1,1,94,0,0.064207,"\int \left(a+b x^2\right) \left(c+d x^2\right)^4 \, dx","Int[(a + b*x^2)*(c + d*x^2)^4,x]","\frac{2}{5} c^2 d x^5 (3 a d+2 b c)+\frac{1}{3} c^3 x^3 (4 a d+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}","\frac{2}{5} c^2 d x^5 (3 a d+2 b c)+\frac{1}{3} c^3 x^3 (4 a d+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,"a*c^4*x + (c^3*(b*c + 4*a*d)*x^3)/3 + (2*c^2*d*(2*b*c + 3*a*d)*x^5)/5 + (2*c*d^2*(3*b*c + 2*a*d)*x^7)/7 + (d^3*(4*b*c + a*d)*x^9)/9 + (b*d^4*x^11)/11","A",2,1,17,0.05882,1,"{373}"
2,1,70,0,0.0420841,"\int \left(a+b x^2\right) \left(c+d x^2\right)^3 \, dx","Int[(a + b*x^2)*(c + d*x^2)^3,x]","\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","\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,"a*c^3*x + (c^2*(b*c + 3*a*d)*x^3)/3 + (3*c*d*(b*c + a*d)*x^5)/5 + (d^2*(3*b*c + a*d)*x^7)/7 + (b*d^3*x^9)/9","A",2,1,17,0.05882,1,"{373}"
3,1,50,0,0.0274234,"\int \left(a+b x^2\right) \left(c+d x^2\right)^2 \, dx","Int[(a + b*x^2)*(c + d*x^2)^2,x]","\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","\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,"a*c^2*x + (c*(b*c + 2*a*d)*x^3)/3 + (d*(2*b*c + a*d)*x^5)/5 + (b*d^2*x^7)/7","A",2,1,17,0.05882,1,"{373}"
4,1,28,0,0.0128864,"\int \left(a+b x^2\right) \left(c+d x^2\right) \, dx","Int[(a + b*x^2)*(c + d*x^2),x]","\frac{1}{3} x^3 (a d+b c)+a c x+\frac{1}{5} b d x^5","\frac{1}{3} x^3 (a d+b c)+a c x+\frac{1}{5} b d x^5",1,"a*c*x + ((b*c + a*d)*x^3)/3 + (b*d*x^5)/5","A",2,1,15,0.06667,1,"{373}"
5,1,40,0,0.0198071,"\int \frac{a+b x^2}{c+d x^2} \, dx","Int[(a + b*x^2)/(c + d*x^2),x]","\frac{b x}{d}-\frac{(b c-a d) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{\sqrt{c} d^{3/2}}","\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",2,2,17,0.1176,1,"{388, 205}"
6,1,63,0,0.0202435,"\int \frac{a+b x^2}{\left(c+d x^2\right)^2} \, dx","Int[(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 (b c-a d)}{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",2,2,17,0.1176,1,"{385, 205}"
7,1,92,0,0.0306019,"\int \frac{a+b x^2}{\left(c+d x^2\right)^3} \, dx","Int[(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 (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}","\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,"-((b*c - a*d)*x)/(4*c*d*(c + d*x^2)^2) + ((b*c + 3*a*d)*x)/(8*c^2*d*(c + d*x^2)) + ((b*c + 3*a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(5/2)*d^(3/2))","A",3,3,17,0.1765,1,"{385, 199, 205}"
8,1,122,0,0.0735117,"\int \left(a+b x^2\right)^2 \left(c+d x^2\right)^3 \, dx","Int[(a + b*x^2)^2*(c + d*x^2)^3,x]","\frac{1}{7} d x^7 \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{1}{5} c x^5 \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+a^2 c^3 x+\frac{1}{3} a c^2 x^3 (3 a d+2 b c)+\frac{1}{9} b d^2 x^9 (2 a d+3 b c)+\frac{1}{11} b^2 d^3 x^{11}","\frac{1}{7} d x^7 \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{1}{5} c x^5 \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+a^2 c^3 x+\frac{1}{3} a c^2 x^3 (3 a d+2 b c)+\frac{1}{9} b d^2 x^9 (2 a d+3 b c)+\frac{1}{11} b^2 d^3 x^{11}",1,"a^2*c^3*x + (a*c^2*(2*b*c + 3*a*d)*x^3)/3 + (c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^5)/5 + (d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^7)/7 + (b*d^2*(3*b*c + 2*a*d)*x^9)/9 + (b^2*d^3*x^11)/11","A",2,1,19,0.05263,1,"{373}"
9,1,82,0,0.0457626,"\int \left(a+b x^2\right)^2 \left(c+d x^2\right)^2 \, dx","Int[(a + b*x^2)^2*(c + d*x^2)^2,x]","\frac{1}{5} x^5 \left(a^2 d^2+4 a b c d+b^2 c^2\right)+a^2 c^2 x+\frac{2}{7} b d x^7 (a d+b c)+\frac{2}{3} a c x^3 (a d+b c)+\frac{1}{9} b^2 d^2 x^9","\frac{1}{5} x^5 \left(a^2 d^2+4 a b c d+b^2 c^2\right)+a^2 c^2 x+\frac{2}{7} b d x^7 (a d+b c)+\frac{2}{3} a c x^3 (a d+b c)+\frac{1}{9} b^2 d^2 x^9",1,"a^2*c^2*x + (2*a*c*(b*c + a*d)*x^3)/3 + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^5)/5 + (2*b*d*(b*c + a*d)*x^7)/7 + (b^2*d^2*x^9)/9","A",2,1,19,0.05263,1,"{373}"
10,1,50,0,0.0279368,"\int \left(a+b x^2\right)^2 \left(c+d x^2\right) \, dx","Int[(a + b*x^2)^2*(c + d*x^2),x]","a^2 c x+\frac{1}{5} b x^5 (2 a d+b c)+\frac{1}{3} a x^3 (a d+2 b c)+\frac{1}{7} b^2 d x^7","a^2 c x+\frac{1}{5} b x^5 (2 a d+b c)+\frac{1}{3} a x^3 (a d+2 b c)+\frac{1}{7} b^2 d x^7",1,"a^2*c*x + (a*(2*b*c + a*d)*x^3)/3 + (b*(b*c + 2*a*d)*x^5)/5 + (b^2*d*x^7)/7","A",2,1,17,0.05882,1,"{373}"
11,1,63,0,0.0428152,"\int \frac{\left(a+b x^2\right)^2}{c+d x^2} \, dx","Int[(a + b*x^2)^2/(c + d*x^2),x]","-\frac{b x (b c-2 a d)}{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^2 x^3}{3 d}","-\frac{b x (b c-2 a d)}{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^2 x^3}{3 d}",1,"-((b*(b*c - 2*a*d)*x)/d^2) + (b^2*x^3)/(3*d) + ((b*c - a*d)^2*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*d^(5/2))","A",3,2,19,0.1053,1,"{390, 205}"
12,1,82,0,0.0988627,"\int \frac{\left(a+b x^2\right)^2}{\left(c+d x^2\right)^2} \, dx","Int[(a + b*x^2)^2/(c + d*x^2)^2,x]","-\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}","-\frac{(b c-a d) (a d+3 b c) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{3/2} d^{5/2}}+\frac{x (b c-a d)^2}{2 c d^2 \left(c+d x^2\right)}+\frac{b^2 x}{d^2}",1,"(b^2*x)/d^2 + ((b*c - a*d)^2*x)/(2*c*d^2*(c + d*x^2)) - ((b*c - a*d)*(3*b*c + a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(3/2)*d^(5/2))","A",4,3,19,0.1579,1,"{390, 385, 205}"
13,1,116,0,0.0722065,"\int \frac{\left(a+b x^2\right)^2}{\left(c+d x^2\right)^3} \, dx","Int[(a + b*x^2)^2/(c + d*x^2)^3,x]","\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}","\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,"-((b*c - a*d)*x*(a + b*x^2))/(4*c*d*(c + d*x^2)^2) + (3*(a^2/c^2 - b^2/d^2)*x)/(8*(c + d*x^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",3,3,19,0.1579,1,"{413, 385, 205}"
14,1,154,0,0.1032678,"\int \left(a+b x^2\right)^3 \left(c+d x^2\right)^3 \, dx","Int[(a + b*x^2)^3*(c + d*x^2)^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)+a^3 c^3 x+\frac{3}{11} b^2 d^2 x^{11} (a d+b c)+\frac{1}{13} b^3 d^3 x^{13}","\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)+a^3 c^3 x+\frac{3}{11} b^2 d^2 x^{11} (a d+b c)+\frac{1}{13} b^3 d^3 x^{13}",1,"a^3*c^3*x + a^2*c^2*(b*c + a*d)*x^3 + (3*a*c*(b^2*c^2 + 3*a*b*c*d + a^2*d^2)*x^5)/5 + ((b*c + a*d)*(b^2*c^2 + 8*a*b*c*d + a^2*d^2)*x^7)/7 + (b*d*(b^2*c^2 + 3*a*b*c*d + a^2*d^2)*x^9)/3 + (3*b^2*d^2*(b*c + a*d)*x^11)/11 + (b^3*d^3*x^13)/13","A",2,1,19,0.05263,1,"{373}"
15,1,122,0,0.0700041,"\int \left(a+b x^2\right)^3 \left(c+d x^2\right)^2 \, dx","Int[(a + b*x^2)^3*(c + d*x^2)^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)+a^3 c^2 x+\frac{1}{9} b^2 d x^9 (3 a d+2 b c)+\frac{1}{11} b^3 d^2 x^{11}","\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)+a^3 c^2 x+\frac{1}{9} b^2 d x^9 (3 a d+2 b c)+\frac{1}{11} b^3 d^2 x^{11}",1,"a^3*c^2*x + (a^2*c*(3*b*c + 2*a*d)*x^3)/3 + (a*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^5)/5 + (b*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^7)/7 + (b^2*d*(2*b*c + 3*a*d)*x^9)/9 + (b^3*d^2*x^11)/11","A",2,1,19,0.05263,1,"{373}"
16,1,70,0,0.0437831,"\int \left(a+b x^2\right)^3 \left(c+d x^2\right) \, dx","Int[(a + b*x^2)^3*(c + d*x^2),x]","\frac{1}{3} a^2 x^3 (a d+3 b c)+a^3 c x+\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","\frac{1}{3} a^2 x^3 (a d+3 b c)+a^3 c x+\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,"a^3*c*x + (a^2*(3*b*c + a*d)*x^3)/3 + (3*a*b*(b*c + a*d)*x^5)/5 + (b^2*(b*c + 3*a*d)*x^7)/7 + (b^3*d*x^9)/9","A",2,1,17,0.05882,1,"{373}"
17,1,98,0,0.0637283,"\int \frac{\left(a+b x^2\right)^3}{c+d x^2} \, dx","Int[(a + b*x^2)^3/(c + d*x^2),x]","\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}","\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*(b^2*c^2 - 3*a*b*c*d + 3*a^2*d^2)*x)/d^3 - (b^2*(b*c - 3*a*d)*x^3)/(3*d^2) + (b^3*x^5)/(5*d) - ((b*c - a*d)^3*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*d^(7/2))","A",3,2,19,0.1053,1,"{390, 205}"
18,1,107,0,0.0961711,"\int \frac{\left(a+b x^2\right)^3}{\left(c+d x^2\right)^2} \, dx","Int[(a + b*x^2)^3/(c + d*x^2)^2,x]","-\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}","-\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,"-((b^2*(2*b*c - 3*a*d)*x)/d^3) + (b^3*x^3)/(3*d^2) - ((b*c - a*d)^3*x)/(2*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",4,3,19,0.1579,1,"{390, 385, 205}"
19,1,130,0,0.1647264,"\int \frac{\left(a+b x^2\right)^3}{\left(c+d x^2\right)^3} \, dx","Int[(a + b*x^2)^3/(c + d*x^2)^3,x]","-\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}","-\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,"(b^3*x)/d^3 - ((b*c - a*d)^3*x)/(4*c*d^3*(c + d*x^2)^2) + (3*(b*c - a*d)^2*(3*b*c + a*d)*x)/(8*c^2*d^3*(c + d*x^2)) - (3*(b*c - a*d)*(4*b^2*c^2 + (b*c + a*d)^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(5/2)*d^(7/2))","A",5,4,19,0.2105,1,"{390, 1157, 385, 205}"
20,1,142,0,0.093036,"\int \frac{\left(c+d x^2\right)^4}{a+b x^2} \, dx","Int[(c + d*x^2)^4/(a + b*x^2),x]","\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{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^3 x^5 (4 b c-a d)}{5 b^2}+\frac{(b c-a d)^4 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} b^{9/2}}+\frac{d^4 x^7}{7 b}","\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{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^3 x^5 (4 b c-a d)}{5 b^2}+\frac{(b c-a d)^4 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} b^{9/2}}+\frac{d^4 x^7}{7 b}",1,"(d*(2*b*c - a*d)*(2*b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x)/b^4 + (d^2*(6*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*x^3)/(3*b^3) + (d^3*(4*b*c - a*d)*x^5)/(5*b^2) + (d^4*x^7)/(7*b) + ((b*c - a*d)^4*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*b^(9/2))","A",3,2,19,0.1053,1,"{390, 205}"
21,1,98,0,0.0594187,"\int \frac{\left(c+d x^2\right)^3}{a+b x^2} \, dx","Int[(c + d*x^2)^3/(a + b*x^2),x]","\frac{d x \left(a^2 d^2-3 a b c d+3 b^2 c^2\right)}{b^3}+\frac{d^2 x^3 (3 b c-a d)}{3 b^2}+\frac{(b c-a d)^3 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} b^{7/2}}+\frac{d^3 x^5}{5 b}","\frac{d x \left(a^2 d^2-3 a b c d+3 b^2 c^2\right)}{b^3}+\frac{d^2 x^3 (3 b c-a d)}{3 b^2}+\frac{(b c-a d)^3 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} b^{7/2}}+\frac{d^3 x^5}{5 b}",1,"(d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x)/b^3 + (d^2*(3*b*c - a*d)*x^3)/(3*b^2) + (d^3*x^5)/(5*b) + ((b*c - a*d)^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*b^(7/2))","A",3,2,19,0.1053,1,"{390, 205}"
22,1,63,0,0.0409947,"\int \frac{\left(c+d x^2\right)^2}{a+b x^2} \, dx","Int[(c + d*x^2)^2/(a + b*x^2),x]","\frac{d x (2 b c-a d)}{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^2 x^3}{3 b}","\frac{d x (2 b c-a d)}{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^2 x^3}{3 b}",1,"(d*(2*b*c - a*d)*x)/b^2 + (d^2*x^3)/(3*b) + ((b*c - a*d)^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*b^(5/2))","A",3,2,19,0.1053,1,"{390, 205}"
23,1,39,0,0.0153834,"\int \frac{c+d x^2}{a+b x^2} \, dx","Int[(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",2,2,17,0.1176,1,"{388, 205}"
24,1,70,0,0.0270253,"\int \frac{1}{\left(a+b x^2\right) \left(c+d x^2\right)} \, dx","Int[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} (b c-a d)}-\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",3,2,19,0.1053,1,"{391, 205}"
25,1,109,0,0.0837109,"\int \frac{1}{\left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","Int[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} (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)}","\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} (b c-a d)^2}-\frac{\sqrt{d} (3 b c-a d) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{3/2} (b c-a d)^2}-\frac{d x}{2 c \left(c+d x^2\right) (b c-a d)}",1,"-(d*x)/(2*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) - (Sqrt[d]*(3*b*c - a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(3/2)*(b*c - a*d)^2)","A",4,3,19,0.1579,1,"{414, 522, 205}"
26,1,160,0,0.1919667,"\int \frac{1}{\left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","Int[1/((a + b*x^2)*(c + d*x^2)^3),x]","-\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)}","-\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,"-(d*x)/(4*c*(b*c - a*d)*(c + d*x^2)^2) - (d*(7*b*c - 3*a*d)*x)/(8*c^2*(b*c - a*d)^2*(c + d*x^2)) + (b^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*(b*c - a*d)^3) - (Sqrt[d]*(15*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(5/2)*(b*c - a*d)^3)","A",5,4,19,0.2105,1,"{414, 527, 522, 205}"
27,1,192,0,0.1625388,"\int \frac{\left(c+d x^2\right)^5}{\left(a+b x^2\right)^2} \, dx","Int[(c + d*x^2)^5/(a + b*x^2)^2,x]","\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(15 a^2 b c d^2-4 a^3 d^3-20 a b^2 c^2 d+10 b^3 c^3\right)}{b^5}+\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^4 x^5 (5 b c-2 a d)}{5 b^3}+\frac{x (b c-a d)^5}{2 a b^5 \left(a+b x^2\right)}+\frac{d^5 x^7}{7 b^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(15 a^2 b c d^2-4 a^3 d^3-20 a b^2 c^2 d+10 b^3 c^3\right)}{b^5}+\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^4 x^5 (5 b c-2 a d)}{5 b^3}+\frac{x (b c-a d)^5}{2 a b^5 \left(a+b x^2\right)}+\frac{d^5 x^7}{7 b^2}",1,"(d^2*(10*b^3*c^3 - 20*a*b^2*c^2*d + 15*a^2*b*c*d^2 - 4*a^3*d^3)*x)/b^5 + (d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*x^3)/(3*b^4) + (d^4*(5*b*c - 2*a*d)*x^5)/(5*b^3) + (d^5*x^7)/(7*b^2) + ((b*c - a*d)^5*x)/(2*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",4,3,19,0.1579,1,"{390, 385, 205}"
28,1,142,0,0.120468,"\int \frac{\left(c+d x^2\right)^4}{\left(a+b x^2\right)^2} \, dx","Int[(c + d*x^2)^4/(a + b*x^2)^2,x]","\frac{d^2 x \left(3 a^2 d^2-8 a b c d+6 b^2 c^2\right)}{b^4}+\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{2 d^3 x^3 (2 b c-a d)}{3 b^3}+\frac{x (b c-a d)^4}{2 a b^4 \left(a+b x^2\right)}+\frac{d^4 x^5}{5 b^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{(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{2 d^3 x^3 (2 b c-a d)}{3 b^3}+\frac{x (b c-a d)^4}{2 a b^4 \left(a+b x^2\right)}+\frac{d^4 x^5}{5 b^2}",1,"(d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*x)/b^4 + (2*d^3*(2*b*c - a*d)*x^3)/(3*b^3) + (d^4*x^5)/(5*b^2) + ((b*c - a*d)^4*x)/(2*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",4,3,19,0.1579,1,"{390, 385, 205}"
29,1,106,0,0.0932229,"\int \frac{\left(c+d x^2\right)^3}{\left(a+b x^2\right)^2} \, dx","Int[(c + d*x^2)^3/(a + b*x^2)^2,x]","\frac{(5 a d+b c) (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} b^{7/2}}+\frac{d^2 x (3 b c-2 a d)}{b^3}+\frac{x (b c-a d)^3}{2 a b^3 \left(a+b x^2\right)}+\frac{d^3 x^3}{3 b^2}","\frac{(5 a d+b c) (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} b^{7/2}}+\frac{d^2 x (3 b c-2 a d)}{b^3}+\frac{x (b c-a d)^3}{2 a b^3 \left(a+b x^2\right)}+\frac{d^3 x^3}{3 b^2}",1,"(d^2*(3*b*c - 2*a*d)*x)/b^3 + (d^3*x^3)/(3*b^2) + ((b*c - a*d)^3*x)/(2*a*b^3*(a + b*x^2)) + ((b*c - a*d)^2*(b*c + 5*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*b^(7/2))","A",4,3,19,0.1579,1,"{390, 385, 205}"
30,1,82,0,0.1036505,"\int \frac{\left(c+d x^2\right)^2}{\left(a+b x^2\right)^2} \, dx","Int[(c + d*x^2)^2/(a + b*x^2)^2,x]","\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}","\frac{(b c-a d) (3 a d+b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} b^{5/2}}+\frac{x (b c-a d)^2}{2 a b^2 \left(a+b x^2\right)}+\frac{d^2 x}{b^2}",1,"(d^2*x)/b^2 + ((b*c - a*d)^2*x)/(2*a*b^2*(a + b*x^2)) + ((b*c - a*d)*(b*c + 3*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*b^(5/2))","A",4,3,19,0.1579,1,"{390, 385, 205}"
31,1,63,0,0.0214587,"\int \frac{c+d x^2}{\left(a+b x^2\right)^2} \, dx","Int[(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 (b c-a d)}{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,"((b*c - a*d)*x)/(2*a*b*(a + b*x^2)) + ((b*c + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*b^(3/2))","A",2,2,17,0.1176,1,"{385, 205}"
32,1,108,0,0.081232,"\int \frac{1}{\left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","Int[1/((a + b*x^2)^2*(c + d*x^2)),x]","\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)}","\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,"(b*x)/(2*a*(b*c - a*d)*(a + b*x^2)) + (Sqrt[b]*(b*c - 3*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*(b*c - a*d)^2) + (d^(3/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*(b*c - a*d)^2)","A",4,3,19,0.1579,1,"{414, 522, 205}"
33,1,167,0,0.2011346,"\int \frac{1}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","Int[1/((a + b*x^2)^2*(c + d*x^2)^2),x]","\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}","\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,"(d*(b*c + a*d)*x)/(2*a*c*(b*c - a*d)^2*(c + d*x^2)) + (b*x)/(2*a*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)) + (b^(3/2)*(b*c - 5*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*(b*c - a*d)^3) + (d^(3/2)*(5*b*c - a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(3/2)*(b*c - a*d)^3)","A",5,4,19,0.2105,1,"{414, 527, 522, 205}"
34,1,230,0,0.3087602,"\int \frac{1}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","Int[1/((a + b*x^2)^2*(c + d*x^2)^3),x]","\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{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 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}","\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{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 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,"(d*(2*b*c + a*d)*x)/(4*a*c*(b*c - a*d)^2*(c + d*x^2)^2) + (b*x)/(2*a*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^2) + (d*(4*b*c - a*d)*(b*c + 3*a*d)*x)/(8*a*c^2*(b*c - a*d)^3*(c + d*x^2)) + (b^(5/2)*(b*c - 7*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*(b*c - a*d)^4) + (d^(3/2)*(35*b^2*c^2 - 14*a*b*c*d + 3*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(5/2)*(b*c - a*d)^4)","A",6,4,19,0.2105,1,"{414, 527, 522, 205}"
35,1,196,0,0.2269191,"\int \frac{\left(c+d x^2\right)^5}{\left(a+b x^2\right)^3} \, dx","Int[(c + d*x^2)^5/(a + b*x^2)^3,x]","\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 (17 a d+3 b c) (b c-a d)^4}{8 a^2 b^5 \left(a+b x^2\right)}+\frac{d^4 x^3 (5 b c-3 a d)}{3 b^4}+\frac{x (b c-a d)^5}{4 a b^5 \left(a+b x^2\right)^2}+\frac{d^5 x^5}{5 b^3}","\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 (17 a d+3 b c) (b c-a d)^4}{8 a^2 b^5 \left(a+b x^2\right)}+\frac{d^4 x^3 (5 b c-3 a d)}{3 b^4}+\frac{x (b c-a d)^5}{4 a b^5 \left(a+b x^2\right)^2}+\frac{d^5 x^5}{5 b^3}",1,"(d^3*(10*b^2*c^2 - 15*a*b*c*d + 6*a^2*d^2)*x)/b^5 + (d^4*(5*b*c - 3*a*d)*x^3)/(3*b^4) + (d^5*x^5)/(5*b^3) + ((b*c - a*d)^5*x)/(4*a*b^5*(a + b*x^2)^2) + ((b*c - a*d)^4*(3*b*c + 17*a*d)*x)/(8*a^2*b^5*(a + b*x^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",5,4,19,0.2105,1,"{390, 1157, 385, 205}"
36,1,160,0,0.1970188,"\int \frac{\left(c+d x^2\right)^4}{\left(a+b x^2\right)^3} \, dx","Int[(c + d*x^2)^4/(a + b*x^2)^3,x]","\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{x (b c-a d)^3 (13 a d+3 b c)}{8 a^2 b^4 \left(a+b x^2\right)}+\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}","\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{x (b c-a d)^3 (13 a d+3 b c)}{8 a^2 b^4 \left(a+b x^2\right)}+\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,"(d^3*(4*b*c - 3*a*d)*x)/b^4 + (d^4*x^3)/(3*b^3) + ((b*c - a*d)^4*x)/(4*a*b^4*(a + b*x^2)^2) + ((b*c - a*d)^3*(3*b*c + 13*a*d)*x)/(8*a^2*b^4*(a + b*x^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",5,4,19,0.2105,1,"{390, 1157, 385, 205}"
37,1,130,0,0.1657626,"\int \frac{\left(c+d x^2\right)^3}{\left(a+b x^2\right)^3} \, dx","Int[(c + d*x^2)^3/(a + b*x^2)^3,x]","\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{3 x (b c-a d)^2 (3 a d+b c)}{8 a^2 b^3 \left(a+b x^2\right)}+\frac{x (b c-a d)^3}{4 a b^3 \left(a+b x^2\right)^2}+\frac{d^3 x}{b^3}","\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{3 x (b c-a d)^2 (3 a d+b c)}{8 a^2 b^3 \left(a+b x^2\right)}+\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,"(d^3*x)/b^3 + ((b*c - a*d)^3*x)/(4*a*b^3*(a + b*x^2)^2) + (3*(b*c - a*d)^2*(b*c + 3*a*d)*x)/(8*a^2*b^3*(a + b*x^2)) + (3*(b*c - a*d)*(4*a^2*d^2 + (b*c + a*d)^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*b^(7/2))","A",5,4,19,0.2105,1,"{390, 1157, 385, 205}"
38,1,116,0,0.0769482,"\int \frac{\left(c+d x^2\right)^2}{\left(a+b x^2\right)^3} \, dx","Int[(c + d*x^2)^2/(a + b*x^2)^3,x]","\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}","\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,"(3*(c^2/a^2 - d^2/b^2)*x)/(8*(a + b*x^2)) + ((b*c - a*d)*x*(c + d*x^2))/(4*a*b*(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",3,3,19,0.1579,1,"{413, 385, 205}"
39,1,92,0,0.0328971,"\int \frac{c+d x^2}{\left(a+b x^2\right)^3} \, dx","Int[(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 (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}","\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,"((b*c - a*d)*x)/(4*a*b*(a + b*x^2)^2) + ((3*b*c + a*d)*x)/(8*a^2*b*(a + b*x^2)) + ((3*b*c + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*b^(3/2))","A",3,3,17,0.1765,1,"{385, 199, 205}"
40,1,161,0,0.1973466,"\int \frac{1}{\left(a+b x^2\right)^3 \left(c+d x^2\right)} \, dx","Int[1/((a + b*x^2)^3*(c + d*x^2)),x]","\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{b x (3 b c-7 a d)}{8 a^2 \left(a+b x^2\right) (b c-a d)^2}-\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)}","\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{b x (3 b c-7 a d)}{8 a^2 \left(a+b x^2\right) (b c-a d)^2}-\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,"(b*x)/(4*a*(b*c - a*d)*(a + b*x^2)^2) + (b*(3*b*c - 7*a*d)*x)/(8*a^2*(b*c - a*d)^2*(a + b*x^2)) + (Sqrt[b]*(3*b^2*c^2 - 10*a*b*c*d + 15*a^2*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",5,4,19,0.2105,1,"{414, 527, 522, 205}"
41,1,236,0,0.3105133,"\int \frac{1}{\left(a+b x^2\right)^3 \left(c+d x^2\right)^2} \, dx","Int[1/((a + b*x^2)^3*(c + d*x^2)^2),x]","\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 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{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)}","\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 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{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,"(d*(b*c - 4*a*d)*(3*b*c + a*d)*x)/(8*a^2*c*(b*c - a*d)^3*(c + d*x^2)) + (b*x)/(4*a*(b*c - a*d)*(a + b*x^2)^2*(c + d*x^2)) + (3*b*(b*c - 3*a*d)*x)/(8*a^2*(b*c - a*d)^2*(a + b*x^2)*(c + d*x^2)) + (b^(3/2)*(3*b^2*c^2 - 14*a*b*c*d + 35*a^2*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*(b*c - a*d)^4) - (d^(5/2)*(7*b*c - a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(3/2)*(b*c - a*d)^4)","A",6,4,19,0.2105,1,"{414, 527, 522, 205}"
42,1,315,0,0.4508652,"\int \frac{1}{\left(a+b x^2\right)^3 \left(c+d x^2\right)^3} \, dx","Int[1/((a + b*x^2)^3*(c + d*x^2)^3),x]","\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 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{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{b x}{4 a \left(a+b x^2\right)^2 \left(c+d x^2\right)^2 (b c-a d)}","\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 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{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{b x}{4 a \left(a+b x^2\right)^2 \left(c+d x^2\right)^2 (b c-a d)}",1,"(d*(3*b^2*c^2 - 13*a*b*c*d - 2*a^2*d^2)*x)/(8*a^2*c*(b*c - a*d)^3*(c + d*x^2)^2) + (b*x)/(4*a*(b*c - a*d)*(a + b*x^2)^2*(c + d*x^2)^2) + (b*(3*b*c - 11*a*d)*x)/(8*a^2*(b*c - a*d)^2*(a + b*x^2)*(c + d*x^2)^2) + (3*d*(b*c + a*d)*(b^2*c^2 - 6*a*b*c*d + a^2*d^2)*x)/(8*a^2*c^2*(b*c - a*d)^4*(c + d*x^2)) + (3*b^(5/2)*(b^2*c^2 - 6*a*b*c*d + 21*a^2*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*(b*c - a*d)^5) - (3*d^(5/2)*(21*b^2*c^2 - 6*a*b*c*d + a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(5/2)*(b*c - a*d)^5)","A",7,4,19,0.2105,1,"{414, 527, 522, 205}"
43,1,34,0,0.0095149,"\int \frac{\left(-1+x^2\right)^3}{\left(1+x^2\right)^4} \, dx","Int[(-1 + x^2)^3/(1 + x^2)^4,x]","-\frac{x \left(1-x^2\right)^2}{3 \left(x^2+1\right)^3}-\frac{2 x}{3 \left(x^2+1\right)}","-\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,"-(x*(1 - x^2)^2)/(3*(1 + x^2)^3) - (2*x)/(3*(1 + x^2))","A",3,3,15,0.2000,1,"{413, 21, 383}"
44,1,47,0,0.0163951,"\int \frac{\left(-1+x^2\right)^4}{\left(1+x^2\right)^5} \, dx","Int[(-1 + x^2)^4/(1 + x^2)^5,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)","\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,"(x*(1 - x^2)^3)/(4*(1 + x^2)^4) + (3*x*(1 - x^2))/(8*(1 + x^2)^2) + (3*ArcTan[x])/8","A",5,3,15,0.2000,1,"{413, 21, 203}"
45,1,231,0,0.178677,"\int \sqrt{a+b x^2} \left(c+d x^2\right)^3 \, dx","Int[Sqrt[a + b*x^2]*(c + d*x^2)^3,x]","\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(24 a^2 b c d^2-5 a^3 d^3-48 a b^2 c^2 d+64 b^3 c^3\right)}{128 b^3}+\frac{a \left(24 a^2 b c d^2-5 a^3 d^3-48 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}","\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(24 a^2 b c d^2-5 a^3 d^3-48 a b^2 c^2 d+64 b^3 c^3\right)}{128 b^3}+\frac{a \left(24 a^2 b c d^2-5 a^3 d^3-48 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,"((64*b^3*c^3 - 48*a*b^2*c^2*d + 24*a^2*b*c*d^2 - 5*a^3*d^3)*x*Sqrt[a + b*x^2])/(128*b^3) + (d*(72*b^2*c^2 - 52*a*b*c*d + 15*a^2*d^2)*x*(a + b*x^2)^(3/2))/(192*b^3) + (d*(12*b*c - 5*a*d)*x*(a + b*x^2)^(3/2)*(c + d*x^2))/(48*b^2) + (d*x*(a + b*x^2)^(3/2)*(c + d*x^2)^2)/(8*b) + (a*(64*b^3*c^3 - 48*a*b^2*c^2*d + 24*a^2*b*c*d^2 - 5*a^3*d^3)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(128*b^(7/2))","A",6,6,21,0.2857,1,"{416, 528, 388, 195, 217, 206}"
