1,1,97,128,0.0750939,"\int \left(a^2+2 a b x^2+b^2 x^4\right)^{3/4} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/4),x]","\frac{\left(\left(a+b x^2\right)^2\right)^{3/4} \left(3 a^{3/2} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)+\sqrt{b} x \left(5 a+2 b x^2\right) \sqrt{\frac{b x^2}{a}+1}\right)}{8 \sqrt{b} \left(a+b x^2\right) \sqrt{\frac{b x^2}{a}+1}}","\frac{3 a x \left(a^2+2 a b x^2+b^2 x^4\right)^{3/4}}{8 \left(a+b x^2\right)}+\frac{1}{4} x \left(a^2+2 a b x^2+b^2 x^4\right)^{3/4}+\frac{3 \sqrt{a} \left(a^2+2 a b x^2+b^2 x^4\right)^{3/4} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 \sqrt{b} \left(\frac{b x^2}{a}+1\right)^{3/2}}",1,"(((a + b*x^2)^2)^(3/4)*(Sqrt[b]*x*(5*a + 2*b*x^2)*Sqrt[1 + (b*x^2)/a] + 3*a^(3/2)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]]))/(8*Sqrt[b]*(a + b*x^2)*Sqrt[1 + (b*x^2)/a])","A",1
2,1,59,91,0.0370087,"\int \sqrt[4]{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/4),x]","\frac{1}{2} \sqrt[4]{\left(a+b x^2\right)^2} \left(\frac{a \log \left(\sqrt{b} \sqrt{a+b x^2}+b x\right)}{\sqrt{b} \sqrt{a+b x^2}}+x\right)","\frac{1}{2} x \sqrt[4]{a^2+2 a b x^2+b^2 x^4}+\frac{\sqrt{a} \sqrt[4]{a^2+2 a b x^2+b^2 x^4} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 \sqrt{b} \sqrt{\frac{b x^2}{a}+1}}",1,"(((a + b*x^2)^2)^(1/4)*(x + (a*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(Sqrt[b]*Sqrt[a + b*x^2])))/2","A",1
3,1,49,60,0.0147255,"\int \frac{1}{\sqrt[4]{a^2+2 a b x^2+b^2 x^4}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(-1/4),x]","\frac{\sqrt{a+b x^2} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{\sqrt{b} \sqrt[4]{\left(a+b x^2\right)^2}}","\frac{\sqrt{a} \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{b} \sqrt[4]{a^2+2 a b x^2+b^2 x^4}}",1,"(Sqrt[a + b*x^2]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*((a + b*x^2)^2)^(1/4))","A",1
4,1,25,34,0.0112811,"\int \frac{1}{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/4}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(-3/4),x]","\frac{x \left(a+b x^2\right)}{a \left(\left(a+b x^2\right)^2\right)^{3/4}}","\frac{x \left(a+b x^2\right)}{a \left(a^2+2 a b x^2+b^2 x^4\right)^{3/4}}",1,"(x*(a + b*x^2))/(a*((a + b*x^2)^2)^(3/4))","A",1
