1,1,103,94,0.0515403,"\int \left(a+b x^2\right) \left(c+d x^2\right)^4 \, dx","IntegrateAlgebraic[(a + b*x^2)*(c + d*x^2)^4,x]","\frac{3465 a c^4 x+4620 a c^3 d x^3+4158 a c^2 d^2 x^5+1980 a c d^3 x^7+385 a d^4 x^9+1155 b c^4 x^3+2772 b c^3 d x^5+2970 b c^2 d^2 x^7+1540 b c d^3 x^9+315 b d^4 x^{11}}{3465}","\frac{1}{3} c^3 x^3 (4 a d+b c)+\frac{2}{5} c^2 d x^5 (3 a d+2 b c)+\frac{1}{9} d^3 x^9 (a d+4 b c)+\frac{2}{7} c d^2 x^7 (2 a d+3 b c)+a c^4 x+\frac{1}{11} b d^4 x^{11}",1,"(3465*a*c^4*x + 1155*b*c^4*x^3 + 4620*a*c^3*d*x^3 + 2772*b*c^3*d*x^5 + 4158*a*c^2*d^2*x^5 + 2970*b*c^2*d^2*x^7 + 1980*a*c*d^3*x^7 + 1540*b*c*d^3*x^9 + 385*a*d^4*x^9 + 315*b*d^4*x^11)/3465","A",1
2,1,79,70,0.0381058,"\int \left(a+b x^2\right) \left(c+d x^2\right)^3 \, dx","IntegrateAlgebraic[(a + b*x^2)*(c + d*x^2)^3,x]","\frac{1}{315} \left(315 a c^3 x+315 a c^2 d x^3+189 a c d^2 x^5+45 a d^3 x^7+105 b c^3 x^3+189 b c^2 d x^5+135 b c d^2 x^7+35 b d^3 x^9\right)","\frac{1}{3} c^2 x^3 (3 a d+b c)+\frac{1}{7} d^2 x^7 (a d+3 b c)+\frac{3}{5} c d x^5 (a d+b c)+a c^3 x+\frac{1}{9} b d^3 x^9",1,"(315*a*c^3*x + 105*b*c^3*x^3 + 315*a*c^2*d*x^3 + 189*b*c^2*d*x^5 + 189*a*c*d^2*x^5 + 135*b*c*d^2*x^7 + 45*a*d^3*x^7 + 35*b*d^3*x^9)/315","A",1
3,1,55,50,0.0276891,"\int \left(a+b x^2\right) \left(c+d x^2\right)^2 \, dx","IntegrateAlgebraic[(a + b*x^2)*(c + d*x^2)^2,x]","\frac{1}{105} \left(105 a c^2 x+70 a c d x^3+21 a d^2 x^5+35 b c^2 x^3+42 b c d x^5+15 b d^2 x^7\right)","\frac{1}{5} d x^5 (a d+2 b c)+\frac{1}{3} c x^3 (2 a d+b c)+a c^2 x+\frac{1}{7} b d^2 x^7",1,"(105*a*c^2*x + 35*b*c^2*x^3 + 70*a*c*d*x^3 + 42*b*c*d*x^5 + 21*a*d^2*x^5 + 15*b*d^2*x^7)/105","A",1
4,1,31,28,0.0142627,"\int \left(a+b x^2\right) \left(c+d x^2\right) \, dx","IntegrateAlgebraic[(a + b*x^2)*(c + d*x^2),x]","\frac{1}{15} \left(15 a c x+5 a d x^3+5 b c x^3+3 b d x^5\right)","\frac{1}{3} x^3 (a d+b c)+a c x+\frac{1}{5} b d x^5",1,"(15*a*c*x + 5*b*c*x^3 + 5*a*d*x^3 + 3*b*d*x^5)/15","A",1
5,1,39,40,0.0372704,"\int \frac{a+b x^2}{c+d x^2} \, dx","IntegrateAlgebraic[(a + b*x^2)/(c + d*x^2),x]","\frac{(a d-b c) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{\sqrt{c} d^{3/2}}+\frac{b x}{d}","\frac{b x}{d}-\frac{(b c-a d) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{\sqrt{c} d^{3/2}}",1,"(b*x)/d + ((-(b*c) + a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*d^(3/2))","A",1
6,1,63,63,0.0661271,"\int \frac{a+b x^2}{\left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[(a + b*x^2)/(c + d*x^2)^2,x]","\frac{(a d+b c) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{3/2} d^{3/2}}+\frac{x (a d-b c)}{2 c d \left(c+d x^2\right)}","\frac{(a d+b c) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{3/2} d^{3/2}}-\frac{x (b c-a d)}{2 c d \left(c+d x^2\right)}",1,"((-(b*c) + a*d)*x)/(2*c*d*(c + d*x^2)) + ((b*c + a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(3/2)*d^(3/2))","A",1
7,1,84,92,0.0866461,"\int \frac{a+b x^2}{\left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[(a + b*x^2)/(c + d*x^2)^3,x]","\frac{(3 a d+b c) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{5/2} d^{3/2}}-\frac{x \left(-5 a c d-3 a d^2 x^2+b c^2-b c d x^2\right)}{8 c^2 d \left(c+d x^2\right)^2}","\frac{(3 a d+b c) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{5/2} d^{3/2}}+\frac{x (3 a d+b c)}{8 c^2 d \left(c+d x^2\right)}-\frac{x (b c-a d)}{4 c d \left(c+d x^2\right)^2}",1,"-1/8*(x*(b*c^2 - 5*a*c*d - b*c*d*x^2 - 3*a*d^2*x^2))/(c^2*d*(c + d*x^2)^2) + ((b*c + 3*a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(5/2)*d^(3/2))","A",1
8,1,137,122,0.0618144,"\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
9,1,96,82,0.0415194,"\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
10,1,55,50,0.0202996,"\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
11,1,59,63,0.0681691,"\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
12,1,102,82,0.0875958,"\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
13,1,131,116,0.1312526,"\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
14,1,195,154,0.0729504,"\int \left(a+b x^2\right)^3 \left(c+d x^2\right)^3 \, dx","IntegrateAlgebraic[(a + b*x^2)^3*(c + d*x^2)^3,x]","\frac{15015 a^3 c^3 x+15015 a^3 c^2 d x^3+9009 a^3 c d^2 x^5+2145 a^3 d^3 x^7+15015 a^2 b c^3 x^3+27027 a^2 b c^2 d x^5+19305 a^2 b c d^2 x^7+5005 a^2 b d^3 x^9+9009 a b^2 c^3 x^5+19305 a b^2 c^2 d x^7+15015 a b^2 c d^2 x^9+4095 a b^2 d^3 x^{11}+2145 b^3 c^3 x^7+5005 b^3 c^2 d x^9+4095 b^3 c d^2 x^{11}+1155 b^3 d^3 x^{13}}{15015}","a^3 c^3 x+\frac{1}{3} b d x^9 \left(a^2 d^2+3 a b c d+b^2 c^2\right)+\frac{1}{7} x^7 (a d+b c) \left(a^2 d^2+8 a b c d+b^2 c^2\right)+\frac{3}{5} a c x^5 \left(a^2 d^2+3 a b c d+b^2 c^2\right)+a^2 c^2 x^3 (a d+b c)+\frac{3}{11} b^2 d^2 x^{11} (a d+b c)+\frac{1}{13} b^3 d^3 x^{13}",1,"(15015*a^3*c^3*x + 15015*a^2*b*c^3*x^3 + 15015*a^3*c^2*d*x^3 + 9009*a*b^2*c^3*x^5 + 27027*a^2*b*c^2*d*x^5 + 9009*a^3*c*d^2*x^5 + 2145*b^3*c^3*x^7 + 19305*a*b^2*c^2*d*x^7 + 19305*a^2*b*c*d^2*x^7 + 2145*a^3*d^3*x^7 + 5005*b^3*c^2*d*x^9 + 15015*a*b^2*c*d^2*x^9 + 5005*a^2*b*d^3*x^9 + 4095*b^3*c*d^2*x^11 + 4095*a*b^2*d^3*x^11 + 1155*b^3*d^3*x^13)/15015","A",1
15,1,137,122,0.0563795,"\int \left(a+b x^2\right)^3 \left(c+d x^2\right)^2 \, dx","IntegrateAlgebraic[(a + b*x^2)^3*(c + d*x^2)^2,x]","\frac{3465 a^3 c^2 x+2310 a^3 c d x^3+693 a^3 d^2 x^5+3465 a^2 b c^2 x^3+4158 a^2 b c d x^5+1485 a^2 b d^2 x^7+2079 a b^2 c^2 x^5+2970 a b^2 c d x^7+1155 a b^2 d^2 x^9+495 b^3 c^2 x^7+770 b^3 c d x^9+315 b^3 d^2 x^{11}}{3465}","a^3 c^2 x+\frac{1}{7} b x^7 \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+\frac{1}{5} a x^5 \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{1}{3} a^2 c x^3 (2 a d+3 b c)+\frac{1}{9} b^2 d x^9 (3 a d+2 b c)+\frac{1}{11} b^3 d^2 x^{11}",1,"(3465*a^3*c^2*x + 3465*a^2*b*c^2*x^3 + 2310*a^3*c*d*x^3 + 2079*a*b^2*c^2*x^5 + 4158*a^2*b*c*d*x^5 + 693*a^3*d^2*x^5 + 495*b^3*c^2*x^7 + 2970*a*b^2*c*d*x^7 + 1485*a^2*b*d^2*x^7 + 770*b^3*c*d*x^9 + 1155*a*b^2*d^2*x^9 + 315*b^3*d^2*x^11)/3465","A",1
16,1,79,70,0.0336394,"\int \left(a+b x^2\right)^3 \left(c+d x^2\right) \, dx","IntegrateAlgebraic[(a + b*x^2)^3*(c + d*x^2),x]","\frac{1}{315} \left(315 a^3 c x+105 a^3 d x^3+315 a^2 b c x^3+189 a^2 b d x^5+189 a b^2 c x^5+135 a b^2 d x^7+45 b^3 c x^7+35 b^3 d x^9\right)","a^3 c x+\frac{1}{3} a^2 x^3 (a d+3 b c)+\frac{1}{7} b^2 x^7 (3 a d+b c)+\frac{3}{5} a b x^5 (a d+b c)+\frac{1}{9} b^3 d x^9",1,"(315*a^3*c*x + 315*a^2*b*c*x^3 + 105*a^3*d*x^3 + 189*a*b^2*c*x^5 + 189*a^2*b*d*x^5 + 45*b^3*c*x^7 + 135*a*b^2*d*x^7 + 35*b^3*d*x^9)/315","A",1
17,1,99,98,0.0882642,"\int \frac{\left(a+b x^2\right)^3}{c+d x^2} \, dx","IntegrateAlgebraic[(a + b*x^2)^3/(c + d*x^2),x]","\frac{b x \left(45 a^2 d^2-45 a b c d+15 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{(b c-a d)^3 \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{\sqrt{c} d^{7/2}}","\frac{b x \left(3 a^2 d^2-3 a b c d+b^2 c^2\right)}{d^3}-\frac{b^2 x^3 (b c-3 a d)}{3 d^2}-\frac{(b c-a d)^3 \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{\sqrt{c} d^{7/2}}+\frac{b^3 x^5}{5 d}",1,"(b*x*(15*b^2*c^2 - 45*a*b*c*d + 45*a^2*d^2 - 5*b^2*c*d*x^2 + 15*a*b*d^2*x^2 + 3*b^2*d^2*x^4))/(15*d^3) - ((b*c - a*d)^3*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*d^(7/2))","A",1
18,1,140,107,0.1011059,"\int \frac{\left(a+b x^2\right)^3}{\left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[(a + b*x^2)^3/(c + d*x^2)^2,x]","\frac{(b c-a d)^2 (a d+5 b c) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{3/2} d^{7/2}}-\frac{x \left(-3 a^3 d^3+9 a^2 b c d^2-27 a b^2 c^2 d-18 a b^2 c d^2 x^2+15 b^3 c^3+10 b^3 c^2 d x^2-2 b^3 c d^2 x^4\right)}{6 c d^3 \left(c+d x^2\right)}","-\frac{b^2 x (2 b c-3 a d)}{d^3}+\frac{(a d+5 b c) (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{3/2} d^{7/2}}-\frac{x (b c-a d)^3}{2 c d^3 \left(c+d x^2\right)}+\frac{b^3 x^3}{3 d^2}",1,"-1/6*(x*(15*b^3*c^3 - 27*a*b^2*c^2*d + 9*a^2*b*c*d^2 - 3*a^3*d^3 + 10*b^3*c^2*d*x^2 - 18*a*b^2*c*d^2*x^2 - 2*b^3*c*d^2*x^4))/(c*d^3*(c + d*x^2)) + ((b*c - a*d)^2*(5*b*c + a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(3/2)*d^(7/2))","A",1
19,1,190,130,0.1286061,"\int \frac{\left(a+b x^2\right)^3}{\left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[(a + b*x^2)^3/(c + d*x^2)^3,x]","\frac{x \left(5 a^3 c d^3+3 a^3 d^4 x^2-3 a^2 b c^2 d^2+3 a^2 b c d^3 x^2-9 a b^2 c^3 d-15 a b^2 c^2 d^2 x^2+15 b^3 c^4+25 b^3 c^3 d x^2+8 b^3 c^2 d^2 x^4\right)}{8 c^2 d^3 \left(c+d x^2\right)^2}-\frac{3 \left(-a^3 d^3-a^2 b c d^2-3 a b^2 c^2 d+5 b^3 c^3\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{5/2} d^{7/2}}","-\frac{3 (b c-a d) \left((a d+b c)^2+4 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{5/2} d^{7/2}}+\frac{3 x (b c-a d)^2 (a d+3 b c)}{8 c^2 d^3 \left(c+d x^2\right)}-\frac{x (b c-a d)^3}{4 c d^3 \left(c+d x^2\right)^2}+\frac{b^3 x}{d^3}",1,"(x*(15*b^3*c^4 - 9*a*b^2*c^3*d - 3*a^2*b*c^2*d^2 + 5*a^3*c*d^3 + 25*b^3*c^3*d*x^2 - 15*a*b^2*c^2*d^2*x^2 + 3*a^2*b*c*d^3*x^2 + 3*a^3*d^4*x^2 + 8*b^3*c^2*d^2*x^4))/(8*c^2*d^3*(c + d*x^2)^2) - (3*(5*b^3*c^3 - 3*a*b^2*c^2*d - a^2*b*c*d^2 - a^3*d^3)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(5/2)*d^(7/2))","A",1
20,1,153,142,0.1267459,"\int \frac{\left(c+d x^2\right)^4}{a+b x^2} \, dx","IntegrateAlgebraic[(c + d*x^2)^4/(a + b*x^2),x]","\frac{d x \left(-105 a^3 d^3+420 a^2 b c d^2+35 a^2 b d^3 x^2-630 a b^2 c^2 d-140 a b^2 c d^2 x^2-21 a b^2 d^3 x^4+420 b^3 c^3+210 b^3 c^2 d x^2+84 b^3 c d^2 x^4+15 b^3 d^3 x^6\right)}{105 b^4}+\frac{(a d-b c)^4 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} b^{9/2}}","\frac{d x (2 b c-a d) \left(a^2 d^2-2 a b c d+2 b^2 c^2\right)}{b^4}+\frac{d^2 x^3 \left(a^2 d^2-4 a b c d+6 b^2 c^2\right)}{3 b^3}+\frac{(b c-a d)^4 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} b^{9/2}}+\frac{d^3 x^5 (4 b c-a d)}{5 b^2}+\frac{d^4 x^7}{7 b}",1,"(d*x*(420*b^3*c^3 - 630*a*b^2*c^2*d + 420*a^2*b*c*d^2 - 105*a^3*d^3 + 210*b^3*c^2*d*x^2 - 140*a*b^2*c*d^2*x^2 + 35*a^2*b*d^3*x^2 + 84*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) + ((-(b*c) + a*d)^4*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*b^(9/2))","A",1
21,1,99,98,0.0928383,"\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
22,1,59,63,0.0662639,"\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
23,1,39,39,0.0344358,"\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
24,1,71,70,0.0718353,"\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
25,1,112,109,0.2210461,"\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
26,1,159,160,0.3121611,"\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
27,1,273,192,0.1632354,"\int \frac{\left(c+d x^2\right)^5}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2)^5/(a + b*x^2)^2,x]","\frac{(a d-b c)^4 (9 a d+b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} b^{11/2}}-\frac{x \left(945 a^5 d^5-3675 a^4 b c d^4+630 a^4 b d^5 x^2+5250 a^3 b^2 c^2 d^3-2450 a^3 b^2 c d^4 x^2-126 a^3 b^2 d^5 x^4-3150 a^2 b^3 c^3 d^2+3500 a^2 b^3 c^2 d^3 x^2+490 a^2 b^3 c d^4 x^4+54 a^2 b^3 d^5 x^6+525 a b^4 c^4 d-2100 a b^4 c^3 d^2 x^2-700 a b^4 c^2 d^3 x^4-210 a b^4 c d^4 x^6-30 a b^4 d^5 x^8-105 b^5 c^5\right)}{210 a b^5 \left(a+b x^2\right)}","\frac{(b c-a d)^4 (9 a d+b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} b^{11/2}}+\frac{d^3 x^3 \left(3 a^2 d^2-10 a b c d+10 b^2 c^2\right)}{3 b^4}+\frac{d^2 x \left(-4 a^3 d^3+15 a^2 b c d^2-20 a b^2 c^2 d+10 b^3 c^3\right)}{b^5}+\frac{x (b c-a d)^5}{2 a b^5 \left(a+b x^2\right)}+\frac{d^4 x^5 (5 b c-2 a d)}{5 b^3}+\frac{d^5 x^7}{7 b^2}",1,"-1/210*(x*(-105*b^5*c^5 + 525*a*b^4*c^4*d - 3150*a^2*b^3*c^3*d^2 + 5250*a^3*b^2*c^2*d^3 - 3675*a^4*b*c*d^4 + 945*a^5*d^5 - 2100*a*b^4*c^3*d^2*x^2 + 3500*a^2*b^3*c^2*d^3*x^2 - 2450*a^3*b^2*c*d^4*x^2 + 630*a^4*b*d^5*x^2 - 700*a*b^4*c^2*d^3*x^4 + 490*a^2*b^3*c*d^4*x^4 - 126*a^3*b^2*d^5*x^4 - 210*a*b^4*c*d^4*x^6 + 54*a^2*b^3*d^5*x^6 - 30*a*b^4*d^5*x^8))/(a*b^5*(a + b*x^2)) + ((-(b*c) + a*d)^4*(b*c + 9*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*b^(11/2))","A",1
28,1,198,142,0.137531,"\int \frac{\left(c+d x^2\right)^4}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2)^4/(a + b*x^2)^2,x]","\frac{x \left(105 a^4 d^4-300 a^3 b c d^3+70 a^3 b d^4 x^2+270 a^2 b^2 c^2 d^2-200 a^2 b^2 c d^3 x^2-14 a^2 b^2 d^4 x^4-60 a b^3 c^3 d+180 a b^3 c^2 d^2 x^2+40 a b^3 c d^3 x^4+6 a b^3 d^4 x^6+15 b^4 c^4\right)}{30 a b^4 \left(a+b x^2\right)}-\frac{(a d-b c)^3 (7 a d+b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} b^{9/2}}","\frac{(b c-a d)^3 (7 a d+b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} b^{9/2}}+\frac{d^2 x \left(3 a^2 d^2-8 a b c d+6 b^2 c^2\right)}{b^4}+\frac{x (b c-a d)^4}{2 a b^4 \left(a+b x^2\right)}+\frac{2 d^3 x^3 (2 b c-a d)}{3 b^3}+\frac{d^4 x^5}{5 b^2}",1,"(x*(15*b^4*c^4 - 60*a*b^3*c^3*d + 270*a^2*b^2*c^2*d^2 - 300*a^3*b*c*d^3 + 105*a^4*d^4 + 180*a*b^3*c^2*d^2*x^2 - 200*a^2*b^2*c*d^3*x^2 + 70*a^3*b*d^4*x^2 + 40*a*b^3*c*d^3*x^4 - 14*a^2*b^2*d^4*x^4 + 6*a*b^3*d^4*x^6))/(30*a*b^4*(a + b*x^2)) - ((-(b*c) + a*d)^3*(b*c + 7*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*b^(9/2))","A",1
29,1,140,106,0.0961243,"\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
30,1,102,82,0.0875715,"\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
31,1,63,63,0.0648316,"\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
32,1,112,108,0.1963481,"\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
33,1,173,167,0.4048635,"\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
34,1,275,230,0.5123027,"\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
35,1,323,196,0.2040102,"\int \frac{\left(c+d x^2\right)^5}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(c + d*x^2)^5/(a + b*x^2)^3,x]","\frac{x \left(945 a^6 d^5-2625 a^5 b c d^4+1575 a^5 b d^5 x^2+2250 a^4 b^2 c^2 d^3-4375 a^4 b^2 c d^4 x^2+504 a^4 b^2 d^5 x^4-450 a^3 b^3 c^3 d^2+3750 a^3 b^3 c^2 d^3 x^2-1400 a^3 b^3 c d^4 x^4-72 a^3 b^3 d^5 x^6-75 a^2 b^4 c^4 d-750 a^2 b^4 c^3 d^2 x^2+1200 a^2 b^4 c^2 d^3 x^4+200 a^2 b^4 c d^4 x^6+24 a^2 b^4 d^5 x^8+75 a b^5 c^5+75 a b^5 c^4 d x^2+45 b^6 c^5 x^2\right)}{120 a^2 b^5 \left(a+b x^2\right)^2}-\frac{(a d-b c)^3 \left(63 a^2 d^2+14 a b c d+3 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} b^{11/2}}","\frac{x (17 a d+3 b c) (b c-a d)^4}{8 a^2 b^5 \left(a+b x^2\right)}+\frac{d^3 x \left(6 a^2 d^2-15 a b c d+10 b^2 c^2\right)}{b^5}+\frac{\left(63 a^2 d^2+14 a b c d+3 b^2 c^2\right) (b c-a d)^3 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} b^{11/2}}+\frac{x (b c-a d)^5}{4 a b^5 \left(a+b x^2\right)^2}+\frac{d^4 x^3 (5 b c-3 a d)}{3 b^4}+\frac{d^5 x^5}{5 b^3}",1,"(x*(75*a*b^5*c^5 - 75*a^2*b^4*c^4*d - 450*a^3*b^3*c^3*d^2 + 2250*a^4*b^2*c^2*d^3 - 2625*a^5*b*c*d^4 + 945*a^6*d^5 + 45*b^6*c^5*x^2 + 75*a*b^5*c^4*d*x^2 - 750*a^2*b^4*c^3*d^2*x^2 + 3750*a^3*b^3*c^2*d^3*x^2 - 4375*a^4*b^2*c*d^4*x^2 + 1575*a^5*b*d^5*x^2 + 1200*a^2*b^4*c^2*d^3*x^4 - 1400*a^3*b^3*c*d^4*x^4 + 504*a^4*b^2*d^5*x^4 + 200*a^2*b^4*c*d^4*x^6 - 72*a^3*b^3*d^5*x^6 + 24*a^2*b^4*d^5*x^8))/(120*a^2*b^5*(a + b*x^2)^2) - ((-(b*c) + a*d)^3*(3*b^2*c^2 + 14*a*b*c*d + 63*a^2*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*b^(11/2))","A",1
36,1,246,160,0.1547251,"\int \frac{\left(c+d x^2\right)^4}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(c + d*x^2)^4/(a + b*x^2)^3,x]","\frac{(a d-b c)^2 \left(35 a^2 d^2+10 a b c d+3 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} b^{9/2}}-\frac{x \left(105 a^5 d^4-180 a^4 b c d^3+175 a^4 b d^4 x^2+54 a^3 b^2 c^2 d^2-300 a^3 b^2 c d^3 x^2+56 a^3 b^2 d^4 x^4+12 a^2 b^3 c^3 d+90 a^2 b^3 c^2 d^2 x^2-96 a^2 b^3 c d^3 x^4-8 a^2 b^3 d^4 x^6-15 a b^4 c^4-12 a b^4 c^3 d x^2-9 b^5 c^4 x^2\right)}{24 a^2 b^4 \left(a+b x^2\right)^2}","\frac{x (b c-a d)^3 (13 a d+3 b c)}{8 a^2 b^4 \left(a+b x^2\right)}+\frac{(b c-a d)^2 \left(35 a^2 d^2+10 a b c d+3 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} b^{9/2}}+\frac{d^3 x (4 b c-3 a d)}{b^4}+\frac{x (b c-a d)^4}{4 a b^4 \left(a+b x^2\right)^2}+\frac{d^4 x^3}{3 b^3}",1,"-1/24*(x*(-15*a*b^4*c^4 + 12*a^2*b^3*c^3*d + 54*a^3*b^2*c^2*d^2 - 180*a^4*b*c*d^3 + 105*a^5*d^4 - 9*b^5*c^4*x^2 - 12*a*b^4*c^3*d*x^2 + 90*a^2*b^3*c^2*d^2*x^2 - 300*a^3*b^2*c*d^3*x^2 + 175*a^4*b*d^4*x^2 - 96*a^2*b^3*c*d^3*x^4 + 56*a^3*b^2*d^4*x^4 - 8*a^2*b^3*d^4*x^6))/(a^2*b^4*(a + b*x^2)^2) + ((-(b*c) + a*d)^2*(3*b^2*c^2 + 10*a*b*c*d + 35*a^2*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*b^(9/2))","A",1
37,1,190,130,0.1304802,"\int \frac{\left(c+d x^2\right)^3}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(c + d*x^2)^3/(a + b*x^2)^3,x]","\frac{x \left(15 a^4 d^3-9 a^3 b c d^2+25 a^3 b d^3 x^2-3 a^2 b^2 c^2 d-15 a^2 b^2 c d^2 x^2+8 a^2 b^2 d^3 x^4+5 a b^3 c^3+3 a b^3 c^2 d x^2+3 b^4 c^3 x^2\right)}{8 a^2 b^3 \left(a+b x^2\right)^2}-\frac{3 \left(5 a^3 d^3-3 a^2 b c d^2-a b^2 c^2 d-b^3 c^3\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} b^{7/2}}","\frac{3 x (b c-a d)^2 (3 a d+b c)}{8 a^2 b^3 \left(a+b x^2\right)}+\frac{3 (b c-a d) \left(4 a^2 d^2+(a d+b c)^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} b^{7/2}}+\frac{x (b c-a d)^3}{4 a b^3 \left(a+b x^2\right)^2}+\frac{d^3 x}{b^3}",1,"(x*(5*a*b^3*c^3 - 3*a^2*b^2*c^2*d - 9*a^3*b*c*d^2 + 15*a^4*d^3 + 3*b^4*c^3*x^2 + 3*a*b^3*c^2*d*x^2 - 15*a^2*b^2*c*d^2*x^2 + 25*a^3*b*d^3*x^2 + 8*a^2*b^2*d^3*x^4))/(8*a^2*b^3*(a + b*x^2)^2) - (3*(-(b^3*c^3) - a*b^2*c^2*d - 3*a^2*b*c*d^2 + 5*a^3*d^3)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*b^(7/2))","A",1
38,1,131,116,0.1283663,"\int \frac{\left(c+d x^2\right)^2}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(c + d*x^2)^2/(a + b*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{b} x}{\sqrt{a}}\right)}{8 a^{5/2} b^{5/2}}-\frac{x \left(3 a^3 d^2+2 a^2 b c d+5 a^2 b d^2 x^2-5 a b^2 c^2-2 a b^2 c d x^2-3 b^3 c^2 x^2\right)}{8 a^2 b^2 \left(a+b x^2\right)^2}","\frac{3 x \left(\frac{c^2}{a^2}-\frac{d^2}{b^2}\right)}{8 \left(a+b 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{b} x}{\sqrt{a}}\right)}{8 a^{5/2} b^{5/2}}+\frac{x \left(c+d x^2\right) (b c-a d)}{4 a b \left(a+b x^2\right)^2}",1,"-1/8*(x*(-5*a*b^2*c^2 + 2*a^2*b*c*d + 3*a^3*d^2 - 3*b^3*c^2*x^2 - 2*a*b^2*c*d*x^2 + 5*a^2*b*d^2*x^2))/(a^2*b^2*(a + b*x^2)^2) + ((3*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*b^(5/2))","A",1
39,1,84,92,0.086954,"\int \frac{c+d x^2}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(c + d*x^2)/(a + b*x^2)^3,x]","\frac{(a d+3 b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} b^{3/2}}-\frac{x \left(a^2 d-5 a b c-a b d x^2-3 b^2 c x^2\right)}{8 a^2 b \left(a+b x^2\right)^2}","\frac{(a d+3 b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} b^{3/2}}+\frac{x (a d+3 b c)}{8 a^2 b \left(a+b x^2\right)}+\frac{x (b c-a d)}{4 a b \left(a+b x^2\right)^2}",1,"-1/8*(x*(-5*a*b*c + a^2*d - 3*b^2*c*x^2 - a*b*d*x^2))/(a^2*b*(a + b*x^2)^2) + ((3*b*c + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*b^(3/2))","A",1
40,1,159,161,0.3425753,"\int \frac{1}{\left(a+b x^2\right)^3 \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^3*(c + d*x^2)),x]","-\frac{b x \left(9 a^2 d-5 a b c+7 a b d x^2-3 b^2 c x^2\right)}{8 a^2 \left(a+b x^2\right)^2 (a d-b c)^2}+\frac{\left(-15 a^2 \sqrt{b} d^2+10 a b^{3/2} c d-3 b^{5/2} c^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} (a d-b c)^3}-\frac{d^{5/2} \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{\sqrt{c} (b c-a d)^3}","\frac{b x (3 b c-7 a d)}{8 a^2 \left(a+b x^2\right) (b c-a d)^2}+\frac{\sqrt{b} \left(15 a^2 d^2-10 a b c d+3 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} (b c-a d)^3}-\frac{d^{5/2} \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{\sqrt{c} (b c-a d)^3}+\frac{b x}{4 a \left(a+b x^2\right)^2 (b c-a d)}",1,"-1/8*(b*x*(-5*a*b*c + 9*a^2*d - 3*b^2*c*x^2 + 7*a*b*d*x^2))/(a^2*(-(b*c) + a*d)^2*(a + b*x^2)^2) + ((-3*b^(5/2)*c^2 + 10*a*b^(3/2)*c*d - 15*a^2*Sqrt[b]*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*(-(b*c) + a*d)^3) - (d^(5/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*(b*c - a*d)^3)","A",1
41,1,275,236,0.5168041,"\int \frac{1}{\left(a+b x^2\right)^3 \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^3*(c + d*x^2)^2),x]","\frac{\left(35 a^2 b^{3/2} d^2-14 a b^{5/2} c d+3 b^{7/2} c^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} (a d-b c)^4}+\frac{x \left(4 a^4 d^3+8 a^3 b d^3 x^2+13 a^2 b^2 c^2 d+13 a^2 b^2 c d^2 x^2+4 a^2 b^2 d^3 x^4-5 a b^3 c^3+6 a b^3 c^2 d x^2+11 a b^3 c d^2 x^4-3 b^4 c^3 x^2-3 b^4 c^2 d x^4\right)}{8 a^2 c \left(a+b x^2\right)^2 \left(c+d x^2\right) (a d-b c)^3}+\frac{\left(a d^{7/2}-7 b c d^{5/2}\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{3/2} (b c-a d)^4}","\frac{d x (b c-4 a d) (a d+3 b c)}{8 a^2 c \left(c+d x^2\right) (b c-a d)^3}+\frac{3 b x (b c-3 a d)}{8 a^2 \left(a+b x^2\right) \left(c+d x^2\right) (b c-a d)^2}+\frac{b^{3/2} \left(35 a^2 d^2-14 a b c d+3 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} (b c-a d)^4}-\frac{d^{5/2} (7 b c-a d) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{3/2} (b c-a d)^4}+\frac{b x}{4 a \left(a+b x^2\right)^2 \left(c+d x^2\right) (b c-a d)}",1,"(x*(-5*a*b^3*c^3 + 13*a^2*b^2*c^2*d + 4*a^4*d^3 - 3*b^4*c^3*x^2 + 6*a*b^3*c^2*d*x^2 + 13*a^2*b^2*c*d^2*x^2 + 8*a^3*b*d^3*x^2 - 3*b^4*c^2*d*x^4 + 11*a*b^3*c*d^2*x^4 + 4*a^2*b^2*d^3*x^4))/(8*a^2*c*(-(b*c) + a*d)^3*(a + b*x^2)^2*(c + d*x^2)) + ((3*b^(7/2)*c^2 - 14*a*b^(5/2)*c*d + 35*a^2*b^(3/2)*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*(-(b*c) + a*d)^4) + ((-7*b*c*d^(5/2) + a*d^(7/2))*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(3/2)*(b*c - a*d)^4)","A",1
42,1,424,315,1.1103388,"\int \frac{1}{\left(a+b x^2\right)^3 \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^3*(c + d*x^2)^3),x]","-\frac{3 \left(a^2 d^{9/2}-6 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^{5/2} (b c-a d)^5}-\frac{3 \left(21 a^2 b^{5/2} d^2-6 a b^{7/2} c d+b^{9/2} c^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} (a d-b c)^5}+\frac{x \left(5 a^5 c d^4+3 a^5 d^5 x^2-17 a^4 b c^2 d^3-5 a^4 b c d^4 x^2+6 a^4 b d^5 x^4-34 a^3 b^2 c^2 d^3 x^2-25 a^3 b^2 c d^4 x^4+3 a^3 b^2 d^5 x^6-17 a^2 b^3 c^4 d-34 a^2 b^3 c^3 d^2 x^2-34 a^2 b^3 c^2 d^3 x^4-15 a^2 b^3 c d^4 x^6+5 a b^4 c^5-5 a b^4 c^4 d x^2-25 a b^4 c^3 d^2 x^4-15 a b^4 c^2 d^3 x^6+3 b^5 c^5 x^2+6 b^5 c^4 d x^4+3 b^5 c^3 d^2 x^6\right)}{8 a^2 c^2 \left(a+b x^2\right)^2 \left(c+d x^2\right)^2 (a d-b c)^4}","\frac{3 d x (a d+b c) \left(a^2 d^2-6 a b c d+b^2 c^2\right)}{8 a^2 c^2 \left(c+d x^2\right) (b c-a d)^4}+\frac{d x \left(-2 a^2 d^2-13 a b c d+3 b^2 c^2\right)}{8 a^2 c \left(c+d x^2\right)^2 (b c-a d)^3}-\frac{3 d^{5/2} \left(a^2 d^2-6 a b c d+21 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{5/2} (b c-a d)^5}+\frac{b x (3 b c-11 a d)}{8 a^2 \left(a+b x^2\right) \left(c+d x^2\right)^2 (b c-a d)^2}+\frac{3 b^{5/2} \left(21 a^2 d^2-6 a b c d+b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} (b c-a d)^5}+\frac{b x}{4 a \left(a+b x^2\right)^2 \left(c+d x^2\right)^2 (b c-a d)}",1,"(x*(5*a*b^4*c^5 - 17*a^2*b^3*c^4*d - 17*a^4*b*c^2*d^3 + 5*a^5*c*d^4 + 3*b^5*c^5*x^2 - 5*a*b^4*c^4*d*x^2 - 34*a^2*b^3*c^3*d^2*x^2 - 34*a^3*b^2*c^2*d^3*x^2 - 5*a^4*b*c*d^4*x^2 + 3*a^5*d^5*x^2 + 6*b^5*c^4*d*x^4 - 25*a*b^4*c^3*d^2*x^4 - 34*a^2*b^3*c^2*d^3*x^4 - 25*a^3*b^2*c*d^4*x^4 + 6*a^4*b*d^5*x^4 + 3*b^5*c^3*d^2*x^6 - 15*a*b^4*c^2*d^3*x^6 - 15*a^2*b^3*c*d^4*x^6 + 3*a^3*b^2*d^5*x^6))/(8*a^2*c^2*(-(b*c) + a*d)^4*(a + b*x^2)^2*(c + d*x^2)^2) - (3*(b^(9/2)*c^2 - 6*a*b^(7/2)*c*d + 21*a^2*b^(5/2)*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*(-(b*c) + a*d)^5) - (3*(21*b^2*c^2*d^(5/2) - 6*a*b*c*d^(7/2) + a^2*d^(9/2))*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(5/2)*(b*c - a*d)^5)","A",1
43,1,24,34,0.0123104,"\int \frac{\left(-1+x^2\right)^3}{\left(1+x^2\right)^4} \, dx","IntegrateAlgebraic[(-1 + x^2)^3/(1 + x^2)^4,x]","-\frac{x \left(3 x^4+2 x^2+3\right)}{3 \left(x^2+1\right)^3}","-\frac{x \left(1-x^2\right)^2}{3 \left(x^2+1\right)^3}-\frac{2 x}{3 \left(x^2+1\right)}",1,"-1/3*(x*(3 + 2*x^2 + 3*x^4))/(1 + x^2)^3","A",1
44,1,37,47,0.0218532,"\int \frac{\left(-1+x^2\right)^4}{\left(1+x^2\right)^5} \, dx","IntegrateAlgebraic[(-1 + x^2)^4/(1 + x^2)^5,x]","\frac{-5 x^7+3 x^5-3 x^3+5 x}{8 \left(x^2+1\right)^4}+\frac{3}{8} \tan ^{-1}(x)","\frac{x \left(1-x^2\right)^3}{4 \left(x^2+1\right)^4}+\frac{3 x \left(1-x^2\right)}{8 \left(x^2+1\right)^2}+\frac{3}{8} \tan ^{-1}(x)",1,"(5*x - 3*x^3 + 3*x^5 - 5*x^7)/(8*(1 + x^2)^4) + (3*ArcTan[x])/8","A",1
45,1,200,231,0.5021529,"\int \sqrt{a+b x^2} \left(c+d x^2\right)^3 \, dx","IntegrateAlgebraic[Sqrt[a + b*x^2]*(c + d*x^2)^3,x]","\frac{\sqrt{a+b x^2} \left(15 a^3 d^3 x-72 a^2 b c d^2 x-10 a^2 b d^3 x^3+144 a b^2 c^2 d x+48 a b^2 c d^2 x^3+8 a b^2 d^3 x^5+192 b^3 c^3 x+288 b^3 c^2 d x^3+192 b^3 c d^2 x^5+48 b^3 d^3 x^7\right)}{384 b^3}+\frac{\left(-5 a^4 d^3+24 a^3 b c d^2-48 a^2 b^2 c^2 d+64 a b^3 c^3\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{128 b^{7/2}}","\frac{d x \left(a+b x^2\right)^{3/2} \left(15 a^2 d^2-52 a b c d+72 b^2 c^2\right)}{192 b^3}+\frac{x \sqrt{a+b x^2} \left(-5 a^3 d^3+24 a^2 b c d^2-48 a b^2 c^2 d+64 b^3 c^3\right)}{128 b^3}+\frac{a \left(-5 a^3 d^3+24 a^2 b c d^2-48 a b^2 c^2 d+64 b^3 c^3\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{128 b^{7/2}}+\frac{d x \left(a+b x^2\right)^{3/2} \left(c+d x^2\right) (12 b c-5 a d)}{48 b^2}+\frac{d x \left(a+b x^2\right)^{3/2} \left(c+d x^2\right)^2}{8 b}",1,"(Sqrt[a + b*x^2]*(192*b^3*c^3*x + 144*a*b^2*c^2*d*x - 72*a^2*b*c*d^2*x + 15*a^3*d^3*x + 288*b^3*c^2*d*x^3 + 48*a*b^2*c*d^2*x^3 - 10*a^2*b*d^3*x^3 + 192*b^3*c*d^2*x^5 + 8*a*b^2*d^3*x^5 + 48*b^3*d^3*x^7))/(384*b^3) + ((64*a*b^3*c^3 - 48*a^2*b^2*c^2*d + 24*a^3*b*c*d^2 - 5*a^4*d^3)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(128*b^(7/2))","A",1
46,1,129,149,0.3359284,"\int \sqrt{a+b x^2} \left(c+d x^2\right)^2 \, dx","IntegrateAlgebraic[Sqrt[a + b*x^2]*(c + d*x^2)^2,x]","\frac{\sqrt{a+b x^2} \left(-3 a^2 d^2 x+12 a b c d x+2 a b d^2 x^3+24 b^2 c^2 x+24 b^2 c d x^3+8 b^2 d^2 x^5\right)}{48 b^2}+\frac{\left(a^3 d^2-4 a^2 b c d+8 a b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{16 b^{5/2}}","\frac{x \sqrt{a+b x^2} \left(a^2 d^2-4 a b c d+8 b^2 c^2\right)}{16 b^2}+\frac{a \left(a^2 d^2-4 a b c d+8 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{16 b^{5/2}}+\frac{d x \left(a+b x^2\right)^{3/2} (8 b c-3 a d)}{24 b^2}+\frac{d x \left(a+b x^2\right)^{3/2} \left(c+d x^2\right)}{6 b}",1,"(Sqrt[a + b*x^2]*(24*b^2*c^2*x + 12*a*b*c*d*x - 3*a^2*d^2*x + 24*b^2*c*d*x^3 + 2*a*b*d^2*x^3 + 8*b^2*d^2*x^5))/(48*b^2) + ((8*a*b^2*c^2 - 4*a^2*b*c*d + a^3*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*b^(5/2))","A",1
