1,1,172,0,0.1922137,"\int \left(a+b x^2\right) \left(c+d x^2\right) \left(e+f x^2\right)^4 \, dx","Int[(a + b*x^2)*(c + d*x^2)*(e + f*x^2)^4,x]","\frac{1}{5} e^2 x^5 (2 a f (3 c f+2 d e)+b e (4 c f+d e))+\frac{1}{3} e^3 x^3 (4 a c f+a d e+b c e)+\frac{1}{11} f^3 x^{11} (a d f+b c f+4 b d e)+\frac{1}{9} f^2 x^9 (a f (c f+4 d e)+2 b e (2 c f+3 d e))+\frac{2}{7} e f x^7 (a f (2 c f+3 d e)+b e (3 c f+2 d e))+a c e^4 x+\frac{1}{13} b d f^4 x^{13}","\frac{1}{5} e^2 x^5 (2 a f (3 c f+2 d e)+b e (4 c f+d e))+\frac{1}{3} e^3 x^3 (4 a c f+a d e+b c e)+\frac{1}{11} f^3 x^{11} (a d f+b c f+4 b d e)+\frac{1}{9} f^2 x^9 (a f (c f+4 d e)+2 b e (2 c f+3 d e))+\frac{2}{7} e f x^7 (a f (2 c f+3 d e)+b e (3 c f+2 d e))+a c e^4 x+\frac{1}{13} b d f^4 x^{13}",1,"a*c*e^4*x + (e^3*(b*c*e + a*d*e + 4*a*c*f)*x^3)/3 + (e^2*(2*a*f*(2*d*e + 3*c*f) + b*e*(d*e + 4*c*f))*x^5)/5 + (2*e*f*(a*f*(3*d*e + 2*c*f) + b*e*(2*d*e + 3*c*f))*x^7)/7 + (f^2*(a*f*(4*d*e + c*f) + 2*b*e*(3*d*e + 2*c*f))*x^9)/9 + (f^3*(4*b*d*e + b*c*f + a*d*f)*x^11)/11 + (b*d*f^4*x^13)/13","A",2,1,24,0.04167,1,"{521}"
2,1,130,0,0.1345802,"\int \left(a+b x^2\right) \left(c+d x^2\right) \left(e+f x^2\right)^3 \, dx","Int[(a + b*x^2)*(c + d*x^2)*(e + f*x^2)^3,x]","\frac{1}{3} e^2 x^3 (3 a c f+a d e+b c e)+\frac{1}{9} f^2 x^9 (a d f+b c f+3 b d e)+\frac{1}{7} f x^7 (a f (c f+3 d e)+3 b e (c f+d e))+\frac{1}{5} e x^5 (3 a f (c f+d e)+b e (3 c f+d e))+a c e^3 x+\frac{1}{11} b d f^3 x^{11}","\frac{1}{3} e^2 x^3 (3 a c f+a d e+b c e)+\frac{1}{9} f^2 x^9 (a d f+b c f+3 b d e)+\frac{1}{7} f x^7 (a f (c f+3 d e)+3 b e (c f+d e))+\frac{1}{5} e x^5 (3 a f (c f+d e)+b e (3 c f+d e))+a c e^3 x+\frac{1}{11} b d f^3 x^{11}",1,"a*c*e^3*x + (e^2*(b*c*e + a*d*e + 3*a*c*f)*x^3)/3 + (e*(3*a*f*(d*e + c*f) + b*e*(d*e + 3*c*f))*x^5)/5 + (f*(3*b*e*(d*e + c*f) + a*f*(3*d*e + c*f))*x^7)/7 + (f^2*(3*b*d*e + b*c*f + a*d*f)*x^9)/9 + (b*d*f^3*x^11)/11","A",2,1,24,0.04167,1,"{521}"
3,1,94,0,0.0826775,"\int \left(a+b x^2\right) \left(c+d x^2\right) \left(e+f x^2\right)^2 \, dx","Int[(a + b*x^2)*(c + d*x^2)*(e + f*x^2)^2,x]","\frac{1}{7} f x^7 (a d f+b c f+2 b d e)+\frac{1}{5} x^5 (a f (c f+2 d e)+b e (2 c f+d e))+\frac{1}{3} e x^3 (2 a c f+a d e+b c e)+a c e^2 x+\frac{1}{9} b d f^2 x^9","\frac{1}{7} f x^7 (a d f+b c f+2 b d e)+\frac{1}{5} x^5 (a f (c f+2 d e)+b e (2 c f+d e))+\frac{1}{3} e x^3 (2 a c f+a d e+b c e)+a c e^2 x+\frac{1}{9} b d f^2 x^9",1,"a*c*e^2*x + (e*(b*c*e + a*d*e + 2*a*c*f)*x^3)/3 + ((a*f*(2*d*e + c*f) + b*e*(d*e + 2*c*f))*x^5)/5 + (f*(2*b*d*e + b*c*f + a*d*f)*x^7)/7 + (b*d*f^2*x^9)/9","A",2,1,24,0.04167,1,"{521}"
4,1,56,0,0.0370527,"\int \left(a+b x^2\right) \left(c+d x^2\right) \left(e+f x^2\right) \, dx","Int[(a + b*x^2)*(c + d*x^2)*(e + f*x^2),x]","\frac{1}{5} x^5 (a d f+b c f+b d e)+\frac{1}{3} x^3 (a c f+a d e+b c e)+a c e x+\frac{1}{7} b d f x^7","\frac{1}{5} x^5 (a d f+b c f+b d e)+\frac{1}{3} x^3 (a c f+a d e+b c e)+a c e x+\frac{1}{7} b d f x^7",1,"a*c*e*x + ((b*c*e + a*d*e + a*c*f)*x^3)/3 + ((b*d*e + b*c*f + a*d*f)*x^5)/5 + (b*d*f*x^7)/7","A",2,1,22,0.04545,1,"{521}"
5,1,81,0,0.0798565,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)}{e+f x^2} \, dx","Int[((a + b*x^2)*(c + d*x^2))/(e + f*x^2),x]","-\frac{x (-2 a d f-3 b c f+3 b d e)}{3 f^2}+\frac{(b e-a f) (d e-c f) \tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)}{\sqrt{e} f^{5/2}}+\frac{d x \left(a+b x^2\right)}{3 f}","-\frac{x (-2 a d f-3 b c f+3 b d e)}{3 f^2}+\frac{(b e-a f) (d e-c f) \tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)}{\sqrt{e} f^{5/2}}+\frac{d x \left(a+b x^2\right)}{3 f}",1,"-((3*b*d*e - 3*b*c*f - 2*a*d*f)*x)/(3*f^2) + (d*x*(a + b*x^2))/(3*f) + ((b*e - a*f)*(d*e - c*f)*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*f^(5/2))","A",3,3,24,0.1250,1,"{528, 388, 205}"
6,1,108,0,0.0849137,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)}{\left(e+f x^2\right)^2} \, dx","Int[((a + b*x^2)*(c + d*x^2))/(e + f*x^2)^2,x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right) (b e (3 d e-c f)-a f (c f+d e))}{2 e^{3/2} f^{5/2}}-\frac{x \left(a+b x^2\right) (d e-c f)}{2 e f \left(e+f x^2\right)}+\frac{b x (3 d e-c f)}{2 e f^2}","-\frac{\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right) (b e (3 d e-c f)-a f (c f+d e))}{2 e^{3/2} f^{5/2}}-\frac{x \left(a+b x^2\right) (d e-c f)}{2 e f \left(e+f x^2\right)}+\frac{b x (3 d e-c f)}{2 e f^2}",1,"(b*(3*d*e - c*f)*x)/(2*e*f^2) - ((d*e - c*f)*x*(a + b*x^2))/(2*e*f*(e + f*x^2)) - ((b*e*(3*d*e - c*f) - a*f*(d*e + c*f))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(2*e^(3/2)*f^(5/2))","A",3,3,24,0.1250,1,"{526, 388, 205}"
7,1,130,0,0.1081638,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)}{\left(e+f x^2\right)^3} \, dx","Int[((a + b*x^2)*(c + d*x^2))/(e + f*x^2)^3,x]","-\frac{x (b e (c f+3 d e)-a f (3 c f+d e))}{8 e^2 f^2 \left(e+f x^2\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right) (a f (3 c f+d e)+b e (c f+3 d e))}{8 e^{5/2} f^{5/2}}-\frac{x \left(a+b x^2\right) (d e-c f)}{4 e f \left(e+f x^2\right)^2}","-\frac{x (b e (c f+3 d e)-a f (3 c f+d e))}{8 e^2 f^2 \left(e+f x^2\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right) (a f (3 c f+d e)+b e (c f+3 d e))}{8 e^{5/2} f^{5/2}}-\frac{x \left(a+b x^2\right) (d e-c f)}{4 e f \left(e+f x^2\right)^2}",1,"-((d*e - c*f)*x*(a + b*x^2))/(4*e*f*(e + f*x^2)^2) - ((b*e*(3*d*e + c*f) - a*f*(d*e + 3*c*f))*x)/(8*e^2*f^2*(e + f*x^2)) + ((b*e*(3*d*e + c*f) + a*f*(d*e + 3*c*f))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(8*e^(5/2)*f^(5/2))","A",3,3,24,0.1250,1,"{526, 385, 205}"
8,1,171,0,0.1634059,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)}{\left(e+f x^2\right)^4} \, dx","Int[((a + b*x^2)*(c + d*x^2))/(e + f*x^2)^4,x]","\frac{x (a f (5 c f+d e)+b e (c f+d e))}{16 e^3 f^2 \left(e+f x^2\right)}-\frac{x (3 b e (c f+d e)-a f (5 c f+d e))}{24 e^2 f^2 \left(e+f x^2\right)^2}+\frac{\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right) (a f (5 c f+d e)+b e (c f+d e))}{16 e^{7/2} f^{5/2}}-\frac{x \left(a+b x^2\right) (d e-c f)}{6 e f \left(e+f x^2\right)^3}","\frac{x (a f (5 c f+d e)+b e (c f+d e))}{16 e^3 f^2 \left(e+f x^2\right)}-\frac{x (3 b e (c f+d e)-a f (5 c f+d e))}{24 e^2 f^2 \left(e+f x^2\right)^2}+\frac{\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right) (a f (5 c f+d e)+b e (c f+d e))}{16 e^{7/2} f^{5/2}}-\frac{x \left(a+b x^2\right) (d e-c f)}{6 e f \left(e+f x^2\right)^3}",1,"-((d*e - c*f)*x*(a + b*x^2))/(6*e*f*(e + f*x^2)^3) - ((3*b*e*(d*e + c*f) - a*f*(d*e + 5*c*f))*x)/(24*e^2*f^2*(e + f*x^2)^2) + ((b*e*(d*e + c*f) + a*f*(d*e + 5*c*f))*x)/(16*e^3*f^2*(e + f*x^2)) + ((b*e*(d*e + c*f) + a*f*(d*e + 5*c*f))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(16*e^(7/2)*f^(5/2))","A",4,4,24,0.1667,1,"{526, 385, 199, 205}"
9,1,226,0,0.2140688,"\int \left(a+b x^2\right) \left(c+d x^2\right)^2 \left(e+f x^2\right)^3 \, dx","Int[(a + b*x^2)*(c + d*x^2)^2*(e + f*x^2)^3,x]","\frac{1}{9} f x^9 \left(a d f (2 c f+3 d e)+b \left(c^2 f^2+6 c d e f+3 d^2 e^2\right)\right)+\frac{1}{7} x^7 \left(a f \left(c^2 f^2+6 c d e f+3 d^2 e^2\right)+b e \left(3 c^2 f^2+6 c d e f+d^2 e^2\right)\right)+\frac{1}{5} e x^5 \left(a \left(3 c^2 f^2+6 c d e f+d^2 e^2\right)+b c e (3 c f+2 d e)\right)+\frac{1}{3} c e^2 x^3 (3 a c f+2 a d e+b c e)+\frac{1}{11} d f^2 x^{11} (a d f+2 b c f+3 b d e)+a c^2 e^3 x+\frac{1}{13} b d^2 f^3 x^{13}","\frac{1}{9} f x^9 \left(a d f (2 c f+3 d e)+b \left(c^2 f^2+6 c d e f+3 d^2 e^2\right)\right)+\frac{1}{7} x^7 \left(a f \left(c^2 f^2+6 c d e f+3 d^2 e^2\right)+b e \left(3 c^2 f^2+6 c d e f+d^2 e^2\right)\right)+\frac{1}{5} e x^5 \left(a \left(3 c^2 f^2+6 c d e f+d^2 e^2\right)+b c e (3 c f+2 d e)\right)+\frac{1}{3} c e^2 x^3 (3 a c f+2 a d e+b c e)+\frac{1}{11} d f^2 x^{11} (a d f+2 b c f+3 b d e)+a c^2 e^3 x+\frac{1}{13} b d^2 f^3 x^{13}",1,"a*c^2*e^3*x + (c*e^2*(b*c*e + 2*a*d*e + 3*a*c*f)*x^3)/3 + (e*(b*c*e*(2*d*e + 3*c*f) + a*(d^2*e^2 + 6*c*d*e*f + 3*c^2*f^2))*x^5)/5 + ((a*f*(3*d^2*e^2 + 6*c*d*e*f + c^2*f^2) + b*e*(d^2*e^2 + 6*c*d*e*f + 3*c^2*f^2))*x^7)/7 + (f*(a*d*f*(3*d*e + 2*c*f) + b*(3*d^2*e^2 + 6*c*d*e*f + c^2*f^2))*x^9)/9 + (d*f^2*(3*b*d*e + 2*b*c*f + a*d*f)*x^11)/11 + (b*d^2*f^3*x^13)/13","A",2,1,26,0.03846,1,"{521}"
10,1,158,0,0.1670843,"\int \left(a+b x^2\right) \left(c+d x^2\right)^2 \left(e+f x^2\right)^2 \, dx","Int[(a + b*x^2)*(c + d*x^2)^2*(e + f*x^2)^2,x]","\frac{1}{7} x^7 \left(2 a d f (c f+d e)+b \left(c^2 f^2+4 c d e f+d^2 e^2\right)\right)+\frac{1}{5} x^5 \left(a \left(c^2 f^2+4 c d e f+d^2 e^2\right)+2 b c e (c f+d e)\right)+\frac{1}{9} d f x^9 (a d f+2 b (c f+d e))+\frac{1}{3} c e x^3 (2 a (c f+d e)+b c e)+a c^2 e^2 x+\frac{1}{11} b d^2 f^2 x^{11}","\frac{1}{7} x^7 \left(2 a d f (c f+d e)+b \left(c^2 f^2+4 c d e f+d^2 e^2\right)\right)+\frac{1}{5} x^5 \left(a \left(c^2 f^2+4 c d e f+d^2 e^2\right)+2 b c e (c f+d e)\right)+\frac{1}{9} d f x^9 (a d f+2 b (c f+d e))+\frac{1}{3} c e x^3 (2 a (c f+d e)+b c e)+a c^2 e^2 x+\frac{1}{11} b d^2 f^2 x^{11}",1,"a*c^2*e^2*x + (c*e*(b*c*e + 2*a*(d*e + c*f))*x^3)/3 + ((2*b*c*e*(d*e + c*f) + a*(d^2*e^2 + 4*c*d*e*f + c^2*f^2))*x^5)/5 + ((2*a*d*f*(d*e + c*f) + b*(d^2*e^2 + 4*c*d*e*f + c^2*f^2))*x^7)/7 + (d*f*(a*d*f + 2*b*(d*e + c*f))*x^9)/9 + (b*d^2*f^2*x^11)/11","A",2,1,26,0.03846,1,"{521}"
11,1,94,0,0.0812315,"\int \left(a+b x^2\right) \left(c+d x^2\right)^2 \left(e+f x^2\right) \, dx","Int[(a + b*x^2)*(c + d*x^2)^2*(e + f*x^2),x]","\frac{1}{7} d x^7 (a d f+2 b c f+b d e)+\frac{1}{5} x^5 (a d (2 c f+d e)+b c (c f+2 d e))+\frac{1}{3} c x^3 (a c f+2 a d e+b c e)+a c^2 e x+\frac{1}{9} b d^2 f x^9","\frac{1}{7} d x^7 (a d f+2 b c f+b d e)+\frac{1}{5} x^5 (a d (2 c f+d e)+b c (c f+2 d e))+\frac{1}{3} c x^3 (a c f+2 a d e+b c e)+a c^2 e x+\frac{1}{9} b d^2 f x^9",1,"a*c^2*e*x + (c*(b*c*e + 2*a*d*e + a*c*f)*x^3)/3 + ((b*c*(2*d*e + c*f) + a*d*(d*e + 2*c*f))*x^5)/5 + (d*(b*d*e + 2*b*c*f + a*d*f)*x^7)/7 + (b*d^2*f*x^9)/9","A",2,1,24,0.04167,1,"{521}"
12,1,142,0,0.2082568,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^2}{e+f x^2} \, dx","Int[((a + b*x^2)*(c + d*x^2)^2)/(e + f*x^2),x]","-\frac{x \left(5 a d f (3 d e-5 c f)-b \left(8 c^2 f^2-25 c d e f+15 d^2 e^2\right)\right)}{15 f^3}-\frac{x \left(c+d x^2\right) (-5 a d f-4 b c f+5 b d e)}{15 f^2}-\frac{(b e-a f) (d e-c f)^2 \tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)}{\sqrt{e} f^{7/2}}+\frac{b x \left(c+d x^2\right)^2}{5 f}","-\frac{x \left(5 a d f (3 d e-5 c f)-b \left(8 c^2 f^2-25 c d e f+15 d^2 e^2\right)\right)}{15 f^3}-\frac{x \left(c+d x^2\right) (-5 a d f-4 b c f+5 b d e)}{15 f^2}-\frac{(b e-a f) (d e-c f)^2 \tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)}{\sqrt{e} f^{7/2}}+\frac{b x \left(c+d x^2\right)^2}{5 f}",1,"-((5*a*d*f*(3*d*e - 5*c*f) - b*(15*d^2*e^2 - 25*c*d*e*f + 8*c^2*f^2))*x)/(15*f^3) - ((5*b*d*e - 4*b*c*f - 5*a*d*f)*x*(c + d*x^2))/(15*f^2) + (b*x*(c + d*x^2)^2)/(5*f) - ((b*e - a*f)*(d*e - c*f)^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*f^(7/2))","A",4,3,26,0.1154,1,"{528, 388, 205}"
13,1,164,0,0.2323707,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^2}{\left(e+f x^2\right)^2} \, dx","Int[((a + b*x^2)*(c + d*x^2)^2)/(e + f*x^2)^2,x]","\frac{(d e-c f) \tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right) (b e (5 d e-c f)-a f (c f+3 d e))}{2 e^{3/2} f^{7/2}}+\frac{d x \left(c+d x^2\right) (5 b e-3 a f)}{6 e f^2}-\frac{d x (b e (15 d e-13 c f)-3 a f (3 d e-c f))}{6 e f^3}-\frac{x \left(c+d x^2\right)^2 (b e-a f)}{2 e f \left(e+f x^2\right)}","\frac{(d e-c f) \tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right) (b e (5 d e-c f)-a f (c f+3 d e))}{2 e^{3/2} f^{7/2}}+\frac{d x \left(c+d x^2\right) (5 b e-3 a f)}{6 e f^2}-\frac{d x (b e (15 d e-13 c f)-3 a f (3 d e-c f))}{6 e f^3}-\frac{x \left(c+d x^2\right)^2 (b e-a f)}{2 e f \left(e+f x^2\right)}",1,"-(d*(b*e*(15*d*e - 13*c*f) - 3*a*f*(3*d*e - c*f))*x)/(6*e*f^3) + (d*(5*b*e - 3*a*f)*x*(c + d*x^2))/(6*e*f^2) - ((b*e - a*f)*x*(c + d*x^2)^2)/(2*e*f*(e + f*x^2)) + ((d*e - c*f)*(b*e*(5*d*e - c*f) - a*f*(3*d*e + c*f))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(2*e^(3/2)*f^(7/2))","A",4,4,26,0.1538,1,"{526, 528, 388, 205}"
14,1,207,0,0.2389582,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^2}{\left(e+f x^2\right)^3} \, dx","Int[((a + b*x^2)*(c + d*x^2)^2)/(e + f*x^2)^3,x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right) \left(b e \left(-c^2 f^2-6 c d e f+15 d^2 e^2\right)-a f \left(3 c^2 f^2+2 c d e f+3 d^2 e^2\right)\right)}{8 e^{5/2} f^{7/2}}-\frac{x \left(c+d x^2\right) (b e (5 d e-c f)-a f (3 c f+d e))}{8 e^2 f^2 \left(e+f x^2\right)}+\frac{d x (b e (15 d e-c f)-3 a f (c f+d e))}{8 e^2 f^3}-\frac{x \left(c+d x^2\right)^2 (b e-a f)}{4 e f \left(e+f x^2\right)^2}","-\frac{\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right) \left(b e \left(-c^2 f^2-6 c d e f+15 d^2 e^2\right)-a f \left(3 c^2 f^2+2 c d e f+3 d^2 e^2\right)\right)}{8 e^{5/2} f^{7/2}}-\frac{x \left(c+d x^2\right) (b e (5 d e-c f)-a f (3 c f+d e))}{8 e^2 f^2 \left(e+f x^2\right)}+\frac{d x (b e (15 d e-c f)-3 a f (c f+d e))}{8 e^2 f^3}-\frac{x \left(c+d x^2\right)^2 (b e-a f)}{4 e f \left(e+f x^2\right)^2}",1,"(d*(b*e*(15*d*e - c*f) - 3*a*f*(d*e + c*f))*x)/(8*e^2*f^3) - ((b*e - a*f)*x*(c + d*x^2)^2)/(4*e*f*(e + f*x^2)^2) - ((b*e*(5*d*e - c*f) - a*f*(d*e + 3*c*f))*x*(c + d*x^2))/(8*e^2*f^2*(e + f*x^2)) - ((b*e*(15*d^2*e^2 - 6*c*d*e*f - c^2*f^2) - a*f*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(8*e^(5/2)*f^(7/2))","A",4,3,26,0.1154,1,"{526, 388, 205}"
15,1,240,0,0.2804371,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^2}{\left(e+f x^2\right)^4} \, dx","Int[((a + b*x^2)*(c + d*x^2)^2)/(e + f*x^2)^4,x]","-\frac{x \left(a f \left(-15 c^2 f^2+4 c d e f+3 d^2 e^2\right)+b e \left(-3 c^2 f^2-4 c d e f+15 d^2 e^2\right)\right)}{48 e^3 f^3 \left(e+f x^2\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right) \left(a f \left(5 c^2 f^2+2 c d e f+d^2 e^2\right)+b e \left(c^2 f^2+2 c d e f+5 d^2 e^2\right)\right)}{16 e^{7/2} f^{7/2}}-\frac{x \left(c+d x^2\right) (d e (a f+5 b e)-c f (5 a f+b e))}{24 e^2 f^2 \left(e+f x^2\right)^2}-\frac{x \left(c+d x^2\right)^2 (b e-a f)}{6 e f \left(e+f x^2\right)^3}","-\frac{x \left(a f \left(-15 c^2 f^2+4 c d e f+3 d^2 e^2\right)+b e \left(-3 c^2 f^2-4 c d e f+15 d^2 e^2\right)\right)}{48 e^3 f^3 \left(e+f x^2\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right) \left(a f \left(5 c^2 f^2+2 c d e f+d^2 e^2\right)+b e \left(c^2 f^2+2 c d e f+5 d^2 e^2\right)\right)}{16 e^{7/2} f^{7/2}}-\frac{x \left(c+d x^2\right) (d e (a f+5 b e)-c f (5 a f+b e))}{24 e^2 f^2 \left(e+f x^2\right)^2}-\frac{x \left(c+d x^2\right)^2 (b e-a f)}{6 e f \left(e+f x^2\right)^3}",1,"-((b*e - a*f)*x*(c + d*x^2)^2)/(6*e*f*(e + f*x^2)^3) - ((d*e*(5*b*e + a*f) - c*f*(b*e + 5*a*f))*x*(c + d*x^2))/(24*e^2*f^2*(e + f*x^2)^2) - ((a*f*(3*d^2*e^2 + 4*c*d*e*f - 15*c^2*f^2) + b*e*(15*d^2*e^2 - 4*c*d*e*f - 3*c^2*f^2))*x)/(48*e^3*f^3*(e + f*x^2)) + ((b*e*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2) + a*f*(d^2*e^2 + 2*c*d*e*f + 5*c^2*f^2))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(16*e^(7/2)*f^(7/2))","A",4,3,26,0.1154,1,"{526, 385, 205}"
16,1,310,0,0.3865706,"\int \left(a+b x^2\right) \left(c+d x^2\right)^3 \left(e+f x^2\right)^3 \, dx","Int[(a + b*x^2)*(c + d*x^2)^3*(e + f*x^2)^3,x]","\frac{3}{11} d f x^{11} \left(a d f (c f+d e)+b \left(c^2 f^2+3 c d e f+d^2 e^2\right)\right)+\frac{1}{9} x^9 \left(3 a d f \left(c^2 f^2+3 c d e f+d^2 e^2\right)+b \left(9 c^2 d e f^2+c^3 f^3+9 c d^2 e^2 f+d^3 e^3\right)\right)+\frac{1}{7} x^7 \left(a \left(9 c^2 d e f^2+c^3 f^3+9 c d^2 e^2 f+d^3 e^3\right)+3 b c e \left(c^2 f^2+3 c d e f+d^2 e^2\right)\right)+\frac{3}{5} c e x^5 \left(a \left(c^2 f^2+3 c d e f+d^2 e^2\right)+b c e (c f+d e)\right)+\frac{1}{3} c^2 e^2 x^3 (3 a (c f+d e)+b c e)+\frac{1}{13} d^2 f^2 x^{13} (a d f+3 b (c f+d e))+a c^3 e^3 x+\frac{1}{15} b d^3 f^3 x^{15}","\frac{3}{11} d f x^{11} \left(a d f (c f+d e)+b \left(c^2 f^2+3 c d e f+d^2 e^2\right)\right)+\frac{1}{9} x^9 \left(3 a d f \left(c^2 f^2+3 c d e f+d^2 e^2\right)+b \left(9 c^2 d e f^2+c^3 f^3+9 c d^2 e^2 f+d^3 e^3\right)\right)+\frac{1}{7} x^7 \left(a \left(9 c^2 d e f^2+c^3 f^3+9 c d^2 e^2 f+d^3 e^3\right)+3 b c e \left(c^2 f^2+3 c d e f+d^2 e^2\right)\right)+\frac{3}{5} c e x^5 \left(a \left(c^2 f^2+3 c d e f+d^2 e^2\right)+b c e (c f+d e)\right)+\frac{1}{3} c^2 e^2 x^3 (3 a (c f+d e)+b c e)+\frac{1}{13} d^2 f^2 x^{13} (a d f+3 b (c f+d e))+a c^3 e^3 x+\frac{1}{15} b d^3 f^3 x^{15}",1,"a*c^3*e^3*x + (c^2*e^2*(b*c*e + 3*a*(d*e + c*f))*x^3)/3 + (3*c*e*(b*c*e*(d*e + c*f) + a*(d^2*e^2 + 3*c*d*e*f + c^2*f^2))*x^5)/5 + ((3*b*c*e*(d^2*e^2 + 3*c*d*e*f + c^2*f^2) + a*(d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + c^3*f^3))*x^7)/7 + ((3*a*d*f*(d^2*e^2 + 3*c*d*e*f + c^2*f^2) + b*(d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + c^3*f^3))*x^9)/9 + (3*d*f*(a*d*f*(d*e + c*f) + b*(d^2*e^2 + 3*c*d*e*f + c^2*f^2))*x^11)/11 + (d^2*f^2*(a*d*f + 3*b*(d*e + c*f))*x^13)/13 + (b*d^3*f^3*x^15)/15","A",2,1,26,0.03846,1,"{521}"
17,1,226,0,0.2162331,"\int \left(a+b x^2\right) \left(c+d x^2\right)^3 \left(e+f x^2\right)^2 \, dx","Int[(a + b*x^2)*(c + d*x^2)^3*(e + f*x^2)^2,x]","\frac{1}{9} d x^9 \left(a d f (3 c f+2 d e)+b \left(3 c^2 f^2+6 c d e f+d^2 e^2\right)\right)+\frac{1}{7} x^7 \left(a d \left(3 c^2 f^2+6 c d e f+d^2 e^2\right)+b c \left(c^2 f^2+6 c d e f+3 d^2 e^2\right)\right)+\frac{1}{5} c x^5 \left(a \left(c^2 f^2+6 c d e f+3 d^2 e^2\right)+b c e (2 c f+3 d e)\right)+\frac{1}{3} c^2 e x^3 (2 a c f+3 a d e+b c e)+\frac{1}{11} d^2 f x^{11} (a d f+3 b c f+2 b d e)+a c^3 e^2 x+\frac{1}{13} b d^3 f^2 x^{13}","\frac{1}{9} d x^9 \left(a d f (3 c f+2 d e)+b \left(3 c^2 f^2+6 c d e f+d^2 e^2\right)\right)+\frac{1}{7} x^7 \left(a d \left(3 c^2 f^2+6 c d e f+d^2 e^2\right)+b c \left(c^2 f^2+6 c d e f+3 d^2 e^2\right)\right)+\frac{1}{5} c x^5 \left(a \left(c^2 f^2+6 c d e f+3 d^2 e^2\right)+b c e (2 c f+3 d e)\right)+\frac{1}{3} c^2 e x^3 (2 a c f+3 a d e+b c e)+\frac{1}{11} d^2 f x^{11} (a d f+3 b c f+2 b d e)+a c^3 e^2 x+\frac{1}{13} b d^3 f^2 x^{13}",1,"a*c^3*e^2*x + (c^2*e*(b*c*e + 3*a*d*e + 2*a*c*f)*x^3)/3 + (c*(b*c*e*(3*d*e + 2*c*f) + a*(3*d^2*e^2 + 6*c*d*e*f + c^2*f^2))*x^5)/5 + ((b*c*(3*d^2*e^2 + 6*c*d*e*f + c^2*f^2) + a*d*(d^2*e^2 + 6*c*d*e*f + 3*c^2*f^2))*x^7)/7 + (d*(a*d*f*(2*d*e + 3*c*f) + b*(d^2*e^2 + 6*c*d*e*f + 3*c^2*f^2))*x^9)/9 + (d^2*f*(2*b*d*e + 3*b*c*f + a*d*f)*x^11)/11 + (b*d^3*f^2*x^13)/13","A",2,1,26,0.03846,1,"{521}"
18,1,130,0,0.1347902,"\int \left(a+b x^2\right) \left(c+d x^2\right)^3 \left(e+f x^2\right) \, dx","Int[(a + b*x^2)*(c + d*x^2)^3*(e + f*x^2),x]","\frac{1}{3} c^2 x^3 (a c f+3 a d e+b c e)+\frac{1}{9} d^2 x^9 (a d f+3 b c f+b d e)+\frac{1}{7} d x^7 (a d (3 c f+d e)+3 b c (c f+d e))+\frac{1}{5} c x^5 (3 a d (c f+d e)+b c (c f+3 d e))+a c^3 e x+\frac{1}{11} b d^3 f x^{11}","\frac{1}{3} c^2 x^3 (a c f+3 a d e+b c e)+\frac{1}{9} d^2 x^9 (a d f+3 b c f+b d e)+\frac{1}{7} d x^7 (a d (3 c f+d e)+3 b c (c f+d e))+\frac{1}{5} c x^5 (3 a d (c f+d e)+b c (c f+3 d e))+a c^3 e x+\frac{1}{11} b d^3 f x^{11}",1,"a*c^3*e*x + (c^2*(b*c*e + 3*a*d*e + a*c*f)*x^3)/3 + (c*(3*a*d*(d*e + c*f) + b*c*(3*d*e + c*f))*x^5)/5 + (d*(3*b*c*(d*e + c*f) + a*d*(d*e + 3*c*f))*x^7)/7 + (d^2*(b*d*e + 3*b*c*f + a*d*f)*x^9)/9 + (b*d^3*f*x^11)/11","A",2,1,24,0.04167,1,"{521}"
19,1,227,0,0.3712852,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^3}{e+f x^2} \, dx","Int[((a + b*x^2)*(c + d*x^2)^3)/(e + f*x^2),x]","-\frac{x \left(c+d x^2\right) \left(7 a d f (5 d e-9 c f)-b \left(24 c^2 f^2-63 c d e f+35 d^2 e^2\right)\right)}{105 f^3}+\frac{x \left(7 a d f \left(33 c^2 f^2-40 c d e f+15 d^2 e^2\right)-b \left(231 c^2 d e f^2-48 c^3 f^3-280 c d^2 e^2 f+105 d^3 e^3\right)\right)}{105 f^4}-\frac{x \left(c+d x^2\right)^2 (-7 a d f-6 b c f+7 b d e)}{35 f^2}+\frac{(b e-a f) (d e-c f)^3 \tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)}{\sqrt{e} f^{9/2}}+\frac{b x \left(c+d x^2\right)^3}{7 f}","-\frac{x \left(c+d x^2\right) \left(7 a d f (5 d e-9 c f)-b \left(24 c^2 f^2-63 c d e f+35 d^2 e^2\right)\right)}{105 f^3}+\frac{x \left(7 a d f \left(33 c^2 f^2-40 c d e f+15 d^2 e^2\right)-b \left(231 c^2 d e f^2-48 c^3 f^3-280 c d^2 e^2 f+105 d^3 e^3\right)\right)}{105 f^4}-\frac{x \left(c+d x^2\right)^2 (-7 a d f-6 b c f+7 b d e)}{35 f^2}+\frac{(b e-a f) (d e-c f)^3 \tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)}{\sqrt{e} f^{9/2}}+\frac{b x \left(c+d x^2\right)^3}{7 f}",1,"((7*a*d*f*(15*d^2*e^2 - 40*c*d*e*f + 33*c^2*f^2) - b*(105*d^3*e^3 - 280*c*d^2*e^2*f + 231*c^2*d*e*f^2 - 48*c^3*f^3))*x)/(105*f^4) - ((7*a*d*f*(5*d*e - 9*c*f) - b*(35*d^2*e^2 - 63*c*d*e*f + 24*c^2*f^2))*x*(c + d*x^2))/(105*f^3) - ((7*b*d*e - 6*b*c*f - 7*a*d*f)*x*(c + d*x^2)^2)/(35*f^2) + (b*x*(c + d*x^2)^3)/(7*f) + ((b*e - a*f)*(d*e - c*f)^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*f^(9/2))","A",5,3,26,0.1154,1,"{528, 388, 205}"
20,1,242,0,0.3986148,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^3}{\left(e+f x^2\right)^2} \, dx","Int[((a + b*x^2)*(c + d*x^2)^3)/(e + f*x^2)^2,x]","-\frac{d x \left(5 a f \left(3 c^2 f^2-22 c d e f+15 d^2 e^2\right)-b e \left(81 c^2 f^2-190 c d e f+105 d^2 e^2\right)\right)}{30 e f^4}-\frac{(d e-c f)^2 \tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right) (b e (7 d e-c f)-a f (c f+5 d e))}{2 e^{3/2} f^{9/2}}+\frac{d x \left(c+d x^2\right)^2 (7 b e-5 a f)}{10 e f^2}-\frac{d x \left(c+d x^2\right) (b e (35 d e-33 c f)-5 a f (5 d e-3 c f))}{30 e f^3}-\frac{x \left(c+d x^2\right)^3 (b e-a f)}{2 e f \left(e+f x^2\right)}","-\frac{d x \left(5 a f \left(3 c^2 f^2-22 c d e f+15 d^2 e^2\right)-b e \left(81 c^2 f^2-190 c d e f+105 d^2 e^2\right)\right)}{30 e f^4}-\frac{(d e-c f)^2 \tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right) (b e (7 d e-c f)-a f (c f+5 d e))}{2 e^{3/2} f^{9/2}}+\frac{d x \left(c+d x^2\right)^2 (7 b e-5 a f)}{10 e f^2}-\frac{d x \left(c+d x^2\right) (b e (35 d e-33 c f)-5 a f (5 d e-3 c f))}{30 e f^3}-\frac{x \left(c+d x^2\right)^3 (b e-a f)}{2 e f \left(e+f x^2\right)}",1,"-(d*(5*a*f*(15*d^2*e^2 - 22*c*d*e*f + 3*c^2*f^2) - b*e*(105*d^2*e^2 - 190*c*d*e*f + 81*c^2*f^2))*x)/(30*e*f^4) - (d*(b*e*(35*d*e - 33*c*f) - 5*a*f*(5*d*e - 3*c*f))*x*(c + d*x^2))/(30*e*f^3) + (d*(7*b*e - 5*a*f)*x*(c + d*x^2)^2)/(10*e*f^2) - ((b*e - a*f)*x*(c + d*x^2)^3)/(2*e*f*(e + f*x^2)) - ((d*e - c*f)^2*(b*e*(7*d*e - c*f) - a*f*(5*d*e + c*f))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(2*e^(3/2)*f^(9/2))","A",5,4,26,0.1538,1,"{526, 528, 388, 205}"
21,1,291,0,0.415631,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^3}{\left(e+f x^2\right)^3} \, dx","Int[((a + b*x^2)*(c + d*x^2)^3)/(e + f*x^2)^3,x]","\frac{d x \left(3 a f \left(-3 c^2 f^2-4 c d e f+15 d^2 e^2\right)-b e \left(3 c^2 f^2-100 c d e f+105 d^2 e^2\right)\right)}{24 e^2 f^4}+\frac{(d e-c f) \tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right) \left(b e \left(-c^2 f^2-10 c d e f+35 d^2 e^2\right)-3 a f \left(c^2 f^2+2 c d e f+5 d^2 e^2\right)\right)}{8 e^{5/2} f^{9/2}}-\frac{x \left(c+d x^2\right)^2 (b e (7 d e-c f)-3 a f (c f+d e))}{8 e^2 f^2 \left(e+f x^2\right)}+\frac{d x \left(c+d x^2\right) (b e (35 d e-3 c f)-3 a f (3 c f+5 d e))}{24 e^2 f^3}-\frac{x \left(c+d x^2\right)^3 (b e-a f)}{4 e f \left(e+f x^2\right)^2}","\frac{d x \left(3 a f \left(-3 c^2 f^2-4 c d e f+15 d^2 e^2\right)-b e \left(3 c^2 f^2-100 c d e f+105 d^2 e^2\right)\right)}{24 e^2 f^4}+\frac{(d e-c f) \tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right) \left(b e \left(-c^2 f^2-10 c d e f+35 d^2 e^2\right)-3 a f \left(c^2 f^2+2 c d e f+5 d^2 e^2\right)\right)}{8 e^{5/2} f^{9/2}}-\frac{x \left(c+d x^2\right)^2 (b e (7 d e-c f)-3 a f (c f+d e))}{8 e^2 f^2 \left(e+f x^2\right)}+\frac{d x \left(c+d x^2\right) (b e (35 d e-3 c f)-3 a f (3 c f+5 d e))}{24 e^2 f^3}-\frac{x \left(c+d x^2\right)^3 (b e-a f)}{4 e f \left(e+f x^2\right)^2}",1,"(d*(3*a*f*(15*d^2*e^2 - 4*c*d*e*f - 3*c^2*f^2) - b*e*(105*d^2*e^2 - 100*c*d*e*f + 3*c^2*f^2))*x)/(24*e^2*f^4) + (d*(b*e*(35*d*e - 3*c*f) - 3*a*f*(5*d*e + 3*c*f))*x*(c + d*x^2))/(24*e^2*f^3) - ((b*e - a*f)*x*(c + d*x^2)^3)/(4*e*f*(e + f*x^2)^2) - ((b*e*(7*d*e - c*f) - 3*a*f*(d*e + c*f))*x*(c + d*x^2)^2)/(8*e^2*f^2*(e + f*x^2)) + ((d*e - c*f)*(b*e*(35*d^2*e^2 - 10*c*d*e*f - c^2*f^2) - 3*a*f*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(8*e^(5/2)*f^(9/2))","A",5,4,26,0.1538,1,"{526, 528, 388, 205}"
