1,1,94,94,0.021079,"\int \left(a+b x^2\right) \left(c+d x^2\right)^4 \, dx","Integrate[(a + b*x^2)*(c + d*x^2)^4,x]","\frac{1}{3} c^3 x^3 (4 a d+b c)+\frac{2}{5} c^2 d x^5 (3 a d+2 b c)+\frac{1}{9} d^3 x^9 (a d+4 b c)+\frac{2}{7} c d^2 x^7 (2 a d+3 b c)+a c^4 x+\frac{1}{11} b d^4 x^{11}","\frac{1}{3} c^3 x^3 (4 a d+b c)+\frac{2}{5} c^2 d x^5 (3 a d+2 b c)+\frac{1}{9} d^3 x^9 (a d+4 b c)+\frac{2}{7} c d^2 x^7 (2 a d+3 b c)+a c^4 x+\frac{1}{11} b d^4 x^{11}",1,"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",1
2,1,70,70,0.0151076,"\int \left(a+b x^2\right) \left(c+d x^2\right)^3 \, dx","Integrate[(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",1
3,1,50,50,0.0114167,"\int \left(a+b x^2\right) \left(c+d x^2\right)^2 \, dx","Integrate[(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",1
4,1,28,28,0.0054941,"\int \left(a+b x^2\right) \left(c+d x^2\right) \, dx","Integrate[(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",1
5,1,40,40,0.0233608,"\int \frac{a+b x^2}{c+d x^2} \, dx","Integrate[(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",1
6,1,63,63,0.0466126,"\int \frac{a+b x^2}{\left(c+d x^2\right)^2} \, dx","Integrate[(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,"-1/2*((b*c - a*d)*x)/(c*d*(c + d*x^2)) + ((b*c + a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(3/2)*d^(3/2))","A",1
7,1,82,92,0.0592755,"\int \frac{a+b x^2}{\left(c+d x^2\right)^3} \, dx","Integrate[(a + b*x^2)/(c + d*x^2)^3,x]","\frac{(3 a d+b c) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{5/2} d^{3/2}}+\frac{x \left(a d \left(5 c+3 d x^2\right)+b c \left(d x^2-c\right)\right)}{8 c^2 d \left(c+d x^2\right)^2}","\frac{(3 a d+b c) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{5/2} d^{3/2}}+\frac{x (3 a d+b c)}{8 c^2 d \left(c+d x^2\right)}-\frac{x (b c-a d)}{4 c d \left(c+d x^2\right)^2}",1,"(x*(b*c*(-c + d*x^2) + a*d*(5*c + 3*d*x^2)))/(8*c^2*d*(c + d*x^2)^2) + ((b*c + 3*a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(5/2)*d^(3/2))","A",1
8,1,122,122,0.0234831,"\int \left(a+b x^2\right)^2 \left(c+d x^2\right)^3 \, dx","Integrate[(a + b*x^2)^2*(c + d*x^2)^3,x]","\frac{1}{7} d x^7 \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{1}{5} c x^5 \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+a^2 c^3 x+\frac{1}{3} a c^2 x^3 (3 a d+2 b c)+\frac{1}{9} b d^2 x^9 (2 a d+3 b c)+\frac{1}{11} b^2 d^3 x^{11}","\frac{1}{7} d x^7 \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{1}{5} c x^5 \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+a^2 c^3 x+\frac{1}{3} a c^2 x^3 (3 a d+2 b c)+\frac{1}{9} b d^2 x^9 (2 a d+3 b c)+\frac{1}{11} b^2 d^3 x^{11}",1,"a^2*c^3*x + (a*c^2*(2*b*c + 3*a*d)*x^3)/3 + (c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^5)/5 + (d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^7)/7 + (b*d^2*(3*b*c + 2*a*d)*x^9)/9 + (b^2*d^3*x^11)/11","A",1
9,1,82,82,0.0164829,"\int \left(a+b x^2\right)^2 \left(c+d x^2\right)^2 \, dx","Integrate[(a + b*x^2)^2*(c + d*x^2)^2,x]","\frac{1}{5} x^5 \left(a^2 d^2+4 a b c d+b^2 c^2\right)+a^2 c^2 x+\frac{2}{7} b d x^7 (a d+b c)+\frac{2}{3} a c x^3 (a d+b c)+\frac{1}{9} b^2 d^2 x^9","\frac{1}{5} x^5 \left(a^2 d^2+4 a b c d+b^2 c^2\right)+a^2 c^2 x+\frac{2}{7} b d x^7 (a d+b c)+\frac{2}{3} a c x^3 (a d+b c)+\frac{1}{9} b^2 d^2 x^9",1,"a^2*c^2*x + (2*a*c*(b*c + a*d)*x^3)/3 + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^5)/5 + (2*b*d*(b*c + a*d)*x^7)/7 + (b^2*d^2*x^9)/9","A",1
10,1,50,50,0.0071434,"\int \left(a+b x^2\right)^2 \left(c+d x^2\right) \, dx","Integrate[(a + b*x^2)^2*(c + d*x^2),x]","a^2 c x+\frac{1}{5} b x^5 (2 a d+b c)+\frac{1}{3} a x^3 (a d+2 b c)+\frac{1}{7} b^2 d x^7","a^2 c x+\frac{1}{5} b x^5 (2 a d+b c)+\frac{1}{3} a x^3 (a d+2 b c)+\frac{1}{7} b^2 d x^7",1,"a^2*c*x + (a*(2*b*c + a*d)*x^3)/3 + (b*(b*c + 2*a*d)*x^5)/5 + (b^2*d*x^7)/7","A",1
11,1,59,63,0.0495312,"\int \frac{\left(a+b x^2\right)^2}{c+d x^2} \, dx","Integrate[(a + b*x^2)^2/(c + d*x^2),x]","\frac{(b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{\sqrt{c} d^{5/2}}+\frac{b x \left(6 a d-3 b c+b d x^2\right)}{3 d^2}","\frac{(b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{\sqrt{c} d^{5/2}}-\frac{b x (b c-2 a d)}{d^2}+\frac{b^2 x^3}{3 d}",1,"(b*x*(-3*b*c + 6*a*d + b*d*x^2))/(3*d^2) + ((b*c - a*d)^2*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*d^(5/2))","A",1
12,1,89,82,0.0604459,"\int \frac{\left(a+b x^2\right)^2}{\left(c+d x^2\right)^2} \, dx","Integrate[(a + b*x^2)^2/(c + d*x^2)^2,x]","-\frac{\left(-a^2 d^2-2 a b c d+3 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{3/2} d^{5/2}}+\frac{x (b c-a d)^2}{2 c d^2 \left(c+d x^2\right)}+\frac{b^2 x}{d^2}","-\frac{(b c-a d) (a d+3 b c) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{3/2} d^{5/2}}+\frac{x (b c-a d)^2}{2 c d^2 \left(c+d x^2\right)}+\frac{b^2 x}{d^2}",1,"(b^2*x)/d^2 + ((b*c - a*d)^2*x)/(2*c*d^2*(c + d*x^2)) - ((3*b^2*c^2 - 2*a*b*c*d - a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(3/2)*d^(5/2))","A",1
13,1,121,116,0.0969923,"\int \frac{\left(a+b x^2\right)^2}{\left(c+d x^2\right)^3} \, dx","Integrate[(a + b*x^2)^2/(c + d*x^2)^3,x]","\frac{x \left(a^2 d^2 \left(5 c+3 d x^2\right)-2 a b c d \left(c-d x^2\right)-b^2 c^2 \left(3 c+5 d x^2\right)\right)}{8 c^2 d^2 \left(c+d x^2\right)^2}+\frac{\left(3 a^2 d^2+2 a b c d+3 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{5/2} d^{5/2}}","\frac{3 x \left(\frac{a^2}{c^2}-\frac{b^2}{d^2}\right)}{8 \left(c+d x^2\right)}+\frac{\left(3 a^2 d^2+2 a b c d+3 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{5/2} d^{5/2}}-\frac{x \left(a+b x^2\right) (b c-a d)}{4 c d \left(c+d x^2\right)^2}",1,"(x*(-2*a*b*c*d*(c - d*x^2) + a^2*d^2*(5*c + 3*d*x^2) - b^2*c^2*(3*c + 5*d*x^2)))/(8*c^2*d^2*(c + d*x^2)^2) + ((3*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(5/2)*d^(5/2))","A",1
14,1,161,154,0.0297967,"\int \left(a+b x^2\right)^3 \left(c+d x^2\right)^3 \, dx","Integrate[(a + b*x^2)^3*(c + d*x^2)^3,x]","a^3 c^3 x+\frac{1}{3} b d x^9 \left(a^2 d^2+3 a b c d+b^2 c^2\right)+\frac{3}{5} a c x^5 \left(a^2 d^2+3 a b c d+b^2 c^2\right)+a^2 c^2 x^3 (a d+b c)+\frac{1}{7} x^7 \left(a^3 d^3+9 a^2 b c d^2+9 a b^2 c^2 d+b^3 c^3\right)+\frac{3}{11} b^2 d^2 x^{11} (a d+b c)+\frac{1}{13} b^3 d^3 x^{13}","a^3 c^3 x+\frac{1}{3} b d x^9 \left(a^2 d^2+3 a b c d+b^2 c^2\right)+\frac{1}{7} x^7 (a d+b c) \left(a^2 d^2+8 a b c d+b^2 c^2\right)+\frac{3}{5} a c x^5 \left(a^2 d^2+3 a b c d+b^2 c^2\right)+a^2 c^2 x^3 (a d+b c)+\frac{3}{11} b^2 d^2 x^{11} (a d+b c)+\frac{1}{13} b^3 d^3 x^{13}",1,"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^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*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",1
15,1,122,122,0.0216626,"\int \left(a+b x^2\right)^3 \left(c+d x^2\right)^2 \, dx","Integrate[(a + b*x^2)^3*(c + d*x^2)^2,x]","a^3 c^2 x+\frac{1}{7} b x^7 \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+\frac{1}{5} a x^5 \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{1}{3} a^2 c x^3 (2 a d+3 b c)+\frac{1}{9} b^2 d x^9 (3 a d+2 b c)+\frac{1}{11} b^3 d^2 x^{11}","a^3 c^2 x+\frac{1}{7} b x^7 \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+\frac{1}{5} a x^5 \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{1}{3} a^2 c x^3 (2 a d+3 b c)+\frac{1}{9} b^2 d x^9 (3 a d+2 b c)+\frac{1}{11} b^3 d^2 x^{11}",1,"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",1
16,1,70,70,0.0122651,"\int \left(a+b x^2\right)^3 \left(c+d x^2\right) \, dx","Integrate[(a + b*x^2)^3*(c + d*x^2),x]","a^3 c x+\frac{1}{3} a^2 x^3 (a d+3 b c)+\frac{1}{7} b^2 x^7 (3 a d+b c)+\frac{3}{5} a b x^5 (a d+b c)+\frac{1}{9} b^3 d x^9","a^3 c x+\frac{1}{3} a^2 x^3 (a d+3 b c)+\frac{1}{7} b^2 x^7 (3 a d+b c)+\frac{3}{5} a b x^5 (a d+b c)+\frac{1}{9} b^3 d x^9",1,"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",1
17,1,93,98,0.0614573,"\int \frac{\left(a+b x^2\right)^3}{c+d x^2} \, dx","Integrate[(a + b*x^2)^3/(c + d*x^2),x]","\frac{b x \left(45 a^2 d^2+15 a b d \left(d x^2-3 c\right)+b^2 \left(15 c^2-5 c d x^2+3 d^2 x^4\right)\right)}{15 d^3}-\frac{(b c-a d)^3 \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{\sqrt{c} d^{7/2}}","\frac{b x \left(3 a^2 d^2-3 a b c d+b^2 c^2\right)}{d^3}-\frac{b^2 x^3 (b c-3 a d)}{3 d^2}-\frac{(b c-a d)^3 \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{\sqrt{c} d^{7/2}}+\frac{b^3 x^5}{5 d}",1,"(b*x*(45*a^2*d^2 + 15*a*b*d*(-3*c + d*x^2) + b^2*(15*c^2 - 5*c*d*x^2 + 3*d^2*x^4)))/(15*d^3) - ((b*c - a*d)^3*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*d^(7/2))","A",1
18,1,107,107,0.0593322,"\int \frac{\left(a+b x^2\right)^3}{\left(c+d x^2\right)^2} \, dx","Integrate[(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",1
19,1,141,130,0.0808508,"\int \frac{\left(a+b x^2\right)^3}{\left(c+d x^2\right)^3} \, dx","Integrate[(a + b*x^2)^3/(c + d*x^2)^3,x]","-\frac{3 \left(-a^3 d^3-a^2 b c d^2-3 a b^2 c^2 d+5 b^3 c^3\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{5/2} d^{7/2}}+\frac{3 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*(5*b^3*c^3 - 3*a*b^2*c^2*d - a^2*b*c*d^2 - a^3*d^3)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(5/2)*d^(7/2))","A",1
20,1,136,142,0.0907271,"\int \frac{\left(c+d x^2\right)^4}{a+b x^2} \, dx","Integrate[(c + d*x^2)^4/(a + b*x^2),x]","\frac{d x \left(-105 a^3 d^3+35 a^2 b d^2 \left(12 c+d x^2\right)-7 a b^2 d \left(90 c^2+20 c d x^2+3 d^2 x^4\right)+3 b^3 \left(140 c^3+70 c^2 d x^2+28 c d^2 x^4+5 d^3 x^6\right)\right)}{105 b^4}+\frac{(b c-a d)^4 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} b^{9/2}}","\frac{d x (2 b c-a d) \left(a^2 d^2-2 a b c d+2 b^2 c^2\right)}{b^4}+\frac{d^2 x^3 \left(a^2 d^2-4 a b c d+6 b^2 c^2\right)}{3 b^3}+\frac{(b c-a d)^4 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} b^{9/2}}+\frac{d^3 x^5 (4 b c-a d)}{5 b^2}+\frac{d^4 x^7}{7 b}",1,"(d*x*(-105*a^3*d^3 + 35*a^2*b*d^2*(12*c + d*x^2) - 7*a*b^2*d*(90*c^2 + 20*c*d*x^2 + 3*d^2*x^4) + 3*b^3*(140*c^3 + 70*c^2*d*x^2 + 28*c*d^2*x^4 + 5*d^3*x^6)))/(105*b^4) + ((b*c - a*d)^4*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*b^(9/2))","A",1
21,1,92,98,0.067489,"\int \frac{\left(c+d x^2\right)^3}{a+b x^2} \, dx","Integrate[(c + d*x^2)^3/(a + b*x^2),x]","\frac{d x \left(15 a^2 d^2-5 a b d \left(9 c+d x^2\right)+3 b^2 \left(15 c^2+5 c d x^2+d^2 x^4\right)\right)}{15 b^3}+\frac{(b c-a d)^3 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} b^{7/2}}","\frac{d x \left(a^2 d^2-3 a b c d+3 b^2 c^2\right)}{b^3}+\frac{(b c-a d)^3 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} b^{7/2}}+\frac{d^2 x^3 (3 b c-a d)}{3 b^2}+\frac{d^3 x^5}{5 b}",1,"(d*x*(15*a^2*d^2 - 5*a*b*d*(9*c + d*x^2) + 3*b^2*(15*c^2 + 5*c*d*x^2 + d^2*x^4)))/(15*b^3) + ((b*c - a*d)^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*b^(7/2))","A",1
22,1,59,63,0.0513163,"\int \frac{\left(c+d x^2\right)^2}{a+b x^2} \, dx","Integrate[(c + d*x^2)^2/(a + b*x^2),x]","\frac{(b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} b^{5/2}}+\frac{d x \left(-3 a d+6 b c+b d x^2\right)}{3 b^2}","\frac{(b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} b^{5/2}}+\frac{d x (2 b c-a d)}{b^2}+\frac{d^2 x^3}{3 b}",1,"(d*x*(6*b*c - 3*a*d + b*d*x^2))/(3*b^2) + ((b*c - a*d)^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*b^(5/2))","A",1
23,1,40,39,0.0259772,"\int \frac{c+d x^2}{a+b x^2} \, dx","Integrate[(c + d*x^2)/(a + b*x^2),x]","\frac{d x}{b}-\frac{(a d-b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} b^{3/2}}","\frac{(b c-a d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} b^{3/2}}+\frac{d x}{b}",1,"(d*x)/b - ((-(b*c) + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*b^(3/2))","A",1
24,1,61,70,0.0453396,"\int \frac{1}{\left(a+b x^2\right) \left(c+d x^2\right)} \, dx","Integrate[1/((a + b*x^2)*(c + d*x^2)),x]","\frac{\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a}}-\frac{\sqrt{d} \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{\sqrt{c}}}{b c-a d}","\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} (b c-a d)}-\frac{\sqrt{d} \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{\sqrt{c} (b c-a d)}",1,"((Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[a] - (Sqrt[d]*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/Sqrt[c])/(b*c - a*d)","A",1
25,1,95,109,0.1712903,"\int \frac{1}{\left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","Integrate[1/((a + b*x^2)*(c + d*x^2)^2),x]","\frac{\frac{2 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a}}+\frac{\sqrt{d} (a d-3 b c) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{3/2}}+\frac{d x (a d-b c)}{c \left(c+d x^2\right)}}{2 (b c-a d)^2}","\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} (b c-a d)^2}-\frac{\sqrt{d} (3 b c-a d) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{3/2} (b c-a d)^2}-\frac{d x}{2 c \left(c+d x^2\right) (b c-a d)}",1,"((d*(-(b*c) + a*d)*x)/(c*(c + d*x^2)) + (2*b^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[a] + (Sqrt[d]*(-3*b*c + a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/c^(3/2))/(2*(b*c - a*d)^2)","A",1
26,1,158,160,0.2379747,"\int \frac{1}{\left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","Integrate[1/((a + b*x^2)*(c + d*x^2)^3),x]","\frac{1}{8} \left(-\frac{\sqrt{d} \left(3 a^2 d^2-10 a b c d+15 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{5/2} (b c-a d)^3}-\frac{8 b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} (a d-b c)^3}+\frac{d x (3 a d-7 b c)}{c^2 \left(c+d x^2\right) (b c-a d)^2}-\frac{2 d x}{c \left(c+d x^2\right)^2 (b c-a d)}\right)","-\frac{\sqrt{d} \left(3 a^2 d^2-10 a b c d+15 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{5/2} (b c-a d)^3}+\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} (b c-a d)^3}-\frac{d x (7 b c-3 a d)}{8 c^2 \left(c+d x^2\right) (b c-a d)^2}-\frac{d x}{4 c \left(c+d x^2\right)^2 (b c-a d)}",1,"((-2*d*x)/(c*(b*c - a*d)*(c + d*x^2)^2) + (d*(-7*b*c + 3*a*d)*x)/(c^2*(b*c - a*d)^2*(c + d*x^2)) - (8*b^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*(-(b*c) + a*d)^3) - (Sqrt[d]*(15*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(5/2)*(b*c - a*d)^3))/8","A",1
27,1,192,192,0.0980223,"\int \frac{\left(c+d x^2\right)^5}{\left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)^5/(a + b*x^2)^2,x]","\frac{(b c-a d)^4 (9 a d+b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} b^{11/2}}+\frac{d^3 x^3 \left(3 a^2 d^2-10 a b c d+10 b^2 c^2\right)}{3 b^4}+\frac{d^2 x \left(-4 a^3 d^3+15 a^2 b c d^2-20 a b^2 c^2 d+10 b^3 c^3\right)}{b^5}+\frac{x (b c-a d)^5}{2 a b^5 \left(a+b x^2\right)}+\frac{d^4 x^5 (5 b c-2 a d)}{5 b^3}+\frac{d^5 x^7}{7 b^2}","\frac{(b c-a d)^4 (9 a d+b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} b^{11/2}}+\frac{d^3 x^3 \left(3 a^2 d^2-10 a b c d+10 b^2 c^2\right)}{3 b^4}+\frac{d^2 x \left(-4 a^3 d^3+15 a^2 b c d^2-20 a b^2 c^2 d+10 b^3 c^3\right)}{b^5}+\frac{x (b c-a d)^5}{2 a b^5 \left(a+b x^2\right)}+\frac{d^4 x^5 (5 b c-2 a d)}{5 b^3}+\frac{d^5 x^7}{7 b^2}",1,"(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",1
28,1,142,142,0.0901688,"\int \frac{\left(c+d x^2\right)^4}{\left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)^4/(a + b*x^2)^2,x]","\frac{(b c-a d)^3 (7 a d+b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} b^{9/2}}+\frac{d^2 x \left(3 a^2 d^2-8 a b c d+6 b^2 c^2\right)}{b^4}+\frac{x (b c-a d)^4}{2 a b^4 \left(a+b x^2\right)}+\frac{2 d^3 x^3 (2 b c-a d)}{3 b^3}+\frac{d^4 x^5}{5 b^2}","\frac{(b c-a d)^3 (7 a d+b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} b^{9/2}}+\frac{d^2 x \left(3 a^2 d^2-8 a b c d+6 b^2 c^2\right)}{b^4}+\frac{x (b c-a d)^4}{2 a b^4 \left(a+b x^2\right)}+\frac{2 d^3 x^3 (2 b c-a d)}{3 b^3}+\frac{d^4 x^5}{5 b^2}",1,"(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",1
29,1,106,106,0.0617295,"\int \frac{\left(c+d x^2\right)^3}{\left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)^3/(a + b*x^2)^2,x]","\frac{(5 a d+b c) (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} b^{7/2}}+\frac{d^2 x (3 b c-2 a d)}{b^3}+\frac{x (b c-a d)^3}{2 a b^3 \left(a+b x^2\right)}+\frac{d^3 x^3}{3 b^2}","\frac{(5 a d+b c) (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} b^{7/2}}+\frac{d^2 x (3 b c-2 a d)}{b^3}+\frac{x (b c-a d)^3}{2 a b^3 \left(a+b x^2\right)}+\frac{d^3 x^3}{3 b^2}",1,"(d^2*(3*b*c - 2*a*d)*x)/b^3 + (d^3*x^3)/(3*b^2) + ((b*c - a*d)^3*x)/(2*a*b^3*(a + b*x^2)) + ((b*c - a*d)^2*(b*c + 5*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*b^(7/2))","A",1
30,1,88,82,0.0629053,"\int \frac{\left(c+d x^2\right)^2}{\left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)^2/(a + b*x^2)^2,x]","\frac{\left(-3 a^2 d^2+2 a b c d+b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} b^{5/2}}+\frac{x (b c-a d)^2}{2 a b^2 \left(a+b x^2\right)}+\frac{d^2 x}{b^2}","\frac{(b c-a d) (3 a d+b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} b^{5/2}}+\frac{x (b c-a d)^2}{2 a b^2 \left(a+b x^2\right)}+\frac{d^2 x}{b^2}",1,"(d^2*x)/b^2 + ((b*c - a*d)^2*x)/(2*a*b^2*(a + b*x^2)) + ((b^2*c^2 + 2*a*b*c*d - 3*a^2*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*b^(5/2))","A",1
31,1,63,63,0.0465959,"\int \frac{c+d x^2}{\left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2)/(a + b*x^2)^2,x]","\frac{(a d+b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} b^{3/2}}-\frac{x (a d-b c)}{2 a b \left(a+b x^2\right)}","\frac{(a d+b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} b^{3/2}}+\frac{x (b c-a d)}{2 a b \left(a+b x^2\right)}",1,"-1/2*((-(b*c) + a*d)*x)/(a*b*(a + b*x^2)) + ((b*c + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*b^(3/2))","A",1
32,1,109,108,0.1464806,"\int \frac{1}{\left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","Integrate[1/((a + b*x^2)^2*(c + d*x^2)),x]","-\frac{\sqrt{b} (3 a d-b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} (a d-b c)^2}+\frac{d^{3/2} \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{\sqrt{c} (b c-a d)^2}-\frac{b x}{2 a \left(a+b x^2\right) (a d-b c)}","\frac{\sqrt{b} (b c-3 a d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} (b c-a d)^2}+\frac{d^{3/2} \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{\sqrt{c} (b c-a d)^2}+\frac{b x}{2 a \left(a+b x^2\right) (b c-a d)}",1,"-1/2*(b*x)/(a*(-(b*c) + a*d)*(a + b*x^2)) - (Sqrt[b]*(-(b*c) + 3*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*(-(b*c) + a*d)^2) + (d^(3/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*(b*c - a*d)^2)","A",1
33,1,136,167,0.318948,"\int \frac{1}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","Integrate[1/((a + b*x^2)^2*(c + d*x^2)^2),x]","\frac{1}{2} \left(\frac{b^{3/2} (5 a d-b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2} (a d-b c)^3}+\frac{x (b c-a d) \left(\frac{b^2}{a^2+a b x^2}+\frac{d^2}{c^2+c d x^2}\right)+\frac{d^{3/2} (5 b c-a d) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{3/2}}}{(b c-a d)^3}\right)","\frac{b^{3/2} (b c-5 a d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} (b c-a d)^3}+\frac{d^{3/2} (5 b c-a d) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{3/2} (b c-a d)^3}+\frac{b x}{2 a \left(a+b x^2\right) \left(c+d x^2\right) (b c-a d)}+\frac{d x (a d+b c)}{2 a c \left(c+d x^2\right) (b c-a d)^2}",1,"((b^(3/2)*(-(b*c) + 5*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*(-(b*c) + a*d)^3) + ((b*c - a*d)*x*(b^2/(a^2 + a*b*x^2) + d^2/(c^2 + c*d*x^2)) + (d^(3/2)*(5*b*c - a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/c^(3/2))/(b*c - a*d)^3)/2","A",1
34,1,197,230,0.4194801,"\int \frac{1}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","Integrate[1/((a + b*x^2)^2*(c + d*x^2)^3),x]","\frac{1}{8} \left(\frac{4 b^{5/2} (b c-7 a d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2} (b c-a d)^4}+\frac{d^{3/2} \left(3 a^2 d^2-14 a b c d+35 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{5/2} (b c-a d)^4}-\frac{4 b^3 x}{a \left(a+b x^2\right) (a d-b c)^3}+\frac{d^2 x (11 b c-3 a d)}{c^2 \left(c+d x^2\right) (b c-a d)^3}+\frac{2 d^2 x}{c \left(c+d x^2\right)^2 (b c-a d)^2}\right)","\frac{b^{5/2} (b c-7 a d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} (b c-a d)^4}+\frac{d^{3/2} \left(3 a^2 d^2-14 a b c d+35 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{5/2} (b c-a d)^4}+\frac{d x (4 b c-a d) (3 a d+b c)}{8 a c^2 \left(c+d x^2\right) (b c-a d)^3}+\frac{b x}{2 a \left(a+b x^2\right) \left(c+d x^2\right)^2 (b c-a d)}+\frac{d x (a d+2 b c)}{4 a c \left(c+d x^2\right)^2 (b c-a d)^2}",1,"((-4*b^3*x)/(a*(-(b*c) + a*d)^3*(a + b*x^2)) + (2*d^2*x)/(c*(b*c - a*d)^2*(c + d*x^2)^2) + (d^2*(11*b*c - 3*a*d)*x)/(c^2*(b*c - a*d)^3*(c + d*x^2)) + (4*b^(5/2)*(b*c - 7*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*(b*c - a*d)^4) + (d^(3/2)*(35*b^2*c^2 - 14*a*b*c*d + 3*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(5/2)*(b*c - a*d)^4))/8","A",1
35,1,196,196,0.1261769,"\int \frac{\left(c+d x^2\right)^5}{\left(a+b x^2\right)^3} \, dx","Integrate[(c + d*x^2)^5/(a + b*x^2)^3,x]","\frac{x (17 a d+3 b c) (b c-a d)^4}{8 a^2 b^5 \left(a+b x^2\right)}+\frac{d^3 x \left(6 a^2 d^2-15 a b c d+10 b^2 c^2\right)}{b^5}+\frac{\left(63 a^2 d^2+14 a b c d+3 b^2 c^2\right) (b c-a d)^3 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} b^{11/2}}+\frac{x (b c-a d)^5}{4 a b^5 \left(a+b x^2\right)^2}+\frac{d^4 x^3 (5 b c-3 a d)}{3 b^4}+\frac{d^5 x^5}{5 b^3}","\frac{x (17 a d+3 b c) (b c-a d)^4}{8 a^2 b^5 \left(a+b x^2\right)}+\frac{d^3 x \left(6 a^2 d^2-15 a b c d+10 b^2 c^2\right)}{b^5}+\frac{\left(63 a^2 d^2+14 a b c d+3 b^2 c^2\right) (b c-a d)^3 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} b^{11/2}}+\frac{x (b c-a d)^5}{4 a b^5 \left(a+b x^2\right)^2}+\frac{d^4 x^3 (5 b c-3 a d)}{3 b^4}+\frac{d^5 x^5}{5 b^3}",1,"(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",1
36,1,160,160,0.0973389,"\int \frac{\left(c+d x^2\right)^4}{\left(a+b x^2\right)^3} \, dx","Integrate[(c + d*x^2)^4/(a + b*x^2)^3,x]","\frac{x (b c-a d)^3 (13 a d+3 b c)}{8 a^2 b^4 \left(a+b x^2\right)}+\frac{(b c-a d)^2 \left(35 a^2 d^2+10 a b c d+3 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} b^{9/2}}+\frac{d^3 x (4 b c-3 a d)}{b^4}+\frac{x (b c-a d)^4}{4 a b^4 \left(a+b x^2\right)^2}+\frac{d^4 x^3}{3 b^3}","\frac{x (b c-a d)^3 (13 a d+3 b c)}{8 a^2 b^4 \left(a+b x^2\right)}+\frac{(b c-a d)^2 \left(35 a^2 d^2+10 a b c d+3 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} b^{9/2}}+\frac{d^3 x (4 b c-3 a d)}{b^4}+\frac{x (b c-a d)^4}{4 a b^4 \left(a+b x^2\right)^2}+\frac{d^4 x^3}{3 b^3}",1,"(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",1
37,1,139,130,0.0843538,"\int \frac{\left(c+d x^2\right)^3}{\left(a+b x^2\right)^3} \, dx","Integrate[(c + d*x^2)^3/(a + b*x^2)^3,x]","\frac{3 x (b c-a d)^2 (3 a d+b c)}{8 a^2 b^3 \left(a+b x^2\right)}+\frac{3 \left(-5 a^3 d^3+3 a^2 b c d^2+a b^2 c^2 d+b^3 c^3\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} b^{7/2}}+\frac{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 x (b c-a d)^2 (3 a d+b c)}{8 a^2 b^3 \left(a+b x^2\right)}+\frac{3 (b c-a d) \left(4 a^2 d^2+(a d+b c)^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} b^{7/2}}+\frac{x (b c-a d)^3}{4 a b^3 \left(a+b x^2\right)^2}+\frac{d^3 x}{b^3}",1,"(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^3*c^3 + a*b^2*c^2*d + 3*a^2*b*c*d^2 - 5*a^3*d^3)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*b^(7/2))","A",1
38,1,124,116,0.0964627,"\int \frac{\left(c+d x^2\right)^2}{\left(a+b x^2\right)^3} \, dx","Integrate[(c + d*x^2)^2/(a + b*x^2)^3,x]","\frac{\left(3 a^2 d^2+2 a b c d+3 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} b^{5/2}}+\frac{x \left(-3 a^3 d^2-a^2 b d \left(2 c+5 d x^2\right)+a b^2 c \left(5 c+2 d x^2\right)+3 b^3 c^2 x^2\right)}{8 a^2 b^2 \left(a+b x^2\right)^2}","\frac{3 x \left(\frac{c^2}{a^2}-\frac{d^2}{b^2}\right)}{8 \left(a+b x^2\right)}+\frac{\left(3 a^2 d^2+2 a b c d+3 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} b^{5/2}}+\frac{x \left(c+d x^2\right) (b c-a d)}{4 a b \left(a+b x^2\right)^2}",1,"(x*(-3*a^3*d^2 + 3*b^3*c^2*x^2 + a*b^2*c*(5*c + 2*d*x^2) - a^2*b*d*(2*c + 5*d*x^2)))/(8*a^2*b^2*(a + b*x^2)^2) + ((3*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*b^(5/2))","A",1
39,1,84,92,0.0641187,"\int \frac{c+d x^2}{\left(a+b x^2\right)^3} \, dx","Integrate[(c + d*x^2)/(a + b*x^2)^3,x]","\frac{(a d+3 b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} b^{3/2}}+\frac{x \left(a^2 (-d)+a b \left(5 c+d x^2\right)+3 b^2 c x^2\right)}{8 a^2 b \left(a+b x^2\right)^2}","\frac{(a d+3 b c) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} b^{3/2}}+\frac{x (a d+3 b c)}{8 a^2 b \left(a+b x^2\right)}+\frac{x (b c-a d)}{4 a b \left(a+b x^2\right)^2}",1,"(x*(-(a^2*d) + 3*b^2*c*x^2 + a*b*(5*c + d*x^2)))/(8*a^2*b*(a + b*x^2)^2) + ((3*b*c + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*b^(3/2))","A",1
40,1,158,161,0.2805737,"\int \frac{1}{\left(a+b x^2\right)^3 \left(c+d x^2\right)} \, dx","Integrate[1/((a + b*x^2)^3*(c + d*x^2)),x]","\frac{1}{8} \left(\frac{b x (3 b c-7 a d)}{a^2 \left(a+b x^2\right) (b c-a d)^2}-\frac{\sqrt{b} \left(15 a^2 d^2-10 a b c d+3 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{5/2} (a d-b c)^3}-\frac{8 d^{5/2} \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{\sqrt{c} (b c-a d)^3}-\frac{2 b x}{a \left(a+b x^2\right)^2 (a d-b c)}\right)","\frac{b x (3 b c-7 a d)}{8 a^2 \left(a+b x^2\right) (b c-a d)^2}+\frac{\sqrt{b} \left(15 a^2 d^2-10 a b c d+3 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} (b c-a d)^3}-\frac{d^{5/2} \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{\sqrt{c} (b c-a d)^3}+\frac{b x}{4 a \left(a+b x^2\right)^2 (b c-a d)}",1,"((-2*b*x)/(a*(-(b*c) + a*d)*(a + b*x^2)^2) + (b*(3*b*c - 7*a*d)*x)/(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]])/(a^(5/2)*(-(b*c) + a*d)^3) - (8*d^(5/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*(b*c - a*d)^3))/8","A",1
41,1,197,236,0.4181223,"\int \frac{1}{\left(a+b x^2\right)^3 \left(c+d x^2\right)^2} \, dx","Integrate[1/((a + b*x^2)^3*(c + d*x^2)^2),x]","\frac{1}{8} \left(\frac{b^2 x (11 a d-3 b c)}{a^2 \left(a+b x^2\right) (a d-b c)^3}+\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)}{a^{5/2} (b c-a d)^4}+\frac{2 b^2 x}{a \left(a+b x^2\right)^2 (b c-a d)^2}+\frac{4 d^{5/2} (a d-7 b c) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{c^{3/2} (b c-a d)^4}-\frac{4 d^3 x}{c \left(c+d x^2\right) (b c-a d)^3}\right)","\frac{d x (b c-4 a d) (a d+3 b c)}{8 a^2 c \left(c+d x^2\right) (b c-a d)^3}+\frac{3 b x (b c-3 a d)}{8 a^2 \left(a+b x^2\right) \left(c+d x^2\right) (b c-a d)^2}+\frac{b^{3/2} \left(35 a^2 d^2-14 a b c d+3 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} (b c-a d)^4}-\frac{d^{5/2} (7 b c-a d) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{3/2} (b c-a d)^4}+\frac{b x}{4 a \left(a+b x^2\right)^2 \left(c+d x^2\right) (b c-a d)}",1,"((2*b^2*x)/(a*(b*c - a*d)^2*(a + b*x^2)^2) + (b^2*(-3*b*c + 11*a*d)*x)/(a^2*(-(b*c) + a*d)^3*(a + b*x^2)) - (4*d^3*x)/(c*(b*c - a*d)^3*(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]])/(a^(5/2)*(b*c - a*d)^4) + (4*d^(5/2)*(-7*b*c + a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(3/2)*(b*c - a*d)^4))/8","A",1
42,1,233,315,0.9284525,"\int \frac{1}{\left(a+b x^2\right)^3 \left(c+d x^2\right)^3} \, dx","Integrate[1/((a + b*x^2)^3*(c + d*x^2)^3),x]","\frac{1}{8} \left(\frac{x (b c-a d) \left(\frac{3 b^4 c}{a^2 \left(a+b x^2\right)}+\frac{b^3 \left(-17 a d+2 b c-15 b d x^2\right)}{a \left(a+b x^2\right)^2}-\frac{d^3 \left(-2 a d+17 b c+15 b d x^2\right)}{c \left(c+d x^2\right)^2}+\frac{3 a d^4}{c^2 \left(c+d x^2\right)}\right)-\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)}{c^{5/2}}}{(b c-a d)^5}-\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)}{a^{5/2} (a d-b c)^5}\right)","\frac{3 d x (a d+b c) \left(a^2 d^2-6 a b c d+b^2 c^2\right)}{8 a^2 c^2 \left(c+d x^2\right) (b c-a d)^4}+\frac{d x \left(-2 a^2 d^2-13 a b c d+3 b^2 c^2\right)}{8 a^2 c \left(c+d x^2\right)^2 (b c-a d)^3}-\frac{3 d^{5/2} \left(a^2 d^2-6 a b c d+21 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{5/2} (b c-a d)^5}+\frac{b x (3 b c-11 a d)}{8 a^2 \left(a+b x^2\right) \left(c+d x^2\right)^2 (b c-a d)^2}+\frac{3 b^{5/2} \left(21 a^2 d^2-6 a b c d+b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} (b c-a d)^5}+\frac{b x}{4 a \left(a+b x^2\right)^2 \left(c+d x^2\right)^2 (b c-a d)}",1,"((-3*b^(5/2)*(b^2*c^2 - 6*a*b*c*d + 21*a^2*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(5/2)*(-(b*c) + a*d)^5) + ((b*c - a*d)*x*((3*b^4*c)/(a^2*(a + b*x^2)) + (3*a*d^4)/(c^2*(c + d*x^2)) + (b^3*(2*b*c - 17*a*d - 15*b*d*x^2))/(a*(a + b*x^2)^2) - (d^3*(17*b*c - 2*a*d + 15*b*d*x^2))/(c*(c + d*x^2)^2)) - (3*d^(5/2)*(21*b^2*c^2 - 6*a*b*c*d + a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/c^(5/2))/(b*c - a*d)^5)/8","A",1
43,1,24,34,0.0076756,"\int \frac{\left(-1+x^2\right)^3}{\left(1+x^2\right)^4} \, dx","Integrate[(-1 + x^2)^3/(1 + x^2)^4,x]","-\frac{x \left(3 x^4+2 x^2+3\right)}{3 \left(x^2+1\right)^3}","-\frac{x \left(1-x^2\right)^2}{3 \left(x^2+1\right)^3}-\frac{2 x}{3 \left(x^2+1\right)}",1,"-1/3*(x*(3 + 2*x^2 + 3*x^4))/(1 + x^2)^3","A",1
44,1,41,47,0.0131774,"\int \frac{\left(-1+x^2\right)^4}{\left(1+x^2\right)^5} \, dx","Integrate[(-1 + x^2)^4/(1 + x^2)^5,x]","\frac{-5 x^7+3 x^5-3 x^3+3 \left(x^2+1\right)^4 \tan ^{-1}(x)+5 x}{8 \left(x^2+1\right)^4}","\frac{x \left(1-x^2\right)^3}{4 \left(x^2+1\right)^4}+\frac{3 x \left(1-x^2\right)}{8 \left(x^2+1\right)^2}+\frac{3}{8} \tan ^{-1}(x)",1,"(5*x - 3*x^3 + 3*x^5 - 5*x^7 + 3*(1 + x^2)^4*ArcTan[x])/(8*(1 + x^2)^4)","A",1
45,1,181,231,5.1142372,"\int \sqrt{a+b x^2} \left(c+d x^2\right)^3 \, dx","Integrate[Sqrt[a + b*x^2]*(c + d*x^2)^3,x]","\frac{\sqrt{b} x \sqrt{a+b x^2} \left(15 a^3 d^3-2 a^2 b d^2 \left(36 c+5 d x^2\right)+8 a b^2 d \left(18 c^2+6 c d x^2+d^2 x^4\right)+48 b^3 \left(4 c^3+6 c^2 d x^2+4 c d^2 x^4+d^3 x^6\right)\right)-3 a \left(5 a^3 d^3-24 a^2 b c d^2+48 a b^2 c^2 d-64 b^3 c^3\right) \log \left(\sqrt{b} \sqrt{a+b x^2}+b x\right)}{384 b^{7/2}}","\frac{d x \left(a+b x^2\right)^{3/2} \left(15 a^2 d^2-52 a b c d+72 b^2 c^2\right)}{192 b^3}+\frac{x \sqrt{a+b x^2} \left(-5 a^3 d^3+24 a^2 b c d^2-48 a b^2 c^2 d+64 b^3 c^3\right)}{128 b^3}+\frac{a \left(-5 a^3 d^3+24 a^2 b c d^2-48 a b^2 c^2 d+64 b^3 c^3\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{128 b^{7/2}}+\frac{d x \left(a+b x^2\right)^{3/2} \left(c+d x^2\right) (12 b c-5 a d)}{48 b^2}+\frac{d x \left(a+b x^2\right)^{3/2} \left(c+d x^2\right)^2}{8 b}",1,"(Sqrt[b]*x*Sqrt[a + b*x^2]*(15*a^3*d^3 - 2*a^2*b*d^2*(36*c + 5*d*x^2) + 8*a*b^2*d*(18*c^2 + 6*c*d*x^2 + d^2*x^4) + 48*b^3*(4*c^3 + 6*c^2*d*x^2 + 4*c*d^2*x^4 + d^3*x^6)) - 3*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)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(384*b^(7/2))","A",1
46,1,160,149,2.672801,"\int \sqrt{a+b x^2} \left(c+d x^2\right)^2 \, dx","Integrate[Sqrt[a + b*x^2]*(c + d*x^2)^2,x]","\frac{x \sqrt{a+b x^2} \left(2 b x^2 \left(c+d x^2\right)^2 \, _3F_2\left(\frac{1}{2},\frac{3}{2},2;1,\frac{9}{2};-\frac{b x^2}{a}\right)+4 b x^2 \left(2 c^2+3 c d x^2+d^2 x^4\right) \, _2F_1\left(\frac{1}{2},\frac{3}{2};\frac{9}{2};-\frac{b x^2}{a}\right)+7 a \left(15 c^2+10 c d x^2+3 d^2 x^4\right) \, _2F_1\left(-\frac{1}{2},\frac{1}{2};\frac{7}{2};-\frac{b x^2}{a}\right)\right)}{105 a \sqrt{\frac{b x^2}{a}+1}}","\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,"(x*Sqrt[a + b*x^2]*(7*a*(15*c^2 + 10*c*d*x^2 + 3*d^2*x^4)*Hypergeometric2F1[-1/2, 1/2, 7/2, -((b*x^2)/a)] + 4*b*x^2*(2*c^2 + 3*c*d*x^2 + d^2*x^4)*Hypergeometric2F1[1/2, 3/2, 9/2, -((b*x^2)/a)] + 2*b*x^2*(c + d*x^2)^2*HypergeometricPFQ[{1/2, 3/2, 2}, {1, 9/2}, -((b*x^2)/a)]))/(105*a*Sqrt[1 + (b*x^2)/a])","C",0
47,1,85,87,0.1573261,"\int \sqrt{a+b x^2} \left(c+d x^2\right) \, dx","Integrate[Sqrt[a + b*x^2]*(c + d*x^2),x]","\frac{\sqrt{a+b x^2} \left(\sqrt{b} x \left(a d+4 b c+2 b d x^2\right)-\frac{\sqrt{a} (a d-4 b c) \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{\frac{b x^2}{a}+1}}\right)}{8 b^{3/2}}","\frac{a (4 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 b^{3/2}}+\frac{x \sqrt{a+b x^2} (4 b c-a d)}{8 b}+\frac{d x \left(a+b x^2\right)^{3/2}}{4 b}",1,"(Sqrt[a + b*x^2]*(Sqrt[b]*x*(4*b*c + a*d + 2*b*d*x^2) - (Sqrt[a]*(-4*b*c + a*d)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[1 + (b*x^2)/a]))/(8*b^(3/2))","A",1
48,1,49,46,0.0205827,"\int \sqrt{a+b x^2} \, dx","Integrate[Sqrt[a + b*x^2],x]","\frac{1}{2} x \sqrt{a+b x^2}+\frac{a \log \left(\sqrt{b} \sqrt{a+b x^2}+b x\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*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(2*Sqrt[b])","A",1
49,1,84,82,0.0441747,"\int \frac{\sqrt{a+b x^2}}{c+d x^2} \, dx","Integrate[Sqrt[a + b*x^2]/(c + d*x^2),x]","\frac{\sqrt{a d-b c} \tan ^{-1}\left(\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{\sqrt{c} d}+\frac{\sqrt{b} \log \left(\sqrt{b} \sqrt{a+b x^2}+b x\right)}{d}","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{d}-\frac{\sqrt{b c-a d} \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{\sqrt{c} d}",1,"(Sqrt[-(b*c) + a*d]*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(Sqrt[c]*d) + (Sqrt[b]*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/d","A",1
50,1,165,82,0.2343818,"\int \frac{\sqrt{a+b x^2}}{\left(c+d x^2\right)^2} \, dx","Integrate[Sqrt[a + b*x^2]/(c + d*x^2)^2,x]","\frac{x \sqrt{a+b x^2} \left(\sqrt{x^2 \left(\frac{d}{c}-\frac{b}{a}\right)} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}+\sqrt{\frac{d x^2}{c}+1} \sin ^{-1}\left(\frac{\sqrt{x^2 \left(\frac{d}{c}-\frac{b}{a}\right)}}{\sqrt{\frac{d x^2}{c}+1}}\right)\right)}{2 c^2 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \sqrt{x^2 \left(\frac{d}{c}-\frac{b}{a}\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]*(Sqrt[(-(b/a) + d/c)*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))] + Sqrt[1 + (d*x^2)/c]*ArcSin[Sqrt[(-(b/a) + d/c)*x^2]/Sqrt[1 + (d*x^2)/c]]))/(2*c^2*Sqrt[(-(b/a) + d/c)*x^2]*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c])","B",0
51,1,176,149,0.5662374,"\int \frac{\sqrt{a+b x^2}}{\left(c+d x^2\right)^3} \, dx","Integrate[Sqrt[a + b*x^2]/(c + d*x^2)^3,x]","\frac{x \left(c \left(a^2 d \left(5 c+3 d x^2\right)+a b \left(-4 c^2+3 c d x^2+3 d^2 x^4\right)-2 b^2 c x^2 \left(2 c+d x^2\right)\right)+\frac{a \left(c+d x^2\right)^2 (3 a d-4 b c) \tanh ^{-1}\left(\sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}\right)}{\sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}}\right)}{8 c^3 \sqrt{a+b x^2} \left(c+d x^2\right)^2 (a d-b c)}","\frac{a (4 b c-3 a d) \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{8 c^{5/2} (b c-a d)^{3/2}}+\frac{x \sqrt{a+b x^2} (4 b c-3 a d)}{8 c^2 \left(c+d x^2\right) (b c-a d)}-\frac{d x \left(a+b x^2\right)^{3/2}}{4 c \left(c+d x^2\right)^2 (b c-a d)}",1,"(x*(c*(-2*b^2*c*x^2*(2*c + d*x^2) + a^2*d*(5*c + 3*d*x^2) + a*b*(-4*c^2 + 3*c*d*x^2 + 3*d^2*x^4)) + (a*(-4*b*c + 3*a*d)*(c + d*x^2)^2*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]))/(8*c^3*(-(b*c) + a*d)*Sqrt[a + b*x^2]*(c + d*x^2)^2)","A",1
52,1,227,208,0.9899338,"\int \frac{\sqrt{a+b x^2}}{\left(c+d x^2\right)^4} \, dx","Integrate[Sqrt[a + b*x^2]/(c + d*x^2)^4,x]","\frac{x \sqrt{a+b x^2} \left(\frac{3 a \left(c+d x^2\right)^3 \left(5 a^2 d^2-12 a b c d+8 b^2 c^2\right) \sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}} \tanh ^{-1}\left(\sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}\right)}{x^2}+(b c-a d) \left(a^2 d^2 \left(33 c^2+40 c d x^2+15 d^2 x^4\right)-2 a b c d \left(30 c^2+35 c d x^2+13 d^2 x^4\right)+8 b^2 c^2 \left(3 c^2+3 c d x^2+d^2 x^4\right)\right)\right)}{48 c^3 \left(c+d x^2\right)^3 (b c-a d)^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]*((b*c - a*d)*(8*b^2*c^2*(3*c^2 + 3*c*d*x^2 + d^2*x^4) - 2*a*b*c*d*(30*c^2 + 35*c*d*x^2 + 13*d^2*x^4) + a^2*d^2*(33*c^2 + 40*c*d*x^2 + 15*d^2*x^4)) + (3*a*(8*b^2*c^2 - 12*a*b*c*d + 5*a^2*d^2)*Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]*(c + d*x^2)^3*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/x^2))/(48*c^3*(b*c - a*d)^3*(c + d*x^2)^3)","A",1
