1,1,172,172,0.0874999,"\int \left(a+b x^2\right) \left(c+d x^2\right) \left(e+f x^2\right)^4 \, dx","Integrate[(a + b*x^2)*(c + d*x^2)*(e + f*x^2)^4,x]","\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}","\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,"a*c*e^4*x + (e^3*(b*c*e + a*d*e + 4*a*c*f)*x^3)/3 + (e^2*(2*a*f*(2*d*e + 3*c*f) + b*e*(d*e + 4*c*f))*x^5)/5 + (2*e*f*(a*f*(3*d*e + 2*c*f) + b*e*(2*d*e + 3*c*f))*x^7)/7 + (f^2*(a*f*(4*d*e + c*f) + 2*b*e*(3*d*e + 2*c*f))*x^9)/9 + (f^3*(4*b*d*e + b*c*f + a*d*f)*x^11)/11 + (b*d*f^4*x^13)/13","A",1
2,1,130,130,0.0703611,"\int \left(a+b x^2\right) \left(c+d x^2\right) \left(e+f x^2\right)^3 \, dx","Integrate[(a + b*x^2)*(c + d*x^2)*(e + f*x^2)^3,x]","\frac{1}{3} e^2 x^3 (3 a c f+a d e+b c e)+\frac{1}{9} f^2 x^9 (a d f+b c f+3 b d e)+\frac{1}{7} f x^7 (a f (c f+3 d e)+3 b e (c f+d e))+\frac{1}{5} e x^5 (3 a f (c f+d e)+b e (3 c f+d e))+a c e^3 x+\frac{1}{11} b d f^3 x^{11}","\frac{1}{3} e^2 x^3 (3 a c f+a d e+b c e)+\frac{1}{9} f^2 x^9 (a d f+b c f+3 b d e)+\frac{1}{7} f x^7 (a f (c f+3 d e)+3 b e (c f+d e))+\frac{1}{5} e x^5 (3 a f (c f+d e)+b e (3 c f+d e))+a c e^3 x+\frac{1}{11} b d f^3 x^{11}",1,"a*c*e^3*x + (e^2*(b*c*e + a*d*e + 3*a*c*f)*x^3)/3 + (e*(3*a*f*(d*e + c*f) + b*e*(d*e + 3*c*f))*x^5)/5 + (f*(3*b*e*(d*e + c*f) + a*f*(3*d*e + c*f))*x^7)/7 + (f^2*(3*b*d*e + b*c*f + a*d*f)*x^9)/9 + (b*d*f^3*x^11)/11","A",1
3,1,96,94,0.0372167,"\int \left(a+b x^2\right) \left(c+d x^2\right) \left(e+f x^2\right)^2 \, dx","Integrate[(a + b*x^2)*(c + d*x^2)*(e + f*x^2)^2,x]","\frac{1}{5} x^5 \left(a c f^2+2 a d e f+2 b c e f+b d e^2\right)+\frac{1}{7} f x^7 (a d f+b c f+2 b d e)+\frac{1}{3} e x^3 (2 a c f+a d e+b c e)+a c e^2 x+\frac{1}{9} b d f^2 x^9","\frac{1}{7} f x^7 (a d f+b c f+2 b d e)+\frac{1}{5} x^5 (a f (c f+2 d e)+b e (2 c f+d e))+\frac{1}{3} e x^3 (2 a c f+a d e+b c e)+a c e^2 x+\frac{1}{9} b d f^2 x^9",1,"a*c*e^2*x + (e*(b*c*e + a*d*e + 2*a*c*f)*x^3)/3 + ((b*d*e^2 + 2*b*c*e*f + 2*a*d*e*f + a*c*f^2)*x^5)/5 + (f*(2*b*d*e + b*c*f + a*d*f)*x^7)/7 + (b*d*f^2*x^9)/9","A",1
4,1,56,56,0.0129629,"\int \left(a+b x^2\right) \left(c+d x^2\right) \left(e+f x^2\right) \, dx","Integrate[(a + b*x^2)*(c + d*x^2)*(e + f*x^2),x]","\frac{1}{5} x^5 (a d f+b c f+b d e)+\frac{1}{3} x^3 (a c f+a d e+b c e)+a c e x+\frac{1}{7} b d f x^7","\frac{1}{5} x^5 (a d f+b c f+b d e)+\frac{1}{3} x^3 (a c f+a d e+b c e)+a c e x+\frac{1}{7} b d f x^7",1,"a*c*e*x + ((b*c*e + a*d*e + a*c*f)*x^3)/3 + ((b*d*e + b*c*f + a*d*f)*x^5)/5 + (b*d*f*x^7)/7","A",1
5,1,72,81,0.0535893,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)}{e+f x^2} \, dx","Integrate[((a + b*x^2)*(c + d*x^2))/(e + f*x^2),x]","\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 (a d f+b c f-b d e)}{f^2}+\frac{b d x^3}{3 f}","\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,"((-(b*d*e) + b*c*f + a*d*f)*x)/f^2 + (b*d*x^3)/(3*f) + ((b*e - a*f)*(d*e - c*f)*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*f^(5/2))","A",1
6,1,95,108,0.0649801,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)}{\left(e+f x^2\right)^2} \, dx","Integrate[((a + b*x^2)*(c + d*x^2))/(e + f*x^2)^2,x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right) (b e (3 d e-c f)-a f (c f+d e))}{2 e^{3/2} f^{5/2}}+\frac{x (b e-a f) (d e-c f)}{2 e f^2 \left(e+f x^2\right)}+\frac{b d x}{f^2}","-\frac{\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right) (b e (3 d e-c f)-a f (c f+d e))}{2 e^{3/2} f^{5/2}}-\frac{x \left(a+b x^2\right) (d e-c f)}{2 e f \left(e+f x^2\right)}+\frac{b x (3 d e-c f)}{2 e f^2}",1,"(b*d*x)/f^2 + ((b*e - a*f)*(d*e - c*f)*x)/(2*e*f^2*(e + f*x^2)) - ((b*e*(3*d*e - c*f) - a*f*(d*e + c*f))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(2*e^(3/2)*f^(5/2))","A",1
7,1,130,130,0.0833575,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)}{\left(e+f x^2\right)^3} \, dx","Integrate[((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) (a f (3 c f+d e)+b e (c f+3 d e))}{8 e^{5/2} f^{5/2}}+\frac{x (a f (3 c f+d e)+b e (c f-5 d e))}{8 e^2 f^2 \left(e+f x^2\right)}+\frac{x (b e-a f) (d e-c f)}{4 e 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,"((b*e - a*f)*(d*e - c*f)*x)/(4*e*f^2*(e + f*x^2)^2) + ((b*e*(-5*d*e + c*f) + a*f*(d*e + 3*c*f))*x)/(8*e^2*f^2*(e + f*x^2)) + ((b*e*(3*d*e + c*f) + a*f*(d*e + 3*c*f))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(8*e^(5/2)*f^(5/2))","A",1
8,1,171,171,0.1039002,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)}{\left(e+f x^2\right)^4} \, dx","Integrate[((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) (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 (a f (5 c f+d e)+b e (c f-7 d e))}{24 e^2 f^2 \left(e+f x^2\right)^2}+\frac{x (b e-a f) (d e-c f)}{6 e 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,"((b*e - a*f)*(d*e - c*f)*x)/(6*e*f^2*(e + f*x^2)^3) + ((b*e*(-7*d*e + c*f) + a*f*(d*e + 5*c*f))*x)/(24*e^2*f^2*(e + f*x^2)^2) + ((b*e*(d*e + c*f) + a*f*(d*e + 5*c*f))*x)/(16*e^3*f^2*(e + f*x^2)) + ((b*e*(d*e + c*f) + a*f*(d*e + 5*c*f))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(16*e^(7/2)*f^(5/2))","A",1
9,1,226,226,0.0849515,"\int \left(a+b x^2\right) \left(c+d x^2\right)^2 \left(e+f x^2\right)^3 \, dx","Integrate[(a + b*x^2)*(c + d*x^2)^2*(e + f*x^2)^3,x]","\frac{1}{9} f x^9 \left(a d f (2 c f+3 d e)+b \left(c^2 f^2+6 c d e f+3 d^2 e^2\right)\right)+\frac{1}{7} x^7 \left(a f \left(c^2 f^2+6 c d e f+3 d^2 e^2\right)+b e \left(3 c^2 f^2+6 c d e f+d^2 e^2\right)\right)+\frac{1}{5} e x^5 \left(a \left(3 c^2 f^2+6 c d e f+d^2 e^2\right)+b c e (3 c f+2 d e)\right)+\frac{1}{3} c e^2 x^3 (3 a c f+2 a d e+b c e)+\frac{1}{11} d f^2 x^{11} (a d f+2 b c f+3 b d e)+a c^2 e^3 x+\frac{1}{13} b d^2 f^3 x^{13}","\frac{1}{9} f x^9 \left(a d f (2 c f+3 d e)+b \left(c^2 f^2+6 c d e f+3 d^2 e^2\right)\right)+\frac{1}{7} x^7 \left(a f \left(c^2 f^2+6 c d e f+3 d^2 e^2\right)+b e \left(3 c^2 f^2+6 c d e f+d^2 e^2\right)\right)+\frac{1}{5} e x^5 \left(a \left(3 c^2 f^2+6 c d e f+d^2 e^2\right)+b c e (3 c f+2 d e)\right)+\frac{1}{3} c e^2 x^3 (3 a c f+2 a d e+b c e)+\frac{1}{11} d f^2 x^{11} (a d f+2 b c f+3 b d e)+a c^2 e^3 x+\frac{1}{13} b d^2 f^3 x^{13}",1,"a*c^2*e^3*x + (c*e^2*(b*c*e + 2*a*d*e + 3*a*c*f)*x^3)/3 + (e*(b*c*e*(2*d*e + 3*c*f) + a*(d^2*e^2 + 6*c*d*e*f + 3*c^2*f^2))*x^5)/5 + ((a*f*(3*d^2*e^2 + 6*c*d*e*f + c^2*f^2) + b*e*(d^2*e^2 + 6*c*d*e*f + 3*c^2*f^2))*x^7)/7 + (f*(a*d*f*(3*d*e + 2*c*f) + b*(3*d^2*e^2 + 6*c*d*e*f + c^2*f^2))*x^9)/9 + (d*f^2*(3*b*d*e + 2*b*c*f + a*d*f)*x^11)/11 + (b*d^2*f^3*x^13)/13","A",1
10,1,158,158,0.0643206,"\int \left(a+b x^2\right) \left(c+d x^2\right)^2 \left(e+f x^2\right)^2 \, dx","Integrate[(a + b*x^2)*(c + d*x^2)^2*(e + f*x^2)^2,x]","\frac{1}{7} x^7 \left(2 a d f (c f+d e)+b \left(c^2 f^2+4 c d e f+d^2 e^2\right)\right)+\frac{1}{5} x^5 \left(a \left(c^2 f^2+4 c d e f+d^2 e^2\right)+2 b c e (c f+d e)\right)+\frac{1}{9} d f x^9 (a d f+2 b (c f+d e))+\frac{1}{3} c e x^3 (2 a (c f+d e)+b c e)+a c^2 e^2 x+\frac{1}{11} b d^2 f^2 x^{11}","\frac{1}{7} x^7 \left(2 a d f (c f+d e)+b \left(c^2 f^2+4 c d e f+d^2 e^2\right)\right)+\frac{1}{5} x^5 \left(a \left(c^2 f^2+4 c d e f+d^2 e^2\right)+2 b c e (c f+d e)\right)+\frac{1}{9} d f x^9 (a d f+2 b (c f+d e))+\frac{1}{3} c e x^3 (2 a (c f+d e)+b c e)+a c^2 e^2 x+\frac{1}{11} b d^2 f^2 x^{11}",1,"a*c^2*e^2*x + (c*e*(b*c*e + 2*a*(d*e + c*f))*x^3)/3 + ((2*b*c*e*(d*e + c*f) + a*(d^2*e^2 + 4*c*d*e*f + c^2*f^2))*x^5)/5 + ((2*a*d*f*(d*e + c*f) + b*(d^2*e^2 + 4*c*d*e*f + c^2*f^2))*x^7)/7 + (d*f*(a*d*f + 2*b*(d*e + c*f))*x^9)/9 + (b*d^2*f^2*x^11)/11","A",1
11,1,96,94,0.0308079,"\int \left(a+b x^2\right) \left(c+d x^2\right)^2 \left(e+f x^2\right) \, dx","Integrate[(a + b*x^2)*(c + d*x^2)^2*(e + f*x^2),x]","\frac{1}{5} x^5 \left(2 a c d f+a d^2 e+b c^2 f+2 b c d e\right)+\frac{1}{7} d x^7 (a d f+2 b c f+b d e)+\frac{1}{3} c x^3 (a c f+2 a d e+b c e)+a c^2 e x+\frac{1}{9} b d^2 f x^9","\frac{1}{7} d x^7 (a d f+2 b c f+b d e)+\frac{1}{5} x^5 (a d (2 c f+d e)+b c (c f+2 d e))+\frac{1}{3} c x^3 (a c f+2 a d e+b c e)+a c^2 e x+\frac{1}{9} b d^2 f x^9",1,"a*c^2*e*x + (c*(b*c*e + 2*a*d*e + a*c*f)*x^3)/3 + ((2*b*c*d*e + a*d^2*e + b*c^2*f + 2*a*c*d*f)*x^5)/5 + (d*(b*d*e + 2*b*c*f + a*d*f)*x^7)/7 + (b*d^2*f*x^9)/9","A",1
12,1,115,142,0.0706388,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^2}{e+f x^2} \, dx","Integrate[((a + b*x^2)*(c + d*x^2)^2)/(e + f*x^2),x]","-\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(a d f (2 c f-d e)+b (d e-c f)^2\right)}{f^3}+\frac{d x^3 (a d f+2 b c f-b d e)}{3 f^2}+\frac{b d^2 x^5}{5 f}","-\frac{x \left(5 a d f (3 d e-5 c f)-b \left(8 c^2 f^2-25 c d e f+15 d^2 e^2\right)\right)}{15 f^3}-\frac{(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,"((b*(d*e - c*f)^2 + a*d*f*(-(d*e) + 2*c*f))*x)/f^3 + (d*(-(b*d*e) + 2*b*c*f + a*d*f)*x^3)/(3*f^2) + (b*d^2*x^5)/(5*f) - ((b*e - a*f)*(d*e - c*f)^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*f^(7/2))","A",1
13,1,134,164,0.0988402,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^2}{\left(e+f x^2\right)^2} \, dx","Integrate[((a + b*x^2)*(c + d*x^2)^2)/(e + f*x^2)^2,x]","\frac{(d e-c f) \tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right) (b e (5 d e-c f)-a f (c f+3 d e))}{2 e^{3/2} f^{7/2}}-\frac{x (b e-a f) (d e-c f)^2}{2 e f^3 \left(e+f x^2\right)}+\frac{d x (a d f+2 b c f-2 b d e)}{f^3}+\frac{b d^2 x^3}{3 f^2}","\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,"(d*(-2*b*d*e + 2*b*c*f + a*d*f)*x)/f^3 + (b*d^2*x^3)/(3*f^2) - ((b*e - a*f)*(d*e - c*f)^2*x)/(2*e*f^3*(e + f*x^2)) + ((d*e - c*f)*(b*e*(5*d*e - c*f) - a*f*(3*d*e + c*f))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(2*e^(3/2)*f^(7/2))","A",1
14,1,183,207,0.129739,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^2}{\left(e+f x^2\right)^3} \, dx","Integrate[((a + b*x^2)*(c + d*x^2)^2)/(e + f*x^2)^3,x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right) \left(b e \left(-c^2 f^2-6 c d e f+15 d^2 e^2\right)-a f \left(3 c^2 f^2+2 c d e f+3 d^2 e^2\right)\right)}{8 e^{5/2} f^{7/2}}+\frac{x (d e-c f) (b e (9 d e-c f)-a f (3 c f+5 d e))}{8 e^2 f^3 \left(e+f x^2\right)}-\frac{x (b e-a f) (d e-c f)^2}{4 e f^3 \left(e+f x^2\right)^2}+\frac{b d^2 x}{f^3}","-\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,"(b*d^2*x)/f^3 - ((b*e - a*f)*(d*e - c*f)^2*x)/(4*e*f^3*(e + f*x^2)^2) + ((d*e - c*f)*(b*e*(9*d*e - c*f) - a*f*(5*d*e + 3*c*f))*x)/(8*e^2*f^3*(e + f*x^2)) - ((b*e*(15*d^2*e^2 - 6*c*d*e*f - c^2*f^2) - a*f*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(8*e^(5/2)*f^(7/2))","A",1
15,1,242,240,0.1769819,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^2}{\left(e+f x^2\right)^4} \, dx","Integrate[((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(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(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-11 d^2 e^2\right)\right)}{16 e^3 f^3 \left(e+f x^2\right)}+\frac{x (d e-c f) (b e (13 d e-c f)-a f (5 c f+7 d e))}{24 e^2 f^3 \left(e+f x^2\right)^2}-\frac{x (b e-a f) (d e-c f)^2}{6 e 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,"-1/6*((b*e - a*f)*(d*e - c*f)^2*x)/(e*f^3*(e + f*x^2)^3) + ((d*e - c*f)*(b*e*(13*d*e - c*f) - a*f*(7*d*e + 5*c*f))*x)/(24*e^2*f^3*(e + f*x^2)^2) + ((b*e*(-11*d^2*e^2 + 2*c*d*e*f + c^2*f^2) + a*f*(d^2*e^2 + 2*c*d*e*f + 5*c^2*f^2))*x)/(16*e^3*f^3*(e + f*x^2)) + ((b*e*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2) + a*f*(d^2*e^2 + 2*c*d*e*f + 5*c^2*f^2))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(16*e^(7/2)*f^(7/2))","A",1
16,1,310,310,0.129538,"\int \left(a+b x^2\right) \left(c+d x^2\right)^3 \left(e+f x^2\right)^3 \, dx","Integrate[(a + b*x^2)*(c + d*x^2)^3*(e + f*x^2)^3,x]","\frac{3}{11} d f x^{11} \left(a d f (c f+d e)+b \left(c^2 f^2+3 c d e f+d^2 e^2\right)\right)+\frac{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}","\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,"a*c^3*e^3*x + (c^2*e^2*(b*c*e + 3*a*(d*e + c*f))*x^3)/3 + (3*c*e*(b*c*e*(d*e + c*f) + a*(d^2*e^2 + 3*c*d*e*f + c^2*f^2))*x^5)/5 + ((3*b*c*e*(d^2*e^2 + 3*c*d*e*f + c^2*f^2) + a*(d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + c^3*f^3))*x^7)/7 + ((3*a*d*f*(d^2*e^2 + 3*c*d*e*f + c^2*f^2) + b*(d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + c^3*f^3))*x^9)/9 + (3*d*f*(a*d*f*(d*e + c*f) + b*(d^2*e^2 + 3*c*d*e*f + c^2*f^2))*x^11)/11 + (d^2*f^2*(a*d*f + 3*b*(d*e + c*f))*x^13)/13 + (b*d^3*f^3*x^15)/15","A",1
17,1,226,226,0.088489,"\int \left(a+b x^2\right) \left(c+d x^2\right)^3 \left(e+f x^2\right)^2 \, dx","Integrate[(a + b*x^2)*(c + d*x^2)^3*(e + f*x^2)^2,x]","\frac{1}{9} d x^9 \left(a d f (3 c f+2 d e)+b \left(3 c^2 f^2+6 c d e f+d^2 e^2\right)\right)+\frac{1}{7} x^7 \left(a d \left(3 c^2 f^2+6 c d e f+d^2 e^2\right)+b c \left(c^2 f^2+6 c d e f+3 d^2 e^2\right)\right)+\frac{1}{5} c x^5 \left(a \left(c^2 f^2+6 c d e f+3 d^2 e^2\right)+b c e (2 c f+3 d e)\right)+\frac{1}{3} c^2 e x^3 (2 a c f+3 a d e+b c e)+\frac{1}{11} d^2 f x^{11} (a d f+3 b c f+2 b d e)+a c^3 e^2 x+\frac{1}{13} b d^3 f^2 x^{13}","\frac{1}{9} d x^9 \left(a d f (3 c f+2 d e)+b \left(3 c^2 f^2+6 c d e f+d^2 e^2\right)\right)+\frac{1}{7} x^7 \left(a d \left(3 c^2 f^2+6 c d e f+d^2 e^2\right)+b c \left(c^2 f^2+6 c d e f+3 d^2 e^2\right)\right)+\frac{1}{5} c x^5 \left(a \left(c^2 f^2+6 c d e f+3 d^2 e^2\right)+b c e (2 c f+3 d e)\right)+\frac{1}{3} c^2 e x^3 (2 a c f+3 a d e+b c e)+\frac{1}{11} d^2 f x^{11} (a d f+3 b c f+2 b d e)+a c^3 e^2 x+\frac{1}{13} b d^3 f^2 x^{13}",1,"a*c^3*e^2*x + (c^2*e*(b*c*e + 3*a*d*e + 2*a*c*f)*x^3)/3 + (c*(b*c*e*(3*d*e + 2*c*f) + a*(3*d^2*e^2 + 6*c*d*e*f + c^2*f^2))*x^5)/5 + ((b*c*(3*d^2*e^2 + 6*c*d*e*f + c^2*f^2) + a*d*(d^2*e^2 + 6*c*d*e*f + 3*c^2*f^2))*x^7)/7 + (d*(a*d*f*(2*d*e + 3*c*f) + b*(d^2*e^2 + 6*c*d*e*f + 3*c^2*f^2))*x^9)/9 + (d^2*f*(2*b*d*e + 3*b*c*f + a*d*f)*x^11)/11 + (b*d^3*f^2*x^13)/13","A",1
18,1,130,130,0.0551423,"\int \left(a+b x^2\right) \left(c+d x^2\right)^3 \left(e+f x^2\right) \, dx","Integrate[(a + b*x^2)*(c + d*x^2)^3*(e + f*x^2),x]","\frac{1}{3} c^2 x^3 (a c f+3 a d e+b c e)+\frac{1}{9} d^2 x^9 (a d f+3 b c f+b d e)+\frac{1}{7} d x^7 (a d (3 c f+d e)+3 b c (c f+d e))+\frac{1}{5} c x^5 (3 a d (c f+d e)+b c (c f+3 d e))+a c^3 e x+\frac{1}{11} b d^3 f x^{11}","\frac{1}{3} c^2 x^3 (a c f+3 a d e+b c e)+\frac{1}{9} d^2 x^9 (a d f+3 b c f+b d e)+\frac{1}{7} d x^7 (a d (3 c f+d e)+3 b c (c f+d e))+\frac{1}{5} c x^5 (3 a d (c f+d e)+b c (c f+3 d e))+a c^3 e x+\frac{1}{11} b d^3 f x^{11}",1,"a*c^3*e*x + (c^2*(b*c*e + 3*a*d*e + a*c*f)*x^3)/3 + (c*(3*a*d*(d*e + c*f) + b*c*(3*d*e + c*f))*x^5)/5 + (d*(3*b*c*(d*e + c*f) + a*d*(d*e + 3*c*f))*x^7)/7 + (d^2*(b*d*e + 3*b*c*f + a*d*f)*x^9)/9 + (b*d^3*f*x^11)/11","A",1
19,1,179,227,0.1076595,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^3}{e+f x^2} \, dx","Integrate[((a + b*x^2)*(c + d*x^2)^3)/(e + f*x^2),x]","\frac{x \left(a d f \left(3 c^2 f^2-3 c d e f+d^2 e^2\right)-b (d e-c f)^3\right)}{f^4}+\frac{d x^3 \left(a d f (3 c f-d e)+b \left(3 c^2 f^2-3 c d e f+d^2 e^2\right)\right)}{3 f^3}+\frac{d^2 x^5 (a d f+3 b c f-b d e)}{5 f^2}+\frac{(b e-a f) (d e-c f)^3 \tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)}{\sqrt{e} f^{9/2}}+\frac{b d^3 x^7}{7 f}","-\frac{x \left(c+d x^2\right) \left(7 a d f (5 d e-9 c f)-b \left(24 c^2 f^2-63 c d e f+35 d^2 e^2\right)\right)}{105 f^3}+\frac{x \left(7 a d f \left(33 c^2 f^2-40 c d e f+15 d^2 e^2\right)-b \left(-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,"((-(b*(d*e - c*f)^3) + a*d*f*(d^2*e^2 - 3*c*d*e*f + 3*c^2*f^2))*x)/f^4 + (d*(a*d*f*(-(d*e) + 3*c*f) + b*(d^2*e^2 - 3*c*d*e*f + 3*c^2*f^2))*x^3)/(3*f^3) + (d^2*(-(b*d*e) + 3*b*c*f + a*d*f)*x^5)/(5*f^2) + (b*d^3*x^7)/(7*f) + ((b*e - a*f)*(d*e - c*f)^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*f^(9/2))","A",1
20,1,176,242,0.1283826,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^3}{\left(e+f x^2\right)^2} \, dx","Integrate[((a + b*x^2)*(c + d*x^2)^3)/(e + f*x^2)^2,x]","\frac{d^2 x^3 (a d f+3 b c f-2 b d e)}{3 f^3}-\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{x (b e-a f) (d e-c f)^3}{2 e f^4 \left(e+f x^2\right)}+\frac{d x \left(a d f (3 c f-2 d e)+3 b (d e-c f)^2\right)}{f^4}+\frac{b d^3 x^5}{5 f^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,"(d*(3*b*(d*e - c*f)^2 + a*d*f*(-2*d*e + 3*c*f))*x)/f^4 + (d^2*(-2*b*d*e + 3*b*c*f + a*d*f)*x^3)/(3*f^3) + (b*d^3*x^5)/(5*f^2) + ((b*e - a*f)*(d*e - c*f)^3*x)/(2*e*f^4*(e + f*x^2)) - ((d*e - c*f)^2*(b*e*(7*d*e - c*f) - a*f*(5*d*e + c*f))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(2*e^(3/2)*f^(9/2))","A",1
