1,1,25,25,0.0023164,"\int x^2 \left(a x^2+b x^3+c x^4\right) \, dx","Integrate[x^2*(a*x^2 + b*x^3 + c*x^4),x]","\frac{a x^5}{5}+\frac{b x^6}{6}+\frac{c x^7}{7}","\frac{a x^5}{5}+\frac{b x^6}{6}+\frac{c x^7}{7}",1,"(a*x^5)/5 + (b*x^6)/6 + (c*x^7)/7","A",1
2,1,25,25,0.0016273,"\int x \left(a x^2+b x^3+c x^4\right) \, dx","Integrate[x*(a*x^2 + b*x^3 + c*x^4),x]","\frac{a x^4}{4}+\frac{b x^5}{5}+\frac{c x^6}{6}","\frac{a x^4}{4}+\frac{b x^5}{5}+\frac{c x^6}{6}",1,"(a*x^4)/4 + (b*x^5)/5 + (c*x^6)/6","A",1
3,1,25,25,0.000076,"\int \left(a x^2+b x^3+c x^4\right) \, dx","Integrate[a*x^2 + b*x^3 + c*x^4,x]","\frac{a x^3}{3}+\frac{b x^4}{4}+\frac{c x^5}{5}","\frac{a x^3}{3}+\frac{b x^4}{4}+\frac{c x^5}{5}",1,"(a*x^3)/3 + (b*x^4)/4 + (c*x^5)/5","A",1
4,1,25,25,0.001494,"\int \frac{a x^2+b x^3+c x^4}{x} \, dx","Integrate[(a*x^2 + b*x^3 + c*x^4)/x,x]","\frac{a x^2}{2}+\frac{b x^3}{3}+\frac{c x^4}{4}","\frac{a x^2}{2}+\frac{b x^3}{3}+\frac{c x^4}{4}",1,"(a*x^2)/2 + (b*x^3)/3 + (c*x^4)/4","A",1
5,1,20,20,0.001181,"\int \frac{a x^2+b x^3+c x^4}{x^2} \, dx","Integrate[(a*x^2 + b*x^3 + c*x^4)/x^2,x]","a x+\frac{b x^2}{2}+\frac{c x^3}{3}","a x+\frac{b x^2}{2}+\frac{c x^3}{3}",1,"a*x + (b*x^2)/2 + (c*x^3)/3","A",1
6,1,54,54,0.0075176,"\int x^2 \left(a x^2+b x^3+c x^4\right)^2 \, dx","Integrate[x^2*(a*x^2 + b*x^3 + c*x^4)^2,x]","\frac{a^2 x^7}{7}+\frac{1}{9} x^9 \left(2 a c+b^2\right)+\frac{1}{4} a b x^8+\frac{1}{5} b c x^{10}+\frac{c^2 x^{11}}{11}","\frac{a^2 x^7}{7}+\frac{1}{9} x^9 \left(2 a c+b^2\right)+\frac{1}{4} a b x^8+\frac{1}{5} b c x^{10}+\frac{c^2 x^{11}}{11}",1,"(a^2*x^7)/7 + (a*b*x^8)/4 + ((b^2 + 2*a*c)*x^9)/9 + (b*c*x^10)/5 + (c^2*x^11)/11","A",1
7,1,54,54,0.0070001,"\int x \left(a x^2+b x^3+c x^4\right)^2 \, dx","Integrate[x*(a*x^2 + b*x^3 + c*x^4)^2,x]","\frac{a^2 x^6}{6}+\frac{1}{8} x^8 \left(2 a c+b^2\right)+\frac{2}{7} a b x^7+\frac{2}{9} b c x^9+\frac{c^2 x^{10}}{10}","\frac{a^2 x^6}{6}+\frac{1}{8} x^8 \left(2 a c+b^2\right)+\frac{2}{7} a b x^7+\frac{2}{9} b c x^9+\frac{c^2 x^{10}}{10}",1,"(a^2*x^6)/6 + (2*a*b*x^7)/7 + ((b^2 + 2*a*c)*x^8)/8 + (2*b*c*x^9)/9 + (c^2*x^10)/10","A",1
8,1,54,54,0.0063083,"\int \left(a x^2+b x^3+c x^4\right)^2 \, dx","Integrate[(a*x^2 + b*x^3 + c*x^4)^2,x]","\frac{a^2 x^5}{5}+\frac{1}{7} x^7 \left(2 a c+b^2\right)+\frac{1}{3} a b x^6+\frac{1}{4} b c x^8+\frac{c^2 x^9}{9}","\frac{a^2 x^5}{5}+\frac{1}{7} x^7 \left(2 a c+b^2\right)+\frac{1}{3} a b x^6+\frac{1}{4} b c x^8+\frac{c^2 x^9}{9}",1,"(a^2*x^5)/5 + (a*b*x^6)/3 + ((b^2 + 2*a*c)*x^7)/7 + (b*c*x^8)/4 + (c^2*x^9)/9","A",1
9,1,54,54,0.007444,"\int \frac{\left(a x^2+b x^3+c x^4\right)^2}{x} \, dx","Integrate[(a*x^2 + b*x^3 + c*x^4)^2/x,x]","\frac{a^2 x^4}{4}+\frac{1}{6} x^6 \left(2 a c+b^2\right)+\frac{2}{5} a b x^5+\frac{2}{7} b c x^7+\frac{c^2 x^8}{8}","\frac{a^2 x^4}{4}+\frac{1}{6} x^6 \left(2 a c+b^2\right)+\frac{2}{5} a b x^5+\frac{2}{7} b c x^7+\frac{c^2 x^8}{8}",1,"(a^2*x^4)/4 + (2*a*b*x^5)/5 + ((b^2 + 2*a*c)*x^6)/6 + (2*b*c*x^7)/7 + (c^2*x^8)/8","A",1
10,1,54,54,0.0092996,"\int \frac{\left(a x^2+b x^3+c x^4\right)^2}{x^2} \, dx","Integrate[(a*x^2 + b*x^3 + c*x^4)^2/x^2,x]","\frac{a^2 x^3}{3}+\frac{1}{5} x^5 \left(2 a c+b^2\right)+\frac{1}{2} a b x^4+\frac{1}{3} b c x^6+\frac{c^2 x^7}{7}","\frac{a^2 x^3}{3}+\frac{1}{5} x^5 \left(2 a c+b^2\right)+\frac{1}{2} a b x^4+\frac{1}{3} b c x^6+\frac{c^2 x^7}{7}",1,"(a^2*x^3)/3 + (a*b*x^4)/2 + ((b^2 + 2*a*c)*x^5)/5 + (b*c*x^6)/3 + (c^2*x^7)/7","A",1
11,1,84,89,0.1090341,"\int \frac{x^5}{a x^2+b x^3+c x^4} \, dx","Integrate[x^5/(a*x^2 + b*x^3 + c*x^4),x]","\frac{\left(b^2-a c\right) \log (a+x (b+c x))-\frac{2 b \left(b^2-3 a c\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+c x (c x-2 b)}{2 c^3}","\frac{\left(b^2-a c\right) \log \left(a+b x+c x^2\right)}{2 c^3}+\frac{b \left(b^2-3 a c\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{c^3 \sqrt{b^2-4 a c}}-\frac{b x}{c^2}+\frac{x^2}{2 c}",1,"(c*x*(-2*b + c*x) - (2*b*(b^2 - 3*a*c)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + (b^2 - a*c)*Log[a + x*(b + c*x)])/(2*c^3)","A",1
12,1,73,70,0.0632303,"\int \frac{x^4}{a x^2+b x^3+c x^4} \, dx","Integrate[x^4/(a*x^2 + b*x^3 + c*x^4),x]","\frac{\left(b^2-2 a c\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{c^2 \sqrt{4 a c-b^2}}-\frac{b \log \left(a+b x+c x^2\right)}{2 c^2}+\frac{x}{c}","-\frac{\left(b^2-2 a c\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{c^2 \sqrt{b^2-4 a c}}-\frac{b \log \left(a+b x+c x^2\right)}{2 c^2}+\frac{x}{c}",1,"x/c + ((b^2 - 2*a*c)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(c^2*Sqrt[-b^2 + 4*a*c]) - (b*Log[a + b*x + c*x^2])/(2*c^2)","A",1
13,1,57,56,0.0298816,"\int \frac{x^3}{a x^2+b x^3+c x^4} \, dx","Integrate[x^3/(a*x^2 + b*x^3 + c*x^4),x]","\frac{\log (a+x (b+c x))-\frac{2 b \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}}{2 c}","\frac{b \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{c \sqrt{b^2-4 a c}}+\frac{\log \left(a+b x+c x^2\right)}{2 c}",1,"((-2*b*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + Log[a + x*(b + c*x)])/(2*c)","A",1
14,1,38,34,0.0064569,"\int \frac{x^2}{a x^2+b x^3+c x^4} \, dx","Integrate[x^2/(a*x^2 + b*x^3 + c*x^4),x]","\frac{2 \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}","-\frac{2 \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c}}",1,"(2*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c]","A",1
15,1,61,62,0.0651048,"\int \frac{x}{a x^2+b x^3+c x^4} \, dx","Integrate[x/(a*x^2 + b*x^3 + c*x^4),x]","-\frac{\frac{2 b \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+\log (a+x (b+c x))-2 \log (x)}{2 a}","\frac{b \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{a \sqrt{b^2-4 a c}}-\frac{\log \left(a+b x+c x^2\right)}{2 a}+\frac{\log (x)}{a}",1,"-1/2*((2*b*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] - 2*Log[x] + Log[a + x*(b + c*x)])/a","A",1
16,1,77,81,0.0805333,"\int \frac{1}{a x^2+b x^3+c x^4} \, dx","Integrate[(a*x^2 + b*x^3 + c*x^4)^(-1),x]","\frac{\frac{2 \left(b^2-2 a c\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+b \log (a+x (b+c x))-\frac{2 a}{x}-2 b \log (x)}{2 a^2}","-\frac{\left(b^2-2 a c\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{a^2 \sqrt{b^2-4 a c}}+\frac{b \log \left(a+b x+c x^2\right)}{2 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{a x}",1,"((-2*a)/x + (2*(b^2 - 2*a*c)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] - 2*b*Log[x] + b*Log[a + x*(b + c*x)])/(2*a^2)","A",1
17,1,102,104,0.1359929,"\int \frac{1}{x \left(a x^2+b x^3+c x^4\right)} \, dx","Integrate[1/(x*(a*x^2 + b*x^3 + c*x^4)),x]","\frac{-\frac{a^2}{x^2}+2 \log (x) \left(b^2-a c\right)+\left(a c-b^2\right) \log (a+x (b+c x))-\frac{2 b \left(b^2-3 a c\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+\frac{2 a b}{x}}{2 a^3}","-\frac{\left(b^2-a c\right) \log \left(a+b x+c x^2\right)}{2 a^3}+\frac{\log (x) \left(b^2-a c\right)}{a^3}+\frac{b \left(b^2-3 a c\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{a^3 \sqrt{b^2-4 a c}}+\frac{b}{a^2 x}-\frac{1}{2 a x^2}",1,"(-(a^2/x^2) + (2*a*b)/x - (2*b*(b^2 - 3*a*c)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + 2*(b^2 - a*c)*Log[x] + (-b^2 + a*c)*Log[a + x*(b + c*x)])/(2*a^3)","A",1
18,1,131,137,0.1044634,"\int \frac{1}{x^2 \left(a x^2+b x^3+c x^4\right)} \, dx","Integrate[1/(x^2*(a*x^2 + b*x^3 + c*x^4)),x]","\frac{-\frac{2 a^3}{x^3}+\frac{6 \left(2 a^2 c^2-4 a b^2 c+b^4\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+\frac{3 a^2 b}{x^2}-6 \log (x) \left(b^3-2 a b c\right)+3 \left(b^3-2 a b c\right) \log (a+x (b+c x))+\frac{6 a \left(a c-b^2\right)}{x}}{6 a^4}","\frac{b \left(b^2-2 a c\right) \log \left(a+b x+c x^2\right)}{2 a^4}-\frac{b \log (x) \left(b^2-2 a c\right)}{a^4}-\frac{b^2-a c}{a^3 x}+\frac{b}{2 a^2 x^2}-\frac{\left(2 a^2 c^2-4 a b^2 c+b^4\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{a^4 \sqrt{b^2-4 a c}}-\frac{1}{3 a x^3}",1,"((-2*a^3)/x^3 + (3*a^2*b)/x^2 + (6*a*(-b^2 + a*c))/x + (6*(b^4 - 4*a*b^2*c + 2*a^2*c^2)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] - 6*(b^3 - 2*a*b*c)*Log[x] + 3*(b^3 - 2*a*b*c)*Log[a + x*(b + c*x)])/(6*a^4)","A",1
19,1,132,150,0.1955833,"\int \frac{x^8}{\left(a x^2+b x^3+c x^4\right)^2} \, dx","Integrate[x^8/(a*x^2 + b*x^3 + c*x^4)^2,x]","\frac{-\frac{2 \left(6 a^2 c^2-6 a b^2 c+b^4\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{3/2}}+\frac{a^2 c (3 b-2 c x)-a b^2 (b-4 c x)+b^4 (-x)}{\left(b^2-4 a c\right) (a+x (b+c x))}-b \log (a+x (b+c x))+c x}{c^3}","-\frac{2 \left(6 a^2 c^2-6 a b^2 c+b^4\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{c^3 \left(b^2-4 a c\right)^{3/2}}+\frac{2 x \left(b^2-3 a c\right)}{c^2 \left(b^2-4 a c\right)}-\frac{b x^2}{c \left(b^2-4 a c\right)}+\frac{x^3 (2 a+b x)}{\left(b^2-4 a c\right) \left(a+b x+c x^2\right)}-\frac{b \log \left(a+b x+c x^2\right)}{c^3}",1,"(c*x + (-(b^4*x) - a*b^2*(b - 4*c*x) + a^2*c*(3*b - 2*c*x))/((b^2 - 4*a*c)*(a + x*(b + c*x))) - (2*(b^4 - 6*a*b^2*c + 6*a^2*c^2)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) - b*Log[a + x*(b + c*x)])/c^3","A",1
20,1,109,114,0.1460261,"\int \frac{x^7}{\left(a x^2+b x^3+c x^4\right)^2} \, dx","Integrate[x^7/(a*x^2 + b*x^3 + c*x^4)^2,x]","\frac{\frac{2 \left(-2 a^2 c+a b (b-3 c x)+b^3 x\right)}{\left(b^2-4 a c\right) (a+x (b+c x))}+\frac{2 b \left(b^2-6 a c\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{3/2}}+\log (a+x (b+c x))}{2 c^2}","\frac{b \left(b^2-6 a c\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{c^2 \left(b^2-4 a c\right)^{3/2}}+\frac{x^2 (2 a+b x)}{\left(b^2-4 a c\right) \left(a+b x+c x^2\right)}-\frac{b x}{c \left(b^2-4 a c\right)}+\frac{\log \left(a+b x+c x^2\right)}{2 c^2}",1,"((2*(-2*a^2*c + b^3*x + a*b*(b - 3*c*x)))/((b^2 - 4*a*c)*(a + x*(b + c*x))) + (2*b*(b^2 - 6*a*c)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + Log[a + x*(b + c*x)])/(2*c^2)","A",1
21,1,81,67,0.0910454,"\int \frac{x^6}{\left(a x^2+b x^3+c x^4\right)^2} \, dx","Integrate[x^6/(a*x^2 + b*x^3 + c*x^4)^2,x]","\frac{a (b-2 c x)+b^2 x}{c \left(4 a c-b^2\right) (a+x (b+c x))}+\frac{4 a \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{3/2}}","\frac{x (2 a+b x)}{\left(b^2-4 a c\right) \left(a+b x+c x^2\right)}+\frac{4 a \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/2}}",1,"(b^2*x + a*(b - 2*c*x))/(c*(-b^2 + 4*a*c)*(a + x*(b + c*x))) + (4*a*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2)","A",1
22,1,69,66,0.0671624,"\int \frac{x^5}{\left(a x^2+b x^3+c x^4\right)^2} \, dx","Integrate[x^5/(a*x^2 + b*x^3 + c*x^4)^2,x]","\frac{2 a+b x}{\left(b^2-4 a c\right) (a+x (b+c x))}-\frac{2 b \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{3/2}}","\frac{2 a+b x}{\left(b^2-4 a c\right) \left(a+b x+c x^2\right)}-\frac{2 b \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/2}}",1,"(2*a + b*x)/((b^2 - 4*a*c)*(a + x*(b + c*x))) - (2*b*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2)","A",1
23,1,70,66,0.0714999,"\int \frac{x^4}{\left(a x^2+b x^3+c x^4\right)^2} \, dx","Integrate[x^4/(a*x^2 + b*x^3 + c*x^4)^2,x]","-\frac{\frac{4 c \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+\frac{b+2 c x}{a+x (b+c x)}}{b^2-4 a c}","\frac{4 c \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/2}}-\frac{b+2 c x}{\left(b^2-4 a c\right) \left(a+b x+c x^2\right)}",1,"-(((b + 2*c*x)/(a + x*(b + c*x)) + (4*c*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(b^2 - 4*a*c))","A",1
24,1,107,108,0.1837975,"\int \frac{x^3}{\left(a x^2+b x^3+c x^4\right)^2} \, dx","Integrate[x^3/(a*x^2 + b*x^3 + c*x^4)^2,x]","\frac{\frac{2 a \left(-2 a c+b^2+b c x\right)}{\left(b^2-4 a c\right) (a+x (b+c x))}+\frac{2 b \left(b^2-6 a c\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{3/2}}-\log (a+x (b+c x))+2 \log (x)}{2 a^2}","\frac{b \left(b^2-6 a c\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{a^2 \left(b^2-4 a c\right)^{3/2}}-\frac{\log \left(a+b x+c x^2\right)}{2 a^2}+\frac{\log (x)}{a^2}+\frac{-2 a c+b^2+b c x}{a \left(b^2-4 a c\right) \left(a+b x+c x^2\right)}",1,"((2*a*(b^2 - 2*a*c + b*c*x))/((b^2 - 4*a*c)*(a + x*(b + c*x))) + (2*b*(b^2 - 6*a*c)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + 2*Log[x] - Log[a + x*(b + c*x)])/(2*a^2)","A",1
25,1,131,148,0.2643269,"\int \frac{x^2}{\left(a x^2+b x^3+c x^4\right)^2} \, dx","Integrate[x^2/(a*x^2 + b*x^3 + c*x^4)^2,x]","-\frac{\frac{2 \left(6 a^2 c^2-6 a b^2 c+b^4\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{3/2}}+\frac{a \left(-3 a b c-2 a c^2 x+b^3+b^2 c x\right)}{\left(b^2-4 a c\right) (a+x (b+c x))}-b \log (a+x (b+c x))+\frac{a}{x}+2 b \log (x)}{a^3}","\frac{b \log \left(a+b x+c x^2\right)}{a^3}-\frac{2 b \log (x)}{a^3}-\frac{2 \left(b^2-3 a c\right)}{a^2 x \left(b^2-4 a c\right)}-\frac{2 \left(6 a^2 c^2-6 a b^2 c+b^4\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{a^3 \left(b^2-4 a c\right)^{3/2}}+\frac{-2 a c+b^2+b c x}{a x \left(b^2-4 a c\right) \left(a+b x+c x^2\right)}",1,"-((a/x + (a*(b^3 - 3*a*b*c + b^2*c*x - 2*a*c^2*x))/((b^2 - 4*a*c)*(a + x*(b + c*x))) + (2*(b^4 - 6*a*b^2*c + 6*a^2*c^2)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + 2*b*Log[x] - b*Log[a + x*(b + c*x)])/a^3)","A",1
26,1,175,202,0.3434779,"\int \frac{x}{\left(a x^2+b x^3+c x^4\right)^2} \, dx","Integrate[x/(a*x^2 + b*x^3 + c*x^4)^2,x]","\frac{\frac{2 b \left(30 a^2 c^2-20 a b^2 c+3 b^4\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{3/2}}+\frac{2 a \left(2 a^2 c^2-4 a b^2 c-3 a b c^2 x+b^4+b^3 c x\right)}{\left(b^2-4 a c\right) (a+x (b+c x))}-\frac{a^2}{x^2}+2 \log (x) \left(3 b^2-2 a c\right)+\left(2 a c-3 b^2\right) \log (a+x (b+c x))+\frac{4 a b}{x}}{2 a^4}","-\frac{\left(3 b^2-2 a c\right) \log \left(a+b x+c x^2\right)}{2 a^4}+\frac{\log (x) \left(3 b^2-2 a c\right)}{a^4}+\frac{b \left(3 b^2-11 a c\right)}{a^3 x \left(b^2-4 a c\right)}-\frac{3 b^2-8 a c}{2 a^2 x^2 \left(b^2-4 a c\right)}+\frac{b \left(30 a^2 c^2-20 a b^2 c+3 b^4\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{a^4 \left(b^2-4 a c\right)^{3/2}}+\frac{-2 a c+b^2+b c x}{a x^2 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)}",1,"(-(a^2/x^2) + (4*a*b)/x + (2*a*(b^4 - 4*a*b^2*c + 2*a^2*c^2 + b^3*c*x - 3*a*b*c^2*x))/((b^2 - 4*a*c)*(a + x*(b + c*x))) + (2*b*(3*b^4 - 20*a*b^2*c + 30*a^2*c^2)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + 2*(3*b^2 - 2*a*c)*Log[x] + (-3*b^2 + 2*a*c)*Log[a + x*(b + c*x)])/(2*a^4)","A",1