53,1,220,272,5.1293719,"\int \left(a+b x^2\right)^{3/2} \left(c+d x^2\right)^3 \, dx","Integrate[(a + b*x^2)^(3/2)*(c + d*x^2)^3,x]","\frac{\sqrt{b} x \sqrt{a+b x^2} \left(15 a^4 d^3-10 a^3 b d^2 \left(9 c+d x^2\right)+4 a^2 b^2 d \left(60 c^2+15 c d x^2+2 d^2 x^4\right)+16 a b^3 \left(50 c^3+70 c^2 d x^2+45 c d^2 x^4+11 d^3 x^6\right)+32 b^4 x^2 \left(10 c^3+20 c^2 d x^2+15 c d^2 x^4+4 d^3 x^6\right)\right)-15 a^2 (a d-4 b c) \left(a^2 d^2-2 a b c d+8 b^2 c^2\right) \log \left(\sqrt{b} \sqrt{a+b x^2}+b x\right)}{1280 b^{7/2}}","\frac{3 a^2 (4 b c-a d) \left(a^2 d^2-2 a b c d+8 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{256 b^{7/2}}+\frac{d x \left(a+b x^2\right)^{5/2} \left(5 a^2 d^2-20 a b c d+36 b^2 c^2\right)}{160 b^3}+\frac{x \left(a+b x^2\right)^{3/2} (4 b c-a d) \left(a^2 d^2-2 a b c d+8 b^2 c^2\right)}{128 b^3}+\frac{3 a x \sqrt{a+b x^2} (4 b c-a d) \left(a^2 d^2-2 a b c d+8 b^2 c^2\right)}{256 b^3}+\frac{d x \left(a+b x^2\right)^{5/2} \left(c+d x^2\right) (14 b c-5 a d)}{80 b^2}+\frac{d x \left(a+b x^2\right)^{5/2} \left(c+d x^2\right)^2}{10 b}",1,"(Sqrt[b]*x*Sqrt[a + b*x^2]*(15*a^4*d^3 - 10*a^3*b*d^2*(9*c + d*x^2) + 4*a^2*b^2*d*(60*c^2 + 15*c*d*x^2 + 2*d^2*x^4) + 32*b^4*x^2*(10*c^3 + 20*c^2*d*x^2 + 15*c*d^2*x^4 + 4*d^3*x^6) + 16*a*b^3*(50*c^3 + 70*c^2*d*x^2 + 45*c*d^2*x^4 + 11*d^3*x^6)) - 15*a^2*(-4*b*c + a*d)*(8*b^2*c^2 - 2*a*b*c*d + a^2*d^2)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(1280*b^(7/2))","A",1
54,1,157,196,2.7123414,"\int \left(a+b x^2\right)^{3/2} \left(c+d x^2\right)^2 \, dx","Integrate[(a + b*x^2)^(3/2)*(c + d*x^2)^2,x]","\frac{x \sqrt{a+b x^2} \left(6 b x^2 \left(c+d x^2\right)^2 \, _3F_2\left(-\frac{1}{2},\frac{3}{2},2;1,\frac{9}{2};-\frac{b x^2}{a}\right)+12 b x^2 \left(2 c^2+3 c d x^2+d^2 x^4\right) \, _2F_1\left(-\frac{1}{2},\frac{3}{2};\frac{9}{2};-\frac{b x^2}{a}\right)+7 a \left(15 c^2+10 c d x^2+3 d^2 x^4\right) \, _2F_1\left(-\frac{3}{2},\frac{1}{2};\frac{7}{2};-\frac{b x^2}{a}\right)\right)}{105 \sqrt{\frac{b x^2}{a}+1}}","\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,"(x*Sqrt[a + b*x^2]*(7*a*(15*c^2 + 10*c*d*x^2 + 3*d^2*x^4)*Hypergeometric2F1[-3/2, 1/2, 7/2, -((b*x^2)/a)] + 12*b*x^2*(2*c^2 + 3*c*d*x^2 + d^2*x^4)*Hypergeometric2F1[-1/2, 3/2, 9/2, -((b*x^2)/a)] + 6*b*x^2*(c + d*x^2)^2*HypergeometricPFQ[{-1/2, 3/2, 2}, {1, 9/2}, -((b*x^2)/a)]))/(105*Sqrt[1 + (b*x^2)/a])","C",0
55,1,109,118,0.2037559,"\int \left(a+b x^2\right)^{3/2} \left(c+d x^2\right) \, dx","Integrate[(a + b*x^2)^(3/2)*(c + d*x^2),x]","\frac{\sqrt{a+b x^2} \left(\sqrt{b} x \left(3 a^2 d+2 a b \left(15 c+7 d x^2\right)+4 b^2 x^2 \left(3 c+2 d x^2\right)\right)-\frac{3 a^{3/2} (a d-6 b c) \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{\frac{b x^2}{a}+1}}\right)}{48 b^{3/2}}","\frac{a^2 (6 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{16 b^{3/2}}+\frac{x \left(a+b x^2\right)^{3/2} (6 b c-a d)}{24 b}+\frac{a x \sqrt{a+b x^2} (6 b c-a d)}{16 b}+\frac{d x \left(a+b x^2\right)^{5/2}}{6 b}",1,"(Sqrt[a + b*x^2]*(Sqrt[b]*x*(3*a^2*d + 4*b^2*x^2*(3*c + 2*d*x^2) + 2*a*b*(15*c + 7*d*x^2)) - (3*a^(3/2)*(-6*b*c + a*d)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[1 + (b*x^2)/a]))/(48*b^(3/2))","A",1
56,1,65,65,0.0898503,"\int \left(a+b x^2\right)^{3/2} \, dx","Integrate[(a + b*x^2)^(3/2),x]","\frac{1}{8} \sqrt{a+b x^2} \left(\frac{3 a^{3/2} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{\frac{b x^2}{a}+1}}+5 a x+2 b x^3\right)","\frac{3 a^2 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 \sqrt{b}}+\frac{3}{8} a x \sqrt{a+b x^2}+\frac{1}{4} x \left(a+b x^2\right)^{3/2}",1,"(Sqrt[a + b*x^2]*(5*a*x + 2*b*x^3 + (3*a^(3/2)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[b]*Sqrt[1 + (b*x^2)/a])))/8","A",1
57,1,110,113,0.1995484,"\int \frac{\left(a+b x^2\right)^{3/2}}{c+d x^2} \, dx","Integrate[(a + b*x^2)^(3/2)/(c + d*x^2),x]","\frac{\sqrt{b} (3 a d-2 b c) \log \left(\sqrt{b} \sqrt{a+b x^2}+b x\right)+\frac{2 (a d-b c)^{3/2} \tan ^{-1}\left(\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{\sqrt{c}}+b d x \sqrt{a+b x^2}}{2 d^2}","\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*d*x*Sqrt[a + b*x^2] + (2*(-(b*c) + a*d)^(3/2)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/Sqrt[c] + Sqrt[b]*(-2*b*c + 3*a*d)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(2*d^2)","A",1
58,1,142,131,0.1282024,"\int \frac{\left(a+b x^2\right)^{3/2}}{\left(c+d x^2\right)^2} \, dx","Integrate[(a + b*x^2)^(3/2)/(c + d*x^2)^2,x]","\frac{\frac{\left(a^2 d^2+a b c d-2 b^2 c^2\right) \tan ^{-1}\left(\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{c^{3/2} \sqrt{a d-b c}}+2 b^{3/2} \log \left(\sqrt{b} \sqrt{a+b x^2}+b x\right)-\frac{d x \sqrt{a+b x^2} (b c-a d)}{c \left(c+d x^2\right)}}{2 d^2}","\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,"(-((d*(b*c - a*d)*x*Sqrt[a + b*x^2])/(c*(c + d*x^2))) + ((-2*b^2*c^2 + a*b*c*d + a^2*d^2)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(c^(3/2)*Sqrt[-(b*c) + a*d]) + 2*b^(3/2)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(2*d^2)","A",1
59,1,163,113,0.6929251,"\int \frac{\left(a+b x^2\right)^{3/2}}{\left(c+d x^2\right)^3} \, dx","Integrate[(a + b*x^2)^(3/2)/(c + d*x^2)^3,x]","\frac{x \sqrt{a+b x^2} \left(\frac{\sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}} \left(5 a c+3 a d x^2+2 b c x^2\right)}{\left(c+d x^2\right) \sqrt{\frac{d x^2}{c}+1}}+\frac{3 a \sin ^{-1}\left(\frac{\sqrt{x^2 \left(\frac{d}{c}-\frac{b}{a}\right)}}{\sqrt{\frac{d x^2}{c}+1}}\right)}{\sqrt{\frac{x^2 (a d-b c)}{a c}}}\right)}{8 c^3 \sqrt{\frac{b x^2}{a}+1}}","\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*Sqrt[a + b*x^2]*((Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*(5*a*c + 2*b*c*x^2 + 3*a*d*x^2))/((c + d*x^2)*Sqrt[1 + (d*x^2)/c]) + (3*a*ArcSin[Sqrt[(-(b/a) + d/c)*x^2]/Sqrt[1 + (d*x^2)/c]])/Sqrt[((-(b*c) + a*d)*x^2)/(a*c)]))/(8*c^3*Sqrt[1 + (b*x^2)/a])","A",0
60,1,247,199,0.8171884,"\int \frac{\left(a+b x^2\right)^{3/2}}{\left(c+d x^2\right)^4} \, dx","Integrate[(a + b*x^2)^(3/2)/(c + d*x^2)^4,x]","\frac{a x \left(\frac{b x^2}{a}+1\right) \left(\frac{3 a^2 \left(c+d x^2\right)^3 (5 a d-6 b c) \tanh ^{-1}\left(\sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}\right)}{\sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}}+c \left(a^3 d \left(33 c^2+40 c d x^2+15 d^2 x^4\right)+a^2 b \left(-30 c^3+11 c^2 d x^2+32 c d^2 x^4+15 d^3 x^6\right)-2 a b^2 c x^2 \left(21 c^2+13 c d x^2+4 d^2 x^4\right)-4 b^3 c^2 x^4 \left(3 c+d x^2\right)\right)\right)}{48 c^4 \left(a+b x^2\right)^{3/2} \left(c+d x^2\right)^3 (a d-b c)}","\frac{a^2 (6 b c-5 a d) \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{16 c^{7/2} (b c-a d)^{3/2}}+\frac{a x \sqrt{a+b x^2} (6 b c-5 a d)}{16 c^3 \left(c+d x^2\right) (b c-a d)}+\frac{x \left(a+b x^2\right)^{3/2} (6 b c-5 a d)}{24 c^2 \left(c+d x^2\right)^2 (b c-a d)}-\frac{d x \left(a+b x^2\right)^{5/2}}{6 c \left(c+d x^2\right)^3 (b c-a d)}",1,"(a*x*(1 + (b*x^2)/a)*(c*(-4*b^3*c^2*x^4*(3*c + d*x^2) - 2*a*b^2*c*x^2*(21*c^2 + 13*c*d*x^2 + 4*d^2*x^4) + a^3*d*(33*c^2 + 40*c*d*x^2 + 15*d^2*x^4) + a^2*b*(-30*c^3 + 11*c^2*d*x^2 + 32*c*d^2*x^4 + 15*d^3*x^6)) + (3*a^2*(-6*b*c + 5*a*d)*(c + d*x^2)^3*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]))/(48*c^4*(-(b*c) + a*d)*(a + b*x^2)^(3/2)*(c + d*x^2)^3)","A",1
61,1,362,300,1.3848557,"\int \frac{\left(a+b x^2\right)^{3/2}}{\left(c+d x^2\right)^5} \, dx","Integrate[(a + b*x^2)^(3/2)/(c + d*x^2)^5,x]","\frac{a x \left(\frac{b x^2}{a}+1\right) \left(\frac{3 a^2 \left(c+d x^2\right)^4 \left(35 a^2 d^2-80 a b c d+48 b^2 c^2\right) \tanh ^{-1}\left(\sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}\right)}{\sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}}+c \left(a^4 d^2 \left(279 c^3+511 c^2 d x^2+385 c d^2 x^4+105 d^3 x^6\right)+a^3 b d \left(-528 c^4-563 c^3 d x^2-117 c^2 d^2 x^4+215 c d^3 x^6+105 d^4 x^8\right)+2 a^2 b^2 c \left(120 c^4-160 c^3 d x^2-345 c^2 d^2 x^4-294 c d^3 x^6-85 d^4 x^8\right)+8 a b^3 c^2 x^2 \left(42 c^3+34 c^2 d x^2+21 c d^2 x^4+5 d^3 x^6\right)+16 b^4 c^3 x^4 \left(6 c^2+4 c d x^2+d^2 x^4\right)\right)\right)}{384 c^5 \left(a+b x^2\right)^{3/2} \left(c+d x^2\right)^4 (b c-a d)^2}","\frac{a^2 \left(35 a^2 d^2-80 a b c d+48 b^2 c^2\right) \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{128 c^{9/2} (b c-a d)^{5/2}}+\frac{x \sqrt{a+b x^2} \left(-35 a^2 d^2+24 a b c d+8 b^2 c^2\right)}{192 c^3 d \left(c+d x^2\right)^2 (b c-a d)}+\frac{x \sqrt{a+b x^2} \left(105 a^3 d^3-170 a^2 b c d^2+40 a b^2 c^2 d+16 b^3 c^3\right)}{384 c^4 d \left(c+d x^2\right) (b c-a d)^2}+\frac{x \sqrt{a+b x^2} (7 a d+2 b c)}{48 c^2 d \left(c+d x^2\right)^3}-\frac{x \sqrt{a+b x^2} (b c-a d)}{8 c d \left(c+d x^2\right)^4}",1,"(a*x*(1 + (b*x^2)/a)*(c*(16*b^4*c^3*x^4*(6*c^2 + 4*c*d*x^2 + d^2*x^4) + 8*a*b^3*c^2*x^2*(42*c^3 + 34*c^2*d*x^2 + 21*c*d^2*x^4 + 5*d^3*x^6) + a^4*d^2*(279*c^3 + 511*c^2*d*x^2 + 385*c*d^2*x^4 + 105*d^3*x^6) + 2*a^2*b^2*c*(120*c^4 - 160*c^3*d*x^2 - 345*c^2*d^2*x^4 - 294*c*d^3*x^6 - 85*d^4*x^8) + a^3*b*d*(-528*c^4 - 563*c^3*d*x^2 - 117*c^2*d^2*x^4 + 215*c*d^3*x^6 + 105*d^4*x^8)) + (3*a^2*(48*b^2*c^2 - 80*a*b*c*d + 35*a^2*d^2)*(c + d*x^2)^4*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]))/(384*c^5*(b*c - a*d)^2*(a + b*x^2)^(3/2)*(c + d*x^2)^4)","A",1
62,1,270,349,5.1785082,"\int \left(a+b x^2\right)^{5/2} \left(c+d x^2\right)^3 \, dx","Integrate[(a + b*x^2)^(5/2)*(c + d*x^2)^3,x]","\frac{\sqrt{b} x \sqrt{a+b x^2} \left(75 a^5 d^3-10 a^4 b d^2 \left(54 c+5 d x^2\right)+40 a^3 b^2 d \left(45 c^2+9 c d x^2+d^2 x^4\right)+48 a^2 b^3 \left(220 c^3+295 c^2 d x^2+186 c d^2 x^4+45 d^3 x^6\right)+64 a b^4 x^2 \left(130 c^3+255 c^2 d x^2+189 c d^2 x^4+50 d^3 x^6\right)+128 b^5 x^4 \left(20 c^3+45 c^2 d x^2+36 c d^2 x^4+10 d^3 x^6\right)\right)-15 a^3 \left(5 a^3 d^3-36 a^2 b c d^2+120 a b^2 c^2 d-320 b^3 c^3\right) \log \left(\sqrt{b} \sqrt{a+b x^2}+b x\right)}{15360 b^{7/2}}","\frac{d x \left(a+b x^2\right)^{7/2} \left(15 a^2 d^2-68 a b c d+152 b^2 c^2\right)}{960 b^3}+\frac{x \left(a+b x^2\right)^{5/2} \left(-5 a^3 d^3+36 a^2 b c d^2-120 a b^2 c^2 d+320 b^3 c^3\right)}{1920 b^3}+\frac{a x \left(a+b x^2\right)^{3/2} \left(-5 a^3 d^3+36 a^2 b c d^2-120 a b^2 c^2 d+320 b^3 c^3\right)}{1536 b^3}+\frac{a^2 x \sqrt{a+b x^2} \left(-5 a^3 d^3+36 a^2 b c d^2-120 a b^2 c^2 d+320 b^3 c^3\right)}{1024 b^3}+\frac{a^3 \left(-5 a^3 d^3+36 a^2 b c d^2-120 a b^2 c^2 d+320 b^3 c^3\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{1024 b^{7/2}}+\frac{d x \left(a+b x^2\right)^{7/2} \left(c+d x^2\right) (16 b c-5 a d)}{120 b^2}+\frac{d x \left(a+b x^2\right)^{7/2} \left(c+d x^2\right)^2}{12 b}",1,"(Sqrt[b]*x*Sqrt[a + b*x^2]*(75*a^5*d^3 - 10*a^4*b*d^2*(54*c + 5*d*x^2) + 40*a^3*b^2*d*(45*c^2 + 9*c*d*x^2 + d^2*x^4) + 128*b^5*x^4*(20*c^3 + 45*c^2*d*x^2 + 36*c*d^2*x^4 + 10*d^3*x^6) + 48*a^2*b^3*(220*c^3 + 295*c^2*d*x^2 + 186*c*d^2*x^4 + 45*d^3*x^6) + 64*a*b^4*x^2*(130*c^3 + 255*c^2*d*x^2 + 189*c*d^2*x^4 + 50*d^3*x^6)) - 15*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)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(15360*b^(7/2))","A",1
63,1,158,241,2.8024371,"\int \left(a+b x^2\right)^{5/2} \left(c+d x^2\right)^2 \, dx","Integrate[(a + b*x^2)^(5/2)*(c + d*x^2)^2,x]","\frac{a x \sqrt{a+b x^2} \left(10 b x^2 \left(c+d x^2\right)^2 \, _3F_2\left(-\frac{3}{2},\frac{3}{2},2;1,\frac{9}{2};-\frac{b x^2}{a}\right)+20 b x^2 \left(2 c^2+3 c d x^2+d^2 x^4\right) \, _2F_1\left(-\frac{3}{2},\frac{3}{2};\frac{9}{2};-\frac{b x^2}{a}\right)+7 a \left(15 c^2+10 c d x^2+3 d^2 x^4\right) \, _2F_1\left(-\frac{5}{2},\frac{1}{2};\frac{7}{2};-\frac{b x^2}{a}\right)\right)}{105 \sqrt{\frac{b x^2}{a}+1}}","\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*x*Sqrt[a + b*x^2]*(7*a*(15*c^2 + 10*c*d*x^2 + 3*d^2*x^4)*Hypergeometric2F1[-5/2, 1/2, 7/2, -((b*x^2)/a)] + 20*b*x^2*(2*c^2 + 3*c*d*x^2 + d^2*x^4)*Hypergeometric2F1[-3/2, 3/2, 9/2, -((b*x^2)/a)] + 10*b*x^2*(c + d*x^2)^2*HypergeometricPFQ[{-3/2, 3/2, 2}, {1, 9/2}, -((b*x^2)/a)]))/(105*Sqrt[1 + (b*x^2)/a])","C",0
64,1,130,149,0.2281882,"\int \left(a+b x^2\right)^{5/2} \left(c+d x^2\right) \, dx","Integrate[(a + b*x^2)^(5/2)*(c + d*x^2),x]","\frac{\sqrt{a+b x^2} \left(\sqrt{b} x \left(15 a^3 d+2 a^2 b \left(132 c+59 d x^2\right)+8 a b^2 x^2 \left(26 c+17 d x^2\right)+16 b^3 x^4 \left(4 c+3 d x^2\right)\right)-\frac{15 a^{5/2} (a d-8 b c) \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{\frac{b x^2}{a}+1}}\right)}{384 b^{3/2}}","\frac{5 a^3 (8 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{128 b^{3/2}}+\frac{5 a^2 x \sqrt{a+b x^2} (8 b c-a d)}{128 b}+\frac{x \left(a+b x^2\right)^{5/2} (8 b c-a d)}{48 b}+\frac{5 a x \left(a+b x^2\right)^{3/2} (8 b c-a d)}{192 b}+\frac{d x \left(a+b x^2\right)^{7/2}}{8 b}",1,"(Sqrt[a + b*x^2]*(Sqrt[b]*x*(15*a^3*d + 16*b^3*x^4*(4*c + 3*d*x^2) + 8*a*b^2*x^2*(26*c + 17*d*x^2) + 2*a^2*b*(132*c + 59*d*x^2)) - (15*a^(5/2)*(-8*b*c + a*d)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[1 + (b*x^2)/a]))/(384*b^(3/2))","A",1
65,1,76,84,0.1118724,"\int \left(a+b x^2\right)^{5/2} \, dx","Integrate[(a + b*x^2)^(5/2),x]","\frac{1}{48} \sqrt{a+b x^2} \left(\frac{15 a^{5/2} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{\frac{b x^2}{a}+1}}+33 a^2 x+26 a b x^3+8 b^2 x^5\right)","\frac{5 a^3 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{16 \sqrt{b}}+\frac{5}{16} a^2 x \sqrt{a+b x^2}+\frac{5}{24} a x \left(a+b x^2\right)^{3/2}+\frac{1}{6} x \left(a+b x^2\right)^{5/2}",1,"(Sqrt[a + b*x^2]*(33*a^2*x + 26*a*b*x^3 + 8*b^2*x^5 + (15*a^(5/2)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[b]*Sqrt[1 + (b*x^2)/a])))/48","A",1
66,1,140,157,0.1216506,"\int \frac{\left(a+b x^2\right)^{5/2}}{c+d x^2} \, dx","Integrate[(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) \log \left(\sqrt{b} \sqrt{a+b x^2}+b x\right)+b d x \sqrt{a+b x^2} \left(9 a d-4 b c+2 b d x^2\right)+\frac{8 (a d-b c)^{5/2} \tan ^{-1}\left(\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{\sqrt{c}}}{8 d^3}","\frac{\sqrt{b} \left(15 a^2 d^2-20 a b c d+8 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 d^3}-\frac{(b c-a d)^{5/2} \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{\sqrt{c} d^3}-\frac{b x \sqrt{a+b x^2} (4 b c-7 a d)}{8 d^2}+\frac{b x \left(a+b x^2\right)^{3/2}}{4 d}",1,"(b*d*x*Sqrt[a + b*x^2]*(-4*b*c + 9*a*d + 2*b*d*x^2) + (8*(-(b*c) + a*d)^(5/2)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/Sqrt[c] + Sqrt[b]*(8*b^2*c^2 - 20*a*b*c*d + 15*a^2*d^2)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(8*d^3)","A",1
67,1,144,175,0.164342,"\int \frac{\left(a+b x^2\right)^{5/2}}{\left(c+d x^2\right)^2} \, dx","Integrate[(a + b*x^2)^(5/2)/(c + d*x^2)^2,x]","\frac{-\left(b^{3/2} (4 b c-5 a d) \log \left(\sqrt{b} \sqrt{a+b x^2}+b x\right)\right)+d x \sqrt{a+b x^2} \left(\frac{(b c-a d)^2}{c \left(c+d x^2\right)}+b^2\right)+\frac{(a d-b c)^{3/2} (a d+4 b c) \tan ^{-1}\left(\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{c^{3/2}}}{2 d^3}","-\frac{b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 d^3}+\frac{(b c-a d)^{3/2} (a d+4 b c) \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{2 c^{3/2} d^3}+\frac{b x \sqrt{a+b x^2} (2 b c-a d)}{2 c d^2}-\frac{x \left(a+b x^2\right)^{3/2} (b c-a d)}{2 c d \left(c+d x^2\right)}",1,"(d*x*Sqrt[a + b*x^2]*(b^2 + (b*c - a*d)^2/(c*(c + d*x^2))) + ((-(b*c) + a*d)^(3/2)*(4*b*c + a*d)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/c^(3/2) - b^(3/2)*(4*b*c - 5*a*d)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(2*d^3)","A",1
68,1,184,194,0.2026315,"\int \frac{\left(a+b x^2\right)^{5/2}}{\left(c+d x^2\right)^3} \, dx","Integrate[(a + b*x^2)^(5/2)/(c + d*x^2)^3,x]","\frac{\frac{\left(3 a^3 d^3+a^2 b c d^2+4 a b^2 c^2 d-8 b^3 c^3\right) \tan ^{-1}\left(\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{c^{5/2} \sqrt{a d-b c}}+8 b^{5/2} \log \left(\sqrt{b} \sqrt{a+b x^2}+b x\right)+\frac{d x \sqrt{a+b x^2} (a d-b c) \left(a d \left(5 c+3 d x^2\right)+2 b c \left(2 c+3 d x^2\right)\right)}{c^2 \left(c+d x^2\right)^2}}{8 d^3}","-\frac{\sqrt{b c-a d} \left(3 a^2 d^2+4 a b c d+8 b^2 c^2\right) \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{8 c^{5/2} d^3}+\frac{b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{d^3}-\frac{x \sqrt{a+b x^2} (b c-a d) (3 a d+4 b c)}{8 c^2 d^2 \left(c+d x^2\right)}-\frac{x \left(a+b x^2\right)^{3/2} (b c-a d)}{4 c d \left(c+d x^2\right)^2}",1,"((d*(-(b*c) + a*d)*x*Sqrt[a + b*x^2]*(2*b*c*(2*c + 3*d*x^2) + a*d*(5*c + 3*d*x^2)))/(c^2*(c + d*x^2)^2) + ((-8*b^3*c^3 + 4*a*b^2*c^2*d + a^2*b*c*d^2 + 3*a^3*d^3)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(c^(5/2)*Sqrt[-(b*c) + a*d]) + 8*b^(5/2)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(8*d^3)","A",1
69,1,201,144,0.8067818,"\int \frac{\left(a+b x^2\right)^{5/2}}{\left(c+d x^2\right)^4} \, dx","Integrate[(a + b*x^2)^(5/2)/(c + d*x^2)^4,x]","\frac{x \sqrt{a+b x^2} \left(\frac{\sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}} \left(a^2 \left(33 c^2+40 c d x^2+15 d^2 x^4\right)+2 a b c x^2 \left(13 c+5 d x^2\right)+8 b^2 c^2 x^4\right)}{\left(c+d x^2\right)^2 \sqrt{\frac{d x^2}{c}+1}}+\frac{15 a^2 \sin ^{-1}\left(\frac{\sqrt{x^2 \left(\frac{d}{c}-\frac{b}{a}\right)}}{\sqrt{\frac{d x^2}{c}+1}}\right)}{\sqrt{\frac{x^2 (a d-b c)}{a c}}}\right)}{48 c^4 \sqrt{\frac{b x^2}{a}+1}}","\frac{5 a^3 \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{16 c^{7/2} \sqrt{b c-a d}}+\frac{5 a^2 x \sqrt{a+b x^2}}{16 c^3 \left(c+d x^2\right)}+\frac{5 a x \left(a+b x^2\right)^{3/2}}{24 c^2 \left(c+d x^2\right)^2}+\frac{x \left(a+b x^2\right)^{5/2}}{6 c \left(c+d x^2\right)^3}",1,"(x*Sqrt[a + b*x^2]*((Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*(8*b^2*c^2*x^4 + 2*a*b*c*x^2*(13*c + 5*d*x^2) + a^2*(33*c^2 + 40*c*d*x^2 + 15*d^2*x^4)))/((c + d*x^2)^2*Sqrt[1 + (d*x^2)/c]) + (15*a^2*ArcSin[Sqrt[(-(b/a) + d/c)*x^2]/Sqrt[1 + (d*x^2)/c]])/Sqrt[((-(b*c) + a*d)*x^2)/(a*c)]))/(48*c^4*Sqrt[1 + (b*x^2)/a])","A",0
70,1,306,249,1.0937984,"\int \frac{\left(a+b x^2\right)^{5/2}}{\left(c+d x^2\right)^5} \, dx","Integrate[(a + b*x^2)^(5/2)/(c + d*x^2)^5,x]","\frac{x \left(\frac{15 a^3 \left(c+d x^2\right)^4 (7 a d-8 b c) \tanh ^{-1}\left(\sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}\right)}{\sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}}-c \left(-a^4 d \left(279 c^3+511 c^2 d x^2+385 c d^2 x^4+105 d^3 x^6\right)+a^3 b \left(264 c^4-21 c^3 d x^2-323 c^2 d^2 x^4-335 c d^3 x^6-105 d^4 x^8\right)+2 a^2 b^2 c x^2 \left(236 c^3+173 c^2 d x^2+106 c d^2 x^4+25 d^3 x^6\right)+8 a b^3 c^2 x^4 \left(34 c^2+13 c d x^2+3 d^2 x^4\right)+16 b^4 c^3 x^6 \left(4 c+d x^2\right)\right)\right)}{384 c^5 \sqrt{a+b x^2} \left(c+d x^2\right)^4 (a d-b c)}","\frac{5 a^3 (8 b c-7 a d) \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{128 c^{9/2} (b c-a d)^{3/2}}+\frac{5 a^2 x \sqrt{a+b x^2} (8 b c-7 a d)}{128 c^4 \left(c+d x^2\right) (b c-a d)}+\frac{5 a x \left(a+b x^2\right)^{3/2} (8 b c-7 a d)}{192 c^3 \left(c+d x^2\right)^2 (b c-a d)}+\frac{x \left(a+b x^2\right)^{5/2} (8 b c-7 a d)}{48 c^2 \left(c+d x^2\right)^3 (b c-a d)}-\frac{d x \left(a+b x^2\right)^{7/2}}{8 c \left(c+d x^2\right)^4 (b c-a d)}",1,"(x*(-(c*(16*b^4*c^3*x^6*(4*c + d*x^2) + 8*a*b^3*c^2*x^4*(34*c^2 + 13*c*d*x^2 + 3*d^2*x^4) + 2*a^2*b^2*c*x^2*(236*c^3 + 173*c^2*d*x^2 + 106*c*d^2*x^4 + 25*d^3*x^6) - a^4*d*(279*c^3 + 511*c^2*d*x^2 + 385*c*d^2*x^4 + 105*d^3*x^6) + a^3*b*(264*c^4 - 21*c^3*d*x^2 - 323*c^2*d^2*x^4 - 335*c*d^3*x^6 - 105*d^4*x^8))) + (15*a^3*(-8*b*c + 7*a*d)*(c + d*x^2)^4*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]))/(384*c^5*(-(b*c) + a*d)*Sqrt[a + b*x^2]*(c + d*x^2)^4)","A",1
71,1,30,30,0.0285926,"\int \frac{\sqrt{1-x^2}}{1+x^2} \, dx","Integrate[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",1
72,1,64,27,0.0260166,"\int \frac{\sqrt{1+x^2}}{-1+x^2} \, dx","Integrate[Sqrt[1 + x^2]/(-1 + x^2),x]","\frac{\log \left(\sqrt{2} \sqrt{x^2+1}-x+1\right)-\log \left(\sqrt{2} \sqrt{x^2+1}+x+1\right)+\log (1-x)-\log (x+1)}{\sqrt{2}}+\sinh ^{-1}(x)","\sinh ^{-1}(x)-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{x^2+1}}\right)",1,"ArcSinh[x] + (Log[1 - x] - Log[1 + x] + Log[1 - x + Sqrt[2]*Sqrt[1 + x^2]] - Log[1 + x + Sqrt[2]*Sqrt[1 + x^2]])/Sqrt[2]","B",1
73,1,25,25,0.0107727,"\int \frac{\sqrt{1-x^2}}{-1+2 x^2} \, dx","Integrate[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,"-1/2*ArcSin[x] - ArcTanh[x/Sqrt[1 - x^2]]/2","A",1
74,1,140,169,5.0947175,"\int \frac{\left(c+d x^2\right)^3}{\sqrt{a+b x^2}} \, dx","Integrate[(c + d*x^2)^3/Sqrt[a + b*x^2],x]","\frac{\sqrt{b} d x \sqrt{a+b x^2} \left(15 a^2 d^2-2 a b d \left(27 c+5 d x^2\right)+4 b^2 \left(18 c^2+9 c d x^2+2 d^2 x^4\right)\right)+3 \left(-5 a^3 d^3+18 a^2 b c d^2-24 a b^2 c^2 d+16 b^3 c^3\right) \log \left(\sqrt{b} \sqrt{a+b x^2}+b x\right)}{48 b^{7/2}}","\frac{(2 b c-a d) \left(5 a^2 d^2-8 a b c d+8 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{16 b^{7/2}}+\frac{d x \sqrt{a+b x^2} \left(15 a^2 d^2-44 a b c d+44 b^2 c^2\right)}{48 b^3}+\frac{5 d x \sqrt{a+b x^2} \left(c+d x^2\right) (2 b c-a d)}{24 b^2}+\frac{d x \sqrt{a+b x^2} \left(c+d x^2\right)^2}{6 b}",1,"(Sqrt[b]*d*x*Sqrt[a + b*x^2]*(15*a^2*d^2 - 2*a*b*d*(27*c + 5*d*x^2) + 4*b^2*(18*c^2 + 9*c*d*x^2 + 2*d^2*x^4)) + 3*(16*b^3*c^3 - 24*a*b^2*c^2*d + 18*a^2*b*c*d^2 - 5*a^3*d^3)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(48*b^(7/2))","A",1
75,1,160,108,2.4890175,"\int \frac{\left(c+d x^2\right)^2}{\sqrt{a+b x^2}} \, dx","Integrate[(c + d*x^2)^2/Sqrt[a + b*x^2],x]","\frac{x \sqrt{\frac{b x^2}{a}+1} \left(-2 b x^2 \left(c+d x^2\right)^2 \, _3F_2\left(\frac{3}{2},\frac{3}{2},2;1,\frac{9}{2};-\frac{b x^2}{a}\right)-4 b x^2 \left(2 c^2+3 c d x^2+d^2 x^4\right) \, _2F_1\left(\frac{3}{2},\frac{3}{2};\frac{9}{2};-\frac{b x^2}{a}\right)+7 a \left(15 c^2+10 c d x^2+3 d^2 x^4\right) \, _2F_1\left(\frac{1}{2},\frac{1}{2};\frac{7}{2};-\frac{b x^2}{a}\right)\right)}{105 a \sqrt{a+b x^2}}","\frac{\left(3 a^2 d^2-8 a b c d+8 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 b^{5/2}}+\frac{3 d x \sqrt{a+b x^2} (2 b c-a d)}{8 b^2}+\frac{d x \sqrt{a+b x^2} \left(c+d x^2\right)}{4 b}",1,"(x*Sqrt[1 + (b*x^2)/a]*(7*a*(15*c^2 + 10*c*d*x^2 + 3*d^2*x^4)*Hypergeometric2F1[1/2, 1/2, 7/2, -((b*x^2)/a)] - 4*b*x^2*(2*c^2 + 3*c*d*x^2 + d^2*x^4)*Hypergeometric2F1[3/2, 3/2, 9/2, -((b*x^2)/a)] - 2*b*x^2*(c + d*x^2)^2*HypergeometricPFQ[{3/2, 3/2, 2}, {1, 9/2}, -((b*x^2)/a)]))/(105*a*Sqrt[a + b*x^2])","C",0
76,1,57,58,0.0207785,"\int \frac{c+d x^2}{\sqrt{a+b x^2}} \, dx","Integrate[(c + d*x^2)/Sqrt[a + b*x^2],x]","\frac{d x \sqrt{a+b x^2}}{2 b}-\frac{(a d-2 b c) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 b^{3/2}}","\frac{(2 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 b^{3/2}}+\frac{d x \sqrt{a+b x^2}}{2 b}",1,"(d*x*Sqrt[a + b*x^2])/(2*b) - ((-2*b*c + a*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(3/2))","A",1
77,1,25,25,0.0042959,"\int \frac{1}{\sqrt{a+b x^2}} \, dx","Integrate[1/Sqrt[a + b*x^2],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{\sqrt{b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{\sqrt{b}}",1,"ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]]/Sqrt[b]","A",1
78,1,49,49,0.0168442,"\int \frac{1}{\sqrt{a+b x^2} \left(c+d x^2\right)} \, dx","Integrate[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",1
79,1,126,101,0.3024758,"\int \frac{1}{\sqrt{a+b x^2} \left(c+d x^2\right)^2} \, dx","Integrate[1/(Sqrt[a + b*x^2]*(c + d*x^2)^2),x]","\frac{x \left(\frac{\left(c+d x^2\right) (2 b c-a d) \tanh ^{-1}\left(\sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}\right)}{c \sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}}-d \left(a+b x^2\right)\right)}{2 c \sqrt{a+b x^2} \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,"(x*(-(d*(a + b*x^2)) + ((2*b*c - a*d)*(c + d*x^2)*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/(c*Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))])))/(2*c*(b*c - a*d)*Sqrt[a + b*x^2]*(c + d*x^2))","A",1
80,1,192,163,0.6681646,"\int \frac{1}{\sqrt{a+b x^2} \left(c+d x^2\right)^3} \, dx","Integrate[1/(Sqrt[a + b*x^2]*(c + d*x^2)^3),x]","\frac{x \left(\frac{\left(c+d x^2\right)^2 \left(3 a^2 d^2-8 a b c d+8 b^2 c^2\right) \tanh ^{-1}\left(\sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}\right)}{\sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}}-c d \left(a^2 (-d) \left(5 c+3 d x^2\right)+a b \left(8 c^2+c d x^2-3 d^2 x^4\right)+2 b^2 c x^2 \left(4 c+3 d x^2\right)\right)\right)}{8 c^3 \sqrt{a+b x^2} \left(c+d x^2\right)^2 (b c-a d)^2}","\frac{\left(3 a^2 d^2-8 a b c d+8 b^2 c^2\right) \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{8 c^{5/2} (b c-a d)^{5/2}}-\frac{3 d x \sqrt{a+b x^2} (2 b c-a d)}{8 c^2 \left(c+d x^2\right) (b c-a d)^2}-\frac{d x \sqrt{a+b x^2}}{4 c \left(c+d x^2\right)^2 (b c-a d)}",1,"(x*(-(c*d*(2*b^2*c*x^2*(4*c + 3*d*x^2) - a^2*d*(5*c + 3*d*x^2) + a*b*(8*c^2 + c*d*x^2 - 3*d^2*x^4))) + ((8*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*(c + d*x^2)^2*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]))/(8*c^3*(b*c - a*d)^2*Sqrt[a + b*x^2]*(c + d*x^2)^2)","A",1
81,1,172,257,5.2027348,"\int \frac{\left(c+d x^2\right)^4}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[(c + d*x^2)^4/(a + b*x^2)^(3/2),x]","\frac{\sqrt{b} x \sqrt{a+b x^2} \left(3 d^2 \left(19 a^2 d^2-56 a b c d+48 b^2 c^2\right)+2 b d^3 x^2 (24 b c-11 a d)+\frac{48 (b c-a d)^4}{a \left(a+b x^2\right)}+8 b^2 d^4 x^4\right)+3 d \left(-35 a^3 d^3+120 a^2 b c d^2-144 a b^2 c^2 d+64 b^3 c^3\right) \log \left(\sqrt{b} \sqrt{a+b x^2}+b x\right)}{48 b^{9/2}}","-\frac{d x \sqrt{a+b x^2} \left(c+d x^2\right) \left(35 a^2 d^2-64 a b c d+24 b^2 c^2\right)}{24 a b^3}+\frac{d \left(-35 a^3 d^3+120 a^2 b c d^2-144 a b^2 c^2 d+64 b^3 c^3\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{16 b^{9/2}}-\frac{d x \sqrt{a+b x^2} \left(-105 a^3 d^3+290 a^2 b c d^2-248 a b^2 c^2 d+48 b^3 c^3\right)}{48 a b^4}-\frac{d x \sqrt{a+b x^2} \left(c+d x^2\right)^2 (6 b c-7 a d)}{6 a b^2}+\frac{x \left(c+d x^2\right)^3 (b c-a d)}{a b \sqrt{a+b x^2}}",1,"(Sqrt[b]*x*Sqrt[a + b*x^2]*(3*d^2*(48*b^2*c^2 - 56*a*b*c*d + 19*a^2*d^2) + 2*b*d^3*(24*b*c - 11*a*d)*x^2 + 8*b^2*d^4*x^4 + (48*(b*c - a*d)^4)/(a*(a + b*x^2))) + 3*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)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(48*b^(9/2))","A",1
82,1,122,169,5.0996783,"\int \frac{\left(c+d x^2\right)^3}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[(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) \log \left(\sqrt{b} \sqrt{a+b x^2}+b x\right)}{8 b^{7/2}}+\frac{x \sqrt{a+b x^2} \left(d^2 (12 b c-7 a d)+\frac{8 (b c-a d)^3}{a \left(a+b x^2\right)}+2 b d^3 x^2\right)}{8 b^3}","\frac{3 d \left(5 a^2 d^2-12 a b c d+8 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 b^{7/2}}-\frac{d x \sqrt{a+b x^2} (2 b c-5 a d) (4 b c-3 a d)}{8 a b^3}-\frac{d x \sqrt{a+b x^2} \left(c+d x^2\right) (4 b c-5 a d)}{4 a b^2}+\frac{x \left(c+d x^2\right)^2 (b c-a d)}{a b \sqrt{a+b x^2}}",1,"(x*Sqrt[a + b*x^2]*(d^2*(12*b*c - 7*a*d) + 2*b*d^3*x^2 + (8*(b*c - a*d)^3)/(a*(a + b*x^2))))/(8*b^3) + (3*d*(8*b^2*c^2 - 12*a*b*c*d + 5*a^2*d^2)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(8*b^(7/2))","A",1
83,1,160,90,2.5291792,"\int \frac{\left(c+d x^2\right)^2}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[(c + d*x^2)^2/(a + b*x^2)^(3/2),x]","\frac{x \sqrt{\frac{b x^2}{a}+1} \left(-6 b x^2 \left(c+d x^2\right)^2 \, _3F_2\left(\frac{3}{2},2,\frac{5}{2};1,\frac{9}{2};-\frac{b x^2}{a}\right)-12 b x^2 \left(2 c^2+3 c d x^2+d^2 x^4\right) \, _2F_1\left(\frac{3}{2},\frac{5}{2};\frac{9}{2};-\frac{b x^2}{a}\right)+7 a \left(15 c^2+10 c d x^2+3 d^2 x^4\right) \, _2F_1\left(\frac{1}{2},\frac{3}{2};\frac{7}{2};-\frac{b x^2}{a}\right)\right)}{105 a^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{x (b c-a d)^2}{a b^2 \sqrt{a+b x^2}}+\frac{d^2 x \sqrt{a+b x^2}}{2 b^2}",1,"(x*Sqrt[1 + (b*x^2)/a]*(7*a*(15*c^2 + 10*c*d*x^2 + 3*d^2*x^4)*Hypergeometric2F1[1/2, 3/2, 7/2, -((b*x^2)/a)] - 12*b*x^2*(2*c^2 + 3*c*d*x^2 + d^2*x^4)*Hypergeometric2F1[3/2, 5/2, 9/2, -((b*x^2)/a)] - 6*b*x^2*(c + d*x^2)^2*HypergeometricPFQ[{3/2, 2, 5/2}, {1, 9/2}, -((b*x^2)/a)]))/(105*a^2*Sqrt[a + b*x^2])","C",0
84,1,70,54,0.0609463,"\int \frac{c+d x^2}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[(c + d*x^2)/(a + b*x^2)^(3/2),x]","\frac{a^{3/2} d \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)+\sqrt{b} x (b c-a d)}{a b^{3/2} \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,"(Sqrt[b]*(b*c - a*d)*x + a^(3/2)*d*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(a*b^(3/2)*Sqrt[a + b*x^2])","A",1
85,1,16,16,0.0038208,"\int \frac{1}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[(a + b*x^2)^(-3/2),x]","\frac{x}{a \sqrt{a+b x^2}}","\frac{x}{a \sqrt{a+b x^2}}",1,"x/(a*Sqrt[a + b*x^2])","A",1
86,1,309,79,0.7204694,"\int \frac{1}{\left(a+b x^2\right)^{3/2} \left(c+d x^2\right)} \, dx","Integrate[1/((a + b*x^2)^(3/2)*(c + d*x^2)),x]","\frac{x \left(2 d x^2 \left(\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}\right)^{5/2} \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)+2 c \left(\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}\right)^{5/2} \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)-10 d x^2 \sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}-15 c \sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}+10 d x^2 \tanh ^{-1}\left(\sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}\right)+15 c \tanh ^{-1}\left(\sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}\right)\right)}{5 c^2 \left(a+b x^2\right)^{3/2} \left(\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}\right)^{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,"(x*(-15*c*Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))] - 10*d*x^2*Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))] + 15*c*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]] + 10*d*x^2*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]] + 2*c*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + 2*d*x^2*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))]))/(5*c^2*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(3/2)*(a + b*x^2)^(3/2))","C",0
87,1,758,143,2.6725197,"\int \frac{1}{\left(a+b x^2\right)^{3/2} \left(c+d x^2\right)^2} \, dx","Integrate[1/((a + b*x^2)^(3/2)*(c + d*x^2)^2),x]","\frac{x \left(\frac{24 d^2 x^4 \left(\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}\right)^{7/2} \, _3F_2\left(2,2,\frac{5}{2};1,\frac{9}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)}{c^2}+\frac{48 d x^2 \left(\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}\right)^{7/2} \, _3F_2\left(2,2,\frac{5}{2};1,\frac{9}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)}{c}+24 \left(\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}\right)^{7/2} \, _3F_2\left(2,2,\frac{5}{2};1,\frac{9}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)+\frac{1050 d^2 x^6 (a d-b c) \tanh ^{-1}\left(\sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}\right)}{c^3 \left(a+b x^2\right)}+\frac{280 d^2 x^4 \left(\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}\right)^{3/2}}{c^2}-\frac{2310 d^2 x^4 \sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}}{c^2}+\frac{2310 d^2 x^4 \tanh ^{-1}\left(\sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}\right)}{c^2}+\frac{2310 d x^4 (a d-b c) \tanh ^{-1}\left(\sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}\right)}{c^2 \left(a+b x^2\right)}+\frac{560 d x^2 \left(\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}\right)^{3/2}}{c}-\frac{5250 d x^2 \sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}}{c}+70 \left(\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}\right)^{3/2}-2625 \sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}+\frac{5250 d x^2 \tanh ^{-1}\left(\sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}\right)}{c}-\frac{945 x^2 (b c-a d) \tanh ^{-1}\left(\sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}\right)}{c \left(a+b x^2\right)}+2625 \tanh ^{-1}\left(\sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}\right)\right)}{210 c \left(a+b x^2\right)^{3/2} \left(c+d x^2\right) \left(\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}\right)^{5/2}}","-\frac{d (4 b c-a d) \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{2 c^{3/2} (b c-a d)^{5/2}}+\frac{b x (a d+2 b c)}{2 a c \sqrt{a+b x^2} (b c-a d)^2}-\frac{d x}{2 c \sqrt{a+b x^2} \left(c+d x^2\right) (b c-a d)}",1,"(x*(-2625*Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))] - (5250*d*x^2*Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))])/c - (2310*d^2*x^4*Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))])/c^2 + 70*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(3/2) + (560*d*x^2*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(3/2))/c + (280*d^2*x^4*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(3/2))/c^2 + 2625*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]] + (5250*d*x^2*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/c + (2310*d^2*x^4*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/c^2 - (945*(b*c - a*d)*x^2*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/(c*(a + b*x^2)) + (2310*d*(-(b*c) + a*d)*x^4*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/(c^2*(a + b*x^2)) + (1050*d^2*(-(b*c) + a*d)*x^6*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/(c^3*(a + b*x^2)) + 24*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(7/2)*HypergeometricPFQ[{2, 2, 5/2}, {1, 9/2}, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + (48*d*x^2*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(7/2)*HypergeometricPFQ[{2, 2, 5/2}, {1, 9/2}, ((b*c - a*d)*x^2)/(c*(a + b*x^2))])/c + (24*d^2*x^4*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(7/2)*HypergeometricPFQ[{2, 2, 5/2}, {1, 9/2}, ((b*c - a*d)*x^2)/(c*(a + b*x^2))])/c^2))/(210*c*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(5/2)*(a + b*x^2)^(3/2)*(c + d*x^2))","C",0