22,1,348,0,0.4495969,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^3}{\left(e+f x^2\right)^4} \, dx","Int[((a + b*x^2)*(c + d*x^2)^3)/(e + f*x^2)^4,x]","-\frac{x \left(c+d x^2\right) \left(b e \left(-3 c^2 f^2-8 c d e f+35 d^2 e^2\right)-a f \left(15 c^2 f^2+4 c d e f+5 d^2 e^2\right)\right)}{48 e^3 f^3 \left(e+f x^2\right)}+\frac{d x \left(b e \left(-3 c^2 f^2-10 c d e f+105 d^2 e^2\right)-a f \left(15 c^2 f^2+14 c d e f+15 d^2 e^2\right)\right)}{48 e^3 f^4}-\frac{\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right) \left(b e \left(-3 c^2 d e f^2-c^3 f^3-15 c d^2 e^2 f+35 d^3 e^3\right)-a f \left(3 c^2 d e f^2+5 c^3 f^3+3 c d^2 e^2 f+5 d^3 e^3\right)\right)}{16 e^{7/2} f^{9/2}}-\frac{x \left(c+d x^2\right)^2 (b e (7 d e-c f)-a f (5 c f+d e))}{24 e^2 f^2 \left(e+f x^2\right)^2}-\frac{x \left(c+d x^2\right)^3 (b e-a f)}{6 e f \left(e+f x^2\right)^3}","-\frac{x \left(c+d x^2\right) \left(b e \left(-3 c^2 f^2-8 c d e f+35 d^2 e^2\right)-a f \left(15 c^2 f^2+4 c d e f+5 d^2 e^2\right)\right)}{48 e^3 f^3 \left(e+f x^2\right)}+\frac{d x \left(b e \left(-3 c^2 f^2-10 c d e f+105 d^2 e^2\right)-a f \left(15 c^2 f^2+14 c d e f+15 d^2 e^2\right)\right)}{48 e^3 f^4}-\frac{\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right) \left(b e \left(-3 c^2 d e f^2-c^3 f^3-15 c d^2 e^2 f+35 d^3 e^3\right)-a f \left(3 c^2 d e f^2+5 c^3 f^3+3 c d^2 e^2 f+5 d^3 e^3\right)\right)}{16 e^{7/2} f^{9/2}}-\frac{x \left(c+d x^2\right)^2 (b e (7 d e-c f)-a f (5 c f+d e))}{24 e^2 f^2 \left(e+f x^2\right)^2}-\frac{x \left(c+d x^2\right)^3 (b e-a f)}{6 e f \left(e+f x^2\right)^3}",1,"(d*(b*e*(105*d^2*e^2 - 10*c*d*e*f - 3*c^2*f^2) - a*f*(15*d^2*e^2 + 14*c*d*e*f + 15*c^2*f^2))*x)/(48*e^3*f^4) - ((b*e - a*f)*x*(c + d*x^2)^3)/(6*e*f*(e + f*x^2)^3) - ((b*e*(7*d*e - c*f) - a*f*(d*e + 5*c*f))*x*(c + d*x^2)^2)/(24*e^2*f^2*(e + f*x^2)^2) - ((b*e*(35*d^2*e^2 - 8*c*d*e*f - 3*c^2*f^2) - a*f*(5*d^2*e^2 + 4*c*d*e*f + 15*c^2*f^2))*x*(c + d*x^2))/(48*e^3*f^3*(e + f*x^2)) - ((b*e*(35*d^3*e^3 - 15*c*d^2*e^2*f - 3*c^2*d*e*f^2 - c^3*f^3) - a*f*(5*d^3*e^3 + 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 5*c^3*f^3))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(16*e^(7/2)*f^(9/2))","A",5,3,26,0.1154,1,"{526, 388, 205}"
23,1,544,0,0.681249,"\int \left(a+b x^2\right) \left(c+d x^2\right)^{3/2} \sqrt{e+f x^2} \, dx","Int[(a + b*x^2)*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2],x]","\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} \left(7 a d f (3 c f+d e)-b \left(6 c^2 f^2-6 c d e f+4 d^2 e^2\right)\right)}{105 d f^2}-\frac{x \sqrt{c+d x^2} \left(7 a d f \left(-3 c^2 f^2-7 c d e f+2 d^2 e^2\right)-b \left(9 c^2 d e f^2-6 c^3 f^3-19 c d^2 e^2 f+8 d^3 e^3\right)\right)}{105 d^2 f^2 \sqrt{e+f x^2}}-\frac{e^{3/2} \sqrt{c+d x^2} \left(7 a d f (d e-9 c f)-b \left(-3 c^2 f^2-9 c d e f+4 d^2 e^2\right)\right) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{105 d f^{5/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{\sqrt{e} \sqrt{c+d x^2} \left(7 a d f \left(-3 c^2 f^2-7 c d e f+2 d^2 e^2\right)-b \left(9 c^2 d e f^2-6 c^3 f^3-19 c d^2 e^2 f+8 d^3 e^3\right)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{105 d^2 f^{5/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{x \left(c+d x^2\right)^{3/2} \sqrt{e+f x^2} (7 a d f-2 b c f+b d e)}{35 d f}+\frac{b x \left(c+d x^2\right)^{5/2} \sqrt{e+f x^2}}{7 d}","\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} \left(7 a d f (3 c f+d e)-b \left(6 c^2 f^2-6 c d e f+4 d^2 e^2\right)\right)}{105 d f^2}-\frac{x \sqrt{c+d x^2} \left(7 a d f \left(-3 c^2 f^2-7 c d e f+2 d^2 e^2\right)-b \left(9 c^2 d e f^2-6 c^3 f^3-19 c d^2 e^2 f+8 d^3 e^3\right)\right)}{105 d^2 f^2 \sqrt{e+f x^2}}-\frac{e^{3/2} \sqrt{c+d x^2} \left(7 a d f (d e-9 c f)-b \left(-3 c^2 f^2-9 c d e f+4 d^2 e^2\right)\right) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{105 d f^{5/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{\sqrt{e} \sqrt{c+d x^2} \left(7 a d f \left(-3 c^2 f^2-7 c d e f+2 d^2 e^2\right)-b \left(9 c^2 d e f^2-6 c^3 f^3-19 c d^2 e^2 f+8 d^3 e^3\right)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{105 d^2 f^{5/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{x \left(c+d x^2\right)^{3/2} \sqrt{e+f x^2} (7 a d f-2 b c f+b d e)}{35 d f}+\frac{b x \left(c+d x^2\right)^{5/2} \sqrt{e+f x^2}}{7 d}",1,"-((7*a*d*f*(2*d^2*e^2 - 7*c*d*e*f - 3*c^2*f^2) - b*(8*d^3*e^3 - 19*c*d^2*e^2*f + 9*c^2*d*e*f^2 - 6*c^3*f^3))*x*Sqrt[c + d*x^2])/(105*d^2*f^2*Sqrt[e + f*x^2]) + ((7*a*d*f*(d*e + 3*c*f) - b*(4*d^2*e^2 - 6*c*d*e*f + 6*c^2*f^2))*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(105*d*f^2) + ((b*d*e - 2*b*c*f + 7*a*d*f)*x*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2])/(35*d*f) + (b*x*(c + d*x^2)^(5/2)*Sqrt[e + f*x^2])/(7*d) + (Sqrt[e]*(7*a*d*f*(2*d^2*e^2 - 7*c*d*e*f - 3*c^2*f^2) - b*(8*d^3*e^3 - 19*c*d^2*e^2*f + 9*c^2*d*e*f^2 - 6*c^3*f^3))*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(105*d^2*f^(5/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (e^(3/2)*(7*a*d*f*(d*e - 9*c*f) - b*(4*d^2*e^2 - 9*c*d*e*f - 3*c^2*f^2))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(105*d*f^(5/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",7,5,30,0.1667,1,"{528, 531, 418, 492, 411}"
24,1,381,0,0.3777957,"\int \left(a+b x^2\right) \sqrt{c+d x^2} \sqrt{e+f x^2} \, dx","Int[(a + b*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2],x]","\frac{x \sqrt{c+d x^2} \left(5 a d f (c f+d e)-2 b \left(c^2 f^2-c d e f+d^2 e^2\right)\right)}{15 d^2 f \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left(5 a d f (c f+d e)-2 b \left(c^2 f^2-c d e f+d^2 e^2\right)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 d^2 f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{e^{3/2} \sqrt{c+d x^2} (-10 a d f+b c f+b d e) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 d f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} (5 a d f-2 b c f+b d e)}{15 d f}+\frac{b x \left(c+d x^2\right)^{3/2} \sqrt{e+f x^2}}{5 d}","\frac{x \sqrt{c+d x^2} \left(5 a d f (c f+d e)-2 b \left(c^2 f^2-c d e f+d^2 e^2\right)\right)}{15 d^2 f \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left(5 a d f (c f+d e)-2 b \left(c^2 f^2-c d e f+d^2 e^2\right)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 d^2 f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{e^{3/2} \sqrt{c+d x^2} (-10 a d f+b c f+b d e) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 d f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} (5 a d f-2 b c f+b d e)}{15 d f}+\frac{b x \left(c+d x^2\right)^{3/2} \sqrt{e+f x^2}}{5 d}",1,"((5*a*d*f*(d*e + c*f) - 2*b*(d^2*e^2 - c*d*e*f + c^2*f^2))*x*Sqrt[c + d*x^2])/(15*d^2*f*Sqrt[e + f*x^2]) + ((b*d*e - 2*b*c*f + 5*a*d*f)*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(15*d*f) + (b*x*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2])/(5*d) - (Sqrt[e]*(5*a*d*f*(d*e + c*f) - 2*b*(d^2*e^2 - c*d*e*f + c^2*f^2))*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*d^2*f^(3/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (e^(3/2)*(b*d*e + b*c*f - 10*a*d*f)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*d*f^(3/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",6,5,30,0.1667,1,"{528, 531, 418, 492, 411}"
25,1,283,0,0.181188,"\int \frac{\left(a+b x^2\right) \sqrt{e+f x^2}}{\sqrt{c+d x^2}} \, dx","Int[((a + b*x^2)*Sqrt[e + f*x^2])/Sqrt[c + d*x^2],x]","\frac{x \sqrt{c+d x^2} (3 a d f-2 b c f+b d e)}{3 d^2 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} (3 a d f-2 b c f+b d e) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 d^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{e^{3/2} \sqrt{c+d x^2} (b c-3 a d) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 c d \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{b x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 d}","\frac{x \sqrt{c+d x^2} (3 a d f-2 b c f+b d e)}{3 d^2 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} (3 a d f-2 b c f+b d e) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 d^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{e^{3/2} \sqrt{c+d x^2} (b c-3 a d) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 c d \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{b x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 d}",1,"((b*d*e - 2*b*c*f + 3*a*d*f)*x*Sqrt[c + d*x^2])/(3*d^2*Sqrt[e + f*x^2]) + (b*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(3*d) - (Sqrt[e]*(b*d*e - 2*b*c*f + 3*a*d*f)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*d^2*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - ((b*c - 3*a*d)*e^(3/2)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*c*d*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",5,5,30,0.1667,1,"{528, 531, 418, 492, 411}"
26,1,271,0,0.1705696,"\int \frac{\left(a+b x^2\right) \sqrt{e+f x^2}}{\left(c+d x^2\right)^{3/2}} \, dx","Int[((a + b*x^2)*Sqrt[e + f*x^2])/(c + d*x^2)^(3/2),x]","\frac{f x \sqrt{c+d x^2} (2 b c-a d)}{c d^2 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (2 b c-a d) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{c d^2 \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{x \sqrt{e+f x^2} (b c-a d)}{c d \sqrt{c+d x^2}}+\frac{b e^{3/2} \sqrt{c+d x^2} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{c d \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}","\frac{f x \sqrt{c+d x^2} (2 b c-a d)}{c d^2 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (2 b c-a d) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{c d^2 \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{x \sqrt{e+f x^2} (b c-a d)}{c d \sqrt{c+d x^2}}+\frac{b e^{3/2} \sqrt{c+d x^2} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{c d \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}",1,"((2*b*c - a*d)*f*x*Sqrt[c + d*x^2])/(c*d^2*Sqrt[e + f*x^2]) - ((b*c - a*d)*x*Sqrt[e + f*x^2])/(c*d*Sqrt[c + d*x^2]) - ((2*b*c - a*d)*Sqrt[e]*Sqrt[f]*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(c*d^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (b*e^(3/2)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(c*d*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",5,5,30,0.1667,1,"{526, 531, 418, 492, 411}"
27,1,274,0,0.2050301,"\int \frac{\left(a+b x^2\right) \sqrt{e+f x^2}}{\left(c+d x^2\right)^{5/2}} \, dx","Int[((a + b*x^2)*Sqrt[e + f*x^2])/(c + d*x^2)^(5/2),x]","\frac{\sqrt{e+f x^2} (d e (2 a d+b c)-c f (a d+2 b c)) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{3 c^{3/2} d^{3/2} \sqrt{c+d x^2} (d e-c f) \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}+\frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} (b c-a d) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 c^2 d \sqrt{e+f x^2} (d e-c f) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{x \sqrt{e+f x^2} (b c-a d)}{3 c d \left(c+d x^2\right)^{3/2}}","\frac{\sqrt{e+f x^2} (d e (2 a d+b c)-c f (a d+2 b c)) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{3 c^{3/2} d^{3/2} \sqrt{c+d x^2} (d e-c f) \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}+\frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} (b c-a d) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 c^2 d \sqrt{e+f x^2} (d e-c f) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{x \sqrt{e+f x^2} (b c-a d)}{3 c d \left(c+d x^2\right)^{3/2}}",1,"-((b*c - a*d)*x*Sqrt[e + f*x^2])/(3*c*d*(c + d*x^2)^(3/2)) + ((d*(b*c + 2*a*d)*e - c*(2*b*c + a*d)*f)*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(3*c^(3/2)*d^(3/2)*(d*e - c*f)*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]) + ((b*c - a*d)*e^(3/2)*Sqrt[f]*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*c^2*d*(d*e - c*f)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",4,4,30,0.1333,1,"{526, 525, 418, 411}"
28,1,385,0,0.3802048,"\int \frac{\left(a+b x^2\right) \sqrt{e+f x^2}}{\left(c+d x^2\right)^{7/2}} \, dx","Int[((a + b*x^2)*Sqrt[e + f*x^2])/(c + d*x^2)^(7/2),x]","\frac{\sqrt{e+f x^2} \left(a d \left(3 c^2 f^2-13 c d e f+8 d^2 e^2\right)+2 b c \left(c^2 f^2-c d e f+d^2 e^2\right)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{15 c^{5/2} d^{3/2} \sqrt{c+d x^2} (d e-c f)^2 \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}-\frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} (2 a d (2 d e-3 c f)+b c (c f+d e)) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 c^3 d \sqrt{e+f x^2} (d e-c f)^2 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{x \sqrt{e+f x^2} (a d (4 d e-3 c f)+b c (d e-2 c f))}{15 c^2 d \left(c+d x^2\right)^{3/2} (d e-c f)}-\frac{x \sqrt{e+f x^2} (b c-a d)}{5 c d \left(c+d x^2\right)^{5/2}}","\frac{\sqrt{e+f x^2} \left(a d \left(3 c^2 f^2-13 c d e f+8 d^2 e^2\right)+2 b c \left(c^2 f^2-c d e f+d^2 e^2\right)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{15 c^{5/2} d^{3/2} \sqrt{c+d x^2} (d e-c f)^2 \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}-\frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} (2 a d (2 d e-3 c f)+b c (c f+d e)) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 c^3 d \sqrt{e+f x^2} (d e-c f)^2 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{x \sqrt{e+f x^2} (a d (4 d e-3 c f)+b c (d e-2 c f))}{15 c^2 d \left(c+d x^2\right)^{3/2} (d e-c f)}-\frac{x \sqrt{e+f x^2} (b c-a d)}{5 c d \left(c+d x^2\right)^{5/2}}",1,"-((b*c - a*d)*x*Sqrt[e + f*x^2])/(5*c*d*(c + d*x^2)^(5/2)) + ((a*d*(4*d*e - 3*c*f) + b*c*(d*e - 2*c*f))*x*Sqrt[e + f*x^2])/(15*c^2*d*(d*e - c*f)*(c + d*x^2)^(3/2)) + ((2*b*c*(d^2*e^2 - c*d*e*f + c^2*f^2) + a*d*(8*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2))*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(15*c^(5/2)*d^(3/2)*(d*e - c*f)^2*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]) - (e^(3/2)*Sqrt[f]*(2*a*d*(2*d*e - 3*c*f) + b*c*(d*e + c*f))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*c^3*d*(d*e - c*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",5,5,30,0.1667,1,"{526, 527, 525, 418, 411}"
29,1,543,0,0.6448795,"\int \left(a+b x^2\right) \sqrt{c+d x^2} \left(e+f x^2\right)^{3/2} \, dx","Int[(a + b*x^2)*Sqrt[c + d*x^2]*(e + f*x^2)^(3/2),x]","\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} \left(14 a d f (3 d e-c f)+b \left(8 c^2 f^2-15 c d e f+3 d^2 e^2\right)\right)}{105 d^2 f}+\frac{x \sqrt{c+d x^2} \left(7 a d f \left(-2 c^2 f^2+7 c d e f+3 d^2 e^2\right)-b \left(19 c^2 d e f^2-8 c^3 f^3-9 c d^2 e^2 f+6 d^3 e^3\right)\right)}{105 d^3 f \sqrt{e+f x^2}}+\frac{e^{3/2} \sqrt{c+d x^2} \left(7 a d f (9 d e-c f)-b \left(-4 c^2 f^2+9 c d e f+3 d^2 e^2\right)\right) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{105 d^2 f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left(7 a d f \left(-2 c^2 f^2+7 c d e f+3 d^2 e^2\right)-b \left(19 c^2 d e f^2-8 c^3 f^3-9 c d^2 e^2 f+6 d^3 e^3\right)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{105 d^3 f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{x \left(c+d x^2\right)^{3/2} \sqrt{e+f x^2} (7 a d f-4 b c f+3 b d e)}{35 d^2}+\frac{b x \left(c+d x^2\right)^{3/2} \left(e+f x^2\right)^{3/2}}{7 d}","\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} \left(14 a d f (3 d e-c f)+b \left(8 c^2 f^2-15 c d e f+3 d^2 e^2\right)\right)}{105 d^2 f}+\frac{x \sqrt{c+d x^2} \left(7 a d f \left(-2 c^2 f^2+7 c d e f+3 d^2 e^2\right)-b \left(19 c^2 d e f^2-8 c^3 f^3-9 c d^2 e^2 f+6 d^3 e^3\right)\right)}{105 d^3 f \sqrt{e+f x^2}}+\frac{e^{3/2} \sqrt{c+d x^2} \left(7 a d f (9 d e-c f)-b \left(-4 c^2 f^2+9 c d e f+3 d^2 e^2\right)\right) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{105 d^2 f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left(7 a d f \left(-2 c^2 f^2+7 c d e f+3 d^2 e^2\right)-b \left(19 c^2 d e f^2-8 c^3 f^3-9 c d^2 e^2 f+6 d^3 e^3\right)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{105 d^3 f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{x \left(c+d x^2\right)^{3/2} \sqrt{e+f x^2} (7 a d f-4 b c f+3 b d e)}{35 d^2}+\frac{b x \left(c+d x^2\right)^{3/2} \left(e+f x^2\right)^{3/2}}{7 d}",1,"((7*a*d*f*(3*d^2*e^2 + 7*c*d*e*f - 2*c^2*f^2) - b*(6*d^3*e^3 - 9*c*d^2*e^2*f + 19*c^2*d*e*f^2 - 8*c^3*f^3))*x*Sqrt[c + d*x^2])/(105*d^3*f*Sqrt[e + f*x^2]) + ((14*a*d*f*(3*d*e - c*f) + b*(3*d^2*e^2 - 15*c*d*e*f + 8*c^2*f^2))*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(105*d^2*f) + ((3*b*d*e - 4*b*c*f + 7*a*d*f)*x*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2])/(35*d^2) + (b*x*(c + d*x^2)^(3/2)*(e + f*x^2)^(3/2))/(7*d) - (Sqrt[e]*(7*a*d*f*(3*d^2*e^2 + 7*c*d*e*f - 2*c^2*f^2) - b*(6*d^3*e^3 - 9*c*d^2*e^2*f + 19*c^2*d*e*f^2 - 8*c^3*f^3))*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(105*d^3*f^(3/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (e^(3/2)*(7*a*d*f*(9*d*e - c*f) - b*(3*d^2*e^2 + 9*c*d*e*f - 4*c^2*f^2))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(105*d^2*f^(3/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",7,5,30,0.1667,1,"{528, 531, 418, 492, 411}"
30,1,400,0,0.4386404,"\int \frac{\left(a+b x^2\right) \left(e+f x^2\right)^{3/2}}{\sqrt{c+d x^2}} \, dx","Int[((a + b*x^2)*(e + f*x^2)^(3/2))/Sqrt[c + d*x^2],x]","\frac{x \sqrt{c+d x^2} \left(10 a d f (2 d e-c f)+b \left(8 c^2 f^2-13 c d e f+3 d^2 e^2\right)\right)}{15 d^3 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left(10 a d f (2 d e-c f)+b \left(8 c^2 f^2-13 c d e f+3 d^2 e^2\right)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 d^3 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{e^{3/2} \sqrt{c+d x^2} \left(5 a d (3 d e-c f)-b \left(6 c d e-4 c^2 f\right)\right) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 c d^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} (5 a d f-4 b c f+3 b d e)}{15 d^2}+\frac{b x \sqrt{c+d x^2} \left(e+f x^2\right)^{3/2}}{5 d}","\frac{x \sqrt{c+d x^2} \left(10 a d f (2 d e-c f)+b \left(8 c^2 f^2-13 c d e f+3 d^2 e^2\right)\right)}{15 d^3 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left(10 a d f (2 d e-c f)+b \left(8 c^2 f^2-13 c d e f+3 d^2 e^2\right)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 d^3 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{e^{3/2} \sqrt{c+d x^2} \left(5 a d (3 d e-c f)-b \left(6 c d e-4 c^2 f\right)\right) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 c d^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} (5 a d f-4 b c f+3 b d e)}{15 d^2}+\frac{b x \sqrt{c+d x^2} \left(e+f x^2\right)^{3/2}}{5 d}",1,"((10*a*d*f*(2*d*e - c*f) + b*(3*d^2*e^2 - 13*c*d*e*f + 8*c^2*f^2))*x*Sqrt[c + d*x^2])/(15*d^3*Sqrt[e + f*x^2]) + ((3*b*d*e - 4*b*c*f + 5*a*d*f)*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(15*d^2) + (b*x*Sqrt[c + d*x^2]*(e + f*x^2)^(3/2))/(5*d) - (Sqrt[e]*(10*a*d*f*(2*d*e - c*f) + b*(3*d^2*e^2 - 13*c*d*e*f + 8*c^2*f^2))*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*d^3*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (e^(3/2)*(5*a*d*(3*d*e - c*f) - b*(6*c*d*e - 4*c^2*f))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*c*d^2*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",6,5,30,0.1667,1,"{528, 531, 418, 492, 411}"
31,1,369,0,0.3999088,"\int \frac{\left(a+b x^2\right) \left(e+f x^2\right)^{3/2}}{\left(c+d x^2\right)^{3/2}} \, dx","Int[((a + b*x^2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(3/2),x]","\frac{e^{3/2} \sqrt{c+d x^2} (3 a d f-4 b c f+3 b d e) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 c d^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{f x \sqrt{c+d x^2} \sqrt{e+f x^2} (4 b c-3 a d)}{3 c d^2}+\frac{f x \sqrt{c+d x^2} (b c (7 d e-8 c f)-3 a d (d e-2 c f))}{3 c d^3 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (b c (7 d e-8 c f)-3 a d (d e-2 c f)) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 c d^3 \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{x \left(e+f x^2\right)^{3/2} (b c-a d)}{c d \sqrt{c+d x^2}}","\frac{e^{3/2} \sqrt{c+d x^2} (3 a d f-4 b c f+3 b d e) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 c d^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{f x \sqrt{c+d x^2} \sqrt{e+f x^2} (4 b c-3 a d)}{3 c d^2}+\frac{f x \sqrt{c+d x^2} (b c (7 d e-8 c f)-3 a d (d e-2 c f))}{3 c d^3 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (b c (7 d e-8 c f)-3 a d (d e-2 c f)) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 c d^3 \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{x \left(e+f x^2\right)^{3/2} (b c-a d)}{c d \sqrt{c+d x^2}}",1,"(f*(b*c*(7*d*e - 8*c*f) - 3*a*d*(d*e - 2*c*f))*x*Sqrt[c + d*x^2])/(3*c*d^3*Sqrt[e + f*x^2]) + ((4*b*c - 3*a*d)*f*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(3*c*d^2) - ((b*c - a*d)*x*(e + f*x^2)^(3/2))/(c*d*Sqrt[c + d*x^2]) - (Sqrt[e]*Sqrt[f]*(b*c*(7*d*e - 8*c*f) - 3*a*d*(d*e - 2*c*f))*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*c*d^3*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (e^(3/2)*(3*b*d*e - 4*b*c*f + 3*a*d*f)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*c*d^2*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",6,6,30,0.2000,1,"{526, 528, 531, 418, 492, 411}"
32,1,373,0,0.3937493,"\int \frac{\left(a+b x^2\right) \left(e+f x^2\right)^{3/2}}{\left(c+d x^2\right)^{5/2}} \, dx","Int[((a + b*x^2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(5/2),x]","\frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} (4 b c-a d) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 c^2 d^2 \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{x \sqrt{e+f x^2} (a d (c f+2 d e)+b c (d e-4 c f))}{3 c^2 d^2 \sqrt{c+d x^2}}-\frac{f x \sqrt{c+d x^2} (2 a d (c f+d e)+b c (d e-8 c f))}{3 c^2 d^3 \sqrt{e+f x^2}}+\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (2 a d (c f+d e)+b c (d e-8 c f)) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 c^2 d^3 \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{x \left(e+f x^2\right)^{3/2} (b c-a d)}{3 c d \left(c+d x^2\right)^{3/2}}","\frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} (4 b c-a d) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 c^2 d^2 \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{x \sqrt{e+f x^2} (a d (c f+2 d e)+b c (d e-4 c f))}{3 c^2 d^2 \sqrt{c+d x^2}}-\frac{f x \sqrt{c+d x^2} (2 a d (c f+d e)+b c (d e-8 c f))}{3 c^2 d^3 \sqrt{e+f x^2}}+\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (2 a d (c f+d e)+b c (d e-8 c f)) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 c^2 d^3 \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{x \left(e+f x^2\right)^{3/2} (b c-a d)}{3 c d \left(c+d x^2\right)^{3/2}}",1,"-(f*(b*c*(d*e - 8*c*f) + 2*a*d*(d*e + c*f))*x*Sqrt[c + d*x^2])/(3*c^2*d^3*Sqrt[e + f*x^2]) + ((b*c*(d*e - 4*c*f) + a*d*(2*d*e + c*f))*x*Sqrt[e + f*x^2])/(3*c^2*d^2*Sqrt[c + d*x^2]) - ((b*c - a*d)*x*(e + f*x^2)^(3/2))/(3*c*d*(c + d*x^2)^(3/2)) + (Sqrt[e]*Sqrt[f]*(b*c*(d*e - 8*c*f) + 2*a*d*(d*e + c*f))*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*c^2*d^3*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + ((4*b*c - a*d)*e^(3/2)*Sqrt[f]*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*c^2*d^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",6,5,30,0.1667,1,"{526, 531, 418, 492, 411}"