27,1,218,252,0.3178786,"\int \frac{1}{\left(a x^2+b x^3+c x^4\right)^2} \, dx","Integrate[(a*x^2 + b*x^3 + c*x^4)^(-2),x]","\frac{-\frac{a^3}{x^3}-\frac{3 a \left(5 a^2 b c^2+2 a^2 c^3 x-5 a b^3 c-4 a b^2 c^2 x+b^5+b^4 c x\right)}{\left(b^2-4 a c\right) (a+x (b+c x))}+\frac{3 a^2 b}{x^2}-\frac{6 \left(-10 a^3 c^3+30 a^2 b^2 c^2-15 a b^4 c+2 b^6\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{3/2}}+6 \log (x) \left(3 a b c-2 b^3\right)+3 \left(2 b^3-3 a b c\right) \log (a+x (b+c x))+\frac{3 a \left(2 a c-3 b^2\right)}{x}}{3 a^5}","\frac{b \left(2 b^2-3 a c\right) \log \left(a+b x+c x^2\right)}{a^5}-\frac{2 b \log (x) \left(2 b^2-3 a c\right)}{a^5}+\frac{b \left(2 b^2-7 a c\right)}{a^3 x^2 \left(b^2-4 a c\right)}-\frac{2 \left(2 b^2-5 a c\right)}{3 a^2 x^3 \left(b^2-4 a c\right)}-\frac{2 \left(5 a^2 c^2-9 a b^2 c+2 b^4\right)}{a^4 x \left(b^2-4 a c\right)}-\frac{2 \left(-10 a^3 c^3+30 a^2 b^2 c^2-15 a b^4 c+2 b^6\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{a^5 \left(b^2-4 a c\right)^{3/2}}+\frac{-2 a c+b^2+b c x}{a x^3 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)}",1,"(-(a^3/x^3) + (3*a^2*b)/x^2 + (3*a*(-3*b^2 + 2*a*c))/x - (3*a*(b^5 - 5*a*b^3*c + 5*a^2*b*c^2 + b^4*c*x - 4*a*b^2*c^2*x + 2*a^2*c^3*x))/((b^2 - 4*a*c)*(a + x*(b + c*x))) - (6*(2*b^6 - 15*a*b^4*c + 30*a^2*b^2*c^2 - 10*a^3*c^3)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + 6*(-2*b^3 + 3*a*b*c)*Log[x] + 3*(2*b^3 - 3*a*b*c)*Log[a + x*(b + c*x)])/(3*a^5)","A",1
28,1,272,318,0.3775792,"\int \frac{1}{x \left(a x^2+b x^3+c x^4\right)^2} \, dx","Integrate[1/(x*(a*x^2 + b*x^3 + c*x^4)^2),x]","\frac{-\frac{3 a^4}{x^4}+\frac{8 a^3 b}{x^3}+\frac{6 a^2 \left(2 a c-3 b^2\right)}{x^2}+12 \log (x) \left(3 a^2 c^2-12 a b^2 c+5 b^4\right)-6 \left(3 a^2 c^2-12 a b^2 c+5 b^4\right) \log (a+x (b+c x))+\frac{12 b \left(-70 a^3 c^3+105 a^2 b^2 c^2-42 a b^4 c+5 b^6\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{3/2}}-\frac{12 a \left(2 a^3 c^3-9 a^2 b^2 c^2-5 a^2 b c^3 x+6 a b^4 c+5 a b^3 c^2 x-b^6-b^5 c x\right)}{\left(b^2-4 a c\right) (a+x (b+c x))}-\frac{24 a b \left(3 a c-2 b^2\right)}{x}}{12 a^6}","\frac{b \left(5 b^2-17 a c\right)}{3 a^3 x^3 \left(b^2-4 a c\right)}-\frac{5 b^2-12 a c}{4 a^2 x^4 \left(b^2-4 a c\right)}-\frac{\left(3 a^2 c^2-12 a b^2 c+5 b^4\right) \log \left(a+b x+c x^2\right)}{2 a^6}+\frac{\log (x) \left(3 a^2 c^2-12 a b^2 c+5 b^4\right)}{a^6}+\frac{b \left(29 a^2 c^2-27 a b^2 c+5 b^4\right)}{a^5 x \left(b^2-4 a c\right)}-\frac{12 a^2 c^2-22 a b^2 c+5 b^4}{2 a^4 x^2 \left(b^2-4 a c\right)}+\frac{b \left(-70 a^3 c^3+105 a^2 b^2 c^2-42 a b^4 c+5 b^6\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{a^6 \left(b^2-4 a c\right)^{3/2}}+\frac{-2 a c+b^2+b c x}{a x^4 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)}",1,"((-3*a^4)/x^4 + (8*a^3*b)/x^3 + (6*a^2*(-3*b^2 + 2*a*c))/x^2 - (24*a*b*(-2*b^2 + 3*a*c))/x - (12*a*(-b^6 + 6*a*b^4*c - 9*a^2*b^2*c^2 + 2*a^3*c^3 - b^5*c*x + 5*a*b^3*c^2*x - 5*a^2*b*c^3*x))/((b^2 - 4*a*c)*(a + x*(b + c*x))) + (12*b*(5*b^6 - 42*a*b^4*c + 105*a^2*b^2*c^2 - 70*a^3*c^3)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + 12*(5*b^4 - 12*a*b^2*c + 3*a^2*c^2)*Log[x] - 6*(5*b^4 - 12*a*b^2*c + 3*a^2*c^2)*Log[a + x*(b + c*x)])/(12*a^6)","A",1
29,1,180,257,0.2368628,"\int x^2 \sqrt{a x^2+b x^3+c x^4} \, dx","Integrate[x^2*Sqrt[a*x^2 + b*x^3 + c*x^4],x]","\frac{15 x \left(48 a^2 b c^2-40 a b^3 c+7 b^5\right) \sqrt{a+x (b+c x)} \log \left(2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right)+2 \sqrt{c} x (a+x (b+c x)) \left(128 c^2 \left(-2 a^2+a c x^2+3 c^2 x^4\right)+4 b^2 c \left(115 a-14 c x^2\right)+8 b c^2 x \left(6 c x^2-29 a\right)-105 b^4+70 b^3 c x\right)}{3840 c^{9/2} \sqrt{x^2 (a+x (b+c x))}}","-\frac{\left(256 a^2 c^2-460 a b^2 c+105 b^4\right) \sqrt{a x^2+b x^3+c x^4}}{1920 c^4 x}+\frac{b x \left(7 b^2-12 a c\right) \left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{256 c^{9/2} \sqrt{a x^2+b x^3+c x^4}}+\frac{b \left(35 b^2-116 a c\right) \sqrt{a x^2+b x^3+c x^4}}{960 c^3}-\frac{x \left(7 b^2-16 a c\right) \sqrt{a x^2+b x^3+c x^4}}{240 c^2}+\frac{x^2 (b+8 c x) \sqrt{a x^2+b x^3+c x^4}}{40 c}",1,"(2*Sqrt[c]*x*(a + x*(b + c*x))*(-105*b^4 + 70*b^3*c*x + 4*b^2*c*(115*a - 14*c*x^2) + 8*b*c^2*x*(-29*a + 6*c*x^2) + 128*c^2*(-2*a^2 + a*c*x^2 + 3*c^2*x^4)) + 15*(7*b^5 - 40*a*b^3*c + 48*a^2*b*c^2)*x*Sqrt[a + x*(b + c*x)]*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(3840*c^(9/2)*Sqrt[x^2*(a + x*(b + c*x))])","A",1
30,1,150,205,0.1607059,"\int x \sqrt{a x^2+b x^3+c x^4} \, dx","Integrate[x*Sqrt[a*x^2 + b*x^3 + c*x^4],x]","\frac{2 \sqrt{c} x (a+x (b+c x)) \left(b \left(8 c^2 x^2-52 a c\right)+24 c^2 x \left(a+2 c x^2\right)+15 b^3-10 b^2 c x\right)-3 x \left(16 a^2 c^2-24 a b^2 c+5 b^4\right) \sqrt{a+x (b+c x)} \log \left(2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right)}{384 c^{7/2} \sqrt{x^2 (a+x (b+c x))}}","-\frac{x \left(b^2-4 a c\right) \left(5 b^2-4 a c\right) \sqrt{a+b x+c x^2} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{128 c^{7/2} \sqrt{a x^2+b x^3+c x^4}}+\frac{b \left(15 b^2-52 a c\right) \sqrt{a x^2+b x^3+c x^4}}{192 c^3 x}-\frac{\left(5 b^2-12 a c\right) \sqrt{a x^2+b x^3+c x^4}}{96 c^2}+\frac{x (b+6 c x) \sqrt{a x^2+b x^3+c x^4}}{24 c}",1,"(2*Sqrt[c]*x*(a + x*(b + c*x))*(15*b^3 - 10*b^2*c*x + 24*c^2*x*(a + 2*c*x^2) + b*(-52*a*c + 8*c^2*x^2)) - 3*(5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*x*Sqrt[a + x*(b + c*x)]*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(384*c^(7/2)*Sqrt[x^2*(a + x*(b + c*x))])","A",1
31,1,119,163,0.2125423,"\int \sqrt{a x^2+b x^3+c x^4} \, dx","Integrate[Sqrt[a*x^2 + b*x^3 + c*x^4],x]","\frac{2 \sqrt{c} x (a+x (b+c x)) \left(8 c \left(a+c x^2\right)-3 b^2+2 b c x\right)+3 b x \left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{48 c^{5/2} \sqrt{x^2 (a+x (b+c x))}}","\frac{b \left(b^2-4 a c\right) \sqrt{a x^2+b x^3+c x^4} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{16 c^{5/2} x \sqrt{a+b x+c x^2}}-\frac{b (b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{8 c^2 x}+\frac{\left(a+b x+c x^2\right) \sqrt{a x^2+b x^3+c x^4}}{3 c x}",1,"(2*Sqrt[c]*x*(a + x*(b + c*x))*(-3*b^2 + 2*b*c*x + 8*c*(a + c*x^2)) + 3*b*(b^2 - 4*a*c)*x*Sqrt[a + x*(b + c*x)]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(48*c^(5/2)*Sqrt[x^2*(a + x*(b + c*x))])","A",1
32,1,100,119,0.1329599,"\int \frac{\sqrt{a x^2+b x^3+c x^4}}{x} \, dx","Integrate[Sqrt[a*x^2 + b*x^3 + c*x^4]/x,x]","\frac{x \left(2 \sqrt{c} (b+2 c x) (a+x (b+c x))-\left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \log \left(2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right)\right)}{8 c^{3/2} \sqrt{x^2 (a+x (b+c x))}}","\frac{(b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{4 c x}-\frac{x \left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{8 c^{3/2} \sqrt{a x^2+b x^3+c x^4}}",1,"(x*(2*Sqrt[c]*(b + 2*c*x)*(a + x*(b + c*x)) - (b^2 - 4*a*c)*Sqrt[a + x*(b + c*x)]*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]]))/(8*c^(3/2)*Sqrt[x^2*(a + x*(b + c*x))])","A",1
33,1,134,173,0.095337,"\int \frac{\sqrt{a x^2+b x^3+c x^4}}{x^2} \, dx","Integrate[Sqrt[a*x^2 + b*x^3 + c*x^4]/x^2,x]","\frac{x \sqrt{a+x (b+c x)} \left(2 \sqrt{c} \sqrt{a+x (b+c x)}-2 \sqrt{a} \sqrt{c} \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)+b \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)\right)}{2 \sqrt{c} \sqrt{x^2 (a+x (b+c x))}}","\frac{\sqrt{a x^2+b x^3+c x^4}}{x}-\frac{\sqrt{a} x \sqrt{a+b x+c x^2} \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{\sqrt{a x^2+b x^3+c x^4}}+\frac{b x \sqrt{a+b x+c x^2} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{2 \sqrt{c} \sqrt{a x^2+b x^3+c x^4}}",1,"(x*Sqrt[a + x*(b + c*x)]*(2*Sqrt[c]*Sqrt[a + x*(b + c*x)] - 2*Sqrt[a]*Sqrt[c]*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])] + b*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/(2*Sqrt[c]*Sqrt[x^2*(a + x*(b + c*x))])","A",1
34,1,131,173,0.1170149,"\int \frac{\sqrt{a x^2+b x^3+c x^4}}{x^3} \, dx","Integrate[Sqrt[a*x^2 + b*x^3 + c*x^4]/x^3,x]","-\frac{\sqrt{a+x (b+c x)} \left(b x \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)+2 \sqrt{a} \left(\sqrt{a+x (b+c x)}-\sqrt{c} x \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)\right)\right)}{2 \sqrt{a} \sqrt{x^2 (a+x (b+c x))}}","-\frac{\sqrt{a x^2+b x^3+c x^4}}{x^2}-\frac{b x \sqrt{a+b x+c x^2} \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}+\frac{\sqrt{c} x \sqrt{a+b x+c x^2} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{\sqrt{a x^2+b x^3+c x^4}}",1,"-1/2*(Sqrt[a + x*(b + c*x)]*(b*x*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])] + 2*Sqrt[a]*(Sqrt[a + x*(b + c*x)] - Sqrt[c]*x*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])))/(Sqrt[a]*Sqrt[x^2*(a + x*(b + c*x))])","A",1
35,1,112,114,0.0964251,"\int \frac{\sqrt{a x^2+b x^3+c x^4}}{x^4} \, dx","Integrate[Sqrt[a*x^2 + b*x^3 + c*x^4]/x^4,x]","\frac{\sqrt{x^2 (a+x (b+c x))} \left(x^2 \left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)-2 \sqrt{a} (2 a+b x) \sqrt{a+x (b+c x)}\right)}{8 a^{3/2} x^3 \sqrt{a+x (b+c x)}}","\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right)}{8 a^{3/2}}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{4 a x^2}-\frac{\sqrt{a x^2+b x^3+c x^4}}{2 x^3}",1,"(Sqrt[x^2*(a + x*(b + c*x))]*(-2*Sqrt[a]*(2*a + b*x)*Sqrt[a + x*(b + c*x)] + (b^2 - 4*a*c)*x^2*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])]))/(8*a^(3/2)*x^3*Sqrt[a + x*(b + c*x)])","A",1
36,1,131,155,0.1228858,"\int \frac{\sqrt{a x^2+b x^3+c x^4}}{x^5} \, dx","Integrate[Sqrt[a*x^2 + b*x^3 + c*x^4]/x^5,x]","\frac{\sqrt{x^2 (a+x (b+c x))} \left(-2 \sqrt{a} \sqrt{a+x (b+c x)} \left(8 a^2+2 a x (b+4 c x)-3 b^2 x^2\right)-3 b x^3 \left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)\right)}{48 a^{5/2} x^4 \sqrt{a+x (b+c x)}}","-\frac{b \left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right)}{16 a^{5/2}}+\frac{\left(3 b^2-8 a c\right) \sqrt{a x^2+b x^3+c x^4}}{24 a^2 x^2}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{12 a x^3}-\frac{\sqrt{a x^2+b x^3+c x^4}}{3 x^4}",1,"(Sqrt[x^2*(a + x*(b + c*x))]*(-2*Sqrt[a]*Sqrt[a + x*(b + c*x)]*(8*a^2 - 3*b^2*x^2 + 2*a*x*(b + 4*c*x)) - 3*b*(b^2 - 4*a*c)*x^3*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])]))/(48*a^(5/2)*x^4*Sqrt[a + x*(b + c*x)])","A",1
37,1,160,205,0.1682379,"\int \frac{\sqrt{a x^2+b x^3+c x^4}}{x^6} \, dx","Integrate[Sqrt[a*x^2 + b*x^3 + c*x^4]/x^6,x]","\frac{\sqrt{x^2 (a+x (b+c x))} \left(3 x^4 \left(16 a^2 c^2-24 a b^2 c+5 b^4\right) \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)-2 \sqrt{a} \sqrt{a+x (b+c x)} \left(48 a^3+8 a^2 x (b+3 c x)-2 a b x^2 (5 b+26 c x)+15 b^3 x^3\right)\right)}{384 a^{7/2} x^5 \sqrt{a+x (b+c x)}}","\frac{\left(b^2-4 a c\right) \left(5 b^2-4 a c\right) \tanh ^{-1}\left(\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right)}{128 a^{7/2}}-\frac{b \left(15 b^2-52 a c\right) \sqrt{a x^2+b x^3+c x^4}}{192 a^3 x^2}+\frac{\left(5 b^2-12 a c\right) \sqrt{a x^2+b x^3+c x^4}}{96 a^2 x^3}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{24 a x^4}-\frac{\sqrt{a x^2+b x^3+c x^4}}{4 x^5}",1,"(Sqrt[x^2*(a + x*(b + c*x))]*(-2*Sqrt[a]*Sqrt[a + x*(b + c*x)]*(48*a^3 + 15*b^3*x^3 + 8*a^2*x*(b + 3*c*x) - 2*a*b*x^2*(5*b + 26*c*x)) + 3*(5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*x^4*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])]))/(384*a^(7/2)*x^5*Sqrt[a + x*(b + c*x)])","A",1
38,1,236,422,0.3838335,"\int x \left(a x^2+b x^3+c x^4\right)^{3/2} \, dx","Integrate[x*(a*x^2 + b*x^3 + c*x^4)^(3/2),x]","\frac{\left(x^2 (a+x (b+c x))\right)^{3/2} \left(\frac{\left(16 a^2 c^2-72 a b^2 c+33 b^4\right) \left(2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)} \left(4 c \left(5 a+2 c x^2\right)-3 b^2+8 b c x\right)+3 \left(b^2-4 a c\right)^2 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)\right)}{4096 c^{11/2} x^3 (a+x (b+c x))^{3/2}}+\frac{\left(372 a b c-280 a c^2 x-231 b^3+330 b^2 c x\right) (a+x (b+c x))}{560 c^3 x^3}-\frac{11 b (a+x (b+c x))}{14 c x}+a+b x+c x^2\right)}{8 c}","\frac{3 x \left(b^2-4 a c\right)^2 \left(16 a^2 c^2-72 a b^2 c+33 b^4\right) \sqrt{a+b x+c x^2} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{32768 c^{13/2} \sqrt{a x^2+b x^3+c x^4}}-\frac{b x \left(2416 a^2 c^2-1560 a b^2 c+231 b^4\right) \sqrt{a x^2+b x^3+c x^4}}{71680 c^4}+\frac{x^2 \left(560 a^2 c^2-568 a b^2 c+99 b^4\right) \sqrt{a x^2+b x^3+c x^4}}{35840 c^3}-\frac{b \left(-58816 a^3 c^3+81648 a^2 b^2 c^2-30660 a b^4 c+3465 b^6\right) \sqrt{a x^2+b x^3+c x^4}}{573440 c^6 x}+\frac{\left(-6720 a^3 c^3+18896 a^2 b^2 c^2-8988 a b^4 c+1155 b^6\right) \sqrt{a x^2+b x^3+c x^4}}{286720 c^5}-\frac{x^3 \left(10 c x \left(11 b^2-28 a c\right)+b \left(68 a c+11 b^2\right)\right) \sqrt{a x^2+b x^3+c x^4}}{4480 c^2}+\frac{x (3 b+14 c x) \left(a x^2+b x^3+c x^4\right)^{3/2}}{112 c}",1,"((x^2*(a + x*(b + c*x)))^(3/2)*(a + b*x + c*x^2 - (11*b*(a + x*(b + c*x)))/(14*c*x) + ((-231*b^3 + 372*a*b*c + 330*b^2*c*x - 280*a*c^2*x)*(a + x*(b + c*x)))/(560*c^3*x^3) + ((33*b^4 - 72*a*b^2*c + 16*a^2*c^2)*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*(-3*b^2 + 8*b*c*x + 4*c*(5*a + 2*c*x^2)) + 3*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/(4096*c^(11/2)*x^3*(a + x*(b + c*x))^(3/2))))/(8*c)","A",1