47,1,76,87,0.1899823,"\int \sqrt{a+b x^2} \left(c+d x^2\right) \, dx","IntegrateAlgebraic[Sqrt[a + b*x^2]*(c + d*x^2),x]","\frac{\left(4 a b c-a^2 d\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 d x+4 b c x+2 b d x^3\right)}{8 b}","\frac{a (4 b c-a d) \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 b c-a d)}{8 b}+\frac{d x \left(a+b x^2\right)^{3/2}}{4 b}",1,"(Sqrt[a + b*x^2]*(4*b*c*x + a*d*x + 2*b*d*x^3))/(8*b) + ((4*a*b*c - a^2*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*b^(3/2))","A",1
48,1,46,46,0.0005705,"\int \sqrt{a+b x^2} \, dx","IntegrateAlgebraic[Sqrt[a + b*x^2],x]","\frac{1}{2} x \sqrt{a+b x^2}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 \sqrt{b}}","\frac{1}{2} x \sqrt{a+b x^2}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 \sqrt{b}}",1,"(x*Sqrt[a + b*x^2])/2 + (a*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*Sqrt[b])","A",1
49,1,81,82,0.276969,"\int \frac{\sqrt{a+b x^2}}{c+d x^2} \, dx","IntegrateAlgebraic[Sqrt[a + b*x^2]/(c + d*x^2),x]","\frac{\sqrt{a d-b c} \tan ^{-1}\left(\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{\sqrt{c} d}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{d}","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{d}-\frac{\sqrt{b c-a d} \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{\sqrt{c} d}",1,"(Sqrt[-(b*c) + a*d]*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(Sqrt[c]*d) + (Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/d","A",1
50,1,82,82,0.2551964,"\int \frac{\sqrt{a+b x^2}}{\left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[Sqrt[a + b*x^2]/(c + d*x^2)^2,x]","\frac{a \tan ^{-1}\left(\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{2 c^{3/2} \sqrt{a d-b c}}+\frac{x \sqrt{a+b x^2}}{2 c \left(c+d x^2\right)}","\frac{a \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{2 c^{3/2} \sqrt{b c-a d}}+\frac{x \sqrt{a+b x^2}}{2 c \left(c+d x^2\right)}",1,"(x*Sqrt[a + b*x^2])/(2*c*(c + d*x^2)) + (a*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(2*c^(3/2)*Sqrt[-(b*c) + a*d])","A",1
51,1,133,149,0.4845079,"\int \frac{\sqrt{a+b x^2}}{\left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[Sqrt[a + b*x^2]/(c + d*x^2)^3,x]","\frac{\left(3 a^2 d-4 a b c\right) \tan ^{-1}\left(\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{8 c^{5/2} (a d-b c)^{3/2}}+\frac{\sqrt{a+b x^2} \left(-5 a c d x-3 a d^2 x^3+4 b c^2 x+2 b c d x^3\right)}{8 c^2 \left(c+d x^2\right)^2 (b c-a d)}","\frac{a (4 b c-3 a d) \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{8 c^{5/2} (b c-a d)^{3/2}}+\frac{x \sqrt{a+b x^2} (4 b c-3 a d)}{8 c^2 \left(c+d x^2\right) (b c-a d)}-\frac{d x \left(a+b x^2\right)^{3/2}}{4 c \left(c+d x^2\right)^2 (b c-a d)}",1,"(Sqrt[a + b*x^2]*(4*b*c^2*x - 5*a*c*d*x + 2*b*c*d*x^3 - 3*a*d^2*x^3))/(8*c^2*(b*c - a*d)*(c + d*x^2)^2) + ((-4*a*b*c + 3*a^2*d)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(8*c^(5/2)*(-(b*c) + a*d)^(3/2))","A",1
52,1,220,208,0.8021284,"\int \frac{\sqrt{a+b x^2}}{\left(c+d x^2\right)^4} \, dx","IntegrateAlgebraic[Sqrt[a + b*x^2]/(c + d*x^2)^4,x]","\frac{\sqrt{a+b x^2} \left(33 a^2 c^2 d^2 x+40 a^2 c d^3 x^3+15 a^2 d^4 x^5-60 a b c^3 d x-70 a b c^2 d^2 x^3-26 a b c d^3 x^5+24 b^2 c^4 x+24 b^2 c^3 d x^3+8 b^2 c^2 d^2 x^5\right)}{48 c^3 \left(c+d x^2\right)^3 (b c-a d)^2}+\frac{\left(5 a^3 d^2-12 a^2 b c d+8 a b^2 c^2\right) \tan ^{-1}\left(\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{16 c^{7/2} (a d-b c)^{5/2}}","\frac{a \left(5 a^2 d^2-12 a b c d+8 b^2 c^2\right) \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{16 c^{7/2} (b c-a d)^{5/2}}+\frac{x \sqrt{a+b x^2} (2 b c-5 a d) (4 b c-3 a d)}{48 c^3 \left(c+d x^2\right) (b c-a d)^2}+\frac{x \sqrt{a+b x^2} (4 b c-5 a d)}{24 c^2 \left(c+d x^2\right)^2 (b c-a d)}+\frac{x \sqrt{a+b x^2}}{6 c \left(c+d x^2\right)^3}",1,"(Sqrt[a + b*x^2]*(24*b^2*c^4*x - 60*a*b*c^3*d*x + 33*a^2*c^2*d^2*x + 24*b^2*c^3*d*x^3 - 70*a*b*c^2*d^2*x^3 + 40*a^2*c*d^3*x^3 + 8*b^2*c^2*d^2*x^5 - 26*a*b*c*d^3*x^5 + 15*a^2*d^4*x^5))/(48*c^3*(b*c - a*d)^2*(c + d*x^2)^3) + ((8*a*b^2*c^2 - 12*a^2*b*c*d + 5*a^3*d^2)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(16*c^(7/2)*(-(b*c) + a*d)^(5/2))","A",1
53,1,257,272,0.6568502,"\int \left(a+b x^2\right)^{3/2} \left(c+d x^2\right)^3 \, dx","IntegrateAlgebraic[(a + b*x^2)^(3/2)*(c + d*x^2)^3,x]","\frac{\sqrt{a+b x^2} \left(15 a^4 d^3 x-90 a^3 b c d^2 x-10 a^3 b d^3 x^3+240 a^2 b^2 c^2 d x+60 a^2 b^2 c d^2 x^3+8 a^2 b^2 d^3 x^5+800 a b^3 c^3 x+1120 a b^3 c^2 d x^3+720 a b^3 c d^2 x^5+176 a b^3 d^3 x^7+320 b^4 c^3 x^3+640 b^4 c^2 d x^5+480 b^4 c d^2 x^7+128 b^4 d^3 x^9\right)}{1280 b^3}-\frac{3 \left(a^5 d^3-6 a^4 b c d^2+16 a^3 b^2 c^2 d-32 a^2 b^3 c^3\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{256 b^{7/2}}","\frac{3 a^2 (4 b c-a d) \left(a^2 d^2-2 a b c d+8 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{256 b^{7/2}}+\frac{d x \left(a+b x^2\right)^{5/2} \left(5 a^2 d^2-20 a b c d+36 b^2 c^2\right)}{160 b^3}+\frac{x \left(a+b x^2\right)^{3/2} (4 b c-a d) \left(a^2 d^2-2 a b c d+8 b^2 c^2\right)}{128 b^3}+\frac{3 a x \sqrt{a+b x^2} (4 b c-a d) \left(a^2 d^2-2 a b c d+8 b^2 c^2\right)}{256 b^3}+\frac{d x \left(a+b x^2\right)^{5/2} \left(c+d x^2\right) (14 b c-5 a d)}{80 b^2}+\frac{d x \left(a+b x^2\right)^{5/2} \left(c+d x^2\right)^2}{10 b}",1,"(Sqrt[a + b*x^2]*(800*a*b^3*c^3*x + 240*a^2*b^2*c^2*d*x - 90*a^3*b*c*d^2*x + 15*a^4*d^3*x + 320*b^4*c^3*x^3 + 1120*a*b^3*c^2*d*x^3 + 60*a^2*b^2*c*d^2*x^3 - 10*a^3*b*d^3*x^3 + 640*b^4*c^2*d*x^5 + 720*a*b^3*c*d^2*x^5 + 8*a^2*b^2*d^3*x^5 + 480*b^4*c*d^2*x^7 + 176*a*b^3*d^3*x^7 + 128*b^4*d^3*x^9))/(1280*b^3) - (3*(-32*a^2*b^3*c^3 + 16*a^3*b^2*c^2*d - 6*a^4*b*c*d^2 + a^5*d^3)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(256*b^(7/2))","A",1
54,1,171,196,0.4222229,"\int \left(a+b x^2\right)^{3/2} \left(c+d x^2\right)^2 \, dx","IntegrateAlgebraic[(a + b*x^2)^(3/2)*(c + d*x^2)^2,x]","\frac{\sqrt{a+b x^2} \left(-9 a^3 d^2 x+48 a^2 b c d x+6 a^2 b d^2 x^3+240 a b^2 c^2 x+224 a b^2 c d x^3+72 a b^2 d^2 x^5+96 b^3 c^2 x^3+128 b^3 c d x^5+48 b^3 d^2 x^7\right)}{384 b^2}+\frac{\left(3 a^4 d^2-16 a^3 b c d+48 a^2 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{128 b^{5/2}}","\frac{x \left(a+b x^2\right)^{3/2} \left(3 a^2 d^2-16 a b c d+48 b^2 c^2\right)}{192 b^2}+\frac{a x \sqrt{a+b x^2} \left(3 a^2 d^2-16 a b c d+48 b^2 c^2\right)}{128 b^2}+\frac{a^2 \left(3 a^2 d^2-16 a b c d+48 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{128 b^{5/2}}+\frac{d x \left(a+b x^2\right)^{5/2} (10 b c-3 a d)}{48 b^2}+\frac{d x \left(a+b x^2\right)^{5/2} \left(c+d x^2\right)}{8 b}",1,"(Sqrt[a + b*x^2]*(240*a*b^2*c^2*x + 48*a^2*b*c*d*x - 9*a^3*d^2*x + 96*b^3*c^2*x^3 + 224*a*b^2*c*d*x^3 + 6*a^2*b*d^2*x^3 + 128*b^3*c*d*x^5 + 72*a*b^2*d^2*x^5 + 48*b^3*d^2*x^7))/(384*b^2) + ((48*a^2*b^2*c^2 - 16*a^3*b*c*d + 3*a^4*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(128*b^(5/2))","A",1
55,1,101,118,0.2680709,"\int \left(a+b x^2\right)^{3/2} \left(c+d x^2\right) \, dx","IntegrateAlgebraic[(a + b*x^2)^(3/2)*(c + d*x^2),x]","\frac{\sqrt{a+b x^2} \left(3 a^2 d x+30 a b c x+14 a b d x^3+12 b^2 c x^3+8 b^2 d x^5\right)}{48 b}+\frac{\left(6 a^2 b c-a^3 d\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{16 b^{3/2}}","\frac{a^2 (6 b c-a d) \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 b c-a d)}{24 b}+\frac{a x \sqrt{a+b x^2} (6 b c-a d)}{16 b}+\frac{d x \left(a+b x^2\right)^{5/2}}{6 b}",1,"(Sqrt[a + b*x^2]*(30*a*b*c*x + 3*a^2*d*x + 12*b^2*c*x^3 + 14*a*b*d*x^3 + 8*b^2*d*x^5))/(48*b) + ((6*a^2*b*c - a^3*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*b^(3/2))","A",1
56,1,58,65,0.0006065,"\int \left(a+b x^2\right)^{3/2} \, dx","IntegrateAlgebraic[(a + b*x^2)^(3/2),x]","\frac{3 a^2 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 \sqrt{b}}+\frac{1}{8} \sqrt{a+b x^2} \left(5 a x+2 b x^3\right)","\frac{3 a^2 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 \sqrt{b}}+\frac{3}{8} a x \sqrt{a+b x^2}+\frac{1}{4} x \left(a+b x^2\right)^{3/2}",1,"(Sqrt[a + b*x^2]*(5*a*x + 2*b*x^3))/8 + (3*a^2*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*Sqrt[b])","A",1
57,1,117,113,0.385242,"\int \frac{\left(a+b x^2\right)^{3/2}}{c+d x^2} \, dx","IntegrateAlgebraic[(a + b*x^2)^(3/2)/(c + d*x^2),x]","\frac{\left(3 a \sqrt{b} d-2 b^{3/2} c\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 d^2}+\frac{(a d-b c)^{3/2} \tan ^{-1}\left(\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{\sqrt{c} d^2}+\frac{b x \sqrt{a+b x^2}}{2 d}","\frac{(b c-a d)^{3/2} \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{\sqrt{c} d^2}-\frac{\sqrt{b} (2 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 d^2}+\frac{b x \sqrt{a+b x^2}}{2 d}",1,"(b*x*Sqrt[a + b*x^2])/(2*d) + ((-(b*c) + a*d)^(3/2)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(Sqrt[c]*d^2) + ((-2*b^(3/2)*c + 3*a*Sqrt[b]*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*d^2)","A",1
58,1,144,131,0.4875707,"\int \frac{\left(a+b x^2\right)^{3/2}}{\left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[(a + b*x^2)^(3/2)/(c + d*x^2)^2,x]","\frac{\left(a^2 d^2+a b c d-2 b^2 c^2\right) \tan ^{-1}\left(\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{2 c^{3/2} d^2 \sqrt{a d-b c}}+\frac{b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{d^2}+\frac{x \sqrt{a+b x^2} (a d-b c)}{2 c d \left(c+d x^2\right)}","\frac{b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{d^2}-\frac{\sqrt{b c-a d} (a d+2 b c) \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{2 c^{3/2} d^2}-\frac{x \sqrt{a+b x^2} (b c-a d)}{2 c d \left(c+d x^2\right)}",1,"((-(b*c) + a*d)*x*Sqrt[a + b*x^2])/(2*c*d*(c + d*x^2)) + ((-2*b^2*c^2 + a*b*c*d + a^2*d^2)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(2*c^(3/2)*d^2*Sqrt[-(b*c) + a*d]) + (b^(3/2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/d^2","A",1
59,1,103,113,0.3169816,"\int \frac{\left(a+b x^2\right)^{3/2}}{\left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[(a + b*x^2)^(3/2)/(c + d*x^2)^3,x]","\frac{3 a^2 \tan ^{-1}\left(\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{8 c^{5/2} \sqrt{a d-b c}}+\frac{\sqrt{a+b x^2} \left(5 a c x+3 a d x^3+2 b c x^3\right)}{8 c^2 \left(c+d x^2\right)^2}","\frac{3 a^2 \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{8 c^{5/2} \sqrt{b c-a d}}+\frac{3 a x \sqrt{a+b x^2}}{8 c^2 \left(c+d x^2\right)}+\frac{x \left(a+b x^2\right)^{3/2}}{4 c \left(c+d x^2\right)^2}",1,"(Sqrt[a + b*x^2]*(5*a*c*x + 2*b*c*x^3 + 3*a*d*x^3))/(8*c^2*(c + d*x^2)^2) + (3*a^2*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(8*c^(5/2)*Sqrt[-(b*c) + a*d])","A",1
60,1,191,199,0.6939755,"\int \frac{\left(a+b x^2\right)^{3/2}}{\left(c+d x^2\right)^4} \, dx","IntegrateAlgebraic[(a + b*x^2)^(3/2)/(c + d*x^2)^4,x]","\frac{\left(5 a^3 d-6 a^2 b c\right) \tan ^{-1}\left(\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{16 c^{7/2} (a d-b c)^{3/2}}-\frac{\sqrt{a+b x^2} \left(33 a^2 c^2 d x+40 a^2 c d^2 x^3+15 a^2 d^3 x^5-30 a b c^3 x-22 a b c^2 d x^3-8 a b c d^2 x^5-12 b^2 c^3 x^3-4 b^2 c^2 d x^5\right)}{48 c^3 \left(c+d x^2\right)^3 (b c-a d)}","\frac{a^2 (6 b c-5 a d) \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{16 c^{7/2} (b c-a d)^{3/2}}+\frac{a x \sqrt{a+b x^2} (6 b c-5 a d)}{16 c^3 \left(c+d x^2\right) (b c-a d)}+\frac{x \left(a+b x^2\right)^{3/2} (6 b c-5 a d)}{24 c^2 \left(c+d x^2\right)^2 (b c-a d)}-\frac{d x \left(a+b x^2\right)^{5/2}}{6 c \left(c+d x^2\right)^3 (b c-a d)}",1,"-1/48*(Sqrt[a + b*x^2]*(-30*a*b*c^3*x + 33*a^2*c^2*d*x - 12*b^2*c^3*x^3 - 22*a*b*c^2*d*x^3 + 40*a^2*c*d^2*x^3 - 4*b^2*c^2*d*x^5 - 8*a*b*c*d^2*x^5 + 15*a^2*d^3*x^5))/(c^3*(b*c - a*d)*(c + d*x^2)^3) + ((-6*a^2*b*c + 5*a^3*d)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(16*c^(7/2)*(-(b*c) + a*d)^(3/2))","A",1
61,1,312,300,1.1028255,"\int \frac{\left(a+b x^2\right)^{3/2}}{\left(c+d x^2\right)^5} \, dx","IntegrateAlgebraic[(a + b*x^2)^(3/2)/(c + d*x^2)^5,x]","\frac{\sqrt{a+b x^2} \left(279 a^3 c^3 d^2 x+511 a^3 c^2 d^3 x^3+385 a^3 c d^4 x^5+105 a^3 d^5 x^7-528 a^2 b c^4 d x-842 a^2 b c^3 d^2 x^3-628 a^2 b c^2 d^3 x^5-170 a^2 b c d^4 x^7+240 a b^2 c^5 x+208 a b^2 c^4 d x^3+152 a b^2 c^3 d^2 x^5+40 a b^2 c^2 d^3 x^7+96 b^3 c^5 x^3+64 b^3 c^4 d x^5+16 b^3 c^3 d^2 x^7\right)}{384 c^4 \left(c+d x^2\right)^4 (b c-a d)^2}+\frac{\left(35 a^4 d^2-80 a^3 b c d+48 a^2 b^2 c^2\right) \tan ^{-1}\left(\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{128 c^{9/2} (a d-b c)^{5/2}}","\frac{a^2 \left(35 a^2 d^2-80 a b c d+48 b^2 c^2\right) \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{128 c^{9/2} (b c-a d)^{5/2}}+\frac{x \sqrt{a+b x^2} \left(-35 a^2 d^2+24 a b c d+8 b^2 c^2\right)}{192 c^3 d \left(c+d x^2\right)^2 (b c-a d)}+\frac{x \sqrt{a+b x^2} \left(105 a^3 d^3-170 a^2 b c d^2+40 a b^2 c^2 d+16 b^3 c^3\right)}{384 c^4 d \left(c+d x^2\right) (b c-a d)^2}+\frac{x \sqrt{a+b x^2} (7 a d+2 b c)}{48 c^2 d \left(c+d x^2\right)^3}-\frac{x \sqrt{a+b x^2} (b c-a d)}{8 c d \left(c+d x^2\right)^4}",1,"(Sqrt[a + b*x^2]*(240*a*b^2*c^5*x - 528*a^2*b*c^4*d*x + 279*a^3*c^3*d^2*x + 96*b^3*c^5*x^3 + 208*a*b^2*c^4*d*x^3 - 842*a^2*b*c^3*d^2*x^3 + 511*a^3*c^2*d^3*x^3 + 64*b^3*c^4*d*x^5 + 152*a*b^2*c^3*d^2*x^5 - 628*a^2*b*c^2*d^3*x^5 + 385*a^3*c*d^4*x^5 + 16*b^3*c^3*d^2*x^7 + 40*a*b^2*c^2*d^3*x^7 - 170*a^2*b*c*d^4*x^7 + 105*a^3*d^5*x^7))/(384*c^4*(b*c - a*d)^2*(c + d*x^2)^4) + ((48*a^2*b^2*c^2 - 80*a^3*b*c*d + 35*a^4*d^2)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(128*c^(9/2)*(-(b*c) + a*d)^(5/2))","A",1
62,1,316,349,0.8284191,"\int \left(a+b x^2\right)^{5/2} \left(c+d x^2\right)^3 \, dx","IntegrateAlgebraic[(a + b*x^2)^(5/2)*(c + d*x^2)^3,x]","\frac{\left(-5 a^6 d^3+36 a^5 b c d^2-120 a^4 b^2 c^2 d+320 a^3 b^3 c^3\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 d^3 x-540 a^4 b c d^2 x-50 a^4 b d^3 x^3+1800 a^3 b^2 c^2 d x+360 a^3 b^2 c d^2 x^3+40 a^3 b^2 d^3 x^5+10560 a^2 b^3 c^3 x+14160 a^2 b^3 c^2 d x^3+8928 a^2 b^3 c d^2 x^5+2160 a^2 b^3 d^3 x^7+8320 a b^4 c^3 x^3+16320 a b^4 c^2 d x^5+12096 a b^4 c d^2 x^7+3200 a b^4 d^3 x^9+2560 b^5 c^3 x^5+5760 b^5 c^2 d x^7+4608 b^5 c d^2 x^9+1280 b^5 d^3 x^{11}\right)}{15360 b^3}","\frac{d x \left(a+b x^2\right)^{7/2} \left(15 a^2 d^2-68 a b c d+152 b^2 c^2\right)}{960 b^3}+\frac{x \left(a+b x^2\right)^{5/2} \left(-5 a^3 d^3+36 a^2 b c d^2-120 a b^2 c^2 d+320 b^3 c^3\right)}{1920 b^3}+\frac{a x \left(a+b x^2\right)^{3/2} \left(-5 a^3 d^3+36 a^2 b c d^2-120 a b^2 c^2 d+320 b^3 c^3\right)}{1536 b^3}+\frac{a^2 x \sqrt{a+b x^2} \left(-5 a^3 d^3+36 a^2 b c d^2-120 a b^2 c^2 d+320 b^3 c^3\right)}{1024 b^3}+\frac{a^3 \left(-5 a^3 d^3+36 a^2 b c d^2-120 a b^2 c^2 d+320 b^3 c^3\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{1024 b^{7/2}}+\frac{d x \left(a+b x^2\right)^{7/2} \left(c+d x^2\right) (16 b c-5 a d)}{120 b^2}+\frac{d x \left(a+b x^2\right)^{7/2} \left(c+d x^2\right)^2}{12 b}",1,"(Sqrt[a + b*x^2]*(10560*a^2*b^3*c^3*x + 1800*a^3*b^2*c^2*d*x - 540*a^4*b*c*d^2*x + 75*a^5*d^3*x + 8320*a*b^4*c^3*x^3 + 14160*a^2*b^3*c^2*d*x^3 + 360*a^3*b^2*c*d^2*x^3 - 50*a^4*b*d^3*x^3 + 2560*b^5*c^3*x^5 + 16320*a*b^4*c^2*d*x^5 + 8928*a^2*b^3*c*d^2*x^5 + 40*a^3*b^2*d^3*x^5 + 5760*b^5*c^2*d*x^7 + 12096*a*b^4*c*d^2*x^7 + 2160*a^2*b^3*d^3*x^7 + 4608*b^5*c*d^2*x^9 + 3200*a*b^4*d^3*x^9 + 1280*b^5*d^3*x^11))/(15360*b^3) + ((320*a^3*b^3*c^3 - 120*a^4*b^2*c^2*d + 36*a^5*b*c*d^2 - 5*a^6*d^3)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(1024*b^(7/2))","A",1
63,1,212,241,0.5579228,"\int \left(a+b x^2\right)^{5/2} \left(c+d x^2\right)^2 \, dx","IntegrateAlgebraic[(a + b*x^2)^(5/2)*(c + d*x^2)^2,x]","\frac{\left(3 a^5 d^2-20 a^4 b c d+80 a^3 b^2 c^2\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 d^2 x+300 a^3 b c d x+30 a^3 b d^2 x^3+2640 a^2 b^2 c^2 x+2360 a^2 b^2 c d x^3+744 a^2 b^2 d^2 x^5+2080 a b^3 c^2 x^3+2720 a b^3 c d x^5+1008 a b^3 d^2 x^7+640 b^4 c^2 x^5+960 b^4 c d x^7+384 b^4 d^2 x^9\right)}{3840 b^2}","\frac{x \left(a+b x^2\right)^{5/2} \left(3 a^2 d^2-20 a b c d+80 b^2 c^2\right)}{480 b^2}+\frac{a x \left(a+b x^2\right)^{3/2} \left(3 a^2 d^2-20 a b c d+80 b^2 c^2\right)}{384 b^2}+\frac{a^2 x \sqrt{a+b x^2} \left(3 a^2 d^2-20 a b c d+80 b^2 c^2\right)}{256 b^2}+\frac{a^3 \left(3 a^2 d^2-20 a b c d+80 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{256 b^{5/2}}+\frac{3 d x \left(a+b x^2\right)^{7/2} (4 b c-a d)}{80 b^2}+\frac{d x \left(a+b x^2\right)^{7/2} \left(c+d x^2\right)}{10 b}",1,"(Sqrt[a + b*x^2]*(2640*a^2*b^2*c^2*x + 300*a^3*b*c*d*x - 45*a^4*d^2*x + 2080*a*b^3*c^2*x^3 + 2360*a^2*b^2*c*d*x^3 + 30*a^3*b*d^2*x^3 + 640*b^4*c^2*x^5 + 2720*a*b^3*c*d*x^5 + 744*a^2*b^2*d^2*x^5 + 960*b^4*c*d*x^7 + 1008*a*b^3*d^2*x^7 + 384*b^4*d^2*x^9))/(3840*b^2) + ((80*a^3*b^2*c^2 - 20*a^4*b*c*d + 3*a^5*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(256*b^(5/2))","A",1
64,1,124,149,0.3497426,"\int \left(a+b x^2\right)^{5/2} \left(c+d x^2\right) \, dx","IntegrateAlgebraic[(a + b*x^2)^(5/2)*(c + d*x^2),x]","\frac{\sqrt{a+b x^2} \left(15 a^3 d x+264 a^2 b c x+118 a^2 b d x^3+208 a b^2 c x^3+136 a b^2 d x^5+64 b^3 c x^5+48 b^3 d x^7\right)}{384 b}-\frac{5 \left(a^4 d-8 a^3 b c\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{128 b^{3/2}}","\frac{5 a^3 (8 b c-a d) \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 b c-a d)}{128 b}+\frac{x \left(a+b x^2\right)^{5/2} (8 b c-a d)}{48 b}+\frac{5 a x \left(a+b x^2\right)^{3/2} (8 b c-a d)}{192 b}+\frac{d x \left(a+b x^2\right)^{7/2}}{8 b}",1,"(Sqrt[a + b*x^2]*(264*a^2*b*c*x + 15*a^3*d*x + 208*a*b^2*c*x^3 + 118*a^2*b*d*x^3 + 64*b^3*c*x^5 + 136*a*b^2*d*x^5 + 48*b^3*d*x^7))/(384*b) - (5*(-8*a^3*b*c + a^4*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(128*b^(3/2))","A",1
65,1,69,84,0.0005788,"\int \left(a+b x^2\right)^{5/2} \, dx","IntegrateAlgebraic[(a + b*x^2)^(5/2),x]","\frac{5 a^3 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{16 \sqrt{b}}+\frac{1}{48} \sqrt{a+b x^2} \left(33 a^2 x+26 a b x^3+8 b^2 x^5\right)","\frac{5 a^3 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{16 \sqrt{b}}+\frac{5}{16} a^2 x \sqrt{a+b x^2}+\frac{5}{24} a x \left(a+b x^2\right)^{3/2}+\frac{1}{6} x \left(a+b x^2\right)^{5/2}",1,"(Sqrt[a + b*x^2]*(33*a^2*x + 26*a*b*x^3 + 8*b^2*x^5))/48 + (5*a^3*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*Sqrt[b])","A",1
66,1,154,157,0.5543552,"\int \frac{\left(a+b x^2\right)^{5/2}}{c+d x^2} \, dx","IntegrateAlgebraic[(a + b*x^2)^(5/2)/(c + d*x^2),x]","\frac{\left(15 a^2 \sqrt{b} d^2-20 a b^{3/2} c d+8 b^{5/2} c^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 d^3}+\frac{\sqrt{a+b x^2} \left(9 a b d x-4 b^2 c x+2 b^2 d x^3\right)}{8 d^2}+\frac{(a d-b c)^{5/2} \tan ^{-1}\left(\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{\sqrt{c} d^3}","\frac{\sqrt{b} \left(15 a^2 d^2-20 a b c d+8 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 d^3}-\frac{(b c-a d)^{5/2} \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{\sqrt{c} d^3}-\frac{b x \sqrt{a+b x^2} (4 b c-7 a d)}{8 d^2}+\frac{b x \left(a+b x^2\right)^{3/2}}{4 d}",1,"(Sqrt[a + b*x^2]*(-4*b^2*c*x + 9*a*b*d*x + 2*b^2*d*x^3))/(8*d^2) + ((-(b*c) + a*d)^(5/2)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(Sqrt[c]*d^3) + ((8*b^(5/2)*c^2 - 20*a*b^(3/2)*c*d + 15*a^2*Sqrt[b]*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*d^3)","A",1
67,1,172,175,0.5614206,"\int \frac{\left(a+b x^2\right)^{5/2}}{\left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[(a + b*x^2)^(5/2)/(c + d*x^2)^2,x]","\frac{\sqrt{a+b x^2} \left(a^2 d^2 x-2 a b c d x+2 b^2 c^2 x+b^2 c d x^3\right)}{2 c d^2 \left(c+d x^2\right)}+\frac{\left(5 a b^{3/2} d-4 b^{5/2} c\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 d^3}+\frac{(a d+4 b c) (a d-b c)^{3/2} \tan ^{-1}\left(\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{2 c^{3/2} d^3}","-\frac{b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 d^3}+\frac{(b c-a d)^{3/2} (a d+4 b c) \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{2 c^{3/2} d^3}+\frac{b x \sqrt{a+b x^2} (2 b c-a d)}{2 c d^2}-\frac{x \left(a+b x^2\right)^{3/2} (b c-a d)}{2 c d \left(c+d x^2\right)}",1,"(Sqrt[a + b*x^2]*(2*b^2*c^2*x - 2*a*b*c*d*x + a^2*d^2*x + b^2*c*d*x^3))/(2*c*d^2*(c + d*x^2)) + ((-(b*c) + a*d)^(3/2)*(4*b*c + a*d)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(2*c^(3/2)*d^3) + ((-4*b^(5/2)*c + 5*a*b^(3/2)*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*d^3)","A",1
68,1,213,194,0.5396674,"\int \frac{\left(a+b x^2\right)^{5/2}}{\left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[(a + b*x^2)^(5/2)/(c + d*x^2)^3,x]","\frac{\sqrt{a+b x^2} \left(5 a^2 c d^2 x+3 a^2 d^3 x^3-a b c^2 d x+3 a b c d^2 x^3-4 b^2 c^3 x-6 b^2 c^2 d x^3\right)}{8 c^2 d^2 \left(c+d x^2\right)^2}+\frac{\left(3 a^3 d^3+a^2 b c d^2+4 a b^2 c^2 d-8 b^3 c^3\right) \tan ^{-1}\left(\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{8 c^{5/2} d^3 \sqrt{a d-b c}}+\frac{b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{d^3}","-\frac{\sqrt{b c-a d} \left(3 a^2 d^2+4 a b c d+8 b^2 c^2\right) \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{8 c^{5/2} d^3}+\frac{b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{d^3}-\frac{x \sqrt{a+b x^2} (b c-a d) (3 a d+4 b c)}{8 c^2 d^2 \left(c+d x^2\right)}-\frac{x \left(a+b x^2\right)^{3/2} (b c-a d)}{4 c d \left(c+d x^2\right)^2}",1,"(Sqrt[a + b*x^2]*(-4*b^2*c^3*x - a*b*c^2*d*x + 5*a^2*c*d^2*x - 6*b^2*c^2*d*x^3 + 3*a*b*c*d^2*x^3 + 3*a^2*d^3*x^3))/(8*c^2*d^2*(c + d*x^2)^2) + ((-8*b^3*c^3 + 4*a*b^2*c^2*d + a^2*b*c*d^2 + 3*a^3*d^3)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(8*c^(5/2)*d^3*Sqrt[-(b*c) + a*d]) + (b^(5/2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/d^3","A",1
69,1,144,144,0.4993787,"\int \frac{\left(a+b x^2\right)^{5/2}}{\left(c+d x^2\right)^4} \, dx","IntegrateAlgebraic[(a + b*x^2)^(5/2)/(c + d*x^2)^4,x]","\frac{5 a^3 \tan ^{-1}\left(\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{16 c^{7/2} \sqrt{a d-b c}}+\frac{\sqrt{a+b x^2} \left(33 a^2 c^2 x+40 a^2 c d x^3+15 a^2 d^2 x^5+26 a b c^2 x^3+10 a b c d x^5+8 b^2 c^2 x^5\right)}{48 c^3 \left(c+d x^2\right)^3}","\frac{5 a^3 \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{16 c^{7/2} \sqrt{b c-a d}}+\frac{5 a^2 x \sqrt{a+b x^2}}{16 c^3 \left(c+d x^2\right)}+\frac{5 a x \left(a+b x^2\right)^{3/2}}{24 c^2 \left(c+d x^2\right)^2}+\frac{x \left(a+b x^2\right)^{5/2}}{6 c \left(c+d x^2\right)^3}",1,"(Sqrt[a + b*x^2]*(33*a^2*c^2*x + 26*a*b*c^2*x^3 + 40*a^2*c*d*x^3 + 8*b^2*c^2*x^5 + 10*a*b*c*d*x^5 + 15*a^2*d^2*x^5))/(48*c^3*(c + d*x^2)^3) + (5*a^3*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(16*c^(7/2)*Sqrt[-(b*c) + a*d])","A",1
70,1,266,249,0.9880312,"\int \frac{\left(a+b x^2\right)^{5/2}}{\left(c+d x^2\right)^5} \, dx","IntegrateAlgebraic[(a + b*x^2)^(5/2)/(c + d*x^2)^5,x]","-\frac{5 \left(8 a^3 b c-7 a^4 d\right) \tan ^{-1}\left(\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{128 c^{9/2} (a d-b c)^{3/2}}-\frac{\sqrt{a+b x^2} \left(279 a^3 c^3 d x+511 a^3 c^2 d^2 x^3+385 a^3 c d^3 x^5+105 a^3 d^4 x^7-264 a^2 b c^4 x-258 a^2 b c^3 d x^3-188 a^2 b c^2 d^2 x^5-50 a^2 b c d^3 x^7-208 a b^2 c^4 x^3-88 a b^2 c^3 d x^5-24 a b^2 c^2 d^2 x^7-64 b^3 c^4 x^5-16 b^3 c^3 d x^7\right)}{384 c^4 \left(c+d x^2\right)^4 (b c-a d)}","\frac{5 a^3 (8 b c-7 a d) \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{128 c^{9/2} (b c-a d)^{3/2}}+\frac{5 a^2 x \sqrt{a+b x^2} (8 b c-7 a d)}{128 c^4 \left(c+d x^2\right) (b c-a d)}+\frac{5 a x \left(a+b x^2\right)^{3/2} (8 b c-7 a d)}{192 c^3 \left(c+d x^2\right)^2 (b c-a d)}+\frac{x \left(a+b x^2\right)^{5/2} (8 b c-7 a d)}{48 c^2 \left(c+d x^2\right)^3 (b c-a d)}-\frac{d x \left(a+b x^2\right)^{7/2}}{8 c \left(c+d x^2\right)^4 (b c-a d)}",1,"-1/384*(Sqrt[a + b*x^2]*(-264*a^2*b*c^4*x + 279*a^3*c^3*d*x - 208*a*b^2*c^4*x^3 - 258*a^2*b*c^3*d*x^3 + 511*a^3*c^2*d^2*x^3 - 64*b^3*c^4*x^5 - 88*a*b^2*c^3*d*x^5 - 188*a^2*b*c^2*d^2*x^5 + 385*a^3*c*d^3*x^5 - 16*b^3*c^3*d*x^7 - 24*a*b^2*c^2*d^2*x^7 - 50*a^2*b*c*d^3*x^7 + 105*a^3*d^4*x^7))/(c^4*(b*c - a*d)*(c + d*x^2)^4) - (5*(8*a^3*b*c - 7*a^4*d)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(128*c^(9/2)*(-(b*c) + a*d)^(3/2))","A",1
71,1,54,30,0.0956676,"\int \frac{\sqrt{1-x^2}}{1+x^2} \, dx","IntegrateAlgebraic[Sqrt[1 - x^2]/(1 + x^2),x]","2 \tan ^{-1}\left(\frac{\sqrt{1-x^2}}{x+1}\right)-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt{1-x^2}}{x^2-1}\right)","\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1-x^2}}\right)-\sin ^{-1}(x)",1,"2*ArcTan[Sqrt[1 - x^2]/(1 + x)] - Sqrt[2]*ArcTan[(Sqrt[2]*x*Sqrt[1 - x^2])/(-1 + x^2)]","A",1
72,1,57,27,0.094422,"\int \frac{\sqrt{1+x^2}}{-1+x^2} \, dx","IntegrateAlgebraic[Sqrt[1 + x^2]/(-1 + x^2),x]","-\log \left(\sqrt{x^2+1}-x\right)-\sqrt{2} \tanh ^{-1}\left(-\frac{x^2}{\sqrt{2}}+\frac{\sqrt{x^2+1} x}{\sqrt{2}}+\frac{1}{\sqrt{2}}\right)","\sinh ^{-1}(x)-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{x^2+1}}\right)",1,"-(Sqrt[2]*ArcTanh[1/Sqrt[2] - x^2/Sqrt[2] + (x*Sqrt[1 + x^2])/Sqrt[2]]) - Log[-x + Sqrt[1 + x^2]]","B",1
73,1,37,25,0.0622477,"\int \frac{\sqrt{1-x^2}}{-1+2 x^2} \, dx","IntegrateAlgebraic[Sqrt[1 - x^2]/(-1 + 2*x^2),x]","\tan ^{-1}\left(\frac{\sqrt{1-x^2}}{x+1}\right)-\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt{1-x^2}}\right)","-\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt{1-x^2}}\right)-\frac{1}{2} \sin ^{-1}(x)",1,"ArcTan[Sqrt[1 - x^2]/(1 + x)] - ArcTanh[x/Sqrt[1 - x^2]]/2","A",1
74,1,146,169,0.409656,"\int \frac{\left(c+d x^2\right)^3}{\sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(c + d*x^2)^3/Sqrt[a + b*x^2],x]","\frac{\sqrt{a+b x^2} \left(15 a^2 d^3 x-54 a b c d^2 x-10 a b d^3 x^3+72 b^2 c^2 d x+36 b^2 c d^2 x^3+8 b^2 d^3 x^5\right)}{48 b^3}+\frac{\left(-5 a^3 d^3+18 a^2 b c d^2-24 a b^2 c^2 d+16 b^3 c^3\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{16 b^{7/2}}","\frac{(2 b c-a d) \left(5 a^2 d^2-8 a b c d+8 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{16 b^{7/2}}+\frac{d x \sqrt{a+b x^2} \left(15 a^2 d^2-44 a b c d+44 b^2 c^2\right)}{48 b^3}+\frac{5 d x \sqrt{a+b x^2} \left(c+d x^2\right) (2 b c-a d)}{24 b^2}+\frac{d x \sqrt{a+b x^2} \left(c+d x^2\right)^2}{6 b}",1,"(Sqrt[a + b*x^2]*(72*b^2*c^2*d*x - 54*a*b*c*d^2*x + 15*a^2*d^3*x + 36*b^2*c*d^2*x^3 - 10*a*b*d^3*x^3 + 8*b^2*d^3*x^5))/(48*b^3) + ((16*b^3*c^3 - 24*a*b^2*c^2*d + 18*a^2*b*c*d^2 - 5*a^3*d^3)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*b^(7/2))","A",1