33,1,376,0,0.4230385,"\int \frac{\left(a+b x^2\right) \left(e+f x^2\right)^{3/2}}{\left(c+d x^2\right)^{7/2}} \, dx","Int[((a + b*x^2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(7/2),x]","\frac{\sqrt{e+f x^2} \left(a d \left(-2 c^2 f^2-3 c d e f+8 d^2 e^2\right)+b c \left(-8 c^2 f^2+3 c d e f+2 d^2 e^2\right)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{15 c^{5/2} d^{5/2} \sqrt{c+d x^2} (d e-c f) \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}-\frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} (a d (4 d e-c f)+b c (d e-4 c f)) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 c^3 d^2 \sqrt{e+f x^2} (d e-c f) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{x \sqrt{e+f x^2} (d e (4 a d+b c)-c f (a d+4 b c))}{15 c^2 d^2 \left(c+d x^2\right)^{3/2}}-\frac{x \left(e+f x^2\right)^{3/2} (b c-a d)}{5 c d \left(c+d x^2\right)^{5/2}}","\frac{\sqrt{e+f x^2} \left(a d \left(-2 c^2 f^2-3 c d e f+8 d^2 e^2\right)+b c \left(-8 c^2 f^2+3 c d e f+2 d^2 e^2\right)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{15 c^{5/2} d^{5/2} \sqrt{c+d x^2} (d e-c f) \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}-\frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} (a d (4 d e-c f)+b c (d e-4 c f)) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 c^3 d^2 \sqrt{e+f x^2} (d e-c f) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{x \sqrt{e+f x^2} (d e (4 a d+b c)-c f (a d+4 b c))}{15 c^2 d^2 \left(c+d x^2\right)^{3/2}}-\frac{x \left(e+f x^2\right)^{3/2} (b c-a d)}{5 c d \left(c+d x^2\right)^{5/2}}",1,"((d*(b*c + 4*a*d)*e - c*(4*b*c + a*d)*f)*x*Sqrt[e + f*x^2])/(15*c^2*d^2*(c + d*x^2)^(3/2)) - ((b*c - a*d)*x*(e + f*x^2)^(3/2))/(5*c*d*(c + d*x^2)^(5/2)) + ((b*c*(2*d^2*e^2 + 3*c*d*e*f - 8*c^2*f^2) + a*d*(8*d^2*e^2 - 3*c*d*e*f - 2*c^2*f^2))*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(15*c^(5/2)*d^(5/2)*(d*e - c*f)*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]) - (e^(3/2)*Sqrt[f]*(b*c*(d*e - 4*c*f) + a*d*(4*d*e - c*f))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*c^3*d^2*(d*e - c*f)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",5,4,30,0.1333,1,"{526, 525, 418, 411}"
34,1,531,0,0.614756,"\int \frac{\left(a+b x^2\right) \left(e+f x^2\right)^{3/2}}{\left(c+d x^2\right)^{9/2}} \, dx","Int[((a + b*x^2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(9/2),x]","\frac{x \sqrt{e+f x^2} \left(3 a d \left(-2 c^2 f^2-5 c d e f+8 d^2 e^2\right)+b c \left(-8 c^2 f^2+c d e f+4 d^2 e^2\right)\right)}{105 c^3 d^2 \left(c+d x^2\right)^{3/2} (d e-c f)}-\frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} \left(3 a d \left(c^2 f^2-11 c d e f+8 d^2 e^2\right)+2 b c \left(2 c^2 f^2-c d e f+2 d^2 e^2\right)\right) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{105 c^4 d^2 \sqrt{e+f x^2} (d e-c f)^2 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{\sqrt{e+f x^2} \left(6 a d \left(2 c^2 d e f^2+c^3 f^3-12 c d^2 e^2 f+8 d^3 e^3\right)+b c \left(-5 c^2 d e f^2+8 c^3 f^3-5 c d^2 e^2 f+8 d^3 e^3\right)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{105 c^{7/2} d^{5/2} \sqrt{c+d x^2} (d e-c f)^2 \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}+\frac{x \sqrt{e+f x^2} (d e (6 a d+b c)-c f (3 a d+4 b c))}{35 c^2 d^2 \left(c+d x^2\right)^{5/2}}-\frac{x \left(e+f x^2\right)^{3/2} (b c-a d)}{7 c d \left(c+d x^2\right)^{7/2}}","\frac{x \sqrt{e+f x^2} \left(3 a d \left(-2 c^2 f^2-5 c d e f+8 d^2 e^2\right)+b c \left(-8 c^2 f^2+c d e f+4 d^2 e^2\right)\right)}{105 c^3 d^2 \left(c+d x^2\right)^{3/2} (d e-c f)}-\frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} \left(3 a d \left(c^2 f^2-11 c d e f+8 d^2 e^2\right)+2 b c \left(2 c^2 f^2-c d e f+2 d^2 e^2\right)\right) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{105 c^4 d^2 \sqrt{e+f x^2} (d e-c f)^2 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{\sqrt{e+f x^2} \left(6 a d \left(2 c^2 d e f^2+c^3 f^3-12 c d^2 e^2 f+8 d^3 e^3\right)+b c \left(-5 c^2 d e f^2+8 c^3 f^3-5 c d^2 e^2 f+8 d^3 e^3\right)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{105 c^{7/2} d^{5/2} \sqrt{c+d x^2} (d e-c f)^2 \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}+\frac{x \sqrt{e+f x^2} (d e (6 a d+b c)-c f (3 a d+4 b c))}{35 c^2 d^2 \left(c+d x^2\right)^{5/2}}-\frac{x \left(e+f x^2\right)^{3/2} (b c-a d)}{7 c d \left(c+d x^2\right)^{7/2}}",1,"((d*(b*c + 6*a*d)*e - c*(4*b*c + 3*a*d)*f)*x*Sqrt[e + f*x^2])/(35*c^2*d^2*(c + d*x^2)^(5/2)) + ((b*c*(4*d^2*e^2 + c*d*e*f - 8*c^2*f^2) + 3*a*d*(8*d^2*e^2 - 5*c*d*e*f - 2*c^2*f^2))*x*Sqrt[e + f*x^2])/(105*c^3*d^2*(d*e - c*f)*(c + d*x^2)^(3/2)) - ((b*c - a*d)*x*(e + f*x^2)^(3/2))/(7*c*d*(c + d*x^2)^(7/2)) + ((6*a*d*(8*d^3*e^3 - 12*c*d^2*e^2*f + 2*c^2*d*e*f^2 + c^3*f^3) + b*c*(8*d^3*e^3 - 5*c*d^2*e^2*f - 5*c^2*d*e*f^2 + 8*c^3*f^3))*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(105*c^(7/2)*d^(5/2)*(d*e - c*f)^2*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]) - (e^(3/2)*Sqrt[f]*(3*a*d*(8*d^2*e^2 - 11*c*d*e*f + c^2*f^2) + 2*b*c*(2*d^2*e^2 - c*d*e*f + 2*c^2*f^2))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(105*c^4*d^2*(d*e - c*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",6,5,30,0.1667,1,"{526, 527, 525, 418, 411}"
35,1,551,0,0.6284995,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^{5/2}}{\sqrt{e+f x^2}} \, dx","Int[((a + b*x^2)*(c + d*x^2)^(5/2))/Sqrt[e + f*x^2],x]","-\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} \left(28 a d f (d e-2 c f)-b \left(15 c^2 f^2-43 c d e f+24 d^2 e^2\right)\right)}{105 f^3}+\frac{x \sqrt{c+d x^2} \left(7 a d f \left(23 c^2 f^2-23 c d e f+8 d^2 e^2\right)-b \left(103 c^2 d e f^2-15 c^3 f^3-128 c d^2 e^2 f+48 d^3 e^3\right)\right)}{105 d f^3 \sqrt{e+f x^2}}+\frac{\sqrt{e} \sqrt{c+d x^2} \left(7 a f \left(15 c^2 f^2-11 c d e f+4 d^2 e^2\right)-b e \left(45 c^2 f^2-61 c d e f+24 d^2 e^2\right)\right) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{105 f^{7/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left(7 a d f \left(23 c^2 f^2-23 c d e f+8 d^2 e^2\right)-b \left(103 c^2 d e f^2-15 c^3 f^3-128 c d^2 e^2 f+48 d^3 e^3\right)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{105 d f^{7/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{x \left(c+d x^2\right)^{3/2} \sqrt{e+f x^2} (-7 a d f-5 b c f+6 b d e)}{35 f^2}+\frac{b x \left(c+d x^2\right)^{5/2} \sqrt{e+f x^2}}{7 f}","-\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} \left(28 a d f (d e-2 c f)-b \left(15 c^2 f^2-43 c d e f+24 d^2 e^2\right)\right)}{105 f^3}+\frac{x \sqrt{c+d x^2} \left(7 a d f \left(23 c^2 f^2-23 c d e f+8 d^2 e^2\right)-b \left(103 c^2 d e f^2-15 c^3 f^3-128 c d^2 e^2 f+48 d^3 e^3\right)\right)}{105 d f^3 \sqrt{e+f x^2}}+\frac{\sqrt{e} \sqrt{c+d x^2} \left(7 a f \left(15 c^2 f^2-11 c d e f+4 d^2 e^2\right)-b e \left(45 c^2 f^2-61 c d e f+24 d^2 e^2\right)\right) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{105 f^{7/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left(7 a d f \left(23 c^2 f^2-23 c d e f+8 d^2 e^2\right)-b \left(103 c^2 d e f^2-15 c^3 f^3-128 c d^2 e^2 f+48 d^3 e^3\right)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{105 d f^{7/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{x \left(c+d x^2\right)^{3/2} \sqrt{e+f x^2} (-7 a d f-5 b c f+6 b d e)}{35 f^2}+\frac{b x \left(c+d x^2\right)^{5/2} \sqrt{e+f x^2}}{7 f}",1,"((7*a*d*f*(8*d^2*e^2 - 23*c*d*e*f + 23*c^2*f^2) - b*(48*d^3*e^3 - 128*c*d^2*e^2*f + 103*c^2*d*e*f^2 - 15*c^3*f^3))*x*Sqrt[c + d*x^2])/(105*d*f^3*Sqrt[e + f*x^2]) - ((28*a*d*f*(d*e - 2*c*f) - b*(24*d^2*e^2 - 43*c*d*e*f + 15*c^2*f^2))*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(105*f^3) - ((6*b*d*e - 5*b*c*f - 7*a*d*f)*x*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2])/(35*f^2) + (b*x*(c + d*x^2)^(5/2)*Sqrt[e + f*x^2])/(7*f) - (Sqrt[e]*(7*a*d*f*(8*d^2*e^2 - 23*c*d*e*f + 23*c^2*f^2) - b*(48*d^3*e^3 - 128*c*d^2*e^2*f + 103*c^2*d*e*f^2 - 15*c^3*f^3))*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(105*d*f^(7/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (Sqrt[e]*(7*a*f*(4*d^2*e^2 - 11*c*d*e*f + 15*c^2*f^2) - b*e*(24*d^2*e^2 - 61*c*d*e*f + 45*c^2*f^2))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(105*f^(7/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",7,5,30,0.1667,1,"{528, 531, 418, 492, 411}"
36,1,396,0,0.4479574,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^{3/2}}{\sqrt{e+f x^2}} \, dx","Int[((a + b*x^2)*(c + d*x^2)^(3/2))/Sqrt[e + f*x^2],x]","-\frac{x \sqrt{c+d x^2} \left(10 a d f (d e-2 c f)-b \left(3 c^2 f^2-13 c d e f+8 d^2 e^2\right)\right)}{15 d f^2 \sqrt{e+f x^2}}+\frac{\sqrt{e} \sqrt{c+d x^2} \left(10 a d f (d e-2 c f)-b \left(3 c^2 f^2-13 c d e f+8 d^2 e^2\right)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 d f^{5/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left(5 a f (d e-3 c f)-b \left(4 d e^2-6 c e f\right)\right) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 f^{5/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} (-5 a d f-3 b c f+4 b d e)}{15 f^2}+\frac{b x \left(c+d x^2\right)^{3/2} \sqrt{e+f x^2}}{5 f}","-\frac{x \sqrt{c+d x^2} \left(10 a d f (d e-2 c f)-b \left(3 c^2 f^2-13 c d e f+8 d^2 e^2\right)\right)}{15 d f^2 \sqrt{e+f x^2}}+\frac{\sqrt{e} \sqrt{c+d x^2} \left(10 a d f (d e-2 c f)-b \left(3 c^2 f^2-13 c d e f+8 d^2 e^2\right)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 d f^{5/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left(5 a f (d e-3 c f)-b \left(4 d e^2-6 c e f\right)\right) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 f^{5/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} (-5 a d f-3 b c f+4 b d e)}{15 f^2}+\frac{b x \left(c+d x^2\right)^{3/2} \sqrt{e+f x^2}}{5 f}",1,"-((10*a*d*f*(d*e - 2*c*f) - b*(8*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2))*x*Sqrt[c + d*x^2])/(15*d*f^2*Sqrt[e + f*x^2]) - ((4*b*d*e - 3*b*c*f - 5*a*d*f)*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(15*f^2) + (b*x*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2])/(5*f) + (Sqrt[e]*(10*a*d*f*(d*e - 2*c*f) - b*(8*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2))*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*d*f^(5/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (Sqrt[e]*(5*a*f*(d*e - 3*c*f) - b*(4*d*e^2 - 6*c*e*f))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*f^(5/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",6,5,30,0.1667,1,"{528, 531, 418, 492, 411}"
37,1,282,0,0.1843897,"\int \frac{\left(a+b x^2\right) \sqrt{c+d x^2}}{\sqrt{e+f x^2}} \, dx","Int[((a + b*x^2)*Sqrt[c + d*x^2])/Sqrt[e + f*x^2],x]","-\frac{\sqrt{e} \sqrt{c+d x^2} (b e-3 a f) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{\sqrt{e} \sqrt{c+d x^2} (-3 a d f-b c f+2 b d e) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 d f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{x \sqrt{c+d x^2} (-3 a d f-b c f+2 b d e)}{3 d f \sqrt{e+f x^2}}+\frac{b x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 f}","-\frac{\sqrt{e} \sqrt{c+d x^2} (b e-3 a f) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{\sqrt{e} \sqrt{c+d x^2} (-3 a d f-b c f+2 b d e) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 d f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{x \sqrt{c+d x^2} (-3 a d f-b c f+2 b d e)}{3 d f \sqrt{e+f x^2}}+\frac{b x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 f}",1,"-((2*b*d*e - b*c*f - 3*a*d*f)*x*Sqrt[c + d*x^2])/(3*d*f*Sqrt[e + f*x^2]) + (b*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(3*f) + (Sqrt[e]*(2*b*d*e - b*c*f - 3*a*d*f)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*d*f^(3/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (Sqrt[e]*(b*e - 3*a*f)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*f^(3/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",5,5,30,0.1667,1,"{528, 531, 418, 492, 411}"
38,1,206,0,0.1018724,"\int \frac{a+b x^2}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx","Int[(a + b*x^2)/(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]","\frac{a \sqrt{e} \sqrt{c+d x^2} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{b x \sqrt{c+d x^2}}{d \sqrt{e+f x^2}}-\frac{b \sqrt{e} \sqrt{c+d x^2} E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{d \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}","\frac{a \sqrt{e} \sqrt{c+d x^2} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{b x \sqrt{c+d x^2}}{d \sqrt{e+f x^2}}-\frac{b \sqrt{e} \sqrt{c+d x^2} E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{d \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}",1,"(b*x*Sqrt[c + d*x^2])/(d*Sqrt[e + f*x^2]) - (b*Sqrt[e]*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(d*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (a*Sqrt[e]*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(c*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",4,4,30,0.1333,1,"{531, 418, 492, 411}"
39,1,209,0,0.0849165,"\int \frac{a+b x^2}{\left(c+d x^2\right)^{3/2} \sqrt{e+f x^2}} \, dx","Int[(a + b*x^2)/((c + d*x^2)^(3/2)*Sqrt[e + f*x^2]),x]","\frac{\sqrt{e} \sqrt{c+d x^2} (b e-a f) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{c \sqrt{f} \sqrt{e+f x^2} (d e-c f) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{\sqrt{e+f x^2} (b c-a d) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{\sqrt{c} \sqrt{d} \sqrt{c+d x^2} (d e-c f) \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}","\frac{\sqrt{e} \sqrt{c+d x^2} (b e-a f) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{c \sqrt{f} \sqrt{e+f x^2} (d e-c f) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{\sqrt{e+f x^2} (b c-a d) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{\sqrt{c} \sqrt{d} \sqrt{c+d x^2} (d e-c f) \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}",1,"-(((b*c - a*d)*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(Sqrt[c]*Sqrt[d]*(d*e - c*f)*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))])) + (Sqrt[e]*(b*e - a*f)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(c*Sqrt[f]*(d*e - c*f)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",3,3,30,0.1000,1,"{525, 418, 411}"
40,1,284,0,0.2108123,"\int \frac{a+b x^2}{\left(c+d x^2\right)^{5/2} \sqrt{e+f x^2}} \, dx","Int[(a + b*x^2)/((c + d*x^2)^(5/2)*Sqrt[e + f*x^2]),x]","-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (-3 a c f+a d e+2 b c e) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 c^2 \sqrt{e+f x^2} (d e-c f)^2 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{\sqrt{e+f x^2} (2 a d (d e-2 c f)+b c (c f+d e)) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{3 c^{3/2} \sqrt{d} \sqrt{c+d x^2} (d e-c f)^2 \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}-\frac{x \sqrt{e+f x^2} (b c-a d)}{3 c \left(c+d x^2\right)^{3/2} (d e-c f)}","-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (-3 a c f+a d e+2 b c e) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 c^2 \sqrt{e+f x^2} (d e-c f)^2 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{\sqrt{e+f x^2} (2 a d (d e-2 c f)+b c (c f+d e)) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{3 c^{3/2} \sqrt{d} \sqrt{c+d x^2} (d e-c f)^2 \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}-\frac{x \sqrt{e+f x^2} (b c-a d)}{3 c \left(c+d x^2\right)^{3/2} (d e-c f)}",1,"-((b*c - a*d)*x*Sqrt[e + f*x^2])/(3*c*(d*e - c*f)*(c + d*x^2)^(3/2)) + ((2*a*d*(d*e - 2*c*f) + b*c*(d*e + c*f))*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(3*c^(3/2)*Sqrt[d]*(d*e - c*f)^2*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]) - (Sqrt[e]*Sqrt[f]*(2*b*c*e + a*d*e - 3*a*c*f)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*c^2*(d*e - c*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",4,4,30,0.1333,1,"{527, 525, 418, 411}"
41,1,401,0,0.4134091,"\int \frac{a+b x^2}{\left(c+d x^2\right)^{7/2} \sqrt{e+f x^2}} \, dx","Int[(a + b*x^2)/((c + d*x^2)^(7/2)*Sqrt[e + f*x^2]),x]","-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} \left(a \left(15 c^2 f^2-11 c d e f+4 d^2 e^2\right)+b c e (d e-9 c f)\right) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 c^3 \sqrt{e+f x^2} (d e-c f)^3 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{\sqrt{e+f x^2} \left(a d \left(23 c^2 f^2-23 c d e f+8 d^2 e^2\right)+b c \left(-3 c^2 f^2-7 c d e f+2 d^2 e^2\right)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{15 c^{5/2} \sqrt{d} \sqrt{c+d x^2} (d e-c f)^3 \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}+\frac{x \sqrt{e+f x^2} (4 a d (d e-2 c f)+b c (3 c f+d e))}{15 c^2 \left(c+d x^2\right)^{3/2} (d e-c f)^2}-\frac{x \sqrt{e+f x^2} (b c-a d)}{5 c \left(c+d x^2\right)^{5/2} (d e-c f)}","-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} \left(a \left(15 c^2 f^2-11 c d e f+4 d^2 e^2\right)+b c e (d e-9 c f)\right) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 c^3 \sqrt{e+f x^2} (d e-c f)^3 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{\sqrt{e+f x^2} \left(a d \left(23 c^2 f^2-23 c d e f+8 d^2 e^2\right)+b c \left(-3 c^2 f^2-7 c d e f+2 d^2 e^2\right)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{15 c^{5/2} \sqrt{d} \sqrt{c+d x^2} (d e-c f)^3 \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}+\frac{x \sqrt{e+f x^2} (4 a d (d e-2 c f)+b c (3 c f+d e))}{15 c^2 \left(c+d x^2\right)^{3/2} (d e-c f)^2}-\frac{x \sqrt{e+f x^2} (b c-a d)}{5 c \left(c+d x^2\right)^{5/2} (d e-c f)}",1,"-((b*c - a*d)*x*Sqrt[e + f*x^2])/(5*c*(d*e - c*f)*(c + d*x^2)^(5/2)) + ((4*a*d*(d*e - 2*c*f) + b*c*(d*e + 3*c*f))*x*Sqrt[e + f*x^2])/(15*c^2*(d*e - c*f)^2*(c + d*x^2)^(3/2)) + ((b*c*(2*d^2*e^2 - 7*c*d*e*f - 3*c^2*f^2) + a*d*(8*d^2*e^2 - 23*c*d*e*f + 23*c^2*f^2))*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(15*c^(5/2)*Sqrt[d]*(d*e - c*f)^3*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]) - (Sqrt[e]*Sqrt[f]*(b*c*e*(d*e - 9*c*f) + a*(4*d^2*e^2 - 11*c*d*e*f + 15*c^2*f^2))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*c^3*(d*e - c*f)^3*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",5,4,30,0.1333,1,"{527, 525, 418, 411}"
42,1,501,0,0.5987059,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^{5/2}}{\left(e+f x^2\right)^{3/2}} \, dx","Int[((a + b*x^2)*(c + d*x^2)^(5/2))/(e + f*x^2)^(3/2),x]","-\frac{x \sqrt{c+d x^2} \left(5 a f \left(3 c^2 f^2-13 c d e f+8 d^2 e^2\right)-2 b e \left(19 c^2 f^2-44 c d e f+24 d^2 e^2\right)\right)}{15 e f^3 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left(10 a d f (2 d e-3 c f)-b \left(15 c^2 f^2-41 c d e f+24 d^2 e^2\right)\right) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 f^{7/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{\sqrt{c+d x^2} \left(5 a f \left(3 c^2 f^2-13 c d e f+8 d^2 e^2\right)-2 b e \left(19 c^2 f^2-44 c d e f+24 d^2 e^2\right)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 \sqrt{e} f^{7/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{d x \left(c+d x^2\right)^{3/2} \sqrt{e+f x^2} (6 b e-5 a f)}{5 e f^2}-\frac{d x \sqrt{c+d x^2} \sqrt{e+f x^2} (b e (24 d e-23 c f)-5 a f (4 d e-3 c f))}{15 e f^3}-\frac{x \left(c+d x^2\right)^{5/2} (b e-a f)}{e f \sqrt{e+f x^2}}","-\frac{x \sqrt{c+d x^2} \left(5 a f \left(3 c^2 f^2-13 c d e f+8 d^2 e^2\right)-2 b e \left(19 c^2 f^2-44 c d e f+24 d^2 e^2\right)\right)}{15 e f^3 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left(10 a d f (2 d e-3 c f)-b \left(15 c^2 f^2-41 c d e f+24 d^2 e^2\right)\right) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 f^{7/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{\sqrt{c+d x^2} \left(5 a f \left(3 c^2 f^2-13 c d e f+8 d^2 e^2\right)-2 b e \left(19 c^2 f^2-44 c d e f+24 d^2 e^2\right)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 \sqrt{e} f^{7/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{d x \left(c+d x^2\right)^{3/2} \sqrt{e+f x^2} (6 b e-5 a f)}{5 e f^2}-\frac{d x \sqrt{c+d x^2} \sqrt{e+f x^2} (b e (24 d e-23 c f)-5 a f (4 d e-3 c f))}{15 e f^3}-\frac{x \left(c+d x^2\right)^{5/2} (b e-a f)}{e f \sqrt{e+f x^2}}",1,"-((5*a*f*(8*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2) - 2*b*e*(24*d^2*e^2 - 44*c*d*e*f + 19*c^2*f^2))*x*Sqrt[c + d*x^2])/(15*e*f^3*Sqrt[e + f*x^2]) - ((b*e - a*f)*x*(c + d*x^2)^(5/2))/(e*f*Sqrt[e + f*x^2]) - (d*(b*e*(24*d*e - 23*c*f) - 5*a*f*(4*d*e - 3*c*f))*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(15*e*f^3) + (d*(6*b*e - 5*a*f)*x*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2])/(5*e*f^2) + ((5*a*f*(8*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2) - 2*b*e*(24*d^2*e^2 - 44*c*d*e*f + 19*c^2*f^2))*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*Sqrt[e]*f^(7/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (Sqrt[e]*(10*a*d*f*(2*d*e - 3*c*f) - b*(24*d^2*e^2 - 41*c*d*e*f + 15*c^2*f^2))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*f^(7/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",7,6,30,0.2000,1,"{526, 528, 531, 418, 492, 411}"
43,1,358,0,0.3631401,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^{3/2}}{\left(e+f x^2\right)^{3/2}} \, dx","Int[((a + b*x^2)*(c + d*x^2)^(3/2))/(e + f*x^2)^(3/2),x]","\frac{d x \sqrt{c+d x^2} \sqrt{e+f x^2} (4 b e-3 a f)}{3 e f^2}-\frac{x \sqrt{c+d x^2} (b e (8 d e-7 c f)-3 a f (2 d e-c f))}{3 e f^2 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} (-3 a d f-3 b c f+4 b d e) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 f^{5/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{\sqrt{c+d x^2} (b e (8 d e-7 c f)-3 a f (2 d e-c f)) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 \sqrt{e} f^{5/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{x \left(c+d x^2\right)^{3/2} (b e-a f)}{e f \sqrt{e+f x^2}}","\frac{d x \sqrt{c+d x^2} \sqrt{e+f x^2} (4 b e-3 a f)}{3 e f^2}-\frac{x \sqrt{c+d x^2} (b e (8 d e-7 c f)-3 a f (2 d e-c f))}{3 e f^2 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} (-3 a d f-3 b c f+4 b d e) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 f^{5/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{\sqrt{c+d x^2} (b e (8 d e-7 c f)-3 a f (2 d e-c f)) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 \sqrt{e} f^{5/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{x \left(c+d x^2\right)^{3/2} (b e-a f)}{e f \sqrt{e+f x^2}}",1,"-((b*e*(8*d*e - 7*c*f) - 3*a*f*(2*d*e - c*f))*x*Sqrt[c + d*x^2])/(3*e*f^2*Sqrt[e + f*x^2]) - ((b*e - a*f)*x*(c + d*x^2)^(3/2))/(e*f*Sqrt[e + f*x^2]) + (d*(4*b*e - 3*a*f)*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(3*e*f^2) + ((b*e*(8*d*e - 7*c*f) - 3*a*f*(2*d*e - c*f))*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*Sqrt[e]*f^(5/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (Sqrt[e]*(4*b*d*e - 3*b*c*f - 3*a*d*f)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*f^(5/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",6,6,30,0.2000,1,"{526, 528, 531, 418, 492, 411}"
44,1,258,0,0.1623621,"\int \frac{\left(a+b x^2\right) \sqrt{c+d x^2}}{\left(e+f x^2\right)^{3/2}} \, dx","Int[((a + b*x^2)*Sqrt[c + d*x^2])/(e + f*x^2)^(3/2),x]","-\frac{\sqrt{c+d x^2} (2 b e-a f) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{\sqrt{e} f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{x \sqrt{c+d x^2} (b e-a f)}{e f \sqrt{e+f x^2}}+\frac{x \sqrt{c+d x^2} (2 b e-a f)}{e f \sqrt{e+f x^2}}+\frac{b \sqrt{e} \sqrt{c+d x^2} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}","-\frac{\sqrt{c+d x^2} (2 b e-a f) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{\sqrt{e} f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{x \sqrt{c+d x^2} (b e-a f)}{e f \sqrt{e+f x^2}}+\frac{x \sqrt{c+d x^2} (2 b e-a f)}{e f \sqrt{e+f x^2}}+\frac{b \sqrt{e} \sqrt{c+d x^2} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}",1,"-(((b*e - a*f)*x*Sqrt[c + d*x^2])/(e*f*Sqrt[e + f*x^2])) + ((2*b*e - a*f)*x*Sqrt[c + d*x^2])/(e*f*Sqrt[e + f*x^2]) - ((2*b*e - a*f)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(Sqrt[e]*f^(3/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (b*Sqrt[e]*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(f^(3/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",5,5,30,0.1667,1,"{526, 531, 418, 492, 411}"
45,1,209,0,0.0845786,"\int \frac{a+b x^2}{\sqrt{c+d x^2} \left(e+f x^2\right)^{3/2}} \, dx","Int[(a + b*x^2)/(Sqrt[c + d*x^2]*(e + f*x^2)^(3/2)),x]","\frac{\sqrt{c+d x^2} (b e-a f) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{\sqrt{e} \sqrt{f} \sqrt{e+f x^2} (d e-c f) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{\sqrt{e} \sqrt{c+d x^2} (b c-a d) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{c \sqrt{f} \sqrt{e+f x^2} (d e-c f) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}","\frac{\sqrt{c+d x^2} (b e-a f) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{\sqrt{e} \sqrt{f} \sqrt{e+f x^2} (d e-c f) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{\sqrt{e} \sqrt{c+d x^2} (b c-a d) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{c \sqrt{f} \sqrt{e+f x^2} (d e-c f) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}",1,"((b*e - a*f)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(Sqrt[e]*Sqrt[f]*(d*e - c*f)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - ((b*c - a*d)*Sqrt[e]*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(c*Sqrt[f]*(d*e - c*f)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",3,3,30,0.1000,1,"{525, 418, 411}"