39,1,197,364,0.2457571,"\int \left(a x^2+b x^3+c x^4\right)^{3/2} \, dx","Integrate[(a*x^2 + b*x^3 + c*x^4)^(3/2),x]","\frac{\left(x^2 (a+x (b+c x))\right)^{3/2} \left(\frac{\left(-16 a c+21 b^2-30 b c x\right) (a+x (b+c x))}{40 c^2}+\frac{7 \left(4 a b c-3 b^3\right) \left(2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)} \left(4 c \left(5 a+2 c x^2\right)-3 b^2+8 b c x\right)+3 \left(b^2-4 a c\right)^2 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)\right)}{2048 c^{9/2} (a+x (b+c x))^{3/2}}+x^2 (a+x (b+c x))\right)}{7 c x^3}","-\frac{b \left(1168 a^2 c^2-728 a b^2 c+105 b^4\right) \sqrt{a x^2+b x^3+c x^4}}{17920 c^4}+\frac{\left(-2048 a^3 c^3+5488 a^2 b^2 c^2-2520 a b^4 c+315 b^6\right) \sqrt{a x^2+b x^3+c x^4}}{35840 c^5 x}-\frac{3 b x \left(b^2-4 a c\right)^2 \left(3 b^2-4 a c\right) \sqrt{a+b x+c x^2} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{2048 c^{11/2} \sqrt{a x^2+b x^3+c x^4}}+\frac{x \left(7 b^2-32 a c\right) \left(3 b^2-4 a c\right) \sqrt{a x^2+b x^3+c x^4}}{4480 c^3}-\frac{b x^2 \left(9 b^2-44 a c\right) \sqrt{a x^2+b x^3+c x^4}}{2240 c^2}+\frac{x^3 \left(24 a c+b^2+10 b c x\right) \sqrt{a x^2+b x^3+c x^4}}{280 c}+\frac{1}{7} x \left(a x^2+b x^3+c x^4\right)^{3/2}",1,"((x^2*(a + x*(b + c*x)))^(3/2)*(x^2*(a + x*(b + c*x)) + ((21*b^2 - 16*a*c - 30*b*c*x)*(a + x*(b + c*x)))/(40*c^2) + (7*(-3*b^3 + 4*a*b*c)*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*(-3*b^2 + 8*b*c*x + 4*c*(5*a + 2*c*x^2)) + 3*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/(2048*c^(9/2)*(a + x*(b + c*x))^(3/2))))/(7*c*x^3)","A",1
40,1,180,288,0.2194627,"\int \frac{\left(a x^2+b x^3+c x^4\right)^{3/2}}{x} \, dx","Integrate[(a*x^2 + b*x^3 + c*x^4)^(3/2)/x,x]","\frac{\left(x^2 (a+x (b+c x))\right)^{3/2} \left(\frac{\left(7 b^2-4 a c\right) \left(2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)} \left(4 c \left(5 a+2 c x^2\right)-3 b^2+8 b c x\right)+3 \left(b^2-4 a c\right)^2 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)\right)}{512 c^{7/2} (a+x (b+c x))^{3/2}}+x (a+x (b+c x))-\frac{7 b (a+x (b+c x))}{10 c}\right)}{6 c x^3}","-\frac{b \left(1296 a^2 c^2-760 a b^2 c+105 b^4\right) \sqrt{a x^2+b x^3+c x^4}}{7680 c^4 x}+\frac{\left(240 a^2 c^2-216 a b^2 c+35 b^4\right) \sqrt{a x^2+b x^3+c x^4}}{3840 c^3}+\frac{x \left(7 b^2-4 a c\right) \left(b^2-4 a c\right)^2 \sqrt{a+b x+c x^2} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{1024 c^{9/2} \sqrt{a x^2+b x^3+c x^4}}-\frac{x \left(6 c x \left(7 b^2-20 a c\right)+b \left(12 a c+7 b^2\right)\right) \sqrt{a x^2+b x^3+c x^4}}{960 c^2}+\frac{(3 b+10 c x) \left(a x^2+b x^3+c x^4\right)^{3/2}}{60 c x}",1,"((x^2*(a + x*(b + c*x)))^(3/2)*((-7*b*(a + x*(b + c*x)))/(10*c) + x*(a + x*(b + c*x)) + ((7*b^2 - 4*a*c)*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*(-3*b^2 + 8*b*c*x + 4*c*(5*a + 2*c*x^2)) + 3*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/(512*c^(7/2)*(a + x*(b + c*x))^(3/2))))/(6*c*x^3)","A",1
41,1,163,198,0.1727829,"\int \frac{\left(a x^2+b x^3+c x^4\right)^{3/2}}{x^2} \, dx","Integrate[(a*x^2 + b*x^3 + c*x^4)^(3/2)/x^2,x]","\frac{x \sqrt{a+x (b+c x)} \left(2 \sqrt{c} \sqrt{a+x (b+c x)} \left(4 b^2 c \left(2 c x^2-25 a\right)+8 b c^2 x \left(7 a+22 c x^2\right)+128 c^2 \left(a+c x^2\right)^2+15 b^4-10 b^3 c x\right)-15 b \left(b^2-4 a c\right)^2 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)\right)}{1280 c^{7/2} \sqrt{x^2 (a+x (b+c x))}}","-\frac{3 b x \left(b^2-4 a c\right)^2 \sqrt{a+b x+c x^2} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{256 c^{7/2} \sqrt{a x^2+b x^3+c x^4}}+\frac{3 b \left(b^2-4 a c\right) (b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{128 c^3 x}-\frac{b (b+2 c x) \left(a x^2+b x^3+c x^4\right)^{3/2}}{16 c^2 x^3}+\frac{\left(a x^2+b x^3+c x^4\right)^{5/2}}{5 c x^5}",1,"(x*Sqrt[a + x*(b + c*x)]*(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(15*b^4 - 10*b^3*c*x + 128*c^2*(a + c*x^2)^2 + 4*b^2*c*(-25*a + 2*c*x^2) + 8*b*c^2*x*(7*a + 22*c*x^2)) - 15*b*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/(1280*c^(7/2)*Sqrt[x^2*(a + x*(b + c*x))])","A",1
42,1,132,165,0.0595792,"\int \frac{\left(a x^2+b x^3+c x^4\right)^{3/2}}{x^3} \, dx","Integrate[(a*x^2 + b*x^3 + c*x^4)^(3/2)/x^3,x]","\frac{x \sqrt{a+x (b+c x)} \left(2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)} \left(4 c \left(5 a+2 c x^2\right)-3 b^2+8 b c x\right)+3 \left(b^2-4 a c\right)^2 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)\right)}{128 c^{5/2} \sqrt{x^2 (a+x (b+c x))}}","\frac{3 x \left(b^2-4 a c\right)^2 \sqrt{a+b x+c x^2} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{128 c^{5/2} \sqrt{a x^2+b x^3+c x^4}}-\frac{3 \left(b^2-4 a c\right) (b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{64 c^2 x}+\frac{(b+2 c x) \left(a x^2+b x^3+c x^4\right)^{3/2}}{8 c x^3}",1,"(x*Sqrt[a + x*(b + c*x)]*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*(-3*b^2 + 8*b*c*x + 4*c*(5*a + 2*c*x^2)) + 3*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/(128*c^(5/2)*Sqrt[x^2*(a + x*(b + c*x))])","A",1
43,1,166,227,0.2235608,"\int \frac{\left(a x^2+b x^3+c x^4\right)^{3/2}}{x^4} \, dx","Integrate[(a*x^2 + b*x^3 + c*x^4)^(3/2)/x^4,x]","\frac{x \sqrt{a+x (b+c x)} \left(-48 a^{3/2} c^{3/2} \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)+2 \sqrt{c} \sqrt{a+x (b+c x)} \left(8 c \left(4 a+c x^2\right)+3 b^2+14 b c x\right)-3 b \left(b^2-12 a c\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)\right)}{48 c^{3/2} \sqrt{x^2 (a+x (b+c x))}}","-\frac{a^{3/2} x \sqrt{a+b x+c x^2} \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{\sqrt{a x^2+b x^3+c x^4}}-\frac{b x \left(b^2-12 a c\right) \sqrt{a+b x+c x^2} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{16 c^{3/2} \sqrt{a x^2+b x^3+c x^4}}+\frac{\left(8 a c+b^2+2 b c x\right) \sqrt{a x^2+b x^3+c x^4}}{8 c x}+\frac{\left(a x^2+b x^3+c x^4\right)^{3/2}}{3 x^3}",1,"(x*Sqrt[a + x*(b + c*x)]*(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(3*b^2 + 14*b*c*x + 8*c*(4*a + c*x^2)) - 48*a^(3/2)*c^(3/2)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])] - 3*b*(b^2 - 12*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/(48*c^(3/2)*Sqrt[x^2*(a + x*(b + c*x))])","A",1
44,1,158,219,0.18016,"\int \frac{\left(a x^2+b x^3+c x^4\right)^{3/2}}{x^5} \, dx","Integrate[(a*x^2 + b*x^3 + c*x^4)^(3/2)/x^5,x]","\frac{\sqrt{a+x (b+c x)} \left(3 x \left(4 a c+b^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)+2 \sqrt{c} \sqrt{a+x (b+c x)} (x (5 b+2 c x)-4 a)-12 \sqrt{a} b \sqrt{c} x \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)\right)}{8 \sqrt{c} \sqrt{x^2 (a+x (b+c x))}}","\frac{3 x \left(4 a c+b^2\right) \sqrt{a+b x+c x^2} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{8 \sqrt{c} \sqrt{a x^2+b x^3+c x^4}}-\frac{\left(a x^2+b x^3+c x^4\right)^{3/2}}{x^4}+\frac{3 (3 b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{4 x}-\frac{3 \sqrt{a} b x \sqrt{a+b x+c x^2} \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{2 \sqrt{a x^2+b x^3+c x^4}}",1,"(Sqrt[a + x*(b + c*x)]*(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-4*a + x*(5*b + 2*c*x)) - 12*Sqrt[a]*b*Sqrt[c]*x*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])] + 3*(b^2 + 4*a*c)*x*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/(8*Sqrt[c]*Sqrt[x^2*(a + x*(b + c*x))])","A",1
45,1,162,219,0.1896661,"\int \frac{\left(a x^2+b x^3+c x^4\right)^{3/2}}{x^6} \, dx","Integrate[(a*x^2 + b*x^3 + c*x^4)^(3/2)/x^6,x]","-\frac{\sqrt{x^2 (a+x (b+c x))} \left(3 x^2 \left(4 a c+b^2\right) \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)+2 \sqrt{a} \left((2 a+x (5 b-4 c x)) \sqrt{a+x (b+c x)}-6 b \sqrt{c} x^2 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)\right)\right)}{8 \sqrt{a} x^3 \sqrt{a+x (b+c x)}}","-\frac{3 x \left(4 a c+b^2\right) \sqrt{a+b x+c x^2} \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{8 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}-\frac{3 (b-2 c x) \sqrt{a x^2+b x^3+c x^4}}{4 x^2}+\frac{3 b \sqrt{c} x \sqrt{a+b x+c x^2} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{2 \sqrt{a x^2+b x^3+c x^4}}-\frac{\left(a x^2+b x^3+c x^4\right)^{3/2}}{2 x^5}",1,"-1/8*(Sqrt[x^2*(a + x*(b + c*x))]*(3*(b^2 + 4*a*c)*x^2*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])] + 2*Sqrt[a]*((2*a + x*(5*b - 4*c*x))*Sqrt[a + x*(b + c*x)] - 6*b*Sqrt[c]*x^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])))/(Sqrt[a]*x^3*Sqrt[a + x*(b + c*x)])","A",1
46,1,175,257,0.2888098,"\int \frac{\left(a x^2+b x^3+c x^4\right)^{3/2}}{x^7} \, dx","Integrate[(a*x^2 + b*x^3 + c*x^4)^(3/2)/x^7,x]","\frac{\sqrt{x^2 (a+x (b+c x))} \left(3 b x^3 \left(b^2-12 a c\right) \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)-2 \sqrt{a} \left(\sqrt{a+x (b+c x)} \left(8 a^2+2 a x (7 b+16 c x)+3 b^2 x^2\right)-24 a c^{3/2} x^3 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)\right)\right)}{48 a^{3/2} x^4 \sqrt{a+x (b+c x)}}","\frac{b x \left(b^2-12 a c\right) \sqrt{a+b x+c x^2} \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{16 a^{3/2} \sqrt{a x^2+b x^3+c x^4}}+\frac{\left(-8 a c+b^2+2 b c x\right) \sqrt{a x^2+b x^3+c x^4}}{8 a x^2}+\frac{c^{3/2} x \sqrt{a+b x+c x^2} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{\sqrt{a x^2+b x^3+c x^4}}-\frac{\left(a x^2+b x^3+c x^4\right)^{3/2}}{3 x^6}-\frac{b \left(a x^2+b x^3+c x^4\right)^{3/2}}{4 a x^5}",1,"(Sqrt[x^2*(a + x*(b + c*x))]*(3*b*(b^2 - 12*a*c)*x^3*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])] - 2*Sqrt[a]*(Sqrt[a + x*(b + c*x)]*(8*a^2 + 3*b^2*x^2 + 2*a*x*(7*b + 16*c*x)) - 24*a*c^(3/2)*x^3*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])))/(48*a^(3/2)*x^4*Sqrt[a + x*(b + c*x)])","A",1
47,1,141,197,0.1153446,"\int \frac{\left(a x^2+b x^3+c x^4\right)^{3/2}}{x^8} \, dx","Integrate[(a*x^2 + b*x^3 + c*x^4)^(3/2)/x^8,x]","-\frac{\sqrt{x^2 (a+x (b+c x))} \left(2 \sqrt{a} (2 a+b x) \sqrt{a+x (b+c x)} \left(8 a^2+4 a x (2 b+5 c x)-3 b^2 x^2\right)+3 x^4 \left(b^2-4 a c\right)^2 \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)\right)}{128 a^{5/2} x^5 \sqrt{a+x (b+c x)}}","-\frac{3 \left(b^2-4 a c\right)^2 \tanh ^{-1}\left(\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right)}{128 a^{5/2}}+\frac{b \left(3 b^2-20 a c\right) \sqrt{a x^2+b x^3+c x^4}}{64 a^2 x^2}-\frac{\left(b^2-12 a c\right) \sqrt{a x^2+b x^3+c x^4}}{32 a x^3}-\frac{(b+6 c x) \sqrt{a x^2+b x^3+c x^4}}{8 x^4}-\frac{\left(a x^2+b x^3+c x^4\right)^{3/2}}{4 x^7}",1,"-1/128*(Sqrt[x^2*(a + x*(b + c*x))]*(2*Sqrt[a]*(2*a + b*x)*Sqrt[a + x*(b + c*x)]*(8*a^2 - 3*b^2*x^2 + 4*a*x*(2*b + 5*c*x)) + 3*(b^2 - 4*a*c)^2*x^4*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])]))/(a^(5/2)*x^5*Sqrt[a + x*(b + c*x)])","A",1
48,1,177,249,0.1655311,"\int \frac{\left(a x^2+b x^3+c x^4\right)^{3/2}}{x^9} \, dx","Integrate[(a*x^2 + b*x^3 + c*x^4)^(3/2)/x^9,x]","\frac{\sqrt{x^2 (a+x (b+c x))} \left(15 b x^5 \left(b^2-4 a c\right)^2 \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)-2 \sqrt{a} \sqrt{a+x (b+c x)} \left(128 a^4+16 a^3 x (11 b+16 c x)+8 a^2 x^2 \left(b^2+7 b c x+16 c^2 x^2\right)-10 a b^2 x^3 (b+10 c x)+15 b^4 x^4\right)\right)}{1280 a^{7/2} x^6 \sqrt{a+x (b+c x)}}","\frac{3 b \left(b^2-4 a c\right)^2 \tanh ^{-1}\left(\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right)}{256 a^{7/2}}+\frac{b \left(5 b^2-28 a c\right) \sqrt{a x^2+b x^3+c x^4}}{320 a^2 x^3}-\frac{\left(128 a^2 c^2-100 a b^2 c+15 b^4\right) \sqrt{a x^2+b x^3+c x^4}}{640 a^3 x^2}-\frac{\left(b^2-8 a c\right) \sqrt{a x^2+b x^3+c x^4}}{80 a x^4}-\frac{\left(a x^2+b x^3+c x^4\right)^{3/2}}{5 x^8}-\frac{3 (b+4 c x) \sqrt{a x^2+b x^3+c x^4}}{40 x^5}",1,"(Sqrt[x^2*(a + x*(b + c*x))]*(-2*Sqrt[a]*Sqrt[a + x*(b + c*x)]*(128*a^4 + 15*b^4*x^4 - 10*a*b^2*x^3*(b + 10*c*x) + 16*a^3*x*(11*b + 16*c*x) + 8*a^2*x^2*(b^2 + 7*b*c*x + 16*c^2*x^2)) + 15*b*(b^2 - 4*a*c)^2*x^5*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])]))/(1280*a^(7/2)*x^6*Sqrt[a + x*(b + c*x)])","A",1
49,1,105,143,0.0934416,"\int \frac{x^3}{\sqrt{a x^2+b x^3+c x^4}} \, dx","Integrate[x^3/Sqrt[a*x^2 + b*x^3 + c*x^4],x]","\frac{x \left(\left(3 b^2-4 a c\right) \sqrt{a+x (b+c x)} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)+2 \sqrt{c} (2 c x-3 b) (a+x (b+c x))\right)}{8 c^{5/2} \sqrt{x^2 (a+x (b+c x))}}","\frac{x \left(3 b^2-4 a c\right) \sqrt{a+b x+c x^2} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{8 c^{5/2} \sqrt{a x^2+b x^3+c x^4}}-\frac{3 b \sqrt{a x^2+b x^3+c x^4}}{4 c^2 x}+\frac{\sqrt{a x^2+b x^3+c x^4}}{2 c}",1,"(x*(2*Sqrt[c]*(-3*b + 2*c*x)*(a + x*(b + c*x)) + (3*b^2 - 4*a*c)*Sqrt[a + x*(b + c*x)]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/(8*c^(5/2)*Sqrt[x^2*(a + x*(b + c*x))])","A",1
50,1,89,103,0.0524219,"\int \frac{x^2}{\sqrt{a x^2+b x^3+c x^4}} \, dx","Integrate[x^2/Sqrt[a*x^2 + b*x^3 + c*x^4],x]","\frac{x \left(2 \sqrt{c} (a+x (b+c x))-b \sqrt{a+x (b+c x)} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)\right)}{2 c^{3/2} \sqrt{x^2 (a+x (b+c x))}}","\frac{\sqrt{a x^2+b x^3+c x^4}}{c x}-\frac{b x \sqrt{a+b x+c x^2} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{2 c^{3/2} \sqrt{a x^2+b x^3+c x^4}}",1,"(x*(2*Sqrt[c]*(a + x*(b + c*x)) - b*Sqrt[a + x*(b + c*x)]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/(2*c^(3/2)*Sqrt[x^2*(a + x*(b + c*x))])","A",1
51,1,66,71,0.0354933,"\int \frac{x}{\sqrt{a x^2+b x^3+c x^4}} \, dx","Integrate[x/Sqrt[a*x^2 + b*x^3 + c*x^4],x]","\frac{x \sqrt{a+b x+c x^2} \log \left(2 \sqrt{c} \sqrt{a+b x+c x^2}+b+2 c x\right)}{\sqrt{c} \sqrt{x^2 (a+x (b+c x))}}","\frac{x \sqrt{a+b x+c x^2} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{\sqrt{c} \sqrt{a x^2+b x^3+c x^4}}",1,"(x*Sqrt[a + b*x + c*x^2]*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + b*x + c*x^2]])/(Sqrt[c]*Sqrt[x^2*(a + x*(b + c*x))])","A",1
52,1,70,45,0.0188203,"\int \frac{1}{\sqrt{a x^2+b x^3+c x^4}} \, dx","Integrate[1/Sqrt[a*x^2 + b*x^3 + c*x^4],x]","-\frac{x \sqrt{a+b x+c x^2} \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{\sqrt{a} \sqrt{x^2 (a+x (b+c x))}}","-\frac{\tanh ^{-1}\left(\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right)}{\sqrt{a}}",1,"-((x*Sqrt[a + b*x + c*x^2]*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(Sqrt[a]*Sqrt[x^2*(a + x*(b + c*x))]))","A",1
53,1,89,77,0.0514245,"\int \frac{1}{x \sqrt{a x^2+b x^3+c x^4}} \, dx","Integrate[1/(x*Sqrt[a*x^2 + b*x^3 + c*x^4]),x]","\frac{b x \sqrt{a+x (b+c x)} \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)-2 \sqrt{a} (a+x (b+c x))}{2 a^{3/2} \sqrt{x^2 (a+x (b+c x))}}","\frac{b \tanh ^{-1}\left(\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right)}{2 a^{3/2}}-\frac{\sqrt{a x^2+b x^3+c x^4}}{a x^2}",1,"(-2*Sqrt[a]*(a + x*(b + c*x)) + b*x*Sqrt[a + x*(b + c*x)]*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])])/(2*a^(3/2)*Sqrt[x^2*(a + x*(b + c*x))])","A",1