88,1,1392,225,5.0172838,"\int \frac{1}{\left(a+b x^2\right)^{3/2} \left(c+d x^2\right)^3} \, dx","Integrate[1/((a + b*x^2)^(3/2)*(c + d*x^2)^3),x]","\frac{x \left(\frac{7560 d^3 (b c-a d)^2 \tanh ^{-1}\left(\sqrt{\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}}\right) x^{10}}{c^5 \left(b x^2+a\right)^2}+\frac{80640 d^3 (a d-b c) \tanh ^{-1}\left(\sqrt{\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}}\right) x^8}{c^4 \left(b x^2+a\right)}+\frac{21735 d^2 (b c-a d)^2 \tanh ^{-1}\left(\sqrt{\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}}\right) x^8}{c^4 \left(b x^2+a\right)^2}-\frac{1344 d^3 \left(\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)^{5/2} x^6}{c^3}+\frac{46200 d^3 \left(\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)^{3/2} x^6}{c^3}+\frac{103320 d^3 \tanh ^{-1}\left(\sqrt{\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}}\right) x^6}{c^3}+\frac{247590 d^2 (a d-b c) \tanh ^{-1}\left(\sqrt{\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}}\right) x^6}{c^3 \left(b x^2+a\right)}+\frac{17955 d (b c-a d)^2 \tanh ^{-1}\left(\sqrt{\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}}\right) x^6}{c^3 \left(b x^2+a\right)^2}+\frac{64 d^3 \left(\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)^{9/2} \, _4F_3\left(2,2,2,\frac{5}{2};1,1,\frac{11}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right) x^6}{c^3}-\frac{103320 d^3 \sqrt{\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}} x^6}{c^3}-\frac{4032 d^2 \left(\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)^{5/2} x^4}{c^2}+\frac{139545 d^2 \left(\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)^{3/2} x^4}{c^2}+\frac{324135 d^2 \tanh ^{-1}\left(\sqrt{\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}}\right) x^4}{c^2}+\frac{236250 d (a d-b c) \tanh ^{-1}\left(\sqrt{\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}}\right) x^4}{c^2 \left(b x^2+a\right)}+\frac{8505 (b c-a d)^2 \tanh ^{-1}\left(\sqrt{\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}}\right) x^4}{c^2 \left(b x^2+a\right)^2}+\frac{192 d^2 \left(\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)^{9/2} \, _4F_3\left(2,2,2,\frac{5}{2};1,1,\frac{11}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right) x^4}{c^2}-\frac{324135 d^2 \sqrt{\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}} x^4}{c^2}-\frac{4032 d \left(\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)^{5/2} x^2}{c}+\frac{128205 d \left(\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)^{3/2} x^2}{c}+\frac{324135 d \tanh ^{-1}\left(\sqrt{\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}}\right) x^2}{c}-\frac{78750 (b c-a d) \tanh ^{-1}\left(\sqrt{\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}}\right) x^2}{c \left(b x^2+a\right)}+\frac{192 d \left(\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)^{9/2} \, _4F_3\left(2,2,2,\frac{5}{2};1,1,\frac{11}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right) x^2}{c}-\frac{324135 d \sqrt{\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}} x^2}{c}-3864 \left(\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)^{5/2}+42735 \left(\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)^{3/2}+108045 \tanh ^{-1}\left(\sqrt{\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}}\right)+64 \left(\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)^{9/2} \, _4F_3\left(2,2,2,\frac{5}{2};1,1,\frac{11}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)-108045 \sqrt{\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}}\right)}{2520 c \left(\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)^{7/2} \left(b x^2+a\right)^{3/2} \left(d x^2+c\right)^2}","-\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,"(x*(-108045*Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))] - (324135*d*x^2*Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))])/c - (324135*d^2*x^4*Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))])/c^2 - (103320*d^3*x^6*Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))])/c^3 + 42735*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(3/2) + (128205*d*x^2*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(3/2))/c + (139545*d^2*x^4*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(3/2))/c^2 + (46200*d^3*x^6*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(3/2))/c^3 - 3864*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(5/2) - (4032*d*x^2*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(5/2))/c - (4032*d^2*x^4*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(5/2))/c^2 - (1344*d^3*x^6*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(5/2))/c^3 + 108045*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]] + (324135*d*x^2*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/c + (324135*d^2*x^4*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/c^2 + (103320*d^3*x^6*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/c^3 + (8505*(b*c - a*d)^2*x^4*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/(c^2*(a + b*x^2)^2) + (17955*d*(b*c - a*d)^2*x^6*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/(c^3*(a + b*x^2)^2) + (21735*d^2*(b*c - a*d)^2*x^8*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/(c^4*(a + b*x^2)^2) + (7560*d^3*(b*c - a*d)^2*x^10*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/(c^5*(a + b*x^2)^2) - (78750*(b*c - a*d)*x^2*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/(c*(a + b*x^2)) + (236250*d*(-(b*c) + a*d)*x^4*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/(c^2*(a + b*x^2)) + (247590*d^2*(-(b*c) + a*d)*x^6*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/(c^3*(a + b*x^2)) + (80640*d^3*(-(b*c) + a*d)*x^8*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/(c^4*(a + b*x^2)) + 64*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(9/2)*HypergeometricPFQ[{2, 2, 2, 5/2}, {1, 1, 11/2}, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + (192*d*x^2*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(9/2)*HypergeometricPFQ[{2, 2, 2, 5/2}, {1, 1, 11/2}, ((b*c - a*d)*x^2)/(c*(a + b*x^2))])/c + (192*d^2*x^4*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(9/2)*HypergeometricPFQ[{2, 2, 2, 5/2}, {1, 1, 11/2}, ((b*c - a*d)*x^2)/(c*(a + b*x^2))])/c^2 + (64*d^3*x^6*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(9/2)*HypergeometricPFQ[{2, 2, 2, 5/2}, {1, 1, 11/2}, ((b*c - a*d)*x^2)/(c*(a + b*x^2))])/c^3))/(2520*c*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(7/2)*(a + b*x^2)^(3/2)*(c + d*x^2)^2)","C",0
89,1,157,255,5.1675662,"\int \frac{\left(c+d x^2\right)^4}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[(c + d*x^2)^4/(a + b*x^2)^(5/2),x]","\frac{x \sqrt{a+b x^2} \left(\frac{16 (b c-a d)^3 (5 a d+b c)}{a^2 \left(a+b x^2\right)}+3 d^3 (16 b c-11 a d)+\frac{8 (b c-a d)^4}{a \left(a+b x^2\right)^2}+6 b d^4 x^2\right)}{24 b^4}+\frac{d^2 \left(35 a^2 d^2-80 a b c d+48 b^2 c^2\right) \log \left(\sqrt{b} \sqrt{a+b x^2}+b x\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{d^2 \left(35 a^2 d^2-80 a b c d+48 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 b^{9/2}}-\frac{d x \sqrt{a+b x^2} \left(c+d x^2\right) \left(-35 a^2 d^2+24 a b c d+8 b^2 c^2\right)}{12 a^2 b^3}-\frac{d x \sqrt{a+b x^2} \left(105 a^3 d^3-170 a^2 b c d^2+40 a b^2 c^2 d+16 b^3 c^3\right)}{24 a^2 b^4}+\frac{x \left(c+d x^2\right)^3 (b c-a d)}{3 a b \left(a+b x^2\right)^{3/2}}",1,"(x*Sqrt[a + b*x^2]*(3*d^3*(16*b*c - 11*a*d) + 6*b*d^4*x^2 + (8*(b*c - a*d)^4)/(a*(a + b*x^2)^2) + (16*(b*c - a*d)^3*(b*c + 5*a*d))/(a^2*(a + b*x^2))))/(24*b^4) + (d^2*(48*b^2*c^2 - 80*a*b*c*d + 35*a^2*d^2)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(8*b^(9/2))","A",1
90,1,125,172,5.1009723,"\int \frac{\left(c+d x^2\right)^3}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[(c + d*x^2)^3/(a + b*x^2)^(5/2),x]","\frac{x \left(3 a^2 d^3 \left(a+b x^2\right)^2+2 \left(a+b x^2\right) (b c-a d)^2 (7 a d+2 b c)+2 a (b c-a d)^3\right)}{6 a^2 b^3 \left(a+b x^2\right)^{3/2}}+\frac{d^2 (6 b c-5 a d) \log \left(\sqrt{b} \sqrt{a+b x^2}+b x\right)}{2 b^{7/2}}","\frac{x \left(c+d x^2\right) (b c-a d) (5 a d+2 b c)}{3 a^2 b^2 \sqrt{a+b x^2}}-\frac{d x \sqrt{a+b x^2} \left(-15 a^2 d^2+8 a b c d+4 b^2 c^2\right)}{6 a^2 b^3}+\frac{d^2 (6 b c-5 a d) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 b^{7/2}}+\frac{x \left(c+d x^2\right)^2 (b c-a d)}{3 a b \left(a+b x^2\right)^{3/2}}",1,"(x*(2*a*(b*c - a*d)^3 + 2*(b*c - a*d)^2*(2*b*c + 7*a*d)*(a + b*x^2) + 3*a^2*d^3*(a + b*x^2)^2))/(6*a^2*b^3*(a + b*x^2)^(3/2)) + (d^2*(6*b*c - 5*a*d)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(2*b^(7/2))","A",1
91,1,214,105,4.1460477,"\int \frac{\left(c+d x^2\right)^2}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[(c + d*x^2)^2/(a + b*x^2)^(5/2),x]","\frac{\sqrt{\frac{b x^2}{a}+1} \left(-16 b^3 x^6 \left(c+d x^2\right)^2 \, _3F_2\left(\frac{3}{2},2,\frac{7}{2};1,\frac{9}{2};-\frac{b x^2}{a}\right)+\frac{7 a^2 \left(15 c^2+10 c d x^2+3 d^2 x^4\right) \left(\sqrt{-\frac{b x^2 \left(a+b x^2\right)}{a^2}} \left(2 b x^2-3 a\right)+3 a \sin ^{-1}\left(\sqrt{-\frac{b x^2}{a}}\right)\right)}{\sqrt{-\frac{b x^2}{a}}}-32 b^3 x^6 \left(2 c^2+3 c d x^2+d^2 x^4\right) \, _2F_1\left(\frac{3}{2},\frac{7}{2};\frac{9}{2};-\frac{b x^2}{a}\right)\right)}{168 a^3 b^2 x^3 \sqrt{a+b x^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,"(Sqrt[1 + (b*x^2)/a]*((7*a^2*(15*c^2 + 10*c*d*x^2 + 3*d^2*x^4)*(Sqrt[-((b*x^2*(a + b*x^2))/a^2)]*(-3*a + 2*b*x^2) + 3*a*ArcSin[Sqrt[-((b*x^2)/a)]]))/Sqrt[-((b*x^2)/a)] - 32*b^3*x^6*(2*c^2 + 3*c*d*x^2 + d^2*x^4)*Hypergeometric2F1[3/2, 7/2, 9/2, -((b*x^2)/a)] - 16*b^3*x^6*(c + d*x^2)^2*HypergeometricPFQ[{3/2, 2, 7/2}, {1, 9/2}, -((b*x^2)/a)]))/(168*a^3*b^2*x^3*Sqrt[a + b*x^2])","C",0
92,1,37,47,0.016321,"\int \frac{c+d x^2}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[(c + d*x^2)/(a + b*x^2)^(5/2),x]","\frac{x \left(3 a c+a d x^2+2 b c x^2\right)}{3 a^2 \left(a+b x^2\right)^{3/2}}","\frac{2 c x}{3 a^2 \sqrt{a+b x^2}}+\frac{x \left(c+d x^2\right)}{3 a \left(a+b x^2\right)^{3/2}}",1,"(x*(3*a*c + 2*b*c*x^2 + a*d*x^2))/(3*a^2*(a + b*x^2)^(3/2))","A",1
93,1,29,39,0.0063904,"\int \frac{1}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[(a + b*x^2)^(-5/2),x]","\frac{x \left(3 a+2 b x^2\right)}{3 a^2 \left(a+b x^2\right)^{3/2}}","\frac{2 x}{3 a^2 \sqrt{a+b x^2}}+\frac{x}{3 a \left(a+b x^2\right)^{3/2}}",1,"(x*(3*a + 2*b*x^2))/(3*a^2*(a + b*x^2)^(3/2))","A",1
94,1,775,122,2.749536,"\int \frac{1}{\left(a+b x^2\right)^{5/2} \left(c+d x^2\right)} \, dx","Integrate[1/((a + b*x^2)^(5/2)*(c + d*x^2)),x]","\frac{x \left(12 c^2 \left(\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}\right)^{7/2} \, _3F_2\left(2,2,\frac{7}{2};1,\frac{9}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)+12 d^2 x^4 \left(\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}\right)^{7/2} \, _3F_2\left(2,2,\frac{7}{2};1,\frac{9}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)+24 c d x^2 \left(\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}\right)^{7/2} \, _3F_2\left(2,2,\frac{7}{2};1,\frac{9}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)+48 c^2 \left(\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}\right)^{7/2} \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)-105 c^2 \left(\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}\right)^{3/2}-315 c^2 \sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}+315 c^2 \tanh ^{-1}\left(\sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}\right)+36 d^2 x^4 \left(\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}\right)^{7/2} \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)-56 d^2 x^4 \left(\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}\right)^{3/2}-168 d^2 x^4 \sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}+168 d^2 x^4 \tanh ^{-1}\left(\sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}\right)+84 c d x^2 \left(\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}\right)^{7/2} \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)-140 c d x^2 \left(\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}\right)^{3/2}-420 c d x^2 \sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}+420 c d x^2 \tanh ^{-1}\left(\sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}\right)\right)}{63 c^3 \left(a+b x^2\right)^{5/2} \left(\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}\right)^{5/2}}","\frac{b x (2 b c-5 a d)}{3 a^2 \sqrt{a+b x^2} (b c-a d)^2}+\frac{d^2 \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{\sqrt{c} (b c-a d)^{5/2}}+\frac{b x}{3 a \left(a+b x^2\right)^{3/2} (b c-a d)}",1,"(x*(-315*c^2*Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))] - 420*c*d*x^2*Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))] - 168*d^2*x^4*Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))] - 105*c^2*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(3/2) - 140*c*d*x^2*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(3/2) - 56*d^2*x^4*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(3/2) + 315*c^2*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]] + 420*c*d*x^2*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]] + 168*d^2*x^4*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]] + 48*c^2*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(7/2)*Hypergeometric2F1[2, 7/2, 9/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + 84*c*d*x^2*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(7/2)*Hypergeometric2F1[2, 7/2, 9/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + 36*d^2*x^4*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(7/2)*Hypergeometric2F1[2, 7/2, 9/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + 12*c^2*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(7/2)*HypergeometricPFQ[{2, 2, 7/2}, {1, 9/2}, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + 24*c*d*x^2*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(7/2)*HypergeometricPFQ[{2, 2, 7/2}, {1, 9/2}, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + 12*d^2*x^4*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(7/2)*HypergeometricPFQ[{2, 2, 7/2}, {1, 9/2}, ((b*c - a*d)*x^2)/(c*(a + b*x^2))]))/(63*c^3*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(5/2)*(a + b*x^2)^(5/2))","C",0
95,1,170,202,5.4976403,"\int \frac{1}{\left(a+b x^2\right)^{5/2} \left(c+d x^2\right)^2} \, dx","Integrate[1/((a + b*x^2)^(5/2)*(c + d*x^2)^2),x]","\frac{1}{6} \left(x \sqrt{a+b x^2} \left(\frac{4 b^2 (4 a d-b c)}{a^2 \left(a+b x^2\right) (a d-b c)^3}+\frac{2 b^2}{a \left(a+b x^2\right)^2 (b c-a d)^2}-\frac{3 d^3}{c \left(c+d x^2\right) (b c-a d)^3}\right)+\frac{3 d^2 (a d-6 b c) \tan ^{-1}\left(\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{c^{3/2} (a d-b c)^{7/2}}\right)","\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,"(x*Sqrt[a + b*x^2]*((2*b^2)/(a*(b*c - a*d)^2*(a + b*x^2)^2) + (4*b^2*(-(b*c) + 4*a*d))/(a^2*(-(b*c) + a*d)^3*(a + b*x^2)) - (3*d^3)/(c*(b*c - a*d)^3*(c + d*x^2))) + (3*d^2*(-6*b*c + a*d)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(c^(3/2)*(-(b*c) + a*d)^(7/2)))/6","A",1
96,1,221,313,5.6620748,"\int \frac{1}{\left(a+b x^2\right)^{5/2} \left(c+d x^2\right)^3} \, dx","Integrate[1/((a + b*x^2)^(5/2)*(c + d*x^2)^3),x]","\frac{1}{24} \left(x \sqrt{a+b x^2} \left(\frac{8 b^3 (2 b c-11 a d)}{a^2 \left(a+b x^2\right) (b c-a d)^4}-\frac{8 b^3}{a \left(a+b x^2\right)^2 (a d-b c)^3}+\frac{3 d^3 (3 a d-14 b c)}{c^2 \left(c+d x^2\right) (b c-a d)^4}-\frac{6 d^3}{c \left(c+d x^2\right)^2 (b c-a d)^3}\right)+\frac{3 d^2 \left(3 a^2 d^2-16 a b c d+48 b^2 c^2\right) \tan ^{-1}\left(\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{c^{5/2} (a d-b c)^{9/2}}\right)","\frac{b x \left(-3 a^2 d^2-40 a b c d+8 b^2 c^2\right)}{12 a^2 c \sqrt{a+b x^2} \left(c+d x^2\right) (b c-a d)^3}+\frac{d^2 \left(3 a^2 d^2-16 a b c d+48 b^2 c^2\right) \tanh ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right)}{8 c^{5/2} (b c-a d)^{9/2}}+\frac{d x \sqrt{a+b x^2} \left(9 a^3 d^3-42 a^2 b c d^2-88 a b^2 c^2 d+16 b^3 c^3\right)}{24 a^2 c^2 \left(c+d x^2\right) (b c-a d)^4}-\frac{d x}{4 c \left(a+b x^2\right)^{3/2} \left(c+d x^2\right)^2 (b c-a d)}+\frac{b x (3 a d+4 b c)}{12 a c \left(a+b x^2\right)^{3/2} \left(c+d x^2\right) (b c-a d)^2}",1,"(x*Sqrt[a + b*x^2]*((-8*b^3)/(a*(-(b*c) + a*d)^3*(a + b*x^2)^2) + (8*b^3*(2*b*c - 11*a*d))/(a^2*(b*c - a*d)^4*(a + b*x^2)) - (6*d^3)/(c*(b*c - a*d)^3*(c + d*x^2)^2) + (3*d^3*(-14*b*c + 3*a*d))/(c^2*(b*c - a*d)^4*(c + d*x^2))) + (3*d^2*(48*b^2*c^2 - 16*a*b*c*d + 3*a^2*d^2)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(c^(5/2)*(-(b*c) + a*d)^(9/2)))/24","A",1
97,1,163,224,0.1029073,"\int \frac{\left(a+b x^2\right)^3}{\left(c+d x^2\right)^{11/2}} \, dx","Integrate[(a + b*x^2)^3/(c + d*x^2)^(11/2),x]","\frac{a^3 \left(315 c^4 x+840 c^3 d x^3+1008 c^2 d^2 x^5+576 c d^3 x^7+128 d^4 x^9\right)+3 a^2 b c x^3 \left(105 c^3+126 c^2 d x^2+72 c d^2 x^4+16 d^3 x^6\right)+3 a b^2 c^2 x^5 \left(63 c^2+36 c d x^2+8 d^2 x^4\right)+5 b^3 c^3 x^7 \left(9 c+2 d x^2\right)}{315 c^5 \left(c+d x^2\right)^{9/2}}","\frac{16 a^3 x (9 b c-8 a d)}{315 c^5 \sqrt{c+d x^2} (b c-a d)}+\frac{8 a^2 x \left(a+b x^2\right) (9 b c-8 a d)}{315 c^4 \left(c+d x^2\right)^{3/2} (b c-a d)}+\frac{2 a x \left(a+b x^2\right)^2 (9 b c-8 a d)}{105 c^3 \left(c+d x^2\right)^{5/2} (b c-a d)}+\frac{x \left(a+b x^2\right)^3 (9 b c-8 a d)}{63 c^2 \left(c+d x^2\right)^{7/2} (b c-a d)}-\frac{d x \left(a+b x^2\right)^4}{9 c \left(c+d x^2\right)^{9/2} (b c-a d)}",1,"(5*b^3*c^3*x^7*(9*c + 2*d*x^2) + 3*a*b^2*c^2*x^5*(63*c^2 + 36*c*d*x^2 + 8*d^2*x^4) + 3*a^2*b*c*x^3*(105*c^3 + 126*c^2*d*x^2 + 72*c*d^2*x^4 + 16*d^3*x^6) + a^3*(315*c^4*x + 840*c^3*d*x^3 + 1008*c^2*d^2*x^5 + 576*c*d^3*x^7 + 128*d^4*x^9))/(315*c^5*(c + d*x^2)^(9/2))","A",1
98,1,107,174,0.0676635,"\int \frac{\left(a+b x^2\right)^2}{\left(c+d x^2\right)^{9/2}} \, dx","Integrate[(a + b*x^2)^2/(c + d*x^2)^(9/2),x]","\frac{3 a^2 \left(35 c^3 x+70 c^2 d x^3+56 c d^2 x^5+16 d^3 x^7\right)+2 a b c x^3 \left(35 c^2+28 c d x^2+8 d^2 x^4\right)+3 b^2 c^2 x^5 \left(7 c+2 d x^2\right)}{105 c^4 \left(c+d x^2\right)^{7/2}}","\frac{8 a^2 x (7 b c-6 a d)}{105 c^4 \sqrt{c+d x^2} (b c-a d)}+\frac{4 a x \left(a+b x^2\right) (7 b c-6 a d)}{105 c^3 \left(c+d x^2\right)^{3/2} (b c-a d)}+\frac{x \left(a+b x^2\right)^2 (7 b c-6 a d)}{35 c^2 \left(c+d x^2\right)^{5/2} (b c-a d)}-\frac{d x \left(a+b x^2\right)^3}{7 c \left(c+d x^2\right)^{7/2} (b c-a d)}",1,"(3*b^2*c^2*x^5*(7*c + 2*d*x^2) + 2*a*b*c*x^3*(35*c^2 + 28*c*d*x^2 + 8*d^2*x^4) + 3*a^2*(35*c^3*x + 70*c^2*d*x^3 + 56*c*d^2*x^5 + 16*d^3*x^7))/(105*c^4*(c + d*x^2)^(7/2))","A",1
99,1,59,91,0.0227806,"\int \frac{a+b x^2}{\left(c+d x^2\right)^{7/2}} \, dx","Integrate[(a + b*x^2)/(c + d*x^2)^(7/2),x]","\frac{a \left(15 c^2 x+20 c d x^3+8 d^2 x^5\right)+b c x^3 \left(5 c+2 d x^2\right)}{15 c^3 \left(c+d x^2\right)^{5/2}}","\frac{2 x (4 a d+b c)}{15 c^3 d \sqrt{c+d x^2}}+\frac{x (4 a d+b c)}{15 c^2 d \left(c+d x^2\right)^{3/2}}-\frac{x (b c-a d)}{5 c d \left(c+d x^2\right)^{5/2}}",1,"(b*c*x^3*(5*c + 2*d*x^2) + a*(15*c^2*x + 20*c*d*x^3 + 8*d^2*x^5))/(15*c^3*(c + d*x^2)^(5/2))","A",1
100,1,29,39,0.0094737,"\int \frac{1}{\left(c+d x^2\right)^{5/2}} \, dx","Integrate[(c + d*x^2)^(-5/2),x]","\frac{x \left(3 c+2 d x^2\right)}{3 c^2 \left(c+d x^2\right)^{3/2}}","\frac{2 x}{3 c^2 \sqrt{c+d x^2}}+\frac{x}{3 c \left(c+d x^2\right)^{3/2}}",1,"(x*(3*c + 2*d*x^2))/(3*c^2*(c + d*x^2)^(3/2))","A",1
101,1,236,79,2.7160826,"\int \frac{1}{\left(a+b x^2\right) \left(c+d x^2\right)^{3/2}} \, dx","Integrate[1/((a + b*x^2)*(c + d*x^2)^(3/2)),x]","-\frac{\frac{15 c \left(3 c+2 d x^2\right) \left(c \left(a+b x^2\right) \sqrt{\frac{a x^2 \left(c+d x^2\right) (b c-a d)}{c^2 \left(a+b x^2\right)^2}}-a \left(c+d x^2\right) \sin ^{-1}\left(\sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}\right)\right)}{\sqrt{\frac{a x^2 \left(c+d x^2\right) (b c-a d)}{c^2 \left(a+b x^2\right)^2}}}+\frac{4 x^4 \left(c+d x^2\right) (b c-a d)^2 \, _2F_1\left(2,2;\frac{7}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)}{a+b x^2}}{15 c^3 x \left(a+b x^2\right) \sqrt{c+d x^2} (a d-b c)}","\frac{b \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{\sqrt{a} (b c-a d)^{3/2}}-\frac{d x}{c \sqrt{c+d x^2} (b c-a d)}",1,"-1/15*((15*c*(3*c + 2*d*x^2)*(c*(a + b*x^2)*Sqrt[(a*(b*c - a*d)*x^2*(c + d*x^2))/(c^2*(a + b*x^2)^2)] - a*(c + d*x^2)*ArcSin[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]]))/Sqrt[(a*(b*c - a*d)*x^2*(c + d*x^2))/(c^2*(a + b*x^2)^2)] + (4*(b*c - a*d)^2*x^4*(c + d*x^2)*Hypergeometric2F1[2, 2, 7/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))])/(a + b*x^2))/(c^3*(-(b*c) + a*d)*x*(a + b*x^2)*Sqrt[c + d*x^2])","C",0
102,1,405,100,0.7793437,"\int \frac{1}{\left(a+b x^2\right)^2 \sqrt{c+d x^2}} \, dx","Integrate[1/((a + b*x^2)^2*Sqrt[c + d*x^2]),x]","\frac{x \sqrt{c+d x^2} \left(-30 d x^2 \sqrt{\frac{a x^2 \left(c+d x^2\right) (b c-a d)}{c^2 \left(a+b x^2\right)^2}}-45 c \sqrt{\frac{a x^2 \left(c+d x^2\right) (b c-a d)}{c^2 \left(a+b x^2\right)^2}}+16 d x^2 \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}} \left(\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}\right)^{5/2} \, _2F_1\left(2,3;\frac{7}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)+16 c \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}} \left(\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}\right)^{5/2} \, _2F_1\left(2,3;\frac{7}{2};\frac{(b c-a d) x^2}{c \left(b x^2+a\right)}\right)+30 d x^2 \sin ^{-1}\left(\sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}\right)+45 c \sin ^{-1}\left(\sqrt{\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}}\right)\right)}{30 c^2 \left(a+b x^2\right)^2 \left(\frac{x^2 (b c-a d)}{c \left(a+b x^2\right)}\right)^{3/2} \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}}}","\frac{(b c-2 a d) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{2 a^{3/2} (b c-a d)^{3/2}}+\frac{b x \sqrt{c+d x^2}}{2 a \left(a+b x^2\right) (b c-a d)}",1,"(x*Sqrt[c + d*x^2]*(-45*c*Sqrt[(a*(b*c - a*d)*x^2*(c + d*x^2))/(c^2*(a + b*x^2)^2)] - 30*d*x^2*Sqrt[(a*(b*c - a*d)*x^2*(c + d*x^2))/(c^2*(a + b*x^2)^2)] + 45*c*ArcSin[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]] + 30*d*x^2*ArcSin[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]] + 16*c*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(5/2)*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*Hypergeometric2F1[2, 3, 7/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + 16*d*x^2*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(5/2)*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*Hypergeometric2F1[2, 3, 7/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))]))/(30*c^2*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(3/2)*(a + b*x^2)^2*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))])","C",0
103,1,130,149,5.1689874,"\int \frac{\sqrt{c+d x^2}}{\left(a+b x^2\right)^3} \, dx","Integrate[Sqrt[c + d*x^2]/(a + b*x^2)^3,x]","\frac{\frac{\sqrt{a} x \sqrt{c+d x^2} \left(-4 a^2 d+a b \left(5 c-2 d x^2\right)+3 b^2 c x^2\right)}{\left(a+b x^2\right)^2 (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)}{(b c-a d)^{3/2}}}{8 a^{5/2}}","\frac{c (3 b c-4 a d) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{8 a^{5/2} (b c-a d)^{3/2}}+\frac{x \sqrt{c+d x^2} (3 b c-4 a d)}{8 a^2 \left(a+b x^2\right) (b c-a d)}+\frac{b x \left(c+d x^2\right)^{3/2}}{4 a \left(a+b x^2\right)^2 (b c-a d)}",1,"((Sqrt[a]*x*Sqrt[c + d*x^2]*(-4*a^2*d + 3*b^2*c*x^2 + a*b*(5*c - 2*d*x^2)))/((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])])/(b*c - a*d)^(3/2))/(8*a^(5/2))","A",1
104,1,179,199,5.2697012,"\int \frac{\left(c+d x^2\right)^{3/2}}{\left(a+b x^2\right)^4} \, dx","Integrate[(c + d*x^2)^(3/2)/(a + b*x^2)^4,x]","\frac{\frac{3 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)}{(b c-a d)^{3/2}}-\frac{\sqrt{a} x \sqrt{c+d x^2} \left(-6 a^3 d \left(5 c+2 d x^2\right)+a^2 b \left(33 c^2-22 c d x^2-4 d^2 x^4\right)+8 a b^2 c x^2 \left(5 c-d x^2\right)+15 b^3 c^2 x^4\right)}{\left(a+b x^2\right)^3 (a d-b c)}}{48 a^{7/2}}","\frac{c^2 (5 b c-6 a d) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{16 a^{7/2} (b c-a d)^{3/2}}+\frac{c x \sqrt{c+d x^2} (5 b c-6 a d)}{16 a^3 \left(a+b x^2\right) (b c-a d)}+\frac{x \left(c+d x^2\right)^{3/2} (5 b c-6 a d)}{24 a^2 \left(a+b x^2\right)^2 (b c-a d)}+\frac{b x \left(c+d x^2\right)^{5/2}}{6 a \left(a+b x^2\right)^3 (b c-a d)}",1,"(-((Sqrt[a]*x*Sqrt[c + d*x^2]*(15*b^3*c^2*x^4 + 8*a*b^2*c*x^2*(5*c - d*x^2) - 6*a^3*d*(5*c + 2*d*x^2) + a^2*b*(33*c^2 - 22*c*d*x^2 - 4*d^2*x^4)))/((-(b*c) + a*d)*(a + b*x^2)^3)) + (3*c^2*(5*b*c - 6*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(b*c - a*d)^(3/2))/(48*a^(7/2))","A",1
105,1,20,20,0.008458,"\int \frac{1}{\left(\frac{b c}{d}+b x^2\right) \sqrt{c+d x^2}} \, dx","Integrate[1/(((b*c)/d + b*x^2)*Sqrt[c + d*x^2]),x]","\frac{d x}{b c \sqrt{c+d x^2}}","\frac{d x}{b c \sqrt{c+d x^2}}",1,"(d*x)/(b*c*Sqrt[c + d*x^2])","A",1
106,1,25,25,0.0056259,"\int \frac{1}{\sqrt{1-x^2} \left(1+x^2\right)} \, dx","Integrate[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",1
107,1,49,49,0.0147436,"\int \frac{1}{\left(a+b x^2\right) \sqrt{c+d x^2}} \, dx","Integrate[1/((a + b*x^2)*Sqrt[c + d*x^2]),x]","\frac{\tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{\sqrt{a} \sqrt{b c-a d}}","\frac{\tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{\sqrt{a} \sqrt{b c-a d}}",1,"ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])]/(Sqrt[a]*Sqrt[b*c - a*d])","A",1
108,1,15,15,0.0131077,"\int \frac{-1+x^2}{\left(1+x^2\right)^{3/2}} \, dx","Integrate[(-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",1
109,1,99,648,5.0507388,"\int \left(a-b x^2\right)^{2/3} \left(3 a+b x^2\right)^3 \, dx","Integrate[(a - b*x^2)^(2/3)*(3*a + b*x^2)^3,x]","-\frac{3 \left(-40320 a^4 x \sqrt[3]{1-\frac{b x^2}{a}} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right)-15255 a^4 x+3390 a^3 b x^3+8992 a^2 b^2 x^5+2626 a b^3 x^7+247 b^4 x^9\right)}{6175 \sqrt[3]{a-b x^2}}","\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}} \operatorname{EllipticF}\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),4 \sqrt{3}-7\right)}{1235 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{36288 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^{13/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{1235 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{72576 a^4 x}{1235 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}+\frac{18144 a^3 x \left(a-b x^2\right)^{2/3}}{1235}-\frac{23544 a^2 x \left(a-b x^2\right)^{5/3}}{6175}-\frac{378}{475} a x \left(a-b x^2\right)^{5/3} \left(3 a+b x^2\right)-\frac{3}{25} x \left(a-b x^2\right)^{5/3} \left(3 a+b x^2\right)^2",1,"(-3*(-15255*a^4*x + 3390*a^3*b*x^3 + 8992*a^2*b^2*x^5 + 2626*a*b^3*x^7 + 247*b^4*x^9 - 40320*a^4*x*(1 - (b*x^2)/a)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, (b*x^2)/a]))/(6175*(a - b*x^2)^(1/3))","C",1
110,1,176,617,3.3778588,"\int \left(a-b x^2\right)^{2/3} \left(3 a+b x^2\right)^2 \, dx","Integrate[(a - b*x^2)^(2/3)*(3*a + b*x^2)^2,x]","\frac{x \left(a-b x^2\right)^{2/3} \left(4 b \operatorname{Gamma}\left(\frac{1}{3}\right) \left(3 a x+b x^3\right)^2 \, _3F_2\left(\frac{1}{3},\frac{3}{2},2;1,\frac{9}{2};\frac{b x^2}{a}\right)+8 b x^2 \operatorname{Gamma}\left(\frac{1}{3}\right) \left(18 a^2+9 a b x^2+b^2 x^4\right) \, _2F_1\left(\frac{1}{3},\frac{3}{2};\frac{9}{2};\frac{b x^2}{a}\right)+21 a \operatorname{Gamma}\left(-\frac{2}{3}\right) \left(45 a^2+10 a b x^2+b^2 x^4\right) \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{7}{2};\frac{b x^2}{a}\right)\right)}{105 a \operatorname{Gamma}\left(-\frac{2}{3}\right) \left(1-\frac{b x^2}{a}\right)^{2/3}}","\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}} \operatorname{EllipticF}\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),4 \sqrt{3}-7\right)}{1729 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{15552 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^{10/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{1729 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{31104 a^3 x}{1729 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}+\frac{7776 a^2 x \left(a-b x^2\right)^{2/3}}{1729}-\frac{252}{247} a x \left(a-b x^2\right)^{5/3}-\frac{3}{19} x \left(a-b x^2\right)^{5/3} \left(3 a+b x^2\right)",1,"(x*(a - b*x^2)^(2/3)*(21*a*(45*a^2 + 10*a*b*x^2 + b^2*x^4)*Gamma[-2/3]*Hypergeometric2F1[-2/3, 1/2, 7/2, (b*x^2)/a] + 8*b*x^2*(18*a^2 + 9*a*b*x^2 + b^2*x^4)*Gamma[1/3]*Hypergeometric2F1[1/3, 3/2, 9/2, (b*x^2)/a] + 4*b*(3*a*x + b*x^3)^2*Gamma[1/3]*HypergeometricPFQ[{1/3, 3/2, 2}, {1, 9/2}, (b*x^2)/a]))/(105*a*(1 - (b*x^2)/a)^(2/3)*Gamma[-2/3])","C",0
111,1,62,588,0.0774943,"\int \left(a-b x^2\right)^{2/3} \left(3 a+b x^2\right) \, dx","Integrate[(a - b*x^2)^(2/3)*(3*a + b*x^2),x]","\frac{3}{13} x \left(a-b x^2\right)^{2/3} \left(\frac{14 a \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right)}{\left(1-\frac{b x^2}{a}\right)^{2/3}}-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}} \operatorname{EllipticF}\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),4 \sqrt{3}-7\right)}{13 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{36 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^{7/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{13 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{72 a^2 x}{13 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}+\frac{18}{13} a x \left(a-b x^2\right)^{2/3}-\frac{3}{13} x \left(a-b x^2\right)^{5/3}",1,"(3*x*(a - b*x^2)^(2/3)*(-a + b*x^2 + (14*a*Hypergeometric2F1[-2/3, 1/2, 3/2, (b*x^2)/a])/(1 - (b*x^2)/a)^(2/3)))/13","C",1
112,1,162,740,0.1599002,"\int \frac{\left(a-b x^2\right)^{2/3}}{3 a+b x^2} \, dx","Integrate[(a - b*x^2)^(2/3)/(3*a + b*x^2),x]","\frac{9 a x \left(a-b x^2\right)^{2/3} F_1\left(\frac{1}{2};-\frac{2}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)}{\left(3 a+b x^2\right) \left(9 a F_1\left(\frac{1}{2};-\frac{2}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)-2 b x^2 \left(F_1\left(\frac{3}{2};-\frac{2}{3},2;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)+2 F_1\left(\frac{3}{2};\frac{1}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)\right)\right)}","-\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}} \operatorname{EllipticF}\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),4 \sqrt{3}-7\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,"(9*a*x*(a - b*x^2)^(2/3)*AppellF1[1/2, -2/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a])/((3*a + b*x^2)*(9*a*AppellF1[1/2, -2/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a] - 2*b*x^2*(AppellF1[3/2, -2/3, 2, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a] + 2*AppellF1[3/2, 1/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a])))","C",0
113,1,86,584,0.0608,"\int \frac{\left(a-b x^2\right)^{2/3}}{\left(3 a+b x^2\right)^2} \, dx","Integrate[(a - b*x^2)^(2/3)/(3*a + b*x^2)^2,x]","\frac{x \sqrt[3]{\frac{a-b x^2}{a}} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right)}{18 a \sqrt[3]{a-b x^2}}+\frac{x \left(a-b x^2\right)^{2/3}}{6 a \left(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}} \operatorname{EllipticF}\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),4 \sqrt{3}-7\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*((a - b*x^2)/a)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, (b*x^2)/a])/(18*a*(a - b*x^2)^(1/3))","C",1
114,1,252,818,0.3249619,"\int \frac{\left(a-b x^2\right)^{2/3}}{\left(3 a+b x^2\right)^3} \, dx","Integrate[(a - b*x^2)^(2/3)/(3*a + b*x^2)^3,x]","\frac{\frac{b x^3 \sqrt[3]{1-\frac{b x^2}{a}} F_1\left(\frac{3}{2};\frac{1}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)}{a^3}+\frac{27 x \left(-\frac{b^2 x^4}{a^2}+\frac{18 \left(3 a+b x^2\right) F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)}{2 b x^2 \left(F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)-F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)\right)+9 a F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)}-\frac{5 b x^2}{a}+6\right)}{\left(3 a+b x^2\right)^2}}{972 \sqrt[3]{a-b x^2}}","\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}} \operatorname{EllipticF}\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,"((b*x^3*(1 - (b*x^2)/a)^(1/3)*AppellF1[3/2, 1/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a])/a^3 + (27*x*(6 - (5*b*x^2)/a - (b^2*x^4)/a^2 + (18*(3*a + b*x^2)*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a])/(9*a*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a] + 2*b*x^2*(-AppellF1[3/2, 1/3, 2, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a] + AppellF1[3/2, 4/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a]))))/(3*a + b*x^2)^2)/(972*(a - b*x^2)^(1/3))","C",0
115,1,265,849,0.1923154,"\int \frac{\left(a-b x^2\right)^{2/3}}{\left(3 a+b x^2\right)^4} \, dx","Integrate[(a - b*x^2)^(2/3)/(3*a + b*x^2)^4,x]","\frac{x \left(\frac{621 a^3 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)}{\left(3 a+b x^2\right) \left(2 b x^2 \left(F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)-F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)\right)+9 a F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)\right)}+\frac{9 a \left(a-b x^2\right) \left(75 a^2+26 a b x^2+3 b^2 x^4\right)}{\left(3 a+b x^2\right)^3}+b x^2 \sqrt[3]{1-\frac{b x^2}{a}} F_1\left(\frac{3}{2};\frac{1}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)\right)}{3888 a^4 \sqrt[3]{a-b x^2}}","\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}} \operatorname{EllipticF}\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*((9*a*(a - b*x^2)*(75*a^2 + 26*a*b*x^2 + 3*b^2*x^4))/(3*a + b*x^2)^3 + b*x^2*(1 - (b*x^2)/a)^(1/3)*AppellF1[3/2, 1/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a] + (621*a^3*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a])/((3*a + b*x^2)*(9*a*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a] + 2*b*x^2*(-AppellF1[3/2, 1/3, 2, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a] + AppellF1[3/2, 4/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a])))))/(3888*a^4*(a - b*x^2)^(1/3))","C",0
116,1,110,668,5.0529741,"\int \left(a-b x^2\right)^{5/3} \left(3 a+b x^2\right)^3 \, dx","Integrate[(a - b*x^2)^(5/3)*(3*a + b*x^2)^3,x]","\frac{3 \left(6243840 a^5 x \sqrt[3]{1-\frac{b x^2}{a}} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right)+5815935 a^5 x-5312355 a^4 b x^3-1675114 a^3 b^2 x^5+749658 a^2 b^3 x^7+378651 a b^4 x^9+43225 b^5 x^{11}\right)}{1339975 \sqrt[3]{a-b x^2}}","\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}} \operatorname{EllipticF}\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),4 \sqrt{3}-7\right)}{267995 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{5619456 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^{16/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{267995 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{11238912 a^5 x}{267995 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}+\frac{2809728 a^4 x \left(a-b x^2\right)^{2/3}}{267995}+\frac{1404864 a^3 x \left(a-b x^2\right)^{5/3}}{191425}-\frac{33264 a^2 x \left(a-b x^2\right)^{8/3}}{14725}-\frac{432}{775} a x \left(a-b x^2\right)^{8/3} \left(3 a+b x^2\right)-\frac{3}{31} x \left(a-b x^2\right)^{8/3} \left(3 a+b x^2\right)^2",1,"(3*(5815935*a^5*x - 5312355*a^4*b*x^3 - 1675114*a^3*b^2*x^5 + 749658*a^2*b^3*x^7 + 378651*a*b^4*x^9 + 43225*b^5*x^11 + 6243840*a^5*x*(1 - (b*x^2)/a)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, (b*x^2)/a]))/(1339975*(a - b*x^2)^(1/3))","C",1
117,1,173,637,3.0074021,"\int \left(a-b x^2\right)^{5/3} \left(3 a+b x^2\right)^2 \, dx","Integrate[(a - b*x^2)^(5/3)*(3*a + b*x^2)^2,x]","\frac{x \left(a-b x^2\right)^{2/3} \left(4 b \operatorname{Gamma}\left(-\frac{2}{3}\right) \left(3 a x+b x^3\right)^2 \, _3F_2\left(-\frac{2}{3},\frac{3}{2},2;1,\frac{9}{2};\frac{b x^2}{a}\right)+8 b x^2 \operatorname{Gamma}\left(-\frac{2}{3}\right) \left(18 a^2+9 a b x^2+b^2 x^4\right) \, _2F_1\left(-\frac{2}{3},\frac{3}{2};\frac{9}{2};\frac{b x^2}{a}\right)+21 a \operatorname{Gamma}\left(-\frac{5}{3}\right) \left(45 a^2+10 a b x^2+b^2 x^4\right) \, _2F_1\left(-\frac{5}{3},\frac{1}{2};\frac{7}{2};\frac{b x^2}{a}\right)\right)}{105 \operatorname{Gamma}\left(-\frac{5}{3}\right) \left(1-\frac{b x^2}{a}\right)^{2/3}}","\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}} \operatorname{EllipticF}\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),4 \sqrt{3}-7\right)}{8645 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{57024 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^{13/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{8645 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{114048 a^4 x}{8645 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}+\frac{28512 a^3 x \left(a-b x^2\right)^{2/3}}{8645}+\frac{14256 a^2 x \left(a-b x^2\right)^{5/3}}{6175}-\frac{306}{475} a x \left(a-b x^2\right)^{8/3}-\frac{3}{25} x \left(a-b x^2\right)^{8/3} \left(3 a+b x^2\right)",1,"(x*(a - b*x^2)^(2/3)*(21*a*(45*a^2 + 10*a*b*x^2 + b^2*x^4)*Gamma[-5/3]*Hypergeometric2F1[-5/3, 1/2, 7/2, (b*x^2)/a] + 8*b*x^2*(18*a^2 + 9*a*b*x^2 + b^2*x^4)*Gamma[-2/3]*Hypergeometric2F1[-2/3, 3/2, 9/2, (b*x^2)/a] + 4*b*(3*a*x + b*x^3)^2*Gamma[-2/3]*HypergeometricPFQ[{-2/3, 3/2, 2}, {1, 9/2}, (b*x^2)/a]))/(105*(1 - (b*x^2)/a)^(2/3)*Gamma[-5/3])","C",0
118,1,68,608,0.0635407,"\int \left(a-b x^2\right)^{5/3} \left(3 a+b x^2\right) \, dx","Integrate[(a - b*x^2)^(5/3)*(3*a + b*x^2),x]","\frac{3}{19} x \left(a-b x^2\right)^{2/3} \left(\frac{20 a^2 \, _2F_1\left(-\frac{5}{3},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right)}{\left(1-\frac{b x^2}{a}\right)^{2/3}}-\left(a-b x^2\right)^2\right)","\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}} \operatorname{EllipticF}\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),4 \sqrt{3}-7\right)}{1729 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{3600 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^{10/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{1729 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{7200 a^3 x}{1729 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}+\frac{1800 a^2 x \left(a-b x^2\right)^{2/3}}{1729}+\frac{180}{247} a x \left(a-b x^2\right)^{5/3}-\frac{3}{19} x \left(a-b x^2\right)^{8/3}",1,"(3*x*(a - b*x^2)^(2/3)*(-(a - b*x^2)^2 + (20*a^2*Hypergeometric2F1[-5/3, 1/2, 3/2, (b*x^2)/a])/(1 - (b*x^2)/a)^(2/3)))/19","C",1
119,1,231,765,0.150802,"\int \frac{\left(a-b x^2\right)^{5/3}}{3 a+b x^2} \, dx","Integrate[(a - b*x^2)^(5/3)/(3*a + b*x^2),x]","\frac{x \left(27 \left(\frac{48 a^3 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)}{\left(3 a+b x^2\right) \left(2 b x^2 \left(F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)-F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)\right)+9 a F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)\right)}-a+b x^2\right)-32 b x^2 \sqrt[3]{1-\frac{b x^2}{a}} F_1\left(\frac{3}{2};\frac{1}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)\right)}{63 \sqrt[3]{a-b x^2}}","-\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}} \operatorname{EllipticF}\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),4 \sqrt{3}-7\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[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{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,"(x*(-32*b*x^2*(1 - (b*x^2)/a)^(1/3)*AppellF1[3/2, 1/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a] + 27*(-a + b*x^2 + (48*a^3*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a])/((3*a + b*x^2)*(9*a*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a] + 2*b*x^2*(-AppellF1[3/2, 1/3, 2, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a] + AppellF1[3/2, 4/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a]))))))/(63*(a - b*x^2)^(1/3))","C",0
