1,1,33,33,0.0156834,"\int x^2 \left(a+b x^2\right) \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^2*(a + b*x^2)*(A + B*x^2),x]","\frac{1}{105} \left(35 a A x^3+21 a B x^5+21 A b x^5+15 b B x^7\right)","\frac{1}{5} x^5 (a B+A b)+\frac{1}{3} a A x^3+\frac{1}{7} b B x^7",1,"(35*a*A*x^3 + 21*A*b*x^5 + 21*a*B*x^5 + 15*b*B*x^7)/105","A",1
2,1,33,33,0.015805,"\int x \left(a+b x^2\right) \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x*(a + b*x^2)*(A + B*x^2),x]","\frac{1}{12} \left(6 a A x^2+3 a B x^4+3 A b x^4+2 b B x^6\right)","\frac{1}{4} x^4 (a B+A b)+\frac{1}{2} a A x^2+\frac{1}{6} b B x^6",1,"(6*a*A*x^2 + 3*A*b*x^4 + 3*a*B*x^4 + 2*b*B*x^6)/12","A",1
3,1,31,28,0.0142412,"\int \left(a+b x^2\right) \left(A+B x^2\right) \, dx","IntegrateAlgebraic[(a + b*x^2)*(A + B*x^2),x]","\frac{1}{15} \left(15 a A x+5 a B x^3+5 A b x^3+3 b B x^5\right)","\frac{1}{3} x^3 (a B+A b)+a A x+\frac{1}{5} b B x^5",1,"(15*a*A*x + 5*A*b*x^3 + 5*a*B*x^3 + 3*b*B*x^5)/15","A",1
4,1,28,29,0.0161759,"\int \frac{\left(a+b x^2\right) \left(A+B x^2\right)}{x} \, dx","IntegrateAlgebraic[((a + b*x^2)*(A + B*x^2))/x,x]","\frac{1}{4} x^2 \left(2 a B+2 A b+b B x^2\right)+a A \log (x)","\frac{1}{2} x^2 (a B+A b)+a A \log (x)+\frac{1}{4} b B x^4",1,"(x^2*(2*A*b + 2*a*B + b*B*x^2))/4 + a*A*Log[x]","A",1
5,1,32,26,0.0148119,"\int \frac{\left(a+b x^2\right) \left(A+B x^2\right)}{x^2} \, dx","IntegrateAlgebraic[((a + b*x^2)*(A + B*x^2))/x^2,x]","\frac{-3 a A+3 a B x^2+3 A b x^2+b B x^4}{3 x}","x (a B+A b)-\frac{a A}{x}+\frac{1}{3} b B x^3",1,"(-3*a*A + 3*A*b*x^2 + 3*a*B*x^2 + b*B*x^4)/(3*x)","A",1
6,1,29,29,0.0181737,"\int \frac{\left(a+b x^2\right) \left(A+B x^2\right)}{x^3} \, dx","IntegrateAlgebraic[((a + b*x^2)*(A + B*x^2))/x^3,x]","\frac{b B x^4-a A}{2 x^2}+\log (x) (a B+A b)","\log (x) (a B+A b)-\frac{a A}{2 x^2}+\frac{1}{2} b B x^2",1,"(-(a*A) + b*B*x^4)/(2*x^2) + (A*b + a*B)*Log[x]","A",1
7,1,33,26,0.0175711,"\int \frac{\left(a+b x^2\right) \left(A+B x^2\right)}{x^4} \, dx","IntegrateAlgebraic[((a + b*x^2)*(A + B*x^2))/x^4,x]","\frac{-a A-3 a B x^2-3 A b x^2+3 b B x^4}{3 x^3}","-\frac{a B+A b}{x}-\frac{a A}{3 x^3}+b B x",1,"(-(a*A) - 3*A*b*x^2 - 3*a*B*x^2 + 3*b*B*x^4)/(3*x^3)","A",1
8,1,32,29,0.0286178,"\int \frac{\left(a+b x^2\right) \left(A+B x^2\right)}{x^5} \, dx","IntegrateAlgebraic[((a + b*x^2)*(A + B*x^2))/x^5,x]","\frac{-a A-2 a B x^2-2 A b x^2}{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,"(-(a*A) - 2*A*b*x^2 - 2*a*B*x^2)/(4*x^4) + b*B*Log[x]","A",1
9,1,33,31,0.0168047,"\int \frac{\left(a+b x^2\right) \left(A+B x^2\right)}{x^6} \, dx","IntegrateAlgebraic[((a + b*x^2)*(A + B*x^2))/x^6,x]","\frac{-3 a A-5 a B x^2-5 A b x^2-15 b B x^4}{15 x^5}","-\frac{a B+A b}{3 x^3}-\frac{a A}{5 x^5}-\frac{b B}{x}",1,"(-3*a*A - 5*A*b*x^2 - 5*a*B*x^2 - 15*b*B*x^4)/(15*x^5)","A",1
10,1,33,33,0.0150273,"\int \frac{\left(a+b x^2\right) \left(A+B x^2\right)}{x^7} \, dx","IntegrateAlgebraic[((a + b*x^2)*(A + B*x^2))/x^7,x]","\frac{-2 a A-3 a B x^2-3 A b x^2-6 b B x^4}{12 x^6}","-\frac{a B+A b}{4 x^4}-\frac{a A}{6 x^6}-\frac{b B}{2 x^2}",1,"(-2*a*A - 3*A*b*x^2 - 3*a*B*x^2 - 6*b*B*x^4)/(12*x^6)","A",1
11,1,57,55,0.0218248,"\int x^2 \left(a+b x^2\right)^2 \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^2*(a + b*x^2)^2*(A + B*x^2),x]","\frac{1}{315} \left(105 a^2 A x^3+63 a^2 B x^5+126 a A b x^5+90 a b B x^7+45 A b^2 x^7+35 b^2 B x^9\right)","\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,"(105*a^2*A*x^3 + 126*a*A*b*x^5 + 63*a^2*B*x^5 + 45*A*b^2*x^7 + 90*a*b*B*x^7 + 35*b^2*B*x^9)/315","A",1
12,1,57,42,0.0270513,"\int x \left(a+b x^2\right)^2 \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x*(a + b*x^2)^2*(A + B*x^2),x]","\frac{1}{24} \left(12 a^2 A x^2+6 a^2 B x^4+12 a A b x^4+8 a b B x^6+4 A b^2 x^6+3 b^2 B x^8\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,"(12*a^2*A*x^2 + 12*a*A*b*x^4 + 6*a^2*B*x^4 + 4*A*b^2*x^6 + 8*a*b*B*x^6 + 3*b^2*B*x^8)/24","A",1
13,1,55,50,0.0195215,"\int \left(a+b x^2\right)^2 \left(A+B x^2\right) \, dx","IntegrateAlgebraic[(a + b*x^2)^2*(A + B*x^2),x]","\frac{1}{105} \left(105 a^2 A x+35 a^2 B x^3+70 a A b x^3+42 a b B x^5+21 A b^2 x^5+15 b^2 B x^7\right)","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,"(105*a^2*A*x + 70*a*A*b*x^3 + 35*a^2*B*x^3 + 21*A*b^2*x^5 + 42*a*b*B*x^5 + 15*b^2*B*x^7)/105","A",1
14,1,53,43,0.0306878,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x^2\right)}{x} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(A + B*x^2))/x,x]","\frac{1}{12} x^2 \left(6 a^2 B+12 a A b+6 a b B x^2+3 A b^2 x^2+2 b^2 B x^4\right)+a^2 A \log (x)","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,"(x^2*(12*a*A*b + 6*a^2*B + 3*A*b^2*x^2 + 6*a*b*B*x^2 + 2*b^2*B*x^4))/12 + a^2*A*Log[x]","A",1
15,1,57,48,0.0229477,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x^2\right)}{x^2} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(A + B*x^2))/x^2,x]","\frac{-15 a^2 A+15 a^2 B x^2+30 a A b x^2+10 a b B x^4+5 A b^2 x^4+3 b^2 B x^6}{15 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",1,"(-15*a^2*A + 30*a*A*b*x^2 + 15*a^2*B*x^2 + 5*A*b^2*x^4 + 10*a*b*B*x^4 + 3*b^2*B*x^6)/(15*x)","A",1
16,1,54,51,0.0381298,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x^2\right)}{x^3} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(A + B*x^2))/x^3,x]","\frac{-2 a^2 A+4 a b B x^4+2 A b^2 x^4+b^2 B x^6}{4 x^2}+\log (x) \left(a^2 B+2 a A b\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 + 2*A*b^2*x^4 + 4*a*b*B*x^4 + b^2*B*x^6)/(4*x^2) + (2*a*A*b + a^2*B)*Log[x]","A",1
17,1,56,48,0.0265306,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x^2\right)}{x^4} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(A + B*x^2))/x^4,x]","\frac{-a^2 A-3 a^2 B x^2-6 a A b x^2+6 a b B x^4+3 A b^2 x^4+b^2 B x^6}{3 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,"(-(a^2*A) - 6*a*A*b*x^2 - 3*a^2*B*x^2 + 3*A*b^2*x^4 + 6*a*b*B*x^4 + b^2*B*x^6)/(3*x^3)","A",1
18,1,55,51,0.0392532,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x^2\right)}{x^5} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(A + B*x^2))/x^5,x]","\frac{-a^2 A-2 a^2 B x^2-4 a A b x^2+2 b^2 B x^6}{4 x^4}+\log (x) \left(2 a b B+A b^2\right)","-\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,"(-(a^2*A) - 4*a*A*b*x^2 - 2*a^2*B*x^2 + 2*b^2*B*x^6)/(4*x^4) + (A*b^2 + 2*a*b*B)*Log[x]","A",1
19,1,57,48,0.0278187,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x^2\right)}{x^6} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(A + B*x^2))/x^6,x]","\frac{-3 a^2 A-5 a^2 B x^2-10 a A b x^2-30 a b B x^4-15 A b^2 x^4+15 b^2 B x^6}{15 x^5}","-\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,"(-3*a^2*A - 10*a*A*b*x^2 - 5*a^2*B*x^2 - 15*A*b^2*x^4 - 30*a*b*B*x^4 + 15*b^2*B*x^6)/(15*x^5)","A",1
20,1,56,51,0.0439757,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x^2\right)}{x^7} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(A + B*x^2))/x^7,x]","\frac{-2 a^2 A-3 a^2 B x^2-6 a A b x^2-12 a b B x^4-6 A b^2 x^4}{12 x^6}+b^2 B \log (x)","-\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,"(-2*a^2*A - 6*a*A*b*x^2 - 3*a^2*B*x^2 - 6*A*b^2*x^4 - 12*a*b*B*x^4)/(12*x^6) + b^2*B*Log[x]","A",1
21,1,57,53,0.0262361,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x^2\right)}{x^8} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(A + B*x^2))/x^8,x]","\frac{-15 a^2 A-21 a^2 B x^2-42 a A b x^2-70 a b B x^4-35 A b^2 x^4-105 b^2 B x^6}{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,"(-15*a^2*A - 42*a*A*b*x^2 - 21*a^2*B*x^2 - 35*A*b^2*x^4 - 70*a*b*B*x^4 - 105*b^2*B*x^6)/(105*x^7)","A",1
22,1,57,48,0.0251944,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x^2\right)}{x^9} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(A + B*x^2))/x^9,x]","\frac{-3 a^2 A-4 a^2 B x^2-8 a A b x^2-12 a b B x^4-6 A b^2 x^4-12 b^2 B x^6}{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,"(-3*a^2*A - 8*a*A*b*x^2 - 4*a^2*B*x^2 - 6*A*b^2*x^4 - 12*a*b*B*x^4 - 12*b^2*B*x^6)/(24*x^8)","A",1
23,1,129,117,0.0496535,"\int x^9 \left(a+b x^2\right)^5 \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^9*(a + b*x^2)^5*(A + B*x^2),x]","\frac{2772 a^5 A x^{10}+2310 a^5 B x^{12}+11550 a^4 A b x^{12}+9900 a^4 b B x^{14}+19800 a^3 A b^2 x^{14}+17325 a^3 b^2 B x^{16}+17325 a^2 A b^3 x^{16}+15400 a^2 b^3 B x^{18}+7700 a A b^4 x^{18}+6930 a b^4 B x^{20}+1386 A b^5 x^{20}+1260 b^5 B x^{22}}{27720}","\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,"(2772*a^5*A*x^10 + 11550*a^4*A*b*x^12 + 2310*a^5*B*x^12 + 19800*a^3*A*b^2*x^14 + 9900*a^4*b*B*x^14 + 17325*a^2*A*b^3*x^16 + 17325*a^3*b^2*B*x^16 + 7700*a*A*b^4*x^18 + 15400*a^2*b^3*B*x^18 + 1386*A*b^5*x^20 + 6930*a*b^4*B*x^20 + 1260*b^5*B*x^22)/27720","A",1
24,1,129,117,0.0478282,"\int x^8 \left(a+b x^2\right)^5 \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^8*(a + b*x^2)^5*(A + B*x^2),x]","\frac{323323 a^5 A x^9+264537 a^5 B x^{11}+1322685 a^4 A b x^{11}+1119195 a^4 b B x^{13}+2238390 a^3 A b^2 x^{13}+1939938 a^3 b^2 B x^{15}+1939938 a^2 A b^3 x^{15}+1711710 a^2 b^3 B x^{17}+855855 a A b^4 x^{17}+765765 a b^4 B x^{19}+153153 A b^5 x^{19}+138567 b^5 B x^{21}}{2909907}","\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,"(323323*a^5*A*x^9 + 1322685*a^4*A*b*x^11 + 264537*a^5*B*x^11 + 2238390*a^3*A*b^2*x^13 + 1119195*a^4*b*B*x^13 + 1939938*a^2*A*b^3*x^15 + 1939938*a^3*b^2*B*x^15 + 855855*a*A*b^4*x^17 + 1711710*a^2*b^3*B*x^17 + 153153*A*b^5*x^19 + 765765*a*b^4*B*x^19 + 138567*b^5*B*x^21)/2909907","A",1
25,1,129,122,0.0416619,"\int x^7 \left(a+b x^2\right)^5 \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^7*(a + b*x^2)^5*(A + B*x^2),x]","\frac{630 a^5 A x^8+504 a^5 B x^{10}+2520 a^4 A b x^{10}+2100 a^4 b B x^{12}+4200 a^3 A b^2 x^{12}+3600 a^3 b^2 B x^{14}+3600 a^2 A b^3 x^{14}+3150 a^2 b^3 B x^{16}+1575 a A b^4 x^{16}+1400 a b^4 B x^{18}+280 A b^5 x^{18}+252 b^5 B x^{20}}{5040}","-\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,"(630*a^5*A*x^8 + 2520*a^4*A*b*x^10 + 504*a^5*B*x^10 + 4200*a^3*A*b^2*x^12 + 2100*a^4*b*B*x^12 + 3600*a^2*A*b^3*x^14 + 3600*a^3*b^2*B*x^14 + 1575*a*A*b^4*x^16 + 3150*a^2*b^3*B*x^16 + 280*A*b^5*x^18 + 1400*a*b^4*B*x^18 + 252*b^5*B*x^20)/5040","A",1
26,1,129,117,0.0413908,"\int x^6 \left(a+b x^2\right)^5 \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^6*(a + b*x^2)^5*(A + B*x^2),x]","\frac{415701 a^5 A x^7+323323 a^5 B x^9+1616615 a^4 A b x^9+1322685 a^4 b B x^{11}+2645370 a^3 A b^2 x^{11}+2238390 a^3 b^2 B x^{13}+2238390 a^2 A b^3 x^{13}+1939938 a^2 b^3 B x^{15}+969969 a A b^4 x^{15}+855855 a b^4 B x^{17}+171171 A b^5 x^{17}+153153 b^5 B x^{19}}{2909907}","\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,"(415701*a^5*A*x^7 + 1616615*a^4*A*b*x^9 + 323323*a^5*B*x^9 + 2645370*a^3*A*b^2*x^11 + 1322685*a^4*b*B*x^11 + 2238390*a^2*A*b^3*x^13 + 2238390*a^3*b^2*B*x^13 + 969969*a*A*b^4*x^15 + 1939938*a^2*b^3*B*x^15 + 171171*A*b^5*x^17 + 855855*a*b^4*B*x^17 + 153153*b^5*B*x^19)/2909907","A",1
27,1,129,95,0.0599723,"\int x^5 \left(a+b x^2\right)^5 \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^5*(a + b*x^2)^5*(A + B*x^2),x]","\frac{x^6 \left(168 a^5 A+126 a^5 B x^2+630 a^4 A b x^2+504 a^4 b B x^4+1008 a^3 A b^2 x^4+840 a^3 b^2 B x^6+840 a^2 A b^3 x^6+720 a^2 b^3 B x^8+360 a A b^4 x^8+315 a b^4 B x^{10}+63 A b^5 x^{10}+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 + 630*a^4*A*b*x^2 + 126*a^5*B*x^2 + 1008*a^3*A*b^2*x^4 + 504*a^4*b*B*x^4 + 840*a^2*A*b^3*x^6 + 840*a^3*b^2*B*x^6 + 360*a*A*b^4*x^8 + 720*a^2*b^3*B*x^8 + 63*A*b^5*x^10 + 315*a*b^4*B*x^10 + 56*b^5*B*x^12))/1008","A",1
28,1,129,117,0.0417414,"\int x^4 \left(a+b x^2\right)^5 \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^4*(a + b*x^2)^5*(A + B*x^2),x]","\frac{153153 a^5 A x^5+109395 a^5 B x^7+546975 a^4 A b x^7+425425 a^4 b B x^9+850850 a^3 A b^2 x^9+696150 a^3 b^2 B x^{11}+696150 a^2 A b^3 x^{11}+589050 a^2 b^3 B x^{13}+294525 a A b^4 x^{13}+255255 a b^4 B x^{15}+51051 A b^5 x^{15}+45045 b^5 B x^{17}}{765765}","\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,"(153153*a^5*A*x^5 + 546975*a^4*A*b*x^7 + 109395*a^5*B*x^7 + 850850*a^3*A*b^2*x^9 + 425425*a^4*b*B*x^9 + 696150*a^2*A*b^3*x^11 + 696150*a^3*b^2*B*x^11 + 294525*a*A*b^4*x^13 + 589050*a^2*b^3*B*x^13 + 51051*A*b^5*x^15 + 255255*a*b^4*B*x^15 + 45045*b^5*B*x^17)/765765","A",1
29,1,129,67,0.0423678,"\int x^3 \left(a+b x^2\right)^5 \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^3*(a + b*x^2)^5*(A + B*x^2),x]","\frac{1}{336} \left(84 a^5 A x^4+56 a^5 B x^6+280 a^4 A b x^6+210 a^4 b B x^8+420 a^3 A b^2 x^8+336 a^3 b^2 B x^{10}+336 a^2 A b^3 x^{10}+280 a^2 b^3 B x^{12}+140 a A b^4 x^{12}+120 a b^4 B x^{14}+24 A b^5 x^{14}+21 b^5 B x^{16}\right)","\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,"(84*a^5*A*x^4 + 280*a^4*A*b*x^6 + 56*a^5*B*x^6 + 420*a^3*A*b^2*x^8 + 210*a^4*b*B*x^8 + 336*a^2*A*b^3*x^10 + 336*a^3*b^2*B*x^10 + 140*a*A*b^4*x^12 + 280*a^2*b^3*B*x^12 + 24*A*b^5*x^14 + 120*a*b^4*B*x^14 + 21*b^5*B*x^16)/336","A",1
30,1,129,117,0.0421833,"\int x^2 \left(a+b x^2\right)^5 \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^2*(a + b*x^2)^5*(A + B*x^2),x]","\frac{15015 a^5 A x^3+9009 a^5 B x^5+45045 a^4 A b x^5+32175 a^4 b B x^7+64350 a^3 A b^2 x^7+50050 a^3 b^2 B x^9+50050 a^2 A b^3 x^9+40950 a^2 b^3 B x^{11}+20475 a A b^4 x^{11}+17325 a b^4 B x^{13}+3465 A b^5 x^{13}+3003 b^5 B x^{15}}{45045}","\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,"(15015*a^5*A*x^3 + 45045*a^4*A*b*x^5 + 9009*a^5*B*x^5 + 64350*a^3*A*b^2*x^7 + 32175*a^4*b*B*x^7 + 50050*a^2*A*b^3*x^9 + 50050*a^3*b^2*B*x^9 + 20475*a*A*b^4*x^11 + 40950*a^2*b^3*B*x^11 + 3465*A*b^5*x^13 + 17325*a*b^4*B*x^13 + 3003*b^5*B*x^15)/45045","A",1
31,1,129,42,0.0511413,"\int x \left(a+b x^2\right)^5 \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x*(a + b*x^2)^5*(A + B*x^2),x]","\frac{1}{84} x^2 \left(42 a^5 A+21 a^5 B x^2+105 a^4 A b x^2+70 a^4 b B x^4+140 a^3 A b^2 x^4+105 a^3 b^2 B x^6+105 a^2 A b^3 x^6+84 a^2 b^3 B x^8+42 a A b^4 x^8+35 a b^4 B x^{10}+7 A b^5 x^{10}+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 + 105*a^4*A*b*x^2 + 21*a^5*B*x^2 + 140*a^3*A*b^2*x^4 + 70*a^4*b*B*x^4 + 105*a^2*A*b^3*x^6 + 105*a^3*b^2*B*x^6 + 42*a*A*b^4*x^8 + 84*a^2*b^3*B*x^8 + 7*A*b^5*x^10 + 35*a*b^4*B*x^10 + 6*b^5*B*x^12))/84","B",1
32,1,127,109,0.0428659,"\int \left(a+b x^2\right)^5 \left(A+B x^2\right) \, dx","IntegrateAlgebraic[(a + b*x^2)^5*(A + B*x^2),x]","\frac{9009 a^5 A x+3003 a^5 B x^3+15015 a^4 A b x^3+9009 a^4 b B x^5+18018 a^3 A b^2 x^5+12870 a^3 b^2 B x^7+12870 a^2 A b^3 x^7+10010 a^2 b^3 B x^9+5005 a A b^4 x^9+4095 a b^4 B x^{11}+819 A b^5 x^{11}+693 b^5 B x^{13}}{9009}","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,"(9009*a^5*A*x + 15015*a^4*A*b*x^3 + 3003*a^5*B*x^3 + 18018*a^3*A*b^2*x^5 + 9009*a^4*b*B*x^5 + 12870*a^2*A*b^3*x^7 + 12870*a^3*b^2*B*x^7 + 5005*a*A*b^4*x^9 + 10010*a^2*b^3*B*x^9 + 819*A*b^5*x^11 + 4095*a*b^4*B*x^11 + 693*b^5*B*x^13)/9009","A",1
33,1,125,88,0.0573556,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x} \, dx","IntegrateAlgebraic[((a + b*x^2)^5*(A + B*x^2))/x,x]","a^5 A \log (x)+\frac{1}{120} x^2 \left(60 a^5 B+300 a^4 A b+150 a^4 b B x^2+300 a^3 A b^2 x^2+200 a^3 b^2 B x^4+200 a^2 A b^3 x^4+150 a^2 b^3 B x^6+75 a A b^4 x^6+60 a b^4 B x^8+12 A b^5 x^8+10 b^5 B x^{10}\right)","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,"(x^2*(300*a^4*A*b + 60*a^5*B + 300*a^3*A*b^2*x^2 + 150*a^4*b*B*x^2 + 200*a^2*A*b^3*x^4 + 200*a^3*b^2*B*x^4 + 75*a*A*b^4*x^6 + 150*a^2*b^3*B*x^6 + 12*A*b^5*x^8 + 60*a*b^4*B*x^8 + 10*b^5*B*x^10))/120 + a^5*A*Log[x]","A",1
34,1,129,108,0.0445528,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^2} \, dx","IntegrateAlgebraic[((a + b*x^2)^5*(A + B*x^2))/x^2,x]","\frac{-693 a^5 A+693 a^5 B x^2+3465 a^4 A b x^2+1155 a^4 b B x^4+2310 a^3 A b^2 x^4+1386 a^3 b^2 B x^6+1386 a^2 A b^3 x^6+990 a^2 b^3 B x^8+495 a A b^4 x^8+385 a b^4 B x^{10}+77 A b^5 x^{10}+63 b^5 B x^{12}}{693 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}",1,"(-693*a^5*A + 3465*a^4*A*b*x^2 + 693*a^5*B*x^2 + 2310*a^3*A*b^2*x^4 + 1155*a^4*b*B*x^4 + 1386*a^2*A*b^3*x^6 + 1386*a^3*b^2*B*x^6 + 495*a*A*b^4*x^8 + 990*a^2*b^3*B*x^8 + 77*A*b^5*x^10 + 385*a*b^4*B*x^10 + 63*b^5*B*x^12)/(693*x)","A",1
35,1,127,113,0.0775903,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^3} \, dx","IntegrateAlgebraic[((a + b*x^2)^5*(A + B*x^2))/x^3,x]","\log (x) \left(a^5 B+5 a^4 A b\right)+\frac{-60 a^5 A+300 a^4 b B x^4+600 a^3 A b^2 x^4+300 a^3 b^2 B x^6+300 a^2 A b^3 x^6+200 a^2 b^3 B x^8+100 a A b^4 x^8+75 a b^4 B x^{10}+15 A b^5 x^{10}+12 b^5 B x^{12}}{120 x^2}","-\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,"(-60*a^5*A + 600*a^3*A*b^2*x^4 + 300*a^4*b*B*x^4 + 300*a^2*A*b^3*x^6 + 300*a^3*b^2*B*x^6 + 100*a*A*b^4*x^8 + 200*a^2*b^3*B*x^8 + 15*A*b^5*x^10 + 75*a*b^4*B*x^10 + 12*b^5*B*x^12)/(120*x^2) + (5*a^4*A*b + a^5*B)*Log[x]","A",1
36,1,129,108,0.0484017,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^4} \, dx","IntegrateAlgebraic[((a + b*x^2)^5*(A + B*x^2))/x^4,x]","\frac{-21 a^5 A-63 a^5 B x^2-315 a^4 A b x^2+315 a^4 b B x^4+630 a^3 A b^2 x^4+210 a^3 b^2 B x^6+210 a^2 A b^3 x^6+126 a^2 b^3 B x^8+63 a A b^4 x^8+45 a b^4 B x^{10}+9 A b^5 x^{10}+7 b^5 B x^{12}}{63 x^3}","-\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,"(-21*a^5*A - 315*a^4*A*b*x^2 - 63*a^5*B*x^2 + 630*a^3*A*b^2*x^4 + 315*a^4*b*B*x^4 + 210*a^2*A*b^3*x^6 + 210*a^3*b^2*B*x^6 + 63*a*A*b^4*x^8 + 126*a^2*b^3*B*x^8 + 9*A*b^5*x^10 + 45*a*b^4*B*x^10 + 7*b^5*B*x^12)/(63*x^3)","A",1
37,1,128,112,0.0665085,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^5} \, dx","IntegrateAlgebraic[((a + b*x^2)^5*(A + B*x^2))/x^5,x]","5 \log (x) \left(a^4 b B+2 a^3 A b^2\right)+\frac{-6 a^5 A-12 a^5 B x^2-60 a^4 A b x^2+120 a^3 b^2 B x^6+120 a^2 A b^3 x^6+60 a^2 b^3 B x^8+30 a A b^4 x^8+20 a b^4 B x^{10}+4 A b^5 x^{10}+3 b^5 B x^{12}}{24 x^4}","-\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,"(-6*a^5*A - 60*a^4*A*b*x^2 - 12*a^5*B*x^2 + 120*a^2*A*b^3*x^6 + 120*a^3*b^2*B*x^6 + 30*a*A*b^4*x^8 + 60*a^2*b^3*B*x^8 + 4*A*b^5*x^10 + 20*a*b^4*B*x^10 + 3*b^5*B*x^12)/(24*x^4) + 5*(2*a^3*A*b^2 + a^4*b*B)*Log[x]","A",1
38,1,129,111,0.0498756,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^6} \, dx","IntegrateAlgebraic[((a + b*x^2)^5*(A + B*x^2))/x^6,x]","\frac{-21 a^5 A-35 a^5 B x^2-175 a^4 A b x^2-525 a^4 b B x^4-1050 a^3 A b^2 x^4+1050 a^3 b^2 B x^6+1050 a^2 A b^3 x^6+350 a^2 b^3 B x^8+175 a A b^4 x^8+105 a b^4 B x^{10}+21 A b^5 x^{10}+15 b^5 B x^{12}}{105 x^5}","-\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,"(-21*a^5*A - 175*a^4*A*b*x^2 - 35*a^5*B*x^2 - 1050*a^3*A*b^2*x^4 - 525*a^4*b*B*x^4 + 1050*a^2*A*b^3*x^6 + 1050*a^3*b^2*B*x^6 + 175*a*A*b^4*x^8 + 350*a^2*b^3*B*x^8 + 21*A*b^5*x^10 + 105*a*b^4*B*x^10 + 15*b^5*B*x^12)/(105*x^5)","A",1
39,1,127,114,0.0645,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^7} \, dx","IntegrateAlgebraic[((a + b*x^2)^5*(A + B*x^2))/x^7,x]","10 \log (x) \left(a^3 b^2 B+a^2 A b^3\right)+\frac{-2 a^5 A-3 a^5 B x^2-15 a^4 A b x^2-30 a^4 b B x^4-60 a^3 A b^2 x^4+60 a^2 b^3 B x^8+30 a A b^4 x^8+15 a b^4 B x^{10}+3 A b^5 x^{10}+2 b^5 B x^{12}}{12 x^6}","-\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,"(-2*a^5*A - 15*a^4*A*b*x^2 - 3*a^5*B*x^2 - 60*a^3*A*b^2*x^4 - 30*a^4*b*B*x^4 + 30*a*A*b^4*x^8 + 60*a^2*b^3*B*x^8 + 3*A*b^5*x^10 + 15*a*b^4*B*x^10 + 2*b^5*B*x^12)/(12*x^6) + 10*(a^2*A*b^3 + a^3*b^2*B)*Log[x]","A",1
40,1,129,111,0.0532637,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^8} \, dx","IntegrateAlgebraic[((a + b*x^2)^5*(A + B*x^2))/x^8,x]","\frac{-15 a^5 A-21 a^5 B x^2-105 a^4 A b x^2-175 a^4 b B x^4-350 a^3 A b^2 x^4-1050 a^3 b^2 B x^6-1050 a^2 A b^3 x^6+1050 a^2 b^3 B x^8+525 a A b^4 x^8+175 a b^4 B x^{10}+35 A b^5 x^{10}+21 b^5 B x^{12}}{105 x^7}","-\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,"(-15*a^5*A - 105*a^4*A*b*x^2 - 21*a^5*B*x^2 - 350*a^3*A*b^2*x^4 - 175*a^4*b*B*x^4 - 1050*a^2*A*b^3*x^6 - 1050*a^3*b^2*B*x^6 + 525*a*A*b^4*x^8 + 1050*a^2*b^3*B*x^8 + 35*A*b^5*x^10 + 175*a*b^4*B*x^10 + 21*b^5*B*x^12)/(105*x^7)","A",1
41,1,128,112,0.0834493,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^9} \, dx","IntegrateAlgebraic[((a + b*x^2)^5*(A + B*x^2))/x^9,x]","5 \log (x) \left(2 a^2 b^3 B+a A b^4\right)+\frac{-3 a^5 A-4 a^5 B x^2-20 a^4 A b x^2-30 a^4 b B x^4-60 a^3 A b^2 x^4-120 a^3 b^2 B x^6-120 a^2 A b^3 x^6+60 a b^4 B x^{10}+12 A b^5 x^{10}+6 b^5 B x^{12}}{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,"(-3*a^5*A - 20*a^4*A*b*x^2 - 4*a^5*B*x^2 - 60*a^3*A*b^2*x^4 - 30*a^4*b*B*x^4 - 120*a^2*A*b^3*x^6 - 120*a^3*b^2*B*x^6 + 12*A*b^5*x^10 + 60*a*b^4*B*x^10 + 6*b^5*B*x^12)/(24*x^8) + 5*(a*A*b^4 + 2*a^2*b^3*B)*Log[x]","A",1
42,1,129,108,0.0487279,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^{10}} \, dx","IntegrateAlgebraic[((a + b*x^2)^5*(A + B*x^2))/x^10,x]","\frac{-7 a^5 A-9 a^5 B x^2-45 a^4 A b x^2-63 a^4 b B x^4-126 a^3 A b^2 x^4-210 a^3 b^2 B x^6-210 a^2 A b^3 x^6-630 a^2 b^3 B x^8-315 a A b^4 x^8+315 a b^4 B x^{10}+63 A b^5 x^{10}+21 b^5 B x^{12}}{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,"(-7*a^5*A - 45*a^4*A*b*x^2 - 9*a^5*B*x^2 - 126*a^3*A*b^2*x^4 - 63*a^4*b*B*x^4 - 210*a^2*A*b^3*x^6 - 210*a^3*b^2*B*x^6 - 315*a*A*b^4*x^8 - 630*a^2*b^3*B*x^8 + 63*A*b^5*x^10 + 315*a*b^4*B*x^10 + 21*b^5*B*x^12)/(63*x^9)","A",1
43,1,127,113,0.0905926,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^{11}} \, dx","IntegrateAlgebraic[((a + b*x^2)^5*(A + B*x^2))/x^11,x]","\frac{-12 a^5 A-15 a^5 B x^2-75 a^4 A b x^2-100 a^4 b B x^4-200 a^3 A b^2 x^4-300 a^3 b^2 B x^6-300 a^2 A b^3 x^6-600 a^2 b^3 B x^8-300 a A b^4 x^8+60 b^5 B x^{12}}{120 x^{10}}+\log (x) \left(5 a b^4 B+A b^5\right)","-\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,"(-12*a^5*A - 75*a^4*A*b*x^2 - 15*a^5*B*x^2 - 200*a^3*A*b^2*x^4 - 100*a^4*b*B*x^4 - 300*a^2*A*b^3*x^6 - 300*a^3*b^2*B*x^6 - 300*a*A*b^4*x^8 - 600*a^2*b^3*B*x^8 + 60*b^5*B*x^12)/(120*x^10) + (A*b^5 + 5*a*b^4*B)*Log[x]","A",1
44,1,129,108,0.0577877,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^{12}} \, dx","IntegrateAlgebraic[((a + b*x^2)^5*(A + B*x^2))/x^12,x]","\frac{-63 a^5 A-77 a^5 B x^2-385 a^4 A b x^2-495 a^4 b B x^4-990 a^3 A b^2 x^4-1386 a^3 b^2 B x^6-1386 a^2 A b^3 x^6-2310 a^2 b^3 B x^8-1155 a A b^4 x^8-3465 a b^4 B x^{10}-693 A b^5 x^{10}+693 b^5 B x^{12}}{693 x^{11}}","-\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,"(-63*a^5*A - 385*a^4*A*b*x^2 - 77*a^5*B*x^2 - 990*a^3*A*b^2*x^4 - 495*a^4*b*B*x^4 - 1386*a^2*A*b^3*x^6 - 1386*a^3*b^2*B*x^6 - 1155*a*A*b^4*x^8 - 2310*a^2*b^3*B*x^8 - 693*A*b^5*x^10 - 3465*a*b^4*B*x^10 + 693*b^5*B*x^12)/(693*x^11)","A",1
45,1,128,91,0.1003483,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^{13}} \, dx","IntegrateAlgebraic[((a + b*x^2)^5*(A + B*x^2))/x^13,x]","\frac{-10 a^5 A-12 a^5 B x^2-60 a^4 A b x^2-75 a^4 b B x^4-150 a^3 A b^2 x^4-200 a^3 b^2 B x^6-200 a^2 A b^3 x^6-300 a^2 b^3 B x^8-150 a A b^4 x^8-300 a b^4 B x^{10}-60 A b^5 x^{10}}{120 x^{12}}+b^5 B \log (x)","-\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,"(-10*a^5*A - 60*a^4*A*b*x^2 - 12*a^5*B*x^2 - 150*a^3*A*b^2*x^4 - 75*a^4*b*B*x^4 - 200*a^2*A*b^3*x^6 - 200*a^3*b^2*B*x^6 - 150*a*A*b^4*x^8 - 300*a^2*b^3*B*x^8 - 60*A*b^5*x^10 - 300*a*b^4*B*x^10)/(120*x^12) + b^5*B*Log[x]","A",1
46,1,129,113,0.0488023,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^{14}} \, dx","IntegrateAlgebraic[((a + b*x^2)^5*(A + B*x^2))/x^14,x]","\frac{-693 a^5 A-819 a^5 B x^2-4095 a^4 A b x^2-5005 a^4 b B x^4-10010 a^3 A b^2 x^4-12870 a^3 b^2 B x^6-12870 a^2 A b^3 x^6-18018 a^2 b^3 B x^8-9009 a A b^4 x^8-15015 a b^4 B x^{10}-3003 A b^5 x^{10}-9009 b^5 B x^{12}}{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,"(-693*a^5*A - 4095*a^4*A*b*x^2 - 819*a^5*B*x^2 - 10010*a^3*A*b^2*x^4 - 5005*a^4*b*B*x^4 - 12870*a^2*A*b^3*x^6 - 12870*a^3*b^2*B*x^6 - 9009*a*A*b^4*x^8 - 18018*a^2*b^3*B*x^8 - 3003*A*b^5*x^10 - 15015*a*b^4*B*x^10 - 9009*b^5*B*x^12)/(9009*x^13)","A",1
47,1,129,48,0.0472394,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^{15}} \, dx","IntegrateAlgebraic[((a + b*x^2)^5*(A + B*x^2))/x^15,x]","\frac{-6 a^5 A-7 a^5 B x^2-35 a^4 A b x^2-42 a^4 b B x^4-84 a^3 A b^2 x^4-105 a^3 b^2 B x^6-105 a^2 A b^3 x^6-140 a^2 b^3 B x^8-70 a A b^4 x^8-105 a b^4 B x^{10}-21 A b^5 x^{10}-42 b^5 B x^{12}}{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,"(-6*a^5*A - 35*a^4*A*b*x^2 - 7*a^5*B*x^2 - 84*a^3*A*b^2*x^4 - 42*a^4*b*B*x^4 - 105*a^2*A*b^3*x^6 - 105*a^3*b^2*B*x^6 - 70*a*A*b^4*x^8 - 140*a^2*b^3*B*x^8 - 21*A*b^5*x^10 - 105*a*b^4*B*x^10 - 42*b^5*B*x^12)/(84*x^14)","B",1
48,1,129,117,0.0493113,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^{16}} \, dx","IntegrateAlgebraic[((a + b*x^2)^5*(A + B*x^2))/x^16,x]","\frac{-3003 a^5 A-3465 a^5 B x^2-17325 a^4 A b x^2-20475 a^4 b B x^4-40950 a^3 A b^2 x^4-50050 a^3 b^2 B x^6-50050 a^2 A b^3 x^6-64350 a^2 b^3 B x^8-32175 a A b^4 x^8-45045 a b^4 B x^{10}-9009 A b^5 x^{10}-15015 b^5 B x^{12}}{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,"(-3003*a^5*A - 17325*a^4*A*b*x^2 - 3465*a^5*B*x^2 - 40950*a^3*A*b^2*x^4 - 20475*a^4*b*B*x^4 - 50050*a^2*A*b^3*x^6 - 50050*a^3*b^2*B*x^6 - 32175*a*A*b^4*x^8 - 64350*a^2*b^3*B*x^8 - 9009*A*b^5*x^10 - 45045*a*b^4*B*x^10 - 15015*b^5*B*x^12)/(45045*x^15)","A",1
49,1,129,76,0.0457407,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^{17}} \, dx","IntegrateAlgebraic[((a + b*x^2)^5*(A + B*x^2))/x^17,x]","\frac{-21 a^5 A-24 a^5 B x^2-120 a^4 A b x^2-140 a^4 b B x^4-280 a^3 A b^2 x^4-336 a^3 b^2 B x^6-336 a^2 A b^3 x^6-420 a^2 b^3 B x^8-210 a A b^4 x^8-280 a b^4 B x^{10}-56 A b^5 x^{10}-84 b^5 B x^{12}}{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,"(-21*a^5*A - 120*a^4*A*b*x^2 - 24*a^5*B*x^2 - 280*a^3*A*b^2*x^4 - 140*a^4*b*B*x^4 - 336*a^2*A*b^3*x^6 - 336*a^3*b^2*B*x^6 - 210*a*A*b^4*x^8 - 420*a^2*b^3*B*x^8 - 56*A*b^5*x^10 - 280*a*b^4*B*x^10 - 84*b^5*B*x^12)/(336*x^16)","A",1
50,1,129,117,0.0631582,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^{18}} \, dx","IntegrateAlgebraic[((a + b*x^2)^5*(A + B*x^2))/x^18,x]","\frac{-45045 a^5 A-51051 a^5 B x^2-255255 a^4 A b x^2-294525 a^4 b B x^4-589050 a^3 A b^2 x^4-696150 a^3 b^2 B x^6-696150 a^2 A b^3 x^6-850850 a^2 b^3 B x^8-425425 a A b^4 x^8-546975 a b^4 B x^{10}-109395 A b^5 x^{10}-153153 b^5 B x^{12}}{765765 x^{17}}","-\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,"(-45045*a^5*A - 255255*a^4*A*b*x^2 - 51051*a^5*B*x^2 - 589050*a^3*A*b^2*x^4 - 294525*a^4*b*B*x^4 - 696150*a^2*A*b^3*x^6 - 696150*a^3*b^2*B*x^6 - 425425*a*A*b^4*x^8 - 850850*a^2*b^3*B*x^8 - 109395*A*b^5*x^10 - 546975*a*b^4*B*x^10 - 153153*b^5*B*x^12)/(765765*x^17)","A",1
51,1,129,117,0.0460835,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^{19}} \, dx","IntegrateAlgebraic[((a + b*x^2)^5*(A + B*x^2))/x^19,x]","\frac{-56 a^5 A-63 a^5 B x^2-315 a^4 A b x^2-360 a^4 b B x^4-720 a^3 A b^2 x^4-840 a^3 b^2 B x^6-840 a^2 A b^3 x^6-1008 a^2 b^3 B x^8-504 a A b^4 x^8-630 a b^4 B x^{10}-126 A b^5 x^{10}-168 b^5 B x^{12}}{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,"(-56*a^5*A - 315*a^4*A*b*x^2 - 63*a^5*B*x^2 - 720*a^3*A*b^2*x^4 - 360*a^4*b*B*x^4 - 840*a^2*A*b^3*x^6 - 840*a^3*b^2*B*x^6 - 504*a*A*b^4*x^8 - 1008*a^2*b^3*B*x^8 - 126*A*b^5*x^10 - 630*a*b^4*B*x^10 - 168*b^5*B*x^12)/(1008*x^18)","A",1
52,1,129,117,0.0580992,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^{20}} \, dx","IntegrateAlgebraic[((a + b*x^2)^5*(A + B*x^2))/x^20,x]","\frac{-153153 a^5 A-171171 a^5 B x^2-855855 a^4 A b x^2-969969 a^4 b B x^4-1939938 a^3 A b^2 x^4-2238390 a^3 b^2 B x^6-2238390 a^2 A b^3 x^6-2645370 a^2 b^3 B x^8-1322685 a A b^4 x^8-1616615 a b^4 B x^{10}-323323 A b^5 x^{10}-415701 b^5 B x^{12}}{2909907 x^{19}}","-\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,"(-153153*a^5*A - 855855*a^4*A*b*x^2 - 171171*a^5*B*x^2 - 1939938*a^3*A*b^2*x^4 - 969969*a^4*b*B*x^4 - 2238390*a^2*A*b^3*x^6 - 2238390*a^3*b^2*B*x^6 - 1322685*a*A*b^4*x^8 - 2645370*a^2*b^3*B*x^8 - 323323*A*b^5*x^10 - 1616615*a*b^4*B*x^10 - 415701*b^5*B*x^12)/(2909907*x^19)","A",1
53,1,129,117,0.0464384,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^{21}} \, dx","IntegrateAlgebraic[((a + b*x^2)^5*(A + B*x^2))/x^21,x]","\frac{-252 a^5 A-280 a^5 B x^2-1400 a^4 A b x^2-1575 a^4 b B x^4-3150 a^3 A b^2 x^4-3600 a^3 b^2 B x^6-3600 a^2 A b^3 x^6-4200 a^2 b^3 B x^8-2100 a A b^4 x^8-2520 a b^4 B x^{10}-504 A b^5 x^{10}-630 b^5 B x^{12}}{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,"(-252*a^5*A - 1400*a^4*A*b*x^2 - 280*a^5*B*x^2 - 3150*a^3*A*b^2*x^4 - 1575*a^4*b*B*x^4 - 3600*a^2*A*b^3*x^6 - 3600*a^3*b^2*B*x^6 - 2100*a*A*b^4*x^8 - 4200*a^2*b^3*B*x^8 - 504*A*b^5*x^10 - 2520*a*b^4*B*x^10 - 630*b^5*B*x^12)/(5040*x^20)","A",1
54,1,129,117,0.0598579,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^{22}} \, dx","IntegrateAlgebraic[((a + b*x^2)^5*(A + B*x^2))/x^22,x]","\frac{-138567 a^5 A-153153 a^5 B x^2-765765 a^4 A b x^2-855855 a^4 b B x^4-1711710 a^3 A b^2 x^4-1939938 a^3 b^2 B x^6-1939938 a^2 A b^3 x^6-2238390 a^2 b^3 B x^8-1119195 a A b^4 x^8-1322685 a b^4 B x^{10}-264537 A b^5 x^{10}-323323 b^5 B x^{12}}{2909907 x^{21}}","-\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,"(-138567*a^5*A - 765765*a^4*A*b*x^2 - 153153*a^5*B*x^2 - 1711710*a^3*A*b^2*x^4 - 855855*a^4*b*B*x^4 - 1939938*a^2*A*b^3*x^6 - 1939938*a^3*b^2*B*x^6 - 1119195*a*A*b^4*x^8 - 2238390*a^2*b^3*B*x^8 - 264537*A*b^5*x^10 - 1322685*a*b^4*B*x^10 - 323323*b^5*B*x^12)/(2909907*x^21)","A",1
55,1,129,117,0.0458674,"\int \frac{\left(a+b x^2\right)^5 \left(A+B x^2\right)}{x^{23}} \, dx","IntegrateAlgebraic[((a + b*x^2)^5*(A + B*x^2))/x^23,x]","\frac{-1260 a^5 A-1386 a^5 B x^2-6930 a^4 A b x^2-7700 a^4 b B x^4-15400 a^3 A b^2 x^4-17325 a^3 b^2 B x^6-17325 a^2 A b^3 x^6-19800 a^2 b^3 B x^8-9900 a A b^4 x^8-11550 a b^4 B x^{10}-2310 A b^5 x^{10}-2772 b^5 B x^{12}}{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,"(-1260*a^5*A - 6930*a^4*A*b*x^2 - 1386*a^5*B*x^2 - 15400*a^3*A*b^2*x^4 - 7700*a^4*b*B*x^4 - 17325*a^2*A*b^3*x^6 - 17325*a^3*b^2*B*x^6 - 9900*a*A*b^4*x^8 - 19800*a^2*b^3*B*x^8 - 2310*A*b^5*x^10 - 11550*a*b^4*B*x^10 - 2772*b^5*B*x^12)/(27720*x^22)","A",1
56,1,108,98,0.0977977,"\int \frac{x^6 \left(A+B x^2\right)}{a+b x^2} \, dx","IntegrateAlgebraic[(x^6*(A + B*x^2))/(a + b*x^2),x]","\frac{\left(a^{7/2} B-a^{5/2} A b\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{9/2}}+\frac{-105 a^3 B x+105 a^2 A b x+35 a^2 b B x^3-35 a A b^2 x^3-21 a b^2 B x^5+21 A b^3 x^5+15 b^3 B x^7}{105 b^4}","-\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,"(105*a^2*A*b*x - 105*a^3*B*x - 35*a*A*b^2*x^3 + 35*a^2*b*B*x^3 + 21*A*b^3*x^5 - 21*a*b^2*B*x^5 + 15*b^3*B*x^7)/(105*b^4) + ((-(a^(5/2)*A*b) + a^(7/2)*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(9/2)","A",1
57,1,77,75,0.0519179,"\int \frac{x^5 \left(A+B x^2\right)}{a+b x^2} \, dx","IntegrateAlgebraic[(x^5*(A + B*x^2))/(a + b*x^2),x]","\frac{x^2 \left(6 a^2 B-6 a A b-3 a b B x^2+3 A b^2 x^2+2 b^2 B x^4\right)}{12 b^3}+\frac{\left(a^2 A b-a^3 B\right) \log \left(a+b x^2\right)}{2 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,"(x^2*(-6*a*A*b + 6*a^2*B + 3*A*b^2*x^2 - 3*a*b*B*x^2 + 2*b^2*B*x^4))/(12*b^3) + ((a^2*A*b - a^3*B)*Log[a + b*x^2])/(2*b^4)","A",1
58,1,83,77,0.0715871,"\int \frac{x^4 \left(A+B x^2\right)}{a+b x^2} \, dx","IntegrateAlgebraic[(x^4*(A + B*x^2))/(a + b*x^2),x]","\frac{\left(a^{3/2} A b-a^{5/2} B\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{7/2}}+\frac{x \left(15 a^2 B-15 a A b-5 a b B x^2+5 A b^2 x^2+3 b^2 B x^4\right)}{15 b^3}","\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,"(x*(-15*a*A*b + 15*a^2*B + 5*A*b^2*x^2 - 5*a*b*B*x^2 + 3*b^2*B*x^4))/(15*b^3) + ((a^(3/2)*A*b - a^(5/2)*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(7/2)","A",1
59,1,52,54,0.0317869,"\int \frac{x^3 \left(A+B x^2\right)}{a+b x^2} \, dx","IntegrateAlgebraic[(x^3*(A + B*x^2))/(a + b*x^2),x]","\frac{\left(a^2 B-a A b\right) \log \left(a+b x^2\right)}{2 b^3}+\frac{x^2 \left(-2 a B+2 A b+b B x^2\right)}{4 b^2}","-\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,"(x^2*(2*A*b - 2*a*B + b*B*x^2))/(4*b^2) + ((-(a*A*b) + a^2*B)*Log[a + b*x^2])/(2*b^3)","A",1
60,1,60,58,0.0610374,"\int \frac{x^2 \left(A+B x^2\right)}{a+b x^2} \, dx","IntegrateAlgebraic[(x^2*(A + B*x^2))/(a + b*x^2),x]","\frac{\left(a^{3/2} B-\sqrt{a} A b\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{5/2}}+\frac{x \left(-3 a B+3 A b+b B x^2\right)}{3 b^2}","-\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,"(x*(3*A*b - 3*a*B + b*B*x^2))/(3*b^2) + ((-(Sqrt[a]*A*b) + a^(3/2)*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(5/2)","A",1
61,1,35,35,0.0179966,"\int \frac{x \left(A+B x^2\right)}{a+b x^2} \, dx","IntegrateAlgebraic[(x*(A + B*x^2))/(a + b*x^2),x]","\frac{(A b-a B) \log \left(a+b x^2\right)}{2 b^2}+\frac{B x^2}{2 b}","\frac{(A b-a B) \log \left(a+b x^2\right)}{2 b^2}+\frac{B x^2}{2 b}",1,"(B*x^2)/(2*b) + ((A*b - a*B)*Log[a + b*x^2])/(2*b^2)","A",1
62,1,39,39,0.0341232,"\int \frac{A+B x^2}{a+b x^2} \, dx","IntegrateAlgebraic[(A + B*x^2)/(a + b*x^2),x]","\frac{(A b-a B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} b^{3/2}}+\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{B x}{b}",1,"(B*x)/b + ((A*b - a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*b^(3/2))","A",1
63,1,34,34,0.0205718,"\int \frac{A+B x^2}{x \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(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
64,1,42,43,0.0368154,"\int \frac{A+B x^2}{x^2 \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(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
65,1,49,50,0.032274,"\int \frac{A+B x^2}{x^3 \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(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
66,1,66,59,0.0708506,"\int \frac{A+B x^2}{x^4 \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^4*(a + b*x^2)),x]","\frac{\left(A b^{3/2}-a \sqrt{b} B\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{5/2}}+\frac{-a A-3 a B x^2+3 A b x^2}{3 a^2 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,"(-(a*A) + 3*A*b*x^2 - 3*a*B*x^2)/(3*a^2*x^3) + ((A*b^(3/2) - a*Sqrt[b]*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(5/2)","A",1
67,1,73,69,0.049446,"\int \frac{A+B x^2}{x^5 \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^5*(a + b*x^2)),x]","\frac{\left(a b B-A b^2\right) \log \left(a+b x^2\right)}{2 a^3}+\frac{\log (x) \left(A b^2-a b B\right)}{a^3}+\frac{-a A-2 a B x^2+2 A b x^2}{4 a^2 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) + 2*A*b*x^2 - 2*a*B*x^2)/(4*a^2*x^4) + ((A*b^2 - a*b*B)*Log[x])/a^3 + ((-(A*b^2) + a*b*B)*Log[a + b*x^2])/(2*a^3)","A",1
68,1,88,80,0.0759823,"\int \frac{A+B x^2}{x^6 \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^6*(a + b*x^2)),x]","\frac{\left(a b^{3/2} B-A b^{5/2}\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{7/2}}+\frac{-3 a^2 A-5 a^2 B x^2+5 a A b x^2+15 a b B x^4-15 A b^2 x^4}{15 a^3 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,"(-3*a^2*A + 5*a*A*b*x^2 - 5*a^2*B*x^2 - 15*A*b^2*x^4 + 15*a*b*B*x^4)/(15*a^3*x^5) + ((-(A*b^(5/2)) + a*b^(3/2)*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(7/2)","A",1
69,1,99,93,0.0634009,"\int \frac{A+B x^2}{x^7 \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(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{-2 a^2 A-3 a^2 B x^2+3 a A b x^2+6 a b B x^4-6 A b^2 x^4}{12 a^3 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,"(-2*a^2*A + 3*a*A*b*x^2 - 3*a^2*B*x^2 - 6*A*b^2*x^4 + 6*a*b*B*x^4)/(12*a^3*x^6) + ((-(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
70,1,112,99,0.0958942,"\int \frac{A+B x^2}{x^8 \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^8*(a + b*x^2)),x]","\frac{\left(A b^{7/2}-a b^{5/2} B\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{9/2}}+\frac{-15 a^3 A-21 a^3 B x^2+21 a^2 A b x^2+35 a^2 b B x^4-35 a A b^2 x^4-105 a b^2 B x^6+105 A b^3 x^6}{105 a^4 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,"(-15*a^3*A + 21*a^2*A*b*x^2 - 21*a^3*B*x^2 - 35*a*A*b^2*x^4 + 35*a^2*b*B*x^4 + 105*A*b^3*x^6 - 105*a*b^2*B*x^6)/(105*a^4*x^7) + ((A*b^(7/2) - a*b^(5/2)*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(9/2)","A",1
71,1,156,126,0.1132342,"\int \frac{x^9 \left(A+B x^2\right)}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^9*(A + B*x^2))/(a + b*x^2)^2,x]","\frac{\left(5 a^4 B-4 a^3 A b\right) \log \left(a+b x^2\right)}{2 b^6}+\frac{12 a^5 B-12 a^4 A b-48 a^4 b B x^2+36 a^3 A b^2 x^2-30 a^3 b^2 B x^4+24 a^2 A b^3 x^4+10 a^2 b^3 B x^6-8 a A b^4 x^6-5 a b^4 B x^8+4 A b^5 x^8+3 b^5 B x^{10}}{24 b^6 \left(a+b x^2\right)}","-\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^4*A*b + 12*a^5*B + 36*a^3*A*b^2*x^2 - 48*a^4*b*B*x^2 + 24*a^2*A*b^3*x^4 - 30*a^3*b^2*B*x^4 - 8*a*A*b^4*x^6 + 10*a^2*b^3*B*x^6 + 4*A*b^5*x^8 - 5*a*b^4*B*x^8 + 3*b^5*B*x^10)/(24*b^6*(a + b*x^2)) + ((-4*a^3*A*b + 5*a^4*B)*Log[a + b*x^2])/(2*b^6)","A",1
72,1,145,131,0.1426411,"\int \frac{x^8 \left(A+B x^2\right)}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^8*(A + B*x^2))/(a + b*x^2)^2,x]","\frac{\left(9 a^{7/2} B-7 a^{5/2} A b\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{11/2}}+\frac{-945 a^4 B x+735 a^3 A b x-630 a^3 b B x^3+490 a^2 A b^2 x^3+126 a^2 b^2 B x^5-98 a A b^3 x^5-54 a b^3 B x^7+42 A b^4 x^7+30 b^4 B x^9}{210 b^5 \left(a+b x^2\right)}","-\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,"(735*a^3*A*b*x - 945*a^4*B*x + 490*a^2*A*b^2*x^3 - 630*a^3*b*B*x^3 - 98*a*A*b^3*x^5 + 126*a^2*b^2*B*x^5 + 42*A*b^4*x^7 - 54*a*b^3*B*x^7 + 30*b^4*B*x^9)/(210*b^5*(a + b*x^2)) + ((-7*a^(5/2)*A*b + 9*a^(7/2)*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(11/2))","A",1
73,1,132,104,0.0970581,"\int \frac{x^7 \left(A+B x^2\right)}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^7*(A + B*x^2))/(a + b*x^2)^2,x]","\frac{\left(3 a^2 A b-4 a^3 B\right) \log \left(a+b x^2\right)}{2 b^5}+\frac{-6 a^4 B+6 a^3 A b+18 a^3 b B x^2-12 a^2 A b^2 x^2+12 a^2 b^2 B x^4-9 a A b^3 x^4-4 a b^3 B x^6+3 A b^4 x^6+2 b^4 B x^8}{12 b^5 \left(a+b x^2\right)}","\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^3*A*b - 6*a^4*B - 12*a^2*A*b^2*x^2 + 18*a^3*b*B*x^2 - 9*a*A*b^3*x^4 + 12*a^2*b^2*B*x^4 + 3*A*b^4*x^6 - 4*a*b^3*B*x^6 + 2*b^4*B*x^8)/(12*b^5*(a + b*x^2)) + ((3*a^2*A*b - 4*a^3*B)*Log[a + b*x^2])/(2*b^5)","A",1
74,1,120,110,0.1156946,"\int \frac{x^6 \left(A+B x^2\right)}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^6*(A + B*x^2))/(a + b*x^2)^2,x]","\frac{\left(5 a^{3/2} A b-7 a^{5/2} B\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{9/2}}+\frac{x \left(105 a^3 B-75 a^2 A b+70 a^2 b B x^2-50 a A b^2 x^2-14 a b^2 B x^4+10 A b^3 x^4+6 b^3 B x^6\right)}{30 b^4 \left(a+b x^2\right)}","\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,"(x*(-75*a^2*A*b + 105*a^3*B - 50*a*A*b^2*x^2 + 70*a^2*b*B*x^2 + 10*A*b^3*x^4 - 14*a*b^2*B*x^4 + 6*b^3*B*x^6))/(30*b^4*(a + b*x^2)) + ((5*a^(3/2)*A*b - 7*a^(5/2)*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(9/2))","A",1
75,1,105,82,0.0878744,"\int \frac{x^5 \left(A+B x^2\right)}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^5*(A + B*x^2))/(a + b*x^2)^2,x]","\frac{\left(3 a^2 B-2 a A b\right) \log \left(a+b x^2\right)}{2 b^4}+\frac{2 a^3 B-2 a^2 A b-4 a^2 b B x^2+2 a A b^2 x^2-3 a b^2 B x^4+2 A b^3 x^4+b^3 B x^6}{4 b^4 \left(a+b x^2\right)}","-\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*a^2*A*b + 2*a^3*B + 2*a*A*b^2*x^2 - 4*a^2*b*B*x^2 + 2*A*b^3*x^4 - 3*a*b^2*B*x^4 + b^3*B*x^6)/(4*b^4*(a + b*x^2)) + ((-2*a*A*b + 3*a^2*B)*Log[a + b*x^2])/(2*b^4)","A",1
76,1,97,87,0.0959005,"\int \frac{x^4 \left(A+B x^2\right)}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^4*(A + B*x^2))/(a + b*x^2)^2,x]","\frac{\left(5 a^{3/2} B-3 \sqrt{a} A b\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{7/2}}+\frac{-15 a^2 B x+9 a A b x-10 a b B x^3+6 A b^2 x^3+2 b^2 B x^5}{6 b^3 \left(a+b x^2\right)}","-\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,"(9*a*A*b*x - 15*a^2*B*x + 6*A*b^2*x^3 - 10*a*b*B*x^3 + 2*b^2*B*x^5)/(6*b^3*(a + b*x^2)) + ((-3*Sqrt[a]*A*b + 5*a^(3/2)*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(7/2))","A",1
77,1,66,60,0.052786,"\int \frac{x^3 \left(A+B x^2\right)}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^3*(A + B*x^2))/(a + b*x^2)^2,x]","\frac{-a^2 B+a A b+a b B x^2+b^2 B x^4}{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{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,"(a*A*b - a^2*B + a*b*B*x^2 + b^2*B*x^4)/(2*b^3*(a + b*x^2)) + ((A*b - 2*a*B)*Log[a + b*x^2])/(2*b^3)","A",1
78,1,69,67,0.0870111,"\int \frac{x^2 \left(A+B x^2\right)}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^2*(A + B*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 \sqrt{a} b^{5/2}}+\frac{x \left(3 a B-A b+2 b B x^2\right)}{2 b^2 \left(a+b x^2\right)}","\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,"(x*(-(A*b) + 3*a*B + 2*b*B*x^2))/(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
79,1,41,41,0.0219854,"\int \frac{x \left(A+B x^2\right)}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(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
80,1,63,63,0.0636847,"\int \frac{A+B x^2}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(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
81,1,51,51,0.0419676,"\int \frac{A+B x^2}{x \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x*(a + b*x^2)^2),x]","-\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)}","-\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*b - a*B)/(2*a*b*(a + b*x^2)) + (A*Log[x])/a^2 - (A*Log[a + b*x^2])/(2*a^2)","A",1
82,1,73,71,0.051321,"\int \frac{A+B x^2}{x^2 \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(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{-2 a A+a B x^2-3 A b x^2}{2 a^2 x \left(a+b x^2\right)}","-\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,"(-2*a*A - 3*A*b*x^2 + a*B*x^2)/(2*a^2*x*(a + b*x^2)) + ((-3*A*b + a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(5/2)*Sqrt[b])","A",1
83,1,76,76,0.0684713,"\int \frac{A+B x^2}{x^3 \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^3*(a + b*x^2)^2),x]","\frac{(2 A b-a B) \log \left(a+b x^2\right)}{2 a^3}+\frac{\log (x) (a B-2 A b)}{a^3}+\frac{-a A+a B x^2-2 A b x^2}{2 a^2 x^2 \left(a+b x^2\right)}","\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) - 2*A*b*x^2 + a*B*x^2)/(2*a^2*x^2*(a + b*x^2)) + ((-2*A*b + a*B)*Log[x])/a^3 + ((2*A*b - a*B)*Log[a + b*x^2])/(2*a^3)","A",1
84,1,101,90,0.1006081,"\int \frac{A+B x^2}{x^4 \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^4*(a + b*x^2)^2),x]","\frac{\left(5 A b^{3/2}-3 a \sqrt{b} B\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{7/2}}+\frac{-2 a^2 A-6 a^2 B x^2+10 a A b x^2-9 a b B x^4+15 A b^2 x^4}{6 a^3 x^3 \left(a+b x^2\right)}","\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,"(-2*a^2*A + 10*a*A*b*x^2 - 6*a^2*B*x^2 + 15*A*b^2*x^4 - 9*a*b*B*x^4)/(6*a^3*x^3*(a + b*x^2)) + ((5*A*b^(3/2) - 3*a*Sqrt[b]*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(7/2))","A",1
85,1,106,97,0.1303036,"\int \frac{A+B x^2}{x^5 \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^5*(a + b*x^2)^2),x]","\frac{\left(2 a b B-3 A b^2\right) \log \left(a+b x^2\right)}{2 a^4}+\frac{\log (x) \left(3 A b^2-2 a b B\right)}{a^4}+\frac{-a^2 A-2 a^2 B x^2+3 a A b x^2-4 a b B x^4+6 A b^2 x^4}{4 a^3 x^4 \left(a+b x^2\right)}","-\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,"(-(a^2*A) + 3*a*A*b*x^2 - 2*a^2*B*x^2 + 6*A*b^2*x^4 - 4*a*b*B*x^4)/(4*a^3*x^4*(a + b*x^2)) + ((3*A*b^2 - 2*a*b*B)*Log[x])/a^4 + ((-3*A*b^2 + 2*a*b*B)*Log[a + b*x^2])/(2*a^4)","A",1
86,1,125,113,0.1090225,"\int \frac{A+B x^2}{x^6 \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^6*(a + b*x^2)^2),x]","\frac{\left(5 a b^{3/2} B-7 A b^{5/2}\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{9/2}}+\frac{-6 a^3 A-10 a^3 B x^2+14 a^2 A b x^2+50 a^2 b B x^4-70 a A b^2 x^4+75 a b^2 B x^6-105 A b^3 x^6}{30 a^4 x^5 \left(a+b x^2\right)}","-\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,"(-6*a^3*A + 14*a^2*A*b*x^2 - 10*a^3*B*x^2 - 70*a*A*b^2*x^4 + 50*a^2*b*B*x^4 - 105*A*b^3*x^6 + 75*a*b^2*B*x^6)/(30*a^4*x^5*(a + b*x^2)) + ((-7*A*b^(5/2) + 5*a*b^(3/2)*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(9/2))","A",1
87,1,134,124,0.1324333,"\int \frac{A+B x^2}{x^7 \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^7*(a + b*x^2)^2),x]","\frac{\left(4 A b^3-3 a b^2 B\right) \log \left(a+b x^2\right)}{2 a^5}+\frac{\log (x) \left(3 a b^2 B-4 A b^3\right)}{a^5}+\frac{-2 a^3 A-3 a^3 B x^2+4 a^2 A b x^2+9 a^2 b B x^4-12 a A b^2 x^4+18 a b^2 B x^6-24 A b^3 x^6}{12 a^4 x^6 \left(a+b x^2\right)}","\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 + 4*a^2*A*b*x^2 - 3*a^3*B*x^2 - 12*a*A*b^2*x^4 + 9*a^2*b*B*x^4 - 24*A*b^3*x^6 + 18*a*b^2*B*x^6)/(12*a^4*x^6*(a + b*x^2)) + ((-4*A*b^3 + 3*a*b^2*B)*Log[x])/a^5 + ((4*A*b^3 - 3*a*b^2*B)*Log[a + b*x^2])/(2*a^5)","A",1
88,1,180,150,0.1262883,"\int \frac{x^{11} \left(A+B x^2\right)}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(x^11*(A + B*x^2))/(a + b*x^2)^3,x]","\frac{5 \left(3 a^4 B-2 a^3 A b\right) \log \left(a+b x^2\right)}{2 b^7}+\frac{66 a^6 B-54 a^5 A b-48 a^5 b B x^2+12 a^4 A b^2 x^2-204 a^4 b^2 B x^4+126 a^3 A b^3 x^4-60 a^3 b^3 B x^6+40 a^2 A b^4 x^6+15 a^2 b^4 B x^8-10 a A b^5 x^8-6 a b^5 B x^{10}+4 A b^6 x^{10}+3 b^6 B x^{12}}{24 b^7 \left(a+b x^2\right)^2}","\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,"(-54*a^5*A*b + 66*a^6*B + 12*a^4*A*b^2*x^2 - 48*a^5*b*B*x^2 + 126*a^3*A*b^3*x^4 - 204*a^4*b^2*B*x^4 + 40*a^2*A*b^4*x^6 - 60*a^3*b^3*B*x^6 - 10*a*A*b^5*x^8 + 15*a^2*b^4*B*x^8 + 4*A*b^6*x^10 - 6*a*b^5*B*x^10 + 3*b^6*B*x^12)/(24*b^7*(a + b*x^2)^2) + (5*(-2*a^3*A*b + 3*a^4*B)*Log[a + b*x^2])/(2*b^7)","A",1
89,1,153,128,0.1112255,"\int \frac{x^9 \left(A+B x^2\right)}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(x^9*(A + B*x^2))/(a + b*x^2)^3,x]","\frac{\left(3 a^2 A b-5 a^3 B\right) \log \left(a+b x^2\right)}{b^6}+\frac{-27 a^5 B+21 a^4 A b+6 a^4 b B x^2+6 a^3 A b^2 x^2+63 a^3 b^2 B x^4-33 a^2 A b^3 x^4+20 a^2 b^3 B x^6-12 a A b^4 x^6-5 a b^4 B x^8+3 A b^5 x^8+2 b^5 B x^{10}}{12 b^6 \left(a+b x^2\right)^2}","-\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,"(21*a^4*A*b - 27*a^5*B + 6*a^3*A*b^2*x^2 + 6*a^4*b*B*x^2 - 33*a^2*A*b^3*x^4 + 63*a^3*b^2*B*x^4 - 12*a*A*b^4*x^6 + 20*a^2*b^3*B*x^6 + 3*A*b^5*x^8 - 5*a*b^4*B*x^8 + 2*b^5*B*x^10)/(12*b^6*(a + b*x^2)^2) + ((3*a^2*A*b - 5*a^3*B)*Log[a + b*x^2])/b^6","A",1
90,1,129,109,0.0962124,"\int \frac{x^7 \left(A+B x^2\right)}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(x^7*(A + B*x^2))/(a + b*x^2)^3,x]","\frac{3 \left(2 a^2 B-a A b\right) \log \left(a+b x^2\right)}{2 b^5}+\frac{7 a^4 B-5 a^3 A b+2 a^3 b B x^2-4 a^2 A b^2 x^2-11 a^2 b^2 B x^4+4 a A b^3 x^4-4 a b^3 B x^6+2 A b^4 x^6+b^4 B x^8}{4 b^5 \left(a+b x^2\right)^2}","\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,"(-5*a^3*A*b + 7*a^4*B - 4*a^2*A*b^2*x^2 + 2*a^3*b*B*x^2 + 4*a*A*b^3*x^4 - 11*a^2*b^2*B*x^4 + 2*A*b^4*x^6 - 4*a*b^3*B*x^6 + b^4*B*x^8)/(4*b^5*(a + b*x^2)^2) + (3*(-(a*A*b) + 2*a^2*B)*Log[a + b*x^2])/(2*b^5)","A",1
91,1,93,88,0.0636601,"\int \frac{x^5 \left(A+B x^2\right)}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(x^5*(A + B*x^2))/(a + b*x^2)^3,x]","\frac{-5 a^3 B+3 a^2 A b-4 a^2 b B x^2+4 a A b^2 x^2+4 a b^2 B x^4+2 b^3 B x^6}{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{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,"(3*a^2*A*b - 5*a^3*B + 4*a*A*b^2*x^2 - 4*a^2*b*B*x^2 + 4*a*b^2*B*x^4 + 2*b^3*B*x^6)/(4*b^4*(a + b*x^2)^2) + ((A*b - 3*a*B)*Log[a + b*x^2])/(2*b^4)","A",1
92,1,62,66,0.0432432,"\int \frac{x^3 \left(A+B x^2\right)}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(x^3*(A + B*x^2))/(a + b*x^2)^3,x]","\frac{3 a^2 B-a A b+4 a b B x^2-2 A b^2 x^2}{4 b^3 \left(a+b x^2\right)^2}+\frac{B \log \left(a+b x^2\right)}{2 b^3}","-\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,"(-(a*A*b) + 3*a^2*B - 2*A*b^2*x^2 + 4*a*b*B*x^2)/(4*b^3*(a + b*x^2)^2) + (B*Log[a + b*x^2])/(2*b^3)","A",1
93,1,32,32,0.0211277,"\int \frac{x \left(A+B x^2\right)}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(x*(A + B*x^2))/(a + b*x^2)^3,x]","\frac{-a B-A b-2 b B x^2}{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,"(-(A*b) - a*B - 2*b*B*x^2)/(4*b^2*(a + b*x^2)^2)","A",1
94,1,64,68,0.0621047,"\int \frac{A+B x^2}{x \left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x*(a + b*x^2)^3),x]","-\frac{A \log \left(a+b x^2\right)}{2 a^3}+\frac{A \log (x)}{a^3}+\frac{a^2 (-B)+3 a A b+2 A b^2 x^2}{4 a^2 b \left(a+b x^2\right)^2}","-\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,"(3*a*A*b - a^2*B + 2*A*b^2*x^2)/(4*a^2*b*(a + b*x^2)^2) + (A*Log[x])/a^3 - (A*Log[a + b*x^2])/(2*a^3)","A",1
95,1,99,101,0.0844434,"\int \frac{A+B x^2}{x^3 \left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^3*(a + b*x^2)^3),x]","\frac{(3 A b-a B) \log \left(a+b x^2\right)}{2 a^4}+\frac{\log (x) (a B-3 A b)}{a^4}+\frac{-2 a^2 A+3 a^2 B x^2-9 a A b x^2+2 a b B x^4-6 A b^2 x^4}{4 a^3 x^2 \left(a+b x^2\right)^2}","\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^2*A - 9*a*A*b*x^2 + 3*a^2*B*x^2 - 6*A*b^2*x^4 + 2*a*b*B*x^4)/(4*a^3*x^2*(a + b*x^2)^2) + ((-3*A*b + a*B)*Log[x])/a^4 + ((3*A*b - a*B)*Log[a + b*x^2])/(2*a^4)","A",1
96,1,129,124,0.1131937,"\int \frac{A+B x^2}{x^5 \left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^5*(a + b*x^2)^3),x]","\frac{3 \left(a b B-2 A b^2\right) \log \left(a+b x^2\right)}{2 a^5}-\frac{3 \log (x) \left(a b B-2 A b^2\right)}{a^5}+\frac{-a^3 A-2 a^3 B x^2+4 a^2 A b x^2-9 a^2 b B x^4+18 a A b^2 x^4-6 a b^2 B x^6+12 A b^3 x^6}{4 a^4 x^4 \left(a+b x^2\right)^2}","-\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^3*A) + 4*a^2*A*b*x^2 - 2*a^3*B*x^2 + 18*a*A*b^2*x^4 - 9*a^2*b*B*x^4 + 12*A*b^3*x^6 - 6*a*b^2*B*x^6)/(4*a^4*x^4*(a + b*x^2)^2) - (3*(-2*A*b^2 + a*b*B)*Log[x])/a^5 + (3*(-2*A*b^2 + a*b*B)*Log[a + b*x^2])/(2*a^5)","A",1
97,1,156,149,0.1645289,"\int \frac{A+B x^2}{x^7 \left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^7*(a + b*x^2)^3),x]","\frac{\left(5 A b^3-3 a b^2 B\right) \log \left(a+b x^2\right)}{a^6}+\frac{2 \log (x) \left(3 a b^2 B-5 A b^3\right)}{a^6}+\frac{-2 a^4 A-3 a^4 B x^2+5 a^3 A b x^2+12 a^3 b B x^4-20 a^2 A b^2 x^4+54 a^2 b^2 B x^6-90 a A b^3 x^6+36 a b^3 B x^8-60 A b^4 x^8}{12 a^5 x^6 \left(a+b x^2\right)^2}","\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^4*A + 5*a^3*A*b*x^2 - 3*a^4*B*x^2 - 20*a^2*A*b^2*x^4 + 12*a^3*b*B*x^4 - 90*a*A*b^3*x^6 + 54*a^2*b^2*B*x^6 - 60*A*b^4*x^8 + 36*a*b^3*B*x^8)/(12*a^5*x^6*(a + b*x^2)^2) + (2*(-5*A*b^3 + 3*a*b^2*B)*Log[x])/a^6 + ((5*A*b^3 - 3*a*b^2*B)*Log[a + b*x^2])/a^6","A",1
98,1,169,158,0.1379904,"\int \frac{x^{10} \left(A+B x^2\right)}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(x^10*(A + B*x^2))/(a + b*x^2)^3,x]","\frac{9 \left(11 a^{7/2} B-7 a^{5/2} A b\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 b^{13/2}}+\frac{-3465 a^5 B x+2205 a^4 A b x-5775 a^4 b B x^3+3675 a^3 A b^2 x^3-1848 a^3 b^2 B x^5+1176 a^2 A b^3 x^5+264 a^2 b^3 B x^7-168 a A b^4 x^7-88 a b^4 B x^9+56 A b^5 x^9+40 b^5 B x^{11}}{280 b^6 \left(a+b x^2\right)^2}","-\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,"(2205*a^4*A*b*x - 3465*a^5*B*x + 3675*a^3*A*b^2*x^3 - 5775*a^4*b*B*x^3 + 1176*a^2*A*b^3*x^5 - 1848*a^3*b^2*B*x^5 - 168*a*A*b^4*x^7 + 264*a^2*b^3*B*x^7 + 56*A*b^5*x^9 - 88*a*b^4*B*x^9 + 40*b^5*B*x^11)/(280*b^6*(a + b*x^2)^2) + (9*(-7*a^(5/2)*A*b + 11*a^(7/2)*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*b^(13/2))","A",1
99,1,144,138,0.1412963,"\int \frac{x^8 \left(A+B x^2\right)}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(x^8*(A + B*x^2))/(a + b*x^2)^3,x]","\frac{x \left(945 a^4 B-525 a^3 A b+1575 a^3 b B x^2-875 a^2 A b^2 x^2+504 a^2 b^2 B x^4-280 a A b^3 x^4-72 a b^3 B x^6+40 A b^4 x^6+24 b^4 B x^8\right)}{120 b^5 \left(a+b x^2\right)^2}-\frac{7 \left(9 a^{5/2} B-5 a^{3/2} A b\right) \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*(-525*a^3*A*b + 945*a^4*B - 875*a^2*A*b^2*x^2 + 1575*a^3*b*B*x^2 - 280*a*A*b^3*x^4 + 504*a^2*b^2*B*x^4 + 40*A*b^4*x^6 - 72*a*b^3*B*x^6 + 24*b^4*B*x^8))/(120*b^5*(a + b*x^2)^2) - (7*(-5*a^(3/2)*A*b + 9*a^(5/2)*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*b^(11/2))","A",1
100,1,121,116,0.1209147,"\int \frac{x^6 \left(A+B x^2\right)}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(x^6*(A + B*x^2))/(a + b*x^2)^3,x]","\frac{5 \left(7 a^{3/2} B-3 \sqrt{a} A b\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 b^{9/2}}+\frac{-105 a^3 B x+45 a^2 A b x-175 a^2 b B x^3+75 a A b^2 x^3-56 a b^2 B x^5+24 A b^3 x^5+8 b^3 B x^7}{24 b^4 \left(a+b x^2\right)^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,"(45*a^2*A*b*x - 105*a^3*B*x + 75*a*A*b^2*x^3 - 175*a^2*b*B*x^3 + 24*A*b^3*x^5 - 56*a*b^2*B*x^5 + 8*b^3*B*x^7)/(24*b^4*(a + b*x^2)^2) + (5*(-3*Sqrt[a]*A*b + 7*a^(3/2)*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*b^(9/2))","A",1
101,1,92,94,0.0987756,"\int \frac{x^4 \left(A+B x^2\right)}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(x^4*(A + B*x^2))/(a + b*x^2)^3,x]","\frac{x \left(15 a^2 B-3 a A b+25 a b B x^2-5 A b^2 x^2+8 b^2 B x^4\right)}{8 b^3 \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 \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*(-3*a*A*b + 15*a^2*B - 5*A*b^2*x^2 + 25*a*b*B*x^2 + 8*b^2*B*x^4))/(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
102,1,84,89,0.1083164,"\int \frac{x^2 \left(A+B x^2\right)}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(x^2*(A + B*x^2))/(a + b*x^2)^3,x]","\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 \left(3 a^2 B+a A b+5 a b B x^2-A b^2 x^2\right)}{8 a b^2 \left(a+b x^2\right)^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,"-1/8*(x*(a*A*b + 3*a^2*B - A*b^2*x^2 + 5*a*b*B*x^2))/(a*b^2*(a + b*x^2)^2) + ((A*b + 3*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(3/2)*b^(5/2))","A",1
103,1,84,92,0.0890966,"\int \frac{A+B x^2}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(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-5 a A b-a b B x^2-3 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,"-1/8*(x*(-5*a*A*b + a^2*B - 3*A*b^2*x^2 - a*b*B*x^2))/(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
104,1,96,97,0.0821625,"\int \frac{A+B x^2}{x^2 \left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(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{-8 a^2 A+5 a^2 B x^2-25 a A b x^2+3 a b B x^4-15 A b^2 x^4}{8 a^3 x \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,"(-8*a^2*A - 25*a*A*b*x^2 + 5*a^2*B*x^2 - 15*A*b^2*x^4 + 3*a*b*B*x^4)/(8*a^3*x*(a + b*x^2)^2) + (3*(-5*A*b + a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(7/2)*Sqrt[b])","A",1
105,1,125,117,0.1239264,"\int \frac{A+B x^2}{x^4 \left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^4*(a + b*x^2)^3),x]","\frac{-8 a^3 A-24 a^3 B x^2+56 a^2 A b x^2-75 a^2 b B x^4+175 a A b^2 x^4-45 a b^2 B x^6+105 A b^3 x^6}{24 a^4 x^3 \left(a+b x^2\right)^2}-\frac{5 \left(3 a \sqrt{b} B-7 A b^{3/2}\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{9/2}}","\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,"(-8*a^3*A + 56*a^2*A*b*x^2 - 24*a^3*B*x^2 + 175*a*A*b^2*x^4 - 75*a^2*b*B*x^4 + 105*A*b^3*x^6 - 45*a*b^2*B*x^6)/(24*a^4*x^3*(a + b*x^2)^2) - (5*(-7*A*b^(3/2) + 3*a*Sqrt[b]*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(9/2))","A",1
106,1,149,142,0.1367011,"\int \frac{A+B x^2}{x^6 \left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^6*(a + b*x^2)^3),x]","\frac{7 \left(5 a b^{3/2} B-9 A b^{5/2}\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{11/2}}+\frac{-24 a^4 A-40 a^4 B x^2+72 a^3 A b x^2+280 a^3 b B x^4-504 a^2 A b^2 x^4+875 a^2 b^2 B x^6-1575 a A b^3 x^6+525 a b^3 B x^8-945 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,"(-24*a^4*A + 72*a^3*A*b*x^2 - 40*a^4*B*x^2 - 504*a^2*A*b^2*x^4 + 280*a^3*b*B*x^4 - 1575*a*A*b^3*x^6 + 875*a^2*b^2*B*x^6 - 945*A*b^4*x^8 + 525*a*b^3*B*x^8)/(120*a^5*x^5*(a + b*x^2)^2) + (7*(-9*A*b^(5/2) + 5*a*b^(3/2)*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(11/2))","A",1
107,1,12,12,0.0100699,"\int \frac{a+b x^2}{1+x^2} \, dx","IntegrateAlgebraic[(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
108,1,11,11,0.0158358,"\int \frac{a+b x^2}{1-x^2} \, dx","IntegrateAlgebraic[(a + b*x^2)/(1 - x^2),x]","(a+b) \tanh ^{-1}(x)-b x","(a+b) \tanh ^{-1}(x)-b x",1,"-(b*x) + (a + b)*ArcTanh[x]","A",1
109,1,10,11,0.0073139,"\int \frac{1+x^2}{\left(-1+x^2\right)^2} \, dx","IntegrateAlgebraic[(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
110,1,9,9,0.0070073,"\int \frac{1-x^2}{\left(1+x^2\right)^2} \, dx","IntegrateAlgebraic[(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
111,1,19,19,0.0130599,"\int \frac{3+2 x^2}{\left(1+x^2\right)^2} \, dx","IntegrateAlgebraic[(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
112,1,19,19,0.0129552,"\int \frac{-2+x^2}{\left(1+x^2\right)^2} \, dx","IntegrateAlgebraic[(-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
113,1,14,14,0.0109232,"\int \frac{3+x^2}{\left(1+x^2\right)^2} \, dx","IntegrateAlgebraic[(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
114,1,14,12,0.0111645,"\int \frac{a+b x^2}{\left(-a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(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
115,1,14,12,0.0091985,"\int \frac{a+b x^2}{\left(a-b x^2\right)^2} \, dx","IntegrateAlgebraic[(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
116,1,39,39,0.0310098,"\int \frac{A+B x^2}{a-b x^2} \, dx","IntegrateAlgebraic[(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
117,1,30,35,0.0168793,"\int \frac{1+x^2}{\left(16+x^2\right)^3} \, dx","IntegrateAlgebraic[(1 + x^2)/(16 + x^2)^3,x]","\frac{x \left(19 x^2-176\right)}{2048 \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,"(x*(-176 + 19*x^2))/(2048*(16 + x^2)^2) + (19*ArcTan[x/4])/8192","A",1
118,1,14,14,0.0097832,"\int \frac{1+2 x^2}{x^5 \left(1+x^2\right)^3} \, dx","IntegrateAlgebraic[(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
119,0,0,1,0.2291495,"\int \frac{\left(1-x^2\right)^2}{\left(-1+x^2\right)^2} \, dx","IntegrateAlgebraic[(1 - x^2)^2/(-1 + x^2)^2,x]","\int \frac{\left(1-x^2\right)^2}{\left(-1+x^2\right)^2} \, dx","x",1,"Could not integrate","F",-1
120,1,8,8,0.0092213,"\int \frac{x^3 \left(a c+b c x^2\right)}{a+b x^2} \, dx","IntegrateAlgebraic[(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
121,0,0,8,0.0728719,"\int \frac{x^2 \left(a c+b c x^2\right)}{a+b x^2} \, dx","IntegrateAlgebraic[(x^2*(a*c + b*c*x^2))/(a + b*x^2),x]","\int \frac{x^2 \left(a c+b c x^2\right)}{a+b x^2} \, dx","\frac{c x^3}{3}",1,"Could not integrate","F",-1
122,1,8,8,0.0078017,"\int \frac{x \left(a c+b c x^2\right)}{a+b x^2} \, dx","IntegrateAlgebraic[(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
123,0,0,3,0.1254811,"\int \frac{a c+b c x^2}{a+b x^2} \, dx","IntegrateAlgebraic[(a*c + b*c*x^2)/(a + b*x^2),x]","\int \frac{a c+b c x^2}{a+b x^2} \, dx","c x",1,"Could not integrate","F",-1
124,1,4,4,0.002863,"\int \frac{a c+b c x^2}{x \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(a*c + b*c*x^2)/(x*(a + b*x^2)),x]","c \log (x)","c \log (x)",1,"c*Log[x]","A",1
125,1,6,6,0.002487,"\int \frac{a c+b c x^2}{x^2 \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(a*c + b*c*x^2)/(x^2*(a + b*x^2)),x]","-\frac{c}{x}","-\frac{c}{x}",1,"-(c/x)","A",1
126,1,8,8,0.0028965,"\int \frac{a c+b c x^2}{x^3 \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(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
127,1,29,29,0.010243,"\int \frac{x^3 \left(a c+b c x^2\right)}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^3*(a*c + b*c*x^2))/(a + b*x^2)^2,x]","\frac{c x^2}{2 b}-\frac{a c \log \left(a+b x^2\right)}{2 b^2}","\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*c*Log[a + b*x^2])/(2*b^2)","A",1
128,1,33,33,0.0155598,"\int \frac{x^2 \left(a c+b c x^2\right)}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^2*(a*c + b*c*x^2))/(a + b*x^2)^2,x]","\frac{c x}{b}-\frac{\sqrt{a} c \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{3/2}}","\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]*c*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(3/2)","A",1
129,1,16,16,0.0060745,"\int \frac{x \left(a c+b c x^2\right)}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(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
130,1,25,25,0.0096361,"\int \frac{a c+b c x^2}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(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
131,1,24,24,0.0103971,"\int \frac{a c+b c x^2}{x \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(a*c + b*c*x^2)/(x*(a + b*x^2)^2),x]","\frac{c \log (x)}{a}-\frac{c \log \left(a+b x^2\right)}{2 a}","\frac{c \log (x)}{a}-\frac{c \log \left(a+b x^2\right)}{2 a}",1,"(c*Log[x])/a - (c*Log[a + b*x^2])/(2*a)","A",1
132,1,36,36,0.0212076,"\int \frac{a c+b c x^2}{x^2 \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(a*c + b*c*x^2)/(x^2*(a + b*x^2)^2),x]","-\frac{\sqrt{b} c \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2}}-\frac{c}{a x}","-\frac{\sqrt{b} c \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2}}-\frac{c}{a x}",1,"-(c/(a*x)) - (Sqrt[b]*c*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(3/2)","A",1
133,1,38,38,0.0140184,"\int \frac{a c+b c x^2}{x^3 \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(a*c + b*c*x^2)/(x^3*(a + b*x^2)^2),x]","\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}","\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,"-1/2*c/(a*x^2) - (b*c*Log[x])/a^2 + (b*c*Log[a + b*x^2])/(2*a^2)","A",1
134,1,35,35,0.0187901,"\int \frac{x^3 \left(a c+b c x^2\right)}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(x^3*(a*c + b*c*x^2))/(a + b*x^2)^3,x]","\frac{a c}{2 b^2 \left(a+b x^2\right)}+\frac{c \log \left(a+b x^2\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,"(a*c)/(2*b^2*(a + b*x^2)) + (c*Log[a + b*x^2])/(2*b^2)","A",1
135,1,47,47,0.0322919,"\int \frac{x^2 \left(a c+b c x^2\right)}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(x^2*(a*c + b*c*x^2))/(a + b*x^2)^3,x]","\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)}","\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,"-1/2*(c*x)/(b*(a + b*x^2)) + (c*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*Sqrt[a]*b^(3/2))","A",1
136,1,17,17,0.0066127,"\int \frac{x \left(a c+b c x^2\right)}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(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
137,1,47,47,0.034882,"\int \frac{a c+b c x^2}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(a*c + b*c*x^2)/(a + b*x^2)^3,x]","\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)}","\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)) + (c*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*Sqrt[b])","A",1
138,1,41,41,0.0258793,"\int \frac{a c+b c x^2}{x \left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(a*c + b*c*x^2)/(x*(a + b*x^2)^3),x]","-\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)}","-\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/(2*a*(a + b*x^2)) + (c*Log[x])/a^2 - (c*Log[a + b*x^2])/(2*a^2)","A",1
139,1,60,60,0.0538524,"\int \frac{a c+b c x^2}{x^2 \left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(a*c + b*c*x^2)/(x^2*(a + b*x^2)^3),x]","\frac{-2 a c-3 b c x^2}{2 a^2 x \left(a+b x^2\right)}-\frac{3 \sqrt{b} c \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{5/2}}","-\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,"(-2*a*c - 3*b*c*x^2)/(2*a^2*x*(a + b*x^2)) - (3*Sqrt[b]*c*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(5/2))","A",1
140,1,52,53,0.0505048,"\int \frac{a c+b c x^2}{x^3 \left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(a*c + b*c*x^2)/(x^3*(a + b*x^2)^3),x]","\frac{b c \log \left(a+b x^2\right)}{a^3}-\frac{2 b c \log (x)}{a^3}-\frac{c \left(a+2 b x^2\right)}{2 a^2 x^2 \left(a+b x^2\right)}","\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 + 2*b*x^2))/(a^2*x^2*(a + b*x^2)) - (2*b*c*Log[x])/a^3 + (b*c*Log[a + b*x^2])/a^3","A",1
141,1,57,55,0.0236158,"\int x^4 \left(a+b x^2\right)^2 \left(c+d x^2\right) \, dx","IntegrateAlgebraic[x^4*(a + b*x^2)^2*(c + d*x^2),x]","\frac{693 a^2 c x^5+495 a^2 d x^7+990 a b c x^7+770 a b d x^9+385 b^2 c x^9+315 b^2 d x^{11}}{3465}","\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,"(693*a^2*c*x^5 + 990*a*b*c*x^7 + 495*a^2*d*x^7 + 385*b^2*c*x^9 + 770*a*b*d*x^9 + 315*b^2*d*x^11)/3465","A",1
142,1,57,55,0.0298014,"\int x^3 \left(a+b x^2\right)^2 \left(c+d x^2\right) \, dx","IntegrateAlgebraic[x^3*(a + b*x^2)^2*(c + d*x^2),x]","\frac{1}{120} x^4 \left(30 a^2 c+20 a^2 d x^2+40 a b c x^2+30 a b d x^4+15 b^2 c x^4+12 b^2 d x^6\right)","\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,"(x^4*(30*a^2*c + 40*a*b*c*x^2 + 20*a^2*d*x^2 + 15*b^2*c*x^4 + 30*a*b*d*x^4 + 12*b^2*d*x^6))/120","A",1
143,1,57,55,0.0210285,"\int x^2 \left(a+b x^2\right)^2 \left(c+d x^2\right) \, dx","IntegrateAlgebraic[x^2*(a + b*x^2)^2*(c + d*x^2),x]","\frac{1}{315} \left(105 a^2 c x^3+63 a^2 d x^5+126 a b c x^5+90 a b d x^7+45 b^2 c x^7+35 b^2 d x^9\right)","\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,"(105*a^2*c*x^3 + 126*a*b*c*x^5 + 63*a^2*d*x^5 + 45*b^2*c*x^7 + 90*a*b*d*x^7 + 35*b^2*d*x^9)/315","A",1
144,1,57,42,0.0264392,"\int x \left(a+b x^2\right)^2 \left(c+d x^2\right) \, dx","IntegrateAlgebraic[x*(a + b*x^2)^2*(c + d*x^2),x]","\frac{1}{24} \left(12 a^2 c x^2+6 a^2 d x^4+12 a b c x^4+8 a b d x^6+4 b^2 c x^6+3 b^2 d x^8\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,"(12*a^2*c*x^2 + 12*a*b*c*x^4 + 6*a^2*d*x^4 + 4*b^2*c*x^6 + 8*a*b*d*x^6 + 3*b^2*d*x^8)/24","A",1
145,1,55,50,0.0005787,"\int \left(a+b x^2\right)^2 \left(c+d x^2\right) \, dx","IntegrateAlgebraic[(a + b*x^2)^2*(c + d*x^2),x]","\frac{1}{105} \left(105 a^2 c x+35 a^2 d x^3+70 a b c x^3+42 a b d x^5+21 b^2 c x^5+15 b^2 d x^7\right)","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,"(105*a^2*c*x + 70*a*b*c*x^3 + 35*a^2*d*x^3 + 21*b^2*c*x^5 + 42*a*b*d*x^5 + 15*b^2*d*x^7)/105","A",1
146,1,53,43,0.0294124,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)}{x} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2))/x,x]","\frac{1}{12} x^2 \left(6 a^2 d+12 a b c+6 a b d x^2+3 b^2 c x^2+2 b^2 d x^4\right)+a^2 c \log (x)","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,"(x^2*(12*a*b*c + 6*a^2*d + 3*b^2*c*x^2 + 6*a*b*d*x^2 + 2*b^2*d*x^4))/12 + a^2*c*Log[x]","A",1
147,1,57,48,0.0247449,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)}{x^2} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2))/x^2,x]","\frac{-15 a^2 c+15 a^2 d x^2+30 a b c x^2+10 a b d x^4+5 b^2 c x^4+3 b^2 d x^6}{15 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",1,"(-15*a^2*c + 30*a*b*c*x^2 + 15*a^2*d*x^2 + 5*b^2*c*x^4 + 10*a*b*d*x^4 + 3*b^2*d*x^6)/(15*x)","A",1
148,1,54,51,0.036097,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)}{x^3} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2))/x^3,x]","\frac{-2 a^2 c+4 a b d x^4+2 b^2 c x^4+b^2 d x^6}{4 x^2}+\log (x) \left(a^2 d+2 a b c\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 + 2*b^2*c*x^4 + 4*a*b*d*x^4 + b^2*d*x^6)/(4*x^2) + (2*a*b*c + a^2*d)*Log[x]","A",1
149,1,56,48,0.0285298,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)}{x^4} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2))/x^4,x]","\frac{-a^2 c-3 a^2 d x^2-6 a b c x^2+6 a b d x^4+3 b^2 c x^4+b^2 d x^6}{3 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,"(-(a^2*c) - 6*a*b*c*x^2 - 3*a^2*d*x^2 + 3*b^2*c*x^4 + 6*a*b*d*x^4 + b^2*d*x^6)/(3*x^3)","A",1
150,1,98,87,0.0398351,"\int x^4 \left(a+b x^2\right)^2 \left(c+d x^2\right)^2 \, dx","IntegrateAlgebraic[x^4*(a + b*x^2)^2*(c + d*x^2)^2,x]","\frac{9009 a^2 c^2 x^5+12870 a^2 c d x^7+5005 a^2 d^2 x^9+12870 a b c^2 x^7+20020 a b c d x^9+8190 a b d^2 x^{11}+5005 b^2 c^2 x^9+8190 b^2 c d x^{11}+3465 b^2 d^2 x^{13}}{45045}","\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,"(9009*a^2*c^2*x^5 + 12870*a*b*c^2*x^7 + 12870*a^2*c*d*x^7 + 5005*b^2*c^2*x^9 + 20020*a*b*c*d*x^9 + 5005*a^2*d^2*x^9 + 8190*b^2*c*d*x^11 + 8190*a*b*d^2*x^11 + 3465*b^2*d^2*x^13)/45045","A",1
151,1,98,87,0.0563691,"\int x^3 \left(a+b x^2\right)^2 \left(c+d x^2\right)^2 \, dx","IntegrateAlgebraic[x^3*(a + b*x^2)^2*(c + d*x^2)^2,x]","\frac{1}{120} x^4 \left(30 a^2 c^2+40 a^2 c d x^2+15 a^2 d^2 x^4+40 a b c^2 x^2+60 a b c d x^4+24 a b d^2 x^6+15 b^2 c^2 x^4+24 b^2 c d x^6+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*b*c^2*x^2 + 40*a^2*c*d*x^2 + 15*b^2*c^2*x^4 + 60*a*b*c*d*x^4 + 15*a^2*d^2*x^4 + 24*b^2*c*d*x^6 + 24*a*b*d^2*x^6 + 10*b^2*d^2*x^8))/120","A",1
152,1,98,87,0.0434096,"\int x^2 \left(a+b x^2\right)^2 \left(c+d x^2\right)^2 \, dx","IntegrateAlgebraic[x^2*(a + b*x^2)^2*(c + d*x^2)^2,x]","\frac{1155 a^2 c^2 x^3+1386 a^2 c d x^5+495 a^2 d^2 x^7+1386 a b c^2 x^5+1980 a b c d x^7+770 a b d^2 x^9+495 b^2 c^2 x^7+770 b^2 c d x^9+315 b^2 d^2 x^{11}}{3465}","\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,"(1155*a^2*c^2*x^3 + 1386*a*b*c^2*x^5 + 1386*a^2*c*d*x^5 + 495*b^2*c^2*x^7 + 1980*a*b*c*d*x^7 + 495*a^2*d^2*x^7 + 770*b^2*c*d*x^9 + 770*a*b*d^2*x^9 + 315*b^2*d^2*x^11)/3465","A",1
153,1,98,71,0.0506977,"\int x \left(a+b x^2\right)^2 \left(c+d x^2\right)^2 \, dx","IntegrateAlgebraic[x*(a + b*x^2)^2*(c + d*x^2)^2,x]","\frac{1}{60} x^2 \left(30 a^2 c^2+30 a^2 c d x^2+10 a^2 d^2 x^4+30 a b c^2 x^2+40 a b c d x^4+15 a b d^2 x^6+10 b^2 c^2 x^4+15 b^2 c d x^6+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*b*c^2*x^2 + 30*a^2*c*d*x^2 + 10*b^2*c^2*x^4 + 40*a*b*c*d*x^4 + 10*a^2*d^2*x^4 + 15*b^2*c*d*x^6 + 15*a*b*d^2*x^6 + 6*b^2*d^2*x^8))/60","A",1
154,1,96,82,0.0006208,"\int \left(a+b x^2\right)^2 \left(c+d x^2\right)^2 \, dx","IntegrateAlgebraic[(a + b*x^2)^2*(c + d*x^2)^2,x]","\frac{1}{315} \left(315 a^2 c^2 x+210 a^2 c d x^3+63 a^2 d^2 x^5+210 a b c^2 x^3+252 a b c d x^5+90 a b d^2 x^7+63 b^2 c^2 x^5+90 b^2 c d x^7+35 b^2 d^2 x^9\right)","\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,"(315*a^2*c^2*x + 210*a*b*c^2*x^3 + 210*a^2*c*d*x^3 + 63*b^2*c^2*x^5 + 252*a*b*c*d*x^5 + 63*a^2*d^2*x^5 + 90*b^2*c*d*x^7 + 90*a*b*d^2*x^7 + 35*b^2*d^2*x^9)/315","A",1
155,1,97,80,0.0489518,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2}{x} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2)^2)/x,x]","\frac{1}{24} \left(24 a^2 c d x^2+6 a^2 d^2 x^4+24 a b c^2 x^2+24 a b c d x^4+8 a b d^2 x^6+6 b^2 c^2 x^4+8 b^2 c d x^6+3 b^2 d^2 x^8\right)+a^2 c^2 \log (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",1,"(24*a*b*c^2*x^2 + 24*a^2*c*d*x^2 + 6*b^2*c^2*x^4 + 24*a*b*c*d*x^4 + 6*a^2*d^2*x^4 + 8*b^2*c*d*x^6 + 8*a*b*d^2*x^6 + 3*b^2*d^2*x^8)/24 + a^2*c^2*Log[x]","A",1
156,1,98,81,0.0521593,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2}{x^2} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2)^2)/x^2,x]","\frac{-105 a^2 c^2+210 a^2 c d x^2+35 a^2 d^2 x^4+210 a b c^2 x^2+140 a b c d x^4+42 a b d^2 x^6+35 b^2 c^2 x^4+42 b^2 c d x^6+15 b^2 d^2 x^8}{105 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",1,"(-105*a^2*c^2 + 210*a*b*c^2*x^2 + 210*a^2*c*d*x^2 + 35*b^2*c^2*x^4 + 140*a*b*c*d*x^4 + 35*a^2*d^2*x^4 + 42*b^2*c*d*x^6 + 42*a*b*d^2*x^6 + 15*b^2*d^2*x^8)/(105*x)","A",1
157,1,95,84,0.0632907,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2}{x^3} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2)^2)/x^3,x]","\frac{-3 a^2 c^2+3 a^2 d^2 x^4+12 a b c d x^4+3 a b d^2 x^6+3 b^2 c^2 x^4+3 b^2 c d x^6+b^2 d^2 x^8}{6 x^2}+2 \log (x) \left(a^2 c d+a b c^2\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^2*c^2 + 3*b^2*c^2*x^4 + 12*a*b*c*d*x^4 + 3*a^2*d^2*x^4 + 3*b^2*c*d*x^6 + 3*a*b*d^2*x^6 + b^2*d^2*x^8)/(6*x^2) + 2*(a*b*c^2 + a^2*c*d)*Log[x]","A",1
158,1,98,80,0.0536593,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2}{x^4} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2)^2)/x^4,x]","\frac{-5 a^2 c^2-30 a^2 c d x^2+15 a^2 d^2 x^4-30 a b c^2 x^2+60 a b c d x^4+10 a b d^2 x^6+15 b^2 c^2 x^4+10 b^2 c d x^6+3 b^2 d^2 x^8}{15 x^3}","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,"(-5*a^2*c^2 - 30*a*b*c^2*x^2 - 30*a^2*c*d*x^2 + 15*b^2*c^2*x^4 + 60*a*b*c*d*x^4 + 15*a^2*d^2*x^4 + 10*b^2*c*d*x^6 + 10*a*b*d^2*x^6 + 3*b^2*d^2*x^8)/(15*x^3)","A",1
159,1,139,127,0.0647811,"\int x^4 \left(a+b x^2\right)^2 \left(c+d x^2\right)^3 \, dx","IntegrateAlgebraic[x^4*(a + b*x^2)^2*(c + d*x^2)^3,x]","\frac{9009 a^2 c^3 x^5+19305 a^2 c^2 d x^7+15015 a^2 c d^2 x^9+4095 a^2 d^3 x^{11}+12870 a b c^3 x^7+30030 a b c^2 d x^9+24570 a b c d^2 x^{11}+6930 a b d^3 x^{13}+5005 b^2 c^3 x^9+12285 b^2 c^2 d x^{11}+10395 b^2 c d^2 x^{13}+3003 b^2 d^3 x^{15}}{45045}","\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,"(9009*a^2*c^3*x^5 + 12870*a*b*c^3*x^7 + 19305*a^2*c^2*d*x^7 + 5005*b^2*c^3*x^9 + 30030*a*b*c^2*d*x^9 + 15015*a^2*c*d^2*x^9 + 12285*b^2*c^2*d*x^11 + 24570*a*b*c*d^2*x^11 + 4095*a^2*d^3*x^11 + 10395*b^2*c*d^2*x^13 + 6930*a*b*d^3*x^13 + 3003*b^2*d^3*x^15)/45045","A",1
160,1,139,106,0.0715406,"\int x^3 \left(a+b x^2\right)^2 \left(c+d x^2\right)^3 \, dx","IntegrateAlgebraic[x^3*(a + b*x^2)^2*(c + d*x^2)^3,x]","\frac{1}{840} x^4 \left(210 a^2 c^3+420 a^2 c^2 d x^2+315 a^2 c d^2 x^4+84 a^2 d^3 x^6+280 a b c^3 x^2+630 a b c^2 d x^4+504 a b c d^2 x^6+140 a b d^3 x^8+105 b^2 c^3 x^4+252 b^2 c^2 d x^6+210 b^2 c d^2 x^8+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 + 280*a*b*c^3*x^2 + 420*a^2*c^2*d*x^2 + 105*b^2*c^3*x^4 + 630*a*b*c^2*d*x^4 + 315*a^2*c*d^2*x^4 + 252*b^2*c^2*d*x^6 + 504*a*b*c*d^2*x^6 + 84*a^2*d^3*x^6 + 210*b^2*c*d^2*x^8 + 140*a*b*d^3*x^8 + 60*b^2*d^3*x^10))/840","A",1
161,1,139,127,0.0539242,"\int x^2 \left(a+b x^2\right)^2 \left(c+d x^2\right)^3 \, dx","IntegrateAlgebraic[x^2*(a + b*x^2)^2*(c + d*x^2)^3,x]","\frac{15015 a^2 c^3 x^3+27027 a^2 c^2 d x^5+19305 a^2 c d^2 x^7+5005 a^2 d^3 x^9+18018 a b c^3 x^5+38610 a b c^2 d x^7+30030 a b c d^2 x^9+8190 a b d^3 x^{11}+6435 b^2 c^3 x^7+15015 b^2 c^2 d x^9+12285 b^2 c d^2 x^{11}+3465 b^2 d^3 x^{13}}{45045}","\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,"(15015*a^2*c^3*x^3 + 18018*a*b*c^3*x^5 + 27027*a^2*c^2*d*x^5 + 6435*b^2*c^3*x^7 + 38610*a*b*c^2*d*x^7 + 19305*a^2*c*d^2*x^7 + 15015*b^2*c^2*d*x^9 + 30030*a*b*c*d^2*x^9 + 5005*a^2*d^3*x^9 + 12285*b^2*c*d^2*x^11 + 8190*a*b*d^3*x^11 + 3465*b^2*d^3*x^13)/45045","A",1
162,1,139,71,0.0685445,"\int x \left(a+b x^2\right)^2 \left(c+d x^2\right)^3 \, dx","IntegrateAlgebraic[x*(a + b*x^2)^2*(c + d*x^2)^3,x]","\frac{1}{120} x^2 \left(60 a^2 c^3+90 a^2 c^2 d x^2+60 a^2 c d^2 x^4+15 a^2 d^3 x^6+60 a b c^3 x^2+120 a b c^2 d x^4+90 a b c d^2 x^6+24 a b d^3 x^8+20 b^2 c^3 x^4+45 b^2 c^2 d x^6+36 b^2 c d^2 x^8+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 + 60*a*b*c^3*x^2 + 90*a^2*c^2*d*x^2 + 20*b^2*c^3*x^4 + 120*a*b*c^2*d*x^4 + 60*a^2*c*d^2*x^4 + 45*b^2*c^2*d*x^6 + 90*a*b*c*d^2*x^6 + 15*a^2*d^3*x^6 + 36*b^2*c*d^2*x^8 + 24*a*b*d^3*x^8 + 10*b^2*d^3*x^10))/120","A",1
163,1,137,122,0.0005965,"\int \left(a+b x^2\right)^2 \left(c+d x^2\right)^3 \, dx","IntegrateAlgebraic[(a + b*x^2)^2*(c + d*x^2)^3,x]","\frac{3465 a^2 c^3 x+3465 a^2 c^2 d x^3+2079 a^2 c d^2 x^5+495 a^2 d^3 x^7+2310 a b c^3 x^3+4158 a b c^2 d x^5+2970 a b c d^2 x^7+770 a b d^3 x^9+693 b^2 c^3 x^5+1485 b^2 c^2 d x^7+1155 b^2 c d^2 x^9+315 b^2 d^3 x^{11}}{3465}","\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,"(3465*a^2*c^3*x + 2310*a*b*c^3*x^3 + 3465*a^2*c^2*d*x^3 + 693*b^2*c^3*x^5 + 4158*a*b*c^2*d*x^5 + 2079*a^2*c*d^2*x^5 + 1485*b^2*c^2*d*x^7 + 2970*a*b*c*d^2*x^7 + 495*a^2*d^3*x^7 + 1155*b^2*c*d^2*x^9 + 770*a*b*d^3*x^9 + 315*b^2*d^3*x^11)/3465","A",1
164,1,135,123,0.0652082,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3}{x} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2)^3)/x,x]","\frac{1}{120} x^2 \left(180 a^2 c^2 d+90 a^2 c d^2 x^2+20 a^2 d^3 x^4+120 a b c^3+180 a b c^2 d x^2+120 a b c d^2 x^4+30 a b d^3 x^6+30 b^2 c^3 x^2+60 b^2 c^2 d x^4+45 b^2 c d^2 x^6+12 b^2 d^3 x^8\right)+a^2 c^3 \log (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}",1,"(x^2*(120*a*b*c^3 + 180*a^2*c^2*d + 30*b^2*c^3*x^2 + 180*a*b*c^2*d*x^2 + 90*a^2*c*d^2*x^2 + 60*b^2*c^2*d*x^4 + 120*a*b*c*d^2*x^4 + 20*a^2*d^3*x^4 + 45*b^2*c*d^2*x^6 + 30*a*b*d^3*x^6 + 12*b^2*d^3*x^8))/120 + a^2*c^3*Log[x]","A",1
165,1,139,120,0.055384,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3}{x^2} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2)^3)/x^2,x]","\frac{-315 a^2 c^3+945 a^2 c^2 d x^2+315 a^2 c d^2 x^4+63 a^2 d^3 x^6+630 a b c^3 x^2+630 a b c^2 d x^4+378 a b c d^2 x^6+90 a b d^3 x^8+105 b^2 c^3 x^4+189 b^2 c^2 d x^6+135 b^2 c d^2 x^8+35 b^2 d^3 x^{10}}{315 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",1,"(-315*a^2*c^3 + 630*a*b*c^3*x^2 + 945*a^2*c^2*d*x^2 + 105*b^2*c^3*x^4 + 630*a*b*c^2*d*x^4 + 315*a^2*c*d^2*x^4 + 189*b^2*c^2*d*x^6 + 378*a*b*c*d^2*x^6 + 63*a^2*d^3*x^6 + 135*b^2*c*d^2*x^8 + 90*a*b*d^3*x^8 + 35*b^2*d^3*x^10)/(315*x)","A",1
166,1,138,123,0.0805406,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3}{x^3} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2)^3)/x^3,x]","\frac{-12 a^2 c^3+36 a^2 c d^2 x^4+6 a^2 d^3 x^6+72 a b c^2 d x^4+36 a b c d^2 x^6+8 a b d^3 x^8+12 b^2 c^3 x^4+18 b^2 c^2 d x^6+12 b^2 c d^2 x^8+3 b^2 d^3 x^{10}}{24 x^2}+\log (x) \left(3 a^2 c^2 d+2 a b c^3\right)","\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,"(-12*a^2*c^3 + 12*b^2*c^3*x^4 + 72*a*b*c^2*d*x^4 + 36*a^2*c*d^2*x^4 + 18*b^2*c^2*d*x^6 + 36*a*b*c*d^2*x^6 + 6*a^2*d^3*x^6 + 12*b^2*c*d^2*x^8 + 8*a*b*d^3*x^8 + 3*b^2*d^3*x^10)/(24*x^2) + (2*a*b*c^3 + 3*a^2*c^2*d)*Log[x]","A",1
167,1,139,120,0.0581862,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3}{x^4} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2)^3)/x^4,x]","\frac{-35 a^2 c^3-315 a^2 c^2 d x^2+315 a^2 c d^2 x^4+35 a^2 d^3 x^6-210 a b c^3 x^2+630 a b c^2 d x^4+210 a b c d^2 x^6+42 a b d^3 x^8+105 b^2 c^3 x^4+105 b^2 c^2 d x^6+63 b^2 c d^2 x^8+15 b^2 d^3 x^{10}}{105 x^3}","\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,"(-35*a^2*c^3 - 210*a*b*c^3*x^2 - 315*a^2*c^2*d*x^2 + 105*b^2*c^3*x^4 + 630*a*b*c^2*d*x^4 + 315*a^2*c*d^2*x^4 + 105*b^2*c^2*d*x^6 + 210*a*b*c*d^2*x^6 + 35*a^2*d^3*x^6 + 63*b^2*c*d^2*x^8 + 42*a*b*d^3*x^8 + 15*b^2*d^3*x^10)/(105*x^3)","A",1
168,1,136,104,0.1287556,"\int \frac{x^4 \left(a+b x^2\right)^2}{c+d x^2} \, dx","IntegrateAlgebraic[(x^4*(a + b*x^2)^2)/(c + d*x^2),x]","\frac{x \left(-105 a^2 c d^2+35 a^2 d^3 x^2+210 a b c^2 d-70 a b c d^2 x^2+42 a b d^3 x^4-105 b^2 c^3+35 b^2 c^2 d x^2-21 b^2 c d^2 x^4+15 b^2 d^3 x^6\right)}{105 d^4}+\frac{c^{3/2} (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{d^{9/2}}","\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,"(x*(-105*b^2*c^3 + 210*a*b*c^2*d - 105*a^2*c*d^2 + 35*b^2*c^2*d*x^2 - 70*a*b*c*d^2*x^2 + 35*a^2*d^3*x^2 - 21*b^2*c*d^2*x^4 + 42*a*b*d^3*x^4 + 15*b^2*d^3*x^6))/(105*d^4) + (c^(3/2)*(b*c - a*d)^2*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/d^(9/2)","A",1
169,1,91,79,0.0652031,"\int \frac{x^3 \left(a+b x^2\right)^2}{c+d x^2} \, dx","IntegrateAlgebraic[(x^3*(a + b*x^2)^2)/(c + d*x^2),x]","\frac{x^2 \left(6 a^2 d^2-12 a b c d+6 a b d^2 x^2+6 b^2 c^2-3 b^2 c d x^2+2 b^2 d^2 x^4\right)}{12 d^3}-\frac{c (b c-a d)^2 \log \left(c+d x^2\right)}{2 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,"(x^2*(6*b^2*c^2 - 12*a*b*c*d + 6*a^2*d^2 - 3*b^2*c*d*x^2 + 6*a*b*d^2*x^2 + 2*b^2*d^2*x^4))/(12*d^3) - (c*(b*c - a*d)^2*Log[c + d*x^2])/(2*d^4)","A",1
170,1,98,83,0.093078,"\int \frac{x^2 \left(a+b x^2\right)^2}{c+d x^2} \, dx","IntegrateAlgebraic[(x^2*(a + b*x^2)^2)/(c + d*x^2),x]","\frac{x \left(15 a^2 d^2-30 a b c d+10 a b d^2 x^2+15 b^2 c^2-5 b^2 c d x^2+3 b^2 d^2 x^4\right)}{15 d^3}-\frac{\sqrt{c} (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{d^{7/2}}","-\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,"(x*(15*b^2*c^2 - 30*a*b*c*d + 15*a^2*d^2 - 5*b^2*c*d*x^2 + 10*a*b*d^2*x^2 + 3*b^2*d^2*x^4))/(15*d^3) - (Sqrt[c]*(b*c - a*d)^2*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/d^(7/2)","A",1
171,1,52,61,0.0352727,"\int \frac{x \left(a+b x^2\right)^2}{c+d x^2} \, dx","IntegrateAlgebraic[(x*(a + b*x^2)^2)/(c + d*x^2),x]","\frac{(a d-b c)^2 \log \left(c+d x^2\right)}{2 d^3}+\frac{b x^2 \left(4 a d-2 b c+b d x^2\right)}{4 d^2}","\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*x^2*(-2*b*c + 4*a*d + b*d*x^2))/(4*d^2) + ((-(b*c) + a*d)^2*Log[c + d*x^2])/(2*d^3)","A",1
172,1,59,63,0.0005881,"\int \frac{\left(a+b x^2\right)^2}{c+d x^2} \, dx","IntegrateAlgebraic[(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
173,1,51,51,0.033692,"\int \frac{\left(a+b x^2\right)^2}{x \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x*(c + d*x^2)),x]","\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}","\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*x^2)/(2*d) + (a^2*Log[x])/c - ((b*c - a*d)^2*Log[c + d*x^2])/(2*c*d^2)","A",1
174,1,61,55,0.0604638,"\int \frac{\left(a+b x^2\right)^2}{x^2 \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^2*(c + d*x^2)),x]","\frac{b^2 c x^2-a^2 d}{c d x}-\frac{(b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{3/2} d^{3/2}}","-\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*d) + b^2*c*x^2)/(c*d*x) - ((b*c - a*d)^2*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(3/2)*d^(3/2))","A",1