54,1,112,119,0.0836794,"\int \frac{1}{x^2 \sqrt{a x^2+b x^3+c x^4}} \, dx","Integrate[1/(x^2*Sqrt[a*x^2 + b*x^3 + c*x^4]),x]","\frac{-\left(x^2 \left(3 b^2-4 a c\right) \sqrt{a+x (b+c x)} \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)\right)-2 \sqrt{a} (2 a-3 b x) (a+x (b+c x))}{8 a^{5/2} x \sqrt{x^2 (a+x (b+c x))}}","-\frac{\left(3 b^2-4 a c\right) \tanh ^{-1}\left(\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right)}{8 a^{5/2}}+\frac{3 b \sqrt{a x^2+b x^3+c x^4}}{4 a^2 x^2}-\frac{\sqrt{a x^2+b x^3+c x^4}}{2 a x^3}",1,"(-2*Sqrt[a]*(2*a - 3*b*x)*(a + x*(b + c*x)) - (3*b^2 - 4*a*c)*x^2*Sqrt[a + x*(b + c*x)]*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])])/(8*a^(5/2)*x*Sqrt[x^2*(a + x*(b + c*x))])","A",1
55,1,183,262,0.2304569,"\int \frac{x^7}{\left(a x^2+b x^3+c x^4\right)^{3/2}} \, dx","Integrate[x^7/(a*x^2 + b*x^3 + c*x^4)^(3/2),x]","\frac{x \left(2 \sqrt{c} \left(4 a^2 c (6 c x-13 b)+a \left(15 b^3-62 b^2 c x-20 b c^2 x^2+8 c^3 x^3\right)+b^2 x \left(15 b^2+5 b c x-2 c^2 x^2\right)\right)-3 \left(16 a^2 c^2-24 a b^2 c+5 b^4\right) \sqrt{a+x (b+c x)} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)\right)}{8 c^{7/2} \left(4 a c-b^2\right) \sqrt{x^2 (a+x (b+c x))}}","\frac{3 x \left(5 b^2-4 a c\right) \sqrt{a+b x+c x^2} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{8 c^{7/2} \sqrt{a x^2+b x^3+c x^4}}-\frac{b \left(15 b^2-52 a c\right) \sqrt{a x^2+b x^3+c x^4}}{4 c^3 x \left(b^2-4 a c\right)}+\frac{\left(5 b^2-12 a c\right) \sqrt{a x^2+b x^3+c x^4}}{2 c^2 \left(b^2-4 a c\right)}+\frac{2 x^4 (2 a+b x)}{\left(b^2-4 a c\right) \sqrt{a x^2+b x^3+c x^4}}-\frac{2 b x \sqrt{a x^2+b x^3+c x^4}}{c \left(b^2-4 a c\right)}",1,"(x*(2*Sqrt[c]*(4*a^2*c*(-13*b + 6*c*x) + b^2*x*(15*b^2 + 5*b*c*x - 2*c^2*x^2) + a*(15*b^3 - 62*b^2*c*x - 20*b*c^2*x^2 + 8*c^3*x^3)) - 3*(5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*Sqrt[a + x*(b + c*x)]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/(8*c^(7/2)*(-b^2 + 4*a*c)*Sqrt[x^2*(a + x*(b + c*x))])","A",1
56,1,141,201,0.1522581,"\int \frac{x^6}{\left(a x^2+b x^3+c x^4\right)^{3/2}} \, dx","Integrate[x^6/(a*x^2 + b*x^3 + c*x^4)^(3/2),x]","\frac{x \left(2 \sqrt{c} \left(8 a^2 c+a \left(-3 b^2+10 b c x+4 c^2 x^2\right)-b^2 x (3 b+c x)\right)+3 b \left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)\right)}{2 c^{5/2} \left(4 a c-b^2\right) \sqrt{x^2 (a+x (b+c x))}}","\frac{\left(3 b^2-8 a c\right) \sqrt{a x^2+b x^3+c x^4}}{c^2 x \left(b^2-4 a c\right)}+\frac{2 x^3 (2 a+b x)}{\left(b^2-4 a c\right) \sqrt{a x^2+b x^3+c x^4}}-\frac{2 b \sqrt{a x^2+b x^3+c x^4}}{c \left(b^2-4 a c\right)}-\frac{3 b x \sqrt{a+b x+c x^2} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{2 c^{5/2} \sqrt{a x^2+b x^3+c x^4}}",1,"(x*(2*Sqrt[c]*(8*a^2*c - b^2*x*(3*b + c*x) + a*(-3*b^2 + 10*b*c*x + 4*c^2*x^2)) + 3*b*(b^2 - 4*a*c)*Sqrt[a + x*(b + c*x)]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/(2*c^(5/2)*(-b^2 + 4*a*c)*Sqrt[x^2*(a + x*(b + c*x))])","A",1
57,1,112,153,0.1192333,"\int \frac{x^5}{\left(a x^2+b x^3+c x^4\right)^{3/2}} \, dx","Integrate[x^5/(a*x^2 + b*x^3 + c*x^4)^(3/2),x]","-\frac{x \left(2 \sqrt{c} \left(-a b+2 a c x+b^2 (-x)\right)+\left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)\right)}{c^{3/2} \left(4 a c-b^2\right) \sqrt{x^2 (a+x (b+c x))}}","\frac{2 x^2 (2 a+b x)}{\left(b^2-4 a c\right) \sqrt{a x^2+b x^3+c x^4}}-\frac{2 b \sqrt{a x^2+b x^3+c x^4}}{c x \left(b^2-4 a c\right)}+\frac{x \sqrt{a+b x+c x^2} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{c^{3/2} \sqrt{a x^2+b x^3+c x^4}}",1,"-((x*(2*Sqrt[c]*(-(a*b) - b^2*x + 2*a*c*x) + (b^2 - 4*a*c)*Sqrt[a + x*(b + c*x)]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/(c^(3/2)*(-b^2 + 4*a*c)*Sqrt[x^2*(a + x*(b + c*x))]))","A",1
58,1,37,40,0.0745254,"\int \frac{x^4}{\left(a x^2+b x^3+c x^4\right)^{3/2}} \, dx","Integrate[x^4/(a*x^2 + b*x^3 + c*x^4)^(3/2),x]","\frac{2 x (2 a+b x)}{\left(b^2-4 a c\right) \sqrt{x^2 (a+x (b+c x))}}","\frac{2 x (2 a+b x)}{\left(b^2-4 a c\right) \sqrt{a x^2+b x^3+c x^4}}",1,"(2*x*(2*a + b*x))/((b^2 - 4*a*c)*Sqrt[x^2*(a + x*(b + c*x))])","A",1
59,1,36,39,0.0243782,"\int \frac{x^3}{\left(a x^2+b x^3+c x^4\right)^{3/2}} \, dx","Integrate[x^3/(a*x^2 + b*x^3 + c*x^4)^(3/2),x]","-\frac{2 x (b+2 c x)}{\left(b^2-4 a c\right) \sqrt{x^2 (a+x (b+c x))}}","-\frac{2 x (b+2 c x)}{\left(b^2-4 a c\right) \sqrt{a x^2+b x^3+c x^4}}",1,"(-2*x*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[x^2*(a + x*(b + c*x))])","A",1
60,1,109,94,0.1277244,"\int \frac{x^2}{\left(a x^2+b x^3+c x^4\right)^{3/2}} \, dx","Integrate[x^2/(a*x^2 + b*x^3 + c*x^4)^(3/2),x]","\frac{x \left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)-2 \sqrt{a} x \left(-2 a c+b^2+b c x\right)}{a^{3/2} \left(4 a c-b^2\right) \sqrt{x^2 (a+x (b+c x))}}","\frac{2 x \left(-2 a c+b^2+b c x\right)}{a \left(b^2-4 a c\right) \sqrt{a x^2+b x^3+c x^4}}-\frac{\tanh ^{-1}\left(\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right)}{a^{3/2}}",1,"(-2*Sqrt[a]*x*(b^2 - 2*a*c + b*c*x) + (b^2 - 4*a*c)*x*Sqrt[a + x*(b + c*x)]*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])])/(a^(3/2)*(-b^2 + 4*a*c)*Sqrt[x^2*(a + x*(b + c*x))])","A",1
61,1,138,144,0.0995041,"\int \frac{x}{\left(a x^2+b x^3+c x^4\right)^{3/2}} \, dx","Integrate[x/(a*x^2 + b*x^3 + c*x^4)^(3/2),x]","\frac{2 \sqrt{a} \left(-4 a^2 c+a \left(b^2-10 b c x-8 c^2 x^2\right)+3 b^2 x (b+c x)\right)-3 b x \left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)}{2 a^{5/2} \left(4 a c-b^2\right) \sqrt{x^2 (a+x (b+c x))}}","\frac{3 b \tanh ^{-1}\left(\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right)}{2 a^{5/2}}-\frac{\left(3 b^2-8 a c\right) \sqrt{a x^2+b x^3+c x^4}}{a^2 x^2 \left(b^2-4 a c\right)}+\frac{2 \left(-2 a c+b^2+b c x\right)}{a \left(b^2-4 a c\right) \sqrt{a x^2+b x^3+c x^4}}",1,"(2*Sqrt[a]*(-4*a^2*c + 3*b^2*x*(b + c*x) + a*(b^2 - 10*b*c*x - 8*c^2*x^2)) - 3*b*(b^2 - 4*a*c)*x*Sqrt[a + x*(b + c*x)]*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])])/(2*a^(5/2)*(-b^2 + 4*a*c)*Sqrt[x^2*(a + x*(b + c*x))])","A",1
62,1,181,209,0.1570431,"\int \frac{1}{\left(a x^2+b x^3+c x^4\right)^{3/2}} \, dx","Integrate[(a*x^2 + b*x^3 + c*x^4)^(-3/2),x]","-\frac{3 x^2 \left(16 a^2 c^2-24 a b^2 c+5 b^4\right) \sqrt{a+x (b+c x)} \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)+2 \sqrt{a} \left(-8 a^3 c+2 a^2 \left(b^2+10 b c x-12 c^2 x^2\right)+a b x \left(-5 b^2+62 b c x+52 c^2 x^2\right)-15 b^3 x^2 (b+c x)\right)}{8 a^{7/2} x \left(b^2-4 a c\right) \sqrt{x^2 (a+x (b+c x))}}","-\frac{3 \left(5 b^2-4 a c\right) \tanh ^{-1}\left(\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right)}{8 a^{7/2}}+\frac{b \left(15 b^2-52 a c\right) \sqrt{a x^2+b x^3+c x^4}}{4 a^3 x^2 \left(b^2-4 a c\right)}-\frac{\left(5 b^2-12 a c\right) \sqrt{a x^2+b x^3+c x^4}}{2 a^2 x^3 \left(b^2-4 a c\right)}+\frac{2 \left(-2 a c+b^2+b c x\right)}{a x \left(b^2-4 a c\right) \sqrt{a x^2+b x^3+c x^4}}",1,"-1/8*(2*Sqrt[a]*(-8*a^3*c - 15*b^3*x^2*(b + c*x) + 2*a^2*(b^2 + 10*b*c*x - 12*c^2*x^2) + a*b*x*(-5*b^2 + 62*b*c*x + 52*c^2*x^2)) + 3*(5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*x^2*Sqrt[a + x*(b + c*x)]*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])])/(a^(7/2)*(b^2 - 4*a*c)*x*Sqrt[x^2*(a + x*(b + c*x))])","A",1
63,1,225,271,0.2027129,"\int \frac{1}{x \left(a x^2+b x^3+c x^4\right)^{3/2}} \, dx","Integrate[1/(x*(a*x^2 + b*x^3 + c*x^4)^(3/2)),x]","\frac{2 \sqrt{a} \left(-32 a^4 c+8 a^3 \left(b^2+7 b c x+16 c^2 x^2\right)+2 a^2 x \left(-7 b^3-86 b^2 c x+244 b c^2 x^2+128 c^3 x^3\right)+5 a b^2 x^2 \left(7 b^2-106 b c x-92 c^2 x^2\right)+105 b^4 x^3 (b+c x)\right)-15 b x^3 \left(48 a^2 c^2-40 a b^2 c+7 b^4\right) \sqrt{a+x (b+c x)} \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)}{48 a^{9/2} x^2 \left(4 a c-b^2\right) \sqrt{x^2 (a+x (b+c x))}}","\frac{5 b \left(7 b^2-12 a c\right) \tanh ^{-1}\left(\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right)}{16 a^{9/2}}+\frac{b \left(35 b^2-116 a c\right) \sqrt{a x^2+b x^3+c x^4}}{12 a^3 x^3 \left(b^2-4 a c\right)}-\frac{\left(7 b^2-16 a c\right) \sqrt{a x^2+b x^3+c x^4}}{3 a^2 x^4 \left(b^2-4 a c\right)}-\frac{\left(256 a^2 c^2-460 a b^2 c+105 b^4\right) \sqrt{a x^2+b x^3+c x^4}}{24 a^4 x^2 \left(b^2-4 a c\right)}+\frac{2 \left(-2 a c+b^2+b c x\right)}{a x^2 \left(b^2-4 a c\right) \sqrt{a x^2+b x^3+c x^4}}",1,"(2*Sqrt[a]*(-32*a^4*c + 105*b^4*x^3*(b + c*x) + 5*a*b^2*x^2*(7*b^2 - 106*b*c*x - 92*c^2*x^2) + 8*a^3*(b^2 + 7*b*c*x + 16*c^2*x^2) + 2*a^2*x*(-7*b^3 - 86*b^2*c*x + 244*b*c^2*x^2 + 128*c^3*x^3)) - 15*b*(7*b^4 - 40*a*b^2*c + 48*a^2*c^2)*x^3*Sqrt[a + x*(b + c*x)]*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])])/(48*a^(9/2)*(-b^2 + 4*a*c)*x^2*Sqrt[x^2*(a + x*(b + c*x))])","A",1
64,1,272,343,0.2494205,"\int \frac{1}{x^2 \left(a x^2+b x^3+c x^4\right)^{3/2}} \, dx","Integrate[1/(x^2*(a*x^2 + b*x^3 + c*x^4)^(3/2)),x]","\frac{15 x^4 \left(-64 a^3 c^3+240 a^2 b^2 c^2-140 a b^4 c+21 b^6\right) \sqrt{a+x (b+c x)} \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)-2 \sqrt{a} \left(64 a^5 c-16 a^4 \left(b^2+6 b c x+10 c^2 x^2\right)+8 a^3 x \left(3 b^3+26 b^2 c x+98 b c^2 x^2-60 c^3 x^3\right)+2 a^2 b x^2 \left(-21 b^3-308 b^2 c x+1352 b c^2 x^2+904 c^3 x^3\right)+105 a b^3 x^3 \left(b^2-18 b c x-16 c^2 x^2\right)+315 b^5 x^4 (b+c x)\right)}{128 a^{11/2} x^3 \left(4 a c-b^2\right) \sqrt{x^2 (a+x (b+c x))}}","\frac{b \left(21 b^2-68 a c\right) \sqrt{a x^2+b x^3+c x^4}}{8 a^3 x^4 \left(b^2-4 a c\right)}-\frac{\left(9 b^2-20 a c\right) \sqrt{a x^2+b x^3+c x^4}}{4 a^2 x^5 \left(b^2-4 a c\right)}-\frac{15 \left(16 a^2 c^2-56 a b^2 c+21 b^4\right) \tanh ^{-1}\left(\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right)}{128 a^{11/2}}+\frac{b \left(1808 a^2 c^2-1680 a b^2 c+315 b^4\right) \sqrt{a x^2+b x^3+c x^4}}{64 a^5 x^2 \left(b^2-4 a c\right)}-\frac{\left(240 a^2 c^2-448 a b^2 c+105 b^4\right) \sqrt{a x^2+b x^3+c x^4}}{32 a^4 x^3 \left(b^2-4 a c\right)}+\frac{2 \left(-2 a c+b^2+b c x\right)}{a x^3 \left(b^2-4 a c\right) \sqrt{a x^2+b x^3+c x^4}}",1,"(-2*Sqrt[a]*(64*a^5*c + 315*b^5*x^4*(b + c*x) + 105*a*b^3*x^3*(b^2 - 18*b*c*x - 16*c^2*x^2) - 16*a^4*(b^2 + 6*b*c*x + 10*c^2*x^2) + 8*a^3*x*(3*b^3 + 26*b^2*c*x + 98*b*c^2*x^2 - 60*c^3*x^3) + 2*a^2*b*x^2*(-21*b^3 - 308*b^2*c*x + 1352*b*c^2*x^2 + 904*c^3*x^3)) + 15*(21*b^6 - 140*a*b^4*c + 240*a^2*b^2*c^2 - 64*a^3*c^3)*x^4*Sqrt[a + x*(b + c*x)]*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])])/(128*a^(11/2)*(-b^2 + 4*a*c)*x^3*Sqrt[x^2*(a + x*(b + c*x))])","A",1
65,1,34,37,0.0257637,"\int x^m \left(a x+b x^3+c x^5\right) \, dx","Integrate[x^m*(a*x + b*x^3 + c*x^5),x]","x^{m+2} \left(\frac{a}{m+2}+\frac{b x^2}{m+4}+\frac{c x^4}{m+6}\right)","\frac{a x^{m+2}}{m+2}+\frac{b x^{m+4}}{m+4}+\frac{c x^{m+6}}{m+6}",1,"x^(2 + m)*(a/(2 + m) + (b*x^2)/(4 + m) + (c*x^4)/(6 + m))","A",1
66,1,25,25,0.0017601,"\int x^2 \left(a x+b x^3+c x^5\right) \, dx","Integrate[x^2*(a*x + b*x^3 + c*x^5),x]","\frac{a x^4}{4}+\frac{b x^6}{6}+\frac{c x^8}{8}","\frac{a x^4}{4}+\frac{b x^6}{6}+\frac{c x^8}{8}",1,"(a*x^4)/4 + (b*x^6)/6 + (c*x^8)/8","A",1
67,1,25,25,0.0019795,"\int x \left(a x+b x^3+c x^5\right) \, dx","Integrate[x*(a*x + b*x^3 + c*x^5),x]","\frac{a x^3}{3}+\frac{b x^5}{5}+\frac{c x^7}{7}","\frac{a x^3}{3}+\frac{b x^5}{5}+\frac{c x^7}{7}",1,"(a*x^3)/3 + (b*x^5)/5 + (c*x^7)/7","A",1
68,1,25,25,0.0000744,"\int \left(a x+b x^3+c x^5\right) \, dx","Integrate[a*x + b*x^3 + c*x^5,x]","\frac{a x^2}{2}+\frac{b x^4}{4}+\frac{c x^6}{6}","\frac{a x^2}{2}+\frac{b x^4}{4}+\frac{c x^6}{6}",1,"(a*x^2)/2 + (b*x^4)/4 + (c*x^6)/6","A",1
69,1,20,20,0.0012913,"\int \frac{a x+b x^3+c x^5}{x} \, dx","Integrate[(a*x + b*x^3 + c*x^5)/x,x]","a x+\frac{b x^3}{3}+\frac{c x^5}{5}","a x+\frac{b x^3}{3}+\frac{c x^5}{5}",1,"a*x + (b*x^3)/3 + (c*x^5)/5","A",1
70,1,21,21,0.0016312,"\int \frac{a x+b x^3+c x^5}{x^2} \, dx","Integrate[(a*x + b*x^3 + c*x^5)/x^2,x]","a \log (x)+\frac{b x^2}{2}+\frac{c x^4}{4}","a \log (x)+\frac{b x^2}{2}+\frac{c x^4}{4}",1,"(b*x^2)/2 + (c*x^4)/4 + a*Log[x]","A",1
71,1,18,18,0.0018991,"\int \frac{a x+b x^3+c x^5}{x^3} \, dx","Integrate[(a*x + b*x^3 + c*x^5)/x^3,x]","-\frac{a}{x}+b x+\frac{c x^3}{3}","-\frac{a}{x}+b x+\frac{c x^3}{3}",1,"-(a/x) + b*x + (c*x^3)/3","A",1
72,1,69,76,0.0710845,"\int x^m \left(a x+b x^3+c x^5\right)^2 \, dx","Integrate[x^m*(a*x + b*x^3 + c*x^5)^2,x]","x^{m+3} \left(\frac{a^2}{m+3}+\frac{x^4 \left(2 a c+b^2\right)}{m+7}+\frac{2 a b x^2}{m+5}+\frac{2 b c x^6}{m+9}+\frac{c^2 x^8}{m+11}\right)","\frac{a^2 x^{m+3}}{m+3}+\frac{x^{m+7} \left(2 a c+b^2\right)}{m+7}+\frac{2 a b x^{m+5}}{m+5}+\frac{2 b c x^{m+9}}{m+9}+\frac{c^2 x^{m+11}}{m+11}",1,"x^(3 + m)*(a^2/(3 + m) + (2*a*b*x^2)/(5 + m) + ((b^2 + 2*a*c)*x^4)/(7 + m) + (2*b*c*x^6)/(9 + m) + (c^2*x^8)/(11 + m))","A",1
73,1,54,54,0.0066674,"\int x^2 \left(a x+b x^3+c x^5\right)^2 \, dx","Integrate[x^2*(a*x + b*x^3 + c*x^5)^2,x]","\frac{a^2 x^5}{5}+\frac{1}{9} x^9 \left(2 a c+b^2\right)+\frac{2}{7} a b x^7+\frac{2}{11} b c x^{11}+\frac{c^2 x^{13}}{13}","\frac{a^2 x^5}{5}+\frac{1}{9} x^9 \left(2 a c+b^2\right)+\frac{2}{7} a b x^7+\frac{2}{11} b c x^{11}+\frac{c^2 x^{13}}{13}",1,"(a^2*x^5)/5 + (2*a*b*x^7)/7 + ((b^2 + 2*a*c)*x^9)/9 + (2*b*c*x^11)/11 + (c^2*x^13)/13","A",1
74,1,48,54,0.0082286,"\int x \left(a x+b x^3+c x^5\right)^2 \, dx","Integrate[x*(a*x + b*x^3 + c*x^5)^2,x]","\frac{1}{120} x^4 \left(30 a^2+15 x^4 \left(2 a c+b^2\right)+40 a b x^2+24 b c x^6+10 c^2 x^8\right)","\frac{a^2 x^4}{4}+\frac{1}{8} x^8 \left(2 a c+b^2\right)+\frac{1}{3} a b x^6+\frac{1}{5} b c x^{10}+\frac{c^2 x^{12}}{12}",1,"(x^4*(30*a^2 + 40*a*b*x^2 + 15*(b^2 + 2*a*c)*x^4 + 24*b*c*x^6 + 10*c^2*x^8))/120","A",1
75,1,54,54,0.0067197,"\int \left(a x+b x^3+c x^5\right)^2 \, dx","Integrate[(a*x + b*x^3 + c*x^5)^2,x]","\frac{a^2 x^3}{3}+\frac{1}{7} x^7 \left(2 a c+b^2\right)+\frac{2}{5} a b x^5+\frac{2}{9} b c x^9+\frac{c^2 x^{11}}{11}","\frac{a^2 x^3}{3}+\frac{1}{7} x^7 \left(2 a c+b^2\right)+\frac{2}{5} a b x^5+\frac{2}{9} b c x^9+\frac{c^2 x^{11}}{11}",1,"(a^2*x^3)/3 + (2*a*b*x^5)/5 + ((b^2 + 2*a*c)*x^7)/7 + (2*b*c*x^9)/9 + (c^2*x^11)/11","A",1