46,1,272,0,0.2206692,"\int \frac{a+b x^2}{\left(c+d x^2\right)^{3/2} \left(e+f x^2\right)^{3/2}} \, dx","Int[(a + b*x^2)/((c + d*x^2)^(3/2)*(e + f*x^2)^(3/2)),x]","-\frac{x (b c-a d)}{c \sqrt{c+d x^2} \sqrt{e+f x^2} (d e-c f)}+\frac{\sqrt{e} \sqrt{c+d x^2} (-2 a d f+b c f+b d e) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{c \sqrt{f} \sqrt{e+f x^2} (d e-c f)^2 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{\sqrt{f} \sqrt{c+d x^2} (-a c f-a d e+2 b c e) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{c \sqrt{e} \sqrt{e+f x^2} (d e-c f)^2 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}","-\frac{x (b c-a d)}{c \sqrt{c+d x^2} \sqrt{e+f x^2} (d e-c f)}+\frac{\sqrt{e} \sqrt{c+d x^2} (-2 a d f+b c f+b d e) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{c \sqrt{f} \sqrt{e+f x^2} (d e-c f)^2 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{\sqrt{f} \sqrt{c+d x^2} (-a c f-a d e+2 b c e) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{c \sqrt{e} \sqrt{e+f x^2} (d e-c f)^2 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}",1,"-(((b*c - a*d)*x)/(c*(d*e - c*f)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])) - (Sqrt[f]*(2*b*c*e - a*d*e - a*c*f)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(c*Sqrt[e]*(d*e - c*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (Sqrt[e]*(b*d*e + b*c*f - 2*a*d*f)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(c*Sqrt[f]*(d*e - c*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",4,4,30,0.1333,1,"{527, 525, 418, 411}"
47,1,375,0,0.3925,"\int \frac{a+b x^2}{\left(c+d x^2\right)^{5/2} \left(e+f x^2\right)^{3/2}} \, dx","Int[(a + b*x^2)/((c + d*x^2)^(5/2)*(e + f*x^2)^(3/2)),x]","\frac{\sqrt{f} \sqrt{c+d x^2} \left(a \left(-3 c^2 f^2-7 c d e f+2 d^2 e^2\right)+b c e (7 c f+d e)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 c^2 \sqrt{e} \sqrt{e+f x^2} (d e-c f)^3 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{x (2 a d (d e-3 c f)+b c (3 c f+d e))}{3 c^2 \sqrt{c+d x^2} \sqrt{e+f x^2} (d e-c f)^2}-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (a d (d e-9 c f)+b c (3 c f+5 d e)) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 c^2 \sqrt{e+f x^2} (d e-c f)^3 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{x (b c-a d)}{3 c \left(c+d x^2\right)^{3/2} \sqrt{e+f x^2} (d e-c f)}","\frac{\sqrt{f} \sqrt{c+d x^2} \left(a \left(-3 c^2 f^2-7 c d e f+2 d^2 e^2\right)+b c e (7 c f+d e)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 c^2 \sqrt{e} \sqrt{e+f x^2} (d e-c f)^3 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{x (2 a d (d e-3 c f)+b c (3 c f+d e))}{3 c^2 \sqrt{c+d x^2} \sqrt{e+f x^2} (d e-c f)^2}-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (a d (d e-9 c f)+b c (3 c f+5 d e)) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 c^2 \sqrt{e+f x^2} (d e-c f)^3 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{x (b c-a d)}{3 c \left(c+d x^2\right)^{3/2} \sqrt{e+f x^2} (d e-c f)}",1,"-((b*c - a*d)*x)/(3*c*(d*e - c*f)*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2]) + ((2*a*d*(d*e - 3*c*f) + b*c*(d*e + 3*c*f))*x)/(3*c^2*(d*e - c*f)^2*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]) + (Sqrt[f]*(b*c*e*(d*e + 7*c*f) + a*(2*d^2*e^2 - 7*c*d*e*f - 3*c^2*f^2))*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*c^2*Sqrt[e]*(d*e - c*f)^3*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (Sqrt[e]*Sqrt[f]*(a*d*(d*e - 9*c*f) + b*c*(5*d*e + 3*c*f))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*c^2*(d*e - c*f)^3*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",5,4,30,0.1333,1,"{527, 525, 418, 411}"
48,1,209,0,0.0863239,"\int \frac{e+f x^2}{\sqrt{a+b x^2} \left(c+d x^2\right)^{3/2}} \, dx","Int[(e + f*x^2)/(Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)),x]","\frac{\sqrt{c} \sqrt{a+b x^2} (b e-a f) F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{a \sqrt{d} \sqrt{c+d x^2} (b c-a d) \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{\sqrt{a+b x^2} (d e-c f) 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} (b c-a d) \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}","\frac{\sqrt{c} \sqrt{a+b x^2} (b e-a f) F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{a \sqrt{d} \sqrt{c+d x^2} (b c-a d) \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{\sqrt{a+b x^2} (d e-c f) 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} (b c-a d) \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}",1,"-(((d*e - c*f)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[c]*Sqrt[d]*(b*c - a*d)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])) + (Sqrt[c]*(b*e - a*f)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*(b*c - a*d)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])","A",3,3,30,0.1000,1,"{525, 418, 411}"
49,1,247,0,0.2398515,"\int \frac{e+f x^2}{\sqrt{a-b x^2} \left(c+d x^2\right)^{3/2}} \, dx","Int[(e + f*x^2)/(Sqrt[a - b*x^2]*(c + d*x^2)^(3/2)),x]","\frac{x \sqrt{a-b x^2} (d e-c f)}{c \sqrt{c+d x^2} (a d+b c)}+\frac{\sqrt{a} \sqrt{b} \sqrt{1-\frac{b x^2}{a}} \sqrt{c+d x^2} (d e-c f) E\left(\sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)|-\frac{a d}{b c}\right)}{c d \sqrt{a-b x^2} \sqrt{\frac{d x^2}{c}+1} (a d+b c)}+\frac{\sqrt{a} f \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} F\left(\sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)|-\frac{a d}{b c}\right)}{\sqrt{b} d \sqrt{a-b x^2} \sqrt{c+d x^2}}","\frac{x \sqrt{a-b x^2} (d e-c f)}{c \sqrt{c+d x^2} (a d+b c)}+\frac{\sqrt{a} \sqrt{b} \sqrt{1-\frac{b x^2}{a}} \sqrt{c+d x^2} (d e-c f) E\left(\sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)|-\frac{a d}{b c}\right)}{c d \sqrt{a-b x^2} \sqrt{\frac{d x^2}{c}+1} (a d+b c)}+\frac{\sqrt{a} f \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} F\left(\sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)|-\frac{a d}{b c}\right)}{\sqrt{b} d \sqrt{a-b x^2} \sqrt{c+d x^2}}",1,"((d*e - c*f)*x*Sqrt[a - b*x^2])/(c*(b*c + a*d)*Sqrt[c + d*x^2]) + (Sqrt[a]*Sqrt[b]*(d*e - c*f)*Sqrt[1 - (b*x^2)/a]*Sqrt[c + d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(c*d*(b*c + a*d)*Sqrt[a - b*x^2]*Sqrt[1 + (d*x^2)/c]) + (Sqrt[a]*f*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[b]*d*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])","A",8,7,31,0.2258,1,"{527, 524, 427, 426, 424, 421, 419}"
50,1,237,0,0.2183236,"\int \frac{e+f x^2}{\sqrt{a+b x^2} \left(c-d x^2\right)^{3/2}} \, dx","Int[(e + f*x^2)/(Sqrt[a + b*x^2]*(c - d*x^2)^(3/2)),x]","\frac{x \sqrt{a+b x^2} (c f+d e)}{c \sqrt{c-d x^2} (a d+b c)}-\frac{\sqrt{a+b x^2} \sqrt{1-\frac{d x^2}{c}} (c f+d e) E\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|-\frac{b c}{a d}\right)}{\sqrt{c} \sqrt{d} \sqrt{\frac{b x^2}{a}+1} \sqrt{c-d x^2} (a d+b c)}+\frac{e \sqrt{\frac{b x^2}{a}+1} \sqrt{1-\frac{d x^2}{c}} F\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|-\frac{b c}{a d}\right)}{\sqrt{c} \sqrt{d} \sqrt{a+b x^2} \sqrt{c-d x^2}}","\frac{x \sqrt{a+b x^2} (c f+d e)}{c \sqrt{c-d x^2} (a d+b c)}-\frac{\sqrt{a+b x^2} \sqrt{1-\frac{d x^2}{c}} (c f+d e) E\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|-\frac{b c}{a d}\right)}{\sqrt{c} \sqrt{d} \sqrt{\frac{b x^2}{a}+1} \sqrt{c-d x^2} (a d+b c)}+\frac{e \sqrt{\frac{b x^2}{a}+1} \sqrt{1-\frac{d x^2}{c}} F\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|-\frac{b c}{a d}\right)}{\sqrt{c} \sqrt{d} \sqrt{a+b x^2} \sqrt{c-d x^2}}",1,"((d*e + c*f)*x*Sqrt[a + b*x^2])/(c*(b*c + a*d)*Sqrt[c - d*x^2]) - ((d*e + c*f)*Sqrt[a + b*x^2]*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(Sqrt[c]*Sqrt[d]*(b*c + a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[c - d*x^2]) + (e*Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(Sqrt[c]*Sqrt[d]*Sqrt[a + b*x^2]*Sqrt[c - d*x^2])","A",8,7,31,0.2258,1,"{527, 524, 427, 426, 424, 421, 419}"
51,1,242,0,0.2374539,"\int \frac{e+f x^2}{\sqrt{a-b x^2} \left(c-d x^2\right)^{3/2}} \, dx","Int[(e + f*x^2)/(Sqrt[a - b*x^2]*(c - d*x^2)^(3/2)),x]","-\frac{x \sqrt{a-b x^2} (c f+d e)}{c \sqrt{c-d x^2} (b c-a d)}+\frac{\sqrt{a-b x^2} \sqrt{1-\frac{d x^2}{c}} (c f+d e) E\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b c}{a d}\right)}{\sqrt{c} \sqrt{d} \sqrt{1-\frac{b x^2}{a}} \sqrt{c-d x^2} (b c-a d)}+\frac{e \sqrt{1-\frac{b x^2}{a}} \sqrt{1-\frac{d x^2}{c}} F\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b c}{a d}\right)}{\sqrt{c} \sqrt{d} \sqrt{a-b x^2} \sqrt{c-d x^2}}","-\frac{x \sqrt{a-b x^2} (c f+d e)}{c \sqrt{c-d x^2} (b c-a d)}+\frac{\sqrt{a-b x^2} \sqrt{1-\frac{d x^2}{c}} (c f+d e) E\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b c}{a d}\right)}{\sqrt{c} \sqrt{d} \sqrt{1-\frac{b x^2}{a}} \sqrt{c-d x^2} (b c-a d)}+\frac{e \sqrt{1-\frac{b x^2}{a}} \sqrt{1-\frac{d x^2}{c}} F\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b c}{a d}\right)}{\sqrt{c} \sqrt{d} \sqrt{a-b x^2} \sqrt{c-d x^2}}",1,"-(((d*e + c*f)*x*Sqrt[a - b*x^2])/(c*(b*c - a*d)*Sqrt[c - d*x^2])) + ((d*e + c*f)*Sqrt[a - b*x^2]*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], (b*c)/(a*d)])/(Sqrt[c]*Sqrt[d]*(b*c - a*d)*Sqrt[1 - (b*x^2)/a]*Sqrt[c - d*x^2]) + (e*Sqrt[1 - (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[d]*x)/Sqrt[c]], (b*c)/(a*d)])/(Sqrt[c]*Sqrt[d]*Sqrt[a - b*x^2]*Sqrt[c - d*x^2])","A",8,7,32,0.2188,1,"{527, 524, 427, 426, 424, 421, 419}"
52,1,191,0,0.0960592,"\int \frac{a+b x^2}{\sqrt{2+d x^2} \sqrt{3+f x^2}} \, dx","Int[(a + b*x^2)/(Sqrt[2 + d*x^2]*Sqrt[3 + f*x^2]),x]","\frac{a \sqrt{d x^2+2} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{3}}\right)|1-\frac{3 d}{2 f}\right)}{\sqrt{2} \sqrt{f} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}+\frac{b x \sqrt{d x^2+2}}{d \sqrt{f x^2+3}}-\frac{\sqrt{2} b \sqrt{d x^2+2} E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{3}}\right)|1-\frac{3 d}{2 f}\right)}{d \sqrt{f} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}","\frac{a \sqrt{d x^2+2} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{3}}\right)|1-\frac{3 d}{2 f}\right)}{\sqrt{2} \sqrt{f} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}+\frac{b x \sqrt{d x^2+2}}{d \sqrt{f x^2+3}}-\frac{\sqrt{2} b \sqrt{d x^2+2} E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{3}}\right)|1-\frac{3 d}{2 f}\right)}{d \sqrt{f} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}",1,"(b*x*Sqrt[2 + d*x^2])/(d*Sqrt[3 + f*x^2]) - (Sqrt[2]*b*Sqrt[2 + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[3]], 1 - (3*d)/(2*f)])/(d*Sqrt[f]*Sqrt[(2 + d*x^2)/(3 + f*x^2)]*Sqrt[3 + f*x^2]) + (a*Sqrt[2 + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[3]], 1 - (3*d)/(2*f)])/(Sqrt[2]*Sqrt[f]*Sqrt[(2 + d*x^2)/(3 + f*x^2)]*Sqrt[3 + f*x^2])","A",4,4,30,0.1333,1,"{531, 418, 492, 411}"
53,1,262,0,0.1675573,"\int \frac{\left(a+b x^2\right) \sqrt{2+d x^2}}{\sqrt{3+f x^2}} \, dx","Int[((a + b*x^2)*Sqrt[2 + d*x^2])/Sqrt[3 + f*x^2],x]","-\frac{\sqrt{2} \sqrt{d x^2+2} (b-a f) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{3}}\right)|1-\frac{3 d}{2 f}\right)}{f^{3/2} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}+\frac{\sqrt{2} \sqrt{d x^2+2} (-3 a d f+6 b d-2 b f) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{3}}\right)|1-\frac{3 d}{2 f}\right)}{3 d f^{3/2} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}-\frac{x \sqrt{d x^2+2} (-3 a d f+6 b d-2 b f)}{3 d f \sqrt{f x^2+3}}+\frac{b x \sqrt{d x^2+2} \sqrt{f x^2+3}}{3 f}","-\frac{\sqrt{2} \sqrt{d x^2+2} (b-a f) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{3}}\right)|1-\frac{3 d}{2 f}\right)}{f^{3/2} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}+\frac{\sqrt{2} \sqrt{d x^2+2} (-3 a d f+6 b d-2 b f) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{3}}\right)|1-\frac{3 d}{2 f}\right)}{3 d f^{3/2} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}-\frac{x \sqrt{d x^2+2} (-3 a d f+6 b d-2 b f)}{3 d f \sqrt{f x^2+3}}+\frac{b x \sqrt{d x^2+2} \sqrt{f x^2+3}}{3 f}",1,"-((6*b*d - 2*b*f - 3*a*d*f)*x*Sqrt[2 + d*x^2])/(3*d*f*Sqrt[3 + f*x^2]) + (b*x*Sqrt[2 + d*x^2]*Sqrt[3 + f*x^2])/(3*f) + (Sqrt[2]*(6*b*d - 2*b*f - 3*a*d*f)*Sqrt[2 + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[3]], 1 - (3*d)/(2*f)])/(3*d*f^(3/2)*Sqrt[(2 + d*x^2)/(3 + f*x^2)]*Sqrt[3 + f*x^2]) - (Sqrt[2]*(b - a*f)*Sqrt[2 + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[3]], 1 - (3*d)/(2*f)])/(f^(3/2)*Sqrt[(2 + d*x^2)/(3 + f*x^2)]*Sqrt[3 + f*x^2])","A",5,5,30,0.1667,1,"{528, 531, 418, 492, 411}"
54,1,356,0,0.3224512,"\int \left(a+b x^2\right) \sqrt{2+d x^2} \sqrt{3+f x^2} \, dx","Int[(a + b*x^2)*Sqrt[2 + d*x^2]*Sqrt[3 + f*x^2],x]","\frac{x \sqrt{d x^2+2} \left(5 a d f (3 d+2 f)-2 b \left(9 d^2-6 d f+4 f^2\right)\right)}{15 d^2 f \sqrt{f x^2+3}}-\frac{\sqrt{2} \sqrt{d x^2+2} \left(5 a d f (3 d+2 f)-2 b \left(9 d^2-6 d f+4 f^2\right)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{3}}\right)|1-\frac{3 d}{2 f}\right)}{15 d^2 f^{3/2} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}-\frac{\sqrt{2} \sqrt{d x^2+2} (-10 a d f+3 b d+2 b f) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{3}}\right)|1-\frac{3 d}{2 f}\right)}{5 d f^{3/2} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}+\frac{x \sqrt{d x^2+2} \sqrt{f x^2+3} (5 a d f+3 b d-4 b f)}{15 d f}+\frac{b x \left(d x^2+2\right)^{3/2} \sqrt{f x^2+3}}{5 d}","\frac{x \sqrt{d x^2+2} \left(5 a d f (3 d+2 f)-2 b \left(9 d^2-6 d f+4 f^2\right)\right)}{15 d^2 f \sqrt{f x^2+3}}-\frac{\sqrt{2} \sqrt{d x^2+2} \left(5 a d f (3 d+2 f)-2 b \left(9 d^2-6 d f+4 f^2\right)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{3}}\right)|1-\frac{3 d}{2 f}\right)}{15 d^2 f^{3/2} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}-\frac{\sqrt{2} \sqrt{d x^2+2} (-10 a d f+3 b d+2 b f) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{3}}\right)|1-\frac{3 d}{2 f}\right)}{5 d f^{3/2} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}+\frac{x \sqrt{d x^2+2} \sqrt{f x^2+3} (5 a d f+3 b d-4 b f)}{15 d f}+\frac{b x \left(d x^2+2\right)^{3/2} \sqrt{f x^2+3}}{5 d}",1,"((5*a*d*f*(3*d + 2*f) - 2*b*(9*d^2 - 6*d*f + 4*f^2))*x*Sqrt[2 + d*x^2])/(15*d^2*f*Sqrt[3 + f*x^2]) + ((3*b*d - 4*b*f + 5*a*d*f)*x*Sqrt[2 + d*x^2]*Sqrt[3 + f*x^2])/(15*d*f) + (b*x*(2 + d*x^2)^(3/2)*Sqrt[3 + f*x^2])/(5*d) - (Sqrt[2]*(5*a*d*f*(3*d + 2*f) - 2*b*(9*d^2 - 6*d*f + 4*f^2))*Sqrt[2 + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[3]], 1 - (3*d)/(2*f)])/(15*d^2*f^(3/2)*Sqrt[(2 + d*x^2)/(3 + f*x^2)]*Sqrt[3 + f*x^2]) - (Sqrt[2]*(3*b*d + 2*b*f - 10*a*d*f)*Sqrt[2 + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[3]], 1 - (3*d)/(2*f)])/(5*d*f^(3/2)*Sqrt[(2 + d*x^2)/(3 + f*x^2)]*Sqrt[3 + f*x^2])","A",6,5,30,0.1667,1,"{528, 531, 418, 492, 411}"
55,1,113,0,0.2038644,"\int \frac{-b-\sqrt{b^2-4 a c}+2 c x^2}{\sqrt{1+\frac{2 c x^2}{-b-\sqrt{b^2-4 a c}}} \sqrt{1+\frac{2 c x^2}{-b+\sqrt{b^2-4 a c}}}} \, dx","Int[(-b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(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{b-\sqrt{b^2-4 a c}} \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}}","-\frac{\sqrt{b-\sqrt{b^2-4 a c}} \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}}",1,"-((Sqrt[b - Sqrt[b^2 - 4*a*c]]*(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",2,2,87,0.02299,1,"{21, 424}"
56,1,526,0,0.6300352,"\int \frac{b-\sqrt{b^2-4 a c}+2 c x^2}{\sqrt{1+\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}} \sqrt{1+\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}}} \, dx","Int[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]),x]","\frac{x \left(b-\sqrt{b^2-4 a c}\right) \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{\left(b-\sqrt{b^2-4 a c}\right) \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} F\left(\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right)|-\frac{2 \sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)}{\sqrt{2} \sqrt{c} \sqrt{\frac{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1}{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}}-\frac{\left(b-\sqrt{b^2-4 a c}\right) \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} E\left(\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right)|-\frac{2 \sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)}{\sqrt{2} \sqrt{c} \sqrt{\frac{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1}{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}}","\frac{x \left(b-\sqrt{b^2-4 a c}\right) \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{\left(b-\sqrt{b^2-4 a c}\right) \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} F\left(\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right)|-\frac{2 \sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)}{\sqrt{2} \sqrt{c} \sqrt{\frac{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1}{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}}-\frac{\left(b-\sqrt{b^2-4 a c}\right) \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,"((b - Sqrt[b^2 - 4*a*c])*x*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])])/Sqrt[1 + (2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])] - ((b - Sqrt[b^2 - 4*a*c])*Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticE[ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]], (-2*Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/(Sqrt[2]*Sqrt[c]*Sqrt[(1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]))/(1 + (2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]))]*Sqrt[1 + (2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]) + ((b - Sqrt[b^2 - 4*a*c])*Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticF[ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]], (-2*Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/(Sqrt[2]*Sqrt[c]*Sqrt[(1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]))/(1 + (2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]))]*Sqrt[1 + (2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])])","A",5,5,81,0.06173,1,"{21, 422, 418, 492, 411}"
57,1,128,0,0.1421787,"\int \frac{\left(a+b x^2\right) \sqrt{c+d x^2}}{e+f x^2} \, dx","Int[((a + b*x^2)*Sqrt[c + d*x^2])/(e + f*x^2),x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right) (-2 a d f-b c f+2 b d e)}{2 \sqrt{d} f^2}+\frac{(b e-a f) \sqrt{d e-c f} \tanh ^{-1}\left(\frac{x \sqrt{d e-c f}}{\sqrt{e} \sqrt{c+d x^2}}\right)}{\sqrt{e} f^2}+\frac{b x \sqrt{c+d x^2}}{2 f}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right) (-2 a d f-b c f+2 b d e)}{2 \sqrt{d} f^2}+\frac{(b e-a f) \sqrt{d e-c f} \tanh ^{-1}\left(\frac{x \sqrt{d e-c f}}{\sqrt{e} \sqrt{c+d x^2}}\right)}{\sqrt{e} f^2}+\frac{b x \sqrt{c+d x^2}}{2 f}",1,"(b*x*Sqrt[c + d*x^2])/(2*f) - ((2*b*d*e - b*c*f - 2*a*d*f)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(2*Sqrt[d]*f^2) + ((b*e - a*f)*Sqrt[d*e - c*f]*ArcTanh[(Sqrt[d*e - c*f]*x)/(Sqrt[e]*Sqrt[c + d*x^2])])/(Sqrt[e]*f^2)","A",6,6,28,0.2143,1,"{528, 523, 217, 206, 377, 208}"
58,1,304,0,0.2994539,"\int \frac{\left(a+b x^2\right)^3}{\left(c+d x^2\right) \sqrt{e+f x^2}} \, dx","Int[(a + b*x^2)^3/((c + d*x^2)*Sqrt[e + f*x^2]),x]","\frac{b \left(8 a^2 f^2-8 a b e f+3 b^2 e^2\right) \tanh ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e+f x^2}}\right)}{8 d f^{5/2}}-\frac{b^2 x \sqrt{e+f x^2} (b c-a d)}{2 d^2 f}-\frac{3 b^2 x \sqrt{e+f x^2} (b e-2 a f)}{8 d f^2}+\frac{b^2 x \left(a+b x^2\right) \sqrt{e+f x^2}}{4 d f}+\frac{b (b c-a d) (b e-2 a f) \tanh ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e+f x^2}}\right)}{2 d^2 f^{3/2}}-\frac{(b c-a d)^3 \tan ^{-1}\left(\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right)}{\sqrt{c} d^3 \sqrt{d e-c f}}+\frac{b (b c-a d)^2 \tanh ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e+f x^2}}\right)}{d^3 \sqrt{f}}","\frac{b \left(8 a^2 f^2-8 a b e f+3 b^2 e^2\right) \tanh ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e+f x^2}}\right)}{8 d f^{5/2}}-\frac{b^2 x \sqrt{e+f x^2} (b c-a d)}{2 d^2 f}-\frac{3 b^2 x \sqrt{e+f x^2} (b e-2 a f)}{8 d f^2}+\frac{b^2 x \left(a+b x^2\right) \sqrt{e+f x^2}}{4 d f}+\frac{b (b c-a d) (b e-2 a f) \tanh ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e+f x^2}}\right)}{2 d^2 f^{3/2}}-\frac{(b c-a d)^3 \tan ^{-1}\left(\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right)}{\sqrt{c} d^3 \sqrt{d e-c f}}+\frac{b (b c-a d)^2 \tanh ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e+f x^2}}\right)}{d^3 \sqrt{f}}",1,"-(b^2*(b*c - a*d)*x*Sqrt[e + f*x^2])/(2*d^2*f) - (3*b^2*(b*e - 2*a*f)*x*Sqrt[e + f*x^2])/(8*d*f^2) + (b^2*x*(a + b*x^2)*Sqrt[e + f*x^2])/(4*d*f) - ((b*c - a*d)^3*ArcTan[(Sqrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])])/(Sqrt[c]*d^3*Sqrt[d*e - c*f]) + (b*(b*c - a*d)^2*ArcTanh[(Sqrt[f]*x)/Sqrt[e + f*x^2]])/(d^3*Sqrt[f]) + (b*(b*c - a*d)*(b*e - 2*a*f)*ArcTanh[(Sqrt[f]*x)/Sqrt[e + f*x^2]])/(2*d^2*f^(3/2)) + (b*(3*b^2*e^2 - 8*a*b*e*f + 8*a^2*f^2)*ArcTanh[(Sqrt[f]*x)/Sqrt[e + f*x^2]])/(8*d*f^(5/2))","A",14,8,30,0.2667,1,"{545, 416, 388, 217, 206, 523, 377, 205}"
59,1,166,0,0.1052435,"\int \frac{\left(a+b x^2\right)^2}{\left(c+d x^2\right) \sqrt{e+f x^2}} \, dx","Int[(a + b*x^2)^2/((c + d*x^2)*Sqrt[e + f*x^2]),x]","\frac{(b c-a d)^2 \tan ^{-1}\left(\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right)}{\sqrt{c} d^2 \sqrt{d e-c f}}-\frac{b (b c-a d) \tanh ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e+f x^2}}\right)}{d^2 \sqrt{f}}-\frac{b (b e-2 a f) \tanh ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e+f x^2}}\right)}{2 d f^{3/2}}+\frac{b^2 x \sqrt{e+f x^2}}{2 d f}","\frac{(b c-a d)^2 \tan ^{-1}\left(\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right)}{\sqrt{c} d^2 \sqrt{d e-c f}}-\frac{b (b c-a d) \tanh ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e+f x^2}}\right)}{d^2 \sqrt{f}}-\frac{b (b e-2 a f) \tanh ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e+f x^2}}\right)}{2 d f^{3/2}}+\frac{b^2 x \sqrt{e+f x^2}}{2 d f}",1,"(b^2*x*Sqrt[e + f*x^2])/(2*d*f) + ((b*c - a*d)^2*ArcTan[(Sqrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])])/(Sqrt[c]*d^2*Sqrt[d*e - c*f]) - (b*(b*c - a*d)*ArcTanh[(Sqrt[f]*x)/Sqrt[e + f*x^2]])/(d^2*Sqrt[f]) - (b*(b*e - 2*a*f)*ArcTanh[(Sqrt[f]*x)/Sqrt[e + f*x^2]])/(2*d*f^(3/2))","A",9,7,30,0.2333,1,"{545, 388, 217, 206, 523, 377, 205}"
60,1,91,0,0.0481703,"\int \frac{a+b x^2}{\left(c+d x^2\right) \sqrt{e+f x^2}} \, dx","Int[(a + b*x^2)/((c + d*x^2)*Sqrt[e + f*x^2]),x]","\frac{b \tanh ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e+f x^2}}\right)}{d \sqrt{f}}-\frac{(b c-a d) \tan ^{-1}\left(\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right)}{\sqrt{c} d \sqrt{d e-c f}}","\frac{b \tanh ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e+f x^2}}\right)}{d \sqrt{f}}-\frac{(b c-a d) \tan ^{-1}\left(\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right)}{\sqrt{c} d \sqrt{d e-c f}}",1,"-(((b*c - a*d)*ArcTan[(Sqrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])])/(Sqrt[c]*d*Sqrt[d*e - c*f])) + (b*ArcTanh[(Sqrt[f]*x)/Sqrt[e + f*x^2]])/(d*Sqrt[f])","A",5,5,28,0.1786,1,"{523, 217, 206, 377, 205}"
61,1,49,0,0.0200121,"\int \frac{1}{\left(c+d x^2\right) \sqrt{e+f x^2}} \, dx","Int[1/((c + d*x^2)*Sqrt[e + f*x^2]),x]","\frac{\tan ^{-1}\left(\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right)}{\sqrt{c} \sqrt{d e-c f}}","\frac{\tan ^{-1}\left(\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right)}{\sqrt{c} \sqrt{d e-c f}}",1,"ArcTan[(Sqrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])]/(Sqrt[c]*Sqrt[d*e - c*f])","A",2,2,21,0.09524,1,"{377, 205}"
62,1,122,0,0.1108276,"\int \frac{1}{\left(a+b x^2\right) \left(c+d x^2\right) \sqrt{e+f x^2}} \, dx","Int[1/((a + b*x^2)*(c + d*x^2)*Sqrt[e + f*x^2]),x]","\frac{b \tan ^{-1}\left(\frac{x \sqrt{b e-a f}}{\sqrt{a} \sqrt{e+f x^2}}\right)}{\sqrt{a} (b c-a d) \sqrt{b e-a f}}-\frac{d \tan ^{-1}\left(\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right)}{\sqrt{c} (b c-a d) \sqrt{d e-c f}}","\frac{b \tan ^{-1}\left(\frac{x \sqrt{b e-a f}}{\sqrt{a} \sqrt{e+f x^2}}\right)}{\sqrt{a} (b c-a d) \sqrt{b e-a f}}-\frac{d \tan ^{-1}\left(\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right)}{\sqrt{c} (b c-a d) \sqrt{d e-c f}}",1,"(b*ArcTan[(Sqrt[b*e - a*f]*x)/(Sqrt[a]*Sqrt[e + f*x^2])])/(Sqrt[a]*(b*c - a*d)*Sqrt[b*e - a*f]) - (d*ArcTan[(Sqrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])])/(Sqrt[c]*(b*c - a*d)*Sqrt[d*e - c*f])","A",5,3,30,0.1000,1,"{532, 377, 205}"