75,1,93,108,0.265766,"\int \frac{\left(c+d x^2\right)^2}{\sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(c + d*x^2)^2/Sqrt[a + b*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{b} x}{\sqrt{a+b x^2}}\right)}{8 b^{5/2}}+\frac{\sqrt{a+b x^2} \left(-3 a d^2 x+8 b c d x+2 b d^2 x^3\right)}{8 b^2}","\frac{\left(3 a^2 d^2-8 a b c d+8 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 b^{5/2}}+\frac{3 d x \sqrt{a+b x^2} (2 b c-a d)}{8 b^2}+\frac{d x \sqrt{a+b x^2} \left(c+d x^2\right)}{4 b}",1,"(Sqrt[a + b*x^2]*(8*b*c*d*x - 3*a*d^2*x + 2*b*d^2*x^3))/(8*b^2) + ((8*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*b^(5/2))","A",1
76,1,58,58,0.1573743,"\int \frac{c+d x^2}{\sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(c + d*x^2)/Sqrt[a + b*x^2],x]","\frac{(2 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 b^{3/2}}+\frac{d x \sqrt{a+b x^2}}{2 b}","\frac{(2 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 b^{3/2}}+\frac{d x \sqrt{a+b x^2}}{2 b}",1,"(d*x*Sqrt[a + b*x^2])/(2*b) + ((2*b*c - a*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(3/2))","A",1
77,1,25,25,0.0005807,"\int \frac{1}{\sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[1/Sqrt[a + b*x^2],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{\sqrt{b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{\sqrt{b}}",1,"ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]]/Sqrt[b]","A",1
78,1,49,49,0.1964749,"\int \frac{1}{\sqrt{a+b x^2} \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[1/(Sqrt[a + b*x^2]*(c + d*x^2)),x]","\frac{\tan ^{-1}\left(\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{\sqrt{c} \sqrt{a d-b c}}","\frac{\tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{\sqrt{c} \sqrt{b c-a d}}",1,"ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])]/(Sqrt[c]*Sqrt[-(b*c) + a*d])","A",1
79,1,100,101,0.3558794,"\int \frac{1}{\sqrt{a+b x^2} \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[1/(Sqrt[a + b*x^2]*(c + d*x^2)^2),x]","\frac{(a d-2 b c) \tan ^{-1}\left(\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{2 c^{3/2} (a d-b c)^{3/2}}-\frac{d x \sqrt{a+b x^2}}{2 c \left(c+d x^2\right) (b c-a d)}","\frac{(2 b c-a d) \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{2 c^{3/2} (b c-a d)^{3/2}}-\frac{d x \sqrt{a+b x^2}}{2 c \left(c+d x^2\right) (b c-a d)}",1,"-1/2*(d*x*Sqrt[a + b*x^2])/(c*(b*c - a*d)*(c + d*x^2)) + ((-2*b*c + a*d)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(2*c^(3/2)*(-(b*c) + a*d)^(3/2))","A",1
80,1,149,163,0.6079475,"\int \frac{1}{\sqrt{a+b x^2} \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[1/(Sqrt[a + b*x^2]*(c + d*x^2)^3),x]","\frac{\left(3 a^2 d^2-8 a b c d+8 b^2 c^2\right) \tan ^{-1}\left(\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{8 c^{5/2} (a d-b c)^{5/2}}+\frac{\sqrt{a+b x^2} \left(5 a c d^2 x+3 a d^3 x^3-8 b c^2 d x-6 b c d^2 x^3\right)}{8 c^2 \left(c+d x^2\right)^2 (b c-a d)^2}","\frac{\left(3 a^2 d^2-8 a b c d+8 b^2 c^2\right) \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{8 c^{5/2} (b c-a d)^{5/2}}-\frac{3 d x \sqrt{a+b x^2} (2 b c-a d)}{8 c^2 \left(c+d x^2\right) (b c-a d)^2}-\frac{d x \sqrt{a+b x^2}}{4 c \left(c+d x^2\right)^2 (b c-a d)}",1,"(Sqrt[a + b*x^2]*(-8*b*c^2*d*x + 5*a*c*d^2*x - 6*b*c*d^2*x^3 + 3*a*d^3*x^3))/(8*c^2*(b*c - a*d)^2*(c + d*x^2)^2) + ((8*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(8*c^(5/2)*(-(b*c) + a*d)^(5/2))","A",1
81,1,227,257,0.8310773,"\int \frac{\left(c+d x^2\right)^4}{\left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(c + d*x^2)^4/(a + b*x^2)^(3/2),x]","\frac{\left(-35 a^3 d^4+120 a^2 b c d^3-144 a b^2 c^2 d^2+64 b^3 c^3 d\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{16 b^{9/2}}+\frac{105 a^4 d^4 x-360 a^3 b c d^3 x+35 a^3 b d^4 x^3+432 a^2 b^2 c^2 d^2 x-120 a^2 b^2 c d^3 x^3-14 a^2 b^2 d^4 x^5-192 a b^3 c^3 d x+144 a b^3 c^2 d^2 x^3+48 a b^3 c d^3 x^5+8 a b^3 d^4 x^7+48 b^4 c^4 x}{48 a b^4 \sqrt{a+b x^2}}","-\frac{d x \sqrt{a+b x^2} \left(c+d x^2\right) \left(35 a^2 d^2-64 a b c d+24 b^2 c^2\right)}{24 a b^3}+\frac{d \left(-35 a^3 d^3+120 a^2 b c d^2-144 a b^2 c^2 d+64 b^3 c^3\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{16 b^{9/2}}-\frac{d x \sqrt{a+b x^2} \left(-105 a^3 d^3+290 a^2 b c d^2-248 a b^2 c^2 d+48 b^3 c^3\right)}{48 a b^4}-\frac{d x \sqrt{a+b x^2} \left(c+d x^2\right)^2 (6 b c-7 a d)}{6 a b^2}+\frac{x \left(c+d x^2\right)^3 (b c-a d)}{a b \sqrt{a+b x^2}}",1,"(48*b^4*c^4*x - 192*a*b^3*c^3*d*x + 432*a^2*b^2*c^2*d^2*x - 360*a^3*b*c*d^3*x + 105*a^4*d^4*x + 144*a*b^3*c^2*d^2*x^3 - 120*a^2*b^2*c*d^3*x^3 + 35*a^3*b*d^4*x^3 + 48*a*b^3*c*d^3*x^5 - 14*a^2*b^2*d^4*x^5 + 8*a*b^3*d^4*x^7)/(48*a*b^4*Sqrt[a + b*x^2]) + ((64*b^3*c^3*d - 144*a*b^2*c^2*d^2 + 120*a^2*b*c*d^3 - 35*a^3*d^4)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*b^(9/2))","A",1
82,1,154,169,0.4543006,"\int \frac{\left(c+d x^2\right)^3}{\left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(c + d*x^2)^3/(a + b*x^2)^(3/2),x]","\frac{3 \left(5 a^2 d^3-12 a b c d^2+8 b^2 c^2 d\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 b^{7/2}}+\frac{-15 a^3 d^3 x+36 a^2 b c d^2 x-5 a^2 b d^3 x^3-24 a b^2 c^2 d x+12 a b^2 c d^2 x^3+2 a b^2 d^3 x^5+8 b^3 c^3 x}{8 a b^3 \sqrt{a+b x^2}}","\frac{3 d \left(5 a^2 d^2-12 a b c d+8 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 b^{7/2}}-\frac{d x \sqrt{a+b x^2} (2 b c-5 a d) (4 b c-3 a d)}{8 a b^3}-\frac{d x \sqrt{a+b x^2} \left(c+d x^2\right) (4 b c-5 a d)}{4 a b^2}+\frac{x \left(c+d x^2\right)^2 (b c-a d)}{a b \sqrt{a+b x^2}}",1,"(8*b^3*c^3*x - 24*a*b^2*c^2*d*x + 36*a^2*b*c*d^2*x - 15*a^3*d^3*x + 12*a*b^2*c*d^2*x^3 - 5*a^2*b*d^3*x^3 + 2*a*b^2*d^3*x^5)/(8*a*b^3*Sqrt[a + b*x^2]) + (3*(8*b^2*c^2*d - 12*a*b*c*d^2 + 5*a^2*d^3)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*b^(7/2))","A",1
83,1,97,90,0.3317046,"\int \frac{\left(c+d x^2\right)^2}{\left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(c + d*x^2)^2/(a + b*x^2)^(3/2),x]","\frac{3 a^2 d^2 x-4 a b c d x+a b d^2 x^3+2 b^2 c^2 x}{2 a b^2 \sqrt{a+b x^2}}+\frac{\left(4 b c d-3 a d^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 b^{5/2}}","\frac{d (4 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 b^{5/2}}+\frac{x (b c-a d)^2}{a b^2 \sqrt{a+b x^2}}+\frac{d^2 x \sqrt{a+b x^2}}{2 b^2}",1,"(2*b^2*c^2*x - 4*a*b*c*d*x + 3*a^2*d^2*x + a*b*d^2*x^3)/(2*a*b^2*Sqrt[a + b*x^2]) + ((4*b*c*d - 3*a*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(5/2))","A",1
84,1,55,54,0.1682625,"\int \frac{c+d x^2}{\left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(c + d*x^2)/(a + b*x^2)^(3/2),x]","\frac{d \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{b^{3/2}}-\frac{x (a d-b c)}{a b \sqrt{a+b x^2}}","\frac{d \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{b^{3/2}}+\frac{x (b c-a d)}{a b \sqrt{a+b x^2}}",1,"-(((-(b*c) + a*d)*x)/(a*b*Sqrt[a + b*x^2])) + (d*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/b^(3/2)","A",1
85,1,16,16,0.0005605,"\int \frac{1}{\left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^(-3/2),x]","\frac{x}{a \sqrt{a+b x^2}}","\frac{x}{a \sqrt{a+b x^2}}",1,"x/(a*Sqrt[a + b*x^2])","A",1
86,1,79,79,0.3415502,"\int \frac{1}{\left(a+b x^2\right)^{3/2} \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^(3/2)*(c + d*x^2)),x]","\frac{d \tan ^{-1}\left(\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{\sqrt{c} (a d-b c)^{3/2}}-\frac{b x}{a \sqrt{a+b x^2} (a d-b c)}","\frac{b x}{a \sqrt{a+b x^2} (b c-a d)}-\frac{d \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{\sqrt{c} (b c-a d)^{3/2}}",1,"-((b*x)/(a*(-(b*c) + a*d)*Sqrt[a + b*x^2])) + (d*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(Sqrt[c]*(-(b*c) + a*d)^(3/2))","A",1
87,1,141,143,0.6184312,"\int \frac{1}{\left(a+b x^2\right)^{3/2} \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^(3/2)*(c + d*x^2)^2),x]","\frac{a^2 d^2 x+a b d^2 x^3+2 b^2 c^2 x+2 b^2 c d x^3}{2 a c \sqrt{a+b x^2} \left(c+d x^2\right) (a d-b c)^2}+\frac{\left(a d^2-4 b c d\right) \tan ^{-1}\left(\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{2 c^{3/2} (a d-b c)^{5/2}}","-\frac{d (4 b c-a d) \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{2 c^{3/2} (b c-a d)^{5/2}}+\frac{b x (a d+2 b c)}{2 a c \sqrt{a+b x^2} (b c-a d)^2}-\frac{d x}{2 c \sqrt{a+b x^2} \left(c+d x^2\right) (b c-a d)}",1,"(2*b^2*c^2*x + a^2*d^2*x + 2*b^2*c*d*x^3 + a*b*d^2*x^3)/(2*a*c*(-(b*c) + a*d)^2*Sqrt[a + b*x^2]*(c + d*x^2)) + ((-4*b*c*d + a*d^2)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(2*c^(3/2)*(-(b*c) + a*d)^(5/2))","A",1
88,1,241,225,1.0736225,"\int \frac{1}{\left(a+b x^2\right)^{3/2} \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^(3/2)*(c + d*x^2)^3),x]","\frac{3 \left(a^2 d^3-4 a b c d^2+8 b^2 c^2 d\right) \tan ^{-1}\left(\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{8 c^{5/2} (a d-b c)^{7/2}}+\frac{5 a^3 c d^3 x+3 a^3 d^4 x^3-12 a^2 b c^2 d^2 x-5 a^2 b c d^3 x^3+3 a^2 b d^4 x^5-12 a b^2 c^2 d^2 x^3-10 a b^2 c d^3 x^5-8 b^3 c^4 x-16 b^3 c^3 d x^3-8 b^3 c^2 d^2 x^5}{8 a c^2 \sqrt{a+b x^2} \left(c+d x^2\right)^2 (a d-b c)^3}","-\frac{3 d \left(a^2 d^2-4 a b c d+8 b^2 c^2\right) \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{8 c^{5/2} (b c-a d)^{7/2}}+\frac{d x \sqrt{a+b x^2} (4 b c-a d) (3 a d+2 b c)}{8 a c^2 \left(c+d x^2\right) (b c-a d)^3}+\frac{b x (a d+4 b c)}{4 a c \sqrt{a+b x^2} \left(c+d x^2\right) (b c-a d)^2}-\frac{d x}{4 c \sqrt{a+b x^2} \left(c+d x^2\right)^2 (b c-a d)}",1,"(-8*b^3*c^4*x - 12*a^2*b*c^2*d^2*x + 5*a^3*c*d^3*x - 16*b^3*c^3*d*x^3 - 12*a*b^2*c^2*d^2*x^3 - 5*a^2*b*c*d^3*x^3 + 3*a^3*d^4*x^3 - 8*b^3*c^2*d^2*x^5 - 10*a*b^2*c*d^3*x^5 + 3*a^2*b*d^4*x^5)/(8*a*c^2*(-(b*c) + a*d)^3*Sqrt[a + b*x^2]*(c + d*x^2)^2) + (3*(8*b^2*c^2*d - 4*a*b*c*d^2 + a^2*d^3)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(8*c^(5/2)*(-(b*c) + a*d)^(7/2))","A",1
89,1,235,255,0.7510505,"\int \frac{\left(c+d x^2\right)^4}{\left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(c + d*x^2)^4/(a + b*x^2)^(5/2),x]","\frac{\left(35 a^2 d^4-80 a b c d^3+48 b^2 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 b^{9/2}}+\frac{-105 a^5 d^4 x+240 a^4 b c d^3 x-140 a^4 b d^4 x^3-144 a^3 b^2 c^2 d^2 x+320 a^3 b^2 c d^3 x^3-21 a^3 b^2 d^4 x^5-192 a^2 b^3 c^2 d^2 x^3+48 a^2 b^3 c d^3 x^5+6 a^2 b^3 d^4 x^7+24 a b^4 c^4 x+32 a b^4 c^3 d x^3+16 b^5 c^4 x^3}{24 a^2 b^4 \left(a+b x^2\right)^{3/2}}","\frac{x \left(c+d x^2\right)^2 (b c-a d) (7 a d+2 b c)}{3 a^2 b^2 \sqrt{a+b x^2}}+\frac{d^2 \left(35 a^2 d^2-80 a b c d+48 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 b^{9/2}}-\frac{d x \sqrt{a+b x^2} \left(c+d x^2\right) \left(-35 a^2 d^2+24 a b c d+8 b^2 c^2\right)}{12 a^2 b^3}-\frac{d x \sqrt{a+b x^2} \left(105 a^3 d^3-170 a^2 b c d^2+40 a b^2 c^2 d+16 b^3 c^3\right)}{24 a^2 b^4}+\frac{x \left(c+d x^2\right)^3 (b c-a d)}{3 a b \left(a+b x^2\right)^{3/2}}",1,"(24*a*b^4*c^4*x - 144*a^3*b^2*c^2*d^2*x + 240*a^4*b*c*d^3*x - 105*a^5*d^4*x + 16*b^5*c^4*x^3 + 32*a*b^4*c^3*d*x^3 - 192*a^2*b^3*c^2*d^2*x^3 + 320*a^3*b^2*c*d^3*x^3 - 140*a^4*b*d^4*x^3 + 48*a^2*b^3*c*d^3*x^5 - 21*a^3*b^2*d^4*x^5 + 6*a^2*b^3*d^4*x^7)/(24*a^2*b^4*(a + b*x^2)^(3/2)) + ((48*b^2*c^2*d^2 - 80*a*b*c*d^3 + 35*a^2*d^4)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*b^(9/2))","A",1
90,1,160,172,0.4725857,"\int \frac{\left(c+d x^2\right)^3}{\left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(c + d*x^2)^3/(a + b*x^2)^(5/2),x]","\frac{15 a^4 d^3 x-18 a^3 b c d^2 x+20 a^3 b d^3 x^3-24 a^2 b^2 c d^2 x^3+3 a^2 b^2 d^3 x^5+6 a b^3 c^3 x+6 a b^3 c^2 d x^3+4 b^4 c^3 x^3}{6 a^2 b^3 \left(a+b x^2\right)^{3/2}}+\frac{\left(6 b c d^2-5 a d^3\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 b^{7/2}}","\frac{x \left(c+d x^2\right) (b c-a d) (5 a d+2 b c)}{3 a^2 b^2 \sqrt{a+b x^2}}-\frac{d x \sqrt{a+b x^2} \left(-15 a^2 d^2+8 a b c d+4 b^2 c^2\right)}{6 a^2 b^3}+\frac{d^2 (6 b c-5 a d) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 b^{7/2}}+\frac{x \left(c+d x^2\right)^2 (b c-a d)}{3 a b \left(a+b x^2\right)^{3/2}}",1,"(6*a*b^3*c^3*x - 18*a^3*b*c*d^2*x + 15*a^4*d^3*x + 4*b^4*c^3*x^3 + 6*a*b^3*c^2*d*x^3 - 24*a^2*b^2*c*d^2*x^3 + 20*a^3*b*d^3*x^3 + 3*a^2*b^2*d^3*x^5)/(6*a^2*b^3*(a + b*x^2)^(3/2)) + ((6*b*c*d^2 - 5*a*d^3)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(7/2))","A",1
91,1,104,105,0.278143,"\int \frac{\left(c+d x^2\right)^2}{\left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(c + d*x^2)^2/(a + b*x^2)^(5/2),x]","\frac{-3 a^3 d^2 x-4 a^2 b d^2 x^3+3 a b^2 c^2 x+2 a b^2 c d x^3+2 b^3 c^2 x^3}{3 a^2 b^2 \left(a+b x^2\right)^{3/2}}+\frac{d^2 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{b^{5/2}}","\frac{x (b c-a d) (3 a d+2 b c)}{3 a^2 b^2 \sqrt{a+b x^2}}+\frac{d^2 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{b^{5/2}}+\frac{x \left(c+d x^2\right) (b c-a d)}{3 a b \left(a+b x^2\right)^{3/2}}",1,"(3*a*b^2*c^2*x - 3*a^3*d^2*x + 2*b^3*c^2*x^3 + 2*a*b^2*c*d*x^3 - 4*a^2*b*d^2*x^3)/(3*a^2*b^2*(a + b*x^2)^(3/2)) + (d^2*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/b^(5/2)","A",1
92,1,37,47,0.1218393,"\int \frac{c+d x^2}{\left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(c + d*x^2)/(a + b*x^2)^(5/2),x]","\frac{x \left(3 a c+a d x^2+2 b c x^2\right)}{3 a^2 \left(a+b x^2\right)^{3/2}}","\frac{2 c x}{3 a^2 \sqrt{a+b x^2}}+\frac{x \left(c+d x^2\right)}{3 a \left(a+b x^2\right)^{3/2}}",1,"(x*(3*a*c + 2*b*c*x^2 + a*d*x^2))/(3*a^2*(a + b*x^2)^(3/2))","A",1
93,1,29,39,0.0006092,"\int \frac{1}{\left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^(-5/2),x]","\frac{x \left(3 a+2 b x^2\right)}{3 a^2 \left(a+b x^2\right)^{3/2}}","\frac{2 x}{3 a^2 \sqrt{a+b x^2}}+\frac{x}{3 a \left(a+b x^2\right)^{3/2}}",1,"(x*(3*a + 2*b*x^2))/(3*a^2*(a + b*x^2)^(3/2))","A",1
94,1,117,122,0.6176806,"\int \frac{1}{\left(a+b x^2\right)^{5/2} \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^(5/2)*(c + d*x^2)),x]","\frac{-6 a^2 b d x+3 a b^2 c x-5 a b^2 d x^3+2 b^3 c x^3}{3 a^2 \left(a+b x^2\right)^{3/2} (a d-b c)^2}+\frac{d^2 \tan ^{-1}\left(\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{\sqrt{c} (a d-b c)^{5/2}}","\frac{b x (2 b c-5 a d)}{3 a^2 \sqrt{a+b x^2} (b c-a d)^2}+\frac{d^2 \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{\sqrt{c} (b c-a d)^{5/2}}+\frac{b x}{3 a \left(a+b x^2\right)^{3/2} (b c-a d)}",1,"(3*a*b^2*c*x - 6*a^2*b*d*x + 2*b^3*c*x^3 - 5*a*b^2*d*x^3)/(3*a^2*(-(b*c) + a*d)^2*(a + b*x^2)^(3/2)) + (d^2*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(Sqrt[c]*(-(b*c) + a*d)^(5/2))","A",1
95,1,229,202,1.077278,"\int \frac{1}{\left(a+b x^2\right)^{5/2} \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^(5/2)*(c + d*x^2)^2),x]","\frac{3 a^4 d^3 x+6 a^3 b d^3 x^3+18 a^2 b^2 c^2 d x+18 a^2 b^2 c d^2 x^3+3 a^2 b^2 d^3 x^5-6 a b^3 c^3 x+10 a b^3 c^2 d x^3+16 a b^3 c d^2 x^5-4 b^4 c^3 x^3-4 b^4 c^2 d x^5}{6 a^2 c \left(a+b x^2\right)^{3/2} \left(c+d x^2\right) (a d-b c)^3}+\frac{\left(a d^3-6 b c d^2\right) \tan ^{-1}\left(\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{2 c^{3/2} (a d-b c)^{7/2}}","\frac{b x \left(-3 a^2 d^2-16 a b c d+4 b^2 c^2\right)}{6 a^2 c \sqrt{a+b x^2} (b c-a d)^3}+\frac{d^2 (6 b c-a d) \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{2 c^{3/2} (b c-a d)^{7/2}}-\frac{d x}{2 c \left(a+b x^2\right)^{3/2} \left(c+d x^2\right) (b c-a d)}+\frac{b x (3 a d+2 b c)}{6 a c \left(a+b x^2\right)^{3/2} (b c-a d)^2}",1,"(-6*a*b^3*c^3*x + 18*a^2*b^2*c^2*d*x + 3*a^4*d^3*x - 4*b^4*c^3*x^3 + 10*a*b^3*c^2*d*x^3 + 18*a^2*b^2*c*d^2*x^3 + 6*a^3*b*d^3*x^3 - 4*b^4*c^2*d*x^5 + 16*a*b^3*c*d^2*x^5 + 3*a^2*b^2*d^3*x^5)/(6*a^2*c*(-(b*c) + a*d)^3*(a + b*x^2)^(3/2)*(c + d*x^2)) + ((-6*b*c*d^2 + a*d^3)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(2*c^(3/2)*(-(b*c) + a*d)^(7/2))","A",1
96,1,379,313,1.8561986,"\int \frac{1}{\left(a+b x^2\right)^{5/2} \left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^(5/2)*(c + d*x^2)^3),x]","\frac{\left(3 a^2 d^4-16 a b c d^3+48 b^2 c^2 d^2\right) \tan ^{-1}\left(\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{8 c^{5/2} (a d-b c)^{9/2}}+\frac{15 a^5 c d^4 x+9 a^5 d^5 x^3-48 a^4 b c^2 d^3 x-12 a^4 b c d^4 x^3+18 a^4 b d^5 x^5-96 a^3 b^2 c^2 d^3 x^3-69 a^3 b^2 c d^4 x^5+9 a^3 b^2 d^5 x^7-96 a^2 b^3 c^4 d x-192 a^2 b^3 c^3 d^2 x^3-144 a^2 b^3 c^2 d^3 x^5-42 a^2 b^3 c d^4 x^7+24 a b^4 c^5 x-40 a b^4 c^4 d x^3-152 a b^4 c^3 d^2 x^5-88 a b^4 c^2 d^3 x^7+16 b^5 c^5 x^3+32 b^5 c^4 d x^5+16 b^5 c^3 d^2 x^7}{24 a^2 c^2 \left(a+b x^2\right)^{3/2} \left(c+d x^2\right)^2 (a d-b c)^4}","\frac{b x \left(-3 a^2 d^2-40 a b c d+8 b^2 c^2\right)}{12 a^2 c \sqrt{a+b x^2} \left(c+d x^2\right) (b c-a d)^3}+\frac{d^2 \left(3 a^2 d^2-16 a b c d+48 b^2 c^2\right) \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{8 c^{5/2} (b c-a d)^{9/2}}+\frac{d x \sqrt{a+b x^2} \left(9 a^3 d^3-42 a^2 b c d^2-88 a b^2 c^2 d+16 b^3 c^3\right)}{24 a^2 c^2 \left(c+d x^2\right) (b c-a d)^4}-\frac{d x}{4 c \left(a+b x^2\right)^{3/2} \left(c+d x^2\right)^2 (b c-a d)}+\frac{b x (3 a d+4 b c)}{12 a c \left(a+b x^2\right)^{3/2} \left(c+d x^2\right) (b c-a d)^2}",1,"(24*a*b^4*c^5*x - 96*a^2*b^3*c^4*d*x - 48*a^4*b*c^2*d^3*x + 15*a^5*c*d^4*x + 16*b^5*c^5*x^3 - 40*a*b^4*c^4*d*x^3 - 192*a^2*b^3*c^3*d^2*x^3 - 96*a^3*b^2*c^2*d^3*x^3 - 12*a^4*b*c*d^4*x^3 + 9*a^5*d^5*x^3 + 32*b^5*c^4*d*x^5 - 152*a*b^4*c^3*d^2*x^5 - 144*a^2*b^3*c^2*d^3*x^5 - 69*a^3*b^2*c*d^4*x^5 + 18*a^4*b*d^5*x^5 + 16*b^5*c^3*d^2*x^7 - 88*a*b^4*c^2*d^3*x^7 - 42*a^2*b^3*c*d^4*x^7 + 9*a^3*b^2*d^5*x^7)/(24*a^2*c^2*(-(b*c) + a*d)^4*(a + b*x^2)^(3/2)*(c + d*x^2)^2) + ((48*b^2*c^2*d^2 - 16*a*b*c*d^3 + 3*a^2*d^4)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(8*c^(5/2)*(-(b*c) + a*d)^(9/2))","A",1
97,1,193,224,0.3925291,"\int \frac{\left(a+b x^2\right)^3}{\left(c+d x^2\right)^{11/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^3/(c + d*x^2)^(11/2),x]","\frac{x \left(315 a^3 c^4+840 a^3 c^3 d x^2+1008 a^3 c^2 d^2 x^4+576 a^3 c d^3 x^6+128 a^3 d^4 x^8+315 a^2 b c^4 x^2+378 a^2 b c^3 d x^4+216 a^2 b c^2 d^2 x^6+48 a^2 b c d^3 x^8+189 a b^2 c^4 x^4+108 a b^2 c^3 d x^6+24 a b^2 c^2 d^2 x^8+45 b^3 c^4 x^6+10 b^3 c^3 d x^8\right)}{315 c^5 \left(c+d x^2\right)^{9/2}}","\frac{16 a^3 x (9 b c-8 a d)}{315 c^5 \sqrt{c+d x^2} (b c-a d)}+\frac{8 a^2 x \left(a+b x^2\right) (9 b c-8 a d)}{315 c^4 \left(c+d x^2\right)^{3/2} (b c-a d)}+\frac{2 a x \left(a+b x^2\right)^2 (9 b c-8 a d)}{105 c^3 \left(c+d x^2\right)^{5/2} (b c-a d)}+\frac{x \left(a+b x^2\right)^3 (9 b c-8 a d)}{63 c^2 \left(c+d x^2\right)^{7/2} (b c-a d)}-\frac{d x \left(a+b x^2\right)^4}{9 c \left(c+d x^2\right)^{9/2} (b c-a d)}",1,"(x*(315*a^3*c^4 + 315*a^2*b*c^4*x^2 + 840*a^3*c^3*d*x^2 + 189*a*b^2*c^4*x^4 + 378*a^2*b*c^3*d*x^4 + 1008*a^3*c^2*d^2*x^4 + 45*b^3*c^4*x^6 + 108*a*b^2*c^3*d*x^6 + 216*a^2*b*c^2*d^2*x^6 + 576*a^3*c*d^3*x^6 + 10*b^3*c^3*d*x^8 + 24*a*b^2*c^2*d^2*x^8 + 48*a^2*b*c*d^3*x^8 + 128*a^3*d^4*x^8))/(315*c^5*(c + d*x^2)^(9/2))","A",1
98,1,118,174,0.2525827,"\int \frac{\left(a+b x^2\right)^2}{\left(c+d x^2\right)^{9/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/(c + d*x^2)^(9/2),x]","\frac{x \left(105 a^2 c^3+210 a^2 c^2 d x^2+168 a^2 c d^2 x^4+48 a^2 d^3 x^6+70 a b c^3 x^2+56 a b c^2 d x^4+16 a b c d^2 x^6+21 b^2 c^3 x^4+6 b^2 c^2 d x^6\right)}{105 c^4 \left(c+d x^2\right)^{7/2}}","\frac{8 a^2 x (7 b c-6 a d)}{105 c^4 \sqrt{c+d x^2} (b c-a d)}+\frac{4 a x \left(a+b x^2\right) (7 b c-6 a d)}{105 c^3 \left(c+d x^2\right)^{3/2} (b c-a d)}+\frac{x \left(a+b x^2\right)^2 (7 b c-6 a d)}{35 c^2 \left(c+d x^2\right)^{5/2} (b c-a d)}-\frac{d x \left(a+b x^2\right)^3}{7 c \left(c+d x^2\right)^{7/2} (b c-a d)}",1,"(x*(105*a^2*c^3 + 70*a*b*c^3*x^2 + 210*a^2*c^2*d*x^2 + 21*b^2*c^3*x^4 + 56*a*b*c^2*d*x^4 + 168*a^2*c*d^2*x^4 + 6*b^2*c^2*d*x^6 + 16*a*b*c*d^2*x^6 + 48*a^2*d^3*x^6))/(105*c^4*(c + d*x^2)^(7/2))","A",1
99,1,60,91,0.1671675,"\int \frac{a+b x^2}{\left(c+d x^2\right)^{7/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)/(c + d*x^2)^(7/2),x]","\frac{x \left(15 a c^2+20 a c d x^2+8 a d^2 x^4+5 b c^2 x^2+2 b c d x^4\right)}{15 c^3 \left(c+d x^2\right)^{5/2}}","\frac{2 x (4 a d+b c)}{15 c^3 d \sqrt{c+d x^2}}+\frac{x (4 a d+b c)}{15 c^2 d \left(c+d x^2\right)^{3/2}}-\frac{x (b c-a d)}{5 c d \left(c+d x^2\right)^{5/2}}",1,"(x*(15*a*c^2 + 5*b*c^2*x^2 + 20*a*c*d*x^2 + 2*b*c*d*x^4 + 8*a*d^2*x^4))/(15*c^3*(c + d*x^2)^(5/2))","A",1
100,1,29,39,0.0903067,"\int \frac{1}{\left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(c + d*x^2)^(-5/2),x]","\frac{x \left(3 c+2 d x^2\right)}{3 c^2 \left(c+d x^2\right)^{3/2}}","\frac{2 x}{3 c^2 \sqrt{c+d x^2}}+\frac{x}{3 c \left(c+d x^2\right)^{3/2}}",1,"(x*(3*c + 2*d*x^2))/(3*c^2*(c + d*x^2)^(3/2))","A",1
101,1,79,79,0.3168912,"\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
102,1,100,100,0.352919,"\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
103,1,133,149,0.4836036,"\int \frac{\sqrt{c+d x^2}}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[Sqrt[c + d*x^2]/(a + b*x^2)^3,x]","\frac{\left(3 b c^2-4 a c d\right) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{8 a^{5/2} (b c-a d)^{3/2}}+\frac{\sqrt{c+d x^2} \left(4 a^2 d x-5 a b c x+2 a b d x^3-3 b^2 c x^3\right)}{8 a^2 \left(a+b x^2\right)^2 (a d-b c)}","\frac{c (3 b c-4 a d) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{8 a^{5/2} (b c-a d)^{3/2}}+\frac{x \sqrt{c+d x^2} (3 b c-4 a d)}{8 a^2 \left(a+b x^2\right) (b c-a d)}+\frac{b x \left(c+d x^2\right)^{3/2}}{4 a \left(a+b x^2\right)^2 (b c-a d)}",1,"(Sqrt[c + d*x^2]*(-5*a*b*c*x + 4*a^2*d*x - 3*b^2*c*x^3 + 2*a*b*d*x^3))/(8*a^2*(-(b*c) + a*d)*(a + b*x^2)^2) + ((3*b*c^2 - 4*a*c*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(8*a^(5/2)*(b*c - a*d)^(3/2))","A",1
104,1,191,199,0.7428988,"\int \frac{\left(c+d x^2\right)^{3/2}}{\left(a+b x^2\right)^4} \, dx","IntegrateAlgebraic[(c + d*x^2)^(3/2)/(a + b*x^2)^4,x]","\frac{\left(5 b c^3-6 a c^2 d\right) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{16 a^{7/2} (b c-a d)^{3/2}}+\frac{\sqrt{c+d x^2} \left(30 a^3 c d x+12 a^3 d^2 x^3-33 a^2 b c^2 x+22 a^2 b c d x^3+4 a^2 b d^2 x^5-40 a b^2 c^2 x^3+8 a b^2 c d x^5-15 b^3 c^2 x^5\right)}{48 a^3 \left(a+b x^2\right)^3 (a d-b c)}","\frac{c^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)}{16 a^{7/2} (b c-a d)^{3/2}}+\frac{c x \sqrt{c+d x^2} (5 b c-6 a d)}{16 a^3 \left(a+b x^2\right) (b c-a d)}+\frac{x \left(c+d x^2\right)^{3/2} (5 b c-6 a d)}{24 a^2 \left(a+b x^2\right)^2 (b c-a d)}+\frac{b x \left(c+d x^2\right)^{5/2}}{6 a \left(a+b x^2\right)^3 (b c-a d)}",1,"(Sqrt[c + d*x^2]*(-33*a^2*b*c^2*x + 30*a^3*c*d*x - 40*a*b^2*c^2*x^3 + 22*a^2*b*c*d*x^3 + 12*a^3*d^2*x^3 - 15*b^3*c^2*x^5 + 8*a*b^2*c*d*x^5 + 4*a^2*b*d^2*x^5))/(48*a^3*(-(b*c) + a*d)*(a + b*x^2)^3) + ((5*b*c^3 - 6*a*c^2*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(16*a^(7/2)*(b*c - a*d)^(3/2))","A",1
105,1,20,20,0.0928023,"\int \frac{1}{\left(\frac{b c}{d}+b x^2\right) \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[1/(((b*c)/d + b*x^2)*Sqrt[c + d*x^2]),x]","\frac{d x}{b c \sqrt{c+d x^2}}","\frac{d x}{b c \sqrt{c+d x^2}}",1,"(d*x)/(b*c*Sqrt[c + d*x^2])","A",1
106,1,33,25,0.0846677,"\int \frac{1}{\sqrt{1-x^2} \left(1+x^2\right)} \, dx","IntegrateAlgebraic[1/(Sqrt[1 - x^2]*(1 + x^2)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x \sqrt{1-x^2}}{x^2-1}\right)}{\sqrt{2}}","\frac{\tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1-x^2}}\right)}{\sqrt{2}}",1,"-(ArcTan[(Sqrt[2]*x*Sqrt[1 - x^2])/(-1 + x^2)]/Sqrt[2])","A",1
107,1,49,49,0.1883237,"\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
108,1,29,15,0.0432275,"\int \frac{-1+x^2}{\left(1+x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(-1 + x^2)/(1 + x^2)^(3/2),x]","-\frac{2 x}{\sqrt{x^2+1}}-\log \left(\sqrt{x^2+1}-x\right)","\sinh ^{-1}(x)-\frac{2 x}{\sqrt{x^2+1}}",1,"(-2*x)/Sqrt[1 + x^2] - Log[-x + Sqrt[1 + x^2]]","A",1
109,0,0,648,5.0205359,"\int \left(a-b x^2\right)^{2/3} \left(3 a+b x^2\right)^3 \, dx","IntegrateAlgebraic[(a - b*x^2)^(2/3)*(3*a + b*x^2)^3,x]","\int \left(a-b x^2\right)^{2/3} \left(3 a+b x^2\right)^3 \, dx","\frac{24192 \sqrt{2} 3^{3/4} a^{13/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{1235 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{36288 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^{13/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{1235 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{72576 a^4 x}{1235 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}+\frac{18144 a^3 x \left(a-b x^2\right)^{2/3}}{1235}-\frac{23544 a^2 x \left(a-b x^2\right)^{5/3}}{6175}-\frac{378}{475} a x \left(a-b x^2\right)^{5/3} \left(3 a+b x^2\right)-\frac{3}{25} x \left(a-b x^2\right)^{5/3} \left(3 a+b x^2\right)^2",1,"Could not integrate","F",-1
110,0,0,617,4.9363982,"\int \left(a-b x^2\right)^{2/3} \left(3 a+b x^2\right)^2 \, dx","IntegrateAlgebraic[(a - b*x^2)^(2/3)*(3*a + b*x^2)^2,x]","\int \left(a-b x^2\right)^{2/3} \left(3 a+b x^2\right)^2 \, dx","\frac{10368 \sqrt{2} 3^{3/4} a^{10/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{1729 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{15552 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^{10/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{1729 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{31104 a^3 x}{1729 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}+\frac{7776 a^2 x \left(a-b x^2\right)^{2/3}}{1729}-\frac{252}{247} a x \left(a-b x^2\right)^{5/3}-\frac{3}{19} x \left(a-b x^2\right)^{5/3} \left(3 a+b x^2\right)",1,"Could not integrate","F",-1
111,0,0,588,4.6493633,"\int \left(a-b x^2\right)^{2/3} \left(3 a+b x^2\right) \, dx","IntegrateAlgebraic[(a - b*x^2)^(2/3)*(3*a + b*x^2),x]","\int \left(a-b x^2\right)^{2/3} \left(3 a+b x^2\right) \, dx","\frac{24 \sqrt{2} 3^{3/4} a^{7/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{13 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{36 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^{7/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{13 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{72 a^2 x}{13 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}+\frac{18}{13} a x \left(a-b x^2\right)^{2/3}-\frac{3}{13} x \left(a-b x^2\right)^{5/3}",1,"Could not integrate","F",-1
