integrate[(a*c + b*c*x^2)/(a + b*x^2),x]","c x","c x",1,"c*x","A",1
124,1,4,4,0.0004013,"\int \frac{a c+b c x^2}{x \left(a+b x^2\right)} \, dx","Integrate[(a*c + b*c*x^2)/(x*(a + b*x^2)),x]","c \log (x)","c \log (x)",1,"c*Log[x]","A",1
125,1,6,6,0.0003745,"\int \frac{a c+b c x^2}{x^2 \left(a+b x^2\right)} \, dx","Integrate[(a*c + b*c*x^2)/(x^2*(a + b*x^2)),x]","-\frac{c}{x}","-\frac{c}{x}",1,"-(c/x)","A",1
126,1,8,8,0.0003455,"\int \frac{a c+b c x^2}{x^3 \left(a+b x^2\right)} \, dx","Integrate[(a*c + b*c*x^2)/(x^3*(a + b*x^2)),x]","-\frac{c}{2 x^2}","-\frac{c}{2 x^2}",1,"-1/2*c/x^2","A",1
127,1,29,29,0.0045635,"\int \frac{x^3 \left(a c+b c x^2\right)}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^3*(a*c + b*c*x^2))/(a + b*x^2)^2,x]","c \left(\frac{x^2}{2 b}-\frac{a \log \left(a+b x^2\right)}{2 b^2}\right)","\frac{c x^2}{2 b}-\frac{a c \log \left(a+b x^2\right)}{2 b^2}",1,"c*(x^2/(2*b) - (a*Log[a + b*x^2])/(2*b^2))","A",1
128,1,33,33,0.0084351,"\int \frac{x^2 \left(a c+b c x^2\right)}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^2*(a*c + b*c*x^2))/(a + b*x^2)^2,x]","c \left(\frac{x}{b}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{3/2}}\right)","\frac{c x}{b}-\frac{\sqrt{a} c \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{3/2}}",1,"c*(x/b - (Sqrt[a]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(3/2))","A",1
129,1,16,16,0.0024052,"\int \frac{x \left(a c+b c x^2\right)}{\left(a+b x^2\right)^2} \, dx","Integrate[(x*(a*c + b*c*x^2))/(a + b*x^2)^2,x]","\frac{c \log \left(a+b x^2\right)}{2 b}","\frac{c \log \left(a+b x^2\right)}{2 b}",1,"(c*Log[a + b*x^2])/(2*b)","A",1
130,1,25,25,0.0047326,"\int \frac{a c+b c x^2}{\left(a+b x^2\right)^2} \, dx","Integrate[(a*c + b*c*x^2)/(a + b*x^2)^2,x]","\frac{c \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}","\frac{c \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}",1,"(c*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b])","A",1
131,1,24,24,0.0050431,"\int \frac{a c+b c x^2}{x \left(a+b x^2\right)^2} \, dx","Integrate[(a*c + b*c*x^2)/(x*(a + b*x^2)^2),x]","c \left(\frac{\log (x)}{a}-\frac{\log \left(a+b x^2\right)}{2 a}\right)","\frac{c \log (x)}{a}-\frac{c \log \left(a+b x^2\right)}{2 a}",1,"c*(Log[x]/a - Log[a + b*x^2]/(2*a))","A",1
132,1,36,36,0.0150633,"\int \frac{a c+b c x^2}{x^2 \left(a+b x^2\right)^2} \, dx","Integrate[(a*c + b*c*x^2)/(x^2*(a + b*x^2)^2),x]","c \left(-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2}}-\frac{1}{a x}\right)","-\frac{\sqrt{b} c \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2}}-\frac{c}{a x}",1,"c*(-(1/(a*x)) - (Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(3/2))","A",1
133,1,37,38,0.0073583,"\int \frac{a c+b c x^2}{x^3 \left(a+b x^2\right)^2} \, dx","Integrate[(a*c + b*c*x^2)/(x^3*(a + b*x^2)^2),x]","c \left(\frac{b \log \left(a+b x^2\right)}{2 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{2 a x^2}\right)","\frac{b c \log \left(a+b x^2\right)}{2 a^2}-\frac{b c \log (x)}{a^2}-\frac{c}{2 a x^2}",1,"c*(-1/2*1/(a*x^2) - (b*Log[x])/a^2 + (b*Log[a + b*x^2])/(2*a^2))","A",1
134,1,28,35,0.0089104,"\int \frac{x^3 \left(a c+b c x^2\right)}{\left(a+b x^2\right)^3} \, dx","Integrate[(x^3*(a*c + b*c*x^2))/(a + b*x^2)^3,x]","\frac{c \left(\frac{a}{a+b x^2}+\log \left(a+b x^2\right)\right)}{2 b^2}","\frac{a c}{2 b^2 \left(a+b x^2\right)}+\frac{c \log \left(a+b x^2\right)}{2 b^2}",1,"(c*(a/(a + b*x^2) + Log[a + b*x^2]))/(2*b^2)","A",1
135,1,47,47,0.0222161,"\int \frac{x^2 \left(a c+b c x^2\right)}{\left(a+b x^2\right)^3} \, dx","Integrate[(x^2*(a*c + b*c*x^2))/(a + b*x^2)^3,x]","c \left(\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 \sqrt{a} b^{3/2}}-\frac{x}{2 b \left(a+b x^2\right)}\right)","\frac{c \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 \sqrt{a} b^{3/2}}-\frac{c x}{2 b \left(a+b x^2\right)}",1,"c*(-1/2*x/(b*(a + b*x^2)) + ArcTan[(Sqrt[b]*x)/Sqrt[a]]/(2*Sqrt[a]*b^(3/2)))","A",1
136,1,17,17,0.0023007,"\int \frac{x \left(a c+b c x^2\right)}{\left(a+b x^2\right)^3} \, dx","Integrate[(x*(a*c + b*c*x^2))/(a + b*x^2)^3,x]","-\frac{c}{2 b \left(a+b x^2\right)}","-\frac{c}{2 b \left(a+b x^2\right)}",1,"-1/2*c/(b*(a + b*x^2))","A",1
137,1,47,47,0.0229976,"\int \frac{a c+b c x^2}{\left(a+b x^2\right)^3} \, dx","Integrate[(a*c + b*c*x^2)/(a + b*x^2)^3,x]","c \left(\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} \sqrt{b}}+\frac{x}{2 a \left(a+b x^2\right)}\right)","\frac{c \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} \sqrt{b}}+\frac{c x}{2 a \left(a+b x^2\right)}",1,"c*(x/(2*a*(a + b*x^2)) + ArcTan[(Sqrt[b]*x)/Sqrt[a]]/(2*a^(3/2)*Sqrt[b]))","A",1
138,1,34,41,0.0143873,"\int \frac{a c+b c x^2}{x \left(a+b x^2\right)^3} \, dx","Integrate[(a*c + b*c*x^2)/(x*(a + b*x^2)^3),x]","\frac{c \left(\frac{a}{a+b x^2}-\log \left(a+b x^2\right)+2 \log (x)\right)}{2 a^2}","-\frac{c \log \left(a+b x^2\right)}{2 a^2}+\frac{c \log (x)}{a^2}+\frac{c}{2 a \left(a+b x^2\right)}",1,"(c*(a/(a + b*x^2) + 2*Log[x] - Log[a + b*x^2]))/(2*a^2)","A",1
139,1,56,60,0.0358577,"\int \frac{a c+b c x^2}{x^2 \left(a+b x^2\right)^3} \, dx","Integrate[(a*c + b*c*x^2)/(x^2*(a + b*x^2)^3),x]","c \left(-\frac{3 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{5/2}}-\frac{b x}{2 a^2 \left(a+b x^2\right)}-\frac{1}{a^2 x}\right)","-\frac{3 \sqrt{b} c \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{5/2}}-\frac{3 c}{2 a^2 x}+\frac{c}{2 a x \left(a+b x^2\right)}",1,"c*(-(1/(a^2*x)) - (b*x)/(2*a^2*(a + b*x^2)) - (3*Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(5/2)))","A",1
140,1,42,53,0.0351427,"\int \frac{a c+b c x^2}{x^3 \left(a+b x^2\right)^3} \, dx","Integrate[(a*c + b*c*x^2)/(x^3*(a + b*x^2)^3),x]","-\frac{c \left(a \left(\frac{b}{a+b x^2}+\frac{1}{x^2}\right)-2 b \log \left(a+b x^2\right)+4 b \log (x)\right)}{2 a^3}","\frac{b c \log \left(a+b x^2\right)}{a^3}-\frac{2 b c \log (x)}{a^3}-\frac{b c}{2 a^2 \left(a+b x^2\right)}-\frac{c}{2 a^2 x^2}",1,"-1/2*(c*(a*(x^(-2) + b/(a + b*x^2)) + 4*b*Log[x] - 2*b*Log[a + b*x^2]))/a^3","A",1
141,1,55,55,0.0092112,"\int x^4 \left(a+b x^2\right)^2 \left(c+d x^2\right) \, dx","Integrate[x^4*(a + b*x^2)^2*(c + d*x^2),x]","\frac{1}{5} a^2 c x^5+\frac{1}{9} b x^9 (2 a d+b c)+\frac{1}{7} a x^7 (a d+2 b c)+\frac{1}{11} b^2 d x^{11}","\frac{1}{5} a^2 c x^5+\frac{1}{9} b x^9 (2 a d+b c)+\frac{1}{7} a x^7 (a d+2 b c)+\frac{1}{11} b^2 d x^{11}",1,"(a^2*c*x^5)/5 + (a*(2*b*c + a*d)*x^7)/7 + (b*(b*c + 2*a*d)*x^9)/9 + (b^2*d*x^11)/11","A",1
142,1,55,55,0.0087707,"\int x^3 \left(a+b x^2\right)^2 \left(c+d x^2\right) \, dx","Integrate[x^3*(a + b*x^2)^2*(c + d*x^2),x]","\frac{1}{4} a^2 c x^4+\frac{1}{8} b x^8 (2 a d+b c)+\frac{1}{6} a x^6 (a d+2 b c)+\frac{1}{10} b^2 d x^{10}","\frac{1}{4} a^2 c x^4+\frac{1}{8} b x^8 (2 a d+b c)+\frac{1}{6} a x^6 (a d+2 b c)+\frac{1}{10} b^2 d x^{10}",1,"(a^2*c*x^4)/4 + (a*(2*b*c + a*d)*x^6)/6 + (b*(b*c + 2*a*d)*x^8)/8 + (b^2*d*x^10)/10","A",1
143,1,55,55,0.0080091,"\int x^2 \left(a+b x^2\right)^2 \left(c+d x^2\right) \, dx","Integrate[x^2*(a + b*x^2)^2*(c + d*x^2),x]","\frac{1}{3} a^2 c x^3+\frac{1}{7} b x^7 (2 a d+b c)+\frac{1}{5} a x^5 (a d+2 b c)+\frac{1}{9} b^2 d x^9","\frac{1}{3} a^2 c x^3+\frac{1}{7} b x^7 (2 a d+b c)+\frac{1}{5} a x^5 (a d+2 b c)+\frac{1}{9} b^2 d x^9",1,"(a^2*c*x^3)/3 + (a*(2*b*c + a*d)*x^5)/5 + (b*(b*c + 2*a*d)*x^7)/7 + (b^2*d*x^9)/9","A",1
144,1,51,42,0.0128748,"\int x \left(a+b x^2\right)^2 \left(c+d x^2\right) \, dx","Integrate[x*(a + b*x^2)^2*(c + d*x^2),x]","\frac{1}{24} x^2 \left(12 a^2 c+4 b x^4 (2 a d+b c)+6 a x^2 (a d+2 b c)+3 b^2 d x^6\right)","\frac{\left(a+b x^2\right)^3 (b c-a d)}{6 b^2}+\frac{d \left(a+b x^2\right)^4}{8 b^2}",1,"(x^2*(12*a^2*c + 6*a*(2*b*c + a*d)*x^2 + 4*b*(b*c + 2*a*d)*x^4 + 3*b^2*d*x^6))/24","A",1
145,1,50,50,0.0075618,"\int \left(a+b x^2\right)^2 \left(c+d x^2\right) \, dx","Integrate[(a + b*x^2)^2*(c + d*x^2),x]","a^2 c x+\frac{1}{5} b x^5 (2 a d+b c)+\frac{1}{3} a x^3 (a d+2 b c)+\frac{1}{7} b^2 d x^7","a^2 c x+\frac{1}{5} b x^5 (2 a d+b c)+\frac{1}{3} a x^3 (a d+2 b c)+\frac{1}{7} b^2 d x^7",1,"a^2*c*x + (a*(2*b*c + a*d)*x^3)/3 + (b*(b*c + 2*a*d)*x^5)/5 + (b^2*d*x^7)/7","A",1
146,1,51,43,0.0155262,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)}{x} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2))/x,x]","a^2 c \log (x)+\frac{1}{4} b x^4 (2 a d+b c)+\frac{1}{2} a x^2 (a d+2 b c)+\frac{1}{6} b^2 d x^6","a^2 c \log (x)+a b c x^2+\frac{d \left(a+b x^2\right)^3}{6 b}+\frac{1}{4} b^2 c x^4",1,"(a*(2*b*c + a*d)*x^2)/2 + (b*(b*c + 2*a*d)*x^4)/4 + (b^2*d*x^6)/6 + a^2*c*Log[x]","A",1
147,1,48,48,0.0163594,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)}{x^2} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2))/x^2,x]","-\frac{a^2 c}{x}+\frac{1}{3} b x^3 (2 a d+b c)+a x (a d+2 b c)+\frac{1}{5} b^2 d x^5","-\frac{a^2 c}{x}+\frac{1}{3} b x^3 (2 a d+b c)+a x (a d+2 b c)+\frac{1}{5} b^2 d x^5",1,"-((a^2*c)/x) + a*(2*b*c + a*d)*x + (b*(b*c + 2*a*d)*x^3)/3 + (b^2*d*x^5)/5","A",1
148,1,49,51,0.0235422,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)}{x^3} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2))/x^3,x]","\frac{1}{4} \left(-\frac{2 a^2 c}{x^2}+2 b x^2 (2 a d+b c)+4 a \log (x) (a d+2 b c)+b^2 d x^4\right)","-\frac{a^2 c}{2 x^2}+\frac{1}{2} b x^2 (2 a d+b c)+a \log (x) (a d+2 b c)+\frac{1}{4} b^2 d x^4",1,"((-2*a^2*c)/x^2 + 2*b*(b*c + 2*a*d)*x^2 + b^2*d*x^4 + 4*a*(2*b*c + a*d)*Log[x])/4","A",1
149,1,50,48,0.0194851,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)}{x^4} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2))/x^4,x]","\frac{a^2 (-d)-2 a b c}{x}-\frac{a^2 c}{3 x^3}+b x (2 a d+b c)+\frac{1}{3} b^2 d x^3","-\frac{a^2 c}{3 x^3}+b x (2 a d+b c)-\frac{a (a d+2 b c)}{x}+\frac{1}{3} b^2 d x^3",1,"-1/3*(a^2*c)/x^3 + (-2*a*b*c - a^2*d)/x + b*(b*c + 2*a*d)*x + (b^2*d*x^3)/3","A",1
150,1,87,87,0.0166488,"\int x^4 \left(a+b x^2\right)^2 \left(c+d x^2\right)^2 \, dx","Integrate[x^4*(a + b*x^2)^2*(c + d*x^2)^2,x]","\frac{1}{9} x^9 \left(a^2 d^2+4 a b c d+b^2 c^2\right)+\frac{1}{5} a^2 c^2 x^5+\frac{2}{11} b d x^{11} (a d+b c)+\frac{2}{7} a c x^7 (a d+b c)+\frac{1}{13} b^2 d^2 x^{13}","\frac{1}{9} x^9 \left(a^2 d^2+4 a b c d+b^2 c^2\right)+\frac{1}{5} a^2 c^2 x^5+\frac{2}{11} b d x^{11} (a d+b c)+\frac{2}{7} a c x^7 (a d+b c)+\frac{1}{13} b^2 d^2 x^{13}",1,"(a^2*c^2*x^5)/5 + (2*a*c*(b*c + a*d)*x^7)/7 + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^9)/9 + (2*b*d*(b*c + a*d)*x^11)/11 + (b^2*d^2*x^13)/13","A",1
151,1,81,87,0.0249758,"\int x^3 \left(a+b x^2\right)^2 \left(c+d x^2\right)^2 \, dx","Integrate[x^3*(a + b*x^2)^2*(c + d*x^2)^2,x]","\frac{1}{120} x^4 \left(15 x^4 \left(a^2 d^2+4 a b c d+b^2 c^2\right)+30 a^2 c^2+24 b d x^6 (a d+b c)+40 a c x^2 (a d+b c)+10 b^2 d^2 x^8\right)","\frac{1}{8} x^8 \left(a^2 d^2+4 a b c d+b^2 c^2\right)+\frac{1}{4} a^2 c^2 x^4+\frac{1}{5} b d x^{10} (a d+b c)+\frac{1}{3} a c x^6 (a d+b c)+\frac{1}{12} b^2 d^2 x^{12}",1,"(x^4*(30*a^2*c^2 + 40*a*c*(b*c + a*d)*x^2 + 15*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^4 + 24*b*d*(b*c + a*d)*x^6 + 10*b^2*d^2*x^8))/120","A",1
152,1,87,87,0.0181491,"\int x^2 \left(a+b x^2\right)^2 \left(c+d x^2\right)^2 \, dx","Integrate[x^2*(a + b*x^2)^2*(c + d*x^2)^2,x]","\frac{1}{7} x^7 \left(a^2 d^2+4 a b c d+b^2 c^2\right)+\frac{1}{3} a^2 c^2 x^3+\frac{2}{9} b d x^9 (a d+b c)+\frac{2}{5} a c x^5 (a d+b c)+\frac{1}{11} b^2 d^2 x^{11}","\frac{1}{7} x^7 \left(a^2 d^2+4 a b c d+b^2 c^2\right)+\frac{1}{3} a^2 c^2 x^3+\frac{2}{9} b d x^9 (a d+b c)+\frac{2}{5} a c x^5 (a d+b c)+\frac{1}{11} b^2 d^2 x^{11}",1,"(a^2*c^2*x^3)/3 + (2*a*c*(b*c + a*d)*x^5)/5 + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^7)/7 + (2*b*d*(b*c + a*d)*x^9)/9 + (b^2*d^2*x^11)/11","A",1
153,1,81,71,0.022381,"\int x \left(a+b x^2\right)^2 \left(c+d x^2\right)^2 \, dx","Integrate[x*(a + b*x^2)^2*(c + d*x^2)^2,x]","\frac{1}{60} x^2 \left(10 x^4 \left(a^2 d^2+4 a b c d+b^2 c^2\right)+30 a^2 c^2+15 b d x^6 (a d+b c)+30 a c x^2 (a d+b c)+6 b^2 d^2 x^8\right)","\frac{d \left(a+b x^2\right)^4 (b c-a d)}{4 b^3}+\frac{\left(a+b x^2\right)^3 (b c-a d)^2}{6 b^3}+\frac{d^2 \left(a+b x^2\right)^5}{10 b^3}",1,"(x^2*(30*a^2*c^2 + 30*a*c*(b*c + a*d)*x^2 + 10*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^4 + 15*b*d*(b*c + a*d)*x^6 + 6*b^2*d^2*x^8))/60","A",1
154,1,82,82,0.0162927,"\int \left(a+b x^2\right)^2 \left(c+d x^2\right)^2 \, dx","Integrate[(a + b*x^2)^2*(c + d*x^2)^2,x]","\frac{1}{5} x^5 \left(a^2 d^2+4 a b c d+b^2 c^2\right)+a^2 c^2 x+\frac{2}{7} b d x^7 (a d+b c)+\frac{2}{3} a c x^3 (a d+b c)+\frac{1}{9} b^2 d^2 x^9","\frac{1}{5} x^5 \left(a^2 d^2+4 a b c d+b^2 c^2\right)+a^2 c^2 x+\frac{2}{7} b d x^7 (a d+b c)+\frac{2}{3} a c x^3 (a d+b c)+\frac{1}{9} b^2 d^2 x^9",1,"a^2*c^2*x + (2*a*c*(b*c + a*d)*x^3)/3 + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^5)/5 + (2*b*d*(b*c + a*d)*x^7)/7 + (b^2*d^2*x^9)/9","A",1
155,1,80,80,0.0259595,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2}{x} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^2)/x,x]","\frac{1}{4} x^4 \left(a^2 d^2+4 a b c d+b^2 c^2\right)+a^2 c^2 \log (x)+\frac{1}{3} b d x^6 (a d+b c)+a c x^2 (a d+b c)+\frac{1}{8} b^2 d^2 x^8","\frac{1}{4} x^4 \left(a^2 d^2+4 a b c d+b^2 c^2\right)+a^2 c^2 \log (x)+\frac{1}{3} b d x^6 (a d+b c)+a c x^2 (a d+b c)+\frac{1}{8} b^2 d^2 x^8",1,"a*c*(b*c + a*d)*x^2 + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^4)/4 + (b*d*(b*c + a*d)*x^6)/3 + (b^2*d^2*x^8)/8 + a^2*c^2*Log[x]","A",1
156,1,81,81,0.0401444,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2}{x^2} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^2)/x^2,x]","\frac{1}{3} x^3 \left(a^2 d^2+4 a b c d+b^2 c^2\right)-\frac{a^2 c^2}{x}+\frac{2}{5} b d x^5 (a d+b c)+2 a c x (a d+b c)+\frac{1}{7} b^2 d^2 x^7","\frac{1}{3} x^3 \left(a^2 d^2+4 a b c d+b^2 c^2\right)-\frac{a^2 c^2}{x}+\frac{2}{5} b d x^5 (a d+b c)+2 a c x (a d+b c)+\frac{1}{7} b^2 d^2 x^7",1,"-((a^2*c^2)/x) + 2*a*c*(b*c + a*d)*x + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^3)/3 + (2*b*d*(b*c + a*d)*x^5)/5 + (b^2*d^2*x^7)/7","A",1
157,1,83,84,0.0413059,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2}{x^3} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^2)/x^3,x]","\frac{1}{6} \left(\frac{3 a^2 \left(d^2 x^4-c^2\right)}{x^2}+3 a b d x^2 \left(4 c+d x^2\right)+12 a c \log (x) (a d+b c)+b^2 x^2 \left(3 c^2+3 c d x^2+d^2 x^4\right)\right)","\frac{1}{2} x^2 \left(a^2 d^2+4 a b c d+b^2 c^2\right)-\frac{a^2 c^2}{2 x^2}+\frac{1}{2} b d x^4 (a d+b c)+2 a c \log (x) (a d+b c)+\frac{1}{6} b^2 d^2 x^6",1,"(3*a*b*d*x^2*(4*c + d*x^2) + (3*a^2*(-c^2 + d^2*x^4))/x^2 + b^2*x^2*(3*c^2 + 3*c*d*x^2 + d^2*x^4) + 12*a*c*(b*c + a*d)*Log[x])/6","A",1
158,1,80,80,0.0413465,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2}{x^4} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^2)/x^4,x]","x \left(a^2 d^2+4 a b c d+b^2 c^2\right)-\frac{a^2 c^2}{3 x^3}+\frac{2}{3} b d x^3 (a d+b c)-\frac{2 a c (a d+b c)}{x}+\frac{1}{5} b^2 d^2 x^5","x \left(a^2 d^2+4 a b c d+b^2 c^2\right)-\frac{a^2 c^2}{3 x^3}+\frac{2}{3} b d x^3 (a d+b c)-\frac{2 a c (a d+b c)}{x}+\frac{1}{5} b^2 d^2 x^5",1,"-1/3*(a^2*c^2)/x^3 - (2*a*c*(b*c + a*d))/x + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x + (2*b*d*(b*c + a*d)*x^3)/3 + (b^2*d^2*x^5)/5","A",1
159,1,127,127,0.027077,"\int x^4 \left(a+b x^2\right)^2 \left(c+d x^2\right)^3 \, dx","Integrate[x^4*(a + b*x^2)^2*(c + d*x^2)^3,x]","\frac{1}{11} d x^{11} \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{1}{9} c x^9 \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+\frac{1}{5} a^2 c^3 x^5+\frac{1}{7} a c^2 x^7 (3 a d+2 b c)+\frac{1}{13} b d^2 x^{13} (2 a d+3 b c)+\frac{1}{15} b^2 d^3 x^{15}","\frac{1}{11} d x^{11} \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{1}{9} c x^9 \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+\frac{1}{5} a^2 c^3 x^5+\frac{1}{7} a c^2 x^7 (3 a d+2 b c)+\frac{1}{13} b d^2 x^{13} (2 a d+3 b c)+\frac{1}{15} b^2 d^3 x^{15}",1,"(a^2*c^3*x^5)/5 + (a*c^2*(2*b*c + 3*a*d)*x^7)/7 + (c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^9)/9 + (d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^11)/11 + (b*d^2*(3*b*c + 2*a*d)*x^13)/13 + (b^2*d^3*x^15)/15","A",1
160,1,119,106,0.0326638,"\int x^3 \left(a+b x^2\right)^2 \left(c+d x^2\right)^3 \, dx","Integrate[x^3*(a + b*x^2)^2*(c + d*x^2)^3,x]","\frac{1}{840} x^4 \left(84 d x^6 \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+105 c x^4 \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+210 a^2 c^3+140 a c^2 x^2 (3 a d+2 b c)+70 b d^2 x^8 (2 a d+3 b c)+60 b^2 d^3 x^{10}\right)","-\frac{b \left(c+d x^2\right)^6 (3 b c-2 a d)}{12 d^4}+\frac{\left(c+d x^2\right)^5 (b c-a d) (3 b c-a d)}{10 d^4}-\frac{c \left(c+d x^2\right)^4 (b c-a d)^2}{8 d^4}+\frac{b^2 \left(c+d x^2\right)^7}{14 d^4}",1,"(x^4*(210*a^2*c^3 + 140*a*c^2*(2*b*c + 3*a*d)*x^2 + 105*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^4 + 84*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^6 + 70*b*d^2*(3*b*c + 2*a*d)*x^8 + 60*b^2*d^3*x^10))/840","A",1
161,1,127,127,0.0221861,"\int x^2 \left(a+b x^2\right)^2 \left(c+d x^2\right)^3 \, dx","Integrate[x^2*(a + b*x^2)^2*(c + d*x^2)^3,x]","\frac{1}{9} d x^9 \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{1}{7} c x^7 \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+\frac{1}{3} a^2 c^3 x^3+\frac{1}{5} a c^2 x^5 (3 a d+2 b c)+\frac{1}{11} b d^2 x^{11} (2 a d+3 b c)+\frac{1}{13} b^2 d^3 x^{13}","\frac{1}{9} d x^9 \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{1}{7} c x^7 \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+\frac{1}{3} a^2 c^3 x^3+\frac{1}{5} a c^2 x^5 (3 a d+2 b c)+\frac{1}{11} b d^2 x^{11} (2 a d+3 b c)+\frac{1}{13} b^2 d^3 x^{13}",1,"(a^2*c^3*x^3)/3 + (a*c^2*(2*b*c + 3*a*d)*x^5)/5 + (c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^7)/7 + (d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^9)/9 + (b*d^2*(3*b*c + 2*a*d)*x^11)/11 + (b^2*d^3*x^13)/13","A",1
162,1,119,71,0.0338803,"\int x \left(a+b x^2\right)^2 \left(c+d x^2\right)^3 \, dx","Integrate[x*(a + b*x^2)^2*(c + d*x^2)^3,x]","\frac{1}{120} x^2 \left(15 d x^6 \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+20 c x^4 \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+60 a^2 c^3+30 a c^2 x^2 (3 a d+2 b c)+12 b d^2 x^8 (2 a d+3 b c)+10 b^2 d^3 x^{10}\right)","-\frac{b \left(c+d x^2\right)^5 (b c-a d)}{5 d^3}+\frac{\left(c+d x^2\right)^4 (b c-a d)^2}{8 d^3}+\frac{b^2 \left(c+d x^2\right)^6}{12 d^3}",1,"(x^2*(60*a^2*c^3 + 30*a*c^2*(2*b*c + 3*a*d)*x^2 + 20*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^4 + 15*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^6 + 12*b*d^2*(3*b*c + 2*a*d)*x^8 + 10*b^2*d^3*x^10))/120","A",1
163,1,122,122,0.0211173,"\int \left(a+b x^2\right)^2 \left(c+d x^2\right)^3 \, dx","Integrate[(a + b*x^2)^2*(c + d*x^2)^3,x]","\frac{1}{7} d x^7 \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{1}{5} c x^5 \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+a^2 c^3 x+\frac{1}{3} a c^2 x^3 (3 a d+2 b c)+\frac{1}{9} b d^2 x^9 (2 a d+3 b c)+\frac{1}{11} b^2 d^3 x^{11}","\frac{1}{7} d x^7 \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{1}{5} c x^5 \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+a^2 c^3 x+\frac{1}{3} a c^2 x^3 (3 a d+2 b c)+\frac{1}{9} b d^2 x^9 (2 a d+3 b c)+\frac{1}{11} b^2 d^3 x^{11}",1,"a^2*c^3*x + (a*c^2*(2*b*c + 3*a*d)*x^3)/3 + (c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^5)/5 + (d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^7)/7 + (b*d^2*(3*b*c + 2*a*d)*x^9)/9 + (b^2*d^3*x^11)/11","A",1
164,1,123,123,0.0316036,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3}{x} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^3)/x,x]","\frac{1}{6} d x^6 \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{1}{4} c x^4 \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+a^2 c^3 \log (x)+\frac{1}{2} a c^2 x^2 (3 a d+2 b c)+\frac{1}{8} b d^2 x^8 (2 a d+3 b c)+\frac{1}{10} b^2 d^3 x^{10}","\frac{1}{6} d x^6 \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{1}{4} c x^4 \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+a^2 c^3 \log (x)+\frac{1}{2} a c^2 x^2 (3 a d+2 b c)+\frac{1}{8} b d^2 x^8 (2 a d+3 b c)+\frac{1}{10} b^2 d^3 x^{10}",1,"(a*c^2*(2*b*c + 3*a*d)*x^2)/2 + (c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^4)/4 + (d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^6)/6 + (b*d^2*(3*b*c + 2*a*d)*x^8)/8 + (b^2*d^3*x^10)/10 + a^2*c^3*Log[x]","A",1
165,1,120,120,0.0380576,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3}{x^2} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^3)/x^2,x]","\frac{1}{5} d x^5 \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{1}{3} c x^3 \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)-\frac{a^2 c^3}{x}+a c^2 x (3 a d+2 b c)+\frac{1}{7} b d^2 x^7 (2 a d+3 b c)+\frac{1}{9} b^2 d^3 x^9","\frac{1}{5} d x^5 \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{1}{3} c x^3 \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)-\frac{a^2 c^3}{x}+a c^2 x (3 a d+2 b c)+\frac{1}{7} b d^2 x^7 (2 a d+3 b c)+\frac{1}{9} b^2 d^3 x^9",1,"-((a^2*c^3)/x) + a*c^2*(2*b*c + 3*a*d)*x + (c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^3)/3 + (d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^5)/5 + (b*d^2*(3*b*c + 2*a*d)*x^7)/7 + (b^2*d^3*x^9)/9","A",1
166,1,120,123,0.0467534,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3}{x^3} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^3)/x^3,x]","\frac{6 a^2 \left(-2 c^3+6 c d^2 x^4+d^3 x^6\right)+4 a b d x^4 \left(18 c^2+9 c d x^2+2 d^2 x^4\right)+3 b^2 x^4 \left(4 c^3+6 c^2 d x^2+4 c d^2 x^4+d^3 x^6\right)}{24 x^2}+a c^2 \log (x) (3 a d+2 b c)","\frac{1}{4} d x^4 \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{1}{2} c x^2 \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)-\frac{a^2 c^3}{2 x^2}+a c^2 \log (x) (3 a d+2 b c)+\frac{1}{6} b d^2 x^6 (2 a d+3 b c)+\frac{1}{8} b^2 d^3 x^8",1,"(4*a*b*d*x^4*(18*c^2 + 9*c*d*x^2 + 2*d^2*x^4) + 3*b^2*x^4*(4*c^3 + 6*c^2*d*x^2 + 4*c*d^2*x^4 + d^3*x^6) + 6*a^2*(-2*c^3 + 6*c*d^2*x^4 + d^3*x^6))/(24*x^2) + a*c^2*(2*b*c + 3*a*d)*Log[x]","A",1
167,1,120,120,0.0444433,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3}{x^4} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^3)/x^4,x]","\frac{1}{3} d x^3 \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+c x \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)-\frac{a^2 c^3}{3 x^3}-\frac{a c^2 (3 a d+2 b c)}{x}+\frac{1}{5} b d^2 x^5 (2 a d+3 b c)+\frac{1}{7} b^2 d^3 x^7","\frac{1}{3} d x^3 \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+c x \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)-\frac{a^2 c^3}{3 x^3}-\frac{a c^2 (3 a d+2 b c)}{x}+\frac{1}{5} b d^2 x^5 (2 a d+3 b c)+\frac{1}{7} b^2 d^3 x^7",1,"-1/3*(a^2*c^3)/x^3 - (a*c^2*(2*b*c + 3*a*d))/x + c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x + (d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^3)/3 + (b*d^2*(3*b*c + 2*a*d)*x^5)/5 + (b^2*d^3*x^7)/7","A",1
168,1,104,104,0.0961346,"\int \frac{x^4 \left(a+b x^2\right)^2}{c+d x^2} \, dx","Integrate[(x^4*(a + b*x^2)^2)/(c + d*x^2),x]","\frac{c^{3/2} (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{d^{9/2}}-\frac{c x (b c-a d)^2}{d^4}+\frac{x^3 (a d-b c)^2}{3 d^3}-\frac{b x^5 (b c-2 a d)}{5 d^2}+\frac{b^2 x^7}{7 d}","\frac{c^{3/2} (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{d^{9/2}}-\frac{c x (b c-a d)^2}{d^4}+\frac{x^3 (b c-a d)^2}{3 d^3}-\frac{b x^5 (b c-2 a d)}{5 d^2}+\frac{b^2 x^7}{7 d}",1,"-((c*(b*c - a*d)^2*x)/d^4) + ((-(b*c) + a*d)^2*x^3)/(3*d^3) - (b*(b*c - 2*a*d)*x^5)/(5*d^2) + (b^2*x^7)/(7*d) + (c^(3/2)*(b*c - a*d)^2*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/d^(9/2)","A",1
169,1,82,79,0.0421644,"\int \frac{x^3 \left(a+b x^2\right)^2}{c+d x^2} \, dx","Integrate[(x^3*(a + b*x^2)^2)/(c + d*x^2),x]","\frac{d x^2 \left(6 a^2 d^2+6 a b d \left(d x^2-2 c\right)+b^2 \left(6 c^2-3 c d x^2+2 d^2 x^4\right)\right)-6 c (b c-a d)^2 \log \left(c+d x^2\right)}{12 d^4}","-\frac{c (b c-a d)^2 \log \left(c+d x^2\right)}{2 d^4}+\frac{x^2 (b c-a d)^2}{2 d^3}-\frac{b x^4 (b c-2 a d)}{4 d^2}+\frac{b^2 x^6}{6 d}",1,"(d*x^2*(6*a^2*d^2 + 6*a*b*d*(-2*c + d*x^2) + b^2*(6*c^2 - 3*c*d*x^2 + 2*d^2*x^4)) - 6*c*(b*c - a*d)^2*Log[c + d*x^2])/(12*d^4)","A",1
170,1,83,83,0.0707953,"\int \frac{x^2 \left(a+b x^2\right)^2}{c+d x^2} \, dx","Integrate[(x^2*(a + b*x^2)^2)/(c + d*x^2),x]","-\frac{\sqrt{c} (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{d^{7/2}}+\frac{x (a d-b c)^2}{d^3}-\frac{b x^3 (b c-2 a d)}{3 d^2}+\frac{b^2 x^5}{5 d}","-\frac{\sqrt{c} (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{d^{7/2}}+\frac{x (b c-a d)^2}{d^3}-\frac{b x^3 (b c-2 a d)}{3 d^2}+\frac{b^2 x^5}{5 d}",1,"((-(b*c) + a*d)^2*x)/d^3 - (b*(b*c - 2*a*d)*x^3)/(3*d^2) + (b^2*x^5)/(5*d) - (Sqrt[c]*(b*c - a*d)^2*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/d^(7/2)","A",1
171,1,49,61,0.023825,"\int \frac{x \left(a+b x^2\right)^2}{c+d x^2} \, dx","Integrate[(x*(a + b*x^2)^2)/(c + d*x^2),x]","\frac{b d x^2 \left(4 a d-2 b c+b d x^2\right)+2 (b c-a d)^2 \log \left(c+d x^2\right)}{4 d^3}","\frac{(b c-a d)^2 \log \left(c+d x^2\right)}{2 d^3}-\frac{b x^2 (b c-a d)}{2 d^2}+\frac{\left(a+b x^2\right)^2}{4 d}",1,"(b*d*x^2*(-2*b*c + 4*a*d + b*d*x^2) + 2*(b*c - a*d)^2*Log[c + d*x^2])/(4*d^3)","A",1
172,1,59,63,0.0505691,"\int \frac{\left(a+b x^2\right)^2}{c+d x^2} \, dx","Integrate[(a + b*x^2)^2/(c + d*x^2),x]","\frac{(b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{\sqrt{c} d^{5/2}}+\frac{b x \left(6 a d-3 b c+b d x^2\right)}{3 d^2}","\frac{(b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{\sqrt{c} d^{5/2}}-\frac{b x (b c-2 a d)}{d^2}+\frac{b^2 x^3}{3 d}",1,"(b*x*(-3*b*c + 6*a*d + b*d*x^2))/(3*d^2) + ((b*c - a*d)^2*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*d^(5/2))","A",1
173,1,50,51,0.0281402,"\int \frac{\left(a+b x^2\right)^2}{x \left(c+d x^2\right)} \, dx","Integrate[(a + b*x^2)^2/(x*(c + d*x^2)),x]","\frac{2 a^2 d^2 \log (x)-(b c-a d)^2 \log \left(c+d x^2\right)+b^2 c d x^2}{2 c d^2}","\frac{a^2 \log (x)}{c}-\frac{(b c-a d)^2 \log \left(c+d x^2\right)}{2 c d^2}+\frac{b^2 x^2}{2 d}",1,"(b^2*c*d*x^2 + 2*a^2*d^2*Log[x] - (b*c - a*d)^2*Log[c + d*x^2])/(2*c*d^2)","A",1
174,1,55,55,0.0467835,"\int \frac{\left(a+b x^2\right)^2}{x^2 \left(c+d x^2\right)} \, dx","Integrate[(a + b*x^2)^2/(x^2*(c + d*x^2)),x]","-\frac{a^2}{c x}-\frac{(b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{3/2} d^{3/2}}+\frac{b^2 x}{d}","-\frac{a^2}{c x}-\frac{(b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{3/2} d^{3/2}}+\frac{b^2 x}{d}",1,"-(a^2/(c*x)) + (b^2*x)/d - ((b*c - a*d)^2*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(3/2)*d^(3/2))","A",1
175,1,60,58,0.0306213,"\int \frac{\left(a+b x^2\right)^2}{x^3 \left(c+d x^2\right)} \, dx","Integrate[(a + b*x^2)^2/(x^3*(c + d*x^2)),x]","\frac{a^2 (-c) d-2 a d x^2 \log (x) (a d-2 b c)+x^2 (b c-a d)^2 \log \left(c+d x^2\right)}{2 c^2 d x^2}","-\frac{a^2}{2 c x^2}+\frac{(b c-a d)^2 \log \left(c+d x^2\right)}{2 c^2 d}+\frac{a \log (x) (2 b c-a d)}{c^2}",1,"(-(a^2*c*d) - 2*a*d*(-2*b*c + a*d)*x^2*Log[x] + (b*c - a*d)^2*x^2*Log[c + d*x^2])/(2*c^2*d*x^2)","A",1
176,1,64,66,0.0556933,"\int \frac{\left(a+b x^2\right)^2}{x^4 \left(c+d x^2\right)} \, dx","Integrate[(a + b*x^2)^2/(x^4*(c + d*x^2)),x]","-\frac{a^2}{3 c x^3}+\frac{(b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{5/2} \sqrt{d}}+\frac{a (a d-2 b c)}{c^2 x}","-\frac{a^2}{3 c x^3}+\frac{(b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{5/2} \sqrt{d}}-\frac{a (2 b c-a d)}{c^2 x}",1,"-1/3*a^2/(c*x^3) + (a*(-2*b*c + a*d))/(c^2*x) + ((b*c - a*d)^2*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(5/2)*Sqrt[d])","A",1
177,1,72,75,0.0449235,"\int \frac{\left(a+b x^2\right)^2}{x^5 \left(c+d x^2\right)} \, dx","Integrate[(a + b*x^2)^2/(x^5*(c + d*x^2)),x]","-\frac{-4 x^4 \log (x) (b c-a d)^2+a c \left(a c-2 a d x^2+4 b c x^2\right)+2 x^4 (b c-a d)^2 \log \left(c+d x^2\right)}{4 c^3 x^4}","-\frac{a^2}{4 c x^4}-\frac{(b c-a d)^2 \log \left(c+d x^2\right)}{2 c^3}+\frac{\log (x) (b c-a d)^2}{c^3}-\frac{a (2 b c-a d)}{2 c^2 x^2}",1,"-1/4*(a*c*(a*c + 4*b*c*x^2 - 2*a*d*x^2) - 4*(b*c - a*d)^2*x^4*Log[x] + 2*(b*c - a*d)^2*x^4*Log[c + d*x^2])/(c^3*x^4)","A",1
178,1,86,87,0.0726921,"\int \frac{\left(a+b x^2\right)^2}{x^6 \left(c+d x^2\right)} \, dx","Integrate[(a + b*x^2)^2/(x^6*(c + d*x^2)),x]","-\frac{a^2}{5 c x^5}-\frac{\sqrt{d} (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{7/2}}-\frac{(b c-a d)^2}{c^3 x}+\frac{a (a d-2 b c)}{3 c^2 x^3}","-\frac{a^2}{5 c x^5}-\frac{\sqrt{d} (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{7/2}}-\frac{(b c-a d)^2}{c^3 x}-\frac{a (2 b c-a d)}{3 c^2 x^3}",1,"-1/5*a^2/(c*x^5) + (a*(-2*b*c + a*d))/(3*c^2*x^3) - (b*c - a*d)^2/(c^3*x) - (Sqrt[d]*(b*c - a*d)^2*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/c^(7/2)","A",1
179,1,108,98,0.0635361,"\int \frac{\left(a+b x^2\right)^2}{x^7 \left(c+d x^2\right)} \, dx","Integrate[(a + b*x^2)^2/(x^7*(c + d*x^2)),x]","-\frac{c \left(a^2 \left(2 c^2-3 c d x^2+6 d^2 x^4\right)+6 a b c x^2 \left(c-2 d x^2\right)+6 b^2 c^2 x^4\right)+12 d x^6 \log (x) (b c-a d)^2-6 d x^6 (b c-a d)^2 \log \left(c+d x^2\right)}{12 c^4 x^6}","-\frac{a^2}{6 c x^6}+\frac{d (b c-a d)^2 \log \left(c+d x^2\right)}{2 c^4}-\frac{d \log (x) (b c-a d)^2}{c^4}-\frac{(b c-a d)^2}{2 c^3 x^2}-\frac{a (2 b c-a d)}{4 c^2 x^4}",1,"-1/12*(c*(6*b^2*c^2*x^4 + 6*a*b*c*x^2*(c - 2*d*x^2) + a^2*(2*c^2 - 3*c*d*x^2 + 6*d^2*x^4)) + 12*d*(b*c - a*d)^2*x^6*Log[x] - 6*d*(b*c - a*d)^2*x^6*Log[c + d*x^2])/(c^4*x^6)","A",1
180,1,138,145,0.0952322,"\int \frac{x^4 \left(a+b x^2\right)^2}{\left(c+d x^2\right)^2} \, dx","Integrate[(x^4*(a + b*x^2)^2)/(c + d*x^2)^2,x]","-\frac{\sqrt{c} \left(3 a^2 d^2-10 a b c d+7 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 d^{9/2}}+\frac{x \left(a^2 d^2-4 a b c d+3 b^2 c^2\right)}{d^4}+\frac{c x (b c-a d)^2}{2 d^4 \left(c+d x^2\right)}-\frac{2 b x^3 (b c-a d)}{3 d^3}+\frac{b^2 x^5}{5 d^2}","-\frac{\sqrt{c} (7 b c-3 a d) (b c-a d) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 d^{9/2}}+\frac{x (7 b c-3 a d) (b c-a d)}{2 d^4}-\frac{x^3 (7 b c-3 a d) (b c-a d)}{6 c d^3}+\frac{x^5 (b c-a d)^2}{2 c d^2 \left(c+d x^2\right)}+\frac{b^2 x^5}{5 d^2}",1,"((3*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*x)/d^4 - (2*b*(b*c - a*d)*x^3)/(3*d^3) + (b^2*x^5)/(5*d^2) + (c*(b*c - a*d)^2*x)/(2*d^4*(c + d*x^2)) - (Sqrt[c]*(7*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*d^(9/2))","A",1
181,1,87,90,0.0625316,"\int \frac{x^3 \left(a+b x^2\right)^2}{\left(c+d x^2\right)^2} \, dx","Integrate[(x^3*(a + b*x^2)^2)/(c + d*x^2)^2,x]","\frac{2 \left(a^2 d^2-4 a b c d+3 b^2 c^2\right) \log \left(c+d x^2\right)+4 b d x^2 (a d-b c)+\frac{2 c (b c-a d)^2}{c+d x^2}+b^2 d^2 x^4}{4 d^4}","\frac{c (b c-a d)^2}{2 d^4 \left(c+d x^2\right)}+\frac{(b c-a d) (3 b c-a d) \log \left(c+d x^2\right)}{2 d^4}-\frac{b x^2 (b c-a d)}{d^3}+\frac{b^2 x^4}{4 d^2}",1,"(4*b*d*(-(b*c) + a*d)*x^2 + b^2*d^2*x^4 + (2*c*(b*c - a*d)^2)/(c + d*x^2) + 2*(3*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*Log[c + d*x^2])/(4*d^4)","A",1
182,1,105,118,0.0721242,"\int \frac{x^2 \left(a+b x^2\right)^2}{\left(c+d x^2\right)^2} \, dx","Integrate[(x^2*(a + b*x^2)^2)/(c + d*x^2)^2,x]","\frac{\left(a^2 d^2-6 a b c d+5 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 \sqrt{c} d^{7/2}}-\frac{x (b c-a d)^2}{2 d^3 \left(c+d x^2\right)}-\frac{2 b x (b c-a d)}{d^3}+\frac{b^2 x^3}{3 d^2}","\frac{(b c-a d) (5 b c-a d) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 \sqrt{c} d^{7/2}}-\frac{x (b c-a d) (5 b c-a d)}{2 c d^3}+\frac{x^3 (b c-a d)^2}{2 c d^2 \left(c+d x^2\right)}+\frac{b^2 x^3}{3 d^2}",1,"(-2*b*(b*c - a*d)*x)/d^3 + (b^2*x^3)/(3*d^2) - ((b*c - a*d)^2*x)/(2*d^3*(c + d*x^2)) + ((5*b^2*c^2 - 6*a*b*c*d + a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*Sqrt[c]*d^(7/2))","A",1
183,1,56,62,0.0448795,"\int \frac{x \left(a+b x^2\right)^2}{\left(c+d x^2\right)^2} \, dx","Integrate[(x*(a + b*x^2)^2)/(c + d*x^2)^2,x]","\frac{-\frac{(b c-a d)^2}{c+d x^2}+2 b (a d-b c) \log \left(c+d x^2\right)+b^2 d x^2}{2 d^3}","-\frac{(b c-a d)^2}{2 d^3 \left(c+d x^2\right)}-\frac{b (b c-a d) \log \left(c+d x^2\right)}{d^3}+\frac{b^2 x^2}{2 d^2}",1,"(b^2*d*x^2 - (b*c - a*d)^2/(c + d*x^2) + 2*b*(-(b*c) + a*d)*Log[c + d*x^2])/(2*d^3)","A",1
184,1,89,82,0.0599042,"\int \frac{\left(a+b x^2\right)^2}{\left(c+d x^2\right)^2} \, dx","Integrate[(a + b*x^2)^2/(c + d*x^2)^2,x]","-\frac{\left(-a^2 d^2-2 a b c d+3 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{3/2} d^{5/2}}+\frac{x (b c-a d)^2}{2 c d^2 \left(c+d x^2\right)}+\frac{b^2 x}{d^2}","-\frac{(b c-a d) (a d+3 b c) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{3/2} d^{5/2}}+\frac{x (b c-a d)^2}{2 c d^2 \left(c+d x^2\right)}+\frac{b^2 x}{d^2}",1,"(b^2*x)/d^2 + ((b*c - a*d)^2*x)/(2*c*d^2*(c + d*x^2)) - ((3*b^2*c^2 - 2*a*b*c*d - a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(3/2)*d^(5/2))","A",1
185,1,70,67,0.0437825,"\int \frac{\left(a+b x^2\right)^2}{x \left(c+d x^2\right)^2} \, dx","Integrate[(a + b*x^2)^2/(x*(c + d*x^2)^2),x]","\frac{2 a^2 \log (x)+\frac{(b c-a d) \left(\left(c+d x^2\right) (a d+b c) \log \left(c+d x^2\right)+c (b c-a d)\right)}{d^2 \left(c+d x^2\right)}}{2 c^2}","-\frac{1}{2} \left(\frac{a^2}{c^2}-\frac{b^2}{d^2}\right) \log \left(c+d x^2\right)+\frac{a^2 \log (x)}{c^2}+\frac{(b c-a d)^2}{2 c d^2 \left(c+d x^2\right)}",1,"(2*a^2*Log[x] + ((b*c - a*d)*(c*(b*c - a*d) + (b*c + a*d)*(c + d*x^2)*Log[c + d*x^2]))/(d^2*(c + d*x^2)))/(2*c^2)","A",1
186,1,91,106,0.0598546,"\int \frac{\left(a+b x^2\right)^2}{x^2 \left(c+d x^2\right)^2} \, dx","Integrate[(a + b*x^2)^2/(x^2*(c + d*x^2)^2),x]","\frac{\left(-3 a^2 d^2+2 a b c d+b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{5/2} d^{3/2}}-\frac{a^2}{c^2 x}-\frac{x (b c-a d)^2}{2 c^2 d \left(c+d x^2\right)}","-\frac{x \left(3 a^2 d^2-2 a b c d+b^2 c^2\right)}{2 c^2 d \left(c+d x^2\right)}-\frac{a^2}{c x \left(c+d x^2\right)}+\frac{(b c-a d) (3 a d+b c) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{5/2} d^{3/2}}",1,"-(a^2/(c^2*x)) - ((b*c - a*d)^2*x)/(2*c^2*d*(c + d*x^2)) + ((b^2*c^2 + 2*a*b*c*d - 3*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(5/2)*d^(3/2))","A",1
187,1,72,81,0.0951443,"\int \frac{\left(a+b x^2\right)^2}{x^3 \left(c+d x^2\right)^2} \, dx","Integrate[(a + b*x^2)^2/(x^3*(c + d*x^2)^2),x]","-\frac{\frac{a^2 c}{x^2}+\frac{c (b c-a d)^2}{d \left(c+d x^2\right)}-2 a (a d-b c) \log \left(c+d x^2\right)+4 a \log (x) (a d-b c)}{2 c^3}","-\frac{a^2}{2 c^2 x^2}-\frac{a (b c-a d) \log \left(c+d x^2\right)}{c^3}+\frac{2 a \log (x) (b c-a d)}{c^3}-\frac{(b c-a d)^2}{2 c^2 d \left(c+d x^2\right)}",1,"-1/2*((a^2*c)/x^2 + (c*(b*c - a*d)^2)/(d*(c + d*x^2)) + 4*a*(-(b*c) + a*d)*Log[x] - 2*a*(-(b*c) + a*d)*Log[c + d*x^2])/c^3","A",1
188,1,107,126,0.065464,"\int \frac{\left(a+b x^2\right)^2}{x^4 \left(c+d x^2\right)^2} \, dx","Integrate[(a + b*x^2)^2/(x^4*(c + d*x^2)^2),x]","\frac{\left(5 a^2 d^2-6 a b c d+b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{7/2} \sqrt{d}}-\frac{a^2}{3 c^2 x^3}+\frac{x (b c-a d)^2}{2 c^3 \left(c+d x^2\right)}+\frac{2 a (a d-b c)}{c^3 x}","\frac{x \left(5 a^2 d^2-6 a b c d+3 b^2 c^2\right)}{6 c^3 \left(c+d x^2\right)}-\frac{a^2}{3 c x^3 \left(c+d x^2\right)}+\frac{(b c-5 a d) (b c-a d) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{7/2} \sqrt{d}}-\frac{a (6 b c-5 a d)}{3 c^3 x}",1,"-1/3*a^2/(c^2*x^3) + (2*a*(-(b*c) + a*d))/(c^3*x) + ((b*c - a*d)^2*x)/(2*c^3*(c + d*x^2)) + ((b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(7/2)*Sqrt[d])","A",1
189,1,148,163,0.0932289,"\int \frac{x^4 \left(a+b x^2\right)^2}{\left(c+d x^2\right)^3} \, dx","Integrate[(x^4*(a + b*x^2)^2)/(c + d*x^2)^3,x]","\frac{\left(3 a^2 d^2-30 a b c d+35 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 \sqrt{c} d^{9/2}}-\frac{x \left(5 a^2 d^2-18 a b c d+13 b^2 c^2\right)}{8 d^4 \left(c+d x^2\right)}+\frac{c x (b c-a d)^2}{4 d^4 \left(c+d x^2\right)^2}-\frac{b x (3 b c-2 a d)}{d^4}+\frac{b^2 x^3}{3 d^3}","\frac{\left(3 a^2 d^2-30 a b c d+35 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 \sqrt{c} d^{9/2}}-\frac{x \left(a^2 d^2-10 a b c d+13 b^2 c^2\right)}{4 c d^4}-\frac{x (b c-a d) (9 b c-a d)}{8 d^4 \left(c+d x^2\right)}+\frac{x^5 (b c-a d)^2}{4 c d^2 \left(c+d x^2\right)^2}+\frac{b^2 x^3}{3 d^3}",1,"-((b*(3*b*c - 2*a*d)*x)/d^4) + (b^2*x^3)/(3*d^3) + (c*(b*c - a*d)^2*x)/(4*d^4*(c + d*x^2)^2) - ((13*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*x)/(8*d^4*(c + d*x^2)) + ((35*b^2*c^2 - 30*a*b*c*d + 3*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*Sqrt[c]*d^(9/2))","A",1
190,1,114,99,0.0503409,"\int \frac{x^3 \left(a+b x^2\right)^2}{\left(c+d x^2\right)^3} \, dx","Integrate[(x^3*(a + b*x^2)^2)/(c + d*x^2)^3,x]","\frac{-a^2 d^2 \left(c+2 d x^2\right)+2 a b c d \left(3 c+4 d x^2\right)-2 b \left(c+d x^2\right)^2 (3 b c-2 a d) \log \left(c+d x^2\right)+b^2 \left(-5 c^3-4 c^2 d x^2+4 c d^2 x^4+2 d^3 x^6\right)}{4 d^4 \left(c+d x^2\right)^2}","\frac{c (b c-a d)^2}{4 d^4 \left(c+d x^2\right)^2}-\frac{(3 b c-a d) (b c-a d)}{2 d^4 \left(c+d x^2\right)}-\frac{b (3 b c-2 a d) \log \left(c+d x^2\right)}{2 d^4}+\frac{b^2 x^2}{2 d^3}",1,"(-(a^2*d^2*(c + 2*d*x^2)) + 2*a*b*c*d*(3*c + 4*d*x^2) + b^2*(-5*c^3 - 4*c^2*d*x^2 + 4*c*d^2*x^4 + 2*d^3*x^6) - 2*b*(3*b*c - 2*a*d)*(c + d*x^2)^2*Log[c + d*x^2])/(4*d^4*(c + d*x^2)^2)","A",1
191,1,130,127,0.1001729,"\int \frac{x^2 \left(a+b x^2\right)^2}{\left(c+d x^2\right)^3} \, dx","Integrate[(x^2*(a + b*x^2)^2)/(c + d*x^2)^3,x]","\frac{x \left(a^2 d^2 \left(d x^2-c\right)-2 a b c d \left(3 c+5 d x^2\right)+b^2 c \left(15 c^2+25 c d x^2+8 d^2 x^4\right)\right)}{8 c d^3 \left(c+d x^2\right)^2}-\frac{\left(-a^2 d^2-6 a b c d+15 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{3/2} d^{7/2}}","-\frac{\left(-a^2 d^2-6 a b c d+15 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{3/2} d^{7/2}}+\frac{x (b c-a d) (a d+7 b c)}{8 c d^3 \left(c+d x^2\right)}+\frac{x^3 (b c-a d)^2}{4 c d^2 \left(c+d x^2\right)^2}+\frac{b^2 x}{d^3}",1,"(x*(a^2*d^2*(-c + d*x^2) - 2*a*b*c*d*(3*c + 5*d*x^2) + b^2*c*(15*c^2 + 25*c*d*x^2 + 8*d^2*x^4)))/(8*c*d^3*(c + d*x^2)^2) - ((15*b^2*c^2 - 6*a*b*c*d - a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(3/2)*d^(7/2))","A",1
192,1,75,67,0.0255301,"\int \frac{x \left(a+b x^2\right)^2}{\left(c+d x^2\right)^3} \, dx","Integrate[(x*(a + b*x^2)^2)/(c + d*x^2)^3,x]","\frac{-a^2 d^2-2 a b d \left(c+2 d x^2\right)+b^2 c \left(3 c+4 d x^2\right)+2 b^2 \left(c+d x^2\right)^2 \log \left(c+d x^2\right)}{4 d^3 \left(c+d x^2\right)^2}","\frac{b (b c-a d)}{d^3 \left(c+d x^2\right)}-\frac{(b c-a d)^2}{4 d^3 \left(c+d x^2\right)^2}+\frac{b^2 \log \left(c+d x^2\right)}{2 d^3}",1,"(-(a^2*d^2) - 2*a*b*d*(c + 2*d*x^2) + b^2*c*(3*c + 4*d*x^2) + 2*b^2*(c + d*x^2)^2*Log[c + d*x^2])/(4*d^3*(c + d*x^2)^2)","A",1
193,1,121,116,0.0973221,"\int \frac{\left(a+b x^2\right)^2}{\left(c+d x^2\right)^3} \, dx","Integrate[(a + b*x^2)^2/(c + d*x^2)^3,x]","\frac{x \left(a^2 d^2 \left(5 c+3 d x^2\right)-2 a b c d \left(c-d x^2\right)-b^2 c^2 \left(3 c+5 d x^2\right)\right)}{8 c^2 d^2 \left(c+d x^2\right)^2}+\frac{\left(3 a^2 d^2+2 a b c d+3 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{5/2} d^{5/2}}","\frac{3 x \left(\frac{a^2}{c^2}-\frac{b^2}{d^2}\right)}{8 \left(c+d x^2\right)}+\frac{\left(3 a^2 d^2+2 a b c d+3 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{5/2} d^{5/2}}-\frac{x \left(a+b x^2\right) (b c-a d)}{4 c d \left(c+d x^2\right)^2}",1,"(x*(-2*a*b*c*d*(c - d*x^2) + a^2*d^2*(5*c + 3*d*x^2) - b^2*c^2*(3*c + 5*d*x^2)))/(8*c^2*d^2*(c + d*x^2)^2) + ((3*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(5/2)*d^(5/2))","A",1
194,1,103,86,0.0467953,"\int \frac{\left(a+b x^2\right)^2}{x \left(c+d x^2\right)^3} \, dx","Integrate[(a + b*x^2)^2/(x*(c + d*x^2)^3),x]","\frac{a^2 d^2-b^2 c^2}{2 c^2 d^2 \left(c+d x^2\right)}+\frac{a^2 d^2-2 a b c d+b^2 c^2}{4 c d^2 \left(c+d x^2\right)^2}-\frac{a^2 \log \left(c+d x^2\right)}{2 c^3}+\frac{a^2 \log (x)}{c^3}","\frac{\frac{a^2}{c^2}-\frac{b^2}{d^2}}{2 \left(c+d x^2\right)}-\frac{a^2 \log \left(c+d x^2\right)}{2 c^3}+\frac{a^2 \log (x)}{c^3}+\frac{(b c-a d)^2}{4 c d^2 \left(c+d x^2\right)^2}",1,"(b^2*c^2 - 2*a*b*c*d + a^2*d^2)/(4*c*d^2*(c + d*x^2)^2) + (-(b^2*c^2) + a^2*d^2)/(2*c^2*d^2*(c + d*x^2)) + (a^2*Log[x])/c^3 - (a^2*Log[c + d*x^2])/(2*c^3)","A",1
195,1,133,152,0.0899252,"\int \frac{\left(a+b x^2\right)^2}{x^2 \left(c+d x^2\right)^3} \, dx","Integrate[(a + b*x^2)^2/(x^2*(c + d*x^2)^3),x]","\frac{\left(-15 a^2 d^2+6 a b c d+b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{7/2} d^{3/2}}+\frac{x \left(-7 a^2 d^2+6 a b c d+b^2 c^2\right)}{8 c^3 d \left(c+d x^2\right)}-\frac{a^2}{c^3 x}-\frac{x (b c-a d)^2}{4 c^2 d \left(c+d x^2\right)^2}","-\frac{x \left(5 a^2 d^2-2 a b c d+b^2 c^2\right)}{4 c^2 d \left(c+d x^2\right)^2}-\frac{a^2}{c x \left(c+d x^2\right)^2}+\frac{\left(3 a d (2 b c-5 a d)+b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{7/2} d^{3/2}}+\frac{x \left(3 a d (2 b c-5 a d)+b^2 c^2\right)}{8 c^3 d \left(c+d x^2\right)}",1,"-(a^2/(c^3*x)) - ((b*c - a*d)^2*x)/(4*c^2*d*(c + d*x^2)^2) + ((b^2*c^2 + 6*a*b*c*d - 7*a^2*d^2)*x)/(8*c^3*d*(c + d*x^2)) + ((b^2*c^2 + 6*a*b*c*d - 15*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(7/2)*d^(3/2))","A",1
196,1,99,106,0.0907294,"\int \frac{\left(a+b x^2\right)^2}{x^3 \left(c+d x^2\right)^3} \, dx","Integrate[(a + b*x^2)^2/(x^3*(c + d*x^2)^3),x]","\frac{-\frac{2 a^2 c}{x^2}-\frac{c^2 (b c-a d)^2}{d \left(c+d x^2\right)^2}+\frac{4 a c (b c-a d)}{c+d x^2}+2 a (3 a d-2 b c) \log \left(c+d x^2\right)+4 a \log (x) (2 b c-3 a d)}{4 c^4}","-\frac{a^2}{2 c^3 x^2}-\frac{a (2 b c-3 a d) \log \left(c+d x^2\right)}{2 c^4}+\frac{a \log (x) (2 b c-3 a d)}{c^4}+\frac{a (b c-a d)}{c^3 \left(c+d x^2\right)}-\frac{(b c-a d)^2}{4 c^2 d \left(c+d x^2\right)^2}",1,"((-2*a^2*c)/x^2 - (c^2*(b*c - a*d)^2)/(d*(c + d*x^2)^2) + (4*a*c*(b*c - a*d))/(c + d*x^2) + 4*a*(2*b*c - 3*a*d)*Log[x] + 2*a*(-2*b*c + 3*a*d)*Log[c + d*x^2])/(4*c^4)","A",1
197,1,148,161,0.0744684,"\int \frac{\left(a+b x^2\right)^2}{x^4 \left(c+d x^2\right)^3} \, dx","Integrate[(a + b*x^2)^2/(x^4*(c + d*x^2)^3),x]","\frac{\left(35 a^2 d^2-30 a b c d+3 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{9/2} \sqrt{d}}+\frac{x \left(11 a^2 d^2-14 a b c d+3 b^2 c^2\right)}{8 c^4 \left(c+d x^2\right)}-\frac{a^2}{3 c^3 x^3}+\frac{a (3 a d-2 b c)}{c^4 x}+\frac{x (b c-a d)^2}{4 c^3 \left(c+d x^2\right)^2}","\frac{\left(35 a^2 d^2-30 a b c d+3 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{9/2} \sqrt{d}}+\frac{x \left(7 a^2 d^2-6 a b c d+3 b^2 c^2\right)}{12 c^3 \left(c+d x^2\right)^2}-\frac{a^2}{3 c x^3 \left(c+d x^2\right)^2}+\frac{x (3 b c-7 a d)^2}{24 c^4 \left(c+d x^2\right)}-\frac{a (6 b c-7 a d)}{3 c^4 x}",1,"-1/3*a^2/(c^3*x^3) + (a*(-2*b*c + 3*a*d))/(c^4*x) + ((b*c - a*d)^2*x)/(4*c^3*(c + d*x^2)^2) + ((3*b^2*c^2 - 14*a*b*c*d + 11*a^2*d^2)*x)/(8*c^4*(c + d*x^2)) + ((3*b^2*c^2 - 30*a*b*c*d + 35*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(9/2)*Sqrt[d])","A",1
198,1,71,75,0.0298406,"\int \frac{x^5 \left(c+d x^2\right)}{a+b x^2} \, dx","Integrate[(x^5*(c + d*x^2))/(a + b*x^2),x]","\frac{b x^2 \left(6 a^2 d-3 a b \left(2 c+d x^2\right)+b^2 x^2 \left(3 c+2 d x^2\right)\right)+6 a^2 (b c-a d) \log \left(a+b x^2\right)}{12 b^4}","\frac{a^2 (b c-a d) \log \left(a+b x^2\right)}{2 b^4}-\frac{a x^2 (b c-a d)}{2 b^3}+\frac{x^4 (b c-a d)}{4 b^2}+\frac{d x^6}{6 b}",1,"(b*x^2*(6*a^2*d - 3*a*b*(2*c + d*x^2) + b^2*x^2*(3*c + 2*d*x^2)) + 6*a^2*(b*c - a*d)*Log[a + b*x^2])/(12*b^4)","A",1
199,1,77,77,0.049848,"\int \frac{x^4 \left(c+d x^2\right)}{a+b x^2} \, dx","Integrate[(x^4*(c + d*x^2))/(a + b*x^2),x]","-\frac{a^{3/2} (a d-b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{7/2}}+\frac{a x (a d-b c)}{b^3}+\frac{x^3 (b c-a d)}{3 b^2}+\frac{d x^5}{5 b}","\frac{a^{3/2} (b c-a d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{7/2}}-\frac{a x (b c-a d)}{b^3}+\frac{x^3 (b c-a d)}{3 b^2}+\frac{d x^5}{5 b}",1,"(a*(-(b*c) + a*d)*x)/b^3 + ((b*c - a*d)*x^3)/(3*b^2) + (d*x^5)/(5*b) - (a^(3/2)*(-(b*c) + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(7/2)","A",1
200,1,47,54,0.019187,"\int \frac{x^3 \left(c+d x^2\right)}{a+b x^2} \, dx","Integrate[(x^3*(c + d*x^2))/(a + b*x^2),x]","\frac{b x^2 \left(-2 a d+2 b c+b d x^2\right)+2 a (a d-b c) \log \left(a+b x^2\right)}{4 b^3}","-\frac{a (b c-a d) \log \left(a+b x^2\right)}{2 b^3}+\frac{x^2 (b c-a d)}{2 b^2}+\frac{d x^4}{4 b}",1,"(b*x^2*(2*b*c - 2*a*d + b*d*x^2) + 2*a*(-(b*c) + a*d)*Log[a + b*x^2])/(4*b^3)","A",1
201,1,57,58,0.0390718,"\int \frac{x^2 \left(c+d x^2\right)}{a+b x^2} \, dx","Integrate[(x^2*(c + d*x^2))/(a + b*x^2),x]","\frac{\sqrt{a} (a d-b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{5/2}}+\frac{x (b c-a d)}{b^2}+\frac{d x^3}{3 b}","-\frac{\sqrt{a} (b c-a d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{5/2}}+\frac{x (b c-a d)}{b^2}+\frac{d x^3}{3 b}",1,"((b*c - a*d)*x)/b^2 + (d*x^3)/(3*b) + (Sqrt[a]*(-(b*c) + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(5/2)","A",1
202,1,31,35,0.0116124,"\int \frac{x \left(c+d x^2\right)}{a+b x^2} \, dx","Integrate[(x*(c + d*x^2))/(a + b*x^2),x]","\frac{(b c-a d) \log \left(a+b x^2\right)+b d x^2}{2 b^2}","\frac{(b c-a d) \log \left(a+b x^2\right)}{2 b^2}+\frac{d x^2}{2 b}",1,"(b*d*x^2 + (b*c - a*d)*Log[a + b*x^2])/(2*b^2)","A",1
203,1,40,39,0.0252243,"\int \frac{c+d x^2}{a+b x^2} \, dx","Integrate[(c + d*x^2)/(a + b*x^2),x]","\frac{d x}{b}-\frac{(a d-b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} b^{3/2}}","\frac{(b c-a d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} b^{3/2}}+\frac{d x}{b}",1,"(d*x)/b - ((-(b*c) + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*b^(3/2))","A",1
204,1,34,34,0.0122024,"\int \frac{c+d x^2}{x \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2)/(x*(a + b*x^2)),x]","\frac{(a d-b c) \log \left(a+b x^2\right)}{2 a b}+\frac{c \log (x)}{a}","\frac{c \log (x)}{a}-\frac{(b c-a d) \log \left(a+b x^2\right)}{2 a b}",1,"(c*Log[x])/a + ((-(b*c) + a*d)*Log[a + b*x^2])/(2*a*b)","A",1
205,1,42,43,0.0246961,"\int \frac{c+d x^2}{x^2 \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2)/(x^2*(a + b*x^2)),x]","\frac{(a d-b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2} \sqrt{b}}-\frac{c}{a x}","-\frac{(b c-a d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2} \sqrt{b}}-\frac{c}{a x}",1,"-(c/(a*x)) + ((-(b*c) + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*Sqrt[b])","A",1
206,1,49,50,0.0221143,"\int \frac{c+d x^2}{x^3 \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2)/(x^3*(a + b*x^2)),x]","\frac{(b c-a d) \log \left(a+b x^2\right)}{2 a^2}+\frac{\log (x) (a d-b c)}{a^2}-\frac{c}{2 a x^2}","\frac{(b c-a d) \log \left(a+b x^2\right)}{2 a^2}-\frac{\log (x) (b c-a d)}{a^2}-\frac{c}{2 a x^2}",1,"-1/2*c/(a*x^2) + ((-(b*c) + a*d)*Log[x])/a^2 + ((b*c - a*d)*Log[a + b*x^2])/(2*a^2)","A",1
207,1,60,59,0.0539545,"\int \frac{c+d x^2}{x^4 \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2)/(x^4*(a + b*x^2)),x]","-\frac{\sqrt{b} (a d-b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{5/2}}+\frac{b c-a d}{a^2 x}-\frac{c}{3 a x^3}","\frac{\sqrt{b} (b c-a d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{5/2}}+\frac{b c-a d}{a^2 x}-\frac{c}{3 a x^3}",1,"-1/3*c/(a*x^3) + (b*c - a*d)/(a^2*x) - (Sqrt[b]*(-(b*c) + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(5/2)","A",1
208,1,116,103,0.0531145,"\int \frac{x^5 \left(c+d x^2\right)^2}{a+b x^2} \, dx","Integrate[(x^5*(c + d*x^2)^2)/(a + b*x^2),x]","\frac{\left(a^4 d^2-2 a^3 b c d+a^2 b^2 c^2\right) \log \left(a+b x^2\right)}{2 b^5}-\frac{a x^2 (a d-b c)^2}{2 b^4}+\frac{x^4 (b c-a d)^2}{4 b^3}+\frac{d x^6 (2 b c-a d)}{6 b^2}+\frac{d^2 x^8}{8 b}","\frac{a^2 (b c-a d)^2 \log \left(a+b x^2\right)}{2 b^5}-\frac{a x^2 (b c-a d)^2}{2 b^4}+\frac{x^4 (b c-a d)^2}{4 b^3}+\frac{d x^6 (2 b c-a d)}{6 b^2}+\frac{d^2 x^8}{8 b}",1,"-1/2*(a*(-(b*c) + a*d)^2*x^2)/b^4 + ((b*c - a*d)^2*x^4)/(4*b^3) + (d*(2*b*c - a*d)*x^6)/(6*b^2) + (d^2*x^8)/(8*b) + ((a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)*Log[a + b*x^2])/(2*b^5)","A",1
209,1,105,105,0.0971402,"\int \frac{x^4 \left(c+d x^2\right)^2}{a+b x^2} \, dx","Integrate[(x^4*(c + d*x^2)^2)/(a + b*x^2),x]","\frac{a^{3/2} (a d-b c)^2 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{9/2}}-\frac{a x (a d-b c)^2}{b^4}+\frac{x^3 (b c-a d)^2}{3 b^3}+\frac{d x^5 (2 b c-a d)}{5 b^2}+\frac{d^2 x^7}{7 b}","\frac{a^{3/2} (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{9/2}}-\frac{a x (b c-a d)^2}{b^4}+\frac{x^3 (b c-a d)^2}{3 b^3}+\frac{d x^5 (2 b c-a d)}{5 b^2}+\frac{d^2 x^7}{7 b}",1,"-((a*(-(b*c) + a*d)^2*x)/b^4) + ((b*c - a*d)^2*x^3)/(3*b^3) + (d*(2*b*c - a*d)*x^5)/(5*b^2) + (d^2*x^7)/(7*b) + (a^(3/2)*(-(b*c) + a*d)^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(9/2)","A",1
210,1,82,80,0.0391461,"\int \frac{x^3 \left(c+d x^2\right)^2}{a+b x^2} \, dx","Integrate[(x^3*(c + d*x^2)^2)/(a + b*x^2),x]","\frac{b x^2 \left(6 a^2 d^2-3 a b d \left(4 c+d x^2\right)+2 b^2 \left(3 c^2+3 c d x^2+d^2 x^4\right)\right)-6 a (b c-a d)^2 \log \left(a+b x^2\right)}{12 b^4}","-\frac{a (b c-a d)^2 \log \left(a+b x^2\right)}{2 b^4}+\frac{x^2 (b c-a d)^2}{2 b^3}+\frac{d x^4 (2 b c-a d)}{4 b^2}+\frac{d^2 x^6}{6 b}",1,"(b*x^2*(6*a^2*d^2 - 3*a*b*d*(4*c + d*x^2) + 2*b^2*(3*c^2 + 3*c*d*x^2 + d^2*x^4)) - 6*a*(b*c - a*d)^2*Log[a + b*x^2])/(12*b^4)","A",1
211,1,84,84,0.074234,"\int \frac{x^2 \left(c+d x^2\right)^2}{a+b x^2} \, dx","Integrate[(x^2*(c + d*x^2)^2)/(a + b*x^2),x]","-\frac{\sqrt{a} (a d-b c)^2 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{7/2}}+\frac{x (b c-a d)^2}{b^3}+\frac{d x^3 (2 b c-a d)}{3 b^2}+\frac{d^2 x^5}{5 b}","-\frac{\sqrt{a} (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{7/2}}+\frac{x (b c-a d)^2}{b^3}+\frac{d x^3 (2 b c-a d)}{3 b^2}+\frac{d^2 x^5}{5 b}",1,"((b*c - a*d)^2*x)/b^3 + (d*(2*b*c - a*d)*x^3)/(3*b^2) + (d^2*x^5)/(5*b) - (Sqrt[a]*(-(b*c) + a*d)^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(7/2)","A",1
212,1,49,61,0.022952,"\int \frac{x \left(c+d x^2\right)^2}{a+b x^2} \, dx","Integrate[(x*(c + d*x^2)^2)/(a + b*x^2),x]","\frac{b d x^2 \left(-2 a d+4 b c+b d x^2\right)+2 (b c-a d)^2 \log \left(a+b x^2\right)}{4 b^3}","\frac{(b c-a d)^2 \log \left(a+b x^2\right)}{2 b^3}+\frac{d x^2 (b c-a d)}{2 b^2}+\frac{\left(c+d x^2\right)^2}{4 b}",1,"(b*d*x^2*(4*b*c - 2*a*d + b*d*x^2) + 2*(b*c - a*d)^2*Log[a + b*x^2])/(4*b^3)","A",1
213,1,59,63,0.0511927,"\int \frac{\left(c+d x^2\right)^2}{a+b x^2} \, dx","Integrate[(c + d*x^2)^2/(a + b*x^2),x]","\frac{(b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} b^{5/2}}+\frac{d x \left(-3 a d+6 b c+b d x^2\right)}{3 b^2}","\frac{(b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} b^{5/2}}+\frac{d x (2 b c-a d)}{b^2}+\frac{d^2 x^3}{3 b}",1,"(d*x*(6*b*c - 3*a*d + b*d*x^2))/(3*b^2) + ((b*c - a*d)^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*b^(5/2))","A",1
214,1,50,51,0.0270622,"\int \frac{\left(c+d x^2\right)^2}{x \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2)^2/(x*(a + b*x^2)),x]","\frac{-(b c-a d)^2 \log \left(a+b x^2\right)+a b d^2 x^2+2 b^2 c^2 \log (x)}{2 a b^2}","-\frac{(b c-a d)^2 \log \left(a+b x^2\right)}{2 a b^2}+\frac{c^2 \log (x)}{a}+\frac{d^2 x^2}{2 b}",1,"(a*b*d^2*x^2 + 2*b^2*c^2*Log[x] - (b*c - a*d)^2*Log[a + b*x^2])/(2*a*b^2)","A",1
215,1,55,55,0.0538579,"\int \frac{\left(c+d x^2\right)^2}{x^2 \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2)^2/(x^2*(a + b*x^2)),x]","-\frac{(a d-b c)^2 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2} b^{3/2}}-\frac{c^2}{a x}+\frac{d^2 x}{b}","-\frac{(b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2} b^{3/2}}-\frac{c^2}{a x}+\frac{d^2 x}{b}",1,"-(c^2/(a*x)) + (d^2*x)/b - ((-(b*c) + a*d)^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*b^(3/2))","A",1
216,1,60,58,0.0278553,"\int \frac{\left(c+d x^2\right)^2}{x^3 \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2)^2/(x^3*(a + b*x^2)),x]","\frac{-a b c^2-2 b c x^2 \log (x) (b c-2 a d)+x^2 (b c-a d)^2 \log \left(a+b x^2\right)}{2 a^2 b x^2}","\frac{(b c-a d)^2 \log \left(a+b x^2\right)}{2 a^2 b}-\frac{c \log (x) (b c-2 a d)}{a^2}-\frac{c^2}{2 a x^2}",1,"(-(a*b*c^2) - 2*b*c*(b*c - 2*a*d)*x^2*Log[x] + (b*c - a*d)^2*x^2*Log[a + b*x^2])/(2*a^2*b*x^2)","A",1
217,1,66,64,0.0630762,"\int \frac{\left(c+d x^2\right)^2}{x^4 \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2)^2/(x^4*(a + b*x^2)),x]","\frac{(a d-b c)^2 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{5/2} \sqrt{b}}-\frac{c (2 a d-b c)}{a^2 x}-\frac{c^2}{3 a x^3}","\frac{(b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{5/2} \sqrt{b}}+\frac{c (b c-2 a d)}{a^2 x}-\frac{c^2}{3 a x^3}",1,"-1/3*c^2/(a*x^3) - (c*(-(b*c) + 2*a*d))/(a^2*x) + ((-(b*c) + a*d)^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(5/2)*Sqrt[b])","A",1
218,1,128,138,0.0708163,"\int \frac{x^5 \left(c+d x^2\right)^3}{a+b x^2} \, dx","Integrate[(x^5*(c + d*x^2)^3)/(a + b*x^2),x]","\frac{20 b^3 d x^6 \left(a^2 d^2-3 a b c d+3 b^2 c^2\right)+60 a^2 (b c-a d)^3 \log \left(a+b x^2\right)+15 b^4 d^2 x^8 (3 b c-a d)+30 b^2 x^4 (b c-a d)^3+60 a b x^2 (a d-b c)^3+12 b^5 d^3 x^{10}}{120 b^6}","\frac{a^2 (b c-a d)^3 \log \left(a+b x^2\right)}{2 b^6}+\frac{d x^6 \left(a^2 d^2-3 a b c d+3 b^2 c^2\right)}{6 b^3}-\frac{a x^2 (b c-a d)^3}{2 b^5}+\frac{x^4 (b c-a d)^3}{4 b^4}+\frac{d^2 x^8 (3 b c-a d)}{8 b^2}+\frac{d^3 x^{10}}{10 b}",1,"(60*a*b*(-(b*c) + a*d)^3*x^2 + 30*b^2*(b*c - a*d)^3*x^4 + 20*b^3*d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^6 + 15*b^4*d^2*(3*b*c - a*d)*x^8 + 12*b^5*d^3*x^10 + 60*a^2*(b*c - a*d)^3*Log[a + b*x^2])/(120*b^6)","A",1
219,1,140,140,0.0474089,"\int \frac{x^4 \left(c+d x^2\right)^3}{a+b x^2} \, dx","Integrate[(x^4*(c + d*x^2)^3)/(a + b*x^2),x]","-\frac{a^{3/2} (a d-b c)^3 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{11/2}}+\frac{d x^5 \left(a^2 d^2-3 a b c d+3 b^2 c^2\right)}{5 b^3}+\frac{a x (a d-b c)^3}{b^5}+\frac{x^3 (b c-a d)^3}{3 b^4}+\frac{d^2 x^7 (3 b c-a d)}{7 b^2}+\frac{d^3 x^9}{9 b}","\frac{a^{3/2} (b c-a d)^3 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{11/2}}+\frac{d x^5 \left(a^2 d^2-3 a b c d+3 b^2 c^2\right)}{5 b^3}-\frac{a x (b c-a d)^3}{b^5}+\frac{x^3 (b c-a d)^3}{3 b^4}+\frac{d^2 x^7 (3 b c-a d)}{7 b^2}+\frac{d^3 x^9}{9 b}",1,"(a*(-(b*c) + a*d)^3*x)/b^5 + ((b*c - a*d)^3*x^3)/(3*b^4) + (d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^5)/(5*b^3) + (d^2*(3*b*c - a*d)*x^7)/(7*b^2) + (d^3*x^9)/(9*b) - (a^(3/2)*(-(b*c) + a*d)^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(11/2)","A",1
220,1,125,115,0.0569363,"\int \frac{x^3 \left(c+d x^2\right)^3}{a+b x^2} \, dx","Integrate[(x^3*(c + d*x^2)^3)/(a + b*x^2),x]","\frac{b x^2 \left(-12 a^3 d^3+6 a^2 b d^2 \left(6 c+d x^2\right)-2 a b^2 d \left(18 c^2+9 c d x^2+2 d^2 x^4\right)+3 b^3 \left(4 c^3+6 c^2 d x^2+4 c d^2 x^4+d^3 x^6\right)\right)+12 a (a d-b c)^3 \log \left(a+b x^2\right)}{24 b^5}","\frac{d x^4 \left(a^2 d^2-3 a b c d+3 b^2 c^2\right)}{4 b^3}-\frac{a (b c-a d)^3 \log \left(a+b x^2\right)}{2 b^5}+\frac{x^2 (b c-a d)^3}{2 b^4}+\frac{d^2 x^6 (3 b c-a d)}{6 b^2}+\frac{d^3 x^8}{8 b}",1,"(b*x^2*(-12*a^3*d^3 + 6*a^2*b*d^2*(6*c + d*x^2) - 2*a*b^2*d*(18*c^2 + 9*c*d*x^2 + 2*d^2*x^4) + 3*b^3*(4*c^3 + 6*c^2*d*x^2 + 4*c*d^2*x^4 + d^3*x^6)) + 12*a*(-(b*c) + a*d)^3*Log[a + b*x^2])/(24*b^5)","A",1
221,1,118,119,0.0420918,"\int \frac{x^2 \left(c+d x^2\right)^3}{a+b x^2} \, dx","Integrate[(x^2*(c + d*x^2)^3)/(a + b*x^2),x]","\frac{d x^3 \left(a^2 d^2-3 a b c d+3 b^2 c^2\right)}{3 b^3}+\frac{\sqrt{a} (a d-b c)^3 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{9/2}}+\frac{x (b c-a d)^3}{b^4}+\frac{d^2 x^5 (3 b c-a d)}{5 b^2}+\frac{d^3 x^7}{7 b}","\frac{d x^3 \left(a^2 d^2-3 a b c d+3 b^2 c^2\right)}{3 b^3}-\frac{\sqrt{a} (b c-a d)^3 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{9/2}}+\frac{x (b c-a d)^3}{b^4}+\frac{d^2 x^5 (3 b c-a d)}{5 b^2}+\frac{d^3 x^7}{7 b}",1,"((b*c - a*d)^3*x)/b^4 + (d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^3)/(3*b^3) + (d^2*(3*b*c - a*d)*x^5)/(5*b^2) + (d^3*x^7)/(7*b) + (Sqrt[a]*(-(b*c) + a*d)^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(9/2)","A",1
222,1,82,87,0.0319079,"\int \frac{x \left(c+d x^2\right)^3}{a+b x^2} \, dx","Integrate[(x*(c + d*x^2)^3)/(a + b*x^2),x]","\frac{b d x^2 \left(6 a^2 d^2-3 a b d \left(6 c+d x^2\right)+b^2 \left(18 c^2+9 c d x^2+2 d^2 x^4\right)\right)+6 (b c-a d)^3 \log \left(a+b x^2\right)}{12 b^4}","\frac{(b c-a d)^3 \log \left(a+b x^2\right)}{2 b^4}+\frac{d x^2 (b c-a d)^2}{2 b^3}+\frac{\left(c+d x^2\right)^2 (b c-a d)}{4 b^2}+\frac{\left(c+d x^2\right)^3}{6 b}",1,"(b*d*x^2*(6*a^2*d^2 - 3*a*b*d*(6*c + d*x^2) + b^2*(18*c^2 + 9*c*d*x^2 + 2*d^2*x^4)) + 6*(b*c - a*d)^3*Log[a + b*x^2])/(12*b^4)","A",1
223,1,92,98,0.0668532,"\int \frac{\left(c+d x^2\right)^3}{a+b x^2} \, dx","Integrate[(c + d*x^2)^3/(a + b*x^2),x]","\frac{d x \left(15 a^2 d^2-5 a b d \left(9 c+d x^2\right)+3 b^2 \left(15 c^2+5 c d x^2+d^2 x^4\right)\right)}{15 b^3}+\frac{(b c-a d)^3 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} b^{7/2}}","\frac{d x \left(a^2 d^2-3 a b c d+3 b^2 c^2\right)}{b^3}+\frac{(b c-a d)^3 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} b^{7/2}}+\frac{d^2 x^3 (3 b c-a d)}{3 b^2}+\frac{d^3 x^5}{5 b}",1,"(d*x*(15*a^2*d^2 - 5*a*b*d*(9*c + d*x^2) + 3*b^2*(15*c^2 + 5*c*d*x^2 + d^2*x^4)))/(15*b^3) + ((b*c - a*d)^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*b^(7/2))","A",1
224,1,65,73,0.0318524,"\int \frac{\left(c+d x^2\right)^3}{x \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2)^3/(x*(a + b*x^2)),x]","\frac{a b d^2 x^2 \left(-2 a d+6 b c+b d x^2\right)-2 (b c-a d)^3 \log \left(a+b x^2\right)+4 b^3 c^3 \log (x)}{4 a b^3}","-\frac{(b c-a d)^3 \log \left(a+b x^2\right)}{2 a b^3}+\frac{d^2 x^2 (3 b c-a d)}{2 b^2}+\frac{c^3 \log (x)}{a}+\frac{d^3 x^4}{4 b}",1,"(a*b*d^2*x^2*(6*b*c - 2*a*d + b*d*x^2) + 4*b^3*c^3*Log[x] - 2*(b*c - a*d)^3*Log[a + b*x^2])/(4*a*b^3)","A",1
225,1,76,77,0.0333543,"\int \frac{\left(c+d x^2\right)^3}{x^2 \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2)^3/(x^2*(a + b*x^2)),x]","\frac{(a d-b c)^3 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2} b^{5/2}}+\frac{d^2 x (3 b c-a d)}{b^2}-\frac{c^3}{a x}+\frac{d^3 x^3}{3 b}","-\frac{(b c-a d)^3 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2} b^{5/2}}+\frac{d^2 x (3 b c-a d)}{b^2}-\frac{c^3}{a x}+\frac{d^3 x^3}{3 b}",1,"-(c^3/(a*x)) + (d^2*(3*b*c - a*d)*x)/b^2 + (d^3*x^3)/(3*b) + ((-(b*c) + a*d)^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*b^(5/2))","A",1
226,1,75,73,0.0381812,"\int \frac{\left(c+d x^2\right)^3}{x^3 \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2)^3/(x^3*(a + b*x^2)),x]","\frac{-2 b^2 c^2 x^2 \log (x) (b c-3 a d)+a b \left(a d^3 x^4-b c^3\right)+x^2 (b c-a d)^3 \log \left(a+b x^2\right)}{2 a^2 b^2 x^2}","\frac{(b c-a d)^3 \log \left(a+b x^2\right)}{2 a^2 b^2}-\frac{c^2 \log (x) (b c-3 a d)}{a^2}-\frac{c^3}{2 a x^2}+\frac{d^3 x^2}{2 b}",1,"(a*b*(-(b*c^3) + a*d^3*x^4) - 2*b^2*c^2*(b*c - 3*a*d)*x^2*Log[x] + (b*c - a*d)^3*x^2*Log[a + b*x^2])/(2*a^2*b^2*x^2)","A",1
227,1,74,74,0.0422197,"\int \frac{\left(c+d x^2\right)^3}{x^4 \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2)^3/(x^4*(a + b*x^2)),x]","\frac{(b c-a d)^3 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{5/2} b^{3/2}}+\frac{c^2 (b c-3 a d)}{a^2 x}-\frac{c^3}{3 a x^3}+\frac{d^3 x}{b}","\frac{(b c-a d)^3 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{5/2} b^{3/2}}+\frac{c^2 (b c-3 a d)}{a^2 x}-\frac{c^3}{3 a x^3}+\frac{d^3 x}{b}",1,"-1/3*c^3/(a*x^3) + (c^2*(b*c - 3*a*d))/(a^2*x) + (d^3*x)/b + ((b*c - a*d)^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(5/2)*b^(3/2))","A",1
228,1,66,70,0.0321303,"\int \frac{x^5}{\left(a+b x^2\right) \left(c+d x^2\right)} \, dx","Integrate[x^5/((a + b*x^2)*(c + d*x^2)),x]","\frac{a^2 d^2 \log \left(a+b x^2\right)-b \left(d x^2 (a d-b c)+b c^2 \log \left(c+d x^2\right)\right)}{2 b^2 d^2 (b c-a d)}","\frac{a^2 \log \left(a+b x^2\right)}{2 b^2 (b c-a d)}-\frac{c^2 \log \left(c+d x^2\right)}{2 d^2 (b c-a d)}+\frac{x^2}{2 b d}",1,"(a^2*d^2*Log[a + b*x^2] - b*(d*(-(b*c) + a*d)*x^2 + b*c^2*Log[c + d*x^2]))/(2*b^2*d^2*(b*c - a*d))","A",1
229,1,74,78,0.0896916,"\int \frac{x^4}{\left(a+b x^2\right) \left(c+d x^2\right)} \, dx","Integrate[x^4/((a + b*x^2)*(c + d*x^2)),x]","\frac{\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{3/2}}-\frac{a x}{b}-\frac{c^{3/2} \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{d^{3/2}}+\frac{c x}{d}}{b c-a d}","\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{3/2} (b c-a d)}-\frac{c^{3/2} \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{d^{3/2} (b c-a d)}+\frac{x}{b d}",1,"(-((a*x)/b) + (c*x)/d + (a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(3/2) - (c^(3/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/d^(3/2))/(b*c - a*d)","A",1
230,1,43,53,0.0259614,"\int \frac{x^3}{\left(a+b x^2\right) \left(c+d x^2\right)} \, dx","Integrate[x^3/((a + b*x^2)*(c + d*x^2)),x]","-\frac{a d \log \left(a+b x^2\right)-b c \log \left(c+d x^2\right)}{2 b^2 c d-2 a b d^2}","\frac{c \log \left(c+d x^2\right)}{2 d (b c-a d)}-\frac{a \log \left(a+b x^2\right)}{2 b (b c-a d)}",1,"-((a*d*Log[a + b*x^2] - b*c*Log[c + d*x^2])/(2*b^2*c*d - 2*a*b*d^2))","A",1
231,1,61,70,0.0526443,"\int \frac{x^2}{\left(a+b x^2\right) \left(c+d x^2\right)} \, dx","Integrate[x^2/((a + b*x^2)*(c + d*x^2)),x]","\frac{\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{\sqrt{d}}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{b}}}{b c-a d}","\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{\sqrt{d} (b c-a d)}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{b} (b c-a d)}",1,"(-((Sqrt[a]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[b]) + (Sqrt[c]*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/Sqrt[d])/(b*c - a*d)","A",1
232,1,31,45,0.0165762,"\int \frac{x}{\left(a+b x^2\right) \left(c+d x^2\right)} \, dx","Integrate[x/((a + b*x^2)*(c + d*x^2)),x]","\frac{\log \left(a+b x^2\right)-\log \left(c+d x^2\right)}{2 b c-2 a d}","\frac{\log \left(a+b x^2\right)}{2 (b c-a d)}-\frac{\log \left(c+d x^2\right)}{2 (b c-a d)}",1,"(Log[a + b*x^2] - Log[c + d*x^2])/(2*b*c - 2*a*d)","A",1
233,1,61,70,0.0432018,"\int \frac{1}{\left(a+b x^2\right) \left(c+d x^2\right)} \, dx","Integrate[1/((a + b*x^2)*(c + d*x^2)),x]","\frac{\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a}}-\frac{\sqrt{d} \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{\sqrt{c}}}{b c-a d}","\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} (b c-a d)}-\frac{\sqrt{d} \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{\sqrt{c} (b c-a d)}",1,"((Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[a] - (Sqrt[d]*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/Sqrt[c])/(b*c - a*d)","A",1
234,1,54,62,0.0276848,"\int \frac{1}{x \left(a+b x^2\right) \left(c+d x^2\right)} \, dx","Integrate[1/(x*(a + b*x^2)*(c + d*x^2)),x]","\frac{-b c \log \left(a+b x^2\right)+a d \log \left(c+d x^2\right)-2 a d \log (x)+2 b c \log (x)}{2 a b c^2-2 a^2 c d}","-\frac{b \log \left(a+b x^2\right)}{2 a (b c-a d)}+\frac{d \log \left(c+d x^2\right)}{2 c (b c-a d)}+\frac{\log (x)}{a c}",1,"(2*b*c*Log[x] - 2*a*d*Log[x] - b*c*Log[a + b*x^2] + a*d*Log[c + d*x^2])/(2*a*b*c^2 - 2*a^2*c*d)","A",1
235,1,76,81,0.0858799,"\int \frac{1}{x^2 \left(a+b x^2\right) \left(c+d x^2\right)} \, dx","Integrate[1/(x^2*(a + b*x^2)*(c + d*x^2)),x]","\frac{-\frac{b^{3/2} x \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2}}-\frac{b}{a}+\frac{d^{3/2} x \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{3/2}}+\frac{d}{c}}{b c x-a d x}","-\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2} (b c-a d)}+\frac{d^{3/2} \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{3/2} (b c-a d)}-\frac{1}{a c x}",1,"(-(b/a) + d/c - (b^(3/2)*x*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(3/2) + (d^(3/2)*x*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/c^(3/2))/(b*c*x - a*d*x)","A",1
236,1,88,87,0.0437062,"\int \frac{1}{x^3 \left(a+b x^2\right) \left(c+d x^2\right)} \, dx","Integrate[1/(x^3*(a + b*x^2)*(c + d*x^2)),x]","-\frac{b^2 \log \left(a+b x^2\right)}{2 a^2 (a d-b c)}+\frac{\log (x) (-a d-b c)}{a^2 c^2}-\frac{d^2 \log \left(c+d x^2\right)}{2 c^2 (b c-a d)}-\frac{1}{2 a c x^2}","\frac{b^2 \log \left(a+b x^2\right)}{2 a^2 (b c-a d)}-\frac{\log (x) (a d+b c)}{a^2 c^2}-\frac{d^2 \log \left(c+d x^2\right)}{2 c^2 (b c-a d)}-\frac{1}{2 a c x^2}",1,"-1/2*1/(a*c*x^2) + ((-(b*c) - a*d)*Log[x])/(a^2*c^2) - (b^2*Log[a + b*x^2])/(2*a^2*(-(b*c) + a*d)) - (d^2*Log[c + d*x^2])/(2*c^2*(b*c - a*d))","A",1
237,1,101,100,0.1311899,"\int \frac{1}{x^4 \left(a+b x^2\right) \left(c+d x^2\right)} \, dx","Integrate[1/(x^4*(a + b*x^2)*(c + d*x^2)),x]","-\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{5/2} (a d-b c)}+\frac{a d+b c}{a^2 c^2 x}-\frac{d^{5/2} \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{5/2} (b c-a d)}-\frac{1}{3 a c x^3}","\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{5/2} (b c-a d)}+\frac{a d+b c}{a^2 c^2 x}-\frac{d^{5/2} \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{5/2} (b c-a d)}-\frac{1}{3 a c x^3}",1,"-1/3*1/(a*c*x^3) + (b*c + a*d)/(a^2*c^2*x) - (b^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(5/2)*(-(b*c) + a*d)) - (d^(5/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(5/2)*(b*c - a*d))","A",1
238,1,119,119,0.0546162,"\int \frac{1}{x^5 \left(a+b x^2\right) \left(c+d x^2\right)} \, dx","Integrate[1/(x^5*(a + b*x^2)*(c + d*x^2)),x]","\frac{b^3 \log \left(a+b x^2\right)}{2 a^3 (a d-b c)}+\frac{a d+b c}{2 a^2 c^2 x^2}+\frac{\log (x) \left(a^2 d^2+a b c d+b^2 c^2\right)}{a^3 c^3}+\frac{d^3 \log \left(c+d x^2\right)}{2 c^3 (b c-a d)}-\frac{1}{4 a c x^4}","-\frac{b^3 \log \left(a+b x^2\right)}{2 a^3 (b c-a d)}+\frac{a d+b c}{2 a^2 c^2 x^2}+\frac{\log (x) \left(a^2 d^2+a b c d+b^2 c^2\right)}{a^3 c^3}+\frac{d^3 \log \left(c+d x^2\right)}{2 c^3 (b c-a d)}-\frac{1}{4 a c x^4}",1,"-1/4*1/(a*c*x^4) + (b*c + a*d)/(2*a^2*c^2*x^2) + ((b^2*c^2 + a*b*c*d + a^2*d^2)*Log[x])/(a^3*c^3) + (b^3*Log[a + b*x^2])/(2*a^3*(-(b*c) + a*d)) + (d^3*Log[c + d*x^2])/(2*c^3*(b*c - a*d))","A",1
239,1,135,134,0.1227555,"\int \frac{1}{x^6 \left(a+b x^2\right) \left(c+d x^2\right)} \, dx","Integrate[1/(x^6*(a + b*x^2)*(c + d*x^2)),x]","\frac{b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{7/2} (a d-b c)}+\frac{a d+b c}{3 a^2 c^2 x^3}+\frac{-a^2 d^2-a b c d-b^2 c^2}{a^3 c^3 x}+\frac{d^{7/2} \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{7/2} (b c-a d)}-\frac{1}{5 a c x^5}","-\frac{b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{7/2} (b c-a d)}+\frac{a d+b c}{3 a^2 c^2 x^3}-\frac{a^2 d^2+a b c d+b^2 c^2}{a^3 c^3 x}+\frac{d^{7/2} \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{7/2} (b c-a d)}-\frac{1}{5 a c x^5}",1,"-1/5*1/(a*c*x^5) + (b*c + a*d)/(3*a^2*c^2*x^3) + (-(b^2*c^2) - a*b*c*d - a^2*d^2)/(a^3*c^3*x) + (b^(7/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(7/2)*(-(b*c) + a*d)) + (d^(7/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(7/2)*(b*c - a*d))","A",1
240,1,147,155,0.0653759,"\int \frac{1}{x^7 \left(a+b x^2\right) \left(c+d x^2\right)} \, dx","Integrate[1/(x^7*(a + b*x^2)*(c + d*x^2)),x]","\frac{12 x^6 \log (x) \left(b^4 c^4-a^4 d^4\right)+a \left(a^3 c d \left(-2 c^2+3 c d x^2-6 d^2 x^4\right)+6 a^3 d^4 x^6 \log \left(c+d x^2\right)+2 a^2 b c^4-3 a b^2 c^4 x^2+6 b^3 c^4 x^4\right)-6 b^4 c^4 x^6 \log \left(a+b x^2\right)}{12 a^4 c^4 x^6 (a d-b c)}","\frac{b^4 \log \left(a+b x^2\right)}{2 a^4 (b c-a d)}+\frac{a d+b c}{4 a^2 c^2 x^4}-\frac{\log (x) (a d+b c) \left(a^2 d^2+b^2 c^2\right)}{a^4 c^4}-\frac{a^2 d^2+a b c d+b^2 c^2}{2 a^3 c^3 x^2}-\frac{d^4 \log \left(c+d x^2\right)}{2 c^4 (b c-a d)}-\frac{1}{6 a c x^6}",1,"(12*(b^4*c^4 - a^4*d^4)*x^6*Log[x] - 6*b^4*c^4*x^6*Log[a + b*x^2] + a*(2*a^2*b*c^4 - 3*a*b^2*c^4*x^2 + 6*b^3*c^4*x^4 + a^3*c*d*(-2*c^2 + 3*c*d*x^2 - 6*d^2*x^4) + 6*a^3*d^4*x^6*Log[c + d*x^2]))/(12*a^4*c^4*(-(b*c) + a*d)*x^6)","A",1
241,1,91,93,0.0485154,"\int \frac{x^5}{\left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","Integrate[x^5/((a + b*x^2)^2*(c + d*x^2)),x]","\frac{a^2 d (a d-b c)+b^2 c^2 \left(a+b x^2\right) \log \left(c+d x^2\right)+a d \left(a+b x^2\right) (a d-2 b c) \log \left(a+b x^2\right)}{2 b^2 d \left(a+b x^2\right) (b c-a d)^2}","-\frac{a^2}{2 b^2 \left(a+b x^2\right) (b c-a d)}-\frac{a (2 b c-a d) \log \left(a+b x^2\right)}{2 b^2 (b c-a d)^2}+\frac{c^2 \log \left(c+d x^2\right)}{2 d (b c-a d)^2}",1,"(a^2*d*(-(b*c) + a*d) + a*d*(-2*b*c + a*d)*(a + b*x^2)*Log[a + b*x^2] + b^2*c^2*(a + b*x^2)*Log[c + d*x^2])/(2*b^2*d*(b*c - a*d)^2*(a + b*x^2))","A",1
242,1,108,108,0.1377594,"\int \frac{x^4}{\left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","Integrate[x^4/((a + b*x^2)*(c + d*x^2)^2),x]","\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{b} (a d-b c)^2}+\frac{\sqrt{c} (b c-3 a d) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 d^{3/2} (b c-a d)^2}+\frac{c x}{2 d \left(c+d x^2\right) (a d-b c)}","\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{b} (b c-a d)^2}+\frac{\sqrt{c} (b c-3 a d) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 d^{3/2} (b c-a d)^2}-\frac{c x}{2 d \left(c+d x^2\right) (b c-a d)}",1,"(c*x)/(2*d*(-(b*c) + a*d)*(c + d*x^2)) + (a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[b]*(-(b*c) + a*d)^2) + (Sqrt[c]*(b*c - 3*a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*d^(3/2)*(b*c - a*d)^2)","A",1
243,1,74,74,0.0400849,"\int \frac{x^3}{\left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","Integrate[x^3/((a + b*x^2)*(c + d*x^2)^2),x]","\frac{c}{2 d \left(c+d x^2\right) (a d-b c)}-\frac{a \log \left(a+b x^2\right)}{2 (b c-a d)^2}+\frac{a \log \left(c+d x^2\right)}{2 (b c-a d)^2}","-\frac{c}{2 d \left(c+d x^2\right) (b c-a d)}-\frac{a \log \left(a+b x^2\right)}{2 (b c-a d)^2}+\frac{a \log \left(c+d x^2\right)}{2 (b c-a d)^2}",1,"c/(2*d*(-(b*c) + a*d)*(c + d*x^2)) - (a*Log[a + b*x^2])/(2*(b*c - a*d)^2) + (a*Log[c + d*x^2])/(2*(b*c - a*d)^2)","A",1
244,1,90,104,0.1271813,"\int \frac{x^2}{\left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","Integrate[x^2/((a + b*x^2)*(c + d*x^2)^2),x]","\frac{\frac{x (b c-a d)}{c+d x^2}+\frac{(a d+b c) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{\sqrt{c} \sqrt{d}}-2 \sqrt{a} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 (b c-a d)^2}","\frac{x}{2 \left(c+d x^2\right) (b c-a d)}-\frac{\sqrt{a} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{(b c-a d)^2}+\frac{(a d+b c) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 \sqrt{c} \sqrt{d} (b c-a d)^2}",1,"(((b*c - a*d)*x)/(c + d*x^2) - 2*Sqrt[a]*Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]] + ((b*c + a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*Sqrt[d]))/(2*(b*c - a*d)^2)","A",1
245,1,66,70,0.0309621,"\int \frac{x}{\left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","Integrate[x/((a + b*x^2)*(c + d*x^2)^2),x]","\frac{b \left(c+d x^2\right) \log \left(a+b x^2\right)-a d-b \left(c+d x^2\right) \log \left(c+d x^2\right)+b c}{2 \left(c+d x^2\right) (b c-a d)^2}","\frac{1}{2 \left(c+d x^2\right) (b c-a d)}+\frac{b \log \left(a+b x^2\right)}{2 (b c-a d)^2}-\frac{b \log \left(c+d x^2\right)}{2 (b c-a d)^2}",1,"(b*c - a*d + b*(c + d*x^2)*Log[a + b*x^2] - b*(c + d*x^2)*Log[c + d*x^2])/(2*(b*c - a*d)^2*(c + d*x^2))","A",1
246,1,95,109,0.1729745,"\int \frac{1}{\left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","Integrate[1/((a + b*x^2)*(c + d*x^2)^2),x]","\frac{\frac{2 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a}}+\frac{\sqrt{d} (a d-3 b c) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{3/2}}+\frac{d x (a d-b c)}{c \left(c+d x^2\right)}}{2 (b c-a d)^2}","\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} (b c-a d)^2}-\frac{\sqrt{d} (3 b c-a d) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{3/2} (b c-a d)^2}-\frac{d x}{2 c \left(c+d x^2\right) (b c-a d)}",1,"((d*(-(b*c) + a*d)*x)/(c*(c + d*x^2)) + (2*b^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[a] + (Sqrt[d]*(-3*b*c + a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/c^(3/2))/(2*(b*c - a*d)^2)","A",1
247,1,98,100,0.0945663,"\int \frac{1}{x \left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","Integrate[1/(x*(a + b*x^2)*(c + d*x^2)^2),x]","\frac{1}{2} \left(-\frac{b^2 \log \left(a+b x^2\right)}{a (b c-a d)^2}+\frac{d (2 b c-a d) \log \left(c+d x^2\right)}{c^2 (b c-a d)^2}-\frac{d}{c \left(c+d x^2\right) (b c-a d)}+\frac{2 \log (x)}{a c^2}\right)","-\frac{b^2 \log \left(a+b x^2\right)}{2 a (b c-a d)^2}+\frac{d (2 b c-a d) \log \left(c+d x^2\right)}{2 c^2 (b c-a d)^2}-\frac{d}{2 c \left(c+d x^2\right) (b c-a d)}+\frac{\log (x)}{a c^2}",1,"(-(d/(c*(b*c - a*d)*(c + d*x^2))) + (2*Log[x])/(a*c^2) - (b^2*Log[a + b*x^2])/(a*(b*c - a*d)^2) + (d*(2*b*c - a*d)*Log[c + d*x^2])/(c^2*(b*c - a*d)^2))/2","A",1
248,1,123,144,0.2608999,"\int \frac{1}{x^2 \left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","Integrate[1/(x^2*(a + b*x^2)*(c + d*x^2)^2),x]","-\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2} (a d-b c)^2}+\frac{d^{3/2} (5 b c-3 a d) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{5/2} (b c-a d)^2}+\frac{d^2 x}{2 c^2 \left(c+d x^2\right) (b c-a d)}-\frac{1}{a c^2 x}","-\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2} (b c-a d)^2}+\frac{d^{3/2} (5 b c-3 a d) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{5/2} (b c-a d)^2}-\frac{2 b c-3 a d}{2 a c^2 x (b c-a d)}-\frac{d}{2 c x \left(c+d x^2\right) (b c-a d)}",1,"-(1/(a*c^2*x)) + (d^2*x)/(2*c^2*(b*c - a*d)*(c + d*x^2)) - (b^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*(-(b*c) + a*d)^2) + (d^(3/2)*(5*b*c - 3*a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(5/2)*(b*c - a*d)^2)","A",1
249,1,117,126,0.2383322,"\int \frac{1}{x^3 \left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","Integrate[1/(x^3*(a + b*x^2)*(c + d*x^2)^2),x]","\frac{1}{2} \left(\frac{b^3 \log \left(a+b x^2\right)}{a^2 (b c-a d)^2}-\frac{2 \log (x) (2 a d+b c)}{a^2 c^3}+\frac{\frac{c d^2}{\left(c+d x^2\right) (b c-a d)}+\frac{d^2 (2 a d-3 b c) \log \left(c+d x^2\right)}{(b c-a d)^2}-\frac{c}{a x^2}}{c^3}\right)","\frac{b^3 \log \left(a+b x^2\right)}{2 a^2 (b c-a d)^2}-\frac{\log (x) (2 a d+b c)}{a^2 c^3}-\frac{d^2 (3 b c-2 a d) \log \left(c+d x^2\right)}{2 c^3 (b c-a d)^2}+\frac{d^2}{2 c^2 \left(c+d x^2\right) (b c-a d)}-\frac{1}{2 a c^2 x^2}",1,"((-2*(b*c + 2*a*d)*Log[x])/(a^2*c^3) + (b^3*Log[a + b*x^2])/(a^2*(b*c - a*d)^2) + (-(c/(a*x^2)) + (c*d^2)/((b*c - a*d)*(c + d*x^2)) + (d^2*(-3*b*c + 2*a*d)*Log[c + d*x^2])/(b*c - a*d)^2)/c^3)/2","A",1
250,1,142,189,0.3866785,"\int \frac{1}{x^4 \left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","Integrate[1/(x^4*(a + b*x^2)*(c + d*x^2)^2),x]","\frac{b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{5/2} (a d-b c)^2}+\frac{2 a d+b c}{a^2 c^3 x}-\frac{d^{5/2} (7 b c-5 a d) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{7/2} (b c-a d)^2}-\frac{d^3 x}{2 c^3 \left(c+d x^2\right) (b c-a d)}-\frac{1}{3 a c^2 x^3}","\frac{b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{5/2} (b c-a d)^2}+\frac{-5 a^2 d^2+2 a b c d+2 b^2 c^2}{2 a^2 c^3 x (b c-a d)}-\frac{d^{5/2} (7 b c-5 a d) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{7/2} (b c-a d)^2}-\frac{2 b c-5 a d}{6 a c^2 x^3 (b c-a d)}-\frac{d}{2 c x^3 \left(c+d x^2\right) (b c-a d)}",1,"-1/3*1/(a*c^2*x^3) + (b*c + 2*a*d)/(a^2*c^3*x) - (d^3*x)/(2*c^3*(b*c - a*d)*(c + d*x^2)) + (b^(7/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(5/2)*(-(b*c) + a*d)^2) - (d^(5/2)*(7*b*c - 5*a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(7/2)*(b*c - a*d)^2)","A",1
251,1,99,116,0.097643,"\int \frac{x^5}{\left(a+b x^2\right)^3 \left(c+d x^2\right)} \, dx","Integrate[x^5/((a + b*x^2)^3*(c + d*x^2)),x]","\frac{-\frac{a^2 (b c-a d)^2}{b^2 \left(a+b x^2\right)^2}+\frac{2 a (a d-2 b c) (a d-b c)}{b^2 \left(a+b x^2\right)}+2 c^2 \log \left(a+b x^2\right)-2 c^2 \log \left(c+d x^2\right)}{4 (b c-a d)^3}","-\frac{a^2}{4 b^2 \left(a+b x^2\right)^2 (b c-a d)}+\frac{a (2 b c-a d)}{2 b^2 \left(a+b x^2\right) (b c-a d)^2}+\frac{c^2 \log \left(a+b x^2\right)}{2 (b c-a d)^3}-\frac{c^2 \log \left(c+d x^2\right)}{2 (b c-a d)^3}",1,"(-((a^2*(b*c - a*d)^2)/(b^2*(a + b*x^2)^2)) + (2*a*(-2*b*c + a*d)*(-(b*c) + a*d))/(b^2*(a + b*x^2)) + 2*c^2*Log[a + b*x^2] - 2*c^2*Log[c + d*x^2])/(4*(b*c - a*d)^3)","A",1
252,1,154,157,0.2728352,"\int \frac{x^4}{\left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","Integrate[x^4/((a + b*x^2)*(c + d*x^2)^3),x]","\frac{1}{8} \left(\frac{8 a^{3/2} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{(b c-a d)^3}+\frac{\left(-3 a^2 d^2-6 a b c d+b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{\sqrt{c} d^{3/2} (b c-a d)^3}+\frac{x (b c-5 a d)}{d \left(c+d x^2\right) (b c-a d)^2}+\frac{2 c x}{d \left(c+d x^2\right)^2 (a d-b c)}\right)","\frac{a^{3/2} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{(b c-a d)^3}+\frac{\left(-3 a^2 d^2-6 a b c d+b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 \sqrt{c} d^{3/2} (b c-a d)^3}+\frac{x (b c-5 a d)}{8 d \left(c+d x^2\right) (b c-a d)^2}-\frac{c x}{4 d \left(c+d x^2\right)^2 (b c-a d)}",1,"((2*c*x)/(d*(-(b*c) + a*d)*(c + d*x^2)^2) + ((b*c - 5*a*d)*x)/(d*(b*c - a*d)^2*(c + d*x^2)) + (8*a^(3/2)*Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(b*c - a*d)^3 + ((b^2*c^2 - 6*a*b*c*d - 3*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*d^(3/2)*(b*c - a*d)^3))/8","A",1
253,1,77,100,0.1119201,"\int \frac{x^3}{\left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","Integrate[x^3/((a + b*x^2)*(c + d*x^2)^3),x]","\frac{\frac{(a d-b c) \left(a d \left(c+2 d x^2\right)+b c^2\right)}{d \left(c+d x^2\right)^2}+2 a b \log \left(c+d x^2\right)-2 a b \log \left(a+b x^2\right)}{4 (b c-a d)^3}","-\frac{a}{2 \left(c+d x^2\right) (b c-a d)^2}-\frac{c}{4 d \left(c+d x^2\right)^2 (b c-a d)}-\frac{a b \log \left(a+b x^2\right)}{2 (b c-a d)^3}+\frac{a b \log \left(c+d x^2\right)}{2 (b c-a d)^3}",1,"(((-(b*c) + a*d)*(b*c^2 + a*d*(c + 2*d*x^2)))/(d*(c + d*x^2)^2) - 2*a*b*Log[a + b*x^2] + 2*a*b*Log[c + d*x^2])/(4*(b*c - a*d)^3)","A",1
254,1,151,155,0.2177487,"\int \frac{x^2}{\left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","Integrate[x^2/((a + b*x^2)*(c + d*x^2)^3),x]","\frac{1}{8} \left(\frac{\left(-a^2 d^2+6 a b c d+3 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{3/2} \sqrt{d} (b c-a d)^3}+\frac{8 \sqrt{a} b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{(a d-b c)^3}+\frac{x (a d+3 b c)}{c \left(c+d x^2\right) (b c-a d)^2}+\frac{2 x}{\left(c+d x^2\right)^2 (b c-a d)}\right)","\frac{\left(-a^2 d^2+6 a b c d+3 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{3/2} \sqrt{d} (b c-a d)^3}-\frac{\sqrt{a} b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{(b c-a d)^3}+\frac{x (a d+3 b c)}{8 c \left(c+d x^2\right) (b c-a d)^2}+\frac{x}{4 \left(c+d x^2\right)^2 (b c-a d)}",1,"((2*x)/((b*c - a*d)*(c + d*x^2)^2) + ((3*b*c + a*d)*x)/(c*(b*c - a*d)^2*(c + d*x^2)) + (8*Sqrt[a]*b^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(-(b*c) + a*d)^3 + ((3*b^2*c^2 + 6*a*b*c*d - a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(3/2)*Sqrt[d]*(b*c - a*d)^3))/8","A",1
255,1,98,98,0.0522948,"\int \frac{x}{\left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","Integrate[x/((a + b*x^2)*(c + d*x^2)^3),x]","\frac{b^2 \log \left(a+b x^2\right)}{2 (b c-a d)^3}-\frac{b^2 \log \left(c+d x^2\right)}{2 (b c-a d)^3}+\frac{b}{2 \left(c+d x^2\right) (b c-a d)^2}-\frac{1}{4 \left(c+d x^2\right)^2 (a d-b c)}","\frac{b^2 \log \left(a+b x^2\right)}{2 (b c-a d)^3}-\frac{b^2 \log \left(c+d x^2\right)}{2 (b c-a d)^3}+\frac{b}{2 \left(c+d x^2\right) (b c-a d)^2}+\frac{1}{4 \left(c+d x^2\right)^2 (b c-a d)}",1,"-1/4*1/((-(b*c) + a*d)*(c + d*x^2)^2) + b/(2*(b*c - a*d)^2*(c + d*x^2)) + (b^2*Log[a + b*x^2])/(2*(b*c - a*d)^3) - (b^2*Log[c + d*x^2])/(2*(b*c - a*d)^3)","A",1
256,1,158,160,0.2379761,"\int \frac{1}{\left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","Integrate[1/((a + b*x^2)*(c + d*x^2)^3),x]","\frac{1}{8} \left(-\frac{\sqrt{d} \left(3 a^2 d^2-10 a b c d+15 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{5/2} (b c-a d)^3}-\frac{8 b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} (a d-b c)^3}+\frac{d x (3 a d-7 b c)}{c^2 \left(c+d x^2\right) (b c-a d)^2}-\frac{2 d x}{c \left(c+d x^2\right)^2 (b c-a d)}\right)","-\frac{\sqrt{d} \left(3 a^2 d^2-10 a b c d+15 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{5/2} (b c-a d)^3}+\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} (b c-a d)^3}-\frac{d x (7 b c-3 a d)}{8 c^2 \left(c+d x^2\right) (b c-a d)^2}-\frac{d x}{4 c \left(c+d x^2\right)^2 (b c-a d)}",1,"((-2*d*x)/(c*(b*c - a*d)*(c + d*x^2)^2) + (d*(-7*b*c + 3*a*d)*x)/(c^2*(b*c - a*d)^2*(c + d*x^2)) - (8*b^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*(-(b*c) + a*d)^3) - (Sqrt[d]*(15*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(5/2)*(b*c - a*d)^3))/8","A",1
257,1,141,149,0.266758,"\int \frac{1}{x \left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","Integrate[1/(x*(a + b*x^2)*(c + d*x^2)^3),x]","\frac{\frac{d \left(\frac{c \left(a^2 d^2 \left(3 c+2 d x^2\right)-2 a b c d \left(4 c+3 d x^2\right)+b^2 c^2 \left(5 c+4 d x^2\right)\right)}{\left(c+d x^2\right)^2}-2 \left(a^2 d^2-3 a b c d+3 b^2 c^2\right) \log \left(c+d x^2\right)\right)}{c^3}+\frac{2 b^3 \log \left(a+b x^2\right)}{a}}{4 (a d-b c)^3}+\frac{\log (x)}{a c^3}","\frac{d \left(a^2 d^2-3 a b c d+3 b^2 c^2\right) \log \left(c+d x^2\right)}{2 c^3 (b c-a d)^3}-\frac{b^3 \log \left(a+b x^2\right)}{2 a (b c-a d)^3}-\frac{d (2 b c-a d)}{2 c^2 \left(c+d x^2\right) (b c-a d)^2}-\frac{d}{4 c \left(c+d x^2\right)^2 (b c-a d)}+\frac{\log (x)}{a c^3}",1,"Log[x]/(a*c^3) + ((2*b^3*Log[a + b*x^2])/a + (d*((c*(a^2*d^2*(3*c + 2*d*x^2) - 2*a*b*c*d*(4*c + 3*d*x^2) + b^2*c^2*(5*c + 4*d*x^2)))/(c + d*x^2)^2 - 2*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*Log[c + d*x^2]))/c^3)/(4*(-(b*c) + a*d)^3)","A",1
258,1,172,211,0.3763086,"\int \frac{1}{x^2 \left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","Integrate[1/(x^2*(a + b*x^2)*(c + d*x^2)^3),x]","\frac{1}{8} \left(\frac{8 b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2} (a d-b c)^3}+\frac{d^{3/2} \left(15 a^2 d^2-42 a b c d+35 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{7/2} (b c-a d)^3}+\frac{d^2 x (11 b c-7 a d)}{c^3 \left(c+d x^2\right) (b c-a d)^2}+\frac{2 d^2 x}{c^2 \left(c+d x^2\right)^2 (b c-a d)}-\frac{8}{a c^3 x}\right)","-\frac{b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2} (b c-a d)^3}+\frac{d^{3/2} \left(15 a^2 d^2-42 a b c d+35 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{7/2} (b c-a d)^3}-\frac{15 a^2 d^2-27 a b c d+8 b^2 c^2}{8 a c^3 x (b c-a d)^2}-\frac{d (9 b c-5 a d)}{8 c^2 x \left(c+d x^2\right) (b c-a d)^2}-\frac{d}{4 c x \left(c+d x^2\right)^2 (b c-a d)}",1,"(-8/(a*c^3*x) + (2*d^2*x)/(c^2*(b*c - a*d)*(c + d*x^2)^2) + (d^2*(11*b*c - 7*a*d)*x)/(c^3*(b*c - a*d)^2*(c + d*x^2)) + (8*b^(7/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*(-(b*c) + a*d)^3) + (d^(3/2)*(35*b^2*c^2 - 42*a*b*c*d + 15*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(7/2)*(b*c - a*d)^3))/8","A",1
259,1,171,178,0.4187828,"\int \frac{1}{x^3 \left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","Integrate[1/(x^3*(a + b*x^2)*(c + d*x^2)^3),x]","\frac{1}{4} \left(-\frac{2 b^4 \log \left(a+b x^2\right)}{a^2 (a d-b c)^3}-\frac{2 d^2 \left(3 a^2 d^2-8 a b c d+6 b^2 c^2\right) \log \left(c+d x^2\right)}{c^4 (b c-a d)^3}-\frac{4 \log (x) (3 a d+b c)}{a^2 c^4}+\frac{2 d^2 (3 b c-2 a d)}{c^3 \left(c+d x^2\right) (b c-a d)^2}+\frac{d^2}{c^2 \left(c+d x^2\right)^2 (b c-a d)}-\frac{2}{a c^3 x^2}\right)","\frac{b^4 \log \left(a+b x^2\right)}{2 a^2 (b c-a d)^3}-\frac{d^2 \left(3 a^2 d^2-8 a b c d+6 b^2 c^2\right) \log \left(c+d x^2\right)}{2 c^4 (b c-a d)^3}-\frac{\log (x) (3 a d+b c)}{a^2 c^4}+\frac{d^2 (3 b c-2 a d)}{2 c^3 \left(c+d x^2\right) (b c-a d)^2}+\frac{d^2}{4 c^2 \left(c+d x^2\right)^2 (b c-a d)}-\frac{1}{2 a c^3 x^2}",1,"(-2/(a*c^3*x^2) + d^2/(c^2*(b*c - a*d)*(c + d*x^2)^2) + (2*d^2*(3*b*c - 2*a*d))/(c^3*(b*c - a*d)^2*(c + d*x^2)) - (4*(b*c + 3*a*d)*Log[x])/(a^2*c^4) - (2*b^4*Log[a + b*x^2])/(a^2*(-(b*c) + a*d)^3) - (2*d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*Log[c + d*x^2])/(c^4*(b*c - a*d)^3))/4","A",1
260,1,196,270,0.431208,"\int \frac{1}{x^4 \left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","Integrate[1/(x^4*(a + b*x^2)*(c + d*x^2)^3),x]","-\frac{b^{9/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{5/2} (a d-b c)^3}-\frac{d^{5/2} \left(35 a^2 d^2-90 a b c d+63 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{9/2} (b c-a d)^3}+\frac{3 a d+b c}{a^2 c^4 x}-\frac{d^3 x (15 b c-11 a d)}{8 c^4 \left(c+d x^2\right) (b c-a d)^2}-\frac{d^3 x}{4 c^3 \left(c+d x^2\right)^2 (b c-a d)}-\frac{1}{3 a c^3 x^3}","\frac{b^{9/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{5/2} (b c-a d)^3}-\frac{d^{5/2} \left(35 a^2 d^2-90 a b c d+63 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{9/2} (b c-a d)^3}-\frac{35 a^2 d^2-55 a b c d+8 b^2 c^2}{24 a c^3 x^3 (b c-a d)^2}+\frac{35 a^3 d^3-55 a^2 b c d^2+8 a b^2 c^2 d+8 b^3 c^3}{8 a^2 c^4 x (b c-a d)^2}-\frac{d (11 b c-7 a d)}{8 c^2 x^3 \left(c+d x^2\right) (b c-a d)^2}-\frac{d}{4 c x^3 \left(c+d x^2\right)^2 (b c-a d)}",1,"-1/3*1/(a*c^3*x^3) + (b*c + 3*a*d)/(a^2*c^4*x) - (d^3*x)/(4*c^3*(b*c - a*d)*(c + d*x^2)^2) - (d^3*(15*b*c - 11*a*d)*x)/(8*c^4*(b*c - a*d)^2*(c + d*x^2)) - (b^(9/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(5/2)*(-(b*c) + a*d)^3) - (d^(5/2)*(63*b^2*c^2 - 90*a*b*c*d + 35*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(9/2)*(b*c - a*d)^3)","A",1
261,1,21,21,0.0043957,"\int \frac{x}{\left(1+x^2\right) \left(4+x^2\right)} \, dx","Integrate[x/((1 + x^2)*(4 + x^2)),x]","\frac{1}{6} \log \left(x^2+1\right)-\frac{1}{6} \log \left(x^2+4\right)","\frac{1}{6} \log \left(x^2+1\right)-\frac{1}{6} \log \left(x^2+4\right)",1,"Log[1 + x^2]/6 - Log[4 + x^2]/6","A",1
262,1,89,87,0.0731393,"\int \frac{x^4 \left(c+d x^2\right)}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^4*(c + d*x^2))/(a + b*x^2)^2,x]","\frac{x \left(a b c-a^2 d\right)}{2 b^3 \left(a+b x^2\right)}+\frac{\sqrt{a} (5 a d-3 b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{7/2}}+\frac{x (b c-2 a d)}{b^3}+\frac{d x^3}{3 b^2}","-\frac{\sqrt{a} (3 b c-5 a d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{7/2}}+\frac{a x (b c-a d)}{2 b^3 \left(a+b x^2\right)}+\frac{x (b c-2 a d)}{b^3}+\frac{d x^3}{3 b^2}",1,"((b*c - 2*a*d)*x)/b^3 + (d*x^3)/(3*b^2) + ((a*b*c - a^2*d)*x)/(2*b^3*(a + b*x^2)) + (Sqrt[a]*(-3*b*c + 5*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(7/2))","A",1
263,1,50,60,0.0350211,"\int \frac{x^3 \left(c+d x^2\right)}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^3*(c + d*x^2))/(a + b*x^2)^2,x]","\frac{\frac{a (b c-a d)}{a+b x^2}+(b c-2 a d) \log \left(a+b x^2\right)+b d x^2}{2 b^3}","\frac{a (b c-a d)}{2 b^3 \left(a+b x^2\right)}+\frac{(b c-2 a d) \log \left(a+b x^2\right)}{2 b^3}+\frac{d x^2}{2 b^2}",1,"(b*d*x^2 + (a*(b*c - a*d))/(a + b*x^2) + (b*c - 2*a*d)*Log[a + b*x^2])/(2*b^3)","A",1
264,1,68,67,0.0670596,"\int \frac{x^2 \left(c+d x^2\right)}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^2*(c + d*x^2))/(a + b*x^2)^2,x]","-\frac{(3 a d-b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 \sqrt{a} b^{5/2}}-\frac{x (b c-a d)}{2 b^2 \left(a+b x^2\right)}+\frac{d x}{b^2}","\frac{(b c-3 a d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 \sqrt{a} b^{5/2}}-\frac{x (b c-a d)}{2 b^2 \left(a+b x^2\right)}+\frac{d x}{b^2}",1,"(d*x)/b^2 - ((b*c - a*d)*x)/(2*b^2*(a + b*x^2)) - ((-(b*c) + 3*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*Sqrt[a]*b^(5/2))","A",1
265,1,41,41,0.0118015,"\int \frac{x \left(c+d x^2\right)}{\left(a+b x^2\right)^2} \, dx","Integrate[(x*(c + d*x^2))/(a + b*x^2)^2,x]","\frac{a d-b c}{2 b^2 \left(a+b x^2\right)}+\frac{d \log \left(a+b x^2\right)}{2 b^2}","\frac{a d-b c}{2 b^2 \left(a+b x^2\right)}+\frac{d \log \left(a+b x^2\right)}{2 b^2}",1,"(-(b*c) + a*d)/(2*b^2*(a + b*x^2)) + (d*Log[a + b*x^2])/(2*b^2)","A",1
266,1,63,63,0.0424642,"\int \frac{c+d x^2}{\left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)/(a + b*x^2)^2,x]","\frac{(a d+b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} b^{3/2}}-\frac{x (a d-b c)}{2 a b \left(a+b x^2\right)}","\frac{(a d+b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} b^{3/2}}+\frac{x (b c-a d)}{2 a b \left(a+b x^2\right)}",1,"-1/2*((-(b*c) + a*d)*x)/(a*b*(a + b*x^2)) + ((b*c + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*b^(3/2))","A",1
267,1,46,51,0.0282636,"\int \frac{c+d x^2}{x \left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)/(x*(a + b*x^2)^2),x]","\frac{\frac{a (b c-a d)}{b \left(a+b x^2\right)}-c \log \left(a+b x^2\right)+2 c \log (x)}{2 a^2}","-\frac{c \log \left(a+b x^2\right)}{2 a^2}+\frac{c \log (x)}{a^2}+\frac{b c-a d}{2 a b \left(a+b x^2\right)}",1,"((a*(b*c - a*d))/(b*(a + b*x^2)) + 2*c*Log[x] - c*Log[a + b*x^2])/(2*a^2)","A",1
268,1,70,71,0.0319068,"\int \frac{c+d x^2}{x^2 \left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)/(x^2*(a + b*x^2)^2),x]","\frac{(a d-3 b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{5/2} \sqrt{b}}+\frac{x (a d-b c)}{2 a^2 \left(a+b x^2\right)}-\frac{c}{a^2 x}","-\frac{(3 b c-a d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{5/2} \sqrt{b}}-\frac{x (b c-a d)}{2 a^2 \left(a+b x^2\right)}-\frac{c}{a^2 x}",1,"-(c/(a^2*x)) + ((-(b*c) + a*d)*x)/(2*a^2*(a + b*x^2)) + ((-3*b*c + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(5/2)*Sqrt[b])","A",1
269,1,64,76,0.0454075,"\int \frac{c+d x^2}{x^3 \left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)/(x^3*(a + b*x^2)^2),x]","\frac{\frac{a (a d-b c)}{a+b x^2}+(2 b c-a d) \log \left(a+b x^2\right)+2 \log (x) (a d-2 b c)-\frac{a c}{x^2}}{2 a^3}","\frac{(2 b c-a d) \log \left(a+b x^2\right)}{2 a^3}-\frac{\log (x) (2 b c-a d)}{a^3}-\frac{b c-a d}{2 a^2 \left(a+b x^2\right)}-\frac{c}{2 a^2 x^2}",1,"(-((a*c)/x^2) + (a*(-(b*c) + a*d))/(a + b*x^2) + 2*(-2*b*c + a*d)*Log[x] + (2*b*c - a*d)*Log[a + b*x^2])/(2*a^3)","A",1
270,1,90,90,0.0682488,"\int \frac{c+d x^2}{x^4 \left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)/(x^4*(a + b*x^2)^2),x]","-\frac{\sqrt{b} (3 a d-5 b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{7/2}}-\frac{b x (a d-b c)}{2 a^3 \left(a+b x^2\right)}+\frac{2 b c-a d}{a^3 x}-\frac{c}{3 a^2 x^3}","\frac{\sqrt{b} (5 b c-3 a d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{7/2}}+\frac{b x (b c-a d)}{2 a^3 \left(a+b x^2\right)}+\frac{2 b c-a d}{a^3 x}-\frac{c}{3 a^2 x^3}",1,"-1/3*c/(a^2*x^3) + (2*b*c - a*d)/(a^3*x) - (b*(-(b*c) + a*d)*x)/(2*a^3*(a + b*x^2)) - (Sqrt[b]*(-5*b*c + 3*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(7/2))","A",1
271,1,138,145,0.092092,"\int \frac{x^4 \left(c+d x^2\right)^2}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^4*(c + d*x^2)^2)/(a + b*x^2)^2,x]","-\frac{\sqrt{a} \left(7 a^2 d^2-10 a b c d+3 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{9/2}}+\frac{x \left(3 a^2 d^2-4 a b c d+b^2 c^2\right)}{b^4}+\frac{a x (b c-a d)^2}{2 b^4 \left(a+b x^2\right)}+\frac{2 d x^3 (b c-a d)}{3 b^3}+\frac{d^2 x^5}{5 b^2}","-\frac{\sqrt{a} (3 b c-7 a d) (b c-a d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{9/2}}+\frac{x (3 b c-7 a d) (b c-a d)}{2 b^4}-\frac{x^3 (3 b c-7 a d) (b c-a d)}{6 a b^3}+\frac{x^5 (b c-a d)^2}{2 a b^2 \left(a+b x^2\right)}+\frac{d^2 x^5}{5 b^2}",1,"((b^2*c^2 - 4*a*b*c*d + 3*a^2*d^2)*x)/b^4 + (2*d*(b*c - a*d)*x^3)/(3*b^3) + (d^2*x^5)/(5*b^2) + (a*(b*c - a*d)^2*x)/(2*b^4*(a + b*x^2)) - (Sqrt[a]*(3*b^2*c^2 - 10*a*b*c*d + 7*a^2*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(9/2))","A",1
272,1,87,88,0.0619378,"\int \frac{x^3 \left(c+d x^2\right)^2}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^3*(c + d*x^2)^2)/(a + b*x^2)^2,x]","\frac{2 \left(3 a^2 d^2-4 a b c d+b^2 c^2\right) \log \left(a+b x^2\right)+4 b d x^2 (b c-a d)+\frac{2 a (b c-a d)^2}{a+b x^2}+b^2 d^2 x^4}{4 b^4}","\frac{a (b c-a d)^2}{2 b^4 \left(a+b x^2\right)}+\frac{(b c-3 a d) (b c-a d) \log \left(a+b x^2\right)}{2 b^4}+\frac{d x^2 (b c-a d)}{b^3}+\frac{d^2 x^4}{4 b^2}",1,"(4*b*d*(b*c - a*d)*x^2 + b^2*d^2*x^4 + (2*a*(b*c - a*d)^2)/(a + b*x^2) + 2*(b^2*c^2 - 4*a*b*c*d + 3*a^2*d^2)*Log[a + b*x^2])/(4*b^4)","A",1
273,1,105,116,0.0708869,"\int \frac{x^2 \left(c+d x^2\right)^2}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^2*(c + d*x^2)^2)/(a + b*x^2)^2,x]","\frac{\left(5 a^2 d^2-6 a b c d+b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 \sqrt{a} b^{7/2}}-\frac{x (b c-a d)^2}{2 b^3 \left(a+b x^2\right)}+\frac{2 d x (b c-a d)}{b^3}+\frac{d^2 x^3}{3 b^2}","\frac{(b c-5 a d) (b c-a d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 \sqrt{a} b^{7/2}}-\frac{x (b c-5 a d) (b c-a d)}{2 a b^3}+\frac{x^3 (b c-a d)^2}{2 a b^2 \left(a+b x^2\right)}+\frac{d^2 x^3}{3 b^2}",1,"(2*d*(b*c - a*d)*x)/b^3 + (d^2*x^3)/(3*b^2) - ((b*c - a*d)^2*x)/(2*b^3*(a + b*x^2)) + ((b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*Sqrt[a]*b^(7/2))","A",1
274,1,56,61,0.0445369,"\int \frac{x \left(c+d x^2\right)^2}{\left(a+b x^2\right)^2} \, dx","Integrate[(x*(c + d*x^2)^2)/(a + b*x^2)^2,x]","\frac{-\frac{(b c-a d)^2}{a+b x^2}+2 d (b c-a d) \log \left(a+b x^2\right)+b d^2 x^2}{2 b^3}","-\frac{(b c-a d)^2}{2 b^3 \left(a+b x^2\right)}+\frac{d (b c-a d) \log \left(a+b x^2\right)}{b^3}+\frac{d^2 x^2}{2 b^2}",1,"(b*d^2*x^2 - (b*c - a*d)^2/(a + b*x^2) + 2*d*(b*c - a*d)*Log[a + b*x^2])/(2*b^3)","A",1
275,1,88,82,0.059058,"\int \frac{\left(c+d x^2\right)^2}{\left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)^2/(a + b*x^2)^2,x]","\frac{\left(-3 a^2 d^2+2 a b c d+b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} b^{5/2}}+\frac{x (b c-a d)^2}{2 a b^2 \left(a+b x^2\right)}+\frac{d^2 x}{b^2}","\frac{(b c-a d) (3 a d+b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} b^{5/2}}+\frac{x (b c-a d)^2}{2 a b^2 \left(a+b x^2\right)}+\frac{d^2 x}{b^2}",1,"(d^2*x)/b^2 + ((b*c - a*d)^2*x)/(2*a*b^2*(a + b*x^2)) + ((b^2*c^2 + 2*a*b*c*d - 3*a^2*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*b^(5/2))","A",1
276,1,70,67,0.0410361,"\int \frac{\left(c+d x^2\right)^2}{x \left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)^2/(x*(a + b*x^2)^2),x]","\frac{\frac{(a d-b c) \left(\left(a+b x^2\right) (a d+b c) \log \left(a+b x^2\right)+a (a d-b c)\right)}{b^2 \left(a+b x^2\right)}+2 c^2 \log (x)}{2 a^2}","-\frac{1}{2} \left(\frac{c^2}{a^2}-\frac{d^2}{b^2}\right) \log \left(a+b x^2\right)+\frac{c^2 \log (x)}{a^2}+\frac{(b c-a d)^2}{2 a b^2 \left(a+b x^2\right)}",1,"(2*c^2*Log[x] + ((-(b*c) + a*d)*(a*(-(b*c) + a*d) + (b*c + a*d)*(a + b*x^2)*Log[a + b*x^2]))/(b^2*(a + b*x^2)))/(2*a^2)","A",1
277,1,91,103,0.0690193,"\int \frac{\left(c+d x^2\right)^2}{x^2 \left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)^2/(x^2*(a + b*x^2)^2),x]","-\frac{x (a d-b c)^2}{2 a^2 b \left(a+b x^2\right)}-\frac{c^2}{a^2 x}+\frac{\left(a^2 d^2+2 a b c d-3 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{5/2} b^{3/2}}","-\frac{(b c-a d) (a d+3 b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{5/2} b^{3/2}}-\frac{x \left(\frac{3 b c^2}{a}+\frac{a d^2}{b}-2 c d\right)}{2 a \left(a+b x^2\right)}-\frac{c^2}{a x \left(a+b x^2\right)}",1,"-(c^2/(a^2*x)) - ((-(b*c) + a*d)^2*x)/(2*a^2*b*(a + b*x^2)) + ((-3*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(5/2)*b^(3/2))","A",1
278,1,72,80,0.1028725,"\int \frac{\left(c+d x^2\right)^2}{x^3 \left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)^2/(x^3*(a + b*x^2)^2),x]","-\frac{\frac{a (b c-a d)^2}{b \left(a+b x^2\right)}-2 c (b c-a d) \log \left(a+b x^2\right)+4 c \log (x) (b c-a d)+\frac{a c^2}{x^2}}{2 a^3}","\frac{c (b c-a d) \log \left(a+b x^2\right)}{a^3}-\frac{2 c \log (x) (b c-a d)}{a^3}-\frac{(b c-a d)^2}{2 a^2 b \left(a+b x^2\right)}-\frac{c^2}{2 a^2 x^2}",1,"-1/2*((a*c^2)/x^2 + (a*(b*c - a*d)^2)/(b*(a + b*x^2)) + 4*c*(b*c - a*d)*Log[x] - 2*c*(b*c - a*d)*Log[a + b*x^2])/a^3","A",1
279,1,107,127,0.0734678,"\int \frac{\left(c+d x^2\right)^2}{x^4 \left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)^2/(x^4*(a + b*x^2)^2),x]","\frac{x (a d-b c)^2}{2 a^3 \left(a+b x^2\right)}-\frac{2 c (a d-b c)}{a^3 x}-\frac{c^2}{3 a^2 x^3}+\frac{\left(a^2 d^2-6 a b c d+5 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{7/2} \sqrt{b}}","\frac{(b c-a d) (5 b c-a d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{7/2} \sqrt{b}}+\frac{c (5 b c-6 a d)}{3 a^3 x}+\frac{x \left(3 a^2 d^2-6 a b c d+5 b^2 c^2\right)}{6 a^3 \left(a+b x^2\right)}-\frac{c^2}{3 a x^3 \left(a+b x^2\right)}",1,"-1/3*c^2/(a^2*x^3) - (2*c*(-(b*c) + a*d))/(a^3*x) + ((-(b*c) + a*d)^2*x)/(2*a^3*(a + b*x^2)) + ((5*b^2*c^2 - 6*a*b*c*d + a^2*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(7/2)*Sqrt[b])","A",1
280,1,151,169,0.0846479,"\int \frac{x^4 \left(c+d x^2\right)^3}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^4*(c + d*x^2)^3)/(a + b*x^2)^2,x]","\frac{3 \sqrt{a} (b c-a d)^2 (3 a d-b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{11/2}}+\frac{a x (b c-a d)^3}{2 b^5 \left(a+b x^2\right)}+\frac{x (b c-4 a d) (b c-a d)^2}{b^5}+\frac{d x^3 (b c-a d)^2}{b^4}+\frac{d^2 x^5 (3 b c-2 a d)}{5 b^3}+\frac{d^3 x^7}{7 b^2}","\frac{d x^3 \left(3 a^2 d^2-7 a b c d+5 b^2 c^2\right)}{2 b^4}-\frac{3 \sqrt{a} (b c-3 a d) (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{11/2}}+\frac{3 x (b c-3 a d) (b c-a d)^2}{2 b^5}+\frac{3 d^2 x^5 (7 b c-3 a d)}{10 b^3}-\frac{x^3 \left(c+d x^2\right)^3}{2 b \left(a+b x^2\right)}+\frac{9 d^3 x^7}{14 b^2}",1,"((b*c - 4*a*d)*(b*c - a*d)^2*x)/b^5 + (d*(b*c - a*d)^2*x^3)/b^4 + (d^2*(3*b*c - 2*a*d)*x^5)/(5*b^3) + (d^3*x^7)/(7*b^2) + (a*(b*c - a*d)^3*x)/(2*b^5*(a + b*x^2)) + (3*Sqrt[a]*(b*c - a*d)^2*(-(b*c) + 3*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(11/2))","A",1
281,1,106,117,0.088966,"\int \frac{x^3 \left(c+d x^2\right)^3}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^3*(c + d*x^2)^3)/(a + b*x^2)^2,x]","\frac{3 b^2 d^2 x^4 (3 b c-2 a d)+18 b d x^2 (b c-a d)^2-\frac{6 a (a d-b c)^3}{a+b x^2}+6 (b c-4 a d) (b c-a d)^2 \log \left(a+b x^2\right)+2 b^3 d^3 x^6}{12 b^5}","\frac{a (b c-a d)^3}{2 b^5 \left(a+b x^2\right)}+\frac{(b c-4 a d) (b c-a d)^2 \log \left(a+b x^2\right)}{2 b^5}+\frac{3 d x^2 (b c-a d)^2}{2 b^4}+\frac{d^2 x^4 (3 b c-2 a d)}{4 b^3}+\frac{d^3 x^6}{6 b^2}",1,"(18*b*d*(b*c - a*d)^2*x^2 + 3*b^2*d^2*(3*b*c - 2*a*d)*x^4 + 2*b^3*d^3*x^6 - (6*a*(-(b*c) + a*d)^3)/(a + b*x^2) + 6*(b*c - 4*a*d)*(b*c - a*d)^2*Log[a + b*x^2])/(12*b^5)","A",1
282,1,125,147,0.0737934,"\int \frac{x^2 \left(c+d x^2\right)^3}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^2*(c + d*x^2)^3)/(a + b*x^2)^2,x]","\frac{(b c-7 a d) (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 \sqrt{a} b^{9/2}}-\frac{x (b c-a d)^3}{2 b^4 \left(a+b x^2\right)}+\frac{3 d x (b c-a d)^2}{b^4}+\frac{d^2 x^3 (3 b c-2 a d)}{3 b^3}+\frac{d^3 x^5}{5 b^2}","\frac{d x \left(105 a^2 d^2-190 a b c d+81 b^2 c^2\right)}{30 b^4}+\frac{(b c-7 a d) (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 \sqrt{a} b^{9/2}}+\frac{d x \left(c+d x^2\right) (33 b c-35 a d)}{30 b^3}-\frac{x \left(c+d x^2\right)^3}{2 b \left(a+b x^2\right)}+\frac{7 d x \left(c+d x^2\right)^2}{10 b^2}",1,"(3*d*(b*c - a*d)^2*x)/b^4 + (d^2*(3*b*c - 2*a*d)*x^3)/(3*b^3) + (d^3*x^5)/(5*b^2) - ((b*c - a*d)^3*x)/(2*b^4*(a + b*x^2)) + ((b*c - 7*a*d)*(b*c - a*d)^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*Sqrt[a]*b^(9/2))","A",1
283,1,127,88,0.0459201,"\int \frac{x \left(c+d x^2\right)^3}{\left(a+b x^2\right)^2} \, dx","Integrate[(x*(c + d*x^2)^3)/(a + b*x^2)^2,x]","\frac{3 \left(a^2 d^3-2 a b c d^2+b^2 c^2 d\right) \log \left(a+b x^2\right)}{2 b^4}+\frac{a^3 d^3-3 a^2 b c d^2+3 a b^2 c^2 d-b^3 c^3}{2 b^4 \left(a+b x^2\right)}+\frac{d^2 x^2 (3 b c-2 a d)}{2 b^3}+\frac{d^3 x^4}{4 b^2}","-\frac{(b c-a d)^3}{2 b^4 \left(a+b x^2\right)}+\frac{3 d (b c-a d)^2 \log \left(a+b x^2\right)}{2 b^4}+\frac{d^2 x^2 (3 b c-2 a d)}{2 b^3}+\frac{d^3 x^4}{4 b^2}",1,"(d^2*(3*b*c - 2*a*d)*x^2)/(2*b^3) + (d^3*x^4)/(4*b^2) + (-(b^3*c^3) + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)/(2*b^4*(a + b*x^2)) + (3*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*Log[a + b*x^2])/(2*b^4)","A",1
284,1,106,106,0.0626234,"\int \frac{\left(c+d x^2\right)^3}{\left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)^3/(a + b*x^2)^2,x]","\frac{(5 a d+b c) (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} b^{7/2}}+\frac{d^2 x (3 b c-2 a d)}{b^3}+\frac{x (b c-a d)^3}{2 a b^3 \left(a+b x^2\right)}+\frac{d^3 x^3}{3 b^2}","\frac{(5 a d+b c) (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} b^{7/2}}+\frac{d^2 x (3 b c-2 a d)}{b^3}+\frac{x (b c-a d)^3}{2 a b^3 \left(a+b x^2\right)}+\frac{d^3 x^3}{3 b^2}",1,"(d^2*(3*b*c - 2*a*d)*x)/b^3 + (d^3*x^3)/(3*b^2) + ((b*c - a*d)^3*x)/(2*a*b^3*(a + b*x^2)) + ((b*c - a*d)^2*(b*c + 5*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*b^(7/2))","A",1
285,1,111,88,0.1012735,"\int \frac{\left(c+d x^2\right)^3}{x \left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)^3/(x*(a + b*x^2)^2),x]","\frac{\frac{\frac{a \left(-a^3 d^3+a^2 b d^2 \left(3 c+d x^2\right)+a b^2 \left(d^3 x^4-3 c^2 d\right)+b^3 c^3\right)}{a+b x^2}-(b c-a d)^2 (2 a d+b c) \log \left(a+b x^2\right)}{b^3}+2 c^3 \log (x)}{2 a^2}","-\frac{(b c-a d)^2 (2 a d+b c) \log \left(a+b x^2\right)}{2 a^2 b^3}+\frac{c^3 \log (x)}{a^2}+\frac{(b c-a d)^3}{2 a b^3 \left(a+b x^2\right)}+\frac{d^3 x^2}{2 b^2}",1,"(2*c^3*Log[x] + ((a*(b^3*c^3 - a^3*d^3 + a^2*b*d^2*(3*c + d*x^2) + a*b^2*(-3*c^2*d + d^3*x^4)))/(a + b*x^2) - (b*c - a*d)^2*(b*c + 2*a*d)*Log[a + b*x^2])/b^3)/(2*a^2)","A",1
286,1,94,131,0.0638745,"\int \frac{\left(c+d x^2\right)^3}{x^2 \left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)^3/(x^2*(a + b*x^2)^2),x]","-\frac{3 (a d-b c)^2 (a d+b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{5/2} b^{5/2}}+\frac{x (a d-b c)^3}{2 a^2 b^2 \left(a+b x^2\right)}-\frac{c^3}{a^2 x}+\frac{d^3 x}{b^2}","-\frac{3 (b c-a d)^2 (a d+b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{5/2} b^{5/2}}-\frac{c^2 (3 b c-a d)}{2 a^2 b x}-\frac{d^2 x (b c-3 a d)}{2 a b^2}+\frac{\left(c+d x^2\right)^2 (b c-a d)}{2 a b x \left(a+b x^2\right)}",1,"-(c^3/(a^2*x)) + (d^3*x)/b^2 + ((-(b*c) + a*d)^3*x)/(2*a^2*b^2*(a + b*x^2)) - (3*(-(b*c) + a*d)^2*(b*c + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(5/2)*b^(5/2))","A",1
287,1,87,98,0.0962176,"\int \frac{\left(c+d x^2\right)^3}{x^3 \left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)^3/(x^3*(a + b*x^2)^2),x]","\frac{\frac{a (a d-b c)^3}{b^2 \left(a+b x^2\right)}+\frac{(b c-a d)^2 (a d+2 b c) \log \left(a+b x^2\right)}{b^2}+2 c^2 \log (x) (3 a d-2 b c)-\frac{a c^3}{x^2}}{2 a^3}","\frac{(b c-a d)^2 (a d+2 b c) \log \left(a+b x^2\right)}{2 a^3 b^2}-\frac{c^2 \log (x) (2 b c-3 a d)}{a^3}-\frac{(b c-a d)^3}{2 a^2 b^2 \left(a+b x^2\right)}-\frac{c^3}{2 a^2 x^2}",1,"(-((a*c^3)/x^2) + (a*(-(b*c) + a*d)^3)/(b^2*(a + b*x^2)) + 2*c^2*(-2*b*c + 3*a*d)*Log[x] + ((b*c - a*d)^2*(2*b*c + a*d)*Log[a + b*x^2])/b^2)/(2*a^3)","A",1
288,1,109,147,0.0682374,"\int \frac{\left(c+d x^2\right)^3}{x^4 \left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)^3/(x^4*(a + b*x^2)^2),x]","\frac{(a d-b c)^2 (a d+5 b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{7/2} b^{3/2}}-\frac{c^2 (3 a d-2 b c)}{a^3 x}-\frac{x (a d-b c)^3}{2 a^3 b \left(a+b x^2\right)}-\frac{c^3}{3 a^2 x^3}","\frac{(b c-a d)^2 (a d+5 b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{7/2} b^{3/2}}-\frac{c^2 (5 b c-3 a d)}{6 a^2 b x^3}+\frac{c \left(2 a^2 d^2-9 a b c d+5 b^2 c^2\right)}{2 a^3 b x}+\frac{\left(c+d x^2\right)^2 (b c-a d)}{2 a b x^3 \left(a+b x^2\right)}",1,"-1/3*c^3/(a^2*x^3) - (c^2*(-2*b*c + 3*a*d))/(a^3*x) - ((-(b*c) + a*d)^3*x)/(2*a^3*b*(a + b*x^2)) + ((-(b*c) + a*d)^2*(5*b*c + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(7/2)*b^(3/2))","A",1
289,1,95,109,0.1477747,"\int \frac{x^4}{\left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","Integrate[x^4/((a + b*x^2)^2*(c + d*x^2)),x]","\frac{\frac{\sqrt{a} (a d-3 b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{3/2}}+\frac{a x (b c-a d)}{b \left(a+b x^2\right)}+\frac{2 c^{3/2} \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{\sqrt{d}}}{2 (b c-a d)^2}","-\frac{\sqrt{a} (3 b c-a d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{3/2} (b c-a d)^2}+\frac{c^{3/2} \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{\sqrt{d} (b c-a d)^2}+\frac{a x}{2 b \left(a+b x^2\right) (b c-a d)}",1,"((a*(b*c - a*d)*x)/(b*(a + b*x^2)) + (Sqrt[a]*(-3*b*c + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(3/2) + (2*c^(3/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/Sqrt[d])/(2*(b*c - a*d)^2)","A",1
290,1,74,74,0.0352388,"\int \frac{x^3}{\left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","Integrate[x^3/((a + b*x^2)^2*(c + d*x^2)),x]","\frac{a}{2 b \left(a+b x^2\right) (b c-a d)}+\frac{c \log \left(a+b x^2\right)}{2 (b c-a d)^2}-\frac{c \log \left(c+d x^2\right)}{2 (b c-a d)^2}","\frac{a}{2 b \left(a+b x^2\right) (b c-a d)}+\frac{c \log \left(a+b x^2\right)}{2 (b c-a d)^2}-\frac{c \log \left(c+d x^2\right)}{2 (b c-a d)^2}",1,"a/(2*b*(b*c - a*d)*(a + b*x^2)) + (c*Log[a + b*x^2])/(2*(b*c - a*d)^2) - (c*Log[c + d*x^2])/(2*(b*c - a*d)^2)","A",1
291,1,104,104,0.1364637,"\int \frac{x^2}{\left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","Integrate[x^2/((a + b*x^2)^2*(c + d*x^2)),x]","\frac{x}{2 \left(a+b x^2\right) (a d-b c)}+\frac{(a d+b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 \sqrt{a} \sqrt{b} (a d-b c)^2}-\frac{\sqrt{c} \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{(b c-a d)^2}","-\frac{x}{2 \left(a+b x^2\right) (b c-a d)}+\frac{(a d+b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 \sqrt{a} \sqrt{b} (b c-a d)^2}-\frac{\sqrt{c} \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{(b c-a d)^2}",1,"x/(2*(-(b*c) + a*d)*(a + b*x^2)) + ((b*c + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*Sqrt[a]*Sqrt[b]*(-(b*c) + a*d)^2) - (Sqrt[c]*Sqrt[d]*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(b*c - a*d)^2","A",1
292,1,66,70,0.0282534,"\int \frac{x}{\left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","Integrate[x/((a + b*x^2)^2*(c + d*x^2)),x]","\frac{d \left(a+b x^2\right) \log \left(c+d x^2\right)-d \left(a+b x^2\right) \log \left(a+b x^2\right)+a d-b c}{2 \left(a+b x^2\right) (b c-a d)^2}","-\frac{1}{2 \left(a+b x^2\right) (b c-a d)}-\frac{d \log \left(a+b x^2\right)}{2 (b c-a d)^2}+\frac{d \log \left(c+d x^2\right)}{2 (b c-a d)^2}",1,"(-(b*c) + a*d - d*(a + b*x^2)*Log[a + b*x^2] + d*(a + b*x^2)*Log[c + d*x^2])/(2*(b*c - a*d)^2*(a + b*x^2))","A",1
293,1,109,108,0.1399025,"\int \frac{1}{\left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","Integrate[1/((a + b*x^2)^2*(c + d*x^2)),x]","-\frac{\sqrt{b} (3 a d-b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} (a d-b c)^2}+\frac{d^{3/2} \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{\sqrt{c} (b c-a d)^2}-\frac{b x}{2 a \left(a+b x^2\right) (a d-b c)}","\frac{\sqrt{b} (b c-3 a d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} (b c-a d)^2}+\frac{d^{3/2} \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{\sqrt{c} (b c-a d)^2}+\frac{b x}{2 a \left(a+b x^2\right) (b c-a d)}",1,"-1/2*(b*x)/(a*(-(b*c) + a*d)*(a + b*x^2)) - (Sqrt[b]*(-(b*c) + 3*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*(-(b*c) + a*d)^2) + (d^(3/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*(b*c - a*d)^2)","A",1
294,1,97,99,0.1161905,"\int \frac{1}{x \left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","Integrate[1/(x*(a + b*x^2)^2*(c + d*x^2)),x]","\frac{2 \log (x)-\frac{a \left(a d^2 \left(a+b x^2\right) \log \left(c+d x^2\right)+b c (a d-b c)\right)+b c \left(a+b x^2\right) (b c-2 a d) \log \left(a+b x^2\right)}{\left(a+b x^2\right) (b c-a d)^2}}{2 a^2 c}","-\frac{b (b c-2 a d) \log \left(a+b x^2\right)}{2 a^2 (b c-a d)^2}+\frac{\log (x)}{a^2 c}-\frac{d^2 \log \left(c+d x^2\right)}{2 c (b c-a d)^2}+\frac{b}{2 a \left(a+b x^2\right) (b c-a d)}",1,"(2*Log[x] - (b*c*(b*c - 2*a*d)*(a + b*x^2)*Log[a + b*x^2] + a*(b*c*(-(b*c) + a*d) + a*d^2*(a + b*x^2)*Log[c + d*x^2]))/((b*c - a*d)^2*(a + b*x^2)))/(2*a^2*c)","A",1
295,1,123,144,0.1801144,"\int \frac{1}{x^2 \left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","Integrate[1/(x^2*(a + b*x^2)^2*(c + d*x^2)),x]","\frac{b^{3/2} (5 a d-3 b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{5/2} (a d-b c)^2}+\frac{b^2 x}{2 a^2 \left(a+b x^2\right) (a d-b c)}-\frac{1}{a^2 c x}-\frac{d^{5/2} \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{3/2} (b c-a d)^2}","-\frac{b^{3/2} (3 b c-5 a d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{5/2} (b c-a d)^2}-\frac{3 b c-2 a d}{2 a^2 c x (b c-a d)}-\frac{d^{5/2} \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{3/2} (b c-a d)^2}+\frac{b}{2 a x \left(a+b x^2\right) (b c-a d)}",1,"-(1/(a^2*c*x)) + (b^2*x)/(2*a^2*(-(b*c) + a*d)*(a + b*x^2)) + (b^(3/2)*(-3*b*c + 5*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(5/2)*(-(b*c) + a*d)^2) - (d^(5/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(3/2)*(b*c - a*d)^2)","A",1
296,1,119,126,0.156191,"\int \frac{1}{x^3 \left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","Integrate[1/(x^3*(a + b*x^2)^2*(c + d*x^2)),x]","\frac{1}{2} \left(\frac{b^2 (2 b c-3 a d) \log \left(a+b x^2\right)}{a^3 (b c-a d)^2}-\frac{2 \log (x) (a d+2 b c)}{a^3 c^2}+\frac{b^2}{a^2 \left(a+b x^2\right) (a d-b c)}-\frac{1}{a^2 c x^2}+\frac{d^3 \log \left(c+d x^2\right)}{c^2 (b c-a d)^2}\right)","\frac{b^2 (2 b c-3 a d) \log \left(a+b x^2\right)}{2 a^3 (b c-a d)^2}-\frac{\log (x) (a d+2 b c)}{a^3 c^2}-\frac{b^2}{2 a^2 \left(a+b x^2\right) (b c-a d)}-\frac{1}{2 a^2 c x^2}+\frac{d^3 \log \left(c+d x^2\right)}{2 c^2 (b c-a d)^2}",1,"(-(1/(a^2*c*x^2)) + b^2/(a^2*(-(b*c) + a*d)*(a + b*x^2)) - (2*(2*b*c + a*d)*Log[x])/(a^3*c^2) + (b^2*(2*b*c - 3*a*d)*Log[a + b*x^2])/(a^3*(b*c - a*d)^2) + (d^3*Log[c + d*x^2])/(c^2*(b*c - a*d)^2))/2","A",1
297,1,142,189,0.2855173,"\int \frac{1}{x^4 \left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","Integrate[1/(x^4*(a + b*x^2)^2*(c + d*x^2)),x]","-\frac{b^{5/2} (7 a d-5 b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{7/2} (a d-b c)^2}-\frac{b^3 x}{2 a^3 \left(a+b x^2\right) (a d-b c)}+\frac{a d+2 b c}{a^3 c^2 x}-\frac{1}{3 a^2 c x^3}+\frac{d^{7/2} \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{5/2} (b c-a d)^2}","\frac{b^{5/2} (5 b c-7 a d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{7/2} (b c-a d)^2}-\frac{5 b c-2 a d}{6 a^2 c x^3 (b c-a d)}+\frac{-2 a^2 d^2-2 a b c d+5 b^2 c^2}{2 a^3 c^2 x (b c-a d)}+\frac{d^{7/2} \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{5/2} (b c-a d)^2}+\frac{b}{2 a x^3 \left(a+b x^2\right) (b c-a d)}",1,"-1/3*1/(a^2*c*x^3) + (2*b*c + a*d)/(a^3*c^2*x) - (b^3*x)/(2*a^3*(-(b*c) + a*d)*(a + b*x^2)) - (b^(5/2)*(-5*b*c + 7*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(7/2)*(-(b*c) + a*d)^2) + (d^(7/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(5/2)*(b*c - a*d)^2)","A",1
298,1,155,160,0.200071,"\int \frac{1}{x^5 \left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","Integrate[1/(x^5*(a + b*x^2)^2*(c + d*x^2)),x]","\frac{1}{4} \left(\frac{2 b^3 (4 a d-3 b c) \log \left(a+b x^2\right)}{a^4 (b c-a d)^2}-\frac{2 b^3}{a^3 \left(a+b x^2\right) (a d-b c)}+\frac{2 a d+4 b c}{a^3 c^2 x^2}-\frac{1}{a^2 c x^4}+\frac{4 \log (x) \left(a^2 d^2+2 a b c d+3 b^2 c^2\right)}{a^4 c^3}-\frac{2 d^4 \log \left(c+d x^2\right)}{c^3 (b c-a d)^2}\right)","-\frac{b^3 (3 b c-4 a d) \log \left(a+b x^2\right)}{2 a^4 (b c-a d)^2}+\frac{b^3}{2 a^3 \left(a+b x^2\right) (b c-a d)}+\frac{a d+2 b c}{2 a^3 c^2 x^2}-\frac{1}{4 a^2 c x^4}+\frac{\log (x) \left(a^2 d^2+2 a b c d+3 b^2 c^2\right)}{a^4 c^3}-\frac{d^4 \log \left(c+d x^2\right)}{2 c^3 (b c-a d)^2}",1,"(-(1/(a^2*c*x^4)) + (4*b*c + 2*a*d)/(a^3*c^2*x^2) - (2*b^3)/(a^3*(-(b*c) + a*d)*(a + b*x^2)) + (4*(3*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*Log[x])/(a^4*c^3) + (2*b^3*(-3*b*c + 4*a*d)*Log[a + b*x^2])/(a^4*(b*c - a*d)^2) - (2*d^4*Log[c + d*x^2])/(c^3*(b*c - a*d)^2))/4","A",1
299,1,179,250,0.3117591,"\int \frac{1}{x^6 \left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","Integrate[1/(x^6*(a + b*x^2)^2*(c + d*x^2)),x]","\frac{b^{7/2} (9 a d-7 b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{9/2} (a d-b c)^2}+\frac{b^4 x}{2 a^4 \left(a+b x^2\right) (a d-b c)}+\frac{a d+2 b c}{3 a^3 c^2 x^3}-\frac{1}{5 a^2 c x^5}+\frac{-a^2 d^2-2 a b c d-3 b^2 c^2}{a^4 c^3 x}-\frac{d^{9/2} \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{7/2} (b c-a d)^2}","-\frac{b^{7/2} (7 b c-9 a d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{9/2} (b c-a d)^2}-\frac{7 b c-2 a d}{10 a^2 c x^5 (b c-a d)}+\frac{-2 a^2 d^2-2 a b c d+7 b^2 c^2}{6 a^3 c^2 x^3 (b c-a d)}-\frac{-2 a^3 d^3-2 a^2 b c d^2-2 a b^2 c^2 d+7 b^3 c^3}{2 a^4 c^3 x (b c-a d)}-\frac{d^{9/2} \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{7/2} (b c-a d)^2}+\frac{b}{2 a x^5 \left(a+b x^2\right) (b c-a d)}",1,"-1/5*1/(a^2*c*x^5) + (2*b*c + a*d)/(3*a^3*c^2*x^3) + (-3*b^2*c^2 - 2*a*b*c*d - a^2*d^2)/(a^4*c^3*x) + (b^4*x)/(2*a^4*(-(b*c) + a*d)*(a + b*x^2)) + (b^(7/2)*(-7*b*c + 9*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(9/2)*(-(b*c) + a*d)^2) - (d^(9/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(7/2)*(b*c - a*d)^2)","A",1
300,1,202,210,0.2766758,"\int \frac{1}{x^7 \left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","Integrate[1/(x^7*(a + b*x^2)^2*(c + d*x^2)),x]","\frac{1}{12} \left(\frac{6 b^4 (4 b c-5 a d) \log \left(a+b x^2\right)}{a^5 (b c-a d)^2}+\frac{6 b^4}{a^4 \left(a+b x^2\right) (a d-b c)}+\frac{3 a d+6 b c}{a^3 c^2 x^4}-\frac{2}{a^2 c x^6}-\frac{6 \left(a^2 d^2+2 a b c d+3 b^2 c^2\right)}{a^4 c^3 x^2}-\frac{12 \log (x) \left(a^3 d^3+2 a^2 b c d^2+3 a b^2 c^2 d+4 b^3 c^3\right)}{a^5 c^4}+\frac{6 d^5 \log \left(c+d x^2\right)}{c^4 (b c-a d)^2}\right)","\frac{b^4 (4 b c-5 a d) \log \left(a+b x^2\right)}{2 a^5 (b c-a d)^2}-\frac{b^4}{2 a^4 \left(a+b x^2\right) (b c-a d)}+\frac{a d+2 b c}{4 a^3 c^2 x^4}-\frac{1}{6 a^2 c x^6}-\frac{a^2 d^2+2 a b c d+3 b^2 c^2}{2 a^4 c^3 x^2}-\frac{\log (x) \left(a^3 d^3+2 a^2 b c d^2+3 a b^2 c^2 d+4 b^3 c^3\right)}{a^5 c^4}+\frac{d^5 \log \left(c+d x^2\right)}{2 c^4 (b c-a d)^2}",1,"(-2/(a^2*c*x^6) + (6*b*c + 3*a*d)/(a^3*c^2*x^4) - (6*(3*b^2*c^2 + 2*a*b*c*d + a^2*d^2))/(a^4*c^3*x^2) + (6*b^4)/(a^4*(-(b*c) + a*d)*(a + b*x^2)) - (12*(4*b^3*c^3 + 3*a*b^2*c^2*d + 2*a^2*b*c*d^2 + a^3*d^3)*Log[x])/(a^5*c^4) + (6*b^4*(4*b*c - 5*a*d)*Log[a + b*x^2])/(a^5*(b*c - a*d)^2) + (6*d^5*Log[c + d*x^2])/(c^4*(b*c - a*d)^2))/12","A",1
301,1,133,162,0.2143402,"\int \frac{x^4}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","Integrate[x^4/((a + b*x^2)^2*(c + d*x^2)^2),x]","\frac{1}{2} \left(\frac{a x}{\left(a+b x^2\right) (b c-a d)^2}+\frac{c x}{\left(c+d x^2\right) (b c-a d)^2}+\frac{\sqrt{a} (a d+3 b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{b} (a d-b c)^3}+\frac{\sqrt{c} (3 a d+b c) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{\sqrt{d} (b c-a d)^3}\right)","\frac{x (a d+b c)}{2 b \left(c+d x^2\right) (b c-a d)^2}+\frac{a x}{2 b \left(a+b x^2\right) \left(c+d x^2\right) (b c-a d)}-\frac{\sqrt{a} (a d+3 b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 \sqrt{b} (b c-a d)^3}+\frac{\sqrt{c} (3 a d+b c) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 \sqrt{d} (b c-a d)^3}",1,"((a*x)/((b*c - a*d)^2*(a + b*x^2)) + (c*x)/((b*c - a*d)^2*(c + d*x^2)) + (Sqrt[a]*(3*b*c + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[b]*(-(b*c) + a*d)^3) + (Sqrt[c]*(b*c + 3*a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[d]*(b*c - a*d)^3))/2","A",1
302,1,86,107,0.0958935,"\int \frac{x^3}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","Integrate[x^3/((a + b*x^2)^2*(c + d*x^2)^2),x]","\frac{\frac{a (b c-a d)}{a+b x^2}+\frac{c (b c-a d)}{c+d x^2}+(a d+b c) \log \left(a+b x^2\right)-(a d+b c) \log \left(c+d x^2\right)}{2 (b c-a d)^3}","\frac{a}{2 \left(a+b x^2\right) (b c-a d)^2}+\frac{c}{2 \left(c+d x^2\right) (b c-a d)^2}+\frac{(a d+b c) \log \left(a+b x^2\right)}{2 (b c-a d)^3}-\frac{(a d+b c) \log \left(c+d x^2\right)}{2 (b c-a d)^3}",1,"((a*(b*c - a*d))/(a + b*x^2) + (c*(b*c - a*d))/(c + d*x^2) + (b*c + a*d)*Log[a + b*x^2] - (b*c + a*d)*Log[c + d*x^2])/(2*(b*c - a*d)^3)","A",1
303,1,137,147,0.1737553,"\int \frac{x^2}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","Integrate[x^2/((a + b*x^2)^2*(c + d*x^2)^2),x]","\frac{1}{2} \left(-\frac{b x}{\left(a+b x^2\right) (b c-a d)^2}-\frac{d x}{\left(c+d x^2\right) (b c-a d)^2}-\frac{\sqrt{b} (3 a d+b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} (a d-b c)^3}-\frac{\sqrt{d} (a d+3 b c) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{\sqrt{c} (b c-a d)^3}\right)","-\frac{x}{2 \left(a+b x^2\right) \left(c+d x^2\right) (b c-a d)}-\frac{d x}{\left(c+d x^2\right) (b c-a d)^2}+\frac{\sqrt{b} (3 a d+b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 \sqrt{a} (b c-a d)^3}-\frac{\sqrt{d} (a d+3 b c) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 \sqrt{c} (b c-a d)^3}",1,"(-((b*x)/((b*c - a*d)^2*(a + b*x^2))) - (d*x)/((b*c - a*d)^2*(c + d*x^2)) - (Sqrt[b]*(b*c + 3*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*(-(b*c) + a*d)^3) - (Sqrt[d]*(3*b*c + a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*(b*c - a*d)^3))/2","A",1
304,1,77,92,0.0787006,"\int \frac{x}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","Integrate[x/((a + b*x^2)^2*(c + d*x^2)^2),x]","\frac{\frac{b (a d-b c)}{a+b x^2}+\frac{d (a d-b c)}{c+d x^2}-2 b d \log \left(a+b x^2\right)+2 b d \log \left(c+d x^2\right)}{2 (b c-a d)^3}","-\frac{b}{2 \left(a+b x^2\right) (b c-a d)^2}-\frac{d}{2 \left(c+d x^2\right) (b c-a d)^2}-\frac{b d \log \left(a+b x^2\right)}{(b c-a d)^3}+\frac{b d \log \left(c+d x^2\right)}{(b c-a d)^3}",1,"((b*(-(b*c) + a*d))/(a + b*x^2) + (d*(-(b*c) + a*d))/(c + d*x^2) - 2*b*d*Log[a + b*x^2] + 2*b*d*Log[c + d*x^2])/(2*(b*c - a*d)^3)","A",1
305,1,136,167,0.3330036,"\int \frac{1}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","Integrate[1/((a + b*x^2)^2*(c + d*x^2)^2),x]","\frac{1}{2} \left(\frac{b^{3/2} (5 a d-b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2} (a d-b c)^3}+\frac{x (b c-a d) \left(\frac{b^2}{a^2+a b x^2}+\frac{d^2}{c^2+c d x^2}\right)+\frac{d^{3/2} (5 b c-a d) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{3/2}}}{(b c-a d)^3}\right)","\frac{b^{3/2} (b c-5 a d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} (b c-a d)^3}+\frac{d^{3/2} (5 b c-a d) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{3/2} (b c-a d)^3}+\frac{b x}{2 a \left(a+b x^2\right) \left(c+d x^2\right) (b c-a d)}+\frac{d x (a d+b c)}{2 a c \left(c+d x^2\right) (b c-a d)^2}",1,"((b^(3/2)*(-(b*c) + 5*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*(-(b*c) + a*d)^3) + ((b*c - a*d)*x*(b^2/(a^2 + a*b*x^2) + d^2/(c^2 + c*d*x^2)) + (d^(3/2)*(5*b*c - a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/c^(3/2))/(b*c - a*d)^3)/2","A",1
306,1,133,141,0.2281207,"\int \frac{1}{x \left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","Integrate[1/(x*(a + b*x^2)^2*(c + d*x^2)^2),x]","\frac{1}{2} \left(\frac{b^2 (b c-3 a d) \log \left(a+b x^2\right)}{a^2 (a d-b c)^3}+\frac{2 \log (x)}{a^2 c^2}+\frac{b^2}{a \left(a+b x^2\right) (b c-a d)^2}+\frac{d^2 (a d-3 b c) \log \left(c+d x^2\right)}{c^2 (b c-a d)^3}+\frac{d^2}{c \left(c+d x^2\right) (b c-a d)^2}\right)","-\frac{b^2 (b c-3 a d) \log \left(a+b x^2\right)}{2 a^2 (b c-a d)^3}+\frac{\log (x)}{a^2 c^2}+\frac{b^2}{2 a \left(a+b x^2\right) (b c-a d)^2}-\frac{d^2 (3 b c-a d) \log \left(c+d x^2\right)}{2 c^2 (b c-a d)^3}+\frac{d^2}{2 c \left(c+d x^2\right) (b c-a d)^2}",1,"(b^2/(a*(b*c - a*d)^2*(a + b*x^2)) + d^2/(c*(b*c - a*d)^2*(c + d*x^2)) + (2*Log[x])/(a^2*c^2) + (b^2*(b*c - 3*a*d)*Log[a + b*x^2])/(a^2*(-(b*c) + a*d)^3) + (d^2*(-3*b*c + a*d)*Log[c + d*x^2])/(c^2*(b*c - a*d)^3))/2","A",1
307,1,158,218,0.2947443,"\int \frac{1}{x^2 \left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","Integrate[1/(x^2*(a + b*x^2)^2*(c + d*x^2)^2),x]","\frac{1}{2} \left(\frac{b^{5/2} (3 b c-7 a d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{5/2} (a d-b c)^3}-\frac{b^3 x}{a^2 \left(a+b x^2\right) (b c-a d)^2}-\frac{2}{a^2 c^2 x}+\frac{d^{5/2} (3 a d-7 b c) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{5/2} (b c-a d)^3}-\frac{d^3 x}{c^2 \left(c+d x^2\right) (b c-a d)^2}\right)","-\frac{b^{5/2} (3 b c-7 a d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{5/2} (b c-a d)^3}-\frac{3 a^2 d^2-4 a b c d+3 b^2 c^2}{2 a^2 c^2 x (b c-a d)^2}-\frac{d^{5/2} (7 b c-3 a d) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{5/2} (b c-a d)^3}+\frac{b}{2 a x \left(a+b x^2\right) \left(c+d x^2\right) (b c-a d)}+\frac{d (a d+b c)}{2 a c x \left(c+d x^2\right) (b c-a d)^2}",1,"(-2/(a^2*c^2*x) - (b^3*x)/(a^2*(b*c - a*d)^2*(a + b*x^2)) - (d^3*x)/(c^2*(b*c - a*d)^2*(c + d*x^2)) + (b^(5/2)*(3*b*c - 7*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(5/2)*(-(b*c) + a*d)^3) + (d^(5/2)*(-7*b*c + 3*a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(5/2)*(b*c - a*d)^3))/2","A",1
308,1,157,156,0.2186993,"\int \frac{1}{x^3 \left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","Integrate[1/(x^3*(a + b*x^2)^2*(c + d*x^2)^2),x]","\frac{1}{2} \left(\frac{2 b^3 (2 a d-b c) \log \left(a+b x^2\right)}{a^3 (a d-b c)^3}-\frac{4 \log (x) (a d+b c)}{a^3 c^3}-\frac{b^3}{a^2 \left(a+b x^2\right) (b c-a d)^2}-\frac{1}{a^2 c^2 x^2}+\frac{2 d^3 (2 b c-a d) \log \left(c+d x^2\right)}{c^3 (b c-a d)^3}-\frac{d^3}{c^2 \left(c+d x^2\right) (b c-a d)^2}\right)","\frac{b^3 (b c-2 a d) \log \left(a+b x^2\right)}{a^3 (b c-a d)^3}-\frac{2 \log (x) (a d+b c)}{a^3 c^3}-\frac{b^3}{2 a^2 \left(a+b x^2\right) (b c-a d)^2}-\frac{1}{2 a^2 c^2 x^2}+\frac{d^3 (2 b c-a d) \log \left(c+d x^2\right)}{c^3 (b c-a d)^3}-\frac{d^3}{2 c^2 \left(c+d x^2\right) (b c-a d)^2}",1,"(-(1/(a^2*c^2*x^2)) - b^3/(a^2*(b*c - a*d)^2*(a + b*x^2)) - d^3/(c^2*(b*c - a*d)^2*(c + d*x^2)) - (4*(b*c + a*d)*Log[x])/(a^3*c^3) + (2*b^3*(-(b*c) + 2*a*d)*Log[a + b*x^2])/(a^3*(-(b*c) + a*d)^3) + (2*d^3*(2*b*c - a*d)*Log[c + d*x^2])/(c^3*(b*c - a*d)^3))/2","A",1
309,1,178,271,0.3945892,"\int \frac{1}{x^4 \left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","Integrate[1/(x^4*(a + b*x^2)^2*(c + d*x^2)^2),x]","\frac{1}{6} \left(\frac{3 b^{7/2} (9 a d-5 b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{7/2} (a d-b c)^3}+\frac{3 b^4 x}{a^3 \left(a+b x^2\right) (b c-a d)^2}+\frac{12 (a d+b c)}{a^3 c^3 x}-\frac{2}{a^2 c^2 x^3}+\frac{3 d^{7/2} (9 b c-5 a d) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{7/2} (b c-a d)^3}+\frac{3 d^4 x}{c^3 \left(c+d x^2\right) (b c-a d)^2}\right)","\frac{b^{7/2} (5 b c-9 a d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{7/2} (b c-a d)^3}-\frac{5 a^2 d^2-4 a b c d+5 b^2 c^2}{6 a^2 c^2 x^3 (b c-a d)^2}+\frac{(a d+b c) \left(5 a^2 d^2-9 a b c d+5 b^2 c^2\right)}{2 a^3 c^3 x (b c-a d)^2}+\frac{d^{7/2} (9 b c-5 a d) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{7/2} (b c-a d)^3}+\frac{b}{2 a x^3 \left(a+b x^2\right) \left(c+d x^2\right) (b c-a d)}+\frac{d (a d+b c)}{2 a c x^3 \left(c+d x^2\right) (b c-a d)^2}",1,"(-2/(a^2*c^2*x^3) + (12*(b*c + a*d))/(a^3*c^3*x) + (3*b^4*x)/(a^3*(b*c - a*d)^2*(a + b*x^2)) + (3*d^4*x)/(c^3*(b*c - a*d)^2*(c + d*x^2)) + (3*b^(7/2)*(-5*b*c + 9*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(7/2)*(-(b*c) + a*d)^3) + (3*d^(7/2)*(9*b*c - 5*a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(7/2)*(b*c - a*d)^3))/6","A",1
310,1,166,207,0.3551474,"\int \frac{x^4}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","Integrate[x^4/((a + b*x^2)^2*(c + d*x^2)^3),x]","\frac{\frac{3 \left(a^2 d^2+6 a b c d+b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{\sqrt{c} \sqrt{d}}+\frac{2 c x (b c-a d)^2}{\left(c+d x^2\right)^2}+\frac{4 a b x (b c-a d)}{a+b x^2}+\frac{x (5 a d+3 b c) (b c-a d)}{c+d x^2}-12 \sqrt{a} \sqrt{b} (a d+b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 (b c-a d)^4}","\frac{3 \left(a^2 d^2+6 a b c d+b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 \sqrt{c} \sqrt{d} (b c-a d)^4}+\frac{3 x (3 a d+b c)}{8 \left(c+d x^2\right) (b c-a d)^3}+\frac{x (2 a d+b c)}{4 b \left(c+d x^2\right)^2 (b c-a d)^2}+\frac{a x}{2 b \left(a+b x^2\right) \left(c+d x^2\right)^2 (b c-a d)}-\frac{3 \sqrt{a} \sqrt{b} (a d+b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 (b c-a d)^4}",1,"((4*a*b*(b*c - a*d)*x)/(a + b*x^2) + (2*c*(b*c - a*d)^2*x)/(c + d*x^2)^2 + ((b*c - a*d)*(3*b*c + 5*a*d)*x)/(c + d*x^2) - 12*Sqrt[a]*Sqrt[b]*(b*c + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]] + (3*(b^2*c^2 + 6*a*b*c*d + a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*Sqrt[d]))/(8*(b*c - a*d)^4)","A",1
311,1,121,142,0.1063387,"\int \frac{x^3}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","Integrate[x^3/((a + b*x^2)^2*(c + d*x^2)^3),x]","\frac{\frac{c (b c-a d)^2}{\left(c+d x^2\right)^2}+\frac{2 a b (b c-a d)}{a+b x^2}+\frac{2 (a d+b c) (b c-a d)}{c+d x^2}+2 b (2 a d+b c) \log \left(a+b x^2\right)-2 b (2 a d+b c) \log \left(c+d x^2\right)}{4 (b c-a d)^4}","\frac{a b}{2 \left(a+b x^2\right) (b c-a d)^3}+\frac{a d+b c}{2 \left(c+d x^2\right) (b c-a d)^3}+\frac{c}{4 \left(c+d x^2\right)^2 (b c-a d)^2}+\frac{b (2 a d+b c) \log \left(a+b x^2\right)}{2 (b c-a d)^4}-\frac{b (2 a d+b c) \log \left(c+d x^2\right)}{2 (b c-a d)^4}",1,"((2*a*b*(b*c - a*d))/(a + b*x^2) + (c*(b*c - a*d)^2)/(c + d*x^2)^2 + (2*(b*c - a*d)*(b*c + a*d))/(c + d*x^2) + 2*b*(b*c + 2*a*d)*Log[a + b*x^2] - 2*b*(b*c + 2*a*d)*Log[c + d*x^2])/(4*(b*c - a*d)^4)","A",1
312,1,171,200,0.4089179,"\int \frac{x^2}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","Integrate[x^2/((a + b*x^2)^2*(c + d*x^2)^3),x]","\frac{\frac{\sqrt{d} \left(a^2 d^2-10 a b c d-15 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{3/2}}+\frac{4 b^{3/2} (5 a d+b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a}}-\frac{4 b^2 x (b c-a d)}{a+b x^2}+\frac{d x (a d-b c) (a d+7 b c)}{c \left(c+d x^2\right)}-\frac{2 d x (b c-a d)^2}{\left(c+d x^2\right)^2}}{8 (b c-a d)^4}","-\frac{\sqrt{d} \left(-a^2 d^2+10 a b c d+15 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{3/2} (b c-a d)^4}+\frac{b^{3/2} (5 a d+b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 \sqrt{a} (b c-a d)^4}-\frac{d x (a d+11 b c)}{8 c \left(c+d x^2\right) (b c-a d)^3}-\frac{3 d x}{4 \left(c+d x^2\right)^2 (b c-a d)^2}-\frac{x}{2 \left(a+b x^2\right) \left(c+d x^2\right)^2 (b c-a d)}",1,"((-4*b^2*(b*c - a*d)*x)/(a + b*x^2) - (2*d*(b*c - a*d)^2*x)/(c + d*x^2)^2 + (d*(-(b*c) + a*d)*(7*b*c + a*d)*x)/(c*(c + d*x^2)) + (4*b^(3/2)*(b*c + 5*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[a] + (Sqrt[d]*(-15*b^2*c^2 - 10*a*b*c*d + a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/c^(3/2))/(8*(b*c - a*d)^4)","A",1
313,1,107,126,0.1339759,"\int \frac{x}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","Integrate[x/((a + b*x^2)^2*(c + d*x^2)^3),x]","-\frac{\frac{2 b^2 (b c-a d)}{a+b x^2}+6 b^2 d \log \left(a+b x^2\right)+\frac{4 b d (b c-a d)}{c+d x^2}+\frac{d (b c-a d)^2}{\left(c+d x^2\right)^2}-6 b^2 d \log \left(c+d x^2\right)}{4 (b c-a d)^4}","-\frac{b^2}{2 \left(a+b x^2\right) (b c-a d)^3}-\frac{3 b^2 d \log \left(a+b x^2\right)}{2 (b c-a d)^4}+\frac{3 b^2 d \log \left(c+d x^2\right)}{2 (b c-a d)^4}-\frac{b d}{\left(c+d x^2\right) (b c-a d)^3}-\frac{d}{4 \left(c+d x^2\right)^2 (b c-a d)^2}",1,"-1/4*((2*b^2*(b*c - a*d))/(a + b*x^2) + (d*(b*c - a*d)^2)/(c + d*x^2)^2 + (4*b*d*(b*c - a*d))/(c + d*x^2) + 6*b^2*d*Log[a + b*x^2] - 6*b^2*d*Log[c + d*x^2])/(b*c - a*d)^4","A",1
314,1,197,230,0.4113801,"\int \frac{1}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","Integrate[1/((a + b*x^2)^2*(c + d*x^2)^3),x]","\frac{1}{8} \left(\frac{4 b^{5/2} (b c-7 a d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2} (b c-a d)^4}+\frac{d^{3/2} \left(3 a^2 d^2-14 a b c d+35 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{5/2} (b c-a d)^4}-\frac{4 b^3 x}{a \left(a+b x^2\right) (a d-b c)^3}+\frac{d^2 x (11 b c-3 a d)}{c^2 \left(c+d x^2\right) (b c-a d)^3}+\frac{2 d^2 x}{c \left(c+d x^2\right)^2 (b c-a d)^2}\right)","\frac{b^{5/2} (b c-7 a d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} (b c-a d)^4}+\frac{d^{3/2} \left(3 a^2 d^2-14 a b c d+35 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{5/2} (b c-a d)^4}+\frac{d x (4 b c-a d) (3 a d+b c)}{8 a c^2 \left(c+d x^2\right) (b c-a d)^3}+\frac{b x}{2 a \left(a+b x^2\right) \left(c+d x^2\right)^2 (b c-a d)}+\frac{d x (a d+2 b c)}{4 a c \left(c+d x^2\right)^2 (b c-a d)^2}",1,"((-4*b^3*x)/(a*(-(b*c) + a*d)^3*(a + b*x^2)) + (2*d^2*x)/(c*(b*c - a*d)^2*(c + d*x^2)^2) + (d^2*(11*b*c - 3*a*d)*x)/(c^2*(b*c - a*d)^3*(c + d*x^2)) + (4*b^(5/2)*(b*c - 7*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*(b*c - a*d)^4) + (d^(3/2)*(35*b^2*c^2 - 14*a*b*c*d + 3*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(5/2)*(b*c - a*d)^4))/8","A",1
315,1,187,192,0.2732969,"\int \frac{1}{x \left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","Integrate[1/(x*(a + b*x^2)^2*(c + d*x^2)^3),x]","\frac{1}{4} \left(\frac{2 b^3 (4 a d-b c) \log \left(a+b x^2\right)}{a^2 (b c-a d)^4}-\frac{2 d^2 \left(a^2 d^2-4 a b c d+6 b^2 c^2\right) \log \left(c+d x^2\right)}{c^3 (b c-a d)^4}+\frac{4 \log (x)}{a^2 c^3}-\frac{2 b^3}{a \left(a+b x^2\right) (a d-b c)^3}+\frac{2 d^2 (3 b c-a d)}{c^2 \left(c+d x^2\right) (b c-a d)^3}+\frac{d^2}{c \left(c+d x^2\right)^2 (b c-a d)^2}\right)","-\frac{b^3 (b c-4 a d) \log \left(a+b x^2\right)}{2 a^2 (b c-a d)^4}-\frac{d^2 \left(a^2 d^2-4 a b c d+6 b^2 c^2\right) \log \left(c+d x^2\right)}{2 c^3 (b c-a d)^4}+\frac{\log (x)}{a^2 c^3}+\frac{b^3}{2 a \left(a+b x^2\right) (b c-a d)^3}+\frac{d^2 (3 b c-a d)}{2 c^2 \left(c+d x^2\right) (b c-a d)^3}+\frac{d^2}{4 c \left(c+d x^2\right)^2 (b c-a d)^2}",1,"((-2*b^3)/(a*(-(b*c) + a*d)^3*(a + b*x^2)) + d^2/(c*(b*c - a*d)^2*(c + d*x^2)^2) + (2*d^2*(3*b*c - a*d))/(c^2*(b*c - a*d)^3*(c + d*x^2)) + (4*Log[x])/(a^2*c^3) + (2*b^3*(-(b*c) + 4*a*d)*Log[a + b*x^2])/(a^2*(b*c - a*d)^4) - (2*d^2*(6*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*Log[c + d*x^2])/(c^3*(b*c - a*d)^4))/4","A",1
316,1,210,297,0.424342,"\int \frac{1}{x^2 \left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","Integrate[1/(x^2*(a + b*x^2)^2*(c + d*x^2)^3),x]","\frac{1}{8} \left(\frac{12 b^{7/2} (3 a d-b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{5/2} (b c-a d)^4}+\frac{4 b^4 x}{a^2 \left(a+b x^2\right) (a d-b c)^3}-\frac{3 d^{5/2} \left(5 a^2 d^2-18 a b c d+21 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{7/2} (b c-a d)^4}-\frac{8}{a^2 c^3 x}+\frac{d^3 x (7 a d-15 b c)}{c^3 \left(c+d x^2\right) (b c-a d)^3}-\frac{2 d^3 x}{c^2 \left(c+d x^2\right)^2 (b c-a d)^2}\right)","-\frac{3 b^{7/2} (b c-3 a d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{5/2} (b c-a d)^4}+\frac{d \left(-5 a^2 d^2+13 a b c d+4 b^2 c^2\right)}{8 a c^2 x \left(c+d x^2\right) (b c-a d)^3}-\frac{3 d^{5/2} \left(5 a^2 d^2-18 a b c d+21 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{7/2} (b c-a d)^4}-\frac{3 (2 b c-a d) \left(5 a^2 d^2-3 a b c d+2 b^2 c^2\right)}{8 a^2 c^3 x (b c-a d)^3}+\frac{b}{2 a x \left(a+b x^2\right) \left(c+d x^2\right)^2 (b c-a d)}+\frac{d (a d+2 b c)}{4 a c x \left(c+d x^2\right)^2 (b c-a d)^2}",1,"(-8/(a^2*c^3*x) + (4*b^4*x)/(a^2*(-(b*c) + a*d)^3*(a + b*x^2)) - (2*d^3*x)/(c^2*(b*c - a*d)^2*(c + d*x^2)^2) + (d^3*(-15*b*c + 7*a*d)*x)/(c^3*(b*c - a*d)^3*(c + d*x^2)) + (12*b^(7/2)*(-(b*c) + 3*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(5/2)*(b*c - a*d)^4) - (3*d^(5/2)*(21*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(7/2)*(b*c - a*d)^4))/8","A",1
317,1,208,215,0.3208883,"\int \frac{1}{x^3 \left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","Integrate[1/(x^3*(a + b*x^2)^2*(c + d*x^2)^3),x]","\frac{1}{4} \left(\frac{2 b^4 (2 b c-5 a d) \log \left(a+b x^2\right)}{a^3 (b c-a d)^4}-\frac{4 \log (x) (3 a d+2 b c)}{a^3 c^4}+\frac{2 b^4}{a^2 \left(a+b x^2\right) (a d-b c)^3}+\frac{2 d^3 \left(3 a^2 d^2-10 a b c d+10 b^2 c^2\right) \log \left(c+d x^2\right)}{c^4 (b c-a d)^4}-\frac{2}{a^2 c^3 x^2}+\frac{4 d^3 (a d-2 b c)}{c^3 \left(c+d x^2\right) (b c-a d)^3}-\frac{d^3}{c^2 \left(c+d x^2\right)^2 (b c-a d)^2}\right)","\frac{b^4 (2 b c-5 a d) \log \left(a+b x^2\right)}{2 a^3 (b c-a d)^4}-\frac{\log (x) (3 a d+2 b c)}{a^3 c^4}-\frac{b^4}{2 a^2 \left(a+b x^2\right) (b c-a d)^3}+\frac{d^3 \left(3 a^2 d^2-10 a b c d+10 b^2 c^2\right) \log \left(c+d x^2\right)}{2 c^4 (b c-a d)^4}-\frac{1}{2 a^2 c^3 x^2}-\frac{d^3 (2 b c-a d)}{c^3 \left(c+d x^2\right) (b c-a d)^3}-\frac{d^3}{4 c^2 \left(c+d x^2\right)^2 (b c-a d)^2}",1,"(-2/(a^2*c^3*x^2) + (2*b^4)/(a^2*(-(b*c) + a*d)^3*(a + b*x^2)) - d^3/(c^2*(b*c - a*d)^2*(c + d*x^2)^2) + (4*d^3*(-2*b*c + a*d))/(c^3*(b*c - a*d)^3*(c + d*x^2)) - (4*(2*b*c + 3*a*d)*Log[x])/(a^3*c^4) + (2*b^4*(2*b*c - 5*a*d)*Log[a + b*x^2])/(a^3*(b*c - a*d)^4) + (2*d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*Log[c + d*x^2])/(c^4*(b*c - a*d)^4))/4","A",1