5,1,40,68,0.0123038,"\int \frac{1}{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/4}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(-5/4),x]","\frac{x \left(3 a+2 b x^2\right)}{3 a^2 \left(a+b x^2\right) \sqrt[4]{\left(a+b x^2\right)^2}}","\frac{2 x}{3 a^2 \sqrt[4]{a^2+2 a b x^2+b^2 x^4}}+\frac{x \left(a+b x^2\right)}{3 a \left(a^2+2 a b x^2+b^2 x^4\right)^{5/4}}",1,"(x*(3*a + 2*b*x^2))/(3*a^2*(a + b*x^2)*((a + b*x^2)^2)^(1/4))","A",1
6,1,51,105,0.0152398,"\int \frac{1}{\left(a^2+2 a b x^2+b^2 x^4\right)^{7/4}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(-7/4),x]","\frac{x \left(15 a^2+20 a b x^2+8 b^2 x^4\right)}{15 a^3 \left(a+b x^2\right) \left(\left(a+b x^2\right)^2\right)^{3/4}}","\frac{4 x}{15 a^2 \left(a^2+2 a b x^2+b^2 x^4\right)^{3/4}}+\frac{x \left(a+b x^2\right)}{5 a \left(a^2+2 a b x^2+b^2 x^4\right)^{7/4}}+\frac{8 x \left(a+b x^2\right)}{15 a^3 \left(a^2+2 a b x^2+b^2 x^4\right)^{3/4}}",1,"(x*(15*a^2 + 20*a*b*x^2 + 8*b^2*x^4))/(15*a^3*(a + b*x^2)*((a + b*x^2)^2)^(3/4))","A",1
7,1,62,135,0.021368,"\int \frac{1}{\left(a^2+2 a b x^2+b^2 x^4\right)^{9/4}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(-9/4),x]","\frac{x \left(35 a^3+70 a^2 b x^2+56 a b^2 x^4+16 b^3 x^6\right)}{35 a^4 \left(a+b x^2\right)^3 \sqrt[4]{\left(a+b x^2\right)^2}}","\frac{6 x}{35 a^2 \left(a^2+2 a b x^2+b^2 x^4\right)^{5/4}}+\frac{x \left(a+b x^2\right)}{7 a \left(a^2+2 a b x^2+b^2 x^4\right)^{9/4}}+\frac{16 x}{35 a^4 \sqrt[4]{a^2+2 a b x^2+b^2 x^4}}+\frac{8 x \left(a+b x^2\right)}{35 a^3 \left(a^2+2 a b x^2+b^2 x^4\right)^{5/4}}",1,"(x*(35*a^3 + 70*a^2*b*x^2 + 56*a*b^2*x^4 + 16*b^3*x^6))/(35*a^4*(a + b*x^2)^3*((a + b*x^2)^2)^(1/4))","A",1
8,1,81,299,0.0433758,"\int \frac{1}{a^2+b+2 a x^2+x^4} \, dx","Integrate[(a^2 + b + 2*a*x^2 + x^4)^(-1),x]","-\frac{i \left(\frac{\tan ^{-1}\left(\frac{x}{\sqrt{a-i \sqrt{b}}}\right)}{\sqrt{a-i \sqrt{b}}}-\frac{\tan ^{-1}\left(\frac{x}{\sqrt{a+i \sqrt{b}}}\right)}{\sqrt{a+i \sqrt{b}}}\right)}{2 \sqrt{b}}","-\frac{\log \left(-\sqrt{2} x \sqrt{\sqrt{a^2+b}-a}+\sqrt{a^2+b}+x^2\right)}{4 \sqrt{2} \sqrt{a^2+b} \sqrt{\sqrt{a^2+b}-a}}+\frac{\log \left(\sqrt{2} x \sqrt{\sqrt{a^2+b}-a}+\sqrt{a^2+b}+x^2\right)}{4 \sqrt{2} \sqrt{a^2+b} \sqrt{\sqrt{a^2+b}-a}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{a^2+b}-a}-\sqrt{2} x}{\sqrt{\sqrt{a^2+b}+a}}\right)}{2 \sqrt{2} \sqrt{a^2+b} \sqrt{\sqrt{a^2+b}+a}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{a^2+b}-a}+\sqrt{2} x}{\sqrt{\sqrt{a^2+b}+a}}\right)}{2 \sqrt{2} \sqrt{a^2+b} \sqrt{\sqrt{a^2+b}+a}}",1,"((-1/2*I)*(ArcTan[x/Sqrt[a - I*Sqrt[b]]]/Sqrt[a - I*Sqrt[b]] - ArcTan[x/Sqrt[a + I*Sqrt[b]]]/Sqrt[a + I*Sqrt[b]]))/Sqrt[b]","C",1