175,1,60,58,0.0429925,"\int \frac{\left(a+b x^2\right)^2}{x^3 \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^3*(c + d*x^2)),x]","\frac{\log (x) \left(2 a b c-a^2 d\right)}{c^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^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,"-1/2*a^2/(c*x^2) + ((2*a*b*c - a^2*d)*Log[x])/c^2 + ((b*c - a*d)^2*Log[c + d*x^2])/(2*c^2*d)","A",1
176,1,64,66,0.0719972,"\int \frac{\left(a+b x^2\right)^2}{x^4 \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^4*(c + d*x^2)),x]","\frac{(b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{5/2} \sqrt{d}}-\frac{a \left(a c-3 a d x^2+6 b c x^2\right)}{3 c^2 x^3}","-\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*(a*c + 6*b*c*x^2 - 3*a*d*x^2))/(c^2*x^3) + ((b*c - a*d)^2*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(5/2)*Sqrt[d])","A",1
177,1,71,75,0.0630446,"\int \frac{\left(a+b x^2\right)^2}{x^5 \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^5*(c + d*x^2)),x]","-\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 \left(a c-2 a d x^2+4 b c x^2\right)}{4 c^2 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*(a*c + 4*b*c*x^2 - 2*a*d*x^2))/(c^2*x^4) + ((b*c - a*d)^2*Log[x])/c^3 - ((b*c - a*d)^2*Log[c + d*x^2])/(2*c^3)","A",1
178,1,106,87,0.0964247,"\int \frac{\left(a+b x^2\right)^2}{x^6 \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^6*(c + d*x^2)),x]","\frac{-3 a^2 c^2+5 a^2 c d x^2-15 a^2 d^2 x^4-10 a b c^2 x^2+30 a b c d x^4-15 b^2 c^2 x^4}{15 c^3 x^5}-\frac{\sqrt{d} (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{7/2}}","-\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,"(-3*a^2*c^2 - 10*a*b*c^2*x^2 + 5*a^2*c*d*x^2 - 15*b^2*c^2*x^4 + 30*a*b*c*d*x^4 - 15*a^2*d^2*x^4)/(15*c^3*x^5) - (Sqrt[d]*(b*c - a*d)^2*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/c^(7/2)","A",1
179,1,115,98,0.0922501,"\int \frac{\left(a+b x^2\right)^2}{x^7 \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^7*(c + d*x^2)),x]","\frac{-2 a^2 c^2+3 a^2 c d x^2-6 a^2 d^2 x^4-6 a b c^2 x^2+12 a b c d x^4-6 b^2 c^2 x^4}{12 c^3 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{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,"(-2*a^2*c^2 - 6*a*b*c^2*x^2 + 3*a^2*c*d*x^2 - 6*b^2*c^2*x^4 + 12*a*b*c*d*x^4 - 6*a^2*d^2*x^4)/(12*c^3*x^6) - (d*(b*c - a*d)^2*Log[x])/c^4 + (d*(b*c - a*d)^2*Log[c + d*x^2])/(2*c^4)","A",1
180,1,167,145,0.1332579,"\int \frac{x^4 \left(a+b x^2\right)^2}{\left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[(x^4*(a + b*x^2)^2)/(c + d*x^2)^2,x]","\frac{\left(-3 a^2 \sqrt{c} d^2+10 a b c^{3/2} d-7 b^2 c^{5/2}\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 d^{9/2}}+\frac{x \left(45 a^2 c d^2+30 a^2 d^3 x^2-150 a b c^2 d-100 a b c d^2 x^2+20 a b d^3 x^4+105 b^2 c^3+70 b^2 c^2 d x^2-14 b^2 c d^2 x^4+6 b^2 d^3 x^6\right)}{30 d^4 \left(c+d x^2\right)}","-\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,"(x*(105*b^2*c^3 - 150*a*b*c^2*d + 45*a^2*c*d^2 + 70*b^2*c^2*d*x^2 - 100*a*b*c*d^2*x^2 + 30*a^2*d^3*x^2 - 14*b^2*c*d^2*x^4 + 20*a*b*d^3*x^4 + 6*b^2*d^3*x^6))/(30*d^4*(c + d*x^2)) + ((-7*b^2*c^(5/2) + 10*a*b*c^(3/2)*d - 3*a^2*Sqrt[c]*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*d^(9/2))","A",1
181,1,135,90,0.0906064,"\int \frac{x^3 \left(a+b x^2\right)^2}{\left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[(x^3*(a + b*x^2)^2)/(c + d*x^2)^2,x]","\frac{\left(a^2 d^2-4 a b c d+3 b^2 c^2\right) \log \left(c+d x^2\right)}{2 d^4}+\frac{2 a^2 c d^2-4 a b c^2 d+4 a b c d^2 x^2+4 a b d^3 x^4+2 b^2 c^3-4 b^2 c^2 d x^2-3 b^2 c d^2 x^4+b^2 d^3 x^6}{4 d^4 \left(c+d x^2\right)}","\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,"(2*b^2*c^3 - 4*a*b*c^2*d + 2*a^2*c*d^2 - 4*b^2*c^2*d*x^2 + 4*a*b*c*d^2*x^2 - 3*b^2*c*d^2*x^4 + 4*a*b*d^3*x^4 + b^2*d^3*x^6)/(4*d^4*(c + d*x^2)) + ((3*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*Log[c + d*x^2])/(2*d^4)","A",1
182,1,121,118,0.0998271,"\int \frac{x^2 \left(a+b x^2\right)^2}{\left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[(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 \left(-3 a^2 d^2+18 a b c d+12 a b d^2 x^2-15 b^2 c^2-10 b^2 c d x^2+2 b^2 d^2 x^4\right)}{6 d^3 \left(c+d x^2\right)}","\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,"(x*(-15*b^2*c^2 + 18*a*b*c*d - 3*a^2*d^2 - 10*b^2*c*d*x^2 + 12*a*b*d^2*x^2 + 2*b^2*d^2*x^4))/(6*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
183,1,82,62,0.0653858,"\int \frac{x \left(a+b x^2\right)^2}{\left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[(x*(a + b*x^2)^2)/(c + d*x^2)^2,x]","\frac{-a^2 d^2+2 a b c d-b^2 c^2+b^2 c d x^2+b^2 d^2 x^4}{2 d^3 \left(c+d x^2\right)}+\frac{\left(a b d-b^2 c\right) \log \left(c+d x^2\right)}{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*c^2) + 2*a*b*c*d - a^2*d^2 + b^2*c*d*x^2 + b^2*d^2*x^4)/(2*d^3*(c + d*x^2)) + ((-(b^2*c) + a*b*d)*Log[c + d*x^2])/d^3","A",1
184,1,102,82,0.0005963,"\int \frac{\left(a+b x^2\right)^2}{\left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(c + d*x^2)^2,x]","\frac{x \left(a^2 d^2-2 a b c d+3 b^2 c^2+2 b^2 c d x^2\right)}{2 c d^2 \left(c+d x^2\right)}+\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{(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,"(x*(3*b^2*c^2 - 2*a*b*c*d + a^2*d^2 + 2*b^2*c*d*x^2))/(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
185,1,73,67,0.0639373,"\int \frac{\left(a+b x^2\right)^2}{x \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x*(c + d*x^2)^2),x]","\frac{\left(b^2 c^2-a^2 d^2\right) \log \left(c+d x^2\right)}{2 c^2 d^2}+\frac{a^2 \log (x)}{c^2}+\frac{(b c-a d)^2}{2 c d^2 \left(c+d x^2\right)}","-\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,"(b*c - a*d)^2/(2*c*d^2*(c + d*x^2)) + (a^2*Log[x])/c^2 + ((b^2*c^2 - a^2*d^2)*Log[c + d*x^2])/(2*c^2*d^2)","A",1
186,1,111,106,0.0888604,"\int \frac{\left(a+b x^2\right)^2}{x^2 \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^2*(c + d*x^2)^2),x]","\frac{-2 a^2 c d-3 a^2 d^2 x^2+2 a b c d x^2-b^2 c^2 x^2}{2 c^2 d x \left(c+d x^2\right)}+\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{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,"(-2*a^2*c*d - b^2*c^2*x^2 + 2*a*b*c*d*x^2 - 3*a^2*d^2*x^2)/(2*c^2*d*x*(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
187,1,103,81,0.1172688,"\int \frac{\left(a+b x^2\right)^2}{x^3 \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^3*(c + d*x^2)^2),x]","\frac{-a^2 c d-2 a^2 d^2 x^2+2 a b c d x^2-b^2 c^2 x^2}{2 c^2 d x^2 \left(c+d x^2\right)}+\frac{\left(a^2 d-a b c\right) \log \left(c+d x^2\right)}{c^3}-\frac{2 \log (x) \left(a^2 d-a b c\right)}{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,"(-(a^2*c*d) - b^2*c^2*x^2 + 2*a*b*c*d*x^2 - 2*a^2*d^2*x^2)/(2*c^2*d*x^2*(c + d*x^2)) - (2*(-(a*b*c) + a^2*d)*Log[x])/c^3 + ((-(a*b*c) + a^2*d)*Log[c + d*x^2])/c^3","A",1
188,1,129,126,0.0905674,"\int \frac{\left(a+b x^2\right)^2}{x^4 \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[(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{-2 a^2 c^2+10 a^2 c d x^2+15 a^2 d^2 x^4-12 a b c^2 x^2-18 a b c d x^4+3 b^2 c^2 x^4}{6 c^3 x^3 \left(c+d x^2\right)}","\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,"(-2*a^2*c^2 - 12*a*b*c^2*x^2 + 10*a^2*c*d*x^2 + 3*b^2*c^2*x^4 - 18*a*b*c*d*x^4 + 15*a^2*d^2*x^4)/(6*c^3*x^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
189,1,161,163,0.1297523,"\int \frac{x^4 \left(a+b x^2\right)^2}{\left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[(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(-9 a^2 c d^2-15 a^2 d^3 x^2+90 a b c^2 d+150 a b c d^2 x^2+48 a b d^3 x^4-105 b^2 c^3-175 b^2 c^2 d x^2-56 b^2 c d^2 x^4+8 b^2 d^3 x^6\right)}{24 d^4 \left(c+d x^2\right)^2}","\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,"(x*(-105*b^2*c^3 + 90*a*b*c^2*d - 9*a^2*c*d^2 - 175*b^2*c^2*d*x^2 + 150*a*b*c*d^2*x^2 - 15*a^2*d^3*x^2 - 56*b^2*c*d^2*x^4 + 48*a*b*d^3*x^4 + 8*b^2*d^3*x^6))/(24*d^4*(c + d*x^2)^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
190,1,127,99,0.0831362,"\int \frac{x^3 \left(a+b x^2\right)^2}{\left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[(x^3*(a + b*x^2)^2)/(c + d*x^2)^3,x]","\frac{-a^2 c d^2-2 a^2 d^3 x^2+6 a b c^2 d+8 a b c d^2 x^2-5 b^2 c^3-4 b^2 c^2 d x^2+4 b^2 c d^2 x^4+2 b^2 d^3 x^6}{4 d^4 \left(c+d x^2\right)^2}+\frac{\left(2 a b d-3 b^2 c\right) \log \left(c+d x^2\right)}{2 d^4}","\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,"(-5*b^2*c^3 + 6*a*b*c^2*d - a^2*c*d^2 - 4*b^2*c^2*d*x^2 + 8*a*b*c*d^2*x^2 - 2*a^2*d^3*x^2 + 4*b^2*c*d^2*x^4 + 2*b^2*d^3*x^6)/(4*d^4*(c + d*x^2)^2) + ((-3*b^2*c + 2*a*b*d)*Log[c + d*x^2])/(2*d^4)","A",1
191,1,141,127,0.1369312,"\int \frac{x^2 \left(a+b x^2\right)^2}{\left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[(x^2*(a + b*x^2)^2)/(c + d*x^2)^3,x]","\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 \left(-a^2 c d^2+a^2 d^3 x^2-6 a b c^2 d-10 a b c d^2 x^2+15 b^2 c^3+25 b^2 c^2 d x^2+8 b^2 c d^2 x^4\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{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*(15*b^2*c^3 - 6*a*b*c^2*d - a^2*c*d^2 + 25*b^2*c^2*d*x^2 - 10*a*b*c*d^2*x^2 + a^2*d^3*x^2 + 8*b^2*c*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
192,1,78,67,0.0467432,"\int \frac{x \left(a+b x^2\right)^2}{\left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[(x*(a + b*x^2)^2)/(c + d*x^2)^3,x]","\frac{-a^2 d^2-2 a b c d-4 a b d^2 x^2+3 b^2 c^2+4 b^2 c d x^2}{4 d^3 \left(c+d x^2\right)^2}+\frac{b^2 \log \left(c+d x^2\right)}{2 d^3}","\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,"(3*b^2*c^2 - 2*a*b*c*d - a^2*d^2 + 4*b^2*c*d*x^2 - 4*a*b*d^2*x^2)/(4*d^3*(c + d*x^2)^2) + (b^2*Log[c + d*x^2])/(2*d^3)","A",1
193,1,131,116,0.0005953,"\int \frac{\left(a+b x^2\right)^2}{\left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(c + d*x^2)^3,x]","\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(-5 a^2 c d^2-3 a^2 d^3 x^2+2 a b c^2 d-2 a b c d^2 x^2+3 b^2 c^3+5 b^2 c^2 d x^2\right)}{8 c^2 d^2 \left(c+d x^2\right)^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,"-1/8*(x*(3*b^2*c^3 + 2*a*b*c^2*d - 5*a^2*c*d^2 + 5*b^2*c^2*d*x^2 - 2*a*b*c*d^2*x^2 - 3*a^2*d^3*x^2))/(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
194,1,96,86,0.0704574,"\int \frac{\left(a+b x^2\right)^2}{x \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x*(c + d*x^2)^3),x]","\frac{3 a^2 c d^2+2 a^2 d^3 x^2-2 a b c^2 d-b^2 c^3-2 b^2 c^2 d x^2}{4 c^2 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^3) - 2*a*b*c^2*d + 3*a^2*c*d^2 - 2*b^2*c^2*d*x^2 + 2*a^2*d^3*x^2)/(4*c^2*d^2*(c + d*x^2)^2) + (a^2*Log[x])/c^3 - (a^2*Log[c + d*x^2])/(2*c^3)","A",1
195,1,149,152,0.1243715,"\int \frac{\left(a+b x^2\right)^2}{x^2 \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[(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{-8 a^2 c^2 d-25 a^2 c d^2 x^2-15 a^2 d^3 x^4+10 a b c^2 d x^2+6 a b c d^2 x^4-b^2 c^3 x^2+b^2 c^2 d x^4}{8 c^3 d x \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,"(-8*a^2*c^2*d - b^2*c^3*x^2 + 10*a*b*c^2*d*x^2 - 25*a^2*c*d^2*x^2 + b^2*c^2*d*x^4 + 6*a*b*c*d^2*x^4 - 15*a^2*d^3*x^4)/(8*c^3*d*x*(c + d*x^2)^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
196,1,134,106,0.1227405,"\int \frac{\left(a+b x^2\right)^2}{x^3 \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^3*(c + d*x^2)^3),x]","\frac{-2 a^2 c^2 d-9 a^2 c d^2 x^2-6 a^2 d^3 x^4+6 a b c^2 d x^2+4 a b c d^2 x^4-b^2 c^3 x^2}{4 c^3 d x^2 \left(c+d x^2\right)^2}+\frac{\left(3 a^2 d-2 a b c\right) \log \left(c+d x^2\right)}{2 c^4}+\frac{\log (x) \left(2 a b c-3 a^2 d\right)}{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^2*d - b^2*c^3*x^2 + 6*a*b*c^2*d*x^2 - 9*a^2*c*d^2*x^2 + 4*a*b*c*d^2*x^4 - 6*a^2*d^3*x^4)/(4*c^3*d*x^2*(c + d*x^2)^2) + ((2*a*b*c - 3*a^2*d)*Log[x])/c^4 + ((-2*a*b*c + 3*a^2*d)*Log[c + d*x^2])/(2*c^4)","A",1
197,1,169,161,0.1085665,"\int \frac{\left(a+b x^2\right)^2}{x^4 \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[(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{-8 a^2 c^3+56 a^2 c^2 d x^2+175 a^2 c d^2 x^4+105 a^2 d^3 x^6-48 a b c^3 x^2-150 a b c^2 d x^4-90 a b c d^2 x^6+15 b^2 c^3 x^4+9 b^2 c^2 d x^6}{24 c^4 x^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,"(-8*a^2*c^3 - 48*a*b*c^3*x^2 + 56*a^2*c^2*d*x^2 + 15*b^2*c^3*x^4 - 150*a*b*c^2*d*x^4 + 175*a^2*c*d^2*x^4 + 9*b^2*c^2*d*x^6 - 90*a*b*c*d^2*x^6 + 105*a^2*d^3*x^6)/(24*c^4*x^3*(c + d*x^2)^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
198,1,77,75,0.0504416,"\int \frac{x^5 \left(c+d x^2\right)}{a+b x^2} \, dx","IntegrateAlgebraic[(x^5*(c + d*x^2))/(a + b*x^2),x]","\frac{x^2 \left(6 a^2 d-6 a b c-3 a b d x^2+3 b^2 c x^2+2 b^2 d x^4\right)}{12 b^3}+\frac{\left(a^2 b c-a^3 d\right) \log \left(a+b x^2\right)}{2 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,"(x^2*(-6*a*b*c + 6*a^2*d + 3*b^2*c*x^2 - 3*a*b*d*x^2 + 2*b^2*d*x^4))/(12*b^3) + ((a^2*b*c - a^3*d)*Log[a + b*x^2])/(2*b^4)","A",1
199,1,83,77,0.0729418,"\int \frac{x^4 \left(c+d x^2\right)}{a+b x^2} \, dx","IntegrateAlgebraic[(x^4*(c + d*x^2))/(a + b*x^2),x]","\frac{\left(a^{3/2} b c-a^{5/2} d\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{7/2}}+\frac{x \left(15 a^2 d-15 a b c-5 a b d x^2+5 b^2 c x^2+3 b^2 d x^4\right)}{15 b^3}","\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,"(x*(-15*a*b*c + 15*a^2*d + 5*b^2*c*x^2 - 5*a*b*d*x^2 + 3*b^2*d*x^4))/(15*b^3) + ((a^(3/2)*b*c - a^(5/2)*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(7/2)","A",1
200,1,52,54,0.0309766,"\int \frac{x^3 \left(c+d x^2\right)}{a+b x^2} \, dx","IntegrateAlgebraic[(x^3*(c + d*x^2))/(a + b*x^2),x]","\frac{\left(a^2 d-a b c\right) \log \left(a+b x^2\right)}{2 b^3}+\frac{x^2 \left(-2 a d+2 b c+b d x^2\right)}{4 b^2}","-\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,"(x^2*(2*b*c - 2*a*d + b*d*x^2))/(4*b^2) + ((-(a*b*c) + a^2*d)*Log[a + b*x^2])/(2*b^3)","A",1
201,1,60,58,0.0565509,"\int \frac{x^2 \left(c+d x^2\right)}{a+b x^2} \, dx","IntegrateAlgebraic[(x^2*(c + d*x^2))/(a + b*x^2),x]","\frac{\left(a^{3/2} d-\sqrt{a} b c\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{5/2}}+\frac{x \left(-3 a d+3 b c+b d x^2\right)}{3 b^2}","-\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,"(x*(3*b*c - 3*a*d + b*d*x^2))/(3*b^2) + ((-(Sqrt[a]*b*c) + a^(3/2)*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(5/2)","A",1
202,1,35,35,0.0174654,"\int \frac{x \left(c+d x^2\right)}{a+b x^2} \, dx","IntegrateAlgebraic[(x*(c + d*x^2))/(a + b*x^2),x]","\frac{(b c-a d) \log \left(a+b x^2\right)}{2 b^2}+\frac{d x^2}{2 b}","\frac{(b c-a d) \log \left(a+b x^2\right)}{2 b^2}+\frac{d x^2}{2 b}",1,"(d*x^2)/(2*b) + ((b*c - a*d)*Log[a + b*x^2])/(2*b^2)","A",1
203,1,39,39,0.0005907,"\int \frac{c+d x^2}{a+b x^2} \, dx","IntegrateAlgebraic[(c + d*x^2)/(a + b*x^2),x]","\frac{(b c-a d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} b^{3/2}}+\frac{d x}{b}","\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
204,1,34,34,0.0198171,"\int \frac{c+d x^2}{x \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(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
205,1,42,43,0.0360671,"\int \frac{c+d x^2}{x^2 \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(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
206,1,49,50,0.0323099,"\int \frac{c+d x^2}{x^3 \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(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
207,1,66,59,0.0709645,"\int \frac{c+d x^2}{x^4 \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(c + d*x^2)/(x^4*(a + b*x^2)),x]","\frac{\left(b^{3/2} c-a \sqrt{b} d\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{5/2}}+\frac{-a c-3 a d x^2+3 b c x^2}{3 a^2 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,"(-(a*c) + 3*b*c*x^2 - 3*a*d*x^2)/(3*a^2*x^3) + ((b^(3/2)*c - a*Sqrt[b]*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(5/2)","A",1
208,1,145,103,0.0851785,"\int \frac{x^5 \left(c+d x^2\right)^2}{a+b x^2} \, dx","IntegrateAlgebraic[(x^5*(c + d*x^2)^2)/(a + b*x^2),x]","\frac{x^2 \left(-12 a^3 d^2+24 a^2 b c d+6 a^2 b d^2 x^2-12 a b^2 c^2-12 a b^2 c d x^2-4 a b^2 d^2 x^4+6 b^3 c^2 x^2+8 b^3 c d x^4+3 b^3 d^2 x^6\right)}{24 b^4}+\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^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,"(x^2*(-12*a*b^2*c^2 + 24*a^2*b*c*d - 12*a^3*d^2 + 6*b^3*c^2*x^2 - 12*a*b^2*c*d*x^2 + 6*a^2*b*d^2*x^2 + 8*b^3*c*d*x^4 - 4*a*b^2*d^2*x^4 + 3*b^3*d^2*x^6))/(24*b^4) + ((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
209,1,136,105,0.1293851,"\int \frac{x^4 \left(c+d x^2\right)^2}{a+b x^2} \, dx","IntegrateAlgebraic[(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{x \left(-105 a^3 d^2+210 a^2 b c d+35 a^2 b d^2 x^2-105 a b^2 c^2-70 a b^2 c d x^2-21 a b^2 d^2 x^4+35 b^3 c^2 x^2+42 b^3 c d x^4+15 b^3 d^2 x^6\right)}{105 b^4}","\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,"(x*(-105*a*b^2*c^2 + 210*a^2*b*c*d - 105*a^3*d^2 + 35*b^3*c^2*x^2 - 70*a*b^2*c*d*x^2 + 35*a^2*b*d^2*x^2 + 42*b^3*c*d*x^4 - 21*a*b^2*d^2*x^4 + 15*b^3*d^2*x^6))/(105*b^4) + (a^(3/2)*(-(b*c) + a*d)^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(9/2)","A",1
210,1,91,80,0.0613566,"\int \frac{x^3 \left(c+d x^2\right)^2}{a+b x^2} \, dx","IntegrateAlgebraic[(x^3*(c + d*x^2)^2)/(a + b*x^2),x]","\frac{x^2 \left(6 a^2 d^2-12 a b c d-3 a b d^2 x^2+6 b^2 c^2+6 b^2 c d x^2+2 b^2 d^2 x^4\right)}{12 b^3}-\frac{a (a d-b c)^2 \log \left(a+b x^2\right)}{2 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,"(x^2*(6*b^2*c^2 - 12*a*b*c*d + 6*a^2*d^2 + 6*b^2*c*d*x^2 - 3*a*b*d^2*x^2 + 2*b^2*d^2*x^4))/(12*b^3) - (a*(-(b*c) + a*d)^2*Log[a + b*x^2])/(2*b^4)","A",1
211,1,98,84,0.0956428,"\int \frac{x^2 \left(c+d x^2\right)^2}{a+b x^2} \, dx","IntegrateAlgebraic[(x^2*(c + d*x^2)^2)/(a + b*x^2),x]","\frac{x \left(15 a^2 d^2-30 a b c d-5 a b d^2 x^2+15 b^2 c^2+10 b^2 c d x^2+3 b^2 d^2 x^4\right)}{15 b^3}-\frac{\sqrt{a} (a d-b c)^2 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{7/2}}","-\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,"(x*(15*b^2*c^2 - 30*a*b*c*d + 15*a^2*d^2 + 10*b^2*c*d*x^2 - 5*a*b*d^2*x^2 + 3*b^2*d^2*x^4))/(15*b^3) - (Sqrt[a]*(-(b*c) + a*d)^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(7/2)","A",1
212,1,52,61,0.033969,"\int \frac{x \left(c+d x^2\right)^2}{a+b x^2} \, dx","IntegrateAlgebraic[(x*(c + d*x^2)^2)/(a + b*x^2),x]","\frac{(b c-a d)^2 \log \left(a+b x^2\right)}{2 b^3}+\frac{d x^2 \left(-2 a d+4 b c+b d x^2\right)}{4 b^2}","\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,"(d*x^2*(4*b*c - 2*a*d + b*d*x^2))/(4*b^2) + ((b*c - a*d)^2*Log[a + b*x^2])/(2*b^3)","A",1
213,1,59,63,0.0005792,"\int \frac{\left(c+d x^2\right)^2}{a+b x^2} \, dx","IntegrateAlgebraic[(c + d*x^2)^2/(a + b*x^2),x]","\frac{(a d-b c)^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
214,1,51,51,0.033042,"\int \frac{\left(c+d x^2\right)^2}{x \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(c + d*x^2)^2/(x*(a + b*x^2)),x]","-\frac{(a d-b c)^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}","-\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,"(d^2*x^2)/(2*b) + (c^2*Log[x])/a - ((-(b*c) + a*d)^2*Log[a + b*x^2])/(2*a*b^2)","A",1
215,1,61,55,0.0674012,"\int \frac{\left(c+d x^2\right)^2}{x^2 \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(c + d*x^2)^2/(x^2*(a + b*x^2)),x]","\frac{a d^2 x^2-b c^2}{a b x}-\frac{(a d-b c)^2 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2} b^{3/2}}","-\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,"(-(b*c^2) + a*d^2*x^2)/(a*b*x) - ((-(b*c) + a*d)^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*b^(3/2))","A",1
216,1,60,58,0.042149,"\int \frac{\left(c+d x^2\right)^2}{x^3 \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(c + d*x^2)^2/(x^3*(a + b*x^2)),x]","\frac{\log (x) \left(2 a c d-b c^2\right)}{a^2}+\frac{(a d-b c)^2 \log \left(a+b x^2\right)}{2 a^2 b}-\frac{c^2}{2 a 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,"-1/2*c^2/(a*x^2) + ((-(b*c^2) + 2*a*c*d)*Log[x])/a^2 + ((-(b*c) + a*d)^2*Log[a + b*x^2])/(2*a^2*b)","A",1
217,1,64,64,0.0877439,"\int \frac{\left(c+d x^2\right)^2}{x^4 \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(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 \left(a c+6 a d x^2-3 b c x^2\right)}{3 a^2 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*(a*c - 3*b*c*x^2 + 6*a*d*x^2))/(a^2*x^3) + ((-(b*c) + a*d)^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(5/2)*Sqrt[b])","A",1
218,1,204,138,0.1191238,"\int \frac{x^5 \left(c+d x^2\right)^3}{a+b x^2} \, dx","IntegrateAlgebraic[(x^5*(c + d*x^2)^3)/(a + b*x^2),x]","\frac{x^2 \left(60 a^4 d^3-180 a^3 b c d^2-30 a^3 b d^3 x^2+180 a^2 b^2 c^2 d+90 a^2 b^2 c d^2 x^2+20 a^2 b^2 d^3 x^4-60 a b^3 c^3-90 a b^3 c^2 d x^2-60 a b^3 c d^2 x^4-15 a b^3 d^3 x^6+30 b^4 c^3 x^2+60 b^4 c^2 d x^4+45 b^4 c d^2 x^6+12 b^4 d^3 x^8\right)}{120 b^5}-\frac{a^2 (a d-b c)^3 \log \left(a+b x^2\right)}{2 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,"(x^2*(-60*a*b^3*c^3 + 180*a^2*b^2*c^2*d - 180*a^3*b*c*d^2 + 60*a^4*d^3 + 30*b^4*c^3*x^2 - 90*a*b^3*c^2*d*x^2 + 90*a^2*b^2*c*d^2*x^2 - 30*a^3*b*d^3*x^2 + 60*b^4*c^2*d*x^4 - 60*a*b^3*c*d^2*x^4 + 20*a^2*b^2*d^3*x^4 + 45*b^4*c*d^2*x^6 - 15*a*b^3*d^3*x^6 + 12*b^4*d^3*x^8))/(120*b^5) - (a^2*(-(b*c) + a*d)^3*Log[a + b*x^2])/(2*b^6)","A",1
219,1,209,140,0.094697,"\int \frac{x^4 \left(c+d x^2\right)^3}{a+b x^2} \, dx","IntegrateAlgebraic[(x^4*(c + d*x^2)^3)/(a + b*x^2),x]","\frac{x \left(315 a^4 d^3-945 a^3 b c d^2-105 a^3 b d^3 x^2+945 a^2 b^2 c^2 d+315 a^2 b^2 c d^2 x^2+63 a^2 b^2 d^3 x^4-315 a b^3 c^3-315 a b^3 c^2 d x^2-189 a b^3 c d^2 x^4-45 a b^3 d^3 x^6+105 b^4 c^3 x^2+189 b^4 c^2 d x^4+135 b^4 c d^2 x^6+35 b^4 d^3 x^8\right)}{315 b^5}-\frac{a^{3/2} (a d-b c)^3 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{11/2}}","\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,"(x*(-315*a*b^3*c^3 + 945*a^2*b^2*c^2*d - 945*a^3*b*c*d^2 + 315*a^4*d^3 + 105*b^4*c^3*x^2 - 315*a*b^3*c^2*d*x^2 + 315*a^2*b^2*c*d^2*x^2 - 105*a^3*b*d^3*x^2 + 189*b^4*c^2*d*x^4 - 189*a*b^3*c*d^2*x^4 + 63*a^2*b^2*d^3*x^4 + 135*b^4*c*d^2*x^6 - 45*a*b^3*d^3*x^6 + 35*b^4*d^3*x^8))/(315*b^5) - (a^(3/2)*(-(b*c) + a*d)^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(11/2)","A",1
220,1,146,115,0.0906631,"\int \frac{x^3 \left(c+d x^2\right)^3}{a+b x^2} \, dx","IntegrateAlgebraic[(x^3*(c + d*x^2)^3)/(a + b*x^2),x]","\frac{x^2 \left(-12 a^3 d^3+36 a^2 b c d^2+6 a^2 b d^3 x^2-36 a b^2 c^2 d-18 a b^2 c d^2 x^2-4 a b^2 d^3 x^4+12 b^3 c^3+18 b^3 c^2 d x^2+12 b^3 c d^2 x^4+3 b^3 d^3 x^6\right)}{24 b^4}+\frac{a (a d-b c)^3 \log \left(a+b x^2\right)}{2 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,"(x^2*(12*b^3*c^3 - 36*a*b^2*c^2*d + 36*a^2*b*c*d^2 - 12*a^3*d^3 + 18*b^3*c^2*d*x^2 - 18*a*b^2*c*d^2*x^2 + 6*a^2*b*d^3*x^2 + 12*b^3*c*d^2*x^4 - 4*a*b^2*d^3*x^4 + 3*b^3*d^3*x^6))/(24*b^4) + (a*(-(b*c) + a*d)^3*Log[a + b*x^2])/(2*b^5)","A",1
221,1,152,119,0.0714523,"\int \frac{x^2 \left(c+d x^2\right)^3}{a+b x^2} \, dx","IntegrateAlgebraic[(x^2*(c + d*x^2)^3)/(a + b*x^2),x]","\frac{x \left(-105 a^3 d^3+315 a^2 b c d^2+35 a^2 b d^3 x^2-315 a b^2 c^2 d-105 a b^2 c d^2 x^2-21 a b^2 d^3 x^4+105 b^3 c^3+105 b^3 c^2 d x^2+63 b^3 c d^2 x^4+15 b^3 d^3 x^6\right)}{105 b^4}+\frac{\sqrt{a} (a d-b c)^3 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{9/2}}","\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,"(x*(105*b^3*c^3 - 315*a*b^2*c^2*d + 315*a^2*b*c*d^2 - 105*a^3*d^3 + 105*b^3*c^2*d*x^2 - 105*a*b^2*c*d^2*x^2 + 35*a^2*b*d^3*x^2 + 63*b^3*c*d^2*x^4 - 21*a*b^2*d^3*x^4 + 15*b^3*d^3*x^6))/(105*b^4) + (Sqrt[a]*(-(b*c) + a*d)^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(9/2)","A",1
222,1,91,87,0.0514404,"\int \frac{x \left(c+d x^2\right)^3}{a+b x^2} \, dx","IntegrateAlgebraic[(x*(c + d*x^2)^3)/(a + b*x^2),x]","\frac{d x^2 \left(6 a^2 d^2-18 a b c d-3 a b d^2 x^2+18 b^2 c^2+9 b^2 c d x^2+2 b^2 d^2 x^4\right)}{12 b^3}+\frac{(b c-a d)^3 \log \left(a+b x^2\right)}{2 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,"(d*x^2*(18*b^2*c^2 - 18*a*b*c*d + 6*a^2*d^2 + 9*b^2*c*d*x^2 - 3*a*b*d^2*x^2 + 2*b^2*d^2*x^4))/(12*b^3) + ((b*c - a*d)^3*Log[a + b*x^2])/(2*b^4)","A",1
223,1,99,98,0.0005878,"\int \frac{\left(c+d x^2\right)^3}{a+b x^2} \, dx","IntegrateAlgebraic[(c + d*x^2)^3/(a + b*x^2),x]","\frac{d x \left(15 a^2 d^2-45 a b c d-5 a b d^2 x^2+45 b^2 c^2+15 b^2 c d x^2+3 b^2 d^2 x^4\right)}{15 b^3}-\frac{(a d-b c)^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*(45*b^2*c^2 - 45*a*b*c*d + 15*a^2*d^2 + 15*b^2*c*d*x^2 - 5*a*b*d^2*x^2 + 3*b^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
224,1,66,73,0.0462059,"\int \frac{\left(c+d x^2\right)^3}{x \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(c + d*x^2)^3/(x*(a + b*x^2)),x]","\frac{(a d-b c)^3 \log \left(a+b x^2\right)}{2 a b^3}+\frac{d^2 x^2 \left(-2 a d+6 b c+b d x^2\right)}{4 b^2}+\frac{c^3 \log (x)}{a}","-\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,"(d^2*x^2*(6*b*c - 2*a*d + b*d*x^2))/(4*b^2) + (c^3*Log[x])/a + ((-(b*c) + a*d)^3*Log[a + b*x^2])/(2*a*b^3)","A",1
225,1,88,77,0.0575184,"\int \frac{\left(c+d x^2\right)^3}{x^2 \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(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{-3 a^2 d^3 x^2+9 a b c d^2 x^2+a b d^3 x^4-3 b^2 c^3}{3 a b^2 x}","-\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,"(-3*b^2*c^3 + 9*a*b*c*d^2*x^2 - 3*a^2*d^3*x^2 + a*b*d^3*x^4)/(3*a*b^2*x) + ((-(b*c) + a*d)^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*b^(5/2))","A",1
226,1,77,73,0.0567139,"\int \frac{\left(c+d x^2\right)^3}{x^3 \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(c + d*x^2)^3/(x^3*(a + b*x^2)),x]","-\frac{(a d-b c)^3 \log \left(a+b x^2\right)}{2 a^2 b^2}+\frac{\log (x) \left(3 a c^2 d-b c^3\right)}{a^2}+\frac{a d^3 x^4-b c^3}{2 a b 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,"(-(b*c^3) + a*d^3*x^4)/(2*a*b*x^2) + ((-(b*c^3) + 3*a*c^2*d)*Log[x])/a^2 - ((-(b*c) + a*d)^3*Log[a + b*x^2])/(2*a^2*b^2)","A",1
227,1,90,74,0.0664375,"\int \frac{\left(c+d x^2\right)^3}{x^4 \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(c + d*x^2)^3/(x^4*(a + b*x^2)),x]","\frac{3 a^2 d^3 x^4-a b c^3-9 a b c^2 d x^2+3 b^2 c^3 x^2}{3 a^2 b x^3}-\frac{(a d-b c)^3 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{5/2} b^{3/2}}","\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,"(-(a*b*c^3) + 3*b^2*c^3*x^2 - 9*a*b*c^2*d*x^2 + 3*a^2*d^3*x^4)/(3*a^2*b*x^3) - ((-(b*c) + a*d)^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(5/2)*b^(3/2))","A",1
228,1,70,70,0.0478735,"\int \frac{x^5}{\left(a+b x^2\right) \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[x^5/((a + b*x^2)*(c + d*x^2)),x]","\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 (a d-b c)}+\frac{x^2}{2 b 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,"x^2/(2*b*d) + (a^2*Log[a + b*x^2])/(2*b^2*(b*c - a*d)) + (c^2*Log[c + d*x^2])/(2*d^2*(-(b*c) + a*d))","A",1
229,1,77,78,0.1250095,"\int \frac{x^4}{\left(a+b x^2\right) \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[x^4/((a + b*x^2)*(c + d*x^2)),x]","\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} (a d-b c)}+\frac{x}{b 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,"x/(b*d) + (a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(b^(3/2)*(b*c - a*d)) + (c^(3/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(d^(3/2)*(-(b*c) + a*d))","A",1
230,1,53,53,0.0332939,"\int \frac{x^3}{\left(a+b x^2\right) \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[x^3/((a + b*x^2)*(c + d*x^2)),x]","-\frac{a \log \left(a+b x^2\right)}{2 b (b c-a d)}-\frac{c \log \left(c+d x^2\right)}{2 d (a d-b c)}","\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,"-1/2*(a*Log[a + b*x^2])/(b*(b*c - a*d)) - (c*Log[c + d*x^2])/(2*d*(-(b*c) + a*d))","A",1
231,1,71,70,0.0692176,"\int \frac{x^2}{\left(a+b x^2\right) \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[x^2/((a + b*x^2)*(c + d*x^2)),x]","-\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} (a d-b c)}","\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]*(b*c - a*d))) - (Sqrt[c]*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[d]*(-(b*c) + a*d))","A",1
232,1,45,45,0.0291744,"\int \frac{x}{\left(a+b x^2\right) \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[x/((a + b*x^2)*(c + d*x^2)),x]","\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)}","\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]/(2*(b*c - a*d)) - Log[c + d*x^2]/(2*(b*c - a*d))","A",1
233,1,71,70,0.0005737,"\int \frac{1}{\left(a+b x^2\right) \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[1/((a + b*x^2)*(c + d*x^2)),x]","-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} (a d-b c)}-\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]*(-(b*c) + a*d))) - (Sqrt[d]*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*(b*c - a*d))","A",1
234,1,62,62,0.0481951,"\int \frac{1}{x \left(a+b x^2\right) \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[1/(x*(a + b*x^2)*(c + d*x^2)),x]","\frac{b \log \left(a+b x^2\right)}{2 a (a d-b c)}+\frac{d \log \left(c+d x^2\right)}{2 c (b c-a d)}+\frac{\log (x)}{a c}","-\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,"Log[x]/(a*c) + (b*Log[a + b*x^2])/(2*a*(-(b*c) + a*d)) + (d*Log[c + d*x^2])/(2*c*(b*c - a*d))","A",1
235,1,80,81,0.1166086,"\int \frac{1}{x^2 \left(a+b x^2\right) \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[1/(x^2*(a + b*x^2)*(c + d*x^2)),x]","\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2} (a d-b c)}+\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}","-\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,"-(1/(a*c*x)) + (b^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*(-(b*c) + a*d)) + (d^(3/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(3/2)*(b*c - a*d))","A",1
236,1,88,87,0.0624487,"\int \frac{1}{x^3 \left(a+b x^2\right) \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[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
237,1,103,100,0.1650358,"\int \frac{1}{x^4 \left(a+b x^2\right) \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[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 c+3 a d x^2+3 b c x^2}{3 a^2 c^2 x^3}-\frac{d^{5/2} \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{5/2} (b c-a d)}","\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,"(-(a*c) + 3*b*c*x^2 + 3*a*d*x^2)/(3*a^2*c^2*x^3) - (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
238,1,118,119,0.0830957,"\int \frac{1}{x^5 \left(a+b x^2\right) \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[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 c+2 a d x^2+2 b c x^2}{4 a^2 c^2 x^4}+\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{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,"(-(a*c) + 2*b*c*x^2 + 2*a*d*x^2)/(4*a^2*c^2*x^4) + ((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
239,1,142,134,0.1751526,"\int \frac{1}{x^6 \left(a+b x^2\right) \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[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{-3 a^2 c^2+5 a^2 c d x^2-15 a^2 d^2 x^4+5 a b c^2 x^2-15 a b c d x^4-15 b^2 c^2 x^4}{15 a^3 c^3 x^5}+\frac{d^{7/2} \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{7/2} (b c-a d)}","-\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,"(-3*a^2*c^2 + 5*a*b*c^2*x^2 + 5*a^2*c*d*x^2 - 15*b^2*c^2*x^4 - 15*a*b*c*d*x^4 - 15*a^2*d^2*x^4)/(15*a^3*c^3*x^5) + (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
240,1,176,155,0.1174958,"\int \frac{1}{x^7 \left(a+b x^2\right) \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[1/(x^7*(a + b*x^2)*(c + d*x^2)),x]","-\frac{b^4 \log \left(a+b x^2\right)}{2 a^4 (a d-b c)}+\frac{-2 a^2 c^2+3 a^2 c d x^2-6 a^2 d^2 x^4+3 a b c^2 x^2-6 a b c d x^4-6 b^2 c^2 x^4}{12 a^3 c^3 x^6}+\frac{\log (x) \left(-a^3 d^3-a^2 b c d^2-a b^2 c^2 d-b^3 c^3\right)}{a^4 c^4}-\frac{d^4 \log \left(c+d x^2\right)}{2 c^4 (b c-a d)}","\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,"(-2*a^2*c^2 + 3*a*b*c^2*x^2 + 3*a^2*c*d*x^2 - 6*b^2*c^2*x^4 - 6*a*b*c*d*x^4 - 6*a^2*d^2*x^4)/(12*a^3*c^3*x^6) + ((-(b^3*c^3) - a*b^2*c^2*d - a^2*b*c*d^2 - a^3*d^3)*Log[x])/(a^4*c^4) - (b^4*Log[a + b*x^2])/(2*a^4*(-(b*c) + a*d)) - (d^4*Log[c + d*x^2])/(2*c^4*(b*c - a*d))","A",1
241,1,94,93,0.0781839,"\int \frac{x^5}{\left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[x^5/((a + b*x^2)^2*(c + d*x^2)),x]","-\frac{a^2}{2 b^2 \left(a+b x^2\right) (b c-a d)}+\frac{\left(a^2 d-2 a b c\right) \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 (a d-b c)^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,"-1/2*a^2/(b^2*(b*c - a*d)*(a + b*x^2)) + ((-2*a*b*c + a^2*d)*Log[a + b*x^2])/(2*b^2*(b*c - a*d)^2) + (c^2*Log[c + d*x^2])/(2*d*(-(b*c) + a*d)^2)","A",1
242,1,112,108,0.1748283,"\int \frac{x^4}{\left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[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} (b c-a d)^2}+\frac{\left(b c^{3/2}-3 a \sqrt{c} d\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 d^{3/2} (a d-b c)^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) + ((b*c^(3/2) - 3*a*Sqrt[c]*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*d^(3/2)*(-(b*c) + a*d)^2)","A",1
243,1,74,74,0.0546161,"\int \frac{x^3}{\left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[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
244,1,104,104,0.1680519,"\int \frac{x^2}{\left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[x^2/((a + b*x^2)*(c + d*x^2)^2),x]","\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}","\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,"x/(2*(b*c - a*d)*(c + d*x^2)) - (Sqrt[a]*Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(b*c - a*d)^2 + ((b*c + a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*Sqrt[c]*Sqrt[d]*(b*c - a*d)^2)","A",1
245,1,70,70,0.0541428,"\int \frac{x}{\left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[x/((a + b*x^2)*(c + d*x^2)^2),x]","\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}","\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,"1/(2*(b*c - a*d)*(c + d*x^2)) + (b*Log[a + b*x^2])/(2*(b*c - a*d)^2) - (b*Log[c + d*x^2])/(2*(b*c - a*d)^2)","A",1
246,1,112,109,0.00058,"\int \frac{1}{\left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[1/((a + b*x^2)*(c + d*x^2)^2),x]","\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} (a d-b c)^2}+\frac{\left(a d^{3/2}-3 b c \sqrt{d}\right) \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)}","\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,"-1/2*(d*x)/(c*(b*c - a*d)*(c + d*x^2)) + (b^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*(-(b*c) + a*d)^2) + ((-3*b*c*Sqrt[d] + a*d^(3/2))*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(3/2)*(b*c - a*d)^2)","A",1
247,1,102,100,0.1203304,"\int \frac{1}{x \left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[1/(x*(a + b*x^2)*(c + d*x^2)^2),x]","-\frac{b^2 \log \left(a+b x^2\right)}{2 a (a d-b c)^2}+\frac{\left(2 b c d-a d^2\right) \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}","-\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,"-1/2*d/(c*(b*c - a*d)*(c + d*x^2)) + Log[x]/(a*c^2) - (b^2*Log[a + b*x^2])/(2*a*(-(b*c) + a*d)^2) + ((2*b*c*d - a*d^2)*Log[c + d*x^2])/(2*c^2*(b*c - a*d)^2)","A",1
248,1,147,144,0.3162939,"\int \frac{1}{x^2 \left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[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{\left(5 b c d^{3/2}-3 a d^{5/2}\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{5/2} (b c-a d)^2}+\frac{-2 a c d-3 a d^2 x^2+2 b c^2+2 b c d x^2}{2 a c^2 x \left(c+d x^2\right) (a d-b c)}","-\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,"(2*b*c^2 - 2*a*c*d + 2*b*c*d*x^2 - 3*a*d^2*x^2)/(2*a*c^2*(-(b*c) + a*d)*x*(c + d*x^2)) - (b^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*(-(b*c) + a*d)^2) + ((5*b*c*d^(3/2) - 3*a*d^(5/2))*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(5/2)*(b*c - a*d)^2)","A",1
249,1,145,126,0.2949456,"\int \frac{1}{x^3 \left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[1/(x^3*(a + b*x^2)*(c + d*x^2)^2),x]","\frac{b^3 \log \left(a+b x^2\right)}{2 a^2 (a d-b c)^2}+\frac{\log (x) (-2 a d-b c)}{a^2 c^3}+\frac{\left(2 a d^3-3 b c d^2\right) \log \left(c+d x^2\right)}{2 c^3 (b c-a d)^2}+\frac{-a c d-2 a d^2 x^2+b c^2+b c d x^2}{2 a c^2 x^2 \left(c+d x^2\right) (a d-b c)}","\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,"(b*c^2 - a*c*d + b*c*d*x^2 - 2*a*d^2*x^2)/(2*a*c^2*(-(b*c) + a*d)*x^2*(c + d*x^2)) + ((-(b*c) - 2*a*d)*Log[x])/(a^2*c^3) + (b^3*Log[a + b*x^2])/(2*a^2*(-(b*c) + a*d)^2) + ((-3*b*c*d^2 + 2*a*d^3)*Log[c + d*x^2])/(2*c^3*(b*c - a*d)^2)","A",1
250,1,202,189,0.4584791,"\int \frac{1}{x^4 \left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[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^2 c^2 d+10 a^2 c d^2 x^2+15 a^2 d^3 x^4+2 a b c^3-4 a b c^2 d x^2-6 a b c d^2 x^4-6 b^2 c^3 x^2-6 b^2 c^2 d x^4}{6 a^2 c^3 x^3 \left(c+d x^2\right) (a d-b c)}+\frac{\left(5 a d^{7/2}-7 b c d^{5/2}\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{7/2} (b c-a d)^2}","\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,"(2*a*b*c^3 - 2*a^2*c^2*d - 6*b^2*c^3*x^2 - 4*a*b*c^2*d*x^2 + 10*a^2*c*d^2*x^2 - 6*b^2*c^2*d*x^4 - 6*a*b*c*d^2*x^4 + 15*a^2*d^3*x^4)/(6*a^2*c^3*(-(b*c) + a*d)*x^3*(c + d*x^2)) + (b^(7/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(5/2)*(-(b*c) + a*d)^2) + ((-7*b*c*d^(5/2) + 5*a*d^(7/2))*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(7/2)*(b*c - a*d)^2)","A",1
251,1,106,116,0.143968,"\int \frac{x^5}{\left(a+b x^2\right)^3 \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[x^5/((a + b*x^2)^3*(c + d*x^2)),x]","-\frac{a \left(a^2 d-3 a b c+2 a b d x^2-4 b^2 c x^2\right)}{4 b^2 \left(a+b x^2\right)^2 (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}","-\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,"-1/4*(a*(-3*a*b*c + a^2*d - 4*b^2*c*x^2 + 2*a*b*d*x^2))/(b^2*(b*c - a*d)^2*(a + b*x^2)^2) + (c^2*Log[a + b*x^2])/(2*(b*c - a*d)^3) - (c^2*Log[c + d*x^2])/(2*(b*c - a*d)^3)","A",1
252,1,149,157,0.3175663,"\int \frac{x^4}{\left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[x^4/((a + b*x^2)*(c + d*x^2)^3),x]","\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 \left(3 a c d+5 a d^2 x^2+b c^2-b c d x^2\right)}{8 d \left(c+d x^2\right)^2 (a d-b c)^2}","\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,"-1/8*(x*(b*c^2 + 3*a*c*d - b*c*d*x^2 + 5*a*d^2*x^2))/(d*(-(b*c) + a*d)^2*(c + d*x^2)^2) + (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]])/(8*Sqrt[c]*d^(3/2)*(b*c - a*d)^3)","A",1
253,1,96,100,0.1489097,"\int \frac{x^3}{\left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[x^3/((a + b*x^2)*(c + d*x^2)^3),x]","\frac{-a c d-2 a d^2 x^2-b c^2}{4 d \left(c+d x^2\right)^2 (a d-b c)^2}-\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}","-\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^2) - a*c*d - 2*a*d^2*x^2)/(4*d*(-(b*c) + a*d)^2*(c + d*x^2)^2) - (a*b*Log[a + b*x^2])/(2*(b*c - a*d)^3) + (a*b*Log[c + d*x^2])/(2*(b*c - a*d)^3)","A",1
254,1,151,155,0.2907514,"\int \frac{x^2}{\left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[x^2/((a + b*x^2)*(c + d*x^2)^3),x]","\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 \left(-a c d+a d^2 x^2+5 b c^2+3 b c d x^2\right)}{8 c \left(c+d x^2\right)^2 (b c-a d)^2}","\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,"(x*(5*b*c^2 - a*c*d + 3*b*c*d*x^2 + a*d^2*x^2))/(8*c*(b*c - a*d)^2*(c + d*x^2)^2) - (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]])/(8*c^(3/2)*Sqrt[d]*(b*c - a*d)^3)","A",1
255,1,90,98,0.0764919,"\int \frac{x}{\left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[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{-a d+3 b c+2 b d x^2}{4 \left(c+d x^2\right)^2 (b c-a d)^2}","\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,"(3*b*c - a*d + 2*b*d*x^2)/(4*(b*c - a*d)^2*(c + d*x^2)^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
256,1,159,160,0.0006119,"\int \frac{1}{\left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[1/((a + b*x^2)*(c + d*x^2)^3),x]","\frac{\left(-3 a^2 d^{5/2}+10 a b c d^{3/2}-15 b^2 c^2 \sqrt{d}\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} (a d-b c)^3}-\frac{d x \left(-5 a c d-3 a d^2 x^2+9 b c^2+7 b c d x^2\right)}{8 c^2 \left(c+d x^2\right)^2 (b c-a d)^2}","-\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,"-1/8*(d*x*(9*b*c^2 - 5*a*c*d + 7*b*c*d*x^2 - 3*a*d^2*x^2))/(c^2*(b*c - a*d)^2*(c + d*x^2)^2) - (b^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*(-(b*c) + a*d)^3) + ((-15*b^2*c^2*Sqrt[d] + 10*a*b*c*d^(3/2) - 3*a^2*d^(5/2))*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(5/2)*(b*c - a*d)^3)","A",1
257,1,144,149,0.3377851,"\int \frac{1}{x \left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[1/(x*(a + b*x^2)*(c + d*x^2)^3),x]","\frac{\left(a^2 d^3-3 a b c d^2+3 b^2 c^2 d\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 (a d-b c)^3}-\frac{d \left(-3 a c d-2 a d^2 x^2+5 b c^2+4 b c d x^2\right)}{4 c^2 \left(c+d x^2\right)^2 (b c-a d)^2}+\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,"-1/4*(d*(5*b*c^2 - 3*a*c*d + 4*b*c*d*x^2 - 2*a*d^2*x^2))/(c^2*(b*c - a*d)^2*(c + d*x^2)^2) + Log[x]/(a*c^3) + (b^3*Log[a + b*x^2])/(2*a*(-(b*c) + a*d)^3) + ((3*b^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3)*Log[c + d*x^2])/(2*c^3*(b*c - a*d)^3)","A",1
258,1,234,211,0.478245,"\int \frac{1}{x^2 \left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[1/(x^2*(a + b*x^2)*(c + d*x^2)^3),x]","\frac{b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2} (a d-b c)^3}+\frac{\left(15 a^2 d^{7/2}-42 a b c d^{5/2}+35 b^2 c^2 d^{3/2}\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{7/2} (b c-a d)^3}+\frac{-8 a^2 c^2 d^2-25 a^2 c d^3 x^2-15 a^2 d^4 x^4+16 a b c^3 d+45 a b c^2 d^2 x^2+27 a b c d^3 x^4-8 b^2 c^4-16 b^2 c^3 d x^2-8 b^2 c^2 d^2 x^4}{8 a c^3 x \left(c+d x^2\right)^2 (a d-b c)^2}","-\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*b^2*c^4 + 16*a*b*c^3*d - 8*a^2*c^2*d^2 - 16*b^2*c^3*d*x^2 + 45*a*b*c^2*d^2*x^2 - 25*a^2*c*d^3*x^2 - 8*b^2*c^2*d^2*x^4 + 27*a*b*c*d^3*x^4 - 15*a^2*d^4*x^4)/(8*a*c^3*(-(b*c) + a*d)^2*x*(c + d*x^2)^2) + (b^(7/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*(-(b*c) + a*d)^3) + ((35*b^2*c^2*d^(3/2) - 42*a*b*c*d^(5/2) + 15*a^2*d^(7/2))*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(7/2)*(b*c - a*d)^3)","A",1
259,1,233,178,0.5166037,"\int \frac{1}{x^3 \left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[1/(x^3*(a + b*x^2)*(c + d*x^2)^3),x]","-\frac{b^4 \log \left(a+b x^2\right)}{2 a^2 (a d-b c)^3}+\frac{\left(-3 a^2 d^4+8 a b c d^3-6 b^2 c^2 d^2\right) \log \left(c+d x^2\right)}{2 c^4 (b c-a d)^3}+\frac{-2 a^2 c^2 d^2-9 a^2 c d^3 x^2-6 a^2 d^4 x^4+4 a b c^3 d+15 a b c^2 d^2 x^2+10 a b c d^3 x^4-2 b^2 c^4-4 b^2 c^3 d x^2-2 b^2 c^2 d^2 x^4}{4 a c^3 x^2 \left(c+d x^2\right)^2 (a d-b c)^2}+\frac{\log (x) (-3 a d-b c)}{a^2 c^4}","\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*b^2*c^4 + 4*a*b*c^3*d - 2*a^2*c^2*d^2 - 4*b^2*c^3*d*x^2 + 15*a*b*c^2*d^2*x^2 - 9*a^2*c*d^3*x^2 - 2*b^2*c^2*d^2*x^4 + 10*a*b*c*d^3*x^4 - 6*a^2*d^4*x^4)/(4*a*c^3*(-(b*c) + a*d)^2*x^2*(c + d*x^2)^2) + ((-(b*c) - 3*a*d)*Log[x])/(a^2*c^4) - (b^4*Log[a + b*x^2])/(2*a^2*(-(b*c) + a*d)^3) + ((-6*b^2*c^2*d^2 + 8*a*b*c*d^3 - 3*a^2*d^4)*Log[c + d*x^2])/(2*c^4*(b*c - a*d)^3)","A",1
260,1,325,270,0.5582949,"\int \frac{1}{x^4 \left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[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{\left(-35 a^2 d^{9/2}+90 a b c d^{7/2}-63 b^2 c^2 d^{5/2}\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{9/2} (b c-a d)^3}+\frac{-8 a^3 c^3 d^2+56 a^3 c^2 d^3 x^2+175 a^3 c d^4 x^4+105 a^3 d^5 x^6+16 a^2 b c^4 d-88 a^2 b c^3 d^2 x^2-275 a^2 b c^2 d^3 x^4-165 a^2 b c d^4 x^6-8 a b^2 c^5+8 a b^2 c^4 d x^2+40 a b^2 c^3 d^2 x^4+24 a b^2 c^2 d^3 x^6+24 b^3 c^5 x^2+48 b^3 c^4 d x^4+24 b^3 c^3 d^2 x^6}{24 a^2 c^4 x^3 \left(c+d x^2\right)^2 (a d-b c)^2}","\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,"(-8*a*b^2*c^5 + 16*a^2*b*c^4*d - 8*a^3*c^3*d^2 + 24*b^3*c^5*x^2 + 8*a*b^2*c^4*d*x^2 - 88*a^2*b*c^3*d^2*x^2 + 56*a^3*c^2*d^3*x^2 + 48*b^3*c^4*d*x^4 + 40*a*b^2*c^3*d^2*x^4 - 275*a^2*b*c^2*d^3*x^4 + 175*a^3*c*d^4*x^4 + 24*b^3*c^3*d^2*x^6 + 24*a*b^2*c^2*d^3*x^6 - 165*a^2*b*c*d^4*x^6 + 105*a^3*d^5*x^6)/(24*a^2*c^4*(-(b*c) + a*d)^2*x^3*(c + d*x^2)^2) - (b^(9/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(5/2)*(-(b*c) + a*d)^3) + ((-63*b^2*c^2*d^(5/2) + 90*a*b*c*d^(7/2) - 35*a^2*d^(9/2))*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(9/2)*(b*c - a*d)^3)","A",1
261,1,16,21,0.0089433,"\int \frac{x}{\left(1+x^2\right) \left(4+x^2\right)} \, dx","IntegrateAlgebraic[x/((1 + x^2)*(4 + x^2)),x]","-\frac{1}{3} \tanh ^{-1}\left(\frac{2 x^2}{3}+\frac{5}{3}\right)","\frac{1}{6} \log \left(x^2+1\right)-\frac{1}{6} \log \left(x^2+4\right)",1,"-1/3*ArcTanh[5/3 + (2*x^2)/3]","A",1
262,1,97,87,0.098916,"\int \frac{x^4 \left(c+d x^2\right)}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^4*(c + d*x^2))/(a + b*x^2)^2,x]","\frac{\left(5 a^{3/2} d-3 \sqrt{a} b c\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{7/2}}+\frac{-15 a^2 d x+9 a b c x-10 a b d x^3+6 b^2 c x^3+2 b^2 d x^5}{6 b^3 \left(a+b x^2\right)}","-\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,"(9*a*b*c*x - 15*a^2*d*x + 6*b^2*c*x^3 - 10*a*b*d*x^3 + 2*b^2*d*x^5)/(6*b^3*(a + b*x^2)) + ((-3*Sqrt[a]*b*c + 5*a^(3/2)*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(7/2))","A",1
263,1,66,60,0.0519589,"\int \frac{x^3 \left(c+d x^2\right)}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^3*(c + d*x^2))/(a + b*x^2)^2,x]","\frac{-a^2 d+a b c+a b d x^2+b^2 d x^4}{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{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,"(a*b*c - a^2*d + a*b*d*x^2 + b^2*d*x^4)/(2*b^3*(a + b*x^2)) + ((b*c - 2*a*d)*Log[a + b*x^2])/(2*b^3)","A",1
264,1,69,67,0.0852643,"\int \frac{x^2 \left(c+d x^2\right)}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^2*(c + d*x^2))/(a + b*x^2)^2,x]","\frac{(b c-3 a d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 \sqrt{a} b^{5/2}}+\frac{x \left(3 a d-b c+2 b d x^2\right)}{2 b^2 \left(a+b x^2\right)}","\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,"(x*(-(b*c) + 3*a*d + 2*b*d*x^2))/(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
265,1,41,41,0.0212891,"\int \frac{x \left(c+d x^2\right)}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(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
266,1,63,63,0.0005687,"\int \frac{c+d x^2}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(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
267,1,51,51,0.0411127,"\int \frac{c+d x^2}{x \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2)/(x*(a + b*x^2)^2),x]","-\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)}","-\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,"(b*c - a*d)/(2*a*b*(a + b*x^2)) + (c*Log[x])/a^2 - (c*Log[a + b*x^2])/(2*a^2)","A",1
268,1,73,71,0.0498059,"\int \frac{c+d x^2}{x^2 \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(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{-2 a c+a d x^2-3 b c x^2}{2 a^2 x \left(a+b x^2\right)}","-\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,"(-2*a*c - 3*b*c*x^2 + a*d*x^2)/(2*a^2*x*(a + b*x^2)) + ((-3*b*c + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(5/2)*Sqrt[b])","A",1
269,1,76,76,0.065908,"\int \frac{c+d x^2}{x^3 \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2)/(x^3*(a + b*x^2)^2),x]","\frac{(2 b c-a d) \log \left(a+b x^2\right)}{2 a^3}+\frac{\log (x) (a d-2 b c)}{a^3}+\frac{-a c+a d x^2-2 b c x^2}{2 a^2 x^2 \left(a+b x^2\right)}","\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) - 2*b*c*x^2 + a*d*x^2)/(2*a^2*x^2*(a + b*x^2)) + ((-2*b*c + a*d)*Log[x])/a^3 + ((2*b*c - a*d)*Log[a + b*x^2])/(2*a^3)","A",1
270,1,101,90,0.0964789,"\int \frac{c+d x^2}{x^4 \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2)/(x^4*(a + b*x^2)^2),x]","\frac{\left(5 b^{3/2} c-3 a \sqrt{b} d\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{7/2}}+\frac{-2 a^2 c-6 a^2 d x^2+10 a b c x^2-9 a b d x^4+15 b^2 c x^4}{6 a^3 x^3 \left(a+b x^2\right)}","\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,"(-2*a^2*c + 10*a*b*c*x^2 - 6*a^2*d*x^2 + 15*b^2*c*x^4 - 9*a*b*d*x^4)/(6*a^3*x^3*(a + b*x^2)) + ((5*b^(3/2)*c - 3*a*Sqrt[b]*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(7/2))","A",1
271,1,167,145,0.1270152,"\int \frac{x^4 \left(c+d x^2\right)^2}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^4*(c + d*x^2)^2)/(a + b*x^2)^2,x]","\frac{\left(10 a^{3/2} b c d-7 a^{5/2} d^2-3 \sqrt{a} b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{9/2}}+\frac{x \left(105 a^3 d^2-150 a^2 b c d+70 a^2 b d^2 x^2+45 a b^2 c^2-100 a b^2 c d x^2-14 a b^2 d^2 x^4+30 b^3 c^2 x^2+20 b^3 c d x^4+6 b^3 d^2 x^6\right)}{30 b^4 \left(a+b x^2\right)}","-\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,"(x*(45*a*b^2*c^2 - 150*a^2*b*c*d + 105*a^3*d^2 + 30*b^3*c^2*x^2 - 100*a*b^2*c*d*x^2 + 70*a^2*b*d^2*x^2 + 20*b^3*c*d*x^4 - 14*a*b^2*d^2*x^4 + 6*b^3*d^2*x^6))/(30*b^4*(a + b*x^2)) + ((-3*Sqrt[a]*b^2*c^2 + 10*a^(3/2)*b*c*d - 7*a^(5/2)*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(9/2))","A",1
272,1,135,88,0.0886617,"\int \frac{x^3 \left(c+d x^2\right)^2}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^3*(c + d*x^2)^2)/(a + b*x^2)^2,x]","\frac{\left(3 a^2 d^2-4 a b c d+b^2 c^2\right) \log \left(a+b x^2\right)}{2 b^4}+\frac{2 a^3 d^2-4 a^2 b c d-4 a^2 b d^2 x^2+2 a b^2 c^2+4 a b^2 c d x^2-3 a b^2 d^2 x^4+4 b^3 c d x^4+b^3 d^2 x^6}{4 b^4 \left(a+b x^2\right)}","\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,"(2*a*b^2*c^2 - 4*a^2*b*c*d + 2*a^3*d^2 + 4*a*b^2*c*d*x^2 - 4*a^2*b*d^2*x^2 + 4*b^3*c*d*x^4 - 3*a*b^2*d^2*x^4 + b^3*d^2*x^6)/(4*b^4*(a + b*x^2)) + ((b^2*c^2 - 4*a*b*c*d + 3*a^2*d^2)*Log[a + b*x^2])/(2*b^4)","A",1
273,1,121,116,0.0996241,"\int \frac{x^2 \left(c+d x^2\right)^2}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(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 \left(-15 a^2 d^2+18 a b c d-10 a b d^2 x^2-3 b^2 c^2+12 b^2 c d x^2+2 b^2 d^2 x^4\right)}{6 b^3 \left(a+b x^2\right)}","\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,"(x*(-3*b^2*c^2 + 18*a*b*c*d - 15*a^2*d^2 + 12*b^2*c*d*x^2 - 10*a*b*d^2*x^2 + 2*b^2*d^2*x^4))/(6*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
274,1,82,61,0.0684987,"\int \frac{x \left(c+d x^2\right)^2}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x*(c + d*x^2)^2)/(a + b*x^2)^2,x]","\frac{-a^2 d^2+2 a b c d+a b d^2 x^2-b^2 c^2+b^2 d^2 x^4}{2 b^3 \left(a+b x^2\right)}+\frac{\left(b c d-a d^2\right) \log \left(a+b x^2\right)}{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^2*c^2) + 2*a*b*c*d - a^2*d^2 + a*b*d^2*x^2 + b^2*d^2*x^4)/(2*b^3*(a + b*x^2)) + ((b*c*d - a*d^2)*Log[a + b*x^2])/b^3","A",1
275,1,102,82,0.000587,"\int \frac{\left(c+d x^2\right)^2}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2)^2/(a + b*x^2)^2,x]","\frac{x \left(3 a^2 d^2-2 a b c d+2 a b d^2 x^2+b^2 c^2\right)}{2 a b^2 \left(a+b x^2\right)}+\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{(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,"(x*(b^2*c^2 - 2*a*b*c*d + 3*a^2*d^2 + 2*a*b*d^2*x^2))/(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
276,1,73,67,0.061652,"\int \frac{\left(c+d x^2\right)^2}{x \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2)^2/(x*(a + b*x^2)^2),x]","\frac{\left(a^2 d^2-b^2 c^2\right) \log \left(a+b x^2\right)}{2 a^2 b^2}+\frac{c^2 \log (x)}{a^2}+\frac{(a d-b c)^2}{2 a b^2 \left(a+b x^2\right)}","-\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,"(-(b*c) + a*d)^2/(2*a*b^2*(a + b*x^2)) + (c^2*Log[x])/a^2 + ((-(b^2*c^2) + a^2*d^2)*Log[a + b*x^2])/(2*a^2*b^2)","A",1
277,1,111,103,0.1059171,"\int \frac{\left(c+d x^2\right)^2}{x^2 \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2)^2/(x^2*(a + b*x^2)^2),x]","\frac{-a^2 d^2 x^2-2 a b c^2+2 a b c d x^2-3 b^2 c^2 x^2}{2 a^2 b x \left(a+b x^2\right)}+\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,"(-2*a*b*c^2 - 3*b^2*c^2*x^2 + 2*a*b*c*d*x^2 - a^2*d^2*x^2)/(2*a^2*b*x*(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
278,1,103,80,0.1396197,"\int \frac{\left(c+d x^2\right)^2}{x^3 \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2)^2/(x^3*(a + b*x^2)^2),x]","\frac{\left(b c^2-a c d\right) \log \left(a+b x^2\right)}{a^3}+\frac{2 \log (x) \left(a c d-b c^2\right)}{a^3}+\frac{-a^2 d^2 x^2-a b c^2+2 a b c d x^2-2 b^2 c^2 x^2}{2 a^2 b x^2 \left(a+b x^2\right)}","\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,"(-(a*b*c^2) - 2*b^2*c^2*x^2 + 2*a*b*c*d*x^2 - a^2*d^2*x^2)/(2*a^2*b*x^2*(a + b*x^2)) + (2*(-(b*c^2) + a*c*d)*Log[x])/a^3 + ((b*c^2 - a*c*d)*Log[a + b*x^2])/a^3","A",1
279,1,129,127,0.0982331,"\int \frac{\left(c+d x^2\right)^2}{x^4 \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2)^2/(x^4*(a + b*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{b} x}{\sqrt{a}}\right)}{2 a^{7/2} \sqrt{b}}+\frac{-2 a^2 c^2-12 a^2 c d x^2+3 a^2 d^2 x^4+10 a b c^2 x^2-18 a b c d x^4+15 b^2 c^2 x^4}{6 a^3 x^3 \left(a+b x^2\right)}","\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,"(-2*a^2*c^2 + 10*a*b*c^2*x^2 - 12*a^2*c*d*x^2 + 15*b^2*c^2*x^4 - 18*a*b*c*d*x^4 + 3*a^2*d^2*x^4)/(6*a^3*x^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
280,1,233,169,0.1374208,"\int \frac{x^4 \left(c+d x^2\right)^3}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^4*(c + d*x^2)^3)/(a + b*x^2)^2,x]","\frac{3 \left(3 a^{3/2} d-\sqrt{a} b c\right) (a d-b c)^2 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{11/2}}+\frac{x \left(-315 a^4 d^3+735 a^3 b c d^2-210 a^3 b d^3 x^2-525 a^2 b^2 c^2 d+490 a^2 b^2 c d^2 x^2+42 a^2 b^2 d^3 x^4+105 a b^3 c^3-350 a b^3 c^2 d x^2-98 a b^3 c d^2 x^4-18 a b^3 d^3 x^6+70 b^4 c^3 x^2+70 b^4 c^2 d x^4+42 b^4 c d^2 x^6+10 b^4 d^3 x^8\right)}{70 b^5 \left(a+b x^2\right)}","\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,"(x*(105*a*b^3*c^3 - 525*a^2*b^2*c^2*d + 735*a^3*b*c*d^2 - 315*a^4*d^3 + 70*b^4*c^3*x^2 - 350*a*b^3*c^2*d*x^2 + 490*a^2*b^2*c*d^2*x^2 - 210*a^3*b*d^3*x^2 + 70*b^4*c^2*d*x^4 - 98*a*b^3*c*d^2*x^4 + 42*a^2*b^2*d^3*x^4 + 42*b^4*c*d^2*x^6 - 18*a*b^3*d^3*x^6 + 10*b^4*d^3*x^8))/(70*b^5*(a + b*x^2)) + (3*(-(b*c) + a*d)^2*(-(Sqrt[a]*b*c) + 3*a^(3/2)*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(11/2))","A",1
281,1,204,117,0.1307435,"\int \frac{x^3 \left(c+d x^2\right)^3}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^3*(c + d*x^2)^3)/(a + b*x^2)^2,x]","\frac{-6 a^4 d^3+18 a^3 b c d^2+18 a^3 b d^3 x^2-18 a^2 b^2 c^2 d-36 a^2 b^2 c d^2 x^2+12 a^2 b^2 d^3 x^4+6 a b^3 c^3+18 a b^3 c^2 d x^2-27 a b^3 c d^2 x^4-4 a b^3 d^3 x^6+18 b^4 c^2 d x^4+9 b^4 c d^2 x^6+2 b^4 d^3 x^8}{12 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{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,"(6*a*b^3*c^3 - 18*a^2*b^2*c^2*d + 18*a^3*b*c*d^2 - 6*a^4*d^3 + 18*a*b^3*c^2*d*x^2 - 36*a^2*b^2*c*d^2*x^2 + 18*a^3*b*d^3*x^2 + 18*b^4*c^2*d*x^4 - 27*a*b^3*c*d^2*x^4 + 12*a^2*b^2*d^3*x^4 + 9*b^4*c*d^2*x^6 - 4*a*b^3*d^3*x^6 + 2*b^4*d^3*x^8)/(12*b^5*(a + b*x^2)) + ((b*c - 4*a*d)*(b*c - a*d)^2*Log[a + b*x^2])/(2*b^5)","A",1
282,1,173,147,0.108756,"\int \frac{x^2 \left(c+d x^2\right)^3}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^2*(c + d*x^2)^3)/(a + b*x^2)^2,x]","\frac{x \left(105 a^3 d^3-225 a^2 b c d^2+70 a^2 b d^3 x^2+135 a b^2 c^2 d-150 a b^2 c d^2 x^2-14 a b^2 d^3 x^4-15 b^3 c^3+90 b^3 c^2 d x^2+30 b^3 c d^2 x^4+6 b^3 d^3 x^6\right)}{30 b^4 \left(a+b x^2\right)}-\frac{(a d-b c)^2 (7 a d-b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 \sqrt{a} b^{9/2}}","\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,"(x*(-15*b^3*c^3 + 135*a*b^2*c^2*d - 225*a^2*b*c*d^2 + 105*a^3*d^3 + 90*b^3*c^2*d*x^2 - 150*a*b^2*c*d^2*x^2 + 70*a^2*b*d^3*x^2 + 30*b^3*c*d^2*x^4 - 14*a*b^2*d^3*x^4 + 6*b^3*d^3*x^6))/(30*b^4*(a + b*x^2)) - ((-(b*c) + a*d)^2*(-(b*c) + 7*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*Sqrt[a]*b^(9/2))","A",1
283,1,152,88,0.0783049,"\int \frac{x \left(c+d x^2\right)^3}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(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{2 a^3 d^3-6 a^2 b c d^2-4 a^2 b d^3 x^2+6 a b^2 c^2 d+6 a b^2 c d^2 x^2-3 a b^2 d^3 x^4-2 b^3 c^3+6 b^3 c d^2 x^4+b^3 d^3 x^6}{4 b^4 \left(a+b x^2\right)}","-\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,"(-2*b^3*c^3 + 6*a*b^2*c^2*d - 6*a^2*b*c*d^2 + 2*a^3*d^3 + 6*a*b^2*c*d^2*x^2 - 4*a^2*b*d^3*x^2 + 6*b^3*c*d^2*x^4 - 3*a*b^2*d^3*x^4 + b^3*d^3*x^6)/(4*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
284,1,140,106,0.0006201,"\int \frac{\left(c+d x^2\right)^3}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2)^3/(a + b*x^2)^2,x]","\frac{(a d-b c)^2 (5 a d+b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} b^{7/2}}-\frac{x \left(15 a^3 d^3-27 a^2 b c d^2+10 a^2 b d^3 x^2+9 a b^2 c^2 d-18 a b^2 c d^2 x^2-2 a b^2 d^3 x^4-3 b^3 c^3\right)}{6 a b^3 \left(a+b x^2\right)}","\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,"-1/6*(x*(-3*b^3*c^3 + 9*a*b^2*c^2*d - 27*a^2*b*c*d^2 + 15*a^3*d^3 - 18*a*b^2*c*d^2*x^2 + 10*a^2*b*d^3*x^2 - 2*a*b^2*d^3*x^4))/(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
285,1,132,88,0.1391721,"\int \frac{\left(c+d x^2\right)^3}{x \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2)^3/(x*(a + b*x^2)^2),x]","\frac{c^3 \log (x)}{a^2}+\frac{\left(-2 a^3 d^3+3 a^2 b c d^2-b^3 c^3\right) \log \left(a+b x^2\right)}{2 a^2 b^3}+\frac{-a^3 d^3+3 a^2 b c d^2+a^2 b d^3 x^2-3 a b^2 c^2 d+a b^2 d^3 x^4+b^3 c^3}{2 a b^3 \left(a+b x^2\right)}","-\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,"(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3 + a^2*b*d^3*x^2 + a*b^2*d^3*x^4)/(2*a*b^3*(a + b*x^2)) + (c^3*Log[x])/a^2 + ((-(b^3*c^3) + 3*a^2*b*c*d^2 - 2*a^3*d^3)*Log[a + b*x^2])/(2*a^2*b^3)","A",1
286,1,137,131,0.106277,"\int \frac{\left(c+d x^2\right)^3}{x^2 \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2)^3/(x^2*(a + b*x^2)^2),x]","\frac{3 a^3 d^3 x^2-3 a^2 b c d^2 x^2+2 a^2 b d^3 x^4-2 a b^2 c^3+3 a b^2 c^2 d x^2-3 b^3 c^3 x^2}{2 a^2 b^2 x \left(a+b x^2\right)}-\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{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,"(-2*a*b^2*c^3 - 3*b^3*c^3*x^2 + 3*a*b^2*c^2*d*x^2 - 3*a^2*b*c*d^2*x^2 + 3*a^3*d^3*x^2 + 2*a^2*b*d^3*x^4)/(2*a^2*b^2*x*(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
287,1,136,98,0.1313027,"\int \frac{\left(c+d x^2\right)^3}{x^3 \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2)^3/(x^3*(a + b*x^2)^2),x]","\frac{(a d+2 b c) (a d-b c)^2 \log \left(a+b x^2\right)}{2 a^3 b^2}+\frac{\log (x) \left(3 a c^2 d-2 b c^3\right)}{a^3}+\frac{a^3 d^3 x^2-3 a^2 b c d^2 x^2-a b^2 c^3+3 a b^2 c^2 d x^2-2 b^3 c^3 x^2}{2 a^2 b^2 x^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)}{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*b^2*c^3) - 2*b^3*c^3*x^2 + 3*a*b^2*c^2*d*x^2 - 3*a^2*b*c*d^2*x^2 + a^3*d^3*x^2)/(2*a^2*b^2*x^2*(a + b*x^2)) + ((-2*b*c^3 + 3*a*c^2*d)*Log[x])/a^3 + ((-(b*c) + a*d)^2*(2*b*c + a*d)*Log[a + b*x^2])/(2*a^3*b^2)","A",1
288,1,151,147,0.0984881,"\int \frac{\left(c+d x^2\right)^3}{x^4 \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2)^3/(x^4*(a + b*x^2)^2),x]","\frac{(a d+5 b c) (a d-b c)^2 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{7/2} b^{3/2}}+\frac{-3 a^3 d^3 x^4-2 a^2 b c^3-18 a^2 b c^2 d x^2+9 a^2 b c d^2 x^4+10 a b^2 c^3 x^2-27 a b^2 c^2 d x^4+15 b^3 c^3 x^4}{6 a^3 b x^3 \left(a+b x^2\right)}","\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,"(-2*a^2*b*c^3 + 10*a*b^2*c^3*x^2 - 18*a^2*b*c^2*d*x^2 + 15*b^3*c^3*x^4 - 27*a*b^2*c^2*d*x^4 + 9*a^2*b*c*d^2*x^4 - 3*a^3*d^3*x^4)/(6*a^3*b*x^3*(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
289,1,112,109,0.1907512,"\int \frac{x^4}{\left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[x^4/((a + b*x^2)^2*(c + d*x^2)),x]","\frac{\left(a^{3/2} d-3 \sqrt{a} b c\right) \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} (a d-b c)^2}+\frac{a x}{2 b \left(a+b x^2\right) (b c-a d)}","-\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*x)/(2*b*(b*c - a*d)*(a + b*x^2)) + ((-3*Sqrt[a]*b*c + a^(3/2)*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(3/2)*(b*c - a*d)^2) + (c^(3/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[d]*(-(b*c) + a*d)^2)","A",1
290,1,74,74,0.0548081,"\int \frac{x^3}{\left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[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
291,1,104,104,0.1772944,"\int \frac{x^2}{\left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[x^2/((a + b*x^2)^2*(c + d*x^2)),x]","-\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} (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,"-1/2*x/((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
292,1,70,70,0.050287,"\int \frac{x}{\left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[x/((a + b*x^2)^2*(c + d*x^2)),x]","-\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}","-\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,"-1/2*1/((b*c - a*d)*(a + b*x^2)) - (d*Log[a + b*x^2])/(2*(b*c - a*d)^2) + (d*Log[c + d*x^2])/(2*(b*c - a*d)^2)","A",1
293,1,112,108,0.000581,"\int \frac{1}{\left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^2*(c + d*x^2)),x]","\frac{\left(b^{3/2} c-3 a \sqrt{b} d\right) \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)) + ((b^(3/2)*c - 3*a*Sqrt[b]*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
294,1,102,99,0.1400547,"\int \frac{1}{x \left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[1/(x*(a + b*x^2)^2*(c + d*x^2)),x]","\frac{\left(2 a b d-b^2 c\right) \log \left(a+b x^2\right)}{2 a^2 (a d-b c)^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) (a d-b 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,"-1/2*b/(a*(-(b*c) + a*d)*(a + b*x^2)) + Log[x]/(a^2*c) + ((-(b^2*c) + 2*a*b*d)*Log[a + b*x^2])/(2*a^2*(-(b*c) + a*d)^2) - (d^2*Log[c + d*x^2])/(2*c*(b*c - a*d)^2)","A",1
295,1,147,144,0.2431914,"\int \frac{1}{x^2 \left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[1/(x^2*(a + b*x^2)^2*(c + d*x^2)),x]","\frac{\left(5 a b^{3/2} d-3 b^{5/2} c\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{5/2} (a d-b c)^2}+\frac{-2 a^2 d+2 a b c-2 a b d x^2+3 b^2 c x^2}{2 a^2 c x \left(a+b x^2\right) (a d-b c)}-\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,"(2*a*b*c - 2*a^2*d + 3*b^2*c*x^2 - 2*a*b*d*x^2)/(2*a^2*c*(-(b*c) + a*d)*x*(a + b*x^2)) + ((-3*b^(5/2)*c + 5*a*b^(3/2)*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
296,1,146,126,0.1957275,"\int \frac{1}{x^3 \left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[1/(x^3*(a + b*x^2)^2*(c + d*x^2)),x]","\frac{\left(2 b^3 c-3 a b^2 d\right) \log \left(a+b x^2\right)}{2 a^3 (a d-b c)^2}+\frac{\log (x) (-a d-2 b c)}{a^3 c^2}+\frac{a^2 (-d)+a b c-a b d x^2+2 b^2 c x^2}{2 a^2 c x^2 \left(a+b x^2\right) (a d-b c)}+\frac{d^3 \log \left(c+d x^2\right)}{2 c^2 (b c-a d)^2}","\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,"(a*b*c - a^2*d + 2*b^2*c*x^2 - a*b*d*x^2)/(2*a^2*c*(-(b*c) + a*d)*x^2*(a + b*x^2)) + ((-2*b*c - a*d)*Log[x])/(a^3*c^2) + ((2*b^3*c - 3*a*b^2*d)*Log[a + b*x^2])/(2*a^3*(-(b*c) + a*d)^2) + (d^3*Log[c + d*x^2])/(2*c^2*(b*c - a*d)^2)","A",1
297,1,202,189,0.3479379,"\int \frac{1}{x^4 \left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[1/(x^4*(a + b*x^2)^2*(c + d*x^2)),x]","\frac{\left(5 b^{7/2} c-7 a b^{5/2} d\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{7/2} (a d-b c)^2}+\frac{-2 a^3 c d+6 a^3 d^2 x^2+2 a^2 b c^2+4 a^2 b c d x^2+6 a^2 b d^2 x^4-10 a b^2 c^2 x^2+6 a b^2 c d x^4-15 b^3 c^2 x^4}{6 a^3 c^2 x^3 \left(a+b x^2\right) (a d-b c)}+\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,"(2*a^2*b*c^2 - 2*a^3*c*d - 10*a*b^2*c^2*x^2 + 4*a^2*b*c*d*x^2 + 6*a^3*d^2*x^2 - 15*b^3*c^2*x^4 + 6*a*b^2*c*d*x^4 + 6*a^2*b*d^2*x^4)/(6*a^3*c^2*(-(b*c) + a*d)*x^3*(a + b*x^2)) + ((5*b^(7/2)*c - 7*a*b^(5/2)*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
298,1,214,160,0.2537262,"\int \frac{1}{x^5 \left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[1/(x^5*(a + b*x^2)^2*(c + d*x^2)),x]","\frac{\left(4 a b^3 d-3 b^4 c\right) \log \left(a+b x^2\right)}{2 a^4 (a d-b c)^2}+\frac{\log (x) \left(a^2 d^2+2 a b c d+3 b^2 c^2\right)}{a^4 c^3}+\frac{-a^3 c d+2 a^3 d^2 x^2+a^2 b c^2+a^2 b c d x^2+2 a^2 b d^2 x^4-3 a b^2 c^2 x^2+2 a b^2 c d x^4-6 b^3 c^2 x^4}{4 a^3 c^2 x^4 \left(a+b x^2\right) (a d-b c)}-\frac{d^4 \log \left(c+d x^2\right)}{2 c^3 (b c-a d)^2}","-\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,"(a^2*b*c^2 - a^3*c*d - 3*a*b^2*c^2*x^2 + a^2*b*c*d*x^2 + 2*a^3*d^2*x^2 - 6*b^3*c^2*x^4 + 2*a*b^2*c*d*x^4 + 2*a^2*b*d^2*x^4)/(4*a^3*c^2*(-(b*c) + a*d)*x^4*(a + b*x^2)) + ((3*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*Log[x])/(a^4*c^3) + ((-3*b^4*c + 4*a*b^3*d)*Log[a + b*x^2])/(2*a^4*(-(b*c) + a*d)^2) - (d^4*Log[c + d*x^2])/(2*c^3*(b*c - a*d)^2)","A",1
299,1,278,250,0.3941697,"\int \frac{1}{x^6 \left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[1/(x^6*(a + b*x^2)^2*(c + d*x^2)),x]","\frac{\left(9 a b^{7/2} d-7 b^{9/2} c\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{9/2} (a d-b c)^2}+\frac{-6 a^4 c^2 d+10 a^4 c d^2 x^2-30 a^4 d^3 x^4+6 a^3 b c^3+4 a^3 b c^2 d x^2-20 a^3 b c d^2 x^4-30 a^3 b d^3 x^6-14 a^2 b^2 c^3 x^2-20 a^2 b^2 c^2 d x^4-30 a^2 b^2 c d^2 x^6+70 a b^3 c^3 x^4-30 a b^3 c^2 d x^6+105 b^4 c^3 x^6}{30 a^4 c^3 x^5 \left(a+b x^2\right) (a d-b c)}-\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,"(6*a^3*b*c^3 - 6*a^4*c^2*d - 14*a^2*b^2*c^3*x^2 + 4*a^3*b*c^2*d*x^2 + 10*a^4*c*d^2*x^2 + 70*a*b^3*c^3*x^4 - 20*a^2*b^2*c^2*d*x^4 - 20*a^3*b*c*d^2*x^4 - 30*a^4*d^3*x^4 + 105*b^4*c^3*x^6 - 30*a*b^3*c^2*d*x^6 - 30*a^2*b^2*c*d^2*x^6 - 30*a^3*b*d^3*x^6)/(30*a^4*c^3*(-(b*c) + a*d)*x^5*(a + b*x^2)) + ((-7*b^(9/2)*c + 9*a*b^(7/2)*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
300,1,305,210,0.3443898,"\int \frac{1}{x^7 \left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[1/(x^7*(a + b*x^2)^2*(c + d*x^2)),x]","\frac{\left(4 b^5 c-5 a b^4 d\right) \log \left(a+b x^2\right)}{2 a^5 (a d-b c)^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{-2 a^4 c^2 d+3 a^4 c d^2 x^2-6 a^4 d^3 x^4+2 a^3 b c^3+a^3 b c^2 d x^2-3 a^3 b c d^2 x^4-6 a^3 b d^3 x^6-4 a^2 b^2 c^3 x^2-3 a^2 b^2 c^2 d x^4-6 a^2 b^2 c d^2 x^6+12 a b^3 c^3 x^4-6 a b^3 c^2 d x^6+24 b^4 c^3 x^6}{12 a^4 c^3 x^6 \left(a+b x^2\right) (a d-b c)}+\frac{d^5 \log \left(c+d x^2\right)}{2 c^4 (b c-a d)^2}","\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^3*b*c^3 - 2*a^4*c^2*d - 4*a^2*b^2*c^3*x^2 + a^3*b*c^2*d*x^2 + 3*a^4*c*d^2*x^2 + 12*a*b^3*c^3*x^4 - 3*a^2*b^2*c^2*d*x^4 - 3*a^3*b*c*d^2*x^4 - 6*a^4*d^3*x^4 + 24*b^4*c^3*x^6 - 6*a*b^3*c^2*d*x^6 - 6*a^2*b^2*c*d^2*x^6 - 6*a^3*b*d^3*x^6)/(12*a^4*c^3*(-(b*c) + a*d)*x^6*(a + b*x^2)) + ((-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) + ((4*b^5*c - 5*a*b^4*d)*Log[a + b*x^2])/(2*a^5*(-(b*c) + a*d)^2) + (d^5*Log[c + d*x^2])/(2*c^4*(b*c - a*d)^2)","A",1
301,1,149,162,0.2458369,"\int \frac{x^4}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[x^4/((a + b*x^2)^2*(c + d*x^2)^2),x]","-\frac{\left(a^{3/2} d+3 \sqrt{a} b c\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 \sqrt{b} (b c-a d)^3}-\frac{\left(3 a \sqrt{c} d+b c^{3/2}\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 \sqrt{d} (a d-b c)^3}+\frac{x \left(2 a c+a d x^2+b c x^2\right)}{2 \left(a+b x^2\right) \left(c+d x^2\right) (b c-a d)^2}","\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,"(x*(2*a*c + b*c*x^2 + a*d*x^2))/(2*(b*c - a*d)^2*(a + b*x^2)*(c + d*x^2)) - ((3*Sqrt[a]*b*c + a^(3/2)*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*Sqrt[b]*(b*c - a*d)^3) - ((b*c^(3/2) + 3*a*Sqrt[c]*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*Sqrt[d]*(-(b*c) + a*d)^3)","A",1
302,1,110,107,0.1077574,"\int \frac{x^3}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[x^3/((a + b*x^2)^2*(c + d*x^2)^2),x]","\frac{2 a c+a d x^2+b c x^2}{2 \left(a+b x^2\right) \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}","\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,"(2*a*c + b*c*x^2 + a*d*x^2)/(2*(b*c - a*d)^2*(a + b*x^2)*(c + d*x^2)) + ((b*c + a*d)*Log[a + b*x^2])/(2*(b*c - a*d)^3) + ((-(b*c) - a*d)*Log[c + d*x^2])/(2*(b*c - a*d)^3)","A",1
303,1,148,147,0.2299244,"\int \frac{x^2}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[x^2/((a + b*x^2)^2*(c + d*x^2)^2),x]","\frac{\left(b^{3/2} (-c)-3 a \sqrt{b} d\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 \sqrt{a} (a d-b c)^3}+\frac{\left(-a d^{3/2}-3 b c \sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 \sqrt{c} (b c-a d)^3}-\frac{x \left(a d+b c+2 b d x^2\right)}{2 \left(a+b x^2\right) \left(c+d x^2\right) (b c-a d)^2}","-\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,"-1/2*(x*(b*c + a*d + 2*b*d*x^2))/((b*c - a*d)^2*(a + b*x^2)*(c + d*x^2)) + ((-(b^(3/2)*c) - 3*a*Sqrt[b]*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*Sqrt[a]*(-(b*c) + a*d)^3) + ((-3*b*c*Sqrt[d] - a*d^(3/2))*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*Sqrt[c]*(b*c - a*d)^3)","A",1
304,1,92,92,0.0950379,"\int \frac{x}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[x/((a + b*x^2)^2*(c + d*x^2)^2),x]","\frac{-a d-b c-2 b d x^2}{2 \left(a+b x^2\right) \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}","-\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*c) - a*d - 2*b*d*x^2)/(2*(b*c - a*d)^2*(a + b*x^2)*(c + d*x^2)) - (b*d*Log[a + b*x^2])/(b*c - a*d)^3 + (b*d*Log[c + d*x^2])/(b*c - a*d)^3","A",1
305,1,173,167,0.0006158,"\int \frac{1}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^2*(c + d*x^2)^2),x]","\frac{\left(5 a b^{3/2} d-b^{5/2} c\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} (a d-b c)^3}+\frac{x \left(a^2 d^2+a b d^2 x^2+b^2 c^2+b^2 c d x^2\right)}{2 a c \left(a+b x^2\right) \left(c+d x^2\right) (a d-b c)^2}+\frac{\left(5 b c d^{3/2}-a d^{5/2}\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{3/2} (b c-a d)^3}","\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,"(x*(b^2*c^2 + a^2*d^2 + b^2*c*d*x^2 + a*b*d^2*x^2))/(2*a*c*(-(b*c) + a*d)^2*(a + b*x^2)*(c + d*x^2)) + ((-(b^(5/2)*c) + 5*a*b^(3/2)*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*(-(b*c) + a*d)^3) + ((5*b*c*d^(3/2) - a*d^(5/2))*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(3/2)*(b*c - a*d)^3)","A",1
306,1,157,141,0.2726706,"\int \frac{1}{x \left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[1/(x*(a + b*x^2)^2*(c + d*x^2)^2),x]","\frac{a^2 d^2+a b d^2 x^2+b^2 c^2+b^2 c d x^2}{2 a c \left(a+b x^2\right) \left(c+d x^2\right) (a d-b c)^2}+\frac{\left(b^3 c-3 a b^2 d\right) \log \left(a+b x^2\right)}{2 a^2 (a d-b c)^3}+\frac{\log (x)}{a^2 c^2}+\frac{\left(a d^3-3 b c d^2\right) \log \left(c+d x^2\right)}{2 c^2 (b c-a d)^3}","-\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*c^2 + a^2*d^2 + b^2*c*d*x^2 + a*b*d^2*x^2)/(2*a*c*(-(b*c) + a*d)^2*(a + b*x^2)*(c + d*x^2)) + Log[x]/(a^2*c^2) + ((b^3*c - 3*a*b^2*d)*Log[a + b*x^2])/(2*a^2*(-(b*c) + a*d)^3) + ((-3*b*c*d^2 + a*d^3)*Log[c + d*x^2])/(2*c^2*(b*c - a*d)^3)","A",1
307,1,256,218,0.3826543,"\int \frac{1}{x^2 \left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[1/(x^2*(a + b*x^2)^2*(c + d*x^2)^2),x]","\frac{\left(3 b^{7/2} c-7 a b^{5/2} d\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{5/2} (a d-b c)^3}+\frac{-2 a^3 c d^2-3 a^3 d^3 x^2+4 a^2 b c^2 d+2 a^2 b c d^2 x^2-3 a^2 b d^3 x^4-2 a b^2 c^3+2 a b^2 c^2 d x^2+4 a b^2 c d^2 x^4-3 b^3 c^3 x^2-3 b^3 c^2 d x^4}{2 a^2 c^2 x \left(a+b x^2\right) \left(c+d x^2\right) (a d-b c)^2}+\frac{\left(3 a d^{7/2}-7 b c d^{5/2}\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{5/2} (b c-a d)^3}","-\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*b^2*c^3 + 4*a^2*b*c^2*d - 2*a^3*c*d^2 - 3*b^3*c^3*x^2 + 2*a*b^2*c^2*d*x^2 + 2*a^2*b*c*d^2*x^2 - 3*a^3*d^3*x^2 - 3*b^3*c^2*d*x^4 + 4*a*b^2*c*d^2*x^4 - 3*a^2*b*d^3*x^4)/(2*a^2*c^2*(-(b*c) + a*d)^2*x*(a + b*x^2)*(c + d*x^2)) + ((3*b^(7/2)*c - 7*a*b^(5/2)*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(5/2)*(-(b*c) + a*d)^3) + ((-7*b*c*d^(5/2) + 3*a*d^(7/2))*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(5/2)*(b*c - a*d)^3)","A",1
308,1,243,156,0.2590104,"\int \frac{1}{x^3 \left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[1/(x^3*(a + b*x^2)^2*(c + d*x^2)^2),x]","\frac{\left(2 a b^3 d-b^4 c\right) \log \left(a+b x^2\right)}{a^3 (a d-b c)^3}-\frac{2 \log (x) (a d+b c)}{a^3 c^3}+\frac{-a^3 c d^2-2 a^3 d^3 x^2+2 a^2 b c^2 d+a^2 b c d^2 x^2-2 a^2 b d^3 x^4-a b^2 c^3+a b^2 c^2 d x^2+2 a b^2 c d^2 x^4-2 b^3 c^3 x^2-2 b^3 c^2 d x^4}{2 a^2 c^2 x^2 \left(a+b x^2\right) \left(c+d x^2\right) (a d-b c)^2}+\frac{\left(2 b c d^3-a d^4\right) \log \left(c+d x^2\right)}{c^3 (b c-a d)^3}","\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,"(-(a*b^2*c^3) + 2*a^2*b*c^2*d - a^3*c*d^2 - 2*b^3*c^3*x^2 + a*b^2*c^2*d*x^2 + a^2*b*c*d^2*x^2 - 2*a^3*d^3*x^2 - 2*b^3*c^2*d*x^4 + 2*a*b^2*c*d^2*x^4 - 2*a^2*b*d^3*x^4)/(2*a^2*c^2*(-(b*c) + a*d)^2*x^2*(a + b*x^2)*(c + d*x^2)) - (2*(b*c + a*d)*Log[x])/(a^3*c^3) + ((-(b^4*c) + 2*a*b^3*d)*Log[a + b*x^2])/(a^3*(-(b*c) + a*d)^3) + ((2*b*c*d^3 - a*d^4)*Log[c + d*x^2])/(c^3*(b*c - a*d)^3)","A",1
309,1,348,271,0.4886576,"\int \frac{1}{x^4 \left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[1/(x^4*(a + b*x^2)^2*(c + d*x^2)^2),x]","\frac{\left(9 a b^{7/2} d-5 b^{9/2} c\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{7/2} (a d-b c)^3}+\frac{-2 a^4 c^2 d^2+10 a^4 c d^3 x^2+15 a^4 d^4 x^4+4 a^3 b c^3 d-10 a^3 b c^2 d^2 x^2-2 a^3 b c d^3 x^4+15 a^3 b d^4 x^6-2 a^2 b^2 c^4-10 a^2 b^2 c^3 d x^2-20 a^2 b^2 c^2 d^2 x^4-12 a^2 b^2 c d^3 x^6+10 a b^3 c^4 x^2-2 a b^3 c^3 d x^4-12 a b^3 c^2 d^2 x^6+15 b^4 c^4 x^4+15 b^4 c^3 d x^6}{6 a^3 c^3 x^3 \left(a+b x^2\right) \left(c+d x^2\right) (a d-b c)^2}+\frac{\left(9 b c d^{7/2}-5 a d^{9/2}\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{7/2} (b c-a d)^3}","\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*b^2*c^4 + 4*a^3*b*c^3*d - 2*a^4*c^2*d^2 + 10*a*b^3*c^4*x^2 - 10*a^2*b^2*c^3*d*x^2 - 10*a^3*b*c^2*d^2*x^2 + 10*a^4*c*d^3*x^2 + 15*b^4*c^4*x^4 - 2*a*b^3*c^3*d*x^4 - 20*a^2*b^2*c^2*d^2*x^4 - 2*a^3*b*c*d^3*x^4 + 15*a^4*d^4*x^4 + 15*b^4*c^3*d*x^6 - 12*a*b^3*c^2*d^2*x^6 - 12*a^2*b^2*c*d^3*x^6 + 15*a^3*b*d^4*x^6)/(6*a^3*c^3*(-(b*c) + a*d)^2*x^3*(a + b*x^2)*(c + d*x^2)) + ((-5*b^(9/2)*c + 9*a*b^(7/2)*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(7/2)*(-(b*c) + a*d)^3) + ((9*b*c*d^(7/2) - 5*a*d^(9/2))*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(7/2)*(b*c - a*d)^3)","A",1
310,1,210,207,0.4225357,"\int \frac{x^4}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[x^4/((a + b*x^2)^2*(c + d*x^2)^3),x]","-\frac{3 \left(a^{3/2} \sqrt{b} d+\sqrt{a} b^{3/2} c\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 (b c-a d)^4}+\frac{x \left(3 a^2 c d+5 a^2 d^2 x^2+9 a b c^2+14 a b c d x^2+9 a b d^2 x^4+5 b^2 c^2 x^2+3 b^2 c d x^4\right)}{8 \left(a+b x^2\right) \left(c+d x^2\right)^2 (b c-a d)^3}+\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 \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,"(x*(9*a*b*c^2 + 3*a^2*c*d + 5*b^2*c^2*x^2 + 14*a*b*c*d*x^2 + 5*a^2*d^2*x^2 + 3*b^2*c*d*x^4 + 9*a*b*d^2*x^4))/(8*(b*c - a*d)^3*(a + b*x^2)*(c + d*x^2)^2) - (3*(Sqrt[a]*b^(3/2)*c + a^(3/2)*Sqrt[b]*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*(b*c - a*d)^4) + (3*(b^2*c^2 + 6*a*b*c*d + a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*Sqrt[c]*Sqrt[d]*(b*c - a*d)^4)","A",1
311,1,165,142,0.1486559,"\int \frac{x^3}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[x^3/((a + b*x^2)^2*(c + d*x^2)^3),x]","\frac{a^2 c d+2 a^2 d^2 x^2+5 a b c^2+7 a b c d x^2+4 a b d^2 x^4+3 b^2 c^2 x^2+2 b^2 c d x^4}{4 \left(a+b x^2\right) \left(c+d x^2\right)^2 (b c-a d)^3}+\frac{\left(2 a b d+b^2 c\right) \log \left(a+b x^2\right)}{2 (b c-a d)^4}+\frac{\left(b^2 (-c)-2 a b d\right) \log \left(c+d x^2\right)}{2 (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,"(5*a*b*c^2 + a^2*c*d + 3*b^2*c^2*x^2 + 7*a*b*c*d*x^2 + 2*a^2*d^2*x^2 + 2*b^2*c*d*x^4 + 4*a*b*d^2*x^4)/(4*(b*c - a*d)^3*(a + b*x^2)*(c + d*x^2)^2) + ((b^2*c + 2*a*b*d)*Log[a + b*x^2])/(2*(b*c - a*d)^4) + ((-(b^2*c) - 2*a*b*d)*Log[c + d*x^2])/(2*(b*c - a*d)^4)","A",1
312,1,231,200,0.4929804,"\int \frac{x^2}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[x^2/((a + b*x^2)^2*(c + d*x^2)^3),x]","\frac{\left(a^2 d^{5/2}-10 a b c d^{3/2}-15 b^2 c^2 \sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{3/2} (b c-a d)^4}-\frac{x \left(-a^2 c d^2+a^2 d^3 x^2+9 a b c^2 d+6 a b c d^2 x^2+a b d^3 x^4+4 b^2 c^3+17 b^2 c^2 d x^2+11 b^2 c d^2 x^4\right)}{8 c \left(a+b x^2\right) \left(c+d x^2\right)^2 (b c-a d)^3}+\frac{\left(5 a b^{3/2} d+b^{5/2} c\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 \sqrt{a} (a d-b c)^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,"-1/8*(x*(4*b^2*c^3 + 9*a*b*c^2*d - a^2*c*d^2 + 17*b^2*c^2*d*x^2 + 6*a*b*c*d^2*x^2 + a^2*d^3*x^2 + 11*b^2*c*d^2*x^4 + a*b*d^3*x^4))/(c*(b*c - a*d)^3*(a + b*x^2)*(c + d*x^2)^2) + ((b^(5/2)*c + 5*a*b^(3/2)*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*Sqrt[a]*(-(b*c) + a*d)^4) + ((-15*b^2*c^2*Sqrt[d] - 10*a*b*c*d^(3/2) + a^2*d^(5/2))*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(3/2)*(b*c - a*d)^4)","A",1
313,1,138,126,0.1750719,"\int \frac{x}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[x/((a + b*x^2)^2*(c + d*x^2)^3),x]","\frac{a^2 d^2-5 a b c d-3 a b d^2 x^2-2 b^2 c^2-9 b^2 c d x^2-6 b^2 d^2 x^4}{4 \left(a+b x^2\right) \left(c+d x^2\right)^2 (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^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,"(-2*b^2*c^2 - 5*a*b*c*d + a^2*d^2 - 9*b^2*c*d*x^2 - 3*a*b*d^2*x^2 - 6*b^2*d^2*x^4)/(4*(b*c - a*d)^3*(a + b*x^2)*(c + d*x^2)^2) - (3*b^2*d*Log[a + b*x^2])/(2*(b*c - a*d)^4) + (3*b^2*d*Log[c + d*x^2])/(2*(b*c - a*d)^4)","A",1
314,1,275,230,0.0006338,"\int \frac{1}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^2*(c + d*x^2)^3),x]","\frac{\left(b^{7/2} c-7 a b^{5/2} d\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} (a d-b c)^4}+\frac{\left(3 a^2 d^{7/2}-14 a b c d^{5/2}+35 b^2 c^2 d^{3/2}\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{5/2} (b c-a d)^4}+\frac{x \left(5 a^3 c d^3+3 a^3 d^4 x^2-13 a^2 b c^2 d^2-6 a^2 b c d^3 x^2+3 a^2 b d^4 x^4-13 a b^2 c^2 d^2 x^2-11 a b^2 c d^3 x^4-4 b^3 c^4-8 b^3 c^3 d x^2-4 b^3 c^2 d^2 x^4\right)}{8 a c^2 \left(a+b x^2\right) \left(c+d x^2\right)^2 (a d-b c)^3}","\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,"(x*(-4*b^3*c^4 - 13*a^2*b*c^2*d^2 + 5*a^3*c*d^3 - 8*b^3*c^3*d*x^2 - 13*a*b^2*c^2*d^2*x^2 - 6*a^2*b*c*d^3*x^2 + 3*a^3*d^4*x^2 - 4*b^3*c^2*d^2*x^4 - 11*a*b^2*c*d^3*x^4 + 3*a^2*b*d^4*x^4))/(8*a*c^2*(-(b*c) + a*d)^3*(a + b*x^2)*(c + d*x^2)^2) + ((b^(7/2)*c - 7*a*b^(5/2)*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*(-(b*c) + a*d)^4) + ((35*b^2*c^2*d^(3/2) - 14*a*b*c*d^(5/2) + 3*a^2*d^(7/2))*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(5/2)*(b*c - a*d)^4)","A",1
315,1,260,192,0.3550973,"\int \frac{1}{x \left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[1/(x*(a + b*x^2)^2*(c + d*x^2)^3),x]","\frac{\left(-a^2 d^4+4 a b c d^3-6 b^2 c^2 d^2\right) \log \left(c+d x^2\right)}{2 c^3 (b c-a d)^4}+\frac{\left(4 a b^3 d-b^4 c\right) \log \left(a+b x^2\right)}{2 a^2 (a d-b c)^4}+\frac{\log (x)}{a^2 c^3}+\frac{3 a^3 c d^3+2 a^3 d^4 x^2-7 a^2 b c^2 d^2-3 a^2 b c d^3 x^2+2 a^2 b d^4 x^4-7 a b^2 c^2 d^2 x^2-6 a b^2 c d^3 x^4-2 b^3 c^4-4 b^3 c^3 d x^2-2 b^3 c^2 d^2 x^4}{4 a c^2 \left(a+b x^2\right) \left(c+d x^2\right)^2 (a d-b c)^3}","-\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*c^4 - 7*a^2*b*c^2*d^2 + 3*a^3*c*d^3 - 4*b^3*c^3*d*x^2 - 7*a*b^2*c^2*d^2*x^2 - 3*a^2*b*c*d^3*x^2 + 2*a^3*d^4*x^2 - 2*b^3*c^2*d^2*x^4 - 6*a*b^2*c*d^3*x^4 + 2*a^2*b*d^4*x^4)/(4*a*c^2*(-(b*c) + a*d)^3*(a + b*x^2)*(c + d*x^2)^2) + Log[x]/(a^2*c^3) + ((-(b^4*c) + 4*a*b^3*d)*Log[a + b*x^2])/(2*a^2*(-(b*c) + a*d)^4) + ((-6*b^2*c^2*d^2 + 4*a*b*c*d^3 - a^2*d^4)*Log[c + d*x^2])/(2*c^3*(b*c - a*d)^4)","A",1
316,1,395,297,0.5469054,"\int \frac{1}{x^2 \left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[1/(x^2*(a + b*x^2)^2*(c + d*x^2)^3),x]","\frac{3 \left(3 a b^{7/2} d-b^{9/2} c\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{5/2} (a d-b c)^4}-\frac{3 \left(5 a^2 d^{9/2}-18 a b c d^{7/2}+21 b^2 c^2 d^{5/2}\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{7/2} (b c-a d)^4}+\frac{-8 a^4 c^2 d^3-25 a^4 c d^4 x^2-15 a^4 d^5 x^4+24 a^3 b c^3 d^2+57 a^3 b c^2 d^3 x^2+14 a^3 b c d^4 x^4-15 a^3 b d^5 x^6-24 a^2 b^2 c^4 d-24 a^2 b^2 c^3 d^2 x^2+41 a^2 b^2 c^2 d^3 x^4+39 a^2 b^2 c d^4 x^6+8 a b^3 c^5-8 a b^3 c^4 d x^2-40 a b^3 c^3 d^2 x^4-24 a b^3 c^2 d^3 x^6+12 b^4 c^5 x^2+24 b^4 c^4 d x^4+12 b^4 c^3 d^2 x^6}{8 a^2 c^3 x \left(a+b x^2\right) \left(c+d x^2\right)^2 (a d-b c)^3}","-\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*b^3*c^5 - 24*a^2*b^2*c^4*d + 24*a^3*b*c^3*d^2 - 8*a^4*c^2*d^3 + 12*b^4*c^5*x^2 - 8*a*b^3*c^4*d*x^2 - 24*a^2*b^2*c^3*d^2*x^2 + 57*a^3*b*c^2*d^3*x^2 - 25*a^4*c*d^4*x^2 + 24*b^4*c^4*d*x^4 - 40*a*b^3*c^3*d^2*x^4 + 41*a^2*b^2*c^2*d^3*x^4 + 14*a^3*b*c*d^4*x^4 - 15*a^4*d^5*x^4 + 12*b^4*c^3*d^2*x^6 - 24*a*b^3*c^2*d^3*x^6 + 39*a^2*b^2*c*d^4*x^6 - 15*a^3*b*d^5*x^6)/(8*a^2*c^3*(-(b*c) + a*d)^3*x*(a + b*x^2)*(c + d*x^2)^2) + (3*(-(b^(9/2)*c) + 3*a*b^(7/2)*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(5/2)*(-(b*c) + a*d)^4) - (3*(21*b^2*c^2*d^(5/2) - 18*a*b*c*d^(7/2) + 5*a^2*d^(9/2))*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(7/2)*(b*c - a*d)^4)","A",1