318,1,230,377,0.4563995,"\int \frac{1}{x^4 \left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","Integrate[1/(x^4*(a + b*x^2)^2*(c + d*x^2)^3),x]","\frac{1}{24} \left(\frac{12 b^{9/2} (5 b c-11 a d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{7/2} (b c-a d)^4}-\frac{12 b^5 x}{a^3 \left(a+b x^2\right) (a d-b c)^3}+\frac{72 a d+48 b c}{a^3 c^4 x}+\frac{3 d^{7/2} \left(35 a^2 d^2-110 a b c d+99 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{9/2} (b c-a d)^4}-\frac{8}{a^2 c^3 x^3}+\frac{3 d^4 x (19 b c-11 a d)}{c^4 \left(c+d x^2\right) (b c-a d)^3}+\frac{6 d^4 x}{c^3 \left(c+d x^2\right)^2 (b c-a d)^2}\right)","\frac{b^{9/2} (5 b c-11 a d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{7/2} (b c-a d)^4}+\frac{d \left(-7 a^2 d^2+15 a b c d+4 b^2 c^2\right)}{8 a c^2 x^3 \left(c+d x^2\right) (b c-a d)^3}+\frac{d^{7/2} \left(35 a^2 d^2-110 a b c d+99 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{9/2} (b c-a d)^4}-\frac{-35 a^3 d^3+75 a^2 b c d^2-24 a b^2 c^2 d+20 b^3 c^3}{24 a^2 c^3 x^3 (b c-a d)^3}+\frac{-35 a^4 d^4+75 a^3 b c d^3-24 a^2 b^2 c^2 d^2-24 a b^3 c^3 d+20 b^4 c^4}{8 a^3 c^4 x (b c-a d)^3}+\frac{b}{2 a x^3 \left(a+b x^2\right) \left(c+d x^2\right)^2 (b c-a d)}+\frac{d (a d+2 b c)}{4 a c x^3 \left(c+d x^2\right)^2 (b c-a d)^2}",1,"(-8/(a^2*c^3*x^3) + (48*b*c + 72*a*d)/(a^3*c^4*x) - (12*b^5*x)/(a^3*(-(b*c) + a*d)^3*(a + b*x^2)) + (6*d^4*x)/(c^3*(b*c - a*d)^2*(c + d*x^2)^2) + (3*d^4*(19*b*c - 11*a*d)*x)/(c^4*(b*c - a*d)^3*(c + d*x^2)) + (12*b^(9/2)*(5*b*c - 11*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(7/2)*(b*c - a*d)^4) + (3*d^(7/2)*(99*b^2*c^2 - 110*a*b*c*d + 35*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(9/2)*(b*c - a*d)^4))/24","A",1
319,1,89,96,0.1152676,"\int x^m \left(a+b x^2\right)^3 \left(A+B x^2\right) \, dx","Integrate[x^m*(a + b*x^2)^3*(A + B*x^2),x]","x^{m+1} \left(\frac{a^3 A}{m+1}+\frac{a^2 x^2 (a B+3 A b)}{m+3}+\frac{b^2 x^6 (3 a B+A b)}{m+7}+\frac{3 a b x^4 (a B+A b)}{m+5}+\frac{b^3 B x^8}{m+9}\right)","\frac{a^3 A x^{m+1}}{m+1}+\frac{a^2 x^{m+3} (a B+3 A b)}{m+3}+\frac{b^2 x^{m+7} (3 a B+A b)}{m+7}+\frac{3 a b x^{m+5} (a B+A b)}{m+5}+\frac{b^3 B x^{m+9}}{m+9}",1,"x^(1 + m)*((a^3*A)/(1 + m) + (a^2*(3*A*b + a*B)*x^2)/(3 + m) + (3*a*b*(A*b + a*B)*x^4)/(5 + m) + (b^2*(A*b + 3*a*B)*x^6)/(7 + m) + (b^3*B*x^8)/(9 + m))","A",1
320,1,66,71,0.0494225,"\int x^m \left(a+b x^2\right)^2 \left(A+B x^2\right) \, dx","Integrate[x^m*(a + b*x^2)^2*(A + B*x^2),x]","x^{m+1} \left(\frac{a^2 A}{m+1}+\frac{b x^4 (2 a B+A b)}{m+5}+\frac{a x^2 (a B+2 A b)}{m+3}+\frac{b^2 B x^6}{m+7}\right)","\frac{a^2 A x^{m+1}}{m+1}+\frac{a x^{m+3} (a B+2 A b)}{m+3}+\frac{b x^{m+5} (2 a B+A b)}{m+5}+\frac{b^2 B x^{m+7}}{m+7}",1,"x^(1 + m)*((a^2*A)/(1 + m) + (a*(2*A*b + a*B)*x^2)/(3 + m) + (b*(A*b + 2*a*B)*x^4)/(5 + m) + (b^2*B*x^6)/(7 + m))","A",1
321,1,42,45,0.0329758,"\int x^m \left(a+b x^2\right) \left(A+B x^2\right) \, dx","Integrate[x^m*(a + b*x^2)*(A + B*x^2),x]","x^{m+1} \left(\frac{x^2 (a B+A b)}{m+3}+\frac{a A}{m+1}+\frac{b B x^4}{m+5}\right)","\frac{x^{m+3} (a B+A b)}{m+3}+\frac{a A x^{m+1}}{m+1}+\frac{b B x^{m+5}}{m+5}",1,"x^(1 + m)*((a*A)/(1 + m) + ((A*b + a*B)*x^2)/(3 + m) + (b*B*x^4)/(5 + m))","A",1
322,1,55,66,0.0812405,"\int \frac{x^m \left(A+B x^2\right)}{a+b x^2} \, dx","Integrate[(x^m*(A + B*x^2))/(a + b*x^2),x]","\frac{x^{m+1} \left((A b-a B) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)+a B\right)}{a b (m+1)}","\frac{x^{m+1} (A b-a B) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a b (m+1)}+\frac{B x^{m+1}}{b (m+1)}",1,"(x^(1 + m)*(a*B + (A*b - a*B)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)]))/(a*b*(1 + m))","A",1
323,1,80,93,0.0556611,"\int \frac{x^m \left(A+B x^2\right)}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^m*(A + B*x^2))/(a + b*x^2)^2,x]","\frac{x^{m+1} \left((A b-a B) \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)+a B \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)\right)}{a^2 b (m+1)}","\frac{x^{m+1} (a B (m+1)+A (b-b m)) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{2 a^2 b (m+1)}+\frac{x^{m+1} (A b-a B)}{2 a b \left(a+b x^2\right)}",1,"(x^(1 + m)*(a*B*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)] + (A*b - a*B)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)]))/(a^2*b*(1 + m))","A",1
324,1,80,93,0.0550685,"\int \frac{x^m \left(A+B x^2\right)}{\left(a+b x^2\right)^3} \, dx","Integrate[(x^m*(A + B*x^2))/(a + b*x^2)^3,x]","\frac{x^{m+1} \left((A b-a B) \, _2F_1\left(3,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)+a B \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)\right)}{a^3 b (m+1)}","\frac{x^{m+1} (a B (m+1)+A b (3-m)) \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{4 a^3 b (m+1)}+\frac{x^{m+1} (A b-a B)}{4 a b \left(a+b x^2\right)^2}",1,"(x^(1 + m)*(a*B*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)] + (A*b - a*B)*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)]))/(a^3*b*(1 + m))","A",1
325,1,141,151,0.1003764,"\int x^m \left(a+b x^2\right)^2 \left(c+d x^2\right)^3 \, dx","Integrate[x^m*(a + b*x^2)^2*(c + d*x^2)^3,x]","x^m \left(\frac{d x^7 \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)}{m+7}+\frac{c x^5 \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)}{m+5}+\frac{a^2 c^3 x}{m+1}+\frac{a c^2 x^3 (3 a d+2 b c)}{m+3}+\frac{b d^2 x^9 (2 a d+3 b c)}{m+9}+\frac{b^2 d^3 x^{11}}{m+11}\right)","\frac{c x^{m+5} \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)}{m+5}+\frac{d x^{m+7} \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)}{m+7}+\frac{a^2 c^3 x^{m+1}}{m+1}+\frac{a c^2 x^{m+3} (3 a d+2 b c)}{m+3}+\frac{b d^2 x^{m+9} (2 a d+3 b c)}{m+9}+\frac{b^2 d^3 x^{m+11}}{m+11}",1,"x^m*((a^2*c^3*x)/(1 + m) + (a*c^2*(2*b*c + 3*a*d)*x^3)/(3 + m) + (c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^5)/(5 + m) + (d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^7)/(7 + m) + (b*d^2*(3*b*c + 2*a*d)*x^9)/(9 + m) + (b^2*d^3*x^11)/(11 + m))","A",1
326,1,101,109,0.0891626,"\int x^m \left(a+b x^2\right)^2 \left(c+d x^2\right)^2 \, dx","Integrate[x^m*(a + b*x^2)^2*(c + d*x^2)^2,x]","x^m \left(\frac{x^5 \left(a^2 d^2+4 a b c d+b^2 c^2\right)}{m+5}+\frac{a^2 c^2 x}{m+1}+\frac{2 b d x^7 (a d+b c)}{m+7}+\frac{2 a c x^3 (a d+b c)}{m+3}+\frac{b^2 d^2 x^9}{m+9}\right)","\frac{x^{m+5} \left(a^2 d^2+4 a b c d+b^2 c^2\right)}{m+5}+\frac{a^2 c^2 x^{m+1}}{m+1}+\frac{2 a c x^{m+3} (a d+b c)}{m+3}+\frac{2 b d x^{m+7} (a d+b c)}{m+7}+\frac{b^2 d^2 x^{m+9}}{m+9}",1,"x^m*((a^2*c^2*x)/(1 + m) + (2*a*c*(b*c + a*d)*x^3)/(3 + m) + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^5)/(5 + m) + (2*b*d*(b*c + a*d)*x^7)/(7 + m) + (b^2*d^2*x^9)/(9 + m))","A",1
327,1,66,71,0.0596489,"\int x^m \left(a+b x^2\right)^2 \left(c+d x^2\right) \, dx","Integrate[x^m*(a + b*x^2)^2*(c + d*x^2),x]","x^{m+1} \left(\frac{a^2 c}{m+1}+\frac{b x^4 (2 a d+b c)}{m+5}+\frac{a x^2 (a d+2 b c)}{m+3}+\frac{b^2 d x^6}{m+7}\right)","\frac{a^2 c x^{m+1}}{m+1}+\frac{a x^{m+3} (a d+2 b c)}{m+3}+\frac{b x^{m+5} (2 a d+b c)}{m+5}+\frac{b^2 d x^{m+7}}{m+7}",1,"x^(1 + m)*((a^2*c)/(1 + m) + (a*(2*b*c + a*d)*x^2)/(3 + m) + (b*(b*c + 2*a*d)*x^4)/(5 + m) + (b^2*d*x^6)/(7 + m))","A",1
328,1,118,94,0.237213,"\int \frac{x^m \left(a+b x^2\right)^2}{c+d x^2} \, dx","Integrate[(x^m*(a + b*x^2)^2)/(c + d*x^2),x]","\frac{x^{m+1} \left(\frac{a^2 \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)}{m+1}+b x^2 \left(\frac{2 a \, _2F_1\left(1,\frac{m+3}{2};\frac{m+5}{2};-\frac{d x^2}{c}\right)}{m+3}+\frac{b x^2 \, _2F_1\left(1,\frac{m+5}{2};\frac{m+7}{2};-\frac{d x^2}{c}\right)}{m+5}\right)\right)}{c}","\frac{x^{m+1} (b c-a d)^2 \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)}{c d^2 (m+1)}-\frac{b x^{m+1} (b c-2 a d)}{d^2 (m+1)}+\frac{b^2 x^{m+3}}{d (m+3)}",1,"(x^(1 + m)*((a^2*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(1 + m) + b*x^2*((2*a*Hypergeometric2F1[1, (3 + m)/2, (5 + m)/2, -((d*x^2)/c)])/(3 + m) + (b*x^2*Hypergeometric2F1[1, (5 + m)/2, (7 + m)/2, -((d*x^2)/c)])/(5 + m))))/c","A",1
329,1,98,120,0.1279728,"\int \frac{x^m \left(a+b x^2\right)^2}{\left(c+d x^2\right)^2} \, dx","Integrate[(x^m*(a + b*x^2)^2)/(c + d*x^2)^2,x]","\frac{x^{m+1} \left(-2 b c (b c-a d) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)+(b c-a d)^2 \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)+b^2 c^2\right)}{c^2 d^2 (m+1)}","-\frac{x^{m+1} (b c-a d) (a d (1-m)+b c (m+3)) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)}{2 c^2 d^2 (m+1)}+\frac{x^{m+1} (b c-a d)^2}{2 c d^2 \left(c+d x^2\right)}+\frac{b^2 x^{m+1}}{d^2 (m+1)}",1,"(x^(1 + m)*(b^2*c^2 - 2*b*c*(b*c - a*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)] + (b*c - a*d)^2*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)]))/(c^2*d^2*(1 + m))","A",1
330,1,124,171,0.1567615,"\int \frac{x^m \left(a+b x^2\right)^2}{\left(c+d x^2\right)^3} \, dx","Integrate[(x^m*(a + b*x^2)^2)/(c + d*x^2)^3,x]","\frac{x^{m+1} \left(b^2 c^2 \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)-(b c-a d) \left((a d-b c) \, _2F_1\left(3,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)+2 b c \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)\right)\right)}{c^3 d^2 (m+1)}","\frac{x^{m+1} \left(a^2 d^2 \left(m^2-4 m+3\right)+2 a b c d \left(1-m^2\right)+b^2 c^2 \left(m^2+4 m+3\right)\right) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)}{8 c^3 d^2 (m+1)}-\frac{x^{m+1} (b c-a d) (a d (3-m)+b c (m+5))}{8 c^2 d^2 \left(c+d x^2\right)}+\frac{x^{m+1} (b c-a d)^2}{4 c d^2 \left(c+d x^2\right)^2}",1,"(x^(1 + m)*(b^2*c^2*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)] - (b*c - a*d)*(2*b*c*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)] + (-(b*c) + a*d)*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])))/(c^3*d^2*(1 + m))","A",1
331,1,114,133,1.8172677,"\int \frac{x^m \left(c+d x^2\right)^3}{a+b x^2} \, dx","Integrate[(x^m*(c + d*x^2)^3)/(a + b*x^2),x]","\frac{x^{m+1} \left(c^3 \Phi \left(-\frac{b x^2}{a},1,\frac{m+1}{2}\right)+d x^2 \left(3 c^2 \Phi \left(-\frac{b x^2}{a},1,\frac{m+3}{2}\right)+d x^2 \left(3 c \Phi \left(-\frac{b x^2}{a},1,\frac{m+5}{2}\right)+d x^2 \Phi \left(-\frac{b x^2}{a},1,\frac{m+7}{2}\right)\right)\right)\right)}{2 a}","\frac{d x^{m+1} \left(a^2 d^2-3 a b c d+3 b^2 c^2\right)}{b^3 (m+1)}+\frac{x^{m+1} (b c-a d)^3 \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a b^3 (m+1)}+\frac{d^2 x^{m+3} (3 b c-a d)}{b^2 (m+3)}+\frac{d^3 x^{m+5}}{b (m+5)}",1,"(x^(1 + m)*(c^3*HurwitzLerchPhi[-((b*x^2)/a), 1, (1 + m)/2] + d*x^2*(3*c^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (3 + m)/2] + d*x^2*(3*c*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + d*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2]))))/(2*a)","C",1
332,1,85,94,0.4416656,"\int \frac{x^m \left(c+d x^2\right)^2}{a+b x^2} \, dx","Integrate[(x^m*(c + d*x^2)^2)/(a + b*x^2),x]","\frac{x^{m+1} \left(c^2 \Phi \left(-\frac{b x^2}{a},1,\frac{m+1}{2}\right)+d x^2 \left(2 c \Phi \left(-\frac{b x^2}{a},1,\frac{m+3}{2}\right)+d x^2 \Phi \left(-\frac{b x^2}{a},1,\frac{m+5}{2}\right)\right)\right)}{2 a}","\frac{x^{m+1} (b c-a d)^2 \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a b^2 (m+1)}+\frac{d x^{m+1} (2 b c-a d)}{b^2 (m+1)}+\frac{d^2 x^{m+3}}{b (m+3)}",1,"(x^(1 + m)*(c^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (1 + m)/2] + d*x^2*(2*c*HurwitzLerchPhi[-((b*x^2)/a), 1, (3 + m)/2] + d*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2])))/(2*a)","C",1
333,1,55,66,0.0556821,"\int \frac{x^m \left(c+d x^2\right)}{a+b x^2} \, dx","Integrate[(x^m*(c + d*x^2))/(a + b*x^2),x]","\frac{x^{m+1} \left((b c-a d) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)+a d\right)}{a b (m+1)}","\frac{x^{m+1} (b c-a d) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a b (m+1)}+\frac{d x^{m+1}}{b (m+1)}",1,"(x^(1 + m)*(a*d + (b*c - a*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)]))/(a*b*(1 + m))","A",1
334,1,85,102,0.0558289,"\int \frac{x^m}{\left(a+b x^2\right) \left(c+d x^2\right)} \, dx","Integrate[x^m/((a + b*x^2)*(c + d*x^2)),x]","\frac{x^{m+1} \left(a d \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)-b c \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)\right)}{a c (m+1) (a d-b c)}","\frac{b x^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a (m+1) (b c-a d)}-\frac{d x^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)}{c (m+1) (b c-a d)}",1,"(x^(1 + m)*(-(b*c*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)]) + a*d*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)]))/(a*c*(-(b*c) + a*d)*(1 + m))","A",1
335,1,127,156,0.0842675,"\int \frac{x^m}{\left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","Integrate[x^m/((a + b*x^2)^2*(c + d*x^2)),x]","\frac{x^{m+1} \left(a^2 d^2 \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)-a b c d \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)+b c (b c-a d) \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)\right)}{a^2 c (m+1) (b c-a d)^2}","\frac{b x^{m+1} (b c (1-m)-a d (3-m)) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{2 a^2 (m+1) (b c-a d)^2}+\frac{d^2 x^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)}{c (m+1) (b c-a d)^2}+\frac{b x^{m+1}}{2 a \left(a+b x^2\right) (b c-a d)}",1,"(x^(1 + m)*(-(a*b*c*d*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)]) + a^2*d^2*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)] + b*c*(b*c - a*d)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)]))/(a^2*c*(b*c - a*d)^2*(1 + m))","A",1
336,1,170,234,0.1912092,"\int \frac{x^m}{\left(a+b x^2\right)^3 \left(c+d x^2\right)} \, dx","Integrate[x^m/((a + b*x^2)^3*(c + d*x^2)),x]","\frac{x^{m+1} \left(a^3 d^3 \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)-a^2 b c d^2 \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)-b c (a d-b c) \left((a d-b c) \, _2F_1\left(3,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)+a d \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)\right)\right)}{a^3 c (m+1) (a d-b c)^3}","\frac{b x^{m+1} (b c (3-m)-a d (7-m))}{8 a^2 \left(a+b x^2\right) (b c-a d)^2}+\frac{b x^{m+1} \left(a^2 d^2 \left(m^2-8 m+15\right)-2 a b c d \left(m^2-6 m+5\right)+b^2 c^2 \left(m^2-4 m+3\right)\right) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{8 a^3 (m+1) (b c-a d)^3}-\frac{d^3 x^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)}{c (m+1) (b c-a d)^3}+\frac{b x^{m+1}}{4 a \left(a+b x^2\right)^2 (b c-a d)}",1,"(x^(1 + m)*(-(a^2*b*c*d^2*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)]) + a^3*d^3*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)] - b*c*(-(b*c) + a*d)*(a*d*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)] + (-(b*c) + a*d)*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])))/(a^3*c*(-(b*c) + a*d)^3*(1 + m))","A",1
337,1,2524,201,5.1298789,"\int \frac{x^m \left(c+d x^2\right)^3}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^m*(c + d*x^2)^3)/(a + b*x^2)^2,x]","\text{Result too large to show}","\frac{x^{m+1} (b c-a d)^2 (a d (m+5)+b (c-c m)) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{2 a^2 b^3 (m+1)}-\frac{d x^{m+1} \left(a^2 d^2 (m+5)-3 a b c d (m+3)+2 b^2 c^2 (m+1)\right)}{2 a b^3 (m+1)}-\frac{d^2 x^{m+3} (b c (m+3)-a d (m+5))}{2 a b^2 (m+3)}+\frac{x^{m+1} \left(c+d x^2\right)^2 (b c-a d)}{2 a b \left(a+b x^2\right)}",1,"-1/192*(x^(1 + m)*(a*(945 + 744*m + 206*m^2 + 24*m^3 + m^4)*(c^3*(-47 + 52*m + 6*m^2 + 4*m^3 + m^4) + 3*c^2*d*(1 + m)^4*x^2 + 3*c*d^2*(1 + m)^4*x^4 + d^3*(1 + m)^4*x^6)*HurwitzLerchPhi[-((b*x^2)/a), 1, (1 + m)/2] - 3*a*(945 + 744*m + 206*m^2 + 24*m^3 + m^4)*(c^3*(3 + m)^4 + 3*c^2*d*(65 + 92*m + 54*m^2 + 12*m^3 + m^4)*x^2 + 3*c*d^2*(3 + m)^4*x^4 + d^3*(3 + m)^4*x^6)*HurwitzLerchPhi[-((b*x^2)/a), 1, (3 + m)/2] + 1771875*a*c^3*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 2812500*a*c^3*m*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 1927500*a*c^3*m^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 745500*a*c^3*m^3*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 178050*a*c^3*m^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 26892*a*c^3*m^5*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 2508*a*c^3*m^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 132*a*c^3*m^7*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 3*a*c^3*m^8*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 5315625*a*c^2*d*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 8437500*a*c^2*d*m*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 5782500*a*c^2*d*m^2*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 2236500*a*c^2*d*m^3*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 534150*a*c^2*d*m^4*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 80676*a*c^2*d*m^5*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 7524*a*c^2*d*m^6*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 396*a*c^2*d*m^7*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 9*a*c^2*d*m^8*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 5723865*a*c*d^2*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 8894988*a*c*d^2*m*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 5978628*a*c*d^2*m^2*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 2276532*a*c*d^2*m^3*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 538038*a*c*d^2*m^4*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 80820*a*c*d^2*m^5*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 7524*a*c*d^2*m^6*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 396*a*c*d^2*m^7*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 9*a*c*d^2*m^8*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 1771875*a*d^3*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 2812500*a*d^3*m*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 1927500*a*d^3*m^2*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 745500*a*d^3*m^3*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 178050*a*d^3*m^4*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 26892*a*d^3*m^5*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 2508*a*d^3*m^6*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 132*a*d^3*m^7*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 3*a*d^3*m^8*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] - 2268945*a*c^3*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 3082884*a*c^3*m*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 1793204*a*c^3*m^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 585452*a*c^3*m^3*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 117670*a*c^3*m^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 14940*a*c^3*m^5*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 1172*a*c^3*m^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 52*a*c^3*m^7*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - a*c^3*m^8*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 6806835*a*c^2*d*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 9248652*a*c^2*d*m*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 5379612*a*c^2*d*m^2*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 1756356*a*c^2*d*m^3*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 353010*a*c^2*d*m^4*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 44820*a*c^2*d*m^5*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 3516*a*c^2*d*m^6*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 156*a*c^2*d*m^7*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 3*a*c^2*d*m^8*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 6806835*a*c*d^2*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 9248652*a*c*d^2*m*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 5379612*a*c*d^2*m^2*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 1756356*a*c*d^2*m^3*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 353010*a*c*d^2*m^4*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 44820*a*c*d^2*m^5*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 3516*a*c*d^2*m^6*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 156*a*c*d^2*m^7*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 3*a*c*d^2*m^8*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 2042145*a*d^3*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 2858964*a*d^3*m*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 1708052*a*d^3*m^2*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 569804*a*d^3*m^3*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 116278*a*d^3*m^4*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 14892*a*d^3*m^5*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 1172*a*d^3*m^6*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 52*a*d^3*m^7*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - a*d^3*m^8*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] + 1536*b*c^3*x^2*HypergeometricPFQ[{2, 2, 2, 2, 3/2 + m/2}, {1, 1, 1, 11/2 + m/2}, -((b*x^2)/a)] + 4608*b*c^2*d*x^4*HypergeometricPFQ[{2, 2, 2, 2, 3/2 + m/2}, {1, 1, 1, 11/2 + m/2}, -((b*x^2)/a)] + 4608*b*c*d^2*x^6*HypergeometricPFQ[{2, 2, 2, 2, 3/2 + m/2}, {1, 1, 1, 11/2 + m/2}, -((b*x^2)/a)] + 1536*b*d^3*x^8*HypergeometricPFQ[{2, 2, 2, 2, 3/2 + m/2}, {1, 1, 1, 11/2 + m/2}, -((b*x^2)/a)]))/(a^3*(3 + m)*(5 + m)*(7 + m)*(9 + m))","C",0
338,1,895,120,2.1552229,"\int \frac{x^m \left(c+d x^2\right)^2}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^m*(c + d*x^2)^2)/(a + b*x^2)^2,x]","\frac{x^{m+1} \left(a d^2 x^4 \Phi \left(-\frac{b x^2}{a},1,\frac{m+5}{2}\right) m^6+a c^2 \Phi \left(-\frac{b x^2}{a},1,\frac{m+5}{2}\right) m^6+2 a c d x^2 \Phi \left(-\frac{b x^2}{a},1,\frac{m+5}{2}\right) m^6+30 a d^2 x^4 \Phi \left(-\frac{b x^2}{a},1,\frac{m+5}{2}\right) m^5+30 a c^2 \Phi \left(-\frac{b x^2}{a},1,\frac{m+5}{2}\right) m^5+60 a c d x^2 \Phi \left(-\frac{b x^2}{a},1,\frac{m+5}{2}\right) m^5+363 a d^2 x^4 \Phi \left(-\frac{b x^2}{a},1,\frac{m+5}{2}\right) m^4+371 a c^2 \Phi \left(-\frac{b x^2}{a},1,\frac{m+5}{2}\right) m^4+742 a c d x^2 \Phi \left(-\frac{b x^2}{a},1,\frac{m+5}{2}\right) m^4+2276 a d^2 x^4 \Phi \left(-\frac{b x^2}{a},1,\frac{m+5}{2}\right) m^3+2420 a c^2 \Phi \left(-\frac{b x^2}{a},1,\frac{m+5}{2}\right) m^3+4840 a c d x^2 \Phi \left(-\frac{b x^2}{a},1,\frac{m+5}{2}\right) m^3+7847 a d^2 x^4 \Phi \left(-\frac{b x^2}{a},1,\frac{m+5}{2}\right) m^2+8775 a c^2 \Phi \left(-\frac{b x^2}{a},1,\frac{m+5}{2}\right) m^2+17550 a c d x^2 \Phi \left(-\frac{b x^2}{a},1,\frac{m+5}{2}\right) m^2+14206 a d^2 x^4 \Phi \left(-\frac{b x^2}{a},1,\frac{m+5}{2}\right) m+16750 a c^2 \Phi \left(-\frac{b x^2}{a},1,\frac{m+5}{2}\right) m+33500 a c d x^2 \Phi \left(-\frac{b x^2}{a},1,\frac{m+5}{2}\right) m+a \left(m^3+15 m^2+71 m+105\right) \left(d^2 (m+1)^3 x^4+2 c d (m+1)^3 x^2+c^2 \left(m^3+3 m^2-5 m+9\right)\right) \Phi \left(-\frac{b x^2}{a},1,\frac{m+1}{2}\right)-2 a \left(m^3+15 m^2+71 m+105\right) \left(d^2 (m+3)^3 x^4+2 c d \left(m^3+9 m^2+31 m+31\right) x^2+c^2 (m+3)^3\right) \Phi \left(-\frac{b x^2}{a},1,\frac{m+3}{2}\right)+10605 a d^2 x^4 \Phi \left(-\frac{b x^2}{a},1,\frac{m+5}{2}\right)+13125 a c^2 \Phi \left(-\frac{b x^2}{a},1,\frac{m+5}{2}\right)+26250 a c d x^2 \Phi \left(-\frac{b x^2}{a},1,\frac{m+5}{2}\right)-128 b d^2 x^6 \, _4F_3\left(2,2,2,\frac{m}{2}+\frac{3}{2};1,1,\frac{m}{2}+\frac{9}{2};-\frac{b x^2}{a}\right)-256 b c d x^4 \, _4F_3\left(2,2,2,\frac{m}{2}+\frac{3}{2};1,1,\frac{m}{2}+\frac{9}{2};-\frac{b x^2}{a}\right)-128 b c^2 x^2 \, _4F_3\left(2,2,2,\frac{m}{2}+\frac{3}{2};1,1,\frac{m}{2}+\frac{9}{2};-\frac{b x^2}{a}\right)\right)}{32 a^3 (m+3) (m+5) (m+7)}","\frac{x^{m+1} (b c-a d) (a d (m+3)+b (c-c m)) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{2 a^2 b^2 (m+1)}+\frac{x^{m+1} (b c-a d)^2}{2 a b^2 \left(a+b x^2\right)}+\frac{d^2 x^{m+1}}{b^2 (m+1)}",1,"(x^(1 + m)*(a*(105 + 71*m + 15*m^2 + m^3)*(c^2*(9 - 5*m + 3*m^2 + m^3) + 2*c*d*(1 + m)^3*x^2 + d^2*(1 + m)^3*x^4)*HurwitzLerchPhi[-((b*x^2)/a), 1, (1 + m)/2] - 2*a*(105 + 71*m + 15*m^2 + m^3)*(c^2*(3 + m)^3 + 2*c*d*(31 + 31*m + 9*m^2 + m^3)*x^2 + d^2*(3 + m)^3*x^4)*HurwitzLerchPhi[-((b*x^2)/a), 1, (3 + m)/2] + 13125*a*c^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 16750*a*c^2*m*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 8775*a*c^2*m^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 2420*a*c^2*m^3*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 371*a*c^2*m^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 30*a*c^2*m^5*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + a*c^2*m^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 26250*a*c*d*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 33500*a*c*d*m*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 17550*a*c*d*m^2*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 4840*a*c*d*m^3*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 742*a*c*d*m^4*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 60*a*c*d*m^5*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 2*a*c*d*m^6*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 10605*a*d^2*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 14206*a*d^2*m*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 7847*a*d^2*m^2*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 2276*a*d^2*m^3*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 363*a*d^2*m^4*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 30*a*d^2*m^5*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + a*d^2*m^6*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] - 128*b*c^2*x^2*HypergeometricPFQ[{2, 2, 2, 3/2 + m/2}, {1, 1, 9/2 + m/2}, -((b*x^2)/a)] - 256*b*c*d*x^4*HypergeometricPFQ[{2, 2, 2, 3/2 + m/2}, {1, 1, 9/2 + m/2}, -((b*x^2)/a)] - 128*b*d^2*x^6*HypergeometricPFQ[{2, 2, 2, 3/2 + m/2}, {1, 1, 9/2 + m/2}, -((b*x^2)/a)]))/(32*a^3*(3 + m)*(5 + m)*(7 + m))","C",0
339,1,80,93,0.0635478,"\int \frac{x^m \left(c+d x^2\right)}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^m*(c + d*x^2))/(a + b*x^2)^2,x]","\frac{x^{m+1} \left((b c-a d) \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)+a d \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)\right)}{a^2 b (m+1)}","\frac{x^{m+1} (a d (m+1)+b (c-c m)) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{2 a^2 b (m+1)}+\frac{x^{m+1} (b c-a d)}{2 a b \left(a+b x^2\right)}",1,"(x^(1 + m)*(a*d*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)] + (b*c - a*d)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)]))/(a^2*b*(1 + m))","A",1
340,1,127,156,0.0719463,"\int \frac{x^m}{\left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","Integrate[x^m/((a + b*x^2)^2*(c + d*x^2)),x]","\frac{x^{m+1} \left(a^2 d^2 \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)-a b c d \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)+b c (b c-a d) \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)\right)}{a^2 c (m+1) (b c-a d)^2}","\frac{b x^{m+1} (b c (1-m)-a d (3-m)) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{2 a^2 (m+1) (b c-a d)^2}+\frac{d^2 x^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)}{c (m+1) (b c-a d)^2}+\frac{b x^{m+1}}{2 a \left(a+b x^2\right) (b c-a d)}",1,"(x^(1 + m)*(-(a*b*c*d*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)]) + a^2*d^2*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)] + b*c*(b*c - a*d)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)]))/(a^2*c*(b*c - a*d)^2*(1 + m))","A",1
341,1,173,230,0.2164018,"\int \frac{x^m}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","Integrate[x^m/((a + b*x^2)^2*(c + d*x^2)^2),x]","\frac{x^{m+1} \left(-(b c-a d) \left(a^2 d^2 \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)+b^2 c^2 \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)\right)-2 a^2 b c d^2 \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)+2 a b^2 c^2 d \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)\right)}{a^2 c^2 (m+1) (a d-b c)^3}","-\frac{b^2 x^{m+1} (a d (5-m)-b (c-c m)) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{2 a^2 (m+1) (b c-a d)^3}-\frac{d^2 x^{m+1} (a d (1-m)-b c (5-m)) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)}{2 c^2 (m+1) (b c-a d)^3}+\frac{d x^{m+1} (a d+b c)}{2 a c \left(c+d x^2\right) (b c-a d)^2}+\frac{b x^{m+1}}{2 a \left(a+b x^2\right) \left(c+d x^2\right) (b c-a d)}",1,"(x^(1 + m)*(2*a*b^2*c^2*d*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)] - 2*a^2*b*c*d^2*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)] - (b*c - a*d)*(b^2*c^2*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)] + a^2*d^2*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])))/(a^2*c^2*(-(b*c) + a*d)^3*(1 + m))","A",1
342,1,215,325,0.3621837,"\int \frac{x^m}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","Integrate[x^m/((a + b*x^2)^2*(c + d*x^2)^3),x]","\frac{x^{m+1} \left((b c-a d) \left(a^2 d^2 \left((b c-a d) \, _2F_1\left(3,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)+2 b c \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)\right)+b^3 c^3 \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)\right)+3 a^2 b^2 c^2 d^2 \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)-3 a b^3 c^3 d \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)\right)}{a^2 c^3 (m+1) (b c-a d)^4}","-\frac{b^3 x^{m+1} (a d (7-m)-b (c-c m)) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{2 a^2 (m+1) (b c-a d)^4}+\frac{d x^{m+1} \left(-a^2 d^2 (3-m)+a b c d (11-m)+4 b^2 c^2\right)}{8 a c^2 \left(c+d x^2\right) (b c-a d)^3}+\frac{d^2 x^{m+1} \left(a^2 d^2 \left(m^2-4 m+3\right)-2 a b c d \left(m^2-8 m+7\right)+b^2 c^2 \left(m^2-12 m+35\right)\right) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)}{8 c^3 (m+1) (b c-a d)^4}+\frac{d x^{m+1} (a d+2 b c)}{4 a c \left(c+d x^2\right)^2 (b c-a d)^2}+\frac{b x^{m+1}}{2 a \left(a+b x^2\right) \left(c+d x^2\right)^2 (b c-a d)}",1,"(x^(1 + m)*(-3*a*b^3*c^3*d*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)] + 3*a^2*b^2*c^2*d^2*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)] + (b*c - a*d)*(b^3*c^3*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)] + a^2*d^2*(2*b*c*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)] + (b*c - a*d)*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)]))))/(a^2*c^3*(b*c - a*d)^4*(1 + m))","A",1
343,1,33,39,0.0212799,"\int x^{7/2} \left(a+b x^2\right) \left(A+B x^2\right) \, dx","Integrate[x^(7/2)*(a + b*x^2)*(A + B*x^2),x]","\frac{2 x^{9/2} \left(153 x^2 (a B+A b)+221 a A+117 b B x^4\right)}{1989}","\frac{2}{13} x^{13/2} (a B+A b)+\frac{2}{9} a A x^{9/2}+\frac{2}{17} b B x^{17/2}",1,"(2*x^(9/2)*(221*a*A + 153*(A*b + a*B)*x^2 + 117*b*B*x^4))/1989","A",1
344,1,33,39,0.0140808,"\int x^{5/2} \left(a+b x^2\right) \left(A+B x^2\right) \, dx","Integrate[x^(5/2)*(a + b*x^2)*(A + B*x^2),x]","\frac{2 x^{7/2} \left(105 x^2 (a B+A b)+165 a A+77 b B x^4\right)}{1155}","\frac{2}{11} x^{11/2} (a B+A b)+\frac{2}{7} a A x^{7/2}+\frac{2}{15} b B x^{15/2}",1,"(2*x^(7/2)*(165*a*A + 105*(A*b + a*B)*x^2 + 77*b*B*x^4))/1155","A",1
345,1,33,39,0.0137788,"\int x^{3/2} \left(a+b x^2\right) \left(A+B x^2\right) \, dx","Integrate[x^(3/2)*(a + b*x^2)*(A + B*x^2),x]","\frac{2}{585} x^{5/2} \left(65 x^2 (a B+A b)+117 a A+45 b B x^4\right)","\frac{2}{9} x^{9/2} (a B+A b)+\frac{2}{5} a A x^{5/2}+\frac{2}{13} b B x^{13/2}",1,"(2*x^(5/2)*(117*a*A + 65*(A*b + a*B)*x^2 + 45*b*B*x^4))/585","A",1
346,1,33,39,0.0135337,"\int \sqrt{x} \left(a+b x^2\right) \left(A+B x^2\right) \, dx","Integrate[Sqrt[x]*(a + b*x^2)*(A + B*x^2),x]","\frac{2}{231} x^{3/2} \left(33 x^2 (a B+A b)+77 a A+21 b B x^4\right)","\frac{2}{7} x^{7/2} (a B+A b)+\frac{2}{3} a A x^{3/2}+\frac{2}{11} b B x^{11/2}",1,"(2*x^(3/2)*(77*a*A + 33*(A*b + a*B)*x^2 + 21*b*B*x^4))/231","A",1
347,1,33,37,0.0139595,"\int \frac{\left(a+b x^2\right) \left(A+B x^2\right)}{\sqrt{x}} \, dx","Integrate[((a + b*x^2)*(A + B*x^2))/Sqrt[x],x]","\frac{2}{45} \sqrt{x} \left(9 x^2 (a B+A b)+45 a A+5 b B x^4\right)","\frac{2}{5} x^{5/2} (a B+A b)+2 a A \sqrt{x}+\frac{2}{9} b B x^{9/2}",1,"(2*Sqrt[x]*(45*a*A + 9*(A*b + a*B)*x^2 + 5*b*B*x^4))/45","A",1
348,1,35,37,0.0097259,"\int \frac{\left(a+b x^2\right) \left(A+B x^2\right)}{x^{3/2}} \, dx","Integrate[((a + b*x^2)*(A + B*x^2))/x^(3/2),x]","\frac{2 \left(-21 a A+7 a B x^2+7 A b x^2+3 b B x^4\right)}{21 \sqrt{x}}","\frac{2}{3} x^{3/2} (a B+A b)-\frac{2 a A}{\sqrt{x}}+\frac{2}{7} b B x^{7/2}",1,"(2*(-21*a*A + 7*A*b*x^2 + 7*a*B*x^2 + 3*b*B*x^4))/(21*Sqrt[x])","A",1
349,1,36,37,0.0128293,"\int \frac{\left(a+b x^2\right) \left(A+B x^2\right)}{x^{5/2}} \, dx","Integrate[((a + b*x^2)*(A + B*x^2))/x^(5/2),x]","\frac{2 \left(3 b x^2 \left(5 A+B x^2\right)-5 a \left(A-3 B x^2\right)\right)}{15 x^{3/2}}","2 \sqrt{x} (a B+A b)-\frac{2 a A}{3 x^{3/2}}+\frac{2}{5} b B x^{5/2}",1,"(2*(-5*a*(A - 3*B*x^2) + 3*b*x^2*(5*A + B*x^2)))/(15*x^(3/2))","A",1
350,1,36,37,0.0131279,"\int \frac{\left(a+b x^2\right) \left(A+B x^2\right)}{x^{7/2}} \, dx","Integrate[((a + b*x^2)*(A + B*x^2))/x^(7/2),x]","\frac{10 b x^2 \left(B x^2-3 A\right)-6 a \left(A+5 B x^2\right)}{15 x^{5/2}}","-\frac{2 (a B+A b)}{\sqrt{x}}-\frac{2 a A}{5 x^{5/2}}+\frac{2}{3} b B x^{3/2}",1,"(10*b*x^2*(-3*A + B*x^2) - 6*a*(A + 5*B*x^2))/(15*x^(5/2))","A",1
351,1,53,63,0.0314932,"\int x^{7/2} \left(a+b x^2\right)^2 \left(A+B x^2\right) \, dx","Integrate[x^(7/2)*(a + b*x^2)^2*(A + B*x^2),x]","\frac{2 x^{9/2} \left(1547 a^2 A+819 b x^4 (2 a B+A b)+1071 a x^2 (a B+2 A b)+663 b^2 B x^6\right)}{13923}","\frac{2}{9} a^2 A x^{9/2}+\frac{2}{17} b x^{17/2} (2 a B+A b)+\frac{2}{13} a x^{13/2} (a B+2 A b)+\frac{2}{21} b^2 B x^{21/2}",1,"(2*x^(9/2)*(1547*a^2*A + 1071*a*(2*A*b + a*B)*x^2 + 819*b*(A*b + 2*a*B)*x^4 + 663*b^2*B*x^6))/13923","A",1
352,1,63,63,0.0283581,"\int x^{5/2} \left(a+b x^2\right)^2 \left(A+B x^2\right) \, dx","Integrate[x^(5/2)*(a + b*x^2)^2*(A + B*x^2),x]","\frac{2}{7} a^2 A x^{7/2}+\frac{2}{15} b x^{15/2} (2 a B+A b)+\frac{2}{11} a x^{11/2} (a B+2 A b)+\frac{2}{19} b^2 B x^{19/2}","\frac{2}{7} a^2 A x^{7/2}+\frac{2}{15} b x^{15/2} (2 a B+A b)+\frac{2}{11} a x^{11/2} (a B+2 A b)+\frac{2}{19} b^2 B x^{19/2}",1,"(2*a^2*A*x^(7/2))/7 + (2*a*(2*A*b + a*B)*x^(11/2))/11 + (2*b*(A*b + 2*a*B)*x^(15/2))/15 + (2*b^2*B*x^(19/2))/19","A",1
353,1,53,63,0.0292021,"\int x^{3/2} \left(a+b x^2\right)^2 \left(A+B x^2\right) \, dx","Integrate[x^(3/2)*(a + b*x^2)^2*(A + B*x^2),x]","\frac{2 x^{5/2} \left(1989 a^2 A+765 b x^4 (2 a B+A b)+1105 a x^2 (a B+2 A b)+585 b^2 B x^6\right)}{9945}","\frac{2}{5} a^2 A x^{5/2}+\frac{2}{13} b x^{13/2} (2 a B+A b)+\frac{2}{9} a x^{9/2} (a B+2 A b)+\frac{2}{17} b^2 B x^{17/2}",1,"(2*x^(5/2)*(1989*a^2*A + 1105*a*(2*A*b + a*B)*x^2 + 765*b*(A*b + 2*a*B)*x^4 + 585*b^2*B*x^6))/9945","A",1
354,1,53,63,0.0281843,"\int \sqrt{x} \left(a+b x^2\right)^2 \left(A+B x^2\right) \, dx","Integrate[Sqrt[x]*(a + b*x^2)^2*(A + B*x^2),x]","\frac{2 x^{3/2} \left(385 a^2 A+105 b x^4 (2 a B+A b)+165 a x^2 (a B+2 A b)+77 b^2 B x^6\right)}{1155}","\frac{2}{3} a^2 A x^{3/2}+\frac{2}{11} b x^{11/2} (2 a B+A b)+\frac{2}{7} a x^{7/2} (a B+2 A b)+\frac{2}{15} b^2 B x^{15/2}",1,"(2*x^(3/2)*(385*a^2*A + 165*a*(2*A*b + a*B)*x^2 + 105*b*(A*b + 2*a*B)*x^4 + 77*b^2*B*x^6))/1155","A",1
355,1,53,61,0.0273761,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x^2\right)}{\sqrt{x}} \, dx","Integrate[((a + b*x^2)^2*(A + B*x^2))/Sqrt[x],x]","\frac{2}{585} \sqrt{x} \left(585 a^2 A+65 b x^4 (2 a B+A b)+117 a x^2 (a B+2 A b)+45 b^2 B x^6\right)","2 a^2 A \sqrt{x}+\frac{2}{9} b x^{9/2} (2 a B+A b)+\frac{2}{5} a x^{5/2} (a B+2 A b)+\frac{2}{13} b^2 B x^{13/2}",1,"(2*Sqrt[x]*(585*a^2*A + 117*a*(2*A*b + a*B)*x^2 + 65*b*(A*b + 2*a*B)*x^4 + 45*b^2*B*x^6))/585","A",1
356,1,60,61,0.0196321,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x^2\right)}{x^{3/2}} \, dx","Integrate[((a + b*x^2)^2*(A + B*x^2))/x^(3/2),x]","\frac{-154 a^2 \left(3 A-B x^2\right)+44 a b x^2 \left(7 A+3 B x^2\right)+6 b^2 x^4 \left(11 A+7 B x^2\right)}{231 \sqrt{x}}","-\frac{2 a^2 A}{\sqrt{x}}+\frac{2}{7} b x^{7/2} (2 a B+A b)+\frac{2}{3} a x^{3/2} (a B+2 A b)+\frac{2}{11} b^2 B x^{11/2}",1,"(-154*a^2*(3*A - B*x^2) + 44*a*b*x^2*(7*A + 3*B*x^2) + 6*b^2*x^4*(11*A + 7*B*x^2))/(231*Sqrt[x])","A",1
357,1,57,61,0.018336,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x^2\right)}{x^{5/2}} \, dx","Integrate[((a + b*x^2)^2*(A + B*x^2))/x^(5/2),x]","\frac{-30 a^2 \left(A-3 B x^2\right)+36 a b x^2 \left(5 A+B x^2\right)+2 b^2 x^4 \left(9 A+5 B x^2\right)}{45 x^{3/2}}","-\frac{2 a^2 A}{3 x^{3/2}}+\frac{2}{5} b x^{5/2} (2 a B+A b)+2 a \sqrt{x} (a B+2 A b)+\frac{2}{9} b^2 B x^{9/2}",1,"(-30*a^2*(A - 3*B*x^2) + 36*a*b*x^2*(5*A + B*x^2) + 2*b^2*x^4*(9*A + 5*B*x^2))/(45*x^(3/2))","A",1
358,1,57,61,0.016776,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x^2\right)}{x^{7/2}} \, dx","Integrate[((a + b*x^2)^2*(A + B*x^2))/x^(7/2),x]","\frac{-42 a^2 \left(A+5 B x^2\right)+140 a b x^2 \left(B x^2-3 A\right)+10 b^2 x^4 \left(7 A+3 B x^2\right)}{105 x^{5/2}}","-\frac{2 a^2 A}{5 x^{5/2}}+\frac{2}{3} b x^{3/2} (2 a B+A b)-\frac{2 a (a B+2 A b)}{\sqrt{x}}+\frac{2}{7} b^2 B x^{7/2}",1,"(140*a*b*x^2*(-3*A + B*x^2) + 10*b^2*x^4*(7*A + 3*B*x^2) - 42*a^2*(A + 5*B*x^2))/(105*x^(5/2))","A",1
359,1,85,85,0.0420926,"\int x^{7/2} \left(a+b x^2\right)^3 \left(A+B x^2\right) \, dx","Integrate[x^(7/2)*(a + b*x^2)^3*(A + B*x^2),x]","\frac{2}{9} a^3 A x^{9/2}+\frac{2}{13} a^2 x^{13/2} (a B+3 A b)+\frac{2}{21} b^2 x^{21/2} (3 a B+A b)+\frac{6}{17} a b x^{17/2} (a B+A b)+\frac{2}{25} b^3 B x^{25/2}","\frac{2}{9} a^3 A x^{9/2}+\frac{2}{13} a^2 x^{13/2} (a B+3 A b)+\frac{2}{21} b^2 x^{21/2} (3 a B+A b)+\frac{6}{17} a b x^{17/2} (a B+A b)+\frac{2}{25} b^3 B x^{25/2}",1,"(2*a^3*A*x^(9/2))/9 + (2*a^2*(3*A*b + a*B)*x^(13/2))/13 + (6*a*b*(A*b + a*B)*x^(17/2))/17 + (2*b^2*(A*b + 3*a*B)*x^(21/2))/21 + (2*b^3*B*x^(25/2))/25","A",1
360,1,85,85,0.0430738,"\int x^{5/2} \left(a+b x^2\right)^3 \left(A+B x^2\right) \, dx","Integrate[x^(5/2)*(a + b*x^2)^3*(A + B*x^2),x]","\frac{2}{7} a^3 A x^{7/2}+\frac{2}{11} a^2 x^{11/2} (a B+3 A b)+\frac{2}{19} b^2 x^{19/2} (3 a B+A b)+\frac{2}{5} a b x^{15/2} (a B+A b)+\frac{2}{23} b^3 B x^{23/2}","\frac{2}{7} a^3 A x^{7/2}+\frac{2}{11} a^2 x^{11/2} (a B+3 A b)+\frac{2}{19} b^2 x^{19/2} (3 a B+A b)+\frac{2}{5} a b x^{15/2} (a B+A b)+\frac{2}{23} b^3 B x^{23/2}",1,"(2*a^3*A*x^(7/2))/7 + (2*a^2*(3*A*b + a*B)*x^(11/2))/11 + (2*a*b*(A*b + a*B)*x^(15/2))/5 + (2*b^2*(A*b + 3*a*B)*x^(19/2))/19 + (2*b^3*B*x^(23/2))/23","A",1
361,1,85,85,0.0386406,"\int x^{3/2} \left(a+b x^2\right)^3 \left(A+B x^2\right) \, dx","Integrate[x^(3/2)*(a + b*x^2)^3*(A + B*x^2),x]","\frac{2}{5} a^3 A x^{5/2}+\frac{2}{9} a^2 x^{9/2} (a B+3 A b)+\frac{2}{17} b^2 x^{17/2} (3 a B+A b)+\frac{6}{13} a b x^{13/2} (a B+A b)+\frac{2}{21} b^3 B x^{21/2}","\frac{2}{5} a^3 A x^{5/2}+\frac{2}{9} a^2 x^{9/2} (a B+3 A b)+\frac{2}{17} b^2 x^{17/2} (3 a B+A b)+\frac{6}{13} a b x^{13/2} (a B+A b)+\frac{2}{21} b^3 B x^{21/2}",1,"(2*a^3*A*x^(5/2))/5 + (2*a^2*(3*A*b + a*B)*x^(9/2))/9 + (6*a*b*(A*b + a*B)*x^(13/2))/13 + (2*b^2*(A*b + 3*a*B)*x^(17/2))/17 + (2*b^3*B*x^(21/2))/21","A",1
362,1,71,85,0.03765,"\int \sqrt{x} \left(a+b x^2\right)^3 \left(A+B x^2\right) \, dx","Integrate[Sqrt[x]*(a + b*x^2)^3*(A + B*x^2),x]","\frac{2 x^{3/2} \left(7315 a^3 A+3135 a^2 x^2 (a B+3 A b)+1463 b^2 x^6 (3 a B+A b)+5985 a b x^4 (a B+A b)+1155 b^3 B x^8\right)}{21945}","\frac{2}{3} a^3 A x^{3/2}+\frac{2}{7} a^2 x^{7/2} (a B+3 A b)+\frac{2}{15} b^2 x^{15/2} (3 a B+A b)+\frac{6}{11} a b x^{11/2} (a B+A b)+\frac{2}{19} b^3 B x^{19/2}",1,"(2*x^(3/2)*(7315*a^3*A + 3135*a^2*(3*A*b + a*B)*x^2 + 5985*a*b*(A*b + a*B)*x^4 + 1463*b^2*(A*b + 3*a*B)*x^6 + 1155*b^3*B*x^8))/21945","A",1
363,1,71,83,0.0393214,"\int \frac{\left(a+b x^2\right)^3 \left(A+B x^2\right)}{\sqrt{x}} \, dx","Integrate[((a + b*x^2)^3*(A + B*x^2))/Sqrt[x],x]","\frac{2 \sqrt{x} \left(3315 a^3 A+663 a^2 x^2 (a B+3 A b)+255 b^2 x^6 (3 a B+A b)+1105 a b x^4 (a B+A b)+195 b^3 B x^8\right)}{3315}","2 a^3 A \sqrt{x}+\frac{2}{5} a^2 x^{5/2} (a B+3 A b)+\frac{2}{13} b^2 x^{13/2} (3 a B+A b)+\frac{2}{3} a b x^{9/2} (a B+A b)+\frac{2}{17} b^3 B x^{17/2}",1,"(2*Sqrt[x]*(3315*a^3*A + 663*a^2*(3*A*b + a*B)*x^2 + 1105*a*b*(A*b + a*B)*x^4 + 255*b^2*(A*b + 3*a*B)*x^6 + 195*b^3*B*x^8))/3315","A",1
364,1,81,83,0.0215224,"\int \frac{\left(a+b x^2\right)^3 \left(A+B x^2\right)}{x^{3/2}} \, dx","Integrate[((a + b*x^2)^3*(A + B*x^2))/x^(3/2),x]","\frac{-770 a^3 \left(3 A-B x^2\right)+330 a^2 b x^2 \left(7 A+3 B x^2\right)+90 a b^2 x^4 \left(11 A+7 B x^2\right)+14 b^3 x^6 \left(15 A+11 B x^2\right)}{1155 \sqrt{x}}","-\frac{2 a^3 A}{\sqrt{x}}+\frac{2}{3} a^2 x^{3/2} (a B+3 A b)+\frac{2}{11} b^2 x^{11/2} (3 a B+A b)+\frac{6}{7} a b x^{7/2} (a B+A b)+\frac{2}{15} b^3 B x^{15/2}",1,"(-770*a^3*(3*A - B*x^2) + 330*a^2*b*x^2*(7*A + 3*B*x^2) + 90*a*b^2*x^4*(11*A + 7*B*x^2) + 14*b^3*x^6*(15*A + 11*B*x^2))/(1155*Sqrt[x])","A",1
365,1,78,83,0.023823,"\int \frac{\left(a+b x^2\right)^3 \left(A+B x^2\right)}{x^{5/2}} \, dx","Integrate[((a + b*x^2)^3*(A + B*x^2))/x^(5/2),x]","\frac{-390 a^3 \left(A-3 B x^2\right)+702 a^2 b x^2 \left(5 A+B x^2\right)+78 a b^2 x^4 \left(9 A+5 B x^2\right)+10 b^3 x^6 \left(13 A+9 B x^2\right)}{585 x^{3/2}}","-\frac{2 a^3 A}{3 x^{3/2}}+2 a^2 \sqrt{x} (a B+3 A b)+\frac{2}{9} b^2 x^{9/2} (3 a B+A b)+\frac{6}{5} a b x^{5/2} (a B+A b)+\frac{2}{13} b^3 B x^{13/2}",1,"(-390*a^3*(A - 3*B*x^2) + 702*a^2*b*x^2*(5*A + B*x^2) + 78*a*b^2*x^4*(9*A + 5*B*x^2) + 10*b^3*x^6*(13*A + 9*B*x^2))/(585*x^(3/2))","A",1
366,1,78,81,0.0225627,"\int \frac{\left(a+b x^2\right)^3 \left(A+B x^2\right)}{x^{7/2}} \, dx","Integrate[((a + b*x^2)^3*(A + B*x^2))/x^(7/2),x]","\frac{2 \left(-77 a^3 \left(A+5 B x^2\right)+385 a^2 b x^2 \left(B x^2-3 A\right)+55 a b^2 x^4 \left(7 A+3 B x^2\right)+5 b^3 x^6 \left(11 A+7 B x^2\right)\right)}{385 x^{5/2}}","-\frac{2 a^3 A}{5 x^{5/2}}-\frac{2 a^2 (a B+3 A b)}{\sqrt{x}}+\frac{2}{7} b^2 x^{7/2} (3 a B+A b)+2 a b x^{3/2} (a B+A b)+\frac{2}{11} b^3 B x^{11/2}",1,"(2*(385*a^2*b*x^2*(-3*A + B*x^2) + 55*a*b^2*x^4*(7*A + 3*B*x^2) - 77*a^3*(A + 5*B*x^2) + 5*b^3*x^6*(11*A + 7*B*x^2)))/(385*x^(5/2))","A",1
367,1,227,276,0.2890688,"\int \frac{x^{7/2} \left(A+B x^2\right)}{a+b x^2} \, dx","Integrate[(x^(7/2)*(A + B*x^2))/(a + b*x^2),x]","\frac{\frac{45 \sqrt{2} a^{5/4} (a B-A b) \left(\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)\right)}{\sqrt[4]{b}}+\frac{90 \sqrt{2} a^{5/4} (a B-A b) \left(\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)-\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)\right)}{\sqrt[4]{b}}+72 b x^{5/2} (A b-a B)+360 a \sqrt{x} (a B-A b)+40 b^2 B x^{9/2}}{180 b^3}","-\frac{a^{5/4} (A b-a B) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} b^{13/4}}+\frac{a^{5/4} (A b-a B) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} b^{13/4}}-\frac{a^{5/4} (A b-a B) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} b^{13/4}}+\frac{a^{5/4} (A b-a B) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} b^{13/4}}-\frac{2 a \sqrt{x} (A b-a B)}{b^3}+\frac{2 x^{5/2} (A b-a B)}{5 b^2}+\frac{2 B x^{9/2}}{9 b}",1,"(360*a*(-(A*b) + a*B)*Sqrt[x] + 72*b*(A*b - a*B)*x^(5/2) + 40*b^2*B*x^(9/2) + (90*Sqrt[2]*a^(5/4)*(-(A*b) + a*B)*(ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]))/b^(1/4) + (45*Sqrt[2]*a^(5/4)*(-(A*b) + a*B)*(Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]))/b^(1/4))/(180*b^3)","A",1
368,1,110,257,0.1327038,"\int \frac{x^{5/2} \left(A+B x^2\right)}{a+b x^2} \, dx","Integrate[(x^(5/2)*(A + B*x^2))/(a + b*x^2),x]","\frac{2 b^{3/4} x^{3/2} \left(-7 a B+7 A b+3 b B x^2\right)-21 (-a)^{3/4} (a B-A b) \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{-a}}\right)+21 (-a)^{3/4} (a B-A b) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{-a}}\right)}{21 b^{11/4}}","-\frac{a^{3/4} (A b-a B) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} b^{11/4}}+\frac{a^{3/4} (A b-a B) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} b^{11/4}}+\frac{a^{3/4} (A b-a B) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} b^{11/4}}-\frac{a^{3/4} (A b-a B) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} b^{11/4}}+\frac{2 x^{3/2} (A b-a B)}{3 b^2}+\frac{2 B x^{7/2}}{7 b}",1,"(2*b^(3/4)*x^(3/2)*(7*A*b - 7*a*B + 3*b*B*x^2) - 21*(-a)^(3/4)*(-(A*b) + a*B)*ArcTan[(b^(1/4)*Sqrt[x])/(-a)^(1/4)] + 21*(-a)^(3/4)*(-(A*b) + a*B)*ArcTanh[(b^(1/4)*Sqrt[x])/(-a)^(1/4)])/(21*b^(11/4))","A",1
369,1,208,255,0.2020343,"\int \frac{x^{3/2} \left(A+B x^2\right)}{a+b x^2} \, dx","Integrate[(x^(3/2)*(A + B*x^2))/(a + b*x^2),x]","\frac{40 \sqrt{x} (A b-a B)+\frac{5 \sqrt{2} \sqrt[4]{a} (A b-a B) \left(\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)\right)}{\sqrt[4]{b}}+\frac{10 \sqrt{2} \sqrt[4]{a} (A b-a B) \left(\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)-\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)\right)}{\sqrt[4]{b}}+8 b B x^{5/2}}{20 b^2}","\frac{\sqrt[4]{a} (A b-a B) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} b^{9/4}}-\frac{\sqrt[4]{a} (A b-a B) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} b^{9/4}}+\frac{\sqrt[4]{a} (A b-a B) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} b^{9/4}}-\frac{\sqrt[4]{a} (A b-a B) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} b^{9/4}}+\frac{2 \sqrt{x} (A b-a B)}{b^2}+\frac{2 B x^{5/2}}{5 b}",1,"(40*(A*b - a*B)*Sqrt[x] + 8*b*B*x^(5/2) + (10*Sqrt[2]*a^(1/4)*(A*b - a*B)*(ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]))/b^(1/4) + (5*Sqrt[2]*a^(1/4)*(A*b - a*B)*(Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]))/b^(1/4))/(20*b^2)","A",1
370,1,95,237,0.0730333,"\int \frac{\sqrt{x} \left(A+B x^2\right)}{a+b x^2} \, dx","Integrate[(Sqrt[x]*(A + B*x^2))/(a + b*x^2),x]","\frac{3 (A b-a B) \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{-a}}\right)+(3 a B-3 A b) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{-a}}\right)+2 \sqrt[4]{-a} b^{3/4} B x^{3/2}}{3 \sqrt[4]{-a} b^{7/4}}","\frac{(A b-a B) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} \sqrt[4]{a} b^{7/4}}-\frac{(A b-a B) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} \sqrt[4]{a} b^{7/4}}-\frac{(A b-a B) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} \sqrt[4]{a} b^{7/4}}+\frac{(A b-a B) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} \sqrt[4]{a} b^{7/4}}+\frac{2 B x^{3/2}}{3 b}",1,"(2*(-a)^(1/4)*b^(3/4)*B*x^(3/2) + 3*(A*b - a*B)*ArcTan[(b^(1/4)*Sqrt[x])/(-a)^(1/4)] + (-3*A*b + 3*a*B)*ArcTanh[(b^(1/4)*Sqrt[x])/(-a)^(1/4)])/(3*(-a)^(1/4)*b^(7/4))","A",1
371,1,166,235,0.13049,"\int \frac{A+B x^2}{\sqrt{x} \left(a+b x^2\right)} \, dx","Integrate[(A + B*x^2)/(Sqrt[x]*(a + b*x^2)),x]","\frac{(a B-A b) \left(\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)\right)}{2 \sqrt{2} a^{3/4} b^{5/4}}+\frac{2 B \sqrt{x}}{b}","-\frac{(A b-a B) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{3/4} b^{5/4}}+\frac{(A b-a B) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{3/4} b^{5/4}}-\frac{(A b-a B) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} a^{3/4} b^{5/4}}+\frac{(A b-a B) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} a^{3/4} b^{5/4}}+\frac{2 B \sqrt{x}}{b}",1,"(2*B*Sqrt[x])/b + ((-(A*b) + a*B)*(2*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]))/(2*Sqrt[2]*a^(3/4)*b^(5/4))","A",1
372,1,74,235,0.0840495,"\int \frac{A+B x^2}{x^{3/2} \left(a+b x^2\right)} \, dx","Integrate[(A + B*x^2)/(x^(3/2)*(a + b*x^2)),x]","\frac{\frac{(a B-A b) \left(\tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{-a}}\right)+\tanh ^{-1}\left(\frac{a \sqrt[4]{b} \sqrt{x}}{(-a)^{5/4}}\right)\right)}{\sqrt[4]{-a} b^{3/4}}-\frac{2 A}{\sqrt{x}}}{a}","-\frac{(A b-a B) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{5/4} b^{3/4}}+\frac{(A b-a B) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{5/4} b^{3/4}}+\frac{(A b-a B) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} a^{5/4} b^{3/4}}-\frac{(A b-a B) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} a^{5/4} b^{3/4}}-\frac{2 A}{a \sqrt{x}}",1,"((-2*A)/Sqrt[x] + ((-(A*b) + a*B)*(ArcTan[(b^(1/4)*Sqrt[x])/(-a)^(1/4)] + ArcTanh[(a*b^(1/4)*Sqrt[x])/(-a)^(5/4)]))/((-a)^(1/4)*b^(3/4)))/a","A",1
373,1,168,237,0.1390556,"\int \frac{A+B x^2}{x^{5/2} \left(a+b x^2\right)} \, dx","Integrate[(A + B*x^2)/(x^(5/2)*(a + b*x^2)),x]","\frac{(A b-a B) \left(\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)\right)}{2 \sqrt{2} a^{7/4} \sqrt[4]{b}}-\frac{2 A}{3 a x^{3/2}}","\frac{(A b-a B) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{7/4} \sqrt[4]{b}}-\frac{(A b-a B) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{7/4} \sqrt[4]{b}}+\frac{(A b-a B) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} a^{7/4} \sqrt[4]{b}}-\frac{(A b-a B) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} a^{7/4} \sqrt[4]{b}}-\frac{2 A}{3 a x^{3/2}}",1,"(-2*A)/(3*a*x^(3/2)) + ((A*b - a*B)*(2*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]))/(2*Sqrt[2]*a^(7/4)*b^(1/4))","A",1
374,1,46,255,0.0146238,"\int \frac{A+B x^2}{x^{7/2} \left(a+b x^2\right)} \, dx","Integrate[(A + B*x^2)/(x^(7/2)*(a + b*x^2)),x]","-\frac{2 \left(5 x^2 (a B-A b) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\frac{b x^2}{a}\right)+a A\right)}{5 a^2 x^{5/2}}","\frac{\sqrt[4]{b} (A b-a B) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{9/4}}-\frac{\sqrt[4]{b} (A b-a B) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{9/4}}-\frac{\sqrt[4]{b} (A b-a B) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} a^{9/4}}+\frac{\sqrt[4]{b} (A b-a B) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} a^{9/4}}+\frac{2 (A b-a B)}{a^2 \sqrt{x}}-\frac{2 A}{5 a x^{5/2}}",1,"(-2*(a*A + 5*(-(A*b) + a*B)*x^2*Hypergeometric2F1[-1/4, 1, 3/4, -((b*x^2)/a)]))/(5*a^2*x^(5/2))","C",1
375,1,385,310,0.427524,"\int \frac{x^{7/2} \left(A+B x^2\right)}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^(7/2)*(A + B*x^2))/(a + b*x^2)^2,x]","\frac{-45 \sqrt{2} a^{5/4} B \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+45 \sqrt{2} a^{5/4} B \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-\frac{40 a^2 \sqrt[4]{b} B \sqrt{x}}{a+b x^2}+\frac{40 a A b^{5/4} \sqrt{x}}{a+b x^2}-10 \sqrt{2} \sqrt[4]{a} (9 a B-5 A b) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)+10 \sqrt{2} \sqrt[4]{a} (9 a B-5 A b) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)+25 \sqrt{2} \sqrt[4]{a} A b \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-25 \sqrt{2} \sqrt[4]{a} A b \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-320 a \sqrt[4]{b} B \sqrt{x}+160 A b^{5/4} \sqrt{x}+32 b^{5/4} B x^{5/2}}{80 b^{13/4}}","\frac{\sqrt[4]{a} (5 A b-9 a B) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} b^{13/4}}-\frac{\sqrt[4]{a} (5 A b-9 a B) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} b^{13/4}}+\frac{\sqrt[4]{a} (5 A b-9 a B) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} b^{13/4}}-\frac{\sqrt[4]{a} (5 A b-9 a B) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} b^{13/4}}+\frac{\sqrt{x} (5 A b-9 a B)}{2 b^3}-\frac{x^{5/2} (5 A b-9 a B)}{10 a b^2}+\frac{x^{9/2} (A b-a B)}{2 a b \left(a+b x^2\right)}",1,"(160*A*b^(5/4)*Sqrt[x] - 320*a*b^(1/4)*B*Sqrt[x] + 32*b^(5/4)*B*x^(5/2) + (40*a*A*b^(5/4)*Sqrt[x])/(a + b*x^2) - (40*a^2*b^(1/4)*B*Sqrt[x])/(a + b*x^2) - 10*Sqrt[2]*a^(1/4)*(-5*A*b + 9*a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 10*Sqrt[2]*a^(1/4)*(-5*A*b + 9*a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 25*Sqrt[2]*a^(1/4)*A*b*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - 45*Sqrt[2]*a^(5/4)*B*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - 25*Sqrt[2]*a^(1/4)*A*b*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 45*Sqrt[2]*a^(5/4)*B*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(80*b^(13/4))","A",1
376,1,136,289,0.1964101,"\int \frac{x^{5/2} \left(A+B x^2\right)}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^(5/2)*(A + B*x^2))/(a + b*x^2)^2,x]","\frac{3 (A b-2 a B) \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{-a}}\right)+(6 a B-3 A b) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{-a}}\right)+2 \sqrt[4]{-a} b^{3/4} B x^{3/2}}{3 \sqrt[4]{-a} b^{11/4}}+\frac{2 x^{3/2} (a B-A b) \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-\frac{b x^2}{a}\right)}{3 a b^2}","\frac{(3 A b-7 a B) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} \sqrt[4]{a} b^{11/4}}-\frac{(3 A b-7 a B) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} \sqrt[4]{a} b^{11/4}}-\frac{(3 A b-7 a B) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} \sqrt[4]{a} b^{11/4}}+\frac{(3 A b-7 a B) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} \sqrt[4]{a} b^{11/4}}-\frac{x^{3/2} (3 A b-7 a B)}{6 a b^2}+\frac{x^{7/2} (A b-a B)}{2 a b \left(a+b x^2\right)}",1,"(2*(-a)^(1/4)*b^(3/4)*B*x^(3/2) + 3*(A*b - 2*a*B)*ArcTan[(b^(1/4)*Sqrt[x])/(-a)^(1/4)] + (-3*A*b + 6*a*B)*ArcTanh[(b^(1/4)*Sqrt[x])/(-a)^(1/4)])/(3*(-a)^(1/4)*b^(11/4)) + (2*(-(A*b) + a*B)*x^(3/2)*Hypergeometric2F1[3/4, 2, 7/4, -((b*x^2)/a)])/(3*a*b^2)","C",1
377,1,353,284,0.3704017,"\int \frac{x^{3/2} \left(A+B x^2\right)}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^(3/2)*(A + B*x^2))/(a + b*x^2)^2,x]","\frac{\frac{2 \sqrt{2} (5 a B-A b) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{a^{3/4}}+\frac{2 \sqrt{2} (A b-5 a B) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{a^{3/4}}-\frac{\sqrt{2} A b \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{3/4}}+\frac{\sqrt{2} A b \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{3/4}}-\frac{8 A b^{5/4} \sqrt{x}}{a+b x^2}+\frac{8 a \sqrt[4]{b} B \sqrt{x}}{a+b x^2}+5 \sqrt{2} \sqrt[4]{a} B \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-5 \sqrt{2} \sqrt[4]{a} B \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+32 \sqrt[4]{b} B \sqrt{x}}{16 b^{9/4}}","-\frac{(A b-5 a B) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{3/4} b^{9/4}}+\frac{(A b-5 a B) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{3/4} b^{9/4}}-\frac{(A b-5 a B) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} a^{3/4} b^{9/4}}+\frac{(A b-5 a B) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} a^{3/4} b^{9/4}}-\frac{\sqrt{x} (A b-5 a B)}{2 a b^2}+\frac{x^{5/2} (A b-a B)}{2 a b \left(a+b x^2\right)}",1,"(32*b^(1/4)*B*Sqrt[x] - (8*A*b^(5/4)*Sqrt[x])/(a + b*x^2) + (8*a*b^(1/4)*B*Sqrt[x])/(a + b*x^2) + (2*Sqrt[2]*(-(A*b) + 5*a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(3/4) + (2*Sqrt[2]*(A*b - 5*a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(3/4) - (Sqrt[2]*A*b*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(3/4) + 5*Sqrt[2]*a^(1/4)*B*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + (Sqrt[2]*A*b*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(3/4) - 5*Sqrt[2]*a^(1/4)*B*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(16*b^(9/4))","A",1
378,1,95,261,0.1206096,"\int \frac{\sqrt{x} \left(A+B x^2\right)}{\left(a+b x^2\right)^2} \, dx","Integrate[(Sqrt[x]*(A + B*x^2))/(a + b*x^2)^2,x]","\frac{2 x^{3/2} (A b-a B) \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-\frac{b x^2}{a}\right)}{3 a^2 b}+\frac{B \left(\tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{-a}}\right)+\tanh ^{-1}\left(\frac{a \sqrt[4]{b} \sqrt{x}}{(-a)^{5/4}}\right)\right)}{\sqrt[4]{-a} b^{7/4}}","\frac{(3 a B+A b) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{5/4} b^{7/4}}-\frac{(3 a B+A b) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{5/4} b^{7/4}}-\frac{(3 a B+A b) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} a^{5/4} b^{7/4}}+\frac{(3 a B+A b) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} a^{5/4} b^{7/4}}+\frac{x^{3/2} (A b-a B)}{2 a b \left(a+b x^2\right)}",1,"(B*(ArcTan[(b^(1/4)*Sqrt[x])/(-a)^(1/4)] + ArcTanh[(a*b^(1/4)*Sqrt[x])/(-a)^(5/4)]))/((-a)^(1/4)*b^(7/4)) + (2*(A*b - a*B)*x^(3/2)*Hypergeometric2F1[3/4, 2, 7/4, -((b*x^2)/a)])/(3*a^2*b)","C",1
379,1,203,261,0.2339706,"\int \frac{A+B x^2}{\sqrt{x} \left(a+b x^2\right)^2} \, dx","Integrate[(A + B*x^2)/(Sqrt[x]*(a + b*x^2)^2),x]","\frac{\frac{(a B+3 A b) \left(8 a^{3/4} \sqrt[4]{b} \sqrt{x}-3 \sqrt{2} \left(a+b x^2\right) \left(\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)\right)\right)}{a^{7/4} \sqrt[4]{b}}-32 B \sqrt{x}}{48 b \left(a+b x^2\right)}","-\frac{(a B+3 A b) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{7/4} b^{5/4}}+\frac{(a B+3 A b) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{7/4} b^{5/4}}-\frac{(a B+3 A b) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} a^{7/4} b^{5/4}}+\frac{(a B+3 A b) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} a^{7/4} b^{5/4}}+\frac{\sqrt{x} (A b-a B)}{2 a b \left(a+b x^2\right)}",1,"(-32*B*Sqrt[x] + ((3*A*b + a*B)*(8*a^(3/4)*b^(1/4)*Sqrt[x] - 3*Sqrt[2]*(a + b*x^2)*(2*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])))/(a^(7/4)*b^(1/4)))/(48*b*(a + b*x^2))","A",1
380,1,117,289,0.2158569,"\int \frac{A+B x^2}{x^{3/2} \left(a+b x^2\right)^2} \, dx","Integrate[(A + B*x^2)/(x^(3/2)*(a + b*x^2)^2),x]","\frac{2 x^{3/2} (a B-A b) \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-\frac{b x^2}{a}\right)+3 A \left((-a)^{3/4} \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{-a}}\right)-(-a)^{3/4} \sqrt[4]{b} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{-a}}\right)-\frac{2 a}{\sqrt{x}}\right)}{3 a^3}","-\frac{(5 A b-a B) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{9/4} b^{3/4}}+\frac{(5 A b-a B) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{9/4} b^{3/4}}+\frac{(5 A b-a B) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} a^{9/4} b^{3/4}}-\frac{(5 A b-a B) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} a^{9/4} b^{3/4}}-\frac{5 A b-a B}{2 a^2 b \sqrt{x}}+\frac{A b-a B}{2 a b \sqrt{x} \left(a+b x^2\right)}",1,"(3*A*((-2*a)/Sqrt[x] + (-a)^(3/4)*b^(1/4)*ArcTan[(b^(1/4)*Sqrt[x])/(-a)^(1/4)] - (-a)^(3/4)*b^(1/4)*ArcTanh[(b^(1/4)*Sqrt[x])/(-a)^(1/4)]) + 2*(-(A*b) + a*B)*x^(3/2)*Hypergeometric2F1[3/4, 2, 7/4, -((b*x^2)/a)])/(3*a^3)","C",1
381,1,355,289,0.3910064,"\int \frac{A+B x^2}{x^{5/2} \left(a+b x^2\right)^2} \, dx","Integrate[(A + B*x^2)/(x^(5/2)*(a + b*x^2)^2),x]","\frac{-\frac{24 a^{3/4} A b \sqrt{x}}{a+b x^2}-\frac{32 a^{3/4} A}{x^{3/2}}+\frac{24 a^{7/4} B \sqrt{x}}{a+b x^2}+21 \sqrt{2} A b^{3/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-21 \sqrt{2} A b^{3/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+\frac{6 \sqrt{2} (7 A b-3 a B) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt[4]{b}}-\frac{6 \sqrt{2} (7 A b-3 a B) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt[4]{b}}-\frac{9 \sqrt{2} a B \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{\sqrt[4]{b}}+\frac{9 \sqrt{2} a B \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{\sqrt[4]{b}}}{48 a^{11/4}}","\frac{(7 A b-3 a B) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{11/4} \sqrt[4]{b}}-\frac{(7 A b-3 a B) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{11/4} \sqrt[4]{b}}+\frac{(7 A b-3 a B) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} a^{11/4} \sqrt[4]{b}}-\frac{(7 A b-3 a B) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} a^{11/4} \sqrt[4]{b}}-\frac{7 A b-3 a B}{6 a^2 b x^{3/2}}+\frac{A b-a B}{2 a b x^{3/2} \left(a+b x^2\right)}",1,"((-32*a^(3/4)*A)/x^(3/2) - (24*a^(3/4)*A*b*Sqrt[x])/(a + b*x^2) + (24*a^(7/4)*B*Sqrt[x])/(a + b*x^2) + (6*Sqrt[2]*(7*A*b - 3*a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(1/4) - (6*Sqrt[2]*(7*A*b - 3*a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(1/4) + 21*Sqrt[2]*A*b^(3/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - (9*Sqrt[2]*a*B*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(1/4) - 21*Sqrt[2]*A*b^(3/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + (9*Sqrt[2]*a*B*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(1/4))/(48*a^(11/4))","A",1
382,1,151,310,0.4342125,"\int \frac{A+B x^2}{x^{7/2} \left(a+b x^2\right)^2} \, dx","Integrate[(A + B*x^2)/(x^(7/2)*(a + b*x^2)^2),x]","\frac{2 b x^{3/2} (A b-a B) \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-\frac{b x^2}{a}\right)}{3 a^4}+\frac{4 A b-2 a B}{a^3 \sqrt{x}}-\frac{2 A}{5 a^2 x^{5/2}}+\frac{\sqrt[4]{b} (a B-2 A b) \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{-a}}\right)}{(-a)^{13/4}}+\frac{\sqrt[4]{b} (2 A b-a B) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{-a}}\right)}{(-a)^{13/4}}","\frac{\sqrt[4]{b} (9 A b-5 a B) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{13/4}}-\frac{\sqrt[4]{b} (9 A b-5 a B) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{13/4}}-\frac{\sqrt[4]{b} (9 A b-5 a B) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} a^{13/4}}+\frac{\sqrt[4]{b} (9 A b-5 a B) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} a^{13/4}}+\frac{9 A b-5 a B}{2 a^3 \sqrt{x}}-\frac{9 A b-5 a B}{10 a^2 b x^{5/2}}+\frac{A b-a B}{2 a b x^{5/2} \left(a+b x^2\right)}",1,"(-2*A)/(5*a^2*x^(5/2)) + (4*A*b - 2*a*B)/(a^3*Sqrt[x]) + (b^(1/4)*(-2*A*b + a*B)*ArcTan[(b^(1/4)*Sqrt[x])/(-a)^(1/4)])/(-a)^(13/4) + (b^(1/4)*(2*A*b - a*B)*ArcTanh[(b^(1/4)*Sqrt[x])/(-a)^(1/4)])/(-a)^(13/4) + (2*b*(A*b - a*B)*x^(3/2)*Hypergeometric2F1[3/4, 2, 7/4, -((b*x^2)/a)])/(3*a^4)","C",1
383,1,402,316,0.4346544,"\int \frac{x^{7/2} \left(A+B x^2\right)}{\left(a+b x^2\right)^3} \, dx","Integrate[(x^(7/2)*(A + B*x^2))/(a + b*x^2)^3,x]","\frac{\frac{10 \sqrt{2} (9 a B-A b) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{a^{3/4}}+\frac{10 \sqrt{2} (A b-9 a B) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{a^{3/4}}-\frac{5 \sqrt{2} A b \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{3/4}}+\frac{5 \sqrt{2} A b \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{3/4}}-\frac{32 a^2 \sqrt[4]{b} B \sqrt{x}}{\left(a+b x^2\right)^2}+\frac{32 a A b^{5/4} \sqrt{x}}{\left(a+b x^2\right)^2}-\frac{72 A b^{5/4} \sqrt{x}}{a+b x^2}+\frac{136 a \sqrt[4]{b} B \sqrt{x}}{a+b x^2}+45 \sqrt{2} \sqrt[4]{a} B \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-45 \sqrt{2} \sqrt[4]{a} B \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+256 \sqrt[4]{b} B \sqrt{x}}{128 b^{13/4}}","-\frac{5 (A b-9 a B) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{3/4} b^{13/4}}+\frac{5 (A b-9 a B) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{3/4} b^{13/4}}-\frac{5 (A b-9 a B) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{32 \sqrt{2} a^{3/4} b^{13/4}}+\frac{5 (A b-9 a B) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{32 \sqrt{2} a^{3/4} b^{13/4}}-\frac{5 \sqrt{x} (A b-9 a B)}{16 a b^3}+\frac{x^{5/2} (A b-9 a B)}{16 a b^2 \left(a+b x^2\right)}+\frac{x^{9/2} (A b-a B)}{4 a b \left(a+b x^2\right)^2}",1,"(256*b^(1/4)*B*Sqrt[x] + (32*a*A*b^(5/4)*Sqrt[x])/(a + b*x^2)^2 - (32*a^2*b^(1/4)*B*Sqrt[x])/(a + b*x^2)^2 - (72*A*b^(5/4)*Sqrt[x])/(a + b*x^2) + (136*a*b^(1/4)*B*Sqrt[x])/(a + b*x^2) + (10*Sqrt[2]*(-(A*b) + 9*a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(3/4) + (10*Sqrt[2]*(A*b - 9*a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(3/4) - (5*Sqrt[2]*A*b*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(3/4) + 45*Sqrt[2]*a^(1/4)*B*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + (5*Sqrt[2]*A*b*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(3/4) - 45*Sqrt[2]*a^(1/4)*B*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(128*b^(13/4))","A",1
384,1,137,293,0.2068609,"\int \frac{x^{5/2} \left(A+B x^2\right)}{\left(a+b x^2\right)^3} \, dx","Integrate[(x^(5/2)*(A + B*x^2))/(a + b*x^2)^3,x]","\frac{2 b^{3/4} x^{3/2} (A b-2 a B) \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-\frac{b x^2}{a}\right)+2 b^{3/4} x^{3/2} (a B-A b) \, _2F_1\left(\frac{3}{4},3;\frac{7}{4};-\frac{b x^2}{a}\right)+3 (-a)^{7/4} B \left(\tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{-a}}\right)+\tanh ^{-1}\left(\frac{a \sqrt[4]{b} \sqrt{x}}{(-a)^{5/4}}\right)\right)}{3 a^2 b^{11/4}}","\frac{3 (7 a B+A b) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{5/4} b^{11/4}}-\frac{3 (7 a B+A b) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{5/4} b^{11/4}}-\frac{3 (7 a B+A b) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{32 \sqrt{2} a^{5/4} b^{11/4}}+\frac{3 (7 a B+A b) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{32 \sqrt{2} a^{5/4} b^{11/4}}-\frac{x^{3/2} (7 a B+A b)}{16 a b^2 \left(a+b x^2\right)}+\frac{x^{7/2} (A b-a B)}{4 a b \left(a+b x^2\right)^2}",1,"(3*(-a)^(7/4)*B*(ArcTan[(b^(1/4)*Sqrt[x])/(-a)^(1/4)] + ArcTanh[(a*b^(1/4)*Sqrt[x])/(-a)^(5/4)]) + 2*b^(3/4)*(A*b - 2*a*B)*x^(3/2)*Hypergeometric2F1[3/4, 2, 7/4, -((b*x^2)/a)] + 2*b^(3/4)*(-(A*b) + a*B)*x^(3/2)*Hypergeometric2F1[3/4, 3, 7/4, -((b*x^2)/a)])/(3*a^2*b^(11/4))","C",1
385,1,389,298,0.4321586,"\int \frac{x^{3/2} \left(A+B x^2\right)}{\left(a+b x^2\right)^3} \, dx","Integrate[(x^(3/2)*(A + B*x^2))/(a + b*x^2)^3,x]","\frac{-\frac{2 \sqrt{2} (5 a B+3 A b) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{a^{7/4}}+\frac{2 \sqrt{2} (5 a B+3 A b) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{a^{7/4}}-\frac{3 \sqrt{2} A b \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{7/4}}+\frac{3 \sqrt{2} A b \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{7/4}}-\frac{5 \sqrt{2} B \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{3/4}}+\frac{5 \sqrt{2} B \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{3/4}}+\frac{8 A b^{5/4} \sqrt{x}}{a^2+a b x^2}-\frac{32 A b^{5/4} \sqrt{x}}{\left(a+b x^2\right)^2}-\frac{72 \sqrt[4]{b} B \sqrt{x}}{a+b x^2}+\frac{32 a \sqrt[4]{b} B \sqrt{x}}{\left(a+b x^2\right)^2}}{128 b^{9/4}}","-\frac{(5 a B+3 A b) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{7/4} b^{9/4}}+\frac{(5 a B+3 A b) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{7/4} b^{9/4}}-\frac{(5 a B+3 A b) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{32 \sqrt{2} a^{7/4} b^{9/4}}+\frac{(5 a B+3 A b) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{32 \sqrt{2} a^{7/4} b^{9/4}}-\frac{\sqrt{x} (5 a B+3 A b)}{16 a b^2 \left(a+b x^2\right)}+\frac{x^{5/2} (A b-a B)}{4 a b \left(a+b x^2\right)^2}",1,"((-32*A*b^(5/4)*Sqrt[x])/(a + b*x^2)^2 + (32*a*b^(1/4)*B*Sqrt[x])/(a + b*x^2)^2 - (72*b^(1/4)*B*Sqrt[x])/(a + b*x^2) + (8*A*b^(5/4)*Sqrt[x])/(a^2 + a*b*x^2) - (2*Sqrt[2]*(3*A*b + 5*a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(7/4) + (2*Sqrt[2]*(3*A*b + 5*a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(7/4) - (3*Sqrt[2]*A*b*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(7/4) - (5*Sqrt[2]*B*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(3/4) + (3*Sqrt[2]*A*b*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(7/4) + (5*Sqrt[2]*B*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(3/4))/(128*b^(9/4))","A",1
386,1,62,298,0.04979,"\int \frac{\sqrt{x} \left(A+B x^2\right)}{\left(a+b x^2\right)^3} \, dx","Integrate[(Sqrt[x]*(A + B*x^2))/(a + b*x^2)^3,x]","\frac{2 x^{3/2} \left((A b-a B) \, _2F_1\left(\frac{3}{4},3;\frac{7}{4};-\frac{b x^2}{a}\right)+a B \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-\frac{b x^2}{a}\right)\right)}{3 a^3 b}","\frac{(3 a B+5 A b) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{9/4} b^{7/4}}-\frac{(3 a B+5 A b) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{9/4} b^{7/4}}-\frac{(3 a B+5 A b) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{32 \sqrt{2} a^{9/4} b^{7/4}}+\frac{(3 a B+5 A b) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{32 \sqrt{2} a^{9/4} b^{7/4}}+\frac{x^{3/2} (3 a B+5 A b)}{16 a^2 b \left(a+b x^2\right)}+\frac{x^{3/2} (A b-a B)}{4 a b \left(a+b x^2\right)^2}",1,"(2*x^(3/2)*(a*B*Hypergeometric2F1[3/4, 2, 7/4, -((b*x^2)/a)] + (A*b - a*B)*Hypergeometric2F1[3/4, 3, 7/4, -((b*x^2)/a)]))/(3*a^3*b)","C",1
387,1,230,293,0.2800832,"\int \frac{A+B x^2}{\sqrt{x} \left(a+b x^2\right)^3} \, dx","Integrate[(A + B*x^2)/(Sqrt[x]*(a + b*x^2)^3),x]","\frac{\frac{(a B+7 A b) \left(7 \left(a+b x^2\right) \left(8 a^{3/4} \sqrt[4]{b} \sqrt{x}-3 \sqrt{2} \left(a+b x^2\right) \left(\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)\right)\right)+32 a^{7/4} \sqrt[4]{b} \sqrt{x}\right)}{a^{11/4} \sqrt[4]{b}}-256 B \sqrt{x}}{896 b \left(a+b x^2\right)^2}","-\frac{3 (a B+7 A b) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{11/4} b^{5/4}}+\frac{3 (a B+7 A b) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{11/4} b^{5/4}}-\frac{3 (a B+7 A b) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{32 \sqrt{2} a^{11/4} b^{5/4}}+\frac{3 (a B+7 A b) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{32 \sqrt{2} a^{11/4} b^{5/4}}+\frac{\sqrt{x} (a B+7 A b)}{16 a^2 b \left(a+b x^2\right)}+\frac{\sqrt{x} (A b-a B)}{4 a b \left(a+b x^2\right)^2}",1,"(-256*B*Sqrt[x] + ((7*A*b + a*B)*(32*a^(7/4)*b^(1/4)*Sqrt[x] + 7*(a + b*x^2)*(8*a^(3/4)*b^(1/4)*Sqrt[x] - 3*Sqrt[2]*(a + b*x^2)*(2*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]))))/(a^(11/4)*b^(1/4)))/(896*b*(a + b*x^2)^2)","A",1
388,1,147,322,0.1805784,"\int \frac{A+B x^2}{x^{3/2} \left(a+b x^2\right)^3} \, dx","Integrate[(A + B*x^2)/(x^(3/2)*(a + b*x^2)^3),x]","\frac{2 x^{3/2} (a B-A b) \, _2F_1\left(\frac{3}{4},3;\frac{7}{4};-\frac{b x^2}{a}\right)}{3 a^4}-\frac{2 A b x^{3/2} \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-\frac{b x^2}{a}\right)}{3 a^4}-\frac{2 A}{a^3 \sqrt{x}}+\frac{A \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{-a}}\right)}{(-a)^{13/4}}+\frac{a A \sqrt[4]{b} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{-a}}\right)}{(-a)^{17/4}}","-\frac{5 (9 A b-a B) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{13/4} b^{3/4}}+\frac{5 (9 A b-a B) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{13/4} b^{3/4}}+\frac{5 (9 A b-a B) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{32 \sqrt{2} a^{13/4} b^{3/4}}-\frac{5 (9 A b-a B) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{32 \sqrt{2} a^{13/4} b^{3/4}}-\frac{5 (9 A b-a B)}{16 a^3 b \sqrt{x}}+\frac{9 A b-a B}{16 a^2 b \sqrt{x} \left(a+b x^2\right)}+\frac{A b-a B}{4 a b \sqrt{x} \left(a+b x^2\right)^2}",1,"(-2*A)/(a^3*Sqrt[x]) + (A*b^(1/4)*ArcTan[(b^(1/4)*Sqrt[x])/(-a)^(1/4)])/(-a)^(13/4) + (a*A*b^(1/4)*ArcTanh[(b^(1/4)*Sqrt[x])/(-a)^(1/4)])/(-a)^(17/4) - (2*A*b*x^(3/2)*Hypergeometric2F1[3/4, 2, 7/4, -((b*x^2)/a)])/(3*a^4) + (2*(-(A*b) + a*B)*x^(3/2)*Hypergeometric2F1[3/4, 3, 7/4, -((b*x^2)/a)])/(3*a^4)","C",1
389,1,400,322,0.4147338,"\int \frac{A+B x^2}{x^{5/2} \left(a+b x^2\right)^3} \, dx","Integrate[(A + B*x^2)/(x^(5/2)*(a + b*x^2)^3),x]","\frac{-\frac{96 a^{7/4} A b \sqrt{x}}{\left(a+b x^2\right)^2}-\frac{360 a^{3/4} A b \sqrt{x}}{a+b x^2}-\frac{256 a^{3/4} A}{x^{3/2}}+\frac{96 a^{11/4} B \sqrt{x}}{\left(a+b x^2\right)^2}+\frac{168 a^{7/4} B \sqrt{x}}{a+b x^2}+231 \sqrt{2} A b^{3/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-231 \sqrt{2} A b^{3/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+\frac{42 \sqrt{2} (11 A b-3 a B) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt[4]{b}}-\frac{42 \sqrt{2} (11 A b-3 a B) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt[4]{b}}-\frac{63 \sqrt{2} a B \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{\sqrt[4]{b}}+\frac{63 \sqrt{2} a B \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{\sqrt[4]{b}}}{384 a^{15/4}}","\frac{7 (11 A b-3 a B) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{15/4} \sqrt[4]{b}}-\frac{7 (11 A b-3 a B) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{15/4} \sqrt[4]{b}}+\frac{7 (11 A b-3 a B) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{32 \sqrt{2} a^{15/4} \sqrt[4]{b}}-\frac{7 (11 A b-3 a B) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{32 \sqrt{2} a^{15/4} \sqrt[4]{b}}-\frac{7 (11 A b-3 a B)}{48 a^3 b x^{3/2}}+\frac{11 A b-3 a B}{16 a^2 b x^{3/2} \left(a+b x^2\right)}+\frac{A b-a B}{4 a b x^{3/2} \left(a+b x^2\right)^2}",1,"((-256*a^(3/4)*A)/x^(3/2) - (96*a^(7/4)*A*b*Sqrt[x])/(a + b*x^2)^2 + (96*a^(11/4)*B*Sqrt[x])/(a + b*x^2)^2 - (360*a^(3/4)*A*b*Sqrt[x])/(a + b*x^2) + (168*a^(7/4)*B*Sqrt[x])/(a + b*x^2) + (42*Sqrt[2]*(11*A*b - 3*a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(1/4) - (42*Sqrt[2]*(11*A*b - 3*a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(1/4) + 231*Sqrt[2]*A*b^(3/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - (63*Sqrt[2]*a*B*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(1/4) - 231*Sqrt[2]*A*b^(3/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + (63*Sqrt[2]*a*B*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(1/4))/(384*a^(15/4))","A",1
390,1,189,343,0.4665148,"\int \frac{A+B x^2}{x^{7/2} \left(a+b x^2\right)^3} \, dx","Integrate[(A + B*x^2)/(x^(7/2)*(a + b*x^2)^3),x]","-\frac{2 b x^{3/2} (a B-2 A b) \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-\frac{b x^2}{a}\right)}{3 a^5}+\frac{2 b x^{3/2} (A b-a B) \, _2F_1\left(\frac{3}{4},3;\frac{7}{4};-\frac{b x^2}{a}\right)}{3 a^5}+\frac{6 A b-2 a B}{a^4 \sqrt{x}}-\frac{2 A}{5 a^3 x^{5/2}}+\frac{\sqrt[4]{b} (3 A b-a B) \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{-a}}\right)}{(-a)^{17/4}}+\frac{\sqrt[4]{b} (a B-3 A b) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{-a}}\right)}{(-a)^{17/4}}","\frac{9 \sqrt[4]{b} (13 A b-5 a B) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{17/4}}-\frac{9 \sqrt[4]{b} (13 A b-5 a B) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{17/4}}-\frac{9 \sqrt[4]{b} (13 A b-5 a B) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{32 \sqrt{2} a^{17/4}}+\frac{9 \sqrt[4]{b} (13 A b-5 a B) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{32 \sqrt{2} a^{17/4}}+\frac{9 (13 A b-5 a B)}{16 a^4 \sqrt{x}}-\frac{9 (13 A b-5 a B)}{80 a^3 b x^{5/2}}+\frac{13 A b-5 a B}{16 a^2 b x^{5/2} \left(a+b x^2\right)}+\frac{A b-a B}{4 a b x^{5/2} \left(a+b x^2\right)^2}",1,"(-2*A)/(5*a^3*x^(5/2)) + (6*A*b - 2*a*B)/(a^4*Sqrt[x]) + (b^(1/4)*(3*A*b - a*B)*ArcTan[(b^(1/4)*Sqrt[x])/(-a)^(1/4)])/(-a)^(17/4) + (b^(1/4)*(-3*A*b + a*B)*ArcTanh[(b^(1/4)*Sqrt[x])/(-a)^(1/4)])/(-a)^(17/4) - (2*b*(-2*A*b + a*B)*x^(3/2)*Hypergeometric2F1[3/4, 2, 7/4, -((b*x^2)/a)])/(3*a^5) + (2*b*(A*b - a*B)*x^(3/2)*Hypergeometric2F1[3/4, 3, 7/4, -((b*x^2)/a)])/(3*a^5)","C",1
391,1,53,63,0.0314379,"\int x^{7/2} \left(a+b x^2\right)^2 \left(c+d x^2\right) \, dx","Integrate[x^(7/2)*(a + b*x^2)^2*(c + d*x^2),x]","\frac{2 x^{9/2} \left(1547 a^2 c+819 b x^4 (2 a d+b c)+1071 a x^2 (a d+2 b c)+663 b^2 d x^6\right)}{13923}","\frac{2}{9} a^2 c x^{9/2}+\frac{2}{17} b x^{17/2} (2 a d+b c)+\frac{2}{13} a x^{13/2} (a d+2 b c)+\frac{2}{21} b^2 d x^{21/2}",1,"(2*x^(9/2)*(1547*a^2*c + 1071*a*(2*b*c + a*d)*x^2 + 819*b*(b*c + 2*a*d)*x^4 + 663*b^2*d*x^6))/13923","A",1
392,1,63,63,0.0287304,"\int x^{5/2} \left(a+b x^2\right)^2 \left(c+d x^2\right) \, dx","Integrate[x^(5/2)*(a + b*x^2)^2*(c + d*x^2),x]","\frac{2}{7} a^2 c x^{7/2}+\frac{2}{15} b x^{15/2} (2 a d+b c)+\frac{2}{11} a x^{11/2} (a d+2 b c)+\frac{2}{19} b^2 d x^{19/2}","\frac{2}{7} a^2 c x^{7/2}+\frac{2}{15} b x^{15/2} (2 a d+b c)+\frac{2}{11} a x^{11/2} (a d+2 b c)+\frac{2}{19} b^2 d x^{19/2}",1,"(2*a^2*c*x^(7/2))/7 + (2*a*(2*b*c + a*d)*x^(11/2))/11 + (2*b*(b*c + 2*a*d)*x^(15/2))/15 + (2*b^2*d*x^(19/2))/19","A",1
393,1,53,63,0.0284588,"\int x^{3/2} \left(a+b x^2\right)^2 \left(c+d x^2\right) \, dx","Integrate[x^(3/2)*(a + b*x^2)^2*(c + d*x^2),x]","\frac{2 x^{5/2} \left(1989 a^2 c+765 b x^4 (2 a d+b c)+1105 a x^2 (a d+2 b c)+585 b^2 d x^6\right)}{9945}","\frac{2}{5} a^2 c x^{5/2}+\frac{2}{13} b x^{13/2} (2 a d+b c)+\frac{2}{9} a x^{9/2} (a d+2 b c)+\frac{2}{17} b^2 d x^{17/2}",1,"(2*x^(5/2)*(1989*a^2*c + 1105*a*(2*b*c + a*d)*x^2 + 765*b*(b*c + 2*a*d)*x^4 + 585*b^2*d*x^6))/9945","A",1
394,1,53,63,0.0289053,"\int \sqrt{x} \left(a+b x^2\right)^2 \left(c+d x^2\right) \, dx","Integrate[Sqrt[x]*(a + b*x^2)^2*(c + d*x^2),x]","\frac{2 x^{3/2} \left(385 a^2 c+105 b x^4 (2 a d+b c)+165 a x^2 (a d+2 b c)+77 b^2 d x^6\right)}{1155}","\frac{2}{3} a^2 c x^{3/2}+\frac{2}{11} b x^{11/2} (2 a d+b c)+\frac{2}{7} a x^{7/2} (a d+2 b c)+\frac{2}{15} b^2 d x^{15/2}",1,"(2*x^(3/2)*(385*a^2*c + 165*a*(2*b*c + a*d)*x^2 + 105*b*(b*c + 2*a*d)*x^4 + 77*b^2*d*x^6))/1155","A",1
395,1,53,61,0.0286389,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)}{\sqrt{x}} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2))/Sqrt[x],x]","\frac{2}{585} \sqrt{x} \left(585 a^2 c+65 b x^4 (2 a d+b c)+117 a x^2 (a d+2 b c)+45 b^2 d x^6\right)","2 a^2 c \sqrt{x}+\frac{2}{9} b x^{9/2} (2 a d+b c)+\frac{2}{5} a x^{5/2} (a d+2 b c)+\frac{2}{13} b^2 d x^{13/2}",1,"(2*Sqrt[x]*(585*a^2*c + 117*a*(2*b*c + a*d)*x^2 + 65*b*(b*c + 2*a*d)*x^4 + 45*b^2*d*x^6))/585","A",1
396,1,60,61,0.0205787,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)}{x^{3/2}} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2))/x^(3/2),x]","\frac{-154 a^2 \left(3 c-d x^2\right)+44 a b x^2 \left(7 c+3 d x^2\right)+6 b^2 x^4 \left(11 c+7 d x^2\right)}{231 \sqrt{x}}","-\frac{2 a^2 c}{\sqrt{x}}+\frac{2}{7} b x^{7/2} (2 a d+b c)+\frac{2}{3} a x^{3/2} (a d+2 b c)+\frac{2}{11} b^2 d x^{11/2}",1,"(-154*a^2*(3*c - d*x^2) + 44*a*b*x^2*(7*c + 3*d*x^2) + 6*b^2*x^4*(11*c + 7*d*x^2))/(231*Sqrt[x])","A",1
397,1,57,61,0.0190388,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)}{x^{5/2}} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2))/x^(5/2),x]","\frac{-30 a^2 \left(c-3 d x^2\right)+36 a b x^2 \left(5 c+d x^2\right)+2 b^2 x^4 \left(9 c+5 d x^2\right)}{45 x^{3/2}}","-\frac{2 a^2 c}{3 x^{3/2}}+\frac{2}{5} b x^{5/2} (2 a d+b c)+2 a \sqrt{x} (a d+2 b c)+\frac{2}{9} b^2 d x^{9/2}",1,"(-30*a^2*(c - 3*d*x^2) + 36*a*b*x^2*(5*c + d*x^2) + 2*b^2*x^4*(9*c + 5*d*x^2))/(45*x^(3/2))","A",1
398,1,57,61,0.0168787,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)}{x^{7/2}} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2))/x^(7/2),x]","\frac{-42 a^2 \left(c+5 d x^2\right)+140 a b x^2 \left(d x^2-3 c\right)+10 b^2 x^4 \left(7 c+3 d x^2\right)}{105 x^{5/2}}","-\frac{2 a^2 c}{5 x^{5/2}}+\frac{2}{3} b x^{3/2} (2 a d+b c)-\frac{2 a (a d+2 b c)}{\sqrt{x}}+\frac{2}{7} b^2 d x^{7/2}",1,"(140*a*b*x^2*(-3*c + d*x^2) + 10*b^2*x^4*(7*c + 3*d*x^2) - 42*a^2*(c + 5*d*x^2))/(105*x^(5/2))","A",1
399,1,97,97,0.0351418,"\int x^{7/2} \left(a+b x^2\right)^2 \left(c+d x^2\right)^2 \, dx","Integrate[x^(7/2)*(a + b*x^2)^2*(c + d*x^2)^2,x]","\frac{2}{17} x^{17/2} \left(a^2 d^2+4 a b c d+b^2 c^2\right)+\frac{2}{9} a^2 c^2 x^{9/2}+\frac{4}{21} b d x^{21/2} (a d+b c)+\frac{4}{13} a c x^{13/2} (a d+b c)+\frac{2}{25} b^2 d^2 x^{25/2}","\frac{2}{17} x^{17/2} \left(a^2 d^2+4 a b c d+b^2 c^2\right)+\frac{2}{9} a^2 c^2 x^{9/2}+\frac{4}{21} b d x^{21/2} (a d+b c)+\frac{4}{13} a c x^{13/2} (a d+b c)+\frac{2}{25} b^2 d^2 x^{25/2}",1,"(2*a^2*c^2*x^(9/2))/9 + (4*a*c*(b*c + a*d)*x^(13/2))/13 + (2*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(17/2))/17 + (4*b*d*(b*c + a*d)*x^(21/2))/21 + (2*b^2*d^2*x^(25/2))/25","A",1
400,1,97,97,0.0344822,"\int x^{5/2} \left(a+b x^2\right)^2 \left(c+d x^2\right)^2 \, dx","Integrate[x^(5/2)*(a + b*x^2)^2*(c + d*x^2)^2,x]","\frac{2}{15} x^{15/2} \left(a^2 d^2+4 a b c d+b^2 c^2\right)+\frac{2}{7} a^2 c^2 x^{7/2}+\frac{4}{19} b d x^{19/2} (a d+b c)+\frac{4}{11} a c x^{11/2} (a d+b c)+\frac{2}{23} b^2 d^2 x^{23/2}","\frac{2}{15} x^{15/2} \left(a^2 d^2+4 a b c d+b^2 c^2\right)+\frac{2}{7} a^2 c^2 x^{7/2}+\frac{4}{19} b d x^{19/2} (a d+b c)+\frac{4}{11} a c x^{11/2} (a d+b c)+\frac{2}{23} b^2 d^2 x^{23/2}",1,"(2*a^2*c^2*x^(7/2))/7 + (4*a*c*(b*c + a*d)*x^(11/2))/11 + (2*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(15/2))/15 + (4*b*d*(b*c + a*d)*x^(19/2))/19 + (2*b^2*d^2*x^(23/2))/23","A",1
401,1,97,97,0.032544,"\int x^{3/2} \left(a+b x^2\right)^2 \left(c+d x^2\right)^2 \, dx","Integrate[x^(3/2)*(a + b*x^2)^2*(c + d*x^2)^2,x]","\frac{2}{13} x^{13/2} \left(a^2 d^2+4 a b c d+b^2 c^2\right)+\frac{2}{5} a^2 c^2 x^{5/2}+\frac{4}{17} b d x^{17/2} (a d+b c)+\frac{4}{9} a c x^{9/2} (a d+b c)+\frac{2}{21} b^2 d^2 x^{21/2}","\frac{2}{13} x^{13/2} \left(a^2 d^2+4 a b c d+b^2 c^2\right)+\frac{2}{5} a^2 c^2 x^{5/2}+\frac{4}{17} b d x^{17/2} (a d+b c)+\frac{4}{9} a c x^{9/2} (a d+b c)+\frac{2}{21} b^2 d^2 x^{21/2}",1,"(2*a^2*c^2*x^(5/2))/5 + (4*a*c*(b*c + a*d)*x^(9/2))/9 + (2*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(13/2))/13 + (4*b*d*(b*c + a*d)*x^(17/2))/17 + (2*b^2*d^2*x^(21/2))/21","A",1
402,1,83,97,0.0348097,"\int \sqrt{x} \left(a+b x^2\right)^2 \left(c+d x^2\right)^2 \, dx","Integrate[Sqrt[x]*(a + b*x^2)^2*(c + d*x^2)^2,x]","\frac{2 x^{3/2} \left(1995 x^4 \left(a^2 d^2+4 a b c d+b^2 c^2\right)+7315 a^2 c^2+2926 b d x^6 (a d+b c)+6270 a c x^2 (a d+b c)+1155 b^2 d^2 x^8\right)}{21945}","\frac{2}{11} x^{11/2} \left(a^2 d^2+4 a b c d+b^2 c^2\right)+\frac{2}{3} a^2 c^2 x^{3/2}+\frac{4}{15} b d x^{15/2} (a d+b c)+\frac{4}{7} a c x^{7/2} (a d+b c)+\frac{2}{19} b^2 d^2 x^{19/2}",1,"(2*x^(3/2)*(7315*a^2*c^2 + 6270*a*c*(b*c + a*d)*x^2 + 1995*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^4 + 2926*b*d*(b*c + a*d)*x^6 + 1155*b^2*d^2*x^8))/21945","A",1
403,1,95,95,0.0314524,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2}{\sqrt{x}} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^2)/Sqrt[x],x]","\frac{2}{9} x^{9/2} \left(a^2 d^2+4 a b c d+b^2 c^2\right)+2 a^2 c^2 \sqrt{x}+\frac{4}{13} b d x^{13/2} (a d+b c)+\frac{4}{5} a c x^{5/2} (a d+b c)+\frac{2}{17} b^2 d^2 x^{17/2}","\frac{2}{9} x^{9/2} \left(a^2 d^2+4 a b c d+b^2 c^2\right)+2 a^2 c^2 \sqrt{x}+\frac{4}{13} b d x^{13/2} (a d+b c)+\frac{4}{5} a c x^{5/2} (a d+b c)+\frac{2}{17} b^2 d^2 x^{17/2}",1,"2*a^2*c^2*Sqrt[x] + (4*a*c*(b*c + a*d)*x^(5/2))/5 + (2*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(9/2))/9 + (4*b*d*(b*c + a*d)*x^(13/2))/13 + (2*b^2*d^2*x^(17/2))/17","A",1
404,1,83,95,0.0326433,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2}{x^{3/2}} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^2)/x^(3/2),x]","\frac{2 \left(165 x^4 \left(a^2 d^2+4 a b c d+b^2 c^2\right)-1155 a^2 c^2+210 b d x^6 (a d+b c)+770 a c x^2 (a d+b c)+77 b^2 d^2 x^8\right)}{1155 \sqrt{x}}","\frac{2}{7} x^{7/2} \left(a^2 d^2+4 a b c d+b^2 c^2\right)-\frac{2 a^2 c^2}{\sqrt{x}}+\frac{4}{11} b d x^{11/2} (a d+b c)+\frac{4}{3} a c x^{3/2} (a d+b c)+\frac{2}{15} b^2 d^2 x^{15/2}",1,"(2*(-1155*a^2*c^2 + 770*a*c*(b*c + a*d)*x^2 + 165*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^4 + 210*b*d*(b*c + a*d)*x^6 + 77*b^2*d^2*x^8))/(1155*Sqrt[x])","A",1
405,1,83,95,0.0347602,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2}{x^{5/2}} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^2)/x^(5/2),x]","\frac{2 \left(117 x^4 \left(a^2 d^2+4 a b c d+b^2 c^2\right)-195 a^2 c^2+130 b d x^6 (a d+b c)+1170 a c x^2 (a d+b c)+45 b^2 d^2 x^8\right)}{585 x^{3/2}}","\frac{2}{5} x^{5/2} \left(a^2 d^2+4 a b c d+b^2 c^2\right)-\frac{2 a^2 c^2}{3 x^{3/2}}+\frac{4}{9} b d x^{9/2} (a d+b c)+4 a c \sqrt{x} (a d+b c)+\frac{2}{13} b^2 d^2 x^{13/2}",1,"(2*(-195*a^2*c^2 + 1170*a*c*(b*c + a*d)*x^2 + 117*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^4 + 130*b*d*(b*c + a*d)*x^6 + 45*b^2*d^2*x^8))/(585*x^(3/2))","A",1
406,1,83,95,0.0386146,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2}{x^{7/2}} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^2)/x^(7/2),x]","\frac{2 \left(385 x^4 \left(a^2 d^2+4 a b c d+b^2 c^2\right)-231 a^2 c^2+330 b d x^6 (a d+b c)-2310 a c x^2 (a d+b c)+105 b^2 d^2 x^8\right)}{1155 x^{5/2}}","\frac{2}{3} x^{3/2} \left(a^2 d^2+4 a b c d+b^2 c^2\right)-\frac{2 a^2 c^2}{5 x^{5/2}}+\frac{4}{7} b d x^{7/2} (a d+b c)-\frac{4 a c (a d+b c)}{\sqrt{x}}+\frac{2}{11} b^2 d^2 x^{11/2}",1,"(2*(-231*a^2*c^2 - 2310*a*c*(b*c + a*d)*x^2 + 385*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^4 + 330*b*d*(b*c + a*d)*x^6 + 105*b^2*d^2*x^8))/(1155*x^(5/2))","A",1
407,1,139,139,0.0415035,"\int x^{7/2} \left(a+b x^2\right)^2 \left(c+d x^2\right)^3 \, dx","Integrate[x^(7/2)*(a + b*x^2)^2*(c + d*x^2)^3,x]","\frac{2}{21} d x^{21/2} \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{2}{17} c x^{17/2} \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+\frac{2}{9} a^2 c^3 x^{9/2}+\frac{2}{13} a c^2 x^{13/2} (3 a d+2 b c)+\frac{2}{25} b d^2 x^{25/2} (2 a d+3 b c)+\frac{2}{29} b^2 d^3 x^{29/2}","\frac{2}{21} d x^{21/2} \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{2}{17} c x^{17/2} \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+\frac{2}{9} a^2 c^3 x^{9/2}+\frac{2}{13} a c^2 x^{13/2} (3 a d+2 b c)+\frac{2}{25} b d^2 x^{25/2} (2 a d+3 b c)+\frac{2}{29} b^2 d^3 x^{29/2}",1,"(2*a^2*c^3*x^(9/2))/9 + (2*a*c^2*(2*b*c + 3*a*d)*x^(13/2))/13 + (2*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(17/2))/17 + (2*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(21/2))/21 + (2*b*d^2*(3*b*c + 2*a*d)*x^(25/2))/25 + (2*b^2*d^3*x^(29/2))/29","A",1
408,1,139,139,0.0406597,"\int x^{5/2} \left(a+b x^2\right)^2 \left(c+d x^2\right)^3 \, dx","Integrate[x^(5/2)*(a + b*x^2)^2*(c + d*x^2)^3,x]","\frac{2}{19} d x^{19/2} \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{2}{15} c x^{15/2} \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+\frac{2}{7} a^2 c^3 x^{7/2}+\frac{2}{11} a c^2 x^{11/2} (3 a d+2 b c)+\frac{2}{23} b d^2 x^{23/2} (2 a d+3 b c)+\frac{2}{27} b^2 d^3 x^{27/2}","\frac{2}{19} d x^{19/2} \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{2}{15} c x^{15/2} \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+\frac{2}{7} a^2 c^3 x^{7/2}+\frac{2}{11} a c^2 x^{11/2} (3 a d+2 b c)+\frac{2}{23} b d^2 x^{23/2} (2 a d+3 b c)+\frac{2}{27} b^2 d^3 x^{27/2}",1,"(2*a^2*c^3*x^(7/2))/7 + (2*a*c^2*(2*b*c + 3*a*d)*x^(11/2))/11 + (2*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(15/2))/15 + (2*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(19/2))/19 + (2*b*d^2*(3*b*c + 2*a*d)*x^(23/2))/23 + (2*b^2*d^3*x^(27/2))/27","A",1
409,1,139,139,0.03785,"\int x^{3/2} \left(a+b x^2\right)^2 \left(c+d x^2\right)^3 \, dx","Integrate[x^(3/2)*(a + b*x^2)^2*(c + d*x^2)^3,x]","\frac{2}{17} d x^{17/2} \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{2}{13} c x^{13/2} \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+\frac{2}{5} a^2 c^3 x^{5/2}+\frac{2}{9} a c^2 x^{9/2} (3 a d+2 b c)+\frac{2}{21} b d^2 x^{21/2} (2 a d+3 b c)+\frac{2}{25} b^2 d^3 x^{25/2}","\frac{2}{17} d x^{17/2} \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{2}{13} c x^{13/2} \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+\frac{2}{5} a^2 c^3 x^{5/2}+\frac{2}{9} a c^2 x^{9/2} (3 a d+2 b c)+\frac{2}{21} b d^2 x^{21/2} (2 a d+3 b c)+\frac{2}{25} b^2 d^3 x^{25/2}",1,"(2*a^2*c^3*x^(5/2))/5 + (2*a*c^2*(2*b*c + 3*a*d)*x^(9/2))/9 + (2*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(13/2))/13 + (2*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(17/2))/17 + (2*b*d^2*(3*b*c + 2*a*d)*x^(21/2))/21 + (2*b^2*d^3*x^(25/2))/25","A",1
410,1,139,139,0.0383471,"\int \sqrt{x} \left(a+b x^2\right)^2 \left(c+d x^2\right)^3 \, dx","Integrate[Sqrt[x]*(a + b*x^2)^2*(c + d*x^2)^3,x]","\frac{2}{15} d x^{15/2} \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{2}{11} c x^{11/2} \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+\frac{2}{3} a^2 c^3 x^{3/2}+\frac{2}{7} a c^2 x^{7/2} (3 a d+2 b c)+\frac{2}{19} b d^2 x^{19/2} (2 a d+3 b c)+\frac{2}{23} b^2 d^3 x^{23/2}","\frac{2}{15} d x^{15/2} \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{2}{11} c x^{11/2} \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+\frac{2}{3} a^2 c^3 x^{3/2}+\frac{2}{7} a c^2 x^{7/2} (3 a d+2 b c)+\frac{2}{19} b d^2 x^{19/2} (2 a d+3 b c)+\frac{2}{23} b^2 d^3 x^{23/2}",1,"(2*a^2*c^3*x^(3/2))/3 + (2*a*c^2*(2*b*c + 3*a*d)*x^(7/2))/7 + (2*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(11/2))/11 + (2*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(15/2))/15 + (2*b*d^2*(3*b*c + 2*a*d)*x^(19/2))/19 + (2*b^2*d^3*x^(23/2))/23","A",1
411,1,137,137,0.0407975,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3}{\sqrt{x}} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^3)/Sqrt[x],x]","\frac{2}{13} d x^{13/2} \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{2}{9} c x^{9/2} \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+2 a^2 c^3 \sqrt{x}+\frac{2}{5} a c^2 x^{5/2} (3 a d+2 b c)+\frac{2}{17} b d^2 x^{17/2} (2 a d+3 b c)+\frac{2}{21} b^2 d^3 x^{21/2}","\frac{2}{13} d x^{13/2} \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{2}{9} c x^{9/2} \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+2 a^2 c^3 \sqrt{x}+\frac{2}{5} a c^2 x^{5/2} (3 a d+2 b c)+\frac{2}{17} b d^2 x^{17/2} (2 a d+3 b c)+\frac{2}{21} b^2 d^3 x^{21/2}",1,"2*a^2*c^3*Sqrt[x] + (2*a*c^2*(2*b*c + 3*a*d)*x^(5/2))/5 + (2*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(9/2))/9 + (2*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(13/2))/13 + (2*b*d^2*(3*b*c + 2*a*d)*x^(17/2))/17 + (2*b^2*d^3*x^(21/2))/21","A",1
412,1,137,137,0.0547837,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3}{x^{3/2}} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^3)/x^(3/2),x]","\frac{2}{11} d x^{11/2} \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{2}{7} c x^{7/2} \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)-\frac{2 a^2 c^3}{\sqrt{x}}+\frac{2}{3} a c^2 x^{3/2} (3 a d+2 b c)+\frac{2}{15} b d^2 x^{15/2} (2 a d+3 b c)+\frac{2}{19} b^2 d^3 x^{19/2}","\frac{2}{11} d x^{11/2} \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{2}{7} c x^{7/2} \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)-\frac{2 a^2 c^3}{\sqrt{x}}+\frac{2}{3} a c^2 x^{3/2} (3 a d+2 b c)+\frac{2}{15} b d^2 x^{15/2} (2 a d+3 b c)+\frac{2}{19} b^2 d^3 x^{19/2}",1,"(-2*a^2*c^3)/Sqrt[x] + (2*a*c^2*(2*b*c + 3*a*d)*x^(3/2))/3 + (2*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(7/2))/7 + (2*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(11/2))/11 + (2*b*d^2*(3*b*c + 2*a*d)*x^(15/2))/15 + (2*b^2*d^3*x^(19/2))/19","A",1
413,1,137,137,0.0544937,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3}{x^{5/2}} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^3)/x^(5/2),x]","\frac{2}{9} d x^{9/2} \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{2}{5} c x^{5/2} \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)-\frac{2 a^2 c^3}{3 x^{3/2}}+2 a c^2 \sqrt{x} (3 a d+2 b c)+\frac{2}{13} b d^2 x^{13/2} (2 a d+3 b c)+\frac{2}{17} b^2 d^3 x^{17/2}","\frac{2}{9} d x^{9/2} \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{2}{5} c x^{5/2} \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)-\frac{2 a^2 c^3}{3 x^{3/2}}+2 a c^2 \sqrt{x} (3 a d+2 b c)+\frac{2}{13} b d^2 x^{13/2} (2 a d+3 b c)+\frac{2}{17} b^2 d^3 x^{17/2}",1,"(-2*a^2*c^3)/(3*x^(3/2)) + 2*a*c^2*(2*b*c + 3*a*d)*Sqrt[x] + (2*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(5/2))/5 + (2*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(9/2))/9 + (2*b*d^2*(3*b*c + 2*a*d)*x^(13/2))/13 + (2*b^2*d^3*x^(17/2))/17","A",1
414,1,121,137,0.0468262,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3}{x^{7/2}} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^3)/x^(7/2),x]","\frac{2 \left(165 d x^6 \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+385 c x^4 \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)-231 a^2 c^3-1155 a c^2 x^2 (3 a d+2 b c)+105 b d^2 x^8 (2 a d+3 b c)+77 b^2 d^3 x^{10}\right)}{1155 x^{5/2}}","\frac{2}{7} d x^{7/2} \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{2}{3} c x^{3/2} \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)-\frac{2 a^2 c^3}{5 x^{5/2}}-\frac{2 a c^2 (3 a d+2 b c)}{\sqrt{x}}+\frac{2}{11} b d^2 x^{11/2} (2 a d+3 b c)+\frac{2}{15} b^2 d^3 x^{15/2}",1,"(2*(-231*a^2*c^3 - 1155*a*c^2*(2*b*c + 3*a*d)*x^2 + 385*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^4 + 165*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^6 + 105*b*d^2*(3*b*c + 2*a*d)*x^8 + 77*b^2*d^3*x^10))/(1155*x^(5/2))","A",1
415,1,299,311,0.1395259,"\int \frac{x^{7/2} \left(a+b x^2\right)^2}{c+d x^2} \, dx","Integrate[(x^(7/2)*(a + b*x^2)^2)/(c + d*x^2),x]","\frac{-585 \sqrt{2} c^{5/4} (b c-a d)^2 \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)+585 \sqrt{2} c^{5/4} (b c-a d)^2 \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)-1170 \sqrt{2} c^{5/4} (b c-a d)^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)+1170 \sqrt{2} c^{5/4} (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)-520 b d^{9/4} x^{9/2} (b c-2 a d)+936 d^{5/4} x^{5/2} (b c-a d)^2-4680 c \sqrt[4]{d} \sqrt{x} (b c-a d)^2+360 b^2 d^{13/4} x^{13/2}}{2340 d^{17/4}}","-\frac{c^{5/4} (b c-a d)^2 \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} d^{17/4}}+\frac{c^{5/4} (b c-a d)^2 \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} d^{17/4}}-\frac{c^{5/4} (b c-a d)^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{\sqrt{2} d^{17/4}}+\frac{c^{5/4} (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{\sqrt{2} d^{17/4}}-\frac{2 c \sqrt{x} (b c-a d)^2}{d^4}+\frac{2 x^{5/2} (b c-a d)^2}{5 d^3}-\frac{2 b x^{9/2} (b c-2 a d)}{9 d^2}+\frac{2 b^2 x^{13/2}}{13 d}",1,"(-4680*c*d^(1/4)*(b*c - a*d)^2*Sqrt[x] + 936*d^(5/4)*(b*c - a*d)^2*x^(5/2) - 520*b*d^(9/4)*(b*c - 2*a*d)*x^(9/2) + 360*b^2*d^(13/4)*x^(13/2) - 1170*Sqrt[2]*c^(5/4)*(b*c - a*d)^2*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + 1170*Sqrt[2]*c^(5/4)*(b*c - a*d)^2*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] - 585*Sqrt[2]*c^(5/4)*(b*c - a*d)^2*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x] + 585*Sqrt[2]*c^(5/4)*(b*c - a*d)^2*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(2340*d^(17/4))","A",1
416,1,276,290,0.1112988,"\int \frac{x^{5/2} \left(a+b x^2\right)^2}{c+d x^2} \, dx","Integrate[(x^(5/2)*(a + b*x^2)^2)/(c + d*x^2),x]","\frac{-231 \sqrt{2} c^{3/4} (b c-a d)^2 \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)+231 \sqrt{2} c^{3/4} (b c-a d)^2 \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)+462 \sqrt{2} c^{3/4} (b c-a d)^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)-462 \sqrt{2} c^{3/4} (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)-264 b d^{7/4} x^{7/2} (b c-2 a d)+616 d^{3/4} x^{3/2} (b c-a d)^2+168 b^2 d^{11/4} x^{11/2}}{924 d^{15/4}}","-\frac{c^{3/4} (b c-a d)^2 \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} d^{15/4}}+\frac{c^{3/4} (b c-a d)^2 \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} d^{15/4}}+\frac{c^{3/4} (b c-a d)^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{\sqrt{2} d^{15/4}}-\frac{c^{3/4} (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{\sqrt{2} d^{15/4}}+\frac{2 x^{3/2} (b c-a d)^2}{3 d^3}-\frac{2 b x^{7/2} (b c-2 a d)}{7 d^2}+\frac{2 b^2 x^{11/2}}{11 d}",1,"(616*d^(3/4)*(b*c - a*d)^2*x^(3/2) - 264*b*d^(7/4)*(b*c - 2*a*d)*x^(7/2) + 168*b^2*d^(11/4)*x^(11/2) + 462*Sqrt[2]*c^(3/4)*(b*c - a*d)^2*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] - 462*Sqrt[2]*c^(3/4)*(b*c - a*d)^2*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] - 231*Sqrt[2]*c^(3/4)*(b*c - a*d)^2*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x] + 231*Sqrt[2]*c^(3/4)*(b*c - a*d)^2*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(924*d^(15/4))","A",1
417,1,276,288,0.10902,"\int \frac{x^{3/2} \left(a+b x^2\right)^2}{c+d x^2} \, dx","Integrate[(x^(3/2)*(a + b*x^2)^2)/(c + d*x^2),x]","\frac{-72 b d^{5/4} x^{5/2} (b c-2 a d)+360 \sqrt[4]{d} \sqrt{x} (b c-a d)^2+45 \sqrt{2} \sqrt[4]{c} (b c-a d)^2 \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)-45 \sqrt{2} \sqrt[4]{c} (b c-a d)^2 \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)+90 \sqrt{2} \sqrt[4]{c} (b c-a d)^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)-90 \sqrt{2} \sqrt[4]{c} (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)+40 b^2 d^{9/4} x^{9/2}}{180 d^{13/4}}","\frac{\sqrt[4]{c} (b c-a d)^2 \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} d^{13/4}}-\frac{\sqrt[4]{c} (b c-a d)^2 \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} d^{13/4}}+\frac{\sqrt[4]{c} (b c-a d)^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{\sqrt{2} d^{13/4}}-\frac{\sqrt[4]{c} (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{\sqrt{2} d^{13/4}}+\frac{2 \sqrt{x} (b c-a d)^2}{d^3}-\frac{2 b x^{5/2} (b c-2 a d)}{5 d^2}+\frac{2 b^2 x^{9/2}}{9 d}",1,"(360*d^(1/4)*(b*c - a*d)^2*Sqrt[x] - 72*b*d^(5/4)*(b*c - 2*a*d)*x^(5/2) + 40*b^2*d^(9/4)*x^(9/2) + 90*Sqrt[2]*c^(1/4)*(b*c - a*d)^2*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] - 90*Sqrt[2]*c^(1/4)*(b*c - a*d)^2*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + 45*Sqrt[2]*c^(1/4)*(b*c - a*d)^2*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x] - 45*Sqrt[2]*c^(1/4)*(b*c - a*d)^2*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(180*d^(13/4))","A",1
418,1,249,268,0.1194737,"\int \frac{\sqrt{x} \left(a+b x^2\right)^2}{c+d x^2} \, dx","Integrate[(Sqrt[x]*(a + b*x^2)^2)/(c + d*x^2),x]","\frac{-56 b \sqrt[4]{c} d^{3/4} x^{3/2} (b c-2 a d)+21 \sqrt{2} (b c-a d)^2 \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)-21 \sqrt{2} (b c-a d)^2 \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)-42 \sqrt{2} (b c-a d)^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)+42 \sqrt{2} (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)+24 b^2 \sqrt[4]{c} d^{7/4} x^{7/2}}{84 \sqrt[4]{c} d^{11/4}}","\frac{(b c-a d)^2 \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} \sqrt[4]{c} d^{11/4}}-\frac{(b c-a d)^2 \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} \sqrt[4]{c} d^{11/4}}-\frac{(b c-a d)^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{\sqrt{2} \sqrt[4]{c} d^{11/4}}+\frac{(b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{\sqrt{2} \sqrt[4]{c} d^{11/4}}-\frac{2 b x^{3/2} (b c-2 a d)}{3 d^2}+\frac{2 b^2 x^{7/2}}{7 d}",1,"(-56*b*c^(1/4)*d^(3/4)*(b*c - 2*a*d)*x^(3/2) + 24*b^2*c^(1/4)*d^(7/4)*x^(7/2) - 42*Sqrt[2]*(b*c - a*d)^2*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + 42*Sqrt[2]*(b*c - a*d)^2*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + 21*Sqrt[2]*(b*c - a*d)^2*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x] - 21*Sqrt[2]*(b*c - a*d)^2*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(84*c^(1/4)*d^(11/4))","A",1
419,1,249,266,0.1158984,"\int \frac{\left(a+b x^2\right)^2}{\sqrt{x} \left(c+d x^2\right)} \, dx","Integrate[(a + b*x^2)^2/(Sqrt[x]*(c + d*x^2)),x]","\frac{-40 b c^{3/4} \sqrt[4]{d} \sqrt{x} (b c-2 a d)-5 \sqrt{2} (b c-a d)^2 \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)+5 \sqrt{2} (b c-a d)^2 \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)-10 \sqrt{2} (b c-a d)^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)+10 \sqrt{2} (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)+8 b^2 c^{3/4} d^{5/4} x^{5/2}}{20 c^{3/4} d^{9/4}}","-\frac{(b c-a d)^2 \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} c^{3/4} d^{9/4}}+\frac{(b c-a d)^2 \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} c^{3/4} d^{9/4}}-\frac{(b c-a d)^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{\sqrt{2} c^{3/4} d^{9/4}}+\frac{(b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{\sqrt{2} c^{3/4} d^{9/4}}-\frac{2 b \sqrt{x} (b c-2 a d)}{d^2}+\frac{2 b^2 x^{5/2}}{5 d}",1,"(-40*b*c^(3/4)*d^(1/4)*(b*c - 2*a*d)*Sqrt[x] + 8*b^2*c^(3/4)*d^(5/4)*x^(5/2) - 10*Sqrt[2]*(b*c - a*d)^2*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + 10*Sqrt[2]*(b*c - a*d)^2*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] - 5*Sqrt[2]*(b*c - a*d)^2*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x] + 5*Sqrt[2]*(b*c - a*d)^2*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(20*c^(3/4)*d^(9/4))","A",1
420,1,261,260,0.1324179,"\int \frac{\left(a+b x^2\right)^2}{x^{3/2} \left(c+d x^2\right)} \, dx","Integrate[(a + b*x^2)^2/(x^(3/2)*(c + d*x^2)),x]","\frac{-24 a^2 \sqrt[4]{c} d^{7/4}-3 \sqrt{2} \sqrt{x} (b c-a d)^2 \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)+3 \sqrt{2} \sqrt{x} (b c-a d)^2 \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)+6 \sqrt{2} \sqrt{x} (b c-a d)^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)-6 \sqrt{2} \sqrt{x} (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)+8 b^2 c^{5/4} d^{3/4} x^2}{12 c^{5/4} d^{7/4} \sqrt{x}}","-\frac{2 a^2}{c \sqrt{x}}-\frac{(b c-a d)^2 \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} c^{5/4} d^{7/4}}+\frac{(b c-a d)^2 \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} c^{5/4} d^{7/4}}+\frac{(b c-a d)^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{\sqrt{2} c^{5/4} d^{7/4}}-\frac{(b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{\sqrt{2} c^{5/4} d^{7/4}}+\frac{2 b^2 x^{3/2}}{3 d}",1,"(-24*a^2*c^(1/4)*d^(7/4) + 8*b^2*c^(5/4)*d^(3/4)*x^2 + 6*Sqrt[2]*(b*c - a*d)^2*Sqrt[x]*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] - 6*Sqrt[2]*(b*c - a*d)^2*Sqrt[x]*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] - 3*Sqrt[2]*(b*c - a*d)^2*Sqrt[x]*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x] + 3*Sqrt[2]*(b*c - a*d)^2*Sqrt[x]*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(12*c^(5/4)*d^(7/4)*Sqrt[x])","A",1
421,1,261,260,0.1269217,"\int \frac{\left(a+b x^2\right)^2}{x^{5/2} \left(c+d x^2\right)} \, dx","Integrate[(a + b*x^2)^2/(x^(5/2)*(c + d*x^2)),x]","\frac{-8 a^2 c^{3/4} d^{5/4}+3 \sqrt{2} x^{3/2} (b c-a d)^2 \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)-3 \sqrt{2} x^{3/2} (b c-a d)^2 \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)+6 \sqrt{2} x^{3/2} (b c-a d)^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)-6 \sqrt{2} x^{3/2} (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)+24 b^2 c^{7/4} \sqrt[4]{d} x^2}{12 c^{7/4} d^{5/4} x^{3/2}}","-\frac{2 a^2}{3 c x^{3/2}}+\frac{(b c-a d)^2 \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} c^{7/4} d^{5/4}}-\frac{(b c-a d)^2 \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} c^{7/4} d^{5/4}}+\frac{(b c-a d)^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{\sqrt{2} c^{7/4} d^{5/4}}-\frac{(b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{\sqrt{2} c^{7/4} d^{5/4}}+\frac{2 b^2 \sqrt{x}}{d}",1,"(-8*a^2*c^(3/4)*d^(5/4) + 24*b^2*c^(7/4)*d^(1/4)*x^2 + 6*Sqrt[2]*(b*c - a*d)^2*x^(3/2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] - 6*Sqrt[2]*(b*c - a*d)^2*x^(3/2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + 3*Sqrt[2]*(b*c - a*d)^2*x^(3/2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x] - 3*Sqrt[2]*(b*c - a*d)^2*x^(3/2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(12*c^(7/4)*d^(5/4)*x^(3/2))","A",1
422,1,254,267,0.1209238,"\int \frac{\left(a+b x^2\right)^2}{x^{7/2} \left(c+d x^2\right)} \, dx","Integrate[(a + b*x^2)^2/(x^(7/2)*(c + d*x^2)),x]","\frac{-\frac{8 a^2 c^{5/4}}{x^{5/2}}+\frac{5 \sqrt{2} (b c-a d)^2 \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{d^{3/4}}-\frac{5 \sqrt{2} (b c-a d)^2 \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{d^{3/4}}-\frac{10 \sqrt{2} (b c-a d)^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{d^{3/4}}+\frac{10 \sqrt{2} (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{d^{3/4}}+\frac{40 a \sqrt[4]{c} (a d-2 b c)}{\sqrt{x}}}{20 c^{9/4}}","-\frac{2 a^2}{5 c x^{5/2}}+\frac{(b c-a d)^2 \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} c^{9/4} d^{3/4}}-\frac{(b c-a d)^2 \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} c^{9/4} d^{3/4}}-\frac{(b c-a d)^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{\sqrt{2} c^{9/4} d^{3/4}}+\frac{(b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{\sqrt{2} c^{9/4} d^{3/4}}-\frac{2 a (2 b c-a d)}{c^2 \sqrt{x}}",1,"((-8*a^2*c^(5/4))/x^(5/2) + (40*a*c^(1/4)*(-2*b*c + a*d))/Sqrt[x] - (10*Sqrt[2]*(b*c - a*d)^2*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/d^(3/4) + (10*Sqrt[2]*(b*c - a*d)^2*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/d^(3/4) + (5*Sqrt[2]*(b*c - a*d)^2*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/d^(3/4) - (5*Sqrt[2]*(b*c - a*d)^2*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/d^(3/4))/(20*c^(9/4))","A",1
423,1,254,269,0.1266194,"\int \frac{\left(a+b x^2\right)^2}{x^{9/2} \left(c+d x^2\right)} \, dx","Integrate[(a + b*x^2)^2/(x^(9/2)*(c + d*x^2)),x]","\frac{-\frac{24 a^2 c^{7/4}}{x^{7/2}}+\frac{56 a c^{3/4} (a d-2 b c)}{x^{3/2}}-\frac{21 \sqrt{2} (b c-a d)^2 \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{\sqrt[4]{d}}+\frac{21 \sqrt{2} (b c-a d)^2 \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{\sqrt[4]{d}}-\frac{42 \sqrt{2} (b c-a d)^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{\sqrt[4]{d}}+\frac{42 \sqrt{2} (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{\sqrt[4]{d}}}{84 c^{11/4}}","-\frac{2 a^2}{7 c x^{7/2}}-\frac{(b c-a d)^2 \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} c^{11/4} \sqrt[4]{d}}+\frac{(b c-a d)^2 \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} c^{11/4} \sqrt[4]{d}}-\frac{(b c-a d)^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{\sqrt{2} c^{11/4} \sqrt[4]{d}}+\frac{(b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{\sqrt{2} c^{11/4} \sqrt[4]{d}}-\frac{2 a (2 b c-a d)}{3 c^2 x^{3/2}}",1,"((-24*a^2*c^(7/4))/x^(7/2) + (56*a*c^(3/4)*(-2*b*c + a*d))/x^(3/2) - (42*Sqrt[2]*(b*c - a*d)^2*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/d^(1/4) + (42*Sqrt[2]*(b*c - a*d)^2*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/d^(1/4) - (21*Sqrt[2]*(b*c - a*d)^2*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/d^(1/4) + (21*Sqrt[2]*(b*c - a*d)^2*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/d^(1/4))/(84*c^(11/4))","A",1
424,1,101,288,0.2098402,"\int \frac{\left(c+d x^2\right)^2}{x^{11/2} \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2)^2/(x^(11/2)*(a + b*x^2)),x]","-\frac{2 \left(a \left(a^2 \left(5 c^2+18 c d x^2+45 d^2 x^4\right)-9 a b c x^2 \left(c+10 d x^2\right)+45 b^2 c^2 x^4\right)+15 b x^6 (b c-a d)^2 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\frac{b x^2}{a}\right)\right)}{45 a^4 x^{9/2}}","-\frac{\sqrt[4]{b} (b c-a d)^2 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{13/4}}+\frac{\sqrt[4]{b} (b c-a d)^2 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{13/4}}+\frac{\sqrt[4]{b} (b c-a d)^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} a^{13/4}}-\frac{\sqrt[4]{b} (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} a^{13/4}}-\frac{2 (b c-a d)^2}{a^3 \sqrt{x}}+\frac{2 c (b c-2 a d)}{5 a^2 x^{5/2}}-\frac{2 c^2}{9 a x^{9/2}}",1,"(-2*(a*(45*b^2*c^2*x^4 - 9*a*b*c*x^2*(c + 10*d*x^2) + a^2*(5*c^2 + 18*c*d*x^2 + 45*d^2*x^4)) + 15*b*(b*c - a*d)^2*x^6*Hypergeometric2F1[3/4, 1, 7/4, -((b*x^2)/a)]))/(45*a^4*x^(9/2))","C",1
425,1,372,375,0.3557028,"\int \frac{x^{7/2} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^2} \, dx","Integrate[(x^(7/2)*(a + b*x^2)^2)/(c + d*x^2)^2,x]","\frac{1440 \sqrt[4]{d} \sqrt{x} \left(a^2 d^2-4 a b c d+3 b^2 c^2\right)+45 \sqrt{2} \sqrt[4]{c} \left(5 a^2 d^2-18 a b c d+13 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)-45 \sqrt{2} \sqrt[4]{c} \left(5 a^2 d^2-18 a b c d+13 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)+90 \sqrt{2} \sqrt[4]{c} \left(5 a^2 d^2-18 a b c d+13 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)-90 \sqrt{2} \sqrt[4]{c} \left(5 a^2 d^2-18 a b c d+13 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)-576 b d^{5/4} x^{5/2} (b c-a d)+\frac{360 c \sqrt[4]{d} \sqrt{x} (b c-a d)^2}{c+d x^2}+160 b^2 d^{9/4} x^{9/2}}{720 d^{17/4}}","\frac{\sqrt[4]{c} (13 b c-5 a d) (b c-a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} d^{17/4}}-\frac{\sqrt[4]{c} (13 b c-5 a d) (b c-a d) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} d^{17/4}}+\frac{\sqrt[4]{c} (13 b c-5 a d) (b c-a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{4 \sqrt{2} d^{17/4}}-\frac{\sqrt[4]{c} (13 b c-5 a d) (b c-a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{4 \sqrt{2} d^{17/4}}+\frac{\sqrt{x} (13 b c-5 a d) (b c-a d)}{2 d^4}-\frac{x^{5/2} (13 b c-5 a d) (b c-a d)}{10 c d^3}+\frac{x^{9/2} (b c-a d)^2}{2 c d^2 \left(c+d x^2\right)}+\frac{2 b^2 x^{9/2}}{9 d^2}",1,"(1440*d^(1/4)*(3*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*Sqrt[x] - 576*b*d^(5/4)*(b*c - a*d)*x^(5/2) + 160*b^2*d^(9/4)*x^(9/2) + (360*c*d^(1/4)*(b*c - a*d)^2*Sqrt[x])/(c + d*x^2) + 90*Sqrt[2]*c^(1/4)*(13*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] - 90*Sqrt[2]*c^(1/4)*(13*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + 45*Sqrt[2]*c^(1/4)*(13*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x] - 45*Sqrt[2]*c^(1/4)*(13*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(720*d^(17/4))","A",1
426,1,337,346,0.1882702,"\int \frac{x^{5/2} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^2} \, dx","Integrate[(x^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^2,x]","\frac{\frac{21 \sqrt{2} \left(3 a^2 d^2-14 a b c d+11 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{\sqrt[4]{c}}-\frac{21 \sqrt{2} \left(3 a^2 d^2-14 a b c d+11 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{\sqrt[4]{c}}-\frac{42 \sqrt{2} \left(3 a^2 d^2-14 a b c d+11 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{\sqrt[4]{c}}+\frac{42 \sqrt{2} \left(3 a^2 d^2-14 a b c d+11 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{\sqrt[4]{c}}-448 b d^{3/4} x^{3/2} (b c-a d)-\frac{168 d^{3/4} x^{3/2} (b c-a d)^2}{c+d x^2}+96 b^2 d^{7/4} x^{7/2}}{336 d^{15/4}}","\frac{(11 b c-3 a d) (b c-a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} \sqrt[4]{c} d^{15/4}}-\frac{(11 b c-3 a d) (b c-a d) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} \sqrt[4]{c} d^{15/4}}-\frac{(11 b c-3 a d) (b c-a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{4 \sqrt{2} \sqrt[4]{c} d^{15/4}}+\frac{(11 b c-3 a d) (b c-a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{4 \sqrt{2} \sqrt[4]{c} d^{15/4}}-\frac{x^{3/2} (11 b c-3 a d) (b c-a d)}{6 c d^3}+\frac{x^{7/2} (b c-a d)^2}{2 c d^2 \left(c+d x^2\right)}+\frac{2 b^2 x^{7/2}}{7 d^2}",1,"(-448*b*d^(3/4)*(b*c - a*d)*x^(3/2) + 96*b^2*d^(7/4)*x^(7/2) - (168*d^(3/4)*(b*c - a*d)^2*x^(3/2))/(c + d*x^2) - (42*Sqrt[2]*(11*b^2*c^2 - 14*a*b*c*d + 3*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(1/4) + (42*Sqrt[2]*(11*b^2*c^2 - 14*a*b*c*d + 3*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(1/4) + (21*Sqrt[2]*(11*b^2*c^2 - 14*a*b*c*d + 3*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(1/4) - (21*Sqrt[2]*(11*b^2*c^2 - 14*a*b*c*d + 3*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(1/4))/(336*d^(15/4))","A",1
427,1,333,346,0.1788175,"\int \frac{x^{3/2} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^2} \, dx","Integrate[(x^(3/2)*(a + b*x^2)^2)/(c + d*x^2)^2,x]","\frac{-\frac{5 \sqrt{2} \left(a^2 d^2-10 a b c d+9 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{3/4}}+\frac{5 \sqrt{2} \left(a^2 d^2-10 a b c d+9 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{3/4}}-\frac{10 \sqrt{2} \left(a^2 d^2-10 a b c d+9 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{c^{3/4}}+\frac{10 \sqrt{2} \left(a^2 d^2-10 a b c d+9 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{c^{3/4}}-\frac{40 \sqrt[4]{d} \sqrt{x} (b c-a d)^2}{c+d x^2}-320 b \sqrt[4]{d} \sqrt{x} (b c-a d)+32 b^2 d^{5/4} x^{5/2}}{80 d^{13/4}}","-\frac{(b c-a d) (9 b c-a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{3/4} d^{13/4}}+\frac{(b c-a d) (9 b c-a d) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{3/4} d^{13/4}}-\frac{(b c-a d) (9 b c-a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{4 \sqrt{2} c^{3/4} d^{13/4}}+\frac{(b c-a d) (9 b c-a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{4 \sqrt{2} c^{3/4} d^{13/4}}-\frac{\sqrt{x} (b c-a d) (9 b c-a d)}{2 c d^3}+\frac{x^{5/2} (b c-a d)^2}{2 c d^2 \left(c+d x^2\right)}+\frac{2 b^2 x^{5/2}}{5 d^2}",1,"(-320*b*d^(1/4)*(b*c - a*d)*Sqrt[x] + 32*b^2*d^(5/4)*x^(5/2) - (40*d^(1/4)*(b*c - a*d)^2*Sqrt[x])/(c + d*x^2) - (10*Sqrt[2]*(9*b^2*c^2 - 10*a*b*c*d + a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(3/4) + (10*Sqrt[2]*(9*b^2*c^2 - 10*a*b*c*d + a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(3/4) - (5*Sqrt[2]*(9*b^2*c^2 - 10*a*b*c*d + a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(3/4) + (5*Sqrt[2]*(9*b^2*c^2 - 10*a*b*c*d + a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(3/4))/(80*d^(13/4))","A",1
428,1,319,310,0.181032,"\int \frac{\sqrt{x} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^2} \, dx","Integrate[(Sqrt[x]*(a + b*x^2)^2)/(c + d*x^2)^2,x]","\frac{-\frac{3 \sqrt{2} \left(-a^2 d^2-6 a b c d+7 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{5/4}}+\frac{3 \sqrt{2} \left(-a^2 d^2-6 a b c d+7 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{5/4}}+\frac{6 \sqrt{2} \left(-a^2 d^2-6 a b c d+7 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{c^{5/4}}-\frac{6 \sqrt{2} \left(-a^2 d^2-6 a b c d+7 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{c^{5/4}}+\frac{24 d^{3/4} x^{3/2} (b c-a d)^2}{c \left(c+d x^2\right)}+32 b^2 d^{3/4} x^{3/2}}{48 d^{11/4}}","-\frac{(b c-a d) (a d+7 b c) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{5/4} d^{11/4}}+\frac{(b c-a d) (a d+7 b c) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{5/4} d^{11/4}}+\frac{(b c-a d) (a d+7 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{4 \sqrt{2} c^{5/4} d^{11/4}}-\frac{(b c-a d) (a d+7 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{4 \sqrt{2} c^{5/4} d^{11/4}}+\frac{x^{3/2} (b c-a d)^2}{2 c d^2 \left(c+d x^2\right)}+\frac{2 b^2 x^{3/2}}{3 d^2}",1,"(32*b^2*d^(3/4)*x^(3/2) + (24*d^(3/4)*(b*c - a*d)^2*x^(3/2))/(c*(c + d*x^2)) + (6*Sqrt[2]*(7*b^2*c^2 - 6*a*b*c*d - a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(5/4) - (6*Sqrt[2]*(7*b^2*c^2 - 6*a*b*c*d - a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(5/4) - (3*Sqrt[2]*(7*b^2*c^2 - 6*a*b*c*d - a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(5/4) + (3*Sqrt[2]*(7*b^2*c^2 - 6*a*b*c*d - a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(5/4))/(48*d^(11/4))","A",1
429,1,318,312,0.1801396,"\int \frac{\left(a+b x^2\right)^2}{\sqrt{x} \left(c+d x^2\right)^2} \, dx","Integrate[(a + b*x^2)^2/(Sqrt[x]*(c + d*x^2)^2),x]","\frac{\frac{\sqrt{2} \left(-3 a^2 d^2-2 a b c d+5 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{7/4}}-\frac{\sqrt{2} \left(-3 a^2 d^2-2 a b c d+5 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{7/4}}+\frac{2 \sqrt{2} \left(-3 a^2 d^2-2 a b c d+5 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{c^{7/4}}-\frac{2 \sqrt{2} \left(-3 a^2 d^2-2 a b c d+5 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{c^{7/4}}+\frac{8 \sqrt[4]{d} \sqrt{x} (b c-a d)^2}{c \left(c+d x^2\right)}+32 b^2 \sqrt[4]{d} \sqrt{x}}{16 d^{9/4}}","\frac{(b c-a d) (3 a d+5 b c) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{7/4} d^{9/4}}-\frac{(b c-a d) (3 a d+5 b c) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{7/4} d^{9/4}}+\frac{(b c-a d) (3 a d+5 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{4 \sqrt{2} c^{7/4} d^{9/4}}-\frac{(b c-a d) (3 a d+5 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{4 \sqrt{2} c^{7/4} d^{9/4}}+\frac{\sqrt{x} (b c-a d)^2}{2 c d^2 \left(c+d x^2\right)}+\frac{2 b^2 \sqrt{x}}{d^2}",1,"(32*b^2*d^(1/4)*Sqrt[x] + (8*d^(1/4)*(b*c - a*d)^2*Sqrt[x])/(c*(c + d*x^2)) + (2*Sqrt[2]*(5*b^2*c^2 - 2*a*b*c*d - 3*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(7/4) - (2*Sqrt[2]*(5*b^2*c^2 - 2*a*b*c*d - 3*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(7/4) + (Sqrt[2]*(5*b^2*c^2 - 2*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(7/4) - (Sqrt[2]*(5*b^2*c^2 - 2*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(7/4))/(16*d^(9/4))","A",1
430,1,317,333,0.1954922,"\int \frac{\left(a+b x^2\right)^2}{x^{3/2} \left(c+d x^2\right)^2} \, dx","Integrate[(a + b*x^2)^2/(x^(3/2)*(c + d*x^2)^2),x]","\frac{\frac{\sqrt{2} \left(-5 a^2 d^2+2 a b c d+3 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{d^{7/4}}+\frac{\sqrt{2} \left(5 a^2 d^2-2 a b c d-3 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{d^{7/4}}+\frac{2 \sqrt{2} \left(5 a^2 d^2-2 a b c d-3 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{d^{7/4}}+\frac{2 \sqrt{2} \left(-5 a^2 d^2+2 a b c d+3 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{d^{7/4}}-\frac{32 a^2 \sqrt[4]{c}}{\sqrt{x}}-\frac{8 \sqrt[4]{c} x^{3/2} (b c-a d)^2}{d \left(c+d x^2\right)}}{16 c^{9/4}}","-\frac{x^{3/2} \left(5 a^2 d^2-2 a b c d+b^2 c^2\right)}{2 c^2 d \left(c+d x^2\right)}-\frac{2 a^2}{c \sqrt{x} \left(c+d x^2\right)}+\frac{(b c-a d) (5 a d+3 b c) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{9/4} d^{7/4}}-\frac{(b c-a d) (5 a d+3 b c) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{9/4} d^{7/4}}-\frac{(b c-a d) (5 a d+3 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{4 \sqrt{2} c^{9/4} d^{7/4}}+\frac{(b c-a d) (5 a d+3 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{4 \sqrt{2} c^{9/4} d^{7/4}}",1,"((-32*a^2*c^(1/4))/Sqrt[x] - (8*c^(1/4)*(b*c - a*d)^2*x^(3/2))/(d*(c + d*x^2)) + (2*Sqrt[2]*(-3*b^2*c^2 - 2*a*b*c*d + 5*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/d^(7/4) + (2*Sqrt[2]*(3*b^2*c^2 + 2*a*b*c*d - 5*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/d^(7/4) + (Sqrt[2]*(3*b^2*c^2 + 2*a*b*c*d - 5*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/d^(7/4) + (Sqrt[2]*(-3*b^2*c^2 - 2*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/d^(7/4))/(16*c^(9/4))","A",1
431,1,315,332,0.1901274,"\int \frac{\left(a+b x^2\right)^2}{x^{5/2} \left(c+d x^2\right)^2} \, dx","Integrate[(a + b*x^2)^2/(x^(5/2)*(c + d*x^2)^2),x]","\frac{-\frac{3 \sqrt{2} \left(-7 a^2 d^2+6 a b c d+b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{d^{5/4}}+\frac{3 \sqrt{2} \left(-7 a^2 d^2+6 a b c d+b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{d^{5/4}}-\frac{6 \sqrt{2} \left(-7 a^2 d^2+6 a b c d+b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{d^{5/4}}+\frac{6 \sqrt{2} \left(-7 a^2 d^2+6 a b c d+b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{d^{5/4}}-\frac{32 a^2 c^{3/4}}{x^{3/2}}-\frac{24 c^{3/4} \sqrt{x} (b c-a d)^2}{d \left(c+d x^2\right)}}{48 c^{11/4}}","-\frac{\sqrt{x} \left(7 a^2 d^2-6 a b c d+3 b^2 c^2\right)}{6 c^2 d \left(c+d x^2\right)}-\frac{2 a^2}{3 c x^{3/2} \left(c+d x^2\right)}-\frac{(b c-a d) (7 a d+b c) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{11/4} d^{5/4}}+\frac{(b c-a d) (7 a d+b c) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{11/4} d^{5/4}}-\frac{(b c-a d) (7 a d+b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{4 \sqrt{2} c^{11/4} d^{5/4}}+\frac{(b c-a d) (7 a d+b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{4 \sqrt{2} c^{11/4} d^{5/4}}",1,"((-32*a^2*c^(3/4))/x^(3/2) - (24*c^(3/4)*(b*c - a*d)^2*Sqrt[x])/(d*(c + d*x^2)) - (6*Sqrt[2]*(b^2*c^2 + 6*a*b*c*d - 7*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/d^(5/4) + (6*Sqrt[2]*(b^2*c^2 + 6*a*b*c*d - 7*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/d^(5/4) - (3*Sqrt[2]*(b^2*c^2 + 6*a*b*c*d - 7*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/d^(5/4) + (3*Sqrt[2]*(b^2*c^2 + 6*a*b*c*d - 7*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/d^(5/4))/(48*c^(11/4))","A",1
432,1,333,363,0.1966457,"\int \frac{\left(a+b x^2\right)^2}{x^{7/2} \left(c+d x^2\right)^2} \, dx","Integrate[(a + b*x^2)^2/(x^(7/2)*(c + d*x^2)^2),x]","\frac{\frac{5 \sqrt{2} \left(9 a^2 d^2-10 a b c d+b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{d^{3/4}}-\frac{5 \sqrt{2} \left(9 a^2 d^2-10 a b c d+b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{d^{3/4}}-\frac{10 \sqrt{2} \left(9 a^2 d^2-10 a b c d+b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{d^{3/4}}+\frac{10 \sqrt{2} \left(9 a^2 d^2-10 a b c d+b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{d^{3/4}}-\frac{32 a^2 c^{5/4}}{x^{5/2}}+\frac{40 \sqrt[4]{c} x^{3/2} (b c-a d)^2}{c+d x^2}+\frac{320 a \sqrt[4]{c} (a d-b c)}{\sqrt{x}}}{80 c^{13/4}}","-\frac{9 a^2 d^2-10 a b c d+5 b^2 c^2}{10 c^2 d \sqrt{x} \left(c+d x^2\right)}-\frac{2 a^2}{5 c x^{5/2} \left(c+d x^2\right)}+\frac{(b c-9 a d) (b c-a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{13/4} d^{3/4}}-\frac{(b c-9 a d) (b c-a d) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{13/4} d^{3/4}}-\frac{(b c-9 a d) (b c-a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{4 \sqrt{2} c^{13/4} d^{3/4}}+\frac{(b c-9 a d) (b c-a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{4 \sqrt{2} c^{13/4} d^{3/4}}+\frac{(b c-9 a d) (b c-a d)}{2 c^3 d \sqrt{x}}",1,"((-32*a^2*c^(5/4))/x^(5/2) + (320*a*c^(1/4)*(-(b*c) + a*d))/Sqrt[x] + (40*c^(1/4)*(b*c - a*d)^2*x^(3/2))/(c + d*x^2) - (10*Sqrt[2]*(b^2*c^2 - 10*a*b*c*d + 9*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/d^(3/4) + (10*Sqrt[2]*(b^2*c^2 - 10*a*b*c*d + 9*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/d^(3/4) + (5*Sqrt[2]*(b^2*c^2 - 10*a*b*c*d + 9*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/d^(3/4) - (5*Sqrt[2]*(b^2*c^2 - 10*a*b*c*d + 9*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/d^(3/4))/(80*c^(13/4))","A",1
433,1,383,440,0.4727504,"\int \frac{x^{7/2} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^3} \, dx","Integrate[(x^(7/2)*(a + b*x^2)^2)/(c + d*x^2)^3,x]","\frac{-\frac{40 \sqrt[4]{d} \sqrt{x} \left(9 a^2 d^2-34 a b c d+25 b^2 c^2\right)}{c+d x^2}-\frac{5 \sqrt{2} \left(5 a^2 d^2-90 a b c d+117 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{3/4}}+\frac{5 \sqrt{2} \left(5 a^2 d^2-90 a b c d+117 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{3/4}}-\frac{10 \sqrt{2} \left(5 a^2 d^2-90 a b c d+117 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{c^{3/4}}+\frac{10 \sqrt{2} \left(5 a^2 d^2-90 a b c d+117 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{c^{3/4}}+\frac{160 c \sqrt[4]{d} \sqrt{x} (b c-a d)^2}{\left(c+d x^2\right)^2}-1280 b \sqrt[4]{d} \sqrt{x} (3 b c-2 a d)+256 b^2 d^{5/4} x^{5/2}}{640 d^{17/4}}","-\frac{\sqrt{x} \left(5 a^2 d^2-90 a b c d+117 b^2 c^2\right)}{16 c d^4}+\frac{x^{5/2} \left(5 a^2 d^2-90 a b c d+117 b^2 c^2\right)}{80 c^2 d^3}-\frac{\left(5 a^2 d^2-90 a b c d+117 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{3/4} d^{17/4}}+\frac{\left(5 a^2 d^2-90 a b c d+117 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{3/4} d^{17/4}}-\frac{\left(5 a^2 d^2-90 a b c d+117 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{32 \sqrt{2} c^{3/4} d^{17/4}}+\frac{\left(5 a^2 d^2-90 a b c d+117 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{32 \sqrt{2} c^{3/4} d^{17/4}}-\frac{x^{9/2} (b c-a d) (17 b c-a d)}{16 c^2 d^2 \left(c+d x^2\right)}+\frac{x^{9/2} (b c-a d)^2}{4 c d^2 \left(c+d x^2\right)^2}",1,"(-1280*b*d^(1/4)*(3*b*c - 2*a*d)*Sqrt[x] + 256*b^2*d^(5/4)*x^(5/2) + (160*c*d^(1/4)*(b*c - a*d)^2*Sqrt[x])/(c + d*x^2)^2 - (40*d^(1/4)*(25*b^2*c^2 - 34*a*b*c*d + 9*a^2*d^2)*Sqrt[x])/(c + d*x^2) - (10*Sqrt[2]*(117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(3/4) + (10*Sqrt[2]*(117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(3/4) - (5*Sqrt[2]*(117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(3/4) + (5*Sqrt[2]*(117*b^2*c^2 - 90*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(3/4))/(640*d^(17/4))","A",1
434,1,363,401,0.2407797,"\int \frac{x^{5/2} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^3} \, dx","Integrate[(x^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^3,x]","\frac{\frac{24 d^{3/4} x^{3/2} \left(3 a^2 d^2-22 a b c d+19 b^2 c^2\right)}{c \left(c+d x^2\right)}-\frac{3 \sqrt{2} \left(-3 a^2 d^2-42 a b c d+77 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{5/4}}+\frac{3 \sqrt{2} \left(-3 a^2 d^2-42 a b c d+77 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{5/4}}+\frac{6 \sqrt{2} \left(-3 a^2 d^2-42 a b c d+77 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{c^{5/4}}-\frac{6 \sqrt{2} \left(-3 a^2 d^2-42 a b c d+77 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{c^{5/4}}-\frac{96 d^{3/4} x^{3/2} (b c-a d)^2}{\left(c+d x^2\right)^2}+256 b^2 d^{3/4} x^{3/2}}{384 d^{15/4}}","-\frac{\left(-3 a^2 d^2-42 a b c d+77 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{5/4} d^{15/4}}+\frac{\left(-3 a^2 d^2-42 a b c d+77 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{5/4} d^{15/4}}+\frac{\left(-3 a^2 d^2-42 a b c d+77 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{32 \sqrt{2} c^{5/4} d^{15/4}}-\frac{\left(-3 a^2 d^2-42 a b c d+77 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{32 \sqrt{2} c^{5/4} d^{15/4}}-\frac{x^{3/2} \left(\frac{3 a^2 d}{c}+42 a b-\frac{77 b^2 c}{d}\right)}{48 c d^2}-\frac{x^{7/2} (b c-a d) (a d+15 b c)}{16 c^2 d^2 \left(c+d x^2\right)}+\frac{x^{7/2} (b c-a d)^2}{4 c d^2 \left(c+d x^2\right)^2}",1,"(256*b^2*d^(3/4)*x^(3/2) - (96*d^(3/4)*(b*c - a*d)^2*x^(3/2))/(c + d*x^2)^2 + (24*d^(3/4)*(19*b^2*c^2 - 22*a*b*c*d + 3*a^2*d^2)*x^(3/2))/(c*(c + d*x^2)) + (6*Sqrt[2]*(77*b^2*c^2 - 42*a*b*c*d - 3*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(5/4) - (6*Sqrt[2]*(77*b^2*c^2 - 42*a*b*c*d - 3*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(5/4) - (3*Sqrt[2]*(77*b^2*c^2 - 42*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(5/4) + (3*Sqrt[2]*(77*b^2*c^2 - 42*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(5/4))/(384*d^(15/4))","A",1
435,1,361,402,0.3858012,"\int \frac{x^{3/2} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^3} \, dx","Integrate[(x^(3/2)*(a + b*x^2)^2)/(c + d*x^2)^3,x]","\frac{\frac{8 \sqrt[4]{d} \sqrt{x} \left(a^2 d^2-18 a b c d+17 b^2 c^2\right)}{c \left(c+d x^2\right)}+\frac{\sqrt{2} \left(-3 a^2 d^2-10 a b c d+45 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{7/4}}-\frac{\sqrt{2} \left(-3 a^2 d^2-10 a b c d+45 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{7/4}}+\frac{2 \sqrt{2} \left(-3 a^2 d^2-10 a b c d+45 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{c^{7/4}}-\frac{2 \sqrt{2} \left(-3 a^2 d^2-10 a b c d+45 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{c^{7/4}}-\frac{32 \sqrt[4]{d} \sqrt{x} (b c-a d)^2}{\left(c+d x^2\right)^2}+256 b^2 \sqrt[4]{d} \sqrt{x}}{128 d^{13/4}}","\frac{\left(-3 a^2 d^2-10 a b c d+45 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{7/4} d^{13/4}}-\frac{\left(-3 a^2 d^2-10 a b c d+45 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{7/4} d^{13/4}}+\frac{\left(-3 a^2 d^2-10 a b c d+45 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{32 \sqrt{2} c^{7/4} d^{13/4}}-\frac{\left(-3 a^2 d^2-10 a b c d+45 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{32 \sqrt{2} c^{7/4} d^{13/4}}-\frac{\sqrt{x} \left(\frac{3 a^2 d}{c}+10 a b-\frac{45 b^2 c}{d}\right)}{16 c d^2}-\frac{x^{5/2} (b c-a d) (3 a d+13 b c)}{16 c^2 d^2 \left(c+d x^2\right)}+\frac{x^{5/2} (b c-a d)^2}{4 c d^2 \left(c+d x^2\right)^2}",1,"(256*b^2*d^(1/4)*Sqrt[x] - (32*d^(1/4)*(b*c - a*d)^2*Sqrt[x])/(c + d*x^2)^2 + (8*d^(1/4)*(17*b^2*c^2 - 18*a*b*c*d + a^2*d^2)*Sqrt[x])/(c*(c + d*x^2)) + (2*Sqrt[2]*(45*b^2*c^2 - 10*a*b*c*d - 3*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(7/4) - (2*Sqrt[2]*(45*b^2*c^2 - 10*a*b*c*d - 3*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(7/4) + (Sqrt[2]*(45*b^2*c^2 - 10*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(7/4) - (Sqrt[2]*(45*b^2*c^2 - 10*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(7/4))/(128*d^(13/4))","A",1
436,1,339,364,0.2069871,"\int \frac{\sqrt{x} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^3} \, dx","Integrate[(Sqrt[x]*(a + b*x^2)^2)/(c + d*x^2)^3,x]","\frac{\sqrt{2} \left(5 a^2 d^2+6 a b c d+21 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)-\sqrt{2} \left(5 a^2 d^2+6 a b c d+21 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)-2 \sqrt{2} \left(5 a^2 d^2+6 a b c d+21 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)+2 \sqrt{2} \left(5 a^2 d^2+6 a b c d+21 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)-\frac{8 \sqrt[4]{c} d^{3/4} x^{3/2} \left(-5 a^2 d^2-6 a b c d+11 b^2 c^2\right)}{c+d x^2}+\frac{32 c^{5/4} d^{3/4} x^{3/2} (b c-a d)^2}{\left(c+d x^2\right)^2}}{128 c^{9/4} d^{11/4}}","\frac{\left(5 a^2 d^2+6 a b c d+21 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{9/4} d^{11/4}}-\frac{\left(5 a^2 d^2+6 a b c d+21 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{9/4} d^{11/4}}-\frac{\left(5 a^2 d^2+6 a b c d+21 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{32 \sqrt{2} c^{9/4} d^{11/4}}+\frac{\left(5 a^2 d^2+6 a b c d+21 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{32 \sqrt{2} c^{9/4} d^{11/4}}-\frac{x^{3/2} (5 a d+11 b c) (b c-a d)}{16 c^2 d^2 \left(c+d x^2\right)}+\frac{x^{3/2} (b c-a d)^2}{4 c d^2 \left(c+d x^2\right)^2}",1,"((32*c^(5/4)*d^(3/4)*(b*c - a*d)^2*x^(3/2))/(c + d*x^2)^2 - (8*c^(1/4)*d^(3/4)*(11*b^2*c^2 - 6*a*b*c*d - 5*a^2*d^2)*x^(3/2))/(c + d*x^2) - 2*Sqrt[2]*(21*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + 2*Sqrt[2]*(21*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + Sqrt[2]*(21*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x] - Sqrt[2]*(21*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(128*c^(9/4)*d^(11/4))","A",1
437,1,339,364,0.2039447,"\int \frac{\left(a+b x^2\right)^2}{\sqrt{x} \left(c+d x^2\right)^3} \, dx","Integrate[(a + b*x^2)^2/(Sqrt[x]*(c + d*x^2)^3),x]","\frac{-\sqrt{2} \left(21 a^2 d^2+6 a b c d+5 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)+\sqrt{2} \left(21 a^2 d^2+6 a b c d+5 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)-2 \sqrt{2} \left(21 a^2 d^2+6 a b c d+5 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)+2 \sqrt{2} \left(21 a^2 d^2+6 a b c d+5 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)-\frac{8 c^{3/4} \sqrt[4]{d} \sqrt{x} \left(-7 a^2 d^2-2 a b c d+9 b^2 c^2\right)}{c+d x^2}+\frac{32 c^{7/4} \sqrt[4]{d} \sqrt{x} (b c-a d)^2}{\left(c+d x^2\right)^2}}{128 c^{11/4} d^{9/4}}","-\frac{\left(21 a^2 d^2+6 a b c d+5 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{11/4} d^{9/4}}+\frac{\left(21 a^2 d^2+6 a b c d+5 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{11/4} d^{9/4}}-\frac{\left(21 a^2 d^2+6 a b c d+5 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{32 \sqrt{2} c^{11/4} d^{9/4}}+\frac{\left(21 a^2 d^2+6 a b c d+5 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{32 \sqrt{2} c^{11/4} d^{9/4}}-\frac{\sqrt{x} (7 a d+9 b c) (b c-a d)}{16 c^2 d^2 \left(c+d x^2\right)}+\frac{\sqrt{x} (b c-a d)^2}{4 c d^2 \left(c+d x^2\right)^2}",1,"((32*c^(7/4)*d^(1/4)*(b*c - a*d)^2*Sqrt[x])/(c + d*x^2)^2 - (8*c^(3/4)*d^(1/4)*(9*b^2*c^2 - 2*a*b*c*d - 7*a^2*d^2)*Sqrt[x])/(c + d*x^2) - 2*Sqrt[2]*(5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + 2*Sqrt[2]*(5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] - Sqrt[2]*(5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x] + Sqrt[2]*(5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(128*c^(11/4)*d^(9/4))","A",1
438,1,364,399,0.4116061,"\int \frac{\left(a+b x^2\right)^2}{x^{3/2} \left(c+d x^2\right)^3} \, dx","Integrate[(a + b*x^2)^2/(x^(3/2)*(c + d*x^2)^3),x]","\frac{\frac{8 \sqrt[4]{c} x^{3/2} \left(-13 a^2 d^2+10 a b c d+3 b^2 c^2\right)}{d \left(c+d x^2\right)}+\frac{\sqrt{2} \left(-45 a^2 d^2+10 a b c d+3 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{d^{7/4}}+\frac{\sqrt{2} \left(45 a^2 d^2-10 a b c d-3 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{d^{7/4}}+\frac{2 \sqrt{2} \left(45 a^2 d^2-10 a b c d-3 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{d^{7/4}}+\frac{2 \sqrt{2} \left(-45 a^2 d^2+10 a b c d+3 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{d^{7/4}}-\frac{256 a^2 \sqrt[4]{c}}{\sqrt{x}}-\frac{32 c^{5/4} x^{3/2} (b c-a d)^2}{d \left(c+d x^2\right)^2}}{128 c^{13/4}}","-\frac{x^{3/2} \left(9 a^2 d^2-2 a b c d+b^2 c^2\right)}{4 c^2 d \left(c+d x^2\right)^2}-\frac{2 a^2}{c \sqrt{x} \left(c+d x^2\right)^2}+\frac{\left(5 a d (2 b c-9 a d)+3 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{13/4} d^{7/4}}-\frac{\left(5 a d (2 b c-9 a d)+3 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{13/4} d^{7/4}}-\frac{\left(5 a d (2 b c-9 a d)+3 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{32 \sqrt{2} c^{13/4} d^{7/4}}+\frac{\left(5 a d (2 b c-9 a d)+3 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{32 \sqrt{2} c^{13/4} d^{7/4}}+\frac{x^{3/2} \left(5 a d (2 b c-9 a d)+3 b^2 c^2\right)}{16 c^3 d \left(c+d x^2\right)}",1,"((-256*a^2*c^(1/4))/Sqrt[x] - (32*c^(5/4)*(b*c - a*d)^2*x^(3/2))/(d*(c + d*x^2)^2) + (8*c^(1/4)*(3*b^2*c^2 + 10*a*b*c*d - 13*a^2*d^2)*x^(3/2))/(d*(c + d*x^2)) + (2*Sqrt[2]*(-3*b^2*c^2 - 10*a*b*c*d + 45*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/d^(7/4) + (2*Sqrt[2]*(3*b^2*c^2 + 10*a*b*c*d - 45*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/d^(7/4) + (Sqrt[2]*(3*b^2*c^2 + 10*a*b*c*d - 45*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/d^(7/4) + (Sqrt[2]*(-3*b^2*c^2 - 10*a*b*c*d + 45*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/d^(7/4))/(128*c^(13/4))","A",1