112,0,0,740,4.9714667,"\int \frac{\left(a-b x^2\right)^{2/3}}{3 a+b x^2} \, dx","IntegrateAlgebraic[(a - b*x^2)^(2/3)/(3*a + b*x^2),x]","\int \frac{\left(a-b x^2\right)^{2/3}}{3 a+b x^2} \, dx","-\frac{\sqrt{2} 3^{3/4} \sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}+\frac{3 \sqrt[4]{3} \sqrt{2+\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{2 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}+\frac{3 x}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}+\frac{\sqrt[3]{2} \sqrt[6]{a} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{2} \sqrt[3]{a-b x^2}\right)}{\sqrt{b} x}\right)}{\sqrt{3} \sqrt{b}}+\frac{\sqrt[3]{2} \sqrt[6]{a} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt[6]{a} \left(\sqrt[3]{2} \sqrt[3]{a-b x^2}+\sqrt[3]{a}\right)}\right)}{\sqrt{b}}+\frac{\sqrt[3]{2} \sqrt[6]{a} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right)}{\sqrt{3} \sqrt{b}}-\frac{\sqrt[3]{2} \sqrt[6]{a} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{3 \sqrt{b}}",1,"Could not integrate","F",-1
113,0,0,584,9.4666125,"\int \frac{\left(a-b x^2\right)^{2/3}}{\left(3 a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(a - b*x^2)^(2/3)/(3*a + b*x^2)^2,x]","\int \frac{\left(a-b x^2\right)^{2/3}}{\left(3 a+b x^2\right)^2} \, dx","\frac{\left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{3 \sqrt{2} \sqrt[4]{3} a^{2/3} b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{\sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{4\ 3^{3/4} a^{2/3} b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}+\frac{x \left(a-b x^2\right)^{2/3}}{6 a \left(3 a+b x^2\right)}-\frac{x}{6 a \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}",1,"Could not integrate","F",-1
114,0,0,818,14.7179457,"\int \frac{\left(a-b x^2\right)^{2/3}}{\left(3 a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(a - b*x^2)^(2/3)/(3*a + b*x^2)^3,x]","\int \frac{\left(a-b x^2\right)^{2/3}}{\left(3 a+b x^2\right)^3} \, dx","\frac{\left(a-b x^2\right)^{2/3} x}{36 a^2 \left(b x^2+3 a\right)}-\frac{x}{36 a^2 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}+\frac{\left(a-b x^2\right)^{2/3} x}{12 a \left(b x^2+3 a\right)^2}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right)}{72\ 2^{2/3} \sqrt{3} a^{11/6} \sqrt{b}}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{2} \sqrt[3]{a-b x^2}\right)}{\sqrt{b} x}\right)}{72\ 2^{2/3} \sqrt{3} a^{11/6} \sqrt{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{216\ 2^{2/3} a^{11/6} \sqrt{b}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{2} \sqrt[3]{a-b x^2}\right)}\right)}{72\ 2^{2/3} a^{11/6} \sqrt{b}}-\frac{\sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a-b x^2} \sqrt[3]{a}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{24\ 3^{3/4} a^{5/3} b \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} x}+\frac{\left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a-b x^2} \sqrt[3]{a}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{18 \sqrt{2} \sqrt[4]{3} a^{5/3} b \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} x}",1,"Could not integrate","F",-1
115,0,0,849,18.8860784,"\int \frac{\left(a-b x^2\right)^{2/3}}{\left(3 a+b x^2\right)^4} \, dx","IntegrateAlgebraic[(a - b*x^2)^(2/3)/(3*a + b*x^2)^4,x]","\int \frac{\left(a-b x^2\right)^{2/3}}{\left(3 a+b x^2\right)^4} \, dx","\frac{\left(a-b x^2\right)^{2/3} x}{144 a^3 \left(b x^2+3 a\right)}-\frac{x}{144 a^3 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}+\frac{\left(a-b x^2\right)^{2/3} x}{54 a^2 \left(b x^2+3 a\right)^2}+\frac{\left(a-b x^2\right)^{2/3} x}{18 a \left(b x^2+3 a\right)^3}+\frac{7 \tan ^{-1}\left(\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right)}{1296\ 2^{2/3} \sqrt{3} a^{17/6} \sqrt{b}}+\frac{7 \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{2} \sqrt[3]{a-b x^2}\right)}{\sqrt{b} x}\right)}{1296\ 2^{2/3} \sqrt{3} a^{17/6} \sqrt{b}}-\frac{7 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{3888\ 2^{2/3} a^{17/6} \sqrt{b}}+\frac{7 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{2} \sqrt[3]{a-b x^2}\right)}\right)}{1296\ 2^{2/3} a^{17/6} \sqrt{b}}-\frac{\sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a-b x^2} \sqrt[3]{a}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{96\ 3^{3/4} a^{8/3} b \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} x}+\frac{\left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a-b x^2} \sqrt[3]{a}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{72 \sqrt{2} \sqrt[4]{3} a^{8/3} b \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} x}",1,"Could not integrate","F",-1
116,0,0,668,5.6362846,"\int \left(a-b x^2\right)^{5/3} \left(3 a+b x^2\right)^3 \, dx","IntegrateAlgebraic[(a - b*x^2)^(5/3)*(3*a + b*x^2)^3,x]","\int \left(a-b x^2\right)^{5/3} \left(3 a+b x^2\right)^3 \, dx","\frac{3746304 \sqrt{2} 3^{3/4} a^{16/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{267995 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{5619456 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^{16/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{267995 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{11238912 a^5 x}{267995 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}+\frac{2809728 a^4 x \left(a-b x^2\right)^{2/3}}{267995}+\frac{1404864 a^3 x \left(a-b x^2\right)^{5/3}}{191425}-\frac{33264 a^2 x \left(a-b x^2\right)^{8/3}}{14725}-\frac{432}{775} a x \left(a-b x^2\right)^{8/3} \left(3 a+b x^2\right)-\frac{3}{31} x \left(a-b x^2\right)^{8/3} \left(3 a+b x^2\right)^2",1,"Could not integrate","F",-1
117,0,0,637,5.3767884,"\int \left(a-b x^2\right)^{5/3} \left(3 a+b x^2\right)^2 \, dx","IntegrateAlgebraic[(a - b*x^2)^(5/3)*(3*a + b*x^2)^2,x]","\int \left(a-b x^2\right)^{5/3} \left(3 a+b x^2\right)^2 \, dx","\frac{38016 \sqrt{2} 3^{3/4} a^{13/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{8645 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{57024 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^{13/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{8645 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{114048 a^4 x}{8645 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}+\frac{28512 a^3 x \left(a-b x^2\right)^{2/3}}{8645}+\frac{14256 a^2 x \left(a-b x^2\right)^{5/3}}{6175}-\frac{306}{475} a x \left(a-b x^2\right)^{8/3}-\frac{3}{25} x \left(a-b x^2\right)^{8/3} \left(3 a+b x^2\right)",1,"Could not integrate","F",-1
118,0,0,608,4.9526651,"\int \left(a-b x^2\right)^{5/3} \left(3 a+b x^2\right) \, dx","IntegrateAlgebraic[(a - b*x^2)^(5/3)*(3*a + b*x^2),x]","\int \left(a-b x^2\right)^{5/3} \left(3 a+b x^2\right) \, dx","\frac{2400 \sqrt{2} 3^{3/4} a^{10/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{1729 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{3600 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^{10/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{1729 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{7200 a^3 x}{1729 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}+\frac{1800 a^2 x \left(a-b x^2\right)^{2/3}}{1729}+\frac{180}{247} a x \left(a-b x^2\right)^{5/3}-\frac{3}{19} x \left(a-b x^2\right)^{8/3}",1,"Could not integrate","F",-1
119,0,0,765,5.5157165,"\int \frac{\left(a-b x^2\right)^{5/3}}{3 a+b x^2} \, dx","IntegrateAlgebraic[(a - b*x^2)^(5/3)/(3*a + b*x^2),x]","\int \frac{\left(a-b x^2\right)^{5/3}}{3 a+b x^2} \, dx","\frac{4 \sqrt[3]{2} a^{7/6} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{2} \sqrt[3]{a-b x^2}\right)}{\sqrt{b} x}\right)}{\sqrt{3} \sqrt{b}}+\frac{4 \sqrt[3]{2} a^{7/6} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt[6]{a} \left(\sqrt[3]{2} \sqrt[3]{a-b x^2}+\sqrt[3]{a}\right)}\right)}{\sqrt{b}}-\frac{32 \sqrt{2} 3^{3/4} a^{4/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{7 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}+\frac{48 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^{4/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{7 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}+\frac{4 \sqrt[3]{2} a^{7/6} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right)}{\sqrt{3} \sqrt{b}}-\frac{4 \sqrt[3]{2} a^{7/6} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{3 \sqrt{b}}+\frac{96 a x}{7 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}-\frac{3}{7} x \left(a-b x^2\right)^{2/3}",1,"Could not integrate","F",-1
120,0,0,775,9.8288838,"\int \frac{\left(a-b x^2\right)^{5/3}}{\left(3 a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(a - b*x^2)^(5/3)/(3*a + b*x^2)^2,x]","\int \frac{\left(a-b x^2\right)^{5/3}}{\left(3 a+b x^2\right)^2} \, dx","\frac{11 \sqrt{2} \sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{3 \sqrt[4]{3} b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{11 \sqrt{2+\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{2\ 3^{3/4} b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}+\frac{2 x \left(a-b x^2\right)^{2/3}}{3 \left(3 a+b x^2\right)}-\frac{11 x}{3 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}-\frac{\sqrt[3]{2} \sqrt[6]{a} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{2} \sqrt[3]{a-b x^2}\right)}{\sqrt{b} x}\right)}{\sqrt{3} \sqrt{b}}-\frac{\sqrt[3]{2} \sqrt[6]{a} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt[6]{a} \left(\sqrt[3]{2} \sqrt[3]{a-b x^2}+\sqrt[3]{a}\right)}\right)}{\sqrt{b}}-\frac{\sqrt[3]{2} \sqrt[6]{a} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right)}{\sqrt{3} \sqrt{b}}+\frac{\sqrt[3]{2} \sqrt[6]{a} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{3 \sqrt{b}}",1,"Could not integrate","F",-1
121,0,0,815,14.718392,"\int \frac{\left(a-b x^2\right)^{5/3}}{\left(3 a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(a - b*x^2)^(5/3)/(3*a + b*x^2)^3,x]","\int \frac{\left(a-b x^2\right)^{5/3}}{\left(3 a+b x^2\right)^3} \, dx","-\frac{\left(a-b x^2\right)^{2/3} x}{18 a \left(b x^2+3 a\right)}+\frac{x}{18 a \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}+\frac{\left(a-b x^2\right)^{2/3} x}{3 \left(b x^2+3 a\right)^2}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right)}{18\ 2^{2/3} \sqrt{3} a^{5/6} \sqrt{b}}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{2} \sqrt[3]{a-b x^2}\right)}{\sqrt{b} x}\right)}{18\ 2^{2/3} \sqrt{3} a^{5/6} \sqrt{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{54\ 2^{2/3} a^{5/6} \sqrt{b}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{2} \sqrt[3]{a-b x^2}\right)}\right)}{18\ 2^{2/3} a^{5/6} \sqrt{b}}+\frac{\sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a-b x^2} \sqrt[3]{a}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{12\ 3^{3/4} a^{2/3} b \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} x}-\frac{\left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a-b x^2} \sqrt[3]{a}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{9 \sqrt{2} \sqrt[4]{3} a^{2/3} b \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} x}",1,"Could not integrate","F",-1
122,0,0,659,4.2651849,"\int \frac{\left(3 a+b x^2\right)^4}{\sqrt[3]{a-b x^2}} \, dx","IntegrateAlgebraic[(3*a + b*x^2)^4/(a - b*x^2)^(1/3),x]","\int \frac{\left(3 a+b x^2\right)^4}{\sqrt[3]{a-b x^2}} \, dx","\frac{1264896 \sqrt{2} 3^{3/4} a^{13/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{8645 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{1897344 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^{13/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{8645 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{3794688 a^4 x}{8645 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}-\frac{1552608 a^3 x \left(a-b x^2\right)^{2/3}}{43225}-\frac{36288 a^2 x \left(a-b x^2\right)^{2/3} \left(3 a+b x^2\right)}{6175}-\frac{18}{19} a x \left(a-b x^2\right)^{2/3} \left(3 a+b x^2\right)^2-\frac{3}{25} x \left(a-b x^2\right)^{2/3} \left(3 a+b x^2\right)^3",1,"Could not integrate","F",-1
123,0,0,628,4.0184051,"\int \frac{\left(3 a+b x^2\right)^3}{\sqrt[3]{a-b x^2}} \, dx","IntegrateAlgebraic[(3*a + b*x^2)^3/(a - b*x^2)^(1/3),x]","\int \frac{\left(3 a+b x^2\right)^3}{\sqrt[3]{a-b x^2}} \, dx","\frac{71712 \sqrt{2} 3^{3/4} a^{10/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{1729 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{107568 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^{10/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{1729 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{215136 a^3 x}{1729 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}-\frac{15768 a^2 x \left(a-b x^2\right)^{2/3}}{1729}-\frac{324}{247} a x \left(a-b x^2\right)^{2/3} \left(3 a+b x^2\right)-\frac{3}{19} x \left(a-b x^2\right)^{2/3} \left(3 a+b x^2\right)^2",1,"Could not integrate","F",-1
124,0,0,597,4.0301763,"\int \frac{\left(3 a+b x^2\right)^2}{\sqrt[3]{a-b x^2}} \, dx","IntegrateAlgebraic[(3*a + b*x^2)^2/(a - b*x^2)^(1/3),x]","\int \frac{\left(3 a+b x^2\right)^2}{\sqrt[3]{a-b x^2}} \, dx","\frac{1080 \sqrt{2} 3^{3/4} a^{7/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{91 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{1620 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^{7/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{91 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{3240 a^2 x}{91 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}-\frac{198}{91} a x \left(a-b x^2\right)^{2/3}-\frac{3}{13} x \left(a-b x^2\right)^{2/3} \left(3 a+b x^2\right)",1,"Could not integrate","F",-1
125,0,0,568,3.9327732,"\int \frac{3 a+b x^2}{\sqrt[3]{a-b x^2}} \, dx","IntegrateAlgebraic[(3*a + b*x^2)/(a - b*x^2)^(1/3),x]","\int \frac{3 a+b x^2}{\sqrt[3]{a-b x^2}} \, dx","\frac{24 \sqrt{2} 3^{3/4} a^{4/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{7 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{36 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^{4/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{7 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{72 a x}{7 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}-\frac{3}{7} x \left(a-b x^2\right)^{2/3}",1,"Could not integrate","F",-1
126,0,0,204,4.5857206,"\int \frac{1}{\sqrt[3]{a-b x^2} \left(3 a+b x^2\right)} \, dx","IntegrateAlgebraic[1/((a - b*x^2)^(1/3)*(3*a + b*x^2)),x]","\int \frac{1}{\sqrt[3]{a-b x^2} \left(3 a+b x^2\right)} \, dx","\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{2} \sqrt[3]{a-b x^2}\right)}{\sqrt{b} x}\right)}{2\ 2^{2/3} \sqrt{3} a^{5/6} \sqrt{b}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt[6]{a} \left(\sqrt[3]{2} \sqrt[3]{a-b x^2}+\sqrt[3]{a}\right)}\right)}{2\ 2^{2/3} a^{5/6} \sqrt{b}}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right)}{2\ 2^{2/3} \sqrt{3} a^{5/6} \sqrt{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{6\ 2^{2/3} a^{5/6} \sqrt{b}}",1,"Could not integrate","F",-1
127,0,0,787,9.4957664,"\int \frac{1}{\sqrt[3]{a-b x^2} \left(3 a+b x^2\right)^2} \, dx","IntegrateAlgebraic[1/((a - b*x^2)^(1/3)*(3*a + b*x^2)^2),x]","\int \frac{1}{\sqrt[3]{a-b x^2} \left(3 a+b x^2\right)^2} \, dx","\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{2} \sqrt[3]{a-b x^2}\right)}{\sqrt{b} x}\right)}{8\ 2^{2/3} \sqrt{3} a^{11/6} \sqrt{b}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt[6]{a} \left(\sqrt[3]{2} \sqrt[3]{a-b x^2}+\sqrt[3]{a}\right)}\right)}{8\ 2^{2/3} a^{11/6} \sqrt{b}}+\frac{\left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{12 \sqrt{2} \sqrt[4]{3} a^{5/3} b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{\sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{16\ 3^{3/4} a^{5/3} b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right)}{8\ 2^{2/3} \sqrt{3} a^{11/6} \sqrt{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{24\ 2^{2/3} a^{11/6} \sqrt{b}}+\frac{x \left(a-b x^2\right)^{2/3}}{24 a^2 \left(3 a+b x^2\right)}-\frac{x}{24 a^2 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}",1,"Could not integrate","F",-1
128,0,0,818,15.5493942,"\int \frac{1}{\sqrt[3]{a-b x^2} \left(3 a+b x^2\right)^3} \, dx","IntegrateAlgebraic[1/((a - b*x^2)^(1/3)*(3*a + b*x^2)^3),x]","\int \frac{1}{\sqrt[3]{a-b x^2} \left(3 a+b x^2\right)^3} \, dx","\frac{5 \left(a-b x^2\right)^{2/3} x}{288 a^3 \left(b x^2+3 a\right)}-\frac{5 x}{288 a^3 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}+\frac{\left(a-b x^2\right)^{2/3} x}{48 a^2 \left(b x^2+3 a\right)^2}+\frac{5 \tan ^{-1}\left(\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right)}{144\ 2^{2/3} \sqrt{3} a^{17/6} \sqrt{b}}+\frac{5 \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{2} \sqrt[3]{a-b x^2}\right)}{\sqrt{b} x}\right)}{144\ 2^{2/3} \sqrt{3} a^{17/6} \sqrt{b}}-\frac{5 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{432\ 2^{2/3} a^{17/6} \sqrt{b}}+\frac{5 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{2} \sqrt[3]{a-b x^2}\right)}\right)}{144\ 2^{2/3} a^{17/6} \sqrt{b}}-\frac{5 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a-b x^2} \sqrt[3]{a}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{192\ 3^{3/4} a^{8/3} b \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} x}+\frac{5 \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a-b x^2} \sqrt[3]{a}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{144 \sqrt{2} \sqrt[4]{3} a^{8/3} b \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} x}",1,"Could not integrate","F",-1
129,0,0,623,4.7708618,"\int \frac{\left(3 a+b x^2\right)^3}{\left(a-b x^2\right)^{4/3}} \, dx","IntegrateAlgebraic[(3*a + b*x^2)^3/(a - b*x^2)^(4/3),x]","\int \frac{\left(3 a+b x^2\right)^3}{\left(a-b x^2\right)^{4/3}} \, dx","-\frac{6696 \sqrt{2} 3^{3/4} a^{7/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{91 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}+\frac{10044 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^{7/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{91 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}+\frac{20088 a^2 x}{91 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}+\frac{2538}{91} a x \left(a-b x^2\right)^{2/3}+\frac{6 x \left(3 a+b x^2\right)^2}{\sqrt[3]{a-b x^2}}+\frac{81}{13} x \left(a-b x^2\right)^{2/3} \left(3 a+b x^2\right)",1,"Could not integrate","F",-1
130,0,0,592,4.883072,"\int \frac{\left(3 a+b x^2\right)^2}{\left(a-b x^2\right)^{4/3}} \, dx","IntegrateAlgebraic[(3*a + b*x^2)^2/(a - b*x^2)^(4/3),x]","\int \frac{\left(3 a+b x^2\right)^2}{\left(a-b x^2\right)^{4/3}} \, dx","-\frac{108 \sqrt{2} 3^{3/4} a^{4/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{7 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}+\frac{162 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^{4/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{7 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}+\frac{324 a x}{7 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}+\frac{6 x \left(3 a+b x^2\right)}{\sqrt[3]{a-b x^2}}+\frac{45}{7} x \left(a-b x^2\right)^{2/3}",1,"Could not integrate","F",-1
131,0,0,561,4.3632879,"\int \frac{3 a+b x^2}{\left(a-b x^2\right)^{4/3}} \, dx","IntegrateAlgebraic[(3*a + b*x^2)/(a - b*x^2)^(4/3),x]","\int \frac{3 a+b x^2}{\left(a-b x^2\right)^{4/3}} \, dx","-\frac{3 \sqrt{2} 3^{3/4} \sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}+\frac{9 \sqrt[4]{3} \sqrt{2+\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{2 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}+\frac{6 x}{\sqrt[3]{a-b x^2}}+\frac{9 x}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}",1,"Could not integrate","F",-1
132,0,0,776,5.3929287,"\int \frac{1}{\left(a-b x^2\right)^{4/3} \left(3 a+b x^2\right)} \, dx","IntegrateAlgebraic[1/((a - b*x^2)^(4/3)*(3*a + b*x^2)),x]","\int \frac{1}{\left(a-b x^2\right)^{4/3} \left(3 a+b x^2\right)} \, dx","\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{2} \sqrt[3]{a-b x^2}\right)}{\sqrt{b} x}\right)}{8\ 2^{2/3} \sqrt{3} a^{11/6} \sqrt{b}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt[6]{a} \left(\sqrt[3]{2} \sqrt[3]{a-b x^2}+\sqrt[3]{a}\right)}\right)}{8\ 2^{2/3} a^{11/6} \sqrt{b}}-\frac{3^{3/4} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{4 \sqrt{2} a^{5/3} b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}+\frac{3 \sqrt[4]{3} \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{16 a^{5/3} b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right)}{8\ 2^{2/3} \sqrt{3} a^{11/6} \sqrt{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{24\ 2^{2/3} a^{11/6} \sqrt{b}}+\frac{3 x}{8 a^2 \sqrt[3]{a-b x^2}}+\frac{3 x}{8 a^2 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}",1,"Could not integrate","F",-1
133,0,0,807,10.070355,"\int \frac{1}{\left(a-b x^2\right)^{4/3} \left(3 a+b x^2\right)^2} \, dx","IntegrateAlgebraic[1/((a - b*x^2)^(4/3)*(3*a + b*x^2)^2),x]","\int \frac{1}{\left(a-b x^2\right)^{4/3} \left(3 a+b x^2\right)^2} \, dx","\frac{x}{12 a^3 \sqrt[3]{a-b x^2}}+\frac{x}{24 a^2 \sqrt[3]{a-b x^2} \left(b x^2+3 a\right)}+\frac{x}{12 a^3 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right)}{16\ 2^{2/3} \sqrt{3} a^{17/6} \sqrt{b}}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{2} \sqrt[3]{a-b x^2}\right)}{\sqrt{b} x}\right)}{16\ 2^{2/3} \sqrt{3} a^{17/6} \sqrt{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{48\ 2^{2/3} a^{17/6} \sqrt{b}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{2} \sqrt[3]{a-b x^2}\right)}\right)}{16\ 2^{2/3} a^{17/6} \sqrt{b}}+\frac{\sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a-b x^2} \sqrt[3]{a}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{8\ 3^{3/4} a^{8/3} b \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} x}-\frac{\left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a-b x^2} \sqrt[3]{a}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{6 \sqrt{2} \sqrt[4]{3} a^{8/3} b \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} x}",1,"Could not integrate","F",-1
134,0,0,849,15.6332294,"\int \frac{1}{\left(a-b x^2\right)^{4/3} \left(3 a+b x^2\right)^3} \, dx","IntegrateAlgebraic[1/((a - b*x^2)^(4/3)*(3*a + b*x^2)^3),x]","\int \frac{1}{\left(a-b x^2\right)^{4/3} \left(3 a+b x^2\right)^3} \, dx","-\frac{19 \left(a-b x^2\right)^{2/3} x}{1152 a^4 \left(b x^2+3 a\right)}+\frac{17 x}{192 a^3 \sqrt[3]{a-b x^2} \left(b x^2+3 a\right)}+\frac{19 x}{1152 a^4 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}+\frac{x}{48 a^2 \sqrt[3]{a-b x^2} \left(b x^2+3 a\right)^2}+\frac{7 \tan ^{-1}\left(\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right)}{288\ 2^{2/3} \sqrt{3} a^{23/6} \sqrt{b}}+\frac{7 \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{2} \sqrt[3]{a-b x^2}\right)}{\sqrt{b} x}\right)}{288\ 2^{2/3} \sqrt{3} a^{23/6} \sqrt{b}}-\frac{7 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{864\ 2^{2/3} a^{23/6} \sqrt{b}}+\frac{7 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{2} \sqrt[3]{a-b x^2}\right)}\right)}{288\ 2^{2/3} a^{23/6} \sqrt{b}}+\frac{19 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a-b x^2} \sqrt[3]{a}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{768\ 3^{3/4} a^{11/3} b \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} x}-\frac{19 \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a-b x^2} \sqrt[3]{a}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{576 \sqrt{2} \sqrt[4]{3} a^{11/3} b \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} x}",1,"Could not integrate","F",-1
135,0,0,653,5.4340851,"\int \frac{\left(3 a+b x^2\right)^4}{\left(a-b x^2\right)^{7/3}} \, dx","IntegrateAlgebraic[(3*a + b*x^2)^4/(a - b*x^2)^(7/3),x]","\int \frac{\left(3 a+b x^2\right)^4}{\left(a-b x^2\right)^{7/3}} \, dx","\frac{12312 \sqrt{2} 3^{3/4} a^{7/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{91 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{18468 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^{7/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{91 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{36936 a^2 x}{91 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}+\frac{3 x \left(3 a+b x^2\right)^3}{2 \left(a-b x^2\right)^{4/3}}-\frac{9 x \left(3 a+b x^2\right)^2}{2 \sqrt[3]{a-b x^2}}-\frac{81}{13} x \left(a-b x^2\right)^{2/3} \left(3 a+b x^2\right)-\frac{3240}{91} a x \left(a-b x^2\right)^{2/3}",1,"Could not integrate","F",-1
136,0,0,596,5.471308,"\int \frac{\left(3 a+b x^2\right)^3}{\left(a-b x^2\right)^{7/3}} \, dx","IntegrateAlgebraic[(3*a + b*x^2)^3/(a - b*x^2)^(7/3),x]","\int \frac{\left(3 a+b x^2\right)^3}{\left(a-b x^2\right)^{7/3}} \, dx","\frac{108 \sqrt{2} 3^{3/4} a^{4/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{7 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{162 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^{4/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{7 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}+\frac{3 x \left(3 a+b x^2\right)^2}{2 \left(a-b x^2\right)^{4/3}}-\frac{27}{14} x \left(a-b x^2\right)^{2/3}-\frac{324 a x}{7 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}",1,"Could not integrate","F",-1
137,0,0,44,5.232399,"\int \frac{\left(3 a+b x^2\right)^2}{\left(a-b x^2\right)^{7/3}} \, dx","IntegrateAlgebraic[(3*a + b*x^2)^2/(a - b*x^2)^(7/3),x]","\int \frac{\left(3 a+b x^2\right)^2}{\left(a-b x^2\right)^{7/3}} \, dx","\frac{3 x \left(3 a+b x^2\right)}{2 \left(a-b x^2\right)^{4/3}}+\frac{9 x}{2 \sqrt[3]{a-b x^2}}",1,"Could not integrate","F",-1
138,0,0,590,4.9073785,"\int \frac{3 a+b x^2}{\left(a-b x^2\right)^{7/3}} \, dx","IntegrateAlgebraic[(3*a + b*x^2)/(a - b*x^2)^(7/3),x]","\int \frac{3 a+b x^2}{\left(a-b x^2\right)^{7/3}} \, dx","-\frac{3\ 3^{3/4} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{2 \sqrt{2} a^{2/3} b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}+\frac{9 \sqrt[4]{3} \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{8 a^{2/3} b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}+\frac{9 x}{4 a \sqrt[3]{a-b x^2}}+\frac{9 x}{4 a \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}+\frac{3 x}{2 \left(a-b x^2\right)^{4/3}}",1,"Could not integrate","F",-1
139,0,0,796,5.5226158,"\int \frac{1}{\left(a-b x^2\right)^{7/3} \left(3 a+b x^2\right)} \, dx","IntegrateAlgebraic[1/((a - b*x^2)^(7/3)*(3*a + b*x^2)),x]","\int \frac{1}{\left(a-b x^2\right)^{7/3} \left(3 a+b x^2\right)} \, dx","\frac{21 x}{64 a^3 \sqrt[3]{a-b x^2}}+\frac{21 x}{64 a^3 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}+\frac{3 x}{32 a^2 \left(a-b x^2\right)^{4/3}}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right)}{32\ 2^{2/3} \sqrt{3} a^{17/6} \sqrt{b}}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{2} \sqrt[3]{a-b x^2}\right)}{\sqrt{b} x}\right)}{32\ 2^{2/3} \sqrt{3} a^{17/6} \sqrt{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{96\ 2^{2/3} a^{17/6} \sqrt{b}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{2} \sqrt[3]{a-b x^2}\right)}\right)}{32\ 2^{2/3} a^{17/6} \sqrt{b}}+\frac{21 \sqrt[4]{3} \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a-b x^2} \sqrt[3]{a}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{128 a^{8/3} b \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} x}-\frac{7\ 3^{3/4} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a-b x^2} \sqrt[3]{a}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{32 \sqrt{2} a^{8/3} b \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} x}",1,"Could not integrate","F",-1