63,1,203,0,0.2651146,"\int \frac{1}{\left(a+b x^2\right)^2 \left(c+d x^2\right) \sqrt{e+f x^2}} \, dx","Int[1/((a + b*x^2)^2*(c + d*x^2)*Sqrt[e + f*x^2]),x]","\frac{b \left(4 a^2 d f-2 a b c f-3 a b d e+b^2 c e\right) \tan ^{-1}\left(\frac{x \sqrt{b e-a f}}{\sqrt{a} \sqrt{e+f x^2}}\right)}{2 a^{3/2} (b c-a d)^2 (b e-a f)^{3/2}}+\frac{b^2 x \sqrt{e+f x^2}}{2 a \left(a+b x^2\right) (b c-a d) (b e-a f)}+\frac{d^2 \tan ^{-1}\left(\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right)}{\sqrt{c} (b c-a d)^2 \sqrt{d e-c f}}","\frac{b \left(4 a^2 d f-2 a b c f-3 a b d e+b^2 c e\right) \tan ^{-1}\left(\frac{x \sqrt{b e-a f}}{\sqrt{a} \sqrt{e+f x^2}}\right)}{2 a^{3/2} (b c-a d)^2 (b e-a f)^{3/2}}+\frac{b^2 x \sqrt{e+f x^2}}{2 a \left(a+b x^2\right) (b c-a d) (b e-a f)}+\frac{d^2 \tan ^{-1}\left(\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right)}{\sqrt{c} (b c-a d)^2 \sqrt{d e-c f}}",1,"(b^2*x*Sqrt[e + f*x^2])/(2*a*(b*c - a*d)*(b*e - a*f)*(a + b*x^2)) + (b*(b^2*c*e - 3*a*b*d*e - 2*a*b*c*f + 4*a^2*d*f)*ArcTan[(Sqrt[b*e - a*f]*x)/(Sqrt[a]*Sqrt[e + f*x^2])])/(2*a^(3/2)*(b*c - a*d)^2*(b*e - a*f)^(3/2)) + (d^2*ArcTan[(Sqrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])])/(Sqrt[c]*(b*c - a*d)^2*Sqrt[d*e - c*f])","A",7,5,30,0.1667,1,"{546, 377, 205, 527, 12}"
64,1,776,0,0.7628761,"\int \frac{\left(c+d x^2\right)^{5/2} \sqrt{e+f x^2}}{a+b x^2} \, dx","Int[((c + d*x^2)^(5/2)*Sqrt[e + f*x^2])/(a + b*x^2),x]","\frac{d e^{3/2} \sqrt{c+d x^2} (5 b c-3 a d) (b c-a d) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 b^3 c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{e^{3/2} \sqrt{c+d x^2} (b c-a d)^3 \Pi \left(1-\frac{b e}{a f};\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{a b^3 c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{d x \sqrt{c+d x^2} \sqrt{e+f x^2} (b c-a d)}{3 b^2}+\frac{x \sqrt{c+d x^2} (b c-a d) (-3 a d f+4 b c f+b d e)}{3 b^3 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} (b c-a d) (-3 a d f+4 b c f+b d e) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 b^3 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{\sqrt{e} \sqrt{c+d x^2} \left(-3 c^2 f^2-7 c d e f+2 d^2 e^2\right) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 b f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{d x \sqrt{c+d x^2} \left(\frac{3 c^2 f}{d}+7 c e-\frac{2 d e^2}{f}\right)}{15 b \sqrt{e+f x^2}}+\frac{d^2 x \sqrt{c+d x^2} \left(e+f x^2\right)^{3/2}}{5 b f}-\frac{d e^{3/2} \sqrt{c+d x^2} (d e-9 c f) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 b f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{2 d x \sqrt{c+d x^2} \sqrt{e+f x^2} (d e-3 c f)}{15 b f}","-\frac{\sqrt{e} \sqrt{c+d x^2} \left(15 a^2 d^2 f^2-5 a b d f (7 c f+d e)+b^2 \left(23 c^2 f^2+12 c d e f-2 d^2 e^2\right)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 b^3 f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{d e^{3/2} \sqrt{c+d x^2} \left(15 a^2 d^2 f-40 a b c d f+b^2 c (34 c f-d e)\right) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 b^3 c f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{e^{3/2} \sqrt{c+d x^2} (b c-a d)^3 \Pi \left(1-\frac{b e}{a f};\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{a b^3 c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{d x \sqrt{c+d x^2} \sqrt{e+f x^2} (b c-a d)}{3 b^2}+\frac{x \sqrt{c+d x^2} (b c-a d) (-3 a d f+4 b c f+b d e)}{3 b^3 \sqrt{e+f x^2}}+\frac{d x \sqrt{c+d x^2} \left(\frac{3 c^2 f}{d}+7 c e-\frac{2 d e^2}{f}\right)}{15 b \sqrt{e+f x^2}}+\frac{d^2 x \sqrt{c+d x^2} \left(e+f x^2\right)^{3/2}}{5 b f}-\frac{2 d x \sqrt{c+d x^2} \sqrt{e+f x^2} (d e-3 c f)}{15 b f}",1,"(d*(7*c*e - (2*d*e^2)/f + (3*c^2*f)/d)*x*Sqrt[c + d*x^2])/(15*b*Sqrt[e + f*x^2]) + ((b*c - a*d)*(b*d*e + 4*b*c*f - 3*a*d*f)*x*Sqrt[c + d*x^2])/(3*b^3*Sqrt[e + f*x^2]) + (d*(b*c - a*d)*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(3*b^2) - (2*d*(d*e - 3*c*f)*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(15*b*f) + (d^2*x*Sqrt[c + d*x^2]*(e + f*x^2)^(3/2))/(5*b*f) - ((b*c - a*d)*Sqrt[e]*(b*d*e + 4*b*c*f - 3*a*d*f)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*b^3*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (Sqrt[e]*(2*d^2*e^2 - 7*c*d*e*f - 3*c^2*f^2)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*b*f^(3/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (d*(5*b*c - 3*a*d)*(b*c - a*d)*e^(3/2)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*b^3*c*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (d*e^(3/2)*(d*e - 9*c*f)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*b*f^(3/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + ((b*c - a*d)^3*e^(3/2)*Sqrt[c + d*x^2]*EllipticPi[1 - (b*e)/(a*f), ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(a*b^3*c*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",14,9,32,0.2812,1,"{545, 416, 528, 531, 418, 492, 411, 543, 539}"
65,1,400,0,0.2891891,"\int \frac{\left(c+d x^2\right)^{3/2} \sqrt{e+f x^2}}{a+b x^2} \, dx","Int[((c + d*x^2)^(3/2)*Sqrt[e + f*x^2])/(a + b*x^2),x]","\frac{d e^{3/2} \sqrt{c+d x^2} (5 b c-3 a d) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 b^2 c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{e^{3/2} \sqrt{c+d x^2} (b c-a d)^2 \Pi \left(1-\frac{b e}{a f};\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{a b^2 c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{x \sqrt{c+d x^2} (-3 a d f+4 b c f+b d e)}{3 b^2 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} (-3 a d f+4 b c f+b d e) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 b^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{d x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 b}","\frac{d e^{3/2} \sqrt{c+d x^2} (5 b c-3 a d) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 b^2 c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{e^{3/2} \sqrt{c+d x^2} (b c-a d)^2 \Pi \left(1-\frac{b e}{a f};\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{a b^2 c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{x \sqrt{c+d x^2} (-3 a d f+4 b c f+b d e)}{3 b^2 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} (-3 a d f+4 b c f+b d e) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 b^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{d x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 b}",1,"((b*d*e + 4*b*c*f - 3*a*d*f)*x*Sqrt[c + d*x^2])/(3*b^2*Sqrt[e + f*x^2]) + (d*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(3*b) - (Sqrt[e]*(b*d*e + 4*b*c*f - 3*a*d*f)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*b^2*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (d*(5*b*c - 3*a*d)*e^(3/2)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*b^2*c*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + ((b*c - a*d)^2*e^(3/2)*Sqrt[c + d*x^2]*EllipticPi[1 - (b*e)/(a*f), ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(a*b^2*c*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",7,7,32,0.2188,1,"{543, 539, 528, 531, 418, 492, 411}"
66,1,321,0,0.1909501,"\int \frac{\sqrt{c+d x^2} \sqrt{e+f x^2}}{a+b x^2} \, dx","Int[(Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(a + b*x^2),x]","\frac{e^{3/2} \sqrt{c+d x^2} (b c-a d) \Pi \left(1-\frac{b e}{a f};\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{a b c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{d e^{3/2} \sqrt{c+d x^2} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{b c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{f x \sqrt{c+d x^2}}{b \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{b \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}","\frac{e^{3/2} \sqrt{c+d x^2} (b c-a d) \Pi \left(1-\frac{b e}{a f};\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{a b c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{d e^{3/2} \sqrt{c+d x^2} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{b c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{f x \sqrt{c+d x^2}}{b \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{b \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}",1,"(f*x*Sqrt[c + d*x^2])/(b*Sqrt[e + f*x^2]) - (Sqrt[e]*Sqrt[f]*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(b*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (d*e^(3/2)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(b*c*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + ((b*c - a*d)*e^(3/2)*Sqrt[c + d*x^2]*EllipticPi[1 - (b*e)/(a*f), ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(a*b*c*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",6,6,32,0.1875,1,"{535, 422, 418, 492, 411, 539}"
67,1,102,0,0.033963,"\int \frac{\sqrt{e+f x^2}}{\left(a+b x^2\right) \sqrt{c+d x^2}} \, dx","Int[Sqrt[e + f*x^2]/((a + b*x^2)*Sqrt[c + d*x^2]),x]","\frac{e^{3/2} \sqrt{c+d x^2} \Pi \left(1-\frac{b e}{a f};\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{a c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}","\frac{e^{3/2} \sqrt{c+d x^2} \Pi \left(1-\frac{b e}{a f};\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{a c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}",1,"(e^(3/2)*Sqrt[c + d*x^2]*EllipticPi[1 - (b*e)/(a*f), ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(a*c*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",1,1,32,0.03125,1,"{539}"
68,1,209,0,0.1051856,"\int \frac{\sqrt{e+f x^2}}{\left(a+b x^2\right) \left(c+d x^2\right)^{3/2}} \, dx","Int[Sqrt[e + f*x^2]/((a + b*x^2)*(c + d*x^2)^(3/2)),x]","\frac{b e^{3/2} \sqrt{c+d x^2} \Pi \left(1-\frac{b e}{a f};\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{a c \sqrt{f} \sqrt{e+f x^2} (b c-a d) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{\sqrt{d} \sqrt{e+f x^2} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{\sqrt{c} \sqrt{c+d x^2} (b c-a d) \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}","\frac{b e^{3/2} \sqrt{c+d x^2} \Pi \left(1-\frac{b e}{a f};\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{a c \sqrt{f} \sqrt{e+f x^2} (b c-a d) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{\sqrt{d} \sqrt{e+f x^2} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{\sqrt{c} \sqrt{c+d x^2} (b c-a d) \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}",1,"-((Sqrt[d]*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(Sqrt[c]*(b*c - a*d)*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))])) + (b*e^(3/2)*Sqrt[c + d*x^2]*EllipticPi[1 - (b*e)/(a*f), ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(a*c*(b*c - a*d)*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",3,3,32,0.09375,1,"{541, 539, 411}"
69,1,401,0,0.3669651,"\int \frac{\sqrt{e+f x^2}}{\left(a+b x^2\right) \left(c+d x^2\right)^{5/2}} \, dx","Int[Sqrt[e + f*x^2]/((a + b*x^2)*(c + d*x^2)^(5/2)),x]","\frac{b^2 e^{3/2} \sqrt{c+d x^2} \Pi \left(1-\frac{b e}{a f};\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{a c \sqrt{f} \sqrt{e+f x^2} (b c-a d)^2 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{d e^{3/2} \sqrt{f} \sqrt{c+d x^2} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 c^2 \sqrt{e+f x^2} (b c-a d) (d e-c f) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{\sqrt{d} \sqrt{e+f x^2} (b c (5 d e-4 c f)-a d (2 d e-c f)) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{3 c^{3/2} \sqrt{c+d x^2} (b c-a d)^2 (d e-c f) \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}-\frac{d x \sqrt{e+f x^2}}{3 c \left(c+d x^2\right)^{3/2} (b c-a d)}","\frac{b^2 e^{3/2} \sqrt{c+d x^2} \Pi \left(1-\frac{b e}{a f};\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{a c \sqrt{f} \sqrt{e+f x^2} (b c-a d)^2 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{d e^{3/2} \sqrt{f} \sqrt{c+d x^2} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 c^2 \sqrt{e+f x^2} (b c-a d) (d e-c f) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{\sqrt{d} \sqrt{e+f x^2} (b c (5 d e-4 c f)-a d (2 d e-c f)) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{3 c^{3/2} \sqrt{c+d x^2} (b c-a d)^2 (d e-c f) \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}-\frac{d x \sqrt{e+f x^2}}{3 c \left(c+d x^2\right)^{3/2} (b c-a d)}",1,"-(d*x*Sqrt[e + f*x^2])/(3*c*(b*c - a*d)*(c + d*x^2)^(3/2)) - (Sqrt[d]*(b*c*(5*d*e - 4*c*f) - a*d*(2*d*e - c*f))*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(3*c^(3/2)*(b*c - a*d)^2*(d*e - c*f)*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]) + (d*e^(3/2)*Sqrt[f]*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*c^2*(b*c - a*d)*(d*e - c*f)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (b^2*e^(3/2)*Sqrt[c + d*x^2]*EllipticPi[1 - (b*e)/(a*f), ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(a*c*(b*c - a*d)^2*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",6,6,32,0.1875,1,"{546, 539, 526, 525, 418, 411}"
70,1,630,0,0.7190614,"\int \frac{\sqrt{e+f x^2}}{\left(a+b x^2\right) \left(c+d x^2\right)^{7/2}} \, dx","Int[Sqrt[e + f*x^2]/((a + b*x^2)*(c + d*x^2)^(7/2)),x]","\frac{b^3 e^{3/2} \sqrt{c+d x^2} \Pi \left(1-\frac{b e}{a f};\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{a c \sqrt{f} \sqrt{e+f x^2} (b c-a d)^3 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{b^2 \sqrt{d} \sqrt{e+f x^2} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{\sqrt{c} \sqrt{c+d x^2} (b c-a d)^3 \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}+\frac{\sqrt{d} \sqrt{e+f x^2} \left(a d \left(3 c^2 f^2-13 c d e f+8 d^2 e^2\right)-2 b c \left(4 c^2 f^2-14 c d e f+9 d^2 e^2\right)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{15 c^{5/2} \sqrt{c+d x^2} (b c-a d)^2 (d e-c f)^2 \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}+\frac{d e^{3/2} \sqrt{f} \sqrt{c+d x^2} (b c (9 d e-11 c f)-2 a d (2 d e-3 c f)) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 c^3 \sqrt{e+f x^2} (b c-a d)^2 (d e-c f)^2 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{d x \sqrt{e+f x^2} (b c (9 d e-8 c f)-a d (4 d e-3 c f))}{15 c^2 \left(c+d x^2\right)^{3/2} (b c-a d)^2 (d e-c f)}-\frac{d x \sqrt{e+f x^2}}{5 c \left(c+d x^2\right)^{5/2} (b c-a d)}","\frac{b^3 e^{3/2} \sqrt{c+d x^2} \Pi \left(1-\frac{b e}{a f};\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{a c \sqrt{f} \sqrt{e+f x^2} (b c-a d)^3 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{b^2 \sqrt{d} \sqrt{e+f x^2} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{\sqrt{c} \sqrt{c+d x^2} (b c-a d)^3 \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}+\frac{\sqrt{d} \sqrt{e+f x^2} \left(a d \left(3 c^2 f^2-13 c d e f+8 d^2 e^2\right)-2 b c \left(4 c^2 f^2-14 c d e f+9 d^2 e^2\right)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{15 c^{5/2} \sqrt{c+d x^2} (b c-a d)^2 (d e-c f)^2 \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}+\frac{d e^{3/2} \sqrt{f} \sqrt{c+d x^2} (b c (9 d e-11 c f)-2 a d (2 d e-3 c f)) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 c^3 \sqrt{e+f x^2} (b c-a d)^2 (d e-c f)^2 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{d x \sqrt{e+f x^2} (b c (9 d e-8 c f)-a d (4 d e-3 c f))}{15 c^2 \left(c+d x^2\right)^{3/2} (b c-a d)^2 (d e-c f)}-\frac{d x \sqrt{e+f x^2}}{5 c \left(c+d x^2\right)^{5/2} (b c-a d)}",1,"-(d*x*Sqrt[e + f*x^2])/(5*c*(b*c - a*d)*(c + d*x^2)^(5/2)) - (d*(b*c*(9*d*e - 8*c*f) - a*d*(4*d*e - 3*c*f))*x*Sqrt[e + f*x^2])/(15*c^2*(b*c - a*d)^2*(d*e - c*f)*(c + d*x^2)^(3/2)) - (b^2*Sqrt[d]*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(Sqrt[c]*(b*c - a*d)^3*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]) + (Sqrt[d]*(a*d*(8*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2) - 2*b*c*(9*d^2*e^2 - 14*c*d*e*f + 4*c^2*f^2))*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(15*c^(5/2)*(b*c - a*d)^2*(d*e - c*f)^2*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]) + (d*e^(3/2)*Sqrt[f]*(b*c*(9*d*e - 11*c*f) - 2*a*d*(2*d*e - 3*c*f))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*c^3*(b*c - a*d)^2*(d*e - c*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (b^3*e^(3/2)*Sqrt[c + d*x^2]*EllipticPi[1 - (b*e)/(a*f), ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(a*c*(b*c - a*d)^3*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",9,8,32,0.2500,1,"{546, 541, 539, 411, 526, 527, 525, 418}"
71,1,784,0,0.7527047,"\int \frac{\left(c+d x^2\right)^{3/2} \left(e+f x^2\right)^{3/2}}{a+b x^2} \, dx","Int[((c + d*x^2)^(3/2)*(e + f*x^2)^(3/2))/(a + b*x^2),x]","\frac{c^{3/2} \sqrt{e+f x^2} (b c-a d) (b e-a f)^2 \Pi \left(1-\frac{b c}{a d};\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{a b^3 \sqrt{d} e \sqrt{c+d x^2} \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}+\frac{f x \sqrt{c+d x^2} \sqrt{e+f x^2} (b c-a d)}{3 b^2}+\frac{f x \sqrt{c+d x^2} (b c-a d) (-3 a d f+b c f+4 b d e)}{3 b^3 d \sqrt{e+f x^2}}+\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (b c-a d) (5 b e-3 a f) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 b^3 \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (b c-a d) (-3 a d f+b c f+4 b d e) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 b^3 d \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{x \sqrt{c+d x^2} \left(-2 c^2 f^2+7 c d e f+3 d^2 e^2\right)}{15 b d \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left(-2 c^2 f^2+7 c d e f+3 d^2 e^2\right) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 b d \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{e^{3/2} \sqrt{c+d x^2} (9 d e-c f) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 b \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{f x \left(c+d x^2\right)^{3/2} \sqrt{e+f x^2}}{5 b}+\frac{2 x \sqrt{c+d x^2} \sqrt{e+f x^2} (3 d e-c f)}{15 b}","-\frac{\sqrt{e} \sqrt{c+d x^2} \left(15 a^2 d^2 f^2-20 a b d f (c f+d e)+3 b^2 \left(c^2 f^2+9 c d e f+d^2 e^2\right)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 b^3 d \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{e^{3/2} \sqrt{c+d x^2} \left(15 a^2 d^2 f-5 a b d (5 c f+3 d e)+3 b^2 c (3 c f+8 d e)\right) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 b^3 c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{e^{3/2} \sqrt{c+d x^2} (b c-a d)^2 (b e-a f) \Pi \left(1-\frac{b e}{a f};\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{a b^3 c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{f^2 x \sqrt{c+d x^2} (b c-a d)^2}{b^3 d \sqrt{e+f x^2}}+\frac{f x \sqrt{c+d x^2} \sqrt{e+f x^2} (b c-a d)}{3 b^2}+\frac{2 f x \sqrt{c+d x^2} (b c-a d) (2 d e-c f)}{3 b^2 d \sqrt{e+f x^2}}+\frac{x \sqrt{c+d x^2} \left(-2 c^2 f^2+7 c d e f+3 d^2 e^2\right)}{15 b d \sqrt{e+f x^2}}+\frac{f x \left(c+d x^2\right)^{3/2} \sqrt{e+f x^2}}{5 b}+\frac{2 x \sqrt{c+d x^2} \sqrt{e+f x^2} (3 d e-c f)}{15 b}",1,"((b*c - a*d)*f*(4*b*d*e + b*c*f - 3*a*d*f)*x*Sqrt[c + d*x^2])/(3*b^3*d*Sqrt[e + f*x^2]) + ((3*d^2*e^2 + 7*c*d*e*f - 2*c^2*f^2)*x*Sqrt[c + d*x^2])/(15*b*d*Sqrt[e + f*x^2]) + ((b*c - a*d)*f*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(3*b^2) + (2*(3*d*e - c*f)*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(15*b) + (f*x*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2])/(5*b) - ((b*c - a*d)*Sqrt[e]*Sqrt[f]*(4*b*d*e + b*c*f - 3*a*d*f)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*b^3*d*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (Sqrt[e]*(3*d^2*e^2 + 7*c*d*e*f - 2*c^2*f^2)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*b*d*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + ((b*c - a*d)*Sqrt[e]*Sqrt[f]*(5*b*e - 3*a*f)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*b^3*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (e^(3/2)*(9*d*e - c*f)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*b*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (c^(3/2)*(b*c - a*d)*(b*e - a*f)^2*Sqrt[e + f*x^2]*EllipticPi[1 - (b*c)/(a*d), ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(a*b^3*Sqrt[d]*e*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))])","A",14,9,32,0.2812,1,"{545, 416, 528, 531, 418, 492, 411, 543, 539}"
72,1,403,0,0.3040382,"\int \frac{\sqrt{c+d x^2} \left(e+f x^2\right)^{3/2}}{a+b x^2} \, dx","Int[(Sqrt[c + d*x^2]*(e + f*x^2)^(3/2))/(a + b*x^2),x]","\frac{c^{3/2} \sqrt{e+f x^2} (b e-a f)^2 \Pi \left(1-\frac{b c}{a d};\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{a b^2 \sqrt{d} e \sqrt{c+d x^2} \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}+\frac{f x \sqrt{c+d x^2} (-3 a d f+b c f+4 b d e)}{3 b^2 d \sqrt{e+f x^2}}+\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (5 b e-3 a f) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 b^2 \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (-3 a d f+b c f+4 b d e) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 b^2 d \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{f x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 b}","\frac{c^{3/2} \sqrt{e+f x^2} (b e-a f)^2 \Pi \left(1-\frac{b c}{a d};\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{a b^2 \sqrt{d} e \sqrt{c+d x^2} \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}+\frac{f x \sqrt{c+d x^2} (-3 a d f+b c f+4 b d e)}{3 b^2 d \sqrt{e+f x^2}}+\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (5 b e-3 a f) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 b^2 \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (-3 a d f+b c f+4 b d e) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 b^2 d \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{f x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 b}",1,"(f*(4*b*d*e + b*c*f - 3*a*d*f)*x*Sqrt[c + d*x^2])/(3*b^2*d*Sqrt[e + f*x^2]) + (f*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(3*b) - (Sqrt[e]*Sqrt[f]*(4*b*d*e + b*c*f - 3*a*d*f)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*b^2*d*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (Sqrt[e]*Sqrt[f]*(5*b*e - 3*a*f)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*b^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (c^(3/2)*(b*e - a*f)^2*Sqrt[e + f*x^2]*EllipticPi[1 - (b*c)/(a*d), ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(a*b^2*Sqrt[d]*e*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))])","A",7,7,32,0.2188,1,"{543, 539, 528, 531, 418, 492, 411}"
73,1,328,0,0.1912378,"\int \frac{\left(e+f x^2\right)^{3/2}}{\left(a+b x^2\right) \sqrt{c+d x^2}} \, dx","Int[(e + f*x^2)^(3/2)/((a + b*x^2)*Sqrt[c + d*x^2]),x]","\frac{e^{3/2} \sqrt{c+d x^2} (b e-a f) \Pi \left(1-\frac{b e}{a f};\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{a b c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{b c \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{f^2 x \sqrt{c+d x^2}}{b d \sqrt{e+f x^2}}-\frac{\sqrt{e} f^{3/2} \sqrt{c+d x^2} E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{b d \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}","\frac{e^{3/2} \sqrt{c+d x^2} (b e-a f) \Pi \left(1-\frac{b e}{a f};\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{a b c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{b c \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{f^2 x \sqrt{c+d x^2}}{b d \sqrt{e+f x^2}}-\frac{\sqrt{e} f^{3/2} \sqrt{c+d x^2} E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{b d \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}",1,"(f^2*x*Sqrt[c + d*x^2])/(b*d*Sqrt[e + f*x^2]) - (Sqrt[e]*f^(3/2)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(b*d*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (e^(3/2)*Sqrt[f]*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(b*c*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (e^(3/2)*(b*e - a*f)*Sqrt[c + d*x^2]*EllipticPi[1 - (b*e)/(a*f), ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(a*b*c*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",6,6,32,0.1875,1,"{545, 422, 418, 492, 411, 539}"