9,1,43,47,0.0225152,"\int \frac{1}{-1+a^2+2 a x^2+x^4} \, dx","Integrate[(-1 + a^2 + 2*a*x^2 + x^4)^(-1),x]","\frac{\tan ^{-1}\left(\frac{x}{\sqrt{a-1}}\right)}{2 \sqrt{a-1}}-\frac{\tan ^{-1}\left(\frac{x}{\sqrt{a+1}}\right)}{2 \sqrt{a+1}}","-\frac{\tan ^{-1}\left(\frac{x}{\sqrt{a+1}}\right)}{2 \sqrt{a+1}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt{1-a}}\right)}{2 \sqrt{1-a}}",1,"ArcTan[x/Sqrt[-1 + a]]/(2*Sqrt[-1 + a]) - ArcTan[x/Sqrt[1 + a]]/(2*Sqrt[1 + a])","A",1
10,1,52,299,0.0327032,"\int \frac{1}{1+a^2+2 a x^2+x^4} \, dx","Integrate[(1 + a^2 + 2*a*x^2 + x^4)^(-1),x]","-\frac{1}{2} i \left(\frac{\tan ^{-1}\left(\frac{x}{\sqrt{a-i}}\right)}{\sqrt{a-i}}-\frac{\tan ^{-1}\left(\frac{x}{\sqrt{a+i}}\right)}{\sqrt{a+i}}\right)","-\frac{\log \left(-\sqrt{2} \sqrt{\sqrt{a^2+1}-a} x+\sqrt{a^2+1}+x^2\right)}{4 \sqrt{2} \sqrt{a^2+1} \sqrt{\sqrt{a^2+1}-a}}+\frac{\log \left(\sqrt{2} \sqrt{\sqrt{a^2+1}-a} x+\sqrt{a^2+1}+x^2\right)}{4 \sqrt{2} \sqrt{a^2+1} \sqrt{\sqrt{a^2+1}-a}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{a^2+1}-a}-\sqrt{2} x}{\sqrt{\sqrt{a^2+1}+a}}\right)}{2 \sqrt{2} \sqrt{a^2+1} \sqrt{\sqrt{a^2+1}+a}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{a^2+1}-a}+\sqrt{2} x}{\sqrt{\sqrt{a^2+1}+a}}\right)}{2 \sqrt{2} \sqrt{a^2+1} \sqrt{\sqrt{a^2+1}+a}}",1,"(-1/2*I)*(ArcTan[x/Sqrt[-I + a]]/Sqrt[-I + a] - ArcTan[x/Sqrt[I + a]]/Sqrt[I + a])","C",1
11,1,37,17,0.0069455,"\int \frac{1}{4-5 x^2+x^4} \, dx","Integrate[(4 - 5*x^2 + x^4)^(-1),x]","-\frac{1}{6} \log (1-x)+\frac{1}{12} \log (2-x)+\frac{1}{6} \log (x+1)-\frac{1}{12} \log (x+2)","\frac{1}{3} \tanh ^{-1}(x)-\frac{1}{6} \tanh ^{-1}\left(\frac{x}{2}\right)",1,"-1/6*Log[1 - x] + Log[2 - x]/12 + Log[1 + x]/6 - Log[2 + x]/12","B",1
12,1,24,24,0.0127351,"\int \frac{1}{3+4 x^2+x^4} \, dx","Integrate[(3 + 4*x^2 + x^4)^(-1),x]","\frac{1}{2} \tan ^{-1}(x)-\frac{\tan ^{-1}\left(\frac{x}{\sqrt{3}}\right)}{2 \sqrt{3}}","\frac{1}{2} \tan ^{-1}(x)-\frac{\tan ^{-1}\left(\frac{x}{\sqrt{3}}\right)}{2 \sqrt{3}}",1,"ArcTan[x]/2 - ArcTan[x/Sqrt[3]]/(2*Sqrt[3])","A",1