76,1,48,54,0.0080126,"\int \frac{\left(a x+b x^3+c x^5\right)^2}{x} \, dx","Integrate[(a*x + b*x^3 + c*x^5)^2/x,x]","\frac{1}{60} x^2 \left(30 a^2+10 x^4 \left(2 a c+b^2\right)+30 a b x^2+15 b c x^6+6 c^2 x^8\right)","\frac{a^2 x^2}{2}+\frac{1}{6} x^6 \left(2 a c+b^2\right)+\frac{1}{2} a b x^4+\frac{1}{4} b c x^8+\frac{c^2 x^{10}}{10}",1,"(x^2*(30*a^2 + 30*a*b*x^2 + 10*(b^2 + 2*a*c)*x^4 + 15*b*c*x^6 + 6*c^2*x^8))/60","A",1
77,1,49,49,0.0051872,"\int \frac{\left(a x+b x^3+c x^5\right)^2}{x^2} \, dx","Integrate[(a*x + b*x^3 + c*x^5)^2/x^2,x]","a^2 x+\frac{1}{5} x^5 \left(2 a c+b^2\right)+\frac{2}{3} a b x^3+\frac{2}{7} b c x^7+\frac{c^2 x^9}{9}","a^2 x+\frac{1}{5} x^5 \left(2 a c+b^2\right)+\frac{2}{3} a b x^3+\frac{2}{7} b c x^7+\frac{c^2 x^9}{9}",1,"a^2*x + (2*a*b*x^3)/3 + ((b^2 + 2*a*c)*x^5)/5 + (2*b*c*x^7)/7 + (c^2*x^9)/9","A",1
78,1,93,100,0.0852365,"\int \frac{x^8}{a x+b x^3+c x^5} \, dx","Integrate[x^8/(a*x + b*x^3 + c*x^5),x]","\frac{-\frac{2 b \left(b^2-3 a c\right) \tan ^{-1}\left(\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+\left(b^2-a c\right) \log \left(a+b x^2+c x^4\right)+c x^2 \left(c x^2-2 b\right)}{4 c^3}","\frac{b \left(b^2-3 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{2 c^3 \sqrt{b^2-4 a c}}+\frac{\left(b^2-a c\right) \log \left(a+b x^2+c x^4\right)}{4 c^3}-\frac{b x^2}{2 c^2}+\frac{x^4}{4 c}",1,"(c*x^2*(-2*b + c*x^2) - (2*b*(b^2 - 3*a*c)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + (b^2 - a*c)*Log[a + b*x^2 + c*x^4])/(4*c^3)","A",1
79,1,250,203,0.1553605,"\int \frac{x^7}{a x+b x^3+c x^5} \, dx","Integrate[x^7/(a*x + b*x^3 + c*x^5),x]","\frac{\left(b^2 \sqrt{b^2-4 a c}-a c \sqrt{b^2-4 a c}+3 a b c-b^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} c^{5/2} \sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left(b^2 \sqrt{b^2-4 a c}-a c \sqrt{b^2-4 a c}-3 a b c+b^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} c^{5/2} \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{b x}{c^2}+\frac{x^3}{3 c}","\frac{\left(-\frac{b \left(b^2-3 a c\right)}{\sqrt{b^2-4 a c}}-a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} c^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left(\frac{b \left(b^2-3 a c\right)}{\sqrt{b^2-4 a c}}-a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} c^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{b x}{c^2}+\frac{x^3}{3 c}",1,"-((b*x)/c^2) + x^3/(3*c) + ((-b^3 + 3*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - a*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(5/2)*Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((b^3 - 3*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - a*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(5/2)*Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]])","A",1
80,1,78,81,0.0444634,"\int \frac{x^6}{a x+b x^3+c x^5} \, dx","Integrate[x^6/(a*x + b*x^3 + c*x^5),x]","\frac{\frac{2 \left(b^2-2 a c\right) \tan ^{-1}\left(\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}-b \log \left(a+b x^2+c x^4\right)+2 c x^2}{4 c^2}","-\frac{\left(b^2-2 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{2 c^2 \sqrt{b^2-4 a c}}-\frac{b \log \left(a+b x^2+c x^4\right)}{4 c^2}+\frac{x^2}{2 c}",1,"(2*c*x^2 + (2*(b^2 - 2*a*c)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] - b*Log[a + b*x^2 + c*x^4])/(4*c^2)","A",1
81,1,202,179,0.1083698,"\int \frac{x^5}{a x+b x^3+c x^5} \, dx","Integrate[x^5/(a*x + b*x^3 + c*x^5),x]","-\frac{\left(b \sqrt{b^2-4 a c}+2 a c-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} c^{3/2} \sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left(b \sqrt{b^2-4 a c}-2 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} c^{3/2} \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x}{c}","-\frac{\left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} c^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} c^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x}{c}",1,"x/c - ((-b^2 + 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(3/2)*Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - ((b^2 - 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(3/2)*Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]])","A",1
82,1,62,63,0.0234188,"\int \frac{x^4}{a x+b x^3+c x^5} \, dx","Integrate[x^4/(a*x + b*x^3 + c*x^5),x]","\frac{\log \left(a+b x^2+c x^4\right)-\frac{2 b \tan ^{-1}\left(\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}}{4 c}","\frac{b \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{2 c \sqrt{b^2-4 a c}}+\frac{\log \left(a+b x^2+c x^4\right)}{4 c}",1,"((-2*b*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + Log[a + b*x^2 + c*x^4])/(4*c)","A",1
83,1,165,150,0.0819848,"\int \frac{x^3}{a x+b x^3+c x^5} \, dx","Integrate[x^3/(a*x + b*x^3 + c*x^5),x]","\frac{\left(\sqrt{b^2-4 a c}-b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{\sqrt{b^2-4 a c}+b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}","\frac{\sqrt{\sqrt{b^2-4 a c}+b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}-\frac{\sqrt{b-\sqrt{b^2-4 a c}} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}",1,"((-b + Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[c]*Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[b + Sqrt[b^2 - 4*a*c]]*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[c]*Sqrt[b^2 - 4*a*c])","A",1
84,1,39,36,0.0083634,"\int \frac{x^2}{a x+b x^3+c x^5} \, dx","Integrate[x^2/(a*x + b*x^3 + c*x^5),x]","\frac{\tan ^{-1}\left(\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}","-\frac{\tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c}}",1,"ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]]/Sqrt[-b^2 + 4*a*c]","A",1
85,1,129,150,0.0793446,"\int \frac{x}{a x+b x^3+c x^5} \, dx","Integrate[x/(a*x + b*x^3 + c*x^5),x]","\frac{\sqrt{2} \sqrt{c} \left(\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{b^2-4 a c}}","\frac{\sqrt{2} \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}",1,"(Sqrt[2]*Sqrt[c]*(ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]]/Sqrt[b - Sqrt[b^2 - 4*a*c]] - ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]]/Sqrt[b + Sqrt[b^2 - 4*a*c]]))/Sqrt[b^2 - 4*a*c]","A",1
86,1,113,69,0.0605667,"\int \frac{1}{a x+b x^3+c x^5} \, dx","Integrate[(a*x + b*x^3 + c*x^5)^(-1),x]","\frac{-\left(\sqrt{b^2-4 a c}+b\right) \log \left(-\sqrt{b^2-4 a c}+b+2 c x^2\right)+\left(b-\sqrt{b^2-4 a c}\right) \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)+4 \log (x) \sqrt{b^2-4 a c}}{4 a \sqrt{b^2-4 a c}}","\frac{b \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{2 a \sqrt{b^2-4 a c}}-\frac{\log \left(a+b x^2+c x^4\right)}{4 a}+\frac{\log (x)}{a}",1,"(4*Sqrt[b^2 - 4*a*c]*Log[x] - (b + Sqrt[b^2 - 4*a*c])*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2] + (b - Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(4*a*Sqrt[b^2 - 4*a*c])","A",1
87,1,191,174,0.4006791,"\int \frac{1}{x \left(a x+b x^3+c x^5\right)} \, dx","Integrate[1/(x*(a*x + b*x^3 + c*x^5)),x]","-\frac{\frac{\sqrt{2} \sqrt{c} \left(\sqrt{b^2-4 a c}+b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left(\sqrt{b^2-4 a c}-b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2}{x}}{2 a}","-\frac{\sqrt{c} \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} a \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} a \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{1}{a x}",1,"-1/2*(2/x + (Sqrt[2]*Sqrt[c]*(b + Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(-b + Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/a","A",1
88,1,135,89,0.1204953,"\int \frac{1}{x^2 \left(a x+b x^3+c x^5\right)} \, dx","Integrate[1/(x^2*(a*x + b*x^3 + c*x^5)),x]","\frac{\frac{\left(b \sqrt{b^2-4 a c}-2 a c+b^2\right) \log \left(-\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\sqrt{b^2-4 a c}}+\frac{\left(b \sqrt{b^2-4 a c}+2 a c-b^2\right) \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\sqrt{b^2-4 a c}}-\frac{2 a}{x^2}-4 b \log (x)}{4 a^2}","-\frac{\left(b^2-2 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{2 a^2 \sqrt{b^2-4 a c}}+\frac{b \log \left(a+b x^2+c x^4\right)}{4 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{2 a x^2}",1,"((-2*a)/x^2 - 4*b*Log[x] + ((b^2 - 2*a*c + b*Sqrt[b^2 - 4*a*c])*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/Sqrt[b^2 - 4*a*c] + ((-b^2 + 2*a*c + b*Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/Sqrt[b^2 - 4*a*c])/(4*a^2)","A",1
89,1,151,166,0.1862485,"\int \frac{x^{11}}{\left(a x+b x^3+c x^5\right)^2} \, dx","Integrate[x^11/(a*x + b*x^3 + c*x^5)^2,x]","\frac{-\frac{2 \left(6 a^2 c^2-6 a b^2 c+b^4\right) \tan ^{-1}\left(\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{3/2}}+\frac{a^2 c \left(3 b-2 c x^2\right)-a b^2 \left(b-4 c x^2\right)+b^4 \left(-x^2\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-b \log \left(a+b x^2+c x^4\right)+c x^2}{2 c^3}","-\frac{\left(6 a^2 c^2-6 a b^2 c+b^4\right) \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{c^3 \left(b^2-4 a c\right)^{3/2}}+\frac{x^2 \left(b^2-3 a c\right)}{c^2 \left(b^2-4 a c\right)}-\frac{b x^4}{2 c \left(b^2-4 a c\right)}+\frac{x^6 \left(2 a+b x^2\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-\frac{b \log \left(a+b x^2+c x^4\right)}{2 c^3}",1,"(c*x^2 + (-(b^4*x^2) - a*b^2*(b - 4*c*x^2) + a^2*c*(3*b - 2*c*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - (2*(b^4 - 6*a*b^2*c + 6*a^2*c^2)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) - b*Log[a + b*x^2 + c*x^4])/(2*c^3)","A",1
90,1,327,331,0.6627731,"\int \frac{x^{10}}{\left(a x+b x^3+c x^5\right)^2} \, dx","Integrate[x^10/(a*x + b*x^3 + c*x^5)^2,x]","\frac{-\frac{2 \sqrt{c} x \left(2 a^2 c-a b \left(b-3 c x^2\right)+b^3 \left(-x^2\right)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-\frac{\sqrt{2} \left(-20 a^2 c^2+19 a b^2 c-13 a b c \sqrt{b^2-4 a c}+3 b^3 \sqrt{b^2-4 a c}-3 b^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \left(20 a^2 c^2-19 a b^2 c-13 a b c \sqrt{b^2-4 a c}+3 b^3 \sqrt{b^2-4 a c}+3 b^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+4 \sqrt{c} x}{4 c^{5/2}}","-\frac{\left(-\frac{20 a^2 c^2-19 a b^2 c+3 b^4}{\sqrt{b^2-4 a c}}-13 a b c+3 b^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} c^{5/2} \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left(\frac{20 a^2 c^2-19 a b^2 c+3 b^4}{\sqrt{b^2-4 a c}}-13 a b c+3 b^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} c^{5/2} \left(b^2-4 a c\right) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x \left(3 b^2-10 a c\right)}{2 c^2 \left(b^2-4 a c\right)}-\frac{b x^3}{2 c \left(b^2-4 a c\right)}+\frac{x^5 \left(2 a+b x^2\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"(4*Sqrt[c]*x - (2*Sqrt[c]*x*(2*a^2*c - b^3*x^2 - a*b*(b - 3*c*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - (Sqrt[2]*(-3*b^4 + 19*a*b^2*c - 20*a^2*c^2 + 3*b^3*Sqrt[b^2 - 4*a*c] - 13*a*b*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*(3*b^4 - 19*a*b^2*c + 20*a^2*c^2 + 3*b^3*Sqrt[b^2 - 4*a*c] - 13*a*b*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(4*c^(5/2))","A",1
91,1,121,132,0.1675553,"\int \frac{x^9}{\left(a x+b x^3+c x^5\right)^2} \, dx","Integrate[x^9/(a*x + b*x^3 + c*x^5)^2,x]","\frac{\frac{2 \left(-2 a^2 c+a b \left(b-3 c x^2\right)+b^3 x^2\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{2 b \left(b^2-6 a c\right) \tan ^{-1}\left(\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{3/2}}+\log \left(a+b x^2+c x^4\right)}{4 c^2}","\frac{b \left(b^2-6 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{2 c^2 \left(b^2-4 a c\right)^{3/2}}-\frac{b x^2}{2 c \left(b^2-4 a c\right)}+\frac{x^4 \left(2 a+b x^2\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\log \left(a+b x^2+c x^4\right)}{4 c^2}",1,"((2*(-2*a^2*c + b^3*x^2 + a*b*(b - 3*c*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (2*b*(b^2 - 6*a*c)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + Log[a + b*x^2 + c*x^4])/(4*c^2)","A",1
92,1,282,271,0.5127601,"\int \frac{x^8}{\left(a x+b x^3+c x^5\right)^2} \, dx","Integrate[x^8/(a*x + b*x^3 + c*x^5)^2,x]","\frac{-\frac{2 \sqrt{c} x \left(a \left(b-2 c x^2\right)+b^2 x^2\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \left(b^2 \sqrt{b^2-4 a c}-6 a c \sqrt{b^2-4 a c}+8 a b c-b^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \left(b^2 \sqrt{b^2-4 a c}-6 a c \sqrt{b^2-4 a c}-8 a b c+b^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}}{4 c^{3/2}}","\frac{\left(-\frac{b \left(b^2-8 a c\right)}{\sqrt{b^2-4 a c}}-6 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} c^{3/2} \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left(\frac{b \left(b^2-8 a c\right)}{\sqrt{b^2-4 a c}}-6 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} c^{3/2} \left(b^2-4 a c\right) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x^3 \left(2 a+b x^2\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-\frac{b x}{2 c \left(b^2-4 a c\right)}",1,"((-2*Sqrt[c]*x*(b^2*x^2 + a*(b - 2*c*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*(-b^3 + 8*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - 6*a*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(b^3 - 8*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - 6*a*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(4*c^(3/2))","A",1
93,1,93,78,0.0857999,"\int \frac{x^7}{\left(a x+b x^3+c x^5\right)^2} \, dx","Integrate[x^7/(a*x + b*x^3 + c*x^5)^2,x]","\frac{2 a \tan ^{-1}\left(\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{3/2}}+\frac{a \left(b-2 c x^2\right)+b^2 x^2}{2 c \left(4 a c-b^2\right) \left(a+b x^2+c x^4\right)}","\frac{2 a \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/2}}+\frac{x^2 \left(2 a+b x^2\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"(b^2*x^2 + a*(b - 2*c*x^2))/(2*c*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)) + (2*a*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2)","A",1