74,1,224,0,0.1117988,"\int \frac{\left(e+f x^2\right)^{3/2}}{\left(a+b x^2\right) \left(c+d x^2\right)^{3/2}} \, dx","Int[(e + f*x^2)^(3/2)/((a + b*x^2)*(c + d*x^2)^(3/2)),x]","\frac{e^{3/2} \sqrt{c+d x^2} (b e-a f) \Pi \left(1-\frac{b e}{a f};\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{a c \sqrt{f} \sqrt{e+f x^2} (b c-a d) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{\sqrt{e+f x^2} (d e-c f) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{\sqrt{c} \sqrt{d} \sqrt{c+d x^2} (b c-a d) \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}","\frac{e^{3/2} \sqrt{c+d x^2} (b e-a f) \Pi \left(1-\frac{b e}{a f};\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{a c \sqrt{f} \sqrt{e+f x^2} (b c-a d) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{\sqrt{e+f x^2} (d e-c f) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{\sqrt{c} \sqrt{d} \sqrt{c+d x^2} (b c-a d) \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}",1,"-(((d*e - c*f)*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(Sqrt[c]*Sqrt[d]*(b*c - a*d)*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))])) + (e^(3/2)*(b*e - a*f)*Sqrt[c + d*x^2]*EllipticPi[1 - (b*e)/(a*f), ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(a*c*(b*c - a*d)*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",3,3,32,0.09375,1,"{542, 539, 411}"
75,1,391,0,0.4041174,"\int \frac{\left(e+f x^2\right)^{3/2}}{\left(a+b x^2\right) \left(c+d x^2\right)^{5/2}} \, dx","Int[(e + f*x^2)^(3/2)/((a + b*x^2)*(c + d*x^2)^(5/2)),x]","\frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 c^2 \sqrt{e+f x^2} (b c-a d) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{\sqrt{e+f x^2} (b c (5 d e-c f)-2 a d (c f+d e)) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{3 c^{3/2} \sqrt{d} \sqrt{c+d x^2} (b c-a d)^2 \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}+\frac{b e^{3/2} \sqrt{c+d x^2} (b e-a f) \Pi \left(1-\frac{b e}{a f};\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{a c \sqrt{f} \sqrt{e+f x^2} (b c-a d)^2 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{x \sqrt{e+f x^2} (d e-c f)}{3 c \left(c+d x^2\right)^{3/2} (b c-a d)}","\frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 c^2 \sqrt{e+f x^2} (b c-a d) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{\sqrt{e+f x^2} (b c (5 d e-c f)-2 a d (c f+d e)) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{3 c^{3/2} \sqrt{d} \sqrt{c+d x^2} (b c-a d)^2 \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}+\frac{b e^{3/2} \sqrt{c+d x^2} (b e-a f) \Pi \left(1-\frac{b e}{a f};\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{a c \sqrt{f} \sqrt{e+f x^2} (b c-a d)^2 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{x \sqrt{e+f x^2} (d e-c f)}{3 c \left(c+d x^2\right)^{3/2} (b c-a d)}",1,"-((d*e - c*f)*x*Sqrt[e + f*x^2])/(3*c*(b*c - a*d)*(c + d*x^2)^(3/2)) - ((b*c*(5*d*e - c*f) - 2*a*d*(d*e + c*f))*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(3*c^(3/2)*Sqrt[d]*(b*c - a*d)^2*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]) + (e^(3/2)*Sqrt[f]*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*c^2*(b*c - a*d)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (b*e^(3/2)*(b*e - a*f)*Sqrt[c + d*x^2]*EllipticPi[1 - (b*e)/(a*f), ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(a*c*(b*c - a*d)^2*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",6,6,32,0.1875,1,"{544, 539, 526, 525, 418, 411}"
76,1,639,0,0.7794073,"\int \frac{\left(e+f x^2\right)^{3/2}}{\left(a+b x^2\right) \left(c+d x^2\right)^{7/2}} \, dx","Int[(e + f*x^2)^(3/2)/((a + b*x^2)*(c + d*x^2)^(7/2)),x]","\frac{b^2 e^{3/2} \sqrt{c+d x^2} (b e-a f) \Pi \left(1-\frac{b e}{a f};\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{a c \sqrt{f} \sqrt{e+f x^2} (b c-a d)^3 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{\sqrt{e+f x^2} \left(a d \left(-2 c^2 f^2-3 c d e f+8 d^2 e^2\right)-3 b c \left(c^2 f^2-6 c d e f+6 d^2 e^2\right)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{15 c^{5/2} \sqrt{d} \sqrt{c+d x^2} (b c-a d)^2 (d e-c f) \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}+\frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} (3 b c (3 d e-2 c f)-a d (4 d e-c f)) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 c^3 \sqrt{e+f x^2} (b c-a d)^2 (d e-c f) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{x \sqrt{e+f x^2} (3 b c (3 d e-c f)-2 a d (c f+2 d e))}{15 c^2 \left(c+d x^2\right)^{3/2} (b c-a d)^2}-\frac{x \sqrt{e+f x^2} (d e-c f)}{5 c \left(c+d x^2\right)^{5/2} (b c-a d)}-\frac{b \sqrt{d} \sqrt{e+f x^2} (b e-a f) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{\sqrt{c} \sqrt{c+d x^2} (b c-a d)^3 \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}","\frac{b^2 e^{3/2} \sqrt{c+d x^2} (b e-a f) \Pi \left(1-\frac{b e}{a f};\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{a c \sqrt{f} \sqrt{e+f x^2} (b c-a d)^3 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{\sqrt{e+f x^2} \left(a d \left(-2 c^2 f^2-3 c d e f+8 d^2 e^2\right)-3 b c \left(c^2 f^2-6 c d e f+6 d^2 e^2\right)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{15 c^{5/2} \sqrt{d} \sqrt{c+d x^2} (b c-a d)^2 (d e-c f) \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}+\frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} (3 b c (3 d e-2 c f)-a d (4 d e-c f)) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 c^3 \sqrt{e+f x^2} (b c-a d)^2 (d e-c f) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{x \sqrt{e+f x^2} (3 b c (3 d e-c f)-2 a d (c f+2 d e))}{15 c^2 \left(c+d x^2\right)^{3/2} (b c-a d)^2}-\frac{x \sqrt{e+f x^2} (d e-c f)}{5 c \left(c+d x^2\right)^{5/2} (b c-a d)}-\frac{b \sqrt{d} \sqrt{e+f x^2} (b e-a f) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{\sqrt{c} \sqrt{c+d x^2} (b c-a d)^3 \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}",1,"-((d*e - c*f)*x*Sqrt[e + f*x^2])/(5*c*(b*c - a*d)*(c + d*x^2)^(5/2)) - ((3*b*c*(3*d*e - c*f) - 2*a*d*(2*d*e + c*f))*x*Sqrt[e + f*x^2])/(15*c^2*(b*c - a*d)^2*(c + d*x^2)^(3/2)) - (b*Sqrt[d]*(b*e - a*f)*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(Sqrt[c]*(b*c - a*d)^3*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]) + ((a*d*(8*d^2*e^2 - 3*c*d*e*f - 2*c^2*f^2) - 3*b*c*(6*d^2*e^2 - 6*c*d*e*f + c^2*f^2))*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(15*c^(5/2)*Sqrt[d]*(b*c - a*d)^2*(d*e - c*f)*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]) + (e^(3/2)*Sqrt[f]*(3*b*c*(3*d*e - 2*c*f) - a*d*(4*d*e - c*f))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*c^3*(b*c - a*d)^2*(d*e - c*f)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (b^2*e^(3/2)*(b*e - a*f)*Sqrt[c + d*x^2]*EllipticPi[1 - (b*e)/(a*f), ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(a*c*(b*c - a*d)^3*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",9,8,32,0.2500,1,"{544, 541, 539, 411, 526, 527, 525, 418}"
77,1,621,0,0.4626584,"\int \frac{\left(c+d x^2\right)^{5/2}}{\left(a+b x^2\right) \sqrt{e+f x^2}} \, dx","Int[(c + d*x^2)^(5/2)/((a + b*x^2)*Sqrt[e + f*x^2]),x]","\frac{c^{3/2} \sqrt{e+f x^2} (b c-a d)^2 \Pi \left(1-\frac{b c}{a d};\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{a b^2 \sqrt{d} e \sqrt{c+d x^2} \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}+\frac{d x \sqrt{c+d x^2} (b c-a d)}{b^2 \sqrt{e+f x^2}}+\frac{d \sqrt{e} \sqrt{c+d x^2} (b c-a d) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{b^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{d \sqrt{e} \sqrt{c+d x^2} (b c-a d) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{b^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{d^2 x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 b f}-\frac{d \sqrt{e} \sqrt{c+d x^2} (d e-3 c f) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 b f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{2 d \sqrt{e} \sqrt{c+d x^2} (d e-2 c f) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 b f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{2 d x \sqrt{c+d x^2} (d e-2 c f)}{3 b f \sqrt{e+f x^2}}","\frac{c^{3/2} \sqrt{e+f x^2} (b c-a d)^2 \Pi \left(1-\frac{b c}{a d};\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{a b^2 \sqrt{d} e \sqrt{c+d x^2} \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}+\frac{d x \sqrt{c+d x^2} (b c-a d)}{b^2 \sqrt{e+f x^2}}+\frac{d \sqrt{e} \sqrt{c+d x^2} (b c-a d) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{b^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{d \sqrt{e} \sqrt{c+d x^2} (b c-a d) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{b^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{d^2 x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 b f}-\frac{d \sqrt{e} \sqrt{c+d x^2} (d e-3 c f) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 b f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{2 d \sqrt{e} \sqrt{c+d x^2} (d e-2 c f) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 b f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{2 d x \sqrt{c+d x^2} (d e-2 c f)}{3 b f \sqrt{e+f x^2}}",1,"(d*(b*c - a*d)*x*Sqrt[c + d*x^2])/(b^2*Sqrt[e + f*x^2]) - (2*d*(d*e - 2*c*f)*x*Sqrt[c + d*x^2])/(3*b*f*Sqrt[e + f*x^2]) + (d^2*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(3*b*f) - (d*(b*c - a*d)*Sqrt[e]*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(b^2*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (2*d*Sqrt[e]*(d*e - 2*c*f)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*b*f^(3/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (d*(b*c - a*d)*Sqrt[e]*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(b^2*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (d*Sqrt[e]*(d*e - 3*c*f)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*b*f^(3/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (c^(3/2)*(b*c - a*d)^2*Sqrt[e + f*x^2]*EllipticPi[1 - (b*c)/(a*d), ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(a*b^2*Sqrt[d]*e*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))])","A",12,8,32,0.2500,1,"{545, 416, 531, 418, 492, 411, 422, 539}"
78,1,319,0,0.1755112,"\int \frac{\left(c+d x^2\right)^{3/2}}{\left(a+b x^2\right) \sqrt{e+f x^2}} \, dx","Int[(c + d*x^2)^(3/2)/((a + b*x^2)*Sqrt[e + f*x^2]),x]","\frac{c^{3/2} \sqrt{e+f x^2} (b c-a d) \Pi \left(1-\frac{b c}{a d};\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{a b \sqrt{d} e \sqrt{c+d x^2} \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}+\frac{d x \sqrt{c+d x^2}}{b \sqrt{e+f x^2}}+\frac{d \sqrt{e} \sqrt{c+d x^2} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{b \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{d \sqrt{e} \sqrt{c+d x^2} E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{b \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}","\frac{c^{3/2} \sqrt{e+f x^2} (b c-a d) \Pi \left(1-\frac{b c}{a d};\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{a b \sqrt{d} e \sqrt{c+d x^2} \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}+\frac{d x \sqrt{c+d x^2}}{b \sqrt{e+f x^2}}+\frac{d \sqrt{e} \sqrt{c+d x^2} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{b \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{d \sqrt{e} \sqrt{c+d x^2} E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{b \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}",1,"(d*x*Sqrt[c + d*x^2])/(b*Sqrt[e + f*x^2]) - (d*Sqrt[e]*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(b*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (d*Sqrt[e]*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(b*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (c^(3/2)*(b*c - a*d)*Sqrt[e + f*x^2]*EllipticPi[1 - (b*c)/(a*d), ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(a*b*Sqrt[d]*e*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))])","A",6,6,32,0.1875,1,"{545, 422, 418, 492, 411, 539}"
79,1,102,0,0.0326949,"\int \frac{\sqrt{c+d x^2}}{\left(a+b x^2\right) \sqrt{e+f x^2}} \, dx","Int[Sqrt[c + d*x^2]/((a + b*x^2)*Sqrt[e + f*x^2]),x]","\frac{c^{3/2} \sqrt{e+f x^2} \Pi \left(1-\frac{b c}{a d};\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{a \sqrt{d} e \sqrt{c+d x^2} \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}","\frac{c^{3/2} \sqrt{e+f x^2} \Pi \left(1-\frac{b c}{a d};\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{a \sqrt{d} e \sqrt{c+d x^2} \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}",1,"(c^(3/2)*Sqrt[e + f*x^2]*EllipticPi[1 - (b*c)/(a*d), ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(a*Sqrt[d]*e*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))])","A",1,1,32,0.03125,1,"{539}"
80,1,100,0,0.1208076,"\int \frac{1}{\left(a+b x^2\right) \sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx","Int[1/((a + b*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]","\frac{\sqrt{-c} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{-c}}\right)|\frac{c f}{d e}\right)}{a \sqrt{d} \sqrt{c+d x^2} \sqrt{e+f x^2}}","\frac{\sqrt{-c} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{-c}}\right)|\frac{c f}{d e}\right)}{a \sqrt{d} \sqrt{c+d x^2} \sqrt{e+f x^2}}",1,"(Sqrt[-c]*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), ArcSin[(Sqrt[d]*x)/Sqrt[-c]], (c*f)/(d*e)])/(a*Sqrt[d]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","A",3,2,32,0.06250,1,"{538, 537}"
81,1,344,0,0.2203749,"\int \frac{1}{\left(a+b x^2\right) \left(c+d x^2\right)^{3/2} \sqrt{e+f x^2}} \, dx","Int[1/((a + b*x^2)*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2]),x]","\frac{b^2 c^{3/2} \sqrt{e+f x^2} \Pi \left(1-\frac{b c}{a d};\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{a \sqrt{d} e \sqrt{c+d x^2} (b c-a d)^2 \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}-\frac{d^{3/2} \sqrt{e+f x^2} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{\sqrt{c} \sqrt{c+d x^2} (b c-a d) (d e-c f) \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}-\frac{d \sqrt{e} \sqrt{c+d x^2} (a d f-2 b c f+b d e) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{c \sqrt{f} \sqrt{e+f x^2} (b c-a d)^2 (d e-c f) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}","\frac{b^2 c^{3/2} \sqrt{e+f x^2} \Pi \left(1-\frac{b c}{a d};\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{a \sqrt{d} e \sqrt{c+d x^2} (b c-a d)^2 \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}-\frac{d^{3/2} \sqrt{e+f x^2} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{\sqrt{c} \sqrt{c+d x^2} (b c-a d) (d e-c f) \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}-\frac{d \sqrt{e} \sqrt{c+d x^2} (a d f-2 b c f+b d e) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{c \sqrt{f} \sqrt{e+f x^2} (b c-a d)^2 (d e-c f) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}",1,"-((d^(3/2)*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(Sqrt[c]*(b*c - a*d)*(d*e - c*f)*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))])) - (d*Sqrt[e]*(b*d*e - 2*b*c*f + a*d*f)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(c*(b*c - a*d)^2*Sqrt[f]*(d*e - c*f)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (b^2*c^(3/2)*Sqrt[e + f*x^2]*EllipticPi[1 - (b*c)/(a*d), ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(a*Sqrt[d]*(b*c - a*d)^2*e*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))])","A",5,5,32,0.1562,1,"{546, 539, 525, 418, 411}"
82,1,435,0,0.5542129,"\int \frac{1}{\left(a+b x^2\right) \left(c+d x^2\right)^{5/2} \sqrt{e+f x^2}} \, dx","Int[1/((a + b*x^2)*(c + d*x^2)^(5/2)*Sqrt[e + f*x^2]),x]","\frac{b^2 \sqrt{-c} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{-c}}\right)|\frac{c f}{d e}\right)}{a \sqrt{d} \sqrt{c+d x^2} \sqrt{e+f x^2} (b c-a d)^2}-\frac{d^{3/2} \sqrt{e+f x^2} (b c (5 d e-7 c f)-2 a d (d e-2 c f)) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{3 c^{3/2} \sqrt{c+d x^2} (b c-a d)^2 (d e-c f)^2 \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}-\frac{d \sqrt{e} \sqrt{f} \sqrt{c+d x^2} (a d (d e-3 c f)-2 b c (2 d e-3 c f)) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 c^2 \sqrt{e+f x^2} (b c-a d)^2 (d e-c f)^2 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{d^2 x \sqrt{e+f x^2}}{3 c \left(c+d x^2\right)^{3/2} (b c-a d) (d e-c f)}","\frac{b^2 \sqrt{-c} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{-c}}\right)|\frac{c f}{d e}\right)}{a \sqrt{d} \sqrt{c+d x^2} \sqrt{e+f x^2} (b c-a d)^2}-\frac{d^{3/2} \sqrt{e+f x^2} (b c (5 d e-7 c f)-2 a d (d e-2 c f)) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{3 c^{3/2} \sqrt{c+d x^2} (b c-a d)^2 (d e-c f)^2 \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}-\frac{d \sqrt{e} \sqrt{f} \sqrt{c+d x^2} (a d (d e-3 c f)-2 b c (2 d e-3 c f)) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 c^2 \sqrt{e+f x^2} (b c-a d)^2 (d e-c f)^2 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{d^2 x \sqrt{e+f x^2}}{3 c \left(c+d x^2\right)^{3/2} (b c-a d) (d e-c f)}",1,"-(d^2*x*Sqrt[e + f*x^2])/(3*c*(b*c - a*d)*(d*e - c*f)*(c + d*x^2)^(3/2)) - (d^(3/2)*(b*c*(5*d*e - 7*c*f) - 2*a*d*(d*e - 2*c*f))*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(3*c^(3/2)*(b*c - a*d)^2*(d*e - c*f)^2*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]) - (d*Sqrt[e]*Sqrt[f]*(a*d*(d*e - 3*c*f) - 2*b*c*(2*d*e - 3*c*f))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*c^2*(b*c - a*d)^2*(d*e - c*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (b^2*Sqrt[-c]*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), ArcSin[(Sqrt[d]*x)/Sqrt[-c]], (c*f)/(d*e)])/(a*Sqrt[d]*(b*c - a*d)^2*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","A",8,7,32,0.2188,1,"{546, 538, 537, 527, 525, 418, 411}"
83,1,980,0,1.1133363,"\int \frac{\left(c+d x^2\right)^{5/2}}{\left(a+b x^2\right) \left(e+f x^2\right)^{3/2}} \, dx","Int[(c + d*x^2)^(5/2)/((a + b*x^2)*(e + f*x^2)^(3/2)),x]","\frac{e^{3/2} \sqrt{d x^2+c} \Pi \left(1-\frac{b e}{a f};\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right) (b c-a d)^3}{a b c \sqrt{f} (b e-a f)^2 \sqrt{\frac{e \left(d x^2+c\right)}{c \left(f x^2+e\right)}} \sqrt{f x^2+e}}-\frac{\sqrt{e} (b d e+4 b c f-3 a d f) \sqrt{d x^2+c} E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right) (b c-a d)}{3 b \sqrt{f} (b e-a f)^2 \sqrt{\frac{e \left(d x^2+c\right)}{c \left(f x^2+e\right)}} \sqrt{f x^2+e}}+\frac{d (5 b c-3 a d) e^{3/2} \sqrt{d x^2+c} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right) (b c-a d)}{3 b c \sqrt{f} (b e-a f)^2 \sqrt{\frac{e \left(d x^2+c\right)}{c \left(f x^2+e\right)}} \sqrt{f x^2+e}}+\frac{d x \sqrt{d x^2+c} \sqrt{f x^2+e} (b c-a d)}{3 (b e-a f)^2}+\frac{(b d e+4 b c f-3 a d f) x \sqrt{d x^2+c} (b c-a d)}{3 b (b e-a f)^2 \sqrt{f x^2+e}}-\frac{\left(b e \left(6 d^2 e^2-7 c d f e-c^2 f^2\right)-a f \left(8 d^2 e^2-13 c d f e+3 c^2 f^2\right)\right) \sqrt{d x^2+c} E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 \sqrt{e} f^{3/2} (b e-a f)^2 \sqrt{\frac{e \left(d x^2+c\right)}{c \left(f x^2+e\right)}} \sqrt{f x^2+e}}-\frac{\sqrt{e} \left(2 a d f (2 d e-3 c f)-b \left(3 d^2 e^2-2 c d f e-3 c^2 f^2\right)\right) \sqrt{d x^2+c} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 f^{3/2} (b e-a f)^2 \sqrt{\frac{e \left(d x^2+c\right)}{c \left(f x^2+e\right)}} \sqrt{f x^2+e}}+\frac{d (a f (4 d e-3 c f)-b e (3 d e-2 c f)) x \sqrt{d x^2+c} \sqrt{f x^2+e}}{3 e f (b e-a f)^2}+\frac{(d e-c f) x \left(d x^2+c\right)^{3/2}}{e (b e-a f) \sqrt{f x^2+e}}+\frac{\left(b e \left(6 d^2 e^2-7 c d f e-c^2 f^2\right)-a f \left(8 d^2 e^2-13 c d f e+3 c^2 f^2\right)\right) x \sqrt{d x^2+c}}{3 e f (b e-a f)^2 \sqrt{f x^2+e}}","\frac{e^{3/2} \sqrt{d x^2+c} \Pi \left(1-\frac{b e}{a f};\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right) (b c-a d)^3}{a b c \sqrt{f} (b e-a f)^2 \sqrt{\frac{e \left(d x^2+c\right)}{c \left(f x^2+e\right)}} \sqrt{f x^2+e}}-\frac{\sqrt{e} (b d e+4 b c f-3 a d f) \sqrt{d x^2+c} E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right) (b c-a d)}{3 b \sqrt{f} (b e-a f)^2 \sqrt{\frac{e \left(d x^2+c\right)}{c \left(f x^2+e\right)}} \sqrt{f x^2+e}}+\frac{d (5 b c-3 a d) e^{3/2} \sqrt{d x^2+c} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right) (b c-a d)}{3 b c \sqrt{f} (b e-a f)^2 \sqrt{\frac{e \left(d x^2+c\right)}{c \left(f x^2+e\right)}} \sqrt{f x^2+e}}+\frac{d x \sqrt{d x^2+c} \sqrt{f x^2+e} (b c-a d)}{3 (b e-a f)^2}+\frac{(b d e+4 b c f-3 a d f) x \sqrt{d x^2+c} (b c-a d)}{3 b (b e-a f)^2 \sqrt{f x^2+e}}-\frac{\left(b e \left(6 d^2 e^2-7 c d f e-c^2 f^2\right)-a f \left(8 d^2 e^2-13 c d f e+3 c^2 f^2\right)\right) \sqrt{d x^2+c} E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 \sqrt{e} f^{3/2} (b e-a f)^2 \sqrt{\frac{e \left(d x^2+c\right)}{c \left(f x^2+e\right)}} \sqrt{f x^2+e}}-\frac{\sqrt{e} \left(2 a d f (2 d e-3 c f)-b \left(3 d^2 e^2-2 c d f e-3 c^2 f^2\right)\right) \sqrt{d x^2+c} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 f^{3/2} (b e-a f)^2 \sqrt{\frac{e \left(d x^2+c\right)}{c \left(f x^2+e\right)}} \sqrt{f x^2+e}}+\frac{d (a f (4 d e-3 c f)-b e (3 d e-2 c f)) x \sqrt{d x^2+c} \sqrt{f x^2+e}}{3 e f (b e-a f)^2}+\frac{(d e-c f) x \left(d x^2+c\right)^{3/2}}{e (b e-a f) \sqrt{f x^2+e}}+\frac{\left(b e \left(6 d^2 e^2-7 c d f e-c^2 f^2\right)-a f \left(8 d^2 e^2-13 c d f e+3 c^2 f^2\right)\right) x \sqrt{d x^2+c}}{3 e f (b e-a f)^2 \sqrt{f x^2+e}}",1,"((b*c - a*d)*(b*d*e + 4*b*c*f - 3*a*d*f)*x*Sqrt[c + d*x^2])/(3*b*(b*e - a*f)^2*Sqrt[e + f*x^2]) + ((b*e*(6*d^2*e^2 - 7*c*d*e*f - c^2*f^2) - a*f*(8*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2))*x*Sqrt[c + d*x^2])/(3*e*f*(b*e - a*f)^2*Sqrt[e + f*x^2]) + ((d*e - c*f)*x*(c + d*x^2)^(3/2))/(e*(b*e - a*f)*Sqrt[e + f*x^2]) + (d*(b*c - a*d)*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(3*(b*e - a*f)^2) + (d*(a*f*(4*d*e - 3*c*f) - b*e*(3*d*e - 2*c*f))*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(3*e*f*(b*e - a*f)^2) - ((b*c - a*d)*Sqrt[e]*(b*d*e + 4*b*c*f - 3*a*d*f)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*b*Sqrt[f]*(b*e - a*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - ((b*e*(6*d^2*e^2 - 7*c*d*e*f - c^2*f^2) - a*f*(8*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2))*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*Sqrt[e]*f^(3/2)*(b*e - a*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (d*(5*b*c - 3*a*d)*(b*c - a*d)*e^(3/2)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*b*c*Sqrt[f]*(b*e - a*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (Sqrt[e]*(2*a*d*f*(2*d*e - 3*c*f) - b*(3*d^2*e^2 - 2*c*d*e*f - 3*c^2*f^2))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*f^(3/2)*(b*e - a*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + ((b*c - a*d)^3*e^(3/2)*Sqrt[c + d*x^2]*EllipticPi[1 - (b*e)/(a*f), ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(a*b*c*Sqrt[f]*(b*e - a*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",14,9,32,0.2812,1,"{544, 543, 539, 528, 531, 418, 492, 411, 526}"
84,1,223,0,0.1203642,"\int \frac{\left(c+d x^2\right)^{3/2}}{\left(a+b x^2\right) \left(e+f x^2\right)^{3/2}} \, dx","Int[(c + d*x^2)^(3/2)/((a + b*x^2)*(e + f*x^2)^(3/2)),x]","\frac{c^{3/2} \sqrt{e+f x^2} (b c-a d) \Pi \left(1-\frac{b c}{a d};\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{a \sqrt{d} e \sqrt{c+d x^2} (b e-a f) \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}+\frac{\sqrt{c+d x^2} (d e-c f) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{\sqrt{e} \sqrt{f} \sqrt{e+f x^2} (b e-a f) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}","\frac{c^{3/2} \sqrt{e+f x^2} (b c-a d) \Pi \left(1-\frac{b c}{a d};\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{a \sqrt{d} e \sqrt{c+d x^2} (b e-a f) \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}+\frac{\sqrt{c+d x^2} (d e-c f) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{\sqrt{e} \sqrt{f} \sqrt{e+f x^2} (b e-a f) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}",1,"((d*e - c*f)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(Sqrt[e]*Sqrt[f]*(b*e - a*f)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (c^(3/2)*(b*c - a*d)*Sqrt[e + f*x^2]*EllipticPi[1 - (b*c)/(a*d), ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(a*Sqrt[d]*e*(b*e - a*f)*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))])","A",3,3,32,0.09375,1,"{542, 539, 411}"