46,1,149,0,0.0879085,"\int \sqrt{a+b x^2} \left(c+d x^2\right)^2 \, dx","Int[Sqrt[a + b*x^2]*(c + d*x^2)^2,x]","\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}","\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,"((8*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*x*Sqrt[a + b*x^2])/(16*b^2) + (d*(8*b*c - 3*a*d)*x*(a + b*x^2)^(3/2))/(24*b^2) + (d*x*(a + b*x^2)^(3/2)*(c + d*x^2))/(6*b) + (a*(8*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*b^(5/2))","A",5,5,21,0.2381,1,"{416, 388, 195, 217, 206}"
47,1,87,0,0.0276762,"\int \sqrt{a+b x^2} \left(c+d x^2\right) \, dx","Int[Sqrt[a + b*x^2]*(c + d*x^2),x]","\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}","\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,"((4*b*c - a*d)*x*Sqrt[a + b*x^2])/(8*b) + (d*x*(a + b*x^2)^(3/2))/(4*b) + (a*(4*b*c - a*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*b^(3/2))","A",4,4,19,0.2105,1,"{388, 195, 217, 206}"
48,1,46,0,0.0103711,"\int \sqrt{a+b x^2} \, dx","Int[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",3,3,11,0.2727,1,"{195, 217, 206}"
49,1,82,0,0.0542935,"\int \frac{\sqrt{a+b x^2}}{c+d x^2} \, dx","Int[Sqrt[a + b*x^2]/(c + d*x^2),x]","\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}","\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]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/d - (Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(Sqrt[c]*d)","A",5,5,21,0.2381,1,"{402, 217, 206, 377, 208}"
50,1,82,0,0.0344839,"\int \frac{\sqrt{a+b x^2}}{\left(c+d x^2\right)^2} \, dx","Int[Sqrt[a + b*x^2]/(c + d*x^2)^2,x]","\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)}","\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*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(2*c^(3/2)*Sqrt[b*c - a*d])","A",3,3,21,0.1429,1,"{378, 377, 208}"
51,1,149,0,0.0940901,"\int \frac{\sqrt{a+b x^2}}{\left(c+d x^2\right)^3} \, dx","Int[Sqrt[a + b*x^2]/(c + d*x^2)^3,x]","\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{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{d x \left(a+b x^2\right)^{3/2}}{4 c \left(c+d x^2\right)^2 (b c-a d)}","\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{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{d x \left(a+b x^2\right)^{3/2}}{4 c \left(c+d x^2\right)^2 (b c-a d)}",1,"-(d*x*(a + b*x^2)^(3/2))/(4*c*(b*c - a*d)*(c + d*x^2)^2) + ((4*b*c - 3*a*d)*x*Sqrt[a + b*x^2])/(8*c^2*(b*c - a*d)*(c + d*x^2)) + (a*(4*b*c - 3*a*d)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(8*c^(5/2)*(b*c - a*d)^(3/2))","A",4,4,21,0.1905,1,"{382, 378, 377, 208}"
52,1,208,0,0.2132173,"\int \frac{\sqrt{a+b x^2}}{\left(c+d x^2\right)^4} \, dx","Int[Sqrt[a + b*x^2]/(c + d*x^2)^4,x]","\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}","\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,"(x*Sqrt[a + b*x^2])/(6*c*(c + d*x^2)^3) + ((4*b*c - 5*a*d)*x*Sqrt[a + b*x^2])/(24*c^2*(b*c - a*d)*(c + d*x^2)^2) + ((2*b*c - 5*a*d)*(4*b*c - 3*a*d)*x*Sqrt[a + b*x^2])/(48*c^3*(b*c - a*d)^2*(c + d*x^2)) + (a*(8*b^2*c^2 - 12*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(16*c^(7/2)*(b*c - a*d)^(5/2))","A",6,5,21,0.2381,1,"{412, 527, 12, 377, 208}"
53,1,272,0,0.2177732,"\int \left(a+b x^2\right)^{3/2} \left(c+d x^2\right)^3 \, dx","Int[(a + b*x^2)^(3/2)*(c + d*x^2)^3,x]","\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{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(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}","\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{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(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,"(3*a*(4*b*c - a*d)*(8*b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x*Sqrt[a + b*x^2])/(256*b^3) + ((4*b*c - a*d)*(8*b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x*(a + b*x^2)^(3/2))/(128*b^3) + (d*(36*b^2*c^2 - 20*a*b*c*d + 5*a^2*d^2)*x*(a + b*x^2)^(5/2))/(160*b^3) + (d*(14*b*c - 5*a*d)*x*(a + b*x^2)^(5/2)*(c + d*x^2))/(80*b^2) + (d*x*(a + b*x^2)^(5/2)*(c + d*x^2)^2)/(10*b) + (3*a^2*(4*b*c - a*d)*(8*b^2*c^2 - 2*a*b*c*d + a^2*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(256*b^(7/2))","A",7,6,21,0.2857,1,"{416, 528, 388, 195, 217, 206}"
54,1,196,0,0.1159345,"\int \left(a+b x^2\right)^{3/2} \left(c+d x^2\right)^2 \, dx","Int[(a + b*x^2)^(3/2)*(c + d*x^2)^2,x]","\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}","\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,"(a*(48*b^2*c^2 - 16*a*b*c*d + 3*a^2*d^2)*x*Sqrt[a + b*x^2])/(128*b^2) + ((48*b^2*c^2 - 16*a*b*c*d + 3*a^2*d^2)*x*(a + b*x^2)^(3/2))/(192*b^2) + (d*(10*b*c - 3*a*d)*x*(a + b*x^2)^(5/2))/(48*b^2) + (d*x*(a + b*x^2)^(5/2)*(c + d*x^2))/(8*b) + (a^2*(48*b^2*c^2 - 16*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(128*b^(5/2))","A",6,5,21,0.2381,1,"{416, 388, 195, 217, 206}"
55,1,118,0,0.0400311,"\int \left(a+b x^2\right)^{3/2} \left(c+d x^2\right) \, dx","Int[(a + b*x^2)^(3/2)*(c + d*x^2),x]","\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}","\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,"(a*(6*b*c - a*d)*x*Sqrt[a + b*x^2])/(16*b) + ((6*b*c - a*d)*x*(a + b*x^2)^(3/2))/(24*b) + (d*x*(a + b*x^2)^(5/2))/(6*b) + (a^2*(6*b*c - a*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*b^(3/2))","A",5,4,19,0.2105,1,"{388, 195, 217, 206}"
56,1,65,0,0.0158949,"\int \left(a+b x^2\right)^{3/2} \, dx","Int[(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{3}{8} a x \sqrt{a+b x^2}+\frac{1}{4} x \left(a+b x^2\right)^{3/2}","\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,"(3*a*x*Sqrt[a + b*x^2])/8 + (x*(a + b*x^2)^(3/2))/4 + (3*a^2*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*Sqrt[b])","A",4,3,11,0.2727,1,"{195, 217, 206}"
57,1,113,0,0.1083912,"\int \frac{\left(a+b x^2\right)^{3/2}}{c+d x^2} \, dx","Int[(a + b*x^2)^(3/2)/(c + d*x^2),x]","\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}","\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) - (Sqrt[b]*(2*b*c - 3*a*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*d^2) + ((b*c - a*d)^(3/2)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(Sqrt[c]*d^2)","A",6,6,21,0.2857,1,"{416, 523, 217, 206, 377, 208}"
58,1,131,0,0.0903201,"\int \frac{\left(a+b x^2\right)^{3/2}}{\left(c+d x^2\right)^2} \, dx","Int[(a + b*x^2)^(3/2)/(c + d*x^2)^2,x]","\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)}","\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)) + (b^(3/2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/d^2 - (Sqrt[b*c - a*d]*(2*b*c + a*d)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(2*c^(3/2)*d^2)","A",6,6,21,0.2857,1,"{413, 523, 217, 206, 377, 208}"
59,1,113,0,0.0574365,"\int \frac{\left(a+b x^2\right)^{3/2}}{\left(c+d x^2\right)^3} \, dx","Int[(a + b*x^2)^(3/2)/(c + d*x^2)^3,x]","\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}","\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,"(x*(a + b*x^2)^(3/2))/(4*c*(c + d*x^2)^2) + (3*a*x*Sqrt[a + b*x^2])/(8*c^2*(c + d*x^2)) + (3*a^2*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(8*c^(5/2)*Sqrt[b*c - a*d])","A",4,3,21,0.1429,1,"{378, 377, 208}"
60,1,199,0,0.1154304,"\int \frac{\left(a+b x^2\right)^{3/2}}{\left(c+d x^2\right)^4} \, dx","Int[(a + b*x^2)^(3/2)/(c + d*x^2)^4,x]","\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{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{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{d x \left(a+b x^2\right)^{5/2}}{6 c \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{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{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{d x \left(a+b x^2\right)^{5/2}}{6 c \left(c+d x^2\right)^3 (b c-a d)}",1,"-(d*x*(a + b*x^2)^(5/2))/(6*c*(b*c - a*d)*(c + d*x^2)^3) + ((6*b*c - 5*a*d)*x*(a + b*x^2)^(3/2))/(24*c^2*(b*c - a*d)*(c + d*x^2)^2) + (a*(6*b*c - 5*a*d)*x*Sqrt[a + b*x^2])/(16*c^3*(b*c - a*d)*(c + d*x^2)) + (a^2*(6*b*c - 5*a*d)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(16*c^(7/2)*(b*c - a*d)^(3/2))","A",5,4,21,0.1905,1,"{382, 378, 377, 208}"
61,1,300,0,0.3651532,"\int \frac{\left(a+b x^2\right)^{3/2}}{\left(c+d x^2\right)^5} \, dx","Int[(a + b*x^2)^(3/2)/(c + d*x^2)^5,x]","\frac{x \sqrt{a+b x^2} \left(-170 a^2 b c d^2+105 a^3 d^3+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} \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{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} (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}","\frac{x \sqrt{a+b x^2} \left(-170 a^2 b c d^2+105 a^3 d^3+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} \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{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} (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,"-((b*c - a*d)*x*Sqrt[a + b*x^2])/(8*c*d*(c + d*x^2)^4) + ((2*b*c + 7*a*d)*x*Sqrt[a + b*x^2])/(48*c^2*d*(c + d*x^2)^3) + ((8*b^2*c^2 + 24*a*b*c*d - 35*a^2*d^2)*x*Sqrt[a + b*x^2])/(192*c^3*d*(b*c - a*d)*(c + d*x^2)^2) + ((16*b^3*c^3 + 40*a*b^2*c^2*d - 170*a^2*b*c*d^2 + 105*a^3*d^3)*x*Sqrt[a + b*x^2])/(384*c^4*d*(b*c - a*d)^2*(c + d*x^2)) + (a^2*(48*b^2*c^2 - 80*a*b*c*d + 35*a^2*d^2)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(128*c^(9/2)*(b*c - a*d)^(5/2))","A",7,5,21,0.2381,1,"{413, 527, 12, 377, 208}"
62,1,349,0,0.2471414,"\int \left(a+b x^2\right)^{5/2} \left(c+d x^2\right)^3 \, dx","Int[(a + b*x^2)^(5/2)*(c + d*x^2)^3,x]","\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(36 a^2 b c d^2-5 a^3 d^3-120 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(36 a^2 b c d^2-5 a^3 d^3-120 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(36 a^2 b c d^2-5 a^3 d^3-120 a b^2 c^2 d+320 b^3 c^3\right)}{1024 b^3}+\frac{a^3 \left(36 a^2 b c d^2-5 a^3 d^3-120 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}","\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(36 a^2 b c d^2-5 a^3 d^3-120 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(36 a^2 b c d^2-5 a^3 d^3-120 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(36 a^2 b c d^2-5 a^3 d^3-120 a b^2 c^2 d+320 b^3 c^3\right)}{1024 b^3}+\frac{a^3 \left(36 a^2 b c d^2-5 a^3 d^3-120 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,"(a^2*(320*b^3*c^3 - 120*a*b^2*c^2*d + 36*a^2*b*c*d^2 - 5*a^3*d^3)*x*Sqrt[a + b*x^2])/(1024*b^3) + (a*(320*b^3*c^3 - 120*a*b^2*c^2*d + 36*a^2*b*c*d^2 - 5*a^3*d^3)*x*(a + b*x^2)^(3/2))/(1536*b^3) + ((320*b^3*c^3 - 120*a*b^2*c^2*d + 36*a^2*b*c*d^2 - 5*a^3*d^3)*x*(a + b*x^2)^(5/2))/(1920*b^3) + (d*(152*b^2*c^2 - 68*a*b*c*d + 15*a^2*d^2)*x*(a + b*x^2)^(7/2))/(960*b^3) + (d*(16*b*c - 5*a*d)*x*(a + b*x^2)^(7/2)*(c + d*x^2))/(120*b^2) + (d*x*(a + b*x^2)^(7/2)*(c + d*x^2)^2)/(12*b) + (a^3*(320*b^3*c^3 - 120*a*b^2*c^2*d + 36*a^2*b*c*d^2 - 5*a^3*d^3)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(1024*b^(7/2))","A",8,6,21,0.2857,1,"{416, 528, 388, 195, 217, 206}"
63,1,241,0,0.1483704,"\int \left(a+b x^2\right)^{5/2} \left(c+d x^2\right)^2 \, dx","Int[(a + b*x^2)^(5/2)*(c + d*x^2)^2,x]","\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}","\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,"(a^2*(80*b^2*c^2 - 20*a*b*c*d + 3*a^2*d^2)*x*Sqrt[a + b*x^2])/(256*b^2) + (a*(80*b^2*c^2 - 20*a*b*c*d + 3*a^2*d^2)*x*(a + b*x^2)^(3/2))/(384*b^2) + ((80*b^2*c^2 - 20*a*b*c*d + 3*a^2*d^2)*x*(a + b*x^2)^(5/2))/(480*b^2) + (3*d*(4*b*c - a*d)*x*(a + b*x^2)^(7/2))/(80*b^2) + (d*x*(a + b*x^2)^(7/2)*(c + d*x^2))/(10*b) + (a^3*(80*b^2*c^2 - 20*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(256*b^(5/2))","A",7,5,21,0.2381,1,"{416, 388, 195, 217, 206}"
64,1,149,0,0.0522127,"\int \left(a+b x^2\right)^{5/2} \left(c+d x^2\right) \, dx","Int[(a + b*x^2)^(5/2)*(c + d*x^2),x]","\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}","\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,"(5*a^2*(8*b*c - a*d)*x*Sqrt[a + b*x^2])/(128*b) + (5*a*(8*b*c - a*d)*x*(a + b*x^2)^(3/2))/(192*b) + ((8*b*c - a*d)*x*(a + b*x^2)^(5/2))/(48*b) + (d*x*(a + b*x^2)^(7/2))/(8*b) + (5*a^3*(8*b*c - a*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(128*b^(3/2))","A",6,4,19,0.2105,1,"{388, 195, 217, 206}"
65,1,84,0,0.023137,"\int \left(a+b x^2\right)^{5/2} \, dx","Int[(a + b*x^2)^(5/2),x]","\frac{5}{16} a^2 x \sqrt{a+b x^2}+\frac{5 a^3 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{16 \sqrt{b}}+\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}","\frac{5}{16} a^2 x \sqrt{a+b x^2}+\frac{5 a^3 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{16 \sqrt{b}}+\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,"(5*a^2*x*Sqrt[a + b*x^2])/16 + (5*a*x*(a + b*x^2)^(3/2))/24 + (x*(a + b*x^2)^(5/2))/6 + (5*a^3*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*Sqrt[b])","A",5,3,11,0.2727,1,"{195, 217, 206}"
66,1,157,0,0.1989565,"\int \frac{\left(a+b x^2\right)^{5/2}}{c+d x^2} \, dx","Int[(a + b*x^2)^(5/2)/(c + d*x^2),x]","\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 x \sqrt{a+b x^2} (4 b c-7 a d)}{8 d^2}-\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 \left(a+b x^2\right)^{3/2}}{4 d}","\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 x \sqrt{a+b x^2} (4 b c-7 a d)}{8 d^2}-\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 \left(a+b x^2\right)^{3/2}}{4 d}",1,"-(b*(4*b*c - 7*a*d)*x*Sqrt[a + b*x^2])/(8*d^2) + (b*x*(a + b*x^2)^(3/2))/(4*d) + (Sqrt[b]*(8*b^2*c^2 - 20*a*b*c*d + 15*a^2*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*d^3) - ((b*c - a*d)^(5/2)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(Sqrt[c]*d^3)","A",7,7,21,0.3333,1,"{416, 528, 523, 217, 206, 377, 208}"
67,1,175,0,0.225999,"\int \frac{\left(a+b x^2\right)^{5/2}}{\left(c+d x^2\right)^2} \, dx","Int[(a + b*x^2)^(5/2)/(c + d*x^2)^2,x]","-\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)}","-\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,"(b*(2*b*c - a*d)*x*Sqrt[a + b*x^2])/(2*c*d^2) - ((b*c - a*d)*x*(a + b*x^2)^(3/2))/(2*c*d*(c + d*x^2)) - (b^(3/2)*(4*b*c - 5*a*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*d^3) + ((b*c - a*d)^(3/2)*(4*b*c + a*d)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(2*c^(3/2)*d^3)","A",7,7,21,0.3333,1,"{413, 528, 523, 217, 206, 377, 208}"
68,1,194,0,0.1944193,"\int \frac{\left(a+b x^2\right)^{5/2}}{\left(c+d x^2\right)^3} \, dx","Int[(a + b*x^2)^(5/2)/(c + d*x^2)^3,x]","-\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}","-\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,"-((b*c - a*d)*x*(a + b*x^2)^(3/2))/(4*c*d*(c + d*x^2)^2) - ((b*c - a*d)*(4*b*c + 3*a*d)*x*Sqrt[a + b*x^2])/(8*c^2*d^2*(c + d*x^2)) + (b^(5/2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/d^3 - (Sqrt[b*c - a*d]*(8*b^2*c^2 + 4*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(8*c^(5/2)*d^3)","A",7,7,21,0.3333,1,"{413, 526, 523, 217, 206, 377, 208}"
69,1,144,0,0.0732243,"\int \frac{\left(a+b x^2\right)^{5/2}}{\left(c+d x^2\right)^4} \, dx","Int[(a + b*x^2)^(5/2)/(c + d*x^2)^4,x]","\frac{5 a^2 x \sqrt{a+b x^2}}{16 c^3 \left(c+d x^2\right)}+\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 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}","\frac{5 a^2 x \sqrt{a+b x^2}}{16 c^3 \left(c+d x^2\right)}+\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 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,"(x*(a + b*x^2)^(5/2))/(6*c*(c + d*x^2)^3) + (5*a*x*(a + b*x^2)^(3/2))/(24*c^2*(c + d*x^2)^2) + (5*a^2*x*Sqrt[a + b*x^2])/(16*c^3*(c + d*x^2)) + (5*a^3*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(16*c^(7/2)*Sqrt[b*c - a*d])","A",5,3,21,0.1429,1,"{378, 377, 208}"
70,1,249,0,0.1373478,"\int \frac{\left(a+b x^2\right)^{5/2}}{\left(c+d x^2\right)^5} \, dx","Int[(a + b*x^2)^(5/2)/(c + d*x^2)^5,x]","\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^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{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{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{d x \left(a+b x^2\right)^{7/2}}{8 c \left(c+d x^2\right)^4 (b c-a d)}","\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^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{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{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{d x \left(a+b x^2\right)^{7/2}}{8 c \left(c+d x^2\right)^4 (b c-a d)}",1,"-(d*x*(a + b*x^2)^(7/2))/(8*c*(b*c - a*d)*(c + d*x^2)^4) + ((8*b*c - 7*a*d)*x*(a + b*x^2)^(5/2))/(48*c^2*(b*c - a*d)*(c + d*x^2)^3) + (5*a*(8*b*c - 7*a*d)*x*(a + b*x^2)^(3/2))/(192*c^3*(b*c - a*d)*(c + d*x^2)^2) + (5*a^2*(8*b*c - 7*a*d)*x*Sqrt[a + b*x^2])/(128*c^4*(b*c - a*d)*(c + d*x^2)) + (5*a^3*(8*b*c - 7*a*d)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(128*c^(9/2)*(b*c - a*d)^(3/2))","A",6,4,21,0.1905,1,"{382, 378, 377, 208}"
71,1,30,0,0.0143253,"\int \frac{\sqrt{1-x^2}}{1+x^2} \, dx","Int[Sqrt[1 - x^2]/(1 + x^2),x]","\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1-x^2}}\right)-\sin ^{-1}(x)","\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1-x^2}}\right)-\sin ^{-1}(x)",1,"-ArcSin[x] + Sqrt[2]*ArcTan[(Sqrt[2]*x)/Sqrt[1 - x^2]]","A",4,4,19,0.2105,1,"{402, 216, 377, 203}"
72,1,27,0,0.0139642,"\int \frac{\sqrt{1+x^2}}{-1+x^2} \, dx","Int[Sqrt[1 + x^2]/(-1 + x^2),x]","\sinh ^{-1}(x)-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{x^2+1}}\right)","\sinh ^{-1}(x)-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{x^2+1}}\right)",1,"ArcSinh[x] - Sqrt[2]*ArcTanh[(Sqrt[2]*x)/Sqrt[1 + x^2]]","A",4,4,17,0.2353,1,"{402, 215, 377, 207}"
73,1,25,0,0.0128449,"\int \frac{\sqrt{1-x^2}}{-1+2 x^2} \, dx","Int[Sqrt[1 - x^2]/(-1 + 2*x^2),x]","-\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt{1-x^2}}\right)-\frac{1}{2} \sin ^{-1}(x)","-\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt{1-x^2}}\right)-\frac{1}{2} \sin ^{-1}(x)",1,"-ArcSin[x]/2 - ArcTanh[x/Sqrt[1 - x^2]]/2","A",4,4,21,0.1905,1,"{402, 216, 377, 207}"
74,1,169,0,0.144695,"\int \frac{\left(c+d x^2\right)^3}{\sqrt{a+b x^2}} \, dx","Int[(c + d*x^2)^3/Sqrt[a + b*x^2],x]","\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{(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{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}","\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{(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{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,"(d*(44*b^2*c^2 - 44*a*b*c*d + 15*a^2*d^2)*x*Sqrt[a + b*x^2])/(48*b^3) + (5*d*(2*b*c - a*d)*x*Sqrt[a + b*x^2]*(c + d*x^2))/(24*b^2) + (d*x*Sqrt[a + b*x^2]*(c + d*x^2)^2)/(6*b) + ((2*b*c - a*d)*(8*b^2*c^2 - 8*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*b^(7/2))","A",5,5,21,0.2381,1,"{416, 528, 388, 217, 206}"
75,1,108,0,0.0556046,"\int \frac{\left(c+d x^2\right)^2}{\sqrt{a+b x^2}} \, dx","Int[(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{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}","\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,"(3*d*(2*b*c - a*d)*x*Sqrt[a + b*x^2])/(8*b^2) + (d*x*Sqrt[a + b*x^2]*(c + d*x^2))/(4*b) + ((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",4,4,21,0.1905,1,"{416, 388, 217, 206}"
76,1,58,0,0.0171933,"\int \frac{c+d x^2}{\sqrt{a+b x^2}} \, dx","Int[(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",3,3,19,0.1579,1,"{388, 217, 206}"
77,1,25,0,0.0062912,"\int \frac{1}{\sqrt{a+b x^2}} \, dx","Int[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",2,2,11,0.1818,1,"{217, 206}"
78,1,49,0,0.0215087,"\int \frac{1}{\sqrt{a+b x^2} \left(c+d x^2\right)} \, dx","Int[1/(Sqrt[a + b*x^2]*(c + d*x^2)),x]","\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}}","\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,"ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])]/(Sqrt[c]*Sqrt[b*c - a*d])","A",2,2,21,0.09524,1,"{377, 208}"
79,1,101,0,0.0489142,"\int \frac{1}{\sqrt{a+b x^2} \left(c+d x^2\right)^2} \, dx","Int[1/(Sqrt[a + b*x^2]*(c + d*x^2)^2),x]","\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)}","\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,"-(d*x*Sqrt[a + b*x^2])/(2*c*(b*c - a*d)*(c + d*x^2)) + ((2*b*c - a*d)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(2*c^(3/2)*(b*c - a*d)^(3/2))","A",3,3,21,0.1429,1,"{382, 377, 208}"
80,1,163,0,0.1191505,"\int \frac{1}{\sqrt{a+b x^2} \left(c+d x^2\right)^3} \, dx","Int[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) \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)}","\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,"-(d*x*Sqrt[a + b*x^2])/(4*c*(b*c - a*d)*(c + d*x^2)^2) - (3*d*(2*b*c - a*d)*x*Sqrt[a + b*x^2])/(8*c^2*(b*c - a*d)^2*(c + d*x^2)) + ((8*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(8*c^(5/2)*(b*c - a*d)^(5/2))","A",5,5,21,0.2381,1,"{414, 527, 12, 377, 208}"
81,1,257,0,0.2592722,"\int \frac{\left(c+d x^2\right)^4}{\left(a+b x^2\right)^{3/2}} \, dx","Int[(c + d*x^2)^4/(a + b*x^2)^(3/2),x]","-\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 x \sqrt{a+b x^2} \left(290 a^2 b c d^2-105 a^3 d^3-248 a b^2 c^2 d+48 b^3 c^3\right)}{48 a b^4}+\frac{d \left(120 a^2 b c d^2-35 a^3 d^3-144 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(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}}","-\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 x \sqrt{a+b x^2} \left(290 a^2 b c d^2-105 a^3 d^3-248 a b^2 c^2 d+48 b^3 c^3\right)}{48 a b^4}+\frac{d \left(120 a^2 b c d^2-35 a^3 d^3-144 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(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,"-(d*(48*b^3*c^3 - 248*a*b^2*c^2*d + 290*a^2*b*c*d^2 - 105*a^3*d^3)*x*Sqrt[a + b*x^2])/(48*a*b^4) - (d*(24*b^2*c^2 - 64*a*b*c*d + 35*a^2*d^2)*x*Sqrt[a + b*x^2]*(c + d*x^2))/(24*a*b^3) - (d*(6*b*c - 7*a*d)*x*Sqrt[a + b*x^2]*(c + d*x^2)^2)/(6*a*b^2) + ((b*c - a*d)*x*(c + d*x^2)^3)/(a*b*Sqrt[a + b*x^2]) + (d*(64*b^3*c^3 - 144*a*b^2*c^2*d + 120*a^2*b*c*d^2 - 35*a^3*d^3)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*b^(9/2))","A",6,5,21,0.2381,1,"{413, 528, 388, 217, 206}"
82,1,169,0,0.1972004,"\int \frac{\left(c+d x^2\right)^3}{\left(a+b x^2\right)^{3/2}} \, dx","Int[(c + d*x^2)^3/(a + b*x^2)^(3/2),x]","\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} \left(c+d x^2\right) (4 b c-5 a d)}{4 a b^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{x \left(c+d x^2\right)^2 (b c-a d)}{a b \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} \left(c+d x^2\right) (4 b c-5 a d)}{4 a b^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{x \left(c+d x^2\right)^2 (b c-a d)}{a b \sqrt{a+b x^2}}",1,"-(d*(2*b*c - 5*a*d)*(4*b*c - 3*a*d)*x*Sqrt[a + b*x^2])/(8*a*b^3) - (d*(4*b*c - 5*a*d)*x*Sqrt[a + b*x^2]*(c + d*x^2))/(4*a*b^2) + ((b*c - a*d)*x*(c + d*x^2)^2)/(a*b*Sqrt[a + b*x^2]) + (3*d*(8*b^2*c^2 - 12*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*b^(7/2))","A",5,5,21,0.2381,1,"{413, 528, 388, 217, 206}"
83,1,105,0,0.0620589,"\int \frac{\left(c+d x^2\right)^2}{\left(a+b x^2\right)^{3/2}} \, dx","Int[(c + d*x^2)^2/(a + b*x^2)^(3/2),x]","-\frac{d x \sqrt{a+b x^2} (2 b c-3 a d)}{2 a b^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 \left(c+d x^2\right) (b c-a d)}{a b \sqrt{a+b x^2}}","\frac{x (b c-a d)^2}{a b^2 \sqrt{a+b x^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{d^2 x \sqrt{a+b x^2}}{2 b^2}",1,"-(d*(2*b*c - 3*a*d)*x*Sqrt[a + b*x^2])/(2*a*b^2) + ((b*c - a*d)*x*(c + d*x^2))/(a*b*Sqrt[a + b*x^2]) + (d*(4*b*c - 3*a*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(5/2))","A",4,4,21,0.1905,1,"{413, 388, 217, 206}"
84,1,54,0,0.0172689,"\int \frac{c+d x^2}{\left(a+b x^2\right)^{3/2}} \, dx","Int[(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 (b c-a d)}{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",3,3,19,0.1579,1,"{385, 217, 206}"
85,1,16,0,0.0020305,"\int \frac{1}{\left(a+b x^2\right)^{3/2}} \, dx","Int[(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",1,1,11,0.09091,1,"{191}"
86,1,79,0,0.0386197,"\int \frac{1}{\left(a+b x^2\right)^{3/2} \left(c+d x^2\right)} \, dx","Int[1/((a + b*x^2)^(3/2)*(c + d*x^2)),x]","\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}}","\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*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(Sqrt[c]*(b*c - a*d)^(3/2))","A",3,3,21,0.1429,1,"{382, 377, 208}"
87,1,143,0,0.1090183,"\int \frac{1}{\left(a+b x^2\right)^{3/2} \left(c+d x^2\right)^2} \, dx","Int[1/((a + b*x^2)^(3/2)*(c + d*x^2)^2),x]","-\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)}","-\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,"(b*(2*b*c + a*d)*x)/(2*a*c*(b*c - a*d)^2*Sqrt[a + b*x^2]) - (d*x)/(2*c*(b*c - a*d)*Sqrt[a + b*x^2]*(c + d*x^2)) - (d*(4*b*c - a*d)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(2*c^(3/2)*(b*c - a*d)^(5/2))","A",5,5,21,0.2381,1,"{414, 527, 12, 377, 208}"
88,1,225,0,0.2433976,"\int \frac{1}{\left(a+b x^2\right)^{3/2} \left(c+d x^2\right)^3} \, dx","Int[1/((a + b*x^2)^(3/2)*(c + d*x^2)^3),x]","-\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)}","-\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,"-(d*x)/(4*c*(b*c - a*d)*Sqrt[a + b*x^2]*(c + d*x^2)^2) + (b*(4*b*c + a*d)*x)/(4*a*c*(b*c - a*d)^2*Sqrt[a + b*x^2]*(c + d*x^2)) + (d*(4*b*c - a*d)*(2*b*c + 3*a*d)*x*Sqrt[a + b*x^2])/(8*a*c^2*(b*c - a*d)^3*(c + d*x^2)) - (3*d*(8*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(8*c^(5/2)*(b*c - a*d)^(7/2))","A",6,5,21,0.2381,1,"{414, 527, 12, 377, 208}"
89,1,255,0,0.2448888,"\int \frac{\left(c+d x^2\right)^4}{\left(a+b x^2\right)^{5/2}} \, dx","Int[(c + d*x^2)^4/(a + b*x^2)^(5/2),x]","-\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(-170 a^2 b c d^2+105 a^3 d^3+40 a b^2 c^2 d+16 b^3 c^3\right)}{24 a^2 b^4}+\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{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{x \left(c+d x^2\right)^3 (b c-a d)}{3 a b \left(a+b x^2\right)^{3/2}}","-\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(-170 a^2 b c d^2+105 a^3 d^3+40 a b^2 c^2 d+16 b^3 c^3\right)}{24 a^2 b^4}+\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{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{x \left(c+d x^2\right)^3 (b c-a d)}{3 a b \left(a+b x^2\right)^{3/2}}",1,"-(d*(16*b^3*c^3 + 40*a*b^2*c^2*d - 170*a^2*b*c*d^2 + 105*a^3*d^3)*x*Sqrt[a + b*x^2])/(24*a^2*b^4) - (d*(8*b^2*c^2 + 24*a*b*c*d - 35*a^2*d^2)*x*Sqrt[a + b*x^2]*(c + d*x^2))/(12*a^2*b^3) + ((b*c - a*d)*(2*b*c + 7*a*d)*x*(c + d*x^2)^2)/(3*a^2*b^2*Sqrt[a + b*x^2]) + ((b*c - a*d)*x*(c + d*x^2)^3)/(3*a*b*(a + b*x^2)^(3/2)) + (d^2*(48*b^2*c^2 - 80*a*b*c*d + 35*a^2*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*b^(9/2))","A",6,6,21,0.2857,1,"{413, 526, 528, 388, 217, 206}"
