1,1,223,172,0.3019852,"\int \left(a+b x^2\right) \left(c+d x^2\right) \left(e+f x^2\right)^4 \, dx","IntegrateAlgebraic[(a + b*x^2)*(c + d*x^2)*(e + f*x^2)^4,x]","\frac{45045 a c e^4 x+60060 a c e^3 f x^3+54054 a c e^2 f^2 x^5+25740 a c e f^3 x^7+5005 a c f^4 x^9+15015 a d e^4 x^3+36036 a d e^3 f x^5+38610 a d e^2 f^2 x^7+20020 a d e f^3 x^9+4095 a d f^4 x^{11}+15015 b c e^4 x^3+36036 b c e^3 f x^5+38610 b c e^2 f^2 x^7+20020 b c e f^3 x^9+4095 b c f^4 x^{11}+9009 b d e^4 x^5+25740 b d e^3 f x^7+30030 b d e^2 f^2 x^9+16380 b d e f^3 x^{11}+3465 b d f^4 x^{13}}{45045}","\frac{1}{3} e^3 x^3 (4 a c f+a d e+b c e)+\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}{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,"(45045*a*c*e^4*x + 15015*b*c*e^4*x^3 + 15015*a*d*e^4*x^3 + 60060*a*c*e^3*f*x^3 + 9009*b*d*e^4*x^5 + 36036*b*c*e^3*f*x^5 + 36036*a*d*e^3*f*x^5 + 54054*a*c*e^2*f^2*x^5 + 25740*b*d*e^3*f*x^7 + 38610*b*c*e^2*f^2*x^7 + 38610*a*d*e^2*f^2*x^7 + 25740*a*c*e*f^3*x^7 + 30030*b*d*e^2*f^2*x^9 + 20020*b*c*e*f^3*x^9 + 20020*a*d*e*f^3*x^9 + 5005*a*c*f^4*x^9 + 16380*b*d*e*f^3*x^11 + 4095*b*c*f^4*x^11 + 4095*a*d*f^4*x^11 + 3465*b*d*f^4*x^13)/45045","A",1
2,1,171,130,0.1508857,"\int \left(a+b x^2\right) \left(c+d x^2\right) \left(e+f x^2\right)^3 \, dx","IntegrateAlgebraic[(a + b*x^2)*(c + d*x^2)*(e + f*x^2)^3,x]","\frac{3465 a c e^3 x+3465 a c e^2 f x^3+2079 a c e f^2 x^5+495 a c f^3 x^7+1155 a d e^3 x^3+2079 a d e^2 f x^5+1485 a d e f^2 x^7+385 a d f^3 x^9+1155 b c e^3 x^3+2079 b c e^2 f x^5+1485 b c e f^2 x^7+385 b c f^3 x^9+693 b d e^3 x^5+1485 b d e^2 f x^7+1155 b d e f^2 x^9+315 b d f^3 x^{11}}{3465}","\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,"(3465*a*c*e^3*x + 1155*b*c*e^3*x^3 + 1155*a*d*e^3*x^3 + 3465*a*c*e^2*f*x^3 + 693*b*d*e^3*x^5 + 2079*b*c*e^2*f*x^5 + 2079*a*d*e^2*f*x^5 + 2079*a*c*e*f^2*x^5 + 1485*b*d*e^2*f*x^7 + 1485*b*c*e*f^2*x^7 + 1485*a*d*e*f^2*x^7 + 495*a*c*f^3*x^7 + 1155*b*d*e*f^2*x^9 + 385*b*c*f^3*x^9 + 385*a*d*f^3*x^9 + 315*b*d*f^3*x^11)/3465","A",1
3,1,119,94,0.397021,"\int \left(a+b x^2\right) \left(c+d x^2\right) \left(e+f x^2\right)^2 \, dx","IntegrateAlgebraic[(a + b*x^2)*(c + d*x^2)*(e + f*x^2)^2,x]","\frac{1}{315} \left(315 a c e^2 x+210 a c e f x^3+63 a c f^2 x^5+105 a d e^2 x^3+126 a d e f x^5+45 a d f^2 x^7+105 b c e^2 x^3+126 b c e f x^5+45 b c f^2 x^7+63 b d e^2 x^5+90 b d e f x^7+35 b d f^2 x^9\right)","\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,"(315*a*c*e^2*x + 105*b*c*e^2*x^3 + 105*a*d*e^2*x^3 + 210*a*c*e*f*x^3 + 63*b*d*e^2*x^5 + 126*b*c*e*f*x^5 + 126*a*d*e*f*x^5 + 63*a*c*f^2*x^5 + 90*b*d*e*f*x^7 + 45*b*c*f^2*x^7 + 45*a*d*f^2*x^7 + 35*b*d*f^2*x^9)/315","A",1
4,1,67,56,0.0308664,"\int \left(a+b x^2\right) \left(c+d x^2\right) \left(e+f x^2\right) \, dx","IntegrateAlgebraic[(a + b*x^2)*(c + d*x^2)*(e + f*x^2),x]","\frac{1}{105} \left(105 a c e x+35 a c f x^3+35 a d e x^3+21 a d f x^5+35 b c e x^3+21 b c f x^5+21 b d e x^5+15 b d f x^7\right)","\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,"(105*a*c*e*x + 35*b*c*e*x^3 + 35*a*d*e*x^3 + 35*a*c*f*x^3 + 21*b*d*e*x^5 + 21*b*c*f*x^5 + 21*a*d*f*x^5 + 15*b*d*f*x^7)/105","A",1
5,1,81,81,0.0774743,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)}{e+f x^2} \, dx","IntegrateAlgebraic[((a + b*x^2)*(c + d*x^2))/(e + f*x^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right) \left(a c f^2-a d e f-b c e f+b d e^2\right)}{\sqrt{e} f^{5/2}}+\frac{x \left(3 a d f+3 b c f-3 b d e+b d f x^2\right)}{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{x (-2 a d f-3 b c f+3 b d e)}{3 f^2}+\frac{d x \left(a+b x^2\right)}{3 f}",1,"(x*(-3*b*d*e + 3*b*c*f + 3*a*d*f + b*d*f*x^2))/(3*f^2) + ((b*d*e^2 - b*c*e*f - a*d*e*f + a*c*f^2)*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*f^(5/2))","A",1
6,1,107,108,0.0998807,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)}{\left(e+f x^2\right)^2} \, dx","IntegrateAlgebraic[((a + b*x^2)*(c + d*x^2))/(e + f*x^2)^2,x]","\frac{x \left(a c f^2-a d e f-b c e f+3 b d e^2+2 b d e f x^2\right)}{2 e f^2 \left(e+f x^2\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right) \left(a c f^2+a d e f+b c e f-3 b d e^2\right)}{2 e^{3/2} f^{5/2}}","-\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,"(x*(3*b*d*e^2 - b*c*e*f - a*d*e*f + a*c*f^2 + 2*b*d*e*f*x^2))/(2*e*f^2*(e + f*x^2)) + ((-3*b*d*e^2 + b*c*e*f + a*d*e*f + a*c*f^2)*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(2*e^(3/2)*f^(5/2))","A",1
7,1,146,130,0.1501301,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)}{\left(e+f x^2\right)^3} \, dx","IntegrateAlgebraic[((a + b*x^2)*(c + d*x^2))/(e + f*x^2)^3,x]","\frac{\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right) \left(3 a c f^2+a d e f+b c e f+3 b d e^2\right)}{8 e^{5/2} f^{5/2}}-\frac{x \left(-5 a c e f^2-3 a c f^3 x^2+a d e^2 f-a d e f^2 x^2+b c e^2 f-b c e f^2 x^2+3 b d e^3+5 b d e^2 f x^2\right)}{8 e^2 f^2 \left(e+f x^2\right)^2}","\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 (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{x \left(a+b x^2\right) (d e-c f)}{4 e f \left(e+f x^2\right)^2}",1,"-1/8*(x*(3*b*d*e^3 + b*c*e^2*f + a*d*e^2*f - 5*a*c*e*f^2 + 5*b*d*e^2*f*x^2 - b*c*e*f^2*x^2 - a*d*e*f^2*x^2 - 3*a*c*f^3*x^2))/(e^2*f^2*(e + f*x^2)^2) + ((3*b*d*e^2 + b*c*e*f + a*d*e*f + 3*a*c*f^2)*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(8*e^(5/2)*f^(5/2))","A",1
8,1,199,171,0.1750198,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)}{\left(e+f x^2\right)^4} \, dx","IntegrateAlgebraic[((a + b*x^2)*(c + d*x^2))/(e + f*x^2)^4,x]","\frac{\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right) \left(5 a c f^2+a d e f+b c e f+b d e^2\right)}{16 e^{7/2} f^{5/2}}+\frac{x \left(33 a c e^2 f^2+40 a c e f^3 x^2+15 a c f^4 x^4-3 a d e^3 f+8 a d e^2 f^2 x^2+3 a d e f^3 x^4-3 b c e^3 f+8 b c e^2 f^2 x^2+3 b c e f^3 x^4-3 b d e^4-8 b d e^3 f x^2+3 b d e^2 f^2 x^4\right)}{48 e^3 f^2 \left(e+f x^2\right)^3}","\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 (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{x \left(a+b x^2\right) (d e-c f)}{6 e f \left(e+f x^2\right)^3}",1,"(x*(-3*b*d*e^4 - 3*b*c*e^3*f - 3*a*d*e^3*f + 33*a*c*e^2*f^2 - 8*b*d*e^3*f*x^2 + 8*b*c*e^2*f^2*x^2 + 8*a*d*e^2*f^2*x^2 + 40*a*c*e*f^3*x^2 + 3*b*d*e^2*f^2*x^4 + 3*b*c*e*f^3*x^4 + 3*a*d*e*f^3*x^4 + 15*a*c*f^4*x^4))/(48*e^3*f^2*(e + f*x^2)^3) + ((b*d*e^2 + b*c*e*f + a*d*e*f + 5*a*c*f^2)*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(16*e^(7/2)*f^(5/2))","A",1
9,1,295,226,0.1930957,"\int \left(a+b x^2\right) \left(c+d x^2\right)^2 \left(e+f x^2\right)^3 \, dx","IntegrateAlgebraic[(a + b*x^2)*(c + d*x^2)^2*(e + f*x^2)^3,x]","\frac{45045 a c^2 e^3 x+45045 a c^2 e^2 f x^3+27027 a c^2 e f^2 x^5+6435 a c^2 f^3 x^7+30030 a c d e^3 x^3+54054 a c d e^2 f x^5+38610 a c d e f^2 x^7+10010 a c d f^3 x^9+9009 a d^2 e^3 x^5+19305 a d^2 e^2 f x^7+15015 a d^2 e f^2 x^9+4095 a d^2 f^3 x^{11}+15015 b c^2 e^3 x^3+27027 b c^2 e^2 f x^5+19305 b c^2 e f^2 x^7+5005 b c^2 f^3 x^9+18018 b c d e^3 x^5+38610 b c d e^2 f x^7+30030 b c d e f^2 x^9+8190 b c d f^3 x^{11}+6435 b d^2 e^3 x^7+15015 b d^2 e^2 f x^9+12285 b d^2 e f^2 x^{11}+3465 b d^2 f^3 x^{13}}{45045}","\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,"(45045*a*c^2*e^3*x + 15015*b*c^2*e^3*x^3 + 30030*a*c*d*e^3*x^3 + 45045*a*c^2*e^2*f*x^3 + 18018*b*c*d*e^3*x^5 + 9009*a*d^2*e^3*x^5 + 27027*b*c^2*e^2*f*x^5 + 54054*a*c*d*e^2*f*x^5 + 27027*a*c^2*e*f^2*x^5 + 6435*b*d^2*e^3*x^7 + 38610*b*c*d*e^2*f*x^7 + 19305*a*d^2*e^2*f*x^7 + 19305*b*c^2*e*f^2*x^7 + 38610*a*c*d*e*f^2*x^7 + 6435*a*c^2*f^3*x^7 + 15015*b*d^2*e^2*f*x^9 + 30030*b*c*d*e*f^2*x^9 + 15015*a*d^2*e*f^2*x^9 + 5005*b*c^2*f^3*x^9 + 10010*a*c*d*f^3*x^9 + 12285*b*d^2*e*f^2*x^11 + 8190*b*c*d*f^3*x^11 + 4095*a*d^2*f^3*x^11 + 3465*b*d^2*f^3*x^13)/45045","A",1