140,0,0,827,10.2090672,"\int \frac{1}{\left(a-b x^2\right)^{7/3} \left(3 a+b x^2\right)^2} \, dx","IntegrateAlgebraic[1/((a - b*x^2)^(7/3)*(3*a + b*x^2)^2),x]","\int \frac{1}{\left(a-b x^2\right)^{7/3} \left(3 a+b x^2\right)^2} \, dx","\frac{79 x}{768 a^4 \sqrt[3]{a-b x^2}}+\frac{x}{24 a^2 \left(a-b x^2\right)^{4/3} \left(b x^2+3 a\right)}+\frac{79 x}{768 a^4 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}+\frac{5 x}{384 a^3 \left(a-b x^2\right)^{4/3}}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right)}{128\ 2^{2/3} a^{23/6} \sqrt{b}}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{2} \sqrt[3]{a-b x^2}\right)}{\sqrt{b} x}\right)}{128\ 2^{2/3} a^{23/6} \sqrt{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{128\ 2^{2/3} a^{23/6} \sqrt{b}}+\frac{3 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{2} \sqrt[3]{a-b x^2}\right)}\right)}{128\ 2^{2/3} a^{23/6} \sqrt{b}}+\frac{79 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a-b x^2} \sqrt[3]{a}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{512\ 3^{3/4} a^{11/3} b \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} x}-\frac{79 \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a-b x^2} \sqrt[3]{a}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{384 \sqrt{2} \sqrt[4]{3} a^{11/3} b \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} x}",1,"Could not integrate","F",-1
141,0,0,252,4.6899949,"\int \frac{1}{\left(-3 a-b x^2\right) \sqrt[3]{-a+b x^2}} \, dx","IntegrateAlgebraic[1/((-3*a - b*x^2)*(-a + b*x^2)^(1/3)),x]","\int \frac{1}{\left(-3 a-b x^2\right) \sqrt[3]{-a+b x^2}} \, dx","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt{a} \left(\sqrt[3]{-a}-\sqrt[3]{2} \sqrt[3]{b x^2-a}\right)}{\sqrt[3]{-a} \sqrt{b} x}\right)}{2\ 2^{2/3} \sqrt{3} \sqrt[3]{-a} \sqrt{a} \sqrt{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[3]{-a} \sqrt{b} x}{\sqrt{a} \left(\sqrt[3]{2} \sqrt[3]{b x^2-a}+\sqrt[3]{-a}\right)}\right)}{2\ 2^{2/3} \sqrt[3]{-a} \sqrt{a} \sqrt{b}}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right)}{2\ 2^{2/3} \sqrt{3} \sqrt[3]{-a} \sqrt{a} \sqrt{b}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{6\ 2^{2/3} \sqrt[3]{-a} \sqrt{a} \sqrt{b}}",1,"Could not integrate","F",-1
142,0,0,202,4.593045,"\int \frac{1}{\left(3 a-b x^2\right) \sqrt[3]{a+b x^2}} \, dx","IntegrateAlgebraic[1/((3*a - b*x^2)*(a + b*x^2)^(1/3)),x]","\int \frac{1}{\left(3 a-b x^2\right) \sqrt[3]{a+b x^2}} \, dx","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt[6]{a} \left(\sqrt[3]{2} \sqrt[3]{a+b x^2}+\sqrt[3]{a}\right)}\right)}{2\ 2^{2/3} a^{5/6} \sqrt{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{2} \sqrt[3]{a+b x^2}\right)}{\sqrt{b} x}\right)}{2\ 2^{2/3} \sqrt{3} a^{5/6} \sqrt{b}}-\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{6\ 2^{2/3} a^{5/6} \sqrt{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right)}{2\ 2^{2/3} \sqrt{3} a^{5/6} \sqrt{b}}",1,"Could not integrate","F",-1
143,0,0,204,4.6110862,"\int \frac{1}{\left(c-d x^2\right) \sqrt[3]{c+3 d x^2}} \, dx","IntegrateAlgebraic[1/((c - d*x^2)*(c + 3*d*x^2)^(1/3)),x]","\int \frac{1}{\left(c-d x^2\right) \sqrt[3]{c+3 d x^2}} \, dx","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt{d} x}{\sqrt[6]{c} \left(\sqrt[3]{2} \sqrt[3]{c+3 d x^2}+\sqrt[3]{c}\right)}\right)}{2\ 2^{2/3} c^{5/6} \sqrt{d}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{c} \left(\sqrt[3]{c}-\sqrt[3]{2} \sqrt[3]{c+3 d x^2}\right)}{\sqrt{d} x}\right)}{2\ 2^{2/3} c^{5/6} \sqrt{d}}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt{d} x}{\sqrt{c}}\right)}{2\ 2^{2/3} \sqrt{3} c^{5/6} \sqrt{d}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c}}{\sqrt{d} x}\right)}{2\ 2^{2/3} c^{5/6} \sqrt{d}}",1,"Could not integrate","F",-1
144,0,0,204,0.0005851,"\int \frac{1}{\sqrt[3]{a-b x^2} \left(3 a+b x^2\right)} \, dx","IntegrateAlgebraic[1/((a - b*x^2)^(1/3)*(3*a + b*x^2)),x]","\int \frac{1}{\sqrt[3]{a-b x^2} \left(3 a+b x^2\right)} \, dx","\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{2} \sqrt[3]{a-b x^2}\right)}{\sqrt{b} x}\right)}{2\ 2^{2/3} \sqrt{3} a^{5/6} \sqrt{b}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt[6]{a} \left(\sqrt[3]{2} \sqrt[3]{a-b x^2}+\sqrt[3]{a}\right)}\right)}{2\ 2^{2/3} a^{5/6} \sqrt{b}}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right)}{2\ 2^{2/3} \sqrt{3} a^{5/6} \sqrt{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{6\ 2^{2/3} a^{5/6} \sqrt{b}}",1,"Could not integrate","F",-1
145,0,0,204,4.6173904,"\int \frac{1}{\sqrt[3]{c-3 d x^2} \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[1/((c - 3*d*x^2)^(1/3)*(c + d*x^2)),x]","\int \frac{1}{\sqrt[3]{c-3 d x^2} \left(c+d x^2\right)} \, dx","\frac{\tan ^{-1}\left(\frac{\sqrt[6]{c} \left(\sqrt[3]{c}-\sqrt[3]{2} \sqrt[3]{c-3 d x^2}\right)}{\sqrt{d} x}\right)}{2\ 2^{2/3} c^{5/6} \sqrt{d}}+\frac{\sqrt{3} \tanh ^{-1}\left(\frac{\sqrt{3} \sqrt{d} x}{\sqrt[6]{c} \left(\sqrt[3]{2} \sqrt[3]{c-3 d x^2}+\sqrt[3]{c}\right)}\right)}{2\ 2^{2/3} c^{5/6} \sqrt{d}}+\frac{\tan ^{-1}\left(\frac{\sqrt{c}}{\sqrt{d} x}\right)}{2\ 2^{2/3} c^{5/6} \sqrt{d}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{3} \sqrt{d} x}{\sqrt{c}}\right)}{2\ 2^{2/3} \sqrt{3} c^{5/6} \sqrt{d}}",1,"Could not integrate","F",-1
146,0,0,113,3.8932753,"\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
147,0,0,109,3.8638161,"\int \frac{1}{\left(3-x^2\right) \sqrt[3]{1+x^2}} \, dx","IntegrateAlgebraic[1/((3 - x^2)*(1 + x^2)^(1/3)),x]","\int \frac{1}{\left(3-x^2\right) \sqrt[3]{1+x^2}} \, dx","\frac{\tan ^{-1}\left(\frac{x}{\sqrt[3]{2} \sqrt[3]{x^2+1}+1}\right)}{2\ 2^{2/3}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{3} \left(1-\sqrt[3]{2} \sqrt[3]{x^2+1}\right)}{x}\right)}{2\ 2^{2/3} \sqrt{3}}-\frac{\tan ^{-1}(x)}{6\ 2^{2/3}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{3}}{x}\right)}{2\ 2^{2/3} \sqrt{3}}",1,"Could not integrate","F",-1
148,1,166,96,0.2283621,"\int \frac{3-x}{\sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","IntegrateAlgebraic[(3 - x)/((1 - x^2)^(1/3)*(3 + x^2)),x]","\frac{\log \left(2 \sqrt[3]{1-x^2}+2^{2/3} x+2^{2/3}\right)}{2^{2/3}}-\frac{\log \left(-\sqrt[3]{2} x^2-2 \left(1-x^2\right)^{2/3}+\left(2^{2/3} x+2^{2/3}\right) \sqrt[3]{1-x^2}-2 \sqrt[3]{2} x-\sqrt[3]{2}\right)}{2\ 2^{2/3}}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{1-x^2}}{\sqrt[3]{1-x^2}-2^{2/3} x-2^{2/3}}\right)}{2^{2/3}}","-\frac{\log \left(x^2+3\right)}{2\ 2^{2/3}}+\frac{3 \log \left((x+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{1-x}\right)}{2\ 2^{2/3}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2^{2/3} (x+1)^{2/3}}{\sqrt{3} \sqrt[3]{1-x}}\right)}{2^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(1 - x^2)^(1/3))/(-2^(2/3) - 2^(2/3)*x + (1 - x^2)^(1/3))])/2^(2/3) + Log[2^(2/3) + 2^(2/3)*x + 2*(1 - x^2)^(1/3)]/2^(2/3) - Log[-2^(1/3) - 2*2^(1/3)*x - 2^(1/3)*x^2 + (2^(2/3) + 2^(2/3)*x)*(1 - x^2)^(1/3) - 2*(1 - x^2)^(2/3)]/(2*2^(2/3))","A",1
149,1,168,95,0.2138528,"\int \frac{3+x}{\sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","IntegrateAlgebraic[(3 + x)/((1 - x^2)^(1/3)*(3 + x^2)),x]","-\frac{\log \left(-2 \sqrt[3]{1-x^2}+2^{2/3} x-2^{2/3}\right)}{2^{2/3}}+\frac{\log \left(\sqrt[3]{2} x^2+2 \left(1-x^2\right)^{2/3}+\left(2^{2/3} x-2^{2/3}\right) \sqrt[3]{1-x^2}-2 \sqrt[3]{2} x+\sqrt[3]{2}\right)}{2\ 2^{2/3}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{1-x^2}}{\sqrt[3]{1-x^2}+2^{2/3} x-2^{2/3}}\right)}{2^{2/3}}","\frac{\log \left(x^2+3\right)}{2\ 2^{2/3}}-\frac{3 \log \left((1-x)^{2/3}+\sqrt[3]{2} \sqrt[3]{x+1}\right)}{2\ 2^{2/3}}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2^{2/3} (1-x)^{2/3}}{\sqrt{3} \sqrt[3]{x+1}}\right)}{2^{2/3}}",1,"-((Sqrt[3]*ArcTan[(Sqrt[3]*(1 - x^2)^(1/3))/(-2^(2/3) + 2^(2/3)*x + (1 - x^2)^(1/3))])/2^(2/3)) - Log[-2^(2/3) + 2^(2/3)*x - 2*(1 - x^2)^(1/3)]/2^(2/3) + Log[2^(1/3) - 2*2^(1/3)*x + 2^(1/3)*x^2 + (-2^(2/3) + 2^(2/3)*x)*(1 - x^2)^(1/3) + 2*(1 - x^2)^(2/3)]/(2*2^(2/3))","A",1
150,0,0,151,4.8416301,"\int \frac{1}{\sqrt[3]{a+b x^2} \left(\frac{9 a d}{b}+d x^2\right)} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^(1/3)*((9*a*d)/b + d*x^2)),x]","\int \frac{1}{\sqrt[3]{a+b x^2} \left(\frac{9 a d}{b}+d x^2\right)} \, dx","\frac{\sqrt{b} \tan ^{-1}\left(\frac{\left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}{3 \sqrt[6]{a} \sqrt{b} x}\right)}{12 a^{5/6} d}-\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\sqrt{b} x}\right)}{4 \sqrt{3} a^{5/6} d}+\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{3 \sqrt{a}}\right)}{12 a^{5/6} d}",1,"Could not integrate","F",-1
151,0,0,153,4.8931798,"\int \frac{1}{\sqrt[3]{a-b x^2} \left(-\frac{9 a d}{b}+d x^2\right)} \, dx","IntegrateAlgebraic[1/((a - b*x^2)^(1/3)*((-9*a*d)/b + d*x^2)),x]","\int \frac{1}{\sqrt[3]{a-b x^2} \left(-\frac{9 a d}{b}+d x^2\right)} \, dx","-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\sqrt{b} x}\right)}{4 \sqrt{3} a^{5/6} d}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}{3 \sqrt[6]{a} \sqrt{b} x}\right)}{12 a^{5/6} d}-\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} x}{3 \sqrt{a}}\right)}{12 a^{5/6} d}",1,"Could not integrate","F",-1
152,0,0,151,4.9235541,"\int \frac{1}{\sqrt[3]{-a+b x^2} \left(-\frac{9 a d}{b}+d x^2\right)} \, dx","IntegrateAlgebraic[1/((-a + b*x^2)^(1/3)*((-9*a*d)/b + d*x^2)),x]","\int \frac{1}{\sqrt[3]{-a+b x^2} \left(-\frac{9 a d}{b}+d x^2\right)} \, dx","\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{b x^2-a}+\sqrt[3]{a}\right)}{\sqrt{b} x}\right)}{4 \sqrt{3} a^{5/6} d}-\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{b x^2-a}+\sqrt[3]{a}\right)^2}{3 \sqrt[6]{a} \sqrt{b} x}\right)}{12 a^{5/6} d}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} x}{3 \sqrt{a}}\right)}{12 a^{5/6} d}",1,"Could not integrate","F",-1
153,0,0,153,4.9059922,"\int \frac{1}{\sqrt[3]{-a-b x^2} \left(\frac{9 a d}{b}+d x^2\right)} \, dx","IntegrateAlgebraic[1/((-a - b*x^2)^(1/3)*((9*a*d)/b + d*x^2)),x]","\int \frac{1}{\sqrt[3]{-a-b x^2} \left(\frac{9 a d}{b}+d x^2\right)} \, dx","-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\left(\sqrt[3]{-a-b x^2}+\sqrt[3]{a}\right)^2}{3 \sqrt[6]{a} \sqrt{b} x}\right)}{12 a^{5/6} d}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{-a-b x^2}+\sqrt[3]{a}\right)}{\sqrt{b} x}\right)}{4 \sqrt{3} a^{5/6} d}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{3 \sqrt{a}}\right)}{12 a^{5/6} d}",1,"Could not integrate","F",-1
154,0,0,151,4.5943311,"\int \frac{1}{\sqrt[3]{2+b x^2} \left(\frac{18 d}{b}+d x^2\right)} \, dx","IntegrateAlgebraic[1/((2 + b*x^2)^(1/3)*((18*d)/b + d*x^2)),x]","\int \frac{1}{\sqrt[3]{2+b x^2} \left(\frac{18 d}{b}+d x^2\right)} \, dx","\frac{\sqrt{b} \tan ^{-1}\left(\frac{\left(\sqrt[3]{2}-\sqrt[3]{b x^2+2}\right)^2}{3 \sqrt[6]{2} \sqrt{b} x}\right)}{12\ 2^{5/6} d}-\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt[6]{2} \sqrt{3} \left(\sqrt[3]{2}-\sqrt[3]{b x^2+2}\right)}{\sqrt{b} x}\right)}{4\ 2^{5/6} \sqrt{3} d}+\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{3 \sqrt{2}}\right)}{12\ 2^{5/6} d}",1,"Could not integrate","F",-1
155,0,0,147,4.5872829,"\int \frac{1}{\sqrt[3]{-2+b x^2} \left(-\frac{18 d}{b}+d x^2\right)} \, dx","IntegrateAlgebraic[1/((-2 + b*x^2)^(1/3)*((-18*d)/b + d*x^2)),x]","\int \frac{1}{\sqrt[3]{-2+b x^2} \left(-\frac{18 d}{b}+d x^2\right)} \, dx","\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt[6]{2} \sqrt{3} \left(\sqrt[3]{b x^2-2}+\sqrt[3]{2}\right)}{\sqrt{b} x}\right)}{4\ 2^{5/6} \sqrt{3} d}-\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{b x^2-2}+\sqrt[3]{2}\right)^2}{3 \sqrt[6]{2} \sqrt{b} x}\right)}{12\ 2^{5/6} d}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} x}{3 \sqrt{2}}\right)}{12\ 2^{5/6} d}",1,"Could not integrate","F",-1
156,0,0,123,4.1688712,"\int \frac{1}{\sqrt[3]{2+3 x^2} \left(6 d+d x^2\right)} \, dx","IntegrateAlgebraic[1/((2 + 3*x^2)^(1/3)*(6*d + d*x^2)),x]","\int \frac{1}{\sqrt[3]{2+3 x^2} \left(6 d+d x^2\right)} \, dx","\frac{\tan ^{-1}\left(\frac{\left(\sqrt[3]{2}-\sqrt[3]{3 x^2+2}\right)^2}{3 \sqrt[6]{2} \sqrt{3} x}\right)}{4\ 2^{5/6} \sqrt{3} d}-\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{2} \left(\sqrt[3]{2}-\sqrt[3]{3 x^2+2}\right)}{x}\right)}{4\ 2^{5/6} d}+\frac{\tan ^{-1}\left(\frac{x}{\sqrt{6}}\right)}{4\ 2^{5/6} \sqrt{3} d}",1,"Could not integrate","F",-1
157,0,0,123,4.154762,"\int \frac{1}{\sqrt[3]{2-3 x^2} \left(-6 d+d x^2\right)} \, dx","IntegrateAlgebraic[1/((2 - 3*x^2)^(1/3)*(-6*d + d*x^2)),x]","\int \frac{1}{\sqrt[3]{2-3 x^2} \left(-6 d+d x^2\right)} \, dx","-\frac{\tan ^{-1}\left(\frac{\sqrt[6]{2} \left(\sqrt[3]{2}-\sqrt[3]{2-3 x^2}\right)}{x}\right)}{4\ 2^{5/6} d}+\frac{\tanh ^{-1}\left(\frac{\left(\sqrt[3]{2}-\sqrt[3]{2-3 x^2}\right)^2}{3 \sqrt[6]{2} \sqrt{3} x}\right)}{4\ 2^{5/6} \sqrt{3} d}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt{6}}\right)}{4\ 2^{5/6} \sqrt{3} d}",1,"Could not integrate","F",-1
158,0,0,119,4.1415161,"\int \frac{1}{\sqrt[3]{-2+3 x^2} \left(-6 d+d x^2\right)} \, dx","IntegrateAlgebraic[1/((-2 + 3*x^2)^(1/3)*(-6*d + d*x^2)),x]","\int \frac{1}{\sqrt[3]{-2+3 x^2} \left(-6 d+d x^2\right)} \, dx","\frac{\tan ^{-1}\left(\frac{\sqrt[6]{2} \left(\sqrt[3]{3 x^2-2}+\sqrt[3]{2}\right)}{x}\right)}{4\ 2^{5/6} d}-\frac{\tanh ^{-1}\left(\frac{\left(\sqrt[3]{3 x^2-2}+\sqrt[3]{2}\right)^2}{3 \sqrt[6]{2} \sqrt{3} x}\right)}{4\ 2^{5/6} \sqrt{3} d}+\frac{\tanh ^{-1}\left(\frac{x}{\sqrt{6}}\right)}{4\ 2^{5/6} \sqrt{3} d}",1,"Could not integrate","F",-1
159,0,0,119,4.2063473,"\int \frac{1}{\sqrt[3]{-2-3 x^2} \left(6 d+d x^2\right)} \, dx","IntegrateAlgebraic[1/((-2 - 3*x^2)^(1/3)*(6*d + d*x^2)),x]","\int \frac{1}{\sqrt[3]{-2-3 x^2} \left(6 d+d x^2\right)} \, dx","-\frac{\tan ^{-1}\left(\frac{\left(\sqrt[3]{-3 x^2-2}+\sqrt[3]{2}\right)^2}{3 \sqrt[6]{2} \sqrt{3} x}\right)}{4\ 2^{5/6} \sqrt{3} d}+\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{2} \left(\sqrt[3]{-3 x^2-2}+\sqrt[3]{2}\right)}{x}\right)}{4\ 2^{5/6} d}-\frac{\tan ^{-1}\left(\frac{x}{\sqrt{6}}\right)}{4\ 2^{5/6} \sqrt{3} d}",1,"Could not integrate","F",-1
160,0,0,70,3.8299115,"\int \frac{1}{\sqrt[3]{1+x^2} \left(9+x^2\right)} \, dx","IntegrateAlgebraic[1/((1 + x^2)^(1/3)*(9 + x^2)),x]","\int \frac{1}{\sqrt[3]{1+x^2} \left(9+x^2\right)} \, dx","\frac{1}{12} \tan ^{-1}\left(\frac{\left(1-\sqrt[3]{x^2+1}\right)^2}{3 x}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt{3} \left(1-\sqrt[3]{x^2+1}\right)}{x}\right)}{4 \sqrt{3}}+\frac{1}{12} \tan ^{-1}\left(\frac{x}{3}\right)",1,"Could not integrate","F",-1
161,0,0,104,4.355618,"\int \frac{1}{\sqrt[3]{1+b x^2} \left(9+b x^2\right)} \, dx","IntegrateAlgebraic[1/((1 + b*x^2)^(1/3)*(9 + b*x^2)),x]","\int \frac{1}{\sqrt[3]{1+b x^2} \left(9+b x^2\right)} \, dx","\frac{\tan ^{-1}\left(\frac{\left(1-\sqrt[3]{b x^2+1}\right)^2}{3 \sqrt{b} x}\right)}{12 \sqrt{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{3} \left(1-\sqrt[3]{b x^2+1}\right)}{\sqrt{b} x}\right)}{4 \sqrt{3} \sqrt{b}}+\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{3}\right)}{12 \sqrt{b}}",1,"Could not integrate","F",-1
162,0,0,74,3.929037,"\int \frac{1}{\sqrt[3]{1-x^2} \left(9-x^2\right)} \, dx","IntegrateAlgebraic[1/((1 - x^2)^(1/3)*(9 - x^2)),x]","\int \frac{1}{\sqrt[3]{1-x^2} \left(9-x^2\right)} \, dx","\frac{\tan ^{-1}\left(\frac{\sqrt{3} \left(1-\sqrt[3]{1-x^2}\right)}{x}\right)}{4 \sqrt{3}}-\frac{1}{12} \tanh ^{-1}\left(\frac{\left(1-\sqrt[3]{1-x^2}\right)^2}{3 x}\right)+\frac{1}{12} \tanh ^{-1}\left(\frac{x}{3}\right)",1,"Could not integrate","F",-1
163,0,0,79,0.5716797,"\int \frac{\sqrt{-1+c^2 x^2}}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[Sqrt[-1 + c^2*x^2]/(d - c^2*d*x^2)^(5/2),x]","\int \frac{\sqrt{-1+c^2 x^2}}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","\frac{\sqrt{c^2 x^2-1} \tanh ^{-1}(c x)}{2 c d^2 \sqrt{d-c^2 d x^2}}+\frac{x \sqrt{c^2 x^2-1}}{2 d \left(d-c^2 d x^2\right)^{3/2}}",1,"Could not integrate","F",-1
164,0,0,74,0.7653224,"\int \frac{1}{\left(-1+c^2 x^2\right)^{3/2} \sqrt{d-c^2 d x^2}} \, dx","IntegrateAlgebraic[1/((-1 + c^2*x^2)^(3/2)*Sqrt[d - c^2*d*x^2]),x]","\int \frac{1}{\left(-1+c^2 x^2\right)^{3/2} \sqrt{d-c^2 d x^2}} \, dx","\frac{d x \sqrt{c^2 x^2-1}}{2 \left(d-c^2 d x^2\right)^{3/2}}+\frac{\sqrt{c^2 x^2-1} \tanh ^{-1}(c x)}{2 c \sqrt{d-c^2 d x^2}}",1,"Could not integrate","F",-1
165,0,0,76,0.8331723,"\int \frac{1}{\sqrt{-1+c^2 x^2} \left(d-c^2 d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[1/(Sqrt[-1 + c^2*x^2]*(d - c^2*d*x^2)^(3/2)),x]","\int \frac{1}{\sqrt{-1+c^2 x^2} \left(d-c^2 d x^2\right)^{3/2}} \, dx","-\frac{x \sqrt{c^2 x^2-1}}{2 \left(d-c^2 d x^2\right)^{3/2}}-\frac{\sqrt{c^2 x^2-1} \tanh ^{-1}(c x)}{2 c d \sqrt{d-c^2 d x^2}}",1,"Could not integrate","F",-1
166,0,0,328,1.1528666,"\int \left(a+b x^2\right)^{3/2} \sqrt{c+d x^2} \, dx","IntegrateAlgebraic[(a + b*x^2)^(3/2)*Sqrt[c + d*x^2],x]","\int \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^2-7 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)}{15 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} \left(\frac{3 a^2 d}{b}+7 a c-\frac{2 b c^2}{d}\right)}{15 \sqrt{c+d x^2}}-\frac{c^{3/2} \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)}{15 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{b x \sqrt{a+b x^2} \left(c+d x^2\right)^{3/2}}{5 d}-\frac{2 x \sqrt{a+b x^2} \sqrt{c+d x^2} (b c-3 a d)}{15 d}",1,"Could not integrate","F",-1
167,0,0,249,0.5810586,"\int \sqrt{a+b x^2} \sqrt{c+d x^2} \, dx","IntegrateAlgebraic[Sqrt[a + b*x^2]*Sqrt[c + d*x^2],x]","\int \sqrt{a+b x^2} \sqrt{c+d x^2} \, dx","\frac{2 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 \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{1}{3} x \sqrt{a+b x^2} \sqrt{c+d x^2}+\frac{x \sqrt{a+b x^2} (a d+b c)}{3 b \sqrt{c+d x^2}}-\frac{\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 \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
168,0,0,204,0.5488206,"\int \frac{\sqrt{c+d x^2}}{\sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[Sqrt[c + d*x^2]/Sqrt[a + b*x^2],x]","\int \frac{\sqrt{c+d x^2}}{\sqrt{a+b 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)}{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{d x \sqrt{a+b x^2}}{b \sqrt{c+d x^2}}-\frac{\sqrt{c} \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)}{b \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
169,0,0,84,0.9811361,"\int \frac{\sqrt{c+d x^2}}{\left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[Sqrt[c + d*x^2]/(a + b*x^2)^(3/2),x]","\int \frac{\sqrt{c+d x^2}}{\left(a+b x^2\right)^{3/2}} \, dx","\frac{\sqrt{c+d x^2} E\left(\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)|1-\frac{a d}{b c}\right)}{\sqrt{a} \sqrt{b} \sqrt{a+b x^2} \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}}}",1,"Could not integrate","F",-1
170,0,0,237,1.1017835,"\int \frac{\sqrt{c+d x^2}}{\left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[Sqrt[c + d*x^2]/(a + b*x^2)^(5/2),x]","\int \frac{\sqrt{c+d x^2}}{\left(a+b x^2\right)^{5/2}} \, dx","\frac{\sqrt{c+d x^2} (2 b c-a d) E\left(\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)|1-\frac{a d}{b c}\right)}{3 a^{3/2} \sqrt{b} \sqrt{a+b x^2} (b c-a d) \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}}}-\frac{c^{3/2} \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+d x^2} (b c-a d) \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{x \sqrt{c+d x^2}}{3 a \left(a+b x^2\right)^{3/2}}",1,"Could not integrate","F",-1
171,0,0,309,1.280686,"\int \frac{\sqrt{c+d x^2}}{\left(a+b x^2\right)^{7/2}} \, dx","IntegrateAlgebraic[Sqrt[c + d*x^2]/(a + b*x^2)^(7/2),x]","\int \frac{\sqrt{c+d x^2}}{\left(a+b x^2\right)^{7/2}} \, dx","-\frac{2 c^{3/2} \sqrt{d} \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 a^3 \sqrt{c+d x^2} (b c-a d)^2 \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{x \sqrt{c+d x^2} (4 b c-3 a d)}{15 a^2 \left(a+b x^2\right)^{3/2} (b c-a d)}+\frac{\sqrt{c+d 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{b} x}{\sqrt{a}}\right)|1-\frac{a d}{b c}\right)}{15 a^{5/2} \sqrt{b} \sqrt{a+b x^2} (b c-a d)^2 \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}}}+\frac{x \sqrt{c+d x^2}}{5 a \left(a+b x^2\right)^{5/2}}",1,"Could not integrate","F",-1
172,0,0,410,1.6538835,"\int \left(a+b x^2\right)^{3/2} \left(c+d x^2\right)^{3/2} \, dx","IntegrateAlgebraic[(a + b*x^2)^(3/2)*(c + d*x^2)^(3/2),x]","\int \left(a+b x^2\right)^{3/2} \left(c+d x^2\right)^{3/2} \, dx","-\frac{2 x \sqrt{a+b x^2} (a d+b c) \left(a^2 d^2-6 a b c d+b^2 c^2\right)}{35 b^2 d \sqrt{c+d x^2}}+\frac{2 \sqrt{c} \sqrt{a+b x^2} (a d+b c) \left(a^2 d^2-6 a b c d+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^{3/2} \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} \left(a^2 d^2-18 a b c d+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^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{1}{35} x \sqrt{a+b x^2} \sqrt{c+d x^2} \left(-\frac{2 a^2 d}{b}+9 a c+\frac{b c^2}{d}\right)+\frac{d x \left(a+b x^2\right)^{5/2} \sqrt{c+d x^2}}{7 b}+\frac{2 x \left(a+b x^2\right)^{3/2} \sqrt{c+d x^2} (4 b c-a d)}{35 b}",1,"Could not integrate","F",-1
173,0,0,336,1.0992823,"\int \sqrt{a+b x^2} \left(c+d x^2\right)^{3/2} \, dx","IntegrateAlgebraic[Sqrt[a + b*x^2]*(c + d*x^2)^(3/2),x]","\int \sqrt{a+b x^2} \left(c+d x^2\right)^{3/2} \, 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)}{15 b^2 \sqrt{c+d x^2}}-\frac{\sqrt{c} \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)}{15 b^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{c^{3/2} \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)}{15 b \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{d x \left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}}{5 b}+\frac{2 x \sqrt{a+b x^2} \sqrt{c+d x^2} (3 b c-a d)}{15 b}",1,"Could not integrate","F",-1
174,0,0,273,0.9741281,"\int \frac{\left(c+d x^2\right)^{3/2}}{\sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(c + d*x^2)^(3/2)/Sqrt[a + b*x^2],x]","\int \frac{\left(c+d x^2\right)^{3/2}}{\sqrt{a+b x^2}} \, dx","\frac{2 d x \sqrt{a+b x^2} (2 b c-a d)}{3 b^2 \sqrt{c+d x^2}}-\frac{2 \sqrt{c} \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 b^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} (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 b \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{d x \sqrt{a+b x^2} \sqrt{c+d x^2}}{3 b}",1,"Could not integrate","F",-1
175,0,0,267,1.3633297,"\int \frac{\left(c+d x^2\right)^{3/2}}{\left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(c + d*x^2)^(3/2)/(a + b*x^2)^(3/2),x]","\int \frac{\left(c+d x^2\right)^{3/2}}{\left(a+b x^2\right)^{3/2}} \, dx","-\frac{d x \sqrt{a+b x^2} (b c-2 a d)}{a b^2 \sqrt{c+d x^2}}+\frac{\sqrt{c} \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)}{a b^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{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)}{a b \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{x \sqrt{c+d x^2} (b c-a d)}{a b \sqrt{a+b x^2}}",1,"Could not integrate","F",-1
176,0,0,229,1.6099609,"\int \frac{\left(c+d x^2\right)^{3/2}}{\left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(c + d*x^2)^(3/2)/(a + b*x^2)^(5/2),x]","\int \frac{\left(c+d x^2\right)^{3/2}}{\left(a+b x^2\right)^{5/2}} \, dx","\frac{2 \sqrt{c+d x^2} (a d+b c) E\left(\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)|1-\frac{a d}{b c}\right)}{3 a^{3/2} b^{3/2} \sqrt{a+b x^2} \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}}}-\frac{c^{3/2} \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 b \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{x \sqrt{c+d x^2} (b c-a d)}{3 a b \left(a+b x^2\right)^{3/2}}",1,"Could not integrate","F",-1
177,0,0,315,1.6804087,"\int \frac{\left(c+d x^2\right)^{3/2}}{\left(a+b x^2\right)^{7/2}} \, dx","IntegrateAlgebraic[(c + d*x^2)^(3/2)/(a + b*x^2)^(7/2),x]","\int \frac{\left(c+d x^2\right)^{3/2}}{\left(a+b x^2\right)^{7/2}} \, dx","-\frac{c^{3/2} \sqrt{d} \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 a^3 b \sqrt{c+d x^2} (b c-a d) \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{2 x \sqrt{c+d x^2} (a d+2 b c)}{15 a^2 b \left(a+b x^2\right)^{3/2}}+\frac{\sqrt{c+d 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{b} x}{\sqrt{a}}\right)|1-\frac{a d}{b c}\right)}{15 a^{5/2} b^{3/2} \sqrt{a+b x^2} (b c-a d) \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}}}+\frac{x \sqrt{c+d x^2} (b c-a d)}{5 a b \left(a+b x^2\right)^{5/2}}",1,"Could not integrate","F",-1
178,0,0,235,0.4583067,"\int \sqrt{2+b x^2} \sqrt{3+d x^2} \, dx","IntegrateAlgebraic[Sqrt[2 + b*x^2]*Sqrt[3 + d*x^2],x]","\int \sqrt{2+b x^2} \sqrt{3+d x^2} \, dx","\frac{1}{3} x \sqrt{b x^2+2} \sqrt{d x^2+3}+\frac{x (3 b+2 d) \sqrt{b x^2+2}}{3 b \sqrt{d x^2+3}}+\frac{2 \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)}{\sqrt{d} \sqrt{d x^2+3} \sqrt{\frac{b x^2+2}{d x^2+3}}}-\frac{\sqrt{2} (3 b+2 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 \sqrt{d} \sqrt{d x^2+3} \sqrt{\frac{b x^2+2}{d x^2+3}}}",1,"Could not integrate","F",-1
179,0,0,38,0.358961,"\int \sqrt{3-6 x^2} \sqrt{2+4 x^2} \, dx","IntegrateAlgebraic[Sqrt[3 - 6*x^2]*Sqrt[2 + 4*x^2],x]","\int \sqrt{3-6 x^2} \sqrt{2+4 x^2} \, dx","\sqrt{\frac{2}{3}} \sqrt{1-4 x^4} x+\frac{2 F\left(\left.\sin ^{-1}\left(\sqrt{2} x\right)\right|-1\right)}{\sqrt{3}}",1,"Could not integrate","F",-1
180,0,0,20,0.2251368,"\int \sqrt{2+4 x^2} \sqrt{3+6 x^2} \, dx","IntegrateAlgebraic[Sqrt[2 + 4*x^2]*Sqrt[3 + 6*x^2],x]","\int \sqrt{2+4 x^2} \sqrt{3+6 x^2} \, dx","2 \sqrt{\frac{2}{3}} x^3+\sqrt{6} x",1,"Could not integrate","F",-1
181,0,0,182,0.5036561,"\int \frac{\sqrt{2+b x^2}}{\sqrt{3+d x^2}} \, dx","IntegrateAlgebraic[Sqrt[2 + b*x^2]/Sqrt[3 + d*x^2],x]","\int \frac{\sqrt{2+b x^2}}{\sqrt{3+d x^2}} \, dx","\frac{x \sqrt{b x^2+2}}{\sqrt{d x^2+3}}+\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)}{\sqrt{d} \sqrt{d x^2+3} \sqrt{\frac{b x^2+2}{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)}{\sqrt{d} \sqrt{d x^2+3} \sqrt{\frac{b x^2+2}{d x^2+3}}}",1,"Could not integrate","F",-1