120,1,235,775,0.1466857,"\int \frac{\left(a-b x^2\right)^{5/3}}{\left(3 a+b x^2\right)^2} \, dx","Integrate[(a - b*x^2)^(5/3)/(3*a + b*x^2)^2,x]","\frac{x \left(\frac{27 \left(-\frac{9 a^2 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)}{2 b x^2 \left(F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)-F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)\right)+9 a F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)}+2 a-2 b x^2\right)}{3 a+b x^2}+\frac{11 b x^2 \sqrt[3]{1-\frac{b x^2}{a}} F_1\left(\frac{3}{2};\frac{1}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)}{a}\right)}{81 \sqrt[3]{a-b x^2}}","\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}} \operatorname{EllipticF}\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),4 \sqrt{3}-7\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,"(x*((11*b*x^2*(1 - (b*x^2)/a)^(1/3)*AppellF1[3/2, 1/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a])/a + (27*(2*a - 2*b*x^2 - (9*a^2*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a])/(9*a*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a] + 2*b*x^2*(-AppellF1[3/2, 1/3, 2, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a] + AppellF1[3/2, 4/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a]))))/(3*a + b*x^2)))/(81*(a - b*x^2)^(1/3))","C",0
121,1,252,815,0.2353087,"\int \frac{\left(a-b x^2\right)^{5/3}}{\left(3 a+b x^2\right)^3} \, dx","Integrate[(a - b*x^2)^(5/3)/(3*a + b*x^2)^3,x]","\frac{\frac{27 x \left(\frac{9 a \left(3 a+b x^2\right) F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)}{2 b x^2 \left(F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)-F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)\right)+9 a F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)}+\frac{b^2 x^4}{a}+3 a-4 b x^2\right)}{\left(3 a+b x^2\right)^2}-\frac{b x^3 \sqrt[3]{1-\frac{b x^2}{a}} F_1\left(\frac{3}{2};\frac{1}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)}{a^2}}{486 \sqrt[3]{a-b x^2}}","-\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}} \operatorname{EllipticF}\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,"(-((b*x^3*(1 - (b*x^2)/a)^(1/3)*AppellF1[3/2, 1/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a])/a^2) + (27*x*(3*a - 4*b*x^2 + (b^2*x^4)/a + (9*a*(3*a + b*x^2)*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a])/(9*a*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a] + 2*b*x^2*(-AppellF1[3/2, 1/3, 2, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a] + AppellF1[3/2, 4/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a]))))/(3*a + b*x^2)^2)/(486*(a - b*x^2)^(1/3))","C",0
122,1,98,659,5.0615829,"\int \frac{\left(3 a+b x^2\right)^4}{\sqrt[3]{a-b x^2}} \, dx","Integrate[(3*a + b*x^2)^4/(a - b*x^2)^(1/3),x]","\frac{3 x \left(2108160 a^4 \sqrt[3]{1-\frac{b x^2}{a}} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right)-941085 a^4+727830 a^3 b x^2+184044 a^2 b^2 x^4+27482 a b^3 x^6+1729 b^4 x^8\right)}{43225 \sqrt[3]{a-b x^2}}","\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}} \operatorname{EllipticF}\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),4 \sqrt{3}-7\right)}{8645 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{1897344 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^{13/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{8645 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{3794688 a^4 x}{8645 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}-\frac{1552608 a^3 x \left(a-b x^2\right)^{2/3}}{43225}-\frac{36288 a^2 x \left(a-b x^2\right)^{2/3} \left(3 a+b x^2\right)}{6175}-\frac{18}{19} a x \left(a-b x^2\right)^{2/3} \left(3 a+b x^2\right)^2-\frac{3}{25} x \left(a-b x^2\right)^{2/3} \left(3 a+b x^2\right)^3",1,"(3*x*(-941085*a^4 + 727830*a^3*b*x^2 + 184044*a^2*b^2*x^4 + 27482*a*b^3*x^6 + 1729*b^4*x^8 + 2108160*a^4*(1 - (b*x^2)/a)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, (b*x^2)/a]))/(43225*(a - b*x^2)^(1/3))","C",1
123,1,88,628,5.0508531,"\int \frac{\left(3 a+b x^2\right)^3}{\sqrt[3]{a-b x^2}} \, dx","Integrate[(3*a + b*x^2)^3/(a - b*x^2)^(1/3),x]","\frac{3 \left(23904 a^3 x \sqrt[3]{1-\frac{b x^2}{a}} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right)-8343 a^3 x+7041 a^2 b x^3+1211 a b^2 x^5+91 b^3 x^7\right)}{1729 \sqrt[3]{a-b x^2}}","\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}} \operatorname{EllipticF}\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),4 \sqrt{3}-7\right)}{1729 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{107568 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^{10/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{1729 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{215136 a^3 x}{1729 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}-\frac{15768 a^2 x \left(a-b x^2\right)^{2/3}}{1729}-\frac{324}{247} a x \left(a-b x^2\right)^{2/3} \left(3 a+b x^2\right)-\frac{3}{19} x \left(a-b x^2\right)^{2/3} \left(3 a+b x^2\right)^2",1,"(3*(-8343*a^3*x + 7041*a^2*b*x^3 + 1211*a*b^2*x^5 + 91*b^3*x^7 + 23904*a^3*x*(1 - (b*x^2)/a)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, (b*x^2)/a]))/(1729*(a - b*x^2)^(1/3))","C",1
124,1,158,597,4.4945301,"\int \frac{\left(3 a+b x^2\right)^2}{\sqrt[3]{a-b x^2}} \, dx","Integrate[(3*a + b*x^2)^2/(a - b*x^2)^(1/3),x]","\frac{x \sqrt[3]{1-\frac{b x^2}{a}} \left(4 b \left(3 a x+b x^3\right)^2 \, _3F_2\left(\frac{4}{3},\frac{3}{2},2;1,\frac{9}{2};\frac{b x^2}{a}\right)+8 b x^2 \left(18 a^2+9 a b x^2+b^2 x^4\right) \, _2F_1\left(\frac{4}{3},\frac{3}{2};\frac{9}{2};\frac{b x^2}{a}\right)+63 a \left(45 a^2+10 a b x^2+b^2 x^4\right) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{7}{2};\frac{b x^2}{a}\right)\right)}{315 a \sqrt[3]{a-b x^2}}","\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}} \operatorname{EllipticF}\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),4 \sqrt{3}-7\right)}{91 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{1620 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^{7/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{91 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{3240 a^2 x}{91 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}-\frac{198}{91} a x \left(a-b x^2\right)^{2/3}-\frac{3}{13} x \left(a-b x^2\right)^{2/3} \left(3 a+b x^2\right)",1,"(x*(1 - (b*x^2)/a)^(1/3)*(63*a*(45*a^2 + 10*a*b*x^2 + b^2*x^4)*Hypergeometric2F1[1/3, 1/2, 7/2, (b*x^2)/a] + 8*b*x^2*(18*a^2 + 9*a*b*x^2 + b^2*x^4)*Hypergeometric2F1[4/3, 3/2, 9/2, (b*x^2)/a] + 4*b*(3*a*x + b*x^3)^2*HypergeometricPFQ[{4/3, 3/2, 2}, {1, 9/2}, (b*x^2)/a]))/(315*a*(a - b*x^2)^(1/3))","C",0
125,1,62,568,0.0271453,"\int \frac{3 a+b x^2}{\sqrt[3]{a-b x^2}} \, dx","Integrate[(3*a + b*x^2)/(a - b*x^2)^(1/3),x]","\frac{3 x \left(8 a \sqrt[3]{1-\frac{b x^2}{a}} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right)-a+b x^2\right)}{7 \sqrt[3]{a-b x^2}}","\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}} \operatorname{EllipticF}\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),4 \sqrt{3}-7\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 + 8*a*(1 - (b*x^2)/a)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, (b*x^2)/a]))/(7*(a - b*x^2)^(1/3))","C",1
126,1,162,204,0.0470164,"\int \frac{1}{\sqrt[3]{a-b x^2} \left(3 a+b x^2\right)} \, dx","Integrate[1/((a - b*x^2)^(1/3)*(3*a + b*x^2)),x]","\frac{9 a x F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)}{\sqrt[3]{a-b x^2} \left(3 a+b x^2\right) \left(2 b x^2 \left(F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)-F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)\right)+9 a F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)\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)}{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,"(9*a*x*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a])/((a - b*x^2)^(1/3)*(3*a + b*x^2)*(9*a*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a] + 2*b*x^2*(-AppellF1[3/2, 1/3, 2, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a] + AppellF1[3/2, 4/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a])))","C",0
127,1,234,787,0.1620844,"\int \frac{1}{\sqrt[3]{a-b x^2} \left(3 a+b x^2\right)^2} \, dx","Integrate[1/((a - b*x^2)^(1/3)*(3*a + b*x^2)^2),x]","\frac{x \left(\frac{b x^2 \sqrt[3]{1-\frac{b x^2}{a}} F_1\left(\frac{3}{2};\frac{1}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)}{a^3}+\frac{27 \left(\frac{a-b x^2}{a^2}+\frac{63 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)}{2 b x^2 \left(F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)-F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)\right)+9 a F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)}\right)}{3 a+b x^2}\right)}{648 \sqrt[3]{a-b x^2}}","\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}} \operatorname{EllipticF}\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),4 \sqrt{3}-7\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{\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{\sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{16\ 3^{3/4} a^{5/3} b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right)}{8\ 2^{2/3} \sqrt{3} a^{11/6} \sqrt{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{24\ 2^{2/3} a^{11/6} \sqrt{b}}+\frac{x \left(a-b x^2\right)^{2/3}}{24 a^2 \left(3 a+b x^2\right)}-\frac{x}{24 a^2 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}",1,"(x*((b*x^2*(1 - (b*x^2)/a)^(1/3)*AppellF1[3/2, 1/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a])/a^3 + (27*((a - b*x^2)/a^2 + (63*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a])/(9*a*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a] + 2*b*x^2*(-AppellF1[3/2, 1/3, 2, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a] + AppellF1[3/2, 4/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a]))))/(3*a + b*x^2)))/(648*(a - b*x^2)^(1/3))","C",0
128,1,255,818,0.172311,"\int \frac{1}{\sqrt[3]{a-b x^2} \left(3 a+b x^2\right)^3} \, dx","Integrate[1/((a - b*x^2)^(1/3)*(3*a + b*x^2)^3),x]","\frac{x \left(\frac{6075 a^3 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)}{\left(3 a+b x^2\right) \left(2 b x^2 \left(F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)-F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)\right)+9 a F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)\right)}+5 b x^2 \sqrt[3]{1-\frac{b x^2}{a}} F_1\left(\frac{3}{2};\frac{1}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)+\frac{27 a \left(a-b x^2\right) \left(21 a+5 b x^2\right)}{\left(3 a+b x^2\right)^2}\right)}{7776 a^4 \sqrt[3]{a-b x^2}}","\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}} \operatorname{EllipticF}\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*((27*a*(a - b*x^2)*(21*a + 5*b*x^2))/(3*a + b*x^2)^2 + 5*b*x^2*(1 - (b*x^2)/a)^(1/3)*AppellF1[3/2, 1/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a] + (6075*a^3*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a])/((3*a + b*x^2)*(9*a*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a] + 2*b*x^2*(-AppellF1[3/2, 1/3, 2, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a] + AppellF1[3/2, 4/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a])))))/(7776*a^4*(a - b*x^2)^(1/3))","C",0
129,1,76,623,5.0568414,"\int \frac{\left(3 a+b x^2\right)^3}{\left(a-b x^2\right)^{4/3}} \, dx","Integrate[(3*a + b*x^2)^3/(a - b*x^2)^(4/3),x]","-\frac{3 x \left(2232 a^2 \sqrt[3]{1-\frac{b x^2}{a}} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right)-3051 a^2+132 a b x^2+7 b^2 x^4\right)}{91 \sqrt[3]{a-b x^2}}","-\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}} \operatorname{EllipticF}\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),4 \sqrt{3}-7\right)}{91 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}+\frac{10044 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^{7/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{91 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}+\frac{20088 a^2 x}{91 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}+\frac{2538}{91} a x \left(a-b x^2\right)^{2/3}+\frac{6 x \left(3 a+b x^2\right)^2}{\sqrt[3]{a-b x^2}}+\frac{81}{13} x \left(a-b x^2\right)^{2/3} \left(3 a+b x^2\right)",1,"(-3*x*(-3051*a^2 + 132*a*b*x^2 + 7*b^2*x^4 + 2232*a^2*(1 - (b*x^2)/a)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, (b*x^2)/a]))/(91*(a - b*x^2)^(1/3))","C",1
130,1,62,592,5.048152,"\int \frac{\left(3 a+b x^2\right)^2}{\left(a-b x^2\right)^{4/3}} \, dx","Integrate[(3*a + b*x^2)^2/(a - b*x^2)^(4/3),x]","-\frac{3 x \left(36 a \sqrt[3]{1-\frac{b x^2}{a}} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right)-57 a+b x^2\right)}{7 \sqrt[3]{a-b x^2}}","-\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}} \operatorname{EllipticF}\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),4 \sqrt{3}-7\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,"(-3*x*(-57*a + b*x^2 + 36*a*(1 - (b*x^2)/a)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, (b*x^2)/a]))/(7*(a - b*x^2)^(1/3))","C",1
131,1,53,561,0.0195962,"\int \frac{3 a+b x^2}{\left(a-b x^2\right)^{4/3}} \, dx","Integrate[(3*a + b*x^2)/(a - b*x^2)^(4/3),x]","\frac{6 x-3 x \sqrt[3]{1-\frac{b x^2}{a}} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right)}{\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}} \operatorname{EllipticF}\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),4 \sqrt{3}-7\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 - 3*x*(1 - (b*x^2)/a)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, (b*x^2)/a])/(a - b*x^2)^(1/3)","C",1
132,1,226,776,0.1422151,"\int \frac{1}{\left(a-b x^2\right)^{4/3} \left(3 a+b x^2\right)} \, dx","Integrate[1/((a - b*x^2)^(4/3)*(3*a + b*x^2)),x]","\frac{x \left(27 \left(\frac{1}{a^2}-\frac{3 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)}{\left(3 a+b x^2\right) \left(2 b x^2 \left(F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)-F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)\right)+9 a F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)\right)}\right)-\frac{b x^2 \sqrt[3]{1-\frac{b x^2}{a}} F_1\left(\frac{3}{2};\frac{1}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)}{a^3}\right)}{72 \sqrt[3]{a-b x^2}}","-\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}} \operatorname{EllipticF}\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),4 \sqrt{3}-7\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{\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 \sqrt[4]{3} \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{16 a^{5/3} b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right)}{8\ 2^{2/3} \sqrt{3} a^{11/6} \sqrt{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{24\ 2^{2/3} a^{11/6} \sqrt{b}}+\frac{3 x}{8 a^2 \sqrt[3]{a-b x^2}}+\frac{3 x}{8 a^2 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}",1,"(x*(-((b*x^2*(1 - (b*x^2)/a)^(1/3)*AppellF1[3/2, 1/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a])/a^3) + 27*(a^(-2) - (3*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a])/((3*a + b*x^2)*(9*a*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a] + 2*b*x^2*(-AppellF1[3/2, 1/3, 2, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a] + AppellF1[3/2, 4/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a]))))))/(72*(a - b*x^2)^(1/3))","C",0
133,1,236,807,0.1801315,"\int \frac{1}{\left(a-b x^2\right)^{4/3} \left(3 a+b x^2\right)^2} \, dx","Integrate[1/((a - b*x^2)^(4/3)*(3*a + b*x^2)^2),x]","\frac{x \left(\frac{27 a \left(\frac{9 a^2 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)}{2 b x^2 \left(F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)-F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)\right)+9 a F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)}+7 a+2 b x^2\right)}{3 a+b x^2}-2 b x^2 \sqrt[3]{1-\frac{b x^2}{a}} F_1\left(\frac{3}{2};\frac{1}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)\right)}{648 a^4 \sqrt[3]{a-b x^2}}","\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}} \operatorname{EllipticF}\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*(-2*b*x^2*(1 - (b*x^2)/a)^(1/3)*AppellF1[3/2, 1/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a] + (27*a*(7*a + 2*b*x^2 + (9*a^2*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a])/(9*a*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a] + 2*b*x^2*(-AppellF1[3/2, 1/3, 2, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a] + AppellF1[3/2, 4/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a]))))/(3*a + b*x^2)))/(648*a^4*(a - b*x^2)^(1/3))","C",0
134,1,256,849,0.2410933,"\int \frac{1}{\left(a-b x^2\right)^{4/3} \left(3 a+b x^2\right)^3} \, dx","Integrate[1/((a - b*x^2)^(4/3)*(3*a + b*x^2)^3),x]","\frac{x \left(\frac{27 a \left(\frac{333 a^2 \left(3 a+b x^2\right) F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)}{2 b x^2 \left(F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)-F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)\right)+9 a F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)}+273 a^2+140 a b x^2+19 b^2 x^4\right)}{\left(3 a+b x^2\right)^2}-19 b x^2 \sqrt[3]{1-\frac{b x^2}{a}} F_1\left(\frac{3}{2};\frac{1}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)\right)}{31104 a^5 \sqrt[3]{a-b x^2}}","-\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}} \operatorname{EllipticF}\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*(-19*b*x^2*(1 - (b*x^2)/a)^(1/3)*AppellF1[3/2, 1/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a] + (27*a*(273*a^2 + 140*a*b*x^2 + 19*b^2*x^4 + (333*a^2*(3*a + b*x^2)*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a])/(9*a*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a] + 2*b*x^2*(-AppellF1[3/2, 1/3, 2, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a] + AppellF1[3/2, 4/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a]))))/(3*a + b*x^2)^2))/(31104*a^5*(a - b*x^2)^(1/3))","C",0
135,1,96,653,5.0787766,"\int \frac{\left(3 a+b x^2\right)^4}{\left(a-b x^2\right)^{7/3}} \, dx","Integrate[(3*a + b*x^2)^4/(a - b*x^2)^(7/3),x]","-\frac{3 \left(1647 a^3 x-4104 a^2 x \left(a-b x^2\right) \sqrt[3]{1-\frac{b x^2}{a}} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right)-4743 a^2 b x^3+177 a b^2 x^5+7 b^3 x^7\right)}{91 \left(a-b x^2\right)^{4/3}}","\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}} \operatorname{EllipticF}\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),4 \sqrt{3}-7\right)}{91 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{18468 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^{7/3} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a-b x^2}+\left(a-b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}}\right)|-7+4 \sqrt{3}\right)}{91 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)^2}}}-\frac{36936 a^2 x}{91 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a-b x^2}\right)}+\frac{3 x \left(3 a+b x^2\right)^3}{2 \left(a-b x^2\right)^{4/3}}-\frac{9 x \left(3 a+b x^2\right)^2}{2 \sqrt[3]{a-b x^2}}-\frac{81}{13} x \left(a-b x^2\right)^{2/3} \left(3 a+b x^2\right)-\frac{3240}{91} a x \left(a-b x^2\right)^{2/3}",1,"(-3*(1647*a^3*x - 4743*a^2*b*x^3 + 177*a*b^2*x^5 + 7*b^3*x^7 - 4104*a^2*x*(a - b*x^2)*(1 - (b*x^2)/a)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, (b*x^2)/a]))/(91*(a - b*x^2)^(4/3))","C",1
136,1,83,596,5.0747833,"\int \frac{\left(3 a+b x^2\right)^3}{\left(a-b x^2\right)^{7/3}} \, dx","Integrate[(3*a + b*x^2)^3/(a - b*x^2)^(7/3),x]","\frac{81 a^2 x+108 a x \left(a-b x^2\right) \sqrt[3]{1-\frac{b x^2}{a}} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right)+90 a b x^3-3 b^2 x^5}{7 \left(a-b x^2\right)^{4/3}}","\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}} \operatorname{EllipticF}\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),4 \sqrt{3}-7\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,"(81*a^2*x + 90*a*b*x^3 - 3*b^2*x^5 + 108*a*x*(a - b*x^2)*(1 - (b*x^2)/a)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, (b*x^2)/a])/(7*(a - b*x^2)^(4/3))","C",1
137,1,24,44,5.0269897,"\int \frac{\left(3 a+b x^2\right)^2}{\left(a-b x^2\right)^{7/3}} \, dx","Integrate[(3*a + b*x^2)^2/(a - b*x^2)^(7/3),x]","\frac{9 a x-3 b x^3}{\left(a-b x^2\right)^{4/3}}","\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*a*x - 3*b*x^3)/(a - b*x^2)^(4/3)","A",1
138,1,74,590,0.0396252,"\int \frac{3 a+b x^2}{\left(a-b x^2\right)^{7/3}} \, dx","Integrate[(3*a + b*x^2)/(a - b*x^2)^(7/3),x]","\frac{-3 x \left(a-b x^2\right) \sqrt[3]{1-\frac{b x^2}{a}} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right)+15 a x-9 b x^3}{4 a \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}} \operatorname{EllipticF}\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),4 \sqrt{3}-7\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,"(15*a*x - 9*b*x^3 - 3*x*(a - b*x^2)*(1 - (b*x^2)/a)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, (b*x^2)/a])/(4*a*(a - b*x^2)^(4/3))","C",1
139,1,248,796,0.2096804,"\int \frac{1}{\left(a-b x^2\right)^{7/3} \left(3 a+b x^2\right)} \, dx","Integrate[1/((a - b*x^2)^(7/3)*(3*a + b*x^2)),x]","\frac{x \left(27 a \left(\frac{9 a-7 b x^2}{a-b x^2}-\frac{51 a^2 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)}{\left(3 a+b x^2\right) \left(2 b x^2 \left(F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)-F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)\right)+9 a F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)\right)}\right)-7 b x^2 \sqrt[3]{1-\frac{b x^2}{a}} F_1\left(\frac{3}{2};\frac{1}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)\right)}{576 a^4 \sqrt[3]{a-b x^2}}","\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}} \operatorname{EllipticF}\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,"(x*(-7*b*x^2*(1 - (b*x^2)/a)^(1/3)*AppellF1[3/2, 1/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a] + 27*a*((9*a - 7*b*x^2)/(a - b*x^2) - (51*a^2*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a])/((3*a + b*x^2)*(9*a*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a] + 2*b*x^2*(-AppellF1[3/2, 1/3, 2, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a] + AppellF1[3/2, 4/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a]))))))/(576*a^4*(a - b*x^2)^(1/3))","C",0
140,1,259,827,0.2433019,"\int \frac{1}{\left(a-b x^2\right)^{7/3} \left(3 a+b x^2\right)^2} \, dx","Integrate[1/((a - b*x^2)^(7/3)*(3*a + b*x^2)^2),x]","\frac{x \left(\frac{27 a \left(\frac{299 a^2-148 a b x^2-79 b^2 x^4}{a-b x^2}-\frac{387 a^2 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)}{2 b x^2 \left(F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)-F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)\right)+9 a F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)}\right)}{3 a+b x^2}-79 b x^2 \sqrt[3]{1-\frac{b x^2}{a}} F_1\left(\frac{3}{2};\frac{1}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)\right)}{20736 a^5 \sqrt[3]{a-b x^2}}","\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}} \operatorname{EllipticF}\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,"(x*(-79*b*x^2*(1 - (b*x^2)/a)^(1/3)*AppellF1[3/2, 1/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a] + (27*a*((299*a^2 - 148*a*b*x^2 - 79*b^2*x^4)/(a - b*x^2) - (387*a^2*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a])/(9*a*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a] + 2*b*x^2*(-AppellF1[3/2, 1/3, 2, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a] + AppellF1[3/2, 4/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a]))))/(3*a + b*x^2)))/(20736*a^5*(a - b*x^2)^(1/3))","C",0
141,1,163,252,0.1382392,"\int \frac{1}{\left(-3 a-b x^2\right) \sqrt[3]{-a+b x^2}} \, dx","Integrate[1/((-3*a - b*x^2)*(-a + b*x^2)^(1/3)),x]","-\frac{9 a x F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)}{\sqrt[3]{b x^2-a} \left(3 a+b x^2\right) \left(2 b x^2 \left(F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)-F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)\right)+9 a F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)\right)}","-\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,"(-9*a*x*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a])/((-a + b*x^2)^(1/3)*(3*a + b*x^2)*(9*a*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a] + 2*b*x^2*(-AppellF1[3/2, 1/3, 2, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a] + AppellF1[3/2, 4/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a])))","C",0
142,1,166,202,0.1561119,"\int \frac{1}{\left(3 a-b x^2\right) \sqrt[3]{a+b x^2}} \, dx","Integrate[1/((3*a - b*x^2)*(a + b*x^2)^(1/3)),x]","\frac{9 a x F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-\frac{b x^2}{a},\frac{b x^2}{3 a}\right)}{\left(3 a-b x^2\right) \sqrt[3]{a+b x^2} \left(2 b x^2 \left(F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};-\frac{b x^2}{a},\frac{b x^2}{3 a}\right)-F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};-\frac{b x^2}{a},\frac{b x^2}{3 a}\right)\right)+9 a F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-\frac{b x^2}{a},\frac{b x^2}{3 a}\right)\right)}","\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,"(9*a*x*AppellF1[1/2, 1/3, 1, 3/2, -((b*x^2)/a), (b*x^2)/(3*a)])/((3*a - b*x^2)*(a + b*x^2)^(1/3)*(9*a*AppellF1[1/2, 1/3, 1, 3/2, -((b*x^2)/a), (b*x^2)/(3*a)] + 2*b*x^2*(AppellF1[3/2, 1/3, 2, 5/2, -((b*x^2)/a), (b*x^2)/(3*a)] - AppellF1[3/2, 4/3, 1, 5/2, -((b*x^2)/a), (b*x^2)/(3*a)])))","C",0
143,1,153,204,0.1513249,"\int \frac{1}{\left(c-d x^2\right) \sqrt[3]{c+3 d x^2}} \, dx","Integrate[1/((c - d*x^2)*(c + 3*d*x^2)^(1/3)),x]","\frac{3 c x F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-\frac{3 d x^2}{c},\frac{d x^2}{c}\right)}{\left(c-d x^2\right) \sqrt[3]{c+3 d x^2} \left(2 d x^2 \left(F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};-\frac{3 d x^2}{c},\frac{d x^2}{c}\right)-F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};-\frac{3 d x^2}{c},\frac{d x^2}{c}\right)\right)+3 c F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-\frac{3 d x^2}{c},\frac{d x^2}{c}\right)\right)}","\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,"(3*c*x*AppellF1[1/2, 1/3, 1, 3/2, (-3*d*x^2)/c, (d*x^2)/c])/((c - d*x^2)*(c + 3*d*x^2)^(1/3)*(3*c*AppellF1[1/2, 1/3, 1, 3/2, (-3*d*x^2)/c, (d*x^2)/c] + 2*d*x^2*(AppellF1[3/2, 1/3, 2, 5/2, (-3*d*x^2)/c, (d*x^2)/c] - AppellF1[3/2, 4/3, 1, 5/2, (-3*d*x^2)/c, (d*x^2)/c])))","C",0
144,1,162,204,0.0415071,"\int \frac{1}{\sqrt[3]{a-b x^2} \left(3 a+b x^2\right)} \, dx","Integrate[1/((a - b*x^2)^(1/3)*(3*a + b*x^2)),x]","\frac{9 a x F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)}{\sqrt[3]{a-b x^2} \left(3 a+b x^2\right) \left(2 b x^2 \left(F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)-F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)\right)+9 a F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right)\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)}{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,"(9*a*x*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a])/((a - b*x^2)^(1/3)*(3*a + b*x^2)*(9*a*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a] + 2*b*x^2*(-AppellF1[3/2, 1/3, 2, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a] + AppellF1[3/2, 4/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a])))","C",0
145,1,156,204,0.1557046,"\int \frac{1}{\sqrt[3]{c-3 d x^2} \left(c+d x^2\right)} \, dx","Integrate[1/((c - 3*d*x^2)^(1/3)*(c + d*x^2)),x]","\frac{3 c x F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{3 d x^2}{c},-\frac{d x^2}{c}\right)}{\sqrt[3]{c-3 d x^2} \left(c+d x^2\right) \left(2 d x^2 \left(F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{3 d x^2}{c},-\frac{d x^2}{c}\right)-F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{3 d x^2}{c},-\frac{d x^2}{c}\right)\right)+3 c F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{3 d x^2}{c},-\frac{d x^2}{c}\right)\right)}","\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,"(3*c*x*AppellF1[1/2, 1/3, 1, 3/2, (3*d*x^2)/c, -((d*x^2)/c)])/((c - 3*d*x^2)^(1/3)*(c + d*x^2)*(3*c*AppellF1[1/2, 1/3, 1, 3/2, (3*d*x^2)/c, -((d*x^2)/c)] + 2*d*x^2*(-AppellF1[3/2, 1/3, 2, 5/2, (3*d*x^2)/c, -((d*x^2)/c)] + AppellF1[3/2, 4/3, 1, 5/2, (3*d*x^2)/c, -((d*x^2)/c)])))","C",0
146,1,118,113,0.0418571,"\int \frac{1}{\sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","Integrate[1/((1 - x^2)^(1/3)*(3 + x^2)),x]","-\frac{9 x F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right)}{\sqrt[3]{1-x^2} \left(x^2+3\right) \left(2 x^2 \left(F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};x^2,-\frac{x^2}{3}\right)-F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right)\right)-9 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right)\right)}","\frac{\tan ^{-1}\left(\frac{\sqrt{3} \left(1-\sqrt[3]{2} \sqrt[3]{1-x^2}\right)}{x}\right)}{2\ 2^{2/3} \sqrt{3}}+\frac{\tanh ^{-1}\left(\frac{x}{\sqrt[3]{2} \sqrt[3]{1-x^2}+1}\right)}{2\ 2^{2/3}}+\frac{\tan ^{-1}\left(\frac{\sqrt{3}}{x}\right)}{2\ 2^{2/3} \sqrt{3}}-\frac{\tanh ^{-1}(x)}{6\ 2^{2/3}}",1,"(-9*x*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1/3*x^2])/((1 - x^2)^(1/3)*(3 + x^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1/3*x^2] + 2*x^2*(AppellF1[3/2, 1/3, 2, 5/2, x^2, -1/3*x^2] - AppellF1[3/2, 4/3, 1, 5/2, x^2, -1/3*x^2])))","C",0
147,1,124,109,0.0392756,"\int \frac{1}{\left(3-x^2\right) \sqrt[3]{1+x^2}} \, dx","Integrate[1/((3 - x^2)*(1 + x^2)^(1/3)),x]","-\frac{9 x F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-x^2,\frac{x^2}{3}\right)}{\left(x^2-3\right) \sqrt[3]{x^2+1} \left(2 x^2 \left(F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};-x^2,\frac{x^2}{3}\right)-F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};-x^2,\frac{x^2}{3}\right)\right)+9 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-x^2,\frac{x^2}{3}\right)\right)}","\frac{\tan ^{-1}\left(\frac{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,"(-9*x*AppellF1[1/2, 1/3, 1, 3/2, -x^2, x^2/3])/((-3 + x^2)*(1 + x^2)^(1/3)*(9*AppellF1[1/2, 1/3, 1, 3/2, -x^2, x^2/3] + 2*x^2*(AppellF1[3/2, 1/3, 2, 5/2, -x^2, x^2/3] - AppellF1[3/2, 4/3, 1, 5/2, -x^2, x^2/3])))","C",0
148,1,143,96,0.1541923,"\int \frac{3-x}{\sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","Integrate[(3 - x)/((1 - x^2)^(1/3)*(3 + x^2)),x]","-\frac{1}{6} x^2 F_1\left(1;\frac{1}{3},1;2;x^2,-\frac{x^2}{3}\right)-\frac{27 x F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right)}{\sqrt[3]{1-x^2} \left(x^2+3\right) \left(2 x^2 \left(F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};x^2,-\frac{x^2}{3}\right)-F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right)\right)-9 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right)\right)}","-\frac{\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,"-1/6*(x^2*AppellF1[1, 1/3, 1, 2, x^2, -1/3*x^2]) - (27*x*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1/3*x^2])/((1 - x^2)^(1/3)*(3 + x^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1/3*x^2] + 2*x^2*(AppellF1[3/2, 1/3, 2, 5/2, x^2, -1/3*x^2] - AppellF1[3/2, 4/3, 1, 5/2, x^2, -1/3*x^2])))","C",0
149,1,143,95,0.1088184,"\int \frac{3+x}{\sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","Integrate[(3 + x)/((1 - x^2)^(1/3)*(3 + x^2)),x]","\frac{1}{6} x^2 F_1\left(1;\frac{1}{3},1;2;x^2,-\frac{x^2}{3}\right)-\frac{27 x F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right)}{\sqrt[3]{1-x^2} \left(x^2+3\right) \left(2 x^2 \left(F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};x^2,-\frac{x^2}{3}\right)-F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right)\right)-9 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right)\right)}","\frac{\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,"(x^2*AppellF1[1, 1/3, 1, 2, x^2, -1/3*x^2])/6 - (27*x*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1/3*x^2])/((1 - x^2)^(1/3)*(3 + x^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1/3*x^2] + 2*x^2*(AppellF1[3/2, 1/3, 2, 5/2, x^2, -1/3*x^2] - AppellF1[3/2, 4/3, 1, 5/2, x^2, -1/3*x^2])))","C",0
150,1,169,151,0.1740259,"\int \frac{1}{\sqrt[3]{a+b x^2} \left(\frac{9 a d}{b}+d x^2\right)} \, dx","Integrate[1/((a + b*x^2)^(1/3)*((9*a*d)/b + d*x^2)),x]","\frac{27 a b x F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{b x^2}{9 a}\right)}{d \sqrt[3]{a+b x^2} \left(9 a+b x^2\right) \left(27 a F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{b x^2}{9 a}\right)-2 b x^2 \left(F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{b x^2}{9 a}\right)+3 F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{b x^2}{9 a}\right)\right)\right)}","\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,"(27*a*b*x*AppellF1[1/2, 1/3, 1, 3/2, -((b*x^2)/a), -1/9*(b*x^2)/a])/(d*(a + b*x^2)^(1/3)*(9*a + b*x^2)*(27*a*AppellF1[1/2, 1/3, 1, 3/2, -((b*x^2)/a), -1/9*(b*x^2)/a] - 2*b*x^2*(AppellF1[3/2, 1/3, 2, 5/2, -((b*x^2)/a), -1/9*(b*x^2)/a] + 3*AppellF1[3/2, 4/3, 1, 5/2, -((b*x^2)/a), -1/9*(b*x^2)/a])))","C",0
151,1,167,153,0.1780855,"\int \frac{1}{\sqrt[3]{a-b x^2} \left(-\frac{9 a d}{b}+d x^2\right)} \, dx","Integrate[1/((a - b*x^2)^(1/3)*((-9*a*d)/b + d*x^2)),x]","-\frac{27 a b x F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},\frac{b x^2}{9 a}\right)}{d \sqrt[3]{a-b x^2} \left(9 a-b x^2\right) \left(2 b x^2 \left(F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{b x^2}{a},\frac{b x^2}{9 a}\right)+3 F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{b x^2}{a},\frac{b x^2}{9 a}\right)\right)+27 a F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},\frac{b x^2}{9 a}\right)\right)}","-\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,"(-27*a*b*x*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, (b*x^2)/(9*a)])/(d*(a - b*x^2)^(1/3)*(9*a - b*x^2)*(27*a*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, (b*x^2)/(9*a)] + 2*b*x^2*(AppellF1[3/2, 1/3, 2, 5/2, (b*x^2)/a, (b*x^2)/(9*a)] + 3*AppellF1[3/2, 4/3, 1, 5/2, (b*x^2)/a, (b*x^2)/(9*a)])))","C",0
152,1,168,151,0.1456579,"\int \frac{1}{\sqrt[3]{-a+b x^2} \left(-\frac{9 a d}{b}+d x^2\right)} \, dx","Integrate[1/((-a + b*x^2)^(1/3)*((-9*a*d)/b + d*x^2)),x]","-\frac{27 a b x F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},\frac{b x^2}{9 a}\right)}{d \left(9 a-b x^2\right) \sqrt[3]{b x^2-a} \left(2 b x^2 \left(F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{b x^2}{a},\frac{b x^2}{9 a}\right)+3 F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{b x^2}{a},\frac{b x^2}{9 a}\right)\right)+27 a F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},\frac{b x^2}{9 a}\right)\right)}","\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,"(-27*a*b*x*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, (b*x^2)/(9*a)])/(d*(9*a - b*x^2)*(-a + b*x^2)^(1/3)*(27*a*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, (b*x^2)/(9*a)] + 2*b*x^2*(AppellF1[3/2, 1/3, 2, 5/2, (b*x^2)/a, (b*x^2)/(9*a)] + 3*AppellF1[3/2, 4/3, 1, 5/2, (b*x^2)/a, (b*x^2)/(9*a)])))","C",0
153,1,172,153,0.1638459,"\int \frac{1}{\sqrt[3]{-a-b x^2} \left(\frac{9 a d}{b}+d x^2\right)} \, dx","Integrate[1/((-a - b*x^2)^(1/3)*((9*a*d)/b + d*x^2)),x]","\frac{27 a b x F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{b x^2}{9 a}\right)}{d \sqrt[3]{-a-b x^2} \left(9 a+b x^2\right) \left(27 a F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{b x^2}{9 a}\right)-2 b x^2 \left(F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{b x^2}{9 a}\right)+3 F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{b x^2}{9 a}\right)\right)\right)}","-\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,"(27*a*b*x*AppellF1[1/2, 1/3, 1, 3/2, -((b*x^2)/a), -1/9*(b*x^2)/a])/(d*(-a - b*x^2)^(1/3)*(9*a + b*x^2)*(27*a*AppellF1[1/2, 1/3, 1, 3/2, -((b*x^2)/a), -1/9*(b*x^2)/a] - 2*b*x^2*(AppellF1[3/2, 1/3, 2, 5/2, -((b*x^2)/a), -1/9*(b*x^2)/a] + 3*AppellF1[3/2, 4/3, 1, 5/2, -((b*x^2)/a), -1/9*(b*x^2)/a])))","C",0