90,1,172,0,0.1565216,"\int \frac{\left(c+d x^2\right)^3}{\left(a+b x^2\right)^{5/2}} \, dx","Int[(c + d*x^2)^3/(a + b*x^2)^(5/2),x]","-\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{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^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}}","-\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{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^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,"-(d*(4*b^2*c^2 + 8*a*b*c*d - 15*a^2*d^2)*x*Sqrt[a + b*x^2])/(6*a^2*b^3) + ((b*c - a*d)*(2*b*c + 5*a*d)*x*(c + d*x^2))/(3*a^2*b^2*Sqrt[a + b*x^2]) + ((b*c - a*d)*x*(c + d*x^2)^2)/(3*a*b*(a + b*x^2)^(3/2)) + (d^2*(6*b*c - 5*a*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(7/2))","A",5,5,21,0.2381,1,"{413, 526, 388, 217, 206}"
91,1,105,0,0.0503894,"\int \frac{\left(c+d x^2\right)^2}{\left(a+b x^2\right)^{5/2}} \, dx","Int[(c + d*x^2)^2/(a + b*x^2)^(5/2),x]","\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}}","\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,"((b*c - a*d)*(2*b*c + 3*a*d)*x)/(3*a^2*b^2*Sqrt[a + b*x^2]) + ((b*c - a*d)*x*(c + d*x^2))/(3*a*b*(a + b*x^2)^(3/2)) + (d^2*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/b^(5/2)","A",4,4,21,0.1905,1,"{413, 385, 217, 206}"
92,1,47,0,0.0096644,"\int \frac{c+d x^2}{\left(a+b x^2\right)^{5/2}} \, dx","Int[(c + d*x^2)/(a + b*x^2)^(5/2),x]","\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}}","\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,"(2*c*x)/(3*a^2*Sqrt[a + b*x^2]) + (x*(c + d*x^2))/(3*a*(a + b*x^2)^(3/2))","A",2,2,19,0.1053,1,"{378, 191}"
93,1,39,0,0.0057249,"\int \frac{1}{\left(a+b x^2\right)^{5/2}} \, dx","Int[(a + b*x^2)^(-5/2),x]","\frac{2 x}{3 a^2 \sqrt{a+b x^2}}+\frac{x}{3 a \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*(a + b*x^2)^(3/2)) + (2*x)/(3*a^2*Sqrt[a + b*x^2])","A",2,2,11,0.1818,1,"{192, 191}"
94,1,122,0,0.102706,"\int \frac{1}{\left(a+b x^2\right)^{5/2} \left(c+d x^2\right)} \, dx","Int[1/((a + b*x^2)^(5/2)*(c + d*x^2)),x]","\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)}","\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,"(b*x)/(3*a*(b*c - a*d)*(a + b*x^2)^(3/2)) + (b*(2*b*c - 5*a*d)*x)/(3*a^2*(b*c - a*d)^2*Sqrt[a + b*x^2]) + (d^2*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(Sqrt[c]*(b*c - a*d)^(5/2))","A",5,5,21,0.2381,1,"{414, 527, 12, 377, 208}"
95,1,202,0,0.2283923,"\int \frac{1}{\left(a+b x^2\right)^{5/2} \left(c+d x^2\right)^2} \, dx","Int[1/((a + b*x^2)^(5/2)*(c + d*x^2)^2),x]","\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}","\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,"(b*(2*b*c + 3*a*d)*x)/(6*a*c*(b*c - a*d)^2*(a + b*x^2)^(3/2)) + (b*(4*b^2*c^2 - 16*a*b*c*d - 3*a^2*d^2)*x)/(6*a^2*c*(b*c - a*d)^3*Sqrt[a + b*x^2]) - (d*x)/(2*c*(b*c - a*d)*(a + b*x^2)^(3/2)*(c + d*x^2)) + (d^2*(6*b*c - a*d)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(2*c^(3/2)*(b*c - a*d)^(7/2))","A",6,5,21,0.2381,1,"{414, 527, 12, 377, 208}"
96,1,313,0,0.3995877,"\int \frac{1}{\left(a+b x^2\right)^{5/2} \left(c+d x^2\right)^3} \, dx","Int[1/((a + b*x^2)^(5/2)*(c + d*x^2)^3),x]","\frac{d x \sqrt{a+b x^2} \left(-42 a^2 b c d^2+9 a^3 d^3-88 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{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}{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}","\frac{d x \sqrt{a+b x^2} \left(-42 a^2 b c d^2+9 a^3 d^3-88 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{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}{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,"-(d*x)/(4*c*(b*c - a*d)*(a + b*x^2)^(3/2)*(c + d*x^2)^2) + (b*(4*b*c + 3*a*d)*x)/(12*a*c*(b*c - a*d)^2*(a + b*x^2)^(3/2)*(c + d*x^2)) + (b*(8*b^2*c^2 - 40*a*b*c*d - 3*a^2*d^2)*x)/(12*a^2*c*(b*c - a*d)^3*Sqrt[a + b*x^2]*(c + d*x^2)) + (d*(16*b^3*c^3 - 88*a*b^2*c^2*d - 42*a^2*b*c*d^2 + 9*a^3*d^3)*x*Sqrt[a + b*x^2])/(24*a^2*c^2*(b*c - a*d)^4*(c + d*x^2)) + (d^2*(48*b^2*c^2 - 16*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(8*c^(5/2)*(b*c - a*d)^(9/2))","A",7,5,21,0.2381,1,"{414, 527, 12, 377, 208}"
97,1,224,0,0.1011626,"\int \frac{\left(a+b x^2\right)^3}{\left(c+d x^2\right)^{11/2}} \, dx","Int[(a + b*x^2)^3/(c + d*x^2)^(11/2),x]","\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{16 a^3 x (9 b c-8 a d)}{315 c^5 \sqrt{c+d x^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{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{d x \left(a+b x^2\right)^4}{9 c \left(c+d x^2\right)^{9/2} (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{16 a^3 x (9 b c-8 a d)}{315 c^5 \sqrt{c+d x^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{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{d x \left(a+b x^2\right)^4}{9 c \left(c+d x^2\right)^{9/2} (b c-a d)}",1,"-(d*x*(a + b*x^2)^4)/(9*c*(b*c - a*d)*(c + d*x^2)^(9/2)) + ((9*b*c - 8*a*d)*x*(a + b*x^2)^3)/(63*c^2*(b*c - a*d)*(c + d*x^2)^(7/2)) + (2*a*(9*b*c - 8*a*d)*x*(a + b*x^2)^2)/(105*c^3*(b*c - a*d)*(c + d*x^2)^(5/2)) + (8*a^2*(9*b*c - 8*a*d)*x*(a + b*x^2))/(315*c^4*(b*c - a*d)*(c + d*x^2)^(3/2)) + (16*a^3*(9*b*c - 8*a*d)*x)/(315*c^5*(b*c - a*d)*Sqrt[c + d*x^2])","A",5,3,21,0.1429,1,"{382, 378, 191}"
98,1,174,0,0.0707433,"\int \frac{\left(a+b x^2\right)^2}{\left(c+d x^2\right)^{9/2}} \, dx","Int[(a + b*x^2)^2/(c + d*x^2)^(9/2),x]","\frac{8 a^2 x (7 b c-6 a d)}{105 c^4 \sqrt{c+d x^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{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{d x \left(a+b x^2\right)^3}{7 c \left(c+d x^2\right)^{7/2} (b c-a d)}","\frac{8 a^2 x (7 b c-6 a d)}{105 c^4 \sqrt{c+d x^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{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{d x \left(a+b x^2\right)^3}{7 c \left(c+d x^2\right)^{7/2} (b c-a d)}",1,"-(d*x*(a + b*x^2)^3)/(7*c*(b*c - a*d)*(c + d*x^2)^(7/2)) + ((7*b*c - 6*a*d)*x*(a + b*x^2)^2)/(35*c^2*(b*c - a*d)*(c + d*x^2)^(5/2)) + (4*a*(7*b*c - 6*a*d)*x*(a + b*x^2))/(105*c^3*(b*c - a*d)*(c + d*x^2)^(3/2)) + (8*a^2*(7*b*c - 6*a*d)*x)/(105*c^4*(b*c - a*d)*Sqrt[c + d*x^2])","A",4,3,21,0.1429,1,"{382, 378, 191}"
99,1,91,0,0.0288756,"\int \frac{a+b x^2}{\left(c+d x^2\right)^{7/2}} \, dx","Int[(a + b*x^2)/(c + d*x^2)^(7/2),x]","\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}}","\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,"-((b*c - a*d)*x)/(5*c*d*(c + d*x^2)^(5/2)) + ((b*c + 4*a*d)*x)/(15*c^2*d*(c + d*x^2)^(3/2)) + (2*(b*c + 4*a*d)*x)/(15*c^3*d*Sqrt[c + d*x^2])","A",3,3,19,0.1579,1,"{385, 192, 191}"
100,1,39,0,0.0056034,"\int \frac{1}{\left(c+d x^2\right)^{5/2}} \, dx","Int[(c + d*x^2)^(-5/2),x]","\frac{2 x}{3 c^2 \sqrt{c+d x^2}}+\frac{x}{3 c \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*(c + d*x^2)^(3/2)) + (2*x)/(3*c^2*Sqrt[c + d*x^2])","A",2,2,11,0.1818,1,"{192, 191}"
101,1,79,0,0.046164,"\int \frac{1}{\left(a+b x^2\right) \left(c+d x^2\right)^{3/2}} \, dx","Int[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",3,3,21,0.1429,1,"{382, 377, 205}"
102,1,100,0,0.0547293,"\int \frac{1}{\left(a+b x^2\right)^2 \sqrt{c+d x^2}} \, dx","Int[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) (b c-a d)}","\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,"(b*x*Sqrt[c + d*x^2])/(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",3,3,21,0.1429,1,"{382, 377, 205}"
103,1,149,0,0.0794741,"\int \frac{\sqrt{c+d x^2}}{\left(a+b x^2\right)^3} \, dx","Int[Sqrt[c + d*x^2]/(a + b*x^2)^3,x]","\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{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{b x \left(c+d x^2\right)^{3/2}}{4 a \left(a+b x^2\right)^2 (b c-a d)}","\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{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{b x \left(c+d x^2\right)^{3/2}}{4 a \left(a+b x^2\right)^2 (b c-a d)}",1,"((3*b*c - 4*a*d)*x*Sqrt[c + d*x^2])/(8*a^2*(b*c - a*d)*(a + b*x^2)) + (b*x*(c + d*x^2)^(3/2))/(4*a*(b*c - a*d)*(a + b*x^2)^2) + (c*(3*b*c - 4*a*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",4,4,21,0.1905,1,"{382, 378, 377, 205}"
104,1,199,0,0.1133643,"\int \frac{\left(c+d x^2\right)^{3/2}}{\left(a+b x^2\right)^4} \, dx","Int[(c + d*x^2)^(3/2)/(a + b*x^2)^4,x]","\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{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{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{b x \left(c+d x^2\right)^{5/2}}{6 a \left(a+b x^2\right)^3 (b c-a d)}","\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{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{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{b x \left(c+d x^2\right)^{5/2}}{6 a \left(a+b x^2\right)^3 (b c-a d)}",1,"(c*(5*b*c - 6*a*d)*x*Sqrt[c + d*x^2])/(16*a^3*(b*c - a*d)*(a + b*x^2)) + ((5*b*c - 6*a*d)*x*(c + d*x^2)^(3/2))/(24*a^2*(b*c - a*d)*(a + b*x^2)^2) + (b*x*(c + d*x^2)^(5/2))/(6*a*(b*c - a*d)*(a + b*x^2)^3) + (c^2*(5*b*c - 6*a*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",5,4,21,0.1905,1,"{382, 378, 377, 205}"
105,1,20,0,0.0046504,"\int \frac{1}{\left(\frac{b c}{d}+b x^2\right) \sqrt{c+d x^2}} \, dx","Int[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",2,2,26,0.07692,1,"{21, 191}"
106,1,25,0,0.0076408,"\int \frac{1}{\sqrt{1-x^2} \left(1+x^2\right)} \, dx","Int[1/(Sqrt[1 - x^2]*(1 + x^2)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1-x^2}}\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]]/Sqrt[2]","A",2,2,19,0.1053,1,"{377, 203}"
107,1,49,0,0.0204205,"\int \frac{1}{\left(a+b x^2\right) \sqrt{c+d x^2}} \, dx","Int[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",2,2,21,0.09524,1,"{377, 205}"
108,1,15,0,0.0040774,"\int \frac{-1+x^2}{\left(1+x^2\right)^{3/2}} \, dx","Int[(-1 + x^2)/(1 + x^2)^(3/2),x]","\sinh ^{-1}(x)-\frac{2 x}{\sqrt{x^2+1}}","\sinh ^{-1}(x)-\frac{2 x}{\sqrt{x^2+1}}",1,"(-2*x)/Sqrt[1 + x^2] + ArcSinh[x]","A",2,2,15,0.1333,1,"{385, 215}"
109,1,648,0,0.5970539,"\int \left(a-b x^2\right)^{2/3} \left(3 a+b x^2\right)^3 \, dx","Int[(a - b*x^2)^(2/3)*(3*a + b*x^2)^3,x]","-\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{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{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","-\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{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{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,"(18144*a^3*x*(a - b*x^2)^(2/3))/1235 - (23544*a^2*x*(a - b*x^2)^(5/3))/6175 - (378*a*x*(a - b*x^2)^(5/3)*(3*a + b*x^2))/475 - (3*x*(a - b*x^2)^(5/3)*(3*a + b*x^2)^2)/25 - (72576*a^4*x)/(1235*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) - (36288*3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(13/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(1235*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) + (24192*Sqrt[2]*3^(3/4)*a^(13/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(1235*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])","A",8,8,24,0.3333,1,"{416, 528, 388, 195, 235, 304, 219, 1879}"
110,1,617,0,0.4257374,"\int \left(a-b x^2\right)^{2/3} \left(3 a+b x^2\right)^2 \, dx","Int[(a - b*x^2)^(2/3)*(3*a + b*x^2)^2,x]","-\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{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{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)","-\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{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{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,"(7776*a^2*x*(a - b*x^2)^(2/3))/1729 - (252*a*x*(a - b*x^2)^(5/3))/247 - (3*x*(a - b*x^2)^(5/3)*(3*a + b*x^2))/19 - (31104*a^3*x)/(1729*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) - (15552*3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(10/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(1729*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) + (10368*Sqrt[2]*3^(3/4)*a^(10/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(1729*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])","A",7,7,24,0.2917,1,"{416, 388, 195, 235, 304, 219, 1879}"
111,1,588,0,0.3693668,"\int \left(a-b x^2\right)^{2/3} \left(3 a+b x^2\right) \, dx","Int[(a - b*x^2)^(2/3)*(3*a + b*x^2),x]","-\frac{72 a^2 x}{13 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}+\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{18}{13} a x \left(a-b x^2\right)^{2/3}-\frac{3}{13} x \left(a-b x^2\right)^{5/3}","-\frac{72 a^2 x}{13 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}+\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{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,"(18*a*x*(a - b*x^2)^(2/3))/13 - (3*x*(a - b*x^2)^(5/3))/13 - (72*a^2*x)/(13*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) - (36*3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(7/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(13*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) + (24*Sqrt[2]*3^(3/4)*a^(7/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(13*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])","A",6,6,22,0.2727,1,"{388, 195, 235, 304, 219, 1879}"
112,1,740,0,0.3209492,"\int \frac{\left(a-b x^2\right)^{2/3}}{3 a+b x^2} \, dx","Int[(a - b*x^2)^(2/3)/(3*a + b*x^2),x]","-\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}}","-\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,"(3*x)/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3)) + (2^(1/3)*a^(1/6)*ArcTan[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)])/(Sqrt[3]*Sqrt[b]) + (2^(1/3)*a^(1/6)*ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(Sqrt[b]*x)])/(Sqrt[3]*Sqrt[b]) - (2^(1/3)*a^(1/6)*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/(3*Sqrt[b]) + (2^(1/3)*a^(1/6)*ArcTanh[(Sqrt[b]*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))])/Sqrt[b] + (3*3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(2*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) - (Sqrt[2]*3^(3/4)*a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])","A",6,6,24,0.2500,1,"{396, 235, 304, 219, 1879, 393}"
113,1,584,0,0.3742119,"\int \frac{\left(a-b x^2\right)^{2/3}}{\left(3 a+b x^2\right)^2} \, dx","Int[(a - b*x^2)^(2/3)/(3*a + b*x^2)^2,x]","\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)}","\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,"(x*(a - b*x^2)^(2/3))/(6*a*(3*a + b*x^2)) - x/(6*a*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) - (Sqrt[2 + Sqrt[3]]*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(4*3^(3/4)*a^(2/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) + ((a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(3*Sqrt[2]*3^(1/4)*a^(2/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])","A",6,6,24,0.2500,1,"{412, 21, 235, 304, 219, 1879}"
114,1,818,0,0.5601256,"\int \frac{\left(a-b x^2\right)^{2/3}}{\left(3 a+b x^2\right)^3} \, dx","Int[(a - b*x^2)^(2/3)/(3*a + b*x^2)^3,x]","\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}","\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,"(x*(a - b*x^2)^(2/3))/(12*a*(3*a + b*x^2)^2) + (x*(a - b*x^2)^(2/3))/(36*a^2*(3*a + b*x^2)) - x/(36*a^2*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) + ArcTan[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)]/(72*2^(2/3)*Sqrt[3]*a^(11/6)*Sqrt[b]) + ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(Sqrt[b]*x)]/(72*2^(2/3)*Sqrt[3]*a^(11/6)*Sqrt[b]) - ArcTanh[(Sqrt[b]*x)/Sqrt[a]]/(216*2^(2/3)*a^(11/6)*Sqrt[b]) + ArcTanh[(Sqrt[b]*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))]/(72*2^(2/3)*a^(11/6)*Sqrt[b]) - (Sqrt[2 + Sqrt[3]]*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(24*3^(3/4)*a^(5/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) + ((a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(18*Sqrt[2]*3^(1/4)*a^(5/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])","A",8,8,24,0.3333,1,"{412, 527, 530, 235, 304, 219, 1879, 393}"
115,1,849,0,0.6506209,"\int \frac{\left(a-b x^2\right)^{2/3}}{\left(3 a+b x^2\right)^4} \, dx","Int[(a - b*x^2)^(2/3)/(3*a + b*x^2)^4,x]","\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}","\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,"(x*(a - b*x^2)^(2/3))/(18*a*(3*a + b*x^2)^3) + (x*(a - b*x^2)^(2/3))/(54*a^2*(3*a + b*x^2)^2) + (x*(a - b*x^2)^(2/3))/(144*a^3*(3*a + b*x^2)) - x/(144*a^3*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) + (7*ArcTan[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)])/(1296*2^(2/3)*Sqrt[3]*a^(17/6)*Sqrt[b]) + (7*ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(Sqrt[b]*x)])/(1296*2^(2/3)*Sqrt[3]*a^(17/6)*Sqrt[b]) - (7*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/(3888*2^(2/3)*a^(17/6)*Sqrt[b]) + (7*ArcTanh[(Sqrt[b]*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))])/(1296*2^(2/3)*a^(17/6)*Sqrt[b]) - (Sqrt[2 + Sqrt[3]]*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(96*3^(3/4)*a^(8/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) + ((a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(72*Sqrt[2]*3^(1/4)*a^(8/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])","A",9,8,24,0.3333,1,"{412, 527, 530, 235, 304, 219, 1879, 393}"
116,1,668,0,0.5707975,"\int \left(a-b x^2\right)^{5/3} \left(3 a+b x^2\right)^3 \, dx","Int[(a - b*x^2)^(5/3)*(3*a + b*x^2)^3,x]","-\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{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{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","-\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{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{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,"(2809728*a^4*x*(a - b*x^2)^(2/3))/267995 + (1404864*a^3*x*(a - b*x^2)^(5/3))/191425 - (33264*a^2*x*(a - b*x^2)^(8/3))/14725 - (432*a*x*(a - b*x^2)^(8/3)*(3*a + b*x^2))/775 - (3*x*(a - b*x^2)^(8/3)*(3*a + b*x^2)^2)/31 - (11238912*a^5*x)/(267995*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) - (5619456*3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(16/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(267995*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) + (3746304*Sqrt[2]*3^(3/4)*a^(16/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(267995*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])","A",9,8,24,0.3333,1,"{416, 528, 388, 195, 235, 304, 219, 1879}"
117,1,637,0,0.4836262,"\int \left(a-b x^2\right)^{5/3} \left(3 a+b x^2\right)^2 \, dx","Int[(a - b*x^2)^(5/3)*(3*a + b*x^2)^2,x]","-\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{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{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)","-\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{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{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,"(28512*a^3*x*(a - b*x^2)^(2/3))/8645 + (14256*a^2*x*(a - b*x^2)^(5/3))/6175 - (306*a*x*(a - b*x^2)^(8/3))/475 - (3*x*(a - b*x^2)^(8/3)*(3*a + b*x^2))/25 - (114048*a^4*x)/(8645*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) - (57024*3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(13/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(8645*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) + (38016*Sqrt[2]*3^(3/4)*a^(13/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(8645*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])","A",8,7,24,0.2917,1,"{416, 388, 195, 235, 304, 219, 1879}"
118,1,608,0,0.420145,"\int \left(a-b x^2\right)^{5/3} \left(3 a+b x^2\right) \, dx","Int[(a - b*x^2)^(5/3)*(3*a + b*x^2),x]","-\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{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{180}{247} a x \left(a-b x^2\right)^{5/3}-\frac{3}{19} x \left(a-b x^2\right)^{8/3}","-\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{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{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,"(1800*a^2*x*(a - b*x^2)^(2/3))/1729 + (180*a*x*(a - b*x^2)^(5/3))/247 - (3*x*(a - b*x^2)^(8/3))/19 - (7200*a^3*x)/(1729*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) - (3600*3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(10/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(1729*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) + (2400*Sqrt[2]*3^(3/4)*a^(10/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(1729*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])","A",7,6,22,0.2727,1,"{388, 195, 235, 304, 219, 1879}"
119,1,765,0,0.4436183,"\int \frac{\left(a-b x^2\right)^{5/3}}{3 a+b x^2} \, dx","Int[(a - b*x^2)^(5/3)/(3*a + b*x^2),x]","\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}","\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,"(-3*x*(a - b*x^2)^(2/3))/7 + (96*a*x)/(7*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) + (4*2^(1/3)*a^(7/6)*ArcTan[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)])/(Sqrt[3]*Sqrt[b]) + (4*2^(1/3)*a^(7/6)*ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(Sqrt[b]*x)])/(Sqrt[3]*Sqrt[b]) - (4*2^(1/3)*a^(7/6)*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/(3*Sqrt[b]) + (4*2^(1/3)*a^(7/6)*ArcTanh[(Sqrt[b]*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))])/Sqrt[b] + (48*3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(4/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(7*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) - (32*Sqrt[2]*3^(3/4)*a^(4/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(7*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])","A",7,7,24,0.2917,1,"{416, 530, 235, 304, 219, 1879, 393}"
120,1,775,0,0.4366307,"\int \frac{\left(a-b x^2\right)^{5/3}}{\left(3 a+b x^2\right)^2} \, dx","Int[(a - b*x^2)^(5/3)/(3*a + b*x^2)^2,x]","\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}}","\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,"(2*x*(a - b*x^2)^(2/3))/(3*(3*a + b*x^2)) - (11*x)/(3*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) - (2^(1/3)*a^(1/6)*ArcTan[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)])/(Sqrt[3]*Sqrt[b]) - (2^(1/3)*a^(1/6)*ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(Sqrt[b]*x)])/(Sqrt[3]*Sqrt[b]) + (2^(1/3)*a^(1/6)*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/(3*Sqrt[b]) - (2^(1/3)*a^(1/6)*ArcTanh[(Sqrt[b]*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))])/Sqrt[b] - (11*Sqrt[2 + Sqrt[3]]*a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(2*3^(3/4)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) + (11*Sqrt[2]*a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(3*3^(1/4)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])","A",7,7,24,0.2917,1,"{413, 530, 235, 304, 219, 1879, 393}"
121,1,815,0,0.6134324,"\int \frac{\left(a-b x^2\right)^{5/3}}{\left(3 a+b x^2\right)^3} \, dx","Int[(a - b*x^2)^(5/3)/(3*a + b*x^2)^3,x]","-\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}","-\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,"(x*(a - b*x^2)^(2/3))/(3*(3*a + b*x^2)^2) - (x*(a - b*x^2)^(2/3))/(18*a*(3*a + b*x^2)) + x/(18*a*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) + ArcTan[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)]/(18*2^(2/3)*Sqrt[3]*a^(5/6)*Sqrt[b]) + ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(Sqrt[b]*x)]/(18*2^(2/3)*Sqrt[3]*a^(5/6)*Sqrt[b]) - ArcTanh[(Sqrt[b]*x)/Sqrt[a]]/(54*2^(2/3)*a^(5/6)*Sqrt[b]) + ArcTanh[(Sqrt[b]*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))]/(18*2^(2/3)*a^(5/6)*Sqrt[b]) + (Sqrt[2 + Sqrt[3]]*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(12*3^(3/4)*a^(2/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) - ((a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(9*Sqrt[2]*3^(1/4)*a^(2/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])","A",9,9,24,0.3750,1,"{413, 12, 471, 530, 235, 304, 219, 1879, 393}"
122,1,659,0,0.5265496,"\int \frac{\left(3 a+b x^2\right)^4}{\sqrt[3]{a-b x^2}} \, dx","Int[(3*a + b*x^2)^4/(a - b*x^2)^(1/3),x]","-\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{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{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","-\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{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{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,"(-1552608*a^3*x*(a - b*x^2)^(2/3))/43225 - (36288*a^2*x*(a - b*x^2)^(2/3)*(3*a + b*x^2))/6175 - (18*a*x*(a - b*x^2)^(2/3)*(3*a + b*x^2)^2)/19 - (3*x*(a - b*x^2)^(2/3)*(3*a + b*x^2)^3)/25 - (3794688*a^4*x)/(8645*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) - (1897344*3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(13/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(8645*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) + (1264896*Sqrt[2]*3^(3/4)*a^(13/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(8645*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])","A",8,7,24,0.2917,1,"{416, 528, 388, 235, 304, 219, 1879}"