21,1,219,291,0.1684449,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^3}{\left(e+f x^2\right)^3} \, dx","Integrate[((a + b*x^2)*(c + d*x^2)^3)/(e + f*x^2)^3,x]","\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^2 x (a d f+3 b c f-3 b d e)}{f^4}-\frac{x (d e-c f)^2 (b e (13 d e-c f)-3 a f (c f+3 d e))}{8 e^2 f^4 \left(e+f x^2\right)}+\frac{x (b e-a f) (d e-c f)^3}{4 e f^4 \left(e+f x^2\right)^2}+\frac{b d^3 x^3}{3 f^3}","\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,"(d^2*(-3*b*d*e + 3*b*c*f + a*d*f)*x)/f^4 + (b*d^3*x^3)/(3*f^3) + ((b*e - a*f)*(d*e - c*f)^3*x)/(4*e*f^4*(e + f*x^2)^2) - ((d*e - c*f)^2*(b*e*(13*d*e - c*f) - 3*a*f*(3*d*e + c*f))*x)/(8*e^2*f^4*(e + f*x^2)) + ((d*e - c*f)*(b*e*(35*d^2*e^2 - 10*c*d*e*f - c^2*f^2) - 3*a*f*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(8*e^(5/2)*f^(9/2))","A",1
22,1,295,348,0.2187212,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^3}{\left(e+f x^2\right)^4} \, dx","Integrate[((a + b*x^2)*(c + d*x^2)^3)/(e + f*x^2)^4,x]","\frac{x (d e-c f) \left(b e \left(-c^2 f^2-4 c d e f+29 d^2 e^2\right)-a f \left(5 c^2 f^2+8 c d e f+11 d^2 e^2\right)\right)}{16 e^3 f^4 \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 (d e-c f)^2 (b e (19 d e-c f)-a f (5 c f+13 d e))}{24 e^2 f^4 \left(e+f x^2\right)^2}+\frac{x (b e-a f) (d e-c f)^3}{6 e f^4 \left(e+f x^2\right)^3}+\frac{b d^3 x}{f^4}","\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,"(b*d^3*x)/f^4 + ((b*e - a*f)*(d*e - c*f)^3*x)/(6*e*f^4*(e + f*x^2)^3) - ((d*e - c*f)^2*(b*e*(19*d*e - c*f) - a*f*(13*d*e + 5*c*f))*x)/(24*e^2*f^4*(e + f*x^2)^2) + ((d*e - c*f)*(b*e*(29*d^2*e^2 - 4*c*d*e*f - c^2*f^2) - a*f*(11*d^2*e^2 + 8*c*d*e*f + 5*c^2*f^2))*x)/(16*e^3*f^4*(e + f*x^2)) - ((b*e*(35*d^3*e^3 - 15*c*d^2*e^2*f - 3*c^2*d*e*f^2 - c^3*f^3) - a*f*(5*d^3*e^3 + 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 5*c^3*f^3))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(16*e^(7/2)*f^(9/2))","A",1
23,1,373,544,1.3345518,"\int \left(a+b x^2\right) \left(c+d x^2\right)^{3/2} \sqrt{e+f x^2} \, dx","Integrate[(a + b*x^2)*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2],x]","\frac{-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) \left(b \left(3 c^2 f^2-15 c d e f+8 d^2 e^2\right)-14 a d f (d e-3 c f)\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+f x \sqrt{\frac{d}{c}} \left(c+d x^2\right) \left(e+f x^2\right) \left(7 a d f \left(6 c f+d \left(e+3 f x^2\right)\right)+b \left(3 c^2 f^2+3 c d f \left(3 e+8 f x^2\right)+d^2 \left(-4 e^2+3 e f x^2+15 f^2 x^4\right)\right)\right)+i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \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(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)}{105 d f^3 \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}}","\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,"(Sqrt[d/c]*f*x*(c + d*x^2)*(e + f*x^2)*(7*a*d*f*(6*c*f + d*(e + 3*f*x^2)) + b*(3*c^2*f^2 + 3*c*d*f*(3*e + 8*f*x^2) + d^2*(-4*e^2 + 3*e*f*x^2 + 15*f^2*x^4))) + I*e*(7*a*d*f*(2*d^2*e^2 - 7*c*d*e*f - 3*c^2*f^2) + b*(-8*d^3*e^3 + 19*c*d^2*e^2*f - 9*c^2*d*e*f^2 + 6*c^3*f^3))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*e*(-(d*e) + c*f)*(-14*a*d*f*(d*e - 3*c*f) + b*(8*d^2*e^2 - 15*c*d*e*f + 3*c^2*f^2))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(105*d*Sqrt[d/c]*f^3*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
24,1,267,381,0.7994717,"\int \left(a+b x^2\right) \sqrt{c+d x^2} \sqrt{e+f x^2} \, dx","Integrate[(a + b*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2],x]","\frac{i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left(2 b \left(c^2 f^2-c d e f+d^2 e^2\right)-5 a d f (c f+d e)\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+f x \sqrt{\frac{d}{c}} \left(c+d x^2\right) \left(e+f x^2\right) \left(5 a d f+b c f+b d \left(e+3 f x^2\right)\right)-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) (5 a d f+b c f-2 b d e) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)}{15 d f^2 \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}}","\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,"(Sqrt[d/c]*f*x*(c + d*x^2)*(e + f*x^2)*(b*c*f + 5*a*d*f + b*d*(e + 3*f*x^2)) + I*e*(-5*a*d*f*(d*e + c*f) + 2*b*(d^2*e^2 - c*d*e*f + c^2*f^2))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*e*(-(d*e) + c*f)*(-2*b*d*e + b*c*f + 5*a*d*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(15*d*Sqrt[d/c]*f^2*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
25,1,212,283,0.4258541,"\int \frac{\left(a+b x^2\right) \sqrt{e+f x^2}}{\sqrt{c+d x^2}} \, dx","Integrate[((a + b*x^2)*Sqrt[e + f*x^2])/Sqrt[c + d*x^2],x]","\frac{i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (-3 a d f+2 b c f-b d e) E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+b f x \sqrt{\frac{d}{c}} \left(c+d x^2\right) \left(e+f x^2\right)-i b e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)}{3 d f \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}}","\frac{x \sqrt{c+d x^2} (3 a d f-2 b c f+b d e)}{3 d^2 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} (3 a d f-2 b c f+b d e) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 d^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{e^{3/2} \sqrt{c+d x^2} (b c-3 a d) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 c d \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{b x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 d}",1,"(b*Sqrt[d/c]*f*x*(c + d*x^2)*(e + f*x^2) + I*e*(-(b*d*e) + 2*b*c*f - 3*a*d*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*b*e*(-(d*e) + c*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(3*d*Sqrt[d/c]*f*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
26,1,192,271,0.3692239,"\int \frac{\left(a+b x^2\right) \sqrt{e+f x^2}}{\left(c+d x^2\right)^{3/2}} \, dx","Integrate[((a + b*x^2)*Sqrt[e + f*x^2])/(c + d*x^2)^(3/2),x]","\frac{-(b c-a d) \left(x \sqrt{\frac{d}{c}} \left(e+f x^2\right)-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)\right)-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (2 b c-a d) E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)}{c^2 \left(\frac{d}{c}\right)^{3/2} \sqrt{c+d x^2} \sqrt{e+f x^2}}","\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,"((-I)*(2*b*c - a*d)*e*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - (b*c - a*d)*(Sqrt[d/c]*x*(e + f*x^2) - I*e*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(c^2*(d/c)^(3/2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
27,1,297,274,1.2671359,"\int \frac{\left(a+b x^2\right) \sqrt{e+f x^2}}{\left(c+d x^2\right)^{5/2}} \, dx","Integrate[((a + b*x^2)*Sqrt[e + f*x^2])/(c + d*x^2)^(5/2),x]","\frac{x \sqrt{\frac{d}{c}} \left(e+f x^2\right) \left(a d \left(2 c^2 f-3 c d e+c d f x^2-2 d^2 e x^2\right)+b c \left(c^2 f+2 c d f x^2-d^2 e x^2\right)\right)-i e \left(c+d x^2\right) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (2 a d+b c) (c f-d e) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+i e \left(c+d x^2\right) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (a d (c f-2 d e)+b c (2 c f-d e)) E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)}{3 c^3 \left(\frac{d}{c}\right)^{3/2} \left(c+d x^2\right)^{3/2} \sqrt{e+f x^2} (c f-d e)}","\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,"(Sqrt[d/c]*x*(e + f*x^2)*(a*d*(-3*c*d*e + 2*c^2*f - 2*d^2*e*x^2 + c*d*f*x^2) + b*c*(c^2*f - d^2*e*x^2 + 2*c*d*f*x^2)) + I*e*(a*d*(-2*d*e + c*f) + b*c*(-(d*e) + 2*c*f))*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*(b*c + 2*a*d)*e*(-(d*e) + c*f)*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(3*c^3*(d/c)^(3/2)*(-(d*e) + c*f)*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2])","C",1
28,1,379,385,1.8430715,"\int \frac{\left(a+b x^2\right) \sqrt{e+f x^2}}{\left(c+d x^2\right)^{7/2}} \, dx","Integrate[((a + b*x^2)*Sqrt[e + f*x^2])/(c + d*x^2)^(7/2),x]","\frac{-x \sqrt{\frac{d}{c}} \left(e+f x^2\right) \left(-\left(c+d x^2\right)^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)+3 c^2 (b c-a d) (d e-c f)^2-c \left(c+d x^2\right) (d e-c f) (a d (4 d e-3 c f)+b c (d e-2 c f))\right)+i e \left(c+d x^2\right)^2 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left(\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(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-(c f-d e) (a d (9 c f-8 d e)+b c (c f-2 d e)) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)\right)}{15 c^4 \left(\frac{d}{c}\right)^{3/2} \left(c+d x^2\right)^{5/2} \sqrt{e+f x^2} (d e-c f)^2}","-\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,"(-(Sqrt[d/c]*x*(e + f*x^2)*(3*c^2*(b*c - a*d)*(d*e - c*f)^2 - c*(d*e - c*f)*(a*d*(4*d*e - 3*c*f) + b*c*(d*e - 2*c*f))*(c + d*x^2) - (2*b*c*(d^2*e^2 - c*d*e*f + c^2*f^2) + a*d*(8*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2))*(c + d*x^2)^2)) + I*e*(c + d*x^2)^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*((2*b*c*(d^2*e^2 - c*d*e*f + c^2*f^2) + a*d*(8*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2))*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - (-(d*e) + c*f)*(b*c*(-2*d*e + c*f) + a*d*(-8*d*e + 9*c*f))*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(15*c^4*(d/c)^(3/2)*(d*e - c*f)^2*(c + d*x^2)^(5/2)*Sqrt[e + f*x^2])","C",1
29,1,372,543,1.7092273,"\int \left(a+b x^2\right) \sqrt{c+d x^2} \left(e+f x^2\right)^{3/2} \, dx","Integrate[(a + b*x^2)*Sqrt[c + d*x^2]*(e + f*x^2)^(3/2),x]","\frac{i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) \left(b \left(4 c^2 f^2-6 c d e f+6 d^2 e^2\right)-7 a d f (c f+3 d e)\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+f x \left(-\sqrt{\frac{d}{c}}\right) \left(c+d x^2\right) \left(e+f x^2\right) \left(-7 a d f \left(c f+6 d e+3 d f x^2\right)+4 b c^2 f^2-3 b c d f \left(3 e+f x^2\right)-3 b d^2 \left(e^2+8 e f x^2+5 f^2 x^4\right)\right)-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \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(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)}{105 c^2 f^2 \left(\frac{d}{c}\right)^{5/2} \sqrt{c+d x^2} \sqrt{e+f x^2}}","\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,"(-(Sqrt[d/c]*f*x*(c + d*x^2)*(e + f*x^2)*(4*b*c^2*f^2 - 3*b*c*d*f*(3*e + f*x^2) - 7*a*d*f*(6*d*e + c*f + 3*d*f*x^2) - 3*b*d^2*(e^2 + 8*e*f*x^2 + 5*f^2*x^4))) - I*e*(7*a*d*f*(3*d^2*e^2 + 7*c*d*e*f - 2*c^2*f^2) + b*(-6*d^3*e^3 + 9*c*d^2*e^2*f - 19*c^2*d*e*f^2 + 8*c^3*f^3))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + I*e*(-(d*e) + c*f)*(-7*a*d*f*(3*d*e + c*f) + b*(6*d^2*e^2 - 6*c*d*e*f + 4*c^2*f^2))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(105*c^2*(d/c)^(5/2)*f^2*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
30,1,275,400,1.1772903,"\int \frac{\left(a+b x^2\right) \left(e+f x^2\right)^{3/2}}{\sqrt{c+d x^2}} \, dx","Integrate[((a + b*x^2)*(e + f*x^2)^(3/2))/Sqrt[c + d*x^2],x]","\frac{-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \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(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+f x \left(-\sqrt{\frac{d}{c}}\right) \left(c+d x^2\right) \left(e+f x^2\right) \left(-5 a d f+4 b c f-3 b d \left(2 e+f x^2\right)\right)+i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) (-5 a d f+4 b c f-3 b d e) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)}{15 c^2 f \left(\frac{d}{c}\right)^{5/2} \sqrt{c+d x^2} \sqrt{e+f x^2}}","\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,"(-(Sqrt[d/c]*f*x*(c + d*x^2)*(e + f*x^2)*(4*b*c*f - 5*a*d*f - 3*b*d*(2*e + f*x^2))) - I*e*(10*a*d*f*(2*d*e - c*f) + b*(3*d^2*e^2 - 13*c*d*e*f + 8*c^2*f^2))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + I*e*(-(d*e) + c*f)*(-3*b*d*e + 4*b*c*f - 5*a*d*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(15*c^2*(d/c)^(5/2)*f*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
31,1,248,369,0.9861313,"\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","Integrate[((a + b*x^2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(3/2),x]","\frac{\sqrt{\frac{d}{c}} \left(x \sqrt{\frac{d}{c}} \left(e+f x^2\right) \left(3 a d (d e-c f)+b c \left(4 c f-3 d e+d f x^2\right)\right)-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (4 b c-3 a d) (c f-d e) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (3 a d (d e-2 c f)+b c (8 c f-7 d e)) E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)\right)}{3 d^3 \sqrt{c+d x^2} \sqrt{e+f x^2}}","\frac{f x \sqrt{c+d x^2} (b c (7 d e-8 c f)-3 a d (d e-2 c f))}{3 c d^3 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (b c (7 d e-8 c f)-3 a d (d e-2 c f)) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 c d^3 \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{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,"(Sqrt[d/c]*(Sqrt[d/c]*x*(e + f*x^2)*(3*a*d*(d*e - c*f) + b*c*(-3*d*e + 4*c*f + d*f*x^2)) + I*e*(3*a*d*(d*e - 2*c*f) + b*c*(-7*d*e + 8*c*f))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*(4*b*c - 3*a*d)*e*(-(d*e) + c*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(3*d^3*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
32,1,296,373,1.2263292,"\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","Integrate[((a + b*x^2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(5/2),x]","\frac{\left(\frac{d}{c}\right)^{3/2} \left(x \sqrt{\frac{d}{c}} \left(e+f x^2\right) \left(a d \left(c^2 f+c d \left(3 e+2 f x^2\right)+2 d^2 e x^2\right)+b c \left(-4 c^2 f-5 c d f x^2+d^2 e x^2\right)\right)+i e \left(c+d x^2\right) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b c (4 c f-d e)-a d (c f+2 d e)) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-i e \left(c+d x^2\right) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b c (8 c f-d e)-2 a d (c f+d e)) E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)\right)}{3 d^4 \left(c+d x^2\right)^{3/2} \sqrt{e+f x^2}}","-\frac{f x \sqrt{c+d x^2} (2 a d (c f+d e)+b c (d e-8 c f))}{3 c^2 d^3 \sqrt{e+f x^2}}+\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (2 a d (c f+d e)+b c (d e-8 c f)) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 c^2 d^3 \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{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,"((d/c)^(3/2)*(Sqrt[d/c]*x*(e + f*x^2)*(b*c*(-4*c^2*f + d^2*e*x^2 - 5*c*d*f*x^2) + a*d*(c^2*f + 2*d^2*e*x^2 + c*d*(3*e + 2*f*x^2))) - I*e*(-2*a*d*(d*e + c*f) + b*c*(-(d*e) + 8*c*f))*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + I*e*(-(a*d*(2*d*e + c*f)) + b*c*(-(d*e) + 4*c*f))*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(3*d^4*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2])","C",1
33,1,382,376,1.6663095,"\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","Integrate[((a + b*x^2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(7/2),x]","\frac{\sqrt{\frac{d}{c}} \left(-x \sqrt{\frac{d}{c}} \left(e+f x^2\right) \left(\left(c+d x^2\right)^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)+3 c^2 (b c-a d) (d e-c f)^2-c \left(c+d x^2\right) (d e-c f) (2 a d (c f+2 d e)+b c (d e-7 c f))\right)-i e \sqrt{\frac{d x^2}{c}+1} \left(c+d x^2\right)^2 \sqrt{\frac{f x^2}{e}+1} \left(\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(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+(d e-c f) (a d (c f+8 d e)+2 b c (2 c f+d e)) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)\right)\right)}{15 c^2 d^3 \left(c+d x^2\right)^{5/2} \sqrt{e+f x^2} (d e-c f)}","-\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,"(Sqrt[d/c]*(-(Sqrt[d/c]*x*(e + f*x^2)*(3*c^2*(b*c - a*d)*(d*e - c*f)^2 - c*(d*e - c*f)*(b*c*(d*e - 7*c*f) + 2*a*d*(2*d*e + c*f))*(c + d*x^2) + (a*d*(-8*d^2*e^2 + 3*c*d*e*f + 2*c^2*f^2) + b*c*(-2*d^2*e^2 - 3*c*d*e*f + 8*c^2*f^2))*(c + d*x^2)^2)) - I*e*(c + d*x^2)^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*((a*d*(-8*d^2*e^2 + 3*c*d*e*f + 2*c^2*f^2) + b*c*(-2*d^2*e^2 - 3*c*d*e*f + 8*c^2*f^2))*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + (d*e - c*f)*(a*d*(8*d*e + c*f) + 2*b*c*(d*e + 2*c*f))*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])))/(15*c^2*d^3*(d*e - c*f)*(c + d*x^2)^(5/2)*Sqrt[e + f*x^2])","C",1