154,1,148,151,0.1499628,"\int \frac{1}{\sqrt[3]{2+b x^2} \left(\frac{18 d}{b}+d x^2\right)} \, dx","Integrate[1/((2 + b*x^2)^(1/3)*((18*d)/b + d*x^2)),x]","-\frac{27 b x F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-\frac{b x^2}{2},-\frac{b x^2}{18}\right)}{d \sqrt[3]{b x^2+2} \left(b x^2+18\right) \left(b x^2 \left(F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};-\frac{b x^2}{2},-\frac{b x^2}{18}\right)+3 F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};-\frac{b x^2}{2},-\frac{b x^2}{18}\right)\right)-27 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-\frac{b x^2}{2},-\frac{b x^2}{18}\right)\right)}","\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,"(-27*b*x*AppellF1[1/2, 1/3, 1, 3/2, -1/2*(b*x^2), -1/18*(b*x^2)])/(d*(2 + b*x^2)^(1/3)*(18 + b*x^2)*(-27*AppellF1[1/2, 1/3, 1, 3/2, -1/2*(b*x^2), -1/18*(b*x^2)] + b*x^2*(AppellF1[3/2, 1/3, 2, 5/2, -1/2*(b*x^2), -1/18*(b*x^2)] + 3*AppellF1[3/2, 4/3, 1, 5/2, -1/2*(b*x^2), -1/18*(b*x^2)])))","C",0
155,1,148,147,0.1677694,"\int \frac{1}{\sqrt[3]{-2+b x^2} \left(-\frac{18 d}{b}+d x^2\right)} \, dx","Integrate[1/((-2 + b*x^2)^(1/3)*((-18*d)/b + d*x^2)),x]","\frac{27 b x F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{2},\frac{b x^2}{18}\right)}{d \left(b x^2-18\right) \sqrt[3]{b x^2-2} \left(b x^2 \left(F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{b x^2}{2},\frac{b x^2}{18}\right)+3 F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{b x^2}{2},\frac{b x^2}{18}\right)\right)+27 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{2},\frac{b x^2}{18}\right)\right)}","\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,"(27*b*x*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/2, (b*x^2)/18])/(d*(-18 + b*x^2)*(-2 + b*x^2)^(1/3)*(27*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/2, (b*x^2)/18] + b*x^2*(AppellF1[3/2, 1/3, 2, 5/2, (b*x^2)/2, (b*x^2)/18] + 3*AppellF1[3/2, 4/3, 1, 5/2, (b*x^2)/2, (b*x^2)/18])))","C",0
156,1,136,123,0.1196998,"\int \frac{1}{\sqrt[3]{2+3 x^2} \left(6 d+d x^2\right)} \, dx","Integrate[1/((2 + 3*x^2)^(1/3)*(6*d + d*x^2)),x]","-\frac{9 x F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-\frac{3 x^2}{2},-\frac{x^2}{6}\right)}{d \left(x^2+6\right) \sqrt[3]{3 x^2+2} \left(x^2 \left(F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};-\frac{3 x^2}{2},-\frac{x^2}{6}\right)+3 F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};-\frac{3 x^2}{2},-\frac{x^2}{6}\right)\right)-9 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-\frac{3 x^2}{2},-\frac{x^2}{6}\right)\right)}","\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,"(-9*x*AppellF1[1/2, 1/3, 1, 3/2, (-3*x^2)/2, -1/6*x^2])/(d*(6 + x^2)*(2 + 3*x^2)^(1/3)*(-9*AppellF1[1/2, 1/3, 1, 3/2, (-3*x^2)/2, -1/6*x^2] + x^2*(AppellF1[3/2, 1/3, 2, 5/2, (-3*x^2)/2, -1/6*x^2] + 3*AppellF1[3/2, 4/3, 1, 5/2, (-3*x^2)/2, -1/6*x^2])))","C",0
157,1,136,123,0.1251995,"\int \frac{1}{\sqrt[3]{2-3 x^2} \left(-6 d+d x^2\right)} \, dx","Integrate[1/((2 - 3*x^2)^(1/3)*(-6*d + d*x^2)),x]","\frac{9 x F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{3 x^2}{2},\frac{x^2}{6}\right)}{d \sqrt[3]{2-3 x^2} \left(x^2-6\right) \left(x^2 \left(F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{3 x^2}{2},\frac{x^2}{6}\right)+3 F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{3 x^2}{2},\frac{x^2}{6}\right)\right)+9 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{3 x^2}{2},\frac{x^2}{6}\right)\right)}","-\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,"(9*x*AppellF1[1/2, 1/3, 1, 3/2, (3*x^2)/2, x^2/6])/(d*(2 - 3*x^2)^(1/3)*(-6 + x^2)*(9*AppellF1[1/2, 1/3, 1, 3/2, (3*x^2)/2, x^2/6] + x^2*(AppellF1[3/2, 1/3, 2, 5/2, (3*x^2)/2, x^2/6] + 3*AppellF1[3/2, 4/3, 1, 5/2, (3*x^2)/2, x^2/6])))","C",0
158,1,136,119,0.1042062,"\int \frac{1}{\sqrt[3]{-2+3 x^2} \left(-6 d+d x^2\right)} \, dx","Integrate[1/((-2 + 3*x^2)^(1/3)*(-6*d + d*x^2)),x]","\frac{9 x F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{3 x^2}{2},\frac{x^2}{6}\right)}{d \left(x^2-6\right) \sqrt[3]{3 x^2-2} \left(x^2 \left(F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{3 x^2}{2},\frac{x^2}{6}\right)+3 F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{3 x^2}{2},\frac{x^2}{6}\right)\right)+9 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{3 x^2}{2},\frac{x^2}{6}\right)\right)}","\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,"(9*x*AppellF1[1/2, 1/3, 1, 3/2, (3*x^2)/2, x^2/6])/(d*(-6 + x^2)*(-2 + 3*x^2)^(1/3)*(9*AppellF1[1/2, 1/3, 1, 3/2, (3*x^2)/2, x^2/6] + x^2*(AppellF1[3/2, 1/3, 2, 5/2, (3*x^2)/2, x^2/6] + 3*AppellF1[3/2, 4/3, 1, 5/2, (3*x^2)/2, x^2/6])))","C",0
159,1,136,119,0.1092974,"\int \frac{1}{\sqrt[3]{-2-3 x^2} \left(6 d+d x^2\right)} \, dx","Integrate[1/((-2 - 3*x^2)^(1/3)*(6*d + d*x^2)),x]","-\frac{9 x F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-\frac{3 x^2}{2},-\frac{x^2}{6}\right)}{d \sqrt[3]{-3 x^2-2} \left(x^2+6\right) \left(x^2 \left(F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};-\frac{3 x^2}{2},-\frac{x^2}{6}\right)+3 F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};-\frac{3 x^2}{2},-\frac{x^2}{6}\right)\right)-9 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-\frac{3 x^2}{2},-\frac{x^2}{6}\right)\right)}","-\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,"(-9*x*AppellF1[1/2, 1/3, 1, 3/2, (-3*x^2)/2, -1/6*x^2])/(d*(-2 - 3*x^2)^(1/3)*(6 + x^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, (-3*x^2)/2, -1/6*x^2] + x^2*(AppellF1[3/2, 1/3, 2, 5/2, (-3*x^2)/2, -1/6*x^2] + 3*AppellF1[3/2, 4/3, 1, 5/2, (-3*x^2)/2, -1/6*x^2])))","C",0
160,1,124,70,0.0954213,"\int \frac{1}{\sqrt[3]{1+x^2} \left(9+x^2\right)} \, dx","Integrate[1/((1 + x^2)^(1/3)*(9 + x^2)),x]","-\frac{27 x F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-x^2,-\frac{x^2}{9}\right)}{\sqrt[3]{x^2+1} \left(x^2+9\right) \left(2 x^2 \left(F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};-x^2,-\frac{x^2}{9}\right)+3 F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};-x^2,-\frac{x^2}{9}\right)\right)-27 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-x^2,-\frac{x^2}{9}\right)\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,"(-27*x*AppellF1[1/2, 1/3, 1, 3/2, -x^2, -1/9*x^2])/((1 + x^2)^(1/3)*(9 + x^2)*(-27*AppellF1[1/2, 1/3, 1, 3/2, -x^2, -1/9*x^2] + 2*x^2*(AppellF1[3/2, 1/3, 2, 5/2, -x^2, -1/9*x^2] + 3*AppellF1[3/2, 4/3, 1, 5/2, -x^2, -1/9*x^2])))","C",0
161,1,137,104,0.1109806,"\int \frac{1}{\sqrt[3]{1+b x^2} \left(9+b x^2\right)} \, dx","Integrate[1/((1 + b*x^2)^(1/3)*(9 + b*x^2)),x]","-\frac{27 x F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-b x^2,-\frac{b x^2}{9}\right)}{\sqrt[3]{b x^2+1} \left(b x^2+9\right) \left(2 b x^2 \left(F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};-b x^2,-\frac{b x^2}{9}\right)+3 F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};-b x^2,-\frac{b x^2}{9}\right)\right)-27 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-b x^2,-\frac{b x^2}{9}\right)\right)}","\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,"(-27*x*AppellF1[1/2, 1/3, 1, 3/2, -(b*x^2), -1/9*(b*x^2)])/((1 + b*x^2)^(1/3)*(9 + b*x^2)*(-27*AppellF1[1/2, 1/3, 1, 3/2, -(b*x^2), -1/9*(b*x^2)] + 2*b*x^2*(AppellF1[3/2, 1/3, 2, 5/2, -(b*x^2), -1/9*(b*x^2)] + 3*AppellF1[3/2, 4/3, 1, 5/2, -(b*x^2), -1/9*(b*x^2)])))","C",0
162,1,125,74,0.0464483,"\int \frac{1}{\sqrt[3]{1-x^2} \left(9-x^2\right)} \, dx","Integrate[1/((1 - x^2)^(1/3)*(9 - x^2)),x]","\frac{\sqrt[3]{\frac{x-1}{x-3}} \sqrt[3]{\frac{x+1}{x-3}} F_1\left(\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};-\frac{4}{x-3},-\frac{2}{x-3}\right)-\sqrt[3]{\frac{x-1}{x+3}} \sqrt[3]{\frac{x+1}{x+3}} F_1\left(\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{2}{x+3},\frac{4}{x+3}\right)}{4 \sqrt[3]{1-x^2}}","\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,"(((-1 + x)/(-3 + x))^(1/3)*((1 + x)/(-3 + x))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, -4/(-3 + x), -2/(-3 + x)] - ((-1 + x)/(3 + x))^(1/3)*((1 + x)/(3 + x))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, 2/(3 + x), 4/(3 + x)])/(4*(1 - x^2)^(1/3))","C",0
163,1,57,79,0.0270065,"\int \frac{\sqrt{-1+c^2 x^2}}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[Sqrt[-1 + c^2*x^2]/(d - c^2*d*x^2)^(5/2),x]","\frac{\left(c^2 x^2-1\right) \tanh ^{-1}(c x)-c x}{2 c d^2 \sqrt{c^2 x^2-1} \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,"(-(c*x) + (-1 + c^2*x^2)*ArcTanh[c*x])/(2*c*d^2*Sqrt[-1 + c^2*x^2]*Sqrt[d - c^2*d*x^2])","A",1
164,1,54,74,0.0194448,"\int \frac{1}{\left(-1+c^2 x^2\right)^{3/2} \sqrt{d-c^2 d x^2}} \, dx","Integrate[1/((-1 + c^2*x^2)^(3/2)*Sqrt[d - c^2*d*x^2]),x]","\frac{\left(c^2 x^2-1\right) \tanh ^{-1}(c x)-c x}{2 c \sqrt{c^2 x^2-1} \sqrt{d-c^2 d x^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,"(-(c*x) + (-1 + c^2*x^2)*ArcTanh[c*x])/(2*c*Sqrt[-1 + c^2*x^2]*Sqrt[d - c^2*d*x^2])","A",1
165,1,57,76,0.0294709,"\int \frac{1}{\sqrt{-1+c^2 x^2} \left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[1/(Sqrt[-1 + c^2*x^2]*(d - c^2*d*x^2)^(3/2)),x]","\frac{\left(1-c^2 x^2\right) \tanh ^{-1}(c x)+c x}{2 c d \sqrt{c^2 x^2-1} \sqrt{d-c^2 d x^2}}","-\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,"(c*x + (1 - c^2*x^2)*ArcTanh[c*x])/(2*c*d*Sqrt[-1 + c^2*x^2]*Sqrt[d - c^2*d*x^2])","A",1
166,1,243,328,0.4639216,"\int \left(a+b x^2\right)^{3/2} \sqrt{c+d x^2} \, dx","Integrate[(a + b*x^2)^(3/2)*Sqrt[c + d*x^2],x]","\frac{-2 i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(3 a^2 d^2-4 a b c d+b^2 c^2\right) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(x \sqrt{\frac{b}{a}}\right),\frac{a d}{b c}\right)-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(3 a^2 d^2+7 a b c d-2 b^2 c^2\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+d x \sqrt{\frac{b}{a}} \left(a+b x^2\right) \left(c+d x^2\right) \left(6 a d+b \left(c+3 d x^2\right)\right)}{15 d^2 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}}","-\frac{c^{3/2} \sqrt{a+b x^2} (b c-9 a d) \operatorname{EllipticF}\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{\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{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,"(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(6*a*d + b*(c + 3*d*x^2)) - I*c*(-2*b^2*c^2 + 7*a*b*c*d + 3*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - (2*I)*c*(b^2*c^2 - 4*a*b*c*d + 3*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(15*Sqrt[b/a]*d^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","C",1
167,1,198,249,0.2505642,"\int \sqrt{a+b x^2} \sqrt{c+d x^2} \, dx","Integrate[Sqrt[a + b*x^2]*Sqrt[c + d*x^2],x]","\frac{-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-b c) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(x \sqrt{\frac{b}{a}}\right),\frac{a d}{b c}\right)+d x \sqrt{\frac{b}{a}} \left(a+b x^2\right) \left(c+d x^2\right)-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d+b c) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{3 d \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}}","\frac{2 c^{3/2} \sqrt{a+b x^2} \operatorname{EllipticF}\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,"(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2) - I*c*(b*c + a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*c*(-(b*c) + a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(3*Sqrt[b/a]*d*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","C",1
168,1,86,204,0.0525343,"\int \frac{\sqrt{c+d x^2}}{\sqrt{a+b x^2}} \, dx","Integrate[Sqrt[c + d*x^2]/Sqrt[a + b*x^2],x]","\frac{\sqrt{\frac{a+b x^2}{a}} \sqrt{c+d x^2} E\left(\sin ^{-1}\left(\sqrt{-\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{\sqrt{-\frac{b}{a}} \sqrt{a+b x^2} \sqrt{\frac{c+d x^2}{c}}}","\frac{c^{3/2} \sqrt{a+b x^2} \operatorname{EllipticF}\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,"(Sqrt[(a + b*x^2)/a]*Sqrt[c + d*x^2]*EllipticE[ArcSin[Sqrt[-(b/a)]*x], (a*d)/(b*c)])/(Sqrt[-(b/a)]*Sqrt[a + b*x^2]*Sqrt[(c + d*x^2)/c])","A",1
169,1,133,84,0.294506,"\int \frac{\sqrt{c+d x^2}}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[Sqrt[c + d*x^2]/(a + b*x^2)^(3/2),x]","\frac{x \left(c+d x^2\right)+\frac{i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)-\operatorname{EllipticF}\left(i \sinh ^{-1}\left(x \sqrt{\frac{b}{a}}\right),\frac{a d}{b c}\right)\right)}{\sqrt{\frac{b}{a}}}}{a \sqrt{a+b x^2} \sqrt{c+d x^2}}","\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,"(x*(c + d*x^2) + (I*c*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*(EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/Sqrt[b/a])/(a*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","C",1
170,1,243,237,0.4664984,"\int \frac{\sqrt{c+d x^2}}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[Sqrt[c + d*x^2]/(a + b*x^2)^(5/2),x]","\frac{-2 i c \left(a+b x^2\right) \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-b c) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(x \sqrt{\frac{b}{a}}\right),\frac{a d}{b c}\right)+x \sqrt{\frac{b}{a}} \left(c+d x^2\right) \left(2 a^2 d+a b \left(d x^2-3 c\right)-2 b^2 c x^2\right)+i c \left(a+b x^2\right) \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-2 b c) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{3 a^2 \sqrt{\frac{b}{a}} \left(a+b x^2\right)^{3/2} \sqrt{c+d x^2} (a d-b c)}","-\frac{c^{3/2} \sqrt{d} \sqrt{a+b x^2} \operatorname{EllipticF}\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,"(Sqrt[b/a]*x*(c + d*x^2)*(2*a^2*d - 2*b^2*c*x^2 + a*b*(-3*c + d*x^2)) + I*c*(-2*b*c + a*d)*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - (2*I)*c*(-(b*c) + a*d)*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(3*a^2*Sqrt[b/a]*(-(b*c) + a*d)*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])","C",1
171,1,285,309,0.5462341,"\int \frac{\sqrt{c+d x^2}}{\left(a+b x^2\right)^{7/2}} \, dx","Integrate[Sqrt[c + d*x^2]/(a + b*x^2)^(7/2),x]","\frac{x \sqrt{\frac{b}{a}} \left(c+d x^2\right) \left(\left(a+b x^2\right)^2 \left(3 a^2 d^2-13 a b c d+8 b^2 c^2\right)+3 a^2 (b c-a d)^2+a \left(a+b x^2\right) (a d-b c) (3 a d-4 b c)\right)+i c \sqrt{\frac{b x^2}{a}+1} \left(a+b x^2\right)^2 \sqrt{\frac{d x^2}{c}+1} \left(\left(-9 a^2 d^2+17 a b c d-8 b^2 c^2\right) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(x \sqrt{\frac{b}{a}}\right),\frac{a d}{b c}\right)+\left(3 a^2 d^2-13 a b c d+8 b^2 c^2\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)\right)}{15 a^3 \sqrt{\frac{b}{a}} \left(a+b x^2\right)^{5/2} \sqrt{c+d x^2} (b c-a d)^2}","-\frac{2 c^{3/2} \sqrt{d} \sqrt{a+b x^2} (2 b c-3 a d) \operatorname{EllipticF}\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right),1-\frac{b c}{a d}\right)}{15 a^3 \sqrt{c+d x^2} (b c-a d)^2 \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{x \sqrt{c+d x^2} (4 b c-3 a d)}{15 a^2 \left(a+b x^2\right)^{3/2} (b c-a d)}+\frac{\sqrt{c+d x^2} \left(3 a^2 d^2-13 a b c d+8 b^2 c^2\right) E\left(\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)|1-\frac{a d}{b c}\right)}{15 a^{5/2} \sqrt{b} \sqrt{a+b x^2} (b c-a d)^2 \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}}}+\frac{x \sqrt{c+d x^2}}{5 a \left(a+b x^2\right)^{5/2}}",1,"(Sqrt[b/a]*x*(c + d*x^2)*(3*a^2*(b*c - a*d)^2 + a*(-(b*c) + a*d)*(-4*b*c + 3*a*d)*(a + b*x^2) + (8*b^2*c^2 - 13*a*b*c*d + 3*a^2*d^2)*(a + b*x^2)^2) + I*c*(a + b*x^2)^2*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*((8*b^2*c^2 - 13*a*b*c*d + 3*a^2*d^2)*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (-8*b^2*c^2 + 17*a*b*c*d - 9*a^2*d^2)*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(15*a^3*Sqrt[b/a]*(b*c - a*d)^2*(a + b*x^2)^(5/2)*Sqrt[c + d*x^2])","C",1
172,1,302,410,0.5990086,"\int \left(a+b x^2\right)^{3/2} \left(c+d x^2\right)^{3/2} \, dx","Integrate[(a + b*x^2)^(3/2)*(c + d*x^2)^(3/2),x]","\frac{-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(a^3 d^3+8 a^2 b c d^2-11 a b^2 c^2 d+2 b^3 c^3\right) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(x \sqrt{\frac{b}{a}}\right),\frac{a d}{b c}\right)+d x \sqrt{\frac{b}{a}} \left(a+b x^2\right) \left(c+d x^2\right) \left(a^2 d^2+a b d \left(17 c+8 d x^2\right)+b^2 \left(c^2+8 c d x^2+5 d^2 x^4\right)\right)+2 i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(a^3 d^3-5 a^2 b c d^2-5 a b^2 c^2 d+b^3 c^3\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{35 b d^2 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \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) \operatorname{EllipticF}\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 x \sqrt{a+b x^2} (a d+b c) \left(a^2 d^2-6 a b c d+b^2 c^2\right)}{35 b^2 d \sqrt{c+d x^2}}+\frac{2 \sqrt{c} \sqrt{a+b x^2} (a d+b c) \left(a^2 d^2-6 a b c d+b^2 c^2\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{35 b^2 d^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{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,"(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(a^2*d^2 + a*b*d*(17*c + 8*d*x^2) + b^2*(c^2 + 8*c*d*x^2 + 5*d^2*x^4)) + (2*I)*c*(b^3*c^3 - 5*a*b^2*c^2*d - 5*a^2*b*c*d^2 + a^3*d^3)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*c*(2*b^3*c^3 - 11*a*b^2*c^2*d + 8*a^2*b*c*d^2 + a^3*d^3)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(35*b*Sqrt[b/a]*d^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","C",1
173,1,246,336,0.4231611,"\int \sqrt{a+b x^2} \left(c+d x^2\right)^{3/2} \, dx","Integrate[Sqrt[a + b*x^2]*(c + d*x^2)^(3/2),x]","\frac{-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(a^2 d^2+2 a b c d-3 b^2 c^2\right) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(x \sqrt{\frac{b}{a}}\right),\frac{a d}{b c}\right)+i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(2 a^2 d^2-7 a b c d-3 b^2 c^2\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+d x \sqrt{\frac{b}{a}} \left(a+b x^2\right) \left(c+d x^2\right) \left(a d+6 b c+3 b d x^2\right)}{15 b d \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}}","\frac{c^{3/2} \sqrt{a+b x^2} (9 b c-a d) \operatorname{EllipticF}\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{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{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,"(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(6*b*c + a*d + 3*b*d*x^2) + I*c*(-3*b^2*c^2 - 7*a*b*c*d + 2*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*c*(-3*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(15*b*Sqrt[b/a]*d*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","C",1
174,1,199,273,0.2800598,"\int \frac{\left(c+d x^2\right)^{3/2}}{\sqrt{a+b x^2}} \, dx","Integrate[(c + d*x^2)^(3/2)/Sqrt[a + b*x^2],x]","\frac{-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-b c) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(x \sqrt{\frac{b}{a}}\right),\frac{a d}{b c}\right)+d x \sqrt{\frac{b}{a}} \left(a+b x^2\right) \left(c+d x^2\right)+2 i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-2 b c) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{3 b \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}}","\frac{c^{3/2} \sqrt{a+b x^2} (3 b c-a d) \operatorname{EllipticF}\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{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{d x \sqrt{a+b x^2} \sqrt{c+d x^2}}{3 b}",1,"(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2) + (2*I)*c*(-2*b*c + a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*c*(-(b*c) + a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(3*b*Sqrt[b/a]*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","C",1
175,1,191,267,0.2994077,"\int \frac{\left(c+d x^2\right)^{3/2}}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[(c + d*x^2)^(3/2)/(a + b*x^2)^(3/2),x]","\frac{(b c-a d) \left(x \sqrt{\frac{b}{a}} \left(c+d x^2\right)-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \operatorname{EllipticF}\left(i \sinh ^{-1}\left(x \sqrt{\frac{b}{a}}\right),\frac{a d}{b c}\right)\right)-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (2 a d-b c) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{a^2 \left(\frac{b}{a}\right)^{3/2} \sqrt{a+b x^2} \sqrt{c+d x^2}}","\frac{c^{3/2} \sqrt{d} \sqrt{a+b x^2} \operatorname{EllipticF}\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{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{x \sqrt{c+d x^2} (b c-a d)}{a b \sqrt{a+b x^2}}",1,"((-I)*c*(-(b*c) + 2*a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (b*c - a*d)*(Sqrt[b/a]*x*(c + d*x^2) - I*c*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(a^2*(b/a)^(3/2)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","C",1
176,1,232,229,0.4845354,"\int \frac{\left(c+d x^2\right)^{3/2}}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[(c + d*x^2)^(3/2)/(a + b*x^2)^(5/2),x]","\frac{-i c \left(a+b x^2\right) \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d+2 b c) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(x \sqrt{\frac{b}{a}}\right),\frac{a d}{b c}\right)+x \sqrt{\frac{b}{a}} \left(c+d x^2\right) \left(a^2 d+a b \left(3 c+2 d x^2\right)+2 b^2 c x^2\right)+2 i c \left(a+b x^2\right) \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d+b c) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{3 a^3 \left(\frac{b}{a}\right)^{3/2} \left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}}","-\frac{c^{3/2} \sqrt{d} \sqrt{a+b x^2} \operatorname{EllipticF}\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{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{x \sqrt{c+d x^2} (b c-a d)}{3 a b \left(a+b x^2\right)^{3/2}}",1,"(Sqrt[b/a]*x*(c + d*x^2)*(a^2*d + 2*b^2*c*x^2 + a*b*(3*c + 2*d*x^2)) + (2*I)*c*(b*c + a*d)*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*c*(2*b*c + a*d)*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(3*a^3*(b/a)^(3/2)*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])","C",1
177,1,285,315,0.6027388,"\int \frac{\left(c+d x^2\right)^{3/2}}{\left(a+b x^2\right)^{7/2}} \, dx","Integrate[(c + d*x^2)^(3/2)/(a + b*x^2)^(7/2),x]","\frac{x \sqrt{\frac{b}{a}} \left(c+d x^2\right) \left(\left(a+b x^2\right)^2 \left(-2 a^2 d^2-3 a b c d+8 b^2 c^2\right)+3 a^2 (b c-a d)^2+2 a \left(a+b x^2\right) (a d+2 b c) (b c-a d)\right)-i c \left(a+b x^2\right)^2 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(\left(-a^2 d^2-7 a b c d+8 b^2 c^2\right) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(x \sqrt{\frac{b}{a}}\right),\frac{a d}{b c}\right)+\left(2 a^2 d^2+3 a b c d-8 b^2 c^2\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)\right)}{15 a^4 \left(\frac{b}{a}\right)^{3/2} \left(a+b x^2\right)^{5/2} \sqrt{c+d x^2} (b c-a d)}","-\frac{c^{3/2} \sqrt{d} \sqrt{a+b x^2} (4 b c-a d) \operatorname{EllipticF}\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right),1-\frac{b c}{a d}\right)}{15 a^3 b \sqrt{c+d x^2} (b c-a d) \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{2 x \sqrt{c+d x^2} (a d+2 b c)}{15 a^2 b \left(a+b x^2\right)^{3/2}}+\frac{\sqrt{c+d x^2} \left(-2 a^2 d^2-3 a b c d+8 b^2 c^2\right) E\left(\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)|1-\frac{a d}{b c}\right)}{15 a^{5/2} b^{3/2} \sqrt{a+b x^2} (b c-a d) \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}}}+\frac{x \sqrt{c+d x^2} (b c-a d)}{5 a b \left(a+b x^2\right)^{5/2}}",1,"(Sqrt[b/a]*x*(c + d*x^2)*(3*a^2*(b*c - a*d)^2 + 2*a*(b*c - a*d)*(2*b*c + a*d)*(a + b*x^2) + (8*b^2*c^2 - 3*a*b*c*d - 2*a^2*d^2)*(a + b*x^2)^2) - I*c*(a + b*x^2)^2*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*((-8*b^2*c^2 + 3*a*b*c*d + 2*a^2*d^2)*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (8*b^2*c^2 - 7*a*b*c*d - a^2*d^2)*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(15*a^4*(b/a)^(3/2)*(b*c - a*d)*(a + b*x^2)^(5/2)*Sqrt[c + d*x^2])","C",1
178,1,127,235,0.1132454,"\int \sqrt{2+b x^2} \sqrt{3+d x^2} \, dx","Integrate[Sqrt[2 + b*x^2]*Sqrt[3 + d*x^2],x]","\frac{i \sqrt{3} (3 b-2 d) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{2}}\right),\frac{2 d}{3 b}\right)+\sqrt{b} d x \sqrt{b x^2+2} \sqrt{d x^2+3}-i \sqrt{3} (3 b+2 d) E\left(i \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{2}}\right)|\frac{2 d}{3 b}\right)}{3 \sqrt{b} d}","\frac{2 \sqrt{2} \sqrt{b x^2+2} \operatorname{EllipticF}\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{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{\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,"(Sqrt[b]*d*x*Sqrt[2 + b*x^2]*Sqrt[3 + d*x^2] - I*Sqrt[3]*(3*b + 2*d)*EllipticE[I*ArcSinh[(Sqrt[b]*x)/Sqrt[2]], (2*d)/(3*b)] + I*Sqrt[3]*(3*b - 2*d)*EllipticF[I*ArcSinh[(Sqrt[b]*x)/Sqrt[2]], (2*d)/(3*b)])/(3*Sqrt[b]*d)","C",1
179,1,22,38,0.0119772,"\int \sqrt{3-6 x^2} \sqrt{2+4 x^2} \, dx","Integrate[Sqrt[3 - 6*x^2]*Sqrt[2 + 4*x^2],x]","\sqrt{6} x \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};4 x^4\right)","\frac{2 \operatorname{EllipticF}\left(\sin ^{-1}\left(\sqrt{2} x\right),-1\right)}{\sqrt{3}}+\sqrt{\frac{2}{3}} \sqrt{1-4 x^4} x",1,"Sqrt[6]*x*Hypergeometric2F1[-1/2, 1/4, 5/4, 4*x^4]","C",1
180,1,15,20,0.0022829,"\int \sqrt{2+4 x^2} \sqrt{3+6 x^2} \, dx","Integrate[Sqrt[2 + 4*x^2]*Sqrt[3 + 6*x^2],x]","\sqrt{6} \left(\frac{2 x^3}{3}+x\right)","2 \sqrt{\frac{2}{3}} x^3+\sqrt{6} x",1,"Sqrt[6]*(x + (2*x^3)/3)","A",1
181,1,37,182,0.008772,"\int \frac{\sqrt{2+b x^2}}{\sqrt{3+d x^2}} \, dx","Integrate[Sqrt[2 + b*x^2]/Sqrt[3 + d*x^2],x]","\frac{\sqrt{2} E\left(\sin ^{-1}\left(\frac{\sqrt{-d} x}{\sqrt{3}}\right)|\frac{3 b}{2 d}\right)}{\sqrt{-d}}","\frac{\sqrt{2} \sqrt{b x^2+2} \operatorname{EllipticF}\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} 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,"(Sqrt[2]*EllipticE[ArcSin[(Sqrt[-d]*x)/Sqrt[3]], (3*b)/(2*d)])/Sqrt[-d]","A",1
182,1,60,91,0.0383711,"\int \frac{\sqrt{4-x^2}}{\sqrt{c+d x^2}} \, dx","Integrate[Sqrt[4 - x^2]/Sqrt[c + d*x^2],x]","\frac{2 \sqrt{\frac{c+d x^2}{c}} E\left(\sin ^{-1}\left(\sqrt{-\frac{d}{c}} x\right)|-\frac{c}{4 d}\right)}{\sqrt{-\frac{d}{c}} \sqrt{c+d x^2}}","\frac{(c+4 d) \sqrt{\frac{d x^2}{c}+1} \operatorname{EllipticF}\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,"(2*Sqrt[(c + d*x^2)/c]*EllipticE[ArcSin[Sqrt[-(d/c)]*x], -1/4*c/d])/(Sqrt[-(d/c)]*Sqrt[c + d*x^2])","A",1
183,1,60,150,0.0332339,"\int \frac{\sqrt{4+x^2}}{\sqrt{c+d x^2}} \, dx","Integrate[Sqrt[4 + x^2]/Sqrt[c + d*x^2],x]","\frac{2 \sqrt{\frac{c+d x^2}{c}} E\left(\sin ^{-1}\left(\sqrt{-\frac{d}{c}} x\right)|\frac{c}{4 d}\right)}{\sqrt{-\frac{d}{c}} \sqrt{c+d x^2}}","\frac{4 \sqrt{c+d x^2} \operatorname{EllipticF}\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{x \sqrt{c+d x^2}}{d \sqrt{x^2+4}}-\frac{\sqrt{c+d x^2} E\left(\tan ^{-1}\left(\frac{x}{2}\right)|1-\frac{4 d}{c}\right)}{d \sqrt{x^2+4} \sqrt{\frac{c+d x^2}{c \left(x^2+4\right)}}}",1,"(2*Sqrt[(c + d*x^2)/c]*EllipticE[ArcSin[Sqrt[-(d/c)]*x], c/(4*d)])/(Sqrt[-(d/c)]*Sqrt[c + d*x^2])","A",1
184,1,20,20,0.0067814,"\int \frac{\sqrt{1-x^2}}{\sqrt{2-3 x^2}} \, dx","Integrate[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
185,1,21,21,0.0045867,"\int \frac{\sqrt{4-x^2}}{\sqrt{2-3 x^2}} \, dx","Integrate[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
186,1,20,20,0.0051438,"\int \frac{\sqrt{1-4 x^2}}{\sqrt{2-3 x^2}} \, dx","Integrate[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
187,1,4,4,0.0034411,"\int \frac{\sqrt{1+x^2}}{\sqrt{1-x^2}} \, dx","Integrate[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
188,1,20,20,0.0037417,"\int \frac{\sqrt{1+x^2}}{\sqrt{2-3 x^2}} \, dx","Integrate[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
189,1,21,21,0.0045036,"\int \frac{\sqrt{4+x^2}}{\sqrt{2-3 x^2}} \, dx","Integrate[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
190,1,20,20,0.0049179,"\int \frac{\sqrt{1+4 x^2}}{\sqrt{2-3 x^2}} \, dx","Integrate[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
191,1,12,13,0.0032852,"\int \frac{\sqrt{1-x^2}}{\sqrt{1+x^2}} \, dx","Integrate[Sqrt[1 - x^2]/Sqrt[1 + x^2],x]","-i E\left(\left.i \sinh ^{-1}(x)\right|-1\right)","2 \operatorname{EllipticF}\left(\sin ^{-1}(x),-1\right)-E\left(\left.\sin ^{-1}(x)\right|-1\right)",1,"(-I)*EllipticE[I*ArcSinh[x], -1]","C",1
192,1,27,31,0.00558,"\int \frac{\sqrt{1-x^2}}{\sqrt{2+3 x^2}} \, dx","Integrate[Sqrt[1 - x^2]/Sqrt[2 + 3*x^2],x]","-\frac{i E\left(i \sinh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|-\frac{2}{3}\right)}{\sqrt{3}}","\frac{5 \operatorname{EllipticF}\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,"((-I)*EllipticE[I*ArcSinh[Sqrt[3/2]*x], -2/3])/Sqrt[3]","C",1
193,1,27,35,0.0046412,"\int \frac{\sqrt{4-x^2}}{\sqrt{2+3 x^2}} \, dx","Integrate[Sqrt[4 - x^2]/Sqrt[2 + 3*x^2],x]","-\frac{2 i E\left(i \sinh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|-\frac{1}{6}\right)}{\sqrt{3}}","\frac{7}{3} \sqrt{2} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{x}{2}\right),-6\right)-\frac{1}{3} \sqrt{2} E\left(\left.\sin ^{-1}\left(\frac{x}{2}\right)\right|-6\right)",1,"((-2*I)*EllipticE[I*ArcSinh[Sqrt[3/2]*x], -1/6])/Sqrt[3]","C",1
194,1,27,35,0.0046863,"\int \frac{\sqrt{1-4 x^2}}{\sqrt{2+3 x^2}} \, dx","Integrate[Sqrt[1 - 4*x^2]/Sqrt[2 + 3*x^2],x]","-\frac{i E\left(i \sinh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|-\frac{8}{3}\right)}{\sqrt{3}}","\frac{11 \operatorname{EllipticF}\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,"((-I)*EllipticE[I*ArcSinh[Sqrt[3/2]*x], -8/3])/Sqrt[3]","C",1
195,1,27,131,0.0040552,"\int \frac{\sqrt{1+x^2}}{\sqrt{2+3 x^2}} \, dx","Integrate[Sqrt[1 + x^2]/Sqrt[2 + 3*x^2],x]","-\frac{i E\left(i \sinh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|\frac{2}{3}\right)}{\sqrt{3}}","\frac{\sqrt{3 x^2+2} \operatorname{EllipticF}\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} x}{3 \sqrt{x^2+1}}-\frac{\sqrt{2} \sqrt{3 x^2+2} E\left(\tan ^{-1}(x)|-\frac{1}{2}\right)}{3 \sqrt{x^2+1} \sqrt{\frac{3 x^2+2}{x^2+1}}}",1,"((-I)*EllipticE[I*ArcSinh[Sqrt[3/2]*x], 2/3])/Sqrt[3]","C",1
196,1,27,136,0.0043161,"\int \frac{\sqrt{4+x^2}}{\sqrt{2+3 x^2}} \, dx","Integrate[Sqrt[4 + x^2]/Sqrt[2 + 3*x^2],x]","-\frac{2 i E\left(i \sinh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|\frac{1}{6}\right)}{\sqrt{3}}","\frac{2 \sqrt{2} \sqrt{3 x^2+2} \operatorname{EllipticF}\left(\tan ^{-1}\left(\frac{x}{2}\right),-5\right)}{\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{\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,"((-2*I)*EllipticE[I*ArcSinh[Sqrt[3/2]*x], 1/6])/Sqrt[3]","C",1
197,1,27,148,0.0046304,"\int \frac{\sqrt{1+4 x^2}}{\sqrt{2+3 x^2}} \, dx","Integrate[Sqrt[1 + 4*x^2]/Sqrt[2 + 3*x^2],x]","-\frac{i E\left(i \sinh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|\frac{8}{3}\right)}{\sqrt{3}}","\frac{\sqrt{3 x^2+2} \operatorname{EllipticF}\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{4 \sqrt{3 x^2+2} x}{3 \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,"((-I)*EllipticE[I*ArcSinh[Sqrt[3/2]*x], 8/3])/Sqrt[3]","C",1
198,1,35,40,0.0252846,"\int \frac{\sqrt{1-x^2}}{\sqrt{-1+2 x^2}} \, dx","Integrate[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{4 x^2-2}}","\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 + 4*x^2]","A",1
199,1,321,423,1.5657632,"\int \frac{\left(a+b x^2\right)^{7/2}}{\sqrt{c+d x^2}} \, dx","Integrate[(a + b*x^2)^(7/2)/Sqrt[c + d*x^2],x]","\frac{-i \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(105 a^4 d^4-298 a^3 b c d^3+353 a^2 b^2 c^2 d^2-208 a b^3 c^3 d+48 b^4 c^4\right) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(x \sqrt{\frac{b}{a}}\right),\frac{a d}{b c}\right)+b d x \sqrt{\frac{b}{a}} \left(a+b x^2\right) \left(c+d x^2\right) \left(122 a^2 d^2+a b d \left(66 d x^2-89 c\right)+3 b^2 \left(8 c^2-6 c d x^2+5 d^2 x^4\right)\right)-8 i b c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(22 a^3 d^3-33 a^2 b c d^2+23 a b^2 c^2 d-6 b^3 c^3\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{105 d^4 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}}","-\frac{\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) \operatorname{EllipticF}\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right),1-\frac{b c}{a d}\right)}{105 d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{8 \sqrt{c} \sqrt{a+b x^2} (b c-2 a d) \left(11 a^2 d^2-11 a b c d+6 b^2 c^2\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{105 d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{b x \sqrt{a+b x^2} \sqrt{c+d x^2} \left(71 a^2 d^2-71 a b c d+24 b^2 c^2\right)}{105 d^3}-\frac{8 x \sqrt{a+b x^2} (b c-2 a d) \left(11 a^2 d^2-11 a b c d+6 b^2 c^2\right)}{105 d^3 \sqrt{c+d x^2}}-\frac{6 b x \left(a+b x^2\right)^{3/2} \sqrt{c+d x^2} (b c-2 a d)}{35 d^2}+\frac{b x \left(a+b x^2\right)^{5/2} \sqrt{c+d x^2}}{7 d}",1,"(b*Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(122*a^2*d^2 + a*b*d*(-89*c + 66*d*x^2) + 3*b^2*(8*c^2 - 6*c*d*x^2 + 5*d^2*x^4)) - (8*I)*b*c*(-6*b^3*c^3 + 23*a*b^2*c^2*d - 33*a^2*b*c*d^2 + 22*a^3*d^3)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*(48*b^4*c^4 - 208*a*b^3*c^3*d + 353*a^2*b^2*c^2*d^2 - 298*a^3*b*c*d^3 + 105*a^4*d^4)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(105*Sqrt[b/a]*d^4*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","C",1
200,1,260,344,0.4946433,"\int \frac{\left(a+b x^2\right)^{5/2}}{\sqrt{c+d x^2}} \, dx","Integrate[(a + b*x^2)^(5/2)/Sqrt[c + d*x^2],x]","\frac{-i \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(15 a^3 d^3-34 a^2 b c d^2+27 a b^2 c^2 d-8 b^3 c^3\right) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(x \sqrt{\frac{b}{a}}\right),\frac{a d}{b c}\right)-i b c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(23 a^2 d^2-23 a b c d+8 b^2 c^2\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+b d x \sqrt{\frac{b}{a}} \left(a+b x^2\right) \left(c+d x^2\right) \left(11 a d-4 b c+3 b d x^2\right)}{15 d^3 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}}","\frac{\sqrt{c} \sqrt{a+b x^2} \left(15 a^2 d^2-11 a b c d+4 b^2 c^2\right) \operatorname{EllipticF}\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{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(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,"(b*Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(-4*b*c + 11*a*d + 3*b*d*x^2) - I*b*c*(8*b^2*c^2 - 23*a*b*c*d + 23*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*(-8*b^3*c^3 + 27*a*b^2*c^2*d - 34*a^2*b*c*d^2 + 15*a^3*d^3)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(15*Sqrt[b/a]*d^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","C",1
201,1,216,260,0.3526134,"\int \frac{\left(a+b x^2\right)^{3/2}}{\sqrt{c+d x^2}} \, dx","Integrate[(a + b*x^2)^(3/2)/Sqrt[c + d*x^2],x]","\frac{-i \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(3 a^2 d^2-5 a b c d+2 b^2 c^2\right) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(x \sqrt{\frac{b}{a}}\right),\frac{a d}{b c}\right)+b d x \sqrt{\frac{b}{a}} \left(a+b x^2\right) \left(c+d x^2\right)-2 i b c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (2 a d-b c) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{3 d^2 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}}","-\frac{\sqrt{c} \sqrt{a+b x^2} (b c-3 a d) \operatorname{EllipticF}\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,"(b*Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2) - (2*I)*b*c*(-(b*c) + 2*a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*(2*b^2*c^2 - 5*a*b*c*d + 3*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(3*Sqrt[b/a]*d^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","C",1
202,1,86,194,0.0533121,"\int \frac{\sqrt{a+b x^2}}{\sqrt{c+d x^2}} \, dx","Integrate[Sqrt[a + b*x^2]/Sqrt[c + d*x^2],x]","\frac{\sqrt{a+b x^2} \sqrt{\frac{c+d x^2}{c}} E\left(\sin ^{-1}\left(\sqrt{-\frac{d}{c}} x\right)|\frac{b c}{a d}\right)}{\sqrt{-\frac{d}{c}} \sqrt{\frac{a+b x^2}{a}} \sqrt{c+d x^2}}","\frac{\sqrt{c} \sqrt{a+b x^2} \operatorname{EllipticF}\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} 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,"(Sqrt[a + b*x^2]*Sqrt[(c + d*x^2)/c]*EllipticE[ArcSin[Sqrt[-(d/c)]*x], (b*c)/(a*d)])/(Sqrt[-(d/c)]*Sqrt[(a + b*x^2)/a]*Sqrt[c + d*x^2])","A",1
203,1,86,87,0.0572882,"\int \frac{1}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","Integrate[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]","\frac{\sqrt{\frac{a+b x^2}{a}} \sqrt{\frac{c+d x^2}{c}} \operatorname{EllipticF}\left(\sin ^{-1}\left(x \sqrt{-\frac{b}{a}}\right),\frac{a d}{b c}\right)}{\sqrt{-\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}}","\frac{\sqrt{c} \sqrt{a+b x^2} \operatorname{EllipticF}\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[(a + b*x^2)/a]*Sqrt[(c + d*x^2)/c]*EllipticF[ArcSin[Sqrt[-(b/a)]*x], (a*d)/(b*c)])/(Sqrt[-(b/a)]*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","A",1