123,1,628,0,0.4396091,"\int \frac{\left(3 a+b x^2\right)^3}{\sqrt[3]{a-b x^2}} \, dx","Int[(3*a + b*x^2)^3/(a - b*x^2)^(1/3),x]","-\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{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{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","-\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{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{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,"(-15768*a^2*x*(a - b*x^2)^(2/3))/1729 - (324*a*x*(a - b*x^2)^(2/3)*(3*a + b*x^2))/247 - (3*x*(a - b*x^2)^(2/3)*(3*a + b*x^2)^2)/19 - (215136*a^3*x)/(1729*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) - (107568*3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(10/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(1729*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) + (71712*Sqrt[2]*3^(3/4)*a^(10/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(1729*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])","A",7,7,24,0.2917,1,"{416, 528, 388, 235, 304, 219, 1879}"
124,1,597,0,0.3809976,"\int \frac{\left(3 a+b x^2\right)^2}{\sqrt[3]{a-b x^2}} \, dx","Int[(3*a + b*x^2)^2/(a - b*x^2)^(1/3),x]","-\frac{3240 a^2 x}{91 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}+\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{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)","-\frac{3240 a^2 x}{91 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}+\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{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,"(-198*a*x*(a - b*x^2)^(2/3))/91 - (3*x*(a - b*x^2)^(2/3)*(3*a + b*x^2))/13 - (3240*a^2*x)/(91*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) - (1620*3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(7/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(91*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) + (1080*Sqrt[2]*3^(3/4)*a^(7/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(91*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])","A",6,6,24,0.2500,1,"{416, 388, 235, 304, 219, 1879}"
125,1,568,0,0.3048619,"\int \frac{3 a+b x^2}{\sqrt[3]{a-b x^2}} \, dx","Int[(3*a + b*x^2)/(a - b*x^2)^(1/3),x]","\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}","\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,"(-3*x*(a - b*x^2)^(2/3))/7 - (72*a*x)/(7*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) - (36*3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(4/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(7*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) + (24*Sqrt[2]*3^(3/4)*a^(4/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(7*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])","A",5,5,22,0.2273,1,"{388, 235, 304, 219, 1879}"
126,1,204,0,0.030783,"\int \frac{1}{\sqrt[3]{a-b x^2} \left(3 a+b x^2\right)} \, dx","Int[1/((a - b*x^2)^(1/3)*(3*a + b*x^2)),x]","\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}}","\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,"ArcTan[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)]/(2*2^(2/3)*Sqrt[3]*a^(5/6)*Sqrt[b]) + ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(Sqrt[b]*x)]/(2*2^(2/3)*Sqrt[3]*a^(5/6)*Sqrt[b]) - ArcTanh[(Sqrt[b]*x)/Sqrt[a]]/(6*2^(2/3)*a^(5/6)*Sqrt[b]) + ArcTanh[(Sqrt[b]*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))]/(2*2^(2/3)*a^(5/6)*Sqrt[b])","A",1,1,24,0.04167,1,"{393}"
127,1,787,0,0.4388981,"\int \frac{1}{\sqrt[3]{a-b x^2} \left(3 a+b x^2\right)^2} \, dx","Int[1/((a - b*x^2)^(1/3)*(3*a + b*x^2)^2),x]","\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)}+\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)}+\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}}",1,"(x*(a - b*x^2)^(2/3))/(24*a^2*(3*a + b*x^2)) - x/(24*a^2*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) + ArcTan[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)]/(8*2^(2/3)*Sqrt[3]*a^(11/6)*Sqrt[b]) + ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(Sqrt[b]*x)]/(8*2^(2/3)*Sqrt[3]*a^(11/6)*Sqrt[b]) - ArcTanh[(Sqrt[b]*x)/Sqrt[a]]/(24*2^(2/3)*a^(11/6)*Sqrt[b]) + ArcTanh[(Sqrt[b]*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))]/(8*2^(2/3)*a^(11/6)*Sqrt[b]) - (Sqrt[2 + Sqrt[3]]*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(16*3^(3/4)*a^(5/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) + ((a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(12*Sqrt[2]*3^(1/4)*a^(5/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])","A",7,7,24,0.2917,1,"{414, 530, 235, 304, 219, 1879, 393}"
128,1,818,0,0.543291,"\int \frac{1}{\sqrt[3]{a-b x^2} \left(3 a+b x^2\right)^3} \, dx","Int[1/((a - b*x^2)^(1/3)*(3*a + b*x^2)^3),x]","\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}","\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,"(x*(a - b*x^2)^(2/3))/(48*a^2*(3*a + b*x^2)^2) + (5*x*(a - b*x^2)^(2/3))/(288*a^3*(3*a + b*x^2)) - (5*x)/(288*a^3*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) + (5*ArcTan[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)])/(144*2^(2/3)*Sqrt[3]*a^(17/6)*Sqrt[b]) + (5*ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(Sqrt[b]*x)])/(144*2^(2/3)*Sqrt[3]*a^(17/6)*Sqrt[b]) - (5*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/(432*2^(2/3)*a^(17/6)*Sqrt[b]) + (5*ArcTanh[(Sqrt[b]*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))])/(144*2^(2/3)*a^(17/6)*Sqrt[b]) - (5*Sqrt[2 + Sqrt[3]]*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(192*3^(3/4)*a^(8/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) + (5*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(144*Sqrt[2]*3^(1/4)*a^(8/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])","A",8,8,24,0.3333,1,"{414, 527, 530, 235, 304, 219, 1879, 393}"
129,1,623,0,0.4346531,"\int \frac{\left(3 a+b x^2\right)^3}{\left(a-b x^2\right)^{4/3}} \, dx","Int[(3*a + b*x^2)^3/(a - b*x^2)^(4/3),x]","\frac{20088 a^2 x}{91 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}-\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{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)","\frac{20088 a^2 x}{91 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}-\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{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,"(2538*a*x*(a - b*x^2)^(2/3))/91 + (81*x*(a - b*x^2)^(2/3)*(3*a + b*x^2))/13 + (6*x*(3*a + b*x^2)^2)/(a - b*x^2)^(1/3) + (20088*a^2*x)/(91*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) + (10044*3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(7/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(91*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) - (6696*Sqrt[2]*3^(3/4)*a^(7/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(91*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])","A",7,7,24,0.2917,1,"{413, 528, 388, 235, 304, 219, 1879}"
130,1,592,0,0.3676317,"\int \frac{\left(3 a+b x^2\right)^2}{\left(a-b x^2\right)^{4/3}} \, dx","Int[(3*a + b*x^2)^2/(a - b*x^2)^(4/3),x]","-\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}","-\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,"(45*x*(a - b*x^2)^(2/3))/7 + (6*x*(3*a + b*x^2))/(a - b*x^2)^(1/3) + (324*a*x)/(7*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) + (162*3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(4/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(7*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) - (108*Sqrt[2]*3^(3/4)*a^(4/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(7*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])","A",6,6,24,0.2500,1,"{413, 388, 235, 304, 219, 1879}"
131,1,561,0,0.3043406,"\int \frac{3 a+b x^2}{\left(a-b x^2\right)^{4/3}} \, dx","Int[(3*a + b*x^2)/(a - b*x^2)^(4/3),x]","-\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}}","-\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,"(6*x)/(a - b*x^2)^(1/3) + (9*x)/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3)) + (9*3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(2*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) - (3*Sqrt[2]*3^(3/4)*a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])","A",5,5,22,0.2273,1,"{385, 235, 304, 219, 1879}"
132,1,776,0,0.4432192,"\int \frac{1}{\left(a-b x^2\right)^{4/3} \left(3 a+b x^2\right)} \, dx","Int[1/((a - b*x^2)^(4/3)*(3*a + b*x^2)),x]","\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)}+\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)}+\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}}",1,"(3*x)/(8*a^2*(a - b*x^2)^(1/3)) + (3*x)/(8*a^2*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) + ArcTan[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)]/(8*2^(2/3)*Sqrt[3]*a^(11/6)*Sqrt[b]) + ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(Sqrt[b]*x)]/(8*2^(2/3)*Sqrt[3]*a^(11/6)*Sqrt[b]) - ArcTanh[(Sqrt[b]*x)/Sqrt[a]]/(24*2^(2/3)*a^(11/6)*Sqrt[b]) + ArcTanh[(Sqrt[b]*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))]/(8*2^(2/3)*a^(11/6)*Sqrt[b]) + (3*3^(1/4)*Sqrt[2 + Sqrt[3]]*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(16*a^(5/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) - (3^(3/4)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(4*Sqrt[2]*a^(5/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])","A",7,7,24,0.2917,1,"{414, 530, 235, 304, 219, 1879, 393}"
133,1,807,0,0.5339812,"\int \frac{1}{\left(a-b x^2\right)^{4/3} \left(3 a+b x^2\right)^2} \, dx","Int[1/((a - b*x^2)^(4/3)*(3*a + b*x^2)^2),x]","\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}","\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,"x/(12*a^3*(a - b*x^2)^(1/3)) + x/(24*a^2*(a - b*x^2)^(1/3)*(3*a + b*x^2)) + x/(12*a^3*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) + ArcTan[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)]/(16*2^(2/3)*Sqrt[3]*a^(17/6)*Sqrt[b]) + ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(Sqrt[b]*x)]/(16*2^(2/3)*Sqrt[3]*a^(17/6)*Sqrt[b]) - ArcTanh[(Sqrt[b]*x)/Sqrt[a]]/(48*2^(2/3)*a^(17/6)*Sqrt[b]) + ArcTanh[(Sqrt[b]*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))]/(16*2^(2/3)*a^(17/6)*Sqrt[b]) + (Sqrt[2 + Sqrt[3]]*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(8*3^(3/4)*a^(8/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) - ((a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(6*Sqrt[2]*3^(1/4)*a^(8/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])","A",8,8,24,0.3333,1,"{414, 527, 530, 235, 304, 219, 1879, 393}"
134,1,849,0,0.6533378,"\int \frac{1}{\left(a-b x^2\right)^{4/3} \left(3 a+b x^2\right)^3} \, dx","Int[1/((a - b*x^2)^(4/3)*(3*a + b*x^2)^3),x]","-\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}","-\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,"x/(48*a^2*(a - b*x^2)^(1/3)*(3*a + b*x^2)^2) + (17*x)/(192*a^3*(a - b*x^2)^(1/3)*(3*a + b*x^2)) - (19*x*(a - b*x^2)^(2/3))/(1152*a^4*(3*a + b*x^2)) + (19*x)/(1152*a^4*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) + (7*ArcTan[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)])/(288*2^(2/3)*Sqrt[3]*a^(23/6)*Sqrt[b]) + (7*ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(Sqrt[b]*x)])/(288*2^(2/3)*Sqrt[3]*a^(23/6)*Sqrt[b]) - (7*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/(864*2^(2/3)*a^(23/6)*Sqrt[b]) + (7*ArcTanh[(Sqrt[b]*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))])/(288*2^(2/3)*a^(23/6)*Sqrt[b]) + (19*Sqrt[2 + Sqrt[3]]*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(768*3^(3/4)*a^(11/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) - (19*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(576*Sqrt[2]*3^(1/4)*a^(11/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])","A",9,8,24,0.3333,1,"{414, 527, 530, 235, 304, 219, 1879, 393}"
135,1,653,0,0.5123368,"\int \frac{\left(3 a+b x^2\right)^4}{\left(a-b x^2\right)^{7/3}} \, dx","Int[(3*a + b*x^2)^4/(a - b*x^2)^(7/3),x]","-\frac{36936 a^2 x}{91 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}+\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{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}","-\frac{36936 a^2 x}{91 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}+\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{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,"(-3240*a*x*(a - b*x^2)^(2/3))/91 - (81*x*(a - b*x^2)^(2/3)*(3*a + b*x^2))/13 - (9*x*(3*a + b*x^2)^2)/(2*(a - b*x^2)^(1/3)) + (3*x*(3*a + b*x^2)^3)/(2*(a - b*x^2)^(4/3)) - (36936*a^2*x)/(91*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) - (18468*3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(7/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(91*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) + (12312*Sqrt[2]*3^(3/4)*a^(7/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(91*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])","A",8,8,24,0.3333,1,"{413, 526, 528, 388, 235, 304, 219, 1879}"
136,1,596,0,0.426372,"\int \frac{\left(3 a+b x^2\right)^3}{\left(a-b x^2\right)^{7/3}} \, dx","Int[(3*a + b*x^2)^3/(a - b*x^2)^(7/3),x]","\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)}","\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,"(-27*x*(a - b*x^2)^(2/3))/14 + (3*x*(3*a + b*x^2)^2)/(2*(a - b*x^2)^(4/3)) - (324*a*x)/(7*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) - (162*3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(4/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(7*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) + (108*Sqrt[2]*3^(3/4)*a^(4/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(7*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])","A",7,7,24,0.2917,1,"{413, 21, 388, 235, 304, 219, 1879}"
137,1,44,0,0.0172769,"\int \frac{\left(3 a+b x^2\right)^2}{\left(a-b x^2\right)^{7/3}} \, dx","Int[(3*a + b*x^2)^2/(a - b*x^2)^(7/3),x]","\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}}","\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,"(9*x)/(2*(a - b*x^2)^(1/3)) + (3*x*(3*a + b*x^2))/(2*(a - b*x^2)^(4/3))","A",2,2,24,0.08333,1,"{413, 383}"
138,1,590,0,0.3664491,"\int \frac{3 a+b x^2}{\left(a-b x^2\right)^{7/3}} \, dx","Int[(3*a + b*x^2)/(a - b*x^2)^(7/3),x]","-\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}}","-\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,"(3*x)/(2*(a - b*x^2)^(4/3)) + (9*x)/(4*a*(a - b*x^2)^(1/3)) + (9*x)/(4*a*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) + (9*3^(1/4)*Sqrt[2 + Sqrt[3]]*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(8*a^(2/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) - (3*3^(3/4)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(2*Sqrt[2]*a^(2/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])","A",6,6,22,0.2727,1,"{385, 199, 235, 304, 219, 1879}"
139,1,796,0,0.5367723,"\int \frac{1}{\left(a-b x^2\right)^{7/3} \left(3 a+b x^2\right)} \, dx","Int[1/((a - b*x^2)^(7/3)*(3*a + b*x^2)),x]","\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}","\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,"(3*x)/(32*a^2*(a - b*x^2)^(4/3)) + (21*x)/(64*a^3*(a - b*x^2)^(1/3)) + (21*x)/(64*a^3*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) + ArcTan[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)]/(32*2^(2/3)*Sqrt[3]*a^(17/6)*Sqrt[b]) + ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(Sqrt[b]*x)]/(32*2^(2/3)*Sqrt[3]*a^(17/6)*Sqrt[b]) - ArcTanh[(Sqrt[b]*x)/Sqrt[a]]/(96*2^(2/3)*a^(17/6)*Sqrt[b]) + ArcTanh[(Sqrt[b]*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))]/(32*2^(2/3)*a^(17/6)*Sqrt[b]) + (21*3^(1/4)*Sqrt[2 + Sqrt[3]]*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(128*a^(8/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) - (7*3^(3/4)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(32*Sqrt[2]*a^(8/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])","A",8,8,24,0.3333,1,"{414, 527, 530, 235, 304, 219, 1879, 393}"
140,1,827,0,0.6343098,"\int \frac{1}{\left(a-b x^2\right)^{7/3} \left(3 a+b x^2\right)^2} \, dx","Int[1/((a - b*x^2)^(7/3)*(3*a + b*x^2)^2),x]","\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}","\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,"(5*x)/(384*a^3*(a - b*x^2)^(4/3)) + (79*x)/(768*a^4*(a - b*x^2)^(1/3)) + x/(24*a^2*(a - b*x^2)^(4/3)*(3*a + b*x^2)) + (79*x)/(768*a^4*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) + (Sqrt[3]*ArcTan[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)])/(128*2^(2/3)*a^(23/6)*Sqrt[b]) + (Sqrt[3]*ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(Sqrt[b]*x)])/(128*2^(2/3)*a^(23/6)*Sqrt[b]) - ArcTanh[(Sqrt[b]*x)/Sqrt[a]]/(128*2^(2/3)*a^(23/6)*Sqrt[b]) + (3*ArcTanh[(Sqrt[b]*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))])/(128*2^(2/3)*a^(23/6)*Sqrt[b]) + (79*Sqrt[2 + Sqrt[3]]*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(512*3^(3/4)*a^(11/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) - (79*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(384*Sqrt[2]*3^(1/4)*a^(11/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])","A",9,8,24,0.3333,1,"{414, 527, 530, 235, 304, 219, 1879, 393}"
141,1,252,0,0.0727463,"\int \frac{1}{\left(-3 a-b x^2\right) \sqrt[3]{-a+b x^2}} \, dx","Int[1/((-3*a - b*x^2)*(-a + b*x^2)^(1/3)),x]","-\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}}","-\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,"-ArcTan[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)]/(2*2^(2/3)*Sqrt[3]*(-a)^(1/3)*Sqrt[a]*Sqrt[b]) - ArcTan[(Sqrt[3]*Sqrt[a]*((-a)^(1/3) - 2^(1/3)*(-a + b*x^2)^(1/3)))/((-a)^(1/3)*Sqrt[b]*x)]/(2*2^(2/3)*Sqrt[3]*(-a)^(1/3)*Sqrt[a]*Sqrt[b]) + ArcTanh[(Sqrt[b]*x)/Sqrt[a]]/(6*2^(2/3)*(-a)^(1/3)*Sqrt[a]*Sqrt[b]) - ArcTanh[((-a)^(1/3)*Sqrt[b]*x)/(Sqrt[a]*((-a)^(1/3) + 2^(1/3)*(-a + b*x^2)^(1/3)))]/(2*2^(2/3)*(-a)^(1/3)*Sqrt[a]*Sqrt[b])","A",1,1,26,0.03846,1,"{393}"
142,1,202,0,0.0287481,"\int \frac{1}{\left(3 a-b x^2\right) \sqrt[3]{a+b x^2}} \, dx","Int[1/((3*a - b*x^2)*(a + b*x^2)^(1/3)),x]","\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}}","\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,"-ArcTan[(Sqrt[b]*x)/Sqrt[a]]/(6*2^(2/3)*a^(5/6)*Sqrt[b]) + ArcTan[(Sqrt[b]*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a + b*x^2)^(1/3)))]/(2*2^(2/3)*a^(5/6)*Sqrt[b]) - ArcTanh[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)]/(2*2^(2/3)*Sqrt[3]*a^(5/6)*Sqrt[b]) - ArcTanh[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3)*(a + b*x^2)^(1/3)))/(Sqrt[b]*x)]/(2*2^(2/3)*Sqrt[3]*a^(5/6)*Sqrt[b])","A",1,1,24,0.04167,1,"{392}"
143,1,204,0,0.0424418,"\int \frac{1}{\left(c-d x^2\right) \sqrt[3]{c+3 d x^2}} \, dx","Int[1/((c - d*x^2)*(c + 3*d*x^2)^(1/3)),x]","\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}}","\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,"-ArcTan[(Sqrt[3]*Sqrt[d]*x)/Sqrt[c]]/(2*2^(2/3)*Sqrt[3]*c^(5/6)*Sqrt[d]) + (Sqrt[3]*ArcTan[(Sqrt[3]*Sqrt[d]*x)/(c^(1/6)*(c^(1/3) + 2^(1/3)*(c + 3*d*x^2)^(1/3)))])/(2*2^(2/3)*c^(5/6)*Sqrt[d]) - ArcTanh[Sqrt[c]/(Sqrt[d]*x)]/(2*2^(2/3)*c^(5/6)*Sqrt[d]) - ArcTanh[(c^(1/6)*(c^(1/3) - 2^(1/3)*(c + 3*d*x^2)^(1/3)))/(Sqrt[d]*x)]/(2*2^(2/3)*c^(5/6)*Sqrt[d])","A",1,1,23,0.04348,1,"{392}"
144,1,204,0,0.0289102,"\int \frac{1}{\sqrt[3]{a-b x^2} \left(3 a+b x^2\right)} \, dx","Int[1/((a - b*x^2)^(1/3)*(3*a + b*x^2)),x]","\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}}","\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,"ArcTan[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)]/(2*2^(2/3)*Sqrt[3]*a^(5/6)*Sqrt[b]) + ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(Sqrt[b]*x)]/(2*2^(2/3)*Sqrt[3]*a^(5/6)*Sqrt[b]) - ArcTanh[(Sqrt[b]*x)/Sqrt[a]]/(6*2^(2/3)*a^(5/6)*Sqrt[b]) + ArcTanh[(Sqrt[b]*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))]/(2*2^(2/3)*a^(5/6)*Sqrt[b])","A",1,1,24,0.04167,1,"{393}"
145,1,204,0,0.0287804,"\int \frac{1}{\sqrt[3]{c-3 d x^2} \left(c+d x^2\right)} \, dx","Int[1/((c - 3*d*x^2)^(1/3)*(c + d*x^2)),x]","\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}}","\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,"ArcTan[Sqrt[c]/(Sqrt[d]*x)]/(2*2^(2/3)*c^(5/6)*Sqrt[d]) + ArcTan[(c^(1/6)*(c^(1/3) - 2^(1/3)*(c - 3*d*x^2)^(1/3)))/(Sqrt[d]*x)]/(2*2^(2/3)*c^(5/6)*Sqrt[d]) - ArcTanh[(Sqrt[3]*Sqrt[d]*x)/Sqrt[c]]/(2*2^(2/3)*Sqrt[3]*c^(5/6)*Sqrt[d]) + (Sqrt[3]*ArcTanh[(Sqrt[3]*Sqrt[d]*x)/(c^(1/6)*(c^(1/3) + 2^(1/3)*(c - 3*d*x^2)^(1/3)))])/(2*2^(2/3)*c^(5/6)*Sqrt[d])","A",1,1,22,0.04545,1,"{393}"
146,1,113,0,0.020868,"\int \frac{1}{\sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","Int[1/((1 - x^2)^(1/3)*(3 + x^2)),x]","\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}}","\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,"ArcTan[Sqrt[3]/x]/(2*2^(2/3)*Sqrt[3]) + ArcTan[(Sqrt[3]*(1 - 2^(1/3)*(1 - x^2)^(1/3)))/x]/(2*2^(2/3)*Sqrt[3]) - ArcTanh[x]/(6*2^(2/3)) + ArcTanh[x/(1 + 2^(1/3)*(1 - x^2)^(1/3))]/(2*2^(2/3))","A",1,1,19,0.05263,1,"{393}"
147,1,109,0,0.0139981,"\int \frac{1}{\left(3-x^2\right) \sqrt[3]{1+x^2}} \, dx","Int[1/((3 - x^2)*(1 + x^2)^(1/3)),x]","\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}}","\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,"-ArcTan[x]/(6*2^(2/3)) + ArcTan[x/(1 + 2^(1/3)*(1 + x^2)^(1/3))]/(2*2^(2/3)) - ArcTanh[Sqrt[3]/x]/(2*2^(2/3)*Sqrt[3]) - ArcTanh[(Sqrt[3]*(1 - 2^(1/3)*(1 + x^2)^(1/3)))/x]/(2*2^(2/3)*Sqrt[3])","A",1,1,19,0.05263,1,"{392}"
148,1,96,0,0.0183386,"\int \frac{3-x}{\sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","Int[(3 - x)/((1 - x^2)^(1/3)*(3 + x^2)),x]","-\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}}","-\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[1/Sqrt[3] - (2^(2/3)*(1 + x)^(2/3))/(Sqrt[3]*(1 - x)^(1/3))])/2^(2/3)) - Log[3 + x^2]/(2*2^(2/3)) + (3*Log[2^(1/3)*(1 - x)^(1/3) + (1 + x)^(2/3)])/(2*2^(2/3))","A",1,1,24,0.04167,1,"{1008}"
149,1,95,0,0.0171041,"\int \frac{3+x}{\sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","Int[(3 + x)/((1 - x^2)^(1/3)*(3 + x^2)),x]","\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}}","\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[1/Sqrt[3] - (2^(2/3)*(1 - x)^(2/3))/(Sqrt[3]*(1 + x)^(1/3))])/2^(2/3) + Log[3 + x^2]/(2*2^(2/3)) - (3*Log[(1 - x)^(2/3) + 2^(1/3)*(1 + x)^(1/3)])/(2*2^(2/3))","A",1,1,22,0.04545,1,"{1008}"
150,1,151,0,0.0270012,"\int \frac{1}{\sqrt[3]{a+b x^2} \left(\frac{9 a d}{b}+d x^2\right)} \, dx","Int[1/((a + b*x^2)^(1/3)*((9*a*d)/b + d*x^2)),x]","\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}","\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,"(Sqrt[b]*ArcTan[(Sqrt[b]*x)/(3*Sqrt[a])])/(12*a^(5/6)*d) + (Sqrt[b]*ArcTan[(a^(1/3) - (a + b*x^2)^(1/3))^2/(3*a^(1/6)*Sqrt[b]*x)])/(12*a^(5/6)*d) - (Sqrt[b]*ArcTanh[(Sqrt[3]*a^(1/6)*(a^(1/3) - (a + b*x^2)^(1/3)))/(Sqrt[b]*x)])/(4*Sqrt[3]*a^(5/6)*d)","A",1,1,27,0.03704,1,"{394}"
151,1,153,0,0.0260154,"\int \frac{1}{\sqrt[3]{a-b x^2} \left(-\frac{9 a d}{b}+d x^2\right)} \, dx","Int[1/((a - b*x^2)^(1/3)*((-9*a*d)/b + d*x^2)),x]","-\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}","-\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,"-(Sqrt[b]*ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) - (a - b*x^2)^(1/3)))/(Sqrt[b]*x)])/(4*Sqrt[3]*a^(5/6)*d) - (Sqrt[b]*ArcTanh[(Sqrt[b]*x)/(3*Sqrt[a])])/(12*a^(5/6)*d) + (Sqrt[b]*ArcTanh[(a^(1/3) - (a - b*x^2)^(1/3))^2/(3*a^(1/6)*Sqrt[b]*x)])/(12*a^(5/6)*d)","A",1,1,28,0.03571,1,"{395}"
152,1,151,0,0.0282874,"\int \frac{1}{\sqrt[3]{-a+b x^2} \left(-\frac{9 a d}{b}+d x^2\right)} \, dx","Int[1/((-a + b*x^2)^(1/3)*((-9*a*d)/b + d*x^2)),x]","\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}","\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,"(Sqrt[b]*ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) + (-a + b*x^2)^(1/3)))/(Sqrt[b]*x)])/(4*Sqrt[3]*a^(5/6)*d) + (Sqrt[b]*ArcTanh[(Sqrt[b]*x)/(3*Sqrt[a])])/(12*a^(5/6)*d) - (Sqrt[b]*ArcTanh[(a^(1/3) + (-a + b*x^2)^(1/3))^2/(3*a^(1/6)*Sqrt[b]*x)])/(12*a^(5/6)*d)","A",1,1,29,0.03448,1,"{395}"
153,1,153,0,0.0288537,"\int \frac{1}{\sqrt[3]{-a-b x^2} \left(\frac{9 a d}{b}+d x^2\right)} \, dx","Int[1/((-a - b*x^2)^(1/3)*((9*a*d)/b + d*x^2)),x]","-\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}","-\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,"-(Sqrt[b]*ArcTan[(Sqrt[b]*x)/(3*Sqrt[a])])/(12*a^(5/6)*d) - (Sqrt[b]*ArcTan[(a^(1/3) + (-a - b*x^2)^(1/3))^2/(3*a^(1/6)*Sqrt[b]*x)])/(12*a^(5/6)*d) + (Sqrt[b]*ArcTanh[(Sqrt[3]*a^(1/6)*(a^(1/3) + (-a - b*x^2)^(1/3)))/(Sqrt[b]*x)])/(4*Sqrt[3]*a^(5/6)*d)","A",1,1,30,0.03333,1,"{394}"
154,1,151,0,0.0338085,"\int \frac{1}{\sqrt[3]{2+b x^2} \left(\frac{18 d}{b}+d x^2\right)} \, dx","Int[1/((2 + b*x^2)^(1/3)*((18*d)/b + d*x^2)),x]","\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}","\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,"(Sqrt[b]*ArcTan[(Sqrt[b]*x)/(3*Sqrt[2])])/(12*2^(5/6)*d) + (Sqrt[b]*ArcTan[(2^(1/3) - (2 + b*x^2)^(1/3))^2/(3*2^(1/6)*Sqrt[b]*x)])/(12*2^(5/6)*d) - (Sqrt[b]*ArcTanh[(2^(1/6)*Sqrt[3]*(2^(1/3) - (2 + b*x^2)^(1/3)))/(Sqrt[b]*x)])/(4*2^(5/6)*Sqrt[3]*d)","A",1,1,26,0.03846,1,"{394}"
155,1,147,0,0.0239209,"\int \frac{1}{\sqrt[3]{-2+b x^2} \left(-\frac{18 d}{b}+d x^2\right)} \, dx","Int[1/((-2 + b*x^2)^(1/3)*((-18*d)/b + d*x^2)),x]","\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}","\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,"(Sqrt[b]*ArcTan[(2^(1/6)*Sqrt[3]*(2^(1/3) + (-2 + b*x^2)^(1/3)))/(Sqrt[b]*x)])/(4*2^(5/6)*Sqrt[3]*d) + (Sqrt[b]*ArcTanh[(Sqrt[b]*x)/(3*Sqrt[2])])/(12*2^(5/6)*d) - (Sqrt[b]*ArcTanh[(2^(1/3) + (-2 + b*x^2)^(1/3))^2/(3*2^(1/6)*Sqrt[b]*x)])/(12*2^(5/6)*d)","A",1,1,26,0.03846,1,"{395}"
156,1,123,0,0.0204762,"\int \frac{1}{\sqrt[3]{2+3 x^2} \left(6 d+d x^2\right)} \, dx","Int[1/((2 + 3*x^2)^(1/3)*(6*d + d*x^2)),x]","\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}","\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,"ArcTan[x/Sqrt[6]]/(4*2^(5/6)*Sqrt[3]*d) + ArcTan[(2^(1/3) - (2 + 3*x^2)^(1/3))^2/(3*2^(1/6)*Sqrt[3]*x)]/(4*2^(5/6)*Sqrt[3]*d) - ArcTanh[(2^(1/6)*(2^(1/3) - (2 + 3*x^2)^(1/3)))/x]/(4*2^(5/6)*d)","A",1,1,23,0.04348,1,"{394}"