10,1,207,158,0.1373757,"\int \left(a+b x^2\right) \left(c+d x^2\right)^2 \left(e+f x^2\right)^2 \, dx","IntegrateAlgebraic[(a + b*x^2)*(c + d*x^2)^2*(e + f*x^2)^2,x]","\frac{3465 a c^2 e^2 x+2310 a c^2 e f x^3+693 a c^2 f^2 x^5+2310 a c d e^2 x^3+2772 a c d e f x^5+990 a c d f^2 x^7+693 a d^2 e^2 x^5+990 a d^2 e f x^7+385 a d^2 f^2 x^9+1155 b c^2 e^2 x^3+1386 b c^2 e f x^5+495 b c^2 f^2 x^7+1386 b c d e^2 x^5+1980 b c d e f x^7+770 b c d f^2 x^9+495 b d^2 e^2 x^7+770 b d^2 e f x^9+315 b d^2 f^2 x^{11}}{3465}","\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,"(3465*a*c^2*e^2*x + 1155*b*c^2*e^2*x^3 + 2310*a*c*d*e^2*x^3 + 2310*a*c^2*e*f*x^3 + 1386*b*c*d*e^2*x^5 + 693*a*d^2*e^2*x^5 + 1386*b*c^2*e*f*x^5 + 2772*a*c*d*e*f*x^5 + 693*a*c^2*f^2*x^5 + 495*b*d^2*e^2*x^7 + 1980*b*c*d*e*f*x^7 + 990*a*d^2*e*f*x^7 + 495*b*c^2*f^2*x^7 + 990*a*c*d*f^2*x^7 + 770*b*d^2*e*f*x^9 + 770*b*c*d*f^2*x^9 + 385*a*d^2*f^2*x^9 + 315*b*d^2*f^2*x^11)/3465","A",1
11,1,119,94,0.071221,"\int \left(a+b x^2\right) \left(c+d x^2\right)^2 \left(e+f x^2\right) \, dx","IntegrateAlgebraic[(a + b*x^2)*(c + d*x^2)^2*(e + f*x^2),x]","\frac{1}{315} \left(315 a c^2 e x+105 a c^2 f x^3+210 a c d e x^3+126 a c d f x^5+63 a d^2 e x^5+45 a d^2 f x^7+105 b c^2 e x^3+63 b c^2 f x^5+126 b c d e x^5+90 b c d f x^7+45 b d^2 e x^7+35 b d^2 f x^9\right)","\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,"(315*a*c^2*e*x + 105*b*c^2*e*x^3 + 210*a*c*d*e*x^3 + 105*a*c^2*f*x^3 + 126*b*c*d*e*x^5 + 63*a*d^2*e*x^5 + 63*b*c^2*f*x^5 + 126*a*c*d*f*x^5 + 45*b*d^2*e*x^7 + 90*b*c*d*f*x^7 + 45*a*d^2*f*x^7 + 35*b*d^2*f*x^9)/315","A",1
12,1,140,142,0.11241,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^2}{e+f x^2} \, dx","IntegrateAlgebraic[((a + b*x^2)*(c + d*x^2)^2)/(e + f*x^2),x]","\frac{x \left(30 a c d f^2-15 a d^2 e f+5 a d^2 f^2 x^2+15 b c^2 f^2-30 b c d e f+10 b c d f^2 x^2+15 b d^2 e^2-5 b d^2 e f x^2+3 b d^2 f^2 x^4\right)}{15 f^3}-\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{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{(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{x \left(c+d x^2\right) (-5 a d f-4 b c f+5 b d e)}{15 f^2}+\frac{b x \left(c+d x^2\right)^2}{5 f}",1,"(x*(15*b*d^2*e^2 - 30*b*c*d*e*f - 15*a*d^2*e*f + 15*b*c^2*f^2 + 30*a*c*d*f^2 - 5*b*d^2*e*f*x^2 + 10*b*c*d*f^2*x^2 + 5*a*d^2*f^2*x^2 + 3*b*d^2*f^2*x^4))/(15*f^3) - ((b*e - a*f)*(d*e - c*f)^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*f^(7/2))","A",1
13,1,211,164,0.1574904,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^2}{\left(e+f x^2\right)^2} \, dx","IntegrateAlgebraic[((a + b*x^2)*(c + d*x^2)^2)/(e + f*x^2)^2,x]","\frac{\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right) \left(a c^2 f^3+2 a c d e f^2-3 a d^2 e^2 f+b c^2 e f^2-6 b c d e^2 f+5 b d^2 e^3\right)}{2 e^{3/2} f^{7/2}}-\frac{x \left(-3 a c^2 f^3+6 a c d e f^2-9 a d^2 e^2 f-6 a d^2 e f^2 x^2+3 b c^2 e f^2-18 b c d e^2 f-12 b c d e f^2 x^2+15 b d^2 e^3+10 b d^2 e^2 f x^2-2 b d^2 e f^2 x^4\right)}{6 e f^3 \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 (b e (15 d e-13 c f)-3 a f (3 d e-c f))}{6 e f^3}+\frac{d x \left(c+d x^2\right) (5 b e-3 a f)}{6 e f^2}-\frac{x \left(c+d x^2\right)^2 (b e-a f)}{2 e f \left(e+f x^2\right)}",1,"-1/6*(x*(15*b*d^2*e^3 - 18*b*c*d*e^2*f - 9*a*d^2*e^2*f + 3*b*c^2*e*f^2 + 6*a*c*d*e*f^2 - 3*a*c^2*f^3 + 10*b*d^2*e^2*f*x^2 - 12*b*c*d*e*f^2*x^2 - 6*a*d^2*e*f^2*x^2 - 2*b*d^2*e*f^2*x^4))/(e*f^3*(e + f*x^2)) + ((5*b*d^2*e^3 - 6*b*c*d*e^2*f - 3*a*d^2*e^2*f + b*c^2*e*f^2 + 2*a*c*d*e*f^2 + a*c^2*f^3)*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(2*e^(3/2)*f^(7/2))","A",1
14,1,259,207,0.2170782,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^2}{\left(e+f x^2\right)^3} \, dx","IntegrateAlgebraic[((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(3 a c^2 f^3+2 a c d e f^2+3 a d^2 e^2 f+b c^2 e f^2+6 b c d e^2 f-15 b d^2 e^3\right)}{8 e^{5/2} f^{7/2}}+\frac{x \left(5 a c^2 e f^3+3 a c^2 f^4 x^2-2 a c d e^2 f^2+2 a c d e f^3 x^2-3 a d^2 e^3 f-5 a d^2 e^2 f^2 x^2-b c^2 e^2 f^2+b c^2 e f^3 x^2-6 b c d e^3 f-10 b c d e^2 f^2 x^2+15 b d^2 e^4+25 b d^2 e^3 f x^2+8 b d^2 e^2 f^2 x^4\right)}{8 e^2 f^3 \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{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) (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{x \left(c+d x^2\right)^2 (b e-a f)}{4 e f \left(e+f x^2\right)^2}",1,"(x*(15*b*d^2*e^4 - 6*b*c*d*e^3*f - 3*a*d^2*e^3*f - b*c^2*e^2*f^2 - 2*a*c*d*e^2*f^2 + 5*a*c^2*e*f^3 + 25*b*d^2*e^3*f*x^2 - 10*b*c*d*e^2*f^2*x^2 - 5*a*d^2*e^2*f^2*x^2 + b*c^2*e*f^3*x^2 + 2*a*c*d*e*f^3*x^2 + 3*a*c^2*f^4*x^2 + 8*b*d^2*e^2*f^2*x^4))/(8*e^2*f^3*(e + f*x^2)^2) + ((-15*b*d^2*e^3 + 6*b*c*d*e^2*f + 3*a*d^2*e^2*f + b*c^2*e*f^2 + 2*a*c*d*e*f^2 + 3*a*c^2*f^3)*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(8*e^(5/2)*f^(7/2))","A",1
15,1,332,240,0.2976048,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^2}{\left(e+f x^2\right)^4} \, dx","IntegrateAlgebraic[((a + b*x^2)*(c + d*x^2)^2)/(e + f*x^2)^4,x]","\frac{\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right) \left(5 a c^2 f^3+2 a c d e f^2+a d^2 e^2 f+b c^2 e f^2+2 b c d e^2 f+5 b d^2 e^3\right)}{16 e^{7/2} f^{7/2}}-\frac{x \left(-33 a c^2 e^2 f^3-40 a c^2 e f^4 x^2-15 a c^2 f^5 x^4+6 a c d e^3 f^2-16 a c d e^2 f^3 x^2-6 a c d e f^4 x^4+3 a d^2 e^4 f+8 a d^2 e^3 f^2 x^2-3 a d^2 e^2 f^3 x^4+3 b c^2 e^3 f^2-8 b c^2 e^2 f^3 x^2-3 b c^2 e f^4 x^4+6 b c d e^4 f+16 b c d e^3 f^2 x^2-6 b c d e^2 f^3 x^4+15 b d^2 e^5+40 b d^2 e^4 f x^2+33 b d^2 e^3 f^2 x^4\right)}{48 e^3 f^3 \left(e+f x^2\right)^3}","\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(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{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,"-1/48*(x*(15*b*d^2*e^5 + 6*b*c*d*e^4*f + 3*a*d^2*e^4*f + 3*b*c^2*e^3*f^2 + 6*a*c*d*e^3*f^2 - 33*a*c^2*e^2*f^3 + 40*b*d^2*e^4*f*x^2 + 16*b*c*d*e^3*f^2*x^2 + 8*a*d^2*e^3*f^2*x^2 - 8*b*c^2*e^2*f^3*x^2 - 16*a*c*d*e^2*f^3*x^2 - 40*a*c^2*e*f^4*x^2 + 33*b*d^2*e^3*f^2*x^4 - 6*b*c*d*e^2*f^3*x^4 - 3*a*d^2*e^2*f^3*x^4 - 3*b*c^2*e*f^4*x^4 - 6*a*c*d*e*f^4*x^4 - 15*a*c^2*f^5*x^4))/(e^3*f^3*(e + f*x^2)^3) + ((5*b*d^2*e^3 + 2*b*c*d*e^2*f + a*d^2*e^2*f + b*c^2*e*f^2 + 2*a*c*d*e*f^2 + 5*a*c^2*f^3)*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(16*e^(7/2)*f^(7/2))","A",1