34,1,545,531,2.5927005,"\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","Integrate[((a + b*x^2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(9/2),x]","\frac{\sqrt{\frac{d}{c}} \left(-x \sqrt{\frac{d}{c}} \left(e+f x^2\right) \left(15 c^3 (b c-a d) (d e-c f)^3-c \left(c+d x^2\right)^2 (d e-c f) \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)-3 c^2 \left(c+d x^2\right) (d e-c f)^2 (2 a d (c f+3 d e)+b c (d e-9 c f))-\left(c+d x^2\right)^3 \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)\right)+i e \left(c+d x^2\right)^3 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left(\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(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-(c f-d e) \left(3 a d \left(c^2 f^2+16 c d e f-16 d^2 e^2\right)+b c \left(4 c^2 f^2+c d e f-8 d^2 e^2\right)\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)\right)\right)}{105 c^3 d^3 \left(c+d x^2\right)^{7/2} \sqrt{e+f x^2} (d e-c f)^2}","\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,"(Sqrt[d/c]*(-(Sqrt[d/c]*x*(e + f*x^2)*(15*c^3*(b*c - a*d)*(d*e - c*f)^3 - 3*c^2*(d*e - c*f)^2*(b*c*(d*e - 9*c*f) + 2*a*d*(3*d*e + c*f))*(c + d*x^2) - c*(d*e - c*f)*(b*c*(4*d^2*e^2 + c*d*e*f - 8*c^2*f^2) + 3*a*d*(8*d^2*e^2 - 5*c*d*e*f - 2*c^2*f^2))*(c + d*x^2)^2 - (6*a*d*(8*d^3*e^3 - 12*c*d^2*e^2*f + 2*c^2*d*e*f^2 + c^3*f^3) + b*c*(8*d^3*e^3 - 5*c*d^2*e^2*f - 5*c^2*d*e*f^2 + 8*c^3*f^3))*(c + d*x^2)^3)) + I*e*(c + d*x^2)^3*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*((6*a*d*(8*d^3*e^3 - 12*c*d^2*e^2*f + 2*c^2*d*e*f^2 + c^3*f^3) + b*c*(8*d^3*e^3 - 5*c*d^2*e^2*f - 5*c^2*d*e*f^2 + 8*c^3*f^3))*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - (-(d*e) + c*f)*(3*a*d*(-16*d^2*e^2 + 16*c*d*e*f + c^2*f^2) + b*c*(-8*d^2*e^2 + c*d*e*f + 4*c^2*f^2))*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])))/(105*c^3*d^3*(d*e - c*f)^2*(c + d*x^2)^(7/2)*Sqrt[e + f*x^2])","C",1
35,1,386,551,1.305787,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^{5/2}}{\sqrt{e+f x^2}} \, dx","Integrate[((a + b*x^2)*(c + d*x^2)^(5/2))/Sqrt[e + f*x^2],x]","\frac{i \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) \left(4 b e \left(15 c^2 f^2-26 c d e f+12 d^2 e^2\right)-7 a f \left(15 c^2 f^2-19 c d e f+8 d^2 e^2\right)\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+f x \sqrt{\frac{d}{c}} \left(c+d x^2\right) \left(e+f x^2\right) \left(7 a d f \left(11 c f-4 d e+3 d f x^2\right)+b \left(45 c^2 f^2+c d f \left(45 f x^2-61 e\right)+3 d^2 \left(8 e^2-6 e f x^2+5 f^2 x^4\right)\right)\right)-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \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(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)}{105 f^4 \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}}","\frac{\sqrt{e} \sqrt{c+d x^2} \left(7 a f \left(15 c^2 f^2-11 c d e f+4 d^2 e^2\right)-b e \left(45 c^2 f^2-61 c d e f+24 d^2 e^2\right)\right) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{105 f^{7/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{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,"(Sqrt[d/c]*f*x*(c + d*x^2)*(e + f*x^2)*(7*a*d*f*(-4*d*e + 11*c*f + 3*d*f*x^2) + b*(45*c^2*f^2 + c*d*f*(-61*e + 45*f*x^2) + 3*d^2*(8*e^2 - 6*e*f*x^2 + 5*f^2*x^4))) - I*e*(7*a*d*f*(8*d^2*e^2 - 23*c*d*e*f + 23*c^2*f^2) + b*(-48*d^3*e^3 + 128*c*d^2*e^2*f - 103*c^2*d*e*f^2 + 15*c^3*f^3))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + I*(-(d*e) + c*f)*(4*b*e*(12*d^2*e^2 - 26*c*d*e*f + 15*c^2*f^2) - 7*a*f*(8*d^2*e^2 - 19*c*d*e*f + 15*c^2*f^2))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(105*Sqrt[d/c]*f^4*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
36,1,279,396,0.9836591,"\int \frac{\left(a+b x^2\right) \left(c+d x^2\right)^{3/2}}{\sqrt{e+f x^2}} \, dx","Integrate[((a + b*x^2)*(c + d*x^2)^(3/2))/Sqrt[e + f*x^2],x]","\frac{-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left(b \left(3 c^2 f^2-13 c d e f+8 d^2 e^2\right)-10 a d f (d e-2 c f)\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+f x \sqrt{\frac{d}{c}} \left(c+d x^2\right) \left(e+f x^2\right) \left(5 a d f+b \left(6 c f-4 d e+3 d f x^2\right)\right)+i \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) (5 a f (2 d e-3 c f)+b e (9 c f-8 d e)) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)}{15 f^3 \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}}","-\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,"(Sqrt[d/c]*f*x*(c + d*x^2)*(e + f*x^2)*(5*a*d*f + b*(-4*d*e + 6*c*f + 3*d*f*x^2)) - I*e*(-10*a*d*f*(d*e - 2*c*f) + b*(8*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + I*(-(d*e) + c*f)*(5*a*f*(2*d*e - 3*c*f) + b*e*(-8*d*e + 9*c*f))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(15*Sqrt[d/c]*f^3*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
37,1,215,282,0.5120231,"\int \frac{\left(a+b x^2\right) \sqrt{c+d x^2}}{\sqrt{e+f x^2}} \, dx","Integrate[((a + b*x^2)*Sqrt[c + d*x^2])/Sqrt[e + f*x^2],x]","\frac{i \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (2 b e-3 a f) (c f-d e) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (3 a d f+b c f-2 b d e) E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+b f x \sqrt{\frac{d}{c}} \left(c+d x^2\right) \left(e+f x^2\right)}{3 f^2 \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}}","-\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,"(b*Sqrt[d/c]*f*x*(c + d*x^2)*(e + f*x^2) - I*e*(-2*b*d*e + b*c*f + 3*a*d*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + I*(2*b*e - 3*a*f)*(-(d*e) + c*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(3*Sqrt[d/c]*f^2*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
38,1,131,206,0.2221668,"\int \frac{a+b x^2}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx","Integrate[(a + b*x^2)/(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]","-\frac{i \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left((a f-b e) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+b e E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)\right)}{f \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}}","\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,"((-I)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*(b*e*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + (-(b*e) + a*f)*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(Sqrt[d/c]*f*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
39,1,206,209,0.7588406,"\int \frac{a+b x^2}{\left(c+d x^2\right)^{3/2} \sqrt{e+f x^2}} \, dx","Integrate[(a + b*x^2)/((c + d*x^2)^(3/2)*Sqrt[e + f*x^2]),x]","\frac{\sqrt{\frac{d}{c}} \left(x \sqrt{\frac{d}{c}} \left(e+f x^2\right) (b c-a d)+i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b c-a d) E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-i a \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)\right)}{d \sqrt{c+d x^2} \sqrt{e+f x^2} (c f-d e)}","\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,"(Sqrt[d/c]*(Sqrt[d/c]*(b*c - a*d)*x*(e + f*x^2) + I*(b*c - a*d)*e*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*a*(-(d*e) + c*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(d*(-(d*e) + c*f)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
40,1,302,284,1.2332715,"\int \frac{a+b x^2}{\left(c+d x^2\right)^{5/2} \sqrt{e+f x^2}} \, dx","Integrate[(a + b*x^2)/((c + d*x^2)^(5/2)*Sqrt[e + f*x^2]),x]","\frac{x \sqrt{\frac{d}{c}} \left(e+f x^2\right) \left(a d \left(-5 c^2 f+c d \left(3 e-4 f x^2\right)+2 d^2 e x^2\right)+b c \left(2 c^2 f+c d f x^2+d^2 e x^2\right)\right)+i \left(c+d x^2\right) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) (-3 a c f+2 a d e+b c e) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+i e \left(c+d x^2\right) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (2 a d (d e-2 c f)+b c (c f+d e)) E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)}{3 c^2 \sqrt{\frac{d}{c}} \left(c+d x^2\right)^{3/2} \sqrt{e+f x^2} (d e-c f)^2}","\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,"(Sqrt[d/c]*x*(e + f*x^2)*(b*c*(2*c^2*f + d^2*e*x^2 + c*d*f*x^2) + a*d*(-5*c^2*f + 2*d^2*e*x^2 + c*d*(3*e - 4*f*x^2))) + I*e*(2*a*d*(d*e - 2*c*f) + b*c*(d*e + c*f))*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + I*(-(d*e) + c*f)*(b*c*e + 2*a*d*e - 3*a*c*f)*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(3*c^2*Sqrt[d/c]*(d*e - c*f)^2*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2])","C",1
41,1,393,401,1.4025079,"\int \frac{a+b x^2}{\left(c+d x^2\right)^{7/2} \sqrt{e+f x^2}} \, dx","Integrate[(a + b*x^2)/((c + d*x^2)^(7/2)*Sqrt[e + f*x^2]),x]","\frac{-x \sqrt{\frac{d}{c}} \left(e+f x^2\right) \left(\left(c+d x^2\right)^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)+3 c^2 (b c-a d) (d e-c f)^2+c \left(c+d x^2\right) (c f-d e) (4 a d (d e-2 c f)+b c (3 c f+d e))\right)-i \sqrt{\frac{d x^2}{c}+1} \left(c+d x^2\right)^2 \sqrt{\frac{f x^2}{e}+1} \left((d e-c f) \left(a \left(15 c^2 f^2-19 c d e f+8 d^2 e^2\right)+2 b c e (d e-3 c f)\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+e \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(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)\right)}{15 c^3 \sqrt{\frac{d}{c}} \left(c+d x^2\right)^{5/2} \sqrt{e+f x^2} (d e-c f)^3}","\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,"(-(Sqrt[d/c]*x*(e + f*x^2)*(3*c^2*(b*c - a*d)*(d*e - c*f)^2 + c*(-(d*e) + c*f)*(4*a*d*(d*e - 2*c*f) + b*c*(d*e + 3*c*f))*(c + d*x^2) + (a*d*(-8*d^2*e^2 + 23*c*d*e*f - 23*c^2*f^2) + b*c*(-2*d^2*e^2 + 7*c*d*e*f + 3*c^2*f^2))*(c + d*x^2)^2)) - I*(c + d*x^2)^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*(e*(a*d*(-8*d^2*e^2 + 23*c*d*e*f - 23*c^2*f^2) + b*c*(-2*d^2*e^2 + 7*c*d*e*f + 3*c^2*f^2))*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + (d*e - c*f)*(2*b*c*e*(d*e - 3*c*f) + a*(8*d^2*e^2 - 19*c*d*e*f + 15*c^2*f^2))*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(15*c^3*Sqrt[d/c]*(d*e - c*f)^3*(c + d*x^2)^(5/2)*Sqrt[e + f*x^2])","C",1
42,1,369,501,1.2312634,"\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","Integrate[((a + b*x^2)*(c + d*x^2)^(5/2))/(e + f*x^2)^(3/2),x]","\frac{-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) \left(5 a d f (9 c f-8 d e)+b \left(15 c^2 f^2-64 c d e f+48 d^2 e^2\right)\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-i d e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left(2 b e \left(19 c^2 f^2-44 c d e f+24 d^2 e^2\right)-5 a f \left(3 c^2 f^2-13 c d e f+8 d^2 e^2\right)\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+f x \sqrt{\frac{d}{c}} \left(c+d x^2\right) \left(5 a f \left(3 c^2 f^2-6 c d e f+d^2 e \left(4 e+f x^2\right)\right)+b e \left(-15 c^2 f^2+c d f \left(41 e+11 f x^2\right)-3 d^2 \left(8 e^2+2 e f x^2-f^2 x^4\right)\right)\right)}{15 e f^4 \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}}","-\frac{\sqrt{e} \sqrt{c+d x^2} \left(10 a d f (2 d e-3 c f)-b \left(15 c^2 f^2-41 c d e f+24 d^2 e^2\right)\right) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 f^{7/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{\sqrt{c+d x^2} \left(5 a f \left(3 c^2 f^2-13 c d e f+8 d^2 e^2\right)-2 b e \left(19 c^2 f^2-44 c d e f+24 d^2 e^2\right)\right) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{15 \sqrt{e} f^{7/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{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,"(Sqrt[d/c]*f*x*(c + d*x^2)*(5*a*f*(-6*c*d*e*f + 3*c^2*f^2 + d^2*e*(4*e + f*x^2)) + b*e*(-15*c^2*f^2 + c*d*f*(41*e + 11*f*x^2) - 3*d^2*(8*e^2 + 2*e*f*x^2 - f^2*x^4))) - I*d*e*(-5*a*f*(8*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2) + 2*b*e*(24*d^2*e^2 - 44*c*d*e*f + 19*c^2*f^2))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*e*(-(d*e) + c*f)*(5*a*d*f*(-8*d*e + 9*c*f) + b*(48*d^2*e^2 - 64*c*d*e*f + 15*c^2*f^2))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(15*Sqrt[d/c]*e*f^4*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
43,1,260,358,0.8266482,"\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","Integrate[((a + b*x^2)*(c + d*x^2)^(3/2))/(e + f*x^2)^(3/2),x]","\frac{f x \sqrt{\frac{d}{c}} \left(c+d x^2\right) \left(3 a f (c f-d e)+b e \left(-3 c f+4 d e+d f x^2\right)\right)-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) (6 a d f+3 b c f-8 b d e) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-i d e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b e (7 c f-8 d e)-3 a f (c f-2 d e)) E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)}{3 e f^3 \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}}","-\frac{\sqrt{e} \sqrt{c+d x^2} (-3 a d f-3 b c f+4 b d e) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 f^{5/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{\sqrt{c+d x^2} (b e (8 d e-7 c f)-3 a f (2 d e-c f)) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 \sqrt{e} f^{5/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{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,"(Sqrt[d/c]*f*x*(c + d*x^2)*(3*a*f*(-(d*e) + c*f) + b*e*(4*d*e - 3*c*f + d*f*x^2)) - I*d*e*(-3*a*f*(-2*d*e + c*f) + b*e*(-8*d*e + 7*c*f))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*e*(-(d*e) + c*f)*(-8*b*d*e + 3*b*c*f + 6*a*d*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(3*Sqrt[d/c]*e*f^3*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
44,1,208,258,0.4056239,"\int \frac{\left(a+b x^2\right) \sqrt{c+d x^2}}{\left(e+f x^2\right)^{3/2}} \, dx","Integrate[((a + b*x^2)*Sqrt[c + d*x^2])/(e + f*x^2)^(3/2),x]","\frac{f x \sqrt{\frac{d}{c}} \left(c+d x^2\right) (a f-b e)-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (a d f+b c f-2 b d e) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-i d e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (2 b e-a f) E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)}{e f^2 \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}}","-\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,"(Sqrt[d/c]*f*(-(b*e) + a*f)*x*(c + d*x^2) - I*d*e*(2*b*e - a*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*e*(-2*b*d*e + b*c*f + a*d*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(Sqrt[d/c]*e*f^2*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
45,1,212,209,0.3843703,"\int \frac{a+b x^2}{\sqrt{c+d x^2} \left(e+f x^2\right)^{3/2}} \, dx","Integrate[(a + b*x^2)/(Sqrt[c + d*x^2]*(e + f*x^2)^(3/2)),x]","\frac{f x \sqrt{\frac{d}{c}} \left(c+d x^2\right) (a f-b e)-i d e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b e-a f) E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-i b e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)}{e f \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2} (c f-d e)}","\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,"(Sqrt[d/c]*f*(-(b*e) + a*f)*x*(c + d*x^2) - I*d*e*(b*e - a*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*b*e*(-(d*e) + c*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(Sqrt[d/c]*e*f*(-(d*e) + c*f)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
46,1,262,272,0.7878623,"\int \frac{a+b x^2}{\left(c+d x^2\right)^{3/2} \left(e+f x^2\right)^{3/2}} \, dx","Integrate[(a + b*x^2)/((c + d*x^2)^(3/2)*(e + f*x^2)^(3/2)),x]","\frac{\sqrt{\frac{d}{c}} \left(x \sqrt{\frac{d}{c}} \left(a \left(c^2 f^2+c d f^2 x^2+d^2 e \left(e+f x^2\right)\right)-b c e \left(c f+d \left(e+2 f x^2\right)\right)\right)-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b c-a d) (c f-d e) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-i d e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (2 b c e-a (c f+d e)) E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)\right)}{d e \sqrt{c+d x^2} \sqrt{e+f x^2} (d e-c f)^2}","-\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,"(Sqrt[d/c]*(Sqrt[d/c]*x*(a*(c^2*f^2 + c*d*f^2*x^2 + d^2*e*(e + f*x^2)) - b*c*e*(c*f + d*(e + 2*f*x^2))) - I*d*e*(2*b*c*e - a*(d*e + c*f))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*(b*c - a*d)*e*(-(d*e) + c*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(d*e*(d*e - c*f)^2*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
47,1,428,375,2.2193497,"\int \frac{a+b x^2}{\left(c+d x^2\right)^{5/2} \left(e+f x^2\right)^{3/2}} \, dx","Integrate[(a + b*x^2)/((c + d*x^2)^(5/2)*(e + f*x^2)^(3/2)),x]","\frac{-i d e \left(c+d x^2\right) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \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(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+x \sqrt{\frac{d}{c}} \left(a \left(3 c^4 f^3+6 c^3 d f^3 x^2+c^2 d^2 f \left(8 e^2+8 e f x^2+3 f^2 x^4\right)+c d^3 e \left(-3 e^2+4 e f x^2+7 f^2 x^4\right)-2 d^4 e^2 x^2 \left(e+f x^2\right)\right)-b c e \left(3 c^3 f^2+c^2 d f \left(5 e+11 f x^2\right)+c d^2 f x^2 \left(4 e+7 f x^2\right)+d^3 e x^2 \left(e+f x^2\right)\right)\right)-i e \left(c+d x^2\right) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) (2 a d (d e-3 c f)+b c (3 c f+d e)) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)}{3 c^2 e \sqrt{\frac{d}{c}} \left(c+d x^2\right)^{3/2} \sqrt{e+f x^2} (c f-d e)^3}","\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,"(Sqrt[d/c]*x*(-(b*c*e*(3*c^3*f^2 + d^3*e*x^2*(e + f*x^2) + c*d^2*f*x^2*(4*e + 7*f*x^2) + c^2*d*f*(5*e + 11*f*x^2))) + a*(3*c^4*f^3 + 6*c^3*d*f^3*x^2 - 2*d^4*e^2*x^2*(e + f*x^2) + c^2*d^2*f*(8*e^2 + 8*e*f*x^2 + 3*f^2*x^4) + c*d^3*e*(-3*e^2 + 4*e*f*x^2 + 7*f^2*x^4))) - I*d*e*(b*c*e*(d*e + 7*c*f) + a*(2*d^2*e^2 - 7*c*d*e*f - 3*c^2*f^2))*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*e*(-(d*e) + c*f)*(2*a*d*(d*e - 3*c*f) + b*c*(d*e + 3*c*f))*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(3*c^2*Sqrt[d/c]*e*(-(d*e) + c*f)^3*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2])","C",1