157,1,123,0,0.0167803,"\int \frac{1}{\sqrt[3]{2-3 x^2} \left(-6 d+d x^2\right)} \, dx","Int[1/((2 - 3*x^2)^(1/3)*(-6*d + d*x^2)),x]","-\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}","-\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,"-ArcTan[(2^(1/6)*(2^(1/3) - (2 - 3*x^2)^(1/3)))/x]/(4*2^(5/6)*d) - ArcTanh[x/Sqrt[6]]/(4*2^(5/6)*Sqrt[3]*d) + ArcTanh[(2^(1/3) - (2 - 3*x^2)^(1/3))^2/(3*2^(1/6)*Sqrt[3]*x)]/(4*2^(5/6)*Sqrt[3]*d)","A",1,1,23,0.04348,1,"{395}"
158,1,119,0,0.0170262,"\int \frac{1}{\sqrt[3]{-2+3 x^2} \left(-6 d+d x^2\right)} \, dx","Int[1/((-2 + 3*x^2)^(1/3)*(-6*d + d*x^2)),x]","\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}","\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,"ArcTan[(2^(1/6)*(2^(1/3) + (-2 + 3*x^2)^(1/3)))/x]/(4*2^(5/6)*d) + ArcTanh[x/Sqrt[6]]/(4*2^(5/6)*Sqrt[3]*d) - ArcTanh[(2^(1/3) + (-2 + 3*x^2)^(1/3))^2/(3*2^(1/6)*Sqrt[3]*x)]/(4*2^(5/6)*Sqrt[3]*d)","A",1,1,23,0.04348,1,"{395}"
159,1,119,0,0.0171224,"\int \frac{1}{\sqrt[3]{-2-3 x^2} \left(6 d+d x^2\right)} \, dx","Int[1/((-2 - 3*x^2)^(1/3)*(6*d + d*x^2)),x]","-\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}","-\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,"-ArcTan[x/Sqrt[6]]/(4*2^(5/6)*Sqrt[3]*d) - ArcTan[(2^(1/3) + (-2 - 3*x^2)^(1/3))^2/(3*2^(1/6)*Sqrt[3]*x)]/(4*2^(5/6)*Sqrt[3]*d) + ArcTanh[(2^(1/6)*(2^(1/3) + (-2 - 3*x^2)^(1/3)))/x]/(4*2^(5/6)*d)","A",1,1,23,0.04348,1,"{394}"
160,1,70,0,0.0088051,"\int \frac{1}{\sqrt[3]{1+x^2} \left(9+x^2\right)} \, dx","Int[1/((1 + x^2)^(1/3)*(9 + x^2)),x]","\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)","\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,"ArcTan[x/3]/12 + ArcTan[(1 - (1 + x^2)^(1/3))^2/(3*x)]/12 - ArcTanh[(Sqrt[3]*(1 - (1 + x^2)^(1/3)))/x]/(4*Sqrt[3])","A",1,1,17,0.05882,1,"{394}"
161,1,104,0,0.0163418,"\int \frac{1}{\sqrt[3]{1+b x^2} \left(9+b x^2\right)} \, dx","Int[1/((1 + b*x^2)^(1/3)*(9 + b*x^2)),x]","\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}}","\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,"ArcTan[(Sqrt[b]*x)/3]/(12*Sqrt[b]) + ArcTan[(1 - (1 + b*x^2)^(1/3))^2/(3*Sqrt[b]*x)]/(12*Sqrt[b]) - ArcTanh[(Sqrt[3]*(1 - (1 + b*x^2)^(1/3)))/(Sqrt[b]*x)]/(4*Sqrt[3]*Sqrt[b])","A",1,1,21,0.04762,1,"{394}"
162,1,74,0,0.0112752,"\int \frac{1}{\sqrt[3]{1-x^2} \left(9-x^2\right)} \, dx","Int[1/((1 - x^2)^(1/3)*(9 - x^2)),x]","\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)","\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,"ArcTan[(Sqrt[3]*(1 - (1 - x^2)^(1/3)))/x]/(4*Sqrt[3]) + ArcTanh[x/3]/12 - ArcTanh[(1 - (1 - x^2)^(1/3))^2/(3*x)]/12","A",1,1,21,0.04762,1,"{395}"
163,1,91,0,0.0192063,"\int \frac{\sqrt{-1+c^2 x^2}}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Int[Sqrt[-1 + c^2*x^2]/(d - c^2*d*x^2)^(5/2),x]","\frac{x \sqrt{c^2 x^2-1}}{2 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}+\frac{\sqrt{c^2 x^2-1} \tanh ^{-1}(c x)}{2 c d^2 \sqrt{d-c^2 d x^2}}","\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,"(x*Sqrt[-1 + c^2*x^2])/(2*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (Sqrt[-1 + c^2*x^2]*ArcTanh[c*x])/(2*c*d^2*Sqrt[d - c^2*d*x^2])","A",3,3,29,0.1034,1,"{23, 199, 208}"
164,1,91,0,0.0176594,"\int \frac{1}{\left(-1+c^2 x^2\right)^{3/2} \sqrt{d-c^2 d x^2}} \, dx","Int[1/((-1 + c^2*x^2)^(3/2)*Sqrt[d - c^2*d*x^2]),x]","\frac{x \left(d-c^2 d x^2\right)^{3/2}}{2 d^2 \left(1-c^2 x^2\right) \left(c^2 x^2-1\right)^{3/2}}+\frac{\left(d-c^2 d x^2\right)^{3/2} \tanh ^{-1}(c x)}{2 c d^2 \left(c^2 x^2-1\right)^{3/2}}","\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,"(x*(d - c^2*d*x^2)^(3/2))/(2*d^2*(1 - c^2*x^2)*(-1 + c^2*x^2)^(3/2)) + ((d - c^2*d*x^2)^(3/2)*ArcTanh[c*x])/(2*c*d^2*(-1 + c^2*x^2)^(3/2))","A",3,3,29,0.1034,1,"{23, 199, 208}"
165,1,91,0,0.0168311,"\int \frac{1}{\sqrt{-1+c^2 x^2} \left(d-c^2 d x^2\right)^{3/2}} \, dx","Int[1/(Sqrt[-1 + c^2*x^2]*(d - c^2*d*x^2)^(3/2)),x]","\frac{x \sqrt{d-c^2 d x^2}}{2 d^2 \left(1-c^2 x^2\right) \sqrt{c^2 x^2-1}}+\frac{\sqrt{d-c^2 d x^2} \tanh ^{-1}(c x)}{2 c d^2 \sqrt{c^2 x^2-1}}","-\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,"(x*Sqrt[d - c^2*d*x^2])/(2*d^2*(1 - c^2*x^2)*Sqrt[-1 + c^2*x^2]) + (Sqrt[d - c^2*d*x^2]*ArcTanh[c*x])/(2*c*d^2*Sqrt[-1 + c^2*x^2])","A",3,3,29,0.1034,1,"{23, 199, 208}"
166,1,328,0,0.3155059,"\int \left(a+b x^2\right)^{3/2} \sqrt{c+d x^2} \, dx","Int[(a + b*x^2)^(3/2)*Sqrt[c + d*x^2],x]","\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}","\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,"((7*a*c - (2*b*c^2)/d + (3*a^2*d)/b)*x*Sqrt[a + b*x^2])/(15*Sqrt[c + d*x^2]) - (2*(b*c - 3*a*d)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(15*d) + (b*x*Sqrt[a + b*x^2]*(c + d*x^2)^(3/2))/(5*d) + (Sqrt[c]*(2*b^2*c^2 - 7*a*b*c*d - 3*a^2*d^2)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*b*d^(3/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (c^(3/2)*(b*c - 9*a*d)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*d^(3/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])","A",6,6,23,0.2609,1,"{416, 528, 531, 418, 492, 411}"
167,1,249,0,0.1785889,"\int \sqrt{a+b x^2} \sqrt{c+d x^2} \, dx","Int[Sqrt[a + b*x^2]*Sqrt[c + d*x^2],x]","\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)}}}","\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,"((b*c + a*d)*x*Sqrt[a + b*x^2])/(3*b*Sqrt[c + d*x^2]) + (x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/3 - (Sqrt[c]*(b*c + a*d)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*b*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (2*c^(3/2)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])","A",5,5,23,0.2174,1,"{417, 531, 418, 492, 411}"
168,1,204,0,0.0925536,"\int \frac{\sqrt{c+d x^2}}{\sqrt{a+b x^2}} \, dx","Int[Sqrt[c + d*x^2]/Sqrt[a + b*x^2],x]","\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)}}}","\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,"(d*x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[d]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (c^(3/2)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])","A",4,4,23,0.1739,1,"{422, 418, 492, 411}"
169,1,84,0,0.0168172,"\int \frac{\sqrt{c+d x^2}}{\left(a+b x^2\right)^{3/2}} \, dx","Int[Sqrt[c + d*x^2]/(a + b*x^2)^(3/2),x]","\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)}}}","\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,"(Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]], 1 - (a*d)/(b*c)])/(Sqrt[a]*Sqrt[b]*Sqrt[a + b*x^2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))])","A",1,1,23,0.04348,1,"{411}"
170,1,237,0,0.120085,"\int \frac{\sqrt{c+d x^2}}{\left(a+b x^2\right)^{5/2}} \, dx","Int[Sqrt[c + d*x^2]/(a + b*x^2)^(5/2),x]","-\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{\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{x \sqrt{c+d x^2}}{3 a \left(a+b x^2\right)^{3/2}}","-\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{\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{x \sqrt{c+d x^2}}{3 a \left(a+b x^2\right)^{3/2}}",1,"(x*Sqrt[c + d*x^2])/(3*a*(a + b*x^2)^(3/2)) + ((2*b*c - a*d)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]], 1 - (a*d)/(b*c)])/(3*a^(3/2)*Sqrt[b]*(b*c - a*d)*Sqrt[a + b*x^2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]) - (c^(3/2)*Sqrt[d]*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*a^2*(b*c - a*d)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])","A",4,4,23,0.1739,1,"{412, 525, 418, 411}"
171,1,309,0,0.2277733,"\int \frac{\sqrt{c+d x^2}}{\left(a+b x^2\right)^{7/2}} \, dx","Int[Sqrt[c + d*x^2]/(a + b*x^2)^(7/2),x]","\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{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{x \sqrt{c+d x^2}}{5 a \left(a+b x^2\right)^{5/2}}","\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{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{x \sqrt{c+d x^2}}{5 a \left(a+b x^2\right)^{5/2}}",1,"(x*Sqrt[c + d*x^2])/(5*a*(a + b*x^2)^(5/2)) + ((4*b*c - 3*a*d)*x*Sqrt[c + d*x^2])/(15*a^2*(b*c - a*d)*(a + b*x^2)^(3/2)) + ((8*b^2*c^2 - 13*a*b*c*d + 3*a^2*d^2)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]], 1 - (a*d)/(b*c)])/(15*a^(5/2)*Sqrt[b]*(b*c - a*d)^2*Sqrt[a + b*x^2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]) - (2*c^(3/2)*Sqrt[d]*(2*b*c - 3*a*d)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*a^3*(b*c - a*d)^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])","A",5,5,23,0.2174,1,"{412, 527, 525, 418, 411}"
172,1,410,0,0.4387853,"\int \left(a+b x^2\right)^{3/2} \left(c+d x^2\right)^{3/2} \, dx","Int[(a + b*x^2)^(3/2)*(c + d*x^2)^(3/2),x]","-\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{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{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{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}","-\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{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{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{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,"(-2*(b*c + a*d)*(b^2*c^2 - 6*a*b*c*d + a^2*d^2)*x*Sqrt[a + b*x^2])/(35*b^2*d*Sqrt[c + d*x^2]) + ((9*a*c + (b*c^2)/d - (2*a^2*d)/b)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/35 + (2*(4*b*c - a*d)*x*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(35*b) + (d*x*(a + b*x^2)^(5/2)*Sqrt[c + d*x^2])/(7*b) + (2*Sqrt[c]*(b*c + a*d)*(b^2*c^2 - 6*a*b*c*d + a^2*d^2)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(35*b^2*d^(3/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (c^(3/2)*(b^2*c^2 - 18*a*b*c*d + a^2*d^2)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(35*b*d^(3/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])","A",7,6,23,0.2609,1,"{416, 528, 531, 418, 492, 411}"
173,1,336,0,0.280561,"\int \sqrt{a+b x^2} \left(c+d x^2\right)^{3/2} \, dx","Int[Sqrt[a + b*x^2]*(c + d*x^2)^(3/2),x]","\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}","\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,"((3*b^2*c^2 + 7*a*b*c*d - 2*a^2*d^2)*x*Sqrt[a + b*x^2])/(15*b^2*Sqrt[c + d*x^2]) + (2*(3*b*c - a*d)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(15*b) + (d*x*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(5*b) - (Sqrt[c]*(3*b^2*c^2 + 7*a*b*c*d - 2*a^2*d^2)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*b^2*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (c^(3/2)*(9*b*c - a*d)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*b*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])","A",6,6,23,0.2609,1,"{416, 528, 531, 418, 492, 411}"
174,1,273,0,0.1648063,"\int \frac{\left(c+d x^2\right)^{3/2}}{\sqrt{a+b x^2}} \, dx","Int[(c + d*x^2)^(3/2)/Sqrt[a + b*x^2],x]","\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}","\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,"(2*d*(2*b*c - a*d)*x*Sqrt[a + b*x^2])/(3*b^2*Sqrt[c + d*x^2]) + (d*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*b) - (2*Sqrt[c]*Sqrt[d]*(2*b*c - a*d)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*b^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (c^(3/2)*(3*b*c - a*d)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*a*b*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])","A",5,5,23,0.2174,1,"{416, 531, 418, 492, 411}"
175,1,267,0,0.1586012,"\int \frac{\left(c+d x^2\right)^{3/2}}{\left(a+b x^2\right)^{3/2}} \, dx","Int[(c + d*x^2)^(3/2)/(a + b*x^2)^(3/2),x]","-\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}}","-\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,"-((d*(b*c - 2*a*d)*x*Sqrt[a + b*x^2])/(a*b^2*Sqrt[c + d*x^2])) + ((b*c - a*d)*x*Sqrt[c + d*x^2])/(a*b*Sqrt[a + b*x^2]) + (Sqrt[c]*Sqrt[d]*(b*c - 2*a*d)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*b^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (c^(3/2)*Sqrt[d]*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*b*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])","A",5,5,23,0.2174,1,"{413, 531, 418, 492, 411}"
176,1,229,0,0.1358412,"\int \frac{\left(c+d x^2\right)^{3/2}}{\left(a+b x^2\right)^{5/2}} \, dx","Int[(c + d*x^2)^(3/2)/(a + b*x^2)^(5/2),x]","\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}}","\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,"((b*c - a*d)*x*Sqrt[c + d*x^2])/(3*a*b*(a + b*x^2)^(3/2)) + (2*(b*c + a*d)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]], 1 - (a*d)/(b*c)])/(3*a^(3/2)*b^(3/2)*Sqrt[a + b*x^2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]) - (c^(3/2)*Sqrt[d]*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*a^2*b*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])","A",4,4,23,0.1739,1,"{413, 525, 418, 411}"
177,1,315,0,0.2823661,"\int \frac{\left(c+d x^2\right)^{3/2}}{\left(a+b x^2\right)^{7/2}} \, dx","Int[(c + d*x^2)^(3/2)/(a + b*x^2)^(7/2),x]","\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{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{x \sqrt{c+d x^2} (b c-a d)}{5 a b \left(a+b x^2\right)^{5/2}}","\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{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{x \sqrt{c+d x^2} (b c-a d)}{5 a b \left(a+b x^2\right)^{5/2}}",1,"((b*c - a*d)*x*Sqrt[c + d*x^2])/(5*a*b*(a + b*x^2)^(5/2)) + (2*(2*b*c + a*d)*x*Sqrt[c + d*x^2])/(15*a^2*b*(a + b*x^2)^(3/2)) + ((8*b^2*c^2 - 3*a*b*c*d - 2*a^2*d^2)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]], 1 - (a*d)/(b*c)])/(15*a^(5/2)*b^(3/2)*(b*c - a*d)*Sqrt[a + b*x^2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]) - (c^(3/2)*Sqrt[d]*(4*b*c - a*d)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*a^3*b*(b*c - a*d)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])","A",5,5,23,0.2174,1,"{413, 527, 525, 418, 411}"
178,1,235,0,0.1347023,"\int \sqrt{2+b x^2} \sqrt{3+d x^2} \, dx","Int[Sqrt[2 + b*x^2]*Sqrt[3 + d*x^2],x]","\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}}}","\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,"((3*b + 2*d)*x*Sqrt[2 + b*x^2])/(3*b*Sqrt[3 + d*x^2]) + (x*Sqrt[2 + b*x^2]*Sqrt[3 + d*x^2])/3 - (Sqrt[2]*(3*b + 2*d)*Sqrt[2 + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[3]], 1 - (3*b)/(2*d)])/(3*b*Sqrt[d]*Sqrt[(2 + b*x^2)/(3 + d*x^2)]*Sqrt[3 + d*x^2]) + (2*Sqrt[2]*Sqrt[2 + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[3]], 1 - (3*b)/(2*d)])/(Sqrt[d]*Sqrt[(2 + b*x^2)/(3 + d*x^2)]*Sqrt[3 + d*x^2])","A",5,5,23,0.2174,1,"{417, 531, 418, 492, 411}"
179,1,38,0,0.009185,"\int \sqrt{3-6 x^2} \sqrt{2+4 x^2} \, dx","Int[Sqrt[3 - 6*x^2]*Sqrt[2 + 4*x^2],x]","\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}}","\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,"Sqrt[2/3]*x*Sqrt[1 - 4*x^4] + (2*EllipticF[ArcSin[Sqrt[2]*x], -1])/Sqrt[3]","A",3,3,23,0.1304,1,"{248, 195, 221}"
180,1,20,0,0.0038428,"\int \sqrt{2+4 x^2} \sqrt{3+6 x^2} \, dx","Int[Sqrt[2 + 4*x^2]*Sqrt[3 + 6*x^2],x]","2 \sqrt{\frac{2}{3}} x^3+\sqrt{6} x","2 \sqrt{\frac{2}{3}} x^3+\sqrt{6} x",1,"Sqrt[6]*x + 2*Sqrt[2/3]*x^3","A",2,1,23,0.04348,1,"{22}"
181,1,182,0,0.0810888,"\int \frac{\sqrt{2+b x^2}}{\sqrt{3+d x^2}} \, dx","Int[Sqrt[2 + b*x^2]/Sqrt[3 + d*x^2],x]","\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}}}","\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,"(x*Sqrt[2 + b*x^2])/Sqrt[3 + d*x^2] - (Sqrt[2]*Sqrt[2 + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[3]], 1 - (3*b)/(2*d)])/(Sqrt[d]*Sqrt[(2 + b*x^2)/(3 + d*x^2)]*Sqrt[3 + d*x^2]) + (Sqrt[2]*Sqrt[2 + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[3]], 1 - (3*b)/(2*d)])/(Sqrt[d]*Sqrt[(2 + b*x^2)/(3 + d*x^2)]*Sqrt[3 + d*x^2])","A",4,4,23,0.1739,1,"{422, 418, 492, 411}"
182,1,91,0,0.0577295,"\int \frac{\sqrt{4-x^2}}{\sqrt{c+d x^2}} \, dx","Int[Sqrt[4 - x^2]/Sqrt[c + d*x^2],x]","\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}}","\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,"-((Sqrt[c + d*x^2]*EllipticE[ArcSin[x/2], (-4*d)/c])/(d*Sqrt[1 + (d*x^2)/c])) + ((c + 4*d)*Sqrt[1 + (d*x^2)/c]*EllipticF[ArcSin[x/2], (-4*d)/c])/(d*Sqrt[c + d*x^2])","A",5,5,23,0.2174,1,"{423, 426, 424, 421, 419}"
183,1,150,0,0.0591805,"\int \frac{\sqrt{4+x^2}}{\sqrt{c+d x^2}} \, dx","Int[Sqrt[4 + x^2]/Sqrt[c + d*x^2],x]","\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)}}}","\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,"(x*Sqrt[c + d*x^2])/(d*Sqrt[4 + x^2]) - (Sqrt[c + d*x^2]*EllipticE[ArcTan[x/2], 1 - (4*d)/c])/(d*Sqrt[4 + x^2]*Sqrt[(c + d*x^2)/(c*(4 + x^2))]) + (4*Sqrt[c + d*x^2]*EllipticF[ArcTan[x/2], 1 - (4*d)/c])/(c*Sqrt[4 + x^2]*Sqrt[(c + d*x^2)/(c*(4 + x^2))])","A",4,4,21,0.1905,1,"{422, 418, 492, 411}"
184,1,20,0,0.0068373,"\int \frac{\sqrt{1-x^2}}{\sqrt{2-3 x^2}} \, dx","Int[Sqrt[1 - x^2]/Sqrt[2 - 3*x^2],x]","\frac{E\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|\frac{2}{3}\right)}{\sqrt{3}}","\frac{E\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|\frac{2}{3}\right)}{\sqrt{3}}",1,"EllipticE[ArcSin[Sqrt[3/2]*x], 2/3]/Sqrt[3]","A",1,1,23,0.04348,1,"{424}"
185,1,21,0,0.0076009,"\int \frac{\sqrt{4-x^2}}{\sqrt{2-3 x^2}} \, dx","Int[Sqrt[4 - x^2]/Sqrt[2 - 3*x^2],x]","\frac{2 E\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|\frac{1}{6}\right)}{\sqrt{3}}","\frac{2 E\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|\frac{1}{6}\right)}{\sqrt{3}}",1,"(2*EllipticE[ArcSin[Sqrt[3/2]*x], 1/6])/Sqrt[3]","A",1,1,23,0.04348,1,"{424}"
186,1,20,0,0.0070226,"\int \frac{\sqrt{1-4 x^2}}{\sqrt{2-3 x^2}} \, dx","Int[Sqrt[1 - 4*x^2]/Sqrt[2 - 3*x^2],x]","\frac{E\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|\frac{8}{3}\right)}{\sqrt{3}}","\frac{E\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|\frac{8}{3}\right)}{\sqrt{3}}",1,"EllipticE[ArcSin[Sqrt[3/2]*x], 8/3]/Sqrt[3]","A",1,1,23,0.04348,1,"{424}"
187,1,4,0,0.0052178,"\int \frac{\sqrt{1+x^2}}{\sqrt{1-x^2}} \, dx","Int[Sqrt[1 + x^2]/Sqrt[1 - x^2],x]","E\left(\left.\sin ^{-1}(x)\right|-1\right)","E\left(\left.\sin ^{-1}(x)\right|-1\right)",1,"EllipticE[ArcSin[x], -1]","A",1,1,21,0.04762,1,"{424}"
188,1,20,0,0.0064151,"\int \frac{\sqrt{1+x^2}}{\sqrt{2-3 x^2}} \, dx","Int[Sqrt[1 + x^2]/Sqrt[2 - 3*x^2],x]","\frac{E\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|-\frac{2}{3}\right)}{\sqrt{3}}","\frac{E\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|-\frac{2}{3}\right)}{\sqrt{3}}",1,"EllipticE[ArcSin[Sqrt[3/2]*x], -2/3]/Sqrt[3]","A",1,1,21,0.04762,1,"{424}"
189,1,21,0,0.0064591,"\int \frac{\sqrt{4+x^2}}{\sqrt{2-3 x^2}} \, dx","Int[Sqrt[4 + x^2]/Sqrt[2 - 3*x^2],x]","\frac{2 E\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|-\frac{1}{6}\right)}{\sqrt{3}}","\frac{2 E\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|-\frac{1}{6}\right)}{\sqrt{3}}",1,"(2*EllipticE[ArcSin[Sqrt[3/2]*x], -1/6])/Sqrt[3]","A",1,1,21,0.04762,1,"{424}"
190,1,20,0,0.0072809,"\int \frac{\sqrt{1+4 x^2}}{\sqrt{2-3 x^2}} \, dx","Int[Sqrt[1 + 4*x^2]/Sqrt[2 - 3*x^2],x]","\frac{E\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|-\frac{8}{3}\right)}{\sqrt{3}}","\frac{E\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|-\frac{8}{3}\right)}{\sqrt{3}}",1,"EllipticE[ArcSin[Sqrt[3/2]*x], -8/3]/Sqrt[3]","A",1,1,23,0.04348,1,"{424}"
191,1,13,0,0.0137832,"\int \frac{\sqrt{1-x^2}}{\sqrt{1+x^2}} \, dx","Int[Sqrt[1 - x^2]/Sqrt[1 + x^2],x]","2 F\left(\left.\sin ^{-1}(x)\right|-1\right)-E\left(\left.\sin ^{-1}(x)\right|-1\right)","2 F\left(\left.\sin ^{-1}(x)\right|-1\right)-E\left(\left.\sin ^{-1}(x)\right|-1\right)",1,"-EllipticE[ArcSin[x], -1] + 2*EllipticF[ArcSin[x], -1]","A",4,4,21,0.1905,1,"{423, 424, 248, 221}"
192,1,31,0,0.0202918,"\int \frac{\sqrt{1-x^2}}{\sqrt{2+3 x^2}} \, dx","Int[Sqrt[1 - x^2]/Sqrt[2 + 3*x^2],x]","\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)","\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,"-(Sqrt[2]*EllipticE[ArcSin[x], -3/2])/3 + (5*EllipticF[ArcSin[x], -3/2])/(3*Sqrt[2])","A",3,3,23,0.1304,1,"{423, 424, 419}"
193,1,35,0,0.0205367,"\int \frac{\sqrt{4-x^2}}{\sqrt{2+3 x^2}} \, dx","Int[Sqrt[4 - x^2]/Sqrt[2 + 3*x^2],x]","\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)","\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,"-(Sqrt[2]*EllipticE[ArcSin[x/2], -6])/3 + (7*Sqrt[2]*EllipticF[ArcSin[x/2], -6])/3","A",3,3,23,0.1304,1,"{423, 424, 419}"
194,1,35,0,0.0199491,"\int \frac{\sqrt{1-4 x^2}}{\sqrt{2+3 x^2}} \, dx","Int[Sqrt[1 - 4*x^2]/Sqrt[2 + 3*x^2],x]","\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)","\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,"(-2*Sqrt[2]*EllipticE[ArcSin[2*x], -3/8])/3 + (11*EllipticF[ArcSin[2*x], -3/8])/(6*Sqrt[2])","A",3,3,23,0.1304,1,"{423, 424, 419}"
195,1,131,0,0.0408642,"\int \frac{\sqrt{1+x^2}}{\sqrt{2+3 x^2}} \, dx","Int[Sqrt[1 + x^2]/Sqrt[2 + 3*x^2],x]","\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}}}","\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,"(x*Sqrt[2 + 3*x^2])/(3*Sqrt[1 + x^2]) - (Sqrt[2]*Sqrt[2 + 3*x^2]*EllipticE[ArcTan[x], -1/2])/(3*Sqrt[1 + x^2]*Sqrt[(2 + 3*x^2)/(1 + x^2)]) + (Sqrt[2 + 3*x^2]*EllipticF[ArcTan[x], -1/2])/(Sqrt[2]*Sqrt[1 + x^2]*Sqrt[(2 + 3*x^2)/(1 + x^2)])","A",4,4,21,0.1905,1,"{422, 418, 492, 411}"
196,1,136,0,0.0442971,"\int \frac{\sqrt{4+x^2}}{\sqrt{2+3 x^2}} \, dx","Int[Sqrt[4 + x^2]/Sqrt[2 + 3*x^2],x]","\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}}}","\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,"(x*Sqrt[2 + 3*x^2])/(3*Sqrt[4 + x^2]) - (Sqrt[2]*Sqrt[2 + 3*x^2]*EllipticE[ArcTan[x/2], -5])/(3*Sqrt[4 + x^2]*Sqrt[(2 + 3*x^2)/(4 + x^2)]) + (2*Sqrt[2]*Sqrt[2 + 3*x^2]*EllipticF[ArcTan[x/2], -5])/(Sqrt[4 + x^2]*Sqrt[(2 + 3*x^2)/(4 + x^2)])","A",4,4,21,0.1905,1,"{422, 418, 492, 411}"
197,1,148,0,0.0497495,"\int \frac{\sqrt{1+4 x^2}}{\sqrt{2+3 x^2}} \, dx","Int[Sqrt[1 + 4*x^2]/Sqrt[2 + 3*x^2],x]","\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}}","\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,"(4*x*Sqrt[2 + 3*x^2])/(3*Sqrt[1 + 4*x^2]) - (2*Sqrt[2]*Sqrt[2 + 3*x^2]*EllipticE[ArcTan[2*x], 5/8])/(3*Sqrt[(2 + 3*x^2)/(1 + 4*x^2)]*Sqrt[1 + 4*x^2]) + (Sqrt[2 + 3*x^2]*EllipticF[ArcTan[2*x], 5/8])/(2*Sqrt[2]*Sqrt[(2 + 3*x^2)/(1 + 4*x^2)]*Sqrt[1 + 4*x^2])","A",4,4,23,0.1739,1,"{422, 418, 492, 411}"
198,1,40,0,0.014525,"\int \frac{\sqrt{1-x^2}}{\sqrt{-1+2 x^2}} \, dx","Int[Sqrt[1 - x^2]/Sqrt[-1 + 2*x^2],x]","\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}}","\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,"(Sqrt[1 - 2*x^2]*EllipticE[ArcSin[Sqrt[2]*x], 1/2])/(Sqrt[2]*Sqrt[-1 + 2*x^2])","A",2,2,23,0.08696,1,"{427, 424}"
199,1,423,0,0.4312953,"\int \frac{\left(a+b x^2\right)^{7/2}}{\sqrt{c+d x^2}} \, dx","Int[(a + b*x^2)^(7/2)/Sqrt[c + d*x^2],x]","\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{\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{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}","\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{\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{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,"(-8*(b*c - 2*a*d)*(6*b^2*c^2 - 11*a*b*c*d + 11*a^2*d^2)*x*Sqrt[a + b*x^2])/(105*d^3*Sqrt[c + d*x^2]) + (b*(24*b^2*c^2 - 71*a*b*c*d + 71*a^2*d^2)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(105*d^3) - (6*b*(b*c - 2*a*d)*x*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(35*d^2) + (b*x*(a + b*x^2)^(5/2)*Sqrt[c + d*x^2])/(7*d) + (8*Sqrt[c]*(b*c - 2*a*d)*(6*b^2*c^2 - 11*a*b*c*d + 11*a^2*d^2)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(105*d^(7/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (Sqrt[c]*(3*b*c - 7*a*d)*(8*b^2*c^2 - 11*a*b*c*d + 15*a^2*d^2)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(105*d^(7/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])","A",7,6,23,0.2609,1,"{416, 528, 531, 418, 492, 411}"
200,1,344,0,0.2826021,"\int \frac{\left(a+b x^2\right)^{5/2}}{\sqrt{c+d x^2}} \, dx","Int[(a + b*x^2)^(5/2)/Sqrt[c + d*x^2],x]","\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}","\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,"((8*b^2*c^2 - 23*a*b*c*d + 23*a^2*d^2)*x*Sqrt[a + b*x^2])/(15*d^2*Sqrt[c + d*x^2]) - (4*b*(b*c - 2*a*d)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(15*d^2) + (b*x*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(5*d) - (Sqrt[c]*(8*b^2*c^2 - 23*a*b*c*d + 23*a^2*d^2)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*d^(5/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (Sqrt[c]*(4*b^2*c^2 - 11*a*b*c*d + 15*a^2*d^2)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*d^(5/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])","A",6,6,23,0.2609,1,"{416, 528, 531, 418, 492, 411}"
201,1,260,0,0.1600535,"\int \frac{\left(a+b x^2\right)^{3/2}}{\sqrt{c+d x^2}} \, dx","Int[(a + b*x^2)^(3/2)/Sqrt[c + d*x^2],x]","-\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}}","-\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,"(-2*(b*c - 2*a*d)*x*Sqrt[a + b*x^2])/(3*d*Sqrt[c + d*x^2]) + (b*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*d) + (2*Sqrt[c]*(b*c - 2*a*d)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*d^(3/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (Sqrt[c]*(b*c - 3*a*d)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*d^(3/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])","A",5,5,23,0.2174,1,"{416, 531, 418, 492, 411}"
202,1,194,0,0.090015,"\int \frac{\sqrt{a+b x^2}}{\sqrt{c+d x^2}} \, dx","Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2],x]","\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)}}}","\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,"(x*Sqrt[a + b*x^2])/Sqrt[c + d*x^2] - (Sqrt[c]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (Sqrt[c]*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])","A",4,4,23,0.1739,1,"{422, 418, 492, 411}"
203,1,87,0,0.0186419,"\int \frac{1}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]","\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)}}}","\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,"(Sqrt[c]*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])","A",1,1,23,0.04348,1,"{418}"