182,0,0,91,0.4025777,"\int \frac{\sqrt{4-x^2}}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[Sqrt[4 - x^2]/Sqrt[c + d*x^2],x]","\int \frac{\sqrt{4-x^2}}{\sqrt{c+d x^2}} \, dx","\frac{(c+4 d) \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}}-\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}}",1,"Could not integrate","F",-1
183,0,0,150,0.3827684,"\int \frac{\sqrt{4+x^2}}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[Sqrt[4 + x^2]/Sqrt[c + d*x^2],x]","\int \frac{\sqrt{4+x^2}}{\sqrt{c+d x^2}} \, dx","\frac{x \sqrt{c+d x^2}}{d \sqrt{x^2+4}}+\frac{4 \sqrt{c+d x^2} F\left(\tan ^{-1}\left(\frac{x}{2}\right)|1-\frac{4 d}{c}\right)}{c \sqrt{x^2+4} \sqrt{\frac{c+d x^2}{c \left(x^2+4\right)}}}-\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
184,0,0,20,0.2205361,"\int \frac{\sqrt{1-x^2}}{\sqrt{2-3 x^2}} \, dx","IntegrateAlgebraic[Sqrt[1 - x^2]/Sqrt[2 - 3*x^2],x]","\int \frac{\sqrt{1-x^2}}{\sqrt{2-3 x^2}} \, dx","\frac{E\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|\frac{2}{3}\right)}{\sqrt{3}}",1,"Could not integrate","F",-1
185,0,0,21,0.2135783,"\int \frac{\sqrt{4-x^2}}{\sqrt{2-3 x^2}} \, dx","IntegrateAlgebraic[Sqrt[4 - x^2]/Sqrt[2 - 3*x^2],x]","\int \frac{\sqrt{4-x^2}}{\sqrt{2-3 x^2}} \, dx","\frac{2 E\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|\frac{1}{6}\right)}{\sqrt{3}}",1,"Could not integrate","F",-1
186,0,0,20,0.2492813,"\int \frac{\sqrt{1-4 x^2}}{\sqrt{2-3 x^2}} \, dx","IntegrateAlgebraic[Sqrt[1 - 4*x^2]/Sqrt[2 - 3*x^2],x]","\int \frac{\sqrt{1-4 x^2}}{\sqrt{2-3 x^2}} \, dx","\frac{E\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|\frac{8}{3}\right)}{\sqrt{3}}",1,"Could not integrate","F",-1
187,0,0,4,0.2383483,"\int \frac{\sqrt{1+x^2}}{\sqrt{1-x^2}} \, dx","IntegrateAlgebraic[Sqrt[1 + x^2]/Sqrt[1 - x^2],x]","\int \frac{\sqrt{1+x^2}}{\sqrt{1-x^2}} \, dx","E\left(\left.\sin ^{-1}(x)\right|-1\right)",1,"Could not integrate","F",-1
188,0,0,20,0.2318778,"\int \frac{\sqrt{1+x^2}}{\sqrt{2-3 x^2}} \, dx","IntegrateAlgebraic[Sqrt[1 + x^2]/Sqrt[2 - 3*x^2],x]","\int \frac{\sqrt{1+x^2}}{\sqrt{2-3 x^2}} \, dx","\frac{E\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|-\frac{2}{3}\right)}{\sqrt{3}}",1,"Could not integrate","F",-1
189,0,0,21,0.2456386,"\int \frac{\sqrt{4+x^2}}{\sqrt{2-3 x^2}} \, dx","IntegrateAlgebraic[Sqrt[4 + x^2]/Sqrt[2 - 3*x^2],x]","\int \frac{\sqrt{4+x^2}}{\sqrt{2-3 x^2}} \, dx","\frac{2 E\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|-\frac{1}{6}\right)}{\sqrt{3}}",1,"Could not integrate","F",-1
190,0,0,20,0.250936,"\int \frac{\sqrt{1+4 x^2}}{\sqrt{2-3 x^2}} \, dx","IntegrateAlgebraic[Sqrt[1 + 4*x^2]/Sqrt[2 - 3*x^2],x]","\int \frac{\sqrt{1+4 x^2}}{\sqrt{2-3 x^2}} \, dx","\frac{E\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|-\frac{8}{3}\right)}{\sqrt{3}}",1,"Could not integrate","F",-1
191,0,0,13,0.2339742,"\int \frac{\sqrt{1-x^2}}{\sqrt{1+x^2}} \, dx","IntegrateAlgebraic[Sqrt[1 - x^2]/Sqrt[1 + x^2],x]","\int \frac{\sqrt{1-x^2}}{\sqrt{1+x^2}} \, dx","2 F\left(\left.\sin ^{-1}(x)\right|-1\right)-E\left(\left.\sin ^{-1}(x)\right|-1\right)",1,"Could not integrate","F",-1
192,0,0,31,0.2390714,"\int \frac{\sqrt{1-x^2}}{\sqrt{2+3 x^2}} \, dx","IntegrateAlgebraic[Sqrt[1 - x^2]/Sqrt[2 + 3*x^2],x]","\int \frac{\sqrt{1-x^2}}{\sqrt{2+3 x^2}} \, dx","\frac{5 F\left(\sin ^{-1}(x)|-\frac{3}{2}\right)}{3 \sqrt{2}}-\frac{1}{3} \sqrt{2} E\left(\sin ^{-1}(x)|-\frac{3}{2}\right)",1,"Could not integrate","F",-1
193,0,0,35,0.2404712,"\int \frac{\sqrt{4-x^2}}{\sqrt{2+3 x^2}} \, dx","IntegrateAlgebraic[Sqrt[4 - x^2]/Sqrt[2 + 3*x^2],x]","\int \frac{\sqrt{4-x^2}}{\sqrt{2+3 x^2}} \, dx","\frac{7}{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
194,0,0,35,0.2610116,"\int \frac{\sqrt{1-4 x^2}}{\sqrt{2+3 x^2}} \, dx","IntegrateAlgebraic[Sqrt[1 - 4*x^2]/Sqrt[2 + 3*x^2],x]","\int \frac{\sqrt{1-4 x^2}}{\sqrt{2+3 x^2}} \, dx","\frac{11 F\left(\sin ^{-1}(2 x)|-\frac{3}{8}\right)}{6 \sqrt{2}}-\frac{2}{3} \sqrt{2} E\left(\sin ^{-1}(2 x)|-\frac{3}{8}\right)",1,"Could not integrate","F",-1
195,0,0,131,0.2424897,"\int \frac{\sqrt{1+x^2}}{\sqrt{2+3 x^2}} \, dx","IntegrateAlgebraic[Sqrt[1 + x^2]/Sqrt[2 + 3*x^2],x]","\int \frac{\sqrt{1+x^2}}{\sqrt{2+3 x^2}} \, dx","\frac{\sqrt{3 x^2+2} x}{3 \sqrt{x^2+1}}+\frac{\sqrt{3 x^2+2} F\left(\tan ^{-1}(x)|-\frac{1}{2}\right)}{\sqrt{2} \sqrt{x^2+1} \sqrt{\frac{3 x^2+2}{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
196,0,0,136,0.2453885,"\int \frac{\sqrt{4+x^2}}{\sqrt{2+3 x^2}} \, dx","IntegrateAlgebraic[Sqrt[4 + x^2]/Sqrt[2 + 3*x^2],x]","\int \frac{\sqrt{4+x^2}}{\sqrt{2+3 x^2}} \, dx","\frac{\sqrt{3 x^2+2} x}{3 \sqrt{x^2+4}}+\frac{2 \sqrt{2} \sqrt{3 x^2+2} F\left(\left.\tan ^{-1}\left(\frac{x}{2}\right)\right|-5\right)}{\sqrt{x^2+4} \sqrt{\frac{3 x^2+2}{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
197,0,0,148,0.2457252,"\int \frac{\sqrt{1+4 x^2}}{\sqrt{2+3 x^2}} \, dx","IntegrateAlgebraic[Sqrt[1 + 4*x^2]/Sqrt[2 + 3*x^2],x]","\int \frac{\sqrt{1+4 x^2}}{\sqrt{2+3 x^2}} \, dx","\frac{4 \sqrt{3 x^2+2} x}{3 \sqrt{4 x^2+1}}+\frac{\sqrt{3 x^2+2} F\left(\tan ^{-1}(2 x)|\frac{5}{8}\right)}{2 \sqrt{2} \sqrt{\frac{3 x^2+2}{4 x^2+1}} \sqrt{4 x^2+1}}-\frac{2 \sqrt{2} \sqrt{3 x^2+2} E\left(\tan ^{-1}(2 x)|\frac{5}{8}\right)}{3 \sqrt{\frac{3 x^2+2}{4 x^2+1}} \sqrt{4 x^2+1}}",1,"Could not integrate","F",-1
198,0,0,40,0.2500204,"\int \frac{\sqrt{1-x^2}}{\sqrt{-1+2 x^2}} \, dx","IntegrateAlgebraic[Sqrt[1 - x^2]/Sqrt[-1 + 2*x^2],x]","\int \frac{\sqrt{1-x^2}}{\sqrt{-1+2 x^2}} \, dx","\frac{\sqrt{1-2 x^2} E\left(\sin ^{-1}\left(\sqrt{2} x\right)|\frac{1}{2}\right)}{\sqrt{2} \sqrt{2 x^2-1}}",1,"Could not integrate","F",-1
199,0,0,423,1.427804,"\int \frac{\left(a+b x^2\right)^{7/2}}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^(7/2)/Sqrt[c + d*x^2],x]","\int \frac{\left(a+b x^2\right)^{7/2}}{\sqrt{c+d x^2}} \, dx","-\frac{\sqrt{c} \sqrt{a+b x^2} (3 b c-7 a d) \left(15 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)}{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{8 \sqrt{c} \sqrt{a+b x^2} (b c-2 a d) \left(11 a^2 d^2-11 a b c d+6 b^2 c^2\right) E\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{b x \sqrt{a+b x^2} \sqrt{c+d x^2} \left(71 a^2 d^2-71 a b c d+24 b^2 c^2\right)}{105 d^3}-\frac{8 x \sqrt{a+b x^2} (b c-2 a d) \left(11 a^2 d^2-11 a b c d+6 b^2 c^2\right)}{105 d^3 \sqrt{c+d x^2}}-\frac{6 b x \left(a+b x^2\right)^{3/2} \sqrt{c+d x^2} (b c-2 a d)}{35 d^2}+\frac{b x \left(a+b x^2\right)^{5/2} \sqrt{c+d x^2}}{7 d}",1,"Could not integrate","F",-1
200,0,0,344,1.1704704,"\int \frac{\left(a+b x^2\right)^{5/2}}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^(5/2)/Sqrt[c + d*x^2],x]","\int \frac{\left(a+b x^2\right)^{5/2}}{\sqrt{c+d x^2}} \, dx","\frac{x \sqrt{a+b x^2} \left(23 a^2 d^2-23 a b c d+8 b^2 c^2\right)}{15 d^2 \sqrt{c+d x^2}}+\frac{\sqrt{c} \sqrt{a+b x^2} \left(15 a^2 d^2-11 a b c d+4 b^2 c^2\right) 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{\sqrt{c} \sqrt{a+b x^2} \left(23 a^2 d^2-23 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 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 b x \sqrt{a+b x^2} \sqrt{c+d x^2} (b c-2 a d)}{15 d^2}+\frac{b x \left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}}{5 d}",1,"Could not integrate","F",-1
201,0,0,260,0.9792781,"\int \frac{\left(a+b x^2\right)^{3/2}}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^(3/2)/Sqrt[c + d*x^2],x]","\int \frac{\left(a+b x^2\right)^{3/2}}{\sqrt{c+d x^2}} \, dx","-\frac{\sqrt{c} \sqrt{a+b x^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)}{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{2 \sqrt{c} \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 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{b x \sqrt{a+b x^2} \sqrt{c+d x^2}}{3 d}-\frac{2 x \sqrt{a+b x^2} (b c-2 a d)}{3 d \sqrt{c+d x^2}}",1,"Could not integrate","F",-1
202,0,0,194,0.4793498,"\int \frac{\sqrt{a+b x^2}}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[Sqrt[a + b*x^2]/Sqrt[c + d*x^2],x]","\int \frac{\sqrt{a+b x^2}}{\sqrt{c+d x^2}} \, dx","\frac{x \sqrt{a+b x^2}}{\sqrt{c+d x^2}}+\frac{\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{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)}{\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
203,0,0,87,0.5397974,"\int \frac{1}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]","\int \frac{1}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","\frac{\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)}}}",1,"Could not integrate","F",-1
204,0,0,273,1.1287276,"\int \frac{1}{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]),x]","\int \frac{1}{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}} \, dx","\frac{b x \sqrt{c+d x^2}}{a \sqrt{a+b x^2} (b c-a d)}-\frac{d x \sqrt{a+b x^2}}{a \sqrt{c+d x^2} (b c-a d)}-\frac{\sqrt{c} \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)}{a \sqrt{c+d x^2} (b c-a d) \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{\sqrt{c} \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+d x^2} (b c-a d) \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}",1,"Could not integrate","F",-1
205,0,0,255,1.4649506,"\int \frac{1}{\left(a+b x^2\right)^{5/2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^(5/2)*Sqrt[c + d*x^2]),x]","\int \frac{1}{\left(a+b x^2\right)^{5/2} \sqrt{c+d x^2}} \, dx","\frac{2 \sqrt{b} \sqrt{c+d x^2} (b c-2 a d) E\left(\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)|1-\frac{a d}{b c}\right)}{3 a^{3/2} \sqrt{a+b x^2} (b c-a d)^2 \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}}}-\frac{\sqrt{c} \sqrt{d} \sqrt{a+b x^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)}{3 a^2 \sqrt{c+d x^2} (b c-a d)^2 \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{b x \sqrt{c+d x^2}}{3 a \left(a+b x^2\right)^{3/2} (b c-a d)}",1,"Could not integrate","F",-1
206,0,0,334,1.6803584,"\int \frac{1}{\left(a+b x^2\right)^{7/2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^(7/2)*Sqrt[c + d*x^2]),x]","\int \frac{1}{\left(a+b x^2\right)^{7/2} \sqrt{c+d x^2}} \, dx","\frac{4 b x \sqrt{c+d x^2} (b c-2 a d)}{15 a^2 \left(a+b x^2\right)^{3/2} (b c-a d)^2}+\frac{\sqrt{b} \sqrt{c+d x^2} \left(23 a^2 d^2-23 a b c d+8 b^2 c^2\right) E\left(\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)|1-\frac{a d}{b c}\right)}{15 a^{5/2} \sqrt{a+b x^2} (b c-a d)^3 \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}}}-\frac{\sqrt{c} \sqrt{d} \sqrt{a+b x^2} \left(15 a^2 d^2-11 a b c d+4 b^2 c^2\right) F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{15 a^3 \sqrt{c+d x^2} (b c-a d)^3 \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{b x \sqrt{c+d x^2}}{5 a \left(a+b x^2\right)^{5/2} (b c-a d)}",1,"Could not integrate","F",-1
207,0,0,445,1.7982384,"\int \frac{\left(a+b x^2\right)^{7/2}}{\left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^(7/2)/(c + d*x^2)^(3/2),x]","\int \frac{\left(a+b x^2\right)^{7/2}}{\left(c+d x^2\right)^{3/2}} \, dx","\frac{b \sqrt{c} \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)}{15 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{b x \sqrt{a+b x^2} \sqrt{c+d x^2} \left(15 a^2 d^2-43 a b c d+24 b^2 c^2\right)}{15 c d^3}+\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)}{15 c d^3 \sqrt{c+d x^2}}-\frac{\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)}{15 \sqrt{c} 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{b x \left(a+b x^2\right)^{3/2} \sqrt{c+d x^2} (6 b c-5 a d)}{5 c d^2}-\frac{x \left(a+b x^2\right)^{5/2} (b c-a d)}{c d \sqrt{c+d x^2}}",1,"Could not integrate","F",-1
208,0,0,346,1.6712547,"\int \frac{\left(a+b x^2\right)^{5/2}}{\left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^(5/2)/(c + d*x^2)^(3/2),x]","\int \frac{\left(a+b x^2\right)^{5/2}}{\left(c+d x^2\right)^{3/2}} \, dx","-\frac{x \sqrt{a+b x^2} \left(3 a^2 d^2-13 a b c d+8 b^2 c^2\right)}{3 c d^2 \sqrt{c+d x^2}}+\frac{\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)}{3 \sqrt{c} 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 b \sqrt{c} \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)}{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{b x \sqrt{a+b x^2} \sqrt{c+d x^2} (4 b c-3 a d)}{3 c d^2}-\frac{x \left(a+b x^2\right)^{3/2} (b c-a d)}{c d \sqrt{c+d x^2}}",1,"Could not integrate","F",-1
209,0,0,258,1.5308157,"\int \frac{\left(a+b x^2\right)^{3/2}}{\left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^(3/2)/(c + d*x^2)^(3/2),x]","\int \frac{\left(a+b x^2\right)^{3/2}}{\left(c+d x^2\right)^{3/2}} \, dx","\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)}{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{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)}{\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} (b c-a d)}{c d \sqrt{c+d x^2}}+\frac{x \sqrt{a+b x^2} (2 b c-a d)}{c d \sqrt{c+d x^2}}",1,"Could not integrate","F",-1
210,0,0,84,0.9514182,"\int \frac{\sqrt{a+b x^2}}{\left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[Sqrt[a + b*x^2]/(c + d*x^2)^(3/2),x]","\int \frac{\sqrt{a+b x^2}}{\left(c+d x^2\right)^{3/2}} \, dx","\frac{\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{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
211,0,0,194,1.191711,"\int \frac{1}{\sqrt{a+b x^2} \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[1/(Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)),x]","\int \frac{1}{\sqrt{a+b x^2} \left(c+d x^2\right)^{3/2}} \, dx","\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} (b c-a d) \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} (b c-a d) \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}",1,"Could not integrate","F",-1
212,0,0,242,1.7978957,"\int \frac{1}{\left(a+b x^2\right)^{3/2} \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)),x]","\int \frac{1}{\left(a+b x^2\right)^{3/2} \left(c+d x^2\right)^{3/2}} \, dx","\frac{b x}{a \sqrt{a+b x^2} \sqrt{c+d x^2} (b c-a d)}-\frac{2 b \sqrt{c} \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)}{a \sqrt{c+d x^2} (b c-a d)^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} (a d+b c) 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} (b c-a d)^2 \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}",1,"Could not integrate","F",-1
213,0,0,323,2.2029699,"\int \frac{1}{\left(a+b x^2\right)^{5/2} \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^(5/2)*(c + d*x^2)^(3/2)),x]","\int \frac{1}{\left(a+b x^2\right)^{5/2} \left(c+d x^2\right)^{3/2}} \, dx","\frac{\sqrt{d} \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 a^2 \sqrt{c} \sqrt{c+d x^2} (b c-a d)^3 \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{2 b x (b c-3 a d)}{3 a^2 \sqrt{a+b x^2} \sqrt{c+d x^2} (b c-a d)^2}-\frac{b \sqrt{c} \sqrt{d} \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 a^2 \sqrt{c+d x^2} (b c-a d)^3 \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{b x}{3 a \left(a+b x^2\right)^{3/2} \sqrt{c+d x^2} (b c-a d)}",1,"Could not integrate","F",-1
214,0,0,87,0.000595,"\int \frac{1}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]","\int \frac{1}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","\frac{\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)}}}",1,"Could not integrate","F",-1
215,0,0,87,0.5830354,"\int \frac{1}{\sqrt{a-b x^2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[a - b*x^2]*Sqrt[c + d*x^2]),x]","\int \frac{1}{\sqrt{a-b x^2} \sqrt{c+d x^2}} \, dx","\frac{\sqrt{a} \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} F\left(\sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)|-\frac{a d}{b c}\right)}{\sqrt{b} \sqrt{a-b x^2} \sqrt{c+d x^2}}",1,"Could not integrate","F",-1
216,0,0,87,0.6501831,"\int \frac{1}{\sqrt{a+b x^2} \sqrt{c-d x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[a + b*x^2]*Sqrt[c - d*x^2]),x]","\int \frac{1}{\sqrt{a+b x^2} \sqrt{c-d x^2}} \, dx","\frac{\sqrt{c} \sqrt{\frac{b x^2}{a}+1} \sqrt{1-\frac{d x^2}{c}} F\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|-\frac{b c}{a d}\right)}{\sqrt{d} \sqrt{a+b x^2} \sqrt{c-d x^2}}",1,"Could not integrate","F",-1
217,0,0,88,0.6420142,"\int \frac{1}{\sqrt{a-b x^2} \sqrt{c-d x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[a - b*x^2]*Sqrt[c - d*x^2]),x]","\int \frac{1}{\sqrt{a-b x^2} \sqrt{c-d x^2}} \, dx","\frac{\sqrt{c} \sqrt{1-\frac{b x^2}{a}} \sqrt{1-\frac{d x^2}{c}} F\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b c}{a d}\right)}{\sqrt{d} \sqrt{a-b x^2} \sqrt{c-d x^2}}",1,"Could not integrate","F",-1
218,0,0,12,0.2588779,"\int \frac{1}{\sqrt{1-x^2} \sqrt{2+5 x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[1 - x^2]*Sqrt[2 + 5*x^2]),x]","\int \frac{1}{\sqrt{1-x^2} \sqrt{2+5 x^2}} \, dx","\frac{F\left(\sin ^{-1}(x)|-\frac{5}{2}\right)}{\sqrt{2}}",1,"Could not integrate","F",-1
219,0,0,10,0.2430139,"\int \frac{1}{\sqrt{1-x^2} \sqrt{2+4 x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[1 - x^2]*Sqrt[2 + 4*x^2]),x]","\int \frac{1}{\sqrt{1-x^2} \sqrt{2+4 x^2}} \, dx","\frac{F\left(\left.\sin ^{-1}(x)\right|-2\right)}{\sqrt{2}}",1,"Could not integrate","F",-1
220,0,0,12,0.2417052,"\int \frac{1}{\sqrt{1-x^2} \sqrt{2+3 x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[1 - x^2]*Sqrt[2 + 3*x^2]),x]","\int \frac{1}{\sqrt{1-x^2} \sqrt{2+3 x^2}} \, dx","\frac{F\left(\sin ^{-1}(x)|-\frac{3}{2}\right)}{\sqrt{2}}",1,"Could not integrate","F",-1
221,0,0,10,0.24608,"\int \frac{1}{\sqrt{1-x^2} \sqrt{2+2 x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[1 - x^2]*Sqrt[2 + 2*x^2]),x]","\int \frac{1}{\sqrt{1-x^2} \sqrt{2+2 x^2}} \, dx","\frac{F\left(\left.\sin ^{-1}(x)\right|-1\right)}{\sqrt{2}}",1,"Could not integrate","F",-1
222,0,0,12,0.2206196,"\int \frac{1}{\sqrt{1-x^2} \sqrt{2+x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[1 - x^2]*Sqrt[2 + x^2]),x]","\int \frac{1}{\sqrt{1-x^2} \sqrt{2+x^2}} \, dx","\frac{F\left(\sin ^{-1}(x)|-\frac{1}{2}\right)}{\sqrt{2}}",1,"Could not integrate","F",-1
223,0,0,12,0.2181491,"\int \frac{1}{\sqrt{1-x^2} \sqrt{2-x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[1 - x^2]*Sqrt[2 - x^2]),x]","\int \frac{1}{\sqrt{1-x^2} \sqrt{2-x^2}} \, dx","\frac{F\left(\sin ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{2}}",1,"Could not integrate","F",-1
224,1,8,8,0.014285,"\int \frac{1}{\sqrt{2-2 x^2} \sqrt{1-x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[2 - 2*x^2]*Sqrt[1 - x^2]),x]","\frac{\tanh ^{-1}(x)}{\sqrt{2}}","\frac{\tanh ^{-1}(x)}{\sqrt{2}}",1,"ArcTanh[x]/Sqrt[2]","A",1
225,0,0,12,0.2944149,"\int \frac{1}{\sqrt{2-3 x^2} \sqrt{1-x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[2 - 3*x^2]*Sqrt[1 - x^2]),x]","\int \frac{1}{\sqrt{2-3 x^2} \sqrt{1-x^2}} \, dx","\frac{F\left(\sin ^{-1}(x)|\frac{3}{2}\right)}{\sqrt{2}}",1,"Could not integrate","F",-1
226,0,0,10,0.280586,"\int \frac{1}{\sqrt{2-4 x^2} \sqrt{1-x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[2 - 4*x^2]*Sqrt[1 - x^2]),x]","\int \frac{1}{\sqrt{2-4 x^2} \sqrt{1-x^2}} \, dx","\frac{F\left(\left.\sin ^{-1}(x)\right|2\right)}{\sqrt{2}}",1,"Could not integrate","F",-1
227,0,0,12,0.2155538,"\int \frac{1}{\sqrt{2-5 x^2} \sqrt{1-x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[2 - 5*x^2]*Sqrt[1 - x^2]),x]","\int \frac{1}{\sqrt{2-5 x^2} \sqrt{1-x^2}} \, dx","\frac{F\left(\sin ^{-1}(x)|\frac{5}{2}\right)}{\sqrt{2}}",1,"Could not integrate","F",-1
228,0,0,51,0.2456754,"\int \frac{1}{\sqrt{1+x^2} \sqrt{2+5 x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[1 + x^2]*Sqrt[2 + 5*x^2]),x]","\int \frac{1}{\sqrt{1+x^2} \sqrt{2+5 x^2}} \, dx","\frac{\sqrt{5 x^2+2} F\left(\tan ^{-1}(x)|-\frac{3}{2}\right)}{\sqrt{2} \sqrt{x^2+1} \sqrt{\frac{5 x^2+2}{x^2+1}}}",1,"Could not integrate","F",-1
229,0,0,49,0.2319687,"\int \frac{1}{\sqrt{1+x^2} \sqrt{2+4 x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[1 + x^2]*Sqrt[2 + 4*x^2]),x]","\int \frac{1}{\sqrt{1+x^2} \sqrt{2+4 x^2}} \, dx","\frac{\sqrt{2 x^2+1} F\left(\left.\tan ^{-1}(x)\right|-1\right)}{\sqrt{2} \sqrt{x^2+1} \sqrt{\frac{2 x^2+1}{x^2+1}}}",1,"Could not integrate","F",-1
230,0,0,51,0.2080109,"\int \frac{1}{\sqrt{1+x^2} \sqrt{2+3 x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]),x]","\int \frac{1}{\sqrt{1+x^2} \sqrt{2+3 x^2}} \, dx","\frac{\sqrt{3 x^2+2} F\left(\tan ^{-1}(x)|-\frac{1}{2}\right)}{\sqrt{2} \sqrt{x^2+1} \sqrt{\frac{3 x^2+2}{x^2+1}}}",1,"Could not integrate","F",-1
231,1,8,8,0.0086039,"\int \frac{1}{\sqrt{1+x^2} \sqrt{2+2 x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[1 + x^2]*Sqrt[2 + 2*x^2]),x]","\frac{\tan ^{-1}(x)}{\sqrt{2}}","\frac{\tan ^{-1}(x)}{\sqrt{2}}",1,"ArcTan[x]/Sqrt[2]","A",1
232,0,0,47,0.2659334,"\int \frac{1}{\sqrt{1+x^2} \sqrt{2+x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[1 + x^2]*Sqrt[2 + x^2]),x]","\int \frac{1}{\sqrt{1+x^2} \sqrt{2+x^2}} \, dx","\frac{\sqrt{x^2+2} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{2} \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}}}",1,"Could not integrate","F",-1
233,0,0,10,0.2305706,"\int \frac{1}{\sqrt{2-x^2} \sqrt{1+x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[2 - x^2]*Sqrt[1 + x^2]),x]","\int \frac{1}{\sqrt{2-x^2} \sqrt{1+x^2}} \, dx","F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)",1,"Could not integrate","F",-1
234,0,0,10,0.2330385,"\int \frac{1}{\sqrt{2-2 x^2} \sqrt{1+x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[2 - 2*x^2]*Sqrt[1 + x^2]),x]","\int \frac{1}{\sqrt{2-2 x^2} \sqrt{1+x^2}} \, dx","\frac{F\left(\left.\sin ^{-1}(x)\right|-1\right)}{\sqrt{2}}",1,"Could not integrate","F",-1
235,0,0,20,0.2223471,"\int \frac{1}{\sqrt{2-3 x^2} \sqrt{1+x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[2 - 3*x^2]*Sqrt[1 + x^2]),x]","\int \frac{1}{\sqrt{2-3 x^2} \sqrt{1+x^2}} \, dx","\frac{F\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|-\frac{2}{3}\right)}{\sqrt{3}}",1,"Could not integrate","F",-1
236,0,0,16,0.2426329,"\int \frac{1}{\sqrt{2-4 x^2} \sqrt{1+x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[2 - 4*x^2]*Sqrt[1 + x^2]),x]","\int \frac{1}{\sqrt{2-4 x^2} \sqrt{1+x^2}} \, dx","\frac{1}{2} F\left(\sin ^{-1}\left(\sqrt{2} x\right)|-\frac{1}{2}\right)",1,"Could not integrate","F",-1
237,0,0,20,0.2190095,"\int \frac{1}{\sqrt{2-5 x^2} \sqrt{1+x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[2 - 5*x^2]*Sqrt[1 + x^2]),x]","\int \frac{1}{\sqrt{2-5 x^2} \sqrt{1+x^2}} \, dx","\frac{F\left(\sin ^{-1}\left(\sqrt{\frac{5}{2}} x\right)|-\frac{2}{5}\right)}{\sqrt{5}}",1,"Could not integrate","F",-1
238,0,0,32,0.2104465,"\int \frac{1}{\sqrt{-1+x^2} \sqrt{2+5 x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[-1 + x^2]*Sqrt[2 + 5*x^2]),x]","\int \frac{1}{\sqrt{-1+x^2} \sqrt{2+5 x^2}} \, dx","\frac{\sqrt{1-x^2} F\left(\sin ^{-1}(x)|-\frac{5}{2}\right)}{\sqrt{2} \sqrt{x^2-1}}",1,"Could not integrate","F",-1
239,0,0,30,0.2229972,"\int \frac{1}{\sqrt{-1+x^2} \sqrt{2+4 x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[-1 + x^2]*Sqrt[2 + 4*x^2]),x]","\int \frac{1}{\sqrt{-1+x^2} \sqrt{2+4 x^2}} \, dx","\frac{\sqrt{1-x^2} F\left(\left.\sin ^{-1}(x)\right|-2\right)}{\sqrt{2} \sqrt{x^2-1}}",1,"Could not integrate","F",-1
240,0,0,32,0.1991258,"\int \frac{1}{\sqrt{-1+x^2} \sqrt{2+3 x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[-1 + x^2]*Sqrt[2 + 3*x^2]),x]","\int \frac{1}{\sqrt{-1+x^2} \sqrt{2+3 x^2}} \, dx","\frac{\sqrt{1-x^2} F\left(\sin ^{-1}(x)|-\frac{3}{2}\right)}{\sqrt{2} \sqrt{x^2-1}}",1,"Could not integrate","F",-1
241,0,0,25,0.2239011,"\int \frac{1}{\sqrt{-1+x^2} \sqrt{2+2 x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[-1 + x^2]*Sqrt[2 + 2*x^2]),x]","\int \frac{1}{\sqrt{-1+x^2} \sqrt{2+2 x^2}} \, dx","\frac{1}{2} F\left(\sin ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right)|\frac{1}{2}\right)",1,"Could not integrate","F",-1
242,0,0,32,0.2002958,"\int \frac{1}{\sqrt{-1+x^2} \sqrt{2+x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[-1 + x^2]*Sqrt[2 + x^2]),x]","\int \frac{1}{\sqrt{-1+x^2} \sqrt{2+x^2}} \, dx","\frac{\sqrt{1-x^2} F\left(\sin ^{-1}(x)|-\frac{1}{2}\right)}{\sqrt{2} \sqrt{x^2-1}}",1,"Could not integrate","F",-1
243,0,0,12,0.2072697,"\int \frac{1}{\sqrt{2-x^2} \sqrt{-1+x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[2 - x^2]*Sqrt[-1 + x^2]),x]","\int \frac{1}{\sqrt{2-x^2} \sqrt{-1+x^2}} \, dx","-F\left(\left.\cos ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)",1,"Could not integrate","F",-1
244,0,0,29,0.217368,"\int \frac{1}{\sqrt{2-2 x^2} \sqrt{-1+x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[2 - 2*x^2]*Sqrt[-1 + x^2]),x]","\int \frac{1}{\sqrt{2-2 x^2} \sqrt{-1+x^2}} \, dx","-\frac{\sqrt{x^2-1} \tanh ^{-1}(x)}{\sqrt{2} \sqrt{1-x^2}}",1,"Could not integrate","F",-1
245,0,0,32,0.1987175,"\int \frac{1}{\sqrt{2-3 x^2} \sqrt{-1+x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[2 - 3*x^2]*Sqrt[-1 + x^2]),x]","\int \frac{1}{\sqrt{2-3 x^2} \sqrt{-1+x^2}} \, dx","\frac{\sqrt{1-x^2} F\left(\sin ^{-1}(x)|\frac{3}{2}\right)}{\sqrt{2} \sqrt{x^2-1}}",1,"Could not integrate","F",-1
246,0,0,30,0.2277024,"\int \frac{1}{\sqrt{2-4 x^2} \sqrt{-1+x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[2 - 4*x^2]*Sqrt[-1 + x^2]),x]","\int \frac{1}{\sqrt{2-4 x^2} \sqrt{-1+x^2}} \, dx","\frac{\sqrt{1-x^2} F\left(\left.\sin ^{-1}(x)\right|2\right)}{\sqrt{2} \sqrt{x^2-1}}",1,"Could not integrate","F",-1
247,0,0,32,0.2023577,"\int \frac{1}{\sqrt{2-5 x^2} \sqrt{-1+x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[2 - 5*x^2]*Sqrt[-1 + x^2]),x]","\int \frac{1}{\sqrt{2-5 x^2} \sqrt{-1+x^2}} \, dx","\frac{\sqrt{1-x^2} F\left(\sin ^{-1}(x)|\frac{5}{2}\right)}{\sqrt{2} \sqrt{x^2-1}}",1,"Could not integrate","F",-1
248,0,0,53,0.221566,"\int \frac{1}{\sqrt{-1-x^2} \sqrt{2+5 x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[-1 - x^2]*Sqrt[2 + 5*x^2]),x]","\int \frac{1}{\sqrt{-1-x^2} \sqrt{2+5 x^2}} \, dx","\frac{\sqrt{5 x^2+2} F\left(\tan ^{-1}(x)|-\frac{3}{2}\right)}{\sqrt{2} \sqrt{-x^2-1} \sqrt{\frac{5 x^2+2}{x^2+1}}}",1,"Could not integrate","F",-1
249,0,0,51,0.2284233,"\int \frac{1}{\sqrt{-1-x^2} \sqrt{2+4 x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[-1 - x^2]*Sqrt[2 + 4*x^2]),x]","\int \frac{1}{\sqrt{-1-x^2} \sqrt{2+4 x^2}} \, dx","\frac{\sqrt{2 x^2+1} F\left(\left.\tan ^{-1}(x)\right|-1\right)}{\sqrt{2} \sqrt{-x^2-1} \sqrt{\frac{2 x^2+1}{x^2+1}}}",1,"Could not integrate","F",-1
250,0,0,53,0.2153691,"\int \frac{1}{\sqrt{-1-x^2} \sqrt{2+3 x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[-1 - x^2]*Sqrt[2 + 3*x^2]),x]","\int \frac{1}{\sqrt{-1-x^2} \sqrt{2+3 x^2}} \, dx","\frac{\sqrt{3 x^2+2} F\left(\tan ^{-1}(x)|-\frac{1}{2}\right)}{\sqrt{2} \sqrt{-x^2-1} \sqrt{\frac{3 x^2+2}{x^2+1}}}",1,"Could not integrate","F",-1