94,1,235,237,0.4070226,"\int \frac{x^6}{\left(a x+b x^3+c x^5\right)^2} \, dx","Integrate[x^6/(a*x + b*x^3 + c*x^5)^2,x]","\frac{1}{4} \left(\frac{2 \left(2 a x+b x^3\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \left(b \sqrt{b^2-4 a c}-4 a c-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{c} \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \left(b \sqrt{b^2-4 a c}+4 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{c} \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}\right)","\frac{x \left(2 a+b x^2\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\left(b-\frac{4 a c+b^2}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} \sqrt{c} \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left(b \sqrt{b^2-4 a c}+4 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} \sqrt{c} \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}",1,"((2*(2*a*x + b*x^3))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*(-b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(b^2 + 4*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/4","A",1
95,1,79,75,0.0655843,"\int \frac{x^5}{\left(a x+b x^3+c x^5\right)^2} \, dx","Integrate[x^5/(a*x + b*x^3 + c*x^5)^2,x]","\frac{2 a+b x^2}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-\frac{b \tan ^{-1}\left(\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{3/2}}","\frac{2 a+b x^2}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-\frac{b \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/2}}",1,"(2*a + b*x^2)/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - (b*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2)","A",1
96,1,222,221,0.4337703,"\int \frac{x^4}{\left(a x+b x^3+c x^5\right)^2} \, dx","Integrate[x^4/(a*x + b*x^3 + c*x^5)^2,x]","\frac{-b x-2 c x^3}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-\frac{\sqrt{c} \left(\sqrt{b^2-4 a c}-2 b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left(\sqrt{b^2-4 a c}+2 b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}","-\frac{x \left(b+2 c x^2\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{c} \left(2 b-\sqrt{b^2-4 a c}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left(\sqrt{b^2-4 a c}+2 b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}",1,"(-(b*x) - 2*c*x^3)/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - (Sqrt[c]*(-2*b + Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[c]*(2*b + Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])","A",1
97,1,79,74,0.0801031,"\int \frac{x^3}{\left(a x+b x^3+c x^5\right)^2} \, dx","Integrate[x^3/(a*x + b*x^3 + c*x^5)^2,x]","-\frac{\frac{4 c \tan ^{-1}\left(\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+\frac{b+2 c x^2}{a+b x^2+c x^4}}{2 \left(b^2-4 a c\right)}","\frac{2 c \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/2}}-\frac{b+2 c x^2}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"-1/2*((b + 2*c*x^2)/(a + b*x^2 + c*x^4) + (4*c*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(b^2 - 4*a*c)","A",1
98,1,243,252,0.4181019,"\int \frac{x^2}{\left(a x+b x^3+c x^5\right)^2} \, dx","Integrate[x^2/(a*x + b*x^3 + c*x^5)^2,x]","\frac{\frac{2 x \left(-2 a c+b^2+b c x^2\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \sqrt{c} \left(b \sqrt{b^2-4 a c}-12 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left(b \sqrt{b^2-4 a c}+12 a c-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}}{4 a}","\frac{x \left(-2 a c+b^2+b c x^2\right)}{2 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{c} \left(b \sqrt{b^2-4 a c}-12 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} a \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left(-b \sqrt{b^2-4 a c}-12 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} a \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}",1,"((2*x*(b^2 - 2*a*c + b*c*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*Sqrt[c]*(b^2 - 12*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(-b^2 + 12*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(4*a)","A",1
99,1,207,122,0.3305321,"\int \frac{x}{\left(a x+b x^3+c x^5\right)^2} \, dx","Integrate[x/(a*x + b*x^3 + c*x^5)^2,x]","\frac{\frac{2 a \left(-2 a c+b^2+b c x^2\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-\frac{\left(b^2 \sqrt{b^2-4 a c}-4 a c \sqrt{b^2-4 a c}-6 a b c+b^3\right) \log \left(-\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}+\frac{\left(-b^2 \sqrt{b^2-4 a c}+4 a c \sqrt{b^2-4 a c}-6 a b c+b^3\right) \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}+4 \log (x)}{4 a^2}","\frac{b \left(b^2-6 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{2 a^2 \left(b^2-4 a c\right)^{3/2}}-\frac{\log \left(a+b x^2+c x^4\right)}{4 a^2}+\frac{\log (x)}{a^2}+\frac{-2 a c+b^2+b c x^2}{2 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"((2*a*(b^2 - 2*a*c + b*c*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + 4*Log[x] - ((b^3 - 6*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - 4*a*c*Sqrt[b^2 - 4*a*c])*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2) + ((b^3 - 6*a*b*c - b^2*Sqrt[b^2 - 4*a*c] + 4*a*c*Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/(4*a^2)","A",1
100,1,302,308,0.6158434,"\int \frac{1}{\left(a x+b x^3+c x^5\right)^2} \, dx","Integrate[(a*x + b*x^3 + c*x^5)^(-2),x]","\frac{-\frac{2 x \left(-3 a b c-2 a c^2 x^2+b^3+b^2 c x^2\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \sqrt{c} \left(-3 b^2 \sqrt{b^2-4 a c}+10 a c \sqrt{b^2-4 a c}+16 a b c-3 b^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left(-3 b^2 \sqrt{b^2-4 a c}+10 a c \sqrt{b^2-4 a c}-16 a b c+3 b^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{4}{x}}{4 a^2}","-\frac{3 b^2-10 a c}{2 a^2 x \left(b^2-4 a c\right)}-\frac{\sqrt{c} \left(\left(3 b^2-10 a c\right) \sqrt{b^2-4 a c}-16 a b c+3 b^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} a^2 \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left(-\left(3 b^2-10 a c\right) \sqrt{b^2-4 a c}-16 a b c+3 b^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} a^2 \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{-2 a c+b^2+b c x^2}{2 a x \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"(-4/x - (2*x*(b^3 - 3*a*b*c + b^2*c*x^2 - 2*a*c^2*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*Sqrt[c]*(-3*b^3 + 16*a*b*c - 3*b^2*Sqrt[b^2 - 4*a*c] + 10*a*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(3*b^3 - 16*a*b*c - 3*b^2*Sqrt[b^2 - 4*a*c] + 10*a*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(4*a^2)","A",1
101,1,248,162,0.2643828,"\int \frac{1}{x \left(a x+b x^3+c x^5\right)^2} \, dx","Integrate[1/(x*(a*x + b*x^3 + c*x^5)^2),x]","\frac{\frac{\left(6 a^2 c^2-6 a b^2 c-4 a b c \sqrt{b^2-4 a c}+b^3 \sqrt{b^2-4 a c}+b^4\right) \log \left(-\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}+\frac{\left(-6 a^2 c^2+6 a b^2 c-4 a b c \sqrt{b^2-4 a c}+b^3 \sqrt{b^2-4 a c}-b^4\right) \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}-\frac{a \left(-3 a b c-2 a c^2 x^2+b^3+b^2 c x^2\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-\frac{a}{x^2}-4 b \log (x)}{2 a^3}","\frac{b \log \left(a+b x^2+c x^4\right)}{2 a^3}-\frac{2 b \log (x)}{a^3}-\frac{b^2-3 a c}{a^2 x^2 \left(b^2-4 a c\right)}-\frac{\left(6 a^2 c^2-6 a b^2 c+b^4\right) \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{a^3 \left(b^2-4 a c\right)^{3/2}}+\frac{-2 a c+b^2+b c x^2}{2 a x^2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"(-(a/x^2) - (a*(b^3 - 3*a*b*c + b^2*c*x^2 - 2*a*c^2*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - 4*b*Log[x] + ((b^4 - 6*a*b^2*c + 6*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 4*a*b*c*Sqrt[b^2 - 4*a*c])*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2) + ((-b^4 + 6*a*b^2*c - 6*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 4*a*b*c*Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/(2*a^3)","A",1
102,1,344,361,0.7146094,"\int \frac{1}{x^2 \left(a x+b x^3+c x^5\right)^2} \, dx","Integrate[1/(x^2*(a*x + b*x^3 + c*x^5)^2),x]","\frac{\frac{6 x \left(2 a^2 c^2-4 a b^2 c-3 a b c^2 x^2+b^4+b^3 c x^2\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{3 \sqrt{2} \sqrt{c} \left(28 a^2 c^2-29 a b^2 c-19 a b c \sqrt{b^2-4 a c}+5 b^3 \sqrt{b^2-4 a c}+5 b^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{2} \sqrt{c} \left(-28 a^2 c^2+29 a b^2 c-19 a b c \sqrt{b^2-4 a c}+5 b^3 \sqrt{b^2-4 a c}-5 b^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{4 a}{x^3}+\frac{24 b}{x}}{12 a^3}","\frac{b \left(5 b^2-19 a c\right)}{2 a^3 x \left(b^2-4 a c\right)}-\frac{5 b^2-14 a c}{6 a^2 x^3 \left(b^2-4 a c\right)}+\frac{\sqrt{c} \left(28 a^2 c^2-29 a b^2 c+b \left(5 b^2-19 a c\right) \sqrt{b^2-4 a c}+5 b^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} a^3 \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left(28 a^2 c^2-29 a b^2 c-b \left(5 b^2-19 a c\right) \sqrt{b^2-4 a c}+5 b^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} a^3 \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{-2 a c+b^2+b c x^2}{2 a x^3 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"((-4*a)/x^3 + (24*b)/x + (6*x*(b^4 - 4*a*b^2*c + 2*a^2*c^2 + b^3*c*x^2 - 3*a*b*c^2*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (3*Sqrt[2]*Sqrt[c]*(5*b^4 - 29*a*b^2*c + 28*a^2*c^2 + 5*b^3*Sqrt[b^2 - 4*a*c] - 19*a*b*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[2]*Sqrt[c]*(-5*b^4 + 29*a*b^2*c - 28*a^2*c^2 + 5*b^3*Sqrt[b^2 - 4*a*c] - 19*a*b*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(12*a^3)","A",1
103,1,328,219,0.3754413,"\int \frac{1}{x^3 \left(a x+b x^3+c x^5\right)^2} \, dx","Integrate[1/(x^3*(a*x + b*x^3 + c*x^5)^2),x]","\frac{\frac{2 a \left(2 a^2 c^2-4 a b^2 c-3 a b c^2 x^2+b^4+b^3 c x^2\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-\frac{\left(8 a^2 c^2 \sqrt{b^2-4 a c}+30 a^2 b c^2-20 a b^3 c-14 a b^2 c \sqrt{b^2-4 a c}+3 b^4 \sqrt{b^2-4 a c}+3 b^5\right) \log \left(-\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}+\frac{\left(-8 a^2 c^2 \sqrt{b^2-4 a c}+30 a^2 b c^2-20 a b^3 c+14 a b^2 c \sqrt{b^2-4 a c}-3 b^4 \sqrt{b^2-4 a c}+3 b^5\right) \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}-\frac{a^2}{x^4}+4 \log (x) \left(3 b^2-2 a c\right)+\frac{4 a b}{x^2}}{4 a^4}","-\frac{\left(3 b^2-2 a c\right) \log \left(a+b x^2+c x^4\right)}{4 a^4}+\frac{\log (x) \left(3 b^2-2 a c\right)}{a^4}+\frac{b \left(3 b^2-11 a c\right)}{2 a^3 x^2 \left(b^2-4 a c\right)}-\frac{3 b^2-8 a c}{4 a^2 x^4 \left(b^2-4 a c\right)}+\frac{b \left(30 a^2 c^2-20 a b^2 c+3 b^4\right) \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{2 a^4 \left(b^2-4 a c\right)^{3/2}}+\frac{-2 a c+b^2+b c x^2}{2 a x^4 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"(-(a^2/x^4) + (4*a*b)/x^2 + (2*a*(b^4 - 4*a*b^2*c + 2*a^2*c^2 + b^3*c*x^2 - 3*a*b*c^2*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + 4*(3*b^2 - 2*a*c)*Log[x] - ((3*b^5 - 20*a*b^3*c + 30*a^2*b*c^2 + 3*b^4*Sqrt[b^2 - 4*a*c] - 14*a*b^2*c*Sqrt[b^2 - 4*a*c] + 8*a^2*c^2*Sqrt[b^2 - 4*a*c])*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2) + ((3*b^5 - 20*a*b^3*c + 30*a^2*b*c^2 - 3*b^4*Sqrt[b^2 - 4*a*c] + 14*a*b^2*c*Sqrt[b^2 - 4*a*c] - 8*a^2*c^2*Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/(4*a^4)","A",1
104,1,170,142,0.0897541,"\int \frac{x}{\sqrt{a x+b x^3+c x^5}} \, dx","Integrate[x/Sqrt[a*x + b*x^3 + c*x^5],x]","\frac{2 x^2 \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{3}{4};\frac{1}{2},\frac{1}{2};\frac{7}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right)}{3 \sqrt{x \left(a+b x^2+c x^4\right)}}","\frac{2 x^2 \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} F_1\left(\frac{3}{4};\frac{1}{2},\frac{1}{2};\frac{7}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right)}{3 \sqrt{a x+b x^3+c x^5}}",1,"(2*x^2*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[3/4, 1/2, 1/2, 7/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/(3*Sqrt[x*(a + b*x^2 + c*x^4)])","A",0
105,1,486,380,1.4810625,"\int x^{3/2} \sqrt{a x+b x^3+c x^5} \, dx","Integrate[x^(3/2)*Sqrt[a*x + b*x^3 + c*x^5],x]","\frac{\sqrt{x} \left(-i \left(b^2-3 a c\right) \left(\sqrt{b^2-4 a c}-b\right) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} 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 c x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left(b+3 c x^2\right) \left(a+b x^2+c x^4\right)+i \left(b^2 \sqrt{b^2-4 a c}-3 a c \sqrt{b^2-4 a c}+4 a b c-b^3\right) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\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)}{30 c^2 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{x \left(a+b x^2+c x^4\right)}}","-\frac{\sqrt[4]{a} \sqrt{x} \left(\sqrt{a} b \sqrt{c}-6 a c+2 b^2\right) \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{30 c^{7/4} \sqrt{a x+b x^3+c x^5}}+\frac{2 \sqrt[4]{a} \sqrt{x} \left(b^2-3 a c\right) \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{15 c^{7/4} \sqrt{a x+b x^3+c x^5}}-\frac{2 x^{3/2} \left(b^2-3 a c\right) \left(a+b x^2+c x^4\right)}{15 c^{3/2} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{a x+b x^3+c x^5}}+\frac{\sqrt{x} \left(b+3 c x^2\right) \sqrt{a x+b x^3+c x^5}}{15 c}",1,"(Sqrt[x]*(2*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*(b + 3*c*x^2)*(a + b*x^2 + c*x^4) - I*(b^2 - 3*a*c)*(-b + Sqrt[b^2 - 4*a*c])*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(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])] + I*(-b^3 + 4*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - 3*a*c*Sqrt[b^2 - 4*a*c])*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - 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])]))/(30*c^2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[x*(a + b*x^2 + c*x^4)])","C",1
106,1,126,129,0.0774802,"\int \sqrt{x} \sqrt{a x+b x^3+c x^5} \, dx","Integrate[Sqrt[x]*Sqrt[a*x + b*x^3 + c*x^5],x]","\frac{\sqrt{x \left(a+b x^2+c x^4\right)} \left(\frac{\left(b+2 c x^2\right) \sqrt{a+b x^2+c x^4}}{4 c}-\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{8 c^{3/2}}\right)}{2 \sqrt{x} \sqrt{a+b x^2+c x^4}}","\frac{\left(b+2 c x^2\right) \sqrt{a x+b x^3+c x^5}}{8 c \sqrt{x}}-\frac{\sqrt{x} \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{16 c^{3/2} \sqrt{a x+b x^3+c x^5}}",1,"(Sqrt[x*(a + b*x^2 + c*x^4)]*(((b + 2*c*x^2)*Sqrt[a + b*x^2 + c*x^4])/(4*c) - ((b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(8*c^(3/2))))/(2*Sqrt[x]*Sqrt[a + b*x^2 + c*x^4])","A",1