439,1,365,402,0.2614794,"\int \frac{\left(a+b x^2\right)^2}{x^{5/2} \left(c+d x^2\right)^3} \, dx","Integrate[(a + b*x^2)^2/(x^(5/2)*(c + d*x^2)^3),x]","\frac{\frac{3 \sqrt{2} \left(77 a^2 d^2-42 a b c d-3 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{d^{5/4}}+\frac{3 \sqrt{2} \left(-77 a^2 d^2+42 a b c d+3 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{d^{5/4}}+\frac{6 \sqrt{2} \left(77 a^2 d^2-42 a b c d-3 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{d^{5/4}}+\frac{6 \sqrt{2} \left(-77 a^2 d^2+42 a b c d+3 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{d^{5/4}}+\frac{24 c^{3/4} \sqrt{x} \left(-15 a^2 d^2+14 a b c d+b^2 c^2\right)}{d \left(c+d x^2\right)}-\frac{256 a^2 c^{3/4}}{x^{3/2}}-\frac{96 c^{7/4} \sqrt{x} (b c-a d)^2}{d \left(c+d x^2\right)^2}}{384 c^{15/4}}","-\frac{\sqrt{x} \left(11 a^2 d^2-6 a b c d+3 b^2 c^2\right)}{12 c^2 d \left(c+d x^2\right)^2}-\frac{2 a^2}{3 c x^{3/2} \left(c+d x^2\right)^2}-\frac{\left(7 a d (6 b c-11 a d)+3 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{15/4} d^{5/4}}+\frac{\left(7 a d (6 b c-11 a d)+3 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{15/4} d^{5/4}}-\frac{\left(7 a d (6 b c-11 a d)+3 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{32 \sqrt{2} c^{15/4} d^{5/4}}+\frac{\left(7 a d (6 b c-11 a d)+3 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{32 \sqrt{2} c^{15/4} d^{5/4}}+\frac{\sqrt{x} \left(7 a d (6 b c-11 a d)+3 b^2 c^2\right)}{48 c^3 d \left(c+d x^2\right)}",1,"((-256*a^2*c^(3/4))/x^(3/2) - (96*c^(7/4)*(b*c - a*d)^2*Sqrt[x])/(d*(c + d*x^2)^2) + (24*c^(3/4)*(b^2*c^2 + 14*a*b*c*d - 15*a^2*d^2)*Sqrt[x])/(d*(c + d*x^2)) + (6*Sqrt[2]*(-3*b^2*c^2 - 42*a*b*c*d + 77*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/d^(5/4) + (6*Sqrt[2]*(3*b^2*c^2 + 42*a*b*c*d - 77*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/d^(5/4) + (3*Sqrt[2]*(-3*b^2*c^2 - 42*a*b*c*d + 77*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/d^(5/4) + (3*Sqrt[2]*(3*b^2*c^2 + 42*a*b*c*d - 77*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/d^(5/4))/(384*c^(15/4))","A",1
440,1,382,439,0.3874622,"\int \frac{\left(a+b x^2\right)^2}{x^{7/2} \left(c+d x^2\right)^3} \, dx","Integrate[(a + b*x^2)^2/(x^(7/2)*(c + d*x^2)^3),x]","\frac{\frac{40 \sqrt[4]{c} x^{3/2} \left(21 a^2 d^2-26 a b c d+5 b^2 c^2\right)}{c+d x^2}+\frac{5 \sqrt{2} \left(117 a^2 d^2-90 a b c d+5 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{d^{3/4}}-\frac{5 \sqrt{2} \left(117 a^2 d^2-90 a b c d+5 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{d^{3/4}}-\frac{10 \sqrt{2} \left(117 a^2 d^2-90 a b c d+5 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{d^{3/4}}+\frac{10 \sqrt{2} \left(117 a^2 d^2-90 a b c d+5 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{d^{3/4}}-\frac{256 a^2 c^{5/4}}{x^{5/2}}+\frac{160 c^{5/4} x^{3/2} (b c-a d)^2}{\left(c+d x^2\right)^2}+\frac{1280 a \sqrt[4]{c} (3 a d-2 b c)}{\sqrt{x}}}{640 c^{17/4}}","-\frac{13 a^2 d^2-10 a b c d+5 b^2 c^2}{20 c^2 d \sqrt{x} \left(c+d x^2\right)^2}-\frac{2 a^2}{5 c x^{5/2} \left(c+d x^2\right)^2}+\frac{\left(5 b^2 c^2-9 a d (10 b c-13 a d)\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{17/4} d^{3/4}}-\frac{\left(5 b^2 c^2-9 a d (10 b c-13 a d)\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{17/4} d^{3/4}}-\frac{\left(5 b^2 c^2-9 a d (10 b c-13 a d)\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{32 \sqrt{2} c^{17/4} d^{3/4}}+\frac{\left(5 b^2 c^2-9 a d (10 b c-13 a d)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{32 \sqrt{2} c^{17/4} d^{3/4}}+\frac{5 b^2 c^2-9 a d (10 b c-13 a d)}{16 c^4 d \sqrt{x}}-\frac{5 b^2 c^2-9 a d (10 b c-13 a d)}{80 c^3 d \sqrt{x} \left(c+d x^2\right)}",1,"((-256*a^2*c^(5/4))/x^(5/2) + (1280*a*c^(1/4)*(-2*b*c + 3*a*d))/Sqrt[x] + (160*c^(5/4)*(b*c - a*d)^2*x^(3/2))/(c + d*x^2)^2 + (40*c^(1/4)*(5*b^2*c^2 - 26*a*b*c*d + 21*a^2*d^2)*x^(3/2))/(c + d*x^2) - (10*Sqrt[2]*(5*b^2*c^2 - 90*a*b*c*d + 117*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/d^(3/4) + (10*Sqrt[2]*(5*b^2*c^2 - 90*a*b*c*d + 117*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/d^(3/4) + (5*Sqrt[2]*(5*b^2*c^2 - 90*a*b*c*d + 117*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/d^(3/4) - (5*Sqrt[2]*(5*b^2*c^2 - 90*a*b*c*d + 117*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/d^(3/4))/(640*c^(17/4))","A",1
441,1,132,328,0.3927722,"\int \frac{x^{5/2} \left(c+d x^2\right)^3}{a+b x^2} \, dx","Integrate[(x^(5/2)*(c + d*x^2)^3)/(a + b*x^2),x]","\frac{2 x^{3/2} \left(-385 a^3 d^3+165 a^2 b d^2 \left(7 c+d x^2\right)-15 a b^2 d \left(77 c^2+33 c d x^2+7 d^2 x^4\right)-385 (b c-a d)^3 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\frac{b x^2}{a}\right)+b^3 \left(385 c^3+495 c^2 d x^2+315 c d^2 x^4+77 d^3 x^6\right)\right)}{1155 b^4}","-\frac{a^{3/4} (b c-a d)^3 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} b^{19/4}}+\frac{a^{3/4} (b c-a d)^3 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} b^{19/4}}+\frac{a^{3/4} (b c-a d)^3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} b^{19/4}}-\frac{a^{3/4} (b c-a d)^3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} b^{19/4}}+\frac{2 d x^{7/2} \left(a^2 d^2-3 a b c d+3 b^2 c^2\right)}{7 b^3}+\frac{2 x^{3/2} (b c-a d)^3}{3 b^4}+\frac{2 d^2 x^{11/2} (3 b c-a d)}{11 b^2}+\frac{2 d^3 x^{15/2}}{15 b}",1,"(2*x^(3/2)*(-385*a^3*d^3 + 165*a^2*b*d^2*(7*c + d*x^2) - 15*a*b^2*d*(77*c^2 + 33*c*d*x^2 + 7*d^2*x^4) + b^3*(385*c^3 + 495*c^2*d*x^2 + 315*c*d^2*x^4 + 77*d^3*x^6) - 385*(b*c - a*d)^3*Hypergeometric2F1[3/4, 1, 7/4, -((b*x^2)/a)]))/(1155*b^4)","C",1
442,1,133,326,0.4008336,"\int \frac{x^{3/2} \left(c+d x^2\right)^3}{a+b x^2} \, dx","Integrate[(x^(3/2)*(c + d*x^2)^3)/(a + b*x^2),x]","\frac{2 \sqrt{x} \left(-585 a^3 d^3+117 a^2 b d^2 \left(15 c+d x^2\right)-13 a b^2 d \left(135 c^2+27 c d x^2+5 d^2 x^4\right)-585 (b c-a d)^3 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};-\frac{b x^2}{a}\right)+3 b^3 \left(195 c^3+117 c^2 d x^2+65 c d^2 x^4+15 d^3 x^6\right)\right)}{585 b^4}","\frac{2 d x^{5/2} \left(a^2 d^2-3 a b c d+3 b^2 c^2\right)}{5 b^3}+\frac{\sqrt[4]{a} (b c-a d)^3 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} b^{17/4}}-\frac{\sqrt[4]{a} (b c-a d)^3 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} b^{17/4}}+\frac{\sqrt[4]{a} (b c-a d)^3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} b^{17/4}}-\frac{\sqrt[4]{a} (b c-a d)^3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} b^{17/4}}+\frac{2 \sqrt{x} (b c-a d)^3}{b^4}+\frac{2 d^2 x^{9/2} (3 b c-a d)}{9 b^2}+\frac{2 d^3 x^{13/2}}{13 b}",1,"(2*Sqrt[x]*(-585*a^3*d^3 + 117*a^2*b*d^2*(15*c + d*x^2) - 13*a*b^2*d*(135*c^2 + 27*c*d*x^2 + 5*d^2*x^4) + 3*b^3*(195*c^3 + 117*c^2*d*x^2 + 65*c*d^2*x^4 + 15*d^3*x^6) - 585*(b*c - a*d)^3*Hypergeometric2F1[1/4, 1, 5/4, -((b*x^2)/a)]))/(585*b^4)","C",1
443,1,97,306,0.3760259,"\int \frac{\sqrt{x} \left(c+d x^2\right)^3}{a+b x^2} \, dx","Integrate[(Sqrt[x]*(c + d*x^2)^3)/(a + b*x^2),x]","\frac{2 x^{3/2} \left(a d \left(77 a^2 d^2-33 a b d \left(7 c+d x^2\right)+3 b^2 \left(77 c^2+33 c d x^2+7 d^2 x^4\right)\right)+77 (b c-a d)^3 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\frac{b x^2}{a}\right)\right)}{231 a b^3}","\frac{2 d x^{3/2} \left(a^2 d^2-3 a b c d+3 b^2 c^2\right)}{3 b^3}+\frac{(b c-a d)^3 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} \sqrt[4]{a} b^{15/4}}-\frac{(b c-a d)^3 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} \sqrt[4]{a} b^{15/4}}-\frac{(b c-a d)^3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} \sqrt[4]{a} b^{15/4}}+\frac{(b c-a d)^3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} \sqrt[4]{a} b^{15/4}}+\frac{2 d^2 x^{7/2} (3 b c-a d)}{7 b^2}+\frac{2 d^3 x^{11/2}}{11 b}",1,"(2*x^(3/2)*(a*d*(77*a^2*d^2 - 33*a*b*d*(7*c + d*x^2) + 3*b^2*(77*c^2 + 33*c*d*x^2 + 7*d^2*x^4)) + 77*(b*c - a*d)^3*Hypergeometric2F1[3/4, 1, 7/4, -((b*x^2)/a)]))/(231*a*b^3)","C",1
444,1,96,304,0.3628936,"\int \frac{\left(c+d x^2\right)^3}{\sqrt{x} \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2)^3/(Sqrt[x]*(a + b*x^2)),x]","\frac{2 \sqrt{x} \left(a d \left(45 a^2 d^2-9 a b d \left(15 c+d x^2\right)+b^2 \left(135 c^2+27 c d x^2+5 d^2 x^4\right)\right)+45 (b c-a d)^3 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};-\frac{b x^2}{a}\right)\right)}{45 a b^3}","-\frac{(b c-a d)^3 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{3/4} b^{13/4}}+\frac{(b c-a d)^3 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{3/4} b^{13/4}}-\frac{(b c-a d)^3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} a^{3/4} b^{13/4}}+\frac{(b c-a d)^3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} a^{3/4} b^{13/4}}+\frac{2 d \sqrt{x} \left(a^2 d^2-3 a b c d+3 b^2 c^2\right)}{b^3}+\frac{2 d^2 x^{5/2} (3 b c-a d)}{5 b^2}+\frac{2 d^3 x^{9/2}}{9 b}",1,"(2*Sqrt[x]*(a*d*(45*a^2*d^2 - 9*a*b*d*(15*c + d*x^2) + b^2*(135*c^2 + 27*c*d*x^2 + 5*d^2*x^4)) + 45*(b*c - a*d)^3*Hypergeometric2F1[1/4, 1, 5/4, -((b*x^2)/a)]))/(45*a*b^3)","C",1
445,1,89,284,0.3521225,"\int \frac{\left(c+d x^2\right)^3}{x^{3/2} \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2)^3/(x^(3/2)*(a + b*x^2)),x]","-\frac{2 \left(a \left(7 a^2 d^3 x^2-3 a b d^2 x^2 \left(7 c+d x^2\right)+21 b^2 c^3\right)+7 x^2 (b c-a d)^3 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\frac{b x^2}{a}\right)\right)}{21 a^2 b^2 \sqrt{x}}","-\frac{(b c-a d)^3 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{5/4} b^{11/4}}+\frac{(b c-a d)^3 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{5/4} b^{11/4}}+\frac{(b c-a d)^3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} a^{5/4} b^{11/4}}-\frac{(b c-a d)^3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} a^{5/4} b^{11/4}}+\frac{2 d^2 x^{3/2} (3 b c-a d)}{3 b^2}-\frac{2 c^3}{a \sqrt{x}}+\frac{2 d^3 x^{7/2}}{7 b}",1,"(-2*(a*(21*b^2*c^3 + 7*a^2*d^3*x^2 - 3*a*b*d^2*x^2*(7*c + d*x^2)) + 7*(b*c - a*d)^3*x^2*Hypergeometric2F1[3/4, 1, 7/4, -((b*x^2)/a)]))/(21*a^2*b^2*Sqrt[x])","C",1
446,1,89,284,0.3801011,"\int \frac{\left(c+d x^2\right)^3}{x^{5/2} \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2)^3/(x^(5/2)*(a + b*x^2)),x]","-\frac{2 \left(a \left(15 a^2 d^3 x^2-3 a b d^2 x^2 \left(15 c+d x^2\right)+5 b^2 c^3\right)+15 x^2 (b c-a d)^3 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};-\frac{b x^2}{a}\right)\right)}{15 a^2 b^2 x^{3/2}}","\frac{(b c-a d)^3 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{7/4} b^{9/4}}-\frac{(b c-a d)^3 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{7/4} b^{9/4}}+\frac{(b c-a d)^3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} a^{7/4} b^{9/4}}-\frac{(b c-a d)^3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} a^{7/4} b^{9/4}}+\frac{2 d^2 \sqrt{x} (3 b c-a d)}{b^2}-\frac{2 c^3}{3 a x^{3/2}}+\frac{2 d^3 x^{5/2}}{5 b}",1,"(-2*(a*(5*b^2*c^3 + 15*a^2*d^3*x^2 - 3*a*b*d^2*x^2*(15*c + d*x^2)) + 15*(b*c - a*d)^3*x^2*Hypergeometric2F1[1/4, 1, 5/4, -((b*x^2)/a)]))/(15*a^2*b^2*x^(3/2))","C",1
447,1,88,283,0.3713471,"\int \frac{\left(c+d x^2\right)^3}{x^{7/2} \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2)^3/(x^(7/2)*(a + b*x^2)),x]","-\frac{2 \left(a \left(-5 a^2 d^3 x^4+3 a b c^2 \left(c+15 d x^2\right)-15 b^2 c^3 x^2\right)-5 x^4 (b c-a d)^3 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\frac{b x^2}{a}\right)\right)}{15 a^3 b x^{5/2}}","\frac{(b c-a d)^3 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{9/4} b^{7/4}}-\frac{(b c-a d)^3 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{9/4} b^{7/4}}-\frac{(b c-a d)^3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} a^{9/4} b^{7/4}}+\frac{(b c-a d)^3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} a^{9/4} b^{7/4}}+\frac{2 c^2 (b c-3 a d)}{a^2 \sqrt{x}}-\frac{2 c^3}{5 a x^{5/2}}+\frac{2 d^3 x^{3/2}}{3 b}",1,"(-2*(a*(-15*b^2*c^3*x^2 - 5*a^2*d^3*x^4 + 3*a*b*c^2*(c + 15*d*x^2)) - 5*(b*c - a*d)^3*x^4*Hypergeometric2F1[3/4, 1, 7/4, -((b*x^2)/a)]))/(15*a^3*b*x^(5/2))","C",1
448,1,88,283,0.3800982,"\int \frac{\left(c+d x^2\right)^3}{x^{9/2} \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2)^3/(x^(9/2)*(a + b*x^2)),x]","-\frac{2 \left(a \left(-21 a^2 d^3 x^4+3 a b c^2 \left(c+7 d x^2\right)-7 b^2 c^3 x^2\right)-21 x^4 (b c-a d)^3 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};-\frac{b x^2}{a}\right)\right)}{21 a^3 b x^{7/2}}","-\frac{(b c-a d)^3 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{11/4} b^{5/4}}+\frac{(b c-a d)^3 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{11/4} b^{5/4}}-\frac{(b c-a d)^3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} a^{11/4} b^{5/4}}+\frac{(b c-a d)^3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} a^{11/4} b^{5/4}}+\frac{2 c^2 (b c-3 a d)}{3 a^2 x^{3/2}}-\frac{2 c^3}{7 a x^{7/2}}+\frac{2 d^3 \sqrt{x}}{b}",1,"(-2*(a*(-7*b^2*c^3*x^2 - 21*a^2*d^3*x^4 + 3*a*b*c^2*(c + 7*d*x^2)) - 21*(b*c - a*d)^3*x^4*Hypergeometric2F1[1/4, 1, 5/4, -((b*x^2)/a)]))/(21*a^3*b*x^(7/2))","C",1
449,1,101,303,0.3842061,"\int \frac{\left(c+d x^2\right)^3}{x^{11/2} \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2)^3/(x^(11/2)*(a + b*x^2)),x]","-\frac{2 \left(a c \left(a^2 \left(5 c^2+27 c d x^2+135 d^2 x^4\right)-9 a b c x^2 \left(c+15 d x^2\right)+45 b^2 c^2 x^4\right)+15 x^6 (b c-a d)^3 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\frac{b x^2}{a}\right)\right)}{45 a^4 x^{9/2}}","-\frac{(b c-a d)^3 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{13/4} b^{3/4}}+\frac{(b c-a d)^3 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{13/4} b^{3/4}}+\frac{(b c-a d)^3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} a^{13/4} b^{3/4}}-\frac{(b c-a d)^3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} a^{13/4} b^{3/4}}+\frac{2 c^2 (b c-3 a d)}{5 a^2 x^{5/2}}-\frac{2 c \left(3 a^2 d^2-3 a b c d+b^2 c^2\right)}{a^3 \sqrt{x}}-\frac{2 c^3}{9 a x^{9/2}}",1,"(-2*(a*c*(45*b^2*c^2*x^4 - 9*a*b*c*x^2*(c + 15*d*x^2) + a^2*(5*c^2 + 27*c*d*x^2 + 135*d^2*x^4)) + 15*(b*c - a*d)^3*x^6*Hypergeometric2F1[3/4, 1, 7/4, -((b*x^2)/a)]))/(45*a^4*x^(9/2))","C",1
450,1,102,305,0.3793113,"\int \frac{\left(c+d x^2\right)^3}{x^{13/2} \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2)^3/(x^(13/2)*(a + b*x^2)),x]","-\frac{2 \left(a c \left(3 a^2 \left(7 c^2+33 c d x^2+77 d^2 x^4\right)-33 a b c x^2 \left(c+7 d x^2\right)+77 b^2 c^2 x^4\right)+231 x^6 (b c-a d)^3 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};-\frac{b x^2}{a}\right)\right)}{231 a^4 x^{11/2}}","\frac{(b c-a d)^3 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{15/4} \sqrt[4]{b}}-\frac{(b c-a d)^3 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{15/4} \sqrt[4]{b}}+\frac{(b c-a d)^3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} a^{15/4} \sqrt[4]{b}}-\frac{(b c-a d)^3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} a^{15/4} \sqrt[4]{b}}+\frac{2 c^2 (b c-3 a d)}{7 a^2 x^{7/2}}-\frac{2 c \left(3 a^2 d^2-3 a b c d+b^2 c^2\right)}{3 a^3 x^{3/2}}-\frac{2 c^3}{11 a x^{11/2}}",1,"(-2*(a*c*(77*b^2*c^2*x^4 - 33*a*b*c*x^2*(c + 7*d*x^2) + 3*a^2*(7*c^2 + 33*c*d*x^2 + 77*d^2*x^4)) + 231*(b*c - a*d)^3*x^6*Hypergeometric2F1[1/4, 1, 5/4, -((b*x^2)/a)]))/(231*a^4*x^(11/2))","C",1
451,1,148,325,0.4252376,"\int \frac{\left(c+d x^2\right)^3}{x^{15/2} \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2)^3/(x^(15/2)*(a + b*x^2)),x]","-\frac{2 \left(a \left(3 a^3 \left(15 c^3+65 c^2 d x^2+117 c d^2 x^4+195 d^3 x^6\right)-13 a^2 b c x^2 \left(5 c^2+27 c d x^2+135 d^2 x^4\right)+117 a b^2 c^2 x^4 \left(c+15 d x^2\right)-585 b^3 c^3 x^6\right)-195 b x^8 (b c-a d)^3 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\frac{b x^2}{a}\right)\right)}{585 a^5 x^{13/2}}","\frac{\sqrt[4]{b} (b c-a d)^3 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{17/4}}-\frac{\sqrt[4]{b} (b c-a d)^3 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{17/4}}-\frac{\sqrt[4]{b} (b c-a d)^3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} a^{17/4}}+\frac{\sqrt[4]{b} (b c-a d)^3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} a^{17/4}}+\frac{2 (b c-a d)^3}{a^4 \sqrt{x}}+\frac{2 c^2 (b c-3 a d)}{9 a^2 x^{9/2}}-\frac{2 c \left(3 a^2 d^2-3 a b c d+b^2 c^2\right)}{5 a^3 x^{5/2}}-\frac{2 c^3}{13 a x^{13/2}}",1,"(-2*(a*(-585*b^3*c^3*x^6 + 117*a*b^2*c^2*x^4*(c + 15*d*x^2) - 13*a^2*b*c*x^2*(5*c^2 + 27*c*d*x^2 + 135*d^2*x^4) + 3*a^3*(15*c^3 + 65*c^2*d*x^2 + 117*c*d^2*x^4 + 195*d^3*x^6)) - 195*b*(b*c - a*d)^3*x^8*Hypergeometric2F1[3/4, 1, 7/4, -((b*x^2)/a)]))/(585*a^5*x^(13/2))","C",1
452,1,419,409,2.6948856,"\int \frac{x^{7/2} \left(c+d x^2\right)^3}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^(7/2)*(c + d*x^2)^3)/(a + b*x^2)^2,x]","\frac{35 a^2 \left(9945 a^4 \left(64827 c^3+194481 c^2 d x^2+194481 c d^2 x^4+62651 d^3 x^6\right)+3978 a^3 b x^2 \left(176389 c^3+529167 c^2 d x^2+541647 c d^2 x^4+177477 d^3 x^6\right)+221 a^2 b^2 x^4 \left(857691 c^3+2417553 c^2 d x^2+2528145 c d^2 x^4+846811 d^3 x^6\right)-9945 a \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};-\frac{b x^2}{a}\right) \left(a^3 \left(64827 c^3+194481 c^2 d x^2+194481 c d^2 x^4+62651 d^3 x^6\right)+a^2 b x^2 \left(83521 c^3+250563 c^2 d x^2+255555 c d^2 x^4+83521 d^3 x^6\right)+a b^2 x^4 \left(28561 c^3+82227 c^2 d x^2+85683 c d^2 x^4+28561 d^3 x^6\right)+b^3 x^6 \left(2827 c^3+6561 c^2 d x^2+6561 c d^2 x^4+2187 d^3 x^6\right)\right)+8704 a b^3 x^6 \left(1609 c^3+3423 c^2 d x^2+3267 c d^2 x^4+1069 d^3 x^6\right)-32768 b^4 x^8 \left(c+d x^2\right)^3\right)-98304 b^6 x^{12} \left(c+d x^2\right)^3 \, _5F_4\left(2,2,2,2,\frac{13}{4};1,1,1,\frac{29}{4};-\frac{b x^2}{a}\right)}{89107200 a^3 b^5 x^{11/2}}","\frac{d x^{5/2} \left(17 a^2 d^2-39 a b c d+27 b^2 c^2\right)}{10 b^4}+\frac{\sqrt[4]{a} (5 b c-17 a d) (b c-a d)^2 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} b^{21/4}}-\frac{\sqrt[4]{a} (5 b c-17 a d) (b c-a d)^2 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} b^{21/4}}+\frac{\sqrt[4]{a} (5 b c-17 a d) (b c-a d)^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} b^{21/4}}-\frac{\sqrt[4]{a} (5 b c-17 a d) (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} b^{21/4}}+\frac{\sqrt{x} (5 b c-17 a d) (b c-a d)^2}{2 b^5}+\frac{d^2 x^{9/2} (39 b c-17 a d)}{18 b^3}-\frac{x^{5/2} \left(c+d x^2\right)^3}{2 b \left(a+b x^2\right)}+\frac{17 d^3 x^{13/2}}{26 b^2}",1,"(35*a^2*(-32768*b^4*x^8*(c + d*x^2)^3 + 8704*a*b^3*x^6*(1609*c^3 + 3423*c^2*d*x^2 + 3267*c*d^2*x^4 + 1069*d^3*x^6) + 9945*a^4*(64827*c^3 + 194481*c^2*d*x^2 + 194481*c*d^2*x^4 + 62651*d^3*x^6) + 3978*a^3*b*x^2*(176389*c^3 + 529167*c^2*d*x^2 + 541647*c*d^2*x^4 + 177477*d^3*x^6) + 221*a^2*b^2*x^4*(857691*c^3 + 2417553*c^2*d*x^2 + 2528145*c*d^2*x^4 + 846811*d^3*x^6) - 9945*a*(b^3*x^6*(2827*c^3 + 6561*c^2*d*x^2 + 6561*c*d^2*x^4 + 2187*d^3*x^6) + a*b^2*x^4*(28561*c^3 + 82227*c^2*d*x^2 + 85683*c*d^2*x^4 + 28561*d^3*x^6) + a^3*(64827*c^3 + 194481*c^2*d*x^2 + 194481*c*d^2*x^4 + 62651*d^3*x^6) + a^2*b*x^2*(83521*c^3 + 250563*c^2*d*x^2 + 255555*c*d^2*x^4 + 83521*d^3*x^6))*Hypergeometric2F1[1/4, 1, 5/4, -((b*x^2)/a)]) - 98304*b^6*x^12*(c + d*x^2)^3*HypergeometricPFQ[{2, 2, 2, 2, 13/4}, {1, 1, 1, 29/4}, -((b*x^2)/a)])/(89107200*a^3*b^5*x^(11/2))","C",0
453,1,377,374,2.2165447,"\int \frac{x^{5/2} \left(c+d x^2\right)^3}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^(5/2)*(c + d*x^2)^3)/(a + b*x^2)^2,x]","\frac{-385 a^3 \left(130321 c^3+390963 c^2 d x^2+390963 c d^2 x^4+124561 d^3 x^6\right)-330 a^2 b x^2 \left(112027 c^3+336081 c^2 d x^2+350865 c d^2 x^4+114907 d^3 x^6\right)+385 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\frac{b x^2}{a}\right) \left(a^3 \left(130321 c^3+390963 c^2 d x^2+390963 c d^2 x^4+124561 d^3 x^6\right)+9 a^2 b x^2 \left(16875 c^3+50625 c^2 d x^2+52033 c d^2 x^4+16875 d^3 x^6\right)+3 a b^2 x^4 \left(14641 c^3+41235 c^2 d x^2+43923 c d^2 x^4+14641 d^3 x^6\right)+b^3 x^6 \left(3553 c^3+7203 c^2 d x^2+7203 c d^2 x^4+2401 d^3 x^6\right)\right)-45 a b^2 x^4 \left(122993 c^3+299987 c^2 d x^2+322515 c d^2 x^4+109553 d^3 x^6\right)-32768 b^3 x^6 \left(c+d x^2\right)^3}{887040 a b^4 x^{9/2}}-\frac{128 b x^{11/2} \left(c+d x^2\right)^3 \, _5F_4\left(2,2,2,2,\frac{11}{4};1,1,1,\frac{27}{4};-\frac{b x^2}{a}\right)}{72105 a^3}","\frac{d x^{3/2} \left(5 a^2 d^2-11 a b c d+7 b^2 c^2\right)}{2 b^4}+\frac{3 (b c-5 a d) (b c-a d)^2 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} \sqrt[4]{a} b^{19/4}}-\frac{3 (b c-5 a d) (b c-a d)^2 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} \sqrt[4]{a} b^{19/4}}-\frac{3 (b c-5 a d) (b c-a d)^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} \sqrt[4]{a} b^{19/4}}+\frac{3 (b c-5 a d) (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} \sqrt[4]{a} b^{19/4}}+\frac{3 d^2 x^{7/2} (11 b c-5 a d)}{14 b^3}-\frac{x^{3/2} \left(c+d x^2\right)^3}{2 b \left(a+b x^2\right)}+\frac{15 d^3 x^{11/2}}{22 b^2}",1,"(-32768*b^3*x^6*(c + d*x^2)^3 - 45*a*b^2*x^4*(122993*c^3 + 299987*c^2*d*x^2 + 322515*c*d^2*x^4 + 109553*d^3*x^6) - 330*a^2*b*x^2*(112027*c^3 + 336081*c^2*d*x^2 + 350865*c*d^2*x^4 + 114907*d^3*x^6) - 385*a^3*(130321*c^3 + 390963*c^2*d*x^2 + 390963*c*d^2*x^4 + 124561*d^3*x^6) + 385*(b^3*x^6*(3553*c^3 + 7203*c^2*d*x^2 + 7203*c*d^2*x^4 + 2401*d^3*x^6) + 3*a*b^2*x^4*(14641*c^3 + 41235*c^2*d*x^2 + 43923*c*d^2*x^4 + 14641*d^3*x^6) + 9*a^2*b*x^2*(16875*c^3 + 50625*c^2*d*x^2 + 52033*c*d^2*x^4 + 16875*d^3*x^6) + a^3*(130321*c^3 + 390963*c^2*d*x^2 + 390963*c*d^2*x^4 + 124561*d^3*x^6))*Hypergeometric2F1[3/4, 1, 7/4, -((b*x^2)/a)])/(887040*a*b^4*x^(9/2)) - (128*b*x^(11/2)*(c + d*x^2)^3*HypergeometricPFQ[{2, 2, 2, 2, 11/4}, {1, 1, 1, 27/4}, -((b*x^2)/a)])/(72105*a^3)","C",0
454,1,377,386,2.6931359,"\int \frac{x^{3/2} \left(c+d x^2\right)^3}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^(3/2)*(c + d*x^2)^3)/(a + b*x^2)^2,x]","\frac{-585 a^3 \left(83521 c^3+250563 c^2 d x^2+250563 c d^2 x^4+78529 d^3 x^6\right)-234 a^2 b x^2 \left(172447 c^3+517341 c^2 d x^2+543261 c d^2 x^4+174943 d^3 x^6\right)+585 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};-\frac{b x^2}{a}\right) \left(a^3 \left(83521 c^3+250563 c^2 d x^2+250563 c d^2 x^4+78529 d^3 x^6\right)+3 a^2 b x^2 \left(28561 c^3+85683 c^2 d x^2+89139 c d^2 x^4+28561 d^3 x^6\right)+9 a b^2 x^4 \left(2187 c^3+5921 c^2 d x^2+6561 c d^2 x^4+2187 d^3 x^6\right)+b^3 x^6 \left(1009 c^3+1875 c^2 d x^2+1875 c d^2 x^4+625 d^3 x^6\right)\right)-13 a b^2 x^4 \left(532193 c^3+1337379 c^2 d x^2+1503267 c d^2 x^4+507233 d^3 x^6\right)-98304 b^3 x^6 \left(c+d x^2\right)^3}{449280 a b^4 x^{11/2}}-\frac{128 b x^{9/2} \left(c+d x^2\right)^3 \, _5F_4\left(2,2,2,2,\frac{9}{4};1,1,1,\frac{25}{4};-\frac{b x^2}{a}\right)}{41769 a^3}","-\frac{(b c-13 a d) (b c-a d)^2 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{3/4} b^{17/4}}+\frac{(b c-13 a d) (b c-a d)^2 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{3/4} b^{17/4}}-\frac{(b c-13 a d) (b c-a d)^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} a^{3/4} b^{17/4}}+\frac{(b c-13 a d) (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} a^{3/4} b^{17/4}}+\frac{d \sqrt{x} \left(585 a^2 d^2-1098 a b c d+497 b^2 c^2\right)}{90 b^4}+\frac{d \sqrt{x} \left(c+d x^2\right) (113 b c-117 a d)}{90 b^3}-\frac{\sqrt{x} \left(c+d x^2\right)^3}{2 b \left(a+b x^2\right)}+\frac{13 d \sqrt{x} \left(c+d x^2\right)^2}{18 b^2}",1,"(-98304*b^3*x^6*(c + d*x^2)^3 - 585*a^3*(83521*c^3 + 250563*c^2*d*x^2 + 250563*c*d^2*x^4 + 78529*d^3*x^6) - 234*a^2*b*x^2*(172447*c^3 + 517341*c^2*d*x^2 + 543261*c*d^2*x^4 + 174943*d^3*x^6) - 13*a*b^2*x^4*(532193*c^3 + 1337379*c^2*d*x^2 + 1503267*c*d^2*x^4 + 507233*d^3*x^6) + 585*(b^3*x^6*(1009*c^3 + 1875*c^2*d*x^2 + 1875*c*d^2*x^4 + 625*d^3*x^6) + 9*a*b^2*x^4*(2187*c^3 + 5921*c^2*d*x^2 + 6561*c*d^2*x^4 + 2187*d^3*x^6) + 3*a^2*b*x^2*(28561*c^3 + 85683*c^2*d*x^2 + 89139*c*d^2*x^4 + 28561*d^3*x^6) + a^3*(83521*c^3 + 250563*c^2*d*x^2 + 250563*c*d^2*x^4 + 78529*d^3*x^6))*Hypergeometric2F1[1/4, 1, 5/4, -((b*x^2)/a)])/(449280*a*b^4*x^(11/2)) - (128*b*x^(9/2)*(c + d*x^2)^3*HypergeometricPFQ[{2, 2, 2, 2, 9/4}, {1, 1, 1, 25/4}, -((b*x^2)/a)])/(41769*a^3)","C",0
455,1,355,376,2.63488,"\int \frac{\sqrt{x} \left(c+d x^2\right)^3}{\left(a+b x^2\right)^2} \, dx","Integrate[(Sqrt[x]*(c + d*x^2)^3)/(a + b*x^2)^2,x]","\frac{95 a \left(a \left(77 a^2 \left(16875 c^3+50625 c^2 d x^2+50625 c d^2 x^4+15467 d^3 x^6\right)+22 a b x^2 \left(25931 c^3+77793 c^2 d x^2+87201 c d^2 x^4+28043 d^3 x^6\right)+b^2 x^4 \left(56099 c^3+79593 c^2 d x^2+79593 c d^2 x^4+26531 d^3 x^6\right)\right)-77 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\frac{b x^2}{a}\right) \left(a^3 \left(16875 c^3+50625 c^2 d x^2+50625 c d^2 x^4+15467 d^3 x^6\right)+a^2 b x^2 \left(14641 c^3+43923 c^2 d x^2+46611 c d^2 x^4+14641 d^3 x^6\right)+a b^2 x^4 \left(2401 c^3+6051 c^2 d x^2+7203 c d^2 x^4+2401 d^3 x^6\right)+b^3 x^6 \left(-101 c^3+81 c^2 d x^2+81 c d^2 x^4+27 d^3 x^6\right)\right)\right)-32768 b^4 x^8 \left(c+d x^2\right)^3 \, _5F_4\left(\frac{7}{4},2,2,2,2;1,1,1,\frac{23}{4};-\frac{b x^2}{a}\right)}{5617920 a^3 b^3 x^{9/2}}","\frac{(b c-a d)^2 (11 a d+b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{5/4} b^{15/4}}-\frac{(b c-a d)^2 (11 a d+b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{5/4} b^{15/4}}-\frac{(b c-a d)^2 (11 a d+b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} a^{5/4} b^{15/4}}+\frac{(b c-a d)^2 (11 a d+b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} a^{5/4} b^{15/4}}-\frac{d x^{3/2} \left(11 a^2 d^2-21 a b c d+6 b^2 c^2\right)}{6 a b^3}-\frac{d^2 x^{7/2} (7 b c-11 a d)}{14 a b^2}+\frac{x^{3/2} \left(c+d x^2\right)^2 (b c-a d)}{2 a b \left(a+b x^2\right)}",1,"(95*a*(a*(77*a^2*(16875*c^3 + 50625*c^2*d*x^2 + 50625*c*d^2*x^4 + 15467*d^3*x^6) + b^2*x^4*(56099*c^3 + 79593*c^2*d*x^2 + 79593*c*d^2*x^4 + 26531*d^3*x^6) + 22*a*b*x^2*(25931*c^3 + 77793*c^2*d*x^2 + 87201*c*d^2*x^4 + 28043*d^3*x^6)) - 77*(b^3*x^6*(-101*c^3 + 81*c^2*d*x^2 + 81*c*d^2*x^4 + 27*d^3*x^6) + a*b^2*x^4*(2401*c^3 + 6051*c^2*d*x^2 + 7203*c*d^2*x^4 + 2401*d^3*x^6) + a^2*b*x^2*(14641*c^3 + 43923*c^2*d*x^2 + 46611*c*d^2*x^4 + 14641*d^3*x^6) + a^3*(16875*c^3 + 50625*c^2*d*x^2 + 50625*c*d^2*x^4 + 15467*d^3*x^6))*Hypergeometric2F1[3/4, 1, 7/4, -((b*x^2)/a)]) - 32768*b^4*x^8*(c + d*x^2)^3*HypergeometricPFQ[{7/4, 2, 2, 2, 2}, {1, 1, 1, 23/4}, -((b*x^2)/a)])/(5617920*a^3*b^3*x^(9/2))","C",0
456,1,358,340,2.6354185,"\int \frac{\left(c+d x^2\right)^3}{\sqrt{x} \left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)^3/(Sqrt[x]*(a + b*x^2)^2),x]","\frac{a \left(45 a^2 \left(28561 c^3+85683 c^2 d x^2+85683 c d^2 x^4+25105 d^3 x^6\right)+18 a b x^2 \left(34927 c^3+104781 c^2 d x^2+119181 c d^2 x^4+36655 d^3 x^6\right)+b^2 x^4 \left(50033 c^3+98259 c^2 d x^2+98259 c d^2 x^4+32753 d^3 x^6\right)\right)-45 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};-\frac{b x^2}{a}\right) \left(a^3 \left(28561 c^3+85683 c^2 d x^2+85683 c d^2 x^4+25105 d^3 x^6\right)+9 a^2 b x^2 \left(2187 c^3+6561 c^2 d x^2+7201 c d^2 x^4+2187 d^3 x^6\right)+3 a b^2 x^4 \left(625 c^3+1491 c^2 d x^2+1875 c d^2 x^4+625 d^3 x^6\right)+b^3 x^6 \left(-1151 c^3+3 c^2 d x^2+3 c d^2 x^4+d^3 x^6\right)\right)}{34560 a^2 b^3 x^{11/2}}-\frac{128 b x^{5/2} \left(c+d x^2\right)^3 \, _5F_4\left(\frac{5}{4},2,2,2,2;1,1,1,\frac{21}{4};-\frac{b x^2}{a}\right)}{9945 a^3}","-\frac{3 (b c-a d)^2 (3 a d+b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{7/4} b^{13/4}}+\frac{3 (b c-a d)^2 (3 a d+b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{7/4} b^{13/4}}-\frac{3 (b c-a d)^2 (3 a d+b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} a^{7/4} b^{13/4}}+\frac{3 (b c-a d)^2 (3 a d+b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} a^{7/4} b^{13/4}}+\frac{2 d^2 \sqrt{x} (3 b c-2 a d)}{b^3}+\frac{\sqrt{x} (b c-a d)^3}{2 a b^3 \left(a+b x^2\right)}+\frac{2 d^3 x^{5/2}}{5 b^2}",1,"(a*(45*a^2*(28561*c^3 + 85683*c^2*d*x^2 + 85683*c*d^2*x^4 + 25105*d^3*x^6) + b^2*x^4*(50033*c^3 + 98259*c^2*d*x^2 + 98259*c*d^2*x^4 + 32753*d^3*x^6) + 18*a*b*x^2*(34927*c^3 + 104781*c^2*d*x^2 + 119181*c*d^2*x^4 + 36655*d^3*x^6)) - 45*(b^3*x^6*(-1151*c^3 + 3*c^2*d*x^2 + 3*c*d^2*x^4 + d^3*x^6) + 3*a*b^2*x^4*(625*c^3 + 1491*c^2*d*x^2 + 1875*c*d^2*x^4 + 625*d^3*x^6) + 9*a^2*b*x^2*(2187*c^3 + 6561*c^2*d*x^2 + 7201*c*d^2*x^4 + 2187*d^3*x^6) + a^3*(28561*c^3 + 85683*c^2*d*x^2 + 85683*c*d^2*x^4 + 25105*d^3*x^6))*Hypergeometric2F1[1/4, 1, 5/4, -((b*x^2)/a)])/(34560*a^2*b^3*x^(11/2)) - (128*b*x^(5/2)*(c + d*x^2)^3*HypergeometricPFQ[{5/4, 2, 2, 2, 2}, {1, 1, 1, 21/4}, -((b*x^2)/a)])/(9945*a^3)","C",0
457,1,355,368,2.3870827,"\int \frac{\left(c+d x^2\right)^3}{x^{3/2} \left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)^3/(x^(3/2)*(a + b*x^2)^2),x]","-\frac{32768 b^3 x^6 \left(c+d x^2\right)^3 \, _5F_4\left(\frac{3}{4},2,2,2,2;1,1,1,\frac{19}{4};-\frac{b x^2}{a}\right)+55 \left(a \left(7 a^2 \left(14641 c^3+43923 c^2 d x^2+43923 c d^2 x^4+11953 d^3 x^6\right)+18 a b x^2 \left(361 c^3+1083 c^2 d x^2+2427 c d^2 x^4+809 d^3 x^6\right)-21 b^2 x^4 \left(-1919 c^3+3 c^2 d x^2+3 c d^2 x^4+d^3 x^6\right)\right)-7 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\frac{b x^2}{a}\right) \left(a^3 \left(14641 c^3+43923 c^2 d x^2+43923 c d^2 x^4+11953 d^3 x^6\right)+3 a^2 b x^2 \left(2401 c^3+7203 c^2 d x^2+8355 c d^2 x^4+2401 d^3 x^6\right)+9 a b^2 x^4 \left(27 c^3+209 c^2 d x^2+81 c d^2 x^4+27 d^3 x^6\right)+b^3 x^6 \left(-1919 c^3+3 c^2 d x^2+3 c d^2 x^4+d^3 x^6\right)\right)\right)}{887040 a^3 b^2 x^{9/2}}","-\frac{(b c-a d)^2 (7 a d+5 b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{9/4} b^{11/4}}+\frac{(b c-a d)^2 (7 a d+5 b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{9/4} b^{11/4}}+\frac{(b c-a d)^2 (7 a d+5 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} a^{9/4} b^{11/4}}-\frac{(b c-a d)^2 (7 a d+5 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} a^{9/4} b^{11/4}}-\frac{c^2 (5 b c-a d)}{2 a^2 b \sqrt{x}}-\frac{d^2 x^{3/2} (3 b c-7 a d)}{6 a b^2}+\frac{\left(c+d x^2\right)^2 (b c-a d)}{2 a b \sqrt{x} \left(a+b x^2\right)}",1,"-1/887040*(55*(a*(-21*b^2*x^4*(-1919*c^3 + 3*c^2*d*x^2 + 3*c*d^2*x^4 + d^3*x^6) + 18*a*b*x^2*(361*c^3 + 1083*c^2*d*x^2 + 2427*c*d^2*x^4 + 809*d^3*x^6) + 7*a^2*(14641*c^3 + 43923*c^2*d*x^2 + 43923*c*d^2*x^4 + 11953*d^3*x^6)) - 7*(b^3*x^6*(-1919*c^3 + 3*c^2*d*x^2 + 3*c*d^2*x^4 + d^3*x^6) + 9*a*b^2*x^4*(27*c^3 + 209*c^2*d*x^2 + 81*c*d^2*x^4 + 27*d^3*x^6) + 3*a^2*b*x^2*(2401*c^3 + 7203*c^2*d*x^2 + 8355*c*d^2*x^4 + 2401*d^3*x^6) + a^3*(14641*c^3 + 43923*c^2*d*x^2 + 43923*c*d^2*x^4 + 11953*d^3*x^6))*Hypergeometric2F1[3/4, 1, 7/4, -((b*x^2)/a)]) + 32768*b^3*x^6*(c + d*x^2)^3*HypergeometricPFQ[{3/4, 2, 2, 2, 2}, {1, 1, 1, 19/4}, -((b*x^2)/a)])/(a^3*b^2*x^(9/2))","C",0
458,1,350,367,2.6719788,"\int \frac{\left(c+d x^2\right)^3}{x^{5/2} \left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)^3/(x^(5/2)*(a + b*x^2)^2),x]","-\frac{32768 b^3 x^6 \left(c+d x^2\right)^3 \, _5F_4\left(\frac{1}{4},2,2,2,2;1,1,1,\frac{17}{4};-\frac{b x^2}{a}\right)+39 a \left(15 a^2 \left(2187 c^3+6561 c^2 d x^2+6561 c d^2 x^4+1547 d^3 x^6\right)+6 a b x^2 \left(469 c^3+1407 c^2 d x^2+2367 c d^2 x^4+789 d^3 x^6\right)-5 b^2 x^4 \left(-869 c^3+81 c^2 d x^2+81 c d^2 x^4+27 d^3 x^6\right)\right)-585 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};-\frac{b x^2}{a}\right) \left(a^3 \left(2187 c^3+6561 c^2 d x^2+6561 c d^2 x^4+1547 d^3 x^6\right)+a^2 b x^2 \left(625 c^3+1875 c^2 d x^2+2259 c d^2 x^4+625 d^3 x^6\right)+a b^2 x^4 \left(c^3+1155 c^2 d x^2+3 c d^2 x^4+d^3 x^6\right)+b^3 x^6 \left(-869 c^3+81 c^2 d x^2+81 c d^2 x^4+27 d^3 x^6\right)\right)}{149760 a^3 b^2 x^{11/2}}","\frac{(b c-a d)^2 (5 a d+7 b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{11/4} b^{9/4}}-\frac{(b c-a d)^2 (5 a d+7 b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{11/4} b^{9/4}}+\frac{(b c-a d)^2 (5 a d+7 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} a^{11/4} b^{9/4}}-\frac{(b c-a d)^2 (5 a d+7 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} a^{11/4} b^{9/4}}-\frac{c^2 (7 b c-3 a d)}{6 a^2 b x^{3/2}}-\frac{d^2 \sqrt{x} (b c-5 a d)}{2 a b^2}+\frac{\left(c+d x^2\right)^2 (b c-a d)}{2 a b x^{3/2} \left(a+b x^2\right)}",1,"-1/149760*(39*a*(-5*b^2*x^4*(-869*c^3 + 81*c^2*d*x^2 + 81*c*d^2*x^4 + 27*d^3*x^6) + 6*a*b*x^2*(469*c^3 + 1407*c^2*d*x^2 + 2367*c*d^2*x^4 + 789*d^3*x^6) + 15*a^2*(2187*c^3 + 6561*c^2*d*x^2 + 6561*c*d^2*x^4 + 1547*d^3*x^6)) - 585*(a*b^2*x^4*(c^3 + 1155*c^2*d*x^2 + 3*c*d^2*x^4 + d^3*x^6) + b^3*x^6*(-869*c^3 + 81*c^2*d*x^2 + 81*c*d^2*x^4 + 27*d^3*x^6) + a^2*b*x^2*(625*c^3 + 1875*c^2*d*x^2 + 2259*c*d^2*x^4 + 625*d^3*x^6) + a^3*(2187*c^3 + 6561*c^2*d*x^2 + 6561*c*d^2*x^4 + 1547*d^3*x^6))*Hypergeometric2F1[1/4, 1, 5/4, -((b*x^2)/a)] + 32768*b^3*x^6*(c + d*x^2)^3*HypergeometricPFQ[{1/4, 2, 2, 2, 2}, {1, 1, 1, 17/4}, -((b*x^2)/a)])/(a^3*b^2*x^(11/2))","C",0
459,1,353,376,2.1876895,"\int \frac{\left(c+d x^2\right)^3}{x^{7/2} \left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)^3/(x^(7/2)*(a + b*x^2)^2),x]","-\frac{-491520 a b^2 x^4 \left(c+d x^2\right)^3 \, _5F_4\left(-\frac{1}{4},2,2,2,2;1,1,1,\frac{15}{4};-\frac{b x^2}{a}\right)-77 a \left(5 a^2 \left(2401 c^3+7203 c^2 d x^2+7203 c d^2 x^4+1249 d^3 x^6\right)+6 a b x^2 \left(-1423 c^3-13485 c^2 d x^2+915 c d^2 x^4+305 d^3 x^6\right)-15 b^2 x^4 \left(-2831 c^3+1875 c^2 d x^2+1875 c d^2 x^4+625 d^3 x^6\right)\right)+385 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\frac{b x^2}{a}\right) \left(a^3 \left(2401 c^3+7203 c^2 d x^2+7203 c d^2 x^4+1249 d^3 x^6\right)+9 a^2 b x^2 \left(27 c^3+81 c^2 d x^2-47 c d^2 x^4+27 d^3 x^6\right)+3 a b^2 x^4 \left(c^3+1923 c^2 d x^2+3 c d^2 x^4+d^3 x^6\right)+b^3 x^6 \left(-2831 c^3+1875 c^2 d x^2+1875 c d^2 x^4+625 d^3 x^6\right)\right)}{887040 a^4 b x^{9/2}}","\frac{3 (b c-a d)^2 (a d+3 b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{13/4} b^{7/4}}-\frac{3 (b c-a d)^2 (a d+3 b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{13/4} b^{7/4}}-\frac{3 (b c-a d)^2 (a d+3 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} a^{13/4} b^{7/4}}+\frac{3 (b c-a d)^2 (a d+3 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} a^{13/4} b^{7/4}}-\frac{c^2 (9 b c-5 a d)}{10 a^2 b x^{5/2}}+\frac{c \left(2 a^2 d^2-15 a b c d+9 b^2 c^2\right)}{2 a^3 b \sqrt{x}}+\frac{\left(c+d x^2\right)^2 (b c-a d)}{2 a b x^{5/2} \left(a+b x^2\right)}",1,"-1/887040*(-77*a*(6*a*b*x^2*(-1423*c^3 - 13485*c^2*d*x^2 + 915*c*d^2*x^4 + 305*d^3*x^6) - 15*b^2*x^4*(-2831*c^3 + 1875*c^2*d*x^2 + 1875*c*d^2*x^4 + 625*d^3*x^6) + 5*a^2*(2401*c^3 + 7203*c^2*d*x^2 + 7203*c*d^2*x^4 + 1249*d^3*x^6)) + 385*(3*a*b^2*x^4*(c^3 + 1923*c^2*d*x^2 + 3*c*d^2*x^4 + d^3*x^6) + 9*a^2*b*x^2*(27*c^3 + 81*c^2*d*x^2 - 47*c*d^2*x^4 + 27*d^3*x^6) + b^3*x^6*(-2831*c^3 + 1875*c^2*d*x^2 + 1875*c*d^2*x^4 + 625*d^3*x^6) + a^3*(2401*c^3 + 7203*c^2*d*x^2 + 7203*c*d^2*x^4 + 1249*d^3*x^6))*Hypergeometric2F1[3/4, 1, 7/4, -((b*x^2)/a)] - 491520*a*b^2*x^4*(c + d*x^2)^3*HypergeometricPFQ[{-1/4, 2, 2, 2, 2}, {1, 1, 1, 15/4}, -((b*x^2)/a)])/(a^4*b*x^(9/2))","C",0
460,1,353,376,2.399614,"\int \frac{\left(c+d x^2\right)^3}{x^{9/2} \left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)^3/(x^(9/2)*(a + b*x^2)^2),x]","-\frac{-229376 a b^2 x^4 \left(c+d x^2\right)^3 \, _5F_4\left(-\frac{3}{4},2,2,2,2;1,1,1,\frac{13}{4};-\frac{b x^2}{a}\right)-15 a \left(21 a^2 \left(625 c^3+1875 c^2 d x^2+1875 c d^2 x^4+241 d^3 x^6\right)+6 a b x^2 \left(-1195 c^3-6657 c^2 d x^2+2751 c d^2 x^4+917 d^3 x^6\right)-7 b^2 x^4 \left(-1823 c^3+7203 c^2 d x^2+7203 c d^2 x^4+2401 d^3 x^6\right)\right)+315 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};-\frac{b x^2}{a}\right) \left(a^3 \left(625 c^3+1875 c^2 d x^2+1875 c d^2 x^4+241 d^3 x^6\right)+3 a^2 b x^2 \left(c^3+3 c^2 d x^2-1149 c d^2 x^4+d^3 x^6\right)+9 a b^2 x^4 \left(27 c^3+977 c^2 d x^2+81 c d^2 x^4+27 d^3 x^6\right)+b^3 x^6 \left(-1823 c^3+7203 c^2 d x^2+7203 c d^2 x^4+2401 d^3 x^6\right)\right)}{241920 a^4 b x^{11/2}}","-\frac{(b c-a d)^2 (a d+11 b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{15/4} b^{5/4}}+\frac{(b c-a d)^2 (a d+11 b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{15/4} b^{5/4}}-\frac{(b c-a d)^2 (a d+11 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} a^{15/4} b^{5/4}}+\frac{(b c-a d)^2 (a d+11 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} a^{15/4} b^{5/4}}-\frac{c^2 (11 b c-7 a d)}{14 a^2 b x^{7/2}}+\frac{c \left(6 a^2 d^2-21 a b c d+11 b^2 c^2\right)}{6 a^3 b x^{3/2}}+\frac{\left(c+d x^2\right)^2 (b c-a d)}{2 a b x^{7/2} \left(a+b x^2\right)}",1,"-1/241920*(-15*a*(21*a^2*(625*c^3 + 1875*c^2*d*x^2 + 1875*c*d^2*x^4 + 241*d^3*x^6) + 6*a*b*x^2*(-1195*c^3 - 6657*c^2*d*x^2 + 2751*c*d^2*x^4 + 917*d^3*x^6) - 7*b^2*x^4*(-1823*c^3 + 7203*c^2*d*x^2 + 7203*c*d^2*x^4 + 2401*d^3*x^6)) + 315*(3*a^2*b*x^2*(c^3 + 3*c^2*d*x^2 - 1149*c*d^2*x^4 + d^3*x^6) + 9*a*b^2*x^4*(27*c^3 + 977*c^2*d*x^2 + 81*c*d^2*x^4 + 27*d^3*x^6) + a^3*(625*c^3 + 1875*c^2*d*x^2 + 1875*c*d^2*x^4 + 241*d^3*x^6) + b^3*x^6*(-1823*c^3 + 7203*c^2*d*x^2 + 7203*c*d^2*x^4 + 2401*d^3*x^6))*Hypergeometric2F1[1/4, 1, 5/4, -((b*x^2)/a)] - 229376*a*b^2*x^4*(c + d*x^2)^3*HypergeometricPFQ[{-3/4, 2, 2, 2, 2}, {1, 1, 1, 13/4}, -((b*x^2)/a)])/(a^4*b*x^(11/2))","C",0
461,1,411,478,0.2331041,"\int \frac{x^{9/2}}{\left(a+b x^2\right) \left(c+d x^2\right)} \, dx","Integrate[x^(9/2)/((a + b*x^2)*(c + d*x^2)),x]","\frac{\frac{3 \sqrt{2} a^{7/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{b^{7/4}}-\frac{3 \sqrt{2} a^{7/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{b^{7/4}}-\frac{6 \sqrt{2} a^{7/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{b^{7/4}}+\frac{6 \sqrt{2} a^{7/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{b^{7/4}}-\frac{8 a x^{3/2}}{b}-\frac{3 \sqrt{2} c^{7/4} \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{d^{7/4}}+\frac{3 \sqrt{2} c^{7/4} \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{d^{7/4}}+\frac{6 \sqrt{2} c^{7/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{d^{7/4}}-\frac{6 \sqrt{2} c^{7/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{d^{7/4}}+\frac{8 c x^{3/2}}{d}}{12 b c-12 a d}","\frac{a^{7/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} b^{7/4} (b c-a d)}-\frac{a^{7/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} b^{7/4} (b c-a d)}-\frac{a^{7/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} b^{7/4} (b c-a d)}+\frac{a^{7/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} b^{7/4} (b c-a d)}-\frac{c^{7/4} \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} d^{7/4} (b c-a d)}+\frac{c^{7/4} \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} d^{7/4} (b c-a d)}+\frac{c^{7/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{\sqrt{2} d^{7/4} (b c-a d)}-\frac{c^{7/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{\sqrt{2} d^{7/4} (b c-a d)}+\frac{2 x^{3/2}}{3 b d}",1,"((-8*a*x^(3/2))/b + (8*c*x^(3/2))/d - (6*Sqrt[2]*a^(7/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(7/4) + (6*Sqrt[2]*a^(7/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(7/4) + (6*Sqrt[2]*c^(7/4)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/d^(7/4) - (6*Sqrt[2]*c^(7/4)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/d^(7/4) + (3*Sqrt[2]*a^(7/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(7/4) - (3*Sqrt[2]*a^(7/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(7/4) - (3*Sqrt[2]*c^(7/4)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/d^(7/4) + (3*Sqrt[2]*c^(7/4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/d^(7/4))/(12*b*c - 12*a*d)","A",1
462,1,409,476,0.1888608,"\int \frac{x^{7/2}}{\left(a+b x^2\right) \left(c+d x^2\right)} \, dx","Integrate[x^(7/2)/((a + b*x^2)*(c + d*x^2)),x]","\frac{-\frac{\sqrt{2} a^{5/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{b^{5/4}}+\frac{\sqrt{2} a^{5/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{b^{5/4}}-\frac{2 \sqrt{2} a^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{b^{5/4}}+\frac{2 \sqrt{2} a^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{b^{5/4}}-\frac{8 a \sqrt{x}}{b}+\frac{\sqrt{2} c^{5/4} \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{d^{5/4}}-\frac{\sqrt{2} c^{5/4} \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{d^{5/4}}+\frac{2 \sqrt{2} c^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{d^{5/4}}-\frac{2 \sqrt{2} c^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{d^{5/4}}+\frac{8 c \sqrt{x}}{d}}{4 b c-4 a d}","-\frac{a^{5/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} b^{5/4} (b c-a d)}+\frac{a^{5/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} b^{5/4} (b c-a d)}-\frac{a^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} b^{5/4} (b c-a d)}+\frac{a^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} b^{5/4} (b c-a d)}+\frac{c^{5/4} \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} d^{5/4} (b c-a d)}-\frac{c^{5/4} \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} d^{5/4} (b c-a d)}+\frac{c^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{\sqrt{2} d^{5/4} (b c-a d)}-\frac{c^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{\sqrt{2} d^{5/4} (b c-a d)}+\frac{2 \sqrt{x}}{b d}",1,"((-8*a*Sqrt[x])/b + (8*c*Sqrt[x])/d - (2*Sqrt[2]*a^(5/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(5/4) + (2*Sqrt[2]*a^(5/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(5/4) + (2*Sqrt[2]*c^(5/4)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/d^(5/4) - (2*Sqrt[2]*c^(5/4)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/d^(5/4) - (Sqrt[2]*a^(5/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(5/4) + (Sqrt[2]*a^(5/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(5/4) + (Sqrt[2]*c^(5/4)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/d^(5/4) - (Sqrt[2]*c^(5/4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/d^(5/4))/(4*b*c - 4*a*d)","A",1
463,1,142,463,0.1130767,"\int \frac{x^{5/2}}{\left(a+b x^2\right) \left(c+d x^2\right)} \, dx","Integrate[x^(5/2)/((a + b*x^2)*(c + d*x^2)),x]","\frac{\frac{(-a)^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{-a}}\right)}{b^{3/4}}-\frac{(-a)^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{-a}}\right)}{b^{3/4}}-\frac{(-c)^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{x}}{\sqrt[4]{-c}}\right)}{d^{3/4}}+\frac{(-c)^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{x}}{\sqrt[4]{-c}}\right)}{d^{3/4}}}{b c-a d}","-\frac{a^{3/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} b^{3/4} (b c-a d)}+\frac{a^{3/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} b^{3/4} (b c-a d)}+\frac{a^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} b^{3/4} (b c-a d)}-\frac{a^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} b^{3/4} (b c-a d)}+\frac{c^{3/4} \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} d^{3/4} (b c-a d)}-\frac{c^{3/4} \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} d^{3/4} (b c-a d)}-\frac{c^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{\sqrt{2} d^{3/4} (b c-a d)}+\frac{c^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{\sqrt{2} d^{3/4} (b c-a d)}",1,"(((-a)^(3/4)*ArcTan[(b^(1/4)*Sqrt[x])/(-a)^(1/4)])/b^(3/4) - ((-c)^(3/4)*ArcTan[(d^(1/4)*Sqrt[x])/(-c)^(1/4)])/d^(3/4) - ((-a)^(3/4)*ArcTanh[(b^(1/4)*Sqrt[x])/(-a)^(1/4)])/b^(3/4) + ((-c)^(3/4)*ArcTanh[(d^(1/4)*Sqrt[x])/(-c)^(1/4)])/d^(3/4))/(b*c - a*d)","A",1
464,1,364,463,0.1098599,"\int \frac{x^{3/2}}{\left(a+b x^2\right) \left(c+d x^2\right)} \, dx","Integrate[x^(3/2)/((a + b*x^2)*(c + d*x^2)),x]","\frac{\sqrt[4]{a} \sqrt[4]{d} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-\sqrt[4]{a} \sqrt[4]{d} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+2 \sqrt[4]{a} \sqrt[4]{d} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)-2 \sqrt[4]{a} \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)-\sqrt[4]{b} \sqrt[4]{c} \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)+\sqrt[4]{b} \sqrt[4]{c} \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)-2 \sqrt[4]{b} \sqrt[4]{c} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)+2 \sqrt[4]{b} \sqrt[4]{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{2 \sqrt{2} \sqrt[4]{b} \sqrt[4]{d} (b c-a d)}","\frac{\sqrt[4]{a} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} \sqrt[4]{b} (b c-a d)}-\frac{\sqrt[4]{a} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} \sqrt[4]{b} (b c-a d)}-\frac{\sqrt[4]{c} \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} \sqrt[4]{d} (b c-a d)}+\frac{\sqrt[4]{c} \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} \sqrt[4]{d} (b c-a d)}+\frac{\sqrt[4]{a} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} \sqrt[4]{b} (b c-a d)}-\frac{\sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} \sqrt[4]{b} (b c-a d)}-\frac{\sqrt[4]{c} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{\sqrt{2} \sqrt[4]{d} (b c-a d)}+\frac{\sqrt[4]{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{\sqrt{2} \sqrt[4]{d} (b c-a d)}",1,"(2*a^(1/4)*d^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 2*a^(1/4)*d^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 2*b^(1/4)*c^(1/4)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + 2*b^(1/4)*c^(1/4)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + a^(1/4)*d^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - a^(1/4)*d^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - b^(1/4)*c^(1/4)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x] + b^(1/4)*c^(1/4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(2*Sqrt[2]*b^(1/4)*d^(1/4)*(b*c - a*d))","A",1
465,1,364,463,0.1390635,"\int \frac{\sqrt{x}}{\left(a+b x^2\right) \left(c+d x^2\right)} \, dx","Integrate[Sqrt[x]/((a + b*x^2)*(c + d*x^2)),x]","\frac{\sqrt[4]{b} \sqrt[4]{c} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-\sqrt[4]{b} \sqrt[4]{c} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-2 \sqrt[4]{b} \sqrt[4]{c} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)+2 \sqrt[4]{b} \sqrt[4]{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)-\sqrt[4]{a} \sqrt[4]{d} \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)+\sqrt[4]{a} \sqrt[4]{d} \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)+2 \sqrt[4]{a} \sqrt[4]{d} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)-2 \sqrt[4]{a} \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{2 \sqrt{2} \sqrt[4]{a} \sqrt[4]{c} (b c-a d)}","\frac{\sqrt[4]{b} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} \sqrt[4]{a} (b c-a d)}-\frac{\sqrt[4]{b} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} \sqrt[4]{a} (b c-a d)}-\frac{\sqrt[4]{d} \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} \sqrt[4]{c} (b c-a d)}+\frac{\sqrt[4]{d} \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} \sqrt[4]{c} (b c-a d)}-\frac{\sqrt[4]{b} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} \sqrt[4]{a} (b c-a d)}+\frac{\sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} \sqrt[4]{a} (b c-a d)}+\frac{\sqrt[4]{d} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{\sqrt{2} \sqrt[4]{c} (b c-a d)}-\frac{\sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{\sqrt{2} \sqrt[4]{c} (b c-a d)}",1,"(-2*b^(1/4)*c^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 2*b^(1/4)*c^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 2*a^(1/4)*d^(1/4)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] - 2*a^(1/4)*d^(1/4)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + b^(1/4)*c^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - b^(1/4)*c^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - a^(1/4)*d^(1/4)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x] + a^(1/4)*d^(1/4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(2*Sqrt[2]*a^(1/4)*c^(1/4)*(b*c - a*d))","A",1
466,1,364,463,0.1218316,"\int \frac{1}{\sqrt{x} \left(a+b x^2\right) \left(c+d x^2\right)} \, dx","Integrate[1/(Sqrt[x]*(a + b*x^2)*(c + d*x^2)),x]","\frac{a^{3/4} d^{3/4} \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)-a^{3/4} d^{3/4} \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)+2 a^{3/4} d^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)-2 a^{3/4} d^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)-b^{3/4} c^{3/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+b^{3/4} c^{3/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-2 b^{3/4} c^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)+2 b^{3/4} c^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} a^{3/4} c^{3/4} (b c-a d)}","-\frac{b^{3/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{3/4} (b c-a d)}+\frac{b^{3/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{3/4} (b c-a d)}-\frac{b^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} a^{3/4} (b c-a d)}+\frac{b^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} a^{3/4} (b c-a d)}+\frac{d^{3/4} \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} c^{3/4} (b c-a d)}-\frac{d^{3/4} \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} c^{3/4} (b c-a d)}+\frac{d^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{\sqrt{2} c^{3/4} (b c-a d)}-\frac{d^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{\sqrt{2} c^{3/4} (b c-a d)}",1,"(-2*b^(3/4)*c^(3/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 2*b^(3/4)*c^(3/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 2*a^(3/4)*d^(3/4)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] - 2*a^(3/4)*d^(3/4)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] - b^(3/4)*c^(3/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + b^(3/4)*c^(3/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + a^(3/4)*d^(3/4)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x] - a^(3/4)*d^(3/4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(2*Sqrt[2]*a^(3/4)*c^(3/4)*(b*c - a*d))","A",1
467,1,409,476,0.2327075,"\int \frac{1}{x^{3/2} \left(a+b x^2\right) \left(c+d x^2\right)} \, dx","Integrate[1/(x^(3/2)*(a + b*x^2)*(c + d*x^2)),x]","\frac{\frac{\sqrt{2} b^{5/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{5/4}}-\frac{\sqrt{2} b^{5/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{5/4}}-\frac{2 \sqrt{2} b^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{a^{5/4}}+\frac{2 \sqrt{2} b^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{a^{5/4}}+\frac{8 b}{a \sqrt{x}}-\frac{\sqrt{2} d^{5/4} \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{5/4}}+\frac{\sqrt{2} d^{5/4} \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{5/4}}+\frac{2 \sqrt{2} d^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{c^{5/4}}-\frac{2 \sqrt{2} d^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{c^{5/4}}-\frac{8 d}{c \sqrt{x}}}{4 a d-4 b c}","-\frac{b^{5/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{5/4} (b c-a d)}+\frac{b^{5/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{5/4} (b c-a d)}+\frac{b^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} a^{5/4} (b c-a d)}-\frac{b^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} a^{5/4} (b c-a d)}+\frac{d^{5/4} \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} c^{5/4} (b c-a d)}-\frac{d^{5/4} \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} c^{5/4} (b c-a d)}-\frac{d^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{\sqrt{2} c^{5/4} (b c-a d)}+\frac{d^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{\sqrt{2} c^{5/4} (b c-a d)}-\frac{2}{a c \sqrt{x}}",1,"((8*b)/(a*Sqrt[x]) - (8*d)/(c*Sqrt[x]) - (2*Sqrt[2]*b^(5/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(5/4) + (2*Sqrt[2]*b^(5/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(5/4) + (2*Sqrt[2]*d^(5/4)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(5/4) - (2*Sqrt[2]*d^(5/4)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(5/4) + (Sqrt[2]*b^(5/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(5/4) - (Sqrt[2]*b^(5/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(5/4) - (Sqrt[2]*d^(5/4)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(5/4) + (Sqrt[2]*d^(5/4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(5/4))/(-4*b*c + 4*a*d)","A",1
468,1,411,478,0.2393429,"\int \frac{1}{x^{5/2} \left(a+b x^2\right) \left(c+d x^2\right)} \, dx","Integrate[1/(x^(5/2)*(a + b*x^2)*(c + d*x^2)),x]","\frac{-\frac{3 \sqrt{2} b^{7/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{7/4}}+\frac{3 \sqrt{2} b^{7/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{7/4}}-\frac{6 \sqrt{2} b^{7/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{a^{7/4}}+\frac{6 \sqrt{2} b^{7/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{a^{7/4}}+\frac{8 b}{a x^{3/2}}+\frac{3 \sqrt{2} d^{7/4} \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{7/4}}-\frac{3 \sqrt{2} d^{7/4} \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{7/4}}+\frac{6 \sqrt{2} d^{7/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{c^{7/4}}-\frac{6 \sqrt{2} d^{7/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{c^{7/4}}-\frac{8 d}{c x^{3/2}}}{12 a d-12 b c}","\frac{b^{7/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{7/4} (b c-a d)}-\frac{b^{7/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{7/4} (b c-a d)}+\frac{b^{7/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} a^{7/4} (b c-a d)}-\frac{b^{7/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} a^{7/4} (b c-a d)}-\frac{d^{7/4} \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} c^{7/4} (b c-a d)}+\frac{d^{7/4} \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} c^{7/4} (b c-a d)}-\frac{d^{7/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{\sqrt{2} c^{7/4} (b c-a d)}+\frac{d^{7/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{\sqrt{2} c^{7/4} (b c-a d)}-\frac{2}{3 a c x^{3/2}}",1,"((8*b)/(a*x^(3/2)) - (8*d)/(c*x^(3/2)) - (6*Sqrt[2]*b^(7/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(7/4) + (6*Sqrt[2]*b^(7/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(7/4) + (6*Sqrt[2]*d^(7/4)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(7/4) - (6*Sqrt[2]*d^(7/4)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(7/4) - (3*Sqrt[2]*b^(7/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(7/4) + (3*Sqrt[2]*b^(7/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(7/4) + (3*Sqrt[2]*d^(7/4)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(7/4) - (3*Sqrt[2]*d^(7/4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(7/4))/(-12*b*c + 12*a*d)","A",1
469,1,437,498,0.2778149,"\int \frac{1}{x^{7/2} \left(a+b x^2\right) \left(c+d x^2\right)} \, dx","Integrate[1/(x^(7/2)*(a + b*x^2)*(c + d*x^2)),x]","\frac{-\frac{5 \sqrt{2} b^{9/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{9/4}}+\frac{5 \sqrt{2} b^{9/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{9/4}}+\frac{10 \sqrt{2} b^{9/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{a^{9/4}}-\frac{10 \sqrt{2} b^{9/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{a^{9/4}}-\frac{40 b^2}{a^2 \sqrt{x}}+\frac{8 b}{a x^{5/2}}+\frac{5 \sqrt{2} d^{9/4} \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{9/4}}-\frac{5 \sqrt{2} d^{9/4} \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{9/4}}-\frac{10 \sqrt{2} d^{9/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{c^{9/4}}+\frac{10 \sqrt{2} d^{9/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{c^{9/4}}+\frac{40 d^2}{c^2 \sqrt{x}}-\frac{8 d}{c x^{5/2}}}{20 a d-20 b c}","\frac{b^{9/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{9/4} (b c-a d)}-\frac{b^{9/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{9/4} (b c-a d)}-\frac{b^{9/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} a^{9/4} (b c-a d)}+\frac{b^{9/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} a^{9/4} (b c-a d)}+\frac{2 (a d+b c)}{a^2 c^2 \sqrt{x}}-\frac{d^{9/4} \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} c^{9/4} (b c-a d)}+\frac{d^{9/4} \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{2 \sqrt{2} c^{9/4} (b c-a d)}+\frac{d^{9/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{\sqrt{2} c^{9/4} (b c-a d)}-\frac{d^{9/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{\sqrt{2} c^{9/4} (b c-a d)}-\frac{2}{5 a c x^{5/2}}",1,"((8*b)/(a*x^(5/2)) - (8*d)/(c*x^(5/2)) - (40*b^2)/(a^2*Sqrt[x]) + (40*d^2)/(c^2*Sqrt[x]) + (10*Sqrt[2]*b^(9/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(9/4) - (10*Sqrt[2]*b^(9/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(9/4) - (10*Sqrt[2]*d^(9/4)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(9/4) + (10*Sqrt[2]*d^(9/4)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(9/4) - (5*Sqrt[2]*b^(9/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(9/4) + (5*Sqrt[2]*b^(9/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(9/4) + (5*Sqrt[2]*d^(9/4)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(9/4) - (5*Sqrt[2]*d^(9/4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(9/4))/(-20*b*c + 20*a*d)","A",1
470,1,563,570,0.357357,"\int \frac{x^{11/2}}{\left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","Integrate[x^(11/2)/((a + b*x^2)*(c + d*x^2)^2),x]","\frac{4 \sqrt{2} a^{9/4} d^{9/4} \left(c+d x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-4 \sqrt{2} a^{9/4} d^{9/4} \left(c+d x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+8 \sqrt{2} a^{9/4} d^{9/4} \left(c+d x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)-8 \sqrt{2} a^{9/4} d^{9/4} \left(c+d x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)+\sqrt{2} b^{5/4} c^{5/4} \left(c+d x^2\right) (5 b c-9 a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)-\sqrt{2} b^{5/4} c^{5/4} \left(c+d x^2\right) (5 b c-9 a d) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)+2 \sqrt{2} b^{5/4} c^{5/4} \left(c+d x^2\right) (5 b c-9 a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)-2 \sqrt{2} b^{5/4} c^{5/4} \left(c+d x^2\right) (5 b c-9 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)+8 b^{5/4} c^2 \sqrt[4]{d} \sqrt{x} (b c-a d)+32 \sqrt[4]{b} \sqrt[4]{d} \sqrt{x} \left(c+d x^2\right) (b c-a d)^2}{16 b^{5/4} d^{9/4} \left(c+d x^2\right) (b c-a d)^2}","\frac{a^{9/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} b^{5/4} (b c-a d)^2}-\frac{a^{9/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} b^{5/4} (b c-a d)^2}+\frac{a^{9/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} b^{5/4} (b c-a d)^2}-\frac{a^{9/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} b^{5/4} (b c-a d)^2}+\frac{c^{5/4} (5 b c-9 a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} d^{9/4} (b c-a d)^2}-\frac{c^{5/4} (5 b c-9 a d) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} d^{9/4} (b c-a d)^2}+\frac{c^{5/4} (5 b c-9 a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{4 \sqrt{2} d^{9/4} (b c-a d)^2}-\frac{c^{5/4} (5 b c-9 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{4 \sqrt{2} d^{9/4} (b c-a d)^2}+\frac{\sqrt{x} (5 b c-4 a d)}{2 b d^2 (b c-a d)}-\frac{c x^{5/2}}{2 d \left(c+d x^2\right) (b c-a d)}",1,"(8*b^(5/4)*c^2*d^(1/4)*(b*c - a*d)*Sqrt[x] + 32*b^(1/4)*d^(1/4)*(b*c - a*d)^2*Sqrt[x]*(c + d*x^2) + 8*Sqrt[2]*a^(9/4)*d^(9/4)*(c + d*x^2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 8*Sqrt[2]*a^(9/4)*d^(9/4)*(c + d*x^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 2*Sqrt[2]*b^(5/4)*c^(5/4)*(5*b*c - 9*a*d)*(c + d*x^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] - 2*Sqrt[2]*b^(5/4)*c^(5/4)*(5*b*c - 9*a*d)*(c + d*x^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + 4*Sqrt[2]*a^(9/4)*d^(9/4)*(c + d*x^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - 4*Sqrt[2]*a^(9/4)*d^(9/4)*(c + d*x^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + Sqrt[2]*b^(5/4)*c^(5/4)*(5*b*c - 9*a*d)*(c + d*x^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x] - Sqrt[2]*b^(5/4)*c^(5/4)*(5*b*c - 9*a*d)*(c + d*x^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(16*b^(5/4)*d^(9/4)*(b*c - a*d)^2*(c + d*x^2))","A",1
471,1,527,536,0.3123895,"\int \frac{x^{9/2}}{\left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","Integrate[x^(9/2)/((a + b*x^2)*(c + d*x^2)^2),x]","\frac{4 \sqrt{2} a^{7/4} d^{7/4} \left(c+d x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-4 \sqrt{2} a^{7/4} d^{7/4} \left(c+d x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-8 \sqrt{2} a^{7/4} d^{7/4} \left(c+d x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)+8 \sqrt{2} a^{7/4} d^{7/4} \left(c+d x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)+\sqrt{2} b^{3/4} c^{3/4} \left(c+d x^2\right) (3 b c-7 a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)-\sqrt{2} b^{3/4} c^{3/4} \left(c+d x^2\right) (3 b c-7 a d) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)-2 \sqrt{2} b^{3/4} c^{3/4} \left(c+d x^2\right) (3 b c-7 a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)+2 \sqrt{2} b^{3/4} c^{3/4} \left(c+d x^2\right) (3 b c-7 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)-8 b^{3/4} c d^{3/4} x^{3/2} (b c-a d)}{16 b^{3/4} d^{7/4} \left(c+d x^2\right) (b c-a d)^2}","\frac{a^{7/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} b^{3/4} (b c-a d)^2}-\frac{a^{7/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} b^{3/4} (b c-a d)^2}-\frac{a^{7/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} b^{3/4} (b c-a d)^2}+\frac{a^{7/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} b^{3/4} (b c-a d)^2}+\frac{c^{3/4} (3 b c-7 a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} d^{7/4} (b c-a d)^2}-\frac{c^{3/4} (3 b c-7 a d) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} d^{7/4} (b c-a d)^2}-\frac{c^{3/4} (3 b c-7 a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{4 \sqrt{2} d^{7/4} (b c-a d)^2}+\frac{c^{3/4} (3 b c-7 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{4 \sqrt{2} d^{7/4} (b c-a d)^2}-\frac{c x^{3/2}}{2 d \left(c+d x^2\right) (b c-a d)}",1,"(-8*b^(3/4)*c*d^(3/4)*(b*c - a*d)*x^(3/2) - 8*Sqrt[2]*a^(7/4)*d^(7/4)*(c + d*x^2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 8*Sqrt[2]*a^(7/4)*d^(7/4)*(c + d*x^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 2*Sqrt[2]*b^(3/4)*c^(3/4)*(3*b*c - 7*a*d)*(c + d*x^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + 2*Sqrt[2]*b^(3/4)*c^(3/4)*(3*b*c - 7*a*d)*(c + d*x^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + 4*Sqrt[2]*a^(7/4)*d^(7/4)*(c + d*x^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - 4*Sqrt[2]*a^(7/4)*d^(7/4)*(c + d*x^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + Sqrt[2]*b^(3/4)*c^(3/4)*(3*b*c - 7*a*d)*(c + d*x^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x] - Sqrt[2]*b^(3/4)*c^(3/4)*(3*b*c - 7*a*d)*(c + d*x^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(16*b^(3/4)*d^(7/4)*(b*c - a*d)^2*(c + d*x^2))","A",1
472,1,523,532,0.3002497,"\int \frac{x^{7/2}}{\left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","Integrate[x^(7/2)/((a + b*x^2)*(c + d*x^2)^2),x]","\frac{-4 \sqrt{2} a^{5/4} d^{5/4} \left(c+d x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+4 \sqrt{2} a^{5/4} d^{5/4} \left(c+d x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-8 \sqrt{2} a^{5/4} d^{5/4} \left(c+d x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)+8 \sqrt{2} a^{5/4} d^{5/4} \left(c+d x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \left(c+d x^2\right) (b c-5 a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)+\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \left(c+d x^2\right) (b c-5 a d) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)-2 \sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \left(c+d x^2\right) (b c-5 a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)+2 \sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \left(c+d x^2\right) (b c-5 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)-8 \sqrt[4]{b} c \sqrt[4]{d} \sqrt{x} (b c-a d)}{16 \sqrt[4]{b} d^{5/4} \left(c+d x^2\right) (b c-a d)^2}","-\frac{a^{5/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} \sqrt[4]{b} (b c-a d)^2}+\frac{a^{5/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} \sqrt[4]{b} (b c-a d)^2}-\frac{a^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} \sqrt[4]{b} (b c-a d)^2}+\frac{a^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} \sqrt[4]{b} (b c-a d)^2}-\frac{\sqrt[4]{c} (b c-5 a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} d^{5/4} (b c-a d)^2}+\frac{\sqrt[4]{c} (b c-5 a d) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} d^{5/4} (b c-a d)^2}-\frac{\sqrt[4]{c} (b c-5 a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{4 \sqrt{2} d^{5/4} (b c-a d)^2}+\frac{\sqrt[4]{c} (b c-5 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{4 \sqrt{2} d^{5/4} (b c-a d)^2}-\frac{c \sqrt{x}}{2 d \left(c+d x^2\right) (b c-a d)}",1,"(-8*b^(1/4)*c*d^(1/4)*(b*c - a*d)*Sqrt[x] - 8*Sqrt[2]*a^(5/4)*d^(5/4)*(c + d*x^2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 8*Sqrt[2]*a^(5/4)*d^(5/4)*(c + d*x^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 2*Sqrt[2]*b^(1/4)*c^(1/4)*(b*c - 5*a*d)*(c + d*x^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + 2*Sqrt[2]*b^(1/4)*c^(1/4)*(b*c - 5*a*d)*(c + d*x^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] - 4*Sqrt[2]*a^(5/4)*d^(5/4)*(c + d*x^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 4*Sqrt[2]*a^(5/4)*d^(5/4)*(c + d*x^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - Sqrt[2]*b^(1/4)*c^(1/4)*(b*c - 5*a*d)*(c + d*x^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x] + Sqrt[2]*b^(1/4)*c^(1/4)*(b*c - 5*a*d)*(c + d*x^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(16*b^(1/4)*d^(5/4)*(b*c - a*d)^2*(c + d*x^2))","A",1
473,1,522,528,0.2906283,"\int \frac{x^{5/2}}{\left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","Integrate[x^(5/2)/((a + b*x^2)*(c + d*x^2)^2),x]","\frac{-4 \sqrt{2} a^{3/4} \sqrt[4]{b} \sqrt[4]{c} d^{3/4} \left(c+d x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+4 \sqrt{2} a^{3/4} \sqrt[4]{b} \sqrt[4]{c} d^{3/4} \left(c+d x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+8 \sqrt{2} a^{3/4} \sqrt[4]{b} \sqrt[4]{c} d^{3/4} \left(c+d x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)-8 \sqrt{2} a^{3/4} \sqrt[4]{b} \sqrt[4]{c} d^{3/4} \left(c+d x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)+8 \sqrt[4]{c} d^{3/4} x^{3/2} (b c-a d)+\sqrt{2} \left(c+d x^2\right) (3 a d+b c) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)-\sqrt{2} \left(c+d x^2\right) (3 a d+b c) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)-2 \sqrt{2} \left(c+d x^2\right) (3 a d+b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)+2 \sqrt{2} \left(c+d x^2\right) (3 a d+b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{16 \sqrt[4]{c} d^{3/4} \left(c+d x^2\right) (b c-a d)^2}","-\frac{a^{3/4} \sqrt[4]{b} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} (b c-a d)^2}+\frac{a^{3/4} \sqrt[4]{b} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} (b c-a d)^2}+\frac{a^{3/4} \sqrt[4]{b} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} (b c-a d)^2}-\frac{a^{3/4} \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} (b c-a d)^2}+\frac{(3 a d+b c) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} \sqrt[4]{c} d^{3/4} (b c-a d)^2}-\frac{(3 a d+b c) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} \sqrt[4]{c} d^{3/4} (b c-a d)^2}-\frac{(3 a d+b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{4 \sqrt{2} \sqrt[4]{c} d^{3/4} (b c-a d)^2}+\frac{(3 a d+b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{4 \sqrt{2} \sqrt[4]{c} d^{3/4} (b c-a d)^2}+\frac{x^{3/2}}{2 \left(c+d x^2\right) (b c-a d)}",1,"(8*c^(1/4)*d^(3/4)*(b*c - a*d)*x^(3/2) + 8*Sqrt[2]*a^(3/4)*b^(1/4)*c^(1/4)*d^(3/4)*(c + d*x^2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 8*Sqrt[2]*a^(3/4)*b^(1/4)*c^(1/4)*d^(3/4)*(c + d*x^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 2*Sqrt[2]*(b*c + 3*a*d)*(c + d*x^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + 2*Sqrt[2]*(b*c + 3*a*d)*(c + d*x^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] - 4*Sqrt[2]*a^(3/4)*b^(1/4)*c^(1/4)*d^(3/4)*(c + d*x^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 4*Sqrt[2]*a^(3/4)*b^(1/4)*c^(1/4)*d^(3/4)*(c + d*x^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + Sqrt[2]*(b*c + 3*a*d)*(c + d*x^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x] - Sqrt[2]*(b*c + 3*a*d)*(c + d*x^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(16*c^(1/4)*d^(3/4)*(b*c - a*d)^2*(c + d*x^2))","A",1
474,1,522,528,0.2974749,"\int \frac{x^{3/2}}{\left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","Integrate[x^(3/2)/((a + b*x^2)*(c + d*x^2)^2),x]","\frac{4 \sqrt{2} \sqrt[4]{a} b^{3/4} c^{3/4} \sqrt[4]{d} \left(c+d x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-4 \sqrt{2} \sqrt[4]{a} b^{3/4} c^{3/4} \sqrt[4]{d} \left(c+d x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+8 \sqrt{2} \sqrt[4]{a} b^{3/4} c^{3/4} \sqrt[4]{d} \left(c+d x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)-8 \sqrt{2} \sqrt[4]{a} b^{3/4} c^{3/4} \sqrt[4]{d} \left(c+d x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)+8 c^{3/4} \sqrt[4]{d} \sqrt{x} (b c-a d)-\sqrt{2} \left(c+d x^2\right) (a d+3 b c) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)+\sqrt{2} \left(c+d x^2\right) (a d+3 b c) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)-2 \sqrt{2} \left(c+d x^2\right) (a d+3 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)+2 \sqrt{2} \left(c+d x^2\right) (a d+3 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{16 c^{3/4} \sqrt[4]{d} \left(c+d x^2\right) (b c-a d)^2}","\frac{\sqrt[4]{a} b^{3/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} (b c-a d)^2}-\frac{\sqrt[4]{a} b^{3/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} (b c-a d)^2}+\frac{\sqrt[4]{a} b^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} (b c-a d)^2}-\frac{\sqrt[4]{a} b^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} (b c-a d)^2}-\frac{(a d+3 b c) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{3/4} \sqrt[4]{d} (b c-a d)^2}+\frac{(a d+3 b c) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{3/4} \sqrt[4]{d} (b c-a d)^2}-\frac{(a d+3 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{4 \sqrt{2} c^{3/4} \sqrt[4]{d} (b c-a d)^2}+\frac{(a d+3 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{4 \sqrt{2} c^{3/4} \sqrt[4]{d} (b c-a d)^2}+\frac{\sqrt{x}}{2 \left(c+d x^2\right) (b c-a d)}",1,"(8*c^(3/4)*d^(1/4)*(b*c - a*d)*Sqrt[x] + 8*Sqrt[2]*a^(1/4)*b^(3/4)*c^(3/4)*d^(1/4)*(c + d*x^2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 8*Sqrt[2]*a^(1/4)*b^(3/4)*c^(3/4)*d^(1/4)*(c + d*x^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 2*Sqrt[2]*(3*b*c + a*d)*(c + d*x^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + 2*Sqrt[2]*(3*b*c + a*d)*(c + d*x^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + 4*Sqrt[2]*a^(1/4)*b^(3/4)*c^(3/4)*d^(1/4)*(c + d*x^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - 4*Sqrt[2]*a^(1/4)*b^(3/4)*c^(3/4)*d^(1/4)*(c + d*x^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - Sqrt[2]*(3*b*c + a*d)*(c + d*x^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x] + Sqrt[2]*(3*b*c + a*d)*(c + d*x^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(16*c^(3/4)*d^(1/4)*(b*c - a*d)^2*(c + d*x^2))","A",1
475,1,523,536,0.4155434,"\int \frac{\sqrt{x}}{\left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","Integrate[Sqrt[x]/((a + b*x^2)*(c + d*x^2)^2),x]","\frac{4 \sqrt{2} b^{5/4} c^{5/4} \left(c+d x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-4 \sqrt{2} b^{5/4} c^{5/4} \left(c+d x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-8 \sqrt{2} b^{5/4} c^{5/4} \left(c+d x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)+8 \sqrt{2} b^{5/4} c^{5/4} \left(c+d x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)+8 \sqrt[4]{a} \sqrt[4]{c} d x^{3/2} (a d-b c)+\sqrt{2} \sqrt[4]{a} \sqrt[4]{d} \left(c+d x^2\right) (a d-5 b c) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)+\sqrt{2} \sqrt[4]{a} \sqrt[4]{d} \left(c+d x^2\right) (5 b c-a d) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)-2 \sqrt{2} \sqrt[4]{a} \sqrt[4]{d} \left(c+d x^2\right) (a d-5 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)+2 \sqrt{2} \sqrt[4]{a} \sqrt[4]{d} \left(c+d x^2\right) (a d-5 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{16 \sqrt[4]{a} c^{5/4} \left(c+d x^2\right) (b c-a d)^2}","\frac{b^{5/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} \sqrt[4]{a} (b c-a d)^2}-\frac{b^{5/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} \sqrt[4]{a} (b c-a d)^2}-\frac{b^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} \sqrt[4]{a} (b c-a d)^2}+\frac{b^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} \sqrt[4]{a} (b c-a d)^2}-\frac{\sqrt[4]{d} (5 b c-a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{5/4} (b c-a d)^2}+\frac{\sqrt[4]{d} (5 b c-a d) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{5/4} (b c-a d)^2}+\frac{\sqrt[4]{d} (5 b c-a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{4 \sqrt{2} c^{5/4} (b c-a d)^2}-\frac{\sqrt[4]{d} (5 b c-a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{4 \sqrt{2} c^{5/4} (b c-a d)^2}-\frac{d x^{3/2}}{2 c \left(c+d x^2\right) (b c-a d)}",1,"(8*a^(1/4)*c^(1/4)*d*(-(b*c) + a*d)*x^(3/2) - 8*Sqrt[2]*b^(5/4)*c^(5/4)*(c + d*x^2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 8*Sqrt[2]*b^(5/4)*c^(5/4)*(c + d*x^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 2*Sqrt[2]*a^(1/4)*d^(1/4)*(-5*b*c + a*d)*(c + d*x^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + 2*Sqrt[2]*a^(1/4)*d^(1/4)*(-5*b*c + a*d)*(c + d*x^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + 4*Sqrt[2]*b^(5/4)*c^(5/4)*(c + d*x^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - 4*Sqrt[2]*b^(5/4)*c^(5/4)*(c + d*x^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + Sqrt[2]*a^(1/4)*d^(1/4)*(-5*b*c + a*d)*(c + d*x^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x] + Sqrt[2]*a^(1/4)*d^(1/4)*(5*b*c - a*d)*(c + d*x^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(16*a^(1/4)*c^(5/4)*(b*c - a*d)^2*(c + d*x^2))","A",1
476,1,526,536,0.4222102,"\int \frac{1}{\sqrt{x} \left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","Integrate[1/(Sqrt[x]*(a + b*x^2)*(c + d*x^2)^2),x]","\frac{8 a^{3/4} c^{3/4} d \sqrt{x} (a d-b c)+\sqrt{2} a^{3/4} d^{3/4} \left(c+d x^2\right) (7 b c-3 a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)+\sqrt{2} a^{3/4} d^{3/4} \left(c+d x^2\right) (3 a d-7 b c) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)-2 \sqrt{2} a^{3/4} d^{3/4} \left(c+d x^2\right) (3 a d-7 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)+2 \sqrt{2} a^{3/4} d^{3/4} \left(c+d x^2\right) (3 a d-7 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)-4 \sqrt{2} b^{7/4} c^{7/4} \left(c+d x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+4 \sqrt{2} b^{7/4} c^{7/4} \left(c+d x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-8 \sqrt{2} b^{7/4} c^{7/4} \left(c+d x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)+8 \sqrt{2} b^{7/4} c^{7/4} \left(c+d x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{16 a^{3/4} c^{7/4} \left(c+d x^2\right) (b c-a d)^2}","-\frac{b^{7/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{3/4} (b c-a d)^2}+\frac{b^{7/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{3/4} (b c-a d)^2}-\frac{b^{7/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} a^{3/4} (b c-a d)^2}+\frac{b^{7/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} a^{3/4} (b c-a d)^2}+\frac{d^{3/4} (7 b c-3 a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{7/4} (b c-a d)^2}-\frac{d^{3/4} (7 b c-3 a d) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{7/4} (b c-a d)^2}+\frac{d^{3/4} (7 b c-3 a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{4 \sqrt{2} c^{7/4} (b c-a d)^2}-\frac{d^{3/4} (7 b c-3 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{4 \sqrt{2} c^{7/4} (b c-a d)^2}-\frac{d \sqrt{x}}{2 c \left(c+d x^2\right) (b c-a d)}",1,"(8*a^(3/4)*c^(3/4)*d*(-(b*c) + a*d)*Sqrt[x] - 8*Sqrt[2]*b^(7/4)*c^(7/4)*(c + d*x^2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 8*Sqrt[2]*b^(7/4)*c^(7/4)*(c + d*x^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 2*Sqrt[2]*a^(3/4)*d^(3/4)*(-7*b*c + 3*a*d)*(c + d*x^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + 2*Sqrt[2]*a^(3/4)*d^(3/4)*(-7*b*c + 3*a*d)*(c + d*x^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] - 4*Sqrt[2]*b^(7/4)*c^(7/4)*(c + d*x^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 4*Sqrt[2]*b^(7/4)*c^(7/4)*(c + d*x^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + Sqrt[2]*a^(3/4)*d^(3/4)*(7*b*c - 3*a*d)*(c + d*x^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x] + Sqrt[2]*a^(3/4)*d^(3/4)*(-7*b*c + 3*a*d)*(c + d*x^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(16*a^(3/4)*c^(7/4)*(b*c - a*d)^2*(c + d*x^2))","A",1
477,1,540,570,2.1736891,"\int \frac{1}{x^{3/2} \left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","Integrate[1/(x^(3/2)*(a + b*x^2)*(c + d*x^2)^2),x]","\frac{1}{16} \left(-\frac{4 \sqrt{2} b^{9/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{5/4} (b c-a d)^2}+\frac{4 \sqrt{2} b^{9/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{5/4} (b c-a d)^2}+\frac{8 \sqrt{2} b^{9/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{a^{5/4} (b c-a d)^2}-\frac{8 \sqrt{2} b^{9/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{a^{5/4} (b c-a d)^2}+\frac{\sqrt{2} d^{5/4} (9 b c-5 a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{9/4} (b c-a d)^2}+\frac{\sqrt{2} d^{5/4} (5 a d-9 b c) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{9/4} (b c-a d)^2}+\frac{2 \sqrt{2} d^{5/4} (5 a d-9 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{c^{9/4} (b c-a d)^2}+\frac{2 \sqrt{2} d^{5/4} (9 b c-5 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{c^{9/4} (b c-a d)^2}+\frac{8 d^2 x^{3/2}}{c^2 \left(c+d x^2\right) (b c-a d)}-\frac{32}{a c^2 \sqrt{x}}\right)","-\frac{b^{9/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{5/4} (b c-a d)^2}+\frac{b^{9/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{5/4} (b c-a d)^2}+\frac{b^{9/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} a^{5/4} (b c-a d)^2}-\frac{b^{9/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} a^{5/4} (b c-a d)^2}+\frac{d^{5/4} (9 b c-5 a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{9/4} (b c-a d)^2}-\frac{d^{5/4} (9 b c-5 a d) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{9/4} (b c-a d)^2}-\frac{d^{5/4} (9 b c-5 a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{4 \sqrt{2} c^{9/4} (b c-a d)^2}+\frac{d^{5/4} (9 b c-5 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{4 \sqrt{2} c^{9/4} (b c-a d)^2}-\frac{4 b c-5 a d}{2 a c^2 \sqrt{x} (b c-a d)}-\frac{d}{2 c \sqrt{x} \left(c+d x^2\right) (b c-a d)}",1,"(-32/(a*c^2*Sqrt[x]) + (8*d^2*x^(3/2))/(c^2*(b*c - a*d)*(c + d*x^2)) + (8*Sqrt[2]*b^(9/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(5/4)*(b*c - a*d)^2) - (8*Sqrt[2]*b^(9/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(5/4)*(b*c - a*d)^2) + (2*Sqrt[2]*d^(5/4)*(-9*b*c + 5*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(9/4)*(b*c - a*d)^2) + (2*Sqrt[2]*d^(5/4)*(9*b*c - 5*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(9/4)*(b*c - a*d)^2) - (4*Sqrt[2]*b^(9/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(5/4)*(b*c - a*d)^2) + (4*Sqrt[2]*b^(9/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(5/4)*(b*c - a*d)^2) + (Sqrt[2]*d^(5/4)*(9*b*c - 5*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(9/4)*(b*c - a*d)^2) + (Sqrt[2]*d^(5/4)*(-9*b*c + 5*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(9/4)*(b*c - a*d)^2))/16","A",1
478,1,602,570,6.1456651,"\int \frac{1}{x^{5/2} \left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","Integrate[1/(x^(5/2)*(a + b*x^2)*(c + d*x^2)^2),x]","\frac{b^{11/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{7/4} (b c-a d)^2}-\frac{b^{11/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{7/4} (b c-a d)^2}-\frac{b^{11/4} \tan ^{-1}\left(\frac{2 \sqrt[4]{b} \sqrt{x}-\sqrt{2} \sqrt[4]{a}}{\sqrt{2} \sqrt[4]{a}}\right)}{\sqrt{2} a^{7/4} (b c-a d)^2}-\frac{b^{11/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a}+2 \sqrt[4]{b} \sqrt{x}}{\sqrt{2} \sqrt[4]{a}}\right)}{\sqrt{2} a^{7/4} (b c-a d)^2}-\frac{d^{7/4} (11 b c-7 a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{11/4} (a d-b c)^2}+\frac{d^{7/4} (11 b c-7 a d) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{11/4} (a d-b c)^2}+\frac{d^{7/4} (11 b c-7 a d) \tan ^{-1}\left(\frac{2 \sqrt[4]{d} \sqrt{x}-\sqrt{2} \sqrt[4]{c}}{\sqrt{2} \sqrt[4]{c}}\right)}{4 \sqrt{2} c^{11/4} (a d-b c)^2}+\frac{d^{7/4} (11 b c-7 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c}+2 \sqrt[4]{d} \sqrt{x}}{\sqrt{2} \sqrt[4]{c}}\right)}{4 \sqrt{2} c^{11/4} (a d-b c)^2}+\frac{d^2 \sqrt{x}}{2 c^2 \left(c+d x^2\right) (b c-a d)}-\frac{2}{3 a c^2 x^{3/2}}","\frac{b^{11/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{7/4} (b c-a d)^2}-\frac{b^{11/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{7/4} (b c-a d)^2}+\frac{b^{11/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} a^{7/4} (b c-a d)^2}-\frac{b^{11/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} a^{7/4} (b c-a d)^2}-\frac{d^{7/4} (11 b c-7 a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{11/4} (b c-a d)^2}+\frac{d^{7/4} (11 b c-7 a d) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{11/4} (b c-a d)^2}-\frac{d^{7/4} (11 b c-7 a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{4 \sqrt{2} c^{11/4} (b c-a d)^2}+\frac{d^{7/4} (11 b c-7 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{4 \sqrt{2} c^{11/4} (b c-a d)^2}-\frac{4 b c-7 a d}{6 a c^2 x^{3/2} (b c-a d)}-\frac{d}{2 c x^{3/2} \left(c+d x^2\right) (b c-a d)}",1,"-2/(3*a*c^2*x^(3/2)) + (d^2*Sqrt[x])/(2*c^2*(b*c - a*d)*(c + d*x^2)) - (b^(11/4)*ArcTan[(-(Sqrt[2]*a^(1/4)) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(Sqrt[2]*a^(7/4)*(b*c - a*d)^2) - (b^(11/4)*ArcTan[(Sqrt[2]*a^(1/4) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(Sqrt[2]*a^(7/4)*(b*c - a*d)^2) + (d^(7/4)*(11*b*c - 7*a*d)*ArcTan[(-(Sqrt[2]*c^(1/4)) + 2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(4*Sqrt[2]*c^(11/4)*(-(b*c) + a*d)^2) + (d^(7/4)*(11*b*c - 7*a*d)*ArcTan[(Sqrt[2]*c^(1/4) + 2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(4*Sqrt[2]*c^(11/4)*(-(b*c) + a*d)^2) + (b^(11/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(7/4)*(b*c - a*d)^2) - (b^(11/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(7/4)*(b*c - a*d)^2) - (d^(7/4)*(11*b*c - 7*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(11/4)*(-(b*c) + a*d)^2) + (d^(7/4)*(11*b*c - 7*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(11/4)*(-(b*c) + a*d)^2)","A",1
479,1,621,618,6.1079206,"\int \frac{1}{x^{7/2} \left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","Integrate[1/(x^(7/2)*(a + b*x^2)*(c + d*x^2)^2),x]","\frac{b^{13/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{9/4} (b c-a d)^2}-\frac{b^{13/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{9/4} (b c-a d)^2}+\frac{b^{13/4} \tan ^{-1}\left(\frac{2 \sqrt[4]{b} \sqrt{x}-\sqrt{2} \sqrt[4]{a}}{\sqrt{2} \sqrt[4]{a}}\right)}{\sqrt{2} a^{9/4} (b c-a d)^2}+\frac{b^{13/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a}+2 \sqrt[4]{b} \sqrt{x}}{\sqrt{2} \sqrt[4]{a}}\right)}{\sqrt{2} a^{9/4} (b c-a d)^2}+\frac{2 (2 a d+b c)}{a^2 c^3 \sqrt{x}}-\frac{d^{9/4} (13 b c-9 a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{13/4} (a d-b c)^2}+\frac{d^{9/4} (13 b c-9 a d) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{13/4} (a d-b c)^2}-\frac{d^{9/4} (13 b c-9 a d) \tan ^{-1}\left(\frac{2 \sqrt[4]{d} \sqrt{x}-\sqrt{2} \sqrt[4]{c}}{\sqrt{2} \sqrt[4]{c}}\right)}{4 \sqrt{2} c^{13/4} (a d-b c)^2}-\frac{d^{9/4} (13 b c-9 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c}+2 \sqrt[4]{d} \sqrt{x}}{\sqrt{2} \sqrt[4]{c}}\right)}{4 \sqrt{2} c^{13/4} (a d-b c)^2}-\frac{d^3 x^{3/2}}{2 c^3 \left(c+d x^2\right) (b c-a d)}-\frac{2}{5 a c^2 x^{5/2}}","\frac{b^{13/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{9/4} (b c-a d)^2}-\frac{b^{13/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{9/4} (b c-a d)^2}-\frac{b^{13/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} a^{9/4} (b c-a d)^2}+\frac{b^{13/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} a^{9/4} (b c-a d)^2}+\frac{-9 a^2 d^2+4 a b c d+4 b^2 c^2}{2 a^2 c^3 \sqrt{x} (b c-a d)}-\frac{d^{9/4} (13 b c-9 a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{13/4} (b c-a d)^2}+\frac{d^{9/4} (13 b c-9 a d) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{13/4} (b c-a d)^2}+\frac{d^{9/4} (13 b c-9 a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{4 \sqrt{2} c^{13/4} (b c-a d)^2}-\frac{d^{9/4} (13 b c-9 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{4 \sqrt{2} c^{13/4} (b c-a d)^2}-\frac{4 b c-9 a d}{10 a c^2 x^{5/2} (b c-a d)}-\frac{d}{2 c x^{5/2} \left(c+d x^2\right) (b c-a d)}",1,"-2/(5*a*c^2*x^(5/2)) + (2*(b*c + 2*a*d))/(a^2*c^3*Sqrt[x]) - (d^3*x^(3/2))/(2*c^3*(b*c - a*d)*(c + d*x^2)) + (b^(13/4)*ArcTan[(-(Sqrt[2]*a^(1/4)) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(Sqrt[2]*a^(9/4)*(b*c - a*d)^2) + (b^(13/4)*ArcTan[(Sqrt[2]*a^(1/4) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(Sqrt[2]*a^(9/4)*(b*c - a*d)^2) - (d^(9/4)*(13*b*c - 9*a*d)*ArcTan[(-(Sqrt[2]*c^(1/4)) + 2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(4*Sqrt[2]*c^(13/4)*(-(b*c) + a*d)^2) - (d^(9/4)*(13*b*c - 9*a*d)*ArcTan[(Sqrt[2]*c^(1/4) + 2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(4*Sqrt[2]*c^(13/4)*(-(b*c) + a*d)^2) + (b^(13/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(9/4)*(b*c - a*d)^2) - (b^(13/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(9/4)*(b*c - a*d)^2) - (d^(9/4)*(13*b*c - 9*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(13/4)*(-(b*c) + a*d)^2) + (d^(9/4)*(13*b*c - 9*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(13/4)*(-(b*c) + a*d)^2)","A",1
480,1,640,631,0.538744,"\int \frac{x^{7/2}}{\left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","Integrate[x^(7/2)/((a + b*x^2)*(c + d*x^2)^3),x]","\frac{-32 \sqrt{2} a^{5/4} b^{3/4} c^{3/4} d^{5/4} \left(c+d x^2\right)^2 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+32 \sqrt{2} a^{5/4} b^{3/4} c^{3/4} d^{5/4} \left(c+d x^2\right)^2 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-64 \sqrt{2} a^{5/4} b^{3/4} c^{3/4} d^{5/4} \left(c+d x^2\right)^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)+64 \sqrt{2} a^{5/4} b^{3/4} c^{3/4} d^{5/4} \left(c+d x^2\right)^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)-\sqrt{2} \left(c+d x^2\right)^2 \left(-5 a^2 d^2-30 a b c d+3 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)+\sqrt{2} \left(c+d x^2\right)^2 \left(-5 a^2 d^2-30 a b c d+3 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)-2 \sqrt{2} \left(c+d x^2\right)^2 \left(-5 a^2 d^2-30 a b c d+3 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)+2 \sqrt{2} \left(c+d x^2\right)^2 \left(-5 a^2 d^2-30 a b c d+3 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)+8 c^{3/4} \sqrt[4]{d} \sqrt{x} \left(c+d x^2\right) (b c-9 a d) (b c-a d)-32 c^{7/4} \sqrt[4]{d} \sqrt{x} (b c-a d)^2}{128 c^{3/4} d^{5/4} \left(c+d x^2\right)^2 (b c-a d)^3}","-\frac{a^{5/4} b^{3/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} (b c-a d)^3}+\frac{a^{5/4} b^{3/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} (b c-a d)^3}-\frac{a^{5/4} b^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} (b c-a d)^3}+\frac{a^{5/4} b^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} (b c-a d)^3}-\frac{\left(-5 a^2 d^2-30 a b c d+3 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{3/4} d^{5/4} (b c-a d)^3}+\frac{\left(-5 a^2 d^2-30 a b c d+3 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{3/4} d^{5/4} (b c-a d)^3}-\frac{\left(-5 a^2 d^2-30 a b c d+3 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{32 \sqrt{2} c^{3/4} d^{5/4} (b c-a d)^3}+\frac{\left(-5 a^2 d^2-30 a b c d+3 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{32 \sqrt{2} c^{3/4} d^{5/4} (b c-a d)^3}+\frac{\sqrt{x} (b c-9 a d)}{16 d \left(c+d x^2\right) (b c-a d)^2}-\frac{c \sqrt{x}}{4 d \left(c+d x^2\right)^2 (b c-a d)}",1,"(-32*c^(7/4)*d^(1/4)*(b*c - a*d)^2*Sqrt[x] + 8*c^(3/4)*d^(1/4)*(b*c - 9*a*d)*(b*c - a*d)*Sqrt[x]*(c + d*x^2) - 64*Sqrt[2]*a^(5/4)*b^(3/4)*c^(3/4)*d^(5/4)*(c + d*x^2)^2*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 64*Sqrt[2]*a^(5/4)*b^(3/4)*c^(3/4)*d^(5/4)*(c + d*x^2)^2*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 2*Sqrt[2]*(3*b^2*c^2 - 30*a*b*c*d - 5*a^2*d^2)*(c + d*x^2)^2*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + 2*Sqrt[2]*(3*b^2*c^2 - 30*a*b*c*d - 5*a^2*d^2)*(c + d*x^2)^2*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] - 32*Sqrt[2]*a^(5/4)*b^(3/4)*c^(3/4)*d^(5/4)*(c + d*x^2)^2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 32*Sqrt[2]*a^(5/4)*b^(3/4)*c^(3/4)*d^(5/4)*(c + d*x^2)^2*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - Sqrt[2]*(3*b^2*c^2 - 30*a*b*c*d - 5*a^2*d^2)*(c + d*x^2)^2*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x] + Sqrt[2]*(3*b^2*c^2 - 30*a*b*c*d - 5*a^2*d^2)*(c + d*x^2)^2*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(128*c^(3/4)*d^(5/4)*(b*c - a*d)^3*(c + d*x^2)^2)","A",1
481,1,544,628,0.7483103,"\int \frac{x^{5/2}}{\left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","Integrate[x^(5/2)/((a + b*x^2)*(c + d*x^2)^3),x]","\frac{-32 \sqrt{2} a^{3/4} b^{5/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+32 \sqrt{2} a^{3/4} b^{5/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+64 \sqrt{2} a^{3/4} b^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)-64 \sqrt{2} a^{3/4} b^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)+\frac{\sqrt{2} \left(-3 a^2 d^2+30 a b c d+5 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{5/4} d^{3/4}}-\frac{\sqrt{2} \left(-3 a^2 d^2+30 a b c d+5 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{5/4} d^{3/4}}-\frac{2 \sqrt{2} \left(-3 a^2 d^2+30 a b c d+5 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{c^{5/4} d^{3/4}}+\frac{2 \sqrt{2} \left(-3 a^2 d^2+30 a b c d+5 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{c^{5/4} d^{3/4}}+\frac{32 x^{3/2} (b c-a d)^2}{\left(c+d x^2\right)^2}+\frac{8 x^{3/2} (3 a d+5 b c) (b c-a d)}{c \left(c+d x^2\right)}}{128 (b c-a d)^3}","-\frac{a^{3/4} b^{5/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} (b c-a d)^3}+\frac{a^{3/4} b^{5/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} (b c-a d)^3}+\frac{a^{3/4} b^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} (b c-a d)^3}-\frac{a^{3/4} b^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} (b c-a d)^3}+\frac{\left(-3 a^2 d^2+30 a b c d+5 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{5/4} d^{3/4} (b c-a d)^3}-\frac{\left(-3 a^2 d^2+30 a b c d+5 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{5/4} d^{3/4} (b c-a d)^3}-\frac{\left(-3 a^2 d^2+30 a b c d+5 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{32 \sqrt{2} c^{5/4} d^{3/4} (b c-a d)^3}+\frac{\left(-3 a^2 d^2+30 a b c d+5 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{32 \sqrt{2} c^{5/4} d^{3/4} (b c-a d)^3}+\frac{x^{3/2} (3 a d+5 b c)}{16 c \left(c+d x^2\right) (b c-a d)^2}+\frac{x^{3/2}}{4 \left(c+d x^2\right)^2 (b c-a d)}",1,"((32*(b*c - a*d)^2*x^(3/2))/(c + d*x^2)^2 + (8*(b*c - a*d)*(5*b*c + 3*a*d)*x^(3/2))/(c*(c + d*x^2)) + 64*Sqrt[2]*a^(3/4)*b^(5/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 64*Sqrt[2]*a^(3/4)*b^(5/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - (2*Sqrt[2]*(5*b^2*c^2 + 30*a*b*c*d - 3*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(5/4)*d^(3/4)) + (2*Sqrt[2]*(5*b^2*c^2 + 30*a*b*c*d - 3*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(5/4)*d^(3/4)) - 32*Sqrt[2]*a^(3/4)*b^(5/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 32*Sqrt[2]*a^(3/4)*b^(5/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + (Sqrt[2]*(5*b^2*c^2 + 30*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(5/4)*d^(3/4)) - (Sqrt[2]*(5*b^2*c^2 + 30*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(5/4)*d^(3/4)))/(128*(b*c - a*d)^3)","A",1
482,1,543,627,0.7295822,"\int \frac{x^{3/2}}{\left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","Integrate[x^(3/2)/((a + b*x^2)*(c + d*x^2)^3),x]","\frac{-\frac{\sqrt{2} \left(-3 a^2 d^2+14 a b c d+21 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{7/4} \sqrt[4]{d}}+\frac{\sqrt{2} \left(-3 a^2 d^2+14 a b c d+21 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{7/4} \sqrt[4]{d}}-\frac{2 \sqrt{2} \left(-3 a^2 d^2+14 a b c d+21 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{c^{7/4} \sqrt[4]{d}}+\frac{2 \sqrt{2} \left(-3 a^2 d^2+14 a b c d+21 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{c^{7/4} \sqrt[4]{d}}+32 \sqrt{2} \sqrt[4]{a} b^{7/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-32 \sqrt{2} \sqrt[4]{a} b^{7/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+64 \sqrt{2} \sqrt[4]{a} b^{7/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)-64 \sqrt{2} \sqrt[4]{a} b^{7/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)+\frac{32 \sqrt{x} (b c-a d)^2}{\left(c+d x^2\right)^2}+\frac{8 \sqrt{x} (a d+7 b c) (b c-a d)}{c \left(c+d x^2\right)}}{128 (b c-a d)^3}","-\frac{\left(-3 a^2 d^2+14 a b c d+21 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{7/4} \sqrt[4]{d} (b c-a d)^3}+\frac{\left(-3 a^2 d^2+14 a b c d+21 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{7/4} \sqrt[4]{d} (b c-a d)^3}-\frac{\left(-3 a^2 d^2+14 a b c d+21 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{32 \sqrt{2} c^{7/4} \sqrt[4]{d} (b c-a d)^3}+\frac{\left(-3 a^2 d^2+14 a b c d+21 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{32 \sqrt{2} c^{7/4} \sqrt[4]{d} (b c-a d)^3}+\frac{\sqrt[4]{a} b^{7/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} (b c-a d)^3}-\frac{\sqrt[4]{a} b^{7/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} (b c-a d)^3}+\frac{\sqrt[4]{a} b^{7/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} (b c-a d)^3}-\frac{\sqrt[4]{a} b^{7/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} (b c-a d)^3}+\frac{\sqrt{x} (a d+7 b c)}{16 c \left(c+d x^2\right) (b c-a d)^2}+\frac{\sqrt{x}}{4 \left(c+d x^2\right)^2 (b c-a d)}",1,"((32*(b*c - a*d)^2*Sqrt[x])/(c + d*x^2)^2 + (8*(b*c - a*d)*(7*b*c + a*d)*Sqrt[x])/(c*(c + d*x^2)) + 64*Sqrt[2]*a^(1/4)*b^(7/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 64*Sqrt[2]*a^(1/4)*b^(7/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - (2*Sqrt[2]*(21*b^2*c^2 + 14*a*b*c*d - 3*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(7/4)*d^(1/4)) + (2*Sqrt[2]*(21*b^2*c^2 + 14*a*b*c*d - 3*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(7/4)*d^(1/4)) + 32*Sqrt[2]*a^(1/4)*b^(7/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - 32*Sqrt[2]*a^(1/4)*b^(7/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - (Sqrt[2]*(21*b^2*c^2 + 14*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(7/4)*d^(1/4)) + (Sqrt[2]*(21*b^2*c^2 + 14*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(7/4)*d^(1/4)))/(128*(b*c - a*d)^3)","A",1
483,1,620,633,0.8596518,"\int \frac{\sqrt{x}}{\left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","Integrate[Sqrt[x]/((a + b*x^2)*(c + d*x^2)^3),x]","\frac{1}{128} \left(\frac{\sqrt{2} \sqrt[4]{d} \left(5 a^2 d^2-18 a b c d+45 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{9/4} (a d-b c)^3}+\frac{\sqrt{2} \sqrt[4]{d} \left(5 a^2 d^2-18 a b c d+45 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{9/4} (b c-a d)^3}+\frac{2 \sqrt{2} \sqrt[4]{d} \left(5 a^2 d^2-18 a b c d+45 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{c^{9/4} (b c-a d)^3}-\frac{2 \sqrt{2} \sqrt[4]{d} \left(5 a^2 d^2-18 a b c d+45 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{c^{9/4} (b c-a d)^3}+\frac{32 \sqrt{2} b^{9/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{\sqrt[4]{a} (b c-a d)^3}+\frac{32 \sqrt{2} b^{9/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{\sqrt[4]{a} (a d-b c)^3}+\frac{64 \sqrt{2} b^{9/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt[4]{a} (a d-b c)^3}-\frac{64 \sqrt{2} b^{9/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt[4]{a} (a d-b c)^3}+\frac{8 d x^{3/2} (5 a d-13 b c)}{c^2 \left(c+d x^2\right) (b c-a d)^2}-\frac{32 d x^{3/2}}{c \left(c+d x^2\right)^2 (b c-a d)}\right)","-\frac{\sqrt[4]{d} \left(5 a^2 d^2-18 a b c d+45 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{9/4} (b c-a d)^3}+\frac{\sqrt[4]{d} \left(5 a^2 d^2-18 a b c d+45 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{9/4} (b c-a d)^3}+\frac{\sqrt[4]{d} \left(5 a^2 d^2-18 a b c d+45 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{32 \sqrt{2} c^{9/4} (b c-a d)^3}-\frac{\sqrt[4]{d} \left(5 a^2 d^2-18 a b c d+45 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{32 \sqrt{2} c^{9/4} (b c-a d)^3}+\frac{b^{9/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} \sqrt[4]{a} (b c-a d)^3}-\frac{b^{9/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} \sqrt[4]{a} (b c-a d)^3}-\frac{b^{9/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} \sqrt[4]{a} (b c-a d)^3}+\frac{b^{9/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} \sqrt[4]{a} (b c-a d)^3}-\frac{d x^{3/2} (13 b c-5 a d)}{16 c^2 \left(c+d x^2\right) (b c-a d)^2}-\frac{d x^{3/2}}{4 c \left(c+d x^2\right)^2 (b c-a d)}",1,"((-32*d*x^(3/2))/(c*(b*c - a*d)*(c + d*x^2)^2) + (8*d*(-13*b*c + 5*a*d)*x^(3/2))/(c^2*(b*c - a*d)^2*(c + d*x^2)) + (64*Sqrt[2]*b^(9/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(1/4)*(-(b*c) + a*d)^3) - (64*Sqrt[2]*b^(9/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(1/4)*(-(b*c) + a*d)^3) + (2*Sqrt[2]*d^(1/4)*(45*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(9/4)*(b*c - a*d)^3) - (2*Sqrt[2]*d^(1/4)*(45*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(9/4)*(b*c - a*d)^3) + (32*Sqrt[2]*b^(9/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(1/4)*(b*c - a*d)^3) + (32*Sqrt[2]*b^(9/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(1/4)*(-(b*c) + a*d)^3) + (Sqrt[2]*d^(1/4)*(45*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(9/4)*(-(b*c) + a*d)^3) + (Sqrt[2]*d^(1/4)*(45*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(9/4)*(b*c - a*d)^3))/128","A",1
484,1,620,633,0.8549658,"\int \frac{1}{\sqrt{x} \left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","Integrate[1/(Sqrt[x]*(a + b*x^2)*(c + d*x^2)^3),x]","\frac{1}{128} \left(\frac{32 \sqrt{2} b^{11/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{3/4} (a d-b c)^3}+\frac{32 \sqrt{2} b^{11/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{3/4} (b c-a d)^3}+\frac{64 \sqrt{2} b^{11/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{a^{3/4} (a d-b c)^3}-\frac{64 \sqrt{2} b^{11/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{a^{3/4} (a d-b c)^3}+\frac{\sqrt{2} d^{3/4} \left(21 a^2 d^2-66 a b c d+77 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{11/4} (b c-a d)^3}+\frac{\sqrt{2} d^{3/4} \left(21 a^2 d^2-66 a b c d+77 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{11/4} (a d-b c)^3}+\frac{2 \sqrt{2} d^{3/4} \left(21 a^2 d^2-66 a b c d+77 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{c^{11/4} (b c-a d)^3}-\frac{2 \sqrt{2} d^{3/4} \left(21 a^2 d^2-66 a b c d+77 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{c^{11/4} (b c-a d)^3}+\frac{8 d \sqrt{x} (7 a d-15 b c)}{c^2 \left(c+d x^2\right) (b c-a d)^2}-\frac{32 d \sqrt{x}}{c \left(c+d x^2\right)^2 (b c-a d)}\right)","-\frac{b^{11/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{3/4} (b c-a d)^3}+\frac{b^{11/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{3/4} (b c-a d)^3}-\frac{b^{11/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} a^{3/4} (b c-a d)^3}+\frac{b^{11/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} a^{3/4} (b c-a d)^3}+\frac{d^{3/4} \left(21 a^2 d^2-66 a b c d+77 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{11/4} (b c-a d)^3}-\frac{d^{3/4} \left(21 a^2 d^2-66 a b c d+77 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{11/4} (b c-a d)^3}+\frac{d^{3/4} \left(21 a^2 d^2-66 a b c d+77 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{32 \sqrt{2} c^{11/4} (b c-a d)^3}-\frac{d^{3/4} \left(21 a^2 d^2-66 a b c d+77 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{32 \sqrt{2} c^{11/4} (b c-a d)^3}-\frac{d \sqrt{x} (15 b c-7 a d)}{16 c^2 \left(c+d x^2\right) (b c-a d)^2}-\frac{d \sqrt{x}}{4 c \left(c+d x^2\right)^2 (b c-a d)}",1,"((-32*d*Sqrt[x])/(c*(b*c - a*d)*(c + d*x^2)^2) + (8*d*(-15*b*c + 7*a*d)*Sqrt[x])/(c^2*(b*c - a*d)^2*(c + d*x^2)) + (64*Sqrt[2]*b^(11/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(3/4)*(-(b*c) + a*d)^3) - (64*Sqrt[2]*b^(11/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(3/4)*(-(b*c) + a*d)^3) + (2*Sqrt[2]*d^(3/4)*(77*b^2*c^2 - 66*a*b*c*d + 21*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(11/4)*(b*c - a*d)^3) - (2*Sqrt[2]*d^(3/4)*(77*b^2*c^2 - 66*a*b*c*d + 21*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(11/4)*(b*c - a*d)^3) + (32*Sqrt[2]*b^(11/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(3/4)*(-(b*c) + a*d)^3) + (32*Sqrt[2]*b^(11/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(3/4)*(b*c - a*d)^3) + (Sqrt[2]*d^(3/4)*(77*b^2*c^2 - 66*a*b*c*d + 21*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(11/4)*(b*c - a*d)^3) + (Sqrt[2]*d^(3/4)*(77*b^2*c^2 - 66*a*b*c*d + 21*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(11/4)*(-(b*c) + a*d)^3))/128","A",1
485,1,699,681,6.1662643,"\int \frac{1}{x^{3/2} \left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","Integrate[1/(x^(3/2)*(a + b*x^2)*(c + d*x^2)^3),x]","-\frac{b^{13/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{5/4} (b c-a d)^3}+\frac{b^{13/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{5/4} (b c-a d)^3}-\frac{b^{13/4} \tan ^{-1}\left(\frac{2 \sqrt[4]{b} \sqrt{x}-\sqrt{2} \sqrt[4]{a}}{\sqrt{2} \sqrt[4]{a}}\right)}{\sqrt{2} a^{5/4} (b c-a d)^3}-\frac{b^{13/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a}+2 \sqrt[4]{b} \sqrt{x}}{\sqrt{2} \sqrt[4]{a}}\right)}{\sqrt{2} a^{5/4} (b c-a d)^3}-\frac{d^{5/4} \left(45 a^2 d^2-130 a b c d+117 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{13/4} (a d-b c)^3}+\frac{d^{5/4} \left(45 a^2 d^2-130 a b c d+117 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{13/4} (a d-b c)^3}-\frac{d^{5/4} \left(45 a^2 d^2-130 a b c d+117 b^2 c^2\right) \tan ^{-1}\left(\frac{2 \sqrt[4]{d} \sqrt{x}-\sqrt{2} \sqrt[4]{c}}{\sqrt{2} \sqrt[4]{c}}\right)}{32 \sqrt{2} c^{13/4} (a d-b c)^3}-\frac{d^{5/4} \left(45 a^2 d^2-130 a b c d+117 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c}+2 \sqrt[4]{d} \sqrt{x}}{\sqrt{2} \sqrt[4]{c}}\right)}{32 \sqrt{2} c^{13/4} (a d-b c)^3}+\frac{d^2 x^{3/2} (21 b c-13 a d)}{16 c^3 \left(c+d x^2\right) (b c-a d)^2}+\frac{d^2 x^{3/2}}{4 c^2 \left(c+d x^2\right)^2 (b c-a d)}-\frac{2}{a c^3 \sqrt{x}}","-\frac{b^{13/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{5/4} (b c-a d)^3}+\frac{b^{13/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{5/4} (b c-a d)^3}+\frac{b^{13/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} a^{5/4} (b c-a d)^3}-\frac{b^{13/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} a^{5/4} (b c-a d)^3}+\frac{d^{5/4} \left(45 a^2 d^2-130 a b c d+117 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{13/4} (b c-a d)^3}-\frac{d^{5/4} \left(45 a^2 d^2-130 a b c d+117 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{13/4} (b c-a d)^3}-\frac{d^{5/4} \left(45 a^2 d^2-130 a b c d+117 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{32 \sqrt{2} c^{13/4} (b c-a d)^3}+\frac{d^{5/4} \left(45 a^2 d^2-130 a b c d+117 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{32 \sqrt{2} c^{13/4} (b c-a d)^3}-\frac{45 a^2 d^2-85 a b c d+32 b^2 c^2}{16 a c^3 \sqrt{x} (b c-a d)^2}-\frac{d (17 b c-9 a d)}{16 c^2 \sqrt{x} \left(c+d x^2\right) (b c-a d)^2}-\frac{d}{4 c \sqrt{x} \left(c+d x^2\right)^2 (b c-a d)}",1,"-2/(a*c^3*Sqrt[x]) + (d^2*x^(3/2))/(4*c^2*(b*c - a*d)*(c + d*x^2)^2) + (d^2*(21*b*c - 13*a*d)*x^(3/2))/(16*c^3*(b*c - a*d)^2*(c + d*x^2)) - (b^(13/4)*ArcTan[(-(Sqrt[2]*a^(1/4)) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(Sqrt[2]*a^(5/4)*(b*c - a*d)^3) - (b^(13/4)*ArcTan[(Sqrt[2]*a^(1/4) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(Sqrt[2]*a^(5/4)*(b*c - a*d)^3) - (d^(5/4)*(117*b^2*c^2 - 130*a*b*c*d + 45*a^2*d^2)*ArcTan[(-(Sqrt[2]*c^(1/4)) + 2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(32*Sqrt[2]*c^(13/4)*(-(b*c) + a*d)^3) - (d^(5/4)*(117*b^2*c^2 - 130*a*b*c*d + 45*a^2*d^2)*ArcTan[(Sqrt[2]*c^(1/4) + 2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(32*Sqrt[2]*c^(13/4)*(-(b*c) + a*d)^3) - (b^(13/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(5/4)*(b*c - a*d)^3) + (b^(13/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(5/4)*(b*c - a*d)^3) - (d^(5/4)*(117*b^2*c^2 - 130*a*b*c*d + 45*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(13/4)*(-(b*c) + a*d)^3) + (d^(5/4)*(117*b^2*c^2 - 130*a*b*c*d + 45*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(13/4)*(-(b*c) + a*d)^3)","A",1
486,1,701,681,6.177553,"\int \frac{1}{x^{5/2} \left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","Integrate[1/(x^(5/2)*(a + b*x^2)*(c + d*x^2)^3),x]","\frac{b^{15/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{b^{15/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{b^{15/4} \tan ^{-1}\left(\frac{2 \sqrt[4]{b} \sqrt{x}-\sqrt{2} \sqrt[4]{a}}{\sqrt{2} \sqrt[4]{a}}\right)}{\sqrt{2} a^{7/4} (b c-a d)^3}-\frac{b^{15/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a}+2 \sqrt[4]{b} \sqrt{x}}{\sqrt{2} \sqrt[4]{a}}\right)}{\sqrt{2} a^{7/4} (b c-a d)^3}+\frac{d^{7/4} \left(77 a^2 d^2-210 a b c d+165 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{15/4} (a d-b c)^3}-\frac{d^{7/4} \left(77 a^2 d^2-210 a b c d+165 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{15/4} (a d-b c)^3}-\frac{d^{7/4} \left(77 a^2 d^2-210 a b c d+165 b^2 c^2\right) \tan ^{-1}\left(\frac{2 \sqrt[4]{d} \sqrt{x}-\sqrt{2} \sqrt[4]{c}}{\sqrt{2} \sqrt[4]{c}}\right)}{32 \sqrt{2} c^{15/4} (a d-b c)^3}-\frac{d^{7/4} \left(77 a^2 d^2-210 a b c d+165 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c}+2 \sqrt[4]{d} \sqrt{x}}{\sqrt{2} \sqrt[4]{c}}\right)}{32 \sqrt{2} c^{15/4} (a d-b c)^3}+\frac{d^2 \sqrt{x} (23 b c-15 a d)}{16 c^3 \left(c+d x^2\right) (b c-a d)^2}+\frac{d^2 \sqrt{x}}{4 c^2 \left(c+d x^2\right)^2 (b c-a d)}-\frac{2}{3 a c^3 x^{3/2}}","\frac{b^{15/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{b^{15/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{7/4} (b c-a d)^3}+\frac{b^{15/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} a^{7/4} (b c-a d)^3}-\frac{b^{15/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} a^{7/4} (b c-a d)^3}-\frac{d^{7/4} \left(77 a^2 d^2-210 a b c d+165 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{15/4} (b c-a d)^3}+\frac{d^{7/4} \left(77 a^2 d^2-210 a b c d+165 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{15/4} (b c-a d)^3}-\frac{d^{7/4} \left(77 a^2 d^2-210 a b c d+165 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{32 \sqrt{2} c^{15/4} (b c-a d)^3}+\frac{d^{7/4} \left(77 a^2 d^2-210 a b c d+165 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{32 \sqrt{2} c^{15/4} (b c-a d)^3}-\frac{77 a^2 d^2-133 a b c d+32 b^2 c^2}{48 a c^3 x^{3/2} (b c-a d)^2}-\frac{d (19 b c-11 a d)}{16 c^2 x^{3/2} \left(c+d x^2\right) (b c-a d)^2}-\frac{d}{4 c x^{3/2} \left(c+d x^2\right)^2 (b c-a d)}",1,"-2/(3*a*c^3*x^(3/2)) + (d^2*Sqrt[x])/(4*c^2*(b*c - a*d)*(c + d*x^2)^2) + (d^2*(23*b*c - 15*a*d)*Sqrt[x])/(16*c^3*(b*c - a*d)^2*(c + d*x^2)) - (b^(15/4)*ArcTan[(-(Sqrt[2]*a^(1/4)) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(Sqrt[2]*a^(7/4)*(b*c - a*d)^3) - (b^(15/4)*ArcTan[(Sqrt[2]*a^(1/4) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(Sqrt[2]*a^(7/4)*(b*c - a*d)^3) - (d^(7/4)*(165*b^2*c^2 - 210*a*b*c*d + 77*a^2*d^2)*ArcTan[(-(Sqrt[2]*c^(1/4)) + 2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(32*Sqrt[2]*c^(15/4)*(-(b*c) + a*d)^3) - (d^(7/4)*(165*b^2*c^2 - 210*a*b*c*d + 77*a^2*d^2)*ArcTan[(Sqrt[2]*c^(1/4) + 2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(32*Sqrt[2]*c^(15/4)*(-(b*c) + a*d)^3) + (b^(15/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(7/4)*(b*c - a*d)^3) - (b^(15/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(7/4)*(b*c - a*d)^3) + (d^(7/4)*(165*b^2*c^2 - 210*a*b*c*d + 77*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(15/4)*(-(b*c) + a*d)^3) - (d^(7/4)*(165*b^2*c^2 - 210*a*b*c*d + 77*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(15/4)*(-(b*c) + a*d)^3)","A",1
487,1,720,743,6.1775512,"\int \frac{1}{x^{7/2} \left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","Integrate[1/(x^(7/2)*(a + b*x^2)*(c + d*x^2)^3),x]","\frac{b^{17/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{9/4} (b c-a d)^3}-\frac{b^{17/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{9/4} (b c-a d)^3}+\frac{b^{17/4} \tan ^{-1}\left(\frac{2 \sqrt[4]{b} \sqrt{x}-\sqrt{2} \sqrt[4]{a}}{\sqrt{2} \sqrt[4]{a}}\right)}{\sqrt{2} a^{9/4} (b c-a d)^3}+\frac{b^{17/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a}+2 \sqrt[4]{b} \sqrt{x}}{\sqrt{2} \sqrt[4]{a}}\right)}{\sqrt{2} a^{9/4} (b c-a d)^3}+\frac{d^{9/4} \left(117 a^2 d^2-306 a b c d+221 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{17/4} (a d-b c)^3}-\frac{d^{9/4} \left(117 a^2 d^2-306 a b c d+221 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{17/4} (a d-b c)^3}+\frac{d^{9/4} \left(117 a^2 d^2-306 a b c d+221 b^2 c^2\right) \tan ^{-1}\left(\frac{2 \sqrt[4]{d} \sqrt{x}-\sqrt{2} \sqrt[4]{c}}{\sqrt{2} \sqrt[4]{c}}\right)}{32 \sqrt{2} c^{17/4} (a d-b c)^3}+\frac{d^{9/4} \left(117 a^2 d^2-306 a b c d+221 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c}+2 \sqrt[4]{d} \sqrt{x}}{\sqrt{2} \sqrt[4]{c}}\right)}{32 \sqrt{2} c^{17/4} (a d-b c)^3}+\frac{2 (3 a d+b c)}{a^2 c^4 \sqrt{x}}-\frac{d^3 x^{3/2} (29 b c-21 a d)}{16 c^4 \left(c+d x^2\right) (b c-a d)^2}-\frac{d^3 x^{3/2}}{4 c^3 \left(c+d x^2\right)^2 (b c-a d)}-\frac{2}{5 a c^3 x^{5/2}}","\frac{b^{17/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{9/4} (b c-a d)^3}-\frac{b^{17/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{9/4} (b c-a d)^3}-\frac{b^{17/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} a^{9/4} (b c-a d)^3}+\frac{b^{17/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} a^{9/4} (b c-a d)^3}-\frac{d^{9/4} \left(117 a^2 d^2-306 a b c d+221 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{17/4} (b c-a d)^3}+\frac{d^{9/4} \left(117 a^2 d^2-306 a b c d+221 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{17/4} (b c-a d)^3}+\frac{d^{9/4} \left(117 a^2 d^2-306 a b c d+221 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{32 \sqrt{2} c^{17/4} (b c-a d)^3}-\frac{d^{9/4} \left(117 a^2 d^2-306 a b c d+221 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{32 \sqrt{2} c^{17/4} (b c-a d)^3}-\frac{117 a^2 d^2-189 a b c d+32 b^2 c^2}{80 a c^3 x^{5/2} (b c-a d)^2}+\frac{117 a^3 d^3-189 a^2 b c d^2+32 a b^2 c^2 d+32 b^3 c^3}{16 a^2 c^4 \sqrt{x} (b c-a d)^2}-\frac{d (21 b c-13 a d)}{16 c^2 x^{5/2} \left(c+d x^2\right) (b c-a d)^2}-\frac{d}{4 c x^{5/2} \left(c+d x^2\right)^2 (b c-a d)}",1,"-2/(5*a*c^3*x^(5/2)) + (2*(b*c + 3*a*d))/(a^2*c^4*Sqrt[x]) - (d^3*x^(3/2))/(4*c^3*(b*c - a*d)*(c + d*x^2)^2) - (d^3*(29*b*c - 21*a*d)*x^(3/2))/(16*c^4*(b*c - a*d)^2*(c + d*x^2)) + (b^(17/4)*ArcTan[(-(Sqrt[2]*a^(1/4)) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(Sqrt[2]*a^(9/4)*(b*c - a*d)^3) + (b^(17/4)*ArcTan[(Sqrt[2]*a^(1/4) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(Sqrt[2]*a^(9/4)*(b*c - a*d)^3) + (d^(9/4)*(221*b^2*c^2 - 306*a*b*c*d + 117*a^2*d^2)*ArcTan[(-(Sqrt[2]*c^(1/4)) + 2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(32*Sqrt[2]*c^(17/4)*(-(b*c) + a*d)^3) + (d^(9/4)*(221*b^2*c^2 - 306*a*b*c*d + 117*a^2*d^2)*ArcTan[(Sqrt[2]*c^(1/4) + 2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(32*Sqrt[2]*c^(17/4)*(-(b*c) + a*d)^3) + (b^(17/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(9/4)*(b*c - a*d)^3) - (b^(17/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(9/4)*(b*c - a*d)^3) + (d^(9/4)*(221*b^2*c^2 - 306*a*b*c*d + 117*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(17/4)*(-(b*c) + a*d)^3) - (d^(9/4)*(221*b^2*c^2 - 306*a*b*c*d + 117*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(17/4)*(-(b*c) + a*d)^3)","A",1
488,1,585,624,0.9237554,"\int \frac{x^{7/2}}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","Integrate[x^(7/2)/((a + b*x^2)^2*(c + d*x^2)^2),x]","\frac{1}{16} \left(\frac{8 a \sqrt{x}}{\left(a+b x^2\right) (b c-a d)^2}+\frac{8 c \sqrt{x}}{\left(c+d x^2\right) (b c-a d)^2}+\frac{\sqrt{2} \sqrt[4]{a} (3 a d+5 b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{\sqrt[4]{b} (b c-a d)^3}-\frac{\sqrt{2} \sqrt[4]{a} (3 a d+5 b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{\sqrt[4]{b} (b c-a d)^3}+\frac{\sqrt{2} \sqrt[4]{c} (5 a d+3 b c) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{\sqrt[4]{d} (a d-b c)^3}-\frac{\sqrt{2} \sqrt[4]{c} (5 a d+3 b c) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{\sqrt[4]{d} (a d-b c)^3}+\frac{2 \sqrt{2} \sqrt[4]{a} (3 a d+5 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt[4]{b} (b c-a d)^3}-\frac{2 \sqrt{2} \sqrt[4]{a} (3 a d+5 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt[4]{b} (b c-a d)^3}+\frac{2 \sqrt{2} \sqrt[4]{c} (5 a d+3 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{\sqrt[4]{d} (a d-b c)^3}+\frac{2 \sqrt{2} \sqrt[4]{c} (5 a d+3 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{\sqrt[4]{d} (b c-a d)^3}\right)","\frac{a \sqrt{x}}{2 b \left(a+b x^2\right) \left(c+d x^2\right) (b c-a d)}+\frac{\sqrt{x} (a d+b c)}{2 b \left(c+d x^2\right) (b c-a d)^2}+\frac{\sqrt[4]{a} (3 a d+5 b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} \sqrt[4]{b} (b c-a d)^3}-\frac{\sqrt[4]{a} (3 a d+5 b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} \sqrt[4]{b} (b c-a d)^3}-\frac{\sqrt[4]{c} (5 a d+3 b c) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} \sqrt[4]{d} (b c-a d)^3}+\frac{\sqrt[4]{c} (5 a d+3 b c) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} \sqrt[4]{d} (b c-a d)^3}+\frac{\sqrt[4]{a} (3 a d+5 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} \sqrt[4]{b} (b c-a d)^3}-\frac{\sqrt[4]{a} (3 a d+5 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} \sqrt[4]{b} (b c-a d)^3}-\frac{\sqrt[4]{c} (5 a d+3 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{4 \sqrt{2} \sqrt[4]{d} (b c-a d)^3}+\frac{\sqrt[4]{c} (5 a d+3 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{4 \sqrt{2} \sqrt[4]{d} (b c-a d)^3}",1,"((8*a*Sqrt[x])/((b*c - a*d)^2*(a + b*x^2)) + (8*c*Sqrt[x])/((b*c - a*d)^2*(c + d*x^2)) + (2*Sqrt[2]*a^(1/4)*(5*b*c + 3*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(b^(1/4)*(b*c - a*d)^3) - (2*Sqrt[2]*a^(1/4)*(5*b*c + 3*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(b^(1/4)*(b*c - a*d)^3) + (2*Sqrt[2]*c^(1/4)*(3*b*c + 5*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(d^(1/4)*(-(b*c) + a*d)^3) + (2*Sqrt[2]*c^(1/4)*(3*b*c + 5*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(d^(1/4)*(b*c - a*d)^3) + (Sqrt[2]*a^(1/4)*(5*b*c + 3*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(b^(1/4)*(b*c - a*d)^3) - (Sqrt[2]*a^(1/4)*(5*b*c + 3*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(b^(1/4)*(b*c - a*d)^3) + (Sqrt[2]*c^(1/4)*(3*b*c + 5*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(d^(1/4)*(-(b*c) + a*d)^3) - (Sqrt[2]*c^(1/4)*(3*b*c + 5*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(d^(1/4)*(-(b*c) + a*d)^3))/16","A",1
489,1,583,609,0.951841,"\int \frac{x^{5/2}}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","Integrate[x^(5/2)/((a + b*x^2)^2*(c + d*x^2)^2),x]","\frac{1}{16} \left(-\frac{8 b x^{3/2}}{\left(a+b x^2\right) (b c-a d)^2}-\frac{8 d x^{3/2}}{\left(c+d x^2\right) (b c-a d)^2}+\frac{\sqrt{2} \sqrt[4]{b} (5 a d+3 b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{\sqrt[4]{a} (b c-a d)^3}+\frac{\sqrt{2} \sqrt[4]{b} (5 a d+3 b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{\sqrt[4]{a} (a d-b c)^3}+\frac{\sqrt{2} \sqrt[4]{d} (3 a d+5 b c) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{\sqrt[4]{c} (a d-b c)^3}+\frac{\sqrt{2} \sqrt[4]{d} (3 a d+5 b c) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{\sqrt[4]{c} (b c-a d)^3}+\frac{2 \sqrt{2} \sqrt[4]{b} (5 a d+3 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt[4]{a} (a d-b c)^3}-\frac{2 \sqrt{2} \sqrt[4]{b} (5 a d+3 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt[4]{a} (a d-b c)^3}+\frac{2 \sqrt{2} \sqrt[4]{d} (3 a d+5 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{\sqrt[4]{c} (b c-a d)^3}-\frac{2 \sqrt{2} \sqrt[4]{d} (3 a d+5 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{\sqrt[4]{c} (b c-a d)^3}\right)","-\frac{x^{3/2}}{2 \left(a+b x^2\right) \left(c+d x^2\right) (b c-a d)}-\frac{d x^{3/2}}{\left(c+d x^2\right) (b c-a d)^2}+\frac{\sqrt[4]{b} (5 a d+3 b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} \sqrt[4]{a} (b c-a d)^3}-\frac{\sqrt[4]{b} (5 a d+3 b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} \sqrt[4]{a} (b c-a d)^3}-\frac{\sqrt[4]{d} (3 a d+5 b c) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} \sqrt[4]{c} (b c-a d)^3}+\frac{\sqrt[4]{d} (3 a d+5 b c) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} \sqrt[4]{c} (b c-a d)^3}-\frac{\sqrt[4]{b} (5 a d+3 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} \sqrt[4]{a} (b c-a d)^3}+\frac{\sqrt[4]{b} (5 a d+3 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} \sqrt[4]{a} (b c-a d)^3}+\frac{\sqrt[4]{d} (3 a d+5 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{4 \sqrt{2} \sqrt[4]{c} (b c-a d)^3}-\frac{\sqrt[4]{d} (3 a d+5 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{4 \sqrt{2} \sqrt[4]{c} (b c-a d)^3}",1,"((-8*b*x^(3/2))/((b*c - a*d)^2*(a + b*x^2)) - (8*d*x^(3/2))/((b*c - a*d)^2*(c + d*x^2)) + (2*Sqrt[2]*b^(1/4)*(3*b*c + 5*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(1/4)*(-(b*c) + a*d)^3) - (2*Sqrt[2]*b^(1/4)*(3*b*c + 5*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(1/4)*(-(b*c) + a*d)^3) + (2*Sqrt[2]*d^(1/4)*(5*b*c + 3*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(1/4)*(b*c - a*d)^3) - (2*Sqrt[2]*d^(1/4)*(5*b*c + 3*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(1/4)*(b*c - a*d)^3) + (Sqrt[2]*b^(1/4)*(3*b*c + 5*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(1/4)*(b*c - a*d)^3) + (Sqrt[2]*b^(1/4)*(3*b*c + 5*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(1/4)*(-(b*c) + a*d)^3) + (Sqrt[2]*d^(1/4)*(5*b*c + 3*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(1/4)*(-(b*c) + a*d)^3) + (Sqrt[2]*d^(1/4)*(5*b*c + 3*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(1/4)*(b*c - a*d)^3))/16","A",1
490,1,575,601,1.0328818,"\int \frac{x^{3/2}}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","Integrate[x^(3/2)/((a + b*x^2)^2*(c + d*x^2)^2),x]","\frac{1}{16} \left(\frac{\sqrt{2} b^{3/4} (7 a d+b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{3/4} (a d-b c)^3}+\frac{\sqrt{2} b^{3/4} (7 a d+b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{3/4} (b c-a d)^3}+\frac{2 \sqrt{2} b^{3/4} (7 a d+b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{a^{3/4} (a d-b c)^3}-\frac{2 \sqrt{2} b^{3/4} (7 a d+b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{a^{3/4} (a d-b c)^3}+\frac{\sqrt{2} d^{3/4} (a d+7 b c) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{3/4} (b c-a d)^3}+\frac{\sqrt{2} d^{3/4} (a d+7 b c) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{3/4} (a d-b c)^3}+\frac{2 \sqrt{2} d^{3/4} (a d+7 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{c^{3/4} (b c-a d)^3}-\frac{2 \sqrt{2} d^{3/4} (a d+7 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{c^{3/4} (b c-a d)^3}-\frac{8 b \sqrt{x}}{\left(a+b x^2\right) (b c-a d)^2}-\frac{8 d \sqrt{x}}{\left(c+d x^2\right) (b c-a d)^2}\right)","-\frac{b^{3/4} (7 a d+b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{3/4} (b c-a d)^3}+\frac{b^{3/4} (7 a d+b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{3/4} (b c-a d)^3}-\frac{b^{3/4} (7 a d+b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} a^{3/4} (b c-a d)^3}+\frac{b^{3/4} (7 a d+b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} a^{3/4} (b c-a d)^3}+\frac{d^{3/4} (a d+7 b c) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{3/4} (b c-a d)^3}-\frac{d^{3/4} (a d+7 b c) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{3/4} (b c-a d)^3}+\frac{d^{3/4} (a d+7 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{4 \sqrt{2} c^{3/4} (b c-a d)^3}-\frac{d^{3/4} (a d+7 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{4 \sqrt{2} c^{3/4} (b c-a d)^3}-\frac{d \sqrt{x}}{\left(c+d x^2\right) (b c-a d)^2}-\frac{\sqrt{x}}{2 \left(a+b x^2\right) \left(c+d x^2\right) (b c-a d)}",1,"((-8*b*Sqrt[x])/((b*c - a*d)^2*(a + b*x^2)) - (8*d*Sqrt[x])/((b*c - a*d)^2*(c + d*x^2)) + (2*Sqrt[2]*b^(3/4)*(b*c + 7*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(3/4)*(-(b*c) + a*d)^3) - (2*Sqrt[2]*b^(3/4)*(b*c + 7*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(3/4)*(-(b*c) + a*d)^3) + (2*Sqrt[2]*d^(3/4)*(7*b*c + a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(3/4)*(b*c - a*d)^3) - (2*Sqrt[2]*d^(3/4)*(7*b*c + a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(3/4)*(b*c - a*d)^3) + (Sqrt[2]*b^(3/4)*(b*c + 7*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(3/4)*(-(b*c) + a*d)^3) + (Sqrt[2]*b^(3/4)*(b*c + 7*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(3/4)*(b*c - a*d)^3) + (Sqrt[2]*d^(3/4)*(7*b*c + a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(3/4)*(b*c - a*d)^3) + (Sqrt[2]*d^(3/4)*(7*b*c + a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(3/4)*(-(b*c) + a*d)^3))/16","A",1
491,1,589,624,2.0930403,"\int \frac{\sqrt{x}}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","Integrate[Sqrt[x]/((a + b*x^2)^2*(c + d*x^2)^2),x]","\frac{1}{16} \left(\frac{\sqrt{2} b^{5/4} (9 a d-b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{5/4} (a d-b c)^3}+\frac{\sqrt{2} b^{5/4} (b c-9 a d) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{5/4} (a d-b c)^3}+\frac{2 \sqrt{2} b^{5/4} (b c-9 a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{a^{5/4} (a d-b c)^3}+\frac{2 \sqrt{2} b^{5/4} (9 a d-b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{a^{5/4} (a d-b c)^3}+\frac{8 b^2 x^{3/2}}{a \left(a+b x^2\right) (b c-a d)^2}+\frac{\sqrt{2} d^{5/4} (9 b c-a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{5/4} (b c-a d)^3}+\frac{\sqrt{2} d^{5/4} (a d-9 b c) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{5/4} (b c-a d)^3}+\frac{2 \sqrt{2} d^{5/4} (a d-9 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{c^{5/4} (b c-a d)^3}+\frac{2 \sqrt{2} d^{5/4} (9 b c-a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{c^{5/4} (b c-a d)^3}+\frac{8 d^2 x^{3/2}}{c \left(c+d x^2\right) (b c-a d)^2}\right)","\frac{b^{5/4} (b c-9 a d) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{5/4} (b c-a d)^3}-\frac{b^{5/4} (b c-9 a d) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{5/4} (b c-a d)^3}-\frac{b^{5/4} (b c-9 a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} a^{5/4} (b c-a d)^3}+\frac{b^{5/4} (b c-9 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} a^{5/4} (b c-a d)^3}+\frac{d^{5/4} (9 b c-a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{5/4} (b c-a d)^3}-\frac{d^{5/4} (9 b c-a d) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{5/4} (b c-a d)^3}-\frac{d^{5/4} (9 b c-a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{4 \sqrt{2} c^{5/4} (b c-a d)^3}+\frac{d^{5/4} (9 b c-a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{4 \sqrt{2} c^{5/4} (b c-a d)^3}+\frac{d x^{3/2} (a d+b c)}{2 a c \left(c+d x^2\right) (b c-a d)^2}+\frac{b x^{3/2}}{2 a \left(a+b x^2\right) \left(c+d x^2\right) (b c-a d)}",1,"((8*b^2*x^(3/2))/(a*(b*c - a*d)^2*(a + b*x^2)) + (8*d^2*x^(3/2))/(c*(b*c - a*d)^2*(c + d*x^2)) + (2*Sqrt[2]*b^(5/4)*(b*c - 9*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(5/4)*(-(b*c) + a*d)^3) + (2*Sqrt[2]*b^(5/4)*(-(b*c) + 9*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(5/4)*(-(b*c) + a*d)^3) + (2*Sqrt[2]*d^(5/4)*(-9*b*c + a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(5/4)*(b*c - a*d)^3) + (2*Sqrt[2]*d^(5/4)*(9*b*c - a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(5/4)*(b*c - a*d)^3) + (Sqrt[2]*b^(5/4)*(-(b*c) + 9*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(5/4)*(-(b*c) + a*d)^3) + (Sqrt[2]*b^(5/4)*(b*c - 9*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(5/4)*(-(b*c) + a*d)^3) + (Sqrt[2]*d^(5/4)*(9*b*c - a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(5/4)*(b*c - a*d)^3) + (Sqrt[2]*d^(5/4)*(-9*b*c + a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(5/4)*(b*c - a*d)^3))/16","A",1
492,1,661,628,6.2111283,"\int \frac{1}{\sqrt{x} \left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","Integrate[1/(Sqrt[x]*(a + b*x^2)^2*(c + d*x^2)^2),x]","\frac{b^{7/4} (11 a d-3 b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{b^{7/4} (11 a d-3 b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{b^{7/4} (11 a d-3 b c) \tan ^{-1}\left(\frac{2 \sqrt[4]{b} \sqrt{x}-\sqrt{2} \sqrt[4]{a}}{\sqrt{2} \sqrt[4]{a}}\right)}{4 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{b^{7/4} (11 a d-3 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a}+2 \sqrt[4]{b} \sqrt{x}}{\sqrt{2} \sqrt[4]{a}}\right)}{4 \sqrt{2} a^{7/4} (b c-a d)^3}+\frac{b^2 \sqrt{x}}{2 a \left(a+b x^2\right) (a d-b c)^2}+\frac{d^{7/4} (11 b c-3 a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{7/4} (a d-b c)^3}-\frac{d^{7/4} (11 b c-3 a d) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{7/4} (a d-b c)^3}-\frac{d^{7/4} (11 b c-3 a d) \tan ^{-1}\left(\frac{2 \sqrt[4]{d} \sqrt{x}-\sqrt{2} \sqrt[4]{c}}{\sqrt{2} \sqrt[4]{c}}\right)}{4 \sqrt{2} c^{7/4} (a d-b c)^3}-\frac{d^{7/4} (11 b c-3 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c}+2 \sqrt[4]{d} \sqrt{x}}{\sqrt{2} \sqrt[4]{c}}\right)}{4 \sqrt{2} c^{7/4} (a d-b c)^3}+\frac{d^2 \sqrt{x}}{2 c \left(c+d x^2\right) (b c-a d)^2}","-\frac{b^{7/4} (3 b c-11 a d) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{7/4} (b c-a d)^3}+\frac{b^{7/4} (3 b c-11 a d) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{b^{7/4} (3 b c-11 a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} a^{7/4} (b c-a d)^3}+\frac{b^{7/4} (3 b c-11 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{d^{7/4} (11 b c-3 a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{7/4} (b c-a d)^3}+\frac{d^{7/4} (11 b c-3 a d) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{7/4} (b c-a d)^3}-\frac{d^{7/4} (11 b c-3 a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{4 \sqrt{2} c^{7/4} (b c-a d)^3}+\frac{d^{7/4} (11 b c-3 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{4 \sqrt{2} c^{7/4} (b c-a d)^3}+\frac{b \sqrt{x}}{2 a \left(a+b x^2\right) \left(c+d x^2\right) (b c-a d)}+\frac{d \sqrt{x} (a d+b c)}{2 a c \left(c+d x^2\right) (b c-a d)^2}",1,"(b^2*Sqrt[x])/(2*a*(-(b*c) + a*d)^2*(a + b*x^2)) + (d^2*Sqrt[x])/(2*c*(b*c - a*d)^2*(c + d*x^2)) - (b^(7/4)*(-3*b*c + 11*a*d)*ArcTan[(-(Sqrt[2]*a^(1/4)) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(4*Sqrt[2]*a^(7/4)*(b*c - a*d)^3) - (b^(7/4)*(-3*b*c + 11*a*d)*ArcTan[(Sqrt[2]*a^(1/4) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(4*Sqrt[2]*a^(7/4)*(b*c - a*d)^3) - (d^(7/4)*(11*b*c - 3*a*d)*ArcTan[(-(Sqrt[2]*c^(1/4)) + 2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(4*Sqrt[2]*c^(7/4)*(-(b*c) + a*d)^3) - (d^(7/4)*(11*b*c - 3*a*d)*ArcTan[(Sqrt[2]*c^(1/4) + 2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(4*Sqrt[2]*c^(7/4)*(-(b*c) + a*d)^3) + (b^(7/4)*(-3*b*c + 11*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(7/4)*(b*c - a*d)^3) - (b^(7/4)*(-3*b*c + 11*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(7/4)*(b*c - a*d)^3) + (d^(7/4)*(11*b*c - 3*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(7/4)*(-(b*c) + a*d)^3) - (d^(7/4)*(11*b*c - 3*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(7/4)*(-(b*c) + a*d)^3)","A",1
493,1,674,676,6.1584442,"\int \frac{1}{x^{3/2} \left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","Integrate[1/(x^(3/2)*(a + b*x^2)^2*(c + d*x^2)^2),x]","\frac{b^{9/4} (13 a d-5 b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{9/4} (b c-a d)^3}-\frac{b^{9/4} (13 a d-5 b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{9/4} (b c-a d)^3}+\frac{b^{9/4} (13 a d-5 b c) \tan ^{-1}\left(\frac{2 \sqrt[4]{b} \sqrt{x}-\sqrt{2} \sqrt[4]{a}}{\sqrt{2} \sqrt[4]{a}}\right)}{4 \sqrt{2} a^{9/4} (b c-a d)^3}+\frac{b^{9/4} (13 a d-5 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a}+2 \sqrt[4]{b} \sqrt{x}}{\sqrt{2} \sqrt[4]{a}}\right)}{4 \sqrt{2} a^{9/4} (b c-a d)^3}-\frac{b^3 x^{3/2}}{2 a^2 \left(a+b x^2\right) (a d-b c)^2}-\frac{2}{a^2 c^2 \sqrt{x}}+\frac{d^{9/4} (13 b c-5 a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{9/4} (a d-b c)^3}-\frac{d^{9/4} (13 b c-5 a d) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{9/4} (a d-b c)^3}+\frac{d^{9/4} (13 b c-5 a d) \tan ^{-1}\left(\frac{2 \sqrt[4]{d} \sqrt{x}-\sqrt{2} \sqrt[4]{c}}{\sqrt{2} \sqrt[4]{c}}\right)}{4 \sqrt{2} c^{9/4} (a d-b c)^3}+\frac{d^{9/4} (13 b c-5 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c}+2 \sqrt[4]{d} \sqrt{x}}{\sqrt{2} \sqrt[4]{c}}\right)}{4 \sqrt{2} c^{9/4} (a d-b c)^3}-\frac{d^3 x^{3/2}}{2 c^2 \left(c+d x^2\right) (b c-a d)^2}","-\frac{b^{9/4} (5 b c-13 a d) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{9/4} (b c-a d)^3}+\frac{b^{9/4} (5 b c-13 a d) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{9/4} (b c-a d)^3}+\frac{b^{9/4} (5 b c-13 a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} a^{9/4} (b c-a d)^3}-\frac{b^{9/4} (5 b c-13 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} a^{9/4} (b c-a d)^3}+\frac{-5 a^2 d^2+8 a b c d-5 b^2 c^2}{2 a^2 c^2 \sqrt{x} (b c-a d)^2}-\frac{d^{9/4} (13 b c-5 a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{9/4} (b c-a d)^3}+\frac{d^{9/4} (13 b c-5 a d) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{9/4} (b c-a d)^3}+\frac{d^{9/4} (13 b c-5 a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{4 \sqrt{2} c^{9/4} (b c-a d)^3}-\frac{d^{9/4} (13 b c-5 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{4 \sqrt{2} c^{9/4} (b c-a d)^3}+\frac{b}{2 a \sqrt{x} \left(a+b x^2\right) \left(c+d x^2\right) (b c-a d)}+\frac{d (a d+b c)}{2 a c \sqrt{x} \left(c+d x^2\right) (b c-a d)^2}",1,"-2/(a^2*c^2*Sqrt[x]) - (b^3*x^(3/2))/(2*a^2*(-(b*c) + a*d)^2*(a + b*x^2)) - (d^3*x^(3/2))/(2*c^2*(b*c - a*d)^2*(c + d*x^2)) + (b^(9/4)*(-5*b*c + 13*a*d)*ArcTan[(-(Sqrt[2]*a^(1/4)) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(4*Sqrt[2]*a^(9/4)*(b*c - a*d)^3) + (b^(9/4)*(-5*b*c + 13*a*d)*ArcTan[(Sqrt[2]*a^(1/4) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(4*Sqrt[2]*a^(9/4)*(b*c - a*d)^3) + (d^(9/4)*(13*b*c - 5*a*d)*ArcTan[(-(Sqrt[2]*c^(1/4)) + 2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(4*Sqrt[2]*c^(9/4)*(-(b*c) + a*d)^3) + (d^(9/4)*(13*b*c - 5*a*d)*ArcTan[(Sqrt[2]*c^(1/4) + 2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(4*Sqrt[2]*c^(9/4)*(-(b*c) + a*d)^3) + (b^(9/4)*(-5*b*c + 13*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(9/4)*(b*c - a*d)^3) - (b^(9/4)*(-5*b*c + 13*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(9/4)*(b*c - a*d)^3) + (d^(9/4)*(13*b*c - 5*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(9/4)*(-(b*c) + a*d)^3) - (d^(9/4)*(13*b*c - 5*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(9/4)*(-(b*c) + a*d)^3)","A",1
494,1,676,676,6.1550013,"\int \frac{1}{x^{5/2} \left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","Integrate[1/(x^(5/2)*(a + b*x^2)^2*(c + d*x^2)^2),x]","-\frac{b^{11/4} (15 a d-7 b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{11/4} (b c-a d)^3}+\frac{b^{11/4} (15 a d-7 b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{11/4} (b c-a d)^3}+\frac{b^{11/4} (15 a d-7 b c) \tan ^{-1}\left(\frac{2 \sqrt[4]{b} \sqrt{x}-\sqrt{2} \sqrt[4]{a}}{\sqrt{2} \sqrt[4]{a}}\right)}{4 \sqrt{2} a^{11/4} (b c-a d)^3}+\frac{b^{11/4} (15 a d-7 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a}+2 \sqrt[4]{b} \sqrt{x}}{\sqrt{2} \sqrt[4]{a}}\right)}{4 \sqrt{2} a^{11/4} (b c-a d)^3}-\frac{b^3 \sqrt{x}}{2 a^2 \left(a+b x^2\right) (a d-b c)^2}-\frac{2}{3 a^2 c^2 x^{3/2}}-\frac{d^{11/4} (15 b c-7 a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{11/4} (a d-b c)^3}+\frac{d^{11/4} (15 b c-7 a d) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{11/4} (a d-b c)^3}+\frac{d^{11/4} (15 b c-7 a d) \tan ^{-1}\left(\frac{2 \sqrt[4]{d} \sqrt{x}-\sqrt{2} \sqrt[4]{c}}{\sqrt{2} \sqrt[4]{c}}\right)}{4 \sqrt{2} c^{11/4} (a d-b c)^3}+\frac{d^{11/4} (15 b c-7 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c}+2 \sqrt[4]{d} \sqrt{x}}{\sqrt{2} \sqrt[4]{c}}\right)}{4 \sqrt{2} c^{11/4} (a d-b c)^3}-\frac{d^3 \sqrt{x}}{2 c^2 \left(c+d x^2\right) (b c-a d)^2}","\frac{b^{11/4} (7 b c-15 a d) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{11/4} (b c-a d)^3}-\frac{b^{11/4} (7 b c-15 a d) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{11/4} (b c-a d)^3}+\frac{b^{11/4} (7 b c-15 a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} a^{11/4} (b c-a d)^3}-\frac{b^{11/4} (7 b c-15 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} a^{11/4} (b c-a d)^3}+\frac{-7 a^2 d^2+8 a b c d-7 b^2 c^2}{6 a^2 c^2 x^{3/2} (b c-a d)^2}+\frac{d^{11/4} (15 b c-7 a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{11/4} (b c-a d)^3}-\frac{d^{11/4} (15 b c-7 a d) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{11/4} (b c-a d)^3}+\frac{d^{11/4} (15 b c-7 a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{4 \sqrt{2} c^{11/4} (b c-a d)^3}-\frac{d^{11/4} (15 b c-7 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{4 \sqrt{2} c^{11/4} (b c-a d)^3}+\frac{b}{2 a x^{3/2} \left(a+b x^2\right) \left(c+d x^2\right) (b c-a d)}+\frac{d (a d+b c)}{2 a c x^{3/2} \left(c+d x^2\right) (b c-a d)^2}",1,"-2/(3*a^2*c^2*x^(3/2)) - (b^3*Sqrt[x])/(2*a^2*(-(b*c) + a*d)^2*(a + b*x^2)) - (d^3*Sqrt[x])/(2*c^2*(b*c - a*d)^2*(c + d*x^2)) + (b^(11/4)*(-7*b*c + 15*a*d)*ArcTan[(-(Sqrt[2]*a^(1/4)) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(4*Sqrt[2]*a^(11/4)*(b*c - a*d)^3) + (b^(11/4)*(-7*b*c + 15*a*d)*ArcTan[(Sqrt[2]*a^(1/4) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(4*Sqrt[2]*a^(11/4)*(b*c - a*d)^3) + (d^(11/4)*(15*b*c - 7*a*d)*ArcTan[(-(Sqrt[2]*c^(1/4)) + 2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(4*Sqrt[2]*c^(11/4)*(-(b*c) + a*d)^3) + (d^(11/4)*(15*b*c - 7*a*d)*ArcTan[(Sqrt[2]*c^(1/4) + 2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(4*Sqrt[2]*c^(11/4)*(-(b*c) + a*d)^3) - (b^(11/4)*(-7*b*c + 15*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(11/4)*(b*c - a*d)^3) + (b^(11/4)*(-7*b*c + 15*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(11/4)*(b*c - a*d)^3) - (d^(11/4)*(15*b*c - 7*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(11/4)*(-(b*c) + a*d)^3) + (d^(11/4)*(15*b*c - 7*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(11/4)*(-(b*c) + a*d)^3)","A",1