251,0,0,28,0.2093131,"\int \frac{1}{\sqrt{-1-x^2} \sqrt{2+2 x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[-1 - x^2]*Sqrt[2 + 2*x^2]),x]","\int \frac{1}{\sqrt{-1-x^2} \sqrt{2+2 x^2}} \, dx","\frac{\sqrt{x^2+1} \tan ^{-1}(x)}{\sqrt{2} \sqrt{-x^2-1}}",1,"Could not integrate","F",-1
252,0,0,49,0.1945998,"\int \frac{1}{\sqrt{-1-x^2} \sqrt{2+x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[-1 - x^2]*Sqrt[2 + x^2]),x]","\int \frac{1}{\sqrt{-1-x^2} \sqrt{2+x^2}} \, dx","\frac{\sqrt{x^2+2} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{2} \sqrt{-x^2-1} \sqrt{\frac{x^2+2}{x^2+1}}}",1,"Could not integrate","F",-1
253,0,0,31,0.1883383,"\int \frac{1}{\sqrt{-1-x^2} \sqrt{2-x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[-1 - x^2]*Sqrt[2 - x^2]),x]","\int \frac{1}{\sqrt{-1-x^2} \sqrt{2-x^2}} \, dx","\frac{\sqrt{x^2+1} F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)}{\sqrt{-x^2-1}}",1,"Could not integrate","F",-1
254,0,0,42,0.2198708,"\int \frac{1}{\sqrt{2-2 x^2} \sqrt{-1-x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[2 - 2*x^2]*Sqrt[-1 - x^2]),x]","\int \frac{1}{\sqrt{2-2 x^2} \sqrt{-1-x^2}} \, dx","-\frac{\sqrt{1-\frac{1}{x^4}} x^2 F\left(\left.\csc ^{-1}(x)\right|-1\right)}{\sqrt{2-2 x^2} \sqrt{-x^2-1}}",1,"Could not integrate","F",-1
255,0,0,40,0.2199069,"\int \frac{1}{\sqrt{2-3 x^2} \sqrt{-1-x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[2 - 3*x^2]*Sqrt[-1 - x^2]),x]","\int \frac{1}{\sqrt{2-3 x^2} \sqrt{-1-x^2}} \, dx","\frac{\sqrt{x^2+1} F\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|-\frac{2}{3}\right)}{\sqrt{3} \sqrt{-x^2-1}}",1,"Could not integrate","F",-1
256,0,0,36,0.2378642,"\int \frac{1}{\sqrt{2-4 x^2} \sqrt{-1-x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[2 - 4*x^2]*Sqrt[-1 - x^2]),x]","\int \frac{1}{\sqrt{2-4 x^2} \sqrt{-1-x^2}} \, dx","\frac{\sqrt{x^2+1} F\left(\sin ^{-1}\left(\sqrt{2} x\right)|-\frac{1}{2}\right)}{2 \sqrt{-x^2-1}}",1,"Could not integrate","F",-1
257,0,0,40,0.2095148,"\int \frac{1}{\sqrt{2-5 x^2} \sqrt{-1-x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[2 - 5*x^2]*Sqrt[-1 - x^2]),x]","\int \frac{1}{\sqrt{2-5 x^2} \sqrt{-1-x^2}} \, dx","\frac{\sqrt{x^2+1} F\left(\sin ^{-1}\left(\sqrt{\frac{5}{2}} x\right)|-\frac{2}{5}\right)}{\sqrt{5} \sqrt{-x^2-1}}",1,"Could not integrate","F",-1
258,0,0,87,0.5692294,"\int \frac{\sqrt{a+b x^2}}{\sqrt{c-d x^2}} \, dx","IntegrateAlgebraic[Sqrt[a + b*x^2]/Sqrt[c - d*x^2],x]","\int \frac{\sqrt{a+b x^2}}{\sqrt{c-d x^2}} \, dx","\frac{\sqrt{c} \sqrt{a+b x^2} \sqrt{1-\frac{d x^2}{c}} E\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|-\frac{b c}{a d}\right)}{\sqrt{d} \sqrt{\frac{b x^2}{a}+1} \sqrt{c-d x^2}}",1,"Could not integrate","F",-1
259,0,0,90,0.5581085,"\int \frac{\sqrt{-a-b x^2}}{\sqrt{c-d x^2}} \, dx","IntegrateAlgebraic[Sqrt[-a - b*x^2]/Sqrt[c - d*x^2],x]","\int \frac{\sqrt{-a-b x^2}}{\sqrt{c-d x^2}} \, dx","\frac{\sqrt{c} \sqrt{-a-b x^2} \sqrt{1-\frac{d x^2}{c}} E\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|-\frac{b c}{a d}\right)}{\sqrt{d} \sqrt{\frac{b x^2}{a}+1} \sqrt{c-d x^2}}",1,"Could not integrate","F",-1
260,0,0,88,0.5342781,"\int \frac{\sqrt{a+b x^2}}{\sqrt{-c+d x^2}} \, dx","IntegrateAlgebraic[Sqrt[a + b*x^2]/Sqrt[-c + d*x^2],x]","\int \frac{\sqrt{a+b x^2}}{\sqrt{-c+d x^2}} \, dx","\frac{\sqrt{c} \sqrt{a+b x^2} \sqrt{1-\frac{d x^2}{c}} E\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|-\frac{b c}{a d}\right)}{\sqrt{d} \sqrt{\frac{b x^2}{a}+1} \sqrt{d x^2-c}}",1,"Could not integrate","F",-1
261,0,0,91,0.5031114,"\int \frac{\sqrt{-a-b x^2}}{\sqrt{-c+d x^2}} \, dx","IntegrateAlgebraic[Sqrt[-a - b*x^2]/Sqrt[-c + d*x^2],x]","\int \frac{\sqrt{-a-b x^2}}{\sqrt{-c+d x^2}} \, dx","\frac{\sqrt{c} \sqrt{-a-b x^2} \sqrt{1-\frac{d x^2}{c}} E\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|-\frac{b c}{a d}\right)}{\sqrt{d} \sqrt{\frac{b x^2}{a}+1} \sqrt{d x^2-c}}",1,"Could not integrate","F",-1
262,0,0,88,0.5554268,"\int \frac{\sqrt{a-b x^2}}{\sqrt{c-d x^2}} \, dx","IntegrateAlgebraic[Sqrt[a - b*x^2]/Sqrt[c - d*x^2],x]","\int \frac{\sqrt{a-b x^2}}{\sqrt{c-d x^2}} \, dx","\frac{\sqrt{c} \sqrt{a-b x^2} \sqrt{1-\frac{d x^2}{c}} E\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b c}{a d}\right)}{\sqrt{d} \sqrt{1-\frac{b x^2}{a}} \sqrt{c-d x^2}}",1,"Could not integrate","F",-1
263,0,0,89,0.5419446,"\int \frac{\sqrt{-a+b x^2}}{\sqrt{c-d x^2}} \, dx","IntegrateAlgebraic[Sqrt[-a + b*x^2]/Sqrt[c - d*x^2],x]","\int \frac{\sqrt{-a+b x^2}}{\sqrt{c-d x^2}} \, dx","\frac{\sqrt{c} \sqrt{b x^2-a} \sqrt{1-\frac{d x^2}{c}} E\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b c}{a d}\right)}{\sqrt{d} \sqrt{1-\frac{b x^2}{a}} \sqrt{c-d x^2}}",1,"Could not integrate","F",-1
264,0,0,89,0.534415,"\int \frac{\sqrt{a-b x^2}}{\sqrt{-c+d x^2}} \, dx","IntegrateAlgebraic[Sqrt[a - b*x^2]/Sqrt[-c + d*x^2],x]","\int \frac{\sqrt{a-b x^2}}{\sqrt{-c+d x^2}} \, dx","\frac{\sqrt{c} \sqrt{a-b x^2} \sqrt{1-\frac{d x^2}{c}} E\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b c}{a d}\right)}{\sqrt{d} \sqrt{1-\frac{b x^2}{a}} \sqrt{d x^2-c}}",1,"Could not integrate","F",-1
265,0,0,90,0.5282514,"\int \frac{\sqrt{-a+b x^2}}{\sqrt{-c+d x^2}} \, dx","IntegrateAlgebraic[Sqrt[-a + b*x^2]/Sqrt[-c + d*x^2],x]","\int \frac{\sqrt{-a+b x^2}}{\sqrt{-c+d x^2}} \, dx","\frac{\sqrt{c} \sqrt{b x^2-a} \sqrt{1-\frac{d x^2}{c}} E\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b c}{a d}\right)}{\sqrt{d} \sqrt{1-\frac{b x^2}{a}} \sqrt{d x^2-c}}",1,"Could not integrate","F",-1
266,0,0,194,0.0005746,"\int \frac{\sqrt{a+b x^2}}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[Sqrt[a + b*x^2]/Sqrt[c + d*x^2],x]","\int \frac{\sqrt{a+b x^2}}{\sqrt{c+d x^2}} \, dx","\frac{x \sqrt{a+b x^2}}{\sqrt{c+d x^2}}+\frac{\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{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)}{\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
267,0,0,203,0.532557,"\int \frac{\sqrt{-a-b x^2}}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[Sqrt[-a - b*x^2]/Sqrt[c + d*x^2],x]","\int \frac{\sqrt{-a-b x^2}}{\sqrt{c+d x^2}} \, dx","\frac{x \sqrt{-a-b x^2}}{\sqrt{c+d x^2}}+\frac{\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{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)}{\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
268,0,0,203,0.5912305,"\int \frac{\sqrt{a+b x^2}}{\sqrt{-c-d x^2}} \, dx","IntegrateAlgebraic[Sqrt[a + b*x^2]/Sqrt[-c - d*x^2],x]","\int \frac{\sqrt{a+b x^2}}{\sqrt{-c-d x^2}} \, dx","\frac{x \sqrt{a+b x^2}}{\sqrt{-c-d x^2}}+\frac{\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{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)}{\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
269,0,0,212,0.5312407,"\int \frac{\sqrt{-a-b x^2}}{\sqrt{-c-d x^2}} \, dx","IntegrateAlgebraic[Sqrt[-a - b*x^2]/Sqrt[-c - d*x^2],x]","\int \frac{\sqrt{-a-b x^2}}{\sqrt{-c-d x^2}} \, dx","\frac{x \sqrt{-a-b x^2}}{\sqrt{-c-d x^2}}+\frac{\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{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)}{\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
270,0,0,189,0.5328837,"\int \frac{\sqrt{a-b x^2}}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[Sqrt[a - b*x^2]/Sqrt[c + d*x^2],x]","\int \frac{\sqrt{a-b x^2}}{\sqrt{c+d x^2}} \, dx","\frac{\sqrt{a} \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} (a d+b c) F\left(\sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)|-\frac{a d}{b c}\right)}{\sqrt{b} d \sqrt{a-b x^2} \sqrt{c+d x^2}}-\frac{\sqrt{a} \sqrt{b} \sqrt{1-\frac{b x^2}{a}} \sqrt{c+d x^2} E\left(\sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)|-\frac{a d}{b c}\right)}{d \sqrt{a-b x^2} \sqrt{\frac{d x^2}{c}+1}}",1,"Could not integrate","F",-1
271,0,0,191,0.5220739,"\int \frac{\sqrt{-a+b x^2}}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[Sqrt[-a + b*x^2]/Sqrt[c + d*x^2],x]","\int \frac{\sqrt{-a+b x^2}}{\sqrt{c+d x^2}} \, dx","\frac{\sqrt{a} \sqrt{b} \sqrt{1-\frac{b x^2}{a}} \sqrt{c+d x^2} E\left(\sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)|-\frac{a d}{b c}\right)}{d \sqrt{b x^2-a} \sqrt{\frac{d x^2}{c}+1}}-\frac{\sqrt{a} \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} (a d+b c) F\left(\sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)|-\frac{a d}{b c}\right)}{\sqrt{b} d \sqrt{b x^2-a} \sqrt{c+d x^2}}",1,"Could not integrate","F",-1
272,0,0,194,0.5346831,"\int \frac{\sqrt{a-b x^2}}{\sqrt{-c-d x^2}} \, dx","IntegrateAlgebraic[Sqrt[a - b*x^2]/Sqrt[-c - d*x^2],x]","\int \frac{\sqrt{a-b x^2}}{\sqrt{-c-d x^2}} \, dx","\frac{\sqrt{a} \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} (a d+b c) F\left(\sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)|-\frac{a d}{b c}\right)}{\sqrt{b} d \sqrt{a-b x^2} \sqrt{-c-d x^2}}+\frac{\sqrt{a} \sqrt{b} \sqrt{1-\frac{b x^2}{a}} \sqrt{-c-d x^2} E\left(\sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)|-\frac{a d}{b c}\right)}{d \sqrt{a-b x^2} \sqrt{\frac{d x^2}{c}+1}}",1,"Could not integrate","F",-1
273,0,0,198,0.5186003,"\int \frac{\sqrt{-a+b x^2}}{\sqrt{-c-d x^2}} \, dx","IntegrateAlgebraic[Sqrt[-a + b*x^2]/Sqrt[-c - d*x^2],x]","\int \frac{\sqrt{-a+b x^2}}{\sqrt{-c-d x^2}} \, dx","-\frac{\sqrt{a} \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} (a d+b c) F\left(\sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)|-\frac{a d}{b c}\right)}{\sqrt{b} d \sqrt{b x^2-a} \sqrt{-c-d x^2}}-\frac{\sqrt{a} \sqrt{b} \sqrt{1-\frac{b x^2}{a}} \sqrt{-c-d x^2} E\left(\sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)|-\frac{a d}{b c}\right)}{d \sqrt{b x^2-a} \sqrt{\frac{d x^2}{c}+1}}",1,"Could not integrate","F",-1
274,0,0,87,0.553573,"\int \frac{\sqrt{c+d x^2}}{\sqrt{a-b x^2}} \, dx","IntegrateAlgebraic[Sqrt[c + d*x^2]/Sqrt[a - b*x^2],x]","\int \frac{\sqrt{c+d x^2}}{\sqrt{a-b x^2}} \, dx","\frac{\sqrt{a} \sqrt{1-\frac{b x^2}{a}} \sqrt{c+d x^2} E\left(\sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)|-\frac{a d}{b c}\right)}{\sqrt{b} \sqrt{a-b x^2} \sqrt{\frac{d x^2}{c}+1}}",1,"Could not integrate","F",-1
275,0,0,90,0.5313612,"\int \frac{\sqrt{-c-d x^2}}{\sqrt{a-b x^2}} \, dx","IntegrateAlgebraic[Sqrt[-c - d*x^2]/Sqrt[a - b*x^2],x]","\int \frac{\sqrt{-c-d x^2}}{\sqrt{a-b x^2}} \, dx","\frac{\sqrt{a} \sqrt{1-\frac{b x^2}{a}} \sqrt{-c-d x^2} E\left(\sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)|-\frac{a d}{b c}\right)}{\sqrt{b} \sqrt{a-b x^2} \sqrt{\frac{d x^2}{c}+1}}",1,"Could not integrate","F",-1
276,0,0,88,0.530604,"\int \frac{\sqrt{c+d x^2}}{\sqrt{-a+b x^2}} \, dx","IntegrateAlgebraic[Sqrt[c + d*x^2]/Sqrt[-a + b*x^2],x]","\int \frac{\sqrt{c+d x^2}}{\sqrt{-a+b x^2}} \, dx","\frac{\sqrt{a} \sqrt{1-\frac{b x^2}{a}} \sqrt{c+d x^2} E\left(\sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)|-\frac{a d}{b c}\right)}{\sqrt{b} \sqrt{b x^2-a} \sqrt{\frac{d x^2}{c}+1}}",1,"Could not integrate","F",-1
277,0,0,91,0.5004593,"\int \frac{\sqrt{-c-d x^2}}{\sqrt{-a+b x^2}} \, dx","IntegrateAlgebraic[Sqrt[-c - d*x^2]/Sqrt[-a + b*x^2],x]","\int \frac{\sqrt{-c-d x^2}}{\sqrt{-a+b x^2}} \, dx","\frac{\sqrt{a} \sqrt{1-\frac{b x^2}{a}} \sqrt{-c-d x^2} E\left(\sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)|-\frac{a d}{b c}\right)}{\sqrt{b} \sqrt{b x^2-a} \sqrt{\frac{d x^2}{c}+1}}",1,"Could not integrate","F",-1
278,0,0,88,0.5624309,"\int \frac{\sqrt{c-d x^2}}{\sqrt{a-b x^2}} \, dx","IntegrateAlgebraic[Sqrt[c - d*x^2]/Sqrt[a - b*x^2],x]","\int \frac{\sqrt{c-d x^2}}{\sqrt{a-b x^2}} \, dx","\frac{\sqrt{a} \sqrt{1-\frac{b x^2}{a}} \sqrt{c-d x^2} E\left(\sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)|\frac{a d}{b c}\right)}{\sqrt{b} \sqrt{a-b x^2} \sqrt{1-\frac{d x^2}{c}}}",1,"Could not integrate","F",-1
279,0,0,89,0.5398727,"\int \frac{\sqrt{-c+d x^2}}{\sqrt{a-b x^2}} \, dx","IntegrateAlgebraic[Sqrt[-c + d*x^2]/Sqrt[a - b*x^2],x]","\int \frac{\sqrt{-c+d x^2}}{\sqrt{a-b x^2}} \, dx","\frac{\sqrt{a} \sqrt{1-\frac{b x^2}{a}} \sqrt{d x^2-c} E\left(\sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)|\frac{a d}{b c}\right)}{\sqrt{b} \sqrt{a-b x^2} \sqrt{1-\frac{d x^2}{c}}}",1,"Could not integrate","F",-1
280,0,0,89,0.5344485,"\int \frac{\sqrt{c-d x^2}}{\sqrt{-a+b x^2}} \, dx","IntegrateAlgebraic[Sqrt[c - d*x^2]/Sqrt[-a + b*x^2],x]","\int \frac{\sqrt{c-d x^2}}{\sqrt{-a+b x^2}} \, dx","\frac{\sqrt{a} \sqrt{1-\frac{b x^2}{a}} \sqrt{c-d x^2} E\left(\sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)|\frac{a d}{b c}\right)}{\sqrt{b} \sqrt{b x^2-a} \sqrt{1-\frac{d x^2}{c}}}",1,"Could not integrate","F",-1
281,0,0,90,0.5070085,"\int \frac{\sqrt{-c+d x^2}}{\sqrt{-a+b x^2}} \, dx","IntegrateAlgebraic[Sqrt[-c + d*x^2]/Sqrt[-a + b*x^2],x]","\int \frac{\sqrt{-c+d x^2}}{\sqrt{-a+b x^2}} \, dx","\frac{\sqrt{a} \sqrt{1-\frac{b x^2}{a}} \sqrt{d x^2-c} E\left(\sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)|\frac{a d}{b c}\right)}{\sqrt{b} \sqrt{b x^2-a} \sqrt{1-\frac{d x^2}{c}}}",1,"Could not integrate","F",-1
282,0,0,204,0.0005704,"\int \frac{\sqrt{c+d x^2}}{\sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[Sqrt[c + d*x^2]/Sqrt[a + b*x^2],x]","\int \frac{\sqrt{c+d x^2}}{\sqrt{a+b 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)}{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{d x \sqrt{a+b x^2}}{b \sqrt{c+d x^2}}-\frac{\sqrt{c} \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)}{b \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
283,0,0,214,0.5168731,"\int \frac{\sqrt{-c-d x^2}}{\sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[Sqrt[-c - d*x^2]/Sqrt[a + b*x^2],x]","\int \frac{\sqrt{-c-d x^2}}{\sqrt{a+b 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)}{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{d x \sqrt{a+b x^2}}{b \sqrt{-c-d x^2}}+\frac{\sqrt{c} \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)}{b \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
284,0,0,214,0.579141,"\int \frac{\sqrt{c+d x^2}}{\sqrt{-a-b x^2}} \, dx","IntegrateAlgebraic[Sqrt[c + d*x^2]/Sqrt[-a - b*x^2],x]","\int \frac{\sqrt{c+d x^2}}{\sqrt{-a-b 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)}{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{d x \sqrt{-a-b x^2}}{b \sqrt{c+d x^2}}+\frac{\sqrt{c} \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)}{b \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
285,0,0,222,0.5122403,"\int \frac{\sqrt{-c-d x^2}}{\sqrt{-a-b x^2}} \, dx","IntegrateAlgebraic[Sqrt[-c - d*x^2]/Sqrt[-a - b*x^2],x]","\int \frac{\sqrt{-c-d x^2}}{\sqrt{-a-b 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)}{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{d x \sqrt{-a-b x^2}}{b \sqrt{-c-d x^2}}-\frac{\sqrt{c} \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)}{b \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
286,0,0,189,0.5469684,"\int \frac{\sqrt{c-d x^2}}{\sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[Sqrt[c - d*x^2]/Sqrt[a + b*x^2],x]","\int \frac{\sqrt{c-d x^2}}{\sqrt{a+b x^2}} \, dx","\frac{\sqrt{c} \sqrt{\frac{b x^2}{a}+1} \sqrt{1-\frac{d x^2}{c}} (a d+b c) F\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|-\frac{b c}{a d}\right)}{b \sqrt{d} \sqrt{a+b x^2} \sqrt{c-d x^2}}-\frac{\sqrt{c} \sqrt{d} \sqrt{a+b x^2} \sqrt{1-\frac{d x^2}{c}} E\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|-\frac{b c}{a d}\right)}{b \sqrt{\frac{b x^2}{a}+1} \sqrt{c-d x^2}}",1,"Could not integrate","F",-1
287,0,0,191,0.5177773,"\int \frac{\sqrt{-c+d x^2}}{\sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[Sqrt[-c + d*x^2]/Sqrt[a + b*x^2],x]","\int \frac{\sqrt{-c+d x^2}}{\sqrt{a+b x^2}} \, dx","\frac{\sqrt{c} \sqrt{d} \sqrt{a+b x^2} \sqrt{1-\frac{d x^2}{c}} E\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|-\frac{b c}{a d}\right)}{b \sqrt{\frac{b x^2}{a}+1} \sqrt{d x^2-c}}-\frac{\sqrt{c} \sqrt{\frac{b x^2}{a}+1} \sqrt{1-\frac{d x^2}{c}} (a d+b c) F\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|-\frac{b c}{a d}\right)}{b \sqrt{d} \sqrt{a+b x^2} \sqrt{d x^2-c}}",1,"Could not integrate","F",-1
288,0,0,194,0.5382585,"\int \frac{\sqrt{c-d x^2}}{\sqrt{-a-b x^2}} \, dx","IntegrateAlgebraic[Sqrt[c - d*x^2]/Sqrt[-a - b*x^2],x]","\int \frac{\sqrt{c-d x^2}}{\sqrt{-a-b x^2}} \, dx","\frac{\sqrt{c} \sqrt{\frac{b x^2}{a}+1} \sqrt{1-\frac{d x^2}{c}} (a d+b c) F\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|-\frac{b c}{a d}\right)}{b \sqrt{d} \sqrt{-a-b x^2} \sqrt{c-d x^2}}+\frac{\sqrt{c} \sqrt{d} \sqrt{-a-b x^2} \sqrt{1-\frac{d x^2}{c}} E\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|-\frac{b c}{a d}\right)}{b \sqrt{\frac{b x^2}{a}+1} \sqrt{c-d x^2}}",1,"Could not integrate","F",-1
289,0,0,198,0.4999289,"\int \frac{\sqrt{-c+d x^2}}{\sqrt{-a-b x^2}} \, dx","IntegrateAlgebraic[Sqrt[-c + d*x^2]/Sqrt[-a - b*x^2],x]","\int \frac{\sqrt{-c+d x^2}}{\sqrt{-a-b x^2}} \, dx","-\frac{\sqrt{c} \sqrt{\frac{b x^2}{a}+1} \sqrt{1-\frac{d x^2}{c}} (a d+b c) F\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|-\frac{b c}{a d}\right)}{b \sqrt{d} \sqrt{-a-b x^2} \sqrt{d x^2-c}}-\frac{\sqrt{c} \sqrt{d} \sqrt{-a-b x^2} \sqrt{1-\frac{d x^2}{c}} E\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|-\frac{b c}{a d}\right)}{b \sqrt{\frac{b x^2}{a}+1} \sqrt{d x^2-c}}",1,"Could not integrate","F",-1
290,0,0,78,0.5371795,"\int \frac{1}{\sqrt{2+b x^2} \sqrt{3+d x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[2 + b*x^2]*Sqrt[3 + d*x^2]),x]","\int \frac{1}{\sqrt{2+b x^2} \sqrt{3+d x^2}} \, dx","\frac{\sqrt{b x^2+2} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{3}}\right)|1-\frac{3 b}{2 d}\right)}{\sqrt{2} \sqrt{d} \sqrt{d x^2+3} \sqrt{\frac{b x^2+2}{d x^2+3}}}",1,"Could not integrate","F",-1
291,0,0,39,0.4461626,"\int \frac{1}{\sqrt{4-x^2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[4 - x^2]*Sqrt[c + d*x^2]),x]","\int \frac{1}{\sqrt{4-x^2} \sqrt{c+d x^2}} \, dx","\frac{\sqrt{\frac{d x^2}{c}+1} F\left(\sin ^{-1}\left(\frac{x}{2}\right)|-\frac{4 d}{c}\right)}{\sqrt{c+d x^2}}",1,"Could not integrate","F",-1
292,0,0,61,0.4256142,"\int \frac{1}{\sqrt{4+x^2} \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[4 + x^2]*Sqrt[c + d*x^2]),x]","\int \frac{1}{\sqrt{4+x^2} \sqrt{c+d x^2}} \, dx","\frac{\sqrt{c+d x^2} F\left(\tan ^{-1}\left(\frac{x}{2}\right)|1-\frac{4 d}{c}\right)}{c \sqrt{x^2+4} \sqrt{\frac{c+d x^2}{c \left(x^2+4\right)}}}",1,"Could not integrate","F",-1
293,0,0,6,0.204314,"\int \frac{1}{\sqrt{1-x^2} \sqrt{-1+2 x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[1 - x^2]*Sqrt[-1 + 2*x^2]),x]","\int \frac{1}{\sqrt{1-x^2} \sqrt{-1+2 x^2}} \, dx","-F\left(\left.\cos ^{-1}(x)\right|2\right)",1,"Could not integrate","F",-1
294,0,0,23,0.4089704,"\int \frac{\sqrt{1-c^2 x^2}}{\sqrt{1+c^2 x^2}} \, dx","IntegrateAlgebraic[Sqrt[1 - c^2*x^2]/Sqrt[1 + c^2*x^2],x]","\int \frac{\sqrt{1-c^2 x^2}}{\sqrt{1+c^2 x^2}} \, dx","\frac{2 F\left(\left.\sin ^{-1}(c x)\right|-1\right)}{c}-\frac{E\left(\left.\sin ^{-1}(c x)\right|-1\right)}{c}",1,"Could not integrate","F",-1
295,0,0,182,0.0005983,"\int \frac{\sqrt{2+b x^2}}{\sqrt{3+d x^2}} \, dx","IntegrateAlgebraic[Sqrt[2 + b*x^2]/Sqrt[3 + d*x^2],x]","\int \frac{\sqrt{2+b x^2}}{\sqrt{3+d x^2}} \, dx","\frac{x \sqrt{b x^2+2}}{\sqrt{d x^2+3}}+\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)}{\sqrt{d} \sqrt{d x^2+3} \sqrt{\frac{b x^2+2}{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)}{\sqrt{d} \sqrt{d x^2+3} \sqrt{\frac{b x^2+2}{d x^2+3}}}",1,"Could not integrate","F",-1
296,0,0,19,0.2842541,"\int \frac{\sqrt{-1+3 x^2}}{\sqrt{2-3 x^2}} \, dx","IntegrateAlgebraic[Sqrt[-1 + 3*x^2]/Sqrt[2 - 3*x^2],x]","\int \frac{\sqrt{-1+3 x^2}}{\sqrt{2-3 x^2}} \, dx","-\frac{E\left(\left.\cos ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)}{\sqrt{3}}",1,"Could not integrate","F",-1
297,0,0,95,1.0405671,"\int \frac{\sqrt{1+\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}}}{\sqrt{1-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}}} \, dx","IntegrateAlgebraic[Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]/Sqrt[1 - (2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])],x]","\int \frac{\sqrt{1+\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}}}{\sqrt{1-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}}} \, dx","\frac{\sqrt{\sqrt{b^2-4 a c}+b} E\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right)|-\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)}{\sqrt{2} \sqrt{c}}",1,"Could not integrate","F",-1
298,0,0,94,1.1052224,"\int \frac{\sqrt{1-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}}}{\sqrt{1-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}}} \, dx","IntegrateAlgebraic[Sqrt[1 - (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]/Sqrt[1 - (2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])],x]","\int \frac{\sqrt{1-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}}}{\sqrt{1-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}}} \, dx","\frac{\sqrt{\sqrt{b^2-4 a c}+b} E\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)}{\sqrt{2} \sqrt{c}}",1,"Could not integrate","F",-1
299,0,0,478,1.0944175,"\int \frac{\sqrt{1+\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}}}{\sqrt{1+\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}}} \, dx","IntegrateAlgebraic[Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]/Sqrt[1 + (2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])],x]","\int \frac{\sqrt{1+\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}}}{\sqrt{1+\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}}} \, dx","\frac{x \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1}}{\sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}}+\frac{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} F\left(\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right)|-\frac{2 \sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)}{\sqrt{2} \sqrt{c} \sqrt{\frac{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1}{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}}-\frac{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} E\left(\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right)|-\frac{2 \sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)}{\sqrt{2} \sqrt{c} \sqrt{\frac{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1}{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}}",1,"Could not integrate","F",-1
300,0,0,215,0.9988853,"\int \frac{\sqrt{1-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}}}{\sqrt{1+\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}}} \, dx","IntegrateAlgebraic[Sqrt[1 - (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]/Sqrt[1 + (2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])],x]","\int \frac{\sqrt{1-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}}}{\sqrt{1+\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}}} \, dx","\frac{\sqrt{2} b F\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)|-\frac{b-\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}\right)}{\sqrt{c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left(\sqrt{b^2-4 a c}+b\right) E\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)|-\frac{b-\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}\right)}{\sqrt{2} \sqrt{c} \sqrt{b-\sqrt{b^2-4 a c}}}",1,"Could not integrate","F",-1
301,0,0,62,0.0367258,"\int \frac{\left(1-2 x^2\right)^m}{\sqrt{1-x^2}} \, dx","IntegrateAlgebraic[(1 - 2*x^2)^m/Sqrt[1 - x^2],x]","\int \frac{\left(1-2 x^2\right)^m}{\sqrt{1-x^2}} \, dx","-\frac{2^{-m-2} \sqrt{x^2} \left(2-4 x^2\right)^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\left(1-2 x^2\right)^2\right)}{(m+1) x}",1,"Could not integrate","F",-1
302,0,0,46,1.1876259,"\int \frac{1}{\sqrt{-1+x^2} \sqrt{7-4 \sqrt{3}+x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[-1 + x^2]*Sqrt[7 - 4*Sqrt[3] + x^2]),x]","\int \frac{1}{\sqrt{-1+x^2} \sqrt{7-4 \sqrt{3}+x^2}} \, dx","\frac{\sqrt{1-x^2} F\left(\sin ^{-1}(x)|-7-4 \sqrt{3}\right)}{\sqrt{7-4 \sqrt{3}} \sqrt{x^2-1}}",1,"Could not integrate","F",-1
303,0,0,47,2.0703306,"\int \frac{1}{\sqrt{3-3 \sqrt{3}+2 \sqrt{3} x^2} \sqrt{3+\left(-3+\sqrt{3}\right) x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[3 - 3*Sqrt[3] + 2*Sqrt[3]*x^2]*Sqrt[3 + (-3 + Sqrt[3])*x^2]),x]","\int \frac{1}{\sqrt{3-3 \sqrt{3}+2 \sqrt{3} x^2} \sqrt{3+\left(-3+\sqrt{3}\right) x^2}} \, dx","-\frac{1}{6} \sqrt{3+\sqrt{3}} F\left(\cos ^{-1}\left(\sqrt{\frac{1}{3} \left(3-\sqrt{3}\right)} x\right)|\frac{1}{2} \left(1+\sqrt{3}\right)\right)",1,"Could not integrate","F",-1
304,1,137,129,0.3000213,"\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{3 x^2+2}}{\sqrt{3}}}{x \sqrt[4]{3 x^2+2}}\right)}{4\ 2^{3/4} \sqrt{3}}+\frac{\tanh ^{-1}\left(\frac{2\ 2^{3/4} \sqrt{3} x \sqrt[4]{3 x^2+2}}{3 \sqrt{2} x^2+4 \sqrt{3 x^2+2}}\right)}{4\ 2^{3/4} \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)}{2\ 2^{3/4} \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)}{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
305,1,137,120,0.2839547,"\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
306,1,137,129,0.3004983,"\int \frac{1}{\sqrt[4]{2+b x^2} \left(4+b x^2\right)} \, dx","IntegrateAlgebraic[1/((2 + b*x^2)^(1/4)*(4 + b*x^2)),x]","\frac{\tan ^{-1}\left(\frac{\frac{\sqrt{b} x^2}{2 \sqrt[4]{2}}-\frac{\sqrt[4]{2} \sqrt{b x^2+2}}{\sqrt{b}}}{x \sqrt[4]{b x^2+2}}\right)}{4\ 2^{3/4} \sqrt{b}}+\frac{\tanh ^{-1}\left(\frac{2\ 2^{3/4} \sqrt{b} x \sqrt[4]{b x^2+2}}{\sqrt{2} b x^2+4 \sqrt{b x^2+2}}\right)}{4\ 2^{3/4} \sqrt{b}}","-\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)}{2\ 2^{3/4} \sqrt{b}}-\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)}{2\ 2^{3/4} \sqrt{b}}",1,"ArcTan[((Sqrt[b]*x^2)/(2*2^(1/4)) - (2^(1/4)*Sqrt[2 + b*x^2])/Sqrt[b])/(x*(2 + b*x^2)^(1/4))]/(4*2^(3/4)*Sqrt[b]) + ArcTanh[(2*2^(3/4)*Sqrt[b]*x*(2 + b*x^2)^(1/4))/(Sqrt[2]*b*x^2 + 4*Sqrt[2 + b*x^2])]/(4*2^(3/4)*Sqrt[b])","A",1
307,1,141,124,0.2638279,"\int \frac{1}{\sqrt[4]{2-b x^2} \left(4-b x^2\right)} \, dx","IntegrateAlgebraic[1/((2 - b*x^2)^(1/4)*(4 - b*x^2)),x]","\frac{\tan ^{-1}\left(\frac{\frac{\sqrt{b} x^2}{2 \sqrt[4]{2}}-\frac{\sqrt[4]{2} \sqrt{2-b x^2}}{\sqrt{b}}}{x \sqrt[4]{2-b x^2}}\right)}{4\ 2^{3/4} \sqrt{b}}+\frac{\tanh ^{-1}\left(\frac{2\ 2^{3/4} \sqrt{b} x \sqrt[4]{2-b x^2}}{\sqrt{2} b x^2+4 \sqrt{2-b x^2}}\right)}{4\ 2^{3/4} \sqrt{b}}","\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)}{2\ 2^{3/4} \sqrt{b}}+\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)}{2\ 2^{3/4} \sqrt{b}}",1,"ArcTan[((Sqrt[b]*x^2)/(2*2^(1/4)) - (2^(1/4)*Sqrt[2 - b*x^2])/Sqrt[b])/(x*(2 - b*x^2)^(1/4))]/(4*2^(3/4)*Sqrt[b]) + ArcTanh[(2*2^(3/4)*Sqrt[b]*x*(2 - b*x^2)^(1/4))/(Sqrt[2]*b*x^2 + 4*Sqrt[2 - b*x^2])]/(4*2^(3/4)*Sqrt[b])","A",1