48,1,212,209,0.4368603,"\int \frac{e+f x^2}{\sqrt{a+b x^2} \left(c+d x^2\right)^{3/2}} \, dx","Integrate[(e + f*x^2)/(Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)),x]","\frac{d x \sqrt{\frac{b}{a}} \left(a+b x^2\right) (d e-c f)-i b c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (c f-d e) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)-i c f \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-b c) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{c d \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2} (a d-b c)}","\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,"(Sqrt[b/a]*d*(d*e - c*f)*x*(a + b*x^2) - I*b*c*(-(d*e) + c*f)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*c*(-(b*c) + a*d)*f*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(Sqrt[b/a]*c*d*(-(b*c) + a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])","C",1
49,1,220,247,0.7713728,"\int \frac{e+f x^2}{\sqrt{a-b x^2} \left(c+d x^2\right)^{3/2}} \, dx","Integrate[(e + f*x^2)/(Sqrt[a - b*x^2]*(c + d*x^2)^(3/2)),x]","\frac{d x \sqrt{-\frac{b}{a}} \left(a-b x^2\right) (d e-c f)+i b c \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} (c f-d e) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{b}{a}} x\right)|-\frac{a d}{b c}\right)-i c f \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} (a d+b c) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{b}{a}} x\right)|-\frac{a d}{b c}\right)}{c d \sqrt{-\frac{b}{a}} \sqrt{a-b x^2} \sqrt{c+d x^2} (a d+b c)}","\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,"(Sqrt[-(b/a)]*d*(d*e - c*f)*x*(a - b*x^2) + I*b*c*(-(d*e) + c*f)*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[-(b/a)]*x], -((a*d)/(b*c))] - I*c*(b*c + a*d)*f*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[-(b/a)]*x], -((a*d)/(b*c))])/(Sqrt[-(b/a)]*c*d*(b*c + a*d)*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])","C",1
50,1,213,237,0.4873369,"\int \frac{e+f x^2}{\sqrt{a+b x^2} \left(c-d x^2\right)^{3/2}} \, dx","Integrate[(e + f*x^2)/(Sqrt[a + b*x^2]*(c - d*x^2)^(3/2)),x]","\frac{d x \sqrt{\frac{b}{a}} \left(a+b x^2\right) (c f+d e)-i b c \sqrt{\frac{b x^2}{a}+1} \sqrt{1-\frac{d x^2}{c}} (c f+d e) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|-\frac{a d}{b c}\right)+i c f \sqrt{\frac{b x^2}{a}+1} \sqrt{1-\frac{d x^2}{c}} (a d+b c) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|-\frac{a d}{b c}\right)}{c d \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c-d x^2} (a d+b c)}","\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,"(Sqrt[b/a]*d*(d*e + c*f)*x*(a + b*x^2) - I*b*c*(d*e + c*f)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], -((a*d)/(b*c))] + I*c*(b*c + a*d)*f*Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], -((a*d)/(b*c))])/(Sqrt[b/a]*c*d*(b*c + a*d)*Sqrt[a + b*x^2]*Sqrt[c - d*x^2])","C",1
51,1,221,242,0.5511467,"\int \frac{e+f x^2}{\sqrt{a-b x^2} \left(c-d x^2\right)^{3/2}} \, dx","Integrate[(e + f*x^2)/(Sqrt[a - b*x^2]*(c - d*x^2)^(3/2)),x]","\frac{d x \sqrt{-\frac{b}{a}} \left(a-b x^2\right) (c f+d e)+i b c \sqrt{1-\frac{b x^2}{a}} \sqrt{1-\frac{d x^2}{c}} (c f+d e) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+i c f \sqrt{1-\frac{b x^2}{a}} \sqrt{1-\frac{d x^2}{c}} (a d-b c) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{c d \sqrt{-\frac{b}{a}} \sqrt{a-b x^2} \sqrt{c-d x^2} (a d-b c)}","-\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,"(Sqrt[-(b/a)]*d*(d*e + c*f)*x*(a - b*x^2) + I*b*c*(d*e + c*f)*Sqrt[1 - (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[-(b/a)]*x], (a*d)/(b*c)] + I*c*(-(b*c) + a*d)*f*Sqrt[1 - (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[-(b/a)]*x], (a*d)/(b*c)])/(Sqrt[-(b/a)]*c*d*(-(b*c) + a*d)*Sqrt[a - b*x^2]*Sqrt[c - d*x^2])","C",1
52,1,81,191,0.1575827,"\int \frac{a+b x^2}{\sqrt{2+d x^2} \sqrt{3+f x^2}} \, dx","Integrate[(a + b*x^2)/(Sqrt[2 + d*x^2]*Sqrt[3 + f*x^2]),x]","-\frac{i \left((a f-3 b) F\left(i \sinh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{2}}\right)|\frac{2 f}{3 d}\right)+3 b E\left(i \sinh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{2}}\right)|\frac{2 f}{3 d}\right)\right)}{\sqrt{3} \sqrt{d} f}","\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,"((-I)*(3*b*EllipticE[I*ArcSinh[(Sqrt[d]*x)/Sqrt[2]], (2*f)/(3*d)] + (-3*b + a*f)*EllipticF[I*ArcSinh[(Sqrt[d]*x)/Sqrt[2]], (2*f)/(3*d)]))/(Sqrt[3]*Sqrt[d]*f)","C",1
53,1,142,262,0.2225904,"\int \frac{\left(a+b x^2\right) \sqrt{2+d x^2}}{\sqrt{3+f x^2}} \, dx","Integrate[((a + b*x^2)*Sqrt[2 + d*x^2])/Sqrt[3 + f*x^2],x]","\frac{i \sqrt{3} (3 d-2 f) (a f-2 b) F\left(i \sinh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{2}}\right)|\frac{2 f}{3 d}\right)+i \sqrt{3} (-3 a d f+6 b d-2 b f) E\left(i \sinh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{2}}\right)|\frac{2 f}{3 d}\right)+b \sqrt{d} f x \sqrt{d x^2+2} \sqrt{f x^2+3}}{3 \sqrt{d} f^2}","-\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,"(b*Sqrt[d]*f*x*Sqrt[2 + d*x^2]*Sqrt[3 + f*x^2] + I*Sqrt[3]*(6*b*d - 2*b*f - 3*a*d*f)*EllipticE[I*ArcSinh[(Sqrt[d]*x)/Sqrt[2]], (2*f)/(3*d)] + I*Sqrt[3]*(3*d - 2*f)*(-2*b + a*f)*EllipticF[I*ArcSinh[(Sqrt[d]*x)/Sqrt[2]], (2*f)/(3*d)])/(3*Sqrt[d]*f^2)","C",1
54,1,186,356,0.3548267,"\int \left(a+b x^2\right) \sqrt{2+d x^2} \sqrt{3+f x^2} \, dx","Integrate[(a + b*x^2)*Sqrt[2 + d*x^2]*Sqrt[3 + f*x^2],x]","\frac{i \sqrt{3} \left(2 b \left(9 d^2-6 d f+4 f^2\right)-5 a d f (3 d+2 f)\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{2}}\right)|\frac{2 f}{3 d}\right)+\sqrt{d} f x \sqrt{d x^2+2} \sqrt{f x^2+3} \left(5 a d f+3 b d \left(f x^2+1\right)+2 b f\right)+i \sqrt{3} (3 d-2 f) (5 a d f-6 b d+2 b f) F\left(i \sinh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{2}}\right)|\frac{2 f}{3 d}\right)}{15 d^{3/2} f^2}","\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,"(Sqrt[d]*f*x*Sqrt[2 + d*x^2]*Sqrt[3 + f*x^2]*(2*b*f + 5*a*d*f + 3*b*d*(1 + f*x^2)) + I*Sqrt[3]*(-5*a*d*f*(3*d + 2*f) + 2*b*(9*d^2 - 6*d*f + 4*f^2))*EllipticE[I*ArcSinh[(Sqrt[d]*x)/Sqrt[2]], (2*f)/(3*d)] + I*Sqrt[3]*(3*d - 2*f)*(-6*b*d + 2*b*f + 5*a*d*f)*EllipticF[I*ArcSinh[(Sqrt[d]*x)/Sqrt[2]], (2*f)/(3*d)])/(15*d^(3/2)*f^2)","C",1
55,1,104,113,0.4376468,"\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","Integrate[(-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]","-2 i \sqrt{2} a \sqrt{\frac{c}{\sqrt{b^2-4 a c}-b}} E\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{\sqrt{b^2-4 a c}-b}} x\right)|\frac{b-\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}\right)","-\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,"(-2*I)*Sqrt[2]*a*Sqrt[c/(-b + Sqrt[b^2 - 4*a*c])]*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(-b + Sqrt[b^2 - 4*a*c])]*x], (b - Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c])]","C",1
56,1,203,526,0.4496829,"\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","Integrate[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]),x]","-\frac{i \left(\left(\sqrt{b^2-4 a c}+b\right) E\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b-\sqrt{b^2-4 a c}}} x\right)|\frac{b-\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}\right)-2 \sqrt{b^2-4 a c} F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b-\sqrt{b^2-4 a c}}} x\right)|\frac{b-\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}\right)\right)}{\sqrt{2} \sqrt{\frac{c}{b-\sqrt{b^2-4 a c}}}}","\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,"((-I)*((b + Sqrt[b^2 - 4*a*c])*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b - Sqrt[b^2 - 4*a*c])]*x], (b - Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c])] - 2*Sqrt[b^2 - 4*a*c]*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b - Sqrt[b^2 - 4*a*c])]*x], (b - Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c])]))/(Sqrt[2]*Sqrt[c/(b - Sqrt[b^2 - 4*a*c])])","C",1
57,1,124,128,0.2654935,"\int \frac{\left(a+b x^2\right) \sqrt{c+d x^2}}{e+f x^2} \, dx","Integrate[((a + b*x^2)*Sqrt[c + d*x^2])/(e + f*x^2),x]","\frac{\frac{\log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right) (2 a d f+b c f-2 b d e)}{\sqrt{d}}-\frac{2 (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}}+b f x \sqrt{c+d x^2}}{2 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 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*f*x*Sqrt[c + d*x^2] - (2*(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] + ((-2*b*d*e + b*c*f + 2*a*d*f)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/Sqrt[d])/(2*f^2)","A",1
58,1,265,304,0.3662687,"\int \frac{\left(a+b x^2\right)^3}{\left(c+d x^2\right) \sqrt{e+f x^2}} \, dx","Integrate[(a + b*x^2)^3/((c + d*x^2)*Sqrt[e + f*x^2]),x]","\frac{\frac{8 b \left(3 a^2 d^2-3 a b c d+b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e+f x^2}}\right)}{\sqrt{f}}+\frac{4 b^2 d e (b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e+f x^2}}\right)}{f^{3/2}}-\frac{4 b^2 d x \sqrt{e+f x^2} (b c-3 a d)}{f}+\frac{8 (a d-b c)^3 \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{3 b^3 d^2 e \left(e \tanh ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e+f x^2}}\right)-\sqrt{f} x \sqrt{e+f x^2}\right)}{f^{5/2}}+\frac{2 b^3 d^2 x^3 \sqrt{e+f x^2}}{f}}{8 d^3}","\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,"((-4*b^2*d*(b*c - 3*a*d)*x*Sqrt[e + f*x^2])/f + (2*b^3*d^2*x^3*Sqrt[e + f*x^2])/f + (8*(-(b*c) + a*d)^3*ArcTan[(Sqrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])])/(Sqrt[c]*Sqrt[d*e - c*f]) + (4*b^2*d*(b*c - 3*a*d)*e*ArcTanh[(Sqrt[f]*x)/Sqrt[e + f*x^2]])/f^(3/2) + (8*b*(b^2*c^2 - 3*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[f]*x)/Sqrt[e + f*x^2]])/Sqrt[f] + (3*b^3*d^2*e*(-(Sqrt[f]*x*Sqrt[e + f*x^2]) + e*ArcTanh[(Sqrt[f]*x)/Sqrt[e + f*x^2]]))/f^(5/2))/(8*d^3)","A",1
59,1,150,166,0.3386778,"\int \frac{\left(a+b x^2\right)^2}{\left(c+d x^2\right) \sqrt{e+f x^2}} \, dx","Integrate[(a + b*x^2)^2/((c + d*x^2)*Sqrt[e + f*x^2]),x]","\frac{\frac{2 f (b c-a d)^2 \tan ^{-1}\left(\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right)+b^2 \sqrt{c} d x \sqrt{e+f x^2} \sqrt{d e-c f}}{\sqrt{c} f \sqrt{d e-c f}}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e+f x^2}}\right) (-4 a d f+2 b c f+b d e)}{f^{3/2}}}{2 d^2}","\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*Sqrt[c]*d*Sqrt[d*e - c*f]*x*Sqrt[e + f*x^2] + 2*(b*c - a*d)^2*f*ArcTan[(Sqrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])])/(Sqrt[c]*f*Sqrt[d*e - c*f]) - (b*(b*d*e + 2*b*c*f - 4*a*d*f)*ArcTanh[(Sqrt[f]*x)/Sqrt[e + f*x^2]])/f^(3/2))/(2*d^2)","A",1
60,1,88,91,0.1161512,"\int \frac{a+b x^2}{\left(c+d x^2\right) \sqrt{e+f x^2}} \, dx","Integrate[(a + b*x^2)/((c + d*x^2)*Sqrt[e + f*x^2]),x]","\frac{\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} \sqrt{d e-c f}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e+f x^2}}\right)}{\sqrt{f}}}{d}","\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]*Sqrt[d*e - c*f]) + (b*ArcTanh[(Sqrt[f]*x)/Sqrt[e + f*x^2]])/Sqrt[f])/d","A",1
61,1,49,49,0.0114798,"\int \frac{1}{\left(c+d x^2\right) \sqrt{e+f x^2}} \, dx","Integrate[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
62,1,113,122,0.2103348,"\int \frac{1}{\left(a+b x^2\right) \left(c+d x^2\right) \sqrt{e+f x^2}} \, dx","Integrate[1/((a + b*x^2)*(c + d*x^2)*Sqrt[e + f*x^2]),x]","\frac{\frac{b \tan ^{-1}\left(\frac{x \sqrt{b e-a f}}{\sqrt{a} \sqrt{e+f x^2}}\right)}{\sqrt{a} \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} \sqrt{d e-c f}}}{b c-a d}","\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]*Sqrt[b*e - a*f]) - (d*ArcTan[(Sqrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])])/(Sqrt[c]*Sqrt[d*e - c*f]))/(b*c - a*d)","A",1
63,1,531,203,2.6393372,"\int \frac{1}{\left(a+b x^2\right)^2 \left(c+d x^2\right) \sqrt{e+f x^2}} \, dx","Integrate[1/((a + b*x^2)^2*(c + d*x^2)*Sqrt[e + f*x^2]),x]","\frac{\frac{b x \sqrt{e+f x^2} (b c-a d) \left(-30 f x^2 \sqrt{\frac{a x^2 \left(e+f x^2\right) (b e-a f)}{e^2 \left(a+b x^2\right)^2}}-45 e \sqrt{\frac{a x^2 \left(e+f x^2\right) (b e-a f)}{e^2 \left(a+b x^2\right)^2}}+16 f x^2 \sqrt{\frac{a \left(e+f x^2\right)}{e \left(a+b x^2\right)}} \left(\frac{x^2 (b e-a f)}{e \left(a+b x^2\right)}\right)^{5/2} \, _2F_1\left(2,3;\frac{7}{2};\frac{(b e-a f) x^2}{e \left(b x^2+a\right)}\right)+16 e \sqrt{\frac{a \left(e+f x^2\right)}{e \left(a+b x^2\right)}} \left(\frac{x^2 (b e-a f)}{e \left(a+b x^2\right)}\right)^{5/2} \, _2F_1\left(2,3;\frac{7}{2};\frac{(b e-a f) x^2}{e \left(b x^2+a\right)}\right)+30 f x^2 \sin ^{-1}\left(\sqrt{\frac{x^2 (b e-a f)}{e \left(a+b x^2\right)}}\right)+45 e \sin ^{-1}\left(\sqrt{\frac{x^2 (b e-a f)}{e \left(a+b x^2\right)}}\right)\right)}{e^2 \left(a+b x^2\right)^2 \left(\frac{x^2 (b e-a f)}{e \left(a+b x^2\right)}\right)^{3/2} \sqrt{\frac{a \left(e+f x^2\right)}{e \left(a+b x^2\right)}}}-\frac{30 b d \tan ^{-1}\left(\frac{x \sqrt{b e-a f}}{\sqrt{a} \sqrt{e+f x^2}}\right)}{\sqrt{a} \sqrt{b e-a f}}+\frac{30 d^2 \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}}}{30 (b c-a d)^2}","\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,"((-30*b*d*ArcTan[(Sqrt[b*e - a*f]*x)/(Sqrt[a]*Sqrt[e + f*x^2])])/(Sqrt[a]*Sqrt[b*e - a*f]) + (30*d^2*ArcTan[(Sqrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])])/(Sqrt[c]*Sqrt[d*e - c*f]) + (b*(b*c - a*d)*x*Sqrt[e + f*x^2]*(-45*e*Sqrt[(a*(b*e - a*f)*x^2*(e + f*x^2))/(e^2*(a + b*x^2)^2)] - 30*f*x^2*Sqrt[(a*(b*e - a*f)*x^2*(e + f*x^2))/(e^2*(a + b*x^2)^2)] + 45*e*ArcSin[Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))]] + 30*f*x^2*ArcSin[Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))]] + 16*e*(((b*e - a*f)*x^2)/(e*(a + b*x^2)))^(5/2)*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]*Hypergeometric2F1[2, 3, 7/2, ((b*e - a*f)*x^2)/(e*(a + b*x^2))] + 16*f*x^2*(((b*e - a*f)*x^2)/(e*(a + b*x^2)))^(5/2)*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]*Hypergeometric2F1[2, 3, 7/2, ((b*e - a*f)*x^2)/(e*(a + b*x^2))]))/(e^2*(((b*e - a*f)*x^2)/(e*(a + b*x^2)))^(3/2)*(a + b*x^2)^2*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]))/(30*(b*c - a*d)^2)","C",0
64,1,456,608,2.8024375,"\int \frac{\left(c+d x^2\right)^{5/2} \sqrt{e+f x^2}}{a+b x^2} \, dx","Integrate[((c + d*x^2)^(5/2)*Sqrt[e + f*x^2])/(a + b*x^2),x]","\frac{-i a b d e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \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(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-i a \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left(-15 a^3 d^3 f^3+45 a^2 b c d^2 f^3+5 a b^2 d f \left(-9 c^2 f^2-c d e f+d^2 e^2\right)+b^3 \left(15 c^3 f^3+11 c^2 d e f^2-13 c d^2 e^2 f+2 d^3 e^3\right)\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+f \left(a b^2 d x \sqrt{\frac{d}{c}} \left(c+d x^2\right) \left(e+f x^2\right) \left(-5 a d f+11 b c f+b d \left(e+3 f x^2\right)\right)-15 i f \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b c-a d)^3 (b e-a f) \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)\right)}{15 a b^4 f^2 \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}}","-\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,"((-I)*a*b*d*e*(15*a^2*d^2*f^2 - 5*a*b*d*f*(d*e + 7*c*f) + b^2*(-2*d^2*e^2 + 12*c*d*e*f + 23*c^2*f^2))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*a*(45*a^2*b*c*d^2*f^3 - 15*a^3*d^3*f^3 + 5*a*b^2*d*f*(d^2*e^2 - c*d*e*f - 9*c^2*f^2) + b^3*(2*d^3*e^3 - 13*c*d^2*e^2*f + 11*c^2*d*e*f^2 + 15*c^3*f^3))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + f*(a*b^2*d*Sqrt[d/c]*x*(c + d*x^2)*(e + f*x^2)*(11*b*c*f - 5*a*d*f + b*d*(e + 3*f*x^2)) - (15*I)*(b*c - a*d)^3*f*(b*e - a*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(15*a*b^4*Sqrt[d/c]*f^2*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