16,1,419,310,0.2670187,"\int \left(a+b x^2\right) \left(c+d x^2\right)^3 \left(e+f x^2\right)^3 \, dx","IntegrateAlgebraic[(a + b*x^2)*(c + d*x^2)^3*(e + f*x^2)^3,x]","\frac{45045 a c^3 e^3 x+45045 a c^3 e^2 f x^3+27027 a c^3 e f^2 x^5+6435 a c^3 f^3 x^7+45045 a c^2 d e^3 x^3+81081 a c^2 d e^2 f x^5+57915 a c^2 d e f^2 x^7+15015 a c^2 d f^3 x^9+27027 a c d^2 e^3 x^5+57915 a c d^2 e^2 f x^7+45045 a c d^2 e f^2 x^9+12285 a c d^2 f^3 x^{11}+6435 a d^3 e^3 x^7+15015 a d^3 e^2 f x^9+12285 a d^3 e f^2 x^{11}+3465 a d^3 f^3 x^{13}+15015 b c^3 e^3 x^3+27027 b c^3 e^2 f x^5+19305 b c^3 e f^2 x^7+5005 b c^3 f^3 x^9+27027 b c^2 d e^3 x^5+57915 b c^2 d e^2 f x^7+45045 b c^2 d e f^2 x^9+12285 b c^2 d f^3 x^{11}+19305 b c d^2 e^3 x^7+45045 b c d^2 e^2 f x^9+36855 b c d^2 e f^2 x^{11}+10395 b c d^2 f^3 x^{13}+5005 b d^3 e^3 x^9+12285 b d^3 e^2 f x^{11}+10395 b d^3 e f^2 x^{13}+3003 b d^3 f^3 x^{15}}{45045}","\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{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}{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(c^3 f^3+9 c^2 d e f^2+9 c d^2 e^2 f+d^3 e^3\right)\right)+\frac{1}{7} x^7 \left(a \left(c^3 f^3+9 c^2 d e f^2+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{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,"(45045*a*c^3*e^3*x + 15015*b*c^3*e^3*x^3 + 45045*a*c^2*d*e^3*x^3 + 45045*a*c^3*e^2*f*x^3 + 27027*b*c^2*d*e^3*x^5 + 27027*a*c*d^2*e^3*x^5 + 27027*b*c^3*e^2*f*x^5 + 81081*a*c^2*d*e^2*f*x^5 + 27027*a*c^3*e*f^2*x^5 + 19305*b*c*d^2*e^3*x^7 + 6435*a*d^3*e^3*x^7 + 57915*b*c^2*d*e^2*f*x^7 + 57915*a*c*d^2*e^2*f*x^7 + 19305*b*c^3*e*f^2*x^7 + 57915*a*c^2*d*e*f^2*x^7 + 6435*a*c^3*f^3*x^7 + 5005*b*d^3*e^3*x^9 + 45045*b*c*d^2*e^2*f*x^9 + 15015*a*d^3*e^2*f*x^9 + 45045*b*c^2*d*e*f^2*x^9 + 45045*a*c*d^2*e*f^2*x^9 + 5005*b*c^3*f^3*x^9 + 15015*a*c^2*d*f^3*x^9 + 12285*b*d^3*e^2*f*x^11 + 36855*b*c*d^2*e*f^2*x^11 + 12285*a*d^3*e*f^2*x^11 + 12285*b*c^2*d*f^3*x^11 + 12285*a*c*d^2*f^3*x^11 + 10395*b*d^3*e*f^2*x^13 + 10395*b*c*d^2*f^3*x^13 + 3465*a*d^3*f^3*x^13 + 3003*b*d^3*f^3*x^15)/45045","A",1
17,1,295,226,0.2030828,"\int \left(a+b x^2\right) \left(c+d x^2\right)^3 \left(e+f x^2\right)^2 \, dx","IntegrateAlgebraic[(a + b*x^2)*(c + d*x^2)^3*(e + f*x^2)^2,x]","\frac{45045 a c^3 e^2 x+30030 a c^3 e f x^3+9009 a c^3 f^2 x^5+45045 a c^2 d e^2 x^3+54054 a c^2 d e f x^5+19305 a c^2 d f^2 x^7+27027 a c d^2 e^2 x^5+38610 a c d^2 e f x^7+15015 a c d^2 f^2 x^9+6435 a d^3 e^2 x^7+10010 a d^3 e f x^9+4095 a d^3 f^2 x^{11}+15015 b c^3 e^2 x^3+18018 b c^3 e f x^5+6435 b c^3 f^2 x^7+27027 b c^2 d e^2 x^5+38610 b c^2 d e f x^7+15015 b c^2 d f^2 x^9+19305 b c d^2 e^2 x^7+30030 b c d^2 e f x^9+12285 b c d^2 f^2 x^{11}+5005 b d^3 e^2 x^9+8190 b d^3 e f x^{11}+3465 b d^3 f^2 x^{13}}{45045}","\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,"(45045*a*c^3*e^2*x + 15015*b*c^3*e^2*x^3 + 45045*a*c^2*d*e^2*x^3 + 30030*a*c^3*e*f*x^3 + 27027*b*c^2*d*e^2*x^5 + 27027*a*c*d^2*e^2*x^5 + 18018*b*c^3*e*f*x^5 + 54054*a*c^2*d*e*f*x^5 + 9009*a*c^3*f^2*x^5 + 19305*b*c*d^2*e^2*x^7 + 6435*a*d^3*e^2*x^7 + 38610*b*c^2*d*e*f*x^7 + 38610*a*c*d^2*e*f*x^7 + 6435*b*c^3*f^2*x^7 + 19305*a*c^2*d*f^2*x^7 + 5005*b*d^3*e^2*x^9 + 30030*b*c*d^2*e*f*x^9 + 10010*a*d^3*e*f*x^9 + 15015*b*c^2*d*f^2*x^9 + 15015*a*c*d^2*f^2*x^9 + 8190*b*d^3*e*f*x^11 + 12285*b*c*d^2*f^2*x^11 + 4095*a*d^3*f^2*x^11 + 3465*b*d^3*f^2*x^13)/45045","A",1
18,1,171,130,0.1182163,"\int \left(a+b x^2\right) \left(c+d x^2\right)^3 \left(e+f x^2\right) \, dx","IntegrateAlgebraic[(a + b*x^2)*(c + d*x^2)^3*(e + f*x^2),x]","\frac{3465 a c^3 e x+1155 a c^3 f x^3+3465 a c^2 d e x^3+2079 a c^2 d f x^5+2079 a c d^2 e x^5+1485 a c d^2 f x^7+495 a d^3 e x^7+385 a d^3 f x^9+1155 b c^3 e x^3+693 b c^3 f x^5+2079 b c^2 d e x^5+1485 b c^2 d f x^7+1485 b c d^2 e x^7+1155 b c d^2 f x^9+385 b d^3 e x^9+315 b d^3 f x^{11}}{3465}","\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,"(3465*a*c^3*e*x + 1155*b*c^3*e*x^3 + 3465*a*c^2*d*e*x^3 + 1155*a*c^3*f*x^3 + 2079*b*c^2*d*e*x^5 + 2079*a*c*d^2*e*x^5 + 693*b*c^3*f*x^5 + 2079*a*c^2*d*f*x^5 + 1485*b*c*d^2*e*x^7 + 495*a*d^3*e*x^7 + 1485*b*c^2*d*f*x^7 + 1485*a*c*d^2*f*x^7 + 385*b*d^3*e*x^9 + 1155*b*c*d^2*f*x^9 + 385*a*d^3*f*x^9 + 315*b*d^3*f*x^11)/3465","A",1
19,1,245,227,0.1720164,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^3}{e+f x^2} \, dx","IntegrateAlgebraic[((a + b*x^2)*(c + d*x^2)^3)/(e + f*x^2),x]","\frac{315 a c^2 d f^3 x-315 a c d^2 e f^2 x+105 a c d^2 f^3 x^3+105 a d^3 e^2 f x-35 a d^3 e f^2 x^3+21 a d^3 f^3 x^5+105 b c^3 f^3 x-315 b c^2 d e f^2 x+105 b c^2 d f^3 x^3+315 b c d^2 e^2 f x-105 b c d^2 e f^2 x^3+63 b c d^2 f^3 x^5-105 b d^3 e^3 x+35 b d^3 e^2 f x^3-21 b d^3 e f^2 x^5+15 b d^3 f^3 x^7}{105 f^4}+\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{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(-48 c^3 f^3+231 c^2 d e f^2-280 c d^2 e^2 f+105 d^3 e^3\right)\right)}{105 f^4}+\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{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 x \left(c+d x^2\right)^3}{7 f}",1,"(-105*b*d^3*e^3*x + 315*b*c*d^2*e^2*f*x + 105*a*d^3*e^2*f*x - 315*b*c^2*d*e*f^2*x - 315*a*c*d^2*e*f^2*x + 105*b*c^3*f^3*x + 315*a*c^2*d*f^3*x + 35*b*d^3*e^2*f*x^3 - 105*b*c*d^2*e*f^2*x^3 - 35*a*d^3*e*f^2*x^3 + 105*b*c^2*d*f^3*x^3 + 105*a*c*d^2*f^3*x^3 - 21*b*d^3*e*f^2*x^5 + 63*b*c*d^2*f^3*x^5 + 21*a*d^3*f^3*x^5 + 15*b*d^3*f^3*x^7)/(105*f^4) + ((b*e - a*f)*(d*e - c*f)^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*f^(9/2))","A",1
20,1,347,242,0.2205602,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^3}{\left(e+f x^2\right)^2} \, dx","IntegrateAlgebraic[((a + b*x^2)*(c + d*x^2)^3)/(e + f*x^2)^2,x]","\frac{x \left(15 a c^3 f^4-45 a c^2 d e f^3+135 a c d^2 e^2 f^2+90 a c d^2 e f^3 x^2-75 a d^3 e^3 f-50 a d^3 e^2 f^2 x^2+10 a d^3 e f^3 x^4-15 b c^3 e f^3+135 b c^2 d e^2 f^2+90 b c^2 d e f^3 x^2-225 b c d^2 e^3 f-150 b c d^2 e^2 f^2 x^2+30 b c d^2 e f^3 x^4+105 b d^3 e^4+70 b d^3 e^3 f x^2-14 b d^3 e^2 f^2 x^4+6 b d^3 e f^3 x^6\right)}{30 e f^4 \left(e+f x^2\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right) \left(a c^3 f^4+3 a c^2 d e f^3-9 a c d^2 e^2 f^2+5 a d^3 e^3 f+b c^3 e f^3-9 b c^2 d e^2 f^2+15 b c d^2 e^3 f-7 b d^3 e^4\right)}{2 e^{3/2} f^{9/2}}","-\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) (b e (35 d e-33 c f)-5 a f (5 d e-3 c f))}{30 e f^3}+\frac{d x \left(c+d x^2\right)^2 (7 b e-5 a f)}{10 e f^2}-\frac{x \left(c+d x^2\right)^3 (b e-a f)}{2 e f \left(e+f x^2\right)}",1,"(x*(105*b*d^3*e^4 - 225*b*c*d^2*e^3*f - 75*a*d^3*e^3*f + 135*b*c^2*d*e^2*f^2 + 135*a*c*d^2*e^2*f^2 - 15*b*c^3*e*f^3 - 45*a*c^2*d*e*f^3 + 15*a*c^3*f^4 + 70*b*d^3*e^3*f*x^2 - 150*b*c*d^2*e^2*f^2*x^2 - 50*a*d^3*e^2*f^2*x^2 + 90*b*c^2*d*e*f^3*x^2 + 90*a*c*d^2*e*f^3*x^2 - 14*b*d^3*e^2*f^2*x^4 + 30*b*c*d^2*e*f^3*x^4 + 10*a*d^3*e*f^3*x^4 + 6*b*d^3*e*f^3*x^6))/(30*e*f^4*(e + f*x^2)) + ((-7*b*d^3*e^4 + 15*b*c*d^2*e^3*f + 5*a*d^3*e^3*f - 9*b*c^2*d*e^2*f^2 - 9*a*c*d^2*e^2*f^2 + b*c^3*e*f^3 + 3*a*c^2*d*e*f^3 + a*c^3*f^4)*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(2*e^(3/2)*f^(9/2))","A",1
21,1,409,291,0.2913424,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^3}{\left(e+f x^2\right)^3} \, dx","IntegrateAlgebraic[((a + b*x^2)*(c + d*x^2)^3)/(e + f*x^2)^3,x]","\frac{\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right) \left(3 a c^3 f^4+3 a c^2 d e f^3+9 a c d^2 e^2 f^2-15 a d^3 e^3 f+b c^3 e f^3+9 b c^2 d e^2 f^2-45 b c d^2 e^3 f+35 b d^3 e^4\right)}{8 e^{5/2} f^{9/2}}+\frac{15 a c^3 e f^4 x+9 a c^3 f^5 x^3-9 a c^2 d e^2 f^3 x+9 a c^2 d e f^4 x^3-27 a c d^2 e^3 f^2 x-45 a c d^2 e^2 f^3 x^3+45 a d^3 e^4 f x+75 a d^3 e^3 f^2 x^3+24 a d^3 e^2 f^3 x^5-3 b c^3 e^2 f^3 x+3 b c^3 e f^4 x^3-27 b c^2 d e^3 f^2 x-45 b c^2 d e^2 f^3 x^3+135 b c d^2 e^4 f x+225 b c d^2 e^3 f^2 x^3+72 b c d^2 e^2 f^3 x^5-105 b d^3 e^5 x-175 b d^3 e^4 f x^3-56 b d^3 e^3 f^2 x^5+8 b d^3 e^2 f^3 x^7}{24 e^2 f^4 \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{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)^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{x \left(c+d x^2\right)^3 (b e-a f)}{4 e f \left(e+f x^2\right)^2}",1,"(-105*b*d^3*e^5*x + 135*b*c*d^2*e^4*f*x + 45*a*d^3*e^4*f*x - 27*b*c^2*d*e^3*f^2*x - 27*a*c*d^2*e^3*f^2*x - 3*b*c^3*e^2*f^3*x - 9*a*c^2*d*e^2*f^3*x + 15*a*c^3*e*f^4*x - 175*b*d^3*e^4*f*x^3 + 225*b*c*d^2*e^3*f^2*x^3 + 75*a*d^3*e^3*f^2*x^3 - 45*b*c^2*d*e^2*f^3*x^3 - 45*a*c*d^2*e^2*f^3*x^3 + 3*b*c^3*e*f^4*x^3 + 9*a*c^2*d*e*f^4*x^3 + 9*a*c^3*f^5*x^3 - 56*b*d^3*e^3*f^2*x^5 + 72*b*c*d^2*e^2*f^3*x^5 + 24*a*d^3*e^2*f^3*x^5 + 8*b*d^3*e^2*f^3*x^7)/(24*e^2*f^4*(e + f*x^2)^2) + ((35*b*d^3*e^4 - 45*b*c*d^2*e^3*f - 15*a*d^3*e^3*f + 9*b*c^2*d*e^2*f^2 + 9*a*c*d^2*e^2*f^2 + b*c^3*e*f^3 + 3*a*c^2*d*e*f^3 + 3*a*c^3*f^4)*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(8*e^(5/2)*f^(9/2))","A",1