204,1,112,273,0.2528318,"\int \frac{1}{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}} \, dx","Integrate[1/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]),x]","\frac{\frac{a d \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} E\left(\sin ^{-1}\left(\sqrt{-\frac{d}{c}} x\right)|\frac{b c}{a d}\right)}{\sqrt{-\frac{d}{c}}}-b x \left(c+d x^2\right)}{a \sqrt{a+b x^2} \sqrt{c+d x^2} (a d-b c)}","-\frac{\sqrt{c} \sqrt{d} \sqrt{a+b x^2} \operatorname{EllipticF}\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} 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,"(-(b*x*(c + d*x^2)) + (a*d*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[ArcSin[Sqrt[-(d/c)]*x], (b*c)/(a*d)])/Sqrt[-(d/c)])/(a*(-(b*c) + a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","A",1
205,1,261,255,0.6128156,"\int \frac{1}{\left(a+b x^2\right)^{5/2} \sqrt{c+d x^2}} \, dx","Integrate[1/((a + b*x^2)^(5/2)*Sqrt[c + d*x^2]),x]","\frac{-i \left(a+b x^2\right) \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(3 a^2 d^2-5 a b c d+2 b^2 c^2\right) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(x \sqrt{\frac{b}{a}}\right),\frac{a d}{b c}\right)+b x \sqrt{\frac{b}{a}} \left(c+d x^2\right) \left(-5 a^2 d+a b \left(3 c-4 d x^2\right)+2 b^2 c x^2\right)-2 i b c \left(a+b x^2\right) \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (2 a d-b c) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{3 a^2 \sqrt{\frac{b}{a}} \left(a+b x^2\right)^{3/2} \sqrt{c+d x^2} (b c-a d)^2}","-\frac{\sqrt{c} \sqrt{d} \sqrt{a+b x^2} (b c-3 a d) \operatorname{EllipticF}\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*Sqrt[b/a]*x*(c + d*x^2)*(-5*a^2*d + 2*b^2*c*x^2 + a*b*(3*c - 4*d*x^2)) - (2*I)*b*c*(-(b*c) + 2*a*d)*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*(2*b^2*c^2 - 5*a*b*c*d + 3*a^2*d^2)*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(3*a^2*Sqrt[b/a]*(b*c - a*d)^2*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])","C",1
206,1,301,334,0.641816,"\int \frac{1}{\left(a+b x^2\right)^{7/2} \sqrt{c+d x^2}} \, dx","Integrate[1/((a + b*x^2)^(7/2)*Sqrt[c + d*x^2]),x]","\frac{b x \sqrt{\frac{b}{a}} \left(c+d x^2\right) \left(\left(a+b x^2\right)^2 \left(23 a^2 d^2-23 a b c d+8 b^2 c^2\right)+3 a^2 (b c-a d)^2+4 a \left(a+b x^2\right) (b c-2 a d) (b c-a d)\right)+i \sqrt{\frac{b x^2}{a}+1} \left(a+b x^2\right)^2 \sqrt{\frac{d x^2}{c}+1} \left(\left(15 a^3 d^3-34 a^2 b c d^2+27 a b^2 c^2 d-8 b^3 c^3\right) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(x \sqrt{\frac{b}{a}}\right),\frac{a d}{b c}\right)+b c \left(23 a^2 d^2-23 a b c d+8 b^2 c^2\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)\right)}{15 a^3 \sqrt{\frac{b}{a}} \left(a+b x^2\right)^{5/2} \sqrt{c+d x^2} (b c-a d)^3}","-\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) \operatorname{EllipticF}\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{4 b x \sqrt{c+d x^2} (b c-2 a d)}{15 a^2 \left(a+b x^2\right)^{3/2} (b c-a d)^2}+\frac{\sqrt{b} \sqrt{c+d x^2} \left(23 a^2 d^2-23 a b c d+8 b^2 c^2\right) E\left(\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)|1-\frac{a d}{b c}\right)}{15 a^{5/2} \sqrt{a+b x^2} (b c-a d)^3 \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}}}+\frac{b x \sqrt{c+d x^2}}{5 a \left(a+b x^2\right)^{5/2} (b c-a d)}",1,"(b*Sqrt[b/a]*x*(c + d*x^2)*(3*a^2*(b*c - a*d)^2 + 4*a*(b*c - 2*a*d)*(b*c - a*d)*(a + b*x^2) + (8*b^2*c^2 - 23*a*b*c*d + 23*a^2*d^2)*(a + b*x^2)^2) + I*(a + b*x^2)^2*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*(b*c*(8*b^2*c^2 - 23*a*b*c*d + 23*a^2*d^2)*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (-8*b^3*c^3 + 27*a*b^2*c^2*d - 34*a^2*b*c*d^2 + 15*a^3*d^3)*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(15*a^3*Sqrt[b/a]*(b*c - a*d)^3*(a + b*x^2)^(5/2)*Sqrt[c + d*x^2])","C",1
207,1,318,445,1.2237713,"\int \frac{\left(a+b x^2\right)^{7/2}}{\left(c+d x^2\right)^{3/2}} \, dx","Integrate[(a + b*x^2)^(7/2)/(c + d*x^2)^(3/2),x]","\frac{4 i b c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(-15 a^3 d^3+41 a^2 b c d^2-38 a b^2 c^2 d+12 b^3 c^3\right) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(x \sqrt{\frac{b}{a}}\right),\frac{a d}{b c}\right)+d x \sqrt{\frac{b}{a}} \left(a+b x^2\right) \left(15 a^3 d^3-45 a^2 b c d^2+a b^2 c d \left(61 c+16 d x^2\right)-3 b^3 c \left(8 c^2+2 c d x^2-d^2 x^4\right)\right)+i b c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(15 a^3 d^3-103 a^2 b c d^2+128 a b^2 c^2 d-48 b^3 c^3\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{15 c d^4 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \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) \operatorname{EllipticF}\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right),1-\frac{b c}{a d}\right)}{15 d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{b x \sqrt{a+b x^2} \sqrt{c+d x^2} \left(15 a^2 d^2-43 a b c d+24 b^2 c^2\right)}{15 c d^3}+\frac{x \sqrt{a+b x^2} \left(-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right)}{15 c d^3 \sqrt{c+d x^2}}-\frac{\sqrt{a+b x^2} \left(-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{15 \sqrt{c} d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{b x \left(a+b x^2\right)^{3/2} \sqrt{c+d x^2} (6 b c-5 a d)}{5 c d^2}-\frac{x \left(a+b x^2\right)^{5/2} (b c-a d)}{c d \sqrt{c+d x^2}}",1,"(Sqrt[b/a]*d*x*(a + b*x^2)*(-45*a^2*b*c*d^2 + 15*a^3*d^3 + a*b^2*c*d*(61*c + 16*d*x^2) - 3*b^3*c*(8*c^2 + 2*c*d*x^2 - d^2*x^4)) + I*b*c*(-48*b^3*c^3 + 128*a*b^2*c^2*d - 103*a^2*b*c*d^2 + 15*a^3*d^3)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (4*I)*b*c*(12*b^3*c^3 - 38*a*b^2*c^2*d + 41*a^2*b*c*d^2 - 15*a^3*d^3)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(15*Sqrt[b/a]*c*d^4*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","C",1
208,1,256,346,0.4749677,"\int \frac{\left(a+b x^2\right)^{5/2}}{\left(c+d x^2\right)^{3/2}} \, dx","Integrate[(a + b*x^2)^(5/2)/(c + d*x^2)^(3/2),x]","\frac{-i b c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(9 a^2 d^2-17 a b c d+8 b^2 c^2\right) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(x \sqrt{\frac{b}{a}}\right),\frac{a d}{b c}\right)+i b c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(3 a^2 d^2-13 a b c d+8 b^2 c^2\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+d x \sqrt{\frac{b}{a}} \left(a+b x^2\right) \left(3 a^2 d^2-6 a b c d+b^2 c \left(4 c+d x^2\right)\right)}{3 c d^3 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}}","-\frac{2 b \sqrt{c} \sqrt{a+b x^2} (2 b c-3 a d) \operatorname{EllipticF}\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 \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{x \left(a+b x^2\right)^{3/2} (b c-a d)}{c d \sqrt{c+d x^2}}",1,"(Sqrt[b/a]*d*x*(a + b*x^2)*(-6*a*b*c*d + 3*a^2*d^2 + b^2*c*(4*c + d*x^2)) + I*b*c*(8*b^2*c^2 - 13*a*b*c*d + 3*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*b*c*(8*b^2*c^2 - 17*a*b*c*d + 9*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(3*Sqrt[b/a]*c*d^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","C",1
209,1,196,258,0.2939894,"\int \frac{\left(a+b x^2\right)^{3/2}}{\left(c+d x^2\right)^{3/2}} \, dx","Integrate[(a + b*x^2)^(3/2)/(c + d*x^2)^(3/2),x]","\frac{(a d-b c) \left(d x \sqrt{\frac{b}{a}} \left(a+b x^2\right)-2 i b c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \operatorname{EllipticF}\left(i \sinh ^{-1}\left(x \sqrt{\frac{b}{a}}\right),\frac{a d}{b c}\right)\right)+i b c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-2 b c) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{c d^2 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}}","\frac{b \sqrt{c} \sqrt{a+b x^2} \operatorname{EllipticF}\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,"(I*b*c*(-2*b*c + a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (-(b*c) + a*d)*(Sqrt[b/a]*d*x*(a + b*x^2) - (2*I)*b*c*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(Sqrt[b/a]*c*d^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","C",1
210,1,136,84,0.3194921,"\int \frac{\sqrt{a+b x^2}}{\left(c+d x^2\right)^{3/2}} \, dx","Integrate[Sqrt[a + b*x^2]/(c + d*x^2)^(3/2),x]","\frac{\frac{x \left(a+b x^2\right)}{c}+\frac{i a \sqrt{\frac{b}{a}} \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)-\operatorname{EllipticF}\left(i \sinh ^{-1}\left(x \sqrt{\frac{b}{a}}\right),\frac{a d}{b c}\right)\right)}{d}}{\sqrt{a+b x^2} \sqrt{c+d x^2}}","\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,"((x*(a + b*x^2))/c + (I*a*Sqrt[b/a]*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*(EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/d)/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","C",1
211,1,112,194,0.2394485,"\int \frac{1}{\sqrt{a+b x^2} \left(c+d x^2\right)^{3/2}} \, dx","Integrate[1/(Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)),x]","\frac{\frac{b c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} E\left(\sin ^{-1}\left(\sqrt{-\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{\sqrt{-\frac{b}{a}}}-d x \left(a+b x^2\right)}{c \sqrt{a+b x^2} \sqrt{c+d x^2} (b c-a d)}","\frac{b \sqrt{c} \sqrt{a+b x^2} \operatorname{EllipticF}\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,"(-(d*x*(a + b*x^2)) + (b*c*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[ArcSin[Sqrt[-(b/a)]*x], (a*d)/(b*c)])/Sqrt[-(b/a)])/(c*(b*c - a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","A",1
212,1,224,242,0.7000416,"\int \frac{1}{\left(a+b x^2\right)^{3/2} \left(c+d x^2\right)^{3/2}} \, dx","Integrate[1/((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)),x]","\frac{\sqrt{\frac{b}{a}} \left(i b c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-b c) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(x \sqrt{\frac{b}{a}}\right),\frac{a d}{b c}\right)+x \sqrt{\frac{b}{a}} \left(a^2 d^2+a b d^2 x^2+b^2 c \left(c+d x^2\right)\right)+i b c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d+b c) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)\right)}{b c \sqrt{a+b x^2} \sqrt{c+d x^2} (b c-a d)^2}","-\frac{2 b \sqrt{c} \sqrt{d} \sqrt{a+b x^2} \operatorname{EllipticF}\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{b x}{a \sqrt{a+b x^2} \sqrt{c+d x^2} (b c-a d)}+\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,"(Sqrt[b/a]*(Sqrt[b/a]*x*(a^2*d^2 + a*b*d^2*x^2 + b^2*c*(c + d*x^2)) + I*b*c*(b*c + a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + I*b*c*(-(b*c) + a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(b*c*(b*c - a*d)^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","C",1
213,1,337,323,1.0664421,"\int \frac{1}{\left(a+b x^2\right)^{5/2} \left(c+d x^2\right)^{3/2}} \, dx","Integrate[1/((a + b*x^2)^(5/2)*(c + d*x^2)^(3/2)),x]","\frac{2 i b c \left(a+b x^2\right) \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(3 a^2 d^2-4 a b c d+b^2 c^2\right) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(x \sqrt{\frac{b}{a}}\right),\frac{a d}{b c}\right)+i b c \left(a+b x^2\right) \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(3 a^2 d^2+7 a b c d-2 b^2 c^2\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+x \sqrt{\frac{b}{a}} \left(3 a^4 d^3+6 a^3 b d^3 x^2+a^2 b^2 d \left(8 c^2+8 c d x^2+3 d^2 x^4\right)+a b^3 c \left(-3 c^2+4 c d x^2+7 d^2 x^4\right)-2 b^4 c^2 x^2 \left(c+d x^2\right)\right)}{3 a^2 c \sqrt{\frac{b}{a}} \left(a+b x^2\right)^{3/2} \sqrt{c+d x^2} (a d-b c)^3}","-\frac{b \sqrt{c} \sqrt{d} \sqrt{a+b x^2} (b c-9 a d) \operatorname{EllipticF}\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{\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 x}{3 a \left(a+b x^2\right)^{3/2} \sqrt{c+d x^2} (b c-a d)}",1,"(Sqrt[b/a]*x*(3*a^4*d^3 + 6*a^3*b*d^3*x^2 - 2*b^4*c^2*x^2*(c + d*x^2) + a^2*b^2*d*(8*c^2 + 8*c*d*x^2 + 3*d^2*x^4) + a*b^3*c*(-3*c^2 + 4*c*d*x^2 + 7*d^2*x^4)) + I*b*c*(-2*b^2*c^2 + 7*a*b*c*d + 3*a^2*d^2)*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (2*I)*b*c*(b^2*c^2 - 4*a*b*c*d + 3*a^2*d^2)*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(3*a^2*Sqrt[b/a]*c*(-(b*c) + a*d)^3*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])","C",1
214,1,86,87,0.0557808,"\int \frac{1}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","Integrate[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]","\frac{\sqrt{\frac{a+b x^2}{a}} \sqrt{\frac{c+d x^2}{c}} \operatorname{EllipticF}\left(\sin ^{-1}\left(x \sqrt{-\frac{b}{a}}\right),\frac{a d}{b c}\right)}{\sqrt{-\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}}","\frac{\sqrt{c} \sqrt{a+b x^2} \operatorname{EllipticF}\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[(a + b*x^2)/a]*Sqrt[(c + d*x^2)/c]*EllipticF[ArcSin[Sqrt[-(b/a)]*x], (a*d)/(b*c)])/(Sqrt[-(b/a)]*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","A",1
215,1,87,87,0.0625884,"\int \frac{1}{\sqrt{a-b x^2} \sqrt{c+d x^2}} \, dx","Integrate[1/(Sqrt[a - b*x^2]*Sqrt[c + d*x^2]),x]","\frac{\sqrt{\frac{a-b x^2}{a}} \sqrt{\frac{c+d x^2}{c}} \operatorname{EllipticF}\left(\sin ^{-1}\left(x \sqrt{\frac{b}{a}}\right),-\frac{a d}{b c}\right)}{\sqrt{\frac{b}{a}} \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} \operatorname{EllipticF}\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 - b*x^2)/a]*Sqrt[(c + d*x^2)/c]*EllipticF[ArcSin[Sqrt[b/a]*x], -((a*d)/(b*c))])/(Sqrt[b/a]*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])","A",1
216,1,89,87,0.0654727,"\int \frac{1}{\sqrt{a+b x^2} \sqrt{c-d x^2}} \, dx","Integrate[1/(Sqrt[a + b*x^2]*Sqrt[c - d*x^2]),x]","\frac{\sqrt{\frac{a+b x^2}{a}} \sqrt{\frac{c-d x^2}{c}} \operatorname{EllipticF}\left(\sin ^{-1}\left(x \sqrt{-\frac{b}{a}}\right),-\frac{a d}{b c}\right)}{\sqrt{-\frac{b}{a}} \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}} \operatorname{EllipticF}\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[(a + b*x^2)/a]*Sqrt[(c - d*x^2)/c]*EllipticF[ArcSin[Sqrt[-(b/a)]*x], -((a*d)/(b*c))])/(Sqrt[-(b/a)]*Sqrt[a + b*x^2]*Sqrt[c - d*x^2])","A",1
217,1,88,88,0.0627389,"\int \frac{1}{\sqrt{a-b x^2} \sqrt{c-d x^2}} \, dx","Integrate[1/(Sqrt[a - b*x^2]*Sqrt[c - d*x^2]),x]","\frac{\sqrt{\frac{a-b x^2}{a}} \sqrt{\frac{c-d x^2}{c}} \operatorname{EllipticF}\left(\sin ^{-1}\left(x \sqrt{\frac{b}{a}}\right),\frac{a d}{b c}\right)}{\sqrt{\frac{b}{a}} \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}} \operatorname{EllipticF}\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[(a - b*x^2)/a]*Sqrt[(c - d*x^2)/c]*EllipticF[ArcSin[Sqrt[b/a]*x], (a*d)/(b*c)])/(Sqrt[b/a]*Sqrt[a - b*x^2]*Sqrt[c - d*x^2])","A",1
218,1,12,12,0.0054766,"\int \frac{1}{\sqrt{1-x^2} \sqrt{2+5 x^2}} \, dx","Integrate[1/(Sqrt[1 - x^2]*Sqrt[2 + 5*x^2]),x]","\frac{\operatorname{EllipticF}\left(\sin ^{-1}(x),-\frac{5}{2}\right)}{\sqrt{2}}","\frac{\operatorname{EllipticF}\left(\sin ^{-1}(x),-\frac{5}{2}\right)}{\sqrt{2}}",1,"EllipticF[ArcSin[x], -5/2]/Sqrt[2]","A",1
219,1,58,10,0.0312183,"\int \frac{1}{\sqrt{1-x^2} \sqrt{2+4 x^2}} \, dx","Integrate[1/(Sqrt[1 - x^2]*Sqrt[2 + 4*x^2]),x]","-\frac{i \sqrt{1-x^2} \sqrt{2 x^2+1} \operatorname{EllipticF}\left(i \sinh ^{-1}\left(\sqrt{2} x\right),-\frac{1}{2}\right)}{2 \sqrt{-2 x^4+x^2+1}}","\frac{\operatorname{EllipticF}\left(\sin ^{-1}(x),-2\right)}{\sqrt{2}}",1,"((-1/2*I)*Sqrt[1 - x^2]*Sqrt[1 + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2]*x], -1/2])/Sqrt[1 + x^2 - 2*x^4]","C",1
220,1,12,12,0.0050267,"\int \frac{1}{\sqrt{1-x^2} \sqrt{2+3 x^2}} \, dx","Integrate[1/(Sqrt[1 - x^2]*Sqrt[2 + 3*x^2]),x]","\frac{\operatorname{EllipticF}\left(\sin ^{-1}(x),-\frac{3}{2}\right)}{\sqrt{2}}","\frac{\operatorname{EllipticF}\left(\sin ^{-1}(x),-\frac{3}{2}\right)}{\sqrt{2}}",1,"EllipticF[ArcSin[x], -3/2]/Sqrt[2]","A",1
221,1,10,10,0.0203348,"\int \frac{1}{\sqrt{1-x^2} \sqrt{2+2 x^2}} \, dx","Integrate[1/(Sqrt[1 - x^2]*Sqrt[2 + 2*x^2]),x]","\frac{\operatorname{EllipticF}\left(\sin ^{-1}(x),-1\right)}{\sqrt{2}}","\frac{\operatorname{EllipticF}\left(\sin ^{-1}(x),-1\right)}{\sqrt{2}}",1,"EllipticF[ArcSin[x], -1]/Sqrt[2]","A",1
222,1,18,12,0.0209657,"\int \frac{1}{\sqrt{1-x^2} \sqrt{2+x^2}} \, dx","Integrate[1/(Sqrt[1 - x^2]*Sqrt[2 + x^2]),x]","-i \operatorname{EllipticF}\left(i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right),-2\right)","\frac{\operatorname{EllipticF}\left(\sin ^{-1}(x),-\frac{1}{2}\right)}{\sqrt{2}}",1,"(-I)*EllipticF[I*ArcSinh[x/Sqrt[2]], -2]","C",1
223,1,12,12,0.0052866,"\int \frac{1}{\sqrt{1-x^2} \sqrt{2-x^2}} \, dx","Integrate[1/(Sqrt[1 - x^2]*Sqrt[2 - x^2]),x]","\frac{\operatorname{EllipticF}\left(\sin ^{-1}(x),\frac{1}{2}\right)}{\sqrt{2}}","\frac{\operatorname{EllipticF}\left(\sin ^{-1}(x),\frac{1}{2}\right)}{\sqrt{2}}",1,"EllipticF[ArcSin[x], 1/2]/Sqrt[2]","A",1
224,1,26,8,0.0050887,"\int \frac{1}{\sqrt{2-2 x^2} \sqrt{1-x^2}} \, dx","Integrate[1/(Sqrt[2 - 2*x^2]*Sqrt[1 - x^2]),x]","-\frac{\frac{1}{2} \log (1-x)-\frac{1}{2} \log (x+1)}{\sqrt{2}}","\frac{\tanh ^{-1}(x)}{\sqrt{2}}",1,"-((Log[1 - x]/2 - Log[1 + x]/2)/Sqrt[2])","B",1
225,1,12,12,0.0052219,"\int \frac{1}{\sqrt{2-3 x^2} \sqrt{1-x^2}} \, dx","Integrate[1/(Sqrt[2 - 3*x^2]*Sqrt[1 - x^2]),x]","\frac{\operatorname{EllipticF}\left(\sin ^{-1}(x),\frac{3}{2}\right)}{\sqrt{2}}","\frac{\operatorname{EllipticF}\left(\sin ^{-1}(x),\frac{3}{2}\right)}{\sqrt{2}}",1,"EllipticF[ArcSin[x], 3/2]/Sqrt[2]","A",1
226,1,10,10,0.0088786,"\int \frac{1}{\sqrt{2-4 x^2} \sqrt{1-x^2}} \, dx","Integrate[1/(Sqrt[2 - 4*x^2]*Sqrt[1 - x^2]),x]","\frac{\operatorname{EllipticF}\left(\sin ^{-1}(x),2\right)}{\sqrt{2}}","\frac{\operatorname{EllipticF}\left(\sin ^{-1}(x),2\right)}{\sqrt{2}}",1,"EllipticF[ArcSin[x], 2]/Sqrt[2]","A",1
227,1,12,12,0.0053224,"\int \frac{1}{\sqrt{2-5 x^2} \sqrt{1-x^2}} \, dx","Integrate[1/(Sqrt[2 - 5*x^2]*Sqrt[1 - x^2]),x]","\frac{\operatorname{EllipticF}\left(\sin ^{-1}(x),\frac{5}{2}\right)}{\sqrt{2}}","\frac{\operatorname{EllipticF}\left(\sin ^{-1}(x),\frac{5}{2}\right)}{\sqrt{2}}",1,"EllipticF[ArcSin[x], 5/2]/Sqrt[2]","A",1
228,1,19,51,0.0215998,"\int \frac{1}{\sqrt{1+x^2} \sqrt{2+5 x^2}} \, dx","Integrate[1/(Sqrt[1 + x^2]*Sqrt[2 + 5*x^2]),x]","-\frac{i \operatorname{EllipticF}\left(i \sinh ^{-1}(x),\frac{5}{2}\right)}{\sqrt{2}}","\frac{\sqrt{5 x^2+2} \operatorname{EllipticF}\left(\tan ^{-1}(x),-\frac{3}{2}\right)}{\sqrt{2} \sqrt{x^2+1} \sqrt{\frac{5 x^2+2}{x^2+1}}}",1,"((-I)*EllipticF[I*ArcSinh[x], 5/2])/Sqrt[2]","C",1
229,1,17,49,0.0291344,"\int \frac{1}{\sqrt{1+x^2} \sqrt{2+4 x^2}} \, dx","Integrate[1/(Sqrt[1 + x^2]*Sqrt[2 + 4*x^2]),x]","-\frac{i \operatorname{EllipticF}\left(i \sinh ^{-1}(x),2\right)}{\sqrt{2}}","\frac{\sqrt{2 x^2+1} \operatorname{EllipticF}\left(\tan ^{-1}(x),-1\right)}{\sqrt{2} \sqrt{x^2+1} \sqrt{\frac{2 x^2+1}{x^2+1}}}",1,"((-I)*EllipticF[I*ArcSinh[x], 2])/Sqrt[2]","C",1
230,1,19,51,0.0212761,"\int \frac{1}{\sqrt{1+x^2} \sqrt{2+3 x^2}} \, dx","Integrate[1/(Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]),x]","-\frac{i \operatorname{EllipticF}\left(i \sinh ^{-1}(x),\frac{3}{2}\right)}{\sqrt{2}}","\frac{\sqrt{3 x^2+2} \operatorname{EllipticF}\left(\tan ^{-1}(x),-\frac{1}{2}\right)}{\sqrt{2} \sqrt{x^2+1} \sqrt{\frac{3 x^2+2}{x^2+1}}}",1,"((-I)*EllipticF[I*ArcSinh[x], 3/2])/Sqrt[2]","C",1
231,1,8,8,0.003266,"\int \frac{1}{\sqrt{1+x^2} \sqrt{2+2 x^2}} \, dx","Integrate[1/(Sqrt[1 + x^2]*Sqrt[2 + 2*x^2]),x]","\frac{\tan ^{-1}(x)}{\sqrt{2}}","\frac{\tan ^{-1}(x)}{\sqrt{2}}",1,"ArcTan[x]/Sqrt[2]","A",1
232,1,19,47,0.0195968,"\int \frac{1}{\sqrt{1+x^2} \sqrt{2+x^2}} \, dx","Integrate[1/(Sqrt[1 + x^2]*Sqrt[2 + x^2]),x]","-\frac{i \operatorname{EllipticF}\left(i \sinh ^{-1}(x),\frac{1}{2}\right)}{\sqrt{2}}","\frac{\sqrt{x^2+2} \operatorname{EllipticF}\left(\tan ^{-1}(x),\frac{1}{2}\right)}{\sqrt{2} \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}}}",1,"((-I)*EllipticF[I*ArcSinh[x], 1/2])/Sqrt[2]","C",1
233,1,19,10,0.0190989,"\int \frac{1}{\sqrt{2-x^2} \sqrt{1+x^2}} \, dx","Integrate[1/(Sqrt[2 - x^2]*Sqrt[1 + x^2]),x]","-\frac{i \operatorname{EllipticF}\left(i \sinh ^{-1}(x),-\frac{1}{2}\right)}{\sqrt{2}}","\operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right),-2\right)",1,"((-I)*EllipticF[I*ArcSinh[x], -1/2])/Sqrt[2]","C",1
234,1,10,10,0.0139975,"\int \frac{1}{\sqrt{2-2 x^2} \sqrt{1+x^2}} \, dx","Integrate[1/(Sqrt[2 - 2*x^2]*Sqrt[1 + x^2]),x]","\frac{\operatorname{EllipticF}\left(\sin ^{-1}(x),-1\right)}{\sqrt{2}}","\frac{\operatorname{EllipticF}\left(\sin ^{-1}(x),-1\right)}{\sqrt{2}}",1,"EllipticF[ArcSin[x], -1]/Sqrt[2]","A",1
235,1,20,20,0.0043279,"\int \frac{1}{\sqrt{2-3 x^2} \sqrt{1+x^2}} \, dx","Integrate[1/(Sqrt[2 - 3*x^2]*Sqrt[1 + x^2]),x]","\frac{\operatorname{EllipticF}\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right),-\frac{2}{3}\right)}{\sqrt{3}}","\frac{\operatorname{EllipticF}\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
236,1,16,16,0.0068589,"\int \frac{1}{\sqrt{2-4 x^2} \sqrt{1+x^2}} \, dx","Integrate[1/(Sqrt[2 - 4*x^2]*Sqrt[1 + x^2]),x]","\frac{1}{2} \operatorname{EllipticF}\left(\sin ^{-1}\left(\sqrt{2} x\right),-\frac{1}{2}\right)","\frac{1}{2} \operatorname{EllipticF}\left(\sin ^{-1}\left(\sqrt{2} x\right),-\frac{1}{2}\right)",1,"EllipticF[ArcSin[Sqrt[2]*x], -1/2]/2","A",1
237,1,20,20,0.0070237,"\int \frac{1}{\sqrt{2-5 x^2} \sqrt{1+x^2}} \, dx","Integrate[1/(Sqrt[2 - 5*x^2]*Sqrt[1 + x^2]),x]","\frac{\operatorname{EllipticF}\left(\sin ^{-1}\left(\sqrt{\frac{5}{2}} x\right),-\frac{2}{5}\right)}{\sqrt{5}}","\frac{\operatorname{EllipticF}\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
238,1,32,32,0.0232765,"\int \frac{1}{\sqrt{-1+x^2} \sqrt{2+5 x^2}} \, dx","Integrate[1/(Sqrt[-1 + x^2]*Sqrt[2 + 5*x^2]),x]","\frac{\sqrt{1-x^2} \operatorname{EllipticF}\left(\sin ^{-1}(x),-\frac{5}{2}\right)}{\sqrt{2} \sqrt{x^2-1}}","\frac{\sqrt{1-x^2} \operatorname{EllipticF}\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",1
239,1,30,30,0.0282228,"\int \frac{1}{\sqrt{-1+x^2} \sqrt{2+4 x^2}} \, dx","Integrate[1/(Sqrt[-1 + x^2]*Sqrt[2 + 4*x^2]),x]","\frac{\sqrt{1-x^2} \operatorname{EllipticF}\left(\sin ^{-1}(x),-2\right)}{\sqrt{2} \sqrt{x^2-1}}","\frac{\sqrt{1-x^2} \operatorname{EllipticF}\left(\sin ^{-1}(x),-2\right)}{\sqrt{2} \sqrt{x^2-1}}",1,"(Sqrt[1 - x^2]*EllipticF[ArcSin[x], -2])/(Sqrt[2]*Sqrt[-1 + x^2])","A",1
240,1,32,32,0.0220036,"\int \frac{1}{\sqrt{-1+x^2} \sqrt{2+3 x^2}} \, dx","Integrate[1/(Sqrt[-1 + x^2]*Sqrt[2 + 3*x^2]),x]","\frac{\sqrt{1-x^2} \operatorname{EllipticF}\left(\sin ^{-1}(x),-\frac{3}{2}\right)}{\sqrt{2} \sqrt{x^2-1}}","\frac{\sqrt{1-x^2} \operatorname{EllipticF}\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",1
241,1,46,25,0.0105233,"\int \frac{1}{\sqrt{-1+x^2} \sqrt{2+2 x^2}} \, dx","Integrate[1/(Sqrt[-1 + x^2]*Sqrt[2 + 2*x^2]),x]","\frac{x \sqrt{1-x^4} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};x^4\right)}{\sqrt{x^2-1} \sqrt{2 x^2+2}}","\frac{1}{2} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right),\frac{1}{2}\right)",1,"(x*Sqrt[1 - x^4]*Hypergeometric2F1[1/4, 1/2, 5/4, x^4])/(Sqrt[-1 + x^2]*Sqrt[2 + 2*x^2])","C",1
242,1,32,32,0.0211392,"\int \frac{1}{\sqrt{-1+x^2} \sqrt{2+x^2}} \, dx","Integrate[1/(Sqrt[-1 + x^2]*Sqrt[2 + x^2]),x]","\frac{\sqrt{1-x^2} \operatorname{EllipticF}\left(\sin ^{-1}(x),-\frac{1}{2}\right)}{\sqrt{2} \sqrt{x^2-1}}","\frac{\sqrt{1-x^2} \operatorname{EllipticF}\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",1
243,1,47,12,0.0242006,"\int \frac{1}{\sqrt{2-x^2} \sqrt{-1+x^2}} \, dx","Integrate[1/(Sqrt[2 - x^2]*Sqrt[-1 + x^2]),x]","\frac{\sqrt{1-x^2} \sqrt{1-\frac{x^2}{2}} \operatorname{EllipticF}\left(\sin ^{-1}(x),\frac{1}{2}\right)}{\sqrt{-x^4+3 x^2-2}}","-\operatorname{EllipticF}\left(\cos ^{-1}\left(\frac{x}{\sqrt{2}}\right),2\right)",1,"(Sqrt[1 - x^2]*Sqrt[1 - x^2/2]*EllipticF[ArcSin[x], 1/2])/Sqrt[-2 + 3*x^2 - x^4]","B",1
244,1,40,29,0.011281,"\int \frac{1}{\sqrt{2-2 x^2} \sqrt{-1+x^2}} \, dx","Integrate[1/(Sqrt[2 - 2*x^2]*Sqrt[-1 + x^2]),x]","\frac{\left(x^2-1\right) (\log (1-x)-\log (x+1))}{2 \sqrt{2} \sqrt{-\left(x^2-1\right)^2}}","-\frac{\sqrt{x^2-1} \tanh ^{-1}(x)}{\sqrt{2} \sqrt{1-x^2}}",1,"((-1 + x^2)*(Log[1 - x] - Log[1 + x]))/(2*Sqrt[2]*Sqrt[-(-1 + x^2)^2])","A",1
245,1,40,32,0.0250329,"\int \frac{1}{\sqrt{2-3 x^2} \sqrt{-1+x^2}} \, dx","Integrate[1/(Sqrt[2 - 3*x^2]*Sqrt[-1 + x^2]),x]","\frac{\sqrt{1-x^2} \operatorname{EllipticF}\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right),\frac{2}{3}\right)}{\sqrt{3} \sqrt{x^2-1}}","\frac{\sqrt{1-x^2} \operatorname{EllipticF}\left(\sin ^{-1}(x),\frac{3}{2}\right)}{\sqrt{2} \sqrt{x^2-1}}",1,"(Sqrt[1 - x^2]*EllipticF[ArcSin[Sqrt[3/2]*x], 2/3])/(Sqrt[3]*Sqrt[-1 + x^2])","A",1
246,1,36,30,0.0294898,"\int \frac{1}{\sqrt{2-4 x^2} \sqrt{-1+x^2}} \, dx","Integrate[1/(Sqrt[2 - 4*x^2]*Sqrt[-1 + x^2]),x]","\frac{\sqrt{1-x^2} \operatorname{EllipticF}\left(\sin ^{-1}\left(\sqrt{2} x\right),\frac{1}{2}\right)}{2 \sqrt{x^2-1}}","\frac{\sqrt{1-x^2} \operatorname{EllipticF}\left(\sin ^{-1}(x),2\right)}{\sqrt{2} \sqrt{x^2-1}}",1,"(Sqrt[1 - x^2]*EllipticF[ArcSin[Sqrt[2]*x], 1/2])/(2*Sqrt[-1 + x^2])","A",1
247,1,40,32,0.025137,"\int \frac{1}{\sqrt{2-5 x^2} \sqrt{-1+x^2}} \, dx","Integrate[1/(Sqrt[2 - 5*x^2]*Sqrt[-1 + x^2]),x]","\frac{\sqrt{1-x^2} \operatorname{EllipticF}\left(\sin ^{-1}\left(\sqrt{\frac{5}{2}} x\right),\frac{2}{5}\right)}{\sqrt{5} \sqrt{x^2-1}}","\frac{\sqrt{1-x^2} \operatorname{EllipticF}\left(\sin ^{-1}(x),\frac{5}{2}\right)}{\sqrt{2} \sqrt{x^2-1}}",1,"(Sqrt[1 - x^2]*EllipticF[ArcSin[Sqrt[5/2]*x], 2/5])/(Sqrt[5]*Sqrt[-1 + x^2])","A",1
248,1,39,53,0.0265557,"\int \frac{1}{\sqrt{-1-x^2} \sqrt{2+5 x^2}} \, dx","Integrate[1/(Sqrt[-1 - x^2]*Sqrt[2 + 5*x^2]),x]","-\frac{i \sqrt{x^2+1} \operatorname{EllipticF}\left(i \sinh ^{-1}(x),\frac{5}{2}\right)}{\sqrt{2} \sqrt{-x^2-1}}","\frac{\sqrt{5 x^2+2} \operatorname{EllipticF}\left(\tan ^{-1}(x),-\frac{3}{2}\right)}{\sqrt{2} \sqrt{-x^2-1} \sqrt{\frac{5 x^2+2}{x^2+1}}}",1,"((-I)*Sqrt[1 + x^2]*EllipticF[I*ArcSinh[x], 5/2])/(Sqrt[2]*Sqrt[-1 - x^2])","C",1
249,1,37,51,0.0296366,"\int \frac{1}{\sqrt{-1-x^2} \sqrt{2+4 x^2}} \, dx","Integrate[1/(Sqrt[-1 - x^2]*Sqrt[2 + 4*x^2]),x]","-\frac{i \sqrt{x^2+1} \operatorname{EllipticF}\left(i \sinh ^{-1}(x),2\right)}{\sqrt{2} \sqrt{-x^2-1}}","\frac{\sqrt{2 x^2+1} \operatorname{EllipticF}\left(\tan ^{-1}(x),-1\right)}{\sqrt{2} \sqrt{-x^2-1} \sqrt{\frac{2 x^2+1}{x^2+1}}}",1,"((-I)*Sqrt[1 + x^2]*EllipticF[I*ArcSinh[x], 2])/(Sqrt[2]*Sqrt[-1 - x^2])","C",1
250,1,39,53,0.0239486,"\int \frac{1}{\sqrt{-1-x^2} \sqrt{2+3 x^2}} \, dx","Integrate[1/(Sqrt[-1 - x^2]*Sqrt[2 + 3*x^2]),x]","-\frac{i \sqrt{x^2+1} \operatorname{EllipticF}\left(i \sinh ^{-1}(x),\frac{3}{2}\right)}{\sqrt{2} \sqrt{-x^2-1}}","\frac{\sqrt{3 x^2+2} \operatorname{EllipticF}\left(\tan ^{-1}(x),-\frac{1}{2}\right)}{\sqrt{2} \sqrt{-x^2-1} \sqrt{\frac{3 x^2+2}{x^2+1}}}",1,"((-I)*Sqrt[1 + x^2]*EllipticF[I*ArcSinh[x], 3/2])/(Sqrt[2]*Sqrt[-1 - x^2])","C",1
251,1,26,28,0.0108345,"\int \frac{1}{\sqrt{-1-x^2} \sqrt{2+2 x^2}} \, dx","Integrate[1/(Sqrt[-1 - x^2]*Sqrt[2 + 2*x^2]),x]","\frac{\left(x^2+1\right) \tan ^{-1}(x)}{\sqrt{2} \sqrt{-\left(x^2+1\right)^2}}","\frac{\sqrt{x^2+1} \tan ^{-1}(x)}{\sqrt{2} \sqrt{-x^2-1}}",1,"((1 + x^2)*ArcTan[x])/(Sqrt[2]*Sqrt[-(1 + x^2)^2])","A",1
252,1,53,49,0.0378048,"\int \frac{1}{\sqrt{-1-x^2} \sqrt{2+x^2}} \, dx","Integrate[1/(Sqrt[-1 - x^2]*Sqrt[2 + x^2]),x]","-\frac{i \sqrt{x^2+1} \sqrt{x^2+2} \operatorname{EllipticF}\left(i \sinh ^{-1}(x),\frac{1}{2}\right)}{\sqrt{2} \sqrt{-\left(\left(x^2+1\right) \left(x^2+2\right)\right)}}","\frac{\sqrt{x^2+2} \operatorname{EllipticF}\left(\tan ^{-1}(x),\frac{1}{2}\right)}{\sqrt{2} \sqrt{-x^2-1} \sqrt{\frac{x^2+2}{x^2+1}}}",1,"((-I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x], 1/2])/(Sqrt[2]*Sqrt[-((1 + x^2)*(2 + x^2))])","C",1
253,1,39,31,0.0250643,"\int \frac{1}{\sqrt{-1-x^2} \sqrt{2-x^2}} \, dx","Integrate[1/(Sqrt[-1 - x^2]*Sqrt[2 - x^2]),x]","-\frac{i \sqrt{x^2+1} \operatorname{EllipticF}\left(i \sinh ^{-1}(x),-\frac{1}{2}\right)}{\sqrt{2} \sqrt{-x^2-1}}","\frac{\sqrt{x^2+1} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right),-2\right)}{\sqrt{-x^2-1}}",1,"((-I)*Sqrt[1 + x^2]*EllipticF[I*ArcSinh[x], -1/2])/(Sqrt[2]*Sqrt[-1 - x^2])","C",1
254,1,48,42,0.0150221,"\int \frac{1}{\sqrt{2-2 x^2} \sqrt{-1-x^2}} \, dx","Integrate[1/(Sqrt[2 - 2*x^2]*Sqrt[-1 - x^2]),x]","\frac{x \sqrt{1-x^4} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};x^4\right)}{\sqrt{2-2 x^2} \sqrt{-x^2-1}}","-\frac{\sqrt{1-\frac{1}{x^4}} x^2 \operatorname{EllipticF}\left(\csc ^{-1}(x),-1\right)}{\sqrt{2-2 x^2} \sqrt{-x^2-1}}",1,"(x*Sqrt[1 - x^4]*Hypergeometric2F1[1/4, 1/2, 5/4, x^4])/(Sqrt[2 - 2*x^2]*Sqrt[-1 - x^2])","C",1
255,1,40,40,0.025081,"\int \frac{1}{\sqrt{2-3 x^2} \sqrt{-1-x^2}} \, dx","Integrate[1/(Sqrt[2 - 3*x^2]*Sqrt[-1 - x^2]),x]","\frac{\sqrt{x^2+1} \operatorname{EllipticF}\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} \operatorname{EllipticF}\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",1
256,1,36,36,0.0309855,"\int \frac{1}{\sqrt{2-4 x^2} \sqrt{-1-x^2}} \, dx","Integrate[1/(Sqrt[2 - 4*x^2]*Sqrt[-1 - x^2]),x]","\frac{\sqrt{x^2+1} \operatorname{EllipticF}\left(\sin ^{-1}\left(\sqrt{2} x\right),-\frac{1}{2}\right)}{2 \sqrt{-x^2-1}}","\frac{\sqrt{x^2+1} \operatorname{EllipticF}\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",1
257,1,40,40,0.0278126,"\int \frac{1}{\sqrt{2-5 x^2} \sqrt{-1-x^2}} \, dx","Integrate[1/(Sqrt[2 - 5*x^2]*Sqrt[-1 - x^2]),x]","\frac{\sqrt{x^2+1} \operatorname{EllipticF}\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} \operatorname{EllipticF}\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",1
258,1,87,87,0.0591488,"\int \frac{\sqrt{a+b x^2}}{\sqrt{c-d x^2}} \, dx","Integrate[Sqrt[a + b*x^2]/Sqrt[c - d*x^2],x]","\frac{\sqrt{a+b x^2} \sqrt{\frac{c-d x^2}{c}} E\left(\sin ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|-\frac{b c}{a d}\right)}{\sqrt{\frac{d}{c}} \sqrt{\frac{a+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{\frac{b x^2}{a}+1} \sqrt{c-d x^2}}",1,"(Sqrt[a + b*x^2]*Sqrt[(c - d*x^2)/c]*EllipticE[ArcSin[Sqrt[d/c]*x], -((b*c)/(a*d))])/(Sqrt[d/c]*Sqrt[(a + b*x^2)/a]*Sqrt[c - d*x^2])","A",1
259,1,90,90,0.0505091,"\int \frac{\sqrt{-a-b x^2}}{\sqrt{c-d x^2}} \, dx","Integrate[Sqrt[-a - b*x^2]/Sqrt[c - d*x^2],x]","\frac{\sqrt{-a-b x^2} \sqrt{\frac{c-d x^2}{c}} E\left(\sin ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|-\frac{b c}{a d}\right)}{\sqrt{\frac{d}{c}} \sqrt{\frac{a+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{\frac{b x^2}{a}+1} \sqrt{c-d x^2}}",1,"(Sqrt[-a - b*x^2]*Sqrt[(c - d*x^2)/c]*EllipticE[ArcSin[Sqrt[d/c]*x], -((b*c)/(a*d))])/(Sqrt[d/c]*Sqrt[(a + b*x^2)/a]*Sqrt[c - d*x^2])","A",1
260,1,88,88,0.0492735,"\int \frac{\sqrt{a+b x^2}}{\sqrt{-c+d x^2}} \, dx","Integrate[Sqrt[a + b*x^2]/Sqrt[-c + d*x^2],x]","\frac{\sqrt{a+b x^2} \sqrt{\frac{c-d x^2}{c}} E\left(\sin ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|-\frac{b c}{a d}\right)}{\sqrt{\frac{d}{c}} \sqrt{\frac{a+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{\frac{b x^2}{a}+1} \sqrt{d x^2-c}}",1,"(Sqrt[a + b*x^2]*Sqrt[(c - d*x^2)/c]*EllipticE[ArcSin[Sqrt[d/c]*x], -((b*c)/(a*d))])/(Sqrt[d/c]*Sqrt[(a + b*x^2)/a]*Sqrt[-c + d*x^2])","A",1
261,1,91,91,0.0488166,"\int \frac{\sqrt{-a-b x^2}}{\sqrt{-c+d x^2}} \, dx","Integrate[Sqrt[-a - b*x^2]/Sqrt[-c + d*x^2],x]","\frac{\sqrt{-a-b x^2} \sqrt{\frac{c-d x^2}{c}} E\left(\sin ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|-\frac{b c}{a d}\right)}{\sqrt{\frac{d}{c}} \sqrt{\frac{a+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{\frac{b x^2}{a}+1} \sqrt{d x^2-c}}",1,"(Sqrt[-a - b*x^2]*Sqrt[(c - d*x^2)/c]*EllipticE[ArcSin[Sqrt[d/c]*x], -((b*c)/(a*d))])/(Sqrt[d/c]*Sqrt[(a + b*x^2)/a]*Sqrt[-c + d*x^2])","A",1
262,1,88,88,0.0576856,"\int \frac{\sqrt{a-b x^2}}{\sqrt{c-d x^2}} \, dx","Integrate[Sqrt[a - b*x^2]/Sqrt[c - d*x^2],x]","\frac{\sqrt{a-b x^2} \sqrt{\frac{c-d x^2}{c}} E\left(\sin ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{b c}{a d}\right)}{\sqrt{\frac{d}{c}} \sqrt{\frac{a-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[a - b*x^2]*Sqrt[(c - d*x^2)/c]*EllipticE[ArcSin[Sqrt[d/c]*x], (b*c)/(a*d)])/(Sqrt[d/c]*Sqrt[(a - b*x^2)/a]*Sqrt[c - d*x^2])","A",1
263,1,89,89,0.0544238,"\int \frac{\sqrt{-a+b x^2}}{\sqrt{c-d x^2}} \, dx","Integrate[Sqrt[-a + b*x^2]/Sqrt[c - d*x^2],x]","\frac{\sqrt{b x^2-a} \sqrt{\frac{c-d x^2}{c}} E\left(\sin ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{b c}{a d}\right)}{\sqrt{\frac{d}{c}} \sqrt{\frac{a-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[-a + b*x^2]*Sqrt[(c - d*x^2)/c]*EllipticE[ArcSin[Sqrt[d/c]*x], (b*c)/(a*d)])/(Sqrt[d/c]*Sqrt[(a - b*x^2)/a]*Sqrt[c - d*x^2])","A",1
264,1,89,89,0.0480919,"\int \frac{\sqrt{a-b x^2}}{\sqrt{-c+d x^2}} \, dx","Integrate[Sqrt[a - b*x^2]/Sqrt[-c + d*x^2],x]","\frac{\sqrt{a-b x^2} \sqrt{\frac{c-d x^2}{c}} E\left(\sin ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{b c}{a d}\right)}{\sqrt{\frac{d}{c}} \sqrt{\frac{a-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[a - b*x^2]*Sqrt[(c - d*x^2)/c]*EllipticE[ArcSin[Sqrt[d/c]*x], (b*c)/(a*d)])/(Sqrt[d/c]*Sqrt[(a - b*x^2)/a]*Sqrt[-c + d*x^2])","A",1
265,1,90,90,0.0482669,"\int \frac{\sqrt{-a+b x^2}}{\sqrt{-c+d x^2}} \, dx","Integrate[Sqrt[-a + b*x^2]/Sqrt[-c + d*x^2],x]","\frac{\sqrt{b x^2-a} \sqrt{\frac{c-d x^2}{c}} E\left(\sin ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{b c}{a d}\right)}{\sqrt{\frac{d}{c}} \sqrt{\frac{a-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[-a + b*x^2]*Sqrt[(c - d*x^2)/c]*EllipticE[ArcSin[Sqrt[d/c]*x], (b*c)/(a*d)])/(Sqrt[d/c]*Sqrt[(a - b*x^2)/a]*Sqrt[-c + d*x^2])","A",1
266,1,86,194,0.0483753,"\int \frac{\sqrt{a+b x^2}}{\sqrt{c+d x^2}} \, dx","Integrate[Sqrt[a + b*x^2]/Sqrt[c + d*x^2],x]","\frac{\sqrt{a+b x^2} \sqrt{\frac{c+d x^2}{c}} E\left(\sin ^{-1}\left(\sqrt{-\frac{d}{c}} x\right)|\frac{b c}{a d}\right)}{\sqrt{-\frac{d}{c}} \sqrt{\frac{a+b x^2}{a}} \sqrt{c+d x^2}}","\frac{\sqrt{c} \sqrt{a+b x^2} \operatorname{EllipticF}\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} 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,"(Sqrt[a + b*x^2]*Sqrt[(c + d*x^2)/c]*EllipticE[ArcSin[Sqrt[-(d/c)]*x], (b*c)/(a*d)])/(Sqrt[-(d/c)]*Sqrt[(a + b*x^2)/a]*Sqrt[c + d*x^2])","A",1
267,1,89,203,0.0472176,"\int \frac{\sqrt{-a-b x^2}}{\sqrt{c+d x^2}} \, dx","Integrate[Sqrt[-a - b*x^2]/Sqrt[c + d*x^2],x]","\frac{\sqrt{-a-b x^2} \sqrt{\frac{c+d x^2}{c}} E\left(\sin ^{-1}\left(\sqrt{-\frac{d}{c}} x\right)|\frac{b c}{a d}\right)}{\sqrt{-\frac{d}{c}} \sqrt{\frac{a+b x^2}{a}} \sqrt{c+d x^2}}","\frac{\sqrt{c} \sqrt{-a-b x^2} \operatorname{EllipticF}\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} 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,"(Sqrt[-a - b*x^2]*Sqrt[(c + d*x^2)/c]*EllipticE[ArcSin[Sqrt[-(d/c)]*x], (b*c)/(a*d)])/(Sqrt[-(d/c)]*Sqrt[(a + b*x^2)/a]*Sqrt[c + d*x^2])","A",1
268,1,89,203,0.0500445,"\int \frac{\sqrt{a+b x^2}}{\sqrt{-c-d x^2}} \, dx","Integrate[Sqrt[a + b*x^2]/Sqrt[-c - d*x^2],x]","\frac{\sqrt{a+b x^2} \sqrt{\frac{c+d x^2}{c}} E\left(\sin ^{-1}\left(\sqrt{-\frac{d}{c}} x\right)|\frac{b c}{a d}\right)}{\sqrt{-\frac{d}{c}} \sqrt{\frac{a+b x^2}{a}} \sqrt{-c-d x^2}}","\frac{\sqrt{c} \sqrt{a+b x^2} \operatorname{EllipticF}\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} 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,"(Sqrt[a + b*x^2]*Sqrt[(c + d*x^2)/c]*EllipticE[ArcSin[Sqrt[-(d/c)]*x], (b*c)/(a*d)])/(Sqrt[-(d/c)]*Sqrt[(a + b*x^2)/a]*Sqrt[-c - d*x^2])","A",1
269,1,92,212,0.0460818,"\int \frac{\sqrt{-a-b x^2}}{\sqrt{-c-d x^2}} \, dx","Integrate[Sqrt[-a - b*x^2]/Sqrt[-c - d*x^2],x]","\frac{\sqrt{-a-b x^2} \sqrt{\frac{c+d x^2}{c}} E\left(\sin ^{-1}\left(\sqrt{-\frac{d}{c}} x\right)|\frac{b c}{a d}\right)}{\sqrt{-\frac{d}{c}} \sqrt{\frac{a+b x^2}{a}} \sqrt{-c-d x^2}}","\frac{\sqrt{c} \sqrt{-a-b x^2} \operatorname{EllipticF}\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} 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,"(Sqrt[-a - b*x^2]*Sqrt[(c + d*x^2)/c]*EllipticE[ArcSin[Sqrt[-(d/c)]*x], (b*c)/(a*d)])/(Sqrt[-(d/c)]*Sqrt[(a + b*x^2)/a]*Sqrt[-c - d*x^2])","A",1
270,1,89,189,0.052442,"\int \frac{\sqrt{a-b x^2}}{\sqrt{c+d x^2}} \, dx","Integrate[Sqrt[a - b*x^2]/Sqrt[c + d*x^2],x]","\frac{\sqrt{a-b x^2} \sqrt{\frac{c+d x^2}{c}} E\left(\sin ^{-1}\left(\sqrt{-\frac{d}{c}} x\right)|-\frac{b c}{a d}\right)}{\sqrt{-\frac{d}{c}} \sqrt{\frac{a-b x^2}{a}} \sqrt{c+d x^2}}","\frac{\sqrt{a} \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} (a d+b c) \operatorname{EllipticF}\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 - b*x^2]*Sqrt[(c + d*x^2)/c]*EllipticE[ArcSin[Sqrt[-(d/c)]*x], -((b*c)/(a*d))])/(Sqrt[-(d/c)]*Sqrt[(a - b*x^2)/a]*Sqrt[c + d*x^2])","A",1