317,1,389,215,0.3977869,"\int \frac{1}{x^3 \left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[1/(x^3*(a + b*x^2)^2*(c + d*x^2)^3),x]","\frac{\left(2 b^5 c-5 a b^4 d\right) \log \left(a+b x^2\right)}{2 a^3 (a d-b c)^4}+\frac{\log (x) (-3 a d-2 b c)}{a^3 c^4}+\frac{\left(3 a^2 d^5-10 a b c d^4+10 b^2 c^2 d^3\right) \log \left(c+d x^2\right)}{2 c^4 (b c-a d)^4}+\frac{-2 a^4 c^2 d^3-9 a^4 c d^4 x^2-6 a^4 d^5 x^4+6 a^3 b c^3 d^2+19 a^3 b c^2 d^3 x^2+5 a^3 b c d^4 x^4-6 a^3 b d^5 x^6-6 a^2 b^2 c^4 d-6 a^2 b^2 c^3 d^2 x^2+15 a^2 b^2 c^2 d^3 x^4+14 a^2 b^2 c d^4 x^6+2 a b^3 c^5-2 a b^3 c^4 d x^2-10 a b^3 c^3 d^2 x^4-6 a b^3 c^2 d^3 x^6+4 b^4 c^5 x^2+8 b^4 c^4 d x^4+4 b^4 c^3 d^2 x^6}{4 a^2 c^3 x^2 \left(a+b x^2\right) \left(c+d x^2\right)^2 (a d-b c)^3}","\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*b^3*c^5 - 6*a^2*b^2*c^4*d + 6*a^3*b*c^3*d^2 - 2*a^4*c^2*d^3 + 4*b^4*c^5*x^2 - 2*a*b^3*c^4*d*x^2 - 6*a^2*b^2*c^3*d^2*x^2 + 19*a^3*b*c^2*d^3*x^2 - 9*a^4*c*d^4*x^2 + 8*b^4*c^4*d*x^4 - 10*a*b^3*c^3*d^2*x^4 + 15*a^2*b^2*c^2*d^3*x^4 + 5*a^3*b*c*d^4*x^4 - 6*a^4*d^5*x^4 + 4*b^4*c^3*d^2*x^6 - 6*a*b^3*c^2*d^3*x^6 + 14*a^2*b^2*c*d^4*x^6 - 6*a^3*b*d^5*x^6)/(4*a^2*c^3*(-(b*c) + a*d)^3*x^2*(a + b*x^2)*(c + d*x^2)^2) + ((-2*b*c - 3*a*d)*Log[x])/(a^3*c^4) + ((2*b^5*c - 5*a*b^4*d)*Log[a + b*x^2])/(2*a^3*(-(b*c) + a*d)^4) + ((10*b^2*c^2*d^3 - 10*a*b*c*d^4 + 3*a^2*d^5)*Log[c + d*x^2])/(2*c^4*(b*c - a*d)^4)","A",1
318,1,504,377,0.5891474,"\int \frac{1}{x^4 \left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[1/(x^4*(a + b*x^2)^2*(c + d*x^2)^3),x]","\frac{\left(5 b^{11/2} c-11 a b^{9/2} d\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{7/2} (a d-b c)^4}+\frac{\left(35 a^2 d^{11/2}-110 a b c d^{9/2}+99 b^2 c^2 d^{7/2}\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{9/2} (b c-a d)^4}+\frac{-8 a^5 c^3 d^3+56 a^5 c^2 d^4 x^2+175 a^5 c d^5 x^4+105 a^5 d^6 x^6+24 a^4 b c^4 d^2-128 a^4 b c^3 d^3 x^2-319 a^4 b c^2 d^4 x^4-50 a^4 b c d^5 x^6+105 a^4 b d^6 x^8-24 a^3 b^2 c^5 d+48 a^3 b^2 c^4 d^2 x^2-303 a^3 b^2 c^2 d^4 x^6-225 a^3 b^2 c d^5 x^8+8 a^2 b^3 c^6+64 a^2 b^3 c^5 d x^2+176 a^2 b^3 c^4 d^2 x^4+192 a^2 b^3 c^3 d^3 x^6+72 a^2 b^3 c^2 d^4 x^8-40 a b^4 c^6 x^2-8 a b^4 c^5 d x^4+104 a b^4 c^4 d^2 x^6+72 a b^4 c^3 d^3 x^8-60 b^5 c^6 x^4-120 b^5 c^5 d x^6-60 b^5 c^4 d^2 x^8}{24 a^3 c^4 x^3 \left(a+b x^2\right) \left(c+d x^2\right)^2 (a d-b c)^3}","\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*b^3*c^6 - 24*a^3*b^2*c^5*d + 24*a^4*b*c^4*d^2 - 8*a^5*c^3*d^3 - 40*a*b^4*c^6*x^2 + 64*a^2*b^3*c^5*d*x^2 + 48*a^3*b^2*c^4*d^2*x^2 - 128*a^4*b*c^3*d^3*x^2 + 56*a^5*c^2*d^4*x^2 - 60*b^5*c^6*x^4 - 8*a*b^4*c^5*d*x^4 + 176*a^2*b^3*c^4*d^2*x^4 - 319*a^4*b*c^2*d^4*x^4 + 175*a^5*c*d^5*x^4 - 120*b^5*c^5*d*x^6 + 104*a*b^4*c^4*d^2*x^6 + 192*a^2*b^3*c^3*d^3*x^6 - 303*a^3*b^2*c^2*d^4*x^6 - 50*a^4*b*c*d^5*x^6 + 105*a^5*d^6*x^6 - 60*b^5*c^4*d^2*x^8 + 72*a*b^4*c^3*d^3*x^8 + 72*a^2*b^3*c^2*d^4*x^8 - 225*a^3*b^2*c*d^5*x^8 + 105*a^4*b*d^6*x^8)/(24*a^3*c^4*(-(b*c) + a*d)^3*x^3*(a + b*x^2)*(c + d*x^2)^2) + ((5*b^(11/2)*c - 11*a*b^(9/2)*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(7/2)*(-(b*c) + a*d)^4) + ((99*b^2*c^2*d^(7/2) - 110*a*b*c*d^(9/2) + 35*a^2*d^(11/2))*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(9/2)*(b*c - a*d)^4)","A",1
319,0,0,96,0.0524072,"\int x^m \left(a+b x^2\right)^3 \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^m*(a + b*x^2)^3*(A + B*x^2),x]","\int x^m \left(a+b x^2\right)^3 \left(A+B x^2\right) \, dx","\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,"Could not integrate","F",-1
320,0,0,71,0.0209696,"\int x^m \left(a+b x^2\right)^2 \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^m*(a + b*x^2)^2*(A + B*x^2),x]","\int x^m \left(a+b x^2\right)^2 \left(A+B x^2\right) \, dx","\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,"Could not integrate","F",-1
321,0,0,45,0.0129462,"\int x^m \left(a+b x^2\right) \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^m*(a + b*x^2)*(A + B*x^2),x]","\int x^m \left(a+b x^2\right) \left(A+B x^2\right) \, dx","\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,"Could not integrate","F",-1
322,0,0,66,0.0227225,"\int \frac{x^m \left(A+B x^2\right)}{a+b x^2} \, dx","IntegrateAlgebraic[(x^m*(A + B*x^2))/(a + b*x^2),x]","\int \frac{x^m \left(A+B x^2\right)}{a+b x^2} \, dx","\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,"Could not integrate","F",-1
323,0,0,93,0.0329556,"\int \frac{x^m \left(A+B x^2\right)}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^m*(A + B*x^2))/(a + b*x^2)^2,x]","\int \frac{x^m \left(A+B x^2\right)}{\left(a+b x^2\right)^2} \, dx","\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,"Could not integrate","F",-1
324,0,0,93,0.0476341,"\int \frac{x^m \left(A+B x^2\right)}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(x^m*(A + B*x^2))/(a + b*x^2)^3,x]","\int \frac{x^m \left(A+B x^2\right)}{\left(a+b x^2\right)^3} \, dx","\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,"Could not integrate","F",-1
325,0,0,151,0.1182767,"\int x^m \left(a+b x^2\right)^2 \left(c+d x^2\right)^3 \, dx","IntegrateAlgebraic[x^m*(a + b*x^2)^2*(c + d*x^2)^3,x]","\int x^m \left(a+b x^2\right)^2 \left(c+d x^2\right)^3 \, dx","\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,"Could not integrate","F",-1
326,0,0,109,0.0431106,"\int x^m \left(a+b x^2\right)^2 \left(c+d x^2\right)^2 \, dx","IntegrateAlgebraic[x^m*(a + b*x^2)^2*(c + d*x^2)^2,x]","\int x^m \left(a+b x^2\right)^2 \left(c+d x^2\right)^2 \, dx","\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,"Could not integrate","F",-1
327,0,0,71,0.0204362,"\int x^m \left(a+b x^2\right)^2 \left(c+d x^2\right) \, dx","IntegrateAlgebraic[x^m*(a + b*x^2)^2*(c + d*x^2),x]","\int x^m \left(a+b x^2\right)^2 \left(c+d x^2\right) \, dx","\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,"Could not integrate","F",-1
328,0,0,94,0.0430277,"\int \frac{x^m \left(a+b x^2\right)^2}{c+d x^2} \, dx","IntegrateAlgebraic[(x^m*(a + b*x^2)^2)/(c + d*x^2),x]","\int \frac{x^m \left(a+b x^2\right)^2}{c+d x^2} \, dx","\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,"Could not integrate","F",-1
329,0,0,120,0.1337111,"\int \frac{x^m \left(a+b x^2\right)^2}{\left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[(x^m*(a + b*x^2)^2)/(c + d*x^2)^2,x]","\int \frac{x^m \left(a+b x^2\right)^2}{\left(c+d x^2\right)^2} \, dx","-\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,"Could not integrate","F",-1
330,0,0,171,0.3862466,"\int \frac{x^m \left(a+b x^2\right)^2}{\left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[(x^m*(a + b*x^2)^2)/(c + d*x^2)^3,x]","\int \frac{x^m \left(a+b x^2\right)^2}{\left(c+d x^2\right)^3} \, dx","\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,"Could not integrate","F",-1
331,0,0,133,0.2068813,"\int \frac{x^m \left(c+d x^2\right)^3}{a+b x^2} \, dx","IntegrateAlgebraic[(x^m*(c + d*x^2)^3)/(a + b*x^2),x]","\int \frac{x^m \left(c+d x^2\right)^3}{a+b x^2} \, dx","\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,"Could not integrate","F",-1
332,0,0,94,0.0434033,"\int \frac{x^m \left(c+d x^2\right)^2}{a+b x^2} \, dx","IntegrateAlgebraic[(x^m*(c + d*x^2)^2)/(a + b*x^2),x]","\int \frac{x^m \left(c+d x^2\right)^2}{a+b x^2} \, dx","\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,"Could not integrate","F",-1
333,0,0,66,0.0226319,"\int \frac{x^m \left(c+d x^2\right)}{a+b x^2} \, dx","IntegrateAlgebraic[(x^m*(c + d*x^2))/(a + b*x^2),x]","\int \frac{x^m \left(c+d x^2\right)}{a+b x^2} \, dx","\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,"Could not integrate","F",-1
334,0,0,102,0.039435,"\int \frac{x^m}{\left(a+b x^2\right) \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[x^m/((a + b*x^2)*(c + d*x^2)),x]","\int \frac{x^m}{\left(a+b x^2\right) \left(c+d x^2\right)} \, dx","\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,"Could not integrate","F",-1
335,0,0,156,0.173078,"\int \frac{x^m}{\left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[x^m/((a + b*x^2)^2*(c + d*x^2)),x]","\int \frac{x^m}{\left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","\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,"Could not integrate","F",-1
336,0,0,234,0.3989641,"\int \frac{x^m}{\left(a+b x^2\right)^3 \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[x^m/((a + b*x^2)^3*(c + d*x^2)),x]","\int \frac{x^m}{\left(a+b x^2\right)^3 \left(c+d x^2\right)} \, dx","\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,"Could not integrate","F",-1
337,0,0,201,0.3339722,"\int \frac{x^m \left(c+d x^2\right)^3}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^m*(c + d*x^2)^3)/(a + b*x^2)^2,x]","\int \frac{x^m \left(c+d x^2\right)^3}{\left(a+b x^2\right)^2} \, dx","\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,"Could not integrate","F",-1
338,0,0,120,0.0940819,"\int \frac{x^m \left(c+d x^2\right)^2}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^m*(c + d*x^2)^2)/(a + b*x^2)^2,x]","\int \frac{x^m \left(c+d x^2\right)^2}{\left(a+b x^2\right)^2} \, dx","\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,"Could not integrate","F",-1
339,0,0,93,0.0327829,"\int \frac{x^m \left(c+d x^2\right)}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^m*(c + d*x^2))/(a + b*x^2)^2,x]","\int \frac{x^m \left(c+d x^2\right)}{\left(a+b x^2\right)^2} \, dx","\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,"Could not integrate","F",-1
340,0,0,156,0.0005758,"\int \frac{x^m}{\left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[x^m/((a + b*x^2)^2*(c + d*x^2)),x]","\int \frac{x^m}{\left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","\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,"Could not integrate","F",-1
341,0,0,230,0.3560997,"\int \frac{x^m}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[x^m/((a + b*x^2)^2*(c + d*x^2)^2),x]","\int \frac{x^m}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","-\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,"Could not integrate","F",-1
342,0,0,325,0.6592653,"\int \frac{x^m}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[x^m/((a + b*x^2)^2*(c + d*x^2)^3),x]","\int \frac{x^m}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","-\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,"Could not integrate","F",-1
343,1,39,39,0.0228652,"\int x^{7/2} \left(a+b x^2\right) \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^(7/2)*(a + b*x^2)*(A + B*x^2),x]","\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}","\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*a*A*x^(9/2))/9 + (2*(A*b + a*B)*x^(13/2))/13 + (2*b*B*x^(17/2))/17","A",1
344,1,39,39,0.0202858,"\int x^{5/2} \left(a+b x^2\right) \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^(5/2)*(a + b*x^2)*(A + B*x^2),x]","\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}","\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*a*A*x^(7/2))/7 + (2*(A*b + a*B)*x^(11/2))/11 + (2*b*B*x^(15/2))/15","A",1
345,1,39,39,0.0192487,"\int x^{3/2} \left(a+b x^2\right) \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^(3/2)*(a + b*x^2)*(A + B*x^2),x]","\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}","\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*a*A*x^(5/2))/5 + (2*(A*b + a*B)*x^(9/2))/9 + (2*b*B*x^(13/2))/13","A",1
346,1,39,39,0.0189107,"\int \sqrt{x} \left(a+b x^2\right) \left(A+B x^2\right) \, dx","IntegrateAlgebraic[Sqrt[x]*(a + b*x^2)*(A + B*x^2),x]","\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}","\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*a*A*x^(3/2))/3 + (2*(A*b + a*B)*x^(7/2))/7 + (2*b*B*x^(11/2))/11","A",1
347,1,37,37,0.0190192,"\int \frac{\left(a+b x^2\right) \left(A+B x^2\right)}{\sqrt{x}} \, dx","IntegrateAlgebraic[((a + b*x^2)*(A + B*x^2))/Sqrt[x],x]","\frac{2}{5} x^{5/2} (a B+A b)+2 a A \sqrt{x}+\frac{2}{9} b B x^{9/2}","\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*a*A*Sqrt[x] + (2*(A*b + a*B)*x^(5/2))/5 + (2*b*B*x^(9/2))/9","A",1
348,1,37,37,0.028773,"\int \frac{\left(a+b x^2\right) \left(A+B x^2\right)}{x^{3/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)*(A + B*x^2))/x^(3/2),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}","\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*a*A)/Sqrt[x] + (2*(A*b + a*B)*x^(3/2))/3 + (2*b*B*x^(7/2))/7","A",1
349,1,37,37,0.0305174,"\int \frac{\left(a+b x^2\right) \left(A+B x^2\right)}{x^{5/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)*(A + B*x^2))/x^(5/2),x]","2 \sqrt{x} (a B+A b)-\frac{2 a A}{3 x^{3/2}}+\frac{2}{5} b B x^{5/2}","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*a*A)/(3*x^(3/2)) + 2*(A*b + a*B)*Sqrt[x] + (2*b*B*x^(5/2))/5","A",1
350,1,37,37,0.0326157,"\int \frac{\left(a+b x^2\right) \left(A+B x^2\right)}{x^{7/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)*(A + B*x^2))/x^(7/2),x]","-\frac{2 (a B+A b)}{\sqrt{x}}-\frac{2 a A}{5 x^{5/2}}+\frac{2}{3} b B x^{3/2}","-\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,"(-2*a*A)/(5*x^(5/2)) - (2*(A*b + a*B))/Sqrt[x] + (2*b*B*x^(3/2))/3","A",1
351,1,67,63,0.0340221,"\int x^{7/2} \left(a+b x^2\right)^2 \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^(7/2)*(a + b*x^2)^2*(A + B*x^2),x]","\frac{2}{13} x^{13/2} \left(a^2 B+2 a A b\right)+\frac{2}{9} a^2 A x^{9/2}+\frac{2}{17} x^{17/2} \left(2 a b B+A b^2\right)+\frac{2}{21} b^2 B x^{21/2}","\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*a^2*A*x^(9/2))/9 + (2*(2*a*A*b + a^2*B)*x^(13/2))/13 + (2*(A*b^2 + 2*a*b*B)*x^(17/2))/17 + (2*b^2*B*x^(21/2))/21","A",1
352,1,67,63,0.0338988,"\int x^{5/2} \left(a+b x^2\right)^2 \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^(5/2)*(a + b*x^2)^2*(A + B*x^2),x]","\frac{2}{11} x^{11/2} \left(a^2 B+2 a A b\right)+\frac{2}{7} a^2 A x^{7/2}+\frac{2}{15} x^{15/2} \left(2 a b B+A b^2\right)+\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*(2*a*A*b + a^2*B)*x^(11/2))/11 + (2*(A*b^2 + 2*a*b*B)*x^(15/2))/15 + (2*b^2*B*x^(19/2))/19","A",1
353,1,67,63,0.0309855,"\int x^{3/2} \left(a+b x^2\right)^2 \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^(3/2)*(a + b*x^2)^2*(A + B*x^2),x]","\frac{2}{9} x^{9/2} \left(a^2 B+2 a A b\right)+\frac{2}{5} a^2 A x^{5/2}+\frac{2}{13} x^{13/2} \left(2 a b B+A b^2\right)+\frac{2}{17} b^2 B x^{17/2}","\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*a^2*A*x^(5/2))/5 + (2*(2*a*A*b + a^2*B)*x^(9/2))/9 + (2*(A*b^2 + 2*a*b*B)*x^(13/2))/13 + (2*b^2*B*x^(17/2))/17","A",1
354,1,67,63,0.0326733,"\int \sqrt{x} \left(a+b x^2\right)^2 \left(A+B x^2\right) \, dx","IntegrateAlgebraic[Sqrt[x]*(a + b*x^2)^2*(A + B*x^2),x]","\frac{2}{7} x^{7/2} \left(a^2 B+2 a A b\right)+\frac{2}{3} a^2 A x^{3/2}+\frac{2}{11} x^{11/2} \left(2 a b B+A b^2\right)+\frac{2}{15} b^2 B x^{15/2}","\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*a^2*A*x^(3/2))/3 + (2*(2*a*A*b + a^2*B)*x^(7/2))/7 + (2*(A*b^2 + 2*a*b*B)*x^(11/2))/11 + (2*b^2*B*x^(15/2))/15","A",1
355,1,65,61,0.0302247,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x^2\right)}{\sqrt{x}} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(A + B*x^2))/Sqrt[x],x]","\frac{2}{5} x^{5/2} \left(a^2 B+2 a A b\right)+2 a^2 A \sqrt{x}+\frac{2}{9} x^{9/2} \left(2 a b B+A b^2\right)+\frac{2}{13} b^2 B x^{13/2}","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*a^2*A*Sqrt[x] + (2*(2*a*A*b + a^2*B)*x^(5/2))/5 + (2*(A*b^2 + 2*a*b*B)*x^(9/2))/9 + (2*b^2*B*x^(13/2))/13","A",1
356,1,65,61,0.0465231,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x^2\right)}{x^{3/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(A + B*x^2))/x^(3/2),x]","\frac{2}{3} x^{3/2} \left(a^2 B+2 a A b\right)-\frac{2 a^2 A}{\sqrt{x}}+\frac{2}{7} x^{7/2} \left(2 a b B+A b^2\right)+\frac{2}{11} b^2 B x^{11/2}","-\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,"(-2*a^2*A)/Sqrt[x] + (2*(2*a*A*b + a^2*B)*x^(3/2))/3 + (2*(A*b^2 + 2*a*b*B)*x^(7/2))/7 + (2*b^2*B*x^(11/2))/11","A",1
357,1,65,61,0.0441803,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x^2\right)}{x^{5/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(A + B*x^2))/x^(5/2),x]","2 \sqrt{x} \left(a^2 B+2 a A b\right)-\frac{2 a^2 A}{3 x^{3/2}}+\frac{2}{5} x^{5/2} \left(2 a b B+A b^2\right)+\frac{2}{9} b^2 B x^{9/2}","-\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,"(-2*a^2*A)/(3*x^(3/2)) + 2*(2*a*A*b + a^2*B)*Sqrt[x] + (2*(A*b^2 + 2*a*b*B)*x^(5/2))/5 + (2*b^2*B*x^(9/2))/9","A",1
358,1,65,61,0.0446835,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x^2\right)}{x^{7/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(A + B*x^2))/x^(7/2),x]","-\frac{2 \left(a^2 B+2 a A b\right)}{\sqrt{x}}-\frac{2 a^2 A}{5 x^{5/2}}+\frac{2}{3} x^{3/2} \left(2 a b B+A b^2\right)+\frac{2}{7} b^2 B x^{7/2}","-\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,"(-2*a^2*A)/(5*x^(5/2)) - (2*(2*a*A*b + a^2*B))/Sqrt[x] + (2*(A*b^2 + 2*a*b*B)*x^(3/2))/3 + (2*b^2*B*x^(7/2))/7","A",1
359,1,93,85,0.0502651,"\int x^{7/2} \left(a+b x^2\right)^3 \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^(7/2)*(a + b*x^2)^3*(A + B*x^2),x]","\frac{2}{9} a^3 A x^{9/2}+\frac{6}{17} x^{17/2} \left(a^2 b B+a A b^2\right)+\frac{2}{13} x^{13/2} \left(a^3 B+3 a^2 A b\right)+\frac{2}{21} x^{21/2} \left(3 a b^2 B+A b^3\right)+\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*(3*a^2*A*b + a^3*B)*x^(13/2))/13 + (6*(a*A*b^2 + a^2*b*B)*x^(17/2))/17 + (2*(A*b^3 + 3*a*b^2*B)*x^(21/2))/21 + (2*b^3*B*x^(25/2))/25","A",1
360,1,93,85,0.0473419,"\int x^{5/2} \left(a+b x^2\right)^3 \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^(5/2)*(a + b*x^2)^3*(A + B*x^2),x]","\frac{2}{7} a^3 A x^{7/2}+\frac{2}{5} x^{15/2} \left(a^2 b B+a A b^2\right)+\frac{2}{11} x^{11/2} \left(a^3 B+3 a^2 A b\right)+\frac{2}{19} x^{19/2} \left(3 a b^2 B+A b^3\right)+\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*(3*a^2*A*b + a^3*B)*x^(11/2))/11 + (2*(a*A*b^2 + a^2*b*B)*x^(15/2))/5 + (2*(A*b^3 + 3*a*b^2*B)*x^(19/2))/19 + (2*b^3*B*x^(23/2))/23","A",1
361,1,93,85,0.0422223,"\int x^{3/2} \left(a+b x^2\right)^3 \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^(3/2)*(a + b*x^2)^3*(A + B*x^2),x]","\frac{2}{5} a^3 A x^{5/2}+\frac{6}{13} x^{13/2} \left(a^2 b B+a A b^2\right)+\frac{2}{9} x^{9/2} \left(a^3 B+3 a^2 A b\right)+\frac{2}{17} x^{17/2} \left(3 a b^2 B+A b^3\right)+\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*(3*a^2*A*b + a^3*B)*x^(9/2))/9 + (6*(a*A*b^2 + a^2*b*B)*x^(13/2))/13 + (2*(A*b^3 + 3*a*b^2*B)*x^(17/2))/17 + (2*b^3*B*x^(21/2))/21","A",1
362,1,93,85,0.0414433,"\int \sqrt{x} \left(a+b x^2\right)^3 \left(A+B x^2\right) \, dx","IntegrateAlgebraic[Sqrt[x]*(a + b*x^2)^3*(A + B*x^2),x]","\frac{2}{3} a^3 A x^{3/2}+\frac{6}{11} x^{11/2} \left(a^2 b B+a A b^2\right)+\frac{2}{7} x^{7/2} \left(a^3 B+3 a^2 A b\right)+\frac{2}{15} x^{15/2} \left(3 a b^2 B+A b^3\right)+\frac{2}{19} b^3 B x^{19/2}","\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*a^3*A*x^(3/2))/3 + (2*(3*a^2*A*b + a^3*B)*x^(7/2))/7 + (6*(a*A*b^2 + a^2*b*B)*x^(11/2))/11 + (2*(A*b^3 + 3*a*b^2*B)*x^(15/2))/15 + (2*b^3*B*x^(19/2))/19","A",1
363,1,91,83,0.0402616,"\int \frac{\left(a+b x^2\right)^3 \left(A+B x^2\right)}{\sqrt{x}} \, dx","IntegrateAlgebraic[((a + b*x^2)^3*(A + B*x^2))/Sqrt[x],x]","2 a^3 A \sqrt{x}+\frac{2}{3} x^{9/2} \left(a^2 b B+a A b^2\right)+\frac{2}{5} x^{5/2} \left(a^3 B+3 a^2 A b\right)+\frac{2}{13} x^{13/2} \left(3 a b^2 B+A b^3\right)+\frac{2}{17} b^3 B x^{17/2}","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*a^3*A*Sqrt[x] + (2*(3*a^2*A*b + a^3*B)*x^(5/2))/5 + (2*(a*A*b^2 + a^2*b*B)*x^(9/2))/3 + (2*(A*b^3 + 3*a*b^2*B)*x^(13/2))/13 + (2*b^3*B*x^(17/2))/17","A",1
364,1,91,83,0.0543605,"\int \frac{\left(a+b x^2\right)^3 \left(A+B x^2\right)}{x^{3/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)^3*(A + B*x^2))/x^(3/2),x]","-\frac{2 a^3 A}{\sqrt{x}}+\frac{6}{7} x^{7/2} \left(a^2 b B+a A b^2\right)+\frac{2}{3} x^{3/2} \left(a^3 B+3 a^2 A b\right)+\frac{2}{11} x^{11/2} \left(3 a b^2 B+A b^3\right)+\frac{2}{15} b^3 B x^{15/2}","-\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,"(-2*a^3*A)/Sqrt[x] + (2*(3*a^2*A*b + a^3*B)*x^(3/2))/3 + (6*(a*A*b^2 + a^2*b*B)*x^(7/2))/7 + (2*(A*b^3 + 3*a*b^2*B)*x^(11/2))/11 + (2*b^3*B*x^(15/2))/15","A",1
365,1,91,83,0.0563724,"\int \frac{\left(a+b x^2\right)^3 \left(A+B x^2\right)}{x^{5/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)^3*(A + B*x^2))/x^(5/2),x]","-\frac{2 a^3 A}{3 x^{3/2}}+\frac{6}{5} x^{5/2} \left(a^2 b B+a A b^2\right)+2 \sqrt{x} \left(a^3 B+3 a^2 A b\right)+\frac{2}{9} x^{9/2} \left(3 a b^2 B+A b^3\right)+\frac{2}{13} b^3 B x^{13/2}","-\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,"(-2*a^3*A)/(3*x^(3/2)) + 2*(3*a^2*A*b + a^3*B)*Sqrt[x] + (6*(a*A*b^2 + a^2*b*B)*x^(5/2))/5 + (2*(A*b^3 + 3*a*b^2*B)*x^(9/2))/9 + (2*b^3*B*x^(13/2))/13","A",1
366,1,89,81,0.0610342,"\int \frac{\left(a+b x^2\right)^3 \left(A+B x^2\right)}{x^{7/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)^3*(A + B*x^2))/x^(7/2),x]","-\frac{2 a^3 A}{5 x^{5/2}}+2 x^{3/2} \left(a^2 b B+a A b^2\right)-\frac{2 \left(a^3 B+3 a^2 A b\right)}{\sqrt{x}}+\frac{2}{7} x^{7/2} \left(3 a b^2 B+A b^3\right)+\frac{2}{11} b^3 B x^{11/2}","-\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*a^3*A)/(5*x^(5/2)) - (2*(3*a^2*A*b + a^3*B))/Sqrt[x] + 2*(a*A*b^2 + a^2*b*B)*x^(3/2) + (2*(A*b^3 + 3*a*b^2*B)*x^(7/2))/7 + (2*b^3*B*x^(11/2))/11","A",1
367,1,185,276,0.2809046,"\int \frac{x^{7/2} \left(A+B x^2\right)}{a+b x^2} \, dx","IntegrateAlgebraic[(x^(7/2)*(A + B*x^2))/(a + b*x^2),x]","\frac{\left(a^{9/4} B-a^{5/4} A b\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)}{\sqrt{2} b^{13/4}}-\frac{\left(a^{9/4} B-a^{5/4} A b\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} b^{13/4}}+\frac{2 \sqrt{x} \left(a^2 B-a A b\right)}{b^3}+\frac{2 x^{5/2} (A b-a B)}{5 b^2}+\frac{2 B x^{9/2}}{9 b}","-\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,"(2*(-(a*A*b) + a^2*B)*Sqrt[x])/b^3 + (2*(A*b - a*B)*x^(5/2))/(5*b^2) + (2*B*x^(9/2))/(9*b) + ((-(a^(5/4)*A*b) + a^(9/4)*B)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(Sqrt[2]*b^(13/4)) - ((-(a^(5/4)*A*b) + a^(9/4)*B)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*b^(13/4))","A",1
368,1,165,257,0.2457643,"\int \frac{x^{5/2} \left(A+B x^2\right)}{a+b x^2} \, dx","IntegrateAlgebraic[(x^(5/2)*(A + B*x^2))/(a + b*x^2),x]","-\frac{\left(a^{7/4} B-a^{3/4} A b\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)}{\sqrt{2} b^{11/4}}-\frac{\left(a^{7/4} B-a^{3/4} A b\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\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}","-\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*(A*b - a*B)*x^(3/2))/(3*b^2) + (2*B*x^(7/2))/(7*b) - ((-(a^(3/4)*A*b) + a^(7/4)*B)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(Sqrt[2]*b^(11/4)) - ((-(a^(3/4)*A*b) + a^(7/4)*B)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*b^(11/4))","A",1
369,1,162,255,0.2297587,"\int \frac{x^{3/2} \left(A+B x^2\right)}{a+b x^2} \, dx","IntegrateAlgebraic[(x^(3/2)*(A + B*x^2))/(a + b*x^2),x]","-\frac{\left(a^{5/4} B-\sqrt[4]{a} A b\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)}{\sqrt{2} b^{9/4}}+\frac{\left(a^{5/4} B-\sqrt[4]{a} A b\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} b^{9/4}}+\frac{2 \sqrt{x} (A b-a B)}{b^2}+\frac{2 B x^{5/2}}{5 b}","\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,"(2*(A*b - a*B)*Sqrt[x])/b^2 + (2*B*x^(5/2))/(5*b) - ((-(a^(1/4)*A*b) + a^(5/4)*B)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(Sqrt[2]*b^(9/4)) + ((-(a^(1/4)*A*b) + a^(5/4)*B)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*b^(9/4))","A",1
370,1,135,237,0.2148686,"\int \frac{\sqrt{x} \left(A+B x^2\right)}{a+b x^2} \, dx","IntegrateAlgebraic[(Sqrt[x]*(A + B*x^2))/(a + b*x^2),x]","\frac{(a B-A b) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)}{\sqrt{2} \sqrt[4]{a} b^{7/4}}+\frac{(a B-A b) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} \sqrt[4]{a} b^{7/4}}+\frac{2 B x^{3/2}}{3 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} \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*B*x^(3/2))/(3*b) + ((-(A*b) + a*B)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(Sqrt[2]*a^(1/4)*b^(7/4)) + ((-(A*b) + a*B)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*a^(1/4)*b^(7/4))","A",1
371,1,134,235,0.2098957,"\int \frac{A+B x^2}{\sqrt{x} \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(A + B*x^2)/(Sqrt[x]*(a + b*x^2)),x]","\frac{(a B-A b) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)}{\sqrt{2} a^{3/4} b^{5/4}}-\frac{(a B-A b) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\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)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(Sqrt[2]*a^(3/4)*b^(5/4)) - ((-(A*b) + a*B)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*a^(3/4)*b^(5/4))","A",1
372,1,135,235,0.2360344,"\int \frac{A+B x^2}{x^{3/2} \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^(3/2)*(a + b*x^2)),x]","-\frac{(a B-A b) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)}{\sqrt{2} a^{5/4} b^{3/4}}-\frac{(a B-A b) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} a^{5/4} b^{3/4}}-\frac{2 A}{a \sqrt{x}}","-\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)/(a*Sqrt[x]) - ((-(A*b) + a*B)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(Sqrt[2]*a^(5/4)*b^(3/4)) - ((-(A*b) + a*B)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*a^(5/4)*b^(3/4))","A",1
373,1,136,237,0.2749505,"\int \frac{A+B x^2}{x^{5/2} \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^(5/2)*(a + b*x^2)),x]","-\frac{(a B-A b) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)}{\sqrt{2} a^{7/4} \sqrt[4]{b}}+\frac{(a B-A b) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\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)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(Sqrt[2]*a^(7/4)*b^(1/4)) + ((-(A*b) + a*B)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*a^(7/4)*b^(1/4))","A",1
374,1,161,255,0.2859834,"\int \frac{A+B x^2}{x^{7/2} \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^(7/2)*(a + b*x^2)),x]","\frac{\left(a \sqrt[4]{b} B-A b^{5/4}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)}{\sqrt{2} a^{9/4}}+\frac{\left(a \sqrt[4]{b} B-A b^{5/4}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} a^{9/4}}-\frac{2 (a B-A b)}{a^2 \sqrt{x}}-\frac{2 A}{5 a 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)/(5*a*x^(5/2)) - (2*(-(A*b) + a*B))/(a^2*Sqrt[x]) + ((-(A*b^(5/4)) + a*b^(1/4)*B)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(Sqrt[2]*a^(9/4)) + ((-(A*b^(5/4)) + a*b^(1/4)*B)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*a^(9/4))","A",1
375,1,222,310,0.2945886,"\int \frac{x^{7/2} \left(A+B x^2\right)}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^(7/2)*(A + B*x^2))/(a + b*x^2)^2,x]","\left(\frac{5 \sqrt[4]{a} A}{4 \sqrt{2} b^{9/4}}-\frac{9 a^{5/4} B}{4 \sqrt{2} b^{13/4}}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(\frac{9 a^{5/4} B}{4 \sqrt{2} b^{13/4}}-\frac{5 \sqrt[4]{a} A}{4 \sqrt{2} b^{9/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)+\frac{\sqrt{x} \left(25 a A b-45 a^2 B\right)+x^{5/2} \left(20 A b^2-36 a b B\right)+4 b^2 B x^{9/2}}{10 b^3 \left(a+b x^2\right)}","\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,"((25*a*A*b - 45*a^2*B)*Sqrt[x] + (20*A*b^2 - 36*a*b*B)*x^(5/2) + 4*b^2*B*x^(9/2))/(10*b^3*(a + b*x^2)) + ((5*a^(1/4)*A)/(4*Sqrt[2]*b^(9/4)) - (9*a^(5/4)*B)/(4*Sqrt[2]*b^(13/4)))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((-5*a^(1/4)*A)/(4*Sqrt[2]*b^(9/4)) + (9*a^(5/4)*B)/(4*Sqrt[2]*b^(13/4)))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]","A",1
376,1,215,289,0.3131541,"\int \frac{x^{5/2} \left(A+B x^2\right)}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^(5/2)*(A + B*x^2))/(a + b*x^2)^2,x]","\left(\frac{7 a^{3/4} B}{4 \sqrt{2} b^{11/4}}-\frac{3 A}{4 \sqrt{2} \sqrt[4]{a} b^{7/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)+\left(\frac{7 a B-3 A b}{8 \sqrt{2} \sqrt[4]{a} b^{11/4}}-\frac{3 A b-7 a B}{8 \sqrt{2} \sqrt[4]{a} b^{11/4}}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\frac{x^{3/2} (7 a B-3 A b)+4 b B x^{7/2}}{6 b^2 \left(a+b x^2\right)}","\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,"((-3*A*b + 7*a*B)*x^(3/2) + 4*b*B*x^(7/2))/(6*b^2*(a + b*x^2)) + (-1/8*(3*A*b - 7*a*B)/(Sqrt[2]*a^(1/4)*b^(11/4)) + (-3*A*b + 7*a*B)/(8*Sqrt[2]*a^(1/4)*b^(11/4)))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((-3*A)/(4*Sqrt[2]*a^(1/4)*b^(7/4)) + (7*a^(3/4)*B)/(4*Sqrt[2]*b^(11/4)))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]","A",1
377,1,214,284,0.3217751,"\int \frac{x^{3/2} \left(A+B x^2\right)}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^(3/2)*(A + B*x^2))/(a + b*x^2)^2,x]","\left(\frac{5 a B-A b}{8 \sqrt{2} a^{3/4} b^{9/4}}-\frac{A b-5 a B}{8 \sqrt{2} a^{3/4} b^{9/4}}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(\frac{A}{4 \sqrt{2} a^{3/4} b^{5/4}}-\frac{5 \sqrt[4]{a} B}{4 \sqrt{2} b^{9/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)+\frac{\sqrt{x} (5 a B-A b)+4 b B x^{5/2}}{2 b^2 \left(a+b x^2\right)}","-\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,"((-(A*b) + 5*a*B)*Sqrt[x] + 4*b*B*x^(5/2))/(2*b^2*(a + b*x^2)) + (-1/8*(A*b - 5*a*B)/(Sqrt[2]*a^(3/4)*b^(9/4)) + (-(A*b) + 5*a*B)/(8*Sqrt[2]*a^(3/4)*b^(9/4)))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + (A/(4*Sqrt[2]*a^(3/4)*b^(5/4)) - (5*a^(1/4)*B)/(4*Sqrt[2]*b^(9/4)))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]","A",1
378,1,190,261,0.246931,"\int \frac{\sqrt{x} \left(A+B x^2\right)}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(Sqrt[x]*(A + B*x^2))/(a + b*x^2)^2,x]","\left(-\frac{A}{4 \sqrt{2} a^{5/4} b^{3/4}}-\frac{3 B}{4 \sqrt{2} \sqrt[4]{a} b^{7/4}}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(-\frac{A}{4 \sqrt{2} a^{5/4} b^{3/4}}-\frac{3 B}{4 \sqrt{2} \sqrt[4]{a} b^{7/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)+\frac{x^{3/2} \left(\frac{A}{2 a}-\frac{B}{2 b}\right)}{a+b x^2}","\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,"((A/(2*a) - B/(2*b))*x^(3/2))/(a + b*x^2) + (-1/4*A/(Sqrt[2]*a^(5/4)*b^(3/4)) - (3*B)/(4*Sqrt[2]*a^(1/4)*b^(7/4)))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + (-1/4*A/(Sqrt[2]*a^(5/4)*b^(3/4)) - (3*B)/(4*Sqrt[2]*a^(1/4)*b^(7/4)))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]","A",1
379,1,190,261,0.2414031,"\int \frac{A+B x^2}{\sqrt{x} \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(A + B*x^2)/(Sqrt[x]*(a + b*x^2)^2),x]","\left(-\frac{3 A}{4 \sqrt{2} a^{7/4} \sqrt[4]{b}}-\frac{B}{4 \sqrt{2} a^{3/4} b^{5/4}}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(\frac{3 A}{4 \sqrt{2} a^{7/4} \sqrt[4]{b}}+\frac{B}{4 \sqrt{2} a^{3/4} b^{5/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)+\frac{\sqrt{x} \left(\frac{A}{2 a}-\frac{B}{2 b}\right)}{a+b x^2}","-\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,"((A/(2*a) - B/(2*b))*Sqrt[x])/(a + b*x^2) + ((-3*A)/(4*Sqrt[2]*a^(7/4)*b^(1/4)) - B/(4*Sqrt[2]*a^(3/4)*b^(5/4)))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((3*A)/(4*Sqrt[2]*a^(7/4)*b^(1/4)) + B/(4*Sqrt[2]*a^(3/4)*b^(5/4)))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]","A",1
380,1,201,289,0.3092869,"\int \frac{A+B x^2}{x^{3/2} \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^(3/2)*(a + b*x^2)^2),x]","\left(\frac{5 A \sqrt[4]{b}}{4 \sqrt{2} a^{9/4}}-\frac{B}{4 \sqrt{2} a^{5/4} b^{3/4}}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(\frac{5 A \sqrt[4]{b}}{4 \sqrt{2} a^{9/4}}-\frac{B}{4 \sqrt{2} a^{5/4} b^{3/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)+\frac{-4 a A+a B x^2-5 A b x^2}{2 a^3 \sqrt{x}+2 a^2 b x^{5/2}}","-\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,"(-4*a*A - 5*A*b*x^2 + a*B*x^2)/(2*a^3*Sqrt[x] + 2*a^2*b*x^(5/2)) + ((5*A*b^(1/4))/(4*Sqrt[2]*a^(9/4)) - B/(4*Sqrt[2]*a^(5/4)*b^(3/4)))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((5*A*b^(1/4))/(4*Sqrt[2]*a^(9/4)) - B/(4*Sqrt[2]*a^(5/4)*b^(3/4)))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]","A",1
381,1,202,289,0.3133375,"\int \frac{A+B x^2}{x^{5/2} \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^(5/2)*(a + b*x^2)^2),x]","\left(\frac{7 A b^{3/4}}{4 \sqrt{2} a^{11/4}}-\frac{3 B}{4 \sqrt{2} a^{7/4} \sqrt[4]{b}}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(\frac{3 B}{4 \sqrt{2} a^{7/4} \sqrt[4]{b}}-\frac{7 A b^{3/4}}{4 \sqrt{2} a^{11/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)+\frac{-4 a A+3 a B x^2-7 A b x^2}{6 a^3 x^{3/2}+6 a^2 b x^{7/2}}","\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,"(-4*a*A - 7*A*b*x^2 + 3*a*B*x^2)/(6*a^3*x^(3/2) + 6*a^2*b*x^(7/2)) + ((7*A*b^(3/4))/(4*Sqrt[2]*a^(11/4)) - (3*B)/(4*Sqrt[2]*a^(7/4)*b^(1/4)))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((-7*A*b^(3/4))/(4*Sqrt[2]*a^(11/4)) + (3*B)/(4*Sqrt[2]*a^(7/4)*b^(1/4)))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]","A",1
382,1,224,310,0.3386304,"\int \frac{A+B x^2}{x^{7/2} \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^(7/2)*(a + b*x^2)^2),x]","\left(\frac{5 \sqrt[4]{b} B}{4 \sqrt{2} a^{9/4}}-\frac{9 A b^{5/4}}{4 \sqrt{2} a^{13/4}}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(\frac{5 \sqrt[4]{b} B}{4 \sqrt{2} a^{9/4}}-\frac{9 A b^{5/4}}{4 \sqrt{2} a^{13/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)+\frac{-4 a^2 A-20 a^2 B x^2+36 a A b x^2-25 a b B x^4+45 A b^2 x^4}{10 a^4 x^{5/2}+10 a^3 b x^{9/2}}","\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,"(-4*a^2*A + 36*a*A*b*x^2 - 20*a^2*B*x^2 + 45*A*b^2*x^4 - 25*a*b*B*x^4)/(10*a^4*x^(5/2) + 10*a^3*b*x^(9/2)) + ((-9*A*b^(5/4))/(4*Sqrt[2]*a^(13/4)) + (5*b^(1/4)*B)/(4*Sqrt[2]*a^(9/4)))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((-9*A*b^(5/4))/(4*Sqrt[2]*a^(13/4)) + (5*b^(1/4)*B)/(4*Sqrt[2]*a^(9/4)))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]","A",1
383,1,237,316,0.3453124,"\int \frac{x^{7/2} \left(A+B x^2\right)}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(x^(7/2)*(A + B*x^2))/(a + b*x^2)^3,x]","\left(\frac{5 (9 a B-A b)}{64 \sqrt{2} a^{3/4} b^{13/4}}-\frac{5 (A b-9 a B)}{64 \sqrt{2} a^{3/4} b^{13/4}}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(\frac{5 A}{32 \sqrt{2} a^{3/4} b^{9/4}}-\frac{45 \sqrt[4]{a} B}{32 \sqrt{2} b^{13/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)+\frac{\sqrt{x} \left(45 a^2 B-5 a A b\right)+x^{5/2} \left(81 a b B-9 A b^2\right)+32 b^2 B x^{9/2}}{16 b^3 \left(a+b x^2\right)^2}","-\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,"((-5*a*A*b + 45*a^2*B)*Sqrt[x] + (-9*A*b^2 + 81*a*b*B)*x^(5/2) + 32*b^2*B*x^(9/2))/(16*b^3*(a + b*x^2)^2) + ((-5*(A*b - 9*a*B))/(64*Sqrt[2]*a^(3/4)*b^(13/4)) + (5*(-(A*b) + 9*a*B))/(64*Sqrt[2]*a^(3/4)*b^(13/4)))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((5*A)/(32*Sqrt[2]*a^(3/4)*b^(9/4)) - (45*a^(1/4)*B)/(32*Sqrt[2]*b^(13/4)))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]","A",1
384,1,214,293,0.3487076,"\int \frac{x^{5/2} \left(A+B x^2\right)}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(x^(5/2)*(A + B*x^2))/(a + b*x^2)^3,x]","\left(-\frac{3 A}{32 \sqrt{2} a^{5/4} b^{7/4}}-\frac{21 B}{32 \sqrt{2} \sqrt[4]{a} b^{11/4}}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(-\frac{3 A}{32 \sqrt{2} a^{5/4} b^{7/4}}-\frac{21 B}{32 \sqrt{2} \sqrt[4]{a} b^{11/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)+\frac{x^{3/2} \left(-7 a^2 B-a A b\right)+x^{7/2} \left(3 A b^2-11 a b B\right)}{16 a b^2 \left(a+b x^2\right)^2}","\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,"((-(a*A*b) - 7*a^2*B)*x^(3/2) + (3*A*b^2 - 11*a*b*B)*x^(7/2))/(16*a*b^2*(a + b*x^2)^2) + ((-3*A)/(32*Sqrt[2]*a^(5/4)*b^(7/4)) - (21*B)/(32*Sqrt[2]*a^(1/4)*b^(11/4)))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((-3*A)/(32*Sqrt[2]*a^(5/4)*b^(7/4)) - (21*B)/(32*Sqrt[2]*a^(1/4)*b^(11/4)))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]","A",1
385,1,213,298,0.3404581,"\int \frac{x^{3/2} \left(A+B x^2\right)}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(x^(3/2)*(A + B*x^2))/(a + b*x^2)^3,x]","\left(-\frac{3 A}{32 \sqrt{2} a^{7/4} b^{5/4}}-\frac{5 B}{32 \sqrt{2} a^{3/4} b^{9/4}}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(\frac{3 A}{32 \sqrt{2} a^{7/4} b^{5/4}}+\frac{5 B}{32 \sqrt{2} a^{3/4} b^{9/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)+\frac{\sqrt{x} \left(-5 a^2 B-3 a A b\right)+x^{5/2} \left(A b^2-9 a b B\right)}{16 a b^2 \left(a+b x^2\right)^2}","-\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,"((-3*a*A*b - 5*a^2*B)*Sqrt[x] + (A*b^2 - 9*a*b*B)*x^(5/2))/(16*a*b^2*(a + b*x^2)^2) + ((-3*A)/(32*Sqrt[2]*a^(7/4)*b^(5/4)) - (5*B)/(32*Sqrt[2]*a^(3/4)*b^(9/4)))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((3*A)/(32*Sqrt[2]*a^(7/4)*b^(5/4)) + (5*B)/(32*Sqrt[2]*a^(3/4)*b^(9/4)))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]","A",1
386,1,214,298,0.2834324,"\int \frac{\sqrt{x} \left(A+B x^2\right)}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(Sqrt[x]*(A + B*x^2))/(a + b*x^2)^3,x]","\left(-\frac{5 A}{32 \sqrt{2} a^{9/4} b^{3/4}}-\frac{3 B}{32 \sqrt{2} a^{5/4} b^{7/4}}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(-\frac{5 A}{32 \sqrt{2} a^{9/4} b^{3/4}}-\frac{3 B}{32 \sqrt{2} a^{5/4} b^{7/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)+\frac{x^{3/2} \left(9 a A b-a^2 B\right)+x^{7/2} \left(3 a b B+5 A b^2\right)}{16 a^2 b \left(a+b x^2\right)^2}","\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,"((9*a*A*b - a^2*B)*x^(3/2) + (5*A*b^2 + 3*a*b*B)*x^(7/2))/(16*a^2*b*(a + b*x^2)^2) + ((-5*A)/(32*Sqrt[2]*a^(9/4)*b^(3/4)) - (3*B)/(32*Sqrt[2]*a^(5/4)*b^(7/4)))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((-5*A)/(32*Sqrt[2]*a^(9/4)*b^(3/4)) - (3*B)/(32*Sqrt[2]*a^(5/4)*b^(7/4)))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]","A",1
387,1,213,293,0.2743141,"\int \frac{A+B x^2}{\sqrt{x} \left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(A + B*x^2)/(Sqrt[x]*(a + b*x^2)^3),x]","\left(-\frac{21 A}{32 \sqrt{2} a^{11/4} \sqrt[4]{b}}-\frac{3 B}{32 \sqrt{2} a^{7/4} b^{5/4}}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(\frac{21 A}{32 \sqrt{2} a^{11/4} \sqrt[4]{b}}+\frac{3 B}{32 \sqrt{2} a^{7/4} b^{5/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)+\frac{\sqrt{x} \left(11 a A b-3 a^2 B\right)+x^{5/2} \left(a b B+7 A b^2\right)}{16 a^2 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,"((11*a*A*b - 3*a^2*B)*Sqrt[x] + (7*A*b^2 + a*b*B)*x^(5/2))/(16*a^2*b*(a + b*x^2)^2) + ((-21*A)/(32*Sqrt[2]*a^(11/4)*b^(1/4)) - (3*B)/(32*Sqrt[2]*a^(7/4)*b^(5/4)))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((21*A)/(32*Sqrt[2]*a^(11/4)*b^(1/4)) + (3*B)/(32*Sqrt[2]*a^(7/4)*b^(5/4)))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]","A",1
388,1,235,322,0.3415418,"\int \frac{A+B x^2}{x^{3/2} \left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^(3/2)*(a + b*x^2)^3),x]","\left(\frac{45 A \sqrt[4]{b}}{32 \sqrt{2} a^{13/4}}-\frac{5 B}{32 \sqrt{2} a^{9/4} b^{3/4}}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(\frac{45 A \sqrt[4]{b}}{32 \sqrt{2} a^{13/4}}-\frac{5 B}{32 \sqrt{2} a^{9/4} b^{3/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)-\frac{2 A}{a^3 \sqrt{x}}+\frac{x^{3/2} \left(\frac{B}{4 a}-\frac{A b}{4 a^2}\right)}{\left(a+b x^2\right)^2}+\frac{x^{3/2} \left(\frac{5 B}{16 a^2}-\frac{13 A b}{16 a^3}\right)}{a+b x^2}","-\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]) + ((-1/4*(A*b)/a^2 + B/(4*a))*x^(3/2))/(a + b*x^2)^2 + (((-13*A*b)/(16*a^3) + (5*B)/(16*a^2))*x^(3/2))/(a + b*x^2) + ((45*A*b^(1/4))/(32*Sqrt[2]*a^(13/4)) - (5*B)/(32*Sqrt[2]*a^(9/4)*b^(3/4)))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((45*A*b^(1/4))/(32*Sqrt[2]*a^(13/4)) - (5*B)/(32*Sqrt[2]*a^(9/4)*b^(3/4)))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]","A",1
389,1,237,322,0.3524349,"\int \frac{A+B x^2}{x^{5/2} \left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^(5/2)*(a + b*x^2)^3),x]","\left(\frac{77 A b^{3/4}}{32 \sqrt{2} a^{15/4}}-\frac{21 B}{32 \sqrt{2} a^{11/4} \sqrt[4]{b}}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(\frac{21 B}{32 \sqrt{2} a^{11/4} \sqrt[4]{b}}-\frac{77 A b^{3/4}}{32 \sqrt{2} a^{15/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)+\frac{-32 a^2 A+33 a^2 B x^2-121 a A b x^2+21 a b B x^4-77 A b^2 x^4}{48 a^5 x^{3/2}+96 a^4 b x^{7/2}+48 a^3 b^2 x^{11/2}}","\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,"(-32*a^2*A - 121*a*A*b*x^2 + 33*a^2*B*x^2 - 77*A*b^2*x^4 + 21*a*b*B*x^4)/(48*a^5*x^(3/2) + 96*a^4*b*x^(7/2) + 48*a^3*b^2*x^(11/2)) + ((77*A*b^(3/4))/(32*Sqrt[2]*a^(15/4)) - (21*B)/(32*Sqrt[2]*a^(11/4)*b^(1/4)))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((-77*A*b^(3/4))/(32*Sqrt[2]*a^(15/4)) + (21*B)/(32*Sqrt[2]*a^(11/4)*b^(1/4)))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]","A",1
390,1,261,343,0.3674329,"\int \frac{A+B x^2}{x^{7/2} \left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^(7/2)*(a + b*x^2)^3),x]","\left(\frac{45 \sqrt[4]{b} B}{32 \sqrt{2} a^{13/4}}-\frac{117 A b^{5/4}}{32 \sqrt{2} a^{17/4}}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(\frac{45 \sqrt[4]{b} B}{32 \sqrt{2} a^{13/4}}-\frac{117 A b^{5/4}}{32 \sqrt{2} a^{17/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)+\frac{-32 a^3 A-160 a^3 B x^2+416 a^2 A b x^2-405 a^2 b B x^4+1053 a A b^2 x^4-225 a b^2 B x^6+585 A b^3 x^6}{80 a^6 x^{5/2}+160 a^5 b x^{9/2}+80 a^4 b^2 x^{13/2}}","\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,"(-32*a^3*A + 416*a^2*A*b*x^2 - 160*a^3*B*x^2 + 1053*a*A*b^2*x^4 - 405*a^2*b*B*x^4 + 585*A*b^3*x^6 - 225*a*b^2*B*x^6)/(80*a^6*x^(5/2) + 160*a^5*b*x^(9/2) + 80*a^4*b^2*x^(13/2)) + ((-117*A*b^(5/4))/(32*Sqrt[2]*a^(17/4)) + (45*b^(1/4)*B)/(32*Sqrt[2]*a^(13/4)))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((-117*A*b^(5/4))/(32*Sqrt[2]*a^(17/4)) + (45*b^(1/4)*B)/(32*Sqrt[2]*a^(13/4)))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]","A",1
391,1,67,63,0.0373357,"\int x^{7/2} \left(a+b x^2\right)^2 \left(c+d x^2\right) \, dx","IntegrateAlgebraic[x^(7/2)*(a + b*x^2)^2*(c + d*x^2),x]","\frac{2}{13} x^{13/2} \left(a^2 d+2 a b c\right)+\frac{2}{9} a^2 c x^{9/2}+\frac{2}{17} x^{17/2} \left(2 a b d+b^2 c\right)+\frac{2}{21} b^2 d x^{21/2}","\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*a^2*c*x^(9/2))/9 + (2*(2*a*b*c + a^2*d)*x^(13/2))/13 + (2*(b^2*c + 2*a*b*d)*x^(17/2))/17 + (2*b^2*d*x^(21/2))/21","A",1
392,1,67,63,0.0358128,"\int x^{5/2} \left(a+b x^2\right)^2 \left(c+d x^2\right) \, dx","IntegrateAlgebraic[x^(5/2)*(a + b*x^2)^2*(c + d*x^2),x]","\frac{2}{11} x^{11/2} \left(a^2 d+2 a b c\right)+\frac{2}{7} a^2 c x^{7/2}+\frac{2}{15} x^{15/2} \left(2 a b d+b^2 c\right)+\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*(2*a*b*c + a^2*d)*x^(11/2))/11 + (2*(b^2*c + 2*a*b*d)*x^(15/2))/15 + (2*b^2*d*x^(19/2))/19","A",1
393,1,67,63,0.031648,"\int x^{3/2} \left(a+b x^2\right)^2 \left(c+d x^2\right) \, dx","IntegrateAlgebraic[x^(3/2)*(a + b*x^2)^2*(c + d*x^2),x]","\frac{2}{9} x^{9/2} \left(a^2 d+2 a b c\right)+\frac{2}{5} a^2 c x^{5/2}+\frac{2}{13} x^{13/2} \left(2 a b d+b^2 c\right)+\frac{2}{17} b^2 d x^{17/2}","\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*a^2*c*x^(5/2))/5 + (2*(2*a*b*c + a^2*d)*x^(9/2))/9 + (2*(b^2*c + 2*a*b*d)*x^(13/2))/13 + (2*b^2*d*x^(17/2))/17","A",1
394,1,67,63,0.0317309,"\int \sqrt{x} \left(a+b x^2\right)^2 \left(c+d x^2\right) \, dx","IntegrateAlgebraic[Sqrt[x]*(a + b*x^2)^2*(c + d*x^2),x]","\frac{2}{7} x^{7/2} \left(a^2 d+2 a b c\right)+\frac{2}{3} a^2 c x^{3/2}+\frac{2}{11} x^{11/2} \left(2 a b d+b^2 c\right)+\frac{2}{15} b^2 d x^{15/2}","\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*a^2*c*x^(3/2))/3 + (2*(2*a*b*c + a^2*d)*x^(7/2))/7 + (2*(b^2*c + 2*a*b*d)*x^(11/2))/11 + (2*b^2*d*x^(15/2))/15","A",1
395,1,65,61,0.032078,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)}{\sqrt{x}} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2))/Sqrt[x],x]","\frac{2}{5} x^{5/2} \left(a^2 d+2 a b c\right)+2 a^2 c \sqrt{x}+\frac{2}{9} x^{9/2} \left(2 a b d+b^2 c\right)+\frac{2}{13} b^2 d x^{13/2}","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*a^2*c*Sqrt[x] + (2*(2*a*b*c + a^2*d)*x^(5/2))/5 + (2*(b^2*c + 2*a*b*d)*x^(9/2))/9 + (2*b^2*d*x^(13/2))/13","A",1
396,1,65,61,0.0469465,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)}{x^{3/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2))/x^(3/2),x]","\frac{2}{3} x^{3/2} \left(a^2 d+2 a b c\right)-\frac{2 a^2 c}{\sqrt{x}}+\frac{2}{7} x^{7/2} \left(2 a b d+b^2 c\right)+\frac{2}{11} b^2 d x^{11/2}","-\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,"(-2*a^2*c)/Sqrt[x] + (2*(2*a*b*c + a^2*d)*x^(3/2))/3 + (2*(b^2*c + 2*a*b*d)*x^(7/2))/7 + (2*b^2*d*x^(11/2))/11","A",1
397,1,65,61,0.0448042,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)}{x^{5/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2))/x^(5/2),x]","2 \sqrt{x} \left(a^2 d+2 a b c\right)-\frac{2 a^2 c}{3 x^{3/2}}+\frac{2}{5} x^{5/2} \left(2 a b d+b^2 c\right)+\frac{2}{9} b^2 d x^{9/2}","-\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,"(-2*a^2*c)/(3*x^(3/2)) + 2*(2*a*b*c + a^2*d)*Sqrt[x] + (2*(b^2*c + 2*a*b*d)*x^(5/2))/5 + (2*b^2*d*x^(9/2))/9","A",1
398,1,65,61,0.0438521,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)}{x^{7/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2))/x^(7/2),x]","-\frac{2 \left(a^2 d+2 a b c\right)}{\sqrt{x}}-\frac{2 a^2 c}{5 x^{5/2}}+\frac{2}{3} x^{3/2} \left(2 a b d+b^2 c\right)+\frac{2}{7} b^2 d x^{7/2}","-\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,"(-2*a^2*c)/(5*x^(5/2)) - (2*(2*a*b*c + a^2*d))/Sqrt[x] + (2*(b^2*c + 2*a*b*d)*x^(3/2))/3 + (2*b^2*d*x^(7/2))/7","A",1
399,1,105,97,0.0589667,"\int x^{7/2} \left(a+b x^2\right)^2 \left(c+d x^2\right)^2 \, dx","IntegrateAlgebraic[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{4}{13} x^{13/2} \left(a^2 c d+a b c^2\right)+\frac{2}{9} a^2 c^2 x^{9/2}+\frac{4}{21} x^{21/2} \left(a b d^2+b^2 c d\right)+\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*b*c^2 + a^2*c*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^2*c*d + a*b*d^2)*x^(21/2))/21 + (2*b^2*d^2*x^(25/2))/25","A",1
400,1,105,97,0.0548576,"\int x^{5/2} \left(a+b x^2\right)^2 \left(c+d x^2\right)^2 \, dx","IntegrateAlgebraic[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{4}{11} x^{11/2} \left(a^2 c d+a b c^2\right)+\frac{2}{7} a^2 c^2 x^{7/2}+\frac{4}{19} x^{19/2} \left(a b d^2+b^2 c d\right)+\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*b*c^2 + a^2*c*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^2*c*d + a*b*d^2)*x^(19/2))/19 + (2*b^2*d^2*x^(23/2))/23","A",1
401,1,105,97,0.0573871,"\int x^{3/2} \left(a+b x^2\right)^2 \left(c+d x^2\right)^2 \, dx","IntegrateAlgebraic[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{4}{9} x^{9/2} \left(a^2 c d+a b c^2\right)+\frac{2}{5} a^2 c^2 x^{5/2}+\frac{4}{17} x^{17/2} \left(a b d^2+b^2 c d\right)+\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*b*c^2 + a^2*c*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^2*c*d + a*b*d^2)*x^(17/2))/17 + (2*b^2*d^2*x^(21/2))/21","A",1
402,1,105,97,0.0518471,"\int \sqrt{x} \left(a+b x^2\right)^2 \left(c+d x^2\right)^2 \, dx","IntegrateAlgebraic[Sqrt[x]*(a + b*x^2)^2*(c + d*x^2)^2,x]","\frac{2}{11} x^{11/2} \left(a^2 d^2+4 a b c d+b^2 c^2\right)+\frac{4}{7} x^{7/2} \left(a^2 c d+a b c^2\right)+\frac{2}{3} a^2 c^2 x^{3/2}+\frac{4}{15} x^{15/2} \left(a b d^2+b^2 c d\right)+\frac{2}{19} b^2 d^2 x^{19/2}","\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*a^2*c^2*x^(3/2))/3 + (4*(a*b*c^2 + a^2*c*d)*x^(7/2))/7 + (2*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(11/2))/11 + (4*(b^2*c*d + a*b*d^2)*x^(15/2))/15 + (2*b^2*d^2*x^(19/2))/19","A",1
403,1,103,95,0.0540741,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2}{\sqrt{x}} \, dx","IntegrateAlgebraic[((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)+\frac{4}{5} x^{5/2} \left(a^2 c d+a b c^2\right)+2 a^2 c^2 \sqrt{x}+\frac{4}{13} x^{13/2} \left(a b d^2+b^2 c d\right)+\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*b*c^2 + a^2*c*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^2*c*d + a*b*d^2)*x^(13/2))/13 + (2*b^2*d^2*x^(17/2))/17","A",1
404,1,103,95,0.0748777,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2}{x^{3/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2)^2)/x^(3/2),x]","\frac{2}{7} x^{7/2} \left(a^2 d^2+4 a b c d+b^2 c^2\right)+\frac{4}{3} x^{3/2} \left(a^2 c d+a b c^2\right)-\frac{2 a^2 c^2}{\sqrt{x}}+\frac{4}{11} x^{11/2} \left(a b d^2+b^2 c d\right)+\frac{2}{15} b^2 d^2 x^{15/2}","\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*a^2*c^2)/Sqrt[x] + (4*(a*b*c^2 + a^2*c*d)*x^(3/2))/3 + (2*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(7/2))/7 + (4*(b^2*c*d + a*b*d^2)*x^(11/2))/11 + (2*b^2*d^2*x^(15/2))/15","A",1
405,1,103,95,0.0784978,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2}{x^{5/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2)^2)/x^(5/2),x]","\frac{2}{5} x^{5/2} \left(a^2 d^2+4 a b c d+b^2 c^2\right)+4 \sqrt{x} \left(a^2 c d+a b c^2\right)-\frac{2 a^2 c^2}{3 x^{3/2}}+\frac{4}{9} x^{9/2} \left(a b d^2+b^2 c d\right)+\frac{2}{13} b^2 d^2 x^{13/2}","\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*a^2*c^2)/(3*x^(3/2)) + 4*(a*b*c^2 + a^2*c*d)*Sqrt[x] + (2*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(5/2))/5 + (4*(b^2*c*d + a*b*d^2)*x^(9/2))/9 + (2*b^2*d^2*x^(13/2))/13","A",1
406,1,103,95,0.0868854,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2}{x^{7/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2)^2)/x^(7/2),x]","\frac{2}{3} x^{3/2} \left(a^2 d^2+4 a b c d+b^2 c^2\right)-\frac{4 \left(a^2 c d+a b c^2\right)}{\sqrt{x}}-\frac{2 a^2 c^2}{5 x^{5/2}}+\frac{4}{7} x^{7/2} \left(a b d^2+b^2 c d\right)+\frac{2}{11} b^2 d^2 x^{11/2}","\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*a^2*c^2)/(5*x^(5/2)) - (4*(a*b*c^2 + a^2*c*d))/Sqrt[x] + (2*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(3/2))/3 + (4*(b^2*c*d + a*b*d^2)*x^(7/2))/7 + (2*b^2*d^2*x^(11/2))/11","A",1
407,1,151,139,0.0860677,"\int x^{7/2} \left(a+b x^2\right)^2 \left(c+d x^2\right)^3 \, dx","IntegrateAlgebraic[x^(7/2)*(a + b*x^2)^2*(c + d*x^2)^3,x]","\frac{2}{21} x^{21/2} \left(a^2 d^3+6 a b c d^2+3 b^2 c^2 d\right)+\frac{2}{17} x^{17/2} \left(3 a^2 c d^2+6 a b c^2 d+b^2 c^3\right)+\frac{2}{13} x^{13/2} \left(3 a^2 c^2 d+2 a b c^3\right)+\frac{2}{9} a^2 c^3 x^{9/2}+\frac{2}{25} x^{25/2} \left(2 a b d^3+3 b^2 c d^2\right)+\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*(2*a*b*c^3 + 3*a^2*c^2*d)*x^(13/2))/13 + (2*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^(17/2))/17 + (2*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x^(21/2))/21 + (2*(3*b^2*c*d^2 + 2*a*b*d^3)*x^(25/2))/25 + (2*b^2*d^3*x^(29/2))/29","A",1
408,1,151,139,0.0805102,"\int x^{5/2} \left(a+b x^2\right)^2 \left(c+d x^2\right)^3 \, dx","IntegrateAlgebraic[x^(5/2)*(a + b*x^2)^2*(c + d*x^2)^3,x]","\frac{2}{19} x^{19/2} \left(a^2 d^3+6 a b c d^2+3 b^2 c^2 d\right)+\frac{2}{15} x^{15/2} \left(3 a^2 c d^2+6 a b c^2 d+b^2 c^3\right)+\frac{2}{11} x^{11/2} \left(3 a^2 c^2 d+2 a b c^3\right)+\frac{2}{7} a^2 c^3 x^{7/2}+\frac{2}{23} x^{23/2} \left(2 a b d^3+3 b^2 c d^2\right)+\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*(2*a*b*c^3 + 3*a^2*c^2*d)*x^(11/2))/11 + (2*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^(15/2))/15 + (2*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x^(19/2))/19 + (2*(3*b^2*c*d^2 + 2*a*b*d^3)*x^(23/2))/23 + (2*b^2*d^3*x^(27/2))/27","A",1
409,1,151,139,0.0708089,"\int x^{3/2} \left(a+b x^2\right)^2 \left(c+d x^2\right)^3 \, dx","IntegrateAlgebraic[x^(3/2)*(a + b*x^2)^2*(c + d*x^2)^3,x]","\frac{2}{17} x^{17/2} \left(a^2 d^3+6 a b c d^2+3 b^2 c^2 d\right)+\frac{2}{13} x^{13/2} \left(3 a^2 c d^2+6 a b c^2 d+b^2 c^3\right)+\frac{2}{9} x^{9/2} \left(3 a^2 c^2 d+2 a b c^3\right)+\frac{2}{5} a^2 c^3 x^{5/2}+\frac{2}{21} x^{21/2} \left(2 a b d^3+3 b^2 c d^2\right)+\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*(2*a*b*c^3 + 3*a^2*c^2*d)*x^(9/2))/9 + (2*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^(13/2))/13 + (2*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x^(17/2))/17 + (2*(3*b^2*c*d^2 + 2*a*b*d^3)*x^(21/2))/21 + (2*b^2*d^3*x^(25/2))/25","A",1
410,1,151,139,0.0722721,"\int \sqrt{x} \left(a+b x^2\right)^2 \left(c+d x^2\right)^3 \, dx","IntegrateAlgebraic[Sqrt[x]*(a + b*x^2)^2*(c + d*x^2)^3,x]","\frac{2}{15} x^{15/2} \left(a^2 d^3+6 a b c d^2+3 b^2 c^2 d\right)+\frac{2}{11} x^{11/2} \left(3 a^2 c d^2+6 a b c^2 d+b^2 c^3\right)+\frac{2}{7} x^{7/2} \left(3 a^2 c^2 d+2 a b c^3\right)+\frac{2}{3} a^2 c^3 x^{3/2}+\frac{2}{19} x^{19/2} \left(2 a b d^3+3 b^2 c d^2\right)+\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*(2*a*b*c^3 + 3*a^2*c^2*d)*x^(7/2))/7 + (2*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^(11/2))/11 + (2*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x^(15/2))/15 + (2*(3*b^2*c*d^2 + 2*a*b*d^3)*x^(19/2))/19 + (2*b^2*d^3*x^(23/2))/23","A",1
411,1,149,137,0.0702359,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3}{\sqrt{x}} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2)^3)/Sqrt[x],x]","\frac{2}{13} x^{13/2} \left(a^2 d^3+6 a b c d^2+3 b^2 c^2 d\right)+\frac{2}{9} x^{9/2} \left(3 a^2 c d^2+6 a b c^2 d+b^2 c^3\right)+\frac{2}{5} x^{5/2} \left(3 a^2 c^2 d+2 a b c^3\right)+2 a^2 c^3 \sqrt{x}+\frac{2}{17} x^{17/2} \left(2 a b d^3+3 b^2 c d^2\right)+\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*(2*a*b*c^3 + 3*a^2*c^2*d)*x^(5/2))/5 + (2*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^(9/2))/9 + (2*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x^(13/2))/13 + (2*(3*b^2*c*d^2 + 2*a*b*d^3)*x^(17/2))/17 + (2*b^2*d^3*x^(21/2))/21","A",1
412,1,149,137,0.0898186,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3}{x^{3/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2)^3)/x^(3/2),x]","\frac{2}{11} x^{11/2} \left(a^2 d^3+6 a b c d^2+3 b^2 c^2 d\right)+\frac{2}{7} x^{7/2} \left(3 a^2 c d^2+6 a b c^2 d+b^2 c^3\right)+\frac{2}{3} x^{3/2} \left(3 a^2 c^2 d+2 a b c^3\right)-\frac{2 a^2 c^3}{\sqrt{x}}+\frac{2}{15} x^{15/2} \left(2 a b d^3+3 b^2 c d^2\right)+\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*(2*a*b*c^3 + 3*a^2*c^2*d)*x^(3/2))/3 + (2*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^(7/2))/7 + (2*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x^(11/2))/11 + (2*(3*b^2*c*d^2 + 2*a*b*d^3)*x^(15/2))/15 + (2*b^2*d^3*x^(19/2))/19","A",1
413,1,149,137,0.0955469,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3}{x^{5/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2)^3)/x^(5/2),x]","\frac{2}{9} x^{9/2} \left(a^2 d^3+6 a b c d^2+3 b^2 c^2 d\right)+\frac{2}{5} x^{5/2} \left(3 a^2 c d^2+6 a b c^2 d+b^2 c^3\right)+2 \sqrt{x} \left(3 a^2 c^2 d+2 a b c^3\right)-\frac{2 a^2 c^3}{3 x^{3/2}}+\frac{2}{13} x^{13/2} \left(2 a b d^3+3 b^2 c d^2\right)+\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*(2*a*b*c^3 + 3*a^2*c^2*d)*Sqrt[x] + (2*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^(5/2))/5 + (2*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x^(9/2))/9 + (2*(3*b^2*c*d^2 + 2*a*b*d^3)*x^(13/2))/13 + (2*b^2*d^3*x^(17/2))/17","A",1
414,1,149,137,0.1010301,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3}{x^{7/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2)^3)/x^(7/2),x]","\frac{2}{7} x^{7/2} \left(a^2 d^3+6 a b c d^2+3 b^2 c^2 d\right)+\frac{2}{3} x^{3/2} \left(3 a^2 c d^2+6 a b c^2 d+b^2 c^3\right)-\frac{2 \left(3 a^2 c^2 d+2 a b c^3\right)}{\sqrt{x}}-\frac{2 a^2 c^3}{5 x^{5/2}}+\frac{2}{11} x^{11/2} \left(2 a b d^3+3 b^2 c d^2\right)+\frac{2}{15} b^2 d^3 x^{15/2}","\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*a^2*c^3)/(5*x^(5/2)) - (2*(2*a*b*c^3 + 3*a^2*c^2*d))/Sqrt[x] + (2*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^(3/2))/3 + (2*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x^(7/2))/7 + (2*(3*b^2*c*d^2 + 2*a*b*d^3)*x^(11/2))/11 + (2*b^2*d^3*x^(15/2))/15","A",1
415,1,266,311,0.2840161,"\int \frac{x^{7/2} \left(a+b x^2\right)^2}{c+d x^2} \, dx","IntegrateAlgebraic[(x^(7/2)*(a + b*x^2)^2)/(c + d*x^2),x]","-\frac{\left(a^2 c^{5/4} d^2-2 a b c^{9/4} d+b^2 c^{13/4}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)}{\sqrt{2} d^{17/4}}+\frac{\left(a^2 c^{5/4} d^2-2 a b c^{9/4} d+b^2 c^{13/4}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)}{\sqrt{2} d^{17/4}}+\frac{2 x^{5/2} \left(a^2 d^2-2 a b c d+b^2 c^2\right)}{5 d^3}-\frac{2 \sqrt{x} \left(a^2 c d^2-2 a b c^2 d+b^2 c^3\right)}{d^4}-\frac{2 x^{9/2} \left(b^2 c-2 a b d\right)}{9 d^2}+\frac{2 b^2 x^{13/2}}{13 d}","-\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,"(-2*(b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*Sqrt[x])/d^4 + (2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x^(5/2))/(5*d^3) - (2*(b^2*c - 2*a*b*d)*x^(9/2))/(9*d^2) + (2*b^2*x^(13/2))/(13*d) - ((b^2*c^(13/4) - 2*a*b*c^(9/4)*d + a^2*c^(5/4)*d^2)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/(Sqrt[2]*d^(17/4)) + ((b^2*c^(13/4) - 2*a*b*c^(9/4)*d + a^2*c^(5/4)*d^2)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(Sqrt[2]*d^(17/4))","A",1
416,1,231,290,0.2407485,"\int \frac{x^{5/2} \left(a+b x^2\right)^2}{c+d x^2} \, dx","IntegrateAlgebraic[(x^(5/2)*(a + b*x^2)^2)/(c + d*x^2),x]","\frac{\left(a^2 c^{3/4} d^2-2 a b c^{7/4} d+b^2 c^{11/4}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)}{\sqrt{2} d^{15/4}}+\frac{\left(a^2 c^{3/4} d^2-2 a b c^{7/4} d+b^2 c^{11/4}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)}{\sqrt{2} d^{15/4}}+\frac{2 x^{3/2} \left(a^2 d^2-2 a b c d+b^2 c^2\right)}{3 d^3}-\frac{2 x^{7/2} \left(b^2 c-2 a b d\right)}{7 d^2}+\frac{2 b^2 x^{11/2}}{11 d}","-\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,"(2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x^(3/2))/(3*d^3) - (2*(b^2*c - 2*a*b*d)*x^(7/2))/(7*d^2) + (2*b^2*x^(11/2))/(11*d) + ((b^2*c^(11/4) - 2*a*b*c^(7/4)*d + a^2*c^(3/4)*d^2)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/(Sqrt[2]*d^(15/4)) + ((b^2*c^(11/4) - 2*a*b*c^(7/4)*d + a^2*c^(3/4)*d^2)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(Sqrt[2]*d^(15/4))","A",1
417,1,230,288,0.2281127,"\int \frac{x^{3/2} \left(a+b x^2\right)^2}{c+d x^2} \, dx","IntegrateAlgebraic[(x^(3/2)*(a + b*x^2)^2)/(c + d*x^2),x]","\frac{\left(a^2 \sqrt[4]{c} d^2-2 a b c^{5/4} d+b^2 c^{9/4}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)}{\sqrt{2} d^{13/4}}-\frac{\left(a^2 \sqrt[4]{c} d^2-2 a b c^{5/4} d+b^2 c^{9/4}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)}{\sqrt{2} d^{13/4}}+\frac{2 \sqrt{x} \left(a^2 d^2-2 a b c d+b^2 c^2\right)}{d^3}-\frac{2 x^{5/2} \left(b^2 c-2 a b d\right)}{5 d^2}+\frac{2 b^2 x^{9/2}}{9 d}","\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,"(2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*Sqrt[x])/d^3 - (2*(b^2*c - 2*a*b*d)*x^(5/2))/(5*d^2) + (2*b^2*x^(9/2))/(9*d) + ((b^2*c^(9/4) - 2*a*b*c^(5/4)*d + a^2*c^(1/4)*d^2)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/(Sqrt[2]*d^(13/4)) - ((b^2*c^(9/4) - 2*a*b*c^(5/4)*d + a^2*c^(1/4)*d^2)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(Sqrt[2]*d^(13/4))","A",1
418,1,188,268,0.2353638,"\int \frac{\sqrt{x} \left(a+b x^2\right)^2}{c+d x^2} \, dx","IntegrateAlgebraic[(Sqrt[x]*(a + b*x^2)^2)/(c + d*x^2),x]","-\frac{\left(a^2 d^2-2 a b c d+b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)}{\sqrt{2} \sqrt[4]{c} d^{11/4}}-\frac{\left(a^2 d^2-2 a b c d+b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)}{\sqrt{2} \sqrt[4]{c} d^{11/4}}-\frac{2 x^{3/2} \left(b^2 c-2 a b d\right)}{3 d^2}+\frac{2 b^2 x^{7/2}}{7 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} \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,"(-2*(b^2*c - 2*a*b*d)*x^(3/2))/(3*d^2) + (2*b^2*x^(7/2))/(7*d) - ((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/(Sqrt[2]*c^(1/4)*d^(11/4)) - ((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(Sqrt[2]*c^(1/4)*d^(11/4))","A",1
419,1,185,266,0.2220103,"\int \frac{\left(a+b x^2\right)^2}{\sqrt{x} \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(Sqrt[x]*(c + d*x^2)),x]","-\frac{\left(a^2 d^2-2 a b c d+b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)}{\sqrt{2} c^{3/4} d^{9/4}}+\frac{\left(a^2 d^2-2 a b c d+b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)}{\sqrt{2} c^{3/4} d^{9/4}}-\frac{2 \sqrt{x} \left(b^2 c-2 a b d\right)}{d^2}+\frac{2 b^2 x^{5/2}}{5 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^{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,"(-2*(b^2*c - 2*a*b*d)*Sqrt[x])/d^2 + (2*b^2*x^(5/2))/(5*d) - ((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/(Sqrt[2]*c^(3/4)*d^(9/4)) + ((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(Sqrt[2]*c^(3/4)*d^(9/4))","A",1
420,1,176,260,0.2421427,"\int \frac{\left(a+b x^2\right)^2}{x^{3/2} \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^(3/2)*(c + d*x^2)),x]","\frac{\left(a^2 d^2-2 a b c d+b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)}{\sqrt{2} c^{5/4} d^{7/4}}+\frac{\left(a^2 d^2-2 a b c d+b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)}{\sqrt{2} c^{5/4} d^{7/4}}-\frac{2 a^2}{c \sqrt{x}}+\frac{2 b^2 x^{3/2}}{3 d}","-\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,"(-2*a^2)/(c*Sqrt[x]) + (2*b^2*x^(3/2))/(3*d) + ((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/(Sqrt[2]*c^(5/4)*d^(7/4)) + ((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(Sqrt[2]*c^(5/4)*d^(7/4))","A",1
421,1,177,260,0.2325815,"\int \frac{\left(a+b x^2\right)^2}{x^{5/2} \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^(5/2)*(c + d*x^2)),x]","\frac{\left(a^2 d^2-2 a b c d+b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)}{\sqrt{2} c^{7/4} d^{5/4}}-\frac{\left(a^2 d^2-2 a b c d+b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)}{\sqrt{2} c^{7/4} d^{5/4}}-\frac{2 a^2}{3 c x^{3/2}}+\frac{2 b^2 \sqrt{x}}{d}","-\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,"(-2*a^2)/(3*c*x^(3/2)) + (2*b^2*Sqrt[x])/d + ((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/(Sqrt[2]*c^(7/4)*d^(5/4)) - ((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(Sqrt[2]*c^(7/4)*d^(5/4))","A",1
422,1,186,267,0.2344286,"\int \frac{\left(a+b x^2\right)^2}{x^{7/2} \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^(7/2)*(c + d*x^2)),x]","-\frac{\left(a^2 d^2-2 a b c d+b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)}{\sqrt{2} c^{9/4} d^{3/4}}-\frac{\left(a^2 d^2-2 a b c d+b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)}{\sqrt{2} c^{9/4} d^{3/4}}+\frac{2 \left(a^2 d-2 a b c\right)}{c^2 \sqrt{x}}-\frac{2 a^2}{5 c x^{5/2}}","-\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,"(-2*a^2)/(5*c*x^(5/2)) + (2*(-2*a*b*c + a^2*d))/(c^2*Sqrt[x]) - ((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/(Sqrt[2]*c^(9/4)*d^(3/4)) - ((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(Sqrt[2]*c^(9/4)*d^(3/4))","A",1
423,1,187,269,0.2310375,"\int \frac{\left(a+b x^2\right)^2}{x^{9/2} \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^(9/2)*(c + d*x^2)),x]","-\frac{\left(a^2 d^2-2 a b c d+b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)}{\sqrt{2} c^{11/4} \sqrt[4]{d}}+\frac{\left(a^2 d^2-2 a b c d+b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)}{\sqrt{2} c^{11/4} \sqrt[4]{d}}+\frac{2 \left(a^2 d-2 a b c\right)}{3 c^2 x^{3/2}}-\frac{2 a^2}{7 c x^{7/2}}","-\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,"(-2*a^2)/(7*c*x^(7/2)) + (2*(-2*a*b*c + a^2*d))/(3*c^2*x^(3/2)) - ((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/(Sqrt[2]*c^(11/4)*d^(1/4)) + ((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(Sqrt[2]*c^(11/4)*d^(1/4))","A",1
424,1,230,288,0.2638666,"\int \frac{\left(c+d x^2\right)^2}{x^{11/2} \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(c + d*x^2)^2/(x^(11/2)*(a + b*x^2)),x]","-\frac{2 \left(2 a c d-b c^2\right)}{5 a^2 x^{5/2}}+\frac{\left(a^2 \sqrt[4]{b} d^2-2 a b^{5/4} c d+b^{9/4} c^2\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)}{\sqrt{2} a^{13/4}}+\frac{\left(a^2 \sqrt[4]{b} d^2-2 a b^{5/4} c d+b^{9/4} c^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} a^{13/4}}-\frac{2 \left(a^2 d^2-2 a b c d+b^2 c^2\right)}{a^3 \sqrt{x}}-\frac{2 c^2}{9 a 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*c^2)/(9*a*x^(9/2)) - (2*(-(b*c^2) + 2*a*c*d))/(5*a^2*x^(5/2)) - (2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2))/(a^3*Sqrt[x]) + ((b^(9/4)*c^2 - 2*a*b^(5/4)*c*d + a^2*b^(1/4)*d^2)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(Sqrt[2]*a^(13/4)) + ((b^(9/4)*c^2 - 2*a*b^(5/4)*c*d + a^2*b^(1/4)*d^2)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*a^(13/4))","A",1
425,1,316,375,0.4566993,"\int \frac{x^{7/2} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[(x^(7/2)*(a + b*x^2)^2)/(c + d*x^2)^2,x]","\left(\frac{5 a^2 \sqrt[4]{c}}{4 \sqrt{2} d^{9/4}}-\frac{9 a b c^{5/4}}{2 \sqrt{2} d^{13/4}}+\frac{13 b^2 c^{9/4}}{4 \sqrt{2} d^{17/4}}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(-\frac{5 a^2 \sqrt[4]{c}}{4 \sqrt{2} d^{9/4}}+\frac{9 a b c^{5/4}}{2 \sqrt{2} d^{13/4}}-\frac{13 b^2 c^{9/4}}{4 \sqrt{2} d^{17/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+\sqrt{x} \left(\frac{2 a^2}{d^2}-\frac{8 a b c}{d^3}+\frac{6 b^2 c^2}{d^4}\right)+x^{5/2} \left(\frac{4 a b}{5 d^2}-\frac{4 b^2 c}{5 d^3}\right)+\frac{c \sqrt{x} (b c-a d)^2}{2 d^4 \left(c+d x^2\right)}+\frac{2 b^2 x^{9/2}}{9 d^2}","\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,"((6*b^2*c^2)/d^4 - (8*a*b*c)/d^3 + (2*a^2)/d^2)*Sqrt[x] + ((-4*b^2*c)/(5*d^3) + (4*a*b)/(5*d^2))*x^(5/2) + (2*b^2*x^(9/2))/(9*d^2) + (c*(b*c - a*d)^2*Sqrt[x])/(2*d^4*(c + d*x^2)) + ((13*b^2*c^(9/4))/(4*Sqrt[2]*d^(17/4)) - (9*a*b*c^(5/4))/(2*Sqrt[2]*d^(13/4)) + (5*a^2*c^(1/4))/(4*Sqrt[2]*d^(9/4)))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + ((-13*b^2*c^(9/4))/(4*Sqrt[2]*d^(17/4)) + (9*a*b*c^(5/4))/(2*Sqrt[2]*d^(13/4)) - (5*a^2*c^(1/4))/(4*Sqrt[2]*d^(9/4)))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
426,1,281,346,0.4503387,"\int \frac{x^{5/2} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[(x^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^2,x]","\left(-\frac{3 a^2}{4 \sqrt{2} \sqrt[4]{c} d^{7/4}}+\frac{7 a b c^{3/4}}{2 \sqrt{2} d^{11/4}}-\frac{11 b^2 c^{7/4}}{4 \sqrt{2} d^{15/4}}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(-\frac{3 a^2}{4 \sqrt{2} \sqrt[4]{c} d^{7/4}}+\frac{7 a b c^{3/4}}{2 \sqrt{2} d^{11/4}}-\frac{11 b^2 c^{7/4}}{4 \sqrt{2} d^{15/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+x^{3/2} \left(\frac{4 a b}{3 d^2}-\frac{4 b^2 c}{3 d^3}\right)-\frac{x^{3/2} (b c-a d)^2}{2 d^3 \left(c+d x^2\right)}+\frac{2 b^2 x^{7/2}}{7 d^2}","\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,"((-4*b^2*c)/(3*d^3) + (4*a*b)/(3*d^2))*x^(3/2) + (2*b^2*x^(7/2))/(7*d^2) - ((b*c - a*d)^2*x^(3/2))/(2*d^3*(c + d*x^2)) + ((-11*b^2*c^(7/4))/(4*Sqrt[2]*d^(15/4)) + (7*a*b*c^(3/4))/(2*Sqrt[2]*d^(11/4)) - (3*a^2)/(4*Sqrt[2]*c^(1/4)*d^(7/4)))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + ((-11*b^2*c^(7/4))/(4*Sqrt[2]*d^(15/4)) + (7*a*b*c^(3/4))/(2*Sqrt[2]*d^(11/4)) - (3*a^2)/(4*Sqrt[2]*c^(1/4)*d^(7/4)))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
427,1,277,346,0.3031103,"\int \frac{x^{3/2} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[(x^(3/2)*(a + b*x^2)^2)/(c + d*x^2)^2,x]","\left(-\frac{a^2}{4 \sqrt{2} c^{3/4} d^{5/4}}+\frac{5 a b \sqrt[4]{c}}{2 \sqrt{2} d^{9/4}}-\frac{9 b^2 c^{5/4}}{4 \sqrt{2} d^{13/4}}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(\frac{a^2}{4 \sqrt{2} c^{3/4} d^{5/4}}-\frac{5 a b \sqrt[4]{c}}{2 \sqrt{2} d^{9/4}}+\frac{9 b^2 c^{5/4}}{4 \sqrt{2} d^{13/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+\sqrt{x} \left(\frac{4 a b}{d^2}-\frac{4 b^2 c}{d^3}\right)-\frac{\sqrt{x} (b c-a d)^2}{2 d^3 \left(c+d x^2\right)}+\frac{2 b^2 x^{5/2}}{5 d^2}","-\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,"((-4*b^2*c)/d^3 + (4*a*b)/d^2)*Sqrt[x] + (2*b^2*x^(5/2))/(5*d^2) - ((b*c - a*d)^2*Sqrt[x])/(2*d^3*(c + d*x^2)) + ((-9*b^2*c^(5/4))/(4*Sqrt[2]*d^(13/4)) + (5*a*b*c^(1/4))/(2*Sqrt[2]*d^(9/4)) - a^2/(4*Sqrt[2]*c^(3/4)*d^(5/4)))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + ((9*b^2*c^(5/4))/(4*Sqrt[2]*d^(13/4)) - (5*a*b*c^(1/4))/(2*Sqrt[2]*d^(9/4)) + a^2/(4*Sqrt[2]*c^(3/4)*d^(5/4)))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
428,1,257,310,0.2856126,"\int \frac{\sqrt{x} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[(Sqrt[x]*(a + b*x^2)^2)/(c + d*x^2)^2,x]","\left(-\frac{a^2}{4 \sqrt{2} c^{5/4} d^{3/4}}-\frac{3 a b}{2 \sqrt{2} \sqrt[4]{c} d^{7/4}}+\frac{7 b^2 c^{3/4}}{4 \sqrt{2} d^{11/4}}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(-\frac{a^2}{4 \sqrt{2} c^{5/4} d^{3/4}}-\frac{3 a b}{2 \sqrt{2} \sqrt[4]{c} d^{7/4}}+\frac{7 b^2 c^{3/4}}{4 \sqrt{2} d^{11/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+\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}","-\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,"(2*b^2*x^(3/2))/(3*d^2) + ((b*c - a*d)^2*x^(3/2))/(2*c*d^2*(c + d*x^2)) + ((7*b^2*c^(3/4))/(4*Sqrt[2]*d^(11/4)) - (3*a*b)/(2*Sqrt[2]*c^(1/4)*d^(7/4)) - a^2/(4*Sqrt[2]*c^(5/4)*d^(3/4)))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + ((7*b^2*c^(3/4))/(4*Sqrt[2]*d^(11/4)) - (3*a*b)/(2*Sqrt[2]*c^(1/4)*d^(7/4)) - a^2/(4*Sqrt[2]*c^(5/4)*d^(3/4)))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
429,1,255,312,0.279047,"\int \frac{\left(a+b x^2\right)^2}{\sqrt{x} \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(Sqrt[x]*(c + d*x^2)^2),x]","\left(-\frac{3 a^2}{4 \sqrt{2} c^{7/4} \sqrt[4]{d}}-\frac{a b}{2 \sqrt{2} c^{3/4} d^{5/4}}+\frac{5 b^2 \sqrt[4]{c}}{4 \sqrt{2} d^{9/4}}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(\frac{3 a^2}{4 \sqrt{2} c^{7/4} \sqrt[4]{d}}+\frac{a b}{2 \sqrt{2} c^{3/4} d^{5/4}}-\frac{5 b^2 \sqrt[4]{c}}{4 \sqrt{2} d^{9/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+\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}","\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,"(2*b^2*Sqrt[x])/d^2 + ((b*c - a*d)^2*Sqrt[x])/(2*c*d^2*(c + d*x^2)) + ((5*b^2*c^(1/4))/(4*Sqrt[2]*d^(9/4)) - (a*b)/(2*Sqrt[2]*c^(3/4)*d^(5/4)) - (3*a^2)/(4*Sqrt[2]*c^(7/4)*d^(1/4)))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + ((-5*b^2*c^(1/4))/(4*Sqrt[2]*d^(9/4)) + (a*b)/(2*Sqrt[2]*c^(3/4)*d^(5/4)) + (3*a^2)/(4*Sqrt[2]*c^(7/4)*d^(1/4)))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
430,1,274,333,0.3002725,"\int \frac{\left(a+b x^2\right)^2}{x^{3/2} \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^(3/2)*(c + d*x^2)^2),x]","\left(\frac{5 a^2 \sqrt[4]{d}}{4 \sqrt{2} c^{9/4}}-\frac{a b}{2 \sqrt{2} c^{5/4} d^{3/4}}-\frac{3 b^2}{4 \sqrt{2} \sqrt[4]{c} d^{7/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+\left(\frac{5 a^2 d^2-2 a b c d-3 b^2 c^2}{8 \sqrt{2} c^{9/4} d^{7/4}}-\frac{-5 a^2 d^2+2 a b c d+3 b^2 c^2}{8 \sqrt{2} c^{9/4} d^{7/4}}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)-\frac{2 a^2}{c^2 \sqrt{x}}-\frac{x^{3/2} (b c-a d)^2}{2 c^2 d \left(c+d x^2\right)}","-\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,"(-2*a^2)/(c^2*Sqrt[x]) - ((b*c - a*d)^2*x^(3/2))/(2*c^2*d*(c + d*x^2)) + (-1/8*(3*b^2*c^2 + 2*a*b*c*d - 5*a^2*d^2)/(Sqrt[2]*c^(9/4)*d^(7/4)) + (-3*b^2*c^2 - 2*a*b*c*d + 5*a^2*d^2)/(8*Sqrt[2]*c^(9/4)*d^(7/4)))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + ((-3*b^2)/(4*Sqrt[2]*c^(1/4)*d^(7/4)) - (a*b)/(2*Sqrt[2]*c^(5/4)*d^(3/4)) + (5*a^2*d^(1/4))/(4*Sqrt[2]*c^(9/4)))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
431,1,257,332,0.2899696,"\int \frac{\left(a+b x^2\right)^2}{x^{5/2} \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^(5/2)*(c + d*x^2)^2),x]","\left(\frac{7 a^2 d^{3/4}}{4 \sqrt{2} c^{11/4}}-\frac{3 a b}{2 \sqrt{2} c^{7/4} \sqrt[4]{d}}-\frac{b^2}{4 \sqrt{2} c^{3/4} d^{5/4}}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(-\frac{7 a^2 d^{3/4}}{4 \sqrt{2} c^{11/4}}+\frac{3 a b}{2 \sqrt{2} c^{7/4} \sqrt[4]{d}}+\frac{b^2}{4 \sqrt{2} c^{3/4} d^{5/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)-\frac{2 a^2}{3 c^2 x^{3/2}}-\frac{\sqrt{x} (b c-a d)^2}{2 c^2 d \left(c+d x^2\right)}","-\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,"(-2*a^2)/(3*c^2*x^(3/2)) - ((b*c - a*d)^2*Sqrt[x])/(2*c^2*d*(c + d*x^2)) + (-1/4*b^2/(Sqrt[2]*c^(3/4)*d^(5/4)) - (3*a*b)/(2*Sqrt[2]*c^(7/4)*d^(1/4)) + (7*a^2*d^(3/4))/(4*Sqrt[2]*c^(11/4)))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + (b^2/(4*Sqrt[2]*c^(3/4)*d^(5/4)) + (3*a*b)/(2*Sqrt[2]*c^(7/4)*d^(1/4)) - (7*a^2*d^(3/4))/(4*Sqrt[2]*c^(11/4)))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
432,1,277,363,0.3152053,"\int \frac{\left(a+b x^2\right)^2}{x^{7/2} \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^(7/2)*(c + d*x^2)^2),x]","\left(-\frac{9 a^2 d^{5/4}}{4 \sqrt{2} c^{13/4}}+\frac{5 a b \sqrt[4]{d}}{2 \sqrt{2} c^{9/4}}-\frac{b^2}{4 \sqrt{2} c^{5/4} d^{3/4}}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(-\frac{9 a^2 d^{5/4}}{4 \sqrt{2} c^{13/4}}+\frac{5 a b \sqrt[4]{d}}{2 \sqrt{2} c^{9/4}}-\frac{b^2}{4 \sqrt{2} c^{5/4} d^{3/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+\frac{\frac{4 a^2 d}{c^3}-\frac{4 a b}{c^2}}{\sqrt{x}}-\frac{2 a^2}{5 c^2 x^{5/2}}+\frac{x^{3/2} (b c-a d)^2}{2 c^3 \left(c+d x^2\right)}","-\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,"(-2*a^2)/(5*c^2*x^(5/2)) + ((-4*a*b)/c^2 + (4*a^2*d)/c^3)/Sqrt[x] + ((b*c - a*d)^2*x^(3/2))/(2*c^3*(c + d*x^2)) + (-1/4*b^2/(Sqrt[2]*c^(5/4)*d^(3/4)) + (5*a*b*d^(1/4))/(2*Sqrt[2]*c^(9/4)) - (9*a^2*d^(5/4))/(4*Sqrt[2]*c^(13/4)))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + (-1/4*b^2/(Sqrt[2]*c^(5/4)*d^(3/4)) + (5*a*b*d^(1/4))/(2*Sqrt[2]*c^(9/4)) - (9*a^2*d^(5/4))/(4*Sqrt[2]*c^(13/4)))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
433,1,325,440,0.4768206,"\int \frac{x^{7/2} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[(x^(7/2)*(a + b*x^2)^2)/(c + d*x^2)^3,x]","\left(-\frac{5 a^2}{32 \sqrt{2} c^{3/4} d^{9/4}}+\frac{45 a b \sqrt[4]{c}}{16 \sqrt{2} d^{13/4}}-\frac{117 b^2 c^{5/4}}{32 \sqrt{2} d^{17/4}}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(\frac{5 a^2}{32 \sqrt{2} c^{3/4} d^{9/4}}-\frac{45 a b \sqrt[4]{c}}{16 \sqrt{2} d^{13/4}}+\frac{117 b^2 c^{5/4}}{32 \sqrt{2} d^{17/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+\frac{x^{5/2} \left(-45 a^2 d^3+810 a b c d^2-1053 b^2 c^2 d\right)+\sqrt{x} \left(-25 a^2 c d^2+450 a b c^2 d-585 b^2 c^3\right)+x^{9/2} \left(320 a b d^3-416 b^2 c d^2\right)+32 b^2 d^3 x^{13/2}}{80 d^4 \left(c+d x^2\right)^2}","-\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,"((-585*b^2*c^3 + 450*a*b*c^2*d - 25*a^2*c*d^2)*Sqrt[x] + (-1053*b^2*c^2*d + 810*a*b*c*d^2 - 45*a^2*d^3)*x^(5/2) + (-416*b^2*c*d^2 + 320*a*b*d^3)*x^(9/2) + 32*b^2*d^3*x^(13/2))/(80*d^4*(c + d*x^2)^2) + ((-117*b^2*c^(5/4))/(32*Sqrt[2]*d^(17/4)) + (45*a*b*c^(1/4))/(16*Sqrt[2]*d^(13/4)) - (5*a^2)/(32*Sqrt[2]*c^(3/4)*d^(9/4)))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + ((117*b^2*c^(5/4))/(32*Sqrt[2]*d^(17/4)) - (45*a*b*c^(1/4))/(16*Sqrt[2]*d^(13/4)) + (5*a^2)/(32*Sqrt[2]*c^(3/4)*d^(9/4)))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
434,1,303,401,0.4956056,"\int \frac{x^{5/2} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[(x^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^3,x]","\left(-\frac{3 a^2}{32 \sqrt{2} c^{5/4} d^{7/4}}-\frac{21 a b}{16 \sqrt{2} \sqrt[4]{c} d^{11/4}}+\frac{77 b^2 c^{3/4}}{32 \sqrt{2} d^{15/4}}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(-\frac{3 a^2}{32 \sqrt{2} c^{5/4} d^{7/4}}-\frac{21 a b}{16 \sqrt{2} \sqrt[4]{c} d^{11/4}}+\frac{77 b^2 c^{3/4}}{32 \sqrt{2} d^{15/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+\frac{x^{3/2} \left(\frac{3 a^2}{16 c d}-\frac{11 a b}{8 d^2}+\frac{19 b^2 c}{16 d^3}\right)}{c+d x^2}-\frac{x^{3/2} (b c-a d)^2}{4 d^3 \left(c+d x^2\right)^2}+\frac{2 b^2 x^{3/2}}{3 d^3}","-\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,"(2*b^2*x^(3/2))/(3*d^3) - ((b*c - a*d)^2*x^(3/2))/(4*d^3*(c + d*x^2)^2) + (((19*b^2*c)/(16*d^3) - (11*a*b)/(8*d^2) + (3*a^2)/(16*c*d))*x^(3/2))/(c + d*x^2) + ((77*b^2*c^(3/4))/(32*Sqrt[2]*d^(15/4)) - (21*a*b)/(16*Sqrt[2]*c^(1/4)*d^(11/4)) - (3*a^2)/(32*Sqrt[2]*c^(5/4)*d^(7/4)))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + ((77*b^2*c^(3/4))/(32*Sqrt[2]*d^(15/4)) - (21*a*b)/(16*Sqrt[2]*c^(1/4)*d^(11/4)) - (3*a^2)/(32*Sqrt[2]*c^(5/4)*d^(7/4)))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
435,1,301,402,0.4748067,"\int \frac{x^{3/2} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[(x^(3/2)*(a + b*x^2)^2)/(c + d*x^2)^3,x]","\left(-\frac{3 a^2}{32 \sqrt{2} c^{7/4} d^{5/4}}-\frac{5 a b}{16 \sqrt{2} c^{3/4} d^{9/4}}+\frac{45 b^2 \sqrt[4]{c}}{32 \sqrt{2} d^{13/4}}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(\frac{3 a^2}{32 \sqrt{2} c^{7/4} d^{5/4}}+\frac{5 a b}{16 \sqrt{2} c^{3/4} d^{9/4}}-\frac{45 b^2 \sqrt[4]{c}}{32 \sqrt{2} d^{13/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+\frac{\sqrt{x} \left(\frac{a^2}{16 c d}-\frac{9 a b}{8 d^2}+\frac{17 b^2 c}{16 d^3}\right)}{c+d x^2}-\frac{\sqrt{x} (b c-a d)^2}{4 d^3 \left(c+d x^2\right)^2}+\frac{2 b^2 \sqrt{x}}{d^3}","\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,"(2*b^2*Sqrt[x])/d^3 - ((b*c - a*d)^2*Sqrt[x])/(4*d^3*(c + d*x^2)^2) + (((17*b^2*c)/(16*d^3) - (9*a*b)/(8*d^2) + a^2/(16*c*d))*Sqrt[x])/(c + d*x^2) + ((45*b^2*c^(1/4))/(32*Sqrt[2]*d^(13/4)) - (5*a*b)/(16*Sqrt[2]*c^(3/4)*d^(9/4)) - (3*a^2)/(32*Sqrt[2]*c^(7/4)*d^(5/4)))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + ((-45*b^2*c^(1/4))/(32*Sqrt[2]*d^(13/4)) + (5*a*b)/(16*Sqrt[2]*c^(3/4)*d^(9/4)) + (3*a^2)/(32*Sqrt[2]*c^(7/4)*d^(5/4)))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
436,1,290,364,0.3078668,"\int \frac{\sqrt{x} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[(Sqrt[x]*(a + b*x^2)^2)/(c + d*x^2)^3,x]","\left(-\frac{5 a^2}{32 \sqrt{2} c^{9/4} d^{3/4}}-\frac{3 a b}{16 \sqrt{2} c^{5/4} d^{7/4}}-\frac{21 b^2}{32 \sqrt{2} \sqrt[4]{c} d^{11/4}}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(-\frac{5 a^2}{32 \sqrt{2} c^{9/4} d^{3/4}}-\frac{3 a b}{16 \sqrt{2} c^{5/4} d^{7/4}}-\frac{21 b^2}{32 \sqrt{2} \sqrt[4]{c} d^{11/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+\frac{x^{3/2} \left(\frac{5 a^2}{16 c^2}+\frac{3 a b}{8 c d}-\frac{11 b^2}{16 d^2}\right)}{c+d x^2}+\frac{x^{3/2} (b c-a d)^2}{4 c d^2 \left(c+d x^2\right)^2}","\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,"((b*c - a*d)^2*x^(3/2))/(4*c*d^2*(c + d*x^2)^2) + (((5*a^2)/(16*c^2) - (11*b^2)/(16*d^2) + (3*a*b)/(8*c*d))*x^(3/2))/(c + d*x^2) + ((-21*b^2)/(32*Sqrt[2]*c^(1/4)*d^(11/4)) - (3*a*b)/(16*Sqrt[2]*c^(5/4)*d^(7/4)) - (5*a^2)/(32*Sqrt[2]*c^(9/4)*d^(3/4)))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + ((-21*b^2)/(32*Sqrt[2]*c^(1/4)*d^(11/4)) - (3*a*b)/(16*Sqrt[2]*c^(5/4)*d^(7/4)) - (5*a^2)/(32*Sqrt[2]*c^(9/4)*d^(3/4)))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
437,1,290,364,0.3092347,"\int \frac{\left(a+b x^2\right)^2}{\sqrt{x} \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(Sqrt[x]*(c + d*x^2)^3),x]","\left(-\frac{21 a^2}{32 \sqrt{2} c^{11/4} \sqrt[4]{d}}-\frac{3 a b}{16 \sqrt{2} c^{7/4} d^{5/4}}-\frac{5 b^2}{32 \sqrt{2} c^{3/4} d^{9/4}}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(\frac{21 a^2}{32 \sqrt{2} c^{11/4} \sqrt[4]{d}}+\frac{3 a b}{16 \sqrt{2} c^{7/4} d^{5/4}}+\frac{5 b^2}{32 \sqrt{2} c^{3/4} d^{9/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+\frac{\sqrt{x} \left(\frac{7 a^2}{16 c^2}+\frac{a b}{8 c d}-\frac{9 b^2}{16 d^2}\right)}{c+d x^2}+\frac{\sqrt{x} (b c-a d)^2}{4 c d^2 \left(c+d x^2\right)^2}","-\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,"((b*c - a*d)^2*Sqrt[x])/(4*c*d^2*(c + d*x^2)^2) + (((7*a^2)/(16*c^2) - (9*b^2)/(16*d^2) + (a*b)/(8*c*d))*Sqrt[x])/(c + d*x^2) + ((-5*b^2)/(32*Sqrt[2]*c^(3/4)*d^(9/4)) - (3*a*b)/(16*Sqrt[2]*c^(7/4)*d^(5/4)) - (21*a^2)/(32*Sqrt[2]*c^(11/4)*d^(1/4)))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + ((5*b^2)/(32*Sqrt[2]*c^(3/4)*d^(9/4)) + (3*a*b)/(16*Sqrt[2]*c^(7/4)*d^(5/4)) + (21*a^2)/(32*Sqrt[2]*c^(11/4)*d^(1/4)))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
438,1,323,399,0.5257457,"\int \frac{\left(a+b x^2\right)^2}{x^{3/2} \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^(3/2)*(c + d*x^2)^3),x]","\left(\frac{45 a^2 \sqrt[4]{d}}{32 \sqrt{2} c^{13/4}}-\frac{5 a b}{16 \sqrt{2} c^{9/4} d^{3/4}}-\frac{3 b^2}{32 \sqrt{2} c^{5/4} d^{7/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+\left(\frac{45 a^2 d^2-10 a b c d-3 b^2 c^2}{64 \sqrt{2} c^{13/4} d^{7/4}}-\frac{-45 a^2 d^2+10 a b c d+3 b^2 c^2}{64 \sqrt{2} c^{13/4} d^{7/4}}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\frac{x^{3/2} \left(-\frac{13 a^2 d}{16 c^3}+\frac{5 a b}{8 c^2}+\frac{3 b^2}{16 c d}\right)}{c+d x^2}-\frac{2 a^2}{c^3 \sqrt{x}}-\frac{x^{3/2} (b c-a d)^2}{4 c^2 d \left(c+d x^2\right)^2}","-\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,"(-2*a^2)/(c^3*Sqrt[x]) - ((b*c - a*d)^2*x^(3/2))/(4*c^2*d*(c + d*x^2)^2) + (((5*a*b)/(8*c^2) + (3*b^2)/(16*c*d) - (13*a^2*d)/(16*c^3))*x^(3/2))/(c + d*x^2) + (-1/64*(3*b^2*c^2 + 10*a*b*c*d - 45*a^2*d^2)/(Sqrt[2]*c^(13/4)*d^(7/4)) + (-3*b^2*c^2 - 10*a*b*c*d + 45*a^2*d^2)/(64*Sqrt[2]*c^(13/4)*d^(7/4)))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + ((-3*b^2)/(32*Sqrt[2]*c^(5/4)*d^(7/4)) - (5*a*b)/(16*Sqrt[2]*c^(9/4)*d^(3/4)) + (45*a^2*d^(1/4))/(32*Sqrt[2]*c^(13/4)))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