495,1,696,731,6.1750762,"\int \frac{1}{x^{7/2} \left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","Integrate[1/(x^(7/2)*(a + b*x^2)^2*(c + d*x^2)^2),x]","-\frac{b^{13/4} (17 a d-9 b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{13/4} (b c-a d)^3}+\frac{b^{13/4} (17 a d-9 b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{13/4} (b c-a d)^3}-\frac{b^{13/4} (17 a d-9 b c) \tan ^{-1}\left(\frac{2 \sqrt[4]{b} \sqrt{x}-\sqrt{2} \sqrt[4]{a}}{\sqrt{2} \sqrt[4]{a}}\right)}{4 \sqrt{2} a^{13/4} (b c-a d)^3}-\frac{b^{13/4} (17 a d-9 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a}+2 \sqrt[4]{b} \sqrt{x}}{\sqrt{2} \sqrt[4]{a}}\right)}{4 \sqrt{2} a^{13/4} (b c-a d)^3}+\frac{b^4 x^{3/2}}{2 a^3 \left(a+b x^2\right) (a d-b c)^2}+\frac{4 (a d+b c)}{a^3 c^3 \sqrt{x}}-\frac{2}{5 a^2 c^2 x^{5/2}}-\frac{d^{13/4} (17 b c-9 a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{13/4} (a d-b c)^3}+\frac{d^{13/4} (17 b c-9 a d) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{13/4} (a d-b c)^3}-\frac{d^{13/4} (17 b c-9 a d) \tan ^{-1}\left(\frac{2 \sqrt[4]{d} \sqrt{x}-\sqrt{2} \sqrt[4]{c}}{\sqrt{2} \sqrt[4]{c}}\right)}{4 \sqrt{2} c^{13/4} (a d-b c)^3}-\frac{d^{13/4} (17 b c-9 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c}+2 \sqrt[4]{d} \sqrt{x}}{\sqrt{2} \sqrt[4]{c}}\right)}{4 \sqrt{2} c^{13/4} (a d-b c)^3}+\frac{d^4 x^{3/2}}{2 c^3 \left(c+d x^2\right) (b c-a d)^2}","\frac{b^{13/4} (9 b c-17 a d) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{13/4} (b c-a d)^3}-\frac{b^{13/4} (9 b c-17 a d) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{13/4} (b c-a d)^3}-\frac{b^{13/4} (9 b c-17 a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} a^{13/4} (b c-a d)^3}+\frac{b^{13/4} (9 b c-17 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} a^{13/4} (b c-a d)^3}+\frac{-9 a^2 d^2+8 a b c d-9 b^2 c^2}{10 a^2 c^2 x^{5/2} (b c-a d)^2}+\frac{(a d+b c) \left(9 a^2 d^2-17 a b c d+9 b^2 c^2\right)}{2 a^3 c^3 \sqrt{x} (b c-a d)^2}+\frac{d^{13/4} (17 b c-9 a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{13/4} (b c-a d)^3}-\frac{d^{13/4} (17 b c-9 a d) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{8 \sqrt{2} c^{13/4} (b c-a d)^3}-\frac{d^{13/4} (17 b c-9 a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{4 \sqrt{2} c^{13/4} (b c-a d)^3}+\frac{d^{13/4} (17 b c-9 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{4 \sqrt{2} c^{13/4} (b c-a d)^3}+\frac{b}{2 a x^{5/2} \left(a+b x^2\right) \left(c+d x^2\right) (b c-a d)}+\frac{d (a d+b c)}{2 a c x^{5/2} \left(c+d x^2\right) (b c-a d)^2}",1,"-2/(5*a^2*c^2*x^(5/2)) + (4*(b*c + a*d))/(a^3*c^3*Sqrt[x]) + (b^4*x^(3/2))/(2*a^3*(-(b*c) + a*d)^2*(a + b*x^2)) + (d^4*x^(3/2))/(2*c^3*(b*c - a*d)^2*(c + d*x^2)) - (b^(13/4)*(-9*b*c + 17*a*d)*ArcTan[(-(Sqrt[2]*a^(1/4)) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(4*Sqrt[2]*a^(13/4)*(b*c - a*d)^3) - (b^(13/4)*(-9*b*c + 17*a*d)*ArcTan[(Sqrt[2]*a^(1/4) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(4*Sqrt[2]*a^(13/4)*(b*c - a*d)^3) - (d^(13/4)*(17*b*c - 9*a*d)*ArcTan[(-(Sqrt[2]*c^(1/4)) + 2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(4*Sqrt[2]*c^(13/4)*(-(b*c) + a*d)^3) - (d^(13/4)*(17*b*c - 9*a*d)*ArcTan[(Sqrt[2]*c^(1/4) + 2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(4*Sqrt[2]*c^(13/4)*(-(b*c) + a*d)^3) - (b^(13/4)*(-9*b*c + 17*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(13/4)*(b*c - a*d)^3) + (b^(13/4)*(-9*b*c + 17*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(13/4)*(b*c - a*d)^3) - (d^(13/4)*(17*b*c - 9*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(13/4)*(-(b*c) + a*d)^3) + (d^(13/4)*(17*b*c - 9*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(13/4)*(-(b*c) + a*d)^3)","A",1
496,1,604,718,1.4086281,"\int \frac{x^{7/2}}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","Integrate[x^(7/2)/((a + b*x^2)^2*(c + d*x^2)^3),x]","\frac{-\frac{\sqrt{2} \left(5 a^2 d^2+70 a b c d+21 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{3/4} \sqrt[4]{d}}+\frac{\sqrt{2} \left(5 a^2 d^2+70 a b c d+21 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{3/4} \sqrt[4]{d}}-\frac{2 \sqrt{2} \left(5 a^2 d^2+70 a b c d+21 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{c^{3/4} \sqrt[4]{d}}+\frac{2 \sqrt{2} \left(5 a^2 d^2+70 a b c d+21 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{c^{3/4} \sqrt[4]{d}}+8 \sqrt{2} \sqrt[4]{a} b^{3/4} (7 a d+5 b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-8 \sqrt{2} \sqrt[4]{a} b^{3/4} (7 a d+5 b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+16 \sqrt{2} \sqrt[4]{a} b^{3/4} (7 a d+5 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)-16 \sqrt{2} \sqrt[4]{a} b^{3/4} (7 a d+5 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)+\frac{32 c \sqrt{x} (b c-a d)^2}{\left(c+d x^2\right)^2}+\frac{64 a b \sqrt{x} (b c-a d)}{a+b x^2}+\frac{8 \sqrt{x} (9 a d+7 b c) (b c-a d)}{c+d x^2}}{128 (b c-a d)^4}","-\frac{\left(5 a^2 d^2+70 a b c d+21 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{3/4} \sqrt[4]{d} (b c-a d)^4}+\frac{\left(5 a^2 d^2+70 a b c d+21 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{3/4} \sqrt[4]{d} (b c-a d)^4}-\frac{\left(5 a^2 d^2+70 a b c d+21 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{32 \sqrt{2} c^{3/4} \sqrt[4]{d} (b c-a d)^4}+\frac{\left(5 a^2 d^2+70 a b c d+21 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{32 \sqrt{2} c^{3/4} \sqrt[4]{d} (b c-a d)^4}+\frac{\sqrt[4]{a} b^{3/4} (7 a d+5 b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} (b c-a d)^4}-\frac{\sqrt[4]{a} b^{3/4} (7 a d+5 b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} (b c-a d)^4}+\frac{\sqrt[4]{a} b^{3/4} (7 a d+5 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} (b c-a d)^4}-\frac{\sqrt[4]{a} b^{3/4} (7 a d+5 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} (b c-a d)^4}+\frac{a \sqrt{x}}{2 b \left(a+b x^2\right) \left(c+d x^2\right)^2 (b c-a d)}+\frac{\sqrt{x} (17 a d+7 b c)}{16 \left(c+d x^2\right) (b c-a d)^3}+\frac{\sqrt{x} (2 a d+b c)}{4 b \left(c+d x^2\right)^2 (b c-a d)^2}",1,"((64*a*b*(b*c - a*d)*Sqrt[x])/(a + b*x^2) + (32*c*(b*c - a*d)^2*Sqrt[x])/(c + d*x^2)^2 + (8*(b*c - a*d)*(7*b*c + 9*a*d)*Sqrt[x])/(c + d*x^2) + 16*Sqrt[2]*a^(1/4)*b^(3/4)*(5*b*c + 7*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 16*Sqrt[2]*a^(1/4)*b^(3/4)*(5*b*c + 7*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - (2*Sqrt[2]*(21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(3/4)*d^(1/4)) + (2*Sqrt[2]*(21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(3/4)*d^(1/4)) + 8*Sqrt[2]*a^(1/4)*b^(3/4)*(5*b*c + 7*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - 8*Sqrt[2]*a^(1/4)*b^(3/4)*(5*b*c + 7*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - (Sqrt[2]*(21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(3/4)*d^(1/4)) + (Sqrt[2]*(21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(3/4)*d^(1/4)))/(128*(b*c - a*d)^4)","A",1
497,1,604,703,2.0071791,"\int \frac{x^{5/2}}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","Integrate[x^(5/2)/((a + b*x^2)^2*(c + d*x^2)^3),x]","\frac{\frac{3 \sqrt{2} \sqrt[4]{d} \left(a^2 d^2-18 a b c d-15 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{5/4}}+\frac{3 \sqrt{2} \sqrt[4]{d} \left(-a^2 d^2+18 a b c d+15 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{5/4}}+\frac{6 \sqrt{2} \sqrt[4]{d} \left(-a^2 d^2+18 a b c d+15 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{c^{5/4}}+\frac{6 \sqrt{2} \sqrt[4]{d} \left(a^2 d^2-18 a b c d-15 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{c^{5/4}}+\frac{24 \sqrt{2} b^{5/4} (3 a d+b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{\sqrt[4]{a}}-\frac{24 \sqrt{2} b^{5/4} (3 a d+b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{\sqrt[4]{a}}-\frac{48 \sqrt{2} b^{5/4} (3 a d+b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt[4]{a}}+\frac{48 \sqrt{2} b^{5/4} (3 a d+b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt[4]{a}}-\frac{64 b^2 x^{3/2} (b c-a d)}{a+b x^2}+\frac{8 d x^{3/2} (a d-b c) (3 a d+13 b c)}{c \left(c+d x^2\right)}-\frac{32 d x^{3/2} (b c-a d)^2}{\left(c+d x^2\right)^2}}{128 (b c-a d)^4}","-\frac{3 \sqrt[4]{d} \left(-a^2 d^2+18 a b c d+15 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{5/4} (b c-a d)^4}+\frac{3 \sqrt[4]{d} \left(-a^2 d^2+18 a b c d+15 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{5/4} (b c-a d)^4}+\frac{3 \sqrt[4]{d} \left(-a^2 d^2+18 a b c d+15 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{32 \sqrt{2} c^{5/4} (b c-a d)^4}-\frac{3 \sqrt[4]{d} \left(-a^2 d^2+18 a b c d+15 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{32 \sqrt{2} c^{5/4} (b c-a d)^4}+\frac{3 b^{5/4} (3 a d+b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} \sqrt[4]{a} (b c-a d)^4}-\frac{3 b^{5/4} (3 a d+b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} \sqrt[4]{a} (b c-a d)^4}-\frac{3 b^{5/4} (3 a d+b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} \sqrt[4]{a} (b c-a d)^4}+\frac{3 b^{5/4} (3 a d+b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} \sqrt[4]{a} (b c-a d)^4}-\frac{3 d x^{3/2} (a d+7 b c)}{16 c \left(c+d x^2\right) (b c-a d)^3}-\frac{x^{3/2}}{2 \left(a+b x^2\right) \left(c+d x^2\right)^2 (b c-a d)}-\frac{3 d x^{3/2}}{4 \left(c+d x^2\right)^2 (b c-a d)^2}",1,"((-64*b^2*(b*c - a*d)*x^(3/2))/(a + b*x^2) - (32*d*(b*c - a*d)^2*x^(3/2))/(c + d*x^2)^2 + (8*d*(-(b*c) + a*d)*(13*b*c + 3*a*d)*x^(3/2))/(c*(c + d*x^2)) - (48*Sqrt[2]*b^(5/4)*(b*c + 3*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(1/4) + (48*Sqrt[2]*b^(5/4)*(b*c + 3*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(1/4) + (6*Sqrt[2]*d^(1/4)*(15*b^2*c^2 + 18*a*b*c*d - a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(5/4) + (6*Sqrt[2]*d^(1/4)*(-15*b^2*c^2 - 18*a*b*c*d + a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(5/4) + (24*Sqrt[2]*b^(5/4)*(b*c + 3*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(1/4) - (24*Sqrt[2]*b^(5/4)*(b*c + 3*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(1/4) + (3*Sqrt[2]*d^(1/4)*(-15*b^2*c^2 - 18*a*b*c*d + a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(5/4) + (3*Sqrt[2]*d^(1/4)*(15*b^2*c^2 + 18*a*b*c*d - a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(5/4))/(128*(b*c - a*d)^4)","A",1
498,1,603,703,2.3941268,"\int \frac{x^{3/2}}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","Integrate[x^(3/2)/((a + b*x^2)^2*(c + d*x^2)^3),x]","\frac{-\frac{8 \sqrt{2} b^{7/4} (11 a d+b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{3/4}}+\frac{8 \sqrt{2} b^{7/4} (11 a d+b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{3/4}}-\frac{16 \sqrt{2} b^{7/4} (11 a d+b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{a^{3/4}}+\frac{16 \sqrt{2} b^{7/4} (11 a d+b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{a^{3/4}}+\frac{\sqrt{2} d^{3/4} \left(-3 a^2 d^2+22 a b c d+77 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{7/4}}+\frac{\sqrt{2} d^{3/4} \left(3 a^2 d^2-22 a b c d-77 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{c^{7/4}}+\frac{2 \sqrt{2} d^{3/4} \left(-3 a^2 d^2+22 a b c d+77 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{c^{7/4}}-\frac{2 \sqrt{2} d^{3/4} \left(-3 a^2 d^2+22 a b c d+77 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{c^{7/4}}-\frac{64 b^2 \sqrt{x} (b c-a d)}{a+b x^2}+\frac{8 d \sqrt{x} (a d-b c) (a d+15 b c)}{c \left(c+d x^2\right)}-\frac{32 d \sqrt{x} (b c-a d)^2}{\left(c+d x^2\right)^2}}{128 (b c-a d)^4}","-\frac{b^{7/4} (11 a d+b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{3/4} (b c-a d)^4}+\frac{b^{7/4} (11 a d+b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{3/4} (b c-a d)^4}-\frac{b^{7/4} (11 a d+b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} a^{3/4} (b c-a d)^4}+\frac{b^{7/4} (11 a d+b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} a^{3/4} (b c-a d)^4}+\frac{d^{3/4} \left(-3 a^2 d^2+22 a b c d+77 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{7/4} (b c-a d)^4}-\frac{d^{3/4} \left(-3 a^2 d^2+22 a b c d+77 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{7/4} (b c-a d)^4}+\frac{d^{3/4} \left(-3 a^2 d^2+22 a b c d+77 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{32 \sqrt{2} c^{7/4} (b c-a d)^4}-\frac{d^{3/4} \left(-3 a^2 d^2+22 a b c d+77 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{32 \sqrt{2} c^{7/4} (b c-a d)^4}-\frac{d \sqrt{x} (a d+23 b c)}{16 c \left(c+d x^2\right) (b c-a d)^3}-\frac{3 d \sqrt{x}}{4 \left(c+d x^2\right)^2 (b c-a d)^2}-\frac{\sqrt{x}}{2 \left(a+b x^2\right) \left(c+d x^2\right)^2 (b c-a d)}",1,"((-64*b^2*(b*c - a*d)*Sqrt[x])/(a + b*x^2) - (32*d*(b*c - a*d)^2*Sqrt[x])/(c + d*x^2)^2 + (8*d*(-(b*c) + a*d)*(15*b*c + a*d)*Sqrt[x])/(c*(c + d*x^2)) - (16*Sqrt[2]*b^(7/4)*(b*c + 11*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(3/4) + (16*Sqrt[2]*b^(7/4)*(b*c + 11*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(3/4) + (2*Sqrt[2]*d^(3/4)*(77*b^2*c^2 + 22*a*b*c*d - 3*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(7/4) - (2*Sqrt[2]*d^(3/4)*(77*b^2*c^2 + 22*a*b*c*d - 3*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(7/4) - (8*Sqrt[2]*b^(7/4)*(b*c + 11*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(3/4) + (8*Sqrt[2]*b^(7/4)*(b*c + 11*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(3/4) + (Sqrt[2]*d^(3/4)*(77*b^2*c^2 + 22*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(7/4) + (Sqrt[2]*d^(3/4)*(-77*b^2*c^2 - 22*a*b*c*d + 3*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(7/4))/(128*(b*c - a*d)^4)","A",1
499,1,760,739,6.2810152,"\int \frac{\sqrt{x}}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","Integrate[Sqrt[x]/((a + b*x^2)^2*(c + d*x^2)^3),x]","-\frac{b^{9/4} (13 a d-b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{5/4} (b c-a d)^4}+\frac{b^{9/4} (13 a d-b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{5/4} (b c-a d)^4}-\frac{b^{9/4} (13 a d-b c) \tan ^{-1}\left(\frac{2 \sqrt[4]{b} \sqrt{x}-\sqrt{2} \sqrt[4]{a}}{\sqrt{2} \sqrt[4]{a}}\right)}{4 \sqrt{2} a^{5/4} (b c-a d)^4}-\frac{b^{9/4} (13 a d-b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a}+2 \sqrt[4]{b} \sqrt{x}}{\sqrt{2} \sqrt[4]{a}}\right)}{4 \sqrt{2} a^{5/4} (b c-a d)^4}+\frac{d^{5/4} \left(5 a^2 d^2-26 a b c d+117 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{9/4} (a d-b c)^4}-\frac{d^{5/4} \left(5 a^2 d^2-26 a b c d+117 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{9/4} (a d-b c)^4}+\frac{d^{5/4} \left(5 a^2 d^2-26 a b c d+117 b^2 c^2\right) \tan ^{-1}\left(\frac{2 \sqrt[4]{d} \sqrt{x}-\sqrt{2} \sqrt[4]{c}}{\sqrt{2} \sqrt[4]{c}}\right)}{32 \sqrt{2} c^{9/4} (a d-b c)^4}+\frac{d^{5/4} \left(5 a^2 d^2-26 a b c d+117 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c}+2 \sqrt[4]{d} \sqrt{x}}{\sqrt{2} \sqrt[4]{c}}\right)}{32 \sqrt{2} c^{9/4} (a d-b c)^4}-\frac{b^3 x^{3/2}}{2 a \left(a+b x^2\right) (a d-b c)^3}+\frac{d^2 x^{3/2} (21 b c-5 a d)}{16 c^2 \left(c+d x^2\right) (b c-a d)^3}+\frac{d^2 x^{3/2}}{4 c \left(c+d x^2\right)^2 (b c-a d)^2}","\frac{b^{9/4} (b c-13 a d) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{5/4} (b c-a d)^4}-\frac{b^{9/4} (b c-13 a d) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{5/4} (b c-a d)^4}-\frac{b^{9/4} (b c-13 a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} a^{5/4} (b c-a d)^4}+\frac{b^{9/4} (b c-13 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} a^{5/4} (b c-a d)^4}+\frac{d x^{3/2} \left(-5 a^2 d^2+21 a b c d+8 b^2 c^2\right)}{16 a c^2 \left(c+d x^2\right) (b c-a d)^3}+\frac{d^{5/4} \left(5 a^2 d^2-26 a b c d+117 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{9/4} (b c-a d)^4}-\frac{d^{5/4} \left(5 a^2 d^2-26 a b c d+117 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{9/4} (b c-a d)^4}-\frac{d^{5/4} \left(5 a^2 d^2-26 a b c d+117 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{32 \sqrt{2} c^{9/4} (b c-a d)^4}+\frac{d^{5/4} \left(5 a^2 d^2-26 a b c d+117 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{32 \sqrt{2} c^{9/4} (b c-a d)^4}+\frac{b x^{3/2}}{2 a \left(a+b x^2\right) \left(c+d x^2\right)^2 (b c-a d)}+\frac{d x^{3/2} (a d+2 b c)}{4 a c \left(c+d x^2\right)^2 (b c-a d)^2}",1,"-1/2*(b^3*x^(3/2))/(a*(-(b*c) + a*d)^3*(a + b*x^2)) + (d^2*x^(3/2))/(4*c*(b*c - a*d)^2*(c + d*x^2)^2) + (d^2*(21*b*c - 5*a*d)*x^(3/2))/(16*c^2*(b*c - a*d)^3*(c + d*x^2)) - (b^(9/4)*(-(b*c) + 13*a*d)*ArcTan[(-(Sqrt[2]*a^(1/4)) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(4*Sqrt[2]*a^(5/4)*(b*c - a*d)^4) - (b^(9/4)*(-(b*c) + 13*a*d)*ArcTan[(Sqrt[2]*a^(1/4) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(4*Sqrt[2]*a^(5/4)*(b*c - a*d)^4) + (d^(5/4)*(117*b^2*c^2 - 26*a*b*c*d + 5*a^2*d^2)*ArcTan[(-(Sqrt[2]*c^(1/4)) + 2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(32*Sqrt[2]*c^(9/4)*(-(b*c) + a*d)^4) + (d^(5/4)*(117*b^2*c^2 - 26*a*b*c*d + 5*a^2*d^2)*ArcTan[(Sqrt[2]*c^(1/4) + 2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(32*Sqrt[2]*c^(9/4)*(-(b*c) + a*d)^4) - (b^(9/4)*(-(b*c) + 13*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(5/4)*(b*c - a*d)^4) + (b^(9/4)*(-(b*c) + 13*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(5/4)*(b*c - a*d)^4) + (d^(5/4)*(117*b^2*c^2 - 26*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(9/4)*(-(b*c) + a*d)^4) - (d^(5/4)*(117*b^2*c^2 - 26*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(9/4)*(-(b*c) + a*d)^4)","A",1
500,1,760,739,6.3292863,"\int \frac{1}{\sqrt{x} \left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","Integrate[1/(Sqrt[x]*(a + b*x^2)^2*(c + d*x^2)^3),x]","\frac{3 b^{11/4} (5 a d-b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{7/4} (b c-a d)^4}-\frac{3 b^{11/4} (5 a d-b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{7/4} (b c-a d)^4}-\frac{3 b^{11/4} (5 a d-b c) \tan ^{-1}\left(\frac{2 \sqrt[4]{b} \sqrt{x}-\sqrt{2} \sqrt[4]{a}}{\sqrt{2} \sqrt[4]{a}}\right)}{4 \sqrt{2} a^{7/4} (b c-a d)^4}-\frac{3 b^{11/4} (5 a d-b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a}+2 \sqrt[4]{b} \sqrt{x}}{\sqrt{2} \sqrt[4]{a}}\right)}{4 \sqrt{2} a^{7/4} (b c-a d)^4}-\frac{3 d^{7/4} \left(7 a^2 d^2-30 a b c d+55 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{11/4} (a d-b c)^4}+\frac{3 d^{7/4} \left(7 a^2 d^2-30 a b c d+55 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{11/4} (a d-b c)^4}+\frac{3 d^{7/4} \left(7 a^2 d^2-30 a b c d+55 b^2 c^2\right) \tan ^{-1}\left(\frac{2 \sqrt[4]{d} \sqrt{x}-\sqrt{2} \sqrt[4]{c}}{\sqrt{2} \sqrt[4]{c}}\right)}{32 \sqrt{2} c^{11/4} (a d-b c)^4}+\frac{3 d^{7/4} \left(7 a^2 d^2-30 a b c d+55 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c}+2 \sqrt[4]{d} \sqrt{x}}{\sqrt{2} \sqrt[4]{c}}\right)}{32 \sqrt{2} c^{11/4} (a d-b c)^4}-\frac{b^3 \sqrt{x}}{2 a \left(a+b x^2\right) (a d-b c)^3}+\frac{d^2 \sqrt{x} (23 b c-7 a d)}{16 c^2 \left(c+d x^2\right) (b c-a d)^3}+\frac{d^2 \sqrt{x}}{4 c \left(c+d x^2\right)^2 (b c-a d)^2}","-\frac{3 b^{11/4} (b c-5 a d) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{7/4} (b c-a d)^4}+\frac{3 b^{11/4} (b c-5 a d) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{7/4} (b c-a d)^4}-\frac{3 b^{11/4} (b c-5 a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} a^{7/4} (b c-a d)^4}+\frac{3 b^{11/4} (b c-5 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} a^{7/4} (b c-a d)^4}+\frac{d \sqrt{x} \left(-7 a^2 d^2+23 a b c d+8 b^2 c^2\right)}{16 a c^2 \left(c+d x^2\right) (b c-a d)^3}-\frac{3 d^{7/4} \left(7 a^2 d^2-30 a b c d+55 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{11/4} (b c-a d)^4}+\frac{3 d^{7/4} \left(7 a^2 d^2-30 a b c d+55 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{11/4} (b c-a d)^4}-\frac{3 d^{7/4} \left(7 a^2 d^2-30 a b c d+55 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{32 \sqrt{2} c^{11/4} (b c-a d)^4}+\frac{3 d^{7/4} \left(7 a^2 d^2-30 a b c d+55 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{32 \sqrt{2} c^{11/4} (b c-a d)^4}+\frac{b \sqrt{x}}{2 a \left(a+b x^2\right) \left(c+d x^2\right)^2 (b c-a d)}+\frac{d \sqrt{x} (a d+2 b c)}{4 a c \left(c+d x^2\right)^2 (b c-a d)^2}",1,"-1/2*(b^3*Sqrt[x])/(a*(-(b*c) + a*d)^3*(a + b*x^2)) + (d^2*Sqrt[x])/(4*c*(b*c - a*d)^2*(c + d*x^2)^2) + (d^2*(23*b*c - 7*a*d)*Sqrt[x])/(16*c^2*(b*c - a*d)^3*(c + d*x^2)) - (3*b^(11/4)*(-(b*c) + 5*a*d)*ArcTan[(-(Sqrt[2]*a^(1/4)) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(4*Sqrt[2]*a^(7/4)*(b*c - a*d)^4) - (3*b^(11/4)*(-(b*c) + 5*a*d)*ArcTan[(Sqrt[2]*a^(1/4) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(4*Sqrt[2]*a^(7/4)*(b*c - a*d)^4) + (3*d^(7/4)*(55*b^2*c^2 - 30*a*b*c*d + 7*a^2*d^2)*ArcTan[(-(Sqrt[2]*c^(1/4)) + 2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(32*Sqrt[2]*c^(11/4)*(-(b*c) + a*d)^4) + (3*d^(7/4)*(55*b^2*c^2 - 30*a*b*c*d + 7*a^2*d^2)*ArcTan[(Sqrt[2]*c^(1/4) + 2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(32*Sqrt[2]*c^(11/4)*(-(b*c) + a*d)^4) + (3*b^(11/4)*(-(b*c) + 5*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(7/4)*(b*c - a*d)^4) - (3*b^(11/4)*(-(b*c) + 5*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(7/4)*(b*c - a*d)^4) - (3*d^(7/4)*(55*b^2*c^2 - 30*a*b*c*d + 7*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(11/4)*(-(b*c) + a*d)^4) + (3*d^(7/4)*(55*b^2*c^2 - 30*a*b*c*d + 7*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(11/4)*(-(b*c) + a*d)^4)","A",1
501,1,773,805,6.2523992,"\int \frac{1}{x^{3/2} \left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","Integrate[1/(x^(3/2)*(a + b*x^2)^2*(c + d*x^2)^3),x]","\frac{b^{13/4} (17 a d-5 b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{9/4} (b c-a d)^4}-\frac{b^{13/4} (17 a d-5 b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{9/4} (b c-a d)^4}+\frac{b^{13/4} (17 a d-5 b c) \tan ^{-1}\left(\frac{2 \sqrt[4]{b} \sqrt{x}-\sqrt{2} \sqrt[4]{a}}{\sqrt{2} \sqrt[4]{a}}\right)}{4 \sqrt{2} a^{9/4} (b c-a d)^4}+\frac{b^{13/4} (17 a d-5 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a}+2 \sqrt[4]{b} \sqrt{x}}{\sqrt{2} \sqrt[4]{a}}\right)}{4 \sqrt{2} a^{9/4} (b c-a d)^4}+\frac{b^4 x^{3/2}}{2 a^2 \left(a+b x^2\right) (a d-b c)^3}-\frac{d^{9/4} \left(45 a^2 d^2-170 a b c d+221 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{13/4} (a d-b c)^4}+\frac{d^{9/4} \left(45 a^2 d^2-170 a b c d+221 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{13/4} (a d-b c)^4}-\frac{d^{9/4} \left(45 a^2 d^2-170 a b c d+221 b^2 c^2\right) \tan ^{-1}\left(\frac{2 \sqrt[4]{d} \sqrt{x}-\sqrt{2} \sqrt[4]{c}}{\sqrt{2} \sqrt[4]{c}}\right)}{32 \sqrt{2} c^{13/4} (a d-b c)^4}-\frac{d^{9/4} \left(45 a^2 d^2-170 a b c d+221 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c}+2 \sqrt[4]{d} \sqrt{x}}{\sqrt{2} \sqrt[4]{c}}\right)}{32 \sqrt{2} c^{13/4} (a d-b c)^4}-\frac{2}{a^2 c^3 \sqrt{x}}-\frac{d^3 x^{3/2} (29 b c-13 a d)}{16 c^3 \left(c+d x^2\right) (b c-a d)^3}-\frac{d^3 x^{3/2}}{4 c^2 \left(c+d x^2\right)^2 (b c-a d)^2}","\frac{(5 b c-17 a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right) b^{13/4}}{4 \sqrt{2} a^{9/4} (b c-a d)^4}-\frac{(5 b c-17 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right) b^{13/4}}{4 \sqrt{2} a^{9/4} (b c-a d)^4}-\frac{(5 b c-17 a d) \log \left(\sqrt{b} x-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}\right) b^{13/4}}{8 \sqrt{2} a^{9/4} (b c-a d)^4}+\frac{(5 b c-17 a d) \log \left(\sqrt{b} x+\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}\right) b^{13/4}}{8 \sqrt{2} a^{9/4} (b c-a d)^4}+\frac{b}{2 a (b c-a d) \sqrt{x} \left(b x^2+a\right) \left(d x^2+c\right)^2}+\frac{d^{9/4} \left(221 b^2 c^2-170 a b d c+45 a^2 d^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{32 \sqrt{2} c^{13/4} (b c-a d)^4}-\frac{d^{9/4} \left(221 b^2 c^2-170 a b d c+45 a^2 d^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{32 \sqrt{2} c^{13/4} (b c-a d)^4}-\frac{d^{9/4} \left(221 b^2 c^2-170 a b d c+45 a^2 d^2\right) \log \left(\sqrt{d} x-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}\right)}{64 \sqrt{2} c^{13/4} (b c-a d)^4}+\frac{d^{9/4} \left(221 b^2 c^2-170 a b d c+45 a^2 d^2\right) \log \left(\sqrt{d} x+\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}\right)}{64 \sqrt{2} c^{13/4} (b c-a d)^4}+\frac{-40 b^3 c^3+96 a b^2 d c^2-125 a^2 b d^2 c+45 a^3 d^3}{16 a^2 c^3 (b c-a d)^3 \sqrt{x}}+\frac{d \left(8 b^2 c^2+25 a b d c-9 a^2 d^2\right)}{16 a c^2 (b c-a d)^3 \sqrt{x} \left(d x^2+c\right)}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 \sqrt{x} \left(d x^2+c\right)^2}",1,"-2/(a^2*c^3*Sqrt[x]) + (b^4*x^(3/2))/(2*a^2*(-(b*c) + a*d)^3*(a + b*x^2)) - (d^3*x^(3/2))/(4*c^2*(b*c - a*d)^2*(c + d*x^2)^2) - (d^3*(29*b*c - 13*a*d)*x^(3/2))/(16*c^3*(b*c - a*d)^3*(c + d*x^2)) + (b^(13/4)*(-5*b*c + 17*a*d)*ArcTan[(-(Sqrt[2]*a^(1/4)) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(4*Sqrt[2]*a^(9/4)*(b*c - a*d)^4) + (b^(13/4)*(-5*b*c + 17*a*d)*ArcTan[(Sqrt[2]*a^(1/4) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(4*Sqrt[2]*a^(9/4)*(b*c - a*d)^4) - (d^(9/4)*(221*b^2*c^2 - 170*a*b*c*d + 45*a^2*d^2)*ArcTan[(-(Sqrt[2]*c^(1/4)) + 2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(32*Sqrt[2]*c^(13/4)*(-(b*c) + a*d)^4) - (d^(9/4)*(221*b^2*c^2 - 170*a*b*c*d + 45*a^2*d^2)*ArcTan[(Sqrt[2]*c^(1/4) + 2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(32*Sqrt[2]*c^(13/4)*(-(b*c) + a*d)^4) + (b^(13/4)*(-5*b*c + 17*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(9/4)*(b*c - a*d)^4) - (b^(13/4)*(-5*b*c + 17*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(9/4)*(b*c - a*d)^4) - (d^(9/4)*(221*b^2*c^2 - 170*a*b*c*d + 45*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(13/4)*(-(b*c) + a*d)^4) + (d^(9/4)*(221*b^2*c^2 - 170*a*b*c*d + 45*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(13/4)*(-(b*c) + a*d)^4)","A",1
502,1,775,805,6.2638534,"\int \frac{1}{x^{5/2} \left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","Integrate[1/(x^(5/2)*(a + b*x^2)^2*(c + d*x^2)^3),x]","-\frac{b^{15/4} (19 a d-7 b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{11/4} (b c-a d)^4}+\frac{b^{15/4} (19 a d-7 b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{11/4} (b c-a d)^4}+\frac{b^{15/4} (19 a d-7 b c) \tan ^{-1}\left(\frac{2 \sqrt[4]{b} \sqrt{x}-\sqrt{2} \sqrt[4]{a}}{\sqrt{2} \sqrt[4]{a}}\right)}{4 \sqrt{2} a^{11/4} (b c-a d)^4}+\frac{b^{15/4} (19 a d-7 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a}+2 \sqrt[4]{b} \sqrt{x}}{\sqrt{2} \sqrt[4]{a}}\right)}{4 \sqrt{2} a^{11/4} (b c-a d)^4}+\frac{b^4 \sqrt{x}}{2 a^2 \left(a+b x^2\right) (a d-b c)^3}+\frac{d^{11/4} \left(77 a^2 d^2-266 a b c d+285 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{15/4} (a d-b c)^4}-\frac{d^{11/4} \left(77 a^2 d^2-266 a b c d+285 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right)}{64 \sqrt{2} c^{15/4} (a d-b c)^4}-\frac{d^{11/4} \left(77 a^2 d^2-266 a b c d+285 b^2 c^2\right) \tan ^{-1}\left(\frac{2 \sqrt[4]{d} \sqrt{x}-\sqrt{2} \sqrt[4]{c}}{\sqrt{2} \sqrt[4]{c}}\right)}{32 \sqrt{2} c^{15/4} (a d-b c)^4}-\frac{d^{11/4} \left(77 a^2 d^2-266 a b c d+285 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c}+2 \sqrt[4]{d} \sqrt{x}}{\sqrt{2} \sqrt[4]{c}}\right)}{32 \sqrt{2} c^{15/4} (a d-b c)^4}-\frac{2}{3 a^2 c^3 x^{3/2}}-\frac{d^3 \sqrt{x} (31 b c-15 a d)}{16 c^3 \left(c+d x^2\right) (b c-a d)^3}-\frac{d^3 \sqrt{x}}{4 c^2 \left(c+d x^2\right)^2 (b c-a d)^2}","\frac{(7 b c-19 a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right) b^{15/4}}{4 \sqrt{2} a^{11/4} (b c-a d)^4}-\frac{(7 b c-19 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right) b^{15/4}}{4 \sqrt{2} a^{11/4} (b c-a d)^4}+\frac{(7 b c-19 a d) \log \left(\sqrt{b} x-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}\right) b^{15/4}}{8 \sqrt{2} a^{11/4} (b c-a d)^4}-\frac{(7 b c-19 a d) \log \left(\sqrt{b} x+\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}\right) b^{15/4}}{8 \sqrt{2} a^{11/4} (b c-a d)^4}+\frac{b}{2 a (b c-a d) x^{3/2} \left(b x^2+a\right) \left(d x^2+c\right)^2}+\frac{d^{11/4} \left(285 b^2 c^2-266 a b d c+77 a^2 d^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{32 \sqrt{2} c^{15/4} (b c-a d)^4}-\frac{d^{11/4} \left(285 b^2 c^2-266 a b d c+77 a^2 d^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{32 \sqrt{2} c^{15/4} (b c-a d)^4}+\frac{d^{11/4} \left(285 b^2 c^2-266 a b d c+77 a^2 d^2\right) \log \left(\sqrt{d} x-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}\right)}{64 \sqrt{2} c^{15/4} (b c-a d)^4}-\frac{d^{11/4} \left(285 b^2 c^2-266 a b d c+77 a^2 d^2\right) \log \left(\sqrt{d} x+\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}\right)}{64 \sqrt{2} c^{15/4} (b c-a d)^4}+\frac{d \left(8 b^2 c^2+27 a b d c-11 a^2 d^2\right)}{16 a c^2 (b c-a d)^3 x^{3/2} \left(d x^2+c\right)}+\frac{-56 b^3 c^3+96 a b^2 d c^2-189 a^2 b d^2 c+77 a^3 d^3}{48 a^2 c^3 (b c-a d)^3 x^{3/2}}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^{3/2} \left(d x^2+c\right)^2}",1,"-2/(3*a^2*c^3*x^(3/2)) + (b^4*Sqrt[x])/(2*a^2*(-(b*c) + a*d)^3*(a + b*x^2)) - (d^3*Sqrt[x])/(4*c^2*(b*c - a*d)^2*(c + d*x^2)^2) - (d^3*(31*b*c - 15*a*d)*Sqrt[x])/(16*c^3*(b*c - a*d)^3*(c + d*x^2)) + (b^(15/4)*(-7*b*c + 19*a*d)*ArcTan[(-(Sqrt[2]*a^(1/4)) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(4*Sqrt[2]*a^(11/4)*(b*c - a*d)^4) + (b^(15/4)*(-7*b*c + 19*a*d)*ArcTan[(Sqrt[2]*a^(1/4) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(4*Sqrt[2]*a^(11/4)*(b*c - a*d)^4) - (d^(11/4)*(285*b^2*c^2 - 266*a*b*c*d + 77*a^2*d^2)*ArcTan[(-(Sqrt[2]*c^(1/4)) + 2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(32*Sqrt[2]*c^(15/4)*(-(b*c) + a*d)^4) - (d^(11/4)*(285*b^2*c^2 - 266*a*b*c*d + 77*a^2*d^2)*ArcTan[(Sqrt[2]*c^(1/4) + 2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(32*Sqrt[2]*c^(15/4)*(-(b*c) + a*d)^4) - (b^(15/4)*(-7*b*c + 19*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(11/4)*(b*c - a*d)^4) + (b^(15/4)*(-7*b*c + 19*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(11/4)*(b*c - a*d)^4) + (d^(11/4)*(285*b^2*c^2 - 266*a*b*c*d + 77*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(15/4)*(-(b*c) + a*d)^4) - (d^(11/4)*(285*b^2*c^2 - 266*a*b*c*d + 77*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(15/4)*(-(b*c) + a*d)^4)","A",1
503,1,797,881,6.2682044,"\int \frac{1}{x^{7/2} \left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","Integrate[1/(x^(7/2)*(a + b*x^2)^2*(c + d*x^2)^3),x]","-\frac{x^{3/2} b^5}{2 a^3 (a d-b c)^3 \left(b x^2+a\right)}-\frac{3 (7 a d-3 b c) \tan ^{-1}\left(\frac{2 \sqrt[4]{b} \sqrt{x}-\sqrt{2} \sqrt[4]{a}}{\sqrt{2} \sqrt[4]{a}}\right) b^{17/4}}{4 \sqrt{2} a^{13/4} (b c-a d)^4}-\frac{3 (7 a d-3 b c) \tan ^{-1}\left(\frac{2 \sqrt[4]{b} \sqrt{x}+\sqrt{2} \sqrt[4]{a}}{\sqrt{2} \sqrt[4]{a}}\right) b^{17/4}}{4 \sqrt{2} a^{13/4} (b c-a d)^4}-\frac{3 (7 a d-3 b c) \log \left(\sqrt{b} x-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}\right) b^{17/4}}{8 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{3 (7 a d-3 b c) \log \left(\sqrt{b} x+\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}\right) b^{17/4}}{8 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{3 d^{13/4} \left(119 b^2 c^2-126 a b d c+39 a^2 d^2\right) \tan ^{-1}\left(\frac{2 \sqrt[4]{d} \sqrt{x}-\sqrt{2} \sqrt[4]{c}}{\sqrt{2} \sqrt[4]{c}}\right)}{32 \sqrt{2} c^{17/4} (a d-b c)^4}+\frac{3 d^{13/4} \left(119 b^2 c^2-126 a b d c+39 a^2 d^2\right) \tan ^{-1}\left(\frac{2 \sqrt[4]{d} \sqrt{x}+\sqrt{2} \sqrt[4]{c}}{\sqrt{2} \sqrt[4]{c}}\right)}{32 \sqrt{2} c^{17/4} (a d-b c)^4}+\frac{3 d^{13/4} \left(119 b^2 c^2-126 a b d c+39 a^2 d^2\right) \log \left(\sqrt{d} x-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}\right)}{64 \sqrt{2} c^{17/4} (a d-b c)^4}-\frac{3 d^{13/4} \left(119 b^2 c^2-126 a b d c+39 a^2 d^2\right) \log \left(\sqrt{d} x+\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}\right)}{64 \sqrt{2} c^{17/4} (a d-b c)^4}+\frac{2 (2 b c+3 a d)}{a^3 c^4 \sqrt{x}}+\frac{d^4 (37 b c-21 a d) x^{3/2}}{16 c^4 (b c-a d)^3 \left(d x^2+c\right)}+\frac{d^4 x^{3/2}}{4 c^3 (b c-a d)^2 \left(d x^2+c\right)^2}-\frac{2}{5 a^2 c^3 x^{5/2}}","-\frac{3 (3 b c-7 a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right) b^{17/4}}{4 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{3 (3 b c-7 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right) b^{17/4}}{4 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{3 (3 b c-7 a d) \log \left(\sqrt{b} x-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}\right) b^{17/4}}{8 \sqrt{2} a^{13/4} (b c-a d)^4}-\frac{3 (3 b c-7 a d) \log \left(\sqrt{b} x+\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}\right) b^{17/4}}{8 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{b}{2 a (b c-a d) x^{5/2} \left(b x^2+a\right) \left(d x^2+c\right)^2}-\frac{3 d^{13/4} \left(119 b^2 c^2-126 a b d c+39 a^2 d^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)}{32 \sqrt{2} c^{17/4} (b c-a d)^4}+\frac{3 d^{13/4} \left(119 b^2 c^2-126 a b d c+39 a^2 d^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}+1\right)}{32 \sqrt{2} c^{17/4} (b c-a d)^4}+\frac{3 d^{13/4} \left(119 b^2 c^2-126 a b d c+39 a^2 d^2\right) \log \left(\sqrt{d} x-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}\right)}{64 \sqrt{2} c^{17/4} (b c-a d)^4}-\frac{3 d^{13/4} \left(119 b^2 c^2-126 a b d c+39 a^2 d^2\right) \log \left(\sqrt{d} x+\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}+\sqrt{c}\right)}{64 \sqrt{2} c^{17/4} (b c-a d)^4}+\frac{3 \left(24 b^4 c^4-32 a b^3 d c^3-32 a^2 b^2 d^2 c^2+87 a^3 b d^3 c-39 a^4 d^4\right)}{16 a^3 c^4 (b c-a d)^3 \sqrt{x}}+\frac{d \left(8 b^2 c^2+29 a b d c-13 a^2 d^2\right)}{16 a c^2 (b c-a d)^3 x^{5/2} \left(d x^2+c\right)}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^{5/2} \left(d x^2+c\right)^2}-\frac{3 \left(24 b^3 c^3-32 a b^2 d c^2+87 a^2 b d^2 c-39 a^3 d^3\right)}{80 a^2 c^3 (b c-a d)^3 x^{5/2}}",1,"-2/(5*a^2*c^3*x^(5/2)) + (2*(2*b*c + 3*a*d))/(a^3*c^4*Sqrt[x]) - (b^5*x^(3/2))/(2*a^3*(-(b*c) + a*d)^3*(a + b*x^2)) + (d^4*x^(3/2))/(4*c^3*(b*c - a*d)^2*(c + d*x^2)^2) + (d^4*(37*b*c - 21*a*d)*x^(3/2))/(16*c^4*(b*c - a*d)^3*(c + d*x^2)) - (3*b^(17/4)*(-3*b*c + 7*a*d)*ArcTan[(-(Sqrt[2]*a^(1/4)) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(4*Sqrt[2]*a^(13/4)*(b*c - a*d)^4) - (3*b^(17/4)*(-3*b*c + 7*a*d)*ArcTan[(Sqrt[2]*a^(1/4) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(4*Sqrt[2]*a^(13/4)*(b*c - a*d)^4) + (3*d^(13/4)*(119*b^2*c^2 - 126*a*b*c*d + 39*a^2*d^2)*ArcTan[(-(Sqrt[2]*c^(1/4)) + 2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(32*Sqrt[2]*c^(17/4)*(-(b*c) + a*d)^4) + (3*d^(13/4)*(119*b^2*c^2 - 126*a*b*c*d + 39*a^2*d^2)*ArcTan[(Sqrt[2]*c^(1/4) + 2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(32*Sqrt[2]*c^(17/4)*(-(b*c) + a*d)^4) - (3*b^(17/4)*(-3*b*c + 7*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(13/4)*(b*c - a*d)^4) + (3*b^(17/4)*(-3*b*c + 7*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(13/4)*(b*c - a*d)^4) + (3*d^(13/4)*(119*b^2*c^2 - 126*a*b*c*d + 39*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(17/4)*(-(b*c) + a*d)^4) - (3*d^(13/4)*(119*b^2*c^2 - 126*a*b*c*d + 39*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(17/4)*(-(b*c) + a*d)^4)","A",1
504,1,75,103,0.056379,"\int x^5 \sqrt{a+b x^2} \left(A+B x^2\right) \, dx","Integrate[x^5*Sqrt[a + b*x^2]*(A + B*x^2),x]","\frac{\left(a+b x^2\right)^{3/2} \left(-16 a^3 B+24 a^2 b \left(A+B x^2\right)-6 a b^2 x^2 \left(6 A+5 B x^2\right)+5 b^3 x^4 \left(9 A+7 B x^2\right)\right)}{315 b^4}","\frac{a^2 \left(a+b x^2\right)^{3/2} (A b-a B)}{3 b^4}+\frac{\left(a+b x^2\right)^{7/2} (A b-3 a B)}{7 b^4}-\frac{a \left(a+b x^2\right)^{5/2} (2 A b-3 a B)}{5 b^4}+\frac{B \left(a+b x^2\right)^{9/2}}{9 b^4}",1,"((a + b*x^2)^(3/2)*(-16*a^3*B + 24*a^2*b*(A + B*x^2) - 6*a*b^2*x^2*(6*A + 5*B*x^2) + 5*b^3*x^4*(9*A + 7*B*x^2)))/(315*b^4)","A",1
505,1,130,155,0.2831504,"\int x^4 \sqrt{a+b x^2} \left(A+B x^2\right) \, dx","Integrate[x^4*Sqrt[a + b*x^2]*(A + B*x^2),x]","\frac{\sqrt{a+b x^2} \left(\sqrt{b} x \left(15 a^3 B-2 a^2 b \left(12 A+5 B x^2\right)+8 a b^2 x^2 \left(2 A+B x^2\right)+16 b^3 x^4 \left(4 A+3 B x^2\right)\right)-\frac{3 a^{5/2} (5 a B-8 A b) \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{\frac{b x^2}{a}+1}}\right)}{384 b^{7/2}}","\frac{a^3 (8 A b-5 a B) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{128 b^{7/2}}-\frac{a^2 x \sqrt{a+b x^2} (8 A b-5 a B)}{128 b^3}+\frac{a x^3 \sqrt{a+b x^2} (8 A b-5 a B)}{192 b^2}+\frac{x^5 \sqrt{a+b x^2} (8 A b-5 a B)}{48 b}+\frac{B x^5 \left(a+b x^2\right)^{3/2}}{8 b}",1,"(Sqrt[a + b*x^2]*(Sqrt[b]*x*(15*a^3*B + 8*a*b^2*x^2*(2*A + B*x^2) + 16*b^3*x^4*(4*A + 3*B*x^2) - 2*a^2*b*(12*A + 5*B*x^2)) - (3*a^(5/2)*(-8*A*b + 5*a*B)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[1 + (b*x^2)/a]))/(384*b^(7/2))","A",1
506,1,57,73,0.0387206,"\int x^3 \sqrt{a+b x^2} \left(A+B x^2\right) \, dx","Integrate[x^3*Sqrt[a + b*x^2]*(A + B*x^2),x]","\frac{\left(a+b x^2\right)^{3/2} \left(8 a^2 B-2 a b \left(7 A+6 B x^2\right)+3 b^2 x^2 \left(7 A+5 B x^2\right)\right)}{105 b^3}","\frac{\left(a+b x^2\right)^{5/2} (A b-2 a B)}{5 b^3}-\frac{a \left(a+b x^2\right)^{3/2} (A b-a B)}{3 b^3}+\frac{B \left(a+b x^2\right)^{7/2}}{7 b^3}",1,"((a + b*x^2)^(3/2)*(8*a^2*B + 3*b^2*x^2*(7*A + 5*B*x^2) - 2*a*b*(7*A + 6*B*x^2)))/(105*b^3)","A",1
507,1,108,122,0.2164454,"\int x^2 \sqrt{a+b x^2} \left(A+B x^2\right) \, dx","Integrate[x^2*Sqrt[a + b*x^2]*(A + B*x^2),x]","\frac{\sqrt{a+b x^2} \left(\frac{3 a^{3/2} (a B-2 A b) \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{\frac{b x^2}{a}+1}}+\sqrt{b} x \left(-3 a^2 B+2 a b \left(3 A+B x^2\right)+4 b^2 x^2 \left(3 A+2 B x^2\right)\right)\right)}{48 b^{5/2}}","-\frac{a^2 (2 A b-a B) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{16 b^{5/2}}+\frac{a x \sqrt{a+b x^2} (2 A b-a B)}{16 b^2}+\frac{x^3 \sqrt{a+b x^2} (2 A b-a B)}{8 b}+\frac{B x^3 \left(a+b x^2\right)^{3/2}}{6 b}",1,"(Sqrt[a + b*x^2]*(Sqrt[b]*x*(-3*a^2*B + 2*a*b*(3*A + B*x^2) + 4*b^2*x^2*(3*A + 2*B*x^2)) + (3*a^(3/2)*(-2*A*b + a*B)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[1 + (b*x^2)/a]))/(48*b^(5/2))","A",1
508,1,34,46,0.0215395,"\int x \sqrt{a+b x^2} \left(A+B x^2\right) \, dx","Integrate[x*Sqrt[a + b*x^2]*(A + B*x^2),x]","\frac{\left(a+b x^2\right)^{3/2} \left(-2 a B+5 A b+3 b B x^2\right)}{15 b^2}","\frac{\left(a+b x^2\right)^{3/2} (A b-a B)}{3 b^2}+\frac{B \left(a+b x^2\right)^{5/2}}{5 b^2}",1,"((a + b*x^2)^(3/2)*(5*A*b - 2*a*B + 3*b*B*x^2))/(15*b^2)","A",1
509,1,85,87,0.1569267,"\int \sqrt{a+b x^2} \left(A+B x^2\right) \, dx","Integrate[Sqrt[a + b*x^2]*(A + B*x^2),x]","\frac{\sqrt{a+b x^2} \left(\sqrt{b} x \left(B \left(a+2 b x^2\right)+4 A b\right)-\frac{\sqrt{a} (a B-4 A b) \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{\frac{b x^2}{a}+1}}\right)}{8 b^{3/2}}","\frac{a (4 A b-a B) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 b^{3/2}}+\frac{x \sqrt{a+b x^2} (4 A b-a B)}{8 b}+\frac{B x \left(a+b x^2\right)^{3/2}}{4 b}",1,"(Sqrt[a + b*x^2]*(Sqrt[b]*x*(4*A*b + B*(a + 2*b*x^2)) - (Sqrt[a]*(-4*A*b + a*B)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[1 + (b*x^2)/a]))/(8*b^(3/2))","A",1
510,1,59,59,0.0339085,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{x} \, dx","Integrate[(Sqrt[a + b*x^2]*(A + B*x^2))/x,x]","\frac{\sqrt{a+b x^2} \left(B \left(a+b x^2\right)+3 A b\right)}{3 b}-\sqrt{a} A \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)","A \sqrt{a+b x^2}-\sqrt{a} A \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)+\frac{B \left(a+b x^2\right)^{3/2}}{3 b}",1,"(Sqrt[a + b*x^2]*(3*A*b + B*(a + b*x^2)))/(3*b) - Sqrt[a]*A*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]","A",1
511,1,71,84,0.1554688,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{x^2} \, dx","Integrate[(Sqrt[a + b*x^2]*(A + B*x^2))/x^2,x]","\frac{1}{2} \sqrt{a+b x^2} \left(\frac{(a B+2 A b) \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b} \sqrt{\frac{b x^2}{a}+1}}-\frac{2 A}{x}+B x\right)","\frac{x \sqrt{a+b x^2} (a B+2 A b)}{2 a}+\frac{(a B+2 A b) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 \sqrt{b}}-\frac{A \left(a+b x^2\right)^{3/2}}{a x}",1,"(Sqrt[a + b*x^2]*((-2*A)/x + B*x + ((2*A*b + a*B)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*Sqrt[1 + (b*x^2)/a])))/2","A",1
512,1,63,84,0.0403203,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{x^3} \, dx","Integrate[(Sqrt[a + b*x^2]*(A + B*x^2))/x^3,x]","\frac{1}{2} \left(\frac{\sqrt{a+b x^2} \left(2 B x^2-A\right)}{x^2}-\frac{(2 a B+A b) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{\sqrt{a}}\right)","\frac{\sqrt{a+b x^2} (2 a B+A b)}{2 a}-\frac{(2 a B+A b) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{2 \sqrt{a}}-\frac{A \left(a+b x^2\right)^{3/2}}{2 a x^2}",1,"((Sqrt[a + b*x^2]*(-A + 2*B*x^2))/x^2 - ((A*b + 2*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/Sqrt[a])/2","A",1
513,1,81,66,0.1729495,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{x^4} \, dx","Integrate[(Sqrt[a + b*x^2]*(A + B*x^2))/x^4,x]","\frac{\sqrt{a+b x^2} \left(\frac{3 \sqrt{a} \sqrt{b} B \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{\frac{b x^2}{a}+1}}-\frac{a A+3 a B x^2+A b x^2}{x^3}\right)}{3 a}","-\frac{A \left(a+b x^2\right)^{3/2}}{3 a x^3}-\frac{B \sqrt{a+b x^2}}{x}+\sqrt{b} B \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)",1,"(Sqrt[a + b*x^2]*(-((a*A + A*b*x^2 + 3*a*B*x^2)/x^3) + (3*Sqrt[a]*Sqrt[b]*B*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[1 + (b*x^2)/a]))/(3*a)","A",1
514,1,93,88,0.0699877,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{x^5} \, dx","Integrate[(Sqrt[a + b*x^2]*(A + B*x^2))/x^5,x]","\frac{-\left(a+b x^2\right) \left(2 a \left(A+2 B x^2\right)+A b x^2\right)-b x^4 \sqrt{\frac{b x^2}{a}+1} (4 a B-A b) \tanh ^{-1}\left(\sqrt{\frac{b x^2}{a}+1}\right)}{8 a x^4 \sqrt{a+b x^2}}","\frac{b (A b-4 a B) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{8 a^{3/2}}+\frac{\sqrt{a+b x^2} (A b-4 a B)}{8 a x^2}-\frac{A \left(a+b x^2\right)^{3/2}}{4 a x^4}",1,"(-((a + b*x^2)*(A*b*x^2 + 2*a*(A + 2*B*x^2))) - b*(-(A*b) + 4*a*B)*x^4*Sqrt[1 + (b*x^2)/a]*ArcTanh[Sqrt[1 + (b*x^2)/a]])/(8*a*x^4*Sqrt[a + b*x^2])","A",1
515,1,40,53,0.0135341,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{x^6} \, dx","Integrate[(Sqrt[a + b*x^2]*(A + B*x^2))/x^6,x]","-\frac{\left(a+b x^2\right)^{3/2} \left(3 a A+5 a B x^2-2 A b x^2\right)}{15 a^2 x^5}","\frac{\left(a+b x^2\right)^{3/2} (2 A b-5 a B)}{15 a^2 x^3}-\frac{A \left(a+b x^2\right)^{3/2}}{5 a x^5}",1,"-1/15*((a + b*x^2)^(3/2)*(3*a*A - 2*A*b*x^2 + 5*a*B*x^2))/(a^2*x^5)","A",1
516,1,61,120,0.0222338,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{x^7} \, dx","Integrate[(Sqrt[a + b*x^2]*(A + B*x^2))/x^7,x]","-\frac{\left(a+b x^2\right)^{3/2} \left(a^3 A+b^2 x^6 (2 a B-A b) \, _2F_1\left(\frac{3}{2},3;\frac{5}{2};\frac{b x^2}{a}+1\right)\right)}{6 a^4 x^6}","-\frac{b^2 (A b-2 a B) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{16 a^{5/2}}+\frac{b \sqrt{a+b x^2} (A b-2 a B)}{16 a^2 x^2}+\frac{\sqrt{a+b x^2} (A b-2 a B)}{8 a x^4}-\frac{A \left(a+b x^2\right)^{3/2}}{6 a x^6}",1,"-1/6*((a + b*x^2)^(3/2)*(a^3*A + b^2*(-(A*b) + 2*a*B)*x^6*Hypergeometric2F1[3/2, 3, 5/2, 1 + (b*x^2)/a]))/(a^4*x^6)","C",1
517,1,63,84,0.0288253,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{x^8} \, dx","Integrate[(Sqrt[a + b*x^2]*(A + B*x^2))/x^8,x]","\frac{\left(a+b x^2\right)^{3/2} \left(-3 a^2 \left(5 A+7 B x^2\right)+2 a b x^2 \left(6 A+7 B x^2\right)-8 A b^2 x^4\right)}{105 a^3 x^7}","-\frac{2 b \left(a+b x^2\right)^{3/2} (4 A b-7 a B)}{105 a^3 x^3}+\frac{\left(a+b x^2\right)^{3/2} (4 A b-7 a B)}{35 a^2 x^5}-\frac{A \left(a+b x^2\right)^{3/2}}{7 a x^7}",1,"((a + b*x^2)^(3/2)*(-8*A*b^2*x^4 - 3*a^2*(5*A + 7*B*x^2) + 2*a*b*x^2*(6*A + 7*B*x^2)))/(105*a^3*x^7)","A",1
518,1,62,156,0.0223749,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{x^9} \, dx","Integrate[(Sqrt[a + b*x^2]*(A + B*x^2))/x^9,x]","-\frac{\left(a+b x^2\right)^{3/2} \left(3 a^4 A+b^3 x^8 (5 A b-8 a B) \, _2F_1\left(\frac{3}{2},4;\frac{5}{2};\frac{b x^2}{a}+1\right)\right)}{24 a^5 x^8}","\frac{b^3 (5 A b-8 a B) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{128 a^{7/2}}-\frac{b^2 \sqrt{a+b x^2} (5 A b-8 a B)}{128 a^3 x^2}+\frac{b \sqrt{a+b x^2} (5 A b-8 a B)}{192 a^2 x^4}+\frac{\sqrt{a+b x^2} (5 A b-8 a B)}{48 a x^6}-\frac{A \left(a+b x^2\right)^{3/2}}{8 a x^8}",1,"-1/24*((a + b*x^2)^(3/2)*(3*a^4*A + b^3*(5*A*b - 8*a*B)*x^8*Hypergeometric2F1[3/2, 4, 5/2, 1 + (b*x^2)/a]))/(a^5*x^8)","C",1
519,1,81,117,0.0362222,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{x^{10}} \, dx","Integrate[(Sqrt[a + b*x^2]*(A + B*x^2))/x^10,x]","\frac{\left(a+b x^2\right)^{3/2} \left(-5 a^3 \left(7 A+9 B x^2\right)+6 a^2 b x^2 \left(5 A+6 B x^2\right)-24 a b^2 x^4 \left(A+B x^2\right)+16 A b^3 x^6\right)}{315 a^4 x^9}","\frac{8 b^2 \left(a+b x^2\right)^{3/2} (2 A b-3 a B)}{315 a^4 x^3}-\frac{4 b \left(a+b x^2\right)^{3/2} (2 A b-3 a B)}{105 a^3 x^5}+\frac{\left(a+b x^2\right)^{3/2} (2 A b-3 a B)}{21 a^2 x^7}-\frac{A \left(a+b x^2\right)^{3/2}}{9 a x^9}",1,"((a + b*x^2)^(3/2)*(16*A*b^3*x^6 - 24*a*b^2*x^4*(A + B*x^2) + 6*a^2*b*x^2*(5*A + 6*B*x^2) - 5*a^3*(7*A + 9*B*x^2)))/(315*a^4*x^9)","A",1
520,1,62,189,0.0289844,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{x^{11}} \, dx","Integrate[(Sqrt[a + b*x^2]*(A + B*x^2))/x^11,x]","-\frac{\left(a+b x^2\right)^{3/2} \left(3 a^5 A+b^4 x^{10} (10 a B-7 A b) \, _2F_1\left(\frac{3}{2},5;\frac{5}{2};\frac{b x^2}{a}+1\right)\right)}{30 a^6 x^{10}}","-\frac{b^4 (7 A b-10 a B) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{256 a^{9/2}}+\frac{b^3 \sqrt{a+b x^2} (7 A b-10 a B)}{256 a^4 x^2}-\frac{b^2 \sqrt{a+b x^2} (7 A b-10 a B)}{384 a^3 x^4}+\frac{b \sqrt{a+b x^2} (7 A b-10 a B)}{480 a^2 x^6}+\frac{\sqrt{a+b x^2} (7 A b-10 a B)}{80 a x^8}-\frac{A \left(a+b x^2\right)^{3/2}}{10 a x^{10}}",1,"-1/30*((a + b*x^2)^(3/2)*(3*a^5*A + b^4*(-7*A*b + 10*a*B)*x^10*Hypergeometric2F1[3/2, 5, 5/2, 1 + (b*x^2)/a]))/(a^6*x^10)","C",1
521,1,78,103,0.0575593,"\int x^5 \left(a+b x^2\right)^{3/2} \left(A+B x^2\right) \, dx","Integrate[x^5*(a + b*x^2)^(3/2)*(A + B*x^2),x]","\frac{\left(a+b x^2\right)^{5/2} \left(-48 a^3 B+8 a^2 b \left(11 A+15 B x^2\right)-10 a b^2 x^2 \left(22 A+21 B x^2\right)+35 b^3 x^4 \left(11 A+9 B x^2\right)\right)}{3465 b^4}","\frac{a^2 \left(a+b x^2\right)^{5/2} (A b-a B)}{5 b^4}+\frac{\left(a+b x^2\right)^{9/2} (A b-3 a B)}{9 b^4}-\frac{a \left(a+b x^2\right)^{7/2} (2 A b-3 a B)}{7 b^4}+\frac{B \left(a+b x^2\right)^{11/2}}{11 b^4}",1,"((a + b*x^2)^(5/2)*(-48*a^3*B + 35*b^3*x^4*(11*A + 9*B*x^2) + 8*a^2*b*(11*A + 15*B*x^2) - 10*a*b^2*x^2*(22*A + 21*B*x^2)))/(3465*b^4)","A",1
522,1,150,188,0.3130159,"\int x^4 \left(a+b x^2\right)^{3/2} \left(A+B x^2\right) \, dx","Integrate[x^4*(a + b*x^2)^(3/2)*(A + B*x^2),x]","\frac{\sqrt{a+b x^2} \left(\sqrt{b} x \left(15 a^4 B-10 a^3 b \left(3 A+B x^2\right)+4 a^2 b^2 x^2 \left(5 A+2 B x^2\right)+16 a b^3 x^4 \left(15 A+11 B x^2\right)+32 b^4 x^6 \left(5 A+4 B x^2\right)\right)-\frac{15 a^{7/2} (a B-2 A b) \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{\frac{b x^2}{a}+1}}\right)}{1280 b^{7/2}}","\frac{3 a^4 (2 A b-a B) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{256 b^{7/2}}-\frac{3 a^3 x \sqrt{a+b x^2} (2 A b-a B)}{256 b^3}+\frac{a^2 x^3 \sqrt{a+b x^2} (2 A b-a B)}{128 b^2}+\frac{x^5 \left(a+b x^2\right)^{3/2} (2 A b-a B)}{16 b}+\frac{a x^5 \sqrt{a+b x^2} (2 A b-a B)}{32 b}+\frac{B x^5 \left(a+b x^2\right)^{5/2}}{10 b}",1,"(Sqrt[a + b*x^2]*(Sqrt[b]*x*(15*a^4*B - 10*a^3*b*(3*A + B*x^2) + 4*a^2*b^2*x^2*(5*A + 2*B*x^2) + 32*b^4*x^6*(5*A + 4*B*x^2) + 16*a*b^3*x^4*(15*A + 11*B*x^2)) - (15*a^(7/2)*(-2*A*b + a*B)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[1 + (b*x^2)/a]))/(1280*b^(7/2))","A",1
523,1,57,73,0.038136,"\int x^3 \left(a+b x^2\right)^{3/2} \left(A+B x^2\right) \, dx","Integrate[x^3*(a + b*x^2)^(3/2)*(A + B*x^2),x]","\frac{\left(a+b x^2\right)^{5/2} \left(8 a^2 B-2 a b \left(9 A+10 B x^2\right)+5 b^2 x^2 \left(9 A+7 B x^2\right)\right)}{315 b^3}","\frac{\left(a+b x^2\right)^{7/2} (A b-2 a B)}{7 b^3}-\frac{a \left(a+b x^2\right)^{5/2} (A b-a B)}{5 b^3}+\frac{B \left(a+b x^2\right)^{9/2}}{9 b^3}",1,"((a + b*x^2)^(5/2)*(8*a^2*B + 5*b^2*x^2*(9*A + 7*B*x^2) - 2*a*b*(9*A + 10*B*x^2)))/(315*b^3)","A",1
524,1,130,155,0.2563061,"\int x^2 \left(a+b x^2\right)^{3/2} \left(A+B x^2\right) \, dx","Integrate[x^2*(a + b*x^2)^(3/2)*(A + B*x^2),x]","\frac{\sqrt{a+b x^2} \left(\frac{3 a^{5/2} (3 a B-8 A b) \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{\frac{b x^2}{a}+1}}+\sqrt{b} x \left(-9 a^3 B+6 a^2 b \left(4 A+B x^2\right)+8 a b^2 x^2 \left(14 A+9 B x^2\right)+16 b^3 x^4 \left(4 A+3 B x^2\right)\right)\right)}{384 b^{5/2}}","-\frac{a^3 (8 A b-3 a B) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{128 b^{5/2}}+\frac{a^2 x \sqrt{a+b x^2} (8 A b-3 a B)}{128 b^2}+\frac{a x^3 \sqrt{a+b x^2} (8 A b-3 a B)}{64 b}+\frac{x^3 \left(a+b x^2\right)^{3/2} (8 A b-3 a B)}{48 b}+\frac{B x^3 \left(a+b x^2\right)^{5/2}}{8 b}",1,"(Sqrt[a + b*x^2]*(Sqrt[b]*x*(-9*a^3*B + 6*a^2*b*(4*A + B*x^2) + 16*b^3*x^4*(4*A + 3*B*x^2) + 8*a*b^2*x^2*(14*A + 9*B*x^2)) + (3*a^(5/2)*(-8*A*b + 3*a*B)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[1 + (b*x^2)/a]))/(384*b^(5/2))","A",1
525,1,34,46,0.0231079,"\int x \left(a+b x^2\right)^{3/2} \left(A+B x^2\right) \, dx","Integrate[x*(a + b*x^2)^(3/2)*(A + B*x^2),x]","\frac{\left(a+b x^2\right)^{5/2} \left(-2 a B+7 A b+5 b B x^2\right)}{35 b^2}","\frac{\left(a+b x^2\right)^{5/2} (A b-a B)}{5 b^2}+\frac{B \left(a+b x^2\right)^{7/2}}{7 b^2}",1,"((a + b*x^2)^(5/2)*(7*A*b - 2*a*B + 5*b*B*x^2))/(35*b^2)","A",1
526,1,109,118,0.2058829,"\int \left(a+b x^2\right)^{3/2} \left(A+B x^2\right) \, dx","Integrate[(a + b*x^2)^(3/2)*(A + B*x^2),x]","\frac{\sqrt{a+b x^2} \left(\sqrt{b} x \left(3 a^2 B+2 a b \left(15 A+7 B x^2\right)+4 b^2 x^2 \left(3 A+2 B x^2\right)\right)-\frac{3 a^{3/2} (a B-6 A b) \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{\frac{b x^2}{a}+1}}\right)}{48 b^{3/2}}","\frac{a^2 (6 A b-a B) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{16 b^{3/2}}+\frac{x \left(a+b x^2\right)^{3/2} (6 A b-a B)}{24 b}+\frac{a x \sqrt{a+b x^2} (6 A b-a B)}{16 b}+\frac{B x \left(a+b x^2\right)^{5/2}}{6 b}",1,"(Sqrt[a + b*x^2]*(Sqrt[b]*x*(3*a^2*B + 4*b^2*x^2*(3*A + 2*B*x^2) + 2*a*b*(15*A + 7*B*x^2)) - (3*a^(3/2)*(-6*A*b + a*B)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[1 + (b*x^2)/a]))/(48*b^(3/2))","A",1
527,1,76,76,0.0647556,"\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{x} \, dx","Integrate[((a + b*x^2)^(3/2)*(A + B*x^2))/x,x]","\frac{1}{3} A \left(a+b x^2\right)^{3/2}+a A \left(\sqrt{a+b x^2}-\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)\right)+\frac{B \left(a+b x^2\right)^{5/2}}{5 b}","-a^{3/2} A \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)+\frac{1}{3} A \left(a+b x^2\right)^{3/2}+a A \sqrt{a+b x^2}+\frac{B \left(a+b x^2\right)^{5/2}}{5 b}",1,"(A*(a + b*x^2)^(3/2))/3 + (B*(a + b*x^2)^(5/2))/(5*b) + a*A*(Sqrt[a + b*x^2] - Sqrt[a]*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])","A",1
528,1,87,109,0.1912738,"\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{x^2} \, dx","Integrate[((a + b*x^2)^(3/2)*(A + B*x^2))/x^2,x]","\frac{1}{8} \sqrt{a+b x^2} \left(\frac{3 \sqrt{a} (a B+4 A b) \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{\frac{b x^2}{a}+1}}-\frac{8 a A}{x}+5 a B x+4 A b x+2 b B x^3\right)","\frac{x \left(a+b x^2\right)^{3/2} (a B+4 A b)}{4 a}+\frac{3}{8} x \sqrt{a+b x^2} (a B+4 A b)+\frac{3 a (a B+4 A b) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 \sqrt{b}}-\frac{A \left(a+b x^2\right)^{5/2}}{a x}",1,"(Sqrt[a + b*x^2]*((-8*a*A)/x + 4*A*b*x + 5*a*B*x + 2*b*B*x^3 + (3*Sqrt[a]*(4*A*b + a*B)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[b]*Sqrt[1 + (b*x^2)/a])))/8","A",1
529,1,80,110,0.0541925,"\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{x^3} \, dx","Integrate[((a + b*x^2)^(3/2)*(A + B*x^2))/x^3,x]","\frac{1}{6} \left(\frac{\sqrt{a+b x^2} \left(-3 a A+8 a B x^2+6 A b x^2+2 b B x^4\right)}{x^2}-3 \sqrt{a} (2 a B+3 A b) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)\right)","\frac{\left(a+b x^2\right)^{3/2} (2 a B+3 A b)}{6 a}+\frac{1}{2} \sqrt{a+b x^2} (2 a B+3 A b)-\frac{1}{2} \sqrt{a} (2 a B+3 A b) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)-\frac{A \left(a+b x^2\right)^{5/2}}{2 a x^2}",1,"((Sqrt[a + b*x^2]*(-3*a*A + 6*A*b*x^2 + 8*a*B*x^2 + 2*b*B*x^4))/x^2 - 3*Sqrt[a]*(3*A*b + 2*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/6","A",1
530,1,83,119,0.0825927,"\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{x^4} \, dx","Integrate[((a + b*x^2)^(3/2)*(A + B*x^2))/x^4,x]","\frac{\sqrt{a+b x^2} (-3 a B-2 A b) \, _2F_1\left(-\frac{3}{2},-\frac{1}{2};\frac{1}{2};-\frac{b x^2}{a}\right)}{3 x \sqrt{\frac{b x^2}{a}+1}}-\frac{A \left(a+b x^2\right)^{5/2}}{3 a x^3}","-\frac{\left(a+b x^2\right)^{3/2} (3 a B+2 A b)}{3 a x}+\frac{b x \sqrt{a+b x^2} (3 a B+2 A b)}{2 a}+\frac{1}{2} \sqrt{b} (3 a B+2 A b) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)-\frac{A \left(a+b x^2\right)^{5/2}}{3 a x^3}",1,"-1/3*(A*(a + b*x^2)^(5/2))/(a*x^3) + ((-2*A*b - 3*a*B)*Sqrt[a + b*x^2]*Hypergeometric2F1[-3/2, -1/2, 1/2, -((b*x^2)/a)])/(3*x*Sqrt[1 + (b*x^2)/a])","C",1
531,1,59,115,0.0290741,"\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{x^5} \, dx","Integrate[((a + b*x^2)^(3/2)*(A + B*x^2))/x^5,x]","\frac{\left(a+b x^2\right)^{5/2} \left(b x^4 (4 a B+A b) \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};\frac{b x^2}{a}+1\right)-5 a^2 A\right)}{20 a^3 x^4}","-\frac{\left(a+b x^2\right)^{3/2} (4 a B+A b)}{8 a x^2}+\frac{3 b \sqrt{a+b x^2} (4 a B+A b)}{8 a}-\frac{3 b (4 a B+A b) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{8 \sqrt{a}}-\frac{A \left(a+b x^2\right)^{5/2}}{4 a x^4}",1,"((a + b*x^2)^(5/2)*(-5*a^2*A + b*(A*b + 4*a*B)*x^4*Hypergeometric2F1[2, 5/2, 7/2, 1 + (b*x^2)/a]))/(20*a^3*x^4)","C",1
532,1,76,86,0.0367268,"\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{x^6} \, dx","Integrate[((a + b*x^2)^(3/2)*(A + B*x^2))/x^6,x]","-\frac{A \left(a+b x^2\right)^{5/2}}{5 a x^5}-\frac{a B \sqrt{a+b x^2} \, _2F_1\left(-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};-\frac{b x^2}{a}\right)}{3 x^3 \sqrt{\frac{b x^2}{a}+1}}","-\frac{A \left(a+b x^2\right)^{5/2}}{5 a x^5}+b^{3/2} B \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)-\frac{b B \sqrt{a+b x^2}}{x}-\frac{B \left(a+b x^2\right)^{3/2}}{3 x^3}",1,"-1/5*(A*(a + b*x^2)^(5/2))/(a*x^5) - (a*B*Sqrt[a + b*x^2]*Hypergeometric2F1[-3/2, -3/2, -1/2, -((b*x^2)/a)])/(3*x^3*Sqrt[1 + (b*x^2)/a])","C",1
533,1,119,120,0.0819629,"\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{x^7} \, dx","Integrate[((a + b*x^2)^(3/2)*(A + B*x^2))/x^7,x]","\frac{-\left(a+b x^2\right) \left(4 a^2 \left(2 A+3 B x^2\right)+2 a b x^2 \left(7 A+15 B x^2\right)+3 A b^2 x^4\right)-3 b^2 x^6 \sqrt{\frac{b x^2}{a}+1} (6 a B-A b) \tanh ^{-1}\left(\sqrt{\frac{b x^2}{a}+1}\right)}{48 a x^6 \sqrt{a+b x^2}}","\frac{b^2 (A b-6 a B) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{16 a^{3/2}}+\frac{b \sqrt{a+b x^2} (A b-6 a B)}{16 a x^2}+\frac{\left(a+b x^2\right)^{3/2} (A b-6 a B)}{24 a x^4}-\frac{A \left(a+b x^2\right)^{5/2}}{6 a x^6}",1,"(-((a + b*x^2)*(3*A*b^2*x^4 + 4*a^2*(2*A + 3*B*x^2) + 2*a*b*x^2*(7*A + 15*B*x^2))) - 3*b^2*(-(A*b) + 6*a*B)*x^6*Sqrt[1 + (b*x^2)/a]*ArcTanh[Sqrt[1 + (b*x^2)/a]])/(48*a*x^6*Sqrt[a + b*x^2])","A",1
534,1,40,53,0.0162036,"\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{x^8} \, dx","Integrate[((a + b*x^2)^(3/2)*(A + B*x^2))/x^8,x]","-\frac{\left(a+b x^2\right)^{5/2} \left(5 a A+7 a B x^2-2 A b x^2\right)}{35 a^2 x^7}","\frac{\left(a+b x^2\right)^{5/2} (2 A b-7 a B)}{35 a^2 x^5}-\frac{A \left(a+b x^2\right)^{5/2}}{7 a x^7}",1,"-1/35*((a + b*x^2)^(5/2)*(5*a*A - 2*A*b*x^2 + 7*a*B*x^2))/(a^2*x^7)","A",1
535,1,62,156,0.0280116,"\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{x^9} \, dx","Integrate[((a + b*x^2)^(3/2)*(A + B*x^2))/x^9,x]","-\frac{\left(a+b x^2\right)^{5/2} \left(5 a^4 A+b^3 x^8 (3 A b-8 a B) \, _2F_1\left(\frac{5}{2},4;\frac{7}{2};\frac{b x^2}{a}+1\right)\right)}{40 a^5 x^8}","-\frac{b^3 (3 A b-8 a B) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{128 a^{5/2}}+\frac{b^2 \sqrt{a+b x^2} (3 A b-8 a B)}{128 a^2 x^2}+\frac{\left(a+b x^2\right)^{3/2} (3 A b-8 a B)}{48 a x^6}+\frac{b \sqrt{a+b x^2} (3 A b-8 a B)}{64 a x^4}-\frac{A \left(a+b x^2\right)^{5/2}}{8 a x^8}",1,"-1/40*((a + b*x^2)^(5/2)*(5*a^4*A + b^3*(3*A*b - 8*a*B)*x^8*Hypergeometric2F1[5/2, 4, 7/2, 1 + (b*x^2)/a]))/(a^5*x^8)","C",1
536,1,63,84,0.0441611,"\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{x^{10}} \, dx","Integrate[((a + b*x^2)^(3/2)*(A + B*x^2))/x^10,x]","\frac{\left(a+b x^2\right)^{5/2} \left(-5 a^2 \left(7 A+9 B x^2\right)+2 a b x^2 \left(10 A+9 B x^2\right)-8 A b^2 x^4\right)}{315 a^3 x^9}","-\frac{2 b \left(a+b x^2\right)^{5/2} (4 A b-9 a B)}{315 a^3 x^5}+\frac{\left(a+b x^2\right)^{5/2} (4 A b-9 a B)}{63 a^2 x^7}-\frac{A \left(a+b x^2\right)^{5/2}}{9 a x^9}",1,"((a + b*x^2)^(5/2)*(-8*A*b^2*x^4 - 5*a^2*(7*A + 9*B*x^2) + 2*a*b*x^2*(10*A + 9*B*x^2)))/(315*a^3*x^9)","A",1
537,1,61,184,0.042974,"\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{x^{11}} \, dx","Integrate[((a + b*x^2)^(3/2)*(A + B*x^2))/x^11,x]","-\frac{\left(a+b x^2\right)^{5/2} \left(a^5 A+b^4 x^{10} (2 a B-A b) \, _2F_1\left(\frac{5}{2},5;\frac{7}{2};\frac{b x^2}{a}+1\right)\right)}{10 a^6 x^{10}}","\frac{3 b^4 (A b-2 a B) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{256 a^{7/2}}-\frac{3 b^3 \sqrt{a+b x^2} (A b-2 a B)}{256 a^3 x^2}+\frac{b^2 \sqrt{a+b x^2} (A b-2 a B)}{128 a^2 x^4}+\frac{\left(a+b x^2\right)^{3/2} (A b-2 a B)}{16 a x^8}+\frac{b \sqrt{a+b x^2} (A b-2 a B)}{32 a x^6}-\frac{A \left(a+b x^2\right)^{5/2}}{10 a x^{10}}",1,"-1/10*((a + b*x^2)^(5/2)*(a^5*A + b^4*(-(A*b) + 2*a*B)*x^10*Hypergeometric2F1[5/2, 5, 7/2, 1 + (b*x^2)/a]))/(a^6*x^10)","C",1
538,1,78,103,0.0692211,"\int x^5 \left(a+b x^2\right)^{5/2} \left(A+B x^2\right) \, dx","Integrate[x^5*(a + b*x^2)^(5/2)*(A + B*x^2),x]","\frac{\left(a+b x^2\right)^{7/2} \left(-48 a^3 B+8 a^2 b \left(13 A+21 B x^2\right)-14 a b^2 x^2 \left(26 A+27 B x^2\right)+63 b^3 x^4 \left(13 A+11 B x^2\right)\right)}{9009 b^4}","\frac{a^2 \left(a+b x^2\right)^{7/2} (A b-a B)}{7 b^4}+\frac{\left(a+b x^2\right)^{11/2} (A b-3 a B)}{11 b^4}-\frac{a \left(a+b x^2\right)^{9/2} (2 A b-3 a B)}{9 b^4}+\frac{B \left(a+b x^2\right)^{13/2}}{13 b^4}",1,"((a + b*x^2)^(7/2)*(-48*a^3*B + 63*b^3*x^4*(13*A + 11*B*x^2) + 8*a^2*b*(13*A + 21*B*x^2) - 14*a*b^2*x^2*(26*A + 27*B*x^2)))/(9009*b^4)","A",1
539,1,172,221,0.4513897,"\int x^4 \left(a+b x^2\right)^{5/2} \left(A+B x^2\right) \, dx","Integrate[x^4*(a + b*x^2)^(5/2)*(A + B*x^2),x]","\frac{\sqrt{a+b x^2} \left(\sqrt{b} x \left(75 a^5 B-10 a^4 b \left(18 A+5 B x^2\right)+40 a^3 b^2 x^2 \left(3 A+B x^2\right)+48 a^2 b^3 x^4 \left(62 A+45 B x^2\right)+64 a b^4 x^6 \left(63 A+50 B x^2\right)+256 b^5 x^8 \left(6 A+5 B x^2\right)\right)-\frac{15 a^{9/2} (5 a B-12 A b) \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{\frac{b x^2}{a}+1}}\right)}{15360 b^{7/2}}","\frac{a^5 (12 A b-5 a B) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{1024 b^{7/2}}-\frac{a^4 x \sqrt{a+b x^2} (12 A b-5 a B)}{1024 b^3}+\frac{a^3 x^3 \sqrt{a+b x^2} (12 A b-5 a B)}{1536 b^2}+\frac{a^2 x^5 \sqrt{a+b x^2} (12 A b-5 a B)}{384 b}+\frac{a x^5 \left(a+b x^2\right)^{3/2} (12 A b-5 a B)}{192 b}+\frac{x^5 \left(a+b x^2\right)^{5/2} (12 A b-5 a B)}{120 b}+\frac{B x^5 \left(a+b x^2\right)^{7/2}}{12 b}",1,"(Sqrt[a + b*x^2]*(Sqrt[b]*x*(75*a^5*B + 40*a^3*b^2*x^2*(3*A + B*x^2) + 256*b^5*x^8*(6*A + 5*B*x^2) - 10*a^4*b*(18*A + 5*B*x^2) + 48*a^2*b^3*x^4*(62*A + 45*B*x^2) + 64*a*b^4*x^6*(63*A + 50*B*x^2)) - (15*a^(9/2)*(-12*A*b + 5*a*B)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[1 + (b*x^2)/a]))/(15360*b^(7/2))","A",1
540,1,57,73,0.0436762,"\int x^3 \left(a+b x^2\right)^{5/2} \left(A+B x^2\right) \, dx","Integrate[x^3*(a + b*x^2)^(5/2)*(A + B*x^2),x]","\frac{\left(a+b x^2\right)^{7/2} \left(8 a^2 B-2 a b \left(11 A+14 B x^2\right)+7 b^2 x^2 \left(11 A+9 B x^2\right)\right)}{693 b^3}","\frac{\left(a+b x^2\right)^{9/2} (A b-2 a B)}{9 b^3}-\frac{a \left(a+b x^2\right)^{7/2} (A b-a B)}{7 b^3}+\frac{B \left(a+b x^2\right)^{11/2}}{11 b^3}",1,"((a + b*x^2)^(7/2)*(8*a^2*B + 7*b^2*x^2*(11*A + 9*B*x^2) - 2*a*b*(11*A + 14*B*x^2)))/(693*b^3)","A",1
541,1,151,188,0.3109362,"\int x^2 \left(a+b x^2\right)^{5/2} \left(A+B x^2\right) \, dx","Integrate[x^2*(a + b*x^2)^(5/2)*(A + B*x^2),x]","\frac{\sqrt{a+b x^2} \left(\frac{15 a^{7/2} (3 a B-10 A b) \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{\frac{b x^2}{a}+1}}+\sqrt{b} x \left(-45 a^4 B+30 a^3 b \left(5 A+B x^2\right)+4 a^2 b^2 x^2 \left(295 A+186 B x^2\right)+16 a b^3 x^4 \left(85 A+63 B x^2\right)+96 b^4 x^6 \left(5 A+4 B x^2\right)\right)\right)}{3840 b^{5/2}}","-\frac{a^4 (10 A b-3 a B) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{256 b^{5/2}}+\frac{a^3 x \sqrt{a+b x^2} (10 A b-3 a B)}{256 b^2}+\frac{a^2 x^3 \sqrt{a+b x^2} (10 A b-3 a B)}{128 b}+\frac{a x^3 \left(a+b x^2\right)^{3/2} (10 A b-3 a B)}{96 b}+\frac{x^3 \left(a+b x^2\right)^{5/2} (10 A b-3 a B)}{80 b}+\frac{B x^3 \left(a+b x^2\right)^{7/2}}{10 b}",1,"(Sqrt[a + b*x^2]*(Sqrt[b]*x*(-45*a^4*B + 30*a^3*b*(5*A + B*x^2) + 96*b^4*x^6*(5*A + 4*B*x^2) + 16*a*b^3*x^4*(85*A + 63*B*x^2) + 4*a^2*b^2*x^2*(295*A + 186*B*x^2)) + (15*a^(7/2)*(-10*A*b + 3*a*B)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[1 + (b*x^2)/a]))/(3840*b^(5/2))","A",1
542,1,34,46,0.0272732,"\int x \left(a+b x^2\right)^{5/2} \left(A+B x^2\right) \, dx","Integrate[x*(a + b*x^2)^(5/2)*(A + B*x^2),x]","\frac{\left(a+b x^2\right)^{7/2} \left(-2 a B+9 A b+7 b B x^2\right)}{63 b^2}","\frac{\left(a+b x^2\right)^{7/2} (A b-a B)}{7 b^2}+\frac{B \left(a+b x^2\right)^{9/2}}{9 b^2}",1,"((a + b*x^2)^(7/2)*(9*A*b - 2*a*B + 7*b*B*x^2))/(63*b^2)","A",1
543,1,130,149,0.2251604,"\int \left(a+b x^2\right)^{5/2} \left(A+B x^2\right) \, dx","Integrate[(a + b*x^2)^(5/2)*(A + B*x^2),x]","\frac{\sqrt{a+b x^2} \left(\sqrt{b} x \left(15 a^3 B+2 a^2 b \left(132 A+59 B x^2\right)+8 a b^2 x^2 \left(26 A+17 B x^2\right)+16 b^3 x^4 \left(4 A+3 B x^2\right)\right)-\frac{15 a^{5/2} (a B-8 A b) \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{\frac{b x^2}{a}+1}}\right)}{384 b^{3/2}}","\frac{5 a^3 (8 A b-a B) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{128 b^{3/2}}+\frac{5 a^2 x \sqrt{a+b x^2} (8 A b-a B)}{128 b}+\frac{x \left(a+b x^2\right)^{5/2} (8 A b-a B)}{48 b}+\frac{5 a x \left(a+b x^2\right)^{3/2} (8 A b-a B)}{192 b}+\frac{B x \left(a+b x^2\right)^{7/2}}{8 b}",1,"(Sqrt[a + b*x^2]*(Sqrt[b]*x*(15*a^3*B + 16*b^3*x^4*(4*A + 3*B*x^2) + 8*a*b^2*x^2*(26*A + 17*B*x^2) + 2*a^2*b*(132*A + 59*B*x^2)) - (15*a^(5/2)*(-8*A*b + a*B)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[1 + (b*x^2)/a]))/(384*b^(3/2))","A",1
544,1,88,95,0.098961,"\int \frac{\left(a+b x^2\right)^{5/2} \left(A+B x^2\right)}{x} \, dx","Integrate[((a + b*x^2)^(5/2)*(A + B*x^2))/x,x]","-a^{5/2} A \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)+\frac{1}{5} A \left(a+b x^2\right)^{5/2}+\frac{1}{3} a A \left(4 a+b x^2\right) \sqrt{a+b x^2}+\frac{B \left(a+b x^2\right)^{7/2}}{7 b}","-a^{5/2} A \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)+a^2 A \sqrt{a+b x^2}+\frac{1}{5} A \left(a+b x^2\right)^{5/2}+\frac{1}{3} a A \left(a+b x^2\right)^{3/2}+\frac{B \left(a+b x^2\right)^{7/2}}{7 b}",1,"(A*(a + b*x^2)^(5/2))/5 + (B*(a + b*x^2)^(7/2))/(7*b) + (a*A*Sqrt[a + b*x^2]*(4*a + b*x^2))/3 - a^(5/2)*A*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]","A",1
545,1,125,136,0.3583119,"\int \frac{\left(a+b x^2\right)^{5/2} \left(A+B x^2\right)}{x^2} \, dx","Integrate[((a + b*x^2)^(5/2)*(A + B*x^2))/x^2,x]","\frac{\sqrt{a+b x^2} \left(\frac{(a B+6 A b) \left(15 a^{5/2} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)+\sqrt{b} x \sqrt{\frac{b x^2}{a}+1} \left(33 a^2+26 a b x^2+8 b^2 x^4\right)\right)}{\sqrt{b} \sqrt{\frac{b x^2}{a}+1}}-\frac{48 A \left(a+b x^2\right)^3}{x}\right)}{48 a}","\frac{5 a^2 (a B+6 A b) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{16 \sqrt{b}}+\frac{x \left(a+b x^2\right)^{5/2} (a B+6 A b)}{6 a}+\frac{5}{24} x \left(a+b x^2\right)^{3/2} (a B+6 A b)+\frac{5}{16} a x \sqrt{a+b x^2} (a B+6 A b)-\frac{A \left(a+b x^2\right)^{7/2}}{a x}",1,"(Sqrt[a + b*x^2]*((-48*A*(a + b*x^2)^3)/x + ((6*A*b + a*B)*(Sqrt[b]*x*Sqrt[1 + (b*x^2)/a]*(33*a^2 + 26*a*b*x^2 + 8*b^2*x^4) + 15*a^(5/2)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]]))/(Sqrt[b]*Sqrt[1 + (b*x^2)/a])))/(48*a)","A",1
546,1,105,135,0.0722244,"\int \frac{\left(a+b x^2\right)^{5/2} \left(A+B x^2\right)}{x^3} \, dx","Integrate[((a + b*x^2)^(5/2)*(A + B*x^2))/x^3,x]","\frac{\sqrt{a+b x^2} \left(a^2 \left(46 B x^2-15 A\right)+a \left(70 A b x^2+22 b B x^4\right)+2 b^2 x^4 \left(5 A+3 B x^2\right)\right)}{30 x^2}-\frac{1}{2} a^{3/2} (2 a B+5 A b) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)","-\frac{1}{2} a^{3/2} (2 a B+5 A b) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)+\frac{\left(a+b x^2\right)^{5/2} (2 a B+5 A b)}{10 a}+\frac{1}{6} \left(a+b x^2\right)^{3/2} (2 a B+5 A b)+\frac{1}{2} a \sqrt{a+b x^2} (2 a B+5 A b)-\frac{A \left(a+b x^2\right)^{7/2}}{2 a x^2}",1,"(Sqrt[a + b*x^2]*(2*b^2*x^4*(5*A + 3*B*x^2) + a^2*(-15*A + 46*B*x^2) + a*(70*A*b*x^2 + 22*b*B*x^4)))/(30*x^2) - (a^(3/2)*(5*A*b + 2*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/2","A",1
547,1,84,146,0.036631,"\int \frac{\left(a+b x^2\right)^{5/2} \left(A+B x^2\right)}{x^4} \, dx","Integrate[((a + b*x^2)^(5/2)*(A + B*x^2))/x^4,x]","\frac{a \sqrt{a+b x^2} (-3 a B-4 A b) \, _2F_1\left(-\frac{5}{2},-\frac{1}{2};\frac{1}{2};-\frac{b x^2}{a}\right)}{3 x \sqrt{\frac{b x^2}{a}+1}}-\frac{A \left(a+b x^2\right)^{7/2}}{3 a x^3}","-\frac{\left(a+b x^2\right)^{5/2} (3 a B+4 A b)}{3 a x}+\frac{5 b x \left(a+b x^2\right)^{3/2} (3 a B+4 A b)}{12 a}+\frac{5}{8} b x \sqrt{a+b x^2} (3 a B+4 A b)+\frac{5}{8} a \sqrt{b} (3 a B+4 A b) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)-\frac{A \left(a+b x^2\right)^{7/2}}{3 a x^3}",1,"-1/3*(A*(a + b*x^2)^(7/2))/(a*x^3) + (a*(-4*A*b - 3*a*B)*Sqrt[a + b*x^2]*Hypergeometric2F1[-5/2, -1/2, 1/2, -((b*x^2)/a)])/(3*x*Sqrt[1 + (b*x^2)/a])","C",1
548,1,60,143,0.0310506,"\int \frac{\left(a+b x^2\right)^{5/2} \left(A+B x^2\right)}{x^5} \, dx","Integrate[((a + b*x^2)^(5/2)*(A + B*x^2))/x^5,x]","\frac{\left(a+b x^2\right)^{7/2} \left(b x^4 (4 a B+3 A b) \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};\frac{b x^2}{a}+1\right)-7 a^2 A\right)}{28 a^3 x^4}","-\frac{\left(a+b x^2\right)^{5/2} (4 a B+3 A b)}{8 a x^2}+\frac{5 b \left(a+b x^2\right)^{3/2} (4 a B+3 A b)}{24 a}+\frac{5}{8} b \sqrt{a+b x^2} (4 a B+3 A b)-\frac{5}{8} \sqrt{a} b (4 a B+3 A b) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)-\frac{A \left(a+b x^2\right)^{7/2}}{4 a x^4}",1,"((a + b*x^2)^(7/2)*(-7*a^2*A + b*(3*A*b + 4*a*B)*x^4*Hypergeometric2F1[2, 7/2, 9/2, 1 + (b*x^2)/a]))/(28*a^3*x^4)","C",1
549,1,84,152,0.0432059,"\int \frac{\left(a+b x^2\right)^{5/2} \left(A+B x^2\right)}{x^6} \, dx","Integrate[((a + b*x^2)^(5/2)*(A + B*x^2))/x^6,x]","\frac{a \sqrt{a+b x^2} (-5 a B-2 A b) \, _2F_1\left(-\frac{5}{2},-\frac{3}{2};-\frac{1}{2};-\frac{b x^2}{a}\right)}{15 x^3 \sqrt{\frac{b x^2}{a}+1}}-\frac{A \left(a+b x^2\right)^{7/2}}{5 a x^5}","\frac{1}{2} b^{3/2} (5 a B+2 A b) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)+\frac{b^2 x \sqrt{a+b x^2} (5 a B+2 A b)}{2 a}-\frac{b \left(a+b x^2\right)^{3/2} (5 a B+2 A b)}{3 a x}-\frac{\left(a+b x^2\right)^{5/2} (5 a B+2 A b)}{15 a x^3}-\frac{A \left(a+b x^2\right)^{7/2}}{5 a x^5}",1,"-1/5*(A*(a + b*x^2)^(7/2))/(a*x^5) + (a*(-2*A*b - 5*a*B)*Sqrt[a + b*x^2]*Hypergeometric2F1[-5/2, -3/2, -1/2, -((b*x^2)/a)])/(15*x^3*Sqrt[1 + (b*x^2)/a])","C",1
550,1,61,149,0.0318367,"\int \frac{\left(a+b x^2\right)^{5/2} \left(A+B x^2\right)}{x^7} \, dx","Integrate[((a + b*x^2)^(5/2)*(A + B*x^2))/x^7,x]","-\frac{\left(a+b x^2\right)^{7/2} \left(7 a^3 A+b^2 x^6 (6 a B+A b) \, _2F_1\left(3,\frac{7}{2};\frac{9}{2};\frac{b x^2}{a}+1\right)\right)}{42 a^4 x^6}","\frac{5 b^2 \sqrt{a+b x^2} (6 a B+A b)}{16 a}-\frac{5 b^2 (6 a B+A b) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{16 \sqrt{a}}-\frac{5 b \left(a+b x^2\right)^{3/2} (6 a B+A b)}{48 a x^2}-\frac{\left(a+b x^2\right)^{5/2} (6 a B+A b)}{24 a x^4}-\frac{A \left(a+b x^2\right)^{7/2}}{6 a x^6}",1,"-1/42*((a + b*x^2)^(7/2)*(7*a^3*A + b^2*(A*b + 6*a*B)*x^6*Hypergeometric2F1[3, 7/2, 9/2, 1 + (b*x^2)/a]))/(a^4*x^6)","C",1
551,1,78,108,0.0918396,"\int \frac{\left(a+b x^2\right)^{5/2} \left(A+B x^2\right)}{x^8} \, dx","Integrate[((a + b*x^2)^(5/2)*(A + B*x^2))/x^8,x]","-\frac{a^2 B \sqrt{a+b x^2} \, _2F_1\left(-\frac{5}{2},-\frac{5}{2};-\frac{3}{2};-\frac{b x^2}{a}\right)}{5 x^5 \sqrt{\frac{b x^2}{a}+1}}-\frac{A \left(a+b x^2\right)^{7/2}}{7 a x^7}","-\frac{A \left(a+b x^2\right)^{7/2}}{7 a x^7}+b^{5/2} B \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)-\frac{b^2 B \sqrt{a+b x^2}}{x}-\frac{B \left(a+b x^2\right)^{5/2}}{5 x^5}-\frac{b B \left(a+b x^2\right)^{3/2}}{3 x^3}",1,"-1/7*(A*(a + b*x^2)^(7/2))/(a*x^7) - (a^2*B*Sqrt[a + b*x^2]*Hypergeometric2F1[-5/2, -5/2, -3/2, -((b*x^2)/a)])/(5*x^5*Sqrt[1 + (b*x^2)/a])","C",1
552,1,140,152,0.1415338,"\int \frac{\left(a+b x^2\right)^{5/2} \left(A+B x^2\right)}{x^9} \, dx","Integrate[((a + b*x^2)^(5/2)*(A + B*x^2))/x^9,x]","\frac{-\left(a+b x^2\right) \left(16 a^3 \left(3 A+4 B x^2\right)+8 a^2 b x^2 \left(17 A+26 B x^2\right)+2 a b^2 x^4 \left(59 A+132 B x^2\right)+15 A b^3 x^6\right)-15 b^3 x^8 \sqrt{\frac{b x^2}{a}+1} (8 a B-A b) \tanh ^{-1}\left(\sqrt{\frac{b x^2}{a}+1}\right)}{384 a x^8 \sqrt{a+b x^2}}","\frac{5 b^3 (A b-8 a B) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{128 a^{3/2}}+\frac{5 b^2 \sqrt{a+b x^2} (A b-8 a B)}{128 a x^2}+\frac{\left(a+b x^2\right)^{5/2} (A b-8 a B)}{48 a x^6}+\frac{5 b \left(a+b x^2\right)^{3/2} (A b-8 a B)}{192 a x^4}-\frac{A \left(a+b x^2\right)^{7/2}}{8 a x^8}",1,"(-((a + b*x^2)*(15*A*b^3*x^6 + 16*a^3*(3*A + 4*B*x^2) + 8*a^2*b*x^2*(17*A + 26*B*x^2) + 2*a*b^2*x^4*(59*A + 132*B*x^2))) - 15*b^3*(-(A*b) + 8*a*B)*x^8*Sqrt[1 + (b*x^2)/a]*ArcTanh[Sqrt[1 + (b*x^2)/a]])/(384*a*x^8*Sqrt[a + b*x^2])","A",1
553,1,40,53,0.0170978,"\int \frac{\left(a+b x^2\right)^{5/2} \left(A+B x^2\right)}{x^{10}} \, dx","Integrate[((a + b*x^2)^(5/2)*(A + B*x^2))/x^10,x]","-\frac{\left(a+b x^2\right)^{7/2} \left(7 a A+9 a B x^2-2 A b x^2\right)}{63 a^2 x^9}","\frac{\left(a+b x^2\right)^{7/2} (2 A b-9 a B)}{63 a^2 x^7}-\frac{A \left(a+b x^2\right)^{7/2}}{9 a x^9}",1,"-1/63*((a + b*x^2)^(7/2)*(7*a*A - 2*A*b*x^2 + 9*a*B*x^2))/(a^2*x^9)","A",1
554,1,62,189,0.0275284,"\int \frac{\left(a+b x^2\right)^{5/2} \left(A+B x^2\right)}{x^{11}} \, dx","Integrate[((a + b*x^2)^(5/2)*(A + B*x^2))/x^11,x]","-\frac{\left(a+b x^2\right)^{7/2} \left(7 a^5 A+b^4 x^{10} (10 a B-3 A b) \, _2F_1\left(\frac{7}{2},5;\frac{9}{2};\frac{b x^2}{a}+1\right)\right)}{70 a^6 x^{10}}","-\frac{b^4 (3 A b-10 a B) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{256 a^{5/2}}+\frac{b^3 \sqrt{a+b x^2} (3 A b-10 a B)}{256 a^2 x^2}+\frac{b^2 \sqrt{a+b x^2} (3 A b-10 a B)}{128 a x^4}+\frac{\left(a+b x^2\right)^{5/2} (3 A b-10 a B)}{80 a x^8}+\frac{b \left(a+b x^2\right)^{3/2} (3 A b-10 a B)}{96 a x^6}-\frac{A \left(a+b x^2\right)^{7/2}}{10 a x^{10}}",1,"-1/70*((a + b*x^2)^(7/2)*(7*a^5*A + b^4*(-3*A*b + 10*a*B)*x^10*Hypergeometric2F1[7/2, 5, 9/2, 1 + (b*x^2)/a]))/(a^6*x^10)","C",1
555,1,78,100,0.0530278,"\int \frac{x^5 \left(A+B x^2\right)}{\sqrt{a+b x^2}} \, dx","Integrate[(x^5*(A + B*x^2))/Sqrt[a + b*x^2],x]","\frac{\sqrt{a+b x^2} \left(-48 a^3 B+8 a^2 b \left(7 A+3 B x^2\right)-2 a b^2 x^2 \left(14 A+9 B x^2\right)+3 b^3 x^4 \left(7 A+5 B x^2\right)\right)}{105 b^4}","\frac{a^2 \sqrt{a+b x^2} (A b-a B)}{b^4}+\frac{\left(a+b x^2\right)^{5/2} (A b-3 a B)}{5 b^4}-\frac{a \left(a+b x^2\right)^{3/2} (2 A b-3 a B)}{3 b^4}+\frac{B \left(a+b x^2\right)^{7/2}}{7 b^4}",1,"(Sqrt[a + b*x^2]*(-48*a^3*B + 8*a^2*b*(7*A + 3*B*x^2) + 3*b^3*x^4*(7*A + 5*B*x^2) - 2*a*b^2*x^2*(14*A + 9*B*x^2)))/(105*b^4)","A",1
556,1,100,122,0.0971817,"\int \frac{x^4 \left(A+B x^2\right)}{\sqrt{a+b x^2}} \, dx","Integrate[(x^4*(A + B*x^2))/Sqrt[a + b*x^2],x]","\frac{\sqrt{b} x \sqrt{a+b x^2} \left(15 a^2 B-2 a b \left(9 A+5 B x^2\right)+4 b^2 x^2 \left(3 A+2 B x^2\right)\right)-3 a^2 (5 a B-6 A b) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{48 b^{7/2}}","\frac{a^2 (6 A b-5 a B) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{16 b^{7/2}}-\frac{a x \sqrt{a+b x^2} (6 A b-5 a B)}{16 b^3}+\frac{x^3 \sqrt{a+b x^2} (6 A b-5 a B)}{24 b^2}+\frac{B x^5 \sqrt{a+b x^2}}{6 b}",1,"(Sqrt[b]*x*Sqrt[a + b*x^2]*(15*a^2*B + 4*b^2*x^2*(3*A + 2*B*x^2) - 2*a*b*(9*A + 5*B*x^2)) - 3*a^2*(-6*A*b + 5*a*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(48*b^(7/2))","A",1
557,1,56,71,0.0363479,"\int \frac{x^3 \left(A+B x^2\right)}{\sqrt{a+b x^2}} \, dx","Integrate[(x^3*(A + B*x^2))/Sqrt[a + b*x^2],x]","\frac{\sqrt{a+b x^2} \left(8 a^2 B-2 a b \left(5 A+2 B x^2\right)+b^2 x^2 \left(5 A+3 B x^2\right)\right)}{15 b^3}","\frac{\left(a+b x^2\right)^{3/2} (A b-2 a B)}{3 b^3}-\frac{a \sqrt{a+b x^2} (A b-a B)}{b^3}+\frac{B \left(a+b x^2\right)^{5/2}}{5 b^3}",1,"(Sqrt[a + b*x^2]*(8*a^2*B - 2*a*b*(5*A + 2*B*x^2) + b^2*x^2*(5*A + 3*B*x^2)))/(15*b^3)","A",1
558,1,74,89,0.0625939,"\int \frac{x^2 \left(A+B x^2\right)}{\sqrt{a+b x^2}} \, dx","Integrate[(x^2*(A + B*x^2))/Sqrt[a + b*x^2],x]","\frac{\sqrt{b} x \sqrt{a+b x^2} \left(-3 a B+4 A b+2 b B x^2\right)+a (3 a B-4 A b) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 b^{5/2}}","-\frac{a (4 A b-3 a B) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 b^{5/2}}+\frac{x \sqrt{a+b x^2} (4 A b-3 a B)}{8 b^2}+\frac{B x^3 \sqrt{a+b x^2}}{4 b}",1,"(Sqrt[b]*x*Sqrt[a + b*x^2]*(4*A*b - 3*a*B + 2*b*B*x^2) + a*(-4*A*b + 3*a*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*b^(5/2))","A",1
559,1,33,43,0.0248742,"\int \frac{x \left(A+B x^2\right)}{\sqrt{a+b x^2}} \, dx","Integrate[(x*(A + B*x^2))/Sqrt[a + b*x^2],x]","\frac{\sqrt{a+b x^2} \left(-2 a B+3 A b+b B x^2\right)}{3 b^2}","\frac{\sqrt{a+b x^2} (A b-a B)}{b^2}+\frac{B \left(a+b x^2\right)^{3/2}}{3 b^2}",1,"(Sqrt[a + b*x^2]*(3*A*b - 2*a*B + b*B*x^2))/(3*b^2)","A",1
560,1,57,58,0.0184176,"\int \frac{A+B x^2}{\sqrt{a+b x^2}} \, dx","Integrate[(A + B*x^2)/Sqrt[a + b*x^2],x]","\frac{B x \sqrt{a+b x^2}}{2 b}-\frac{(a B-2 A b) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 b^{3/2}}","\frac{(2 A b-a B) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 b^{3/2}}+\frac{B x \sqrt{a+b x^2}}{2 b}",1,"(B*x*Sqrt[a + b*x^2])/(2*b) - ((-2*A*b + a*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(3/2))","A",1
561,1,43,43,0.0217931,"\int \frac{A+B x^2}{x \sqrt{a+b x^2}} \, dx","Integrate[(A + B*x^2)/(x*Sqrt[a + b*x^2]),x]","\frac{B \sqrt{a+b x^2}}{b}-\frac{A \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{\sqrt{a}}","\frac{B \sqrt{a+b x^2}}{b}-\frac{A \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{\sqrt{a}}",1,"(B*Sqrt[a + b*x^2])/b - (A*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/Sqrt[a]","A",1
562,1,47,47,0.0177442,"\int \frac{A+B x^2}{x^2 \sqrt{a+b x^2}} \, dx","Integrate[(A + B*x^2)/(x^2*Sqrt[a + b*x^2]),x]","\frac{B \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{\sqrt{b}}-\frac{A \sqrt{a+b x^2}}{a x}","\frac{B \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{\sqrt{b}}-\frac{A \sqrt{a+b x^2}}{a x}",1,"-((A*Sqrt[a + b*x^2])/(a*x)) + (B*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/Sqrt[b]","A",1
563,1,60,58,0.0323434,"\int \frac{A+B x^2}{x^3 \sqrt{a+b x^2}} \, dx","Integrate[(A + B*x^2)/(x^3*Sqrt[a + b*x^2]),x]","\frac{1}{2} \left(-\frac{2 \left(a B-\frac{A b}{2}\right) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{a^{3/2}}-\frac{A \sqrt{a+b x^2}}{a x^2}\right)","\frac{(A b-2 a B) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{2 a^{3/2}}-\frac{A \sqrt{a+b x^2}}{2 a x^2}",1,"(-((A*Sqrt[a + b*x^2])/(a*x^2)) - (2*(-1/2*(A*b) + a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/a^(3/2))/2","A",1
564,1,39,53,0.0139249,"\int \frac{A+B x^2}{x^4 \sqrt{a+b x^2}} \, dx","Integrate[(A + B*x^2)/(x^4*Sqrt[a + b*x^2]),x]","-\frac{\sqrt{a+b x^2} \left(a \left(A+3 B x^2\right)-2 A b x^2\right)}{3 a^2 x^3}","\frac{\sqrt{a+b x^2} (2 A b-3 a B)}{3 a^2 x}-\frac{A \sqrt{a+b x^2}}{3 a x^3}",1,"-1/3*(Sqrt[a + b*x^2]*(-2*A*b*x^2 + a*(A + 3*B*x^2)))/(a^2*x^3)","A",1
565,1,83,90,0.1705091,"\int \frac{A+B x^2}{x^5 \sqrt{a+b x^2}} \, dx","Integrate[(A + B*x^2)/(x^5*Sqrt[a + b*x^2]),x]","\frac{\sqrt{a+b x^2} \left(-\frac{2 a^2 \left(A+2 B x^2\right)}{x^4}+\frac{b (4 a B-3 A b) \tanh ^{-1}\left(\sqrt{\frac{b x^2}{a}+1}\right)}{\sqrt{\frac{b x^2}{a}+1}}+\frac{3 a A b}{x^2}\right)}{8 a^3}","-\frac{b (3 A b-4 a B) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{8 a^{5/2}}+\frac{\sqrt{a+b x^2} (3 A b-4 a B)}{8 a^2 x^2}-\frac{A \sqrt{a+b x^2}}{4 a x^4}",1,"(Sqrt[a + b*x^2]*((3*a*A*b)/x^2 - (2*a^2*(A + 2*B*x^2))/x^4 + (b*(-3*A*b + 4*a*B)*ArcTanh[Sqrt[1 + (b*x^2)/a]])/Sqrt[1 + (b*x^2)/a]))/(8*a^3)","A",1
566,1,62,84,0.0197465,"\int \frac{A+B x^2}{x^6 \sqrt{a+b x^2}} \, dx","Integrate[(A + B*x^2)/(x^6*Sqrt[a + b*x^2]),x]","-\frac{\sqrt{a+b x^2} \left(a^2 \left(3 A+5 B x^2\right)-2 a b x^2 \left(2 A+5 B x^2\right)+8 A b^2 x^4\right)}{15 a^3 x^5}","-\frac{2 b \sqrt{a+b x^2} (4 A b-5 a B)}{15 a^3 x}+\frac{\sqrt{a+b x^2} (4 A b-5 a B)}{15 a^2 x^3}-\frac{A \sqrt{a+b x^2}}{5 a x^5}",1,"-1/15*(Sqrt[a + b*x^2]*(8*A*b^2*x^4 - 2*a*b*x^2*(2*A + 5*B*x^2) + a^2*(3*A + 5*B*x^2)))/(a^3*x^5)","A",1
567,1,61,123,0.0203753,"\int \frac{A+B x^2}{x^7 \sqrt{a+b x^2}} \, dx","Integrate[(A + B*x^2)/(x^7*Sqrt[a + b*x^2]),x]","-\frac{\sqrt{a+b x^2} \left(a^3 A+b^2 x^6 (6 a B-5 A b) \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};\frac{b x^2}{a}+1\right)\right)}{6 a^4 x^6}","\frac{b^2 (5 A b-6 a B) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{16 a^{7/2}}-\frac{b \sqrt{a+b x^2} (5 A b-6 a B)}{16 a^3 x^2}+\frac{\sqrt{a+b x^2} (5 A b-6 a B)}{24 a^2 x^4}-\frac{A \sqrt{a+b x^2}}{6 a x^6}",1,"-1/6*(Sqrt[a + b*x^2]*(a^3*A + b^2*(-5*A*b + 6*a*B)*x^6*Hypergeometric2F1[1/2, 3, 3/2, 1 + (b*x^2)/a]))/(a^4*x^6)","C",1
568,1,84,117,0.0300996,"\int \frac{A+B x^2}{x^8 \sqrt{a+b x^2}} \, dx","Integrate[(A + B*x^2)/(x^8*Sqrt[a + b*x^2]),x]","\frac{\left(\frac{b x^2}{a}+1\right) \left(3 a^2-4 a b x^2+8 b^2 x^4\right) (6 A b-7 a B)}{105 a^3 x^5 \sqrt{a+b x^2}}-\frac{A \sqrt{a+b x^2}}{7 a x^7}","\frac{8 b^2 \sqrt{a+b x^2} (6 A b-7 a B)}{105 a^4 x}-\frac{4 b \sqrt{a+b x^2} (6 A b-7 a B)}{105 a^3 x^3}+\frac{\sqrt{a+b x^2} (6 A b-7 a B)}{35 a^2 x^5}-\frac{A \sqrt{a+b x^2}}{7 a x^7}",1,"-1/7*(A*Sqrt[a + b*x^2])/(a*x^7) + ((6*A*b - 7*a*B)*(1 + (b*x^2)/a)*(3*a^2 - 4*a*b*x^2 + 8*b^2*x^4))/(105*a^3*x^5*Sqrt[a + b*x^2])","A",1
569,1,131,152,0.1641245,"\int \frac{x^6 \left(A+B x^2\right)}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[(x^6*(A + B*x^2))/(a + b*x^2)^(3/2),x]","\frac{\sqrt{b} x \left(105 a^3 B+a^2 \left(35 b B x^2-90 A b\right)-2 a b^2 x^2 \left(15 A+7 B x^2\right)+4 b^3 x^4 \left(3 A+2 B x^2\right)\right)-15 a^{5/2} \sqrt{\frac{b x^2}{a}+1} (7 a B-6 A b) \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{48 b^{9/2} \sqrt{a+b x^2}}","\frac{5 a^2 (6 A b-7 a B) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{16 b^{9/2}}-\frac{5 a x \sqrt{a+b x^2} (6 A b-7 a B)}{16 b^4}+\frac{5 x^3 \sqrt{a+b x^2} (6 A b-7 a B)}{24 b^3}-\frac{x^5 (6 A b-7 a B)}{6 b^2 \sqrt{a+b x^2}}+\frac{B x^7}{6 b \sqrt{a+b x^2}}",1,"(Sqrt[b]*x*(105*a^3*B + 4*b^3*x^4*(3*A + 2*B*x^2) - 2*a*b^2*x^2*(15*A + 7*B*x^2) + a^2*(-90*A*b + 35*b*B*x^2)) - 15*a^(5/2)*(-6*A*b + 7*a*B)*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(48*b^(9/2)*Sqrt[a + b*x^2])","A",1
570,1,77,99,0.0445934,"\int \frac{x^5 \left(A+B x^2\right)}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[(x^5*(A + B*x^2))/(a + b*x^2)^(3/2),x]","\frac{48 a^3 B-8 a^2 b \left(5 A-3 B x^2\right)-2 a b^2 x^2 \left(10 A+3 B x^2\right)+b^3 x^4 \left(5 A+3 B x^2\right)}{15 b^4 \sqrt{a+b x^2}}","-\frac{a^2 (A b-a B)}{b^4 \sqrt{a+b x^2}}+\frac{\left(a+b x^2\right)^{3/2} (A b-3 a B)}{3 b^4}-\frac{a \sqrt{a+b x^2} (2 A b-3 a B)}{b^4}+\frac{B \left(a+b x^2\right)^{5/2}}{5 b^4}",1,"(48*a^3*B - 8*a^2*b*(5*A - 3*B*x^2) + b^3*x^4*(5*A + 3*B*x^2) - 2*a*b^2*x^2*(10*A + 3*B*x^2))/(15*b^4*Sqrt[a + b*x^2])","A",1
571,1,108,119,0.1173978,"\int \frac{x^4 \left(A+B x^2\right)}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[(x^4*(A + B*x^2))/(a + b*x^2)^(3/2),x]","\frac{3 a^{3/2} \sqrt{\frac{b x^2}{a}+1} (5 a B-4 A b) \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)+\sqrt{b} x \left(-15 a^2 B+a b \left(12 A-5 B x^2\right)+2 b^2 x^2 \left(2 A+B x^2\right)\right)}{8 b^{7/2} \sqrt{a+b x^2}}","-\frac{3 a (4 A b-5 a B) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 b^{7/2}}+\frac{3 x \sqrt{a+b x^2} (4 A b-5 a B)}{8 b^3}-\frac{x^3 (4 A b-5 a B)}{4 b^2 \sqrt{a+b x^2}}+\frac{B x^5}{4 b \sqrt{a+b x^2}}",1,"(Sqrt[b]*x*(-15*a^2*B + a*b*(12*A - 5*B*x^2) + 2*b^2*x^2*(2*A + B*x^2)) + 3*a^(3/2)*(-4*A*b + 5*a*B)*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(8*b^(7/2)*Sqrt[a + b*x^2])","A",1
572,1,55,67,0.0309932,"\int \frac{x^3 \left(A+B x^2\right)}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[(x^3*(A + B*x^2))/(a + b*x^2)^(3/2),x]","\frac{-8 a^2 B+a \left(6 A b-4 b B x^2\right)+b^2 x^2 \left(3 A+B x^2\right)}{3 b^3 \sqrt{a+b x^2}}","\frac{\sqrt{a+b x^2} (A b-2 a B)}{b^3}+\frac{a (A b-a B)}{b^3 \sqrt{a+b x^2}}+\frac{B \left(a+b x^2\right)^{3/2}}{3 b^3}",1,"(-8*a^2*B + b^2*x^2*(3*A + B*x^2) + a*(6*A*b - 4*b*B*x^2))/(3*b^3*Sqrt[a + b*x^2])","A",1
573,1,86,83,0.0998795,"\int \frac{x^2 \left(A+B x^2\right)}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[(x^2*(A + B*x^2))/(a + b*x^2)^(3/2),x]","\frac{\sqrt{b} x \left(3 a B-2 A b+b B x^2\right)-\sqrt{a} \sqrt{\frac{b x^2}{a}+1} (3 a B-2 A b) \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{5/2} \sqrt{a+b x^2}}","\frac{(2 A b-3 a B) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 b^{5/2}}-\frac{x (A b-a B)}{b^2 \sqrt{a+b x^2}}+\frac{B x \sqrt{a+b x^2}}{2 b^2}",1,"(Sqrt[b]*x*(-2*A*b + 3*a*B + b*B*x^2) - Sqrt[a]*(-2*A*b + 3*a*B)*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(5/2)*Sqrt[a + b*x^2])","A",1
574,1,30,41,0.0196602,"\int \frac{x \left(A+B x^2\right)}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[(x*(A + B*x^2))/(a + b*x^2)^(3/2),x]","\frac{2 a B-A b+b B x^2}{b^2 \sqrt{a+b x^2}}","\frac{B \sqrt{a+b x^2}}{b^2}-\frac{A b-a B}{b^2 \sqrt{a+b x^2}}",1,"(-(A*b) + 2*a*B + b*B*x^2)/(b^2*Sqrt[a + b*x^2])","A",1
575,1,70,54,0.0528153,"\int \frac{A+B x^2}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[(A + B*x^2)/(a + b*x^2)^(3/2),x]","\frac{a^{3/2} B \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)+\sqrt{b} x (A b-a B)}{a b^{3/2} \sqrt{a+b x^2}}","\frac{x (A b-a B)}{a b \sqrt{a+b x^2}}+\frac{B \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{b^{3/2}}",1,"(Sqrt[b]*(A*b - a*B)*x + a^(3/2)*B*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(a*b^(3/2)*Sqrt[a + b*x^2])","A",1
576,1,53,53,0.0272083,"\int \frac{A+B x^2}{x \left(a+b x^2\right)^{3/2}} \, dx","Integrate[(A + B*x^2)/(x*(a + b*x^2)^(3/2)),x]","\frac{A b-a B}{a b \sqrt{a+b x^2}}-\frac{A \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{a^{3/2}}","\frac{A b-a B}{a b \sqrt{a+b x^2}}-\frac{A \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{a^{3/2}}",1,"(A*b - a*B)/(a*b*Sqrt[a + b*x^2]) - (A*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/a^(3/2)","A",1
577,1,36,47,0.0124522,"\int \frac{A+B x^2}{x^2 \left(a+b x^2\right)^{3/2}} \, dx","Integrate[(A + B*x^2)/(x^2*(a + b*x^2)^(3/2)),x]","\frac{-a A+a B x^2-2 A b x^2}{a^2 x \sqrt{a+b x^2}}","-\frac{x (2 A b-a B)}{a^2 \sqrt{a+b x^2}}-\frac{A}{a x \sqrt{a+b x^2}}",1,"(-(a*A) - 2*A*b*x^2 + a*B*x^2)/(a^2*x*Sqrt[a + b*x^2])","A",1
578,1,57,86,0.0193531,"\int \frac{A+B x^2}{x^3 \left(a+b x^2\right)^{3/2}} \, dx","Integrate[(A + B*x^2)/(x^3*(a + b*x^2)^(3/2)),x]","\frac{x^2 (2 a B-3 A b) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b x^2}{a}+1\right)-a A}{2 a^2 x^2 \sqrt{a+b x^2}}","\frac{(3 A b-2 a B) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{2 a^{5/2}}-\frac{3 A b-2 a B}{2 a^2 \sqrt{a+b x^2}}-\frac{A}{2 a x^2 \sqrt{a+b x^2}}",1,"(-(a*A) + (-3*A*b + 2*a*B)*x^2*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*x^2)/a])/(2*a^2*x^2*Sqrt[a + b*x^2])","C",1
579,1,61,82,0.0175624,"\int \frac{A+B x^2}{x^4 \left(a+b x^2\right)^{3/2}} \, dx","Integrate[(A + B*x^2)/(x^4*(a + b*x^2)^(3/2)),x]","\frac{\left(a+2 b x^2\right) (4 A b-3 a B)}{3 a^3 x \sqrt{a+b x^2}}-\frac{A}{3 a x^3 \sqrt{a+b x^2}}","\frac{2 b x (4 A b-3 a B)}{3 a^3 \sqrt{a+b x^2}}+\frac{4 A b-3 a B}{3 a^2 x \sqrt{a+b x^2}}-\frac{A}{3 a x^3 \sqrt{a+b x^2}}",1,"-1/3*A/(a*x^3*Sqrt[a + b*x^2]) + ((4*A*b - 3*a*B)*(a + 2*b*x^2))/(3*a^3*x*Sqrt[a + b*x^2])","A",1
580,1,60,118,0.0728141,"\int \frac{A+B x^2}{x^5 \left(a+b x^2\right)^{3/2}} \, dx","Integrate[(A + B*x^2)/(x^5*(a + b*x^2)^(3/2)),x]","\frac{b x^4 (5 A b-4 a B) \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};\frac{b x^2}{a}+1\right)-a^2 A}{4 a^3 x^4 \sqrt{a+b x^2}}","-\frac{3 b (5 A b-4 a B) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{8 a^{7/2}}+\frac{3 b (5 A b-4 a B)}{8 a^3 \sqrt{a+b x^2}}+\frac{5 A b-4 a B}{8 a^2 x^2 \sqrt{a+b x^2}}-\frac{A}{4 a x^4 \sqrt{a+b x^2}}",1,"(-(a^2*A) + b*(5*A*b - 4*a*B)*x^4*Hypergeometric2F1[-1/2, 2, 1/2, 1 + (b*x^2)/a])/(4*a^3*x^4*Sqrt[a + b*x^2])","C",1
581,1,60,115,0.0236342,"\int \frac{A+B x^2}{x^6 \left(a+b x^2\right)^{3/2}} \, dx","Integrate[(A + B*x^2)/(x^6*(a + b*x^2)^(3/2)),x]","\frac{x^2 \left(a^2-4 a b x^2-8 b^2 x^4\right) (6 A b-5 a B)-3 a^3 A}{15 a^4 x^5 \sqrt{a+b x^2}}","-\frac{8 b^2 x (6 A b-5 a B)}{15 a^4 \sqrt{a+b x^2}}-\frac{4 b (6 A b-5 a B)}{15 a^3 x \sqrt{a+b x^2}}+\frac{6 A b-5 a B}{15 a^2 x^3 \sqrt{a+b x^2}}-\frac{A}{5 a x^5 \sqrt{a+b x^2}}",1,"(-3*a^3*A + (6*A*b - 5*a*B)*x^2*(a^2 - 4*a*b*x^2 - 8*b^2*x^4))/(15*a^4*x^5*Sqrt[a + b*x^2])","A",1
582,1,62,153,0.0215738,"\int \frac{A+B x^2}{x^7 \left(a+b x^2\right)^{3/2}} \, dx","Integrate[(A + B*x^2)/(x^7*(a + b*x^2)^(3/2)),x]","\frac{b^2 x^6 (6 a B-7 A b) \, _2F_1\left(-\frac{1}{2},3;\frac{1}{2};\frac{b x^2}{a}+1\right)-a^3 A}{6 a^4 x^6 \sqrt{a+b x^2}}","\frac{5 b^2 (7 A b-6 a B) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{16 a^{9/2}}-\frac{5 b^2 (7 A b-6 a B)}{16 a^4 \sqrt{a+b x^2}}-\frac{5 b (7 A b-6 a B)}{48 a^3 x^2 \sqrt{a+b x^2}}+\frac{7 A b-6 a B}{24 a^2 x^4 \sqrt{a+b x^2}}-\frac{A}{6 a x^6 \sqrt{a+b x^2}}",1,"(-(a^3*A) + b^2*(-7*A*b + 6*a*B)*x^6*Hypergeometric2F1[-1/2, 3, 1/2, 1 + (b*x^2)/a])/(6*a^4*x^6*Sqrt[a + b*x^2])","C",1
583,1,71,148,0.0279861,"\int \frac{A+B x^2}{x^8 \left(a+b x^2\right)^{3/2}} \, dx","Integrate[(A + B*x^2)/(x^8*(a + b*x^2)^(3/2)),x]","\frac{x^2 \left(a^3-2 a^2 b x^2+8 a b^2 x^4+16 b^3 x^6\right) (8 A b-7 a B)-5 a^4 A}{35 a^5 x^7 \sqrt{a+b x^2}}","\frac{16 b^3 x (8 A b-7 a B)}{35 a^5 \sqrt{a+b x^2}}+\frac{8 b^2 (8 A b-7 a B)}{35 a^4 x \sqrt{a+b x^2}}-\frac{2 b (8 A b-7 a B)}{35 a^3 x^3 \sqrt{a+b x^2}}+\frac{8 A b-7 a B}{35 a^2 x^5 \sqrt{a+b x^2}}-\frac{A}{7 a x^7 \sqrt{a+b x^2}}",1,"(-5*a^4*A + (8*A*b - 7*a*B)*x^2*(a^3 - 2*a^2*b*x^2 + 8*a*b^2*x^4 + 16*b^3*x^6))/(35*a^5*x^7*Sqrt[a + b*x^2])","A",1
584,1,98,128,0.0687896,"\int \frac{x^7 \left(A+B x^2\right)}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[(x^7*(A + B*x^2))/(a + b*x^2)^(5/2),x]","\frac{128 a^4 B+a^3 \left(192 b B x^2-80 A b\right)+24 a^2 b^2 x^2 \left(2 B x^2-5 A\right)-2 a b^3 x^4 \left(15 A+4 B x^2\right)+b^4 x^6 \left(5 A+3 B x^2\right)}{15 b^5 \left(a+b x^2\right)^{3/2}}","\frac{a^3 (A b-a B)}{3 b^5 \left(a+b x^2\right)^{3/2}}-\frac{a^2 (3 A b-4 a B)}{b^5 \sqrt{a+b x^2}}-\frac{3 a \sqrt{a+b x^2} (A b-2 a B)}{b^5}+\frac{\left(a+b x^2\right)^{3/2} (A b-4 a B)}{3 b^5}+\frac{B \left(a+b x^2\right)^{5/2}}{5 b^5}",1,"(128*a^4*B + 24*a^2*b^2*x^2*(-5*A + 2*B*x^2) + b^4*x^6*(5*A + 3*B*x^2) - 2*a*b^3*x^4*(15*A + 4*B*x^2) + a^3*(-80*A*b + 192*b*B*x^2))/(15*b^5*(a + b*x^2)^(3/2))","A",1
585,1,139,149,0.2349718,"\int \frac{x^6 \left(A+B x^2\right)}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[(x^6*(A + B*x^2))/(a + b*x^2)^(5/2),x]","\frac{x \left(-105 a^3 B+20 a^2 b \left(3 A-7 B x^2\right)+a b^2 x^2 \left(80 A-21 B x^2\right)+6 b^3 x^4 \left(2 A+B x^2\right)\right)}{24 b^4 \left(a+b x^2\right)^{3/2}}+\frac{5 \sqrt{a} \sqrt{a+b x^2} (7 a B-4 A b) \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 b^{9/2} \sqrt{\frac{b x^2}{a}+1}}","-\frac{5 a (4 A b-7 a B) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 b^{9/2}}+\frac{5 x \sqrt{a+b x^2} (4 A b-7 a B)}{8 b^4}-\frac{5 x^3 (4 A b-7 a B)}{12 b^3 \sqrt{a+b x^2}}-\frac{x^5 (4 A b-7 a B)}{12 b^2 \left(a+b x^2\right)^{3/2}}+\frac{B x^7}{4 b \left(a+b x^2\right)^{3/2}}",1,"(x*(-105*a^3*B + a*b^2*x^2*(80*A - 21*B*x^2) + 20*a^2*b*(3*A - 7*B*x^2) + 6*b^3*x^4*(2*A + B*x^2)))/(24*b^4*(a + b*x^2)^(3/2)) + (5*Sqrt[a]*(-4*A*b + 7*a*B)*Sqrt[a + b*x^2]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(8*b^(9/2)*Sqrt[1 + (b*x^2)/a])","A",1
586,1,73,97,0.0491789,"\int \frac{x^5 \left(A+B x^2\right)}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[(x^5*(A + B*x^2))/(a + b*x^2)^(5/2),x]","\frac{-16 a^3 B+8 a^2 b \left(A-3 B x^2\right)-6 a b^2 x^2 \left(B x^2-2 A\right)+b^3 x^4 \left(3 A+B x^2\right)}{3 b^4 \left(a+b x^2\right)^{3/2}}","-\frac{a^2 (A b-a B)}{3 b^4 \left(a+b x^2\right)^{3/2}}+\frac{a (2 A b-3 a B)}{b^4 \sqrt{a+b x^2}}+\frac{\sqrt{a+b x^2} (A b-3 a B)}{b^4}+\frac{B \left(a+b x^2\right)^{3/2}}{3 b^4}",1,"(-16*a^3*B + 8*a^2*b*(A - 3*B*x^2) - 6*a*b^2*x^2*(-2*A + B*x^2) + b^3*x^4*(3*A + B*x^2))/(3*b^4*(a + b*x^2)^(3/2))","A",1
587,1,116,114,0.2600514,"\int \frac{x^4 \left(A+B x^2\right)}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[(x^4*(A + B*x^2))/(a + b*x^2)^(5/2),x]","\frac{\sqrt{b} x \left(15 a^2 B+a \left(20 b B x^2-6 A b\right)+b^2 x^2 \left(3 B x^2-8 A\right)\right)-3 \sqrt{a} \left(a+b x^2\right) \sqrt{\frac{b x^2}{a}+1} (5 a B-2 A b) \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{6 b^{7/2} \left(a+b x^2\right)^{3/2}}","\frac{(2 A b-5 a B) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 b^{7/2}}-\frac{x (4 A b-7 a B)}{3 b^3 \sqrt{a+b x^2}}+\frac{a x (A b-a B)}{3 b^3 \left(a+b x^2\right)^{3/2}}+\frac{B x \sqrt{a+b x^2}}{2 b^3}",1,"(Sqrt[b]*x*(15*a^2*B + b^2*x^2*(-8*A + 3*B*x^2) + a*(-6*A*b + 20*b*B*x^2)) - 3*Sqrt[a]*(-2*A*b + 5*a*B)*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(6*b^(7/2)*(a + b*x^2)^(3/2))","A",1
588,1,54,68,0.0344663,"\int \frac{x^3 \left(A+B x^2\right)}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[(x^3*(A + B*x^2))/(a + b*x^2)^(5/2),x]","\frac{8 a^2 B-2 a b \left(A-6 B x^2\right)+3 b^2 x^2 \left(B x^2-A\right)}{3 b^3 \left(a+b x^2\right)^{3/2}}","-\frac{A b-2 a B}{b^3 \sqrt{a+b x^2}}+\frac{a (A b-a B)}{3 b^3 \left(a+b x^2\right)^{3/2}}+\frac{B \sqrt{a+b x^2}}{b^3}",1,"(8*a^2*B - 2*a*b*(A - 6*B*x^2) + 3*b^2*x^2*(-A + B*x^2))/(3*b^3*(a + b*x^2)^(3/2))","A",1
589,1,96,77,0.1617966,"\int \frac{x^2 \left(A+B x^2\right)}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[(x^2*(A + B*x^2))/(a + b*x^2)^(5/2),x]","\frac{3 a^{3/2} B \left(a+b x^2\right) \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)+\sqrt{b} x \left(-3 a^2 B-4 a b B x^2+A b^2 x^2\right)}{3 a b^{5/2} \left(a+b x^2\right)^{3/2}}","\frac{x^3 (A b-a B)}{3 a b \left(a+b x^2\right)^{3/2}}+\frac{B \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{b^{5/2}}-\frac{B x}{b^2 \sqrt{a+b x^2}}",1,"(Sqrt[b]*x*(-3*a^2*B + A*b^2*x^2 - 4*a*b*B*x^2) + 3*a^(3/2)*B*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(3*a*b^(5/2)*(a + b*x^2)^(3/2))","A",1
590,1,34,44,0.0217535,"\int \frac{x \left(A+B x^2\right)}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[(x*(A + B*x^2))/(a + b*x^2)^(5/2),x]","\frac{-2 a B-A b-3 b B x^2}{3 b^2 \left(a+b x^2\right)^{3/2}}","\frac{a B-A b}{3 b^2 \left(a+b x^2\right)^{3/2}}-\frac{B}{b^2 \sqrt{a+b x^2}}",1,"(-(A*b) - 2*a*B - 3*b*B*x^2)/(3*b^2*(a + b*x^2)^(3/2))","A",1
591,1,37,47,0.0142313,"\int \frac{A+B x^2}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[(A + B*x^2)/(a + b*x^2)^(5/2),x]","\frac{x \left(3 a A+a B x^2+2 A b x^2\right)}{3 a^2 \left(a+b x^2\right)^{3/2}}","\frac{2 A x}{3 a^2 \sqrt{a+b x^2}}+\frac{x \left(A+B x^2\right)}{3 a \left(a+b x^2\right)^{3/2}}",1,"(x*(3*a*A + 2*A*b*x^2 + a*B*x^2))/(3*a^2*(a + b*x^2)^(3/2))","A",1
592,1,61,72,0.0221351,"\int \frac{A+B x^2}{x \left(a+b x^2\right)^{5/2}} \, dx","Integrate[(A + B*x^2)/(x*(a + b*x^2)^(5/2)),x]","\frac{a (A b-a B)+3 A b \left(a+b x^2\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b x^2}{a}+1\right)}{3 a^2 b \left(a+b x^2\right)^{3/2}}","-\frac{A \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{a^{5/2}}+\frac{A}{a^2 \sqrt{a+b x^2}}+\frac{A b-a B}{3 a b \left(a+b x^2\right)^{3/2}}",1,"(a*(A*b - a*B) + 3*A*b*(a + b*x^2)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*x^2)/a])/(3*a^2*b*(a + b*x^2)^(3/2))","C",1
593,1,60,77,0.0372911,"\int \frac{A+B x^2}{x^2 \left(a+b x^2\right)^{5/2}} \, dx","Integrate[(A + B*x^2)/(x^2*(a + b*x^2)^(5/2)),x]","\frac{-3 a^2 \left(A-B x^2\right)+2 a b x^2 \left(B x^2-6 A\right)-8 A b^2 x^4}{3 a^3 x \left(a+b x^2\right)^{3/2}}","-\frac{2 x (4 A b-a B)}{3 a^3 \sqrt{a+b x^2}}-\frac{x (4 A b-a B)}{3 a^2 \left(a+b x^2\right)^{3/2}}-\frac{A}{a x \left(a+b x^2\right)^{3/2}}",1,"(-8*A*b^2*x^4 - 3*a^2*(A - B*x^2) + 2*a*b*x^2*(-6*A + B*x^2))/(3*a^3*x*(a + b*x^2)^(3/2))","A",1
594,1,57,113,0.0387262,"\int \frac{A+B x^2}{x^3 \left(a+b x^2\right)^{5/2}} \, dx","Integrate[(A + B*x^2)/(x^3*(a + b*x^2)^(5/2)),x]","\frac{x^2 (2 a B-5 A b) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b x^2}{a}+1\right)-3 a A}{6 a^2 x^2 \left(a+b x^2\right)^{3/2}}","\frac{(5 A b-2 a B) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{2 a^{7/2}}-\frac{5 A b-2 a B}{2 a^3 \sqrt{a+b x^2}}+\frac{2 a B-5 A b}{6 a^2 \left(a+b x^2\right)^{3/2}}-\frac{A}{2 a x^2 \left(a+b x^2\right)^{3/2}}",1,"(-3*a*A + (-5*A*b + 2*a*B)*x^2*Hypergeometric2F1[-3/2, 1, -1/2, 1 + (b*x^2)/a])/(6*a^2*x^2*(a + b*x^2)^(3/2))","C",1
595,1,79,108,0.0246483,"\int \frac{A+B x^2}{x^4 \left(a+b x^2\right)^{5/2}} \, dx","Integrate[(A + B*x^2)/(x^4*(a + b*x^2)^(5/2)),x]","\frac{-a^3 \left(A+3 B x^2\right)+6 a^2 b x^2 \left(A-2 B x^2\right)-8 a b^2 x^4 \left(B x^2-3 A\right)+16 A b^3 x^6}{3 a^4 x^3 \left(a+b x^2\right)^{3/2}}","\frac{8 b x (2 A b-a B)}{3 a^4 \sqrt{a+b x^2}}+\frac{4 b x (2 A b-a B)}{3 a^3 \left(a+b x^2\right)^{3/2}}+\frac{2 A b-a B}{a^2 x \left(a+b x^2\right)^{3/2}}-\frac{A}{3 a x^3 \left(a+b x^2\right)^{3/2}}",1,"(16*A*b^3*x^6 + 6*a^2*b*x^2*(A - 2*B*x^2) - 8*a*b^2*x^4*(-3*A + B*x^2) - a^3*(A + 3*B*x^2))/(3*a^4*x^3*(a + b*x^2)^(3/2))","A",1
596,1,60,146,0.0220665,"\int \frac{A+B x^2}{x^5 \left(a+b x^2\right)^{5/2}} \, dx","Integrate[(A + B*x^2)/(x^5*(a + b*x^2)^(5/2)),x]","\frac{b x^4 (7 A b-4 a B) \, _2F_1\left(-\frac{3}{2},2;-\frac{1}{2};\frac{b x^2}{a}+1\right)-3 a^2 A}{12 a^3 x^4 \left(a+b x^2\right)^{3/2}}","-\frac{5 b (7 A b-4 a B) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{8 a^{9/2}}+\frac{5 b (7 A b-4 a B)}{8 a^4 \sqrt{a+b x^2}}+\frac{5 b (7 A b-4 a B)}{24 a^3 \left(a+b x^2\right)^{3/2}}+\frac{7 A b-4 a B}{8 a^2 x^2 \left(a+b x^2\right)^{3/2}}-\frac{A}{4 a x^4 \left(a+b x^2\right)^{3/2}}",1,"(-3*a^2*A + b*(7*A*b - 4*a*B)*x^4*Hypergeometric2F1[-3/2, 2, -1/2, 1 + (b*x^2)/a])/(12*a^3*x^4*(a + b*x^2)^(3/2))","C",1
597,1,72,146,0.0429038,"\int \frac{A+B x^2}{x^6 \left(a+b x^2\right)^{5/2}} \, dx","Integrate[(A + B*x^2)/(x^6*(a + b*x^2)^(5/2)),x]","\frac{a x^2 \left(a^3-6 a^2 b x^2-24 a b^2 x^4-16 b^3 x^6\right) (8 A b-5 a B)-3 a^5 A}{15 a^6 x^5 \left(a+b x^2\right)^{3/2}}","-\frac{16 b^2 x (8 A b-5 a B)}{15 a^5 \sqrt{a+b x^2}}-\frac{8 b^2 x (8 A b-5 a B)}{15 a^4 \left(a+b x^2\right)^{3/2}}-\frac{2 b (8 A b-5 a B)}{5 a^3 x \left(a+b x^2\right)^{3/2}}+\frac{8 A b-5 a B}{15 a^2 x^3 \left(a+b x^2\right)^{3/2}}-\frac{A}{5 a x^5 \left(a+b x^2\right)^{3/2}}",1,"(-3*a^5*A + a*(8*A*b - 5*a*B)*x^2*(a^3 - 6*a^2*b*x^2 - 24*a*b^2*x^4 - 16*b^3*x^6))/(15*a^6*x^5*(a + b*x^2)^(3/2))","A",1
598,1,132,157,0.1322237,"\int x^5 \left(a+b x^2\right)^2 \sqrt{c+d x^2} \, dx","Integrate[x^5*(a + b*x^2)^2*Sqrt[c + d*x^2],x]","\frac{\left(c+d x^2\right)^{3/2} \left(33 a^2 d^2 \left(8 c^2-12 c d x^2+15 d^2 x^4\right)+22 a b d \left(-16 c^3+24 c^2 d x^2-30 c d^2 x^4+35 d^3 x^6\right)+b^2 \left(128 c^4-192 c^3 d x^2+240 c^2 d^2 x^4-280 c d^3 x^6+315 d^4 x^8\right)\right)}{3465 d^5}","\frac{\left(c+d x^2\right)^{7/2} \left(a^2 d^2-6 a b c d+6 b^2 c^2\right)}{7 d^5}+\frac{c^2 \left(c+d x^2\right)^{3/2} (b c-a d)^2}{3 d^5}-\frac{2 b \left(c+d x^2\right)^{9/2} (2 b c-a d)}{9 d^5}-\frac{2 c \left(c+d x^2\right)^{5/2} (b c-a d) (2 b c-a d)}{5 d^5}+\frac{b^2 \left(c+d x^2\right)^{11/2}}{11 d^5}",1,"((c + d*x^2)^(3/2)*(33*a^2*d^2*(8*c^2 - 12*c*d*x^2 + 15*d^2*x^4) + 22*a*b*d*(-16*c^3 + 24*c^2*d*x^2 - 30*c*d^2*x^4 + 35*d^3*x^6) + b^2*(128*c^4 - 192*c^3*d*x^2 + 240*c^2*d^2*x^4 - 280*c*d^3*x^6 + 315*d^4*x^8)))/(3465*d^5)","A",1
599,1,99,114,0.1116469,"\int x^3 \left(a+b x^2\right)^2 \sqrt{c+d x^2} \, dx","Integrate[x^3*(a + b*x^2)^2*Sqrt[c + d*x^2],x]","\frac{\left(c+d x^2\right)^{3/2} \left(21 a^2 d^2 \left(3 d x^2-2 c\right)+6 a b d \left(8 c^2-12 c d x^2+15 d^2 x^4\right)+b^2 \left(-16 c^3+24 c^2 d x^2-30 c d^2 x^4+35 d^3 x^6\right)\right)}{315 d^4}","-\frac{b \left(c+d x^2\right)^{7/2} (3 b c-2 a d)}{7 d^4}+\frac{\left(c+d x^2\right)^{5/2} (b c-a d) (3 b c-a d)}{5 d^4}-\frac{c \left(c+d x^2\right)^{3/2} (b c-a d)^2}{3 d^4}+\frac{b^2 \left(c+d x^2\right)^{9/2}}{9 d^4}",1,"((c + d*x^2)^(3/2)*(21*a^2*d^2*(-2*c + 3*d*x^2) + 6*a*b*d*(8*c^2 - 12*c*d*x^2 + 15*d^2*x^4) + b^2*(-16*c^3 + 24*c^2*d*x^2 - 30*c*d^2*x^4 + 35*d^3*x^6)))/(315*d^4)","A",1
600,1,67,77,0.0661584,"\int x \left(a+b x^2\right)^2 \sqrt{c+d x^2} \, dx","Integrate[x*(a + b*x^2)^2*Sqrt[c + d*x^2],x]","\frac{\left(c+d x^2\right)^{3/2} \left(35 a^2 d^2+14 a b d \left(3 d x^2-2 c\right)+b^2 \left(8 c^2-12 c d x^2+15 d^2 x^4\right)\right)}{105 d^3}","-\frac{2 b \left(c+d x^2\right)^{5/2} (b c-a d)}{5 d^3}+\frac{\left(c+d x^2\right)^{3/2} (b c-a d)^2}{3 d^3}+\frac{b^2 \left(c+d x^2\right)^{7/2}}{7 d^3}",1,"((c + d*x^2)^(3/2)*(35*a^2*d^2 + 14*a*b*d*(-2*c + 3*d*x^2) + b^2*(8*c^2 - 12*c*d*x^2 + 15*d^2*x^4)))/(105*d^3)","A",1
601,1,93,92,0.1373728,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{x} \, dx","Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/x,x]","a^2 \sqrt{c+d x^2}-a^2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)+\frac{b \left(c+d x^2\right)^{3/2} (2 a d-b c)}{3 d^2}+\frac{b^2 \left(c+d x^2\right)^{5/2}}{5 d^2}","a^2 \sqrt{c+d x^2}-a^2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)-\frac{b \left(c+d x^2\right)^{3/2} (b c-2 a d)}{3 d^2}+\frac{b^2 \left(c+d x^2\right)^{5/2}}{5 d^2}",1,"a^2*Sqrt[c + d*x^2] + (b*(-(b*c) + 2*a*d)*(c + d*x^2)^(3/2))/(3*d^2) + (b^2*(c + d*x^2)^(5/2))/(5*d^2) - a^2*Sqrt[c]*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]]","A",1
602,1,87,109,0.1135089,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{x^3} \, dx","Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^3,x]","\frac{1}{6} \left(\frac{\sqrt{c+d x^2} \left(-3 a^2 d+12 a b d x^2+2 b^2 x^2 \left(c+d x^2\right)\right)}{d x^2}-\frac{3 a (a d+4 b c) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{\sqrt{c}}\right)","-\frac{a^2 \left(c+d x^2\right)^{3/2}}{2 c x^2}+\frac{a \sqrt{c+d x^2} (a d+4 b c)}{2 c}-\frac{a (a d+4 b c) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{2 \sqrt{c}}+\frac{b^2 \left(c+d x^2\right)^{3/2}}{3 d}",1,"((Sqrt[c + d*x^2]*(-3*a^2*d + 12*a*b*d*x^2 + 2*b^2*x^2*(c + d*x^2)))/(d*x^2) - (3*a*(4*b*c + a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/Sqrt[c])/6","A",1
603,1,104,143,0.1207627,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{x^5} \, dx","Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^5,x]","\frac{\sqrt{c+d x^2} \left(-a^2 \left(2 c+d x^2\right)-8 a b c x^2+8 b^2 c x^4\right)}{8 c x^4}-\frac{\left(-a^2 d^2+8 a b c d+8 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{8 c^{3/2}}","-\frac{a^2 \left(c+d x^2\right)^{3/2}}{4 c x^4}+\frac{\sqrt{c+d x^2} \left(a d (8 b c-a d)+8 b^2 c^2\right)}{8 c^2}-\frac{\left(a d (8 b c-a d)+8 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{8 c^{3/2}}-\frac{a \left(c+d x^2\right)^{3/2} (8 b c-a d)}{8 c^2 x^2}",1,"(Sqrt[c + d*x^2]*(-8*a*b*c*x^2 + 8*b^2*c*x^4 - a^2*(2*c + d*x^2)))/(8*c*x^4) - ((8*b^2*c^2 + 8*a*b*c*d - a^2*d^2)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(8*c^(3/2))","A",1
604,1,142,149,0.1736338,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{x^7} \, dx","Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^7,x]","\frac{-3 d x^6 \sqrt{\frac{d x^2}{c}+1} \left(a^2 d^2-4 a b c d+8 b^2 c^2\right) \tanh ^{-1}\left(\sqrt{\frac{d x^2}{c}+1}\right)-\left(c+d x^2\right) \left(a^2 \left(8 c^2+2 c d x^2-3 d^2 x^4\right)+12 a b c x^2 \left(2 c+d x^2\right)+24 b^2 c^2 x^4\right)}{48 c^2 x^6 \sqrt{c+d x^2}}","-\frac{\sqrt{c+d x^2} \left(a^2 d^2-4 a b c d+8 b^2 c^2\right)}{16 c^2 x^2}-\frac{d \left(a^2 d^2-4 a b c d+8 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{16 c^{5/2}}-\frac{a^2 \left(c+d x^2\right)^{3/2}}{6 c x^6}-\frac{a \left(c+d x^2\right)^{3/2} (4 b c-a d)}{8 c^2 x^4}",1,"(-((c + d*x^2)*(24*b^2*c^2*x^4 + 12*a*b*c*x^2*(2*c + d*x^2) + a^2*(8*c^2 + 2*c*d*x^2 - 3*d^2*x^4))) - 3*d*(8*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*x^6*Sqrt[1 + (d*x^2)/c]*ArcTanh[Sqrt[1 + (d*x^2)/c]])/(48*c^2*x^6*Sqrt[c + d*x^2])","A",1
605,1,157,191,0.1621668,"\int x^2 \left(a+b x^2\right)^2 \sqrt{c+d x^2} \, dx","Integrate[x^2*(a + b*x^2)^2*Sqrt[c + d*x^2],x]","\frac{\sqrt{d} x \sqrt{c+d x^2} \left(48 a^2 d^2 \left(c+2 d x^2\right)+16 a b d \left(-3 c^2+2 c d x^2+8 d^2 x^4\right)+b^2 \left(15 c^3-10 c^2 d x^2+8 c d^2 x^4+48 d^3 x^6\right)\right)-3 c^2 \left(16 a^2 d^2-16 a b c d+5 b^2 c^2\right) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{384 d^{7/2}}","-\frac{c^2 \left(16 a^2 d^2+b c (5 b c-16 a d)\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{128 d^{7/2}}+\frac{x^3 \sqrt{c+d x^2} \left(16 a^2 d^2+b c (5 b c-16 a d)\right)}{64 d^2}+\frac{c x \sqrt{c+d x^2} \left(16 a^2 d^2+b c (5 b c-16 a d)\right)}{128 d^3}-\frac{b x^3 \left(c+d x^2\right)^{3/2} (5 b c-16 a d)}{48 d^2}+\frac{b^2 x^5 \left(c+d x^2\right)^{3/2}}{8 d}",1,"(Sqrt[d]*x*Sqrt[c + d*x^2]*(48*a^2*d^2*(c + 2*d*x^2) + 16*a*b*d*(-3*c^2 + 2*c*d*x^2 + 8*d^2*x^4) + b^2*(15*c^3 - 10*c^2*d*x^2 + 8*c*d^2*x^4 + 48*d^3*x^6)) - 3*c^2*(5*b^2*c^2 - 16*a*b*c*d + 16*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(384*d^(7/2))","A",1
606,1,122,149,0.1322627,"\int \left(a+b x^2\right)^2 \sqrt{c+d x^2} \, dx","Integrate[(a + b*x^2)^2*Sqrt[c + d*x^2],x]","\frac{3 c \left(8 a^2 d^2-4 a b c d+b^2 c^2\right) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)+\sqrt{d} x \sqrt{c+d x^2} \left(24 a^2 d^2+12 a b d \left(c+2 d x^2\right)+b^2 \left(-3 c^2+2 c d x^2+8 d^2 x^4\right)\right)}{48 d^{5/2}}","\frac{x \sqrt{c+d x^2} \left(8 a^2 d^2-4 a b c d+b^2 c^2\right)}{16 d^2}+\frac{c \left(8 a^2 d^2-4 a b c d+b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{16 d^{5/2}}-\frac{b x \left(c+d x^2\right)^{3/2} (3 b c-8 a d)}{24 d^2}+\frac{b x \left(a+b x^2\right) \left(c+d x^2\right)^{3/2}}{6 d}",1,"(Sqrt[d]*x*Sqrt[c + d*x^2]*(24*a^2*d^2 + 12*a*b*d*(c + 2*d*x^2) + b^2*(-3*c^2 + 2*c*d*x^2 + 8*d^2*x^4)) + 3*c*(b^2*c^2 - 4*a*b*c*d + 8*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(48*d^(5/2))","A",1
607,1,99,133,0.1760325,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{x^2} \, dx","Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^2,x]","\frac{\left(8 a^2 d^2+8 a b c d-b^2 c^2\right) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{8 d^{3/2}}+\sqrt{c+d x^2} \left(-\frac{a^2}{x}+a b x+\frac{b^2 x \left(c+2 d x^2\right)}{8 d}\right)","-\frac{a^2 \left(c+d x^2\right)^{3/2}}{c x}-\frac{\left(b^2 c^2-8 a d (a d+b c)\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{8 d^{3/2}}-\frac{x \sqrt{c+d x^2} \left(b^2 c^2-8 a d (a d+b c)\right)}{8 c d}+\frac{b^2 x \left(c+d x^2\right)^{3/2}}{4 d}",1,"Sqrt[c + d*x^2]*(-(a^2/x) + a*b*x + (b^2*x*(c + 2*d*x^2))/(8*d)) + ((-(b^2*c^2) + 8*a*b*c*d + 8*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(8*d^(3/2))","A",1
608,1,91,111,0.1742914,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{x^4} \, dx","Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^4,x]","\sqrt{c+d x^2} \left(-\frac{a^2}{3 x^3}-\frac{a (a d+6 b c)}{3 c x}+\frac{b^2 x}{2}\right)+\frac{b (4 a d+b c) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{2 \sqrt{d}}","-\frac{a^2 \left(c+d x^2\right)^{3/2}}{3 c x^3}-\frac{2 a b \left(c+d x^2\right)^{3/2}}{c x}+\frac{b x \sqrt{c+d x^2} (4 a d+b c)}{2 c}+\frac{b (4 a d+b c) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{2 \sqrt{d}}",1,"(-1/3*a^2/x^3 - (a*(6*b*c + a*d))/(3*c*x) + (b^2*x)/2)*Sqrt[c + d*x^2] + (b*(b*c + 4*a*d)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(2*Sqrt[d])","A",1
609,1,104,103,0.1473949,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{x^6} \, dx","Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^6,x]","b^2 \sqrt{d} \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)-\frac{\sqrt{c+d x^2} \left(a^2 \left(3 c^2+c d x^2-2 d^2 x^4\right)+10 a b c x^2 \left(c+d x^2\right)+15 b^2 c^2 x^4\right)}{15 c^2 x^5}","-\frac{a^2 \left(c+d x^2\right)^{3/2}}{5 c x^5}-\frac{2 a \left(c+d x^2\right)^{3/2} (5 b c-a d)}{15 c^2 x^3}-\frac{b^2 \sqrt{c+d x^2}}{x}+b^2 \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)",1,"-1/15*(Sqrt[c + d*x^2]*(15*b^2*c^2*x^4 + 10*a*b*c*x^2*(c + d*x^2) + a^2*(3*c^2 + c*d*x^2 - 2*d^2*x^4)))/(c^2*x^5) + b^2*Sqrt[d]*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]]","A",1
610,1,76,99,0.0523183,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{x^8} \, dx","Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^8,x]","-\frac{\left(c+d x^2\right)^{3/2} \left(a^2 \left(15 c^2-12 c d x^2+8 d^2 x^4\right)+14 a b c x^2 \left(3 c-2 d x^2\right)+35 b^2 c^2 x^4\right)}{105 c^3 x^7}","-\frac{a^2 \left(c+d x^2\right)^{3/2}}{7 c x^7}-\frac{\left(c+d x^2\right)^{3/2} \left(35 b^2 c^2-4 a d (7 b c-2 a d)\right)}{105 c^3 x^3}-\frac{2 a \left(c+d x^2\right)^{3/2} (7 b c-2 a d)}{35 c^2 x^5}",1,"-1/105*((c + d*x^2)^(3/2)*(35*b^2*c^2*x^4 + 14*a*b*c*x^2*(3*c - 2*d*x^2) + a^2*(15*c^2 - 12*c*d*x^2 + 8*d^2*x^4)))/(c^3*x^7)","A",1
611,1,108,143,0.1476885,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{x^{10}} \, dx","Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^10,x]","-\frac{\left(c+d x^2\right)^{3/2} \left(a^2 \left(35 c^3-30 c^2 d x^2+24 c d^2 x^4-16 d^3 x^6\right)+6 a b c x^2 \left(15 c^2-12 c d x^2+8 d^2 x^4\right)+21 b^2 c^2 x^4 \left(3 c-2 d x^2\right)\right)}{315 c^4 x^9}","-\frac{a^2 \left(c+d x^2\right)^{3/2}}{9 c x^9}+\frac{2 d \left(c+d x^2\right)^{3/2} \left(21 b^2 c^2-8 a d (3 b c-a d)\right)}{315 c^4 x^3}-\frac{\left(c+d x^2\right)^{3/2} \left(21 b^2 c^2-8 a d (3 b c-a d)\right)}{105 c^3 x^5}-\frac{2 a \left(c+d x^2\right)^{3/2} (3 b c-a d)}{21 c^2 x^7}",1,"-1/315*((c + d*x^2)^(3/2)*(21*b^2*c^2*x^4*(3*c - 2*d*x^2) + 6*a*b*c*x^2*(15*c^2 - 12*c*d*x^2 + 8*d^2*x^4) + a^2*(35*c^3 - 30*c^2*d*x^2 + 24*c*d^2*x^4 - 16*d^3*x^6)))/(c^4*x^9)","A",1
612,1,141,189,0.1382763,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{x^{12}} \, dx","Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^12,x]","-\frac{\left(c+d x^2\right)^{3/2} \left(a^2 \left(315 c^4-280 c^3 d x^2+240 c^2 d^2 x^4-192 c d^3 x^6+128 d^4 x^8\right)+22 a b c x^2 \left(35 c^3-30 c^2 d x^2+24 c d^2 x^4-16 d^3 x^6\right)+33 b^2 c^2 x^4 \left(15 c^2-12 c d x^2+8 d^2 x^4\right)\right)}{3465 c^5 x^{11}}","-\frac{a^2 \left(c+d x^2\right)^{3/2}}{11 c x^{11}}-\frac{8 d^2 \left(c+d x^2\right)^{3/2} \left(33 b^2 c^2-4 a d (11 b c-4 a d)\right)}{3465 c^5 x^3}+\frac{4 d \left(c+d x^2\right)^{3/2} \left(33 b^2 c^2-4 a d (11 b c-4 a d)\right)}{1155 c^4 x^5}-\frac{\left(c+d x^2\right)^{3/2} \left(33 b^2 c^2-4 a d (11 b c-4 a d)\right)}{231 c^3 x^7}-\frac{2 a \left(c+d x^2\right)^{3/2} (11 b c-4 a d)}{99 c^2 x^9}",1,"-1/3465*((c + d*x^2)^(3/2)*(33*b^2*c^2*x^4*(15*c^2 - 12*c*d*x^2 + 8*d^2*x^4) + 22*a*b*c*x^2*(35*c^3 - 30*c^2*d*x^2 + 24*c*d^2*x^4 - 16*d^3*x^6) + a^2*(315*c^4 - 280*c^3*d*x^2 + 240*c^2*d^2*x^4 - 192*c*d^3*x^6 + 128*d^4*x^8)))/(c^5*x^11)","A",1
613,1,225,281,0.2124681,"\int x^4 \left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2} \, dx","Integrate[x^4*(a + b*x^2)^2*(c + d*x^2)^(3/2),x]","\frac{15 c^4 \left(24 a^2 d^2-24 a b c d+7 b^2 c^2\right) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)+\sqrt{d} x \sqrt{c+d x^2} \left(120 a^2 d^2 \left(-3 c^3+2 c^2 d x^2+24 c d^2 x^4+16 d^3 x^6\right)+24 a b d \left(15 c^4-10 c^3 d x^2+8 c^2 d^2 x^4+176 c d^3 x^6+128 d^4 x^8\right)+b^2 \left(-105 c^5+70 c^4 d x^2-56 c^3 d^2 x^4+48 c^2 d^3 x^6+1664 c d^4 x^8+1280 d^5 x^{10}\right)\right)}{15360 d^{9/2}}","\frac{c^4 \left(24 a^2 d^2+b c (7 b c-24 a d)\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{1024 d^{9/2}}-\frac{c^3 x \sqrt{c+d x^2} \left(24 a^2 d^2+b c (7 b c-24 a d)\right)}{1024 d^4}+\frac{c^2 x^3 \sqrt{c+d x^2} \left(24 a^2 d^2+b c (7 b c-24 a d)\right)}{1536 d^3}+\frac{x^5 \left(c+d x^2\right)^{3/2} \left(24 a^2 d^2+b c (7 b c-24 a d)\right)}{192 d^2}+\frac{c x^5 \sqrt{c+d x^2} \left(24 a^2 d^2+b c (7 b c-24 a d)\right)}{384 d^2}-\frac{b x^5 \left(c+d x^2\right)^{5/2} (7 b c-24 a d)}{120 d^2}+\frac{b^2 x^7 \left(c+d x^2\right)^{5/2}}{12 d}",1,"(Sqrt[d]*x*Sqrt[c + d*x^2]*(120*a^2*d^2*(-3*c^3 + 2*c^2*d*x^2 + 24*c*d^2*x^4 + 16*d^3*x^6) + 24*a*b*d*(15*c^4 - 10*c^3*d*x^2 + 8*c^2*d^2*x^4 + 176*c*d^3*x^6 + 128*d^4*x^8) + b^2*(-105*c^5 + 70*c^4*d*x^2 - 56*c^3*d^2*x^4 + 48*c^2*d^3*x^6 + 1664*c*d^4*x^8 + 1280*d^5*x^10)) + 15*c^4*(7*b^2*c^2 - 24*a*b*c*d + 24*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(15360*d^(9/2))","A",1
614,1,100,114,0.1660046,"\int x^3 \left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2} \, dx","Integrate[x^3*(a + b*x^2)^2*(c + d*x^2)^(3/2),x]","\frac{\left(c+d x^2\right)^{5/2} \left(99 a^2 d^2 \left(5 d x^2-2 c\right)+22 a b d \left(8 c^2-20 c d x^2+35 d^2 x^4\right)-3 b^2 \left(16 c^3-40 c^2 d x^2+70 c d^2 x^4-105 d^3 x^6\right)\right)}{3465 d^4}","-\frac{b \left(c+d x^2\right)^{9/2} (3 b c-2 a d)}{9 d^4}+\frac{\left(c+d x^2\right)^{7/2} (b c-a d) (3 b c-a d)}{7 d^4}-\frac{c \left(c+d x^2\right)^{5/2} (b c-a d)^2}{5 d^4}+\frac{b^2 \left(c+d x^2\right)^{11/2}}{11 d^4}",1,"((c + d*x^2)^(5/2)*(99*a^2*d^2*(-2*c + 5*d*x^2) + 22*a*b*d*(8*c^2 - 20*c*d*x^2 + 35*d^2*x^4) - 3*b^2*(16*c^3 - 40*c^2*d*x^2 + 70*c*d^2*x^4 - 105*d^3*x^6)))/(3465*d^4)","A",1
615,1,193,235,0.1780899,"\int x^2 \left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2} \, dx","Integrate[x^2*(a + b*x^2)^2*(c + d*x^2)^(3/2),x]","\frac{\sqrt{d} x \sqrt{c+d x^2} \left(80 a^2 d^2 \left(3 c^2+14 c d x^2+8 d^2 x^4\right)+60 a b d \left(-3 c^3+2 c^2 d x^2+24 c d^2 x^4+16 d^3 x^6\right)+3 b^2 \left(15 c^4-10 c^3 d x^2+8 c^2 d^2 x^4+176 c d^3 x^6+128 d^4 x^8\right)\right)-15 c^3 \left(16 a^2 d^2-12 a b c d+3 b^2 c^2\right) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{3840 d^{7/2}}","-\frac{c^3 \left(16 a^2 d^2+3 b c (b c-4 a d)\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{256 d^{7/2}}+\frac{c^2 x \sqrt{c+d x^2} \left(16 a^2 d^2+3 b c (b c-4 a d)\right)}{256 d^3}+\frac{x^3 \left(c+d x^2\right)^{3/2} \left(16 a^2 d^2+3 b c (b c-4 a d)\right)}{96 d^2}+\frac{c x^3 \sqrt{c+d x^2} \left(16 a^2 d^2+3 b c (b c-4 a d)\right)}{128 d^2}-\frac{b x^3 \left(c+d x^2\right)^{5/2} (b c-4 a d)}{16 d^2}+\frac{b^2 x^5 \left(c+d x^2\right)^{5/2}}{10 d}",1,"(Sqrt[d]*x*Sqrt[c + d*x^2]*(80*a^2*d^2*(3*c^2 + 14*c*d*x^2 + 8*d^2*x^4) + 60*a*b*d*(-3*c^3 + 2*c^2*d*x^2 + 24*c*d^2*x^4 + 16*d^3*x^6) + 3*b^2*(15*c^4 - 10*c^3*d*x^2 + 8*c^2*d^2*x^4 + 176*c*d^3*x^6 + 128*d^4*x^8)) - 15*c^3*(3*b^2*c^2 - 12*a*b*c*d + 16*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(3840*d^(7/2))","A",1
616,1,67,77,0.0472806,"\int x \left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2} \, dx","Integrate[x*(a + b*x^2)^2*(c + d*x^2)^(3/2),x]","\frac{\left(c+d x^2\right)^{5/2} \left(63 a^2 d^2+18 a b d \left(5 d x^2-2 c\right)+b^2 \left(8 c^2-20 c d x^2+35 d^2 x^4\right)\right)}{315 d^3}","-\frac{2 b \left(c+d x^2\right)^{7/2} (b c-a d)}{7 d^3}+\frac{\left(c+d x^2\right)^{5/2} (b c-a d)^2}{5 d^3}+\frac{b^2 \left(c+d x^2\right)^{9/2}}{9 d^3}",1,"((c + d*x^2)^(5/2)*(63*a^2*d^2 + 18*a*b*d*(-2*c + 5*d*x^2) + b^2*(8*c^2 - 20*c*d*x^2 + 35*d^2*x^4)))/(315*d^3)","A",1
617,1,159,196,0.0953419,"\int \left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2} \, dx","Integrate[(a + b*x^2)^2*(c + d*x^2)^(3/2),x]","\frac{3 c^2 \left(48 a^2 d^2-16 a b c d+3 b^2 c^2\right) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)+\sqrt{d} x \sqrt{c+d x^2} \left(48 a^2 d^2 \left(5 c+2 d x^2\right)+16 a b d \left(3 c^2+14 c d x^2+8 d^2 x^4\right)+b^2 \left(-9 c^3+6 c^2 d x^2+72 c d^2 x^4+48 d^3 x^6\right)\right)}{384 d^{5/2}}","\frac{x \left(c+d x^2\right)^{3/2} \left(48 a^2 d^2-16 a b c d+3 b^2 c^2\right)}{192 d^2}+\frac{c x \sqrt{c+d x^2} \left(48 a^2 d^2-16 a b c d+3 b^2 c^2\right)}{128 d^2}+\frac{c^2 \left(48 a^2 d^2-16 a b c d+3 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{128 d^{5/2}}-\frac{b x \left(c+d x^2\right)^{5/2} (3 b c-10 a d)}{48 d^2}+\frac{b x \left(a+b x^2\right) \left(c+d x^2\right)^{5/2}}{8 d}",1,"(Sqrt[d]*x*Sqrt[c + d*x^2]*(48*a^2*d^2*(5*c + 2*d*x^2) + 16*a*b*d*(3*c^2 + 14*c*d*x^2 + 8*d^2*x^4) + b^2*(-9*c^3 + 6*c^2*d*x^2 + 72*c*d^2*x^4 + 48*d^3*x^6)) + 3*c^2*(3*b^2*c^2 - 16*a*b*c*d + 48*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(384*d^(5/2))","A",1
618,1,110,111,0.0767197,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}}{x} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^(3/2))/x,x]","\frac{1}{3} a^2 \left(c+d x^2\right)^{3/2}+a^2 c \left(\sqrt{c+d x^2}-\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)\right)+\frac{b \left(c+d x^2\right)^{5/2} (2 a d-b c)}{5 d^2}+\frac{b^2 \left(c+d x^2\right)^{7/2}}{7 d^2}","-a^2 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)+\frac{1}{3} a^2 \left(c+d x^2\right)^{3/2}+a^2 c \sqrt{c+d x^2}-\frac{b \left(c+d x^2\right)^{5/2} (b c-2 a d)}{5 d^2}+\frac{b^2 \left(c+d x^2\right)^{7/2}}{7 d^2}",1,"(a^2*(c + d*x^2)^(3/2))/3 + (b*(-(b*c) + 2*a*d)*(c + d*x^2)^(5/2))/(5*d^2) + (b^2*(c + d*x^2)^(7/2))/(7*d^2) + a^2*c*(Sqrt[c + d*x^2] - Sqrt[c]*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])","A",1
619,1,135,175,0.1486583,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}}{x^2} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^2,x]","\sqrt{c+d x^2} \left(\frac{x \left(8 a^2 d^2+20 a b c d+b^2 c^2\right)}{16 d}-\frac{a^2 c}{x}+\frac{1}{24} b x^3 (12 a d+7 b c)+\frac{1}{6} b^2 d x^5\right)-\frac{c \left(-24 a^2 d^2-12 a b c d+b^2 c^2\right) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{16 d^{3/2}}","-\frac{a^2 \left(c+d x^2\right)^{5/2}}{c x}-\frac{c \left(b^2 c^2-12 a d (2 a d+b c)\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{16 d^{3/2}}-\frac{x \left(c+d x^2\right)^{3/2} \left(b^2 c^2-12 a d (2 a d+b c)\right)}{24 c d}-\frac{x \sqrt{c+d x^2} \left(b^2 c^2-12 a d (2 a d+b c)\right)}{16 d}+\frac{b^2 x \left(c+d x^2\right)^{5/2}}{6 d}",1,"Sqrt[c + d*x^2]*(-((a^2*c)/x) + ((b^2*c^2 + 20*a*b*c*d + 8*a^2*d^2)*x)/(16*d) + (b*(7*b*c + 12*a*d)*x^3)/24 + (b^2*d*x^5)/6) - (c*(b^2*c^2 - 12*a*b*c*d - 24*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(16*d^(3/2))","A",1
620,1,108,136,0.0936284,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}}{x^3} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^3,x]","\frac{\sqrt{c+d x^2} \left(-15 a^2 d \left(c-2 d x^2\right)+20 a b d x^2 \left(4 c+d x^2\right)+6 b^2 x^2 \left(c+d x^2\right)^2\right)}{30 d x^2}-\frac{1}{2} a \sqrt{c} (3 a d+4 b c) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)","-\frac{a^2 \left(c+d x^2\right)^{5/2}}{2 c x^2}+\frac{a \left(c+d x^2\right)^{3/2} (3 a d+4 b c)}{6 c}+\frac{1}{2} a \sqrt{c+d x^2} (3 a d+4 b c)-\frac{1}{2} a \sqrt{c} (3 a d+4 b c) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)+\frac{b^2 \left(c+d x^2\right)^{5/2}}{5 d}",1,"(Sqrt[c + d*x^2]*(-15*a^2*d*(c - 2*d*x^2) + 6*b^2*x^2*(c + d*x^2)^2 + 20*a*b*d*x^2*(4*c + d*x^2)))/(30*d*x^2) - (a*Sqrt[c]*(4*b*c + 3*a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/2","A",1
621,1,118,184,0.104052,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}}{x^4} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^4,x]","\frac{1}{24} \left(\frac{3 \left(8 a^2 d^2+24 a b c d+3 b^2 c^2\right) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{\sqrt{d}}+\frac{\sqrt{c+d x^2} \left(-8 a^2 c+3 b x^4 (8 a d+5 b c)-16 a x^2 (2 a d+3 b c)+6 b^2 d x^6\right)}{x^3}\right)","-\frac{a^2 \left(c+d x^2\right)^{5/2}}{3 c x^3}+\frac{x \left(c+d x^2\right)^{3/2} \left(8 a d (a d+3 b c)+3 b^2 c^2\right)}{12 c^2}+\frac{x \sqrt{c+d x^2} \left(8 a d (a d+3 b c)+3 b^2 c^2\right)}{8 c}+\frac{\left(8 a d (a d+3 b c)+3 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{8 \sqrt{d}}-\frac{2 a \left(c+d x^2\right)^{5/2} (a d+3 b c)}{3 c^2 x}",1,"((Sqrt[c + d*x^2]*(-8*a^2*c - 16*a*(3*b*c + 2*a*d)*x^2 + 3*b*(5*b*c + 8*a*d)*x^4 + 6*b^2*d*x^6))/x^3 + (3*(3*b^2*c^2 + 24*a*b*c*d + 8*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/Sqrt[d])/24","A",1
622,1,116,181,0.1123805,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}}{x^5} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^5,x]","\frac{1}{24} \left(\frac{\sqrt{c+d x^2} \left(-3 a^2 \left(2 c+5 d x^2\right)-24 a b x^2 \left(c-2 d x^2\right)+8 b^2 x^4 \left(4 c+d x^2\right)\right)}{x^4}-\frac{3 \left(3 a^2 d^2+24 a b c d+8 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{\sqrt{c}}\right)","-\frac{a^2 \left(c+d x^2\right)^{5/2}}{4 c x^4}+\frac{\left(c+d x^2\right)^{3/2} \left(3 a d (a d+8 b c)+8 b^2 c^2\right)}{24 c^2}+\frac{\sqrt{c+d x^2} \left(3 a d (a d+8 b c)+8 b^2 c^2\right)}{8 c}-\frac{\left(3 a d (a d+8 b c)+8 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{8 \sqrt{c}}-\frac{a \left(c+d x^2\right)^{5/2} (a d+8 b c)}{8 c^2 x^2}",1,"((Sqrt[c + d*x^2]*(-24*a*b*x^2*(c - 2*d*x^2) + 8*b^2*x^4*(4*c + d*x^2) - 3*a^2*(2*c + 5*d*x^2)))/x^4 - (3*(8*b^2*c^2 + 24*a*b*c*d + 3*a^2*d^2)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/Sqrt[c])/24","A",1
623,1,113,147,0.1110843,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}}{x^6} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^6,x]","\frac{1}{2} b \sqrt{d} (4 a d+3 b c) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)-\frac{\sqrt{c+d x^2} \left(6 a^2 \left(c+d x^2\right)^2+20 a b c x^2 \left(c+4 d x^2\right)+15 b^2 c x^4 \left(2 c-d x^2\right)\right)}{30 c x^5}","-\frac{a^2 \left(c+d x^2\right)^{5/2}}{5 c x^5}-\frac{b \left(c+d x^2\right)^{3/2} (4 a d+3 b c)}{3 c x}+\frac{b d x \sqrt{c+d x^2} (4 a d+3 b c)}{2 c}+\frac{1}{2} b \sqrt{d} (4 a d+3 b c) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)-\frac{2 a b \left(c+d x^2\right)^{5/2}}{3 c x^3}",1,"-1/30*(Sqrt[c + d*x^2]*(15*b^2*c*x^4*(2*c - d*x^2) + 6*a^2*(c + d*x^2)^2 + 20*a*b*c*x^2*(c + 4*d*x^2)))/(c*x^5) + (b*Sqrt[d]*(3*b*c + 4*a*d)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/2","A",1
624,1,92,187,0.0476753,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}}{x^7} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^7,x]","-\frac{\left(c+d x^2\right)^{5/2} \left(d x^6 \left(a^2 d^2-12 a b c d-24 b^2 c^2\right) \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};\frac{d x^2}{c}+1\right)+5 a c^2 \left(4 a c-a d x^2+12 b c x^2\right)\right)}{120 c^4 x^6}","-\frac{a^2 \left(c+d x^2\right)^{5/2}}{6 c x^6}-\frac{\left(c+d x^2\right)^{3/2} \left(a d (12 b c-a d)+24 b^2 c^2\right)}{48 c^2 x^2}+\frac{d \sqrt{c+d x^2} \left(a d (12 b c-a d)+24 b^2 c^2\right)}{16 c^2}-\frac{d \left(a d (12 b c-a d)+24 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{16 c^{3/2}}-\frac{a \left(c+d x^2\right)^{5/2} (12 b c-a d)}{24 c^2 x^4}",1,"-1/120*((c + d*x^2)^(5/2)*(5*a*c^2*(4*a*c + 12*b*c*x^2 - a*d*x^2) + d*(-24*b^2*c^2 - 12*a*b*c*d + a^2*d^2)*x^6*Hypergeometric2F1[2, 5/2, 7/2, 1 + (d*x^2)/c]))/(c^4*x^6)","C",1
625,1,99,114,0.1091572,"\int x^3 \left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2} \, dx","Integrate[x^3*(a + b*x^2)^2*(c + d*x^2)^(5/2),x]","\frac{\left(c+d x^2\right)^{7/2} \left(143 a^2 d^2 \left(7 d x^2-2 c\right)+26 a b d \left(8 c^2-28 c d x^2+63 d^2 x^4\right)+b^2 \left(-48 c^3+168 c^2 d x^2-378 c d^2 x^4+693 d^3 x^6\right)\right)}{9009 d^4}","-\frac{b \left(c+d x^2\right)^{11/2} (3 b c-2 a d)}{11 d^4}+\frac{\left(c+d x^2\right)^{9/2} (b c-a d) (3 b c-a d)}{9 d^4}-\frac{c \left(c+d x^2\right)^{7/2} (b c-a d)^2}{7 d^4}+\frac{b^2 \left(c+d x^2\right)^{13/2}}{13 d^4}",1,"((c + d*x^2)^(7/2)*(143*a^2*d^2*(-2*c + 7*d*x^2) + 26*a*b*d*(8*c^2 - 28*c*d*x^2 + 63*d^2*x^4) + b^2*(-48*c^3 + 168*c^2*d*x^2 - 378*c*d^2*x^4 + 693*d^3*x^6)))/(9009*d^4)","A",1
626,1,226,281,0.1650692,"\int x^2 \left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2} \, dx","Integrate[x^2*(a + b*x^2)^2*(c + d*x^2)^(5/2),x]","\frac{\sqrt{d} x \sqrt{c+d x^2} \left(40 a^2 d^2 \left(15 c^3+118 c^2 d x^2+136 c d^2 x^4+48 d^3 x^6\right)+24 a b d \left(-15 c^4+10 c^3 d x^2+248 c^2 d^2 x^4+336 c d^3 x^6+128 d^4 x^8\right)+5 b^2 \left(15 c^5-10 c^4 d x^2+8 c^3 d^2 x^4+432 c^2 d^3 x^6+640 c d^4 x^8+256 d^5 x^{10}\right)\right)-15 c^4 \left(40 a^2 d^2-24 a b c d+5 b^2 c^2\right) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{15360 d^{7/2}}","-\frac{c^4 \left(40 a^2 d^2+b c (5 b c-24 a d)\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{1024 d^{7/2}}+\frac{c^3 x \sqrt{c+d x^2} \left(40 a^2 d^2+b c (5 b c-24 a d)\right)}{1024 d^3}+\frac{c^2 x^3 \sqrt{c+d x^2} \left(40 a^2 d^2+b c (5 b c-24 a d)\right)}{512 d^2}+\frac{x^3 \left(c+d x^2\right)^{5/2} \left(40 a^2 d^2+b c (5 b c-24 a d)\right)}{320 d^2}+\frac{c x^3 \left(c+d x^2\right)^{3/2} \left(40 a^2 d^2+b c (5 b c-24 a d)\right)}{384 d^2}-\frac{b x^3 \left(c+d x^2\right)^{7/2} (5 b c-24 a d)}{120 d^2}+\frac{b^2 x^5 \left(c+d x^2\right)^{7/2}}{12 d}",1,"(Sqrt[d]*x*Sqrt[c + d*x^2]*(40*a^2*d^2*(15*c^3 + 118*c^2*d*x^2 + 136*c*d^2*x^4 + 48*d^3*x^6) + 24*a*b*d*(-15*c^4 + 10*c^3*d*x^2 + 248*c^2*d^2*x^4 + 336*c*d^3*x^6 + 128*d^4*x^8) + 5*b^2*(15*c^5 - 10*c^4*d*x^2 + 8*c^3*d^2*x^4 + 432*c^2*d^3*x^6 + 640*c*d^4*x^8 + 256*d^5*x^10)) - 15*c^4*(5*b^2*c^2 - 24*a*b*c*d + 40*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(15360*d^(7/2))","A",1
627,1,67,77,0.0502663,"\int x \left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2} \, dx","Integrate[x*(a + b*x^2)^2*(c + d*x^2)^(5/2),x]","\frac{\left(c+d x^2\right)^{7/2} \left(99 a^2 d^2+22 a b d \left(7 d x^2-2 c\right)+b^2 \left(8 c^2-28 c d x^2+63 d^2 x^4\right)\right)}{693 d^3}","-\frac{2 b \left(c+d x^2\right)^{9/2} (b c-a d)}{9 d^3}+\frac{\left(c+d x^2\right)^{7/2} (b c-a d)^2}{7 d^3}+\frac{b^2 \left(c+d x^2\right)^{11/2}}{11 d^3}",1,"((c + d*x^2)^(7/2)*(99*a^2*d^2 + 22*a*b*d*(-2*c + 7*d*x^2) + b^2*(8*c^2 - 28*c*d*x^2 + 63*d^2*x^4)))/(693*d^3)","A",1
628,1,192,240,0.1377288,"\int \left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2} \, dx","Integrate[(a + b*x^2)^2*(c + d*x^2)^(5/2),x]","\frac{15 c^3 \left(80 a^2 d^2-20 a b c d+3 b^2 c^2\right) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)+\sqrt{d} x \sqrt{c+d x^2} \left(80 a^2 d^2 \left(33 c^2+26 c d x^2+8 d^2 x^4\right)+20 a b d \left(15 c^3+118 c^2 d x^2+136 c d^2 x^4+48 d^3 x^6\right)+b^2 \left(-45 c^4+30 c^3 d x^2+744 c^2 d^2 x^4+1008 c d^3 x^6+384 d^4 x^8\right)\right)}{3840 d^{5/2}}","\frac{x \left(c+d x^2\right)^{5/2} \left(80 a^2 d^2-20 a b c d+3 b^2 c^2\right)}{480 d^2}+\frac{c x \left(c+d x^2\right)^{3/2} \left(80 a^2 d^2-20 a b c d+3 b^2 c^2\right)}{384 d^2}+\frac{c^2 x \sqrt{c+d x^2} \left(80 a^2 d^2-20 a b c d+3 b^2 c^2\right)}{256 d^2}+\frac{c^3 \left(80 a^2 d^2-20 a b c d+3 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{256 d^{5/2}}-\frac{3 b x \left(c+d x^2\right)^{7/2} (b c-4 a d)}{80 d^2}+\frac{b x \left(a+b x^2\right) \left(c+d x^2\right)^{7/2}}{10 d}",1,"(Sqrt[d]*x*Sqrt[c + d*x^2]*(80*a^2*d^2*(33*c^2 + 26*c*d*x^2 + 8*d^2*x^4) + 20*a*b*d*(15*c^3 + 118*c^2*d*x^2 + 136*c*d^2*x^4 + 48*d^3*x^6) + b^2*(-45*c^4 + 30*c^3*d*x^2 + 744*c^2*d^2*x^4 + 1008*c*d^3*x^6 + 384*d^4*x^8)) + 15*c^3*(3*b^2*c^2 - 20*a*b*c*d + 80*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(3840*d^(5/2))","A",1
629,1,123,132,0.1193081,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2}}{x} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^(5/2))/x,x]","\frac{1}{3} a^2 c \left(\sqrt{c+d x^2} \left(4 c+d x^2\right)-3 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)\right)+\frac{1}{5} a^2 \left(c+d x^2\right)^{5/2}+\frac{b \left(c+d x^2\right)^{7/2} (2 a d-b c)}{7 d^2}+\frac{b^2 \left(c+d x^2\right)^{9/2}}{9 d^2}","-a^2 c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)+a^2 c^2 \sqrt{c+d x^2}+\frac{1}{5} a^2 \left(c+d x^2\right)^{5/2}+\frac{1}{3} a^2 c \left(c+d x^2\right)^{3/2}-\frac{b \left(c+d x^2\right)^{7/2} (b c-2 a d)}{7 d^2}+\frac{b^2 \left(c+d x^2\right)^{9/2}}{9 d^2}",1,"(a^2*(c + d*x^2)^(5/2))/5 + (b*(-(b*c) + 2*a*d)*(c + d*x^2)^(7/2))/(7*d^2) + (b^2*(c + d*x^2)^(9/2))/(9*d^2) + (a^2*c*(Sqrt[c + d*x^2]*(4*c + d*x^2) - 3*c^(3/2)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]]))/3","A",1
630,1,174,217,0.129608,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2}}{x^2} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^(5/2))/x^2,x]","\sqrt{c+d x^2} \left(\frac{1}{192} x^3 \left(48 a^2 d^2+208 a b c d+59 b^2 c^2\right)+\frac{c x \left(144 a^2 d^2+176 a b c d+5 b^2 c^2\right)}{128 d}-\frac{a^2 c^2}{x}+\frac{1}{48} b d x^5 (16 a d+17 b c)+\frac{1}{8} b^2 d^2 x^7\right)-\frac{5 c^2 \left(-48 a^2 d^2-16 a b c d+b^2 c^2\right) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{128 d^{3/2}}","-\frac{a^2 \left(c+d x^2\right)^{7/2}}{c x}-\frac{5 c^2 \left(b^2 c^2-16 a d (3 a d+b c)\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{128 d^{3/2}}-\frac{x \left(c+d x^2\right)^{5/2} \left(b^2 c^2-16 a d (3 a d+b c)\right)}{48 c d}-\frac{5 x \left(c+d x^2\right)^{3/2} \left(b^2 c^2-16 a d (3 a d+b c)\right)}{192 d}-\frac{5 c x \sqrt{c+d x^2} \left(b^2 c^2-16 a d (3 a d+b c)\right)}{128 d}+\frac{b^2 x \left(c+d x^2\right)^{7/2}}{8 d}",1,"Sqrt[c + d*x^2]*(-((a^2*c^2)/x) + (c*(5*b^2*c^2 + 176*a*b*c*d + 144*a^2*d^2)*x)/(128*d) + ((59*b^2*c^2 + 208*a*b*c*d + 48*a^2*d^2)*x^3)/192 + (b*d*(17*b*c + 16*a*d)*x^5)/48 + (b^2*d^2*x^7)/8) - (5*c^2*(b^2*c^2 - 16*a*b*c*d - 48*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(128*d^(3/2))","A",1
631,1,122,162,0.2440918,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2}}{x^3} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^(5/2))/x^3,x]","-\frac{\frac{a^2 \left(c+d x^2\right)^{7/2}}{x^2}+\frac{1}{15} a (5 a d+4 b c) \left(15 c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)-\sqrt{c+d x^2} \left(23 c^2+11 c d x^2+3 d^2 x^4\right)\right)-\frac{2 b^2 c \left(c+d x^2\right)^{7/2}}{7 d}}{2 c}","-\frac{a^2 \left(c+d x^2\right)^{7/2}}{2 c x^2}-\frac{1}{2} a c^{3/2} (5 a d+4 b c) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)+\frac{a \left(c+d x^2\right)^{5/2} (5 a d+4 b c)}{10 c}+\frac{1}{6} a \left(c+d x^2\right)^{3/2} (5 a d+4 b c)+\frac{1}{2} a c \sqrt{c+d x^2} (5 a d+4 b c)+\frac{b^2 \left(c+d x^2\right)^{7/2}}{7 d}",1,"-1/2*((-2*b^2*c*(c + d*x^2)^(7/2))/(7*d) + (a^2*(c + d*x^2)^(7/2))/x^2 + (a*(4*b*c + 5*a*d)*(-(Sqrt[c + d*x^2]*(23*c^2 + 11*c*d*x^2 + 3*d^2*x^4)) + 15*c^(5/2)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]]))/15)/c","A",1
632,1,155,223,0.1282551,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2}}{x^4} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^(5/2))/x^4,x]","\frac{1}{48} \left(\frac{15 c \left(8 a^2 d^2+12 a b c d+b^2 c^2\right) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{\sqrt{d}}+\frac{\sqrt{c+d x^2} \left(-8 a^2 \left(2 c^2+14 c d x^2-3 d^2 x^4\right)+12 a b x^2 \left(-8 c^2+9 c d x^2+2 d^2 x^4\right)+b^2 x^4 \left(33 c^2+26 c d x^2+8 d^2 x^4\right)\right)}{x^3}\right)","-\frac{a^2 \left(c+d x^2\right)^{7/2}}{3 c x^3}+\frac{x \left(c+d x^2\right)^{5/2} \left(4 a d (2 a d+3 b c)+b^2 c^2\right)}{6 c^2}+\frac{5 x \left(c+d x^2\right)^{3/2} \left(4 a d (2 a d+3 b c)+b^2 c^2\right)}{24 c}+\frac{5}{16} x \sqrt{c+d x^2} \left(4 a d (2 a d+3 b c)+b^2 c^2\right)+\frac{5 c \left(4 a d (2 a d+3 b c)+b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{16 \sqrt{d}}-\frac{2 a \left(c+d x^2\right)^{7/2} (2 a d+3 b c)}{3 c^2 x}",1,"((Sqrt[c + d*x^2]*(-8*a^2*(2*c^2 + 14*c*d*x^2 - 3*d^2*x^4) + 12*a*b*x^2*(-8*c^2 + 9*c*d*x^2 + 2*d^2*x^4) + b^2*x^4*(33*c^2 + 26*c*d*x^2 + 8*d^2*x^4)))/x^3 + (15*c*(b^2*c^2 + 12*a*b*c*d + 8*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/Sqrt[d])/48","A",1
633,1,153,222,0.1132994,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2}}{x^5} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^(5/2))/x^5,x]","\frac{\sqrt{c+d x^2} \left(-15 a^2 \left(2 c^2+9 c d x^2-8 d^2 x^4\right)+40 a b x^2 \left(-3 c^2+14 c d x^2+2 d^2 x^4\right)+8 b^2 x^4 \left(23 c^2+11 c d x^2+3 d^2 x^4\right)\right)}{120 x^4}-\frac{1}{8} \sqrt{c} \left(15 a^2 d^2+40 a b c d+8 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)","-\frac{a^2 \left(c+d x^2\right)^{7/2}}{4 c x^4}+\frac{\left(c+d x^2\right)^{5/2} \left(5 a d (3 a d+8 b c)+8 b^2 c^2\right)}{40 c^2}+\frac{\left(c+d x^2\right)^{3/2} \left(5 a d (3 a d+8 b c)+8 b^2 c^2\right)}{24 c}+\frac{1}{8} \sqrt{c+d x^2} \left(5 a d (3 a d+8 b c)+8 b^2 c^2\right)-\frac{1}{8} \sqrt{c} \left(5 a d (3 a d+8 b c)+8 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)-\frac{a \left(c+d x^2\right)^{7/2} (3 a d+8 b c)}{8 c^2 x^2}",1,"(Sqrt[c + d*x^2]*(-15*a^2*(2*c^2 + 9*c*d*x^2 - 8*d^2*x^4) + 40*a*b*x^2*(-3*c^2 + 14*c*d*x^2 + 2*d^2*x^4) + 8*b^2*x^4*(23*c^2 + 11*c*d*x^2 + 3*d^2*x^4)))/(120*x^4) - (Sqrt[c]*(8*b^2*c^2 + 40*a*b*c*d + 15*a^2*d^2)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/8","A",1
634,1,158,228,0.172111,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2}}{x^6} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^(5/2))/x^6,x]","\frac{1}{8} \sqrt{d} \left(8 a^2 d^2+40 a b c d+15 b^2 c^2\right) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)+\sqrt{c+d x^2} \left(\frac{-23 a^2 d^2-70 a b c d-15 b^2 c^2}{15 x}-\frac{a^2 c^2}{5 x^5}-\frac{a c (11 a d+10 b c)}{15 x^3}+\frac{1}{8} b d x (8 a d+9 b c)+\frac{1}{4} b^2 d^2 x^3\right)","-\frac{a^2 \left(c+d x^2\right)^{7/2}}{5 c x^5}-\frac{\left(c+d x^2\right)^{5/2} \left(8 a d (a d+5 b c)+15 b^2 c^2\right)}{15 c^2 x}+\frac{d x \left(c+d x^2\right)^{3/2} \left(8 a d (a d+5 b c)+15 b^2 c^2\right)}{12 c^2}+\frac{d x \sqrt{c+d x^2} \left(8 a d (a d+5 b c)+15 b^2 c^2\right)}{8 c}+\frac{1}{8} \sqrt{d} \left(8 a d (a d+5 b c)+15 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)-\frac{2 a \left(c+d x^2\right)^{7/2} (a d+5 b c)}{15 c^2 x^3}",1,"Sqrt[c + d*x^2]*(-1/5*(a^2*c^2)/x^5 - (a*c*(10*b*c + 11*a*d))/(15*x^3) + (-15*b^2*c^2 - 70*a*b*c*d - 23*a^2*d^2)/(15*x) + (b*d*(9*b*c + 8*a*d)*x)/8 + (b^2*d^2*x^3)/4) + (Sqrt[d]*(15*b^2*c^2 + 40*a*b*c*d + 8*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/8","A",1
635,1,92,222,0.0607205,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2}}{x^7} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^(5/2))/x^7,x]","\frac{\left(c+d x^2\right)^{7/2} \left(3 d x^6 \left(a^2 d^2+12 a b c d+8 b^2 c^2\right) \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};\frac{d x^2}{c}+1\right)-7 a c^2 \left(4 a c+a d x^2+12 b c x^2\right)\right)}{168 c^4 x^6}","-\frac{a^2 \left(c+d x^2\right)^{7/2}}{6 c x^6}-\frac{\left(c+d x^2\right)^{5/2} \left(a d (a d+12 b c)+8 b^2 c^2\right)}{16 c^2 x^2}+\frac{5 d \left(c+d x^2\right)^{3/2} \left(a d (a d+12 b c)+8 b^2 c^2\right)}{48 c^2}+\frac{5 d \sqrt{c+d x^2} \left(a d (a d+12 b c)+8 b^2 c^2\right)}{16 c}-\frac{5 d \left(a d (a d+12 b c)+8 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{16 \sqrt{c}}-\frac{a \left(c+d x^2\right)^{7/2} (a d+12 b c)}{24 c^2 x^4}",1,"((c + d*x^2)^(7/2)*(-7*a*c^2*(4*a*c + 12*b*c*x^2 + a*d*x^2) + 3*d*(8*b^2*c^2 + 12*a*b*c*d + a^2*d^2)*x^6*Hypergeometric2F1[2, 7/2, 9/2, 1 + (d*x^2)/c]))/(168*c^4*x^6)","C",1