22,1,480,348,0.3664571,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^3}{\left(e+f x^2\right)^4} \, dx","IntegrateAlgebraic[((a + b*x^2)*(c + d*x^2)^3)/(e + f*x^2)^4,x]","\frac{\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right) \left(5 a c^3 f^4+3 a c^2 d e f^3+3 a c d^2 e^2 f^2+5 a d^3 e^3 f+b c^3 e f^3+3 b c^2 d e^2 f^2+15 b c d^2 e^3 f-35 b d^3 e^4\right)}{16 e^{7/2} f^{9/2}}+\frac{x \left(33 a c^3 e^2 f^4+40 a c^3 e f^5 x^2+15 a c^3 f^6 x^4-9 a c^2 d e^3 f^3+24 a c^2 d e^2 f^4 x^2+9 a c^2 d e f^5 x^4-9 a c d^2 e^4 f^2-24 a c d^2 e^3 f^3 x^2+9 a c d^2 e^2 f^4 x^4-15 a d^3 e^5 f-40 a d^3 e^4 f^2 x^2-33 a d^3 e^3 f^3 x^4-3 b c^3 e^3 f^3+8 b c^3 e^2 f^4 x^2+3 b c^3 e f^5 x^4-9 b c^2 d e^4 f^2-24 b c^2 d e^3 f^3 x^2+9 b c^2 d e^2 f^4 x^4-45 b c d^2 e^5 f-120 b c d^2 e^4 f^2 x^2-99 b c d^2 e^3 f^3 x^4+105 b d^3 e^6+280 b d^3 e^5 f x^2+231 b d^3 e^4 f^2 x^4+48 b d^3 e^3 f^3 x^6\right)}{48 e^3 f^4 \left(e+f x^2\right)^3}","\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{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{\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right) \left(b e \left(-c^3 f^3-3 c^2 d e f^2-15 c d^2 e^2 f+35 d^3 e^3\right)-a f \left(5 c^3 f^3+3 c^2 d e f^2+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,"(x*(105*b*d^3*e^6 - 45*b*c*d^2*e^5*f - 15*a*d^3*e^5*f - 9*b*c^2*d*e^4*f^2 - 9*a*c*d^2*e^4*f^2 - 3*b*c^3*e^3*f^3 - 9*a*c^2*d*e^3*f^3 + 33*a*c^3*e^2*f^4 + 280*b*d^3*e^5*f*x^2 - 120*b*c*d^2*e^4*f^2*x^2 - 40*a*d^3*e^4*f^2*x^2 - 24*b*c^2*d*e^3*f^3*x^2 - 24*a*c*d^2*e^3*f^3*x^2 + 8*b*c^3*e^2*f^4*x^2 + 24*a*c^2*d*e^2*f^4*x^2 + 40*a*c^3*e*f^5*x^2 + 231*b*d^3*e^4*f^2*x^4 - 99*b*c*d^2*e^3*f^3*x^4 - 33*a*d^3*e^3*f^3*x^4 + 9*b*c^2*d*e^2*f^4*x^4 + 9*a*c*d^2*e^2*f^4*x^4 + 3*b*c^3*e*f^5*x^4 + 9*a*c^2*d*e*f^5*x^4 + 15*a*c^3*f^6*x^4 + 48*b*d^3*e^3*f^3*x^6))/(48*e^3*f^4*(e + f*x^2)^3) + ((-35*b*d^3*e^4 + 15*b*c*d^2*e^3*f + 5*a*d^3*e^3*f + 3*b*c^2*d*e^2*f^2 + 3*a*c*d^2*e^2*f^2 + b*c^3*e*f^3 + 3*a*c^2*d*e*f^3 + 5*a*c^3*f^4)*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(16*e^(7/2)*f^(9/2))","A",1
23,0,0,544,1.5195501,"\int \left(a+b x^2\right) \left(c+d x^2\right)^{3/2} \sqrt{e+f x^2} \, dx","IntegrateAlgebraic[(a + b*x^2)*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2],x]","\int \left(a+b x^2\right) \left(c+d x^2\right)^{3/2} \sqrt{e+f x^2} \, dx","\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{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{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(-6 c^3 f^3+9 c^2 d e f^2-19 c d^2 e^2 f+8 d^3 e^3\right)\right)}{105 d^2 f^2 \sqrt{e+f x^2}}+\frac{\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(-6 c^3 f^3+9 c^2 d e f^2-19 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,"Could not integrate","F",-1
24,0,0,381,0.9493556,"\int \left(a+b x^2\right) \sqrt{c+d x^2} \sqrt{e+f x^2} \, dx","IntegrateAlgebraic[(a + b*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2],x]","\int \left(a+b x^2\right) \sqrt{c+d x^2} \sqrt{e+f x^2} \, dx","\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,"Could not integrate","F",-1
25,0,0,283,1.0209113,"\int \frac{\left(a+b x^2\right) \sqrt{e+f x^2}}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[((a + b*x^2)*Sqrt[e + f*x^2])/Sqrt[c + d*x^2],x]","\int \frac{\left(a+b x^2\right) \sqrt{e+f x^2}}{\sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
26,0,0,271,1.5425595,"\int \frac{\left(a+b x^2\right) \sqrt{e+f x^2}}{\left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)*Sqrt[e + f*x^2])/(c + d*x^2)^(3/2),x]","\int \frac{\left(a+b x^2\right) \sqrt{e+f x^2}}{\left(c+d x^2\right)^{3/2}} \, dx","\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,"Could not integrate","F",-1
27,0,0,274,1.8160374,"\int \frac{\left(a+b x^2\right) \sqrt{e+f x^2}}{\left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)*Sqrt[e + f*x^2])/(c + d*x^2)^(5/2),x]","\int \frac{\left(a+b x^2\right) \sqrt{e+f x^2}}{\left(c+d x^2\right)^{5/2}} \, dx","\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,"Could not integrate","F",-1
28,0,0,385,1.812518,"\int \frac{\left(a+b x^2\right) \sqrt{e+f x^2}}{\left(c+d x^2\right)^{7/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)*Sqrt[e + f*x^2])/(c + d*x^2)^(7/2),x]","\int \frac{\left(a+b x^2\right) \sqrt{e+f x^2}}{\left(c+d x^2\right)^{7/2}} \, dx","-\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{\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{x \sqrt{e+f x^2} (b c-a d)}{5 c d \left(c+d x^2\right)^{5/2}}",1,"Could not integrate","F",-1
29,0,0,543,1.5189063,"\int \left(a+b x^2\right) \sqrt{c+d x^2} \left(e+f x^2\right)^{3/2} \, dx","IntegrateAlgebraic[(a + b*x^2)*Sqrt[c + d*x^2]*(e + f*x^2)^(3/2),x]","\int \left(a+b x^2\right) \sqrt{c+d x^2} \left(e+f x^2\right)^{3/2} \, dx","\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{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{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(-8 c^3 f^3+19 c^2 d e f^2-9 c d^2 e^2 f+6 d^3 e^3\right)\right)}{105 d^3 f \sqrt{e+f x^2}}-\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(-8 c^3 f^3+19 c^2 d e f^2-9 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,"Could not integrate","F",-1
30,0,0,400,1.5431827,"\int \frac{\left(a+b x^2\right) \left(e+f x^2\right)^{3/2}}{\sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[((a + b*x^2)*(e + f*x^2)^(3/2))/Sqrt[c + d*x^2],x]","\int \frac{\left(a+b x^2\right) \left(e+f x^2\right)^{3/2}}{\sqrt{c+d x^2}} \, dx","\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} \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{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,"Could not integrate","F",-1
31,0,0,369,2.1674491,"\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","IntegrateAlgebraic[((a + b*x^2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(3/2),x]","\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","\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{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{x \left(e+f x^2\right)^{3/2} (b c-a d)}{c d \sqrt{c+d x^2}}",1,"Could not integrate","F",-1
32,0,0,373,2.3674685,"\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","IntegrateAlgebraic[((a + b*x^2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(5/2),x]","\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","-\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{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{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,"Could not integrate","F",-1
33,0,0,376,2.4545522,"\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","IntegrateAlgebraic[((a + b*x^2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(7/2),x]","\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","-\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{\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{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,"Could not integrate","F",-1