107,1,452,347,0.9887964,"\int \frac{\sqrt{a x+b x^3+c x^5}}{\sqrt{x}} \, dx","Integrate[Sqrt[a*x + b*x^3 + c*x^5]/Sqrt[x],x]","\frac{\sqrt{x} \left(-i \left(b \sqrt{b^2-4 a c}+4 a c-b^2\right) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\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)+i b \left(\sqrt{b^2-4 a c}-b\right) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} 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)+4 c x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left(a+b x^2+c x^4\right)\right)}{12 c \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{x \left(a+b x^2+c x^4\right)}}","\frac{\sqrt[4]{a} \sqrt{x} \left(2 \sqrt{a} \sqrt{c}+b\right) \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{6 c^{3/4} \sqrt{a x+b x^3+c x^5}}-\frac{\sqrt[4]{a} b \sqrt{x} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{3 c^{3/4} \sqrt{a x+b x^3+c x^5}}+\frac{1}{3} \sqrt{x} \sqrt{a x+b x^3+c x^5}+\frac{b x^{3/2} \left(a+b x^2+c x^4\right)}{3 \sqrt{c} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{a x+b x^3+c x^5}}",1,"(Sqrt[x]*(4*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*(a + b*x^2 + c*x^4) + I*b*(-b + Sqrt[b^2 - 4*a*c])*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(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])] - I*(-b^2 + 4*a*c + b*Sqrt[b^2 - 4*a*c])*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - 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])]))/(12*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[x*(a + b*x^2 + c*x^4)])","C",1
108,1,155,194,0.05969,"\int \frac{\sqrt{a x+b x^3+c x^5}}{x^{3/2}} \, dx","Integrate[Sqrt[a*x + b*x^3 + c*x^5]/x^(3/2),x]","\frac{\sqrt{x} \sqrt{a+b x^2+c x^4} \left(2 \sqrt{c} \sqrt{a+b x^2+c x^4}-2 \sqrt{a} \sqrt{c} \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)+b \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)\right)}{4 \sqrt{c} \sqrt{x \left(a+b x^2+c x^4\right)}}","\frac{\sqrt{a x+b x^3+c x^5}}{2 \sqrt{x}}-\frac{\sqrt{a} \sqrt{x} \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{2 \sqrt{a x+b x^3+c x^5}}+\frac{b \sqrt{x} \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{4 \sqrt{c} \sqrt{a x+b x^3+c x^5}}",1,"(Sqrt[x]*Sqrt[a + b*x^2 + c*x^4]*(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4] - 2*Sqrt[a]*Sqrt[c]*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])] + b*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])]))/(4*Sqrt[c]*Sqrt[x*(a + b*x^2 + c*x^4)])","A",1
109,1,192,244,0.2036769,"\int x^{3/2} \left(a x+b x^3+c x^5\right)^{3/2} \, dx","Integrate[x^(3/2)*(a*x + b*x^3 + c*x^5)^(3/2),x]","\frac{\left(x \left(a+b x^2+c x^4\right)\right)^{3/2} \left(-\frac{3 b \left(b^2-4 a c\right) \left(\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)-2 \sqrt{c} \left(b+2 c x^2\right) \sqrt{a+b x^2+c x^4}\right)}{256 c^{7/2}}-\frac{b \left(b+2 c x^2\right) \left(a+b x^2+c x^4\right)^{3/2}}{16 c^2}+\frac{\left(a+b x^2+c x^4\right)^{5/2}}{5 c}\right)}{2 x^{3/2} \left(a+b x^2+c x^4\right)^{3/2}}","\frac{\left(128 a^2 c^2-100 a b^2 c+15 b^4\right) \sqrt{a x+b x^3+c x^5}}{1280 c^3 \sqrt{x}}-\frac{3 b \sqrt{x} \left(b^2-4 a c\right)^2 \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{512 c^{7/2} \sqrt{a x+b x^3+c x^5}}-\frac{x^{3/2} \left(4 c x^2 \left(5 b^2-16 a c\right)+b \left(5 b^2-4 a c\right)\right) \sqrt{a x+b x^3+c x^5}}{640 c^2}+\frac{\sqrt{x} \left(3 b+8 c x^2\right) \left(a x+b x^3+c x^5\right)^{3/2}}{80 c}",1,"((x*(a + b*x^2 + c*x^4))^(3/2)*(-1/16*(b*(b + 2*c*x^2)*(a + b*x^2 + c*x^4)^(3/2))/c^2 + (a + b*x^2 + c*x^4)^(5/2)/(5*c) - (3*b*(b^2 - 4*a*c)*(-2*Sqrt[c]*(b + 2*c*x^2)*Sqrt[a + b*x^2 + c*x^4] + (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])]))/(256*c^(7/2))))/(2*x^(3/2)*(a + b*x^2 + c*x^4)^(3/2))","A",1
110,1,609,487,2.2911104,"\int \sqrt{x} \left(a x+b x^3+c x^5\right)^{3/2} \, dx","Integrate[Sqrt[x]*(a*x + b*x^3 + c*x^5)^(3/2),x]","\frac{\sqrt{x} \left(i \left(84 a^2 c^2-57 a b^2 c+8 b^4\right) \left(\sqrt{b^2-4 a c}-b\right) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} 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)+4 c x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left(a^2 c \left(24 b+77 c x^2\right)+a \left(-4 b^3+27 b^2 c x^2+151 b c^2 x^4+112 c^3 x^6\right)-4 b^4 x^2-b^3 c x^4+53 b^2 c^2 x^6+85 b c^3 x^8+35 c^4 x^{10}\right)-i \left(84 a^2 c^2 \sqrt{b^2-4 a c}-132 a^2 b c^2+65 a b^3 c-57 a b^2 c \sqrt{b^2-4 a c}+8 b^4 \sqrt{b^2-4 a c}-8 b^5\right) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\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)}{1260 c^3 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{x \left(a+b x^2+c x^4\right)}}","\frac{\sqrt[4]{a} \sqrt{x} \left(84 a^2 c^2-57 a b^2 c+4 \sqrt{a} b \sqrt{c} \left(b^2-6 a c\right)+8 b^4\right) \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{630 c^{11/4} \sqrt{a x+b x^3+c x^5}}-\frac{\sqrt[4]{a} \sqrt{x} \left(84 a^2 c^2-57 a b^2 c+8 b^4\right) \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{315 c^{11/4} \sqrt{a x+b x^3+c x^5}}+\frac{x^{3/2} \left(84 a^2 c^2-57 a b^2 c+8 b^4\right) \left(a+b x^2+c x^4\right)}{315 c^{5/2} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{a x+b x^3+c x^5}}-\frac{\sqrt{x} \left(6 c x^2 \left(2 b^2-7 a c\right)+b \left(4 b^2-9 a c\right)\right) \sqrt{a x+b x^3+c x^5}}{315 c^2}+\frac{\left(3 b+7 c x^2\right) \left(a x+b x^3+c x^5\right)^{3/2}}{63 c \sqrt{x}}",1,"(Sqrt[x]*(4*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*(-4*b^4*x^2 - b^3*c*x^4 + 53*b^2*c^2*x^6 + 85*b*c^3*x^8 + 35*c^4*x^10 + a^2*c*(24*b + 77*c*x^2) + a*(-4*b^3 + 27*b^2*c*x^2 + 151*b*c^2*x^4 + 112*c^3*x^6)) + I*(8*b^4 - 57*a*b^2*c + 84*a^2*c^2)*(-b + Sqrt[b^2 - 4*a*c])*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(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])] - I*(-8*b^5 + 65*a*b^3*c - 132*a^2*b*c^2 + 8*b^4*Sqrt[b^2 - 4*a*c] - 57*a*b^2*c*Sqrt[b^2 - 4*a*c] + 84*a^2*c^2*Sqrt[b^2 - 4*a*c])*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - 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])]))/(1260*c^3*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[x*(a + b*x^2 + c*x^4)])","C",1
111,1,152,177,0.1114183,"\int \frac{\left(a x+b x^3+c x^5\right)^{3/2}}{\sqrt{x}} \, dx","Integrate[(a*x + b*x^3 + c*x^5)^(3/2)/Sqrt[x],x]","\frac{\sqrt{x} \sqrt{a+b x^2+c x^4} \left(2 \sqrt{c} \left(b+2 c x^2\right) \sqrt{a+b x^2+c x^4} \left(4 c \left(5 a+2 c x^4\right)-3 b^2+8 b c x^2\right)+3 \left(b^2-4 a c\right)^2 \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)\right)}{256 c^{5/2} \sqrt{x \left(a+b x^2+c x^4\right)}}","\frac{3 \sqrt{x} \left(b^2-4 a c\right)^2 \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{256 c^{5/2} \sqrt{a x+b x^3+c x^5}}-\frac{3 \left(b^2-4 a c\right) \left(b+2 c x^2\right) \sqrt{a x+b x^3+c x^5}}{128 c^2 \sqrt{x}}+\frac{\left(b+2 c x^2\right) \left(a x+b x^3+c x^5\right)^{3/2}}{16 c x^{3/2}}",1,"(Sqrt[x]*Sqrt[a + b*x^2 + c*x^4]*(2*Sqrt[c]*(b + 2*c*x^2)*Sqrt[a + b*x^2 + c*x^4]*(-3*b^2 + 8*b*c*x^2 + 4*c*(5*a + 2*c*x^4)) + 3*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])]))/(256*c^(5/2)*Sqrt[x*(a + b*x^2 + c*x^4)])","A",1
112,1,540,425,1.7356029,"\int \frac{\left(a x+b x^3+c x^5\right)^{3/2}}{x^{3/2}} \, dx","Integrate[(a*x + b*x^3 + c*x^5)^(3/2)/x^(3/2),x]","\frac{\sqrt{x} \left(2 c x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left(15 a^2 c+a \left(b^2+23 b c x^2+20 c^2 x^4\right)+x^2 \left(b^3+9 b^2 c x^2+13 b c^2 x^4+5 c^3 x^6\right)\right)+i \left(-20 a^2 c^2+9 a b^2 c-8 a b c \sqrt{b^2-4 a c}+b^3 \sqrt{b^2-4 a c}-b^4\right) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\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)-i b \left(b^2-8 a c\right) \left(\sqrt{b^2-4 a c}-b\right) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} 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)\right)}{70 c^2 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{x \left(a+b x^2+c x^4\right)}}","-\frac{\sqrt[4]{a} \sqrt{x} \left(\sqrt{a} \sqrt{c} \left(b^2-20 a c\right)+2 b \left(b^2-8 a c\right)\right) \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{70 c^{7/4} \sqrt{a x+b x^3+c x^5}}+\frac{2 \sqrt[4]{a} b \sqrt{x} \left(b^2-8 a c\right) \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{35 c^{7/4} \sqrt{a x+b x^3+c x^5}}-\frac{2 b x^{3/2} \left(b^2-8 a c\right) \left(a+b x^2+c x^4\right)}{35 c^{3/2} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{a x+b x^3+c x^5}}+\frac{\sqrt{x} \left(10 a c+b^2+3 b c x^2\right) \sqrt{a x+b x^3+c x^5}}{35 c}+\frac{\left(a x+b x^3+c x^5\right)^{3/2}}{7 \sqrt{x}}",1,"(Sqrt[x]*(2*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*(15*a^2*c + a*(b^2 + 23*b*c*x^2 + 20*c^2*x^4) + x^2*(b^3 + 9*b^2*c*x^2 + 13*b*c^2*x^4 + 5*c^3*x^6)) - I*b*(b^2 - 8*a*c)*(-b + Sqrt[b^2 - 4*a*c])*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(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])] + I*(-b^4 + 9*a*b^2*c - 20*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 8*a*b*c*Sqrt[b^2 - 4*a*c])*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - 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])]))/(70*c^2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[x*(a + b*x^2 + c*x^4)])","C",1
113,1,82,82,0.0177111,"\int \frac{x^{3/2}}{\sqrt{a x+b x^3+c x^5}} \, dx","Integrate[x^(3/2)/Sqrt[a*x + b*x^3 + c*x^5],x]","\frac{\sqrt{x} \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{2 \sqrt{c} \sqrt{x \left(a+b x^2+c x^4\right)}}","\frac{\sqrt{x} \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{2 \sqrt{c} \sqrt{a x+b x^3+c x^5}}",1,"(Sqrt[x]*Sqrt[a + b*x^2 + c*x^4]*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(2*Sqrt[c]*Sqrt[x*(a + b*x^2 + c*x^4)])","A",1
114,1,193,121,0.1182984,"\int \frac{\sqrt{x}}{\sqrt{a x+b x^3+c x^5}} \, dx","Integrate[Sqrt[x]/Sqrt[a*x + b*x^3 + c*x^5],x]","-\frac{i \sqrt{x} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} 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)}{\sqrt{2} \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{x \left(a+b x^2+c x^4\right)}}","\frac{\sqrt{x} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{2 \sqrt[4]{a} \sqrt[4]{c} \sqrt{a x+b x^3+c x^5}}",1,"((-I)*Sqrt[x]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - 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])]*Sqrt[x*(a + b*x^2 + c*x^4)])","C",1
115,1,83,51,0.0194141,"\int \frac{1}{\sqrt{x} \sqrt{a x+b x^3+c x^5}} \, dx","Integrate[1/(Sqrt[x]*Sqrt[a*x + b*x^3 + c*x^5]),x]","-\frac{\sqrt{x} \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{2 \sqrt{a} \sqrt{x \left(a+b x^2+c x^4\right)}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{x} \left(2 a+b x^2\right)}{2 \sqrt{a} \sqrt{a x+b x^3+c x^5}}\right)}{2 \sqrt{a}}",1,"-1/2*(Sqrt[x]*Sqrt[a + b*x^2 + c*x^4]*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/(Sqrt[a]*Sqrt[x*(a + b*x^2 + c*x^4)])","A",1
116,1,303,330,0.4712066,"\int \frac{1}{x^{3/2} \sqrt{a x+b x^3+c x^5}} \, dx","Integrate[1/(x^(3/2)*Sqrt[a*x + b*x^3 + c*x^5]),x]","\frac{-4 \left(a+b x^2+c x^4\right)+\frac{i \sqrt{2} x \left(\sqrt{b^2-4 a c}-b\right) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left(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)-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{\frac{c}{\sqrt{b^2-4 a c}+b}}}}{4 a \sqrt{x} \sqrt{x \left(a+b x^2+c x^4\right)}}","\frac{\sqrt[4]{c} \sqrt{x} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{2 a^{3/4} \sqrt{a x+b x^3+c x^5}}-\frac{\sqrt[4]{c} \sqrt{x} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{a^{3/4} \sqrt{a x+b x^3+c x^5}}-\frac{\sqrt{a x+b x^3+c x^5}}{a x^{3/2}}+\frac{\sqrt{c} x^{3/2} \left(a+b x^2+c x^4\right)}{a \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{a x+b x^3+c x^5}}",1,"(-4*(a + b*x^2 + c*x^4) + (I*Sqrt[2]*(-b + Sqrt[b^2 - 4*a*c])*x*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(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])] - 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[c/(b + Sqrt[b^2 - 4*a*c])])/(4*a*Sqrt[x]*Sqrt[x*(a + b*x^2 + c*x^4)])","C",1
117,1,463,391,1.0272842,"\int \frac{x^{3/2}}{\left(a x+b x^3+c x^5\right)^{3/2}} \, dx","Integrate[x^(3/2)/(a*x + b*x^3 + c*x^5)^(3/2),x]","-\frac{\sqrt{x} \left(-4 x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left(-2 a c+b^2+b c x^2\right)-i \left(b \sqrt{b^2-4 a c}+4 a c-b^2\right) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\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)+i b \left(\sqrt{b^2-4 a c}-b\right) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} 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)\right)}{4 a \left(b^2-4 a c\right) \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{x \left(a+b x^2+c x^4\right)}}","\frac{b \sqrt[4]{c} \sqrt{x} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{a^{3/4} \left(b^2-4 a c\right) \sqrt{a x+b x^3+c x^5}}-\frac{\sqrt[4]{c} \sqrt{x} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{2 a^{3/4} \left(b-2 \sqrt{a} \sqrt{c}\right) \sqrt{a x+b x^3+c x^5}}+\frac{x^{3/2} \left(-2 a c+b^2+b c x^2\right)}{a \left(b^2-4 a c\right) \sqrt{a x+b x^3+c x^5}}-\frac{b \sqrt{c} x^{3/2} \left(a+b x^2+c x^4\right)}{a \left(b^2-4 a c\right) \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{a x+b x^3+c x^5}}",1,"-1/4*(Sqrt[x]*(-4*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*(b^2 - 2*a*c + b*c*x^2) + I*b*(-b + Sqrt[b^2 - 4*a*c])*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(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])] - I*(-b^2 + 4*a*c + b*Sqrt[b^2 - 4*a*c])*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - 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])]))/(a*(b^2 - 4*a*c)*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[x*(a + b*x^2 + c*x^4)])","C",1