271,1,90,191,0.053714,"\int \frac{\sqrt{-a+b x^2}}{\sqrt{c+d x^2}} \, dx","Integrate[Sqrt[-a + b*x^2]/Sqrt[c + d*x^2],x]","\frac{\sqrt{b x^2-a} \sqrt{\frac{c+d x^2}{c}} E\left(\sin ^{-1}\left(\sqrt{-\frac{d}{c}} x\right)|-\frac{b c}{a d}\right)}{\sqrt{-\frac{d}{c}} \sqrt{\frac{a-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) \operatorname{EllipticF}\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 + b*x^2]*Sqrt[(c + d*x^2)/c]*EllipticE[ArcSin[Sqrt[-(d/c)]*x], -((b*c)/(a*d))])/(Sqrt[-(d/c)]*Sqrt[(a - b*x^2)/a]*Sqrt[c + d*x^2])","A",1
272,1,92,194,0.0519208,"\int \frac{\sqrt{a-b x^2}}{\sqrt{-c-d x^2}} \, dx","Integrate[Sqrt[a - b*x^2]/Sqrt[-c - d*x^2],x]","\frac{\sqrt{a-b x^2} \sqrt{\frac{c+d x^2}{c}} E\left(\sin ^{-1}\left(\sqrt{-\frac{d}{c}} x\right)|-\frac{b c}{a d}\right)}{\sqrt{-\frac{d}{c}} \sqrt{\frac{a-b x^2}{a}} \sqrt{-c-d x^2}}","\frac{\sqrt{a} \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} (a d+b c) \operatorname{EllipticF}\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 - b*x^2]*Sqrt[(c + d*x^2)/c]*EllipticE[ArcSin[Sqrt[-(d/c)]*x], -((b*c)/(a*d))])/(Sqrt[-(d/c)]*Sqrt[(a - b*x^2)/a]*Sqrt[-c - d*x^2])","A",1
273,1,93,198,0.0487704,"\int \frac{\sqrt{-a+b x^2}}{\sqrt{-c-d x^2}} \, dx","Integrate[Sqrt[-a + b*x^2]/Sqrt[-c - d*x^2],x]","\frac{\sqrt{b x^2-a} \sqrt{\frac{c+d x^2}{c}} E\left(\sin ^{-1}\left(\sqrt{-\frac{d}{c}} x\right)|-\frac{b c}{a d}\right)}{\sqrt{-\frac{d}{c}} \sqrt{\frac{a-b x^2}{a}} \sqrt{-c-d x^2}}","-\frac{\sqrt{a} \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} (a d+b c) \operatorname{EllipticF}\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 + b*x^2]*Sqrt[(c + d*x^2)/c]*EllipticE[ArcSin[Sqrt[-(d/c)]*x], -((b*c)/(a*d))])/(Sqrt[-(d/c)]*Sqrt[(a - b*x^2)/a]*Sqrt[-c - d*x^2])","A",1
274,1,87,87,0.056416,"\int \frac{\sqrt{c+d x^2}}{\sqrt{a-b x^2}} \, dx","Integrate[Sqrt[c + d*x^2]/Sqrt[a - b*x^2],x]","\frac{\sqrt{\frac{a-b x^2}{a}} \sqrt{c+d x^2} E\left(\sin ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|-\frac{a d}{b c}\right)}{\sqrt{\frac{b}{a}} \sqrt{a-b x^2} \sqrt{\frac{c+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{\frac{d x^2}{c}+1}}",1,"(Sqrt[(a - b*x^2)/a]*Sqrt[c + d*x^2]*EllipticE[ArcSin[Sqrt[b/a]*x], -((a*d)/(b*c))])/(Sqrt[b/a]*Sqrt[a - b*x^2]*Sqrt[(c + d*x^2)/c])","A",1
275,1,90,90,0.0489401,"\int \frac{\sqrt{-c-d x^2}}{\sqrt{a-b x^2}} \, dx","Integrate[Sqrt[-c - d*x^2]/Sqrt[a - b*x^2],x]","\frac{\sqrt{\frac{a-b x^2}{a}} \sqrt{-c-d x^2} E\left(\sin ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|-\frac{a d}{b c}\right)}{\sqrt{\frac{b}{a}} \sqrt{a-b x^2} \sqrt{\frac{c+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{\frac{d x^2}{c}+1}}",1,"(Sqrt[(a - b*x^2)/a]*Sqrt[-c - d*x^2]*EllipticE[ArcSin[Sqrt[b/a]*x], -((a*d)/(b*c))])/(Sqrt[b/a]*Sqrt[a - b*x^2]*Sqrt[(c + d*x^2)/c])","A",1
276,1,88,88,0.0471753,"\int \frac{\sqrt{c+d x^2}}{\sqrt{-a+b x^2}} \, dx","Integrate[Sqrt[c + d*x^2]/Sqrt[-a + b*x^2],x]","\frac{\sqrt{\frac{a-b x^2}{a}} \sqrt{c+d x^2} E\left(\sin ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|-\frac{a d}{b c}\right)}{\sqrt{\frac{b}{a}} \sqrt{b x^2-a} \sqrt{\frac{c+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{\frac{d x^2}{c}+1}}",1,"(Sqrt[(a - b*x^2)/a]*Sqrt[c + d*x^2]*EllipticE[ArcSin[Sqrt[b/a]*x], -((a*d)/(b*c))])/(Sqrt[b/a]*Sqrt[-a + b*x^2]*Sqrt[(c + d*x^2)/c])","A",1
277,1,91,91,0.0425977,"\int \frac{\sqrt{-c-d x^2}}{\sqrt{-a+b x^2}} \, dx","Integrate[Sqrt[-c - d*x^2]/Sqrt[-a + b*x^2],x]","\frac{\sqrt{\frac{a-b x^2}{a}} \sqrt{-c-d x^2} E\left(\sin ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|-\frac{a d}{b c}\right)}{\sqrt{\frac{b}{a}} \sqrt{b x^2-a} \sqrt{\frac{c+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{\frac{d x^2}{c}+1}}",1,"(Sqrt[(a - b*x^2)/a]*Sqrt[-c - d*x^2]*EllipticE[ArcSin[Sqrt[b/a]*x], -((a*d)/(b*c))])/(Sqrt[b/a]*Sqrt[-a + b*x^2]*Sqrt[(c + d*x^2)/c])","A",1
278,1,88,88,0.0560604,"\int \frac{\sqrt{c-d x^2}}{\sqrt{a-b x^2}} \, dx","Integrate[Sqrt[c - d*x^2]/Sqrt[a - b*x^2],x]","\frac{\sqrt{\frac{a-b x^2}{a}} \sqrt{c-d x^2} E\left(\sin ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{\sqrt{\frac{b}{a}} \sqrt{a-b x^2} \sqrt{\frac{c-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 - b*x^2)/a]*Sqrt[c - d*x^2]*EllipticE[ArcSin[Sqrt[b/a]*x], (a*d)/(b*c)])/(Sqrt[b/a]*Sqrt[a - b*x^2]*Sqrt[(c - d*x^2)/c])","A",1
279,1,89,89,0.0489922,"\int \frac{\sqrt{-c+d x^2}}{\sqrt{a-b x^2}} \, dx","Integrate[Sqrt[-c + d*x^2]/Sqrt[a - b*x^2],x]","\frac{\sqrt{\frac{a-b x^2}{a}} \sqrt{d x^2-c} E\left(\sin ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{\sqrt{\frac{b}{a}} \sqrt{a-b x^2} \sqrt{\frac{c-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 - b*x^2)/a]*Sqrt[-c + d*x^2]*EllipticE[ArcSin[Sqrt[b/a]*x], (a*d)/(b*c)])/(Sqrt[b/a]*Sqrt[a - b*x^2]*Sqrt[(c - d*x^2)/c])","A",1
280,1,89,89,0.0491745,"\int \frac{\sqrt{c-d x^2}}{\sqrt{-a+b x^2}} \, dx","Integrate[Sqrt[c - d*x^2]/Sqrt[-a + b*x^2],x]","\frac{\sqrt{\frac{a-b x^2}{a}} \sqrt{c-d x^2} E\left(\sin ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{\sqrt{\frac{b}{a}} \sqrt{b x^2-a} \sqrt{\frac{c-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 - b*x^2)/a]*Sqrt[c - d*x^2]*EllipticE[ArcSin[Sqrt[b/a]*x], (a*d)/(b*c)])/(Sqrt[b/a]*Sqrt[-a + b*x^2]*Sqrt[(c - d*x^2)/c])","A",1
281,1,90,90,0.0429365,"\int \frac{\sqrt{-c+d x^2}}{\sqrt{-a+b x^2}} \, dx","Integrate[Sqrt[-c + d*x^2]/Sqrt[-a + b*x^2],x]","\frac{\sqrt{\frac{a-b x^2}{a}} \sqrt{d x^2-c} E\left(\sin ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{\sqrt{\frac{b}{a}} \sqrt{b x^2-a} \sqrt{\frac{c-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 - b*x^2)/a]*Sqrt[-c + d*x^2]*EllipticE[ArcSin[Sqrt[b/a]*x], (a*d)/(b*c)])/(Sqrt[b/a]*Sqrt[-a + b*x^2]*Sqrt[(c - d*x^2)/c])","A",1
282,1,86,204,0.0445412,"\int \frac{\sqrt{c+d x^2}}{\sqrt{a+b x^2}} \, dx","Integrate[Sqrt[c + d*x^2]/Sqrt[a + b*x^2],x]","\frac{\sqrt{\frac{a+b x^2}{a}} \sqrt{c+d x^2} E\left(\sin ^{-1}\left(\sqrt{-\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{\sqrt{-\frac{b}{a}} \sqrt{a+b x^2} \sqrt{\frac{c+d x^2}{c}}}","\frac{c^{3/2} \sqrt{a+b x^2} \operatorname{EllipticF}\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,"(Sqrt[(a + b*x^2)/a]*Sqrt[c + d*x^2]*EllipticE[ArcSin[Sqrt[-(b/a)]*x], (a*d)/(b*c)])/(Sqrt[-(b/a)]*Sqrt[a + b*x^2]*Sqrt[(c + d*x^2)/c])","A",1
283,1,89,214,0.0461276,"\int \frac{\sqrt{-c-d x^2}}{\sqrt{a+b x^2}} \, dx","Integrate[Sqrt[-c - d*x^2]/Sqrt[a + b*x^2],x]","\frac{\sqrt{\frac{a+b x^2}{a}} \sqrt{-c-d x^2} E\left(\sin ^{-1}\left(\sqrt{-\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{\sqrt{-\frac{b}{a}} \sqrt{a+b x^2} \sqrt{\frac{c+d x^2}{c}}}","-\frac{c^{3/2} \sqrt{a+b x^2} \operatorname{EllipticF}\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,"(Sqrt[(a + b*x^2)/a]*Sqrt[-c - d*x^2]*EllipticE[ArcSin[Sqrt[-(b/a)]*x], (a*d)/(b*c)])/(Sqrt[-(b/a)]*Sqrt[a + b*x^2]*Sqrt[(c + d*x^2)/c])","A",1
284,1,89,214,0.0506764,"\int \frac{\sqrt{c+d x^2}}{\sqrt{-a-b x^2}} \, dx","Integrate[Sqrt[c + d*x^2]/Sqrt[-a - b*x^2],x]","\frac{\sqrt{\frac{a+b x^2}{a}} \sqrt{c+d x^2} E\left(\sin ^{-1}\left(\sqrt{-\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{\sqrt{-\frac{b}{a}} \sqrt{-a-b x^2} \sqrt{\frac{c+d x^2}{c}}}","-\frac{c^{3/2} \sqrt{-a-b x^2} \operatorname{EllipticF}\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,"(Sqrt[(a + b*x^2)/a]*Sqrt[c + d*x^2]*EllipticE[ArcSin[Sqrt[-(b/a)]*x], (a*d)/(b*c)])/(Sqrt[-(b/a)]*Sqrt[-a - b*x^2]*Sqrt[(c + d*x^2)/c])","A",1
285,1,92,222,0.0515558,"\int \frac{\sqrt{-c-d x^2}}{\sqrt{-a-b x^2}} \, dx","Integrate[Sqrt[-c - d*x^2]/Sqrt[-a - b*x^2],x]","\frac{\sqrt{\frac{a+b x^2}{a}} \sqrt{-c-d x^2} E\left(\sin ^{-1}\left(\sqrt{-\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{\sqrt{-\frac{b}{a}} \sqrt{-a-b x^2} \sqrt{\frac{c+d x^2}{c}}}","\frac{c^{3/2} \sqrt{-a-b x^2} \operatorname{EllipticF}\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,"(Sqrt[(a + b*x^2)/a]*Sqrt[-c - d*x^2]*EllipticE[ArcSin[Sqrt[-(b/a)]*x], (a*d)/(b*c)])/(Sqrt[-(b/a)]*Sqrt[-a - b*x^2]*Sqrt[(c + d*x^2)/c])","A",1
286,1,89,189,0.0527196,"\int \frac{\sqrt{c-d x^2}}{\sqrt{a+b x^2}} \, dx","Integrate[Sqrt[c - d*x^2]/Sqrt[a + b*x^2],x]","\frac{\sqrt{\frac{a+b x^2}{a}} \sqrt{c-d x^2} E\left(\sin ^{-1}\left(\sqrt{-\frac{b}{a}} x\right)|-\frac{a d}{b c}\right)}{\sqrt{-\frac{b}{a}} \sqrt{a+b x^2} \sqrt{\frac{c-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) \operatorname{EllipticF}\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[(a + b*x^2)/a]*Sqrt[c - d*x^2]*EllipticE[ArcSin[Sqrt[-(b/a)]*x], -((a*d)/(b*c))])/(Sqrt[-(b/a)]*Sqrt[a + b*x^2]*Sqrt[(c - d*x^2)/c])","A",1
287,1,90,191,0.0467027,"\int \frac{\sqrt{-c+d x^2}}{\sqrt{a+b x^2}} \, dx","Integrate[Sqrt[-c + d*x^2]/Sqrt[a + b*x^2],x]","\frac{\sqrt{\frac{a+b x^2}{a}} \sqrt{d x^2-c} E\left(\sin ^{-1}\left(\sqrt{-\frac{b}{a}} x\right)|-\frac{a d}{b c}\right)}{\sqrt{-\frac{b}{a}} \sqrt{a+b x^2} \sqrt{\frac{c-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) \operatorname{EllipticF}\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[(a + b*x^2)/a]*Sqrt[-c + d*x^2]*EllipticE[ArcSin[Sqrt[-(b/a)]*x], -((a*d)/(b*c))])/(Sqrt[-(b/a)]*Sqrt[a + b*x^2]*Sqrt[(c - d*x^2)/c])","A",1
288,1,92,194,0.0513024,"\int \frac{\sqrt{c-d x^2}}{\sqrt{-a-b x^2}} \, dx","Integrate[Sqrt[c - d*x^2]/Sqrt[-a - b*x^2],x]","\frac{\sqrt{\frac{a+b x^2}{a}} \sqrt{c-d x^2} E\left(\sin ^{-1}\left(\sqrt{-\frac{b}{a}} x\right)|-\frac{a d}{b c}\right)}{\sqrt{-\frac{b}{a}} \sqrt{-a-b x^2} \sqrt{\frac{c-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) \operatorname{EllipticF}\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[(a + b*x^2)/a]*Sqrt[c - d*x^2]*EllipticE[ArcSin[Sqrt[-(b/a)]*x], -((a*d)/(b*c))])/(Sqrt[-(b/a)]*Sqrt[-a - b*x^2]*Sqrt[(c - d*x^2)/c])","A",1
289,1,93,198,0.0514067,"\int \frac{\sqrt{-c+d x^2}}{\sqrt{-a-b x^2}} \, dx","Integrate[Sqrt[-c + d*x^2]/Sqrt[-a - b*x^2],x]","\frac{\sqrt{\frac{a+b x^2}{a}} \sqrt{d x^2-c} E\left(\sin ^{-1}\left(\sqrt{-\frac{b}{a}} x\right)|-\frac{a d}{b c}\right)}{\sqrt{-\frac{b}{a}} \sqrt{-a-b x^2} \sqrt{\frac{c-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) \operatorname{EllipticF}\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[(a + b*x^2)/a]*Sqrt[-c + d*x^2]*EllipticE[ArcSin[Sqrt[-(b/a)]*x], -((a*d)/(b*c))])/(Sqrt[-(b/a)]*Sqrt[-a - b*x^2]*Sqrt[(c - d*x^2)/c])","A",1
290,1,37,78,0.012702,"\int \frac{1}{\sqrt{2+b x^2} \sqrt{3+d x^2}} \, dx","Integrate[1/(Sqrt[2 + b*x^2]*Sqrt[3 + d*x^2]),x]","\frac{\operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{-b} x}{\sqrt{2}}\right),\frac{2 d}{3 b}\right)}{\sqrt{3} \sqrt{-b}}","\frac{\sqrt{b x^2+2} \operatorname{EllipticF}\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,"EllipticF[ArcSin[(Sqrt[-b]*x)/Sqrt[2]], (2*d)/(3*b)]/(Sqrt[3]*Sqrt[-b])","A",1
291,1,40,39,0.0402473,"\int \frac{1}{\sqrt{4-x^2} \sqrt{c+d x^2}} \, dx","Integrate[1/(Sqrt[4 - x^2]*Sqrt[c + d*x^2]),x]","\frac{\sqrt{\frac{c+d x^2}{c}} \operatorname{EllipticF}\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} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{x}{2}\right),-\frac{4 d}{c}\right)}{\sqrt{c+d x^2}}",1,"(Sqrt[(c + d*x^2)/c]*EllipticF[ArcSin[x/2], (-4*d)/c])/Sqrt[c + d*x^2]","A",1
292,1,47,61,0.0365831,"\int \frac{1}{\sqrt{4+x^2} \sqrt{c+d x^2}} \, dx","Integrate[1/(Sqrt[4 + x^2]*Sqrt[c + d*x^2]),x]","-\frac{i \sqrt{\frac{c+d x^2}{c}} \operatorname{EllipticF}\left(i \sinh ^{-1}\left(\frac{x}{2}\right),\frac{4 d}{c}\right)}{\sqrt{c+d x^2}}","\frac{\sqrt{c+d x^2} \operatorname{EllipticF}\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,"((-I)*Sqrt[(c + d*x^2)/c]*EllipticF[I*ArcSinh[x/2], (4*d)/c])/Sqrt[c + d*x^2]","C",1
293,1,27,6,0.0254522,"\int \frac{1}{\sqrt{1-x^2} \sqrt{-1+2 x^2}} \, dx","Integrate[1/(Sqrt[1 - x^2]*Sqrt[-1 + 2*x^2]),x]","\frac{\sqrt{1-2 x^2} \operatorname{EllipticF}\left(\sin ^{-1}(x),2\right)}{\sqrt{2 x^2-1}}","-\operatorname{EllipticF}\left(\cos ^{-1}(x),2\right)",1,"(Sqrt[1 - 2*x^2]*EllipticF[ArcSin[x], 2])/Sqrt[-1 + 2*x^2]","B",1
294,1,24,23,0.0088772,"\int \frac{\sqrt{1-c^2 x^2}}{\sqrt{1+c^2 x^2}} \, dx","Integrate[Sqrt[1 - c^2*x^2]/Sqrt[1 + c^2*x^2],x]","\frac{E\left(\left.\sin ^{-1}\left(\sqrt{-c^2} x\right)\right|-1\right)}{\sqrt{-c^2}}","\frac{2 \operatorname{EllipticF}\left(\sin ^{-1}(c x),-1\right)}{c}-\frac{E\left(\left.\sin ^{-1}(c x)\right|-1\right)}{c}",1,"EllipticE[ArcSin[Sqrt[-c^2]*x], -1]/Sqrt[-c^2]","A",1
295,1,37,182,0.0088409,"\int \frac{\sqrt{2+b x^2}}{\sqrt{3+d x^2}} \, dx","Integrate[Sqrt[2 + b*x^2]/Sqrt[3 + d*x^2],x]","\frac{\sqrt{2} E\left(\sin ^{-1}\left(\frac{\sqrt{-d} x}{\sqrt{3}}\right)|\frac{3 b}{2 d}\right)}{\sqrt{-d}}","\frac{\sqrt{2} \sqrt{b x^2+2} \operatorname{EllipticF}\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} 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,"(Sqrt[2]*EllipticE[ArcSin[(Sqrt[-d]*x)/Sqrt[3]], (3*b)/(2*d)])/Sqrt[-d]","A",1
296,1,35,19,0.0254915,"\int \frac{\sqrt{-1+3 x^2}}{\sqrt{2-3 x^2}} \, dx","Integrate[Sqrt[-1 + 3*x^2]/Sqrt[2 - 3*x^2],x]","\frac{\sqrt{3 x^2-1} E\left(\left.\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)}{\sqrt{3-9 x^2}}","-\frac{E\left(\left.\cos ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)}{\sqrt{3}}",1,"(Sqrt[-1 + 3*x^2]*EllipticE[ArcSin[Sqrt[3/2]*x], 2])/Sqrt[3 - 9*x^2]","A",1
297,1,95,95,0.1121813,"\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","Integrate[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
298,1,95,94,0.0948017,"\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","Integrate[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}}{\sqrt{b^2-4 a c}-b}\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
299,1,102,478,0.1209974,"\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","Integrate[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} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \operatorname{EllipticF}\left(\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\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} 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,"(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
300,1,102,215,0.0934008,"\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","Integrate[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}}{\sqrt{b^2-4 a c}-b}\right)}{\sqrt{2} \sqrt{c}}","\frac{\sqrt{2} b \operatorname{EllipticF}\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}}{\sqrt{b^2-4 a c}+b}\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,"(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
301,1,122,62,0.1286962,"\int \frac{\left(1-2 x^2\right)^m}{\sqrt{1-x^2}} \, dx","Integrate[(1 - 2*x^2)^m/Sqrt[1 - x^2],x]","\frac{3 x \left(1-2 x^2\right)^m F_1\left(\frac{1}{2};-m,\frac{1}{2};\frac{3}{2};2 x^2,x^2\right)}{\sqrt{1-x^2} \left(x^2 \left(F_1\left(\frac{3}{2};-m,\frac{3}{2};\frac{5}{2};2 x^2,x^2\right)-4 m F_1\left(\frac{3}{2};1-m,\frac{1}{2};\frac{5}{2};2 x^2,x^2\right)\right)+3 F_1\left(\frac{1}{2};-m,\frac{1}{2};\frac{3}{2};2 x^2,x^2\right)\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,"(3*x*(1 - 2*x^2)^m*AppellF1[1/2, -m, 1/2, 3/2, 2*x^2, x^2])/(Sqrt[1 - x^2]*(3*AppellF1[1/2, -m, 1/2, 3/2, 2*x^2, x^2] + x^2*(-4*m*AppellF1[3/2, 1 - m, 1/2, 5/2, 2*x^2, x^2] + AppellF1[3/2, -m, 3/2, 5/2, 2*x^2, x^2])))","C",0
302,1,48,46,0.0846733,"\int \frac{1}{\sqrt{-1+x^2} \sqrt{7-4 \sqrt{3}+x^2}} \, dx","Integrate[1/(Sqrt[-1 + x^2]*Sqrt[7 - 4*Sqrt[3] + x^2]),x]","\frac{\sqrt{1-x^2} \operatorname{EllipticF}\left(\sin ^{-1}(x),\frac{1}{4 \sqrt{3}-7}\right)}{\sqrt{7-4 \sqrt{3}} \sqrt{x^2-1}}","\frac{\sqrt{1-x^2} \operatorname{EllipticF}\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])^(-1)])/(Sqrt[7 - 4*Sqrt[3]]*Sqrt[-1 + x^2])","A",1
303,1,81,47,0.1542862,"\int \frac{1}{\sqrt{3-3 \sqrt{3}+2 \sqrt{3} x^2} \sqrt{3+\left(-3+\sqrt{3}\right) x^2}} \, dx","Integrate[1/(Sqrt[3 - 3*Sqrt[3] + 2*Sqrt[3]*x^2]*Sqrt[3 + (-3 + Sqrt[3])*x^2]),x]","\frac{\sqrt{-2 \sqrt{3} x^2+3 \sqrt{3}-3} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{1+\sqrt{3}} x}{\sqrt[4]{3}}\right),2-\sqrt{3}\right)}{3^{3/4} \sqrt{4 \sqrt{3} x^2-6 \sqrt{3}+6}}","-\frac{1}{6} \sqrt{3+\sqrt{3}} \operatorname{EllipticF}\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 + 3*Sqrt[3] - 2*Sqrt[3]*x^2]*EllipticF[ArcSin[(Sqrt[1 + Sqrt[3]]*x)/3^(1/4)], 2 - Sqrt[3]])/(3^(3/4)*Sqrt[6 - 6*Sqrt[3] + 4*Sqrt[3]*x^2])","A",0
304,1,135,129,0.1127355,"\int \frac{1}{\sqrt[4]{2+3 x^2} \left(4+3 x^2\right)} \, dx","Integrate[1/((2 + 3*x^2)^(1/4)*(4 + 3*x^2)),x]","-\frac{4 x F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{3 x^2}{2},-\frac{3 x^2}{4}\right)}{\sqrt[4]{3 x^2+2} \left(3 x^2+4\right) \left(x^2 \left(2 F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{3 x^2}{2},-\frac{3 x^2}{4}\right)+F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{3 x^2}{2},-\frac{3 x^2}{4}\right)\right)-4 F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{3 x^2}{2},-\frac{3 x^2}{4}\right)\right)}","-\frac{\tan ^{-1}\left(\frac{2 \sqrt[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,"(-4*x*AppellF1[1/2, 1/4, 1, 3/2, (-3*x^2)/2, (-3*x^2)/4])/((2 + 3*x^2)^(1/4)*(4 + 3*x^2)*(-4*AppellF1[1/2, 1/4, 1, 3/2, (-3*x^2)/2, (-3*x^2)/4] + x^2*(2*AppellF1[3/2, 1/4, 2, 5/2, (-3*x^2)/2, (-3*x^2)/4] + AppellF1[3/2, 5/4, 1, 5/2, (-3*x^2)/2, (-3*x^2)/4])))","C",0
305,1,135,120,0.1282028,"\int \frac{1}{\sqrt[4]{2-3 x^2} \left(4-3 x^2\right)} \, dx","Integrate[1/((2 - 3*x^2)^(1/4)*(4 - 3*x^2)),x]","-\frac{4 x F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right)}{\sqrt[4]{2-3 x^2} \left(3 x^2-4\right) \left(x^2 \left(2 F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right)+F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right)\right)+4 F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right)\right)}","\frac{\tan ^{-1}\left(\frac{2-\sqrt{2} \sqrt{2-3 x^2}}{\sqrt[4]{2} \sqrt{3} x \sqrt[4]{2-3 x^2}}\right)}{2\ 2^{3/4} \sqrt{3}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{2-3 x^2}+2}{\sqrt[4]{2} \sqrt{3} x \sqrt[4]{2-3 x^2}}\right)}{2\ 2^{3/4} \sqrt{3}}",1,"(-4*x*AppellF1[1/2, 1/4, 1, 3/2, (3*x^2)/2, (3*x^2)/4])/((2 - 3*x^2)^(1/4)*(-4 + 3*x^2)*(4*AppellF1[1/2, 1/4, 1, 3/2, (3*x^2)/2, (3*x^2)/4] + x^2*(2*AppellF1[3/2, 1/4, 2, 5/2, (3*x^2)/2, (3*x^2)/4] + AppellF1[3/2, 5/4, 1, 5/2, (3*x^2)/2, (3*x^2)/4])))","C",0
306,1,144,129,0.138057,"\int \frac{1}{\sqrt[4]{2+b x^2} \left(4+b x^2\right)} \, dx","Integrate[1/((2 + b*x^2)^(1/4)*(4 + b*x^2)),x]","-\frac{12 x F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{2},-\frac{b x^2}{4}\right)}{\sqrt[4]{b x^2+2} \left(b x^2+4\right) \left(b x^2 \left(2 F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{2},-\frac{b x^2}{4}\right)+F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{2},-\frac{b x^2}{4}\right)\right)-12 F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{2},-\frac{b x^2}{4}\right)\right)}","-\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,"(-12*x*AppellF1[1/2, 1/4, 1, 3/2, -1/2*(b*x^2), -1/4*(b*x^2)])/((2 + b*x^2)^(1/4)*(4 + b*x^2)*(-12*AppellF1[1/2, 1/4, 1, 3/2, -1/2*(b*x^2), -1/4*(b*x^2)] + b*x^2*(2*AppellF1[3/2, 1/4, 2, 5/2, -1/2*(b*x^2), -1/4*(b*x^2)] + AppellF1[3/2, 5/4, 1, 5/2, -1/2*(b*x^2), -1/4*(b*x^2)])))","C",0
307,1,145,124,0.1332265,"\int \frac{1}{\sqrt[4]{2-b x^2} \left(4-b x^2\right)} \, dx","Integrate[1/((2 - b*x^2)^(1/4)*(4 - b*x^2)),x]","-\frac{12 x F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{b x^2}{2},\frac{b x^2}{4}\right)}{\sqrt[4]{2-b x^2} \left(b x^2-4\right) \left(b x^2 \left(2 F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};\frac{b x^2}{2},\frac{b x^2}{4}\right)+F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};\frac{b x^2}{2},\frac{b x^2}{4}\right)\right)+12 F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{b x^2}{2},\frac{b x^2}{4}\right)\right)}","\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,"(-12*x*AppellF1[1/2, 1/4, 1, 3/2, (b*x^2)/2, (b*x^2)/4])/((2 - b*x^2)^(1/4)*(-4 + b*x^2)*(12*AppellF1[1/2, 1/4, 1, 3/2, (b*x^2)/2, (b*x^2)/4] + b*x^2*(2*AppellF1[3/2, 1/4, 2, 5/2, (b*x^2)/2, (b*x^2)/4] + AppellF1[3/2, 5/4, 1, 5/2, (b*x^2)/2, (b*x^2)/4])))","C",0
308,1,155,120,0.1450595,"\int \frac{1}{\sqrt[4]{a+3 x^2} \left(2 a+3 x^2\right)} \, dx","Integrate[1/((a + 3*x^2)^(1/4)*(2*a + 3*x^2)),x]","-\frac{2 a x F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{3 x^2}{a},-\frac{3 x^2}{2 a}\right)}{\sqrt[4]{a+3 x^2} \left(2 a+3 x^2\right) \left(x^2 \left(2 F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{3 x^2}{a},-\frac{3 x^2}{2 a}\right)+F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{3 x^2}{a},-\frac{3 x^2}{2 a}\right)\right)-2 a F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{3 x^2}{a},-\frac{3 x^2}{2 a}\right)\right)}","-\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,"(-2*a*x*AppellF1[1/2, 1/4, 1, 3/2, (-3*x^2)/a, (-3*x^2)/(2*a)])/((a + 3*x^2)^(1/4)*(2*a + 3*x^2)*(-2*a*AppellF1[1/2, 1/4, 1, 3/2, (-3*x^2)/a, (-3*x^2)/(2*a)] + x^2*(2*AppellF1[3/2, 1/4, 2, 5/2, (-3*x^2)/a, (-3*x^2)/(2*a)] + AppellF1[3/2, 5/4, 1, 5/2, (-3*x^2)/a, (-3*x^2)/(2*a)])))","C",0
309,1,155,120,0.1618941,"\int \frac{1}{\sqrt[4]{a-3 x^2} \left(2 a-3 x^2\right)} \, dx","Integrate[1/((a - 3*x^2)^(1/4)*(2*a - 3*x^2)),x]","-\frac{2 a x F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{3 x^2}{a},\frac{3 x^2}{2 a}\right)}{\sqrt[4]{a-3 x^2} \left(3 x^2-2 a\right) \left(x^2 \left(2 F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};\frac{3 x^2}{a},\frac{3 x^2}{2 a}\right)+F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};\frac{3 x^2}{a},\frac{3 x^2}{2 a}\right)\right)+2 a F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{3 x^2}{a},\frac{3 x^2}{2 a}\right)\right)}","\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,"(-2*a*x*AppellF1[1/2, 1/4, 1, 3/2, (3*x^2)/a, (3*x^2)/(2*a)])/((a - 3*x^2)^(1/4)*(-2*a + 3*x^2)*(2*a*AppellF1[1/2, 1/4, 1, 3/2, (3*x^2)/a, (3*x^2)/(2*a)] + x^2*(2*AppellF1[3/2, 1/4, 2, 5/2, (3*x^2)/a, (3*x^2)/(2*a)] + AppellF1[3/2, 5/4, 1, 5/2, (3*x^2)/a, (3*x^2)/(2*a)])))","C",0
310,1,165,120,0.1545846,"\int \frac{1}{\sqrt[4]{a+b x^2} \left(2 a+b x^2\right)} \, dx","Integrate[1/((a + b*x^2)^(1/4)*(2*a + b*x^2)),x]","\frac{6 a x F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{b x^2}{2 a}\right)}{\sqrt[4]{a+b x^2} \left(2 a+b x^2\right) \left(6 a F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{b x^2}{2 a}\right)-b x^2 \left(2 F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{b x^2}{2 a}\right)+F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{b x^2}{2 a}\right)\right)\right)}","-\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,"(6*a*x*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -1/2*(b*x^2)/a])/((a + b*x^2)^(1/4)*(2*a + b*x^2)*(6*a*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -1/2*(b*x^2)/a] - b*x^2*(2*AppellF1[3/2, 1/4, 2, 5/2, -((b*x^2)/a), -1/2*(b*x^2)/a] + AppellF1[3/2, 5/4, 1, 5/2, -((b*x^2)/a), -1/2*(b*x^2)/a])))","C",0
311,1,162,124,0.1639253,"\int \frac{1}{\sqrt[4]{a-b x^2} \left(2 a-b x^2\right)} \, dx","Integrate[1/((a - b*x^2)^(1/4)*(2*a - b*x^2)),x]","\frac{6 a x F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{b x^2}{a},\frac{b x^2}{2 a}\right)}{\sqrt[4]{a-b x^2} \left(2 a-b x^2\right) \left(b x^2 \left(2 F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};\frac{b x^2}{a},\frac{b x^2}{2 a}\right)+F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};\frac{b x^2}{a},\frac{b x^2}{2 a}\right)\right)+6 a F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{b x^2}{a},\frac{b x^2}{2 a}\right)\right)}","\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,"(6*a*x*AppellF1[1/2, 1/4, 1, 3/2, (b*x^2)/a, (b*x^2)/(2*a)])/((a - b*x^2)^(1/4)*(2*a - b*x^2)*(6*a*AppellF1[1/2, 1/4, 1, 3/2, (b*x^2)/a, (b*x^2)/(2*a)] + b*x^2*(2*AppellF1[3/2, 1/4, 2, 5/2, (b*x^2)/a, (b*x^2)/(2*a)] + AppellF1[3/2, 5/4, 1, 5/2, (b*x^2)/a, (b*x^2)/(2*a)])))","C",0
312,1,127,61,0.1477133,"\int \frac{1}{\left(-2+3 x^2\right) \sqrt[4]{-1+3 x^2}} \, dx","Integrate[1/((-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)),x]","\frac{2 x F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};3 x^2,\frac{3 x^2}{2}\right)}{\left(3 x^2-2\right) \sqrt[4]{3 x^2-1} \left(x^2 \left(2 F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right)+F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right)\right)+2 F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};3 x^2,\frac{3 x^2}{2}\right)\right)}","-\frac{\tan ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{3 x^2-1}}\right)}{2 \sqrt{6}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{3 x^2-1}}\right)}{2 \sqrt{6}}",1,"(2*x*AppellF1[1/2, 1/4, 1, 3/2, 3*x^2, (3*x^2)/2])/((-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)*(2*AppellF1[1/2, 1/4, 1, 3/2, 3*x^2, (3*x^2)/2] + x^2*(2*AppellF1[3/2, 1/4, 2, 5/2, 3*x^2, (3*x^2)/2] + AppellF1[3/2, 5/4, 1, 5/2, 3*x^2, (3*x^2)/2])))","C",0
313,1,127,61,0.128751,"\int \frac{1}{\left(-2-3 x^2\right) \sqrt[4]{-1-3 x^2}} \, dx","Integrate[1/((-2 - 3*x^2)*(-1 - 3*x^2)^(1/4)),x]","\frac{2 x F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-3 x^2,-\frac{3 x^2}{2}\right)}{\sqrt[4]{-3 x^2-1} \left(3 x^2+2\right) \left(x^2 \left(2 F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-3 x^2,-\frac{3 x^2}{2}\right)+F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-3 x^2,-\frac{3 x^2}{2}\right)\right)-2 F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-3 x^2,-\frac{3 x^2}{2}\right)\right)}","-\frac{\tan ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{-3 x^2-1}}\right)}{2 \sqrt{6}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt[4]{-3 x^2-1}}\right)}{2 \sqrt{6}}",1,"(2*x*AppellF1[1/2, 1/4, 1, 3/2, -3*x^2, (-3*x^2)/2])/((-1 - 3*x^2)^(1/4)*(2 + 3*x^2)*(-2*AppellF1[1/2, 1/4, 1, 3/2, -3*x^2, (-3*x^2)/2] + x^2*(2*AppellF1[3/2, 1/4, 2, 5/2, -3*x^2, (-3*x^2)/2] + AppellF1[3/2, 5/4, 1, 5/2, -3*x^2, (-3*x^2)/2])))","C",0
314,1,132,77,0.1663052,"\int \frac{1}{\left(-2+b x^2\right) \sqrt[4]{-1+b x^2}} \, dx","Integrate[1/((-2 + b*x^2)*(-1 + b*x^2)^(1/4)),x]","\frac{6 x F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};b x^2,\frac{b x^2}{2}\right)}{\left(b x^2-2\right) \sqrt[4]{b x^2-1} \left(b x^2 \left(2 F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};b x^2,\frac{b x^2}{2}\right)+F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};b x^2,\frac{b x^2}{2}\right)\right)+6 F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};b x^2,\frac{b x^2}{2}\right)\right)}","-\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,"(6*x*AppellF1[1/2, 1/4, 1, 3/2, b*x^2, (b*x^2)/2])/((-2 + b*x^2)*(-1 + b*x^2)^(1/4)*(6*AppellF1[1/2, 1/4, 1, 3/2, b*x^2, (b*x^2)/2] + b*x^2*(2*AppellF1[3/2, 1/4, 2, 5/2, b*x^2, (b*x^2)/2] + AppellF1[3/2, 5/4, 1, 5/2, b*x^2, (b*x^2)/2])))","C",0
315,1,137,79,0.1477039,"\int \frac{1}{\left(-2-b x^2\right) \sqrt[4]{-1-b x^2}} \, dx","Integrate[1/((-2 - b*x^2)*(-1 - b*x^2)^(1/4)),x]","\frac{6 x F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-b x^2,-\frac{b x^2}{2}\right)}{\sqrt[4]{-b x^2-1} \left(b x^2+2\right) \left(b x^2 \left(2 F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-b x^2,-\frac{b x^2}{2}\right)+F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-b x^2,-\frac{b x^2}{2}\right)\right)-6 F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-b x^2,-\frac{b x^2}{2}\right)\right)}","-\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,"(6*x*AppellF1[1/2, 1/4, 1, 3/2, -(b*x^2), -1/2*(b*x^2)])/((-1 - b*x^2)^(1/4)*(2 + b*x^2)*(-6*AppellF1[1/2, 1/4, 1, 3/2, -(b*x^2), -1/2*(b*x^2)] + b*x^2*(2*AppellF1[3/2, 1/4, 2, 5/2, -(b*x^2), -1/2*(b*x^2)] + AppellF1[3/2, 5/4, 1, 5/2, -(b*x^2), -1/2*(b*x^2)])))","C",0
316,1,157,85,0.1712158,"\int \frac{1}{\left(-2 a+3 x^2\right) \sqrt[4]{-a+3 x^2}} \, dx","Integrate[1/((-2*a + 3*x^2)*(-a + 3*x^2)^(1/4)),x]","\frac{2 a x F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{3 x^2}{a},\frac{3 x^2}{2 a}\right)}{\left(3 x^2-2 a\right) \sqrt[4]{3 x^2-a} \left(x^2 \left(2 F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};\frac{3 x^2}{a},\frac{3 x^2}{2 a}\right)+F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};\frac{3 x^2}{a},\frac{3 x^2}{2 a}\right)\right)+2 a F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{3 x^2}{a},\frac{3 x^2}{2 a}\right)\right)}","-\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,"(2*a*x*AppellF1[1/2, 1/4, 1, 3/2, (3*x^2)/a, (3*x^2)/(2*a)])/((-2*a + 3*x^2)*(-a + 3*x^2)^(1/4)*(2*a*AppellF1[1/2, 1/4, 1, 3/2, (3*x^2)/a, (3*x^2)/(2*a)] + x^2*(2*AppellF1[3/2, 1/4, 2, 5/2, (3*x^2)/a, (3*x^2)/(2*a)] + AppellF1[3/2, 5/4, 1, 5/2, (3*x^2)/a, (3*x^2)/(2*a)])))","C",0
317,1,157,85,0.1479893,"\int \frac{1}{\left(-2 a-3 x^2\right) \sqrt[4]{-a-3 x^2}} \, dx","Integrate[1/((-2*a - 3*x^2)*(-a - 3*x^2)^(1/4)),x]","\frac{2 a x F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{3 x^2}{a},-\frac{3 x^2}{2 a}\right)}{\sqrt[4]{-a-3 x^2} \left(2 a+3 x^2\right) \left(x^2 \left(2 F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{3 x^2}{a},-\frac{3 x^2}{2 a}\right)+F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{3 x^2}{a},-\frac{3 x^2}{2 a}\right)\right)-2 a F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{3 x^2}{a},-\frac{3 x^2}{2 a}\right)\right)}","-\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,"(2*a*x*AppellF1[1/2, 1/4, 1, 3/2, (-3*x^2)/a, (-3*x^2)/(2*a)])/((-a - 3*x^2)^(1/4)*(2*a + 3*x^2)*(-2*a*AppellF1[1/2, 1/4, 1, 3/2, (-3*x^2)/a, (-3*x^2)/(2*a)] + x^2*(2*AppellF1[3/2, 1/4, 2, 5/2, (-3*x^2)/a, (-3*x^2)/(2*a)] + AppellF1[3/2, 5/4, 1, 5/2, (-3*x^2)/a, (-3*x^2)/(2*a)])))","C",0
318,1,163,101,0.1643132,"\int \frac{1}{\left(-2 a+b x^2\right) \sqrt[4]{-a+b x^2}} \, dx","Integrate[1/((-2*a + b*x^2)*(-a + b*x^2)^(1/4)),x]","-\frac{6 a x F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{b x^2}{a},\frac{b x^2}{2 a}\right)}{\left(2 a-b x^2\right) \sqrt[4]{b x^2-a} \left(b x^2 \left(2 F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};\frac{b x^2}{a},\frac{b x^2}{2 a}\right)+F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};\frac{b x^2}{a},\frac{b x^2}{2 a}\right)\right)+6 a F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{b x^2}{a},\frac{b x^2}{2 a}\right)\right)}","-\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,"(-6*a*x*AppellF1[1/2, 1/4, 1, 3/2, (b*x^2)/a, (b*x^2)/(2*a)])/((2*a - b*x^2)*(-a + b*x^2)^(1/4)*(6*a*AppellF1[1/2, 1/4, 1, 3/2, (b*x^2)/a, (b*x^2)/(2*a)] + b*x^2*(2*AppellF1[3/2, 1/4, 2, 5/2, (b*x^2)/a, (b*x^2)/(2*a)] + AppellF1[3/2, 5/4, 1, 5/2, (b*x^2)/a, (b*x^2)/(2*a)])))","C",0
319,1,168,103,0.1627156,"\int \frac{1}{\left(-2 a-b x^2\right) \sqrt[4]{-a-b x^2}} \, dx","Integrate[1/((-2*a - b*x^2)*(-a - b*x^2)^(1/4)),x]","-\frac{6 a x F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{b x^2}{2 a}\right)}{\sqrt[4]{-a-b x^2} \left(2 a+b x^2\right) \left(6 a F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{b x^2}{2 a}\right)-b x^2 \left(2 F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{b x^2}{2 a}\right)+F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{b x^2}{2 a}\right)\right)\right)}","-\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,"(-6*a*x*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -1/2*(b*x^2)/a])/((-a - b*x^2)^(1/4)*(2*a + b*x^2)*(6*a*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -1/2*(b*x^2)/a] - b*x^2*(2*AppellF1[3/2, 1/4, 2, 5/2, -((b*x^2)/a), -1/2*(b*x^2)/a] + AppellF1[3/2, 5/4, 1, 5/2, -((b*x^2)/a), -1/2*(b*x^2)/a])))","C",0
320,1,115,53,0.1406446,"\int \frac{1}{\left(2-x^2\right) \sqrt[4]{-1+x^2}} \, dx","Integrate[1/((2 - x^2)*(-1 + x^2)^(1/4)),x]","-\frac{6 x F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};x^2,\frac{x^2}{2}\right)}{\left(x^2-2\right) \sqrt[4]{x^2-1} \left(x^2 \left(2 F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};x^2,\frac{x^2}{2}\right)+F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};x^2,\frac{x^2}{2}\right)\right)+6 F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};x^2,\frac{x^2}{2}\right)\right)}","\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,"(-6*x*AppellF1[1/2, 1/4, 1, 3/2, x^2, x^2/2])/((-2 + x^2)*(-1 + x^2)^(1/4)*(6*AppellF1[1/2, 1/4, 1, 3/2, x^2, x^2/2] + x^2*(2*AppellF1[3/2, 1/4, 2, 5/2, x^2, x^2/2] + AppellF1[3/2, 5/4, 1, 5/2, x^2, x^2/2])))","C",0
321,1,346,362,0.4610175,"\int \frac{\left(a+b x^2\right)^{7/4}}{c+d x^2} \, dx","Integrate[(a + b*x^2)^(7/4)/(c + d*x^2),x]","\frac{x \left(\frac{6 \left(b x^2 \left(a+b x^2\right) \left(c+d x^2\right) \left(4 a d F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+b c F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)-3 a c \left(5 a^2 d+2 a b d x^2+2 b^2 x^2 \left(c+d x^2\right)\right) F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)}{\left(c+d x^2\right) \left(x^2 \left(4 a d F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+b c F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)-6 a c F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)}+\frac{b x^2 \sqrt[4]{\frac{b x^2}{a}+1} (8 a d-5 b c) F_1\left(\frac{3}{2};\frac{1}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{c}\right)}{15 d \sqrt[4]{a+b x^2}}","-\frac{6 a^{3/2} \sqrt{b} \sqrt[4]{\frac{b x^2}{a}+1} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{5 d \sqrt[4]{a+b x^2}}+\frac{\sqrt[4]{a} \sqrt{-\frac{b x^2}{a}} (a d-b c)^{3/2} \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{d^{5/2} x}-\frac{\sqrt[4]{a} \sqrt{-\frac{b x^2}{a}} (a d-b c)^{3/2} \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{d^{5/2} x}-\frac{2 b x (b c-a d)}{d^2 \sqrt[4]{a+b x^2}}+\frac{2 \sqrt{a} \sqrt{b} \sqrt[4]{\frac{b x^2}{a}+1} (b c-a d) E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{d^2 \sqrt[4]{a+b x^2}}+\frac{6 a b x}{5 d \sqrt[4]{a+b x^2}}+\frac{2 b x \left(a+b x^2\right)^{3/4}}{5 d}",1,"(x*((b*(-5*b*c + 8*a*d)*x^2*(1 + (b*x^2)/a)^(1/4)*AppellF1[3/2, 1/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])/c + (6*(-3*a*c*(5*a^2*d + 2*a*b*d*x^2 + 2*b^2*x^2*(c + d*x^2))*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + b*x^2*(a + b*x^2)*(c + d*x^2)*(4*a*d*AppellF1[3/2, 1/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + b*c*AppellF1[3/2, 5/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))/((c + d*x^2)*(-6*a*c*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + x^2*(4*a*d*AppellF1[3/2, 1/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + b*c*AppellF1[3/2, 5/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))))/(15*d*(a + b*x^2)^(1/4))","C",0
322,1,348,302,0.4713353,"\int \frac{\left(a+b x^2\right)^{5/4}}{c+d x^2} \, dx","Integrate[(a + b*x^2)^(5/4)/(c + d*x^2),x]","\frac{x \left(\frac{6 \left(b x^2 \left(a+b x^2\right) \left(c+d x^2\right) \left(4 a d F_1\left(\frac{3}{2};\frac{3}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+3 b c F_1\left(\frac{3}{2};\frac{7}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)-3 a c \left(3 a^2 d+2 a b d x^2+2 b^2 x^2 \left(c+d x^2\right)\right) F_1\left(\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)}{\left(c+d x^2\right) \left(x^2 \left(4 a d F_1\left(\frac{3}{2};\frac{3}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+3 b c F_1\left(\frac{3}{2};\frac{7}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)-6 a c F_1\left(\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)}+\frac{b x^2 \left(\frac{b x^2}{a}+1\right)^{3/4} (4 a d-3 b c) F_1\left(\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{c}\right)}{9 d \left(a+b x^2\right)^{3/4}}","\frac{2 a^{3/2} \sqrt{b} \left(\frac{b x^2}{a}+1\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\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) \operatorname{EllipticF}\left(\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\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,"(x*((b*(-3*b*c + 4*a*d)*x^2*(1 + (b*x^2)/a)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])/c + (6*(-3*a*c*(3*a^2*d + 2*a*b*d*x^2 + 2*b^2*x^2*(c + d*x^2))*AppellF1[1/2, 3/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + b*x^2*(a + b*x^2)*(c + d*x^2)*(4*a*d*AppellF1[3/2, 3/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + 3*b*c*AppellF1[3/2, 7/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))/((c + d*x^2)*(-6*a*c*AppellF1[1/2, 3/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + x^2*(4*a*d*AppellF1[3/2, 3/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + 3*b*c*AppellF1[3/2, 7/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))))/(9*d*(a + b*x^2)^(3/4))","C",0