13,1,91,67,0.0709983,"\int \frac{1}{9+5 x^2+x^4} \, dx","Integrate[(9 + 5*x^2 + x^4)^(-1),x]","\frac{i \tan ^{-1}\left(\frac{x}{\sqrt{\frac{1}{2} \left(5+i \sqrt{11}\right)}}\right)}{\sqrt{\frac{11}{2} \left(5+i \sqrt{11}\right)}}-\frac{i \tan ^{-1}\left(\frac{x}{\sqrt{\frac{1}{2} \left(5-i \sqrt{11}\right)}}\right)}{\sqrt{\frac{11}{2} \left(5-i \sqrt{11}\right)}}","-\frac{1}{12} \log \left(x^2-x+3\right)+\frac{1}{12} \log \left(x^2+x+3\right)-\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{11}}\right)}{6 \sqrt{11}}+\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{11}}\right)}{6 \sqrt{11}}",1,"((-I)*ArcTan[x/Sqrt[(5 - I*Sqrt[11])/2]])/Sqrt[(11*(5 - I*Sqrt[11]))/2] + (I*ArcTan[x/Sqrt[(5 + I*Sqrt[11])/2]])/Sqrt[(11*(5 + I*Sqrt[11]))/2]","C",1
14,1,77,74,0.0703887,"\int \frac{1}{1-x^2+x^4} \, dx","Integrate[(1 - x^2 + x^4)^(-1),x]","\frac{i \left(\sqrt{-1-i \sqrt{3}} \tan ^{-1}\left(\frac{1}{2} \left(1-i \sqrt{3}\right) x\right)-\sqrt{-1+i \sqrt{3}} \tan ^{-1}\left(\frac{1}{2} \left(1+i \sqrt{3}\right) x\right)\right)}{\sqrt{6}}","-\frac{\log \left(x^2-\sqrt{3} x+1\right)}{4 \sqrt{3}}+\frac{\log \left(x^2+\sqrt{3} x+1\right)}{4 \sqrt{3}}-\frac{1}{2} \tan ^{-1}\left(\sqrt{3}-2 x\right)+\frac{1}{2} \tan ^{-1}\left(2 x+\sqrt{3}\right)",1,"(I*(Sqrt[-1 - I*Sqrt[3]]*ArcTan[((1 - I*Sqrt[3])*x)/2] - Sqrt[-1 + I*Sqrt[3]]*ArcTan[((1 + I*Sqrt[3])*x)/2]))/Sqrt[6]","C",0
15,1,41,176,0.0375578,"\int \frac{1}{2+2 x^2+x^4} \, dx","Integrate[(2 + 2*x^2 + x^4)^(-1),x]","\frac{1}{4} \left((1-i)^{3/2} \tan ^{-1}\left(\frac{x}{\sqrt{1-i}}\right)+(1+i)^{3/2} \tan ^{-1}\left(\frac{x}{\sqrt{1+i}}\right)\right)","-\frac{\log \left(x^2-\sqrt{2 \left(\sqrt{2}-1\right)} x+\sqrt{2}\right)}{8 \sqrt{\sqrt{2}-1}}+\frac{\log \left(x^2+\sqrt{2 \left(\sqrt{2}-1\right)} x+\sqrt{2}\right)}{8 \sqrt{\sqrt{2}-1}}-\frac{1}{4} \sqrt{\sqrt{2}-1} \tan ^{-1}\left(\frac{\sqrt{2 \left(\sqrt{2}-1\right)}-2 x}{\sqrt{2 \left(1+\sqrt{2}\right)}}\right)+\frac{1}{4} \sqrt{\sqrt{2}-1} \tan ^{-1}\left(\frac{2 x+\sqrt{2 \left(\sqrt{2}-1\right)}}{\sqrt{2 \left(1+\sqrt{2}\right)}}\right)",1,"((1 - I)^(3/2)*ArcTan[x/Sqrt[1 - I]] + (1 + I)^(3/2)*ArcTan[x/Sqrt[1 + I]])/4","C",1