439,1,325,402,0.5194271,"\int \frac{\left(a+b x^2\right)^2}{x^{5/2} \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^(5/2)*(c + d*x^2)^3),x]","\left(-\frac{77 a^2 d^{3/4}}{32 \sqrt{2} c^{15/4}}+\frac{21 a b}{16 \sqrt{2} c^{11/4} \sqrt[4]{d}}+\frac{3 b^2}{32 \sqrt{2} c^{7/4} d^{5/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+\left(\frac{77 a^2 d^2-42 a b c d-3 b^2 c^2}{64 \sqrt{2} c^{15/4} d^{5/4}}-\frac{-77 a^2 d^2+42 a b c d+3 b^2 c^2}{64 \sqrt{2} c^{15/4} d^{5/4}}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\frac{\sqrt{x} \left(-\frac{15 a^2 d}{16 c^3}+\frac{7 a b}{8 c^2}+\frac{b^2}{16 c d}\right)}{c+d x^2}-\frac{2 a^2}{3 c^3 x^{3/2}}-\frac{\sqrt{x} (b c-a d)^2}{4 c^2 d \left(c+d x^2\right)^2}","-\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,"(-2*a^2)/(3*c^3*x^(3/2)) - ((b*c - a*d)^2*Sqrt[x])/(4*c^2*d*(c + d*x^2)^2) + (((7*a*b)/(8*c^2) + b^2/(16*c*d) - (15*a^2*d)/(16*c^3))*Sqrt[x])/(c + d*x^2) + (-1/64*(3*b^2*c^2 + 42*a*b*c*d - 77*a^2*d^2)/(Sqrt[2]*c^(15/4)*d^(5/4)) + (-3*b^2*c^2 - 42*a*b*c*d + 77*a^2*d^2)/(64*Sqrt[2]*c^(15/4)*d^(5/4)))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + ((3*b^2)/(32*Sqrt[2]*c^(7/4)*d^(5/4)) + (21*a*b)/(16*Sqrt[2]*c^(11/4)*d^(1/4)) - (77*a^2*d^(3/4))/(32*Sqrt[2]*c^(15/4)))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
440,1,326,439,0.4915879,"\int \frac{\left(a+b x^2\right)^2}{x^{7/2} \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^(7/2)*(c + d*x^2)^3),x]","\left(-\frac{117 a^2 d^{5/4}}{32 \sqrt{2} c^{17/4}}+\frac{45 a b \sqrt[4]{d}}{16 \sqrt{2} c^{13/4}}-\frac{5 b^2}{32 \sqrt{2} c^{9/4} d^{3/4}}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(-\frac{117 a^2 d^{5/4}}{32 \sqrt{2} c^{17/4}}+\frac{45 a b \sqrt[4]{d}}{16 \sqrt{2} c^{13/4}}-\frac{5 b^2}{32 \sqrt{2} c^{9/4} d^{3/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+\frac{x^{3/2} \left(\frac{21 a^2 d^2}{16 c^4}-\frac{13 a b d}{8 c^3}+\frac{5 b^2}{16 c^2}\right)}{c+d x^2}+\frac{\frac{6 a^2 d}{c^4}-\frac{4 a b}{c^3}}{\sqrt{x}}-\frac{2 a^2}{5 c^3 x^{5/2}}+\frac{x^{3/2} (b c-a d)^2}{4 c^3 \left(c+d x^2\right)^2}","-\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,"(-2*a^2)/(5*c^3*x^(5/2)) + ((-4*a*b)/c^3 + (6*a^2*d)/c^4)/Sqrt[x] + ((b*c - a*d)^2*x^(3/2))/(4*c^3*(c + d*x^2)^2) + (((5*b^2)/(16*c^2) - (13*a*b*d)/(8*c^3) + (21*a^2*d^2)/(16*c^4))*x^(3/2))/(c + d*x^2) + ((-5*b^2)/(32*Sqrt[2]*c^(9/4)*d^(3/4)) + (45*a*b*d^(1/4))/(16*Sqrt[2]*c^(13/4)) - (117*a^2*d^(5/4))/(32*Sqrt[2]*c^(17/4)))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + ((-5*b^2)/(32*Sqrt[2]*c^(9/4)*d^(3/4)) + (45*a*b*d^(1/4))/(16*Sqrt[2]*c^(13/4)) - (117*a^2*d^(5/4))/(32*Sqrt[2]*c^(17/4)))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
441,1,322,328,0.3193749,"\int \frac{x^{5/2} \left(c+d x^2\right)^3}{a+b x^2} \, dx","IntegrateAlgebraic[(x^(5/2)*(c + d*x^2)^3)/(a + b*x^2),x]","-\frac{\left(-a^{3/4} b^3 c^3+3 a^{7/4} b^2 c^2 d-3 a^{11/4} b c d^2+a^{15/4} d^3\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)}{\sqrt{2} b^{19/4}}-\frac{\left(-a^{3/4} b^3 c^3+3 a^{7/4} b^2 c^2 d-3 a^{11/4} b c d^2+a^{15/4} d^3\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} b^{19/4}}+\frac{2 x^{7/2} \left(a^2 d^3-3 a b c d^2+3 b^2 c^2 d\right)}{7 b^3}+\frac{2 x^{3/2} \left(-a^3 d^3+3 a^2 b c d^2-3 a b^2 c^2 d+b^3 c^3\right)}{3 b^4}+\frac{2 x^{11/2} \left(3 b c d^2-a d^3\right)}{11 b^2}+\frac{2 d^3 x^{15/2}}{15 b}","-\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*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*x^(3/2))/(3*b^4) + (2*(3*b^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3)*x^(7/2))/(7*b^3) + (2*(3*b*c*d^2 - a*d^3)*x^(11/2))/(11*b^2) + (2*d^3*x^(15/2))/(15*b) - ((-(a^(3/4)*b^3*c^3) + 3*a^(7/4)*b^2*c^2*d - 3*a^(11/4)*b*c*d^2 + a^(15/4)*d^3)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(Sqrt[2]*b^(19/4)) - ((-(a^(3/4)*b^3*c^3) + 3*a^(7/4)*b^2*c^2*d - 3*a^(11/4)*b*c*d^2 + a^(15/4)*d^3)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*b^(19/4))","A",1
442,1,319,326,0.2919947,"\int \frac{x^{3/2} \left(c+d x^2\right)^3}{a+b x^2} \, dx","IntegrateAlgebraic[(x^(3/2)*(c + d*x^2)^3)/(a + b*x^2),x]","-\frac{\left(3 a^{5/4} b^2 c^2 d-3 a^{9/4} b c d^2+a^{13/4} d^3-\sqrt[4]{a} b^3 c^3\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)}{\sqrt{2} b^{17/4}}+\frac{\left(3 a^{5/4} b^2 c^2 d-3 a^{9/4} b c d^2+a^{13/4} d^3-\sqrt[4]{a} b^3 c^3\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} b^{17/4}}+\frac{2 x^{5/2} \left(a^2 d^3-3 a b c d^2+3 b^2 c^2 d\right)}{5 b^3}+\frac{2 \sqrt{x} \left(-a^3 d^3+3 a^2 b c d^2-3 a b^2 c^2 d+b^3 c^3\right)}{b^4}+\frac{2 x^{9/2} \left(3 b c d^2-a d^3\right)}{9 b^2}+\frac{2 d^3 x^{13/2}}{13 b}","\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*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*Sqrt[x])/b^4 + (2*(3*b^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3)*x^(5/2))/(5*b^3) + (2*(3*b*c*d^2 - a*d^3)*x^(9/2))/(9*b^2) + (2*d^3*x^(13/2))/(13*b) - ((-(a^(1/4)*b^3*c^3) + 3*a^(5/4)*b^2*c^2*d - 3*a^(9/4)*b*c*d^2 + a^(13/4)*d^3)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(Sqrt[2]*b^(17/4)) + ((-(a^(1/4)*b^3*c^3) + 3*a^(5/4)*b^2*c^2*d - 3*a^(9/4)*b*c*d^2 + a^(13/4)*d^3)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*b^(17/4))","A",1
443,1,256,306,0.2976814,"\int \frac{\sqrt{x} \left(c+d x^2\right)^3}{a+b x^2} \, dx","IntegrateAlgebraic[(Sqrt[x]*(c + d*x^2)^3)/(a + b*x^2),x]","\frac{2 x^{3/2} \left(a^2 d^3-3 a b c d^2+3 b^2 c^2 d\right)}{3 b^3}+\frac{\left(a^3 d^3-3 a^2 b c d^2+3 a b^2 c^2 d-b^3 c^3\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)}{\sqrt{2} \sqrt[4]{a} b^{15/4}}+\frac{\left(a^3 d^3-3 a^2 b c d^2+3 a b^2 c^2 d-b^3 c^3\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} \sqrt[4]{a} b^{15/4}}+\frac{2 x^{7/2} \left(3 b c d^2-a d^3\right)}{7 b^2}+\frac{2 d^3 x^{11/2}}{11 b}","\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*(3*b^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3)*x^(3/2))/(3*b^3) + (2*(3*b*c*d^2 - a*d^3)*x^(7/2))/(7*b^2) + (2*d^3*x^(11/2))/(11*b) + ((-(b^3*c^3) + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(Sqrt[2]*a^(1/4)*b^(15/4)) + ((-(b^3*c^3) + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*a^(1/4)*b^(15/4))","A",1
444,1,255,304,0.2924481,"\int \frac{\left(c+d x^2\right)^3}{\sqrt{x} \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(c + d*x^2)^3/(Sqrt[x]*(a + b*x^2)),x]","\frac{2 \sqrt{x} \left(a^2 d^3-3 a b c d^2+3 b^2 c^2 d\right)}{b^3}+\frac{\left(a^3 d^3-3 a^2 b c d^2+3 a b^2 c^2 d-b^3 c^3\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)}{\sqrt{2} a^{3/4} b^{13/4}}-\frac{\left(a^3 d^3-3 a^2 b c d^2+3 a b^2 c^2 d-b^3 c^3\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} a^{3/4} b^{13/4}}+\frac{2 x^{5/2} \left(3 b c d^2-a d^3\right)}{5 b^2}+\frac{2 d^3 x^{9/2}}{9 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^{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*(3*b^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3)*Sqrt[x])/b^3 + (2*(3*b*c*d^2 - a*d^3)*x^(5/2))/(5*b^2) + (2*d^3*x^(9/2))/(9*b) + ((-(b^3*c^3) + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(Sqrt[2]*a^(3/4)*b^(13/4)) - ((-(b^3*c^3) + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*a^(3/4)*b^(13/4))","A",1
445,1,234,284,0.3048694,"\int \frac{\left(c+d x^2\right)^3}{x^{3/2} \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(c + d*x^2)^3/(x^(3/2)*(a + b*x^2)),x]","-\frac{\left(a^3 d^3-3 a^2 b c d^2+3 a b^2 c^2 d-b^3 c^3\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)}{\sqrt{2} a^{5/4} b^{11/4}}-\frac{\left(a^3 d^3-3 a^2 b c d^2+3 a b^2 c^2 d-b^3 c^3\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} a^{5/4} b^{11/4}}+\frac{2 x^{3/2} \left(3 b c d^2-a d^3\right)}{3 b^2}-\frac{2 c^3}{a \sqrt{x}}+\frac{2 d^3 x^{7/2}}{7 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^{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*c^3)/(a*Sqrt[x]) + (2*(3*b*c*d^2 - a*d^3)*x^(3/2))/(3*b^2) + (2*d^3*x^(7/2))/(7*b) - ((-(b^3*c^3) + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(Sqrt[2]*a^(5/4)*b^(11/4)) - ((-(b^3*c^3) + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*a^(5/4)*b^(11/4))","A",1
446,1,233,284,0.3031897,"\int \frac{\left(c+d x^2\right)^3}{x^{5/2} \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(c + d*x^2)^3/(x^(5/2)*(a + b*x^2)),x]","-\frac{\left(a^3 d^3-3 a^2 b c d^2+3 a b^2 c^2 d-b^3 c^3\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)}{\sqrt{2} a^{7/4} b^{9/4}}+\frac{\left(a^3 d^3-3 a^2 b c d^2+3 a b^2 c^2 d-b^3 c^3\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} a^{7/4} b^{9/4}}+\frac{2 \sqrt{x} \left(3 b c d^2-a d^3\right)}{b^2}-\frac{2 c^3}{3 a x^{3/2}}+\frac{2 d^3 x^{5/2}}{5 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^{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*c^3)/(3*a*x^(3/2)) + (2*(3*b*c*d^2 - a*d^3)*Sqrt[x])/b^2 + (2*d^3*x^(5/2))/(5*b) - ((-(b^3*c^3) + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(Sqrt[2]*a^(7/4)*b^(9/4)) + ((-(b^3*c^3) + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*a^(7/4)*b^(9/4))","A",1
447,1,232,283,0.3040349,"\int \frac{\left(c+d x^2\right)^3}{x^{7/2} \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(c + d*x^2)^3/(x^(7/2)*(a + b*x^2)),x]","-\frac{2 \left(3 a c^2 d-b c^3\right)}{a^2 \sqrt{x}}+\frac{\left(a^3 d^3-3 a^2 b c d^2+3 a b^2 c^2 d-b^3 c^3\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)}{\sqrt{2} a^{9/4} b^{7/4}}+\frac{\left(a^3 d^3-3 a^2 b c d^2+3 a b^2 c^2 d-b^3 c^3\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} a^{9/4} b^{7/4}}-\frac{2 c^3}{5 a x^{5/2}}+\frac{2 d^3 x^{3/2}}{3 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^{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*c^3)/(5*a*x^(5/2)) - (2*(-(b*c^3) + 3*a*c^2*d))/(a^2*Sqrt[x]) + (2*d^3*x^(3/2))/(3*b) + ((-(b^3*c^3) + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(Sqrt[2]*a^(9/4)*b^(7/4)) + ((-(b^3*c^3) + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*a^(9/4)*b^(7/4))","A",1
448,1,233,283,0.2993151,"\int \frac{\left(c+d x^2\right)^3}{x^{9/2} \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(c + d*x^2)^3/(x^(9/2)*(a + b*x^2)),x]","-\frac{2 \left(3 a c^2 d-b c^3\right)}{3 a^2 x^{3/2}}+\frac{\left(a^3 d^3-3 a^2 b c d^2+3 a b^2 c^2 d-b^3 c^3\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)}{\sqrt{2} a^{11/4} b^{5/4}}-\frac{\left(a^3 d^3-3 a^2 b c d^2+3 a b^2 c^2 d-b^3 c^3\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} a^{11/4} b^{5/4}}-\frac{2 c^3}{7 a x^{7/2}}+\frac{2 d^3 \sqrt{x}}{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^{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*c^3)/(7*a*x^(7/2)) - (2*(-(b*c^3) + 3*a*c^2*d))/(3*a^2*x^(3/2)) + (2*d^3*Sqrt[x])/b + ((-(b^3*c^3) + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(Sqrt[2]*a^(11/4)*b^(5/4)) - ((-(b^3*c^3) + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*a^(11/4)*b^(5/4))","A",1
449,1,256,303,0.2983478,"\int \frac{\left(c+d x^2\right)^3}{x^{11/2} \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(c + d*x^2)^3/(x^(11/2)*(a + b*x^2)),x]","-\frac{2 \left(3 a c^2 d-b c^3\right)}{5 a^2 x^{5/2}}-\frac{2 \left(3 a^2 c d^2-3 a b c^2 d+b^2 c^3\right)}{a^3 \sqrt{x}}-\frac{\left(a^3 d^3-3 a^2 b c d^2+3 a b^2 c^2 d-b^3 c^3\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)}{\sqrt{2} a^{13/4} b^{3/4}}-\frac{\left(a^3 d^3-3 a^2 b c d^2+3 a b^2 c^2 d-b^3 c^3\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} a^{13/4} b^{3/4}}-\frac{2 c^3}{9 a 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*c^3)/(9*a*x^(9/2)) - (2*(-(b*c^3) + 3*a*c^2*d))/(5*a^2*x^(5/2)) - (2*(b^2*c^3 - 3*a*b*c^2*d + 3*a^2*c*d^2))/(a^3*Sqrt[x]) - ((-(b^3*c^3) + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(Sqrt[2]*a^(13/4)*b^(3/4)) - ((-(b^3*c^3) + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*a^(13/4)*b^(3/4))","A",1
450,1,257,305,0.3039436,"\int \frac{\left(c+d x^2\right)^3}{x^{13/2} \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(c + d*x^2)^3/(x^(13/2)*(a + b*x^2)),x]","-\frac{2 \left(3 a c^2 d-b c^3\right)}{7 a^2 x^{7/2}}-\frac{2 \left(3 a^2 c d^2-3 a b c^2 d+b^2 c^3\right)}{3 a^3 x^{3/2}}-\frac{\left(a^3 d^3-3 a^2 b c d^2+3 a b^2 c^2 d-b^3 c^3\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)}{\sqrt{2} a^{15/4} \sqrt[4]{b}}+\frac{\left(a^3 d^3-3 a^2 b c d^2+3 a b^2 c^2 d-b^3 c^3\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} a^{15/4} \sqrt[4]{b}}-\frac{2 c^3}{11 a 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*c^3)/(11*a*x^(11/2)) - (2*(-(b*c^3) + 3*a*c^2*d))/(7*a^2*x^(7/2)) - (2*(b^2*c^3 - 3*a*b*c^2*d + 3*a^2*c*d^2))/(3*a^3*x^(3/2)) - ((-(b^3*c^3) + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(Sqrt[2]*a^(15/4)*b^(1/4)) + ((-(b^3*c^3) + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*a^(15/4)*b^(1/4))","A",1
451,1,318,325,0.3298919,"\int \frac{\left(c+d x^2\right)^3}{x^{15/2} \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(c + d*x^2)^3/(x^(15/2)*(a + b*x^2)),x]","-\frac{2 \left(3 a c^2 d-b c^3\right)}{9 a^2 x^{9/2}}-\frac{2 \left(3 a^2 c d^2-3 a b c^2 d+b^2 c^3\right)}{5 a^3 x^{5/2}}+\frac{\left(a^3 \sqrt[4]{b} d^3-3 a^2 b^{5/4} c d^2+3 a b^{9/4} c^2 d+b^{13/4} \left(-c^3\right)\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)}{\sqrt{2} a^{17/4}}+\frac{\left(a^3 \sqrt[4]{b} d^3-3 a^2 b^{5/4} c d^2+3 a b^{9/4} c^2 d+b^{13/4} \left(-c^3\right)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} a^{17/4}}-\frac{2 \left(a^3 d^3-3 a^2 b c d^2+3 a b^2 c^2 d-b^3 c^3\right)}{a^4 \sqrt{x}}-\frac{2 c^3}{13 a 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*c^3)/(13*a*x^(13/2)) - (2*(-(b*c^3) + 3*a*c^2*d))/(9*a^2*x^(9/2)) - (2*(b^2*c^3 - 3*a*b*c^2*d + 3*a^2*c*d^2))/(5*a^3*x^(5/2)) - (2*(-(b^3*c^3) + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3))/(a^4*Sqrt[x]) + ((-(b^(13/4)*c^3) + 3*a*b^(9/4)*c^2*d - 3*a^2*b^(5/4)*c*d^2 + a^3*b^(1/4)*d^3)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(Sqrt[2]*a^(17/4)) + ((-(b^(13/4)*c^3) + 3*a*b^(9/4)*c^2*d - 3*a^2*b^(5/4)*c*d^2 + a^3*b^(1/4)*d^3)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*a^(17/4))","A",1
452,1,337,409,0.3835549,"\int \frac{x^{7/2} \left(c+d x^2\right)^3}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^(7/2)*(c + d*x^2)^3)/(a + b*x^2)^2,x]","\left(\frac{5 \sqrt[4]{a} c (b c-a d)^2}{4 \sqrt{2} b^{17/4}}-\frac{17 a^{5/4} d (b c-a d)^2}{4 \sqrt{2} b^{21/4}}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(\frac{17 a^{5/4} d (b c-a d)^2}{4 \sqrt{2} b^{21/4}}-\frac{5 \sqrt[4]{a} c (b c-a d)^2}{4 \sqrt{2} b^{17/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)+\frac{a \sqrt{x} (b c-a d)^3}{2 b^5 \left(a+b x^2\right)}+\frac{6 d x^{5/2} (b c-a d)^2}{5 b^4}+\sqrt{x} \left(\frac{2 c (b c-a d)^2}{b^4}-\frac{8 a d (b c-a d)^2}{b^5}\right)+x^{9/2} \left(\frac{2 c d^2}{3 b^2}-\frac{4 a d^3}{9 b^3}\right)+\frac{2 d^3 x^{13/2}}{13 b^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,"((2*c*(b*c - a*d)^2)/b^4 - (8*a*d*(b*c - a*d)^2)/b^5)*Sqrt[x] + (6*d*(b*c - a*d)^2*x^(5/2))/(5*b^4) + ((2*c*d^2)/(3*b^2) - (4*a*d^3)/(9*b^3))*x^(9/2) + (2*d^3*x^(13/2))/(13*b^2) + (a*(b*c - a*d)^3*Sqrt[x])/(2*b^5*(a + b*x^2)) + ((5*a^(1/4)*c*(b*c - a*d)^2)/(4*Sqrt[2]*b^(17/4)) - (17*a^(5/4)*d*(b*c - a*d)^2)/(4*Sqrt[2]*b^(21/4)))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((-5*a^(1/4)*c*(b*c - a*d)^2)/(4*Sqrt[2]*b^(17/4)) + (17*a^(5/4)*d*(b*c - a*d)^2)/(4*Sqrt[2]*b^(21/4)))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]","A",1
453,1,309,374,0.3806903,"\int \frac{x^{5/2} \left(c+d x^2\right)^3}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^(5/2)*(c + d*x^2)^3)/(a + b*x^2)^2,x]","\left(\frac{15 a^{3/4} d (b c-a d)^2}{4 \sqrt{2} b^{19/4}}-\frac{3 c (b c-a d)^2}{4 \sqrt{2} \sqrt[4]{a} b^{15/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)+\left(\frac{3 (b c-a d)^2 (5 a d-b c)}{8 \sqrt{2} \sqrt[4]{a} b^{19/4}}-\frac{3 (b c-5 a d) (b c-a d)^2}{8 \sqrt{2} \sqrt[4]{a} b^{19/4}}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\frac{2 d x^{3/2} (b c-a d)^2}{b^4}-\frac{x^{3/2} (b c-a d)^3}{2 b^4 \left(a+b x^2\right)}+x^{7/2} \left(\frac{6 c d^2}{7 b^2}-\frac{4 a d^3}{7 b^3}\right)+\frac{2 d^3 x^{11/2}}{11 b^2}","\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,"(2*d*(b*c - a*d)^2*x^(3/2))/b^4 + ((6*c*d^2)/(7*b^2) - (4*a*d^3)/(7*b^3))*x^(7/2) + (2*d^3*x^(11/2))/(11*b^2) - ((b*c - a*d)^3*x^(3/2))/(2*b^4*(a + b*x^2)) + ((-3*(b*c - 5*a*d)*(b*c - a*d)^2)/(8*Sqrt[2]*a^(1/4)*b^(19/4)) + (3*(b*c - a*d)^2*(-(b*c) + 5*a*d))/(8*Sqrt[2]*a^(1/4)*b^(19/4)))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((-3*c*(b*c - a*d)^2)/(4*Sqrt[2]*a^(1/4)*b^(15/4)) + (15*a^(3/4)*d*(b*c - a*d)^2)/(4*Sqrt[2]*b^(19/4)))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]","A",1
454,1,309,386,0.420282,"\int \frac{x^{3/2} \left(c+d x^2\right)^3}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^(3/2)*(c + d*x^2)^3)/(a + b*x^2)^2,x]","\left(\frac{(b c-a d)^2 (13 a d-b c)}{8 \sqrt{2} a^{3/4} b^{17/4}}-\frac{(b c-13 a d) (b c-a d)^2}{8 \sqrt{2} a^{3/4} b^{17/4}}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(\frac{c (b c-a d)^2}{4 \sqrt{2} a^{3/4} b^{13/4}}-\frac{13 \sqrt[4]{a} d (b c-a d)^2}{4 \sqrt{2} b^{17/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)-\frac{\sqrt{x} (b c-a d)^3}{2 b^4 \left(a+b x^2\right)}+\frac{6 d \sqrt{x} (b c-a d)^2}{b^4}+x^{5/2} \left(\frac{6 c d^2}{5 b^2}-\frac{4 a d^3}{5 b^3}\right)+\frac{2 d^3 x^{9/2}}{9 b^2}","-\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,"(6*d*(b*c - a*d)^2*Sqrt[x])/b^4 + ((6*c*d^2)/(5*b^2) - (4*a*d^3)/(5*b^3))*x^(5/2) + (2*d^3*x^(9/2))/(9*b^2) - ((b*c - a*d)^3*Sqrt[x])/(2*b^4*(a + b*x^2)) + (-1/8*((b*c - 13*a*d)*(b*c - a*d)^2)/(Sqrt[2]*a^(3/4)*b^(17/4)) + ((b*c - a*d)^2*(-(b*c) + 13*a*d))/(8*Sqrt[2]*a^(3/4)*b^(17/4)))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((c*(b*c - a*d)^2)/(4*Sqrt[2]*a^(3/4)*b^(13/4)) - (13*a^(1/4)*d*(b*c - a*d)^2)/(4*Sqrt[2]*b^(17/4)))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]","A",1
455,1,274,376,0.4133217,"\int \frac{\sqrt{x} \left(c+d x^2\right)^3}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(Sqrt[x]*(c + d*x^2)^3)/(a + b*x^2)^2,x]","\left(-\frac{c (b c-a d)^2}{4 \sqrt{2} a^{5/4} b^{11/4}}-\frac{11 d (b c-a d)^2}{4 \sqrt{2} \sqrt[4]{a} b^{15/4}}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(-\frac{c (b c-a d)^2}{4 \sqrt{2} a^{5/4} b^{11/4}}-\frac{11 d (b c-a d)^2}{4 \sqrt{2} \sqrt[4]{a} b^{15/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)+\frac{x^{3/2} (b c-a d)^3}{2 a b^3 \left(a+b x^2\right)}+x^{3/2} \left(\frac{2 c d^2}{b^2}-\frac{4 a d^3}{3 b^3}\right)+\frac{2 d^3 x^{7/2}}{7 b^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,"((2*c*d^2)/b^2 - (4*a*d^3)/(3*b^3))*x^(3/2) + (2*d^3*x^(7/2))/(7*b^2) + ((b*c - a*d)^3*x^(3/2))/(2*a*b^3*(a + b*x^2)) + (-1/4*(c*(b*c - a*d)^2)/(Sqrt[2]*a^(5/4)*b^(11/4)) - (11*d*(b*c - a*d)^2)/(4*Sqrt[2]*a^(1/4)*b^(15/4)))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + (-1/4*(c*(b*c - a*d)^2)/(Sqrt[2]*a^(5/4)*b^(11/4)) - (11*d*(b*c - a*d)^2)/(4*Sqrt[2]*a^(1/4)*b^(15/4)))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]","A",1
456,1,272,340,0.3548254,"\int \frac{\left(c+d x^2\right)^3}{\sqrt{x} \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2)^3/(Sqrt[x]*(a + b*x^2)^2),x]","\left(-\frac{3 c (b c-a d)^2}{4 \sqrt{2} a^{7/4} b^{9/4}}-\frac{9 d (b c-a d)^2}{4 \sqrt{2} a^{3/4} b^{13/4}}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(\frac{3 c (b c-a d)^2}{4 \sqrt{2} a^{7/4} b^{9/4}}+\frac{9 d (b c-a d)^2}{4 \sqrt{2} a^{3/4} b^{13/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)+\frac{\sqrt{x} (b c-a d)^3}{2 a b^3 \left(a+b x^2\right)}+\sqrt{x} \left(\frac{6 c d^2}{b^2}-\frac{4 a d^3}{b^3}\right)+\frac{2 d^3 x^{5/2}}{5 b^2}","-\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,"((6*c*d^2)/b^2 - (4*a*d^3)/b^3)*Sqrt[x] + (2*d^3*x^(5/2))/(5*b^2) + ((b*c - a*d)^3*Sqrt[x])/(2*a*b^3*(a + b*x^2)) + ((-3*c*(b*c - a*d)^2)/(4*Sqrt[2]*a^(7/4)*b^(9/4)) - (9*d*(b*c - a*d)^2)/(4*Sqrt[2]*a^(3/4)*b^(13/4)))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((3*c*(b*c - a*d)^2)/(4*Sqrt[2]*a^(7/4)*b^(9/4)) + (9*d*(b*c - a*d)^2)/(4*Sqrt[2]*a^(3/4)*b^(13/4)))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]","A",1
457,1,260,368,0.3442012,"\int \frac{\left(c+d x^2\right)^3}{x^{3/2} \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2)^3/(x^(3/2)*(a + b*x^2)^2),x]","\left(\frac{5 c (b c-a d)^2}{4 \sqrt{2} a^{9/4} b^{7/4}}+\frac{7 d (b c-a d)^2}{4 \sqrt{2} a^{5/4} b^{11/4}}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(\frac{5 c (b c-a d)^2}{4 \sqrt{2} a^{9/4} b^{7/4}}+\frac{7 d (b c-a d)^2}{4 \sqrt{2} a^{5/4} b^{11/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)-\frac{x^{3/2} (b c-a d)^3}{2 a^2 b^2 \left(a+b x^2\right)}-\frac{2 c^3}{a^2 \sqrt{x}}+\frac{2 d^3 x^{3/2}}{3 b^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,"(-2*c^3)/(a^2*Sqrt[x]) + (2*d^3*x^(3/2))/(3*b^2) - ((b*c - a*d)^3*x^(3/2))/(2*a^2*b^2*(a + b*x^2)) + ((5*c*(b*c - a*d)^2)/(4*Sqrt[2]*a^(9/4)*b^(7/4)) + (7*d*(b*c - a*d)^2)/(4*Sqrt[2]*a^(5/4)*b^(11/4)))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((5*c*(b*c - a*d)^2)/(4*Sqrt[2]*a^(9/4)*b^(7/4)) + (7*d*(b*c - a*d)^2)/(4*Sqrt[2]*a^(5/4)*b^(11/4)))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]","A",1
458,1,260,367,0.3499941,"\int \frac{\left(c+d x^2\right)^3}{x^{5/2} \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2)^3/(x^(5/2)*(a + b*x^2)^2),x]","\left(\frac{7 c (b c-a d)^2}{4 \sqrt{2} a^{11/4} b^{5/4}}+\frac{5 d (b c-a d)^2}{4 \sqrt{2} a^{7/4} b^{9/4}}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(-\frac{7 c (b c-a d)^2}{4 \sqrt{2} a^{11/4} b^{5/4}}-\frac{5 d (b c-a d)^2}{4 \sqrt{2} a^{7/4} b^{9/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)-\frac{\sqrt{x} (b c-a d)^3}{2 a^2 b^2 \left(a+b x^2\right)}-\frac{2 c^3}{3 a^2 x^{3/2}}+\frac{2 d^3 \sqrt{x}}{b^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,"(-2*c^3)/(3*a^2*x^(3/2)) + (2*d^3*Sqrt[x])/b^2 - ((b*c - a*d)^3*Sqrt[x])/(2*a^2*b^2*(a + b*x^2)) + ((7*c*(b*c - a*d)^2)/(4*Sqrt[2]*a^(11/4)*b^(5/4)) + (5*d*(b*c - a*d)^2)/(4*Sqrt[2]*a^(7/4)*b^(9/4)))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((-7*c*(b*c - a*d)^2)/(4*Sqrt[2]*a^(11/4)*b^(5/4)) - (5*d*(b*c - a*d)^2)/(4*Sqrt[2]*a^(7/4)*b^(9/4)))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]","A",1
459,1,272,376,0.367758,"\int \frac{\left(c+d x^2\right)^3}{x^{7/2} \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2)^3/(x^(7/2)*(a + b*x^2)^2),x]","\left(-\frac{9 c (b c-a d)^2}{4 \sqrt{2} a^{13/4} b^{3/4}}-\frac{3 d (b c-a d)^2}{4 \sqrt{2} a^{9/4} b^{7/4}}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(-\frac{9 c (b c-a d)^2}{4 \sqrt{2} a^{13/4} b^{3/4}}-\frac{3 d (b c-a d)^2}{4 \sqrt{2} a^{9/4} b^{7/4}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)+\frac{x^{3/2} (b c-a d)^3}{2 a^3 b \left(a+b x^2\right)}-\frac{2 c^3}{5 a^2 x^{5/2}}+\frac{\frac{4 b c^3}{a^3}-\frac{6 c^2 d}{a^2}}{\sqrt{x}}","\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,"(-2*c^3)/(5*a^2*x^(5/2)) + ((4*b*c^3)/a^3 - (6*c^2*d)/a^2)/Sqrt[x] + ((b*c - a*d)^3*x^(3/2))/(2*a^3*b*(a + b*x^2)) + ((-9*c*(b*c - a*d)^2)/(4*Sqrt[2]*a^(13/4)*b^(3/4)) - (3*d*(b*c - a*d)^2)/(4*Sqrt[2]*a^(9/4)*b^(7/4)))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((-9*c*(b*c - a*d)^2)/(4*Sqrt[2]*a^(13/4)*b^(3/4)) - (3*d*(b*c - a*d)^2)/(4*Sqrt[2]*a^(9/4)*b^(7/4)))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]","A",1
460,1,274,376,0.3573546,"\int \frac{\left(c+d x^2\right)^3}{x^{9/2} \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2)^3/(x^(9/2)*(a + b*x^2)^2),x]","\left(-\frac{d (b c-a d)^2}{4 \sqrt{2} a^{11/4} b^{5/4}}-\frac{11 c (b c-a d)^2}{4 \sqrt{2} a^{15/4} \sqrt[4]{b}}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(\frac{d (b c-a d)^2}{4 \sqrt{2} a^{11/4} b^{5/4}}+\frac{11 c (b c-a d)^2}{4 \sqrt{2} a^{15/4} \sqrt[4]{b}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)+\frac{\sqrt{x} (b c-a d)^3}{2 a^3 b \left(a+b x^2\right)}-\frac{2 c^3}{7 a^2 x^{7/2}}+\frac{\frac{4 b c^3}{3 a^3}-\frac{2 c^2 d}{a^2}}{x^{3/2}}","-\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,"(-2*c^3)/(7*a^2*x^(7/2)) + ((4*b*c^3)/(3*a^3) - (2*c^2*d)/a^2)/x^(3/2) + ((b*c - a*d)^3*Sqrt[x])/(2*a^3*b*(a + b*x^2)) + ((-11*c*(b*c - a*d)^2)/(4*Sqrt[2]*a^(15/4)*b^(1/4)) - (d*(b*c - a*d)^2)/(4*Sqrt[2]*a^(11/4)*b^(5/4)))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((11*c*(b*c - a*d)^2)/(4*Sqrt[2]*a^(15/4)*b^(1/4)) + (d*(b*c - a*d)^2)/(4*Sqrt[2]*a^(11/4)*b^(5/4)))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]","A",1
461,1,282,478,0.5045759,"\int \frac{x^{9/2}}{\left(a+b x^2\right) \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[x^(9/2)/((a + b*x^2)*(c + d*x^2)),x]","-\frac{a^{7/4} \tan ^{-1}\left(\frac{\frac{\sqrt[4]{a}}{\sqrt{2} \sqrt[4]{b}}-\frac{\sqrt[4]{b} x}{\sqrt{2} \sqrt[4]{a}}}{\sqrt{x}}\right)}{\sqrt{2} b^{7/4} (b c-a d)}-\frac{a^{7/4} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} b^{7/4} (b c-a d)}-\frac{c^{7/4} \tan ^{-1}\left(\frac{\frac{\sqrt[4]{c}}{\sqrt{2} \sqrt[4]{d}}-\frac{\sqrt[4]{d} x}{\sqrt{2} \sqrt[4]{c}}}{\sqrt{x}}\right)}{\sqrt{2} d^{7/4} (a d-b c)}-\frac{c^{7/4} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)}{\sqrt{2} d^{7/4} (a d-b c)}+\frac{2 x^{3/2}}{3 b 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,"(2*x^(3/2))/(3*b*d) - (a^(7/4)*ArcTan[(a^(1/4)/(Sqrt[2]*b^(1/4)) - (b^(1/4)*x)/(Sqrt[2]*a^(1/4)))/Sqrt[x]])/(Sqrt[2]*b^(7/4)*(b*c - a*d)) - (c^(7/4)*ArcTan[(c^(1/4)/(Sqrt[2]*d^(1/4)) - (d^(1/4)*x)/(Sqrt[2]*c^(1/4)))/Sqrt[x]])/(Sqrt[2]*d^(7/4)*(-(b*c) + a*d)) - (a^(7/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*b^(7/4)*(b*c - a*d)) - (c^(7/4)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(Sqrt[2]*d^(7/4)*(-(b*c) + a*d))","A",1
462,1,278,476,0.4203719,"\int \frac{x^{7/2}}{\left(a+b x^2\right) \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[x^(7/2)/((a + b*x^2)*(c + d*x^2)),x]","-\frac{a^{5/4} \tan ^{-1}\left(\frac{\frac{\sqrt[4]{a}}{\sqrt{2} \sqrt[4]{b}}-\frac{\sqrt[4]{b} x}{\sqrt{2} \sqrt[4]{a}}}{\sqrt{x}}\right)}{\sqrt{2} b^{5/4} (b c-a d)}+\frac{a^{5/4} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} b^{5/4} (b c-a d)}-\frac{c^{5/4} \tan ^{-1}\left(\frac{\frac{\sqrt[4]{c}}{\sqrt{2} \sqrt[4]{d}}-\frac{\sqrt[4]{d} x}{\sqrt{2} \sqrt[4]{c}}}{\sqrt{x}}\right)}{\sqrt{2} d^{5/4} (a d-b c)}+\frac{c^{5/4} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)}{\sqrt{2} d^{5/4} (a d-b c)}+\frac{2 \sqrt{x}}{b 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,"(2*Sqrt[x])/(b*d) - (a^(5/4)*ArcTan[(a^(1/4)/(Sqrt[2]*b^(1/4)) - (b^(1/4)*x)/(Sqrt[2]*a^(1/4)))/Sqrt[x]])/(Sqrt[2]*b^(5/4)*(b*c - a*d)) - (c^(5/4)*ArcTan[(c^(1/4)/(Sqrt[2]*d^(1/4)) - (d^(1/4)*x)/(Sqrt[2]*c^(1/4)))/Sqrt[x]])/(Sqrt[2]*d^(5/4)*(-(b*c) + a*d)) + (a^(5/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*b^(5/4)*(b*c - a*d)) + (c^(5/4)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(Sqrt[2]*d^(5/4)*(-(b*c) + a*d))","A",1
463,1,263,463,0.343145,"\int \frac{x^{5/2}}{\left(a+b x^2\right) \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[x^(5/2)/((a + b*x^2)*(c + d*x^2)),x]","\frac{a^{3/4} \tan ^{-1}\left(\frac{\frac{\sqrt[4]{a}}{\sqrt{2} \sqrt[4]{b}}-\frac{\sqrt[4]{b} x}{\sqrt{2} \sqrt[4]{a}}}{\sqrt{x}}\right)}{\sqrt{2} b^{3/4} (b c-a d)}+\frac{a^{3/4} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} b^{3/4} (b c-a d)}+\frac{c^{3/4} \tan ^{-1}\left(\frac{\frac{\sqrt[4]{c}}{\sqrt{2} \sqrt[4]{d}}-\frac{\sqrt[4]{d} x}{\sqrt{2} \sqrt[4]{c}}}{\sqrt{x}}\right)}{\sqrt{2} d^{3/4} (a d-b c)}+\frac{c^{3/4} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)}{\sqrt{2} d^{3/4} (a d-b c)}","-\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[(a^(1/4)/(Sqrt[2]*b^(1/4)) - (b^(1/4)*x)/(Sqrt[2]*a^(1/4)))/Sqrt[x]])/(Sqrt[2]*b^(3/4)*(b*c - a*d)) + (c^(3/4)*ArcTan[(c^(1/4)/(Sqrt[2]*d^(1/4)) - (d^(1/4)*x)/(Sqrt[2]*c^(1/4)))/Sqrt[x]])/(Sqrt[2]*d^(3/4)*(-(b*c) + a*d)) + (a^(3/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*b^(3/4)*(b*c - a*d)) + (c^(3/4)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(Sqrt[2]*d^(3/4)*(-(b*c) + a*d))","A",1
464,1,265,463,0.3007722,"\int \frac{x^{3/2}}{\left(a+b x^2\right) \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[x^(3/2)/((a + b*x^2)*(c + d*x^2)),x]","\frac{\sqrt[4]{a} \tan ^{-1}\left(\frac{\frac{\sqrt[4]{a}}{\sqrt{2} \sqrt[4]{b}}-\frac{\sqrt[4]{b} x}{\sqrt{2} \sqrt[4]{a}}}{\sqrt{x}}\right)}{\sqrt{2} \sqrt[4]{b} (b c-a d)}+\frac{\sqrt[4]{c} \tan ^{-1}\left(\frac{\frac{\sqrt[4]{c}}{\sqrt{2} \sqrt[4]{d}}-\frac{\sqrt[4]{d} x}{\sqrt{2} \sqrt[4]{c}}}{\sqrt{x}}\right)}{\sqrt{2} \sqrt[4]{d} (a d-b c)}-\frac{\sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} \sqrt[4]{b} (b c-a d)}-\frac{\sqrt[4]{c} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)}{\sqrt{2} \sqrt[4]{d} (a d-b c)}","\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,"(a^(1/4)*ArcTan[(a^(1/4)/(Sqrt[2]*b^(1/4)) - (b^(1/4)*x)/(Sqrt[2]*a^(1/4)))/Sqrt[x]])/(Sqrt[2]*b^(1/4)*(b*c - a*d)) + (c^(1/4)*ArcTan[(c^(1/4)/(Sqrt[2]*d^(1/4)) - (d^(1/4)*x)/(Sqrt[2]*c^(1/4)))/Sqrt[x]])/(Sqrt[2]*d^(1/4)*(-(b*c) + a*d)) - (a^(1/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*b^(1/4)*(b*c - a*d)) - (c^(1/4)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(Sqrt[2]*d^(1/4)*(-(b*c) + a*d))","A",1
465,1,263,463,0.3415682,"\int \frac{\sqrt{x}}{\left(a+b x^2\right) \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[Sqrt[x]/((a + b*x^2)*(c + d*x^2)),x]","\frac{\sqrt[4]{b} \tan ^{-1}\left(\frac{\frac{\sqrt[4]{a}}{\sqrt{2} \sqrt[4]{b}}-\frac{\sqrt[4]{b} x}{\sqrt{2} \sqrt[4]{a}}}{\sqrt{x}}\right)}{\sqrt{2} \sqrt[4]{a} (a d-b c)}+\frac{\sqrt[4]{d} \tan ^{-1}\left(\frac{\frac{\sqrt[4]{c}}{\sqrt{2} \sqrt[4]{d}}-\frac{\sqrt[4]{d} x}{\sqrt{2} \sqrt[4]{c}}}{\sqrt{x}}\right)}{\sqrt{2} \sqrt[4]{c} (b c-a d)}+\frac{\sqrt[4]{b} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} \sqrt[4]{a} (a d-b c)}+\frac{\sqrt[4]{d} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)}{\sqrt{2} \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,"(b^(1/4)*ArcTan[(a^(1/4)/(Sqrt[2]*b^(1/4)) - (b^(1/4)*x)/(Sqrt[2]*a^(1/4)))/Sqrt[x]])/(Sqrt[2]*a^(1/4)*(-(b*c) + a*d)) + (d^(1/4)*ArcTan[(c^(1/4)/(Sqrt[2]*d^(1/4)) - (d^(1/4)*x)/(Sqrt[2]*c^(1/4)))/Sqrt[x]])/(Sqrt[2]*c^(1/4)*(b*c - a*d)) + (b^(1/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*a^(1/4)*(-(b*c) + a*d)) + (d^(1/4)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(Sqrt[2]*c^(1/4)*(b*c - a*d))","A",1
466,1,265,463,0.3519575,"\int \frac{1}{\sqrt{x} \left(a+b x^2\right) \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[1/(Sqrt[x]*(a + b*x^2)*(c + d*x^2)),x]","\frac{b^{3/4} \tan ^{-1}\left(\frac{\frac{\sqrt[4]{a}}{\sqrt{2} \sqrt[4]{b}}-\frac{\sqrt[4]{b} x}{\sqrt{2} \sqrt[4]{a}}}{\sqrt{x}}\right)}{\sqrt{2} a^{3/4} (a d-b c)}-\frac{b^{3/4} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} a^{3/4} (a d-b c)}+\frac{d^{3/4} \tan ^{-1}\left(\frac{\frac{\sqrt[4]{c}}{\sqrt{2} \sqrt[4]{d}}-\frac{\sqrt[4]{d} x}{\sqrt{2} \sqrt[4]{c}}}{\sqrt{x}}\right)}{\sqrt{2} c^{3/4} (b c-a d)}-\frac{d^{3/4} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)}{\sqrt{2} 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,"(b^(3/4)*ArcTan[(a^(1/4)/(Sqrt[2]*b^(1/4)) - (b^(1/4)*x)/(Sqrt[2]*a^(1/4)))/Sqrt[x]])/(Sqrt[2]*a^(3/4)*(-(b*c) + a*d)) + (d^(3/4)*ArcTan[(c^(1/4)/(Sqrt[2]*d^(1/4)) - (d^(1/4)*x)/(Sqrt[2]*c^(1/4)))/Sqrt[x]])/(Sqrt[2]*c^(3/4)*(b*c - a*d)) - (b^(3/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*a^(3/4)*(-(b*c) + a*d)) - (d^(3/4)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(Sqrt[2]*c^(3/4)*(b*c - a*d))","A",1
467,1,266,476,0.3827089,"\int \frac{1}{x^{3/2} \left(a+b x^2\right) \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[1/(x^(3/2)*(a + b*x^2)*(c + d*x^2)),x]","\frac{\frac{b^{5/4} \sqrt{x} \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)}{\sqrt{2} a^{5/4}}+\frac{b^{5/4} \sqrt{x} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} a^{5/4}}-\frac{2 b}{a}-\frac{d^{5/4} \sqrt{x} \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)}{\sqrt{2} c^{5/4}}-\frac{d^{5/4} \sqrt{x} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)}{\sqrt{2} c^{5/4}}+\frac{2 d}{c}}{b c \sqrt{x}-a d \sqrt{x}}","-\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,"((-2*b)/a + (2*d)/c + (b^(5/4)*Sqrt[x]*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(Sqrt[2]*a^(5/4)) - (d^(5/4)*Sqrt[x]*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/(Sqrt[2]*c^(5/4)) + (b^(5/4)*Sqrt[x]*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*a^(5/4)) - (d^(5/4)*Sqrt[x]*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(Sqrt[2]*c^(5/4)))/(b*c*Sqrt[x] - a*d*Sqrt[x])","A",1
468,1,280,478,0.5515885,"\int \frac{1}{x^{5/2} \left(a+b x^2\right) \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[1/(x^(5/2)*(a + b*x^2)*(c + d*x^2)),x]","-\frac{b^{7/4} \tan ^{-1}\left(\frac{\frac{\sqrt[4]{a}}{\sqrt{2} \sqrt[4]{b}}-\frac{\sqrt[4]{b} x}{\sqrt{2} \sqrt[4]{a}}}{\sqrt{x}}\right)}{\sqrt{2} a^{7/4} (a d-b c)}+\frac{b^{7/4} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} a^{7/4} (a d-b c)}-\frac{d^{7/4} \tan ^{-1}\left(\frac{\frac{\sqrt[4]{c}}{\sqrt{2} \sqrt[4]{d}}-\frac{\sqrt[4]{d} x}{\sqrt{2} \sqrt[4]{c}}}{\sqrt{x}}\right)}{\sqrt{2} c^{7/4} (b c-a d)}+\frac{d^{7/4} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)}{\sqrt{2} c^{7/4} (b c-a d)}-\frac{2}{3 a c x^{3/2}}","\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^{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,"-2/(3*a*c*x^(3/2)) - (b^(7/4)*ArcTan[(a^(1/4)/(Sqrt[2]*b^(1/4)) - (b^(1/4)*x)/(Sqrt[2]*a^(1/4)))/Sqrt[x]])/(Sqrt[2]*a^(7/4)*(-(b*c) + a*d)) - (d^(7/4)*ArcTan[(c^(1/4)/(Sqrt[2]*d^(1/4)) - (d^(1/4)*x)/(Sqrt[2]*c^(1/4)))/Sqrt[x]])/(Sqrt[2]*c^(7/4)*(b*c - a*d)) + (b^(7/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*a^(7/4)*(-(b*c) + a*d)) + (d^(7/4)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(Sqrt[2]*c^(7/4)*(b*c - a*d))","A",1
469,1,298,498,0.6087333,"\int \frac{1}{x^{7/2} \left(a+b x^2\right) \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[1/(x^(7/2)*(a + b*x^2)*(c + d*x^2)),x]","\frac{b^{9/4} \tan ^{-1}\left(\frac{\frac{\sqrt[4]{a}}{\sqrt{2} \sqrt[4]{b}}-\frac{\sqrt[4]{b} x}{\sqrt{2} \sqrt[4]{a}}}{\sqrt{x}}\right)}{\sqrt{2} a^{9/4} (a d-b c)}+\frac{b^{9/4} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} a^{9/4} (a d-b c)}+\frac{2 (a d+b c)}{a^2 c^2 \sqrt{x}}+\frac{d^{9/4} \tan ^{-1}\left(\frac{\frac{\sqrt[4]{c}}{\sqrt{2} \sqrt[4]{d}}-\frac{\sqrt[4]{d} x}{\sqrt{2} \sqrt[4]{c}}}{\sqrt{x}}\right)}{\sqrt{2} c^{9/4} (b c-a d)}+\frac{d^{9/4} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)}{\sqrt{2} c^{9/4} (b c-a d)}-\frac{2}{5 a c x^{5/2}}","\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^{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,"-2/(5*a*c*x^(5/2)) + (2*(b*c + a*d))/(a^2*c^2*Sqrt[x]) + (b^(9/4)*ArcTan[(a^(1/4)/(Sqrt[2]*b^(1/4)) - (b^(1/4)*x)/(Sqrt[2]*a^(1/4)))/Sqrt[x]])/(Sqrt[2]*a^(9/4)*(-(b*c) + a*d)) + (d^(9/4)*ArcTan[(c^(1/4)/(Sqrt[2]*d^(1/4)) - (d^(1/4)*x)/(Sqrt[2]*c^(1/4)))/Sqrt[x]])/(Sqrt[2]*c^(9/4)*(b*c - a*d)) + (b^(9/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*a^(9/4)*(-(b*c) + a*d)) + (d^(9/4)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(Sqrt[2]*c^(9/4)*(b*c - a*d))","A",1
470,1,585,570,0.7141624,"\int \frac{x^{11/2}}{\left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[x^(11/2)/((a + b*x^2)*(c + d*x^2)^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right) \left(\frac{a^{9/4} d x^2}{\sqrt{2} b^{5/4} (b c-a d)^2}+\frac{a^{9/4} c}{\sqrt{2} b^{5/4} (b c-a d)^2}\right)+\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right) \left(-\frac{a^{9/4} d x^2}{\sqrt{2} b^{5/4} (b c-a d)^2}-\frac{a^{9/4} c}{\sqrt{2} b^{5/4} (b c-a d)^2}\right)+\tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right) \left(\frac{5 b c^{9/4} x^2}{4 \sqrt{2} d^{5/4} (a d-b c)^2}+\frac{5 b c^{13/4}}{4 \sqrt{2} d^{9/4} (a d-b c)^2}-\frac{9 a c^{9/4}}{4 \sqrt{2} d^{5/4} (a d-b c)^2}-\frac{9 a c^{5/4} x^2}{4 \sqrt{2} \sqrt[4]{d} (a d-b c)^2}\right)+\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right) \left(-\frac{5 b c^{9/4} x^2}{4 \sqrt{2} d^{5/4} (a d-b c)^2}-\frac{5 b c^{13/4}}{4 \sqrt{2} d^{9/4} (a d-b c)^2}+\frac{9 a c^{9/4}}{4 \sqrt{2} d^{5/4} (a d-b c)^2}+\frac{9 a c^{5/4} x^2}{4 \sqrt{2} \sqrt[4]{d} (a d-b c)^2}\right)-\frac{c^2 \sqrt{x}}{2 d^2 (a d-b c)}+\sqrt{x} \left(\frac{2 c}{b d^2}+\frac{2 x^2}{b d}\right)}{c+d x^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,"(-1/2*(c^2*Sqrt[x])/(d^2*(-(b*c) + a*d)) + Sqrt[x]*((2*c)/(b*d^2) + (2*x^2)/(b*d)) + ((a^(9/4)*c)/(Sqrt[2]*b^(5/4)*(b*c - a*d)^2) + (a^(9/4)*d*x^2)/(Sqrt[2]*b^(5/4)*(b*c - a*d)^2))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((5*b*c^(13/4))/(4*Sqrt[2]*d^(9/4)*(-(b*c) + a*d)^2) - (9*a*c^(9/4))/(4*Sqrt[2]*d^(5/4)*(-(b*c) + a*d)^2) + (5*b*c^(9/4)*x^2)/(4*Sqrt[2]*d^(5/4)*(-(b*c) + a*d)^2) - (9*a*c^(5/4)*x^2)/(4*Sqrt[2]*d^(1/4)*(-(b*c) + a*d)^2))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + (-((a^(9/4)*c)/(Sqrt[2]*b^(5/4)*(b*c - a*d)^2)) - (a^(9/4)*d*x^2)/(Sqrt[2]*b^(5/4)*(b*c - a*d)^2))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)] + ((-5*b*c^(13/4))/(4*Sqrt[2]*d^(9/4)*(-(b*c) + a*d)^2) + (9*a*c^(9/4))/(4*Sqrt[2]*d^(5/4)*(-(b*c) + a*d)^2) - (5*b*c^(9/4)*x^2)/(4*Sqrt[2]*d^(5/4)*(-(b*c) + a*d)^2) + (9*a*c^(5/4)*x^2)/(4*Sqrt[2]*d^(1/4)*(-(b*c) + a*d)^2))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(c + d*x^2)","A",1
471,1,558,536,0.6285435,"\int \frac{x^{9/2}}{\left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[x^(9/2)/((a + b*x^2)*(c + d*x^2)^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right) \left(-\frac{a^{7/4} d x^2}{\sqrt{2} b^{3/4} (b c-a d)^2}-\frac{a^{7/4} c}{\sqrt{2} b^{3/4} (b c-a d)^2}\right)+\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right) \left(-\frac{a^{7/4} d x^2}{\sqrt{2} b^{3/4} (b c-a d)^2}-\frac{a^{7/4} c}{\sqrt{2} b^{3/4} (b c-a d)^2}\right)+\tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right) \left(-\frac{3 b c^{7/4} x^2}{4 \sqrt{2} d^{3/4} (a d-b c)^2}-\frac{3 b c^{11/4}}{4 \sqrt{2} d^{7/4} (a d-b c)^2}+\frac{7 a c^{7/4}}{4 \sqrt{2} d^{3/4} (a d-b c)^2}+\frac{7 a c^{3/4} \sqrt[4]{d} x^2}{4 \sqrt{2} (a d-b c)^2}\right)+\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right) \left(-\frac{3 b c^{7/4} x^2}{4 \sqrt{2} d^{3/4} (a d-b c)^2}-\frac{3 b c^{11/4}}{4 \sqrt{2} d^{7/4} (a d-b c)^2}+\frac{7 a c^{7/4}}{4 \sqrt{2} d^{3/4} (a d-b c)^2}+\frac{7 a c^{3/4} \sqrt[4]{d} x^2}{4 \sqrt{2} (a d-b c)^2}\right)+\frac{c x^{3/2}}{2 d (a d-b c)}}{c+d x^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,"((c*x^(3/2))/(2*d*(-(b*c) + a*d)) + (-((a^(7/4)*c)/(Sqrt[2]*b^(3/4)*(b*c - a*d)^2)) - (a^(7/4)*d*x^2)/(Sqrt[2]*b^(3/4)*(b*c - a*d)^2))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((-3*b*c^(11/4))/(4*Sqrt[2]*d^(7/4)*(-(b*c) + a*d)^2) + (7*a*c^(7/4))/(4*Sqrt[2]*d^(3/4)*(-(b*c) + a*d)^2) - (3*b*c^(7/4)*x^2)/(4*Sqrt[2]*d^(3/4)*(-(b*c) + a*d)^2) + (7*a*c^(3/4)*d^(1/4)*x^2)/(4*Sqrt[2]*(-(b*c) + a*d)^2))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + (-((a^(7/4)*c)/(Sqrt[2]*b^(3/4)*(b*c - a*d)^2)) - (a^(7/4)*d*x^2)/(Sqrt[2]*b^(3/4)*(b*c - a*d)^2))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)] + ((-3*b*c^(11/4))/(4*Sqrt[2]*d^(7/4)*(-(b*c) + a*d)^2) + (7*a*c^(7/4))/(4*Sqrt[2]*d^(3/4)*(-(b*c) + a*d)^2) - (3*b*c^(7/4)*x^2)/(4*Sqrt[2]*d^(3/4)*(-(b*c) + a*d)^2) + (7*a*c^(3/4)*d^(1/4)*x^2)/(4*Sqrt[2]*(-(b*c) + a*d)^2))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(c + d*x^2)","A",1
472,1,556,532,0.9102596,"\int \frac{x^{7/2}}{\left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[x^(7/2)/((a + b*x^2)*(c + d*x^2)^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right) \left(-\frac{a^{5/4} d x^2}{\sqrt{2} \sqrt[4]{b} (b c-a d)^2}-\frac{a^{5/4} c}{\sqrt{2} \sqrt[4]{b} (b c-a d)^2}\right)+\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right) \left(\frac{a^{5/4} d x^2}{\sqrt{2} \sqrt[4]{b} (b c-a d)^2}+\frac{a^{5/4} c}{\sqrt{2} \sqrt[4]{b} (b c-a d)^2}\right)+\tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right) \left(-\frac{b c^{9/4}}{4 \sqrt{2} d^{5/4} (a d-b c)^2}-\frac{b c^{5/4} x^2}{4 \sqrt{2} \sqrt[4]{d} (a d-b c)^2}+\frac{5 a c^{5/4}}{4 \sqrt{2} \sqrt[4]{d} (a d-b c)^2}+\frac{5 a \sqrt[4]{c} d^{3/4} x^2}{4 \sqrt{2} (a d-b c)^2}\right)+\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right) \left(\frac{b c^{9/4}}{4 \sqrt{2} d^{5/4} (a d-b c)^2}+\frac{b c^{5/4} x^2}{4 \sqrt{2} \sqrt[4]{d} (a d-b c)^2}-\frac{5 a c^{5/4}}{4 \sqrt{2} \sqrt[4]{d} (a d-b c)^2}-\frac{5 a \sqrt[4]{c} d^{3/4} x^2}{4 \sqrt{2} (a d-b c)^2}\right)+\frac{c \sqrt{x}}{2 d (a d-b c)}}{c+d x^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,"((c*Sqrt[x])/(2*d*(-(b*c) + a*d)) + (-((a^(5/4)*c)/(Sqrt[2]*b^(1/4)*(b*c - a*d)^2)) - (a^(5/4)*d*x^2)/(Sqrt[2]*b^(1/4)*(b*c - a*d)^2))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + (-1/4*(b*c^(9/4))/(Sqrt[2]*d^(5/4)*(-(b*c) + a*d)^2) + (5*a*c^(5/4))/(4*Sqrt[2]*d^(1/4)*(-(b*c) + a*d)^2) - (b*c^(5/4)*x^2)/(4*Sqrt[2]*d^(1/4)*(-(b*c) + a*d)^2) + (5*a*c^(1/4)*d^(3/4)*x^2)/(4*Sqrt[2]*(-(b*c) + a*d)^2))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + ((a^(5/4)*c)/(Sqrt[2]*b^(1/4)*(b*c - a*d)^2) + (a^(5/4)*d*x^2)/(Sqrt[2]*b^(1/4)*(b*c - a*d)^2))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)] + ((b*c^(9/4))/(4*Sqrt[2]*d^(5/4)*(-(b*c) + a*d)^2) - (5*a*c^(5/4))/(4*Sqrt[2]*d^(1/4)*(-(b*c) + a*d)^2) + (b*c^(5/4)*x^2)/(4*Sqrt[2]*d^(1/4)*(-(b*c) + a*d)^2) - (5*a*c^(1/4)*d^(3/4)*x^2)/(4*Sqrt[2]*(-(b*c) + a*d)^2))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(c + d*x^2)","A",1
473,1,550,528,0.6152322,"\int \frac{x^{5/2}}{\left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[x^(5/2)/((a + b*x^2)*(c + d*x^2)^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right) \left(\frac{a^{3/4} \sqrt[4]{b} d x^2}{\sqrt{2} (b c-a d)^2}+\frac{a^{3/4} \sqrt[4]{b} c}{\sqrt{2} (b c-a d)^2}\right)+\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right) \left(\frac{a^{3/4} \sqrt[4]{b} d x^2}{\sqrt{2} (b c-a d)^2}+\frac{a^{3/4} \sqrt[4]{b} c}{\sqrt{2} (b c-a d)^2}\right)+\tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right) \left(-\frac{b c^{7/4}}{4 \sqrt{2} d^{3/4} (b c-a d)^2}-\frac{b c^{3/4} \sqrt[4]{d} x^2}{4 \sqrt{2} (b c-a d)^2}-\frac{3 a c^{3/4} \sqrt[4]{d}}{4 \sqrt{2} (b c-a d)^2}-\frac{3 a d^{5/4} x^2}{4 \sqrt{2} \sqrt[4]{c} (b c-a d)^2}\right)+\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right) \left(-\frac{b c^{7/4}}{4 \sqrt{2} d^{3/4} (b c-a d)^2}-\frac{b c^{3/4} \sqrt[4]{d} x^2}{4 \sqrt{2} (b c-a d)^2}-\frac{3 a c^{3/4} \sqrt[4]{d}}{4 \sqrt{2} (b c-a d)^2}-\frac{3 a d^{5/4} x^2}{4 \sqrt{2} \sqrt[4]{c} (b c-a d)^2}\right)+\frac{x^{3/2}}{2 (b c-a d)}}{c+d x^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,"(x^(3/2)/(2*(b*c - a*d)) + ((a^(3/4)*b^(1/4)*c)/(Sqrt[2]*(b*c - a*d)^2) + (a^(3/4)*b^(1/4)*d*x^2)/(Sqrt[2]*(b*c - a*d)^2))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + (-1/4*(b*c^(7/4))/(Sqrt[2]*d^(3/4)*(b*c - a*d)^2) - (3*a*c^(3/4)*d^(1/4))/(4*Sqrt[2]*(b*c - a*d)^2) - (b*c^(3/4)*d^(1/4)*x^2)/(4*Sqrt[2]*(b*c - a*d)^2) - (3*a*d^(5/4)*x^2)/(4*Sqrt[2]*c^(1/4)*(b*c - a*d)^2))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + ((a^(3/4)*b^(1/4)*c)/(Sqrt[2]*(b*c - a*d)^2) + (a^(3/4)*b^(1/4)*d*x^2)/(Sqrt[2]*(b*c - a*d)^2))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)] + (-1/4*(b*c^(7/4))/(Sqrt[2]*d^(3/4)*(b*c - a*d)^2) - (3*a*c^(3/4)*d^(1/4))/(4*Sqrt[2]*(b*c - a*d)^2) - (b*c^(3/4)*d^(1/4)*x^2)/(4*Sqrt[2]*(b*c - a*d)^2) - (3*a*d^(5/4)*x^2)/(4*Sqrt[2]*c^(1/4)*(b*c - a*d)^2))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(c + d*x^2)","A",1
474,1,552,528,0.7325154,"\int \frac{x^{3/2}}{\left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[x^(3/2)/((a + b*x^2)*(c + d*x^2)^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right) \left(\frac{\sqrt[4]{a} b^{3/4} d x^2}{\sqrt{2} (b c-a d)^2}+\frac{\sqrt[4]{a} b^{3/4} c}{\sqrt{2} (b c-a d)^2}\right)+\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right) \left(-\frac{\sqrt[4]{a} b^{3/4} d x^2}{\sqrt{2} (b c-a d)^2}-\frac{\sqrt[4]{a} b^{3/4} c}{\sqrt{2} (b c-a d)^2}\right)+\tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right) \left(-\frac{a d^{7/4} x^2}{4 \sqrt{2} c^{3/4} (b c-a d)^2}-\frac{3 b c^{5/4}}{4 \sqrt{2} \sqrt[4]{d} (b c-a d)^2}-\frac{3 b \sqrt[4]{c} d^{3/4} x^2}{4 \sqrt{2} (b c-a d)^2}-\frac{a \sqrt[4]{c} d^{3/4}}{4 \sqrt{2} (b c-a d)^2}\right)+\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right) \left(\frac{a d^{7/4} x^2}{4 \sqrt{2} c^{3/4} (b c-a d)^2}+\frac{3 b c^{5/4}}{4 \sqrt{2} \sqrt[4]{d} (b c-a d)^2}+\frac{3 b \sqrt[4]{c} d^{3/4} x^2}{4 \sqrt{2} (b c-a d)^2}+\frac{a \sqrt[4]{c} d^{3/4}}{4 \sqrt{2} (b c-a d)^2}\right)+\frac{\sqrt{x}}{2 (b c-a d)}}{c+d x^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,"(Sqrt[x]/(2*(b*c - a*d)) + ((a^(1/4)*b^(3/4)*c)/(Sqrt[2]*(b*c - a*d)^2) + (a^(1/4)*b^(3/4)*d*x^2)/(Sqrt[2]*(b*c - a*d)^2))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((-3*b*c^(5/4))/(4*Sqrt[2]*d^(1/4)*(b*c - a*d)^2) - (a*c^(1/4)*d^(3/4))/(4*Sqrt[2]*(b*c - a*d)^2) - (3*b*c^(1/4)*d^(3/4)*x^2)/(4*Sqrt[2]*(b*c - a*d)^2) - (a*d^(7/4)*x^2)/(4*Sqrt[2]*c^(3/4)*(b*c - a*d)^2))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + (-((a^(1/4)*b^(3/4)*c)/(Sqrt[2]*(b*c - a*d)^2)) - (a^(1/4)*b^(3/4)*d*x^2)/(Sqrt[2]*(b*c - a*d)^2))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)] + ((3*b*c^(5/4))/(4*Sqrt[2]*d^(1/4)*(b*c - a*d)^2) + (a*c^(1/4)*d^(3/4))/(4*Sqrt[2]*(b*c - a*d)^2) + (3*b*c^(1/4)*d^(3/4)*x^2)/(4*Sqrt[2]*(b*c - a*d)^2) + (a*d^(7/4)*x^2)/(4*Sqrt[2]*c^(3/4)*(b*c - a*d)^2))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(c + d*x^2)","A",1
475,1,558,536,0.6548467,"\int \frac{\sqrt{x}}{\left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[Sqrt[x]/((a + b*x^2)*(c + d*x^2)^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right) \left(-\frac{b^{5/4} d x^2}{\sqrt{2} \sqrt[4]{a} (a d-b c)^2}-\frac{b^{5/4} c}{\sqrt{2} \sqrt[4]{a} (a d-b c)^2}\right)+\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right) \left(-\frac{b^{5/4} d x^2}{\sqrt{2} \sqrt[4]{a} (a d-b c)^2}-\frac{b^{5/4} c}{\sqrt{2} \sqrt[4]{a} (a d-b c)^2}\right)+\tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right) \left(-\frac{a d^{9/4} x^2}{4 \sqrt{2} c^{5/4} (b c-a d)^2}+\frac{5 b c^{3/4} \sqrt[4]{d}}{4 \sqrt{2} (b c-a d)^2}+\frac{5 b d^{5/4} x^2}{4 \sqrt{2} \sqrt[4]{c} (b c-a d)^2}-\frac{a d^{5/4}}{4 \sqrt{2} \sqrt[4]{c} (b c-a d)^2}\right)+\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right) \left(-\frac{a d^{9/4} x^2}{4 \sqrt{2} c^{5/4} (b c-a d)^2}+\frac{5 b c^{3/4} \sqrt[4]{d}}{4 \sqrt{2} (b c-a d)^2}+\frac{5 b d^{5/4} x^2}{4 \sqrt{2} \sqrt[4]{c} (b c-a d)^2}-\frac{a d^{5/4}}{4 \sqrt{2} \sqrt[4]{c} (b c-a d)^2}\right)-\frac{d x^{3/2}}{2 c (b c-a d)}}{c+d x^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,"(-1/2*(d*x^(3/2))/(c*(b*c - a*d)) + (-((b^(5/4)*c)/(Sqrt[2]*a^(1/4)*(-(b*c) + a*d)^2)) - (b^(5/4)*d*x^2)/(Sqrt[2]*a^(1/4)*(-(b*c) + a*d)^2))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((5*b*c^(3/4)*d^(1/4))/(4*Sqrt[2]*(b*c - a*d)^2) - (a*d^(5/4))/(4*Sqrt[2]*c^(1/4)*(b*c - a*d)^2) + (5*b*d^(5/4)*x^2)/(4*Sqrt[2]*c^(1/4)*(b*c - a*d)^2) - (a*d^(9/4)*x^2)/(4*Sqrt[2]*c^(5/4)*(b*c - a*d)^2))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + (-((b^(5/4)*c)/(Sqrt[2]*a^(1/4)*(-(b*c) + a*d)^2)) - (b^(5/4)*d*x^2)/(Sqrt[2]*a^(1/4)*(-(b*c) + a*d)^2))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)] + ((5*b*c^(3/4)*d^(1/4))/(4*Sqrt[2]*(b*c - a*d)^2) - (a*d^(5/4))/(4*Sqrt[2]*c^(1/4)*(b*c - a*d)^2) + (5*b*d^(5/4)*x^2)/(4*Sqrt[2]*c^(1/4)*(b*c - a*d)^2) - (a*d^(9/4)*x^2)/(4*Sqrt[2]*c^(5/4)*(b*c - a*d)^2))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(c + d*x^2)","A",1
476,1,556,536,0.6872624,"\int \frac{1}{\sqrt{x} \left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[1/(Sqrt[x]*(a + b*x^2)*(c + d*x^2)^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right) \left(-\frac{b^{7/4} d x^2}{\sqrt{2} a^{3/4} (a d-b c)^2}-\frac{b^{7/4} c}{\sqrt{2} a^{3/4} (a d-b c)^2}\right)+\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right) \left(\frac{b^{7/4} d x^2}{\sqrt{2} a^{3/4} (a d-b c)^2}+\frac{b^{7/4} c}{\sqrt{2} a^{3/4} (a d-b c)^2}\right)+\tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right) \left(-\frac{3 a d^{11/4} x^2}{4 \sqrt{2} c^{7/4} (b c-a d)^2}+\frac{7 b d^{7/4} x^2}{4 \sqrt{2} c^{3/4} (b c-a d)^2}-\frac{3 a d^{7/4}}{4 \sqrt{2} c^{3/4} (b c-a d)^2}+\frac{7 b \sqrt[4]{c} d^{3/4}}{4 \sqrt{2} (b c-a d)^2}\right)+\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right) \left(\frac{3 a d^{11/4} x^2}{4 \sqrt{2} c^{7/4} (b c-a d)^2}-\frac{7 b d^{7/4} x^2}{4 \sqrt{2} c^{3/4} (b c-a d)^2}+\frac{3 a d^{7/4}}{4 \sqrt{2} c^{3/4} (b c-a d)^2}-\frac{7 b \sqrt[4]{c} d^{3/4}}{4 \sqrt{2} (b c-a d)^2}\right)-\frac{d \sqrt{x}}{2 c (b c-a d)}}{c+d x^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,"(-1/2*(d*Sqrt[x])/(c*(b*c - a*d)) + (-((b^(7/4)*c)/(Sqrt[2]*a^(3/4)*(-(b*c) + a*d)^2)) - (b^(7/4)*d*x^2)/(Sqrt[2]*a^(3/4)*(-(b*c) + a*d)^2))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((7*b*c^(1/4)*d^(3/4))/(4*Sqrt[2]*(b*c - a*d)^2) - (3*a*d^(7/4))/(4*Sqrt[2]*c^(3/4)*(b*c - a*d)^2) + (7*b*d^(7/4)*x^2)/(4*Sqrt[2]*c^(3/4)*(b*c - a*d)^2) - (3*a*d^(11/4)*x^2)/(4*Sqrt[2]*c^(7/4)*(b*c - a*d)^2))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + ((b^(7/4)*c)/(Sqrt[2]*a^(3/4)*(-(b*c) + a*d)^2) + (b^(7/4)*d*x^2)/(Sqrt[2]*a^(3/4)*(-(b*c) + a*d)^2))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)] + ((-7*b*c^(1/4)*d^(3/4))/(4*Sqrt[2]*(b*c - a*d)^2) + (3*a*d^(7/4))/(4*Sqrt[2]*c^(3/4)*(b*c - a*d)^2) - (7*b*d^(7/4)*x^2)/(4*Sqrt[2]*c^(3/4)*(b*c - a*d)^2) + (3*a*d^(11/4)*x^2)/(4*Sqrt[2]*c^(7/4)*(b*c - a*d)^2))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(c + d*x^2)","A",1
477,1,392,570,3.0517132,"\int \frac{1}{x^{3/2} \left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[1/(x^(3/2)*(a + b*x^2)*(c + d*x^2)^2),x]","\frac{b^{9/4} \tan ^{-1}\left(\frac{\frac{\sqrt[4]{a}}{\sqrt{2} \sqrt[4]{b}}-\frac{\sqrt[4]{b} x}{\sqrt{2} \sqrt[4]{a}}}{\sqrt{x}}\right)}{\sqrt{2} a^{5/4} (b c-a d)^2}+\frac{b^{9/4} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} a^{5/4} (b c-a d)^2}+\left(\frac{d^{5/4} (5 a d-9 b c)}{8 \sqrt{2} c^{9/4} (b c-a d)^2}-\frac{d^{5/4} (9 b c-5 a d)}{8 \sqrt{2} c^{9/4} (b c-a d)^2}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(\frac{5 a d^{9/4}}{4 \sqrt{2} c^{9/4} (b c-a d)^2}-\frac{9 b d^{5/4}}{4 \sqrt{2} c^{5/4} (b c-a d)^2}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+\frac{d^2 x^{3/2}}{2 c^2 \left(c+d x^2\right) (b c-a d)}-\frac{2}{a c^2 \sqrt{x}}","-\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,"-2/(a*c^2*Sqrt[x]) + (d^2*x^(3/2))/(2*c^2*(b*c - a*d)*(c + d*x^2)) + (b^(9/4)*ArcTan[(a^(1/4)/(Sqrt[2]*b^(1/4)) - (b^(1/4)*x)/(Sqrt[2]*a^(1/4)))/Sqrt[x]])/(Sqrt[2]*a^(5/4)*(b*c - a*d)^2) + (-1/8*(d^(5/4)*(9*b*c - 5*a*d))/(Sqrt[2]*c^(9/4)*(b*c - a*d)^2) + (d^(5/4)*(-9*b*c + 5*a*d))/(8*Sqrt[2]*c^(9/4)*(b*c - a*d)^2))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + (b^(9/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*a^(5/4)*(b*c - a*d)^2) + ((-9*b*d^(5/4))/(4*Sqrt[2]*c^(5/4)*(b*c - a*d)^2) + (5*a*d^(9/4))/(4*Sqrt[2]*c^(9/4)*(b*c - a*d)^2))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
478,1,379,570,6.3781412,"\int \frac{1}{x^{5/2} \left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[1/(x^(5/2)*(a + b*x^2)*(c + d*x^2)^2),x]","\frac{b^{11/4} \tan ^{-1}\left(\frac{\frac{\sqrt[4]{a}}{\sqrt{2} \sqrt[4]{b}}-\frac{\sqrt[4]{b} x}{\sqrt{2} \sqrt[4]{a}}}{\sqrt{x}}\right)}{\sqrt{2} a^{7/4} (b c-a d)^2}-\frac{b^{11/4} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} a^{7/4} (b c-a d)^2}+\left(\frac{7 a d^{11/4}}{4 \sqrt{2} c^{11/4} (b c-a d)^2}-\frac{11 b d^{7/4}}{4 \sqrt{2} c^{7/4} (b c-a d)^2}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(\frac{11 b d^{7/4}}{4 \sqrt{2} c^{7/4} (b c-a d)^2}-\frac{7 a d^{11/4}}{4 \sqrt{2} c^{11/4} (b c-a d)^2}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+\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[(a^(1/4)/(Sqrt[2]*b^(1/4)) - (b^(1/4)*x)/(Sqrt[2]*a^(1/4)))/Sqrt[x]])/(Sqrt[2]*a^(7/4)*(b*c - a*d)^2) + ((-11*b*d^(7/4))/(4*Sqrt[2]*c^(7/4)*(b*c - a*d)^2) + (7*a*d^(11/4))/(4*Sqrt[2]*c^(11/4)*(b*c - a*d)^2))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] - (b^(11/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*a^(7/4)*(b*c - a*d)^2) + ((11*b*d^(7/4))/(4*Sqrt[2]*c^(7/4)*(b*c - a*d)^2) - (7*a*d^(11/4))/(4*Sqrt[2]*c^(11/4)*(b*c - a*d)^2))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
479,1,405,618,6.434259,"\int \frac{1}{x^{7/2} \left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[1/(x^(7/2)*(a + b*x^2)*(c + d*x^2)^2),x]","-\frac{b^{13/4} \tan ^{-1}\left(\frac{\frac{\sqrt[4]{a}}{\sqrt{2} \sqrt[4]{b}}-\frac{\sqrt[4]{b} x}{\sqrt{2} \sqrt[4]{a}}}{\sqrt{x}}\right)}{\sqrt{2} a^{9/4} (b c-a d)^2}-\frac{b^{13/4} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} a^{9/4} (b c-a d)^2}+\frac{\frac{2 b}{a^2 c^2}+\frac{4 d}{a c^3}}{\sqrt{x}}+\left(\frac{13 b d^{9/4}}{4 \sqrt{2} c^{9/4} (b c-a d)^2}-\frac{9 a d^{13/4}}{4 \sqrt{2} c^{13/4} (b c-a d)^2}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(\frac{13 b d^{9/4}}{4 \sqrt{2} c^{9/4} (b c-a d)^2}-\frac{9 a d^{13/4}}{4 \sqrt{2} c^{13/4} (b c-a d)^2}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)-\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)/(a^2*c^2) + (4*d)/(a*c^3))/Sqrt[x] - (d^3*x^(3/2))/(2*c^3*(b*c - a*d)*(c + d*x^2)) - (b^(13/4)*ArcTan[(a^(1/4)/(Sqrt[2]*b^(1/4)) - (b^(1/4)*x)/(Sqrt[2]*a^(1/4)))/Sqrt[x]])/(Sqrt[2]*a^(9/4)*(b*c - a*d)^2) + ((13*b*d^(9/4))/(4*Sqrt[2]*c^(9/4)*(b*c - a*d)^2) - (9*a*d^(13/4))/(4*Sqrt[2]*c^(13/4)*(b*c - a*d)^2))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] - (b^(13/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*a^(9/4)*(b*c - a*d)^2) + ((13*b*d^(9/4))/(4*Sqrt[2]*c^(9/4)*(b*c - a*d)^2) - (9*a*d^(13/4))/(4*Sqrt[2]*c^(13/4)*(b*c - a*d)^2))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
480,1,1063,631,0.9660925,"\int \frac{x^{7/2}}{\left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[x^(7/2)/((a + b*x^2)*(c + d*x^2)^3),x]","\frac{\frac{\sqrt{x} c}{4 d (a d-b c)}+\sqrt{x} \left(\frac{b c^2}{16 d (a d-b c)^2}+\frac{b x^2 c}{16 (a d-b c)^2}-\frac{9 a c}{16 (a d-b c)^2}-\frac{9 a d x^2}{16 (a d-b c)^2}\right)+\left(-\frac{a^{5/4} b^{3/4} d^2 x^4}{\sqrt{2} (b c-a d)^3}-\frac{\sqrt{2} a^{5/4} b^{3/4} c d x^2}{(b c-a d)^3}-\frac{a^{5/4} b^{3/4} c^2}{\sqrt{2} (b c-a d)^3}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(\frac{5 a^2 d^{11/4} x^4}{32 \sqrt{2} c^{3/4} (b c-a d)^3}+\frac{15 a b \sqrt[4]{c} d^{7/4} x^4}{16 \sqrt{2} (b c-a d)^3}-\frac{3 b^2 c^{5/4} d^{3/4} x^4}{32 \sqrt{2} (b c-a d)^3}+\frac{5 a^2 \sqrt[4]{c} d^{7/4} x^2}{16 \sqrt{2} (b c-a d)^3}+\frac{15 a b c^{5/4} d^{3/4} x^2}{8 \sqrt{2} (b c-a d)^3}-\frac{3 b^2 c^{9/4} x^2}{16 \sqrt{2} \sqrt[4]{d} (b c-a d)^3}+\frac{5 a^2 c^{5/4} d^{3/4}}{32 \sqrt{2} (b c-a d)^3}+\frac{15 a b c^{9/4}}{16 \sqrt{2} \sqrt[4]{d} (b c-a d)^3}-\frac{3 b^2 c^{13/4}}{32 \sqrt{2} d^{5/4} (b c-a d)^3}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(\frac{a^{5/4} b^{3/4} d^2 x^4}{\sqrt{2} (b c-a d)^3}+\frac{\sqrt{2} a^{5/4} b^{3/4} c d x^2}{(b c-a d)^3}+\frac{a^{5/4} b^{3/4} c^2}{\sqrt{2} (b c-a d)^3}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{b} x+\sqrt{a}}\right)+\left(-\frac{5 a^2 d^{11/4} x^4}{32 \sqrt{2} c^{3/4} (b c-a d)^3}-\frac{15 a b \sqrt[4]{c} d^{7/4} x^4}{16 \sqrt{2} (b c-a d)^3}+\frac{3 b^2 c^{5/4} d^{3/4} x^4}{32 \sqrt{2} (b c-a d)^3}-\frac{5 a^2 \sqrt[4]{c} d^{7/4} x^2}{16 \sqrt{2} (b c-a d)^3}-\frac{15 a b c^{5/4} d^{3/4} x^2}{8 \sqrt{2} (b c-a d)^3}+\frac{3 b^2 c^{9/4} x^2}{16 \sqrt{2} \sqrt[4]{d} (b c-a d)^3}-\frac{5 a^2 c^{5/4} d^{3/4}}{32 \sqrt{2} (b c-a d)^3}-\frac{15 a b c^{9/4}}{16 \sqrt{2} \sqrt[4]{d} (b c-a d)^3}+\frac{3 b^2 c^{13/4}}{32 \sqrt{2} d^{5/4} (b c-a d)^3}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{d} x+\sqrt{c}}\right)}{\left(d x^2+c\right)^2}","-\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,"((c*Sqrt[x])/(4*d*(-(b*c) + a*d)) + Sqrt[x]*((-9*a*c)/(16*(-(b*c) + a*d)^2) + (b*c^2)/(16*d*(-(b*c) + a*d)^2) + (b*c*x^2)/(16*(-(b*c) + a*d)^2) - (9*a*d*x^2)/(16*(-(b*c) + a*d)^2)) + (-((a^(5/4)*b^(3/4)*c^2)/(Sqrt[2]*(b*c - a*d)^3)) - (Sqrt[2]*a^(5/4)*b^(3/4)*c*d*x^2)/(b*c - a*d)^3 - (a^(5/4)*b^(3/4)*d^2*x^4)/(Sqrt[2]*(b*c - a*d)^3))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((-3*b^2*c^(13/4))/(32*Sqrt[2]*d^(5/4)*(b*c - a*d)^3) + (15*a*b*c^(9/4))/(16*Sqrt[2]*d^(1/4)*(b*c - a*d)^3) + (5*a^2*c^(5/4)*d^(3/4))/(32*Sqrt[2]*(b*c - a*d)^3) - (3*b^2*c^(9/4)*x^2)/(16*Sqrt[2]*d^(1/4)*(b*c - a*d)^3) + (15*a*b*c^(5/4)*d^(3/4)*x^2)/(8*Sqrt[2]*(b*c - a*d)^3) + (5*a^2*c^(1/4)*d^(7/4)*x^2)/(16*Sqrt[2]*(b*c - a*d)^3) - (3*b^2*c^(5/4)*d^(3/4)*x^4)/(32*Sqrt[2]*(b*c - a*d)^3) + (15*a*b*c^(1/4)*d^(7/4)*x^4)/(16*Sqrt[2]*(b*c - a*d)^3) + (5*a^2*d^(11/4)*x^4)/(32*Sqrt[2]*c^(3/4)*(b*c - a*d)^3))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + ((a^(5/4)*b^(3/4)*c^2)/(Sqrt[2]*(b*c - a*d)^3) + (Sqrt[2]*a^(5/4)*b^(3/4)*c*d*x^2)/(b*c - a*d)^3 + (a^(5/4)*b^(3/4)*d^2*x^4)/(Sqrt[2]*(b*c - a*d)^3))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)] + ((3*b^2*c^(13/4))/(32*Sqrt[2]*d^(5/4)*(b*c - a*d)^3) - (15*a*b*c^(9/4))/(16*Sqrt[2]*d^(1/4)*(b*c - a*d)^3) - (5*a^2*c^(5/4)*d^(3/4))/(32*Sqrt[2]*(b*c - a*d)^3) + (3*b^2*c^(9/4)*x^2)/(16*Sqrt[2]*d^(1/4)*(b*c - a*d)^3) - (15*a*b*c^(5/4)*d^(3/4)*x^2)/(8*Sqrt[2]*(b*c - a*d)^3) - (5*a^2*c^(1/4)*d^(7/4)*x^2)/(16*Sqrt[2]*(b*c - a*d)^3) + (3*b^2*c^(5/4)*d^(3/4)*x^4)/(32*Sqrt[2]*(b*c - a*d)^3) - (15*a*b*c^(1/4)*d^(7/4)*x^4)/(16*Sqrt[2]*(b*c - a*d)^3) - (5*a^2*d^(11/4)*x^4)/(32*Sqrt[2]*c^(3/4)*(b*c - a*d)^3))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(c + d*x^2)^2","A",1
481,1,462,628,1.0422337,"\int \frac{x^{5/2}}{\left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[x^(5/2)/((a + b*x^2)*(c + d*x^2)^3),x]","\frac{a^{3/4} b^{5/4} \tan ^{-1}\left(\frac{\frac{\sqrt[4]{a}}{\sqrt{2} \sqrt[4]{b}}-\frac{\sqrt[4]{b} x}{\sqrt{2} \sqrt[4]{a}}}{\sqrt{x}}\right)}{\sqrt{2} (b c-a d)^3}+\frac{a^{3/4} b^{5/4} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} (b c-a d)^3}+\left(\frac{3 a^2 d^{5/4}}{32 \sqrt{2} c^{5/4} (b c-a d)^3}-\frac{5 b^2 c^{3/4}}{32 \sqrt{2} d^{3/4} (b c-a d)^3}-\frac{15 a b \sqrt[4]{d}}{16 \sqrt{2} \sqrt[4]{c} (b c-a d)^3}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(\frac{3 a^2 d^{5/4}}{32 \sqrt{2} c^{5/4} (b c-a d)^3}-\frac{5 b^2 c^{3/4}}{32 \sqrt{2} d^{3/4} (b c-a d)^3}-\frac{15 a b \sqrt[4]{d}}{16 \sqrt{2} \sqrt[4]{c} (b c-a d)^3}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+\frac{x^{3/2} \left(9 b c^2-a c d\right)+x^{7/2} \left(3 a d^2+5 b c d\right)}{16 c \left(c+d x^2\right)^2 (b c-a d)^2}","-\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,"((9*b*c^2 - a*c*d)*x^(3/2) + (5*b*c*d + 3*a*d^2)*x^(7/2))/(16*c*(b*c - a*d)^2*(c + d*x^2)^2) + (a^(3/4)*b^(5/4)*ArcTan[(a^(1/4)/(Sqrt[2]*b^(1/4)) - (b^(1/4)*x)/(Sqrt[2]*a^(1/4)))/Sqrt[x]])/(Sqrt[2]*(b*c - a*d)^3) + ((-5*b^2*c^(3/4))/(32*Sqrt[2]*d^(3/4)*(b*c - a*d)^3) - (15*a*b*d^(1/4))/(16*Sqrt[2]*c^(1/4)*(b*c - a*d)^3) + (3*a^2*d^(5/4))/(32*Sqrt[2]*c^(5/4)*(b*c - a*d)^3))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + (a^(3/4)*b^(5/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*(b*c - a*d)^3) + ((-5*b^2*c^(3/4))/(32*Sqrt[2]*d^(3/4)*(b*c - a*d)^3) - (15*a*b*d^(1/4))/(16*Sqrt[2]*c^(1/4)*(b*c - a*d)^3) + (3*a^2*d^(5/4))/(32*Sqrt[2]*c^(5/4)*(b*c - a*d)^3))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
482,1,462,627,1.0395551,"\int \frac{x^{3/2}}{\left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[x^(3/2)/((a + b*x^2)*(c + d*x^2)^3),x]","\left(\frac{3 a^2 d^{7/4}}{32 \sqrt{2} c^{7/4} (b c-a d)^3}-\frac{21 b^2 \sqrt[4]{c}}{32 \sqrt{2} \sqrt[4]{d} (b c-a d)^3}-\frac{7 a b d^{3/4}}{16 \sqrt{2} c^{3/4} (b c-a d)^3}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(-\frac{3 a^2 d^{7/4}}{32 \sqrt{2} c^{7/4} (b c-a d)^3}+\frac{21 b^2 \sqrt[4]{c}}{32 \sqrt{2} \sqrt[4]{d} (b c-a d)^3}+\frac{7 a b d^{3/4}}{16 \sqrt{2} c^{3/4} (b c-a d)^3}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+\frac{\sqrt[4]{a} b^{7/4} \tan ^{-1}\left(\frac{\frac{\sqrt[4]{a}}{\sqrt{2} \sqrt[4]{b}}-\frac{\sqrt[4]{b} x}{\sqrt{2} \sqrt[4]{a}}}{\sqrt{x}}\right)}{\sqrt{2} (b c-a d)^3}-\frac{\sqrt[4]{a} b^{7/4} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} (b c-a d)^3}+\frac{\sqrt{x} \left(11 b c^2-3 a c d\right)+x^{5/2} \left(a d^2+7 b c d\right)}{16 c \left(c+d x^2\right)^2 (b c-a d)^2}","-\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,"((11*b*c^2 - 3*a*c*d)*Sqrt[x] + (7*b*c*d + a*d^2)*x^(5/2))/(16*c*(b*c - a*d)^2*(c + d*x^2)^2) + (a^(1/4)*b^(7/4)*ArcTan[(a^(1/4)/(Sqrt[2]*b^(1/4)) - (b^(1/4)*x)/(Sqrt[2]*a^(1/4)))/Sqrt[x]])/(Sqrt[2]*(b*c - a*d)^3) + ((-21*b^2*c^(1/4))/(32*Sqrt[2]*d^(1/4)*(b*c - a*d)^3) - (7*a*b*d^(3/4))/(16*Sqrt[2]*c^(3/4)*(b*c - a*d)^3) + (3*a^2*d^(7/4))/(32*Sqrt[2]*c^(7/4)*(b*c - a*d)^3))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] - (a^(1/4)*b^(7/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*(b*c - a*d)^3) + ((21*b^2*c^(1/4))/(32*Sqrt[2]*d^(1/4)*(b*c - a*d)^3) + (7*a*b*d^(3/4))/(16*Sqrt[2]*c^(3/4)*(b*c - a*d)^3) - (3*a^2*d^(7/4))/(32*Sqrt[2]*c^(7/4)*(b*c - a*d)^3))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
483,1,468,633,1.1867089,"\int \frac{\sqrt{x}}{\left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[Sqrt[x]/((a + b*x^2)*(c + d*x^2)^3),x]","\left(\frac{5 a^2 d^{9/4}}{32 \sqrt{2} c^{9/4} (b c-a d)^3}+\frac{45 b^2 \sqrt[4]{d}}{32 \sqrt{2} \sqrt[4]{c} (b c-a d)^3}-\frac{9 a b d^{5/4}}{16 \sqrt{2} c^{5/4} (b c-a d)^3}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(\frac{5 a^2 d^{9/4}}{32 \sqrt{2} c^{9/4} (b c-a d)^3}+\frac{45 b^2 \sqrt[4]{d}}{32 \sqrt{2} \sqrt[4]{c} (b c-a d)^3}-\frac{9 a b d^{5/4}}{16 \sqrt{2} c^{5/4} (b c-a d)^3}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)-\frac{b^{9/4} \tan ^{-1}\left(\frac{\frac{\sqrt[4]{a}}{\sqrt{2} \sqrt[4]{b}}-\frac{\sqrt[4]{b} x}{\sqrt{2} \sqrt[4]{a}}}{\sqrt{x}}\right)}{\sqrt{2} \sqrt[4]{a} (b c-a d)^3}-\frac{b^{9/4} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} \sqrt[4]{a} (b c-a d)^3}+\frac{-d x^{3/2} \left(17 b c^2-9 a c d\right)-d x^{7/2} \left(13 b c d-5 a d^2\right)}{16 c^2 \left(c+d x^2\right)^2 (b c-a d)^2}","-\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,"(-(d*(17*b*c^2 - 9*a*c*d)*x^(3/2)) - d*(13*b*c*d - 5*a*d^2)*x^(7/2))/(16*c^2*(b*c - a*d)^2*(c + d*x^2)^2) - (b^(9/4)*ArcTan[(a^(1/4)/(Sqrt[2]*b^(1/4)) - (b^(1/4)*x)/(Sqrt[2]*a^(1/4)))/Sqrt[x]])/(Sqrt[2]*a^(1/4)*(b*c - a*d)^3) + ((45*b^2*d^(1/4))/(32*Sqrt[2]*c^(1/4)*(b*c - a*d)^3) - (9*a*b*d^(5/4))/(16*Sqrt[2]*c^(5/4)*(b*c - a*d)^3) + (5*a^2*d^(9/4))/(32*Sqrt[2]*c^(9/4)*(b*c - a*d)^3))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] - (b^(9/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*a^(1/4)*(b*c - a*d)^3) + ((45*b^2*d^(1/4))/(32*Sqrt[2]*c^(1/4)*(b*c - a*d)^3) - (9*a*b*d^(5/4))/(16*Sqrt[2]*c^(5/4)*(b*c - a*d)^3) + (5*a^2*d^(9/4))/(32*Sqrt[2]*c^(9/4)*(b*c - a*d)^3))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
484,1,467,633,1.1331434,"\int \frac{1}{\sqrt{x} \left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[1/(Sqrt[x]*(a + b*x^2)*(c + d*x^2)^3),x]","-\frac{b^{11/4} \tan ^{-1}\left(\frac{\frac{\sqrt[4]{a}}{\sqrt{2} \sqrt[4]{b}}-\frac{\sqrt[4]{b} x}{\sqrt{2} \sqrt[4]{a}}}{\sqrt{x}}\right)}{\sqrt{2} a^{3/4} (b c-a d)^3}+\frac{b^{11/4} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} a^{3/4} (b c-a d)^3}+\left(\frac{21 a^2 d^{11/4}}{32 \sqrt{2} c^{11/4} (b c-a d)^3}+\frac{77 b^2 d^{3/4}}{32 \sqrt{2} c^{3/4} (b c-a d)^3}-\frac{33 a b d^{7/4}}{16 \sqrt{2} c^{7/4} (b c-a d)^3}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(-\frac{21 a^2 d^{11/4}}{32 \sqrt{2} c^{11/4} (b c-a d)^3}-\frac{77 b^2 d^{3/4}}{32 \sqrt{2} c^{3/4} (b c-a d)^3}+\frac{33 a b d^{7/4}}{16 \sqrt{2} c^{7/4} (b c-a d)^3}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+\frac{-d \sqrt{x} \left(19 b c^2-11 a c d\right)-d x^{5/2} \left(15 b c d-7 a d^2\right)}{16 c^2 \left(c+d x^2\right)^2 (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^{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,"(-(d*(19*b*c^2 - 11*a*c*d)*Sqrt[x]) - d*(15*b*c*d - 7*a*d^2)*x^(5/2))/(16*c^2*(b*c - a*d)^2*(c + d*x^2)^2) - (b^(11/4)*ArcTan[(a^(1/4)/(Sqrt[2]*b^(1/4)) - (b^(1/4)*x)/(Sqrt[2]*a^(1/4)))/Sqrt[x]])/(Sqrt[2]*a^(3/4)*(b*c - a*d)^3) + ((77*b^2*d^(3/4))/(32*Sqrt[2]*c^(3/4)*(b*c - a*d)^3) - (33*a*b*d^(7/4))/(16*Sqrt[2]*c^(7/4)*(b*c - a*d)^3) + (21*a^2*d^(11/4))/(32*Sqrt[2]*c^(11/4)*(b*c - a*d)^3))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + (b^(11/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*a^(3/4)*(b*c - a*d)^3) + ((-77*b^2*d^(3/4))/(32*Sqrt[2]*c^(3/4)*(b*c - a*d)^3) + (33*a*b*d^(7/4))/(16*Sqrt[2]*c^(7/4)*(b*c - a*d)^3) - (21*a^2*d^(11/4))/(32*Sqrt[2]*c^(11/4)*(b*c - a*d)^3))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
485,1,504,681,6.4942191,"\int \frac{1}{x^{3/2} \left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[1/(x^(3/2)*(a + b*x^2)*(c + d*x^2)^3),x]","\frac{b^{13/4} \tan ^{-1}\left(\frac{\frac{\sqrt[4]{a}}{\sqrt{2} \sqrt[4]{b}}-\frac{\sqrt[4]{b} x}{\sqrt{2} \sqrt[4]{a}}}{\sqrt{x}}\right)}{\sqrt{2} a^{5/4} (b c-a d)^3}+\frac{b^{13/4} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} a^{5/4} (b c-a d)^3}+\left(-\frac{45 a^2 d^{13/4}}{32 \sqrt{2} c^{13/4} (b c-a d)^3}-\frac{117 b^2 d^{5/4}}{32 \sqrt{2} c^{5/4} (b c-a d)^3}+\frac{65 a b d^{9/4}}{16 \sqrt{2} c^{9/4} (b c-a d)^3}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(-\frac{45 a^2 d^{13/4}}{32 \sqrt{2} c^{13/4} (b c-a d)^3}-\frac{117 b^2 d^{5/4}}{32 \sqrt{2} c^{5/4} (b c-a d)^3}+\frac{65 a b d^{9/4}}{16 \sqrt{2} c^{9/4} (b c-a d)^3}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+\frac{d^2 x^{3/2}}{4 c^2 \left(c+d x^2\right)^2 (b c-a d)}+\frac{x^{3/2} \left(\frac{21 b d^2}{16 c^2 (b c-a d)^2}-\frac{13 a d^3}{16 c^3 (b c-a d)^2}\right)}{c+d x^2}-\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) + (((21*b*d^2)/(16*c^2*(b*c - a*d)^2) - (13*a*d^3)/(16*c^3*(b*c - a*d)^2))*x^(3/2))/(c + d*x^2) + (b^(13/4)*ArcTan[(a^(1/4)/(Sqrt[2]*b^(1/4)) - (b^(1/4)*x)/(Sqrt[2]*a^(1/4)))/Sqrt[x]])/(Sqrt[2]*a^(5/4)*(b*c - a*d)^3) + ((-117*b^2*d^(5/4))/(32*Sqrt[2]*c^(5/4)*(b*c - a*d)^3) + (65*a*b*d^(9/4))/(16*Sqrt[2]*c^(9/4)*(b*c - a*d)^3) - (45*a^2*d^(13/4))/(32*Sqrt[2]*c^(13/4)*(b*c - a*d)^3))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + (b^(13/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*a^(5/4)*(b*c - a*d)^3) + ((-117*b^2*d^(5/4))/(32*Sqrt[2]*c^(5/4)*(b*c - a*d)^3) + (65*a*b*d^(9/4))/(16*Sqrt[2]*c^(9/4)*(b*c - a*d)^3) - (45*a^2*d^(13/4))/(32*Sqrt[2]*c^(13/4)*(b*c - a*d)^3))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
486,1,507,681,6.5099909,"\int \frac{1}{x^{5/2} \left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[1/(x^(5/2)*(a + b*x^2)*(c + d*x^2)^3),x]","\frac{b^{15/4} \tan ^{-1}\left(\frac{\frac{\sqrt[4]{a}}{\sqrt{2} \sqrt[4]{b}}-\frac{\sqrt[4]{b} x}{\sqrt{2} \sqrt[4]{a}}}{\sqrt{x}}\right)}{\sqrt{2} a^{7/4} (b c-a d)^3}-\frac{b^{15/4} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} a^{7/4} (b c-a d)^3}+\left(-\frac{77 a^2 d^{15/4}}{32 \sqrt{2} c^{15/4} (b c-a d)^3}-\frac{165 b^2 d^{7/4}}{32 \sqrt{2} c^{7/4} (b c-a d)^3}+\frac{105 a b d^{11/4}}{16 \sqrt{2} c^{11/4} (b c-a d)^3}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(\frac{77 a^2 d^{15/4}}{32 \sqrt{2} c^{15/4} (b c-a d)^3}+\frac{165 b^2 d^{7/4}}{32 \sqrt{2} c^{7/4} (b c-a d)^3}-\frac{105 a b d^{11/4}}{16 \sqrt{2} c^{11/4} (b c-a d)^3}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+\frac{d^2 \sqrt{x}}{4 c^2 \left(c+d x^2\right)^2 (b c-a d)}+\frac{\sqrt{x} \left(\frac{23 b d^2}{16 c^2 (b c-a d)^2}-\frac{15 a d^3}{16 c^3 (b c-a d)^2}\right)}{c+d x^2}-\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) + (((23*b*d^2)/(16*c^2*(b*c - a*d)^2) - (15*a*d^3)/(16*c^3*(b*c - a*d)^2))*Sqrt[x])/(c + d*x^2) + (b^(15/4)*ArcTan[(a^(1/4)/(Sqrt[2]*b^(1/4)) - (b^(1/4)*x)/(Sqrt[2]*a^(1/4)))/Sqrt[x]])/(Sqrt[2]*a^(7/4)*(b*c - a*d)^3) + ((-165*b^2*d^(7/4))/(32*Sqrt[2]*c^(7/4)*(b*c - a*d)^3) + (105*a*b*d^(11/4))/(16*Sqrt[2]*c^(11/4)*(b*c - a*d)^3) - (77*a^2*d^(15/4))/(32*Sqrt[2]*c^(15/4)*(b*c - a*d)^3))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] - (b^(15/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*a^(7/4)*(b*c - a*d)^3) + ((165*b^2*d^(7/4))/(32*Sqrt[2]*c^(7/4)*(b*c - a*d)^3) - (105*a*b*d^(11/4))/(16*Sqrt[2]*c^(11/4)*(b*c - a*d)^3) + (77*a^2*d^(15/4))/(32*Sqrt[2]*c^(15/4)*(b*c - a*d)^3))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
487,1,533,743,6.5618557,"\int \frac{1}{x^{7/2} \left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[1/(x^(7/2)*(a + b*x^2)*(c + d*x^2)^3),x]","-\frac{b^{17/4} \tan ^{-1}\left(\frac{\frac{\sqrt[4]{a}}{\sqrt{2} \sqrt[4]{b}}-\frac{\sqrt[4]{b} x}{\sqrt{2} \sqrt[4]{a}}}{\sqrt{x}}\right)}{\sqrt{2} a^{9/4} (b c-a d)^3}-\frac{b^{17/4} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)}{\sqrt{2} a^{9/4} (b c-a d)^3}+\left(\frac{117 a^2 d^{17/4}}{32 \sqrt{2} c^{17/4} (b c-a d)^3}+\frac{221 b^2 d^{9/4}}{32 \sqrt{2} c^{9/4} (b c-a d)^3}-\frac{153 a b d^{13/4}}{16 \sqrt{2} c^{13/4} (b c-a d)^3}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(\frac{117 a^2 d^{17/4}}{32 \sqrt{2} c^{17/4} (b c-a d)^3}+\frac{221 b^2 d^{9/4}}{32 \sqrt{2} c^{9/4} (b c-a d)^3}-\frac{153 a b d^{13/4}}{16 \sqrt{2} c^{13/4} (b c-a d)^3}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+\frac{\frac{2 b}{a^2 c^3}+\frac{6 d}{a c^4}}{\sqrt{x}}-\frac{d^3 x^{3/2}}{4 c^3 \left(c+d x^2\right)^2 (b c-a d)}+\frac{x^{3/2} \left(\frac{21 a d^4}{16 c^4 (b c-a d)^2}-\frac{29 b d^3}{16 c^3 (b c-a d)^2}\right)}{c+d x^2}-\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)/(a^2*c^3) + (6*d)/(a*c^4))/Sqrt[x] - (d^3*x^(3/2))/(4*c^3*(b*c - a*d)*(c + d*x^2)^2) + (((-29*b*d^3)/(16*c^3*(b*c - a*d)^2) + (21*a*d^4)/(16*c^4*(b*c - a*d)^2))*x^(3/2))/(c + d*x^2) - (b^(17/4)*ArcTan[(a^(1/4)/(Sqrt[2]*b^(1/4)) - (b^(1/4)*x)/(Sqrt[2]*a^(1/4)))/Sqrt[x]])/(Sqrt[2]*a^(9/4)*(b*c - a*d)^3) + ((221*b^2*d^(9/4))/(32*Sqrt[2]*c^(9/4)*(b*c - a*d)^3) - (153*a*b*d^(13/4))/(16*Sqrt[2]*c^(13/4)*(b*c - a*d)^3) + (117*a^2*d^(17/4))/(32*Sqrt[2]*c^(17/4)*(b*c - a*d)^3))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] - (b^(17/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*a^(9/4)*(b*c - a*d)^3) + ((221*b^2*d^(9/4))/(32*Sqrt[2]*c^(9/4)*(b*c - a*d)^3) - (153*a*b*d^(13/4))/(16*Sqrt[2]*c^(13/4)*(b*c - a*d)^3) + (117*a^2*d^(17/4))/(32*Sqrt[2]*c^(17/4)*(b*c - a*d)^3))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
488,1,450,624,1.1771247,"\int \frac{x^{7/2}}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[x^(7/2)/((a + b*x^2)^2*(c + d*x^2)^2),x]","\left(\frac{3 a^{5/4} d}{4 \sqrt{2} \sqrt[4]{b} (b c-a d)^3}+\frac{5 \sqrt[4]{a} b^{3/4} c}{4 \sqrt{2} (b c-a d)^3}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(-\frac{3 a^{5/4} d}{4 \sqrt{2} \sqrt[4]{b} (b c-a d)^3}-\frac{5 \sqrt[4]{a} b^{3/4} c}{4 \sqrt{2} (b c-a d)^3}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)+\left(-\frac{3 b c^{5/4}}{4 \sqrt{2} \sqrt[4]{d} (b c-a d)^3}-\frac{5 a \sqrt[4]{c} d^{3/4}}{4 \sqrt{2} (b c-a d)^3}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(\frac{3 b c^{5/4}}{4 \sqrt{2} \sqrt[4]{d} (b c-a d)^3}+\frac{5 a \sqrt[4]{c} d^{3/4}}{4 \sqrt{2} (b c-a d)^3}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+\frac{x^{5/2} (a d+b c)+2 a c \sqrt{x}}{2 \left(a+b x^2\right) \left(c+d x^2\right) (b c-a d)^2}","\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,"(2*a*c*Sqrt[x] + (b*c + a*d)*x^(5/2))/(2*(b*c - a*d)^2*(a + b*x^2)*(c + d*x^2)) + ((5*a^(1/4)*b^(3/4)*c)/(4*Sqrt[2]*(b*c - a*d)^3) + (3*a^(5/4)*d)/(4*Sqrt[2]*b^(1/4)*(b*c - a*d)^3))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((-3*b*c^(5/4))/(4*Sqrt[2]*d^(1/4)*(b*c - a*d)^3) - (5*a*c^(1/4)*d^(3/4))/(4*Sqrt[2]*(b*c - a*d)^3))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + ((-5*a^(1/4)*b^(3/4)*c)/(4*Sqrt[2]*(b*c - a*d)^3) - (3*a^(5/4)*d)/(4*Sqrt[2]*b^(1/4)*(b*c - a*d)^3))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)] + ((3*b*c^(5/4))/(4*Sqrt[2]*d^(1/4)*(b*c - a*d)^3) + (5*a*c^(1/4)*d^(3/4))/(4*Sqrt[2]*(b*c - a*d)^3))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
489,1,452,609,1.2487447,"\int \frac{x^{5/2}}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[x^(5/2)/((a + b*x^2)^2*(c + d*x^2)^2),x]","\left(-\frac{5 a^{3/4} \sqrt[4]{b} d}{4 \sqrt{2} (b c-a d)^3}-\frac{3 b^{5/4} c}{4 \sqrt{2} \sqrt[4]{a} (b c-a d)^3}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(-\frac{5 a^{3/4} \sqrt[4]{b} d}{4 \sqrt{2} (b c-a d)^3}-\frac{3 b^{5/4} c}{4 \sqrt{2} \sqrt[4]{a} (b c-a d)^3}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)+\left(\frac{5 b c^{3/4} \sqrt[4]{d}}{4 \sqrt{2} (b c-a d)^3}+\frac{3 a d^{5/4}}{4 \sqrt{2} \sqrt[4]{c} (b c-a d)^3}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(\frac{5 b c^{3/4} \sqrt[4]{d}}{4 \sqrt{2} (b c-a d)^3}+\frac{3 a d^{5/4}}{4 \sqrt{2} \sqrt[4]{c} (b c-a d)^3}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+\frac{x^{3/2} (-a d-b c)-2 b d x^{7/2}}{2 \left(a+b x^2\right) \left(c+d x^2\right) (b c-a d)^2}","-\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,"((-(b*c) - a*d)*x^(3/2) - 2*b*d*x^(7/2))/(2*(b*c - a*d)^2*(a + b*x^2)*(c + d*x^2)) + ((-3*b^(5/4)*c)/(4*Sqrt[2]*a^(1/4)*(b*c - a*d)^3) - (5*a^(3/4)*b^(1/4)*d)/(4*Sqrt[2]*(b*c - a*d)^3))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((5*b*c^(3/4)*d^(1/4))/(4*Sqrt[2]*(b*c - a*d)^3) + (3*a*d^(5/4))/(4*Sqrt[2]*c^(1/4)*(b*c - a*d)^3))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + ((-3*b^(5/4)*c)/(4*Sqrt[2]*a^(1/4)*(b*c - a*d)^3) - (5*a^(3/4)*b^(1/4)*d)/(4*Sqrt[2]*(b*c - a*d)^3))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)] + ((5*b*c^(3/4)*d^(1/4))/(4*Sqrt[2]*(b*c - a*d)^3) + (3*a*d^(5/4))/(4*Sqrt[2]*c^(1/4)*(b*c - a*d)^3))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
490,1,452,601,1.3114695,"\int \frac{x^{3/2}}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[x^(3/2)/((a + b*x^2)^2*(c + d*x^2)^2),x]","\left(-\frac{b^{7/4} c}{4 \sqrt{2} a^{3/4} (b c-a d)^3}-\frac{7 \sqrt[4]{a} b^{3/4} d}{4 \sqrt{2} (b c-a d)^3}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(\frac{b^{7/4} c}{4 \sqrt{2} a^{3/4} (b c-a d)^3}+\frac{7 \sqrt[4]{a} b^{3/4} d}{4 \sqrt{2} (b c-a d)^3}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)+\left(\frac{a d^{7/4}}{4 \sqrt{2} c^{3/4} (b c-a d)^3}+\frac{7 b \sqrt[4]{c} d^{3/4}}{4 \sqrt{2} (b c-a d)^3}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(-\frac{a d^{7/4}}{4 \sqrt{2} c^{3/4} (b c-a d)^3}-\frac{7 b \sqrt[4]{c} d^{3/4}}{4 \sqrt{2} (b c-a d)^3}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+\frac{\sqrt{x} (-a d-b c)-2 b d x^{5/2}}{2 \left(a+b x^2\right) \left(c+d x^2\right) (b c-a d)^2}","-\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,"((-(b*c) - a*d)*Sqrt[x] - 2*b*d*x^(5/2))/(2*(b*c - a*d)^2*(a + b*x^2)*(c + d*x^2)) + (-1/4*(b^(7/4)*c)/(Sqrt[2]*a^(3/4)*(b*c - a*d)^3) - (7*a^(1/4)*b^(3/4)*d)/(4*Sqrt[2]*(b*c - a*d)^3))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((7*b*c^(1/4)*d^(3/4))/(4*Sqrt[2]*(b*c - a*d)^3) + (a*d^(7/4))/(4*Sqrt[2]*c^(3/4)*(b*c - a*d)^3))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + ((b^(7/4)*c)/(4*Sqrt[2]*a^(3/4)*(b*c - a*d)^3) + (7*a^(1/4)*b^(3/4)*d)/(4*Sqrt[2]*(b*c - a*d)^3))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)] + ((-7*b*c^(1/4)*d^(3/4))/(4*Sqrt[2]*(b*c - a*d)^3) - (a*d^(7/4))/(4*Sqrt[2]*c^(3/4)*(b*c - a*d)^3))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
491,1,493,624,3.0989577,"\int \frac{\sqrt{x}}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[Sqrt[x]/((a + b*x^2)^2*(c + d*x^2)^2),x]","\left(\frac{b^{5/4} (9 a d-b c)}{8 \sqrt{2} a^{5/4} (b c-a d)^3}-\frac{b^{5/4} (b c-9 a d)}{8 \sqrt{2} a^{5/4} (b c-a d)^3}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(\frac{9 b^{5/4} d}{4 \sqrt{2} \sqrt[4]{a} (b c-a d)^3}-\frac{b^{9/4} c}{4 \sqrt{2} a^{5/4} (b c-a d)^3}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)+\frac{b^2 x^{3/2}}{2 a \left(a+b x^2\right) (b c-a d)^2}+\left(\frac{d^{5/4} (a d-9 b c)}{8 \sqrt{2} c^{5/4} (b c-a d)^3}-\frac{d^{5/4} (9 b c-a d)}{8 \sqrt{2} c^{5/4} (b c-a d)^3}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(\frac{a d^{9/4}}{4 \sqrt{2} c^{5/4} (b c-a d)^3}-\frac{9 b d^{5/4}}{4 \sqrt{2} \sqrt[4]{c} (b c-a d)^3}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+\frac{d^2 x^{3/2}}{2 c \left(c+d x^2\right) (b c-a d)^2}","\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,"(b^2*x^(3/2))/(2*a*(b*c - a*d)^2*(a + b*x^2)) + (d^2*x^(3/2))/(2*c*(b*c - a*d)^2*(c + d*x^2)) + (-1/8*(b^(5/4)*(b*c - 9*a*d))/(Sqrt[2]*a^(5/4)*(b*c - a*d)^3) + (b^(5/4)*(-(b*c) + 9*a*d))/(8*Sqrt[2]*a^(5/4)*(b*c - a*d)^3))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + (-1/8*(d^(5/4)*(9*b*c - a*d))/(Sqrt[2]*c^(5/4)*(b*c - a*d)^3) + (d^(5/4)*(-9*b*c + a*d))/(8*Sqrt[2]*c^(5/4)*(b*c - a*d)^3))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + (-1/4*(b^(9/4)*c)/(Sqrt[2]*a^(5/4)*(b*c - a*d)^3) + (9*b^(5/4)*d)/(4*Sqrt[2]*a^(1/4)*(b*c - a*d)^3))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)] + ((-9*b*d^(5/4))/(4*Sqrt[2]*c^(1/4)*(b*c - a*d)^3) + (a*d^(9/4))/(4*Sqrt[2]*c^(5/4)*(b*c - a*d)^3))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
492,1,463,628,6.4406089,"\int \frac{1}{\sqrt{x} \left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[1/(Sqrt[x]*(a + b*x^2)^2*(c + d*x^2)^2),x]","\left(\frac{11 b^{7/4} d}{4 \sqrt{2} a^{3/4} (b c-a d)^3}-\frac{3 b^{11/4} c}{4 \sqrt{2} a^{7/4} (b c-a d)^3}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(\frac{3 b^{11/4} c}{4 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{11 b^{7/4} d}{4 \sqrt{2} a^{3/4} (b c-a d)^3}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)+\frac{b^2 \sqrt{x}}{2 a \left(a+b x^2\right) (b c-a d)^2}+\left(\frac{3 a d^{11/4}}{4 \sqrt{2} c^{7/4} (b c-a d)^3}-\frac{11 b d^{7/4}}{4 \sqrt{2} c^{3/4} (b c-a d)^3}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(\frac{11 b d^{7/4}}{4 \sqrt{2} c^{3/4} (b c-a d)^3}-\frac{3 a d^{11/4}}{4 \sqrt{2} c^{7/4} (b c-a d)^3}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+\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)) + ((-3*b^(11/4)*c)/(4*Sqrt[2]*a^(7/4)*(b*c - a*d)^3) + (11*b^(7/4)*d)/(4*Sqrt[2]*a^(3/4)*(b*c - a*d)^3))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((-11*b*d^(7/4))/(4*Sqrt[2]*c^(3/4)*(b*c - a*d)^3) + (3*a*d^(11/4))/(4*Sqrt[2]*c^(7/4)*(b*c - a*d)^3))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + ((3*b^(11/4)*c)/(4*Sqrt[2]*a^(7/4)*(b*c - a*d)^3) - (11*b^(7/4)*d)/(4*Sqrt[2]*a^(3/4)*(b*c - a*d)^3))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)] + ((11*b*d^(7/4))/(4*Sqrt[2]*c^(3/4)*(b*c - a*d)^3) - (3*a*d^(11/4))/(4*Sqrt[2]*c^(7/4)*(b*c - a*d)^3))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
493,1,476,676,6.5281561,"\int \frac{1}{x^{3/2} \left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[1/(x^(3/2)*(a + b*x^2)^2*(c + d*x^2)^2),x]","\left(\frac{5 b^{13/4} c}{4 \sqrt{2} a^{9/4} (b c-a d)^3}-\frac{13 b^{9/4} d}{4 \sqrt{2} a^{5/4} (b c-a d)^3}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(\frac{5 b^{13/4} c}{4 \sqrt{2} a^{9/4} (b c-a d)^3}-\frac{13 b^{9/4} d}{4 \sqrt{2} a^{5/4} (b c-a d)^3}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)-\frac{b^3 x^{3/2}}{2 a^2 \left(a+b x^2\right) (b c-a d)^2}-\frac{2}{a^2 c^2 \sqrt{x}}+\left(\frac{13 b d^{9/4}}{4 \sqrt{2} c^{5/4} (b c-a d)^3}-\frac{5 a d^{13/4}}{4 \sqrt{2} c^{9/4} (b c-a d)^3}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(\frac{13 b d^{9/4}}{4 \sqrt{2} c^{5/4} (b c-a d)^3}-\frac{5 a d^{13/4}}{4 \sqrt{2} c^{9/4} (b c-a d)^3}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)-\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)) + ((5*b^(13/4)*c)/(4*Sqrt[2]*a^(9/4)*(b*c - a*d)^3) - (13*b^(9/4)*d)/(4*Sqrt[2]*a^(5/4)*(b*c - a*d)^3))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((13*b*d^(9/4))/(4*Sqrt[2]*c^(5/4)*(b*c - a*d)^3) - (5*a*d^(13/4))/(4*Sqrt[2]*c^(9/4)*(b*c - a*d)^3))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + ((5*b^(13/4)*c)/(4*Sqrt[2]*a^(9/4)*(b*c - a*d)^3) - (13*b^(9/4)*d)/(4*Sqrt[2]*a^(5/4)*(b*c - a*d)^3))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)] + ((13*b*d^(9/4))/(4*Sqrt[2]*c^(5/4)*(b*c - a*d)^3) - (5*a*d^(13/4))/(4*Sqrt[2]*c^(9/4)*(b*c - a*d)^3))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