204,1,273,0,0.1430313,"\int \frac{1}{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}} \, dx","Int[1/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]),x]","\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)}}}","\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,"-((d*x*Sqrt[a + b*x^2])/(a*(b*c - a*d)*Sqrt[c + d*x^2])) + (b*x*Sqrt[c + d*x^2])/(a*(b*c - a*d)*Sqrt[a + b*x^2]) + (Sqrt[c]*Sqrt[d]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*(b*c - a*d)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[d]*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*(b*c - a*d)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])","A",6,6,23,0.2609,1,"{414, 21, 422, 418, 492, 411}"
205,1,255,0,0.1463156,"\int \frac{1}{\left(a+b x^2\right)^{5/2} \sqrt{c+d x^2}} \, dx","Int[1/((a + b*x^2)^(5/2)*Sqrt[c + d*x^2]),x]","-\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{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{b x \sqrt{c+d x^2}}{3 a \left(a+b x^2\right)^{3/2} (b c-a d)}","-\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{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{b x \sqrt{c+d x^2}}{3 a \left(a+b x^2\right)^{3/2} (b c-a d)}",1,"(b*x*Sqrt[c + d*x^2])/(3*a*(b*c - a*d)*(a + b*x^2)^(3/2)) + (2*Sqrt[b]*(b*c - 2*a*d)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]], 1 - (a*d)/(b*c)])/(3*a^(3/2)*(b*c - a*d)^2*Sqrt[a + b*x^2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]) - (Sqrt[c]*Sqrt[d]*(b*c - 3*a*d)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*a^2*(b*c - a*d)^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])","A",4,4,23,0.1739,1,"{414, 525, 418, 411}"
206,1,334,0,0.2775281,"\int \frac{1}{\left(a+b x^2\right)^{7/2} \sqrt{c+d x^2}} \, dx","Int[1/((a + b*x^2)^(7/2)*Sqrt[c + d*x^2]),x]","-\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{\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{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{b x \sqrt{c+d x^2}}{5 a \left(a+b x^2\right)^{5/2} (b c-a d)}","-\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{\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{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{b x \sqrt{c+d x^2}}{5 a \left(a+b x^2\right)^{5/2} (b c-a d)}",1,"(b*x*Sqrt[c + d*x^2])/(5*a*(b*c - a*d)*(a + b*x^2)^(5/2)) + (4*b*(b*c - 2*a*d)*x*Sqrt[c + d*x^2])/(15*a^2*(b*c - a*d)^2*(a + b*x^2)^(3/2)) + (Sqrt[b]*(8*b^2*c^2 - 23*a*b*c*d + 23*a^2*d^2)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]], 1 - (a*d)/(b*c)])/(15*a^(5/2)*(b*c - a*d)^3*Sqrt[a + b*x^2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]) - (Sqrt[c]*Sqrt[d]*(4*b^2*c^2 - 11*a*b*c*d + 15*a^2*d^2)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*a^3*(b*c - a*d)^3*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])","A",5,5,23,0.2174,1,"{414, 527, 525, 418, 411}"
207,1,445,0,0.405257,"\int \frac{\left(a+b x^2\right)^{7/2}}{\left(c+d x^2\right)^{3/2}} \, dx","Int[(a + b*x^2)^(7/2)/(c + d*x^2)^(3/2),x]","-\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(103 a^2 b c d^2-15 a^3 d^3-128 a b^2 c^2 d+48 b^3 c^3\right)}{15 c d^3 \sqrt{c+d x^2}}+\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{\sqrt{a+b x^2} \left(103 a^2 b c d^2-15 a^3 d^3-128 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}}","-\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(103 a^2 b c d^2-15 a^3 d^3-128 a b^2 c^2 d+48 b^3 c^3\right)}{15 c d^3 \sqrt{c+d x^2}}+\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{\sqrt{a+b x^2} \left(103 a^2 b c d^2-15 a^3 d^3-128 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,"((48*b^3*c^3 - 128*a*b^2*c^2*d + 103*a^2*b*c*d^2 - 15*a^3*d^3)*x*Sqrt[a + b*x^2])/(15*c*d^3*Sqrt[c + d*x^2]) - ((b*c - a*d)*x*(a + b*x^2)^(5/2))/(c*d*Sqrt[c + d*x^2]) - (b*(24*b^2*c^2 - 43*a*b*c*d + 15*a^2*d^2)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(15*c*d^3) + (b*(6*b*c - 5*a*d)*x*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(5*c*d^2) - ((48*b^3*c^3 - 128*a*b^2*c^2*d + 103*a^2*b*c*d^2 - 15*a^3*d^3)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*Sqrt[c]*d^(7/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (b*Sqrt[c]*(24*b^2*c^2 - 61*a*b*c*d + 45*a^2*d^2)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*d^(7/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])","A",7,6,23,0.2609,1,"{413, 528, 531, 418, 492, 411}"
208,1,346,0,0.266402,"\int \frac{\left(a+b x^2\right)^{5/2}}{\left(c+d x^2\right)^{3/2}} \, dx","Int[(a + b*x^2)^(5/2)/(c + d*x^2)^(3/2),x]","-\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{b x \sqrt{a+b x^2} \sqrt{c+d x^2} (4 b c-3 a d)}{3 c d^2}-\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{x \left(a+b x^2\right)^{3/2} (b c-a d)}{c d \sqrt{c+d x^2}}","-\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{b x \sqrt{a+b x^2} \sqrt{c+d x^2} (4 b c-3 a d)}{3 c d^2}-\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{x \left(a+b x^2\right)^{3/2} (b c-a d)}{c d \sqrt{c+d x^2}}",1,"-((8*b^2*c^2 - 13*a*b*c*d + 3*a^2*d^2)*x*Sqrt[a + b*x^2])/(3*c*d^2*Sqrt[c + d*x^2]) - ((b*c - a*d)*x*(a + b*x^2)^(3/2))/(c*d*Sqrt[c + d*x^2]) + (b*(4*b*c - 3*a*d)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*c*d^2) + ((8*b^2*c^2 - 13*a*b*c*d + 3*a^2*d^2)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*Sqrt[c]*d^(5/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (2*b*Sqrt[c]*(2*b*c - 3*a*d)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*d^(5/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])","A",6,6,23,0.2609,1,"{413, 528, 531, 418, 492, 411}"
209,1,258,0,0.15181,"\int \frac{\left(a+b x^2\right)^{3/2}}{\left(c+d x^2\right)^{3/2}} \, dx","Int[(a + b*x^2)^(3/2)/(c + d*x^2)^(3/2),x]","\frac{b \sqrt{c} \sqrt{a+b x^2} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{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}}","\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,"-(((b*c - a*d)*x*Sqrt[a + b*x^2])/(c*d*Sqrt[c + d*x^2])) + ((2*b*c - a*d)*x*Sqrt[a + b*x^2])/(c*d*Sqrt[c + d*x^2]) - ((2*b*c - a*d)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[c]*d^(3/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (b*Sqrt[c]*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(d^(3/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])","A",5,5,23,0.2174,1,"{413, 531, 418, 492, 411}"
210,1,84,0,0.0165276,"\int \frac{\sqrt{a+b x^2}}{\left(c+d x^2\right)^{3/2}} \, dx","Int[Sqrt[a + b*x^2]/(c + d*x^2)^(3/2),x]","\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)}}}","\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,"(Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[c]*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])","A",1,1,23,0.04348,1,"{411}"
211,1,194,0,0.1519669,"\int \frac{1}{\sqrt{a+b x^2} \left(c+d x^2\right)^{3/2}} \, dx","Int[1/(Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)),x]","\frac{b \sqrt{c} \sqrt{a+b x^2} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{a \sqrt{d} \sqrt{c+d x^2} (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)}}}","\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,"-((Sqrt[d]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[c]*(b*c - a*d)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])) + (b*Sqrt[c]*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*(b*c - a*d)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])","A",6,6,23,0.2609,1,"{414, 21, 422, 418, 492, 411}"
212,1,242,0,0.1198303,"\int \frac{1}{\left(a+b x^2\right)^{3/2} \left(c+d x^2\right)^{3/2}} \, dx","Int[1/((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)),x]","\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)}}}","\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,"(b*x)/(a*(b*c - a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]) + (Sqrt[d]*(b*c + a*d)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[c]*(b*c - a*d)^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (2*b*Sqrt[c]*Sqrt[d]*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*(b*c - a*d)^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])","A",4,4,23,0.1739,1,"{414, 525, 418, 411}"
213,1,323,0,0.2644756,"\int \frac{1}{\left(a+b x^2\right)^{5/2} \left(c+d x^2\right)^{3/2}} \, dx","Int[1/((a + b*x^2)^(5/2)*(c + d*x^2)^(3/2)),x]","\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)}","\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,"(b*x)/(3*a*(b*c - a*d)*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2]) + (2*b*(b*c - 3*a*d)*x)/(3*a^2*(b*c - a*d)^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]) + (Sqrt[d]*(2*b^2*c^2 - 7*a*b*c*d - 3*a^2*d^2)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*a^2*Sqrt[c]*(b*c - a*d)^3*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (b*Sqrt[c]*Sqrt[d]*(b*c - 9*a*d)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*a^2*(b*c - a*d)^3*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])","A",5,5,23,0.2174,1,"{414, 527, 525, 418, 411}"
214,1,87,0,0.0190631,"\int \frac{1}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]","\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)}}}","\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,"(Sqrt[c]*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])","A",1,1,23,0.04348,1,"{418}"
215,1,87,0,0.0537777,"\int \frac{1}{\sqrt{a-b x^2} \sqrt{c+d x^2}} \, dx","Int[1/(Sqrt[a - b*x^2]*Sqrt[c + d*x^2]),x]","\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}}","\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,"(Sqrt[a]*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[b]*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])","A",3,2,24,0.08333,1,"{421, 419}"
216,1,87,0,0.0524197,"\int \frac{1}{\sqrt{a+b x^2} \sqrt{c-d x^2}} \, dx","Int[1/(Sqrt[a + b*x^2]*Sqrt[c - d*x^2]),x]","\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}}","\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,"(Sqrt[c]*Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(Sqrt[d]*Sqrt[a + b*x^2]*Sqrt[c - d*x^2])","A",3,2,24,0.08333,1,"{421, 419}"
217,1,88,0,0.0570123,"\int \frac{1}{\sqrt{a-b x^2} \sqrt{c-d x^2}} \, dx","Int[1/(Sqrt[a - b*x^2]*Sqrt[c - d*x^2]),x]","\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}}","\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,"(Sqrt[c]*Sqrt[1 - (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[d]*x)/Sqrt[c]], (b*c)/(a*d)])/(Sqrt[d]*Sqrt[a - b*x^2]*Sqrt[c - d*x^2])","A",3,2,25,0.08000,1,"{421, 419}"
218,1,12,0,0.0070182,"\int \frac{1}{\sqrt{1-x^2} \sqrt{2+5 x^2}} \, dx","Int[1/(Sqrt[1 - x^2]*Sqrt[2 + 5*x^2]),x]","\frac{F\left(\sin ^{-1}(x)|-\frac{5}{2}\right)}{\sqrt{2}}","\frac{F\left(\sin ^{-1}(x)|-\frac{5}{2}\right)}{\sqrt{2}}",1,"EllipticF[ArcSin[x], -5/2]/Sqrt[2]","A",1,1,23,0.04348,1,"{419}"
219,1,10,0,0.007433,"\int \frac{1}{\sqrt{1-x^2} \sqrt{2+4 x^2}} \, dx","Int[1/(Sqrt[1 - x^2]*Sqrt[2 + 4*x^2]),x]","\frac{F\left(\left.\sin ^{-1}(x)\right|-2\right)}{\sqrt{2}}","\frac{F\left(\left.\sin ^{-1}(x)\right|-2\right)}{\sqrt{2}}",1,"EllipticF[ArcSin[x], -2]/Sqrt[2]","A",1,1,23,0.04348,1,"{419}"
220,1,12,0,0.0068817,"\int \frac{1}{\sqrt{1-x^2} \sqrt{2+3 x^2}} \, dx","Int[1/(Sqrt[1 - x^2]*Sqrt[2 + 3*x^2]),x]","\frac{F\left(\sin ^{-1}(x)|-\frac{3}{2}\right)}{\sqrt{2}}","\frac{F\left(\sin ^{-1}(x)|-\frac{3}{2}\right)}{\sqrt{2}}",1,"EllipticF[ArcSin[x], -3/2]/Sqrt[2]","A",1,1,23,0.04348,1,"{419}"
221,1,10,0,0.0052985,"\int \frac{1}{\sqrt{1-x^2} \sqrt{2+2 x^2}} \, dx","Int[1/(Sqrt[1 - x^2]*Sqrt[2 + 2*x^2]),x]","\frac{F\left(\left.\sin ^{-1}(x)\right|-1\right)}{\sqrt{2}}","\frac{F\left(\left.\sin ^{-1}(x)\right|-1\right)}{\sqrt{2}}",1,"EllipticF[ArcSin[x], -1]/Sqrt[2]","A",2,2,23,0.08696,1,"{248, 221}"
222,1,12,0,0.0065566,"\int \frac{1}{\sqrt{1-x^2} \sqrt{2+x^2}} \, dx","Int[1/(Sqrt[1 - x^2]*Sqrt[2 + x^2]),x]","\frac{F\left(\sin ^{-1}(x)|-\frac{1}{2}\right)}{\sqrt{2}}","\frac{F\left(\sin ^{-1}(x)|-\frac{1}{2}\right)}{\sqrt{2}}",1,"EllipticF[ArcSin[x], -1/2]/Sqrt[2]","A",1,1,21,0.04762,1,"{419}"
223,1,12,0,0.0074622,"\int \frac{1}{\sqrt{1-x^2} \sqrt{2-x^2}} \, dx","Int[1/(Sqrt[1 - x^2]*Sqrt[2 - x^2]),x]","\frac{F\left(\sin ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{2}}","\frac{F\left(\sin ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{2}}",1,"EllipticF[ArcSin[x], 1/2]/Sqrt[2]","A",1,1,23,0.04348,1,"{419}"
224,1,8,0,0.0022928,"\int \frac{1}{\sqrt{2-2 x^2} \sqrt{1-x^2}} \, dx","Int[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",2,2,23,0.08696,1,"{22, 206}"
225,1,12,0,0.0068764,"\int \frac{1}{\sqrt{2-3 x^2} \sqrt{1-x^2}} \, dx","Int[1/(Sqrt[2 - 3*x^2]*Sqrt[1 - x^2]),x]","\frac{F\left(\sin ^{-1}(x)|\frac{3}{2}\right)}{\sqrt{2}}","\frac{F\left(\sin ^{-1}(x)|\frac{3}{2}\right)}{\sqrt{2}}",1,"EllipticF[ArcSin[x], 3/2]/Sqrt[2]","A",1,1,23,0.04348,1,"{419}"
226,1,10,0,0.0073752,"\int \frac{1}{\sqrt{2-4 x^2} \sqrt{1-x^2}} \, dx","Int[1/(Sqrt[2 - 4*x^2]*Sqrt[1 - x^2]),x]","\frac{F\left(\left.\sin ^{-1}(x)\right|2\right)}{\sqrt{2}}","\frac{F\left(\left.\sin ^{-1}(x)\right|2\right)}{\sqrt{2}}",1,"EllipticF[ArcSin[x], 2]/Sqrt[2]","A",1,1,23,0.04348,1,"{419}"
227,1,12,0,0.006778,"\int \frac{1}{\sqrt{2-5 x^2} \sqrt{1-x^2}} \, dx","Int[1/(Sqrt[2 - 5*x^2]*Sqrt[1 - x^2]),x]","\frac{F\left(\sin ^{-1}(x)|\frac{5}{2}\right)}{\sqrt{2}}","\frac{F\left(\sin ^{-1}(x)|\frac{5}{2}\right)}{\sqrt{2}}",1,"EllipticF[ArcSin[x], 5/2]/Sqrt[2]","A",1,1,23,0.04348,1,"{419}"
228,1,51,0,0.0094425,"\int \frac{1}{\sqrt{1+x^2} \sqrt{2+5 x^2}} \, dx","Int[1/(Sqrt[1 + x^2]*Sqrt[2 + 5*x^2]),x]","\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}}}","\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,"(Sqrt[2 + 5*x^2]*EllipticF[ArcTan[x], -3/2])/(Sqrt[2]*Sqrt[1 + x^2]*Sqrt[(2 + 5*x^2)/(1 + x^2)])","A",1,1,21,0.04762,1,"{418}"
229,1,49,0,0.0092925,"\int \frac{1}{\sqrt{1+x^2} \sqrt{2+4 x^2}} \, dx","Int[1/(Sqrt[1 + x^2]*Sqrt[2 + 4*x^2]),x]","\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}}}","\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,"(Sqrt[1 + 2*x^2]*EllipticF[ArcTan[x], -1])/(Sqrt[2]*Sqrt[1 + x^2]*Sqrt[(1 + 2*x^2)/(1 + x^2)])","A",1,1,21,0.04762,1,"{418}"
230,1,51,0,0.0087415,"\int \frac{1}{\sqrt{1+x^2} \sqrt{2+3 x^2}} \, dx","Int[1/(Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]),x]","\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{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,"(Sqrt[2 + 3*x^2]*EllipticF[ArcTan[x], -1/2])/(Sqrt[2]*Sqrt[1 + x^2]*Sqrt[(2 + 3*x^2)/(1 + x^2)])","A",1,1,21,0.04762,1,"{418}"
231,1,8,0,0.0023531,"\int \frac{1}{\sqrt{1+x^2} \sqrt{2+2 x^2}} \, dx","Int[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",2,2,21,0.09524,1,"{22, 203}"
232,1,47,0,0.00898,"\int \frac{1}{\sqrt{1+x^2} \sqrt{2+x^2}} \, dx","Int[1/(Sqrt[1 + x^2]*Sqrt[2 + x^2]),x]","\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}}}","\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,"(Sqrt[2 + x^2]*EllipticF[ArcTan[x], 1/2])/(Sqrt[2]*Sqrt[1 + x^2]*Sqrt[(2 + x^2)/(1 + x^2)])","A",1,1,19,0.05263,1,"{418}"
233,1,10,0,0.0061151,"\int \frac{1}{\sqrt{2-x^2} \sqrt{1+x^2}} \, dx","Int[1/(Sqrt[2 - x^2]*Sqrt[1 + x^2]),x]","F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)","F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)",1,"EllipticF[ArcSin[x/Sqrt[2]], -2]","A",1,1,21,0.04762,1,"{419}"
234,1,10,0,0.0047369,"\int \frac{1}{\sqrt{2-2 x^2} \sqrt{1+x^2}} \, dx","Int[1/(Sqrt[2 - 2*x^2]*Sqrt[1 + x^2]),x]","\frac{F\left(\left.\sin ^{-1}(x)\right|-1\right)}{\sqrt{2}}","\frac{F\left(\left.\sin ^{-1}(x)\right|-1\right)}{\sqrt{2}}",1,"EllipticF[ArcSin[x], -1]/Sqrt[2]","A",2,2,21,0.09524,1,"{248, 221}"
235,1,20,0,0.0073399,"\int \frac{1}{\sqrt{2-3 x^2} \sqrt{1+x^2}} \, dx","Int[1/(Sqrt[2 - 3*x^2]*Sqrt[1 + x^2]),x]","\frac{F\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|-\frac{2}{3}\right)}{\sqrt{3}}","\frac{F\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|-\frac{2}{3}\right)}{\sqrt{3}}",1,"EllipticF[ArcSin[Sqrt[3/2]*x], -2/3]/Sqrt[3]","A",1,1,21,0.04762,1,"{419}"
236,1,16,0,0.0066735,"\int \frac{1}{\sqrt{2-4 x^2} \sqrt{1+x^2}} \, dx","Int[1/(Sqrt[2 - 4*x^2]*Sqrt[1 + x^2]),x]","\frac{1}{2} F\left(\sin ^{-1}\left(\sqrt{2} x\right)|-\frac{1}{2}\right)","\frac{1}{2} F\left(\sin ^{-1}\left(\sqrt{2} x\right)|-\frac{1}{2}\right)",1,"EllipticF[ArcSin[Sqrt[2]*x], -1/2]/2","A",1,1,21,0.04762,1,"{419}"
237,1,20,0,0.0091487,"\int \frac{1}{\sqrt{2-5 x^2} \sqrt{1+x^2}} \, dx","Int[1/(Sqrt[2 - 5*x^2]*Sqrt[1 + x^2]),x]","\frac{F\left(\sin ^{-1}\left(\sqrt{\frac{5}{2}} x\right)|-\frac{2}{5}\right)}{\sqrt{5}}","\frac{F\left(\sin ^{-1}\left(\sqrt{\frac{5}{2}} x\right)|-\frac{2}{5}\right)}{\sqrt{5}}",1,"EllipticF[ArcSin[Sqrt[5/2]*x], -2/5]/Sqrt[5]","A",1,1,21,0.04762,1,"{419}"
238,1,32,0,0.0144636,"\int \frac{1}{\sqrt{-1+x^2} \sqrt{2+5 x^2}} \, dx","Int[1/(Sqrt[-1 + x^2]*Sqrt[2 + 5*x^2]),x]","\frac{\sqrt{1-x^2} F\left(\sin ^{-1}(x)|-\frac{5}{2}\right)}{\sqrt{2} \sqrt{x^2-1}}","\frac{\sqrt{1-x^2} F\left(\sin ^{-1}(x)|-\frac{5}{2}\right)}{\sqrt{2} \sqrt{x^2-1}}",1,"(Sqrt[1 - x^2]*EllipticF[ArcSin[x], -5/2])/(Sqrt[2]*Sqrt[-1 + x^2])","A",2,2,21,0.09524,1,"{421, 419}"
239,1,30,0,0.0146774,"\int \frac{1}{\sqrt{-1+x^2} \sqrt{2+4 x^2}} \, dx","Int[1/(Sqrt[-1 + x^2]*Sqrt[2 + 4*x^2]),x]","\frac{\sqrt{1-x^2} F\left(\left.\sin ^{-1}(x)\right|-2\right)}{\sqrt{2} \sqrt{x^2-1}}","\frac{\sqrt{1-x^2} F\left(\left.\sin ^{-1}(x)\right|-2\right)}{\sqrt{2} \sqrt{x^2-1}}",1,"(Sqrt[1 - x^2]*EllipticF[ArcSin[x], -2])/(Sqrt[2]*Sqrt[-1 + x^2])","A",2,2,21,0.09524,1,"{421, 419}"
240,1,32,0,0.0132576,"\int \frac{1}{\sqrt{-1+x^2} \sqrt{2+3 x^2}} \, dx","Int[1/(Sqrt[-1 + x^2]*Sqrt[2 + 3*x^2]),x]","\frac{\sqrt{1-x^2} F\left(\sin ^{-1}(x)|-\frac{3}{2}\right)}{\sqrt{2} \sqrt{x^2-1}}","\frac{\sqrt{1-x^2} F\left(\sin ^{-1}(x)|-\frac{3}{2}\right)}{\sqrt{2} \sqrt{x^2-1}}",1,"(Sqrt[1 - x^2]*EllipticF[ArcSin[x], -3/2])/(Sqrt[2]*Sqrt[-1 + x^2])","A",2,2,21,0.09524,1,"{421, 419}"
241,1,25,0,0.0109653,"\int \frac{1}{\sqrt{-1+x^2} \sqrt{2+2 x^2}} \, dx","Int[1/(Sqrt[-1 + x^2]*Sqrt[2 + 2*x^2]),x]","\frac{1}{2} F\left(\sin ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right)|\frac{1}{2}\right)","\frac{1}{2} F\left(\sin ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right)|\frac{1}{2}\right)",1,"EllipticF[ArcSin[(Sqrt[2]*x)/Sqrt[-1 + x^2]], 1/2]/2","A",2,2,21,0.09524,1,"{253, 222}"
242,1,32,0,0.0131919,"\int \frac{1}{\sqrt{-1+x^2} \sqrt{2+x^2}} \, dx","Int[1/(Sqrt[-1 + x^2]*Sqrt[2 + x^2]),x]","\frac{\sqrt{1-x^2} F\left(\sin ^{-1}(x)|-\frac{1}{2}\right)}{\sqrt{2} \sqrt{x^2-1}}","\frac{\sqrt{1-x^2} F\left(\sin ^{-1}(x)|-\frac{1}{2}\right)}{\sqrt{2} \sqrt{x^2-1}}",1,"(Sqrt[1 - x^2]*EllipticF[ArcSin[x], -1/2])/(Sqrt[2]*Sqrt[-1 + x^2])","A",2,2,19,0.1053,1,"{421, 419}"
243,1,12,0,0.0065863,"\int \frac{1}{\sqrt{2-x^2} \sqrt{-1+x^2}} \, dx","Int[1/(Sqrt[2 - x^2]*Sqrt[-1 + x^2]),x]","-F\left(\left.\cos ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)","-F\left(\left.\cos ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)",1,"-EllipticF[ArcCos[x/Sqrt[2]], 2]","A",1,1,21,0.04762,1,"{420}"
244,1,29,0,0.0033172,"\int \frac{1}{\sqrt{2-2 x^2} \sqrt{-1+x^2}} \, dx","Int[1/(Sqrt[2 - 2*x^2]*Sqrt[-1 + x^2]),x]","-\frac{\sqrt{x^2-1} \tanh ^{-1}(x)}{\sqrt{2} \sqrt{1-x^2}}","-\frac{\sqrt{x^2-1} \tanh ^{-1}(x)}{\sqrt{2} \sqrt{1-x^2}}",1,"-((Sqrt[-1 + x^2]*ArcTanh[x])/(Sqrt[2]*Sqrt[1 - x^2]))","A",2,2,21,0.09524,1,"{23, 207}"
245,1,32,0,0.0147927,"\int \frac{1}{\sqrt{2-3 x^2} \sqrt{-1+x^2}} \, dx","Int[1/(Sqrt[2 - 3*x^2]*Sqrt[-1 + x^2]),x]","\frac{\sqrt{1-x^2} F\left(\sin ^{-1}(x)|\frac{3}{2}\right)}{\sqrt{2} \sqrt{x^2-1}}","\frac{\sqrt{1-x^2} F\left(\sin ^{-1}(x)|\frac{3}{2}\right)}{\sqrt{2} \sqrt{x^2-1}}",1,"(Sqrt[1 - x^2]*EllipticF[ArcSin[x], 3/2])/(Sqrt[2]*Sqrt[-1 + x^2])","A",2,2,21,0.09524,1,"{421, 419}"
246,1,30,0,0.0150831,"\int \frac{1}{\sqrt{2-4 x^2} \sqrt{-1+x^2}} \, dx","Int[1/(Sqrt[2 - 4*x^2]*Sqrt[-1 + x^2]),x]","\frac{\sqrt{1-x^2} F\left(\left.\sin ^{-1}(x)\right|2\right)}{\sqrt{2} \sqrt{x^2-1}}","\frac{\sqrt{1-x^2} F\left(\left.\sin ^{-1}(x)\right|2\right)}{\sqrt{2} \sqrt{x^2-1}}",1,"(Sqrt[1 - x^2]*EllipticF[ArcSin[x], 2])/(Sqrt[2]*Sqrt[-1 + x^2])","A",2,2,21,0.09524,1,"{421, 419}"
247,1,32,0,0.0141692,"\int \frac{1}{\sqrt{2-5 x^2} \sqrt{-1+x^2}} \, dx","Int[1/(Sqrt[2 - 5*x^2]*Sqrt[-1 + x^2]),x]","\frac{\sqrt{1-x^2} F\left(\sin ^{-1}(x)|\frac{5}{2}\right)}{\sqrt{2} \sqrt{x^2-1}}","\frac{\sqrt{1-x^2} F\left(\sin ^{-1}(x)|\frac{5}{2}\right)}{\sqrt{2} \sqrt{x^2-1}}",1,"(Sqrt[1 - x^2]*EllipticF[ArcSin[x], 5/2])/(Sqrt[2]*Sqrt[-1 + x^2])","A",2,2,21,0.09524,1,"{421, 419}"
248,1,53,0,0.0104685,"\int \frac{1}{\sqrt{-1-x^2} \sqrt{2+5 x^2}} \, dx","Int[1/(Sqrt[-1 - x^2]*Sqrt[2 + 5*x^2]),x]","\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}}}","\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,"(Sqrt[2 + 5*x^2]*EllipticF[ArcTan[x], -3/2])/(Sqrt[2]*Sqrt[-1 - x^2]*Sqrt[(2 + 5*x^2)/(1 + x^2)])","A",1,1,23,0.04348,1,"{418}"
249,1,51,0,0.0101914,"\int \frac{1}{\sqrt{-1-x^2} \sqrt{2+4 x^2}} \, dx","Int[1/(Sqrt[-1 - x^2]*Sqrt[2 + 4*x^2]),x]","\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}}}","\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,"(Sqrt[1 + 2*x^2]*EllipticF[ArcTan[x], -1])/(Sqrt[2]*Sqrt[-1 - x^2]*Sqrt[(1 + 2*x^2)/(1 + x^2)])","A",1,1,23,0.04348,1,"{418}"
250,1,53,0,0.0094579,"\int \frac{1}{\sqrt{-1-x^2} \sqrt{2+3 x^2}} \, dx","Int[1/(Sqrt[-1 - x^2]*Sqrt[2 + 3*x^2]),x]","\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{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,"(Sqrt[2 + 3*x^2]*EllipticF[ArcTan[x], -1/2])/(Sqrt[2]*Sqrt[-1 - x^2]*Sqrt[(2 + 3*x^2)/(1 + x^2)])","A",1,1,23,0.04348,1,"{418}"
251,1,28,0,0.0035139,"\int \frac{1}{\sqrt{-1-x^2} \sqrt{2+2 x^2}} \, dx","Int[1/(Sqrt[-1 - x^2]*Sqrt[2 + 2*x^2]),x]","\frac{\sqrt{x^2+1} \tan ^{-1}(x)}{\sqrt{2} \sqrt{-x^2-1}}","\frac{\sqrt{x^2+1} \tan ^{-1}(x)}{\sqrt{2} \sqrt{-x^2-1}}",1,"(Sqrt[1 + x^2]*ArcTan[x])/(Sqrt[2]*Sqrt[-1 - x^2])","A",2,2,23,0.08696,1,"{23, 203}"
252,1,49,0,0.0087737,"\int \frac{1}{\sqrt{-1-x^2} \sqrt{2+x^2}} \, dx","Int[1/(Sqrt[-1 - x^2]*Sqrt[2 + x^2]),x]","\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}}}","\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,"(Sqrt[2 + x^2]*EllipticF[ArcTan[x], 1/2])/(Sqrt[2]*Sqrt[-1 - x^2]*Sqrt[(2 + x^2)/(1 + x^2)])","A",1,1,21,0.04762,1,"{418}"
253,1,31,0,0.0156464,"\int \frac{1}{\sqrt{-1-x^2} \sqrt{2-x^2}} \, dx","Int[1/(Sqrt[-1 - x^2]*Sqrt[2 - x^2]),x]","\frac{\sqrt{x^2+1} F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)}{\sqrt{-x^2-1}}","\frac{\sqrt{x^2+1} F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)}{\sqrt{-x^2-1}}",1,"(Sqrt[1 + x^2]*EllipticF[ArcSin[x/Sqrt[2]], -2])/Sqrt[-1 - x^2]","A",2,2,23,0.08696,1,"{421, 419}"
254,1,65,0,0.0122466,"\int \frac{1}{\sqrt{2-2 x^2} \sqrt{-1-x^2}} \, dx","Int[1/(Sqrt[2 - 2*x^2]*Sqrt[-1 - x^2]),x]","\frac{\sqrt{x^2-1} \sqrt{x^2+1} F\left(\sin ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right)|\frac{1}{2}\right)}{2 \sqrt{-x^2-1} \sqrt{1-x^2}}","-\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,"(Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[(Sqrt[2]*x)/Sqrt[-1 + x^2]], 1/2])/(2*Sqrt[-1 - x^2]*Sqrt[1 - x^2])","A",2,2,23,0.08696,1,"{253, 222}"
255,1,40,0,0.0156102,"\int \frac{1}{\sqrt{2-3 x^2} \sqrt{-1-x^2}} \, dx","Int[1/(Sqrt[2 - 3*x^2]*Sqrt[-1 - x^2]),x]","\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}}","\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,"(Sqrt[1 + x^2]*EllipticF[ArcSin[Sqrt[3/2]*x], -2/3])/(Sqrt[3]*Sqrt[-1 - x^2])","A",2,2,23,0.08696,1,"{421, 419}"
256,1,36,0,0.0152354,"\int \frac{1}{\sqrt{2-4 x^2} \sqrt{-1-x^2}} \, dx","Int[1/(Sqrt[2 - 4*x^2]*Sqrt[-1 - x^2]),x]","\frac{\sqrt{x^2+1} F\left(\sin ^{-1}\left(\sqrt{2} x\right)|-\frac{1}{2}\right)}{2 \sqrt{-x^2-1}}","\frac{\sqrt{x^2+1} F\left(\sin ^{-1}\left(\sqrt{2} x\right)|-\frac{1}{2}\right)}{2 \sqrt{-x^2-1}}",1,"(Sqrt[1 + x^2]*EllipticF[ArcSin[Sqrt[2]*x], -1/2])/(2*Sqrt[-1 - x^2])","A",2,2,23,0.08696,1,"{421, 419}"
257,1,40,0,0.0150192,"\int \frac{1}{\sqrt{2-5 x^2} \sqrt{-1-x^2}} \, dx","Int[1/(Sqrt[2 - 5*x^2]*Sqrt[-1 - x^2]),x]","\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}}","\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,"(Sqrt[1 + x^2]*EllipticF[ArcSin[Sqrt[5/2]*x], -2/5])/(Sqrt[5]*Sqrt[-1 - x^2])","A",2,2,23,0.08696,1,"{421, 419}"
258,1,87,0,0.0534156,"\int \frac{\sqrt{a+b x^2}}{\sqrt{c-d x^2}} \, dx","Int[Sqrt[a + b*x^2]/Sqrt[c - d*x^2],x]","\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}}","\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,"(Sqrt[c]*Sqrt[a + b*x^2]*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(Sqrt[d]*Sqrt[1 + (b*x^2)/a]*Sqrt[c - d*x^2])","A",3,3,24,0.1250,1,"{427, 426, 424}"
259,1,90,0,0.0524295,"\int \frac{\sqrt{-a-b x^2}}{\sqrt{c-d x^2}} \, dx","Int[Sqrt[-a - b*x^2]/Sqrt[c - d*x^2],x]","\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}}","\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,"(Sqrt[c]*Sqrt[-a - b*x^2]*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(Sqrt[d]*Sqrt[1 + (b*x^2)/a]*Sqrt[c - d*x^2])","A",3,3,27,0.1111,1,"{427, 426, 424}"
260,1,88,0,0.0493846,"\int \frac{\sqrt{a+b x^2}}{\sqrt{-c+d x^2}} \, dx","Int[Sqrt[a + b*x^2]/Sqrt[-c + d*x^2],x]","\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}}","\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,"(Sqrt[c]*Sqrt[a + b*x^2]*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(Sqrt[d]*Sqrt[1 + (b*x^2)/a]*Sqrt[-c + d*x^2])","A",3,3,25,0.1200,1,"{427, 426, 424}"
261,1,91,0,0.0538916,"\int \frac{\sqrt{-a-b x^2}}{\sqrt{-c+d x^2}} \, dx","Int[Sqrt[-a - b*x^2]/Sqrt[-c + d*x^2],x]","\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}}","\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,"(Sqrt[c]*Sqrt[-a - b*x^2]*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(Sqrt[d]*Sqrt[1 + (b*x^2)/a]*Sqrt[-c + d*x^2])","A",3,3,28,0.1071,1,"{427, 426, 424}"