16,1,65,10,0.0272828,"\int \frac{1}{\sqrt{2+5 x^2-3 x^4}} \, dx","Integrate[1/Sqrt[2 + 5*x^2 - 3*x^4],x]","-\frac{i \sqrt{1-\frac{x^2}{2}} \sqrt{3 x^2+1} F\left(i \sinh ^{-1}\left(\sqrt{3} x\right)|-\frac{1}{6}\right)}{\sqrt{3} \sqrt{-3 x^4+5 x^2+2}}","F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-6\right)",1,"((-I)*Sqrt[1 - x^2/2]*Sqrt[1 + 3*x^2]*EllipticF[I*ArcSinh[Sqrt[3]*x], -1/6])/(Sqrt[3]*Sqrt[2 + 5*x^2 - 3*x^4])","C",1
17,1,49,48,0.0581231,"\int \frac{1}{\sqrt{2+4 x^2-3 x^4}} \, dx","Integrate[1/Sqrt[2 + 4*x^2 - 3*x^4],x]","-\frac{i F\left(i \sinh ^{-1}\left(\sqrt{1+\sqrt{\frac{5}{2}}} x\right)|\frac{1}{3} \left(-7+2 \sqrt{10}\right)\right)}{\sqrt{2+\sqrt{10}}}","\sqrt{\frac{1}{6} \left(2+\sqrt{10}\right)} F\left(\sin ^{-1}\left(\sqrt{\frac{1}{2} \left(-2+\sqrt{10}\right)} x\right)|\frac{1}{3} \left(-7-2 \sqrt{10}\right)\right)",1,"((-I)*EllipticF[I*ArcSinh[Sqrt[1 + Sqrt[5/2]]*x], (-7 + 2*Sqrt[10])/3])/Sqrt[2 + Sqrt[10]]","C",0
18,1,53,48,0.0607334,"\int \frac{1}{\sqrt{2+3 x^2-3 x^4}} \, dx","Integrate[1/Sqrt[2 + 3*x^2 - 3*x^4],x]","-i \sqrt{\frac{2}{3+\sqrt{33}}} F\left(i \sinh ^{-1}\left(\sqrt{\frac{6}{-3+\sqrt{33}}} x\right)|\frac{1}{4} \left(-7+\sqrt{33}\right)\right)","\sqrt{\frac{2}{\sqrt{33}-3}} F\left(\sin ^{-1}\left(\sqrt{\frac{6}{3+\sqrt{33}}} x\right)|\frac{1}{4} \left(-7-\sqrt{33}\right)\right)",1,"(-I)*Sqrt[2/(3 + Sqrt[33])]*EllipticF[I*ArcSinh[Sqrt[6/(-3 + Sqrt[33])]*x], (-7 + Sqrt[33])/4]","C",0
19,1,49,44,0.046134,"\int \frac{1}{\sqrt{2+2 x^2-3 x^4}} \, dx","Integrate[1/Sqrt[2 + 2*x^2 - 3*x^4],x]","-\frac{i F\left(i \sinh ^{-1}\left(\sqrt{\frac{3}{-1+\sqrt{7}}} x\right)|\frac{1}{3} \left(-4+\sqrt{7}\right)\right)}{\sqrt{1+\sqrt{7}}}","\frac{F\left(\sin ^{-1}\left(\sqrt{\frac{3}{1+\sqrt{7}}} x\right)|\frac{1}{3} \left(-4-\sqrt{7}\right)\right)}{\sqrt{\sqrt{7}-1}}",1,"((-I)*EllipticF[I*ArcSinh[Sqrt[3/(-1 + Sqrt[7])]*x], (-4 + Sqrt[7])/3])/Sqrt[1 + Sqrt[7]]","C",0
20,1,63,12,0.0241489,"\int \frac{1}{\sqrt{2+x^2-3 x^4}} \, dx","Integrate[1/Sqrt[2 + x^2 - 3*x^4],x]","-\frac{i \sqrt{1-x^2} \sqrt{3 x^2+2} F\left(i \sinh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|-\frac{2}{3}\right)}{\sqrt{3} \sqrt{-3 x^4+x^2+2}}","\frac{F\left(\sin ^{-1}(x)|-\frac{3}{2}\right)}{\sqrt{2}}",1,"((-I)*Sqrt[1 - x^2]*Sqrt[2 + 3*x^2]*EllipticF[I*ArcSinh[Sqrt[3/2]*x], -2/3])/(Sqrt[3]*Sqrt[2 + x^2 - 3*x^4])","C",1