118,1,126,103,0.0787642,"\int \frac{\sqrt{x}}{\left(a x+b x^3+c x^5\right)^{3/2}} \, dx","Integrate[Sqrt[x]/(a*x + b*x^3 + c*x^5)^(3/2),x]","\frac{\sqrt{x} \left(\left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)-2 \sqrt{a} \left(-2 a c+b^2+b c x^2\right)\right)}{2 a^{3/2} \left(4 a c-b^2\right) \sqrt{x \left(a+b x^2+c x^4\right)}}","\frac{\sqrt{x} \left(-2 a c+b^2+b c x^2\right)}{a \left(b^2-4 a c\right) \sqrt{a x+b x^3+c x^5}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{x} \left(2 a+b x^2\right)}{2 \sqrt{a} \sqrt{a x+b x^3+c x^5}}\right)}{2 a^{3/2}}",1,"(Sqrt[x]*(-2*Sqrt[a]*(b^2 - 2*a*c + b*c*x^2) + (b^2 - 4*a*c)*Sqrt[a + b*x^2 + c*x^4]*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])]))/(2*a^(3/2)*(-b^2 + 4*a*c)*Sqrt[x*(a + b*x^2 + c*x^4)])","A",1
119,1,519,468,1.3465546,"\int \frac{1}{\sqrt{x} \left(a x+b x^3+c x^5\right)^{3/2}} \, dx","Integrate[1/(Sqrt[x]*(a*x + b*x^3 + c*x^5)^(3/2)),x]","-\frac{2 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left(-4 a^2 c+a \left(b^2-7 b c x^2-6 c^2 x^4\right)+2 b^2 x^2 \left(b+c x^2\right)\right)-i x \left(b^2-3 a c\right) \left(\sqrt{b^2-4 a c}-b\right) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} 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)+i x \left(b^2 \sqrt{b^2-4 a c}-3 a c \sqrt{b^2-4 a c}+4 a b c-b^3\right) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\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)}{2 a^2 \sqrt{x} \left(b^2-4 a c\right) \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{x \left(a+b x^2+c x^4\right)}}","\frac{\sqrt[4]{c} \sqrt{x} \left(\sqrt{a} b \sqrt{c}-6 a c+2 b^2\right) \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{2 a^{7/4} \left(b^2-4 a c\right) \sqrt{a x+b x^3+c x^5}}-\frac{2 \sqrt[4]{c} \sqrt{x} \left(b^2-3 a c\right) \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{a^{7/4} \left(b^2-4 a c\right) \sqrt{a x+b x^3+c x^5}}-\frac{2 \left(b^2-3 a c\right) \sqrt{a x+b x^3+c x^5}}{a^2 x^{3/2} \left(b^2-4 a c\right)}+\frac{2 \sqrt{c} x^{3/2} \left(b^2-3 a c\right) \left(a+b x^2+c x^4\right)}{a^2 \left(b^2-4 a c\right) \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{a x+b x^3+c x^5}}+\frac{-2 a c+b^2+b c x^2}{a \sqrt{x} \left(b^2-4 a c\right) \sqrt{a x+b x^3+c x^5}}",1,"-1/2*(2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*(-4*a^2*c + 2*b^2*x^2*(b + c*x^2) + a*(b^2 - 7*b*c*x^2 - 6*c^2*x^4)) - I*(b^2 - 3*a*c)*(-b + Sqrt[b^2 - 4*a*c])*x*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(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])] + I*(-b^3 + 4*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - 3*a*c*Sqrt[b^2 - 4*a*c])*x*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - 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])])/(a^2*(b^2 - 4*a*c)*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[x]*Sqrt[x*(a + b*x^2 + c*x^4)])","C",1
120,1,160,154,0.0742563,"\int \frac{1}{x^{3/2} \left(a x+b x^3+c x^5\right)^{3/2}} \, dx","Integrate[1/(x^(3/2)*(a*x + b*x^3 + c*x^5)^(3/2)),x]","\frac{2 \sqrt{a} \left(-4 a^2 c+a \left(b^2-10 b c x^2-8 c^2 x^4\right)+3 b^2 x^2 \left(b+c x^2\right)\right)-3 b x^2 \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{4 a^{5/2} x^{3/2} \left(4 a c-b^2\right) \sqrt{x \left(a+b x^2+c x^4\right)}}","\frac{3 b \tanh ^{-1}\left(\frac{\sqrt{x} \left(2 a+b x^2\right)}{2 \sqrt{a} \sqrt{a x+b x^3+c x^5}}\right)}{4 a^{5/2}}-\frac{\left(3 b^2-8 a c\right) \sqrt{a x+b x^3+c x^5}}{2 a^2 x^{5/2} \left(b^2-4 a c\right)}+\frac{-2 a c+b^2+b c x^2}{a x^{3/2} \left(b^2-4 a c\right) \sqrt{a x+b x^3+c x^5}}",1,"(2*Sqrt[a]*(-4*a^2*c + 3*b^2*x^2*(b + c*x^2) + a*(b^2 - 10*b*c*x^2 - 8*c^2*x^4)) - 3*b*(b^2 - 4*a*c)*x^2*Sqrt[a + b*x^2 + c*x^4]*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/(4*a^(5/2)*(-b^2 + 4*a*c)*x^(3/2)*Sqrt[x*(a + b*x^2 + c*x^4)])","A",1
121,1,46,51,0.0857323,"\int \frac{x^{\frac{3}{2} (-1+n)}}{\left(a x^{-1+n}+b x^n+c x^{1+n}\right)^{3/2}} \, dx","Integrate[x^((3*(-1 + n))/2)/(a*x^(-1 + n) + b*x^n + c*x^(1 + n))^(3/2),x]","-\frac{2 x^{\frac{n-1}{2}} (b+2 c x)}{\left(b^2-4 a c\right) \sqrt{x^{n-1} (a+x (b+c x))}}","-\frac{2 x^{\frac{n-1}{2}} (b+2 c x)}{\left(b^2-4 a c\right) \sqrt{a x^{n-1}+b x^n+c x^{n+1}}}",1,"(-2*x^((-1 + n)/2)*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[x^(-1 + n)*(a + x*(b + c*x))])","A",1
122,1,239,287,5.1247688,"\int \frac{x \left(d+e x^2\right)}{\sqrt{a x+b x^3+c x^5}} \, dx","Integrate[(x*(d + e*x^2))/Sqrt[a*x + b*x^3 + c*x^5],x]","\frac{2 \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \left(7 d x^2 F_1\left(\frac{3}{4};\frac{1}{2},\frac{1}{2};\frac{7}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right)+3 e x^4 F_1\left(\frac{7}{4};\frac{1}{2},\frac{1}{2};\frac{11}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right)\right)}{21 \sqrt{x \left(a+b x^2+c x^4\right)}}","\frac{2 d x^2 \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} F_1\left(\frac{3}{4};\frac{1}{2},\frac{1}{2};\frac{7}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right)}{3 \sqrt{a x+b x^3+c x^5}}+\frac{2 e x^4 \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} F_1\left(\frac{7}{4};\frac{1}{2},\frac{1}{2};\frac{11}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right)}{7 \sqrt{a x+b x^3+c x^5}}",1,"(2*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*(7*d*x^2*AppellF1[3/4, 1/2, 1/2, 7/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])] + 3*e*x^4*AppellF1[7/4, 1/2, 1/2, 11/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])]))/(21*Sqrt[x*(a + b*x^2 + c*x^4)])","A",0
123,1,73,45,0.020235,"\int \frac{1}{\sqrt{3 x^2-3 x^4+x^6}} \, dx","Integrate[1/Sqrt[3*x^2 - 3*x^4 + x^6],x]","-\frac{x \sqrt{x^4-3 x^2+3} \tanh ^{-1}\left(\frac{6-3 x^2}{2 \sqrt{3} \sqrt{x^4-3 x^2+3}}\right)}{2 \sqrt{3} \sqrt{x^2 \left(x^4-3 x^2+3\right)}}","-\frac{\tanh ^{-1}\left(\frac{x \left(6-3 x^2\right)}{2 \sqrt{3} \sqrt{x^6-3 x^4+3 x^2}}\right)}{2 \sqrt{3}}",1,"-1/2*(x*Sqrt[3 - 3*x^2 + x^4]*ArcTanh[(6 - 3*x^2)/(2*Sqrt[3]*Sqrt[3 - 3*x^2 + x^4])])/(Sqrt[3]*Sqrt[x^2*(3 - 3*x^2 + x^4)])","A",1
124,1,73,45,0.0033994,"\int \frac{1}{\sqrt{x^2 \left(3-3 x^2+x^4\right)}} \, dx","Integrate[1/Sqrt[x^2*(3 - 3*x^2 + x^4)],x]","-\frac{x \sqrt{x^4-3 x^2+3} \tanh ^{-1}\left(\frac{6-3 x^2}{2 \sqrt{3} \sqrt{x^4-3 x^2+3}}\right)}{2 \sqrt{3} \sqrt{x^2 \left(x^4-3 x^2+3\right)}}","-\frac{\tanh ^{-1}\left(\frac{x \left(6-3 x^2\right)}{2 \sqrt{3} \sqrt{x^6-3 x^4+3 x^2}}\right)}{2 \sqrt{3}}",1,"-1/2*(x*Sqrt[3 - 3*x^2 + x^4]*ArcTanh[(6 - 3*x^2)/(2*Sqrt[3]*Sqrt[3 - 3*x^2 + x^4])])/(Sqrt[3]*Sqrt[x^2*(3 - 3*x^2 + x^4)])","A",1
125,1,73,45,0.003717,"\int \frac{1}{\sqrt{1-\left(1-x^2\right)^3}} \, dx","Integrate[1/Sqrt[1 - (1 - x^2)^3],x]","-\frac{x \sqrt{x^4-3 x^2+3} \tanh ^{-1}\left(\frac{6-3 x^2}{2 \sqrt{3} \sqrt{x^4-3 x^2+3}}\right)}{2 \sqrt{3} \sqrt{x^2 \left(x^4-3 x^2+3\right)}}","-\frac{\tanh ^{-1}\left(\frac{x \left(6-3 x^2\right)}{2 \sqrt{3} \sqrt{x^6-3 x^4+3 x^2}}\right)}{2 \sqrt{3}}",1,"-1/2*(x*Sqrt[3 - 3*x^2 + x^4]*ArcTanh[(6 - 3*x^2)/(2*Sqrt[3]*Sqrt[3 - 3*x^2 + x^4])])/(Sqrt[3]*Sqrt[x^2*(3 - 3*x^2 + x^4)])","A",1
126,1,70,86,0.0334171,"\int \sqrt{3 x^2-3 x^4+x^6} \, dx","Integrate[Sqrt[3*x^2 - 3*x^4 + x^6],x]","\frac{x \left(4 x^6-18 x^4+30 x^2+3 \sqrt{x^4-3 x^2+3} \sinh ^{-1}\left(\frac{2 x^2-3}{\sqrt{3}}\right)-18\right)}{16 \sqrt{x^2 \left(x^4-3 x^2+3\right)}}","-\frac{\sqrt{x^6-3 x^4+3 x^2} \left(3-2 x^2\right)}{8 x}-\frac{3 \sqrt{x^6-3 x^4+3 x^2} \sinh ^{-1}\left(\frac{3-2 x^2}{\sqrt{3}}\right)}{16 x \sqrt{x^4-3 x^2+3}}",1,"(x*(-18 + 30*x^2 - 18*x^4 + 4*x^6 + 3*Sqrt[3 - 3*x^2 + x^4]*ArcSinh[(-3 + 2*x^2)/Sqrt[3]]))/(16*Sqrt[x^2*(3 - 3*x^2 + x^4)])","A",1
127,1,70,86,0.0097043,"\int \sqrt{x^2 \left(3-3 x^2+x^4\right)} \, dx","Integrate[Sqrt[x^2*(3 - 3*x^2 + x^4)],x]","\frac{x \left(4 x^6-18 x^4+30 x^2+3 \sqrt{x^4-3 x^2+3} \sinh ^{-1}\left(\frac{2 x^2-3}{\sqrt{3}}\right)-18\right)}{16 \sqrt{x^2 \left(x^4-3 x^2+3\right)}}","-\frac{\sqrt{x^6-3 x^4+3 x^2} \left(3-2 x^2\right)}{8 x}-\frac{3 \sqrt{x^6-3 x^4+3 x^2} \sinh ^{-1}\left(\frac{3-2 x^2}{\sqrt{3}}\right)}{16 x \sqrt{x^4-3 x^2+3}}",1,"(x*(-18 + 30*x^2 - 18*x^4 + 4*x^6 + 3*Sqrt[3 - 3*x^2 + x^4]*ArcSinh[(-3 + 2*x^2)/Sqrt[3]]))/(16*Sqrt[x^2*(3 - 3*x^2 + x^4)])","A",1
128,1,70,86,0.0034727,"\int \sqrt{1-\left(1-x^2\right)^3} \, dx","Integrate[Sqrt[1 - (1 - x^2)^3],x]","\frac{x \left(4 x^6-18 x^4+30 x^2+3 \sqrt{x^4-3 x^2+3} \sinh ^{-1}\left(\frac{2 x^2-3}{\sqrt{3}}\right)-18\right)}{16 \sqrt{x^2 \left(x^4-3 x^2+3\right)}}","-\frac{\sqrt{x^6-3 x^4+3 x^2} \left(3-2 x^2\right)}{8 x}-\frac{3 \sqrt{x^6-3 x^4+3 x^2} \sinh ^{-1}\left(\frac{3-2 x^2}{\sqrt{3}}\right)}{16 x \sqrt{x^4-3 x^2+3}}",1,"(x*(-18 + 30*x^2 - 18*x^4 + 4*x^6 + 3*Sqrt[3 - 3*x^2 + x^4]*ArcSinh[(-3 + 2*x^2)/Sqrt[3]]))/(16*Sqrt[x^2*(3 - 3*x^2 + x^4)])","A",1
129,1,37,38,0.0195908,"\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx","Integrate[1/(x*Sqrt[a + b*x + c*x^2]),x]","-\frac{\tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)}{\sqrt{a}}","-\frac{\tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{\sqrt{a}}",1,"-(ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])]/Sqrt[a])","A",1
130,1,70,45,0.0235631,"\int \frac{1}{\sqrt{x^2 \left(a+b x+c x^2\right)}} \, dx","Integrate[1/Sqrt[x^2*(a + b*x + c*x^2)],x]","-\frac{x \sqrt{a+b x+c x^2} \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{\sqrt{a} \sqrt{x^2 (a+x (b+c x))}}","-\frac{\tanh ^{-1}\left(\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right)}{\sqrt{a}}",1,"-((x*Sqrt[a + b*x + c*x^2]*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(Sqrt[a]*Sqrt[x^2*(a + x*(b + c*x))]))","A",1
131,1,72,47,0.0335452,"\int \frac{1}{\sqrt{x} \sqrt{x \left(a+b x+c x^2\right)}} \, dx","Integrate[1/(Sqrt[x]*Sqrt[x*(a + b*x + c*x^2)]),x]","-\frac{\sqrt{x} \sqrt{a+b x+c x^2} \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{\sqrt{a} \sqrt{x (a+x (b+c x))}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{x} (2 a+b x)}{2 \sqrt{a} \sqrt{a x+b x^2+c x^3}}\right)}{\sqrt{a}}",1,"-((Sqrt[x]*Sqrt[a + b*x + c*x^2]*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(Sqrt[a]*Sqrt[x*(a + x*(b + c*x))]))","A",1
132,1,74,49,0.0284688,"\int \frac{\sqrt{x}}{\sqrt{x^3 \left(a+b x+c x^2\right)}} \, dx","Integrate[Sqrt[x]/Sqrt[x^3*(a + b*x + c*x^2)],x]","-\frac{x^{3/2} \sqrt{a+b x+c x^2} \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{\sqrt{a} \sqrt{x^3 (a+x (b+c x))}}","-\frac{\tanh ^{-1}\left(\frac{x^{3/2} (2 a+b x)}{2 \sqrt{a} \sqrt{a x^3+b x^4+c x^5}}\right)}{\sqrt{a}}",1,"-((x^(3/2)*Sqrt[a + b*x + c*x^2]*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(Sqrt[a]*Sqrt[x^3*(a + x*(b + c*x))]))","A",1
133,1,44,44,0.0066333,"\int \frac{1}{x \sqrt{a+b x^2+c x^4}} \, dx","Integrate[1/(x*Sqrt[a + b*x^2 + c*x^4]),x]","-\frac{\tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{2 \sqrt{a}}","-\frac{\tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{2 \sqrt{a}}",1,"-1/2*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])]/Sqrt[a]","A",1
134,1,81,49,0.0166349,"\int \frac{1}{\sqrt{x^2 \left(a+b x^2+c x^4\right)}} \, dx","Integrate[1/Sqrt[x^2*(a + b*x^2 + c*x^4)],x]","-\frac{x \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{2 \sqrt{a} \sqrt{x^2 \left(a+b x^2+c x^4\right)}}","-\frac{\tanh ^{-1}\left(\frac{x \left(2 a+b x^2\right)}{2 \sqrt{a} \sqrt{a x^2+b x^4+c x^6}}\right)}{2 \sqrt{a}}",1,"-1/2*(x*Sqrt[a + b*x^2 + c*x^4]*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/(Sqrt[a]*Sqrt[x^2*(a + b*x^2 + c*x^4)])","A",1
135,1,83,51,0.0181605,"\int \frac{1}{\sqrt{x} \sqrt{x \left(a+b x^2+c x^4\right)}} \, dx","Integrate[1/(Sqrt[x]*Sqrt[x*(a + b*x^2 + c*x^4)]),x]","-\frac{\sqrt{x} \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{2 \sqrt{a} \sqrt{x \left(a+b x^2+c x^4\right)}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{x} \left(2 a+b x^2\right)}{2 \sqrt{a} \sqrt{a x+b x^3+c x^5}}\right)}{2 \sqrt{a}}",1,"-1/2*(Sqrt[x]*Sqrt[a + b*x^2 + c*x^4]*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/(Sqrt[a]*Sqrt[x*(a + b*x^2 + c*x^4)])","A",1
136,1,85,53,0.019131,"\int \frac{\sqrt{x}}{\sqrt{x^3 \left(a+b x^2+c x^4\right)}} \, dx","Integrate[Sqrt[x]/Sqrt[x^3*(a + b*x^2 + c*x^4)],x]","-\frac{x^{3/2} \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{2 \sqrt{a} \sqrt{x^3 \left(a+b x^2+c x^4\right)}}","-\frac{\tanh ^{-1}\left(\frac{x^{3/2} \left(2 a+b x^2\right)}{2 \sqrt{a} \sqrt{a x^3+b x^5+c x^7}}\right)}{2 \sqrt{a}}",1,"-1/2*(x^(3/2)*Sqrt[a + b*x^2 + c*x^4]*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/(Sqrt[a]*Sqrt[x^3*(a + b*x^2 + c*x^4)])","A",1
137,1,40,40,0.0084883,"\int \frac{1}{x \sqrt{3-3 x^2+x^4}} \, dx","Integrate[1/(x*Sqrt[3 - 3*x^2 + x^4]),x]","-\frac{\tanh ^{-1}\left(\frac{6-3 x^2}{2 \sqrt{3} \sqrt{x^4-3 x^2+3}}\right)}{2 \sqrt{3}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{3} \left(2-x^2\right)}{2 \sqrt{x^4-3 x^2+3}}\right)}{2 \sqrt{3}}",1,"-1/2*ArcTanh[(6 - 3*x^2)/(2*Sqrt[3]*Sqrt[3 - 3*x^2 + x^4])]/Sqrt[3]","A",1
138,1,73,45,0.007583,"\int \frac{1}{\sqrt{x^2 \left(3-3 x^2+x^4\right)}} \, dx","Integrate[1/Sqrt[x^2*(3 - 3*x^2 + x^4)],x]","-\frac{x \sqrt{x^4-3 x^2+3} \tanh ^{-1}\left(\frac{6-3 x^2}{2 \sqrt{3} \sqrt{x^4-3 x^2+3}}\right)}{2 \sqrt{3} \sqrt{x^2 \left(x^4-3 x^2+3\right)}}","-\frac{\tanh ^{-1}\left(\frac{x \left(6-3 x^2\right)}{2 \sqrt{3} \sqrt{x^6-3 x^4+3 x^2}}\right)}{2 \sqrt{3}}",1,"-1/2*(x*Sqrt[3 - 3*x^2 + x^4]*ArcTanh[(6 - 3*x^2)/(2*Sqrt[3]*Sqrt[3 - 3*x^2 + x^4])])/(Sqrt[3]*Sqrt[x^2*(3 - 3*x^2 + x^4)])","A",1
139,1,62,43,0.019721,"\int \frac{1}{\sqrt{x} \sqrt{x \left(3-3 x+x^2\right)}} \, dx","Integrate[1/(Sqrt[x]*Sqrt[x*(3 - 3*x + x^2)]),x]","\frac{\sqrt{x} \sqrt{x^2-3 x+3} \tanh ^{-1}\left(\frac{\sqrt{3} (x-2)}{2 \sqrt{x^2-3 x+3}}\right)}{\sqrt{3} \sqrt{x \left(x^2-3 x+3\right)}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{3} (2-x) \sqrt{x}}{2 \sqrt{x^3-3 x^2+3 x}}\right)}{\sqrt{3}}",1,"(Sqrt[x]*Sqrt[3 - 3*x + x^2]*ArcTanh[(Sqrt[3]*(-2 + x))/(2*Sqrt[3 - 3*x + x^2])])/(Sqrt[3]*Sqrt[x*(3 - 3*x + x^2)])","A",1
140,0,0,70,0.3954067,"\int \frac{x^{-1+\frac{q}{2}}}{\sqrt{b x^n+c x^{2 n-q}+a x^q}} \, dx","Integrate[x^(-1 + q/2)/Sqrt[b*x^n + c*x^(2*n - q) + a*x^q],x]","\int \frac{x^{-1+\frac{q}{2}}}{\sqrt{b x^n+c x^{2 n-q}+a x^q}} \, dx","-\frac{\tanh ^{-1}\left(\frac{x^{q/2} \left(2 a+b x^{n-q}\right)}{2 \sqrt{a} \sqrt{a x^q+b x^n+c x^{2 n-q}}}\right)}{\sqrt{a} (n-q)}",1,"Integrate[x^(-1 + q/2)/Sqrt[b*x^n + c*x^(2*n - q) + a*x^q], x]","F",-1