494,1,478,676,6.4995757,"\int \frac{1}{x^{5/2} \left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[1/(x^(5/2)*(a + b*x^2)^2*(c + d*x^2)^2),x]","\left(\frac{7 b^{15/4} c}{4 \sqrt{2} a^{11/4} (b c-a d)^3}-\frac{15 b^{11/4} d}{4 \sqrt{2} a^{7/4} (b c-a d)^3}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(\frac{15 b^{11/4} d}{4 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{7 b^{15/4} c}{4 \sqrt{2} a^{11/4} (b c-a d)^3}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)-\frac{b^3 \sqrt{x}}{2 a^2 \left(a+b x^2\right) (b c-a d)^2}-\frac{2}{3 a^2 c^2 x^{3/2}}+\left(\frac{15 b d^{11/4}}{4 \sqrt{2} c^{7/4} (b c-a d)^3}-\frac{7 a d^{15/4}}{4 \sqrt{2} c^{11/4} (b c-a d)^3}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(\frac{7 a d^{15/4}}{4 \sqrt{2} c^{11/4} (b c-a d)^3}-\frac{15 b d^{11/4}}{4 \sqrt{2} c^{7/4} (b c-a d)^3}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)-\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)) + ((7*b^(15/4)*c)/(4*Sqrt[2]*a^(11/4)*(b*c - a*d)^3) - (15*b^(11/4)*d)/(4*Sqrt[2]*a^(7/4)*(b*c - a*d)^3))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((15*b*d^(11/4))/(4*Sqrt[2]*c^(7/4)*(b*c - a*d)^3) - (7*a*d^(15/4))/(4*Sqrt[2]*c^(11/4)*(b*c - a*d)^3))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + ((-7*b^(15/4)*c)/(4*Sqrt[2]*a^(11/4)*(b*c - a*d)^3) + (15*b^(11/4)*d)/(4*Sqrt[2]*a^(7/4)*(b*c - a*d)^3))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)] + ((-15*b*d^(11/4))/(4*Sqrt[2]*c^(7/4)*(b*c - a*d)^3) + (7*a*d^(15/4))/(4*Sqrt[2]*c^(11/4)*(b*c - a*d)^3))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
495,1,503,731,6.5349547,"\int \frac{1}{x^{7/2} \left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[1/(x^(7/2)*(a + b*x^2)^2*(c + d*x^2)^2),x]","\left(\frac{17 b^{13/4} d}{4 \sqrt{2} a^{9/4} (b c-a d)^3}-\frac{9 b^{17/4} c}{4 \sqrt{2} a^{13/4} (b c-a d)^3}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(\frac{17 b^{13/4} d}{4 \sqrt{2} a^{9/4} (b c-a d)^3}-\frac{9 b^{17/4} c}{4 \sqrt{2} a^{13/4} (b c-a d)^3}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)+\frac{b^4 x^{3/2}}{2 a^3 \left(a+b x^2\right) (b c-a d)^2}-\frac{2}{5 a^2 c^2 x^{5/2}}+\frac{\frac{4 b}{a^3 c^2}+\frac{4 d}{a^2 c^3}}{\sqrt{x}}+\left(\frac{9 a d^{17/4}}{4 \sqrt{2} c^{13/4} (b c-a d)^3}-\frac{17 b d^{13/4}}{4 \sqrt{2} c^{9/4} (b c-a d)^3}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(\frac{9 a d^{17/4}}{4 \sqrt{2} c^{13/4} (b c-a d)^3}-\frac{17 b d^{13/4}}{4 \sqrt{2} c^{9/4} (b c-a d)^3}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+\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)/(a^3*c^2) + (4*d)/(a^2*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)) + ((-9*b^(17/4)*c)/(4*Sqrt[2]*a^(13/4)*(b*c - a*d)^3) + (17*b^(13/4)*d)/(4*Sqrt[2]*a^(9/4)*(b*c - a*d)^3))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((-17*b*d^(13/4))/(4*Sqrt[2]*c^(9/4)*(b*c - a*d)^3) + (9*a*d^(17/4))/(4*Sqrt[2]*c^(13/4)*(b*c - a*d)^3))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + ((-9*b^(17/4)*c)/(4*Sqrt[2]*a^(13/4)*(b*c - a*d)^3) + (17*b^(13/4)*d)/(4*Sqrt[2]*a^(9/4)*(b*c - a*d)^3))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)] + ((-17*b*d^(13/4))/(4*Sqrt[2]*c^(9/4)*(b*c - a*d)^3) + (9*a*d^(17/4))/(4*Sqrt[2]*c^(13/4)*(b*c - a*d)^3))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
496,1,569,718,1.6405789,"\int \frac{x^{7/2}}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[x^(7/2)/((a + b*x^2)^2*(c + d*x^2)^3),x]","\left(\frac{7 a^{5/4} b^{3/4} d}{4 \sqrt{2} (b c-a d)^4}+\frac{5 \sqrt[4]{a} b^{7/4} c}{4 \sqrt{2} (b c-a d)^4}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(-\frac{7 a^{5/4} b^{3/4} d}{4 \sqrt{2} (b c-a d)^4}-\frac{5 \sqrt[4]{a} b^{7/4} c}{4 \sqrt{2} (b c-a d)^4}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)+\left(-\frac{5 a^2 d^{7/4}}{32 \sqrt{2} c^{3/4} (b c-a d)^4}-\frac{21 b^2 c^{5/4}}{32 \sqrt{2} \sqrt[4]{d} (b c-a d)^4}-\frac{35 a b \sqrt[4]{c} d^{3/4}}{16 \sqrt{2} (b c-a d)^4}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(\frac{5 a^2 d^{7/4}}{32 \sqrt{2} c^{3/4} (b c-a d)^4}+\frac{21 b^2 c^{5/4}}{32 \sqrt{2} \sqrt[4]{d} (b c-a d)^4}+\frac{35 a b \sqrt[4]{c} d^{3/4}}{16 \sqrt{2} (b c-a d)^4}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+\frac{x^{5/2} \left(9 a^2 d^2+28 a b c d+11 b^2 c^2\right)+\sqrt{x} \left(5 a^2 c d+19 a b c^2\right)+x^{9/2} \left(17 a b d^2+7 b^2 c d\right)}{16 \left(a+b x^2\right) \left(c+d x^2\right)^2 (b c-a d)^3}","-\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,"((19*a*b*c^2 + 5*a^2*c*d)*Sqrt[x] + (11*b^2*c^2 + 28*a*b*c*d + 9*a^2*d^2)*x^(5/2) + (7*b^2*c*d + 17*a*b*d^2)*x^(9/2))/(16*(b*c - a*d)^3*(a + b*x^2)*(c + d*x^2)^2) + ((5*a^(1/4)*b^(7/4)*c)/(4*Sqrt[2]*(b*c - a*d)^4) + (7*a^(5/4)*b^(3/4)*d)/(4*Sqrt[2]*(b*c - a*d)^4))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((-21*b^2*c^(5/4))/(32*Sqrt[2]*d^(1/4)*(b*c - a*d)^4) - (35*a*b*c^(1/4)*d^(3/4))/(16*Sqrt[2]*(b*c - a*d)^4) - (5*a^2*d^(7/4))/(32*Sqrt[2]*c^(3/4)*(b*c - a*d)^4))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + ((-5*a^(1/4)*b^(7/4)*c)/(4*Sqrt[2]*(b*c - a*d)^4) - (7*a^(5/4)*b^(3/4)*d)/(4*Sqrt[2]*(b*c - a*d)^4))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)] + ((21*b^2*c^(5/4))/(32*Sqrt[2]*d^(1/4)*(b*c - a*d)^4) + (35*a*b*c^(1/4)*d^(3/4))/(16*Sqrt[2]*(b*c - a*d)^4) + (5*a^2*d^(7/4))/(32*Sqrt[2]*c^(3/4)*(b*c - a*d)^4))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
497,1,595,703,2.0385694,"\int \frac{x^{5/2}}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[x^(5/2)/((a + b*x^2)^2*(c + d*x^2)^3),x]","\left(-\frac{9 a^{3/4} b^{5/4} d}{4 \sqrt{2} (b c-a d)^4}-\frac{3 b^{9/4} c}{4 \sqrt{2} \sqrt[4]{a} (b c-a d)^4}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(-\frac{9 a^{3/4} b^{5/4} d}{4 \sqrt{2} (b c-a d)^4}-\frac{3 b^{9/4} c}{4 \sqrt{2} \sqrt[4]{a} (b c-a d)^4}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)+\left(-\frac{3 a^2 d^{9/4}}{32 \sqrt{2} c^{5/4} (b c-a d)^4}+\frac{45 b^2 c^{3/4} \sqrt[4]{d}}{32 \sqrt{2} (b c-a d)^4}+\frac{27 a b d^{5/4}}{16 \sqrt{2} \sqrt[4]{c} (b c-a d)^4}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+\left(\frac{3 \sqrt[4]{d} \left(-a^2 d^2+18 a b c d+15 b^2 c^2\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)}{64 \sqrt{2} c^{5/4} (b c-a d)^4}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\frac{x^{7/2} \left(-3 a^2 d^3-12 a b c d^2-33 b^2 c^2 d\right)+x^{3/2} \left(a^2 c d^2-17 a b c^2 d-8 b^2 c^3\right)+x^{11/2} \left(-3 a b d^3-21 b^2 c d^2\right)}{16 c \left(a+b x^2\right) \left(c+d x^2\right)^2 (b c-a d)^3}","-\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,"((-8*b^2*c^3 - 17*a*b*c^2*d + a^2*c*d^2)*x^(3/2) + (-33*b^2*c^2*d - 12*a*b*c*d^2 - 3*a^2*d^3)*x^(7/2) + (-21*b^2*c*d^2 - 3*a*b*d^3)*x^(11/2))/(16*c*(b*c - a*d)^3*(a + b*x^2)*(c + d*x^2)^2) + ((-3*b^(9/4)*c)/(4*Sqrt[2]*a^(1/4)*(b*c - a*d)^4) - (9*a^(3/4)*b^(5/4)*d)/(4*Sqrt[2]*(b*c - a*d)^4))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((3*d^(1/4)*(15*b^2*c^2 + 18*a*b*c*d - a^2*d^2))/(64*Sqrt[2]*c^(5/4)*(b*c - a*d)^4) - (3*d^(1/4)*(-15*b^2*c^2 - 18*a*b*c*d + a^2*d^2))/(64*Sqrt[2]*c^(5/4)*(b*c - a*d)^4))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + ((-3*b^(9/4)*c)/(4*Sqrt[2]*a^(1/4)*(b*c - a*d)^4) - (9*a^(3/4)*b^(5/4)*d)/(4*Sqrt[2]*(b*c - a*d)^4))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)] + ((45*b^2*c^(3/4)*d^(1/4))/(32*Sqrt[2]*(b*c - a*d)^4) + (27*a*b*d^(5/4))/(16*Sqrt[2]*c^(1/4)*(b*c - a*d)^4) - (3*a^2*d^(9/4))/(32*Sqrt[2]*c^(5/4)*(b*c - a*d)^4))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
498,1,588,703,2.2063222,"\int \frac{x^{3/2}}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[x^(3/2)/((a + b*x^2)^2*(c + d*x^2)^3),x]","\left(-\frac{b^{11/4} c}{4 \sqrt{2} a^{3/4} (b c-a d)^4}-\frac{11 \sqrt[4]{a} b^{7/4} d}{4 \sqrt{2} (b c-a d)^4}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(\frac{b^{11/4} c}{4 \sqrt{2} a^{3/4} (b c-a d)^4}+\frac{11 \sqrt[4]{a} b^{7/4} d}{4 \sqrt{2} (b c-a d)^4}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)+\left(-\frac{3 a^2 d^{11/4}}{32 \sqrt{2} c^{7/4} (b c-a d)^4}+\frac{77 b^2 \sqrt[4]{c} d^{3/4}}{32 \sqrt{2} (b c-a d)^4}+\frac{11 a b d^{7/4}}{16 \sqrt{2} c^{3/4} (b c-a d)^4}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(\frac{3 a^2 d^{11/4}}{32 \sqrt{2} c^{7/4} (b c-a d)^4}-\frac{77 b^2 \sqrt[4]{c} d^{3/4}}{32 \sqrt{2} (b c-a d)^4}-\frac{11 a b d^{7/4}}{16 \sqrt{2} c^{3/4} (b c-a d)^4}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+\frac{x^{5/2} \left(-a^2 d^3-12 a b c d^2-35 b^2 c^2 d\right)+\sqrt{x} \left(3 a^2 c d^2-19 a b c^2 d-8 b^2 c^3\right)+x^{9/2} \left(-a b d^3-23 b^2 c d^2\right)}{16 c \left(a+b x^2\right) \left(c+d x^2\right)^2 (b c-a d)^3}","-\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,"((-8*b^2*c^3 - 19*a*b*c^2*d + 3*a^2*c*d^2)*Sqrt[x] + (-35*b^2*c^2*d - 12*a*b*c*d^2 - a^2*d^3)*x^(5/2) + (-23*b^2*c*d^2 - a*b*d^3)*x^(9/2))/(16*c*(b*c - a*d)^3*(a + b*x^2)*(c + d*x^2)^2) + (-1/4*(b^(11/4)*c)/(Sqrt[2]*a^(3/4)*(b*c - a*d)^4) - (11*a^(1/4)*b^(7/4)*d)/(4*Sqrt[2]*(b*c - a*d)^4))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((77*b^2*c^(1/4)*d^(3/4))/(32*Sqrt[2]*(b*c - a*d)^4) + (11*a*b*d^(7/4))/(16*Sqrt[2]*c^(3/4)*(b*c - a*d)^4) - (3*a^2*d^(11/4))/(32*Sqrt[2]*c^(7/4)*(b*c - a*d)^4))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + ((b^(11/4)*c)/(4*Sqrt[2]*a^(3/4)*(b*c - a*d)^4) + (11*a^(1/4)*b^(7/4)*d)/(4*Sqrt[2]*(b*c - a*d)^4))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)] + ((-77*b^2*c^(1/4)*d^(3/4))/(32*Sqrt[2]*(b*c - a*d)^4) - (11*a*b*d^(7/4))/(16*Sqrt[2]*c^(3/4)*(b*c - a*d)^4) + (3*a^2*d^(11/4))/(32*Sqrt[2]*c^(7/4)*(b*c - a*d)^4))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
499,1,591,739,6.5584455,"\int \frac{\sqrt{x}}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[Sqrt[x]/((a + b*x^2)^2*(c + d*x^2)^3),x]","\left(\frac{13 b^{9/4} d}{4 \sqrt{2} \sqrt[4]{a} (b c-a d)^4}-\frac{b^{13/4} c}{4 \sqrt{2} a^{5/4} (b c-a d)^4}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(\frac{13 b^{9/4} d}{4 \sqrt{2} \sqrt[4]{a} (b c-a d)^4}-\frac{b^{13/4} c}{4 \sqrt{2} a^{5/4} (b c-a d)^4}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)+\left(-\frac{5 a^2 d^{13/4}}{32 \sqrt{2} c^{9/4} (b c-a d)^4}-\frac{117 b^2 d^{5/4}}{32 \sqrt{2} \sqrt[4]{c} (b c-a d)^4}+\frac{13 a b d^{9/4}}{16 \sqrt{2} c^{5/4} (b c-a d)^4}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(-\frac{5 a^2 d^{13/4}}{32 \sqrt{2} c^{9/4} (b c-a d)^4}-\frac{117 b^2 d^{5/4}}{32 \sqrt{2} \sqrt[4]{c} (b c-a d)^4}+\frac{13 a b d^{9/4}}{16 \sqrt{2} c^{5/4} (b c-a d)^4}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+\frac{b^3 x^{3/2}}{2 a \left(a+b x^2\right) (b c-a d)^3}+\frac{x^{3/2} \left(\frac{21 b d^2}{16 c (b c-a d)^3}-\frac{5 a d^3}{16 c^2 (b c-a d)^3}\right)}{c+d x^2}+\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,"(b^3*x^(3/2))/(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) + (((21*b*d^2)/(16*c*(b*c - a*d)^3) - (5*a*d^3)/(16*c^2*(b*c - a*d)^3))*x^(3/2))/(c + d*x^2) + (-1/4*(b^(13/4)*c)/(Sqrt[2]*a^(5/4)*(b*c - a*d)^4) + (13*b^(9/4)*d)/(4*Sqrt[2]*a^(1/4)*(b*c - a*d)^4))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((-117*b^2*d^(5/4))/(32*Sqrt[2]*c^(1/4)*(b*c - a*d)^4) + (13*a*b*d^(9/4))/(16*Sqrt[2]*c^(5/4)*(b*c - a*d)^4) - (5*a^2*d^(13/4))/(32*Sqrt[2]*c^(9/4)*(b*c - a*d)^4))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + (-1/4*(b^(13/4)*c)/(Sqrt[2]*a^(5/4)*(b*c - a*d)^4) + (13*b^(9/4)*d)/(4*Sqrt[2]*a^(1/4)*(b*c - a*d)^4))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)] + ((-117*b^2*d^(5/4))/(32*Sqrt[2]*c^(1/4)*(b*c - a*d)^4) + (13*a*b*d^(9/4))/(16*Sqrt[2]*c^(5/4)*(b*c - a*d)^4) - (5*a^2*d^(13/4))/(32*Sqrt[2]*c^(9/4)*(b*c - a*d)^4))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
500,1,591,739,6.5527584,"\int \frac{1}{\sqrt{x} \left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[1/(Sqrt[x]*(a + b*x^2)^2*(c + d*x^2)^3),x]","\left(\frac{15 b^{11/4} d}{4 \sqrt{2} a^{3/4} (b c-a d)^4}-\frac{3 b^{15/4} c}{4 \sqrt{2} a^{7/4} (b c-a d)^4}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(\frac{3 b^{15/4} c}{4 \sqrt{2} a^{7/4} (b c-a d)^4}-\frac{15 b^{11/4} d}{4 \sqrt{2} a^{3/4} (b c-a d)^4}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)+\left(-\frac{21 a^2 d^{15/4}}{32 \sqrt{2} c^{11/4} (b c-a d)^4}-\frac{165 b^2 d^{7/4}}{32 \sqrt{2} c^{3/4} (b c-a d)^4}+\frac{45 a b d^{11/4}}{16 \sqrt{2} c^{7/4} (b c-a d)^4}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(\frac{21 a^2 d^{15/4}}{32 \sqrt{2} c^{11/4} (b c-a d)^4}+\frac{165 b^2 d^{7/4}}{32 \sqrt{2} c^{3/4} (b c-a d)^4}-\frac{45 a b d^{11/4}}{16 \sqrt{2} c^{7/4} (b c-a d)^4}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)+\frac{b^3 \sqrt{x}}{2 a \left(a+b x^2\right) (b c-a d)^3}+\frac{\sqrt{x} \left(\frac{23 b d^2}{16 c (b c-a d)^3}-\frac{7 a d^3}{16 c^2 (b c-a d)^3}\right)}{c+d x^2}+\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,"(b^3*Sqrt[x])/(2*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) + (((23*b*d^2)/(16*c*(b*c - a*d)^3) - (7*a*d^3)/(16*c^2*(b*c - a*d)^3))*Sqrt[x])/(c + d*x^2) + ((-3*b^(15/4)*c)/(4*Sqrt[2]*a^(7/4)*(b*c - a*d)^4) + (15*b^(11/4)*d)/(4*Sqrt[2]*a^(3/4)*(b*c - a*d)^4))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((-165*b^2*d^(7/4))/(32*Sqrt[2]*c^(3/4)*(b*c - a*d)^4) + (45*a*b*d^(11/4))/(16*Sqrt[2]*c^(7/4)*(b*c - a*d)^4) - (21*a^2*d^(15/4))/(32*Sqrt[2]*c^(11/4)*(b*c - a*d)^4))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + ((3*b^(15/4)*c)/(4*Sqrt[2]*a^(7/4)*(b*c - a*d)^4) - (15*b^(11/4)*d)/(4*Sqrt[2]*a^(3/4)*(b*c - a*d)^4))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)] + ((165*b^2*d^(7/4))/(32*Sqrt[2]*c^(3/4)*(b*c - a*d)^4) - (45*a*b*d^(11/4))/(16*Sqrt[2]*c^(7/4)*(b*c - a*d)^4) + (21*a^2*d^(15/4))/(32*Sqrt[2]*c^(11/4)*(b*c - a*d)^4))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
501,1,604,805,6.6909856,"\int \frac{1}{x^{3/2} \left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[1/(x^(3/2)*(a + b*x^2)^2*(c + d*x^2)^3),x]","\left(\frac{5 b^{17/4} c}{4 \sqrt{2} a^{9/4} (b c-a d)^4}-\frac{17 b^{13/4} d}{4 \sqrt{2} a^{5/4} (b c-a d)^4}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(\frac{5 b^{17/4} c}{4 \sqrt{2} a^{9/4} (b c-a d)^4}-\frac{17 b^{13/4} d}{4 \sqrt{2} a^{5/4} (b c-a d)^4}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)-\frac{b^4 x^{3/2}}{2 a^2 \left(a+b x^2\right) (b c-a d)^3}+\left(\frac{45 a^2 d^{17/4}}{32 \sqrt{2} c^{13/4} (b c-a d)^4}+\frac{221 b^2 d^{9/4}}{32 \sqrt{2} c^{5/4} (b c-a d)^4}-\frac{85 a b d^{13/4}}{16 \sqrt{2} c^{9/4} (b c-a d)^4}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(\frac{45 a^2 d^{17/4}}{32 \sqrt{2} c^{13/4} (b c-a d)^4}+\frac{221 b^2 d^{9/4}}{32 \sqrt{2} c^{5/4} (b c-a d)^4}-\frac{85 a b d^{13/4}}{16 \sqrt{2} c^{9/4} (b c-a d)^4}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)-\frac{2}{a^2 c^3 \sqrt{x}}-\frac{d^3 x^{3/2}}{4 c^2 \left(c+d x^2\right)^2 (b c-a d)^2}+\frac{x^{3/2} \left(\frac{13 a d^4}{16 c^3 (b c-a d)^3}-\frac{29 b d^3}{16 c^2 (b c-a d)^3}\right)}{c+d x^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) + (((-29*b*d^3)/(16*c^2*(b*c - a*d)^3) + (13*a*d^4)/(16*c^3*(b*c - a*d)^3))*x^(3/2))/(c + d*x^2) + ((5*b^(17/4)*c)/(4*Sqrt[2]*a^(9/4)*(b*c - a*d)^4) - (17*b^(13/4)*d)/(4*Sqrt[2]*a^(5/4)*(b*c - a*d)^4))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((221*b^2*d^(9/4))/(32*Sqrt[2]*c^(5/4)*(b*c - a*d)^4) - (85*a*b*d^(13/4))/(16*Sqrt[2]*c^(9/4)*(b*c - a*d)^4) + (45*a^2*d^(17/4))/(32*Sqrt[2]*c^(13/4)*(b*c - a*d)^4))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + ((5*b^(17/4)*c)/(4*Sqrt[2]*a^(9/4)*(b*c - a*d)^4) - (17*b^(13/4)*d)/(4*Sqrt[2]*a^(5/4)*(b*c - a*d)^4))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)] + ((221*b^2*d^(9/4))/(32*Sqrt[2]*c^(5/4)*(b*c - a*d)^4) - (85*a*b*d^(13/4))/(16*Sqrt[2]*c^(9/4)*(b*c - a*d)^4) + (45*a^2*d^(17/4))/(32*Sqrt[2]*c^(13/4)*(b*c - a*d)^4))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
502,1,606,805,6.6891316,"\int \frac{1}{x^{5/2} \left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[1/(x^(5/2)*(a + b*x^2)^2*(c + d*x^2)^3),x]","\left(\frac{7 b^{19/4} c}{4 \sqrt{2} a^{11/4} (b c-a d)^4}-\frac{19 b^{15/4} d}{4 \sqrt{2} a^{7/4} (b c-a d)^4}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(\frac{19 b^{15/4} d}{4 \sqrt{2} a^{7/4} (b c-a d)^4}-\frac{7 b^{19/4} c}{4 \sqrt{2} a^{11/4} (b c-a d)^4}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)-\frac{b^4 \sqrt{x}}{2 a^2 \left(a+b x^2\right) (b c-a d)^3}+\left(\frac{77 a^2 d^{19/4}}{32 \sqrt{2} c^{15/4} (b c-a d)^4}+\frac{285 b^2 d^{11/4}}{32 \sqrt{2} c^{7/4} (b c-a d)^4}-\frac{133 a b d^{15/4}}{16 \sqrt{2} c^{11/4} (b c-a d)^4}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(-\frac{77 a^2 d^{19/4}}{32 \sqrt{2} c^{15/4} (b c-a d)^4}-\frac{285 b^2 d^{11/4}}{32 \sqrt{2} c^{7/4} (b c-a d)^4}+\frac{133 a b d^{15/4}}{16 \sqrt{2} c^{11/4} (b c-a d)^4}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)-\frac{2}{3 a^2 c^3 x^{3/2}}-\frac{d^3 \sqrt{x}}{4 c^2 \left(c+d x^2\right)^2 (b c-a d)^2}+\frac{\sqrt{x} \left(\frac{15 a d^4}{16 c^3 (b c-a d)^3}-\frac{31 b d^3}{16 c^2 (b c-a d)^3}\right)}{c+d x^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) + (((-31*b*d^3)/(16*c^2*(b*c - a*d)^3) + (15*a*d^4)/(16*c^3*(b*c - a*d)^3))*Sqrt[x])/(c + d*x^2) + ((7*b^(19/4)*c)/(4*Sqrt[2]*a^(11/4)*(b*c - a*d)^4) - (19*b^(15/4)*d)/(4*Sqrt[2]*a^(7/4)*(b*c - a*d)^4))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((285*b^2*d^(11/4))/(32*Sqrt[2]*c^(7/4)*(b*c - a*d)^4) - (133*a*b*d^(15/4))/(16*Sqrt[2]*c^(11/4)*(b*c - a*d)^4) + (77*a^2*d^(19/4))/(32*Sqrt[2]*c^(15/4)*(b*c - a*d)^4))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + ((-7*b^(19/4)*c)/(4*Sqrt[2]*a^(11/4)*(b*c - a*d)^4) + (19*b^(15/4)*d)/(4*Sqrt[2]*a^(7/4)*(b*c - a*d)^4))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)] + ((-285*b^2*d^(11/4))/(32*Sqrt[2]*c^(7/4)*(b*c - a*d)^4) + (133*a*b*d^(15/4))/(16*Sqrt[2]*c^(11/4)*(b*c - a*d)^4) - (77*a^2*d^(19/4))/(32*Sqrt[2]*c^(15/4)*(b*c - a*d)^4))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
503,1,631,881,6.6923885,"\int \frac{1}{x^{7/2} \left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[1/(x^(7/2)*(a + b*x^2)^2*(c + d*x^2)^3),x]","\left(\frac{21 b^{17/4} d}{4 \sqrt{2} a^{9/4} (b c-a d)^4}-\frac{9 b^{21/4} c}{4 \sqrt{2} a^{13/4} (b c-a d)^4}\right) \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}\right)+\left(\frac{21 b^{17/4} d}{4 \sqrt{2} a^{9/4} (b c-a d)^4}-\frac{9 b^{21/4} c}{4 \sqrt{2} a^{13/4} (b c-a d)^4}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}}{\sqrt{a}+\sqrt{b} x}\right)+\frac{b^5 x^{3/2}}{2 a^3 \left(a+b x^2\right) (b c-a d)^3}+\left(-\frac{117 a^2 d^{21/4}}{32 \sqrt{2} c^{17/4} (b c-a d)^4}-\frac{357 b^2 d^{13/4}}{32 \sqrt{2} c^{9/4} (b c-a d)^4}+\frac{189 a b d^{17/4}}{16 \sqrt{2} c^{13/4} (b c-a d)^4}\right) \tan ^{-1}\left(\frac{\sqrt{c}-\sqrt{d} x}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}\right)+\left(-\frac{117 a^2 d^{21/4}}{32 \sqrt{2} c^{17/4} (b c-a d)^4}-\frac{357 b^2 d^{13/4}}{32 \sqrt{2} c^{9/4} (b c-a d)^4}+\frac{189 a b d^{17/4}}{16 \sqrt{2} c^{13/4} (b c-a d)^4}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{x}}{\sqrt{c}+\sqrt{d} x}\right)-\frac{2}{5 a^2 c^3 x^{5/2}}+\frac{\frac{4 b}{a^3 c^3}+\frac{6 d}{a^2 c^4}}{\sqrt{x}}+\frac{d^4 x^{3/2}}{4 c^3 \left(c+d x^2\right)^2 (b c-a d)^2}+\frac{x^{3/2} \left(\frac{37 b d^4}{16 c^3 (b c-a d)^3}-\frac{21 a d^5}{16 c^4 (b c-a d)^3}\right)}{c+d x^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)) + ((4*b)/(a^3*c^3) + (6*d)/(a^2*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) + (((37*b*d^4)/(16*c^3*(b*c - a*d)^3) - (21*a*d^5)/(16*c^4*(b*c - a*d)^3))*x^(3/2))/(c + d*x^2) + ((-9*b^(21/4)*c)/(4*Sqrt[2]*a^(13/4)*(b*c - a*d)^4) + (21*b^(17/4)*d)/(4*Sqrt[2]*a^(9/4)*(b*c - a*d)^4))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + ((-357*b^2*d^(13/4))/(32*Sqrt[2]*c^(9/4)*(b*c - a*d)^4) + (189*a*b*d^(17/4))/(16*Sqrt[2]*c^(13/4)*(b*c - a*d)^4) - (117*a^2*d^(21/4))/(32*Sqrt[2]*c^(17/4)*(b*c - a*d)^4))*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + ((-9*b^(21/4)*c)/(4*Sqrt[2]*a^(13/4)*(b*c - a*d)^4) + (21*b^(17/4)*d)/(4*Sqrt[2]*a^(9/4)*(b*c - a*d)^4))*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)] + ((-357*b^2*d^(13/4))/(32*Sqrt[2]*c^(9/4)*(b*c - a*d)^4) + (189*a*b*d^(17/4))/(16*Sqrt[2]*c^(13/4)*(b*c - a*d)^4) - (117*a^2*d^(21/4))/(32*Sqrt[2]*c^(17/4)*(b*c - a*d)^4))*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]","A",1
504,1,104,103,0.0693376,"\int x^5 \sqrt{a+b x^2} \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^5*Sqrt[a + b*x^2]*(A + B*x^2),x]","\frac{\sqrt{a+b x^2} \left(-16 a^4 B+24 a^3 A b+8 a^3 b B x^2-12 a^2 A b^2 x^2-6 a^2 b^2 B x^4+9 a A b^3 x^4+5 a b^3 B x^6+45 A b^4 x^6+35 b^4 B x^8\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,"(Sqrt[a + b*x^2]*(24*a^3*A*b - 16*a^4*B - 12*a^2*A*b^2*x^2 + 8*a^3*b*B*x^2 + 9*a*A*b^3*x^4 - 6*a^2*b^2*B*x^4 + 45*A*b^4*x^6 + 5*a*b^3*B*x^6 + 35*b^4*B*x^8))/(315*b^4)","A",1
505,1,125,155,0.3683333,"\int x^4 \sqrt{a+b x^2} \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^4*Sqrt[a + b*x^2]*(A + B*x^2),x]","\frac{\left(8 a^3 A b-5 a^4 B\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{128 b^{7/2}}+\frac{\sqrt{a+b x^2} \left(15 a^3 B x-24 a^2 A b x-10 a^2 b B x^3+16 a A b^2 x^3+8 a b^2 B x^5+64 A b^3 x^5+48 b^3 B x^7\right)}{384 b^3}","\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]*(-24*a^2*A*b*x + 15*a^3*B*x + 16*a*A*b^2*x^3 - 10*a^2*b*B*x^3 + 64*A*b^3*x^5 + 8*a*b^2*B*x^5 + 48*b^3*B*x^7))/(384*b^3) + ((8*a^3*A*b - 5*a^4*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(128*b^(7/2))","A",1
506,1,80,73,0.0552705,"\int x^3 \sqrt{a+b x^2} \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^3*Sqrt[a + b*x^2]*(A + B*x^2),x]","\frac{\sqrt{a+b x^2} \left(8 a^3 B-14 a^2 A b-4 a^2 b B x^2+7 a A b^2 x^2+3 a b^2 B x^4+21 A b^3 x^4+15 b^3 B x^6\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,"(Sqrt[a + b*x^2]*(-14*a^2*A*b + 8*a^3*B + 7*a*A*b^2*x^2 - 4*a^2*b*B*x^2 + 21*A*b^3*x^4 + 3*a*b^2*B*x^4 + 15*b^3*B*x^6))/(105*b^3)","A",1
507,1,100,122,0.2665576,"\int x^2 \sqrt{a+b x^2} \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^2*Sqrt[a + b*x^2]*(A + B*x^2),x]","\frac{\sqrt{a+b x^2} \left(-3 a^2 B x+6 a A b x+2 a b B x^3+12 A b^2 x^3+8 b^2 B x^5\right)}{48 b^2}+\frac{\left(a^3 B-2 a^2 A b\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{16 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]*(6*a*A*b*x - 3*a^2*B*x + 12*A*b^2*x^3 + 2*a*b*B*x^3 + 8*b^2*B*x^5))/(48*b^2) + ((-2*a^2*A*b + a^3*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*b^(5/2))","A",1
508,1,55,46,0.0450074,"\int x \sqrt{a+b x^2} \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x*Sqrt[a + b*x^2]*(A + B*x^2),x]","\frac{\sqrt{a+b x^2} \left(-2 a^2 B+5 a A b+a b B x^2+5 A b^2 x^2+3 b^2 B x^4\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,"(Sqrt[a + b*x^2]*(5*a*A*b - 2*a^2*B + 5*A*b^2*x^2 + a*b*B*x^2 + 3*b^2*B*x^4))/(15*b^2)","A",1
509,1,76,87,0.2206021,"\int \sqrt{a+b x^2} \left(A+B x^2\right) \, dx","IntegrateAlgebraic[Sqrt[a + b*x^2]*(A + B*x^2),x]","\frac{\left(4 a A b-a^2 B\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 b^{3/2}}+\frac{\sqrt{a+b x^2} \left(a B x+4 A b x+2 b B x^3\right)}{8 b}","\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]*(4*A*b*x + a*B*x + 2*b*B*x^3))/(8*b) + ((4*a*A*b - a^2*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*b^(3/2))","A",1
510,1,59,59,0.0873217,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{x} \, dx","IntegrateAlgebraic[(Sqrt[a + b*x^2]*(A + B*x^2))/x,x]","\frac{\sqrt{a+b x^2} \left(a B+3 A b+b B x^2\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 + a*B + b*B*x^2))/(3*b) - Sqrt[a]*A*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]","A",1
511,1,64,84,0.1661978,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{x^2} \, dx","IntegrateAlgebraic[(Sqrt[a + b*x^2]*(A + B*x^2))/x^2,x]","\frac{\sqrt{a+b x^2} \left(B x^2-2 A\right)}{2 x}+\frac{(a B+2 A b) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 \sqrt{b}}","\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 + B*x^2))/(2*x) + ((2*A*b + a*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*Sqrt[b])","A",1
512,1,65,84,0.1047493,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{x^3} \, dx","IntegrateAlgebraic[(Sqrt[a + b*x^2]*(A + B*x^2))/x^3,x]","\frac{\sqrt{a+b x^2} \left(2 B x^2-A\right)}{2 x^2}+\frac{(-2 a B-A b) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{2 \sqrt{a}}","\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))/(2*x^2) + ((-(A*b) - 2*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(2*Sqrt[a])","A",1
513,1,67,66,0.1420895,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{x^4} \, dx","IntegrateAlgebraic[(Sqrt[a + b*x^2]*(A + B*x^2))/x^4,x]","\frac{\sqrt{a+b x^2} \left(-a A-3 a B x^2-A b x^2\right)}{3 a x^3}+\sqrt{b} B \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)","-\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))/(3*a*x^3) + Sqrt[b]*B*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]]","A",1
514,1,79,88,0.1460057,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{x^5} \, dx","IntegrateAlgebraic[(Sqrt[a + b*x^2]*(A + B*x^2))/x^5,x]","\frac{\left(A b^2-4 a b B\right) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{8 a^{3/2}}+\frac{\sqrt{a+b x^2} \left(-2 a A-4 a B x^2-A b x^2\right)}{8 a x^4}","\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,"(Sqrt[a + b*x^2]*(-2*a*A - A*b*x^2 - 4*a*B*x^2))/(8*a*x^4) + ((A*b^2 - 4*a*b*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(8*a^(3/2))","A",1
515,1,62,53,0.1001497,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{x^6} \, dx","IntegrateAlgebraic[(Sqrt[a + b*x^2]*(A + B*x^2))/x^6,x]","\frac{\sqrt{a+b x^2} \left(-3 a^2 A-5 a^2 B x^2-a A b x^2-5 a b B x^4+2 A b^2 x^4\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,"(Sqrt[a + b*x^2]*(-3*a^2*A - a*A*b*x^2 - 5*a^2*B*x^2 + 2*A*b^2*x^4 - 5*a*b*B*x^4))/(15*a^2*x^5)","A",1
516,1,104,120,0.1716703,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{x^7} \, dx","IntegrateAlgebraic[(Sqrt[a + b*x^2]*(A + B*x^2))/x^7,x]","\frac{\left(2 a b^2 B-A b^3\right) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{16 a^{5/2}}+\frac{\sqrt{a+b x^2} \left(-8 a^2 A-12 a^2 B x^2-2 a A b x^2-6 a b B x^4+3 A b^2 x^4\right)}{48 a^2 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,"(Sqrt[a + b*x^2]*(-8*a^2*A - 2*a*A*b*x^2 - 12*a^2*B*x^2 + 3*A*b^2*x^4 - 6*a*b*B*x^4))/(48*a^2*x^6) + ((-(A*b^3) + 2*a*b^2*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(16*a^(5/2))","A",1
517,1,86,84,0.1292612,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{x^8} \, dx","IntegrateAlgebraic[(Sqrt[a + b*x^2]*(A + B*x^2))/x^8,x]","\frac{\sqrt{a+b x^2} \left(-15 a^3 A-21 a^3 B x^2-3 a^2 A b x^2-7 a^2 b B x^4+4 a A b^2 x^4+14 a b^2 B x^6-8 A b^3 x^6\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,"(Sqrt[a + b*x^2]*(-15*a^3*A - 3*a^2*A*b*x^2 - 21*a^3*B*x^2 + 4*a*A*b^2*x^4 - 7*a^2*b*B*x^4 - 8*A*b^3*x^6 + 14*a*b^2*B*x^6))/(105*a^3*x^7)","A",1
518,1,128,156,0.2529648,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{x^9} \, dx","IntegrateAlgebraic[(Sqrt[a + b*x^2]*(A + B*x^2))/x^9,x]","\frac{\left(5 A b^4-8 a b^3 B\right) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{128 a^{7/2}}+\frac{\sqrt{a+b x^2} \left(-48 a^3 A-64 a^3 B x^2-8 a^2 A b x^2-16 a^2 b B x^4+10 a A b^2 x^4+24 a b^2 B x^6-15 A b^3 x^6\right)}{384 a^3 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,"(Sqrt[a + b*x^2]*(-48*a^3*A - 8*a^2*A*b*x^2 - 64*a^3*B*x^2 + 10*a*A*b^2*x^4 - 16*a^2*b*B*x^4 - 15*A*b^3*x^6 + 24*a*b^2*B*x^6))/(384*a^3*x^8) + ((5*A*b^4 - 8*a*b^3*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(128*a^(7/2))","A",1
519,1,110,117,0.1533864,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{x^{10}} \, dx","IntegrateAlgebraic[(Sqrt[a + b*x^2]*(A + B*x^2))/x^10,x]","\frac{\sqrt{a+b x^2} \left(-35 a^4 A-45 a^4 B x^2-5 a^3 A b x^2-9 a^3 b B x^4+6 a^2 A b^2 x^4+12 a^2 b^2 B x^6-8 a A b^3 x^6-24 a b^3 B x^8+16 A b^4 x^8\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,"(Sqrt[a + b*x^2]*(-35*a^4*A - 5*a^3*A*b*x^2 - 45*a^4*B*x^2 + 6*a^2*A*b^2*x^4 - 9*a^3*b*B*x^4 - 8*a*A*b^3*x^6 + 12*a^2*b^2*B*x^6 + 16*A*b^4*x^8 - 24*a*b^3*B*x^8))/(315*a^4*x^9)","A",1
520,1,152,189,0.3018955,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{x^{11}} \, dx","IntegrateAlgebraic[(Sqrt[a + b*x^2]*(A + B*x^2))/x^11,x]","\frac{\left(10 a b^4 B-7 A b^5\right) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{256 a^{9/2}}+\frac{\sqrt{a+b x^2} \left(-384 a^4 A-480 a^4 B x^2-48 a^3 A b x^2-80 a^3 b B x^4+56 a^2 A b^2 x^4+100 a^2 b^2 B x^6-70 a A b^3 x^6-150 a b^3 B x^8+105 A b^4 x^8\right)}{3840 a^4 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,"(Sqrt[a + b*x^2]*(-384*a^4*A - 48*a^3*A*b*x^2 - 480*a^4*B*x^2 + 56*a^2*A*b^2*x^4 - 80*a^3*b*B*x^4 - 70*a*A*b^3*x^6 + 100*a^2*b^2*B*x^6 + 105*A*b^4*x^8 - 150*a*b^3*B*x^8))/(3840*a^4*x^10) + ((-7*A*b^5 + 10*a*b^4*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(256*a^(9/2))","A",1
521,1,128,103,0.0724048,"\int x^5 \left(a+b x^2\right)^{3/2} \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^5*(a + b*x^2)^(3/2)*(A + B*x^2),x]","\frac{\sqrt{a+b x^2} \left(-48 a^5 B+88 a^4 A b+24 a^4 b B x^2-44 a^3 A b^2 x^2-18 a^3 b^2 B x^4+33 a^2 A b^3 x^4+15 a^2 b^3 B x^6+550 a A b^4 x^6+420 a b^4 B x^8+385 A b^5 x^8+315 b^5 B x^{10}\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,"(Sqrt[a + b*x^2]*(88*a^4*A*b - 48*a^5*B - 44*a^3*A*b^2*x^2 + 24*a^4*b*B*x^2 + 33*a^2*A*b^3*x^4 - 18*a^3*b^2*B*x^4 + 550*a*A*b^4*x^6 + 15*a^2*b^3*B*x^6 + 385*A*b^5*x^8 + 420*a*b^4*B*x^8 + 315*b^5*B*x^10))/(3465*b^4)","A",1
522,1,148,188,0.4310486,"\int x^4 \left(a+b x^2\right)^{3/2} \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^4*(a + b*x^2)^(3/2)*(A + B*x^2),x]","\frac{\sqrt{a+b x^2} \left(15 a^4 B x-30 a^3 A b x-10 a^3 b B x^3+20 a^2 A b^2 x^3+8 a^2 b^2 B x^5+240 a A b^3 x^5+176 a b^3 B x^7+160 A b^4 x^7+128 b^4 B x^9\right)}{1280 b^3}-\frac{3 \left(a^5 B-2 a^4 A b\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{256 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]*(-30*a^3*A*b*x + 15*a^4*B*x + 20*a^2*A*b^2*x^3 - 10*a^3*b*B*x^3 + 240*a*A*b^3*x^5 + 8*a^2*b^2*B*x^5 + 160*A*b^4*x^7 + 176*a*b^3*B*x^7 + 128*b^4*B*x^9))/(1280*b^3) - (3*(-2*a^4*A*b + a^5*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(256*b^(7/2))","A",1
523,1,104,73,0.058831,"\int x^3 \left(a+b x^2\right)^{3/2} \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^3*(a + b*x^2)^(3/2)*(A + B*x^2),x]","\frac{\sqrt{a+b x^2} \left(8 a^4 B-18 a^3 A b-4 a^3 b B x^2+9 a^2 A b^2 x^2+3 a^2 b^2 B x^4+72 a A b^3 x^4+50 a b^3 B x^6+45 A b^4 x^6+35 b^4 B x^8\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,"(Sqrt[a + b*x^2]*(-18*a^3*A*b + 8*a^4*B + 9*a^2*A*b^2*x^2 - 4*a^3*b*B*x^2 + 72*a*A*b^3*x^4 + 3*a^2*b^2*B*x^4 + 45*A*b^4*x^6 + 50*a*b^3*B*x^6 + 35*b^4*B*x^8))/(315*b^3)","A",1
524,1,125,155,0.3491354,"\int x^2 \left(a+b x^2\right)^{3/2} \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^2*(a + b*x^2)^(3/2)*(A + B*x^2),x]","\frac{\left(3 a^4 B-8 a^3 A b\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{128 b^{5/2}}+\frac{\sqrt{a+b x^2} \left(-9 a^3 B x+24 a^2 A b x+6 a^2 b B x^3+112 a A b^2 x^3+72 a b^2 B x^5+64 A b^3 x^5+48 b^3 B x^7\right)}{384 b^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]*(24*a^2*A*b*x - 9*a^3*B*x + 112*a*A*b^2*x^3 + 6*a^2*b*B*x^3 + 64*A*b^3*x^5 + 72*a*b^2*B*x^5 + 48*b^3*B*x^7))/(384*b^2) + ((-8*a^3*A*b + 3*a^4*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(128*b^(5/2))","A",1
525,1,79,46,0.0492866,"\int x \left(a+b x^2\right)^{3/2} \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x*(a + b*x^2)^(3/2)*(A + B*x^2),x]","\frac{\sqrt{a+b x^2} \left(-2 a^3 B+7 a^2 A b+a^2 b B x^2+14 a A b^2 x^2+8 a b^2 B x^4+7 A b^3 x^4+5 b^3 B x^6\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,"(Sqrt[a + b*x^2]*(7*a^2*A*b - 2*a^3*B + 14*a*A*b^2*x^2 + a^2*b*B*x^2 + 7*A*b^3*x^4 + 8*a*b^2*B*x^4 + 5*b^3*B*x^6))/(35*b^2)","A",1
526,1,101,118,0.2763862,"\int \left(a+b x^2\right)^{3/2} \left(A+B x^2\right) \, dx","IntegrateAlgebraic[(a + b*x^2)^(3/2)*(A + B*x^2),x]","\frac{\sqrt{a+b x^2} \left(3 a^2 B x+30 a A b x+14 a b B x^3+12 A b^2 x^3+8 b^2 B x^5\right)}{48 b}+\frac{\left(6 a^2 A b-a^3 B\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{16 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]*(30*a*A*b*x + 3*a^2*B*x + 12*A*b^2*x^3 + 14*a*b*B*x^3 + 8*b^2*B*x^5))/(48*b) + ((6*a^2*A*b - a^3*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*b^(3/2))","A",1
527,1,83,76,0.1024136,"\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{x} \, dx","IntegrateAlgebraic[((a + b*x^2)^(3/2)*(A + B*x^2))/x,x]","\frac{\sqrt{a+b x^2} \left(3 a^2 B+20 a A b+6 a b B x^2+5 A b^2 x^2+3 b^2 B x^4\right)}{15 b}-a^{3/2} A \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)","-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,"(Sqrt[a + b*x^2]*(20*a*A*b + 3*a^2*B + 5*A*b^2*x^2 + 6*a*b*B*x^2 + 3*b^2*B*x^4))/(15*b) - a^(3/2)*A*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]","A",1
528,1,84,109,0.2084134,"\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{x^2} \, dx","IntegrateAlgebraic[((a + b*x^2)^(3/2)*(A + B*x^2))/x^2,x]","\frac{3 \left(a^2 B+4 a A b\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 \sqrt{b}}+\frac{\sqrt{a+b x^2} \left(-8 a A+5 a B x^2+4 A b x^2+2 b B x^4\right)}{8 x}","\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 + 4*A*b*x^2 + 5*a*B*x^2 + 2*b*B*x^4))/(8*x) + (3*(4*a*A*b + a^2*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*Sqrt[b])","A",1
529,1,85,110,0.1563981,"\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{x^3} \, dx","IntegrateAlgebraic[((a + b*x^2)^(3/2)*(A + B*x^2))/x^3,x]","\frac{1}{2} \left(-2 a^{3/2} B-3 \sqrt{a} A b\right) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)+\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)}{6 x^2}","\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))/(6*x^2) + ((-3*Sqrt[a]*A*b - 2*a^(3/2)*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/2","A",1
530,1,86,119,0.2303287,"\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{x^4} \, dx","IntegrateAlgebraic[((a + b*x^2)^(3/2)*(A + B*x^2))/x^4,x]","\frac{1}{2} \left(3 a \sqrt{b} B+2 A b^{3/2}\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)+\frac{\sqrt{a+b x^2} \left(-2 a A-6 a B x^2-8 A b x^2+3 b B x^4\right)}{6 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,"(Sqrt[a + b*x^2]*(-2*a*A - 8*A*b*x^2 - 6*a*B*x^2 + 3*b*B*x^4))/(6*x^3) + ((2*A*b^(3/2) + 3*a*Sqrt[b]*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/2","A",1
531,1,83,115,0.1398227,"\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{x^5} \, dx","IntegrateAlgebraic[((a + b*x^2)^(3/2)*(A + B*x^2))/x^5,x]","\frac{\sqrt{a+b x^2} \left(-2 a A-4 a B x^2-5 A b x^2+8 b B x^4\right)}{8 x^4}-\frac{3 \left(4 a b B+A b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{8 \sqrt{a}}","-\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,"(Sqrt[a + b*x^2]*(-2*a*A - 5*A*b*x^2 - 4*a*B*x^2 + 8*b*B*x^4))/(8*x^4) - (3*(A*b^2 + 4*a*b*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(8*Sqrt[a])","A",1
532,1,89,86,0.1708352,"\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{x^6} \, dx","IntegrateAlgebraic[((a + b*x^2)^(3/2)*(A + B*x^2))/x^6,x]","\frac{\sqrt{a+b x^2} \left(-3 a^2 A-5 a^2 B x^2-6 a A b x^2-20 a b B x^4-3 A b^2 x^4\right)}{15 a x^5}+b^{3/2} B \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)","-\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,"(Sqrt[a + b*x^2]*(-3*a^2*A - 6*a*A*b*x^2 - 5*a^2*B*x^2 - 3*A*b^2*x^4 - 20*a*b*B*x^4))/(15*a*x^5) + b^(3/2)*B*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]]","A",1
533,1,103,120,0.198053,"\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{x^7} \, dx","IntegrateAlgebraic[((a + b*x^2)^(3/2)*(A + B*x^2))/x^7,x]","\frac{\left(A b^3-6 a b^2 B\right) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{16 a^{3/2}}+\frac{\sqrt{a+b x^2} \left(-8 a^2 A-12 a^2 B x^2-14 a A b x^2-30 a b B x^4-3 A b^2 x^4\right)}{48 a x^6}","\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,"(Sqrt[a + b*x^2]*(-8*a^2*A - 14*a*A*b*x^2 - 12*a^2*B*x^2 - 3*A*b^2*x^4 - 30*a*b*B*x^4))/(48*a*x^6) + ((A*b^3 - 6*a*b^2*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(16*a^(3/2))","A",1
534,1,86,53,0.1138284,"\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{x^8} \, dx","IntegrateAlgebraic[((a + b*x^2)^(3/2)*(A + B*x^2))/x^8,x]","\frac{\sqrt{a+b x^2} \left(-5 a^3 A-7 a^3 B x^2-8 a^2 A b x^2-14 a^2 b B x^4-a A b^2 x^4-7 a b^2 B x^6+2 A b^3 x^6\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,"(Sqrt[a + b*x^2]*(-5*a^3*A - 8*a^2*A*b*x^2 - 7*a^3*B*x^2 - a*A*b^2*x^4 - 14*a^2*b*B*x^4 + 2*A*b^3*x^6 - 7*a*b^2*B*x^6))/(35*a^2*x^7)","A",1
535,1,128,156,0.24645,"\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{x^9} \, dx","IntegrateAlgebraic[((a + b*x^2)^(3/2)*(A + B*x^2))/x^9,x]","\frac{\left(8 a b^3 B-3 A b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{128 a^{5/2}}+\frac{\sqrt{a+b x^2} \left(-48 a^3 A-64 a^3 B x^2-72 a^2 A b x^2-112 a^2 b B x^4-6 a A b^2 x^4-24 a b^2 B x^6+9 A b^3 x^6\right)}{384 a^2 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,"(Sqrt[a + b*x^2]*(-48*a^3*A - 72*a^2*A*b*x^2 - 64*a^3*B*x^2 - 6*a*A*b^2*x^4 - 112*a^2*b*B*x^4 + 9*A*b^3*x^6 - 24*a*b^2*B*x^6))/(384*a^2*x^8) + ((-3*A*b^4 + 8*a*b^3*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(128*a^(5/2))","A",1
536,1,110,84,0.1489212,"\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{x^{10}} \, dx","IntegrateAlgebraic[((a + b*x^2)^(3/2)*(A + B*x^2))/x^10,x]","\frac{\sqrt{a+b x^2} \left(-35 a^4 A-45 a^4 B x^2-50 a^3 A b x^2-72 a^3 b B x^4-3 a^2 A b^2 x^4-9 a^2 b^2 B x^6+4 a A b^3 x^6+18 a b^3 B x^8-8 A b^4 x^8\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,"(Sqrt[a + b*x^2]*(-35*a^4*A - 50*a^3*A*b*x^2 - 45*a^4*B*x^2 - 3*a^2*A*b^2*x^4 - 72*a^3*b*B*x^4 + 4*a*A*b^3*x^6 - 9*a^2*b^2*B*x^6 - 8*A*b^4*x^8 + 18*a*b^3*B*x^8))/(315*a^3*x^9)","A",1
537,1,152,184,0.2884375,"\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{x^{11}} \, dx","IntegrateAlgebraic[((a + b*x^2)^(3/2)*(A + B*x^2))/x^11,x]","\frac{\sqrt{a+b x^2} \left(-128 a^4 A-160 a^4 B x^2-176 a^3 A b x^2-240 a^3 b B x^4-8 a^2 A b^2 x^4-20 a^2 b^2 B x^6+10 a A b^3 x^6+30 a b^3 B x^8-15 A b^4 x^8\right)}{1280 a^3 x^{10}}-\frac{3 \left(2 a b^4 B-A b^5\right) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{256 a^{7/2}}","\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,"(Sqrt[a + b*x^2]*(-128*a^4*A - 176*a^3*A*b*x^2 - 160*a^4*B*x^2 - 8*a^2*A*b^2*x^4 - 240*a^3*b*B*x^4 + 10*a*A*b^3*x^6 - 20*a^2*b^2*B*x^6 - 15*A*b^4*x^8 + 30*a*b^3*B*x^8))/(1280*a^3*x^10) - (3*(-(A*b^5) + 2*a*b^4*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(256*a^(7/2))","A",1
538,1,152,103,0.0768198,"\int x^5 \left(a+b x^2\right)^{5/2} \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^5*(a + b*x^2)^(5/2)*(A + B*x^2),x]","\frac{\sqrt{a+b x^2} \left(-48 a^6 B+104 a^5 A b+24 a^5 b B x^2-52 a^4 A b^2 x^2-18 a^4 b^2 B x^4+39 a^3 A b^3 x^4+15 a^3 b^3 B x^6+1469 a^2 A b^4 x^6+1113 a^2 b^4 B x^8+2093 a A b^5 x^8+1701 a b^5 B x^{10}+819 A b^6 x^{10}+693 b^6 B x^{12}\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,"(Sqrt[a + b*x^2]*(104*a^5*A*b - 48*a^6*B - 52*a^4*A*b^2*x^2 + 24*a^5*b*B*x^2 + 39*a^3*A*b^3*x^4 - 18*a^4*b^2*B*x^4 + 1469*a^2*A*b^4*x^6 + 15*a^3*b^3*B*x^6 + 2093*a*A*b^5*x^8 + 1113*a^2*b^4*B*x^8 + 819*A*b^6*x^10 + 1701*a*b^5*B*x^10 + 693*b^6*B*x^12))/(9009*b^4)","A",1
539,1,173,221,0.5268158,"\int x^4 \left(a+b x^2\right)^{5/2} \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^4*(a + b*x^2)^(5/2)*(A + B*x^2),x]","\frac{\left(12 a^5 A b-5 a^6 B\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{1024 b^{7/2}}+\frac{\sqrt{a+b x^2} \left(75 a^5 B x-180 a^4 A b x-50 a^4 b B x^3+120 a^3 A b^2 x^3+40 a^3 b^2 B x^5+2976 a^2 A b^3 x^5+2160 a^2 b^3 B x^7+4032 a A b^4 x^7+3200 a b^4 B x^9+1536 A b^5 x^9+1280 b^5 B x^{11}\right)}{15360 b^3}","\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]*(-180*a^4*A*b*x + 75*a^5*B*x + 120*a^3*A*b^2*x^3 - 50*a^4*b*B*x^3 + 2976*a^2*A*b^3*x^5 + 40*a^3*b^2*B*x^5 + 4032*a*A*b^4*x^7 + 2160*a^2*b^3*B*x^7 + 1536*A*b^5*x^9 + 3200*a*b^4*B*x^9 + 1280*b^5*B*x^11))/(15360*b^3) + ((12*a^5*A*b - 5*a^6*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(1024*b^(7/2))","A",1
540,1,128,73,0.0662161,"\int x^3 \left(a+b x^2\right)^{5/2} \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^3*(a + b*x^2)^(5/2)*(A + B*x^2),x]","\frac{\sqrt{a+b x^2} \left(8 a^5 B-22 a^4 A b-4 a^4 b B x^2+11 a^3 A b^2 x^2+3 a^3 b^2 B x^4+165 a^2 A b^3 x^4+113 a^2 b^3 B x^6+209 a A b^4 x^6+161 a b^4 B x^8+77 A b^5 x^8+63 b^5 B x^{10}\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,"(Sqrt[a + b*x^2]*(-22*a^4*A*b + 8*a^5*B + 11*a^3*A*b^2*x^2 - 4*a^4*b*B*x^2 + 165*a^2*A*b^3*x^4 + 3*a^3*b^2*B*x^4 + 209*a*A*b^4*x^6 + 113*a^2*b^3*B*x^6 + 77*A*b^5*x^8 + 161*a*b^4*B*x^8 + 63*b^5*B*x^10))/(693*b^3)","A",1
541,1,149,188,0.4663999,"\int x^2 \left(a+b x^2\right)^{5/2} \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x^2*(a + b*x^2)^(5/2)*(A + B*x^2),x]","\frac{\left(3 a^5 B-10 a^4 A b\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{256 b^{5/2}}+\frac{\sqrt{a+b x^2} \left(-45 a^4 B x+150 a^3 A b x+30 a^3 b B x^3+1180 a^2 A b^2 x^3+744 a^2 b^2 B x^5+1360 a A b^3 x^5+1008 a b^3 B x^7+480 A b^4 x^7+384 b^4 B x^9\right)}{3840 b^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]*(150*a^3*A*b*x - 45*a^4*B*x + 1180*a^2*A*b^2*x^3 + 30*a^3*b*B*x^3 + 1360*a*A*b^3*x^5 + 744*a^2*b^2*B*x^5 + 480*A*b^4*x^7 + 1008*a*b^3*B*x^7 + 384*b^4*B*x^9))/(3840*b^2) + ((-10*a^4*A*b + 3*a^5*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(256*b^(5/2))","A",1
542,1,103,46,0.0608898,"\int x \left(a+b x^2\right)^{5/2} \left(A+B x^2\right) \, dx","IntegrateAlgebraic[x*(a + b*x^2)^(5/2)*(A + B*x^2),x]","\frac{\sqrt{a+b x^2} \left(-2 a^4 B+9 a^3 A b+a^3 b B x^2+27 a^2 A b^2 x^2+15 a^2 b^2 B x^4+27 a A b^3 x^4+19 a b^3 B x^6+9 A b^4 x^6+7 b^4 B x^8\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,"(Sqrt[a + b*x^2]*(9*a^3*A*b - 2*a^4*B + 27*a^2*A*b^2*x^2 + a^3*b*B*x^2 + 27*a*A*b^3*x^4 + 15*a^2*b^2*B*x^4 + 9*A*b^4*x^6 + 19*a*b^3*B*x^6 + 7*b^4*B*x^8))/(63*b^2)","B",1
543,1,124,149,0.4058684,"\int \left(a+b x^2\right)^{5/2} \left(A+B x^2\right) \, dx","IntegrateAlgebraic[(a + b*x^2)^(5/2)*(A + B*x^2),x]","\frac{\sqrt{a+b x^2} \left(15 a^3 B x+264 a^2 A b x+118 a^2 b B x^3+208 a A b^2 x^3+136 a b^2 B x^5+64 A b^3 x^5+48 b^3 B x^7\right)}{384 b}-\frac{5 \left(a^4 B-8 a^3 A b\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{128 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]*(264*a^2*A*b*x + 15*a^3*B*x + 208*a*A*b^2*x^3 + 118*a^2*b*B*x^3 + 64*A*b^3*x^5 + 136*a*b^2*B*x^5 + 48*b^3*B*x^7))/(384*b) - (5*(-8*a^3*A*b + a^4*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(128*b^(3/2))","A",1
544,1,107,95,0.1332705,"\int \frac{\left(a+b x^2\right)^{5/2} \left(A+B x^2\right)}{x} \, dx","IntegrateAlgebraic[((a + b*x^2)^(5/2)*(A + B*x^2))/x,x]","\frac{\sqrt{a+b x^2} \left(15 a^3 B+161 a^2 A b+45 a^2 b B x^2+77 a A b^2 x^2+45 a b^2 B x^4+21 A b^3 x^4+15 b^3 B x^6\right)}{105 b}-a^{5/2} A \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)","-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,"(Sqrt[a + b*x^2]*(161*a^2*A*b + 15*a^3*B + 77*a*A*b^2*x^2 + 45*a^2*b*B*x^2 + 21*A*b^3*x^4 + 45*a*b^2*B*x^4 + 15*b^3*B*x^6))/(105*b) - a^(5/2)*A*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]","A",1
545,1,110,136,0.292987,"\int \frac{\left(a+b x^2\right)^{5/2} \left(A+B x^2\right)}{x^2} \, dx","IntegrateAlgebraic[((a + b*x^2)^(5/2)*(A + B*x^2))/x^2,x]","\frac{\sqrt{a+b x^2} \left(-48 a^2 A+33 a^2 B x^2+54 a A b x^2+26 a b B x^4+12 A b^2 x^4+8 b^2 B x^6\right)}{48 x}+\frac{5 \left(a^3 B+6 a^2 A b\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{16 \sqrt{b}}","\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^2*A + 54*a*A*b*x^2 + 33*a^2*B*x^2 + 12*A*b^2*x^4 + 26*a*b*B*x^4 + 8*b^2*B*x^6))/(48*x) + (5*(6*a^2*A*b + a^3*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*Sqrt[b])","A",1
546,1,109,135,0.1789873,"\int \frac{\left(a+b x^2\right)^{5/2} \left(A+B x^2\right)}{x^3} \, dx","IntegrateAlgebraic[((a + b*x^2)^(5/2)*(A + B*x^2))/x^3,x]","\frac{1}{2} \left(-5 a^{3/2} A b-2 a^{5/2} B\right) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)+\frac{\sqrt{a+b x^2} \left(-15 a^2 A+46 a^2 B x^2+70 a A b x^2+22 a b B x^4+10 A b^2 x^4+6 b^2 B x^6\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{\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]*(-15*a^2*A + 70*a*A*b*x^2 + 46*a^2*B*x^2 + 10*A*b^2*x^4 + 22*a*b*B*x^4 + 6*b^2*B*x^6))/(30*x^2) + ((-5*a^(3/2)*A*b - 2*a^(5/2)*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/2","A",1
547,1,113,146,0.2573547,"\int \frac{\left(a+b x^2\right)^{5/2} \left(A+B x^2\right)}{x^4} \, dx","IntegrateAlgebraic[((a + b*x^2)^(5/2)*(A + B*x^2))/x^4,x]","\frac{5}{8} \left(3 a^2 \sqrt{b} B+4 a A b^{3/2}\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)+\frac{\sqrt{a+b x^2} \left(-8 a^2 A-24 a^2 B x^2-56 a A b x^2+27 a b B x^4+12 A b^2 x^4+6 b^2 B x^6\right)}{24 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,"(Sqrt[a + b*x^2]*(-8*a^2*A - 56*a*A*b*x^2 - 24*a^2*B*x^2 + 12*A*b^2*x^4 + 27*a*b*B*x^4 + 6*b^2*B*x^6))/(24*x^3) + (5*(4*a*A*b^(3/2) + 3*a^2*Sqrt[b]*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/8","A",1
548,1,112,143,0.1885916,"\int \frac{\left(a+b x^2\right)^{5/2} \left(A+B x^2\right)}{x^5} \, dx","IntegrateAlgebraic[((a + b*x^2)^(5/2)*(A + B*x^2))/x^5,x]","\frac{\sqrt{a+b x^2} \left(-6 a^2 A-12 a^2 B x^2-27 a A b x^2+56 a b B x^4+24 A b^2 x^4+8 b^2 B x^6\right)}{24 x^4}-\frac{5}{8} \left(4 a^{3/2} b B+3 \sqrt{a} A b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)","-\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,"(Sqrt[a + b*x^2]*(-6*a^2*A - 27*a*A*b*x^2 - 12*a^2*B*x^2 + 24*A*b^2*x^4 + 56*a*b*B*x^4 + 8*b^2*B*x^6))/(24*x^4) - (5*(3*Sqrt[a]*A*b^2 + 4*a^(3/2)*b*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/8","A",1
549,1,110,152,0.3125115,"\int \frac{\left(a+b x^2\right)^{5/2} \left(A+B x^2\right)}{x^6} \, dx","IntegrateAlgebraic[((a + b*x^2)^(5/2)*(A + B*x^2))/x^6,x]","\frac{\sqrt{a+b x^2} \left(-6 a^2 A-10 a^2 B x^2-22 a A b x^2-70 a b B x^4-46 A b^2 x^4+15 b^2 B x^6\right)}{30 x^5}+\frac{1}{2} \left(5 a b^{3/2} B+2 A b^{5/2}\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)","\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,"(Sqrt[a + b*x^2]*(-6*a^2*A - 22*a*A*b*x^2 - 10*a^2*B*x^2 - 46*A*b^2*x^4 - 70*a*b*B*x^4 + 15*b^2*B*x^6))/(30*x^5) + ((2*A*b^(5/2) + 5*a*b^(3/2)*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/2","A",1
550,1,109,149,0.2151171,"\int \frac{\left(a+b x^2\right)^{5/2} \left(A+B x^2\right)}{x^7} \, dx","IntegrateAlgebraic[((a + b*x^2)^(5/2)*(A + B*x^2))/x^7,x]","\frac{\sqrt{a+b x^2} \left(-8 a^2 A-12 a^2 B x^2-26 a A b x^2-54 a b B x^4-33 A b^2 x^4+48 b^2 B x^6\right)}{48 x^6}-\frac{5 \left(6 a b^2 B+A b^3\right) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{16 \sqrt{a}}","\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,"(Sqrt[a + b*x^2]*(-8*a^2*A - 26*a*A*b*x^2 - 12*a^2*B*x^2 - 33*A*b^2*x^4 - 54*a*b*B*x^4 + 48*b^2*B*x^6))/(48*x^6) - (5*(A*b^3 + 6*a*b^2*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(16*Sqrt[a])","A",1
551,1,113,108,0.2238649,"\int \frac{\left(a+b x^2\right)^{5/2} \left(A+B x^2\right)}{x^8} \, dx","IntegrateAlgebraic[((a + b*x^2)^(5/2)*(A + B*x^2))/x^8,x]","\frac{\sqrt{a+b x^2} \left(-15 a^3 A-21 a^3 B x^2-45 a^2 A b x^2-77 a^2 b B x^4-45 a A b^2 x^4-161 a b^2 B x^6-15 A b^3 x^6\right)}{105 a x^7}+b^{5/2} B \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)","-\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,"(Sqrt[a + b*x^2]*(-15*a^3*A - 45*a^2*A*b*x^2 - 21*a^3*B*x^2 - 45*a*A*b^2*x^4 - 77*a^2*b*B*x^4 - 15*A*b^3*x^6 - 161*a*b^2*B*x^6))/(105*a*x^7) + b^(5/2)*B*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]]","A",1
552,1,128,152,0.3040013,"\int \frac{\left(a+b x^2\right)^{5/2} \left(A+B x^2\right)}{x^9} \, dx","IntegrateAlgebraic[((a + b*x^2)^(5/2)*(A + B*x^2))/x^9,x]","\frac{\sqrt{a+b x^2} \left(-48 a^3 A-64 a^3 B x^2-136 a^2 A b x^2-208 a^2 b B x^4-118 a A b^2 x^4-264 a b^2 B x^6-15 A b^3 x^6\right)}{384 a x^8}-\frac{5 \left(8 a b^3 B-A b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{128 a^{3/2}}","\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,"(Sqrt[a + b*x^2]*(-48*a^3*A - 136*a^2*A*b*x^2 - 64*a^3*B*x^2 - 118*a*A*b^2*x^4 - 208*a^2*b*B*x^4 - 15*A*b^3*x^6 - 264*a*b^2*B*x^6))/(384*a*x^8) - (5*(-(A*b^4) + 8*a*b^3*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(128*a^(3/2))","A",1
553,1,110,53,0.1429795,"\int \frac{\left(a+b x^2\right)^{5/2} \left(A+B x^2\right)}{x^{10}} \, dx","IntegrateAlgebraic[((a + b*x^2)^(5/2)*(A + B*x^2))/x^10,x]","\frac{\sqrt{a+b x^2} \left(-7 a^4 A-9 a^4 B x^2-19 a^3 A b x^2-27 a^3 b B x^4-15 a^2 A b^2 x^4-27 a^2 b^2 B x^6-a A b^3 x^6-9 a b^3 B x^8+2 A b^4 x^8\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,"(Sqrt[a + b*x^2]*(-7*a^4*A - 19*a^3*A*b*x^2 - 9*a^4*B*x^2 - 15*a^2*A*b^2*x^4 - 27*a^3*b*B*x^4 - a*A*b^3*x^6 - 27*a^2*b^2*B*x^6 + 2*A*b^4*x^8 - 9*a*b^3*B*x^8))/(63*a^2*x^9)","B",1
554,1,152,189,0.341657,"\int \frac{\left(a+b x^2\right)^{5/2} \left(A+B x^2\right)}{x^{11}} \, dx","IntegrateAlgebraic[((a + b*x^2)^(5/2)*(A + B*x^2))/x^11,x]","\frac{\left(10 a b^4 B-3 A b^5\right) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{256 a^{5/2}}+\frac{\sqrt{a+b x^2} \left(-384 a^4 A-480 a^4 B x^2-1008 a^3 A b x^2-1360 a^3 b B x^4-744 a^2 A b^2 x^4-1180 a^2 b^2 B x^6-30 a A b^3 x^6-150 a b^3 B x^8+45 A b^4 x^8\right)}{3840 a^2 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,"(Sqrt[a + b*x^2]*(-384*a^4*A - 1008*a^3*A*b*x^2 - 480*a^4*B*x^2 - 744*a^2*A*b^2*x^4 - 1360*a^3*b*B*x^4 - 30*a*A*b^3*x^6 - 1180*a^2*b^2*B*x^6 + 45*A*b^4*x^8 - 150*a*b^3*B*x^8))/(3840*a^2*x^10) + ((-3*A*b^5 + 10*a*b^4*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(256*a^(5/2))","A",1
555,1,80,100,0.0692848,"\int \frac{x^5 \left(A+B x^2\right)}{\sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(x^5*(A + B*x^2))/Sqrt[a + b*x^2],x]","\frac{\sqrt{a+b x^2} \left(-48 a^3 B+56 a^2 A b+24 a^2 b B x^2-28 a A b^2 x^2-18 a b^2 B x^4+21 A b^3 x^4+15 b^3 B x^6\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]*(56*a^2*A*b - 48*a^3*B - 28*a*A*b^2*x^2 + 24*a^2*b*B*x^2 + 21*A*b^3*x^4 - 18*a*b^2*B*x^4 + 15*b^3*B*x^6))/(105*b^4)","A",1
556,1,101,122,0.3408103,"\int \frac{x^4 \left(A+B x^2\right)}{\sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(x^4*(A + B*x^2))/Sqrt[a + b*x^2],x]","\frac{\sqrt{a+b x^2} \left(15 a^2 B x-18 a A b x-10 a b B x^3+12 A b^2 x^3+8 b^2 B x^5\right)}{48 b^3}+\frac{\left(6 a^2 A b-5 a^3 B\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{16 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[a + b*x^2]*(-18*a*A*b*x + 15*a^2*B*x + 12*A*b^2*x^3 - 10*a*b*B*x^3 + 8*b^2*B*x^5))/(48*b^3) + ((6*a^2*A*b - 5*a^3*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*b^(7/2))","A",1
557,1,56,71,0.0556548,"\int \frac{x^3 \left(A+B x^2\right)}{\sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(x^3*(A + B*x^2))/Sqrt[a + b*x^2],x]","\frac{\sqrt{a+b x^2} \left(8 a^2 B-10 a A b-4 a b B x^2+5 A b^2 x^2+3 b^2 B x^4\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]*(-10*a*A*b + 8*a^2*B + 5*A*b^2*x^2 - 4*a*b*B*x^2 + 3*b^2*B*x^4))/(15*b^3)","A",1
558,1,77,89,0.238702,"\int \frac{x^2 \left(A+B x^2\right)}{\sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(x^2*(A + B*x^2))/Sqrt[a + b*x^2],x]","\frac{\left(3 a^2 B-4 a A b\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 b^{5/2}}+\frac{\sqrt{a+b x^2} \left(-3 a B x+4 A b x+2 b B x^3\right)}{8 b^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[a + b*x^2]*(4*A*b*x - 3*a*B*x + 2*b*B*x^3))/(8*b^2) + ((-4*a*A*b + 3*a^2*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*b^(5/2))","A",1
559,1,33,43,0.0415294,"\int \frac{x \left(A+B x^2\right)}{\sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(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
560,1,58,58,0.2026321,"\int \frac{A+B x^2}{\sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(A + B*x^2)/Sqrt[a + b*x^2],x]","\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}","\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
561,1,43,43,0.0685824,"\int \frac{A+B x^2}{x \sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(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
562,1,47,47,0.1416025,"\int \frac{A+B x^2}{x^2 \sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(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
563,1,58,58,0.1067115,"\int \frac{A+B x^2}{x^3 \sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^3*Sqrt[a + b*x^2]),x]","\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}","\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,"-1/2*(A*Sqrt[a + b*x^2])/(a*x^2) + ((A*b - 2*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(2*a^(3/2))","A",1
564,1,40,53,0.1045747,"\int \frac{A+B x^2}{x^4 \sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^4*Sqrt[a + b*x^2]),x]","\frac{\sqrt{a+b x^2} \left(-a A-3 a B x^2+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,"(Sqrt[a + b*x^2]*(-(a*A) + 2*A*b*x^2 - 3*a*B*x^2))/(3*a^2*x^3)","A",1
565,1,80,90,0.1890642,"\int \frac{A+B x^2}{x^5 \sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^5*Sqrt[a + b*x^2]),x]","\frac{\left(4 a b B-3 A b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{8 a^{5/2}}+\frac{\sqrt{a+b x^2} \left(-2 a A-4 a B x^2+3 A b x^2\right)}{8 a^2 x^4}","-\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]*(-2*a*A + 3*A*b*x^2 - 4*a*B*x^2))/(8*a^2*x^4) + ((-3*A*b^2 + 4*a*b*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(8*a^(5/2))","A",1
566,1,62,84,0.1274696,"\int \frac{A+B x^2}{x^6 \sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^6*Sqrt[a + b*x^2]),x]","\frac{\sqrt{a+b x^2} \left(-3 a^2 A-5 a^2 B x^2+4 a A b x^2+10 a b B x^4-8 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,"(Sqrt[a + b*x^2]*(-3*a^2*A + 4*a*A*b*x^2 - 5*a^2*B*x^2 - 8*A*b^2*x^4 + 10*a*b*B*x^4))/(15*a^3*x^5)","A",1
567,1,104,123,0.2287444,"\int \frac{A+B x^2}{x^7 \sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^7*Sqrt[a + b*x^2]),x]","\frac{\left(5 A b^3-6 a b^2 B\right) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{16 a^{7/2}}+\frac{\sqrt{a+b x^2} \left(-8 a^2 A-12 a^2 B x^2+10 a A b x^2+18 a b B x^4-15 A b^2 x^4\right)}{48 a^3 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,"(Sqrt[a + b*x^2]*(-8*a^2*A + 10*a*A*b*x^2 - 12*a^2*B*x^2 - 15*A*b^2*x^4 + 18*a*b*B*x^4))/(48*a^3*x^6) + ((5*A*b^3 - 6*a*b^2*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(16*a^(7/2))","A",1
568,1,86,117,0.1776709,"\int \frac{A+B x^2}{x^8 \sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^8*Sqrt[a + b*x^2]),x]","\frac{\sqrt{a+b x^2} \left(-15 a^3 A-21 a^3 B x^2+18 a^2 A b x^2+28 a^2 b B x^4-24 a A b^2 x^4-56 a b^2 B x^6+48 A b^3 x^6\right)}{105 a^4 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,"(Sqrt[a + b*x^2]*(-15*a^3*A + 18*a^2*A*b*x^2 - 21*a^3*B*x^2 - 24*a*A*b^2*x^4 + 28*a^2*b*B*x^4 + 48*A*b^3*x^6 - 56*a*b^2*B*x^6))/(105*a^4*x^7)","A",1
569,1,125,152,0.4503128,"\int \frac{x^6 \left(A+B x^2\right)}{\left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(x^6*(A + B*x^2))/(a + b*x^2)^(3/2),x]","\frac{105 a^3 B x-90 a^2 A b x+35 a^2 b B x^3-30 a A b^2 x^3-14 a b^2 B x^5+12 A b^3 x^5+8 b^3 B x^7}{48 b^4 \sqrt{a+b x^2}}-\frac{5 \left(7 a^3 B-6 a^2 A b\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{16 b^{9/2}}","\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,"(-90*a^2*A*b*x + 105*a^3*B*x - 30*a*A*b^2*x^3 + 35*a^2*b*B*x^3 + 12*A*b^3*x^5 - 14*a*b^2*B*x^5 + 8*b^3*B*x^7)/(48*b^4*Sqrt[a + b*x^2]) - (5*(-6*a^2*A*b + 7*a^3*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*b^(9/2))","A",1
570,1,80,99,0.0867782,"\int \frac{x^5 \left(A+B x^2\right)}{\left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(x^5*(A + B*x^2))/(a + b*x^2)^(3/2),x]","\frac{48 a^3 B-40 a^2 A b+24 a^2 b B x^2-20 a A b^2 x^2-6 a b^2 B x^4+5 A b^3 x^4+3 b^3 B x^6}{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,"(-40*a^2*A*b + 48*a^3*B - 20*a*A*b^2*x^2 + 24*a^2*b*B*x^2 + 5*A*b^3*x^4 - 6*a*b^2*B*x^4 + 3*b^3*B*x^6)/(15*b^4*Sqrt[a + b*x^2])","A",1
571,1,99,119,0.318338,"\int \frac{x^4 \left(A+B x^2\right)}{\left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(x^4*(A + B*x^2))/(a + b*x^2)^(3/2),x]","\frac{3 \left(5 a^2 B-4 a A b\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 b^{7/2}}+\frac{-15 a^2 B x+12 a A b x-5 a b B x^3+4 A b^2 x^3+2 b^2 B x^5}{8 b^3 \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,"(12*a*A*b*x - 15*a^2*B*x + 4*A*b^2*x^3 - 5*a*b*B*x^3 + 2*b^2*B*x^5)/(8*b^3*Sqrt[a + b*x^2]) + (3*(-4*a*A*b + 5*a^2*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*b^(7/2))","A",1
572,1,55,67,0.0646219,"\int \frac{x^3 \left(A+B x^2\right)}{\left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(x^3*(A + B*x^2))/(a + b*x^2)^(3/2),x]","\frac{-8 a^2 B+6 a A b-4 a b B x^2+3 A b^2 x^2+b^2 B x^4}{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,"(6*a*A*b - 8*a^2*B + 3*A*b^2*x^2 - 4*a*b*B*x^2 + b^2*B*x^4)/(3*b^3*Sqrt[a + b*x^2])","A",1
573,1,73,83,0.2659874,"\int \frac{x^2 \left(A+B x^2\right)}{\left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(x^2*(A + B*x^2))/(a + b*x^2)^(3/2),x]","\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{3 a B x-2 A b x+b B x^3}{2 b^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,"(-2*A*b*x + 3*a*B*x + b*B*x^3)/(2*b^2*Sqrt[a + b*x^2]) + ((2*A*b - 3*a*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(5/2))","A",1
574,1,30,41,0.0456279,"\int \frac{x \left(A+B x^2\right)}{\left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(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
575,1,55,54,0.168433,"\int \frac{A+B x^2}{\left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(A + B*x^2)/(a + b*x^2)^(3/2),x]","\frac{B \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{b^{3/2}}-\frac{x (a B-A b)}{a b \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,"-(((-(A*b) + a*B)*x)/(a*b*Sqrt[a + b*x^2])) + (B*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/b^(3/2)","A",1
576,1,53,53,0.0898881,"\int \frac{A+B x^2}{x \left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(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
577,1,36,47,0.1145452,"\int \frac{A+B x^2}{x^2 \left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(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
578,1,77,86,0.1576507,"\int \frac{A+B x^2}{x^3 \left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^3*(a + b*x^2)^(3/2)),x]","\frac{(3 A b-2 a B) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{2 a^{5/2}}+\frac{-a A+2 a B x^2-3 A b x^2}{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*x^2 + 2*a*B*x^2)/(2*a^2*x^2*Sqrt[a + b*x^2]) + ((3*A*b - 2*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(2*a^(5/2))","A",1
579,1,62,82,0.1527215,"\int \frac{A+B x^2}{x^4 \left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^4*(a + b*x^2)^(3/2)),x]","\frac{-a^2 A-3 a^2 B x^2+4 a A b x^2-6 a b B x^4+8 A b^2 x^4}{3 a^3 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,"(-(a^2*A) + 4*a*A*b*x^2 - 3*a^2*B*x^2 + 8*A*b^2*x^4 - 6*a*b*B*x^4)/(3*a^3*x^3*Sqrt[a + b*x^2])","A",1
580,1,102,118,0.1923515,"\int \frac{A+B x^2}{x^5 \left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^5*(a + b*x^2)^(3/2)),x]","\frac{3 \left(4 a b B-5 A b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{8 a^{7/2}}+\frac{-2 a^2 A-4 a^2 B x^2+5 a A b x^2-12 a b B x^4+15 A b^2 x^4}{8 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,"(-2*a^2*A + 5*a*A*b*x^2 - 4*a^2*B*x^2 + 15*A*b^2*x^4 - 12*a*b*B*x^4)/(8*a^3*x^4*Sqrt[a + b*x^2]) + (3*(-5*A*b^2 + 4*a*b*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(8*a^(7/2))","A",1
581,1,86,115,0.1999138,"\int \frac{A+B x^2}{x^6 \left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^6*(a + b*x^2)^(3/2)),x]","\frac{-3 a^3 A-5 a^3 B x^2+6 a^2 A b x^2+20 a^2 b B x^4-24 a A b^2 x^4+40 a b^2 B x^6-48 A b^3 x^6}{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^2*A*b*x^2 - 5*a^3*B*x^2 - 24*a*A*b^2*x^4 + 20*a^2*b*B*x^4 - 48*A*b^3*x^6 + 40*a*b^2*B*x^6)/(15*a^4*x^5*Sqrt[a + b*x^2])","A",1
582,1,128,153,0.2748789,"\int \frac{A+B x^2}{x^7 \left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^7*(a + b*x^2)^(3/2)),x]","\frac{-8 a^3 A-12 a^3 B x^2+14 a^2 A b x^2+30 a^2 b B x^4-35 a A b^2 x^4+90 a b^2 B x^6-105 A b^3 x^6}{48 a^4 x^6 \sqrt{a+b x^2}}-\frac{5 \left(6 a b^2 B-7 A b^3\right) \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) \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,"(-8*a^3*A + 14*a^2*A*b*x^2 - 12*a^3*B*x^2 - 35*a*A*b^2*x^4 + 30*a^2*b*B*x^4 - 105*A*b^3*x^6 + 90*a*b^2*B*x^6)/(48*a^4*x^6*Sqrt[a + b*x^2]) - (5*(-7*A*b^3 + 6*a*b^2*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(16*a^(9/2))","A",1
583,1,110,148,0.2501904,"\int \frac{A+B x^2}{x^8 \left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^8*(a + b*x^2)^(3/2)),x]","\frac{-5 a^4 A-7 a^4 B x^2+8 a^3 A b x^2+14 a^3 b B x^4-16 a^2 A b^2 x^4-56 a^2 b^2 B x^6+64 a A b^3 x^6-112 a b^3 B x^8+128 A b^4 x^8}{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^3*A*b*x^2 - 7*a^4*B*x^2 - 16*a^2*A*b^2*x^4 + 14*a^3*b*B*x^4 + 64*a*A*b^3*x^6 - 56*a^2*b^2*B*x^6 + 128*A*b^4*x^8 - 112*a*b^3*B*x^8)/(35*a^5*x^7*Sqrt[a + b*x^2])","A",1
584,1,104,128,0.1061893,"\int \frac{x^7 \left(A+B x^2\right)}{\left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(x^7*(A + B*x^2))/(a + b*x^2)^(5/2),x]","\frac{128 a^4 B-80 a^3 A b+192 a^3 b B x^2-120 a^2 A b^2 x^2+48 a^2 b^2 B x^4-30 a A b^3 x^4-8 a b^3 B x^6+5 A b^4 x^6+3 b^4 B x^8}{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,"(-80*a^3*A*b + 128*a^4*B - 120*a^2*A*b^2*x^2 + 192*a^3*b*B*x^2 - 30*a*A*b^3*x^4 + 48*a^2*b^2*B*x^4 + 5*A*b^4*x^6 - 8*a*b^3*B*x^6 + 3*b^4*B*x^8)/(15*b^5*(a + b*x^2)^(3/2))","A",1
585,1,123,149,0.4418251,"\int \frac{x^6 \left(A+B x^2\right)}{\left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(x^6*(A + B*x^2))/(a + b*x^2)^(5/2),x]","\frac{5 \left(7 a^2 B-4 a A b\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 b^{9/2}}+\frac{-105 a^3 B x+60 a^2 A b x-140 a^2 b B x^3+80 a A b^2 x^3-21 a b^2 B x^5+12 A b^3 x^5+6 b^3 B x^7}{24 b^4 \left(a+b x^2\right)^{3/2}}","-\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,"(60*a^2*A*b*x - 105*a^3*B*x + 80*a*A*b^2*x^3 - 140*a^2*b*B*x^3 + 12*A*b^3*x^5 - 21*a*b^2*B*x^5 + 6*b^3*B*x^7)/(24*b^4*(a + b*x^2)^(3/2)) + (5*(-4*a*A*b + 7*a^2*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*b^(9/2))","A",1
586,1,79,97,0.0812749,"\int \frac{x^5 \left(A+B x^2\right)}{\left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(x^5*(A + B*x^2))/(a + b*x^2)^(5/2),x]","\frac{-16 a^3 B+8 a^2 A b-24 a^2 b B x^2+12 a A b^2 x^2-6 a b^2 B x^4+3 A b^3 x^4+b^3 B x^6}{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,"(8*a^2*A*b - 16*a^3*B + 12*a*A*b^2*x^2 - 24*a^2*b*B*x^2 + 3*A*b^3*x^4 - 6*a*b^2*B*x^4 + b^3*B*x^6)/(3*b^4*(a + b*x^2)^(3/2))","A",1
587,1,96,114,0.3229356,"\int \frac{x^4 \left(A+B x^2\right)}{\left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(x^4*(A + B*x^2))/(a + b*x^2)^(5/2),x]","\frac{15 a^2 B x-6 a A b x+20 a b B x^3-8 A b^2 x^3+3 b^2 B x^5}{6 b^3 \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{(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,"(-6*a*A*b*x + 15*a^2*B*x - 8*A*b^2*x^3 + 20*a*b*B*x^3 + 3*b^2*B*x^5)/(6*b^3*(a + b*x^2)^(3/2)) + ((2*A*b - 5*a*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(7/2))","A",1
588,1,56,68,0.0676168,"\int \frac{x^3 \left(A+B x^2\right)}{\left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(x^3*(A + B*x^2))/(a + b*x^2)^(5/2),x]","\frac{8 a^2 B-2 a A b+12 a b B x^2-3 A b^2 x^2+3 b^2 B x^4}{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,"(-2*a*A*b + 8*a^2*B - 3*A*b^2*x^2 + 12*a*b*B*x^2 + 3*b^2*B*x^4)/(3*b^3*(a + b*x^2)^(3/2))","A",1
589,1,72,77,0.2347158,"\int \frac{x^2 \left(A+B x^2\right)}{\left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(x^2*(A + B*x^2))/(a + b*x^2)^(5/2),x]","\frac{-3 a^2 B x-4 a b B x^3+A b^2 x^3}{3 a b^2 \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{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,"(-3*a^2*B*x + A*b^2*x^3 - 4*a*b*B*x^3)/(3*a*b^2*(a + b*x^2)^(3/2)) + (B*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/b^(5/2)","A",1
590,1,34,44,0.0471665,"\int \frac{x \left(A+B x^2\right)}{\left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(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
591,1,37,47,0.1201677,"\int \frac{A+B x^2}{\left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(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
592,1,69,72,0.1206737,"\int \frac{A+B x^2}{x \left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x*(a + b*x^2)^(5/2)),x]","\frac{a^2 (-B)+4 a A b+3 A b^2 x^2}{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 \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,"(4*a*A*b - a^2*B + 3*A*b^2*x^2)/(3*a^2*b*(a + b*x^2)^(3/2)) - (A*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/a^(5/2)","A",1
593,1,62,77,0.1550658,"\int \frac{A+B x^2}{x^2 \left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^2*(a + b*x^2)^(5/2)),x]","\frac{-3 a^2 A+3 a^2 B x^2-12 a A b x^2+2 a b B x^4-8 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,"(-3*a^2*A - 12*a*A*b*x^2 + 3*a^2*B*x^2 - 8*A*b^2*x^4 + 2*a*b*B*x^4)/(3*a^3*x*(a + b*x^2)^(3/2))","A",1
594,1,99,113,0.197178,"\int \frac{A+B x^2}{x^3 \left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^3*(a + b*x^2)^(5/2)),x]","\frac{(5 A b-2 a B) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{2 a^{7/2}}+\frac{-3 a^2 A+8 a^2 B x^2-20 a A b x^2+6 a b B x^4-15 A b^2 x^4}{6 a^3 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^2*A - 20*a*A*b*x^2 + 8*a^2*B*x^2 - 15*A*b^2*x^4 + 6*a*b*B*x^4)/(6*a^3*x^2*(a + b*x^2)^(3/2)) + ((5*A*b - 2*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(2*a^(7/2))","A",1
595,1,86,108,0.2061235,"\int \frac{A+B x^2}{x^4 \left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^4*(a + b*x^2)^(5/2)),x]","\frac{-a^3 A-3 a^3 B x^2+6 a^2 A b x^2-12 a^2 b B x^4+24 a A b^2 x^4-8 a b^2 B x^6+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,"(-(a^3*A) + 6*a^2*A*b*x^2 - 3*a^3*B*x^2 + 24*a*A*b^2*x^4 - 12*a^2*b*B*x^4 + 16*A*b^3*x^6 - 8*a*b^2*B*x^6)/(3*a^4*x^3*(a + b*x^2)^(3/2))","A",1
596,1,126,146,0.3032006,"\int \frac{A+B x^2}{x^5 \left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^5*(a + b*x^2)^(5/2)),x]","\frac{5 \left(4 a b B-7 A b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{8 a^{9/2}}+\frac{-6 a^3 A-12 a^3 B x^2+21 a^2 A b x^2-80 a^2 b B x^4+140 a A b^2 x^4-60 a b^2 B x^6+105 A b^3 x^6}{24 a^4 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,"(-6*a^3*A + 21*a^2*A*b*x^2 - 12*a^3*B*x^2 + 140*a*A*b^2*x^4 - 80*a^2*b*B*x^4 + 105*A*b^3*x^6 - 60*a*b^2*B*x^6)/(24*a^4*x^4*(a + b*x^2)^(3/2)) + (5*(-7*A*b^2 + 4*a*b*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(8*a^(9/2))","A",1
597,1,110,146,0.3105902,"\int \frac{A+B x^2}{x^6 \left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(A + B*x^2)/(x^6*(a + b*x^2)^(5/2)),x]","\frac{-3 a^4 A-5 a^4 B x^2+8 a^3 A b x^2+30 a^3 b B x^4-48 a^2 A b^2 x^4+120 a^2 b^2 B x^6-192 a A b^3 x^6+80 a b^3 B x^8-128 A b^4 x^8}{15 a^5 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^4*A + 8*a^3*A*b*x^2 - 5*a^4*B*x^2 - 48*a^2*A*b^2*x^4 + 30*a^3*b*B*x^4 - 192*a*A*b^3*x^6 + 120*a^2*b^2*B*x^6 - 128*A*b^4*x^8 + 80*a*b^3*B*x^8)/(15*a^5*x^5*(a + b*x^2)^(3/2))","A",1
598,1,193,157,0.1013944,"\int x^5 \left(a+b x^2\right)^2 \sqrt{c+d x^2} \, dx","IntegrateAlgebraic[x^5*(a + b*x^2)^2*Sqrt[c + d*x^2],x]","\frac{\sqrt{c+d x^2} \left(264 a^2 c^3 d^2-132 a^2 c^2 d^3 x^2+99 a^2 c d^4 x^4+495 a^2 d^5 x^6-352 a b c^4 d+176 a b c^3 d^2 x^2-132 a b c^2 d^3 x^4+110 a b c d^4 x^6+770 a b d^5 x^8+128 b^2 c^5-64 b^2 c^4 d x^2+48 b^2 c^3 d^2 x^4-40 b^2 c^2 d^3 x^6+35 b^2 c d^4 x^8+315 b^2 d^5 x^{10}\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,"(Sqrt[c + d*x^2]*(128*b^2*c^5 - 352*a*b*c^4*d + 264*a^2*c^3*d^2 - 64*b^2*c^4*d*x^2 + 176*a*b*c^3*d^2*x^2 - 132*a^2*c^2*d^3*x^2 + 48*b^2*c^3*d^2*x^4 - 132*a*b*c^2*d^3*x^4 + 99*a^2*c*d^4*x^4 - 40*b^2*c^2*d^3*x^6 + 110*a*b*c*d^4*x^6 + 495*a^2*d^5*x^6 + 35*b^2*c*d^4*x^8 + 770*a*b*d^5*x^8 + 315*b^2*d^5*x^10))/(3465*d^5)","A",1
599,1,152,114,0.0824957,"\int x^3 \left(a+b x^2\right)^2 \sqrt{c+d x^2} \, dx","IntegrateAlgebraic[x^3*(a + b*x^2)^2*Sqrt[c + d*x^2],x]","\frac{\sqrt{c+d x^2} \left(-42 a^2 c^2 d^2+21 a^2 c d^3 x^2+63 a^2 d^4 x^4+48 a b c^3 d-24 a b c^2 d^2 x^2+18 a b c d^3 x^4+90 a b d^4 x^6-16 b^2 c^4+8 b^2 c^3 d x^2-6 b^2 c^2 d^2 x^4+5 b^2 c d^3 x^6+35 b^2 d^4 x^8\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,"(Sqrt[c + d*x^2]*(-16*b^2*c^4 + 48*a*b*c^3*d - 42*a^2*c^2*d^2 + 8*b^2*c^3*d*x^2 - 24*a*b*c^2*d^2*x^2 + 21*a^2*c*d^3*x^2 - 6*b^2*c^2*d^2*x^4 + 18*a*b*c*d^3*x^4 + 63*a^2*d^4*x^4 + 5*b^2*c*d^3*x^6 + 90*a*b*d^4*x^6 + 35*b^2*d^4*x^8))/(315*d^4)","A",1
600,1,111,77,0.0661118,"\int x \left(a+b x^2\right)^2 \sqrt{c+d x^2} \, dx","IntegrateAlgebraic[x*(a + b*x^2)^2*Sqrt[c + d*x^2],x]","\frac{\sqrt{c+d x^2} \left(35 a^2 c d^2+35 a^2 d^3 x^2-28 a b c^2 d+14 a b c d^2 x^2+42 a b d^3 x^4+8 b^2 c^3-4 b^2 c^2 d x^2+3 b^2 c d^2 x^4+15 b^2 d^3 x^6\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,"(Sqrt[c + d*x^2]*(8*b^2*c^3 - 28*a*b*c^2*d + 35*a^2*c*d^2 - 4*b^2*c^2*d*x^2 + 14*a*b*c*d^2*x^2 + 35*a^2*d^3*x^2 + 3*b^2*c*d^2*x^4 + 42*a*b*d^3*x^4 + 15*b^2*d^3*x^6))/(105*d^3)","A",1
601,1,100,92,0.1313365,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{x} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*Sqrt[c + d*x^2])/x,x]","\frac{\sqrt{c+d x^2} \left(15 a^2 d^2+10 a b c d+10 a b d^2 x^2-2 b^2 c^2+b^2 c d x^2+3 b^2 d^2 x^4\right)}{15 d^2}-a^2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)","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,"(Sqrt[c + d*x^2]*(-2*b^2*c^2 + 10*a*b*c*d + 15*a^2*d^2 + b^2*c*d*x^2 + 10*a*b*d^2*x^2 + 3*b^2*d^2*x^4))/(15*d^2) - a^2*Sqrt[c]*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]]","A",1
602,1,94,109,0.1565966,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{x^3} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^3,x]","\frac{\sqrt{c+d x^2} \left(-3 a^2 d+12 a b d x^2+2 b^2 c x^2+2 b^2 d x^4\right)}{6 d x^2}+\frac{\left(a^2 (-d)-4 a b c\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{2 \sqrt{c}}","-\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 + 2*b^2*c*x^2 + 12*a*b*d*x^2 + 2*b^2*d*x^4))/(6*d*x^2) + ((-4*a*b*c - a^2*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*Sqrt[c])","A",1
603,1,104,143,0.2240204,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{x^5} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^5,x]","\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{\sqrt{c+d x^2} \left(-2 a^2 c-a^2 d x^2-8 a b c x^2+8 b^2 c x^4\right)}{8 c x^4}","-\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]*(-2*a^2*c - 8*a*b*c*x^2 - a^2*d*x^2 + 8*b^2*c*x^4))/(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
604,1,135,149,0.2581516,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{x^7} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^7,x]","\frac{\sqrt{c+d x^2} \left(-8 a^2 c^2-2 a^2 c d x^2+3 a^2 d^2 x^4-24 a b c^2 x^2-12 a b c d x^4-24 b^2 c^2 x^4\right)}{48 c^2 x^6}+\frac{\left(-a^2 d^3+4 a b c d^2-8 b^2 c^2 d\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{16 c^{5/2}}","-\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,"(Sqrt[c + d*x^2]*(-8*a^2*c^2 - 24*a*b*c^2*x^2 - 2*a^2*c*d*x^2 - 24*b^2*c^2*x^4 - 12*a*b*c*d*x^4 + 3*a^2*d^2*x^4))/(48*c^2*x^6) + ((-8*b^2*c^2*d + 4*a*b*c*d^2 - a^2*d^3)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(16*c^(5/2))","A",1
605,1,171,191,0.4609972,"\int x^2 \left(a+b x^2\right)^2 \sqrt{c+d x^2} \, dx","IntegrateAlgebraic[x^2*(a + b*x^2)^2*Sqrt[c + d*x^2],x]","\frac{\sqrt{c+d x^2} \left(48 a^2 c d^2 x+96 a^2 d^3 x^3-48 a b c^2 d x+32 a b c d^2 x^3+128 a b d^3 x^5+15 b^2 c^3 x-10 b^2 c^2 d x^3+8 b^2 c d^2 x^5+48 b^2 d^3 x^7\right)}{384 d^3}+\frac{\left(-16 a^2 c^2 d^2+16 a b c^3 d-5 b^2 c^4\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{128 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[c + d*x^2]*(15*b^2*c^3*x - 48*a*b*c^2*d*x + 48*a^2*c*d^2*x - 10*b^2*c^2*d*x^3 + 32*a*b*c*d^2*x^3 + 96*a^2*d^3*x^3 + 8*b^2*c*d^2*x^5 + 128*a*b*d^3*x^5 + 48*b^2*d^3*x^7))/(384*d^3) + ((-5*b^2*c^4 + 16*a*b*c^3*d - 16*a^2*c^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(128*d^(7/2))","A",1
606,1,129,149,0.3560419,"\int \left(a+b x^2\right)^2 \sqrt{c+d x^2} \, dx","IntegrateAlgebraic[(a + b*x^2)^2*Sqrt[c + d*x^2],x]","\frac{\sqrt{c+d x^2} \left(24 a^2 d^2 x+12 a b c d x+24 a b d^2 x^3-3 b^2 c^2 x+2 b^2 c d x^3+8 b^2 d^2 x^5\right)}{48 d^2}+\frac{\left(8 a^2 c d^2-4 a b c^2 d+b^2 c^3\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{16 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[c + d*x^2]*(-3*b^2*c^2*x + 12*a*b*c*d*x + 24*a^2*d^2*x + 2*b^2*c*d*x^3 + 24*a*b*d^2*x^3 + 8*b^2*d^2*x^5))/(48*d^2) + ((b^2*c^3 - 4*a*b*c^2*d + 8*a^2*c*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(16*d^(5/2))","A",1
607,1,105,133,0.2631014,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{x^2} \, dx","IntegrateAlgebraic[((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) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{8 d^{3/2}}+\frac{\sqrt{c+d x^2} \left(-8 a^2 d+8 a b d x^2+b^2 c x^2+2 b^2 d x^4\right)}{8 d x}","-\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]*(-8*a^2*d + b^2*c*x^2 + 8*a*b*d*x^2 + 2*b^2*d*x^4))/(8*d*x) + ((-(b^2*c^2) + 8*a*b*c*d + 8*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(8*d^(3/2))","A",1
608,1,94,111,0.2344455,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{x^4} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^4,x]","\frac{\sqrt{c+d x^2} \left(-2 a^2 c-2 a^2 d x^2-12 a b c x^2+3 b^2 c x^4\right)}{6 c x^3}+\frac{\left(4 a b d+b^2 c\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\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,"(Sqrt[c + d*x^2]*(-2*a^2*c - 12*a*b*c*x^2 - 2*a^2*d*x^2 + 3*b^2*c*x^4))/(6*c*x^3) + ((b^2*c + 4*a*b*d)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(2*Sqrt[d])","A",1
609,1,110,103,0.2026915,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{x^6} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^6,x]","\frac{\sqrt{c+d x^2} \left(-3 a^2 c^2-a^2 c d x^2+2 a^2 d^2 x^4-10 a b c^2 x^2-10 a b c d x^4-15 b^2 c^2 x^4\right)}{15 c^2 x^5}+b^2 \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)","-\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,"(Sqrt[c + d*x^2]*(-3*a^2*c^2 - 10*a*b*c^2*x^2 - a^2*c*d*x^2 - 15*b^2*c^2*x^4 - 10*a*b*c*d*x^4 + 2*a^2*d^2*x^4))/(15*c^2*x^5) + b^2*Sqrt[d]*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]]","A",1
610,1,120,99,0.1349835,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{x^8} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^8,x]","\frac{\sqrt{c+d x^2} \left(-15 a^2 c^3-3 a^2 c^2 d x^2+4 a^2 c d^2 x^4-8 a^2 d^3 x^6-42 a b c^3 x^2-14 a b c^2 d x^4+28 a b c d^2 x^6-35 b^2 c^3 x^4-35 b^2 c^2 d x^6\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,"(Sqrt[c + d*x^2]*(-15*a^2*c^3 - 42*a*b*c^3*x^2 - 3*a^2*c^2*d*x^2 - 35*b^2*c^3*x^4 - 14*a*b*c^2*d*x^4 + 4*a^2*c*d^2*x^4 - 35*b^2*c^2*d*x^6 + 28*a*b*c*d^2*x^6 - 8*a^2*d^3*x^6))/(105*c^3*x^7)","A",1
611,1,161,143,0.1859738,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{x^{10}} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^10,x]","\frac{\sqrt{c+d x^2} \left(-35 a^2 c^4-5 a^2 c^3 d x^2+6 a^2 c^2 d^2 x^4-8 a^2 c d^3 x^6+16 a^2 d^4 x^8-90 a b c^4 x^2-18 a b c^3 d x^4+24 a b c^2 d^2 x^6-48 a b c d^3 x^8-63 b^2 c^4 x^4-21 b^2 c^3 d x^6+42 b^2 c^2 d^2 x^8\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,"(Sqrt[c + d*x^2]*(-35*a^2*c^4 - 90*a*b*c^4*x^2 - 5*a^2*c^3*d*x^2 - 63*b^2*c^4*x^4 - 18*a*b*c^3*d*x^4 + 6*a^2*c^2*d^2*x^4 - 21*b^2*c^3*d*x^6 + 24*a*b*c^2*d^2*x^6 - 8*a^2*c*d^3*x^6 + 42*b^2*c^2*d^2*x^8 - 48*a*b*c*d^3*x^8 + 16*a^2*d^4*x^8))/(315*c^4*x^9)","A",1
612,1,202,189,0.2333201,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{x^{12}} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^12,x]","\frac{\sqrt{c+d x^2} \left(-315 a^2 c^5-35 a^2 c^4 d x^2+40 a^2 c^3 d^2 x^4-48 a^2 c^2 d^3 x^6+64 a^2 c d^4 x^8-128 a^2 d^5 x^{10}-770 a b c^5 x^2-110 a b c^4 d x^4+132 a b c^3 d^2 x^6-176 a b c^2 d^3 x^8+352 a b c d^4 x^{10}-495 b^2 c^5 x^4-99 b^2 c^4 d x^6+132 b^2 c^3 d^2 x^8-264 b^2 c^2 d^3 x^{10}\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,"(Sqrt[c + d*x^2]*(-315*a^2*c^5 - 770*a*b*c^5*x^2 - 35*a^2*c^4*d*x^2 - 495*b^2*c^5*x^4 - 110*a*b*c^4*d*x^4 + 40*a^2*c^3*d^2*x^4 - 99*b^2*c^4*d*x^6 + 132*a*b*c^3*d^2*x^6 - 48*a^2*c^2*d^3*x^6 + 132*b^2*c^3*d^2*x^8 - 176*a*b*c^2*d^3*x^8 + 64*a^2*c*d^4*x^8 - 264*b^2*c^2*d^3*x^10 + 352*a*b*c*d^4*x^10 - 128*a^2*d^5*x^10))/(3465*c^5*x^11)","A",1
613,1,253,281,0.7817311,"\int x^4 \left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2} \, dx","IntegrateAlgebraic[x^4*(a + b*x^2)^2*(c + d*x^2)^(3/2),x]","\frac{\left(24 a^2 c^4 d^2-24 a b c^5 d+7 b^2 c^6\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{1024 d^{9/2}}+\frac{\sqrt{c+d x^2} \left(-360 a^2 c^3 d^2 x+240 a^2 c^2 d^3 x^3+2880 a^2 c d^4 x^5+1920 a^2 d^5 x^7+360 a b c^4 d x-240 a b c^3 d^2 x^3+192 a b c^2 d^3 x^5+4224 a b c d^4 x^7+3072 a b d^5 x^9-105 b^2 c^5 x+70 b^2 c^4 d x^3-56 b^2 c^3 d^2 x^5+48 b^2 c^2 d^3 x^7+1664 b^2 c d^4 x^9+1280 b^2 d^5 x^{11}\right)}{15360 d^4}","\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[c + d*x^2]*(-105*b^2*c^5*x + 360*a*b*c^4*d*x - 360*a^2*c^3*d^2*x + 70*b^2*c^4*d*x^3 - 240*a*b*c^3*d^2*x^3 + 240*a^2*c^2*d^3*x^3 - 56*b^2*c^3*d^2*x^5 + 192*a*b*c^2*d^3*x^5 + 2880*a^2*c*d^4*x^5 + 48*b^2*c^2*d^3*x^7 + 4224*a*b*c*d^4*x^7 + 1920*a^2*d^5*x^7 + 1664*b^2*c*d^4*x^9 + 3072*a*b*d^5*x^9 + 1280*b^2*d^5*x^11))/(15360*d^4) + ((7*b^2*c^6 - 24*a*b*c^5*d + 24*a^2*c^4*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(1024*d^(9/2))","A",1
614,1,193,114,0.1064884,"\int x^3 \left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2} \, dx","IntegrateAlgebraic[x^3*(a + b*x^2)^2*(c + d*x^2)^(3/2),x]","\frac{\sqrt{c+d x^2} \left(-198 a^2 c^3 d^2+99 a^2 c^2 d^3 x^2+792 a^2 c d^4 x^4+495 a^2 d^5 x^6+176 a b c^4 d-88 a b c^3 d^2 x^2+66 a b c^2 d^3 x^4+1100 a b c d^4 x^6+770 a b d^5 x^8-48 b^2 c^5+24 b^2 c^4 d x^2-18 b^2 c^3 d^2 x^4+15 b^2 c^2 d^3 x^6+420 b^2 c d^4 x^8+315 b^2 d^5 x^{10}\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,"(Sqrt[c + d*x^2]*(-48*b^2*c^5 + 176*a*b*c^4*d - 198*a^2*c^3*d^2 + 24*b^2*c^4*d*x^2 - 88*a*b*c^3*d^2*x^2 + 99*a^2*c^2*d^3*x^2 - 18*b^2*c^3*d^2*x^4 + 66*a*b*c^2*d^3*x^4 + 792*a^2*c*d^4*x^4 + 15*b^2*c^2*d^3*x^6 + 1100*a*b*c*d^4*x^6 + 495*a^2*d^5*x^6 + 420*b^2*c*d^4*x^8 + 770*a*b*d^5*x^8 + 315*b^2*d^5*x^10))/(3465*d^4)","A",1
615,1,212,235,0.7065606,"\int x^2 \left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2} \, dx","IntegrateAlgebraic[x^2*(a + b*x^2)^2*(c + d*x^2)^(3/2),x]","\frac{\left(-16 a^2 c^3 d^2+12 a b c^4 d-3 b^2 c^5\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{256 d^{7/2}}+\frac{\sqrt{c+d x^2} \left(240 a^2 c^2 d^2 x+1120 a^2 c d^3 x^3+640 a^2 d^4 x^5-180 a b c^3 d x+120 a b c^2 d^2 x^3+1440 a b c d^3 x^5+960 a b d^4 x^7+45 b^2 c^4 x-30 b^2 c^3 d x^3+24 b^2 c^2 d^2 x^5+528 b^2 c d^3 x^7+384 b^2 d^4 x^9\right)}{3840 d^3}","-\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[c + d*x^2]*(45*b^2*c^4*x - 180*a*b*c^3*d*x + 240*a^2*c^2*d^2*x - 30*b^2*c^3*d*x^3 + 120*a*b*c^2*d^2*x^3 + 1120*a^2*c*d^3*x^3 + 24*b^2*c^2*d^2*x^5 + 1440*a*b*c*d^3*x^5 + 640*a^2*d^4*x^5 + 528*b^2*c*d^3*x^7 + 960*a*b*d^4*x^7 + 384*b^2*d^4*x^9))/(3840*d^3) + ((-3*b^2*c^5 + 12*a*b*c^4*d - 16*a^2*c^3*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(256*d^(7/2))","A",1
616,1,152,77,0.0845737,"\int x \left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2} \, dx","IntegrateAlgebraic[x*(a + b*x^2)^2*(c + d*x^2)^(3/2),x]","\frac{\sqrt{c+d x^2} \left(63 a^2 c^2 d^2+126 a^2 c d^3 x^2+63 a^2 d^4 x^4-36 a b c^3 d+18 a b c^2 d^2 x^2+144 a b c d^3 x^4+90 a b d^4 x^6+8 b^2 c^4-4 b^2 c^3 d x^2+3 b^2 c^2 d^2 x^4+50 b^2 c d^3 x^6+35 b^2 d^4 x^8\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,"(Sqrt[c + d*x^2]*(8*b^2*c^4 - 36*a*b*c^3*d + 63*a^2*c^2*d^2 - 4*b^2*c^3*d*x^2 + 18*a*b*c^2*d^2*x^2 + 126*a^2*c*d^3*x^2 + 3*b^2*c^2*d^2*x^4 + 144*a*b*c*d^3*x^4 + 63*a^2*d^4*x^4 + 50*b^2*c*d^3*x^6 + 90*a*b*d^4*x^6 + 35*b^2*d^4*x^8))/(315*d^3)","A",1
617,1,171,196,0.5069541,"\int \left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2} \, dx","IntegrateAlgebraic[(a + b*x^2)^2*(c + d*x^2)^(3/2),x]","\frac{\sqrt{c+d x^2} \left(240 a^2 c d^2 x+96 a^2 d^3 x^3+48 a b c^2 d x+224 a b c d^2 x^3+128 a b d^3 x^5-9 b^2 c^3 x+6 b^2 c^2 d x^3+72 b^2 c d^2 x^5+48 b^2 d^3 x^7\right)}{384 d^2}+\frac{\left(48 a^2 c^2 d^2-16 a b c^3 d+3 b^2 c^4\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{128 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[c + d*x^2]*(-9*b^2*c^3*x + 48*a*b*c^2*d*x + 240*a^2*c*d^2*x + 6*b^2*c^2*d*x^3 + 224*a*b*c*d^2*x^3 + 96*a^2*d^3*x^3 + 72*b^2*c*d^2*x^5 + 128*a*b*d^3*x^5 + 48*b^2*d^3*x^7))/(384*d^2) + ((3*b^2*c^4 - 16*a*b*c^3*d + 48*a^2*c^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(128*d^(5/2))","A",1
618,1,140,111,0.1800674,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}}{x} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2)^(3/2))/x,x]","\frac{\sqrt{c+d x^2} \left(140 a^2 c d^2+35 a^2 d^3 x^2+42 a b c^2 d+84 a b c d^2 x^2+42 a b d^3 x^4-6 b^2 c^3+3 b^2 c^2 d x^2+24 b^2 c d^2 x^4+15 b^2 d^3 x^6\right)}{105 d^2}-a^2 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)","-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,"(Sqrt[c + d*x^2]*(-6*b^2*c^3 + 42*a*b*c^2*d + 140*a^2*c*d^2 + 3*b^2*c^2*d*x^2 + 84*a*b*c*d^2*x^2 + 35*a^2*d^3*x^2 + 24*b^2*c*d^2*x^4 + 42*a*b*d^3*x^4 + 15*b^2*d^3*x^6))/(105*d^2) - a^2*c^(3/2)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]]","A",1