21,1,18,18,0.0222529,"\int \frac{1}{\sqrt{2-3 x^4}} \, dx","Integrate[1/Sqrt[2 - 3*x^4],x]","\frac{F\left(\left.\sin ^{-1}\left(\sqrt[4]{\frac{3}{2}} x\right)\right|-1\right)}{\sqrt[4]{6}}","\frac{F\left(\left.\sin ^{-1}\left(\sqrt[4]{\frac{3}{2}} x\right)\right|-1\right)}{\sqrt[4]{6}}",1,"EllipticF[ArcSin[(3/2)^(1/4)*x], -1]/6^(1/4)","A",1
22,1,20,20,0.0256907,"\int \frac{1}{\sqrt{2-x^2-3 x^4}} \, dx","Integrate[1/Sqrt[2 - x^2 - 3*x^4],x]","\frac{F\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|-\frac{2}{3}\right)}{\sqrt{3}}","\frac{F\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|-\frac{2}{3}\right)}{\sqrt{3}}",1,"EllipticF[ArcSin[Sqrt[3/2]*x], -2/3]/Sqrt[3]","A",1
23,1,51,42,0.0442681,"\int \frac{1}{\sqrt{2-2 x^2-3 x^4}} \, dx","Integrate[1/Sqrt[2 - 2*x^2 - 3*x^4],x]","-\frac{i F\left(i \sinh ^{-1}\left(\sqrt{\frac{3}{1+\sqrt{7}}} x\right)|-\frac{4}{3}-\frac{\sqrt{7}}{3}\right)}{\sqrt{\sqrt{7}-1}}","\frac{F\left(\sin ^{-1}\left(\sqrt{\frac{3}{-1+\sqrt{7}}} x\right)|\frac{1}{3} \left(-4+\sqrt{7}\right)\right)}{\sqrt{1+\sqrt{7}}}",1,"((-I)*EllipticF[I*ArcSinh[Sqrt[3/(1 + Sqrt[7])]*x], -4/3 - Sqrt[7]/3])/Sqrt[-1 + Sqrt[7]]","C",0
24,1,55,46,0.0615579,"\int \frac{1}{\sqrt{2-3 x^2-3 x^4}} \, dx","Integrate[1/Sqrt[2 - 3*x^2 - 3*x^4],x]","-i \sqrt{\frac{2}{\sqrt{33}-3}} F\left(i \sinh ^{-1}\left(\sqrt{\frac{6}{3+\sqrt{33}}} x\right)|-\frac{7}{4}-\frac{\sqrt{33}}{4}\right)","\sqrt{\frac{2}{3+\sqrt{33}}} F\left(\sin ^{-1}\left(\sqrt{\frac{6}{-3+\sqrt{33}}} x\right)|\frac{1}{4} \left(-7+\sqrt{33}\right)\right)",1,"(-I)*Sqrt[2/(-3 + Sqrt[33])]*EllipticF[I*ArcSinh[Sqrt[6/(3 + Sqrt[33])]*x], -7/4 - Sqrt[33]/4]","C",0
25,1,49,48,0.0604667,"\int \frac{1}{\sqrt{2-4 x^2-3 x^4}} \, dx","Integrate[1/Sqrt[2 - 4*x^2 - 3*x^4],x]","-\frac{i F\left(i \sinh ^{-1}\left(\sqrt{-1+\sqrt{\frac{5}{2}}} x\right)|\frac{1}{3} \left(-7-2 \sqrt{10}\right)\right)}{\sqrt{\sqrt{10}-2}}","\sqrt{\frac{1}{6} \left(\sqrt{10}-2\right)} F\left(\sin ^{-1}\left(\sqrt{\frac{1}{2} \left(2+\sqrt{10}\right)} x\right)|\frac{1}{3} \left(-7+2 \sqrt{10}\right)\right)",1,"((-I)*EllipticF[I*ArcSinh[Sqrt[-1 + Sqrt[5/2]]*x], (-7 - 2*Sqrt[10])/3])/Sqrt[-2 + Sqrt[10]]","C",0
26,1,54,18,0.0229453,"\int \frac{1}{\sqrt{2-5 x^2-3 x^4}} \, dx","Integrate[1/Sqrt[2 - 5*x^2 - 3*x^4],x]","\frac{\sqrt{1-3 x^2} \sqrt{x^2+2} F\left(\sin ^{-1}\left(\sqrt{3} x\right)|-\frac{1}{6}\right)}{\sqrt{6} \sqrt{-3 x^4-5 x^2+2}}","\frac{F\left(\sin ^{-1}\left(\sqrt{3} x\right)|-\frac{1}{6}\right)}{\sqrt{6}}",1,"(Sqrt[1 - 3*x^2]*Sqrt[2 + x^2]*EllipticF[ArcSin[Sqrt[3]*x], -1/6])/(Sqrt[6]*Sqrt[2 - 5*x^2 - 3*x^4])","B",1