85,1,209,0,0.1052844,"\int \frac{\sqrt{c+d x^2}}{\left(a+b x^2\right) \left(e+f x^2\right)^{3/2}} \, dx","Int[Sqrt[c + d*x^2]/((a + b*x^2)*(e + f*x^2)^(3/2)),x]","\frac{b c^{3/2} \sqrt{e+f x^2} \Pi \left(1-\frac{b c}{a d};\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{a \sqrt{d} e \sqrt{c+d x^2} (b e-a f) \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}-\frac{\sqrt{f} \sqrt{c+d x^2} E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{\sqrt{e} \sqrt{e+f x^2} (b e-a f) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}","\frac{b c^{3/2} \sqrt{e+f x^2} \Pi \left(1-\frac{b c}{a d};\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{a \sqrt{d} e \sqrt{c+d x^2} (b e-a f) \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}-\frac{\sqrt{f} \sqrt{c+d x^2} E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{\sqrt{e} \sqrt{e+f x^2} (b e-a f) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}",1,"-((Sqrt[f]*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(Sqrt[e]*(b*e - a*f)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])) + (b*c^(3/2)*Sqrt[e + f*x^2]*EllipticPi[1 - (b*c)/(a*d), ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(a*Sqrt[d]*e*(b*e - a*f)*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))])","A",3,3,32,0.09375,1,"{541, 539, 411}"
86,1,344,0,0.2224209,"\int \frac{1}{\left(a+b x^2\right) \sqrt{c+d x^2} \left(e+f x^2\right)^{3/2}} \, dx","Int[1/((a + b*x^2)*Sqrt[c + d*x^2]*(e + f*x^2)^(3/2)),x]","\frac{b^2 e^{3/2} \sqrt{c+d x^2} \Pi \left(1-\frac{b e}{a f};\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{a c \sqrt{f} \sqrt{e+f x^2} (b e-a f)^2 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{f^{3/2} \sqrt{c+d x^2} E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{\sqrt{e} \sqrt{e+f x^2} (b e-a f) (d e-c f) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (-a d f-b c f+2 b d e) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{c \sqrt{e+f x^2} (b e-a f)^2 (d e-c f) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}","\frac{b^2 e^{3/2} \sqrt{c+d x^2} \Pi \left(1-\frac{b e}{a f};\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{a c \sqrt{f} \sqrt{e+f x^2} (b e-a f)^2 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{f^{3/2} \sqrt{c+d x^2} E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{\sqrt{e} \sqrt{e+f x^2} (b e-a f) (d e-c f) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (-a d f-b c f+2 b d e) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{c \sqrt{e+f x^2} (b e-a f)^2 (d e-c f) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}",1,"(f^(3/2)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(Sqrt[e]*(b*e - a*f)*(d*e - c*f)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (Sqrt[e]*Sqrt[f]*(2*b*d*e - b*c*f - a*d*f)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(c*(b*e - a*f)^2*(d*e - c*f)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (b^2*e^(3/2)*Sqrt[c + d*x^2]*EllipticPi[1 - (b*e)/(a*f), ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(a*c*Sqrt[f]*(b*e - a*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",5,5,32,0.1562,1,"{546, 539, 525, 418, 411}"
87,1,539,0,0.4792773,"\int \frac{1}{\left(a+b x^2\right) \left(c+d x^2\right)^{3/2} \left(e+f x^2\right)^{3/2}} \, dx","Int[1/((a + b*x^2)*(c + d*x^2)^(3/2)*(e + f*x^2)^(3/2)),x]","\frac{b^3 c^{3/2} \sqrt{e+f x^2} \Pi \left(1-\frac{b c}{a d};\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{a \sqrt{d} e \sqrt{c+d x^2} (b c-a d)^2 (b e-a f) \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}-\frac{b^2 \sqrt{f} \sqrt{c+d x^2} E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{\sqrt{e} \sqrt{e+f x^2} (b c-a d)^2 (b e-a f) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{d \sqrt{f} \sqrt{c+d x^2} \left(2 b c^2 f-a d (c f+d e)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{c \sqrt{e} \sqrt{e+f x^2} (b c-a d)^2 (d e-c f)^2 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{d^2 x}{c \sqrt{c+d x^2} \sqrt{e+f x^2} (b c-a d) (d e-c f)}-\frac{d^2 \sqrt{e} \sqrt{c+d x^2} (2 a d f-3 b c f+b d e) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{c \sqrt{f} \sqrt{e+f x^2} (b c-a d)^2 (d e-c f)^2 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}","\frac{b^3 c^{3/2} \sqrt{e+f x^2} \Pi \left(1-\frac{b c}{a d};\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{a \sqrt{d} e \sqrt{c+d x^2} (b c-a d)^2 (b e-a f) \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}-\frac{b^2 \sqrt{f} \sqrt{c+d x^2} E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{\sqrt{e} \sqrt{e+f x^2} (b c-a d)^2 (b e-a f) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{d \sqrt{f} \sqrt{c+d x^2} \left(2 b c^2 f-a d (c f+d e)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{c \sqrt{e} \sqrt{e+f x^2} (b c-a d)^2 (d e-c f)^2 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{d^2 x}{c \sqrt{c+d x^2} \sqrt{e+f x^2} (b c-a d) (d e-c f)}-\frac{d^2 \sqrt{e} \sqrt{c+d x^2} (2 a d f-3 b c f+b d e) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{c \sqrt{f} \sqrt{e+f x^2} (b c-a d)^2 (d e-c f)^2 \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}",1,"-((d^2*x)/(c*(b*c - a*d)*(d*e - c*f)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])) - (b^2*Sqrt[f]*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/((b*c - a*d)^2*Sqrt[e]*(b*e - a*f)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (d*Sqrt[f]*(2*b*c^2*f - a*d*(d*e + c*f))*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(c*(b*c - a*d)^2*Sqrt[e]*(d*e - c*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (d^2*Sqrt[e]*(b*d*e - 3*b*c*f + 2*a*d*f)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(c*(b*c - a*d)^2*Sqrt[f]*(d*e - c*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (b^3*c^(3/2)*Sqrt[e + f*x^2]*EllipticPi[1 - (b*c)/(a*d), ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(a*Sqrt[d]*(b*c - a*d)^2*e*(b*e - a*f)*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))])","A",8,7,32,0.2188,1,"{546, 541, 539, 411, 527, 525, 418}"
88,1,814,0,0.9579962,"\int \frac{1}{\left(a+b x^2\right) \left(c+d x^2\right)^{5/2} \left(e+f x^2\right)^{3/2}} \, dx","Int[1/((a + b*x^2)*(c + d*x^2)^(5/2)*(e + f*x^2)^(3/2)),x]","\frac{e^{3/2} \sqrt{d x^2+c} \Pi \left(1-\frac{b e}{a f};\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right) b^4}{a c (b c-a d)^2 \sqrt{f} (b e-a f)^2 \sqrt{\frac{e \left(d x^2+c\right)}{c \left(f x^2+e\right)}} \sqrt{f x^2+e}}+\frac{f^{3/2} \sqrt{d x^2+c} E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right) b^2}{(b c-a d)^2 \sqrt{e} (b e-a f) (d e-c f) \sqrt{\frac{e \left(d x^2+c\right)}{c \left(f x^2+e\right)}} \sqrt{f x^2+e}}-\frac{\sqrt{e} \sqrt{f} (2 b d e-b c f-a d f) \sqrt{d x^2+c} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right) b^2}{c (b c-a d)^2 (b e-a f)^2 (d e-c f) \sqrt{\frac{e \left(d x^2+c\right)}{c \left(f x^2+e\right)}} \sqrt{f x^2+e}}-\frac{d \sqrt{f} \left(b c \left(5 d^2 e^2-7 c d f e-6 c^2 f^2\right)-a d \left(2 d^2 e^2-7 c d f e-3 c^2 f^2\right)\right) \sqrt{d x^2+c} E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 c^2 (b c-a d)^2 \sqrt{e} (d e-c f)^3 \sqrt{\frac{e \left(d x^2+c\right)}{c \left(f x^2+e\right)}} \sqrt{f x^2+e}}+\frac{d^2 \sqrt{e} \sqrt{f} (b c (7 d e-15 c f)-a d (d e-9 c f)) \sqrt{d x^2+c} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 c^2 (b c-a d)^2 (d e-c f)^3 \sqrt{\frac{e \left(d x^2+c\right)}{c \left(f x^2+e\right)}} \sqrt{f x^2+e}}-\frac{d^2 (b c (5 d e-9 c f)-2 a d (d e-3 c f)) x}{3 c^2 (b c-a d)^2 (d e-c f)^2 \sqrt{d x^2+c} \sqrt{f x^2+e}}-\frac{d^2 x}{3 c (b c-a d) (d e-c f) \left(d x^2+c\right)^{3/2} \sqrt{f x^2+e}}","\frac{e^{3/2} \sqrt{d x^2+c} \Pi \left(1-\frac{b e}{a f};\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right) b^4}{a c (b c-a d)^2 \sqrt{f} (b e-a f)^2 \sqrt{\frac{e \left(d x^2+c\right)}{c \left(f x^2+e\right)}} \sqrt{f x^2+e}}+\frac{f^{3/2} \sqrt{d x^2+c} E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right) b^2}{(b c-a d)^2 \sqrt{e} (b e-a f) (d e-c f) \sqrt{\frac{e \left(d x^2+c\right)}{c \left(f x^2+e\right)}} \sqrt{f x^2+e}}-\frac{\sqrt{e} \sqrt{f} (2 b d e-b c f-a d f) \sqrt{d x^2+c} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right) b^2}{c (b c-a d)^2 (b e-a f)^2 (d e-c f) \sqrt{\frac{e \left(d x^2+c\right)}{c \left(f x^2+e\right)}} \sqrt{f x^2+e}}-\frac{d \sqrt{f} \left(b c \left(5 d^2 e^2-7 c d f e-6 c^2 f^2\right)-a d \left(2 d^2 e^2-7 c d f e-3 c^2 f^2\right)\right) \sqrt{d x^2+c} E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 c^2 (b c-a d)^2 \sqrt{e} (d e-c f)^3 \sqrt{\frac{e \left(d x^2+c\right)}{c \left(f x^2+e\right)}} \sqrt{f x^2+e}}+\frac{d^2 \sqrt{e} \sqrt{f} (b c (7 d e-15 c f)-a d (d e-9 c f)) \sqrt{d x^2+c} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 c^2 (b c-a d)^2 (d e-c f)^3 \sqrt{\frac{e \left(d x^2+c\right)}{c \left(f x^2+e\right)}} \sqrt{f x^2+e}}-\frac{d^2 (b c (5 d e-9 c f)-2 a d (d e-3 c f)) x}{3 c^2 (b c-a d)^2 (d e-c f)^2 \sqrt{d x^2+c} \sqrt{f x^2+e}}-\frac{d^2 x}{3 c (b c-a d) (d e-c f) \left(d x^2+c\right)^{3/2} \sqrt{f x^2+e}}",1,"-(d^2*x)/(3*c*(b*c - a*d)*(d*e - c*f)*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2]) - (d^2*(b*c*(5*d*e - 9*c*f) - 2*a*d*(d*e - 3*c*f))*x)/(3*c^2*(b*c - a*d)^2*(d*e - c*f)^2*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]) + (b^2*f^(3/2)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/((b*c - a*d)^2*Sqrt[e]*(b*e - a*f)*(d*e - c*f)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (d*Sqrt[f]*(b*c*(5*d^2*e^2 - 7*c*d*e*f - 6*c^2*f^2) - a*d*(2*d^2*e^2 - 7*c*d*e*f - 3*c^2*f^2))*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*c^2*(b*c - a*d)^2*Sqrt[e]*(d*e - c*f)^3*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (b^2*Sqrt[e]*Sqrt[f]*(2*b*d*e - b*c*f - a*d*f)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(c*(b*c - a*d)^2*(b*e - a*f)^2*(d*e - c*f)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (d^2*Sqrt[e]*Sqrt[f]*(b*c*(7*d*e - 15*c*f) - a*d*(d*e - 9*c*f))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*c^2*(b*c - a*d)^2*(d*e - c*f)^3*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (b^4*e^(3/2)*Sqrt[c + d*x^2]*EllipticPi[1 - (b*e)/(a*f), ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(a*c*(b*c - a*d)^2*Sqrt[f]*(b*e - a*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])","A",11,6,32,0.1875,1,"{546, 539, 525, 418, 411, 527}"
89,1,239,0,0.1474567,"\int \frac{\left(1+x^2\right)^{3/2} \sqrt{2+x^2}}{a+b x^2} \, dx","Int[((1 + x^2)^(3/2)*Sqrt[2 + x^2])/(a + b*x^2),x]","-\frac{x \sqrt{x^2+2} (a-2 b)}{b^2 \sqrt{x^2+1}}-\frac{\sqrt{2} \sqrt{x^2+2} (3 a-5 b) F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{3 b^2 \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}}}+\frac{\sqrt{2} \sqrt{x^2+2} (a-2 b) E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{b^2 \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}}}+\frac{2 \sqrt{x^2+1} (a-b)^2 \Pi \left(1-\frac{2 b}{a};\left.\tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-1\right)}{a b^2 \sqrt{\frac{x^2+1}{x^2+2}} \sqrt{x^2+2}}+\frac{x \sqrt{x^2+1} \sqrt{x^2+2}}{3 b}","-\frac{\sqrt{x^2+2} x (a-2 b)}{b^2 \sqrt{x^2+1}}-\frac{\sqrt{x^2+2} (3 a-7 b) F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{3 \sqrt{2} b^2 \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}}}+\frac{\sqrt{2} \sqrt{x^2+2} (a-2 b) E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{b^2 \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}}}+\frac{\sqrt{x^2+2} (a-2 b) (a-b) \Pi \left(1-\frac{b}{a};\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{2} a b^2 \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}}}+\frac{\sqrt{x^2+1} \sqrt{x^2+2} x}{3 b}",1,"-(((a - 2*b)*x*Sqrt[2 + x^2])/(b^2*Sqrt[1 + x^2])) + (x*Sqrt[1 + x^2]*Sqrt[2 + x^2])/(3*b) + (Sqrt[2]*(a - 2*b)*Sqrt[2 + x^2]*EllipticE[ArcTan[x], 1/2])/(b^2*Sqrt[1 + x^2]*Sqrt[(2 + x^2)/(1 + x^2)]) - (Sqrt[2]*(3*a - 5*b)*Sqrt[2 + x^2]*EllipticF[ArcTan[x], 1/2])/(3*b^2*Sqrt[1 + x^2]*Sqrt[(2 + x^2)/(1 + x^2)]) + (2*(a - b)^2*Sqrt[1 + x^2]*EllipticPi[1 - (2*b)/a, ArcTan[x/Sqrt[2]], -1])/(a*b^2*Sqrt[(1 + x^2)/(2 + x^2)]*Sqrt[2 + x^2])","A",7,7,28,0.2500,1,"{543, 539, 528, 531, 418, 492, 411}"
90,1,192,0,0.0918102,"\int \frac{\sqrt{1+x^2} \sqrt{2+x^2}}{a+b x^2} \, dx","Int[(Sqrt[1 + x^2]*Sqrt[2 + x^2])/(a + b*x^2),x]","-\frac{\sqrt{x^2+2} (a-2 b) \Pi \left(1-\frac{b}{a};\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{2} a b \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}}}+\frac{\sqrt{x^2+2} x}{b \sqrt{x^2+1}}+\frac{\sqrt{x^2+2} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{2} b \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}}}-\frac{\sqrt{2} \sqrt{x^2+2} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{b \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}}}","-\frac{\sqrt{x^2+2} (a-2 b) \Pi \left(1-\frac{b}{a};\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{2} a b \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}}}+\frac{\sqrt{x^2+2} x}{b \sqrt{x^2+1}}+\frac{\sqrt{x^2+2} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{2} b \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}}}-\frac{\sqrt{2} \sqrt{x^2+2} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{b \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}}}",1,"(x*Sqrt[2 + x^2])/(b*Sqrt[1 + x^2]) - (Sqrt[2]*Sqrt[2 + x^2]*EllipticE[ArcTan[x], 1/2])/(b*Sqrt[1 + x^2]*Sqrt[(2 + x^2)/(1 + x^2)]) + (Sqrt[2 + x^2]*EllipticF[ArcTan[x], 1/2])/(Sqrt[2]*b*Sqrt[1 + x^2]*Sqrt[(2 + x^2)/(1 + x^2)]) - ((a - 2*b)*Sqrt[2 + x^2]*EllipticPi[1 - b/a, ArcTan[x], 1/2])/(Sqrt[2]*a*b*Sqrt[1 + x^2]*Sqrt[(2 + x^2)/(1 + x^2)])","A",6,6,28,0.2143,1,"{534, 422, 418, 492, 411, 539}"
91,1,58,0,0.0217733,"\int \frac{\sqrt{2+x^2}}{\sqrt{1+x^2} \left(a+b x^2\right)} \, dx","Int[Sqrt[2 + x^2]/(Sqrt[1 + x^2]*(a + b*x^2)),x]","\frac{2 \sqrt{x^2+1} \Pi \left(1-\frac{2 b}{a};\left.\tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-1\right)}{a \sqrt{\frac{x^2+1}{x^2+2}} \sqrt{x^2+2}}","\frac{2 \sqrt{x^2+1} \Pi \left(1-\frac{2 b}{a};\left.\tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-1\right)}{a \sqrt{\frac{x^2+1}{x^2+2}} \sqrt{x^2+2}}",1,"(2*Sqrt[1 + x^2]*EllipticPi[1 - (2*b)/a, ArcTan[x/Sqrt[2]], -1])/(a*Sqrt[(1 + x^2)/(2 + x^2)]*Sqrt[2 + x^2])","A",1,1,28,0.03571,1,"{539}"
92,1,121,0,0.0597335,"\int \frac{\sqrt{2+x^2}}{\left(1+x^2\right)^{3/2} \left(a+b x^2\right)} \, dx","Int[Sqrt[2 + x^2]/((1 + x^2)^(3/2)*(a + b*x^2)),x]","\frac{\sqrt{2} \sqrt{x^2+2} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}} (a-b)}-\frac{2 b \sqrt{x^2+1} \Pi \left(1-\frac{2 b}{a};\left.\tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-1\right)}{a \sqrt{\frac{x^2+1}{x^2+2}} \sqrt{x^2+2} (a-b)}","\frac{\sqrt{2} \sqrt{x^2+2} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}} (a-b)}-\frac{2 b \sqrt{x^2+1} \Pi \left(1-\frac{2 b}{a};\left.\tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-1\right)}{a \sqrt{\frac{x^2+1}{x^2+2}} \sqrt{x^2+2} (a-b)}",1,"(Sqrt[2]*Sqrt[2 + x^2]*EllipticE[ArcTan[x], 1/2])/((a - b)*Sqrt[1 + x^2]*Sqrt[(2 + x^2)/(1 + x^2)]) - (2*b*Sqrt[1 + x^2]*EllipticPi[1 - (2*b)/a, ArcTan[x/Sqrt[2]], -1])/(a*(a - b)*Sqrt[(1 + x^2)/(2 + x^2)]*Sqrt[2 + x^2])","A",3,3,28,0.1071,1,"{541, 539, 411}"
93,1,215,0,0.1449946,"\int \frac{\sqrt{2+x^2}}{\left(1+x^2\right)^{5/2} \left(a+b x^2\right)} \, dx","Int[Sqrt[2 + x^2]/((1 + x^2)^(5/2)*(a + b*x^2)),x]","\frac{2 b^2 \sqrt{x^2+1} \Pi \left(1-\frac{2 b}{a};\left.\tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-1\right)}{a \sqrt{\frac{x^2+1}{x^2+2}} \sqrt{x^2+2} (a-b)^2}+\frac{x \sqrt{x^2+2}}{3 \left(x^2+1\right)^{3/2} (a-b)}-\frac{\sqrt{2} \sqrt{x^2+2} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{3 \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}} (a-b)}+\frac{\sqrt{2} \sqrt{x^2+2} (a-2 b) E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}} (a-b)^2}","\frac{2 b^2 \sqrt{x^2+1} \Pi \left(1-\frac{2 b}{a};\left.\tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-1\right)}{a \sqrt{\frac{x^2+1}{x^2+2}} \sqrt{x^2+2} (a-b)^2}+\frac{x \sqrt{x^2+2}}{3 \left(x^2+1\right)^{3/2} (a-b)}-\frac{\sqrt{2} \sqrt{x^2+2} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{3 \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}} (a-b)}+\frac{\sqrt{2} \sqrt{x^2+2} (a-2 b) E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}} (a-b)^2}",1,"(x*Sqrt[2 + x^2])/(3*(a - b)*(1 + x^2)^(3/2)) + (Sqrt[2]*(a - 2*b)*Sqrt[2 + x^2]*EllipticE[ArcTan[x], 1/2])/((a - b)^2*Sqrt[1 + x^2]*Sqrt[(2 + x^2)/(1 + x^2)]) - (Sqrt[2]*Sqrt[2 + x^2]*EllipticF[ArcTan[x], 1/2])/(3*(a - b)*Sqrt[1 + x^2]*Sqrt[(2 + x^2)/(1 + x^2)]) + (2*b^2*Sqrt[1 + x^2]*EllipticPi[1 - (2*b)/a, ArcTan[x/Sqrt[2]], -1])/(a*(a - b)^2*Sqrt[(1 + x^2)/(2 + x^2)]*Sqrt[2 + x^2])","A",6,6,28,0.2143,1,"{546, 539, 526, 525, 418, 411}"
94,1,298,0,0.1790368,"\int \frac{\sqrt{2+d x^2} \sqrt{3+f x^2}}{a+b x^2} \, dx","Int[(Sqrt[2 + d*x^2]*Sqrt[3 + f*x^2])/(a + b*x^2),x]","\frac{3 \sqrt{d x^2+2} (2 b-a d) \Pi \left(1-\frac{3 b}{a f};\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{3}}\right)|1-\frac{3 d}{2 f}\right)}{\sqrt{2} a b \sqrt{f} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}+\frac{f x \sqrt{d x^2+2}}{b \sqrt{f x^2+3}}+\frac{3 d \sqrt{d x^2+2} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{3}}\right)|1-\frac{3 d}{2 f}\right)}{\sqrt{2} b \sqrt{f} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}-\frac{\sqrt{2} \sqrt{f} \sqrt{d x^2+2} E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{3}}\right)|1-\frac{3 d}{2 f}\right)}{b \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}","\frac{3 \sqrt{d x^2+2} (2 b-a d) \Pi \left(1-\frac{3 b}{a f};\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{3}}\right)|1-\frac{3 d}{2 f}\right)}{\sqrt{2} a b \sqrt{f} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}+\frac{f x \sqrt{d x^2+2}}{b \sqrt{f x^2+3}}+\frac{3 d \sqrt{d x^2+2} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{3}}\right)|1-\frac{3 d}{2 f}\right)}{\sqrt{2} b \sqrt{f} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}-\frac{\sqrt{2} \sqrt{f} \sqrt{d x^2+2} E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{3}}\right)|1-\frac{3 d}{2 f}\right)}{b \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}",1,"(f*x*Sqrt[2 + d*x^2])/(b*Sqrt[3 + f*x^2]) - (Sqrt[2]*Sqrt[f]*Sqrt[2 + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[3]], 1 - (3*d)/(2*f)])/(b*Sqrt[(2 + d*x^2)/(3 + f*x^2)]*Sqrt[3 + f*x^2]) + (3*d*Sqrt[2 + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[3]], 1 - (3*d)/(2*f)])/(Sqrt[2]*b*Sqrt[f]*Sqrt[(2 + d*x^2)/(3 + f*x^2)]*Sqrt[3 + f*x^2]) + (3*(2*b - a*d)*Sqrt[2 + d*x^2]*EllipticPi[1 - (3*b)/(a*f), ArcTan[(Sqrt[f]*x)/Sqrt[3]], 1 - (3*d)/(2*f)])/(Sqrt[2]*a*b*Sqrt[f]*Sqrt[(2 + d*x^2)/(3 + f*x^2)]*Sqrt[3 + f*x^2])","A",6,6,32,0.1875,1,"{535, 422, 418, 492, 411, 539}"
95,1,93,0,0.0361127,"\int \frac{\sqrt{2+d x^2}}{\left(a+b x^2\right) \sqrt{3+f x^2}} \, dx","Int[Sqrt[2 + d*x^2]/((a + b*x^2)*Sqrt[3 + f*x^2]),x]","\frac{2 \sqrt{f x^2+3} \Pi \left(1-\frac{2 b}{a d};\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{2}}\right)|1-\frac{2 f}{3 d}\right)}{\sqrt{3} a \sqrt{d} \sqrt{d x^2+2} \sqrt{\frac{f x^2+3}{d x^2+2}}}","\frac{2 \sqrt{f x^2+3} \Pi \left(1-\frac{2 b}{a d};\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{2}}\right)|1-\frac{2 f}{3 d}\right)}{\sqrt{3} a \sqrt{d} \sqrt{d x^2+2} \sqrt{\frac{f x^2+3}{d x^2+2}}}",1,"(2*Sqrt[3 + f*x^2]*EllipticPi[1 - (2*b)/(a*d), ArcTan[(Sqrt[d]*x)/Sqrt[2]], 1 - (2*f)/(3*d)])/(Sqrt[3]*a*Sqrt[d]*Sqrt[2 + d*x^2]*Sqrt[(3 + f*x^2)/(2 + d*x^2)])","A",1,1,32,0.03125,1,"{539}"
96,1,49,0,0.0415932,"\int \frac{1}{\left(a+b x^2\right) \sqrt{2+d x^2} \sqrt{3+f x^2}} \, dx","Int[1/((a + b*x^2)*Sqrt[2 + d*x^2]*Sqrt[3 + f*x^2]),x]","\frac{\Pi \left(\frac{2 b}{a d};\sin ^{-1}\left(\frac{\sqrt{-d} x}{\sqrt{2}}\right)|\frac{2 f}{3 d}\right)}{\sqrt{3} a \sqrt{-d}}","\frac{\Pi \left(\frac{2 b}{a d};\sin ^{-1}\left(\frac{\sqrt{-d} x}{\sqrt{2}}\right)|\frac{2 f}{3 d}\right)}{\sqrt{3} a \sqrt{-d}}",1,"EllipticPi[(2*b)/(a*d), ArcSin[(Sqrt[-d]*x)/Sqrt[2]], (2*f)/(3*d)]/(Sqrt[3]*a*Sqrt[-d])","A",1,1,32,0.03125,1,"{537}"
97,1,36,0,0.0248665,"\int \frac{\sqrt{1-x^2}}{\left(-1+x^2\right) \sqrt{a+b x^2}} \, dx","Int[Sqrt[1 - x^2]/((-1 + x^2)*Sqrt[a + b*x^2]),x]","-\frac{\sqrt{\frac{b x^2}{a}+1} F\left(\sin ^{-1}(x)|-\frac{b}{a}\right)}{\sqrt{a+b x^2}}","-\frac{\sqrt{\frac{b x^2}{a}+1} F\left(\sin ^{-1}(x)|-\frac{b}{a}\right)}{\sqrt{a+b x^2}}",1,"-((Sqrt[1 + (b*x^2)/a]*EllipticF[ArcSin[x], -(b/a)])/Sqrt[a + b*x^2])","A",3,3,30,0.1000,1,"{21, 421, 419}"
98,1,113,0,0.1188087,"\int \frac{a+b x^2}{\sqrt{c+d x^2} \left(e+f x^2\right)^2} \, dx","Int[(a + b*x^2)/(Sqrt[c + d*x^2]*(e + f*x^2)^2),x]","\frac{x \sqrt{c+d x^2} (b e-a f)}{2 e \left(e+f x^2\right) (d e-c f)}-\frac{(a c f-2 a d e+b c e) \tanh ^{-1}\left(\frac{x \sqrt{d e-c f}}{\sqrt{e} \sqrt{c+d x^2}}\right)}{2 e^{3/2} (d e-c f)^{3/2}}","\frac{x \sqrt{c+d x^2} (b e-a f)}{2 e \left(e+f x^2\right) (d e-c f)}-\frac{(a c f-2 a d e+b c e) \tanh ^{-1}\left(\frac{x \sqrt{d e-c f}}{\sqrt{e} \sqrt{c+d x^2}}\right)}{2 e^{3/2} (d e-c f)^{3/2}}",1,"((b*e - a*f)*x*Sqrt[c + d*x^2])/(2*e*(d*e - c*f)*(e + f*x^2)) - ((b*c*e - 2*a*d*e + a*c*f)*ArcTanh[(Sqrt[d*e - c*f]*x)/(Sqrt[e]*Sqrt[c + d*x^2])])/(2*e^(3/2)*(d*e - c*f)^(3/2))","A",4,4,28,0.1429,1,"{527, 12, 377, 208}"
99,1,359,0,0.3236447,"\int \frac{\sqrt{c-d x^2} \sqrt{e+f x^2}}{\left(a+b x^2\right)^2} \, dx","Int[(Sqrt[c - d*x^2]*Sqrt[e + f*x^2])/(a + b*x^2)^2,x]","\frac{\sqrt{c} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \left(a^2 d f+b^2 c e\right) \Pi \left(-\frac{b c}{a d};\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|-\frac{c f}{d e}\right)}{2 a^2 b^2 \sqrt{d} \sqrt{c-d x^2} \sqrt{e+f x^2}}-\frac{\sqrt{c} \sqrt{d} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} (a f+b e) F\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|-\frac{c f}{d e}\right)}{2 a b^2 \sqrt{c-d x^2} \sqrt{e+f x^2}}+\frac{x \sqrt{c-d x^2} \sqrt{e+f x^2}}{2 a \left(a+b x^2\right)}+\frac{\sqrt{c} \sqrt{d} \sqrt{1-\frac{d x^2}{c}} \sqrt{e+f x^2} E\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|-\frac{c f}{d e}\right)}{2 a b \sqrt{c-d x^2} \sqrt{\frac{f x^2}{e}+1}}","\frac{\sqrt{c} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \left(a^2 d f+b^2 c e\right) \Pi \left(-\frac{b c}{a d};\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|-\frac{c f}{d e}\right)}{2 a^2 b^2 \sqrt{d} \sqrt{c-d x^2} \sqrt{e+f x^2}}-\frac{\sqrt{c} \sqrt{d} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} (a f+b e) F\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|-\frac{c f}{d e}\right)}{2 a b^2 \sqrt{c-d x^2} \sqrt{e+f x^2}}+\frac{x \sqrt{c-d x^2} \sqrt{e+f x^2}}{2 a \left(a+b x^2\right)}+\frac{\sqrt{c} \sqrt{d} \sqrt{1-\frac{d x^2}{c}} \sqrt{e+f x^2} E\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|-\frac{c f}{d e}\right)}{2 a b \sqrt{c-d x^2} \sqrt{\frac{f x^2}{e}+1}}",1,"(x*Sqrt[c - d*x^2]*Sqrt[e + f*x^2])/(2*a*(a + b*x^2)) + (Sqrt[c]*Sqrt[d]*Sqrt[1 - (d*x^2)/c]*Sqrt[e + f*x^2]*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((c*f)/(d*e))])/(2*a*b*Sqrt[c - d*x^2]*Sqrt[1 + (f*x^2)/e]) - (Sqrt[c]*Sqrt[d]*(b*e + a*f)*Sqrt[1 - (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((c*f)/(d*e))])/(2*a*b^2*Sqrt[c - d*x^2]*Sqrt[e + f*x^2]) + (Sqrt[c]*(b^2*c*e + a^2*d*f)*Sqrt[1 - (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[-((b*c)/(a*d)), ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((c*f)/(d*e))])/(2*a^2*b^2*Sqrt[d]*Sqrt[c - d*x^2]*Sqrt[e + f*x^2])","A",11,9,33,0.2727,1,"{548, 524, 427, 426, 424, 421, 419, 538, 537}"
100,1,381,0,0.2913338,"\int \frac{\sqrt{c+d x^2} \sqrt{e+f x^2}}{\left(a+b x^2\right)^2} \, dx","Int[(Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(a + b*x^2)^2,x]","\frac{\sqrt{-c} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left(b^2 c e-a^2 d f\right) \Pi \left(\frac{b c}{a d};\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{-c}}\right)|\frac{c f}{d e}\right)}{2 a^2 b^2 \sqrt{d} \sqrt{c+d x^2} \sqrt{e+f x^2}}+\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2}}{2 a \left(a+b x^2\right)}-\frac{f x \sqrt{c+d x^2}}{2 a b \sqrt{e+f x^2}}+\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{2 a b \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{d \sqrt{e} \sqrt{f} \sqrt{c+d x^2} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{2 b^2 c \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}","\frac{\sqrt{-c} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left(b^2 c e-a^2 d f\right) \Pi \left(\frac{b c}{a d};\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{-c}}\right)|\frac{c f}{d e}\right)}{2 a^2 b^2 \sqrt{d} \sqrt{c+d x^2} \sqrt{e+f x^2}}+\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2}}{2 a \left(a+b x^2\right)}-\frac{f x \sqrt{c+d x^2}}{2 a b \sqrt{e+f x^2}}+\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{2 a b \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{d \sqrt{e} \sqrt{f} \sqrt{c+d x^2} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{2 b^2 c \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}",1,"-(f*x*Sqrt[c + d*x^2])/(2*a*b*Sqrt[e + f*x^2]) + (x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(2*a*(a + b*x^2)) + (Sqrt[e]*Sqrt[f]*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(2*a*b*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (d*Sqrt[e]*Sqrt[f]*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(2*b^2*c*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (Sqrt[-c]*(b^2*c*e - a^2*d*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), ArcSin[(Sqrt[d]*x)/Sqrt[-c]], (c*f)/(d*e)])/(2*a^2*b^2*Sqrt[d]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","A",8,7,32,0.2188,1,"{548, 531, 418, 492, 411, 538, 537}"