636,1,159,194,0.1576088,"\int \frac{x^4 \left(a+b x^2\right)^2}{\sqrt{c+d x^2}} \, dx","Integrate[(x^4*(a + b*x^2)^2)/Sqrt[c + d*x^2],x]","\frac{3 c^2 \left(48 a^2 d^2-80 a b c d+35 b^2 c^2\right) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)+\sqrt{d} x \sqrt{c+d x^2} \left(48 a^2 d^2 \left(2 d x^2-3 c\right)+16 a b d \left(15 c^2-10 c d x^2+8 d^2 x^4\right)+b^2 \left(-105 c^3+70 c^2 d x^2-56 c d^2 x^4+48 d^3 x^6\right)\right)}{384 d^{9/2}}","\frac{c^2 \left(48 a^2 d^2+5 b c (7 b c-16 a d)\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{128 d^{9/2}}-\frac{c x \sqrt{c+d x^2} \left(48 a^2 d^2+5 b c (7 b c-16 a d)\right)}{128 d^4}+\frac{x^3 \sqrt{c+d x^2} \left(48 a^2 d^2+5 b c (7 b c-16 a d)\right)}{192 d^3}-\frac{b x^5 \sqrt{c+d x^2} (7 b c-16 a d)}{48 d^2}+\frac{b^2 x^7 \sqrt{c+d x^2}}{8 d}",1,"(Sqrt[d]*x*Sqrt[c + d*x^2]*(48*a^2*d^2*(-3*c + 2*d*x^2) + 16*a*b*d*(15*c^2 - 10*c*d*x^2 + 8*d^2*x^4) + b^2*(-105*c^3 + 70*c^2*d*x^2 - 56*c*d^2*x^4 + 48*d^3*x^6)) + 3*c^2*(35*b^2*c^2 - 80*a*b*c*d + 48*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(384*d^(9/2))","A",1
637,1,99,112,0.100976,"\int \frac{x^3 \left(a+b x^2\right)^2}{\sqrt{c+d x^2}} \, dx","Integrate[(x^3*(a + b*x^2)^2)/Sqrt[c + d*x^2],x]","\frac{\sqrt{c+d x^2} \left(35 a^2 d^2 \left(d x^2-2 c\right)+14 a b d \left(8 c^2-4 c d x^2+3 d^2 x^4\right)-3 b^2 \left(16 c^3-8 c^2 d x^2+6 c d^2 x^4-5 d^3 x^6\right)\right)}{105 d^4}","-\frac{b \left(c+d x^2\right)^{5/2} (3 b c-2 a d)}{5 d^4}+\frac{\left(c+d x^2\right)^{3/2} (b c-a d) (3 b c-a d)}{3 d^4}-\frac{c \sqrt{c+d x^2} (b c-a d)^2}{d^4}+\frac{b^2 \left(c+d x^2\right)^{7/2}}{7 d^4}",1,"(Sqrt[c + d*x^2]*(35*a^2*d^2*(-2*c + d*x^2) + 14*a*b*d*(8*c^2 - 4*c*d*x^2 + 3*d^2*x^4) - 3*b^2*(16*c^3 - 8*c^2*d*x^2 + 6*c*d^2*x^4 - 5*d^3*x^6)))/(105*d^4)","A",1
638,1,125,146,0.1291549,"\int \frac{x^2 \left(a+b x^2\right)^2}{\sqrt{c+d x^2}} \, dx","Integrate[(x^2*(a + b*x^2)^2)/Sqrt[c + d*x^2],x]","\frac{\sqrt{d} x \sqrt{c+d x^2} \left(24 a^2 d^2+12 a b d \left(2 d x^2-3 c\right)+b^2 \left(15 c^2-10 c d x^2+8 d^2 x^4\right)\right)-3 c \left(8 a^2 d^2-12 a b c d+5 b^2 c^2\right) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{48 d^{7/2}}","-\frac{c \left(8 a^2 d^2+b c (5 b c-12 a d)\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{16 d^{7/2}}+\frac{x \sqrt{c+d x^2} \left(8 a^2 d^2+b c (5 b c-12 a d)\right)}{16 d^3}-\frac{b x^3 \sqrt{c+d x^2} (5 b c-12 a d)}{24 d^2}+\frac{b^2 x^5 \sqrt{c+d x^2}}{6 d}",1,"(Sqrt[d]*x*Sqrt[c + d*x^2]*(24*a^2*d^2 + 12*a*b*d*(-3*c + 2*d*x^2) + b^2*(15*c^2 - 10*c*d*x^2 + 8*d^2*x^4)) - 3*c*(5*b^2*c^2 - 12*a*b*c*d + 8*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(48*d^(7/2))","A",1
639,1,66,74,0.0517616,"\int \frac{x \left(a+b x^2\right)^2}{\sqrt{c+d x^2}} \, dx","Integrate[(x*(a + b*x^2)^2)/Sqrt[c + d*x^2],x]","\frac{\sqrt{c+d x^2} \left(15 a^2 d^2+10 a b d \left(d x^2-2 c\right)+b^2 \left(8 c^2-4 c d x^2+3 d^2 x^4\right)\right)}{15 d^3}","-\frac{2 b \left(c+d x^2\right)^{3/2} (b c-a d)}{3 d^3}+\frac{\sqrt{c+d x^2} (b c-a d)^2}{d^3}+\frac{b^2 \left(c+d x^2\right)^{5/2}}{5 d^3}",1,"(Sqrt[c + d*x^2]*(15*a^2*d^2 + 10*a*b*d*(-2*c + d*x^2) + b^2*(8*c^2 - 4*c*d*x^2 + 3*d^2*x^4)))/(15*d^3)","A",1
640,1,91,107,0.0594367,"\int \frac{\left(a+b x^2\right)^2}{\sqrt{c+d x^2}} \, dx","Integrate[(a + b*x^2)^2/Sqrt[c + d*x^2],x]","\frac{\left(8 a^2 d^2-8 a b c d+3 b^2 c^2\right) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)+b \sqrt{d} x \sqrt{c+d x^2} \left(8 a d-3 b c+2 b d x^2\right)}{8 d^{5/2}}","\frac{\left(8 a^2 d^2-8 a b c d+3 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{8 d^{5/2}}-\frac{3 b x \sqrt{c+d x^2} (b c-2 a d)}{8 d^2}+\frac{b x \left(a+b x^2\right) \sqrt{c+d x^2}}{4 d}",1,"(b*Sqrt[d]*x*Sqrt[c + d*x^2]*(-3*b*c + 8*a*d + 2*b*d*x^2) + (3*b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(8*d^(5/2))","A",1
641,1,63,75,0.0782943,"\int \frac{\left(a+b x^2\right)^2}{x \sqrt{c+d x^2}} \, dx","Integrate[(a + b*x^2)^2/(x*Sqrt[c + d*x^2]),x]","\frac{b \sqrt{c+d x^2} \left(6 a d-2 b c+b d x^2\right)}{3 d^2}-\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{\sqrt{c}}","-\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{\sqrt{c}}-\frac{b \sqrt{c+d x^2} (b c-2 a d)}{d^2}+\frac{b^2 \left(c+d x^2\right)^{3/2}}{3 d^2}",1,"(b*Sqrt[c + d*x^2]*(-2*b*c + 6*a*d + b*d*x^2))/(3*d^2) - (a^2*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/Sqrt[c]","A",1
642,1,76,82,0.0791157,"\int \frac{\left(a+b x^2\right)^2}{x^2 \sqrt{c+d x^2}} \, dx","Integrate[(a + b*x^2)^2/(x^2*Sqrt[c + d*x^2]),x]","\sqrt{c+d x^2} \left(\frac{b^2 x}{2 d}-\frac{a^2}{c x}\right)-\frac{b (b c-4 a d) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{2 d^{3/2}}","-\frac{a^2 \sqrt{c+d x^2}}{c x}-\frac{b (b c-4 a d) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{2 d^{3/2}}+\frac{b^2 x \sqrt{c+d x^2}}{2 d}",1,"(-(a^2/(c*x)) + (b^2*x)/(2*d))*Sqrt[c + d*x^2] - (b*(b*c - 4*a*d)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(2*d^(3/2))","A",1
643,1,77,80,0.06507,"\int \frac{\left(a+b x^2\right)^2}{x^3 \sqrt{c+d x^2}} \, dx","Integrate[(a + b*x^2)^2/(x^3*Sqrt[c + d*x^2]),x]","\frac{\frac{\sqrt{c} \sqrt{c+d x^2} \left(2 b^2 c x^2-a^2 d\right)}{d x^2}+a (a d-4 b c) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{2 c^{3/2}}","-\frac{a^2 \sqrt{c+d x^2}}{2 c x^2}-\frac{a (4 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{2 c^{3/2}}+\frac{b^2 \sqrt{c+d x^2}}{d}",1,"((Sqrt[c]*(-(a^2*d) + 2*b^2*c*x^2)*Sqrt[c + d*x^2])/(d*x^2) + a*(-4*b*c + a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*c^(3/2))","A",1
644,1,72,84,0.1091573,"\int \frac{\left(a+b x^2\right)^2}{x^4 \sqrt{c+d x^2}} \, dx","Integrate[(a + b*x^2)^2/(x^4*Sqrt[c + d*x^2]),x]","\frac{b^2 \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{\sqrt{d}}-\frac{a \sqrt{c+d x^2} \left(a \left(c-2 d x^2\right)+6 b c x^2\right)}{3 c^2 x^3}","-\frac{a^2 \sqrt{c+d x^2}}{3 c x^3}-\frac{2 a \sqrt{c+d x^2} (3 b c-a d)}{3 c^2 x}+\frac{b^2 \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{\sqrt{d}}",1,"-1/3*(a*Sqrt[c + d*x^2]*(6*b*c*x^2 + a*(c - 2*d*x^2)))/(c^2*x^3) + (b^2*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/Sqrt[d]","A",1
645,1,92,106,0.1125848,"\int \frac{\left(a+b x^2\right)^2}{x^5 \sqrt{c+d x^2}} \, dx","Integrate[(a + b*x^2)^2/(x^5*Sqrt[c + d*x^2]),x]","-\frac{\left(3 a^2 d^2-8 a b c d+8 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{8 c^{5/2}}-\frac{a \sqrt{c+d x^2} \left(2 a c-3 a d x^2+8 b c x^2\right)}{8 c^2 x^4}","-\frac{\left(3 a^2 d^2-8 a b c d+8 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{8 c^{5/2}}-\frac{a^2 \sqrt{c+d x^2}}{4 c x^4}-\frac{a \sqrt{c+d x^2} (8 b c-3 a d)}{8 c^2 x^2}",1,"-1/8*(a*Sqrt[c + d*x^2]*(2*a*c + 8*b*c*x^2 - 3*a*d*x^2))/(c^2*x^4) - ((8*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(8*c^(5/2))","A",1
646,1,74,99,0.0224717,"\int \frac{\left(a+b x^2\right)^2}{x^6 \sqrt{c+d x^2}} \, dx","Integrate[(a + b*x^2)^2/(x^6*Sqrt[c + d*x^2]),x]","-\frac{\sqrt{c+d x^2} \left(a^2 \left(3 c^2-4 c d x^2+8 d^2 x^4\right)+10 a b c x^2 \left(c-2 d x^2\right)+15 b^2 c^2 x^4\right)}{15 c^3 x^5}","-\frac{a^2 \sqrt{c+d x^2}}{5 c x^5}-\frac{\sqrt{c+d x^2} \left(15 b^2 c^2-4 a d (5 b c-2 a d)\right)}{15 c^3 x}-\frac{2 a \sqrt{c+d x^2} (5 b c-2 a d)}{15 c^2 x^3}",1,"-1/15*(Sqrt[c + d*x^2]*(15*b^2*c^2*x^4 + 10*a*b*c*x^2*(c - 2*d*x^2) + a^2*(3*c^2 - 4*c*d*x^2 + 8*d^2*x^4)))/(c^3*x^5)","A",1
647,1,135,151,0.2492019,"\int \frac{\left(a+b x^2\right)^2}{x^7 \sqrt{c+d x^2}} \, dx","Integrate[(a + b*x^2)^2/(x^7*Sqrt[c + d*x^2]),x]","\frac{\sqrt{c+d x^2} \left(\frac{3 d \left(5 a^2 d^2-12 a b c d+8 b^2 c^2\right) \tanh ^{-1}\left(\sqrt{\frac{d x^2}{c}+1}\right)}{\sqrt{\frac{d x^2}{c}+1}}-\frac{c \left(a^2 \left(8 c^2-10 c d x^2+15 d^2 x^4\right)+12 a b c x^2 \left(2 c-3 d x^2\right)+24 b^2 c^2 x^4\right)}{x^6}\right)}{48 c^4}","\frac{d \left(5 a^2 d^2-12 a b c d+8 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{16 c^{7/2}}-\frac{\sqrt{c+d x^2} \left(5 a^2 d^2-12 a b c d+8 b^2 c^2\right)}{16 c^3 x^2}-\frac{a^2 \sqrt{c+d x^2}}{6 c x^6}-\frac{a \sqrt{c+d x^2} (12 b c-5 a d)}{24 c^2 x^4}",1,"(Sqrt[c + d*x^2]*(-((c*(24*b^2*c^2*x^4 + 12*a*b*c*x^2*(2*c - 3*d*x^2) + a^2*(8*c^2 - 10*c*d*x^2 + 15*d^2*x^4)))/x^6) + (3*d*(8*b^2*c^2 - 12*a*b*c*d + 5*a^2*d^2)*ArcTanh[Sqrt[1 + (d*x^2)/c]])/Sqrt[1 + (d*x^2)/c]))/(48*c^4)","A",1
648,1,158,197,0.1584537,"\int \frac{x^4 \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{3/2}} \, dx","Integrate[(x^4*(a + b*x^2)^2)/(c + d*x^2)^(3/2),x]","\sqrt{c+d x^2} \left(\frac{x \left(8 a^2 d^2-28 a b c d+19 b^2 c^2\right)}{16 d^4}+\frac{c x (b c-a d)^2}{d^4 \left(c+d x^2\right)}-\frac{b x^3 (11 b c-12 a d)}{24 d^3}+\frac{b^2 x^5}{6 d^2}\right)-\frac{c \left(24 a^2 d^2-60 a b c d+35 b^2 c^2\right) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{16 d^{9/2}}","-\frac{c \left(24 a^2 d^2-60 a b c d+35 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{16 d^{9/2}}+\frac{x \sqrt{c+d x^2} \left(24 a^2 d^2-60 a b c d+35 b^2 c^2\right)}{16 d^4}-\frac{x^3 \sqrt{c+d x^2} \left(24 a^2 d^2-60 a b c d+35 b^2 c^2\right)}{24 c d^3}+\frac{x^5 (b c-a d)^2}{c d^2 \sqrt{c+d x^2}}+\frac{b^2 x^5 \sqrt{c+d x^2}}{6 d^2}",1,"Sqrt[c + d*x^2]*(((19*b^2*c^2 - 28*a*b*c*d + 8*a^2*d^2)*x)/(16*d^4) - (b*(11*b*c - 12*a*d)*x^3)/(24*d^3) + (b^2*x^5)/(6*d^2) + (c*(b*c - a*d)^2*x)/(d^4*(c + d*x^2))) - (c*(35*b^2*c^2 - 60*a*b*c*d + 24*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(16*d^(9/2))","A",1
649,1,97,108,0.0590902,"\int \frac{x^3 \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{3/2}} \, dx","Integrate[(x^3*(a + b*x^2)^2)/(c + d*x^2)^(3/2),x]","\frac{15 a^2 d^2 \left(2 c+d x^2\right)+10 a b d \left(-8 c^2-4 c d x^2+d^2 x^4\right)+3 b^2 \left(16 c^3+8 c^2 d x^2-2 c d^2 x^4+d^3 x^6\right)}{15 d^4 \sqrt{c+d x^2}}","-\frac{b \left(c+d x^2\right)^{3/2} (3 b c-2 a d)}{3 d^4}+\frac{\sqrt{c+d x^2} (b c-a d) (3 b c-a d)}{d^4}+\frac{c (b c-a d)^2}{d^4 \sqrt{c+d x^2}}+\frac{b^2 \left(c+d x^2\right)^{5/2}}{5 d^4}",1,"(15*a^2*d^2*(2*c + d*x^2) + 10*a*b*d*(-8*c^2 - 4*c*d*x^2 + d^2*x^4) + 3*b^2*(16*c^3 + 8*c^2*d*x^2 - 2*c*d^2*x^4 + d^3*x^6))/(15*d^4*Sqrt[c + d*x^2])","A",1
650,1,124,152,0.1181918,"\int \frac{x^2 \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{3/2}} \, dx","Integrate[(x^2*(a + b*x^2)^2)/(c + d*x^2)^(3/2),x]","\frac{\left(8 a^2 d^2-24 a b c d+15 b^2 c^2\right) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{8 d^{7/2}}+\sqrt{c+d x^2} \left(-\frac{x (a d-b c)^2}{d^3 \left(c+d x^2\right)}-\frac{b x (7 b c-8 a d)}{8 d^3}+\frac{b^2 x^3}{4 d^2}\right)","\frac{\left(8 a^2 d^2-24 a b c d+15 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{8 d^{7/2}}-\frac{x \sqrt{c+d x^2} \left(8 a^2 d^2-24 a b c d+15 b^2 c^2\right)}{8 c d^3}+\frac{x^3 (b c-a d)^2}{c d^2 \sqrt{c+d x^2}}+\frac{b^2 x^3 \sqrt{c+d x^2}}{4 d^2}",1,"Sqrt[c + d*x^2]*(-1/8*(b*(7*b*c - 8*a*d)*x)/d^3 + (b^2*x^3)/(4*d^2) - ((-(b*c) + a*d)^2*x)/(d^3*(c + d*x^2))) + ((15*b^2*c^2 - 24*a*b*c*d + 8*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(8*d^(7/2))","A",1
651,1,65,73,0.0370644,"\int \frac{x \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{3/2}} \, dx","Integrate[(x*(a + b*x^2)^2)/(c + d*x^2)^(3/2),x]","\frac{-3 a^2 d^2+6 a b d \left(2 c+d x^2\right)+b^2 \left(-8 c^2-4 c d x^2+d^2 x^4\right)}{3 d^3 \sqrt{c+d x^2}}","-\frac{2 b \sqrt{c+d x^2} (b c-a d)}{d^3}-\frac{(b c-a d)^2}{d^3 \sqrt{c+d x^2}}+\frac{b^2 \left(c+d x^2\right)^{3/2}}{3 d^3}",1,"(-3*a^2*d^2 + 6*a*b*d*(2*c + d*x^2) + b^2*(-8*c^2 - 4*c*d*x^2 + d^2*x^4))/(3*d^3*Sqrt[c + d*x^2])","A",1
652,1,93,106,0.1101037,"\int \frac{\left(a+b x^2\right)^2}{\left(c+d x^2\right)^{3/2}} \, dx","Integrate[(a + b*x^2)^2/(c + d*x^2)^(3/2),x]","\sqrt{c+d x^2} \left(\frac{x (b c-a d)^2}{c d^2 \left(c+d x^2\right)}+\frac{b^2 x}{2 d^2}\right)-\frac{b (3 b c-4 a d) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{2 d^{5/2}}","-\frac{b (3 b c-4 a d) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{2 d^{5/2}}+\frac{b x \sqrt{c+d x^2} (3 b c-2 a d)}{2 c d^2}-\frac{x \left(a+b x^2\right) (b c-a d)}{c d \sqrt{c+d x^2}}",1,"Sqrt[c + d*x^2]*((b^2*x)/(2*d^2) + ((b*c - a*d)^2*x)/(c*d^2*(c + d*x^2))) - (b*(3*b*c - 4*a*d)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(2*d^(5/2))","A",1
653,1,62,75,0.0418983,"\int \frac{\left(a+b x^2\right)^2}{x \left(c+d x^2\right)^{3/2}} \, dx","Integrate[(a + b*x^2)^2/(x*(c + d*x^2)^(3/2)),x]","\frac{a^2 d^2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{d x^2}{c}+1\right)+b c \left(-2 a d+2 b c+b d x^2\right)}{c d^2 \sqrt{c+d x^2}}","-\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{c^{3/2}}+\frac{(b c-a d)^2}{c d^2 \sqrt{c+d x^2}}+\frac{b^2 \sqrt{c+d x^2}}{d^2}",1,"(b*c*(2*b*c - 2*a*d + b*d*x^2) + a^2*d^2*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (d*x^2)/c])/(c*d^2*Sqrt[c + d*x^2])","C",1
654,1,81,91,0.1036189,"\int \frac{\left(a+b x^2\right)^2}{x^2 \left(c+d x^2\right)^{3/2}} \, dx","Integrate[(a + b*x^2)^2/(x^2*(c + d*x^2)^(3/2)),x]","\frac{b^2 \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{d^{3/2}}-\frac{\sqrt{c+d x^2} \left(a^2+\frac{x^2 (b c-a d)^2}{d \left(c+d x^2\right)}\right)}{c^2 x}","-\frac{a^2}{c x \sqrt{c+d x^2}}-\frac{x \left(b^2 c^2-2 a d (b c-a d)\right)}{c^2 d \sqrt{c+d x^2}}+\frac{b^2 \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{d^{3/2}}",1,"-((Sqrt[c + d*x^2]*(a^2 + ((b*c - a*d)^2*x^2)/(d*(c + d*x^2))))/(c^2*x)) + (b^2*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/d^(3/2)","A",1
655,1,96,103,0.0845505,"\int \frac{\left(a+b x^2\right)^2}{x^3 \left(c+d x^2\right)^{3/2}} \, dx","Integrate[(a + b*x^2)^2/(x^3*(c + d*x^2)^(3/2)),x]","\frac{a (3 a d-4 b c) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{2 c^{5/2}}-\frac{a^2 d \left(c+3 d x^2\right)-4 a b c d x^2+2 b^2 c^2 x^2}{2 c^2 d x^2 \sqrt{c+d x^2}}","\frac{-\frac{3 a^2 d}{c}+4 a b-\frac{2 b^2 c}{d}}{2 c \sqrt{c+d x^2}}-\frac{a^2}{2 c x^2 \sqrt{c+d x^2}}-\frac{a (4 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{2 c^{5/2}}",1,"-1/2*(2*b^2*c^2*x^2 - 4*a*b*c*d*x^2 + a^2*d*(c + 3*d*x^2))/(c^2*d*x^2*Sqrt[c + d*x^2]) + (a*(-4*b*c + 3*a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*c^(5/2))","A",1
656,1,74,97,0.0243567,"\int \frac{\left(a+b x^2\right)^2}{x^4 \left(c+d x^2\right)^{3/2}} \, dx","Integrate[(a + b*x^2)^2/(x^4*(c + d*x^2)^(3/2)),x]","\frac{a^2 \left(-c^2+4 c d x^2+8 d^2 x^4\right)-6 a b c x^2 \left(c+2 d x^2\right)+3 b^2 c^2 x^4}{3 c^3 x^3 \sqrt{c+d x^2}}","-\frac{a^2}{3 c x^3 \sqrt{c+d x^2}}+\frac{x \left(3 b^2 c^2-4 a d (3 b c-2 a d)\right)}{3 c^3 \sqrt{c+d x^2}}-\frac{2 a (3 b c-2 a d)}{3 c^2 x \sqrt{c+d x^2}}",1,"(3*b^2*c^2*x^4 - 6*a*b*c*x^2*(c + 2*d*x^2) + a^2*(-c^2 + 4*c*d*x^2 + 8*d^2*x^4))/(3*c^3*x^3*Sqrt[c + d*x^2])","A",1
657,1,89,145,0.0367852,"\int \frac{\left(a+b x^2\right)^2}{x^5 \left(c+d x^2\right)^{3/2}} \, dx","Integrate[(a + b*x^2)^2/(x^5*(c + d*x^2)^(3/2)),x]","\frac{x^4 \left(15 a^2 d^2-24 a b c d+8 b^2 c^2\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{d x^2}{c}+1\right)+a c \left(-2 a c+5 a d x^2-8 b c x^2\right)}{8 c^3 x^4 \sqrt{c+d x^2}}","-\frac{a^2}{4 c x^4 \sqrt{c+d x^2}}-\frac{\left(8 b^2 c^2-3 a d (8 b c-5 a d)\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{8 c^{7/2}}+\frac{8 b^2 c^2-3 a d (8 b c-5 a d)}{8 c^3 \sqrt{c+d x^2}}-\frac{a (8 b c-5 a d)}{8 c^2 x^2 \sqrt{c+d x^2}}",1,"(a*c*(-2*a*c - 8*b*c*x^2 + 5*a*d*x^2) + (8*b^2*c^2 - 24*a*b*c*d + 15*a^2*d^2)*x^4*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (d*x^2)/c])/(8*c^3*x^4*Sqrt[c + d*x^2])","C",1
658,1,105,141,0.0859356,"\int \frac{\left(a+b x^2\right)^2}{x^6 \left(c+d x^2\right)^{3/2}} \, dx","Integrate[(a + b*x^2)^2/(x^6*(c + d*x^2)^(3/2)),x]","\sqrt{c+d x^2} \left(\frac{-33 a^2 d^2+50 a b c d-15 b^2 c^2}{15 c^4 x}-\frac{a^2}{5 c^2 x^5}-\frac{d x (b c-a d)^2}{c^4 \left(c+d x^2\right)}+\frac{a (9 a d-10 b c)}{15 c^3 x^3}\right)","-\frac{a^2}{5 c x^5 \sqrt{c+d x^2}}-\frac{2 d x \left(15 b^2 c^2-8 a d (5 b c-3 a d)\right)}{15 c^4 \sqrt{c+d x^2}}-\frac{15 b^2 c^2-8 a d (5 b c-3 a d)}{15 c^3 x \sqrt{c+d x^2}}-\frac{2 a (5 b c-3 a d)}{15 c^2 x^3 \sqrt{c+d x^2}}",1,"Sqrt[c + d*x^2]*(-1/5*a^2/(c^2*x^5) + (a*(-10*b*c + 9*a*d))/(15*c^3*x^3) + (-15*b^2*c^2 + 50*a*b*c*d - 33*a^2*d^2)/(15*c^4*x) - (d*(b*c - a*d)^2*x)/(c^4*(c + d*x^2)))","A",1
659,1,92,190,0.037652,"\int \frac{\left(a+b x^2\right)^2}{x^7 \left(c+d x^2\right)^{3/2}} \, dx","Integrate[(a + b*x^2)^2/(x^7*(c + d*x^2)^(3/2)),x]","\frac{d x^6 \left(-35 a^2 d^2+60 a b c d-24 b^2 c^2\right) \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};\frac{d x^2}{c}+1\right)+a c^2 \left(-4 a c+7 a d x^2-12 b c x^2\right)}{24 c^4 x^6 \sqrt{c+d x^2}}","-\frac{a^2}{6 c x^6 \sqrt{c+d x^2}}+\frac{d \left(24 b^2 c^2-5 a d (12 b c-7 a d)\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{16 c^{9/2}}-\frac{d \left(24 b^2 c^2-5 a d (12 b c-7 a d)\right)}{16 c^4 \sqrt{c+d x^2}}-\frac{24 b^2 c^2-5 a d (12 b c-7 a d)}{48 c^3 x^2 \sqrt{c+d x^2}}-\frac{a (12 b c-7 a d)}{24 c^2 x^4 \sqrt{c+d x^2}}",1,"(a*c^2*(-4*a*c - 12*b*c*x^2 + 7*a*d*x^2) + d*(-24*b^2*c^2 + 60*a*b*c*d - 35*a^2*d^2)*x^6*Hypergeometric2F1[-1/2, 2, 1/2, 1 + (d*x^2)/c])/(24*c^4*x^6*Sqrt[c + d*x^2])","C",1
660,1,156,202,0.1804041,"\int \frac{x^4 \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{5/2}} \, dx","Integrate[(x^4*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x]","\frac{\left(8 a^2 d^2-40 a b c d+35 b^2 c^2\right) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{8 d^{9/2}}+\frac{x \left(-8 a^2 d^2 \left(3 c+4 d x^2\right)+8 a b d \left(15 c^2+20 c d x^2+3 d^2 x^4\right)-\left(b^2 \left(105 c^3+140 c^2 d x^2+21 c d^2 x^4-6 d^3 x^6\right)\right)\right)}{24 d^4 \left(c+d x^2\right)^{3/2}}","\frac{\left(8 a^2 d^2-40 a b c d+35 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{8 d^{9/2}}-\frac{x \sqrt{c+d x^2} \left(8 a^2 d^2-40 a b c d+35 b^2 c^2\right)}{8 c d^4}+\frac{x^3 \left(8 a^2 d^2-40 a b c d+35 b^2 c^2\right)}{12 c d^3 \sqrt{c+d x^2}}+\frac{x^5 (b c-a d)^2}{3 c d^2 \left(c+d x^2\right)^{3/2}}+\frac{b^2 x^5}{4 d^2 \sqrt{c+d x^2}}",1,"(x*(-8*a^2*d^2*(3*c + 4*d*x^2) + 8*a*b*d*(15*c^2 + 20*c*d*x^2 + 3*d^2*x^4) - b^2*(105*c^3 + 140*c^2*d*x^2 + 21*c*d^2*x^4 - 6*d^3*x^6)))/(24*d^4*(c + d*x^2)^(3/2)) + ((35*b^2*c^2 - 40*a*b*c*d + 8*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(8*d^(9/2))","A",1
661,1,98,110,0.0577283,"\int \frac{x^3 \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{5/2}} \, dx","Integrate[(x^3*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x]","\frac{-a^2 d^2 \left(2 c+3 d x^2\right)+2 a b d \left(8 c^2+12 c d x^2+3 d^2 x^4\right)+b^2 \left(-16 c^3-24 c^2 d x^2-6 c d^2 x^4+d^3 x^6\right)}{3 d^4 \left(c+d x^2\right)^{3/2}}","-\frac{b \sqrt{c+d x^2} (3 b c-2 a d)}{d^4}-\frac{(b c-a d) (3 b c-a d)}{d^4 \sqrt{c+d x^2}}+\frac{c (b c-a d)^2}{3 d^4 \left(c+d x^2\right)^{3/2}}+\frac{b^2 \left(c+d x^2\right)^{3/2}}{3 d^4}",1,"(-(a^2*d^2*(2*c + 3*d*x^2)) + 2*a*b*d*(8*c^2 + 12*c*d*x^2 + 3*d^2*x^4) + b^2*(-16*c^3 - 24*c^2*d*x^2 - 6*c*d^2*x^4 + d^3*x^6))/(3*d^4*(c + d*x^2)^(3/2))","A",1
662,1,118,121,0.1152401,"\int \frac{x^2 \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{5/2}} \, dx","Integrate[(x^2*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x]","\frac{x \left(2 a^2 d^3 x^2-4 a b c d \left(3 c+4 d x^2\right)+b^2 c \left(15 c^2+20 c d x^2+3 d^2 x^4\right)\right)}{6 c d^3 \left(c+d x^2\right)^{3/2}}+\frac{b (4 a d-5 b c) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{2 d^{7/2}}","-\frac{b (5 b c-4 a d) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{2 d^{7/2}}+\frac{2 b x (b c-a d)}{d^3 \sqrt{c+d x^2}}+\frac{x^3 (b c-a d)^2}{3 c d^2 \left(c+d x^2\right)^{3/2}}+\frac{b^2 x \sqrt{c+d x^2}}{2 d^3}",1,"(x*(2*a^2*d^3*x^2 - 4*a*b*c*d*(3*c + 4*d*x^2) + b^2*c*(15*c^2 + 20*c*d*x^2 + 3*d^2*x^4)))/(6*c*d^3*(c + d*x^2)^(3/2)) + (b*(-5*b*c + 4*a*d)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(2*d^(7/2))","A",1
663,1,67,72,0.0390637,"\int \frac{x \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{5/2}} \, dx","Integrate[(x*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x]","\frac{-a^2 d^2-2 a b d \left(2 c+3 d x^2\right)+b^2 \left(8 c^2+12 c d x^2+3 d^2 x^4\right)}{3 d^3 \left(c+d x^2\right)^{3/2}}","\frac{2 b (b c-a d)}{d^3 \sqrt{c+d x^2}}-\frac{(b c-a d)^2}{3 d^3 \left(c+d x^2\right)^{3/2}}+\frac{b^2 \sqrt{c+d x^2}}{d^3}",1,"(-(a^2*d^2) - 2*a*b*d*(2*c + 3*d*x^2) + b^2*(8*c^2 + 12*c*d*x^2 + 3*d^2*x^4))/(3*d^3*(c + d*x^2)^(3/2))","A",1
664,1,101,105,0.1412713,"\int \frac{\left(a+b x^2\right)^2}{\left(c+d x^2\right)^{5/2}} \, dx","Integrate[(a + b*x^2)^2/(c + d*x^2)^(5/2),x]","\frac{x \left(a^2 d^2 \left(3 c+2 d x^2\right)+2 a b c d^2 x^2-b^2 c^2 \left(3 c+4 d x^2\right)\right)}{3 c^2 d^2 \left(c+d x^2\right)^{3/2}}+\frac{b^2 \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{d^{5/2}}","-\frac{x (b c-a d) (2 a d+3 b c)}{3 c^2 d^2 \sqrt{c+d x^2}}-\frac{x \left(a+b x^2\right) (b c-a d)}{3 c d \left(c+d x^2\right)^{3/2}}+\frac{b^2 \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{d^{5/2}}",1,"(x*(2*a*b*c*d^2*x^2 + a^2*d^2*(3*c + 2*d*x^2) - b^2*c^2*(3*c + 4*d*x^2)))/(3*c^2*d^2*(c + d*x^2)^(3/2)) + (b^2*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/d^(5/2)","A",1
665,1,67,88,0.0444405,"\int \frac{\left(a+b x^2\right)^2}{x \left(c+d x^2\right)^{5/2}} \, dx","Integrate[(a + b*x^2)^2/(x*(c + d*x^2)^(5/2)),x]","\frac{a^2 d^2 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{d x^2}{c}+1\right)-b c \left(2 a d+2 b c+3 b d x^2\right)}{3 c d^2 \left(c+d x^2\right)^{3/2}}","\frac{\frac{a^2}{c^2}-\frac{b^2}{d^2}}{\sqrt{c+d x^2}}-\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{c^{5/2}}+\frac{(b c-a d)^2}{3 c d^2 \left(c+d x^2\right)^{3/2}}",1,"(-(b*c*(2*b*c + 2*a*d + 3*b*d*x^2)) + a^2*d^2*Hypergeometric2F1[-3/2, 1, -1/2, 1 + (d*x^2)/c])/(3*c*d^2*(c + d*x^2)^(3/2))","C",1
666,1,76,90,0.0271467,"\int \frac{\left(a+b x^2\right)^2}{x^2 \left(c+d x^2\right)^{5/2}} \, dx","Integrate[(a + b*x^2)^2/(x^2*(c + d*x^2)^(5/2)),x]","\frac{-a^2 \left(3 c^2+12 c d x^2+8 d^2 x^4\right)+2 a b c x^2 \left(3 c+2 d x^2\right)+b^2 c^2 x^4}{3 c^3 x \left(c+d x^2\right)^{3/2}}","-\frac{a^2}{c x \left(c+d x^2\right)^{3/2}}+\frac{x \left(2 a (b c-2 a d)+b^2 c x^2\right)}{3 c^2 \left(c+d x^2\right)^{3/2}}+\frac{4 a x (b c-2 a d)}{3 c^3 \sqrt{c+d x^2}}",1,"(b^2*c^2*x^4 + 2*a*b*c*x^2*(3*c + 2*d*x^2) - a^2*(3*c^2 + 12*c*d*x^2 + 8*d^2*x^4))/(3*c^3*x*(c + d*x^2)^(3/2))","A",1
667,1,105,131,0.0463933,"\int \frac{\left(a+b x^2\right)^2}{x^3 \left(c+d x^2\right)^{5/2}} \, dx","Integrate[(a + b*x^2)^2/(x^3*(c + d*x^2)^(5/2)),x]","\frac{-c \left(a^2 d \left(3 c+5 d x^2\right)-4 a b c d x^2+2 b^2 c^2 x^2\right)-3 a d x^2 \left(c+d x^2\right) (5 a d-4 b c) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{d x^2}{c}+1\right)}{6 c^3 d x^2 \left(c+d x^2\right)^{3/2}}","\frac{-\frac{5 a^2 d}{c}+4 a b-\frac{2 b^2 c}{d}}{6 c \left(c+d x^2\right)^{3/2}}-\frac{a^2}{2 c x^2 \left(c+d x^2\right)^{3/2}}-\frac{a (4 b c-5 a d) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{2 c^{7/2}}+\frac{a (4 b c-5 a d)}{2 c^3 \sqrt{c+d x^2}}",1,"(-(c*(2*b^2*c^2*x^2 - 4*a*b*c*d*x^2 + a^2*d*(3*c + 5*d*x^2))) - 3*a*d*(-4*b*c + 5*a*d)*x^2*(c + d*x^2)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (d*x^2)/c])/(6*c^3*d*x^2*(c + d*x^2)^(3/2))","C",1
668,1,107,131,0.0725249,"\int \frac{\left(a+b x^2\right)^2}{x^4 \left(c+d x^2\right)^{5/2}} \, dx","Integrate[(a + b*x^2)^2/(x^4*(c + d*x^2)^(5/2)),x]","\frac{a^2 \left(-c^3+6 c^2 d x^2+24 c d^2 x^4+16 d^3 x^6\right)-2 a b c x^2 \left(3 c^2+12 c d x^2+8 d^2 x^4\right)+b^2 c^2 x^4 \left(3 c+2 d x^2\right)}{3 c^4 x^3 \left(c+d x^2\right)^{3/2}}","-\frac{a^2}{3 c x^3 \left(c+d x^2\right)^{3/2}}+\frac{2 x \left(b^2 c^2-8 a d (b c-a d)\right)}{3 c^4 \sqrt{c+d x^2}}+\frac{x \left(b^2 c^2-8 a d (b c-a d)\right)}{3 c^3 \left(c+d x^2\right)^{3/2}}-\frac{2 a (b c-a d)}{c^2 x \left(c+d x^2\right)^{3/2}}",1,"(b^2*c^2*x^4*(3*c + 2*d*x^2) - 2*a*b*c*x^2*(3*c^2 + 12*c*d*x^2 + 8*d^2*x^4) + a^2*(-c^3 + 6*c^2*d*x^2 + 24*c*d^2*x^4 + 16*d^3*x^6))/(3*c^4*x^3*(c + d*x^2)^(3/2))","A",1
669,1,90,185,0.0365525,"\int \frac{\left(a+b x^2\right)^2}{x^5 \left(c+d x^2\right)^{5/2}} \, dx","Integrate[(a + b*x^2)^2/(x^5*(c + d*x^2)^(5/2)),x]","\frac{x^4 \left(35 a^2 d^2-40 a b c d+8 b^2 c^2\right) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{d x^2}{c}+1\right)-3 a c \left(2 a c-7 a d x^2+8 b c x^2\right)}{24 c^3 x^4 \left(c+d x^2\right)^{3/2}}","-\frac{a^2}{4 c x^4 \left(c+d x^2\right)^{3/2}}-\frac{\left(8 b^2 c^2-5 a d (8 b c-7 a d)\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{8 c^{9/2}}+\frac{8 b^2 c^2-5 a d (8 b c-7 a d)}{8 c^4 \sqrt{c+d x^2}}+\frac{8 b^2 c^2-5 a d (8 b c-7 a d)}{24 c^3 \left(c+d x^2\right)^{3/2}}-\frac{a (8 b c-7 a d)}{8 c^2 x^2 \left(c+d x^2\right)^{3/2}}",1,"(-3*a*c*(2*a*c + 8*b*c*x^2 - 7*a*d*x^2) + (8*b^2*c^2 - 40*a*b*c*d + 35*a^2*d^2)*x^4*Hypergeometric2F1[-3/2, 1, -1/2, 1 + (d*x^2)/c])/(24*c^3*x^4*(c + d*x^2)^(3/2))","C",1
670,1,142,183,0.1820277,"\int \frac{\left(a+b x^2\right)^2}{x^6 \left(c+d x^2\right)^{5/2}} \, dx","Integrate[(a + b*x^2)^2/(x^6*(c + d*x^2)^(5/2)),x]","\frac{-a^2 \left(3 c^4-8 c^3 d x^2+48 c^2 d^2 x^4+192 c d^3 x^6+128 d^4 x^8\right)+10 a b c x^2 \left(-c^3+6 c^2 d x^2+24 c d^2 x^4+16 d^3 x^6\right)-5 b^2 c^2 x^4 \left(3 c^2+12 c d x^2+8 d^2 x^4\right)}{15 c^5 x^5 \left(c+d x^2\right)^{3/2}}","-\frac{a^2}{5 c x^5 \left(c+d x^2\right)^{3/2}}-\frac{8 d x \left(5 b^2 c^2-4 a d (5 b c-4 a d)\right)}{15 c^5 \sqrt{c+d x^2}}-\frac{4 d x \left(5 b^2 c^2-4 a d (5 b c-4 a d)\right)}{15 c^4 \left(c+d x^2\right)^{3/2}}-\frac{5 b^2 c^2-4 a d (5 b c-4 a d)}{5 c^3 x \left(c+d x^2\right)^{3/2}}-\frac{2 a (5 b c-4 a d)}{15 c^2 x^3 \left(c+d x^2\right)^{3/2}}",1,"(-5*b^2*c^2*x^4*(3*c^2 + 12*c*d*x^2 + 8*d^2*x^4) + 10*a*b*c*x^2*(-c^3 + 6*c^2*d*x^2 + 24*c*d^2*x^4 + 16*d^3*x^6) - a^2*(3*c^4 - 8*c^3*d*x^2 + 48*c^2*d^2*x^4 + 192*c*d^3*x^6 + 128*d^4*x^8))/(15*c^5*x^5*(c + d*x^2)^(3/2))","A",1
671,1,56,72,0.0704776,"\int \frac{x^5}{\sqrt{d x^2} \left(a+b x^2\right)} \, dx","Integrate[x^5/(Sqrt[d*x^2]*(a + b*x^2)),x]","\frac{x \left(3 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)+\sqrt{b} x \left(b x^2-3 a\right)\right)}{3 b^{5/2} \sqrt{d x^2}}","\frac{a^{3/2} x \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{5/2} \sqrt{d x^2}}-\frac{a x^2}{b^2 \sqrt{d x^2}}+\frac{x^4}{3 b \sqrt{d x^2}}",1,"(x*(Sqrt[b]*x*(-3*a + b*x^2) + 3*a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]]))/(3*b^(5/2)*Sqrt[d*x^2])","A",1
672,1,44,52,0.0174728,"\int \frac{x^3}{\sqrt{d x^2} \left(a+b x^2\right)} \, dx","Integrate[x^3/(Sqrt[d*x^2]*(a + b*x^2)),x]","\frac{x \left(\sqrt{b} x-\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right)}{b^{3/2} \sqrt{d x^2}}","\frac{x^2}{b \sqrt{d x^2}}-\frac{\sqrt{a} x \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{3/2} \sqrt{d x^2}}",1,"(x*(Sqrt[b]*x - Sqrt[a]*ArcTan[(Sqrt[b]*x)/Sqrt[a]]))/(b^(3/2)*Sqrt[d*x^2])","A",1
673,1,34,34,0.0078708,"\int \frac{x}{\sqrt{d x^2} \left(a+b x^2\right)} \, dx","Integrate[x/(Sqrt[d*x^2]*(a + b*x^2)),x]","\frac{x \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b} \sqrt{d x^2}}","\frac{x \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b} \sqrt{d x^2}}",1,"(x*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*Sqrt[d*x^2])","A",1
674,1,46,50,0.0184303,"\int \frac{1}{x \sqrt{d x^2} \left(a+b x^2\right)} \, dx","Integrate[1/(x*Sqrt[d*x^2]*(a + b*x^2)),x]","-\frac{d x^2 \left(\sqrt{b} x \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)+\sqrt{a}\right)}{a^{3/2} \left(d x^2\right)^{3/2}}","-\frac{\sqrt{b} x \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2} \sqrt{d x^2}}-\frac{1}{a \sqrt{d x^2}}",1,"-((d*x^2*(Sqrt[a] + Sqrt[b]*x*ArcTan[(Sqrt[b]*x)/Sqrt[a]]))/(a^(3/2)*(d*x^2)^(3/2)))","A",1
675,1,58,68,0.0236603,"\int \frac{1}{x^3 \sqrt{d x^2} \left(a+b x^2\right)} \, dx","Integrate[1/(x^3*Sqrt[d*x^2]*(a + b*x^2)),x]","\frac{d \left(3 b^{3/2} x^3 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)-\sqrt{a} \left(a-3 b x^2\right)\right)}{3 a^{5/2} \left(d x^2\right)^{3/2}}","\frac{b^{3/2} x \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{5/2} \sqrt{d x^2}}+\frac{b}{a^2 \sqrt{d x^2}}-\frac{1}{3 a x^2 \sqrt{d x^2}}",1,"(d*(-(Sqrt[a]*(a - 3*b*x^2)) + 3*b^(3/2)*x^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]]))/(3*a^(5/2)*(d*x^2)^(3/2))","A",1
676,1,148,157,0.2298251,"\int \frac{x^4 \sqrt{c+d x^2}}{a+b x^2} \, dx","Integrate[(x^4*Sqrt[c + d*x^2])/(a + b*x^2),x]","\frac{8 a^{3/2} d^{3/2} \sqrt{b c-a d} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)-\left(-8 a^2 d^2+4 a b c d+b^2 c^2\right) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)+b \sqrt{d} x \sqrt{c+d x^2} \left(-4 a d+b c+2 b d x^2\right)}{8 b^3 d^{3/2}}","\frac{a^{3/2} \sqrt{b c-a d} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{b^3}-\frac{\left(-8 a^2 d^2+4 a b c d+b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{8 b^3 d^{3/2}}+\frac{x \sqrt{c+d x^2} (b c-4 a d)}{8 b^2 d}+\frac{x^3 \sqrt{c+d x^2}}{4 b}",1,"(b*Sqrt[d]*x*Sqrt[c + d*x^2]*(b*c - 4*a*d + 2*b*d*x^2) + 8*a^(3/2)*d^(3/2)*Sqrt[b*c - a*d]*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])] - (b^2*c^2 + 4*a*b*c*d - 8*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(8*b^3*d^(3/2))","A",1
677,1,85,88,0.0677459,"\int \frac{x^3 \sqrt{c+d x^2}}{a+b x^2} \, dx","Integrate[(x^3*Sqrt[c + d*x^2])/(a + b*x^2),x]","\frac{a \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{b^{5/2}}+\frac{\sqrt{c+d x^2} \left(b \left(c+d x^2\right)-3 a d\right)}{3 b^2 d}","\frac{a \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{b^{5/2}}-\frac{a \sqrt{c+d x^2}}{b^2}+\frac{\left(c+d x^2\right)^{3/2}}{3 b d}",1,"(Sqrt[c + d*x^2]*(-3*a*d + b*(c + d*x^2)))/(3*b^2*d) + (a*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/b^(5/2)","A",1
678,1,108,112,0.1304065,"\int \frac{x^2 \sqrt{c+d x^2}}{a+b x^2} \, dx","Integrate[(x^2*Sqrt[c + d*x^2])/(a + b*x^2),x]","\frac{\frac{(b c-2 a d) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{\sqrt{d}}-2 \sqrt{a} \sqrt{b c-a d} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)+b x \sqrt{c+d x^2}}{2 b^2}","-\frac{\sqrt{a} \sqrt{b c-a d} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{b^2}+\frac{(b c-2 a d) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{2 b^2 \sqrt{d}}+\frac{x \sqrt{c+d x^2}}{2 b}",1,"(b*x*Sqrt[c + d*x^2] - 2*Sqrt[a]*Sqrt[b*c - a*d]*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])] + ((b*c - 2*a*d)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/Sqrt[d])/(2*b^2)","A",1
679,1,65,65,0.0270687,"\int \frac{x \sqrt{c+d x^2}}{a+b x^2} \, dx","Integrate[(x*Sqrt[c + d*x^2])/(a + b*x^2),x]","\frac{\sqrt{c+d x^2}}{b}-\frac{\sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{b^{3/2}}","\frac{\sqrt{c+d x^2}}{b}-\frac{\sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{b^{3/2}}",1,"Sqrt[c + d*x^2]/b - (Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/b^(3/2)","A",1
680,1,84,81,0.034722,"\int \frac{\sqrt{c+d x^2}}{a+b x^2} \, dx","Integrate[Sqrt[c + d*x^2]/(a + b*x^2),x]","\frac{\sqrt{b c-a d} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{\sqrt{a} b}+\frac{\sqrt{d} \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{b}","\frac{\sqrt{b c-a d} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{\sqrt{a} b}+\frac{\sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{b}",1,"(Sqrt[b*c - a*d]*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(Sqrt[a]*b) + (Sqrt[d]*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/b","A",1
681,1,78,80,0.034399,"\int \frac{\sqrt{c+d x^2}}{x \left(a+b x^2\right)} \, dx","Integrate[Sqrt[c + d*x^2]/(x*(a + b*x^2)),x]","\frac{\frac{\sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{\sqrt{b}}-\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{a}","\frac{\sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{a \sqrt{b}}-\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{a}",1,"(-(Sqrt[c]*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]]) + (Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/Sqrt[b])/a","A",1
682,1,51,70,0.0192456,"\int \frac{\sqrt{c+d x^2}}{x^2 \left(a+b x^2\right)} \, dx","Integrate[Sqrt[c + d*x^2]/(x^2*(a + b*x^2)),x]","-\frac{\sqrt{c+d x^2} \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{(a d-b c) x^2}{a \left(d x^2+c\right)}\right)}{a x}","-\frac{\sqrt{b c-a d} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{a^{3/2}}-\frac{\sqrt{c+d x^2}}{a x}",1,"-((Sqrt[c + d*x^2]*Hypergeometric2F1[-1/2, 1, 1/2, ((-(b*c) + a*d)*x^2)/(a*(c + d*x^2))])/(a*x))","C",1
683,1,107,113,0.1737923,"\int \frac{\sqrt{c+d x^2}}{x^3 \left(a+b x^2\right)} \, dx","Integrate[Sqrt[c + d*x^2]/(x^3*(a + b*x^2)),x]","\frac{\frac{(2 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{\sqrt{c}}-2 \sqrt{b} \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)-\frac{a \sqrt{c+d x^2}}{x^2}}{2 a^2}","\frac{(2 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{2 a^2 \sqrt{c}}-\frac{\sqrt{b} \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{a^2}-\frac{\sqrt{c+d x^2}}{2 a x^2}",1,"(-((a*Sqrt[c + d*x^2])/x^2) + ((2*b*c - a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/Sqrt[c] - 2*Sqrt[b]*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(2*a^2)","A",1
684,1,93,105,5.1596098,"\int \frac{\sqrt{c+d x^2}}{x^4 \left(a+b x^2\right)} \, dx","Integrate[Sqrt[c + d*x^2]/(x^4*(a + b*x^2)),x]","\frac{b \sqrt{b c-a d} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{a^{5/2}}+\frac{\sqrt{c+d x^2} \left(3 b c x^2-a \left(c+d x^2\right)\right)}{3 a^2 c x^3}","\frac{b \sqrt{b c-a d} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{a^{5/2}}+\frac{\sqrt{c+d x^2} (3 b c-a d)}{3 a^2 c x}-\frac{\sqrt{c+d x^2}}{3 a x^3}",1,"(Sqrt[c + d*x^2]*(3*b*c*x^2 - a*(c + d*x^2)))/(3*a^2*c*x^3) + (b*Sqrt[b*c - a*d]*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/a^(5/2)","A",1
685,1,196,210,0.3102408,"\int \frac{x^4 \left(c+d x^2\right)^{3/2}}{a+b x^2} \, dx","Integrate[(x^4*(c + d*x^2)^(3/2))/(a + b*x^2),x]","\frac{48 a^{3/2} d^{3/2} (b c-a d)^{3/2} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)+b \sqrt{d} x \sqrt{c+d x^2} \left(24 a^2 d^2-6 a b d \left(5 c+2 d x^2\right)+b^2 \left(3 c^2+14 c d x^2+8 d^2 x^4\right)\right)-3 \left(16 a^3 d^3-24 a^2 b c d^2+6 a b^2 c^2 d+b^3 c^3\right) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{48 b^4 d^{3/2}}","\frac{a^{3/2} (b c-a d)^{3/2} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{b^4}-\frac{(b c-2 a d) \left(-8 a^2 d^2+8 a b c d+b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{16 b^4 d^{3/2}}+\frac{x \sqrt{c+d x^2} \left(8 a^2 d^2-10 a b c d+b^2 c^2\right)}{16 b^3 d}+\frac{x^3 \sqrt{c+d x^2} (7 b c-6 a d)}{24 b^2}+\frac{d x^5 \sqrt{c+d x^2}}{6 b}",1,"(b*Sqrt[d]*x*Sqrt[c + d*x^2]*(24*a^2*d^2 - 6*a*b*d*(5*c + 2*d*x^2) + b^2*(3*c^2 + 14*c*d*x^2 + 8*d^2*x^4)) + 48*a^(3/2)*d^(3/2)*(b*c - a*d)^(3/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])] - 3*(b^3*c^3 + 6*a*b^2*c^2*d - 24*a^2*b*c*d^2 + 16*a^3*d^3)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(48*b^4*d^(3/2))","A",1
686,1,108,115,0.1239896,"\int \frac{x^3 \left(c+d x^2\right)^{3/2}}{a+b x^2} \, dx","Integrate[(x^3*(c + d*x^2)^(3/2))/(a + b*x^2),x]","\frac{\sqrt{c+d x^2} \left(15 a^2 d^2-5 a b d \left(4 c+d x^2\right)+3 b^2 \left(c+d x^2\right)^2\right)}{15 b^3 d}+\frac{a (b c-a d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{b^{7/2}}","\frac{a (b c-a d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{b^{7/2}}-\frac{a \sqrt{c+d x^2} (b c-a d)}{b^3}-\frac{a \left(c+d x^2\right)^{3/2}}{3 b^2}+\frac{\left(c+d x^2\right)^{5/2}}{5 b d}",1,"(Sqrt[c + d*x^2]*(15*a^2*d^2 + 3*b^2*(c + d*x^2)^2 - 5*a*b*d*(4*c + d*x^2)))/(15*b^3*d) + (a*(b*c - a*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/b^(7/2)","A",1
687,1,139,158,0.2350975,"\int \frac{x^2 \left(c+d x^2\right)^{3/2}}{a+b x^2} \, dx","Integrate[(x^2*(c + d*x^2)^(3/2))/(a + b*x^2),x]","\frac{\frac{\left(8 a^2 d^2-12 a b c d+3 b^2 c^2\right) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{\sqrt{d}}+b x \sqrt{c+d x^2} \left(-4 a d+5 b c+2 b d x^2\right)-8 \sqrt{a} (b c-a d)^{3/2} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{8 b^3}","\frac{\left(8 a^2 d^2-12 a b c d+3 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{8 b^3 \sqrt{d}}-\frac{\sqrt{a} (b c-a d)^{3/2} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{b^3}+\frac{x \sqrt{c+d x^2} (5 b c-4 a d)}{8 b^2}+\frac{d x^3 \sqrt{c+d x^2}}{4 b}",1,"(b*x*Sqrt[c + d*x^2]*(5*b*c - 4*a*d + 2*b*d*x^2) - 8*Sqrt[a]*(b*c - a*d)^(3/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])] + ((3*b^2*c^2 - 12*a*b*c*d + 8*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/Sqrt[d])/(8*b^3)","A",1
688,1,83,91,0.0934135,"\int \frac{x \left(c+d x^2\right)^{3/2}}{a+b x^2} \, dx","Integrate[(x*(c + d*x^2)^(3/2))/(a + b*x^2),x]","\frac{\sqrt{c+d x^2} \left(-3 a d+4 b c+b d x^2\right)}{3 b^2}-\frac{(b c-a d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{b^{5/2}}","-\frac{(b c-a d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{b^{5/2}}+\frac{\sqrt{c+d x^2} (b c-a d)}{b^2}+\frac{\left(c+d x^2\right)^{3/2}}{3 b}",1,"(Sqrt[c + d*x^2]*(4*b*c - 3*a*d + b*d*x^2))/(3*b^2) - ((b*c - a*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/b^(5/2)","A",1
689,1,110,113,0.2355128,"\int \frac{\left(c+d x^2\right)^{3/2}}{a+b x^2} \, dx","Integrate[(c + d*x^2)^(3/2)/(a + b*x^2),x]","\frac{\sqrt{d} (3 b c-2 a d) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)+\frac{2 (b c-a d)^{3/2} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{\sqrt{a}}+b d x \sqrt{c+d x^2}}{2 b^2}","\frac{(b c-a d)^{3/2} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{\sqrt{a} b^2}+\frac{\sqrt{d} (3 b c-2 a d) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{2 b^2}+\frac{d x \sqrt{c+d x^2}}{2 b}",1,"(b*d*x*Sqrt[c + d*x^2] + (2*(b*c - a*d)^(3/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/Sqrt[a] + Sqrt[d]*(3*b*c - 2*a*d)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(2*b^2)","A",1
690,1,102,96,0.0724727,"\int \frac{\left(c+d x^2\right)^{3/2}}{x \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2)^(3/2)/(x*(a + b*x^2)),x]","\frac{a \sqrt{b} d \sqrt{c+d x^2}+(b c-a d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)-b^{3/2} c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{a b^{3/2}}","\frac{(b c-a d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{a b^{3/2}}-\frac{c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{a}+\frac{d \sqrt{c+d x^2}}{b}",1,"(a*Sqrt[b]*d*Sqrt[c + d*x^2] - b^(3/2)*c^(3/2)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]] + (b*c - a*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(a*b^(3/2))","A",1
691,1,105,102,0.2059255,"\int \frac{\left(c+d x^2\right)^{3/2}}{x^2 \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2)^(3/2)/(x^2*(a + b*x^2)),x]","-\frac{(b c-a d)^{3/2} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{a^{3/2} b}-\frac{c \sqrt{c+d x^2}}{a x}+\frac{d^{3/2} \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{b}","-\frac{(b c-a d)^{3/2} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{a^{3/2} b}-\frac{c \sqrt{c+d x^2}}{a x}+\frac{d^{3/2} \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{b}",1,"-((c*Sqrt[c + d*x^2])/(a*x)) - ((b*c - a*d)^(3/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(a^(3/2)*b) + (d^(3/2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/b","A",1
692,1,108,114,0.1655502,"\int \frac{\left(c+d x^2\right)^{3/2}}{x^3 \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2)^(3/2)/(x^3*(a + b*x^2)),x]","\frac{-\frac{2 (b c-a d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{\sqrt{b}}+\sqrt{c} (2 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)-\frac{a c \sqrt{c+d x^2}}{x^2}}{2 a^2}","-\frac{(b c-a d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{a^2 \sqrt{b}}+\frac{\sqrt{c} (2 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{2 a^2}-\frac{c \sqrt{c+d x^2}}{2 a x^2}",1,"(-((a*c*Sqrt[c + d*x^2])/x^2) + Sqrt[c]*(2*b*c - 3*a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]] - (2*(b*c - a*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/Sqrt[b])/(2*a^2)","A",1
693,1,53,102,0.0177834,"\int \frac{\left(c+d x^2\right)^{3/2}}{x^4 \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2)^(3/2)/(x^4*(a + b*x^2)),x]","-\frac{\left(c+d x^2\right)^{3/2} \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{(a d-b c) x^2}{a \left(d x^2+c\right)}\right)}{3 a x^3}","\frac{(b c-a d)^{3/2} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{a^{5/2}}+\frac{\sqrt{c+d x^2} (3 b c-4 a d)}{3 a^2 x}-\frac{c \sqrt{c+d x^2}}{3 a x^3}",1,"-1/3*((c + d*x^2)^(3/2)*Hypergeometric2F1[-3/2, 1, -1/2, ((-(b*c) + a*d)*x^2)/(a*(c + d*x^2))])/(a*x^3)","C",1
694,1,247,291,0.3725231,"\int \frac{x^4 \left(c+d x^2\right)^{5/2}}{a+b x^2} \, dx","Integrate[(x^4*(c + d*x^2)^(5/2))/(a + b*x^2),x]","\frac{384 a^{3/2} (b c-a d)^{5/2} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)+\frac{b x \sqrt{c+d x^2} \left(-192 a^3 d^3+48 a^2 b d^2 \left(9 c+2 d x^2\right)-8 a b^2 d \left(33 c^2+26 c d x^2+8 d^2 x^4\right)+b^3 \left(15 c^3+118 c^2 d x^2+136 c d^2 x^4+48 d^3 x^6\right)\right)}{d}+\frac{3 \left(128 a^4 d^4-320 a^3 b c d^3+240 a^2 b^2 c^2 d^2-40 a b^3 c^3 d-5 b^4 c^4\right) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{d^{3/2}}}{384 b^5}","\frac{a^{3/2} (b c-a d)^{5/2} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{b^5}+\frac{x^3 \sqrt{c+d x^2} \left(48 a^2 d^2-104 a b c d+59 b^2 c^2\right)}{192 b^3}+\frac{x \sqrt{c+d x^2} \left(-64 a^3 d^3+144 a^2 b c d^2-88 a b^2 c^2 d+5 b^3 c^3\right)}{128 b^4 d}-\frac{\left(-128 a^4 d^4+320 a^3 b c d^3-240 a^2 b^2 c^2 d^2+40 a b^3 c^3 d+5 b^4 c^4\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{128 b^5 d^{3/2}}+\frac{d x^5 \sqrt{c+d x^2} (11 b c-8 a d)}{48 b^2}+\frac{d x^5 \left(c+d x^2\right)^{3/2}}{8 b}",1,"((b*x*Sqrt[c + d*x^2]*(-192*a^3*d^3 + 48*a^2*b*d^2*(9*c + 2*d*x^2) - 8*a*b^2*d*(33*c^2 + 26*c*d*x^2 + 8*d^2*x^4) + b^3*(15*c^3 + 118*c^2*d*x^2 + 136*c*d^2*x^4 + 48*d^3*x^6)))/d + 384*a^(3/2)*(b*c - a*d)^(5/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])] + (3*(-5*b^4*c^4 - 40*a*b^3*c^3*d + 240*a^2*b^2*c^2*d^2 - 320*a^3*b*c*d^3 + 128*a^4*d^4)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/d^(3/2))/(384*b^5)","A",1
695,1,136,144,0.3081717,"\int \frac{x^3 \left(c+d x^2\right)^{5/2}}{a+b x^2} \, dx","Integrate[(x^3*(c + d*x^2)^(5/2))/(a + b*x^2),x]","-\frac{a (b c-a d) \left(\sqrt{b} \sqrt{c+d x^2} \left(-3 a d+4 b c+b d x^2\right)-3 (b c-a d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)\right)}{3 b^{9/2}}-\frac{a \left(c+d x^2\right)^{5/2}}{5 b^2}+\frac{\left(c+d x^2\right)^{7/2}}{7 b d}","\frac{a (b c-a d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{b^{9/2}}-\frac{a \sqrt{c+d x^2} (b c-a d)^2}{b^4}-\frac{a \left(c+d x^2\right)^{3/2} (b c-a d)}{3 b^3}-\frac{a \left(c+d x^2\right)^{5/2}}{5 b^2}+\frac{\left(c+d x^2\right)^{7/2}}{7 b d}",1,"-1/5*(a*(c + d*x^2)^(5/2))/b^2 + (c + d*x^2)^(7/2)/(7*b*d) - (a*(b*c - a*d)*(Sqrt[b]*Sqrt[c + d*x^2]*(4*b*c - 3*a*d + b*d*x^2) - 3*(b*c - a*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]]))/(3*b^(9/2))","A",1
696,1,187,217,0.1578503,"\int \frac{x^2 \left(c+d x^2\right)^{5/2}}{a+b x^2} \, dx","Integrate[(x^2*(c + d*x^2)^(5/2))/(a + b*x^2),x]","\frac{b x \sqrt{c+d x^2} \left(24 a^2 d^2-6 a b d \left(9 c+2 d x^2\right)+b^2 \left(33 c^2+26 c d x^2+8 d^2 x^4\right)\right)+\frac{3 \left(-16 a^3 d^3+40 a^2 b c d^2-30 a b^2 c^2 d+5 b^3 c^3\right) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{\sqrt{d}}-48 \sqrt{a} (b c-a d)^{5/2} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{48 b^4}","\frac{x \sqrt{c+d x^2} \left(8 a^2 d^2-18 a b c d+11 b^2 c^2\right)}{16 b^3}+\frac{\left(-16 a^3 d^3+40 a^2 b c d^2-30 a b^2 c^2 d+5 b^3 c^3\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{16 b^4 \sqrt{d}}-\frac{\sqrt{a} (b c-a d)^{5/2} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{b^4}+\frac{d x^3 \sqrt{c+d x^2} (3 b c-2 a d)}{8 b^2}+\frac{d x^3 \left(c+d x^2\right)^{3/2}}{6 b}",1,"(b*x*Sqrt[c + d*x^2]*(24*a^2*d^2 - 6*a*b*d*(9*c + 2*d*x^2) + b^2*(33*c^2 + 26*c*d*x^2 + 8*d^2*x^4)) - 48*Sqrt[a]*(b*c - a*d)^(5/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])] + (3*(5*b^3*c^3 - 30*a*b^2*c^2*d + 40*a^2*b*c*d^2 - 16*a^3*d^3)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/Sqrt[d])/(48*b^4)","A",1
697,1,113,119,0.1604037,"\int \frac{x \left(c+d x^2\right)^{5/2}}{a+b x^2} \, dx","Integrate[(x*(c + d*x^2)^(5/2))/(a + b*x^2),x]","\frac{(b c-a d) \left(\sqrt{b} \sqrt{c+d x^2} \left(-3 a d+4 b c+b d x^2\right)-3 (b c-a d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)\right)}{3 b^{7/2}}+\frac{\left(c+d x^2\right)^{5/2}}{5 b}","-\frac{(b c-a d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{b^{7/2}}+\frac{\sqrt{c+d x^2} (b c-a d)^2}{b^3}+\frac{\left(c+d x^2\right)^{3/2} (b c-a d)}{3 b^2}+\frac{\left(c+d x^2\right)^{5/2}}{5 b}",1,"(c + d*x^2)^(5/2)/(5*b) + ((b*c - a*d)*(Sqrt[b]*Sqrt[c + d*x^2]*(4*b*c - 3*a*d + b*d*x^2) - 3*(b*c - a*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]]))/(3*b^(7/2))","A",1
698,1,140,156,0.1145882,"\int \frac{\left(c+d x^2\right)^{5/2}}{a+b x^2} \, dx","Integrate[(c + d*x^2)^(5/2)/(a + b*x^2),x]","\frac{\sqrt{d} \left(8 a^2 d^2-20 a b c d+15 b^2 c^2\right) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)+b d x \sqrt{c+d x^2} \left(-4 a d+9 b c+2 b d x^2\right)+\frac{8 (b c-a d)^{5/2} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{\sqrt{a}}}{8 b^3}","\frac{\sqrt{d} \left(8 a^2 d^2-20 a b c d+15 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{8 b^3}+\frac{(b c-a d)^{5/2} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{\sqrt{a} b^3}+\frac{d x \sqrt{c+d x^2} (7 b c-4 a d)}{8 b^2}+\frac{d x \left(c+d x^2\right)^{3/2}}{4 b}",1,"(b*d*x*Sqrt[c + d*x^2]*(9*b*c - 4*a*d + 2*b*d*x^2) + (8*(b*c - a*d)^(5/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/Sqrt[a] + Sqrt[d]*(15*b^2*c^2 - 20*a*b*c*d + 8*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(8*b^3)","A",1
699,1,114,124,0.1178152,"\int \frac{\left(c+d x^2\right)^{5/2}}{x \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2)^(5/2)/(x*(a + b*x^2)),x]","\frac{(b c-a d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{a b^{5/2}}+\frac{d \sqrt{c+d x^2} \left(-3 a d+7 b c+b d x^2\right)}{3 b^2}-\frac{c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{a}","\frac{(b c-a d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{a b^{5/2}}+\frac{d \sqrt{c+d x^2} (2 b c-a d)}{b^2}-\frac{c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{a}+\frac{d \left(c+d x^2\right)^{3/2}}{3 b}",1,"(d*Sqrt[c + d*x^2]*(7*b*c - 3*a*d + b*d*x^2))/(3*b^2) - (c^(5/2)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/a + ((b*c - a*d)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(a*b^(5/2))","A",1
700,1,132,145,0.0917401,"\int \frac{\left(c+d x^2\right)^{5/2}}{x^2 \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2)^(5/2)/(x^2*(a + b*x^2)),x]","-\frac{(b c-a d)^{5/2} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{a^{3/2} b^2}+\frac{d^{3/2} (5 b c-2 a d) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{2 b^2}+\sqrt{c+d x^2} \left(\frac{d^2 x}{2 b}-\frac{c^2}{a x}\right)","-\frac{(b c-a d)^{5/2} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{a^{3/2} b^2}+\frac{d^{3/2} (5 b c-2 a d) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{2 b^2}+\frac{d x \sqrt{c+d x^2} (a d+2 b c)}{2 a b}-\frac{c \left(c+d x^2\right)^{3/2}}{a x}",1,"(-(c^2/(a*x)) + (d^2*x)/(2*b))*Sqrt[c + d*x^2] - ((b*c - a*d)^(5/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(a^(3/2)*b^2) + (d^(3/2)*(5*b*c - 2*a*d)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(2*b^2)","A",1
701,1,125,144,0.1415648,"\int \frac{\left(c+d x^2\right)^{5/2}}{x^3 \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2)^(5/2)/(x^3*(a + b*x^2)),x]","\frac{-\frac{2 (b c-a d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{b^{3/2}}+c^{3/2} (2 b c-5 a d) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)+\frac{a \sqrt{c+d x^2} \left(2 a d^2 x^2-b c^2\right)}{b x^2}}{2 a^2}","-\frac{(b c-a d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{a^2 b^{3/2}}+\frac{c^{3/2} (2 b c-5 a d) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{2 a^2}+\frac{d \sqrt{c+d x^2} (2 a d+b c)}{2 a b}-\frac{c \left(c+d x^2\right)^{3/2}}{2 a x^2}",1,"((a*Sqrt[c + d*x^2]*(-(b*c^2) + 2*a*d^2*x^2))/(b*x^2) + c^(3/2)*(2*b*c - 5*a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]] - (2*(b*c - a*d)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/b^(3/2))/(2*a^2)","A",1
702,1,125,130,0.1016602,"\int \frac{\left(c+d x^2\right)^{5/2}}{x^4 \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2)^(5/2)/(x^4*(a + b*x^2)),x]","\frac{(b c-a d)^{5/2} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{a^{5/2} b}+\frac{c \sqrt{c+d x^2} \left(3 b c x^2-a \left(c+7 d x^2\right)\right)}{3 a^2 x^3}+\frac{d^{5/2} \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{b}","\frac{(b c-a d)^{5/2} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{a^{5/2} b}+\frac{c \sqrt{c+d x^2} (b c-2 a d)}{a^2 x}-\frac{c \left(c+d x^2\right)^{3/2}}{3 a x^3}+\frac{d^{5/2} \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{b}",1,"(c*Sqrt[c + d*x^2]*(3*b*c*x^2 - a*(c + 7*d*x^2)))/(3*a^2*x^3) + ((b*c - a*d)^(5/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(a^(5/2)*b) + (d^(5/2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/b","A",1
703,1,89,100,0.130968,"\int \frac{x^5}{\left(a+b x^2\right) \sqrt{c+d x^2}} \, dx","Integrate[x^5/((a + b*x^2)*Sqrt[c + d*x^2]),x]","\frac{\sqrt{c+d x^2} \left(-3 a d-2 b c+b d x^2\right)}{3 b^2 d^2}-\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{b^{5/2} \sqrt{b c-a d}}","-\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{b^{5/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x^2} (a d+b c)}{b^2 d^2}+\frac{\left(c+d x^2\right)^{3/2}}{3 b d^2}",1,"(Sqrt[c + d*x^2]*(-2*b*c - 3*a*d + b*d*x^2))/(3*b^2*d^2) - (a^2*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(b^(5/2)*Sqrt[b*c - a*d])","A",1
704,1,68,68,0.0610632,"\int \frac{x^3}{\left(a+b x^2\right) \sqrt{c+d x^2}} \, dx","Integrate[x^3/((a + b*x^2)*Sqrt[c + d*x^2]),x]","\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{b^{3/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^2}}{b d}","\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{b^{3/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^2}}{b d}",1,"Sqrt[c + d*x^2]/(b*d) + (a*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(b^(3/2)*Sqrt[b*c - a*d])","A",1
705,1,49,49,0.0153244,"\int \frac{x}{\left(a+b x^2\right) \sqrt{c+d x^2}} \, dx","Integrate[x/((a + b*x^2)*Sqrt[c + d*x^2]),x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{\sqrt{b} \sqrt{b c-a d}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{\sqrt{b} \sqrt{b c-a d}}",1,"-(ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]]/(Sqrt[b]*Sqrt[b*c - a*d]))","A",1
706,1,78,80,0.0853494,"\int \frac{1}{x \left(a+b x^2\right) \sqrt{c+d x^2}} \, dx","Integrate[1/(x*(a + b*x^2)*Sqrt[c + d*x^2]),x]","\frac{\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{\sqrt{b c-a d}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{\sqrt{c}}}{a}","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{a \sqrt{b c-a d}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{a \sqrt{c}}",1,"(-(ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]]/Sqrt[c]) + (Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/Sqrt[b*c - a*d])/a","A",1
707,1,109,115,0.3796771,"\int \frac{1}{x^3 \left(a+b x^2\right) \sqrt{c+d x^2}} \, dx","Integrate[1/(x^3*(a + b*x^2)*Sqrt[c + d*x^2]),x]","\frac{-\frac{2 b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{\sqrt{b c-a d}}+\frac{(a d+2 b c) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{c^{3/2}}-\frac{a \sqrt{c+d x^2}}{c x^2}}{2 a^2}","-\frac{b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{a^2 \sqrt{b c-a d}}+\frac{(a d+2 b c) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{2 a^2 c^{3/2}}-\frac{\sqrt{c+d x^2}}{2 a c x^2}",1,"(-((a*Sqrt[c + d*x^2])/(c*x^2)) + ((2*b*c + a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/c^(3/2) - (2*b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/Sqrt[b*c - a*d])/(2*a^2)","A",1
708,1,112,114,0.1791321,"\int \frac{x^4}{\left(a+b x^2\right) \sqrt{c+d x^2}} \, dx","Integrate[x^4/((a + b*x^2)*Sqrt[c + d*x^2]),x]","\frac{\frac{2 a^{3/2} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{\sqrt{b c-a d}}-\frac{(2 a d+b c) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{d^{3/2}}+\frac{b x \sqrt{c+d x^2}}{d}}{2 b^2}","\frac{a^{3/2} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{b^2 \sqrt{b c-a d}}-\frac{(2 a d+b c) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{2 b^2 d^{3/2}}+\frac{x \sqrt{c+d x^2}}{2 b d}",1,"((b*x*Sqrt[c + d*x^2])/d + (2*a^(3/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/Sqrt[b*c - a*d] - ((b*c + 2*a*d)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/d^(3/2))/(2*b^2)","A",1
709,1,85,82,0.0680595,"\int \frac{x^2}{\left(a+b x^2\right) \sqrt{c+d x^2}} \, dx","Integrate[x^2/((a + b*x^2)*Sqrt[c + d*x^2]),x]","\frac{\log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{b \sqrt{d}}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{b \sqrt{b c-a d}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{b \sqrt{d}}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{b \sqrt{b c-a d}}",1,"-((Sqrt[a]*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(b*Sqrt[b*c - a*d])) + Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]]/(b*Sqrt[d])","A",1
710,1,49,49,0.0151283,"\int \frac{1}{\left(a+b x^2\right) \sqrt{c+d x^2}} \, dx","Integrate[1/((a + b*x^2)*Sqrt[c + d*x^2]),x]","\frac{\tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{\sqrt{a} \sqrt{b c-a d}}","\frac{\tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{\sqrt{a} \sqrt{b c-a d}}",1,"ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])]/(Sqrt[a]*Sqrt[b*c - a*d])","A",1
711,1,177,74,3.943245,"\int \frac{1}{x^2 \left(a+b x^2\right) \sqrt{c+d x^2}} \, dx","Integrate[1/(x^2*(a + b*x^2)*Sqrt[c + d*x^2]),x]","-\frac{\left(\frac{d x^2}{c}+1\right) \left(\frac{4 x^2 \left(c+d x^2\right) (b c-a d) \, _2F_1\left(2,2;\frac{5}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)}{3 c^2 \left(a+b x^2\right)}+\frac{\left(c+2 d x^2\right) \sin ^{-1}\left(\sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}\right)}{c \sqrt{\frac{a x^2 \left(c+d x^2\right) (b c-a d)}{c^2 \left(a+b x^2\right)^2}}}\right)}{x \left(a+b x^2\right) \sqrt{c+d x^2}}","-\frac{b \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{a^{3/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x^2}}{a c x}",1,"-(((1 + (d*x^2)/c)*(((c + 2*d*x^2)*ArcSin[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/(c*Sqrt[(a*(b*c - a*d)*x^2*(c + d*x^2))/(c^2*(a + b*x^2)^2)]) + (4*(b*c - a*d)*x^2*(c + d*x^2)*Hypergeometric2F1[2, 2, 5/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))])/(3*c^2*(a + b*x^2))))/(x*(a + b*x^2)*Sqrt[c + d*x^2]))","C",0
712,1,96,110,5.1228512,"\int \frac{1}{x^4 \left(a+b x^2\right) \sqrt{c+d x^2}} \, dx","Integrate[1/(x^4*(a + b*x^2)*Sqrt[c + d*x^2]),x]","\frac{b^2 \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{a^{5/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^2} \left(-a c+2 a d x^2+3 b c x^2\right)}{3 a^2 c^2 x^3}","\frac{b^2 \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{a^{5/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^2} (2 a d+3 b c)}{3 a^2 c^2 x}-\frac{\sqrt{c+d x^2}}{3 a c x^3}",1,"(Sqrt[c + d*x^2]*(-(a*c) + 3*b*c*x^2 + 2*a*d*x^2))/(3*a^2*c^2*x^3) + (b^2*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(a^(5/2)*Sqrt[b*c - a*d])","A",1
713,1,111,109,0.1762572,"\int \frac{x^4}{\left(a+b x^2\right) \left(c+d x^2\right)^{3/2}} \, dx","Integrate[x^4/((a + b*x^2)*(c + d*x^2)^(3/2)),x]","\frac{a^{3/2} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{b (b c-a d)^{3/2}}+\frac{c x}{d \sqrt{c+d x^2} (a d-b c)}+\frac{\log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{b d^{3/2}}","\frac{a^{3/2} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{b (b c-a d)^{3/2}}-\frac{c x}{d \sqrt{c+d x^2} (b c-a d)}+\frac{\tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{b d^{3/2}}",1,"(c*x)/(d*(-(b*c) + a*d)*Sqrt[c + d*x^2]) + (a^(3/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(b*(b*c - a*d)^(3/2)) + Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]]/(b*d^(3/2))","A",1
714,1,86,77,0.0749843,"\int \frac{x^3}{\left(a+b x^2\right) \left(c+d x^2\right)^{3/2}} \, dx","Integrate[x^3/((a + b*x^2)*(c + d*x^2)^(3/2)),x]","\frac{c}{d \sqrt{c+d x^2} (a d-b c)}+\frac{a \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{\sqrt{b} (a d-b c)^2}","\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{\sqrt{b} (b c-a d)^{3/2}}-\frac{c}{d \sqrt{c+d x^2} (b c-a d)}",1,"c/(d*(-(b*c) + a*d)*Sqrt[c + d*x^2]) + (a*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(Sqrt[b]*(-(b*c) + a*d)^2)","A",1
715,1,111,74,0.3805698,"\int \frac{x^2}{\left(a+b x^2\right) \left(c+d x^2\right)^{3/2}} \, dx","Integrate[x^2/((a + b*x^2)*(c + d*x^2)^(3/2)),x]","\frac{x^2 (b c-a d)+a c \sqrt{\frac{d x^2}{c}+1} \sqrt{x^2 \left(\frac{d}{c}-\frac{b}{a}\right)} \tanh ^{-1}\left(\frac{\sqrt{x^2 \left(\frac{d}{c}-\frac{b}{a}\right)}}{\sqrt{\frac{d x^2}{c}+1}}\right)}{x \sqrt{c+d x^2} (b c-a d)^2}","\frac{x}{\sqrt{c+d x^2} (b c-a d)}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{(b c-a d)^{3/2}}",1,"((b*c - a*d)*x^2 + a*c*Sqrt[(-(b/a) + d/c)*x^2]*Sqrt[1 + (d*x^2)/c]*ArcTanh[Sqrt[(-(b/a) + d/c)*x^2]/Sqrt[1 + (d*x^2)/c]])/((b*c - a*d)^2*x*Sqrt[c + d*x^2])","A",1
716,1,50,72,0.0175957,"\int \frac{x}{\left(a+b x^2\right) \left(c+d x^2\right)^{3/2}} \, dx","Integrate[x/((a + b*x^2)*(c + d*x^2)^(3/2)),x]","-\frac{\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \left(d x^2+c\right)}{b c-a d}\right)}{\sqrt{c+d x^2} (a d-b c)}","\frac{1}{\sqrt{c+d x^2} (b c-a d)}-\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{(b c-a d)^{3/2}}",1,"-(Hypergeometric2F1[-1/2, 1, 1/2, (b*(c + d*x^2))/(b*c - a*d)]/((-(b*c) + a*d)*Sqrt[c + d*x^2]))","C",1
717,1,236,79,2.9612054,"\int \frac{1}{\left(a+b x^2\right) \left(c+d x^2\right)^{3/2}} \, dx","Integrate[1/((a + b*x^2)*(c + d*x^2)^(3/2)),x]","-\frac{\frac{15 c \left(3 c+2 d x^2\right) \left(c \left(a+b x^2\right) \sqrt{\frac{a x^2 \left(c+d x^2\right) (b c-a d)}{c^2 \left(a+b x^2\right)^2}}-a \left(c+d x^2\right) \sin ^{-1}\left(\sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}\right)\right)}{\sqrt{\frac{a x^2 \left(c+d x^2\right) (b c-a d)}{c^2 \left(a+b x^2\right)^2}}}+\frac{4 x^4 \left(c+d x^2\right) (b c-a d)^2 \, _2F_1\left(2,2;\frac{7}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)}{a+b x^2}}{15 c^3 x \left(a+b x^2\right) \sqrt{c+d x^2} (a d-b c)}","\frac{b \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{\sqrt{a} (b c-a d)^{3/2}}-\frac{d x}{c \sqrt{c+d x^2} (b c-a d)}",1,"-1/15*((15*c*(3*c + 2*d*x^2)*(c*(a + b*x^2)*Sqrt[(a*(b*c - a*d)*x^2*(c + d*x^2))/(c^2*(a + b*x^2)^2)] - a*(c + d*x^2)*ArcSin[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]]))/Sqrt[(a*(b*c - a*d)*x^2*(c + d*x^2))/(c^2*(a + b*x^2)^2)] + (4*(b*c - a*d)^2*x^4*(c + d*x^2)*Hypergeometric2F1[2, 2, 7/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))])/(a + b*x^2))/(c^3*(-(b*c) + a*d)*x*(a + b*x^2)*Sqrt[c + d*x^2])","C",0
718,1,87,107,0.036753,"\int \frac{1}{x \left(a+b x^2\right) \left(c+d x^2\right)^{3/2}} \, dx","Integrate[1/(x*(a + b*x^2)*(c + d*x^2)^(3/2)),x]","\frac{(b c-a d) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{d x^2}{c}+1\right)-b c \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \left(d x^2+c\right)}{b c-a d}\right)}{a c \sqrt{c+d x^2} (b c-a d)}","\frac{b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{a (b c-a d)^{3/2}}-\frac{d}{c \sqrt{c+d x^2} (b c-a d)}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{a c^{3/2}}",1,"(-(b*c*Hypergeometric2F1[-1/2, 1, 1/2, (b*(c + d*x^2))/(b*c - a*d)]) + (b*c - a*d)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (d*x^2)/c])/(a*c*(b*c - a*d)*Sqrt[c + d*x^2])","C",1
719,1,102,124,5.2073769,"\int \frac{1}{x^2 \left(a+b x^2\right) \left(c+d x^2\right)^{3/2}} \, dx","Integrate[1/(x^2*(a + b*x^2)*(c + d*x^2)^(3/2)),x]","\frac{\frac{d^2 x^2}{b c-a d}-\frac{c+d x^2}{a}}{c^2 x \sqrt{c+d x^2}}-\frac{b^2 \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{a^{3/2} (b c-a d)^{3/2}}","-\frac{b^2 \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{a^{3/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^2} (b c-2 a d)}{a c^2 x (b c-a d)}-\frac{d}{c x \sqrt{c+d x^2} (b c-a d)}",1,"((d^2*x^2)/(b*c - a*d) - (c + d*x^2)/a)/(c^2*x*Sqrt[c + d*x^2]) - (b^2*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(a^(3/2)*(b*c - a*d)^(3/2))","A",1
720,1,117,156,0.043571,"\int \frac{1}{x^3 \left(a+b x^2\right) \left(c+d x^2\right)^{3/2}} \, dx","Integrate[1/(x^3*(a + b*x^2)*(c + d*x^2)^(3/2)),x]","\frac{2 b^2 c^2 x^2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \left(d x^2+c\right)}{b c-a d}\right)+(a d-b c) \left(x^2 (3 a d+2 b c) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{d x^2}{c}+1\right)+a c\right)}{2 a^2 c^2 x^2 \sqrt{c+d x^2} (b c-a d)}","-\frac{b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{a^2 (b c-a d)^{3/2}}+\frac{(3 a d+2 b c) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{2 a^2 c^{5/2}}-\frac{d (b c-3 a d)}{2 a c^2 \sqrt{c+d x^2} (b c-a d)}-\frac{1}{2 a c x^2 \sqrt{c+d x^2}}",1,"(2*b^2*c^2*x^2*Hypergeometric2F1[-1/2, 1, 1/2, (b*(c + d*x^2))/(b*c - a*d)] + (-(b*c) + a*d)*(a*c + (2*b*c + 3*a*d)*x^2*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (d*x^2)/c]))/(2*a^2*c^2*(b*c - a*d)*x^2*Sqrt[c + d*x^2])","C",1
721,1,124,176,5.2483534,"\int \frac{1}{x^4 \left(a+b x^2\right) \left(c+d x^2\right)^{3/2}} \, dx","Integrate[1/(x^4*(a + b*x^2)*(c + d*x^2)^(3/2)),x]","\frac{b^3 \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{a^{5/2} (b c-a d)^{3/2}}+\frac{\sqrt{c+d x^2} \left(\frac{x^2 (5 a d+3 b c)}{a^2}+\frac{3 d^3 x^4}{\left(c+d x^2\right) (a d-b c)}-\frac{c}{a}\right)}{3 c^3 x^3}","\frac{b^3 \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{a^{5/2} (b c-a d)^{3/2}}+\frac{\sqrt{c+d x^2} (3 b c-4 a d) (2 a d+b c)}{3 a^2 c^3 x (b c-a d)}-\frac{\sqrt{c+d x^2} (b c-4 a d)}{3 a c^2 x^3 (b c-a d)}-\frac{d}{c x^3 \sqrt{c+d x^2} (b c-a d)}",1,"(Sqrt[c + d*x^2]*(-(c/a) + ((3*b*c + 5*a*d)*x^2)/a^2 + (3*d^3*x^4)/((-(b*c) + a*d)*(c + d*x^2))))/(3*c^3*x^3) + (b^3*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(a^(5/2)*(b*c - a*d)^(3/2))","A",1
722,1,160,117,0.2948955,"\int \frac{x^4}{\left(a+b x^2\right) \left(c+d x^2\right)^{5/2}} \, dx","Integrate[x^4/((a + b*x^2)*(c + d*x^2)^(5/2)),x]","\frac{x^2 \left(a^2 d \left(3 c+4 d x^2\right)-a b c \left(3 c+5 d x^2\right)+b^2 c^2 x^2\right)-\frac{3 a^2 \left(c+d x^2\right)^2 \sqrt{\frac{x^2 (a d-b c)}{a c}} \tanh ^{-1}\left(\frac{\sqrt{x^2 \left(\frac{d}{c}-\frac{b}{a}\right)}}{\sqrt{\frac{d x^2}{c}+1}}\right)}{\sqrt{\frac{d x^2}{c}+1}}}{3 x \left(c+d x^2\right)^{3/2} (b c-a d)^3}","\frac{a^{3/2} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{(b c-a d)^{5/2}}+\frac{x (b c-4 a d)}{3 d \sqrt{c+d x^2} (b c-a d)^2}-\frac{c x}{3 d \left(c+d x^2\right)^{3/2} (b c-a d)}",1,"(x^2*(b^2*c^2*x^2 + a^2*d*(3*c + 4*d*x^2) - a*b*c*(3*c + 5*d*x^2)) - (3*a^2*Sqrt[((-(b*c) + a*d)*x^2)/(a*c)]*(c + d*x^2)^2*ArcTanh[Sqrt[(-(b/a) + d/c)*x^2]/Sqrt[1 + (d*x^2)/c]])/Sqrt[1 + (d*x^2)/c])/(3*(b*c - a*d)^3*x*(c + d*x^2)^(3/2))","A",1
723,1,77,103,0.028543,"\int \frac{x^3}{\left(a+b x^2\right) \left(c+d x^2\right)^{5/2}} \, dx","Integrate[x^3/((a + b*x^2)*(c + d*x^2)^(5/2)),x]","\frac{c (a d-b c)-3 a d \left(c+d x^2\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \left(d x^2+c\right)}{b c-a d}\right)}{3 d \left(c+d x^2\right)^{3/2} (b c-a d)^2}","-\frac{a}{\sqrt{c+d x^2} (b c-a d)^2}-\frac{c}{3 d \left(c+d x^2\right)^{3/2} (b c-a d)}+\frac{a \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{(b c-a d)^{5/2}}",1,"(c*(-(b*c) + a*d) - 3*a*d*(c + d*x^2)*Hypergeometric2F1[-1/2, 1, 1/2, (b*(c + d*x^2))/(b*c - a*d)])/(3*d*(b*c - a*d)^2*(c + d*x^2)^(3/2))","C",1
724,1,257,115,3.0811263,"\int \frac{x^2}{\left(a+b x^2\right) \left(c+d x^2\right)^{5/2}} \, dx","Integrate[x^2/((a + b*x^2)*(c + d*x^2)^(5/2)),x]","\frac{12 x^6 \left(c+d x^2\right) (b c-a d)^3 \, _2F_1\left(2,2;\frac{9}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)-\frac{35 c \left(a+b x^2\right) \left(5 c+2 d x^2\right) \left(-3 a^2 \left(c+d x^2\right)^2 \sin ^{-1}\left(\sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}\right)-c \left(a+b x^2\right) \sqrt{\frac{a x^2 \left(c+d x^2\right) (b c-a d)}{c^2 \left(a+b x^2\right)^2}} \left(-3 a c-4 a d x^2+b c x^2\right)\right)}{\sqrt{\frac{a x^2 \left(c+d x^2\right) (b c-a d)}{c^2 \left(a+b x^2\right)^2}}}}{315 c^3 x \left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2} (b c-a d)^2}","\frac{x (a d+2 b c)}{3 c \sqrt{c+d x^2} (b c-a d)^2}+\frac{x}{3 \left(c+d x^2\right)^{3/2} (b c-a d)}-\frac{\sqrt{a} b \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{(b c-a d)^{5/2}}",1,"((-35*c*(a + b*x^2)*(5*c + 2*d*x^2)*(-(c*(a + b*x^2)*Sqrt[(a*(b*c - a*d)*x^2*(c + d*x^2))/(c^2*(a + b*x^2)^2)]*(-3*a*c + b*c*x^2 - 4*a*d*x^2)) - 3*a^2*(c + d*x^2)^2*ArcSin[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]]))/Sqrt[(a*(b*c - a*d)*x^2*(c + d*x^2))/(c^2*(a + b*x^2)^2)] + 12*(b*c - a*d)^3*x^6*(c + d*x^2)*Hypergeometric2F1[2, 2, 9/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))])/(315*c^3*(b*c - a*d)^2*x*(a + b*x^2)^2*(c + d*x^2)^(3/2))","C",0
725,1,52,98,0.0223441,"\int \frac{x}{\left(a+b x^2\right) \left(c+d x^2\right)^{5/2}} \, dx","Integrate[x/((a + b*x^2)*(c + d*x^2)^(5/2)),x]","\frac{\, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \left(d x^2+c\right)}{b c-a d}\right)}{3 \left(c+d x^2\right)^{3/2} (b c-a d)}","-\frac{b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{(b c-a d)^{5/2}}+\frac{b}{\sqrt{c+d x^2} (b c-a d)^2}+\frac{1}{3 \left(c+d x^2\right)^{3/2} (b c-a d)}",1,"Hypergeometric2F1[-3/2, 1, -1/2, (b*(c + d*x^2))/(b*c - a*d)]/(3*(b*c - a*d)*(c + d*x^2)^(3/2))","C",1
726,1,1385,122,6.260631,"\int \frac{1}{\left(a+b x^2\right) \left(c+d x^2\right)^{5/2}} \, dx","Integrate[1/((a + b*x^2)*(c + d*x^2)^(5/2)),x]","\frac{x \sqrt{\frac{a \left(d x^2+c\right)}{c \left(b x^2+a\right)}} \left(\frac{840 d^2 (b c-a d)^2 \sin ^{-1}\left(\sqrt{\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}}\right) x^8}{c^4 \left(b x^2+a\right)^2}+\frac{1680 d^2 (a d-b c) \sin ^{-1}\left(\sqrt{\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}}\right) x^6}{c^3 \left(b x^2+a\right)}+\frac{2100 d (b c-a d)^2 \sin ^{-1}\left(\sqrt{\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}}\right) x^6}{c^3 \left(b x^2+a\right)^2}+\frac{840 d^2 \sin ^{-1}\left(\sqrt{\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}}\right) x^4}{c^2}+\frac{4200 d (a d-b c) \sin ^{-1}\left(\sqrt{\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}}\right) x^4}{c^2 \left(b x^2+a\right)}+\frac{1575 (b c-a d)^2 \sin ^{-1}\left(\sqrt{\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}}\right) x^4}{c^2 \left(b x^2+a\right)^2}+\frac{72 d^2 \left(\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)^{7/2} \sqrt{\frac{a \left(d x^2+c\right)}{c \left(b x^2+a\right)}} \, _2F_1\left(2,2;\frac{9}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right) x^4}{c^2}+\frac{24 d^2 \left(\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)^{7/2} \sqrt{\frac{a \left(d x^2+c\right)}{c \left(b x^2+a\right)}} \, _3F_2\left(2,2,2;1,\frac{9}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right) x^4}{c^2}-\frac{840 d^2 \sqrt{\frac{a (b c-a d) x^2 \left(d x^2+c\right)}{c^2 \left(b x^2+a\right)^2}} x^4}{c^2}+\frac{1120 d^2 \left(\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)^{3/2} \sqrt{\frac{a \left(d x^2+c\right)}{c \left(b x^2+a\right)}} x^4}{c^2}+\frac{2100 d \sin ^{-1}\left(\sqrt{\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}}\right) x^2}{c}-\frac{3150 (b c-a d) \sin ^{-1}\left(\sqrt{\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}}\right) x^2}{c \left(b x^2+a\right)}+\frac{168 d \left(\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)^{7/2} \sqrt{\frac{a \left(d x^2+c\right)}{c \left(b x^2+a\right)}} \, _2F_1\left(2,2;\frac{9}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right) x^2}{c}+\frac{48 d \left(\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)^{7/2} \sqrt{\frac{a \left(d x^2+c\right)}{c \left(b x^2+a\right)}} \, _3F_2\left(2,2,2;1,\frac{9}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right) x^2}{c}-\frac{2100 d \sqrt{\frac{a (b c-a d) x^2 \left(d x^2+c\right)}{c^2 \left(b x^2+a\right)^2}} x^2}{c}+\frac{2800 d \left(\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)^{3/2} \sqrt{\frac{a \left(d x^2+c\right)}{c \left(b x^2+a\right)}} x^2}{c}+1575 \sin ^{-1}\left(\sqrt{\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}}\right)+96 \left(\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)^{7/2} \sqrt{\frac{a \left(d x^2+c\right)}{c \left(b x^2+a\right)}} \, _2F_1\left(2,2;\frac{9}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)+24 \left(\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)^{7/2} \sqrt{\frac{a \left(d x^2+c\right)}{c \left(b x^2+a\right)}} \, _3F_2\left(2,2,2;1,\frac{9}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)-1575 \sqrt{\frac{a (b c-a d) x^2 \left(d x^2+c\right)}{c^2 \left(b x^2+a\right)^2}}+2100 \left(\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)^{3/2} \sqrt{\frac{a \left(d x^2+c\right)}{c \left(b x^2+a\right)}}\right)}{315 a \left(\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)^{5/2} \left(d x^2+c\right)^{5/2}}","\frac{b^2 \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{\sqrt{a} (b c-a d)^{5/2}}-\frac{d x (5 b c-2 a d)}{3 c^2 \sqrt{c+d x^2} (b c-a d)^2}-\frac{d x}{3 c \left(c+d x^2\right)^{3/2} (b c-a d)}",1,"(x*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*(-1575*Sqrt[(a*(b*c - a*d)*x^2*(c + d*x^2))/(c^2*(a + b*x^2)^2)] - (2100*d*x^2*Sqrt[(a*(b*c - a*d)*x^2*(c + d*x^2))/(c^2*(a + b*x^2)^2)])/c - (840*d^2*x^4*Sqrt[(a*(b*c - a*d)*x^2*(c + d*x^2))/(c^2*(a + b*x^2)^2)])/c^2 + 2100*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(3/2)*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))] + (2800*d*x^2*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(3/2)*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))])/c + (1120*d^2*x^4*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(3/2)*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))])/c^2 + 1575*ArcSin[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]] + (2100*d*x^2*ArcSin[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/c + (840*d^2*x^4*ArcSin[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/c^2 + (1575*(b*c - a*d)^2*x^4*ArcSin[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/(c^2*(a + b*x^2)^2) + (2100*d*(b*c - a*d)^2*x^6*ArcSin[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/(c^3*(a + b*x^2)^2) + (840*d^2*(b*c - a*d)^2*x^8*ArcSin[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/(c^4*(a + b*x^2)^2) - (3150*(b*c - a*d)*x^2*ArcSin[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/(c*(a + b*x^2)) + (4200*d*(-(b*c) + a*d)*x^4*ArcSin[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/(c^2*(a + b*x^2)) + (1680*d^2*(-(b*c) + a*d)*x^6*ArcSin[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/(c^3*(a + b*x^2)) + 96*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(7/2)*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*Hypergeometric2F1[2, 2, 9/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + (168*d*x^2*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(7/2)*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*Hypergeometric2F1[2, 2, 9/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))])/c + (72*d^2*x^4*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(7/2)*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*Hypergeometric2F1[2, 2, 9/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))])/c^2 + 24*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(7/2)*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + (48*d*x^2*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(7/2)*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, ((b*c - a*d)*x^2)/(c*(a + b*x^2))])/c + (24*d^2*x^4*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(7/2)*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, ((b*c - a*d)*x^2)/(c*(a + b*x^2))])/c^2))/(315*a*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(5/2)*(c + d*x^2)^(5/2))","C",0
727,1,90,145,0.0324117,"\int \frac{1}{x \left(a+b x^2\right) \left(c+d x^2\right)^{5/2}} \, dx","Integrate[1/(x*(a + b*x^2)*(c + d*x^2)^(5/2)),x]","\frac{(b c-a d) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{d x^2}{c}+1\right)-b c \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \left(d x^2+c\right)}{b c-a d}\right)}{3 a c \left(c+d x^2\right)^{3/2} (b c-a d)}","\frac{b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{a (b c-a d)^{5/2}}-\frac{d (2 b c-a d)}{c^2 \sqrt{c+d x^2} (b c-a d)^2}-\frac{d}{3 c \left(c+d x^2\right)^{3/2} (b c-a d)}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{a c^{5/2}}",1,"(-(b*c*Hypergeometric2F1[-3/2, 1, -1/2, (b*(c + d*x^2))/(b*c - a*d)]) + (b*c - a*d)*Hypergeometric2F1[-3/2, 1, -1/2, 1 + (d*x^2)/c])/(3*a*c*(b*c - a*d)*(c + d*x^2)^(3/2))","C",1
728,1,143,178,5.2675433,"\int \frac{1}{x^2 \left(a+b x^2\right) \left(c+d x^2\right)^{5/2}} \, dx","Integrate[1/(x^2*(a + b*x^2)*(c + d*x^2)^(5/2)),x]","\frac{\sqrt{c+d x^2} \left(\frac{d^2 x^2 (8 b c-5 a d)}{\left(c+d x^2\right) (b c-a d)^2}+\frac{c d^2 x^2}{\left(c+d x^2\right)^2 (b c-a d)}-\frac{3}{a}\right)}{3 c^3 x}-\frac{b^3 \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{a^{3/2} (b c-a d)^{5/2}}","-\frac{b^3 \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{a^{3/2} (b c-a d)^{5/2}}-\frac{\sqrt{c+d x^2} (b c-4 a d) (3 b c-2 a d)}{3 a c^3 x (b c-a d)^2}-\frac{d (7 b c-4 a d)}{3 c^2 x \sqrt{c+d x^2} (b c-a d)^2}-\frac{d}{3 c x \left(c+d x^2\right)^{3/2} (b c-a d)}",1,"(Sqrt[c + d*x^2]*(-3/a + (c*d^2*x^2)/((b*c - a*d)*(c + d*x^2)^2) + (d^2*(8*b*c - 5*a*d)*x^2)/((b*c - a*d)^2*(c + d*x^2))))/(3*c^3*x) - (b^3*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(a^(3/2)*(b*c - a*d)^(5/2))","A",1
729,1,118,211,0.0528417,"\int \frac{1}{x^3 \left(a+b x^2\right) \left(c+d x^2\right)^{5/2}} \, dx","Integrate[1/(x^3*(a + b*x^2)*(c + d*x^2)^(5/2)),x]","\frac{2 b^2 c^2 x^2 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \left(d x^2+c\right)}{b c-a d}\right)+(a d-b c) \left(x^2 (5 a d+2 b c) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{d x^2}{c}+1\right)+3 a c\right)}{6 a^2 c^2 x^2 \left(c+d x^2\right)^{3/2} (b c-a d)}","-\frac{b^{7/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{a^2 (b c-a d)^{5/2}}-\frac{d \left(5 a^2 d^2-8 a b c d+b^2 c^2\right)}{2 a c^3 \sqrt{c+d x^2} (b c-a d)^2}+\frac{(5 a d+2 b c) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{2 a^2 c^{7/2}}-\frac{d (3 b c-5 a d)}{6 a c^2 \left(c+d x^2\right)^{3/2} (b c-a d)}-\frac{1}{2 a c x^2 \left(c+d x^2\right)^{3/2}}",1,"(2*b^2*c^2*x^2*Hypergeometric2F1[-3/2, 1, -1/2, (b*(c + d*x^2))/(b*c - a*d)] + (-(b*c) + a*d)*(3*a*c + (2*b*c + 5*a*d)*x^2*Hypergeometric2F1[-3/2, 1, -1/2, 1 + (d*x^2)/c]))/(6*a^2*c^2*(b*c - a*d)*x^2*(c + d*x^2)^(3/2))","C",1
730,1,160,245,5.3616975,"\int \frac{1}{x^4 \left(a+b x^2\right) \left(c+d x^2\right)^{5/2}} \, dx","Integrate[1/(x^4*(a + b*x^2)*(c + d*x^2)^(5/2)),x]","\frac{b^4 \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{a^{5/2} (b c-a d)^{5/2}}+\frac{\sqrt{c+d x^2} \left(\frac{x^2 (8 a d+3 b c)}{a^2}+\frac{d^3 x^4 (8 a d-11 b c)}{\left(c+d x^2\right) (b c-a d)^2}-\frac{c d^3 x^4}{\left(c+d x^2\right)^2 (b c-a d)}-\frac{c}{a}\right)}{3 c^4 x^3}","\frac{b^4 \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{a^{5/2} (b c-a d)^{5/2}}+\frac{\sqrt{c+d x^2} (b c-2 a d) \left(-8 a^2 d^2+8 a b c d+3 b^2 c^2\right)}{3 a^2 c^4 x (b c-a d)^2}-\frac{\sqrt{c+d x^2} \left(8 a^2 d^2-12 a b c d+b^2 c^2\right)}{3 a c^3 x^3 (b c-a d)^2}-\frac{d (3 b c-2 a d)}{c^2 x^3 \sqrt{c+d x^2} (b c-a d)^2}-\frac{d}{3 c x^3 \left(c+d x^2\right)^{3/2} (b c-a d)}",1,"(Sqrt[c + d*x^2]*(-(c/a) + ((3*b*c + 8*a*d)*x^2)/a^2 - (c*d^3*x^4)/((b*c - a*d)*(c + d*x^2)^2) + (d^3*(-11*b*c + 8*a*d)*x^4)/((b*c - a*d)^2*(c + d*x^2))))/(3*c^4*x^3) + (b^4*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(a^(5/2)*(b*c - a*d)^(5/2))","A",1
731,1,134,150,0.1803049,"\int \frac{x^4 \sqrt{c+d x^2}}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^4*Sqrt[c + d*x^2])/(a + b*x^2)^2,x]","\frac{\frac{b x \left(2 a+b x^2\right) \sqrt{c+d x^2}}{a+b x^2}+\frac{(b c-4 a d) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{\sqrt{d}}+\frac{\sqrt{a} (4 a d-3 b c) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{\sqrt{b c-a d}}}{2 b^3}","-\frac{\sqrt{a} (3 b c-4 a d) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 b^3 \sqrt{b c-a d}}+\frac{(b c-4 a d) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{2 b^3 \sqrt{d}}-\frac{x^3 \sqrt{c+d x^2}}{2 b \left(a+b x^2\right)}+\frac{x \sqrt{c+d x^2}}{b^2}",1,"((b*x*(2*a + b*x^2)*Sqrt[c + d*x^2])/(a + b*x^2) + (Sqrt[a]*(-3*b*c + 4*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/Sqrt[b*c - a*d] + ((b*c - 4*a*d)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/Sqrt[d])/(2*b^3)","A",1
732,1,117,136,0.1449794,"\int \frac{x^3 \sqrt{c+d x^2}}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^3*Sqrt[c + d*x^2])/(a + b*x^2)^2,x]","\frac{\frac{(2 b c-3 a d) \left(\sqrt{b} \sqrt{c+d x^2}-\sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)\right)}{b^{3/2}}+\frac{a \left(c+d x^2\right)^{3/2}}{a+b x^2}}{2 b (b c-a d)}","-\frac{(2 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{2 b^{5/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^2} (2 b c-3 a d)}{2 b^2 (b c-a d)}+\frac{a \left(c+d x^2\right)^{3/2}}{2 b \left(a+b x^2\right) (b c-a d)}",1,"((a*(c + d*x^2)^(3/2))/(a + b*x^2) + ((2*b*c - 3*a*d)*(Sqrt[b]*Sqrt[c + d*x^2] - Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]]))/b^(3/2))/(2*b*(b*c - a*d))","A",1
733,1,118,120,0.0828808,"\int \frac{x^2 \sqrt{c+d x^2}}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^2*Sqrt[c + d*x^2])/(a + b*x^2)^2,x]","\frac{-\frac{b x \sqrt{c+d x^2}}{a+b x^2}+\frac{(b c-2 a d) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{\sqrt{a} \sqrt{b c-a d}}+2 \sqrt{d} \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{2 b^2}","\frac{(b c-2 a d) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 \sqrt{a} b^2 \sqrt{b c-a d}}-\frac{x \sqrt{c+d x^2}}{2 b \left(a+b x^2\right)}+\frac{\sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{b^2}",1,"(-((b*x*Sqrt[c + d*x^2])/(a + b*x^2)) + ((b*c - 2*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(Sqrt[a]*Sqrt[b*c - a*d]) + 2*Sqrt[d]*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(2*b^2)","A",1
734,1,80,80,0.0840124,"\int \frac{x \sqrt{c+d x^2}}{\left(a+b x^2\right)^2} \, dx","Integrate[(x*Sqrt[c + d*x^2])/(a + b*x^2)^2,x]","\frac{d \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{a d-b c}}\right)}{2 b^{3/2} \sqrt{a d-b c}}-\frac{\sqrt{c+d x^2}}{2 b \left(a+b x^2\right)}","-\frac{d \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{2 b^{3/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x^2}}{2 b \left(a+b x^2\right)}",1,"-1/2*Sqrt[c + d*x^2]/(b*(a + b*x^2)) + (d*ArcTan[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[-(b*c) + a*d]])/(2*b^(3/2)*Sqrt[-(b*c) + a*d])","A",1
735,1,112,82,0.2934368,"\int \frac{\sqrt{c+d x^2}}{\left(a+b x^2\right)^2} \, dx","Integrate[Sqrt[c + d*x^2]/(a + b*x^2)^2,x]","\frac{x \left(\frac{a \left(c+d x^2\right)}{a+b x^2}+\frac{c \sqrt{\frac{d x^2}{c}+1} \tanh ^{-1}\left(\frac{\sqrt{x^2 \left(\frac{d}{c}-\frac{b}{a}\right)}}{\sqrt{\frac{d x^2}{c}+1}}\right)}{\sqrt{\frac{x^2 (a d-b c)}{a c}}}\right)}{2 a^2 \sqrt{c+d x^2}}","\frac{c \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 a^{3/2} \sqrt{b c-a d}}+\frac{x \sqrt{c+d x^2}}{2 a \left(a+b x^2\right)}",1,"(x*((a*(c + d*x^2))/(a + b*x^2) + (c*Sqrt[1 + (d*x^2)/c]*ArcTanh[Sqrt[(-(b/a) + d/c)*x^2]/Sqrt[1 + (d*x^2)/c]])/Sqrt[((-(b*c) + a*d)*x^2)/(a*c)]))/(2*a^2*Sqrt[c + d*x^2])","A",1
736,1,112,119,0.2189785,"\int \frac{\sqrt{c+d x^2}}{x \left(a+b x^2\right)^2} \, dx","Integrate[Sqrt[c + d*x^2]/(x*(a + b*x^2)^2),x]","\frac{\frac{a \sqrt{c+d x^2}}{a+b x^2}+\frac{(2 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{\sqrt{b} \sqrt{b c-a d}}-2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{2 a^2}","\frac{(2 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{2 a^2 \sqrt{b} \sqrt{b c-a d}}-\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{a^2}+\frac{\sqrt{c+d x^2}}{2 a \left(a+b x^2\right)}",1,"((a*Sqrt[c + d*x^2])/(a + b*x^2) - 2*Sqrt[c]*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]] + ((2*b*c - a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(Sqrt[b]*Sqrt[b*c - a*d]))/(2*a^2)","A",1
737,1,101,113,5.0729725,"\int \frac{\sqrt{c+d x^2}}{x^2 \left(a+b x^2\right)^2} \, dx","Integrate[Sqrt[c + d*x^2]/(x^2*(a + b*x^2)^2),x]","\frac{(2 a d-3 b c) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 a^{5/2} \sqrt{b c-a d}}+\left(-\frac{b x}{2 a^2 \left(a+b x^2\right)}-\frac{1}{a^2 x}\right) \sqrt{c+d x^2}","-\frac{(3 b c-2 a d) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 a^{5/2} \sqrt{b c-a d}}-\frac{3 \sqrt{c+d x^2}}{2 a^2 x}+\frac{\sqrt{c+d x^2}}{2 a x \left(a+b x^2\right)}",1,"Sqrt[c + d*x^2]*(-(1/(a^2*x)) - (b*x)/(2*a^2*(a + b*x^2))) + ((-3*b*c + 2*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(5/2)*Sqrt[b*c - a*d])","A",1
738,1,190,159,0.2328817,"\int \frac{\sqrt{c+d x^2}}{x^3 \left(a+b x^2\right)^2} \, dx","Integrate[Sqrt[c + d*x^2]/(x^3*(a + b*x^2)^2),x]","\frac{\sqrt{c} \left(a \left(a+2 b x^2\right) \sqrt{c+d x^2} (b c-a d)+\sqrt{b} x^2 \left(a+b x^2\right) (4 b c-3 a d) \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)\right)-x^2 \left(a+b x^2\right) \left(a^2 d^2-5 a b c d+4 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{2 a^3 \sqrt{c} x^2 \left(a+b x^2\right) (a d-b c)}","-\frac{\sqrt{b} (4 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{2 a^3 \sqrt{b c-a d}}+\frac{(4 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{2 a^3 \sqrt{c}}-\frac{b \sqrt{c+d x^2}}{a^2 \left(a+b x^2\right)}-\frac{\sqrt{c+d x^2}}{2 a x^2 \left(a+b x^2\right)}",1,"(-((4*b^2*c^2 - 5*a*b*c*d + a^2*d^2)*x^2*(a + b*x^2)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]]) + Sqrt[c]*(a*(b*c - a*d)*(a + 2*b*x^2)*Sqrt[c + d*x^2] + Sqrt[b]*(4*b*c - 3*a*d)*Sqrt[b*c - a*d]*x^2*(a + b*x^2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]]))/(2*a^3*Sqrt[c]*(-(b*c) + a*d)*x^2*(a + b*x^2))","A",1
739,1,120,147,5.1439318,"\int \frac{\sqrt{c+d x^2}}{x^4 \left(a+b x^2\right)^2} \, dx","Integrate[Sqrt[c + d*x^2]/(x^4*(a + b*x^2)^2),x]","\frac{b (5 b c-4 a d) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 a^{7/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^2} \left(3 b x^2 \left(\frac{b x^2}{a+b x^2}+4\right)-\frac{2 a \left(c+d x^2\right)}{c}\right)}{6 a^3 x^3}","\frac{b (5 b c-4 a d) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 a^{7/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^2} (15 b c-2 a d)}{6 a^3 c x}-\frac{5 \sqrt{c+d x^2}}{6 a^2 x^3}+\frac{\sqrt{c+d x^2}}{2 a x^3 \left(a+b x^2\right)}",1,"(Sqrt[c + d*x^2]*((-2*a*(c + d*x^2))/c + 3*b*x^2*(4 + (b*x^2)/(a + b*x^2))))/(6*a^3*x^3) + (b*(5*b*c - 4*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(7/2)*Sqrt[b*c - a*d])","A",1
740,1,192,197,0.1682958,"\int \frac{x^4 \left(c+d x^2\right)^{3/2}}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^4*(c + d*x^2)^(3/2))/(a + b*x^2)^2,x]","\frac{\frac{3 \left(8 a^2 d^2-8 a b c d+b^2 c^2\right) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{\sqrt{d}}-\frac{12 \sqrt{a} \left(2 a^2 d^2-3 a b c d+b^2 c^2\right) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{\sqrt{b c-a d}}+\frac{b \sqrt{c+d x^2} \left(-12 a^2 d x+a b \left(9 c x-6 d x^3\right)+b^2 x^3 \left(5 c+2 d x^2\right)\right)}{a+b x^2}}{8 b^4}","\frac{3 \left(8 a^2 d^2-8 a b c d+b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{8 b^4 \sqrt{d}}-\frac{3 \sqrt{a} (b c-2 a d) \sqrt{b c-a d} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 b^4}+\frac{3 x \sqrt{c+d x^2} (3 b c-4 a d)}{8 b^3}-\frac{x^3 \left(c+d x^2\right)^{3/2}}{2 b \left(a+b x^2\right)}+\frac{3 d x^3 \sqrt{c+d x^2}}{4 b^2}",1,"((b*Sqrt[c + d*x^2]*(-12*a^2*d*x + b^2*x^3*(5*c + 2*d*x^2) + a*b*(9*c*x - 6*d*x^3)))/(a + b*x^2) - (12*Sqrt[a]*(b^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/Sqrt[b*c - a*d] + (3*(b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/Sqrt[d])/(8*b^4)","A",1
741,1,125,163,0.1085988,"\int \frac{x^3 \left(c+d x^2\right)^{3/2}}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^3*(c + d*x^2)^(3/2))/(a + b*x^2)^2,x]","\frac{\sqrt{c+d x^2} \left(-15 a^2 d+a b \left(11 c-10 d x^2\right)+2 b^2 x^2 \left(4 c+d x^2\right)\right)}{6 b^3 \left(a+b x^2\right)}-\frac{(2 b c-5 a d) \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{2 b^{7/2}}","-\frac{(2 b c-5 a d) \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{2 b^{7/2}}+\frac{\sqrt{c+d x^2} (2 b c-5 a d)}{2 b^3}+\frac{\left(c+d x^2\right)^{3/2} (2 b c-5 a d)}{6 b^2 (b c-a d)}+\frac{a \left(c+d x^2\right)^{5/2}}{2 b \left(a+b x^2\right) (b c-a d)}",1,"(Sqrt[c + d*x^2]*(-15*a^2*d + a*b*(11*c - 10*d*x^2) + 2*b^2*x^2*(4*c + d*x^2)))/(6*b^3*(a + b*x^2)) - ((2*b*c - 5*a*d)*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(2*b^(7/2))","A",1
742,1,155,149,0.1765647,"\int \frac{x^2 \left(c+d x^2\right)^{3/2}}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^2*(c + d*x^2)^(3/2))/(a + b*x^2)^2,x]","\frac{\frac{\left(4 a^2 d^2-5 a b c d+b^2 c^2\right) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{\sqrt{a} \sqrt{b c-a d}}-\frac{b x \sqrt{c+d x^2} \left(b \left(c-d x^2\right)-2 a d\right)}{a+b x^2}+\sqrt{d} (3 b c-4 a d) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{2 b^3}","\frac{(b c-4 a d) \sqrt{b c-a d} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 \sqrt{a} b^3}+\frac{\sqrt{d} (3 b c-4 a d) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{2 b^3}-\frac{x \left(c+d x^2\right)^{3/2}}{2 b \left(a+b x^2\right)}+\frac{d x \sqrt{c+d x^2}}{b^2}",1,"(-((b*x*Sqrt[c + d*x^2]*(-2*a*d + b*(c - d*x^2)))/(a + b*x^2)) + ((b^2*c^2 - 5*a*b*c*d + 4*a^2*d^2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(Sqrt[a]*Sqrt[b*c - a*d]) + Sqrt[d]*(3*b*c - 4*a*d)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(2*b^3)","A",1
743,1,54,99,0.0209746,"\int \frac{x \left(c+d x^2\right)^{3/2}}{\left(a+b x^2\right)^2} \, dx","Integrate[(x*(c + d*x^2)^(3/2))/(a + b*x^2)^2,x]","\frac{d \left(c+d x^2\right)^{5/2} \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};-\frac{b \left(d x^2+c\right)}{a d-b c}\right)}{5 (a d-b c)^2}","-\frac{3 d \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{2 b^{5/2}}-\frac{\left(c+d x^2\right)^{3/2}}{2 b \left(a+b x^2\right)}+\frac{3 d \sqrt{c+d x^2}}{2 b^2}",1,"(d*(c + d*x^2)^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, -((b*(c + d*x^2))/(-(b*c) + a*d))])/(5*(-(b*c) + a*d)^2)","C",1
744,1,141,131,0.1552709,"\int \frac{\left(c+d x^2\right)^{3/2}}{\left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)^(3/2)/(a + b*x^2)^2,x]","\frac{\frac{\left(-2 a^2 d^2+a b c d+b^2 c^2\right) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{a^{3/2} \sqrt{b c-a d}}+\frac{b x \sqrt{c+d x^2} (b c-a d)}{a \left(a+b x^2\right)}+2 d^{3/2} \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{2 b^2}","\frac{\sqrt{b c-a d} (2 a d+b c) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 a^{3/2} b^2}+\frac{x \sqrt{c+d x^2} (b c-a d)}{2 a b \left(a+b x^2\right)}+\frac{d^{3/2} \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{b^2}",1,"((b*(b*c - a*d)*x*Sqrt[c + d*x^2])/(a*(a + b*x^2)) + ((b^2*c^2 + a*b*c*d - 2*a^2*d^2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(a^(3/2)*Sqrt[b*c - a*d]) + 2*d^(3/2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(2*b^2)","A",1
745,1,122,129,0.1623658,"\int \frac{\left(c+d x^2\right)^{3/2}}{x \left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)^(3/2)/(x*(a + b*x^2)^2),x]","\frac{\frac{\sqrt{b c-a d} (a d+2 b c) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{b^{3/2}}+\frac{a \sqrt{c+d x^2} (b c-a d)}{b \left(a+b x^2\right)}-2 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{2 a^2}","\frac{\sqrt{b c-a d} (a d+2 b c) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{2 a^2 b^{3/2}}-\frac{c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{a^2}+\frac{\sqrt{c+d x^2} (b c-a d)}{2 a b \left(a+b x^2\right)}",1,"((a*(b*c - a*d)*Sqrt[c + d*x^2])/(b*(a + b*x^2)) - 2*c^(3/2)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]] + (Sqrt[b*c - a*d]*(2*b*c + a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/b^(3/2))/(2*a^2)","A",1
746,1,52,128,0.0143419,"\int \frac{\left(c+d x^2\right)^{3/2}}{x^2 \left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)^(3/2)/(x^2*(a + b*x^2)^2),x]","-\frac{c \sqrt{c+d x^2} \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};\frac{(a d-b c) x^2}{a \left(d x^2+c\right)}\right)}{a^2 x}","-\frac{3 c \sqrt{b c-a d} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 a^{5/2}}-\frac{\sqrt{c+d x^2} (3 b c-a d)}{2 a^2 b x}+\frac{\sqrt{c+d x^2} (b c-a d)}{2 a b x \left(a+b x^2\right)}",1,"-((c*Sqrt[c + d*x^2]*Hypergeometric2F1[-1/2, 2, 1/2, ((-(b*c) + a*d)*x^2)/(a*(c + d*x^2))])/(a^2*x))","C",1
747,1,142,170,0.3236586,"\int \frac{\left(c+d x^2\right)^{3/2}}{x^3 \left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)^(3/2)/(x^3*(a + b*x^2)^2),x]","\frac{\frac{a \sqrt{c+d x^2} \left(-a c+a d x^2-2 b c x^2\right)}{x^2 \left(a+b x^2\right)}+\sqrt{c} (4 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)-\frac{\sqrt{b c-a d} (4 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{\sqrt{b}}}{2 a^3}","\frac{\sqrt{c} (4 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{2 a^3}-\frac{\sqrt{b c-a d} (4 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{2 a^3 \sqrt{b}}-\frac{\sqrt{c+d x^2} (2 b c-a d)}{2 a^2 \left(a+b x^2\right)}-\frac{c \sqrt{c+d x^2}}{2 a x^2 \left(a+b x^2\right)}",1,"((a*Sqrt[c + d*x^2]*(-(a*c) - 2*b*c*x^2 + a*d*x^2))/(x^2*(a + b*x^2)) + Sqrt[c]*(4*b*c - 3*a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]] - (Sqrt[b*c - a*d]*(4*b*c - a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/Sqrt[b])/(2*a^3)","A",1
748,1,131,166,5.1596883,"\int \frac{\left(c+d x^2\right)^{3/2}}{x^4 \left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)^(3/2)/(x^4*(a + b*x^2)^2),x]","\frac{(5 b c-2 a d) \sqrt{b c-a d} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 a^{7/2}}+\frac{\sqrt{c+d x^2} \left(-2 a^2 \left(c+4 d x^2\right)+a b x^2 \left(10 c-11 d x^2\right)+15 b^2 c x^4\right)}{6 a^3 x^3 \left(a+b x^2\right)}","\frac{(5 b c-2 a d) \sqrt{b c-a d} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 a^{7/2}}+\frac{\sqrt{c+d x^2} (15 b c-11 a d)}{6 a^3 x}-\frac{\sqrt{c+d x^2} (5 b c-3 a d)}{6 a^2 b x^3}+\frac{\sqrt{c+d x^2} (b c-a d)}{2 a b x^3 \left(a+b x^2\right)}",1,"(Sqrt[c + d*x^2]*(15*b^2*c*x^4 + a*b*x^2*(10*c - 11*d*x^2) - 2*a^2*(c + 4*d*x^2)))/(6*a^3*x^3*(a + b*x^2)) + ((5*b*c - 2*a*d)*Sqrt[b*c - a*d]*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(7/2))","A",1
749,1,219,258,0.2920871,"\int \frac{x^4 \left(c+d x^2\right)^{5/2}}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^4*(c + d*x^2)^(5/2))/(a + b*x^2)^2,x]","\frac{b x \sqrt{c+d x^2} \left(72 a^2 d^2+2 b d x^2 (13 b c-12 a d)+\frac{24 a (b c-a d)^2}{a+b x^2}-108 a b c d+33 b^2 c^2+8 b^2 d^2 x^4\right)+\frac{3 \left(-64 a^3 d^3+120 a^2 b c d^2-60 a b^2 c^2 d+5 b^3 c^3\right) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{\sqrt{d}}+24 \sqrt{a} (8 a d-3 b c) (b c-a d)^{3/2} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{48 b^5}","\frac{x \sqrt{c+d x^2} \left(32 a^2 d^2-52 a b c d+19 b^2 c^2\right)}{16 b^4}+\frac{\left(-64 a^3 d^3+120 a^2 b c d^2-60 a b^2 c^2 d+5 b^3 c^3\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{16 b^5 \sqrt{d}}-\frac{\sqrt{a} (3 b c-8 a d) (b c-a d)^{3/2} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 b^5}+\frac{d x^3 \sqrt{c+d x^2} (7 b c-8 a d)}{8 b^3}-\frac{x^3 \left(c+d x^2\right)^{5/2}}{2 b \left(a+b x^2\right)}+\frac{2 d x^3 \left(c+d x^2\right)^{3/2}}{3 b^2}",1,"(b*x*Sqrt[c + d*x^2]*(33*b^2*c^2 - 108*a*b*c*d + 72*a^2*d^2 + 2*b*d*(13*b*c - 12*a*d)*x^2 + 8*b^2*d^2*x^4 + (24*a*(b*c - a*d)^2)/(a + b*x^2)) + 24*Sqrt[a]*(b*c - a*d)^(3/2)*(-3*b*c + 8*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])] + (3*(5*b^3*c^3 - 60*a*b^2*c^2*d + 120*a^2*b*c*d^2 - 64*a^3*d^3)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/Sqrt[d])/(48*b^5)","A",1
750,1,164,198,0.431631,"\int \frac{x^3 \left(c+d x^2\right)^{5/2}}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^3*(c + d*x^2)^(5/2))/(a + b*x^2)^2,x]","\frac{\left(b c-\frac{7 a d}{2}\right) \left(\frac{2 (b c-a d) \left(\sqrt{b} \sqrt{c+d x^2} \left(-3 a d+4 b c+b d x^2\right)-3 (b c-a d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)\right)}{3 b^{7/2}}+\frac{2 \left(c+d x^2\right)^{5/2}}{5 b}\right)+\frac{a \left(c+d x^2\right)^{7/2}}{a+b x^2}}{2 b (b c-a d)}","-\frac{(2 b c-7 a d) (b c-a d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{2 b^{9/2}}+\frac{\sqrt{c+d x^2} (2 b c-7 a d) (b c-a d)}{2 b^4}+\frac{\left(c+d x^2\right)^{3/2} (2 b c-7 a d)}{6 b^3}+\frac{\left(c+d x^2\right)^{5/2} (2 b c-7 a d)}{10 b^2 (b c-a d)}+\frac{a \left(c+d x^2\right)^{7/2}}{2 b \left(a+b x^2\right) (b c-a d)}",1,"((a*(c + d*x^2)^(7/2))/(a + b*x^2) + (b*c - (7*a*d)/2)*((2*(c + d*x^2)^(5/2))/(5*b) + (2*(b*c - a*d)*(Sqrt[b]*Sqrt[c + d*x^2]*(4*b*c - 3*a*d + b*d*x^2) - 3*(b*c - a*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]]))/(3*b^(7/2))))/(2*b*(b*c - a*d))","A",1
751,1,173,195,0.2295254,"\int \frac{x^2 \left(c+d x^2\right)^{5/2}}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^2*(c + d*x^2)^(5/2))/(a + b*x^2)^2,x]","\frac{\sqrt{d} \left(24 a^2 d^2-40 a b c d+15 b^2 c^2\right) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)+b x \sqrt{c+d x^2} \left(-\frac{4 (b c-a d)^2}{a+b x^2}+d (9 b c-8 a d)+2 b d^2 x^2\right)+\frac{4 (b c-6 a d) (b c-a d)^{3/2} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{\sqrt{a}}}{8 b^4}","\frac{\sqrt{d} \left(24 a^2 d^2-40 a b c d+15 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{8 b^4}+\frac{(b c-6 a d) (b c-a d)^{3/2} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 \sqrt{a} b^4}+\frac{d x \sqrt{c+d x^2} (11 b c-12 a d)}{8 b^3}-\frac{x \left(c+d x^2\right)^{5/2}}{2 b \left(a+b x^2\right)}+\frac{3 d x \left(c+d x^2\right)^{3/2}}{4 b^2}",1,"(b*x*Sqrt[c + d*x^2]*(d*(9*b*c - 8*a*d) + 2*b*d^2*x^2 - (4*(b*c - a*d)^2)/(a + b*x^2)) + (4*(b*c - 6*a*d)*(b*c - a*d)^(3/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/Sqrt[a] + Sqrt[d]*(15*b^2*c^2 - 40*a*b*c*d + 24*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(8*b^4)","A",1
752,1,54,126,0.0206433,"\int \frac{x \left(c+d x^2\right)^{5/2}}{\left(a+b x^2\right)^2} \, dx","Integrate[(x*(c + d*x^2)^(5/2))/(a + b*x^2)^2,x]","\frac{d \left(c+d x^2\right)^{7/2} \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};-\frac{b \left(d x^2+c\right)}{a d-b c}\right)}{7 (a d-b c)^2}","-\frac{5 d (b c-a d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{2 b^{7/2}}+\frac{5 d \sqrt{c+d x^2} (b c-a d)}{2 b^3}-\frac{\left(c+d x^2\right)^{5/2}}{2 b \left(a+b x^2\right)}+\frac{5 d \left(c+d x^2\right)^{3/2}}{6 b^2}",1,"(d*(c + d*x^2)^(7/2)*Hypergeometric2F1[2, 7/2, 9/2, -((b*(c + d*x^2))/(-(b*c) + a*d))])/(7*(-(b*c) + a*d)^2)","C",1
753,1,143,174,0.2080507,"\int \frac{\left(c+d x^2\right)^{5/2}}{\left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)^(5/2)/(a + b*x^2)^2,x]","\frac{\frac{(b c-a d)^{3/2} (4 a d+b c) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{a^{3/2}}+d^{3/2} (5 b c-4 a d) \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)+b x \sqrt{c+d x^2} \left(\frac{(b c-a d)^2}{a \left(a+b x^2\right)}+d^2\right)}{2 b^3}","\frac{(b c-a d)^{3/2} (4 a d+b c) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 a^{3/2} b^3}+\frac{d^{3/2} (5 b c-4 a d) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{2 b^3}-\frac{d x \sqrt{c+d x^2} (b c-2 a d)}{2 a b^2}+\frac{x \left(c+d x^2\right)^{3/2} (b c-a d)}{2 a b \left(a+b x^2\right)}",1,"(b*x*Sqrt[c + d*x^2]*(d^2 + (b*c - a*d)^2/(a*(a + b*x^2))) + ((b*c - a*d)^(3/2)*(b*c + 4*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/a^(3/2) + d^(3/2)*(5*b*c - 4*a*d)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(2*b^3)","A",1
754,1,158,160,0.2103961,"\int \frac{\left(c+d x^2\right)^{5/2}}{x \left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)^(5/2)/(x*(a + b*x^2)^2),x]","\frac{\frac{a \sqrt{c+d x^2} \left(3 a^2 d^2+2 a b d \left(d x^2-c\right)+b^2 c^2\right)}{b^2 \left(a+b x^2\right)}+\frac{\sqrt{b c-a d} \left(-3 a^2 d^2+a b c d+2 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{b^{5/2}}-2 c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{2 a^2}","\frac{(b c-a d)^{3/2} (3 a d+2 b c) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{2 a^2 b^{5/2}}-\frac{c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{a^2}-\frac{d \sqrt{c+d x^2} (b c-3 a d)}{2 a b^2}+\frac{\left(c+d x^2\right)^{3/2} (b c-a d)}{2 a b \left(a+b x^2\right)}",1,"((a*Sqrt[c + d*x^2]*(b^2*c^2 + 3*a^2*d^2 + 2*a*b*d*(-c + d*x^2)))/(b^2*(a + b*x^2)) - 2*c^(5/2)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]] + (Sqrt[b*c - a*d]*(2*b^2*c^2 + a*b*c*d - 3*a^2*d^2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/b^(5/2))/(2*a^2)","A",1
755,1,150,168,0.1488196,"\int \frac{\left(c+d x^2\right)^{5/2}}{x^2 \left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)^(5/2)/(x^2*(a + b*x^2)^2),x]","-\frac{(b c-a d)^{3/2} (2 a d+3 b c) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 a^{5/2} b^2}+\sqrt{c+d x^2} \left(-\frac{x (b c-a d)^2}{2 a^2 b \left(a+b x^2\right)}-\frac{c^2}{a^2 x}\right)+\frac{d^{5/2} \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{b^2}","-\frac{(b c-a d)^{3/2} (2 a d+3 b c) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 a^{5/2} b^2}-\frac{c \sqrt{c+d x^2} (3 b c-a d)}{2 a^2 b x}+\frac{\left(c+d x^2\right)^{3/2} (b c-a d)}{2 a b x \left(a+b x^2\right)}+\frac{d^{5/2} \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{b^2}",1,"Sqrt[c + d*x^2]*(-(c^2/(a^2*x)) - ((b*c - a*d)^2*x)/(2*a^2*b*(a + b*x^2))) - ((b*c - a*d)^(3/2)*(3*b*c + 2*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(5/2)*b^2) + (d^(5/2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/b^2","A",1
756,1,175,180,0.4529883,"\int \frac{\left(c+d x^2\right)^{5/2}}{x^3 \left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)^(5/2)/(x^3*(a + b*x^2)^2),x]","-\frac{\frac{a \sqrt{c+d x^2} \left(a^2 d^2 x^2+a b c \left(c-2 d x^2\right)+2 b^2 c^2 x^2\right)}{b x^2 \left(a+b x^2\right)}+\frac{\sqrt{b c-a d} \left(-a^2 d^2-3 a b c d+4 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{b^{3/2}}-\left(c^{3/2} (4 b c-5 a d) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)\right)}{2 a^3}","-\frac{(b c-a d)^{3/2} (a d+4 b c) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{2 a^3 b^{3/2}}+\frac{c^{3/2} (4 b c-5 a d) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{2 a^3}-\frac{\sqrt{c+d x^2} (b c-a d) (2 b c-a d)}{2 a^2 b \left(a+b x^2\right)}-\frac{c \left(c+d x^2\right)^{3/2}}{2 a x^2 \left(a+b x^2\right)}",1,"-1/2*((a*Sqrt[c + d*x^2]*(2*b^2*c^2*x^2 + a^2*d^2*x^2 + a*b*c*(c - 2*d*x^2)))/(b*x^2*(a + b*x^2)) - c^(3/2)*(4*b*c - 5*a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]] + (Sqrt[b*c - a*d]*(4*b^2*c^2 - 3*a*b*c*d - a^2*d^2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/b^(3/2))/a^3","A",1
757,1,54,176,0.0176138,"\int \frac{\left(c+d x^2\right)^{5/2}}{x^4 \left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)^(5/2)/(x^4*(a + b*x^2)^2),x]","-\frac{c \left(c+d x^2\right)^{3/2} \, _2F_1\left(-\frac{3}{2},2;-\frac{1}{2};\frac{(a d-b c) x^2}{a \left(d x^2+c\right)}\right)}{3 a^2 x^3}","\frac{5 c (b c-a d)^{3/2} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 a^{7/2}}-\frac{c \sqrt{c+d x^2} (5 b c-3 a d)}{6 a^2 b x^3}+\frac{\sqrt{c+d x^2} \left(3 a^2 d^2-20 a b c d+15 b^2 c^2\right)}{6 a^3 b x}+\frac{\left(c+d x^2\right)^{3/2} (b c-a d)}{2 a b x^3 \left(a+b x^2\right)}",1,"-1/3*(c*(c + d*x^2)^(3/2)*Hypergeometric2F1[-3/2, 2, -1/2, ((-(b*c) + a*d)*x^2)/(a*(c + d*x^2))])/(a^2*x^3)","C",1
758,1,129,132,0.1940743,"\int \frac{x^4}{\left(a+b x^2\right)^2 \sqrt{c+d x^2}} \, dx","Integrate[x^4/((a + b*x^2)^2*Sqrt[c + d*x^2]),x]","\frac{\frac{a b x \sqrt{c+d x^2}}{\left(a+b x^2\right) (b c-a d)}+\frac{\sqrt{a} (2 a d-3 b c) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{(b c-a d)^{3/2}}+\frac{2 \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)}{\sqrt{d}}}{2 b^2}","-\frac{\sqrt{a} (3 b c-2 a d) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 b^2 (b c-a d)^{3/2}}+\frac{a x \sqrt{c+d x^2}}{2 b \left(a+b x^2\right) (b c-a d)}+\frac{\tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{b^2 \sqrt{d}}",1,"((a*b*x*Sqrt[c + d*x^2])/((b*c - a*d)*(a + b*x^2)) + (Sqrt[a]*(-3*b*c + 2*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(b*c - a*d)^(3/2) + (2*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/Sqrt[d])/(2*b^2)","A",1
759,1,98,99,0.0851799,"\int \frac{x^3}{\left(a+b x^2\right)^2 \sqrt{c+d x^2}} \, dx","Integrate[x^3/((a + b*x^2)^2*Sqrt[c + d*x^2]),x]","\frac{\frac{a \sqrt{b} \sqrt{c+d x^2}}{\left(a+b x^2\right) (b c-a d)}+\frac{(a d-2 b c) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{(b c-a d)^{3/2}}}{2 b^{3/2}}","\frac{a \sqrt{c+d x^2}}{2 b \left(a+b x^2\right) (b c-a d)}-\frac{(2 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{2 b^{3/2} (b c-a d)^{3/2}}",1,"((a*Sqrt[b]*Sqrt[c + d*x^2])/((b*c - a*d)*(a + b*x^2)) + ((-2*b*c + a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(b*c - a*d)^(3/2))/(2*b^(3/2))","A",1
760,1,124,89,0.4428234,"\int \frac{x^2}{\left(a+b x^2\right)^2 \sqrt{c+d x^2}} \, dx","Integrate[x^2/((a + b*x^2)^2*Sqrt[c + d*x^2]),x]","\frac{\sqrt{c+d x^2} \left(-\frac{x^2 (b c-a d)}{a+b x^2}-\frac{c \sqrt{x^2 \left(\frac{d}{c}-\frac{b}{a}\right)} \tanh ^{-1}\left(\frac{\sqrt{x^2 \left(\frac{d}{c}-\frac{b}{a}\right)}}{\sqrt{\frac{d x^2}{c}+1}}\right)}{\sqrt{\frac{d x^2}{c}+1}}\right)}{2 x (b c-a d)^2}","\frac{c \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 \sqrt{a} (b c-a d)^{3/2}}-\frac{x \sqrt{c+d x^2}}{2 \left(a+b x^2\right) (b c-a d)}",1,"(Sqrt[c + d*x^2]*(-(((b*c - a*d)*x^2)/(a + b*x^2)) - (c*Sqrt[(-(b/a) + d/c)*x^2]*ArcTanh[Sqrt[(-(b/a) + d/c)*x^2]/Sqrt[1 + (d*x^2)/c]])/Sqrt[1 + (d*x^2)/c]))/(2*(b*c - a*d)^2*x)","A",1
761,1,85,87,0.0884762,"\int \frac{x}{\left(a+b x^2\right)^2 \sqrt{c+d x^2}} \, dx","Integrate[x/((a + b*x^2)^2*Sqrt[c + d*x^2]),x]","\frac{1}{2} \left(\frac{\sqrt{c+d x^2}}{\left(a+b x^2\right) (a d-b c)}+\frac{d \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{a d-b c}}\right)}{\sqrt{b} (a d-b c)^{3/2}}\right)","\frac{d \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{2 \sqrt{b} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^2}}{2 \left(a+b x^2\right) (b c-a d)}",1,"(Sqrt[c + d*x^2]/((-(b*c) + a*d)*(a + b*x^2)) + (d*ArcTan[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[-(b*c) + a*d]])/(Sqrt[b]*(-(b*c) + a*d)^(3/2)))/2","A",1
762,1,405,100,4.086509,"\int \frac{1}{\left(a+b x^2\right)^2 \sqrt{c+d x^2}} \, dx","Integrate[1/((a + b*x^2)^2*Sqrt[c + d*x^2]),x]","\frac{x \sqrt{c+d x^2} \left(-30 d x^2 \sqrt{\frac{a x^2 \left(c+d x^2\right) (b c-a d)}{c^2 \left(a+b x^2\right)^2}}-45 c \sqrt{\frac{a x^2 \left(c+d x^2\right) (b c-a d)}{c^2 \left(a+b x^2\right)^2}}+16 d x^2 \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}} \left(\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}\right)^{5/2} \, _2F_1\left(2,3;\frac{7}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)+16 c \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}} \left(\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}\right)^{5/2} \, _2F_1\left(2,3;\frac{7}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)+30 d x^2 \sin ^{-1}\left(\sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}\right)+45 c \sin ^{-1}\left(\sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}\right)\right)}{30 c^2 \left(a+b x^2\right)^2 \left(\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}\right)^{3/2} \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}}}","\frac{(b c-2 a d) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 a^{3/2} (b c-a d)^{3/2}}+\frac{b x \sqrt{c+d x^2}}{2 a \left(a+b x^2\right) (b c-a d)}",1,"(x*Sqrt[c + d*x^2]*(-45*c*Sqrt[(a*(b*c - a*d)*x^2*(c + d*x^2))/(c^2*(a + b*x^2)^2)] - 30*d*x^2*Sqrt[(a*(b*c - a*d)*x^2*(c + d*x^2))/(c^2*(a + b*x^2)^2)] + 45*c*ArcSin[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]] + 30*d*x^2*ArcSin[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]] + 16*c*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(5/2)*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*Hypergeometric2F1[2, 3, 7/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + 16*d*x^2*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(5/2)*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*Hypergeometric2F1[2, 3, 7/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))]))/(30*c^2*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(3/2)*(a + b*x^2)^2*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))])","C",0
763,1,123,130,0.2418963,"\int \frac{1}{x \left(a+b x^2\right)^2 \sqrt{c+d x^2}} \, dx","Integrate[1/(x*(a + b*x^2)^2*Sqrt[c + d*x^2]),x]","\frac{\frac{a b \sqrt{c+d x^2}}{\left(a+b x^2\right) (b c-a d)}+\frac{\sqrt{b} (2 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{(b c-a d)^{3/2}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{\sqrt{c}}}{2 a^2}","\frac{\sqrt{b} (2 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{2 a^2 (b c-a d)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{a^2 \sqrt{c}}+\frac{b \sqrt{c+d x^2}}{2 a \left(a+b x^2\right) (b c-a d)}",1,"((a*b*Sqrt[c + d*x^2])/((b*c - a*d)*(a + b*x^2)) - (2*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/Sqrt[c] + (Sqrt[b]*(2*b*c - 3*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(b*c - a*d)^(3/2))/(2*a^2)","A",1
764,1,116,147,5.1611777,"\int \frac{1}{x^2 \left(a+b x^2\right)^2 \sqrt{c+d x^2}} \, dx","Integrate[1/(x^2*(a + b*x^2)^2*Sqrt[c + d*x^2]),x]","\frac{\sqrt{c+d x^2} \left(\frac{b^2 x^2}{\left(a+b x^2\right) (a d-b c)}-\frac{2}{c}\right)}{2 a^2 x}-\frac{b (3 b c-4 a d) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 a^{5/2} (b c-a d)^{3/2}}","-\frac{b (3 b c-4 a d) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 a^{5/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^2} (3 b c-2 a d)}{2 a^2 c x (b c-a d)}+\frac{b \sqrt{c+d x^2}}{2 a x \left(a+b x^2\right) (b c-a d)}",1,"(Sqrt[c + d*x^2]*(-2/c + (b^2*x^2)/((-(b*c) + a*d)*(a + b*x^2))))/(2*a^2*x) - (b*(3*b*c - 4*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(5/2)*(b*c - a*d)^(3/2))","A",1
765,1,163,185,0.5696016,"\int \frac{1}{x^3 \left(a+b x^2\right)^2 \sqrt{c+d x^2}} \, dx","Integrate[1/(x^3*(a + b*x^2)^2*Sqrt[c + d*x^2]),x]","\frac{\frac{a \sqrt{c+d x^2} \left(a^2 d+a b \left(d x^2-c\right)-2 b^2 c x^2\right)}{x^2 \left(a+b x^2\right) (b c-a d)}+\frac{b^{3/2} c (5 a d-4 b c) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{(b c-a d)^{3/2}}+\frac{(a d+4 b c) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{\sqrt{c}}}{2 a^3 c}","-\frac{b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{2 a^3 (b c-a d)^{3/2}}+\frac{(a d+4 b c) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{2 a^3 c^{3/2}}-\frac{b \sqrt{c+d x^2} (2 b c-a d)}{2 a^2 c \left(a+b x^2\right) (b c-a d)}-\frac{\sqrt{c+d x^2}}{2 a c x^2 \left(a+b x^2\right)}",1,"((a*Sqrt[c + d*x^2]*(a^2*d - 2*b^2*c*x^2 + a*b*(-c + d*x^2)))/((b*c - a*d)*x^2*(a + b*x^2)) + ((4*b*c + a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/Sqrt[c] + (b^(3/2)*c*(-4*b*c + 5*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(b*c - a*d)^(3/2))/(2*a^3*c)","A",1
766,1,136,206,5.248803,"\int \frac{1}{x^4 \left(a+b x^2\right)^2 \sqrt{c+d x^2}} \, dx","Integrate[1/(x^4*(a + b*x^2)^2*Sqrt[c + d*x^2]),x]","\frac{b^2 (5 b c-6 a d) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 a^{7/2} (b c-a d)^{3/2}}+\frac{\sqrt{c+d x^2} \left(\frac{3 b^3 x^4}{\left(a+b x^2\right) (b c-a d)}+\frac{4 x^2 (a d+3 b c)}{c^2}-\frac{2 a}{c}\right)}{6 a^3 x^3}","\frac{b^2 (5 b c-6 a d) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 a^{7/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^2} (5 b c-2 a d)}{6 a^2 c x^3 (b c-a d)}+\frac{\sqrt{c+d x^2} \left(-4 a^2 d^2-8 a b c d+15 b^2 c^2\right)}{6 a^3 c^2 x (b c-a d)}+\frac{b \sqrt{c+d x^2}}{2 a x^3 \left(a+b x^2\right) (b c-a d)}",1,"(Sqrt[c + d*x^2]*((-2*a)/c + (4*(3*b*c + a*d)*x^2)/c^2 + (3*b^3*x^4)/((b*c - a*d)*(a + b*x^2))))/(6*a^3*x^3) + (b^2*(5*b*c - 6*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(7/2)*(b*c - a*d)^(3/2))","A",1
767,1,54,130,0.0191333,"\int \frac{x^4}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}} \, dx","Integrate[x^4/((a + b*x^2)^2*(c + d*x^2)^(3/2)),x]","\frac{c x^5 \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};\frac{(a d-b c) x^2}{a \left(d x^2+c\right)}\right)}{5 a^2 \left(c+d x^2\right)^{5/2}}","\frac{x (a d+2 b c)}{2 b \sqrt{c+d x^2} (b c-a d)^2}+\frac{a x}{2 b \left(a+b x^2\right) \sqrt{c+d x^2} (b c-a d)}-\frac{3 \sqrt{a} c \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 (b c-a d)^{5/2}}",1,"(c*x^5*Hypergeometric2F1[2, 5/2, 7/2, ((-(b*c) + a*d)*x^2)/(a*(c + d*x^2))])/(5*a^2*(c + d*x^2)^(5/2))","C",1
768,1,91,134,0.0329439,"\int \frac{x^3}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}} \, dx","Integrate[x^3/((a + b*x^2)^2*(c + d*x^2)^(3/2)),x]","\frac{\left(a+b x^2\right) (a d+2 b c) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \left(d x^2+c\right)}{b c-a d}\right)+a (b c-a d)}{2 b \left(a+b x^2\right) \sqrt{c+d x^2} (b c-a d)^2}","\frac{a}{2 b \left(a+b x^2\right) \sqrt{c+d x^2} (b c-a d)}+\frac{a d+2 b c}{2 b \sqrt{c+d x^2} (b c-a d)^2}-\frac{(a d+2 b c) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{2 \sqrt{b} (b c-a d)^{5/2}}",1,"(a*(b*c - a*d) + (2*b*c + a*d)*(a + b*x^2)*Hypergeometric2F1[-1/2, 1, 1/2, (b*(c + d*x^2))/(b*c - a*d)])/(2*b*(b*c - a*d)^2*(a + b*x^2)*Sqrt[c + d*x^2])","C",1
769,1,133,123,1.1764965,"\int \frac{x^2}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}} \, dx","Integrate[x^2/((a + b*x^2)^2*(c + d*x^2)^(3/2)),x]","\frac{x^3 \left(\frac{8 x^2 \left(c+d x^2\right) (b c-a d) \, _2F_1\left(2,3;\frac{9}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)}{a+b x^2}+7 c \left(5 c+2 d x^2\right) \, _2F_1\left(1,2;\frac{7}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)\right)}{105 c^3 \left(a+b x^2\right)^2 \sqrt{c+d x^2}}","-\frac{x}{2 \left(a+b x^2\right) \sqrt{c+d x^2} (b c-a d)}-\frac{3 d x}{2 \sqrt{c+d x^2} (b c-a d)^2}+\frac{(2 a d+b c) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 \sqrt{a} (b c-a d)^{5/2}}",1,"(x^3*(7*c*(5*c + 2*d*x^2)*Hypergeometric2F1[1, 2, 7/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + (8*(b*c - a*d)*x^2*(c + d*x^2)*Hypergeometric2F1[2, 3, 9/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))])/(a + b*x^2)))/(105*c^3*(a + b*x^2)^2*Sqrt[c + d*x^2])","C",0
770,1,52,113,0.0160853,"\int \frac{x}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}} \, dx","Integrate[x/((a + b*x^2)^2*(c + d*x^2)^(3/2)),x]","-\frac{d \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};-\frac{b \left(d x^2+c\right)}{a d-b c}\right)}{\sqrt{c+d x^2} (a d-b c)^2}","-\frac{3 d}{2 \sqrt{c+d x^2} (b c-a d)^2}-\frac{1}{2 \left(a+b x^2\right) \sqrt{c+d x^2} (b c-a d)}+\frac{3 \sqrt{b} d \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{2 (b c-a d)^{5/2}}",1,"-((d*Hypergeometric2F1[-1/2, 2, 1/2, -((b*(c + d*x^2))/(-(b*c) + a*d))])/((-(b*c) + a*d)^2*Sqrt[c + d*x^2]))","C",1
771,1,120,142,5.2451702,"\int \frac{1}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}} \, dx","Integrate[1/((a + b*x^2)^2*(c + d*x^2)^(3/2)),x]","\frac{1}{2} \left(\frac{b (b c-4 a d) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{a^{3/2} (b c-a d)^{5/2}}+\frac{x \sqrt{c+d x^2} \left(\frac{b^2}{a^2+a b x^2}+\frac{2 d^2}{c^2+c d x^2}\right)}{(b c-a d)^2}\right)","\frac{b (b c-4 a d) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 a^{3/2} (b c-a d)^{5/2}}+\frac{d x (2 a d+b c)}{2 a c \sqrt{c+d x^2} (b c-a d)^2}+\frac{b x}{2 a \left(a+b x^2\right) \sqrt{c+d x^2} (b c-a d)}",1,"((x*Sqrt[c + d*x^2]*(b^2/(a^2 + a*b*x^2) + (2*d^2)/(c^2 + c*d*x^2)))/(b*c - a*d)^2 + (b*(b*c - 4*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(a^(3/2)*(b*c - a*d)^(5/2)))/2","A",1
772,1,123,170,0.1315754,"\int \frac{1}{x \left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}} \, dx","Integrate[1/(x*(a + b*x^2)^2*(c + d*x^2)^(3/2)),x]","\frac{\frac{b (2 b c-5 a d) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \left(d x^2+c\right)}{b c-a d}\right)}{a (a d-b c)}+\left(\frac{2 b}{a}-\frac{2 d}{c}\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{d x^2}{c}+1\right)+\frac{b}{a+b x^2}}{2 a \sqrt{c+d x^2} (b c-a d)}","\frac{b^{3/2} (2 b c-5 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{2 a^2 (b c-a d)^{5/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{a^2 c^{3/2}}+\frac{b}{2 a \left(a+b x^2\right) \sqrt{c+d x^2} (b c-a d)}+\frac{d (2 a d+b c)}{2 a c \sqrt{c+d x^2} (b c-a d)^2}",1,"(b/(a + b*x^2) + (b*(2*b*c - 5*a*d)*Hypergeometric2F1[-1/2, 1, 1/2, (b*(c + d*x^2))/(b*c - a*d)])/(a*(-(b*c) + a*d)) + ((2*b)/a - (2*d)/c)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (d*x^2)/c])/(2*a*(b*c - a*d)*Sqrt[c + d*x^2])","C",1
773,1,145,205,5.367908,"\int \frac{1}{x^2 \left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}} \, dx","Integrate[1/(x^2*(a + b*x^2)^2*(c + d*x^2)^(3/2)),x]","\sqrt{c+d x^2} \left(-\frac{\frac{b^3 x}{2 \left(a+b x^2\right) (b c-a d)^2}+\frac{1}{c^2 x}}{a^2}-\frac{d^3 x}{c^2 \left(c+d x^2\right) (b c-a d)^2}\right)-\frac{3 b^2 (b c-2 a d) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 a^{5/2} (b c-a d)^{5/2}}","-\frac{3 b^2 (b c-2 a d) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 a^{5/2} (b c-a d)^{5/2}}-\frac{\sqrt{c+d x^2} \left(4 a^2 d^2-4 a b c d+3 b^2 c^2\right)}{2 a^2 c^2 x (b c-a d)^2}+\frac{b}{2 a x \left(a+b x^2\right) \sqrt{c+d x^2} (b c-a d)}+\frac{d (2 a d+b c)}{2 a c x \sqrt{c+d x^2} (b c-a d)^2}",1,"Sqrt[c + d*x^2]*(-((d^3*x)/(c^2*(b*c - a*d)^2*(c + d*x^2))) - (1/(c^2*x) + (b^3*x)/(2*(b*c - a*d)^2*(a + b*x^2)))/a^2) - (3*b^2*(b*c - 2*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(5/2)*(b*c - a*d)^(5/2))","A",1
774,1,189,241,0.1309576,"\int \frac{1}{x^3 \left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}} \, dx","Integrate[1/(x^3*(a + b*x^2)^2*(c + d*x^2)^(3/2)),x]","\frac{b^2 c^2 x^2 \left(a+b x^2\right) (4 b c-7 a d) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \left(d x^2+c\right)}{b c-a d}\right)-(a d-b c) \left(x^2 \left(a+b x^2\right) \left(3 a^2 d^2+a b c d-4 b^2 c^2\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{d x^2}{c}+1\right)+a c \left(a^2 d+a b \left(d x^2-c\right)-2 b^2 c x^2\right)\right)}{2 a^3 c^2 x^2 \left(a+b x^2\right) \sqrt{c+d x^2} (b c-a d)^2}","-\frac{b^{5/2} (4 b c-7 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{2 a^3 (b c-a d)^{5/2}}+\frac{(3 a d+4 b c) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{2 a^3 c^{5/2}}-\frac{d \left(3 a^2 d^2-2 a b c d+2 b^2 c^2\right)}{2 a^2 c^2 \sqrt{c+d x^2} (b c-a d)^2}-\frac{b (2 b c-a d)}{2 a^2 c \left(a+b x^2\right) \sqrt{c+d x^2} (b c-a d)}-\frac{1}{2 a c x^2 \left(a+b x^2\right) \sqrt{c+d x^2}}",1,"(b^2*c^2*(4*b*c - 7*a*d)*x^2*(a + b*x^2)*Hypergeometric2F1[-1/2, 1, 1/2, (b*(c + d*x^2))/(b*c - a*d)] - (-(b*c) + a*d)*(a*c*(a^2*d - 2*b^2*c*x^2 + a*b*(-c + d*x^2)) + (-4*b^2*c^2 + a*b*c*d + 3*a^2*d^2)*x^2*(a + b*x^2)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (d*x^2)/c]))/(2*a^3*c^2*(b*c - a*d)^2*x^2*(a + b*x^2)*Sqrt[c + d*x^2])","C",1
775,1,167,277,5.4865568,"\int \frac{1}{x^4 \left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}} \, dx","Integrate[1/(x^4*(a + b*x^2)^2*(c + d*x^2)^(3/2)),x]","\frac{b^3 (5 b c-8 a d) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 a^{7/2} (b c-a d)^{5/2}}+\sqrt{c+d x^2} \left(\frac{\frac{b^4 x}{2 \left(a+b x^2\right) (b c-a d)^2}+\frac{2 b}{c^2 x}}{a^3}-\frac{c-5 d x^2}{3 a^2 c^3 x^3}+\frac{d^4 x}{c^3 \left(c+d x^2\right) (b c-a d)^2}\right)","\frac{b^3 (5 b c-8 a d) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 a^{7/2} (b c-a d)^{5/2}}-\frac{\sqrt{c+d x^2} \left(8 a^2 d^2-4 a b c d+5 b^2 c^2\right)}{6 a^2 c^2 x^3 (b c-a d)^2}+\frac{\sqrt{c+d x^2} \left(16 a^3 d^3-8 a^2 b c d^2-14 a b^2 c^2 d+15 b^3 c^3\right)}{6 a^3 c^3 x (b c-a d)^2}+\frac{b}{2 a x^3 \left(a+b x^2\right) \sqrt{c+d x^2} (b c-a d)}+\frac{d (2 a d+b c)}{2 a c x^3 \sqrt{c+d x^2} (b c-a d)^2}",1,"Sqrt[c + d*x^2]*(-1/3*(c - 5*d*x^2)/(a^2*c^3*x^3) + (d^4*x)/(c^3*(b*c - a*d)^2*(c + d*x^2)) + ((2*b)/(c^2*x) + (b^4*x)/(2*(b*c - a*d)^2*(a + b*x^2)))/a^3) + (b^3*(5*b*c - 8*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(7/2)*(b*c - a*d)^(5/2))","A",1
776,1,133,174,1.1662507,"\int \frac{x^4}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2}} \, dx","Integrate[x^4/((a + b*x^2)^2*(c + d*x^2)^(5/2)),x]","\frac{x^5 \left(\frac{8 x^2 \left(c+d x^2\right) (b c-a d) \, _2F_1\left(2,3;\frac{11}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)}{a+b x^2}+9 c \left(7 c+2 d x^2\right) \, _2F_1\left(1,2;\frac{9}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)\right)}{315 c^3 \left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}}","\frac{x (11 a d+4 b c)}{6 \sqrt{c+d x^2} (b c-a d)^3}+\frac{x (3 a d+2 b c)}{6 b \left(c+d x^2\right)^{3/2} (b c-a d)^2}+\frac{a x}{2 b \left(a+b x^2\right) \left(c+d x^2\right)^{3/2} (b c-a d)}-\frac{\sqrt{a} (2 a d+3 b c) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 (b c-a d)^{7/2}}",1,"(x^5*(9*c*(7*c + 2*d*x^2)*Hypergeometric2F1[1, 2, 9/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + (8*(b*c - a*d)*x^2*(c + d*x^2)*Hypergeometric2F1[2, 3, 11/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))])/(a + b*x^2)))/(315*c^3*(a + b*x^2)^2*(c + d*x^2)^(3/2))","C",0
777,1,93,170,0.0335043,"\int \frac{x^3}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2}} \, dx","Integrate[x^3/((a + b*x^2)^2*(c + d*x^2)^(5/2)),x]","\frac{\left(a+b x^2\right) (3 a d+2 b c) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \left(d x^2+c\right)}{b c-a d}\right)+3 a (b c-a d)}{6 b \left(a+b x^2\right) \left(c+d x^2\right)^{3/2} (b c-a d)^2}","\frac{a}{2 b \left(a+b x^2\right) \left(c+d x^2\right)^{3/2} (b c-a d)}+\frac{3 a d+2 b c}{2 \sqrt{c+d x^2} (b c-a d)^3}+\frac{3 a d+2 b c}{6 b \left(c+d x^2\right)^{3/2} (b c-a d)^2}-\frac{\sqrt{b} (3 a d+2 b c) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{2 (b c-a d)^{7/2}}",1,"(3*a*(b*c - a*d) + (2*b*c + 3*a*d)*(a + b*x^2)*Hypergeometric2F1[-3/2, 1, -1/2, (b*(c + d*x^2))/(b*c - a*d)])/(6*b*(b*c - a*d)^2*(a + b*x^2)*(c + d*x^2)^(3/2))","C",1
778,1,211,163,2.4275798,"\int \frac{x^2}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2}} \, dx","Integrate[x^2/((a + b*x^2)^2*(c + d*x^2)^(5/2)),x]","\frac{x^3 \left(16 x^2 \left(c+d x^2\right)^2 (b c-a d) \, _3F_2\left(2,2,3;1,\frac{11}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)+48 x^2 \left(2 c^2+3 c d x^2+d^2 x^4\right) (b c-a d) \, _2F_1\left(2,3;\frac{11}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)+9 c \left(a+b x^2\right) \left(35 c^2+28 c d x^2+8 d^2 x^4\right) \, _2F_1\left(1,2;\frac{9}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)\right)}{945 c^4 \left(a+b x^2\right)^3 \left(c+d x^2\right)^{3/2}}","-\frac{d x (2 a d+13 b c)}{6 c \sqrt{c+d x^2} (b c-a d)^3}-\frac{x}{2 \left(a+b x^2\right) \left(c+d x^2\right)^{3/2} (b c-a d)}-\frac{5 d x}{6 \left(c+d x^2\right)^{3/2} (b c-a d)^2}+\frac{b (4 a d+b c) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 \sqrt{a} (b c-a d)^{7/2}}",1,"(x^3*(9*c*(a + b*x^2)*(35*c^2 + 28*c*d*x^2 + 8*d^2*x^4)*Hypergeometric2F1[1, 2, 9/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + 48*(b*c - a*d)*x^2*(2*c^2 + 3*c*d*x^2 + d^2*x^4)*Hypergeometric2F1[2, 3, 11/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + 16*(b*c - a*d)*x^2*(c + d*x^2)^2*HypergeometricPFQ[{2, 2, 3}, {1, 11/2}, ((b*c - a*d)*x^2)/(c*(a + b*x^2))]))/(945*c^4*(a + b*x^2)^3*(c + d*x^2)^(3/2))","C",0
779,1,54,140,0.0187812,"\int \frac{x}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2}} \, dx","Integrate[x/((a + b*x^2)^2*(c + d*x^2)^(5/2)),x]","-\frac{d \, _2F_1\left(-\frac{3}{2},2;-\frac{1}{2};-\frac{b \left(d x^2+c\right)}{a d-b c}\right)}{3 \left(c+d x^2\right)^{3/2} (a d-b c)^2}","\frac{5 b^{3/2} d \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{2 (b c-a d)^{7/2}}-\frac{5 b d}{2 \sqrt{c+d x^2} (b c-a d)^3}-\frac{1}{2 \left(a+b x^2\right) \left(c+d x^2\right)^{3/2} (b c-a d)}-\frac{5 d}{6 \left(c+d x^2\right)^{3/2} (b c-a d)^2}",1,"-1/3*(d*Hypergeometric2F1[-3/2, 2, -1/2, -((b*(c + d*x^2))/(-(b*c) + a*d))])/((-(b*c) + a*d)^2*(c + d*x^2)^(3/2))","C",1
780,1,170,201,5.5086997,"\int \frac{1}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2}} \, dx","Integrate[1/((a + b*x^2)^2*(c + d*x^2)^(5/2)),x]","\frac{1}{6} \left(\frac{3 b^2 (b c-6 a d) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{a^{3/2} (b c-a d)^{7/2}}+x \sqrt{c+d x^2} \left(-\frac{3 b^3}{a \left(a+b x^2\right) (a d-b c)^3}+\frac{4 d^2 (4 b c-a d)}{c^2 \left(c+d x^2\right) (b c-a d)^3}+\frac{2 d^2}{c \left(c+d x^2\right)^2 (b c-a d)^2}\right)\right)","\frac{b^2 (b c-6 a d) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 a^{3/2} (b c-a d)^{7/2}}+\frac{d x \left(-4 a^2 d^2+16 a b c d+3 b^2 c^2\right)}{6 a c^2 \sqrt{c+d x^2} (b c-a d)^3}+\frac{b x}{2 a \left(a+b x^2\right) \left(c+d x^2\right)^{3/2} (b c-a d)}+\frac{d x (2 a d+3 b c)}{6 a c \left(c+d x^2\right)^{3/2} (b c-a d)^2}",1,"(x*Sqrt[c + d*x^2]*((-3*b^3)/(a*(-(b*c) + a*d)^3*(a + b*x^2)) + (2*d^2)/(c*(b*c - a*d)^2*(c + d*x^2)^2) + (4*d^2*(4*b*c - a*d))/(c^2*(b*c - a*d)^3*(c + d*x^2))) + (3*b^2*(b*c - 6*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(a^(3/2)*(b*c - a*d)^(7/2)))/6","A",1
781,1,114,225,0.1176019,"\int \frac{1}{x \left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2}} \, dx","Integrate[1/(x*(a + b*x^2)^2*(c + d*x^2)^(5/2)),x]","\frac{-\frac{b (2 b c-7 a d) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \left(d x^2+c\right)}{b c-a d}\right)}{(b c-a d)^2}+\frac{3 a b}{\left(a+b x^2\right) (b c-a d)}+\frac{2 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{d x^2}{c}+1\right)}{c}}{6 a^2 \left(c+d x^2\right)^{3/2}}","\frac{b^{5/2} (2 b c-7 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{2 a^2 (b c-a d)^{7/2}}+\frac{d \left(-2 a^2 d^2+6 a b c d+b^2 c^2\right)}{2 a c^2 \sqrt{c+d x^2} (b c-a d)^3}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{a^2 c^{5/2}}+\frac{b}{2 a \left(a+b x^2\right) \left(c+d x^2\right)^{3/2} (b c-a d)}+\frac{d (2 a d+3 b c)}{6 a c \left(c+d x^2\right)^{3/2} (b c-a d)^2}",1,"((3*a*b)/((b*c - a*d)*(a + b*x^2)) - (b*(2*b*c - 7*a*d)*Hypergeometric2F1[-3/2, 1, -1/2, (b*(c + d*x^2))/(b*c - a*d)])/(b*c - a*d)^2 + (2*Hypergeometric2F1[-3/2, 1, -1/2, 1 + (d*x^2)/c])/c)/(6*a^2*(c + d*x^2)^(3/2))","C",1
782,1,188,279,5.5900983,"\int \frac{1}{x^2 \left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2}} \, dx","Integrate[1/(x^2*(a + b*x^2)^2*(c + d*x^2)^(5/2)),x]","\sqrt{c+d x^2} \left(\frac{b^4 x}{2 a^2 \left(a+b x^2\right) (a d-b c)^3}-\frac{1}{a^2 c^3 x}+\frac{d^3 x (5 a d-11 b c)}{3 c^3 \left(c+d x^2\right) (b c-a d)^3}-\frac{d^3 x}{3 c^2 \left(c+d x^2\right)^2 (b c-a d)^2}\right)-\frac{b^3 (3 b c-8 a d) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 a^{5/2} (b c-a d)^{7/2}}","-\frac{b^3 (3 b c-8 a d) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 a^{5/2} (b c-a d)^{7/2}}+\frac{d \left(-8 a^2 d^2+20 a b c d+3 b^2 c^2\right)}{6 a c^2 x \sqrt{c+d x^2} (b c-a d)^3}-\frac{\sqrt{c+d x^2} \left(-16 a^3 d^3+40 a^2 b c d^2-18 a b^2 c^2 d+9 b^3 c^3\right)}{6 a^2 c^3 x (b c-a d)^3}+\frac{b}{2 a x \left(a+b x^2\right) \left(c+d x^2\right)^{3/2} (b c-a d)}+\frac{d (2 a d+3 b c)}{6 a c x \left(c+d x^2\right)^{3/2} (b c-a d)^2}",1,"Sqrt[c + d*x^2]*(-(1/(a^2*c^3*x)) + (b^4*x)/(2*a^2*(-(b*c) + a*d)^3*(a + b*x^2)) - (d^3*x)/(3*c^2*(b*c - a*d)^2*(c + d*x^2)^2) + (d^3*(-11*b*c + 5*a*d)*x)/(3*c^3*(b*c - a*d)^3*(c + d*x^2))) - (b^3*(3*b*c - 8*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(5/2)*(b*c - a*d)^(7/2))","A",1
783,1,190,304,0.1380112,"\int \frac{1}{x^3 \left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2}} \, dx","Integrate[1/(x^3*(a + b*x^2)^2*(c + d*x^2)^(5/2)),x]","\frac{b^2 c^2 x^2 \left(a+b x^2\right) (4 b c-9 a d) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \left(d x^2+c\right)}{b c-a d}\right)-(b c-a d) \left(x^2 \left(a+b x^2\right) \left(-5 a^2 d^2+a b c d+4 b^2 c^2\right) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{d x^2}{c}+1\right)-3 a c \left(a^2 d+a b \left(d x^2-c\right)-2 b^2 c x^2\right)\right)}{6 a^3 c^2 x^2 \left(a+b x^2\right) \left(c+d x^2\right)^{3/2} (b c-a d)^2}","-\frac{b^{7/2} (4 b c-9 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{2 a^3 (b c-a d)^{7/2}}+\frac{(5 a d+4 b c) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{2 a^3 c^{7/2}}-\frac{d \left(5 a^2 d^2-6 a b c d+6 b^2 c^2\right)}{6 a^2 c^2 \left(c+d x^2\right)^{3/2} (b c-a d)^2}-\frac{d (2 b c-a d) \left(5 a^2 d^2-a b c d+b^2 c^2\right)}{2 a^2 c^3 \sqrt{c+d x^2} (b c-a d)^3}-\frac{b (2 b c-a d)}{2 a^2 c \left(a+b x^2\right) \left(c+d x^2\right)^{3/2} (b c-a d)}-\frac{1}{2 a c x^2 \left(a+b x^2\right) \left(c+d x^2\right)^{3/2}}",1,"(b^2*c^2*(4*b*c - 9*a*d)*x^2*(a + b*x^2)*Hypergeometric2F1[-3/2, 1, -1/2, (b*(c + d*x^2))/(b*c - a*d)] - (b*c - a*d)*(-3*a*c*(a^2*d - 2*b^2*c*x^2 + a*b*(-c + d*x^2)) + (4*b^2*c^2 + a*b*c*d - 5*a^2*d^2)*x^2*(a + b*x^2)*Hypergeometric2F1[-3/2, 1, -1/2, 1 + (d*x^2)/c]))/(6*a^3*c^2*(b*c - a*d)^2*x^2*(a + b*x^2)*(c + d*x^2)^(3/2))","C",1
784,1,210,362,5.867511,"\int \frac{1}{x^4 \left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2}} \, dx","Integrate[1/(x^4*(a + b*x^2)^2*(c + d*x^2)^(5/2)),x]","\frac{5 b^4 (b c-2 a d) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 a^{7/2} (b c-a d)^{7/2}}+\frac{\sqrt{c+d x^2} \left(-\frac{3 b^5 x^4}{a^3 \left(a+b x^2\right) (a d-b c)^3}+\frac{4 x^2 (4 a d+3 b c)}{a^3 c^4}-\frac{2}{a^2 c^3}+\frac{4 d^4 x^4 (7 b c-4 a d)}{c^4 \left(c+d x^2\right) (b c-a d)^3}+\frac{2 d^4 x^4}{c^3 \left(c+d x^2\right)^2 (b c-a d)^2}\right)}{6 x^3}","\frac{5 b^4 (b c-2 a d) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 a^{7/2} (b c-a d)^{7/2}}+\frac{d \left(-4 a^2 d^2+8 a b c d+b^2 c^2\right)}{2 a c^2 x^3 \sqrt{c+d x^2} (b c-a d)^3}-\frac{\sqrt{c+d x^2} \left(-16 a^3 d^3+32 a^2 b c d^2-6 a b^2 c^2 d+5 b^3 c^3\right)}{6 a^2 c^3 x^3 (b c-a d)^3}+\frac{\sqrt{c+d x^2} \left(-32 a^4 d^4+64 a^3 b c d^3-12 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+15 b^4 c^4\right)}{6 a^3 c^4 x (b c-a d)^3}+\frac{b}{2 a x^3 \left(a+b x^2\right) \left(c+d x^2\right)^{3/2} (b c-a d)}+\frac{d (2 a d+3 b c)}{6 a c x^3 \left(c+d x^2\right)^{3/2} (b c-a d)^2}",1,"(Sqrt[c + d*x^2]*(-2/(a^2*c^3) + (4*(3*b*c + 4*a*d)*x^2)/(a^3*c^4) - (3*b^5*x^4)/(a^3*(-(b*c) + a*d)^3*(a + b*x^2)) + (2*d^4*x^4)/(c^3*(b*c - a*d)^2*(c + d*x^2)^2) + (4*d^4*(7*b*c - 4*a*d)*x^4)/(c^4*(b*c - a*d)^3*(c + d*x^2))))/(6*x^3) + (5*b^4*(b*c - 2*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(7/2)*(b*c - a*d)^(7/2))","A",1
785,1,110,212,0.1574132,"\int (e x)^{3/2} \sqrt{a+b x^2} \left(A+B x^2\right) \, dx","Integrate[(e*x)^(3/2)*Sqrt[a + b*x^2]*(A + B*x^2),x]","\frac{2 e \sqrt{e x} \sqrt{a+b x^2} \left(a (5 a B-11 A b) \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^2}{a}\right)-\left(a+b x^2\right) \sqrt{\frac{b x^2}{a}+1} \left(5 a B-11 A b-7 b B x^2\right)\right)}{77 b^2 \sqrt{\frac{b x^2}{a}+1}}","-\frac{2 a^{7/4} e^{3/2} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (11 A b-5 a B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{231 b^{9/4} \sqrt{a+b x^2}}+\frac{4 a e \sqrt{e x} \sqrt{a+b x^2} (11 A b-5 a B)}{231 b^2}+\frac{2 (e x)^{5/2} \sqrt{a+b x^2} (11 A b-5 a B)}{77 b e}+\frac{2 B (e x)^{5/2} \left(a+b x^2\right)^{3/2}}{11 b e}",1,"(2*e*Sqrt[e*x]*Sqrt[a + b*x^2]*(-((a + b*x^2)*Sqrt[1 + (b*x^2)/a]*(-11*A*b + 5*a*B - 7*b*B*x^2)) + a*(-11*A*b + 5*a*B)*Hypergeometric2F1[-1/2, 1/4, 5/4, -((b*x^2)/a)]))/(77*b^2*Sqrt[1 + (b*x^2)/a])","C",1
786,1,93,337,0.0936108,"\int \sqrt{e x} \sqrt{a+b x^2} \left(A+B x^2\right) \, dx","Integrate[Sqrt[e*x]*Sqrt[a + b*x^2]*(A + B*x^2),x]","\frac{2 x \sqrt{e x} \sqrt{a+b x^2} \left((3 A b-a B) \, _2F_1\left(-\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right)+B \sqrt{\frac{b x^2}{a}+1} \left(a+b x^2\right)\right)}{9 b \sqrt{\frac{b x^2}{a}+1}}","\frac{2 a^{5/4} \sqrt{e} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (3 A b-a B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{15 b^{7/4} \sqrt{a+b x^2}}-\frac{4 a^{5/4} \sqrt{e} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (3 A b-a B) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{15 b^{7/4} \sqrt{a+b x^2}}+\frac{4 a \sqrt{e x} \sqrt{a+b x^2} (3 A b-a B)}{15 b^{3/2} \left(\sqrt{a}+\sqrt{b} x\right)}+\frac{2 (e x)^{3/2} \sqrt{a+b x^2} (3 A b-a B)}{15 b e}+\frac{2 B (e x)^{3/2} \left(a+b x^2\right)^{3/2}}{9 b e}",1,"(2*x*Sqrt[e*x]*Sqrt[a + b*x^2]*(B*(a + b*x^2)*Sqrt[1 + (b*x^2)/a] + (3*A*b - a*B)*Hypergeometric2F1[-1/2, 3/4, 7/4, -((b*x^2)/a)]))/(9*b*Sqrt[1 + (b*x^2)/a])","C",1
787,1,93,176,0.0589974,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{\sqrt{e x}} \, dx","Integrate[(Sqrt[a + b*x^2]*(A + B*x^2))/Sqrt[e*x],x]","\frac{2 x \sqrt{a+b x^2} \left((7 A b-a B) \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^2}{a}\right)+B \sqrt{\frac{b x^2}{a}+1} \left(a+b x^2\right)\right)}{7 b \sqrt{e x} \sqrt{\frac{b x^2}{a}+1}}","\frac{2 a^{3/4} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (7 A b-a B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{21 b^{5/4} \sqrt{e} \sqrt{a+b x^2}}+\frac{2 \sqrt{e x} \sqrt{a+b x^2} (7 A b-a B)}{21 b e}+\frac{2 B \sqrt{e x} \left(a+b x^2\right)^{3/2}}{7 b e}",1,"(2*x*Sqrt[a + b*x^2]*(B*(a + b*x^2)*Sqrt[1 + (b*x^2)/a] + (7*A*b - a*B)*Hypergeometric2F1[-1/2, 1/4, 5/4, -((b*x^2)/a)]))/(7*b*Sqrt[e*x]*Sqrt[1 + (b*x^2)/a])","C",1
788,1,96,333,0.0579182,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{(e x)^{3/2}} \, dx","Integrate[(Sqrt[a + b*x^2]*(A + B*x^2))/(e*x)^(3/2),x]","\frac{2 x \sqrt{a+b x^2} \left(x^2 (a B+5 A b) \, _2F_1\left(-\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right)-3 A \left(a+b x^2\right) \sqrt{\frac{b x^2}{a}+1}\right)}{3 a (e x)^{3/2} \sqrt{\frac{b x^2}{a}+1}}","\frac{2 \sqrt[4]{a} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (a B+5 A b) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{5 b^{3/4} e^{3/2} \sqrt{a+b x^2}}-\frac{4 \sqrt[4]{a} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (a B+5 A b) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{5 b^{3/4} e^{3/2} \sqrt{a+b x^2}}+\frac{2 (e x)^{3/2} \sqrt{a+b x^2} (a B+5 A b)}{5 a e^3}+\frac{4 \sqrt{e x} \sqrt{a+b x^2} (a B+5 A b)}{5 \sqrt{b} e^2 \left(\sqrt{a}+\sqrt{b} x\right)}-\frac{2 A \left(a+b x^2\right)^{3/2}}{a e \sqrt{e x}}",1,"(2*x*Sqrt[a + b*x^2]*(-3*A*(a + b*x^2)*Sqrt[1 + (b*x^2)/a] + (5*A*b + a*B)*x^2*Hypergeometric2F1[-1/2, 3/4, 7/4, -((b*x^2)/a)]))/(3*a*(e*x)^(3/2)*Sqrt[1 + (b*x^2)/a])","C",1
789,1,82,172,0.0827417,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{(e x)^{5/2}} \, dx","Integrate[(Sqrt[a + b*x^2]*(A + B*x^2))/(e*x)^(5/2),x]","\frac{2 x \sqrt{a+b x^2} \left(\frac{3 x^2 (a B+A b) \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^2}{a}\right)}{\sqrt{\frac{b x^2}{a}+1}}-A \left(a+b x^2\right)\right)}{3 a (e x)^{5/2}}","\frac{2 \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (a B+A b) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{3 \sqrt[4]{a} \sqrt[4]{b} e^{5/2} \sqrt{a+b x^2}}+\frac{2 \sqrt{e x} \sqrt{a+b x^2} (a B+A b)}{3 a e^3}-\frac{2 A \left(a+b x^2\right)^{3/2}}{3 a e (e x)^{3/2}}",1,"(2*x*Sqrt[a + b*x^2]*(-(A*(a + b*x^2)) + (3*(A*b + a*B)*x^2*Hypergeometric2F1[-1/2, 1/4, 5/4, -((b*x^2)/a)])/Sqrt[1 + (b*x^2)/a]))/(3*a*(e*x)^(5/2))","C",1
790,1,95,338,0.0655219,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{(e x)^{7/2}} \, dx","Integrate[(Sqrt[a + b*x^2]*(A + B*x^2))/(e*x)^(7/2),x]","-\frac{2 x \sqrt{a+b x^2} \left(x^2 (5 a B+A b) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};-\frac{b x^2}{a}\right)+A \left(a+b x^2\right) \sqrt{\frac{b x^2}{a}+1}\right)}{5 a (e x)^{7/2} \sqrt{\frac{b x^2}{a}+1}}","\frac{2 \sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (5 a B+A b) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{5 a^{3/4} e^{7/2} \sqrt{a+b x^2}}-\frac{4 \sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (5 a B+A b) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{5 a^{3/4} e^{7/2} \sqrt{a+b x^2}}+\frac{4 \sqrt{b} \sqrt{e x} \sqrt{a+b x^2} (5 a B+A b)}{5 a e^4 \left(\sqrt{a}+\sqrt{b} x\right)}-\frac{2 \sqrt{a+b x^2} (5 a B+A b)}{5 a e^3 \sqrt{e x}}-\frac{2 A \left(a+b x^2\right)^{3/2}}{5 a e (e x)^{5/2}}",1,"(-2*x*Sqrt[a + b*x^2]*(A*(a + b*x^2)*Sqrt[1 + (b*x^2)/a] + (A*b + 5*a*B)*x^2*Hypergeometric2F1[-1/2, -1/4, 3/4, -((b*x^2)/a)]))/(5*a*(e*x)^(7/2)*Sqrt[1 + (b*x^2)/a])","C",1