34,0,0,531,2.4922826,"\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","IntegrateAlgebraic[((a + b*x^2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(9/2),x]","\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","\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{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{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{\sqrt{e+f x^2} \left(6 a d \left(c^3 f^3+2 c^2 d e f^2-12 c d^2 e^2 f+8 d^3 e^3\right)+b c \left(8 c^3 f^3-5 c^2 d e f^2-5 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 \left(e+f x^2\right)^{3/2} (b c-a d)}{7 c d \left(c+d x^2\right)^{7/2}}",1,"Could not integrate","F",-1
35,0,0,551,2.0982939,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^{5/2}}{\sqrt{e+f x^2}} \, dx","IntegrateAlgebraic[((a + b*x^2)*(c + d*x^2)^(5/2))/Sqrt[e + f*x^2],x]","\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^{5/2}}{\sqrt{e+f x^2}} \, dx","\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{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(-15 c^3 f^3+103 c^2 d e f^2-128 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 d f \left(23 c^2 f^2-23 c d e f+8 d^2 e^2\right)-b \left(-15 c^3 f^3+103 c^2 d e f^2-128 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,"Could not integrate","F",-1
36,0,0,396,1.535387,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^{3/2}}{\sqrt{e+f x^2}} \, dx","IntegrateAlgebraic[((a + b*x^2)*(c + d*x^2)^(3/2))/Sqrt[e + f*x^2],x]","\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^{3/2}}{\sqrt{e+f x^2}} \, dx","-\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,"Could not integrate","F",-1
37,0,0,282,0.9676891,"\int \frac{\left(a+b x^2\right) \sqrt{c+d x^2}}{\sqrt{e+f x^2}} \, dx","IntegrateAlgebraic[((a + b*x^2)*Sqrt[c + d*x^2])/Sqrt[e + f*x^2],x]","\int \frac{\left(a+b x^2\right) \sqrt{c+d x^2}}{\sqrt{e+f x^2}} \, dx","-\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,"Could not integrate","F",-1
38,0,0,206,1.1499342,"\int \frac{a+b x^2}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)/(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]","\int \frac{a+b x^2}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx","\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,"Could not integrate","F",-1
39,0,0,209,1.8666113,"\int \frac{a+b x^2}{\left(c+d x^2\right)^{3/2} \sqrt{e+f x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)/((c + d*x^2)^(3/2)*Sqrt[e + f*x^2]),x]","\int \frac{a+b x^2}{\left(c+d x^2\right)^{3/2} \sqrt{e+f x^2}} \, dx","\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,"Could not integrate","F",-1
40,0,0,284,2.1890335,"\int \frac{a+b x^2}{\left(c+d x^2\right)^{5/2} \sqrt{e+f x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)/((c + d*x^2)^(5/2)*Sqrt[e + f*x^2]),x]","\int \frac{a+b x^2}{\left(c+d x^2\right)^{5/2} \sqrt{e+f x^2}} \, dx","\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{\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{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,"Could not integrate","F",-1
41,0,0,401,2.3335049,"\int \frac{a+b x^2}{\left(c+d x^2\right)^{7/2} \sqrt{e+f x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)/((c + d*x^2)^(7/2)*Sqrt[e + f*x^2]),x]","\int \frac{a+b x^2}{\left(c+d x^2\right)^{7/2} \sqrt{e+f x^2}} \, dx","\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{\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{\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{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,"Could not integrate","F",-1
42,0,0,501,2.4614034,"\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","IntegrateAlgebraic[((a + b*x^2)*(c + d*x^2)^(5/2))/(e + f*x^2)^(3/2),x]","\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","-\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{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{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{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{x \left(c+d x^2\right)^{5/2} (b e-a f)}{e f \sqrt{e+f x^2}}",1,"Could not integrate","F",-1
43,0,0,358,2.1704501,"\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","IntegrateAlgebraic[((a + b*x^2)*(c + d*x^2)^(3/2))/(e + f*x^2)^(3/2),x]","\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","-\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{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{x \left(c+d x^2\right)^{3/2} (b e-a f)}{e f \sqrt{e+f x^2}}",1,"Could not integrate","F",-1
44,0,0,258,1.5568134,"\int \frac{\left(a+b x^2\right) \sqrt{c+d x^2}}{\left(e+f x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[((a + b*x^2)*Sqrt[c + d*x^2])/(e + f*x^2)^(3/2),x]","\int \frac{\left(a+b x^2\right) \sqrt{c+d x^2}}{\left(e+f x^2\right)^{3/2}} \, dx","-\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,"Could not integrate","F",-1
45,0,0,209,1.8427412,"\int \frac{a+b x^2}{\sqrt{c+d x^2} \left(e+f x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)/(Sqrt[c + d*x^2]*(e + f*x^2)^(3/2)),x]","\int \frac{a+b x^2}{\sqrt{c+d x^2} \left(e+f x^2\right)^{3/2}} \, dx","\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,"Could not integrate","F",-1
46,0,0,272,2.4847144,"\int \frac{a+b x^2}{\left(c+d x^2\right)^{3/2} \left(e+f x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)/((c + d*x^2)^(3/2)*(e + f*x^2)^(3/2)),x]","\int \frac{a+b x^2}{\left(c+d x^2\right)^{3/2} \left(e+f x^2\right)^{3/2}} \, dx","-\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,"Could not integrate","F",-1
47,0,0,375,2.5791568,"\int \frac{a+b x^2}{\left(c+d x^2\right)^{5/2} \left(e+f x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(a + b*x^2)/((c + d*x^2)^(5/2)*(e + f*x^2)^(3/2)),x]","\int \frac{a+b x^2}{\left(c+d x^2\right)^{5/2} \left(e+f x^2\right)^{3/2}} \, dx","\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,"Could not integrate","F",-1
48,0,0,209,2.0273714,"\int \frac{e+f x^2}{\sqrt{a+b x^2} \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(e + f*x^2)/(Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)),x]","\int \frac{e+f x^2}{\sqrt{a+b x^2} \left(c+d x^2\right)^{3/2}} \, dx","\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,"Could not integrate","F",-1
49,0,0,247,2.010958,"\int \frac{e+f x^2}{\sqrt{a-b x^2} \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(e + f*x^2)/(Sqrt[a - b*x^2]*(c + d*x^2)^(3/2)),x]","\int \frac{e+f x^2}{\sqrt{a-b x^2} \left(c+d x^2\right)^{3/2}} \, dx","\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,"Could not integrate","F",-1
50,0,0,237,1.9188379,"\int \frac{e+f x^2}{\sqrt{a+b x^2} \left(c-d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(e + f*x^2)/(Sqrt[a + b*x^2]*(c - d*x^2)^(3/2)),x]","\int \frac{e+f x^2}{\sqrt{a+b x^2} \left(c-d x^2\right)^{3/2}} \, dx","\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,"Could not integrate","F",-1
51,0,0,242,2.033498,"\int \frac{e+f x^2}{\sqrt{a-b x^2} \left(c-d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(e + f*x^2)/(Sqrt[a - b*x^2]*(c - d*x^2)^(3/2)),x]","\int \frac{e+f x^2}{\sqrt{a-b x^2} \left(c-d x^2\right)^{3/2}} \, dx","-\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,"Could not integrate","F",-1
52,0,0,191,1.1499791,"\int \frac{a+b x^2}{\sqrt{2+d x^2} \sqrt{3+f x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)/(Sqrt[2 + d*x^2]*Sqrt[3 + f*x^2]),x]","\int \frac{a+b x^2}{\sqrt{2+d x^2} \sqrt{3+f x^2}} \, dx","\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,"Could not integrate","F",-1