65,1,346,400,1.5840688,"\int \frac{\left(c+d x^2\right)^{3/2} \sqrt{e+f x^2}}{a+b x^2} \, dx","Integrate[((c + d*x^2)^(3/2)*Sqrt[e + f*x^2])/(a + b*x^2),x]","\frac{-i a \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left(3 a^2 d^2 f^2-6 a b c d f^2+b^2 \left(3 c^2 f^2+c d e f-d^2 e^2\right)\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+f \left(a b^2 d x \sqrt{\frac{d}{c}} \left(c+d x^2\right) \left(e+f x^2\right)-3 i \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b c-a d)^2 (b e-a f) \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)\right)-i a b d e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (-3 a d f+4 b c f+b d e) E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)}{3 a b^3 f \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}}","\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,"((-I)*a*b*d*e*(b*d*e + 4*b*c*f - 3*a*d*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*a*(-6*a*b*c*d*f^2 + 3*a^2*d^2*f^2 + b^2*(-(d^2*e^2) + c*d*e*f + 3*c^2*f^2))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + f*(a*b^2*d*Sqrt[d/c]*x*(c + d*x^2)*(e + f*x^2) - (3*I)*(b*c - a*d)^2*(b*e - a*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(3*a*b^3*Sqrt[d/c]*f*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
66,1,184,321,0.3866807,"\int \frac{\sqrt{c+d x^2} \sqrt{e+f x^2}}{a+b x^2} \, dx","Integrate[(Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(a + b*x^2),x]","-\frac{i \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left(a b d e E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+(b c-a d) \left((b e-a f) \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+a f F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)\right)\right)}{a b^2 \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}}","\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,"((-I)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*(a*b*d*e*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + (b*c - a*d)*(a*f*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + (b*e - a*f)*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])))/(a*b^2*Sqrt[d/c]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
67,1,143,102,0.2440582,"\int \frac{\sqrt{e+f x^2}}{\left(a+b x^2\right) \sqrt{c+d x^2}} \, dx","Integrate[Sqrt[e + f*x^2]/((a + b*x^2)*Sqrt[c + d*x^2]),x]","-\frac{i \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left((b e-a f) \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+a f F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)\right)}{a b \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}}","\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,"((-I)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*(a*f*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + (b*e - a*f)*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(a*b*Sqrt[d/c]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
68,1,347,209,0.8040581,"\int \frac{\sqrt{e+f x^2}}{\left(a+b x^2\right) \left(c+d x^2\right)^{3/2}} \, dx","Integrate[Sqrt[e + f*x^2]/((a + b*x^2)*(c + d*x^2)^(3/2)),x]","\frac{\sqrt{\frac{d}{c}} \left(i b c e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-i a c f \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+i a \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+i a d e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+a d e x \sqrt{\frac{d}{c}}+a d f x^3 \sqrt{\frac{d}{c}}\right)}{a d \sqrt{c+d x^2} \sqrt{e+f x^2} (a d-b c)}","\frac{b e^{3/2} \sqrt{c+d x^2} \Pi \left(1-\frac{b e}{a f};\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{a c \sqrt{f} \sqrt{e+f x^2} (b c-a d) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{\sqrt{d} \sqrt{e+f x^2} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{\sqrt{c} \sqrt{c+d x^2} (b c-a d) \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}",1,"(Sqrt[d/c]*(a*d*Sqrt[d/c]*e*x + a*d*Sqrt[d/c]*f*x^3 + I*a*d*e*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + I*a*(-(d*e) + c*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + I*b*c*e*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*a*c*f*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(a*d*(-(b*c) + a*d)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
69,1,427,401,3.6270787,"\int \frac{\sqrt{e+f x^2}}{\left(a+b x^2\right) \left(c+d x^2\right)^{5/2}} \, dx","Integrate[Sqrt[e + f*x^2]/((a + b*x^2)*(c + d*x^2)^(5/2)),x]","\frac{a c x \left(\frac{d}{c}\right)^{3/2} \left(e+f x^2\right) \left(a d \left(2 c^2 f-3 c d e+c d f x^2-2 d^2 e x^2\right)+b c \left(-5 c^2 f+6 c d e-4 c d f x^2+5 d^2 e x^2\right)\right)-3 i b c^2 \left(c+d x^2\right) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b e-a f) (c f-d e) \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-i a \left(c+d x^2\right) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) \left(2 a d^2 e+b c (3 c f-5 d e)\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-i a d e \left(c+d x^2\right) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (a d (2 d e-c f)+b c (4 c f-5 d e)) E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)}{3 a c^2 \sqrt{\frac{d}{c}} \left(c+d x^2\right)^{3/2} \sqrt{e+f x^2} (b c-a d)^2 (c f-d e)}","\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,"(a*c*(d/c)^(3/2)*x*(e + f*x^2)*(b*c*(6*c*d*e - 5*c^2*f + 5*d^2*e*x^2 - 4*c*d*f*x^2) + a*d*(-3*c*d*e + 2*c^2*f - 2*d^2*e*x^2 + c*d*f*x^2)) - I*a*d*e*(a*d*(2*d*e - c*f) + b*c*(-5*d*e + 4*c*f))*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*a*(-(d*e) + c*f)*(2*a*d^2*e + b*c*(-5*d*e + 3*c*f))*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - (3*I)*b*c^2*(b*e - a*f)*(-(d*e) + c*f)*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(3*a*c^2*Sqrt[d/c]*(b*c - a*d)^2*(-(d*e) + c*f)*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2])","C",1
70,1,584,630,3.0825538,"\int \frac{\sqrt{e+f x^2}}{\left(a+b x^2\right) \left(c+d x^2\right)^{7/2}} \, dx","Integrate[Sqrt[e + f*x^2]/((a + b*x^2)*(c + d*x^2)^(7/2)),x]","\frac{-a d x \sqrt{\frac{d}{c}} \left(e+f x^2\right) \left(\left(c+d x^2\right)^2 \left(a^2 d^2 \left(3 c^2 f^2-13 c d e f+8 d^2 e^2\right)+a b c d \left(-11 c^2 f^2+41 c d e f-26 d^2 e^2\right)+b^2 c^2 \left(23 c^2 f^2-58 c d e f+33 d^2 e^2\right)\right)+3 c^2 (b c-a d)^2 (d e-c f)^2+c \left(c+d x^2\right) (b c-a d) (c f-d e) (a d (4 d e-3 c f)+b c (8 c f-9 d e))\right)-i \sqrt{\frac{d x^2}{c}+1} \left(c+d x^2\right)^2 \sqrt{\frac{f x^2}{e}+1} \left(a d e \left(a^2 d^2 \left(3 c^2 f^2-13 c d e f+8 d^2 e^2\right)+a b c d \left(-11 c^2 f^2+41 c d e f-26 d^2 e^2\right)+b^2 c^2 \left(23 c^2 f^2-58 c d e f+33 d^2 e^2\right)\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-(d e-c f) \left(15 b^2 c^3 (b e-a f) (c f-d e) \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-a \left(a^2 d^3 e (9 c f-8 d e)+2 a b c d^2 e (13 d e-14 c f)+b^2 c^2 \left(-15 c^2 f^2+49 c d e f-33 d^2 e^2\right)\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)\right)\right)}{15 a c^3 \sqrt{\frac{d}{c}} \left(c+d x^2\right)^{5/2} \sqrt{e+f x^2} (b c-a d)^3 (d e-c f)^2}","\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,"(-(a*d*Sqrt[d/c]*x*(e + f*x^2)*(3*c^2*(b*c - a*d)^2*(d*e - c*f)^2 + c*(b*c - a*d)*(-(d*e) + c*f)*(a*d*(4*d*e - 3*c*f) + b*c*(-9*d*e + 8*c*f))*(c + d*x^2) + (a*b*c*d*(-26*d^2*e^2 + 41*c*d*e*f - 11*c^2*f^2) + a^2*d^2*(8*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2) + b^2*c^2*(33*d^2*e^2 - 58*c*d*e*f + 23*c^2*f^2))*(c + d*x^2)^2)) - I*(c + d*x^2)^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*(a*d*e*(a*b*c*d*(-26*d^2*e^2 + 41*c*d*e*f - 11*c^2*f^2) + a^2*d^2*(8*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2) + b^2*c^2*(33*d^2*e^2 - 58*c*d*e*f + 23*c^2*f^2))*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - (d*e - c*f)*(-(a*(2*a*b*c*d^2*e*(13*d*e - 14*c*f) + a^2*d^3*e*(-8*d*e + 9*c*f) + b^2*c^2*(-33*d^2*e^2 + 49*c*d*e*f - 15*c^2*f^2))*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]) + 15*b^2*c^3*(b*e - a*f)*(-(d*e) + c*f)*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])))/(15*a*c^3*Sqrt[d/c]*(b*c - a*d)^3*(d*e - c*f)^2*(c + d*x^2)^(5/2)*Sqrt[e + f*x^2])","C",1
71,1,445,659,2.6588973,"\int \frac{\left(c+d x^2\right)^{3/2} \left(e+f x^2\right)^{3/2}}{a+b x^2} \, dx","Integrate[((c + d*x^2)^(3/2)*(e + f*x^2)^(3/2))/(a + b*x^2),x]","\frac{-i a b e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \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(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-i a \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left(-15 a^3 d^2 f^3+15 a^2 b d f^2 (2 c f+d e)+5 a b^2 f \left(-3 c^2 f^2-7 c d e f+d^2 e^2\right)-3 b^3 e \left(-7 c^2 f^2+c d e f+d^2 e^2\right)\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+f \left(a b^2 x \sqrt{\frac{d}{c}} \left(c+d x^2\right) \left(e+f x^2\right) \left(3 b \left(2 c f+2 d e+d f x^2\right)-5 a d f\right)-15 i \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b c-a d)^2 (b e-a f)^2 \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)\right)}{15 a b^4 f \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}}","-\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,"((-I)*a*b*e*(15*a^2*d^2*f^2 - 20*a*b*d*f*(d*e + c*f) + 3*b^2*(d^2*e^2 + 9*c*d*e*f + c^2*f^2))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*a*(-15*a^3*d^2*f^3 + 15*a^2*b*d*f^2*(d*e + 2*c*f) - 3*b^3*e*(d^2*e^2 + c*d*e*f - 7*c^2*f^2) + 5*a*b^2*f*(d^2*e^2 - 7*c*d*e*f - 3*c^2*f^2))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + f*(a*b^2*Sqrt[d/c]*x*(c + d*x^2)*(e + f*x^2)*(-5*a*d*f + 3*b*(2*d*e + 2*c*f + d*f*x^2)) - (15*I)*(b*c - a*d)^2*(b*e - a*f)^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(15*a*b^4*Sqrt[d/c]*f*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
72,1,739,403,1.1481221,"\int \frac{\sqrt{c+d x^2} \left(e+f x^2\right)^{3/2}}{a+b x^2} \, dx","Integrate[(Sqrt[c + d*x^2]*(e + f*x^2)^(3/2))/(a + b*x^2),x]","\frac{3 i a^3 d f^2 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-i a \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left(3 a^2 d f^2-3 a b f (c f+d e)+b^2 e (4 c f-d e)\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-3 i a^2 b c f^2 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-6 i a^2 b d e f \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-3 i b^3 c e^2 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+3 i a b^2 d e^2 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+a b^2 d e f x^3 \sqrt{\frac{d}{c}}+6 i a b^2 c e f \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+a b^2 c e f x \sqrt{\frac{d}{c}}+a b^2 d f^2 x^5 \sqrt{\frac{d}{c}}+a b^2 c f^2 x^3 \sqrt{\frac{d}{c}}-i a b e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (-3 a d f+b c f+4 b d e) E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)}{3 a b^3 \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}}","\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,"(a*b^2*c*Sqrt[d/c]*e*f*x + a*b^2*d*Sqrt[d/c]*e*f*x^3 + a*b^2*c*Sqrt[d/c]*f^2*x^3 + a*b^2*d*Sqrt[d/c]*f^2*x^5 - I*a*b*e*(4*b*d*e + b*c*f - 3*a*d*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*a*(3*a^2*d*f^2 - 3*a*b*f*(d*e + c*f) + b^2*e*(-(d*e) + 4*c*f))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - (3*I)*b^3*c*e^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + (3*I)*a*b^2*d*e^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + (6*I)*a*b^2*c*e*f*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - (6*I)*a^2*b*d*e*f*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - (3*I)*a^2*b*c*f^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + (3*I)*a^3*d*f^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(3*a*b^3*Sqrt[d/c]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
73,1,184,328,0.3594107,"\int \frac{\left(e+f x^2\right)^{3/2}}{\left(a+b x^2\right) \sqrt{c+d x^2}} \, dx","Integrate[(e + f*x^2)^(3/2)/((a + b*x^2)*Sqrt[c + d*x^2]),x]","-\frac{i \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left(a b e f E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+(b e-a f) \left((b e-a f) \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+a f F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)\right)\right)}{a b^2 \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}}","\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,"((-I)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*(a*b*e*f*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + (b*e - a*f)*(a*f*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + (b*e - a*f)*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])))/(a*b^2*Sqrt[d/c]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
74,1,492,224,1.5178568,"\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","Integrate[(e + f*x^2)^(3/2)/((a + b*x^2)*(c + d*x^2)^(3/2)),x]","\frac{\sqrt{\frac{d}{c}} \left(i a^2 c f^2 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+i b^2 c e^2 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+a b d e^2 x \sqrt{\frac{d}{c}}+i a \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left(b e (2 c f-d e)-a c f^2\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+a b d e f x^3 \sqrt{\frac{d}{c}}-i a b e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-2 i a b c e f \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-a b c e f x \sqrt{\frac{d}{c}}-a b c f^2 x^3 \sqrt{\frac{d}{c}}\right)}{a b d \sqrt{c+d x^2} \sqrt{e+f x^2} (a d-b c)}","\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,"(Sqrt[d/c]*(a*b*d*Sqrt[d/c]*e^2*x - a*b*c*Sqrt[d/c]*e*f*x + a*b*d*Sqrt[d/c]*e*f*x^3 - a*b*c*Sqrt[d/c]*f^2*x^3 - I*a*b*e*(-(d*e) + c*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + I*a*(-(a*c*f^2) + b*e*(-(d*e) + 2*c*f))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + I*b^2*c*e^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - (2*I)*a*b*c*e*f*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + I*a^2*c*f^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(a*b*d*(-(b*c) + a*d)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
75,1,999,391,1.8637675,"\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","Integrate[(e + f*x^2)^(3/2)/((a + b*x^2)*(c + d*x^2)^(5/2)),x]","\frac{a b c^3 \left(\frac{d}{c}\right)^{3/2} f^2 x^5+2 a^2 c d^2 \sqrt{\frac{d}{c}} f^2 x^5+2 a^2 d^3 \sqrt{\frac{d}{c}} e f x^5-5 a b c d^2 \sqrt{\frac{d}{c}} e f x^5+2 a^2 d^3 \sqrt{\frac{d}{c}} e^2 x^3-5 a b c d^2 \sqrt{\frac{d}{c}} e^2 x^3+a^2 c^3 \left(\frac{d}{c}\right)^{3/2} f^2 x^3+2 a b c^3 \sqrt{\frac{d}{c}} f^2 x^3-5 a b c^3 \left(\frac{d}{c}\right)^{3/2} e f x^3+5 a^2 c d^2 \sqrt{\frac{d}{c}} e f x^3-3 i b^2 c^2 d e^2 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right) x^2-3 i a^2 c^2 d f^2 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right) x^2+6 i a b c^2 d e f \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right) x^2-6 a b c^3 \left(\frac{d}{c}\right)^{3/2} e^2 x+3 a^2 c d^2 \sqrt{\frac{d}{c}} e^2 x+a^2 c^3 \left(\frac{d}{c}\right)^{3/2} e f x+2 a b c^3 \sqrt{\frac{d}{c}} e f x+i a e (b c (c f-5 d e)+2 a d (d e+c f)) \left(d x^2+c\right) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-i a (c f-d e) (5 b c e-2 a d e-3 a c f) \left(d x^2+c\right) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-3 i b^2 c^3 e^2 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-3 i a^2 c^3 f^2 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+6 i a b c^3 e f \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)}{3 a c^2 \sqrt{\frac{d}{c}} (b c-a d)^2 \left(d x^2+c\right)^{3/2} \sqrt{f x^2+e}}","-\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,"(3*a^2*c*d^2*Sqrt[d/c]*e^2*x - 6*a*b*c^3*(d/c)^(3/2)*e^2*x + 2*a*b*c^3*Sqrt[d/c]*e*f*x + a^2*c^3*(d/c)^(3/2)*e*f*x - 5*a*b*c*d^2*Sqrt[d/c]*e^2*x^3 + 2*a^2*d^3*Sqrt[d/c]*e^2*x^3 + 5*a^2*c*d^2*Sqrt[d/c]*e*f*x^3 - 5*a*b*c^3*(d/c)^(3/2)*e*f*x^3 + 2*a*b*c^3*Sqrt[d/c]*f^2*x^3 + a^2*c^3*(d/c)^(3/2)*f^2*x^3 - 5*a*b*c*d^2*Sqrt[d/c]*e*f*x^5 + 2*a^2*d^3*Sqrt[d/c]*e*f*x^5 + 2*a^2*c*d^2*Sqrt[d/c]*f^2*x^5 + a*b*c^3*(d/c)^(3/2)*f^2*x^5 + I*a*e*(b*c*(-5*d*e + c*f) + 2*a*d*(d*e + c*f))*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*a*(-(d*e) + c*f)*(5*b*c*e - 2*a*d*e - 3*a*c*f)*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - (3*I)*b^2*c^3*e^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + (6*I)*a*b*c^3*e*f*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - (3*I)*a^2*c^3*f^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - (3*I)*b^2*c^2*d*e^2*x^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + (6*I)*a*b*c^2*d*e*f*x^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - (3*I)*a^2*c^2*d*f^2*x^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(3*a*c^2*Sqrt[d/c]*(b*c - a*d)^2*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2])","C",1