791,1,79,152,0.0933322,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{x^{9/2}} \, dx","Integrate[(Sqrt[a + b*x^2]*(A + B*x^2))/x^(9/2),x]","\frac{2 \sqrt{a+b x^2} \left(\frac{x^2 (A b-7 a B) \, _2F_1\left(-\frac{3}{4},-\frac{1}{2};\frac{1}{4};-\frac{b x^2}{a}\right)}{\sqrt{\frac{b x^2}{a}+1}}-3 A \left(a+b x^2\right)\right)}{21 a x^{7/2}}","-\frac{2 b^{3/4} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (A b-7 a B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{21 a^{5/4} \sqrt{a+b x^2}}+\frac{2 \sqrt{a+b x^2} (A b-7 a B)}{21 a x^{3/2}}-\frac{2 A \left(a+b x^2\right)^{3/2}}{7 a x^{7/2}}",1,"(2*Sqrt[a + b*x^2]*(-3*A*(a + b*x^2) + ((A*b - 7*a*B)*x^2*Hypergeometric2F1[-3/4, -1/2, 1/4, -((b*x^2)/a)])/Sqrt[1 + (b*x^2)/a]))/(21*a*x^(7/2))","C",1
792,1,80,331,0.117623,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{x^{11/2}} \, dx","Integrate[(Sqrt[a + b*x^2]*(A + B*x^2))/x^(11/2),x]","\frac{2 \sqrt{a+b x^2} \left(\frac{3 x^2 (A b-3 a B) \, _2F_1\left(-\frac{5}{4},-\frac{1}{2};-\frac{1}{4};-\frac{b x^2}{a}\right)}{\sqrt{\frac{b x^2}{a}+1}}-5 A \left(a+b x^2\right)\right)}{45 a x^{9/2}}","-\frac{2 b^{5/4} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (A b-3 a B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{15 a^{7/4} \sqrt{a+b x^2}}+\frac{4 b^{5/4} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (A b-3 a B) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{15 a^{7/4} \sqrt{a+b x^2}}-\frac{4 b^{3/2} \sqrt{x} \sqrt{a+b x^2} (A b-3 a B)}{15 a^2 \left(\sqrt{a}+\sqrt{b} x\right)}+\frac{4 b \sqrt{a+b x^2} (A b-3 a B)}{15 a^2 \sqrt{x}}+\frac{2 \sqrt{a+b x^2} (A b-3 a B)}{15 a x^{5/2}}-\frac{2 A \left(a+b x^2\right)^{3/2}}{9 a x^{9/2}}",1,"(2*Sqrt[a + b*x^2]*(-5*A*(a + b*x^2) + (3*(A*b - 3*a*B)*x^2*Hypergeometric2F1[-5/4, -1/2, -1/4, -((b*x^2)/a)])/Sqrt[1 + (b*x^2)/a]))/(45*a*x^(9/2))","C",1
793,1,80,187,0.1082528,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{x^{13/2}} \, dx","Integrate[(Sqrt[a + b*x^2]*(A + B*x^2))/x^(13/2),x]","\frac{2 \sqrt{a+b x^2} \left(\frac{x^2 (5 A b-11 a B) \, _2F_1\left(-\frac{7}{4},-\frac{1}{2};-\frac{3}{4};-\frac{b x^2}{a}\right)}{\sqrt{\frac{b x^2}{a}+1}}-7 A \left(a+b x^2\right)\right)}{77 a x^{11/2}}","\frac{2 b^{7/4} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (5 A b-11 a B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{231 a^{9/4} \sqrt{a+b x^2}}+\frac{4 b \sqrt{a+b x^2} (5 A b-11 a B)}{231 a^2 x^{3/2}}+\frac{2 \sqrt{a+b x^2} (5 A b-11 a B)}{77 a x^{7/2}}-\frac{2 A \left(a+b x^2\right)^{3/2}}{11 a x^{11/2}}",1,"(2*Sqrt[a + b*x^2]*(-7*A*(a + b*x^2) + ((5*A*b - 11*a*B)*x^2*Hypergeometric2F1[-7/4, -1/2, -3/4, -((b*x^2)/a)])/Sqrt[1 + (b*x^2)/a]))/(77*a*x^(11/2))","C",1
794,1,114,252,0.1693344,"\int (e x)^{3/2} \left(a+b x^2\right)^{3/2} \left(A+B x^2\right) \, dx","Integrate[(e*x)^(3/2)*(a + b*x^2)^(3/2)*(A + B*x^2),x]","\frac{2 e \sqrt{e x} \sqrt{a+b x^2} \left(5 a^2 (a B-3 A b) \, _2F_1\left(-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^2}{a}\right)-\left(a+b x^2\right)^2 \sqrt{\frac{b x^2}{a}+1} \left(5 a B-15 A b-11 b B x^2\right)\right)}{165 b^2 \sqrt{\frac{b x^2}{a}+1}}","-\frac{4 a^{11/4} e^{3/2} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (3 A b-a B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{231 b^{9/4} \sqrt{a+b x^2}}+\frac{8 a^2 e \sqrt{e x} \sqrt{a+b x^2} (3 A b-a B)}{231 b^2}+\frac{4 a (e x)^{5/2} \sqrt{a+b x^2} (3 A b-a B)}{77 b e}+\frac{2 (e x)^{5/2} \left(a+b x^2\right)^{3/2} (3 A b-a B)}{33 b e}+\frac{2 B (e x)^{5/2} \left(a+b x^2\right)^{5/2}}{15 b e}",1,"(2*e*Sqrt[e*x]*Sqrt[a + b*x^2]*(-((a + b*x^2)^2*Sqrt[1 + (b*x^2)/a]*(-15*A*b + 5*a*B - 11*b*B*x^2)) + 5*a^2*(-3*A*b + a*B)*Hypergeometric2F1[-3/2, 1/4, 5/4, -((b*x^2)/a)]))/(165*b^2*Sqrt[1 + (b*x^2)/a])","C",1
795,1,97,377,0.1130177,"\int \sqrt{e x} \left(a+b x^2\right)^{3/2} \left(A+B x^2\right) \, dx","Integrate[Sqrt[e*x]*(a + b*x^2)^(3/2)*(A + B*x^2),x]","\frac{2 x \sqrt{e x} \sqrt{a+b x^2} \left(a (13 A b-3 a B) \, _2F_1\left(-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right)+3 B \sqrt{\frac{b x^2}{a}+1} \left(a+b x^2\right)^2\right)}{39 b \sqrt{\frac{b x^2}{a}+1}}","\frac{4 a^{9/4} \sqrt{e} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (13 A b-3 a B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{195 b^{7/4} \sqrt{a+b x^2}}-\frac{8 a^{9/4} \sqrt{e} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (13 A b-3 a B) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{195 b^{7/4} \sqrt{a+b x^2}}+\frac{8 a^2 \sqrt{e x} \sqrt{a+b x^2} (13 A b-3 a B)}{195 b^{3/2} \left(\sqrt{a}+\sqrt{b} x\right)}+\frac{2 (e x)^{3/2} \left(a+b x^2\right)^{3/2} (13 A b-3 a B)}{117 b e}+\frac{4 a (e x)^{3/2} \sqrt{a+b x^2} (13 A b-3 a B)}{195 b e}+\frac{2 B (e x)^{3/2} \left(a+b x^2\right)^{5/2}}{13 b e}",1,"(2*x*Sqrt[e*x]*Sqrt[a + b*x^2]*(3*B*(a + b*x^2)^2*Sqrt[1 + (b*x^2)/a] + a*(13*A*b - 3*a*B)*Hypergeometric2F1[-3/2, 3/4, 7/4, -((b*x^2)/a)]))/(39*b*Sqrt[1 + (b*x^2)/a])","C",1
796,1,96,214,0.0793128,"\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{\sqrt{e x}} \, dx","Integrate[((a + b*x^2)^(3/2)*(A + B*x^2))/Sqrt[e*x],x]","\frac{2 x \sqrt{a+b x^2} \left(a (11 A b-a B) \, _2F_1\left(-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^2}{a}\right)+B \sqrt{\frac{b x^2}{a}+1} \left(a+b x^2\right)^2\right)}{11 b \sqrt{e x} \sqrt{\frac{b x^2}{a}+1}}","\frac{4 a^{7/4} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (11 A b-a B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{77 b^{5/4} \sqrt{e} \sqrt{a+b x^2}}+\frac{2 \sqrt{e x} \left(a+b x^2\right)^{3/2} (11 A b-a B)}{77 b e}+\frac{4 a \sqrt{e x} \sqrt{a+b x^2} (11 A b-a B)}{77 b e}+\frac{2 B \sqrt{e x} \left(a+b x^2\right)^{5/2}}{11 b e}",1,"(2*x*Sqrt[a + b*x^2]*(B*(a + b*x^2)^2*Sqrt[1 + (b*x^2)/a] + a*(11*A*b - a*B)*Hypergeometric2F1[-3/2, 1/4, 5/4, -((b*x^2)/a)]))/(11*b*Sqrt[e*x]*Sqrt[1 + (b*x^2)/a])","C",1
797,1,84,367,0.0795573,"\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{(e x)^{3/2}} \, dx","Integrate[((a + b*x^2)^(3/2)*(A + B*x^2))/(e*x)^(3/2),x]","\frac{2 x \sqrt{a+b x^2} \left(\frac{x^2 (a B+9 A b) \, _2F_1\left(-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right)}{\sqrt{\frac{b x^2}{a}+1}}-\frac{3 A \left(a+b x^2\right)^2}{a}\right)}{3 (e x)^{3/2}}","\frac{4 a^{5/4} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (a B+9 A b) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{15 b^{3/4} e^{3/2} \sqrt{a+b x^2}}-\frac{8 a^{5/4} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (a B+9 A b) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{15 b^{3/4} e^{3/2} \sqrt{a+b x^2}}+\frac{2 (e x)^{3/2} \left(a+b x^2\right)^{3/2} (a B+9 A b)}{9 a e^3}+\frac{4 (e x)^{3/2} \sqrt{a+b x^2} (a B+9 A b)}{15 e^3}+\frac{8 a \sqrt{e x} \sqrt{a+b x^2} (a B+9 A b)}{15 \sqrt{b} e^2 \left(\sqrt{a}+\sqrt{b} x\right)}-\frac{2 A \left(a+b x^2\right)^{5/2}}{a e \sqrt{e x}}",1,"(2*x*Sqrt[a + b*x^2]*((-3*A*(a + b*x^2)^2)/a + ((9*A*b + a*B)*x^2*Hypergeometric2F1[-3/2, 3/4, 7/4, -((b*x^2)/a)])/Sqrt[1 + (b*x^2)/a]))/(3*(e*x)^(3/2))","C",1
798,1,85,210,0.0714205,"\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{(e x)^{5/2}} \, dx","Integrate[((a + b*x^2)^(3/2)*(A + B*x^2))/(e*x)^(5/2),x]","\frac{2 x \sqrt{a+b x^2} \left(\frac{x^2 (3 a B+7 A b) \, _2F_1\left(-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^2}{a}\right)}{\sqrt{\frac{b x^2}{a}+1}}-\frac{A \left(a+b x^2\right)^2}{a}\right)}{3 (e x)^{5/2}}","\frac{4 a^{3/4} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (3 a B+7 A b) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{21 \sqrt[4]{b} e^{5/2} \sqrt{a+b x^2}}+\frac{2 \sqrt{e x} \left(a+b x^2\right)^{3/2} (3 a B+7 A b)}{21 a e^3}+\frac{4 \sqrt{e x} \sqrt{a+b x^2} (3 a B+7 A b)}{21 e^3}-\frac{2 A \left(a+b x^2\right)^{5/2}}{3 a e (e x)^{3/2}}",1,"(2*x*Sqrt[a + b*x^2]*(-((A*(a + b*x^2)^2)/a) + ((7*A*b + 3*a*B)*x^2*Hypergeometric2F1[-3/2, 1/4, 5/4, -((b*x^2)/a)])/Sqrt[1 + (b*x^2)/a]))/(3*(e*x)^(5/2))","C",1
799,1,84,365,0.071232,"\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{(e x)^{7/2}} \, dx","Integrate[((a + b*x^2)^(3/2)*(A + B*x^2))/(e*x)^(7/2),x]","\frac{2 x \sqrt{a+b x^2} \left(-\frac{5 x^2 (a B+A b) \, _2F_1\left(-\frac{3}{2},-\frac{1}{4};\frac{3}{4};-\frac{b x^2}{a}\right)}{\sqrt{\frac{b x^2}{a}+1}}-\frac{A \left(a+b x^2\right)^2}{a}\right)}{5 (e x)^{7/2}}","\frac{12 \sqrt[4]{a} \sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (a B+A b) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{5 e^{7/2} \sqrt{a+b x^2}}-\frac{24 \sqrt[4]{a} \sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (a B+A b) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{5 e^{7/2} \sqrt{a+b x^2}}+\frac{12 b (e x)^{3/2} \sqrt{a+b x^2} (a B+A b)}{5 a e^5}+\frac{24 \sqrt{b} \sqrt{e x} \sqrt{a+b x^2} (a B+A b)}{5 e^4 \left(\sqrt{a}+\sqrt{b} x\right)}-\frac{2 \left(a+b x^2\right)^{3/2} (a B+A b)}{a e^3 \sqrt{e x}}-\frac{2 A \left(a+b x^2\right)^{5/2}}{5 a e (e x)^{5/2}}",1,"(2*x*Sqrt[a + b*x^2]*(-((A*(a + b*x^2)^2)/a) - (5*(A*b + a*B)*x^2*Hypergeometric2F1[-3/2, -1/4, 3/4, -((b*x^2)/a)])/Sqrt[1 + (b*x^2)/a]))/(5*(e*x)^(7/2))","C",1
800,1,96,338,0.1191555,"\int \frac{(e x)^{5/2} \left(A+B x^2\right)}{\sqrt{a+b x^2}} \, dx","Integrate[((e*x)^(5/2)*(A + B*x^2))/Sqrt[a + b*x^2],x]","\frac{2 e (e x)^{3/2} \left(a \sqrt{\frac{b x^2}{a}+1} (7 a B-9 A b) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right)-\left(a+b x^2\right) \left(7 a B-9 A b-5 b B x^2\right)\right)}{45 b^2 \sqrt{a+b x^2}}","-\frac{a^{5/4} e^{5/2} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (9 A b-7 a B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{15 b^{11/4} \sqrt{a+b x^2}}+\frac{2 a^{5/4} e^{5/2} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (9 A b-7 a B) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{15 b^{11/4} \sqrt{a+b x^2}}-\frac{2 a e^2 \sqrt{e x} \sqrt{a+b x^2} (9 A b-7 a B)}{15 b^{5/2} \left(\sqrt{a}+\sqrt{b} x\right)}+\frac{2 e (e x)^{3/2} \sqrt{a+b x^2} (9 A b-7 a B)}{45 b^2}+\frac{2 B (e x)^{7/2} \sqrt{a+b x^2}}{9 b e}",1,"(2*e*(e*x)^(3/2)*(-((a + b*x^2)*(-9*A*b + 7*a*B - 5*b*B*x^2)) + a*(-9*A*b + 7*a*B)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, -((b*x^2)/a)]))/(45*b^2*Sqrt[a + b*x^2])","C",1
801,1,96,174,0.1046633,"\int \frac{(e x)^{3/2} \left(A+B x^2\right)}{\sqrt{a+b x^2}} \, dx","Integrate[((e*x)^(3/2)*(A + B*x^2))/Sqrt[a + b*x^2],x]","\frac{2 e \sqrt{e x} \left(a \sqrt{\frac{b x^2}{a}+1} (5 a B-7 A b) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^2}{a}\right)-\left(a+b x^2\right) \left(5 a B-7 A b-3 b B x^2\right)\right)}{21 b^2 \sqrt{a+b x^2}}","-\frac{a^{3/4} e^{3/2} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (7 A b-5 a B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{21 b^{9/4} \sqrt{a+b x^2}}+\frac{2 e \sqrt{e x} \sqrt{a+b x^2} (7 A b-5 a B)}{21 b^2}+\frac{2 B (e x)^{5/2} \sqrt{a+b x^2}}{7 b e}",1,"(2*e*Sqrt[e*x]*(-((a + b*x^2)*(-7*A*b + 5*a*B - 3*b*B*x^2)) + a*(-7*A*b + 5*a*B)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^2)/a)]))/(21*b^2*Sqrt[a + b*x^2])","C",1
802,1,80,299,0.0909087,"\int \frac{\sqrt{e x} \left(A+B x^2\right)}{\sqrt{a+b x^2}} \, dx","Integrate[(Sqrt[e*x]*(A + B*x^2))/Sqrt[a + b*x^2],x]","\frac{2 x \sqrt{e x} \left(\sqrt{\frac{b x^2}{a}+1} (5 A b-3 a B) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right)+3 B \left(a+b x^2\right)\right)}{15 b \sqrt{a+b x^2}}","\frac{2 \sqrt{e x} \sqrt{a+b x^2} (5 A b-3 a B)}{5 b^{3/2} \left(\sqrt{a}+\sqrt{b} x\right)}+\frac{\sqrt[4]{a} \sqrt{e} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (5 A b-3 a B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{5 b^{7/4} \sqrt{a+b x^2}}-\frac{2 \sqrt[4]{a} \sqrt{e} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (5 A b-3 a B) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{5 b^{7/4} \sqrt{a+b x^2}}+\frac{2 B (e x)^{3/2} \sqrt{a+b x^2}}{5 b e}",1,"(2*x*Sqrt[e*x]*(3*B*(a + b*x^2) + (5*A*b - 3*a*B)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, -((b*x^2)/a)]))/(15*b*Sqrt[a + b*x^2])","C",1
803,1,79,139,0.0824602,"\int \frac{A+B x^2}{\sqrt{e x} \sqrt{a+b x^2}} \, dx","Integrate[(A + B*x^2)/(Sqrt[e*x]*Sqrt[a + b*x^2]),x]","\frac{2 x \left(\sqrt{\frac{b x^2}{a}+1} (3 A b-a B) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^2}{a}\right)+B \left(a+b x^2\right)\right)}{3 b \sqrt{e x} \sqrt{a+b x^2}}","\frac{\left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (3 A b-a B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{3 \sqrt[4]{a} b^{5/4} \sqrt{e} \sqrt{a+b x^2}}+\frac{2 B \sqrt{e x} \sqrt{a+b x^2}}{3 b e}",1,"(2*x*(B*(a + b*x^2) + (3*A*b - a*B)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^2)/a)]))/(3*b*Sqrt[e*x]*Sqrt[a + b*x^2])","C",1
804,1,82,290,0.0651928,"\int \frac{A+B x^2}{(e x)^{3/2} \sqrt{a+b x^2}} \, dx","Integrate[(A + B*x^2)/((e*x)^(3/2)*Sqrt[a + b*x^2]),x]","\frac{x \left(2 x^2 \sqrt{\frac{b x^2}{a}+1} (a B+A b) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right)-6 A \left(a+b x^2\right)\right)}{3 a (e x)^{3/2} \sqrt{a+b x^2}}","\frac{\left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (a B+A b) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{a^{3/4} b^{3/4} e^{3/2} \sqrt{a+b x^2}}-\frac{2 \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (a B+A b) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{a^{3/4} b^{3/4} e^{3/2} \sqrt{a+b x^2}}+\frac{2 \sqrt{e x} \sqrt{a+b x^2} (a B+A b)}{a \sqrt{b} e^2 \left(\sqrt{a}+\sqrt{b} x\right)}-\frac{2 A \sqrt{a+b x^2}}{a e \sqrt{e x}}",1,"(x*(-6*A*(a + b*x^2) + 2*(A*b + a*B)*x^2*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, -((b*x^2)/a)]))/(3*a*(e*x)^(3/2)*Sqrt[a + b*x^2])","C",1
805,1,81,138,0.0476687,"\int \frac{A+B x^2}{(e x)^{5/2} \sqrt{a+b x^2}} \, dx","Integrate[(A + B*x^2)/((e*x)^(5/2)*Sqrt[a + b*x^2]),x]","-\frac{2 x \left(x^2 \sqrt{\frac{b x^2}{a}+1} (A b-3 a B) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^2}{a}\right)+A \left(a+b x^2\right)\right)}{3 a (e x)^{5/2} \sqrt{a+b x^2}}","-\frac{\left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (A b-3 a B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{3 a^{5/4} \sqrt[4]{b} e^{5/2} \sqrt{a+b x^2}}-\frac{2 A \sqrt{a+b x^2}}{3 a e (e x)^{3/2}}",1,"(-2*x*(A*(a + b*x^2) + (A*b - 3*a*B)*x^2*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^2)/a)]))/(3*a*(e*x)^(5/2)*Sqrt[a + b*x^2])","C",1
806,1,82,342,0.0499073,"\int \frac{A+B x^2}{(e x)^{7/2} \sqrt{a+b x^2}} \, dx","Integrate[(A + B*x^2)/((e*x)^(7/2)*Sqrt[a + b*x^2]),x]","-\frac{2 x \left(x^2 \sqrt{\frac{b x^2}{a}+1} (5 a B-3 A b) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{b x^2}{a}\right)+A \left(a+b x^2\right)\right)}{5 a (e x)^{7/2} \sqrt{a+b x^2}}","-\frac{\sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (3 A b-5 a B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{5 a^{7/4} e^{7/2} \sqrt{a+b x^2}}+\frac{2 \sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (3 A b-5 a B) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{5 a^{7/4} e^{7/2} \sqrt{a+b x^2}}-\frac{2 \sqrt{b} \sqrt{e x} \sqrt{a+b x^2} (3 A b-5 a B)}{5 a^2 e^4 \left(\sqrt{a}+\sqrt{b} x\right)}+\frac{2 \sqrt{a+b x^2} (3 A b-5 a B)}{5 a^2 e^3 \sqrt{e x}}-\frac{2 A \sqrt{a+b x^2}}{5 a e (e x)^{5/2}}",1,"(-2*x*(A*(a + b*x^2) + (-3*A*b + 5*a*B)*x^2*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[-1/4, 1/2, 3/4, -((b*x^2)/a)]))/(5*a*(e*x)^(7/2)*Sqrt[a + b*x^2])","C",1
807,1,111,211,0.1509002,"\int \frac{(e x)^{7/2} \left(A+B x^2\right)}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[((e*x)^(7/2)*(A + B*x^2))/(a + b*x^2)^(3/2),x]","\frac{e^3 \sqrt{e x} \left(-45 a^2 B+5 a \sqrt{\frac{b x^2}{a}+1} (9 a B-7 A b) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^2}{a}\right)+a b \left(35 A-18 B x^2\right)+2 b^2 x^2 \left(7 A+3 B x^2\right)\right)}{21 b^3 \sqrt{a+b x^2}}","-\frac{5 a^{3/4} e^{7/2} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (7 A b-9 a B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{42 b^{13/4} \sqrt{a+b x^2}}+\frac{5 e^3 \sqrt{e x} \sqrt{a+b x^2} (7 A b-9 a B)}{21 b^3}-\frac{e (e x)^{5/2} (7 A b-9 a B)}{7 b^2 \sqrt{a+b x^2}}+\frac{2 B (e x)^{9/2}}{7 b e \sqrt{a+b x^2}}",1,"(e^3*Sqrt[e*x]*(-45*a^2*B + a*b*(35*A - 18*B*x^2) + 2*b^2*x^2*(7*A + 3*B*x^2) + 5*a*(-7*A*b + 9*a*B)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^2)/a)]))/(21*b^3*Sqrt[a + b*x^2])","C",1
808,1,84,337,0.120712,"\int \frac{(e x)^{5/2} \left(A+B x^2\right)}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[((e*x)^(5/2)*(A + B*x^2))/(a + b*x^2)^(3/2),x]","\frac{2 e (e x)^{3/2} \left(\sqrt{\frac{b x^2}{a}+1} (7 a B-5 A b) \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{b x^2}{a}\right)-7 a B+5 A b+b B x^2\right)}{5 b^2 \sqrt{a+b x^2}}","\frac{3 \sqrt[4]{a} e^{5/2} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (5 A b-7 a B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{10 b^{11/4} \sqrt{a+b x^2}}-\frac{3 \sqrt[4]{a} e^{5/2} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (5 A b-7 a B) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{5 b^{11/4} \sqrt{a+b x^2}}+\frac{3 e^2 \sqrt{e x} \sqrt{a+b x^2} (5 A b-7 a B)}{5 b^{5/2} \left(\sqrt{a}+\sqrt{b} x\right)}-\frac{e (e x)^{3/2} (5 A b-7 a B)}{5 b^2 \sqrt{a+b x^2}}+\frac{2 B (e x)^{7/2}}{5 b e \sqrt{a+b x^2}}",1,"(2*e*(e*x)^(3/2)*(5*A*b - 7*a*B + b*B*x^2 + (-5*A*b + 7*a*B)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[3/4, 3/2, 7/4, -((b*x^2)/a)]))/(5*b^2*Sqrt[a + b*x^2])","C",1
809,1,85,174,0.1169877,"\int \frac{(e x)^{3/2} \left(A+B x^2\right)}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[((e*x)^(3/2)*(A + B*x^2))/(a + b*x^2)^(3/2),x]","\frac{e \sqrt{e x} \left(\sqrt{\frac{b x^2}{a}+1} (3 A b-5 a B) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^2}{a}\right)+5 a B-3 A b+2 b B x^2\right)}{3 b^2 \sqrt{a+b x^2}}","\frac{e^{3/2} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (3 A b-5 a B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{6 \sqrt[4]{a} b^{9/4} \sqrt{a+b x^2}}-\frac{e \sqrt{e x} (3 A b-5 a B)}{3 b^2 \sqrt{a+b x^2}}+\frac{2 B (e x)^{5/2}}{3 b e \sqrt{a+b x^2}}",1,"(e*Sqrt[e*x]*(-3*A*b + 5*a*B + 2*b*B*x^2 + (3*A*b - 5*a*B)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^2)/a)]))/(3*b^2*Sqrt[a + b*x^2])","C",1
810,1,76,301,0.1101193,"\int \frac{\sqrt{e x} \left(A+B x^2\right)}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[(Sqrt[e*x]*(A + B*x^2))/(a + b*x^2)^(3/2),x]","\frac{2 x \sqrt{e x} \left(\sqrt{\frac{b x^2}{a}+1} (A b-3 a B) \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{b x^2}{a}\right)+3 a B\right)}{3 a b \sqrt{a+b x^2}}","-\frac{\sqrt{e} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (A b-3 a B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{2 a^{3/4} b^{7/4} \sqrt{a+b x^2}}+\frac{\sqrt{e} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (A b-3 a B) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{a^{3/4} b^{7/4} \sqrt{a+b x^2}}-\frac{\sqrt{e x} \sqrt{a+b x^2} (A b-3 a B)}{a b^{3/2} \left(\sqrt{a}+\sqrt{b} x\right)}+\frac{(e x)^{3/2} (A b-a B)}{a b e \sqrt{a+b x^2}}",1,"(2*x*Sqrt[e*x]*(3*a*B + (A*b - 3*a*B)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[3/4, 3/2, 7/4, -((b*x^2)/a)]))/(3*a*b*Sqrt[a + b*x^2])","C",1
811,1,75,144,0.0665228,"\int \frac{A+B x^2}{\sqrt{e x} \left(a+b x^2\right)^{3/2}} \, dx","Integrate[(A + B*x^2)/(Sqrt[e*x]*(a + b*x^2)^(3/2)),x]","\frac{x \left(\sqrt{\frac{b x^2}{a}+1} (a B+A b) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^2}{a}\right)-a B+A b\right)}{a b \sqrt{e x} \sqrt{a+b x^2}}","\frac{\left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (a B+A b) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{2 a^{5/4} b^{5/4} \sqrt{e} \sqrt{a+b x^2}}+\frac{\sqrt{e x} (A b-a B)}{a b e \sqrt{a+b x^2}}",1,"(x*(A*b - a*B + (A*b + a*B)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^2)/a)]))/(a*b*Sqrt[e*x]*Sqrt[a + b*x^2])","C",1
812,1,77,333,0.0676307,"\int \frac{A+B x^2}{(e x)^{3/2} \left(a+b x^2\right)^{3/2}} \, dx","Integrate[(A + B*x^2)/((e*x)^(3/2)*(a + b*x^2)^(3/2)),x]","\frac{x \left(2 x^2 \sqrt{\frac{b x^2}{a}+1} (a B-3 A b) \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{b x^2}{a}\right)-6 a A\right)}{3 a^2 (e x)^{3/2} \sqrt{a+b x^2}}","\frac{\left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (3 A b-a B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{2 a^{7/4} b^{3/4} e^{3/2} \sqrt{a+b x^2}}-\frac{\left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (3 A b-a B) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{a^{7/4} b^{3/4} e^{3/2} \sqrt{a+b x^2}}-\frac{(e x)^{3/2} (3 A b-a B)}{a^2 e^3 \sqrt{a+b x^2}}+\frac{\sqrt{e x} \sqrt{a+b x^2} (3 A b-a B)}{a^2 \sqrt{b} e^2 \left(\sqrt{a}+\sqrt{b} x\right)}-\frac{2 A}{a e \sqrt{e x} \sqrt{a+b x^2}}",1,"(x*(-6*a*A + 2*(-3*A*b + a*B)*x^2*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[3/4, 3/2, 7/4, -((b*x^2)/a)]))/(3*a^2*(e*x)^(3/2)*Sqrt[a + b*x^2])","C",1
813,1,91,176,0.0511941,"\int \frac{A+B x^2}{(e x)^{5/2} \left(a+b x^2\right)^{3/2}} \, dx","Integrate[(A + B*x^2)/((e*x)^(5/2)*(a + b*x^2)^(3/2)),x]","\frac{x \left(x^2 \sqrt{\frac{b x^2}{a}+1} (3 a B-5 A b) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^2}{a}\right)-2 a A+3 a B x^2-5 A b x^2\right)}{3 a^2 (e x)^{5/2} \sqrt{a+b x^2}}","-\frac{\left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (5 A b-3 a B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{6 a^{9/4} \sqrt[4]{b} e^{5/2} \sqrt{a+b x^2}}-\frac{\sqrt{e x} (5 A b-3 a B)}{3 a^2 e^3 \sqrt{a+b x^2}}-\frac{2 A}{3 a e (e x)^{3/2} \sqrt{a+b x^2}}",1,"(x*(-2*a*A - 5*A*b*x^2 + 3*a*B*x^2 + (-5*A*b + 3*a*B)*x^2*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^2)/a)]))/(3*a^2*(e*x)^(5/2)*Sqrt[a + b*x^2])","C",1
814,1,78,379,0.0480058,"\int \frac{A+B x^2}{(e x)^{7/2} \left(a+b x^2\right)^{3/2}} \, dx","Integrate[(A + B*x^2)/((e*x)^(7/2)*(a + b*x^2)^(3/2)),x]","\frac{x \left(2 x^2 \sqrt{\frac{b x^2}{a}+1} (7 A b-5 a B) \, _2F_1\left(-\frac{1}{4},\frac{3}{2};\frac{3}{4};-\frac{b x^2}{a}\right)-2 a A\right)}{5 a^2 (e x)^{7/2} \sqrt{a+b x^2}}","-\frac{3 \sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (7 A b-5 a B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{10 a^{11/4} e^{7/2} \sqrt{a+b x^2}}+\frac{3 \sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (7 A b-5 a B) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{5 a^{11/4} e^{7/2} \sqrt{a+b x^2}}-\frac{3 \sqrt{b} \sqrt{e x} \sqrt{a+b x^2} (7 A b-5 a B)}{5 a^3 e^4 \left(\sqrt{a}+\sqrt{b} x\right)}+\frac{3 \sqrt{a+b x^2} (7 A b-5 a B)}{5 a^3 e^3 \sqrt{e x}}-\frac{7 A b-5 a B}{5 a^2 e^3 \sqrt{e x} \sqrt{a+b x^2}}-\frac{2 A}{5 a e (e x)^{5/2} \sqrt{a+b x^2}}",1,"(x*(-2*a*A + 2*(7*A*b - 5*a*B)*x^2*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[-1/4, 3/2, 3/4, -((b*x^2)/a)]))/(5*a^2*(e*x)^(7/2)*Sqrt[a + b*x^2])","C",1
815,1,116,208,0.1942832,"\int \frac{(e x)^{7/2} \left(A+B x^2\right)}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[((e*x)^(7/2)*(A + B*x^2))/(a + b*x^2)^(5/2),x]","\frac{e^3 \sqrt{e x} \left(15 a^2 B+5 \left(a+b x^2\right) \sqrt{\frac{b x^2}{a}+1} (A b-3 a B) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^2}{a}\right)+a \left(21 b B x^2-5 A b\right)+b^2 x^2 \left(4 B x^2-7 A\right)\right)}{6 b^3 \left(a+b x^2\right)^{3/2}}","\frac{5 e^{7/2} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (A b-3 a B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{12 \sqrt[4]{a} b^{13/4} \sqrt{a+b x^2}}-\frac{5 e^3 \sqrt{e x} (A b-3 a B)}{6 b^3 \sqrt{a+b x^2}}-\frac{e (e x)^{5/2} (A b-3 a B)}{3 b^2 \left(a+b x^2\right)^{3/2}}+\frac{2 B (e x)^{9/2}}{3 b e \left(a+b x^2\right)^{3/2}}",1,"(e^3*Sqrt[e*x]*(15*a^2*B + b^2*x^2*(-7*A + 4*B*x^2) + a*(-5*A*b + 21*b*B*x^2) + 5*(A*b - 3*a*B)*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^2)/a)]))/(6*b^3*(a + b*x^2)^(3/2))","C",1
816,1,97,349,0.1474815,"\int \frac{(e x)^{5/2} \left(A+B x^2\right)}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[((e*x)^(5/2)*(A + B*x^2))/(a + b*x^2)^(5/2),x]","-\frac{2 e (e x)^{3/2} \left(\left(a+b x^2\right) \sqrt{\frac{b x^2}{a}+1} (7 a B-A b) \, _2F_1\left(\frac{3}{4},\frac{5}{2};\frac{7}{4};-\frac{b x^2}{a}\right)+a \left(-7 a B+A b-3 b B x^2\right)\right)}{3 a b^2 \left(a+b x^2\right)^{3/2}}","-\frac{e^{5/2} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (A b-7 a B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{4 a^{3/4} b^{11/4} \sqrt{a+b x^2}}+\frac{e^{5/2} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (A b-7 a B) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{2 a^{3/4} b^{11/4} \sqrt{a+b x^2}}-\frac{e^2 \sqrt{e x} \sqrt{a+b x^2} (A b-7 a B)}{2 a b^{5/2} \left(\sqrt{a}+\sqrt{b} x\right)}+\frac{e (e x)^{3/2} (A b-7 a B)}{6 a b^2 \sqrt{a+b x^2}}+\frac{(e x)^{7/2} (A b-a B)}{3 a b e \left(a+b x^2\right)^{3/2}}",1,"(-2*e*(e*x)^(3/2)*(a*(A*b - 7*a*B - 3*b*B*x^2) + (-(A*b) + 7*a*B)*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[3/4, 5/2, 7/4, -((b*x^2)/a)]))/(3*a*b^2*(a + b*x^2)^(3/2))","C",1
817,1,105,185,0.1721134,"\int \frac{(e x)^{3/2} \left(A+B x^2\right)}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[((e*x)^(3/2)*(A + B*x^2))/(a + b*x^2)^(5/2),x]","\frac{e \sqrt{e x} \left(-5 a^2 B+\left(a+b x^2\right) \sqrt{\frac{b x^2}{a}+1} (5 a B+A b) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^2}{a}\right)-a b \left(A+7 B x^2\right)+A b^2 x^2\right)}{6 a b^2 \left(a+b x^2\right)^{3/2}}","\frac{e^{3/2} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (5 a B+A b) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{12 a^{5/4} b^{9/4} \sqrt{a+b x^2}}-\frac{e \sqrt{e x} (5 a B+A b)}{6 a b^2 \sqrt{a+b x^2}}+\frac{(e x)^{5/2} (A b-a B)}{3 a b e \left(a+b x^2\right)^{3/2}}",1,"(e*Sqrt[e*x]*(-5*a^2*B + A*b^2*x^2 - a*b*(A + 7*B*x^2) + (A*b + 5*a*B)*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^2)/a)]))/(6*a*b^2*(a + b*x^2)^(3/2))","C",1
818,1,84,344,0.1120344,"\int \frac{\sqrt{e x} \left(A+B x^2\right)}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[(Sqrt[e*x]*(A + B*x^2))/(a + b*x^2)^(5/2),x]","\frac{2 x \sqrt{e x} \left(\left(a+b x^2\right) \sqrt{\frac{b x^2}{a}+1} (a B+A b) \, _2F_1\left(\frac{3}{4},\frac{5}{2};\frac{7}{4};-\frac{b x^2}{a}\right)-a^2 B\right)}{3 a^2 b \left(a+b x^2\right)^{3/2}}","-\frac{\sqrt{e} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (a B+A b) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{4 a^{7/4} b^{7/4} \sqrt{a+b x^2}}+\frac{\sqrt{e} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (a B+A b) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{2 a^{7/4} b^{7/4} \sqrt{a+b x^2}}-\frac{\sqrt{e x} \sqrt{a+b x^2} (a B+A b)}{2 a^2 b^{3/2} \left(\sqrt{a}+\sqrt{b} x\right)}+\frac{(e x)^{3/2} (a B+A b)}{2 a^2 b e \sqrt{a+b x^2}}+\frac{(e x)^{3/2} (A b-a B)}{3 a b e \left(a+b x^2\right)^{3/2}}",1,"(2*x*Sqrt[e*x]*(-(a^2*B) + (A*b + a*B)*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[3/4, 5/2, 7/4, -((b*x^2)/a)]))/(3*a^2*b*(a + b*x^2)^(3/2))","C",1
819,1,108,187,0.0919933,"\int \frac{A+B x^2}{\sqrt{e x} \left(a+b x^2\right)^{5/2}} \, dx","Integrate[(A + B*x^2)/(Sqrt[e*x]*(a + b*x^2)^(5/2)),x]","\frac{-a^2 B x+x \left(a+b x^2\right) \sqrt{\frac{b x^2}{a}+1} (a B+5 A b) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^2}{a}\right)+a b x \left(7 A+B x^2\right)+5 A b^2 x^3}{6 a^2 b \sqrt{e x} \left(a+b x^2\right)^{3/2}}","\frac{\left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (a B+5 A b) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{12 a^{9/4} b^{5/4} \sqrt{e} \sqrt{a+b x^2}}+\frac{\sqrt{e x} (a B+5 A b)}{6 a^2 b e \sqrt{a+b x^2}}+\frac{\sqrt{e x} (A b-a B)}{3 a b e \left(a+b x^2\right)^{3/2}}",1,"(-(a^2*B*x) + 5*A*b^2*x^3 + a*b*x*(7*A + B*x^2) + (5*A*b + a*B)*x*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^2)/a)])/(6*a^2*b*Sqrt[e*x]*(a + b*x^2)^(3/2))","C",1
820,1,86,377,0.0676915,"\int \frac{A+B x^2}{(e x)^{3/2} \left(a+b x^2\right)^{5/2}} \, dx","Integrate[(A + B*x^2)/((e*x)^(3/2)*(a + b*x^2)^(5/2)),x]","\frac{x \left(2 x^2 \left(a+b x^2\right) \sqrt{\frac{b x^2}{a}+1} (a B-7 A b) \, _2F_1\left(\frac{3}{4},\frac{5}{2};\frac{7}{4};-\frac{b x^2}{a}\right)-6 a^2 A\right)}{3 a^3 (e x)^{3/2} \left(a+b x^2\right)^{3/2}}","\frac{\left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (7 A b-a B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{4 a^{11/4} b^{3/4} e^{3/2} \sqrt{a+b x^2}}-\frac{\left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (7 A b-a B) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{2 a^{11/4} b^{3/4} e^{3/2} \sqrt{a+b x^2}}-\frac{(e x)^{3/2} (7 A b-a B)}{2 a^3 e^3 \sqrt{a+b x^2}}+\frac{\sqrt{e x} \sqrt{a+b x^2} (7 A b-a B)}{2 a^3 \sqrt{b} e^2 \left(\sqrt{a}+\sqrt{b} x\right)}-\frac{(e x)^{3/2} (7 A b-a B)}{3 a^2 e^3 \left(a+b x^2\right)^{3/2}}-\frac{2 A}{a e \sqrt{e x} \left(a+b x^2\right)^{3/2}}",1,"(x*(-6*a^2*A + 2*(-7*A*b + a*B)*x^2*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[3/4, 5/2, 7/4, -((b*x^2)/a)]))/(3*a^3*(e*x)^(3/2)*(a + b*x^2)^(3/2))","C",1
821,1,120,213,0.0856071,"\int \frac{A+B x^2}{(e x)^{5/2} \left(a+b x^2\right)^{5/2}} \, dx","Integrate[(A + B*x^2)/((e*x)^(5/2)*(a + b*x^2)^(5/2)),x]","\frac{x \left(a^2 \left(7 B x^2-4 A\right)+5 x^2 \left(a+b x^2\right) \sqrt{\frac{b x^2}{a}+1} (a B-3 A b) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^2}{a}\right)+a \left(5 b B x^4-21 A b x^2\right)-15 A b^2 x^4\right)}{6 a^3 (e x)^{5/2} \left(a+b x^2\right)^{3/2}}","-\frac{5 \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} (3 A b-a B) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt[4]{a} \sqrt{e}}\right)|\frac{1}{2}\right)}{12 a^{13/4} \sqrt[4]{b} e^{5/2} \sqrt{a+b x^2}}-\frac{5 \sqrt{e x} (3 A b-a B)}{6 a^3 e^3 \sqrt{a+b x^2}}-\frac{\sqrt{e x} (3 A b-a B)}{3 a^2 e^3 \left(a+b x^2\right)^{3/2}}-\frac{2 A}{3 a e (e x)^{3/2} \left(a+b x^2\right)^{3/2}}",1,"(x*(-15*A*b^2*x^4 + a^2*(-4*A + 7*B*x^2) + a*(-21*A*b*x^2 + 5*b*B*x^4) + 5*(-3*A*b + a*B)*x^2*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^2)/a)]))/(6*a^3*(e*x)^(5/2)*(a + b*x^2)^(3/2))","C",1
822,1,225,288,0.3473779,"\int (e x)^{3/2} \left(a+b x^2\right)^2 \sqrt{c+d x^2} \, dx","Integrate[(e*x)^(3/2)*(a + b*x^2)^2*Sqrt[c + d*x^2],x]","\frac{(e x)^{3/2} \left(\frac{2 \sqrt{x} \left(c+d x^2\right) \left(55 a^2 d^2 \left(2 c+3 d x^2\right)+10 a b d \left(-10 c^2+6 c d x^2+21 d^2 x^4\right)+b^2 \left(30 c^3-18 c^2 d x^2+14 c d^2 x^4+77 d^3 x^6\right)\right)}{5 d^3}-\frac{4 i c^2 x \sqrt{\frac{c}{d x^2}+1} \left(11 a^2 d^2-10 a b c d+3 b^2 c^2\right) F\left(\left.i \sinh ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right)\right|-1\right)}{d^3 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right)}{231 x^{3/2} \sqrt{c+d x^2}}","-\frac{2 c^{7/4} e^{3/2} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(11 a^2 d^2+b c (3 b c-10 a d)\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{231 d^{13/4} \sqrt{c+d x^2}}+\frac{2 (e x)^{5/2} \sqrt{c+d x^2} \left(11 a^2 d^2+b c (3 b c-10 a d)\right)}{77 d^2 e}+\frac{4 c e \sqrt{e x} \sqrt{c+d x^2} \left(11 a^2 d^2+b c (3 b c-10 a d)\right)}{231 d^3}-\frac{2 b (e x)^{5/2} \left(c+d x^2\right)^{3/2} (3 b c-10 a d)}{55 d^2 e}+\frac{2 b^2 (e x)^{9/2} \left(c+d x^2\right)^{3/2}}{15 d e^3}",1,"((e*x)^(3/2)*((2*Sqrt[x]*(c + d*x^2)*(55*a^2*d^2*(2*c + 3*d*x^2) + 10*a*b*d*(-10*c^2 + 6*c*d*x^2 + 21*d^2*x^4) + b^2*(30*c^3 - 18*c^2*d*x^2 + 14*c*d^2*x^4 + 77*d^3*x^6)))/(5*d^3) - ((4*I)*c^2*(3*b^2*c^2 - 10*a*b*c*d + 11*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/(Sqrt[(I*Sqrt[c])/Sqrt[d]]*d^3)))/(231*x^(3/2)*Sqrt[c + d*x^2])","C",1
823,1,145,425,0.1486222,"\int \sqrt{e x} \left(a+b x^2\right)^2 \sqrt{c+d x^2} \, dx","Integrate[Sqrt[e*x]*(a + b*x^2)^2*Sqrt[c + d*x^2],x]","\frac{2 \sqrt{e x} \left(6 c x \sqrt{\frac{c}{d x^2}+1} \left(39 a^2 d^2-26 a b c d+7 b^2 c^2\right) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{c}{d x^2}\right)-x \left(c+d x^2\right) \left(-117 a^2 d^2-26 a b d \left(2 c+5 d x^2\right)+b^2 \left(14 c^2-10 c d x^2-45 d^2 x^4\right)\right)\right)}{585 d^2 \sqrt{c+d x^2}}","\frac{2 c^{5/4} \sqrt{e} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(39 a^2 d^2+b c (7 b c-26 a d)\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{195 d^{11/4} \sqrt{c+d x^2}}-\frac{4 c^{5/4} \sqrt{e} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(39 a^2 d^2+b c (7 b c-26 a d)\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{195 d^{11/4} \sqrt{c+d x^2}}+\frac{2 (e x)^{3/2} \sqrt{c+d x^2} \left(39 a^2 d^2+b c (7 b c-26 a d)\right)}{195 d^2 e}+\frac{4 c \sqrt{e x} \sqrt{c+d x^2} \left(39 a^2 d^2+b c (7 b c-26 a d)\right)}{195 d^{5/2} \left(\sqrt{c}+\sqrt{d} x\right)}-\frac{2 b (e x)^{3/2} \left(c+d x^2\right)^{3/2} (7 b c-26 a d)}{117 d^2 e}+\frac{2 b^2 (e x)^{7/2} \left(c+d x^2\right)^{3/2}}{13 d e^3}",1,"(2*Sqrt[e*x]*(-(x*(c + d*x^2)*(-117*a^2*d^2 - 26*a*b*d*(2*c + 5*d*x^2) + b^2*(14*c^2 - 10*c*d*x^2 - 45*d^2*x^4))) + 6*c*(7*b^2*c^2 - 26*a*b*c*d + 39*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*Hypergeometric2F1[-1/4, 1/2, 3/4, -(c/(d*x^2))]))/(585*d^2*Sqrt[c + d*x^2])","C",1
824,1,189,244,0.2304726,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{\sqrt{e x}} \, dx","Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/Sqrt[e*x],x]","\frac{\sqrt{x} \left(\frac{2 \sqrt{x} \left(c+d x^2\right) \left(77 a^2 d^2+22 a b d \left(2 c+3 d x^2\right)+b^2 \left(-10 c^2+6 c d x^2+21 d^2 x^4\right)\right)}{d^2}+\frac{4 i c x \sqrt{\frac{c}{d x^2}+1} \left(77 a^2 d^2-22 a b c d+5 b^2 c^2\right) F\left(\left.i \sinh ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right)\right|-1\right)}{d^2 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right)}{231 \sqrt{e x} \sqrt{c+d x^2}}","\frac{2 \sqrt{e x} \sqrt{c+d x^2} \left(77 a^2 d^2-22 a b c d+5 b^2 c^2\right)}{231 d^2 e}+\frac{2 c^{3/4} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(77 a^2 d^2-22 a b c d+5 b^2 c^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{231 d^{9/4} \sqrt{e} \sqrt{c+d x^2}}-\frac{2 b \sqrt{e x} \left(c+d x^2\right)^{3/2} (5 b c-22 a d)}{77 d^2 e}+\frac{2 b^2 (e x)^{5/2} \left(c+d x^2\right)^{3/2}}{11 d e^3}",1,"(Sqrt[x]*((2*Sqrt[x]*(c + d*x^2)*(77*a^2*d^2 + 22*a*b*d*(2*c + 3*d*x^2) + b^2*(-10*c^2 + 6*c*d*x^2 + 21*d^2*x^4)))/d^2 + ((4*I)*c*(5*b^2*c^2 - 22*a*b*c*d + 77*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/(Sqrt[(I*Sqrt[c])/Sqrt[d]]*d^2)))/(231*Sqrt[e*x]*Sqrt[c + d*x^2])","C",1
825,1,129,421,0.1492257,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{(e x)^{3/2}} \, dx","Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/(e*x)^(3/2),x]","\frac{x \left(12 x^2 \sqrt{\frac{c}{d x^2}+1} \left(15 a^2 d^2+6 a b c d-b^2 c^2\right) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{c}{d x^2}\right)+2 \left(c+d x^2\right) \left(-45 a^2 d+18 a b d x^2+b^2 x^2 \left(2 c+5 d x^2\right)\right)\right)}{45 d (e x)^{3/2} \sqrt{c+d x^2}}","-\frac{2 a^2 \left(c+d x^2\right)^{3/2}}{c e \sqrt{e x}}-\frac{2 \sqrt[4]{c} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(b^2 c^2-3 a d (5 a d+2 b c)\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{15 d^{7/4} e^{3/2} \sqrt{c+d x^2}}+\frac{4 \sqrt[4]{c} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(b^2 c^2-3 a d (5 a d+2 b c)\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{15 d^{7/4} e^{3/2} \sqrt{c+d x^2}}-\frac{4 \sqrt{e x} \sqrt{c+d x^2} \left(b^2 c^2-3 a d (5 a d+2 b c)\right)}{15 d^{3/2} e^2 \left(\sqrt{c}+\sqrt{d} x\right)}-\frac{2 (e x)^{3/2} \sqrt{c+d x^2} \left(b^2 c^2-3 a d (5 a d+2 b c)\right)}{15 c d e^3}+\frac{2 b^2 (e x)^{3/2} \left(c+d x^2\right)^{3/2}}{9 d e^3}",1,"(x*(2*(c + d*x^2)*(-45*a^2*d + 18*a*b*d*x^2 + b^2*x^2*(2*c + 5*d*x^2)) + 12*(-(b^2*c^2) + 6*a*b*c*d + 15*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x^2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(c/(d*x^2))]))/(45*d*(e*x)^(3/2)*Sqrt[c + d*x^2])","C",1
826,1,171,234,0.2273806,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{(e x)^{5/2}} \, dx","Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/(e*x)^(5/2),x]","\frac{x^{5/2} \left(\frac{2 \left(c+d x^2\right) \left(-7 a^2 d+14 a b d x^2+b^2 x^2 \left(2 c+3 d x^2\right)\right)}{d x^{3/2}}+\frac{4 i x \sqrt{\frac{c}{d x^2}+1} \left(7 a^2 d^2+14 a b c d-b^2 c^2\right) F\left(\left.i \sinh ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right)\right|-1\right)}{d \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right)}{21 (e x)^{5/2} \sqrt{c+d x^2}}","-\frac{2 a^2 \left(c+d x^2\right)^{3/2}}{3 c e (e x)^{3/2}}-\frac{2 \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(b^2 c^2-7 a d (a d+2 b c)\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{21 \sqrt[4]{c} d^{5/4} e^{5/2} \sqrt{c+d x^2}}-\frac{2 \sqrt{e x} \sqrt{c+d x^2} \left(b^2 c^2-7 a d (a d+2 b c)\right)}{21 c d e^3}+\frac{2 b^2 \sqrt{e x} \left(c+d x^2\right)^{3/2}}{7 d e^3}",1,"(x^(5/2)*((2*(c + d*x^2)*(-7*a^2*d + 14*a*b*d*x^2 + b^2*x^2*(2*c + 3*d*x^2)))/(d*x^(3/2)) + ((4*I)*(-(b^2*c^2) + 14*a*b*c*d + 7*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/(Sqrt[(I*Sqrt[c])/Sqrt[d]]*d)))/(21*(e*x)^(5/2)*Sqrt[c + d*x^2])","C",1
827,1,125,421,0.1321886,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{(e x)^{7/2}} \, dx","Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/(e*x)^(7/2),x]","\frac{x \left(4 x^4 \sqrt{\frac{c}{d x^2}+1} \left(a^2 d^2+10 a b c d+b^2 c^2\right) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{c}{d x^2}\right)-2 \left(c+d x^2\right) \left(a^2 \left(c+2 d x^2\right)+10 a b c x^2-b^2 c x^4\right)\right)}{5 c (e x)^{7/2} \sqrt{c+d x^2}}","-\frac{2 a^2 \left(c+d x^2\right)^{3/2}}{5 c e (e x)^{5/2}}+\frac{2 (e x)^{3/2} \sqrt{c+d x^2} \left(a d (a d+10 b c)+b^2 c^2\right)}{5 c^2 e^5}+\frac{4 \sqrt{e x} \sqrt{c+d x^2} \left(a d (a d+10 b c)+b^2 c^2\right)}{5 c \sqrt{d} e^4 \left(\sqrt{c}+\sqrt{d} x\right)}+\frac{2 \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(a d (a d+10 b c)+b^2 c^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{5 c^{3/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}-\frac{4 \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(a d (a d+10 b c)+b^2 c^2\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{5 c^{3/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}-\frac{2 a \left(c+d x^2\right)^{3/2} (a d+10 b c)}{5 c^2 e^3 \sqrt{e x}}",1,"(x*(-2*(c + d*x^2)*(10*a*b*c*x^2 - b^2*c*x^4 + a^2*(c + 2*d*x^2)) + 4*(b^2*c^2 + 10*a*b*c*d + a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x^4*Hypergeometric2F1[-1/4, 1/2, 3/4, -(c/(d*x^2))]))/(5*c*(e*x)^(7/2)*Sqrt[c + d*x^2])","C",1
828,1,160,213,0.2487606,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{x^{9/2}} \, dx","Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^(9/2),x]","\frac{2 \left(\left(c+d x^2\right) \left(-a^2 \left(3 c+2 d x^2\right)-14 a b c x^2+7 b^2 c x^4\right)+\frac{2 i x^{9/2} \sqrt{\frac{c}{d x^2}+1} \left(-a^2 d^2+14 a b c d+7 b^2 c^2\right) F\left(\left.i \sinh ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right)\right|-1\right)}{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right)}{21 c x^{7/2} \sqrt{c+d x^2}}","-\frac{2 a^2 \left(c+d x^2\right)^{3/2}}{7 c x^{7/2}}+\frac{2 \sqrt{x} \sqrt{c+d x^2} \left(a d (14 b c-a d)+7 b^2 c^2\right)}{21 c^2}+\frac{2 \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(a d (14 b c-a d)+7 b^2 c^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{21 c^{5/4} \sqrt[4]{d} \sqrt{c+d x^2}}-\frac{2 a \left(c+d x^2\right)^{3/2} (14 b c-a d)}{21 c^2 x^{3/2}}",1,"(2*((c + d*x^2)*(-14*a*b*c*x^2 + 7*b^2*c*x^4 - a^2*(3*c + 2*d*x^2)) + ((2*I)*(7*b^2*c^2 + 14*a*b*c*d - a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x^(9/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/Sqrt[(I*Sqrt[c])/Sqrt[d]]))/(21*c*x^(7/2)*Sqrt[c + d*x^2])","C",1
829,1,148,386,0.145055,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{x^{11/2}} \, dx","Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^(11/2),x]","\frac{12 d x^6 \sqrt{\frac{c}{d x^2}+1} \left(-a^2 d^2+6 a b c d+15 b^2 c^2\right) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{c}{d x^2}\right)-2 \left(c+d x^2\right) \left(a^2 \left(5 c^2+2 c d x^2-6 d^2 x^4\right)+18 a b c x^2 \left(c+2 d x^2\right)+45 b^2 c^2 x^4\right)}{45 c^2 x^{9/2} \sqrt{c+d x^2}}","-\frac{2 a^2 \left(c+d x^2\right)^{3/2}}{9 c x^{9/2}}-\frac{2 \sqrt{c+d x^2} \left(a d (6 b c-a d)+15 b^2 c^2\right)}{15 c^2 \sqrt{x}}+\frac{4 \sqrt{d} \sqrt{x} \sqrt{c+d x^2} \left(a d (6 b c-a d)+15 b^2 c^2\right)}{15 c^2 \left(\sqrt{c}+\sqrt{d} x\right)}+\frac{2 \sqrt[4]{d} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(a d (6 b c-a d)+15 b^2 c^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{15 c^{7/4} \sqrt{c+d x^2}}-\frac{4 \sqrt[4]{d} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(a d (6 b c-a d)+15 b^2 c^2\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{15 c^{7/4} \sqrt{c+d x^2}}-\frac{2 a \left(c+d x^2\right)^{3/2} (6 b c-a d)}{15 c^2 x^{5/2}}",1,"(-2*(c + d*x^2)*(45*b^2*c^2*x^4 + 18*a*b*c*x^2*(c + 2*d*x^2) + a^2*(5*c^2 + 2*c*d*x^2 - 6*d^2*x^4)) + 12*d*(15*b^2*c^2 + 6*a*b*c*d - a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x^6*Hypergeometric2F1[-1/4, 1/2, 3/4, -(c/(d*x^2))])/(45*c^2*x^(9/2)*Sqrt[c + d*x^2])","C",1
830,1,187,217,0.1708645,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{x^{13/2}} \, dx","Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^(13/2),x]","-\frac{2 \sqrt{c+d x^2} \left(a^2 \left(21 c^2+6 c d x^2-10 d^2 x^4\right)+22 a b c x^2 \left(3 c+2 d x^2\right)+77 b^2 c^2 x^4\right)}{231 c^2 x^{11/2}}+\frac{4 i d x \sqrt{\frac{c}{d x^2}+1} \left(5 a^2 d^2-22 a b c d+77 b^2 c^2\right) F\left(\left.i \sinh ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right)\right|-1\right)}{231 c^2 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \sqrt{c+d x^2}}","-\frac{2 \sqrt{c+d x^2} \left(5 a^2 d^2-22 a b c d+77 b^2 c^2\right)}{231 c^2 x^{3/2}}+\frac{2 d^{3/4} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(5 a^2 d^2-22 a b c d+77 b^2 c^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{231 c^{9/4} \sqrt{c+d x^2}}-\frac{2 a^2 \left(c+d x^2\right)^{3/2}}{11 c x^{11/2}}-\frac{2 a \left(c+d x^2\right)^{3/2} (22 b c-5 a d)}{77 c^2 x^{7/2}}",1,"(-2*Sqrt[c + d*x^2]*(77*b^2*c^2*x^4 + 22*a*b*c*x^2*(3*c + 2*d*x^2) + a^2*(21*c^2 + 6*c*d*x^2 - 10*d^2*x^4)))/(231*c^2*x^(11/2)) + (((4*I)/231)*d*(77*b^2*c^2 - 22*a*b*c*d + 5*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/(c^2*Sqrt[(I*Sqrt[c])/Sqrt[d]]*Sqrt[c + d*x^2])","C",1
831,1,182,441,0.1989444,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{x^{15/2}} \, dx","Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^(15/2),x]","\frac{4 d^2 x^8 \sqrt{\frac{d x^2}{c}+1} \left(7 a^2 d^2-26 a b c d+39 b^2 c^2\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{d x^2}{c}\right)-2 \left(c+d x^2\right) \left(a^2 \left(45 c^3+10 c^2 d x^2-14 c d^2 x^4+42 d^3 x^6\right)+26 a b c x^2 \left(5 c^2+2 c d x^2-6 d^2 x^4\right)+117 b^2 c^2 x^4 \left(c+2 d x^2\right)\right)}{585 c^3 x^{13/2} \sqrt{c+d x^2}}","-\frac{2 \sqrt{c+d x^2} \left(7 a^2 d^2-26 a b c d+39 b^2 c^2\right)}{195 c^2 x^{5/2}}+\frac{2 d^{5/4} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(7 a^2 d^2-26 a b c d+39 b^2 c^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{195 c^{11/4} \sqrt{c+d x^2}}-\frac{4 d^{5/4} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(7 a^2 d^2-26 a b c d+39 b^2 c^2\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{x}}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{195 c^{11/4} \sqrt{c+d x^2}}-\frac{4 d \sqrt{c+d x^2} \left(7 a^2 d^2-26 a b c d+39 b^2 c^2\right)}{195 c^3 \sqrt{x}}+\frac{4 d^{3/2} \sqrt{x} \sqrt{c+d x^2} \left(7 a^2 d^2-26 a b c d+39 b^2 c^2\right)}{195 c^3 \left(\sqrt{c}+\sqrt{d} x\right)}-\frac{2 a^2 \left(c+d x^2\right)^{3/2}}{13 c x^{13/2}}-\frac{2 a \left(c+d x^2\right)^{3/2} (26 b c-7 a d)}{117 c^2 x^{9/2}}",1,"(-2*(c + d*x^2)*(117*b^2*c^2*x^4*(c + 2*d*x^2) + 26*a*b*c*x^2*(5*c^2 + 2*c*d*x^2 - 6*d^2*x^4) + a^2*(45*c^3 + 10*c^2*d*x^2 - 14*c*d^2*x^4 + 42*d^3*x^6)) + 4*d^2*(39*b^2*c^2 - 26*a*b*c*d + 7*a^2*d^2)*x^8*Sqrt[1 + (d*x^2)/c]*Hypergeometric2F1[1/2, 3/4, 7/4, -((d*x^2)/c)])/(585*c^3*x^(13/2)*Sqrt[c + d*x^2])","C",1
832,1,210,530,0.1795163,"\int (e x)^{5/2} \left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2} \, dx","Integrate[(e*x)^(5/2)*(a + b*x^2)^2*(c + d*x^2)^(3/2),x]","\frac{2 e (e x)^{3/2} \left(\left(c+d x^2\right) \left(357 a^2 d^2 \left(4 c^2+25 c d x^2+15 d^2 x^4\right)+42 a b d \left(-28 c^3+20 c^2 d x^2+285 c d^2 x^4+195 d^3 x^6\right)+b^2 \left(308 c^4-220 c^3 d x^2+180 c^2 d^2 x^4+4485 c d^3 x^6+3315 d^4 x^8\right)\right)-84 c^3 \sqrt{\frac{c}{d x^2}+1} \left(51 a^2 d^2-42 a b c d+11 b^2 c^2\right) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{c}{d x^2}\right)\right)}{69615 d^3 \sqrt{c+d x^2}}","-\frac{4 c^{13/4} e^{5/2} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(51 a^2 d^2+b c (11 b c-42 a d)\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{3315 d^{15/4} \sqrt{c+d x^2}}+\frac{8 c^{13/4} e^{5/2} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(51 a^2 d^2+b c (11 b c-42 a d)\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{3315 d^{15/4} \sqrt{c+d x^2}}-\frac{8 c^3 e^2 \sqrt{e x} \sqrt{c+d x^2} \left(51 a^2 d^2+b c (11 b c-42 a d)\right)}{3315 d^{7/2} \left(\sqrt{c}+\sqrt{d} x\right)}+\frac{8 c^2 e (e x)^{3/2} \sqrt{c+d x^2} \left(51 a^2 d^2+b c (11 b c-42 a d)\right)}{9945 d^3}+\frac{2 (e x)^{7/2} \left(c+d x^2\right)^{3/2} \left(51 a^2 d^2+b c (11 b c-42 a d)\right)}{663 d^2 e}+\frac{4 c (e x)^{7/2} \sqrt{c+d x^2} \left(51 a^2 d^2+b c (11 b c-42 a d)\right)}{1989 d^2 e}-\frac{2 b (e x)^{7/2} \left(c+d x^2\right)^{5/2} (11 b c-42 a d)}{357 d^2 e}+\frac{2 b^2 (e x)^{11/2} \left(c+d x^2\right)^{5/2}}{21 d e^3}",1,"(2*e*(e*x)^(3/2)*((c + d*x^2)*(357*a^2*d^2*(4*c^2 + 25*c*d*x^2 + 15*d^2*x^4) + 42*a*b*d*(-28*c^3 + 20*c^2*d*x^2 + 285*c*d^2*x^4 + 195*d^3*x^6) + b^2*(308*c^4 - 220*c^3*d*x^2 + 180*c^2*d^2*x^4 + 4485*c*d^3*x^6 + 3315*d^4*x^8)) - 84*c^3*(11*b^2*c^2 - 42*a*b*c*d + 51*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*Hypergeometric2F1[-1/4, 1/2, 3/4, -(c/(d*x^2))]))/(69615*d^3*Sqrt[c + d*x^2])","C",1
833,1,259,340,0.2823783,"\int (e x)^{3/2} \left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2} \, dx","Integrate[(e*x)^(3/2)*(a + b*x^2)^2*(c + d*x^2)^(3/2),x]","\frac{(e x)^{3/2} \left(\frac{2 \sqrt{x} \left(c+d x^2\right) \left(285 a^2 d^2 \left(4 c^2+13 c d x^2+7 d^2 x^4\right)+38 a b d \left(-20 c^3+12 c^2 d x^2+119 c d^2 x^4+77 d^3 x^6\right)+3 b^2 \left(60 c^4-36 c^3 d x^2+28 c^2 d^2 x^4+539 c d^3 x^6+385 d^4 x^8\right)\right)}{5 d^3}-\frac{8 i c^3 x \sqrt{\frac{c}{d x^2}+1} \left(57 a^2 d^2-38 a b c d+9 b^2 c^2\right) F\left(\left.i \sinh ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right)\right|-1\right)}{d^3 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right)}{4389 x^{3/2} \sqrt{c+d x^2}}","-\frac{4 c^{11/4} e^{3/2} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(57 a^2 d^2+b c (9 b c-38 a d)\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{4389 d^{13/4} \sqrt{c+d x^2}}+\frac{8 c^2 e \sqrt{e x} \sqrt{c+d x^2} \left(57 a^2 d^2+b c (9 b c-38 a d)\right)}{4389 d^3}+\frac{2 (e x)^{5/2} \left(c+d x^2\right)^{3/2} \left(57 a^2 d^2+b c (9 b c-38 a d)\right)}{627 d^2 e}+\frac{4 c (e x)^{5/2} \sqrt{c+d x^2} \left(57 a^2 d^2+b c (9 b c-38 a d)\right)}{1463 d^2 e}-\frac{2 b (e x)^{5/2} \left(c+d x^2\right)^{5/2} (9 b c-38 a d)}{285 d^2 e}+\frac{2 b^2 (e x)^{9/2} \left(c+d x^2\right)^{5/2}}{19 d e^3}",1,"((e*x)^(3/2)*((2*Sqrt[x]*(c + d*x^2)*(285*a^2*d^2*(4*c^2 + 13*c*d*x^2 + 7*d^2*x^4) + 38*a*b*d*(-20*c^3 + 12*c^2*d*x^2 + 119*c*d^2*x^4 + 77*d^3*x^6) + 3*b^2*(60*c^4 - 36*c^3*d*x^2 + 28*c^2*d^2*x^4 + 539*c*d^3*x^6 + 385*d^4*x^8)))/(5*d^3) - ((8*I)*c^3*(9*b^2*c^2 - 38*a*b*c*d + 57*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/(Sqrt[(I*Sqrt[c])/Sqrt[d]]*d^3)))/(4389*x^(3/2)*Sqrt[c + d*x^2])","C",1
834,1,179,482,0.1537289,"\int \sqrt{e x} \left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2} \, dx","Integrate[Sqrt[e*x]*(a + b*x^2)^2*(c + d*x^2)^(3/2),x]","\frac{2 \sqrt{e x} \left(12 c^2 x \sqrt{\frac{c}{d x^2}+1} \left(221 a^2 d^2-102 a b c d+21 b^2 c^2\right) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{c}{d x^2}\right)-x \left(c+d x^2\right) \left(-221 a^2 d^2 \left(11 c+5 d x^2\right)-102 a b d \left(4 c^2+25 c d x^2+15 d^2 x^4\right)+b^2 \left(84 c^3-60 c^2 d x^2-855 c d^2 x^4-585 d^3 x^6\right)\right)\right)}{9945 d^2 \sqrt{c+d x^2}}","\frac{4 c^{9/4} \sqrt{e} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(221 a^2 d^2+3 b c (7 b c-34 a d)\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{3315 d^{11/4} \sqrt{c+d x^2}}-\frac{8 c^{9/4} \sqrt{e} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(221 a^2 d^2+3 b c (7 b c-34 a d)\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{3315 d^{11/4} \sqrt{c+d x^2}}+\frac{8 c^2 \sqrt{e x} \sqrt{c+d x^2} \left(221 a^2 d^2+3 b c (7 b c-34 a d)\right)}{3315 d^{5/2} \left(\sqrt{c}+\sqrt{d} x\right)}+\frac{2 (e x)^{3/2} \left(c+d x^2\right)^{3/2} \left(221 a^2 d^2+3 b c (7 b c-34 a d)\right)}{1989 d^2 e}+\frac{4 c (e x)^{3/2} \sqrt{c+d x^2} \left(221 a^2 d^2+3 b c (7 b c-34 a d)\right)}{3315 d^2 e}-\frac{2 b (e x)^{3/2} \left(c+d x^2\right)^{5/2} (7 b c-34 a d)}{221 d^2 e}+\frac{2 b^2 (e x)^{7/2} \left(c+d x^2\right)^{5/2}}{17 d e^3}",1,"(2*Sqrt[e*x]*(-(x*(c + d*x^2)*(-221*a^2*d^2*(11*c + 5*d*x^2) - 102*a*b*d*(4*c^2 + 25*c*d*x^2 + 15*d^2*x^4) + b^2*(84*c^3 - 60*c^2*d*x^2 - 855*c*d^2*x^4 - 585*d^3*x^6))) + 12*c^2*(21*b^2*c^2 - 102*a*b*c*d + 221*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*Hypergeometric2F1[-1/4, 1/2, 3/4, -(c/(d*x^2))]))/(9945*d^2*Sqrt[c + d*x^2])","C",1
835,1,223,286,0.2503354,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}}{\sqrt{e x}} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^(3/2))/Sqrt[e*x],x]","\frac{\sqrt{x} \left(\frac{2 \sqrt{x} \left(c+d x^2\right) \left(165 a^2 d^2 \left(3 c+d x^2\right)+30 a b d \left(4 c^2+13 c d x^2+7 d^2 x^4\right)+b^2 \left(-20 c^3+12 c^2 d x^2+119 c d^2 x^4+77 d^3 x^6\right)\right)}{5 d^2}+\frac{8 i c^2 x \sqrt{\frac{c}{d x^2}+1} \left(33 a^2 d^2-6 a b c d+b^2 c^2\right) F\left(\left.i \sinh ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right)\right|-1\right)}{d^2 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right)}{231 \sqrt{e x} \sqrt{c+d x^2}}","\frac{4 c^{7/4} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(33 a^2 d^2+b c (b c-6 a d)\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{231 d^{9/4} \sqrt{e} \sqrt{c+d x^2}}+\frac{2 \sqrt{e x} \left(c+d x^2\right)^{3/2} \left(33 a^2 d^2+b c (b c-6 a d)\right)}{231 d^2 e}+\frac{4 c \sqrt{e x} \sqrt{c+d x^2} \left(33 a^2 d^2+b c (b c-6 a d)\right)}{231 d^2 e}-\frac{2 b \sqrt{e x} \left(c+d x^2\right)^{5/2} (b c-6 a d)}{33 d^2 e}+\frac{2 b^2 (e x)^{5/2} \left(c+d x^2\right)^{5/2}}{15 d e^3}",1,"(Sqrt[x]*((2*Sqrt[x]*(c + d*x^2)*(165*a^2*d^2*(3*c + d*x^2) + 30*a*b*d*(4*c^2 + 13*c*d*x^2 + 7*d^2*x^4) + b^2*(-20*c^3 + 12*c^2*d*x^2 + 119*c*d^2*x^4 + 77*d^3*x^6)))/(5*d^2) + ((8*I)*c^2*(b^2*c^2 - 6*a*b*c*d + 33*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/(Sqrt[(I*Sqrt[c])/Sqrt[d]]*d^2)))/(231*Sqrt[e*x]*Sqrt[c + d*x^2])","C",1
836,1,161,476,0.1482616,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}}{(e x)^{3/2}} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^(3/2))/(e*x)^(3/2),x]","\frac{x \left(24 c x^2 \sqrt{\frac{c}{d x^2}+1} \left(117 a^2 d^2+26 a b c d-3 b^2 c^2\right) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{c}{d x^2}\right)+2 \left(c+d x^2\right) \left(117 a^2 d \left(d x^2-5 c\right)+26 a b d x^2 \left(11 c+5 d x^2\right)+3 b^2 x^2 \left(4 c^2+25 c d x^2+15 d^2 x^4\right)\right)\right)}{585 d (e x)^{3/2} \sqrt{c+d x^2}}","-\frac{2 a^2 \left(c+d x^2\right)^{5/2}}{c e \sqrt{e x}}-\frac{8 c \sqrt{e x} \sqrt{c+d x^2} \left(3 b^2 c^2-13 a d (9 a d+2 b c)\right)}{195 d^{3/2} e^2 \left(\sqrt{c}+\sqrt{d} x\right)}-\frac{2 (e x)^{3/2} \left(c+d x^2\right)^{3/2} \left(3 b^2 c^2-13 a d (9 a d+2 b c)\right)}{117 c d e^3}-\frac{4 (e x)^{3/2} \sqrt{c+d x^2} \left(3 b^2 c^2-13 a d (9 a d+2 b c)\right)}{195 d e^3}-\frac{4 c^{5/4} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(3 b^2 c^2-13 a d (9 a d+2 b c)\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{195 d^{7/4} e^{3/2} \sqrt{c+d x^2}}+\frac{8 c^{5/4} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(3 b^2 c^2-13 a d (9 a d+2 b c)\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{195 d^{7/4} e^{3/2} \sqrt{c+d x^2}}+\frac{2 b^2 (e x)^{3/2} \left(c+d x^2\right)^{5/2}}{13 d e^3}",1,"(x*(2*(c + d*x^2)*(117*a^2*d*(-5*c + d*x^2) + 26*a*b*d*x^2*(11*c + 5*d*x^2) + 3*b^2*x^2*(4*c^2 + 25*c*d*x^2 + 15*d^2*x^4)) + 24*c*(-3*b^2*c^2 + 26*a*b*c*d + 117*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x^2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(c/(d*x^2))]))/(585*d*(e*x)^(3/2)*Sqrt[c + d*x^2])","C",1
837,1,202,288,0.2921157,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}}{(e x)^{5/2}} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^(3/2))/(e*x)^(5/2),x]","\frac{x^{5/2} \left(\frac{2 \left(c+d x^2\right) \left(77 a^2 d \left(d x^2-c\right)+66 a b d x^2 \left(3 c+d x^2\right)+3 b^2 x^2 \left(4 c^2+13 c d x^2+7 d^2 x^4\right)\right)}{d x^{3/2}}+\frac{8 i c x \sqrt{\frac{c}{d x^2}+1} \left(77 a^2 d^2+66 a b c d-3 b^2 c^2\right) F\left(\left.i \sinh ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right)\right|-1\right)}{d \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right)}{231 (e x)^{5/2} \sqrt{c+d x^2}}","-\frac{2 a^2 \left(c+d x^2\right)^{5/2}}{3 c e (e x)^{3/2}}-\frac{2 \sqrt{e x} \left(c+d x^2\right)^{3/2} \left(3 b^2 c^2-11 a d (7 a d+6 b c)\right)}{231 c d e^3}-\frac{4 \sqrt{e x} \sqrt{c+d x^2} \left(3 b^2 c^2-11 a d (7 a d+6 b c)\right)}{231 d e^3}-\frac{4 c^{3/4} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(3 b^2 c^2-11 a d (7 a d+6 b c)\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{231 d^{5/4} e^{5/2} \sqrt{c+d x^2}}+\frac{2 b^2 \sqrt{e x} \left(c+d x^2\right)^{5/2}}{11 d e^3}",1,"(x^(5/2)*((2*(c + d*x^2)*(77*a^2*d*(-c + d*x^2) + 66*a*b*d*x^2*(3*c + d*x^2) + 3*b^2*x^2*(4*c^2 + 13*c*d*x^2 + 7*d^2*x^4)))/(d*x^(3/2)) + ((8*I)*c*(-3*b^2*c^2 + 66*a*b*c*d + 77*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/(Sqrt[(I*Sqrt[c])/Sqrt[d]]*d)))/(231*(e*x)^(5/2)*Sqrt[c + d*x^2])","C",1
838,1,141,468,0.1755544,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}}{(e x)^{7/2}} \, dx","Integrate[((a + b*x^2)^2*(c + d*x^2)^(3/2))/(e*x)^(7/2),x]","\frac{x \left(24 x^4 \sqrt{\frac{c}{d x^2}+1} \left(9 a^2 d^2+18 a b c d+b^2 c^2\right) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{c}{d x^2}\right)-2 \left(c+d x^2\right) \left(9 a^2 \left(c+7 d x^2\right)-18 a b x^2 \left(d x^2-5 c\right)-b^2 x^4 \left(11 c+5 d x^2\right)\right)\right)}{45 (e x)^{7/2} \sqrt{c+d x^2}}","-\frac{2 a^2 \left(c+d x^2\right)^{5/2}}{5 c e (e x)^{5/2}}+\frac{4 \sqrt[4]{c} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(9 a d (a d+2 b c)+b^2 c^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{15 d^{3/4} e^{7/2} \sqrt{c+d x^2}}-\frac{8 \sqrt[4]{c} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(9 a d (a d+2 b c)+b^2 c^2\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{15 d^{3/4} e^{7/2} \sqrt{c+d x^2}}+\frac{2 (e x)^{3/2} \left(c+d x^2\right)^{3/2} \left(9 a d (a d+2 b c)+b^2 c^2\right)}{9 c^2 e^5}+\frac{4 (e x)^{3/2} \sqrt{c+d x^2} \left(9 a d (a d+2 b c)+b^2 c^2\right)}{15 c e^5}+\frac{8 \sqrt{e x} \sqrt{c+d x^2} \left(9 a d (a d+2 b c)+b^2 c^2\right)}{15 \sqrt{d} e^4 \left(\sqrt{c}+\sqrt{d} x\right)}-\frac{2 a \left(c+d x^2\right)^{5/2} (a d+2 b c)}{c^2 e^3 \sqrt{e x}}",1,"(x*(-2*(c + d*x^2)*(-18*a*b*x^2*(-5*c + d*x^2) - b^2*x^4*(11*c + 5*d*x^2) + 9*a^2*(c + 7*d*x^2)) + 24*(b^2*c^2 + 18*a*b*c*d + 9*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x^4*Hypergeometric2F1[-1/4, 1/2, 3/4, -(c/(d*x^2))]))/(45*(e*x)^(7/2)*Sqrt[c + d*x^2])","C",1
839,1,143,430,0.1558522,"\int \frac{(e x)^{5/2} \left(a+b x^2\right)^2}{\sqrt{c+d x^2}} \, dx","Integrate[((e*x)^(5/2)*(a + b*x^2)^2)/Sqrt[c + d*x^2],x]","\frac{2 e (e x)^{3/2} \left(\left(c+d x^2\right) \left(117 a^2 d^2+26 a b d \left(5 d x^2-7 c\right)+b^2 \left(77 c^2-55 c d x^2+45 d^2 x^4\right)\right)-3 c \sqrt{\frac{c}{d x^2}+1} \left(117 a^2 d^2-182 a b c d+77 b^2 c^2\right) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{c}{d x^2}\right)\right)}{585 d^3 \sqrt{c+d x^2}}","-\frac{c^{5/4} e^{5/2} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(117 a^2 d^2+7 b c (11 b c-26 a d)\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{195 d^{15/4} \sqrt{c+d x^2}}+\frac{2 c^{5/4} e^{5/2} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(117 a^2 d^2+7 b c (11 b c-26 a d)\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{195 d^{15/4} \sqrt{c+d x^2}}-\frac{2 c e^2 \sqrt{e x} \sqrt{c+d x^2} \left(117 a^2 d^2+7 b c (11 b c-26 a d)\right)}{195 d^{7/2} \left(\sqrt{c}+\sqrt{d} x\right)}+\frac{2 e (e x)^{3/2} \sqrt{c+d x^2} \left(117 a^2 d^2+7 b c (11 b c-26 a d)\right)}{585 d^3}-\frac{2 b (e x)^{7/2} \sqrt{c+d x^2} (11 b c-26 a d)}{117 d^2 e}+\frac{2 b^2 (e x)^{11/2} \sqrt{c+d x^2}}{13 d e^3}",1,"(2*e*(e*x)^(3/2)*((c + d*x^2)*(117*a^2*d^2 + 26*a*b*d*(-7*c + 5*d*x^2) + b^2*(77*c^2 - 55*c*d*x^2 + 45*d^2*x^4)) - 3*c*(77*b^2*c^2 - 182*a*b*c*d + 117*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*Hypergeometric2F1[-1/4, 1/2, 3/4, -(c/(d*x^2))]))/(585*d^3*Sqrt[c + d*x^2])","C",1
840,1,190,240,0.2363325,"\int \frac{(e x)^{3/2} \left(a+b x^2\right)^2}{\sqrt{c+d x^2}} \, dx","Integrate[((e*x)^(3/2)*(a + b*x^2)^2)/Sqrt[c + d*x^2],x]","\frac{(e x)^{3/2} \left(\frac{2 \sqrt{x} \left(c+d x^2\right) \left(77 a^2 d^2+22 a b d \left(3 d x^2-5 c\right)+3 b^2 \left(15 c^2-9 c d x^2+7 d^2 x^4\right)\right)}{d^3}-\frac{2 i c x \sqrt{\frac{c}{d x^2}+1} \left(77 a^2 d^2-110 a b c d+45 b^2 c^2\right) F\left(\left.i \sinh ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right)\right|-1\right)}{d^3 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right)}{231 x^{3/2} \sqrt{c+d x^2}}","-\frac{c^{3/4} e^{3/2} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(77 a^2 d^2+5 b c (9 b c-22 a d)\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{231 d^{13/4} \sqrt{c+d x^2}}+\frac{2 e \sqrt{e x} \sqrt{c+d x^2} \left(77 a^2 d^2+5 b c (9 b c-22 a d)\right)}{231 d^3}-\frac{2 b (e x)^{5/2} \sqrt{c+d x^2} (9 b c-22 a d)}{77 d^2 e}+\frac{2 b^2 (e x)^{9/2} \sqrt{c+d x^2}}{11 d e^3}",1,"((e*x)^(3/2)*((2*Sqrt[x]*(c + d*x^2)*(77*a^2*d^2 + 22*a*b*d*(-5*c + 3*d*x^2) + 3*b^2*(15*c^2 - 9*c*d*x^2 + 7*d^2*x^4)))/d^3 - ((2*I)*c*(45*b^2*c^2 - 110*a*b*c*d + 77*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/(Sqrt[(I*Sqrt[c])/Sqrt[d]]*d^3)))/(231*x^(3/2)*Sqrt[c + d*x^2])","C",1
841,1,111,375,0.1152218,"\int \frac{\sqrt{e x} \left(a+b x^2\right)^2}{\sqrt{c+d x^2}} \, dx","Integrate[(Sqrt[e*x]*(a + b*x^2)^2)/Sqrt[c + d*x^2],x]","\frac{2 \sqrt{e x} \left(3 x \sqrt{\frac{c}{d x^2}+1} \left(15 a^2 d^2-18 a b c d+7 b^2 c^2\right) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{c}{d x^2}\right)+b x \left(c+d x^2\right) \left(18 a d-7 b c+5 b d x^2\right)\right)}{45 d^2 \sqrt{c+d x^2}}","\frac{2 \sqrt{e x} \sqrt{c+d x^2} \left(15 a^2 d^2+b c (7 b c-18 a d)\right)}{15 d^{5/2} \left(\sqrt{c}+\sqrt{d} x\right)}+\frac{\sqrt[4]{c} \sqrt{e} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(15 a^2 d^2+b c (7 b c-18 a d)\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{15 d^{11/4} \sqrt{c+d x^2}}-\frac{2 \sqrt[4]{c} \sqrt{e} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(15 a^2 d^2+b c (7 b c-18 a d)\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{15 d^{11/4} \sqrt{c+d x^2}}-\frac{2 b (e x)^{3/2} \sqrt{c+d x^2} (7 b c-18 a d)}{45 d^2 e}+\frac{2 b^2 (e x)^{7/2} \sqrt{c+d x^2}}{9 d e^3}",1,"(2*Sqrt[e*x]*(b*x*(c + d*x^2)*(-7*b*c + 18*a*d + 5*b*d*x^2) + 3*(7*b^2*c^2 - 18*a*b*c*d + 15*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*Hypergeometric2F1[-1/4, 1/2, 3/4, -(c/(d*x^2))]))/(45*d^2*Sqrt[c + d*x^2])","C",1
842,1,148,193,0.2510411,"\int \frac{\left(a+b x^2\right)^2}{\sqrt{e x} \sqrt{c+d x^2}} \, dx","Integrate[(a + b*x^2)^2/(Sqrt[e*x]*Sqrt[c + d*x^2]),x]","\frac{2 x \left(-b \left(c+d x^2\right) \left(-14 a d+5 b c-3 b d x^2\right)+\frac{i \sqrt{x} \sqrt{\frac{c}{d x^2}+1} \left(21 a^2 d^2-14 a b c d+5 b^2 c^2\right) F\left(\left.i \sinh ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right)\right|-1\right)}{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right)}{21 d^2 \sqrt{e x} \sqrt{c+d x^2}}","\frac{\left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(21 a^2 d^2-14 a b c d+5 b^2 c^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{21 \sqrt[4]{c} d^{9/4} \sqrt{e} \sqrt{c+d x^2}}-\frac{2 b \sqrt{e x} \sqrt{c+d x^2} (5 b c-14 a d)}{21 d^2 e}+\frac{2 b^2 (e x)^{5/2} \sqrt{c+d x^2}}{7 d e^3}",1,"(2*x*(-(b*(c + d*x^2)*(5*b*c - 14*a*d - 3*b*d*x^2)) + (I*(5*b^2*c^2 - 14*a*b*c*d + 21*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*Sqrt[x]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/Sqrt[(I*Sqrt[c])/Sqrt[d]]))/(21*d^2*Sqrt[e*x]*Sqrt[c + d*x^2])","C",1
843,1,115,372,0.1165701,"\int \frac{\left(a+b x^2\right)^2}{(e x)^{3/2} \sqrt{c+d x^2}} \, dx","Integrate[(a + b*x^2)^2/((e*x)^(3/2)*Sqrt[c + d*x^2]),x]","\frac{x \left(2 x^2 \sqrt{\frac{c}{d x^2}+1} \left(5 a^2 d^2+10 a b c d-3 b^2 c^2\right) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{c}{d x^2}\right)+2 \left(c+d x^2\right) \left(b^2 c x^2-5 a^2 d\right)\right)}{5 c d (e x)^{3/2} \sqrt{c+d x^2}}","-\frac{2 a^2 \sqrt{c+d x^2}}{c e \sqrt{e x}}-\frac{2 \sqrt{e x} \sqrt{c+d x^2} \left(3 b^2 c^2-5 a d (a d+2 b c)\right)}{5 c d^{3/2} e^2 \left(\sqrt{c}+\sqrt{d} x\right)}-\frac{\left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(3 b^2 c^2-5 a d (a d+2 b c)\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{5 c^{3/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}+\frac{2 \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(3 b^2 c^2-5 a d (a d+2 b c)\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{5 c^{3/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}+\frac{2 b^2 (e x)^{3/2} \sqrt{c+d x^2}}{5 d e^3}",1,"(x*(2*(-5*a^2*d + b^2*c*x^2)*(c + d*x^2) + 2*(-3*b^2*c^2 + 10*a*b*c*d + 5*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x^2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(c/(d*x^2))]))/(5*c*d*(e*x)^(3/2)*Sqrt[c + d*x^2])","C",1
844,1,165,184,0.1870334,"\int \frac{\left(a+b x^2\right)^2}{(e x)^{5/2} \sqrt{c+d x^2}} \, dx","Integrate[(a + b*x^2)^2/((e*x)^(5/2)*Sqrt[c + d*x^2]),x]","\frac{x \left(2 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \left(c+d x^2\right) \left(b^2 c x^2-a^2 d\right)-2 i x^{5/2} \sqrt{\frac{c}{d x^2}+1} \left(a^2 d^2-6 a b c d+b^2 c^2\right) F\left(\left.i \sinh ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right)\right|-1\right)\right)}{3 c d \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} (e x)^{5/2} \sqrt{c+d x^2}}","-\frac{\left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(a^2 d^2-6 a b c d+b^2 c^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{3 c^{5/4} d^{5/4} e^{5/2} \sqrt{c+d x^2}}-\frac{2 a^2 \sqrt{c+d x^2}}{3 c e (e x)^{3/2}}+\frac{2 b^2 \sqrt{e x} \sqrt{c+d x^2}}{3 d e^3}",1,"(x*(2*Sqrt[(I*Sqrt[c])/Sqrt[d]]*(-(a^2*d) + b^2*c*x^2)*(c + d*x^2) - (2*I)*(b^2*c^2 - 6*a*b*c*d + a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x^(5/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1]))/(3*c*Sqrt[(I*Sqrt[c])/Sqrt[d]]*d*(e*x)^(5/2)*Sqrt[c + d*x^2])","C",1
845,1,116,387,0.1265332,"\int \frac{\left(a+b x^2\right)^2}{(e x)^{7/2} \sqrt{c+d x^2}} \, dx","Integrate[(a + b*x^2)^2/((e*x)^(7/2)*Sqrt[c + d*x^2]),x]","\frac{x \left(2 x^4 \sqrt{\frac{c}{d x^2}+1} \left(-3 a^2 d^2+10 a b c d+5 b^2 c^2\right) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{c}{d x^2}\right)-2 a \left(c+d x^2\right) \left(a \left(c-3 d x^2\right)+10 b c x^2\right)\right)}{5 c^2 (e x)^{7/2} \sqrt{c+d x^2}}","\frac{2 \sqrt{e x} \sqrt{c+d x^2} \left(-3 a^2 d^2+10 a b c d+5 b^2 c^2\right)}{5 c^2 \sqrt{d} e^4 \left(\sqrt{c}+\sqrt{d} x\right)}+\frac{\left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(-3 a^2 d^2+10 a b c d+5 b^2 c^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{5 c^{7/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}-\frac{2 \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(-3 a^2 d^2+10 a b c d+5 b^2 c^2\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{5 c^{7/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}-\frac{2 a^2 \sqrt{c+d x^2}}{5 c e (e x)^{5/2}}-\frac{2 a \sqrt{c+d x^2} (10 b c-3 a d)}{5 c^2 e^3 \sqrt{e x}}",1,"(x*(-2*a*(c + d*x^2)*(10*b*c*x^2 + a*(c - 3*d*x^2)) + 2*(5*b^2*c^2 + 10*a*b*c*d - 3*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x^4*Hypergeometric2F1[-1/4, 1/2, 3/4, -(c/(d*x^2))]))/(5*c^2*(e*x)^(7/2)*Sqrt[c + d*x^2])","C",1
846,1,159,193,0.217717,"\int \frac{\left(a+b x^2\right)^2}{(e x)^{9/2} \sqrt{c+d x^2}} \, dx","Integrate[(a + b*x^2)^2/((e*x)^(9/2)*Sqrt[c + d*x^2]),x]","\frac{x^{9/2} \left(\frac{2 a \left(c+d x^2\right) \left(-3 a c+5 a d x^2-14 b c x^2\right)}{c^2 x^{7/2}}+\frac{2 i x \sqrt{\frac{c}{d x^2}+1} \left(5 a^2 d^2-14 a b c d+21 b^2 c^2\right) F\left(\left.i \sinh ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right)\right|-1\right)}{c^2 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right)}{21 (e x)^{9/2} \sqrt{c+d x^2}}","\frac{\left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(5 a^2 d^2-14 a b c d+21 b^2 c^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{21 c^{9/4} \sqrt[4]{d} e^{9/2} \sqrt{c+d x^2}}-\frac{2 a^2 \sqrt{c+d x^2}}{7 c e (e x)^{7/2}}-\frac{2 a \sqrt{c+d x^2} (14 b c-5 a d)}{21 c^2 e^3 (e x)^{3/2}}",1,"(x^(9/2)*((2*a*(c + d*x^2)*(-3*a*c - 14*b*c*x^2 + 5*a*d*x^2))/(c^2*x^(7/2)) + ((2*I)*(21*b^2*c^2 - 14*a*b*c*d + 5*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/(c^2*Sqrt[(I*Sqrt[c])/Sqrt[d]])))/(21*(e*x)^(9/2)*Sqrt[c + d*x^2])","C",1
847,1,155,438,0.2014067,"\int \frac{\left(a+b x^2\right)^2}{(e x)^{11/2} \sqrt{c+d x^2}} \, dx","Integrate[(a + b*x^2)^2/((e*x)^(11/2)*Sqrt[c + d*x^2]),x]","\frac{2 \sqrt{e x} \left(d x^6 \sqrt{\frac{d x^2}{c}+1} \left(7 a^2 d^2-18 a b c d+15 b^2 c^2\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{d x^2}{c}\right)-\left(c+d x^2\right) \left(a^2 \left(5 c^2-7 c d x^2+21 d^2 x^4\right)+18 a b c x^2 \left(c-3 d x^2\right)+45 b^2 c^2 x^4\right)\right)}{45 c^3 e^6 x^5 \sqrt{c+d x^2}}","\frac{\sqrt[4]{d} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(7 a^2 d^2-18 a b c d+15 b^2 c^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{15 c^{11/4} e^{11/2} \sqrt{c+d x^2}}-\frac{2 \sqrt[4]{d} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(7 a^2 d^2-18 a b c d+15 b^2 c^2\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{15 c^{11/4} e^{11/2} \sqrt{c+d x^2}}+\frac{2 \sqrt{d} \sqrt{e x} \sqrt{c+d x^2} \left(7 a^2 d^2-18 a b c d+15 b^2 c^2\right)}{15 c^3 e^6 \left(\sqrt{c}+\sqrt{d} x\right)}-\frac{2 \sqrt{c+d x^2} \left(7 a^2 d^2-18 a b c d+15 b^2 c^2\right)}{15 c^3 e^5 \sqrt{e x}}-\frac{2 a^2 \sqrt{c+d x^2}}{9 c e (e x)^{9/2}}-\frac{2 a \sqrt{c+d x^2} (18 b c-7 a d)}{45 c^2 e^3 (e x)^{5/2}}",1,"(2*Sqrt[e*x]*(-((c + d*x^2)*(45*b^2*c^2*x^4 + 18*a*b*c*x^2*(c - 3*d*x^2) + a^2*(5*c^2 - 7*c*d*x^2 + 21*d^2*x^4))) + d*(15*b^2*c^2 - 18*a*b*c*d + 7*a^2*d^2)*x^6*Sqrt[1 + (d*x^2)/c]*Hypergeometric2F1[1/2, 3/4, 7/4, -((d*x^2)/c)]))/(45*c^3*e^6*x^5*Sqrt[c + d*x^2])","C",1
848,1,196,242,0.2648731,"\int \frac{\left(a+b x^2\right)^2}{(e x)^{13/2} \sqrt{c+d x^2}} \, dx","Integrate[(a + b*x^2)^2/((e*x)^(13/2)*Sqrt[c + d*x^2]),x]","\frac{x^{13/2} \left(-\frac{2 \left(c+d x^2\right) \left(3 a^2 \left(7 c^2-9 c d x^2+15 d^2 x^4\right)+22 a b c x^2 \left(3 c-5 d x^2\right)+77 b^2 c^2 x^4\right)}{c^3 x^{11/2}}-\frac{2 i d x \sqrt{\frac{c}{d x^2}+1} \left(45 a^2 d^2-110 a b c d+77 b^2 c^2\right) F\left(\left.i \sinh ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right)\right|-1\right)}{c^3 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right)}{231 (e x)^{13/2} \sqrt{c+d x^2}}","-\frac{2 a^2 \sqrt{c+d x^2}}{11 c e (e x)^{11/2}}-\frac{d^{3/4} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(77 b^2 c^2-5 a d (22 b c-9 a d)\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{231 c^{13/4} e^{13/2} \sqrt{c+d x^2}}-\frac{2 \sqrt{c+d x^2} \left(77 b^2 c^2-5 a d (22 b c-9 a d)\right)}{231 c^3 e^5 (e x)^{3/2}}-\frac{2 a \sqrt{c+d x^2} (22 b c-9 a d)}{77 c^2 e^3 (e x)^{7/2}}",1,"(x^(13/2)*((-2*(c + d*x^2)*(77*b^2*c^2*x^4 + 22*a*b*c*x^2*(3*c - 5*d*x^2) + 3*a^2*(7*c^2 - 9*c*d*x^2 + 15*d^2*x^4)))/(c^3*x^(11/2)) - ((2*I)*d*(77*b^2*c^2 - 110*a*b*c*d + 45*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/(c^3*Sqrt[(I*Sqrt[c])/Sqrt[d]])))/(231*(e*x)^(13/2)*Sqrt[c + d*x^2])","C",1
849,1,226,296,0.2733938,"\int \frac{(e x)^{7/2} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{3/2}} \, dx","Integrate[((e*x)^(7/2)*(a + b*x^2)^2)/(c + d*x^2)^(3/2),x]","\frac{e^3 \sqrt{e x} \left(\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \left(77 a^2 d^2 \left(5 c+2 d x^2\right)+66 a b d \left(-15 c^2-6 c d x^2+2 d^2 x^4\right)+3 b^2 \left(195 c^3+78 c^2 d x^2-26 c d^2 x^4+14 d^3 x^6\right)\right)-5 i c \sqrt{x} \sqrt{\frac{c}{d x^2}+1} \left(77 a^2 d^2-198 a b c d+117 b^2 c^2\right) F\left(\left.i \sinh ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right)\right|-1\right)\right)}{231 d^4 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \sqrt{c+d x^2}}","\frac{5 e^3 \sqrt{e x} \sqrt{c+d x^2} \left(77 a^2 d^2-198 a b c d+117 b^2 c^2\right)}{231 d^4}-\frac{e (e x)^{5/2} \sqrt{c+d x^2} \left(77 a^2 d^2-198 a b c d+117 b^2 c^2\right)}{77 c d^3}-\frac{5 c^{3/4} e^{7/2} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(77 a^2 d^2-198 a b c d+117 b^2 c^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{462 d^{17/4} \sqrt{c+d x^2}}+\frac{(e x)^{9/2} (b c-a d)^2}{c d^2 e \sqrt{c+d x^2}}+\frac{2 b^2 (e x)^{9/2} \sqrt{c+d x^2}}{11 d^2 e}",1,"(e^3*Sqrt[e*x]*(Sqrt[(I*Sqrt[c])/Sqrt[d]]*(77*a^2*d^2*(5*c + 2*d*x^2) + 66*a*b*d*(-15*c^2 - 6*c*d*x^2 + 2*d^2*x^4) + 3*b^2*(195*c^3 + 78*c^2*d*x^2 - 26*c*d^2*x^4 + 14*d^3*x^6)) - (5*I)*c*(117*b^2*c^2 - 198*a*b*c*d + 77*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*Sqrt[x]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1]))/(231*Sqrt[(I*Sqrt[c])/Sqrt[d]]*d^4*Sqrt[c + d*x^2])","C",1
850,1,133,436,0.1435684,"\int \frac{(e x)^{5/2} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{3/2}} \, dx","Integrate[((e*x)^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^(3/2),x]","\frac{e (e x)^{3/2} \left(3 \sqrt{\frac{c}{d x^2}+1} \left(45 a^2 d^2-126 a b c d+77 b^2 c^2\right) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{c}{d x^2}\right)-45 a^2 d^2+18 a b d \left(7 c+2 d x^2\right)+b^2 \left(-77 c^2-22 c d x^2+10 d^2 x^4\right)\right)}{45 d^3 \sqrt{c+d x^2}}","\frac{\sqrt[4]{c} e^{5/2} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(45 a^2 d^2-126 a b c d+77 b^2 c^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{30 d^{15/4} \sqrt{c+d x^2}}-\frac{\sqrt[4]{c} e^{5/2} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(45 a^2 d^2-126 a b c d+77 b^2 c^2\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{15 d^{15/4} \sqrt{c+d x^2}}+\frac{e^2 \sqrt{e x} \sqrt{c+d x^2} \left(45 a^2 d^2-126 a b c d+77 b^2 c^2\right)}{15 d^{7/2} \left(\sqrt{c}+\sqrt{d} x\right)}-\frac{e (e x)^{3/2} \sqrt{c+d x^2} \left(45 a^2 d^2-126 a b c d+77 b^2 c^2\right)}{45 c d^3}+\frac{(e x)^{7/2} (b c-a d)^2}{c d^2 e \sqrt{c+d x^2}}+\frac{2 b^2 (e x)^{7/2} \sqrt{c+d x^2}}{9 d^2 e}",1,"(e*(e*x)^(3/2)*(-45*a^2*d^2 + 18*a*b*d*(7*c + 2*d*x^2) + b^2*(-77*c^2 - 22*c*d*x^2 + 10*d^2*x^4) + 3*(77*b^2*c^2 - 126*a*b*c*d + 45*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*Hypergeometric2F1[-1/4, 1/2, 3/4, -(c/(d*x^2))]))/(45*d^3*Sqrt[c + d*x^2])","C",1
851,1,191,245,0.2147705,"\int \frac{(e x)^{3/2} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{3/2}} \, dx","Integrate[((e*x)^(3/2)*(a + b*x^2)^2)/(c + d*x^2)^(3/2),x]","\frac{e \sqrt{e x} \left(i \sqrt{x} \sqrt{\frac{c}{d x^2}+1} \left(21 a^2 d^2-70 a b c d+45 b^2 c^2\right) F\left(\left.i \sinh ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right)\right|-1\right)+\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \left(-21 a^2 d^2+14 a b d \left(5 c+2 d x^2\right)-3 b^2 \left(15 c^2+6 c d x^2-2 d^2 x^4\right)\right)\right)}{21 d^3 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \sqrt{c+d x^2}}","\frac{e^{3/2} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(21 a^2 d^2-70 a b c d+45 b^2 c^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{42 \sqrt[4]{c} d^{13/4} \sqrt{c+d x^2}}-\frac{e \sqrt{e x} \sqrt{c+d x^2} \left(21 a^2 d^2-70 a b c d+45 b^2 c^2\right)}{21 c d^3}+\frac{(e x)^{5/2} (b c-a d)^2}{c d^2 e \sqrt{c+d x^2}}+\frac{2 b^2 (e x)^{5/2} \sqrt{c+d x^2}}{7 d^2 e}",1,"(e*Sqrt[e*x]*(Sqrt[(I*Sqrt[c])/Sqrt[d]]*(-21*a^2*d^2 + 14*a*b*d*(5*c + 2*d*x^2) - 3*b^2*(15*c^2 + 6*c*d*x^2 - 2*d^2*x^4)) + I*(45*b^2*c^2 - 70*a*b*c*d + 21*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*Sqrt[x]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1]))/(21*Sqrt[(I*Sqrt[c])/Sqrt[d]]*d^3*Sqrt[c + d*x^2])","C",1
852,1,119,384,0.127011,"\int \frac{\sqrt{e x} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{3/2}} \, dx","Integrate[(Sqrt[e*x]*(a + b*x^2)^2)/(c + d*x^2)^(3/2),x]","\frac{\sqrt{e x} \left(x \sqrt{\frac{c}{d x^2}+1} \left(-5 a^2 d^2+30 a b c d-21 b^2 c^2\right) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{c}{d x^2}\right)+x \left(5 a^2 d^2-10 a b c d+b^2 c \left(7 c+2 d x^2\right)\right)\right)}{5 c d^2 \sqrt{c+d x^2}}","-\frac{\sqrt{e x} \sqrt{c+d x^2} \left(5 a^2 d^2-30 a b c d+21 b^2 c^2\right)}{5 c d^{5/2} \left(\sqrt{c}+\sqrt{d} x\right)}-\frac{\sqrt{e} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(5 a^2 d^2-30 a b c d+21 b^2 c^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{10 c^{3/4} d^{11/4} \sqrt{c+d x^2}}+\frac{\sqrt{e} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(5 a^2 d^2-30 a b c d+21 b^2 c^2\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{5 c^{3/4} d^{11/4} \sqrt{c+d x^2}}+\frac{(e x)^{3/2} (b c-a d)^2}{c d^2 e \sqrt{c+d x^2}}+\frac{2 b^2 (e x)^{3/2} \sqrt{c+d x^2}}{5 d^2 e}",1,"(Sqrt[e*x]*(x*(-10*a*b*c*d + 5*a^2*d^2 + b^2*c*(7*c + 2*d*x^2)) + (-21*b^2*c^2 + 30*a*b*c*d - 5*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*Hypergeometric2F1[-1/4, 1/2, 3/4, -(c/(d*x^2))]))/(5*c*d^2*Sqrt[c + d*x^2])","C",1
853,1,174,193,0.1680762,"\int \frac{\left(a+b x^2\right)^2}{\sqrt{e x} \left(c+d x^2\right)^{3/2}} \, dx","Integrate[(a + b*x^2)^2/(Sqrt[e*x]*(c + d*x^2)^(3/2)),x]","\frac{i x^{3/2} \sqrt{\frac{c}{d x^2}+1} \left(3 a^2 d^2+6 a b c d-5 b^2 c^2\right) F\left(\left.i \sinh ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right)\right|-1\right)+x \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \left(3 a^2 d^2-6 a b c d+b^2 c \left(5 c+2 d x^2\right)\right)}{3 c d^2 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \sqrt{e x} \sqrt{c+d x^2}}","-\frac{\left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(-3 a^2 d^2-6 a b c d+5 b^2 c^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{6 c^{5/4} d^{9/4} \sqrt{e} \sqrt{c+d x^2}}+\frac{\sqrt{e x} (b c-a d)^2}{c d^2 e \sqrt{c+d x^2}}+\frac{2 b^2 \sqrt{e x} \sqrt{c+d x^2}}{3 d^2 e}",1,"(Sqrt[(I*Sqrt[c])/Sqrt[d]]*x*(-6*a*b*c*d + 3*a^2*d^2 + b^2*c*(5*c + 2*d*x^2)) + I*(-5*b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/(3*c*Sqrt[(I*Sqrt[c])/Sqrt[d]]*d^2*Sqrt[e*x]*Sqrt[c + d*x^2])","C",1
854,1,126,393,0.1151494,"\int \frac{\left(a+b x^2\right)^2}{(e x)^{3/2} \left(c+d x^2\right)^{3/2}} \, dx","Integrate[(a + b*x^2)^2/((e*x)^(3/2)*(c + d*x^2)^(3/2)),x]","\frac{x \left(x^2 \sqrt{\frac{d x^2}{c}+1} \left(3 a^2 d^2-2 a b c d+3 b^2 c^2\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{d x^2}{c}\right)-3 a^2 d \left(2 c+3 d x^2\right)+6 a b c d x^2-3 b^2 c^2 x^2\right)}{3 c^2 d (e x)^{3/2} \sqrt{c+d x^2}}","-\frac{(e x)^{3/2} \left(3 a^2 d^2-2 a b c d+b^2 c^2\right)}{c^2 d e^3 \sqrt{c+d x^2}}+\frac{\sqrt{e x} \sqrt{c+d x^2} \left(3 a^2 d^2-2 a b c d+3 b^2 c^2\right)}{c^2 d^{3/2} e^2 \left(\sqrt{c}+\sqrt{d} x\right)}+\frac{\left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(3 a^2 d^2-2 a b c d+3 b^2 c^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{2 c^{7/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}-\frac{\left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(3 a^2 d^2-2 a b c d+3 b^2 c^2\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{c^{7/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}-\frac{2 a^2}{c e \sqrt{e x} \sqrt{c+d x^2}}",1,"(x*(-3*b^2*c^2*x^2 + 6*a*b*c*d*x^2 - 3*a^2*d*(2*c + 3*d*x^2) + (3*b^2*c^2 - 2*a*b*c*d + 3*a^2*d^2)*x^2*Sqrt[1 + (d*x^2)/c]*Hypergeometric2F1[1/2, 3/4, 7/4, -((d*x^2)/c)]))/(3*c^2*d*(e*x)^(3/2)*Sqrt[c + d*x^2])","C",1
855,1,181,207,0.1985393,"\int \frac{\left(a+b x^2\right)^2}{(e x)^{5/2} \left(c+d x^2\right)^{3/2}} \, dx","Integrate[(a + b*x^2)^2/((e*x)^(5/2)*(c + d*x^2)^(3/2)),x]","\frac{x \left(-i x^{5/2} \sqrt{\frac{c}{d x^2}+1} \left(5 a^2 d^2-6 a b c d-3 b^2 c^2\right) F\left(\left.i \sinh ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right)\right|-1\right)-\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \left(a^2 d \left(2 c+5 d x^2\right)-6 a b c d x^2+3 b^2 c^2 x^2\right)\right)}{3 c^2 d \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} (e x)^{5/2} \sqrt{c+d x^2}}","-\frac{\sqrt{e x} \left(5 a^2 d^2-6 a b c d+3 b^2 c^2\right)}{3 c^2 d e^3 \sqrt{c+d x^2}}-\frac{2 a^2}{3 c e (e x)^{3/2} \sqrt{c+d x^2}}+\frac{\left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(a d (6 b c-5 a d)+3 b^2 c^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{6 c^{9/4} d^{5/4} e^{5/2} \sqrt{c+d x^2}}",1,"(x*(-(Sqrt[(I*Sqrt[c])/Sqrt[d]]*(3*b^2*c^2*x^2 - 6*a*b*c*d*x^2 + a^2*d*(2*c + 5*d*x^2))) - I*(-3*b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x^(5/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1]))/(3*c^2*Sqrt[(I*Sqrt[c])/Sqrt[d]]*d*(e*x)^(5/2)*Sqrt[c + d*x^2])","C",1
856,1,141,434,0.1410915,"\int \frac{\left(a+b x^2\right)^2}{(e x)^{7/2} \left(c+d x^2\right)^{3/2}} \, dx","Integrate[(a + b*x^2)^2/((e*x)^(7/2)*(c + d*x^2)^(3/2)),x]","\frac{x \left(x^4 \sqrt{\frac{d x^2}{c}+1} \left(-21 a^2 d^2+30 a b c d-5 b^2 c^2\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{d x^2}{c}\right)+a^2 \left(-6 c^2+42 c d x^2+63 d^2 x^4\right)-30 a b c x^2 \left(2 c+3 d x^2\right)+15 b^2 c^2 x^4\right)}{15 c^3 (e x)^{7/2} \sqrt{c+d x^2}}","-\frac{2 a^2}{5 c e (e x)^{5/2} \sqrt{c+d x^2}}-\frac{\left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(5 b^2 c^2-3 a d (10 b c-7 a d)\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{10 c^{11/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}+\frac{\left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(5 b^2 c^2-3 a d (10 b c-7 a d)\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{5 c^{11/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}+\frac{(e x)^{3/2} \left(5 b^2 c^2-3 a d (10 b c-7 a d)\right)}{5 c^3 e^5 \sqrt{c+d x^2}}-\frac{\sqrt{e x} \sqrt{c+d x^2} \left(5 b^2 c^2-3 a d (10 b c-7 a d)\right)}{5 c^3 \sqrt{d} e^4 \left(\sqrt{c}+\sqrt{d} x\right)}-\frac{2 a (10 b c-7 a d)}{5 c^2 e^3 \sqrt{e x} \sqrt{c+d x^2}}",1,"(x*(15*b^2*c^2*x^4 - 30*a*b*c*x^2*(2*c + 3*d*x^2) + a^2*(-6*c^2 + 42*c*d*x^2 + 63*d^2*x^4) + (-5*b^2*c^2 + 30*a*b*c*d - 21*a^2*d^2)*x^4*Sqrt[1 + (d*x^2)/c]*Hypergeometric2F1[1/2, 3/4, 7/4, -((d*x^2)/c)]))/(15*c^3*(e*x)^(7/2)*Sqrt[c + d*x^2])","C",1
857,1,222,302,0.3204696,"\int \frac{(e x)^{7/2} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{5/2}} \, dx","Integrate[((e*x)^(7/2)*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x]","\frac{(e x)^{7/2} \left(\frac{\sqrt{x} \left(-7 a^2 d^2 \left(5 c+7 d x^2\right)+14 a b d \left(15 c^2+21 c d x^2+4 d^2 x^4\right)-\left(b^2 \left(195 c^3+273 c^2 d x^2+52 c d^2 x^4-12 d^3 x^6\right)\right)\right)}{d^4 \left(c+d x^2\right)}+\frac{5 i x \sqrt{\frac{c}{d x^2}+1} \left(7 a^2 d^2-42 a b c d+39 b^2 c^2\right) F\left(\left.i \sinh ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right)\right|-1\right)}{d^4 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right)}{42 x^{7/2} \sqrt{c+d x^2}}","\frac{5 e^{7/2} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(7 a^2 d^2-42 a b c d+39 b^2 c^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{84 \sqrt[4]{c} d^{17/4} \sqrt{c+d x^2}}-\frac{5 e^3 \sqrt{e x} \sqrt{c+d x^2} \left(7 a^2 d^2-42 a b c d+39 b^2 c^2\right)}{42 c d^4}+\frac{e (e x)^{5/2} \left(7 a^2 d^2-42 a b c d+39 b^2 c^2\right)}{14 c d^3 \sqrt{c+d x^2}}+\frac{(e x)^{9/2} (b c-a d)^2}{3 c d^2 e \left(c+d x^2\right)^{3/2}}+\frac{2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt{c+d x^2}}",1,"((e*x)^(7/2)*((Sqrt[x]*(-7*a^2*d^2*(5*c + 7*d*x^2) + 14*a*b*d*(15*c^2 + 21*c*d*x^2 + 4*d^2*x^4) - b^2*(195*c^3 + 273*c^2*d*x^2 + 52*c*d^2*x^4 - 12*d^3*x^6)))/(d^4*(c + d*x^2)) + ((5*I)*(39*b^2*c^2 - 42*a*b*c*d + 7*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/(Sqrt[(I*Sqrt[c])/Sqrt[d]]*d^4)))/(42*x^(7/2)*Sqrt[c + d*x^2])","C",1
858,1,153,442,0.2014448,"\int \frac{(e x)^{5/2} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{5/2}} \, dx","Integrate[((e*x)^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x]","\frac{e (e x)^{3/2} \left(-3 \sqrt{\frac{c}{d x^2}+1} \left(c+d x^2\right) \left(5 a^2 d^2-70 a b c d+77 b^2 c^2\right) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{c}{d x^2}\right)+5 a^2 d^2 \left(c+3 d x^2\right)-10 a b c d \left(7 c+9 d x^2\right)+b^2 c \left(77 c^2+99 c d x^2+12 d^2 x^4\right)\right)}{30 c d^3 \left(c+d x^2\right)^{3/2}}","-\frac{e^2 \sqrt{e x} \sqrt{c+d x^2} \left(5 a^2 d^2-70 a b c d+77 b^2 c^2\right)}{10 c d^{7/2} \left(\sqrt{c}+\sqrt{d} x\right)}+\frac{e (e x)^{3/2} \left(5 a^2 d^2-70 a b c d+77 b^2 c^2\right)}{30 c d^3 \sqrt{c+d x^2}}-\frac{e^{5/2} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(5 a^2 d^2-70 a b c d+77 b^2 c^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{20 c^{3/4} d^{15/4} \sqrt{c+d x^2}}+\frac{e^{5/2} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(5 a^2 d^2-70 a b c d+77 b^2 c^2\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{10 c^{3/4} d^{15/4} \sqrt{c+d x^2}}+\frac{(e x)^{7/2} (b c-a d)^2}{3 c d^2 e \left(c+d x^2\right)^{3/2}}+\frac{2 b^2 (e x)^{7/2}}{5 d^2 e \sqrt{c+d x^2}}",1,"(e*(e*x)^(3/2)*(5*a^2*d^2*(c + 3*d*x^2) - 10*a*b*c*d*(7*c + 9*d*x^2) + b^2*c*(77*c^2 + 99*c*d*x^2 + 12*d^2*x^4) - 3*(77*b^2*c^2 - 70*a*b*c*d + 5*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*(c + d*x^2)*Hypergeometric2F1[-1/4, 1/2, 3/4, -(c/(d*x^2))]))/(30*c*d^3*(c + d*x^2)^(3/2))","C",1
859,1,204,248,0.2897407,"\int \frac{(e x)^{3/2} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{5/2}} \, dx","Integrate[((e*x)^(3/2)*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x]","\frac{(e x)^{3/2} \left(\frac{\sqrt{x} \left(a^2 d^2 \left(d x^2-c\right)-2 a b c d \left(5 c+7 d x^2\right)+b^2 c \left(15 c^2+21 c d x^2+4 d^2 x^4\right)\right)}{c d^3 \left(c+d x^2\right)}+\frac{i x \sqrt{\frac{c}{d x^2}+1} \left(a^2 d^2+10 a b c d-15 b^2 c^2\right) F\left(\left.i \sinh ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right)\right|-1\right)}{c d^3 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right)}{6 x^{3/2} \sqrt{c+d x^2}}","\frac{e \sqrt{e x} \left(-a^2 d^2-10 a b c d+15 b^2 c^2\right)}{6 c d^3 \sqrt{c+d x^2}}-\frac{e^{3/2} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(-a^2 d^2-10 a b c d+15 b^2 c^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{12 c^{5/4} d^{13/4} \sqrt{c+d x^2}}+\frac{(e x)^{5/2} (b c-a d)^2}{3 c d^2 e \left(c+d x^2\right)^{3/2}}+\frac{2 b^2 (e x)^{5/2}}{3 d^2 e \sqrt{c+d x^2}}",1,"((e*x)^(3/2)*((Sqrt[x]*(a^2*d^2*(-c + d*x^2) - 2*a*b*c*d*(5*c + 7*d*x^2) + b^2*c*(15*c^2 + 21*c*d*x^2 + 4*d^2*x^4)))/(c*d^3*(c + d*x^2)) + (I*(-15*b^2*c^2 + 10*a*b*c*d + a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/(c*Sqrt[(I*Sqrt[c])/Sqrt[d]]*d^3)))/(6*x^(3/2)*Sqrt[c + d*x^2])","C",1
860,1,147,403,0.188064,"\int \frac{\sqrt{e x} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{5/2}} \, dx","Integrate[(Sqrt[e*x]*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x]","\frac{\sqrt{e x} \left(3 x \sqrt{\frac{c}{d x^2}+1} \left(c+d x^2\right) \left(-a^2 d^2-2 a b c d+7 b^2 c^2\right) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{c}{d x^2}\right)+x \left(a^2 d^2 \left(5 c+3 d x^2\right)+2 a b c d \left(c+3 d x^2\right)-b^2 c^2 \left(7 c+9 d x^2\right)\right)\right)}{6 c^2 d^2 \left(c+d x^2\right)^{3/2}}","\frac{\sqrt{e x} \sqrt{c+d x^2} \left(-a^2 d^2-2 a b c d+7 b^2 c^2\right)}{2 c^2 d^{5/2} \left(\sqrt{c}+\sqrt{d} x\right)}+\frac{\sqrt{e} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(-a^2 d^2-2 a b c d+7 b^2 c^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{4 c^{7/4} d^{11/4} \sqrt{c+d x^2}}-\frac{\sqrt{e} \left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(-a^2 d^2-2 a b c d+7 b^2 c^2\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{2 c^{7/4} d^{11/4} \sqrt{c+d x^2}}-\frac{(e x)^{3/2} (a d+3 b c) (b c-a d)}{2 c^2 d^2 e \sqrt{c+d x^2}}+\frac{(e x)^{3/2} (b c-a d)^2}{3 c d^2 e \left(c+d x^2\right)^{3/2}}",1,"(Sqrt[e*x]*(x*(2*a*b*c*d*(c + 3*d*x^2) + a^2*d^2*(5*c + 3*d*x^2) - b^2*c^2*(7*c + 9*d*x^2)) + 3*(7*b^2*c^2 - 2*a*b*c*d - a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*(c + d*x^2)*Hypergeometric2F1[-1/4, 1/2, 3/4, -(c/(d*x^2))]))/(6*c^2*d^2*(c + d*x^2)^(3/2))","C",1
861,1,169,213,0.29607,"\int \frac{\left(a+b x^2\right)^2}{\sqrt{e x} \left(c+d x^2\right)^{5/2}} \, dx","Integrate[(a + b*x^2)^2/(Sqrt[e*x]*(c + d*x^2)^(5/2)),x]","\frac{x \left(\frac{i \sqrt{x} \sqrt{\frac{c}{d x^2}+1} \left(5 a^2 d^2+2 a b c d+5 b^2 c^2\right) F\left(\left.i \sinh ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right)\right|-1\right)}{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}+5 a^2 d^2+\frac{2 c (b c-a d)^2}{c+d x^2}+2 a b c d-7 b^2 c^2\right)}{6 c^2 d^2 \sqrt{e x} \sqrt{c+d x^2}}","\frac{\left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(5 a^2 d^2+2 a b c d+5 b^2 c^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{12 c^{9/4} d^{9/4} \sqrt{e} \sqrt{c+d x^2}}-\frac{\sqrt{e x} (5 a d+7 b c) (b c-a d)}{6 c^2 d^2 e \sqrt{c+d x^2}}+\frac{\sqrt{e x} (b c-a d)^2}{3 c d^2 e \left(c+d x^2\right)^{3/2}}",1,"(x*(-7*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2 + (2*c*(b*c - a*d)^2)/(c + d*x^2) + (I*(5*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*Sqrt[x]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/Sqrt[(I*Sqrt[c])/Sqrt[d]]))/(6*c^2*d^2*Sqrt[e*x]*Sqrt[c + d*x^2])","C",1
862,1,161,442,0.175499,"\int \frac{\left(a+b x^2\right)^2}{(e x)^{3/2} \left(c+d x^2\right)^{5/2}} \, dx","Integrate[(a + b*x^2)^2/((e*x)^(3/2)*(c + d*x^2)^(5/2)),x]","\frac{x \left(-x^2 \left(c+d x^2\right) \sqrt{\frac{d x^2}{c}+1} \left(-7 a^2 d^2+2 a b c d+b^2 c^2\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{d x^2}{c}\right)+a^2 (-d) \left(12 c^2+35 c d x^2+21 d^2 x^4\right)+2 a b c d x^2 \left(5 c+3 d x^2\right)+b^2 c^2 x^2 \left(c+3 d x^2\right)\right)}{6 c^3 d (e x)^{3/2} \left(c+d x^2\right)^{3/2}}","-\frac{(e x)^{3/2} \left(7 a^2 d^2-2 a b c d+b^2 c^2\right)}{3 c^2 d e^3 \left(c+d x^2\right)^{3/2}}-\frac{2 a^2}{c e \sqrt{e x} \left(c+d x^2\right)^{3/2}}-\frac{\left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(a d (2 b c-7 a d)+b^2 c^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{4 c^{11/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}+\frac{\left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(a d (2 b c-7 a d)+b^2 c^2\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{2 c^{11/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}-\frac{\sqrt{e x} \sqrt{c+d x^2} \left(a d (2 b c-7 a d)+b^2 c^2\right)}{2 c^3 d^{3/2} e^2 \left(\sqrt{c}+\sqrt{d} x\right)}+\frac{(e x)^{3/2} \left(a d (2 b c-7 a d)+b^2 c^2\right)}{2 c^3 d e^3 \sqrt{c+d x^2}}",1,"(x*(b^2*c^2*x^2*(c + 3*d*x^2) + 2*a*b*c*d*x^2*(5*c + 3*d*x^2) - a^2*d*(12*c^2 + 35*c*d*x^2 + 21*d^2*x^4) - (b^2*c^2 + 2*a*b*c*d - 7*a^2*d^2)*x^2*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Hypergeometric2F1[1/2, 3/4, 7/4, -((d*x^2)/c)]))/(6*c^3*d*(e*x)^(3/2)*(c + d*x^2)^(3/2))","C",1
863,1,211,258,0.3059992,"\int \frac{\left(a+b x^2\right)^2}{(e x)^{5/2} \left(c+d x^2\right)^{5/2}} \, dx","Integrate[(a + b*x^2)^2/((e*x)^(5/2)*(c + d*x^2)^(5/2)),x]","\frac{x^{5/2} \left(\frac{a^2 (-d) \left(4 c^2+21 c d x^2+15 d^2 x^4\right)+2 a b c d x^2 \left(7 c+5 d x^2\right)+b^2 c^2 x^2 \left(d x^2-c\right)}{c^3 d x^{3/2} \left(c+d x^2\right)}+\frac{i x \sqrt{\frac{c}{d x^2}+1} \left(-15 a^2 d^2+10 a b c d+b^2 c^2\right) F\left(\left.i \sinh ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right)\right|-1\right)}{c^3 d \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right)}{6 (e x)^{5/2} \sqrt{c+d x^2}}","-\frac{\sqrt{e x} \left(3 a^2 d^2-2 a b c d+b^2 c^2\right)}{3 c^2 d e^3 \left(c+d x^2\right)^{3/2}}-\frac{2 a^2}{3 c e (e x)^{3/2} \left(c+d x^2\right)^{3/2}}+\frac{\left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(5 a d (2 b c-3 a d)+b^2 c^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{12 c^{13/4} d^{5/4} e^{5/2} \sqrt{c+d x^2}}+\frac{\sqrt{e x} \left(5 a d (2 b c-3 a d)+b^2 c^2\right)}{6 c^3 d e^3 \sqrt{c+d x^2}}",1,"(x^(5/2)*((b^2*c^2*x^2*(-c + d*x^2) + 2*a*b*c*d*x^2*(7*c + 5*d*x^2) - a^2*d*(4*c^2 + 21*c*d*x^2 + 15*d^2*x^4))/(c^3*d*x^(3/2)*(c + d*x^2)) + (I*(b^2*c^2 + 10*a*b*c*d - 15*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/(c^3*Sqrt[(I*Sqrt[c])/Sqrt[d]]*d)))/(6*(e*x)^(5/2)*Sqrt[c + d*x^2])","C",1
864,1,181,489,0.2039392,"\int \frac{\left(a+b x^2\right)^2}{(e x)^{7/2} \left(c+d x^2\right)^{5/2}} \, dx","Integrate[(a + b*x^2)^2/((e*x)^(7/2)*(c + d*x^2)^(5/2)),x]","\frac{x \left(-x^4 \left(c+d x^2\right) \sqrt{\frac{d x^2}{c}+1} \left(77 a^2 d^2-70 a b c d+5 b^2 c^2\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{d x^2}{c}\right)+a^2 \left(-12 c^3+132 c^2 d x^2+385 c d^2 x^4+231 d^3 x^6\right)-10 a b c x^2 \left(12 c^2+35 c d x^2+21 d^2 x^4\right)+5 b^2 c^2 x^4 \left(5 c+3 d x^2\right)\right)}{30 c^4 (e x)^{7/2} \left(c+d x^2\right)^{3/2}}","-\frac{\left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(77 a^2 d^2-70 a b c d+5 b^2 c^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{20 c^{15/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}+\frac{\left(\sqrt{c}+\sqrt{d} x\right) \sqrt{\frac{c+d x^2}{\left(\sqrt{c}+\sqrt{d} x\right)^2}} \left(77 a^2 d^2-70 a b c d+5 b^2 c^2\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)|\frac{1}{2}\right)}{10 c^{15/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}+\frac{(e x)^{3/2} \left(77 a^2 d^2-70 a b c d+5 b^2 c^2\right)}{10 c^4 e^5 \sqrt{c+d x^2}}-\frac{\sqrt{e x} \sqrt{c+d x^2} \left(77 a^2 d^2-70 a b c d+5 b^2 c^2\right)}{10 c^4 \sqrt{d} e^4 \left(\sqrt{c}+\sqrt{d} x\right)}+\frac{(e x)^{3/2} \left(77 a^2 d^2-70 a b c d+5 b^2 c^2\right)}{15 c^3 e^5 \left(c+d x^2\right)^{3/2}}-\frac{2 a^2}{5 c e (e x)^{5/2} \left(c+d x^2\right)^{3/2}}-\frac{2 a (10 b c-11 a d)}{5 c^2 e^3 \sqrt{e x} \left(c+d x^2\right)^{3/2}}",1,"(x*(5*b^2*c^2*x^4*(5*c + 3*d*x^2) - 10*a*b*c*x^2*(12*c^2 + 35*c*d*x^2 + 21*d^2*x^4) + a^2*(-12*c^3 + 132*c^2*d*x^2 + 385*c*d^2*x^4 + 231*d^3*x^6) - (5*b^2*c^2 - 70*a*b*c*d + 77*a^2*d^2)*x^4*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Hypergeometric2F1[1/2, 3/4, 7/4, -((d*x^2)/c)]))/(30*c^4*(e*x)^(7/2)*(c + d*x^2)^(3/2))","C",1
865,1,187,372,0.2236065,"\int \frac{(e x)^{7/2} \sqrt{c-d x^2}}{a-b x^2} \, dx","Integrate[((e*x)^(7/2)*Sqrt[c - d*x^2])/(a - b*x^2),x]","\frac{2 e^3 \sqrt{e x} \left(x^2 \sqrt{1-\frac{d x^2}{c}} \left(-21 a^2 d^2+14 a b c d+2 b^2 c^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+5 a c \sqrt{1-\frac{d x^2}{c}} (7 a d-2 b c) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-5 a \left(c-d x^2\right) \left(7 a d-2 b c+3 b d x^2\right)\right)}{105 a b^2 d \sqrt{c-d x^2}}","-\frac{2 \sqrt[4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} \left(-21 a^2 d^2+14 a b c d+2 b^2 c^2\right) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{21 b^3 d^{5/4} \sqrt{c-d x^2}}+\frac{a \sqrt[4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{b^3 \sqrt[4]{d} \sqrt{c-d x^2}}+\frac{a \sqrt[4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{b^3 \sqrt[4]{d} \sqrt{c-d x^2}}+\frac{2 e^3 \sqrt{e x} \sqrt{c-d x^2} (2 b c-7 a d)}{21 b^2 d}-\frac{2 e (e x)^{5/2} \sqrt{c-d x^2}}{7 b}",1,"(2*e^3*Sqrt[e*x]*(-5*a*(c - d*x^2)*(-2*b*c + 7*a*d + 3*b*d*x^2) + 5*a*c*(-2*b*c + 7*a*d)*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + (2*b^2*c^2 + 14*a*b*c*d - 21*a^2*d^2)*x^2*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(105*a*b^2*d*Sqrt[c - d*x^2])","C",0
866,1,143,414,0.1497928,"\int \frac{(e x)^{5/2} \sqrt{c-d x^2}}{a-b x^2} \, dx","Integrate[((e*x)^(5/2)*Sqrt[c - d*x^2])/(a - b*x^2),x]","\frac{2 e (e x)^{3/2} \left(x^2 \sqrt{1-\frac{d x^2}{c}} (2 b c-5 a d) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+7 a c \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-7 a \left(c-d x^2\right)\right)}{35 a b \sqrt{c-d x^2}}","-\frac{\sqrt{a} \sqrt[4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{b^{5/2} \sqrt[4]{d} \sqrt{c-d x^2}}+\frac{\sqrt{a} \sqrt[4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{b^{5/2} \sqrt[4]{d} \sqrt{c-d x^2}}+\frac{2 c^{3/4} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (2 b c-5 a d) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{5 b^2 d^{3/4} \sqrt{c-d x^2}}-\frac{2 c^{3/4} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (2 b c-5 a d) E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{5 b^2 d^{3/4} \sqrt{c-d x^2}}-\frac{2 e (e x)^{3/2} \sqrt{c-d x^2}}{5 b}",1,"(2*e*(e*x)^(3/2)*(-7*a*(c - d*x^2) + 7*a*c*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + (2*b*c - 5*a*d)*x^2*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(35*a*b*Sqrt[c - d*x^2])","C",0
867,1,143,315,0.1197898,"\int \frac{(e x)^{3/2} \sqrt{c-d x^2}}{a-b x^2} \, dx","Integrate[((e*x)^(3/2)*Sqrt[c - d*x^2])/(a - b*x^2),x]","\frac{2 e \sqrt{e x} \left(x^2 \sqrt{1-\frac{d x^2}{c}} (2 b c-3 a d) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+5 a c \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-5 a \left(c-d x^2\right)\right)}{15 a b \sqrt{c-d x^2}}","-\frac{2 \sqrt[4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (2 b c-3 a d) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{3 b^2 \sqrt[4]{d} \sqrt{c-d x^2}}+\frac{\sqrt[4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{b^2 \sqrt[4]{d} \sqrt{c-d x^2}}+\frac{\sqrt[4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{b^2 \sqrt[4]{d} \sqrt{c-d x^2}}-\frac{2 e \sqrt{e x} \sqrt{c-d x^2}}{3 b}",1,"(2*e*Sqrt[e*x]*(-5*a*(c - d*x^2) + 5*a*c*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + (2*b*c - 3*a*d)*x^2*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(15*a*b*Sqrt[c - d*x^2])","C",0
868,1,69,365,0.0445134,"\int \frac{\sqrt{e x} \sqrt{c-d x^2}}{a-b x^2} \, dx","Integrate[(Sqrt[e*x]*Sqrt[c - d*x^2])/(a - b*x^2),x]","\frac{2 x \sqrt{e x} \sqrt{c-d x^2} F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)}{3 a \sqrt{1-\frac{d x^2}{c}}}","-\frac{\sqrt[4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{\sqrt{a} b^{3/2} \sqrt[4]{d} \sqrt{c-d x^2}}+\frac{\sqrt[4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{\sqrt{a} b^{3/2} \sqrt[4]{d} \sqrt{c-d x^2}}-\frac{2 c^{3/4} \sqrt[4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{b \sqrt{c-d x^2}}+\frac{2 c^{3/4} \sqrt[4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{b \sqrt{c-d x^2}}",1,"(2*x*Sqrt[e*x]*Sqrt[c - d*x^2]*AppellF1[3/4, -1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a])/(3*a*Sqrt[1 - (d*x^2)/c])","C",0
869,1,67,283,0.0399242,"\int \frac{\sqrt{c-d x^2}}{\sqrt{e x} \left(a-b x^2\right)} \, dx","Integrate[Sqrt[c - d*x^2]/(Sqrt[e*x]*(a - b*x^2)),x]","\frac{2 x \sqrt{c-d x^2} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)}{a \sqrt{e x} \sqrt{1-\frac{d x^2}{c}}}","\frac{\sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a b \sqrt[4]{d} \sqrt{e} \sqrt{c-d x^2}}+\frac{\sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a b \sqrt[4]{d} \sqrt{e} \sqrt{c-d x^2}}+\frac{2 \sqrt[4]{c} d^{3/4} \sqrt{1-\frac{d x^2}{c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{b \sqrt{e} \sqrt{c-d x^2}}",1,"(2*x*Sqrt[c - d*x^2]*AppellF1[1/4, -1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a])/(a*Sqrt[e*x]*Sqrt[1 - (d*x^2)/c])","C",0
870,1,143,392,0.1342751,"\int \frac{\sqrt{c-d x^2}}{(e x)^{3/2} \left(a-b x^2\right)} \, dx","Integrate[Sqrt[c - d*x^2]/((e*x)^(3/2)*(a - b*x^2)),x]","\frac{x \left(14 x^2 \sqrt{1-\frac{d x^2}{c}} (b c-2 a d) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+6 b d x^4 \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-42 a \left(c-d x^2\right)\right)}{21 a^2 (e x)^{3/2} \sqrt{c-d x^2}}","-\frac{\sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a^{3/2} \sqrt{b} \sqrt[4]{d} e^{3/2} \sqrt{c-d x^2}}+\frac{\sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a^{3/2} \sqrt{b} \sqrt[4]{d} e^{3/2} \sqrt{c-d x^2}}+\frac{2 c^{3/4} \sqrt[4]{d} \sqrt{1-\frac{d x^2}{c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a e^{3/2} \sqrt{c-d x^2}}-\frac{2 c^{3/4} \sqrt[4]{d} \sqrt{1-\frac{d x^2}{c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a e^{3/2} \sqrt{c-d x^2}}-\frac{2 \sqrt{c-d x^2}}{a e \sqrt{e x}}",1,"(x*(-42*a*(c - d*x^2) + 14*(b*c - 2*a*d)*x^2*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 6*b*d*x^4*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(21*a^2*(e*x)^(3/2)*Sqrt[c - d*x^2])","C",0
871,1,146,308,0.1427678,"\int \frac{\sqrt{c-d x^2}}{(e x)^{5/2} \left(a-b x^2\right)} \, dx","Integrate[Sqrt[c - d*x^2]/((e*x)^(5/2)*(a - b*x^2)),x]","\frac{x \left(10 x^2 \sqrt{1-\frac{d x^2}{c}} (3 b c-2 a d) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-2 \left(b d x^4 \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+5 a \left(c-d x^2\right)\right)\right)}{15 a^2 (e x)^{5/2} \sqrt{c-d x^2}}","\frac{\sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a^2 \sqrt[4]{d} e^{5/2} \sqrt{c-d x^2}}+\frac{\sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a^2 \sqrt[4]{d} e^{5/2} \sqrt{c-d x^2}}+\frac{2 \sqrt[4]{c} d^{3/4} \sqrt{1-\frac{d x^2}{c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{3 a e^{5/2} \sqrt{c-d x^2}}-\frac{2 \sqrt{c-d x^2}}{3 a e (e x)^{3/2}}",1,"(x*(10*(3*b*c - 2*a*d)*x^2*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] - 2*(5*a*(c - d*x^2) + b*d*x^4*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a])))/(15*a^2*(e*x)^(5/2)*Sqrt[c - d*x^2])","C",0
872,1,190,457,0.2360267,"\int \frac{\sqrt{c-d x^2}}{(e x)^{7/2} \left(a-b x^2\right)} \, dx","Integrate[Sqrt[c - d*x^2]/((e*x)^(7/2)*(a - b*x^2)),x]","\frac{x \left(14 x^4 \sqrt{1-\frac{d x^2}{c}} \left(2 a^2 d^2-10 a b c d+5 b^2 c^2\right) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-6 \left(b d x^6 \sqrt{1-\frac{d x^2}{c}} (2 a d-5 b c) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+7 a \left(c-d x^2\right) \left(a c-2 a d x^2+5 b c x^2\right)\right)\right)}{105 a^3 c (e x)^{7/2} \sqrt{c-d x^2}}","-\frac{\sqrt{b} \sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a^{5/2} \sqrt[4]{d} e^{7/2} \sqrt{c-d x^2}}+\frac{\sqrt{b} \sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a^{5/2} \sqrt[4]{d} e^{7/2} \sqrt{c-d x^2}}+\frac{2 \sqrt[4]{d} \sqrt{1-\frac{d x^2}{c}} (5 b c-2 a d) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{5 a^2 \sqrt[4]{c} e^{7/2} \sqrt{c-d x^2}}-\frac{2 \sqrt[4]{d} \sqrt{1-\frac{d x^2}{c}} (5 b c-2 a d) E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{5 a^2 \sqrt[4]{c} e^{7/2} \sqrt{c-d x^2}}-\frac{2 \sqrt{c-d x^2} (5 b c-2 a d)}{5 a^2 c e^3 \sqrt{e x}}-\frac{2 \sqrt{c-d x^2}}{5 a e (e x)^{5/2}}",1,"(x*(14*(5*b^2*c^2 - 10*a*b*c*d + 2*a^2*d^2)*x^4*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] - 6*(7*a*(c - d*x^2)*(a*c + 5*b*c*x^2 - 2*a*d*x^2) + b*d*(-5*b*c + 2*a*d)*x^6*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a])))/(105*a^3*c*(e*x)^(7/2)*Sqrt[c - d*x^2])","C",0
873,1,183,485,0.2151617,"\int \frac{(e x)^{5/2} \left(c-d x^2\right)^{3/2}}{a-b x^2} \, dx","Integrate[((e*x)^(5/2)*(c - d*x^2)^(3/2))/(a - b*x^2),x]","-\frac{2 e (e x)^{3/2} \left(-3 x^2 \sqrt{1-\frac{d x^2}{c}} \left(15 a^2 d^2-21 a b c d+4 b^2 c^2\right) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+7 a c \sqrt{1-\frac{d x^2}{c}} (9 a d-11 b c) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-7 a \left(c-d x^2\right) \left(9 a d-11 b c+5 b d x^2\right)\right)}{315 a b^2 \sqrt{c-d x^2}}","\frac{2 c^{3/4} e^{5/2} \sqrt{1-\frac{d x^2}{c}} \left(15 a^2 d^2-21 a b c d+4 b^2 c^2\right) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{15 b^3 d^{3/4} \sqrt{c-d x^2}}-\frac{2 c^{3/4} e^{5/2} \sqrt{1-\frac{d x^2}{c}} \left(15 a^2 d^2-21 a b c d+4 b^2 c^2\right) E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{15 b^3 d^{3/4} \sqrt{c-d x^2}}-\frac{\sqrt{a} \sqrt[4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (b c-a d)^2 \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{b^{7/2} \sqrt[4]{d} \sqrt{c-d x^2}}+\frac{\sqrt{a} \sqrt[4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (b c-a d)^2 \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{b^{7/2} \sqrt[4]{d} \sqrt{c-d x^2}}-\frac{2 e (e x)^{3/2} \sqrt{c-d x^2} (11 b c-9 a d)}{45 b^2}+\frac{2 d (e x)^{7/2} \sqrt{c-d x^2}}{9 b e}",1,"(-2*e*(e*x)^(3/2)*(-7*a*(c - d*x^2)*(-11*b*c + 9*a*d + 5*b*d*x^2) + 7*a*c*(-11*b*c + 9*a*d)*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] - 3*(4*b^2*c^2 - 21*a*b*c*d + 15*a^2*d^2)*x^2*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(315*a*b^2*Sqrt[c - d*x^2])","C",0
874,1,182,372,0.2012485,"\int \frac{(e x)^{3/2} \left(c-d x^2\right)^{3/2}}{a-b x^2} \, dx","Integrate[((e*x)^(3/2)*(c - d*x^2)^(3/2))/(a - b*x^2),x]","-\frac{2 e \sqrt{e x} \left(x^2 \sqrt{1-\frac{d x^2}{c}} \left(-21 a^2 d^2+35 a b c d-12 b^2 c^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+5 a c \sqrt{1-\frac{d x^2}{c}} (7 a d-9 b c) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-5 a \left(c-d x^2\right) \left(7 a d-9 b c+3 b d x^2\right)\right)}{105 a b^2 \sqrt{c-d x^2}}","-\frac{2 \sqrt[4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} \left(21 a^2 d^2-35 a b c d+12 b^2 c^2\right) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{21 b^3 \sqrt[4]{d} \sqrt{c-d x^2}}+\frac{\sqrt[4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (b c-a d)^2 \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{b^3 \sqrt[4]{d} \sqrt{c-d x^2}}+\frac{\sqrt[4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (b c-a d)^2 \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{b^3 \sqrt[4]{d} \sqrt{c-d x^2}}-\frac{2 e \sqrt{e x} \sqrt{c-d x^2} (9 b c-7 a d)}{21 b^2}+\frac{2 d (e x)^{5/2} \sqrt{c-d x^2}}{7 b e}",1,"(-2*e*Sqrt[e*x]*(-5*a*(c - d*x^2)*(-9*b*c + 7*a*d + 3*b*d*x^2) + 5*a*c*(-9*b*c + 7*a*d)*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + (-12*b^2*c^2 + 35*a*b*c*d - 21*a^2*d^2)*x^2*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(105*a*b^2*Sqrt[c - d*x^2])","C",0
875,1,155,421,0.2284076,"\int \frac{\sqrt{e x} \left(c-d x^2\right)^{3/2}}{a-b x^2} \, dx","Integrate[(Sqrt[e*x]*(c - d*x^2)^(3/2))/(a - b*x^2),x]","\frac{2 x \sqrt{e x} \left(7 c \sqrt{1-\frac{d x^2}{c}} (5 b c-3 a d) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+3 d \left(x^2 \sqrt{1-\frac{d x^2}{c}} (5 a d-7 b c) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+7 a \left(c-d x^2\right)\right)\right)}{105 a b \sqrt{c-d x^2}}","-\frac{\sqrt[4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (b c-a d)^2 \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{\sqrt{a} b^{5/2} \sqrt[4]{d} \sqrt{c-d x^2}}+\frac{\sqrt[4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (b c-a d)^2 \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{\sqrt{a} b^{5/2} \sqrt[4]{d} \sqrt{c-d x^2}}-\frac{2 c^{3/4} \sqrt[4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (7 b c-5 a d) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{5 b^2 \sqrt{c-d x^2}}+\frac{2 c^{3/4} \sqrt[4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (7 b c-5 a d) E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{5 b^2 \sqrt{c-d x^2}}+\frac{2 d (e x)^{3/2} \sqrt{c-d x^2}}{5 b e}",1,"(2*x*Sqrt[e*x]*(7*c*(5*b*c - 3*a*d)*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 3*d*(7*a*(c - d*x^2) + (-7*b*c + 5*a*d)*x^2*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a])))/(105*a*b*Sqrt[c - d*x^2])","C",0
876,1,153,328,0.1747646,"\int \frac{\left(c-d x^2\right)^{3/2}}{\sqrt{e x} \left(a-b x^2\right)} \, dx","Integrate[(c - d*x^2)^(3/2)/(Sqrt[e*x]*(a - b*x^2)),x]","\frac{2 x \left(d x^2 \sqrt{1-\frac{d x^2}{c}} (3 a d-5 b c) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+5 c \sqrt{1-\frac{d x^2}{c}} (3 b c-a d) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+5 a d \left(c-d x^2\right)\right)}{15 a b \sqrt{e x} \sqrt{c-d x^2}}","\frac{2 \sqrt[4]{c} d^{3/4} \sqrt{1-\frac{d x^2}{c}} (5 b c-3 a d) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{3 b^2 \sqrt{e} \sqrt{c-d x^2}}+\frac{\sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d)^2 \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a b^2 \sqrt[4]{d} \sqrt{e} \sqrt{c-d x^2}}+\frac{\sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d)^2 \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a b^2 \sqrt[4]{d} \sqrt{e} \sqrt{c-d x^2}}+\frac{2 d \sqrt{e x} \sqrt{c-d x^2}}{3 b e}",1,"(2*x*(5*a*d*(c - d*x^2) + 5*c*(3*b*c - a*d)*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + d*(-5*b*c + 3*a*d)*x^2*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(15*a*b*Sqrt[e*x]*Sqrt[c - d*x^2])","C",0
877,1,151,417,0.1434588,"\int \frac{\left(c-d x^2\right)^{3/2}}{(e x)^{3/2} \left(a-b x^2\right)} \, dx","Integrate[(c - d*x^2)^(3/2)/((e*x)^(3/2)*(a - b*x^2)),x]","\frac{x \left(14 c x^2 \sqrt{1-\frac{d x^2}{c}} (b c-3 a d) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+6 d x^4 \sqrt{1-\frac{d x^2}{c}} (a d+b c) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-42 a c \left(c-d x^2\right)\right)}{21 a^2 (e x)^{3/2} \sqrt{c-d x^2}}","-\frac{\sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d)^2 \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a^{3/2} b^{3/2} \sqrt[4]{d} e^{3/2} \sqrt{c-d x^2}}+\frac{\sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d)^2 \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a^{3/2} b^{3/2} \sqrt[4]{d} e^{3/2} \sqrt{c-d x^2}}+\frac{2 c^{3/4} \sqrt[4]{d} \sqrt{1-\frac{d x^2}{c}} (a d+b c) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a b e^{3/2} \sqrt{c-d x^2}}-\frac{2 c^{3/4} \sqrt[4]{d} \sqrt{1-\frac{d x^2}{c}} (a d+b c) E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a b e^{3/2} \sqrt{c-d x^2}}-\frac{2 c \sqrt{c-d x^2}}{a e \sqrt{e x}}",1,"(x*(-42*a*c*(c - d*x^2) + 14*c*(b*c - 3*a*d)*x^2*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 6*d*(b*c + a*d)*x^4*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(21*a^2*(e*x)^(3/2)*Sqrt[c - d*x^2])","C",0
878,1,153,330,0.1607735,"\int \frac{\left(c-d x^2\right)^{3/2}}{(e x)^{5/2} \left(a-b x^2\right)} \, dx","Integrate[(c - d*x^2)^(3/2)/((e*x)^(5/2)*(a - b*x^2)),x]","\frac{x \left(10 c x^2 \sqrt{1-\frac{d x^2}{c}} (3 b c-5 a d) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-2 d x^4 \sqrt{1-\frac{d x^2}{c}} (b c-3 a d) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-10 a c \left(c-d x^2\right)\right)}{15 a^2 (e x)^{5/2} \sqrt{c-d x^2}}","\frac{\sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d)^2 \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a^2 b \sqrt[4]{d} e^{5/2} \sqrt{c-d x^2}}+\frac{\sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d)^2 \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a^2 b \sqrt[4]{d} e^{5/2} \sqrt{c-d x^2}}+\frac{2 \sqrt[4]{c} d^{3/4} \sqrt{1-\frac{d x^2}{c}} (b c-3 a d) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{3 a b e^{5/2} \sqrt{c-d x^2}}-\frac{2 c \sqrt{c-d x^2}}{3 a e (e x)^{3/2}}",1,"(x*(-10*a*c*(c - d*x^2) + 10*c*(3*b*c - 5*a*d)*x^2*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] - 2*d*(b*c - 3*a*d)*x^4*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(15*a^2*(e*x)^(5/2)*Sqrt[c - d*x^2])","C",0
879,1,187,459,0.2551584,"\int \frac{\left(c-d x^2\right)^{3/2}}{(e x)^{7/2} \left(a-b x^2\right)} \, dx","Integrate[(c - d*x^2)^(3/2)/((e*x)^(7/2)*(a - b*x^2)),x]","\frac{x \left(14 x^4 \sqrt{1-\frac{d x^2}{c}} \left(12 a^2 d^2-15 a b c d+5 b^2 c^2\right) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-6 \left(b d x^6 \sqrt{1-\frac{d x^2}{c}} (7 a d-5 b c) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+7 a \left(c-d x^2\right) \left(a c-7 a d x^2+5 b c x^2\right)\right)\right)}{105 a^3 (e x)^{7/2} \sqrt{c-d x^2}}","-\frac{\sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d)^2 \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a^{5/2} \sqrt{b} \sqrt[4]{d} e^{7/2} \sqrt{c-d x^2}}+\frac{\sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d)^2 \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a^{5/2} \sqrt{b} \sqrt[4]{d} e^{7/2} \sqrt{c-d x^2}}+\frac{2 c^{3/4} \sqrt[4]{d} \sqrt{1-\frac{d x^2}{c}} (5 b c-7 a d) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{5 a^2 e^{7/2} \sqrt{c-d x^2}}-\frac{2 c^{3/4} \sqrt[4]{d} \sqrt{1-\frac{d x^2}{c}} (5 b c-7 a d) E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{5 a^2 e^{7/2} \sqrt{c-d x^2}}-\frac{2 \sqrt{c-d x^2} (5 b c-7 a d)}{5 a^2 e^3 \sqrt{e x}}-\frac{2 c \sqrt{c-d x^2}}{5 a e (e x)^{5/2}}",1,"(x*(14*(5*b^2*c^2 - 15*a*b*c*d + 12*a^2*d^2)*x^4*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] - 6*(7*a*(c - d*x^2)*(a*c + 5*b*c*x^2 - 7*a*d*x^2) + b*d*(-5*b*c + 7*a*d)*x^6*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a])))/(105*a^3*(e*x)^(7/2)*Sqrt[c - d*x^2])","C",0
880,1,147,305,0.1623235,"\int \frac{(e x)^{7/2}}{\left(a-b x^2\right) \sqrt{c-d x^2}} \, dx","Integrate[(e*x)^(7/2)/((a - b*x^2)*Sqrt[c - d*x^2]),x]","\frac{2 e^3 \sqrt{e x} \left(x^2 \sqrt{1-\frac{d x^2}{c}} (3 a d+b c) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-5 a c \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+5 a \left(c-d x^2\right)\right)}{15 a b d \sqrt{c-d x^2}}","-\frac{2 \sqrt[4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (3 a d+b c) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{3 b^2 d^{5/4} \sqrt{c-d x^2}}+\frac{a \sqrt[4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{b^2 \sqrt[4]{d} \sqrt{c-d x^2}}+\frac{a \sqrt[4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{b^2 \sqrt[4]{d} \sqrt{c-d x^2}}+\frac{2 e^3 \sqrt{e x} \sqrt{c-d x^2}}{3 b d}",1,"(2*e^3*Sqrt[e*x]*(5*a*(c - d*x^2) - 5*a*c*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + (b*c + 3*a*d)*x^2*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(15*a*b*d*Sqrt[c - d*x^2])","C",0
881,1,70,349,0.046001,"\int \frac{(e x)^{5/2}}{\left(a-b x^2\right) \sqrt{c-d x^2}} \, dx","Integrate[(e*x)^(5/2)/((a - b*x^2)*Sqrt[c - d*x^2]),x]","\frac{2 x (e x)^{5/2} \sqrt{\frac{c-d x^2}{c}} F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)}{7 a \sqrt{c-d x^2}}","-\frac{\sqrt{a} \sqrt[4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{b^{3/2} \sqrt[4]{d} \sqrt{c-d x^2}}+\frac{\sqrt{a} \sqrt[4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{b^{3/2} \sqrt[4]{d} \sqrt{c-d x^2}}+\frac{2 c^{3/4} e^{5/2} \sqrt{1-\frac{d x^2}{c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{b d^{3/4} \sqrt{c-d x^2}}-\frac{2 c^{3/4} e^{5/2} \sqrt{1-\frac{d x^2}{c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{b d^{3/4} \sqrt{c-d x^2}}",1,"(2*x*(e*x)^(5/2)*Sqrt[(c - d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a])/(7*a*Sqrt[c - d*x^2])","C",0
882,1,70,261,0.0423725,"\int \frac{(e x)^{3/2}}{\left(a-b x^2\right) \sqrt{c-d x^2}} \, dx","Integrate[(e*x)^(3/2)/((a - b*x^2)*Sqrt[c - d*x^2]),x]","\frac{2 x (e x)^{3/2} \sqrt{\frac{c-d x^2}{c}} F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)}{5 a \sqrt{c-d x^2}}","\frac{\sqrt[4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{b \sqrt[4]{d} \sqrt{c-d x^2}}+\frac{\sqrt[4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{b \sqrt[4]{d} \sqrt{c-d x^2}}-\frac{2 \sqrt[4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{b \sqrt[4]{d} \sqrt{c-d x^2}}",1,"(2*x*(e*x)^(3/2)*Sqrt[(c - d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a])/(5*a*Sqrt[c - d*x^2])","C",0
883,1,70,203,0.0428587,"\int \frac{\sqrt{e x}}{\left(a-b x^2\right) \sqrt{c-d x^2}} \, dx","Integrate[Sqrt[e*x]/((a - b*x^2)*Sqrt[c - d*x^2]),x]","\frac{2 x \sqrt{e x} \sqrt{\frac{c-d x^2}{c}} F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)}{3 a \sqrt{c-d x^2}}","\frac{\sqrt[4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{\sqrt{a} \sqrt{b} \sqrt[4]{d} \sqrt{c-d x^2}}-\frac{\sqrt[4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{\sqrt{a} \sqrt{b} \sqrt[4]{d} \sqrt{c-d x^2}}",1,"(2*x*Sqrt[e*x]*Sqrt[(c - d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a])/(3*a*Sqrt[c - d*x^2])","C",0
884,1,68,188,0.0428242,"\int \frac{1}{\sqrt{e x} \left(a-b x^2\right) \sqrt{c-d x^2}} \, dx","Integrate[1/(Sqrt[e*x]*(a - b*x^2)*Sqrt[c - d*x^2]),x]","\frac{2 x \sqrt{\frac{c-d x^2}{c}} F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)}{a \sqrt{e x} \sqrt{c-d x^2}}","\frac{\sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a \sqrt[4]{d} \sqrt{e} \sqrt{c-d x^2}}+\frac{\sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a \sqrt[4]{d} \sqrt{e} \sqrt{c-d x^2}}",1,"(2*x*Sqrt[(c - d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a])/(a*Sqrt[e*x]*Sqrt[c - d*x^2])","C",0
885,1,146,379,0.1169262,"\int \frac{1}{(e x)^{3/2} \left(a-b x^2\right) \sqrt{c-d x^2}} \, dx","Integrate[1/((e*x)^(3/2)*(a - b*x^2)*Sqrt[c - d*x^2]),x]","\frac{x \left(14 x^2 \sqrt{1-\frac{d x^2}{c}} (b c-a d) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+6 b d x^4 \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-42 a \left(c-d x^2\right)\right)}{21 a^2 c (e x)^{3/2} \sqrt{c-d x^2}}","-\frac{\sqrt{b} \sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a^{3/2} \sqrt[4]{d} e^{3/2} \sqrt{c-d x^2}}+\frac{\sqrt{b} \sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a^{3/2} \sqrt[4]{d} e^{3/2} \sqrt{c-d x^2}}+\frac{2 \sqrt[4]{d} \sqrt{1-\frac{d x^2}{c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a \sqrt[4]{c} e^{3/2} \sqrt{c-d x^2}}-\frac{2 \sqrt[4]{d} \sqrt{1-\frac{d x^2}{c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a \sqrt[4]{c} e^{3/2} \sqrt{c-d x^2}}-\frac{2 \sqrt{c-d x^2}}{a c e \sqrt{e x}}",1,"(x*(-42*a*(c - d*x^2) + 14*(b*c - a*d)*x^2*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 6*b*d*x^4*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(21*a^2*c*(e*x)^(3/2)*Sqrt[c - d*x^2])","C",0
886,1,148,297,0.1377207,"\int \frac{1}{(e x)^{5/2} \left(a-b x^2\right) \sqrt{c-d x^2}} \, dx","Integrate[1/((e*x)^(5/2)*(a - b*x^2)*Sqrt[c - d*x^2]),x]","\frac{x \left(10 x^2 \sqrt{1-\frac{d x^2}{c}} (a d+3 b c) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-2 \left(b d x^4 \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+5 a \left(c-d x^2\right)\right)\right)}{15 a^2 c (e x)^{5/2} \sqrt{c-d x^2}}","\frac{b \sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a^2 \sqrt[4]{d} e^{5/2} \sqrt{c-d x^2}}+\frac{b \sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a^2 \sqrt[4]{d} e^{5/2} \sqrt{c-d x^2}}+\frac{2 d^{3/4} \sqrt{1-\frac{d x^2}{c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{3 a c^{3/4} e^{5/2} \sqrt{c-d x^2}}-\frac{2 \sqrt{c-d x^2}}{3 a c e (e x)^{3/2}}",1,"(x*(10*(3*b*c + a*d)*x^2*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] - 2*(5*a*(c - d*x^2) + b*d*x^4*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a])))/(15*a^2*c*(e*x)^(5/2)*Sqrt[c - d*x^2])","C",0
887,1,188,444,0.2417482,"\int \frac{1}{(e x)^{7/2} \left(a-b x^2\right) \sqrt{c-d x^2}} \, dx","Integrate[1/((e*x)^(7/2)*(a - b*x^2)*Sqrt[c - d*x^2]),x]","\frac{x \left(14 x^4 \sqrt{1-\frac{d x^2}{c}} \left(-3 a^2 d^2-5 a b c d+5 b^2 c^2\right) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+6 b d x^6 \sqrt{1-\frac{d x^2}{c}} (3 a d+5 b c) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-42 a \left(c-d x^2\right) \left(a \left(c+3 d x^2\right)+5 b c x^2\right)\right)}{105 a^3 c^2 (e x)^{7/2} \sqrt{c-d x^2}}","-\frac{b^{3/2} \sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a^{5/2} \sqrt[4]{d} e^{7/2} \sqrt{c-d x^2}}+\frac{b^{3/2} \sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a^{5/2} \sqrt[4]{d} e^{7/2} \sqrt{c-d x^2}}+\frac{2 \sqrt[4]{d} \sqrt{1-\frac{d x^2}{c}} (3 a d+5 b c) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{5 a^2 c^{5/4} e^{7/2} \sqrt{c-d x^2}}-\frac{2 \sqrt[4]{d} \sqrt{1-\frac{d x^2}{c}} (3 a d+5 b c) E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{5 a^2 c^{5/4} e^{7/2} \sqrt{c-d x^2}}-\frac{2 \sqrt{c-d x^2} (3 a d+5 b c)}{5 a^2 c^2 e^3 \sqrt{e x}}-\frac{2 \sqrt{c-d x^2}}{5 a c e (e x)^{5/2}}",1,"(x*(-42*a*(c - d*x^2)*(5*b*c*x^2 + a*(c + 3*d*x^2)) + 14*(5*b^2*c^2 - 5*a*b*c*d - 3*a^2*d^2)*x^4*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 6*b*d*(5*b*c + 3*a*d)*x^6*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(105*a^3*c^2*(e*x)^(7/2)*Sqrt[c - d*x^2])","C",0
888,1,148,444,0.1523928,"\int \frac{(e x)^{9/2}}{\left(a-b x^2\right) \left(c-d x^2\right)^{3/2}} \, dx","Integrate[(e*x)^(9/2)/((a - b*x^2)*(c - d*x^2)^(3/2)),x]","-\frac{e^3 (e x)^{3/2} \left(x^2 \sqrt{1-\frac{d x^2}{c}} (2 a d-3 b c) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+7 a c \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-7 a c\right)}{7 a d \sqrt{c-d x^2} (a d-b c)}","-\frac{a^{3/2} \sqrt[4]{c} e^{9/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{b^{3/2} \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)}+\frac{a^{3/2} \sqrt[4]{c} e^{9/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{b^{3/2} \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)}-\frac{c^{3/4} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (3 b c-2 a d) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{b d^{7/4} \sqrt{c-d x^2} (b c-a d)}+\frac{c^{3/4} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (3 b c-2 a d) E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{b d^{7/4} \sqrt{c-d x^2} (b c-a d)}-\frac{c e^3 (e x)^{3/2}}{d \sqrt{c-d x^2} (b c-a d)}",1,"-1/7*(e^3*(e*x)^(3/2)*(-7*a*c + 7*a*c*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + (-3*b*c + 2*a*d)*x^2*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(a*d*(-(b*c) + a*d)*Sqrt[c - d*x^2])","C",0
889,1,148,338,0.1475728,"\int \frac{(e x)^{7/2}}{\left(a-b x^2\right) \left(c-d x^2\right)^{3/2}} \, dx","Integrate[(e*x)^(7/2)/((a - b*x^2)*(c - d*x^2)^(3/2)),x]","-\frac{e^3 \sqrt{e x} \left(x^2 \sqrt{1-\frac{d x^2}{c}} (2 a d-b c) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+5 a c \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-5 a c\right)}{5 a d \sqrt{c-d x^2} (a d-b c)}","\frac{\sqrt[4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (b c-2 a d) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{b d^{5/4} \sqrt{c-d x^2} (b c-a d)}+\frac{a \sqrt[4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{b \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)}+\frac{a \sqrt[4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{b \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)}-\frac{c e^3 \sqrt{e x}}{d \sqrt{c-d x^2} (b c-a d)}",1,"-1/5*(e^3*Sqrt[e*x]*(-5*a*c + 5*a*c*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + (-(b*c) + 2*a*d)*x^2*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(a*d*(-(b*c) + a*d)*Sqrt[c - d*x^2])","C",0
890,1,133,414,0.1014648,"\int \frac{(e x)^{5/2}}{\left(a-b x^2\right) \left(c-d x^2\right)^{3/2}} \, dx","Integrate[(e*x)^(5/2)/((a - b*x^2)*(c - d*x^2)^(3/2)),x]","-\frac{e (e x)^{3/2} \left(b x^2 \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-7 a \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+7 a\right)}{7 a \sqrt{c-d x^2} (b c-a d)}","-\frac{c^{3/4} e^{5/2} \sqrt{1-\frac{d x^2}{c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{d^{3/4} \sqrt{c-d x^2} (b c-a d)}+\frac{c^{3/4} e^{5/2} \sqrt{1-\frac{d x^2}{c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{d^{3/4} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt{a} \sqrt[4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{\sqrt{b} \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)}+\frac{\sqrt{a} \sqrt[4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{\sqrt{b} \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)}-\frac{e (e x)^{3/2}}{\sqrt{c-d x^2} (b c-a d)}",1,"-1/7*(e*(e*x)^(3/2)*(7*a - 7*a*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + b*x^2*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(a*(b*c - a*d)*Sqrt[c - d*x^2])","C",0
891,1,133,314,0.0956588,"\int \frac{(e x)^{3/2}}{\left(a-b x^2\right) \left(c-d x^2\right)^{3/2}} \, dx","Integrate[(e*x)^(3/2)/((a - b*x^2)*(c - d*x^2)^(3/2)),x]","\frac{e \sqrt{e x} \left(b x^2 \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+5 a \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-5 a\right)}{5 a \sqrt{c-d x^2} (b c-a d)}","-\frac{\sqrt[4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{\sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)}+\frac{\sqrt[4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{\sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)}+\frac{\sqrt[4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{\sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)}-\frac{e \sqrt{e x}}{\sqrt{c-d x^2} (b c-a d)}",1,"(e*Sqrt[e*x]*(-5*a + 5*a*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + b*x^2*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(5*a*(b*c - a*d)*Sqrt[c - d*x^2])","C",0
892,1,148,420,0.2431261,"\int \frac{\sqrt{e x}}{\left(a-b x^2\right) \left(c-d x^2\right)^{3/2}} \, dx","Integrate[Sqrt[e*x]/((a - b*x^2)*(c - d*x^2)^(3/2)),x]","\frac{\sqrt{e x} \left(7 x \sqrt{1-\frac{d x^2}{c}} (a d+2 b c) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-3 d x \left(b x^2 \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+7 a\right)\right)}{21 a c \sqrt{c-d x^2} (b c-a d)}","-\frac{d (e x)^{3/2}}{c e \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt[4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{\sqrt[4]{c} \sqrt{c-d x^2} (b c-a d)}+\frac{\sqrt[4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{\sqrt[4]{c} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt{b} \sqrt[4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{\sqrt{a} \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)}+\frac{\sqrt{b} \sqrt[4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{\sqrt{a} \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)}",1,"(Sqrt[e*x]*(7*(2*b*c + a*d)*x*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] - 3*d*x*(7*a + b*x^2*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a])))/(21*a*c*(b*c - a*d)*Sqrt[c - d*x^2])","C",0
893,1,147,328,0.1313749,"\int \frac{1}{\sqrt{e x} \left(a-b x^2\right) \left(c-d x^2\right)^{3/2}} \, dx","Integrate[1/(Sqrt[e*x]*(a - b*x^2)*(c - d*x^2)^(3/2)),x]","\frac{5 x \sqrt{1-\frac{d x^2}{c}} (2 b c-a d) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+b d x^3 \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-5 a d x}{5 a c \sqrt{e x} \sqrt{c-d x^2} (b c-a d)}","-\frac{d^{3/4} \sqrt{1-\frac{d x^2}{c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{c^{3/4} \sqrt{e} \sqrt{c-d x^2} (b c-a d)}-\frac{d \sqrt{e x}}{c e \sqrt{c-d x^2} (b c-a d)}+\frac{b \sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a \sqrt[4]{d} \sqrt{e} \sqrt{c-d x^2} (b c-a d)}+\frac{b \sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a \sqrt[4]{d} \sqrt{e} \sqrt{c-d x^2} (b c-a d)}",1,"(-5*a*d*x + 5*(2*b*c - a*d)*x*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + b*d*x^3*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a])/(5*a*c*(b*c - a*d)*Sqrt[e*x]*Sqrt[c - d*x^2])","C",0
894,1,198,493,0.237219,"\int \frac{1}{(e x)^{3/2} \left(a-b x^2\right) \left(c-d x^2\right)^{3/2}} \, dx","Integrate[1/((e*x)^(3/2)*(a - b*x^2)*(c - d*x^2)^(3/2)),x]","\frac{x \left(7 x^2 \sqrt{1-\frac{d x^2}{c}} \left(3 a^2 d^2-2 a b c d+2 b^2 c^2\right) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+3 b d x^4 \sqrt{1-\frac{d x^2}{c}} (2 b c-3 a d) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+21 a \left(a d \left(2 c-3 d x^2\right)-2 b c \left(c-d x^2\right)\right)\right)}{21 a^2 c^2 (e x)^{3/2} \sqrt{c-d x^2} (b c-a d)}","-\frac{b^{3/2} \sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a^{3/2} \sqrt[4]{d} e^{3/2} \sqrt{c-d x^2} (b c-a d)}+\frac{b^{3/2} \sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a^{3/2} \sqrt[4]{d} e^{3/2} \sqrt{c-d x^2} (b c-a d)}+\frac{\sqrt[4]{d} \sqrt{1-\frac{d x^2}{c}} (2 b c-3 a d) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a c^{5/4} e^{3/2} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt[4]{d} \sqrt{1-\frac{d x^2}{c}} (2 b c-3 a d) E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a c^{5/4} e^{3/2} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt{c-d x^2} (2 b c-3 a d)}{a c^2 e \sqrt{e x} (b c-a d)}-\frac{d}{c e \sqrt{e x} \sqrt{c-d x^2} (b c-a d)}",1,"(x*(21*a*(a*d*(2*c - 3*d*x^2) - 2*b*c*(c - d*x^2)) + 7*(2*b^2*c^2 - 2*a*b*c*d + 3*a^2*d^2)*x^2*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 3*b*d*(2*b*c - 3*a*d)*x^4*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(21*a^2*c^2*(b*c - a*d)*(e*x)^(3/2)*Sqrt[c - d*x^2])","C",0
895,1,197,397,0.2598498,"\int \frac{1}{(e x)^{5/2} \left(a-b x^2\right) \left(c-d x^2\right)^{3/2}} \, dx","Integrate[1/((e*x)^(5/2)*(a - b*x^2)*(c - d*x^2)^(3/2)),x]","\frac{x \left(5 x^2 \sqrt{1-\frac{d x^2}{c}} \left(-5 a^2 d^2+2 a b c d+6 b^2 c^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+b d x^4 \sqrt{1-\frac{d x^2}{c}} (5 a d-2 b c) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+5 a \left(a d \left(2 c-5 d x^2\right)-2 b c \left(c-d x^2\right)\right)\right)}{15 a^2 c^2 (e x)^{5/2} \sqrt{c-d x^2} (b c-a d)}","\frac{b^2 \sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a^2 \sqrt[4]{d} e^{5/2} \sqrt{c-d x^2} (b c-a d)}+\frac{b^2 \sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{a^2 \sqrt[4]{d} e^{5/2} \sqrt{c-d x^2} (b c-a d)}+\frac{d^{3/4} \sqrt{1-\frac{d x^2}{c}} (2 b c-5 a d) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{3 a c^{7/4} e^{5/2} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt{c-d x^2} (2 b c-5 a d)}{3 a c^2 e (e x)^{3/2} (b c-a d)}-\frac{d}{c e (e x)^{3/2} \sqrt{c-d x^2} (b c-a d)}",1,"(x*(5*a*(a*d*(2*c - 5*d*x^2) - 2*b*c*(c - d*x^2)) + 5*(6*b^2*c^2 + 2*a*b*c*d - 5*a^2*d^2)*x^2*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + b*d*(-2*b*c + 5*a*d)*x^4*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(15*a^2*c^2*(b*c - a*d)*(e*x)^(5/2)*Sqrt[c - d*x^2])","C",0
896,1,184,362,0.1807197,"\int \frac{(e x)^{7/2} \sqrt{c-d x^2}}{\left(a-b x^2\right)^2} \, dx","Integrate[((e*x)^(7/2)*Sqrt[c - d*x^2])/(a - b*x^2)^2,x]","\frac{e^3 \sqrt{e x} \left(-\left(x^2 \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} (21 a d-8 b c) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)\right)+35 a c \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+5 a \left(7 a-4 b x^2\right) \left(d x^2-c\right)\right)}{30 a b^2 \left(b x^2-a\right) \sqrt{c-d x^2}}","\frac{\sqrt[4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (8 b c-21 a d) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{6 b^3 \sqrt[4]{d} \sqrt{c-d x^2}}-\frac{\sqrt[4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (5 b c-7 a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 b^3 \sqrt[4]{d} \sqrt{c-d x^2}}-\frac{\sqrt[4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (5 b c-7 a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 b^3 \sqrt[4]{d} \sqrt{c-d x^2}}+\frac{e (e x)^{5/2} \sqrt{c-d x^2}}{2 b \left(a-b x^2\right)}+\frac{7 e^3 \sqrt{e x} \sqrt{c-d x^2}}{6 b^2}",1,"(e^3*Sqrt[e*x]*(5*a*(7*a - 4*b*x^2)*(-c + d*x^2) + 35*a*c*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] - (-8*b*c + 21*a*d)*x^2*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(30*a*b^2*(-a + b*x^2)*Sqrt[c - d*x^2])","C",0
897,1,163,413,0.1525477,"\int \frac{(e x)^{5/2} \sqrt{c-d x^2}}{\left(a-b x^2\right)^2} \, dx","Integrate[((e*x)^(5/2)*Sqrt[c - d*x^2])/(a - b*x^2)^2,x]","\frac{e (e x)^{3/2} \left(5 d x^2 \left(b x^2-a\right) \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+7 c \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-7 a \left(c-d x^2\right)\right)}{14 a b \left(b x^2-a\right) \sqrt{c-d x^2}}","\frac{\sqrt[4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (3 b c-5 a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 \sqrt{a} b^{5/2} \sqrt[4]{d} \sqrt{c-d x^2}}-\frac{\sqrt[4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (3 b c-5 a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 \sqrt{a} b^{5/2} \sqrt[4]{d} \sqrt{c-d x^2}}+\frac{e (e x)^{3/2} \sqrt{c-d x^2}}{2 b \left(a-b x^2\right)}+\frac{5 c^{3/4} \sqrt[4]{d} e^{5/2} \sqrt{1-\frac{d x^2}{c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 b^2 \sqrt{c-d x^2}}-\frac{5 c^{3/4} \sqrt[4]{d} e^{5/2} \sqrt{1-\frac{d x^2}{c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 b^2 \sqrt{c-d x^2}}",1,"(e*(e*x)^(3/2)*(-7*a*(c - d*x^2) + 7*c*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 5*d*x^2*(-a + b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(14*a*b*(-a + b*x^2)*Sqrt[c - d*x^2])","C",0
898,1,163,328,0.128197,"\int \frac{(e x)^{3/2} \sqrt{c-d x^2}}{\left(a-b x^2\right)^2} \, dx","Integrate[((e*x)^(3/2)*Sqrt[c - d*x^2])/(a - b*x^2)^2,x]","\frac{e \sqrt{e x} \left(3 d x^2 \left(b x^2-a\right) \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+5 c \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-5 a \left(c-d x^2\right)\right)}{10 a b \left(b x^2-a\right) \sqrt{c-d x^2}}","-\frac{\sqrt[4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (b c-3 a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a b^2 \sqrt[4]{d} \sqrt{c-d x^2}}-\frac{\sqrt[4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (b c-3 a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a b^2 \sqrt[4]{d} \sqrt{c-d x^2}}+\frac{e \sqrt{e x} \sqrt{c-d x^2}}{2 b \left(a-b x^2\right)}-\frac{3 \sqrt[4]{c} d^{3/4} e^{3/2} \sqrt{1-\frac{d x^2}{c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 b^2 \sqrt{c-d x^2}}",1,"(e*Sqrt[e*x]*(-5*a*(c - d*x^2) + 5*c*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + 3*d*x^2*(-a + b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(10*a*b*(-a + b*x^2)*Sqrt[c - d*x^2])","C",0
899,1,163,417,0.1345792,"\int \frac{\sqrt{e x} \sqrt{c-d x^2}}{\left(a-b x^2\right)^2} \, dx","Integrate[(Sqrt[e*x]*Sqrt[c - d*x^2])/(a - b*x^2)^2,x]","\frac{\sqrt{e x} \left(7 c x \left(b x^2-a\right) \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+3 d x^3 \left(b x^2-a\right) \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+21 a x \left(d x^2-c\right)\right)}{42 a^2 \left(b x^2-a\right) \sqrt{c-d x^2}}","-\frac{\sqrt[4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (a d+b c) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^{3/2} b^{3/2} \sqrt[4]{d} \sqrt{c-d x^2}}+\frac{\sqrt[4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (a d+b c) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^{3/2} b^{3/2} \sqrt[4]{d} \sqrt{c-d x^2}}+\frac{c^{3/4} \sqrt[4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 a b \sqrt{c-d x^2}}-\frac{c^{3/4} \sqrt[4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 a b \sqrt{c-d x^2}}+\frac{(e x)^{3/2} \sqrt{c-d x^2}}{2 a e \left(a-b x^2\right)}",1,"(Sqrt[e*x]*(21*a*x*(-c + d*x^2) + 7*c*x*(-a + b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 3*d*x^3*(-a + b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(42*a^2*(-a + b*x^2)*Sqrt[c - d*x^2])","C",0
900,1,161,335,0.1498839,"\int \frac{\sqrt{c-d x^2}}{\sqrt{e x} \left(a-b x^2\right)^2} \, dx","Integrate[Sqrt[c - d*x^2]/(Sqrt[e*x]*(a - b*x^2)^2),x]","\frac{15 c x \left(b x^2-a\right) \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+d x^3 \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+5 a x \left(d x^2-c\right)}{10 a^2 \sqrt{e x} \left(b x^2-a\right) \sqrt{c-d x^2}}","\frac{\sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (3 b c-a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^2 b \sqrt[4]{d} \sqrt{e} \sqrt{c-d x^2}}+\frac{\sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (3 b c-a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^2 b \sqrt[4]{d} \sqrt{e} \sqrt{c-d x^2}}+\frac{\sqrt[4]{c} d^{3/4} \sqrt{1-\frac{d x^2}{c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 a b \sqrt{e} \sqrt{c-d x^2}}+\frac{\sqrt{e x} \sqrt{c-d x^2}}{2 a e \left(a-b x^2\right)}",1,"(5*a*x*(-c + d*x^2) + 15*c*x*(-a + b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + d*x^3*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a])/(10*a^2*Sqrt[e*x]*(-a + b*x^2)*Sqrt[c - d*x^2])","C",0