262,1,88,0,0.0521836,"\int \frac{\sqrt{a-b x^2}}{\sqrt{c-d x^2}} \, dx","Int[Sqrt[a - b*x^2]/Sqrt[c - d*x^2],x]","\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}}","\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,"(Sqrt[c]*Sqrt[a - b*x^2]*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], (b*c)/(a*d)])/(Sqrt[d]*Sqrt[1 - (b*x^2)/a]*Sqrt[c - d*x^2])","A",3,3,25,0.1200,1,"{427, 426, 424}"
263,1,89,0,0.0528436,"\int \frac{\sqrt{-a+b x^2}}{\sqrt{c-d x^2}} \, dx","Int[Sqrt[-a + b*x^2]/Sqrt[c - d*x^2],x]","\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}}","\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,"(Sqrt[c]*Sqrt[-a + b*x^2]*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], (b*c)/(a*d)])/(Sqrt[d]*Sqrt[1 - (b*x^2)/a]*Sqrt[c - d*x^2])","A",3,3,26,0.1154,1,"{427, 426, 424}"
264,1,89,0,0.054621,"\int \frac{\sqrt{a-b x^2}}{\sqrt{-c+d x^2}} \, dx","Int[Sqrt[a - b*x^2]/Sqrt[-c + d*x^2],x]","\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}}","\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,"(Sqrt[c]*Sqrt[a - b*x^2]*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], (b*c)/(a*d)])/(Sqrt[d]*Sqrt[1 - (b*x^2)/a]*Sqrt[-c + d*x^2])","A",3,3,26,0.1154,1,"{427, 426, 424}"
265,1,90,0,0.0519834,"\int \frac{\sqrt{-a+b x^2}}{\sqrt{-c+d x^2}} \, dx","Int[Sqrt[-a + b*x^2]/Sqrt[-c + d*x^2],x]","\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}}","\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,"(Sqrt[c]*Sqrt[-a + b*x^2]*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], (b*c)/(a*d)])/(Sqrt[d]*Sqrt[1 - (b*x^2)/a]*Sqrt[-c + d*x^2])","A",3,3,27,0.1111,1,"{427, 426, 424}"
266,1,194,0,0.0844847,"\int \frac{\sqrt{a+b x^2}}{\sqrt{c+d x^2}} \, dx","Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2],x]","\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)}}}","\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,"(x*Sqrt[a + b*x^2])/Sqrt[c + d*x^2] - (Sqrt[c]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (Sqrt[c]*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])","A",4,4,23,0.1739,1,"{422, 418, 492, 411}"
267,1,203,0,0.0886766,"\int \frac{\sqrt{-a-b x^2}}{\sqrt{c+d x^2}} \, dx","Int[Sqrt[-a - b*x^2]/Sqrt[c + d*x^2],x]","\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)}}}","\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,"(x*Sqrt[-a - b*x^2])/Sqrt[c + d*x^2] - (Sqrt[c]*Sqrt[-a - b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (Sqrt[c]*Sqrt[-a - b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])","A",4,4,26,0.1538,1,"{422, 418, 492, 411}"
268,1,203,0,0.0918711,"\int \frac{\sqrt{a+b x^2}}{\sqrt{-c-d x^2}} \, dx","Int[Sqrt[a + b*x^2]/Sqrt[-c - d*x^2],x]","\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)}}}","\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,"(x*Sqrt[a + b*x^2])/Sqrt[-c - d*x^2] - (Sqrt[c]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*Sqrt[-c - d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]) + (Sqrt[c]*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*Sqrt[-c - d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))])","A",4,4,26,0.1538,1,"{422, 418, 492, 411}"
269,1,212,0,0.1033423,"\int \frac{\sqrt{-a-b x^2}}{\sqrt{-c-d x^2}} \, dx","Int[Sqrt[-a - b*x^2]/Sqrt[-c - d*x^2],x]","\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)}}}","\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,"(x*Sqrt[-a - b*x^2])/Sqrt[-c - d*x^2] - (Sqrt[c]*Sqrt[-a - b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*Sqrt[-c - d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]) + (Sqrt[c]*Sqrt[-a - b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*Sqrt[-c - d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))])","A",4,4,29,0.1379,1,"{422, 418, 492, 411}"
270,1,189,0,0.1249484,"\int \frac{\sqrt{a-b x^2}}{\sqrt{c+d x^2}} \, dx","Int[Sqrt[a - b*x^2]/Sqrt[c + d*x^2],x]","\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}}","\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,"-((Sqrt[a]*Sqrt[b]*Sqrt[1 - (b*x^2)/a]*Sqrt[c + d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(d*Sqrt[a - b*x^2]*Sqrt[1 + (d*x^2)/c])) + (Sqrt[a]*(b*c + a*d)*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[b]*d*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])","A",7,6,24,0.2500,1,"{423, 427, 426, 424, 421, 419}"
271,1,191,0,0.1229703,"\int \frac{\sqrt{-a+b x^2}}{\sqrt{c+d x^2}} \, dx","Int[Sqrt[-a + b*x^2]/Sqrt[c + d*x^2],x]","\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}}","\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,"(Sqrt[a]*Sqrt[b]*Sqrt[1 - (b*x^2)/a]*Sqrt[c + d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(d*Sqrt[-a + b*x^2]*Sqrt[1 + (d*x^2)/c]) - (Sqrt[a]*(b*c + a*d)*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[b]*d*Sqrt[-a + b*x^2]*Sqrt[c + d*x^2])","A",7,6,25,0.2400,1,"{423, 427, 426, 424, 421, 419}"
272,1,194,0,0.128482,"\int \frac{\sqrt{a-b x^2}}{\sqrt{-c-d x^2}} \, dx","Int[Sqrt[a - b*x^2]/Sqrt[-c - d*x^2],x]","\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}}","\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,"(Sqrt[a]*Sqrt[b]*Sqrt[1 - (b*x^2)/a]*Sqrt[-c - d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(d*Sqrt[a - b*x^2]*Sqrt[1 + (d*x^2)/c]) + (Sqrt[a]*(b*c + a*d)*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[b]*d*Sqrt[a - b*x^2]*Sqrt[-c - d*x^2])","A",7,6,27,0.2222,1,"{423, 427, 426, 424, 421, 419}"
273,1,198,0,0.1309313,"\int \frac{\sqrt{-a+b x^2}}{\sqrt{-c-d x^2}} \, dx","Int[Sqrt[-a + b*x^2]/Sqrt[-c - d*x^2],x]","-\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}}","-\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,"-((Sqrt[a]*Sqrt[b]*Sqrt[1 - (b*x^2)/a]*Sqrt[-c - d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(d*Sqrt[-a + b*x^2]*Sqrt[1 + (d*x^2)/c])) - (Sqrt[a]*(b*c + a*d)*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[b]*d*Sqrt[-a + b*x^2]*Sqrt[-c - d*x^2])","A",7,6,28,0.2143,1,"{423, 427, 426, 424, 421, 419}"
274,1,87,0,0.0478082,"\int \frac{\sqrt{c+d x^2}}{\sqrt{a-b x^2}} \, dx","Int[Sqrt[c + d*x^2]/Sqrt[a - b*x^2],x]","\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}}","\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,"(Sqrt[a]*Sqrt[1 - (b*x^2)/a]*Sqrt[c + d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[b]*Sqrt[a - b*x^2]*Sqrt[1 + (d*x^2)/c])","A",3,3,24,0.1250,1,"{427, 426, 424}"
275,1,90,0,0.0517018,"\int \frac{\sqrt{-c-d x^2}}{\sqrt{a-b x^2}} \, dx","Int[Sqrt[-c - d*x^2]/Sqrt[a - b*x^2],x]","\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}}","\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,"(Sqrt[a]*Sqrt[1 - (b*x^2)/a]*Sqrt[-c - d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[b]*Sqrt[a - b*x^2]*Sqrt[1 + (d*x^2)/c])","A",3,3,27,0.1111,1,"{427, 426, 424}"
276,1,88,0,0.0467954,"\int \frac{\sqrt{c+d x^2}}{\sqrt{-a+b x^2}} \, dx","Int[Sqrt[c + d*x^2]/Sqrt[-a + b*x^2],x]","\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}}","\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,"(Sqrt[a]*Sqrt[1 - (b*x^2)/a]*Sqrt[c + d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[b]*Sqrt[-a + b*x^2]*Sqrt[1 + (d*x^2)/c])","A",3,3,25,0.1200,1,"{427, 426, 424}"
277,1,91,0,0.0500475,"\int \frac{\sqrt{-c-d x^2}}{\sqrt{-a+b x^2}} \, dx","Int[Sqrt[-c - d*x^2]/Sqrt[-a + b*x^2],x]","\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}}","\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,"(Sqrt[a]*Sqrt[1 - (b*x^2)/a]*Sqrt[-c - d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[b]*Sqrt[-a + b*x^2]*Sqrt[1 + (d*x^2)/c])","A",3,3,28,0.1071,1,"{427, 426, 424}"
278,1,88,0,0.0517505,"\int \frac{\sqrt{c-d x^2}}{\sqrt{a-b x^2}} \, dx","Int[Sqrt[c - d*x^2]/Sqrt[a - b*x^2],x]","\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}}}","\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,"(Sqrt[a]*Sqrt[1 - (b*x^2)/a]*Sqrt[c - d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], (a*d)/(b*c)])/(Sqrt[b]*Sqrt[a - b*x^2]*Sqrt[1 - (d*x^2)/c])","A",3,3,25,0.1200,1,"{427, 426, 424}"
279,1,89,0,0.0506423,"\int \frac{\sqrt{-c+d x^2}}{\sqrt{a-b x^2}} \, dx","Int[Sqrt[-c + d*x^2]/Sqrt[a - b*x^2],x]","\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}}}","\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,"(Sqrt[a]*Sqrt[1 - (b*x^2)/a]*Sqrt[-c + d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], (a*d)/(b*c)])/(Sqrt[b]*Sqrt[a - b*x^2]*Sqrt[1 - (d*x^2)/c])","A",3,3,26,0.1154,1,"{427, 426, 424}"
280,1,89,0,0.0507299,"\int \frac{\sqrt{c-d x^2}}{\sqrt{-a+b x^2}} \, dx","Int[Sqrt[c - d*x^2]/Sqrt[-a + b*x^2],x]","\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}}}","\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,"(Sqrt[a]*Sqrt[1 - (b*x^2)/a]*Sqrt[c - d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], (a*d)/(b*c)])/(Sqrt[b]*Sqrt[-a + b*x^2]*Sqrt[1 - (d*x^2)/c])","A",3,3,26,0.1154,1,"{427, 426, 424}"
281,1,90,0,0.0519902,"\int \frac{\sqrt{-c+d x^2}}{\sqrt{-a+b x^2}} \, dx","Int[Sqrt[-c + d*x^2]/Sqrt[-a + b*x^2],x]","\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}}}","\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,"(Sqrt[a]*Sqrt[1 - (b*x^2)/a]*Sqrt[-c + d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], (a*d)/(b*c)])/(Sqrt[b]*Sqrt[-a + b*x^2]*Sqrt[1 - (d*x^2)/c])","A",3,3,27,0.1111,1,"{427, 426, 424}"
282,1,204,0,0.0881219,"\int \frac{\sqrt{c+d x^2}}{\sqrt{a+b x^2}} \, dx","Int[Sqrt[c + d*x^2]/Sqrt[a + b*x^2],x]","\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)}}}","\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,"(d*x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[d]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (c^(3/2)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])","A",4,4,23,0.1739,1,"{422, 418, 492, 411}"
283,1,214,0,0.0955676,"\int \frac{\sqrt{-c-d x^2}}{\sqrt{a+b x^2}} \, dx","Int[Sqrt[-c - d*x^2]/Sqrt[a + b*x^2],x]","-\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)}}}","-\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,"-((d*x*Sqrt[a + b*x^2])/(b*Sqrt[-c - d*x^2])) + (Sqrt[c]*Sqrt[d]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[-c - d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]) - (c^(3/2)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*Sqrt[-c - d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))])","A",4,4,26,0.1538,1,"{422, 418, 492, 411}"
284,1,214,0,0.0968212,"\int \frac{\sqrt{c+d x^2}}{\sqrt{-a-b x^2}} \, dx","Int[Sqrt[c + d*x^2]/Sqrt[-a - b*x^2],x]","-\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)}}}","-\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,"-((d*x*Sqrt[-a - b*x^2])/(b*Sqrt[c + d*x^2])) + (Sqrt[c]*Sqrt[d]*Sqrt[-a - b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (c^(3/2)*Sqrt[-a - b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])","A",4,4,26,0.1538,1,"{422, 418, 492, 411}"
285,1,222,0,0.1026502,"\int \frac{\sqrt{-c-d x^2}}{\sqrt{-a-b x^2}} \, dx","Int[Sqrt[-c - d*x^2]/Sqrt[-a - b*x^2],x]","\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)}}}","\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,"(d*x*Sqrt[-a - b*x^2])/(b*Sqrt[-c - d*x^2]) - (Sqrt[c]*Sqrt[d]*Sqrt[-a - b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[-c - d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]) + (c^(3/2)*Sqrt[-a - b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*Sqrt[-c - d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))])","A",4,4,29,0.1379,1,"{422, 418, 492, 411}"
286,1,189,0,0.1206461,"\int \frac{\sqrt{c-d x^2}}{\sqrt{a+b x^2}} \, dx","Int[Sqrt[c - d*x^2]/Sqrt[a + b*x^2],x]","\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}}","\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,"-((Sqrt[c]*Sqrt[d]*Sqrt[a + b*x^2]*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(b*Sqrt[1 + (b*x^2)/a]*Sqrt[c - d*x^2])) + (Sqrt[c]*(b*c + a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(b*Sqrt[d]*Sqrt[a + b*x^2]*Sqrt[c - d*x^2])","A",7,6,24,0.2500,1,"{423, 427, 426, 424, 421, 419}"
287,1,191,0,0.1215492,"\int \frac{\sqrt{-c+d x^2}}{\sqrt{a+b x^2}} \, dx","Int[Sqrt[-c + d*x^2]/Sqrt[a + b*x^2],x]","\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}}","\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,"(Sqrt[c]*Sqrt[d]*Sqrt[a + b*x^2]*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(b*Sqrt[1 + (b*x^2)/a]*Sqrt[-c + d*x^2]) - (Sqrt[c]*(b*c + a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(b*Sqrt[d]*Sqrt[a + b*x^2]*Sqrt[-c + d*x^2])","A",7,6,25,0.2400,1,"{423, 427, 426, 424, 421, 419}"
288,1,194,0,0.1253587,"\int \frac{\sqrt{c-d x^2}}{\sqrt{-a-b x^2}} \, dx","Int[Sqrt[c - d*x^2]/Sqrt[-a - b*x^2],x]","\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}}","\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,"(Sqrt[c]*Sqrt[d]*Sqrt[-a - b*x^2]*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(b*Sqrt[1 + (b*x^2)/a]*Sqrt[c - d*x^2]) + (Sqrt[c]*(b*c + a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(b*Sqrt[d]*Sqrt[-a - b*x^2]*Sqrt[c - d*x^2])","A",7,6,27,0.2222,1,"{423, 427, 426, 424, 421, 419}"
289,1,198,0,0.1294123,"\int \frac{\sqrt{-c+d x^2}}{\sqrt{-a-b x^2}} \, dx","Int[Sqrt[-c + d*x^2]/Sqrt[-a - b*x^2],x]","-\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}}","-\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,"-((Sqrt[c]*Sqrt[d]*Sqrt[-a - b*x^2]*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(b*Sqrt[1 + (b*x^2)/a]*Sqrt[-c + d*x^2])) - (Sqrt[c]*(b*c + a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(b*Sqrt[d]*Sqrt[-a - b*x^2]*Sqrt[-c + d*x^2])","A",7,6,28,0.2143,1,"{423, 427, 426, 424, 421, 419}"
290,1,78,0,0.0137813,"\int \frac{1}{\sqrt{2+b x^2} \sqrt{3+d x^2}} \, dx","Int[1/(Sqrt[2 + b*x^2]*Sqrt[3 + d*x^2]),x]","\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}}}","\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,"(Sqrt[2 + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[3]], 1 - (3*b)/(2*d)])/(Sqrt[2]*Sqrt[d]*Sqrt[(2 + b*x^2)/(3 + d*x^2)]*Sqrt[3 + d*x^2])","A",1,1,23,0.04348,1,"{418}"
291,1,39,0,0.0200224,"\int \frac{1}{\sqrt{4-x^2} \sqrt{c+d x^2}} \, dx","Int[1/(Sqrt[4 - x^2]*Sqrt[c + d*x^2]),x]","\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}}","\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,"(Sqrt[1 + (d*x^2)/c]*EllipticF[ArcSin[x/2], (-4*d)/c])/Sqrt[c + d*x^2]","A",2,2,23,0.08696,1,"{421, 419}"
292,1,61,0,0.0124724,"\int \frac{1}{\sqrt{4+x^2} \sqrt{c+d x^2}} \, dx","Int[1/(Sqrt[4 + x^2]*Sqrt[c + d*x^2]),x]","\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)}}}","\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,"(Sqrt[c + d*x^2]*EllipticF[ArcTan[x/2], 1 - (4*d)/c])/(c*Sqrt[4 + x^2]*Sqrt[(c + d*x^2)/(c*(4 + x^2))])","A",1,1,21,0.04762,1,"{418}"
293,1,6,0,0.0060251,"\int \frac{1}{\sqrt{1-x^2} \sqrt{-1+2 x^2}} \, dx","Int[1/(Sqrt[1 - x^2]*Sqrt[-1 + 2*x^2]),x]","-F\left(\left.\cos ^{-1}(x)\right|2\right)","-F\left(\left.\cos ^{-1}(x)\right|2\right)",1,"-EllipticF[ArcCos[x], 2]","A",1,1,23,0.04348,1,"{420}"
294,1,23,0,0.02485,"\int \frac{\sqrt{1-c^2 x^2}}{\sqrt{1+c^2 x^2}} \, dx","Int[Sqrt[1 - c^2*x^2]/Sqrt[1 + c^2*x^2],x]","\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}","\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,"-(EllipticE[ArcSin[c*x], -1]/c) + (2*EllipticF[ArcSin[c*x], -1])/c","A",4,4,28,0.1429,1,"{423, 424, 248, 221}"
295,1,182,0,0.0752468,"\int \frac{\sqrt{2+b x^2}}{\sqrt{3+d x^2}} \, dx","Int[Sqrt[2 + b*x^2]/Sqrt[3 + d*x^2],x]","\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}}}","\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,"(x*Sqrt[2 + b*x^2])/Sqrt[3 + d*x^2] - (Sqrt[2]*Sqrt[2 + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[3]], 1 - (3*b)/(2*d)])/(Sqrt[d]*Sqrt[(2 + b*x^2)/(3 + d*x^2)]*Sqrt[3 + d*x^2]) + (Sqrt[2]*Sqrt[2 + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[3]], 1 - (3*b)/(2*d)])/(Sqrt[d]*Sqrt[(2 + b*x^2)/(3 + d*x^2)]*Sqrt[3 + d*x^2])","A",4,4,23,0.1739,1,"{422, 418, 492, 411}"
296,1,19,0,0.0065772,"\int \frac{\sqrt{-1+3 x^2}}{\sqrt{2-3 x^2}} \, dx","Int[Sqrt[-1 + 3*x^2]/Sqrt[2 - 3*x^2],x]","-\frac{E\left(\left.\cos ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)}{\sqrt{3}}","-\frac{E\left(\left.\cos ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)}{\sqrt{3}}",1,"-(EllipticE[ArcCos[Sqrt[3/2]*x], 2]/Sqrt[3])","A",1,1,23,0.04348,1,"{425}"
297,1,95,0,0.1322172,"\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","Int[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]","\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}}","\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,"(Sqrt[b + Sqrt[b^2 - 4*a*c]]*EllipticE[ArcSin[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]], -((b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c]))])/(Sqrt[2]*Sqrt[c])","A",1,1,59,0.01695,1,"{424}"
298,1,94,0,0.1019948,"\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","Int[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]","\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}}","\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,"(Sqrt[b + Sqrt[b^2 - 4*a*c]]*EllipticE[ArcSin[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/(Sqrt[2]*Sqrt[c])","A",1,1,59,0.01695,1,"{424}"
299,1,478,0,0.3667398,"\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","Int[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]","\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}}","\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,"(x*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])] - (Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticE[ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]], (-2*Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/(Sqrt[2]*Sqrt[c]*Sqrt[(1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]))/(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])]) + (Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticF[ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]], (-2*Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/(Sqrt[2]*Sqrt[c]*Sqrt[(1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]))/(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])])","A",4,4,59,0.06780,1,"{422, 418, 492, 411}"
300,1,215,0,0.2076678,"\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","Int[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]","\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}}}","\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,"-(((b + Sqrt[b^2 - 4*a*c])*EllipticE[ArcSin[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]], -((b - Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c]))])/(Sqrt[2]*Sqrt[c]*Sqrt[b - Sqrt[b^2 - 4*a*c]])) + (Sqrt[2]*b*EllipticF[ArcSin[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]], -((b - Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c]))])/(Sqrt[c]*Sqrt[b - Sqrt[b^2 - 4*a*c]])","A",3,3,59,0.05085,1,"{423, 424, 419}"
301,1,23,0,0.010505,"\int \frac{\left(1-2 x^2\right)^m}{\sqrt{1-x^2}} \, dx","Int[(1 - 2*x^2)^m/Sqrt[1 - x^2],x]","x F_1\left(\frac{1}{2};-m,\frac{1}{2};\frac{3}{2};2 x^2,x^2\right)","-\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,"x*AppellF1[1/2, -m, 1/2, 3/2, 2*x^2, x^2]","C",1,1,21,0.04762,0,"{429}"
302,1,46,0,0.0407638,"\int \frac{1}{\sqrt{-1+x^2} \sqrt{7-4 \sqrt{3}+x^2}} \, dx","Int[1/(Sqrt[-1 + x^2]*Sqrt[7 - 4*Sqrt[3] + x^2]),x]","\frac{\sqrt{1-x^2} F\left(\sin ^{-1}(x)|-7-4 \sqrt{3}\right)}{\sqrt{7-4 \sqrt{3}} \sqrt{x^2-1}}","\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,"(Sqrt[1 - x^2]*EllipticF[ArcSin[x], -7 - 4*Sqrt[3]])/(Sqrt[7 - 4*Sqrt[3]]*Sqrt[-1 + x^2])","A",2,2,26,0.07692,1,"{421, 419}"
303,1,47,0,0.0616624,"\int \frac{1}{\sqrt{3-3 \sqrt{3}+2 \sqrt{3} x^2} \sqrt{3+\left(-3+\sqrt{3}\right) x^2}} \, dx","Int[1/(Sqrt[3 - 3*Sqrt[3] + 2*Sqrt[3]*x^2]*Sqrt[3 + (-3 + Sqrt[3])*x^2]),x]","-\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)","-\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,"-(Sqrt[3 + Sqrt[3]]*EllipticF[ArcCos[Sqrt[(3 - Sqrt[3])/3]*x], (1 + Sqrt[3])/2])/6","A",1,1,41,0.02439,1,"{420}"
304,1,129,0,0.0191751,"\int \frac{1}{\sqrt[4]{2+3 x^2} \left(4+3 x^2\right)} \, dx","Int[1/((2 + 3*x^2)^(1/4)*(4 + 3*x^2)),x]","-\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}}","-\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[(2*2^(3/4) + 2*2^(1/4)*Sqrt[2 + 3*x^2])/(2*Sqrt[3]*x*(2 + 3*x^2)^(1/4))]/(2*2^(3/4)*Sqrt[3]) - ArcTanh[(2*2^(3/4) - 2*2^(1/4)*Sqrt[2 + 3*x^2])/(2*Sqrt[3]*x*(2 + 3*x^2)^(1/4))]/(2*2^(3/4)*Sqrt[3])","A",1,1,21,0.04762,1,"{397}"
305,1,120,0,0.0158787,"\int \frac{1}{\sqrt[4]{2-3 x^2} \left(4-3 x^2\right)} \, dx","Int[1/((2 - 3*x^2)^(1/4)*(4 - 3*x^2)),x]","\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}}","\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[(2 - Sqrt[2]*Sqrt[2 - 3*x^2])/(2^(1/4)*Sqrt[3]*x*(2 - 3*x^2)^(1/4))]/(2*2^(3/4)*Sqrt[3]) + ArcTanh[(2 + Sqrt[2]*Sqrt[2 - 3*x^2])/(2^(1/4)*Sqrt[3]*x*(2 - 3*x^2)^(1/4))]/(2*2^(3/4)*Sqrt[3])","A",1,1,21,0.04762,1,"{397}"
306,1,129,0,0.0228716,"\int \frac{1}{\sqrt[4]{2+b x^2} \left(4+b x^2\right)} \, dx","Int[1/((2 + b*x^2)^(1/4)*(4 + b*x^2)),x]","-\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}}","-\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[(2*2^(3/4) + 2*2^(1/4)*Sqrt[2 + b*x^2])/(2*Sqrt[b]*x*(2 + b*x^2)^(1/4))]/(2*2^(3/4)*Sqrt[b]) - ArcTanh[(2*2^(3/4) - 2*2^(1/4)*Sqrt[2 + b*x^2])/(2*Sqrt[b]*x*(2 + b*x^2)^(1/4))]/(2*2^(3/4)*Sqrt[b])","A",1,1,21,0.04762,1,"{397}"
307,1,124,0,0.0204322,"\int \frac{1}{\sqrt[4]{2-b x^2} \left(4-b x^2\right)} \, dx","Int[1/((2 - b*x^2)^(1/4)*(4 - b*x^2)),x]","\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}}","\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[(2 - Sqrt[2]*Sqrt[2 - b*x^2])/(2^(1/4)*Sqrt[b]*x*(2 - b*x^2)^(1/4))]/(2*2^(3/4)*Sqrt[b]) + ArcTanh[(2 + Sqrt[2]*Sqrt[2 - b*x^2])/(2^(1/4)*Sqrt[b]*x*(2 - b*x^2)^(1/4))]/(2*2^(3/4)*Sqrt[b])","A",1,1,23,0.04348,1,"{397}"
308,1,120,0,0.0198623,"\int \frac{1}{\sqrt[4]{a+3 x^2} \left(2 a+3 x^2\right)} \, dx","Int[1/((a + 3*x^2)^(1/4)*(2*a + 3*x^2)),x]","-\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}}","-\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,"-ArcTan[(a^(3/4)*(1 + Sqrt[a + 3*x^2]/Sqrt[a]))/(Sqrt[3]*x*(a + 3*x^2)^(1/4))]/(2*Sqrt[3]*a^(3/4)) - ArcTanh[(a^(3/4)*(1 - Sqrt[a + 3*x^2]/Sqrt[a]))/(Sqrt[3]*x*(a + 3*x^2)^(1/4))]/(2*Sqrt[3]*a^(3/4))","A",1,1,23,0.04348,1,"{397}"
309,1,120,0,0.0171325,"\int \frac{1}{\sqrt[4]{a-3 x^2} \left(2 a-3 x^2\right)} \, dx","Int[1/((a - 3*x^2)^(1/4)*(2*a - 3*x^2)),x]","\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}}","\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,"ArcTan[(a^(3/4)*(1 - Sqrt[a - 3*x^2]/Sqrt[a]))/(Sqrt[3]*x*(a - 3*x^2)^(1/4))]/(2*Sqrt[3]*a^(3/4)) + ArcTanh[(a^(3/4)*(1 + Sqrt[a - 3*x^2]/Sqrt[a]))/(Sqrt[3]*x*(a - 3*x^2)^(1/4))]/(2*Sqrt[3]*a^(3/4))","A",1,1,23,0.04348,1,"{397}"
310,1,120,0,0.0214657,"\int \frac{1}{\sqrt[4]{a+b x^2} \left(2 a+b x^2\right)} \, dx","Int[1/((a + b*x^2)^(1/4)*(2*a + b*x^2)),x]","-\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}}","-\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,"-ArcTan[(a^(3/4)*(1 + Sqrt[a + b*x^2]/Sqrt[a]))/(Sqrt[b]*x*(a + b*x^2)^(1/4))]/(2*a^(3/4)*Sqrt[b]) - ArcTanh[(a^(3/4)*(1 - Sqrt[a + b*x^2]/Sqrt[a]))/(Sqrt[b]*x*(a + b*x^2)^(1/4))]/(2*a^(3/4)*Sqrt[b])","A",1,1,23,0.04348,1,"{397}"
311,1,124,0,0.020752,"\int \frac{1}{\sqrt[4]{a-b x^2} \left(2 a-b x^2\right)} \, dx","Int[1/((a - b*x^2)^(1/4)*(2*a - b*x^2)),x]","\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}}","\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,"ArcTan[(a^(3/4)*(1 - Sqrt[a - b*x^2]/Sqrt[a]))/(Sqrt[b]*x*(a - b*x^2)^(1/4))]/(2*a^(3/4)*Sqrt[b]) + ArcTanh[(a^(3/4)*(1 + Sqrt[a - b*x^2]/Sqrt[a]))/(Sqrt[b]*x*(a - b*x^2)^(1/4))]/(2*a^(3/4)*Sqrt[b])","A",1,1,25,0.04000,1,"{397}"
312,1,61,0,0.0088104,"\int \frac{1}{\left(-2+3 x^2\right) \sqrt[4]{-1+3 x^2}} \, dx","Int[1/((-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{3 x^2-1}}\right)}{2 \sqrt{6}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{3 x^2-1}}\right)}{2 \sqrt{6}}","-\frac{\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[3/2]*x)/(-1 + 3*x^2)^(1/4)]/(2*Sqrt[6]) - ArcTanh[(Sqrt[3/2]*x)/(-1 + 3*x^2)^(1/4)]/(2*Sqrt[6])","A",1,1,21,0.04762,1,"{398}"
313,1,61,0,0.0098623,"\int \frac{1}{\left(-2-3 x^2\right) \sqrt[4]{-1-3 x^2}} \, dx","Int[1/((-2 - 3*x^2)*(-1 - 3*x^2)^(1/4)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{-3 x^2-1}}\right)}{2 \sqrt{6}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{-3 x^2-1}}\right)}{2 \sqrt{6}}","-\frac{\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[3/2]*x)/(-1 - 3*x^2)^(1/4)]/(2*Sqrt[6]) - ArcTanh[(Sqrt[3/2]*x)/(-1 - 3*x^2)^(1/4)]/(2*Sqrt[6])","A",1,1,21,0.04762,1,"{398}"
314,1,77,0,0.013133,"\int \frac{1}{\left(-2+b x^2\right) \sqrt[4]{-1+b x^2}} \, dx","Int[1/((-2 + b*x^2)*(-1 + b*x^2)^(1/4)),x]","-\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}}","-\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[b]*x)/(Sqrt[2]*(-1 + b*x^2)^(1/4))]/(2*Sqrt[2]*Sqrt[b]) - ArcTanh[(Sqrt[b]*x)/(Sqrt[2]*(-1 + b*x^2)^(1/4))]/(2*Sqrt[2]*Sqrt[b])","A",1,1,21,0.04762,1,"{398}"
315,1,79,0,0.0137537,"\int \frac{1}{\left(-2-b x^2\right) \sqrt[4]{-1-b x^2}} \, dx","Int[1/((-2 - b*x^2)*(-1 - b*x^2)^(1/4)),x]","-\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}}","-\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[b]*x)/(Sqrt[2]*(-1 - b*x^2)^(1/4))]/(2*Sqrt[2]*Sqrt[b]) - ArcTanh[(Sqrt[b]*x)/(Sqrt[2]*(-1 - b*x^2)^(1/4))]/(2*Sqrt[2]*Sqrt[b])","A",1,1,23,0.04348,1,"{398}"