308,1,136,120,0.2400618,"\int \frac{1}{\sqrt[4]{a+3 x^2} \left(2 a+3 x^2\right)} \, dx","IntegrateAlgebraic[1/((a + 3*x^2)^(1/4)*(2*a + 3*x^2)),x]","\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)}{4 \sqrt{3} a^{3/4}}-\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)}{4 \sqrt{3} a^{3/4}}","-\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)}{2 \sqrt{3} a^{3/4}}-\frac{\tanh ^{-1}\left(\frac{a^{3/4} \left(1-\frac{\sqrt{a+3 x^2}}{\sqrt{a}}\right)}{\sqrt{3} x \sqrt[4]{a+3 x^2}}\right)}{2 \sqrt{3} a^{3/4}}",1,"-1/4*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^(3/4)) + 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])]/(4*Sqrt[3]*a^(3/4))","A",1
309,1,136,120,0.248855,"\int \frac{1}{\sqrt[4]{a-3 x^2} \left(2 a-3 x^2\right)} \, dx","IntegrateAlgebraic[1/((a - 3*x^2)^(1/4)*(2*a - 3*x^2)),x]","\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)}{4 \sqrt{3} a^{3/4}}-\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)}{4 \sqrt{3} a^{3/4}}","\frac{\tan ^{-1}\left(\frac{a^{3/4} \left(1-\frac{\sqrt{a-3 x^2}}{\sqrt{a}}\right)}{\sqrt{3} x \sqrt[4]{a-3 x^2}}\right)}{2 \sqrt{3} a^{3/4}}+\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)}{2 \sqrt{3} a^{3/4}}",1,"-1/4*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^(3/4)) + 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])]/(4*Sqrt[3]*a^(3/4))","A",1
310,1,136,120,0.2503842,"\int \frac{1}{\sqrt[4]{a+b x^2} \left(2 a+b x^2\right)} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^(1/4)*(2*a + b*x^2)),x]","\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)}{4 a^{3/4} \sqrt{b}}-\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)}{4 a^{3/4} \sqrt{b}}","-\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)}{2 a^{3/4} \sqrt{b}}-\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)}{2 a^{3/4} \sqrt{b}}",1,"-1/4*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^(3/4)*Sqrt[b]) + 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])]/(4*a^(3/4)*Sqrt[b])","A",1
311,1,140,124,0.2607816,"\int \frac{1}{\sqrt[4]{a-b x^2} \left(2 a-b x^2\right)} \, dx","IntegrateAlgebraic[1/((a - b*x^2)^(1/4)*(2*a - b*x^2)),x]","\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)}{4 a^{3/4} \sqrt{b}}-\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)}{4 a^{3/4} \sqrt{b}}","\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)}{2 a^{3/4} \sqrt{b}}+\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)}{2 a^{3/4} \sqrt{b}}",1,"-1/4*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^(3/4)*Sqrt[b]) + 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])]/(4*a^(3/4)*Sqrt[b])","A",1
312,1,63,61,0.11767,"\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
313,1,72,61,0.1188609,"\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 \left(-3 x^2-1\right)^{3/4}}{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)^(3/4))/(1 + 3*x^2)]/(2*Sqrt[6])","A",1
314,1,79,77,0.1354273,"\int \frac{1}{\left(-2+b x^2\right) \sqrt[4]{-1+b x^2}} \, dx","IntegrateAlgebraic[1/((-2 + b*x^2)*(-1 + b*x^2)^(1/4)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b x^2-1}}{\sqrt{b} x}\right)}{2 \sqrt{2} \sqrt{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{2} \sqrt[4]{b x^2-1}}\right)}{2 \sqrt{2} \sqrt{b}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{2} \sqrt[4]{b x^2-1}}\right)}{2 \sqrt{2} \sqrt{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{2} \sqrt[4]{b x^2-1}}\right)}{2 \sqrt{2} \sqrt{b}}",1,"ArcTan[(Sqrt[2]*(-1 + b*x^2)^(1/4))/(Sqrt[b]*x)]/(2*Sqrt[2]*Sqrt[b]) - ArcTanh[(Sqrt[b]*x)/(Sqrt[2]*(-1 + b*x^2)^(1/4))]/(2*Sqrt[2]*Sqrt[b])","A",1
315,1,90,79,0.1350448,"\int \frac{1}{\left(-2-b x^2\right) \sqrt[4]{-1-b x^2}} \, dx","IntegrateAlgebraic[1/((-2 - b*x^2)*(-1 - b*x^2)^(1/4)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-b x^2-1}}{\sqrt{b} x}\right)}{2 \sqrt{2} \sqrt{b}}+\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)}{2 \sqrt{2} \sqrt{b}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{2} \sqrt[4]{-b x^2-1}}\right)}{2 \sqrt{2} \sqrt{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{2} \sqrt[4]{-b x^2-1}}\right)}{2 \sqrt{2} \sqrt{b}}",1,"ArcTan[(Sqrt[2]*(-1 - b*x^2)^(1/4))/(Sqrt[b]*x)]/(2*Sqrt[2]*Sqrt[b]) + ArcTanh[(Sqrt[b]*x*(-1 - b*x^2)^(3/4))/(Sqrt[2]*(1 + b*x^2))]/(2*Sqrt[2]*Sqrt[b])","A",1
316,1,89,85,0.1541121,"\int \frac{1}{\left(-2 a+3 x^2\right) \sqrt[4]{-a+3 x^2}} \, dx","IntegrateAlgebraic[1/((-2*a + 3*x^2)*(-a + 3*x^2)^(1/4)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{\frac{2}{3}} \sqrt[4]{a} \sqrt[4]{3 x^2-a}}{x}\right)}{2 \sqrt{6} a^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{\frac{2}{3}} \sqrt[4]{a} \sqrt[4]{3 x^2-a}}{x}\right)}{2 \sqrt{6} a^{3/4}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{a} \sqrt[4]{3 x^2-a}}\right)}{2 \sqrt{6} a^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{a} \sqrt[4]{3 x^2-a}}\right)}{2 \sqrt{6} a^{3/4}}",1,"ArcTan[(Sqrt[2/3]*a^(1/4)*(-a + 3*x^2)^(1/4))/x]/(2*Sqrt[6]*a^(3/4)) - ArcTanh[(Sqrt[2/3]*a^(1/4)*(-a + 3*x^2)^(1/4))/x]/(2*Sqrt[6]*a^(3/4))","A",1
317,1,89,85,0.14746,"\int \frac{1}{\left(-2 a-3 x^2\right) \sqrt[4]{-a-3 x^2}} \, dx","IntegrateAlgebraic[1/((-2*a - 3*x^2)*(-a - 3*x^2)^(1/4)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{\frac{2}{3}} \sqrt[4]{a} \sqrt[4]{-a-3 x^2}}{x}\right)}{2 \sqrt{6} a^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{\frac{2}{3}} \sqrt[4]{a} \sqrt[4]{-a-3 x^2}}{x}\right)}{2 \sqrt{6} a^{3/4}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{a} \sqrt[4]{-a-3 x^2}}\right)}{2 \sqrt{6} a^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{a} \sqrt[4]{-a-3 x^2}}\right)}{2 \sqrt{6} a^{3/4}}",1,"ArcTan[(Sqrt[2/3]*a^(1/4)*(-a - 3*x^2)^(1/4))/x]/(2*Sqrt[6]*a^(3/4)) - ArcTanh[(Sqrt[2/3]*a^(1/4)*(-a - 3*x^2)^(1/4))/x]/(2*Sqrt[6]*a^(3/4))","A",1
318,1,105,101,0.1708487,"\int \frac{1}{\left(-2 a+b x^2\right) \sqrt[4]{-a+b x^2}} \, dx","IntegrateAlgebraic[1/((-2*a + b*x^2)*(-a + b*x^2)^(1/4)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b x^2-a}}{\sqrt{b} x}\right)}{2 \sqrt{2} a^{3/4} \sqrt{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b x^2-a}}{\sqrt{b} x}\right)}{2 \sqrt{2} a^{3/4} \sqrt{b}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b x^2-a}}\right)}{2 \sqrt{2} a^{3/4} \sqrt{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b x^2-a}}\right)}{2 \sqrt{2} a^{3/4} \sqrt{b}}",1,"ArcTan[(Sqrt[2]*a^(1/4)*(-a + b*x^2)^(1/4))/(Sqrt[b]*x)]/(2*Sqrt[2]*a^(3/4)*Sqrt[b]) - ArcTanh[(Sqrt[2]*a^(1/4)*(-a + b*x^2)^(1/4))/(Sqrt[b]*x)]/(2*Sqrt[2]*a^(3/4)*Sqrt[b])","A",1
319,1,107,103,0.1638848,"\int \frac{1}{\left(-2 a-b x^2\right) \sqrt[4]{-a-b x^2}} \, dx","IntegrateAlgebraic[1/((-2*a - b*x^2)*(-a - b*x^2)^(1/4)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{-a-b x^2}}{\sqrt{b} x}\right)}{2 \sqrt{2} a^{3/4} \sqrt{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{-a-b x^2}}{\sqrt{b} x}\right)}{2 \sqrt{2} a^{3/4} \sqrt{b}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{-a-b x^2}}\right)}{2 \sqrt{2} a^{3/4} \sqrt{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{-a-b x^2}}\right)}{2 \sqrt{2} a^{3/4} \sqrt{b}}",1,"ArcTan[(Sqrt[2]*a^(1/4)*(-a - b*x^2)^(1/4))/(Sqrt[b]*x)]/(2*Sqrt[2]*a^(3/4)*Sqrt[b]) - ArcTanh[(Sqrt[2]*a^(1/4)*(-a - b*x^2)^(1/4))/(Sqrt[b]*x)]/(2*Sqrt[2]*a^(3/4)*Sqrt[b])","A",1
320,1,55,53,0.0921114,"\int \frac{1}{\left(2-x^2\right) \sqrt[4]{-1+x^2}} \, dx","IntegrateAlgebraic[1/((2 - x^2)*(-1 + x^2)^(1/4)),x]","\frac{\tanh ^{-1}\left(\frac{x}{\sqrt{2} \sqrt[4]{x^2-1}}\right)}{2 \sqrt{2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{x^2-1}}{x}\right)}{2 \sqrt{2}}","\frac{\tan ^{-1}\left(\frac{x}{\sqrt{2} \sqrt[4]{x^2-1}}\right)}{2 \sqrt{2}}+\frac{\tanh ^{-1}\left(\frac{x}{\sqrt{2} \sqrt[4]{x^2-1}}\right)}{2 \sqrt{2}}",1,"-1/2*ArcTan[(Sqrt[2]*(-1 + x^2)^(1/4))/x]/Sqrt[2] + ArcTanh[x/(Sqrt[2]*(-1 + x^2)^(1/4))]/(2*Sqrt[2])","A",1
321,0,0,362,6.3871228,"\int \frac{\left(a+b x^2\right)^{7/4}}{c+d x^2} \, dx","IntegrateAlgebraic[(a + b*x^2)^(7/4)/(c + d*x^2),x]","\int \frac{\left(a+b x^2\right)^{7/4}}{c+d x^2} \, dx","-\frac{6 a^{3/2} \sqrt{b} \sqrt[4]{\frac{b x^2}{a}+1} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{5 d \sqrt[4]{a+b x^2}}+\frac{\sqrt[4]{a} \sqrt{-\frac{b x^2}{a}} (a d-b c)^{3/2} \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{d^{5/2} x}-\frac{\sqrt[4]{a} \sqrt{-\frac{b x^2}{a}} (a d-b c)^{3/2} \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{d^{5/2} x}-\frac{2 b x (b c-a d)}{d^2 \sqrt[4]{a+b x^2}}+\frac{2 \sqrt{a} \sqrt{b} \sqrt[4]{\frac{b x^2}{a}+1} (b c-a d) E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{d^2 \sqrt[4]{a+b x^2}}+\frac{6 a b x}{5 d \sqrt[4]{a+b x^2}}+\frac{2 b x \left(a+b x^2\right)^{3/4}}{5 d}",1,"Could not integrate","F",-1
322,0,0,302,6.3477352,"\int \frac{\left(a+b x^2\right)^{5/4}}{c+d x^2} \, dx","IntegrateAlgebraic[(a + b*x^2)^(5/4)/(c + d*x^2),x]","\int \frac{\left(a+b x^2\right)^{5/4}}{c+d x^2} \, dx","\frac{2 a^{3/2} \sqrt{b} \left(\frac{b x^2}{a}+1\right)^{3/4} F\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{3 d \left(a+b x^2\right)^{3/4}}-\frac{2 \sqrt{a} \sqrt{b} \left(\frac{b x^2}{a}+1\right)^{3/4} (b c-a d) F\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{d^2 \left(a+b x^2\right)^{3/4}}+\frac{\sqrt[4]{a} \sqrt{-\frac{b x^2}{a}} (b c-a d) \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{d^2 x}+\frac{\sqrt[4]{a} \sqrt{-\frac{b x^2}{a}} (b c-a d) \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{d^2 x}+\frac{2 b x \sqrt[4]{a+b x^2}}{3 d}",1,"Could not integrate","F",-1
323,0,0,244,5.7056928,"\int \frac{\left(a+b x^2\right)^{3/4}}{c+d x^2} \, dx","IntegrateAlgebraic[(a + b*x^2)^(3/4)/(c + d*x^2),x]","\int \frac{\left(a+b x^2\right)^{3/4}}{c+d x^2} \, dx","\frac{\sqrt[4]{a} \sqrt{-\frac{b x^2}{a}} \sqrt{a d-b c} \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{d^{3/2} x}-\frac{\sqrt[4]{a} \sqrt{-\frac{b x^2}{a}} \sqrt{a d-b c} \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{d^{3/2} x}+\frac{2 b x}{d \sqrt[4]{a+b x^2}}-\frac{2 \sqrt{a} \sqrt{b} \sqrt[4]{\frac{b x^2}{a}+1} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{d \sqrt[4]{a+b x^2}}",1,"Could not integrate","F",-1
324,0,0,199,5.1521701,"\int \frac{\sqrt[4]{a+b x^2}}{c+d x^2} \, dx","IntegrateAlgebraic[(a + b*x^2)^(1/4)/(c + d*x^2),x]","\int \frac{\sqrt[4]{a+b x^2}}{c+d x^2} \, dx","-\frac{\sqrt[4]{a} \sqrt{-\frac{b x^2}{a}} \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{d x}-\frac{\sqrt[4]{a} \sqrt{-\frac{b x^2}{a}} \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{d x}+\frac{2 \sqrt{a} \sqrt{b} \left(\frac{b x^2}{a}+1\right)^{3/4} F\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{d \left(a+b x^2\right)^{3/4}}",1,"Could not integrate","F",-1
325,0,0,167,5.0778424,"\int \frac{1}{\sqrt[4]{a+b x^2} \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^(1/4)*(c + d*x^2)),x]","\int \frac{1}{\sqrt[4]{a+b x^2} \left(c+d x^2\right)} \, dx","\frac{\sqrt[4]{a} \sqrt{-\frac{b x^2}{a}} \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{\sqrt{d} x \sqrt{a d-b c}}-\frac{\sqrt[4]{a} \sqrt{-\frac{b x^2}{a}} \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{\sqrt{d} x \sqrt{a d-b c}}",1,"Could not integrate","F",-1
326,0,0,152,5.4572504,"\int \frac{1}{\left(a+b x^2\right)^{3/4} \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^(3/4)*(c + d*x^2)),x]","\int \frac{1}{\left(a+b x^2\right)^{3/4} \left(c+d x^2\right)} \, dx","\frac{\sqrt[4]{a} \sqrt{-\frac{b x^2}{a}} \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{x (b c-a d)}+\frac{\sqrt[4]{a} \sqrt{-\frac{b x^2}{a}} \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{x (b c-a d)}",1,"Could not integrate","F",-1
327,0,0,233,5.9839723,"\int \frac{1}{\left(a+b x^2\right)^{5/4} \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^(5/4)*(c + d*x^2)),x]","\int \frac{1}{\left(a+b x^2\right)^{5/4} \left(c+d x^2\right)} \, dx","\frac{2 \sqrt{b} \sqrt[4]{\frac{b x^2}{a}+1} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{\sqrt{a} \sqrt[4]{a+b x^2} (b c-a d)}+\frac{\sqrt[4]{a} \sqrt{d} \sqrt{-\frac{b x^2}{a}} \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{x (a d-b c)^{3/2}}-\frac{\sqrt[4]{a} \sqrt{d} \sqrt{-\frac{b x^2}{a}} \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{x (a d-b c)^{3/2}}",1,"Could not integrate","F",-1
328,0,0,254,6.3308164,"\int \frac{1}{\left(a+b x^2\right)^{7/4} \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^(7/4)*(c + d*x^2)),x]","\int \frac{1}{\left(a+b x^2\right)^{7/4} \left(c+d x^2\right)} \, dx","\frac{2 b x}{3 a \left(a+b x^2\right)^{3/4} (b c-a d)}+\frac{2 \sqrt{b} \left(\frac{b x^2}{a}+1\right)^{3/4} F\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{3 \sqrt{a} \left(a+b x^2\right)^{3/4} (b c-a d)}-\frac{\sqrt[4]{a} d \sqrt{-\frac{b x^2}{a}} \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{x (b c-a d)^2}-\frac{\sqrt[4]{a} d \sqrt{-\frac{b x^2}{a}} \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{x (b c-a d)^2}",1,"Could not integrate","F",-1
329,0,0,274,6.6908195,"\int \frac{1}{\left(a+b x^2\right)^{9/4} \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^(9/4)*(c + d*x^2)),x]","\int \frac{1}{\left(a+b x^2\right)^{9/4} \left(c+d x^2\right)} \, dx","\frac{2 \sqrt{b} \sqrt[4]{\frac{b x^2}{a}+1} (3 b c-8 a d) E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{5 a^{3/2} \sqrt[4]{a+b x^2} (b c-a d)^2}+\frac{\sqrt[4]{a} d^{3/2} \sqrt{-\frac{b x^2}{a}} \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{x (a d-b c)^{5/2}}-\frac{\sqrt[4]{a} d^{3/2} \sqrt{-\frac{b x^2}{a}} \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{x (a d-b c)^{5/2}}+\frac{2 b x}{5 a \left(a+b x^2\right)^{5/4} (b c-a d)}",1,"Could not integrate","F",-1
330,0,0,304,6.7043635,"\int \frac{1}{\left(a+b x^2\right)^{11/4} \left(c+d x^2\right)} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^(11/4)*(c + d*x^2)),x]","\int \frac{1}{\left(a+b x^2\right)^{11/4} \left(c+d x^2\right)} \, dx","\frac{2 \sqrt{b} \left(\frac{b x^2}{a}+1\right)^{3/4} (5 b c-12 a d) F\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{21 a^{3/2} \left(a+b x^2\right)^{3/4} (b c-a d)^2}+\frac{2 b x (5 b c-12 a d)}{21 a^2 \left(a+b x^2\right)^{3/4} (b c-a d)^2}+\frac{\sqrt[4]{a} d^2 \sqrt{-\frac{b x^2}{a}} \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{x (b c-a d)^3}+\frac{\sqrt[4]{a} d^2 \sqrt{-\frac{b x^2}{a}} \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{x (b c-a d)^3}+\frac{2 b x}{7 a \left(a+b x^2\right)^{7/4} (b c-a d)}",1,"Could not integrate","F",-1
331,0,0,340,11.0541772,"\int \frac{\left(a+b x^2\right)^{7/4}}{\left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[(a + b*x^2)^(7/4)/(c + d*x^2)^2,x]","\int \frac{\left(a+b x^2\right)^{7/4}}{\left(c+d x^2\right)^2} \, dx","\frac{\sqrt[4]{a} \sqrt{-\frac{b x^2}{a}} \sqrt{a d-b c} (2 a d+5 b c) \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{4 c d^{5/2} x}-\frac{\sqrt[4]{a} \sqrt{-\frac{b x^2}{a}} \sqrt{a d-b c} (2 a d+5 b c) \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{4 c d^{5/2} x}+\frac{b x (5 b c-a d)}{2 c d^2 \sqrt[4]{a+b x^2}}-\frac{\sqrt{a} \sqrt{b} \sqrt[4]{\frac{b x^2}{a}+1} (5 b c-a d) E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{2 c d^2 \sqrt[4]{a+b x^2}}-\frac{x \left(a+b x^2\right)^{3/4} (b c-a d)}{2 c d \left(c+d x^2\right)}",1,"Could not integrate","F",-1
332,0,0,279,11.1492753,"\int \frac{\left(a+b x^2\right)^{5/4}}{\left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[(a + b*x^2)^(5/4)/(c + d*x^2)^2,x]","\int \frac{\left(a+b x^2\right)^{5/4}}{\left(c+d x^2\right)^2} \, dx","\frac{\sqrt{a} \sqrt{b} \left(\frac{b x^2}{a}+1\right)^{3/4} (a d+3 b c) F\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{2 c d^2 \left(a+b x^2\right)^{3/4}}-\frac{\sqrt[4]{a} \sqrt{-\frac{b x^2}{a}} (2 a d+3 b c) \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{4 c d^2 x}-\frac{\sqrt[4]{a} \sqrt{-\frac{b x^2}{a}} (2 a d+3 b c) \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{4 c d^2 x}-\frac{x \sqrt[4]{a+b x^2} (b c-a d)}{2 c d \left(c+d x^2\right)}",1,"Could not integrate","F",-1
333,0,0,309,10.8205565,"\int \frac{\left(a+b x^2\right)^{3/4}}{\left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[(a + b*x^2)^(3/4)/(c + d*x^2)^2,x]","\int \frac{\left(a+b x^2\right)^{3/4}}{\left(c+d x^2\right)^2} \, dx","\frac{\sqrt[4]{a} \sqrt{-\frac{b x^2}{a}} (2 a d+b c) \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{4 c d^{3/2} x \sqrt{a d-b c}}-\frac{\sqrt[4]{a} \sqrt{-\frac{b x^2}{a}} (2 a d+b c) \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{4 c d^{3/2} x \sqrt{a d-b c}}-\frac{b x}{2 c d \sqrt[4]{a+b x^2}}+\frac{x \left(a+b x^2\right)^{3/4}}{2 c \left(c+d x^2\right)}+\frac{\sqrt{a} \sqrt{b} \sqrt[4]{\frac{b x^2}{a}+1} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{2 c d \sqrt[4]{a+b x^2}}",1,"Could not integrate","F",-1
334,0,0,278,10.0502628,"\int \frac{\sqrt[4]{a+b x^2}}{\left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[(a + b*x^2)^(1/4)/(c + d*x^2)^2,x]","\int \frac{\sqrt[4]{a+b x^2}}{\left(c+d x^2\right)^2} \, dx","\frac{x \sqrt[4]{a+b x^2}}{2 c \left(c+d x^2\right)}+\frac{\sqrt{a} \sqrt{b} \left(\frac{b x^2}{a}+1\right)^{3/4} F\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{2 c d \left(a+b x^2\right)^{3/4}}-\frac{\sqrt[4]{a} \sqrt{-\frac{b x^2}{a}} (b c-2 a d) \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{4 c d x (b c-a d)}-\frac{\sqrt[4]{a} \sqrt{-\frac{b x^2}{a}} (b c-2 a d) \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{4 c d x (b c-a d)}",1,"Could not integrate","F",-1
335,0,0,336,10.5075107,"\int \frac{1}{\sqrt[4]{a+b x^2} \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^(1/4)*(c + d*x^2)^2),x]","\int \frac{1}{\sqrt[4]{a+b x^2} \left(c+d x^2\right)^2} \, dx","\frac{b x}{2 c \sqrt[4]{a+b x^2} (b c-a d)}-\frac{d x \left(a+b x^2\right)^{3/4}}{2 c \left(c+d x^2\right) (b c-a d)}-\frac{\sqrt{a} \sqrt{b} \sqrt[4]{\frac{b x^2}{a}+1} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{2 c \sqrt[4]{a+b x^2} (b c-a d)}-\frac{\sqrt[4]{a} \sqrt{-\frac{b x^2}{a}} (3 b c-2 a d) \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{4 c \sqrt{d} x (a d-b c)^{3/2}}+\frac{\sqrt[4]{a} \sqrt{-\frac{b x^2}{a}} (3 b c-2 a d) \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{4 c \sqrt{d} x (a d-b c)^{3/2}}",1,"Could not integrate","F",-1
336,0,0,292,10.8549055,"\int \frac{1}{\left(a+b x^2\right)^{3/4} \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^(3/4)*(c + d*x^2)^2),x]","\int \frac{1}{\left(a+b x^2\right)^{3/4} \left(c+d x^2\right)^2} \, dx","-\frac{d x \sqrt[4]{a+b x^2}}{2 c \left(c+d x^2\right) (b c-a d)}-\frac{\sqrt{a} \sqrt{b} \left(\frac{b x^2}{a}+1\right)^{3/4} F\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{2 c \left(a+b x^2\right)^{3/4} (b c-a d)}+\frac{\sqrt[4]{a} \sqrt{-\frac{b x^2}{a}} (5 b c-2 a d) \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{4 c x (b c-a d)^2}+\frac{\sqrt[4]{a} \sqrt{-\frac{b x^2}{a}} (5 b c-2 a d) \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{4 c x (b c-a d)^2}",1,"Could not integrate","F",-1
337,0,0,314,11.569054,"\int \frac{1}{\left(a+b x^2\right)^{5/4} \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^(5/4)*(c + d*x^2)^2),x]","\int \frac{1}{\left(a+b x^2\right)^{5/4} \left(c+d x^2\right)^2} \, dx","-\frac{d x}{2 c \sqrt[4]{a+b x^2} \left(c+d x^2\right) (b c-a d)}+\frac{\sqrt{b} \sqrt[4]{\frac{b x^2}{a}+1} (a d+4 b c) E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{2 \sqrt{a} c \sqrt[4]{a+b x^2} (b c-a d)^2}-\frac{\sqrt[4]{a} \sqrt{d} \sqrt{-\frac{b x^2}{a}} (7 b c-2 a d) \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{4 c x (a d-b c)^{5/2}}+\frac{\sqrt[4]{a} \sqrt{d} \sqrt{-\frac{b x^2}{a}} (7 b c-2 a d) \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{4 c x (a d-b c)^{5/2}}",1,"Could not integrate","F",-1
338,0,0,345,11.7090203,"\int \frac{1}{\left(a+b x^2\right)^{7/4} \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^(7/4)*(c + d*x^2)^2),x]","\int \frac{1}{\left(a+b x^2\right)^{7/4} \left(c+d x^2\right)^2} \, dx","\frac{b x (3 a d+4 b c)}{6 a c \left(a+b x^2\right)^{3/4} (b c-a d)^2}-\frac{d x}{2 c \left(a+b x^2\right)^{3/4} \left(c+d x^2\right) (b c-a d)}+\frac{\sqrt{b} \left(\frac{b x^2}{a}+1\right)^{3/4} (3 a d+4 b c) F\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{6 \sqrt{a} c \left(a+b x^2\right)^{3/4} (b c-a d)^2}-\frac{\sqrt[4]{a} d \sqrt{-\frac{b x^2}{a}} (9 b c-2 a d) \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{4 c x (b c-a d)^3}-\frac{\sqrt[4]{a} d \sqrt{-\frac{b x^2}{a}} (9 b c-2 a d) \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{4 c x (b c-a d)^3}",1,"Could not integrate","F",-1
339,0,0,371,11.8875805,"\int \frac{1}{\left(a+b x^2\right)^{9/4} \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^(9/4)*(c + d*x^2)^2),x]","\int \frac{1}{\left(a+b x^2\right)^{9/4} \left(c+d x^2\right)^2} \, dx","\frac{\sqrt{b} \sqrt[4]{\frac{b x^2}{a}+1} \left(-5 a^2 d^2-52 a b c d+12 b^2 c^2\right) E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{10 a^{3/2} c \sqrt[4]{a+b x^2} (b c-a d)^3}-\frac{\sqrt[4]{a} d^{3/2} \sqrt{-\frac{b x^2}{a}} (11 b c-2 a d) \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{4 c x (a d-b c)^{7/2}}+\frac{\sqrt[4]{a} d^{3/2} \sqrt{-\frac{b x^2}{a}} (11 b c-2 a d) \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{4 c x (a d-b c)^{7/2}}-\frac{d x}{2 c \left(a+b x^2\right)^{5/4} \left(c+d x^2\right) (b c-a d)}+\frac{b x (5 a d+4 b c)}{10 a c \left(a+b x^2\right)^{5/4} (b c-a d)^2}",1,"Could not integrate","F",-1
340,0,0,419,11.8621531,"\int \frac{1}{\left(a+b x^2\right)^{11/4} \left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^(11/4)*(c + d*x^2)^2),x]","\int \frac{1}{\left(a+b x^2\right)^{11/4} \left(c+d x^2\right)^2} \, dx","\frac{b x \left(-21 a^2 d^2-76 a b c d+20 b^2 c^2\right)}{42 a^2 c \left(a+b x^2\right)^{3/4} (b c-a d)^3}+\frac{\sqrt{b} \left(\frac{b x^2}{a}+1\right)^{3/4} \left(-21 a^2 d^2-76 a b c d+20 b^2 c^2\right) F\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{42 a^{3/2} c \left(a+b x^2\right)^{3/4} (b c-a d)^3}+\frac{\sqrt[4]{a} d^2 \sqrt{-\frac{b x^2}{a}} (13 b c-2 a d) \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{4 c x (b c-a d)^4}+\frac{\sqrt[4]{a} d^2 \sqrt{-\frac{b x^2}{a}} (13 b c-2 a d) \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{4 c x (b c-a d)^4}-\frac{d x}{2 c \left(a+b x^2\right)^{7/4} \left(c+d x^2\right) (b c-a d)}+\frac{b x (7 a d+4 b c)}{14 a c \left(a+b x^2\right)^{7/4} (b c-a d)^2}",1,"Could not integrate","F",-1
341,0,0,79,0.0297232,"\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
342,0,0,296,0.0962104,"\int \left(a+b x^2\right)^p \left(c+d x^2\right)^3 \, dx","IntegrateAlgebraic[(a + b*x^2)^p*(c + d*x^2)^3,x]","\int \left(a+b x^2\right)^p \left(c+d x^2\right)^3 \, dx","\frac{d x \left(a+b x^2\right)^{p+1} \left(15 a^2 d^2-8 a b c d (p+6)+b^2 c^2 \left(4 p^2+28 p+57\right)\right)}{b^3 (2 p+3) (2 p+5) (2 p+7)}-\frac{x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(15 a^3 d^3-9 a^2 b c d^2 (2 p+7)+3 a b^2 c^2 d \left(4 p^2+24 p+35\right)-b^3 c^3 \left(8 p^3+60 p^2+142 p+105\right)\right) \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)}{b^3 (2 p+3) (2 p+5) (2 p+7)}-\frac{d x \left(c+d x^2\right) \left(a+b x^2\right)^{p+1} (5 a d-b c (2 p+11))}{b^2 (2 p+5) (2 p+7)}+\frac{d x \left(c+d x^2\right)^2 \left(a+b x^2\right)^{p+1}}{b (2 p+7)}",1,"Could not integrate","F",-1
343,0,0,176,0.0736697,"\int \left(a+b x^2\right)^p \left(c+d x^2\right)^2 \, dx","IntegrateAlgebraic[(a + b*x^2)^p*(c + d*x^2)^2,x]","\int \left(a+b x^2\right)^p \left(c+d x^2\right)^2 \, dx","\frac{x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(3 a^2 d^2-2 a b c d (2 p+5)+b^2 c^2 \left(4 p^2+16 p+15\right)\right) \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)}{b^2 (2 p+3) (2 p+5)}-\frac{d x \left(a+b x^2\right)^{p+1} (3 a d-b c (2 p+7))}{b^2 (2 p+3) (2 p+5)}+\frac{d x \left(c+d x^2\right) \left(a+b x^2\right)^{p+1}}{b (2 p+5)}",1,"Could not integrate","F",-1
344,0,0,93,0.0582777,"\int \left(a+b x^2\right)^p \left(c+d x^2\right) \, dx","IntegrateAlgebraic[(a + b*x^2)^p*(c + d*x^2),x]","\int \left(a+b x^2\right)^p \left(c+d x^2\right) \, dx","\frac{d x \left(a+b x^2\right)^{p+1}}{b (2 p+3)}-\frac{x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} (a d-b c (2 p+3)) \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)}{b (2 p+3)}",1,"Could not integrate","F",-1
345,0,0,44,0.0005806,"\int \left(a+b x^2\right)^p \, dx","IntegrateAlgebraic[(a + b*x^2)^p,x]","\int \left(a+b x^2\right)^p \, dx","x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)",1,"Could not integrate","F",-1
346,0,0,57,0.1076099,"\int \frac{\left(a+b x^2\right)^p}{c+d x^2} \, dx","IntegrateAlgebraic[(a + b*x^2)^p/(c + d*x^2),x]","\int \frac{\left(a+b x^2\right)^p}{c+d x^2} \, dx","\frac{x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{c}",1,"Could not integrate","F",-1
347,0,0,57,0.1841926,"\int \frac{\left(a+b x^2\right)^p}{\left(c+d x^2\right)^2} \, dx","IntegrateAlgebraic[(a + b*x^2)^p/(c + d*x^2)^2,x]","\int \frac{\left(a+b x^2\right)^p}{\left(c+d x^2\right)^2} \, dx","\frac{x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} F_1\left(\frac{1}{2};-p,2;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{c^2}",1,"Could not integrate","F",-1
348,0,0,57,0.1970079,"\int \frac{\left(a+b x^2\right)^p}{\left(c+d x^2\right)^3} \, dx","IntegrateAlgebraic[(a + b*x^2)^p/(c + d*x^2)^3,x]","\int \frac{\left(a+b x^2\right)^p}{\left(c+d x^2\right)^3} \, dx","\frac{x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} F_1\left(\frac{1}{2};-p,3;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{c^3}",1,"Could not integrate","F",-1
349,0,0,53,0.1021643,"\int \left(a+b x^2\right)^{-1-\frac{b c}{2 b c-2 a d}} \left(c+d x^2\right)^{-1+\frac{a d}{2 b c-2 a d}} \, dx","IntegrateAlgebraic[(a + b*x^2)^(-1 - (b*c)/(2*b*c - 2*a*d))*(c + d*x^2)^(-1 + (a*d)/(2*b*c - 2*a*d)),x]","\int \left(a+b x^2\right)^{-1-\frac{b c}{2 b c-2 a d}} \left(c+d x^2\right)^{-1+\frac{a d}{2 b c-2 a d}} \, dx","\frac{x \left(a+b x^2\right)^{-\frac{b c}{2 b c-2 a d}} \left(c+d x^2\right)^{\frac{a d}{2 b c-2 a d}}}{a c}",1,"Could not integrate","F",-1