901,1,182,444,0.182695,"\int \frac{\sqrt{c-d x^2}}{(e x)^{3/2} \left(a-b x^2\right)^2} \, dx","Integrate[Sqrt[c - d*x^2]/((e*x)^(3/2)*(a - b*x^2)^2),x]","\frac{x \left(7 x^2 \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} (8 a d-5 b c) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+15 b d x^4 \left(b x^2-a\right) \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+21 a \left(4 a-5 b x^2\right) \left(c-d x^2\right)\right)}{42 a^3 (e x)^{3/2} \left(b x^2-a\right) \sqrt{c-d x^2}}","-\frac{\sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (5 b c-3 a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^{5/2} \sqrt{b} \sqrt[4]{d} e^{3/2} \sqrt{c-d x^2}}+\frac{\sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (5 b c-3 a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^{5/2} \sqrt{b} \sqrt[4]{d} e^{3/2} \sqrt{c-d x^2}}+\frac{5 c^{3/4} \sqrt[4]{d} \sqrt{1-\frac{d x^2}{c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 a^2 e^{3/2} \sqrt{c-d x^2}}-\frac{5 c^{3/4} \sqrt[4]{d} \sqrt{1-\frac{d x^2}{c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 a^2 e^{3/2} \sqrt{c-d x^2}}-\frac{5 \sqrt{c-d x^2}}{2 a^2 e \sqrt{e x}}+\frac{\sqrt{c-d x^2}}{2 a e \sqrt{e x} \left(a-b x^2\right)}",1,"(x*(21*a*(4*a - 5*b*x^2)*(c - d*x^2) + 7*(-5*b*c + 8*a*d)*x^2*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 15*b*d*x^4*(-a + b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(42*a^3*(e*x)^(3/2)*(-a + b*x^2)*Sqrt[c - d*x^2])","C",0
902,1,181,355,0.1869625,"\int \frac{\sqrt{c-d x^2}}{(e x)^{5/2} \left(a-b x^2\right)^2} \, dx","Integrate[Sqrt[c - d*x^2]/((e*x)^(5/2)*(a - b*x^2)^2),x]","\frac{x \left(5 x^2 \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} (8 a d-21 b c) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+7 b d x^4 \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+5 a \left(4 a-7 b x^2\right) \left(c-d x^2\right)\right)}{30 a^3 (e x)^{5/2} \left(b x^2-a\right) \sqrt{c-d x^2}}","\frac{\sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (7 b c-5 a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^3 \sqrt[4]{d} e^{5/2} \sqrt{c-d x^2}}+\frac{\sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (7 b c-5 a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^3 \sqrt[4]{d} e^{5/2} \sqrt{c-d x^2}}+\frac{7 \sqrt[4]{c} d^{3/4} \sqrt{1-\frac{d x^2}{c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{6 a^2 e^{5/2} \sqrt{c-d x^2}}-\frac{7 \sqrt{c-d x^2}}{6 a^2 e (e x)^{3/2}}+\frac{\sqrt{c-d x^2}}{2 a e (e x)^{3/2} \left(a-b x^2\right)}",1,"(x*(5*a*(4*a - 7*b*x^2)*(c - d*x^2) + 5*(-21*b*c + 8*a*d)*x^2*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + 7*b*d*x^4*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(30*a^3*(e*x)^(5/2)*(-a + b*x^2)*Sqrt[c - d*x^2])","C",0
903,1,233,429,0.2971896,"\int \frac{(e x)^{7/2} \left(c-d x^2\right)^{3/2}}{\left(a-b x^2\right)^2} \, dx","Integrate[((e*x)^(7/2)*(c - d*x^2)^(3/2))/(a - b*x^2)^2,x]","\frac{e^3 \sqrt{e x} \left(x^2 \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} \left(231 a^2 d^2-259 a b c d+48 b^2 c^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+5 a \left(c-d x^2\right) \left(77 a^2 d-a b \left(57 c+44 d x^2\right)-12 b^2 x^2 \left(d x^2-3 c\right)\right)-5 a c \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} (77 a d-57 b c) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)\right)}{210 a b^3 \left(b x^2-a\right) \sqrt{c-d x^2}}","\frac{\sqrt[4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} \left(231 a^2 d^2-259 a b c d+48 b^2 c^2\right) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{42 b^4 \sqrt[4]{d} \sqrt{c-d x^2}}-\frac{\sqrt[4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (5 b c-11 a d) (b c-a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 b^4 \sqrt[4]{d} \sqrt{c-d x^2}}-\frac{\sqrt[4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (5 b c-11 a d) (b c-a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 b^4 \sqrt[4]{d} \sqrt{c-d x^2}}+\frac{e^3 \sqrt{e x} \sqrt{c-d x^2} (57 b c-77 a d)}{42 b^3}+\frac{e (e x)^{5/2} \left(c-d x^2\right)^{3/2}}{2 b \left(a-b x^2\right)}-\frac{11 d e (e x)^{5/2} \sqrt{c-d x^2}}{14 b^2}",1,"(e^3*Sqrt[e*x]*(5*a*(c - d*x^2)*(77*a^2*d - 12*b^2*x^2*(-3*c + d*x^2) - a*b*(57*c + 44*d*x^2)) - 5*a*c*(-57*b*c + 77*a*d)*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + (48*b^2*c^2 - 259*a*b*c*d + 231*a^2*d^2)*x^2*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(210*a*b^3*(-a + b*x^2)*Sqrt[c - d*x^2])","C",0
904,1,196,485,0.3095327,"\int \frac{(e x)^{5/2} \left(c-d x^2\right)^{3/2}}{\left(a-b x^2\right)^2} \, dx","Integrate[((e*x)^(5/2)*(c - d*x^2)^(3/2))/(a - b*x^2)^2,x]","\frac{e (e x)^{3/2} \left(3 d x^2 \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} (15 a d-11 b c) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-7 c \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} (9 a d-5 b c) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+7 a \left(c-d x^2\right) \left(9 a d-5 b c-4 b d x^2\right)\right)}{70 a b^2 \left(b x^2-a\right) \sqrt{c-d x^2}}","\frac{3 \sqrt[4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} \left(3 a^2 d^2-4 a b c d+b^2 c^2\right) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 \sqrt{a} b^{7/2} \sqrt[4]{d} \sqrt{c-d x^2}}-\frac{3 \sqrt[4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} \left(3 a^2 d^2-4 a b c d+b^2 c^2\right) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 \sqrt{a} b^{7/2} \sqrt[4]{d} \sqrt{c-d x^2}}+\frac{3 c^{3/4} \sqrt[4]{d} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (11 b c-15 a d) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{10 b^3 \sqrt{c-d x^2}}-\frac{3 c^{3/4} \sqrt[4]{d} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (11 b c-15 a d) E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{10 b^3 \sqrt{c-d x^2}}+\frac{e (e x)^{3/2} \left(c-d x^2\right)^{3/2}}{2 b \left(a-b x^2\right)}-\frac{9 d e (e x)^{3/2} \sqrt{c-d x^2}}{10 b^2}",1,"(e*(e*x)^(3/2)*(7*a*(c - d*x^2)*(-5*b*c + 9*a*d - 4*b*d*x^2) - 7*c*(-5*b*c + 9*a*d)*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 3*d*(-11*b*c + 15*a*d)*x^2*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(70*a*b^2*(-a + b*x^2)*Sqrt[c - d*x^2])","C",0
905,1,195,381,0.232046,"\int \frac{(e x)^{3/2} \left(c-d x^2\right)^{3/2}}{\left(a-b x^2\right)^2} \, dx","Integrate[((e*x)^(3/2)*(c - d*x^2)^(3/2))/(a - b*x^2)^2,x]","\frac{e \sqrt{e x} \left(d x^2 \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} (21 a d-17 b c) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-5 c \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} (7 a d-3 b c) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+5 a \left(c-d x^2\right) \left(7 a d-3 b c-4 b d x^2\right)\right)}{30 a b^2 \left(b x^2-a\right) \sqrt{c-d x^2}}","-\frac{\sqrt[4]{c} d^{3/4} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (17 b c-21 a d) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{6 b^3 \sqrt{c-d x^2}}-\frac{\sqrt[4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (b c-7 a d) (b c-a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a b^3 \sqrt[4]{d} \sqrt{c-d x^2}}-\frac{\sqrt[4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (b c-7 a d) (b c-a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a b^3 \sqrt[4]{d} \sqrt{c-d x^2}}+\frac{e \sqrt{e x} \left(c-d x^2\right)^{3/2}}{2 b \left(a-b x^2\right)}-\frac{7 d e \sqrt{e x} \sqrt{c-d x^2}}{6 b^2}",1,"(e*Sqrt[e*x]*(5*a*(c - d*x^2)*(-3*b*c + 7*a*d - 4*b*d*x^2) - 5*c*(-3*b*c + 7*a*d)*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + d*(-17*b*c + 21*a*d)*x^2*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(30*a*b^2*(-a + b*x^2)*Sqrt[c - d*x^2])","C",0
906,1,189,474,0.2312892,"\int \frac{\sqrt{e x} \left(c-d x^2\right)^{3/2}}{\left(a-b x^2\right)^2} \, dx","Integrate[(Sqrt[e*x]*(c - d*x^2)^(3/2))/(a - b*x^2)^2,x]","\frac{\sqrt{e x} \left(7 c x \left(b x^2-a\right) \sqrt{1-\frac{d x^2}{c}} (3 a d+b c) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+3 d x^3 \left(b x^2-a\right) \sqrt{1-\frac{d x^2}{c}} (b c-5 a d) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+21 a x \left(c-d x^2\right) (a d-b c)\right)}{42 a^2 b \left(b x^2-a\right) \sqrt{c-d x^2}}","-\frac{\sqrt[4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} \left(-5 a^2 d^2+4 a b c d+b^2 c^2\right) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^{3/2} b^{5/2} \sqrt[4]{d} \sqrt{c-d x^2}}+\frac{\sqrt[4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} \left(-5 a^2 d^2+4 a b c d+b^2 c^2\right) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^{3/2} b^{5/2} \sqrt[4]{d} \sqrt{c-d x^2}}+\frac{c^{3/4} \sqrt[4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (b c-5 a d) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 a b^2 \sqrt{c-d x^2}}-\frac{c^{3/4} \sqrt[4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (b c-5 a d) E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 a b^2 \sqrt{c-d x^2}}+\frac{(e x)^{3/2} \sqrt{c-d x^2} (b c-a d)}{2 a b e \left(a-b x^2\right)}",1,"(Sqrt[e*x]*(21*a*(-(b*c) + a*d)*x*(c - d*x^2) + 7*c*(b*c + 3*a*d)*x*(-a + b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 3*d*(b*c - 5*a*d)*x^3*(-a + b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(42*a^2*b*(-a + b*x^2)*Sqrt[c - d*x^2])","C",0
907,1,187,366,0.1971144,"\int \frac{\left(c-d x^2\right)^{3/2}}{\sqrt{e x} \left(a-b x^2\right)^2} \, dx","Integrate[(c - d*x^2)^(3/2)/(Sqrt[e*x]*(a - b*x^2)^2),x]","\frac{5 c x \left(b x^2-a\right) \sqrt{1-\frac{d x^2}{c}} (a d+3 b c) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+d x^3 \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} (3 a d+b c) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+5 a x \left(c-d x^2\right) (a d-b c)}{10 a^2 b \sqrt{e x} \left(b x^2-a\right) \sqrt{c-d x^2}}","\frac{3 \sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (a d+b c) (b c-a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^2 b^2 \sqrt[4]{d} \sqrt{e} \sqrt{c-d x^2}}+\frac{3 \sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (a d+b c) (b c-a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^2 b^2 \sqrt[4]{d} \sqrt{e} \sqrt{c-d x^2}}+\frac{\sqrt[4]{c} d^{3/4} \sqrt{1-\frac{d x^2}{c}} (3 a d+b c) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 a b^2 \sqrt{e} \sqrt{c-d x^2}}+\frac{\sqrt{e x} \sqrt{c-d x^2} (b c-a d)}{2 a b e \left(a-b x^2\right)}",1,"(5*a*(-(b*c) + a*d)*x*(c - d*x^2) + 5*c*(3*b*c + a*d)*x*(-a + b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + d*(b*c + 3*a*d)*x^3*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a])/(10*a^2*b*Sqrt[e*x]*(-a + b*x^2)*Sqrt[c - d*x^2])","C",0
908,1,197,519,0.3106207,"\int \frac{\left(c-d x^2\right)^{3/2}}{(e x)^{3/2} \left(a-b x^2\right)^2} \, dx","Integrate[(c - d*x^2)^(3/2)/((e*x)^(3/2)*(a - b*x^2)^2),x]","\frac{x \left(7 c x^2 \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} (9 a d-5 b c) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+3 d x^4 \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} (a d-5 b c) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+21 a \left(c-d x^2\right) \left(4 a c+a d x^2-5 b c x^2\right)\right)}{42 a^3 (e x)^{3/2} \left(b x^2-a\right) \sqrt{c-d x^2}}","\frac{c^{3/4} \sqrt[4]{d} \sqrt{1-\frac{d x^2}{c}} (5 b c-a d) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 a^2 b e^{3/2} \sqrt{c-d x^2}}-\frac{c^{3/4} \sqrt[4]{d} \sqrt{1-\frac{d x^2}{c}} (5 b c-a d) E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 a^2 b e^{3/2} \sqrt{c-d x^2}}-\frac{\sqrt{c-d x^2} (5 b c-a d)}{2 a^2 b e \sqrt{e x}}-\frac{\sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} \left(-a^2 d^2-4 a b c d+5 b^2 c^2\right) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^{5/2} b^{3/2} \sqrt[4]{d} e^{3/2} \sqrt{c-d x^2}}+\frac{\sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} \left(-a^2 d^2-4 a b c d+5 b^2 c^2\right) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^{5/2} b^{3/2} \sqrt[4]{d} e^{3/2} \sqrt{c-d x^2}}+\frac{\sqrt{c-d x^2} (b c-a d)}{2 a b e \sqrt{e x} \left(a-b x^2\right)}",1,"(x*(21*a*(c - d*x^2)*(4*a*c - 5*b*c*x^2 + a*d*x^2) + 7*c*(-5*b*c + 9*a*d)*x^2*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 3*d*(-5*b*c + a*d)*x^4*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(42*a^3*(e*x)^(3/2)*(-a + b*x^2)*Sqrt[c - d*x^2])","C",0
909,1,199,412,0.234634,"\int \frac{\left(c-d x^2\right)^{3/2}}{(e x)^{5/2} \left(a-b x^2\right)^2} \, dx","Integrate[(c - d*x^2)^(3/2)/((e*x)^(5/2)*(a - b*x^2)^2),x]","\frac{x \left(5 c x^2 \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} (17 a d-21 b c) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-d x^4 \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} (3 a d-7 b c) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+5 a \left(c-d x^2\right) \left(4 a c+3 a d x^2-7 b c x^2\right)\right)}{30 a^3 (e x)^{5/2} \left(b x^2-a\right) \sqrt{c-d x^2}}","\frac{\sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d) (7 b c-a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^3 b \sqrt[4]{d} e^{5/2} \sqrt{c-d x^2}}+\frac{\sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d) (7 b c-a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^3 b \sqrt[4]{d} e^{5/2} \sqrt{c-d x^2}}+\frac{\sqrt[4]{c} d^{3/4} \sqrt{1-\frac{d x^2}{c}} (7 b c-3 a d) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{6 a^2 b e^{5/2} \sqrt{c-d x^2}}-\frac{\sqrt{c-d x^2} (7 b c-3 a d)}{6 a^2 b e (e x)^{3/2}}+\frac{\sqrt{c-d x^2} (b c-a d)}{2 a b e (e x)^{3/2} \left(a-b x^2\right)}",1,"(x*(5*a*(c - d*x^2)*(4*a*c - 7*b*c*x^2 + 3*a*d*x^2) + 5*c*(-21*b*c + 17*a*d)*x^2*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] - d*(-7*b*c + 3*a*d)*x^4*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(30*a^3*(e*x)^(5/2)*(-a + b*x^2)*Sqrt[c - d*x^2])","C",0
910,1,184,484,0.208813,"\int \frac{(e x)^{9/2}}{\left(a-b x^2\right)^2 \sqrt{c-d x^2}} \, dx","Integrate[(e*x)^(9/2)/((a - b*x^2)^2*Sqrt[c - d*x^2]),x]","\frac{e^3 (e x)^{3/2} \left(-7 a^2 \left(c-d x^2\right)+7 a c \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-x^2 \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} (5 a d-4 b c) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)\right)}{14 a b \left(a-b x^2\right) \sqrt{c-d x^2} (a d-b c)}","\frac{\sqrt{a} \sqrt[4]{c} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (7 b c-5 a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 b^{5/2} \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt{a} \sqrt[4]{c} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (7 b c-5 a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 b^{5/2} \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)}-\frac{c^{3/4} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (4 b c-5 a d) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 b^2 d^{3/4} \sqrt{c-d x^2} (b c-a d)}+\frac{c^{3/4} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (4 b c-5 a d) E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 b^2 d^{3/4} \sqrt{c-d x^2} (b c-a d)}+\frac{a e^3 (e x)^{3/2} \sqrt{c-d x^2}}{2 b \left(a-b x^2\right) (b c-a d)}",1,"(e^3*(e*x)^(3/2)*(-7*a^2*(c - d*x^2) + 7*a*c*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] - (-4*b*c + 5*a*d)*x^2*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(14*a*b*(-(b*c) + a*d)*(a - b*x^2)*Sqrt[c - d*x^2])","C",0
911,1,184,376,0.1989624,"\int \frac{(e x)^{7/2}}{\left(a-b x^2\right)^2 \sqrt{c-d x^2}} \, dx","Integrate[(e*x)^(7/2)/((a - b*x^2)^2*Sqrt[c - d*x^2]),x]","\frac{e^3 \sqrt{e x} \left(-5 a^2 \left(c-d x^2\right)+5 a c \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-x^2 \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} (3 a d-4 b c) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)\right)}{10 a b \left(a-b x^2\right) \sqrt{c-d x^2} (a d-b c)}","\frac{\sqrt[4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (4 b c-3 a d) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 b^2 \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt[4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (5 b c-3 a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 b^2 \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt[4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (5 b c-3 a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 b^2 \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)}+\frac{a e^3 \sqrt{e x} \sqrt{c-d x^2}}{2 b \left(a-b x^2\right) (b c-a d)}",1,"(e^3*Sqrt[e*x]*(-5*a^2*(c - d*x^2) + 5*a*c*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] - (-4*b*c + 3*a*d)*x^2*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(10*a*b*(-(b*c) + a*d)*(a - b*x^2)*Sqrt[c - d*x^2])","C",0
912,1,168,460,0.1586648,"\int \frac{(e x)^{5/2}}{\left(a-b x^2\right)^2 \sqrt{c-d x^2}} \, dx","Integrate[(e*x)^(5/2)/((a - b*x^2)^2*Sqrt[c - d*x^2]),x]","\frac{e (e x)^{3/2} \left(d x^2 \left(b x^2-a\right) \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+7 c \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-7 a \left(c-d x^2\right)\right)}{14 a \left(a-b x^2\right) \sqrt{c-d x^2} (a d-b c)}","\frac{\sqrt[4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (3 b c-a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 \sqrt{a} b^{3/2} \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt[4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (3 b c-a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 \sqrt{a} b^{3/2} \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)}+\frac{c^{3/4} \sqrt[4]{d} e^{5/2} \sqrt{1-\frac{d x^2}{c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 b \sqrt{c-d x^2} (b c-a d)}-\frac{c^{3/4} \sqrt[4]{d} e^{5/2} \sqrt{1-\frac{d x^2}{c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 b \sqrt{c-d x^2} (b c-a d)}+\frac{e (e x)^{3/2} \sqrt{c-d x^2}}{2 \left(a-b x^2\right) (b c-a d)}",1,"(e*(e*x)^(3/2)*(-7*a*(c - d*x^2) + 7*c*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + d*x^2*(-a + b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(14*a*(-(b*c) + a*d)*(a - b*x^2)*Sqrt[c - d*x^2])","C",0
913,1,169,363,0.1344958,"\int \frac{(e x)^{3/2}}{\left(a-b x^2\right)^2 \sqrt{c-d x^2}} \, dx","Integrate[(e*x)^(3/2)/((a - b*x^2)^2*Sqrt[c - d*x^2]),x]","-\frac{e \sqrt{e x} \left(d x^2 \left(b x^2-a\right) \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+5 c \left(b x^2-a\right) \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+5 a \left(c-d x^2\right)\right)}{10 a \left(a-b x^2\right) \sqrt{c-d x^2} (a d-b c)}","\frac{\sqrt[4]{c} d^{3/4} e^{3/2} \sqrt{1-\frac{d x^2}{c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 b \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt[4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (a d+b c) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a b \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt[4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (a d+b c) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a b \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)}+\frac{e \sqrt{e x} \sqrt{c-d x^2}}{2 \left(a-b x^2\right) (b c-a d)}",1,"-1/10*(e*Sqrt[e*x]*(5*a*(c - d*x^2) + 5*c*(-a + b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + d*x^2*(-a + b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(a*(-(b*c) + a*d)*(a - b*x^2)*Sqrt[c - d*x^2])","C",0
914,1,181,464,0.2368721,"\int \frac{\sqrt{e x}}{\left(a-b x^2\right)^2 \sqrt{c-d x^2}} \, dx","Integrate[Sqrt[e*x]/((a - b*x^2)^2*Sqrt[c - d*x^2]),x]","\frac{x \sqrt{e x} \left(7 \left(b x^2-a\right) \sqrt{1-\frac{d x^2}{c}} (b c-4 a d) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+3 b \left(d x^2 \left(b x^2-a\right) \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-7 a \left(c-d x^2\right)\right)\right)}{42 a^2 \left(b x^2-a\right) \sqrt{c-d x^2} (b c-a d)}","-\frac{\sqrt[4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (b c-3 a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^{3/2} \sqrt{b} \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)}+\frac{\sqrt[4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (b c-3 a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^{3/2} \sqrt{b} \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)}+\frac{c^{3/4} \sqrt[4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 a \sqrt{c-d x^2} (b c-a d)}-\frac{c^{3/4} \sqrt[4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 a \sqrt{c-d x^2} (b c-a d)}+\frac{b (e x)^{3/2} \sqrt{c-d x^2}}{2 a e \left(a-b x^2\right) (b c-a d)}",1,"(x*Sqrt[e*x]*(7*(b*c - 4*a*d)*(-a + b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 3*b*(-7*a*(c - d*x^2) + d*x^2*(-a + b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a])))/(42*a^2*(b*c - a*d)*(-a + b*x^2)*Sqrt[c - d*x^2])","C",0
915,1,180,367,0.190333,"\int \frac{1}{\sqrt{e x} \left(a-b x^2\right)^2 \sqrt{c-d x^2}} \, dx","Integrate[1/(Sqrt[e*x]*(a - b*x^2)^2*Sqrt[c - d*x^2]),x]","\frac{5 x \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} (4 a d-3 b c) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+b d x^3 \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+5 a b x \left(d x^2-c\right)}{10 a^2 \sqrt{e x} \left(b x^2-a\right) \sqrt{c-d x^2} (b c-a d)}","\frac{\sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (3 b c-5 a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^2 \sqrt[4]{d} \sqrt{e} \sqrt{c-d x^2} (b c-a d)}+\frac{\sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (3 b c-5 a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^2 \sqrt[4]{d} \sqrt{e} \sqrt{c-d x^2} (b c-a d)}+\frac{\sqrt[4]{c} d^{3/4} \sqrt{1-\frac{d x^2}{c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 a \sqrt{e} \sqrt{c-d x^2} (b c-a d)}+\frac{b \sqrt{e x} \sqrt{c-d x^2}}{2 a e \left(a-b x^2\right) (b c-a d)}",1,"(5*a*b*x*(-c + d*x^2) + 5*(-3*b*c + 4*a*d)*x*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + b*d*x^3*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a])/(10*a^2*(b*c - a*d)*Sqrt[e*x]*(-a + b*x^2)*Sqrt[c - d*x^2])","C",0
916,1,235,535,0.3246703,"\int \frac{1}{(e x)^{3/2} \left(a-b x^2\right)^2 \sqrt{c-d x^2}} \, dx","Integrate[1/((e*x)^(3/2)*(a - b*x^2)^2*Sqrt[c - d*x^2]),x]","\frac{x \left(7 x^2 \left(b x^2-a\right) \sqrt{1-\frac{d x^2}{c}} \left(4 a^2 d^2-12 a b c d+5 b^2 c^2\right) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-21 a \left(c-d x^2\right) \left(4 a^2 d-4 a b \left(c+d x^2\right)+5 b^2 c x^2\right)+3 b d x^4 \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} (4 a d-5 b c) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)\right)}{42 a^3 c (e x)^{3/2} \left(b x^2-a\right) \sqrt{c-d x^2} (b c-a d)}","-\frac{\sqrt{b} \sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (5 b c-7 a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^{5/2} \sqrt[4]{d} e^{3/2} \sqrt{c-d x^2} (b c-a d)}+\frac{\sqrt{b} \sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (5 b c-7 a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^{5/2} \sqrt[4]{d} e^{3/2} \sqrt{c-d x^2} (b c-a d)}+\frac{\sqrt[4]{d} \sqrt{1-\frac{d x^2}{c}} (5 b c-4 a d) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 a^2 \sqrt[4]{c} e^{3/2} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt[4]{d} \sqrt{1-\frac{d x^2}{c}} (5 b c-4 a d) E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 a^2 \sqrt[4]{c} e^{3/2} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt{c-d x^2} (5 b c-4 a d)}{2 a^2 c e \sqrt{e x} (b c-a d)}+\frac{b \sqrt{c-d x^2}}{2 a e \sqrt{e x} \left(a-b x^2\right) (b c-a d)}",1,"(x*(-21*a*(c - d*x^2)*(4*a^2*d + 5*b^2*c*x^2 - 4*a*b*(c + d*x^2)) + 7*(5*b^2*c^2 - 12*a*b*c*d + 4*a^2*d^2)*x^2*(-a + b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 3*b*d*(-5*b*c + 4*a*d)*x^4*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(42*a^3*c*(b*c - a*d)*(e*x)^(3/2)*(-a + b*x^2)*Sqrt[c - d*x^2])","C",0
917,1,234,429,0.3169461,"\int \frac{1}{(e x)^{5/2} \left(a-b x^2\right)^2 \sqrt{c-d x^2}} \, dx","Integrate[1/((e*x)^(5/2)*(a - b*x^2)^2*Sqrt[c - d*x^2]),x]","\frac{x \left(5 x^2 \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} \left(4 a^2 d^2+20 a b c d-21 b^2 c^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-5 a \left(c-d x^2\right) \left(4 a^2 d-4 a b \left(c+d x^2\right)+7 b^2 c x^2\right)-b d x^4 \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} (4 a d-7 b c) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)\right)}{30 a^3 c (e x)^{5/2} \left(b x^2-a\right) \sqrt{c-d x^2} (b c-a d)}","\frac{b \sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (7 b c-9 a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^3 \sqrt[4]{d} e^{5/2} \sqrt{c-d x^2} (b c-a d)}+\frac{b \sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (7 b c-9 a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^3 \sqrt[4]{d} e^{5/2} \sqrt{c-d x^2} (b c-a d)}+\frac{d^{3/4} \sqrt{1-\frac{d x^2}{c}} (7 b c-4 a d) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{6 a^2 c^{3/4} e^{5/2} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt{c-d x^2} (7 b c-4 a d)}{6 a^2 c e (e x)^{3/2} (b c-a d)}+\frac{b \sqrt{c-d x^2}}{2 a e (e x)^{3/2} \left(a-b x^2\right) (b c-a d)}",1,"(x*(-5*a*(c - d*x^2)*(4*a^2*d + 7*b^2*c*x^2 - 4*a*b*(c + d*x^2)) + 5*(-21*b^2*c^2 + 20*a*b*c*d + 4*a^2*d^2)*x^2*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] - b*d*(-7*b*c + 4*a*d)*x^4*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(30*a^3*c*(b*c - a*d)*(e*x)^(5/2)*(-a + b*x^2)*Sqrt[c - d*x^2])","C",0
918,1,189,529,0.228476,"\int \frac{(e x)^{9/2}}{\left(a-b x^2\right)^2 \left(c-d x^2\right)^{3/2}} \, dx","Integrate[(e*x)^(9/2)/((a - b*x^2)^2*(c - d*x^2)^(3/2)),x]","\frac{e^3 (e x)^{3/2} \left(x^2 \left(b x^2-a\right) \sqrt{1-\frac{d x^2}{c}} (a d+2 b c) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+21 a c \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+7 a \left(-3 a c+a d x^2+2 b c x^2\right)\right)}{14 a \left(b x^2-a\right) \sqrt{c-d x^2} (b c-a d)^2}","\frac{\sqrt{a} \sqrt[4]{c} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (7 b c-a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 b^{3/2} \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)^2}-\frac{\sqrt{a} \sqrt[4]{c} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (7 b c-a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 b^{3/2} \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)^2}+\frac{c^{3/4} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (a d+2 b c) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 b d^{3/4} \sqrt{c-d x^2} (b c-a d)^2}-\frac{c^{3/4} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (a d+2 b c) E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 b d^{3/4} \sqrt{c-d x^2} (b c-a d)^2}+\frac{e^3 (e x)^{3/2} (a d+2 b c)}{2 b \sqrt{c-d x^2} (b c-a d)^2}+\frac{a e^3 (e x)^{3/2}}{2 b \left(a-b x^2\right) \sqrt{c-d x^2} (b c-a d)}",1,"(e^3*(e*x)^(3/2)*(7*a*(-3*a*c + 2*b*c*x^2 + a*d*x^2) + 21*a*c*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + (2*b*c + a*d)*x^2*(-a + b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(14*a*(b*c - a*d)^2*(-a + b*x^2)*Sqrt[c - d*x^2])","C",0
919,1,191,420,0.2130866,"\int \frac{(e x)^{7/2}}{\left(a-b x^2\right)^2 \left(c-d x^2\right)^{3/2}} \, dx","Integrate[(e*x)^(7/2)/((a - b*x^2)^2*(c - d*x^2)^(3/2)),x]","-\frac{e^3 \sqrt{e x} \left(x^2 \left(b x^2-a\right) \sqrt{1-\frac{d x^2}{c}} (a d+2 b c) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+15 a c \left(b x^2-a\right) \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+5 a \left(3 a c-a d x^2-2 b c x^2\right)\right)}{10 a \left(b x^2-a\right) \sqrt{c-d x^2} (b c-a d)^2}","\frac{\sqrt[4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (a d+2 b c) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 b \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)^2}-\frac{\sqrt[4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (a d+5 b c) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 b \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)^2}-\frac{\sqrt[4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (a d+5 b c) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 b \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)^2}+\frac{e^3 \sqrt{e x} (a d+2 b c)}{2 b \sqrt{c-d x^2} (b c-a d)^2}+\frac{a e^3 \sqrt{e x}}{2 b \left(a-b x^2\right) \sqrt{c-d x^2} (b c-a d)}",1,"-1/10*(e^3*Sqrt[e*x]*(5*a*(3*a*c - 2*b*c*x^2 - a*d*x^2) + 15*a*c*(-a + b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + (2*b*c + a*d)*x^2*(-a + b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(a*(b*c - a*d)^2*(-a + b*x^2)*Sqrt[c - d*x^2])","C",0
920,1,185,485,0.2258233,"\int \frac{(e x)^{5/2}}{\left(a-b x^2\right)^2 \left(c-d x^2\right)^{3/2}} \, dx","Integrate[(e*x)^(5/2)/((a - b*x^2)^2*(c - d*x^2)^(3/2)),x]","\frac{e (e x)^{3/2} \left(3 b d x^2 \left(b x^2-a\right) \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+7 \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} (2 a d+b c) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-7 a \left(2 a d+b \left(c-3 d x^2\right)\right)\right)}{14 a \left(b x^2-a\right) \sqrt{c-d x^2} (b c-a d)^2}","\frac{3 c^{3/4} \sqrt[4]{d} e^{5/2} \sqrt{1-\frac{d x^2}{c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 \sqrt{c-d x^2} (b c-a d)^2}-\frac{3 c^{3/4} \sqrt[4]{d} e^{5/2} \sqrt{1-\frac{d x^2}{c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 \sqrt{c-d x^2} (b c-a d)^2}+\frac{3 \sqrt[4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (a d+b c) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 \sqrt{a} \sqrt{b} \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)^2}-\frac{3 \sqrt[4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (a d+b c) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 \sqrt{a} \sqrt{b} \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)^2}+\frac{3 d e (e x)^{3/2}}{2 \sqrt{c-d x^2} (b c-a d)^2}+\frac{e (e x)^{3/2}}{2 \left(a-b x^2\right) \sqrt{c-d x^2} (b c-a d)}",1,"(e*(e*x)^(3/2)*(-7*a*(2*a*d + b*(c - 3*d*x^2)) + 7*(b*c + 2*a*d)*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 3*b*d*x^2*(-a + b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(14*a*(b*c - a*d)^2*(-a + b*x^2)*Sqrt[c - d*x^2])","C",0
921,1,186,391,0.2141756,"\int \frac{(e x)^{3/2}}{\left(a-b x^2\right)^2 \left(c-d x^2\right)^{3/2}} \, dx","Integrate[(e*x)^(3/2)/((a - b*x^2)^2*(c - d*x^2)^(3/2)),x]","-\frac{e \sqrt{e x} \left(3 b d x^2 \left(b x^2-a\right) \sqrt{1-\frac{d x^2}{c}} F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+5 \left(b x^2-a\right) \sqrt{1-\frac{d x^2}{c}} (2 a d+b c) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+5 a \left(2 a d+b \left(c-3 d x^2\right)\right)\right)}{10 a \left(b x^2-a\right) \sqrt{c-d x^2} (b c-a d)^2}","\frac{3 \sqrt[4]{c} d^{3/4} e^{3/2} \sqrt{1-\frac{d x^2}{c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 \sqrt{c-d x^2} (b c-a d)^2}-\frac{\sqrt[4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (5 a d+b c) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)^2}-\frac{\sqrt[4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (5 a d+b c) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)^2}+\frac{3 d e \sqrt{e x}}{2 \sqrt{c-d x^2} (b c-a d)^2}+\frac{e \sqrt{e x}}{2 \left(a-b x^2\right) \sqrt{c-d x^2} (b c-a d)}",1,"-1/10*(e*Sqrt[e*x]*(5*a*(2*a*d + b*(c - 3*d*x^2)) + 5*(b*c + 2*a*d)*(-a + b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + 3*b*d*x^2*(-a + b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(a*(b*c - a*d)^2*(-a + b*x^2)*Sqrt[c - d*x^2])","C",0
922,1,230,531,0.3445762,"\int \frac{\sqrt{e x}}{\left(a-b x^2\right)^2 \left(c-d x^2\right)^{3/2}} \, dx","Integrate[Sqrt[e*x]/((a - b*x^2)^2*(c - d*x^2)^(3/2)),x]","\frac{\sqrt{e x} \left(7 x \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} \left(2 a^2 d^2+8 a b c d-b^2 c^2\right) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+21 a x \left(-2 a^2 d^2+2 a b d^2 x^2+b^2 c \left(d x^2-c\right)\right)+3 b d x^3 \left(b x^2-a\right) \sqrt{1-\frac{d x^2}{c}} (2 a d+b c) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)\right)}{42 a^2 c \left(b x^2-a\right) \sqrt{c-d x^2} (b c-a d)^2}","-\frac{\sqrt{b} \sqrt[4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (b c-7 a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^{3/2} \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)^2}+\frac{\sqrt{b} \sqrt[4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (b c-7 a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^{3/2} \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)^2}+\frac{d (e x)^{3/2} (2 a d+b c)}{2 a c e \sqrt{c-d x^2} (b c-a d)^2}+\frac{b (e x)^{3/2}}{2 a e \left(a-b x^2\right) \sqrt{c-d x^2} (b c-a d)}+\frac{\sqrt[4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (2 a d+b c) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 a \sqrt[4]{c} \sqrt{c-d x^2} (b c-a d)^2}-\frac{\sqrt[4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (2 a d+b c) E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 a \sqrt[4]{c} \sqrt{c-d x^2} (b c-a d)^2}",1,"(Sqrt[e*x]*(21*a*x*(-2*a^2*d^2 + 2*a*b*d^2*x^2 + b^2*c*(-c + d*x^2)) + 7*(-(b^2*c^2) + 8*a*b*c*d + 2*a^2*d^2)*x*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 3*b*d*(b*c + 2*a*d)*x^3*(-a + b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(42*a^2*c*(b*c - a*d)^2*(-a + b*x^2)*Sqrt[c - d*x^2])","C",0
923,1,229,426,0.2920767,"\int \frac{1}{\sqrt{e x} \left(a-b x^2\right)^2 \left(c-d x^2\right)^{3/2}} \, dx","Integrate[1/(Sqrt[e*x]*(a - b*x^2)^2*(c - d*x^2)^(3/2)),x]","\frac{5 x \left(b x^2-a\right) \sqrt{1-\frac{d x^2}{c}} \left(2 a^2 d^2-8 a b c d+3 b^2 c^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+5 a x \left(-2 a^2 d^2+2 a b d^2 x^2+b^2 c \left(d x^2-c\right)\right)+b d x^3 \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} (2 a d+b c) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)}{10 a^2 c \sqrt{e x} \left(b x^2-a\right) \sqrt{c-d x^2} (b c-a d)^2}","\frac{3 b \sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-3 a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^2 \sqrt[4]{d} \sqrt{e} \sqrt{c-d x^2} (b c-a d)^2}+\frac{3 b \sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-3 a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^2 \sqrt[4]{d} \sqrt{e} \sqrt{c-d x^2} (b c-a d)^2}+\frac{d^{3/4} \sqrt{1-\frac{d x^2}{c}} (2 a d+b c) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 a c^{3/4} \sqrt{e} \sqrt{c-d x^2} (b c-a d)^2}+\frac{b \sqrt{e x}}{2 a e \left(a-b x^2\right) \sqrt{c-d x^2} (b c-a d)}+\frac{d \sqrt{e x} (2 a d+b c)}{2 a c e \sqrt{c-d x^2} (b c-a d)^2}",1,"(5*a*x*(-2*a^2*d^2 + 2*a*b*d^2*x^2 + b^2*c*(-c + d*x^2)) + 5*(3*b^2*c^2 - 8*a*b*c*d + 2*a^2*d^2)*x*(-a + b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + b*d*(b*c + 2*a*d)*x^3*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a])/(10*a^2*c*(b*c - a*d)^2*Sqrt[e*x]*(-a + b*x^2)*Sqrt[c - d*x^2])","C",0
924,1,319,628,0.5541122,"\int \frac{1}{(e x)^{3/2} \left(a-b x^2\right)^2 \left(c-d x^2\right)^{3/2}} \, dx","Integrate[1/((e*x)^(3/2)*(a - b*x^2)^2*(c - d*x^2)^(3/2)),x]","\frac{x \left(3 b d x^4 \left(b x^2-a\right) \sqrt{1-\frac{d x^2}{c}} \left(6 a^2 d^2-8 a b c d+5 b^2 c^2\right) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+7 x^2 \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} \left(6 a^3 d^3-8 a^2 b c d^2+16 a b^2 c^2 d-5 b^3 c^3\right) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+21 a \left(2 a^3 d^2 \left(2 c-3 d x^2\right)+2 a^2 b d \left(-4 c^2+2 c d x^2+3 d^2 x^4\right)+4 a b^2 c \left(c^2+c d x^2-2 d^2 x^4\right)-5 b^3 c^2 x^2 \left(c-d x^2\right)\right)\right)}{42 a^3 c^2 (e x)^{3/2} \left(b x^2-a\right) \sqrt{c-d x^2} (b c-a d)^2}","-\frac{b^{3/2} \sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (5 b c-11 a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^{5/2} \sqrt[4]{d} e^{3/2} \sqrt{c-d x^2} (b c-a d)^2}+\frac{b^{3/2} \sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (5 b c-11 a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^{5/2} \sqrt[4]{d} e^{3/2} \sqrt{c-d x^2} (b c-a d)^2}-\frac{\sqrt{c-d x^2} \left(6 a^2 d^2-8 a b c d+5 b^2 c^2\right)}{2 a^2 c^2 e \sqrt{e x} (b c-a d)^2}+\frac{\sqrt[4]{d} \sqrt{1-\frac{d x^2}{c}} \left(6 a^2 d^2-8 a b c d+5 b^2 c^2\right) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 a^2 c^{5/4} e^{3/2} \sqrt{c-d x^2} (b c-a d)^2}-\frac{\sqrt[4]{d} \sqrt{1-\frac{d x^2}{c}} \left(6 a^2 d^2-8 a b c d+5 b^2 c^2\right) E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 a^2 c^{5/4} e^{3/2} \sqrt{c-d x^2} (b c-a d)^2}+\frac{b}{2 a e \sqrt{e x} \left(a-b x^2\right) \sqrt{c-d x^2} (b c-a d)}+\frac{d (2 a d+b c)}{2 a c e \sqrt{e x} \sqrt{c-d x^2} (b c-a d)^2}",1,"(x*(21*a*(2*a^3*d^2*(2*c - 3*d*x^2) - 5*b^3*c^2*x^2*(c - d*x^2) + 4*a*b^2*c*(c^2 + c*d*x^2 - 2*d^2*x^4) + 2*a^2*b*d*(-4*c^2 + 2*c*d*x^2 + 3*d^2*x^4)) + 7*(-5*b^3*c^3 + 16*a*b^2*c^2*d - 8*a^2*b*c*d^2 + 6*a^3*d^3)*x^2*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 3*b*d*(5*b^2*c^2 - 8*a*b*c*d + 6*a^2*d^2)*x^4*(-a + b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(42*a^3*c^2*(b*c - a*d)^2*(e*x)^(3/2)*(-a + b*x^2)*Sqrt[c - d*x^2])","C",0
925,1,318,512,0.5039678,"\int \frac{1}{(e x)^{5/2} \left(a-b x^2\right)^2 \left(c-d x^2\right)^{3/2}} \, dx","Integrate[1/((e*x)^(5/2)*(a - b*x^2)^2*(c - d*x^2)^(3/2)),x]","\frac{x \left(b d x^4 \left(a-b x^2\right) \sqrt{1-\frac{d x^2}{c}} \left(10 a^2 d^2-8 a b c d+7 b^2 c^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+5 x^2 \left(b x^2-a\right) \sqrt{1-\frac{d x^2}{c}} \left(10 a^3 d^3-8 a^2 b c d^2-32 a b^2 c^2 d+21 b^3 c^3\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+5 a \left(2 a^3 d^2 \left(2 c-5 d x^2\right)+2 a^2 b d \left(-4 c^2+2 c d x^2+5 d^2 x^4\right)+4 a b^2 c \left(c^2+c d x^2-2 d^2 x^4\right)-7 b^3 c^2 x^2 \left(c-d x^2\right)\right)\right)}{30 a^3 c^2 (e x)^{5/2} \left(b x^2-a\right) \sqrt{c-d x^2} (b c-a d)^2}","\frac{b^2 \sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (7 b c-13 a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^3 \sqrt[4]{d} e^{5/2} \sqrt{c-d x^2} (b c-a d)^2}+\frac{b^2 \sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (7 b c-13 a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^3 \sqrt[4]{d} e^{5/2} \sqrt{c-d x^2} (b c-a d)^2}-\frac{\sqrt{c-d x^2} \left(10 a^2 d^2-8 a b c d+7 b^2 c^2\right)}{6 a^2 c^2 e (e x)^{3/2} (b c-a d)^2}+\frac{d^{3/4} \sqrt{1-\frac{d x^2}{c}} \left(10 a^2 d^2-8 a b c d+7 b^2 c^2\right) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{6 a^2 c^{7/4} e^{5/2} \sqrt{c-d x^2} (b c-a d)^2}+\frac{b}{2 a e (e x)^{3/2} \left(a-b x^2\right) \sqrt{c-d x^2} (b c-a d)}+\frac{d (2 a d+b c)}{2 a c e (e x)^{3/2} \sqrt{c-d x^2} (b c-a d)^2}",1,"(x*(5*a*(2*a^3*d^2*(2*c - 5*d*x^2) - 7*b^3*c^2*x^2*(c - d*x^2) + 4*a*b^2*c*(c^2 + c*d*x^2 - 2*d^2*x^4) + 2*a^2*b*d*(-4*c^2 + 2*c*d*x^2 + 5*d^2*x^4)) + 5*(21*b^3*c^3 - 32*a*b^2*c^2*d - 8*a^2*b*c*d^2 + 10*a^3*d^3)*x^2*(-a + b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + b*d*(7*b^2*c^2 - 8*a*b*c*d + 10*a^2*d^2)*x^4*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(30*a^3*c^2*(b*c - a*d)^2*(e*x)^(5/2)*(-a + b*x^2)*Sqrt[c - d*x^2])","C",0
926,1,256,568,0.5074074,"\int \frac{(e x)^{9/2}}{\left(a-b x^2\right)^2 \left(c-d x^2\right)^{5/2}} \, dx","Integrate[(e*x)^(9/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)),x]","-\frac{e^3 (e x)^{3/2} \left(7 a \left(a^2 d \left(7 c-9 d x^2\right)+4 a b \left(2 c^2-4 c d x^2+3 d^2 x^4\right)+b^2 c x^2 \left(3 d x^2-5 c\right)\right)+3 b x^2 \left(a-b x^2\right) \left(c-d x^2\right) \sqrt{1-\frac{d x^2}{c}} (4 a d+b c) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+7 a \left(b x^2-a\right) \left(c-d x^2\right) \sqrt{1-\frac{d x^2}{c}} (7 a d+8 b c) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)\right)}{42 a \left(b x^2-a\right) \left(c-d x^2\right)^{3/2} (b c-a d)^3}","\frac{c^{3/4} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (4 a d+b c) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 d^{3/4} \sqrt{c-d x^2} (b c-a d)^3}-\frac{c^{3/4} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (4 a d+b c) E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 d^{3/4} \sqrt{c-d x^2} (b c-a d)^3}+\frac{\sqrt{a} \sqrt[4]{c} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (3 a d+7 b c) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 \sqrt{b} \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)^3}-\frac{\sqrt{a} \sqrt[4]{c} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (3 a d+7 b c) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 \sqrt{b} \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)^3}+\frac{e^3 (e x)^{3/2} (4 a d+b c)}{2 \sqrt{c-d x^2} (b c-a d)^3}+\frac{e^3 (e x)^{3/2} (3 a d+2 b c)}{6 b \left(c-d x^2\right)^{3/2} (b c-a d)^2}+\frac{a e^3 (e x)^{3/2}}{2 b \left(a-b x^2\right) \left(c-d x^2\right)^{3/2} (b c-a d)}",1,"-1/42*(e^3*(e*x)^(3/2)*(7*a*(a^2*d*(7*c - 9*d*x^2) + b^2*c*x^2*(-5*c + 3*d*x^2) + 4*a*b*(2*c^2 - 4*c*d*x^2 + 3*d^2*x^4)) + 7*a*(8*b*c + 7*a*d)*(-a + b*x^2)*(c - d*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 3*b*(b*c + 4*a*d)*x^2*(a - b*x^2)*(c - d*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(a*(b*c - a*d)^3*(-a + b*x^2)*(c - d*x^2)^(3/2))","C",0
927,1,252,454,0.5046717,"\int \frac{(e x)^{7/2}}{\left(a-b x^2\right)^2 \left(c-d x^2\right)^{5/2}} \, dx","Integrate[(e*x)^(7/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)),x]","\frac{e^3 \sqrt{e x} \left(a \left(a^2 d \left(7 d x^2-5 c\right)-2 a b \left(5 c^2-8 c d x^2+5 d^2 x^4\right)+b^2 c x^2 \left(7 c-5 d x^2\right)\right)+b x^2 \left(a-b x^2\right) \left(c-d x^2\right) \sqrt{1-\frac{d x^2}{c}} (2 a d+b c) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+5 a \left(a-b x^2\right) \left(c-d x^2\right) \sqrt{1-\frac{d x^2}{c}} (a d+2 b c) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)\right)}{6 a \left(b x^2-a\right) \left(c-d x^2\right)^{3/2} (b c-a d)^3}","\frac{5 \sqrt[4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (2 a d+b c) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{6 \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)^3}-\frac{5 \sqrt[4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (a d+b c) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)^3}-\frac{5 \sqrt[4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (a d+b c) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)^3}+\frac{5 e^3 \sqrt{e x} (2 a d+b c)}{6 \sqrt{c-d x^2} (b c-a d)^3}+\frac{e^3 \sqrt{e x} (3 a d+2 b c)}{6 b \left(c-d x^2\right)^{3/2} (b c-a d)^2}+\frac{a e^3 \sqrt{e x}}{2 b \left(a-b x^2\right) \left(c-d x^2\right)^{3/2} (b c-a d)}",1,"(e^3*Sqrt[e*x]*(a*(b^2*c*x^2*(7*c - 5*d*x^2) + a^2*d*(-5*c + 7*d*x^2) - 2*a*b*(5*c^2 - 8*c*d*x^2 + 5*d^2*x^4)) + 5*a*(2*b*c + a*d)*(a - b*x^2)*(c - d*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + b*(b*c + 2*a*d)*x^2*(a - b*x^2)*(c - d*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(6*a*(b*c - a*d)^3*(-a + b*x^2)*(c - d*x^2)^(3/2))","C",0
928,1,278,551,0.5345992,"\int \frac{(e x)^{5/2}}{\left(a-b x^2\right)^2 \left(c-d x^2\right)^{5/2}} \, dx","Integrate[(e*x)^(5/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)),x]","-\frac{e (e x)^{3/2} \left(7 \left(b x^2-a\right) \left(c-d x^2\right) \sqrt{1-\frac{d x^2}{c}} \left(a^2 d^2+11 a b c d+3 b^2 c^2\right) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+7 a \left(a^2 d^2 \left(c-3 d x^2\right)+a b d \left(11 c^2-10 c d x^2+3 d^2 x^4\right)+b^2 c \left(3 c^2-17 c d x^2+12 d^2 x^4\right)\right)+3 b d x^2 \left(a-b x^2\right) \left(c-d x^2\right) \sqrt{1-\frac{d x^2}{c}} (a d+4 b c) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)\right)}{42 a c \left(b x^2-a\right) \left(c-d x^2\right)^{3/2} (b c-a d)^3}","\frac{\sqrt[4]{d} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (a d+4 b c) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 \sqrt[4]{c} \sqrt{c-d x^2} (b c-a d)^3}-\frac{\sqrt[4]{d} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (a d+4 b c) E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 \sqrt[4]{c} \sqrt{c-d x^2} (b c-a d)^3}+\frac{\sqrt{b} \sqrt[4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (7 a d+3 b c) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 \sqrt{a} \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)^3}-\frac{\sqrt{b} \sqrt[4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (7 a d+3 b c) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 \sqrt{a} \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)^3}+\frac{d e (e x)^{3/2} (a d+4 b c)}{2 c \sqrt{c-d x^2} (b c-a d)^3}+\frac{5 d e (e x)^{3/2}}{6 \left(c-d x^2\right)^{3/2} (b c-a d)^2}+\frac{e (e x)^{3/2}}{2 \left(a-b x^2\right) \left(c-d x^2\right)^{3/2} (b c-a d)}",1,"-1/42*(e*(e*x)^(3/2)*(7*a*(a^2*d^2*(c - 3*d*x^2) + a*b*d*(11*c^2 - 10*c*d*x^2 + 3*d^2*x^4) + b^2*c*(3*c^2 - 17*c*d*x^2 + 12*d^2*x^4)) + 7*(3*b^2*c^2 + 11*a*b*c*d + a^2*d^2)*(-a + b*x^2)*(c - d*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 3*b*d*(4*b*c + a*d)*x^2*(a - b*x^2)*(c - d*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(a*c*(b*c - a*d)^3*(-a + b*x^2)*(c - d*x^2)^(3/2))","C",0
929,1,275,447,0.5322975,"\int \frac{(e x)^{3/2}}{\left(a-b x^2\right)^2 \left(c-d x^2\right)^{5/2}} \, dx","Integrate[(e*x)^(3/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)),x]","\frac{e \sqrt{e x} \left(-5 \left(a-b x^2\right) \left(c-d x^2\right) \sqrt{1-\frac{d x^2}{c}} \left(a^2 d^2-13 a b c d-3 b^2 c^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+5 a \left(a^2 d^2 \left(c+d x^2\right)-a b d \left(13 c^2-10 c d x^2+d^2 x^4\right)+b^2 c \left(-3 c^2+19 c d x^2-14 d^2 x^4\right)\right)+b d x^2 \left(a-b x^2\right) \left(c-d x^2\right) \sqrt{1-\frac{d x^2}{c}} (a d+14 b c) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)\right)}{30 a c \left(b x^2-a\right) \left(c-d x^2\right)^{3/2} (b c-a d)^3}","\frac{d^{3/4} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (a d+14 b c) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{6 c^{3/4} \sqrt{c-d x^2} (b c-a d)^3}-\frac{b \sqrt[4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (9 a d+b c) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)^3}-\frac{b \sqrt[4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (9 a d+b c) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)^3}+\frac{d e \sqrt{e x} (a d+14 b c)}{6 c \sqrt{c-d x^2} (b c-a d)^3}+\frac{5 d e \sqrt{e x}}{6 \left(c-d x^2\right)^{3/2} (b c-a d)^2}+\frac{e \sqrt{e x}}{2 \left(a-b x^2\right) \left(c-d x^2\right)^{3/2} (b c-a d)}",1,"(e*Sqrt[e*x]*(5*a*(a^2*d^2*(c + d*x^2) + b^2*c*(-3*c^2 + 19*c*d*x^2 - 14*d^2*x^4) - a*b*d*(13*c^2 - 10*c*d*x^2 + d^2*x^4)) - 5*(-3*b^2*c^2 - 13*a*b*c*d + a^2*d^2)*(a - b*x^2)*(c - d*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + b*d*(14*b*c + a*d)*x^2*(a - b*x^2)*(c - d*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(30*a*c*(b*c - a*d)^3*(-a + b*x^2)*(c - d*x^2)^(3/2))","C",0
930,1,327,625,0.566493,"\int \frac{\sqrt{e x}}{\left(a-b x^2\right)^2 \left(c-d x^2\right)^{5/2}} \, dx","Integrate[Sqrt[e*x]/((a - b*x^2)^2*(c - d*x^2)^(5/2)),x]","\frac{\sqrt{e x} \left(3 b d x^3 \left(b x^2-a\right) \left(c-d x^2\right) \sqrt{1-\frac{d x^2}{c}} \left(-a^2 d^2+5 a b c d+b^2 c^2\right) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+7 x \left(b x^2-a\right) \left(c-d x^2\right) \sqrt{1-\frac{d x^2}{c}} \left(a^3 d^3-5 a^2 b c d^2-12 a b^2 c^2 d+b^3 c^3\right) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+7 a x \left(a^3 d^3 \left(5 c-3 d x^2\right)+a^2 b d^2 \left(-17 c^2+10 c d x^2+3 d^2 x^4\right)+a b^2 c d^2 x^2 \left(17 c-15 d x^2\right)-3 b^3 c^2 \left(c-d x^2\right)^2\right)\right)}{42 a^2 c^2 \left(b x^2-a\right) \left(c-d x^2\right)^{3/2} (b c-a d)^3}","-\frac{b^{3/2} \sqrt[4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (b c-11 a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^{3/2} \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b^{3/2} \sqrt[4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (b c-11 a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^{3/2} \sqrt[4]{d} \sqrt{c-d x^2} (b c-a d)^3}+\frac{d (e x)^{3/2} \left(-a^2 d^2+5 a b c d+b^2 c^2\right)}{2 a c^2 e \sqrt{c-d x^2} (b c-a d)^3}+\frac{\sqrt[4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} \left(-a^2 d^2+5 a b c d+b^2 c^2\right) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 a c^{5/4} \sqrt{c-d x^2} (b c-a d)^3}-\frac{\sqrt[4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} \left(-a^2 d^2+5 a b c d+b^2 c^2\right) E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 a c^{5/4} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b (e x)^{3/2}}{2 a e \left(a-b x^2\right) \left(c-d x^2\right)^{3/2} (b c-a d)}+\frac{d (e x)^{3/2} (2 a d+3 b c)}{6 a c e \left(c-d x^2\right)^{3/2} (b c-a d)^2}",1,"(Sqrt[e*x]*(7*a*x*(a*b^2*c*d^2*x^2*(17*c - 15*d*x^2) + a^3*d^3*(5*c - 3*d*x^2) - 3*b^3*c^2*(c - d*x^2)^2 + a^2*b*d^2*(-17*c^2 + 10*c*d*x^2 + 3*d^2*x^4)) + 7*(b^3*c^3 - 12*a*b^2*c^2*d - 5*a^2*b*c*d^2 + a^3*d^3)*x*(-a + b*x^2)*(c - d*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 3*b*d*(b^2*c^2 + 5*a*b*c*d - a^2*d^2)*x^3*(-a + b*x^2)*(c - d*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(42*a^2*c^2*(b*c - a*d)^3*(-a + b*x^2)*(c - d*x^2)^(3/2))","C",0
931,1,328,514,0.5513662,"\int \frac{1}{\sqrt{e x} \left(a-b x^2\right)^2 \left(c-d x^2\right)^{5/2}} \, dx","Integrate[1/(Sqrt[e*x]*(a - b*x^2)^2*(c - d*x^2)^(5/2)),x]","-\frac{b d x^3 \left(b x^2-a\right) \left(c-d x^2\right) \sqrt{1-\frac{d x^2}{c}} \left(-5 a^2 d^2+17 a b c d+3 b^2 c^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-5 x \left(a-b x^2\right) \left(c-d x^2\right) \sqrt{1-\frac{d x^2}{c}} \left(5 a^3 d^3-17 a^2 b c d^2+36 a b^2 c^2 d-9 b^3 c^3\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)+5 a x \left(a^3 d^3 \left(5 d x^2-7 c\right)+a^2 b d^2 \left(19 c^2-10 c d x^2-5 d^2 x^4\right)+a b^2 c d^2 x^2 \left(17 d x^2-19 c\right)+3 b^3 c^2 \left(c-d x^2\right)^2\right)}{30 a^2 c^2 \sqrt{e x} \left(b x^2-a\right) \left(c-d x^2\right)^{3/2} (b c-a d)^3}","\frac{d \sqrt{e x} \left(-5 a^2 d^2+17 a b c d+3 b^2 c^2\right)}{6 a c^2 e \sqrt{c-d x^2} (b c-a d)^3}+\frac{d^{3/4} \sqrt{1-\frac{d x^2}{c}} \left(-5 a^2 d^2+17 a b c d+3 b^2 c^2\right) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{6 a c^{7/4} \sqrt{e} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b^2 \sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (3 b c-13 a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^2 \sqrt[4]{d} \sqrt{e} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b^2 \sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (3 b c-13 a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^2 \sqrt[4]{d} \sqrt{e} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b \sqrt{e x}}{2 a e \left(a-b x^2\right) \left(c-d x^2\right)^{3/2} (b c-a d)}+\frac{d \sqrt{e x} (2 a d+3 b c)}{6 a c e \left(c-d x^2\right)^{3/2} (b c-a d)^2}",1,"-1/30*(5*a*x*(3*b^3*c^2*(c - d*x^2)^2 + a^3*d^3*(-7*c + 5*d*x^2) + a*b^2*c*d^2*x^2*(-19*c + 17*d*x^2) + a^2*b*d^2*(19*c^2 - 10*c*d*x^2 - 5*d^2*x^4)) - 5*(-9*b^3*c^3 + 36*a*b^2*c^2*d - 17*a^2*b*c*d^2 + 5*a^3*d^3)*x*(a - b*x^2)*(c - d*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + b*d*(3*b^2*c^2 + 17*a*b*c*d - 5*a^2*d^2)*x^3*(-a + b*x^2)*(c - d*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a])/(a^2*c^2*(b*c - a*d)^3*Sqrt[e*x]*(-a + b*x^2)*(c - d*x^2)^(3/2))","C",0
932,1,407,735,1.4065293,"\int \frac{1}{(e x)^{3/2} \left(a-b x^2\right)^2 \left(c-d x^2\right)^{5/2}} \, dx","Integrate[1/((e*x)^(3/2)*(a - b*x^2)^2*(c - d*x^2)^(5/2)),x]","\frac{x \left(3 b d x^4 \sqrt{1-\frac{d x^2}{c}} \left(7 a^3 d^3-19 a^2 b c d^2+12 a b^2 c^2 d-5 b^3 c^3\right) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-7 x^2 \sqrt{1-\frac{d x^2}{c}} \left(7 a^4 d^4-19 a^3 b c d^3+12 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+5 b^4 c^4\right) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)-\frac{7 a \left(a^4 d^3 \left(12 c^2-35 c d x^2+21 d^2 x^4\right)-a^3 b d^2 \left(36 c^3-83 c^2 d x^2+22 c d^2 x^4+21 d^3 x^6\right)+a^2 b^2 c d \left(36 c^3-36 c^2 d x^2-59 c d^2 x^4+57 d^3 x^6\right)-12 a b^3 c^2 \left(c-d x^2\right)^2 \left(c+3 d x^2\right)+15 b^4 c^3 x^2 \left(c-d x^2\right)^2\right)}{\left(a-b x^2\right) \left(c-d x^2\right)}\right)}{42 a^3 c^3 (e x)^{3/2} \sqrt{c-d x^2} (a d-b c)^3}","-\frac{5 b^{5/2} \sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-3 a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^{5/2} \sqrt[4]{d} e^{3/2} \sqrt{c-d x^2} (b c-a d)^3}+\frac{5 b^{5/2} \sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-3 a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^{5/2} \sqrt[4]{d} e^{3/2} \sqrt{c-d x^2} (b c-a d)^3}+\frac{d \left(-7 a^2 d^2+19 a b c d+3 b^2 c^2\right)}{6 a c^2 e \sqrt{e x} \sqrt{c-d x^2} (b c-a d)^3}-\frac{\sqrt{c-d x^2} \left(-7 a^3 d^3+19 a^2 b c d^2-12 a b^2 c^2 d+5 b^3 c^3\right)}{2 a^2 c^3 e \sqrt{e x} (b c-a d)^3}+\frac{\sqrt[4]{d} \sqrt{1-\frac{d x^2}{c}} \left(-7 a^3 d^3+19 a^2 b c d^2-12 a b^2 c^2 d+5 b^3 c^3\right) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 a^2 c^{9/4} e^{3/2} \sqrt{c-d x^2} (b c-a d)^3}-\frac{\sqrt[4]{d} \sqrt{1-\frac{d x^2}{c}} \left(-7 a^3 d^3+19 a^2 b c d^2-12 a b^2 c^2 d+5 b^3 c^3\right) E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{2 a^2 c^{9/4} e^{3/2} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b}{2 a e \sqrt{e x} \left(a-b x^2\right) \left(c-d x^2\right)^{3/2} (b c-a d)}+\frac{d (2 a d+3 b c)}{6 a c e \sqrt{e x} \left(c-d x^2\right)^{3/2} (b c-a d)^2}",1,"(x*((-7*a*(15*b^4*c^3*x^2*(c - d*x^2)^2 - 12*a*b^3*c^2*(c - d*x^2)^2*(c + 3*d*x^2) + a^4*d^3*(12*c^2 - 35*c*d*x^2 + 21*d^2*x^4) - a^3*b*d^2*(36*c^3 - 83*c^2*d*x^2 + 22*c*d^2*x^4 + 21*d^3*x^6) + a^2*b^2*c*d*(36*c^3 - 36*c^2*d*x^2 - 59*c*d^2*x^4 + 57*d^3*x^6)))/((a - b*x^2)*(c - d*x^2)) - 7*(5*b^4*c^4 - 20*a*b^3*c^3*d + 12*a^2*b^2*c^2*d^2 - 19*a^3*b*c*d^3 + 7*a^4*d^4)*x^2*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 3*b*d*(-5*b^3*c^3 + 12*a*b^2*c^2*d - 19*a^2*b*c*d^2 + 7*a^3*d^3)*x^4*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(42*a^3*c^3*(-(b*c) + a*d)^3*(e*x)^(3/2)*Sqrt[c - d*x^2])","C",0
933,1,427,606,1.4659192,"\int \frac{1}{(e x)^{5/2} \left(a-b x^2\right)^2 \left(c-d x^2\right)^{5/2}} \, dx","Integrate[1/((e*x)^(5/2)*(a - b*x^2)^2*(c - d*x^2)^(5/2)),x]","\frac{x \left(-\frac{b d x^4 \sqrt{1-\frac{d x^2}{c}} \left(-15 a^3 d^3+35 a^2 b c d^2-12 a b^2 c^2 d+7 b^3 c^3\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)}{(b c-a d)^3}+\frac{5 x^2 \sqrt{1-\frac{d x^2}{c}} \left(-15 a^4 d^4+35 a^3 b c d^3-12 a^2 b^2 c^2 d^2-44 a b^3 c^3 d+21 b^4 c^4\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right)}{(b c-a d)^3}-\frac{5 a \left(a^4 d^3 \left(4 c^2-21 c d x^2+15 d^2 x^4\right)-a^3 b d^2 \left(12 c^3-45 c^2 d x^2+14 c d^2 x^4+15 d^3 x^6\right)+a^2 b^2 c d \left(12 c^3-12 c^2 d x^2-37 c d^2 x^4+35 d^3 x^6\right)-4 a b^3 c^2 \left(c-d x^2\right)^2 \left(c+3 d x^2\right)+7 b^4 c^3 x^2 \left(c-d x^2\right)^2\right)}{\left(a-b x^2\right) \left(c-d x^2\right) (a d-b c)^3}\right)}{30 a^3 c^3 (e x)^{5/2} \sqrt{c-d x^2}}","\frac{b^3 \sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (7 b c-17 a d) \Pi \left(-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^3 \sqrt[4]{d} e^{5/2} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b^3 \sqrt[4]{c} \sqrt{1-\frac{d x^2}{c}} (7 b c-17 a d) \Pi \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{4 a^3 \sqrt[4]{d} e^{5/2} \sqrt{c-d x^2} (b c-a d)^3}+\frac{d \left(-3 a^2 d^2+7 a b c d+b^2 c^2\right)}{2 a c^2 e (e x)^{3/2} \sqrt{c-d x^2} (b c-a d)^3}-\frac{\sqrt{c-d x^2} \left(-15 a^3 d^3+35 a^2 b c d^2-12 a b^2 c^2 d+7 b^3 c^3\right)}{6 a^2 c^3 e (e x)^{3/2} (b c-a d)^3}+\frac{d^{3/4} \sqrt{1-\frac{d x^2}{c}} \left(-15 a^3 d^3+35 a^2 b c d^2-12 a b^2 c^2 d+7 b^3 c^3\right) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{e x}}{\sqrt[4]{c} \sqrt{e}}\right)\right|-1\right)}{6 a^2 c^{11/4} e^{5/2} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b}{2 a e (e x)^{3/2} \left(a-b x^2\right) \left(c-d x^2\right)^{3/2} (b c-a d)}+\frac{d (2 a d+3 b c)}{6 a c e (e x)^{3/2} \left(c-d x^2\right)^{3/2} (b c-a d)^2}",1,"(x*((-5*a*(7*b^4*c^3*x^2*(c - d*x^2)^2 - 4*a*b^3*c^2*(c - d*x^2)^2*(c + 3*d*x^2) + a^4*d^3*(4*c^2 - 21*c*d*x^2 + 15*d^2*x^4) - a^3*b*d^2*(12*c^3 - 45*c^2*d*x^2 + 14*c*d^2*x^4 + 15*d^3*x^6) + a^2*b^2*c*d*(12*c^3 - 12*c^2*d*x^2 - 37*c*d^2*x^4 + 35*d^3*x^6)))/((-(b*c) + a*d)^3*(a - b*x^2)*(c - d*x^2)) + (5*(21*b^4*c^4 - 44*a*b^3*c^3*d - 12*a^2*b^2*c^2*d^2 + 35*a^3*b*c*d^3 - 15*a^4*d^4)*x^2*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a])/(b*c - a*d)^3 - (b*d*(7*b^3*c^3 - 12*a*b^2*c^2*d + 35*a^2*b*c*d^2 - 15*a^3*d^3)*x^4*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a])/(b*c - a*d)^3))/(30*a^3*c^3*(e*x)^(5/2)*Sqrt[c - d*x^2])","C",0
934,1,187,209,0.4515481,"\int \frac{x^5 \sqrt{a+b x^2}}{\sqrt{c+d x^2}} \, dx","Integrate[(x^5*Sqrt[a + b*x^2])/Sqrt[c + d*x^2],x]","\frac{-3 \left(a^2 d^2+2 a b c d+5 b^2 c^2\right) (b c-a d)^{3/2} \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)-b \sqrt{d} \sqrt{a+b x^2} \left(c+d x^2\right) \left(3 a^2 d^2-2 a b d \left(d x^2-2 c\right)+b^2 \left(-15 c^2+10 c d x^2-8 d^2 x^4\right)\right)}{48 b^3 d^{7/2} \sqrt{c+d x^2}}","\frac{\sqrt{a+b x^2} \sqrt{c+d x^2} \left(a^2 d^2+2 a b c d+5 b^2 c^2\right)}{16 b^2 d^3}-\frac{(b c-a d) \left(a^2 d^2+2 a b c d+5 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right)}{16 b^{5/2} d^{7/2}}-\frac{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2} (3 a d+5 b c)}{24 b^2 d^2}+\frac{x^2 \left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}}{6 b d}",1,"(-(b*Sqrt[d]*Sqrt[a + b*x^2]*(c + d*x^2)*(3*a^2*d^2 - 2*a*b*d*(-2*c + d*x^2) + b^2*(-15*c^2 + 10*c*d*x^2 - 8*d^2*x^4))) - 3*(b*c - a*d)^(3/2)*(5*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]])/(48*b^3*d^(7/2)*Sqrt[c + d*x^2])","A",1
935,1,138,137,0.3085984,"\int \frac{x^3 \sqrt{a+b x^2}}{\sqrt{c+d x^2}} \, dx","Integrate[(x^3*Sqrt[a + b*x^2])/Sqrt[c + d*x^2],x]","\frac{b \sqrt{d} \sqrt{a+b x^2} \left(c+d x^2\right) \left(a d-3 b c+2 b d x^2\right)+(a d+3 b c) (b c-a d)^{3/2} \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)}{8 b^2 d^{5/2} \sqrt{c+d x^2}}","\frac{(b c-a d) (a d+3 b c) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right)}{8 b^{3/2} d^{5/2}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2} (a d+3 b c)}{8 b d^2}+\frac{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}}{4 b d}",1,"(b*Sqrt[d]*Sqrt[a + b*x^2]*(c + d*x^2)*(-3*b*c + a*d + 2*b*d*x^2) + (b*c - a*d)^(3/2)*(3*b*c + a*d)*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]])/(8*b^2*d^(5/2)*Sqrt[c + d*x^2])","A",1
936,1,116,86,0.2860685,"\int \frac{x \sqrt{a+b x^2}}{\sqrt{c+d x^2}} \, dx","Integrate[(x*Sqrt[a + b*x^2])/Sqrt[c + d*x^2],x]","\frac{b \sqrt{d} \sqrt{a+b x^2} \left(c+d x^2\right)-(b c-a d)^{3/2} \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)}{2 b d^{3/2} \sqrt{c+d x^2}}","\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{2 d}-\frac{(b c-a d) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right)}{2 \sqrt{b} d^{3/2}}",1,"(b*Sqrt[d]*Sqrt[a + b*x^2]*(c + d*x^2) - (b*c - a*d)^(3/2)*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]])/(2*b*d^(3/2)*Sqrt[c + d*x^2])","A",1
937,1,129,92,0.1417322,"\int \frac{\sqrt{a+b x^2}}{x \sqrt{c+d x^2}} \, dx","Integrate[Sqrt[a + b*x^2]/(x*Sqrt[c + d*x^2]),x]","\frac{\sqrt{b c-a d} \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)}{\sqrt{d} \sqrt{c+d x^2}}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{\sqrt{c}}","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right)}{\sqrt{d}}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{\sqrt{c}}",1,"(Sqrt[b*c - a*d]*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]])/(Sqrt[d]*Sqrt[c + d*x^2]) - (Sqrt[a]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])])/Sqrt[c]","A",1
938,1,89,89,0.0649456,"\int \frac{\sqrt{a+b x^2}}{x^3 \sqrt{c+d x^2}} \, dx","Integrate[Sqrt[a + b*x^2]/(x^3*Sqrt[c + d*x^2]),x]","\frac{1}{2} \left(-\frac{(b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{\sqrt{a} c^{3/2}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{c x^2}\right)","-\frac{(b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 \sqrt{a} c^{3/2}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{2 c x^2}",1,"(-((Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(c*x^2)) - ((b*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])])/(Sqrt[a]*c^(3/2)))/2","A",1
939,1,125,143,0.0776362,"\int \frac{\sqrt{a+b x^2}}{x^5 \sqrt{c+d x^2}} \, dx","Integrate[Sqrt[a + b*x^2]/(x^5*Sqrt[c + d*x^2]),x]","\frac{\left(-3 a^2 d^2+2 a b c d+b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{8 a^{3/2} c^{5/2}}+\frac{\sqrt{a+b x^2} \sqrt{c+d x^2} \left(-2 a c+3 a d x^2-b c x^2\right)}{8 a c^2 x^4}","\frac{(b c-a d) (3 a d+b c) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{8 a^{3/2} c^{5/2}}+\frac{\sqrt{a+b x^2} \sqrt{c+d x^2} (3 a d+b c)}{8 a c^2 x^2}-\frac{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}}{4 a c x^4}",1,"(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(-2*a*c - b*c*x^2 + 3*a*d*x^2))/(8*a*c^2*x^4) + ((b^2*c^2 + 2*a*b*c*d - 3*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])])/(8*a^(3/2)*c^(5/2))","A",1
940,1,246,343,0.4893459,"\int \frac{x^4 \sqrt{a+b x^2}}{\sqrt{c+d x^2}} \, dx","Integrate[(x^4*Sqrt[a + b*x^2])/Sqrt[c + d*x^2],x]","\frac{-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(a^2 d^2+7 a b c d-8 b^2 c^2\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(2 a^2 d^2+3 a b c d-8 b^2 c^2\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+d x \sqrt{\frac{b}{a}} \left(a+b x^2\right) \left(c+d x^2\right) \left(a d-4 b c+3 b d x^2\right)}{15 b d^3 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}}","\frac{x \sqrt{a+b x^2} \left(-2 a^2 d^2-3 a b c d+8 b^2 c^2\right)}{15 b^2 d^2 \sqrt{c+d x^2}}-\frac{\sqrt{c} \sqrt{a+b x^2} \left(-2 a^2 d^2-3 a b c d+8 b^2 c^2\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{15 b^2 d^{5/2} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{c^{3/2} \sqrt{a+b x^2} (4 b c-a d) F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{15 b d^{5/2} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{x \sqrt{a+b x^2} \sqrt{c+d x^2} (4 b c-a d)}{15 b d^2}+\frac{x^3 \sqrt{a+b x^2} \sqrt{c+d x^2}}{5 d}",1,"(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(-4*b*c + a*d + 3*b*d*x^2) + I*c*(-8*b^2*c^2 + 3*a*b*c*d + 2*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*c*(-8*b^2*c^2 + 7*a*b*c*d + a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(15*b*Sqrt[b/a]*d^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","C",1
941,1,199,259,0.2742835,"\int \frac{x^2 \sqrt{a+b x^2}}{\sqrt{c+d x^2}} \, dx","Integrate[(x^2*Sqrt[a + b*x^2])/Sqrt[c + d*x^2],x]","\frac{d x \sqrt{\frac{b}{a}} \left(a+b x^2\right) \left(c+d x^2\right)+2 i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-b c) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-2 b c) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{3 d^2 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}}","-\frac{c^{3/2} \sqrt{a+b x^2} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{3 d^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{\sqrt{c} \sqrt{a+b x^2} (2 b c-a d) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{3 b d^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{x \sqrt{a+b x^2} \sqrt{c+d x^2}}{3 d}-\frac{x \sqrt{a+b x^2} (2 b c-a d)}{3 b d \sqrt{c+d x^2}}",1,"(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2) - I*c*(-2*b*c + a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (2*I)*c*(-(b*c) + a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(3*Sqrt[b/a]*d^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","C",1
942,1,111,232,0.2772852,"\int \frac{\sqrt{a+b x^2}}{x^2 \sqrt{c+d x^2}} \, dx","Integrate[Sqrt[a + b*x^2]/(x^2*Sqrt[c + d*x^2]),x]","\frac{\frac{b c x \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} E\left(\sin ^{-1}\left(\sqrt{-\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{\sqrt{-\frac{b}{a}}}-\left(a+b x^2\right) \left(c+d x^2\right)}{c x \sqrt{a+b x^2} \sqrt{c+d x^2}}","\frac{d x \sqrt{a+b x^2}}{c \sqrt{c+d x^2}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{c x}+\frac{b \sqrt{c} \sqrt{a+b x^2} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{a \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{\sqrt{d} \sqrt{a+b x^2} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{\sqrt{c} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}",1,"(-((a + b*x^2)*(c + d*x^2)) + (b*c*x*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[ArcSin[Sqrt[-(b/a)]*x], (a*d)/(b*c)])/Sqrt[-(b/a)])/(c*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","A",1
943,1,228,307,0.8036312,"\int \frac{\sqrt{a+b x^2}}{x^4 \sqrt{c+d x^2}} \, dx","Integrate[Sqrt[a + b*x^2]/(x^4*Sqrt[c + d*x^2]),x]","\frac{-\frac{\left(a+b x^2\right) \left(c+d x^2\right) \left(a c-2 a d x^2+b c x^2\right)}{a}+i c x^3 \sqrt{\frac{b}{a}} \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (b c-a d) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+i c x^3 \sqrt{\frac{b}{a}} \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (2 a d-b c) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{3 c^2 x^3 \sqrt{a+b x^2} \sqrt{c+d x^2}}","-\frac{\sqrt{d} \sqrt{a+b x^2} (b c-2 a d) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{3 a c^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2} (b c-2 a d)}{3 a c^2 x}+\frac{d x \sqrt{a+b x^2} (b c-2 a d)}{3 a c^2 \sqrt{c+d x^2}}-\frac{b \sqrt{d} \sqrt{a+b x^2} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{3 a \sqrt{c} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{3 c x^3}",1,"(-(((a + b*x^2)*(c + d*x^2)*(a*c + b*c*x^2 - 2*a*d*x^2))/a) + I*Sqrt[b/a]*c*(-(b*c) + 2*a*d)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + I*Sqrt[b/a]*c*(b*c - a*d)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(3*c^2*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","C",1
944,1,231,276,0.6000101,"\int \frac{x^5 \left(a+b x^2\right)^{3/2}}{\sqrt{c+d x^2}} \, dx","Integrate[(x^5*(a + b*x^2)^(3/2))/Sqrt[c + d*x^2],x]","\frac{3 (b c-a d)^{5/2} \left(3 a^2 d^2+10 a b c d+35 b^2 c^2\right) \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)-b \sqrt{d} \sqrt{a+b x^2} \left(c+d x^2\right) \left(9 a^3 d^3+3 a^2 b d^2 \left(5 c-2 d x^2\right)+a b^2 d \left(-145 c^2+92 c d x^2-72 d^2 x^4\right)+b^3 \left(105 c^3-70 c^2 d x^2+56 c d^2 x^4-48 d^3 x^6\right)\right)}{384 b^3 d^{9/2} \sqrt{c+d x^2}}","-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2} (b c-a d) \left(3 a^2 d^2+10 a b c d+35 b^2 c^2\right)}{128 b^2 d^4}+\frac{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2} \left(3 a^2 d^2+10 a b c d+35 b^2 c^2\right)}{192 b^2 d^3}+\frac{(b c-a d)^2 \left(3 a^2 d^2+10 a b c d+35 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right)}{128 b^{5/2} d^{9/2}}-\frac{\left(a+b x^2\right)^{5/2} \sqrt{c+d x^2} (3 a d+7 b c)}{48 b^2 d^2}+\frac{x^2 \left(a+b x^2\right)^{5/2} \sqrt{c+d x^2}}{8 b d}",1,"(-(b*Sqrt[d]*Sqrt[a + b*x^2]*(c + d*x^2)*(9*a^3*d^3 + 3*a^2*b*d^2*(5*c - 2*d*x^2) + a*b^2*d*(-145*c^2 + 92*c*d*x^2 - 72*d^2*x^4) + b^3*(105*c^3 - 70*c^2*d*x^2 + 56*c*d^2*x^4 - 48*d^3*x^6))) + 3*(b*c - a*d)^(5/2)*(35*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]])/(384*b^3*d^(9/2)*Sqrt[c + d*x^2])","A",1
945,1,173,187,0.4436802,"\int \frac{x^3 \left(a+b x^2\right)^{3/2}}{\sqrt{c+d x^2}} \, dx","Integrate[(x^3*(a + b*x^2)^(3/2))/Sqrt[c + d*x^2],x]","\frac{b \sqrt{d} \sqrt{a+b x^2} \left(c+d x^2\right) \left(3 a^2 d^2+2 a b d \left(7 d x^2-11 c\right)+b^2 \left(15 c^2-10 c d x^2+8 d^2 x^4\right)\right)-3 (b c-a d)^{5/2} (a d+5 b c) \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)}{48 b^2 d^{7/2} \sqrt{c+d x^2}}","-\frac{(b c-a d)^2 (a d+5 b c) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right)}{16 b^{3/2} d^{7/2}}+\frac{\sqrt{a+b x^2} \sqrt{c+d x^2} (b c-a d) (a d+5 b c)}{16 b d^3}-\frac{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2} (a d+5 b c)}{24 b d^2}+\frac{\left(a+b x^2\right)^{5/2} \sqrt{c+d x^2}}{6 b d}",1,"(b*Sqrt[d]*Sqrt[a + b*x^2]*(c + d*x^2)*(3*a^2*d^2 + 2*a*b*d*(-11*c + 7*d*x^2) + b^2*(15*c^2 - 10*c*d*x^2 + 8*d^2*x^4)) - 3*(b*c - a*d)^(5/2)*(5*b*c + a*d)*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]])/(48*b^2*d^(7/2)*Sqrt[c + d*x^2])","A",1
946,1,131,125,0.3837559,"\int \frac{x \left(a+b x^2\right)^{3/2}}{\sqrt{c+d x^2}} \, dx","Integrate[(x*(a + b*x^2)^(3/2))/Sqrt[c + d*x^2],x]","\frac{\sqrt{d} \sqrt{a+b x^2} \left(c+d x^2\right) \left(5 a d-3 b c+2 b d x^2\right)+\frac{3 (b c-a d)^{5/2} \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)}{b}}{8 d^{5/2} \sqrt{c+d x^2}}","\frac{3 (b c-a d)^2 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right)}{8 \sqrt{b} d^{5/2}}-\frac{3 \sqrt{a+b x^2} \sqrt{c+d x^2} (b c-a d)}{8 d^2}+\frac{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}}{4 d}",1,"(Sqrt[d]*Sqrt[a + b*x^2]*(c + d*x^2)*(-3*b*c + 5*a*d + 2*b*d*x^2) + (3*(b*c - a*d)^(5/2)*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]])/b)/(8*d^(5/2)*Sqrt[c + d*x^2])","A",1
947,1,195,133,0.7679298,"\int \frac{\left(a+b x^2\right)^{3/2}}{x \sqrt{c+d x^2}} \, dx","Integrate[(a + b*x^2)^(3/2)/(x*Sqrt[c + d*x^2]),x]","\frac{\sqrt{d} \left(b \sqrt{a+b x^2} \left(c+d x^2\right)-\frac{2 a^{3/2} d \sqrt{c+d x^2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{\sqrt{c}}\right)-\frac{\left(3 a^2 d^2-4 a b c d+b^2 c^2\right) \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)}{\sqrt{b c-a d}}}{2 d^{3/2} \sqrt{c+d x^2}}","-\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{\sqrt{c}}-\frac{\sqrt{b} (b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right)}{2 d^{3/2}}+\frac{b \sqrt{a+b x^2} \sqrt{c+d x^2}}{2 d}",1,"(-(((b^2*c^2 - 4*a*b*c*d + 3*a^2*d^2)*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]])/Sqrt[b*c - a*d]) + Sqrt[d]*(b*Sqrt[a + b*x^2]*(c + d*x^2) - (2*a^(3/2)*d*Sqrt[c + d*x^2]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])])/Sqrt[c]))/(2*d^(3/2)*Sqrt[c + d*x^2])","A",1
948,1,172,136,1.013774,"\int \frac{\left(a+b x^2\right)^{3/2}}{x^3 \sqrt{c+d x^2}} \, dx","Integrate[(a + b*x^2)^(3/2)/(x^3*Sqrt[c + d*x^2]),x]","\frac{\sqrt{a} (a d-3 b c) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 c^{3/2}}-\frac{a \sqrt{a+b x^2} \sqrt{c+d x^2}}{2 c x^2}+\frac{(b c-a d)^{3/2} \left(\frac{b \left(c+d x^2\right)}{b c-a d}\right)^{3/2} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)}{\sqrt{d} \left(c+d x^2\right)^{3/2}}","\frac{b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right)}{\sqrt{d}}-\frac{\sqrt{a} (3 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 c^{3/2}}-\frac{a \sqrt{a+b x^2} \sqrt{c+d x^2}}{2 c x^2}",1,"-1/2*(a*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(c*x^2) + ((b*c - a*d)^(3/2)*((b*(c + d*x^2))/(b*c - a*d))^(3/2)*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]])/(Sqrt[d]*(c + d*x^2)^(3/2)) + (Sqrt[a]*(-3*b*c + a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*c^(3/2))","A",1
949,1,110,131,0.0722611,"\int \frac{\left(a+b x^2\right)^{3/2}}{x^5 \sqrt{c+d x^2}} \, dx","Integrate[(a + b*x^2)^(3/2)/(x^5*Sqrt[c + d*x^2]),x]","\frac{\sqrt{a+b x^2} \sqrt{c+d x^2} \left(-2 a c+3 a d x^2-5 b c x^2\right)}{8 c^2 x^4}-\frac{3 (b c-a d)^2 \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{8 \sqrt{a} c^{5/2}}","-\frac{3 (b c-a d)^2 \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{8 \sqrt{a} c^{5/2}}-\frac{3 \sqrt{a+b x^2} \sqrt{c+d x^2} (b c-a d)}{8 c^2 x^2}-\frac{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}}{4 c x^4}",1,"(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(-2*a*c - 5*b*c*x^2 + 3*a*d*x^2))/(8*c^2*x^4) - (3*(b*c - a*d)^2*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])])/(8*Sqrt[a]*c^(5/2))","A",1
950,1,305,429,0.6451699,"\int \frac{x^4 \left(a+b x^2\right)^{3/2}}{\sqrt{c+d x^2}} \, dx","Integrate[(x^4*(a + b*x^2)^(3/2))/Sqrt[c + d*x^2],x]","\frac{d x \sqrt{\frac{b}{a}} \left(a+b x^2\right) \left(c+d x^2\right) \left(a^2 d^2+a b d \left(8 d x^2-11 c\right)+b^2 \left(8 c^2-6 c d x^2+5 d^2 x^4\right)\right)-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(a^3 d^3+15 a^2 b c d^2-32 a b^2 c^2 d+16 b^3 c^3\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+2 i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(a^3 d^3+2 a^2 b c d^2-12 a b^2 c^2 d+8 b^3 c^3\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{35 b d^4 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}}","\frac{2 \sqrt{c} \sqrt{a+b x^2} (2 b c-a d) \left(-a^2 d^2-4 a b c d+4 b^2 c^2\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{35 b^2 d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{x \sqrt{a+b x^2} \sqrt{c+d x^2} \left(a^2 d^2-11 a b c d+8 b^2 c^2\right)}{35 b d^3}-\frac{2 x \sqrt{a+b x^2} (2 b c-a d) \left(-a^2 d^2-4 a b c d+4 b^2 c^2\right)}{35 b^2 d^3 \sqrt{c+d x^2}}-\frac{c^{3/2} \sqrt{a+b x^2} \left(a^2 d^2-11 a b c d+8 b^2 c^2\right) F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{35 b d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{2 x^3 \sqrt{a+b x^2} \sqrt{c+d x^2} (3 b c-4 a d)}{35 d^2}+\frac{b x^5 \sqrt{a+b x^2} \sqrt{c+d x^2}}{7 d}",1,"(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(a^2*d^2 + a*b*d*(-11*c + 8*d*x^2) + b^2*(8*c^2 - 6*c*d*x^2 + 5*d^2*x^4)) + (2*I)*c*(8*b^3*c^3 - 12*a*b^2*c^2*d + 2*a^2*b*c*d^2 + a^3*d^3)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*c*(16*b^3*c^3 - 32*a*b^2*c^2*d + 15*a^2*b*c*d^2 + a^3*d^3)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(35*b*Sqrt[b/a]*d^4*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","C",1
951,1,245,335,0.4937933,"\int \frac{x^2 \left(a+b x^2\right)^{3/2}}{\sqrt{c+d x^2}} \, dx","Integrate[(x^2*(a + b*x^2)^(3/2))/Sqrt[c + d*x^2],x]","\frac{i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(9 a^2 d^2-17 a b c d+8 b^2 c^2\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(3 a^2 d^2-13 a b c d+8 b^2 c^2\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+d x \sqrt{\frac{b}{a}} \left(a+b x^2\right) \left(c+d x^2\right) \left(6 a d-4 b c+3 b d x^2\right)}{15 d^3 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}}","-\frac{\sqrt{c} \sqrt{a+b x^2} \left(3 a^2 d^2-13 a b c d+8 b^2 c^2\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{15 b d^{5/2} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{x \sqrt{a+b x^2} \left(-\frac{3 a^2 d}{b}+13 a c-\frac{8 b c^2}{d}\right)}{15 d \sqrt{c+d x^2}}+\frac{2 c^{3/2} \sqrt{a+b x^2} (2 b c-3 a d) F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{15 d^{5/2} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{2 x \sqrt{a+b x^2} \sqrt{c+d x^2} (2 b c-3 a d)}{15 d^2}+\frac{b x^3 \sqrt{a+b x^2} \sqrt{c+d x^2}}{5 d}",1,"(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(-4*b*c + 6*a*d + 3*b*d*x^2) - I*c*(8*b^2*c^2 - 13*a*b*c*d + 3*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + I*c*(8*b^2*c^2 - 17*a*b*c*d + 9*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(15*Sqrt[b/a]*d^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","C",1
952,1,206,244,0.3032486,"\int \frac{\left(a+b x^2\right)^{3/2}}{x^2 \sqrt{c+d x^2}} \, dx","Integrate[(a + b*x^2)^(3/2)/(x^2*Sqrt[c + d*x^2]),x]","\frac{-a d \sqrt{\frac{b}{a}} \left(a+b x^2\right) \left(c+d x^2\right)-i b c x \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-b c) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)-i b c x \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d+b c) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{c d x \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}}","-\frac{a \sqrt{a+b x^2} \sqrt{c+d x^2}}{c x}+\frac{x \sqrt{a+b x^2} (a d+b c)}{c \sqrt{c+d x^2}}+\frac{2 b \sqrt{c} \sqrt{a+b x^2} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{\sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{\sqrt{a+b x^2} (a d+b c) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{\sqrt{c} \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}",1,"(-(a*Sqrt[b/a]*d*(a + b*x^2)*(c + d*x^2)) - I*b*c*(b*c + a*d)*x*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*b*c*(-(b*c) + a*d)*x*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(Sqrt[b/a]*c*d*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","C",1
953,1,227,311,0.3834154,"\int \frac{\left(a+b x^2\right)^{3/2}}{x^4 \sqrt{c+d x^2}} \, dx","Integrate[(a + b*x^2)^(3/2)/(x^4*Sqrt[c + d*x^2]),x]","\frac{\sqrt{\frac{b}{a}} \left(a+b x^2\right) \left(c+d x^2\right) \left(-a c+2 a d x^2-4 b c x^2\right)-i b c x^3 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-b c) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+2 i b c x^3 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-2 b c) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{3 c^2 x^3 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}}","-\frac{2 \sqrt{d} \sqrt{a+b x^2} (2 b c-a d) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{3 c^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{2 \sqrt{a+b x^2} \sqrt{c+d x^2} (2 b c-a d)}{3 c^2 x}+\frac{2 d x \sqrt{a+b x^2} (2 b c-a d)}{3 c^2 \sqrt{c+d x^2}}+\frac{b \sqrt{a+b x^2} (3 b c-a d) F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{3 a \sqrt{c} \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{a \sqrt{a+b x^2} \sqrt{c+d x^2}}{3 c x^3}",1,"(Sqrt[b/a]*(a + b*x^2)*(c + d*x^2)*(-(a*c) - 4*b*c*x^2 + 2*a*d*x^2) + (2*I)*b*c*(-2*b*c + a*d)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*b*c*(-(b*c) + a*d)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(3*Sqrt[b/a]*c^2*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","C",1
954,1,271,340,1.2013793,"\int \frac{x^5 \left(a+b x^2\right)^{5/2}}{\sqrt{c+d x^2}} \, dx","Integrate[(x^5*(a + b*x^2)^(5/2))/Sqrt[c + d*x^2],x]","\frac{\sqrt{c+d x^2} \left(\frac{5 (b c-a d)^3 \left(3 a^2 d^2+14 a b c d+63 b^2 c^2\right) \left(\frac{16 d^3 \left(a+b x^2\right)^3}{15 (b c-a d)^3}-\frac{4 d^2 \left(a+b x^2\right)^2}{3 (b c-a d)^2}+\frac{2 d \left(a+b x^2\right)}{b c-a d}-\frac{2 \sqrt{d} \sqrt{a+b x^2} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)}{\sqrt{b c-a d} \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}}}\right)}{4 b d^5}-\frac{24 \left(a+b x^2\right)^4 (a d+3 b c)}{b d}+64 x^2 \left(a+b x^2\right)^4\right)}{640 b d \sqrt{a+b x^2}}","\frac{\sqrt{a+b x^2} \sqrt{c+d x^2} (b c-a d)^2 \left(3 a^2 d^2+14 a b c d+63 b^2 c^2\right)}{256 b^2 d^5}-\frac{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2} (b c-a d) \left(3 a^2 d^2+14 a b c d+63 b^2 c^2\right)}{384 b^2 d^4}+\frac{\left(a+b x^2\right)^{5/2} \sqrt{c+d x^2} \left(3 a^2 d^2+14 a b c d+63 b^2 c^2\right)}{480 b^2 d^3}-\frac{(b c-a d)^3 \left(3 a^2 d^2+14 a b c d+63 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right)}{256 b^{5/2} d^{11/2}}-\frac{3 \left(a+b x^2\right)^{7/2} \sqrt{c+d x^2} (a d+3 b c)}{80 b^2 d^2}+\frac{x^2 \left(a+b x^2\right)^{7/2} \sqrt{c+d x^2}}{10 b d}",1,"(Sqrt[c + d*x^2]*((-24*(3*b*c + a*d)*(a + b*x^2)^4)/(b*d) + 64*x^2*(a + b*x^2)^4 + (5*(b*c - a*d)^3*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*((2*d*(a + b*x^2))/(b*c - a*d) - (4*d^2*(a + b*x^2)^2)/(3*(b*c - a*d)^2) + (16*d^3*(a + b*x^2)^3)/(15*(b*c - a*d)^3) - (2*Sqrt[d]*Sqrt[a + b*x^2]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]])/(Sqrt[b*c - a*d]*Sqrt[(b*(c + d*x^2))/(b*c - a*d)])))/(4*b*d^5)))/(640*b*d*Sqrt[a + b*x^2])","A",1
955,1,214,237,0.585006,"\int \frac{x^3 \left(a+b x^2\right)^{5/2}}{\sqrt{c+d x^2}} \, dx","Integrate[(x^3*(a + b*x^2)^(5/2))/Sqrt[c + d*x^2],x]","\frac{b \sqrt{d} \sqrt{a+b x^2} \left(c+d x^2\right) \left(15 a^3 d^3+a^2 b d^2 \left(118 d x^2-191 c\right)+a b^2 d \left(265 c^2-172 c d x^2+136 d^2 x^4\right)+b^3 \left(-105 c^3+70 c^2 d x^2-56 c d^2 x^4+48 d^3 x^6\right)\right)+15 (a d+7 b c) (b c-a d)^{7/2} \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)}{384 b^2 d^{9/2} \sqrt{c+d x^2}}","\frac{5 (b c-a d)^3 (a d+7 b c) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right)}{128 b^{3/2} d^{9/2}}-\frac{5 \sqrt{a+b x^2} \sqrt{c+d x^2} (b c-a d)^2 (a d+7 b c)}{128 b d^4}+\frac{5 \left(a+b x^2\right)^{3/2} \sqrt{c+d x^2} (b c-a d) (a d+7 b c)}{192 b d^3}-\frac{\left(a+b x^2\right)^{5/2} \sqrt{c+d x^2} (a d+7 b c)}{48 b d^2}+\frac{\left(a+b x^2\right)^{7/2} \sqrt{c+d x^2}}{8 b d}",1,"(b*Sqrt[d]*Sqrt[a + b*x^2]*(c + d*x^2)*(15*a^3*d^3 + a^2*b*d^2*(-191*c + 118*d*x^2) + a*b^2*d*(265*c^2 - 172*c*d*x^2 + 136*d^2*x^4) + b^3*(-105*c^3 + 70*c^2*d*x^2 - 56*c*d^2*x^4 + 48*d^3*x^6)) + 15*(b*c - a*d)^(7/2)*(7*b*c + a*d)*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]])/(384*b^2*d^(9/2)*Sqrt[c + d*x^2])","A",1
956,1,164,164,0.56002,"\int \frac{x \left(a+b x^2\right)^{5/2}}{\sqrt{c+d x^2}} \, dx","Integrate[(x*(a + b*x^2)^(5/2))/Sqrt[c + d*x^2],x]","\frac{\sqrt{d} \sqrt{a+b x^2} \left(c+d x^2\right) \left(33 a^2 d^2+2 a b d \left(13 d x^2-20 c\right)+b^2 \left(15 c^2-10 c d x^2+8 d^2 x^4\right)\right)-\frac{15 (b c-a d)^{7/2} \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)}{b}}{48 d^{7/2} \sqrt{c+d x^2}}","-\frac{5 (b c-a d)^3 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right)}{16 \sqrt{b} d^{7/2}}+\frac{5 \sqrt{a+b x^2} \sqrt{c+d x^2} (b c-a d)^2}{16 d^3}-\frac{5 \left(a+b x^2\right)^{3/2} \sqrt{c+d x^2} (b c-a d)}{24 d^2}+\frac{\left(a+b x^2\right)^{5/2} \sqrt{c+d x^2}}{6 d}",1,"(Sqrt[d]*Sqrt[a + b*x^2]*(c + d*x^2)*(33*a^2*d^2 + 2*a*b*d*(-20*c + 13*d*x^2) + b^2*(15*c^2 - 10*c*d*x^2 + 8*d^2*x^4)) - (15*(b*c - a*d)^(7/2)*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]])/b)/(48*d^(7/2)*Sqrt[c + d*x^2])","A",1
957,1,213,187,0.6661025,"\int \frac{\left(a+b x^2\right)^{5/2}}{x \sqrt{c+d x^2}} \, dx","Integrate[(a + b*x^2)^(5/2)/(x*Sqrt[c + d*x^2]),x]","\frac{1}{8} \left(-\frac{8 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{\sqrt{c}}+\frac{\left(-15 a^3 d^3+25 a^2 b c d^2-13 a b^2 c^2 d+3 b^3 c^3\right) \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)}{d^{5/2} \sqrt{c+d x^2} \sqrt{b c-a d}}+\frac{b \sqrt{a+b x^2} \sqrt{c+d x^2} \left(9 a d-3 b c+2 b d x^2\right)}{d^2}\right)","-\frac{a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{\sqrt{c}}+\frac{\sqrt{b} \left(15 a^2 d^2-10 a b c d+3 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right)}{8 d^{5/2}}-\frac{b \sqrt{a+b x^2} \sqrt{c+d x^2} (3 b c-7 a d)}{8 d^2}+\frac{b \left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}}{4 d}",1,"((b*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(-3*b*c + 9*a*d + 2*b*d*x^2))/d^2 + ((3*b^3*c^3 - 13*a*b^2*c^2*d + 25*a^2*b*c*d^2 - 15*a^3*d^3)*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]])/(d^(5/2)*Sqrt[b*c - a*d]*Sqrt[c + d*x^2]) - (8*a^(5/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])])/Sqrt[c])/8","A",1
958,1,196,187,1.0603034,"\int \frac{\left(a+b x^2\right)^{5/2}}{x^3 \sqrt{c+d x^2}} \, dx","Integrate[(a + b*x^2)^(5/2)/(x^3*Sqrt[c + d*x^2]),x]","\frac{1}{2} \left(\frac{a^{3/2} (a d-5 b c) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{c^{3/2}}+\frac{\sqrt{a+b x^2} \sqrt{c+d x^2} \left(b^2 c x^2-a^2 d\right)}{c d x^2}-\frac{(b c-5 a d) (b c-a d)^{3/2} \left(\frac{b \left(c+d x^2\right)}{b c-a d}\right)^{3/2} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)}{d^{3/2} \left(c+d x^2\right)^{3/2}}\right)","-\frac{a^{3/2} (5 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 c^{3/2}}-\frac{b^{3/2} (b c-5 a d) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right)}{2 d^{3/2}}-\frac{a \left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}}{2 c x^2}+\frac{b \sqrt{a+b x^2} \sqrt{c+d x^2} (a d+b c)}{2 c d}",1,"((Sqrt[a + b*x^2]*(-(a^2*d) + b^2*c*x^2)*Sqrt[c + d*x^2])/(c*d*x^2) - ((b*c - 5*a*d)*(b*c - a*d)^(3/2)*((b*(c + d*x^2))/(b*c - a*d))^(3/2)*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]])/(d^(3/2)*(c + d*x^2)^(3/2)) + (a^(3/2)*(-5*b*c + a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])])/c^(3/2))/2","A",1
959,1,206,192,1.2479634,"\int \frac{\left(a+b x^2\right)^{5/2}}{x^5 \sqrt{c+d x^2}} \, dx","Integrate[(a + b*x^2)^(5/2)/(x^5*Sqrt[c + d*x^2]),x]","-\frac{\sqrt{a} \left(3 a^2 d^2-10 a b c d+15 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{8 c^{5/2}}+\frac{a \sqrt{a+b x^2} \sqrt{c+d x^2} \left(-2 a c+3 a d x^2-9 b c x^2\right)}{8 c^2 x^4}+\frac{(b c-a d)^{5/2} \left(\frac{b \left(c+d x^2\right)}{b c-a d}\right)^{5/2} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)}{\sqrt{d} \left(c+d x^2\right)^{5/2}}","-\frac{\sqrt{a} \left(3 a^2 d^2-10 a b c d+15 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{8 c^{5/2}}+\frac{b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right)}{\sqrt{d}}-\frac{a \sqrt{a+b x^2} \sqrt{c+d x^2} (7 b c-3 a d)}{8 c^2 x^2}-\frac{a \left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}}{4 c x^4}",1,"(a*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(-2*a*c - 9*b*c*x^2 + 3*a*d*x^2))/(8*c^2*x^4) + ((b*c - a*d)^(5/2)*((b*(c + d*x^2))/(b*c - a*d))^(5/2)*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]])/(Sqrt[d]*(c + d*x^2)^(5/2)) - (Sqrt[a]*(15*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])])/(8*c^(5/2))","A",1
960,1,379,553,1.9316303,"\int \frac{x^4 \left(a+b x^2\right)^{5/2}}{\sqrt{c+d x^2}} \, dx","Integrate[(x^4*(a + b*x^2)^(5/2))/Sqrt[c + d*x^2],x]","\frac{d x \sqrt{\frac{b}{a}} \left(a+b x^2\right) \left(c+d x^2\right) \left(5 a^3 d^3+15 a^2 b d^2 \left(5 d x^2-7 c\right)+a b^2 d \left(156 c^2-115 c d x^2+95 d^2 x^4\right)+b^3 \left(-64 c^3+48 c^2 d x^2-40 c d^2 x^4+35 d^3 x^6\right)\right)-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(5 a^4 d^4+130 a^3 b c d^3-399 a^2 b^2 c^2 d^2+392 a b^3 c^3 d-128 b^4 c^4\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(10 a^4 d^4+25 a^3 b c d^3-243 a^2 b^2 c^2 d^2+328 a b^3 c^3 d-128 b^4 c^4\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{315 b d^5 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}}","\frac{x^3 \sqrt{a+b x^2} \sqrt{c+d x^2} \left(75 a^2 d^2-115 a b c d+48 b^2 c^2\right)}{315 d^3}-\frac{x \sqrt{a+b x^2} \sqrt{c+d x^2} \left(-5 a^3 d^3+105 a^2 b c d^2-156 a b^2 c^2 d+64 b^3 c^3\right)}{315 b d^4}+\frac{c^{3/2} \sqrt{a+b x^2} \left(-5 a^3 d^3+105 a^2 b c d^2-156 a b^2 c^2 d+64 b^3 c^3\right) F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{315 b d^{9/2} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{x \sqrt{a+b x^2} \left(-10 a^4 d^4-25 a^3 b c d^3+243 a^2 b^2 c^2 d^2-328 a b^3 c^3 d+128 b^4 c^4\right)}{315 b^2 d^4 \sqrt{c+d x^2}}-\frac{\sqrt{c} \sqrt{a+b x^2} \left(-10 a^4 d^4-25 a^3 b c d^3+243 a^2 b^2 c^2 d^2-328 a b^3 c^3 d+128 b^4 c^4\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{315 b^2 d^{9/2} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{4 b x^5 \sqrt{a+b x^2} \sqrt{c+d x^2} (2 b c-3 a d)}{63 d^2}+\frac{b x^5 \left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}}{9 d}",1,"(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(5*a^3*d^3 + 15*a^2*b*d^2*(-7*c + 5*d*x^2) + a*b^2*d*(156*c^2 - 115*c*d*x^2 + 95*d^2*x^4) + b^3*(-64*c^3 + 48*c^2*d*x^2 - 40*c*d^2*x^4 + 35*d^3*x^6)) + I*c*(-128*b^4*c^4 + 328*a*b^3*c^3*d - 243*a^2*b^2*c^2*d^2 + 25*a^3*b*c*d^3 + 10*a^4*d^4)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*c*(-128*b^4*c^4 + 392*a*b^3*c^3*d - 399*a^2*b^2*c^2*d^2 + 130*a^3*b*c*d^3 + 5*a^4*d^4)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(315*b*Sqrt[b/a]*d^5*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","C",1
961,1,306,436,0.6407882,"\int \frac{x^2 \left(a+b x^2\right)^{5/2}}{\sqrt{c+d x^2}} \, dx","Integrate[(x^2*(a + b*x^2)^(5/2))/Sqrt[c + d*x^2],x]","\frac{d x \sqrt{\frac{b}{a}} \left(a+b x^2\right) \left(c+d x^2\right) \left(45 a^2 d^2+a b d \left(45 d x^2-61 c\right)+3 b^2 \left(8 c^2-6 c d x^2+5 d^2 x^4\right)\right)+4 i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(15 a^3 d^3-41 a^2 b c d^2+38 a b^2 c^2 d-12 b^3 c^3\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(15 a^3 d^3-103 a^2 b c d^2+128 a b^2 c^2 d-48 b^3 c^3\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{105 d^4 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}}","\frac{x \sqrt{a+b x^2} \sqrt{c+d x^2} \left(45 a^2 d^2-61 a b c d+24 b^2 c^2\right)}{105 d^3}-\frac{c^{3/2} \sqrt{a+b x^2} \left(45 a^2 d^2-61 a b c d+24 b^2 c^2\right) F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{105 d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{x \sqrt{a+b x^2} \left(-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right)}{105 b d^3 \sqrt{c+d x^2}}+\frac{\sqrt{c} \sqrt{a+b x^2} \left(-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{105 b d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{2 b x^3 \sqrt{a+b x^2} \sqrt{c+d x^2} (3 b c-5 a d)}{35 d^2}+\frac{b x^3 \left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}}{7 d}",1,"(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(45*a^2*d^2 + a*b*d*(-61*c + 45*d*x^2) + 3*b^2*(8*c^2 - 6*c*d*x^2 + 5*d^2*x^4)) - I*c*(-48*b^3*c^3 + 128*a*b^2*c^2*d - 103*a^2*b*c*d^2 + 15*a^3*d^3)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (4*I)*c*(-12*b^3*c^3 + 38*a*b^2*c^2*d - 41*a^2*b*c*d^2 + 15*a^3*d^3)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(105*Sqrt[b/a]*d^4*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","C",1
962,1,254,330,0.4769097,"\int \frac{\left(a+b x^2\right)^{5/2}}{x^2 \sqrt{c+d x^2}} \, dx","Integrate[(a + b*x^2)^(5/2)/(x^2*Sqrt[c + d*x^2]),x]","\frac{-2 i b c x \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(3 a^2 d^2-4 a b c d+b^2 c^2\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)-i b c x \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(3 a^2 d^2+7 a b c d-2 b^2 c^2\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+d \left(-\sqrt{\frac{b}{a}}\right) \left(a+b x^2\right) \left(c+d x^2\right) \left(3 a^2 d-b^2 c x^2\right)}{3 c d^2 x \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}}","\frac{\sqrt{a+b x^2} \left(-3 a^2 d^2-7 a b c d+2 b^2 c^2\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{3 \sqrt{c} d^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{x \sqrt{a+b x^2} \left(\frac{3 a^2 d}{c}+7 a b-\frac{2 b^2 c}{d}\right)}{3 \sqrt{c+d x^2}}-\frac{b \sqrt{c} \sqrt{a+b x^2} (b c-9 a d) F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{3 d^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{a \left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}}{c x}+\frac{b x \sqrt{a+b x^2} \sqrt{c+d x^2} (3 a d+b c)}{3 c d}",1,"(-(Sqrt[b/a]*d*(a + b*x^2)*(3*a^2*d - b^2*c*x^2)*(c + d*x^2)) - I*b*c*(-2*b^2*c^2 + 7*a*b*c*d + 3*a^2*d^2)*x*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - (2*I)*b*c*(b^2*c^2 - 4*a*b*c*d + 3*a^2*d^2)*x*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(3*Sqrt[b/a]*c*d^2*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","C",1
963,1,261,336,0.4823433,"\int \frac{\left(a+b x^2\right)^{5/2}}{x^4 \sqrt{c+d x^2}} \, dx","Integrate[(a + b*x^2)^(5/2)/(x^4*Sqrt[c + d*x^2]),x]","\frac{-i b c x^3 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(a^2 d^2+2 a b c d-3 b^2 c^2\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+i b c x^3 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(2 a^2 d^2-7 a b c d-3 b^2 c^2\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+a d \sqrt{\frac{b}{a}} \left(a+b x^2\right) \left(c+d x^2\right) \left(-a c+2 a d x^2-7 b c x^2\right)}{3 c^2 d x^3 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}}","\frac{x \sqrt{a+b x^2} \left(-2 a^2 d^2+7 a b c d+3 b^2 c^2\right)}{3 c^2 \sqrt{c+d x^2}}-\frac{\sqrt{a+b x^2} \left(-2 a^2 d^2+7 a b c d+3 b^2 c^2\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{3 c^{3/2} \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{2 a \sqrt{a+b x^2} \sqrt{c+d x^2} (3 b c-a d)}{3 c^2 x}+\frac{b \sqrt{a+b x^2} (9 b c-a d) F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{3 \sqrt{c} \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{a \left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}}{3 c x^3}",1,"(a*Sqrt[b/a]*d*(a + b*x^2)*(c + d*x^2)*(-(a*c) - 7*b*c*x^2 + 2*a*d*x^2) + I*b*c*(-3*b^2*c^2 - 7*a*b*c*d + 2*a^2*d^2)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*b*c*(-3*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(3*Sqrt[b/a]*c^2*d*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","C",1
964,1,92,99,0.0971433,"\int \frac{x^4 \sqrt{-1+3 x^2}}{\sqrt{2-3 x^2}} \, dx","Integrate[(x^4*Sqrt[-1 + 3*x^2])/Sqrt[2 - 3*x^2],x]","\frac{10 \sqrt{3-9 x^2} F\left(\left.\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)-24 \sqrt{3-9 x^2} E\left(\left.\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)-3 x \sqrt{2-3 x^2} \left(27 x^4+12 x^2-7\right)}{405 \sqrt{3 x^2-1}}","-\frac{7}{135} \sqrt{2-3 x^2} \sqrt{3 x^2-1} x-\frac{1}{15} \sqrt{2-3 x^2} \sqrt{3 x^2-1} x^3-\frac{2 F\left(\left.\cos ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)}{27 \sqrt{3}}-\frac{8 E\left(\left.\cos ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)}{45 \sqrt{3}}",1,"(-3*x*Sqrt[2 - 3*x^2]*(-7 + 12*x^2 + 27*x^4) - 24*Sqrt[3 - 9*x^2]*EllipticE[ArcSin[Sqrt[3/2]*x], 2] + 10*Sqrt[3 - 9*x^2]*EllipticF[ArcSin[Sqrt[3/2]*x], 2])/(405*Sqrt[-1 + 3*x^2])","A",1
965,1,44,65,0.026253,"\int \frac{x^3 \sqrt{-1+3 x^2}}{\sqrt{2-3 x^2}} \, dx","Integrate[(x^3*Sqrt[-1 + 3*x^2])/Sqrt[2 - 3*x^2],x]","\frac{1}{72} \left(-7 \sin ^{-1}\left(\sqrt{2-3 x^2}\right)-\sqrt{-9 x^4+9 x^2-2} \left(6 x^2+5\right)\right)","-\frac{1}{36} \sqrt{2-3 x^2} \left(3 x^2-1\right)^{3/2}-\frac{7}{72} \sqrt{2-3 x^2} \sqrt{3 x^2-1}-\frac{7}{144} \sin ^{-1}\left(3-6 x^2\right)",1,"(-((5 + 6*x^2)*Sqrt[-2 + 9*x^2 - 9*x^4]) - 7*ArcSin[Sqrt[2 - 3*x^2]])/72","A",1
966,1,86,70,0.0737505,"\int \frac{x^2 \sqrt{-1+3 x^2}}{\sqrt{2-3 x^2}} \, dx","Integrate[(x^2*Sqrt[-1 + 3*x^2])/Sqrt[2 - 3*x^2],x]","\frac{3 x \sqrt{2-3 x^2} \left(1-3 x^2\right)+\sqrt{3-9 x^2} F\left(\left.\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)-3 \sqrt{3-9 x^2} E\left(\left.\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)}{27 \sqrt{3 x^2-1}}","-\frac{1}{9} \sqrt{2-3 x^2} \sqrt{3 x^2-1} x-\frac{F\left(\left.\cos ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)}{9 \sqrt{3}}-\frac{E\left(\left.\cos ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)}{3 \sqrt{3}}",1,"(3*x*(1 - 3*x^2)*Sqrt[2 - 3*x^2] - 3*Sqrt[3 - 9*x^2]*EllipticE[ArcSin[Sqrt[3/2]*x], 2] + Sqrt[3 - 9*x^2]*EllipticF[ArcSin[Sqrt[3/2]*x], 2])/(27*Sqrt[-1 + 3*x^2])","A",1
967,1,37,39,0.0249885,"\int \frac{x \sqrt{-1+3 x^2}}{\sqrt{2-3 x^2}} \, dx","Integrate[(x*Sqrt[-1 + 3*x^2])/Sqrt[2 - 3*x^2],x]","\frac{1}{6} \left(-\sin ^{-1}\left(\sqrt{2-3 x^2}\right)-\sqrt{-9 x^4+9 x^2-2}\right)","-\frac{1}{6} \sqrt{2-3 x^2} \sqrt{3 x^2-1}-\frac{1}{12} \sin ^{-1}\left(3-6 x^2\right)",1,"(-Sqrt[-2 + 9*x^2 - 9*x^4] - ArcSin[Sqrt[2 - 3*x^2]])/6","A",1
968,1,127,241,0.1329735,"\int \frac{x^2 \sqrt{2+b x^2}}{\sqrt{3+d x^2}} \, dx","Integrate[(x^2*Sqrt[2 + b*x^2])/Sqrt[3 + d*x^2],x]","\frac{\sqrt{b} d x \sqrt{b x^2+2} \sqrt{d x^2+3}-2 i \sqrt{3} (3 b-2 d) F\left(i \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{2}}\right)|\frac{2 d}{3 b}\right)+2 i \sqrt{3} (3 b-d) E\left(i \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{2}}\right)|\frac{2 d}{3 b}\right)}{3 \sqrt{b} d^2}","-\frac{\sqrt{2} \sqrt{b x^2+2} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{3}}\right)|1-\frac{3 b}{2 d}\right)}{d^{3/2} \sqrt{d x^2+3} \sqrt{\frac{b x^2+2}{d x^2+3}}}+\frac{2 \sqrt{2} (3 b-d) \sqrt{b x^2+2} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{3}}\right)|1-\frac{3 b}{2 d}\right)}{3 b d^{3/2} \sqrt{d x^2+3} \sqrt{\frac{b x^2+2}{d x^2+3}}}+\frac{x \sqrt{b x^2+2} \sqrt{d x^2+3}}{3 d}-\frac{2 x (3 b-d) \sqrt{b x^2+2}}{3 b d \sqrt{d x^2+3}}",1,"(Sqrt[b]*d*x*Sqrt[2 + b*x^2]*Sqrt[3 + d*x^2] + (2*I)*Sqrt[3]*(3*b - d)*EllipticE[I*ArcSinh[(Sqrt[b]*x)/Sqrt[2]], (2*d)/(3*b)] - (2*I)*Sqrt[3]*(3*b - 2*d)*EllipticF[I*ArcSinh[(Sqrt[b]*x)/Sqrt[2]], (2*d)/(3*b)])/(3*Sqrt[b]*d^2)","C",1
969,1,154,141,0.2748117,"\int \frac{x^5}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","Integrate[x^5/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]","\frac{\sqrt{b c-a d} \left(3 a^2 d^2+2 a b c d+3 b^2 c^2\right) \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)+b \sqrt{d} \sqrt{a+b x^2} \left(c+d x^2\right) \left(-3 a d-3 b c+2 b d x^2\right)}{8 b^3 d^{5/2} \sqrt{c+d x^2}}","-\frac{\left(4 a b c d-3 (a d+b c)^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right)}{8 b^{5/2} d^{5/2}}-\frac{3 \sqrt{a+b x^2} \sqrt{c+d x^2} (a d+b c)}{8 b^2 d^2}+\frac{x^2 \sqrt{a+b x^2} \sqrt{c+d x^2}}{4 b d}",1,"(b*Sqrt[d]*Sqrt[a + b*x^2]*(c + d*x^2)*(-3*b*c - 3*a*d + 2*b*d*x^2) + Sqrt[b*c - a*d]*(3*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]])/(8*b^3*d^(5/2)*Sqrt[c + d*x^2])","A",1
970,1,123,88,0.1786165,"\int \frac{x^3}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","Integrate[x^3/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]","\frac{b \sqrt{d} \sqrt{a+b x^2} \left(c+d x^2\right)-\sqrt{b c-a d} (a d+b c) \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)}{2 b^2 d^{3/2} \sqrt{c+d x^2}}","\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{2 b d}-\frac{(a d+b c) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right)}{2 b^{3/2} d^{3/2}}",1,"(b*Sqrt[d]*Sqrt[a + b*x^2]*(c + d*x^2) - Sqrt[b*c - a*d]*(b*c + a*d)*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]])/(2*b^2*d^(3/2)*Sqrt[c + d*x^2])","A",1
971,1,82,45,0.0659234,"\int \frac{x}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","Integrate[x/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]","\frac{\sqrt{c+d x^2} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)}{\sqrt{d} \sqrt{b c-a d} \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right)}{\sqrt{b} \sqrt{d}}",1,"(Sqrt[c + d*x^2]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]])/(Sqrt[d]*Sqrt[b*c - a*d]*Sqrt[(b*(c + d*x^2))/(b*c - a*d)])","A",1
972,1,46,46,0.0126788,"\int \frac{1}{x \sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","Integrate[1/(x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{\sqrt{a} \sqrt{c}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{\sqrt{a} \sqrt{c}}",1,"-(ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])]/(Sqrt[a]*Sqrt[c]))","A",1
973,1,91,91,0.0528628,"\int \frac{1}{x^3 \sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","Integrate[1/(x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]","\frac{(a d+b c) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 a^{3/2} c^{3/2}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{2 a c x^2}","\frac{(a d+b c) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 a^{3/2} c^{3/2}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{2 a c x^2}",1,"-1/2*(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(a*c*x^2) + ((b*c + a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(3/2)*c^(3/2))","A",1
974,1,126,149,0.0885745,"\int \frac{1}{x^5 \sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","Integrate[1/(x^5*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]","\frac{\sqrt{a+b x^2} \sqrt{c+d x^2} \left(-2 a c+3 a d x^2+3 b c x^2\right)}{8 a^2 c^2 x^4}-\frac{\left(3 a^2 d^2+2 a b c d+3 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{8 a^{5/2} c^{5/2}}","\frac{3 \sqrt{a+b x^2} \sqrt{c+d x^2} (a d+b c)}{8 a^2 c^2 x^2}-\frac{\left(3 a^2 d^2+2 a b c d+3 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{8 a^{5/2} c^{5/2}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{4 a c x^4}",1,"(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(-2*a*c + 3*b*c*x^2 + 3*a*d*x^2))/(8*a^2*c^2*x^4) - ((3*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])])/(8*a^(5/2)*c^(5/2))","A",1
975,1,249,342,0.5080872,"\int \frac{x^6}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","Integrate[x^6/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]","\frac{i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(4 a^2 d^2+3 a b c d+8 b^2 c^2\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(8 a^2 d^2+7 a b c d+8 b^2 c^2\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+d x \left(-\sqrt{\frac{b}{a}}\right) \left(a+b x^2\right) \left(c+d x^2\right) \left(4 a d+4 b c-3 b d x^2\right)}{15 a^2 d^3 \left(\frac{b}{a}\right)^{5/2} \sqrt{a+b x^2} \sqrt{c+d x^2}}","\frac{x \sqrt{a+b x^2} \left(8 a^2 d^2+7 a b c d+8 b^2 c^2\right)}{15 b^3 d^2 \sqrt{c+d x^2}}-\frac{\sqrt{c} \sqrt{a+b x^2} \left(8 a^2 d^2+7 a b c d+8 b^2 c^2\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{15 b^3 d^{5/2} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{4 c^{3/2} \sqrt{a+b x^2} (a d+b c) F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{15 b^2 d^{5/2} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{4 x \sqrt{a+b x^2} \sqrt{c+d x^2} (a d+b c)}{15 b^2 d^2}+\frac{x^3 \sqrt{a+b x^2} \sqrt{c+d x^2}}{5 b d}",1,"(-(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(4*b*c + 4*a*d - 3*b*d*x^2)) - I*c*(8*b^2*c^2 + 7*a*b*c*d + 8*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + I*c*(8*b^2*c^2 + 3*a*b*c*d + 4*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(15*a^2*(b/a)^(5/2)*d^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","C",1
976,1,201,261,0.293637,"\int \frac{x^4}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","Integrate[x^4/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]","\frac{d x \sqrt{\frac{b}{a}} \left(a+b x^2\right) \left(c+d x^2\right)-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d+2 b c) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+2 i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d+b c) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{3 b d^2 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}}","\frac{2 \sqrt{c} \sqrt{a+b x^2} (a d+b c) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{3 b^2 d^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{2 x \sqrt{a+b x^2} (a d+b c)}{3 b^2 d \sqrt{c+d x^2}}-\frac{c^{3/2} \sqrt{a+b x^2} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{3 b d^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{x \sqrt{a+b x^2} \sqrt{c+d x^2}}{3 b d}",1,"(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2) + (2*I)*c*(b*c + a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*c*(2*b*c + a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(3*b*Sqrt[b/a]*d^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","C",1
977,1,122,116,0.0737141,"\int \frac{x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","Integrate[x^2/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]","-\frac{i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)-F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)\right)}{d \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}}","\frac{x \sqrt{a+b x^2}}{b \sqrt{c+d x^2}}-\frac{\sqrt{c} \sqrt{a+b x^2} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{b \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}",1,"((-I)*c*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*(EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(Sqrt[b/a]*d*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","C",1
978,1,146,153,0.3487492,"\int \frac{1}{x^2 \sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","Integrate[1/(x^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]","\frac{-\frac{\left(a+b x^2\right) \left(c+d x^2\right)}{c x}-i a \sqrt{\frac{b}{a}} \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)-F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)\right)}{a \sqrt{a+b x^2} \sqrt{c+d x^2}}","\frac{d x \sqrt{a+b x^2}}{a c \sqrt{c+d x^2}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{a c x}-\frac{\sqrt{d} \sqrt{a+b x^2} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{a \sqrt{c} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}",1,"(-(((a + b*x^2)*(c + d*x^2))/(c*x)) - I*a*Sqrt[b/a]*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*(EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(a*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","C",1
979,1,229,307,0.3581431,"\int \frac{1}{x^4 \sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","Integrate[1/(x^4*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]","\frac{\sqrt{\frac{b}{a}} \left(a+b x^2\right) \left(c+d x^2\right) \left(-a c+2 a d x^2+2 b c x^2\right)-i b c x^3 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d+2 b c) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+2 i b c x^3 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d+b c) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{3 a^2 c^2 x^3 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}}","\frac{2 \sqrt{d} \sqrt{a+b x^2} (a d+b c) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{3 a^2 c^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{2 \sqrt{a+b x^2} \sqrt{c+d x^2} (a d+b c)}{3 a^2 c^2 x}-\frac{2 d x \sqrt{a+b x^2} (a d+b c)}{3 a^2 c^2 \sqrt{c+d x^2}}-\frac{b \sqrt{d} \sqrt{a+b x^2} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{3 a^2 \sqrt{c} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{3 a c x^3}",1,"(Sqrt[b/a]*(a + b*x^2)*(c + d*x^2)*(-(a*c) + 2*b*c*x^2 + 2*a*d*x^2) + (2*I)*b*c*(b*c + a*d)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*b*c*(2*b*c + a*d)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(3*a^2*Sqrt[b/a]*c^2*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","C",1
980,1,185,129,0.3539342,"\int \frac{x^5}{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}} \, dx","Integrate[x^5/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]),x]","\frac{\sqrt{a+b x^2} \sqrt{b c-a d} \left(-3 a^2 d^2+2 a b c d+b^2 c^2\right) \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)-b \sqrt{d} \left(c+d x^2\right) \left(-3 a^2 d+a b \left(c-d x^2\right)+b^2 c x^2\right)}{2 b^3 d^{3/2} \sqrt{a+b x^2} \sqrt{c+d x^2} (a d-b c)}","-\frac{a^2 \sqrt{c+d x^2}}{b^2 \sqrt{a+b x^2} (b c-a d)}-\frac{(3 a d+b c) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right)}{2 b^{5/2} d^{3/2}}+\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{2 b^2 d}",1,"(-(b*Sqrt[d]*(c + d*x^2)*(-3*a^2*d + b^2*c*x^2 + a*b*(c - d*x^2))) + Sqrt[b*c - a*d]*(b^2*c^2 + 2*a*b*c*d - 3*a^2*d^2)*Sqrt[a + b*x^2]*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]])/(2*b^3*d^(3/2)*(-(b*c) + a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","A",1
981,1,118,83,0.5178793,"\int \frac{x^3}{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}} \, dx","Integrate[x^3/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]),x]","\frac{\frac{a b \left(c+d x^2\right)}{\sqrt{a+b x^2} (b c-a d)}+\frac{\sqrt{b c-a d} \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)}{\sqrt{d}}}{b^2 \sqrt{c+d x^2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right)}{b^{3/2} \sqrt{d}}+\frac{a \sqrt{c+d x^2}}{b \sqrt{a+b x^2} (b c-a d)}",1,"((a*b*(c + d*x^2))/((b*c - a*d)*Sqrt[a + b*x^2]) + (Sqrt[b*c - a*d]*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]])/Sqrt[d])/(b^2*Sqrt[c + d*x^2])","A",1
982,1,33,34,0.0110963,"\int \frac{x}{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}} \, dx","Integrate[x/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]),x]","\frac{\sqrt{c+d x^2}}{\sqrt{a+b x^2} (a d-b c)}","-\frac{\sqrt{c+d x^2}}{\sqrt{a+b x^2} (b c-a d)}",1,"Sqrt[c + d*x^2]/((-(b*c) + a*d)*Sqrt[a + b*x^2])","A",1
983,1,214,137,0.4833871,"\int \frac{x^5}{\left(a+b x^2\right)^{5/2} \sqrt{c+d x^2}} \, dx","Integrate[x^5/((a + b*x^2)^(5/2)*Sqrt[c + d*x^2]),x]","\frac{\sqrt{c+d x^2} \left(\frac{\left(a+b x^2\right) \left(3 b^2 c^2-a^2 d^2\right)}{d (b c-a d)^2}+\frac{a^2}{a d-b c}-\frac{3 \left(a+b x^2\right) \left(\sqrt{b c-a d} \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}}-\sqrt{d} \sqrt{a+b x^2} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)\right)}{d \sqrt{b c-a d} \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}}}\right)}{3 b^2 \left(a+b x^2\right)^{3/2}}","-\frac{a^2 \sqrt{c+d x^2}}{3 b^2 \left(a+b x^2\right)^{3/2} (b c-a d)}+\frac{\tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right)}{b^{5/2} \sqrt{d}}+\frac{2 a \sqrt{c+d x^2} (3 b c-2 a d)}{3 b^2 \sqrt{a+b x^2} (b c-a d)^2}",1,"(Sqrt[c + d*x^2]*(a^2/(-(b*c) + a*d) + ((3*b^2*c^2 - a^2*d^2)*(a + b*x^2))/(d*(b*c - a*d)^2) - (3*(a + b*x^2)*(Sqrt[b*c - a*d]*Sqrt[(b*(c + d*x^2))/(b*c - a*d)] - Sqrt[d]*Sqrt[a + b*x^2]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]]))/(d*Sqrt[b*c - a*d]*Sqrt[(b*(c + d*x^2))/(b*c - a*d)])))/(3*b^2*(a + b*x^2)^(3/2))","A",1
984,1,54,89,0.0273843,"\int \frac{x^3}{\left(a+b x^2\right)^{5/2} \sqrt{c+d x^2}} \, dx","Integrate[x^3/((a + b*x^2)^(5/2)*Sqrt[c + d*x^2]),x]","\frac{\sqrt{c+d x^2} \left(-2 a c+a d x^2-3 b c x^2\right)}{3 \left(a+b x^2\right)^{3/2} (b c-a d)^2}","\frac{a \sqrt{c+d x^2}}{3 b \left(a+b x^2\right)^{3/2} (b c-a d)}-\frac{\sqrt{c+d x^2} (3 b c-a d)}{3 b \sqrt{a+b x^2} (b c-a d)^2}",1,"(Sqrt[c + d*x^2]*(-2*a*c - 3*b*c*x^2 + a*d*x^2))/(3*(b*c - a*d)^2*(a + b*x^2)^(3/2))","A",1
985,1,52,74,0.0164243,"\int \frac{x}{\left(a+b x^2\right)^{5/2} \sqrt{c+d x^2}} \, dx","Integrate[x/((a + b*x^2)^(5/2)*Sqrt[c + d*x^2]),x]","\frac{\sqrt{c+d x^2} \left(3 a d-b c+2 b d x^2\right)}{3 \left(a+b x^2\right)^{3/2} (b c-a d)^2}","\frac{2 d \sqrt{c+d x^2}}{3 \sqrt{a+b x^2} (b c-a d)^2}-\frac{\sqrt{c+d x^2}}{3 \left(a+b x^2\right)^{3/2} (b c-a d)}",1,"(Sqrt[c + d*x^2]*(-(b*c) + 3*a*d + 2*b*d*x^2))/(3*(b*c - a*d)^2*(a + b*x^2)^(3/2))","A",1
986,1,91,154,0.0565763,"\int \frac{x^5}{\left(a+b x^2\right)^{7/2} \sqrt{c+d x^2}} \, dx","Integrate[x^5/((a + b*x^2)^(7/2)*Sqrt[c + d*x^2]),x]","-\frac{\sqrt{c+d x^2} \left(a^2 \left(8 c^2-4 c d x^2+3 d^2 x^4\right)+10 a b c x^2 \left(2 c-d x^2\right)+15 b^2 c^2 x^4\right)}{15 \left(a+b x^2\right)^{5/2} (b c-a d)^3}","-\frac{\sqrt{c+d x^2} \left(3 a^2 d^2-10 a b c d+15 b^2 c^2\right)}{15 b^2 \sqrt{a+b x^2} (b c-a d)^3}-\frac{a^2 \sqrt{c+d x^2}}{5 b^2 \left(a+b x^2\right)^{5/2} (b c-a d)}+\frac{2 a \sqrt{c+d x^2} (5 b c-3 a d)}{15 b^2 \left(a+b x^2\right)^{3/2} (b c-a d)^2}",1,"-1/15*(Sqrt[c + d*x^2]*(15*b^2*c^2*x^4 + 10*a*b*c*x^2*(2*c - d*x^2) + a^2*(8*c^2 - 4*c*d*x^2 + 3*d^2*x^4)))/((b*c - a*d)^3*(a + b*x^2)^(5/2))","A",1
987,1,91,138,0.0423332,"\int \frac{x^3}{\left(a+b x^2\right)^{7/2} \sqrt{c+d x^2}} \, dx","Integrate[x^3/((a + b*x^2)^(7/2)*Sqrt[c + d*x^2]),x]","\frac{\sqrt{c+d x^2} \left(-5 a^2 d \left(d x^2-2 c\right)-2 a b \left(c^2-13 c d x^2+d^2 x^4\right)-5 b^2 c x^2 \left(c-2 d x^2\right)\right)}{15 \left(a+b x^2\right)^{5/2} (b c-a d)^3}","\frac{a \sqrt{c+d x^2}}{5 b \left(a+b x^2\right)^{5/2} (b c-a d)}+\frac{2 d \sqrt{c+d x^2} (5 b c-a d)}{15 b \sqrt{a+b x^2} (b c-a d)^3}-\frac{\sqrt{c+d x^2} (5 b c-a d)}{15 b \left(a+b x^2\right)^{3/2} (b c-a d)^2}",1,"(Sqrt[c + d*x^2]*(-5*b^2*c*x^2*(c - 2*d*x^2) - 5*a^2*d*(-2*c + d*x^2) - 2*a*b*(c^2 - 13*c*d*x^2 + d^2*x^4)))/(15*(b*c - a*d)^3*(a + b*x^2)^(5/2))","A",1
988,1,83,113,0.0291874,"\int \frac{x}{\left(a+b x^2\right)^{7/2} \sqrt{c+d x^2}} \, dx","Integrate[x/((a + b*x^2)^(7/2)*Sqrt[c + d*x^2]),x]","-\frac{\sqrt{c+d x^2} \left(15 a^2 d^2-10 a b d \left(c-2 d x^2\right)+b^2 \left(3 c^2-4 c d x^2+8 d^2 x^4\right)\right)}{15 \left(a+b x^2\right)^{5/2} (b c-a d)^3}","-\frac{8 d^2 \sqrt{c+d x^2}}{15 \sqrt{a+b x^2} (b c-a d)^3}+\frac{4 d \sqrt{c+d x^2}}{15 \left(a+b x^2\right)^{3/2} (b c-a d)^2}-\frac{\sqrt{c+d x^2}}{5 \left(a+b x^2\right)^{5/2} (b c-a d)}",1,"-1/15*(Sqrt[c + d*x^2]*(15*a^2*d^2 - 10*a*b*d*(c - 2*d*x^2) + b^2*(3*c^2 - 4*c*d*x^2 + 8*d^2*x^4)))/((b*c - a*d)^3*(a + b*x^2)^(5/2))","A",1
989,1,151,217,0.0800133,"\int \frac{x^5}{\left(a+b x^2\right)^{9/2} \sqrt{c+d x^2}} \, dx","Integrate[x^5/((a + b*x^2)^(9/2)*Sqrt[c + d*x^2]),x]","\frac{\sqrt{c+d x^2} \left(7 a^3 d \left(8 c^2-4 c d x^2+3 d^2 x^4\right)+a^2 b \left(-8 c^3+200 c^2 d x^2-101 c d^2 x^4+6 d^3 x^6\right)-7 a b^2 c x^2 \left(4 c^2-37 c d x^2+4 d^2 x^4\right)-35 b^3 c^2 x^4 \left(c-2 d x^2\right)\right)}{105 \left(a+b x^2\right)^{7/2} (b c-a d)^4}","\frac{2 d \sqrt{c+d x^2} \left(3 a^2 d^2-14 a b c d+35 b^2 c^2\right)}{105 b^2 \sqrt{a+b x^2} (b c-a d)^4}-\frac{\sqrt{c+d x^2} \left(3 a^2 d^2-14 a b c d+35 b^2 c^2\right)}{105 b^2 \left(a+b x^2\right)^{3/2} (b c-a d)^3}-\frac{a^2 \sqrt{c+d x^2}}{7 b^2 \left(a+b x^2\right)^{7/2} (b c-a d)}+\frac{2 a \sqrt{c+d x^2} (7 b c-4 a d)}{35 b^2 \left(a+b x^2\right)^{5/2} (b c-a d)^2}",1,"(Sqrt[c + d*x^2]*(-35*b^3*c^2*x^4*(c - 2*d*x^2) + 7*a^3*d*(8*c^2 - 4*c*d*x^2 + 3*d^2*x^4) - 7*a*b^2*c*x^2*(4*c^2 - 37*c*d*x^2 + 4*d^2*x^4) + a^2*b*(-8*c^3 + 200*c^2*d*x^2 - 101*c*d^2*x^4 + 6*d^3*x^6)))/(105*(b*c - a*d)^4*(a + b*x^2)^(7/2))","A",1
990,1,108,47,0.0920671,"\int \frac{x}{\sqrt{a-b x^2} \sqrt{c+d x^2}} \, dx","Integrate[x/(Sqrt[a - b*x^2]*Sqrt[c + d*x^2]),x]","\frac{\sqrt{-b} \sqrt{-a d-b c} \sqrt{\frac{b \left(c+d x^2\right)}{a d+b c}} \sin ^{-1}\left(\frac{\sqrt{-b} \sqrt{d} \sqrt{a-b x^2}}{\sqrt{b} \sqrt{-a d-b c}}\right)}{b^{3/2} \sqrt{d} \sqrt{c+d x^2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a-b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right)}{\sqrt{b} \sqrt{d}}",1,"(Sqrt[-b]*Sqrt[-(b*c) - a*d]*Sqrt[(b*(c + d*x^2))/(b*c + a*d)]*ArcSin[(Sqrt[-b]*Sqrt[d]*Sqrt[a - b*x^2])/(Sqrt[b]*Sqrt[-(b*c) - a*d])])/(b^(3/2)*Sqrt[d]*Sqrt[c + d*x^2])","B",1
991,1,109,48,0.092375,"\int \frac{x}{\sqrt{a-b x^2} \sqrt{c-d x^2}} \, dx","Integrate[x/(Sqrt[a - b*x^2]*Sqrt[c - d*x^2]),x]","\frac{\sqrt{-b} \sqrt{a d-b c} \sqrt{\frac{b \left(c-d x^2\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{-b} \sqrt{d} \sqrt{a-b x^2}}{\sqrt{b} \sqrt{a d-b c}}\right)}{b^{3/2} \sqrt{d} \sqrt{c-d x^2}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-b x^2}}{\sqrt{b} \sqrt{c-d x^2}}\right)}{\sqrt{b} \sqrt{d}}",1,"(Sqrt[-b]*Sqrt[-(b*c) + a*d]*Sqrt[(b*(c - d*x^2))/(b*c - a*d)]*ArcSinh[(Sqrt[-b]*Sqrt[d]*Sqrt[a - b*x^2])/(Sqrt[b]*Sqrt[-(b*c) + a*d])])/(b^(3/2)*Sqrt[d]*Sqrt[c - d*x^2])","B",1
992,1,72,110,0.0500866,"\int \frac{x^2}{\sqrt{2+b x^2} \sqrt{3+d x^2}} \, dx","Integrate[x^2/(Sqrt[2 + b*x^2]*Sqrt[3 + d*x^2]),x]","-\frac{i \sqrt{3} \left(E\left(i \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{2}}\right)|\frac{2 d}{3 b}\right)-F\left(i \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{2}}\right)|\frac{2 d}{3 b}\right)\right)}{\sqrt{b} d}","\frac{x \sqrt{b x^2+2}}{b \sqrt{d x^2+3}}-\frac{\sqrt{2} \sqrt{b x^2+2} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{3}}\right)|1-\frac{3 b}{2 d}\right)}{b \sqrt{d} \sqrt{d x^2+3} \sqrt{\frac{b x^2+2}{d x^2+3}}}",1,"((-I)*Sqrt[3]*(EllipticE[I*ArcSinh[(Sqrt[b]*x)/Sqrt[2]], (2*d)/(3*b)] - EllipticF[I*ArcSinh[(Sqrt[b]*x)/Sqrt[2]], (2*d)/(3*b)]))/(Sqrt[b]*d)","C",1
993,1,59,87,0.0535811,"\int \frac{x^2}{\sqrt{4-x^2} \sqrt{c+d x^2}} \, dx","Integrate[x^2/(Sqrt[4 - x^2]*Sqrt[c + d*x^2]),x]","\frac{c \sqrt{\frac{d x^2}{c}+1} \left(E\left(\sin ^{-1}\left(\frac{x}{2}\right)|-\frac{4 d}{c}\right)-F\left(\sin ^{-1}\left(\frac{x}{2}\right)|-\frac{4 d}{c}\right)\right)}{d \sqrt{c+d x^2}}","\frac{\sqrt{c+d x^2} E\left(\sin ^{-1}\left(\frac{x}{2}\right)|-\frac{4 d}{c}\right)}{d \sqrt{\frac{d x^2}{c}+1}}-\frac{c \sqrt{\frac{d x^2}{c}+1} F\left(\sin ^{-1}\left(\frac{x}{2}\right)|-\frac{4 d}{c}\right)}{d \sqrt{c+d x^2}}",1,"(c*Sqrt[1 + (d*x^2)/c]*(EllipticE[ArcSin[x/2], (-4*d)/c] - EllipticF[ArcSin[x/2], (-4*d)/c]))/(d*Sqrt[c + d*x^2])","A",1
994,1,70,88,0.0579491,"\int \frac{x^2}{\sqrt{4+x^2} \sqrt{c+d x^2}} \, dx","Integrate[x^2/(Sqrt[4 + x^2]*Sqrt[c + d*x^2]),x]","-\frac{i c \sqrt{\frac{d x^2}{c}+1} \left(E\left(i \sinh ^{-1}\left(\frac{x}{2}\right)|\frac{4 d}{c}\right)-F\left(i \sinh ^{-1}\left(\frac{x}{2}\right)|\frac{4 d}{c}\right)\right)}{d \sqrt{c+d x^2}}","\frac{x \sqrt{c+d x^2}}{d \sqrt{x^2+4}}-\frac{\sqrt{c+d x^2} E\left(\tan ^{-1}\left(\frac{x}{2}\right)|1-\frac{4 d}{c}\right)}{d \sqrt{x^2+4} \sqrt{\frac{c+d x^2}{c \left(x^2+4\right)}}}",1,"((-I)*c*Sqrt[1 + (d*x^2)/c]*(EllipticE[I*ArcSinh[x/2], (4*d)/c] - EllipticF[I*ArcSinh[x/2], (4*d)/c]))/(d*Sqrt[c + d*x^2])","C",1
995,1,24,31,0.0336401,"\int \frac{x^2}{\sqrt{1-x^2} \sqrt{2+3 x^2}} \, dx","Integrate[x^2/(Sqrt[1 - x^2]*Sqrt[2 + 3*x^2]),x]","\frac{1}{3} \sqrt{2} \left(E\left(\sin ^{-1}(x)|-\frac{3}{2}\right)-F\left(\sin ^{-1}(x)|-\frac{3}{2}\right)\right)","\frac{1}{3} \sqrt{2} E\left(\sin ^{-1}(x)|-\frac{3}{2}\right)-\frac{1}{3} \sqrt{2} F\left(\sin ^{-1}(x)|-\frac{3}{2}\right)",1,"(Sqrt[2]*(EllipticE[ArcSin[x], -3/2] - EllipticF[ArcSin[x], -3/2]))/3","A",1
996,1,37,31,0.0380832,"\int \frac{x^2}{\sqrt{2-3 x^2} \sqrt{1-x^2}} \, dx","Integrate[x^2/(Sqrt[2 - 3*x^2]*Sqrt[1 - x^2]),x]","\frac{F\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|\frac{2}{3}\right)-E\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|\frac{2}{3}\right)}{\sqrt{3}}","\frac{1}{3} \sqrt{2} F\left(\sin ^{-1}(x)|\frac{3}{2}\right)-\frac{1}{3} \sqrt{2} E\left(\sin ^{-1}(x)|\frac{3}{2}\right)",1,"(-EllipticE[ArcSin[Sqrt[3/2]*x], 2/3] + EllipticF[ArcSin[Sqrt[3/2]*x], 2/3])/Sqrt[3]","A",1
997,1,28,35,0.0314627,"\int \frac{x^2}{\sqrt{4-x^2} \sqrt{2+3 x^2}} \, dx","Integrate[x^2/(Sqrt[4 - x^2]*Sqrt[2 + 3*x^2]),x]","\frac{1}{3} \sqrt{2} \left(E\left(\left.\sin ^{-1}\left(\frac{x}{2}\right)\right|-6\right)-F\left(\left.\sin ^{-1}\left(\frac{x}{2}\right)\right|-6\right)\right)","\frac{1}{3} \sqrt{2} E\left(\left.\sin ^{-1}\left(\frac{x}{2}\right)\right|-6\right)-\frac{1}{3} \sqrt{2} F\left(\left.\sin ^{-1}\left(\frac{x}{2}\right)\right|-6\right)",1,"(Sqrt[2]*(EllipticE[ArcSin[x/2], -6] - EllipticF[ArcSin[x/2], -6]))/3","A",1
998,1,38,35,0.0537908,"\int \frac{x^2}{\sqrt{2-3 x^2} \sqrt{4-x^2}} \, dx","Integrate[x^2/(Sqrt[2 - 3*x^2]*Sqrt[4 - x^2]),x]","-\frac{2 \left(E\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|\frac{1}{6}\right)-F\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|\frac{1}{6}\right)\right)}{\sqrt{3}}","\frac{1}{3} \sqrt{2} F\left(\left.\sin ^{-1}\left(\frac{x}{2}\right)\right|6\right)-\frac{1}{3} \sqrt{2} E\left(\left.\sin ^{-1}\left(\frac{x}{2}\right)\right|6\right)",1,"(-2*(EllipticE[ArcSin[Sqrt[3/2]*x], 1/6] - EllipticF[ArcSin[Sqrt[3/2]*x], 1/6]))/Sqrt[3]","A",1
999,1,28,35,0.0344189,"\int \frac{x^2}{\sqrt{1-4 x^2} \sqrt{2+3 x^2}} \, dx","Integrate[x^2/(Sqrt[1 - 4*x^2]*Sqrt[2 + 3*x^2]),x]","\frac{E\left(\sin ^{-1}(2 x)|-\frac{3}{8}\right)-F\left(\sin ^{-1}(2 x)|-\frac{3}{8}\right)}{3 \sqrt{2}}","\frac{E\left(\sin ^{-1}(2 x)|-\frac{3}{8}\right)}{3 \sqrt{2}}-\frac{F\left(\sin ^{-1}(2 x)|-\frac{3}{8}\right)}{3 \sqrt{2}}",1,"(EllipticE[ArcSin[2*x], -3/8] - EllipticF[ArcSin[2*x], -3/8])/(3*Sqrt[2])","A",1
1000,1,28,35,0.0553225,"\int \frac{x^2}{\sqrt{1-4 x^2} \sqrt{2-3 x^2}} \, dx","Integrate[x^2/(Sqrt[1 - 4*x^2]*Sqrt[2 - 3*x^2]),x]","\frac{F\left(\sin ^{-1}(2 x)|\frac{3}{8}\right)-E\left(\sin ^{-1}(2 x)|\frac{3}{8}\right)}{3 \sqrt{2}}","\frac{F\left(\sin ^{-1}(2 x)|\frac{3}{8}\right)}{3 \sqrt{2}}-\frac{E\left(\sin ^{-1}(2 x)|\frac{3}{8}\right)}{3 \sqrt{2}}",1,"(-EllipticE[ArcSin[2*x], 3/8] + EllipticF[ArcSin[2*x], 3/8])/(3*Sqrt[2])","A",1
1001,1,37,42,0.0315228,"\int \frac{x^2}{\sqrt{2-3 x^2} \sqrt{1+x^2}} \, dx","Integrate[x^2/(Sqrt[2 - 3*x^2]*Sqrt[1 + x^2]),x]","\frac{E\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|-\frac{2}{3}\right)-F\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|-\frac{2}{3}\right)}{\sqrt{3}}","\frac{E\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|-\frac{2}{3}\right)}{\sqrt{3}}-\frac{F\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|-\frac{2}{3}\right)}{\sqrt{3}}",1,"(EllipticE[ArcSin[Sqrt[3/2]*x], -2/3] - EllipticF[ArcSin[Sqrt[3/2]*x], -2/3])/Sqrt[3]","A",1
1002,1,38,43,0.0345448,"\int \frac{x^2}{\sqrt{2-3 x^2} \sqrt{4+x^2}} \, dx","Integrate[x^2/(Sqrt[2 - 3*x^2]*Sqrt[4 + x^2]),x]","\frac{2 \left(E\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|-\frac{1}{6}\right)-F\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|-\frac{1}{6}\right)\right)}{\sqrt{3}}","\frac{2 E\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|-\frac{1}{6}\right)}{\sqrt{3}}-\frac{2 F\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|-\frac{1}{6}\right)}{\sqrt{3}}",1,"(2*(EllipticE[ArcSin[Sqrt[3/2]*x], -1/6] - EllipticF[ArcSin[Sqrt[3/2]*x], -1/6]))/Sqrt[3]","A",1
1003,1,40,47,0.0341096,"\int \frac{x^2}{\sqrt{2-3 x^2} \sqrt{1+4 x^2}} \, dx","Integrate[x^2/(Sqrt[2 - 3*x^2]*Sqrt[1 + 4*x^2]),x]","\frac{E\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|-\frac{8}{3}\right)-F\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|-\frac{8}{3}\right)}{4 \sqrt{3}}","\frac{E\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|-\frac{8}{3}\right)}{4 \sqrt{3}}-\frac{F\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|-\frac{8}{3}\right)}{4 \sqrt{3}}",1,"(EllipticE[ArcSin[Sqrt[3/2]*x], -8/3] - EllipticF[ArcSin[Sqrt[3/2]*x], -8/3])/(4*Sqrt[3])","A",1
1004,1,34,80,0.029596,"\int \frac{x^2}{\sqrt{1+x^2} \sqrt{2+3 x^2}} \, dx","Integrate[x^2/(Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]),x]","-\frac{1}{3} i \sqrt{2} \left(E\left(i \sinh ^{-1}(x)|\frac{3}{2}\right)-F\left(i \sinh ^{-1}(x)|\frac{3}{2}\right)\right)","\frac{x \sqrt{3 x^2+2}}{3 \sqrt{x^2+1}}-\frac{\sqrt{2} \sqrt{3 x^2+2} E\left(\tan ^{-1}(x)|-\frac{1}{2}\right)}{3 \sqrt{x^2+1} \sqrt{\frac{3 x^2+2}{x^2+1}}}",1,"(-1/3*I)*Sqrt[2]*(EllipticE[I*ArcSinh[x], 3/2] - EllipticF[I*ArcSinh[x], 3/2])","C",1
1005,1,38,82,0.0303737,"\int \frac{x^2}{\sqrt{4+x^2} \sqrt{2+3 x^2}} \, dx","Integrate[x^2/(Sqrt[4 + x^2]*Sqrt[2 + 3*x^2]),x]","-\frac{1}{3} i \sqrt{2} \left(E\left(\left.i \sinh ^{-1}\left(\frac{x}{2}\right)\right|6\right)-F\left(\left.i \sinh ^{-1}\left(\frac{x}{2}\right)\right|6\right)\right)","\frac{x \sqrt{3 x^2+2}}{3 \sqrt{x^2+4}}-\frac{\sqrt{2} \sqrt{3 x^2+2} E\left(\left.\tan ^{-1}\left(\frac{x}{2}\right)\right|-5\right)}{3 \sqrt{x^2+4} \sqrt{\frac{3 x^2+2}{x^2+4}}}",1,"(-1/3*I)*Sqrt[2]*(EllipticE[I*ArcSinh[x/2], 6] - EllipticF[I*ArcSinh[x/2], 6])","C",1
1006,1,50,88,0.0366874,"\int \frac{x^2}{\sqrt{2+3 x^2} \sqrt{1+4 x^2}} \, dx","Integrate[x^2/(Sqrt[2 + 3*x^2]*Sqrt[1 + 4*x^2]),x]","-\frac{i \left(E\left(i \sinh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|\frac{8}{3}\right)-F\left(i \sinh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|\frac{8}{3}\right)\right)}{4 \sqrt{3}}","\frac{x \sqrt{3 x^2+2}}{3 \sqrt{4 x^2+1}}-\frac{\sqrt{3 x^2+2} E\left(\tan ^{-1}(2 x)|\frac{5}{8}\right)}{3 \sqrt{2} \sqrt{\frac{3 x^2+2}{4 x^2+1}} \sqrt{4 x^2+1}}",1,"((-1/4*I)*(EllipticE[I*ArcSinh[Sqrt[3/2]*x], 8/3] - EllipticF[I*ArcSinh[Sqrt[3/2]*x], 8/3]))/Sqrt[3]","C",1
1007,1,37,17,0.0369054,"\int \frac{x^2}{\sqrt{1-x^2} \sqrt{-1+2 x^2}} \, dx","Integrate[x^2/(Sqrt[1 - x^2]*Sqrt[-1 + 2*x^2]),x]","\frac{\sqrt{1-2 x^2} \left(F\left(\left.\sin ^{-1}(x)\right|2\right)-E\left(\left.\sin ^{-1}(x)\right|2\right)\right)}{2 \sqrt{2 x^2-1}}","-\frac{1}{2} F\left(\left.\cos ^{-1}(x)\right|2\right)-\frac{1}{2} E\left(\left.\cos ^{-1}(x)\right|2\right)",1,"(Sqrt[1 - 2*x^2]*(-EllipticE[ArcSin[x], 2] + EllipticF[ArcSin[x], 2]))/(2*Sqrt[-1 + 2*x^2])","B",1
1008,1,106,109,0.1327881,"\int \frac{x^5}{\sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","Integrate[x^5/((1 - x^2)^(1/3)*(3 + x^2)),x]","-\frac{3}{40} \left(4 \left(1-x^2\right)^{2/3} x^2-24 \left(1-x^2\right)^{2/3}+15 \sqrt[3]{2} \log \left(x^2+3\right)-45 \sqrt[3]{2} \log \left(2^{2/3}-\sqrt[3]{1-x^2}\right)-30 \sqrt[3]{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{2-2 x^2}+1}{\sqrt{3}}\right)\right)","\frac{3}{10} \left(1-x^2\right)^{5/3}+\frac{3}{2} \left(1-x^2\right)^{2/3}-\frac{9 \log \left(x^2+3\right)}{4\ 2^{2/3}}+\frac{27 \log \left(2^{2/3}-\sqrt[3]{1-x^2}\right)}{4\ 2^{2/3}}+\frac{9 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{2-2 x^2}+1}{\sqrt{3}}\right)}{2\ 2^{2/3}}",1,"(-3*(-24*(1 - x^2)^(2/3) + 4*x^2*(1 - x^2)^(2/3) - 30*2^(1/3)*Sqrt[3]*ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]] + 15*2^(1/3)*Log[3 + x^2] - 45*2^(1/3)*Log[2^(2/3) - (1 - x^2)^(1/3)]))/40","A",1
1009,1,90,94,0.0333908,"\int \frac{x^3}{\sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","Integrate[x^3/((1 - x^2)^(1/3)*(3 + x^2)),x]","-\frac{3}{8} \left(2 \left(1-x^2\right)^{2/3}-\sqrt[3]{2} \log \left(x^2+3\right)+3 \sqrt[3]{2} \log \left(2^{2/3}-\sqrt[3]{1-x^2}\right)+2 \sqrt[3]{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{2-2 x^2}+1}{\sqrt{3}}\right)\right)","-\frac{3}{4} \left(1-x^2\right)^{2/3}+\frac{3 \log \left(x^2+3\right)}{4\ 2^{2/3}}-\frac{9 \log \left(2^{2/3}-\sqrt[3]{1-x^2}\right)}{4\ 2^{2/3}}-\frac{3 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{2-2 x^2}+1}{\sqrt{3}}\right)}{2\ 2^{2/3}}",1,"(-3*(2*(1 - x^2)^(2/3) + 2*2^(1/3)*Sqrt[3]*ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]] - 2^(1/3)*Log[3 + x^2] + 3*2^(1/3)*Log[2^(2/3) - (1 - x^2)^(1/3)]))/8","A",1
1010,1,67,79,0.0184405,"\int \frac{x}{\sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","Integrate[x/((1 - x^2)^(1/3)*(3 + x^2)),x]","\frac{-\log \left(x^2+3\right)+3 \log \left(2^{2/3}-\sqrt[3]{1-x^2}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{2-2 x^2}+1}{\sqrt{3}}\right)}{4\ 2^{2/3}}","-\frac{\log \left(x^2+3\right)}{4\ 2^{2/3}}+\frac{3 \log \left(2^{2/3}-\sqrt[3]{1-x^2}\right)}{4\ 2^{2/3}}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{2-2 x^2}+1}{\sqrt{3}}\right)}{2\ 2^{2/3}}",1,"(2*Sqrt[3]*ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]] - Log[3 + x^2] + 3*Log[2^(2/3) - (1 - x^2)^(1/3)])/(4*2^(2/3))","A",1
1011,1,127,136,0.0395124,"\int \frac{1}{x \sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","Integrate[1/(x*(1 - x^2)^(1/3)*(3 + x^2)),x]","\frac{1}{24} \left(\sqrt[3]{2} \log \left(x^2+3\right)+6 \log \left(1-\sqrt[3]{1-x^2}\right)-3 \sqrt[3]{2} \log \left(2^{2/3}-\sqrt[3]{1-x^2}\right)-2 \sqrt[3]{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{2-2 x^2}+1}{\sqrt{3}}\right)+4 \sqrt{3} \tan ^{-1}\left(\frac{2 \sqrt[3]{1-x^2}+1}{\sqrt{3}}\right)-4 \log (x)\right)","\frac{\log \left(x^2+3\right)}{12\ 2^{2/3}}+\frac{1}{4} \log \left(1-\sqrt[3]{1-x^2}\right)-\frac{\log \left(2^{2/3}-\sqrt[3]{1-x^2}\right)}{4\ 2^{2/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{2-2 x^2}+1}{\sqrt{3}}\right)}{2\ 2^{2/3} \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{2 \sqrt[3]{1-x^2}+1}{\sqrt{3}}\right)}{2 \sqrt{3}}-\frac{\log (x)}{6}",1,"(-2*2^(1/3)*Sqrt[3]*ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]] + 4*Sqrt[3]*ArcTan[(1 + 2*(1 - x^2)^(1/3))/Sqrt[3]] - 4*Log[x] + 2^(1/3)*Log[3 + x^2] + 6*Log[1 - (1 - x^2)^(1/3)] - 3*2^(1/3)*Log[2^(2/3) - (1 - x^2)^(1/3)])/24","A",1
1012,1,93,97,0.0456067,"\int \frac{1}{x^3 \sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","Integrate[1/(x^3*(1 - x^2)^(1/3)*(3 + x^2)),x]","\frac{1}{72} \left(-\frac{12 \left(1-x^2\right)^{2/3}}{x^2}-\sqrt[3]{2} \log \left(x^2+3\right)+3 \sqrt[3]{2} \log \left(2^{2/3}-\sqrt[3]{1-x^2}\right)+2 \sqrt[3]{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{2-2 x^2}+1}{\sqrt{3}}\right)\right)","-\frac{\left(1-x^2\right)^{2/3}}{6 x^2}-\frac{\log \left(x^2+3\right)}{36\ 2^{2/3}}+\frac{\log \left(2^{2/3}-\sqrt[3]{1-x^2}\right)}{12\ 2^{2/3}}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{2-2 x^2}+1}{\sqrt{3}}\right)}{6\ 2^{2/3} \sqrt{3}}",1,"((-12*(1 - x^2)^(2/3))/x^2 + 2*2^(1/3)*Sqrt[3]*ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]] - 2^(1/3)*Log[3 + x^2] + 3*2^(1/3)*Log[2^(2/3) - (1 - x^2)^(1/3)])/72","A",1
1013,1,178,172,0.0769474,"\int \frac{1}{x^5 \sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","Integrate[1/(x^5*(1 - x^2)^(1/3)*(3 + x^2)),x]","-\frac{8 x^4 \log (x)+12 \left(1-x^2\right)^{2/3} x^2+18 \left(1-x^2\right)^{2/3}-\sqrt[3]{2} x^4 \log \left(x^2+3\right)-12 x^4 \log \left(1-\sqrt[3]{1-x^2}\right)+3 \sqrt[3]{2} x^4 \log \left(2^{2/3}-\sqrt[3]{1-x^2}\right)+2 \sqrt[3]{2} \sqrt{3} x^4 \tan ^{-1}\left(\frac{\sqrt[3]{2-2 x^2}+1}{\sqrt{3}}\right)-8 \sqrt{3} x^4 \tan ^{-1}\left(\frac{2 \sqrt[3]{1-x^2}+1}{\sqrt{3}}\right)}{216 x^4}","-\frac{\left(1-x^2\right)^{2/3}}{18 x^2}+\frac{\log \left(x^2+3\right)}{108\ 2^{2/3}}+\frac{1}{18} \log \left(1-\sqrt[3]{1-x^2}\right)-\frac{\log \left(2^{2/3}-\sqrt[3]{1-x^2}\right)}{36\ 2^{2/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{2-2 x^2}+1}{\sqrt{3}}\right)}{18\ 2^{2/3} \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{2 \sqrt[3]{1-x^2}+1}{\sqrt{3}}\right)}{9 \sqrt{3}}-\frac{\left(1-x^2\right)^{2/3}}{12 x^4}-\frac{\log (x)}{27}",1,"-1/216*(18*(1 - x^2)^(2/3) + 12*x^2*(1 - x^2)^(2/3) + 2*2^(1/3)*Sqrt[3]*x^4*ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]] - 8*Sqrt[3]*x^4*ArcTan[(1 + 2*(1 - x^2)^(1/3))/Sqrt[3]] + 8*x^4*Log[x] - 2^(1/3)*x^4*Log[3 + x^2] - 12*x^4*Log[1 - (1 - x^2)^(1/3)] + 3*2^(1/3)*x^4*Log[2^(2/3) - (1 - x^2)^(1/3)])/x^4","A",1
1014,1,156,536,0.1950293,"\int \frac{x^4}{\sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","Integrate[x^4/((1 - x^2)^(1/3)*(3 + x^2)),x]","\frac{1}{7} x \left(\frac{3 \left(-\frac{27 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right)}{\left(x^2+3\right) \left(2 x^2 \left(F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};x^2,-\frac{x^2}{3}\right)-F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right)\right)-9 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right)\right)}+x^2-1\right)}{\sqrt[3]{1-x^2}}-2 x^2 F_1\left(\frac{3}{2};\frac{1}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right)\right)","-\frac{3}{7} \left(1-x^2\right)^{2/3} x+\frac{54 x}{7 \left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)}+\frac{3 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \left(1-\sqrt[3]{2} \sqrt[3]{1-x^2}\right)}{x}\right)}{2\ 2^{2/3}}+\frac{9 \tanh ^{-1}\left(\frac{x}{\sqrt[3]{2} \sqrt[3]{1-x^2}+1}\right)}{2\ 2^{2/3}}-\frac{18 \sqrt{2} 3^{3/4} \left(1-\sqrt[3]{1-x^2}\right) \sqrt{\frac{\left(1-x^2\right)^{2/3}+\sqrt[3]{1-x^2}+1}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-\sqrt[3]{1-x^2}+\sqrt{3}+1}{-\sqrt[3]{1-x^2}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{7 \sqrt{-\frac{1-\sqrt[3]{1-x^2}}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} x}+\frac{27 \sqrt[4]{3} \sqrt{2+\sqrt{3}} \left(1-\sqrt[3]{1-x^2}\right) \sqrt{\frac{\left(1-x^2\right)^{2/3}+\sqrt[3]{1-x^2}+1}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-\sqrt[3]{1-x^2}+\sqrt{3}+1}{-\sqrt[3]{1-x^2}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{7 \sqrt{-\frac{1-\sqrt[3]{1-x^2}}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} x}+\frac{3 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3}}{x}\right)}{2\ 2^{2/3}}-\frac{3 \tanh ^{-1}(x)}{2\ 2^{2/3}}",1,"(x*(-2*x^2*AppellF1[3/2, 1/3, 1, 5/2, x^2, -1/3*x^2] + (3*(-1 + x^2 - (27*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1/3*x^2])/((3 + x^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1/3*x^2] + 2*x^2*(AppellF1[3/2, 1/3, 2, 5/2, x^2, -1/3*x^2] - AppellF1[3/2, 4/3, 1, 5/2, x^2, -1/3*x^2])))))/(1 - x^2)^(1/3)))/7","C",0
1015,1,28,515,0.0183316,"\int \frac{x^2}{\sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","Integrate[x^2/((1 - x^2)^(1/3)*(3 + x^2)),x]","\frac{1}{9} x^3 F_1\left(\frac{3}{2};\frac{1}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right)","-\frac{3 x}{-\sqrt[3]{1-x^2}-\sqrt{3}+1}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \left(1-\sqrt[3]{2} \sqrt[3]{1-x^2}\right)}{x}\right)}{2\ 2^{2/3}}-\frac{3 \tanh ^{-1}\left(\frac{x}{\sqrt[3]{2} \sqrt[3]{1-x^2}+1}\right)}{2\ 2^{2/3}}+\frac{\sqrt{2} 3^{3/4} \left(1-\sqrt[3]{1-x^2}\right) \sqrt{\frac{\left(1-x^2\right)^{2/3}+\sqrt[3]{1-x^2}+1}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-\sqrt[3]{1-x^2}+\sqrt{3}+1}{-\sqrt[3]{1-x^2}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt{-\frac{1-\sqrt[3]{1-x^2}}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} x}-\frac{3 \sqrt[4]{3} \sqrt{2+\sqrt{3}} \left(1-\sqrt[3]{1-x^2}\right) \sqrt{\frac{\left(1-x^2\right)^{2/3}+\sqrt[3]{1-x^2}+1}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-\sqrt[3]{1-x^2}+\sqrt{3}+1}{-\sqrt[3]{1-x^2}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{2 \sqrt{-\frac{1-\sqrt[3]{1-x^2}}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} x}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3}}{x}\right)}{2\ 2^{2/3}}+\frac{\tanh ^{-1}(x)}{2\ 2^{2/3}}",1,"(x^3*AppellF1[3/2, 1/3, 1, 5/2, x^2, -1/3*x^2])/9","C",0
1016,1,118,113,0.0362076,"\int \frac{1}{\sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","Integrate[1/((1 - x^2)^(1/3)*(3 + x^2)),x]","-\frac{9 x F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right)}{\sqrt[3]{1-x^2} \left(x^2+3\right) \left(2 x^2 \left(F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};x^2,-\frac{x^2}{3}\right)-F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right)\right)-9 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right)\right)}","\frac{\tan ^{-1}\left(\frac{\sqrt{3} \left(1-\sqrt[3]{2} \sqrt[3]{1-x^2}\right)}{x}\right)}{2\ 2^{2/3} \sqrt{3}}+\frac{\tanh ^{-1}\left(\frac{x}{\sqrt[3]{2} \sqrt[3]{1-x^2}+1}\right)}{2\ 2^{2/3}}+\frac{\tan ^{-1}\left(\frac{\sqrt{3}}{x}\right)}{2\ 2^{2/3} \sqrt{3}}-\frac{\tanh ^{-1}(x)}{6\ 2^{2/3}}",1,"(-9*x*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1/3*x^2])/((1 - x^2)^(1/3)*(3 + x^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1/3*x^2] + 2*x^2*(AppellF1[3/2, 1/3, 2, 5/2, x^2, -1/3*x^2] - AppellF1[3/2, 4/3, 1, 5/2, x^2, -1/3*x^2])))","C",0