27,1,52,45,0.0456747,"\int \frac{1}{\sqrt{3+7 x^2-2 x^4}} \, dx","Integrate[1/Sqrt[3 + 7*x^2 - 2*x^4],x]","-i \sqrt{\frac{2}{7+\sqrt{73}}} F\left(i \sinh ^{-1}\left(\frac{2 x}{\sqrt{-7+\sqrt{73}}}\right)|\frac{1}{12} \left(-61+7 \sqrt{73}\right)\right)","\sqrt{\frac{2}{\sqrt{73}-7}} F\left(\sin ^{-1}\left(\frac{2 x}{\sqrt{7+\sqrt{73}}}\right)|\frac{1}{12} \left(-61-7 \sqrt{73}\right)\right)",1,"(-I)*Sqrt[2/(7 + Sqrt[73])]*EllipticF[I*ArcSinh[(2*x)/Sqrt[-7 + Sqrt[73]]], (-61 + 7*Sqrt[73])/12]","C",0
28,1,43,44,0.0521842,"\int \frac{1}{\sqrt{3+6 x^2-2 x^4}} \, dx","Integrate[1/Sqrt[3 + 6*x^2 - 2*x^4],x]","-\frac{i F\left(i \sinh ^{-1}\left(\sqrt{1+\sqrt{\frac{5}{3}}} x\right)|-4+\sqrt{15}\right)}{\sqrt{3+\sqrt{15}}}","\sqrt{\frac{1}{6} \left(3+\sqrt{15}\right)} F\left(\sin ^{-1}\left(\sqrt{\frac{1}{3} \left(-3+\sqrt{15}\right)} x\right)|-4-\sqrt{15}\right)",1,"((-I)*EllipticF[I*ArcSinh[Sqrt[1 + Sqrt[5/3]]*x], -4 + Sqrt[15]])/Sqrt[3 + Sqrt[15]]","C",0
29,1,65,10,0.0265455,"\int \frac{1}{\sqrt{3+5 x^2-2 x^4}} \, dx","Integrate[1/Sqrt[3 + 5*x^2 - 2*x^4],x]","-\frac{i \sqrt{1-\frac{x^2}{3}} \sqrt{2 x^2+1} F\left(i \sinh ^{-1}\left(\sqrt{2} x\right)|-\frac{1}{6}\right)}{\sqrt{2} \sqrt{-2 x^4+5 x^2+3}}","F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{3}}\right)\right|-6\right)",1,"((-I)*Sqrt[1 - x^2/3]*Sqrt[1 + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2]*x], -1/6])/(Sqrt[2]*Sqrt[3 + 5*x^2 - 2*x^4])","C",1
30,1,51,44,0.055462,"\int \frac{1}{\sqrt{3+4 x^2-2 x^4}} \, dx","Integrate[1/Sqrt[3 + 4*x^2 - 2*x^4],x]","-\frac{i F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{-2+\sqrt{10}}} x\right)|-\frac{7}{3}+\frac{2 \sqrt{10}}{3}\right)}{\sqrt{2+\sqrt{10}}}","\frac{F\left(\sin ^{-1}\left(\sqrt{\frac{2}{2+\sqrt{10}}} x\right)|\frac{1}{3} \left(-7-2 \sqrt{10}\right)\right)}{\sqrt{\sqrt{10}-2}}",1,"((-I)*EllipticF[I*ArcSinh[Sqrt[2/(-2 + Sqrt[10])]*x], -7/3 + (2*Sqrt[10])/3])/Sqrt[2 + Sqrt[10]]","C",0