53,0,0,262,0.8244286,"\int \frac{\left(a+b x^2\right) \sqrt{2+d x^2}}{\sqrt{3+f x^2}} \, dx","IntegrateAlgebraic[((a + b*x^2)*Sqrt[2 + d*x^2])/Sqrt[3 + f*x^2],x]","\int \frac{\left(a+b x^2\right) \sqrt{2+d x^2}}{\sqrt{3+f x^2}} \, dx","-\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,"Could not integrate","F",-1
54,0,0,356,0.8052026,"\int \left(a+b x^2\right) \sqrt{2+d x^2} \sqrt{3+f x^2} \, dx","IntegrateAlgebraic[(a + b*x^2)*Sqrt[2 + d*x^2]*Sqrt[3 + f*x^2],x]","\int \left(a+b x^2\right) \sqrt{2+d x^2} \sqrt{3+f x^2} \, dx","\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,"Could not integrate","F",-1
55,0,0,113,1.7820641,"\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","IntegrateAlgebraic[(-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]","\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","-\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,"Could not integrate","F",-1
56,0,0,526,1.6413989,"\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","IntegrateAlgebraic[(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]","\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","\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,"Could not integrate","F",-1
57,1,128,128,0.4733831,"\int \frac{\left(a+b x^2\right) \sqrt{c+d x^2}}{e+f x^2} \, dx","IntegrateAlgebraic[((a + b*x^2)*Sqrt[c + d*x^2])/(e + f*x^2),x]","-\frac{(b e-a f) \sqrt{c f-d e} \tan ^{-1}\left(\frac{x \sqrt{c f-d e}}{\sqrt{e} \sqrt{c+d x^2}}\right)}{\sqrt{e} f^2}+\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 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) - ((b*e - a*f)*Sqrt[-(d*e) + c*f]*ArcTan[(Sqrt[-(d*e) + c*f]*x)/(Sqrt[e]*Sqrt[c + d*x^2])])/(Sqrt[e]*f^2) + ((-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)","A",1
58,1,243,304,1.1936042,"\int \frac{\left(a+b x^2\right)^3}{\left(c+d x^2\right) \sqrt{e+f x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^3/((c + d*x^2)*Sqrt[e + f*x^2]),x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e+f x^2}}\right) \left(24 a^2 b d^2 f^2-24 a b^2 c d f^2-12 a b^2 d^2 e f+8 b^3 c^2 f^2+4 b^3 c d e f+3 b^3 d^2 e^2\right)}{8 d^3 f^{5/2}}+\frac{\left(a^3 d^3-3 a^2 b c d^2+3 a b^2 c^2 d-b^3 c^3\right) \tan ^{-1}\left(\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right)}{\sqrt{c} d^3 \sqrt{d e-c f}}+\frac{\sqrt{e+f x^2} \left(12 a b^2 d f x-4 b^3 c f x-3 b^3 d e x+2 b^3 d f x^3\right)}{8 d^2 f^2}","\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 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 (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}}",1,"(Sqrt[e + f*x^2]*(-3*b^3*d*e*x - 4*b^3*c*f*x + 12*a*b^2*d*f*x + 2*b^3*d*f*x^3))/(8*d^2*f^2) + ((-(b^3*c^3) + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*ArcTan[(Sqrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])])/(Sqrt[c]*d^3*Sqrt[d*e - c*f]) + ((3*b^3*d^2*e^2 + 4*b^3*c*d*e*f - 12*a*b^2*d^2*e*f + 8*b^3*c^2*f^2 - 24*a*b^2*c*d*f^2 + 24*a^2*b*d^2*f^2)*ArcTanh[(Sqrt[f]*x)/Sqrt[e + f*x^2]])/(8*d^3*f^(5/2))","A",1
59,1,151,166,0.5602036,"\int \frac{\left(a+b x^2\right)^2}{\left(c+d x^2\right) \sqrt{e+f x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)^2/((c + d*x^2)*Sqrt[e + f*x^2]),x]","\frac{\left(a^2 d^2-2 a b c d+b^2 c^2\right) \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{\tanh ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e+f x^2}}\right) \left(4 a b d f-2 b^2 c f+b^2 (-d) e\right)}{2 d^2 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^2*c^2 - 2*a*b*c*d + a^2*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^2*d*e) - 2*b^2*c*f + 4*a*b*d*f)*ArcTanh[(Sqrt[f]*x)/Sqrt[e + f*x^2]])/(2*d^2*f^(3/2))","A",1
60,1,90,91,0.3454043,"\int \frac{a+b x^2}{\left(c+d x^2\right) \sqrt{e+f x^2}} \, dx","IntegrateAlgebraic[(a + b*x^2)/((c + d*x^2)*Sqrt[e + f*x^2]),x]","\frac{(a d-b c) \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 \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",1
61,1,49,49,0.1868896,"\int \frac{1}{\left(c+d x^2\right) \sqrt{e+f x^2}} \, dx","IntegrateAlgebraic[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",1
62,1,121,122,0.5589347,"\int \frac{1}{\left(a+b x^2\right) \left(c+d x^2\right) \sqrt{e+f x^2}} \, dx","IntegrateAlgebraic[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} (a d-b c) \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",1
63,1,207,203,1.2917562,"\int \frac{1}{\left(a+b x^2\right)^2 \left(c+d x^2\right) \sqrt{e+f x^2}} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^2*(c + d*x^2)*Sqrt[e + f*x^2]),x]","\frac{\left(4 a^2 b d f-2 a b^2 c f-3 a b^2 d e+b^3 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) (a d-b c) (a f-b e)}+\frac{d^2 \tan ^{-1}\left(\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right)}{\sqrt{c} (a d-b c)^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^3*c*e - 3*a*b^2*d*e - 2*a*b^2*c*f + 4*a^2*b*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",1
64,0,0,608,2.1600293,"\int \frac{\left(c+d x^2\right)^{5/2} \sqrt{e+f x^2}}{a+b x^2} \, dx","IntegrateAlgebraic[((c + d*x^2)^(5/2)*Sqrt[e + f*x^2])/(a + b*x^2),x]","\int \frac{\left(c+d x^2\right)^{5/2} \sqrt{e+f x^2}}{a+b x^2} \, dx","-\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{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} \sqrt{e+f x^2} (b c-a d)}{3 b^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,"Could not integrate","F",-1
65,0,0,400,1.8966789,"\int \frac{\left(c+d x^2\right)^{3/2} \sqrt{e+f x^2}}{a+b x^2} \, dx","IntegrateAlgebraic[((c + d*x^2)^(3/2)*Sqrt[e + f*x^2])/(a + b*x^2),x]","\int \frac{\left(c+d x^2\right)^{3/2} \sqrt{e+f x^2}}{a+b x^2} \, dx","\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,"Could not integrate","F",-1
66,0,0,321,1.2786634,"\int \frac{\sqrt{c+d x^2} \sqrt{e+f x^2}}{a+b x^2} \, dx","IntegrateAlgebraic[(Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(a + b*x^2),x]","\int \frac{\sqrt{c+d x^2} \sqrt{e+f x^2}}{a+b x^2} \, dx","\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,"Could not integrate","F",-1
67,0,0,102,1.4578577,"\int \frac{\sqrt{e+f x^2}}{\left(a+b x^2\right) \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[Sqrt[e + f*x^2]/((a + b*x^2)*Sqrt[c + d*x^2]),x]","\int \frac{\sqrt{e+f x^2}}{\left(a+b x^2\right) \sqrt{c+d x^2}} \, dx","\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,"Could not integrate","F",-1
68,0,0,209,2.308522,"\int \frac{\sqrt{e+f x^2}}{\left(a+b x^2\right) \left(c+d x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[Sqrt[e + f*x^2]/((a + b*x^2)*(c + d*x^2)^(3/2)),x]","\int \frac{\sqrt{e+f x^2}}{\left(a+b x^2\right) \left(c+d x^2\right)^{3/2}} \, dx","\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,"Could not integrate","F",-1
69,0,0,401,2.5609999,"\int \frac{\sqrt{e+f x^2}}{\left(a+b x^2\right) \left(c+d x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[Sqrt[e + f*x^2]/((a + b*x^2)*(c + d*x^2)^(5/2)),x]","\int \frac{\sqrt{e+f x^2}}{\left(a+b x^2\right) \left(c+d x^2\right)^{5/2}} \, dx","\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{\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 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{d x \sqrt{e+f x^2}}{3 c \left(c+d x^2\right)^{3/2} (b c-a d)}",1,"Could not integrate","F",-1
70,0,0,630,2.6549701,"\int \frac{\sqrt{e+f x^2}}{\left(a+b x^2\right) \left(c+d x^2\right)^{7/2}} \, dx","IntegrateAlgebraic[Sqrt[e + f*x^2]/((a + b*x^2)*(c + d*x^2)^(7/2)),x]","\int \frac{\sqrt{e+f x^2}}{\left(a+b x^2\right) \left(c+d x^2\right)^{7/2}} \, dx","\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{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{\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 x \sqrt{e+f x^2}}{5 c \left(c+d x^2\right)^{5/2} (b c-a d)}",1,"Could not integrate","F",-1