101,1,426,0,0.3730774,"\int \frac{1}{\left(a+b x^2\right)^2 \sqrt{c-d x^2} \sqrt{e+f x^2}} \, dx","Int[1/((a + b*x^2)^2*Sqrt[c - d*x^2]*Sqrt[e + f*x^2]),x]","\frac{\sqrt{c} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \left(-3 a^2 d f+a b (2 d e-2 c f)+b^2 c e\right) \Pi \left(-\frac{b c}{a d};\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|-\frac{c f}{d e}\right)}{2 a^2 \sqrt{d} \sqrt{c-d x^2} \sqrt{e+f x^2} (a d+b c) (b e-a f)}+\frac{b^2 x \sqrt{c-d x^2} \sqrt{e+f x^2}}{2 a \left(a+b x^2\right) (a d+b c) (b e-a f)}-\frac{\sqrt{c} \sqrt{d} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} F\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|-\frac{c f}{d e}\right)}{2 a \sqrt{c-d x^2} \sqrt{e+f x^2} (a d+b c)}+\frac{b \sqrt{c} \sqrt{d} \sqrt{1-\frac{d x^2}{c}} \sqrt{e+f x^2} E\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|-\frac{c f}{d e}\right)}{2 a \sqrt{c-d x^2} \sqrt{\frac{f x^2}{e}+1} (a d+b c) (b e-a f)}","\frac{\sqrt{c} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \left(-3 a^2 d f+a b (2 d e-2 c f)+b^2 c e\right) \Pi \left(-\frac{b c}{a d};\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|-\frac{c f}{d e}\right)}{2 a^2 \sqrt{d} \sqrt{c-d x^2} \sqrt{e+f x^2} (a d+b c) (b e-a f)}+\frac{b^2 x \sqrt{c-d x^2} \sqrt{e+f x^2}}{2 a \left(a+b x^2\right) (a d+b c) (b e-a f)}-\frac{\sqrt{c} \sqrt{d} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} F\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|-\frac{c f}{d e}\right)}{2 a \sqrt{c-d x^2} \sqrt{e+f x^2} (a d+b c)}+\frac{b \sqrt{c} \sqrt{d} \sqrt{1-\frac{d x^2}{c}} \sqrt{e+f x^2} E\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|-\frac{c f}{d e}\right)}{2 a \sqrt{c-d x^2} \sqrt{\frac{f x^2}{e}+1} (a d+b c) (b e-a f)}",1,"(b^2*x*Sqrt[c - d*x^2]*Sqrt[e + f*x^2])/(2*a*(b*c + a*d)*(b*e - a*f)*(a + b*x^2)) + (b*Sqrt[c]*Sqrt[d]*Sqrt[1 - (d*x^2)/c]*Sqrt[e + f*x^2]*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((c*f)/(d*e))])/(2*a*(b*c + a*d)*(b*e - a*f)*Sqrt[c - d*x^2]*Sqrt[1 + (f*x^2)/e]) - (Sqrt[c]*Sqrt[d]*Sqrt[1 - (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((c*f)/(d*e))])/(2*a*(b*c + a*d)*Sqrt[c - d*x^2]*Sqrt[e + f*x^2]) + (Sqrt[c]*(b^2*c*e - 3*a^2*d*f + a*b*(2*d*e - 2*c*f))*Sqrt[1 - (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[-((b*c)/(a*d)), ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((c*f)/(d*e))])/(2*a^2*Sqrt[d]*(b*c + a*d)*(b*e - a*f)*Sqrt[c - d*x^2]*Sqrt[e + f*x^2])","A",11,9,33,0.2727,1,"{549, 524, 427, 426, 424, 421, 419, 538, 537}"
102,1,485,0,0.3471215,"\int \frac{1}{\left(a+b x^2\right)^2 \sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx","Int[1/((a + b*x^2)^2*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]","\frac{\sqrt{-c} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left(3 a^2 d f-2 a b (c f+d e)+b^2 c e\right) \Pi \left(\frac{b c}{a d};\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{-c}}\right)|\frac{c f}{d e}\right)}{2 a^2 \sqrt{d} \sqrt{c+d x^2} \sqrt{e+f x^2} (b c-a d) (b e-a f)}+\frac{b^2 x \sqrt{c+d x^2} \sqrt{e+f x^2}}{2 a \left(a+b x^2\right) (b c-a d) (b e-a f)}-\frac{b f x \sqrt{c+d x^2}}{2 a \sqrt{e+f x^2} (b c-a d) (b e-a f)}-\frac{d \sqrt{e} \sqrt{f} \sqrt{c+d x^2} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{2 c \sqrt{e+f x^2} (b c-a d) (b e-a f) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{b \sqrt{e} \sqrt{f} \sqrt{c+d x^2} E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{2 a \sqrt{e+f x^2} (b c-a d) (b e-a f) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}","\frac{\sqrt{-c} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left(3 a^2 d f-2 a b (c f+d e)+b^2 c e\right) \Pi \left(\frac{b c}{a d};\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{-c}}\right)|\frac{c f}{d e}\right)}{2 a^2 \sqrt{d} \sqrt{c+d x^2} \sqrt{e+f x^2} (b c-a d) (b e-a f)}+\frac{b^2 x \sqrt{c+d x^2} \sqrt{e+f x^2}}{2 a \left(a+b x^2\right) (b c-a d) (b e-a f)}-\frac{b f x \sqrt{c+d x^2}}{2 a \sqrt{e+f x^2} (b c-a d) (b e-a f)}-\frac{d \sqrt{e} \sqrt{f} \sqrt{c+d x^2} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{2 c \sqrt{e+f x^2} (b c-a d) (b e-a f) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{b \sqrt{e} \sqrt{f} \sqrt{c+d x^2} E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{2 a \sqrt{e+f x^2} (b c-a d) (b e-a f) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}",1,"-(b*f*x*Sqrt[c + d*x^2])/(2*a*(b*c - a*d)*(b*e - a*f)*Sqrt[e + f*x^2]) + (b^2*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(2*a*(b*c - a*d)*(b*e - a*f)*(a + b*x^2)) + (b*Sqrt[e]*Sqrt[f]*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(2*a*(b*c - a*d)*(b*e - a*f)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (d*Sqrt[e]*Sqrt[f]*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(2*c*(b*c - a*d)*(b*e - a*f)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (Sqrt[-c]*(b^2*c*e + 3*a^2*d*f - 2*a*b*(d*e + c*f))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), ArcSin[(Sqrt[d]*x)/Sqrt[-c]], (c*f)/(d*e)])/(2*a^2*Sqrt[d]*(b*c - a*d)*(b*e - a*f)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","A",8,7,32,0.2188,1,"{549, 531, 418, 492, 411, 538, 537}"
103,0,0,0,0.0571239,"\int \frac{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}}{\sqrt{e+f x^2}} \, dx","Int[((a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/Sqrt[e + f*x^2],x]","\int \frac{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}}{\sqrt{e+f x^2}} \, dx","\text{Int}\left(\frac{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}}{\sqrt{e+f x^2}},x\right)",0,"Defer[Int][((a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/Sqrt[e + f*x^2], x]","A",0,0,0,0,-1,"{}"
104,1,545,0,0.4865765,"\int \frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{\sqrt{e+f x^2}} \, dx","Int[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/Sqrt[e + f*x^2],x]","\frac{d x \sqrt{a+b x^2} \sqrt{e+f x^2}}{2 f \sqrt{c+d x^2}}+\frac{b \sqrt{e} \sqrt{c+d x^2} (d e-c f) \sqrt{\frac{a \left(e+f x^2\right)}{e \left(a+b x^2\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{b e-a f} x}{\sqrt{e} \sqrt{b x^2+a}}\right)|\frac{(b c-a d) e}{c (b e-a f)}\right)}{2 d f \sqrt{e+f x^2} \sqrt{b e-a f} \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}}}-\frac{\sqrt{e} \sqrt{a+b x^2} \sqrt{d e-c f} \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}} E\left(\sin ^{-1}\left(\frac{\sqrt{d e-c f} x}{\sqrt{e} \sqrt{d x^2+c}}\right)|-\frac{(b c-a d) e}{a (d e-c f)}\right)}{2 f \sqrt{e+f x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{c \sqrt{e} \sqrt{a+b x^2} \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}} (-a d f-b c f+b d e) \Pi \left(\frac{d e}{d e-c f};\sin ^{-1}\left(\frac{\sqrt{d e-c f} x}{\sqrt{e} \sqrt{d x^2+c}}\right)|-\frac{(b c-a d) e}{a (d e-c f)}\right)}{2 a d f \sqrt{e+f x^2} \sqrt{d e-c f} \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{e+f x^2}}{2 f \sqrt{c+d x^2}}+\frac{b \sqrt{e} \sqrt{c+d x^2} (d e-c f) \sqrt{\frac{a \left(e+f x^2\right)}{e \left(a+b x^2\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{b e-a f} x}{\sqrt{e} \sqrt{b x^2+a}}\right)|\frac{(b c-a d) e}{c (b e-a f)}\right)}{2 d f \sqrt{e+f x^2} \sqrt{b e-a f} \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}}}-\frac{\sqrt{e} \sqrt{a+b x^2} \sqrt{d e-c f} \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}} E\left(\sin ^{-1}\left(\frac{\sqrt{d e-c f} x}{\sqrt{e} \sqrt{d x^2+c}}\right)|-\frac{(b c-a d) e}{a (d e-c f)}\right)}{2 f \sqrt{e+f x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{c \sqrt{e} \sqrt{a+b x^2} \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}} (-a d f-b c f+b d e) \Pi \left(\frac{d e}{d e-c f};\sin ^{-1}\left(\frac{\sqrt{d e-c f} x}{\sqrt{e} \sqrt{d x^2+c}}\right)|-\frac{(b c-a d) e}{a (d e-c f)}\right)}{2 a d f \sqrt{e+f x^2} \sqrt{d e-c f} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}",1,"(d*x*Sqrt[a + b*x^2]*Sqrt[e + f*x^2])/(2*f*Sqrt[c + d*x^2]) - (Sqrt[e]*Sqrt[d*e - c*f]*Sqrt[a + b*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]*EllipticE[ArcSin[(Sqrt[d*e - c*f]*x)/(Sqrt[e]*Sqrt[c + d*x^2])], -(((b*c - a*d)*e)/(a*(d*e - c*f)))])/(2*f*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[e + f*x^2]) + (b*Sqrt[e]*(d*e - c*f)*Sqrt[c + d*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]*EllipticF[ArcSin[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])], ((b*c - a*d)*e)/(c*(b*e - a*f))])/(2*d*f*Sqrt[b*e - a*f]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*Sqrt[e + f*x^2]) - (c*Sqrt[e]*(b*d*e - b*c*f - a*d*f)*Sqrt[a + b*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]*EllipticPi[(d*e)/(d*e - c*f), ArcSin[(Sqrt[d*e - c*f]*x)/(Sqrt[e]*Sqrt[c + d*x^2])], -(((b*c - a*d)*e)/(a*(d*e - c*f)))])/(2*a*d*f*Sqrt[d*e - c*f]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[e + f*x^2])","A",7,7,34,0.2059,1,"{555, 554, 424, 552, 419, 553, 537}"
105,1,163,0,0.1338385,"\int \frac{\sqrt{c+d x^2}}{\sqrt{a+b x^2} \sqrt{e+f x^2}} \, dx","Int[Sqrt[c + d*x^2]/(Sqrt[a + b*x^2]*Sqrt[e + f*x^2]),x]","\frac{c \sqrt{e} \sqrt{a+b x^2} \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}} \Pi \left(\frac{d e}{d e-c f};\sin ^{-1}\left(\frac{\sqrt{d e-c f} x}{\sqrt{e} \sqrt{d x^2+c}}\right)|-\frac{(b c-a d) e}{a (d e-c f)}\right)}{a \sqrt{e+f x^2} \sqrt{d e-c f} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}","\frac{c \sqrt{e} \sqrt{a+b x^2} \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}} \Pi \left(\frac{d e}{d e-c f};\sin ^{-1}\left(\frac{\sqrt{d e-c f} x}{\sqrt{e} \sqrt{d x^2+c}}\right)|-\frac{(b c-a d) e}{a (d e-c f)}\right)}{a \sqrt{e+f x^2} \sqrt{d e-c f} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}",1,"(c*Sqrt[e]*Sqrt[a + b*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]*EllipticPi[(d*e)/(d*e - c*f), ArcSin[(Sqrt[d*e - c*f]*x)/(Sqrt[e]*Sqrt[c + d*x^2])], -(((b*c - a*d)*e)/(a*(d*e - c*f)))])/(a*Sqrt[d*e - c*f]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[e + f*x^2])","A",2,2,34,0.05882,1,"{553, 537}"
106,1,148,0,0.0846888,"\int \frac{\sqrt{c+d x^2}}{\left(a+b x^2\right)^{3/2} \sqrt{e+f x^2}} \, dx","Int[Sqrt[c + d*x^2]/((a + b*x^2)^(3/2)*Sqrt[e + f*x^2]),x]","\frac{\sqrt{e} \sqrt{c+d x^2} \sqrt{\frac{a \left(e+f x^2\right)}{e \left(a+b x^2\right)}} E\left(\sin ^{-1}\left(\frac{\sqrt{b e-a f} x}{\sqrt{e} \sqrt{b x^2+a}}\right)|\frac{(b c-a d) e}{c (b e-a f)}\right)}{a \sqrt{e+f x^2} \sqrt{b e-a f} \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}}}","\frac{\sqrt{e} \sqrt{c+d x^2} \sqrt{\frac{a \left(e+f x^2\right)}{e \left(a+b x^2\right)}} E\left(\sin ^{-1}\left(\frac{\sqrt{b e-a f} x}{\sqrt{e} \sqrt{b x^2+a}}\right)|\frac{(b c-a d) e}{c (b e-a f)}\right)}{a \sqrt{e+f x^2} \sqrt{b e-a f} \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}}}",1,"(Sqrt[e]*Sqrt[c + d*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]*EllipticE[ArcSin[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])], ((b*c - a*d)*e)/(c*(b*e - a*f))])/(a*Sqrt[b*e - a*f]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*Sqrt[e + f*x^2])","A",2,2,34,0.05882,1,"{554, 424}"
107,0,0,0,0.0611418,"\int \frac{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}}{\left(e+f x^2\right)^{3/2}} \, dx","Int[((a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(e + f*x^2)^(3/2),x]","\int \frac{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}}{\left(e+f x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}}{\left(e+f x^2\right)^{3/2}},x\right)",0,"Defer[Int][((a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(e + f*x^2)^(3/2), x]","A",0,0,0,0,-1,"{}"
108,1,484,0,0.7534752,"\int \frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{\left(e+f x^2\right)^{3/2}} \, dx","Int[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(e + f*x^2)^(3/2),x]","-\frac{c^{3/2} \sqrt{a+b x^2} (b e-a f) F\left(\tan ^{-1}\left(\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{f x^2+e}}\right)|-\frac{(b c-a d) e}{a (d e-c f)}\right)}{a e f \sqrt{c+d x^2} \sqrt{d e-c f} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{x \sqrt{a+b x^2} (d e-c f)}{e f \sqrt{c+d x^2} \sqrt{e+f x^2}}+\frac{\sqrt{c} \sqrt{a+b x^2} \sqrt{d e-c f} E\left(\tan ^{-1}\left(\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{f x^2+e}}\right)|-\frac{(b c-a d) e}{a (d e-c f)}\right)}{e f \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{b c \sqrt{e} \sqrt{a+b x^2} \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}} \Pi \left(\frac{d e}{d e-c f};\sin ^{-1}\left(\frac{\sqrt{d e-c f} x}{\sqrt{e} \sqrt{d x^2+c}}\right)|-\frac{(b c-a d) e}{a (d e-c f)}\right)}{a f \sqrt{e+f x^2} \sqrt{d e-c f} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}","-\frac{c^{3/2} \sqrt{a+b x^2} (b e-a f) F\left(\tan ^{-1}\left(\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{f x^2+e}}\right)|-\frac{(b c-a d) e}{a (d e-c f)}\right)}{a e f \sqrt{c+d x^2} \sqrt{d e-c f} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{x \sqrt{a+b x^2} (d e-c f)}{e f \sqrt{c+d x^2} \sqrt{e+f x^2}}+\frac{\sqrt{c} \sqrt{a+b x^2} \sqrt{d e-c f} E\left(\tan ^{-1}\left(\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{f x^2+e}}\right)|-\frac{(b c-a d) e}{a (d e-c f)}\right)}{e f \sqrt{c+d x^2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{b c \sqrt{e} \sqrt{a+b x^2} \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}} \Pi \left(\frac{d e}{d e-c f};\sin ^{-1}\left(\frac{\sqrt{d e-c f} x}{\sqrt{e} \sqrt{d x^2+c}}\right)|-\frac{(b c-a d) e}{a (d e-c f)}\right)}{a f \sqrt{e+f x^2} \sqrt{d e-c f} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}",1,"-(((d*e - c*f)*x*Sqrt[a + b*x^2])/(e*f*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])) + (Sqrt[c]*Sqrt[d*e - c*f]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])], -(((b*c - a*d)*e)/(a*(d*e - c*f)))])/(e*f*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (c^(3/2)*(b*e - a*f)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])], -(((b*c - a*d)*e)/(a*(d*e - c*f)))])/(a*e*f*Sqrt[d*e - c*f]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (b*c*Sqrt[e]*Sqrt[a + b*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]*EllipticPi[(d*e)/(d*e - c*f), ArcSin[(Sqrt[d*e - c*f]*x)/(Sqrt[e]*Sqrt[c + d*x^2])], -(((b*c - a*d)*e)/(a*(d*e - c*f)))])/(a*f*Sqrt[d*e - c*f]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[e + f*x^2])","A",8,8,34,0.2353,1,"{557, 553, 537, 554, 422, 418, 492, 411}"
109,1,319,0,0.4666698,"\int \frac{\sqrt{c+d x^2}}{\sqrt{a+b x^2} \left(e+f x^2\right)^{3/2}} \, dx","Int[Sqrt[c + d*x^2]/(Sqrt[a + b*x^2]*(e + f*x^2)^(3/2)),x]","\frac{c^{3/2} \sqrt{a+b x^2} F\left(\tan ^{-1}\left(\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{f x^2+e}}\right)|-\frac{(b c-a d) e}{a (d e-c f)}\right)}{a e \sqrt{c+d x^2} \sqrt{d e-c f} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{x \sqrt{a+b x^2} (d e-c f)}{e \sqrt{c+d x^2} \sqrt{e+f x^2} (b e-a f)}-\frac{\sqrt{c} \sqrt{a+b x^2} \sqrt{d e-c f} E\left(\tan ^{-1}\left(\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{f x^2+e}}\right)|-\frac{(b c-a d) e}{a (d e-c f)}\right)}{e \sqrt{c+d x^2} (b e-a f) \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}","\frac{c^{3/2} \sqrt{a+b x^2} F\left(\tan ^{-1}\left(\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{f x^2+e}}\right)|-\frac{(b c-a d) e}{a (d e-c f)}\right)}{a e \sqrt{c+d x^2} \sqrt{d e-c f} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{x \sqrt{a+b x^2} (d e-c f)}{e \sqrt{c+d x^2} \sqrt{e+f x^2} (b e-a f)}-\frac{\sqrt{c} \sqrt{a+b x^2} \sqrt{d e-c f} E\left(\tan ^{-1}\left(\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{f x^2+e}}\right)|-\frac{(b c-a d) e}{a (d e-c f)}\right)}{e \sqrt{c+d x^2} (b e-a f) \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}",1,"((d*e - c*f)*x*Sqrt[a + b*x^2])/(e*(b*e - a*f)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]) - (Sqrt[c]*Sqrt[d*e - c*f]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])], -(((b*c - a*d)*e)/(a*(d*e - c*f)))])/(e*(b*e - a*f)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (c^(3/2)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])], -(((b*c - a*d)*e)/(a*(d*e - c*f)))])/(a*e*Sqrt[d*e - c*f]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])","A",5,5,34,0.1471,1,"{554, 422, 418, 492, 411}"
110,0,0,0,0.0600382,"\int \frac{\sqrt{c+d x^2}}{\left(a+b x^2\right)^{3/2} \left(e+f x^2\right)^{3/2}} \, dx","Int[Sqrt[c + d*x^2]/((a + b*x^2)^(3/2)*(e + f*x^2)^(3/2)),x]","\int \frac{\sqrt{c+d x^2}}{\left(a+b x^2\right)^{3/2} \left(e+f x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{\sqrt{c+d x^2}}{\left(a+b x^2\right)^{3/2} \left(e+f x^2\right)^{3/2}},x\right)",0,"Defer[Int][Sqrt[c + d*x^2]/((a + b*x^2)^(3/2)*(e + f*x^2)^(3/2)), x]","A",0,0,0,0,-1,"{}"
111,1,541,0,0.4360348,"\int \frac{\sqrt{c+d x^2} \sqrt{e+f x^2}}{\sqrt{a+b x^2}} \, dx","Int[(Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/Sqrt[a + b*x^2],x]","\frac{\sqrt{e} \sqrt{c+d x^2} (b c-a d) (2 b e-a f) \sqrt{\frac{a \left(e+f x^2\right)}{e \left(a+b x^2\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{b e-a f} x}{\sqrt{e} \sqrt{b x^2+a}}\right)|\frac{(b c-a d) e}{c (b e-a f)}\right)}{2 b^2 c \sqrt{e+f x^2} \sqrt{b e-a f} \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}}}-\frac{a \sqrt{c+d x^2} \sqrt{\frac{a \left(e+f x^2\right)}{e \left(a+b x^2\right)}} (a d f-b (c f+d e)) \Pi \left(\frac{b c}{b c-a d};\sin ^{-1}\left(\frac{\sqrt{b c-a d} x}{\sqrt{c} \sqrt{b x^2+a}}\right)|\frac{c (b e-a f)}{(b c-a d) e}\right)}{2 b^2 \sqrt{c} \sqrt{e+f x^2} \sqrt{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} \sqrt{e+f x^2}}{2 \sqrt{a+b x^2}}-\frac{\sqrt{c} \sqrt{e+f x^2} \sqrt{b c-a d} \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}} E\left(\sin ^{-1}\left(\frac{\sqrt{b c-a d} x}{\sqrt{c} \sqrt{b x^2+a}}\right)|\frac{c (b e-a f)}{(b c-a d) e}\right)}{2 b \sqrt{c+d x^2} \sqrt{\frac{a \left(e+f x^2\right)}{e \left(a+b x^2\right)}}}","\frac{\sqrt{e} \sqrt{c+d x^2} (b c-a d) (2 b e-a f) \sqrt{\frac{a \left(e+f x^2\right)}{e \left(a+b x^2\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{b e-a f} x}{\sqrt{e} \sqrt{b x^2+a}}\right)|\frac{(b c-a d) e}{c (b e-a f)}\right)}{2 b^2 c \sqrt{e+f x^2} \sqrt{b e-a f} \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}}}-\frac{a \sqrt{c+d x^2} \sqrt{\frac{a \left(e+f x^2\right)}{e \left(a+b x^2\right)}} (a d f-b (c f+d e)) \Pi \left(\frac{b c}{b c-a d};\sin ^{-1}\left(\frac{\sqrt{b c-a d} x}{\sqrt{c} \sqrt{b x^2+a}}\right)|\frac{c (b e-a f)}{(b c-a d) e}\right)}{2 b^2 \sqrt{c} \sqrt{e+f x^2} \sqrt{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} \sqrt{e+f x^2}}{2 \sqrt{a+b x^2}}-\frac{\sqrt{c} \sqrt{e+f x^2} \sqrt{b c-a d} \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}} E\left(\sin ^{-1}\left(\frac{\sqrt{b c-a d} x}{\sqrt{c} \sqrt{b x^2+a}}\right)|\frac{c (b e-a f)}{(b c-a d) e}\right)}{2 b \sqrt{c+d x^2} \sqrt{\frac{a \left(e+f x^2\right)}{e \left(a+b x^2\right)}}}",1,"(x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(2*Sqrt[a + b*x^2]) - (Sqrt[c]*Sqrt[b*c - a*d]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*Sqrt[e + f*x^2]*EllipticE[ArcSin[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])], (c*(b*e - a*f))/((b*c - a*d)*e)])/(2*b*Sqrt[c + d*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]) + ((b*c - a*d)*Sqrt[e]*(2*b*e - a*f)*Sqrt[c + d*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]*EllipticF[ArcSin[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])], ((b*c - a*d)*e)/(c*(b*e - a*f))])/(2*b^2*c*Sqrt[b*e - a*f]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*Sqrt[e + f*x^2]) - (a*(a*d*f - b*(d*e + c*f))*Sqrt[c + d*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]*EllipticPi[(b*c)/(b*c - a*d), ArcSin[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])], (c*(b*e - a*f))/((b*c - a*d)*e)])/(2*b^2*Sqrt[c]*Sqrt[b*c - a*d]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*Sqrt[e + f*x^2])","A",7,7,34,0.2059,1,"{556, 554, 424, 552, 419, 553, 537}"
112,0,0,0,0.0559183,"\int \frac{\left(a+b x^2\right)^{3/2}}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx","Int[(a + b*x^2)^(3/2)/(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]","\int \frac{\left(a+b x^2\right)^{3/2}}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx","\text{Int}\left(\frac{\left(a+b x^2\right)^{3/2}}{\sqrt{c+d x^2} \sqrt{e+f x^2}},x\right)",0,"Defer[Int][(a + b*x^2)^(3/2)/(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x]","A",0,0,0,0,-1,"{}"
113,1,159,0,0.1352288,"\int \frac{\sqrt{a+b x^2}}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx","Int[Sqrt[a + b*x^2]/(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]","\frac{a \sqrt{c+d x^2} \sqrt{\frac{a \left(e+f x^2\right)}{e \left(a+b x^2\right)}} \Pi \left(\frac{b c}{b c-a d};\sin ^{-1}\left(\frac{\sqrt{b c-a d} x}{\sqrt{c} \sqrt{b x^2+a}}\right)|\frac{c (b e-a f)}{(b c-a d) e}\right)}{\sqrt{c} \sqrt{e+f x^2} \sqrt{b c-a d} \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}}}","\frac{a \sqrt{c+d x^2} \sqrt{\frac{a \left(e+f x^2\right)}{e \left(a+b x^2\right)}} \Pi \left(\frac{b c}{b c-a d};\sin ^{-1}\left(\frac{\sqrt{b c-a d} x}{\sqrt{c} \sqrt{b x^2+a}}\right)|\frac{c (b e-a f)}{(b c-a d) e}\right)}{\sqrt{c} \sqrt{e+f x^2} \sqrt{b c-a d} \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}}}",1,"(a*Sqrt[c + d*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]*EllipticPi[(b*c)/(b*c - a*d), ArcSin[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])], (c*(b*e - a*f))/((b*c - a*d)*e)])/(Sqrt[c]*Sqrt[b*c - a*d]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*Sqrt[e + f*x^2])","A",2,2,34,0.05882,1,"{553, 537}"
114,1,148,0,0.0799099,"\int \frac{1}{\sqrt{a+b x^2} \sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx","Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]","\frac{\sqrt{e} \sqrt{c+d x^2} \sqrt{\frac{a \left(e+f x^2\right)}{e \left(a+b x^2\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{b e-a f} x}{\sqrt{e} \sqrt{b x^2+a}}\right)|\frac{(b c-a d) e}{c (b e-a f)}\right)}{c \sqrt{e+f x^2} \sqrt{b e-a f} \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}}}","\frac{\sqrt{e} \sqrt{c+d x^2} \sqrt{\frac{a \left(e+f x^2\right)}{e \left(a+b x^2\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{b e-a f} x}{\sqrt{e} \sqrt{b x^2+a}}\right)|\frac{(b c-a d) e}{c (b e-a f)}\right)}{c \sqrt{e+f x^2} \sqrt{b e-a f} \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}}}",1,"(Sqrt[e]*Sqrt[c + d*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]*EllipticF[ArcSin[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])], ((b*c - a*d)*e)/(c*(b*e - a*f))])/(c*Sqrt[b*e - a*f]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*Sqrt[e + f*x^2])","A",2,2,34,0.05882,1,"{552, 419}"
115,0,0,0,0.0552669,"\int \frac{1}{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx","Int[1/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]","\int \frac{1}{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx","\text{Int}\left(\frac{1}{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2} \sqrt{e+f x^2}},x\right)",0,"Defer[Int][1/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x]","A",0,0,0,0,-1,"{}"