619,1,146,175,0.3972179,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}}{x^2} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^2,x]","\frac{\sqrt{c+d x^2} \left(-48 a^2 c d+24 a^2 d^2 x^2+60 a b c d x^2+24 a b d^2 x^4+3 b^2 c^2 x^2+14 b^2 c d x^4+8 b^2 d^2 x^6\right)}{48 d x}+\frac{\left(24 a^2 c d^2+12 a b c^2 d-b^2 c^3\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\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]*(-48*a^2*c*d + 3*b^2*c^2*x^2 + 60*a*b*c*d*x^2 + 24*a^2*d^2*x^2 + 14*b^2*c*d*x^4 + 24*a*b*d^2*x^4 + 8*b^2*d^2*x^6))/(48*d*x) + ((-(b^2*c^3) + 12*a*b*c^2*d + 24*a^2*c*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(16*d^(3/2))","A",1
620,1,135,136,0.2227656,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}}{x^3} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^3,x]","\frac{\sqrt{c+d x^2} \left(-15 a^2 c d+30 a^2 d^2 x^2+80 a b c d x^2+20 a b d^2 x^4+6 b^2 c^2 x^2+12 b^2 c d x^4+6 b^2 d^2 x^6\right)}{30 d x^2}+\frac{1}{2} \left(-3 a^2 \sqrt{c} d-4 a b c^{3/2}\right) \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*c*d + 6*b^2*c^2*x^2 + 80*a*b*c*d*x^2 + 30*a^2*d^2*x^2 + 12*b^2*c*d*x^4 + 20*a*b*d^2*x^4 + 6*b^2*d^2*x^6))/(30*d*x^2) + ((-4*a*b*c^(3/2) - 3*a^2*Sqrt[c]*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/2","A",1
621,1,120,184,0.2944589,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}}{x^4} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^4,x]","\frac{\left(8 a^2 d^2+24 a b c d+3 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{8 \sqrt{d}}+\frac{\sqrt{c+d x^2} \left(-8 a^2 c-32 a^2 d x^2-48 a b c x^2+24 a b d x^4+15 b^2 c x^4+6 b^2 d x^6\right)}{24 x^3}","-\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 - 48*a*b*c*x^2 - 32*a^2*d*x^2 + 15*b^2*c*x^4 + 24*a*b*d*x^4 + 6*b^2*d*x^6))/(24*x^3) + ((3*b^2*c^2 + 24*a*b*c*d + 8*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(8*Sqrt[d])","A",1
622,1,119,181,0.204049,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}}{x^5} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^5,x]","\frac{\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)}{8 \sqrt{c}}+\frac{\sqrt{c+d x^2} \left(-6 a^2 c-15 a^2 d x^2-24 a b c x^2+48 a b d x^4+32 b^2 c x^4+8 b^2 d x^6\right)}{24 x^4}","-\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]*(-6*a^2*c - 24*a*b*c*x^2 - 15*a^2*d*x^2 + 32*b^2*c*x^4 + 48*a*b*d*x^4 + 8*b^2*d*x^6))/(24*x^4) + ((-8*b^2*c^2 - 24*a*b*c*d - 3*a^2*d^2)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(8*Sqrt[c])","A",1
623,1,136,147,0.3318385,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}}{x^6} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^6,x]","\frac{\sqrt{c+d x^2} \left(-6 a^2 c^2-12 a^2 c d x^2-6 a^2 d^2 x^4-20 a b c^2 x^2-80 a b c d x^4-30 b^2 c^2 x^4+15 b^2 c d x^6\right)}{30 c x^5}+\frac{1}{2} \left(4 a b d^{3/2}+3 b^2 c \sqrt{d}\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)","-\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,"(Sqrt[c + d*x^2]*(-6*a^2*c^2 - 20*a*b*c^2*x^2 - 12*a^2*c*d*x^2 - 30*b^2*c^2*x^4 - 80*a*b*c*d*x^4 - 6*a^2*d^2*x^4 + 15*b^2*c*d*x^6))/(30*c*x^5) + ((3*b^2*c*Sqrt[d] + 4*a*b*d^(3/2))*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/2","A",1
624,1,144,187,0.2907878,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}}{x^7} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^7,x]","\frac{\sqrt{c+d x^2} \left(-8 a^2 c^2-14 a^2 c d x^2-3 a^2 d^2 x^4-24 a b c^2 x^2-60 a b c d x^4-24 b^2 c^2 x^4+48 b^2 c d x^6\right)}{48 c x^6}+\frac{\left(a^2 d^3-12 a b c d^2-24 b^2 c^2 d\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{16 c^{3/2}}","-\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,"(Sqrt[c + d*x^2]*(-8*a^2*c^2 - 24*a*b*c^2*x^2 - 14*a^2*c*d*x^2 - 24*b^2*c^2*x^4 - 60*a*b*c*d*x^4 - 3*a^2*d^2*x^4 + 48*b^2*c*d*x^6))/(48*c*x^6) + ((-24*b^2*c^2*d - 12*a*b*c*d^2 + a^2*d^3)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(16*c^(3/2))","A",1
625,1,234,114,0.0996958,"\int x^3 \left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2} \, dx","IntegrateAlgebraic[x^3*(a + b*x^2)^2*(c + d*x^2)^(5/2),x]","\frac{\sqrt{c+d x^2} \left(-286 a^2 c^4 d^2+143 a^2 c^3 d^3 x^2+2145 a^2 c^2 d^4 x^4+2717 a^2 c d^5 x^6+1001 a^2 d^6 x^8+208 a b c^5 d-104 a b c^4 d^2 x^2+78 a b c^3 d^3 x^4+2938 a b c^2 d^4 x^6+4186 a b c d^5 x^8+1638 a b d^6 x^{10}-48 b^2 c^6+24 b^2 c^5 d x^2-18 b^2 c^4 d^2 x^4+15 b^2 c^3 d^3 x^6+1113 b^2 c^2 d^4 x^8+1701 b^2 c d^5 x^{10}+693 b^2 d^6 x^{12}\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,"(Sqrt[c + d*x^2]*(-48*b^2*c^6 + 208*a*b*c^5*d - 286*a^2*c^4*d^2 + 24*b^2*c^5*d*x^2 - 104*a*b*c^4*d^2*x^2 + 143*a^2*c^3*d^3*x^2 - 18*b^2*c^4*d^2*x^4 + 78*a*b*c^3*d^3*x^4 + 2145*a^2*c^2*d^4*x^4 + 15*b^2*c^3*d^3*x^6 + 2938*a*b*c^2*d^4*x^6 + 2717*a^2*c*d^5*x^6 + 1113*b^2*c^2*d^4*x^8 + 4186*a*b*c*d^5*x^8 + 1001*a^2*d^6*x^8 + 1701*b^2*c*d^5*x^10 + 1638*a*b*d^6*x^10 + 693*b^2*d^6*x^12))/(9009*d^4)","B",1
626,1,253,281,0.7307191,"\int x^2 \left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2} \, dx","IntegrateAlgebraic[x^2*(a + b*x^2)^2*(c + d*x^2)^(5/2),x]","\frac{\left(-40 a^2 c^4 d^2+24 a b c^5 d-5 b^2 c^6\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{1024 d^{7/2}}+\frac{\sqrt{c+d x^2} \left(600 a^2 c^3 d^2 x+4720 a^2 c^2 d^3 x^3+5440 a^2 c d^4 x^5+1920 a^2 d^5 x^7-360 a b c^4 d x+240 a b c^3 d^2 x^3+5952 a b c^2 d^3 x^5+8064 a b c d^4 x^7+3072 a b d^5 x^9+75 b^2 c^5 x-50 b^2 c^4 d x^3+40 b^2 c^3 d^2 x^5+2160 b^2 c^2 d^3 x^7+3200 b^2 c d^4 x^9+1280 b^2 d^5 x^{11}\right)}{15360 d^3}","-\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[c + d*x^2]*(75*b^2*c^5*x - 360*a*b*c^4*d*x + 600*a^2*c^3*d^2*x - 50*b^2*c^4*d*x^3 + 240*a*b*c^3*d^2*x^3 + 4720*a^2*c^2*d^3*x^3 + 40*b^2*c^3*d^2*x^5 + 5952*a*b*c^2*d^3*x^5 + 5440*a^2*c*d^4*x^5 + 2160*b^2*c^2*d^3*x^7 + 8064*a*b*c*d^4*x^7 + 1920*a^2*d^5*x^7 + 3200*b^2*c*d^4*x^9 + 3072*a*b*d^5*x^9 + 1280*b^2*d^5*x^11))/(15360*d^3) + ((-5*b^2*c^6 + 24*a*b*c^5*d - 40*a^2*c^4*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(1024*d^(7/2))","A",1
627,1,193,77,0.0914076,"\int x \left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2} \, dx","IntegrateAlgebraic[x*(a + b*x^2)^2*(c + d*x^2)^(5/2),x]","\frac{\sqrt{c+d x^2} \left(99 a^2 c^3 d^2+297 a^2 c^2 d^3 x^2+297 a^2 c d^4 x^4+99 a^2 d^5 x^6-44 a b c^4 d+22 a b c^3 d^2 x^2+330 a b c^2 d^3 x^4+418 a b c d^4 x^6+154 a b d^5 x^8+8 b^2 c^5-4 b^2 c^4 d x^2+3 b^2 c^3 d^2 x^4+113 b^2 c^2 d^3 x^6+161 b^2 c d^4 x^8+63 b^2 d^5 x^{10}\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,"(Sqrt[c + d*x^2]*(8*b^2*c^5 - 44*a*b*c^4*d + 99*a^2*c^3*d^2 - 4*b^2*c^4*d*x^2 + 22*a*b*c^3*d^2*x^2 + 297*a^2*c^2*d^3*x^2 + 3*b^2*c^3*d^2*x^4 + 330*a*b*c^2*d^3*x^4 + 297*a^2*c*d^4*x^4 + 113*b^2*c^2*d^3*x^6 + 418*a*b*c*d^4*x^6 + 99*a^2*d^5*x^6 + 161*b^2*c*d^4*x^8 + 154*a*b*d^5*x^8 + 63*b^2*d^5*x^10))/(693*d^3)","B",1
628,1,212,240,0.6259282,"\int \left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2} \, dx","IntegrateAlgebraic[(a + b*x^2)^2*(c + d*x^2)^(5/2),x]","\frac{\left(80 a^2 c^3 d^2-20 a b c^4 d+3 b^2 c^5\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{256 d^{5/2}}+\frac{\sqrt{c+d x^2} \left(2640 a^2 c^2 d^2 x+2080 a^2 c d^3 x^3+640 a^2 d^4 x^5+300 a b c^3 d x+2360 a b c^2 d^2 x^3+2720 a b c d^3 x^5+960 a b d^4 x^7-45 b^2 c^4 x+30 b^2 c^3 d x^3+744 b^2 c^2 d^2 x^5+1008 b^2 c d^3 x^7+384 b^2 d^4 x^9\right)}{3840 d^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[c + d*x^2]*(-45*b^2*c^4*x + 300*a*b*c^3*d*x + 2640*a^2*c^2*d^2*x + 30*b^2*c^3*d*x^3 + 2360*a*b*c^2*d^2*x^3 + 2080*a^2*c*d^3*x^3 + 744*b^2*c^2*d^2*x^5 + 2720*a*b*c*d^3*x^5 + 640*a^2*d^4*x^5 + 1008*b^2*c*d^3*x^7 + 960*a*b*d^4*x^7 + 384*b^2*d^4*x^9))/(3840*d^2) + ((3*b^2*c^5 - 20*a*b*c^4*d + 80*a^2*c^3*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(256*d^(5/2))","A",1
629,1,181,132,0.1972889,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2}}{x} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2)^(5/2))/x,x]","\frac{\sqrt{c+d x^2} \left(483 a^2 c^2 d^2+231 a^2 c d^3 x^2+63 a^2 d^4 x^4+90 a b c^3 d+270 a b c^2 d^2 x^2+270 a b c d^3 x^4+90 a b d^4 x^6-10 b^2 c^4+5 b^2 c^3 d x^2+75 b^2 c^2 d^2 x^4+95 b^2 c d^3 x^6+35 b^2 d^4 x^8\right)}{315 d^2}-a^2 c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)","-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,"(Sqrt[c + d*x^2]*(-10*b^2*c^4 + 90*a*b*c^3*d + 483*a^2*c^2*d^2 + 5*b^2*c^3*d*x^2 + 270*a*b*c^2*d^2*x^2 + 231*a^2*c*d^3*x^2 + 75*b^2*c^2*d^2*x^4 + 270*a*b*c*d^3*x^4 + 63*a^2*d^4*x^4 + 95*b^2*c*d^3*x^6 + 90*a*b*d^4*x^6 + 35*b^2*d^4*x^8))/(315*d^2) - a^2*c^(5/2)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]]","A",1
630,1,188,217,0.4882629,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2}}{x^2} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2)^(5/2))/x^2,x]","\frac{\sqrt{c+d x^2} \left(-384 a^2 c^2 d+432 a^2 c d^2 x^2+96 a^2 d^3 x^4+528 a b c^2 d x^2+416 a b c d^2 x^4+128 a b d^3 x^6+15 b^2 c^3 x^2+118 b^2 c^2 d x^4+136 b^2 c d^2 x^6+48 b^2 d^3 x^8\right)}{384 d x}-\frac{5 \left(-48 a^2 c^2 d^2-16 a b c^3 d+b^2 c^4\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\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]*(-384*a^2*c^2*d + 15*b^2*c^3*x^2 + 528*a*b*c^2*d*x^2 + 432*a^2*c*d^2*x^2 + 118*b^2*c^2*d*x^4 + 416*a*b*c*d^2*x^4 + 96*a^2*d^3*x^4 + 136*b^2*c*d^2*x^6 + 128*a*b*d^3*x^6 + 48*b^2*d^3*x^8))/(384*d*x) - (5*(b^2*c^4 - 16*a*b*c^3*d - 48*a^2*c^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(128*d^(3/2))","A",1
631,1,176,162,0.2481372,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2}}{x^3} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2)^(5/2))/x^3,x]","\frac{\sqrt{c+d x^2} \left(-105 a^2 c^2 d+490 a^2 c d^2 x^2+70 a^2 d^3 x^4+644 a b c^2 d x^2+308 a b c d^2 x^4+84 a b d^3 x^6+30 b^2 c^3 x^2+90 b^2 c^2 d x^4+90 b^2 c d^2 x^6+30 b^2 d^3 x^8\right)}{210 d x^2}+\frac{1}{2} \left(-5 a^2 c^{3/2} d-4 a b c^{5/2}\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)","-\frac{a^2 \left(c+d x^2\right)^{7/2}}{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,"(Sqrt[c + d*x^2]*(-105*a^2*c^2*d + 30*b^2*c^3*x^2 + 644*a*b*c^2*d*x^2 + 490*a^2*c*d^2*x^2 + 90*b^2*c^2*d*x^4 + 308*a*b*c*d^2*x^4 + 70*a^2*d^3*x^4 + 90*b^2*c*d^2*x^6 + 84*a*b*d^3*x^6 + 30*b^2*d^3*x^8))/(210*d*x^2) + ((-4*a*b*c^(5/2) - 5*a^2*c^(3/2)*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/2","A",1
632,1,163,223,0.4014512,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2}}{x^4} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2)^(5/2))/x^4,x]","\frac{\sqrt{c+d x^2} \left(-16 a^2 c^2-112 a^2 c d x^2+24 a^2 d^2 x^4-96 a b c^2 x^2+108 a b c d x^4+24 a b d^2 x^6+33 b^2 c^2 x^4+26 b^2 c d x^6+8 b^2 d^2 x^8\right)}{48 x^3}+\frac{5 \left(8 a^2 c d^2+12 a b c^2 d+b^2 c^3\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{16 \sqrt{d}}","-\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]*(-16*a^2*c^2 - 96*a*b*c^2*x^2 - 112*a^2*c*d*x^2 + 33*b^2*c^2*x^4 + 108*a*b*c*d*x^4 + 24*a^2*d^2*x^4 + 26*b^2*c*d*x^6 + 24*a*b*d^2*x^6 + 8*b^2*d^2*x^8))/(48*x^3) + (5*(b^2*c^3 + 12*a*b*c^2*d + 8*a^2*c*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(16*Sqrt[d])","A",1
633,1,166,222,0.2242901,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2}}{x^5} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2)^(5/2))/x^5,x]","\frac{1}{8} \left(-15 a^2 \sqrt{c} d^2-40 a b c^{3/2} d-8 b^2 c^{5/2}\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)+\frac{\sqrt{c+d x^2} \left(-30 a^2 c^2-135 a^2 c d x^2+120 a^2 d^2 x^4-120 a b c^2 x^2+560 a b c d x^4+80 a b d^2 x^6+184 b^2 c^2 x^4+88 b^2 c d x^6+24 b^2 d^2 x^8\right)}{120 x^4}","-\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]*(-30*a^2*c^2 - 120*a*b*c^2*x^2 - 135*a^2*c*d*x^2 + 184*b^2*c^2*x^4 + 560*a*b*c*d*x^4 + 120*a^2*d^2*x^4 + 88*b^2*c*d*x^6 + 80*a*b*d^2*x^6 + 24*b^2*d^2*x^8))/(120*x^4) + ((-8*b^2*c^(5/2) - 40*a*b*c^(3/2)*d - 15*a^2*Sqrt[c]*d^2)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/8","A",1
634,1,167,228,0.3684978,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2}}{x^6} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2)^(5/2))/x^6,x]","\frac{1}{8} \left(8 a^2 d^{5/2}+40 a b c d^{3/2}+15 b^2 c^2 \sqrt{d}\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)+\frac{\sqrt{c+d x^2} \left(-24 a^2 c^2-88 a^2 c d x^2-184 a^2 d^2 x^4-80 a b c^2 x^2-560 a b c d x^4+120 a b d^2 x^6-120 b^2 c^2 x^4+135 b^2 c d x^6+30 b^2 d^2 x^8\right)}{120 x^5}","-\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]*(-24*a^2*c^2 - 80*a*b*c^2*x^2 - 88*a^2*c*d*x^2 - 120*b^2*c^2*x^4 - 560*a*b*c*d*x^4 - 184*a^2*d^2*x^4 + 135*b^2*c*d*x^6 + 120*a*b*d^2*x^6 + 30*b^2*d^2*x^8))/(120*x^5) + ((15*b^2*c^2*Sqrt[d] + 40*a*b*c*d^(3/2) + 8*a^2*d^(5/2))*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/8","A",1
635,1,162,222,0.2858506,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2}}{x^7} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2)^(5/2))/x^7,x]","\frac{\sqrt{c+d x^2} \left(-8 a^2 c^2-26 a^2 c d x^2-33 a^2 d^2 x^4-24 a b c^2 x^2-108 a b c d x^4+96 a b d^2 x^6-24 b^2 c^2 x^4+112 b^2 c d x^6+16 b^2 d^2 x^8\right)}{48 x^6}-\frac{5 \left(a^2 d^3+12 a b c d^2+8 b^2 c^2 d\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{16 \sqrt{c}}","-\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,"(Sqrt[c + d*x^2]*(-8*a^2*c^2 - 24*a*b*c^2*x^2 - 26*a^2*c*d*x^2 - 24*b^2*c^2*x^4 - 108*a*b*c*d*x^4 - 33*a^2*d^2*x^4 + 112*b^2*c*d*x^6 + 96*a*b*d^2*x^6 + 16*b^2*d^2*x^8))/(48*x^6) - (5*(8*b^2*c^2*d + 12*a*b*c*d^2 + a^2*d^3)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(16*Sqrt[c])","A",1
636,1,171,194,0.5748715,"\int \frac{x^4 \left(a+b x^2\right)^2}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(x^4*(a + b*x^2)^2)/Sqrt[c + d*x^2],x]","\frac{\sqrt{c+d x^2} \left(-144 a^2 c d^2 x+96 a^2 d^3 x^3+240 a b c^2 d x-160 a b c d^2 x^3+128 a b d^3 x^5-105 b^2 c^3 x+70 b^2 c^2 d x^3-56 b^2 c d^2 x^5+48 b^2 d^3 x^7\right)}{384 d^4}+\frac{\left(48 a^2 c^2 d^2-80 a b c^3 d+35 b^2 c^4\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{128 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[c + d*x^2]*(-105*b^2*c^3*x + 240*a*b*c^2*d*x - 144*a^2*c*d^2*x + 70*b^2*c^2*d*x^3 - 160*a*b*c*d^2*x^3 + 96*a^2*d^3*x^3 - 56*b^2*c*d^2*x^5 + 128*a*b*d^3*x^5 + 48*b^2*d^3*x^7))/(384*d^4) + ((35*b^2*c^4 - 80*a*b*c^3*d + 48*a^2*c^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(128*d^(9/2))","A",1
637,1,111,112,0.0799401,"\int \frac{x^3 \left(a+b x^2\right)^2}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(x^3*(a + b*x^2)^2)/Sqrt[c + d*x^2],x]","\frac{\sqrt{c+d x^2} \left(-70 a^2 c d^2+35 a^2 d^3 x^2+112 a b c^2 d-56 a b c d^2 x^2+42 a b d^3 x^4-48 b^2 c^3+24 b^2 c^2 d x^2-18 b^2 c d^2 x^4+15 b^2 d^3 x^6\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]*(-48*b^2*c^3 + 112*a*b*c^2*d - 70*a^2*c*d^2 + 24*b^2*c^2*d*x^2 - 56*a*b*c*d^2*x^2 + 35*a^2*d^3*x^2 - 18*b^2*c*d^2*x^4 + 42*a*b*d^3*x^4 + 15*b^2*d^3*x^6))/(105*d^4)","A",1
638,1,130,146,0.4115761,"\int \frac{x^2 \left(a+b x^2\right)^2}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(x^2*(a + b*x^2)^2)/Sqrt[c + d*x^2],x]","\frac{\sqrt{c+d x^2} \left(24 a^2 d^2 x-36 a b c d x+24 a b d^2 x^3+15 b^2 c^2 x-10 b^2 c d x^3+8 b^2 d^2 x^5\right)}{48 d^3}+\frac{\left(-8 a^2 c d^2+12 a b c^2 d-5 b^2 c^3\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{16 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[c + d*x^2]*(15*b^2*c^2*x - 36*a*b*c*d*x + 24*a^2*d^2*x - 10*b^2*c*d*x^3 + 24*a*b*d^2*x^3 + 8*b^2*d^2*x^5))/(48*d^3) + ((-5*b^2*c^3 + 12*a*b*c^2*d - 8*a^2*c*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(16*d^(7/2))","A",1
639,1,72,74,0.0630997,"\int \frac{x \left(a+b x^2\right)^2}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(x*(a + b*x^2)^2)/Sqrt[c + d*x^2],x]","\frac{\sqrt{c+d x^2} \left(15 a^2 d^2-20 a b c d+10 a b d^2 x^2+8 b^2 c^2-4 b^2 c d x^2+3 b^2 d^2 x^4\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]*(8*b^2*c^2 - 20*a*b*c*d + 15*a^2*d^2 - 4*b^2*c*d*x^2 + 10*a*b*d^2*x^2 + 3*b^2*d^2*x^4))/(15*d^3)","A",1
640,1,93,107,0.2789326,"\int \frac{\left(a+b x^2\right)^2}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(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) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{8 d^{5/2}}+\frac{\sqrt{c+d x^2} \left(8 a b d x-3 b^2 c x+2 b^2 d x^3\right)}{8 d^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,"(Sqrt[c + d*x^2]*(-3*b^2*c*x + 8*a*b*d*x + 2*b^2*d*x^3))/(8*d^2) + ((3*b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(8*d^(5/2))","A",1
641,1,67,75,0.1112091,"\int \frac{\left(a+b x^2\right)^2}{x \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x*Sqrt[c + d*x^2]),x]","\frac{\sqrt{c+d x^2} \left(6 a b d-2 b^2 c+b^2 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,"(Sqrt[c + d*x^2]*(-2*b^2*c + 6*a*b*d + b^2*d*x^2))/(3*d^2) - (a^2*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/Sqrt[c]","A",1
642,1,80,82,0.2278814,"\int \frac{\left(a+b x^2\right)^2}{x^2 \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^2*Sqrt[c + d*x^2]),x]","\frac{\sqrt{c+d x^2} \left(b^2 c x^2-2 a^2 d\right)}{2 c d x}+\frac{\left(4 a b d-b^2 c\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\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,"((-2*a^2*d + b^2*c*x^2)*Sqrt[c + d*x^2])/(2*c*d*x) + ((-(b^2*c) + 4*a*b*d)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(2*d^(3/2))","A",1
643,1,79,80,0.1414792,"\int \frac{\left(a+b x^2\right)^2}{x^3 \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^3*Sqrt[c + d*x^2]),x]","\frac{\sqrt{c+d x^2} \left(2 b^2 c x^2-a^2 d\right)}{2 c d x^2}+\frac{\left(a^2 d-4 a b c\right) \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,"((-(a^2*d) + 2*b^2*c*x^2)*Sqrt[c + d*x^2])/(2*c*d*x^2) + ((-4*a*b*c + a^2*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*c^(3/2))","A",1
644,1,74,84,0.1897951,"\int \frac{\left(a+b x^2\right)^2}{x^4 \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^4*Sqrt[c + d*x^2]),x]","\frac{\sqrt{c+d x^2} \left(-a^2 c+2 a^2 d x^2-6 a b c x^2\right)}{3 c^2 x^3}+\frac{b^2 \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{\sqrt{d}}","-\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,"(Sqrt[c + d*x^2]*(-(a^2*c) - 6*a*b*c*x^2 + 2*a^2*d*x^2))/(3*c^2*x^3) + (b^2*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/Sqrt[d]","A",1
645,1,96,106,0.1887209,"\int \frac{\left(a+b x^2\right)^2}{x^5 \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(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{\sqrt{c+d x^2} \left(-2 a^2 c+3 a^2 d x^2-8 a 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,"(Sqrt[c + d*x^2]*(-2*a^2*c - 8*a*b*c*x^2 + 3*a^2*d*x^2))/(8*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
646,1,81,99,0.134055,"\int \frac{\left(a+b x^2\right)^2}{x^6 \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^6*Sqrt[c + d*x^2]),x]","\frac{\sqrt{c+d x^2} \left(-3 a^2 c^2+4 a^2 c d x^2-8 a^2 d^2 x^4-10 a b c^2 x^2+20 a b c d x^4-15 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,"(Sqrt[c + d*x^2]*(-3*a^2*c^2 - 10*a*b*c^2*x^2 + 4*a^2*c*d*x^2 - 15*b^2*c^2*x^4 + 20*a*b*c*d*x^4 - 8*a^2*d^2*x^4))/(15*c^3*x^5)","A",1
647,1,135,151,0.2804242,"\int \frac{\left(a+b x^2\right)^2}{x^7 \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^7*Sqrt[c + d*x^2]),x]","\frac{\left(5 a^2 d^3-12 a b c d^2+8 b^2 c^2 d\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{16 c^{7/2}}+\frac{\sqrt{c+d x^2} \left(-8 a^2 c^2+10 a^2 c d x^2-15 a^2 d^2 x^4-24 a b c^2 x^2+36 a b c d x^4-24 b^2 c^2 x^4\right)}{48 c^3 x^6}","\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]*(-8*a^2*c^2 - 24*a*b*c^2*x^2 + 10*a^2*c*d*x^2 - 24*b^2*c^2*x^4 + 36*a*b*c*d*x^4 - 15*a^2*d^2*x^4))/(48*c^3*x^6) + ((8*b^2*c^2*d - 12*a*b*c*d^2 + 5*a^2*d^3)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(16*c^(7/2))","A",1
648,1,169,197,0.5171947,"\int \frac{x^4 \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(x^4*(a + b*x^2)^2)/(c + d*x^2)^(3/2),x]","\frac{\left(-24 a^2 c d^2+60 a b c^2 d-35 b^2 c^3\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{16 d^{9/2}}+\frac{72 a^2 c d^2 x+24 a^2 d^3 x^3-180 a b c^2 d x-60 a b c d^2 x^3+24 a b d^3 x^5+105 b^2 c^3 x+35 b^2 c^2 d x^3-14 b^2 c d^2 x^5+8 b^2 d^3 x^7}{48 d^4 \sqrt{c+d x^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,"(105*b^2*c^3*x - 180*a*b*c^2*d*x + 72*a^2*c*d^2*x + 35*b^2*c^2*d*x^3 - 60*a*b*c*d^2*x^3 + 24*a^2*d^3*x^3 - 14*b^2*c*d^2*x^5 + 24*a*b*d^3*x^5 + 8*b^2*d^3*x^7)/(48*d^4*Sqrt[c + d*x^2]) + ((-35*b^2*c^3 + 60*a*b*c^2*d - 24*a^2*c*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(16*d^(9/2))","A",1
649,1,111,108,0.0914714,"\int \frac{x^3 \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(x^3*(a + b*x^2)^2)/(c + d*x^2)^(3/2),x]","\frac{30 a^2 c d^2+15 a^2 d^3 x^2-80 a b c^2 d-40 a b c d^2 x^2+10 a b d^3 x^4+48 b^2 c^3+24 b^2 c^2 d x^2-6 b^2 c d^2 x^4+3 b^2 d^3 x^6}{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,"(48*b^2*c^3 - 80*a*b*c^2*d + 30*a^2*c*d^2 + 24*b^2*c^2*d*x^2 - 40*a*b*c*d^2*x^2 + 15*a^2*d^3*x^2 - 6*b^2*c*d^2*x^4 + 10*a*b*d^3*x^4 + 3*b^2*d^3*x^6)/(15*d^4*Sqrt[c + d*x^2])","A",1
650,1,127,152,0.3558786,"\int \frac{x^2 \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(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) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{8 d^{7/2}}+\frac{-8 a^2 d^2 x+24 a b c d x+8 a b d^2 x^3-15 b^2 c^2 x-5 b^2 c d x^3+2 b^2 d^2 x^5}{8 d^3 \sqrt{c+d x^2}}","\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,"(-15*b^2*c^2*x + 24*a*b*c*d*x - 8*a^2*d^2*x - 5*b^2*c*d*x^3 + 8*a*b*d^2*x^3 + 2*b^2*d^2*x^5)/(8*d^3*Sqrt[c + d*x^2]) + ((15*b^2*c^2 - 24*a*b*c*d + 8*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(8*d^(7/2))","A",1
651,1,71,73,0.0712264,"\int \frac{x \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(x*(a + b*x^2)^2)/(c + d*x^2)^(3/2),x]","\frac{-3 a^2 d^2+12 a b c d+6 a b d^2 x^2-8 b^2 c^2-4 b^2 c d x^2+b^2 d^2 x^4}{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,"(-8*b^2*c^2 + 12*a*b*c*d - 3*a^2*d^2 - 4*b^2*c*d*x^2 + 6*a*b*d^2*x^2 + b^2*d^2*x^4)/(3*d^3*Sqrt[c + d*x^2])","A",1
652,1,97,106,0.3358754,"\int \frac{\left(a+b x^2\right)^2}{\left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(c + d*x^2)^(3/2),x]","\frac{2 a^2 d^2 x-4 a b c d x+3 b^2 c^2 x+b^2 c d x^3}{2 c d^2 \sqrt{c+d x^2}}+\frac{\left(4 a b d-3 b^2 c\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\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,"(3*b^2*c^2*x - 4*a*b*c*d*x + 2*a^2*d^2*x + b^2*c*d*x^3)/(2*c*d^2*Sqrt[c + d*x^2]) + ((-3*b^2*c + 4*a*b*d)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(2*d^(5/2))","A",1
653,1,78,75,0.1329714,"\int \frac{\left(a+b x^2\right)^2}{x \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x*(c + d*x^2)^(3/2)),x]","\frac{a^2 d^2-2 a b c d+2 b^2 c^2+b^2 c d x^2}{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{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,"(2*b^2*c^2 - 2*a*b*c*d + a^2*d^2 + b^2*c*d*x^2)/(c*d^2*Sqrt[c + d*x^2]) - (a^2*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/c^(3/2)","A",1
654,1,89,91,0.2343676,"\int \frac{\left(a+b x^2\right)^2}{x^2 \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^2*(c + d*x^2)^(3/2)),x]","\frac{-a^2 c d-2 a^2 d^2 x^2+2 a b c d x^2-b^2 c^2 x^2}{c^2 d x \sqrt{c+d x^2}}+\frac{b^2 \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{d^{3/2}}","-\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,"(-(a^2*c*d) - b^2*c^2*x^2 + 2*a*b*c*d*x^2 - 2*a^2*d^2*x^2)/(c^2*d*x*Sqrt[c + d*x^2]) + (b^2*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/d^(3/2)","A",1
655,1,103,103,0.234552,"\int \frac{\left(a+b x^2\right)^2}{x^3 \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^3*(c + d*x^2)^(3/2)),x]","\frac{-a^2 c d-3 a^2 d^2 x^2+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{\left(3 a^2 d-4 a b c\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{2 c^{5/2}}","\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,"(-(a^2*c*d) - 2*b^2*c^2*x^2 + 4*a*b*c*d*x^2 - 3*a^2*d^2*x^2)/(2*c^2*d*x^2*Sqrt[c + d*x^2]) + ((-4*a*b*c + 3*a^2*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*c^(5/2))","A",1
656,1,81,97,0.1804155,"\int \frac{\left(a+b x^2\right)^2}{x^4 \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^4*(c + d*x^2)^(3/2)),x]","\frac{-a^2 c^2+4 a^2 c d x^2+8 a^2 d^2 x^4-6 a b c^2 x^2-12 a b c d x^4+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,"(-(a^2*c^2) - 6*a*b*c^2*x^2 + 4*a^2*c*d*x^2 + 3*b^2*c^2*x^4 - 12*a*b*c*d*x^4 + 8*a^2*d^2*x^4)/(3*c^3*x^3*Sqrt[c + d*x^2])","A",1
657,1,132,145,0.2306746,"\int \frac{\left(a+b x^2\right)^2}{x^5 \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^5*(c + d*x^2)^(3/2)),x]","\frac{\left(-15 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)}{8 c^{7/2}}+\frac{-2 a^2 c^2+5 a^2 c d x^2+15 a^2 d^2 x^4-8 a b c^2 x^2-24 a b c d x^4+8 b^2 c^2 x^4}{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,"(-2*a^2*c^2 - 8*a*b*c^2*x^2 + 5*a^2*c*d*x^2 + 8*b^2*c^2*x^4 - 24*a*b*c*d*x^4 + 15*a^2*d^2*x^4)/(8*c^3*x^4*Sqrt[c + d*x^2]) + ((-8*b^2*c^2 + 24*a*b*c*d - 15*a^2*d^2)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(8*c^(7/2))","A",1
658,1,120,141,0.2400296,"\int \frac{\left(a+b x^2\right)^2}{x^6 \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^6*(c + d*x^2)^(3/2)),x]","\frac{-3 a^2 c^3+6 a^2 c^2 d x^2-24 a^2 c d^2 x^4-48 a^2 d^3 x^6-10 a b c^3 x^2+40 a b c^2 d x^4+80 a b c d^2 x^6-15 b^2 c^3 x^4-30 b^2 c^2 d x^6}{15 c^4 x^5 \sqrt{c+d x^2}}","-\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,"(-3*a^2*c^3 - 10*a*b*c^3*x^2 + 6*a^2*c^2*d*x^2 - 15*b^2*c^3*x^4 + 40*a*b*c^2*d*x^4 - 24*a^2*c*d^2*x^4 - 30*b^2*c^2*d*x^6 + 80*a*b*c*d^2*x^6 - 48*a^2*d^3*x^6)/(15*c^4*x^5*Sqrt[c + d*x^2])","A",1
659,1,174,190,0.3469896,"\int \frac{\left(a+b x^2\right)^2}{x^7 \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^7*(c + d*x^2)^(3/2)),x]","\frac{\left(35 a^2 d^3-60 a b c d^2+24 b^2 c^2 d\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{16 c^{9/2}}+\frac{-8 a^2 c^3+14 a^2 c^2 d x^2-35 a^2 c d^2 x^4-105 a^2 d^3 x^6-24 a b c^3 x^2+60 a b c^2 d x^4+180 a b c d^2 x^6-24 b^2 c^3 x^4-72 b^2 c^2 d x^6}{48 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,"(-8*a^2*c^3 - 24*a*b*c^3*x^2 + 14*a^2*c^2*d*x^2 - 24*b^2*c^3*x^4 + 60*a*b*c^2*d*x^4 - 35*a^2*c*d^2*x^4 - 72*b^2*c^2*d*x^6 + 180*a*b*c*d^2*x^6 - 105*a^2*d^3*x^6)/(48*c^4*x^6*Sqrt[c + d*x^2]) + ((24*b^2*c^2*d - 60*a*b*c*d^2 + 35*a^2*d^3)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(16*c^(9/2))","A",1
660,1,166,202,0.4848306,"\int \frac{x^4 \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(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) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{8 d^{9/2}}+\frac{-24 a^2 c d^2 x-32 a^2 d^3 x^3+120 a b c^2 d x+160 a b c d^2 x^3+24 a b d^3 x^5-105 b^2 c^3 x-140 b^2 c^2 d x^3-21 b^2 c d^2 x^5+6 b^2 d^3 x^7}{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,"(-105*b^2*c^3*x + 120*a*b*c^2*d*x - 24*a^2*c*d^2*x - 140*b^2*c^2*d*x^3 + 160*a*b*c*d^2*x^3 - 32*a^2*d^3*x^3 - 21*b^2*c*d^2*x^5 + 24*a*b*d^3*x^5 + 6*b^2*d^3*x^7)/(24*d^4*(c + d*x^2)^(3/2)) + ((35*b^2*c^2 - 40*a*b*c*d + 8*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(8*d^(9/2))","A",1
661,1,110,110,0.0951274,"\int \frac{x^3 \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(x^3*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x]","\frac{-2 a^2 c d^2-3 a^2 d^3 x^2+16 a b c^2 d+24 a b c d^2 x^2+6 a b d^3 x^4-16 b^2 c^3-24 b^2 c^2 d x^2-6 b^2 c d^2 x^4+b^2 d^3 x^6}{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,"(-16*b^2*c^3 + 16*a*b*c^2*d - 2*a^2*c*d^2 - 24*b^2*c^2*d*x^2 + 24*a*b*c*d^2*x^2 - 3*a^2*d^3*x^2 - 6*b^2*c*d^2*x^4 + 6*a*b*d^3*x^4 + b^2*d^3*x^6)/(3*d^4*(c + d*x^2)^(3/2))","A",1
662,1,127,121,0.3936264,"\int \frac{x^2 \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(x^2*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x]","\frac{2 a^2 d^3 x^3-12 a b c^2 d x-16 a b c d^2 x^3+15 b^2 c^3 x+20 b^2 c^2 d x^3+3 b^2 c d^2 x^5}{6 c d^3 \left(c+d x^2\right)^{3/2}}+\frac{\left(4 a b d-5 b^2 c\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\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,"(15*b^2*c^3*x - 12*a*b*c^2*d*x + 20*b^2*c^2*d*x^3 - 16*a*b*c*d^2*x^3 + 2*a^2*d^3*x^3 + 3*b^2*c*d^2*x^5)/(6*c*d^3*(c + d*x^2)^(3/2)) + ((-5*b^2*c + 4*a*b*d)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(2*d^(7/2))","A",1
663,1,72,72,0.0733213,"\int \frac{x \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(x*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x]","\frac{-a^2 d^2-4 a b c d-6 a b d^2 x^2+8 b^2 c^2+12 b^2 c d x^2+3 b^2 d^2 x^4}{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,"(8*b^2*c^2 - 4*a*b*c*d - a^2*d^2 + 12*b^2*c*d*x^2 - 6*a*b*d^2*x^2 + 3*b^2*d^2*x^4)/(3*d^3*(c + d*x^2)^(3/2))","A",1
664,1,104,105,0.2796628,"\int \frac{\left(a+b x^2\right)^2}{\left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(c + d*x^2)^(5/2),x]","\frac{3 a^2 c d^2 x+2 a^2 d^3 x^3+2 a b c d^2 x^3-3 b^2 c^3 x-4 b^2 c^2 d x^3}{3 c^2 d^2 \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}}","-\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,"(-3*b^2*c^3*x + 3*a^2*c*d^2*x - 4*b^2*c^2*d*x^3 + 2*a*b*c*d^2*x^3 + 2*a^2*d^3*x^3)/(3*c^2*d^2*(c + d*x^2)^(3/2)) + (b^2*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/d^(5/2)","A",1
665,1,99,88,0.161029,"\int \frac{\left(a+b x^2\right)^2}{x \left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x*(c + d*x^2)^(5/2)),x]","\frac{4 a^2 c d^2+3 a^2 d^3 x^2-2 a b c^2 d-2 b^2 c^3-3 b^2 c^2 d x^2}{3 c^2 d^2 \left(c+d x^2\right)^{3/2}}-\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{c^{5/2}}","\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,"(-2*b^2*c^3 - 2*a*b*c^2*d + 4*a^2*c*d^2 - 3*b^2*c^2*d*x^2 + 3*a^2*d^3*x^2)/(3*c^2*d^2*(c + d*x^2)^(3/2)) - (a^2*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/c^(5/2)","A",1
666,1,80,90,0.1822672,"\int \frac{\left(a+b x^2\right)^2}{x^2 \left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^2*(c + d*x^2)^(5/2)),x]","\frac{-3 a^2 c^2-12 a^2 c d x^2-8 a^2 d^2 x^4+6 a b c^2 x^2+4 a b c d x^4+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,"(-3*a^2*c^2 + 6*a*b*c^2*x^2 - 12*a^2*c*d*x^2 + b^2*c^2*x^4 + 4*a*b*c*d*x^4 - 8*a^2*d^2*x^4)/(3*c^3*x*(c + d*x^2)^(3/2))","A",1
667,1,130,131,0.2391732,"\int \frac{\left(a+b x^2\right)^2}{x^3 \left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^3*(c + d*x^2)^(5/2)),x]","\frac{-3 a^2 c^2 d-20 a^2 c d^2 x^2-15 a^2 d^3 x^4+16 a b c^2 d x^2+12 a b c d^2 x^4-2 b^2 c^3 x^2}{6 c^3 d x^2 \left(c+d x^2\right)^{3/2}}+\frac{\left(5 a^2 d-4 a b c\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{2 c^{7/2}}","\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,"(-3*a^2*c^2*d - 2*b^2*c^3*x^2 + 16*a*b*c^2*d*x^2 - 20*a^2*c*d^2*x^2 + 12*a*b*c*d^2*x^4 - 15*a^2*d^3*x^4)/(6*c^3*d*x^2*(c + d*x^2)^(3/2)) + ((-4*a*b*c + 5*a^2*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*c^(7/2))","A",1
668,1,120,131,0.2346981,"\int \frac{\left(a+b x^2\right)^2}{x^4 \left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^4*(c + d*x^2)^(5/2)),x]","\frac{-a^2 c^3+6 a^2 c^2 d x^2+24 a^2 c d^2 x^4+16 a^2 d^3 x^6-6 a b c^3 x^2-24 a b c^2 d x^4-16 a b c d^2 x^6+3 b^2 c^3 x^4+2 b^2 c^2 d x^6}{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,"(-(a^2*c^3) - 6*a*b*c^3*x^2 + 6*a^2*c^2*d*x^2 + 3*b^2*c^3*x^4 - 24*a*b*c^2*d*x^4 + 24*a^2*c*d^2*x^4 + 2*b^2*c^2*d*x^6 - 16*a*b*c*d^2*x^6 + 16*a^2*d^3*x^6)/(3*c^4*x^3*(c + d*x^2)^(3/2))","A",1
669,1,171,185,0.2879979,"\int \frac{\left(a+b x^2\right)^2}{x^5 \left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^5*(c + d*x^2)^(5/2)),x]","\frac{\left(-35 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)}{8 c^{9/2}}+\frac{-6 a^2 c^3+21 a^2 c^2 d x^2+140 a^2 c d^2 x^4+105 a^2 d^3 x^6-24 a b c^3 x^2-160 a b c^2 d x^4-120 a b c d^2 x^6+32 b^2 c^3 x^4+24 b^2 c^2 d x^6}{24 c^4 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,"(-6*a^2*c^3 - 24*a*b*c^3*x^2 + 21*a^2*c^2*d*x^2 + 32*b^2*c^3*x^4 - 160*a*b*c^2*d*x^4 + 140*a^2*c*d^2*x^4 + 24*b^2*c^2*d*x^6 - 120*a*b*c*d^2*x^6 + 105*a^2*d^3*x^6)/(24*c^4*x^4*(c + d*x^2)^(3/2)) + ((-8*b^2*c^2 + 40*a*b*c*d - 35*a^2*d^2)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(8*c^(9/2))","A",1
670,1,161,183,0.37092,"\int \frac{\left(a+b x^2\right)^2}{x^6 \left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(x^6*(c + d*x^2)^(5/2)),x]","\frac{-3 a^2 c^4+8 a^2 c^3 d x^2-48 a^2 c^2 d^2 x^4-192 a^2 c d^3 x^6-128 a^2 d^4 x^8-10 a b c^4 x^2+60 a b c^3 d x^4+240 a b c^2 d^2 x^6+160 a b c d^3 x^8-15 b^2 c^4 x^4-60 b^2 c^3 d x^6-40 b^2 c^2 d^2 x^8}{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,"(-3*a^2*c^4 - 10*a*b*c^4*x^2 + 8*a^2*c^3*d*x^2 - 15*b^2*c^4*x^4 + 60*a*b*c^3*d*x^4 - 48*a^2*c^2*d^2*x^4 - 60*b^2*c^3*d*x^6 + 240*a*b*c^2*d^2*x^6 - 192*a^2*c*d^3*x^6 - 40*b^2*c^2*d^2*x^8 + 160*a*b*c*d^3*x^8 - 128*a^2*d^4*x^8)/(15*c^5*x^5*(c + d*x^2)^(3/2))","A",1
671,1,60,72,0.0551058,"\int \frac{x^5}{\sqrt{d x^2} \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[x^5/(Sqrt[d*x^2]*(a + b*x^2)),x]","\sqrt{d x^2} \left(\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{5/2} d x}+\frac{b x^2-3 a}{3 b^2 d}\right)","\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,"Sqrt[d*x^2]*((-3*a + b*x^2)/(3*b^2*d) + (a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(b^(5/2)*d*x))","A",1
672,1,49,52,0.0344189,"\int \frac{x^3}{\sqrt{d x^2} \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[x^3/(Sqrt[d*x^2]*(a + b*x^2)),x]","\sqrt{d x^2} \left(\frac{1}{b d}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{3/2} d x}\right)","\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,"Sqrt[d*x^2]*(1/(b*d) - (Sqrt[a]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(b^(3/2)*d*x))","A",1
673,1,39,34,0.0232493,"\int \frac{x}{\sqrt{d x^2} \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[x/(Sqrt[d*x^2]*(a + b*x^2)),x]","\frac{\sqrt{d x^2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b} d x}","\frac{x \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b} \sqrt{d x^2}}",1,"(Sqrt[d*x^2]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*d*x)","A",1
674,1,53,50,0.039592,"\int \frac{1}{x \sqrt{d x^2} \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[1/(x*Sqrt[d*x^2]*(a + b*x^2)),x]","\sqrt{d x^2} \left(-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2} d x}-\frac{1}{a d x^2}\right)","-\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,"Sqrt[d*x^2]*(-(1/(a*d*x^2)) - (Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*d*x))","A",1
675,1,64,68,0.0555961,"\int \frac{1}{x^3 \sqrt{d x^2} \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[1/(x^3*Sqrt[d*x^2]*(a + b*x^2)),x]","\sqrt{d x^2} \left(\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{5/2} d x}+\frac{3 b x^2-a}{3 a^2 d x^4}\right)","\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,"Sqrt[d*x^2]*((-a + 3*b*x^2)/(3*a^2*d*x^4) + (b^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(5/2)*d*x))","A",1
676,1,145,157,0.4925735,"\int \frac{x^4 \sqrt{c+d x^2}}{a+b x^2} \, dx","IntegrateAlgebraic[(x^4*Sqrt[c + d*x^2])/(a + b*x^2),x]","\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{\sqrt{c+d x^2} \left(-4 a d x+b c x+2 b d x^3\right)}{8 b^2 d}","\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,"(Sqrt[c + d*x^2]*(b*c*x - 4*a*d*x + 2*b*d*x^3))/(8*b^2*d) + (a^(3/2)*Sqrt[b*c - a*d]*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/b^3 + ((-(b^2*c^2) - 4*a*b*c*d + 8*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(8*b^3*d^(3/2))","A",1
677,1,96,88,0.1816862,"\int \frac{x^3 \sqrt{c+d x^2}}{a+b x^2} \, dx","IntegrateAlgebraic[(x^3*Sqrt[c + d*x^2])/(a + b*x^2),x]","\frac{\sqrt{c+d x^2} \left(-3 a d+b c+b d x^2\right)}{3 b^2 d}-\frac{a \sqrt{a d-b c} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{b^{5/2}}","\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]*(b*c - 3*a*d + b*d*x^2))/(3*b^2*d) - (a*Sqrt[-(b*c) + a*d]*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/b^(5/2)","A",1
678,1,112,112,0.3243322,"\int \frac{x^2 \sqrt{c+d x^2}}{a+b x^2} \, dx","IntegrateAlgebraic[(x^2*Sqrt[c + d*x^2])/(a + b*x^2),x]","-\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}","-\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,"(x*Sqrt[c + d*x^2])/(2*b) - (Sqrt[a]*Sqrt[b*c - a*d]*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/b^2 + ((b*c - 2*a*d)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(2*b^2*Sqrt[d])","A",1
679,1,74,65,0.118689,"\int \frac{x \sqrt{c+d x^2}}{a+b x^2} \, dx","IntegrateAlgebraic[(x*Sqrt[c + d*x^2])/(a + b*x^2),x]","\frac{\sqrt{a d-b c} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{b^{3/2}}+\frac{\sqrt{c+d x^2}}{b}","\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]*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/b^(3/2)","A",1
680,1,81,81,0.2422637,"\int \frac{\sqrt{c+d x^2}}{a+b x^2} \, dx","IntegrateAlgebraic[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} \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\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]*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/b","A",1
681,1,91,80,0.1600891,"\int \frac{\sqrt{c+d x^2}}{x \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[Sqrt[c + d*x^2]/(x*(a + b*x^2)),x]","-\frac{\sqrt{a d-b c} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{a \sqrt{b}}-\frac{\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[-(b*c) + a*d]*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(a*Sqrt[b])) - (Sqrt[c]*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/a","A",1
682,1,70,70,0.198384,"\int \frac{\sqrt{c+d x^2}}{x^2 \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[Sqrt[c + d*x^2]/(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)}{a^{3/2}}-\frac{\sqrt{c+d x^2}}{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]/(a*x)) - (Sqrt[b*c - a*d]*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/a^(3/2)","A",1
683,1,122,113,0.218628,"\int \frac{\sqrt{c+d x^2}}{x^3 \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[Sqrt[c + d*x^2]/(x^3*(a + b*x^2)),x]","\frac{\sqrt{b} \sqrt{a d-b c} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{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{c+d x^2}}{2 a x^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,"-1/2*Sqrt[c + d*x^2]/(a*x^2) + (Sqrt[b]*Sqrt[-(b*c) + a*d]*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/a^2 + ((2*b*c - a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*a^2*Sqrt[c])","A",1
684,1,94,105,0.2694788,"\int \frac{\sqrt{c+d x^2}}{x^4 \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[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(-a c-a d x^2+3 b c x^2\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]*(-(a*c) + 3*b*c*x^2 - a*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
685,1,199,210,0.7125278,"\int \frac{x^4 \left(c+d x^2\right)^{3/2}}{a+b x^2} \, dx","IntegrateAlgebraic[(x^4*(c + d*x^2)^(3/2))/(a + b*x^2),x]","\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{\sqrt{c+d x^2} \left(24 a^2 d^2 x-30 a b c d x-12 a b d^2 x^3+3 b^2 c^2 x+14 b^2 c d x^3+8 b^2 d^2 x^5\right)}{48 b^3 d}+\frac{\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) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{16 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,"(Sqrt[c + d*x^2]*(3*b^2*c^2*x - 30*a*b*c*d*x + 24*a^2*d^2*x + 14*b^2*c*d*x^3 - 12*a*b*d^2*x^3 + 8*b^2*d^2*x^5))/(48*b^3*d) + (a^(3/2)*(b*c - a*d)^(3/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/b^4 + ((-(b^3*c^3) - 6*a*b^2*c^2*d + 24*a^2*b*c*d^2 - 16*a^3*d^3)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(16*b^4*d^(3/2))","A",1
686,1,135,115,0.2664159,"\int \frac{x^3 \left(c+d x^2\right)^{3/2}}{a+b x^2} \, dx","IntegrateAlgebraic[(x^3*(c + d*x^2)^(3/2))/(a + b*x^2),x]","\frac{\sqrt{c+d x^2} \left(15 a^2 d^2-20 a b c d-5 a b d^2 x^2+3 b^2 c^2+6 b^2 c d x^2+3 b^2 d^2 x^4\right)}{15 b^3 d}+\frac{a (a d-b c)^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{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]*(3*b^2*c^2 - 20*a*b*c*d + 15*a^2*d^2 + 6*b^2*c*d*x^2 - 5*a*b*d^2*x^2 + 3*b^2*d^2*x^4))/(15*b^3*d) + (a*(-(b*c) + a*d)^(3/2)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/b^(7/2)","A",1
687,1,144,158,0.6101331,"\int \frac{x^2 \left(c+d x^2\right)^{3/2}}{a+b x^2} \, dx","IntegrateAlgebraic[(x^2*(c + d*x^2)^(3/2))/(a + b*x^2),x]","\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{\sqrt{c+d x^2} \left(-4 a d x+5 b c x+2 b d x^3\right)}{8 b^2}","\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,"(Sqrt[c + d*x^2]*(5*b*c*x - 4*a*d*x + 2*b*d*x^3))/(8*b^2) - (Sqrt[a]*(b*c - a*d)^(3/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/b^3 + ((3*b^2*c^2 - 12*a*b*c*d + 8*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(8*b^3*Sqrt[d])","A",1
688,1,93,91,0.1843732,"\int \frac{x \left(c+d x^2\right)^{3/2}}{a+b x^2} \, dx","IntegrateAlgebraic[(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{(a d-b c)^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{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)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/b^(5/2)","A",1
689,1,117,113,0.3838465,"\int \frac{\left(c+d x^2\right)^{3/2}}{a+b x^2} \, dx","IntegrateAlgebraic[(c + d*x^2)^(3/2)/(a + b*x^2),x]","\frac{\left(3 b c \sqrt{d}-2 a d^{3/2}\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{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{d x \sqrt{c+d x^2}}{2 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)}{\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,"(d*x*Sqrt[c + d*x^2])/(2*b) + ((b*c - a*d)^(3/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(Sqrt[a]*b^2) + ((3*b*c*Sqrt[d] - 2*a*d^(3/2))*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(2*b^2)","A",1
690,1,106,96,0.226224,"\int \frac{\left(c+d x^2\right)^{3/2}}{x \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(c + d*x^2)^(3/2)/(x*(a + b*x^2)),x]","\frac{(a d-b c)^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{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}","\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,"(d*Sqrt[c + d*x^2])/b + ((-(b*c) + a*d)^(3/2)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(a*b^(3/2)) - (c^(3/2)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/a","A",1
691,1,102,102,0.3215816,"\int \frac{\left(c+d x^2\right)^{3/2}}{x^2 \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(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} \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\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)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/b","A",1
692,1,128,114,0.2596172,"\int \frac{\left(c+d x^2\right)^{3/2}}{x^3 \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(c + d*x^2)^(3/2)/(x^3*(a + b*x^2)),x]","\frac{\left(2 b c^{3/2}-3 a \sqrt{c} d\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{2 a^2}-\frac{(a d-b c)^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{a^2 \sqrt{b}}-\frac{c \sqrt{c+d x^2}}{2 a x^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,"-1/2*(c*Sqrt[c + d*x^2])/(a*x^2) - ((-(b*c) + a*d)^(3/2)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(a^2*Sqrt[b]) + ((2*b*c^(3/2) - 3*a*Sqrt[c]*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*a^2)","A",1
693,1,90,102,0.2555276,"\int \frac{\left(c+d x^2\right)^{3/2}}{x^4 \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(c + d*x^2)^(3/2)/(x^4*(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^{5/2}}+\frac{\sqrt{c+d x^2} \left(-a c-4 a d x^2+3 b c x^2\right)}{3 a^2 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,"(Sqrt[c + d*x^2]*(-(a*c) + 3*b*c*x^2 - 4*a*d*x^2))/(3*a^2*x^3) + ((b*c - a*d)^(3/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/a^(5/2)","A",1
694,1,269,291,1.0192933,"\int \frac{x^4 \left(c+d x^2\right)^{5/2}}{a+b x^2} \, dx","IntegrateAlgebraic[(x^4*(c + d*x^2)^(5/2))/(a + b*x^2),x]","\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{\sqrt{c+d x^2} \left(-192 a^3 d^3 x+432 a^2 b c d^2 x+96 a^2 b d^3 x^3-264 a b^2 c^2 d x-208 a b^2 c d^2 x^3-64 a b^2 d^3 x^5+15 b^3 c^3 x+118 b^3 c^2 d x^3+136 b^3 c d^2 x^5+48 b^3 d^3 x^7\right)}{384 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{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,"(Sqrt[c + d*x^2]*(15*b^3*c^3*x - 264*a*b^2*c^2*d*x + 432*a^2*b*c*d^2*x - 192*a^3*d^3*x + 118*b^3*c^2*d*x^3 - 208*a*b^2*c*d^2*x^3 + 96*a^2*b*d^3*x^3 + 136*b^3*c*d^2*x^5 - 64*a*b^2*d^3*x^5 + 48*b^3*d^3*x^7))/(384*b^4*d) + (a^(3/2)*(b*c - a*d)^(5/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/b^5 + ((-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)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(128*b^5*d^(3/2))","A",1
695,1,191,144,0.3233766,"\int \frac{x^3 \left(c+d x^2\right)^{5/2}}{a+b x^2} \, dx","IntegrateAlgebraic[(x^3*(c + d*x^2)^(5/2))/(a + b*x^2),x]","\frac{\sqrt{c+d x^2} \left(-105 a^3 d^3+245 a^2 b c d^2+35 a^2 b d^3 x^2-161 a b^2 c^2 d-77 a b^2 c d^2 x^2-21 a b^2 d^3 x^4+15 b^3 c^3+45 b^3 c^2 d x^2+45 b^3 c d^2 x^4+15 b^3 d^3 x^6\right)}{105 b^4 d}-\frac{a (a d-b c)^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{b^{9/2}}","\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,"(Sqrt[c + d*x^2]*(15*b^3*c^3 - 161*a*b^2*c^2*d + 245*a^2*b*c*d^2 - 105*a^3*d^3 + 45*b^3*c^2*d*x^2 - 77*a*b^2*c*d^2*x^2 + 35*a^2*b*d^3*x^2 + 45*b^3*c*d^2*x^4 - 21*a*b^2*d^3*x^4 + 15*b^3*d^3*x^6))/(105*b^4*d) - (a*(-(b*c) + a*d)^(5/2)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/b^(9/2)","A",1
696,1,197,217,0.8099319,"\int \frac{x^2 \left(c+d x^2\right)^{5/2}}{a+b x^2} \, dx","IntegrateAlgebraic[(x^2*(c + d*x^2)^(5/2))/(a + b*x^2),x]","\frac{\sqrt{c+d x^2} \left(24 a^2 d^2 x-54 a b c d x-12 a b d^2 x^3+33 b^2 c^2 x+26 b^2 c d x^3+8 b^2 d^2 x^5\right)}{48 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{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,"(Sqrt[c + d*x^2]*(33*b^2*c^2*x - 54*a*b*c*d*x + 24*a^2*d^2*x + 26*b^2*c*d*x^3 - 12*a*b*d^2*x^3 + 8*b^2*d^2*x^5))/(48*b^3) - (Sqrt[a]*(b*c - a*d)^(5/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/b^4 + ((5*b^3*c^3 - 30*a*b^2*c^2*d + 40*a^2*b*c*d^2 - 16*a^3*d^3)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(16*b^4*Sqrt[d])","A",1
697,1,131,119,0.2351532,"\int \frac{x \left(c+d x^2\right)^{5/2}}{a+b x^2} \, dx","IntegrateAlgebraic[(x*(c + d*x^2)^(5/2))/(a + b*x^2),x]","\frac{\sqrt{c+d x^2} \left(15 a^2 d^2-35 a b c d-5 a b d^2 x^2+23 b^2 c^2+11 b^2 c d x^2+3 b^2 d^2 x^4\right)}{15 b^3}+\frac{(a d-b c)^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{b^{7/2}}","-\frac{(b c-a d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right)}{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,"(Sqrt[c + d*x^2]*(23*b^2*c^2 - 35*a*b*c*d + 15*a^2*d^2 + 11*b^2*c*d*x^2 - 5*a*b*d^2*x^2 + 3*b^2*d^2*x^4))/(15*b^3) + ((-(b*c) + a*d)^(5/2)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/b^(7/2)","A",1
698,1,154,156,0.5721788,"\int \frac{\left(c+d x^2\right)^{5/2}}{a+b x^2} \, dx","IntegrateAlgebraic[(c + d*x^2)^(5/2)/(a + b*x^2),x]","\frac{\left(8 a^2 d^{5/2}-20 a b c d^{3/2}+15 b^2 c^2 \sqrt{d}\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{\sqrt{c+d x^2} \left(-4 a d^2 x+9 b c d x+2 b d^2 x^3\right)}{8 b^2}","\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,"(Sqrt[c + d*x^2]*(9*b*c*d*x - 4*a*d^2*x + 2*b*d^2*x^3))/(8*b^2) + ((b*c - a*d)^(5/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(Sqrt[a]*b^3) + ((15*b^2*c^2*Sqrt[d] - 20*a*b*c*d^(3/2) + 8*a^2*d^(5/2))*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(8*b^3)","A",1
699,1,129,124,0.282337,"\int \frac{\left(c+d x^2\right)^{5/2}}{x \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(c + d*x^2)^(5/2)/(x*(a + b*x^2)),x]","-\frac{(a d-b c)^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{a b^{5/2}}+\frac{\sqrt{c+d x^2} \left(-3 a d^2+7 b c d+b d^2 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,"(Sqrt[c + d*x^2]*(7*b*c*d - 3*a*d^2 + b*d^2*x^2))/(3*b^2) - ((-(b*c) + a*d)^(5/2)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(a*b^(5/2)) - (c^(5/2)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/a","A",1
700,1,137,145,0.4596069,"\int \frac{\left(c+d x^2\right)^{5/2}}{x^2 \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(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{\left(5 b c d^{3/2}-2 a d^{5/2}\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{2 b^2}+\frac{\sqrt{c+d x^2} \left(a d^2 x^2-2 b c^2\right)}{2 a b 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) \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,"(Sqrt[c + d*x^2]*(-2*b*c^2 + a*d^2*x^2))/(2*a*b*x) - ((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) + ((5*b*c*d^(3/2) - 2*a*d^(5/2))*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(2*b^2)","A",1
701,1,145,144,0.3127544,"\int \frac{\left(c+d x^2\right)^{5/2}}{x^3 \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(c + d*x^2)^(5/2)/(x^3*(a + b*x^2)),x]","\frac{(a d-b c)^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{a^2 b^{3/2}}+\frac{\left(2 b c^{5/2}-5 a c^{3/2} d\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{2 a^2}+\frac{\sqrt{c+d x^2} \left(2 a d^2 x^2-b c^2\right)}{2 a b x^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,"(Sqrt[c + d*x^2]*(-(b*c^2) + 2*a*d^2*x^2))/(2*a*b*x^2) + ((-(b*c) + a*d)^(5/2)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(a^2*b^(3/2)) + ((2*b*c^(5/2) - 5*a*c^(3/2)*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*a^2)","A",1
702,1,126,130,0.3780664,"\int \frac{\left(c+d x^2\right)^{5/2}}{x^4 \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(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{\sqrt{c+d x^2} \left(-a c^2-7 a c d x^2+3 b c^2 x^2\right)}{3 a^2 x^3}+\frac{d^{5/2} \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\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,"(Sqrt[c + d*x^2]*(-(a*c^2) + 3*b*c^2*x^2 - 7*a*c*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)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/b","A",1
703,1,99,100,0.1940426,"\int \frac{x^5}{\left(a+b x^2\right) \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[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 \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{b^{5/2} \sqrt{a d-b c}}","-\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*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(b^(5/2)*Sqrt[-(b*c) + a*d])","A",1
704,1,78,68,0.1161964,"\int \frac{x^3}{\left(a+b x^2\right) \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[x^3/((a + b*x^2)*Sqrt[c + d*x^2]),x]","\frac{a \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{b^{3/2} \sqrt{a d-b c}}+\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*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(b^(3/2)*Sqrt[-(b*c) + a*d])","A",1
705,1,59,49,0.0760136,"\int \frac{x}{\left(a+b x^2\right) \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[x/((a + b*x^2)*Sqrt[c + d*x^2]),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{\sqrt{b} \sqrt{a d-b c}}","-\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,"-(ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)]/(Sqrt[b]*Sqrt[-(b*c) + a*d]))","A",1
706,1,90,80,0.1515126,"\int \frac{1}{x \left(a+b x^2\right) \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[1/(x*(a + b*x^2)*Sqrt[c + d*x^2]),x]","\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{a \sqrt{a d-b c}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{a \sqrt{c}}","\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,"(Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(a*Sqrt[-(b*c) + a*d]) - ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]]/(a*Sqrt[c])","A",1
707,1,125,115,0.203683,"\int \frac{1}{x^3 \left(a+b x^2\right) \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[1/(x^3*(a + b*x^2)*Sqrt[c + d*x^2]),x]","-\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{a^2 \sqrt{a d-b c}}+\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}","-\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,"-1/2*Sqrt[c + d*x^2]/(a*c*x^2) - (b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(a^2*Sqrt[-(b*c) + a*d]) + ((2*b*c + a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*a^2*c^(3/2))","A",1
708,1,115,114,0.3956559,"\int \frac{x^4}{\left(a+b x^2\right) \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[x^4/((a + b*x^2)*Sqrt[c + d*x^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^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}","\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,"(x*Sqrt[c + d*x^2])/(2*b*d) + (a^(3/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(b^2*Sqrt[b*c - a*d]) + ((-(b*c) - 2*a*d)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(2*b^2*d^(3/2))","A",1
709,1,82,82,0.2743938,"\int \frac{x^2}{\left(a+b x^2\right) \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[x^2/((a + b*x^2)*Sqrt[c + d*x^2]),x]","\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}}","\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])) + ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]]/(b*Sqrt[d])","A",1
710,1,49,49,0.0006134,"\int \frac{1}{\left(a+b x^2\right) \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[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
711,1,74,74,0.2358354,"\int \frac{1}{x^2 \left(a+b x^2\right) \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[1/(x^2*(a + b*x^2)*Sqrt[c + d*x^2]),x]","-\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}","-\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,"-(Sqrt[c + d*x^2]/(a*c*x)) - (b*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(a^(3/2)*Sqrt[b*c - a*d])","A",1
712,1,96,110,0.34906,"\int \frac{1}{x^4 \left(a+b x^2\right) \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[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
713,1,108,109,0.4939772,"\int \frac{x^4}{\left(a+b x^2\right) \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[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{\tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\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)) + ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]]/(b*d^(3/2))","A",1
714,1,87,77,0.1744174,"\int \frac{x^3}{\left(a+b x^2\right) \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[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 \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{\sqrt{b} (a d-b c)^{3/2}}","\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*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(Sqrt[b]*(-(b*c) + a*d)^(3/2))","A",1
715,1,74,74,0.3361314,"\int \frac{x^2}{\left(a+b x^2\right) \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[x^2/((a + b*x^2)*(c + d*x^2)^(3/2)),x]","\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}}","\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,"x/((b*c - a*d)*Sqrt[c + d*x^2]) - (Sqrt[a]*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(b*c - a*d)^(3/2)","A",1
716,1,81,72,0.1419233,"\int \frac{x}{\left(a+b x^2\right) \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[x/((a + b*x^2)*(c + d*x^2)^(3/2)),x]","\frac{1}{\sqrt{c+d x^2} (b c-a d)}+\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{(a d-b c)^{3/2}}","\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,"1/((b*c - a*d)*Sqrt[c + d*x^2]) + (Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(-(b*c) + a*d)^(3/2)","A",1
717,1,79,79,0.0005966,"\int \frac{1}{\left(a+b x^2\right) \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[1/((a + b*x^2)*(c + d*x^2)^(3/2)),x]","\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)}","\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,"-((d*x)/(c*(b*c - a*d)*Sqrt[c + d*x^2])) + (b*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(Sqrt[a]*(b*c - a*d)^(3/2))","A",1
718,1,118,107,0.2643964,"\int \frac{1}{x \left(a+b x^2\right) \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[1/(x*(a + b*x^2)*(c + d*x^2)^(3/2)),x]","-\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{a (a d-b c)^{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}}","\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,"-(d/(c*(b*c - a*d)*Sqrt[c + d*x^2])) - (b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(a*(-(b*c) + a*d)^(3/2)) - ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]]/(a*c^(3/2))","A",1
719,1,112,124,0.466835,"\int \frac{1}{x^2 \left(a+b x^2\right) \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[1/(x^2*(a + b*x^2)*(c + d*x^2)^(3/2)),x]","\frac{-a c d-2 a d^2 x^2+b c^2+b c d x^2}{a c^2 x \sqrt{c+d x^2} (a d-b c)}-\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,"(b*c^2 - a*c*d + b*c*d*x^2 - 2*a*d^2*x^2)/(a*c^2*(-(b*c) + a*d)*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
720,1,162,156,0.3678306,"\int \frac{1}{x^3 \left(a+b x^2\right) \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[1/(x^3*(a + b*x^2)*(c + d*x^2)^(3/2)),x]","\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{a^2 (a d-b c)^{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{-a c d-3 a d^2 x^2+b c^2+b c d x^2}{2 a c^2 x^2 \sqrt{c+d x^2} (a d-b c)}","-\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,"(b*c^2 - a*c*d + b*c*d*x^2 - 3*a*d^2*x^2)/(2*a*c^2*(-(b*c) + a*d)*x^2*Sqrt[c + d*x^2]) + (b^(5/2)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(a^2*(-(b*c) + a*d)^(3/2)) + ((2*b*c + 3*a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*a^2*c^(5/2))","A",1
721,1,171,176,0.6846866,"\int \frac{1}{x^4 \left(a+b x^2\right) \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[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{-a^2 c^2 d+4 a^2 c d^2 x^2+8 a^2 d^3 x^4+a b c^3-a b c^2 d x^2-2 a b c d^2 x^4-3 b^2 c^3 x^2-3 b^2 c^2 d x^4}{3 a^2 c^3 x^3 \sqrt{c+d x^2} (a d-b c)}","\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,"(a*b*c^3 - a^2*c^2*d - 3*b^2*c^3*x^2 - a*b*c^2*d*x^2 + 4*a^2*c*d^2*x^2 - 3*b^2*c^2*d*x^4 - 2*a*b*c*d^2*x^4 + 8*a^2*d^3*x^4)/(3*a^2*c^3*(-(b*c) + a*d)*x^3*Sqrt[c + d*x^2]) + (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
722,1,94,117,0.5535942,"\int \frac{x^4}{\left(a+b x^2\right) \left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[x^4/((a + b*x^2)*(c + d*x^2)^(5/2)),x]","\frac{a^{3/2} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{(b c-a d)^{5/2}}+\frac{-3 a c x-4 a d x^3+b c x^3}{3 \left(c+d x^2\right)^{3/2} (b c-a d)^2}","\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,"(-3*a*c*x + b*c*x^3 - 4*a*d*x^3)/(3*(b*c - a*d)^2*(c + d*x^2)^(3/2)) + (a^(3/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(b*c - a*d)^(5/2)","A",1
723,1,109,103,0.2585254,"\int \frac{x^3}{\left(a+b x^2\right) \left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[x^3/((a + b*x^2)*(c + d*x^2)^(5/2)),x]","\frac{-2 a c d-3 a d^2 x^2-b c^2}{3 d \left(c+d x^2\right)^{3/2} (a d-b c)^2}+\frac{a \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{(a d-b c)^{5/2}}","-\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,"(-(b*c^2) - 2*a*c*d - 3*a*d^2*x^2)/(3*d*(-(b*c) + a*d)^2*(c + d*x^2)^(3/2)) + (a*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(-(b*c) + a*d)^(5/2)","A",1
724,1,104,115,0.4623334,"\int \frac{x^2}{\left(a+b x^2\right) \left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[x^2/((a + b*x^2)*(c + d*x^2)^(5/2)),x]","\frac{a d^2 x^3+3 b c^2 x+2 b c d x^3}{3 c \left(c+d x^2\right)^{3/2} (b c-a d)^2}-\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}}","\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,"(3*b*c^2*x + 2*b*c*d*x^3 + a*d^2*x^3)/(3*c*(b*c - a*d)^2*(c + d*x^2)^(3/2)) - (Sqrt[a]*b*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(b*c - a*d)^(5/2)","A",1
725,1,101,98,0.2146881,"\int \frac{x}{\left(a+b x^2\right) \left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[x/((a + b*x^2)*(c + d*x^2)^(5/2)),x]","\frac{-a d+4 b c+3 b d x^2}{3 \left(c+d x^2\right)^{3/2} (b c-a d)^2}-\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{(a d-b c)^{5/2}}","-\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,"(4*b*c - a*d + 3*b*d*x^2)/(3*(b*c - a*d)^2*(c + d*x^2)^(3/2)) - (b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(-(b*c) + a*d)^(5/2)","A",1
726,1,117,122,0.5470061,"\int \frac{1}{\left(a+b x^2\right) \left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[1/((a + b*x^2)*(c + d*x^2)^(5/2)),x]","\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{3 a c d^2 x+2 a d^3 x^3-6 b c^2 d x-5 b c d^2 x^3}{3 c^2 \left(c+d x^2\right)^{3/2} (b c-a d)^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,"(-6*b*c^2*d*x + 3*a*c*d^2*x - 5*b*c*d^2*x^3 + 2*a*d^3*x^3)/(3*c^2*(b*c - a*d)^2*(c + d*x^2)^(3/2)) + (b^2*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(Sqrt[a]*(b*c - a*d)^(5/2))","A",1
727,1,152,145,0.414453,"\int \frac{1}{x \left(a+b x^2\right) \left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[1/(x*(a + b*x^2)*(c + d*x^2)^(5/2)),x]","\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{a (a d-b c)^{5/2}}+\frac{4 a c d^2+3 a d^3 x^2-7 b c^2 d-6 b c d^2 x^2}{3 c^2 \left(c+d x^2\right)^{3/2} (b c-a d)^2}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{a c^{5/2}}","\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,"(-7*b*c^2*d + 4*a*c*d^2 - 6*b*c*d^2*x^2 + 3*a*d^3*x^2)/(3*c^2*(b*c - a*d)^2*(c + d*x^2)^(3/2)) + (b^(5/2)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(a*(-(b*c) + a*d)^(5/2)) - ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]]/(a*c^(5/2))","A",1
728,1,189,178,0.7412932,"\int \frac{1}{x^2 \left(a+b x^2\right) \left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[1/(x^2*(a + b*x^2)*(c + d*x^2)^(5/2)),x]","\frac{-3 a^2 c^2 d^2-12 a^2 c d^3 x^2-8 a^2 d^4 x^4+6 a b c^3 d+21 a b c^2 d^2 x^2+14 a b c d^3 x^4-3 b^2 c^4-6 b^2 c^3 d x^2-3 b^2 c^2 d^2 x^4}{3 a c^3 x \left(c+d x^2\right)^{3/2} (a d-b c)^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{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,"(-3*b^2*c^4 + 6*a*b*c^3*d - 3*a^2*c^2*d^2 - 6*b^2*c^3*d*x^2 + 21*a*b*c^2*d^2*x^2 - 12*a^2*c*d^3*x^2 - 3*b^2*c^2*d^2*x^4 + 14*a*b*c*d^3*x^4 - 8*a^2*d^4*x^4)/(3*a*c^3*(-(b*c) + a*d)^2*x*(c + d*x^2)^(3/2)) - (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
729,1,237,211,0.6398687,"\int \frac{1}{x^3 \left(a+b x^2\right) \left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[1/(x^3*(a + b*x^2)*(c + d*x^2)^(5/2)),x]","-\frac{b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{a^2 (a d-b c)^{5/2}}+\frac{-3 a^2 c^2 d^2-20 a^2 c d^3 x^2-15 a^2 d^4 x^4+6 a b c^3 d+32 a b c^2 d^2 x^2+24 a b c d^3 x^4-3 b^2 c^4-6 b^2 c^3 d x^2-3 b^2 c^2 d^2 x^4}{6 a c^3 x^2 \left(c+d x^2\right)^{3/2} (a d-b c)^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{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,"(-3*b^2*c^4 + 6*a*b*c^3*d - 3*a^2*c^2*d^2 - 6*b^2*c^3*d*x^2 + 32*a*b*c^2*d^2*x^2 - 20*a^2*c*d^3*x^2 - 3*b^2*c^2*d^2*x^4 + 24*a*b*c*d^3*x^4 - 15*a^2*d^4*x^4)/(6*a*c^3*(-(b*c) + a*d)^2*x^2*(c + d*x^2)^(3/2)) - (b^(7/2)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(a^2*(-(b*c) + a*d)^(5/2)) + ((2*b*c + 5*a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*a^2*c^(7/2))","A",1
730,1,265,245,1.3607659,"\int \frac{1}{x^4 \left(a+b x^2\right) \left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[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{-a^3 c^3 d^2+6 a^3 c^2 d^3 x^2+24 a^3 c d^4 x^4+16 a^3 d^5 x^6+2 a^2 b c^4 d-9 a^2 b c^3 d^2 x^2-36 a^2 b c^2 d^3 x^4-24 a^2 b c d^4 x^6-a b^2 c^5+3 a b^2 c^3 d^2 x^4+2 a b^2 c^2 d^3 x^6+3 b^3 c^5 x^2+6 b^3 c^4 d x^4+3 b^3 c^3 d^2 x^6}{3 a^2 c^4 x^3 \left(c+d x^2\right)^{3/2} (a d-b c)^2}","\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,"(-(a*b^2*c^5) + 2*a^2*b*c^4*d - a^3*c^3*d^2 + 3*b^3*c^5*x^2 - 9*a^2*b*c^3*d^2*x^2 + 6*a^3*c^2*d^3*x^2 + 6*b^3*c^4*d*x^4 + 3*a*b^2*c^3*d^2*x^4 - 36*a^2*b*c^2*d^3*x^4 + 24*a^3*c*d^4*x^4 + 3*b^3*c^3*d^2*x^6 + 2*a*b^2*c^2*d^3*x^6 - 24*a^2*b*c*d^4*x^6 + 16*a^3*d^5*x^6)/(3*a^2*c^4*(-(b*c) + a*d)^2*x^3*(c + d*x^2)^(3/2)) + (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
731,1,145,150,0.5410807,"\int \frac{x^4 \sqrt{c+d x^2}}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^4*Sqrt[c + d*x^2])/(a + b*x^2)^2,x]","\frac{\left(4 a^{3/2} d-3 \sqrt{a} b c\right) \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{\left(2 a x+b x^3\right) \sqrt{c+d x^2}}{2 b^2 \left(a+b x^2\right)}","-\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,"(Sqrt[c + d*x^2]*(2*a*x + b*x^3))/(2*b^2*(a + b*x^2)) + ((-3*Sqrt[a]*b*c + 4*a^(3/2)*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*b^3*Sqrt[b*c - a*d]) + ((b*c - 4*a*d)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(2*b^3*Sqrt[d])","A",1
732,1,108,136,0.22766,"\int \frac{x^3 \sqrt{c+d x^2}}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^3*Sqrt[c + d*x^2])/(a + b*x^2)^2,x]","\frac{(3 a d-2 b c) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{2 b^{5/2} \sqrt{a d-b c}}+\frac{\left(3 a+2 b x^2\right) \sqrt{c+d x^2}}{2 b^2 \left(a+b x^2\right)}","-\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,"((3*a + 2*b*x^2)*Sqrt[c + d*x^2])/(2*b^2*(a + b*x^2)) + ((-2*b*c + 3*a*d)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(2*b^(5/2)*Sqrt[-(b*c) + a*d])","A",1
733,1,120,120,0.3768386,"\int \frac{x^2 \sqrt{c+d x^2}}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^2*Sqrt[c + d*x^2])/(a + b*x^2)^2,x]","\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}","\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,"-1/2*(x*Sqrt[c + d*x^2])/(b*(a + b*x^2)) + ((b*c - 2*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*Sqrt[a]*b^2*Sqrt[b*c - a*d]) + (Sqrt[d]*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/b^2","A",1
734,1,90,80,0.1580039,"\int \frac{x \sqrt{c+d x^2}}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(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}}{b c-a d}\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[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(2*b^(3/2)*Sqrt[-(b*c) + a*d])","A",1
735,1,82,82,0.2477483,"\int \frac{\sqrt{c+d x^2}}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[Sqrt[c + d*x^2]/(a + b*x^2)^2,x]","\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)}","\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*Sqrt[c + d*x^2])/(2*a*(a + b*x^2)) + (c*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(3/2)*Sqrt[b*c - a*d])","A",1
736,1,129,119,0.2852138,"\int \frac{\sqrt{c+d x^2}}{x \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[Sqrt[c + d*x^2]/(x*(a + b*x^2)^2),x]","\frac{(2 b c-a d) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{2 a^2 \sqrt{b} \sqrt{a d-b c}}-\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)}","\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,"Sqrt[c + d*x^2]/(2*a*(a + b*x^2)) + ((2*b*c - a*d)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(2*a^2*Sqrt[b]*Sqrt[-(b*c) + a*d]) - (Sqrt[c]*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/a^2","A",1
737,1,102,113,0.3365918,"\int \frac{\sqrt{c+d x^2}}{x^2 \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[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}}+\frac{\left(-2 a-3 b x^2\right) \sqrt{c+d x^2}}{2 a^2 x \left(a+b x^2\right)}","-\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,"((-2*a - 3*b*x^2)*Sqrt[c + d*x^2])/(2*a^2*x*(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
738,1,157,159,0.407633,"\int \frac{\sqrt{c+d x^2}}{x^3 \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[Sqrt[c + d*x^2]/(x^3*(a + b*x^2)^2),x]","\frac{\left(3 a \sqrt{b} d-4 b^{3/2} c\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{2 a^3 \sqrt{a d-b c}}+\frac{(4 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{2 a^3 \sqrt{c}}+\frac{\left(-a-2 b x^2\right) \sqrt{c+d x^2}}{2 a^2 x^2 \left(a+b x^2\right)}","-\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,"((-a - 2*b*x^2)*Sqrt[c + d*x^2])/(2*a^2*x^2*(a + b*x^2)) + ((-4*b^(3/2)*c + 3*a*Sqrt[b]*d)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(2*a^3*Sqrt[-(b*c) + a*d]) + ((4*b*c - a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*a^3*Sqrt[c])","A",1
739,1,139,147,0.4317465,"\int \frac{\sqrt{c+d x^2}}{x^4 \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[Sqrt[c + d*x^2]/(x^4*(a + b*x^2)^2),x]","\frac{\left(5 b^2 c-4 a b d\right) \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(-2 a^2 c-2 a^2 d x^2+10 a b c x^2-2 a b d x^4+15 b^2 c x^4\right)}{6 a^3 c x^3 \left(a+b x^2\right)}","\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^2*c + 10*a*b*c*x^2 - 2*a^2*d*x^2 + 15*b^2*c*x^4 - 2*a*b*d*x^4))/(6*a^3*c*x^3*(a + b*x^2)) + ((5*b^2*c - 4*a*b*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
740,1,204,197,0.8549468,"\int \frac{x^4 \left(c+d x^2\right)^{3/2}}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^4*(c + d*x^2)^(3/2))/(a + b*x^2)^2,x]","-\frac{3 \left(-3 a^{3/2} b c d+2 a^{5/2} d^2+\sqrt{a} b^2 c^2\right) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 b^4 \sqrt{b c-a d}}+\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{\sqrt{c+d x^2} \left(-12 a^2 d x+9 a b c x-6 a b d x^3+5 b^2 c x^3+2 b^2 d x^5\right)}{8 b^3 \left(a+b x^2\right)}","\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,"(Sqrt[c + d*x^2]*(9*a*b*c*x - 12*a^2*d*x + 5*b^2*c*x^3 - 6*a*b*d*x^3 + 2*b^2*d*x^5))/(8*b^3*(a + b*x^2)) - (3*(Sqrt[a]*b^2*c^2 - 3*a^(3/2)*b*c*d + 2*a^(5/2)*d^2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*b^4*Sqrt[b*c - a*d]) + (3*(b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(8*b^4*Sqrt[d])","A",1
741,1,150,163,0.2892566,"\int \frac{x^3 \left(c+d x^2\right)^{3/2}}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^3*(c + d*x^2)^(3/2))/(a + b*x^2)^2,x]","\frac{\left(-5 a^2 d^2+7 a b c d-2 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{2 b^{7/2} \sqrt{a d-b c}}+\frac{\sqrt{c+d x^2} \left(-15 a^2 d+11 a b c-10 a b d x^2+8 b^2 c x^2+2 b^2 d x^4\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{\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]*(11*a*b*c - 15*a^2*d + 8*b^2*c*x^2 - 10*a*b*d*x^2 + 2*b^2*d*x^4))/(6*b^3*(a + b*x^2)) + ((-2*b^2*c^2 + 7*a*b*c*d - 5*a^2*d^2)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(2*b^(7/2)*Sqrt[-(b*c) + a*d])","A",1
742,1,166,149,0.5734594,"\int \frac{x^2 \left(c+d x^2\right)^{3/2}}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^2*(c + d*x^2)^(3/2))/(a + b*x^2)^2,x]","\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)}{2 \sqrt{a} b^3 \sqrt{b c-a d}}+\frac{\left(3 b c \sqrt{d}-4 a d^{3/2}\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{2 b^3}+\frac{\sqrt{c+d x^2} \left(2 a d x-b c x+b d x^3\right)}{2 b^2 \left(a+b x^2\right)}","\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,"(Sqrt[c + d*x^2]*(-(b*c*x) + 2*a*d*x + b*d*x^3))/(2*b^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])])/(2*Sqrt[a]*b^3*Sqrt[b*c - a*d]) + ((3*b*c*Sqrt[d] - 4*a*d^(3/2))*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(2*b^3)","A",1
743,1,106,99,0.2058012,"\int \frac{x \left(c+d x^2\right)^{3/2}}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x*(c + d*x^2)^(3/2))/(a + b*x^2)^2,x]","\frac{3 d \sqrt{a d-b c} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{2 b^{5/2}}+\frac{\sqrt{c+d x^2} \left(3 a d-b c+2 b d x^2\right)}{2 b^2 \left(a+b x^2\right)}","-\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,"(Sqrt[c + d*x^2]*(-(b*c) + 3*a*d + 2*b*d*x^2))/(2*b^2*(a + b*x^2)) + (3*d*Sqrt[-(b*c) + a*d]*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(2*b^(5/2))","A",1
744,1,144,131,0.4825341,"\int \frac{\left(c+d x^2\right)^{3/2}}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2)^(3/2)/(a + b*x^2)^2,x]","\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)}{2 a^{3/2} b^2 \sqrt{b c-a d}}-\frac{x \sqrt{c+d x^2} (a d-b c)}{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}","\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,"-1/2*((-(b*c) + a*d)*x*Sqrt[c + d*x^2])/(a*b*(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])])/(2*a^(3/2)*b^2*Sqrt[b*c - a*d]) + (d^(3/2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/b^2","A",1
745,1,139,129,0.4688531,"\int \frac{\left(c+d x^2\right)^{3/2}}{x \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2)^(3/2)/(x*(a + b*x^2)^2),x]","-\frac{\sqrt{a d-b c} (a d+2 b c) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{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)}","\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,"((b*c - a*d)*Sqrt[c + d*x^2])/(2*a*b*(a + b*x^2)) - (Sqrt[-(b*c) + a*d]*(2*b*c + a*d)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(2*a^2*b^(3/2)) - (c^(3/2)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/a^2","A",1
746,1,102,128,0.3049517,"\int \frac{\left(c+d x^2\right)^{3/2}}{x^2 \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2)^(3/2)/(x^2*(a + b*x^2)^2),x]","\frac{\sqrt{c+d x^2} \left(-2 a c+a d x^2-3 b c x^2\right)}{2 a^2 x \left(a+b x^2\right)}-\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{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,"(Sqrt[c + d*x^2]*(-2*a*c - 3*b*c*x^2 + a*d*x^2))/(2*a^2*x*(a + b*x^2)) - (3*c*Sqrt[b*c - a*d]*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(5/2))","A",1
747,1,179,170,0.5014142,"\int \frac{\left(c+d x^2\right)^{3/2}}{x^3 \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2)^(3/2)/(x^3*(a + b*x^2)^2),x]","\frac{\left(4 b c^{3/2}-3 a \sqrt{c} d\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{2 a^3}+\frac{\sqrt{c+d x^2} \left(-a c+a d x^2-2 b c x^2\right)}{2 a^2 x^2 \left(a+b x^2\right)}+\frac{\left(-a^2 d^2+5 a b c d-4 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{2 a^3 \sqrt{b} \sqrt{a d-b c}}","\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,"(Sqrt[c + d*x^2]*(-(a*c) - 2*b*c*x^2 + a*d*x^2))/(2*a^2*x^2*(a + b*x^2)) + ((-4*b^2*c^2 + 5*a*b*c*d - a^2*d^2)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(2*a^3*Sqrt[b]*Sqrt[-(b*c) + a*d]) + ((4*b*c^(3/2) - 3*a*Sqrt[c]*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*a^3)","A",1
748,1,147,166,0.4679826,"\int \frac{\left(c+d x^2\right)^{3/2}}{x^4 \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2)^(3/2)/(x^4*(a + b*x^2)^2),x]","\frac{\left(2 a^2 d^2-7 a b c d+5 b^2 c^2\right) \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(-2 a^2 c-8 a^2 d x^2+10 a b c x^2-11 a b d x^4+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]*(-2*a^2*c + 10*a*b*c*x^2 - 8*a^2*d*x^2 + 15*b^2*c*x^4 - 11*a*b*d*x^4))/(6*a^3*x^3*(a + b*x^2)) + ((5*b^2*c^2 - 7*a*b*c*d + 2*a^2*d^2)*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
749,1,260,258,1.3090447,"\int \frac{x^4 \left(c+d x^2\right)^{5/2}}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^4*(c + d*x^2)^(5/2))/(a + b*x^2)^2,x]","-\frac{\left(3 \sqrt{a} b c-8 a^{3/2} d\right) (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{\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{c+d x^2} \left(96 a^3 d^2 x-156 a^2 b c d x+48 a^2 b d^2 x^3+57 a b^2 c^2 x-82 a b^2 c d x^3-16 a b^2 d^2 x^5+33 b^3 c^2 x^3+26 b^3 c d x^5+8 b^3 d^2 x^7\right)}{48 b^4 \left(a+b x^2\right)}","\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,"(Sqrt[c + d*x^2]*(57*a*b^2*c^2*x - 156*a^2*b*c*d*x + 96*a^3*d^2*x + 33*b^3*c^2*x^3 - 82*a*b^2*c*d*x^3 + 48*a^2*b*d^2*x^3 + 26*b^3*c*d*x^5 - 16*a*b^2*d^2*x^5 + 8*b^3*d^2*x^7))/(48*b^4*(a + b*x^2)) - ((b*c - a*d)^(3/2)*(3*Sqrt[a]*b*c - 8*a^(3/2)*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*b^5) + ((5*b^3*c^3 - 60*a*b^2*c^2*d + 120*a^2*b*c*d^2 - 64*a^3*d^3)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(16*b^5*Sqrt[d])","A",1
750,1,191,198,0.317441,"\int \frac{x^3 \left(c+d x^2\right)^{5/2}}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^3*(c + d*x^2)^(5/2))/(a + b*x^2)^2,x]","\frac{\sqrt{c+d x^2} \left(105 a^3 d^2-170 a^2 b c d+70 a^2 b d^2 x^2+61 a b^2 c^2-118 a b^2 c d x^2-14 a b^2 d^2 x^4+46 b^3 c^2 x^2+22 b^3 c d x^4+6 b^3 d^2 x^6\right)}{30 b^4 \left(a+b x^2\right)}-\frac{(2 b c-7 a d) (a d-b c)^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{2 b^{9/2}}","-\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,"(Sqrt[c + d*x^2]*(61*a*b^2*c^2 - 170*a^2*b*c*d + 105*a^3*d^2 + 46*b^3*c^2*x^2 - 118*a*b^2*c*d*x^2 + 70*a^2*b*d^2*x^2 + 22*b^3*c*d*x^4 - 14*a*b^2*d^2*x^4 + 6*b^3*d^2*x^6))/(30*b^4*(a + b*x^2)) - ((2*b*c - 7*a*d)*(-(b*c) + a*d)^(3/2)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(2*b^(9/2))","A",1
751,1,208,195,0.9350311,"\int \frac{x^2 \left(c+d x^2\right)^{5/2}}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^2*(c + d*x^2)^(5/2))/(a + b*x^2)^2,x]","\frac{\left(24 a^2 d^{5/2}-40 a b c d^{3/2}+15 b^2 c^2 \sqrt{d}\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{8 b^4}+\frac{\sqrt{c+d x^2} \left(-12 a^2 d^2 x+17 a b c d x-6 a b d^2 x^3-4 b^2 c^2 x+9 b^2 c d x^3+2 b^2 d^2 x^5\right)}{8 b^3 \left(a+b x^2\right)}+\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{\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,"(Sqrt[c + d*x^2]*(-4*b^2*c^2*x + 17*a*b*c*d*x - 12*a^2*d^2*x + 9*b^2*c*d*x^3 - 6*a*b*d^2*x^3 + 2*b^2*d^2*x^5))/(8*b^3*(a + b*x^2)) + ((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])])/(2*Sqrt[a]*b^4) + ((15*b^2*c^2*Sqrt[d] - 40*a*b*c*d^(3/2) + 24*a^2*d^(5/2))*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(8*b^4)","A",1
752,1,144,126,0.2910338,"\int \frac{x \left(c+d x^2\right)^{5/2}}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x*(c + d*x^2)^(5/2))/(a + b*x^2)^2,x]","\frac{\sqrt{c+d x^2} \left(-15 a^2 d^2+20 a b c d-10 a b d^2 x^2-3 b^2 c^2+14 b^2 c d x^2+2 b^2 d^2 x^4\right)}{6 b^3 \left(a+b x^2\right)}-\frac{5 d (a d-b c)^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{2 b^{7/2}}","-\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,"(Sqrt[c + d*x^2]*(-3*b^2*c^2 + 20*a*b*c*d - 15*a^2*d^2 + 14*b^2*c*d*x^2 - 10*a*b*d^2*x^2 + 2*b^2*d^2*x^4))/(6*b^3*(a + b*x^2)) - (5*d*(-(b*c) + a*d)^(3/2)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(2*b^(7/2))","A",1
753,1,172,174,0.7788309,"\int \frac{\left(c+d x^2\right)^{5/2}}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2)^(5/2)/(a + b*x^2)^2,x]","\frac{(4 a d+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)}{2 a^{3/2} b^3}+\frac{\sqrt{c+d x^2} \left(2 a^2 d^2 x-2 a b c d x+a b d^2 x^3+b^2 c^2 x\right)}{2 a b^2 \left(a+b x^2\right)}+\frac{\left(5 b c d^{3/2}-4 a d^{5/2}\right) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^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,"(Sqrt[c + d*x^2]*(b^2*c^2*x - 2*a*b*c*d*x + 2*a^2*d^2*x + a*b*d^2*x^3))/(2*a*b^2*(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])])/(2*a^(3/2)*b^3) + ((5*b*c*d^(3/2) - 4*a*d^(5/2))*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(2*b^3)","A",1
754,1,164,160,0.4457092,"\int \frac{\left(c+d x^2\right)^{5/2}}{x \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2)^(5/2)/(x*(a + b*x^2)^2),x]","\frac{(a d-b c)^{3/2} (3 a d+2 b c) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{2 a^2 b^{5/2}}+\frac{\sqrt{c+d x^2} \left(3 a^2 d^2-2 a b c d+2 a b d^2 x^2+b^2 c^2\right)}{2 a b^2 \left(a+b x^2\right)}-\frac{c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{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,"(Sqrt[c + d*x^2]*(b^2*c^2 - 2*a*b*c*d + 3*a^2*d^2 + 2*a*b*d^2*x^2))/(2*a*b^2*(a + b*x^2)) + ((-(b*c) + a*d)^(3/2)*(2*b*c + 3*a*d)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(2*a^2*b^(5/2)) - (c^(5/2)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/a^2","A",1
755,1,165,168,0.5669996,"\int \frac{\left(c+d x^2\right)^{5/2}}{x^2 \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(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}+\frac{\sqrt{c+d x^2} \left(-a^2 d^2 x^2-2 a b c^2+2 a b c d x^2-3 b^2 c^2 x^2\right)}{2 a^2 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}","-\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]*(-2*a*b*c^2 - 3*b^2*c^2*x^2 + 2*a*b*c*d*x^2 - a^2*d^2*x^2))/(2*a^2*b*x*(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)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/b^2","A",1
756,1,188,180,0.4897391,"\int \frac{\left(c+d x^2\right)^{5/2}}{x^3 \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2)^(5/2)/(x^3*(a + b*x^2)^2),x]","-\frac{(a d+4 b c) (a d-b c)^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{2 a^3 b^{3/2}}+\frac{\left(4 b c^{5/2}-5 a c^{3/2} d\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{2 a^3}+\frac{\sqrt{c+d x^2} \left(-a^2 d^2 x^2-a b c^2+2 a b c d x^2-2 b^2 c^2 x^2\right)}{2 a^2 b x^2 \left(a+b x^2\right)}","-\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,"(Sqrt[c + d*x^2]*(-(a*b*c^2) - 2*b^2*c^2*x^2 + 2*a*b*c*d*x^2 - a^2*d^2*x^2))/(2*a^2*b*x^2*(a + b*x^2)) - ((-(b*c) + a*d)^(3/2)*(4*b*c + a*d)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(2*a^3*b^(3/2)) + ((4*b*c^(5/2) - 5*a*c^(3/2)*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*a^3)","A",1