316,1,85,0,0.0178728,"\int \frac{1}{\left(-2 a+3 x^2\right) \sqrt[4]{-a+3 x^2}} \, dx","Int[1/((-2*a + 3*x^2)*(-a + 3*x^2)^(1/4)),x]","-\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}}","-\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[3/2]*x)/(a^(1/4)*(-a + 3*x^2)^(1/4))]/(2*Sqrt[6]*a^(3/4)) - ArcTanh[(Sqrt[3/2]*x)/(a^(1/4)*(-a + 3*x^2)^(1/4))]/(2*Sqrt[6]*a^(3/4))","A",1,1,25,0.04000,1,"{398}"
317,1,85,0,0.0139391,"\int \frac{1}{\left(-2 a-3 x^2\right) \sqrt[4]{-a-3 x^2}} \, dx","Int[1/((-2*a - 3*x^2)*(-a - 3*x^2)^(1/4)),x]","-\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}}","-\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[3/2]*x)/(a^(1/4)*(-a - 3*x^2)^(1/4))]/(2*Sqrt[6]*a^(3/4)) - ArcTanh[(Sqrt[3/2]*x)/(a^(1/4)*(-a - 3*x^2)^(1/4))]/(2*Sqrt[6]*a^(3/4))","A",1,1,25,0.04000,1,"{398}"
318,1,101,0,0.0193968,"\int \frac{1}{\left(-2 a+b x^2\right) \sqrt[4]{-a+b x^2}} \, dx","Int[1/((-2*a + b*x^2)*(-a + b*x^2)^(1/4)),x]","-\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}}","-\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[b]*x)/(Sqrt[2]*a^(1/4)*(-a + b*x^2)^(1/4))]/(2*Sqrt[2]*a^(3/4)*Sqrt[b]) - ArcTanh[(Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*(-a + b*x^2)^(1/4))]/(2*Sqrt[2]*a^(3/4)*Sqrt[b])","A",1,1,25,0.04000,1,"{398}"
319,1,103,0,0.0212074,"\int \frac{1}{\left(-2 a-b x^2\right) \sqrt[4]{-a-b x^2}} \, dx","Int[1/((-2*a - b*x^2)*(-a - b*x^2)^(1/4)),x]","-\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}}","-\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[b]*x)/(Sqrt[2]*a^(1/4)*(-a - b*x^2)^(1/4))]/(2*Sqrt[2]*a^(3/4)*Sqrt[b]) - ArcTanh[(Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*(-a - b*x^2)^(1/4))]/(2*Sqrt[2]*a^(3/4)*Sqrt[b])","A",1,1,27,0.03704,1,"{398}"
320,1,53,0,0.0081616,"\int \frac{1}{\left(2-x^2\right) \sqrt[4]{-1+x^2}} \, dx","Int[1/((2 - x^2)*(-1 + x^2)^(1/4)),x]","\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}}","\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,"ArcTan[x/(Sqrt[2]*(-1 + x^2)^(1/4))]/(2*Sqrt[2]) + ArcTanh[x/(Sqrt[2]*(-1 + x^2)^(1/4))]/(2*Sqrt[2])","A",1,1,19,0.05263,1,"{398}"
321,1,362,0,0.2653661,"\int \frac{\left(a+b x^2\right)^{7/4}}{c+d x^2} \, dx","Int[(a + b*x^2)^(7/4)/(c + d*x^2),x]","-\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{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{\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{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}","-\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{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{\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{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,"(6*a*b*x)/(5*d*(a + b*x^2)^(1/4)) - (2*b*(b*c - a*d)*x)/(d^2*(a + b*x^2)^(1/4)) + (2*b*x*(a + b*x^2)^(3/4))/(5*d) - (6*a^(3/2)*Sqrt[b]*(1 + (b*x^2)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(5*d*(a + b*x^2)^(1/4)) + (2*Sqrt[a]*Sqrt[b]*(b*c - a*d)*(1 + (b*x^2)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(d^2*(a + b*x^2)^(1/4)) + (a^(1/4)*(-(b*c) + a*d)^(3/2)*Sqrt[-((b*x^2)/a)]*EllipticPi[-((Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]), ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(d^(5/2)*x) - (a^(1/4)*(-(b*c) + a*d)^(3/2)*Sqrt[-((b*x^2)/a)]*EllipticPi[(Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d], ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(d^(5/2)*x)","A",13,8,21,0.3810,1,"{402, 195, 229, 227, 196, 399, 490, 1218}"
322,1,302,0,0.2066928,"\int \frac{\left(a+b x^2\right)^{5/4}}{c+d x^2} \, dx","Int[(a + b*x^2)^(5/4)/(c + d*x^2),x]","\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}","\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,"(2*b*x*(a + b*x^2)^(1/4))/(3*d) + (2*a^(3/2)*Sqrt[b]*(1 + (b*x^2)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(3*d*(a + b*x^2)^(3/4)) - (2*Sqrt[a]*Sqrt[b]*(b*c - a*d)*(1 + (b*x^2)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(d^2*(a + b*x^2)^(3/4)) + (a^(1/4)*(b*c - a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi[-((Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]), ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(d^2*x) + (a^(1/4)*(b*c - a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi[(Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d], ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(d^2*x)","A",12,8,21,0.3810,1,"{402, 195, 233, 231, 401, 108, 409, 1218}"
323,1,244,0,0.1503411,"\int \frac{\left(a+b x^2\right)^{3/4}}{c+d x^2} \, dx","Int[(a + b*x^2)^(3/4)/(c + d*x^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{\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}}","\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,"(2*b*x)/(d*(a + b*x^2)^(1/4)) - (2*Sqrt[a]*Sqrt[b]*(1 + (b*x^2)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(d*(a + b*x^2)^(1/4)) + (a^(1/4)*Sqrt[-(b*c) + a*d]*Sqrt[-((b*x^2)/a)]*EllipticPi[-((Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]), ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(d^(3/2)*x) - (a^(1/4)*Sqrt[-(b*c) + a*d]*Sqrt[-((b*x^2)/a)]*EllipticPi[(Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d], ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(d^(3/2)*x)","A",8,7,21,0.3333,1,"{402, 229, 227, 196, 399, 490, 1218}"
324,1,199,0,0.1474815,"\int \frac{\sqrt[4]{a+b x^2}}{c+d x^2} \, dx","Int[(a + b*x^2)^(1/4)/(c + d*x^2),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{\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}}","-\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,"(2*Sqrt[a]*Sqrt[b]*(1 + (b*x^2)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(d*(a + b*x^2)^(3/4)) - (a^(1/4)*Sqrt[-((b*x^2)/a)]*EllipticPi[-((Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]), ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(d*x) - (a^(1/4)*Sqrt[-((b*x^2)/a)]*EllipticPi[(Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d], ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(d*x)","A",8,7,21,0.3333,1,"{402, 233, 231, 401, 108, 409, 1218}"
325,1,167,0,0.1082756,"\int \frac{1}{\sqrt[4]{a+b x^2} \left(c+d x^2\right)} \, dx","Int[1/((a + b*x^2)^(1/4)*(c + d*x^2)),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)}{\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}}","\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,"(a^(1/4)*Sqrt[-((b*x^2)/a)]*EllipticPi[-((Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]), ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(Sqrt[d]*Sqrt[-(b*c) + a*d]*x) - (a^(1/4)*Sqrt[-((b*x^2)/a)]*EllipticPi[(Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d], ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(Sqrt[d]*Sqrt[-(b*c) + a*d]*x)","A",4,3,21,0.1429,1,"{399, 490, 1218}"
326,1,152,0,0.1179274,"\int \frac{1}{\left(a+b x^2\right)^{3/4} \left(c+d x^2\right)} \, dx","Int[1/((a + b*x^2)^(3/4)*(c + d*x^2)),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)}{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)}","\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,"(a^(1/4)*Sqrt[-((b*x^2)/a)]*EllipticPi[-((Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]), ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/((b*c - a*d)*x) + (a^(1/4)*Sqrt[-((b*x^2)/a)]*EllipticPi[(Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d], ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/((b*c - a*d)*x)","A",5,4,21,0.1905,1,"{401, 108, 409, 1218}"
327,1,233,0,0.1600153,"\int \frac{1}{\left(a+b x^2\right)^{5/4} \left(c+d x^2\right)} \, dx","Int[1/((a + b*x^2)^(5/4)*(c + d*x^2)),x]","\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}}","\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,"(2*Sqrt[b]*(1 + (b*x^2)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(Sqrt[a]*(b*c - a*d)*(a + b*x^2)^(1/4)) + (a^(1/4)*Sqrt[d]*Sqrt[-((b*x^2)/a)]*EllipticPi[-((Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]), ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/((-(b*c) + a*d)^(3/2)*x) - (a^(1/4)*Sqrt[d]*Sqrt[-((b*x^2)/a)]*EllipticPi[(Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d], ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/((-(b*c) + a*d)^(3/2)*x)","A",7,6,21,0.2857,1,"{403, 197, 196, 399, 490, 1218}"
328,1,254,0,0.1570102,"\int \frac{1}{\left(a+b x^2\right)^{7/4} \left(c+d x^2\right)} \, dx","Int[1/((a + b*x^2)^(7/4)*(c + d*x^2)),x]","\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}","\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,"(2*b*x)/(3*a*(b*c - a*d)*(a + b*x^2)^(3/4)) + (2*Sqrt[b]*(1 + (b*x^2)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(3*Sqrt[a]*(b*c - a*d)*(a + b*x^2)^(3/4)) - (a^(1/4)*d*Sqrt[-((b*x^2)/a)]*EllipticPi[-((Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]), ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/((b*c - a*d)^2*x) - (a^(1/4)*d*Sqrt[-((b*x^2)/a)]*EllipticPi[(Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d], ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/((b*c - a*d)^2*x)","A",9,8,21,0.3810,1,"{403, 199, 233, 231, 401, 108, 409, 1218}"
329,1,274,0,0.3817079,"\int \frac{1}{\left(a+b x^2\right)^{9/4} \left(c+d x^2\right)} \, dx","Int[1/((a + b*x^2)^(9/4)*(c + d*x^2)),x]","\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)}","\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,"(2*b*x)/(5*a*(b*c - a*d)*(a + b*x^2)^(5/4)) + (2*Sqrt[b]*(3*b*c - 8*a*d)*(1 + (b*x^2)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(5*a^(3/2)*(b*c - a*d)^2*(a + b*x^2)^(1/4)) + (a^(1/4)*d^(3/2)*Sqrt[-((b*x^2)/a)]*EllipticPi[-((Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]), ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/((-(b*c) + a*d)^(5/2)*x) - (a^(1/4)*d^(3/2)*Sqrt[-((b*x^2)/a)]*EllipticPi[(Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d], ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/((-(b*c) + a*d)^(5/2)*x)","A",10,9,21,0.4286,1,"{414, 527, 530, 229, 227, 196, 399, 490, 1218}"
330,1,304,0,0.3426442,"\int \frac{1}{\left(a+b x^2\right)^{11/4} \left(c+d x^2\right)} \, dx","Int[1/((a + b*x^2)^(11/4)*(c + d*x^2)),x]","\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{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{\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)}","\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{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{\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,"(2*b*x)/(7*a*(b*c - a*d)*(a + b*x^2)^(7/4)) + (2*b*(5*b*c - 12*a*d)*x)/(21*a^2*(b*c - a*d)^2*(a + b*x^2)^(3/4)) + (2*Sqrt[b]*(5*b*c - 12*a*d)*(1 + (b*x^2)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(21*a^(3/2)*(b*c - a*d)^2*(a + b*x^2)^(3/4)) + (a^(1/4)*d^2*Sqrt[-((b*x^2)/a)]*EllipticPi[-((Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]), ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/((b*c - a*d)^3*x) + (a^(1/4)*d^2*Sqrt[-((b*x^2)/a)]*EllipticPi[(Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d], ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/((b*c - a*d)^3*x)","A",10,9,21,0.4286,1,"{414, 527, 530, 233, 231, 401, 108, 409, 1218}"
331,1,340,0,0.2944285,"\int \frac{\left(a+b x^2\right)^{7/4}}{\left(c+d x^2\right)^2} \, dx","Int[(a + b*x^2)^(7/4)/(c + d*x^2)^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{\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{x \left(a+b x^2\right)^{3/4} (b c-a d)}{2 c d \left(c+d x^2\right)}","\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{\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{x \left(a+b x^2\right)^{3/4} (b c-a d)}{2 c d \left(c+d x^2\right)}",1,"(b*(5*b*c - a*d)*x)/(2*c*d^2*(a + b*x^2)^(1/4)) - ((b*c - a*d)*x*(a + b*x^2)^(3/4))/(2*c*d*(c + d*x^2)) - (Sqrt[a]*Sqrt[b]*(5*b*c - a*d)*(1 + (b*x^2)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(2*c*d^2*(a + b*x^2)^(1/4)) + (a^(1/4)*Sqrt[-(b*c) + a*d]*(5*b*c + 2*a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi[-((Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]), ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(4*c*d^(5/2)*x) - (a^(1/4)*Sqrt[-(b*c) + a*d]*(5*b*c + 2*a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi[(Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d], ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(4*c*d^(5/2)*x)","A",9,8,21,0.3810,1,"{413, 530, 229, 227, 196, 399, 490, 1218}"
332,1,279,0,0.2707671,"\int \frac{\left(a+b x^2\right)^{5/4}}{\left(c+d x^2\right)^2} \, dx","Int[(a + b*x^2)^(5/4)/(c + d*x^2)^2,x]","\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)}","\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,"-((b*c - a*d)*x*(a + b*x^2)^(1/4))/(2*c*d*(c + d*x^2)) + (Sqrt[a]*Sqrt[b]*(3*b*c + a*d)*(1 + (b*x^2)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(2*c*d^2*(a + b*x^2)^(3/4)) - (a^(1/4)*(3*b*c + 2*a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi[-((Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]), ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(4*c*d^2*x) - (a^(1/4)*(3*b*c + 2*a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi[(Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d], ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(4*c*d^2*x)","A",9,8,21,0.3810,1,"{413, 530, 233, 231, 401, 108, 409, 1218}"
333,1,309,0,0.2223817,"\int \frac{\left(a+b x^2\right)^{3/4}}{\left(c+d x^2\right)^2} \, dx","Int[(a + b*x^2)^(3/4)/(c + d*x^2)^2,x]","\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}}","\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,"-(b*x)/(2*c*d*(a + b*x^2)^(1/4)) + (x*(a + b*x^2)^(3/4))/(2*c*(c + d*x^2)) + (Sqrt[a]*Sqrt[b]*(1 + (b*x^2)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(2*c*d*(a + b*x^2)^(1/4)) + (a^(1/4)*(b*c + 2*a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi[-((Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]), ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(4*c*d^(3/2)*Sqrt[-(b*c) + a*d]*x) - (a^(1/4)*(b*c + 2*a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi[(Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d], ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(4*c*d^(3/2)*Sqrt[-(b*c) + a*d]*x)","A",9,8,21,0.3810,1,"{412, 530, 229, 227, 196, 399, 490, 1218}"
334,1,278,0,0.2055186,"\int \frac{\sqrt[4]{a+b x^2}}{\left(c+d x^2\right)^2} \, dx","Int[(a + b*x^2)^(1/4)/(c + d*x^2)^2,x]","\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)}","\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,"(x*(a + b*x^2)^(1/4))/(2*c*(c + d*x^2)) + (Sqrt[a]*Sqrt[b]*(1 + (b*x^2)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(2*c*d*(a + b*x^2)^(3/4)) - (a^(1/4)*(b*c - 2*a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi[-((Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]), ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(4*c*d*(b*c - a*d)*x) - (a^(1/4)*(b*c - 2*a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi[(Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d], ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(4*c*d*(b*c - a*d)*x)","A",9,8,21,0.3810,1,"{412, 530, 233, 231, 401, 108, 409, 1218}"
335,1,336,0,0.2549609,"\int \frac{1}{\sqrt[4]{a+b x^2} \left(c+d x^2\right)^2} \, dx","Int[1/((a + b*x^2)^(1/4)*(c + d*x^2)^2),x]","\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}}","\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,"(b*x)/(2*c*(b*c - a*d)*(a + b*x^2)^(1/4)) - (d*x*(a + b*x^2)^(3/4))/(2*c*(b*c - a*d)*(c + d*x^2)) - (Sqrt[a]*Sqrt[b]*(1 + (b*x^2)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(2*c*(b*c - a*d)*(a + b*x^2)^(1/4)) - (a^(1/4)*(3*b*c - 2*a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi[-((Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]), ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(4*c*Sqrt[d]*(-(b*c) + a*d)^(3/2)*x) + (a^(1/4)*(3*b*c - 2*a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi[(Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d], ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(4*c*Sqrt[d]*(-(b*c) + a*d)^(3/2)*x)","A",9,8,21,0.3810,1,"{414, 530, 229, 227, 196, 399, 490, 1218}"
336,1,292,0,0.23078,"\int \frac{1}{\left(a+b x^2\right)^{3/4} \left(c+d x^2\right)^2} \, dx","Int[1/((a + b*x^2)^(3/4)*(c + d*x^2)^2),x]","-\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}","-\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,"-(d*x*(a + b*x^2)^(1/4))/(2*c*(b*c - a*d)*(c + d*x^2)) - (Sqrt[a]*Sqrt[b]*(1 + (b*x^2)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(2*c*(b*c - a*d)*(a + b*x^2)^(3/4)) + (a^(1/4)*(5*b*c - 2*a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi[-((Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]), ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(4*c*(b*c - a*d)^2*x) + (a^(1/4)*(5*b*c - 2*a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi[(Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d], ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(4*c*(b*c - a*d)^2*x)","A",9,8,21,0.3810,1,"{414, 530, 233, 231, 401, 108, 409, 1218}"
337,1,314,0,0.3951988,"\int \frac{1}{\left(a+b x^2\right)^{5/4} \left(c+d x^2\right)^2} \, dx","Int[1/((a + b*x^2)^(5/4)*(c + d*x^2)^2),x]","-\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}}","-\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,"-(d*x)/(2*c*(b*c - a*d)*(a + b*x^2)^(1/4)*(c + d*x^2)) + (Sqrt[b]*(4*b*c + a*d)*(1 + (b*x^2)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(2*Sqrt[a]*c*(b*c - a*d)^2*(a + b*x^2)^(1/4)) - (a^(1/4)*Sqrt[d]*(7*b*c - 2*a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi[-((Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]), ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(4*c*(-(b*c) + a*d)^(5/2)*x) + (a^(1/4)*Sqrt[d]*(7*b*c - 2*a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi[(Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d], ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(4*c*(-(b*c) + a*d)^(5/2)*x)","A",10,9,21,0.4286,1,"{414, 527, 530, 229, 227, 196, 399, 490, 1218}"
338,1,345,0,0.3823579,"\int \frac{1}{\left(a+b x^2\right)^{7/4} \left(c+d x^2\right)^2} \, dx","Int[1/((a + b*x^2)^(7/4)*(c + d*x^2)^2),x]","\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}","\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,"(b*(4*b*c + 3*a*d)*x)/(6*a*c*(b*c - a*d)^2*(a + b*x^2)^(3/4)) - (d*x)/(2*c*(b*c - a*d)*(a + b*x^2)^(3/4)*(c + d*x^2)) + (Sqrt[b]*(4*b*c + 3*a*d)*(1 + (b*x^2)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(6*Sqrt[a]*c*(b*c - a*d)^2*(a + b*x^2)^(3/4)) - (a^(1/4)*d*(9*b*c - 2*a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi[-((Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]), ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(4*c*(b*c - a*d)^3*x) - (a^(1/4)*d*(9*b*c - 2*a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi[(Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d], ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(4*c*(b*c - a*d)^3*x)","A",10,9,21,0.4286,1,"{414, 527, 530, 233, 231, 401, 108, 409, 1218}"
339,1,371,0,0.551699,"\int \frac{1}{\left(a+b x^2\right)^{9/4} \left(c+d x^2\right)^2} \, dx","Int[1/((a + b*x^2)^(9/4)*(c + d*x^2)^2),x]","\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}","\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,"(b*(4*b*c + 5*a*d)*x)/(10*a*c*(b*c - a*d)^2*(a + b*x^2)^(5/4)) - (d*x)/(2*c*(b*c - a*d)*(a + b*x^2)^(5/4)*(c + d*x^2)) + (Sqrt[b]*(12*b^2*c^2 - 52*a*b*c*d - 5*a^2*d^2)*(1 + (b*x^2)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(10*a^(3/2)*c*(b*c - a*d)^3*(a + b*x^2)^(1/4)) - (a^(1/4)*d^(3/2)*(11*b*c - 2*a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi[-((Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]), ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(4*c*(-(b*c) + a*d)^(7/2)*x) + (a^(1/4)*d^(3/2)*(11*b*c - 2*a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi[(Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d], ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(4*c*(-(b*c) + a*d)^(7/2)*x)","A",11,9,21,0.4286,1,"{414, 527, 530, 229, 227, 196, 399, 490, 1218}"
340,1,419,0,0.484125,"\int \frac{1}{\left(a+b x^2\right)^{11/4} \left(c+d x^2\right)^2} \, dx","Int[1/((a + b*x^2)^(11/4)*(c + d*x^2)^2),x]","\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}","\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,"(b*(4*b*c + 7*a*d)*x)/(14*a*c*(b*c - a*d)^2*(a + b*x^2)^(7/4)) + (b*(20*b^2*c^2 - 76*a*b*c*d - 21*a^2*d^2)*x)/(42*a^2*c*(b*c - a*d)^3*(a + b*x^2)^(3/4)) - (d*x)/(2*c*(b*c - a*d)*(a + b*x^2)^(7/4)*(c + d*x^2)) + (Sqrt[b]*(20*b^2*c^2 - 76*a*b*c*d - 21*a^2*d^2)*(1 + (b*x^2)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(42*a^(3/2)*c*(b*c - a*d)^3*(a + b*x^2)^(3/4)) + (a^(1/4)*d^2*(13*b*c - 2*a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi[-((Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]), ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(4*c*(b*c - a*d)^4*x) + (a^(1/4)*d^2*(13*b*c - 2*a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi[(Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d], ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(4*c*(b*c - a*d)^4*x)","A",11,9,21,0.4286,1,"{414, 527, 530, 233, 231, 401, 108, 409, 1218}"
341,1,79,0,0.0429968,"\int \left(a+b x^2\right)^p \left(c+d x^2\right)^q \, dx","Int[(a + b*x^2)^p*(c + d*x^2)^q,x]","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)","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,"(x*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1[1/2, -p, -q, 3/2, -((b*x^2)/a), -((d*x^2)/c)])/((1 + (b*x^2)/a)^p*(1 + (d*x^2)/c)^q)","A",3,2,19,0.1053,1,"{430, 429}"
342,1,296,0,0.2763197,"\int \left(a+b x^2\right)^p \left(c+d x^2\right)^3 \, dx","Int[(a + b*x^2)^p*(c + d*x^2)^3,x]","-\frac{x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(-9 a^2 b c d^2 (2 p+7)+15 a^3 d^3+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(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{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)}","-\frac{x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(-9 a^2 b c d^2 (2 p+7)+15 a^3 d^3+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(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{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,"(d*(15*a^2*d^2 - 8*a*b*c*d*(6 + p) + b^2*c^2*(57 + 28*p + 4*p^2))*x*(a + b*x^2)^(1 + p))/(b^3*(3 + 2*p)*(5 + 2*p)*(7 + 2*p)) - (d*(5*a*d - b*c*(11 + 2*p))*x*(a + b*x^2)^(1 + p)*(c + d*x^2))/(b^2*(5 + 2*p)*(7 + 2*p)) + (d*x*(a + b*x^2)^(1 + p)*(c + d*x^2)^2)/(b*(7 + 2*p)) - ((15*a^3*d^3 - 9*a^2*b*c*d^2*(7 + 2*p) + 3*a*b^2*c^2*d*(35 + 24*p + 4*p^2) - b^3*c^3*(105 + 142*p + 60*p^2 + 8*p^3))*x*(a + b*x^2)^p*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2)/a)])/(b^3*(3 + 2*p)*(5 + 2*p)*(7 + 2*p)*(1 + (b*x^2)/a)^p)","A",5,5,19,0.2632,1,"{416, 528, 388, 246, 245}"
343,1,176,0,0.1208523,"\int \left(a+b x^2\right)^p \left(c+d x^2\right)^2 \, dx","Int[(a + b*x^2)^p*(c + d*x^2)^2,x]","\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)}","\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,"-((d*(3*a*d - b*c*(7 + 2*p))*x*(a + b*x^2)^(1 + p))/(b^2*(3 + 2*p)*(5 + 2*p))) + (d*x*(a + b*x^2)^(1 + p)*(c + d*x^2))/(b*(5 + 2*p)) + ((3*a^2*d^2 - 2*a*b*c*d*(5 + 2*p) + b^2*c^2*(15 + 16*p + 4*p^2))*x*(a + b*x^2)^p*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2)/a)])/(b^2*(3 + 2*p)*(5 + 2*p)*(1 + (b*x^2)/a)^p)","A",4,4,19,0.2105,1,"{416, 388, 246, 245}"
344,1,85,0,0.0390779,"\int \left(a+b x^2\right)^p \left(c+d x^2\right) \, dx","Int[(a + b*x^2)^p*(c + d*x^2),x]","x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(c-\frac{a d}{2 b p+3 b}\right) \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)+\frac{d x \left(a+b x^2\right)^{p+1}}{b (2 p+3)}","\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,"(d*x*(a + b*x^2)^(1 + p))/(b*(3 + 2*p)) + ((c - (a*d)/(3*b + 2*b*p))*x*(a + b*x^2)^p*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p","A",3,3,17,0.1765,1,"{388, 246, 245}"
345,1,44,0,0.0092847,"\int \left(a+b x^2\right)^p \, dx","Int[(a + b*x^2)^p,x]","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)","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,"(x*(a + b*x^2)^p*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p","A",2,2,9,0.2222,1,"{246, 245}"
346,1,57,0,0.026394,"\int \frac{\left(a+b x^2\right)^p}{c+d x^2} \, dx","Int[(a + b*x^2)^p/(c + d*x^2),x]","\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}","\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,"(x*(a + b*x^2)^p*AppellF1[1/2, -p, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)])/(c*(1 + (b*x^2)/a)^p)","A",2,2,19,0.1053,1,"{430, 429}"
347,1,57,0,0.0251872,"\int \frac{\left(a+b x^2\right)^p}{\left(c+d x^2\right)^2} \, dx","Int[(a + b*x^2)^p/(c + d*x^2)^2,x]","\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}","\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,"(x*(a + b*x^2)^p*AppellF1[1/2, -p, 2, 3/2, -((b*x^2)/a), -((d*x^2)/c)])/(c^2*(1 + (b*x^2)/a)^p)","A",2,2,19,0.1053,1,"{430, 429}"
348,1,57,0,0.0251613,"\int \frac{\left(a+b x^2\right)^p}{\left(c+d x^2\right)^3} \, dx","Int[(a + b*x^2)^p/(c + d*x^2)^3,x]","\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}","\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,"(x*(a + b*x^2)^p*AppellF1[1/2, -p, 3, 3/2, -((b*x^2)/a), -((d*x^2)/c)])/(c^3*(1 + (b*x^2)/a)^p)","A",2,2,19,0.1053,1,"{430, 429}"
349,1,53,0,0.0200724,"\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","Int[(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]","\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}","\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,"(x*(c + d*x^2)^((a*d)/(2*b*c - 2*a*d)))/(a*c*(a + b*x^2)^((b*c)/(2*b*c - 2*a*d)))","A",1,1,50,0.02000,1,"{381}"