76,1,570,639,3.2389735,"\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","Integrate[(e + f*x^2)^(3/2)/((a + b*x^2)*(c + d*x^2)^(7/2)),x]","\frac{-a x \sqrt{\frac{d}{c}} \left(e+f x^2\right) \left(\left(c+d x^2\right)^2 \left(a^2 d^2 \left(-2 c^2 f^2-3 c d e f+8 d^2 e^2\right)+2 a b c d \left(7 c^2 f^2+3 c d e f-13 d^2 e^2\right)+3 b^2 c^2 \left(c^2 f^2-11 c d e f+11 d^2 e^2\right)\right)+3 c^2 (b c-a d)^2 (d e-c f)^2+c \left(c+d x^2\right) (b c-a d) (c f-d e) (2 a d (c f+2 d e)+3 b c (c f-3 d e))\right)+i \left(c+d x^2\right)^2 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left(a e \left(a^2 d^2 \left(2 c^2 f^2+3 c d e f-8 d^2 e^2\right)-2 a b c d \left(7 c^2 f^2+3 c d e f-13 d^2 e^2\right)-3 b^2 c^2 \left(c^2 f^2-11 c d e f+11 d^2 e^2\right)\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+(d e-c f) \left(a \left(a^2 d^2 e (c f+8 d e)+a b c \left(15 c^2 f^2-7 c d e f-26 d^2 e^2\right)+3 b^2 c^2 e (11 d e-8 c f)\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-15 b c^3 (b e-a f)^2 \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)\right)\right)}{15 a c^3 \sqrt{\frac{d}{c}} \left(c+d x^2\right)^{5/2} \sqrt{e+f x^2} (b c-a d)^3 (d e-c f)}","\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,"(-(a*Sqrt[d/c]*x*(e + f*x^2)*(3*c^2*(b*c - a*d)^2*(d*e - c*f)^2 + c*(b*c - a*d)*(-(d*e) + c*f)*(3*b*c*(-3*d*e + c*f) + 2*a*d*(2*d*e + c*f))*(c + d*x^2) + (a^2*d^2*(8*d^2*e^2 - 3*c*d*e*f - 2*c^2*f^2) + 3*b^2*c^2*(11*d^2*e^2 - 11*c*d*e*f + c^2*f^2) + 2*a*b*c*d*(-13*d^2*e^2 + 3*c*d*e*f + 7*c^2*f^2))*(c + d*x^2)^2)) + I*(c + d*x^2)^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*(a*e*(-3*b^2*c^2*(11*d^2*e^2 - 11*c*d*e*f + c^2*f^2) + a^2*d^2*(-8*d^2*e^2 + 3*c*d*e*f + 2*c^2*f^2) - 2*a*b*c*d*(-13*d^2*e^2 + 3*c*d*e*f + 7*c^2*f^2))*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + (d*e - c*f)*(a*(3*b^2*c^2*e*(11*d*e - 8*c*f) + a^2*d^2*e*(8*d*e + c*f) + a*b*c*(-26*d^2*e^2 - 7*c*d*e*f + 15*c^2*f^2))*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - 15*b*c^3*(b*e - a*f)^2*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])))/(15*a*c^3*Sqrt[d/c]*(b*c - a*d)^3*(d*e - c*f)*(c + d*x^2)^(5/2)*Sqrt[e + f*x^2])","C",1
77,1,350,621,1.6185197,"\int \frac{\left(c+d x^2\right)^{5/2}}{\left(a+b x^2\right) \sqrt{e+f x^2}} \, dx","Integrate[(c + d*x^2)^(5/2)/((a + b*x^2)*Sqrt[e + f*x^2]),x]","\frac{-i a d \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left(3 a^2 d^2 f^2+3 a b d f (d e-3 c f)+b^2 \left(9 c^2 f^2-8 c d e f+2 d^2 e^2\right)\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+f \left(a b^2 c d x \left(\frac{d}{c}\right)^{3/2} \left(c+d x^2\right) \left(e+f x^2\right)-3 i f \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b c-a d)^3 \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)\right)-i a b d^2 e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (-3 a d f+7 b c f-2 b d e) E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)}{3 a b^3 f^2 \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}}","\frac{c^{3/2} \sqrt{e+f x^2} (b c-a d)^2 \Pi \left(1-\frac{b c}{a d};\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{c f}{d e}\right)}{a b^2 \sqrt{d} e \sqrt{c+d x^2} \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}}}+\frac{d x \sqrt{c+d x^2} (b c-a d)}{b^2 \sqrt{e+f x^2}}+\frac{d \sqrt{e} \sqrt{c+d x^2} (b c-a d) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{b^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{d \sqrt{e} \sqrt{c+d x^2} (b c-a d) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{b^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{d^2 x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 b f}-\frac{d \sqrt{e} \sqrt{c+d x^2} (d e-3 c f) F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 b f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{2 d \sqrt{e} \sqrt{c+d x^2} (d e-2 c f) E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 b f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}-\frac{2 d x \sqrt{c+d x^2} (d e-2 c f)}{3 b f \sqrt{e+f x^2}}",1,"((-I)*a*b*d^2*e*(-2*b*d*e + 7*b*c*f - 3*a*d*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*a*d*(3*a^2*d^2*f^2 + 3*a*b*d*f*(d*e - 3*c*f) + b^2*(2*d^2*e^2 - 8*c*d*e*f + 9*c^2*f^2))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + f*(a*b^2*c*d*(d/c)^(3/2)*x*(c + d*x^2)*(e + f*x^2) - (3*I)*(b*c - a*d)^3*f*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(3*a*b^3*Sqrt[d/c]*f^2*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
78,1,197,319,0.4592913,"\int \frac{\left(c+d x^2\right)^{3/2}}{\left(a+b x^2\right) \sqrt{e+f x^2}} \, dx","Integrate[(c + d*x^2)^(3/2)/((a + b*x^2)*Sqrt[e + f*x^2]),x]","-\frac{i \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left(a b d^2 e E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-a d (a d f-2 b c f+b d e) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+f (b c-a d)^2 \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)\right)}{a b^2 f \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}}","\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,"((-I)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*(a*b*d^2*e*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - a*d*(b*d*e - 2*b*c*f + a*d*f)*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + (b*c - a*d)^2*f*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(a*b^2*Sqrt[d/c]*f*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
79,1,143,102,0.245112,"\int \frac{\sqrt{c+d x^2}}{\left(a+b x^2\right) \sqrt{e+f x^2}} \, dx","Integrate[Sqrt[c + d*x^2]/((a + b*x^2)*Sqrt[e + f*x^2]),x]","-\frac{i \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left((b c-a d) \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+a d F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)\right)}{a b \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}}","\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,"((-I)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*(a*d*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + (b*c - a*d)*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(a*b*Sqrt[d/c]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
80,1,101,100,0.2256868,"\int \frac{1}{\left(a+b x^2\right) \sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx","Integrate[1/((a + b*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]","-\frac{i \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)}{a \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}}","\frac{\sqrt{-c} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{-c}}\right)|\frac{c f}{d e}\right)}{a \sqrt{d} \sqrt{c+d x^2} \sqrt{e+f x^2}}",1,"((-I)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(a*Sqrt[d/c]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
81,1,365,344,0.9535235,"\int \frac{1}{\left(a+b x^2\right) \left(c+d x^2\right)^{3/2} \sqrt{e+f x^2}} \, dx","Integrate[1/((a + b*x^2)*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2]),x]","\frac{\sqrt{\frac{d}{c}} \left(-i b c^2 f \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+i b c d e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+i a d^2 e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+i a d \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+a c d e x \left(\frac{d}{c}\right)^{3/2}+a c d f x^3 \left(\frac{d}{c}\right)^{3/2}\right)}{a d \sqrt{c+d x^2} \sqrt{e+f x^2} (a d-b c) (d e-c f)}","\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,"(Sqrt[d/c]*(a*c*d*(d/c)^(3/2)*e*x + a*c*d*(d/c)^(3/2)*f*x^3 + I*a*d^2*e*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + I*a*d*(-(d*e) + c*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + I*b*c*d*e*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*b*c^2*f*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(a*d*(-(b*c) + a*d)*(d*e - c*f)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
82,1,433,435,4.1253906,"\int \frac{1}{\left(a+b x^2\right) \left(c+d x^2\right)^{5/2} \sqrt{e+f x^2}} \, dx","Integrate[1/((a + b*x^2)*(c + d*x^2)^(5/2)*Sqrt[e + f*x^2]),x]","\frac{-3 i b^2 c^2 \left(c+d x^2\right) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (d e-c f)^2 \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+a c d x \left(\frac{d}{c}\right)^{3/2} \left(e+f x^2\right) \left(a d \left(-5 c^2 f+c d \left(3 e-4 f x^2\right)+2 d^2 e x^2\right)+b c \left(8 c^2 f-6 c d e+7 c d f x^2-5 d^2 e x^2\right)\right)+i a d^2 e \left(c+d x^2\right) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (2 a d (d e-2 c f)+b c (7 c f-5 d e)) E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+i a d \left(c+d x^2\right) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) (a d (2 d e-3 c f)+b c (6 c f-5 d e)) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)}{3 a c^2 \sqrt{\frac{d}{c}} \left(c+d x^2\right)^{3/2} \sqrt{e+f x^2} (b c-a d)^2 (d e-c f)^2}","\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,"(a*c*d*(d/c)^(3/2)*x*(e + f*x^2)*(b*c*(-6*c*d*e + 8*c^2*f - 5*d^2*e*x^2 + 7*c*d*f*x^2) + a*d*(-5*c^2*f + 2*d^2*e*x^2 + c*d*(3*e - 4*f*x^2))) + I*a*d^2*e*(2*a*d*(d*e - 2*c*f) + b*c*(-5*d*e + 7*c*f))*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + I*a*d*(-(d*e) + c*f)*(a*d*(2*d*e - 3*c*f) + b*c*(-5*d*e + 6*c*f))*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - (3*I)*b^2*c^2*(d*e - c*f)^2*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(3*a*c^2*Sqrt[d/c]*(b*c - a*d)^2*(d*e - c*f)^2*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2])","C",1
83,1,352,980,1.6748615,"\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","Integrate[(c + d*x^2)^(5/2)/((a + b*x^2)*(e + f*x^2)^(3/2)),x]","\frac{-f \left(a b^2 x \sqrt{\frac{d}{c}} \left(c+d x^2\right) (d e-c f)^2+i e f \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b c-a d)^3 \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)\right)-i a b d e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left(b \left(c^2 f^2-2 c d e f+2 d^2 e^2\right)-a d^2 e f\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-i a d^2 e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b e-a f) (-a d f+3 b c f-2 b d e) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)}{a b^2 e f^2 \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2} (b e-a f)}","\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,"((-I)*a*b*d*e*(-(a*d^2*e*f) + b*(2*d^2*e^2 - 2*c*d*e*f + c^2*f^2))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*a*d^2*e*(b*e - a*f)*(-2*b*d*e + 3*b*c*f - a*d*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - f*(a*b^2*Sqrt[d/c]*(d*e - c*f)^2*x*(c + d*x^2) + I*(b*c - a*d)^3*e*f*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(a*b^2*Sqrt[d/c]*e*f^2*(b*e - a*f)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
84,1,304,223,1.109676,"\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","Integrate[(c + d*x^2)^(3/2)/((a + b*x^2)*(e + f*x^2)^(3/2)),x]","\frac{-i a d^2 e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b e-a f) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+a b f x \sqrt{\frac{d}{c}} \left(c+d x^2\right) (d e-c f)-i a b d e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-i e f \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b c-a d)^2 \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)}{a b e f \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2} (b e-a f)}","\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,"(a*b*Sqrt[d/c]*f*(d*e - c*f)*x*(c + d*x^2) - I*a*b*d*e*(-(d*e) + c*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*a*d^2*e*(b*e - a*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*(b*c - a*d)^2*e*f*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(a*b*Sqrt[d/c]*e*f*(b*e - a*f)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
85,1,207,209,0.6654453,"\int \frac{\sqrt{c+d x^2}}{\left(a+b x^2\right) \left(e+f x^2\right)^{3/2}} \, dx","Integrate[Sqrt[c + d*x^2]/((a + b*x^2)*(e + f*x^2)^(3/2)),x]","\frac{-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b c-a d) \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-i a d e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-a f x \sqrt{\frac{d}{c}} \left(c+d x^2\right)}{a e \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2} (b e-a f)}","\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,"(-(a*Sqrt[d/c]*f*x*(c + d*x^2)) - I*a*d*e*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*(b*c - a*d)*e*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(a*Sqrt[d/c]*e*(b*e - a*f)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
86,1,221,344,0.642902,"\int \frac{1}{\left(a+b x^2\right) \sqrt{c+d x^2} \left(e+f x^2\right)^{3/2}} \, dx","Integrate[1/((a + b*x^2)*Sqrt[c + d*x^2]*(e + f*x^2)^(3/2)),x]","\frac{-i b e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-i a d e f \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-a f^2 x \sqrt{\frac{d}{c}} \left(c+d x^2\right)}{a e \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2} (a f-b e) (d e-c f)}","\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,"(-(a*Sqrt[d/c]*f^2*x*(c + d*x^2)) - I*a*d*e*f*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*b*e*(-(d*e) + c*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(a*Sqrt[d/c]*e*(-(b*e) + a*f)*(d*e - c*f)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
87,1,418,539,4.3993275,"\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","Integrate[1/((a + b*x^2)*(c + d*x^2)^(3/2)*(e + f*x^2)^(3/2)),x]","\frac{i b^2 c e \sqrt{\frac{d}{c}} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (d e-c f)^2 \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+i a d e \sqrt{\frac{d}{c}} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left(b \left(c^2 f^2+d^2 e^2\right)-a d f (c f+d e)\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+\frac{a d x \left(b \left(c^3 f^3+c^2 d f^3 x^2+d^3 e^2 \left(e+f x^2\right)\right)-a d f \left(c^2 f^2+c d f^2 x^2+d^2 e \left(e+f x^2\right)\right)\right)}{c}+i a c d e \left(\frac{d}{c}\right)^{3/2} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b e-a f) (c f-d e) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)}{a d e \sqrt{c+d x^2} \sqrt{e+f x^2} (a d-b c) (b e-a f) (d e-c f)^2}","\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,"((a*d*x*(-(a*d*f*(c^2*f^2 + c*d*f^2*x^2 + d^2*e*(e + f*x^2))) + b*(c^3*f^3 + c^2*d*f^3*x^2 + d^3*e^2*(e + f*x^2))))/c + I*a*d*Sqrt[d/c]*e*(-(a*d*f*(d*e + c*f)) + b*(d^2*e^2 + c^2*f^2))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + I*a*c*d*(d/c)^(3/2)*e*(b*e - a*f)*(-(d*e) + c*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + I*b^2*c*Sqrt[d/c]*e*(d*e - c*f)^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(a*d*(-(b*c) + a*d)*e*(b*e - a*f)*(d*e - c*f)^2*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