757,1,144,176,0.3935963,"\int \frac{\left(c+d x^2\right)^{5/2}}{x^4 \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2)^(5/2)/(x^4*(a + b*x^2)^2),x]","\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{\sqrt{c+d x^2} \left(-2 a^2 c^2-14 a^2 c d x^2+3 a^2 d^2 x^4+10 a b c^2 x^2-20 a b c d x^4+15 b^2 c^2 x^4\right)}{6 a^3 x^3 \left(a+b x^2\right)}","\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,"(Sqrt[c + d*x^2]*(-2*a^2*c^2 + 10*a*b*c^2*x^2 - 14*a^2*c*d*x^2 + 15*b^2*c^2*x^4 - 20*a*b*c*d*x^4 + 3*a^2*d^2*x^4))/(6*a^3*x^3*(a + b*x^2)) + (5*c*(b*c - a*d)^(3/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(7/2))","A",1
758,1,136,132,0.6158918,"\int \frac{x^4}{\left(a+b x^2\right)^2 \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[x^4/((a + b*x^2)^2*Sqrt[c + d*x^2]),x]","\frac{\left(2 a^{3/2} d-3 \sqrt{a} b c\right) \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}}","-\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*x*Sqrt[c + d*x^2])/(2*b*(b*c - a*d)*(a + b*x^2)) + ((-3*Sqrt[a]*b*c + 2*a^(3/2)*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*b^2*(b*c - a*d)^(3/2)) + ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]]/(b^2*Sqrt[d])","A",1
759,1,109,99,0.219763,"\int \frac{x^3}{\left(a+b x^2\right)^2 \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[x^3/((a + b*x^2)^2*Sqrt[c + d*x^2]),x]","\frac{(2 b c-a d) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{2 b^{3/2} (a d-b c)^{3/2}}+\frac{a \sqrt{c+d x^2}}{2 b \left(a+b x^2\right) (b c-a d)}","\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[c + d*x^2])/(2*b*(b*c - a*d)*(a + b*x^2)) + ((2*b*c - a*d)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(2*b^(3/2)*(-(b*c) + a*d)^(3/2))","A",1
760,1,89,89,0.3407621,"\int \frac{x^2}{\left(a+b x^2\right)^2 \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[x^2/((a + b*x^2)^2*Sqrt[c + d*x^2]),x]","\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)}","\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,"-1/2*(x*Sqrt[c + d*x^2])/((b*c - a*d)*(a + b*x^2)) + (c*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*Sqrt[a]*(b*c - a*d)^(3/2))","A",1
761,1,97,87,0.1927417,"\int \frac{x}{\left(a+b x^2\right)^2 \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[x/((a + b*x^2)^2*Sqrt[c + d*x^2]),x]","-\frac{\sqrt{c+d x^2}}{2 \left(a+b x^2\right) (b c-a d)}-\frac{d \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{2 \sqrt{b} (a d-b c)^{3/2}}","\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,"-1/2*Sqrt[c + d*x^2]/((b*c - a*d)*(a + b*x^2)) - (d*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(2*Sqrt[b]*(-(b*c) + a*d)^(3/2))","A",1
762,1,100,100,0.0005962,"\int \frac{1}{\left(a+b x^2\right)^2 \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^2*Sqrt[c + d*x^2]),x]","\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) (a d-b c)}","\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,"-1/2*(b*x*Sqrt[c + d*x^2])/(a*(-(b*c) + a*d)*(a + b*x^2)) + ((b*c - 2*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(3/2)*(b*c - a*d)^(3/2))","A",1
763,1,144,130,0.4223923,"\int \frac{1}{x \left(a+b x^2\right)^2 \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[1/(x*(a + b*x^2)^2*Sqrt[c + d*x^2]),x]","\frac{\left(3 a \sqrt{b} d-2 b^{3/2} c\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{2 a^2 (a d-b c)^{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) (a d-b c)}","\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,"-1/2*(b*Sqrt[c + d*x^2])/(a*(-(b*c) + a*d)*(a + b*x^2)) + ((-2*b^(3/2)*c + 3*a*Sqrt[b]*d)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(2*a^2*(-(b*c) + a*d)^(3/2)) - ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]]/(a^2*Sqrt[c])","A",1
764,1,137,147,0.5131948,"\int \frac{1}{x^2 \left(a+b x^2\right)^2 \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[1/(x^2*(a + b*x^2)^2*Sqrt[c + d*x^2]),x]","\frac{\left(4 a b d-3 b^2 c\right) \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} \left(-2 a^2 d+2 a b c-2 a b d x^2+3 b^2 c x^2\right)}{2 a^2 c x \left(a+b x^2\right) (a d-b c)}","-\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*a*b*c - 2*a^2*d + 3*b^2*c*x^2 - 2*a*b*d*x^2))/(2*a^2*c*(-(b*c) + a*d)*x*(a + b*x^2)) + ((-3*b^2*c + 4*a*b*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
765,1,187,185,0.7072751,"\int \frac{1}{x^3 \left(a+b x^2\right)^2 \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[1/(x^3*(a + b*x^2)^2*Sqrt[c + d*x^2]),x]","\frac{\left(4 b^{5/2} c-5 a b^{3/2} d\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{2 a^3 (a d-b c)^{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{\sqrt{c+d x^2} \left(a^2 (-d)+a b c-a b d x^2+2 b^2 c x^2\right)}{2 a^2 c x^2 \left(a+b x^2\right) (a d-b 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,"(Sqrt[c + d*x^2]*(a*b*c - a^2*d + 2*b^2*c*x^2 - a*b*d*x^2))/(2*a^2*c*(-(b*c) + a*d)*x^2*(a + b*x^2)) + ((4*b^(5/2)*c - 5*a*b^(3/2)*d)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(2*a^3*(-(b*c) + a*d)^(3/2)) + ((4*b*c + a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*a^3*c^(3/2))","A",1
766,1,195,206,0.6693378,"\int \frac{1}{x^4 \left(a+b x^2\right)^2 \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[1/(x^4*(a + b*x^2)^2*Sqrt[c + d*x^2]),x]","\frac{\left(5 b^3 c-6 a b^2 d\right) \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(-2 a^3 c d+4 a^3 d^2 x^2+2 a^2 b c^2+6 a^2 b c d x^2+4 a^2 b d^2 x^4-10 a b^2 c^2 x^2+8 a b^2 c d x^4-15 b^3 c^2 x^4\right)}{6 a^3 c^2 x^3 \left(a+b x^2\right) (a d-b c)}","\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^2*b*c^2 - 2*a^3*c*d - 10*a*b^2*c^2*x^2 + 6*a^2*b*c*d*x^2 + 4*a^3*d^2*x^2 - 15*b^3*c^2*x^4 + 8*a*b^2*c*d*x^4 + 4*a^2*b*d^2*x^4))/(6*a^3*c^2*(-(b*c) + a*d)*x^3*(a + b*x^2)) + ((5*b^3*c - 6*a*b^2*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
767,1,107,130,0.7213901,"\int \frac{x^4}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[x^4/((a + b*x^2)^2*(c + d*x^2)^(3/2)),x]","\frac{3 a c x+a d x^3+2 b c x^3}{2 \left(a+b x^2\right) \sqrt{c+d x^2} (b c-a d)^2}-\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}}","\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,"(3*a*c*x + 2*b*c*x^3 + a*d*x^3)/(2*(b*c - a*d)^2*(a + b*x^2)*Sqrt[c + d*x^2]) - (3*Sqrt[a]*c*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*(b*c - a*d)^(5/2))","A",1
768,1,123,134,0.3589888,"\int \frac{x^3}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[x^3/((a + b*x^2)^2*(c + d*x^2)^(3/2)),x]","\frac{3 a c+a d x^2+2 b c x^2}{2 \left(a+b x^2\right) \sqrt{c+d x^2} (b c-a d)^2}+\frac{(-a d-2 b c) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{2 \sqrt{b} (a d-b c)^{5/2}}","\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,"(3*a*c + 2*b*c*x^2 + a*d*x^2)/(2*(b*c - a*d)^2*(a + b*x^2)*Sqrt[c + d*x^2]) + ((-2*b*c - a*d)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(2*Sqrt[b]*(-(b*c) + a*d)^(5/2))","A",1
769,1,113,123,0.6070052,"\int \frac{x^2}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[x^2/((a + b*x^2)^2*(c + d*x^2)^(3/2)),x]","\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}}+\frac{-2 a d x-b c x-3 b d x^3}{2 \left(a+b x^2\right) \sqrt{c+d x^2} (b c-a d)^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,"(-(b*c*x) - 2*a*d*x - 3*b*d*x^3)/(2*(b*c - a*d)^2*(a + b*x^2)*Sqrt[c + d*x^2]) + ((b*c + 2*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*Sqrt[a]*(b*c - a*d)^(5/2))","A",1
770,1,113,113,0.2502809,"\int \frac{x}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[x/((a + b*x^2)^2*(c + d*x^2)^(3/2)),x]","\frac{-2 a d-b c-3 b d x^2}{2 \left(a+b x^2\right) \sqrt{c+d x^2} (b c-a d)^2}+\frac{3 \sqrt{b} d \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{2 (a d-b c)^{5/2}}","-\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,"(-(b*c) - 2*a*d - 3*b*d*x^2)/(2*(b*c - a*d)^2*(a + b*x^2)*Sqrt[c + d*x^2]) + (3*Sqrt[b]*d*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(2*(-(b*c) + a*d)^(5/2))","A",1
771,1,141,142,0.6826491,"\int \frac{1}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^2*(c + d*x^2)^(3/2)),x]","\frac{\left(b^2 c-4 a b d\right) \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{2 a^2 d^2 x+2 a b d^2 x^3+b^2 c^2 x+b^2 c d x^3}{2 a c \left(a+b x^2\right) \sqrt{c+d x^2} (a d-b c)^2}","\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,"(b^2*c^2*x + 2*a^2*d^2*x + b^2*c*d*x^3 + 2*a*b*d^2*x^3)/(2*a*c*(-(b*c) + a*d)^2*(a + b*x^2)*Sqrt[c + d*x^2]) + ((b^2*c - 4*a*b*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(3/2)*(b*c - a*d)^(5/2))","A",1
772,1,181,170,0.6136225,"\int \frac{1}{x \left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[1/(x*(a + b*x^2)^2*(c + d*x^2)^(3/2)),x]","\frac{\left(2 b^{5/2} c-5 a b^{3/2} d\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{2 a^2 (a d-b c)^{5/2}}+\frac{2 a^2 d^2+2 a b d^2 x^2+b^2 c^2+b^2 c d x^2}{2 a c \left(a+b x^2\right) \sqrt{c+d x^2} (a d-b c)^2}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{a^2 c^{3/2}}","\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^2*c^2 + 2*a^2*d^2 + b^2*c*d*x^2 + 2*a*b*d^2*x^2)/(2*a*c*(-(b*c) + a*d)^2*(a + b*x^2)*Sqrt[c + d*x^2]) + ((2*b^(5/2)*c - 5*a*b^(3/2)*d)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(2*a^2*(-(b*c) + a*d)^(5/2)) - ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]]/(a^2*c^(3/2))","A",1
773,1,223,205,0.9752425,"\int \frac{1}{x^2 \left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[1/(x^2*(a + b*x^2)^2*(c + d*x^2)^(3/2)),x]","\frac{-2 a^3 c d^2-4 a^3 d^3 x^2+4 a^2 b c^2 d+2 a^2 b c d^2 x^2-4 a^2 b d^3 x^4-2 a b^2 c^3+2 a b^2 c^2 d x^2+4 a b^2 c d^2 x^4-3 b^3 c^3 x^2-3 b^3 c^2 d x^4}{2 a^2 c^2 x \left(a+b x^2\right) \sqrt{c+d x^2} (a d-b c)^2}-\frac{3 \left(b^3 c-2 a b^2 d\right) \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,"(-2*a*b^2*c^3 + 4*a^2*b*c^2*d - 2*a^3*c*d^2 - 3*b^3*c^3*x^2 + 2*a*b^2*c^2*d*x^2 + 2*a^2*b*c*d^2*x^2 - 4*a^3*d^3*x^2 - 3*b^3*c^2*d*x^4 + 4*a*b^2*c*d^2*x^4 - 4*a^2*b*d^3*x^4)/(2*a^2*c^2*(-(b*c) + a*d)^2*x*(a + b*x^2)*Sqrt[c + d*x^2]) - (3*(b^3*c - 2*a*b^2*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
774,1,272,241,1.0495983,"\int \frac{1}{x^3 \left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[1/(x^3*(a + b*x^2)^2*(c + d*x^2)^(3/2)),x]","\frac{\left(7 a b^{5/2} d-4 b^{7/2} c\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{2 a^3 (a d-b c)^{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{-a^3 c d^2-3 a^3 d^3 x^2+2 a^2 b c^2 d+a^2 b c d^2 x^2-3 a^2 b d^3 x^4-a b^2 c^3+a b^2 c^2 d x^2+2 a b^2 c d^2 x^4-2 b^3 c^3 x^2-2 b^3 c^2 d x^4}{2 a^2 c^2 x^2 \left(a+b x^2\right) \sqrt{c+d x^2} (a d-b c)^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,"(-(a*b^2*c^3) + 2*a^2*b*c^2*d - a^3*c*d^2 - 2*b^3*c^3*x^2 + a*b^2*c^2*d*x^2 + a^2*b*c*d^2*x^2 - 3*a^3*d^3*x^2 - 2*b^3*c^2*d*x^4 + 2*a*b^2*c*d^2*x^4 - 3*a^2*b*d^3*x^4)/(2*a^2*c^2*(-(b*c) + a*d)^2*x^2*(a + b*x^2)*Sqrt[c + d*x^2]) + ((-4*b^(7/2)*c + 7*a*b^(5/2)*d)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(2*a^3*(-(b*c) + a*d)^(5/2)) + ((4*b*c + 3*a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*a^3*c^(5/2))","A",1
775,1,303,277,1.5648826,"\int \frac{1}{x^4 \left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[1/(x^4*(a + b*x^2)^2*(c + d*x^2)^(3/2)),x]","\frac{\left(5 b^4 c-8 a b^3 d\right) \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{-2 a^4 c^2 d^2+8 a^4 c d^3 x^2+16 a^4 d^4 x^4+4 a^3 b c^3 d-6 a^3 b c^2 d^2 x^2+16 a^3 b d^4 x^6-2 a^2 b^2 c^4-12 a^2 b^2 c^3 d x^2-18 a^2 b^2 c^2 d^2 x^4-8 a^2 b^2 c d^3 x^6+10 a b^3 c^4 x^2-4 a b^3 c^3 d x^4-14 a b^3 c^2 d^2 x^6+15 b^4 c^4 x^4+15 b^4 c^3 d x^6}{6 a^3 c^3 x^3 \left(a+b x^2\right) \sqrt{c+d x^2} (a d-b c)^2}","\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,"(-2*a^2*b^2*c^4 + 4*a^3*b*c^3*d - 2*a^4*c^2*d^2 + 10*a*b^3*c^4*x^2 - 12*a^2*b^2*c^3*d*x^2 - 6*a^3*b*c^2*d^2*x^2 + 8*a^4*c*d^3*x^2 + 15*b^4*c^4*x^4 - 4*a*b^3*c^3*d*x^4 - 18*a^2*b^2*c^2*d^2*x^4 + 16*a^4*d^4*x^4 + 15*b^4*c^3*d*x^6 - 14*a*b^3*c^2*d^2*x^6 - 8*a^2*b^2*c*d^3*x^6 + 16*a^3*b*d^4*x^6)/(6*a^3*c^3*(-(b*c) + a*d)^2*x^3*(a + b*x^2)*Sqrt[c + d*x^2]) + ((5*b^4*c - 8*a*b^3*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
776,1,168,174,1.1623359,"\int \frac{x^4}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[x^4/((a + b*x^2)^2*(c + d*x^2)^(5/2)),x]","\frac{\left(-2 a^{3/2} d-3 \sqrt{a} b c\right) \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}}-\frac{-6 a^2 c d x-8 a^2 d^2 x^3-9 a b c^2 x-16 a b c d x^3-11 a b d^2 x^5-6 b^2 c^2 x^3-4 b^2 c d x^5}{6 \left(a+b x^2\right) \left(c+d x^2\right)^{3/2} (b c-a d)^3}","\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,"-1/6*(-9*a*b*c^2*x - 6*a^2*c*d*x - 6*b^2*c^2*x^3 - 16*a*b*c*d*x^3 - 8*a^2*d^2*x^3 - 4*b^2*c*d*x^5 - 11*a*b*d^2*x^5)/((b*c - a*d)^3*(a + b*x^2)*(c + d*x^2)^(3/2)) + ((-3*Sqrt[a]*b*c - 2*a^(3/2)*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*(b*c - a*d)^(7/2))","A",1
777,1,175,170,0.6076842,"\int \frac{x^3}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[x^3/((a + b*x^2)^2*(c + d*x^2)^(5/2)),x]","\frac{\left(3 a \sqrt{b} d+2 b^{3/2} c\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{2 (a d-b c)^{7/2}}-\frac{-4 a^2 c d-6 a^2 d^2 x^2-11 a b c^2-16 a b c d x^2-9 a b d^2 x^4-8 b^2 c^2 x^2-6 b^2 c d x^4}{6 \left(a+b x^2\right) \left(c+d x^2\right)^{3/2} (b c-a d)^3}","\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,"-1/6*(-11*a*b*c^2 - 4*a^2*c*d - 8*b^2*c^2*x^2 - 16*a*b*c*d*x^2 - 6*a^2*d^2*x^2 - 6*b^2*c*d*x^4 - 9*a*b*d^2*x^4)/((b*c - a*d)^3*(a + b*x^2)*(c + d*x^2)^(3/2)) + ((2*b^(3/2)*c + 3*a*Sqrt[b]*d)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(2*(-(b*c) + a*d)^(7/2))","A",1
778,1,176,163,1.2726268,"\int \frac{x^2}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[x^2/((a + b*x^2)^2*(c + d*x^2)^(5/2)),x]","\frac{-2 a^2 d^3 x^3-12 a b c^2 d x-10 a b c d^2 x^3-2 a b d^3 x^5-3 b^2 c^3 x-18 b^2 c^2 d x^3-13 b^2 c d^2 x^5}{6 c \left(a+b x^2\right) \left(c+d x^2\right)^{3/2} (b c-a d)^3}+\frac{\left(4 a b d+b^2 c\right) \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}}","-\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,"(-3*b^2*c^3*x - 12*a*b*c^2*d*x - 18*b^2*c^2*d*x^3 - 10*a*b*c*d^2*x^3 - 2*a^2*d^3*x^3 - 13*b^2*c*d^2*x^5 - 2*a*b*d^3*x^5)/(6*c*(b*c - a*d)^3*(a + b*x^2)*(c + d*x^2)^(3/2)) + ((b^2*c + 4*a*b*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*Sqrt[a]*(b*c - a*d)^(7/2))","A",1
779,1,151,140,0.3986249,"\int \frac{x}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[x/((a + b*x^2)^2*(c + d*x^2)^(5/2)),x]","\frac{2 a^2 d^2-14 a b c d-10 a b d^2 x^2-3 b^2 c^2-20 b^2 c d x^2-15 b^2 d^2 x^4}{6 \left(a+b x^2\right) \left(c+d x^2\right)^{3/2} (b c-a d)^3}-\frac{5 b^{3/2} d \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{2 (a d-b c)^{7/2}}","\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,"(-3*b^2*c^2 - 14*a*b*c*d + 2*a^2*d^2 - 20*b^2*c*d*x^2 - 10*a*b*d^2*x^2 - 15*b^2*d^2*x^4)/(6*(b*c - a*d)^3*(a + b*x^2)*(c + d*x^2)^(3/2)) - (5*b^(3/2)*d*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(2*(-(b*c) + a*d)^(7/2))","A",1
780,1,229,201,1.3157852,"\int \frac{1}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^2*(c + d*x^2)^(5/2)),x]","\frac{\left(b^3 c-6 a b^2 d\right) \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{6 a^3 c d^3 x+4 a^3 d^4 x^3-18 a^2 b c^2 d^2 x-10 a^2 b c d^3 x^3+4 a^2 b d^4 x^5-18 a b^2 c^2 d^2 x^3-16 a b^2 c d^3 x^5-3 b^3 c^4 x-6 b^3 c^3 d x^3-3 b^3 c^2 d^2 x^5}{6 a c^2 \left(a+b x^2\right) \left(c+d x^2\right)^{3/2} (a d-b c)^3}","\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,"(-3*b^3*c^4*x - 18*a^2*b*c^2*d^2*x + 6*a^3*c*d^3*x - 6*b^3*c^3*d*x^3 - 18*a*b^2*c^2*d^2*x^3 - 10*a^2*b*c*d^3*x^3 + 4*a^3*d^4*x^3 - 3*b^3*c^2*d^2*x^5 - 16*a*b^2*c*d^3*x^5 + 4*a^2*b*d^4*x^5)/(6*a*c^2*(-(b*c) + a*d)^3*(a + b*x^2)*(c + d*x^2)^(3/2)) + ((b^3*c - 6*a*b^2*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(3/2)*(b*c - a*d)^(7/2))","A",1
781,1,266,225,0.87363,"\int \frac{1}{x \left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[1/(x*(a + b*x^2)^2*(c + d*x^2)^(5/2)),x]","\frac{\left(7 a b^{5/2} d-2 b^{7/2} c\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{2 a^2 (a d-b c)^{7/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{a^2 c^{5/2}}+\frac{8 a^3 c d^3+6 a^3 d^4 x^2-20 a^2 b c^2 d^2-10 a^2 b c d^3 x^2+6 a^2 b d^4 x^4-20 a b^2 c^2 d^2 x^2-18 a b^2 c d^3 x^4-3 b^3 c^4-6 b^3 c^3 d x^2-3 b^3 c^2 d^2 x^4}{6 a c^2 \left(a+b x^2\right) \left(c+d x^2\right)^{3/2} (a d-b c)^3}","\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*b^3*c^4 - 20*a^2*b*c^2*d^2 + 8*a^3*c*d^3 - 6*b^3*c^3*d*x^2 - 20*a*b^2*c^2*d^2*x^2 - 10*a^2*b*c*d^3*x^2 + 6*a^3*d^4*x^2 - 3*b^3*c^2*d^2*x^4 - 18*a*b^2*c*d^3*x^4 + 6*a^2*b*d^4*x^4)/(6*a*c^2*(-(b*c) + a*d)^3*(a + b*x^2)*(c + d*x^2)^(3/2)) + ((-2*b^(7/2)*c + 7*a*b^(5/2)*d)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(2*a^2*(-(b*c) + a*d)^(7/2)) - ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]]/(a^2*c^(5/2))","A",1
782,1,347,279,1.5011705,"\int \frac{1}{x^2 \left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[1/(x^2*(a + b*x^2)^2*(c + d*x^2)^(5/2)),x]","\frac{\left(8 a b^3 d-3 b^4 c\right) \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{-6 a^4 c^2 d^3-24 a^4 c d^4 x^2-16 a^4 d^5 x^4+18 a^3 b c^3 d^2+54 a^3 b c^2 d^3 x^2+16 a^3 b c d^4 x^4-16 a^3 b d^5 x^6-18 a^2 b^2 c^4 d-18 a^2 b^2 c^3 d^2 x^2+42 a^2 b^2 c^2 d^3 x^4+40 a^2 b^2 c d^4 x^6+6 a b^3 c^5-6 a b^3 c^4 d x^2-30 a b^3 c^3 d^2 x^4-18 a b^3 c^2 d^3 x^6+9 b^4 c^5 x^2+18 b^4 c^4 d x^4+9 b^4 c^3 d^2 x^6}{6 a^2 c^3 x \left(a+b x^2\right) \left(c+d x^2\right)^{3/2} (a d-b c)^3}","-\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,"(6*a*b^3*c^5 - 18*a^2*b^2*c^4*d + 18*a^3*b*c^3*d^2 - 6*a^4*c^2*d^3 + 9*b^4*c^5*x^2 - 6*a*b^3*c^4*d*x^2 - 18*a^2*b^2*c^3*d^2*x^2 + 54*a^3*b*c^2*d^3*x^2 - 24*a^4*c*d^4*x^2 + 18*b^4*c^4*d*x^4 - 30*a*b^3*c^3*d^2*x^4 + 42*a^2*b^2*c^2*d^3*x^4 + 16*a^3*b*c*d^4*x^4 - 16*a^4*d^5*x^4 + 9*b^4*c^3*d^2*x^6 - 18*a*b^3*c^2*d^3*x^6 + 40*a^2*b^2*c*d^4*x^6 - 16*a^3*b*d^5*x^6)/(6*a^2*c^3*(-(b*c) + a*d)^3*x*(a + b*x^2)*(c + d*x^2)^(3/2)) + ((-3*b^4*c + 8*a*b^3*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
783,1,397,304,1.5048986,"\int \frac{1}{x^3 \left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[1/(x^3*(a + b*x^2)^2*(c + d*x^2)^(5/2)),x]","\frac{\left(4 b^{9/2} c-9 a b^{7/2} d\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^2} \sqrt{a d-b c}}{b c-a d}\right)}{2 a^3 (a d-b c)^{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{-3 a^4 c^2 d^3-20 a^4 c d^4 x^2-15 a^4 d^5 x^4+9 a^3 b c^3 d^2+41 a^3 b c^2 d^3 x^2+13 a^3 b c d^4 x^4-15 a^3 b d^5 x^6-9 a^2 b^2 c^4 d-9 a^2 b^2 c^3 d^2 x^2+35 a^2 b^2 c^2 d^3 x^4+33 a^2 b^2 c d^4 x^6+3 a b^3 c^5-3 a b^3 c^4 d x^2-15 a b^3 c^3 d^2 x^4-9 a b^3 c^2 d^3 x^6+6 b^4 c^5 x^2+12 b^4 c^4 d x^4+6 b^4 c^3 d^2 x^6}{6 a^2 c^3 x^2 \left(a+b x^2\right) \left(c+d x^2\right)^{3/2} (a d-b c)^3}","-\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,"(3*a*b^3*c^5 - 9*a^2*b^2*c^4*d + 9*a^3*b*c^3*d^2 - 3*a^4*c^2*d^3 + 6*b^4*c^5*x^2 - 3*a*b^3*c^4*d*x^2 - 9*a^2*b^2*c^3*d^2*x^2 + 41*a^3*b*c^2*d^3*x^2 - 20*a^4*c*d^4*x^2 + 12*b^4*c^4*d*x^4 - 15*a*b^3*c^3*d^2*x^4 + 35*a^2*b^2*c^2*d^3*x^4 + 13*a^3*b*c*d^4*x^4 - 15*a^4*d^5*x^4 + 6*b^4*c^3*d^2*x^6 - 9*a*b^3*c^2*d^3*x^6 + 33*a^2*b^2*c*d^4*x^6 - 15*a^3*b*d^5*x^6)/(6*a^2*c^3*(-(b*c) + a*d)^3*x^2*(a + b*x^2)*(c + d*x^2)^(3/2)) + ((4*b^(9/2)*c - 9*a*b^(7/2)*d)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(2*a^3*(-(b*c) + a*d)^(7/2)) + ((4*b*c + 5*a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*a^3*c^(7/2))","A",1
784,1,459,362,2.4914522,"\int \frac{1}{x^4 \left(a+b x^2\right)^2 \left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[1/(x^4*(a + b*x^2)^2*(c + d*x^2)^(5/2)),x]","\frac{5 \left(b^5 c-2 a b^4 d\right) \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{-2 a^5 c^3 d^3+12 a^5 c^2 d^4 x^2+48 a^5 c d^5 x^4+32 a^5 d^6 x^6+6 a^4 b c^4 d^2-26 a^4 b c^3 d^3 x^2-84 a^4 b c^2 d^4 x^4-16 a^4 b c d^5 x^6+32 a^4 b d^6 x^8-6 a^3 b^2 c^5 d+6 a^3 b^2 c^4 d^2 x^2-6 a^3 b^2 c^3 d^3 x^4-84 a^3 b^2 c^2 d^4 x^6-64 a^3 b^2 c d^5 x^8+2 a^2 b^3 c^6+18 a^2 b^3 c^5 d x^2+42 a^2 b^3 c^4 d^2 x^4+38 a^2 b^3 c^3 d^3 x^6+12 a^2 b^3 c^2 d^4 x^8-10 a b^4 c^6 x^2+30 a b^4 c^4 d^2 x^6+20 a b^4 c^3 d^3 x^8-15 b^5 c^6 x^4-30 b^5 c^5 d x^6-15 b^5 c^4 d^2 x^8}{6 a^3 c^4 x^3 \left(a+b x^2\right) \left(c+d x^2\right)^{3/2} (a d-b c)^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,"(2*a^2*b^3*c^6 - 6*a^3*b^2*c^5*d + 6*a^4*b*c^4*d^2 - 2*a^5*c^3*d^3 - 10*a*b^4*c^6*x^2 + 18*a^2*b^3*c^5*d*x^2 + 6*a^3*b^2*c^4*d^2*x^2 - 26*a^4*b*c^3*d^3*x^2 + 12*a^5*c^2*d^4*x^2 - 15*b^5*c^6*x^4 + 42*a^2*b^3*c^4*d^2*x^4 - 6*a^3*b^2*c^3*d^3*x^4 - 84*a^4*b*c^2*d^4*x^4 + 48*a^5*c*d^5*x^4 - 30*b^5*c^5*d*x^6 + 30*a*b^4*c^4*d^2*x^6 + 38*a^2*b^3*c^3*d^3*x^6 - 84*a^3*b^2*c^2*d^4*x^6 - 16*a^4*b*c*d^5*x^6 + 32*a^5*d^6*x^6 - 15*b^5*c^4*d^2*x^8 + 20*a*b^4*c^3*d^3*x^8 + 12*a^2*b^3*c^2*d^4*x^8 - 64*a^3*b^2*c*d^5*x^8 + 32*a^4*b*d^6*x^8)/(6*a^3*c^4*(-(b*c) + a*d)^3*x^3*(a + b*x^2)*(c + d*x^2)^(3/2)) + (5*(b^5*c - 2*a*b^4*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
785,0,0,212,12.9636025,"\int (e x)^{3/2} \sqrt{a+b x^2} \left(A+B x^2\right) \, dx","IntegrateAlgebraic[(e*x)^(3/2)*Sqrt[a + b*x^2]*(A + B*x^2),x]","\int (e x)^{3/2} \sqrt{a+b x^2} \left(A+B x^2\right) \, dx","-\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,"Could not integrate","F",-1
786,0,0,337,12.3959891,"\int \sqrt{e x} \sqrt{a+b x^2} \left(A+B x^2\right) \, dx","IntegrateAlgebraic[Sqrt[e*x]*Sqrt[a + b*x^2]*(A + B*x^2),x]","\int \sqrt{e x} \sqrt{a+b x^2} \left(A+B x^2\right) \, dx","\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,"Could not integrate","F",-1
787,0,0,176,12.0698891,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{\sqrt{e x}} \, dx","IntegrateAlgebraic[(Sqrt[a + b*x^2]*(A + B*x^2))/Sqrt[e*x],x]","\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{\sqrt{e x}} \, dx","\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,"Could not integrate","F",-1
788,0,0,333,10.7729141,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{(e x)^{3/2}} \, dx","IntegrateAlgebraic[(Sqrt[a + b*x^2]*(A + B*x^2))/(e*x)^(3/2),x]","\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{(e x)^{3/2}} \, dx","\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,"Could not integrate","F",-1
789,0,0,172,9.0359266,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{(e x)^{5/2}} \, dx","IntegrateAlgebraic[(Sqrt[a + b*x^2]*(A + B*x^2))/(e*x)^(5/2),x]","\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{(e x)^{5/2}} \, dx","\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,"Could not integrate","F",-1
790,0,0,338,15.8450065,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{(e x)^{7/2}} \, dx","IntegrateAlgebraic[(Sqrt[a + b*x^2]*(A + B*x^2))/(e*x)^(7/2),x]","\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{(e x)^{7/2}} \, dx","\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,"Could not integrate","F",-1
791,0,0,152,19.82646,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{x^{9/2}} \, dx","IntegrateAlgebraic[(Sqrt[a + b*x^2]*(A + B*x^2))/x^(9/2),x]","\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{x^{9/2}} \, dx","-\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,"Could not integrate","F",-1
792,0,0,331,26.9027656,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{x^{11/2}} \, dx","IntegrateAlgebraic[(Sqrt[a + b*x^2]*(A + B*x^2))/x^(11/2),x]","\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{x^{11/2}} \, dx","-\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,"Could not integrate","F",-1
793,0,0,187,31.1124692,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{x^{13/2}} \, dx","IntegrateAlgebraic[(Sqrt[a + b*x^2]*(A + B*x^2))/x^(13/2),x]","\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right)}{x^{13/2}} \, dx","\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,"Could not integrate","F",-1
794,0,0,252,15.7264839,"\int (e x)^{3/2} \left(a+b x^2\right)^{3/2} \left(A+B x^2\right) \, dx","IntegrateAlgebraic[(e*x)^(3/2)*(a + b*x^2)^(3/2)*(A + B*x^2),x]","\int (e x)^{3/2} \left(a+b x^2\right)^{3/2} \left(A+B x^2\right) \, dx","-\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,"Could not integrate","F",-1
795,0,0,377,14.1498778,"\int \sqrt{e x} \left(a+b x^2\right)^{3/2} \left(A+B x^2\right) \, dx","IntegrateAlgebraic[Sqrt[e*x]*(a + b*x^2)^(3/2)*(A + B*x^2),x]","\int \sqrt{e x} \left(a+b x^2\right)^{3/2} \left(A+B x^2\right) \, dx","\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,"Could not integrate","F",-1
796,0,0,214,13.4298597,"\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{\sqrt{e x}} \, dx","IntegrateAlgebraic[((a + b*x^2)^(3/2)*(A + B*x^2))/Sqrt[e*x],x]","\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{\sqrt{e x}} \, dx","\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,"Could not integrate","F",-1
797,0,0,367,13.0524518,"\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{(e x)^{3/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)^(3/2)*(A + B*x^2))/(e*x)^(3/2),x]","\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{(e x)^{3/2}} \, dx","\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,"Could not integrate","F",-1
798,0,0,210,11.7791107,"\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{(e x)^{5/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)^(3/2)*(A + B*x^2))/(e*x)^(5/2),x]","\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{(e x)^{5/2}} \, dx","\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,"Could not integrate","F",-1
799,0,0,365,13.4496524,"\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{(e x)^{7/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)^(3/2)*(A + B*x^2))/(e*x)^(7/2),x]","\int \frac{\left(a+b x^2\right)^{3/2} \left(A+B x^2\right)}{(e x)^{7/2}} \, dx","\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,"Could not integrate","F",-1
800,0,0,338,31.0046016,"\int \frac{(e x)^{5/2} \left(A+B x^2\right)}{\sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[((e*x)^(5/2)*(A + B*x^2))/Sqrt[a + b*x^2],x]","\int \frac{(e x)^{5/2} \left(A+B x^2\right)}{\sqrt{a+b x^2}} \, dx","-\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,"Could not integrate","F",-1
801,0,0,174,27.071151,"\int \frac{(e x)^{3/2} \left(A+B x^2\right)}{\sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[((e*x)^(3/2)*(A + B*x^2))/Sqrt[a + b*x^2],x]","\int \frac{(e x)^{3/2} \left(A+B x^2\right)}{\sqrt{a+b x^2}} \, dx","-\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,"Could not integrate","F",-1
802,0,0,299,26.2091724,"\int \frac{\sqrt{e x} \left(A+B x^2\right)}{\sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(Sqrt[e*x]*(A + B*x^2))/Sqrt[a + b*x^2],x]","\int \frac{\sqrt{e x} \left(A+B x^2\right)}{\sqrt{a+b x^2}} \, dx","\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,"Could not integrate","F",-1
803,0,0,139,25.7959678,"\int \frac{A+B x^2}{\sqrt{e x} \sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(A + B*x^2)/(Sqrt[e*x]*Sqrt[a + b*x^2]),x]","\int \frac{A+B x^2}{\sqrt{e x} \sqrt{a+b x^2}} \, dx","\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,"Could not integrate","F",-1
804,0,0,290,27.0284666,"\int \frac{A+B x^2}{(e x)^{3/2} \sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(A + B*x^2)/((e*x)^(3/2)*Sqrt[a + b*x^2]),x]","\int \frac{A+B x^2}{(e x)^{3/2} \sqrt{a+b x^2}} \, dx","\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,"Could not integrate","F",-1
805,0,0,138,26.0431209,"\int \frac{A+B x^2}{(e x)^{5/2} \sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(A + B*x^2)/((e*x)^(5/2)*Sqrt[a + b*x^2]),x]","\int \frac{A+B x^2}{(e x)^{5/2} \sqrt{a+b x^2}} \, dx","-\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,"Could not integrate","F",-1
806,0,0,342,33.4388594,"\int \frac{A+B x^2}{(e x)^{7/2} \sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(A + B*x^2)/((e*x)^(7/2)*Sqrt[a + b*x^2]),x]","\int \frac{A+B x^2}{(e x)^{7/2} \sqrt{a+b x^2}} \, dx","-\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,"Could not integrate","F",-1
807,0,0,211,25.5593502,"\int \frac{(e x)^{7/2} \left(A+B x^2\right)}{\left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[((e*x)^(7/2)*(A + B*x^2))/(a + b*x^2)^(3/2),x]","\int \frac{(e x)^{7/2} \left(A+B x^2\right)}{\left(a+b x^2\right)^{3/2}} \, dx","-\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,"Could not integrate","F",-1
808,0,0,337,22.0552418,"\int \frac{(e x)^{5/2} \left(A+B x^2\right)}{\left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[((e*x)^(5/2)*(A + B*x^2))/(a + b*x^2)^(3/2),x]","\int \frac{(e x)^{5/2} \left(A+B x^2\right)}{\left(a+b x^2\right)^{3/2}} \, dx","\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,"Could not integrate","F",-1
809,0,0,174,21.7736088,"\int \frac{(e x)^{3/2} \left(A+B x^2\right)}{\left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[((e*x)^(3/2)*(A + B*x^2))/(a + b*x^2)^(3/2),x]","\int \frac{(e x)^{3/2} \left(A+B x^2\right)}{\left(a+b x^2\right)^{3/2}} \, dx","\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,"Could not integrate","F",-1
810,0,0,301,20.9622092,"\int \frac{\sqrt{e x} \left(A+B x^2\right)}{\left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(Sqrt[e*x]*(A + B*x^2))/(a + b*x^2)^(3/2),x]","\int \frac{\sqrt{e x} \left(A+B x^2\right)}{\left(a+b x^2\right)^{3/2}} \, dx","-\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,"Could not integrate","F",-1
811,0,0,144,20.7270548,"\int \frac{A+B x^2}{\sqrt{e x} \left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(A + B*x^2)/(Sqrt[e*x]*(a + b*x^2)^(3/2)),x]","\int \frac{A+B x^2}{\sqrt{e x} \left(a+b x^2\right)^{3/2}} \, dx","\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,"Could not integrate","F",-1
812,0,0,333,21.4793451,"\int \frac{A+B x^2}{(e x)^{3/2} \left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(A + B*x^2)/((e*x)^(3/2)*(a + b*x^2)^(3/2)),x]","\int \frac{A+B x^2}{(e x)^{3/2} \left(a+b x^2\right)^{3/2}} \, dx","\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,"Could not integrate","F",-1
813,0,0,176,23.2539876,"\int \frac{A+B x^2}{(e x)^{5/2} \left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(A + B*x^2)/((e*x)^(5/2)*(a + b*x^2)^(3/2)),x]","\int \frac{A+B x^2}{(e x)^{5/2} \left(a+b x^2\right)^{3/2}} \, dx","-\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,"Could not integrate","F",-1
814,0,0,379,27.7174474,"\int \frac{A+B x^2}{(e x)^{7/2} \left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(A + B*x^2)/((e*x)^(7/2)*(a + b*x^2)^(3/2)),x]","\int \frac{A+B x^2}{(e x)^{7/2} \left(a+b x^2\right)^{3/2}} \, dx","-\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,"Could not integrate","F",-1
815,0,0,208,25.6625859,"\int \frac{(e x)^{7/2} \left(A+B x^2\right)}{\left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[((e*x)^(7/2)*(A + B*x^2))/(a + b*x^2)^(5/2),x]","\int \frac{(e x)^{7/2} \left(A+B x^2\right)}{\left(a+b x^2\right)^{5/2}} \, dx","\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,"Could not integrate","F",-1
816,0,0,349,22.2609757,"\int \frac{(e x)^{5/2} \left(A+B x^2\right)}{\left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[((e*x)^(5/2)*(A + B*x^2))/(a + b*x^2)^(5/2),x]","\int \frac{(e x)^{5/2} \left(A+B x^2\right)}{\left(a+b x^2\right)^{5/2}} \, dx","-\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,"Could not integrate","F",-1
817,0,0,185,21.8252213,"\int \frac{(e x)^{3/2} \left(A+B x^2\right)}{\left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[((e*x)^(3/2)*(A + B*x^2))/(a + b*x^2)^(5/2),x]","\int \frac{(e x)^{3/2} \left(A+B x^2\right)}{\left(a+b x^2\right)^{5/2}} \, dx","\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,"Could not integrate","F",-1
818,0,0,344,21.2883913,"\int \frac{\sqrt{e x} \left(A+B x^2\right)}{\left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(Sqrt[e*x]*(A + B*x^2))/(a + b*x^2)^(5/2),x]","\int \frac{\sqrt{e x} \left(A+B x^2\right)}{\left(a+b x^2\right)^{5/2}} \, dx","-\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,"Could not integrate","F",-1
819,0,0,187,21.186373,"\int \frac{A+B x^2}{\sqrt{e x} \left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(A + B*x^2)/(Sqrt[e*x]*(a + b*x^2)^(5/2)),x]","\int \frac{A+B x^2}{\sqrt{e x} \left(a+b x^2\right)^{5/2}} \, dx","\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,"Could not integrate","F",-1
820,0,0,377,21.9412882,"\int \frac{A+B x^2}{(e x)^{3/2} \left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(A + B*x^2)/((e*x)^(3/2)*(a + b*x^2)^(5/2)),x]","\int \frac{A+B x^2}{(e x)^{3/2} \left(a+b x^2\right)^{5/2}} \, dx","\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,"Could not integrate","F",-1
821,0,0,213,22.9691462,"\int \frac{A+B x^2}{(e x)^{5/2} \left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(A + B*x^2)/((e*x)^(5/2)*(a + b*x^2)^(5/2)),x]","\int \frac{A+B x^2}{(e x)^{5/2} \left(a+b x^2\right)^{5/2}} \, dx","-\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,"Could not integrate","F",-1
822,0,0,288,19.3378344,"\int (e x)^{3/2} \left(a+b x^2\right)^2 \sqrt{c+d x^2} \, dx","IntegrateAlgebraic[(e*x)^(3/2)*(a + b*x^2)^2*Sqrt[c + d*x^2],x]","\int (e x)^{3/2} \left(a+b x^2\right)^2 \sqrt{c+d x^2} \, dx","-\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,"Could not integrate","F",-1
823,0,0,425,16.8867043,"\int \sqrt{e x} \left(a+b x^2\right)^2 \sqrt{c+d x^2} \, dx","IntegrateAlgebraic[Sqrt[e*x]*(a + b*x^2)^2*Sqrt[c + d*x^2],x]","\int \sqrt{e x} \left(a+b x^2\right)^2 \sqrt{c+d x^2} \, dx","\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,"Could not integrate","F",-1
824,0,0,244,15.7996651,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{\sqrt{e x}} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*Sqrt[c + d*x^2])/Sqrt[e*x],x]","\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{\sqrt{e x}} \, dx","\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,"Could not integrate","F",-1
825,0,0,421,14.2843073,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{(e x)^{3/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*Sqrt[c + d*x^2])/(e*x)^(3/2),x]","\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{(e x)^{3/2}} \, dx","-\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,"Could not integrate","F",-1
826,0,0,234,16.598633,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{(e x)^{5/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*Sqrt[c + d*x^2])/(e*x)^(5/2),x]","\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{(e x)^{5/2}} \, dx","-\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,"Could not integrate","F",-1
827,0,0,421,14.0007371,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{(e x)^{7/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*Sqrt[c + d*x^2])/(e*x)^(7/2),x]","\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{(e x)^{7/2}} \, dx","-\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,"Could not integrate","F",-1
828,0,0,213,12.5175916,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{x^{9/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^(9/2),x]","\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{x^{9/2}} \, dx","-\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,"Could not integrate","F",-1
829,0,0,386,17.4243039,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{x^{11/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^(11/2),x]","\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{x^{11/2}} \, dx","-\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,"Could not integrate","F",-1
830,0,0,217,27.3846947,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{x^{13/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^(13/2),x]","\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{x^{13/2}} \, dx","-\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,"Could not integrate","F",-1
831,0,0,441,29.3769007,"\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{x^{15/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^(15/2),x]","\int \frac{\left(a+b x^2\right)^2 \sqrt{c+d x^2}}{x^{15/2}} \, dx","-\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,"Could not integrate","F",-1
832,0,0,530,24.8280994,"\int (e x)^{5/2} \left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2} \, dx","IntegrateAlgebraic[(e*x)^(5/2)*(a + b*x^2)^2*(c + d*x^2)^(3/2),x]","\int (e x)^{5/2} \left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2} \, dx","-\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,"Could not integrate","F",-1
833,0,0,340,22.8029067,"\int (e x)^{3/2} \left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2} \, dx","IntegrateAlgebraic[(e*x)^(3/2)*(a + b*x^2)^2*(c + d*x^2)^(3/2),x]","\int (e x)^{3/2} \left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2} \, dx","-\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,"Could not integrate","F",-1
834,0,0,482,20.4904122,"\int \sqrt{e x} \left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2} \, dx","IntegrateAlgebraic[Sqrt[e*x]*(a + b*x^2)^2*(c + d*x^2)^(3/2),x]","\int \sqrt{e x} \left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2} \, dx","\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,"Could not integrate","F",-1
835,0,0,286,18.9400458,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}}{\sqrt{e x}} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2)^(3/2))/Sqrt[e*x],x]","\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}}{\sqrt{e x}} \, dx","\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,"Could not integrate","F",-1
836,0,0,476,16.9087202,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}}{(e x)^{3/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2)^(3/2))/(e*x)^(3/2),x]","\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}}{(e x)^{3/2}} \, dx","-\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,"Could not integrate","F",-1
837,0,0,288,17.219414,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}}{(e x)^{5/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2)^(3/2))/(e*x)^(5/2),x]","\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}}{(e x)^{5/2}} \, dx","-\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,"Could not integrate","F",-1
838,0,0,468,16.2601247,"\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}}{(e x)^{7/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2)^(3/2))/(e*x)^(7/2),x]","\int \frac{\left(a+b x^2\right)^2 \left(c+d x^2\right)^{3/2}}{(e x)^{7/2}} \, dx","-\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,"Could not integrate","F",-1
839,0,0,430,34.3029061,"\int \frac{(e x)^{5/2} \left(a+b x^2\right)^2}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[((e*x)^(5/2)*(a + b*x^2)^2)/Sqrt[c + d*x^2],x]","\int \frac{(e x)^{5/2} \left(a+b x^2\right)^2}{\sqrt{c+d x^2}} \, dx","-\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,"Could not integrate","F",-1
840,0,0,240,34.2222449,"\int \frac{(e x)^{3/2} \left(a+b x^2\right)^2}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[((e*x)^(3/2)*(a + b*x^2)^2)/Sqrt[c + d*x^2],x]","\int \frac{(e x)^{3/2} \left(a+b x^2\right)^2}{\sqrt{c+d x^2}} \, dx","-\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,"Could not integrate","F",-1
841,0,0,375,32.1026559,"\int \frac{\sqrt{e x} \left(a+b x^2\right)^2}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(Sqrt[e*x]*(a + b*x^2)^2)/Sqrt[c + d*x^2],x]","\int \frac{\sqrt{e x} \left(a+b x^2\right)^2}{\sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
842,0,0,193,29.657858,"\int \frac{\left(a+b x^2\right)^2}{\sqrt{e x} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(Sqrt[e*x]*Sqrt[c + d*x^2]),x]","\int \frac{\left(a+b x^2\right)^2}{\sqrt{e x} \sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
843,0,0,372,29.8862329,"\int \frac{\left(a+b x^2\right)^2}{(e x)^{3/2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/((e*x)^(3/2)*Sqrt[c + d*x^2]),x]","\int \frac{\left(a+b x^2\right)^2}{(e x)^{3/2} \sqrt{c+d x^2}} \, dx","-\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,"Could not integrate","F",-1
844,0,0,184,31.9168413,"\int \frac{\left(a+b x^2\right)^2}{(e x)^{5/2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/((e*x)^(5/2)*Sqrt[c + d*x^2]),x]","\int \frac{\left(a+b x^2\right)^2}{(e x)^{5/2} \sqrt{c+d x^2}} \, dx","-\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,"Could not integrate","F",-1
845,0,0,387,34.5553787,"\int \frac{\left(a+b x^2\right)^2}{(e x)^{7/2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/((e*x)^(7/2)*Sqrt[c + d*x^2]),x]","\int \frac{\left(a+b x^2\right)^2}{(e x)^{7/2} \sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
846,0,0,193,40.3831418,"\int \frac{\left(a+b x^2\right)^2}{(e x)^{9/2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/((e*x)^(9/2)*Sqrt[c + d*x^2]),x]","\int \frac{\left(a+b x^2\right)^2}{(e x)^{9/2} \sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
847,0,0,438,48.0905401,"\int \frac{\left(a+b x^2\right)^2}{(e x)^{11/2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/((e*x)^(11/2)*Sqrt[c + d*x^2]),x]","\int \frac{\left(a+b x^2\right)^2}{(e x)^{11/2} \sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
848,0,0,242,55.7904368,"\int \frac{\left(a+b x^2\right)^2}{(e x)^{13/2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/((e*x)^(13/2)*Sqrt[c + d*x^2]),x]","\int \frac{\left(a+b x^2\right)^2}{(e x)^{13/2} \sqrt{c+d x^2}} \, dx","-\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,"Could not integrate","F",-1
849,0,0,296,26.5489181,"\int \frac{(e x)^{7/2} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[((e*x)^(7/2)*(a + b*x^2)^2)/(c + d*x^2)^(3/2),x]","\int \frac{(e x)^{7/2} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{3/2}} \, dx","\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,"Could not integrate","F",-1
850,0,0,436,25.7847983,"\int \frac{(e x)^{5/2} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[((e*x)^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^(3/2),x]","\int \frac{(e x)^{5/2} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{3/2}} \, dx","\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,"Could not integrate","F",-1
851,0,0,245,24.0885406,"\int \frac{(e x)^{3/2} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[((e*x)^(3/2)*(a + b*x^2)^2)/(c + d*x^2)^(3/2),x]","\int \frac{(e x)^{3/2} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{3/2}} \, dx","\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,"Could not integrate","F",-1
852,0,0,384,22.5956926,"\int \frac{\sqrt{e x} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(Sqrt[e*x]*(a + b*x^2)^2)/(c + d*x^2)^(3/2),x]","\int \frac{\sqrt{e x} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{3/2}} \, dx","-\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,"Could not integrate","F",-1
853,0,0,193,22.8551899,"\int \frac{\left(a+b x^2\right)^2}{\sqrt{e x} \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(Sqrt[e*x]*(c + d*x^2)^(3/2)),x]","\int \frac{\left(a+b x^2\right)^2}{\sqrt{e x} \left(c+d x^2\right)^{3/2}} \, dx","-\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,"Could not integrate","F",-1
854,0,0,393,24.00098,"\int \frac{\left(a+b x^2\right)^2}{(e x)^{3/2} \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/((e*x)^(3/2)*(c + d*x^2)^(3/2)),x]","\int \frac{\left(a+b x^2\right)^2}{(e x)^{3/2} \left(c+d x^2\right)^{3/2}} \, dx","-\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,"Could not integrate","F",-1
855,0,0,207,28.3793191,"\int \frac{\left(a+b x^2\right)^2}{(e x)^{5/2} \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/((e*x)^(5/2)*(c + d*x^2)^(3/2)),x]","\int \frac{\left(a+b x^2\right)^2}{(e x)^{5/2} \left(c+d x^2\right)^{3/2}} \, dx","-\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,"Could not integrate","F",-1
856,0,0,434,29.4002515,"\int \frac{\left(a+b x^2\right)^2}{(e x)^{7/2} \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/((e*x)^(7/2)*(c + d*x^2)^(3/2)),x]","\int \frac{\left(a+b x^2\right)^2}{(e x)^{7/2} \left(c+d x^2\right)^{3/2}} \, dx","-\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,"Could not integrate","F",-1
857,0,0,302,29.1066862,"\int \frac{(e x)^{7/2} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[((e*x)^(7/2)*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x]","\int \frac{(e x)^{7/2} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{5/2}} \, dx","\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,"Could not integrate","F",-1
858,0,0,442,26.752049,"\int \frac{(e x)^{5/2} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[((e*x)^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x]","\int \frac{(e x)^{5/2} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{5/2}} \, dx","-\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,"Could not integrate","F",-1
859,0,0,248,24.6200182,"\int \frac{(e x)^{3/2} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[((e*x)^(3/2)*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x]","\int \frac{(e x)^{3/2} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{5/2}} \, dx","\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,"Could not integrate","F",-1
860,0,0,403,25.4888477,"\int \frac{\sqrt{e x} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(Sqrt[e*x]*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x]","\int \frac{\sqrt{e x} \left(a+b x^2\right)^2}{\left(c+d x^2\right)^{5/2}} \, dx","\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,"Could not integrate","F",-1
861,0,0,213,23.1856652,"\int \frac{\left(a+b x^2\right)^2}{\sqrt{e x} \left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(Sqrt[e*x]*(c + d*x^2)^(5/2)),x]","\int \frac{\left(a+b x^2\right)^2}{\sqrt{e x} \left(c+d x^2\right)^{5/2}} \, dx","\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,"Could not integrate","F",-1
862,0,0,442,24.8149361,"\int \frac{\left(a+b x^2\right)^2}{(e x)^{3/2} \left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/((e*x)^(3/2)*(c + d*x^2)^(5/2)),x]","\int \frac{\left(a+b x^2\right)^2}{(e x)^{3/2} \left(c+d x^2\right)^{5/2}} \, dx","-\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,"Could not integrate","F",-1
863,0,0,258,28.8007631,"\int \frac{\left(a+b x^2\right)^2}{(e x)^{5/2} \left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/((e*x)^(5/2)*(c + d*x^2)^(5/2)),x]","\int \frac{\left(a+b x^2\right)^2}{(e x)^{5/2} \left(c+d x^2\right)^{5/2}} \, dx","-\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,"Could not integrate","F",-1
864,0,0,489,30.4855691,"\int \frac{\left(a+b x^2\right)^2}{(e x)^{7/2} \left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/((e*x)^(7/2)*(c + d*x^2)^(5/2)),x]","\int \frac{\left(a+b x^2\right)^2}{(e x)^{7/2} \left(c+d x^2\right)^{5/2}} \, dx","-\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,"Could not integrate","F",-1
865,0,0,372,18.4770567,"\int \frac{(e x)^{7/2} \sqrt{c-d x^2}}{a-b x^2} \, dx","IntegrateAlgebraic[((e*x)^(7/2)*Sqrt[c - d*x^2])/(a - b*x^2),x]","\int \frac{(e x)^{7/2} \sqrt{c-d x^2}}{a-b x^2} \, dx","-\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,"Could not integrate","F",-1
866,0,0,414,16.9236986,"\int \frac{(e x)^{5/2} \sqrt{c-d x^2}}{a-b x^2} \, dx","IntegrateAlgebraic[((e*x)^(5/2)*Sqrt[c - d*x^2])/(a - b*x^2),x]","\int \frac{(e x)^{5/2} \sqrt{c-d x^2}}{a-b x^2} \, dx","-\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,"Could not integrate","F",-1
867,0,0,315,13.2169442,"\int \frac{(e x)^{3/2} \sqrt{c-d x^2}}{a-b x^2} \, dx","IntegrateAlgebraic[((e*x)^(3/2)*Sqrt[c - d*x^2])/(a - b*x^2),x]","\int \frac{(e x)^{3/2} \sqrt{c-d x^2}}{a-b x^2} \, dx","-\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,"Could not integrate","F",-1
868,0,0,365,16.7825073,"\int \frac{\sqrt{e x} \sqrt{c-d x^2}}{a-b x^2} \, dx","IntegrateAlgebraic[(Sqrt[e*x]*Sqrt[c - d*x^2])/(a - b*x^2),x]","\int \frac{\sqrt{e x} \sqrt{c-d x^2}}{a-b x^2} \, dx","-\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,"Could not integrate","F",-1
869,0,0,283,19.756271,"\int \frac{\sqrt{c-d x^2}}{\sqrt{e x} \left(a-b x^2\right)} \, dx","IntegrateAlgebraic[Sqrt[c - d*x^2]/(Sqrt[e*x]*(a - b*x^2)),x]","\int \frac{\sqrt{c-d x^2}}{\sqrt{e x} \left(a-b x^2\right)} \, dx","\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,"Could not integrate","F",-1
870,0,0,392,22.1920651,"\int \frac{\sqrt{c-d x^2}}{(e x)^{3/2} \left(a-b x^2\right)} \, dx","IntegrateAlgebraic[Sqrt[c - d*x^2]/((e*x)^(3/2)*(a - b*x^2)),x]","\int \frac{\sqrt{c-d x^2}}{(e x)^{3/2} \left(a-b x^2\right)} \, dx","-\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,"Could not integrate","F",-1
871,0,0,308,26.1375229,"\int \frac{\sqrt{c-d x^2}}{(e x)^{5/2} \left(a-b x^2\right)} \, dx","IntegrateAlgebraic[Sqrt[c - d*x^2]/((e*x)^(5/2)*(a - b*x^2)),x]","\int \frac{\sqrt{c-d x^2}}{(e x)^{5/2} \left(a-b x^2\right)} \, dx","\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,"Could not integrate","F",-1
872,0,0,457,29.0008222,"\int \frac{\sqrt{c-d x^2}}{(e x)^{7/2} \left(a-b x^2\right)} \, dx","IntegrateAlgebraic[Sqrt[c - d*x^2]/((e*x)^(7/2)*(a - b*x^2)),x]","\int \frac{\sqrt{c-d x^2}}{(e x)^{7/2} \left(a-b x^2\right)} \, dx","-\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,"Could not integrate","F",-1
873,0,0,485,18.4313267,"\int \frac{(e x)^{5/2} \left(c-d x^2\right)^{3/2}}{a-b x^2} \, dx","IntegrateAlgebraic[((e*x)^(5/2)*(c - d*x^2)^(3/2))/(a - b*x^2),x]","\int \frac{(e x)^{5/2} \left(c-d x^2\right)^{3/2}}{a-b x^2} \, dx","\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,"Could not integrate","F",-1
874,0,0,372,17.4227469,"\int \frac{(e x)^{3/2} \left(c-d x^2\right)^{3/2}}{a-b x^2} \, dx","IntegrateAlgebraic[((e*x)^(3/2)*(c - d*x^2)^(3/2))/(a - b*x^2),x]","\int \frac{(e x)^{3/2} \left(c-d x^2\right)^{3/2}}{a-b x^2} \, dx","-\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,"Could not integrate","F",-1
875,0,0,421,18.0607733,"\int \frac{\sqrt{e x} \left(c-d x^2\right)^{3/2}}{a-b x^2} \, dx","IntegrateAlgebraic[(Sqrt[e*x]*(c - d*x^2)^(3/2))/(a - b*x^2),x]","\int \frac{\sqrt{e x} \left(c-d x^2\right)^{3/2}}{a-b x^2} \, dx","-\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,"Could not integrate","F",-1
876,0,0,328,21.8371733,"\int \frac{\left(c-d x^2\right)^{3/2}}{\sqrt{e x} \left(a-b x^2\right)} \, dx","IntegrateAlgebraic[(c - d*x^2)^(3/2)/(Sqrt[e*x]*(a - b*x^2)),x]","\int \frac{\left(c-d x^2\right)^{3/2}}{\sqrt{e x} \left(a-b x^2\right)} \, dx","\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,"Could not integrate","F",-1
877,0,0,417,23.0653935,"\int \frac{\left(c-d x^2\right)^{3/2}}{(e x)^{3/2} \left(a-b x^2\right)} \, dx","IntegrateAlgebraic[(c - d*x^2)^(3/2)/((e*x)^(3/2)*(a - b*x^2)),x]","\int \frac{\left(c-d x^2\right)^{3/2}}{(e x)^{3/2} \left(a-b x^2\right)} \, dx","-\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,"Could not integrate","F",-1
878,0,0,330,22.7828849,"\int \frac{\left(c-d x^2\right)^{3/2}}{(e x)^{5/2} \left(a-b x^2\right)} \, dx","IntegrateAlgebraic[(c - d*x^2)^(3/2)/((e*x)^(5/2)*(a - b*x^2)),x]","\int \frac{\left(c-d x^2\right)^{3/2}}{(e x)^{5/2} \left(a-b x^2\right)} \, dx","\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,"Could not integrate","F",-1
879,0,0,459,23.4812664,"\int \frac{\left(c-d x^2\right)^{3/2}}{(e x)^{7/2} \left(a-b x^2\right)} \, dx","IntegrateAlgebraic[(c - d*x^2)^(3/2)/((e*x)^(7/2)*(a - b*x^2)),x]","\int \frac{\left(c-d x^2\right)^{3/2}}{(e x)^{7/2} \left(a-b x^2\right)} \, dx","-\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,"Could not integrate","F",-1
880,0,0,305,41.1867789,"\int \frac{(e x)^{7/2}}{\left(a-b x^2\right) \sqrt{c-d x^2}} \, dx","IntegrateAlgebraic[(e*x)^(7/2)/((a - b*x^2)*Sqrt[c - d*x^2]),x]","\int \frac{(e x)^{7/2}}{\left(a-b x^2\right) \sqrt{c-d x^2}} \, dx","-\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,"Could not integrate","F",-1
881,0,0,349,34.0385676,"\int \frac{(e x)^{5/2}}{\left(a-b x^2\right) \sqrt{c-d x^2}} \, dx","IntegrateAlgebraic[(e*x)^(5/2)/((a - b*x^2)*Sqrt[c - d*x^2]),x]","\int \frac{(e x)^{5/2}}{\left(a-b x^2\right) \sqrt{c-d x^2}} \, dx","-\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,"Could not integrate","F",-1
882,0,0,261,30.6623163,"\int \frac{(e x)^{3/2}}{\left(a-b x^2\right) \sqrt{c-d x^2}} \, dx","IntegrateAlgebraic[(e*x)^(3/2)/((a - b*x^2)*Sqrt[c - d*x^2]),x]","\int \frac{(e x)^{3/2}}{\left(a-b x^2\right) \sqrt{c-d x^2}} \, dx","\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,"Could not integrate","F",-1
883,0,0,203,34.0812978,"\int \frac{\sqrt{e x}}{\left(a-b x^2\right) \sqrt{c-d x^2}} \, dx","IntegrateAlgebraic[Sqrt[e*x]/((a - b*x^2)*Sqrt[c - d*x^2]),x]","\int \frac{\sqrt{e x}}{\left(a-b x^2\right) \sqrt{c-d x^2}} \, dx","\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,"Could not integrate","F",-1
884,0,0,188,36.9134907,"\int \frac{1}{\sqrt{e x} \left(a-b x^2\right) \sqrt{c-d x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[e*x]*(a - b*x^2)*Sqrt[c - d*x^2]),x]","\int \frac{1}{\sqrt{e x} \left(a-b x^2\right) \sqrt{c-d x^2}} \, dx","\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,"Could not integrate","F",-1
885,0,0,379,41.3900206,"\int \frac{1}{(e x)^{3/2} \left(a-b x^2\right) \sqrt{c-d x^2}} \, dx","IntegrateAlgebraic[1/((e*x)^(3/2)*(a - b*x^2)*Sqrt[c - d*x^2]),x]","\int \frac{1}{(e x)^{3/2} \left(a-b x^2\right) \sqrt{c-d x^2}} \, dx","-\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,"Could not integrate","F",-1
886,0,0,297,49.6699269,"\int \frac{1}{(e x)^{5/2} \left(a-b x^2\right) \sqrt{c-d x^2}} \, dx","IntegrateAlgebraic[1/((e*x)^(5/2)*(a - b*x^2)*Sqrt[c - d*x^2]),x]","\int \frac{1}{(e x)^{5/2} \left(a-b x^2\right) \sqrt{c-d x^2}} \, dx","\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,"Could not integrate","F",-1
887,0,0,444,52.7759121,"\int \frac{1}{(e x)^{7/2} \left(a-b x^2\right) \sqrt{c-d x^2}} \, dx","IntegrateAlgebraic[1/((e*x)^(7/2)*(a - b*x^2)*Sqrt[c - d*x^2]),x]","\int \frac{1}{(e x)^{7/2} \left(a-b x^2\right) \sqrt{c-d x^2}} \, dx","-\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,"Could not integrate","F",-1
888,0,0,444,34.4058954,"\int \frac{(e x)^{9/2}}{\left(a-b x^2\right) \left(c-d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(e*x)^(9/2)/((a - b*x^2)*(c - d*x^2)^(3/2)),x]","\int \frac{(e x)^{9/2}}{\left(a-b x^2\right) \left(c-d x^2\right)^{3/2}} \, dx","-\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,"Could not integrate","F",-1
889,0,0,338,32.280764,"\int \frac{(e x)^{7/2}}{\left(a-b x^2\right) \left(c-d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(e*x)^(7/2)/((a - b*x^2)*(c - d*x^2)^(3/2)),x]","\int \frac{(e x)^{7/2}}{\left(a-b x^2\right) \left(c-d x^2\right)^{3/2}} \, dx","\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,"Could not integrate","F",-1
890,0,0,414,28.9401363,"\int \frac{(e x)^{5/2}}{\left(a-b x^2\right) \left(c-d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(e*x)^(5/2)/((a - b*x^2)*(c - d*x^2)^(3/2)),x]","\int \frac{(e x)^{5/2}}{\left(a-b x^2\right) \left(c-d x^2\right)^{3/2}} \, dx","-\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,"Could not integrate","F",-1
891,0,0,314,26.8508456,"\int \frac{(e x)^{3/2}}{\left(a-b x^2\right) \left(c-d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(e*x)^(3/2)/((a - b*x^2)*(c - d*x^2)^(3/2)),x]","\int \frac{(e x)^{3/2}}{\left(a-b x^2\right) \left(c-d x^2\right)^{3/2}} \, dx","-\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,"Could not integrate","F",-1
892,0,0,420,30.0080993,"\int \frac{\sqrt{e x}}{\left(a-b x^2\right) \left(c-d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[Sqrt[e*x]/((a - b*x^2)*(c - d*x^2)^(3/2)),x]","\int \frac{\sqrt{e x}}{\left(a-b x^2\right) \left(c-d x^2\right)^{3/2}} \, dx","-\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,"Could not integrate","F",-1
893,0,0,328,33.6027508,"\int \frac{1}{\sqrt{e x} \left(a-b x^2\right) \left(c-d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[1/(Sqrt[e*x]*(a - b*x^2)*(c - d*x^2)^(3/2)),x]","\int \frac{1}{\sqrt{e x} \left(a-b x^2\right) \left(c-d x^2\right)^{3/2}} \, dx","-\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,"Could not integrate","F",-1
894,0,0,493,38.0837108,"\int \frac{1}{(e x)^{3/2} \left(a-b x^2\right) \left(c-d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[1/((e*x)^(3/2)*(a - b*x^2)*(c - d*x^2)^(3/2)),x]","\int \frac{1}{(e x)^{3/2} \left(a-b x^2\right) \left(c-d x^2\right)^{3/2}} \, dx","-\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,"Could not integrate","F",-1
895,0,0,397,44.9850442,"\int \frac{1}{(e x)^{5/2} \left(a-b x^2\right) \left(c-d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[1/((e*x)^(5/2)*(a - b*x^2)*(c - d*x^2)^(3/2)),x]","\int \frac{1}{(e x)^{5/2} \left(a-b x^2\right) \left(c-d x^2\right)^{3/2}} \, dx","\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,"Could not integrate","F",-1
896,0,0,362,24.0962023,"\int \frac{(e x)^{7/2} \sqrt{c-d x^2}}{\left(a-b x^2\right)^2} \, dx","IntegrateAlgebraic[((e*x)^(7/2)*Sqrt[c - d*x^2])/(a - b*x^2)^2,x]","\int \frac{(e x)^{7/2} \sqrt{c-d x^2}}{\left(a-b x^2\right)^2} \, dx","\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,"Could not integrate","F",-1
897,0,0,413,25.4766795,"\int \frac{(e x)^{5/2} \sqrt{c-d x^2}}{\left(a-b x^2\right)^2} \, dx","IntegrateAlgebraic[((e*x)^(5/2)*Sqrt[c - d*x^2])/(a - b*x^2)^2,x]","\int \frac{(e x)^{5/2} \sqrt{c-d x^2}}{\left(a-b x^2\right)^2} \, dx","\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,"Could not integrate","F",-1
898,0,0,328,28.0939858,"\int \frac{(e x)^{3/2} \sqrt{c-d x^2}}{\left(a-b x^2\right)^2} \, dx","IntegrateAlgebraic[((e*x)^(3/2)*Sqrt[c - d*x^2])/(a - b*x^2)^2,x]","\int \frac{(e x)^{3/2} \sqrt{c-d x^2}}{\left(a-b x^2\right)^2} \, dx","-\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,"Could not integrate","F",-1
899,0,0,417,30.6975581,"\int \frac{\sqrt{e x} \sqrt{c-d x^2}}{\left(a-b x^2\right)^2} \, dx","IntegrateAlgebraic[(Sqrt[e*x]*Sqrt[c - d*x^2])/(a - b*x^2)^2,x]","\int \frac{\sqrt{e x} \sqrt{c-d x^2}}{\left(a-b x^2\right)^2} \, dx","-\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,"Could not integrate","F",-1
900,0,0,335,28.2516352,"\int \frac{\sqrt{c-d x^2}}{\sqrt{e x} \left(a-b x^2\right)^2} \, dx","IntegrateAlgebraic[Sqrt[c - d*x^2]/(Sqrt[e*x]*(a - b*x^2)^2),x]","\int \frac{\sqrt{c-d x^2}}{\sqrt{e x} \left(a-b x^2\right)^2} \, dx","\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,"Could not integrate","F",-1
901,0,0,444,30.4635248,"\int \frac{\sqrt{c-d x^2}}{(e x)^{3/2} \left(a-b x^2\right)^2} \, dx","IntegrateAlgebraic[Sqrt[c - d*x^2]/((e*x)^(3/2)*(a - b*x^2)^2),x]","\int \frac{\sqrt{c-d x^2}}{(e x)^{3/2} \left(a-b x^2\right)^2} \, dx","-\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,"Could not integrate","F",-1
902,0,0,355,35.449275,"\int \frac{\sqrt{c-d x^2}}{(e x)^{5/2} \left(a-b x^2\right)^2} \, dx","IntegrateAlgebraic[Sqrt[c - d*x^2]/((e*x)^(5/2)*(a - b*x^2)^2),x]","\int \frac{\sqrt{c-d x^2}}{(e x)^{5/2} \left(a-b x^2\right)^2} \, dx","\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,"Could not integrate","F",-1
903,0,0,429,25.7488243,"\int \frac{(e x)^{7/2} \left(c-d x^2\right)^{3/2}}{\left(a-b x^2\right)^2} \, dx","IntegrateAlgebraic[((e*x)^(7/2)*(c - d*x^2)^(3/2))/(a - b*x^2)^2,x]","\int \frac{(e x)^{7/2} \left(c-d x^2\right)^{3/2}}{\left(a-b x^2\right)^2} \, dx","\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,"Could not integrate","F",-1
904,0,0,485,24.5622051,"\int \frac{(e x)^{5/2} \left(c-d x^2\right)^{3/2}}{\left(a-b x^2\right)^2} \, dx","IntegrateAlgebraic[((e*x)^(5/2)*(c - d*x^2)^(3/2))/(a - b*x^2)^2,x]","\int \frac{(e x)^{5/2} \left(c-d x^2\right)^{3/2}}{\left(a-b x^2\right)^2} \, dx","\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,"Could not integrate","F",-1
905,0,0,381,27.8131317,"\int \frac{(e x)^{3/2} \left(c-d x^2\right)^{3/2}}{\left(a-b x^2\right)^2} \, dx","IntegrateAlgebraic[((e*x)^(3/2)*(c - d*x^2)^(3/2))/(a - b*x^2)^2,x]","\int \frac{(e x)^{3/2} \left(c-d x^2\right)^{3/2}}{\left(a-b x^2\right)^2} \, dx","-\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,"Could not integrate","F",-1
906,0,0,474,28.2193723,"\int \frac{\sqrt{e x} \left(c-d x^2\right)^{3/2}}{\left(a-b x^2\right)^2} \, dx","IntegrateAlgebraic[(Sqrt[e*x]*(c - d*x^2)^(3/2))/(a - b*x^2)^2,x]","\int \frac{\sqrt{e x} \left(c-d x^2\right)^{3/2}}{\left(a-b x^2\right)^2} \, dx","-\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,"Could not integrate","F",-1
907,0,0,366,27.0059949,"\int \frac{\left(c-d x^2\right)^{3/2}}{\sqrt{e x} \left(a-b x^2\right)^2} \, dx","IntegrateAlgebraic[(c - d*x^2)^(3/2)/(Sqrt[e*x]*(a - b*x^2)^2),x]","\int \frac{\left(c-d x^2\right)^{3/2}}{\sqrt{e x} \left(a-b x^2\right)^2} \, dx","\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,"Could not integrate","F",-1
908,0,0,519,26.0022388,"\int \frac{\left(c-d x^2\right)^{3/2}}{(e x)^{3/2} \left(a-b x^2\right)^2} \, dx","IntegrateAlgebraic[(c - d*x^2)^(3/2)/((e*x)^(3/2)*(a - b*x^2)^2),x]","\int \frac{\left(c-d x^2\right)^{3/2}}{(e x)^{3/2} \left(a-b x^2\right)^2} \, dx","\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,"Could not integrate","F",-1
909,0,0,412,28.6009493,"\int \frac{\left(c-d x^2\right)^{3/2}}{(e x)^{5/2} \left(a-b x^2\right)^2} \, dx","IntegrateAlgebraic[(c - d*x^2)^(3/2)/((e*x)^(5/2)*(a - b*x^2)^2),x]","\int \frac{\left(c-d x^2\right)^{3/2}}{(e x)^{5/2} \left(a-b x^2\right)^2} \, dx","\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,"Could not integrate","F",-1
910,0,0,484,47.9434326,"\int \frac{(e x)^{9/2}}{\left(a-b x^2\right)^2 \sqrt{c-d x^2}} \, dx","IntegrateAlgebraic[(e*x)^(9/2)/((a - b*x^2)^2*Sqrt[c - d*x^2]),x]","\int \frac{(e x)^{9/2}}{\left(a-b x^2\right)^2 \sqrt{c-d x^2}} \, dx","\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,"Could not integrate","F",-1
911,0,0,376,49.8718875,"\int \frac{(e x)^{7/2}}{\left(a-b x^2\right)^2 \sqrt{c-d x^2}} \, dx","IntegrateAlgebraic[(e*x)^(7/2)/((a - b*x^2)^2*Sqrt[c - d*x^2]),x]","\int \frac{(e x)^{7/2}}{\left(a-b x^2\right)^2 \sqrt{c-d x^2}} \, dx","\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,"Could not integrate","F",-1
912,0,0,460,54.6095395,"\int \frac{(e x)^{5/2}}{\left(a-b x^2\right)^2 \sqrt{c-d x^2}} \, dx","IntegrateAlgebraic[(e*x)^(5/2)/((a - b*x^2)^2*Sqrt[c - d*x^2]),x]","\int \frac{(e x)^{5/2}}{\left(a-b x^2\right)^2 \sqrt{c-d x^2}} \, dx","\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,"Could not integrate","F",-1
913,0,0,363,56.4561889,"\int \frac{(e x)^{3/2}}{\left(a-b x^2\right)^2 \sqrt{c-d x^2}} \, dx","IntegrateAlgebraic[(e*x)^(3/2)/((a - b*x^2)^2*Sqrt[c - d*x^2]),x]","\int \frac{(e x)^{3/2}}{\left(a-b x^2\right)^2 \sqrt{c-d x^2}} \, dx","\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,"Could not integrate","F",-1
914,0,0,464,58.1653939,"\int \frac{\sqrt{e x}}{\left(a-b x^2\right)^2 \sqrt{c-d x^2}} \, dx","IntegrateAlgebraic[Sqrt[e*x]/((a - b*x^2)^2*Sqrt[c - d*x^2]),x]","\int \frac{\sqrt{e x}}{\left(a-b x^2\right)^2 \sqrt{c-d x^2}} \, dx","-\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,"Could not integrate","F",-1
915,0,0,367,55.7616829,"\int \frac{1}{\sqrt{e x} \left(a-b x^2\right)^2 \sqrt{c-d x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[e*x]*(a - b*x^2)^2*Sqrt[c - d*x^2]),x]","\int \frac{1}{\sqrt{e x} \left(a-b x^2\right)^2 \sqrt{c-d x^2}} \, dx","\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,"Could not integrate","F",-1
916,0,0,535,56.2722698,"\int \frac{1}{(e x)^{3/2} \left(a-b x^2\right)^2 \sqrt{c-d x^2}} \, dx","IntegrateAlgebraic[1/((e*x)^(3/2)*(a - b*x^2)^2*Sqrt[c - d*x^2]),x]","\int \frac{1}{(e x)^{3/2} \left(a-b x^2\right)^2 \sqrt{c-d x^2}} \, dx","-\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,"Could not integrate","F",-1
917,0,0,429,56.7255109,"\int \frac{1}{(e x)^{5/2} \left(a-b x^2\right)^2 \sqrt{c-d x^2}} \, dx","IntegrateAlgebraic[1/((e*x)^(5/2)*(a - b*x^2)^2*Sqrt[c - d*x^2]),x]","\int \frac{1}{(e x)^{5/2} \left(a-b x^2\right)^2 \sqrt{c-d x^2}} \, dx","\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,"Could not integrate","F",-1
918,0,0,529,37.5269993,"\int \frac{(e x)^{9/2}}{\left(a-b x^2\right)^2 \left(c-d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(e*x)^(9/2)/((a - b*x^2)^2*(c - d*x^2)^(3/2)),x]","\int \frac{(e x)^{9/2}}{\left(a-b x^2\right)^2 \left(c-d x^2\right)^{3/2}} \, dx","\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,"Could not integrate","F",-1
919,0,0,420,37.4040792,"\int \frac{(e x)^{7/2}}{\left(a-b x^2\right)^2 \left(c-d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(e*x)^(7/2)/((a - b*x^2)^2*(c - d*x^2)^(3/2)),x]","\int \frac{(e x)^{7/2}}{\left(a-b x^2\right)^2 \left(c-d x^2\right)^{3/2}} \, dx","\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,"Could not integrate","F",-1
920,0,0,485,40.9910875,"\int \frac{(e x)^{5/2}}{\left(a-b x^2\right)^2 \left(c-d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(e*x)^(5/2)/((a - b*x^2)^2*(c - d*x^2)^(3/2)),x]","\int \frac{(e x)^{5/2}}{\left(a-b x^2\right)^2 \left(c-d x^2\right)^{3/2}} \, dx","\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,"Could not integrate","F",-1
921,0,0,391,45.9584184,"\int \frac{(e x)^{3/2}}{\left(a-b x^2\right)^2 \left(c-d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(e*x)^(3/2)/((a - b*x^2)^2*(c - d*x^2)^(3/2)),x]","\int \frac{(e x)^{3/2}}{\left(a-b x^2\right)^2 \left(c-d x^2\right)^{3/2}} \, dx","\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,"Could not integrate","F",-1
922,0,0,531,45.3931942,"\int \frac{\sqrt{e x}}{\left(a-b x^2\right)^2 \left(c-d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[Sqrt[e*x]/((a - b*x^2)^2*(c - d*x^2)^(3/2)),x]","\int \frac{\sqrt{e x}}{\left(a-b x^2\right)^2 \left(c-d x^2\right)^{3/2}} \, dx","-\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,"Could not integrate","F",-1
923,0,0,426,43.5836554,"\int \frac{1}{\sqrt{e x} \left(a-b x^2\right)^2 \left(c-d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[1/(Sqrt[e*x]*(a - b*x^2)^2*(c - d*x^2)^(3/2)),x]","\int \frac{1}{\sqrt{e x} \left(a-b x^2\right)^2 \left(c-d x^2\right)^{3/2}} \, dx","\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,"Could not integrate","F",-1
924,0,0,628,44.1301389,"\int \frac{1}{(e x)^{3/2} \left(a-b x^2\right)^2 \left(c-d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[1/((e*x)^(3/2)*(a - b*x^2)^2*(c - d*x^2)^(3/2)),x]","\int \frac{1}{(e x)^{3/2} \left(a-b x^2\right)^2 \left(c-d x^2\right)^{3/2}} \, dx","-\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,"Could not integrate","F",-1
925,0,0,512,49.0274573,"\int \frac{1}{(e x)^{5/2} \left(a-b x^2\right)^2 \left(c-d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[1/((e*x)^(5/2)*(a - b*x^2)^2*(c - d*x^2)^(3/2)),x]","\int \frac{1}{(e x)^{5/2} \left(a-b x^2\right)^2 \left(c-d x^2\right)^{3/2}} \, dx","\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,"Could not integrate","F",-1
926,0,0,568,33.5257275,"\int \frac{(e x)^{9/2}}{\left(a-b x^2\right)^2 \left(c-d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(e*x)^(9/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)),x]","\int \frac{(e x)^{9/2}}{\left(a-b x^2\right)^2 \left(c-d x^2\right)^{5/2}} \, dx","\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,"Could not integrate","F",-1
927,0,0,454,33.2387943,"\int \frac{(e x)^{7/2}}{\left(a-b x^2\right)^2 \left(c-d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(e*x)^(7/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)),x]","\int \frac{(e x)^{7/2}}{\left(a-b x^2\right)^2 \left(c-d x^2\right)^{5/2}} \, dx","\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,"Could not integrate","F",-1
928,0,0,551,38.3688183,"\int \frac{(e x)^{5/2}}{\left(a-b x^2\right)^2 \left(c-d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(e*x)^(5/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)),x]","\int \frac{(e x)^{5/2}}{\left(a-b x^2\right)^2 \left(c-d x^2\right)^{5/2}} \, dx","\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,"Could not integrate","F",-1
929,0,0,447,43.694052,"\int \frac{(e x)^{3/2}}{\left(a-b x^2\right)^2 \left(c-d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(e*x)^(3/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)),x]","\int \frac{(e x)^{3/2}}{\left(a-b x^2\right)^2 \left(c-d x^2\right)^{5/2}} \, dx","\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,"Could not integrate","F",-1
930,0,0,625,44.4602086,"\int \frac{\sqrt{e x}}{\left(a-b x^2\right)^2 \left(c-d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[Sqrt[e*x]/((a - b*x^2)^2*(c - d*x^2)^(5/2)),x]","\int \frac{\sqrt{e x}}{\left(a-b x^2\right)^2 \left(c-d x^2\right)^{5/2}} \, dx","-\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,"Could not integrate","F",-1
931,0,0,514,41.7913858,"\int \frac{1}{\sqrt{e x} \left(a-b x^2\right)^2 \left(c-d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[1/(Sqrt[e*x]*(a - b*x^2)^2*(c - d*x^2)^(5/2)),x]","\int \frac{1}{\sqrt{e x} \left(a-b x^2\right)^2 \left(c-d x^2\right)^{5/2}} \, dx","\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,"Could not integrate","F",-1
932,0,0,735,44.234463,"\int \frac{1}{(e x)^{3/2} \left(a-b x^2\right)^2 \left(c-d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[1/((e*x)^(3/2)*(a - b*x^2)^2*(c - d*x^2)^(5/2)),x]","\int \frac{1}{(e x)^{3/2} \left(a-b x^2\right)^2 \left(c-d x^2\right)^{5/2}} \, dx","-\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,"Could not integrate","F",-1
933,0,0,606,46.6431016,"\int \frac{1}{(e x)^{5/2} \left(a-b x^2\right)^2 \left(c-d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[1/((e*x)^(5/2)*(a - b*x^2)^2*(c - d*x^2)^(5/2)),x]","\int \frac{1}{(e x)^{5/2} \left(a-b x^2\right)^2 \left(c-d x^2\right)^{5/2}} \, dx","\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,"Could not integrate","F",-1
934,0,0,209,0.9184878,"\int \frac{x^5 \sqrt{a+b x^2}}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(x^5*Sqrt[a + b*x^2])/Sqrt[c + d*x^2],x]","\int \frac{x^5 \sqrt{a+b x^2}}{\sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
935,0,0,137,0.7404618,"\int \frac{x^3 \sqrt{a+b x^2}}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(x^3*Sqrt[a + b*x^2])/Sqrt[c + d*x^2],x]","\int \frac{x^3 \sqrt{a+b x^2}}{\sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
936,0,0,86,0.4755157,"\int \frac{x \sqrt{a+b x^2}}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(x*Sqrt[a + b*x^2])/Sqrt[c + d*x^2],x]","\int \frac{x \sqrt{a+b x^2}}{\sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
937,0,0,92,0.6197467,"\int \frac{\sqrt{a+b x^2}}{x \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[Sqrt[a + b*x^2]/(x*Sqrt[c + d*x^2]),x]","\int \frac{\sqrt{a+b x^2}}{x \sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
938,0,0,89,0.7666312,"\int \frac{\sqrt{a+b x^2}}{x^3 \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[Sqrt[a + b*x^2]/(x^3*Sqrt[c + d*x^2]),x]","\int \frac{\sqrt{a+b x^2}}{x^3 \sqrt{c+d x^2}} \, dx","-\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,"Could not integrate","F",-1
939,0,0,143,1.0800721,"\int \frac{\sqrt{a+b x^2}}{x^5 \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[Sqrt[a + b*x^2]/(x^5*Sqrt[c + d*x^2]),x]","\int \frac{\sqrt{a+b x^2}}{x^5 \sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
940,0,0,343,0.9438108,"\int \frac{x^4 \sqrt{a+b x^2}}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(x^4*Sqrt[a + b*x^2])/Sqrt[c + d*x^2],x]","\int \frac{x^4 \sqrt{a+b x^2}}{\sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
941,0,0,259,0.4946396,"\int \frac{x^2 \sqrt{a+b x^2}}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(x^2*Sqrt[a + b*x^2])/Sqrt[c + d*x^2],x]","\int \frac{x^2 \sqrt{a+b x^2}}{\sqrt{c+d x^2}} \, dx","-\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,"Could not integrate","F",-1
942,0,0,232,0.8094171,"\int \frac{\sqrt{a+b x^2}}{x^2 \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[Sqrt[a + b*x^2]/(x^2*Sqrt[c + d*x^2]),x]","\int \frac{\sqrt{a+b x^2}}{x^2 \sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
943,0,0,307,0.8831433,"\int \frac{\sqrt{a+b x^2}}{x^4 \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[Sqrt[a + b*x^2]/(x^4*Sqrt[c + d*x^2]),x]","\int \frac{\sqrt{a+b x^2}}{x^4 \sqrt{c+d x^2}} \, dx","-\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,"Could not integrate","F",-1
944,0,0,276,1.5472567,"\int \frac{x^5 \left(a+b x^2\right)^{3/2}}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(x^5*(a + b*x^2)^(3/2))/Sqrt[c + d*x^2],x]","\int \frac{x^5 \left(a+b x^2\right)^{3/2}}{\sqrt{c+d x^2}} \, dx","-\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,"Could not integrate","F",-1
945,0,0,187,1.3253031,"\int \frac{x^3 \left(a+b x^2\right)^{3/2}}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(x^3*(a + b*x^2)^(3/2))/Sqrt[c + d*x^2],x]","\int \frac{x^3 \left(a+b x^2\right)^{3/2}}{\sqrt{c+d x^2}} \, dx","-\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,"Could not integrate","F",-1
946,0,0,125,0.8519688,"\int \frac{x \left(a+b x^2\right)^{3/2}}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(x*(a + b*x^2)^(3/2))/Sqrt[c + d*x^2],x]","\int \frac{x \left(a+b x^2\right)^{3/2}}{\sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
947,0,0,133,1.1180321,"\int \frac{\left(a+b x^2\right)^{3/2}}{x \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^(3/2)/(x*Sqrt[c + d*x^2]),x]","\int \frac{\left(a+b x^2\right)^{3/2}}{x \sqrt{c+d x^2}} \, dx","-\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,"Could not integrate","F",-1
948,0,0,136,1.0859972,"\int \frac{\left(a+b x^2\right)^{3/2}}{x^3 \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^(3/2)/(x^3*Sqrt[c + d*x^2]),x]","\int \frac{\left(a+b x^2\right)^{3/2}}{x^3 \sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
949,0,0,131,1.2314096,"\int \frac{\left(a+b x^2\right)^{3/2}}{x^5 \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^(3/2)/(x^5*Sqrt[c + d*x^2]),x]","\int \frac{\left(a+b x^2\right)^{3/2}}{x^5 \sqrt{c+d x^2}} \, dx","-\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,"Could not integrate","F",-1
950,0,0,429,1.5938956,"\int \frac{x^4 \left(a+b x^2\right)^{3/2}}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(x^4*(a + b*x^2)^(3/2))/Sqrt[c + d*x^2],x]","\int \frac{x^4 \left(a+b x^2\right)^{3/2}}{\sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
951,0,0,335,0.9828852,"\int \frac{x^2 \left(a+b x^2\right)^{3/2}}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(x^2*(a + b*x^2)^(3/2))/Sqrt[c + d*x^2],x]","\int \frac{x^2 \left(a+b x^2\right)^{3/2}}{\sqrt{c+d x^2}} \, dx","-\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,"Could not integrate","F",-1
952,0,0,244,1.0867248,"\int \frac{\left(a+b x^2\right)^{3/2}}{x^2 \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^(3/2)/(x^2*Sqrt[c + d*x^2]),x]","\int \frac{\left(a+b x^2\right)^{3/2}}{x^2 \sqrt{c+d x^2}} \, dx","-\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,"Could not integrate","F",-1
953,0,0,311,1.3161193,"\int \frac{\left(a+b x^2\right)^{3/2}}{x^4 \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^(3/2)/(x^4*Sqrt[c + d*x^2]),x]","\int \frac{\left(a+b x^2\right)^{3/2}}{x^4 \sqrt{c+d x^2}} \, dx","-\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,"Could not integrate","F",-1
954,0,0,340,1.7842118,"\int \frac{x^5 \left(a+b x^2\right)^{5/2}}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(x^5*(a + b*x^2)^(5/2))/Sqrt[c + d*x^2],x]","\int \frac{x^5 \left(a+b x^2\right)^{5/2}}{\sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
955,0,0,237,1.4565688,"\int \frac{x^3 \left(a+b x^2\right)^{5/2}}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(x^3*(a + b*x^2)^(5/2))/Sqrt[c + d*x^2],x]","\int \frac{x^3 \left(a+b x^2\right)^{5/2}}{\sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
956,0,0,164,1.06884,"\int \frac{x \left(a+b x^2\right)^{5/2}}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(x*(a + b*x^2)^(5/2))/Sqrt[c + d*x^2],x]","\int \frac{x \left(a+b x^2\right)^{5/2}}{\sqrt{c+d x^2}} \, dx","-\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,"Could not integrate","F",-1
957,0,0,187,1.2039725,"\int \frac{\left(a+b x^2\right)^{5/2}}{x \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^(5/2)/(x*Sqrt[c + d*x^2]),x]","\int \frac{\left(a+b x^2\right)^{5/2}}{x \sqrt{c+d x^2}} \, dx","-\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,"Could not integrate","F",-1
958,0,0,187,1.3238156,"\int \frac{\left(a+b x^2\right)^{5/2}}{x^3 \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^(5/2)/(x^3*Sqrt[c + d*x^2]),x]","\int \frac{\left(a+b x^2\right)^{5/2}}{x^3 \sqrt{c+d x^2}} \, dx","-\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,"Could not integrate","F",-1
959,0,0,192,1.4001945,"\int \frac{\left(a+b x^2\right)^{5/2}}{x^5 \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^(5/2)/(x^5*Sqrt[c + d*x^2]),x]","\int \frac{\left(a+b x^2\right)^{5/2}}{x^5 \sqrt{c+d x^2}} \, dx","-\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,"Could not integrate","F",-1
960,0,0,553,1.7292502,"\int \frac{x^4 \left(a+b x^2\right)^{5/2}}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(x^4*(a + b*x^2)^(5/2))/Sqrt[c + d*x^2],x]","\int \frac{x^4 \left(a+b x^2\right)^{5/2}}{\sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
961,0,0,436,1.1894851,"\int \frac{x^2 \left(a+b x^2\right)^{5/2}}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(x^2*(a + b*x^2)^(5/2))/Sqrt[c + d*x^2],x]","\int \frac{x^2 \left(a+b x^2\right)^{5/2}}{\sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
962,0,0,330,1.2637676,"\int \frac{\left(a+b x^2\right)^{5/2}}{x^2 \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^(5/2)/(x^2*Sqrt[c + d*x^2]),x]","\int \frac{\left(a+b x^2\right)^{5/2}}{x^2 \sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
963,0,0,336,1.4540417,"\int \frac{\left(a+b x^2\right)^{5/2}}{x^4 \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^(5/2)/(x^4*Sqrt[c + d*x^2]),x]","\int \frac{\left(a+b x^2\right)^{5/2}}{x^4 \sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
964,0,0,99,0.4234289,"\int \frac{x^4 \sqrt{-1+3 x^2}}{\sqrt{2-3 x^2}} \, dx","IntegrateAlgebraic[(x^4*Sqrt[-1 + 3*x^2])/Sqrt[2 - 3*x^2],x]","\int \frac{x^4 \sqrt{-1+3 x^2}}{\sqrt{2-3 x^2}} \, dx","-\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,"Could not integrate","F",-1
965,0,0,65,0.506526,"\int \frac{x^3 \sqrt{-1+3 x^2}}{\sqrt{2-3 x^2}} \, dx","IntegrateAlgebraic[(x^3*Sqrt[-1 + 3*x^2])/Sqrt[2 - 3*x^2],x]","\int \frac{x^3 \sqrt{-1+3 x^2}}{\sqrt{2-3 x^2}} \, dx","-\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,"Could not integrate","F",-1
966,0,0,70,0.226393,"\int \frac{x^2 \sqrt{-1+3 x^2}}{\sqrt{2-3 x^2}} \, dx","IntegrateAlgebraic[(x^2*Sqrt[-1 + 3*x^2])/Sqrt[2 - 3*x^2],x]","\int \frac{x^2 \sqrt{-1+3 x^2}}{\sqrt{2-3 x^2}} \, dx","-\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,"Could not integrate","F",-1
967,1,59,39,0.1237815,"\int \frac{x \sqrt{-1+3 x^2}}{\sqrt{2-3 x^2}} \, dx","IntegrateAlgebraic[(x*Sqrt[-1 + 3*x^2])/Sqrt[2 - 3*x^2],x]","\frac{1}{3} \tan ^{-1}\left(\frac{\sqrt{3 x^2-1}}{\sqrt{2-3 x^2}-1}\right)-\frac{1}{6} \sqrt{2-3 x^2} \sqrt{3 x^2-1}","-\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,"-1/6*(Sqrt[2 - 3*x^2]*Sqrt[-1 + 3*x^2]) + ArcTan[Sqrt[-1 + 3*x^2]/(-1 + Sqrt[2 - 3*x^2])]/3","A",1
968,0,0,241,0.5004699,"\int \frac{x^2 \sqrt{2+b x^2}}{\sqrt{3+d x^2}} \, dx","IntegrateAlgebraic[(x^2*Sqrt[2 + b*x^2])/Sqrt[3 + d*x^2],x]","\int \frac{x^2 \sqrt{2+b x^2}}{\sqrt{3+d x^2}} \, dx","-\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,"Could not integrate","F",-1
969,0,0,141,1.1273049,"\int \frac{x^5}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[x^5/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]","\int \frac{x^5}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","-\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,"Could not integrate","F",-1
970,0,0,88,0.8985107,"\int \frac{x^3}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[x^3/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]","\int \frac{x^3}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
971,0,0,45,0.7107315,"\int \frac{x}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[x/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]","\int \frac{x}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
972,0,0,46,0.7924118,"\int \frac{1}{x \sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[1/(x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]","\int \frac{1}{x \sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","-\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,"Could not integrate","F",-1
973,0,0,91,1.0736761,"\int \frac{1}{x^3 \sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[1/(x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]","\int \frac{1}{x^3 \sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
974,0,0,149,1.1248797,"\int \frac{1}{x^5 \sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[1/(x^5*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]","\int \frac{1}{x^5 \sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
975,0,0,342,1.5735497,"\int \frac{x^6}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[x^6/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]","\int \frac{x^6}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