71,0,0,659,2.5165238,"\int \frac{\left(c+d x^2\right)^{3/2} \left(e+f x^2\right)^{3/2}}{a+b x^2} \, dx","IntegrateAlgebraic[((c + d*x^2)^(3/2)*(e + f*x^2)^(3/2))/(a + b*x^2),x]","\int \frac{\left(c+d x^2\right)^{3/2} \left(e+f x^2\right)^{3/2}}{a+b x^2} \, dx","-\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,"Could not integrate","F",-1
72,0,0,403,1.8077151,"\int \frac{\sqrt{c+d x^2} \left(e+f x^2\right)^{3/2}}{a+b x^2} \, dx","IntegrateAlgebraic[(Sqrt[c + d*x^2]*(e + f*x^2)^(3/2))/(a + b*x^2),x]","\int \frac{\sqrt{c+d x^2} \left(e+f x^2\right)^{3/2}}{a+b x^2} \, dx","\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,"Could not integrate","F",-1
73,0,0,328,2.1675351,"\int \frac{\left(e+f x^2\right)^{3/2}}{\left(a+b x^2\right) \sqrt{c+d x^2}} \, dx","IntegrateAlgebraic[(e + f*x^2)^(3/2)/((a + b*x^2)*Sqrt[c + d*x^2]),x]","\int \frac{\left(e+f x^2\right)^{3/2}}{\left(a+b x^2\right) \sqrt{c+d x^2}} \, dx","\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{\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{f^2 x \sqrt{c+d x^2}}{b d \sqrt{e+f x^2}}",1,"Could not integrate","F",-1
74,0,0,224,2.8568617,"\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","IntegrateAlgebraic[(e + f*x^2)^(3/2)/((a + b*x^2)*(c + d*x^2)^(3/2)),x]","\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","\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,"Could not integrate","F",-1
75,0,0,391,2.878076,"\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","IntegrateAlgebraic[(e + f*x^2)^(3/2)/((a + b*x^2)*(c + d*x^2)^(5/2)),x]","\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","-\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{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{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,"Could not integrate","F",-1
76,0,0,639,2.9820387,"\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","IntegrateAlgebraic[(e + f*x^2)^(3/2)/((a + b*x^2)*(c + d*x^2)^(7/2)),x]","\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","\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{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{\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{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,"Could not integrate","F",-1
77,0,0,621,2.6017309,"\int \frac{\left(c+d x^2\right)^{5/2}}{\left(a+b x^2\right) \sqrt{e+f x^2}} \, dx","IntegrateAlgebraic[(c + d*x^2)^(5/2)/((a + b*x^2)*Sqrt[e + f*x^2]),x]","\int \frac{\left(c+d x^2\right)^{5/2}}{\left(a+b x^2\right) \sqrt{e+f x^2}} \, dx","\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,"Could not integrate","F",-1
78,0,0,319,2.3923426,"\int \frac{\left(c+d x^2\right)^{3/2}}{\left(a+b x^2\right) \sqrt{e+f x^2}} \, dx","IntegrateAlgebraic[(c + d*x^2)^(3/2)/((a + b*x^2)*Sqrt[e + f*x^2]),x]","\int \frac{\left(c+d x^2\right)^{3/2}}{\left(a+b x^2\right) \sqrt{e+f x^2}} \, dx","\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,"Could not integrate","F",-1
79,0,0,102,1.4585687,"\int \frac{\sqrt{c+d x^2}}{\left(a+b x^2\right) \sqrt{e+f x^2}} \, dx","IntegrateAlgebraic[Sqrt[c + d*x^2]/((a + b*x^2)*Sqrt[e + f*x^2]),x]","\int \frac{\sqrt{c+d x^2}}{\left(a+b x^2\right) \sqrt{e+f x^2}} \, dx","\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,"Could not integrate","F",-1
80,0,0,100,2.0049216,"\int \frac{1}{\left(a+b x^2\right) \sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx","IntegrateAlgebraic[1/((a + b*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]","\int \frac{1}{\left(a+b x^2\right) \sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx","\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,"Could not integrate","F",-1
81,0,0,344,2.5913708,"\int \frac{1}{\left(a+b x^2\right) \left(c+d x^2\right)^{3/2} \sqrt{e+f x^2}} \, dx","IntegrateAlgebraic[1/((a + b*x^2)*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2]),x]","\int \frac{1}{\left(a+b x^2\right) \left(c+d x^2\right)^{3/2} \sqrt{e+f x^2}} \, dx","\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,"Could not integrate","F",-1
82,0,0,435,2.6878842,"\int \frac{1}{\left(a+b x^2\right) \left(c+d x^2\right)^{5/2} \sqrt{e+f x^2}} \, dx","IntegrateAlgebraic[1/((a + b*x^2)*(c + d*x^2)^(5/2)*Sqrt[e + f*x^2]),x]","\int \frac{1}{\left(a+b x^2\right) \left(c+d x^2\right)^{5/2} \sqrt{e+f x^2}} \, dx","\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,"Could not integrate","F",-1
83,0,0,980,2.9601596,"\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","IntegrateAlgebraic[(c + d*x^2)^(5/2)/((a + b*x^2)*(e + f*x^2)^(3/2)),x]","\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","\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,"Could not integrate","F",-1
84,0,0,223,2.9350014,"\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","IntegrateAlgebraic[(c + d*x^2)^(3/2)/((a + b*x^2)*(e + f*x^2)^(3/2)),x]","\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","\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,"Could not integrate","F",-1
85,0,0,209,2.275923,"\int \frac{\sqrt{c+d x^2}}{\left(a+b x^2\right) \left(e+f x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[Sqrt[c + d*x^2]/((a + b*x^2)*(e + f*x^2)^(3/2)),x]","\int \frac{\sqrt{c+d x^2}}{\left(a+b x^2\right) \left(e+f x^2\right)^{3/2}} \, dx","\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,"Could not integrate","F",-1
86,0,0,344,2.5960757,"\int \frac{1}{\left(a+b x^2\right) \sqrt{c+d x^2} \left(e+f x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[1/((a + b*x^2)*Sqrt[c + d*x^2]*(e + f*x^2)^(3/2)),x]","\int \frac{1}{\left(a+b x^2\right) \sqrt{c+d x^2} \left(e+f x^2\right)^{3/2}} \, dx","\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,"Could not integrate","F",-1
87,0,0,539,2.9619009,"\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","IntegrateAlgebraic[1/((a + b*x^2)*(c + d*x^2)^(3/2)*(e + f*x^2)^(3/2)),x]","\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","\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,"Could not integrate","F",-1
88,0,0,814,2.9966252,"\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","IntegrateAlgebraic[1/((a + b*x^2)*(c + d*x^2)^(5/2)*(e + f*x^2)^(3/2)),x]","\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","\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,"Could not integrate","F",-1
89,0,0,242,1.2360608,"\int \frac{\left(1+x^2\right)^{3/2} \sqrt{2+x^2}}{a+b x^2} \, dx","IntegrateAlgebraic[((1 + x^2)^(3/2)*Sqrt[2 + x^2])/(a + b*x^2),x]","\int \frac{\left(1+x^2\right)^{3/2} \sqrt{2+x^2}}{a+b x^2} \, dx","-\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,"Could not integrate","F",-1
90,0,0,192,0.7273384,"\int \frac{\sqrt{1+x^2} \sqrt{2+x^2}}{a+b x^2} \, dx","IntegrateAlgebraic[(Sqrt[1 + x^2]*Sqrt[2 + x^2])/(a + b*x^2),x]","\int \frac{\sqrt{1+x^2} \sqrt{2+x^2}}{a+b x^2} \, dx","-\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,"Could not integrate","F",-1
91,0,0,58,0.8685964,"\int \frac{\sqrt{2+x^2}}{\sqrt{1+x^2} \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[Sqrt[2 + x^2]/(Sqrt[1 + x^2]*(a + b*x^2)),x]","\int \frac{\sqrt{2+x^2}}{\sqrt{1+x^2} \left(a+b x^2\right)} \, dx","\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,"Could not integrate","F",-1
92,0,0,121,1.446137,"\int \frac{\sqrt{2+x^2}}{\left(1+x^2\right)^{3/2} \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[Sqrt[2 + x^2]/((1 + x^2)^(3/2)*(a + b*x^2)),x]","\int \frac{\sqrt{2+x^2}}{\left(1+x^2\right)^{3/2} \left(a+b x^2\right)} \, dx","\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,"Could not integrate","F",-1