88,1,1645,814,7.2726589,"\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","Integrate[1/((a + b*x^2)*(c + d*x^2)^(5/2)*(e + f*x^2)^(3/2)),x]","\frac{\frac{\sqrt{\frac{d}{c}} \left(-3 i b^3 \sqrt{\frac{d}{c}} e f^3 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right) c^7+9 i b^3 \left(\frac{d}{c}\right)^{3/2} e^2 f^2 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right) c^7-3 i b^3 \left(\frac{d}{c}\right)^{3/2} e f^3 x^2 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right) c^7+9 i b^3 \left(\frac{d}{c}\right)^{5/2} e^2 f^2 x^2 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right) c^7-9 i b^3 \left(\frac{d}{c}\right)^{5/2} e^3 f \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right) c^7-3 a b^2 d f^4 x c^6-6 a b^2 d^2 f^4 x^3 c^5+6 a^2 b d^2 f^4 x c^5-3 a b^2 d^3 f^4 x^5 c^4+12 a^2 b d^3 f^4 x^3 c^4-3 a^3 d^3 f^4 x c^4+3 i b^3 d^3 \sqrt{\frac{d}{c}} e^4 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right) c^4-9 i b^3 d^3 \sqrt{\frac{d}{c}} e^3 f x^2 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right) c^4+6 a^2 b d^4 f^4 x^5 c^3-6 a^3 d^4 f^4 x^3 c^3+11 a^2 b d^4 e f^3 x^3 c^3-11 a b^2 d^4 e^2 f^2 x^3 c^3+11 a^2 b d^4 e^2 f^2 x c^3-11 a b^2 d^4 e^3 f x c^3+3 i b^3 d^4 \sqrt{\frac{d}{c}} e^4 x^2 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right) c^3-3 a^3 d^5 f^4 x^5 c^2+10 a^2 b d^5 e f^3 x^5 c^2-10 a b^2 d^5 e^2 f^2 x^5 c^2-8 a^3 d^5 e f^3 x^3 c^2+12 a^2 b d^5 e^2 f^2 x^3 c^2-4 a b^2 d^5 e^3 f x^3 c^2+6 a b^2 d^5 e^4 x c^2-8 a^3 d^5 e^2 f^2 x c^2+2 a^2 b d^5 e^3 f x c^2-7 a^3 d^6 e f^3 x^5 c+2 a^2 b d^6 e^2 f^2 x^5 c+5 a b^2 d^6 e^3 f x^5 c+5 a b^2 d^6 e^4 x^3 c-4 a^3 d^6 e^2 f^2 x^3 c-a^2 b d^6 e^3 f x^3 c-3 a^2 b d^6 e^4 x c+3 a^3 d^6 e^3 f x c-i a d^2 \sqrt{\frac{d}{c}} e (b e-a f) (c f-d e) (2 a d (d e-3 c f)+b c (9 c f-5 d e)) \left(d x^2+c\right) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right) c+2 a^3 d^7 e^2 f^2 x^5-2 a^2 b d^7 e^3 f x^5-2 a^2 b d^7 e^4 x^3+2 a^3 d^7 e^3 f x^3\right)}{d}-i a d e \left(c \left(-5 d^3 e^3+10 c d^2 f e^2+3 c^3 f^3\right) b^2+2 a d (d e-3 c f) (d e+c f)^2 b+a^2 d^2 f \left(-2 d^2 e^2+7 c d f e+3 c^2 f^2\right)\right) \left(d x^2+c\right) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)}{3 a c^2 \sqrt{\frac{d}{c}} (b c-a d)^2 e (b e-a f) (c f-d e)^3 \left(d x^2+c\right)^{3/2} \sqrt{f x^2+e}}","\frac{e^{3/2} \sqrt{d x^2+c} \Pi \left(1-\frac{b e}{a f};\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right) b^4}{a c (b c-a d)^2 \sqrt{f} (b e-a f)^2 \sqrt{\frac{e \left(d x^2+c\right)}{c \left(f x^2+e\right)}} \sqrt{f x^2+e}}+\frac{f^{3/2} \sqrt{d x^2+c} E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right) b^2}{(b c-a d)^2 \sqrt{e} (b e-a f) (d e-c f) \sqrt{\frac{e \left(d x^2+c\right)}{c \left(f x^2+e\right)}} \sqrt{f x^2+e}}-\frac{\sqrt{e} \sqrt{f} (2 b d e-b c f-a d f) \sqrt{d x^2+c} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right) b^2}{c (b c-a d)^2 (b e-a f)^2 (d e-c f) \sqrt{\frac{e \left(d x^2+c\right)}{c \left(f x^2+e\right)}} \sqrt{f x^2+e}}-\frac{d \sqrt{f} \left(b c \left(5 d^2 e^2-7 c d f e-6 c^2 f^2\right)-a d \left(2 d^2 e^2-7 c d f e-3 c^2 f^2\right)\right) \sqrt{d x^2+c} E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 c^2 (b c-a d)^2 \sqrt{e} (d e-c f)^3 \sqrt{\frac{e \left(d x^2+c\right)}{c \left(f x^2+e\right)}} \sqrt{f x^2+e}}+\frac{d^2 \sqrt{e} \sqrt{f} (b c (7 d e-15 c f)-a d (d e-9 c f)) \sqrt{d x^2+c} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{3 c^2 (b c-a d)^2 (d e-c f)^3 \sqrt{\frac{e \left(d x^2+c\right)}{c \left(f x^2+e\right)}} \sqrt{f x^2+e}}-\frac{d^2 (b c (5 d e-9 c f)-2 a d (d e-3 c f)) x}{3 c^2 (b c-a d)^2 (d e-c f)^2 \sqrt{d x^2+c} \sqrt{f x^2+e}}-\frac{d^2 x}{3 c (b c-a d) (d e-c f) \left(d x^2+c\right)^{3/2} \sqrt{f x^2+e}}",1,"((-I)*a*d*e*(2*a*b*d*(d*e - 3*c*f)*(d*e + c*f)^2 + a^2*d^2*f*(-2*d^2*e^2 + 7*c*d*e*f + 3*c^2*f^2) + b^2*c*(-5*d^3*e^3 + 10*c*d^2*e^2*f + 3*c^3*f^3))*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + (Sqrt[d/c]*(6*a*b^2*c^2*d^5*e^4*x - 3*a^2*b*c*d^6*e^4*x - 11*a*b^2*c^3*d^4*e^3*f*x + 2*a^2*b*c^2*d^5*e^3*f*x + 3*a^3*c*d^6*e^3*f*x + 11*a^2*b*c^3*d^4*e^2*f^2*x - 8*a^3*c^2*d^5*e^2*f^2*x - 3*a*b^2*c^6*d*f^4*x + 6*a^2*b*c^5*d^2*f^4*x - 3*a^3*c^4*d^3*f^4*x + 5*a*b^2*c*d^6*e^4*x^3 - 2*a^2*b*d^7*e^4*x^3 - 4*a*b^2*c^2*d^5*e^3*f*x^3 - a^2*b*c*d^6*e^3*f*x^3 + 2*a^3*d^7*e^3*f*x^3 - 11*a*b^2*c^3*d^4*e^2*f^2*x^3 + 12*a^2*b*c^2*d^5*e^2*f^2*x^3 - 4*a^3*c*d^6*e^2*f^2*x^3 + 11*a^2*b*c^3*d^4*e*f^3*x^3 - 8*a^3*c^2*d^5*e*f^3*x^3 - 6*a*b^2*c^5*d^2*f^4*x^3 + 12*a^2*b*c^4*d^3*f^4*x^3 - 6*a^3*c^3*d^4*f^4*x^3 + 5*a*b^2*c*d^6*e^3*f*x^5 - 2*a^2*b*d^7*e^3*f*x^5 - 10*a*b^2*c^2*d^5*e^2*f^2*x^5 + 2*a^2*b*c*d^6*e^2*f^2*x^5 + 2*a^3*d^7*e^2*f^2*x^5 + 10*a^2*b*c^2*d^5*e*f^3*x^5 - 7*a^3*c*d^6*e*f^3*x^5 - 3*a*b^2*c^4*d^3*f^4*x^5 + 6*a^2*b*c^3*d^4*f^4*x^5 - 3*a^3*c^2*d^5*f^4*x^5 - I*a*c*d^2*Sqrt[d/c]*e*(b*e - a*f)*(-(d*e) + c*f)*(2*a*d*(d*e - 3*c*f) + b*c*(-5*d*e + 9*c*f))*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + (3*I)*b^3*c^4*d^3*Sqrt[d/c]*e^4*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - (9*I)*b^3*c^7*(d/c)^(5/2)*e^3*f*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + (9*I)*b^3*c^7*(d/c)^(3/2)*e^2*f^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - (3*I)*b^3*c^7*Sqrt[d/c]*e*f^3*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + (3*I)*b^3*c^3*d^4*Sqrt[d/c]*e^4*x^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - (9*I)*b^3*c^4*d^3*Sqrt[d/c]*e^3*f*x^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + (9*I)*b^3*c^7*(d/c)^(5/2)*e^2*f^2*x^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - (3*I)*b^3*c^7*(d/c)^(3/2)*e*f^3*x^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/d)/(3*a*c^2*Sqrt[d/c]*(b*c - a*d)^2*e*(b*e - a*f)*(-(d*e) + c*f)^3*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2])","C",1
89,1,204,242,0.5076481,"\int \frac{\left(1+x^2\right)^{3/2} \sqrt{2+x^2}}{a+b x^2} \, dx","Integrate[((1 + x^2)^(3/2)*Sqrt[2 + x^2])/(a + b*x^2),x]","\frac{3 i a^3 \Pi \left(\frac{2 b}{a};\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)-i a \left(3 a^2-9 a b+7 b^2\right) F\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)-12 i a^2 b \Pi \left(\frac{2 b}{a};\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)-6 i b^3 \Pi \left(\frac{2 b}{a};\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+a b^2 x \sqrt{x^2+1} \sqrt{x^2+2}+15 i a b^2 \Pi \left(\frac{2 b}{a};\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+3 i a b (a-2 b) E\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)}{3 a b^3}","-\frac{\sqrt{x^2+2} x (a-2 b)}{b^2 \sqrt{x^2+1}}-\frac{\sqrt{x^2+2} (3 a-7 b) F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{3 \sqrt{2} b^2 \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}}}+\frac{\sqrt{2} \sqrt{x^2+2} (a-2 b) E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{b^2 \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}}}+\frac{\sqrt{x^2+2} (a-2 b) (a-b) \Pi \left(1-\frac{b}{a};\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{2} a b^2 \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}}}+\frac{\sqrt{x^2+1} \sqrt{x^2+2} x}{3 b}",1,"(a*b^2*x*Sqrt[1 + x^2]*Sqrt[2 + x^2] + (3*I)*a*(a - 2*b)*b*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - I*a*(3*a^2 - 9*a*b + 7*b^2)*EllipticF[I*ArcSinh[x/Sqrt[2]], 2] + (3*I)*a^3*EllipticPi[(2*b)/a, I*ArcSinh[x/Sqrt[2]], 2] - (12*I)*a^2*b*EllipticPi[(2*b)/a, I*ArcSinh[x/Sqrt[2]], 2] + (15*I)*a*b^2*EllipticPi[(2*b)/a, I*ArcSinh[x/Sqrt[2]], 2] - (6*I)*b^3*EllipticPi[(2*b)/a, I*ArcSinh[x/Sqrt[2]], 2])/(3*a*b^3)","C",1
90,1,71,192,0.2501351,"\int \frac{\sqrt{1+x^2} \sqrt{2+x^2}}{a+b x^2} \, dx","Integrate[(Sqrt[1 + x^2]*Sqrt[2 + x^2])/(a + b*x^2),x]","\frac{i \left((a-b) \left(a F\left(i \sinh ^{-1}(x)|\frac{1}{2}\right)-(a-2 b) \Pi \left(\frac{b}{a};i \sinh ^{-1}(x)|\frac{1}{2}\right)\right)-2 a b E\left(i \sinh ^{-1}(x)|\frac{1}{2}\right)\right)}{\sqrt{2} a b^2}","-\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,"(I*(-2*a*b*EllipticE[I*ArcSinh[x], 1/2] + (a - b)*(a*EllipticF[I*ArcSinh[x], 1/2] - (a - 2*b)*EllipticPi[b/a, I*ArcSinh[x], 1/2])))/(Sqrt[2]*a*b^2)","C",1
91,1,50,58,0.1760118,"\int \frac{\sqrt{2+x^2}}{\sqrt{1+x^2} \left(a+b x^2\right)} \, dx","Integrate[Sqrt[2 + x^2]/(Sqrt[1 + x^2]*(a + b*x^2)),x]","-\frac{i \left(a F\left(i \sinh ^{-1}(x)|\frac{1}{2}\right)-(a-2 b) \Pi \left(\frac{b}{a};i \sinh ^{-1}(x)|\frac{1}{2}\right)\right)}{\sqrt{2} a b}","\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,"((-I)*(a*EllipticF[I*ArcSinh[x], 1/2] - (a - 2*b)*EllipticPi[b/a, I*ArcSinh[x], 1/2]))/(Sqrt[2]*a*b)","C",1
92,1,122,121,0.4675936,"\int \frac{\sqrt{2+x^2}}{\left(1+x^2\right)^{3/2} \left(a+b x^2\right)} \, dx","Integrate[Sqrt[2 + x^2]/((1 + x^2)^(3/2)*(a + b*x^2)),x]","\frac{\frac{2 i \sqrt{2} b \Pi \left(\frac{b}{a};i \sinh ^{-1}(x)|\frac{1}{2}\right)}{a}-i \sqrt{2} \Pi \left(\frac{b}{a};i \sinh ^{-1}(x)|\frac{1}{2}\right)+\frac{2 \sqrt{x^2+2} x}{\sqrt{x^2+1}}-i \sqrt{2} F\left(i \sinh ^{-1}(x)|\frac{1}{2}\right)+2 i \sqrt{2} E\left(i \sinh ^{-1}(x)|\frac{1}{2}\right)}{2 a-2 b}","\frac{\sqrt{2} \sqrt{x^2+2} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}} (a-b)}-\frac{2 b \sqrt{x^2+1} \Pi \left(1-\frac{2 b}{a};\left.\tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-1\right)}{a \sqrt{\frac{x^2+1}{x^2+2}} \sqrt{x^2+2} (a-b)}",1,"((2*x*Sqrt[2 + x^2])/Sqrt[1 + x^2] + (2*I)*Sqrt[2]*EllipticE[I*ArcSinh[x], 1/2] - I*Sqrt[2]*EllipticF[I*ArcSinh[x], 1/2] - I*Sqrt[2]*EllipticPi[b/a, I*ArcSinh[x], 1/2] + ((2*I)*Sqrt[2]*b*EllipticPi[b/a, I*ArcSinh[x], 1/2])/a)/(2*a - 2*b)","C",1
93,1,357,215,0.4971478,"\int \frac{\sqrt{2+x^2}}{\left(1+x^2\right)^{5/2} \left(a+b x^2\right)} \, dx","Integrate[Sqrt[2 + x^2]/((1 + x^2)^(5/2)*(a + b*x^2)),x]","\frac{8 a^2 \sqrt{x^2+1} \sqrt{x^2+2} x+6 a^2 \sqrt{x^2+1} \sqrt{x^2+2} x^3-6 i \sqrt{2} b^2 x^4 \Pi \left(\frac{b}{a};i \sinh ^{-1}(x)|\frac{1}{2}\right)-12 i \sqrt{2} b^2 x^2 \Pi \left(\frac{b}{a};i \sinh ^{-1}(x)|\frac{1}{2}\right)-6 i \sqrt{2} b^2 \Pi \left(\frac{b}{a};i \sinh ^{-1}(x)|\frac{1}{2}\right)+3 i \sqrt{2} a b x^4 \Pi \left(\frac{b}{a};i \sinh ^{-1}(x)|\frac{1}{2}\right)-14 a b \sqrt{x^2+1} \sqrt{x^2+2} x-i \sqrt{2} a \left(x^2+1\right)^2 (4 a-7 b) F\left(i \sinh ^{-1}(x)|\frac{1}{2}\right)+6 i \sqrt{2} a \left(x^2+1\right)^2 (a-2 b) E\left(i \sinh ^{-1}(x)|\frac{1}{2}\right)+6 i \sqrt{2} a b x^2 \Pi \left(\frac{b}{a};i \sinh ^{-1}(x)|\frac{1}{2}\right)-12 a b \sqrt{x^2+1} \sqrt{x^2+2} x^3+3 i \sqrt{2} a b \Pi \left(\frac{b}{a};i \sinh ^{-1}(x)|\frac{1}{2}\right)}{6 a \left(x^2+1\right)^2 (a-b)^2}","\frac{2 b^2 \sqrt{x^2+1} \Pi \left(1-\frac{2 b}{a};\left.\tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-1\right)}{a \sqrt{\frac{x^2+1}{x^2+2}} \sqrt{x^2+2} (a-b)^2}+\frac{x \sqrt{x^2+2}}{3 \left(x^2+1\right)^{3/2} (a-b)}-\frac{\sqrt{2} \sqrt{x^2+2} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{3 \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}} (a-b)}+\frac{\sqrt{2} \sqrt{x^2+2} (a-2 b) E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}} (a-b)^2}",1,"(8*a^2*x*Sqrt[1 + x^2]*Sqrt[2 + x^2] - 14*a*b*x*Sqrt[1 + x^2]*Sqrt[2 + x^2] + 6*a^2*x^3*Sqrt[1 + x^2]*Sqrt[2 + x^2] - 12*a*b*x^3*Sqrt[1 + x^2]*Sqrt[2 + x^2] + (6*I)*Sqrt[2]*a*(a - 2*b)*(1 + x^2)^2*EllipticE[I*ArcSinh[x], 1/2] - I*Sqrt[2]*a*(4*a - 7*b)*(1 + x^2)^2*EllipticF[I*ArcSinh[x], 1/2] + (3*I)*Sqrt[2]*a*b*EllipticPi[b/a, I*ArcSinh[x], 1/2] - (6*I)*Sqrt[2]*b^2*EllipticPi[b/a, I*ArcSinh[x], 1/2] + (6*I)*Sqrt[2]*a*b*x^2*EllipticPi[b/a, I*ArcSinh[x], 1/2] - (12*I)*Sqrt[2]*b^2*x^2*EllipticPi[b/a, I*ArcSinh[x], 1/2] + (3*I)*Sqrt[2]*a*b*x^4*EllipticPi[b/a, I*ArcSinh[x], 1/2] - (6*I)*Sqrt[2]*b^2*x^4*EllipticPi[b/a, I*ArcSinh[x], 1/2])/(6*a*(a - b)^2*(1 + x^2)^2)","C",1
94,1,134,298,0.397917,"\int \frac{\sqrt{2+d x^2} \sqrt{3+f x^2}}{a+b x^2} \, dx","Integrate[(Sqrt[2 + d*x^2]*Sqrt[3 + f*x^2])/(a + b*x^2),x]","\frac{i \left((a d-2 b) \left((3 b-a f) \Pi \left(\frac{2 b}{a d};i \sinh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{2}}\right)|\frac{2 f}{3 d}\right)+a f F\left(i \sinh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{2}}\right)|\frac{2 f}{3 d}\right)\right)-3 a b d E\left(i \sinh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{2}}\right)|\frac{2 f}{3 d}\right)\right)}{\sqrt{3} a b^2 \sqrt{d}}","\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,"(I*(-3*a*b*d*EllipticE[I*ArcSinh[(Sqrt[d]*x)/Sqrt[2]], (2*f)/(3*d)] + (-2*b + a*d)*(a*f*EllipticF[I*ArcSinh[(Sqrt[d]*x)/Sqrt[2]], (2*f)/(3*d)] + (3*b - a*f)*EllipticPi[(2*b)/(a*d), I*ArcSinh[(Sqrt[d]*x)/Sqrt[2]], (2*f)/(3*d)])))/(Sqrt[3]*a*b^2*Sqrt[d])","C",1
95,1,94,93,0.2704028,"\int \frac{\sqrt{2+d x^2}}{\left(a+b x^2\right) \sqrt{3+f x^2}} \, dx","Integrate[Sqrt[2 + d*x^2]/((a + b*x^2)*Sqrt[3 + f*x^2]),x]","-\frac{i \left((2 b-a d) \Pi \left(\frac{2 b}{a d};i \sinh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{2}}\right)|\frac{2 f}{3 d}\right)+a d F\left(i \sinh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{2}}\right)|\frac{2 f}{3 d}\right)\right)}{\sqrt{3} a b \sqrt{d}}","\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,"((-I)*(a*d*EllipticF[I*ArcSinh[(Sqrt[d]*x)/Sqrt[2]], (2*f)/(3*d)] + (2*b - a*d)*EllipticPi[(2*b)/(a*d), I*ArcSinh[(Sqrt[d]*x)/Sqrt[2]], (2*f)/(3*d)]))/(Sqrt[3]*a*b*Sqrt[d])","C",1
96,1,49,49,0.0175625,"\int \frac{1}{\left(a+b x^2\right) \sqrt{2+d x^2} \sqrt{3+f x^2}} \, dx","Integrate[1/((a + b*x^2)*Sqrt[2 + d*x^2]*Sqrt[3 + f*x^2]),x]","\frac{\Pi \left(\frac{2 b}{a d};\sin ^{-1}\left(\frac{\sqrt{-d} x}{\sqrt{2}}\right)|\frac{2 f}{3 d}\right)}{\sqrt{3} a \sqrt{-d}}","\frac{\Pi \left(\frac{2 b}{a d};\sin ^{-1}\left(\frac{\sqrt{-d} x}{\sqrt{2}}\right)|\frac{2 f}{3 d}\right)}{\sqrt{3} a \sqrt{-d}}",1,"EllipticPi[(2*b)/(a*d), ArcSin[(Sqrt[-d]*x)/Sqrt[2]], (2*f)/(3*d)]/(Sqrt[3]*a*Sqrt[-d])","A",1
97,1,37,36,0.0579692,"\int \frac{\sqrt{1-x^2}}{\left(-1+x^2\right) \sqrt{a+b x^2}} \, dx","Integrate[Sqrt[1 - x^2]/((-1 + x^2)*Sqrt[a + b*x^2]),x]","-\frac{\sqrt{\frac{a+b x^2}{a}} F\left(\sin ^{-1}(x)|-\frac{b}{a}\right)}{\sqrt{a+b x^2}}","-\frac{\sqrt{\frac{b x^2}{a}+1} F\left(\sin ^{-1}(x)|-\frac{b}{a}\right)}{\sqrt{a+b x^2}}",1,"-((Sqrt[(a + b*x^2)/a]*EllipticF[ArcSin[x], -(b/a)])/Sqrt[a + b*x^2])","A",1
98,1,191,113,0.8084936,"\int \frac{a+b x^2}{\sqrt{c+d x^2} \left(e+f x^2\right)^2} \, dx","Integrate[(a + b*x^2)/(Sqrt[c + d*x^2]*(e + f*x^2)^2),x]","\frac{x (b e-a f) \left(f \left(c+d x^2\right)-\frac{\left(e+f x^2\right) (2 d e-c f) \tanh ^{-1}\left(\sqrt{\frac{x^2 (d e-c f)}{e \left(c+d x^2\right)}}\right)}{e \sqrt{\frac{x^2 (d e-c f)}{e \left(c+d x^2\right)}}}\right)}{2 e f \sqrt{c+d x^2} \left(e+f x^2\right) (d e-c f)}+\frac{b \tanh ^{-1}\left(\frac{x \sqrt{d e-c f}}{\sqrt{e} \sqrt{c+d x^2}}\right)}{\sqrt{e} f \sqrt{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*(f*(c + d*x^2) - ((2*d*e - c*f)*(e + f*x^2)*ArcTanh[Sqrt[((d*e - c*f)*x^2)/(e*(c + d*x^2))]])/(e*Sqrt[((d*e - c*f)*x^2)/(e*(c + d*x^2))])))/(2*e*f*(d*e - c*f)*Sqrt[c + d*x^2]*(e + f*x^2)) + (b*ArcTanh[(Sqrt[d*e - c*f]*x)/(Sqrt[e]*Sqrt[c + d*x^2])])/(Sqrt[e]*f*Sqrt[d*e - c*f])","A",1
99,1,422,359,2.8194418,"\int \frac{\sqrt{c-d x^2} \sqrt{e+f x^2}}{\left(a+b x^2\right)^2} \, dx","Integrate[(Sqrt[c - d*x^2]*Sqrt[e + f*x^2])/(a + b*x^2)^2,x]","\frac{-\frac{i c \sqrt{-\frac{d}{c}} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} (a f+b e) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{d}{c}} x\right)|-\frac{c f}{d e}\right)}{b^2}+\frac{i a c f \sqrt{-\frac{d}{c}} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \Pi \left(-\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{-\frac{d}{c}} x\right)|-\frac{c f}{d e}\right)}{b^2}+\frac{i d e \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \Pi \left(-\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{-\frac{d}{c}} x\right)|-\frac{c f}{d e}\right)}{a \left(-\frac{d}{c}\right)^{3/2}}+\frac{c e x}{a+b x^2}+\frac{c f x^3}{a+b x^2}-\frac{d e x^3}{a+b x^2}-\frac{d f x^5}{a+b x^2}+\frac{i c e \sqrt{-\frac{d}{c}} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} E\left(i \sinh ^{-1}\left(\sqrt{-\frac{d}{c}} x\right)|-\frac{c f}{d e}\right)}{b}}{2 a \sqrt{c-d x^2} \sqrt{e+f x^2}}","\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,"((c*e*x)/(a + b*x^2) - (d*e*x^3)/(a + b*x^2) + (c*f*x^3)/(a + b*x^2) - (d*f*x^5)/(a + b*x^2) + (I*c*Sqrt[-(d/c)]*e*Sqrt[1 - (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[-(d/c)]*x], -((c*f)/(d*e))])/b - (I*c*Sqrt[-(d/c)]*(b*e + a*f)*Sqrt[1 - (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[-(d/c)]*x], -((c*f)/(d*e))])/b^2 + (I*d*e*Sqrt[1 - (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[-((b*c)/(a*d)), I*ArcSinh[Sqrt[-(d/c)]*x], -((c*f)/(d*e))])/(a*(-(d/c))^(3/2)) + (I*a*c*Sqrt[-(d/c)]*f*Sqrt[1 - (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[-((b*c)/(a*d)), I*ArcSinh[Sqrt[-(d/c)]*x], -((c*f)/(d*e))])/b^2)/(2*a*Sqrt[c - d*x^2]*Sqrt[e + f*x^2])","C",1
100,1,401,381,2.1636194,"\int \frac{\sqrt{c+d x^2} \sqrt{e+f x^2}}{\left(a+b x^2\right)^2} \, dx","Integrate[(Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(a + b*x^2)^2,x]","\frac{-\frac{i c \sqrt{\frac{d}{c}} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (a f+b e) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)}{b^2}+\frac{i a c f \sqrt{\frac{d}{c}} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)}{b^2}-\frac{i c e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)}{a \sqrt{\frac{d}{c}}}+\frac{c e x}{a+b x^2}+\frac{c f x^3}{a+b x^2}+\frac{d e x^3}{a+b x^2}+\frac{d f x^5}{a+b x^2}+\frac{i c e \sqrt{\frac{d}{c}} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)}{b}}{2 a \sqrt{c+d x^2} \sqrt{e+f x^2}}","\frac{\sqrt{-c} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \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,"((c*e*x)/(a + b*x^2) + (d*e*x^3)/(a + b*x^2) + (c*f*x^3)/(a + b*x^2) + (d*f*x^5)/(a + b*x^2) + (I*c*Sqrt[d/c]*e*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/b - (I*c*Sqrt[d/c]*(b*e + a*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/b^2 - (I*c*e*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(a*Sqrt[d/c]) + (I*a*c*Sqrt[d/c]*f*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/b^2)/(2*a*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