976,0,0,261,1.1679131,"\int \frac{x^4}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[x^4/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]","\int \frac{x^4}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
977,0,0,116,0.5687153,"\int \frac{x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[x^2/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]","\int \frac{x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
978,0,0,153,1.100626,"\int \frac{1}{x^2 \sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[1/(x^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]","\int \frac{1}{x^2 \sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
979,0,0,307,1.3094001,"\int \frac{1}{x^4 \sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[1/(x^4*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]","\int \frac{1}{x^4 \sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
980,0,0,129,1.7433438,"\int \frac{x^5}{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[x^5/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]),x]","\int \frac{x^5}{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}} \, dx","-\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,"Could not integrate","F",-1
981,0,0,83,1.4071488,"\int \frac{x^3}{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[x^3/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]),x]","\int \frac{x^3}{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
982,0,0,34,1.0726553,"\int \frac{x}{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[x/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]),x]","\int \frac{x}{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}} \, dx","-\frac{\sqrt{c+d x^2}}{\sqrt{a+b x^2} (b c-a d)}",1,"Could not integrate","F",-1
983,0,0,137,1.9455719,"\int \frac{x^5}{\left(a+b x^2\right)^{5/2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[x^5/((a + b*x^2)^(5/2)*Sqrt[c + d*x^2]),x]","\int \frac{x^5}{\left(a+b x^2\right)^{5/2} \sqrt{c+d x^2}} \, dx","-\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,"Could not integrate","F",-1
984,0,0,89,1.5947996,"\int \frac{x^3}{\left(a+b x^2\right)^{5/2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[x^3/((a + b*x^2)^(5/2)*Sqrt[c + d*x^2]),x]","\int \frac{x^3}{\left(a+b x^2\right)^{5/2} \sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
985,0,0,74,1.1688249,"\int \frac{x}{\left(a+b x^2\right)^{5/2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[x/((a + b*x^2)^(5/2)*Sqrt[c + d*x^2]),x]","\int \frac{x}{\left(a+b x^2\right)^{5/2} \sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
986,0,0,154,2.0475822,"\int \frac{x^5}{\left(a+b x^2\right)^{7/2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[x^5/((a + b*x^2)^(7/2)*Sqrt[c + d*x^2]),x]","\int \frac{x^5}{\left(a+b x^2\right)^{7/2} \sqrt{c+d x^2}} \, dx","-\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,"Could not integrate","F",-1
987,0,0,138,1.853559,"\int \frac{x^3}{\left(a+b x^2\right)^{7/2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[x^3/((a + b*x^2)^(7/2)*Sqrt[c + d*x^2]),x]","\int \frac{x^3}{\left(a+b x^2\right)^{7/2} \sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
988,0,0,113,1.4843927,"\int \frac{x}{\left(a+b x^2\right)^{7/2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[x/((a + b*x^2)^(7/2)*Sqrt[c + d*x^2]),x]","\int \frac{x}{\left(a+b x^2\right)^{7/2} \sqrt{c+d x^2}} \, dx","-\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,"Could not integrate","F",-1
989,0,0,217,2.1746562,"\int \frac{x^5}{\left(a+b x^2\right)^{9/2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[x^5/((a + b*x^2)^(9/2)*Sqrt[c + d*x^2]),x]","\int \frac{x^5}{\left(a+b x^2\right)^{9/2} \sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
990,0,0,47,0.4984514,"\int \frac{x}{\sqrt{a-b x^2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[x/(Sqrt[a - b*x^2]*Sqrt[c + d*x^2]),x]","\int \frac{x}{\sqrt{a-b x^2} \sqrt{c+d x^2}} \, dx","-\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,"Could not integrate","F",-1
991,0,0,48,0.5930652,"\int \frac{x}{\sqrt{a-b x^2} \sqrt{c-d x^2}} \, dx","IntegrateAlgebraic[x/(Sqrt[a - b*x^2]*Sqrt[c - d*x^2]),x]","\int \frac{x}{\sqrt{a-b x^2} \sqrt{c-d x^2}} \, dx","-\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,"Could not integrate","F",-1
992,0,0,110,0.5669188,"\int \frac{x^2}{\sqrt{2+b x^2} \sqrt{3+d x^2}} \, dx","IntegrateAlgebraic[x^2/(Sqrt[2 + b*x^2]*Sqrt[3 + d*x^2]),x]","\int \frac{x^2}{\sqrt{2+b x^2} \sqrt{3+d x^2}} \, dx","\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,"Could not integrate","F",-1
993,0,0,87,0.4989491,"\int \frac{x^2}{\sqrt{4-x^2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[x^2/(Sqrt[4 - x^2]*Sqrt[c + d*x^2]),x]","\int \frac{x^2}{\sqrt{4-x^2} \sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
994,0,0,88,0.4930671,"\int \frac{x^2}{\sqrt{4+x^2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[x^2/(Sqrt[4 + x^2]*Sqrt[c + d*x^2]),x]","\int \frac{x^2}{\sqrt{4+x^2} \sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
995,0,0,31,0.3324972,"\int \frac{x^2}{\sqrt{1-x^2} \sqrt{2+3 x^2}} \, dx","IntegrateAlgebraic[x^2/(Sqrt[1 - x^2]*Sqrt[2 + 3*x^2]),x]","\int \frac{x^2}{\sqrt{1-x^2} \sqrt{2+3 x^2}} \, dx","\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,"Could not integrate","F",-1
996,0,0,31,0.2330019,"\int \frac{x^2}{\sqrt{2-3 x^2} \sqrt{1-x^2}} \, dx","IntegrateAlgebraic[x^2/(Sqrt[2 - 3*x^2]*Sqrt[1 - x^2]),x]","\int \frac{x^2}{\sqrt{2-3 x^2} \sqrt{1-x^2}} \, dx","\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,"Could not integrate","F",-1
997,0,0,35,0.2394664,"\int \frac{x^2}{\sqrt{4-x^2} \sqrt{2+3 x^2}} \, dx","IntegrateAlgebraic[x^2/(Sqrt[4 - x^2]*Sqrt[2 + 3*x^2]),x]","\int \frac{x^2}{\sqrt{4-x^2} \sqrt{2+3 x^2}} \, dx","\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,"Could not integrate","F",-1
998,0,0,35,0.3118821,"\int \frac{x^2}{\sqrt{2-3 x^2} \sqrt{4-x^2}} \, dx","IntegrateAlgebraic[x^2/(Sqrt[2 - 3*x^2]*Sqrt[4 - x^2]),x]","\int \frac{x^2}{\sqrt{2-3 x^2} \sqrt{4-x^2}} \, dx","\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,"Could not integrate","F",-1
999,0,0,35,0.3359894,"\int \frac{x^2}{\sqrt{1-4 x^2} \sqrt{2+3 x^2}} \, dx","IntegrateAlgebraic[x^2/(Sqrt[1 - 4*x^2]*Sqrt[2 + 3*x^2]),x]","\int \frac{x^2}{\sqrt{1-4 x^2} \sqrt{2+3 x^2}} \, dx","\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,"Could not integrate","F",-1
1000,0,0,35,0.2310475,"\int \frac{x^2}{\sqrt{1-4 x^2} \sqrt{2-3 x^2}} \, dx","IntegrateAlgebraic[x^2/(Sqrt[1 - 4*x^2]*Sqrt[2 - 3*x^2]),x]","\int \frac{x^2}{\sqrt{1-4 x^2} \sqrt{2-3 x^2}} \, dx","\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,"Could not integrate","F",-1
1001,0,0,42,0.2432627,"\int \frac{x^2}{\sqrt{2-3 x^2} \sqrt{1+x^2}} \, dx","IntegrateAlgebraic[x^2/(Sqrt[2 - 3*x^2]*Sqrt[1 + x^2]),x]","\int \frac{x^2}{\sqrt{2-3 x^2} \sqrt{1+x^2}} \, dx","\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,"Could not integrate","F",-1
1002,0,0,43,0.3246009,"\int \frac{x^2}{\sqrt{2-3 x^2} \sqrt{4+x^2}} \, dx","IntegrateAlgebraic[x^2/(Sqrt[2 - 3*x^2]*Sqrt[4 + x^2]),x]","\int \frac{x^2}{\sqrt{2-3 x^2} \sqrt{4+x^2}} \, dx","\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,"Could not integrate","F",-1
1003,0,0,47,0.3273837,"\int \frac{x^2}{\sqrt{2-3 x^2} \sqrt{1+4 x^2}} \, dx","IntegrateAlgebraic[x^2/(Sqrt[2 - 3*x^2]*Sqrt[1 + 4*x^2]),x]","\int \frac{x^2}{\sqrt{2-3 x^2} \sqrt{1+4 x^2}} \, dx","\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,"Could not integrate","F",-1
1004,0,0,80,0.2301029,"\int \frac{x^2}{\sqrt{1+x^2} \sqrt{2+3 x^2}} \, dx","IntegrateAlgebraic[x^2/(Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]),x]","\int \frac{x^2}{\sqrt{1+x^2} \sqrt{2+3 x^2}} \, dx","\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,"Could not integrate","F",-1
1005,0,0,82,0.2257179,"\int \frac{x^2}{\sqrt{4+x^2} \sqrt{2+3 x^2}} \, dx","IntegrateAlgebraic[x^2/(Sqrt[4 + x^2]*Sqrt[2 + 3*x^2]),x]","\int \frac{x^2}{\sqrt{4+x^2} \sqrt{2+3 x^2}} \, dx","\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,"Could not integrate","F",-1
1006,0,0,88,0.2339083,"\int \frac{x^2}{\sqrt{2+3 x^2} \sqrt{1+4 x^2}} \, dx","IntegrateAlgebraic[x^2/(Sqrt[2 + 3*x^2]*Sqrt[1 + 4*x^2]),x]","\int \frac{x^2}{\sqrt{2+3 x^2} \sqrt{1+4 x^2}} \, dx","\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,"Could not integrate","F",-1
1007,0,0,17,0.3312276,"\int \frac{x^2}{\sqrt{1-x^2} \sqrt{-1+2 x^2}} \, dx","IntegrateAlgebraic[x^2/(Sqrt[1 - x^2]*Sqrt[-1 + 2*x^2]),x]","\int \frac{x^2}{\sqrt{1-x^2} \sqrt{-1+2 x^2}} \, dx","-\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,"Could not integrate","F",-1
1008,1,140,109,0.2240239,"\int \frac{x^5}{\sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","IntegrateAlgebraic[x^5/((1 - x^2)^(1/3)*(3 + x^2)),x]","-\frac{3}{10} \left(1-x^2\right)^{2/3} \left(x^2-6\right)+\frac{9 \log \left(\sqrt[3]{2} \sqrt[3]{1-x^2}-2\right)}{2\ 2^{2/3}}-\frac{9 \log \left(2^{2/3} \left(1-x^2\right)^{2/3}+2 \sqrt[3]{2} \sqrt[3]{1-x^2}+4\right)}{4\ 2^{2/3}}+\frac{9 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{2} \sqrt[3]{1-x^2}}{\sqrt{3}}+\frac{1}{\sqrt{3}}\right)}{2\ 2^{2/3}}","\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*(1 - x^2)^(2/3)*(-6 + x^2))/10 + (9*Sqrt[3]*ArcTan[1/Sqrt[3] + (2^(1/3)*(1 - x^2)^(1/3))/Sqrt[3]])/(2*2^(2/3)) + (9*Log[-2 + 2^(1/3)*(1 - x^2)^(1/3)])/(2*2^(2/3)) - (9*Log[4 + 2*2^(1/3)*(1 - x^2)^(1/3) + 2^(2/3)*(1 - x^2)^(2/3)])/(4*2^(2/3))","A",1
1009,1,135,94,0.182761,"\int \frac{x^3}{\sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","IntegrateAlgebraic[x^3/((1 - x^2)^(1/3)*(3 + x^2)),x]","-\frac{3}{4} \left(1-x^2\right)^{2/3}-\frac{3 \log \left(\sqrt[3]{2} \sqrt[3]{1-x^2}-2\right)}{2\ 2^{2/3}}+\frac{3 \log \left(2^{2/3} \left(1-x^2\right)^{2/3}+2 \sqrt[3]{2} \sqrt[3]{1-x^2}+4\right)}{4\ 2^{2/3}}-\frac{3 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{2} \sqrt[3]{1-x^2}}{\sqrt{3}}+\frac{1}{\sqrt{3}}\right)}{2\ 2^{2/3}}","-\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*(1 - x^2)^(2/3))/4 - (3*Sqrt[3]*ArcTan[1/Sqrt[3] + (2^(1/3)*(1 - x^2)^(1/3))/Sqrt[3]])/(2*2^(2/3)) - (3*Log[-2 + 2^(1/3)*(1 - x^2)^(1/3)])/(2*2^(2/3)) + (3*Log[4 + 2*2^(1/3)*(1 - x^2)^(1/3) + 2^(2/3)*(1 - x^2)^(2/3)])/(4*2^(2/3))","A",1
1010,1,120,79,0.3423189,"\int \frac{x}{\sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","IntegrateAlgebraic[x/((1 - x^2)^(1/3)*(3 + x^2)),x]","\frac{\log \left(\sqrt[3]{2} \sqrt[3]{1-x^2}-2\right)}{2\ 2^{2/3}}-\frac{\log \left(2^{2/3} \left(1-x^2\right)^{2/3}+2 \sqrt[3]{2} \sqrt[3]{1-x^2}+4\right)}{4\ 2^{2/3}}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{2} \sqrt[3]{1-x^2}}{\sqrt{3}}+\frac{1}{\sqrt{3}}\right)}{2\ 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,"(Sqrt[3]*ArcTan[1/Sqrt[3] + (2^(1/3)*(1 - x^2)^(1/3))/Sqrt[3]])/(2*2^(2/3)) + Log[-2 + 2^(1/3)*(1 - x^2)^(1/3)]/(2*2^(2/3)) - Log[4 + 2*2^(1/3)*(1 - x^2)^(1/3) + 2^(2/3)*(1 - x^2)^(2/3)]/(4*2^(2/3))","A",1
1011,1,201,136,0.3145089,"\int \frac{1}{x \sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","IntegrateAlgebraic[1/(x*(1 - x^2)^(1/3)*(3 + x^2)),x]","\frac{1}{6} \log \left(\sqrt[3]{1-x^2}-1\right)-\frac{\log \left(\sqrt[3]{2} \sqrt[3]{1-x^2}-2\right)}{6\ 2^{2/3}}-\frac{1}{12} \log \left(\left(1-x^2\right)^{2/3}+\sqrt[3]{1-x^2}+1\right)+\frac{\log \left(2^{2/3} \left(1-x^2\right)^{2/3}+2 \sqrt[3]{2} \sqrt[3]{1-x^2}+4\right)}{12\ 2^{2/3}}+\frac{\tan ^{-1}\left(\frac{2 \sqrt[3]{1-x^2}}{\sqrt{3}}+\frac{1}{\sqrt{3}}\right)}{2 \sqrt{3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{2} \sqrt[3]{1-x^2}}{\sqrt{3}}+\frac{1}{\sqrt{3}}\right)}{2\ 2^{2/3} \sqrt{3}}","\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,"ArcTan[1/Sqrt[3] + (2*(1 - x^2)^(1/3))/Sqrt[3]]/(2*Sqrt[3]) - ArcTan[1/Sqrt[3] + (2^(1/3)*(1 - x^2)^(1/3))/Sqrt[3]]/(2*2^(2/3)*Sqrt[3]) + Log[-1 + (1 - x^2)^(1/3)]/6 - Log[-2 + 2^(1/3)*(1 - x^2)^(1/3)]/(6*2^(2/3)) - Log[1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3)]/12 + Log[4 + 2*2^(1/3)*(1 - x^2)^(1/3) + 2^(2/3)*(1 - x^2)^(2/3)]/(12*2^(2/3))","A",1
1012,1,138,97,0.2257867,"\int \frac{1}{x^3 \sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","IntegrateAlgebraic[1/(x^3*(1 - x^2)^(1/3)*(3 + x^2)),x]","-\frac{\left(1-x^2\right)^{2/3}}{6 x^2}+\frac{\log \left(\sqrt[3]{2} \sqrt[3]{1-x^2}-2\right)}{18\ 2^{2/3}}-\frac{\log \left(2^{2/3} \left(1-x^2\right)^{2/3}+2 \sqrt[3]{2} \sqrt[3]{1-x^2}+4\right)}{36\ 2^{2/3}}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{2} \sqrt[3]{1-x^2}}{\sqrt{3}}+\frac{1}{\sqrt{3}}\right)}{6\ 2^{2/3} \sqrt{3}}","-\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,"-1/6*(1 - x^2)^(2/3)/x^2 + ArcTan[1/Sqrt[3] + (2^(1/3)*(1 - x^2)^(1/3))/Sqrt[3]]/(6*2^(2/3)*Sqrt[3]) + Log[-2 + 2^(1/3)*(1 - x^2)^(1/3)]/(18*2^(2/3)) - Log[4 + 2*2^(1/3)*(1 - x^2)^(1/3) + 2^(2/3)*(1 - x^2)^(2/3)]/(36*2^(2/3))","A",1
1013,1,226,172,0.3579808,"\int \frac{1}{x^5 \sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","IntegrateAlgebraic[1/(x^5*(1 - x^2)^(1/3)*(3 + x^2)),x]","\frac{1}{27} \log \left(\sqrt[3]{1-x^2}-1\right)-\frac{\log \left(\sqrt[3]{2} \sqrt[3]{1-x^2}-2\right)}{54\ 2^{2/3}}-\frac{1}{54} \log \left(\left(1-x^2\right)^{2/3}+\sqrt[3]{1-x^2}+1\right)+\frac{\log \left(2^{2/3} \left(1-x^2\right)^{2/3}+2 \sqrt[3]{2} \sqrt[3]{1-x^2}+4\right)}{108\ 2^{2/3}}+\frac{\tan ^{-1}\left(\frac{2 \sqrt[3]{1-x^2}}{\sqrt{3}}+\frac{1}{\sqrt{3}}\right)}{9 \sqrt{3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{2} \sqrt[3]{1-x^2}}{\sqrt{3}}+\frac{1}{\sqrt{3}}\right)}{18\ 2^{2/3} \sqrt{3}}+\frac{\left(1-x^2\right)^{2/3} \left(-2 x^2-3\right)}{36 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,"((-3 - 2*x^2)*(1 - x^2)^(2/3))/(36*x^4) + ArcTan[1/Sqrt[3] + (2*(1 - x^2)^(1/3))/Sqrt[3]]/(9*Sqrt[3]) - ArcTan[1/Sqrt[3] + (2^(1/3)*(1 - x^2)^(1/3))/Sqrt[3]]/(18*2^(2/3)*Sqrt[3]) + Log[-1 + (1 - x^2)^(1/3)]/27 - Log[-2 + 2^(1/3)*(1 - x^2)^(1/3)]/(54*2^(2/3)) - Log[1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3)]/54 + Log[4 + 2*2^(1/3)*(1 - x^2)^(1/3) + 2^(2/3)*(1 - x^2)^(2/3)]/(108*2^(2/3))","A",1
1014,0,0,536,4.3125801,"\int \frac{x^4}{\sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","IntegrateAlgebraic[x^4/((1 - x^2)^(1/3)*(3 + x^2)),x]","\int \frac{x^4}{\sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","-\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,"Could not integrate","F",-1
1015,0,0,515,3.8952759,"\int \frac{x^2}{\sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","IntegrateAlgebraic[x^2/((1 - x^2)^(1/3)*(3 + x^2)),x]","\int \frac{x^2}{\sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","-\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,"Could not integrate","F",-1
1016,0,0,113,0.0005771,"\int \frac{1}{\sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","IntegrateAlgebraic[1/((1 - x^2)^(1/3)*(3 + x^2)),x]","\int \frac{1}{\sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","\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,"Could not integrate","F",-1
1017,0,0,538,8.5782095,"\int \frac{1}{x^2 \sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","IntegrateAlgebraic[1/(x^2*(1 - x^2)^(1/3)*(3 + x^2)),x]","\int \frac{1}{x^2 \sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","\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,"Could not integrate","F",-1
1018,0,0,556,15.3066165,"\int \frac{1}{x^4 \sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","IntegrateAlgebraic[1/(x^4*(1 - x^2)^(1/3)*(3 + x^2)),x]","\int \frac{1}{x^4 \sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","\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,"Could not integrate","F",-1
1019,1,154,133,0.2488091,"\int \frac{x^7}{\sqrt[3]{1-x^2} \left(3+x^2\right)^2} \, dx","IntegrateAlgebraic[x^7/((1 - x^2)^(1/3)*(3 + x^2)^2),x]","\frac{99 \log \left(\sqrt[3]{2} \sqrt[3]{1-x^2}-2\right)}{8\ 2^{2/3}}-\frac{99 \log \left(2^{2/3} \left(1-x^2\right)^{2/3}+2 \sqrt[3]{2} \sqrt[3]{1-x^2}+4\right)}{16\ 2^{2/3}}+\frac{99 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{2} \sqrt[3]{1-x^2}}{\sqrt{3}}+\frac{1}{\sqrt{3}}\right)}{8\ 2^{2/3}}-\frac{3 \left(1-x^2\right)^{2/3} \left(4 x^4-42 x^2-207\right)}{40 \left(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*(1 - x^2)^(2/3)*(-207 - 42*x^2 + 4*x^4))/(40*(3 + x^2)) + (99*Sqrt[3]*ArcTan[1/Sqrt[3] + (2^(1/3)*(1 - x^2)^(1/3))/Sqrt[3]])/(8*2^(2/3)) + (99*Log[-2 + 2^(1/3)*(1 - x^2)^(1/3)])/(8*2^(2/3)) - (99*Log[4 + 2*2^(1/3)*(1 - x^2)^(1/3) + 2^(2/3)*(1 - x^2)^(2/3)])/(16*2^(2/3))","A",1
1020,1,149,116,0.2285105,"\int \frac{x^5}{\sqrt[3]{1-x^2} \left(3+x^2\right)^2} \, dx","IntegrateAlgebraic[x^5/((1 - x^2)^(1/3)*(3 + x^2)^2),x]","-\frac{3 \left(1-x^2\right)^{2/3} \left(2 x^2+9\right)}{8 \left(x^2+3\right)}-\frac{21 \log \left(\sqrt[3]{2} \sqrt[3]{1-x^2}-2\right)}{8\ 2^{2/3}}+\frac{21 \log \left(2^{2/3} \left(1-x^2\right)^{2/3}+2 \sqrt[3]{2} \sqrt[3]{1-x^2}+4\right)}{16\ 2^{2/3}}-\frac{21 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{2} \sqrt[3]{1-x^2}}{\sqrt{3}}+\frac{1}{\sqrt{3}}\right)}{8\ 2^{2/3}}","-\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*(1 - x^2)^(2/3)*(9 + 2*x^2))/(8*(3 + x^2)) - (21*Sqrt[3]*ArcTan[1/Sqrt[3] + (2^(1/3)*(1 - x^2)^(1/3))/Sqrt[3]])/(8*2^(2/3)) - (21*Log[-2 + 2^(1/3)*(1 - x^2)^(1/3)])/(8*2^(2/3)) + (21*Log[4 + 2*2^(1/3)*(1 - x^2)^(1/3) + 2^(2/3)*(1 - x^2)^(2/3)])/(16*2^(2/3))","A",1
1021,1,142,101,0.2158104,"\int \frac{x^3}{\sqrt[3]{1-x^2} \left(3+x^2\right)^2} \, dx","IntegrateAlgebraic[x^3/((1 - x^2)^(1/3)*(3 + x^2)^2),x]","\frac{3 \left(1-x^2\right)^{2/3}}{8 \left(x^2+3\right)}+\frac{3 \log \left(\sqrt[3]{2} \sqrt[3]{1-x^2}-2\right)}{8\ 2^{2/3}}-\frac{3 \log \left(2^{2/3} \left(1-x^2\right)^{2/3}+2 \sqrt[3]{2} \sqrt[3]{1-x^2}+4\right)}{16\ 2^{2/3}}+\frac{3 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{2} \sqrt[3]{1-x^2}}{\sqrt{3}}+\frac{1}{\sqrt{3}}\right)}{8\ 2^{2/3}}","\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*(1 - x^2)^(2/3))/(8*(3 + x^2)) + (3*Sqrt[3]*ArcTan[1/Sqrt[3] + (2^(1/3)*(1 - x^2)^(1/3))/Sqrt[3]])/(8*2^(2/3)) + (3*Log[-2 + 2^(1/3)*(1 - x^2)^(1/3)])/(8*2^(2/3)) - (3*Log[4 + 2*2^(1/3)*(1 - x^2)^(1/3) + 2^(2/3)*(1 - x^2)^(2/3)])/(16*2^(2/3))","A",1
1022,1,142,101,0.2152769,"\int \frac{x}{\sqrt[3]{1-x^2} \left(3+x^2\right)^2} \, dx","IntegrateAlgebraic[x/((1 - x^2)^(1/3)*(3 + x^2)^2),x]","-\frac{\left(1-x^2\right)^{2/3}}{8 \left(x^2+3\right)}+\frac{\log \left(\sqrt[3]{2} \sqrt[3]{1-x^2}-2\right)}{24\ 2^{2/3}}-\frac{\log \left(2^{2/3} \left(1-x^2\right)^{2/3}+2 \sqrt[3]{2} \sqrt[3]{1-x^2}+4\right)}{48\ 2^{2/3}}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{2} \sqrt[3]{1-x^2}}{\sqrt{3}}+\frac{1}{\sqrt{3}}\right)}{8\ 2^{2/3} \sqrt{3}}","-\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,"-1/8*(1 - x^2)^(2/3)/(3 + x^2) + ArcTan[1/Sqrt[3] + (2^(1/3)*(1 - x^2)^(1/3))/Sqrt[3]]/(8*2^(2/3)*Sqrt[3]) + Log[-2 + 2^(1/3)*(1 - x^2)^(1/3)]/(24*2^(2/3)) - Log[4 + 2*2^(1/3)*(1 - x^2)^(1/3) + 2^(2/3)*(1 - x^2)^(2/3)]/(48*2^(2/3))","A",1
1023,1,223,158,0.3649574,"\int \frac{1}{x \sqrt[3]{1-x^2} \left(3+x^2\right)^2} \, dx","IntegrateAlgebraic[1/(x*(1 - x^2)^(1/3)*(3 + x^2)^2),x]","\frac{\left(1-x^2\right)^{2/3}}{24 \left(x^2+3\right)}+\frac{1}{18} \log \left(\sqrt[3]{1-x^2}-1\right)-\frac{5 \log \left(\sqrt[3]{2} \sqrt[3]{1-x^2}-2\right)}{72\ 2^{2/3}}-\frac{1}{36} \log \left(\left(1-x^2\right)^{2/3}+\sqrt[3]{1-x^2}+1\right)+\frac{5 \log \left(2^{2/3} \left(1-x^2\right)^{2/3}+2 \sqrt[3]{2} \sqrt[3]{1-x^2}+4\right)}{144\ 2^{2/3}}+\frac{\tan ^{-1}\left(\frac{2 \sqrt[3]{1-x^2}}{\sqrt{3}}+\frac{1}{\sqrt{3}}\right)}{6 \sqrt{3}}-\frac{5 \tan ^{-1}\left(\frac{\sqrt[3]{2} \sqrt[3]{1-x^2}}{\sqrt{3}}+\frac{1}{\sqrt{3}}\right)}{24\ 2^{2/3} \sqrt{3}}","\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,"(1 - x^2)^(2/3)/(24*(3 + x^2)) + ArcTan[1/Sqrt[3] + (2*(1 - x^2)^(1/3))/Sqrt[3]]/(6*Sqrt[3]) - (5*ArcTan[1/Sqrt[3] + (2^(1/3)*(1 - x^2)^(1/3))/Sqrt[3]])/(24*2^(2/3)*Sqrt[3]) + Log[-1 + (1 - x^2)^(1/3)]/18 - (5*Log[-2 + 2^(1/3)*(1 - x^2)^(1/3)])/(72*2^(2/3)) - Log[1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3)]/36 + (5*Log[4 + 2*2^(1/3)*(1 - x^2)^(1/3) + 2^(2/3)*(1 - x^2)^(2/3)])/(144*2^(2/3))","A",1
1024,1,233,183,0.3838247,"\int \frac{1}{x^3 \sqrt[3]{1-x^2} \left(3+x^2\right)^2} \, dx","IntegrateAlgebraic[1/(x^3*(1 - x^2)^(1/3)*(3 + x^2)^2),x]","\frac{\left(1-x^2\right)^{2/3} \left(-5 x^2-12\right)}{72 x^2 \left(x^2+3\right)}-\frac{1}{54} \log \left(\sqrt[3]{1-x^2}-1\right)+\frac{\log \left(\sqrt[3]{2} \sqrt[3]{1-x^2}-2\right)}{24\ 2^{2/3}}+\frac{1}{108} \log \left(\left(1-x^2\right)^{2/3}+\sqrt[3]{1-x^2}+1\right)-\frac{\log \left(2^{2/3} \left(1-x^2\right)^{2/3}+2 \sqrt[3]{2} \sqrt[3]{1-x^2}+4\right)}{48\ 2^{2/3}}-\frac{\tan ^{-1}\left(\frac{2 \sqrt[3]{1-x^2}}{\sqrt{3}}+\frac{1}{\sqrt{3}}\right)}{18 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{2} \sqrt[3]{1-x^2}}{\sqrt{3}}+\frac{1}{\sqrt{3}}\right)}{8\ 2^{2/3} \sqrt{3}}","-\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,"((-12 - 5*x^2)*(1 - x^2)^(2/3))/(72*x^2*(3 + x^2)) - ArcTan[1/Sqrt[3] + (2*(1 - x^2)^(1/3))/Sqrt[3]]/(18*Sqrt[3]) + ArcTan[1/Sqrt[3] + (2^(1/3)*(1 - x^2)^(1/3))/Sqrt[3]]/(8*2^(2/3)*Sqrt[3]) - Log[-1 + (1 - x^2)^(1/3)]/54 + Log[-2 + 2^(1/3)*(1 - x^2)^(1/3)]/(24*2^(2/3)) + Log[1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3)]/108 - Log[4 + 2*2^(1/3)*(1 - x^2)^(1/3) + 2^(2/3)*(1 - x^2)^(2/3)]/(48*2^(2/3))","A",1
1025,1,236,208,0.4101079,"\int \frac{1}{x^5 \sqrt[3]{1-x^2} \left(3+x^2\right)^2} \, dx","IntegrateAlgebraic[1/(x^5*(1 - x^2)^(1/3)*(3 + x^2)^2),x]","\frac{1}{54} \log \left(\sqrt[3]{1-x^2}-1\right)-\frac{13 \log \left(\sqrt[3]{2} \sqrt[3]{1-x^2}-2\right)}{648\ 2^{2/3}}-\frac{1}{108} \log \left(\left(1-x^2\right)^{2/3}+\sqrt[3]{1-x^2}+1\right)+\frac{13 \log \left(2^{2/3} \left(1-x^2\right)^{2/3}+2 \sqrt[3]{2} \sqrt[3]{1-x^2}+4\right)}{1296\ 2^{2/3}}+\frac{\tan ^{-1}\left(\frac{2 \sqrt[3]{1-x^2}}{\sqrt{3}}+\frac{1}{\sqrt{3}}\right)}{18 \sqrt{3}}-\frac{13 \tan ^{-1}\left(\frac{\sqrt[3]{2} \sqrt[3]{1-x^2}}{\sqrt{3}}+\frac{1}{\sqrt{3}}\right)}{216\ 2^{2/3} \sqrt{3}}+\frac{\left(1-x^2\right)^{2/3} \left(x^4-6 x^2-18\right)}{216 x^4 \left(x^2+3\right)}","-\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,"((1 - x^2)^(2/3)*(-18 - 6*x^2 + x^4))/(216*x^4*(3 + x^2)) + ArcTan[1/Sqrt[3] + (2*(1 - x^2)^(1/3))/Sqrt[3]]/(18*Sqrt[3]) - (13*ArcTan[1/Sqrt[3] + (2^(1/3)*(1 - x^2)^(1/3))/Sqrt[3]])/(216*2^(2/3)*Sqrt[3]) + Log[-1 + (1 - x^2)^(1/3)]/54 - (13*Log[-2 + 2^(1/3)*(1 - x^2)^(1/3)])/(648*2^(2/3)) - Log[1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3)]/108 + (13*Log[4 + 2*2^(1/3)*(1 - x^2)^(1/3) + 2^(2/3)*(1 - x^2)^(2/3)])/(1296*2^(2/3))","A",1
1026,0,0,543,4.8073216,"\int \frac{x^4}{\sqrt[3]{1-x^2} \left(3+x^2\right)^2} \, dx","IntegrateAlgebraic[x^4/((1 - x^2)^(1/3)*(3 + x^2)^2),x]","\int \frac{x^4}{\sqrt[3]{1-x^2} \left(3+x^2\right)^2} \, dx","\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,"Could not integrate","F",-1
1027,0,0,543,4.4665744,"\int \frac{x^2}{\sqrt[3]{1-x^2} \left(3+x^2\right)^2} \, dx","IntegrateAlgebraic[x^2/((1 - x^2)^(1/3)*(3 + x^2)^2),x]","\int \frac{x^2}{\sqrt[3]{1-x^2} \left(3+x^2\right)^2} \, dx","-\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,"Could not integrate","F",-1
1028,0,0,543,8.5743628,"\int \frac{1}{\sqrt[3]{1-x^2} \left(3+x^2\right)^2} \, dx","IntegrateAlgebraic[1/((1 - x^2)^(1/3)*(3 + x^2)^2),x]","\int \frac{1}{\sqrt[3]{1-x^2} \left(3+x^2\right)^2} \, dx","\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,"Could not integrate","F",-1
1029,0,0,563,14.6284092,"\int \frac{1}{x^2 \sqrt[3]{1-x^2} \left(3+x^2\right)^2} \, dx","IntegrateAlgebraic[1/(x^2*(1 - x^2)^(1/3)*(3 + x^2)^2),x]","\int \frac{1}{x^2 \sqrt[3]{1-x^2} \left(3+x^2\right)^2} \, dx","\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,"Could not integrate","F",-1
1030,0,0,581,18.1342231,"\int \frac{1}{x^4 \sqrt[3]{1-x^2} \left(3+x^2\right)^2} \, dx","IntegrateAlgebraic[1/(x^4*(1 - x^2)^(1/3)*(3 + x^2)^2),x]","\int \frac{1}{x^4 \sqrt[3]{1-x^2} \left(3+x^2\right)^2} \, dx","-\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,"Could not integrate","F",-1
1031,1,125,136,0.2124009,"\int \frac{x^7}{\sqrt[4]{2-3 x^2} \left(4-3 x^2\right)} \, dx","IntegrateAlgebraic[x^7/((2 - 3*x^2)^(1/4)*(4 - 3*x^2)),x]","-\frac{32}{81} \sqrt[4]{2} \tan ^{-1}\left(\frac{2^{3/4} \sqrt[4]{2-3 x^2}}{\sqrt{2}-\sqrt{2-3 x^2}}\right)+\frac{32}{81} \sqrt[4]{2} \tanh ^{-1}\left(\frac{2 \sqrt[4]{2} \sqrt[4]{2-3 x^2}}{\sqrt{2} \sqrt{2-3 x^2}+2}\right)+\frac{2 \left(2-3 x^2\right)^{3/4} \left(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))/18711 - (32*2^(1/4)*ArcTan[(2^(3/4)*(2 - 3*x^2)^(1/4))/(Sqrt[2] - Sqrt[2 - 3*x^2])])/81 + (32*2^(1/4)*ArcTanh[(2*2^(1/4)*(2 - 3*x^2)^(1/4))/(2 + Sqrt[2]*Sqrt[2 - 3*x^2])])/81","A",1
1032,1,118,121,0.1940721,"\int \frac{x^5}{\sqrt[4]{2-3 x^2} \left(4-3 x^2\right)} \, dx","IntegrateAlgebraic[x^5/((2 - 3*x^2)^(1/4)*(4 - 3*x^2)),x]","\frac{2}{63} \left(2-3 x^2\right)^{3/4} \left(x^2+4\right)-\frac{8}{27} \sqrt[4]{2} \tan ^{-1}\left(\frac{2^{3/4} \sqrt[4]{2-3 x^2}}{\sqrt{2}-\sqrt{2-3 x^2}}\right)+\frac{8}{27} \sqrt[4]{2} \tanh ^{-1}\left(\frac{2 \sqrt[4]{2} \sqrt[4]{2-3 x^2}}{\sqrt{2} \sqrt{2-3 x^2}+2}\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)*(4 + x^2))/63 - (8*2^(1/4)*ArcTan[(2^(3/4)*(2 - 3*x^2)^(1/4))/(Sqrt[2] - Sqrt[2 - 3*x^2])])/27 + (8*2^(1/4)*ArcTanh[(2*2^(1/4)*(2 - 3*x^2)^(1/4))/(2 + Sqrt[2]*Sqrt[2 - 3*x^2])])/27","A",1
1033,1,113,106,0.1925796,"\int \frac{x^3}{\sqrt[4]{2-3 x^2} \left(4-3 x^2\right)} \, dx","IntegrateAlgebraic[x^3/((2 - 3*x^2)^(1/4)*(4 - 3*x^2)),x]","\frac{2}{27} \left(2-3 x^2\right)^{3/4}-\frac{2}{9} \sqrt[4]{2} \tan ^{-1}\left(\frac{2^{3/4} \sqrt[4]{2-3 x^2}}{\sqrt{2}-\sqrt{2-3 x^2}}\right)+\frac{2}{9} \sqrt[4]{2} \tanh ^{-1}\left(\frac{2 \sqrt[4]{2} \sqrt[4]{2-3 x^2}}{\sqrt{2} \sqrt{2-3 x^2}+2}\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))/27 - (2*2^(1/4)*ArcTan[(2^(3/4)*(2 - 3*x^2)^(1/4))/(Sqrt[2] - Sqrt[2 - 3*x^2])])/9 + (2*2^(1/4)*ArcTanh[(2*2^(1/4)*(2 - 3*x^2)^(1/4))/(2 + Sqrt[2]*Sqrt[2 - 3*x^2])])/9","A",1
1034,1,98,91,0.1784807,"\int \frac{x}{\sqrt[4]{2-3 x^2} \left(4-3 x^2\right)} \, dx","IntegrateAlgebraic[x/((2 - 3*x^2)^(1/4)*(4 - 3*x^2)),x]","\frac{\tanh ^{-1}\left(\frac{2 \sqrt[4]{2} \sqrt[4]{2-3 x^2}}{\sqrt{2} \sqrt{2-3 x^2}+2}\right)}{3\ 2^{3/4}}-\frac{\tan ^{-1}\left(\frac{2^{3/4} \sqrt[4]{2-3 x^2}}{\sqrt{2}-\sqrt{2-3 x^2}}\right)}{3\ 2^{3/4}}","\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,"-1/3*ArcTan[(2^(3/4)*(2 - 3*x^2)^(1/4))/(Sqrt[2] - Sqrt[2 - 3*x^2])]/2^(3/4) + ArcTanh[(2*2^(1/4)*(2 - 3*x^2)^(1/4))/(2 + Sqrt[2]*Sqrt[2 - 3*x^2])]/(3*2^(3/4))","A",1
1035,1,152,145,0.2581416,"\int \frac{1}{x \sqrt[4]{2-3 x^2} \left(4-3 x^2\right)} \, dx","IntegrateAlgebraic[1/(x*(2 - 3*x^2)^(1/4)*(4 - 3*x^2)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2-3 x^2}}{\sqrt[4]{2}}\right)}{4 \sqrt[4]{2}}-\frac{\tan ^{-1}\left(\frac{2^{3/4} \sqrt[4]{2-3 x^2}}{\sqrt{2}-\sqrt{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{2 \sqrt[4]{2} \sqrt[4]{2-3 x^2}}{\sqrt{2} \sqrt{2-3 x^2}+2}\right)}{4\ 2^{3/4}}","\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,"ArcTan[(2 - 3*x^2)^(1/4)/2^(1/4)]/(4*2^(1/4)) - ArcTan[(2^(3/4)*(2 - 3*x^2)^(1/4))/(Sqrt[2] - Sqrt[2 - 3*x^2])]/(4*2^(3/4)) - ArcTanh[(2 - 3*x^2)^(1/4)/2^(1/4)]/(4*2^(1/4)) + ArcTanh[(2*2^(1/4)*(2 - 3*x^2)^(1/4))/(2 + Sqrt[2]*Sqrt[2 - 3*x^2])]/(4*2^(3/4))","A",1
1036,1,170,163,0.3443905,"\int \frac{1}{x^3 \sqrt[4]{2-3 x^2} \left(4-3 x^2\right)} \, dx","IntegrateAlgebraic[1/(x^3*(2 - 3*x^2)^(1/4)*(4 - 3*x^2)),x]","-\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{2^{3/4} \sqrt[4]{2-3 x^2}}{\sqrt{2}-\sqrt{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{2 \sqrt[4]{2} \sqrt[4]{2-3 x^2}}{\sqrt{2} \sqrt{2-3 x^2}+2}\right)}{16\ 2^{3/4}}","-\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,"-1/16*(2 - 3*x^2)^(3/4)/x^2 + (9*ArcTan[(2 - 3*x^2)^(1/4)/2^(1/4)])/(32*2^(1/4)) - (3*ArcTan[(2^(3/4)*(2 - 3*x^2)^(1/4))/(Sqrt[2] - Sqrt[2 - 3*x^2])])/(16*2^(3/4)) - (9*ArcTanh[(2 - 3*x^2)^(1/4)/2^(1/4)])/(32*2^(1/4)) + (3*ArcTanh[(2*2^(1/4)*(2 - 3*x^2)^(1/4))/(2 + Sqrt[2]*Sqrt[2 - 3*x^2])])/(16*2^(3/4))","A",1
1037,0,0,164,4.7043146,"\int \frac{x^4}{\sqrt[4]{2-3 x^2} \left(4-3 x^2\right)} \, dx","IntegrateAlgebraic[x^4/((2 - 3*x^2)^(1/4)*(4 - 3*x^2)),x]","\int \frac{x^4}{\sqrt[4]{2-3 x^2} \left(4-3 x^2\right)} \, dx","\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,"Could not integrate","F",-1
1038,0,0,148,4.4754531,"\int \frac{x^2}{\sqrt[4]{2-3 x^2} \left(4-3 x^2\right)} \, dx","IntegrateAlgebraic[x^2/((2 - 3*x^2)^(1/4)*(4 - 3*x^2)),x]","\int \frac{x^2}{\sqrt[4]{2-3 x^2} \left(4-3 x^2\right)} \, dx","\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,"Could not integrate","F",-1
1039,1,137,120,0.0006141,"\int \frac{1}{\sqrt[4]{2-3 x^2} \left(4-3 x^2\right)} \, dx","IntegrateAlgebraic[1/((2 - 3*x^2)^(1/4)*(4 - 3*x^2)),x]","\frac{\tan ^{-1}\left(\frac{\frac{\sqrt{3} x^2}{2 \sqrt[4]{2}}-\frac{\sqrt[4]{2} \sqrt{2-3 x^2}}{\sqrt{3}}}{x \sqrt[4]{2-3 x^2}}\right)}{4\ 2^{3/4} \sqrt{3}}+\frac{\tanh ^{-1}\left(\frac{2\ 2^{3/4} \sqrt{3} x \sqrt[4]{2-3 x^2}}{3 \sqrt{2} x^2+4 \sqrt{2-3 x^2}}\right)}{4\ 2^{3/4} \sqrt{3}}","\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,"ArcTan[((Sqrt[3]*x^2)/(2*2^(1/4)) - (2^(1/4)*Sqrt[2 - 3*x^2])/Sqrt[3])/(x*(2 - 3*x^2)^(1/4))]/(4*2^(3/4)*Sqrt[3]) + ArcTanh[(2*2^(3/4)*Sqrt[3]*x*(2 - 3*x^2)^(1/4))/(3*Sqrt[2]*x^2 + 4*Sqrt[2 - 3*x^2])]/(4*2^(3/4)*Sqrt[3])","A",1
1040,0,0,166,9.3341103,"\int \frac{1}{x^2 \sqrt[4]{2-3 x^2} \left(4-3 x^2\right)} \, dx","IntegrateAlgebraic[1/(x^2*(2 - 3*x^2)^(1/4)*(4 - 3*x^2)),x]","\int \frac{1}{x^2 \sqrt[4]{2-3 x^2} \left(4-3 x^2\right)} \, dx","-\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,"Could not integrate","F",-1
1041,0,0,184,14.5075979,"\int \frac{1}{x^4 \sqrt[4]{2-3 x^2} \left(4-3 x^2\right)} \, dx","IntegrateAlgebraic[1/(x^4*(2 - 3*x^2)^(1/4)*(4 - 3*x^2)),x]","\int \frac{1}{x^4 \sqrt[4]{2-3 x^2} \left(4-3 x^2\right)} \, dx","-\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,"Could not integrate","F",-1
1042,1,60,78,0.0765792,"\int \frac{x^7}{\left(-2+3 x^2\right) \sqrt[4]{-1+3 x^2}} \, dx","IntegrateAlgebraic[x^7/((-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)),x]","\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)+\frac{2 \left(3 x^2-1\right)^{3/4} \left(189 x^4+270 x^2+428\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))/18711 + (8*ArcTan[(-1 + 3*x^2)^(1/4)])/81 - (8*ArcTanh[(-1 + 3*x^2)^(1/4)])/81","A",1
1043,1,53,63,0.0650272,"\int \frac{x^5}{\left(-2+3 x^2\right) \sqrt[4]{-1+3 x^2}} \, dx","IntegrateAlgebraic[x^5/((-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)),x]","\frac{2}{63} \left(3 x^2-1\right)^{3/4} \left(x^2+2\right)+\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)","\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*(2 + x^2)*(-1 + 3*x^2)^(3/4))/63 + (4*ArcTan[(-1 + 3*x^2)^(1/4)])/27 - (4*ArcTanh[(-1 + 3*x^2)^(1/4)])/27","A",1
1044,1,48,48,0.0479206,"\int \frac{x^3}{\left(-2+3 x^2\right) \sqrt[4]{-1+3 x^2}} \, dx","IntegrateAlgebraic[x^3/((-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)),x]","\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)","\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))/27 + (2*ArcTan[(-1 + 3*x^2)^(1/4)])/9 - (2*ArcTanh[(-1 + 3*x^2)^(1/4)])/9","A",1
1045,1,33,33,0.0375429,"\int \frac{x}{\left(-2+3 x^2\right) \sqrt[4]{-1+3 x^2}} \, dx","IntegrateAlgebraic[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
1046,1,117,173,0.1736873,"\int \frac{1}{x \left(-2+3 x^2\right) \sqrt[4]{-1+3 x^2}} \, dx","IntegrateAlgebraic[1/(x*(-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)),x]","\frac{1}{2} \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{3 x^2-1}}{\sqrt{3 x^2-1}-1}\right)}{2 \sqrt{2}}-\frac{1}{2} \tanh ^{-1}\left(\sqrt[4]{3 x^2-1}\right)+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{3 x^2-1}}{\sqrt{3 x^2-1}+1}\right)}{2 \sqrt{2}}","-\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 + ArcTan[(Sqrt[2]*(-1 + 3*x^2)^(1/4))/(-1 + Sqrt[-1 + 3*x^2])]/(2*Sqrt[2]) - ArcTanh[(-1 + 3*x^2)^(1/4)]/2 + ArcTanh[(Sqrt[2]*(-1 + 3*x^2)^(1/4))/(1 + Sqrt[-1 + 3*x^2])]/(2*Sqrt[2])","A",1
1047,1,135,191,0.2500895,"\int \frac{1}{x^3 \left(-2+3 x^2\right) \sqrt[4]{-1+3 x^2}} \, dx","IntegrateAlgebraic[1/(x^3*(-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)),x]","-\frac{\left(3 x^2-1\right)^{3/4}}{4 x^2}+\frac{3}{4} \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)+\frac{9 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{3 x^2-1}}{\sqrt{3 x^2-1}-1}\right)}{8 \sqrt{2}}-\frac{3}{4} \tanh ^{-1}\left(\sqrt[4]{3 x^2-1}\right)+\frac{9 \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{3 x^2-1}}{\sqrt{3 x^2-1}+1}\right)}{8 \sqrt{2}}","-\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,"-1/4*(-1 + 3*x^2)^(3/4)/x^2 + (3*ArcTan[(-1 + 3*x^2)^(1/4)])/4 + (9*ArcTan[(Sqrt[2]*(-1 + 3*x^2)^(1/4))/(-1 + Sqrt[-1 + 3*x^2])])/(8*Sqrt[2]) - (3*ArcTanh[(-1 + 3*x^2)^(1/4)])/4 + (9*ArcTanh[(Sqrt[2]*(-1 + 3*x^2)^(1/4))/(1 + Sqrt[-1 + 3*x^2])])/(8*Sqrt[2])","A",1
1048,0,0,244,4.6413922,"\int \frac{x^4}{\left(-2+3 x^2\right) \sqrt[4]{-1+3 x^2}} \, dx","IntegrateAlgebraic[x^4/((-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)),x]","\int \frac{x^4}{\left(-2+3 x^2\right) \sqrt[4]{-1+3 x^2}} \, dx","\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,"Could not integrate","F",-1
1049,0,0,224,4.4033664,"\int \frac{x^2}{\left(-2+3 x^2\right) \sqrt[4]{-1+3 x^2}} \, dx","IntegrateAlgebraic[x^2/((-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)),x]","\int \frac{x^2}{\left(-2+3 x^2\right) \sqrt[4]{-1+3 x^2}} \, dx","\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,"Could not integrate","F",-1
1050,1,63,61,0.0006003,"\int \frac{1}{\left(-2+3 x^2\right) \sqrt[4]{-1+3 x^2}} \, dx","IntegrateAlgebraic[1/((-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{\frac{2}{3}} \sqrt[4]{3 x^2-1}}{x}\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{\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,"ArcTan[(Sqrt[2/3]*(-1 + 3*x^2)^(1/4))/x]/(2*Sqrt[6]) - ArcTanh[(Sqrt[3/2]*x)/(-1 + 3*x^2)^(1/4)]/(2*Sqrt[6])","A",1
1051,0,0,246,9.2918271,"\int \frac{1}{x^2 \left(-2+3 x^2\right) \sqrt[4]{-1+3 x^2}} \, dx","IntegrateAlgebraic[1/(x^2*(-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)),x]","\int \frac{1}{x^2 \left(-2+3 x^2\right) \sqrt[4]{-1+3 x^2}} \, dx","\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,"Could not integrate","F",-1
1052,0,0,264,14.3539995,"\int \frac{1}{x^4 \left(-2+3 x^2\right) \sqrt[4]{-1+3 x^2}} \, dx","IntegrateAlgebraic[1/(x^4*(-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)),x]","\int \frac{1}{x^4 \left(-2+3 x^2\right) \sqrt[4]{-1+3 x^2}} \, dx","\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,"Could not integrate","F",-1
1053,1,136,129,2.2859748,"\int \frac{x^2}{\left(2+3 x^2\right)^{3/4} \left(4+3 x^2\right)} \, dx","IntegrateAlgebraic[x^2/((2 + 3*x^2)^(3/4)*(4 + 3*x^2)),x]","-\frac{\tan ^{-1}\left(\frac{\frac{\sqrt[4]{2} \sqrt{3 x^2+2}}{\sqrt{3}}-\frac{\sqrt{3} x^2}{2 \sqrt[4]{2}}}{x \sqrt[4]{3 x^2+2}}\right)}{6 \sqrt[4]{2} \sqrt{3}}-\frac{\tanh ^{-1}\left(\frac{2 \sqrt[4]{2} \sqrt{3} x \sqrt[4]{3 x^2+2}}{3 x^2+2 \sqrt{2} \sqrt{3 x^2+2}}\right)}{6 \sqrt[4]{2} \sqrt{3}}","\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,"-1/6*ArcTan[(-1/2*(Sqrt[3]*x^2)/2^(1/4) + (2^(1/4)*Sqrt[2 + 3*x^2])/Sqrt[3])/(x*(2 + 3*x^2)^(1/4))]/(2^(1/4)*Sqrt[3]) - ArcTanh[(2*2^(1/4)*Sqrt[3]*x*(2 + 3*x^2)^(1/4))/(3*x^2 + 2*Sqrt[2]*Sqrt[2 + 3*x^2])]/(6*2^(1/4)*Sqrt[3])","A",1
1054,1,136,120,2.2347983,"\int \frac{x^2}{\left(2-3 x^2\right)^{3/4} \left(4-3 x^2\right)} \, dx","IntegrateAlgebraic[x^2/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]","-\frac{\tan ^{-1}\left(\frac{\frac{\sqrt[4]{2} \sqrt{2-3 x^2}}{\sqrt{3}}-\frac{\sqrt{3} x^2}{2 \sqrt[4]{2}}}{x \sqrt[4]{2-3 x^2}}\right)}{6 \sqrt[4]{2} \sqrt{3}}-\frac{\tanh ^{-1}\left(\frac{2 \sqrt[4]{2} \sqrt{3} x \sqrt[4]{2-3 x^2}}{3 x^2+2 \sqrt{2} \sqrt{2-3 x^2}}\right)}{6 \sqrt[4]{2} \sqrt{3}}","\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,"-1/6*ArcTan[(-1/2*(Sqrt[3]*x^2)/2^(1/4) + (2^(1/4)*Sqrt[2 - 3*x^2])/Sqrt[3])/(x*(2 - 3*x^2)^(1/4))]/(2^(1/4)*Sqrt[3]) - ArcTanh[(2*2^(1/4)*Sqrt[3]*x*(2 - 3*x^2)^(1/4))/(3*x^2 + 2*Sqrt[2]*Sqrt[2 - 3*x^2])]/(6*2^(1/4)*Sqrt[3])","A",1
1055,1,136,124,2.3026743,"\int \frac{x^2}{\left(2+b x^2\right)^{3/4} \left(4+b x^2\right)} \, dx","IntegrateAlgebraic[x^2/((2 + b*x^2)^(3/4)*(4 + b*x^2)),x]","-\frac{\tan ^{-1}\left(\frac{\frac{\sqrt[4]{2} \sqrt{b x^2+2}}{\sqrt{b}}-\frac{\sqrt{b} x^2}{2 \sqrt[4]{2}}}{x \sqrt[4]{b x^2+2}}\right)}{2 \sqrt[4]{2} b^{3/2}}-\frac{\tanh ^{-1}\left(\frac{2 \sqrt[4]{2} \sqrt{b} x \sqrt[4]{b x^2+2}}{b x^2+2 \sqrt{2} \sqrt{b x^2+2}}\right)}{2 \sqrt[4]{2} b^{3/2}}","\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,"-1/2*ArcTan[(-1/2*(Sqrt[b]*x^2)/2^(1/4) + (2^(1/4)*Sqrt[2 + b*x^2])/Sqrt[b])/(x*(2 + b*x^2)^(1/4))]/(2^(1/4)*b^(3/2)) - ArcTanh[(2*2^(1/4)*Sqrt[b]*x*(2 + b*x^2)^(1/4))/(b*x^2 + 2*Sqrt[2]*Sqrt[2 + b*x^2])]/(2*2^(1/4)*b^(3/2))","A",1
1056,1,140,119,2.3226106,"\int \frac{x^2}{\left(2-b x^2\right)^{3/4} \left(4-b x^2\right)} \, dx","IntegrateAlgebraic[x^2/((2 - b*x^2)^(3/4)*(4 - b*x^2)),x]","-\frac{\tan ^{-1}\left(\frac{\frac{\sqrt[4]{2} \sqrt{2-b x^2}}{\sqrt{b}}-\frac{\sqrt{b} x^2}{2 \sqrt[4]{2}}}{x \sqrt[4]{2-b x^2}}\right)}{2 \sqrt[4]{2} b^{3/2}}-\frac{\tanh ^{-1}\left(\frac{2 \sqrt[4]{2} \sqrt{b} x \sqrt[4]{2-b x^2}}{b x^2+2 \sqrt{2} \sqrt{2-b x^2}}\right)}{2 \sqrt[4]{2} b^{3/2}}","\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,"-1/2*ArcTan[(-1/2*(Sqrt[b]*x^2)/2^(1/4) + (2^(1/4)*Sqrt[2 - b*x^2])/Sqrt[b])/(x*(2 - b*x^2)^(1/4))]/(2^(1/4)*b^(3/2)) - ArcTanh[(2*2^(1/4)*Sqrt[b]*x*(2 - b*x^2)^(1/4))/(b*x^2 + 2*Sqrt[2]*Sqrt[2 - b*x^2])]/(2*2^(1/4)*b^(3/2))","A",1
1057,1,136,120,2.2490169,"\int \frac{x^2}{\left(a+3 x^2\right)^{3/4} \left(2 a+3 x^2\right)} \, dx","IntegrateAlgebraic[x^2/((a + 3*x^2)^(3/4)*(2*a + 3*x^2)),x]","-\frac{\tan ^{-1}\left(\frac{\frac{\sqrt[4]{a} \sqrt{a+3 x^2}}{\sqrt{3}}-\frac{\sqrt{3} x^2}{2 \sqrt[4]{a}}}{x \sqrt[4]{a+3 x^2}}\right)}{6 \sqrt{3} \sqrt[4]{a}}-\frac{\tanh ^{-1}\left(\frac{2 \sqrt{3} \sqrt[4]{a} x \sqrt[4]{a+3 x^2}}{2 \sqrt{a} \sqrt{a+3 x^2}+3 x^2}\right)}{6 \sqrt{3} \sqrt[4]{a}}","\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,"-1/6*ArcTan[(-1/2*(Sqrt[3]*x^2)/a^(1/4) + (a^(1/4)*Sqrt[a + 3*x^2])/Sqrt[3])/(x*(a + 3*x^2)^(1/4))]/(Sqrt[3]*a^(1/4)) - ArcTanh[(2*Sqrt[3]*a^(1/4)*x*(a + 3*x^2)^(1/4))/(3*x^2 + 2*Sqrt[a]*Sqrt[a + 3*x^2])]/(6*Sqrt[3]*a^(1/4))","A",1
1058,1,136,120,2.2521246,"\int \frac{x^2}{\left(a-3 x^2\right)^{3/4} \left(2 a-3 x^2\right)} \, dx","IntegrateAlgebraic[x^2/((a - 3*x^2)^(3/4)*(2*a - 3*x^2)),x]","-\frac{\tan ^{-1}\left(\frac{\frac{\sqrt[4]{a} \sqrt{a-3 x^2}}{\sqrt{3}}-\frac{\sqrt{3} x^2}{2 \sqrt[4]{a}}}{x \sqrt[4]{a-3 x^2}}\right)}{6 \sqrt{3} \sqrt[4]{a}}-\frac{\tanh ^{-1}\left(\frac{2 \sqrt{3} \sqrt[4]{a} x \sqrt[4]{a-3 x^2}}{2 \sqrt{a} \sqrt{a-3 x^2}+3 x^2}\right)}{6 \sqrt{3} \sqrt[4]{a}}","\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,"-1/6*ArcTan[(-1/2*(Sqrt[3]*x^2)/a^(1/4) + (a^(1/4)*Sqrt[a - 3*x^2])/Sqrt[3])/(x*(a - 3*x^2)^(1/4))]/(Sqrt[3]*a^(1/4)) - ArcTanh[(2*Sqrt[3]*a^(1/4)*x*(a - 3*x^2)^(1/4))/(3*x^2 + 2*Sqrt[a]*Sqrt[a - 3*x^2])]/(6*Sqrt[3]*a^(1/4))","A",1
1059,1,136,115,2.3901082,"\int \frac{x^2}{\left(a+b x^2\right)^{3/4} \left(2 a+b x^2\right)} \, dx","IntegrateAlgebraic[x^2/((a + b*x^2)^(3/4)*(2*a + b*x^2)),x]","-\frac{\tan ^{-1}\left(\frac{\frac{\sqrt[4]{a} \sqrt{a+b x^2}}{\sqrt{b}}-\frac{\sqrt{b} x^2}{2 \sqrt[4]{a}}}{x \sqrt[4]{a+b x^2}}\right)}{2 \sqrt[4]{a} b^{3/2}}-\frac{\tanh ^{-1}\left(\frac{2 \sqrt[4]{a} \sqrt{b} x \sqrt[4]{a+b x^2}}{2 \sqrt{a} \sqrt{a+b x^2}+b x^2}\right)}{2 \sqrt[4]{a} b^{3/2}}","\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,"-1/2*ArcTan[(-1/2*(Sqrt[b]*x^2)/a^(1/4) + (a^(1/4)*Sqrt[a + b*x^2])/Sqrt[b])/(x*(a + b*x^2)^(1/4))]/(a^(1/4)*b^(3/2)) - ArcTanh[(2*a^(1/4)*Sqrt[b]*x*(a + b*x^2)^(1/4))/(b*x^2 + 2*Sqrt[a]*Sqrt[a + b*x^2])]/(2*a^(1/4)*b^(3/2))","A",1
1060,1,140,119,2.4002666,"\int \frac{x^2}{\left(a-b x^2\right)^{3/4} \left(2 a-b x^2\right)} \, dx","IntegrateAlgebraic[x^2/((a - b*x^2)^(3/4)*(2*a - b*x^2)),x]","-\frac{\tan ^{-1}\left(\frac{\frac{\sqrt[4]{a} \sqrt{a-b x^2}}{\sqrt{b}}-\frac{\sqrt{b} x^2}{2 \sqrt[4]{a}}}{x \sqrt[4]{a-b x^2}}\right)}{2 \sqrt[4]{a} b^{3/2}}-\frac{\tanh ^{-1}\left(\frac{2 \sqrt[4]{a} \sqrt{b} x \sqrt[4]{a-b x^2}}{2 \sqrt{a} \sqrt{a-b x^2}+b x^2}\right)}{2 \sqrt[4]{a} b^{3/2}}","\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,"-1/2*ArcTan[(-1/2*(Sqrt[b]*x^2)/a^(1/4) + (a^(1/4)*Sqrt[a - b*x^2])/Sqrt[b])/(x*(a - b*x^2)^(1/4))]/(a^(1/4)*b^(3/2)) - ArcTanh[(2*a^(1/4)*Sqrt[b]*x*(a - b*x^2)^(1/4))/(b*x^2 + 2*Sqrt[a]*Sqrt[a - b*x^2])]/(2*a^(1/4)*b^(3/2))","A",1
1061,1,125,188,0.2226584,"\int \frac{x^7}{\left(2-3 x^2\right)^{3/4} \left(4-3 x^2\right)} \, dx","IntegrateAlgebraic[x^7/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]","-\frac{16}{81} 2^{3/4} \tan ^{-1}\left(\frac{2^{3/4} \sqrt[4]{2-3 x^2}}{\sqrt{2}-\sqrt{2-3 x^2}}\right)-\frac{16}{81} 2^{3/4} \tanh ^{-1}\left(\frac{2 \sqrt[4]{2} \sqrt[4]{2-3 x^2}}{\sqrt{2} \sqrt{2-3 x^2}+2}\right)+\frac{2 \sqrt[4]{2-3 x^2} \left(45 x^4+156 x^2+1136\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*(2 - 3*x^2)^(1/4)*(1136 + 156*x^2 + 45*x^4))/3645 - (16*2^(3/4)*ArcTan[(2^(3/4)*(2 - 3*x^2)^(1/4))/(Sqrt[2] - Sqrt[2 - 3*x^2])])/81 - (16*2^(3/4)*ArcTanh[(2*2^(1/4)*(2 - 3*x^2)^(1/4))/(2 + Sqrt[2]*Sqrt[2 - 3*x^2])])/81","A",1
1062,1,120,173,0.2027512,"\int \frac{x^5}{\left(2-3 x^2\right)^{3/4} \left(4-3 x^2\right)} \, dx","IntegrateAlgebraic[x^5/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]","\frac{2}{135} \sqrt[4]{2-3 x^2} \left(3 x^2+28\right)-\frac{4}{27} 2^{3/4} \tan ^{-1}\left(\frac{2^{3/4} \sqrt[4]{2-3 x^2}}{\sqrt{2}-\sqrt{2-3 x^2}}\right)-\frac{4}{27} 2^{3/4} \tanh ^{-1}\left(\frac{2 \sqrt[4]{2} \sqrt[4]{2-3 x^2}}{\sqrt{2} \sqrt{2-3 x^2}+2}\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*(2 - 3*x^2)^(1/4)*(28 + 3*x^2))/135 - (4*2^(3/4)*ArcTan[(2^(3/4)*(2 - 3*x^2)^(1/4))/(Sqrt[2] - Sqrt[2 - 3*x^2])])/27 - (4*2^(3/4)*ArcTanh[(2*2^(1/4)*(2 - 3*x^2)^(1/4))/(2 + Sqrt[2]*Sqrt[2 - 3*x^2])])/27","A",1
1063,1,113,158,0.1869293,"\int \frac{x^3}{\left(2-3 x^2\right)^{3/4} \left(4-3 x^2\right)} \, dx","IntegrateAlgebraic[x^3/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]","\frac{2}{9} \sqrt[4]{2-3 x^2}-\frac{1}{9} 2^{3/4} \tan ^{-1}\left(\frac{2^{3/4} \sqrt[4]{2-3 x^2}}{\sqrt{2}-\sqrt{2-3 x^2}}\right)-\frac{1}{9} 2^{3/4} \tanh ^{-1}\left(\frac{2 \sqrt[4]{2} \sqrt[4]{2-3 x^2}}{\sqrt{2} \sqrt{2-3 x^2}+2}\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,"(2*(2 - 3*x^2)^(1/4))/9 - (2^(3/4)*ArcTan[(2^(3/4)*(2 - 3*x^2)^(1/4))/(Sqrt[2] - Sqrt[2 - 3*x^2])])/9 - (2^(3/4)*ArcTanh[(2*2^(1/4)*(2 - 3*x^2)^(1/4))/(2 + Sqrt[2]*Sqrt[2 - 3*x^2])])/9","A",1
1064,1,98,143,0.178229,"\int \frac{x}{\left(2-3 x^2\right)^{3/4} \left(4-3 x^2\right)} \, dx","IntegrateAlgebraic[x/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]","-\frac{\tan ^{-1}\left(\frac{2^{3/4} \sqrt[4]{2-3 x^2}}{\sqrt{2}-\sqrt{2-3 x^2}}\right)}{6 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{2 \sqrt[4]{2} \sqrt[4]{2-3 x^2}}{\sqrt{2} \sqrt{2-3 x^2}+2}\right)}{6 \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,"-1/6*ArcTan[(2^(3/4)*(2 - 3*x^2)^(1/4))/(Sqrt[2] - Sqrt[2 - 3*x^2])]/2^(1/4) - ArcTanh[(2*2^(1/4)*(2 - 3*x^2)^(1/4))/(2 + Sqrt[2]*Sqrt[2 - 3*x^2])]/(6*2^(1/4))","A",1
1065,1,152,197,0.2614938,"\int \frac{1}{x \left(2-3 x^2\right)^{3/4} \left(4-3 x^2\right)} \, dx","IntegrateAlgebraic[1/(x*(2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2-3 x^2}}{\sqrt[4]{2}}\right)}{4\ 2^{3/4}}-\frac{\tan ^{-1}\left(\frac{2^{3/4} \sqrt[4]{2-3 x^2}}{\sqrt{2}-\sqrt{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}}-\frac{\tanh ^{-1}\left(\frac{2 \sqrt[4]{2} \sqrt[4]{2-3 x^2}}{\sqrt{2} \sqrt{2-3 x^2}+2}\right)}{8 \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{\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,"-1/4*ArcTan[(2 - 3*x^2)^(1/4)/2^(1/4)]/2^(3/4) - ArcTan[(2^(3/4)*(2 - 3*x^2)^(1/4))/(Sqrt[2] - Sqrt[2 - 3*x^2])]/(8*2^(1/4)) - ArcTanh[(2 - 3*x^2)^(1/4)/2^(1/4)]/(4*2^(3/4)) - ArcTanh[(2*2^(1/4)*(2 - 3*x^2)^(1/4))/(2 + Sqrt[2]*Sqrt[2 - 3*x^2])]/(8*2^(1/4))","A",1
1066,1,170,215,0.3527638,"\int \frac{1}{x^3 \left(2-3 x^2\right)^{3/4} \left(4-3 x^2\right)} \, dx","IntegrateAlgebraic[1/(x^3*(2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]","-\frac{\sqrt[4]{2-3 x^2}}{16 x^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(\frac{2^{3/4} \sqrt[4]{2-3 x^2}}{\sqrt{2}-\sqrt{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}}-\frac{3 \tanh ^{-1}\left(\frac{2 \sqrt[4]{2} \sqrt[4]{2-3 x^2}}{\sqrt{2} \sqrt{2-3 x^2}+2}\right)}{32 \sqrt[4]{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,"-1/16*(2 - 3*x^2)^(1/4)/x^2 - (15*ArcTan[(2 - 3*x^2)^(1/4)/2^(1/4)])/(32*2^(3/4)) - (3*ArcTan[(2^(3/4)*(2 - 3*x^2)^(1/4))/(Sqrt[2] - Sqrt[2 - 3*x^2])])/(32*2^(1/4)) - (15*ArcTanh[(2 - 3*x^2)^(1/4)/2^(1/4)])/(32*2^(3/4)) - (3*ArcTanh[(2*2^(1/4)*(2 - 3*x^2)^(1/4))/(2 + Sqrt[2]*Sqrt[2 - 3*x^2])])/(32*2^(1/4))","A",1
1067,0,0,182,5.0618025,"\int \frac{x^6}{\left(2-3 x^2\right)^{3/4} \left(4-3 x^2\right)} \, dx","IntegrateAlgebraic[x^6/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]","\int \frac{x^6}{\left(2-3 x^2\right)^{3/4} \left(4-3 x^2\right)} \, dx","\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,"Could not integrate","F",-1
1068,0,0,164,5.2813557,"\int \frac{x^4}{\left(2-3 x^2\right)^{3/4} \left(4-3 x^2\right)} \, dx","IntegrateAlgebraic[x^4/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]","\int \frac{x^4}{\left(2-3 x^2\right)^{3/4} \left(4-3 x^2\right)} \, dx","\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,"Could not integrate","F",-1
1069,1,136,120,0.0005976,"\int \frac{x^2}{\left(2-3 x^2\right)^{3/4} \left(4-3 x^2\right)} \, dx","IntegrateAlgebraic[x^2/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]","-\frac{\tan ^{-1}\left(\frac{\frac{\sqrt[4]{2} \sqrt{2-3 x^2}}{\sqrt{3}}-\frac{\sqrt{3} x^2}{2 \sqrt[4]{2}}}{x \sqrt[4]{2-3 x^2}}\right)}{6 \sqrt[4]{2} \sqrt{3}}-\frac{\tanh ^{-1}\left(\frac{2 \sqrt[4]{2} \sqrt{3} x \sqrt[4]{2-3 x^2}}{3 x^2+2 \sqrt{2} \sqrt{2-3 x^2}}\right)}{6 \sqrt[4]{2} \sqrt{3}}","\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,"-1/6*ArcTan[(-1/2*(Sqrt[3]*x^2)/2^(1/4) + (2^(1/4)*Sqrt[2 - 3*x^2])/Sqrt[3])/(x*(2 - 3*x^2)^(1/4))]/(2^(1/4)*Sqrt[3]) - ArcTanh[(2*2^(1/4)*Sqrt[3]*x*(2 - 3*x^2)^(1/4))/(3*x^2 + 2*Sqrt[2]*Sqrt[2 - 3*x^2])]/(6*2^(1/4)*Sqrt[3])","A",1
1070,0,0,148,4.802425,"\int \frac{1}{\left(2-3 x^2\right)^{3/4} \left(4-3 x^2\right)} \, dx","IntegrateAlgebraic[1/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]","\int \frac{1}{\left(2-3 x^2\right)^{3/4} \left(4-3 x^2\right)} \, dx","\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,"Could not integrate","F",-1
1071,0,0,166,9.5745465,"\int \frac{1}{x^2 \left(2-3 x^2\right)^{3/4} \left(4-3 x^2\right)} \, dx","IntegrateAlgebraic[1/(x^2*(2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]","\int \frac{1}{x^2 \left(2-3 x^2\right)^{3/4} \left(4-3 x^2\right)} \, dx","-\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,"Could not integrate","F",-1
1072,0,0,184,14.7272269,"\int \frac{1}{x^4 \left(2-3 x^2\right)^{3/4} \left(4-3 x^2\right)} \, dx","IntegrateAlgebraic[1/(x^4*(2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]","\int \frac{1}{x^4 \left(2-3 x^2\right)^{3/4} \left(4-3 x^2\right)} \, dx","-\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,"Could not integrate","F",-1
1073,1,61,61,2.1204081,"\int \frac{x^2}{\left(-2+3 x^2\right) \left(-1+3 x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[x^2/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]","\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{\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,"ArcTan[(Sqrt[3/2]*x)/(-1 + 3*x^2)^(1/4)]/(3*Sqrt[6]) - ArcTanh[(Sqrt[3/2]*x)/(-1 + 3*x^2)^(1/4)]/(3*Sqrt[6])","A",1
1074,1,79,61,2.1091437,"\int \frac{x^2}{\left(-2-3 x^2\right) \left(-1-3 x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[x^2/((-2 - 3*x^2)*(-1 - 3*x^2)^(3/4)),x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x \left(-3 x^2-1\right)^{3/4}}{3 x^2+1}\right)}{3 \sqrt{6}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x \left(-3 x^2-1\right)^{3/4}}{3 x^2+1}\right)}{3 \sqrt{6}}","\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,"-1/3*ArcTan[(Sqrt[3/2]*x*(-1 - 3*x^2)^(3/4))/(1 + 3*x^2)]/Sqrt[6] + ArcTanh[(Sqrt[3/2]*x*(-1 - 3*x^2)^(3/4))/(1 + 3*x^2)]/(3*Sqrt[6])","A",1
1075,1,72,72,2.1762358,"\int \frac{x^2}{\left(-2+b x^2\right) \left(-1+b x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[x^2/((-2 + b*x^2)*(-1 + b*x^2)^(3/4)),x]","\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}}","\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,"ArcTan[(Sqrt[b]*x)/(Sqrt[2]*(-1 + b*x^2)^(1/4))]/(Sqrt[2]*b^(3/2)) - ArcTanh[(Sqrt[b]*x)/(Sqrt[2]*(-1 + b*x^2)^(1/4))]/(Sqrt[2]*b^(3/2))","A",1
1076,1,92,74,2.2086171,"\int \frac{x^2}{\left(-2-b x^2\right) \left(-1-b x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[x^2/((-2 - b*x^2)*(-1 - b*x^2)^(3/4)),x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x \left(-b x^2-1\right)^{3/4}}{\sqrt{2} \left(b x^2+1\right)}\right)}{\sqrt{2} b^{3/2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{b} x \left(-b x^2-1\right)^{3/4}}{\sqrt{2} \left(b x^2+1\right)}\right)}{\sqrt{2} b^{3/2}}","\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,"-(ArcTan[(Sqrt[b]*x*(-1 - b*x^2)^(3/4))/(Sqrt[2]*(1 + b*x^2))]/(Sqrt[2]*b^(3/2))) + ArcTanh[(Sqrt[b]*x*(-1 - b*x^2)^(3/4))/(Sqrt[2]*(1 + b*x^2))]/(Sqrt[2]*b^(3/2))","A",1
1077,1,87,85,2.21118,"\int \frac{x^2}{\left(-2 a+3 x^2\right) \left(-a+3 x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[x^2/((-2*a + 3*x^2)*(-a + 3*x^2)^(3/4)),x]","\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{2}{3}} \sqrt[4]{a} \sqrt[4]{3 x^2-a}}{x}\right)}{3 \sqrt{6} \sqrt[4]{a}}","\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,"ArcTan[(Sqrt[3/2]*x)/(a^(1/4)*(-a + 3*x^2)^(1/4))]/(3*Sqrt[6]*a^(1/4)) - ArcTanh[(Sqrt[2/3]*a^(1/4)*(-a + 3*x^2)^(1/4))/x]/(3*Sqrt[6]*a^(1/4))","A",1
1078,1,87,85,2.2059399,"\int \frac{x^2}{\left(-2 a-3 x^2\right) \left(-a-3 x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[x^2/((-2*a - 3*x^2)*(-a - 3*x^2)^(3/4)),x]","\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{2}{3}} \sqrt[4]{a} \sqrt[4]{-a-3 x^2}}{x}\right)}{3 \sqrt{6} \sqrt[4]{a}}","\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,"ArcTan[(Sqrt[3/2]*x)/(a^(1/4)*(-a - 3*x^2)^(1/4))]/(3*Sqrt[6]*a^(1/4)) - ArcTanh[(Sqrt[2/3]*a^(1/4)*(-a - 3*x^2)^(1/4))/x]/(3*Sqrt[6]*a^(1/4))","A",1
1079,1,98,96,2.3432138,"\int \frac{x^2}{\left(-2 a+b x^2\right) \left(-a+b x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[x^2/((-2*a + b*x^2)*(-a + b*x^2)^(3/4)),x]","\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{2} \sqrt[4]{a} \sqrt[4]{b x^2-a}}{\sqrt{b} x}\right)}{\sqrt{2} \sqrt[4]{a} b^{3/2}}","\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,"ArcTan[(Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*(-a + b*x^2)^(1/4))]/(Sqrt[2]*a^(1/4)*b^(3/2)) - ArcTanh[(Sqrt[2]*a^(1/4)*(-a + b*x^2)^(1/4))/(Sqrt[b]*x)]/(Sqrt[2]*a^(1/4)*b^(3/2))","A",1
1080,1,100,98,2.3425528,"\int \frac{x^2}{\left(-2 a-b x^2\right) \left(-a-b x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[x^2/((-2*a - b*x^2)*(-a - b*x^2)^(3/4)),x]","\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{2} \sqrt[4]{a} \sqrt[4]{-a-b x^2}}{\sqrt{b} x}\right)}{\sqrt{2} \sqrt[4]{a} b^{3/2}}","\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,"ArcTan[(Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*(-a - b*x^2)^(1/4))]/(Sqrt[2]*a^(1/4)*b^(3/2)) - ArcTanh[(Sqrt[2]*a^(1/4)*(-a - b*x^2)^(1/4))/(Sqrt[b]*x)]/(Sqrt[2]*a^(1/4)*b^(3/2))","A",1
1081,1,60,78,0.0648057,"\int \frac{x^7}{\left(-2+3 x^2\right) \left(-1+3 x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[x^7/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]","-\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)+\frac{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))/3645 - (8*ArcTan[(-1 + 3*x^2)^(1/4)])/81 - (8*ArcTanh[(-1 + 3*x^2)^(1/4)])/81","A",1
1082,1,55,63,0.0546898,"\int \frac{x^5}{\left(-2+3 x^2\right) \left(-1+3 x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[x^5/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]","\frac{2}{135} \sqrt[4]{3 x^2-1} \left(3 x^2+14\right)-\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)","\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))/135 - (4*ArcTan[(-1 + 3*x^2)^(1/4)])/27 - (4*ArcTanh[(-1 + 3*x^2)^(1/4)])/27","A",1
1083,1,48,48,0.0476572,"\int \frac{x^3}{\left(-2+3 x^2\right) \left(-1+3 x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[x^3/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]","\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)","\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))/9 - (2*ArcTan[(-1 + 3*x^2)^(1/4)])/9 - (2*ArcTanh[(-1 + 3*x^2)^(1/4)])/9","A",1
1084,1,33,33,0.0389983,"\int \frac{x}{\left(-2+3 x^2\right) \left(-1+3 x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[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
1085,1,117,173,0.1690926,"\int \frac{1}{x \left(-2+3 x^2\right) \left(-1+3 x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x*(-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]","-\frac{1}{2} \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{3 x^2-1}}{\sqrt{3 x^2-1}-1}\right)}{2 \sqrt{2}}-\frac{1}{2} \tanh ^{-1}\left(\sqrt[4]{3 x^2-1}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{3 x^2-1}}{\sqrt{3 x^2-1}+1}\right)}{2 \sqrt{2}}","\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,"-1/2*ArcTan[(-1 + 3*x^2)^(1/4)] + ArcTan[(Sqrt[2]*(-1 + 3*x^2)^(1/4))/(-1 + Sqrt[-1 + 3*x^2])]/(2*Sqrt[2]) - ArcTanh[(-1 + 3*x^2)^(1/4)]/2 - ArcTanh[(Sqrt[2]*(-1 + 3*x^2)^(1/4))/(1 + Sqrt[-1 + 3*x^2])]/(2*Sqrt[2])","A",1
1086,1,135,191,0.2367091,"\int \frac{1}{x^3 \left(-2+3 x^2\right) \left(-1+3 x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x^3*(-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]","-\frac{\sqrt[4]{3 x^2-1}}{4 x^2}-\frac{3}{4} \tan ^{-1}\left(\sqrt[4]{3 x^2-1}\right)+\frac{15 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{3 x^2-1}}{\sqrt{3 x^2-1}-1}\right)}{8 \sqrt{2}}-\frac{3}{4} \tanh ^{-1}\left(\sqrt[4]{3 x^2-1}\right)-\frac{15 \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{3 x^2-1}}{\sqrt{3 x^2-1}+1}\right)}{8 \sqrt{2}}","-\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,"-1/4*(-1 + 3*x^2)^(1/4)/x^2 - (3*ArcTan[(-1 + 3*x^2)^(1/4)])/4 + (15*ArcTan[(Sqrt[2]*(-1 + 3*x^2)^(1/4))/(-1 + Sqrt[-1 + 3*x^2])])/(8*Sqrt[2]) - (3*ArcTanh[(-1 + 3*x^2)^(1/4)])/4 - (15*ArcTanh[(Sqrt[2]*(-1 + 3*x^2)^(1/4))/(1 + Sqrt[-1 + 3*x^2])])/(8*Sqrt[2])","A",1
1087,0,0,165,5.0130911,"\int \frac{x^6}{\left(-2+3 x^2\right) \left(-1+3 x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[x^6/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]","\int \frac{x^6}{\left(-2+3 x^2\right) \left(-1+3 x^2\right)^{3/4}} \, dx","\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,"Could not integrate","F",-1
1088,0,0,147,5.0613413,"\int \frac{x^4}{\left(-2+3 x^2\right) \left(-1+3 x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[x^4/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]","\int \frac{x^4}{\left(-2+3 x^2\right) \left(-1+3 x^2\right)^{3/4}} \, dx","\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,"Could not integrate","F",-1
1089,1,61,61,0.0006178,"\int \frac{x^2}{\left(-2+3 x^2\right) \left(-1+3 x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[x^2/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]","\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{\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,"ArcTan[(Sqrt[3/2]*x)/(-1 + 3*x^2)^(1/4)]/(3*Sqrt[6]) - ArcTanh[(Sqrt[3/2]*x)/(-1 + 3*x^2)^(1/4)]/(3*Sqrt[6])","A",1
1090,0,0,127,4.701962,"\int \frac{1}{\left(-2+3 x^2\right) \left(-1+3 x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[1/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]","\int \frac{1}{\left(-2+3 x^2\right) \left(-1+3 x^2\right)^{3/4}} \, dx","\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,"Could not integrate","F",-1
1091,0,0,149,9.5692929,"\int \frac{1}{x^2 \left(-2+3 x^2\right) \left(-1+3 x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x^2*(-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]","\int \frac{1}{x^2 \left(-2+3 x^2\right) \left(-1+3 x^2\right)^{3/4}} \, dx","-\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,"Could not integrate","F",-1
1092,0,0,165,14.5778695,"\int \frac{1}{x^4 \left(-2+3 x^2\right) \left(-1+3 x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x^4*(-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]","\int \frac{1}{x^4 \left(-2+3 x^2\right) \left(-1+3 x^2\right)^{3/4}} \, dx","-\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,"Could not integrate","F",-1
1093,1,206,173,2.1469516,"\int \frac{(e x)^{5/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[((e*x)^(5/2)*(c + d*x^2))/(a + b*x^2)^(3/4),x]","-\frac{3 e^{5/2} \left(7 a^2 d-8 a b c\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} e^{3/2} \sqrt{e x} \left(a+b x^2\right)^{3/4}}{a e^2+b e^2 x^2}\right)}{32 b^{11/4}}+\frac{3 e^{5/2} \left(7 a^2 d-8 a b c\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} e^{3/2} \sqrt{e x} \left(a+b x^2\right)^{3/4}}{a e^2+b e^2 x^2}\right)}{32 b^{11/4}}+\frac{\sqrt[4]{a+b x^2} \left(-7 a d e^2 (e x)^{3/2}+8 b c e^2 (e x)^{3/2}+4 b d (e x)^{7/2}\right)}{16 b^2 e}","\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,"((a + b*x^2)^(1/4)*(8*b*c*e^2*(e*x)^(3/2) - 7*a*d*e^2*(e*x)^(3/2) + 4*b*d*(e*x)^(7/2)))/(16*b^2*e) - (3*(-8*a*b*c + 7*a^2*d)*e^(5/2)*ArcTan[(b^(1/4)*e^(3/2)*Sqrt[e*x]*(a + b*x^2)^(3/4))/(a*e^2 + b*e^2*x^2)])/(32*b^(11/4)) + (3*(-8*a*b*c + 7*a^2*d)*e^(5/2)*ArcTanh[(b^(1/4)*e^(3/2)*Sqrt[e*x]*(a + b*x^2)^(3/4))/(a*e^2 + b*e^2*x^2)])/(32*b^(11/4))","A",1
1094,1,168,136,1.1830921,"\int \frac{\sqrt{e x} \left(c+d x^2\right)}{\left(a+b x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[(Sqrt[e*x]*(c + d*x^2))/(a + b*x^2)^(3/4),x]","-\frac{\sqrt{e} (4 b c-3 a d) \tan ^{-1}\left(\frac{\sqrt[4]{b} e^{3/2} \sqrt{e x} \left(a+b x^2\right)^{3/4}}{a e^2+b e^2 x^2}\right)}{4 b^{7/4}}+\frac{\sqrt{e} (4 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt[4]{b} e^{3/2} \sqrt{e x} \left(a+b x^2\right)^{3/4}}{a e^2+b e^2 x^2}\right)}{4 b^{7/4}}+\frac{d (e x)^{3/2} \sqrt[4]{a+b x^2}}{2 b e}","-\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,"(d*(e*x)^(3/2)*(a + b*x^2)^(1/4))/(2*b*e) - ((4*b*c - 3*a*d)*Sqrt[e]*ArcTan[(b^(1/4)*e^(3/2)*Sqrt[e*x]*(a + b*x^2)^(3/4))/(a*e^2 + b*e^2*x^2)])/(4*b^(7/4)) + ((4*b*c - 3*a*d)*Sqrt[e]*ArcTanh[(b^(1/4)*e^(3/2)*Sqrt[e*x]*(a + b*x^2)^(3/4))/(a*e^2 + b*e^2*x^2)])/(4*b^(7/4))","A",1
1095,1,145,113,0.9662371,"\int \frac{c+d x^2}{(e x)^{3/2} \left(a+b x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[(c + d*x^2)/((e*x)^(3/2)*(a + b*x^2)^(3/4)),x]","-\frac{d \tan ^{-1}\left(\frac{\sqrt[4]{b} e^{3/2} \sqrt{e x} \left(a+b x^2\right)^{3/4}}{a e^2+b e^2 x^2}\right)}{b^{3/4} e^{3/2}}+\frac{d \tanh ^{-1}\left(\frac{\sqrt[4]{b} e^{3/2} \sqrt{e x} \left(a+b x^2\right)^{3/4}}{a e^2+b e^2 x^2}\right)}{b^{3/4} e^{3/2}}-\frac{2 c \sqrt[4]{a+b x^2}}{a e \sqrt{e x}}","-\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,"(-2*c*(a + b*x^2)^(1/4))/(a*e*Sqrt[e*x]) - (d*ArcTan[(b^(1/4)*e^(3/2)*Sqrt[e*x]*(a + b*x^2)^(3/4))/(a*e^2 + b*e^2*x^2)])/(b^(3/4)*e^(3/2)) + (d*ArcTanh[(b^(1/4)*e^(3/2)*Sqrt[e*x]*(a + b*x^2)^(3/4))/(a*e^2 + b*e^2*x^2)])/(b^(3/4)*e^(3/2))","A",1
1096,1,55,67,11.2974002,"\int \frac{c+d x^2}{(e x)^{7/2} \left(a+b x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[(c + d*x^2)/((e*x)^(7/2)*(a + b*x^2)^(3/4)),x]","-\frac{2 \sqrt[4]{a+b x^2} \left(a c e^2+5 a d e^2 x^2-4 b c e^2 x^2\right)}{5 a^2 e^3 (e x)^{5/2}}","\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*(a + b*x^2)^(1/4)*(a*c*e^2 - 4*b*c*e^2*x^2 + 5*a*d*e^2*x^2))/(5*a^2*e^3*(e*x)^(5/2))","A",1
1097,1,84,104,1.338523,"\int \frac{c+d x^2}{(e x)^{11/2} \left(a+b x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[(c + d*x^2)/((e*x)^(11/2)*(a + b*x^2)^(3/4)),x]","\frac{2 \sqrt[4]{a+b x^2} \left(-5 a^2 c e^4-9 a^2 d e^4 x^2+8 a b c e^4 x^2+36 a b d e^4 x^4-32 b^2 c e^4 x^4\right)}{45 a^3 e^5 (e x)^{9/2}}","-\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*(a + b*x^2)^(1/4)*(-5*a^2*c*e^4 + 8*a*b*c*e^4*x^2 - 9*a^2*d*e^4*x^2 - 32*b^2*c*e^4*x^4 + 36*a*b*d*e^4*x^4))/(45*a^3*e^5*(e*x)^(9/2))","A",1
1098,1,114,141,2.2277171,"\int \frac{c+d x^2}{(e x)^{15/2} \left(a+b x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[(c + d*x^2)/((e*x)^(15/2)*(a + b*x^2)^(3/4)),x]","-\frac{2 \sqrt[4]{a+b x^2} \left(45 a^3 c e^6+65 a^3 d e^6 x^2-60 a^2 b c e^6 x^2-104 a^2 b d e^6 x^4+96 a b^2 c e^6 x^4+416 a b^2 d e^6 x^6-384 b^3 c e^6 x^6\right)}{585 a^4 e^7 (e x)^{13/2}}","\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*(a + b*x^2)^(1/4)*(45*a^3*c*e^6 - 60*a^2*b*c*e^6*x^2 + 65*a^3*d*e^6*x^2 + 96*a*b^2*c*e^6*x^4 - 104*a^2*b*d*e^6*x^4 - 384*b^3*c*e^6*x^6 + 416*a*b^2*d*e^6*x^6))/(585*a^4*e^7*(e*x)^(13/2))","A",1
1099,0,0,180,13.2038019,"\int \frac{(e x)^{7/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[((e*x)^(7/2)*(c + d*x^2))/(a + b*x^2)^(3/4),x]","\int \frac{(e x)^{7/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{3/4}} \, dx","-\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,"Could not integrate","F",-1
1100,0,0,139,11.5251224,"\int \frac{(e x)^{3/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[((e*x)^(3/2)*(c + d*x^2))/(a + b*x^2)^(3/4),x]","\int \frac{(e x)^{3/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{3/4}} \, dx","\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,"Could not integrate","F",-1
1101,0,0,102,10.3565396,"\int \frac{c+d x^2}{\sqrt{e x} \left(a+b x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[(c + d*x^2)/(Sqrt[e*x]*(a + b*x^2)^(3/4)),x]","\int \frac{c+d x^2}{\sqrt{e x} \left(a+b x^2\right)^{3/4}} \, dx","\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,"Could not integrate","F",-1
1102,0,0,107,10.9678772,"\int \frac{c+d x^2}{(e x)^{5/2} \left(a+b x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[(c + d*x^2)/((e*x)^(5/2)*(a + b*x^2)^(3/4)),x]","\int \frac{c+d x^2}{(e x)^{5/2} \left(a+b x^2\right)^{3/4}} \, dx","\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,"Could not integrate","F",-1
1103,0,0,144,22.4479561,"\int \frac{c+d x^2}{(e x)^{9/2} \left(a+b x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[(c + d*x^2)/((e*x)^(9/2)*(a + b*x^2)^(3/4)),x]","\int \frac{c+d x^2}{(e x)^{9/2} \left(a+b x^2\right)^{3/4}} \, dx","-\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,"Could not integrate","F",-1
1104,0,0,182,23.8525831,"\int \frac{c+d x^2}{(e x)^{13/2} \left(a+b x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[(c + d*x^2)/((e*x)^(13/2)*(a + b*x^2)^(3/4)),x]","\int \frac{c+d x^2}{(e x)^{13/2} \left(a+b x^2\right)^{3/4}} \, dx","\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,"Could not integrate","F",-1
1105,0,0,171,11.9687005,"\int \frac{(e x)^{3/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{5/4}} \, dx","IntegrateAlgebraic[((e*x)^(3/2)*(c + d*x^2))/(a + b*x^2)^(5/4),x]","\int \frac{(e x)^{3/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{5/4}} \, dx","\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,"Could not integrate","F",-1
1106,1,137,122,7.7717976,"\int \frac{c+d x^2}{\sqrt{e x} \left(a+b x^2\right)^{5/4}} \, dx","IntegrateAlgebraic[(c + d*x^2)/(Sqrt[e*x]*(a + b*x^2)^(5/4)),x]","-\frac{d \tan ^{-1}\left(\frac{\sqrt{e} \sqrt[4]{a+b x^2}}{\sqrt[4]{b} \sqrt{e x}}\right)}{b^{5/4} \sqrt{e}}+\frac{d \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt[4]{a+b x^2}}{\sqrt[4]{b} \sqrt{e x}}\right)}{b^{5/4} \sqrt{e}}-\frac{2 e \sqrt{e x} \left(a+b x^2\right)^{3/4} (a d-b c)}{a b \left(a e^2+b e^2 x^2\right)}","\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*(-(b*c) + a*d)*e*Sqrt[e*x]*(a + b*x^2)^(3/4))/(a*b*(a*e^2 + b*e^2*x^2)) - (d*ArcTan[(Sqrt[e]*(a + b*x^2)^(1/4))/(b^(1/4)*Sqrt[e*x])])/(b^(5/4)*Sqrt[e]) + (d*ArcTanh[(Sqrt[e]*(a + b*x^2)^(1/4))/(b^(1/4)*Sqrt[e*x])])/(b^(5/4)*Sqrt[e])","A",1
1107,1,72,67,7.7921427,"\int \frac{c+d x^2}{(e x)^{5/2} \left(a+b x^2\right)^{5/4}} \, dx","IntegrateAlgebraic[(c + d*x^2)/((e*x)^(5/2)*(a + b*x^2)^(5/4)),x]","\frac{2 \left(a+b x^2\right)^{3/4} \left(-a c e^2+3 a d e^2 x^2-4 b c e^2 x^2\right)}{3 a^2 e (e x)^{3/2} \left(a e^2+b e^2 x^2\right)}","-\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,"(2*(a + b*x^2)^(3/4)*(-(a*c*e^2) - 4*b*c*e^2*x^2 + 3*a*d*e^2*x^2))/(3*a^2*e*(e*x)^(3/2)*(a*e^2 + b*e^2*x^2))","A",1
1108,1,100,104,17.2760544,"\int \frac{c+d x^2}{(e x)^{9/2} \left(a+b x^2\right)^{5/4}} \, dx","IntegrateAlgebraic[(c + d*x^2)/((e*x)^(9/2)*(a + b*x^2)^(5/4)),x]","-\frac{2 \left(a+b x^2\right)^{3/4} \left(3 a^2 c e^4+7 a^2 d e^4 x^2-8 a b c e^4 x^2+28 a b d e^4 x^4-32 b^2 c e^4 x^4\right)}{21 a^3 e^3 (e x)^{7/2} \left(a e^2+b e^2 x^2\right)}","\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*(a + b*x^2)^(3/4)*(3*a^2*c*e^4 - 8*a*b*c*e^4*x^2 + 7*a^2*d*e^4*x^2 - 32*b^2*c*e^4*x^4 + 28*a*b*d*e^4*x^4))/(21*a^3*e^3*(e*x)^(7/2)*(a*e^2 + b*e^2*x^2))","A",1
1109,1,130,141,4.208898,"\int \frac{c+d x^2}{(e x)^{13/2} \left(a+b x^2\right)^{5/4}} \, dx","IntegrateAlgebraic[(c + d*x^2)/((e*x)^(13/2)*(a + b*x^2)^(5/4)),x]","\frac{2 \left(a+b x^2\right)^{3/4} \left(-21 a^3 c e^6-33 a^3 d e^6 x^2+36 a^2 b c e^6 x^2+88 a^2 b d e^6 x^4-96 a b^2 c e^6 x^4+352 a b^2 d e^6 x^6-384 b^3 c e^6 x^6\right)}{231 a^4 e^5 (e x)^{11/2} \left(a e^2+b e^2 x^2\right)}","-\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*(a + b*x^2)^(3/4)*(-21*a^3*c*e^6 + 36*a^2*b*c*e^6*x^2 - 33*a^3*d*e^6*x^2 - 96*a*b^2*c*e^6*x^4 + 88*a^2*b*d*e^6*x^4 - 384*b^3*c*e^6*x^6 + 352*a*b^2*d*e^6*x^6))/(231*a^4*e^5*(e*x)^(11/2)*(a*e^2 + b*e^2*x^2))","A",1
1110,0,0,180,12.7499218,"\int \frac{(e x)^{9/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{5/4}} \, dx","IntegrateAlgebraic[((e*x)^(9/2)*(c + d*x^2))/(a + b*x^2)^(5/4),x]","\int \frac{(e x)^{9/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{5/4}} \, dx","-\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,"Could not integrate","F",-1
1111,0,0,142,11.7533322,"\int \frac{(e x)^{5/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{5/4}} \, dx","IntegrateAlgebraic[((e*x)^(5/2)*(c + d*x^2))/(a + b*x^2)^(5/4),x]","\int \frac{(e x)^{5/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{5/4}} \, dx","\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,"Could not integrate","F",-1
1112,0,0,99,11.6468772,"\int \frac{\sqrt{e x} \left(c+d x^2\right)}{\left(a+b x^2\right)^{5/4}} \, dx","IntegrateAlgebraic[(Sqrt[e*x]*(c + d*x^2))/(a + b*x^2)^(5/4),x]","\int \frac{\sqrt{e x} \left(c+d x^2\right)}{\left(a+b x^2\right)^{5/4}} \, dx","\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,"Could not integrate","F",-1
1113,0,0,103,11.5196517,"\int \frac{c+d x^2}{(e x)^{3/2} \left(a+b x^2\right)^{5/4}} \, dx","IntegrateAlgebraic[(c + d*x^2)/((e*x)^(3/2)*(a + b*x^2)^(5/4)),x]","\int \frac{c+d x^2}{(e x)^{3/2} \left(a+b x^2\right)^{5/4}} \, dx","\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,"Could not integrate","F",-1
1114,0,0,144,17.0120974,"\int \frac{c+d x^2}{(e x)^{7/2} \left(a+b x^2\right)^{5/4}} \, dx","IntegrateAlgebraic[(c + d*x^2)/((e*x)^(7/2)*(a + b*x^2)^(5/4)),x]","\int \frac{c+d x^2}{(e x)^{7/2} \left(a+b x^2\right)^{5/4}} \, dx","-\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,"Could not integrate","F",-1
1115,0,0,182,26.2302884,"\int \frac{c+d x^2}{(e x)^{11/2} \left(a+b x^2\right)^{5/4}} \, dx","IntegrateAlgebraic[(c + d*x^2)/((e*x)^(11/2)*(a + b*x^2)^(5/4)),x]","\int \frac{c+d x^2}{(e x)^{11/2} \left(a+b x^2\right)^{5/4}} \, dx","\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,"Could not integrate","F",-1
1116,0,0,184,12.1787989,"\int \frac{(e x)^{5/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{7/4}} \, dx","IntegrateAlgebraic[((e*x)^(5/2)*(c + d*x^2))/(a + b*x^2)^(7/4),x]","\int \frac{(e x)^{5/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{7/4}} \, dx","-\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,"Could not integrate","F",-1
1117,0,0,125,11.8232259,"\int \frac{\sqrt{e x} \left(c+d x^2\right)}{\left(a+b x^2\right)^{7/4}} \, dx","IntegrateAlgebraic[(Sqrt[e*x]*(c + d*x^2))/(a + b*x^2)^(7/4),x]","\int \frac{\sqrt{e x} \left(c+d x^2\right)}{\left(a+b x^2\right)^{7/4}} \, dx","-\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,"Could not integrate","F",-1
1118,0,0,65,11.6784495,"\int \frac{c+d x^2}{(e x)^{3/2} \left(a+b x^2\right)^{7/4}} \, dx","IntegrateAlgebraic[(c + d*x^2)/((e*x)^(3/2)*(a + b*x^2)^(7/4)),x]","\int \frac{c+d x^2}{(e x)^{3/2} \left(a+b x^2\right)^{7/4}} \, dx","-\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,"Could not integrate","F",-1
1119,0,0,104,16.6362754,"\int \frac{c+d x^2}{(e x)^{7/2} \left(a+b x^2\right)^{7/4}} \, dx","IntegrateAlgebraic[(c + d*x^2)/((e*x)^(7/2)*(a + b*x^2)^(7/4)),x]","\int \frac{c+d x^2}{(e x)^{7/2} \left(a+b x^2\right)^{7/4}} \, dx","\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,"Could not integrate","F",-1
1120,0,0,141,27.3905248,"\int \frac{c+d x^2}{(e x)^{11/2} \left(a+b x^2\right)^{7/4}} \, dx","IntegrateAlgebraic[(c + d*x^2)/((e*x)^(11/2)*(a + b*x^2)^(7/4)),x]","\int \frac{c+d x^2}{(e x)^{11/2} \left(a+b x^2\right)^{7/4}} \, dx","-\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,"Could not integrate","F",-1
1121,0,0,192,14.4995372,"\int \frac{(e x)^{7/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{7/4}} \, dx","IntegrateAlgebraic[((e*x)^(7/2)*(c + d*x^2))/(a + b*x^2)^(7/4),x]","\int \frac{(e x)^{7/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{7/4}} \, dx","\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,"Could not integrate","F",-1
1122,0,0,152,12.3826518,"\int \frac{(e x)^{3/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{7/4}} \, dx","IntegrateAlgebraic[((e*x)^(3/2)*(c + d*x^2))/(a + b*x^2)^(7/4),x]","\int \frac{(e x)^{3/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{7/4}} \, dx","-\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,"Could not integrate","F",-1
1123,0,0,116,11.6560172,"\int \frac{c+d x^2}{\sqrt{e x} \left(a+b x^2\right)^{7/4}} \, dx","IntegrateAlgebraic[(c + d*x^2)/(Sqrt[e*x]*(a + b*x^2)^(7/4)),x]","\int \frac{c+d x^2}{\sqrt{e x} \left(a+b x^2\right)^{7/4}} \, dx","\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,"Could not integrate","F",-1
1124,0,0,144,11.7542073,"\int \frac{c+d x^2}{(e x)^{5/2} \left(a+b x^2\right)^{7/4}} \, dx","IntegrateAlgebraic[(c + d*x^2)/((e*x)^(5/2)*(a + b*x^2)^(7/4)),x]","\int \frac{c+d x^2}{(e x)^{5/2} \left(a+b x^2\right)^{7/4}} \, dx","\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,"Could not integrate","F",-1
1125,0,0,181,23.2691697,"\int \frac{c+d x^2}{(e x)^{9/2} \left(a+b x^2\right)^{7/4}} \, dx","IntegrateAlgebraic[(c + d*x^2)/((e*x)^(9/2)*(a + b*x^2)^(7/4)),x]","\int \frac{c+d x^2}{(e x)^{9/2} \left(a+b x^2\right)^{7/4}} \, dx","-\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,"Could not integrate","F",-1
1126,0,0,221,15.3509679,"\int \frac{(e x)^{7/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{9/4}} \, dx","IntegrateAlgebraic[((e*x)^(7/2)*(c + d*x^2))/(a + b*x^2)^(9/4),x]","\int \frac{(e x)^{7/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{9/4}} \, dx","\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,"Could not integrate","F",-1
1127,0,0,149,13.0011077,"\int \frac{(e x)^{3/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{9/4}} \, dx","IntegrateAlgebraic[((e*x)^(3/2)*(c + d*x^2))/(a + b*x^2)^(9/4),x]","\int \frac{(e x)^{3/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{9/4}} \, dx","\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,"Could not integrate","F",-1
1128,0,0,79,11.9991444,"\int \frac{c+d x^2}{\sqrt{e x} \left(a+b x^2\right)^{9/4}} \, dx","IntegrateAlgebraic[(c + d*x^2)/(Sqrt[e*x]*(a + b*x^2)^(9/4)),x]","\int \frac{c+d x^2}{\sqrt{e x} \left(a+b x^2\right)^{9/4}} \, dx","\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,"Could not integrate","F",-1
1129,0,0,104,12.1408435,"\int \frac{c+d x^2}{(e x)^{5/2} \left(a+b x^2\right)^{9/4}} \, dx","IntegrateAlgebraic[(c + d*x^2)/((e*x)^(5/2)*(a + b*x^2)^(9/4)),x]","\int \frac{c+d x^2}{(e x)^{5/2} \left(a+b x^2\right)^{9/4}} \, dx","-\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,"Could not integrate","F",-1
1130,0,0,141,23.0480888,"\int \frac{c+d x^2}{(e x)^{9/2} \left(a+b x^2\right)^{9/4}} \, dx","IntegrateAlgebraic[(c + d*x^2)/((e*x)^(9/2)*(a + b*x^2)^(9/4)),x]","\int \frac{c+d x^2}{(e x)^{9/2} \left(a+b x^2\right)^{9/4}} \, dx","\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,"Could not integrate","F",-1
1131,0,0,178,27.8655077,"\int \frac{c+d x^2}{(e x)^{13/2} \left(a+b x^2\right)^{9/4}} \, dx","IntegrateAlgebraic[(c + d*x^2)/((e*x)^(13/2)*(a + b*x^2)^(9/4)),x]","\int \frac{c+d x^2}{(e x)^{13/2} \left(a+b x^2\right)^{9/4}} \, dx","-\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,"Could not integrate","F",-1
1132,0,0,230,14.242576,"\int \frac{(e x)^{13/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{9/4}} \, dx","IntegrateAlgebraic[((e*x)^(13/2)*(c + d*x^2))/(a + b*x^2)^(9/4),x]","\int \frac{(e x)^{13/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{9/4}} \, dx","-\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,"Could not integrate","F",-1
1133,0,0,192,13.2500583,"\int \frac{(e x)^{9/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{9/4}} \, dx","IntegrateAlgebraic[((e*x)^(9/2)*(c + d*x^2))/(a + b*x^2)^(9/4),x]","\int \frac{(e x)^{9/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{9/4}} \, dx","\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,"Could not integrate","F",-1
1134,0,0,155,12.6356713,"\int \frac{(e x)^{5/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{9/4}} \, dx","IntegrateAlgebraic[((e*x)^(5/2)*(c + d*x^2))/(a + b*x^2)^(9/4),x]","\int \frac{(e x)^{5/2} \left(c+d x^2\right)}{\left(a+b x^2\right)^{9/4}} \, dx","-\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,"Could not integrate","F",-1
1135,0,0,114,12.1465422,"\int \frac{\sqrt{e x} \left(c+d x^2\right)}{\left(a+b x^2\right)^{9/4}} \, dx","IntegrateAlgebraic[(Sqrt[e*x]*(c + d*x^2))/(a + b*x^2)^(9/4),x]","\int \frac{\sqrt{e x} \left(c+d x^2\right)}{\left(a+b x^2\right)^{9/4}} \, dx","\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,"Could not integrate","F",-1
1136,0,0,142,12.033442,"\int \frac{c+d x^2}{(e x)^{3/2} \left(a+b x^2\right)^{9/4}} \, dx","IntegrateAlgebraic[(c + d*x^2)/((e*x)^(3/2)*(a + b*x^2)^(9/4)),x]","\int \frac{c+d x^2}{(e x)^{3/2} \left(a+b x^2\right)^{9/4}} \, dx","\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,"Could not integrate","F",-1
1137,0,0,181,17.1907939,"\int \frac{c+d x^2}{(e x)^{7/2} \left(a+b x^2\right)^{9/4}} \, dx","IntegrateAlgebraic[(c + d*x^2)/((e*x)^(7/2)*(a + b*x^2)^(9/4)),x]","\int \frac{c+d x^2}{(e x)^{7/2} \left(a+b x^2\right)^{9/4}} \, dx","-\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,"Could not integrate","F",-1
1138,0,0,219,26.367673,"\int \frac{c+d x^2}{(e x)^{11/2} \left(a+b x^2\right)^{9/4}} \, dx","IntegrateAlgebraic[(c + d*x^2)/((e*x)^(11/2)*(a + b*x^2)^(9/4)),x]","\int \frac{c+d x^2}{(e x)^{11/2} \left(a+b x^2\right)^{9/4}} \, dx","\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,"Could not integrate","F",-1
1139,0,0,101,0.0252457,"\int (e x)^m \left(a+b x^2\right)^p \left(c+d x^2\right)^q \, dx","IntegrateAlgebraic[(e*x)^m*(a + b*x^2)^p*(c + d*x^2)^q,x]","\int (e x)^m \left(a+b x^2\right)^p \left(c+d x^2\right)^q \, dx","\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,"Could not integrate","F",-1
1140,0,0,84,0.0368419,"\int x^4 \left(a+b x^2\right)^p \left(c+d x^2\right)^q \, dx","IntegrateAlgebraic[x^4*(a + b*x^2)^p*(c + d*x^2)^q,x]","\int x^4 \left(a+b x^2\right)^p \left(c+d x^2\right)^q \, dx","\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,"Could not integrate","F",-1
1141,0,0,84,0.0286137,"\int x^2 \left(a+b x^2\right)^p \left(c+d x^2\right)^q \, dx","IntegrateAlgebraic[x^2*(a + b*x^2)^p*(c + d*x^2)^q,x]","\int x^2 \left(a+b x^2\right)^p \left(c+d x^2\right)^q \, dx","\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,"Could not integrate","F",-1
1142,0,0,79,0.0006413,"\int \left(a+b x^2\right)^p \left(c+d x^2\right)^q \, dx","IntegrateAlgebraic[(a + b*x^2)^p*(c + d*x^2)^q,x]","\int \left(a+b x^2\right)^p \left(c+d x^2\right)^q \, dx","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,"Could not integrate","F",-1
1143,0,0,82,0.0349148,"\int \frac{\left(a+b x^2\right)^p \left(c+d x^2\right)^q}{x^2} \, dx","IntegrateAlgebraic[((a + b*x^2)^p*(c + d*x^2)^q)/x^2,x]","\int \frac{\left(a+b x^2\right)^p \left(c+d x^2\right)^q}{x^2} \, dx","-\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,"Could not integrate","F",-1
1144,0,0,84,0.03687,"\int \frac{\left(a+b x^2\right)^p \left(c+d x^2\right)^q}{x^4} \, dx","IntegrateAlgebraic[((a + b*x^2)^p*(c + d*x^2)^q)/x^4,x]","\int \frac{\left(a+b x^2\right)^p \left(c+d x^2\right)^q}{x^4} \, dx","-\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,"Could not integrate","F",-1
1145,0,0,242,0.0349422,"\int x^5 \left(a+b x^2\right)^p \left(c+d x^2\right)^q \, dx","IntegrateAlgebraic[x^5*(a + b*x^2)^p*(c + d*x^2)^q,x]","\int x^5 \left(a+b x^2\right)^p \left(c+d x^2\right)^q \, dx","\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,"Could not integrate","F",-1
1146,0,0,146,0.0398696,"\int x^3 \left(a+b x^2\right)^p \left(c+d x^2\right)^q \, dx","IntegrateAlgebraic[x^3*(a + b*x^2)^p*(c + d*x^2)^q,x]","\int x^3 \left(a+b x^2\right)^p \left(c+d x^2\right)^q \, dx","\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,"Could not integrate","F",-1
1147,0,0,85,0.0286418,"\int x \left(a+b x^2\right)^p \left(c+d x^2\right)^q \, dx","IntegrateAlgebraic[x*(a + b*x^2)^p*(c + d*x^2)^q,x]","\int x \left(a+b x^2\right)^p \left(c+d x^2\right)^q \, dx","\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,"Could not integrate","F",-1
1148,0,0,97,0.0306625,"\int \frac{\left(a+b x^2\right)^p \left(c+d x^2\right)^q}{x} \, dx","IntegrateAlgebraic[((a + b*x^2)^p*(c + d*x^2)^q)/x,x]","\int \frac{\left(a+b x^2\right)^p \left(c+d x^2\right)^q}{x} \, dx","-\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,"Could not integrate","F",-1
1149,0,0,98,0.0329144,"\int \frac{\left(a+b x^2\right)^p \left(c+d x^2\right)^q}{x^3} \, dx","IntegrateAlgebraic[((a + b*x^2)^p*(c + d*x^2)^q)/x^3,x]","\int \frac{\left(a+b x^2\right)^p \left(c+d x^2\right)^q}{x^3} \, dx","\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,"Could not integrate","F",-1
1150,0,0,100,0.0338887,"\int \frac{\left(a+b x^2\right)^p \left(c+d x^2\right)^q}{x^5} \, dx","IntegrateAlgebraic[((a + b*x^2)^p*(c + d*x^2)^q)/x^5,x]","\int \frac{\left(a+b x^2\right)^p \left(c+d x^2\right)^q}{x^5} \, dx","-\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,"Could not integrate","F",-1
1151,0,0,91,0.081815,"\int (e x)^{5/2} \left(a+b x^2\right)^p \left(c+d x^2\right)^q \, dx","IntegrateAlgebraic[(e*x)^(5/2)*(a + b*x^2)^p*(c + d*x^2)^q,x]","\int (e x)^{5/2} \left(a+b x^2\right)^p \left(c+d x^2\right)^q \, dx","\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,"Could not integrate","F",-1
1152,0,0,91,0.074978,"\int (e x)^{3/2} \left(a+b x^2\right)^p \left(c+d x^2\right)^q \, dx","IntegrateAlgebraic[(e*x)^(3/2)*(a + b*x^2)^p*(c + d*x^2)^q,x]","\int (e x)^{3/2} \left(a+b x^2\right)^p \left(c+d x^2\right)^q \, dx","\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,"Could not integrate","F",-1
1153,0,0,91,0.0716752,"\int \sqrt{e x} \left(a+b x^2\right)^p \left(c+d x^2\right)^q \, dx","IntegrateAlgebraic[Sqrt[e*x]*(a + b*x^2)^p*(c + d*x^2)^q,x]","\int \sqrt{e x} \left(a+b x^2\right)^p \left(c+d x^2\right)^q \, dx","\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,"Could not integrate","F",-1
1154,0,0,89,0.0847716,"\int \frac{\left(a+b x^2\right)^p \left(c+d x^2\right)^q}{\sqrt{e x}} \, dx","IntegrateAlgebraic[((a + b*x^2)^p*(c + d*x^2)^q)/Sqrt[e*x],x]","\int \frac{\left(a+b x^2\right)^p \left(c+d x^2\right)^q}{\sqrt{e x}} \, dx","\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,"Could not integrate","F",-1
1155,0,0,89,0.0984264,"\int \frac{\left(a+b x^2\right)^p \left(c+d x^2\right)^q}{(e x)^{3/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)^p*(c + d*x^2)^q)/(e*x)^(3/2),x]","\int \frac{\left(a+b x^2\right)^p \left(c+d x^2\right)^q}{(e x)^{3/2}} \, dx","-\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,"Could not integrate","F",-1
1156,0,0,91,0.1019959,"\int \frac{\left(a+b x^2\right)^p \left(c+d x^2\right)^q}{(e x)^{5/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)^p*(c + d*x^2)^q)/(e*x)^(5/2),x]","\int \frac{\left(a+b x^2\right)^p \left(c+d x^2\right)^q}{(e x)^{5/2}} \, dx","-\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,"Could not integrate","F",-1