93,0,0,215,1.5836035,"\int \frac{\sqrt{2+x^2}}{\left(1+x^2\right)^{5/2} \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[Sqrt[2 + x^2]/((1 + x^2)^(5/2)*(a + b*x^2)),x]","\int \frac{\sqrt{2+x^2}}{\left(1+x^2\right)^{5/2} \left(a+b x^2\right)} \, dx","\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,"Could not integrate","F",-1
94,0,0,298,1.1923095,"\int \frac{\sqrt{2+d x^2} \sqrt{3+f x^2}}{a+b x^2} \, dx","IntegrateAlgebraic[(Sqrt[2 + d*x^2]*Sqrt[3 + f*x^2])/(a + b*x^2),x]","\int \frac{\sqrt{2+d x^2} \sqrt{3+f x^2}}{a+b x^2} \, dx","\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,"Could not integrate","F",-1
95,0,0,93,1.332722,"\int \frac{\sqrt{2+d x^2}}{\left(a+b x^2\right) \sqrt{3+f x^2}} \, dx","IntegrateAlgebraic[Sqrt[2 + d*x^2]/((a + b*x^2)*Sqrt[3 + f*x^2]),x]","\int \frac{\sqrt{2+d x^2}}{\left(a+b x^2\right) \sqrt{3+f x^2}} \, dx","\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,"Could not integrate","F",-1
96,0,0,49,1.8848717,"\int \frac{1}{\left(a+b x^2\right) \sqrt{2+d x^2} \sqrt{3+f x^2}} \, dx","IntegrateAlgebraic[1/((a + b*x^2)*Sqrt[2 + d*x^2]*Sqrt[3 + f*x^2]),x]","\int \frac{1}{\left(a+b x^2\right) \sqrt{2+d x^2} \sqrt{3+f x^2}} \, dx","\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,"Could not integrate","F",-1
97,0,0,36,0.6435886,"\int \frac{\sqrt{1-x^2}}{\left(-1+x^2\right) \sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[Sqrt[1 - x^2]/((-1 + x^2)*Sqrt[a + b*x^2]),x]","\int \frac{\sqrt{1-x^2}}{\left(-1+x^2\right) \sqrt{a+b x^2}} \, dx","-\frac{\sqrt{\frac{b x^2}{a}+1} F\left(\sin ^{-1}(x)|-\frac{b}{a}\right)}{\sqrt{a+b x^2}}",1,"Could not integrate","F",-1
98,1,113,113,0.6443742,"\int \frac{a+b x^2}{\sqrt{c+d x^2} \left(e+f x^2\right)^2} \, dx","IntegrateAlgebraic[(a + b*x^2)/(Sqrt[c + d*x^2]*(e + f*x^2)^2),x]","\frac{(a c f-2 a d e+b c e) \tan ^{-1}\left(\frac{x \sqrt{c f-d e}}{\sqrt{e} \sqrt{c+d x^2}}\right)}{2 e^{3/2} (c f-d e)^{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{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)*ArcTan[(Sqrt[-(d*e) + c*f]*x)/(Sqrt[e]*Sqrt[c + d*x^2])])/(2*e^(3/2)*(-(d*e) + c*f)^(3/2))","A",1
99,0,0,359,1.6146386,"\int \frac{\sqrt{c-d x^2} \sqrt{e+f x^2}}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(Sqrt[c - d*x^2]*Sqrt[e + f*x^2])/(a + b*x^2)^2,x]","\int \frac{\sqrt{c-d x^2} \sqrt{e+f x^2}}{\left(a+b x^2\right)^2} \, dx","\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,"Could not integrate","F",-1
100,0,0,381,1.5135929,"\int \frac{\sqrt{c+d x^2} \sqrt{e+f x^2}}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(a + b*x^2)^2,x]","\int \frac{\sqrt{c+d x^2} \sqrt{e+f x^2}}{\left(a+b x^2\right)^2} \, dx","\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,"Could not integrate","F",-1
101,0,0,426,2.3239907,"\int \frac{1}{\left(a+b x^2\right)^2 \sqrt{c-d x^2} \sqrt{e+f x^2}} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^2*Sqrt[c - d*x^2]*Sqrt[e + f*x^2]),x]","\int \frac{1}{\left(a+b x^2\right)^2 \sqrt{c-d x^2} \sqrt{e+f x^2}} \, dx","\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,"Could not integrate","F",-1
102,0,0,485,2.3721781,"\int \frac{1}{\left(a+b x^2\right)^2 \sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx","IntegrateAlgebraic[1/((a + b*x^2)^2*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]","\int \frac{1}{\left(a+b x^2\right)^2 \sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx","\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,"Could not integrate","F",-1
103,0,0,37,2.3114216,"\int \frac{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}}{\sqrt{e+f x^2}} \, dx","IntegrateAlgebraic[((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,"Could not integrate","A",-1
104,0,0,545,1.3899549,"\int \frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{\sqrt{e+f x^2}} \, dx","IntegrateAlgebraic[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/Sqrt[e + f*x^2],x]","\int \frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{\sqrt{e+f x^2}} \, dx","\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,"Could not integrate","F",-1
105,0,0,163,1.5213349,"\int \frac{\sqrt{c+d x^2}}{\sqrt{a+b x^2} \sqrt{e+f x^2}} \, dx","IntegrateAlgebraic[Sqrt[c + d*x^2]/(Sqrt[a + b*x^2]*Sqrt[e + f*x^2]),x]","\int \frac{\sqrt{c+d x^2}}{\sqrt{a+b x^2} \sqrt{e+f x^2}} \, dx","\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,"Could not integrate","F",-1
106,0,0,148,2.387503,"\int \frac{\sqrt{c+d x^2}}{\left(a+b x^2\right)^{3/2} \sqrt{e+f x^2}} \, dx","IntegrateAlgebraic[Sqrt[c + d*x^2]/((a + b*x^2)^(3/2)*Sqrt[e + f*x^2]),x]","\int \frac{\sqrt{c+d x^2}}{\left(a+b x^2\right)^{3/2} \sqrt{e+f x^2}} \, dx","\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,"Could not integrate","F",-1
107,0,0,37,2.8186523,"\int \frac{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}}{\left(e+f x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[((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,"Could not integrate","A",-1
108,0,0,484,2.0163676,"\int \frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{\left(e+f x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(e + f*x^2)^(3/2),x]","\int \frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{\left(e+f x^2\right)^{3/2}} \, dx","-\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,"Could not integrate","F",-1
109,0,0,319,2.5497064,"\int \frac{\sqrt{c+d x^2}}{\sqrt{a+b x^2} \left(e+f x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[Sqrt[c + d*x^2]/(Sqrt[a + b*x^2]*(e + f*x^2)^(3/2)),x]","\int \frac{\sqrt{c+d x^2}}{\sqrt{a+b x^2} \left(e+f x^2\right)^{3/2}} \, dx","\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,"Could not integrate","F",-1
110,0,0,37,3.0937621,"\int \frac{\sqrt{c+d x^2}}{\left(a+b x^2\right)^{3/2} \left(e+f x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[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,"Could not integrate","A",-1
111,0,0,541,1.3385827,"\int \frac{\sqrt{c+d x^2} \sqrt{e+f x^2}}{\sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/Sqrt[a + b*x^2],x]","\int \frac{\sqrt{c+d x^2} \sqrt{e+f x^2}}{\sqrt{a+b x^2}} \, dx","\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,"Could not integrate","F",-1
112,0,0,37,2.4104029,"\int \frac{\left(a+b x^2\right)^{3/2}}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx","IntegrateAlgebraic[(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,"Could not integrate","A",-1
113,0,0,159,1.5915136,"\int \frac{\sqrt{a+b x^2}}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx","IntegrateAlgebraic[Sqrt[a + b*x^2]/(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]","\int \frac{\sqrt{a+b x^2}}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx","\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,"Could not integrate","F",-1
114,0,0,148,2.0505176,"\int \frac{1}{\sqrt{a+b x^2} \sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx","IntegrateAlgebraic[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]","\int \frac{1}{\sqrt{a+b x^2} \sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx","\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,"Could not integrate","F",-1
115,0,0,37,2.7938371,"\int \frac{1}{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx","IntegrateAlgebraic[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,"Could not integrate","A",-1