101,1,617,426,6.0661716,"\int \frac{1}{\left(a+b x^2\right)^2 \sqrt{c-d x^2} \sqrt{e+f x^2}} \, dx","Integrate[1/((a + b*x^2)^2*Sqrt[c - d*x^2]*Sqrt[e + f*x^2]),x]","\frac{\frac{i b^2 c e \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \Pi \left(-\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{-\frac{d}{c}} x\right)|-\frac{c f}{d e}\right)}{a \sqrt{-\frac{d}{c}}}-\frac{b^2 c e x}{a+b x^2}-\frac{b^2 c f x^3}{a+b x^2}+\frac{b^2 d e x^3}{a+b x^2}+\frac{b^2 d f x^5}{a+b x^2}+i c \sqrt{-\frac{d}{c}} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} (b e-a f) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{d}{c}} x\right)|-\frac{c f}{d e}\right)-2 i b c e \sqrt{-\frac{d}{c}} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \Pi \left(-\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{-\frac{d}{c}} x\right)|-\frac{c f}{d e}\right)+3 i a c f \sqrt{-\frac{d}{c}} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \Pi \left(-\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{-\frac{d}{c}} x\right)|-\frac{c f}{d e}\right)+\frac{2 i b d f \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \Pi \left(-\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{-\frac{d}{c}} x\right)|-\frac{c f}{d e}\right)}{\left(-\frac{d}{c}\right)^{3/2}}-i b c e \sqrt{-\frac{d}{c}} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} E\left(i \sinh ^{-1}\left(\sqrt{-\frac{d}{c}} x\right)|-\frac{c f}{d e}\right)}{2 a \sqrt{c-d x^2} \sqrt{e+f x^2} (a d+b c) (a f-b e)}","\frac{\sqrt{c} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \left(-3 a^2 d f+a b (2 d e-2 c f)+b^2 c e\right) \Pi \left(-\frac{b c}{a d};\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|-\frac{c f}{d e}\right)}{2 a^2 \sqrt{d} \sqrt{c-d x^2} \sqrt{e+f x^2} (a d+b c) (b e-a f)}+\frac{b^2 x \sqrt{c-d x^2} \sqrt{e+f x^2}}{2 a \left(a+b x^2\right) (a d+b c) (b e-a f)}-\frac{\sqrt{c} \sqrt{d} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} F\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|-\frac{c f}{d e}\right)}{2 a \sqrt{c-d x^2} \sqrt{e+f x^2} (a d+b c)}+\frac{b \sqrt{c} \sqrt{d} \sqrt{1-\frac{d x^2}{c}} \sqrt{e+f x^2} E\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|-\frac{c f}{d e}\right)}{2 a \sqrt{c-d x^2} \sqrt{\frac{f x^2}{e}+1} (a d+b c) (b e-a f)}",1,"(-((b^2*c*e*x)/(a + b*x^2)) + (b^2*d*e*x^3)/(a + b*x^2) - (b^2*c*f*x^3)/(a + b*x^2) + (b^2*d*f*x^5)/(a + b*x^2) - I*b*c*Sqrt[-(d/c)]*e*Sqrt[1 - (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[-(d/c)]*x], -((c*f)/(d*e))] + I*c*Sqrt[-(d/c)]*(b*e - a*f)*Sqrt[1 - (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[-(d/c)]*x], -((c*f)/(d*e))] + (I*b^2*c*e*Sqrt[1 - (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[-((b*c)/(a*d)), I*ArcSinh[Sqrt[-(d/c)]*x], -((c*f)/(d*e))])/(a*Sqrt[-(d/c)]) - (2*I)*b*c*Sqrt[-(d/c)]*e*Sqrt[1 - (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[-((b*c)/(a*d)), I*ArcSinh[Sqrt[-(d/c)]*x], -((c*f)/(d*e))] + ((2*I)*b*d*f*Sqrt[1 - (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[-((b*c)/(a*d)), I*ArcSinh[Sqrt[-(d/c)]*x], -((c*f)/(d*e))])/(-(d/c))^(3/2) + (3*I)*a*c*Sqrt[-(d/c)]*f*Sqrt[1 - (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[-((b*c)/(a*d)), I*ArcSinh[Sqrt[-(d/c)]*x], -((c*f)/(d*e))])/(2*a*(b*c + a*d)*(-(b*e) + a*f)*Sqrt[c - d*x^2]*Sqrt[e + f*x^2])","C",1
102,1,587,485,2.8845542,"\int \frac{1}{\left(a+b x^2\right)^2 \sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx","Integrate[1/((a + b*x^2)^2*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]","\frac{-\frac{i b^2 c e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)}{a \sqrt{\frac{d}{c}}}+\frac{b^2 c e x}{a+b x^2}+\frac{b^2 c f x^3}{a+b x^2}+\frac{b^2 d e x^3}{a+b x^2}+\frac{b^2 d f x^5}{a+b x^2}-i c \sqrt{\frac{d}{c}} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b e-a f) F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+2 i b c e \sqrt{\frac{d}{c}} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)-3 i a c f \sqrt{\frac{d}{c}} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)+\frac{2 i b c f \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(\frac{b c}{a d};i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)}{\sqrt{\frac{d}{c}}}+i b c e \sqrt{\frac{d}{c}} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c}} x\right)|\frac{c f}{d e}\right)}{2 a \sqrt{c+d x^2} \sqrt{e+f x^2} (a d-b c) (a f-b e)}","\frac{\sqrt{-c} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left(3 a^2 d f-2 a b (c f+d e)+b^2 c e\right) \Pi \left(\frac{b c}{a d};\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{-c}}\right)|\frac{c f}{d e}\right)}{2 a^2 \sqrt{d} \sqrt{c+d x^2} \sqrt{e+f x^2} (b c-a d) (b e-a f)}+\frac{b^2 x \sqrt{c+d x^2} \sqrt{e+f x^2}}{2 a \left(a+b x^2\right) (b c-a d) (b e-a f)}-\frac{b f x \sqrt{c+d x^2}}{2 a \sqrt{e+f x^2} (b c-a d) (b e-a f)}-\frac{d \sqrt{e} \sqrt{f} \sqrt{c+d x^2} F\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{2 c \sqrt{e+f x^2} (b c-a d) (b e-a f) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}+\frac{b \sqrt{e} \sqrt{f} \sqrt{c+d x^2} E\left(\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{e}}\right)|1-\frac{d e}{c f}\right)}{2 a \sqrt{e+f x^2} (b c-a d) (b e-a f) \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}",1,"((b^2*c*e*x)/(a + b*x^2) + (b^2*d*e*x^3)/(a + b*x^2) + (b^2*c*f*x^3)/(a + b*x^2) + (b^2*d*f*x^5)/(a + b*x^2) + I*b*c*Sqrt[d/c]*e*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*c*Sqrt[d/c]*(b*e - a*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - (I*b^2*c*e*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(a*Sqrt[d/c]) + (2*I)*b*c*Sqrt[d/c]*e*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + ((2*I)*b*c*f*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/Sqrt[d/c] - (3*I)*a*c*Sqrt[d/c]*f*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(2*a*(-(b*c) + a*d)*(-(b*e) + a*f)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
103,0,0,37,0.7482785,"\int \frac{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}}{\sqrt{e+f x^2}} \, dx","Integrate[((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,"Integrate[((a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/Sqrt[e + f*x^2], x]","A",-1
104,1,503,545,1.4694631,"\int \frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{\sqrt{e+f x^2}} \, dx","Integrate[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/Sqrt[e + f*x^2],x]","\frac{\frac{\sqrt{e+f x^2} (b e-2 a f) (d e-c f) \sqrt{\frac{a \left(c+d x^2\right)}{c \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 e-a d e}{b c e-a c f}\right)}{\sqrt{e} f^2 \sqrt{b e-a f} \sqrt{\frac{a \left(e+f x^2\right)}{e \left(a+b x^2\right)}}}+\frac{e \sqrt{a+b x^2} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}} (a d f+b c f-b d e) \Pi \left(\frac{a f}{a f-b e};\sin ^{-1}\left(\frac{\sqrt{a f-b e} x}{\sqrt{a} \sqrt{f x^2+e}}\right)|\frac{a d e-a c f}{b c e-a c f}\right)}{\sqrt{a} f^2 \sqrt{a f-b e} \sqrt{\frac{e \left(a+b x^2\right)}{a \left(e+f x^2\right)}}}+\frac{x \sqrt{a+b x^2} \left(c+d x^2\right)}{\sqrt{e+f x^2}}-\frac{\sqrt{c} \sqrt{a+b x^2} \sqrt{c f-d e} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}} E\left(\sin ^{-1}\left(\frac{\sqrt{c f-d e} x}{\sqrt{c} \sqrt{f x^2+e}}\right)|\frac{b c e-a c f}{a d e-a c f}\right)}{f \sqrt{\frac{e \left(a+b x^2\right)}{a \left(e+f x^2\right)}}}}{2 \sqrt{c+d x^2}}","\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,"((x*Sqrt[a + b*x^2]*(c + d*x^2))/Sqrt[e + f*x^2] - (Sqrt[c]*Sqrt[-(d*e) + c*f]*Sqrt[a + b*x^2]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*EllipticE[ArcSin[(Sqrt[-(d*e) + c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])], (b*c*e - a*c*f)/(a*d*e - a*c*f)])/(f*Sqrt[(e*(a + b*x^2))/(a*(e + f*x^2))]) + ((b*e - 2*a*f)*(d*e - c*f)*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*Sqrt[e + f*x^2]*EllipticF[ArcSin[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])], (b*c*e - a*d*e)/(b*c*e - a*c*f)])/(Sqrt[e]*f^2*Sqrt[b*e - a*f]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]) + (e*(-(b*d*e) + b*c*f + a*d*f)*Sqrt[a + b*x^2]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*EllipticPi[(a*f)/(-(b*e) + a*f), ArcSin[(Sqrt[-(b*e) + a*f]*x)/(Sqrt[a]*Sqrt[e + f*x^2])], (a*d*e - a*c*f)/(b*c*e - a*c*f)])/(Sqrt[a]*f^2*Sqrt[-(b*e) + a*f]*Sqrt[(e*(a + b*x^2))/(a*(e + f*x^2))]))/(2*Sqrt[c + d*x^2])","A",1
105,1,162,163,0.1030463,"\int \frac{\sqrt{c+d x^2}}{\sqrt{a+b x^2} \sqrt{e+f x^2}} \, dx","Integrate[Sqrt[c + d*x^2]/(Sqrt[a + b*x^2]*Sqrt[e + f*x^2]),x]","\frac{c \sqrt{e} \sqrt{a+b x^2} \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}} \Pi \left(\frac{d e}{d e-c f};\sin ^{-1}\left(\frac{\sqrt{d e-c f} x}{\sqrt{e} \sqrt{d x^2+c}}\right)|\frac{(a d-b c) e}{a (d e-c f)}\right)}{a \sqrt{e+f x^2} \sqrt{d e-c f} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}","\frac{c \sqrt{e} \sqrt{a+b x^2} \sqrt{\frac{c \left(e+f x^2\right)}{e \left(c+d x^2\right)}} \Pi \left(\frac{d e}{d e-c f};\sin ^{-1}\left(\frac{\sqrt{d e-c f} x}{\sqrt{e} \sqrt{d x^2+c}}\right)|-\frac{(b c-a d) e}{a (d e-c f)}\right)}{a \sqrt{e+f x^2} \sqrt{d e-c f} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}",1,"(c*Sqrt[e]*Sqrt[a + b*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]*EllipticPi[(d*e)/(d*e - c*f), ArcSin[(Sqrt[d*e - c*f]*x)/(Sqrt[e]*Sqrt[c + d*x^2])], ((-(b*c) + a*d)*e)/(a*(d*e - c*f))])/(a*Sqrt[d*e - c*f]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[e + f*x^2])","A",1
106,1,148,148,0.1171452,"\int \frac{\sqrt{c+d x^2}}{\left(a+b x^2\right)^{3/2} \sqrt{e+f x^2}} \, dx","Integrate[Sqrt[c + d*x^2]/((a + b*x^2)^(3/2)*Sqrt[e + f*x^2]),x]","\frac{\sqrt{e} \sqrt{c+d x^2} \sqrt{\frac{a \left(e+f x^2\right)}{e \left(a+b x^2\right)}} E\left(\sin ^{-1}\left(\frac{\sqrt{b e-a f} x}{\sqrt{e} \sqrt{b x^2+a}}\right)|\frac{(b c-a d) e}{c (b e-a f)}\right)}{a \sqrt{e+f x^2} \sqrt{b e-a f} \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}}}","\frac{\sqrt{e} \sqrt{c+d x^2} \sqrt{\frac{a \left(e+f x^2\right)}{e \left(a+b x^2\right)}} E\left(\sin ^{-1}\left(\frac{\sqrt{b e-a f} x}{\sqrt{e} \sqrt{b x^2+a}}\right)|\frac{(b c-a d) e}{c (b e-a f)}\right)}{a \sqrt{e+f x^2} \sqrt{b e-a f} \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}}}",1,"(Sqrt[e]*Sqrt[c + d*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]*EllipticE[ArcSin[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])], ((b*c - a*d)*e)/(c*(b*e - a*f))])/(a*Sqrt[b*e - a*f]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*Sqrt[e + f*x^2])","A",1
107,0,0,37,0.9106054,"\int \frac{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2}}{\left(e+f x^2\right)^{3/2}} \, dx","Integrate[((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,"Integrate[((a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(e + f*x^2)^(3/2), x]","A",-1
108,0,0,484,0.8848307,"\int \frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{\left(e+f x^2\right)^{3/2}} \, dx","Integrate[(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,"Integrate[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(e + f*x^2)^(3/2), x]","F",-1
109,1,148,319,0.1176978,"\int \frac{\sqrt{c+d x^2}}{\sqrt{a+b x^2} \left(e+f x^2\right)^{3/2}} \, dx","Integrate[Sqrt[c + d*x^2]/(Sqrt[a + b*x^2]*(e + f*x^2)^(3/2)),x]","\frac{\sqrt{a} \sqrt{c+d x^2} \sqrt{\frac{e \left(a+b x^2\right)}{a \left(e+f x^2\right)}} E\left(\sin ^{-1}\left(\frac{\sqrt{a f-b e} x}{\sqrt{a} \sqrt{f x^2+e}}\right)|\frac{a (c f-d e)}{c (a f-b e)}\right)}{e \sqrt{a+b x^2} \sqrt{a f-b e} \sqrt{\frac{e \left(c+d x^2\right)}{c \left(e+f x^2\right)}}}","\frac{c^{3/2} \sqrt{a+b x^2} F\left(\tan ^{-1}\left(\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{f x^2+e}}\right)|-\frac{(b c-a d) e}{a (d e-c f)}\right)}{a e \sqrt{c+d x^2} \sqrt{d e-c f} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{x \sqrt{a+b x^2} (d e-c f)}{e \sqrt{c+d x^2} \sqrt{e+f x^2} (b e-a f)}-\frac{\sqrt{c} \sqrt{a+b x^2} \sqrt{d e-c f} E\left(\tan ^{-1}\left(\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{f x^2+e}}\right)|-\frac{(b c-a d) e}{a (d e-c f)}\right)}{e \sqrt{c+d x^2} (b e-a f) \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}",1,"(Sqrt[a]*Sqrt[c + d*x^2]*Sqrt[(e*(a + b*x^2))/(a*(e + f*x^2))]*EllipticE[ArcSin[(Sqrt[-(b*e) + a*f]*x)/(Sqrt[a]*Sqrt[e + f*x^2])], (a*(-(d*e) + c*f))/(c*(-(b*e) + a*f))])/(e*Sqrt[-(b*e) + a*f]*Sqrt[a + b*x^2]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))])","A",1
110,0,0,37,1.1128137,"\int \frac{\sqrt{c+d x^2}}{\left(a+b x^2\right)^{3/2} \left(e+f x^2\right)^{3/2}} \, dx","Integrate[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,"Integrate[Sqrt[c + d*x^2]/((a + b*x^2)^(3/2)*(e + f*x^2)^(3/2)), x]","A",-1
111,1,512,541,2.1726317,"\int \frac{\sqrt{c+d x^2} \sqrt{e+f x^2}}{\sqrt{a+b x^2}} \, dx","Integrate[(Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/Sqrt[a + b*x^2],x]","\frac{\sqrt{a+b x^2} \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}} \left(b^2 c x \left(e+f x^2\right) \sqrt{b c-a d} \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}}+\sqrt{e} \sqrt{a+b x^2} \sqrt{b c-a d} (2 b c-a d) \sqrt{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 e-a d e}{b c e-a c f}\right)-b c \sqrt{e} \sqrt{a+b x^2} \sqrt{b c-a d} \sqrt{b e-a f} \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 e-a d e}{b c e-a c f}\right)-a \sqrt{c} \sqrt{a+b 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{b c e-a c f}{b c e-a d e}\right)\right)}{2 a b^2 \sqrt{c+d x^2} \sqrt{e+f x^2} \sqrt{b c-a d}}","\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,"(Sqrt[a + b*x^2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*(b^2*c*Sqrt[b*c - a*d]*x*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*(e + f*x^2) - b*c*Sqrt[b*c - a*d]*Sqrt[e]*Sqrt[b*e - a*f]*Sqrt[a + b*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]*EllipticE[ArcSin[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])], (b*c*e - a*d*e)/(b*c*e - a*c*f)] + Sqrt[b*c - a*d]*(2*b*c - a*d)*Sqrt[e]*Sqrt[b*e - a*f]*Sqrt[a + b*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]*EllipticF[ArcSin[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])], (b*c*e - a*d*e)/(b*c*e - a*c*f)] - a*Sqrt[c]*(a*d*f - b*(d*e + c*f))*Sqrt[a + b*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]*EllipticPi[(b*c)/(b*c - a*d), ArcSin[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])], (b*c*e - a*c*f)/(b*c*e - a*d*e)]))/(2*a*b^2*Sqrt[b*c - a*d]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","A",1
112,0,0,37,0.1759204,"\int \frac{\left(a+b x^2\right)^{3/2}}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx","Integrate[(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,"Integrate[(a + b*x^2)^(3/2)/(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x]","A",-1
113,1,159,159,0.1002201,"\int \frac{\sqrt{a+b x^2}}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx","Integrate[Sqrt[a + b*x^2]/(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]","\frac{a \sqrt{c+d x^2} \sqrt{\frac{a \left(e+f x^2\right)}{e \left(a+b x^2\right)}} \Pi \left(\frac{b c}{b c-a d};\sin ^{-1}\left(\frac{\sqrt{b c-a d} x}{\sqrt{c} \sqrt{b x^2+a}}\right)|\frac{c (b e-a f)}{(b c-a d) e}\right)}{\sqrt{c} \sqrt{e+f x^2} \sqrt{b c-a d} \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}}}","\frac{a \sqrt{c+d x^2} \sqrt{\frac{a \left(e+f x^2\right)}{e \left(a+b x^2\right)}} \Pi \left(\frac{b c}{b c-a d};\sin ^{-1}\left(\frac{\sqrt{b c-a d} x}{\sqrt{c} \sqrt{b x^2+a}}\right)|\frac{c (b e-a f)}{(b c-a d) e}\right)}{\sqrt{c} \sqrt{e+f x^2} \sqrt{b c-a d} \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}}}",1,"(a*Sqrt[c + d*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]*EllipticPi[(b*c)/(b*c - a*d), ArcSin[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])], (c*(b*e - a*f))/((b*c - a*d)*e)])/(Sqrt[c]*Sqrt[b*c - a*d]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*Sqrt[e + f*x^2])","A",1
114,1,148,148,0.1207513,"\int \frac{1}{\sqrt{a+b x^2} \sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx","Integrate[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]","\frac{\sqrt{e} \sqrt{c+d x^2} \sqrt{\frac{a \left(e+f x^2\right)}{e \left(a+b x^2\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{b e-a f} x}{\sqrt{e} \sqrt{b x^2+a}}\right)|\frac{(b c-a d) e}{c (b e-a f)}\right)}{c \sqrt{e+f x^2} \sqrt{b e-a f} \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}}}","\frac{\sqrt{e} \sqrt{c+d x^2} \sqrt{\frac{a \left(e+f x^2\right)}{e \left(a+b x^2\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{b e-a f} x}{\sqrt{e} \sqrt{b x^2+a}}\right)|\frac{(b c-a d) e}{c (b e-a f)}\right)}{c \sqrt{e+f x^2} \sqrt{b e-a f} \sqrt{\frac{a \left(c+d x^2\right)}{c \left(a+b x^2\right)}}}",1,"(Sqrt[e]*Sqrt[c + d*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]*EllipticF[ArcSin[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])], ((b*c - a*d)*e)/(c*(b*e - a*f))])/(c*Sqrt[b*e - a*f]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*Sqrt[e + f*x^2])","A",1
115,0,0,37,0.9975443,"\int \frac{1}{\left(a+b x^2\right)^{3/2} \sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx","Integrate[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,"Integrate[1/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x]","A",-1