1017,1,161,538,0.0947656,"\int \frac{1}{x^2 \sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","Integrate[1/(x^2*(1 - x^2)^(1/3)*(3 + x^2)),x]","\frac{\frac{18 x^2 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right)}{\left(x^2+3\right) \left(2 x^2 \left(F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};x^2,-\frac{x^2}{3}\right)-F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right)\right)-9 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right)\right)}+x^2-1}{3 x \sqrt[3]{1-x^2}}-\frac{1}{81} x^3 F_1\left(\frac{3}{2};\frac{1}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right)","\frac{x}{3 \left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)}-\frac{\left(1-x^2\right)^{2/3}}{3 x}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} \left(1-\sqrt[3]{2} \sqrt[3]{1-x^2}\right)}{x}\right)}{6\ 2^{2/3} \sqrt{3}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt[3]{2} \sqrt[3]{1-x^2}+1}\right)}{6\ 2^{2/3}}-\frac{\sqrt{2} \left(1-\sqrt[3]{1-x^2}\right) \sqrt{\frac{\left(1-x^2\right)^{2/3}+\sqrt[3]{1-x^2}+1}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-\sqrt[3]{1-x^2}+\sqrt{3}+1}{-\sqrt[3]{1-x^2}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt{-\frac{1-\sqrt[3]{1-x^2}}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} x}+\frac{\sqrt{2+\sqrt{3}} \left(1-\sqrt[3]{1-x^2}\right) \sqrt{\frac{\left(1-x^2\right)^{2/3}+\sqrt[3]{1-x^2}+1}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-\sqrt[3]{1-x^2}+\sqrt{3}+1}{-\sqrt[3]{1-x^2}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{2\ 3^{3/4} \sqrt{-\frac{1-\sqrt[3]{1-x^2}}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} x}-\frac{\tan ^{-1}\left(\frac{\sqrt{3}}{x}\right)}{6\ 2^{2/3} \sqrt{3}}+\frac{\tanh ^{-1}(x)}{18\ 2^{2/3}}",1,"-1/81*(x^3*AppellF1[3/2, 1/3, 1, 5/2, x^2, -1/3*x^2]) + (-1 + x^2 + (18*x^2*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1/3*x^2])/((3 + x^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1/3*x^2] + 2*x^2*(AppellF1[3/2, 1/3, 2, 5/2, x^2, -1/3*x^2] - AppellF1[3/2, 4/3, 1, 5/2, x^2, -1/3*x^2]))))/(3*x*(1 - x^2)^(1/3))","C",0
1018,1,166,556,0.1106841,"\int \frac{1}{x^4 \sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","Integrate[1/(x^4*(1 - x^2)^(1/3)*(3 + x^2)),x]","\frac{-\frac{9 x^4 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right)}{\left(x^2+3\right) \left(2 x^2 \left(F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};x^2,-\frac{x^2}{3}\right)-F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right)\right)-9 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right)\right)}+2 x^4+x^2-3}{27 x^3 \sqrt[3]{1-x^2}}-\frac{2}{729} x^3 F_1\left(\frac{3}{2};\frac{1}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right)","\frac{2 x}{27 \left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)}-\frac{2 \left(1-x^2\right)^{2/3}}{27 x}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} \left(1-\sqrt[3]{2} \sqrt[3]{1-x^2}\right)}{x}\right)}{18\ 2^{2/3} \sqrt{3}}+\frac{\tanh ^{-1}\left(\frac{x}{\sqrt[3]{2} \sqrt[3]{1-x^2}+1}\right)}{18\ 2^{2/3}}-\frac{2 \sqrt{2} \left(1-\sqrt[3]{1-x^2}\right) \sqrt{\frac{\left(1-x^2\right)^{2/3}+\sqrt[3]{1-x^2}+1}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-\sqrt[3]{1-x^2}+\sqrt{3}+1}{-\sqrt[3]{1-x^2}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{27 \sqrt[4]{3} \sqrt{-\frac{1-\sqrt[3]{1-x^2}}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} x}+\frac{\sqrt{2+\sqrt{3}} \left(1-\sqrt[3]{1-x^2}\right) \sqrt{\frac{\left(1-x^2\right)^{2/3}+\sqrt[3]{1-x^2}+1}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-\sqrt[3]{1-x^2}+\sqrt{3}+1}{-\sqrt[3]{1-x^2}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{9\ 3^{3/4} \sqrt{-\frac{1-\sqrt[3]{1-x^2}}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} x}-\frac{\left(1-x^2\right)^{2/3}}{9 x^3}+\frac{\tan ^{-1}\left(\frac{\sqrt{3}}{x}\right)}{18\ 2^{2/3} \sqrt{3}}-\frac{\tanh ^{-1}(x)}{54\ 2^{2/3}}",1,"(-2*x^3*AppellF1[3/2, 1/3, 1, 5/2, x^2, -1/3*x^2])/729 + (-3 + x^2 + 2*x^4 - (9*x^4*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1/3*x^2])/((3 + x^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1/3*x^2] + 2*x^2*(AppellF1[3/2, 1/3, 2, 5/2, x^2, -1/3*x^2] - AppellF1[3/2, 4/3, 1, 5/2, x^2, -1/3*x^2]))))/(27*x^3*(1 - x^2)^(1/3))","C",0
1019,1,120,133,0.1909038,"\int \frac{x^7}{\sqrt[3]{1-x^2} \left(3+x^2\right)^2} \, dx","Integrate[x^7/((1 - x^2)^(1/3)*(3 + x^2)^2),x]","\frac{3}{80} \left(\frac{6 \left(1-x^2\right)^{2/3} \left(14 x^2+69\right)}{x^2+3}+\frac{165 \left(-\log \left(x^2+3\right)+3 \log \left(2^{2/3}-\sqrt[3]{1-x^2}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{2-2 x^2}+1}{\sqrt{3}}\right)\right)}{2^{2/3}}-\frac{8 \left(1-x^2\right)^{2/3} x^4}{x^2+3}\right)","\frac{9 \left(1-x^2\right)^{2/3} \left(14 x^2+69\right)}{40 \left(x^2+3\right)}-\frac{99 \log \left(x^2+3\right)}{16\ 2^{2/3}}+\frac{297 \log \left(2^{2/3}-\sqrt[3]{1-x^2}\right)}{16\ 2^{2/3}}+\frac{99 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{2-2 x^2}+1}{\sqrt{3}}\right)}{8\ 2^{2/3}}-\frac{3 \left(1-x^2\right)^{2/3} x^4}{10 \left(x^2+3\right)}",1,"(3*((-8*x^4*(1 - x^2)^(2/3))/(3 + x^2) + (6*(1 - x^2)^(2/3)*(69 + 14*x^2))/(3 + x^2) + (165*(2*Sqrt[3]*ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]] - Log[3 + x^2] + 3*Log[2^(2/3) - (1 - x^2)^(1/3)]))/2^(2/3)))/80","A",1
1020,1,110,116,0.1687795,"\int \frac{x^5}{\sqrt[3]{1-x^2} \left(3+x^2\right)^2} \, dx","Integrate[x^5/((1 - x^2)^(1/3)*(3 + x^2)^2),x]","\frac{3}{32} \left(-8 \left(1-x^2\right)^{2/3}-\frac{12 \left(1-x^2\right)^{2/3}}{x^2+3}+7 \sqrt[3]{2} \log \left(x^2+3\right)-21 \sqrt[3]{2} \log \left(2^{2/3}-\sqrt[3]{1-x^2}\right)-14 \sqrt[3]{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{2-2 x^2}+1}{\sqrt{3}}\right)\right)","-\frac{3}{4} \left(1-x^2\right)^{2/3}-\frac{9 \left(1-x^2\right)^{2/3}}{8 \left(x^2+3\right)}+\frac{21 \log \left(x^2+3\right)}{16\ 2^{2/3}}-\frac{63 \log \left(2^{2/3}-\sqrt[3]{1-x^2}\right)}{16\ 2^{2/3}}-\frac{21 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{2-2 x^2}+1}{\sqrt{3}}\right)}{8\ 2^{2/3}}",1,"(3*(-8*(1 - x^2)^(2/3) - (12*(1 - x^2)^(2/3))/(3 + x^2) - 14*2^(1/3)*Sqrt[3]*ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]] + 7*2^(1/3)*Log[3 + x^2] - 21*2^(1/3)*Log[2^(2/3) - (1 - x^2)^(1/3)]))/32","A",1
1021,1,97,101,0.1076161,"\int \frac{x^3}{\sqrt[3]{1-x^2} \left(3+x^2\right)^2} \, dx","Integrate[x^3/((1 - x^2)^(1/3)*(3 + x^2)^2),x]","\frac{3}{32} \left(\frac{4 \left(1-x^2\right)^{2/3}}{x^2+3}-\sqrt[3]{2} \log \left(x^2+3\right)+3 \sqrt[3]{2} \log \left(2^{2/3}-\sqrt[3]{1-x^2}\right)+2 \sqrt[3]{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{2-2 x^2}+1}{\sqrt{3}}\right)\right)","\frac{3 \left(1-x^2\right)^{2/3}}{8 \left(x^2+3\right)}-\frac{3 \log \left(x^2+3\right)}{16\ 2^{2/3}}+\frac{9 \log \left(2^{2/3}-\sqrt[3]{1-x^2}\right)}{16\ 2^{2/3}}+\frac{3 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{2-2 x^2}+1}{\sqrt{3}}\right)}{8\ 2^{2/3}}",1,"(3*((4*(1 - x^2)^(2/3))/(3 + x^2) + 2*2^(1/3)*Sqrt[3]*ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]] - 2^(1/3)*Log[3 + x^2] + 3*2^(1/3)*Log[2^(2/3) - (1 - x^2)^(1/3)]))/32","A",1
1022,1,34,101,0.0083259,"\int \frac{x}{\sqrt[3]{1-x^2} \left(3+x^2\right)^2} \, dx","Integrate[x/((1 - x^2)^(1/3)*(3 + x^2)^2),x]","-\frac{3}{64} \left(1-x^2\right)^{2/3} \, _2F_1\left(\frac{2}{3},2;\frac{5}{3};\frac{1}{4} \left(1-x^2\right)\right)","-\frac{\left(1-x^2\right)^{2/3}}{8 \left(x^2+3\right)}-\frac{\log \left(x^2+3\right)}{48\ 2^{2/3}}+\frac{\log \left(2^{2/3}-\sqrt[3]{1-x^2}\right)}{16\ 2^{2/3}}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{2-2 x^2}+1}{\sqrt{3}}\right)}{8\ 2^{2/3} \sqrt{3}}",1,"(-3*(1 - x^2)^(2/3)*Hypergeometric2F1[2/3, 2, 5/3, (1 - x^2)/4])/64","C",1
1023,1,148,158,0.2375916,"\int \frac{1}{x \sqrt[3]{1-x^2} \left(3+x^2\right)^2} \, dx","Integrate[1/(x*(1 - x^2)^(1/3)*(3 + x^2)^2),x]","\frac{1}{288} \left(\frac{12 \left(1-x^2\right)^{2/3}}{x^2+3}+5 \sqrt[3]{2} \log \left(x^2+3\right)+24 \log \left(1-\sqrt[3]{1-x^2}\right)-15 \sqrt[3]{2} \log \left(2^{2/3}-\sqrt[3]{1-x^2}\right)-10 \sqrt[3]{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{2-2 x^2}+1}{\sqrt{3}}\right)+16 \sqrt{3} \tan ^{-1}\left(\frac{2 \sqrt[3]{1-x^2}+1}{\sqrt{3}}\right)-16 \log (x)\right)","\frac{\left(1-x^2\right)^{2/3}}{24 \left(x^2+3\right)}+\frac{5 \log \left(x^2+3\right)}{144\ 2^{2/3}}+\frac{1}{12} \log \left(1-\sqrt[3]{1-x^2}\right)-\frac{5 \log \left(2^{2/3}-\sqrt[3]{1-x^2}\right)}{48\ 2^{2/3}}-\frac{5 \tan ^{-1}\left(\frac{\sqrt[3]{2-2 x^2}+1}{\sqrt{3}}\right)}{24\ 2^{2/3} \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{2 \sqrt[3]{1-x^2}+1}{\sqrt{3}}\right)}{6 \sqrt{3}}-\frac{\log (x)}{18}",1,"((12*(1 - x^2)^(2/3))/(3 + x^2) - 10*2^(1/3)*Sqrt[3]*ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]] + 16*Sqrt[3]*ArcTan[(1 + 2*(1 - x^2)^(1/3))/Sqrt[3]] - 16*Log[x] + 5*2^(1/3)*Log[3 + x^2] + 24*Log[1 - (1 - x^2)^(1/3)] - 15*2^(1/3)*Log[2^(2/3) - (1 - x^2)^(1/3)])/288","A",1
1024,1,171,183,0.1797822,"\int \frac{1}{x^3 \sqrt[3]{1-x^2} \left(3+x^2\right)^2} \, dx","Integrate[1/(x^3*(1 - x^2)^(1/3)*(3 + x^2)^2),x]","\frac{1}{864} \left(-\frac{144 \left(1-x^2\right)^{2/3}}{x^2 \left(x^2+3\right)}-\frac{60 \left(1-x^2\right)^{2/3}}{x^2+3}-9 \sqrt[3]{2} \log \left(x^2+3\right)-24 \log \left(1-\sqrt[3]{1-x^2}\right)+27 \sqrt[3]{2} \log \left(2^{2/3}-\sqrt[3]{1-x^2}\right)+18 \sqrt[3]{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{2-2 x^2}+1}{\sqrt{3}}\right)-16 \sqrt{3} \tan ^{-1}\left(\frac{2 \sqrt[3]{1-x^2}+1}{\sqrt{3}}\right)+16 \log (x)\right)","-\frac{\left(1-x^2\right)^{2/3}}{6 x^2 \left(x^2+3\right)}-\frac{5 \left(1-x^2\right)^{2/3}}{72 \left(x^2+3\right)}-\frac{\log \left(x^2+3\right)}{48\ 2^{2/3}}-\frac{1}{36} \log \left(1-\sqrt[3]{1-x^2}\right)+\frac{\log \left(2^{2/3}-\sqrt[3]{1-x^2}\right)}{16\ 2^{2/3}}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{2-2 x^2}+1}{\sqrt{3}}\right)}{8\ 2^{2/3} \sqrt{3}}-\frac{\tan ^{-1}\left(\frac{2 \sqrt[3]{1-x^2}+1}{\sqrt{3}}\right)}{18 \sqrt{3}}+\frac{\log (x)}{54}",1,"((-60*(1 - x^2)^(2/3))/(3 + x^2) - (144*(1 - x^2)^(2/3))/(x^2*(3 + x^2)) + 18*2^(1/3)*Sqrt[3]*ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]] - 16*Sqrt[3]*ArcTan[(1 + 2*(1 - x^2)^(1/3))/Sqrt[3]] + 16*Log[x] - 9*2^(1/3)*Log[3 + x^2] - 24*Log[1 - (1 - x^2)^(1/3)] + 27*2^(1/3)*Log[2^(2/3) - (1 - x^2)^(1/3)])/864","A",1
1025,1,194,208,0.1793324,"\int \frac{1}{x^5 \sqrt[3]{1-x^2} \left(3+x^2\right)^2} \, dx","Integrate[1/(x^5*(1 - x^2)^(1/3)*(3 + x^2)^2),x]","\frac{-\frac{72 \left(1-x^2\right)^{2/3}}{x^2 \left(x^2+3\right)}+\frac{12 \left(1-x^2\right)^{2/3}}{x^2+3}+13 \sqrt[3]{2} \log \left(x^2+3\right)+72 \log \left(1-\sqrt[3]{1-x^2}\right)-39 \sqrt[3]{2} \log \left(2^{2/3}-\sqrt[3]{1-x^2}\right)-26 \sqrt[3]{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{2-2 x^2}+1}{\sqrt{3}}\right)+48 \sqrt{3} \tan ^{-1}\left(\frac{2 \sqrt[3]{1-x^2}+1}{\sqrt{3}}\right)-\frac{216 \left(1-x^2\right)^{2/3}}{x^4 \left(x^2+3\right)}-48 \log (x)}{2592}","-\frac{\left(1-x^2\right)^{2/3}}{36 x^2 \left(x^2+3\right)}+\frac{\left(1-x^2\right)^{2/3}}{216 \left(x^2+3\right)}+\frac{13 \log \left(x^2+3\right)}{1296\ 2^{2/3}}+\frac{1}{36} \log \left(1-\sqrt[3]{1-x^2}\right)-\frac{13 \log \left(2^{2/3}-\sqrt[3]{1-x^2}\right)}{432\ 2^{2/3}}-\frac{13 \tan ^{-1}\left(\frac{\sqrt[3]{2-2 x^2}+1}{\sqrt{3}}\right)}{216\ 2^{2/3} \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{2 \sqrt[3]{1-x^2}+1}{\sqrt{3}}\right)}{18 \sqrt{3}}-\frac{\left(1-x^2\right)^{2/3}}{12 x^4 \left(x^2+3\right)}-\frac{\log (x)}{54}",1,"((12*(1 - x^2)^(2/3))/(3 + x^2) - (216*(1 - x^2)^(2/3))/(x^4*(3 + x^2)) - (72*(1 - x^2)^(2/3))/(x^2*(3 + x^2)) - 26*2^(1/3)*Sqrt[3]*ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]] + 48*Sqrt[3]*ArcTan[(1 + 2*(1 - x^2)^(1/3))/Sqrt[3]] - 48*Log[x] + 13*2^(1/3)*Log[3 + x^2] + 72*Log[1 - (1 - x^2)^(1/3)] - 39*2^(1/3)*Log[2^(2/3) - (1 - x^2)^(1/3)])/2592","A",1
1026,1,157,543,0.1121416,"\int \frac{x^4}{\sqrt[3]{1-x^2} \left(3+x^2\right)^2} \, dx","Integrate[x^4/((1 - x^2)^(1/3)*(3 + x^2)^2),x]","\frac{1}{8} x \left(x^2 F_1\left(\frac{3}{2};\frac{1}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right)+\frac{3 \left(\frac{9 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right)}{2 x^2 \left(F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};x^2,-\frac{x^2}{3}\right)-F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right)\right)-9 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right)}-x^2+1\right)}{\sqrt[3]{1-x^2} \left(x^2+3\right)}\right)","\frac{3 \left(1-x^2\right)^{2/3} x}{8 \left(x^2+3\right)}-\frac{27 x}{8 \left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)}-\frac{5 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \left(1-\sqrt[3]{2} \sqrt[3]{1-x^2}\right)}{x}\right)}{8\ 2^{2/3}}-\frac{15 \tanh ^{-1}\left(\frac{x}{\sqrt[3]{2} \sqrt[3]{1-x^2}+1}\right)}{8\ 2^{2/3}}+\frac{9\ 3^{3/4} \left(1-\sqrt[3]{1-x^2}\right) \sqrt{\frac{\left(1-x^2\right)^{2/3}+\sqrt[3]{1-x^2}+1}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-\sqrt[3]{1-x^2}+\sqrt{3}+1}{-\sqrt[3]{1-x^2}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{4 \sqrt{2} \sqrt{-\frac{1-\sqrt[3]{1-x^2}}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} x}-\frac{27 \sqrt[4]{3} \sqrt{2+\sqrt{3}} \left(1-\sqrt[3]{1-x^2}\right) \sqrt{\frac{\left(1-x^2\right)^{2/3}+\sqrt[3]{1-x^2}+1}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-\sqrt[3]{1-x^2}+\sqrt{3}+1}{-\sqrt[3]{1-x^2}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{16 \sqrt{-\frac{1-\sqrt[3]{1-x^2}}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} x}-\frac{5 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3}}{x}\right)}{8\ 2^{2/3}}+\frac{5 \tanh ^{-1}(x)}{8\ 2^{2/3}}",1,"(x*(x^2*AppellF1[3/2, 1/3, 1, 5/2, x^2, -1/3*x^2] + (3*(1 - x^2 + (9*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1/3*x^2])/(-9*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1/3*x^2] + 2*x^2*(AppellF1[3/2, 1/3, 2, 5/2, x^2, -1/3*x^2] - AppellF1[3/2, 4/3, 1, 5/2, x^2, -1/3*x^2]))))/((1 - x^2)^(1/3)*(3 + x^2))))/8","C",0
1027,1,156,543,0.0916609,"\int \frac{x^2}{\sqrt[3]{1-x^2} \left(3+x^2\right)^2} \, dx","Integrate[x^2/((1 - x^2)^(1/3)*(3 + x^2)^2),x]","\frac{x \left(\frac{9 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right)}{2 x^2 \left(F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right)-F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};x^2,-\frac{x^2}{3}\right)\right)+9 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right)}+x^2-1\right)}{8 \sqrt[3]{1-x^2} \left(x^2+3\right)}-\frac{1}{216} x^3 F_1\left(\frac{3}{2};\frac{1}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right)","-\frac{\left(1-x^2\right)^{2/3} x}{8 \left(x^2+3\right)}+\frac{x}{8 \left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} \left(1-\sqrt[3]{2} \sqrt[3]{1-x^2}\right)}{x}\right)}{8\ 2^{2/3} \sqrt{3}}+\frac{\tanh ^{-1}\left(\frac{x}{\sqrt[3]{2} \sqrt[3]{1-x^2}+1}\right)}{8\ 2^{2/3}}-\frac{\left(1-\sqrt[3]{1-x^2}\right) \sqrt{\frac{\left(1-x^2\right)^{2/3}+\sqrt[3]{1-x^2}+1}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-\sqrt[3]{1-x^2}+\sqrt{3}+1}{-\sqrt[3]{1-x^2}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{4 \sqrt{2} \sqrt[4]{3} \sqrt{-\frac{1-\sqrt[3]{1-x^2}}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} x}+\frac{\sqrt[4]{3} \sqrt{2+\sqrt{3}} \left(1-\sqrt[3]{1-x^2}\right) \sqrt{\frac{\left(1-x^2\right)^{2/3}+\sqrt[3]{1-x^2}+1}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-\sqrt[3]{1-x^2}+\sqrt{3}+1}{-\sqrt[3]{1-x^2}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{16 \sqrt{-\frac{1-\sqrt[3]{1-x^2}}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} x}+\frac{\tan ^{-1}\left(\frac{\sqrt{3}}{x}\right)}{8\ 2^{2/3} \sqrt{3}}-\frac{\tanh ^{-1}(x)}{24\ 2^{2/3}}",1,"-1/216*(x^3*AppellF1[3/2, 1/3, 1, 5/2, x^2, -1/3*x^2]) + (x*(-1 + x^2 + (9*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1/3*x^2])/(9*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1/3*x^2] + 2*x^2*(-AppellF1[3/2, 1/3, 2, 5/2, x^2, -1/3*x^2] + AppellF1[3/2, 4/3, 1, 5/2, x^2, -1/3*x^2]))))/(8*(1 - x^2)^(1/3)*(3 + x^2))","C",0
1028,1,157,543,0.1032929,"\int \frac{1}{\sqrt[3]{1-x^2} \left(3+x^2\right)^2} \, dx","Integrate[1/((1 - x^2)^(1/3)*(3 + x^2)^2),x]","\frac{1}{648} x \left(x^2 F_1\left(\frac{3}{2};\frac{1}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right)+\frac{27 \left(\frac{63 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right)}{2 x^2 \left(F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right)-F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};x^2,-\frac{x^2}{3}\right)\right)+9 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right)}-x^2+1\right)}{\sqrt[3]{1-x^2} \left(x^2+3\right)}\right)","\frac{\left(1-x^2\right)^{2/3} x}{24 \left(x^2+3\right)}-\frac{x}{24 \left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} \left(1-\sqrt[3]{2} \sqrt[3]{1-x^2}\right)}{x}\right)}{8\ 2^{2/3} \sqrt{3}}+\frac{\tanh ^{-1}\left(\frac{x}{\sqrt[3]{2} \sqrt[3]{1-x^2}+1}\right)}{8\ 2^{2/3}}+\frac{\left(1-\sqrt[3]{1-x^2}\right) \sqrt{\frac{\left(1-x^2\right)^{2/3}+\sqrt[3]{1-x^2}+1}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-\sqrt[3]{1-x^2}+\sqrt{3}+1}{-\sqrt[3]{1-x^2}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{12 \sqrt{2} \sqrt[4]{3} \sqrt{-\frac{1-\sqrt[3]{1-x^2}}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} x}-\frac{\sqrt{2+\sqrt{3}} \left(1-\sqrt[3]{1-x^2}\right) \sqrt{\frac{\left(1-x^2\right)^{2/3}+\sqrt[3]{1-x^2}+1}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-\sqrt[3]{1-x^2}+\sqrt{3}+1}{-\sqrt[3]{1-x^2}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{16\ 3^{3/4} \sqrt{-\frac{1-\sqrt[3]{1-x^2}}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} x}+\frac{\tan ^{-1}\left(\frac{\sqrt{3}}{x}\right)}{8\ 2^{2/3} \sqrt{3}}-\frac{\tanh ^{-1}(x)}{24\ 2^{2/3}}",1,"(x*(x^2*AppellF1[3/2, 1/3, 1, 5/2, x^2, -1/3*x^2] + (27*(1 - x^2 + (63*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1/3*x^2])/(9*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1/3*x^2] + 2*x^2*(-AppellF1[3/2, 1/3, 2, 5/2, x^2, -1/3*x^2] + AppellF1[3/2, 4/3, 1, 5/2, x^2, -1/3*x^2]))))/((1 - x^2)^(1/3)*(3 + x^2))))/648","C",0
1029,1,168,563,0.1096645,"\int \frac{1}{x^2 \sqrt[3]{1-x^2} \left(3+x^2\right)^2} \, dx","Integrate[1/(x^2*(1 - x^2)^(1/3)*(3 + x^2)^2),x]","\frac{\frac{9 \left(\frac{69 x^2 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right)}{2 x^2 \left(F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};x^2,-\frac{x^2}{3}\right)-F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right)\right)-9 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right)}+3 x^4+5 x^2-8\right)}{\sqrt[3]{1-x^2} \left(x^2+3\right)}-x^4 F_1\left(\frac{3}{2};\frac{1}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right)}{216 x}","\frac{x}{8 \left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)}-\frac{\left(1-x^2\right)^{2/3}}{8 x}+\frac{\left(1-x^2\right)^{2/3}}{24 \left(x^2+3\right) x}-\frac{7 \tan ^{-1}\left(\frac{\sqrt{3} \left(1-\sqrt[3]{2} \sqrt[3]{1-x^2}\right)}{x}\right)}{72\ 2^{2/3} \sqrt{3}}-\frac{7 \tanh ^{-1}\left(\frac{x}{\sqrt[3]{2} \sqrt[3]{1-x^2}+1}\right)}{72\ 2^{2/3}}-\frac{\left(1-\sqrt[3]{1-x^2}\right) \sqrt{\frac{\left(1-x^2\right)^{2/3}+\sqrt[3]{1-x^2}+1}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-\sqrt[3]{1-x^2}+\sqrt{3}+1}{-\sqrt[3]{1-x^2}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{4 \sqrt{2} \sqrt[4]{3} \sqrt{-\frac{1-\sqrt[3]{1-x^2}}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} x}+\frac{\sqrt[4]{3} \sqrt{2+\sqrt{3}} \left(1-\sqrt[3]{1-x^2}\right) \sqrt{\frac{\left(1-x^2\right)^{2/3}+\sqrt[3]{1-x^2}+1}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-\sqrt[3]{1-x^2}+\sqrt{3}+1}{-\sqrt[3]{1-x^2}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{16 \sqrt{-\frac{1-\sqrt[3]{1-x^2}}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} x}-\frac{7 \tan ^{-1}\left(\frac{\sqrt{3}}{x}\right)}{72\ 2^{2/3} \sqrt{3}}+\frac{7 \tanh ^{-1}(x)}{216\ 2^{2/3}}",1,"(-(x^4*AppellF1[3/2, 1/3, 1, 5/2, x^2, -1/3*x^2]) + (9*(-8 + 5*x^2 + 3*x^4 + (69*x^2*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1/3*x^2])/(-9*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1/3*x^2] + 2*x^2*(AppellF1[3/2, 1/3, 2, 5/2, x^2, -1/3*x^2] - AppellF1[3/2, 4/3, 1, 5/2, x^2, -1/3*x^2]))))/((1 - x^2)^(1/3)*(3 + x^2)))/(216*x)","C",0
1030,1,173,581,0.1300027,"\int \frac{1}{x^4 \sqrt[3]{1-x^2} \left(3+x^2\right)^2} \, dx","Integrate[1/(x^4*(1 - x^2)^(1/3)*(3 + x^2)^2),x]","\frac{11 x^6 F_1\left(\frac{3}{2};\frac{1}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right)+\frac{27 \left(\frac{693 x^4 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right)}{2 x^2 \left(F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right)-F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};x^2,-\frac{x^2}{3}\right)\right)+9 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right)}-11 x^6+11 x^4+72 x^2-72\right)}{\sqrt[3]{1-x^2} \left(x^2+3\right)}}{17496 x^3}","-\frac{11 x}{648 \left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)}+\frac{11 \left(1-x^2\right)^{2/3}}{648 x}+\frac{11 \tan ^{-1}\left(\frac{\sqrt{3} \left(1-\sqrt[3]{2} \sqrt[3]{1-x^2}\right)}{x}\right)}{216\ 2^{2/3} \sqrt{3}}+\frac{11 \tanh ^{-1}\left(\frac{x}{\sqrt[3]{2} \sqrt[3]{1-x^2}+1}\right)}{216\ 2^{2/3}}+\frac{11 \left(1-\sqrt[3]{1-x^2}\right) \sqrt{\frac{\left(1-x^2\right)^{2/3}+\sqrt[3]{1-x^2}+1}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-\sqrt[3]{1-x^2}+\sqrt{3}+1}{-\sqrt[3]{1-x^2}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{324 \sqrt{2} \sqrt[4]{3} \sqrt{-\frac{1-\sqrt[3]{1-x^2}}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} x}-\frac{11 \sqrt{2+\sqrt{3}} \left(1-\sqrt[3]{1-x^2}\right) \sqrt{\frac{\left(1-x^2\right)^{2/3}+\sqrt[3]{1-x^2}+1}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-\sqrt[3]{1-x^2}+\sqrt{3}+1}{-\sqrt[3]{1-x^2}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{432\ 3^{3/4} \sqrt{-\frac{1-\sqrt[3]{1-x^2}}{\left(-\sqrt[3]{1-x^2}-\sqrt{3}+1\right)^2}} x}-\frac{11 \left(1-x^2\right)^{2/3}}{216 x^3}+\frac{\left(1-x^2\right)^{2/3}}{24 \left(x^2+3\right) x^3}+\frac{11 \tan ^{-1}\left(\frac{\sqrt{3}}{x}\right)}{216\ 2^{2/3} \sqrt{3}}-\frac{11 \tanh ^{-1}(x)}{648\ 2^{2/3}}",1,"(11*x^6*AppellF1[3/2, 1/3, 1, 5/2, x^2, -1/3*x^2] + (27*(-72 + 72*x^2 + 11*x^4 - 11*x^6 + (693*x^4*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1/3*x^2])/(9*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1/3*x^2] + 2*x^2*(-AppellF1[3/2, 1/3, 2, 5/2, x^2, -1/3*x^2] + AppellF1[3/2, 4/3, 1, 5/2, x^2, -1/3*x^2]))))/((1 - x^2)^(1/3)*(3 + x^2)))/(17496*x^3)","C",0
1031,1,46,136,0.0302866,"\int \frac{x^7}{\sqrt[4]{2-3 x^2} \left(4-3 x^2\right)} \, dx","Integrate[x^7/((2 - 3*x^2)^(1/4)*(4 - 3*x^2)),x]","\frac{2 \left(2-3 x^2\right)^{3/4} \left(-2464 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};\frac{3 x^2}{2}-1\right)+189 x^4+540 x^2+1712\right)}{18711}","\frac{2}{891} \left(2-3 x^2\right)^{11/4}-\frac{16}{567} \left(2-3 x^2\right)^{7/4}+\frac{56}{243} \left(2-3 x^2\right)^{3/4}+\frac{32}{81} \sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt{2}-\sqrt{2-3 x^2}}{2^{3/4} \sqrt[4]{2-3 x^2}}\right)+\frac{32}{81} \sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt{2-3 x^2}+\sqrt{2}}{2^{3/4} \sqrt[4]{2-3 x^2}}\right)",1,"(2*(2 - 3*x^2)^(3/4)*(1712 + 540*x^2 + 189*x^4 - 2464*Hypergeometric2F1[3/4, 1, 7/4, -1 + (3*x^2)/2]))/18711","C",1
1032,1,42,121,0.0226114,"\int \frac{x^5}{\sqrt[4]{2-3 x^2} \left(4-3 x^2\right)} \, dx","Integrate[x^5/((2 - 3*x^2)^(1/4)*(4 - 3*x^2)),x]","\frac{2}{567} \left(2-3 x^2\right)^{3/4} \left(9 \left(x^2+4\right)-56 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};\frac{3 x^2}{2}-1\right)\right)","-\frac{2}{189} \left(2-3 x^2\right)^{7/4}+\frac{4}{27} \left(2-3 x^2\right)^{3/4}+\frac{8}{27} \sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt{2}-\sqrt{2-3 x^2}}{2^{3/4} \sqrt[4]{2-3 x^2}}\right)+\frac{8}{27} \sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt{2-3 x^2}+\sqrt{2}}{2^{3/4} \sqrt[4]{2-3 x^2}}\right)",1,"(2*(2 - 3*x^2)^(3/4)*(9*(4 + x^2) - 56*Hypergeometric2F1[3/4, 1, 7/4, -1 + (3*x^2)/2]))/567","C",1
1033,1,36,106,0.0122874,"\int \frac{x^3}{\sqrt[4]{2-3 x^2} \left(4-3 x^2\right)} \, dx","Integrate[x^3/((2 - 3*x^2)^(1/4)*(4 - 3*x^2)),x]","-\frac{2}{27} \left(2-3 x^2\right)^{3/4} \left(2 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};\frac{3 x^2}{2}-1\right)-1\right)","\frac{2}{27} \left(2-3 x^2\right)^{3/4}+\frac{2}{9} \sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt{2}-\sqrt{2-3 x^2}}{2^{3/4} \sqrt[4]{2-3 x^2}}\right)+\frac{2}{9} \sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt{2-3 x^2}+\sqrt{2}}{2^{3/4} \sqrt[4]{2-3 x^2}}\right)",1,"(-2*(2 - 3*x^2)^(3/4)*(-1 + 2*Hypergeometric2F1[3/4, 1, 7/4, -1 + (3*x^2)/2]))/27","C",1
1034,1,34,91,0.0077991,"\int \frac{x}{\sqrt[4]{2-3 x^2} \left(4-3 x^2\right)} \, dx","Integrate[x/((2 - 3*x^2)^(1/4)*(4 - 3*x^2)),x]","-\frac{1}{9} \left(2-3 x^2\right)^{3/4} \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};\frac{1}{2} \left(3 x^2-2\right)\right)","\frac{\tan ^{-1}\left(\frac{\sqrt{2}-\sqrt{2-3 x^2}}{2^{3/4} \sqrt[4]{2-3 x^2}}\right)}{3\ 2^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2-3 x^2}+\sqrt{2}}{2^{3/4} \sqrt[4]{2-3 x^2}}\right)}{3\ 2^{3/4}}",1,"-1/9*((2 - 3*x^2)^(3/4)*Hypergeometric2F1[3/4, 1, 7/4, (-2 + 3*x^2)/2])","C",1
1035,1,73,145,0.0954005,"\int \frac{1}{x \sqrt[4]{2-3 x^2} \left(4-3 x^2\right)} \, dx","Integrate[1/(x*(2 - 3*x^2)^(1/4)*(4 - 3*x^2)),x]","\frac{1}{24} \left(3\ 2^{3/4} \left(\tan ^{-1}\left(\sqrt[4]{1-\frac{3 x^2}{2}}\right)-\tanh ^{-1}\left(\sqrt[4]{1-\frac{3 x^2}{2}}\right)\right)-2 \left(2-3 x^2\right)^{3/4} \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};\frac{3 x^2}{2}-1\right)\right)","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2-3 x^2}}{\sqrt[4]{2}}\right)}{4 \sqrt[4]{2}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2}-\sqrt{2-3 x^2}}{2^{3/4} \sqrt[4]{2-3 x^2}}\right)}{4\ 2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2-3 x^2}}{\sqrt[4]{2}}\right)}{4 \sqrt[4]{2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2-3 x^2}+\sqrt{2}}{2^{3/4} \sqrt[4]{2-3 x^2}}\right)}{4\ 2^{3/4}}",1,"(3*2^(3/4)*(ArcTan[(1 - (3*x^2)/2)^(1/4)] - ArcTanh[(1 - (3*x^2)/2)^(1/4)]) - 2*(2 - 3*x^2)^(3/4)*Hypergeometric2F1[3/4, 1, 7/4, -1 + (3*x^2)/2])/24","C",1
1036,1,102,163,0.0538587,"\int \frac{1}{x^3 \sqrt[4]{2-3 x^2} \left(4-3 x^2\right)} \, dx","Integrate[1/(x^3*(2 - 3*x^2)^(1/4)*(4 - 3*x^2)),x]","-\frac{4 \left(2-3 x^2\right)^{3/4} x^2 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};\frac{3 x^2}{2}-1\right)+4 \left(2-3 x^2\right)^{3/4}-9\ 2^{3/4} x^2 \tan ^{-1}\left(\sqrt[4]{1-\frac{3 x^2}{2}}\right)+9\ 2^{3/4} x^2 \tanh ^{-1}\left(\sqrt[4]{1-\frac{3 x^2}{2}}\right)}{64 x^2}","-\frac{\left(2-3 x^2\right)^{3/4}}{16 x^2}+\frac{9 \tan ^{-1}\left(\frac{\sqrt[4]{2-3 x^2}}{\sqrt[4]{2}}\right)}{32 \sqrt[4]{2}}+\frac{3 \tan ^{-1}\left(\frac{\sqrt{2}-\sqrt{2-3 x^2}}{2^{3/4} \sqrt[4]{2-3 x^2}}\right)}{16\ 2^{3/4}}-\frac{9 \tanh ^{-1}\left(\frac{\sqrt[4]{2-3 x^2}}{\sqrt[4]{2}}\right)}{32 \sqrt[4]{2}}+\frac{3 \tanh ^{-1}\left(\frac{\sqrt{2-3 x^2}+\sqrt{2}}{2^{3/4} \sqrt[4]{2-3 x^2}}\right)}{16\ 2^{3/4}}",1,"-1/64*(4*(2 - 3*x^2)^(3/4) - 9*2^(3/4)*x^2*ArcTan[(1 - (3*x^2)/2)^(1/4)] + 9*2^(3/4)*x^2*ArcTanh[(1 - (3*x^2)/2)^(1/4)] + 4*x^2*(2 - 3*x^2)^(3/4)*Hypergeometric2F1[3/4, 1, 7/4, -1 + (3*x^2)/2])/x^2","C",1
1037,1,184,164,0.2643766,"\int \frac{x^4}{\sqrt[4]{2-3 x^2} \left(4-3 x^2\right)} \, dx","Integrate[x^4/((2 - 3*x^2)^(1/4)*(4 - 3*x^2)),x]","\frac{1}{45} x \left(3\ 2^{3/4} x^2 F_1\left(\frac{3}{2};\frac{1}{4},1;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right)+\frac{2 \left(\frac{32 F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right)}{\left(3 x^2-4\right) \left(x^2 \left(2 F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right)+F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right)\right)+4 F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right)\right)}-3 x^2+2\right)}{\sqrt[4]{2-3 x^2}}\right)","\frac{2}{45} \left(2-3 x^2\right)^{3/4} x+\frac{4 \sqrt[4]{2} \tan ^{-1}\left(\frac{2^{3/4}-\sqrt[4]{2} \sqrt{2-3 x^2}}{\sqrt{3} x \sqrt[4]{2-3 x^2}}\right)}{9 \sqrt{3}}+\frac{4 \sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt{2-3 x^2}+2^{3/4}}{\sqrt{3} x \sqrt[4]{2-3 x^2}}\right)}{9 \sqrt{3}}-\frac{16 \sqrt[4]{2} E\left(\left.\frac{1}{2} \sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)}{15 \sqrt{3}}",1,"(x*(3*2^(3/4)*x^2*AppellF1[3/2, 1/4, 1, 5/2, (3*x^2)/2, (3*x^2)/4] + (2*(2 - 3*x^2 + (32*AppellF1[1/2, 1/4, 1, 3/2, (3*x^2)/2, (3*x^2)/4])/((-4 + 3*x^2)*(4*AppellF1[1/2, 1/4, 1, 3/2, (3*x^2)/2, (3*x^2)/4] + x^2*(2*AppellF1[3/2, 1/4, 2, 5/2, (3*x^2)/2, (3*x^2)/4] + AppellF1[3/2, 5/4, 1, 5/2, (3*x^2)/2, (3*x^2)/4])))))/(2 - 3*x^2)^(1/4)))/45","C",0
1038,1,37,148,0.0222566,"\int \frac{x^2}{\sqrt[4]{2-3 x^2} \left(4-3 x^2\right)} \, dx","Integrate[x^2/((2 - 3*x^2)^(1/4)*(4 - 3*x^2)),x]","\frac{x^3 F_1\left(\frac{3}{2};\frac{1}{4},1;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right)}{12 \sqrt[4]{2}}","\frac{\sqrt[4]{2} \tan ^{-1}\left(\frac{2^{3/4}-\sqrt[4]{2} \sqrt{2-3 x^2}}{\sqrt{3} x \sqrt[4]{2-3 x^2}}\right)}{3 \sqrt{3}}+\frac{\sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt{2-3 x^2}+2^{3/4}}{\sqrt{3} x \sqrt[4]{2-3 x^2}}\right)}{3 \sqrt{3}}-\frac{2 \sqrt[4]{2} E\left(\left.\frac{1}{2} \sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)}{3 \sqrt{3}}",1,"(x^3*AppellF1[3/2, 1/4, 1, 5/2, (3*x^2)/2, (3*x^2)/4])/(12*2^(1/4))","C",0
1039,1,135,120,0.0326172,"\int \frac{1}{\sqrt[4]{2-3 x^2} \left(4-3 x^2\right)} \, dx","Integrate[1/((2 - 3*x^2)^(1/4)*(4 - 3*x^2)),x]","-\frac{4 x F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right)}{\sqrt[4]{2-3 x^2} \left(3 x^2-4\right) \left(x^2 \left(2 F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right)+F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right)\right)+4 F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right)\right)}","\frac{\tan ^{-1}\left(\frac{2-\sqrt{2} \sqrt{2-3 x^2}}{\sqrt[4]{2} \sqrt{3} x \sqrt[4]{2-3 x^2}}\right)}{2\ 2^{3/4} \sqrt{3}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{2-3 x^2}+2}{\sqrt[4]{2} \sqrt{3} x \sqrt[4]{2-3 x^2}}\right)}{2\ 2^{3/4} \sqrt{3}}",1,"(-4*x*AppellF1[1/2, 1/4, 1, 3/2, (3*x^2)/2, (3*x^2)/4])/((2 - 3*x^2)^(1/4)*(-4 + 3*x^2)*(4*AppellF1[1/2, 1/4, 1, 3/2, (3*x^2)/2, (3*x^2)/4] + x^2*(2*AppellF1[3/2, 1/4, 2, 5/2, (3*x^2)/2, (3*x^2)/4] + AppellF1[3/2, 5/4, 1, 5/2, (3*x^2)/2, (3*x^2)/4])))","C",0
1040,1,56,166,0.0418451,"\int \frac{1}{x^2 \sqrt[4]{2-3 x^2} \left(4-3 x^2\right)} \, dx","Integrate[1/(x^2*(2 - 3*x^2)^(1/4)*(4 - 3*x^2)),x]","\frac{3 x^3 F_1\left(\frac{3}{2};\frac{1}{4},1;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right)}{64 \sqrt[4]{2}}-\frac{\left(2-3 x^2\right)^{3/4}}{8 x}","-\frac{\left(2-3 x^2\right)^{3/4}}{8 x}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{2^{3/4}-\sqrt[4]{2} \sqrt{2-3 x^2}}{\sqrt{3} x \sqrt[4]{2-3 x^2}}\right)}{8\ 2^{3/4}}+\frac{\sqrt{3} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt{2-3 x^2}+2^{3/4}}{\sqrt{3} x \sqrt[4]{2-3 x^2}}\right)}{8\ 2^{3/4}}-\frac{\sqrt{3} E\left(\left.\frac{1}{2} \sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)}{4\ 2^{3/4}}",1,"-1/8*(2 - 3*x^2)^(3/4)/x + (3*x^3*AppellF1[3/2, 1/4, 1, 5/2, (3*x^2)/2, (3*x^2)/4])/(64*2^(1/4))","C",0
1041,1,156,184,0.1651095,"\int \frac{1}{x^4 \sqrt[4]{2-3 x^2} \left(4-3 x^2\right)} \, dx","Integrate[1/(x^4*(2 - 3*x^2)^(1/4)*(4 - 3*x^2)),x]","\frac{1}{8} \left(2-3 x^2\right)^{3/4} \left(\frac{9 x F_1\left(\frac{1}{2};-\frac{3}{4},1;\frac{3}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right)}{\left(3 x^2-4\right) \left(x^2 \left(2 F_1\left(\frac{3}{2};-\frac{3}{4},2;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right)-3 F_1\left(\frac{3}{2};\frac{1}{4},1;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right)\right)+4 F_1\left(\frac{1}{2};-\frac{3}{4},1;\frac{3}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right)\right)}-\frac{9 x^2+2}{6 x^3}\right)","-\frac{3 \left(2-3 x^2\right)^{3/4}}{16 x}+\frac{3 \sqrt{3} \tan ^{-1}\left(\frac{2^{3/4}-\sqrt[4]{2} \sqrt{2-3 x^2}}{\sqrt{3} x \sqrt[4]{2-3 x^2}}\right)}{32\ 2^{3/4}}+\frac{3 \sqrt{3} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt{2-3 x^2}+2^{3/4}}{\sqrt{3} x \sqrt[4]{2-3 x^2}}\right)}{32\ 2^{3/4}}-\frac{\left(2-3 x^2\right)^{3/4}}{24 x^3}-\frac{3 \sqrt{3} E\left(\left.\frac{1}{2} \sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)}{8\ 2^{3/4}}",1,"((2 - 3*x^2)^(3/4)*(-1/6*(2 + 9*x^2)/x^3 + (9*x*AppellF1[1/2, -3/4, 1, 3/2, (3*x^2)/2, (3*x^2)/4])/((-4 + 3*x^2)*(4*AppellF1[1/2, -3/4, 1, 3/2, (3*x^2)/2, (3*x^2)/4] + x^2*(2*AppellF1[3/2, -3/4, 2, 5/2, (3*x^2)/2, (3*x^2)/4] - 3*AppellF1[3/2, 1/4, 1, 5/2, (3*x^2)/2, (3*x^2)/4])))))/8","C",0
1042,1,57,78,0.0433112,"\int \frac{x^7}{\left(-2+3 x^2\right) \sqrt[4]{-1+3 x^2}} \, dx","Integrate[x^7/((-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)),x]","\frac{2 \left(924 \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)-924 \tanh ^{-1}\left(\sqrt[4]{3 x^2-1}\right)+\left(3 x^2-1\right)^{3/4} \left(189 x^4+270 x^2+428\right)\right)}{18711}","\frac{2}{891} \left(3 x^2-1\right)^{11/4}+\frac{8}{567} \left(3 x^2-1\right)^{7/4}+\frac{14}{243} \left(3 x^2-1\right)^{3/4}+\frac{8}{81} \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)-\frac{8}{81} \tanh ^{-1}\left(\sqrt[4]{3 x^2-1}\right)",1,"(2*((-1 + 3*x^2)^(3/4)*(428 + 270*x^2 + 189*x^4) + 924*ArcTan[(-1 + 3*x^2)^(1/4)] - 924*ArcTanh[(-1 + 3*x^2)^(1/4)]))/18711","A",1
1043,1,51,63,0.0406545,"\int \frac{x^5}{\left(-2+3 x^2\right) \sqrt[4]{-1+3 x^2}} \, dx","Integrate[x^5/((-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)),x]","\frac{2}{189} \left(3 \left(3 x^2-1\right)^{3/4} \left(x^2+2\right)+14 \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)-14 \tanh ^{-1}\left(\sqrt[4]{3 x^2-1}\right)\right)","\frac{2}{189} \left(3 x^2-1\right)^{7/4}+\frac{2}{27} \left(3 x^2-1\right)^{3/4}+\frac{4}{27} \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)-\frac{4}{27} \tanh ^{-1}\left(\sqrt[4]{3 x^2-1}\right)",1,"(2*(3*(2 + x^2)*(-1 + 3*x^2)^(3/4) + 14*ArcTan[(-1 + 3*x^2)^(1/4)] - 14*ArcTanh[(-1 + 3*x^2)^(1/4)]))/189","A",1
1044,1,44,48,0.0107913,"\int \frac{x^3}{\left(-2+3 x^2\right) \sqrt[4]{-1+3 x^2}} \, dx","Integrate[x^3/((-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)),x]","\frac{2}{27} \left(\left(3 x^2-1\right)^{3/4}+3 \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)-3 \tanh ^{-1}\left(\sqrt[4]{3 x^2-1}\right)\right)","\frac{2}{27} \left(3 x^2-1\right)^{3/4}+\frac{2}{9} \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)-\frac{2}{9} \tanh ^{-1}\left(\sqrt[4]{3 x^2-1}\right)",1,"(2*((-1 + 3*x^2)^(3/4) + 3*ArcTan[(-1 + 3*x^2)^(1/4)] - 3*ArcTanh[(-1 + 3*x^2)^(1/4)]))/27","A",1
1045,1,33,33,0.0064173,"\int \frac{x}{\left(-2+3 x^2\right) \sqrt[4]{-1+3 x^2}} \, dx","Integrate[x/((-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)),x]","\frac{1}{3} \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)-\frac{1}{3} \tanh ^{-1}\left(\sqrt[4]{3 x^2-1}\right)","\frac{1}{3} \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)-\frac{1}{3} \tanh ^{-1}\left(\sqrt[4]{3 x^2-1}\right)",1,"ArcTan[(-1 + 3*x^2)^(1/4)]/3 - ArcTanh[(-1 + 3*x^2)^(1/4)]/3","A",1
1046,1,63,173,0.0170982,"\int \frac{1}{x \left(-2+3 x^2\right) \sqrt[4]{-1+3 x^2}} \, dx","Integrate[1/(x*(-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)),x]","-\frac{1}{3} \left(3 x^2-1\right)^{3/4} \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};1-3 x^2\right)+\frac{1}{2} \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)-\frac{1}{2} \tanh ^{-1}\left(\sqrt[4]{3 x^2-1}\right)","-\frac{\log \left(\sqrt{3 x^2-1}-\sqrt{2} \sqrt[4]{3 x^2-1}+1\right)}{4 \sqrt{2}}+\frac{\log \left(\sqrt{3 x^2-1}+\sqrt{2} \sqrt[4]{3 x^2-1}+1\right)}{4 \sqrt{2}}+\frac{1}{2} \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)+\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt[4]{3 x^2-1}\right)}{2 \sqrt{2}}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt[4]{3 x^2-1}+1\right)}{2 \sqrt{2}}-\frac{1}{2} \tanh ^{-1}\left(\sqrt[4]{3 x^2-1}\right)",1,"ArcTan[(-1 + 3*x^2)^(1/4)]/2 - ArcTanh[(-1 + 3*x^2)^(1/4)]/2 - ((-1 + 3*x^2)^(3/4)*Hypergeometric2F1[3/4, 1, 7/4, 1 - 3*x^2])/3","C",1
1047,1,77,191,0.0333587,"\int \frac{1}{x^3 \left(-2+3 x^2\right) \sqrt[4]{-1+3 x^2}} \, dx","Integrate[1/(x^3*(-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)),x]","\frac{1}{4} \left(-3 \left(3 x^2-1\right)^{3/4} \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};1-3 x^2\right)-\frac{\left(3 x^2-1\right)^{3/4}}{x^2}+3 \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)-3 \tanh ^{-1}\left(\sqrt[4]{3 x^2-1}\right)\right)","-\frac{\left(3 x^2-1\right)^{3/4}}{4 x^2}-\frac{9 \log \left(\sqrt{3 x^2-1}-\sqrt{2} \sqrt[4]{3 x^2-1}+1\right)}{16 \sqrt{2}}+\frac{9 \log \left(\sqrt{3 x^2-1}+\sqrt{2} \sqrt[4]{3 x^2-1}+1\right)}{16 \sqrt{2}}+\frac{3}{4} \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)+\frac{9 \tan ^{-1}\left(1-\sqrt{2} \sqrt[4]{3 x^2-1}\right)}{8 \sqrt{2}}-\frac{9 \tan ^{-1}\left(\sqrt{2} \sqrt[4]{3 x^2-1}+1\right)}{8 \sqrt{2}}-\frac{3}{4} \tanh ^{-1}\left(\sqrt[4]{3 x^2-1}\right)",1,"(-((-1 + 3*x^2)^(3/4)/x^2) + 3*ArcTan[(-1 + 3*x^2)^(1/4)] - 3*ArcTanh[(-1 + 3*x^2)^(1/4)] - 3*(-1 + 3*x^2)^(3/4)*Hypergeometric2F1[3/4, 1, 7/4, 1 - 3*x^2])/4","C",1
1048,1,177,244,0.2119613,"\int \frac{x^4}{\left(-2+3 x^2\right) \sqrt[4]{-1+3 x^2}} \, dx","Integrate[x^4/((-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)),x]","\frac{2 x \left(-3 \sqrt[4]{1-3 x^2} x^2 F_1\left(\frac{3}{2};\frac{1}{4},1;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right)-\frac{4 F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};3 x^2,\frac{3 x^2}{2}\right)}{\left(3 x^2-2\right) \left(x^2 \left(2 F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right)+F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right)\right)+2 F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};3 x^2,\frac{3 x^2}{2}\right)\right)}+3 x^2-1\right)}{45 \sqrt[4]{3 x^2-1}}","\frac{2}{45} \left(3 x^2-1\right)^{3/4} x+\frac{8 \sqrt[4]{3 x^2-1} x}{15 \left(\sqrt{3 x^2-1}+1\right)}-\frac{1}{9} \sqrt{\frac{2}{3}} \tan ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{3 x^2-1}}\right)-\frac{1}{9} \sqrt{\frac{2}{3}} \tanh ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{3 x^2-1}}\right)+\frac{4 \sqrt{\frac{x^2}{\left(\sqrt{3 x^2-1}+1\right)^2}} \left(\sqrt{3 x^2-1}+1\right) F\left(2 \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)|\frac{1}{2}\right)}{15 \sqrt{3} x}-\frac{8 \sqrt{\frac{x^2}{\left(\sqrt{3 x^2-1}+1\right)^2}} \left(\sqrt{3 x^2-1}+1\right) E\left(2 \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)|\frac{1}{2}\right)}{15 \sqrt{3} x}",1,"(2*x*(-1 + 3*x^2 - 3*x^2*(1 - 3*x^2)^(1/4)*AppellF1[3/2, 1/4, 1, 5/2, 3*x^2, (3*x^2)/2] - (4*AppellF1[1/2, 1/4, 1, 3/2, 3*x^2, (3*x^2)/2])/((-2 + 3*x^2)*(2*AppellF1[1/2, 1/4, 1, 3/2, 3*x^2, (3*x^2)/2] + x^2*(2*AppellF1[3/2, 1/4, 2, 5/2, 3*x^2, (3*x^2)/2] + AppellF1[3/2, 5/4, 1, 5/2, 3*x^2, (3*x^2)/2])))))/(45*(-1 + 3*x^2)^(1/4))","C",0
1049,1,52,224,0.0199116,"\int \frac{x^2}{\left(-2+3 x^2\right) \sqrt[4]{-1+3 x^2}} \, dx","Integrate[x^2/((-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)),x]","-\frac{x^3 \sqrt[4]{1-3 x^2} F_1\left(\frac{3}{2};\frac{1}{4},1;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right)}{6 \sqrt[4]{3 x^2-1}}","\frac{2 \sqrt[4]{3 x^2-1} x}{3 \left(\sqrt{3 x^2-1}+1\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{3 x^2-1}}\right)}{3 \sqrt{6}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{3 x^2-1}}\right)}{3 \sqrt{6}}+\frac{\sqrt{\frac{x^2}{\left(\sqrt{3 x^2-1}+1\right)^2}} \left(\sqrt{3 x^2-1}+1\right) F\left(2 \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)|\frac{1}{2}\right)}{3 \sqrt{3} x}-\frac{2 \sqrt{\frac{x^2}{\left(\sqrt{3 x^2-1}+1\right)^2}} \left(\sqrt{3 x^2-1}+1\right) E\left(2 \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)|\frac{1}{2}\right)}{3 \sqrt{3} x}",1,"-1/6*(x^3*(1 - 3*x^2)^(1/4)*AppellF1[3/2, 1/4, 1, 5/2, 3*x^2, (3*x^2)/2])/(-1 + 3*x^2)^(1/4)","C",0
1050,1,127,61,0.0215162,"\int \frac{1}{\left(-2+3 x^2\right) \sqrt[4]{-1+3 x^2}} \, dx","Integrate[1/((-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)),x]","\frac{2 x F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};3 x^2,\frac{3 x^2}{2}\right)}{\left(3 x^2-2\right) \sqrt[4]{3 x^2-1} \left(x^2 \left(2 F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right)+F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right)\right)+2 F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};3 x^2,\frac{3 x^2}{2}\right)\right)}","-\frac{\tan ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{3 x^2-1}}\right)}{2 \sqrt{6}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{3 x^2-1}}\right)}{2 \sqrt{6}}",1,"(2*x*AppellF1[1/2, 1/4, 1, 3/2, 3*x^2, (3*x^2)/2])/((-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)*(2*AppellF1[1/2, 1/4, 1, 3/2, 3*x^2, (3*x^2)/2] + x^2*(2*AppellF1[3/2, 1/4, 2, 5/2, 3*x^2, (3*x^2)/2] + AppellF1[3/2, 5/4, 1, 5/2, 3*x^2, (3*x^2)/2])))","C",0
1051,1,64,246,0.0294321,"\int \frac{1}{x^2 \left(-2+3 x^2\right) \sqrt[4]{-1+3 x^2}} \, dx","Integrate[1/(x^2*(-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)),x]","\frac{-3 \sqrt[4]{1-3 x^2} x^4 F_1\left(\frac{3}{2};\frac{1}{4},1;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right)-12 x^2+4}{8 x \sqrt[4]{3 x^2-1}}","\frac{3 \sqrt[4]{3 x^2-1} x}{2 \left(\sqrt{3 x^2-1}+1\right)}-\frac{\left(3 x^2-1\right)^{3/4}}{2 x}-\frac{1}{4} \sqrt{\frac{3}{2}} \tan ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{3 x^2-1}}\right)-\frac{1}{4} \sqrt{\frac{3}{2}} \tanh ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{3 x^2-1}}\right)+\frac{\sqrt{3} \sqrt{\frac{x^2}{\left(\sqrt{3 x^2-1}+1\right)^2}} \left(\sqrt{3 x^2-1}+1\right) F\left(2 \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)|\frac{1}{2}\right)}{4 x}-\frac{\sqrt{3} \sqrt{\frac{x^2}{\left(\sqrt{3 x^2-1}+1\right)^2}} \left(\sqrt{3 x^2-1}+1\right) E\left(2 \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)|\frac{1}{2}\right)}{2 x}",1,"(4 - 12*x^2 - 3*x^4*(1 - 3*x^2)^(1/4)*AppellF1[3/2, 1/4, 1, 5/2, 3*x^2, (3*x^2)/2])/(8*x*(-1 + 3*x^2)^(1/4))","C",0
1052,1,148,264,0.1550729,"\int \frac{1}{x^4 \left(-2+3 x^2\right) \sqrt[4]{-1+3 x^2}} \, dx","Integrate[1/(x^4*(-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)),x]","\frac{1}{2} \left(3 x^2-1\right)^{3/4} \left(\frac{9 x F_1\left(\frac{1}{2};-\frac{3}{4},1;\frac{3}{2};3 x^2,\frac{3 x^2}{2}\right)}{\left(3 x^2-2\right) \left(x^2 \left(2 F_1\left(\frac{3}{2};-\frac{3}{4},2;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right)-3 F_1\left(\frac{3}{2};\frac{1}{4},1;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right)\right)+2 F_1\left(\frac{1}{2};-\frac{3}{4},1;\frac{3}{2};3 x^2,\frac{3 x^2}{2}\right)\right)}-\frac{9 x^2+1}{3 x^3}\right)","\frac{9 \sqrt[4]{3 x^2-1} x}{2 \left(\sqrt{3 x^2-1}+1\right)}-\frac{3 \left(3 x^2-1\right)^{3/4}}{2 x}-\frac{3}{8} \sqrt{\frac{3}{2}} \tan ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{3 x^2-1}}\right)-\frac{3}{8} \sqrt{\frac{3}{2}} \tanh ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{3 x^2-1}}\right)+\frac{3 \sqrt{3} \sqrt{\frac{x^2}{\left(\sqrt{3 x^2-1}+1\right)^2}} \left(\sqrt{3 x^2-1}+1\right) F\left(2 \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)|\frac{1}{2}\right)}{4 x}-\frac{3 \sqrt{3} \sqrt{\frac{x^2}{\left(\sqrt{3 x^2-1}+1\right)^2}} \left(\sqrt{3 x^2-1}+1\right) E\left(2 \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)|\frac{1}{2}\right)}{2 x}-\frac{\left(3 x^2-1\right)^{3/4}}{6 x^3}",1,"((-1 + 3*x^2)^(3/4)*(-1/3*(1 + 9*x^2)/x^3 + (9*x*AppellF1[1/2, -3/4, 1, 3/2, 3*x^2, (3*x^2)/2])/((-2 + 3*x^2)*(2*AppellF1[1/2, -3/4, 1, 3/2, 3*x^2, (3*x^2)/2] + x^2*(2*AppellF1[3/2, -3/4, 2, 5/2, 3*x^2, (3*x^2)/2] - 3*AppellF1[3/2, 1/4, 1, 5/2, 3*x^2, (3*x^2)/2])))))/2","C",0
1053,1,37,129,0.04188,"\int \frac{x^2}{\left(2+3 x^2\right)^{3/4} \left(4+3 x^2\right)} \, dx","Integrate[x^2/((2 + 3*x^2)^(3/4)*(4 + 3*x^2)),x]","\frac{x^3 F_1\left(\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{3 x^2}{2},-\frac{3 x^2}{4}\right)}{12\ 2^{3/4}}","\frac{\tanh ^{-1}\left(\frac{2\ 2^{3/4}-2 \sqrt[4]{2} \sqrt{3 x^2+2}}{2 \sqrt{3} x \sqrt[4]{3 x^2+2}}\right)}{3 \sqrt[4]{2} \sqrt{3}}-\frac{\tan ^{-1}\left(\frac{2 \sqrt[4]{2} \sqrt{3 x^2+2}+2\ 2^{3/4}}{2 \sqrt{3} x \sqrt[4]{3 x^2+2}}\right)}{3 \sqrt[4]{2} \sqrt{3}}",1,"(x^3*AppellF1[3/2, 3/4, 1, 5/2, (-3*x^2)/2, (-3*x^2)/4])/(12*2^(3/4))","C",0
1054,1,37,120,0.0412973,"\int \frac{x^2}{\left(2-3 x^2\right)^{3/4} \left(4-3 x^2\right)} \, dx","Integrate[x^2/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]","\frac{x^3 F_1\left(\frac{3}{2};\frac{3}{4},1;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right)}{12\ 2^{3/4}}","\frac{\tan ^{-1}\left(\frac{2-\sqrt{2} \sqrt{2-3 x^2}}{\sqrt[4]{2} \sqrt{3} x \sqrt[4]{2-3 x^2}}\right)}{3 \sqrt[4]{2} \sqrt{3}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{2-3 x^2}+2}{\sqrt[4]{2} \sqrt{3} x \sqrt[4]{2-3 x^2}}\right)}{3 \sqrt[4]{2} \sqrt{3}}",1,"(x^3*AppellF1[3/2, 3/4, 1, 5/2, (3*x^2)/2, (3*x^2)/4])/(12*2^(3/4))","C",0
1055,1,39,124,0.0484107,"\int \frac{x^2}{\left(2+b x^2\right)^{3/4} \left(4+b x^2\right)} \, dx","Integrate[x^2/((2 + b*x^2)^(3/4)*(4 + b*x^2)),x]","\frac{x^3 F_1\left(\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{2},-\frac{b x^2}{4}\right)}{12\ 2^{3/4}}","\frac{\tanh ^{-1}\left(\frac{2\ 2^{3/4}-2 \sqrt[4]{2} \sqrt{b x^2+2}}{2 \sqrt{b} x \sqrt[4]{b x^2+2}}\right)}{\sqrt[4]{2} b^{3/2}}-\frac{\tan ^{-1}\left(\frac{2 \sqrt[4]{2} \sqrt{b x^2+2}+2\ 2^{3/4}}{2 \sqrt{b} x \sqrt[4]{b x^2+2}}\right)}{\sqrt[4]{2} b^{3/2}}",1,"(x^3*AppellF1[3/2, 3/4, 1, 5/2, -1/2*(b*x^2), -1/4*(b*x^2)])/(12*2^(3/4))","C",0
1056,1,39,119,0.0545706,"\int \frac{x^2}{\left(2-b x^2\right)^{3/4} \left(4-b x^2\right)} \, dx","Integrate[x^2/((2 - b*x^2)^(3/4)*(4 - b*x^2)),x]","\frac{x^3 F_1\left(\frac{3}{2};\frac{3}{4},1;\frac{5}{2};\frac{b x^2}{2},\frac{b x^2}{4}\right)}{12\ 2^{3/4}}","\frac{\tan ^{-1}\left(\frac{2-\sqrt{2} \sqrt{2-b x^2}}{\sqrt[4]{2} \sqrt{b} x \sqrt[4]{2-b x^2}}\right)}{\sqrt[4]{2} b^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{2-b x^2}+2}{\sqrt[4]{2} \sqrt{b} x \sqrt[4]{2-b x^2}}\right)}{\sqrt[4]{2} b^{3/2}}",1,"(x^3*AppellF1[3/2, 3/4, 1, 5/2, (b*x^2)/2, (b*x^2)/4])/(12*2^(3/4))","C",0
1057,1,65,120,0.063102,"\int \frac{x^2}{\left(a+3 x^2\right)^{3/4} \left(2 a+3 x^2\right)} \, dx","Integrate[x^2/((a + 3*x^2)^(3/4)*(2*a + 3*x^2)),x]","\frac{x^3 \left(\frac{a+3 x^2}{a}\right)^{3/4} F_1\left(\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{3 x^2}{a},-\frac{3 x^2}{2 a}\right)}{6 a \left(a+3 x^2\right)^{3/4}}","\frac{\tanh ^{-1}\left(\frac{a^{3/4} \left(1-\frac{\sqrt{a+3 x^2}}{\sqrt{a}}\right)}{\sqrt{3} x \sqrt[4]{a+3 x^2}}\right)}{3 \sqrt{3} \sqrt[4]{a}}-\frac{\tan ^{-1}\left(\frac{a^{3/4} \left(\frac{\sqrt{a+3 x^2}}{\sqrt{a}}+1\right)}{\sqrt{3} x \sqrt[4]{a+3 x^2}}\right)}{3 \sqrt{3} \sqrt[4]{a}}",1,"(x^3*((a + 3*x^2)/a)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, (-3*x^2)/a, (-3*x^2)/(2*a)])/(6*a*(a + 3*x^2)^(3/4))","C",0
1058,1,65,120,0.0579914,"\int \frac{x^2}{\left(a-3 x^2\right)^{3/4} \left(2 a-3 x^2\right)} \, dx","Integrate[x^2/((a - 3*x^2)^(3/4)*(2*a - 3*x^2)),x]","\frac{x^3 \left(\frac{a-3 x^2}{a}\right)^{3/4} F_1\left(\frac{3}{2};\frac{3}{4},1;\frac{5}{2};\frac{3 x^2}{a},\frac{3 x^2}{2 a}\right)}{6 a \left(a-3 x^2\right)^{3/4}}","\frac{\tan ^{-1}\left(\frac{a^{3/4} \left(1-\frac{\sqrt{a-3 x^2}}{\sqrt{a}}\right)}{\sqrt{3} x \sqrt[4]{a-3 x^2}}\right)}{3 \sqrt{3} \sqrt[4]{a}}-\frac{\tanh ^{-1}\left(\frac{a^{3/4} \left(\frac{\sqrt{a-3 x^2}}{\sqrt{a}}+1\right)}{\sqrt{3} x \sqrt[4]{a-3 x^2}}\right)}{3 \sqrt{3} \sqrt[4]{a}}",1,"(x^3*((a - 3*x^2)/a)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, (3*x^2)/a, (3*x^2)/(2*a)])/(6*a*(a - 3*x^2)^(3/4))","C",0
1059,1,67,115,0.0622998,"\int \frac{x^2}{\left(a+b x^2\right)^{3/4} \left(2 a+b x^2\right)} \, dx","Integrate[x^2/((a + b*x^2)^(3/4)*(2*a + b*x^2)),x]","\frac{x^3 \left(\frac{a+b x^2}{a}\right)^{3/4} F_1\left(\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{b x^2}{2 a}\right)}{6 a \left(a+b x^2\right)^{3/4}}","\frac{\tanh ^{-1}\left(\frac{a^{3/4} \left(1-\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{\sqrt{b} x \sqrt[4]{a+b x^2}}\right)}{\sqrt[4]{a} b^{3/2}}-\frac{\tan ^{-1}\left(\frac{a^{3/4} \left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}+1\right)}{\sqrt{b} x \sqrt[4]{a+b x^2}}\right)}{\sqrt[4]{a} b^{3/2}}",1,"(x^3*((a + b*x^2)/a)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, -((b*x^2)/a), -1/2*(b*x^2)/a])/(6*a*(a + b*x^2)^(3/4))","C",0
1060,1,68,119,0.0722773,"\int \frac{x^2}{\left(a-b x^2\right)^{3/4} \left(2 a-b x^2\right)} \, dx","Integrate[x^2/((a - b*x^2)^(3/4)*(2*a - b*x^2)),x]","\frac{x^3 \left(\frac{a-b x^2}{a}\right)^{3/4} F_1\left(\frac{3}{2};\frac{3}{4},1;\frac{5}{2};\frac{b x^2}{a},\frac{b x^2}{2 a}\right)}{6 a \left(a-b x^2\right)^{3/4}}","\frac{\tan ^{-1}\left(\frac{a^{3/4} \left(1-\frac{\sqrt{a-b x^2}}{\sqrt{a}}\right)}{\sqrt{b} x \sqrt[4]{a-b x^2}}\right)}{\sqrt[4]{a} b^{3/2}}-\frac{\tanh ^{-1}\left(\frac{a^{3/4} \left(\frac{\sqrt{a-b x^2}}{\sqrt{a}}+1\right)}{\sqrt{b} x \sqrt[4]{a-b x^2}}\right)}{\sqrt[4]{a} b^{3/2}}",1,"(x^3*((a - b*x^2)/a)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, (b*x^2)/a, (b*x^2)/(2*a)])/(6*a*(a - b*x^2)^(3/4))","C",0
1061,1,179,188,0.1102027,"\int \frac{x^7}{\left(2-3 x^2\right)^{3/4} \left(4-3 x^2\right)} \, dx","Integrate[x^7/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]","\frac{2 \left(156 \sqrt[4]{2-3 x^2} x^2+1136 \sqrt[4]{2-3 x^2}+180\ 2^{3/4} \log \left(\sqrt{2-3 x^2}-2^{3/4} \sqrt[4]{2-3 x^2}+\sqrt{2}\right)-180\ 2^{3/4} \log \left(\sqrt{2-3 x^2}+2^{3/4} \sqrt[4]{2-3 x^2}+\sqrt{2}\right)+360\ 2^{3/4} \tan ^{-1}\left(1-\sqrt[4]{4-6 x^2}\right)-360\ 2^{3/4} \tan ^{-1}\left(\sqrt[4]{4-6 x^2}+1\right)+45 \sqrt[4]{2-3 x^2} x^4\right)}{3645}","\frac{2}{729} \left(2-3 x^2\right)^{9/4}-\frac{16}{405} \left(2-3 x^2\right)^{5/4}+\frac{56}{81} \sqrt[4]{2-3 x^2}+\frac{8}{81} 2^{3/4} \log \left(\sqrt{2-3 x^2}-2^{3/4} \sqrt[4]{2-3 x^2}+\sqrt{2}\right)-\frac{8}{81} 2^{3/4} \log \left(\sqrt{2-3 x^2}+2^{3/4} \sqrt[4]{2-3 x^2}+\sqrt{2}\right)-\frac{16}{81} 2^{3/4} \tan ^{-1}\left(\sqrt[4]{4-6 x^2}+1\right)+\frac{16}{81} 2^{3/4} \tan ^{-1}\left(1-\sqrt[4]{2} \sqrt[4]{2-3 x^2}\right)",1,"(2*(1136*(2 - 3*x^2)^(1/4) + 156*x^2*(2 - 3*x^2)^(1/4) + 45*x^4*(2 - 3*x^2)^(1/4) + 360*2^(3/4)*ArcTan[1 - (4 - 6*x^2)^(1/4)] - 360*2^(3/4)*ArcTan[1 + (4 - 6*x^2)^(1/4)] + 180*2^(3/4)*Log[Sqrt[2] - 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqrt[2 - 3*x^2]] - 180*2^(3/4)*Log[Sqrt[2] + 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqrt[2 - 3*x^2]]))/3645","A",1
1062,1,163,173,0.0663851,"\int \frac{x^5}{\left(2-3 x^2\right)^{3/4} \left(4-3 x^2\right)} \, dx","Integrate[x^5/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]","\frac{2}{135} \left(3 \sqrt[4]{2-3 x^2} x^2+28 \sqrt[4]{2-3 x^2}+5\ 2^{3/4} \log \left(\sqrt{2-3 x^2}-2^{3/4} \sqrt[4]{2-3 x^2}+\sqrt{2}\right)-5\ 2^{3/4} \log \left(\sqrt{2-3 x^2}+2^{3/4} \sqrt[4]{2-3 x^2}+\sqrt{2}\right)+10\ 2^{3/4} \tan ^{-1}\left(1-\sqrt[4]{4-6 x^2}\right)-10\ 2^{3/4} \tan ^{-1}\left(\sqrt[4]{4-6 x^2}+1\right)\right)","-\frac{2}{135} \left(2-3 x^2\right)^{5/4}+\frac{4}{9} \sqrt[4]{2-3 x^2}+\frac{2}{27} 2^{3/4} \log \left(\sqrt{2-3 x^2}-2^{3/4} \sqrt[4]{2-3 x^2}+\sqrt{2}\right)-\frac{2}{27} 2^{3/4} \log \left(\sqrt{2-3 x^2}+2^{3/4} \sqrt[4]{2-3 x^2}+\sqrt{2}\right)-\frac{4}{27} 2^{3/4} \tan ^{-1}\left(\sqrt[4]{4-6 x^2}+1\right)+\frac{4}{27} 2^{3/4} \tan ^{-1}\left(1-\sqrt[4]{2} \sqrt[4]{2-3 x^2}\right)",1,"(2*(28*(2 - 3*x^2)^(1/4) + 3*x^2*(2 - 3*x^2)^(1/4) + 10*2^(3/4)*ArcTan[1 - (4 - 6*x^2)^(1/4)] - 10*2^(3/4)*ArcTan[1 + (4 - 6*x^2)^(1/4)] + 5*2^(3/4)*Log[Sqrt[2] - 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqrt[2 - 3*x^2]] - 5*2^(3/4)*Log[Sqrt[2] + 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqrt[2 - 3*x^2]]))/135","A",1
1063,1,146,158,0.0404275,"\int \frac{x^3}{\left(2-3 x^2\right)^{3/4} \left(4-3 x^2\right)} \, dx","Integrate[x^3/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]","\frac{1}{18} \left(4 \sqrt[4]{2-3 x^2}+2^{3/4} \log \left(\sqrt{2-3 x^2}-2^{3/4} \sqrt[4]{2-3 x^2}+\sqrt{2}\right)-2^{3/4} \log \left(\sqrt{2-3 x^2}+2^{3/4} \sqrt[4]{2-3 x^2}+\sqrt{2}\right)+2\ 2^{3/4} \tan ^{-1}\left(1-\sqrt[4]{4-6 x^2}\right)-2\ 2^{3/4} \tan ^{-1}\left(\sqrt[4]{4-6 x^2}+1\right)\right)","\frac{2}{9} \sqrt[4]{2-3 x^2}+\frac{\log \left(\sqrt{2-3 x^2}-2^{3/4} \sqrt[4]{2-3 x^2}+\sqrt{2}\right)}{9 \sqrt[4]{2}}-\frac{\log \left(\sqrt{2-3 x^2}+2^{3/4} \sqrt[4]{2-3 x^2}+\sqrt{2}\right)}{9 \sqrt[4]{2}}-\frac{1}{9} 2^{3/4} \tan ^{-1}\left(\sqrt[4]{4-6 x^2}+1\right)+\frac{1}{9} 2^{3/4} \tan ^{-1}\left(1-\sqrt[4]{2} \sqrt[4]{2-3 x^2}\right)",1,"(4*(2 - 3*x^2)^(1/4) + 2*2^(3/4)*ArcTan[1 - (4 - 6*x^2)^(1/4)] - 2*2^(3/4)*ArcTan[1 + (4 - 6*x^2)^(1/4)] + 2^(3/4)*Log[Sqrt[2] - 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqrt[2 - 3*x^2]] - 2^(3/4)*Log[Sqrt[2] + 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqrt[2 - 3*x^2]])/18","A",1
1064,1,117,143,0.027356,"\int \frac{x}{\left(2-3 x^2\right)^{3/4} \left(4-3 x^2\right)} \, dx","Integrate[x/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]","\frac{\log \left(\sqrt{2-3 x^2}-2^{3/4} \sqrt[4]{2-3 x^2}+\sqrt{2}\right)-\log \left(\sqrt{2-3 x^2}+2^{3/4} \sqrt[4]{2-3 x^2}+\sqrt{2}\right)+2 \tan ^{-1}\left(1-\sqrt[4]{4-6 x^2}\right)-2 \tan ^{-1}\left(\sqrt[4]{4-6 x^2}+1\right)}{12 \sqrt[4]{2}}","\frac{\log \left(\sqrt{2-3 x^2}-2^{3/4} \sqrt[4]{2-3 x^2}+\sqrt{2}\right)}{12 \sqrt[4]{2}}-\frac{\log \left(\sqrt{2-3 x^2}+2^{3/4} \sqrt[4]{2-3 x^2}+\sqrt{2}\right)}{12 \sqrt[4]{2}}-\frac{\tan ^{-1}\left(\sqrt[4]{4-6 x^2}+1\right)}{6 \sqrt[4]{2}}+\frac{\tan ^{-1}\left(1-\sqrt[4]{2} \sqrt[4]{2-3 x^2}\right)}{6 \sqrt[4]{2}}",1,"(2*ArcTan[1 - (4 - 6*x^2)^(1/4)] - 2*ArcTan[1 + (4 - 6*x^2)^(1/4)] + Log[Sqrt[2] - 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqrt[2 - 3*x^2]] - Log[Sqrt[2] + 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqrt[2 - 3*x^2]])/(12*2^(1/4))","A",1
1065,1,156,197,0.0598348,"\int \frac{1}{x \left(2-3 x^2\right)^{3/4} \left(4-3 x^2\right)} \, dx","Integrate[1/(x*(2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]","\frac{-4 \tan ^{-1}\left(\sqrt[4]{1-\frac{3 x^2}{2}}\right)-4 \tanh ^{-1}\left(\sqrt[4]{1-\frac{3 x^2}{2}}\right)+\sqrt{2} \left(\log \left(\sqrt{2-3 x^2}-2^{3/4} \sqrt[4]{2-3 x^2}+\sqrt{2}\right)-\log \left(\sqrt{2-3 x^2}+2^{3/4} \sqrt[4]{2-3 x^2}+\sqrt{2}\right)+2 \tan ^{-1}\left(1-\sqrt[4]{4-6 x^2}\right)-2 \tan ^{-1}\left(\sqrt[4]{4-6 x^2}+1\right)\right)}{16\ 2^{3/4}}","\frac{\log \left(\sqrt{2-3 x^2}-2^{3/4} \sqrt[4]{2-3 x^2}+\sqrt{2}\right)}{16 \sqrt[4]{2}}-\frac{\log \left(\sqrt{2-3 x^2}+2^{3/4} \sqrt[4]{2-3 x^2}+\sqrt{2}\right)}{16 \sqrt[4]{2}}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2-3 x^2}}{\sqrt[4]{2}}\right)}{4\ 2^{3/4}}-\frac{\tan ^{-1}\left(\sqrt[4]{4-6 x^2}+1\right)}{8 \sqrt[4]{2}}+\frac{\tan ^{-1}\left(1-\sqrt[4]{2} \sqrt[4]{2-3 x^2}\right)}{8 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2-3 x^2}}{\sqrt[4]{2}}\right)}{4\ 2^{3/4}}",1,"(-4*ArcTan[(1 - (3*x^2)/2)^(1/4)] - 4*ArcTanh[(1 - (3*x^2)/2)^(1/4)] + Sqrt[2]*(2*ArcTan[1 - (4 - 6*x^2)^(1/4)] - 2*ArcTan[1 + (4 - 6*x^2)^(1/4)] + Log[Sqrt[2] - 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqrt[2 - 3*x^2]] - Log[Sqrt[2] + 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqrt[2 - 3*x^2]]))/(16*2^(3/4))","A",1
1066,1,210,215,0.089055,"\int \frac{1}{x^3 \left(2-3 x^2\right)^{3/4} \left(4-3 x^2\right)} \, dx","Integrate[1/(x^3*(2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]","-\frac{8 \sqrt[4]{2-3 x^2}-3\ 2^{3/4} x^2 \log \left(\sqrt{2-3 x^2}-2^{3/4} \sqrt[4]{2-3 x^2}+\sqrt{2}\right)+3\ 2^{3/4} x^2 \log \left(\sqrt{2-3 x^2}+2^{3/4} \sqrt[4]{2-3 x^2}+\sqrt{2}\right)+30 \sqrt[4]{2} x^2 \tan ^{-1}\left(\sqrt[4]{1-\frac{3 x^2}{2}}\right)-6\ 2^{3/4} x^2 \tan ^{-1}\left(1-\sqrt[4]{4-6 x^2}\right)+6\ 2^{3/4} x^2 \tan ^{-1}\left(\sqrt[4]{4-6 x^2}+1\right)+30 \sqrt[4]{2} x^2 \tanh ^{-1}\left(\sqrt[4]{1-\frac{3 x^2}{2}}\right)}{128 x^2}","-\frac{\sqrt[4]{2-3 x^2}}{16 x^2}+\frac{3 \log \left(\sqrt{2-3 x^2}-2^{3/4} \sqrt[4]{2-3 x^2}+\sqrt{2}\right)}{64 \sqrt[4]{2}}-\frac{3 \log \left(\sqrt{2-3 x^2}+2^{3/4} \sqrt[4]{2-3 x^2}+\sqrt{2}\right)}{64 \sqrt[4]{2}}-\frac{15 \tan ^{-1}\left(\frac{\sqrt[4]{2-3 x^2}}{\sqrt[4]{2}}\right)}{32\ 2^{3/4}}-\frac{3 \tan ^{-1}\left(\sqrt[4]{4-6 x^2}+1\right)}{32 \sqrt[4]{2}}+\frac{3 \tan ^{-1}\left(1-\sqrt[4]{2} \sqrt[4]{2-3 x^2}\right)}{32 \sqrt[4]{2}}-\frac{15 \tanh ^{-1}\left(\frac{\sqrt[4]{2-3 x^2}}{\sqrt[4]{2}}\right)}{32\ 2^{3/4}}",1,"-1/128*(8*(2 - 3*x^2)^(1/4) + 30*2^(1/4)*x^2*ArcTan[(1 - (3*x^2)/2)^(1/4)] - 6*2^(3/4)*x^2*ArcTan[1 - (4 - 6*x^2)^(1/4)] + 6*2^(3/4)*x^2*ArcTan[1 + (4 - 6*x^2)^(1/4)] + 30*2^(1/4)*x^2*ArcTanh[(1 - (3*x^2)/2)^(1/4)] - 3*2^(3/4)*x^2*Log[Sqrt[2] - 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqrt[2 - 3*x^2]] + 3*2^(3/4)*x^2*Log[Sqrt[2] + 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqrt[2 - 3*x^2]])/x^2","A",1
1067,1,190,182,0.2484344,"\int \frac{x^6}{\left(2-3 x^2\right)^{3/4} \left(4-3 x^2\right)} \, dx","Integrate[x^6/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]","\frac{2}{567} x \left(31 \sqrt[4]{2} x^2 F_1\left(\frac{3}{2};\frac{3}{4},1;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right)+\frac{\frac{1280 F_1\left(\frac{1}{2};\frac{3}{4},1;\frac{3}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right)}{\left(3 x^2-4\right) \left(x^2 \left(2 F_1\left(\frac{3}{2};\frac{3}{4},2;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right)+3 F_1\left(\frac{3}{2};\frac{7}{4},1;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right)\right)+4 F_1\left(\frac{1}{2};\frac{3}{4},1;\frac{3}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right)\right)}-27 x^4-102 x^2+80}{\left(2-3 x^2\right)^{3/4}}\right)","\frac{80}{567} \sqrt[4]{2-3 x^2} x+\frac{8\ 2^{3/4} \tan ^{-1}\left(\frac{2^{3/4}-\sqrt[4]{2} \sqrt{2-3 x^2}}{\sqrt{3} x \sqrt[4]{2-3 x^2}}\right)}{27 \sqrt{3}}-\frac{8\ 2^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt{2-3 x^2}+2^{3/4}}{\sqrt{3} x \sqrt[4]{2-3 x^2}}\right)}{27 \sqrt{3}}+\frac{2}{63} \sqrt[4]{2-3 x^2} x^3-\frac{160\ 2^{3/4} F\left(\left.\frac{1}{2} \sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)}{567 \sqrt{3}}",1,"(2*x*(31*2^(1/4)*x^2*AppellF1[3/2, 3/4, 1, 5/2, (3*x^2)/2, (3*x^2)/4] + (80 - 102*x^2 - 27*x^4 + (1280*AppellF1[1/2, 3/4, 1, 3/2, (3*x^2)/2, (3*x^2)/4])/((-4 + 3*x^2)*(4*AppellF1[1/2, 3/4, 1, 3/2, (3*x^2)/2, (3*x^2)/4] + x^2*(2*AppellF1[3/2, 3/4, 2, 5/2, (3*x^2)/2, (3*x^2)/4] + 3*AppellF1[3/2, 7/4, 1, 5/2, (3*x^2)/2, (3*x^2)/4]))))/(2 - 3*x^2)^(3/4)))/567","C",0
1068,1,184,164,0.1271069,"\int \frac{x^4}{\left(2-3 x^2\right)^{3/4} \left(4-3 x^2\right)} \, dx","Integrate[x^4/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]","\frac{2}{27} x \left(\sqrt[4]{2} x^2 F_1\left(\frac{3}{2};\frac{3}{4},1;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right)+\frac{\frac{32 F_1\left(\frac{1}{2};\frac{3}{4},1;\frac{3}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right)}{\left(3 x^2-4\right) \left(x^2 \left(2 F_1\left(\frac{3}{2};\frac{3}{4},2;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right)+3 F_1\left(\frac{3}{2};\frac{7}{4},1;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right)\right)+4 F_1\left(\frac{1}{2};\frac{3}{4},1;\frac{3}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right)\right)}-3 x^2+2}{\left(2-3 x^2\right)^{3/4}}\right)","\frac{2}{27} \sqrt[4]{2-3 x^2} x+\frac{2\ 2^{3/4} \tan ^{-1}\left(\frac{2^{3/4}-\sqrt[4]{2} \sqrt{2-3 x^2}}{\sqrt{3} x \sqrt[4]{2-3 x^2}}\right)}{9 \sqrt{3}}-\frac{2\ 2^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt{2-3 x^2}+2^{3/4}}{\sqrt{3} x \sqrt[4]{2-3 x^2}}\right)}{9 \sqrt{3}}-\frac{4\ 2^{3/4} F\left(\left.\frac{1}{2} \sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)}{27 \sqrt{3}}",1,"(2*x*(2^(1/4)*x^2*AppellF1[3/2, 3/4, 1, 5/2, (3*x^2)/2, (3*x^2)/4] + (2 - 3*x^2 + (32*AppellF1[1/2, 3/4, 1, 3/2, (3*x^2)/2, (3*x^2)/4])/((-4 + 3*x^2)*(4*AppellF1[1/2, 3/4, 1, 3/2, (3*x^2)/2, (3*x^2)/4] + x^2*(2*AppellF1[3/2, 3/4, 2, 5/2, (3*x^2)/2, (3*x^2)/4] + 3*AppellF1[3/2, 7/4, 1, 5/2, (3*x^2)/2, (3*x^2)/4]))))/(2 - 3*x^2)^(3/4)))/27","C",0
1069,1,37,120,0.0318798,"\int \frac{x^2}{\left(2-3 x^2\right)^{3/4} \left(4-3 x^2\right)} \, dx","Integrate[x^2/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]","\frac{x^3 F_1\left(\frac{3}{2};\frac{3}{4},1;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right)}{12\ 2^{3/4}}","\frac{\tan ^{-1}\left(\frac{2-\sqrt{2} \sqrt{2-3 x^2}}{\sqrt[4]{2} \sqrt{3} x \sqrt[4]{2-3 x^2}}\right)}{3 \sqrt[4]{2} \sqrt{3}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{2-3 x^2}+2}{\sqrt[4]{2} \sqrt{3} x \sqrt[4]{2-3 x^2}}\right)}{3 \sqrt[4]{2} \sqrt{3}}",1,"(x^3*AppellF1[3/2, 3/4, 1, 5/2, (3*x^2)/2, (3*x^2)/4])/(12*2^(3/4))","C",0
1070,1,63,148,0.0206594,"\int \frac{1}{\left(2-3 x^2\right)^{3/4} \left(4-3 x^2\right)} \, dx","Integrate[1/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]","-\frac{\sqrt{x^2} \left(\Pi \left(-i;\left.\sin ^{-1}\left(\sqrt[4]{1-\frac{3 x^2}{2}}\right)\right|-1\right)+\Pi \left(i;\left.\sin ^{-1}\left(\sqrt[4]{1-\frac{3 x^2}{2}}\right)\right|-1\right)\right)}{2 \sqrt[4]{2} \sqrt{3} x}","\frac{\tan ^{-1}\left(\frac{2^{3/4}-\sqrt[4]{2} \sqrt{2-3 x^2}}{\sqrt{3} x \sqrt[4]{2-3 x^2}}\right)}{4 \sqrt[4]{2} \sqrt{3}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt{2-3 x^2}+2^{3/4}}{\sqrt{3} x \sqrt[4]{2-3 x^2}}\right)}{4 \sqrt[4]{2} \sqrt{3}}+\frac{F\left(\left.\frac{1}{2} \sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)}{2 \sqrt[4]{2} \sqrt{3}}",1,"-1/2*(Sqrt[x^2]*(EllipticPi[-I, ArcSin[(1 - (3*x^2)/2)^(1/4)], -1] + EllipticPi[I, ArcSin[(1 - (3*x^2)/2)^(1/4)], -1]))/(2^(1/4)*Sqrt[3]*x)","C",1
1071,1,37,166,0.0910095,"\int \frac{1}{x^2 \left(2-3 x^2\right)^{3/4} \left(4-3 x^2\right)} \, dx","Integrate[1/(x^2*(2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]","-\frac{F_1\left(-\frac{1}{2};\frac{3}{4},1;\frac{1}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right)}{4\ 2^{3/4} x}","-\frac{\sqrt[4]{2-3 x^2}}{8 x}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{2^{3/4}-\sqrt[4]{2} \sqrt{2-3 x^2}}{\sqrt{3} x \sqrt[4]{2-3 x^2}}\right)}{16 \sqrt[4]{2}}-\frac{\sqrt{3} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt{2-3 x^2}+2^{3/4}}{\sqrt{3} x \sqrt[4]{2-3 x^2}}\right)}{16 \sqrt[4]{2}}+\frac{\sqrt{3} F\left(\left.\frac{1}{2} \sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)}{4 \sqrt[4]{2}}",1,"-1/4*AppellF1[-1/2, 3/4, 1, 1/2, (3*x^2)/2, (3*x^2)/4]/(2^(3/4)*x)","C",0
1072,1,37,184,0.061129,"\int \frac{1}{x^4 \left(2-3 x^2\right)^{3/4} \left(4-3 x^2\right)} \, dx","Integrate[1/(x^4*(2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]","-\frac{F_1\left(-\frac{3}{2};\frac{3}{4},1;-\frac{1}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right)}{12\ 2^{3/4} x^3}","-\frac{\sqrt[4]{2-3 x^2}}{4 x}+\frac{3 \sqrt{3} \tan ^{-1}\left(\frac{2^{3/4}-\sqrt[4]{2} \sqrt{2-3 x^2}}{\sqrt{3} x \sqrt[4]{2-3 x^2}}\right)}{64 \sqrt[4]{2}}-\frac{3 \sqrt{3} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt{2-3 x^2}+2^{3/4}}{\sqrt{3} x \sqrt[4]{2-3 x^2}}\right)}{64 \sqrt[4]{2}}-\frac{\sqrt[4]{2-3 x^2}}{24 x^3}+\frac{11 \sqrt{3} F\left(\left.\frac{1}{2} \sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)}{32 \sqrt[4]{2}}",1,"-1/12*AppellF1[-3/2, 3/4, 1, -1/2, (3*x^2)/2, (3*x^2)/4]/(2^(3/4)*x^3)","C",0
1073,1,52,61,0.0412146,"\int \frac{x^2}{\left(-2+3 x^2\right) \left(-1+3 x^2\right)^{3/4}} \, dx","Integrate[x^2/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]","-\frac{x^3 \left(1-3 x^2\right)^{3/4} F_1\left(\frac{3}{2};\frac{3}{4},1;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right)}{6 \left(3 x^2-1\right)^{3/4}}","\frac{\tan ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{3 x^2-1}}\right)}{3 \sqrt{6}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{3 x^2-1}}\right)}{3 \sqrt{6}}",1,"-1/6*(x^3*(1 - 3*x^2)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, 3*x^2, (3*x^2)/2])/(-1 + 3*x^2)^(3/4)","C",0
1074,1,52,61,0.0972385,"\int \frac{x^2}{\left(-2-3 x^2\right) \left(-1-3 x^2\right)^{3/4}} \, dx","Integrate[x^2/((-2 - 3*x^2)*(-1 - 3*x^2)^(3/4)),x]","-\frac{x^3 \left(3 x^2+1\right)^{3/4} F_1\left(\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-3 x^2,-\frac{3 x^2}{2}\right)}{6 \left(-3 x^2-1\right)^{3/4}}","\frac{\tan ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{-3 x^2-1}}\right)}{3 \sqrt{6}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{-3 x^2-1}}\right)}{3 \sqrt{6}}",1,"-1/6*(x^3*(1 + 3*x^2)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, -3*x^2, (-3*x^2)/2])/(-1 - 3*x^2)^(3/4)","C",0
1075,1,54,72,0.053291,"\int \frac{x^2}{\left(-2+b x^2\right) \left(-1+b x^2\right)^{3/4}} \, dx","Integrate[x^2/((-2 + b*x^2)*(-1 + b*x^2)^(3/4)),x]","-\frac{x^3 \left(1-b x^2\right)^{3/4} F_1\left(\frac{3}{2};\frac{3}{4},1;\frac{5}{2};b x^2,\frac{b x^2}{2}\right)}{6 \left(b x^2-1\right)^{3/4}}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{2} \sqrt[4]{b x^2-1}}\right)}{\sqrt{2} b^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{2} \sqrt[4]{b x^2-1}}\right)}{\sqrt{2} b^{3/2}}",1,"-1/6*(x^3*(1 - b*x^2)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, b*x^2, (b*x^2)/2])/(-1 + b*x^2)^(3/4)","C",0
1076,1,55,74,0.0611247,"\int \frac{x^2}{\left(-2-b x^2\right) \left(-1-b x^2\right)^{3/4}} \, dx","Integrate[x^2/((-2 - b*x^2)*(-1 - b*x^2)^(3/4)),x]","-\frac{x^3 \left(b x^2+1\right)^{3/4} F_1\left(\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-b x^2,-\frac{b x^2}{2}\right)}{6 \left(-b x^2-1\right)^{3/4}}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{2} \sqrt[4]{-b x^2-1}}\right)}{\sqrt{2} b^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{2} \sqrt[4]{-b x^2-1}}\right)}{\sqrt{2} b^{3/2}}",1,"-1/6*(x^3*(1 + b*x^2)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, -(b*x^2), -1/2*(b*x^2)])/(-1 - b*x^2)^(3/4)","C",0
1077,1,66,85,0.1056704,"\int \frac{x^2}{\left(-2 a+3 x^2\right) \left(-a+3 x^2\right)^{3/4}} \, dx","Integrate[x^2/((-2*a + 3*x^2)*(-a + 3*x^2)^(3/4)),x]","-\frac{x^3 \left(1-\frac{3 x^2}{a}\right)^{3/4} F_1\left(\frac{3}{2};\frac{3}{4},1;\frac{5}{2};\frac{3 x^2}{a},\frac{3 x^2}{2 a}\right)}{6 a \left(3 x^2-a\right)^{3/4}}","\frac{\tan ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{a} \sqrt[4]{3 x^2-a}}\right)}{3 \sqrt{6} \sqrt[4]{a}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{a} \sqrt[4]{3 x^2-a}}\right)}{3 \sqrt{6} \sqrt[4]{a}}",1,"-1/6*(x^3*(1 - (3*x^2)/a)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, (3*x^2)/a, (3*x^2)/(2*a)])/(a*(-a + 3*x^2)^(3/4))","C",0
1078,1,67,85,0.0586689,"\int \frac{x^2}{\left(-2 a-3 x^2\right) \left(-a-3 x^2\right)^{3/4}} \, dx","Integrate[x^2/((-2*a - 3*x^2)*(-a - 3*x^2)^(3/4)),x]","-\frac{x^3 \left(\frac{a+3 x^2}{a}\right)^{3/4} F_1\left(\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{3 x^2}{a},-\frac{3 x^2}{2 a}\right)}{6 a \left(-a-3 x^2\right)^{3/4}}","\frac{\tan ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{a} \sqrt[4]{-a-3 x^2}}\right)}{3 \sqrt{6} \sqrt[4]{a}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{a} \sqrt[4]{-a-3 x^2}}\right)}{3 \sqrt{6} \sqrt[4]{a}}",1,"-1/6*(x^3*((a + 3*x^2)/a)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, (-3*x^2)/a, (-3*x^2)/(2*a)])/(a*(-a - 3*x^2)^(3/4))","C",0
1079,1,68,96,0.0667393,"\int \frac{x^2}{\left(-2 a+b x^2\right) \left(-a+b x^2\right)^{3/4}} \, dx","Integrate[x^2/((-2*a + b*x^2)*(-a + b*x^2)^(3/4)),x]","-\frac{x^3 \left(1-\frac{b x^2}{a}\right)^{3/4} F_1\left(\frac{3}{2};\frac{3}{4},1;\frac{5}{2};\frac{b x^2}{a},\frac{b x^2}{2 a}\right)}{6 a \left(b x^2-a\right)^{3/4}}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b x^2-a}}\right)}{\sqrt{2} \sqrt[4]{a} b^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b x^2-a}}\right)}{\sqrt{2} \sqrt[4]{a} b^{3/2}}",1,"-1/6*(x^3*(1 - (b*x^2)/a)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, (b*x^2)/a, (b*x^2)/(2*a)])/(a*(-a + b*x^2)^(3/4))","C",0
1080,1,70,98,0.0643262,"\int \frac{x^2}{\left(-2 a-b x^2\right) \left(-a-b x^2\right)^{3/4}} \, dx","Integrate[x^2/((-2*a - b*x^2)*(-a - b*x^2)^(3/4)),x]","-\frac{x^3 \left(\frac{a+b x^2}{a}\right)^{3/4} F_1\left(\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{b x^2}{2 a}\right)}{6 a \left(-a-b x^2\right)^{3/4}}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{-a-b x^2}}\right)}{\sqrt{2} \sqrt[4]{a} b^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{-a-b x^2}}\right)}{\sqrt{2} \sqrt[4]{a} b^{3/2}}",1,"-1/6*(x^3*((a + b*x^2)/a)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, -((b*x^2)/a), -1/2*(b*x^2)/a])/(a*(-a - b*x^2)^(3/4))","C",0
1081,1,58,78,0.0365347,"\int \frac{x^7}{\left(-2+3 x^2\right) \left(-1+3 x^2\right)^{3/4}} \, dx","Integrate[x^7/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]","\frac{-360 \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)-360 \tanh ^{-1}\left(\sqrt[4]{3 x^2-1}\right)+2 \sqrt[4]{3 x^2-1} \left(45 x^4+78 x^2+284\right)}{3645}","\frac{2}{729} \left(3 x^2-1\right)^{9/4}+\frac{8}{405} \left(3 x^2-1\right)^{5/4}+\frac{14}{81} \sqrt[4]{3 x^2-1}-\frac{8}{81} \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)-\frac{8}{81} \tanh ^{-1}\left(\sqrt[4]{3 x^2-1}\right)",1,"(2*(-1 + 3*x^2)^(1/4)*(284 + 78*x^2 + 45*x^4) - 360*ArcTan[(-1 + 3*x^2)^(1/4)] - 360*ArcTanh[(-1 + 3*x^2)^(1/4)])/3645","A",1
1082,1,53,63,0.027727,"\int \frac{x^5}{\left(-2+3 x^2\right) \left(-1+3 x^2\right)^{3/4}} \, dx","Integrate[x^5/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]","\frac{1}{135} \left(2 \sqrt[4]{3 x^2-1} \left(3 x^2+14\right)-20 \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)-20 \tanh ^{-1}\left(\sqrt[4]{3 x^2-1}\right)\right)","\frac{2}{135} \left(3 x^2-1\right)^{5/4}+\frac{2}{9} \sqrt[4]{3 x^2-1}-\frac{4}{27} \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)-\frac{4}{27} \tanh ^{-1}\left(\sqrt[4]{3 x^2-1}\right)",1,"(2*(-1 + 3*x^2)^(1/4)*(14 + 3*x^2) - 20*ArcTan[(-1 + 3*x^2)^(1/4)] - 20*ArcTanh[(-1 + 3*x^2)^(1/4)])/135","A",1
1083,1,44,48,0.0109865,"\int \frac{x^3}{\left(-2+3 x^2\right) \left(-1+3 x^2\right)^{3/4}} \, dx","Integrate[x^3/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]","\frac{2}{9} \left(\sqrt[4]{3 x^2-1}-\tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)-\tanh ^{-1}\left(\sqrt[4]{3 x^2-1}\right)\right)","\frac{2}{9} \sqrt[4]{3 x^2-1}-\frac{2}{9} \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)-\frac{2}{9} \tanh ^{-1}\left(\sqrt[4]{3 x^2-1}\right)",1,"(2*((-1 + 3*x^2)^(1/4) - ArcTan[(-1 + 3*x^2)^(1/4)] - ArcTanh[(-1 + 3*x^2)^(1/4)]))/9","A",1
1084,1,33,33,0.0060132,"\int \frac{x}{\left(-2+3 x^2\right) \left(-1+3 x^2\right)^{3/4}} \, dx","Integrate[x/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]","-\frac{1}{3} \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)-\frac{1}{3} \tanh ^{-1}\left(\sqrt[4]{3 x^2-1}\right)","-\frac{1}{3} \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)-\frac{1}{3} \tanh ^{-1}\left(\sqrt[4]{3 x^2-1}\right)",1,"-1/3*ArcTan[(-1 + 3*x^2)^(1/4)] - ArcTanh[(-1 + 3*x^2)^(1/4)]/3","A",1
1085,1,150,173,0.0514268,"\int \frac{1}{x \left(-2+3 x^2\right) \left(-1+3 x^2\right)^{3/4}} \, dx","Integrate[1/(x*(-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]","\frac{1}{8} \left(-4 \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)-4 \tanh ^{-1}\left(\sqrt[4]{3 x^2-1}\right)+\sqrt{2} \left(\log \left(\sqrt{3 x^2-1}-\sqrt{2} \sqrt[4]{3 x^2-1}+1\right)-\log \left(\sqrt{3 x^2-1}+\sqrt{2} \sqrt[4]{3 x^2-1}+1\right)+2 \tan ^{-1}\left(1-\sqrt{2} \sqrt[4]{3 x^2-1}\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt[4]{3 x^2-1}+1\right)\right)\right)","\frac{\log \left(\sqrt{3 x^2-1}-\sqrt{2} \sqrt[4]{3 x^2-1}+1\right)}{4 \sqrt{2}}-\frac{\log \left(\sqrt{3 x^2-1}+\sqrt{2} \sqrt[4]{3 x^2-1}+1\right)}{4 \sqrt{2}}-\frac{1}{2} \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)+\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt[4]{3 x^2-1}\right)}{2 \sqrt{2}}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt[4]{3 x^2-1}+1\right)}{2 \sqrt{2}}-\frac{1}{2} \tanh ^{-1}\left(\sqrt[4]{3 x^2-1}\right)",1,"(-4*ArcTan[(-1 + 3*x^2)^(1/4)] - 4*ArcTanh[(-1 + 3*x^2)^(1/4)] + Sqrt[2]*(2*ArcTan[1 - Sqrt[2]*(-1 + 3*x^2)^(1/4)] - 2*ArcTan[1 + Sqrt[2]*(-1 + 3*x^2)^(1/4)] + Log[1 - Sqrt[2]*(-1 + 3*x^2)^(1/4) + Sqrt[-1 + 3*x^2]] - Log[1 + Sqrt[2]*(-1 + 3*x^2)^(1/4) + Sqrt[-1 + 3*x^2]]))/8","A",1
1086,1,181,191,0.0540329,"\int \frac{1}{x^3 \left(-2+3 x^2\right) \left(-1+3 x^2\right)^{3/4}} \, dx","Integrate[1/(x^3*(-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]","\frac{1}{32} \left(-\frac{8 \sqrt[4]{3 x^2-1}}{x^2}+15 \sqrt{2} \log \left(\sqrt{3 x^2-1}-\sqrt{2} \sqrt[4]{3 x^2-1}+1\right)-15 \sqrt{2} \log \left(\sqrt{3 x^2-1}+\sqrt{2} \sqrt[4]{3 x^2-1}+1\right)-24 \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)+30 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt[4]{3 x^2-1}\right)-30 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt[4]{3 x^2-1}+1\right)-24 \tanh ^{-1}\left(\sqrt[4]{3 x^2-1}\right)\right)","-\frac{\sqrt[4]{3 x^2-1}}{4 x^2}+\frac{15 \log \left(\sqrt{3 x^2-1}-\sqrt{2} \sqrt[4]{3 x^2-1}+1\right)}{16 \sqrt{2}}-\frac{15 \log \left(\sqrt{3 x^2-1}+\sqrt{2} \sqrt[4]{3 x^2-1}+1\right)}{16 \sqrt{2}}-\frac{3}{4} \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)+\frac{15 \tan ^{-1}\left(1-\sqrt{2} \sqrt[4]{3 x^2-1}\right)}{8 \sqrt{2}}-\frac{15 \tan ^{-1}\left(\sqrt{2} \sqrt[4]{3 x^2-1}+1\right)}{8 \sqrt{2}}-\frac{3}{4} \tanh ^{-1}\left(\sqrt[4]{3 x^2-1}\right)",1,"((-8*(-1 + 3*x^2)^(1/4))/x^2 - 24*ArcTan[(-1 + 3*x^2)^(1/4)] + 30*Sqrt[2]*ArcTan[1 - Sqrt[2]*(-1 + 3*x^2)^(1/4)] - 30*Sqrt[2]*ArcTan[1 + Sqrt[2]*(-1 + 3*x^2)^(1/4)] - 24*ArcTanh[(-1 + 3*x^2)^(1/4)] + 15*Sqrt[2]*Log[1 - Sqrt[2]*(-1 + 3*x^2)^(1/4) + Sqrt[-1 + 3*x^2]] - 15*Sqrt[2]*Log[1 + Sqrt[2]*(-1 + 3*x^2)^(1/4) + Sqrt[-1 + 3*x^2]])/32","A",1
1087,1,184,165,0.2178457,"\int \frac{x^6}{\left(-2+3 x^2\right) \left(-1+3 x^2\right)^{3/4}} \, dx","Integrate[x^6/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]","\frac{2 x \left(-31 \left(1-3 x^2\right)^{3/4} x^2 F_1\left(\frac{3}{2};\frac{3}{4},1;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right)-\frac{80 F_1\left(\frac{1}{2};\frac{3}{4},1;\frac{3}{2};3 x^2,\frac{3 x^2}{2}\right)}{\left(3 x^2-2\right) \left(x^2 \left(2 F_1\left(\frac{3}{2};\frac{3}{4},2;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right)+3 F_1\left(\frac{3}{2};\frac{7}{4},1;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right)\right)+2 F_1\left(\frac{1}{2};\frac{3}{4},1;\frac{3}{2};3 x^2,\frac{3 x^2}{2}\right)\right)}+27 x^4+51 x^2-20\right)}{567 \left(3 x^2-1\right)^{3/4}}","\frac{40}{567} \sqrt[4]{3 x^2-1} x+\frac{2}{27} \sqrt{\frac{2}{3}} \tan ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{3 x^2-1}}\right)-\frac{2}{27} \sqrt{\frac{2}{3}} \tanh ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{3 x^2-1}}\right)+\frac{40 \sqrt{\frac{x^2}{\left(\sqrt{3 x^2-1}+1\right)^2}} \left(\sqrt{3 x^2-1}+1\right) F\left(2 \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)|\frac{1}{2}\right)}{567 \sqrt{3} x}+\frac{2}{63} \sqrt[4]{3 x^2-1} x^3",1,"(2*x*(-20 + 51*x^2 + 27*x^4 - 31*x^2*(1 - 3*x^2)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, 3*x^2, (3*x^2)/2] - (80*AppellF1[1/2, 3/4, 1, 3/2, 3*x^2, (3*x^2)/2])/((-2 + 3*x^2)*(2*AppellF1[1/2, 3/4, 1, 3/2, 3*x^2, (3*x^2)/2] + x^2*(2*AppellF1[3/2, 3/4, 2, 5/2, 3*x^2, (3*x^2)/2] + 3*AppellF1[3/2, 7/4, 1, 5/2, 3*x^2, (3*x^2)/2])))))/(567*(-1 + 3*x^2)^(3/4))","C",0
1088,1,179,147,0.0752164,"\int \frac{x^4}{\left(-2+3 x^2\right) \left(-1+3 x^2\right)^{3/4}} \, dx","Integrate[x^4/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]","\frac{2 x \left(-2 \left(1-3 x^2\right)^{3/4} x^2 F_1\left(\frac{3}{2};\frac{3}{4},1;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right)-\frac{4 F_1\left(\frac{1}{2};\frac{3}{4},1;\frac{3}{2};3 x^2,\frac{3 x^2}{2}\right)}{\left(3 x^2-2\right) \left(x^2 \left(2 F_1\left(\frac{3}{2};\frac{3}{4},2;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right)+3 F_1\left(\frac{3}{2};\frac{7}{4},1;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right)\right)+2 F_1\left(\frac{1}{2};\frac{3}{4},1;\frac{3}{2};3 x^2,\frac{3 x^2}{2}\right)\right)}+3 x^2-1\right)}{27 \left(3 x^2-1\right)^{3/4}}","\frac{2}{27} \sqrt[4]{3 x^2-1} x+\frac{1}{9} \sqrt{\frac{2}{3}} \tan ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{3 x^2-1}}\right)-\frac{1}{9} \sqrt{\frac{2}{3}} \tanh ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{3 x^2-1}}\right)+\frac{2 \sqrt{\frac{x^2}{\left(\sqrt{3 x^2-1}+1\right)^2}} \left(\sqrt{3 x^2-1}+1\right) F\left(2 \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)|\frac{1}{2}\right)}{27 \sqrt{3} x}",1,"(2*x*(-1 + 3*x^2 - 2*x^2*(1 - 3*x^2)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, 3*x^2, (3*x^2)/2] - (4*AppellF1[1/2, 3/4, 1, 3/2, 3*x^2, (3*x^2)/2])/((-2 + 3*x^2)*(2*AppellF1[1/2, 3/4, 1, 3/2, 3*x^2, (3*x^2)/2] + x^2*(2*AppellF1[3/2, 3/4, 2, 5/2, 3*x^2, (3*x^2)/2] + 3*AppellF1[3/2, 7/4, 1, 5/2, 3*x^2, (3*x^2)/2])))))/(27*(-1 + 3*x^2)^(3/4))","C",0
1089,1,52,61,0.03083,"\int \frac{x^2}{\left(-2+3 x^2\right) \left(-1+3 x^2\right)^{3/4}} \, dx","Integrate[x^2/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]","-\frac{x^3 \left(1-3 x^2\right)^{3/4} F_1\left(\frac{3}{2};\frac{3}{4},1;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right)}{6 \left(3 x^2-1\right)^{3/4}}","\frac{\tan ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{3 x^2-1}}\right)}{3 \sqrt{6}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{3 x^2-1}}\right)}{3 \sqrt{6}}",1,"-1/6*(x^3*(1 - 3*x^2)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, 3*x^2, (3*x^2)/2])/(-1 + 3*x^2)^(3/4)","C",0
1090,1,68,127,0.01721,"\int \frac{1}{\left(-2+3 x^2\right) \left(-1+3 x^2\right)^{3/4}} \, dx","Integrate[1/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]","\frac{\sqrt[4]{-1} \sqrt{x^2} \left(\Pi \left(-i;\left.\sin ^{-1}\left((-1)^{3/4} \sqrt[4]{3 x^2-1}\right)\right|-1\right)+\Pi \left(i;\left.\sin ^{-1}\left((-1)^{3/4} \sqrt[4]{3 x^2-1}\right)\right|-1\right)\right)}{\sqrt{3} x}","\frac{\tan ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{3 x^2-1}}\right)}{2 \sqrt{6}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{3 x^2-1}}\right)}{2 \sqrt{6}}-\frac{\sqrt{\frac{x^2}{\left(\sqrt{3 x^2-1}+1\right)^2}} \left(\sqrt{3 x^2-1}+1\right) F\left(2 \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)|\frac{1}{2}\right)}{2 \sqrt{3} x}",1,"((-1)^(1/4)*Sqrt[x^2]*(EllipticPi[-I, ArcSin[(-1)^(3/4)*(-1 + 3*x^2)^(1/4)], -1] + EllipticPi[I, ArcSin[(-1)^(3/4)*(-1 + 3*x^2)^(1/4)], -1]))/(Sqrt[3]*x)","C",1
1091,1,52,149,0.0560976,"\int \frac{1}{x^2 \left(-2+3 x^2\right) \left(-1+3 x^2\right)^{3/4}} \, dx","Integrate[1/(x^2*(-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]","\frac{\left(1-3 x^2\right)^{3/4} F_1\left(-\frac{1}{2};\frac{3}{4},1;\frac{1}{2};3 x^2,\frac{3 x^2}{2}\right)}{2 x \left(3 x^2-1\right)^{3/4}}","-\frac{\sqrt[4]{3 x^2-1}}{2 x}+\frac{1}{4} \sqrt{\frac{3}{2}} \tan ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{3 x^2-1}}\right)-\frac{1}{4} \sqrt{\frac{3}{2}} \tanh ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{3 x^2-1}}\right)-\frac{\sqrt{3} \sqrt{\frac{x^2}{\left(\sqrt{3 x^2-1}+1\right)^2}} \left(\sqrt{3 x^2-1}+1\right) F\left(2 \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)|\frac{1}{2}\right)}{2 x}",1,"((1 - 3*x^2)^(3/4)*AppellF1[-1/2, 3/4, 1, 1/2, 3*x^2, (3*x^2)/2])/(2*x*(-1 + 3*x^2)^(3/4))","C",0
1092,1,52,165,0.0584455,"\int \frac{1}{x^4 \left(-2+3 x^2\right) \left(-1+3 x^2\right)^{3/4}} \, dx","Integrate[1/(x^4*(-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]","\frac{\left(1-3 x^2\right)^{3/4} F_1\left(-\frac{3}{2};\frac{3}{4},1;-\frac{1}{2};3 x^2,\frac{3 x^2}{2}\right)}{6 x^3 \left(3 x^2-1\right)^{3/4}}","-\frac{2 \sqrt[4]{3 x^2-1}}{x}+\frac{3}{8} \sqrt{\frac{3}{2}} \tan ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{3 x^2-1}}\right)-\frac{3}{8} \sqrt{\frac{3}{2}} \tanh ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{3 x^2-1}}\right)-\frac{11 \sqrt{3} \sqrt{\frac{x^2}{\left(\sqrt{3 x^2-1}+1\right)^2}} \left(\sqrt{3 x^2-1}+1\right) F\left(2 \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)|\frac{1}{2}\right)}{8 x}-\frac{\sqrt[4]{3 x^2-1}}{6 x^3}",1,"((1 - 3*x^2)^(3/4)*AppellF1[-3/2, 3/4, 1, -1/2, 3*x^2, (3*x^2)/2])/(6*x^3*(-1 + 3*x^2)^(3/4))","C",0
1093,1,131,173,0.1746673,"\int \frac{(e x)^{5/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{3/4}} \, dx","Integrate[((e*x)^(5/2)*(c + d*x^2))/(a + b*x^2)^(3/4),x]","\frac{(e x)^{5/2} \left(2 b^{3/4} x^{3/2} \sqrt[4]{a+b x^2} \left(-7 a d+8 b c+4 b d x^2\right)-3 a (7 a d-8 b c) \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a+b x^2}}\right)+3 a (7 a d-8 b c) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a+b x^2}}\right)\right)}{32 b^{11/4} x^{5/2}}","\frac{3 a e^{5/2} (8 b c-7 a d) \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt{e} \sqrt[4]{a+b x^2}}\right)}{32 b^{11/4}}-\frac{3 a e^{5/2} (8 b c-7 a d) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt{e} \sqrt[4]{a+b x^2}}\right)}{32 b^{11/4}}+\frac{e (e x)^{3/2} \sqrt[4]{a+b x^2} (8 b c-7 a d)}{16 b^2}+\frac{d (e x)^{7/2} \sqrt[4]{a+b x^2}}{4 b e}",1,"((e*x)^(5/2)*(2*b^(3/4)*x^(3/2)*(a + b*x^2)^(1/4)*(8*b*c - 7*a*d + 4*b*d*x^2) - 3*a*(-8*b*c + 7*a*d)*ArcTan[(b^(1/4)*Sqrt[x])/(a + b*x^2)^(1/4)] + 3*a*(-8*b*c + 7*a*d)*ArcTanh[(b^(1/4)*Sqrt[x])/(a + b*x^2)^(1/4)]))/(32*b^(11/4)*x^(5/2))","A",1
1094,1,112,136,0.0999289,"\int \frac{\sqrt{e x} \left(c+d x^2\right)}{\left(a+b x^2\right)^{3/4}} \, dx","Integrate[(Sqrt[e*x]*(c + d*x^2))/(a + b*x^2)^(3/4),x]","\frac{\sqrt{e x} \left(2 b^{3/4} d x^{3/2} \sqrt[4]{a+b x^2}+(3 a d-4 b c) \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a+b x^2}}\right)+(4 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a+b x^2}}\right)\right)}{4 b^{7/4} \sqrt{x}}","-\frac{\sqrt{e} (4 b c-3 a d) \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt{e} \sqrt[4]{a+b x^2}}\right)}{4 b^{7/4}}+\frac{\sqrt{e} (4 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt{e} \sqrt[4]{a+b x^2}}\right)}{4 b^{7/4}}+\frac{d (e x)^{3/2} \sqrt[4]{a+b x^2}}{2 b e}",1,"(Sqrt[e*x]*(2*b^(3/4)*d*x^(3/2)*(a + b*x^2)^(1/4) + (-4*b*c + 3*a*d)*ArcTan[(b^(1/4)*Sqrt[x])/(a + b*x^2)^(1/4)] + (4*b*c - 3*a*d)*ArcTanh[(b^(1/4)*Sqrt[x])/(a + b*x^2)^(1/4)]))/(4*b^(7/4)*Sqrt[x])","A",1
1095,1,100,113,0.0316305,"\int \frac{c+d x^2}{(e x)^{3/2} \left(a+b x^2\right)^{3/4}} \, dx","Integrate[(c + d*x^2)/((e*x)^(3/2)*(a + b*x^2)^(3/4)),x]","\frac{x \left(-2 b^{3/4} c \sqrt[4]{a+b x^2}-a d \sqrt{x} \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a+b x^2}}\right)+a d \sqrt{x} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a+b x^2}}\right)\right)}{a b^{3/4} (e x)^{3/2}}","-\frac{d \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt{e} \sqrt[4]{a+b x^2}}\right)}{b^{3/4} e^{3/2}}+\frac{d \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt{e} \sqrt[4]{a+b x^2}}\right)}{b^{3/4} e^{3/2}}-\frac{2 c \sqrt[4]{a+b x^2}}{a e \sqrt{e x}}",1,"(x*(-2*b^(3/4)*c*(a + b*x^2)^(1/4) - a*d*Sqrt[x]*ArcTan[(b^(1/4)*Sqrt[x])/(a + b*x^2)^(1/4)] + a*d*Sqrt[x]*ArcTanh[(b^(1/4)*Sqrt[x])/(a + b*x^2)^(1/4)]))/(a*b^(3/4)*(e*x)^(3/2))","A",1
1096,1,44,67,0.0193099,"\int \frac{c+d x^2}{(e x)^{7/2} \left(a+b x^2\right)^{3/4}} \, dx","Integrate[(c + d*x^2)/((e*x)^(7/2)*(a + b*x^2)^(3/4)),x]","-\frac{2 x \sqrt[4]{a+b x^2} \left(a \left(c+5 d x^2\right)-4 b c x^2\right)}{5 a^2 (e x)^{7/2}}","\frac{2 \sqrt[4]{a+b x^2} (4 b c-5 a d)}{5 a^2 e^3 \sqrt{e x}}-\frac{2 c \sqrt[4]{a+b x^2}}{5 a e (e x)^{5/2}}",1,"(-2*x*(a + b*x^2)^(1/4)*(-4*b*c*x^2 + a*(c + 5*d*x^2)))/(5*a^2*(e*x)^(7/2))","A",1
1097,1,72,104,0.0413796,"\int \frac{c+d x^2}{(e x)^{11/2} \left(a+b x^2\right)^{3/4}} \, dx","Integrate[(c + d*x^2)/((e*x)^(11/2)*(a + b*x^2)^(3/4)),x]","-\frac{2 \sqrt{e x} \sqrt[4]{a+b x^2} \left(a^2 \left(5 c+9 d x^2\right)-4 a b x^2 \left(2 c+9 d x^2\right)+32 b^2 c x^4\right)}{45 a^3 e^6 x^5}","-\frac{8 \left(a+b x^2\right)^{5/4} (8 b c-9 a d)}{45 a^3 e^3 (e x)^{5/2}}+\frac{2 \sqrt[4]{a+b x^2} (8 b c-9 a d)}{9 a^2 e^3 (e x)^{5/2}}-\frac{2 c \sqrt[4]{a+b x^2}}{9 a e (e x)^{9/2}}",1,"(-2*Sqrt[e*x]*(a + b*x^2)^(1/4)*(32*b^2*c*x^4 - 4*a*b*x^2*(2*c + 9*d*x^2) + a^2*(5*c + 9*d*x^2)))/(45*a^3*e^6*x^5)","A",1
1098,1,94,141,0.05151,"\int \frac{c+d x^2}{(e x)^{15/2} \left(a+b x^2\right)^{3/4}} \, dx","Integrate[(c + d*x^2)/((e*x)^(15/2)*(a + b*x^2)^(3/4)),x]","-\frac{2 \sqrt{e x} \sqrt[4]{a+b x^2} \left(5 a^3 \left(9 c+13 d x^2\right)-4 a^2 b x^2 \left(15 c+26 d x^2\right)+32 a b^2 x^4 \left(3 c+13 d x^2\right)-384 b^3 c x^6\right)}{585 a^4 e^8 x^7}","\frac{64 \left(a+b x^2\right)^{9/4} (12 b c-13 a d)}{585 a^4 e^3 (e x)^{9/2}}-\frac{16 \left(a+b x^2\right)^{5/4} (12 b c-13 a d)}{65 a^3 e^3 (e x)^{9/2}}+\frac{2 \sqrt[4]{a+b x^2} (12 b c-13 a d)}{13 a^2 e^3 (e x)^{9/2}}-\frac{2 c \sqrt[4]{a+b x^2}}{13 a e (e x)^{13/2}}",1,"(-2*Sqrt[e*x]*(a + b*x^2)^(1/4)*(-384*b^3*c*x^6 + 32*a*b^2*x^4*(3*c + 13*d*x^2) + 5*a^3*(9*c + 13*d*x^2) - 4*a^2*b*x^2*(15*c + 26*d*x^2)))/(585*a^4*e^8*x^7)","A",1
1099,1,123,180,0.1385489,"\int \frac{(e x)^{7/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{3/4}} \, dx","Integrate[((e*x)^(7/2)*(c + d*x^2))/(a + b*x^2)^(3/4),x]","\frac{e^3 \sqrt{e x} \left(\left(a+b x^2\right) \left(45 a^2 d-2 a b \left(25 c+9 d x^2\right)+4 b^2 x^2 \left(5 c+3 d x^2\right)\right)+5 a^2 \left(\frac{b x^2}{a}+1\right)^{3/4} (10 b c-9 a d) \, _2F_1\left(\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right)\right)}{60 b^3 \left(a+b x^2\right)^{3/4}}","-\frac{a^{3/2} e^2 (e x)^{3/2} \left(\frac{a}{b x^2}+1\right)^{3/4} (10 b c-9 a d) F\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{12 b^{5/2} \left(a+b x^2\right)^{3/4}}-\frac{a e^3 \sqrt{e x} \sqrt[4]{a+b x^2} (10 b c-9 a d)}{12 b^3}+\frac{e (e x)^{5/2} \sqrt[4]{a+b x^2} (10 b c-9 a d)}{30 b^2}+\frac{d (e x)^{9/2} \sqrt[4]{a+b x^2}}{5 b e}",1,"(e^3*Sqrt[e*x]*((a + b*x^2)*(45*a^2*d + 4*b^2*x^2*(5*c + 3*d*x^2) - 2*a*b*(25*c + 9*d*x^2)) + 5*a^2*(10*b*c - 9*a*d)*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, -((b*x^2)/a)]))/(60*b^3*(a + b*x^2)^(3/4))","C",1
1100,1,97,139,0.1103555,"\int \frac{(e x)^{3/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{3/4}} \, dx","Integrate[((e*x)^(3/2)*(c + d*x^2))/(a + b*x^2)^(3/4),x]","\frac{e \sqrt{e x} \left(a \left(\frac{b x^2}{a}+1\right)^{3/4} (5 a d-6 b c) \, _2F_1\left(\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right)-\left(a+b x^2\right) \left(5 a d-2 b \left(3 c+d x^2\right)\right)\right)}{6 b^2 \left(a+b x^2\right)^{3/4}}","\frac{\sqrt{a} (e x)^{3/2} \left(\frac{a}{b x^2}+1\right)^{3/4} (6 b c-5 a d) F\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{6 b^{3/2} \left(a+b x^2\right)^{3/4}}+\frac{e \sqrt{e x} \sqrt[4]{a+b x^2} (6 b c-5 a d)}{6 b^2}+\frac{d (e x)^{5/2} \sqrt[4]{a+b x^2}}{3 b e}",1,"(e*Sqrt[e*x]*(-((a + b*x^2)*(5*a*d - 2*b*(3*c + d*x^2))) + a*(-6*b*c + 5*a*d)*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, -((b*x^2)/a)]))/(6*b^2*(a + b*x^2)^(3/4))","C",1
1101,1,77,102,0.062758,"\int \frac{c+d x^2}{\sqrt{e x} \left(a+b x^2\right)^{3/4}} \, dx","Integrate[(c + d*x^2)/(Sqrt[e*x]*(a + b*x^2)^(3/4)),x]","\frac{x \left(\frac{b x^2}{a}+1\right)^{3/4} (2 b c-a d) \, _2F_1\left(\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right)+d x \left(a+b x^2\right)}{b \sqrt{e x} \left(a+b x^2\right)^{3/4}}","\frac{d \sqrt{e x} \sqrt[4]{a+b x^2}}{b e}-\frac{(e x)^{3/2} \left(\frac{a}{b x^2}+1\right)^{3/4} (2 b c-a d) F\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{\sqrt{a} \sqrt{b} e^2 \left(a+b x^2\right)^{3/4}}",1,"(d*x*(a + b*x^2) + (2*b*c - a*d)*x*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, -((b*x^2)/a)])/(b*Sqrt[e*x]*(a + b*x^2)^(3/4))","C",1
1102,1,84,107,0.0445949,"\int \frac{c+d x^2}{(e x)^{5/2} \left(a+b x^2\right)^{3/4}} \, dx","Integrate[(c + d*x^2)/((e*x)^(5/2)*(a + b*x^2)^(3/4)),x]","\frac{x \left(2 x^2 \left(\frac{b x^2}{a}+1\right)^{3/4} (3 a d-2 b c) \, _2F_1\left(\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right)-2 c \left(a+b x^2\right)\right)}{3 a (e x)^{5/2} \left(a+b x^2\right)^{3/4}}","\frac{2 \sqrt{b} (e x)^{3/2} \left(\frac{a}{b x^2}+1\right)^{3/4} (2 b c-3 a d) F\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{3 a^{3/2} e^4 \left(a+b x^2\right)^{3/4}}-\frac{2 c \sqrt[4]{a+b x^2}}{3 a e (e x)^{3/2}}",1,"(x*(-2*c*(a + b*x^2) + 2*(-2*b*c + 3*a*d)*x^2*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, -((b*x^2)/a)]))/(3*a*(e*x)^(5/2)*(a + b*x^2)^(3/4))","C",1
1103,1,88,144,0.0801847,"\int \frac{c+d x^2}{(e x)^{9/2} \left(a+b x^2\right)^{3/4}} \, dx","Integrate[(c + d*x^2)/((e*x)^(9/2)*(a + b*x^2)^(3/4)),x]","-\frac{2 \sqrt{e x} \left(x^2 \left(\frac{b x^2}{a}+1\right)^{3/4} (7 a d-6 b c) \, _2F_1\left(-\frac{3}{4},\frac{3}{4};\frac{1}{4};-\frac{b x^2}{a}\right)+3 c \left(a+b x^2\right)\right)}{21 a e^5 x^4 \left(a+b x^2\right)^{3/4}}","-\frac{4 b^{3/2} (e x)^{3/2} \left(\frac{a}{b x^2}+1\right)^{3/4} (6 b c-7 a d) F\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{21 a^{5/2} e^6 \left(a+b x^2\right)^{3/4}}+\frac{2 \sqrt[4]{a+b x^2} (6 b c-7 a d)}{21 a^2 e^3 (e x)^{3/2}}-\frac{2 c \sqrt[4]{a+b x^2}}{7 a e (e x)^{7/2}}",1,"(-2*Sqrt[e*x]*(3*c*(a + b*x^2) + (-6*b*c + 7*a*d)*x^2*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[-3/4, 3/4, 1/4, -((b*x^2)/a)]))/(21*a*e^5*x^4*(a + b*x^2)^(3/4))","C",1
1104,1,88,182,0.0816503,"\int \frac{c+d x^2}{(e x)^{13/2} \left(a+b x^2\right)^{3/4}} \, dx","Integrate[(c + d*x^2)/((e*x)^(13/2)*(a + b*x^2)^(3/4)),x]","-\frac{2 \sqrt{e x} \left(x^2 \left(\frac{b x^2}{a}+1\right)^{3/4} (11 a d-10 b c) \, _2F_1\left(-\frac{7}{4},\frac{3}{4};-\frac{3}{4};-\frac{b x^2}{a}\right)+7 c \left(a+b x^2\right)\right)}{77 a e^7 x^6 \left(a+b x^2\right)^{3/4}}","\frac{8 b^{5/2} (e x)^{3/2} \left(\frac{a}{b x^2}+1\right)^{3/4} (10 b c-11 a d) F\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{77 a^{7/2} e^8 \left(a+b x^2\right)^{3/4}}-\frac{4 b \sqrt[4]{a+b x^2} (10 b c-11 a d)}{77 a^3 e^5 (e x)^{3/2}}+\frac{2 \sqrt[4]{a+b x^2} (10 b c-11 a d)}{77 a^2 e^3 (e x)^{7/2}}-\frac{2 c \sqrt[4]{a+b x^2}}{11 a e (e x)^{11/2}}",1,"(-2*Sqrt[e*x]*(7*c*(a + b*x^2) + (-10*b*c + 11*a*d)*x^2*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[-7/4, 3/4, -3/4, -((b*x^2)/a)]))/(77*a*e^7*x^6*(a + b*x^2)^(3/4))","C",1
1105,1,77,171,0.1222413,"\int \frac{(e x)^{3/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{5/4}} \, dx","Integrate[((e*x)^(3/2)*(c + d*x^2))/(a + b*x^2)^(5/4),x]","\frac{x (e x)^{3/2} \left(\sqrt[4]{\frac{b x^2}{a}+1} (4 b c-5 a d) \, _2F_1\left(\frac{5}{4},\frac{5}{4};\frac{9}{4};-\frac{b x^2}{a}\right)+5 a d\right)}{10 a b \sqrt[4]{a+b x^2}}","\frac{e^{3/2} (4 b c-5 a d) \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt{e} \sqrt[4]{a+b x^2}}\right)}{4 b^{9/4}}+\frac{e^{3/2} (4 b c-5 a d) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt{e} \sqrt[4]{a+b x^2}}\right)}{4 b^{9/4}}-\frac{e \sqrt{e x} (4 b c-5 a d)}{2 b^2 \sqrt[4]{a+b x^2}}+\frac{d (e x)^{5/2}}{2 b e \sqrt[4]{a+b x^2}}",1,"(x*(e*x)^(3/2)*(5*a*d + (4*b*c - 5*a*d)*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[5/4, 5/4, 9/4, -((b*x^2)/a)]))/(10*a*b*(a + b*x^2)^(1/4))","C",1
1106,1,68,122,0.0627067,"\int \frac{c+d x^2}{\sqrt{e x} \left(a+b x^2\right)^{5/4}} \, dx","Integrate[(c + d*x^2)/(Sqrt[e*x]*(a + b*x^2)^(5/4)),x]","\frac{2 \left(d x^3 \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{5}{4},\frac{5}{4};\frac{9}{4};-\frac{b x^2}{a}\right)+5 c x\right)}{5 a \sqrt{e x} \sqrt[4]{a+b x^2}}","\frac{d \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt{e} \sqrt[4]{a+b x^2}}\right)}{b^{5/4} \sqrt{e}}+\frac{d \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt{e} \sqrt[4]{a+b x^2}}\right)}{b^{5/4} \sqrt{e}}+\frac{2 \sqrt{e x} (b c-a d)}{a b e \sqrt[4]{a+b x^2}}",1,"(2*(5*c*x + d*x^3*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[5/4, 5/4, 9/4, -((b*x^2)/a)]))/(5*a*Sqrt[e*x]*(a + b*x^2)^(1/4))","C",1
1107,1,45,67,0.0218033,"\int \frac{c+d x^2}{(e x)^{5/2} \left(a+b x^2\right)^{5/4}} \, dx","Integrate[(c + d*x^2)/((e*x)^(5/2)*(a + b*x^2)^(5/4)),x]","\frac{x \left(-2 a c+6 a d x^2-8 b c x^2\right)}{3 a^2 (e x)^{5/2} \sqrt[4]{a+b x^2}}","-\frac{2 \sqrt{e x} (4 b c-3 a d)}{3 a^2 e^3 \sqrt[4]{a+b x^2}}-\frac{2 c}{3 a e (e x)^{3/2} \sqrt[4]{a+b x^2}}",1,"(x*(-2*a*c - 8*b*c*x^2 + 6*a*d*x^2))/(3*a^2*(e*x)^(5/2)*(a + b*x^2)^(1/4))","A",1
1108,1,71,104,0.0300918,"\int \frac{c+d x^2}{(e x)^{9/2} \left(a+b x^2\right)^{5/4}} \, dx","Integrate[(c + d*x^2)/((e*x)^(9/2)*(a + b*x^2)^(5/4)),x]","-\frac{2 \sqrt{e x} \left(a^2 \left(3 c+7 d x^2\right)+a b \left(28 d x^4-8 c x^2\right)-32 b^2 c x^4\right)}{21 a^3 e^5 x^4 \sqrt[4]{a+b x^2}}","\frac{8 \left(a+b x^2\right)^{3/4} (8 b c-7 a d)}{21 a^3 e^3 (e x)^{3/2}}-\frac{2 (8 b c-7 a d)}{7 a^2 e^3 (e x)^{3/2} \sqrt[4]{a+b x^2}}-\frac{2 c}{7 a e (e x)^{7/2} \sqrt[4]{a+b x^2}}",1,"(-2*Sqrt[e*x]*(-32*b^2*c*x^4 + a^2*(3*c + 7*d*x^2) + a*b*(-8*c*x^2 + 28*d*x^4)))/(21*a^3*e^5*x^4*(a + b*x^2)^(1/4))","A",1
1109,1,68,141,0.0422256,"\int \frac{c+d x^2}{(e x)^{13/2} \left(a+b x^2\right)^{5/4}} \, dx","Integrate[(c + d*x^2)/((e*x)^(13/2)*(a + b*x^2)^(5/4)),x]","\frac{2 x \left(-21 a^3 c-x^2 \left(-3 a^2+8 a b x^2+32 b^2 x^4\right) (12 b c-11 a d)\right)}{231 a^4 (e x)^{13/2} \sqrt[4]{a+b x^2}}","-\frac{64 \left(a+b x^2\right)^{7/4} (12 b c-11 a d)}{231 a^4 e^3 (e x)^{7/2}}+\frac{16 \left(a+b x^2\right)^{3/4} (12 b c-11 a d)}{33 a^3 e^3 (e x)^{7/2}}-\frac{2 (12 b c-11 a d)}{11 a^2 e^3 (e x)^{7/2} \sqrt[4]{a+b x^2}}-\frac{2 c}{11 a e (e x)^{11/2} \sqrt[4]{a+b x^2}}",1,"(2*x*(-21*a^3*c - (12*b*c - 11*a*d)*x^2*(-3*a^2 + 8*a*b*x^2 + 32*b^2*x^4)))/(231*a^4*(e*x)^(13/2)*(a + b*x^2)^(1/4))","A",1
1110,1,112,180,0.15114,"\int \frac{(e x)^{9/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{5/4}} \, dx","Integrate[((e*x)^(9/2)*(c + d*x^2))/(a + b*x^2)^(5/4),x]","\frac{e^3 (e x)^{3/2} \left(77 a^2 d+7 a \sqrt[4]{\frac{b x^2}{a}+1} (10 b c-11 a d) \, _2F_1\left(\frac{3}{4},\frac{5}{4};\frac{7}{4};-\frac{b x^2}{a}\right)-2 a b \left(35 c+11 d x^2\right)+4 b^2 x^2 \left(5 c+3 d x^2\right)\right)}{60 b^3 \sqrt[4]{a+b x^2}}","-\frac{7 a^{3/2} e^4 \sqrt{e x} \sqrt[4]{\frac{a}{b x^2}+1} (10 b c-11 a d) E\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{20 b^{7/2} \sqrt[4]{a+b x^2}}-\frac{7 a e^3 (e x)^{3/2} (10 b c-11 a d)}{60 b^3 \sqrt[4]{a+b x^2}}+\frac{e (e x)^{7/2} (10 b c-11 a d)}{30 b^2 \sqrt[4]{a+b x^2}}+\frac{d (e x)^{11/2}}{5 b e \sqrt[4]{a+b x^2}}",1,"(e^3*(e*x)^(3/2)*(77*a^2*d + 4*b^2*x^2*(5*c + 3*d*x^2) - 2*a*b*(35*c + 11*d*x^2) + 7*a*(10*b*c - 11*a*d)*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[3/4, 5/4, 7/4, -((b*x^2)/a)]))/(60*b^3*(a + b*x^2)^(1/4))","C",1
1111,1,85,142,0.1194202,"\int \frac{(e x)^{5/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{5/4}} \, dx","Integrate[((e*x)^(5/2)*(c + d*x^2))/(a + b*x^2)^(5/4),x]","\frac{e (e x)^{3/2} \left(\sqrt[4]{\frac{b x^2}{a}+1} (7 a d-6 b c) \, _2F_1\left(\frac{3}{4},\frac{5}{4};\frac{7}{4};-\frac{b x^2}{a}\right)-7 a d+6 b c+2 b d x^2\right)}{6 b^2 \sqrt[4]{a+b x^2}}","\frac{\sqrt{a} e^2 \sqrt{e x} \sqrt[4]{\frac{a}{b x^2}+1} (6 b c-7 a d) E\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{2 b^{5/2} \sqrt[4]{a+b x^2}}+\frac{e (e x)^{3/2} (6 b c-7 a d)}{6 b^2 \sqrt[4]{a+b x^2}}+\frac{d (e x)^{7/2}}{3 b e \sqrt[4]{a+b x^2}}",1,"(e*(e*x)^(3/2)*(6*b*c - 7*a*d + 2*b*d*x^2 + (-6*b*c + 7*a*d)*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[3/4, 5/4, 7/4, -((b*x^2)/a)]))/(6*b^2*(a + b*x^2)^(1/4))","C",1
1112,1,77,99,0.1036586,"\int \frac{\sqrt{e x} \left(c+d x^2\right)}{\left(a+b x^2\right)^{5/4}} \, dx","Integrate[(Sqrt[e*x]*(c + d*x^2))/(a + b*x^2)^(5/4),x]","\frac{x \sqrt{e x} \left(\sqrt[4]{\frac{b x^2}{a}+1} (2 b c-3 a d) \, _2F_1\left(\frac{3}{4},\frac{5}{4};\frac{7}{4};-\frac{b x^2}{a}\right)+3 a d\right)}{3 a b \sqrt[4]{a+b x^2}}","\frac{d (e x)^{3/2}}{b e \sqrt[4]{a+b x^2}}-\frac{\sqrt{e x} \sqrt[4]{\frac{a}{b x^2}+1} (2 b c-3 a d) E\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{\sqrt{a} b^{3/2} \sqrt[4]{a+b x^2}}",1,"(x*Sqrt[e*x]*(3*a*d + (2*b*c - 3*a*d)*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[3/4, 5/4, 7/4, -((b*x^2)/a)]))/(3*a*b*(a + b*x^2)^(1/4))","C",1
1113,1,77,103,0.0433752,"\int \frac{c+d x^2}{(e x)^{3/2} \left(a+b x^2\right)^{5/4}} \, dx","Integrate[(c + d*x^2)/((e*x)^(3/2)*(a + b*x^2)^(5/4)),x]","\frac{x \left(2 x^2 \sqrt[4]{\frac{b x^2}{a}+1} (a d-2 b c) \, _2F_1\left(\frac{3}{4},\frac{5}{4};\frac{7}{4};-\frac{b x^2}{a}\right)-6 a c\right)}{3 a^2 (e x)^{3/2} \sqrt[4]{a+b x^2}}","\frac{2 \sqrt{e x} \sqrt[4]{\frac{a}{b x^2}+1} (2 b c-a d) E\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{a^{3/2} \sqrt{b} e^2 \sqrt[4]{a+b x^2}}-\frac{2 c}{a e \sqrt{e x} \sqrt[4]{a+b x^2}}",1,"(x*(-6*a*c + 2*(-2*b*c + a*d)*x^2*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[3/4, 5/4, 7/4, -((b*x^2)/a)]))/(3*a^2*(e*x)^(3/2)*(a + b*x^2)^(1/4))","C",1
1114,1,78,144,0.0458749,"\int \frac{c+d x^2}{(e x)^{7/2} \left(a+b x^2\right)^{5/4}} \, dx","Integrate[(c + d*x^2)/((e*x)^(7/2)*(a + b*x^2)^(5/4)),x]","\frac{x \left(2 x^2 \sqrt[4]{\frac{b x^2}{a}+1} (6 b c-5 a d) \, _2F_1\left(-\frac{1}{4},\frac{5}{4};\frac{3}{4};-\frac{b x^2}{a}\right)-2 a c\right)}{5 a^2 (e x)^{7/2} \sqrt[4]{a+b x^2}}","-\frac{4 \sqrt{b} \sqrt{e x} \sqrt[4]{\frac{a}{b x^2}+1} (6 b c-5 a d) E\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{5 a^{5/2} e^4 \sqrt[4]{a+b x^2}}+\frac{2 (6 b c-5 a d)}{5 a^2 e^3 \sqrt{e x} \sqrt[4]{a+b x^2}}-\frac{2 c}{5 a e (e x)^{5/2} \sqrt[4]{a+b x^2}}",1,"(x*(-2*a*c + 2*(6*b*c - 5*a*d)*x^2*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[-1/4, 5/4, 3/4, -((b*x^2)/a)]))/(5*a^2*(e*x)^(7/2)*(a + b*x^2)^(1/4))","C",1
1115,1,82,182,0.0686616,"\int \frac{c+d x^2}{(e x)^{11/2} \left(a+b x^2\right)^{5/4}} \, dx","Integrate[(c + d*x^2)/((e*x)^(11/2)*(a + b*x^2)^(5/4)),x]","-\frac{2 \sqrt{e x} \left(x^2 \sqrt[4]{\frac{b x^2}{a}+1} (9 a d-10 b c) \, _2F_1\left(-\frac{5}{4},\frac{5}{4};-\frac{1}{4};-\frac{b x^2}{a}\right)+5 a c\right)}{45 a^2 e^6 x^5 \sqrt[4]{a+b x^2}}","\frac{8 b^{3/2} \sqrt{e x} \sqrt[4]{\frac{a}{b x^2}+1} (10 b c-9 a d) E\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{15 a^{7/2} e^6 \sqrt[4]{a+b x^2}}-\frac{4 b (10 b c-9 a d)}{15 a^3 e^5 \sqrt{e x} \sqrt[4]{a+b x^2}}+\frac{2 (10 b c-9 a d)}{45 a^2 e^3 (e x)^{5/2} \sqrt[4]{a+b x^2}}-\frac{2 c}{9 a e (e x)^{9/2} \sqrt[4]{a+b x^2}}",1,"(-2*Sqrt[e*x]*(5*a*c + (-10*b*c + 9*a*d)*x^2*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[-5/4, 5/4, -1/4, -((b*x^2)/a)]))/(45*a^2*e^6*x^5*(a + b*x^2)^(1/4))","C",1
1116,1,77,184,0.1256714,"\int \frac{(e x)^{5/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{7/4}} \, dx","Integrate[((e*x)^(5/2)*(c + d*x^2))/(a + b*x^2)^(7/4),x]","\frac{x (e x)^{5/2} \left(\left(\frac{b x^2}{a}+1\right)^{3/4} (4 b c-7 a d) \, _2F_1\left(\frac{7}{4},\frac{7}{4};\frac{11}{4};-\frac{b x^2}{a}\right)+7 a d\right)}{14 a b \left(a+b x^2\right)^{3/4}}","-\frac{e^{5/2} (4 b c-7 a d) \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt{e} \sqrt[4]{a+b x^2}}\right)}{4 b^{11/4}}+\frac{e^{5/2} (4 b c-7 a d) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt{e} \sqrt[4]{a+b x^2}}\right)}{4 b^{11/4}}-\frac{e (e x)^{3/2} \sqrt[4]{a+b x^2} (4 b c-7 a d)}{6 a b^2}+\frac{2 (e x)^{7/2} (b c-a d)}{3 a b e \left(a+b x^2\right)^{3/4}}",1,"(x*(e*x)^(5/2)*(7*a*d + (4*b*c - 7*a*d)*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[7/4, 7/4, 11/4, -((b*x^2)/a)]))/(14*a*b*(a + b*x^2)^(3/4))","C",1
1117,1,69,125,0.0597881,"\int \frac{\sqrt{e x} \left(c+d x^2\right)}{\left(a+b x^2\right)^{7/4}} \, dx","Integrate[(Sqrt[e*x]*(c + d*x^2))/(a + b*x^2)^(7/4),x]","\frac{2 \sqrt{e x} \left(3 d x^3 \left(\frac{b x^2}{a}+1\right)^{3/4} \, _2F_1\left(\frac{7}{4},\frac{7}{4};\frac{11}{4};-\frac{b x^2}{a}\right)+7 c x\right)}{21 a \left(a+b x^2\right)^{3/4}}","-\frac{d \sqrt{e} \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt{e} \sqrt[4]{a+b x^2}}\right)}{b^{7/4}}+\frac{d \sqrt{e} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt{e} \sqrt[4]{a+b x^2}}\right)}{b^{7/4}}+\frac{2 (e x)^{3/2} (b c-a d)}{3 a b e \left(a+b x^2\right)^{3/4}}",1,"(2*Sqrt[e*x]*(7*c*x + 3*d*x^3*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[7/4, 7/4, 11/4, -((b*x^2)/a)]))/(21*a*(a + b*x^2)^(3/4))","C",1
1118,1,44,65,0.0198266,"\int \frac{c+d x^2}{(e x)^{3/2} \left(a+b x^2\right)^{7/4}} \, dx","Integrate[(c + d*x^2)/((e*x)^(3/2)*(a + b*x^2)^(7/4)),x]","\frac{2 x \left(-3 a c+a d x^2-4 b c x^2\right)}{3 a^2 (e x)^{3/2} \left(a+b x^2\right)^{3/4}}","-\frac{2 (e x)^{3/2} (4 b c-a d)}{3 a^2 e^3 \left(a+b x^2\right)^{3/4}}-\frac{2 c}{a e \sqrt{e x} \left(a+b x^2\right)^{3/4}}",1,"(2*x*(-3*a*c - 4*b*c*x^2 + a*d*x^2))/(3*a^2*(e*x)^(3/2)*(a + b*x^2)^(3/4))","A",1
1119,1,66,104,0.0285209,"\int \frac{c+d x^2}{(e x)^{7/2} \left(a+b x^2\right)^{7/4}} \, dx","Integrate[(c + d*x^2)/((e*x)^(7/2)*(a + b*x^2)^(7/4)),x]","\frac{x \left(-6 a^2 \left(c+5 d x^2\right)+8 a b x^2 \left(6 c-5 d x^2\right)+64 b^2 c x^4\right)}{15 a^3 (e x)^{7/2} \left(a+b x^2\right)^{3/4}}","\frac{8 \sqrt[4]{a+b x^2} (8 b c-5 a d)}{15 a^3 e^3 \sqrt{e x}}-\frac{2 (8 b c-5 a d)}{15 a^2 e^3 \sqrt{e x} \left(a+b x^2\right)^{3/4}}-\frac{2 c}{5 a e (e x)^{5/2} \left(a+b x^2\right)^{3/4}}",1,"(x*(64*b^2*c*x^4 + 8*a*b*x^2*(6*c - 5*d*x^2) - 6*a^2*(c + 5*d*x^2)))/(15*a^3*(e*x)^(7/2)*(a + b*x^2)^(3/4))","A",1
1120,1,88,141,0.0418041,"\int \frac{c+d x^2}{(e x)^{11/2} \left(a+b x^2\right)^{7/4}} \, dx","Integrate[(c + d*x^2)/((e*x)^(11/2)*(a + b*x^2)^(7/4)),x]","\frac{4 x^3 \left(3 a^2-24 a b x^2-32 b^2 x^4\right) \left(6 b c-\frac{9 a d}{2}\right)}{135 a^4 (e x)^{11/2} \left(a+b x^2\right)^{3/4}}-\frac{2 c x}{9 a (e x)^{11/2} \left(a+b x^2\right)^{3/4}}","-\frac{64 \left(a+b x^2\right)^{5/4} (4 b c-3 a d)}{45 a^4 e^3 (e x)^{5/2}}+\frac{16 \sqrt[4]{a+b x^2} (4 b c-3 a d)}{9 a^3 e^3 (e x)^{5/2}}-\frac{2 (4 b c-3 a d)}{9 a^2 e^3 (e x)^{5/2} \left(a+b x^2\right)^{3/4}}-\frac{2 c}{9 a e (e x)^{9/2} \left(a+b x^2\right)^{3/4}}",1,"(-2*c*x)/(9*a*(e*x)^(11/2)*(a + b*x^2)^(3/4)) + (4*(6*b*c - (9*a*d)/2)*x^3*(3*a^2 - 24*a*b*x^2 - 32*b^2*x^4))/(135*a^4*(e*x)^(11/2)*(a + b*x^2)^(3/4))","A",1
1121,1,110,192,0.1480056,"\int \frac{(e x)^{7/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{7/4}} \, dx","Integrate[((e*x)^(7/2)*(c + d*x^2))/(a + b*x^2)^(7/4),x]","\frac{e^3 \sqrt{e x} \left(-15 a^2 d+5 a \left(\frac{b x^2}{a}+1\right)^{3/4} (3 a d-2 b c) \, _2F_1\left(\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right)+a b \left(10 c-9 d x^2\right)+2 b^2 x^2 \left(3 c+d x^2\right)\right)}{6 b^3 \left(a+b x^2\right)^{3/4}}","\frac{5 \sqrt{a} e^2 (e x)^{3/2} \left(\frac{a}{b x^2}+1\right)^{3/4} (2 b c-3 a d) F\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{6 b^{5/2} \left(a+b x^2\right)^{3/4}}+\frac{5 e^3 \sqrt{e x} \sqrt[4]{a+b x^2} (2 b c-3 a d)}{6 b^3}-\frac{e (e x)^{5/2} \sqrt[4]{a+b x^2} (2 b c-3 a d)}{3 a b^2}+\frac{2 (e x)^{9/2} (b c-a d)}{3 a b e \left(a+b x^2\right)^{3/4}}",1,"(e^3*Sqrt[e*x]*(-15*a^2*d + a*b*(10*c - 9*d*x^2) + 2*b^2*x^2*(3*c + d*x^2) + 5*a*(-2*b*c + 3*a*d)*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, -((b*x^2)/a)]))/(6*b^3*(a + b*x^2)^(3/4))","C",1
1122,1,85,152,0.1262723,"\int \frac{(e x)^{3/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{7/4}} \, dx","Integrate[((e*x)^(3/2)*(c + d*x^2))/(a + b*x^2)^(7/4),x]","\frac{e \sqrt{e x} \left(\left(\frac{b x^2}{a}+1\right)^{3/4} (2 b c-5 a d) \, _2F_1\left(\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right)+5 a d-2 b c+3 b d x^2\right)}{3 b^2 \left(a+b x^2\right)^{3/4}}","-\frac{(e x)^{3/2} \left(\frac{a}{b x^2}+1\right)^{3/4} (2 b c-5 a d) F\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{3 \sqrt{a} b^{3/2} \left(a+b x^2\right)^{3/4}}-\frac{e \sqrt{e x} \sqrt[4]{a+b x^2} (2 b c-5 a d)}{3 a b^2}+\frac{2 (e x)^{5/2} (b c-a d)}{3 a b e \left(a+b x^2\right)^{3/4}}",1,"(e*Sqrt[e*x]*(-2*b*c + 5*a*d + 3*b*d*x^2 + (2*b*c - 5*a*d)*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, -((b*x^2)/a)]))/(3*b^2*(a + b*x^2)^(3/4))","C",1
1123,1,79,116,0.058061,"\int \frac{c+d x^2}{\sqrt{e x} \left(a+b x^2\right)^{7/4}} \, dx","Integrate[(c + d*x^2)/(Sqrt[e*x]*(a + b*x^2)^(7/4)),x]","\frac{2 x \left(\left(\frac{b x^2}{a}+1\right)^{3/4} (a d+2 b c) \, _2F_1\left(\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right)-a d+b c\right)}{3 a b \sqrt{e x} \left(a+b x^2\right)^{3/4}}","\frac{2 \sqrt{e x} (b c-a d)}{3 a b e \left(a+b x^2\right)^{3/4}}-\frac{2 (e x)^{3/2} \left(\frac{a}{b x^2}+1\right)^{3/4} (a d+2 b c) F\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{3 a^{3/2} \sqrt{b} e^2 \left(a+b x^2\right)^{3/4}}",1,"(2*x*(b*c - a*d + (2*b*c + a*d)*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, -((b*x^2)/a)]))/(3*a*b*Sqrt[e*x]*(a + b*x^2)^(3/4))","C",1
1124,1,91,144,0.0482899,"\int \frac{c+d x^2}{(e x)^{5/2} \left(a+b x^2\right)^{7/4}} \, dx","Integrate[(c + d*x^2)/((e*x)^(5/2)*(a + b*x^2)^(7/4)),x]","\frac{x \left(4 x^2 \left(\frac{b x^2}{a}+1\right)^{3/4} (a d-2 b c) \, _2F_1\left(\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right)-2 a c+2 a d x^2-4 b c x^2\right)}{3 a^2 (e x)^{5/2} \left(a+b x^2\right)^{3/4}}","\frac{4 \sqrt{b} (e x)^{3/2} \left(\frac{a}{b x^2}+1\right)^{3/4} (2 b c-a d) F\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{3 a^{5/2} e^4 \left(a+b x^2\right)^{3/4}}-\frac{2 \sqrt{e x} (2 b c-a d)}{3 a^2 e^3 \left(a+b x^2\right)^{3/4}}-\frac{2 c}{3 a e (e x)^{3/2} \left(a+b x^2\right)^{3/4}}",1,"(x*(-2*a*c - 4*b*c*x^2 + 2*a*d*x^2 + 4*(-2*b*c + a*d)*x^2*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, -((b*x^2)/a)]))/(3*a^2*(e*x)^(5/2)*(a + b*x^2)^(3/4))","C",1
1125,1,82,181,0.0704942,"\int \frac{c+d x^2}{(e x)^{9/2} \left(a+b x^2\right)^{7/4}} \, dx","Integrate[(c + d*x^2)/((e*x)^(9/2)*(a + b*x^2)^(7/4)),x]","-\frac{2 \sqrt{e x} \left(x^2 \left(\frac{b x^2}{a}+1\right)^{3/4} (7 a d-10 b c) \, _2F_1\left(-\frac{3}{4},\frac{7}{4};\frac{1}{4};-\frac{b x^2}{a}\right)+3 a c\right)}{21 a^2 e^5 x^4 \left(a+b x^2\right)^{3/4}}","-\frac{8 b^{3/2} (e x)^{3/2} \left(\frac{a}{b x^2}+1\right)^{3/4} (10 b c-7 a d) F\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{21 a^{7/2} e^6 \left(a+b x^2\right)^{3/4}}+\frac{4 \sqrt[4]{a+b x^2} (10 b c-7 a d)}{21 a^3 e^3 (e x)^{3/2}}-\frac{2 (10 b c-7 a d)}{21 a^2 e^3 (e x)^{3/2} \left(a+b x^2\right)^{3/4}}-\frac{2 c}{7 a e (e x)^{7/2} \left(a+b x^2\right)^{3/4}}",1,"(-2*Sqrt[e*x]*(3*a*c + (-10*b*c + 7*a*d)*x^2*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[-3/4, 7/4, 1/4, -((b*x^2)/a)]))/(21*a^2*e^5*x^4*(a + b*x^2)^(3/4))","C",1
1126,1,91,221,0.1242082,"\int \frac{(e x)^{7/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{9/4}} \, dx","Integrate[((e*x)^(7/2)*(c + d*x^2))/(a + b*x^2)^(9/4),x]","\frac{e^3 x^4 \sqrt{e x} \left(9 a^2 d+\left(a+b x^2\right) \sqrt[4]{\frac{b x^2}{a}+1} (4 b c-9 a d) \, _2F_1\left(\frac{9}{4},\frac{9}{4};\frac{13}{4};-\frac{b x^2}{a}\right)\right)}{18 a^2 b \left(a+b x^2\right)^{5/4}}","\frac{e^{7/2} (4 b c-9 a d) \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt{e} \sqrt[4]{a+b x^2}}\right)}{4 b^{13/4}}+\frac{e^{7/2} (4 b c-9 a d) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt{e} \sqrt[4]{a+b x^2}}\right)}{4 b^{13/4}}-\frac{e^3 \sqrt{e x} (4 b c-9 a d)}{2 b^3 \sqrt[4]{a+b x^2}}-\frac{e (e x)^{5/2} (4 b c-9 a d)}{10 a b^2 \sqrt[4]{a+b x^2}}+\frac{2 (e x)^{9/2} (b c-a d)}{5 a b e \left(a+b x^2\right)^{5/4}}",1,"(e^3*x^4*Sqrt[e*x]*(9*a^2*d + (4*b*c - 9*a*d)*(a + b*x^2)*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[9/4, 9/4, 13/4, -((b*x^2)/a)]))/(18*a^2*b*(a + b*x^2)^(5/4))","C",1
1127,1,77,149,0.0811432,"\int \frac{(e x)^{3/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{9/4}} \, dx","Integrate[((e*x)^(3/2)*(c + d*x^2))/(a + b*x^2)^(9/4),x]","\frac{2 x (e x)^{3/2} \left(5 d x^2 \left(a+b x^2\right) \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{9}{4},\frac{9}{4};\frac{13}{4};-\frac{b x^2}{a}\right)+9 a c\right)}{45 a^2 \left(a+b x^2\right)^{5/4}}","\frac{d e^{3/2} \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt{e} \sqrt[4]{a+b x^2}}\right)}{b^{9/4}}+\frac{d e^{3/2} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{e x}}{\sqrt{e} \sqrt[4]{a+b x^2}}\right)}{b^{9/4}}-\frac{2 d e \sqrt{e x}}{b^2 \sqrt[4]{a+b x^2}}+\frac{2 (e x)^{5/2} (b c-a d)}{5 a b e \left(a+b x^2\right)^{5/4}}",1,"(2*x*(e*x)^(3/2)*(9*a*c + 5*d*x^2*(a + b*x^2)*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[9/4, 9/4, 13/4, -((b*x^2)/a)]))/(45*a^2*(a + b*x^2)^(5/4))","C",1
1128,1,44,79,0.0460925,"\int \frac{c+d x^2}{\sqrt{e x} \left(a+b x^2\right)^{9/4}} \, dx","Integrate[(c + d*x^2)/(Sqrt[e*x]*(a + b*x^2)^(9/4)),x]","\frac{2 x \left(5 a c+a d x^2+4 b c x^2\right)}{5 a^2 \sqrt{e x} \left(a+b x^2\right)^{5/4}}","\frac{2 \sqrt{e x} (a d+4 b c)}{5 a^2 b e \sqrt[4]{a+b x^2}}+\frac{2 \sqrt{e x} (b c-a d)}{5 a b e \left(a+b x^2\right)^{5/4}}",1,"(2*x*(5*a*c + 4*b*c*x^2 + a*d*x^2))/(5*a^2*Sqrt[e*x]*(a + b*x^2)^(5/4))","A",1
1129,1,65,104,0.0431561,"\int \frac{c+d x^2}{(e x)^{5/2} \left(a+b x^2\right)^{9/4}} \, dx","Integrate[(c + d*x^2)/((e*x)^(5/2)*(a + b*x^2)^(9/4)),x]","\frac{x \left(-10 a^2 \left(c-3 d x^2\right)+a b \left(24 d x^4-80 c x^2\right)-64 b^2 c x^4\right)}{15 a^3 (e x)^{5/2} \left(a+b x^2\right)^{5/4}}","-\frac{8 \sqrt{e x} (8 b c-3 a d)}{15 a^3 e^3 \sqrt[4]{a+b x^2}}-\frac{2 \sqrt{e x} (8 b c-3 a d)}{15 a^2 e^3 \left(a+b x^2\right)^{5/4}}-\frac{2 c}{3 a e (e x)^{3/2} \left(a+b x^2\right)^{5/4}}",1,"(x*(-64*b^2*c*x^4 - 10*a^2*(c - 3*d*x^2) + a*b*(-80*c*x^2 + 24*d*x^4)))/(15*a^3*(e*x)^(5/2)*(a + b*x^2)^(5/4))","A",1
1130,1,94,141,0.050568,"\int \frac{c+d x^2}{(e x)^{9/2} \left(a+b x^2\right)^{9/4}} \, dx","Integrate[(c + d*x^2)/((e*x)^(9/2)*(a + b*x^2)^(9/4)),x]","\frac{\sqrt{e x} \left(-10 a^3 \left(3 c+7 d x^2\right)+40 a^2 b x^2 \left(3 c-14 d x^2\right)+64 a b^2 x^4 \left(15 c-7 d x^2\right)+768 b^3 c x^6\right)}{105 a^4 e^5 x^4 \left(a+b x^2\right)^{5/4}}","\frac{64 \left(a+b x^2\right)^{3/4} (12 b c-7 a d)}{105 a^4 e^3 (e x)^{3/2}}-\frac{16 (12 b c-7 a d)}{35 a^3 e^3 (e x)^{3/2} \sqrt[4]{a+b x^2}}-\frac{2 (12 b c-7 a d)}{35 a^2 e^3 (e x)^{3/2} \left(a+b x^2\right)^{5/4}}-\frac{2 c}{7 a e (e x)^{7/2} \left(a+b x^2\right)^{5/4}}",1,"(Sqrt[e*x]*(768*b^3*c*x^6 + 40*a^2*b*x^2*(3*c - 14*d*x^2) + 64*a*b^2*x^4*(15*c - 7*d*x^2) - 10*a^3*(3*c + 7*d*x^2)))/(105*a^4*e^5*x^4*(a + b*x^2)^(5/4))","A",1
1131,1,79,178,0.0580748,"\int \frac{c+d x^2}{(e x)^{13/2} \left(a+b x^2\right)^{9/4}} \, dx","Integrate[(c + d*x^2)/((e*x)^(13/2)*(a + b*x^2)^(9/4)),x]","\frac{2 x \left(a x^2 \left(-5 a^3+20 a^2 b x^2+160 a b^2 x^4+128 b^3 x^6\right) (11 a d-16 b c)-35 a^5 c\right)}{385 a^6 (e x)^{13/2} \left(a+b x^2\right)^{5/4}}","-\frac{256 \left(a+b x^2\right)^{7/4} (16 b c-11 a d)}{385 a^5 e^3 (e x)^{7/2}}+\frac{64 \left(a+b x^2\right)^{3/4} (16 b c-11 a d)}{55 a^4 e^3 (e x)^{7/2}}-\frac{24 (16 b c-11 a d)}{55 a^3 e^3 (e x)^{7/2} \sqrt[4]{a+b x^2}}-\frac{2 (16 b c-11 a d)}{55 a^2 e^3 (e x)^{7/2} \left(a+b x^2\right)^{5/4}}-\frac{2 c}{11 a e (e x)^{11/2} \left(a+b x^2\right)^{5/4}}",1,"(2*x*(-35*a^5*c + a*(-16*b*c + 11*a*d)*x^2*(-5*a^3 + 20*a^2*b*x^2 + 160*a*b^2*x^4 + 128*b^3*x^6)))/(385*a^6*(e*x)^(13/2)*(a + b*x^2)^(5/4))","A",1
1132,1,140,230,0.1862259,"\int \frac{(e x)^{13/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{9/4}} \, dx","Integrate[((e*x)^(13/2)*(c + d*x^2))/(a + b*x^2)^(9/4),x]","\frac{e^5 (e x)^{3/2} \left(1155 a^3 d-110 a^2 b \left(7 c-3 d x^2\right)-20 a b^2 x^2 \left(11 c+3 d x^2\right)-385 a \left(a+b x^2\right) \sqrt[4]{\frac{b x^2}{a}+1} (3 a d-2 b c) \, _2F_1\left(\frac{3}{4},\frac{9}{4};\frac{7}{4};-\frac{b x^2}{a}\right)+8 b^3 x^4 \left(5 c+3 d x^2\right)\right)}{120 b^4 \left(a+b x^2\right)^{5/4}}","-\frac{77 a^{3/2} e^6 \sqrt{e x} \sqrt[4]{\frac{a}{b x^2}+1} (2 b c-3 a d) E\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{20 b^{9/2} \sqrt[4]{a+b x^2}}-\frac{77 a e^5 (e x)^{3/2} (2 b c-3 a d)}{60 b^4 \sqrt[4]{a+b x^2}}+\frac{11 e^3 (e x)^{7/2} (2 b c-3 a d)}{30 b^3 \sqrt[4]{a+b x^2}}-\frac{e (e x)^{11/2} (2 b c-3 a d)}{5 a b^2 \sqrt[4]{a+b x^2}}+\frac{2 (e x)^{15/2} (b c-a d)}{5 a b e \left(a+b x^2\right)^{5/4}}",1,"(e^5*(e*x)^(3/2)*(1155*a^3*d - 110*a^2*b*(7*c - 3*d*x^2) + 8*b^3*x^4*(5*c + 3*d*x^2) - 20*a*b^2*x^2*(11*c + 3*d*x^2) - 385*a*(-2*b*c + 3*a*d)*(a + b*x^2)*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[3/4, 9/4, 7/4, -((b*x^2)/a)]))/(120*b^4*(a + b*x^2)^(5/4))","C",1
1133,1,116,192,0.153145,"\int \frac{(e x)^{9/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{9/4}} \, dx","Integrate[((e*x)^(9/2)*(c + d*x^2))/(a + b*x^2)^(9/4),x]","\frac{e^3 (e x)^{3/2} \left(-77 a^2 d+7 \left(a+b x^2\right) \sqrt[4]{\frac{b x^2}{a}+1} (11 a d-6 b c) \, _2F_1\left(\frac{3}{4},\frac{9}{4};\frac{7}{4};-\frac{b x^2}{a}\right)+a b \left(42 c-22 d x^2\right)+4 b^2 x^2 \left(3 c+d x^2\right)\right)}{12 b^3 \left(a+b x^2\right)^{5/4}}","\frac{7 \sqrt{a} e^4 \sqrt{e x} \sqrt[4]{\frac{a}{b x^2}+1} (6 b c-11 a d) E\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{10 b^{7/2} \sqrt[4]{a+b x^2}}+\frac{7 e^3 (e x)^{3/2} (6 b c-11 a d)}{30 b^3 \sqrt[4]{a+b x^2}}-\frac{e (e x)^{7/2} (6 b c-11 a d)}{15 a b^2 \sqrt[4]{a+b x^2}}+\frac{2 (e x)^{11/2} (b c-a d)}{5 a b e \left(a+b x^2\right)^{5/4}}",1,"(e^3*(e*x)^(3/2)*(-77*a^2*d + a*b*(42*c - 22*d*x^2) + 4*b^2*x^2*(3*c + d*x^2) + 7*(-6*b*c + 11*a*d)*(a + b*x^2)*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[3/4, 9/4, 7/4, -((b*x^2)/a)]))/(12*b^3*(a + b*x^2)^(5/4))","C",1
1134,1,98,155,0.134073,"\int \frac{(e x)^{5/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{9/4}} \, dx","Integrate[((e*x)^(5/2)*(c + d*x^2))/(a + b*x^2)^(9/4),x]","\frac{e (e x)^{3/2} \left(\left(a+b x^2\right) \sqrt[4]{\frac{b x^2}{a}+1} (2 b c-7 a d) \, _2F_1\left(\frac{3}{4},\frac{9}{4};\frac{7}{4};-\frac{b x^2}{a}\right)+a \left(7 a d-2 b c+2 b d x^2\right)\right)}{2 a b^2 \left(a+b x^2\right)^{5/4}}","-\frac{3 e^2 \sqrt{e x} \sqrt[4]{\frac{a}{b x^2}+1} (2 b c-7 a d) E\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{5 \sqrt{a} b^{5/2} \sqrt[4]{a+b x^2}}-\frac{e (e x)^{3/2} (2 b c-7 a d)}{5 a b^2 \sqrt[4]{a+b x^2}}+\frac{2 (e x)^{7/2} (b c-a d)}{5 a b e \left(a+b x^2\right)^{5/4}}",1,"(e*(e*x)^(3/2)*(a*(-2*b*c + 7*a*d + 2*b*d*x^2) + (2*b*c - 7*a*d)*(a + b*x^2)*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[3/4, 9/4, 7/4, -((b*x^2)/a)]))/(2*a*b^2*(a + b*x^2)^(5/4))","C",1
1135,1,86,114,0.1001695,"\int \frac{\sqrt{e x} \left(c+d x^2\right)}{\left(a+b x^2\right)^{9/4}} \, dx","Integrate[(Sqrt[e*x]*(c + d*x^2))/(a + b*x^2)^(9/4),x]","\frac{x \sqrt{e x} \left(\left(a+b x^2\right) \sqrt[4]{\frac{b x^2}{a}+1} (3 a d+2 b c) \, _2F_1\left(\frac{3}{4},\frac{9}{4};\frac{7}{4};-\frac{b x^2}{a}\right)-3 a^2 d\right)}{3 a^2 b \left(a+b x^2\right)^{5/4}}","\frac{2 (e x)^{3/2} (b c-a d)}{5 a b e \left(a+b x^2\right)^{5/4}}-\frac{2 \sqrt{e x} \sqrt[4]{\frac{a}{b x^2}+1} (3 a d+2 b c) E\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{5 a^{3/2} b^{3/2} \sqrt[4]{a+b x^2}}",1,"(x*Sqrt[e*x]*(-3*a^2*d + (2*b*c + 3*a*d)*(a + b*x^2)*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[3/4, 9/4, 7/4, -((b*x^2)/a)]))/(3*a^2*b*(a + b*x^2)^(5/4))","C",1
1136,1,85,142,0.0451959,"\int \frac{c+d x^2}{(e x)^{3/2} \left(a+b x^2\right)^{9/4}} \, dx","Integrate[(c + d*x^2)/((e*x)^(3/2)*(a + b*x^2)^(9/4)),x]","\frac{2 x \left(x^2 \left(a+b x^2\right) \sqrt[4]{\frac{b x^2}{a}+1} (a d-6 b c) \, _2F_1\left(\frac{3}{4},\frac{9}{4};\frac{7}{4};-\frac{b x^2}{a}\right)-3 a^2 c\right)}{3 a^3 (e x)^{3/2} \left(a+b x^2\right)^{5/4}}","\frac{4 \sqrt{e x} \sqrt[4]{\frac{a}{b x^2}+1} (6 b c-a d) E\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{5 a^{5/2} \sqrt{b} e^2 \sqrt[4]{a+b x^2}}-\frac{2 (e x)^{3/2} (6 b c-a d)}{5 a^2 e^3 \left(a+b x^2\right)^{5/4}}-\frac{2 c}{a e \sqrt{e x} \left(a+b x^2\right)^{5/4}}",1,"(2*x*(-3*a^2*c + (-6*b*c + a*d)*x^2*(a + b*x^2)*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[3/4, 9/4, 7/4, -((b*x^2)/a)]))/(3*a^3*(e*x)^(3/2)*(a + b*x^2)^(5/4))","C",1
1137,1,86,181,0.0444326,"\int \frac{c+d x^2}{(e x)^{7/2} \left(a+b x^2\right)^{9/4}} \, dx","Integrate[(c + d*x^2)/((e*x)^(7/2)*(a + b*x^2)^(9/4)),x]","\frac{2 x \left(a^2 (-c)-5 x^2 \left(a+b x^2\right) \sqrt[4]{\frac{b x^2}{a}+1} (a d-2 b c) \, _2F_1\left(-\frac{1}{4},\frac{9}{4};\frac{3}{4};-\frac{b x^2}{a}\right)\right)}{5 a^3 (e x)^{7/2} \left(a+b x^2\right)^{5/4}}","-\frac{24 \sqrt{b} \sqrt{e x} \sqrt[4]{\frac{a}{b x^2}+1} (2 b c-a d) E\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{5 a^{7/2} e^4 \sqrt[4]{a+b x^2}}+\frac{12 (2 b c-a d)}{5 a^3 e^3 \sqrt{e x} \sqrt[4]{a+b x^2}}-\frac{2 (2 b c-a d)}{5 a^2 e^3 \sqrt{e x} \left(a+b x^2\right)^{5/4}}-\frac{2 c}{5 a e (e x)^{5/2} \left(a+b x^2\right)^{5/4}}",1,"(2*x*(-(a^2*c) - 5*(-2*b*c + a*d)*x^2*(a + b*x^2)*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[-1/4, 9/4, 3/4, -((b*x^2)/a)]))/(5*a^3*(e*x)^(7/2)*(a + b*x^2)^(5/4))","C",1
1138,1,87,219,0.0482366,"\int \frac{c+d x^2}{(e x)^{11/2} \left(a+b x^2\right)^{9/4}} \, dx","Integrate[(c + d*x^2)/((e*x)^(11/2)*(a + b*x^2)^(9/4)),x]","\frac{2 x \left(-5 a^2 c-x^2 \left(a+b x^2\right) \sqrt[4]{\frac{b x^2}{a}+1} (9 a d-14 b c) \, _2F_1\left(-\frac{5}{4},\frac{9}{4};-\frac{1}{4};-\frac{b x^2}{a}\right)\right)}{45 a^3 (e x)^{11/2} \left(a+b x^2\right)^{5/4}}","\frac{16 b^{3/2} \sqrt{e x} \sqrt[4]{\frac{a}{b x^2}+1} (14 b c-9 a d) E\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{15 a^{9/2} e^6 \sqrt[4]{a+b x^2}}-\frac{8 b (14 b c-9 a d)}{15 a^4 e^5 \sqrt{e x} \sqrt[4]{a+b x^2}}+\frac{4 (14 b c-9 a d)}{45 a^3 e^3 (e x)^{5/2} \sqrt[4]{a+b x^2}}-\frac{2 (14 b c-9 a d)}{45 a^2 e^3 (e x)^{5/2} \left(a+b x^2\right)^{5/4}}-\frac{2 c}{9 a e (e x)^{9/2} \left(a+b x^2\right)^{5/4}}",1,"(2*x*(-5*a^2*c - (-14*b*c + 9*a*d)*x^2*(a + b*x^2)*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[-5/4, 9/4, -1/4, -((b*x^2)/a)]))/(45*a^3*(e*x)^(11/2)*(a + b*x^2)^(5/4))","C",1
1139,1,97,101,0.1201463,"\int (e x)^m \left(a+b x^2\right)^p \left(c+d x^2\right)^q \, dx","Integrate[(e*x)^m*(a + b*x^2)^p*(c + d*x^2)^q,x]","\frac{x (e x)^m \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(c+d x^2\right)^q \left(\frac{d x^2}{c}+1\right)^{-q} F_1\left(\frac{m+1}{2};-p,-q;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{m+1}","\frac{(e x)^{m+1} \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(c+d x^2\right)^q \left(\frac{d x^2}{c}+1\right)^{-q} F_1\left(\frac{m+1}{2};-p,-q;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{e (m+1)}",1,"(x*(e*x)^m*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1[(1 + m)/2, -p, -q, (3 + m)/2, -((b*x^2)/a), -((d*x^2)/c)])/((1 + m)*(1 + (b*x^2)/a)^p*(1 + (d*x^2)/c)^q)","A",1
1140,1,86,84,0.0603487,"\int x^4 \left(a+b x^2\right)^p \left(c+d x^2\right)^q \, dx","Integrate[x^4*(a + b*x^2)^p*(c + d*x^2)^q,x]","\frac{1}{5} x^5 \left(a+b x^2\right)^p \left(\frac{a+b x^2}{a}\right)^{-p} \left(c+d x^2\right)^q \left(\frac{c+d x^2}{c}\right)^{-q} F_1\left(\frac{5}{2};-p,-q;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)","\frac{1}{5} x^5 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(c+d x^2\right)^q \left(\frac{d x^2}{c}+1\right)^{-q} F_1\left(\frac{5}{2};-p,-q;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)",1,"(x^5*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1[5/2, -p, -q, 7/2, -((b*x^2)/a), -((d*x^2)/c)])/(5*((a + b*x^2)/a)^p*((c + d*x^2)/c)^q)","A",1
1141,1,86,84,0.0623141,"\int x^2 \left(a+b x^2\right)^p \left(c+d x^2\right)^q \, dx","Integrate[x^2*(a + b*x^2)^p*(c + d*x^2)^q,x]","\frac{1}{3} x^3 \left(a+b x^2\right)^p \left(\frac{a+b x^2}{a}\right)^{-p} \left(c+d x^2\right)^q \left(\frac{c+d x^2}{c}\right)^{-q} F_1\left(\frac{3}{2};-p,-q;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)","\frac{1}{3} x^3 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(c+d x^2\right)^q \left(\frac{d x^2}{c}+1\right)^{-q} F_1\left(\frac{3}{2};-p,-q;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)",1,"(x^3*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1[3/2, -p, -q, 5/2, -((b*x^2)/a), -((d*x^2)/c)])/(3*((a + b*x^2)/a)^p*((c + d*x^2)/c)^q)","A",1
1142,1,172,79,0.2206542,"\int \left(a+b x^2\right)^p \left(c+d x^2\right)^q \, dx","Integrate[(a + b*x^2)^p*(c + d*x^2)^q,x]","\frac{3 a c x \left(a+b x^2\right)^p \left(c+d x^2\right)^q F_1\left(\frac{1}{2};-p,-q;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{2 x^2 \left(b c p F_1\left(\frac{3}{2};1-p,-q;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+a d q F_1\left(\frac{3}{2};-p,1-q;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)+3 a c F_1\left(\frac{1}{2};-p,-q;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}","x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(c+d x^2\right)^q \left(\frac{d x^2}{c}+1\right)^{-q} F_1\left(\frac{1}{2};-p,-q;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)",1,"(3*a*c*x*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1[1/2, -p, -q, 3/2, -((b*x^2)/a), -((d*x^2)/c)])/(3*a*c*AppellF1[1/2, -p, -q, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + 2*x^2*(b*c*p*AppellF1[3/2, 1 - p, -q, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + a*d*q*AppellF1[3/2, -p, 1 - q, 5/2, -((b*x^2)/a), -((d*x^2)/c)]))","B",0
1143,1,84,82,0.0567139,"\int \frac{\left(a+b x^2\right)^p \left(c+d x^2\right)^q}{x^2} \, dx","Integrate[((a + b*x^2)^p*(c + d*x^2)^q)/x^2,x]","-\frac{\left(a+b x^2\right)^p \left(\frac{a+b x^2}{a}\right)^{-p} \left(c+d x^2\right)^q \left(\frac{c+d x^2}{c}\right)^{-q} F_1\left(-\frac{1}{2};-p,-q;\frac{1}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{x}","-\frac{\left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(c+d x^2\right)^q \left(\frac{d x^2}{c}+1\right)^{-q} F_1\left(-\frac{1}{2};-p,-q;\frac{1}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{x}",1,"-(((a + b*x^2)^p*(c + d*x^2)^q*AppellF1[-1/2, -p, -q, 1/2, -((b*x^2)/a), -((d*x^2)/c)])/(x*((a + b*x^2)/a)^p*((c + d*x^2)/c)^q))","A",1
1144,1,86,84,0.0982423,"\int \frac{\left(a+b x^2\right)^p \left(c+d x^2\right)^q}{x^4} \, dx","Integrate[((a + b*x^2)^p*(c + d*x^2)^q)/x^4,x]","-\frac{\left(a+b x^2\right)^p \left(\frac{a+b x^2}{a}\right)^{-p} \left(c+d x^2\right)^q \left(\frac{c+d x^2}{c}\right)^{-q} F_1\left(-\frac{3}{2};-p,-q;-\frac{1}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{3 x^3}","-\frac{\left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(c+d x^2\right)^q \left(\frac{d x^2}{c}+1\right)^{-q} F_1\left(-\frac{3}{2};-p,-q;-\frac{1}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{3 x^3}",1,"-1/3*((a + b*x^2)^p*(c + d*x^2)^q*AppellF1[-3/2, -p, -q, -1/2, -((b*x^2)/a), -((d*x^2)/c)])/(x^3*((a + b*x^2)/a)^p*((c + d*x^2)/c)^q)","A",1
1145,1,195,242,0.3029744,"\int x^5 \left(a+b x^2\right)^p \left(c+d x^2\right)^q \, dx","Integrate[x^5*(a + b*x^2)^p*(c + d*x^2)^q,x]","\frac{\left(a+b x^2\right)^{p+1} \left(c+d x^2\right)^q \left(\frac{\left(a^2 d^2 \left(q^2+3 q+2\right)+2 a b c d (p+1) (q+1)+b^2 c^2 \left(p^2+3 p+2\right)\right) \left(\frac{b \left(c+d x^2\right)}{b c-a d}\right)^{-q} \, _2F_1\left(p+1,-q;p+2;\frac{d \left(b x^2+a\right)}{a d-b c}\right)}{b^2 d (p+1) (p+q+2)}-\frac{\left(c+d x^2\right) (a d (q+2)+b c (p+2))}{b d (p+q+2)}+x^2 \left(c+d x^2\right)\right)}{2 b d (p+q+3)}","\frac{\left(a+b x^2\right)^{p+1} \left(c+d x^2\right)^q \left(a^2 d^2 \left(q^2+3 q+2\right)+2 a b c d (p+1) (q+1)+b^2 c^2 \left(p^2+3 p+2\right)\right) \left(\frac{b \left(c+d x^2\right)}{b c-a d}\right)^{-q} \, _2F_1\left(p+1,-q;p+2;-\frac{d \left(b x^2+a\right)}{b c-a d}\right)}{2 b^3 d^2 (p+1) (p+q+2) (p+q+3)}-\frac{\left(a+b x^2\right)^{p+1} \left(c+d x^2\right)^{q+1} (a d (q+2)+b c (p+2))}{2 b^2 d^2 (p+q+2) (p+q+3)}+\frac{x^2 \left(a+b x^2\right)^{p+1} \left(c+d x^2\right)^{q+1}}{2 b d (p+q+3)}",1,"((a + b*x^2)^(1 + p)*(c + d*x^2)^q*(-(((b*c*(2 + p) + a*d*(2 + q))*(c + d*x^2))/(b*d*(2 + p + q))) + x^2*(c + d*x^2) + ((b^2*c^2*(2 + 3*p + p^2) + 2*a*b*c*d*(1 + p)*(1 + q) + a^2*d^2*(2 + 3*q + q^2))*Hypergeometric2F1[1 + p, -q, 2 + p, (d*(a + b*x^2))/(-(b*c) + a*d)])/(b^2*d*(1 + p)*(2 + p + q)*((b*(c + d*x^2))/(b*c - a*d))^q)))/(2*b*d*(3 + p + q))","A",1
1146,1,118,146,0.0997945,"\int x^3 \left(a+b x^2\right)^p \left(c+d x^2\right)^q \, dx","Integrate[x^3*(a + b*x^2)^p*(c + d*x^2)^q,x]","\frac{\left(a+b x^2\right)^{p+1} \left(c+d x^2\right)^q \left(b \left(c+d x^2\right)-\frac{(a d (q+1)+b c (p+1)) \left(\frac{b \left(c+d x^2\right)}{b c-a d}\right)^{-q} \, _2F_1\left(p+1,-q;p+2;\frac{d \left(b x^2+a\right)}{a d-b c}\right)}{p+1}\right)}{2 b^2 d (p+q+2)}","\frac{\left(a+b x^2\right)^{p+1} \left(c+d x^2\right)^{q+1}}{2 b d (p+q+2)}-\frac{\left(a+b x^2\right)^{p+1} \left(c+d x^2\right)^q (a d (q+1)+b c (p+1)) \left(\frac{b \left(c+d x^2\right)}{b c-a d}\right)^{-q} \, _2F_1\left(p+1,-q;p+2;-\frac{d \left(b x^2+a\right)}{b c-a d}\right)}{2 b^2 d (p+1) (p+q+2)}",1,"((a + b*x^2)^(1 + p)*(c + d*x^2)^q*(b*(c + d*x^2) - ((b*c*(1 + p) + a*d*(1 + q))*Hypergeometric2F1[1 + p, -q, 2 + p, (d*(a + b*x^2))/(-(b*c) + a*d)])/((1 + p)*((b*(c + d*x^2))/(b*c - a*d))^q)))/(2*b^2*d*(2 + p + q))","A",1
1147,1,84,85,0.0241214,"\int x \left(a+b x^2\right)^p \left(c+d x^2\right)^q \, dx","Integrate[x*(a + b*x^2)^p*(c + d*x^2)^q,x]","\frac{\left(a+b x^2\right)^{p+1} \left(c+d x^2\right)^q \left(\frac{b \left(c+d x^2\right)}{b c-a d}\right)^{-q} \, _2F_1\left(p+1,-q;p+2;\frac{d \left(b x^2+a\right)}{a d-b c}\right)}{2 b (p+1)}","\frac{\left(a+b x^2\right)^{p+1} \left(c+d x^2\right)^q \left(\frac{b \left(c+d x^2\right)}{b c-a d}\right)^{-q} \, _2F_1\left(p+1,-q;p+2;-\frac{d \left(b x^2+a\right)}{b c-a d}\right)}{2 b (p+1)}",1,"((a + b*x^2)^(1 + p)*(c + d*x^2)^q*Hypergeometric2F1[1 + p, -q, 2 + p, (d*(a + b*x^2))/(-(b*c) + a*d)])/(2*b*(1 + p)*((b*(c + d*x^2))/(b*c - a*d))^q)","A",1
1148,1,95,97,0.0959757,"\int \frac{\left(a+b x^2\right)^p \left(c+d x^2\right)^q}{x} \, dx","Integrate[((a + b*x^2)^p*(c + d*x^2)^q)/x,x]","\frac{\left(\frac{a}{b x^2}+1\right)^{-p} \left(a+b x^2\right)^p \left(\frac{c}{d x^2}+1\right)^{-q} \left(c+d x^2\right)^q F_1\left(-p-q;-p,-q;-p-q+1;-\frac{a}{b x^2},-\frac{c}{d x^2}\right)}{2 (p+q)}","-\frac{\left(a+b x^2\right)^{p+1} \left(c+d x^2\right)^q \left(\frac{b \left(c+d x^2\right)}{b c-a d}\right)^{-q} F_1\left(p+1;-q,1;p+2;-\frac{d \left(b x^2+a\right)}{b c-a d},\frac{b x^2+a}{a}\right)}{2 a (p+1)}",1,"((a + b*x^2)^p*(c + d*x^2)^q*AppellF1[-p - q, -p, -q, 1 - p - q, -(a/(b*x^2)), -(c/(d*x^2))])/(2*(p + q)*(1 + a/(b*x^2))^p*(1 + c/(d*x^2))^q)","A",0
1149,1,100,98,0.0731315,"\int \frac{\left(a+b x^2\right)^p \left(c+d x^2\right)^q}{x^3} \, dx","Integrate[((a + b*x^2)^p*(c + d*x^2)^q)/x^3,x]","\frac{\left(\frac{a}{b x^2}+1\right)^{-p} \left(a+b x^2\right)^p \left(\frac{c}{d x^2}+1\right)^{-q} \left(c+d x^2\right)^q F_1\left(-p-q+1;-p,-q;-p-q+2;-\frac{a}{b x^2},-\frac{c}{d x^2}\right)}{2 x^2 (p+q-1)}","\frac{b \left(a+b x^2\right)^{p+1} \left(c+d x^2\right)^q \left(\frac{b \left(c+d x^2\right)}{b c-a d}\right)^{-q} F_1\left(p+1;-q,2;p+2;-\frac{d \left(b x^2+a\right)}{b c-a d},\frac{b x^2+a}{a}\right)}{2 a^2 (p+1)}",1,"((a + b*x^2)^p*(c + d*x^2)^q*AppellF1[1 - p - q, -p, -q, 2 - p - q, -(a/(b*x^2)), -(c/(d*x^2))])/(2*(-1 + p + q)*(1 + a/(b*x^2))^p*(1 + c/(d*x^2))^q*x^2)","A",0
1150,1,100,100,0.1153453,"\int \frac{\left(a+b x^2\right)^p \left(c+d x^2\right)^q}{x^5} \, dx","Integrate[((a + b*x^2)^p*(c + d*x^2)^q)/x^5,x]","\frac{\left(\frac{a}{b x^2}+1\right)^{-p} \left(a+b x^2\right)^p \left(\frac{c}{d x^2}+1\right)^{-q} \left(c+d x^2\right)^q F_1\left(-p-q+2;-p,-q;-p-q+3;-\frac{a}{b x^2},-\frac{c}{d x^2}\right)}{2 x^4 (p+q-2)}","-\frac{b^2 \left(a+b x^2\right)^{p+1} \left(c+d x^2\right)^q \left(\frac{b \left(c+d x^2\right)}{b c-a d}\right)^{-q} F_1\left(p+1;-q,3;p+2;-\frac{d \left(b x^2+a\right)}{b c-a d},\frac{b x^2+a}{a}\right)}{2 a^3 (p+1)}",1,"((a + b*x^2)^p*(c + d*x^2)^q*AppellF1[2 - p - q, -p, -q, 3 - p - q, -(a/(b*x^2)), -(c/(d*x^2))])/(2*(-2 + p + q)*(1 + a/(b*x^2))^p*(1 + c/(d*x^2))^q*x^4)","A",0
1151,1,91,91,0.0709155,"\int (e x)^{5/2} \left(a+b x^2\right)^p \left(c+d x^2\right)^q \, dx","Integrate[(e*x)^(5/2)*(a + b*x^2)^p*(c + d*x^2)^q,x]","\frac{2}{7} x (e x)^{5/2} \left(a+b x^2\right)^p \left(\frac{a+b x^2}{a}\right)^{-p} \left(c+d x^2\right)^q \left(\frac{c+d x^2}{c}\right)^{-q} F_1\left(\frac{7}{4};-p,-q;\frac{11}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)","\frac{2 (e x)^{7/2} \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(c+d x^2\right)^q \left(\frac{d x^2}{c}+1\right)^{-q} F_1\left(\frac{7}{4};-p,-q;\frac{11}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{7 e}",1,"(2*x*(e*x)^(5/2)*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1[7/4, -p, -q, 11/4, -((b*x^2)/a), -((d*x^2)/c)])/(7*((a + b*x^2)/a)^p*((c + d*x^2)/c)^q)","A",1
1152,1,91,91,0.0565868,"\int (e x)^{3/2} \left(a+b x^2\right)^p \left(c+d x^2\right)^q \, dx","Integrate[(e*x)^(3/2)*(a + b*x^2)^p*(c + d*x^2)^q,x]","\frac{2}{5} x (e x)^{3/2} \left(a+b x^2\right)^p \left(\frac{a+b x^2}{a}\right)^{-p} \left(c+d x^2\right)^q \left(\frac{c+d x^2}{c}\right)^{-q} F_1\left(\frac{5}{4};-p,-q;\frac{9}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)","\frac{2 (e x)^{5/2} \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(c+d x^2\right)^q \left(\frac{d x^2}{c}+1\right)^{-q} F_1\left(\frac{5}{4};-p,-q;\frac{9}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{5 e}",1,"(2*x*(e*x)^(3/2)*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1[5/4, -p, -q, 9/4, -((b*x^2)/a), -((d*x^2)/c)])/(5*((a + b*x^2)/a)^p*((c + d*x^2)/c)^q)","A",1
1153,1,91,91,0.0555087,"\int \sqrt{e x} \left(a+b x^2\right)^p \left(c+d x^2\right)^q \, dx","Integrate[Sqrt[e*x]*(a + b*x^2)^p*(c + d*x^2)^q,x]","\frac{2}{3} x \sqrt{e x} \left(a+b x^2\right)^p \left(\frac{a+b x^2}{a}\right)^{-p} \left(c+d x^2\right)^q \left(\frac{c+d x^2}{c}\right)^{-q} F_1\left(\frac{3}{4};-p,-q;\frac{7}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)","\frac{2 (e x)^{3/2} \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(c+d x^2\right)^q \left(\frac{d x^2}{c}+1\right)^{-q} F_1\left(\frac{3}{4};-p,-q;\frac{7}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{3 e}",1,"(2*x*Sqrt[e*x]*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1[3/4, -p, -q, 7/4, -((b*x^2)/a), -((d*x^2)/c)])/(3*((a + b*x^2)/a)^p*((c + d*x^2)/c)^q)","A",1
1154,1,89,89,0.0595741,"\int \frac{\left(a+b x^2\right)^p \left(c+d x^2\right)^q}{\sqrt{e x}} \, dx","Integrate[((a + b*x^2)^p*(c + d*x^2)^q)/Sqrt[e*x],x]","\frac{2 x \left(a+b x^2\right)^p \left(\frac{a+b x^2}{a}\right)^{-p} \left(c+d x^2\right)^q \left(\frac{c+d x^2}{c}\right)^{-q} F_1\left(\frac{1}{4};-p,-q;\frac{5}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{\sqrt{e x}}","\frac{2 \sqrt{e x} \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(c+d x^2\right)^q \left(\frac{d x^2}{c}+1\right)^{-q} F_1\left(\frac{1}{4};-p,-q;\frac{5}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{e}",1,"(2*x*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1[1/4, -p, -q, 5/4, -((b*x^2)/a), -((d*x^2)/c)])/(Sqrt[e*x]*((a + b*x^2)/a)^p*((c + d*x^2)/c)^q)","A",1
1155,1,89,89,0.0597089,"\int \frac{\left(a+b x^2\right)^p \left(c+d x^2\right)^q}{(e x)^{3/2}} \, dx","Integrate[((a + b*x^2)^p*(c + d*x^2)^q)/(e*x)^(3/2),x]","-\frac{2 x \left(a+b x^2\right)^p \left(\frac{a+b x^2}{a}\right)^{-p} \left(c+d x^2\right)^q \left(\frac{c+d x^2}{c}\right)^{-q} F_1\left(-\frac{1}{4};-p,-q;\frac{3}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{(e x)^{3/2}}","-\frac{2 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(c+d x^2\right)^q \left(\frac{d x^2}{c}+1\right)^{-q} F_1\left(-\frac{1}{4};-p,-q;\frac{3}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{e \sqrt{e x}}",1,"(-2*x*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1[-1/4, -p, -q, 3/4, -((b*x^2)/a), -((d*x^2)/c)])/((e*x)^(3/2)*((a + b*x^2)/a)^p*((c + d*x^2)/c)^q)","A",1
1156,1,91,91,0.0612039,"\int \frac{\left(a+b x^2\right)^p \left(c+d x^2\right)^q}{(e x)^{5/2}} \, dx","Integrate[((a + b*x^2)^p*(c + d*x^2)^q)/(e*x)^(5/2),x]","-\frac{2 x \left(a+b x^2\right)^p \left(\frac{a+b x^2}{a}\right)^{-p} \left(c+d x^2\right)^q \left(\frac{c+d x^2}{c}\right)^{-q} F_1\left(-\frac{3}{4};-p,-q;\frac{1}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{3 (e x)^{5/2}}","-\frac{2 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(c+d x^2\right)^q \left(\frac{d x^2}{c}+1\right)^{-q} F_1\left(-\frac{3}{4};-p,-q;\frac{1}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{3 e (e x)^{3/2}}",1,"(-2*x*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1[-3/4, -p, -q, 1/4, -((b*x^2)/a), -((d*x^2)/c)])/(3*(e*x)^(5/2)*((a + b*x^2)/a)^p*((c + d*x^2)/c)^q)","A",1