323,1,161,244,0.1628922,"\int \frac{\left(a+b x^2\right)^{3/4}}{c+d x^2} \, dx","Integrate[(a + b*x^2)^(3/4)/(c + d*x^2),x]","\frac{6 a c x \left(a+b x^2\right)^{3/4} F_1\left(\frac{1}{2};-\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{\left(c+d x^2\right) \left(x^2 \left(3 b c F_1\left(\frac{3}{2};\frac{1}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)-4 a d F_1\left(\frac{3}{2};-\frac{3}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)+6 a c F_1\left(\frac{1}{2};-\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)}","\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,"(6*a*c*x*(a + b*x^2)^(3/4)*AppellF1[1/2, -3/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)])/((c + d*x^2)*(6*a*c*AppellF1[1/2, -3/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + x^2*(-4*a*d*AppellF1[3/2, -3/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + 3*b*c*AppellF1[3/2, 1/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))","C",0
324,1,160,199,0.158286,"\int \frac{\sqrt[4]{a+b x^2}}{c+d x^2} \, dx","Integrate[(a + b*x^2)^(1/4)/(c + d*x^2),x]","\frac{6 a c x \sqrt[4]{a+b x^2} F_1\left(\frac{1}{2};-\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{\left(c+d x^2\right) \left(x^2 \left(b c F_1\left(\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)-4 a d F_1\left(\frac{3}{2};-\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)+6 a c F_1\left(\frac{1}{2};-\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)}","\frac{2 \sqrt{a} \sqrt{b} \left(\frac{b x^2}{a}+1\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\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}",1,"(6*a*c*x*(a + b*x^2)^(1/4)*AppellF1[1/2, -1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)])/((c + d*x^2)*(6*a*c*AppellF1[1/2, -1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + x^2*(-4*a*d*AppellF1[3/2, -1/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + b*c*AppellF1[3/2, 3/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))","C",0
325,1,160,167,0.055544,"\int \frac{1}{\sqrt[4]{a+b x^2} \left(c+d x^2\right)} \, dx","Integrate[1/((a + b*x^2)^(1/4)*(c + d*x^2)),x]","-\frac{6 a c x F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{\sqrt[4]{a+b x^2} \left(c+d x^2\right) \left(x^2 \left(4 a d F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+b c F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)-6 a c F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)}","\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,"(-6*a*c*x*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)])/((a + b*x^2)^(1/4)*(c + d*x^2)*(-6*a*c*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + x^2*(4*a*d*AppellF1[3/2, 1/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + b*c*AppellF1[3/2, 5/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))","C",0
326,1,120,152,0.031789,"\int \frac{1}{\left(a+b x^2\right)^{3/4} \left(c+d x^2\right)} \, dx","Integrate[1/((a + b*x^2)^(3/4)*(c + d*x^2)),x]","\frac{\sqrt[4]{a} \sqrt{-\frac{b x^2}{a}} \left(\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)+\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)\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] + 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
327,1,327,233,0.2328876,"\int \frac{1}{\left(a+b x^2\right)^{5/4} \left(c+d x^2\right)} \, dx","Integrate[1/((a + b*x^2)^(5/4)*(c + d*x^2)),x]","\frac{x \left(\frac{b d x^2 \sqrt[4]{\frac{b x^2}{a}+1} F_1\left(\frac{3}{2};\frac{1}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{c}+\frac{6 \left(b x^2 \left(c+d x^2\right) \left(4 a d F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+b c F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)+3 a c \left(a d-b \left(c+2 d x^2\right)\right) F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)}{\left(c+d x^2\right) \left(6 a c F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)-x^2 \left(4 a d F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+b c F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)\right)}\right)}{3 a \sqrt[4]{a+b x^2} (a d-b c)}","\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,"(x*((b*d*x^2*(1 + (b*x^2)/a)^(1/4)*AppellF1[3/2, 1/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])/c + (6*(3*a*c*(a*d - b*(c + 2*d*x^2))*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + b*x^2*(c + d*x^2)*(4*a*d*AppellF1[3/2, 1/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + b*c*AppellF1[3/2, 5/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))/((c + d*x^2)*(6*a*c*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] - x^2*(4*a*d*AppellF1[3/2, 1/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + b*c*AppellF1[3/2, 5/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))))/(3*a*(-(b*c) + a*d)*(a + b*x^2)^(1/4))","C",0
328,1,331,254,0.2617548,"\int \frac{1}{\left(a+b x^2\right)^{7/4} \left(c+d x^2\right)} \, dx","Integrate[1/((a + b*x^2)^(7/4)*(c + d*x^2)),x]","\frac{x \left(\frac{6 \left(b x^2 \left(c+d x^2\right) \left(4 a d F_1\left(\frac{3}{2};\frac{3}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+3 b c F_1\left(\frac{3}{2};\frac{7}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)+3 a c \left(3 a d-3 b c-2 b d x^2\right) F_1\left(\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)}{\left(c+d x^2\right) \left(6 a c F_1\left(\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)-x^2 \left(4 a d F_1\left(\frac{3}{2};\frac{3}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+3 b c F_1\left(\frac{3}{2};\frac{7}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)\right)}-\frac{b d x^2 \left(\frac{b x^2}{a}+1\right)^{3/4} F_1\left(\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{c}\right)}{9 a \left(a+b x^2\right)^{3/4} (a d-b c)}","\frac{2 \sqrt{b} \left(\frac{b x^2}{a}+1\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{3 \sqrt{a} \left(a+b x^2\right)^{3/4} (b c-a d)}+\frac{2 b x}{3 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,"(x*(-((b*d*x^2*(1 + (b*x^2)/a)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])/c) + (6*(3*a*c*(-3*b*c + 3*a*d - 2*b*d*x^2)*AppellF1[1/2, 3/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + b*x^2*(c + d*x^2)*(4*a*d*AppellF1[3/2, 3/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + 3*b*c*AppellF1[3/2, 7/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))/((c + d*x^2)*(6*a*c*AppellF1[1/2, 3/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] - x^2*(4*a*d*AppellF1[3/2, 3/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + 3*b*c*AppellF1[3/2, 7/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))))/(9*a*(-(b*c) + a*d)*(a + b*x^2)^(3/4))","C",0
329,1,419,274,0.6603756,"\int \frac{1}{\left(a+b x^2\right)^{9/4} \left(c+d x^2\right)} \, dx","Integrate[1/((a + b*x^2)^(9/4)*(c + d*x^2)),x]","\frac{x \left(\frac{b d x^2 \sqrt[4]{\frac{b x^2}{a}+1} (8 a d-3 b c) F_1\left(\frac{3}{2};\frac{1}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{c}-\frac{6 \left(b x^2 \left(c+d x^2\right) \left(9 a^2 d-4 a b \left(c-2 d x^2\right)-3 b^2 c x^2\right) \left(4 a d F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+b c F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)+3 a c \left(5 a^3 d^2-a^2 b d \left(10 c+13 d x^2\right)+a b^2 \left(5 c^2-16 d^2 x^4\right)+3 b^3 c x^2 \left(c+2 d x^2\right)\right) F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)}{\left(a+b x^2\right) \left(c+d x^2\right) \left(x^2 \left(4 a d F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+b c F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)-6 a c F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)}\right)}{15 a^2 \sqrt[4]{a+b x^2} (b c-a d)^2}","\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,"(x*((b*d*(-3*b*c + 8*a*d)*x^2*(1 + (b*x^2)/a)^(1/4)*AppellF1[3/2, 1/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])/c - (6*(3*a*c*(5*a^3*d^2 + 3*b^3*c*x^2*(c + 2*d*x^2) - a^2*b*d*(10*c + 13*d*x^2) + a*b^2*(5*c^2 - 16*d^2*x^4))*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + b*x^2*(c + d*x^2)*(9*a^2*d - 3*b^2*c*x^2 - 4*a*b*(c - 2*d*x^2))*(4*a*d*AppellF1[3/2, 1/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + b*c*AppellF1[3/2, 5/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))/((a + b*x^2)*(c + d*x^2)*(-6*a*c*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + x^2*(4*a*d*AppellF1[3/2, 1/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + b*c*AppellF1[3/2, 5/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))))/(15*a^2*(b*c - a*d)^2*(a + b*x^2)^(1/4))","C",0
330,1,431,304,0.7761562,"\int \frac{1}{\left(a+b x^2\right)^{11/4} \left(c+d x^2\right)} \, dx","Integrate[1/((a + b*x^2)^(11/4)*(c + d*x^2)),x]","-\frac{x \left(\frac{6 \left(b x^2 \left(c+d x^2\right) \left(15 a^2 d+a b \left(12 d x^2-8 c\right)-5 b^2 c x^2\right) \left(4 a d F_1\left(\frac{3}{2};\frac{3}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+3 b c F_1\left(\frac{3}{2};\frac{7}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)+3 a c \left(21 a^3 d^2-3 a^2 b d \left(14 c+3 d x^2\right)+a b^2 \left(21 c^2-20 c d x^2-24 d^2 x^4\right)+5 b^3 c x^2 \left(3 c+2 d x^2\right)\right) F_1\left(\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)}{\left(a+b x^2\right) \left(c+d x^2\right) \left(x^2 \left(4 a d F_1\left(\frac{3}{2};\frac{3}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+3 b c F_1\left(\frac{3}{2};\frac{7}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)-6 a c F_1\left(\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)}+\frac{b d x^2 \left(\frac{b x^2}{a}+1\right)^{3/4} (12 a d-5 b c) F_1\left(\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{c}\right)}{63 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) \operatorname{EllipticF}\left(\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{21 a^{3/2} \left(a+b x^2\right)^{3/4} (b c-a d)^2}+\frac{2 b x (5 b c-12 a d)}{21 a^2 \left(a+b x^2\right)^{3/4} (b c-a d)^2}+\frac{\sqrt[4]{a} d^2 \sqrt{-\frac{b x^2}{a}} \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{x (b c-a d)^3}+\frac{\sqrt[4]{a} d^2 \sqrt{-\frac{b x^2}{a}} \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b x^2+a}}{\sqrt[4]{a}}\right)\right|-1\right)}{x (b c-a d)^3}+\frac{2 b x}{7 a \left(a+b x^2\right)^{7/4} (b c-a d)}",1,"-1/63*(x*((b*d*(-5*b*c + 12*a*d)*x^2*(1 + (b*x^2)/a)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])/c + (6*(3*a*c*(21*a^3*d^2 + 5*b^3*c*x^2*(3*c + 2*d*x^2) - 3*a^2*b*d*(14*c + 3*d*x^2) + a*b^2*(21*c^2 - 20*c*d*x^2 - 24*d^2*x^4))*AppellF1[1/2, 3/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + b*x^2*(c + d*x^2)*(15*a^2*d - 5*b^2*c*x^2 + a*b*(-8*c + 12*d*x^2))*(4*a*d*AppellF1[3/2, 3/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + 3*b*c*AppellF1[3/2, 7/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))/((a + b*x^2)*(c + d*x^2)*(-6*a*c*AppellF1[1/2, 3/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + x^2*(4*a*d*AppellF1[3/2, 3/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + 3*b*c*AppellF1[3/2, 7/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))))/(a^2*(b*c - a*d)^2*(a + b*x^2)^(3/4))","C",0
331,1,340,340,0.3381756,"\int \frac{\left(a+b x^2\right)^{7/4}}{\left(c+d x^2\right)^2} \, dx","Integrate[(a + b*x^2)^(7/4)/(c + d*x^2)^2,x]","\frac{x \left(\frac{6 c \left(x^2 \left(a+b x^2\right) (a d-b c) \left(4 a d F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+b c F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)-6 a c \left(2 a^2 d+a b d x^2-b^2 c x^2\right) F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)}{\left(c+d x^2\right) \left(x^2 \left(4 a d F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+b c F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)-6 a c F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)}-b x^2 \sqrt[4]{\frac{b x^2}{a}+1} (a d-5 b c) F_1\left(\frac{3}{2};\frac{1}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)}{12 c^2 d \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{b x (5 b c-a d)}{2 c d^2 \sqrt[4]{a+b x^2}}-\frac{\sqrt{a} \sqrt{b} \sqrt[4]{\frac{b x^2}{a}+1} (5 b c-a d) E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{2 c d^2 \sqrt[4]{a+b x^2}}-\frac{x \left(a+b x^2\right)^{3/4} (b c-a d)}{2 c d \left(c+d x^2\right)}",1,"(x*(-(b*(-5*b*c + a*d)*x^2*(1 + (b*x^2)/a)^(1/4)*AppellF1[3/2, 1/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)]) + (6*c*(-6*a*c*(2*a^2*d - b^2*c*x^2 + a*b*d*x^2)*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + (-(b*c) + a*d)*x^2*(a + b*x^2)*(4*a*d*AppellF1[3/2, 1/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + b*c*AppellF1[3/2, 5/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))/((c + d*x^2)*(-6*a*c*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + x^2*(4*a*d*AppellF1[3/2, 1/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + b*c*AppellF1[3/2, 5/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))))/(12*c^2*d*(a + b*x^2)^(1/4))","C",0
332,1,341,279,0.3319052,"\int \frac{\left(a+b x^2\right)^{5/4}}{\left(c+d x^2\right)^2} \, dx","Integrate[(a + b*x^2)^(5/4)/(c + d*x^2)^2,x]","\frac{x \left(\frac{6 c \left(x^2 \left(a+b x^2\right) (a d-b c) \left(4 a d F_1\left(\frac{3}{2};\frac{3}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+3 b c F_1\left(\frac{3}{2};\frac{7}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)-6 a c \left(2 a^2 d+a b d x^2-b^2 c x^2\right) F_1\left(\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)}{\left(c+d x^2\right) \left(x^2 \left(4 a d F_1\left(\frac{3}{2};\frac{3}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+3 b c F_1\left(\frac{3}{2};\frac{7}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)-6 a c F_1\left(\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)}+b x^2 \left(\frac{b x^2}{a}+1\right)^{3/4} (a d+3 b c) F_1\left(\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)}{12 c^2 d \left(a+b x^2\right)^{3/4}}","\frac{\sqrt{a} \sqrt{b} \left(\frac{b x^2}{a}+1\right)^{3/4} (a d+3 b c) \operatorname{EllipticF}\left(\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\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,"(x*(b*(3*b*c + a*d)*x^2*(1 + (b*x^2)/a)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + (6*c*(-6*a*c*(2*a^2*d - b^2*c*x^2 + a*b*d*x^2)*AppellF1[1/2, 3/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + (-(b*c) + a*d)*x^2*(a + b*x^2)*(4*a*d*AppellF1[3/2, 3/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + 3*b*c*AppellF1[3/2, 7/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))/((c + d*x^2)*(-6*a*c*AppellF1[1/2, 3/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + x^2*(4*a*d*AppellF1[3/2, 3/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + 3*b*c*AppellF1[3/2, 7/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))))/(12*c^2*d*(a + b*x^2)^(3/4))","C",0
333,1,232,309,0.2027476,"\int \frac{\left(a+b x^2\right)^{3/4}}{\left(c+d x^2\right)^2} \, dx","Integrate[(a + b*x^2)^(3/4)/(c + d*x^2)^2,x]","\frac{x \left(\frac{6 \left(\frac{a+b x^2}{c}-\frac{6 a^2 F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{x^2 \left(4 a d F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+b c F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)-6 a c F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}\right)}{c+d x^2}-\frac{b x^2 \sqrt[4]{\frac{b x^2}{a}+1} F_1\left(\frac{3}{2};\frac{1}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{c^2}\right)}{12 \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,"(x*(-((b*x^2*(1 + (b*x^2)/a)^(1/4)*AppellF1[3/2, 1/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])/c^2) + (6*((a + b*x^2)/c - (6*a^2*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)])/(-6*a*c*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + x^2*(4*a*d*AppellF1[3/2, 1/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + b*c*AppellF1[3/2, 5/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)]))))/(c + d*x^2)))/(12*(a + b*x^2)^(1/4))","C",0
334,1,232,278,0.1890367,"\int \frac{\sqrt[4]{a+b x^2}}{\left(c+d x^2\right)^2} \, dx","Integrate[(a + b*x^2)^(1/4)/(c + d*x^2)^2,x]","\frac{x \left(\frac{6 \left(\frac{a+b x^2}{c}-\frac{6 a^2 F_1\left(\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{x^2 \left(4 a d F_1\left(\frac{3}{2};\frac{3}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+3 b c F_1\left(\frac{3}{2};\frac{7}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)-6 a c F_1\left(\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}\right)}{c+d x^2}+\frac{b x^2 \left(\frac{b x^2}{a}+1\right)^{3/4} F_1\left(\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{c^2}\right)}{12 \left(a+b x^2\right)^{3/4}}","\frac{\sqrt{a} \sqrt{b} \left(\frac{b x^2}{a}+1\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{2 c d \left(a+b x^2\right)^{3/4}}+\frac{x \sqrt[4]{a+b x^2}}{2 c \left(c+d x^2\right)}-\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*((b*x^2*(1 + (b*x^2)/a)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])/c^2 + (6*((a + b*x^2)/c - (6*a^2*AppellF1[1/2, 3/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)])/(-6*a*c*AppellF1[1/2, 3/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + x^2*(4*a*d*AppellF1[3/2, 3/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + 3*b*c*AppellF1[3/2, 7/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)]))))/(c + d*x^2)))/(12*(a + b*x^2)^(3/4))","C",0
335,1,392,336,0.2526331,"\int \frac{1}{\sqrt[4]{a+b x^2} \left(c+d x^2\right)^2} \, dx","Integrate[1/((a + b*x^2)^(1/4)*(c + d*x^2)^2),x]","\frac{-6 a c x F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right) \left(b d x^2 \sqrt[4]{\frac{b x^2}{a}+1} \left(c+d x^2\right) F_1\left(\frac{3}{2};\frac{1}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)-6 c \left(2 a d-2 b c+b d x^2\right)\right)-d x^3 \left(6 c \left(a+b x^2\right)-b x^2 \sqrt[4]{\frac{b x^2}{a}+1} \left(c+d x^2\right) F_1\left(\frac{3}{2};\frac{1}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right) \left(4 a d F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+b c F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)}{12 c^2 \sqrt[4]{a+b x^2} \left(c+d x^2\right) (b c-a d) \left(x^2 \left(4 a d F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+b c F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)-6 a c F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)}","\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,"(-6*a*c*x*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)]*(-6*c*(-2*b*c + 2*a*d + b*d*x^2) + b*d*x^2*(1 + (b*x^2)/a)^(1/4)*(c + d*x^2)*AppellF1[3/2, 1/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)]) - d*x^3*(6*c*(a + b*x^2) - b*x^2*(1 + (b*x^2)/a)^(1/4)*(c + d*x^2)*AppellF1[3/2, 1/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])*(4*a*d*AppellF1[3/2, 1/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + b*c*AppellF1[3/2, 5/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)]))/(12*c^2*(b*c - a*d)*(a + b*x^2)^(1/4)*(c + d*x^2)*(-6*a*c*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + x^2*(4*a*d*AppellF1[3/2, 1/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + b*c*AppellF1[3/2, 5/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))","C",0
336,1,336,292,0.320139,"\int \frac{1}{\left(a+b x^2\right)^{3/4} \left(c+d x^2\right)^2} \, dx","Integrate[1/((a + b*x^2)^(3/4)*(c + d*x^2)^2),x]","\frac{x \left(\frac{b d x^2 \left(\frac{b x^2}{a}+1\right)^{3/4} F_1\left(\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{a d-b c}+\frac{c \left(36 a c \left(2 a d-2 b c+b d x^2\right) F_1\left(\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)-6 d x^2 \left(a+b x^2\right) \left(4 a d F_1\left(\frac{3}{2};\frac{3}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+3 b c F_1\left(\frac{3}{2};\frac{7}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)\right)}{\left(c+d x^2\right) (b c-a d) \left(x^2 \left(4 a d F_1\left(\frac{3}{2};\frac{3}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+3 b c F_1\left(\frac{3}{2};\frac{7}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)-6 a c F_1\left(\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)}\right)}{12 c^2 \left(a+b x^2\right)^{3/4}}","-\frac{\sqrt{a} \sqrt{b} \left(\frac{b x^2}{a}+1\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{2 c \left(a+b x^2\right)^{3/4} (b c-a d)}-\frac{d x \sqrt[4]{a+b x^2}}{2 c \left(c+d x^2\right) (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,"(x*((b*d*x^2*(1 + (b*x^2)/a)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])/(-(b*c) + a*d) + (c*(36*a*c*(-2*b*c + 2*a*d + b*d*x^2)*AppellF1[1/2, 3/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] - 6*d*x^2*(a + b*x^2)*(4*a*d*AppellF1[3/2, 3/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + 3*b*c*AppellF1[3/2, 7/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))/((b*c - a*d)*(c + d*x^2)*(-6*a*c*AppellF1[1/2, 3/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + x^2*(4*a*d*AppellF1[3/2, 3/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + 3*b*c*AppellF1[3/2, 7/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))))/(12*c^2*(a + b*x^2)^(3/4))","C",0
337,1,380,314,0.4907517,"\int \frac{1}{\left(a+b x^2\right)^{5/4} \left(c+d x^2\right)^2} \, dx","Integrate[1/((a + b*x^2)^(5/4)*(c + d*x^2)^2),x]","\frac{x \left(\frac{c \left(36 a c \left(2 a^2 d^2+a b d \left(d x^2-4 c\right)+2 b^2 c \left(c+2 d x^2\right)\right) F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)-6 x^2 \left(a^2 d^2+a b d^2 x^2+4 b^2 c \left(c+d x^2\right)\right) \left(4 a d F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+b c F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)\right)}{\left(c+d x^2\right) \left(6 a c F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)-x^2 \left(4 a d F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+b c F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)\right)}-b d x^2 \sqrt[4]{\frac{b x^2}{a}+1} (a d+4 b c) F_1\left(\frac{3}{2};\frac{1}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)}{12 a c^2 \sqrt[4]{a+b x^2} (b c-a d)^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,"(x*(-(b*d*(4*b*c + a*d)*x^2*(1 + (b*x^2)/a)^(1/4)*AppellF1[3/2, 1/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)]) + (c*(36*a*c*(2*a^2*d^2 + a*b*d*(-4*c + d*x^2) + 2*b^2*c*(c + 2*d*x^2))*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] - 6*x^2*(a^2*d^2 + a*b*d^2*x^2 + 4*b^2*c*(c + d*x^2))*(4*a*d*AppellF1[3/2, 1/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + b*c*AppellF1[3/2, 5/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))/((c + d*x^2)*(6*a*c*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] - x^2*(4*a*d*AppellF1[3/2, 1/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + b*c*AppellF1[3/2, 5/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))))/(12*a*c^2*(b*c - a*d)^2*(a + b*x^2)^(1/4))","C",0
338,1,387,345,0.51233,"\int \frac{1}{\left(a+b x^2\right)^{7/4} \left(c+d x^2\right)^2} \, dx","Integrate[1/((a + b*x^2)^(7/4)*(c + d*x^2)^2),x]","\frac{x \left(\frac{c \left(36 a c \left(6 a^2 d^2+3 a b d \left(d x^2-4 c\right)+2 b^2 c \left(3 c+2 d x^2\right)\right) F_1\left(\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)-6 x^2 \left(3 a^2 d^2+3 a b d^2 x^2+4 b^2 c \left(c+d x^2\right)\right) \left(4 a d F_1\left(\frac{3}{2};\frac{3}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+3 b c F_1\left(\frac{3}{2};\frac{7}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)\right)}{\left(c+d x^2\right) \left(6 a c F_1\left(\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)-x^2 \left(4 a d F_1\left(\frac{3}{2};\frac{3}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+3 b c F_1\left(\frac{3}{2};\frac{7}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)\right)}+b d x^2 \left(\frac{b x^2}{a}+1\right)^{3/4} (3 a d+4 b c) F_1\left(\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)}{36 a c^2 \left(a+b x^2\right)^{3/4} (b c-a d)^2}","\frac{\sqrt{b} \left(\frac{b x^2}{a}+1\right)^{3/4} (3 a d+4 b c) \operatorname{EllipticF}\left(\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{6 \sqrt{a} c \left(a+b x^2\right)^{3/4} (b c-a d)^2}+\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[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,"(x*(b*d*(4*b*c + 3*a*d)*x^2*(1 + (b*x^2)/a)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + (c*(36*a*c*(6*a^2*d^2 + 3*a*b*d*(-4*c + d*x^2) + 2*b^2*c*(3*c + 2*d*x^2))*AppellF1[1/2, 3/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] - 6*x^2*(3*a^2*d^2 + 3*a*b*d^2*x^2 + 4*b^2*c*(c + d*x^2))*(4*a*d*AppellF1[3/2, 3/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + 3*b*c*AppellF1[3/2, 7/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))/((c + d*x^2)*(6*a*c*AppellF1[1/2, 3/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] - x^2*(4*a*d*AppellF1[3/2, 3/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + 3*b*c*AppellF1[3/2, 7/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))))/(36*a*c^2*(b*c - a*d)^2*(a + b*x^2)^(3/4))","C",0
339,1,536,371,1.0476955,"\int \frac{1}{\left(a+b x^2\right)^{9/4} \left(c+d x^2\right)^2} \, dx","Integrate[1/((a + b*x^2)^(9/4)*(c + d*x^2)^2),x]","\frac{b d x^3 \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) F_1\left(\frac{3}{2};\frac{1}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+\frac{6 c \left(x^3 \left(5 a^4 d^3+10 a^3 b d^3 x^2+a^2 b^2 d \left(56 c^2+56 c d x^2+5 d^2 x^4\right)+4 a b^3 c \left(-4 c^2+9 c d x^2+13 d^2 x^4\right)-12 b^4 c^2 x^2 \left(c+d x^2\right)\right) \left(4 a d F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+b c F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)-6 a c x \left(10 a^4 d^3+15 a^3 b d^2 \left(d x^2-2 c\right)+a^2 b^2 d \left(30 c^2+26 c d x^2+5 d^2 x^4\right)+2 a b^3 c \left(-5 c^2+5 c d x^2+26 d^2 x^4\right)-6 b^4 c^2 x^2 \left(c+2 d x^2\right)\right) F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)}{\left(a+b x^2\right) \left(c+d x^2\right) \left(6 a c F_1\left(\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)-x^2 \left(4 a d F_1\left(\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+b c F_1\left(\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)\right)}}{60 a^2 c^2 \sqrt[4]{a+b x^2} (b c-a d)^3}","\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*d*(-12*b^2*c^2 + 52*a*b*c*d + 5*a^2*d^2)*x^3*(1 + (b*x^2)/a)^(1/4)*AppellF1[3/2, 1/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + (6*c*(-6*a*c*x*(10*a^4*d^3 + 15*a^3*b*d^2*(-2*c + d*x^2) - 6*b^4*c^2*x^2*(c + 2*d*x^2) + a^2*b^2*d*(30*c^2 + 26*c*d*x^2 + 5*d^2*x^4) + 2*a*b^3*c*(-5*c^2 + 5*c*d*x^2 + 26*d^2*x^4))*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + x^3*(5*a^4*d^3 + 10*a^3*b*d^3*x^2 - 12*b^4*c^2*x^2*(c + d*x^2) + a^2*b^2*d*(56*c^2 + 56*c*d*x^2 + 5*d^2*x^4) + 4*a*b^3*c*(-4*c^2 + 9*c*d*x^2 + 13*d^2*x^4))*(4*a*d*AppellF1[3/2, 1/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + b*c*AppellF1[3/2, 5/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))/((a + b*x^2)*(c + d*x^2)*(6*a*c*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] - x^2*(4*a*d*AppellF1[3/2, 1/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + b*c*AppellF1[3/2, 5/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)]))))/(60*a^2*c^2*(b*c - a*d)^3*(a + b*x^2)^(1/4))","C",0
340,1,550,419,0.9202578,"\int \frac{1}{\left(a+b x^2\right)^{11/4} \left(c+d x^2\right)^2} \, dx","Integrate[1/((a + b*x^2)^(11/4)*(c + d*x^2)^2),x]","\frac{\frac{b d x^3 \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_1\left(\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{(a d-b c)^3}+\frac{6 c \left(x^3 \left(21 a^4 d^3+42 a^3 b d^3 x^2+a^2 b^2 d \left(88 c^2+88 c d x^2+21 d^2 x^4\right)+4 a b^3 c \left(-8 c^2+11 c d x^2+19 d^2 x^4\right)-20 b^4 c^2 x^2 \left(c+d x^2\right)\right) \left(4 a d F_1\left(\frac{3}{2};\frac{3}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+3 b c F_1\left(\frac{3}{2};\frac{7}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)-6 a c x \left(42 a^4 d^3+63 a^3 b d^2 \left(d x^2-2 c\right)+a^2 b^2 d \left(126 c^2-38 c d x^2+21 d^2 x^4\right)+2 a b^3 c \left(-21 c^2+41 c d x^2+38 d^2 x^4\right)-10 b^4 c^2 x^2 \left(3 c+2 d x^2\right)\right) F_1\left(\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)}{\left(a+b x^2\right) \left(c+d x^2\right) (b c-a d)^3 \left(6 a c F_1\left(\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)-x^2 \left(4 a d F_1\left(\frac{3}{2};\frac{3}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+3 b c F_1\left(\frac{3}{2};\frac{7}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)\right)}}{252 a^2 c^2 \left(a+b x^2\right)^{3/4}}","\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) \operatorname{EllipticF}\left(\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{42 a^{3/2} c \left(a+b x^2\right)^{3/4} (b c-a d)^3}+\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[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*d*(-20*b^2*c^2 + 76*a*b*c*d + 21*a^2*d^2)*x^3*(1 + (b*x^2)/a)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])/(-(b*c) + a*d)^3 + (6*c*(-6*a*c*x*(42*a^4*d^3 + 63*a^3*b*d^2*(-2*c + d*x^2) - 10*b^4*c^2*x^2*(3*c + 2*d*x^2) + a^2*b^2*d*(126*c^2 - 38*c*d*x^2 + 21*d^2*x^4) + 2*a*b^3*c*(-21*c^2 + 41*c*d*x^2 + 38*d^2*x^4))*AppellF1[1/2, 3/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + x^3*(21*a^4*d^3 + 42*a^3*b*d^3*x^2 - 20*b^4*c^2*x^2*(c + d*x^2) + 4*a*b^3*c*(-8*c^2 + 11*c*d*x^2 + 19*d^2*x^4) + a^2*b^2*d*(88*c^2 + 88*c*d*x^2 + 21*d^2*x^4))*(4*a*d*AppellF1[3/2, 3/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + 3*b*c*AppellF1[3/2, 7/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))/((b*c - a*d)^3*(a + b*x^2)*(c + d*x^2)*(6*a*c*AppellF1[1/2, 3/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] - x^2*(4*a*d*AppellF1[3/2, 3/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + 3*b*c*AppellF1[3/2, 7/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)]))))/(252*a^2*c^2*(a + b*x^2)^(3/4))","C",0
341,1,172,79,0.2250123,"\int \left(a+b x^2\right)^p \left(c+d x^2\right)^q \, dx","Integrate[(a + b*x^2)^p*(c + d*x^2)^q,x]","\frac{3 a c x \left(a+b x^2\right)^p \left(c+d x^2\right)^q F_1\left(\frac{1}{2};-p,-q;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{2 x^2 \left(b c p F_1\left(\frac{3}{2};1-p,-q;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+a d q F_1\left(\frac{3}{2};-p,1-q;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)+3 a c F_1\left(\frac{1}{2};-p,-q;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}","x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(c+d x^2\right)^q \left(\frac{d x^2}{c}+1\right)^{-q} F_1\left(\frac{1}{2};-p,-q;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)",1,"(3*a*c*x*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1[1/2, -p, -q, 3/2, -((b*x^2)/a), -((d*x^2)/c)])/(3*a*c*AppellF1[1/2, -p, -q, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + 2*x^2*(b*c*p*AppellF1[3/2, 1 - p, -q, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + a*d*q*AppellF1[3/2, -p, 1 - q, 5/2, -((b*x^2)/a), -((d*x^2)/c)]))","B",0
342,1,136,296,5.0672607,"\int \left(a+b x^2\right)^p \left(c+d x^2\right)^3 \, dx","Integrate[(a + b*x^2)^p*(c + d*x^2)^3,x]","\frac{1}{35} x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(35 c^3 \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)+d x^2 \left(35 c^2 \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right)+d x^2 \left(21 c \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};-\frac{b x^2}{a}\right)+5 d x^2 \, _2F_1\left(\frac{7}{2},-p;\frac{9}{2};-\frac{b x^2}{a}\right)\right)\right)\right)","\frac{d x \left(a+b x^2\right)^{p+1} \left(15 a^2 d^2-8 a b c d (p+6)+b^2 c^2 \left(4 p^2+28 p+57\right)\right)}{b^3 (2 p+3) (2 p+5) (2 p+7)}-\frac{x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(15 a^3 d^3-9 a^2 b c d^2 (2 p+7)+3 a b^2 c^2 d \left(4 p^2+24 p+35\right)-b^3 c^3 \left(8 p^3+60 p^2+142 p+105\right)\right) \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)}{b^3 (2 p+3) (2 p+5) (2 p+7)}-\frac{d x \left(c+d x^2\right) \left(a+b x^2\right)^{p+1} (5 a d-b c (2 p+11))}{b^2 (2 p+5) (2 p+7)}+\frac{d x \left(c+d x^2\right)^2 \left(a+b x^2\right)^{p+1}}{b (2 p+7)}",1,"(x*(a + b*x^2)^p*(35*c^3*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2)/a)] + d*x^2*(35*c^2*Hypergeometric2F1[3/2, -p, 5/2, -((b*x^2)/a)] + d*x^2*(21*c*Hypergeometric2F1[5/2, -p, 7/2, -((b*x^2)/a)] + 5*d*x^2*Hypergeometric2F1[7/2, -p, 9/2, -((b*x^2)/a)]))))/(35*(1 + (b*x^2)/a)^p)","A",1
343,1,106,176,5.0480164,"\int \left(a+b x^2\right)^p \left(c+d x^2\right)^2 \, dx","Integrate[(a + b*x^2)^p*(c + d*x^2)^2,x]","\frac{1}{15} x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(15 c^2 \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)+d x^2 \left(10 c \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right)+3 d x^2 \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};-\frac{b x^2}{a}\right)\right)\right)","\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,"(x*(a + b*x^2)^p*(15*c^2*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2)/a)] + d*x^2*(10*c*Hypergeometric2F1[3/2, -p, 5/2, -((b*x^2)/a)] + 3*d*x^2*Hypergeometric2F1[5/2, -p, 7/2, -((b*x^2)/a)])))/(15*(1 + (b*x^2)/a)^p)","A",1
344,1,90,93,0.0320109,"\int \left(a+b x^2\right)^p \left(c+d x^2\right) \, dx","Integrate[(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} \left((b c (2 p+3)-a d) \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)+d \left(a+b x^2\right) \left(\frac{b x^2}{a}+1\right)^p\right)}{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,"(x*(a + b*x^2)^p*(d*(a + b*x^2)*(1 + (b*x^2)/a)^p + (-(a*d) + b*c*(3 + 2*p))*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2)/a)]))/(b*(3 + 2*p)*(1 + (b*x^2)/a)^p)","A",1
345,1,44,44,0.0042715,"\int \left(a+b x^2\right)^p \, dx","Integrate[(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",1
346,1,162,57,0.1824073,"\int \frac{\left(a+b x^2\right)^p}{c+d x^2} \, dx","Integrate[(a + b*x^2)^p/(c + d*x^2),x]","-\frac{3 a c x \left(a+b x^2\right)^p F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{\left(c+d x^2\right) \left(2 x^2 \left(a d F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)-b c p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)-3 a c F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)}","\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,"(-3*a*c*x*(a + b*x^2)^p*AppellF1[1/2, -p, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)])/((c + d*x^2)*(-3*a*c*AppellF1[1/2, -p, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + 2*x^2*(-(b*c*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)]) + a*d*AppellF1[3/2, -p, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))","B",0
347,1,162,57,0.1932302,"\int \frac{\left(a+b x^2\right)^p}{\left(c+d x^2\right)^2} \, dx","Integrate[(a + b*x^2)^p/(c + d*x^2)^2,x]","-\frac{3 a c x \left(a+b x^2\right)^p F_1\left(\frac{1}{2};-p,2;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{\left(c+d x^2\right)^2 \left(-2 x^2 \left(b c p F_1\left(\frac{3}{2};1-p,2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)-2 a d F_1\left(\frac{3}{2};-p,3;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)-3 a c F_1\left(\frac{1}{2};-p,2;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)}","\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,"(-3*a*c*x*(a + b*x^2)^p*AppellF1[1/2, -p, 2, 3/2, -((b*x^2)/a), -((d*x^2)/c)])/((c + d*x^2)^2*(-3*a*c*AppellF1[1/2, -p, 2, 3/2, -((b*x^2)/a), -((d*x^2)/c)] - 2*x^2*(b*c*p*AppellF1[3/2, 1 - p, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] - 2*a*d*AppellF1[3/2, -p, 3, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))","B",0
348,1,162,57,0.2454667,"\int \frac{\left(a+b x^2\right)^p}{\left(c+d x^2\right)^3} \, dx","Integrate[(a + b*x^2)^p/(c + d*x^2)^3,x]","-\frac{3 a c x \left(a+b x^2\right)^p F_1\left(\frac{1}{2};-p,3;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{\left(c+d x^2\right)^3 \left(-2 x^2 \left(b c p F_1\left(\frac{3}{2};1-p,3;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)-3 a d F_1\left(\frac{3}{2};-p,4;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)-3 a c F_1\left(\frac{1}{2};-p,3;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)}","\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,"(-3*a*c*x*(a + b*x^2)^p*AppellF1[1/2, -p, 3, 3/2, -((b*x^2)/a), -((d*x^2)/c)])/((c + d*x^2)^3*(-3*a*c*AppellF1[1/2, -p, 3, 3/2, -((b*x^2)/a), -((d*x^2)/c)] - 2*x^2*(b*c*p*AppellF1[3/2, 1 - p, 3, 5/2, -((b*x^2)/a), -((d*x^2)/c)] - 3*a*d*AppellF1[3/2, -p, 4, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))","B",0
349,1,52,53,0.0323283,"\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","Integrate[(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 a d-2 b c}} \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*(a + b*x^2)^((b*c)/(-2*b*c + 2*a*d))*(c + d*x^2)^((a*d)/(2*b*c - 2*a*d)))/(a*c)","A",1