31,1,50,45,0.0509085,"\int \frac{1}{\sqrt{3+3 x^2-2 x^4}} \, dx","Integrate[1/Sqrt[3 + 3*x^2 - 2*x^4],x]","-i \sqrt{\frac{2}{3+\sqrt{33}}} F\left(i \sinh ^{-1}\left(\frac{2 x}{\sqrt{-3+\sqrt{33}}}\right)|\frac{1}{4} \left(-7+\sqrt{33}\right)\right)","\sqrt{\frac{2}{\sqrt{33}-3}} F\left(\sin ^{-1}\left(\frac{2 x}{\sqrt{3+\sqrt{33}}}\right)|\frac{1}{4} \left(-7-\sqrt{33}\right)\right)",1,"(-I)*Sqrt[2/(3 + Sqrt[33])]*EllipticF[I*ArcSinh[(2*x)/Sqrt[-3 + Sqrt[33]]], (-7 + Sqrt[33])/4]","C",0
32,1,49,44,0.0447926,"\int \frac{1}{\sqrt{3+2 x^2-2 x^4}} \, dx","Integrate[1/Sqrt[3 + 2*x^2 - 2*x^4],x]","-\frac{i F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{-1+\sqrt{7}}} x\right)|\frac{1}{3} \left(-4+\sqrt{7}\right)\right)}{\sqrt{1+\sqrt{7}}}","\frac{F\left(\sin ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{7}}} x\right)|\frac{1}{3} \left(-4-\sqrt{7}\right)\right)}{\sqrt{\sqrt{7}-1}}",1,"((-I)*EllipticF[I*ArcSinh[Sqrt[2/(-1 + Sqrt[7])]*x], (-4 + Sqrt[7])/3])/Sqrt[1 + Sqrt[7]]","C",0
33,1,20,20,0.0277618,"\int \frac{1}{\sqrt{3+x^2-2 x^4}} \, dx","Integrate[1/Sqrt[3 + x^2 - 2*x^4],x]","\frac{F\left(\sin ^{-1}\left(\sqrt{\frac{2}{3}} x\right)|-\frac{3}{2}\right)}{\sqrt{2}}","\frac{F\left(\sin ^{-1}\left(\sqrt{\frac{2}{3}} x\right)|-\frac{3}{2}\right)}{\sqrt{2}}",1,"EllipticF[ArcSin[Sqrt[2/3]*x], -3/2]/Sqrt[2]","A",1
34,1,18,18,0.022167,"\int \frac{1}{\sqrt{3-2 x^4}} \, dx","Integrate[1/Sqrt[3 - 2*x^4],x]","\frac{F\left(\left.\sin ^{-1}\left(\sqrt[4]{\frac{2}{3}} x\right)\right|-1\right)}{\sqrt[4]{6}}","\frac{F\left(\left.\sin ^{-1}\left(\sqrt[4]{\frac{2}{3}} x\right)\right|-1\right)}{\sqrt[4]{6}}",1,"EllipticF[ArcSin[(2/3)^(1/4)*x], -1]/6^(1/4)","A",1
35,1,65,12,0.0225955,"\int \frac{1}{\sqrt{3-x^2-2 x^4}} \, dx","Integrate[1/Sqrt[3 - x^2 - 2*x^4],x]","-\frac{i \sqrt{1-x^2} \sqrt{2 x^2+3} F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{3}} x\right)|-\frac{3}{2}\right)}{\sqrt{2} \sqrt{-2 x^4-x^2+3}}","\frac{F\left(\sin ^{-1}(x)|-\frac{2}{3}\right)}{\sqrt{3}}",1,"((-I)*Sqrt[1 - x^2]*Sqrt[3 + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/3]*x], -3/2])/(Sqrt[2]*Sqrt[3 - x^2 - 2*x^4])","C",1
36,1,51,42,0.0392856,"\int \frac{1}{\sqrt{3-2 x^2-2 x^4}} \, dx","Integrate[1/Sqrt[3 - 2*x^2 - 2*x^4],x]","-\frac{i F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{7}}} x\right)|-\frac{4}{3}-\frac{\sqrt{7}}{3}\right)}{\sqrt{\sqrt{7}-1}}","\frac{F\left(\sin ^{-1}\left(\sqrt{\frac{2}{-1+\sqrt{7}}} x\right)|\frac{1}{3} \left(-4+\sqrt{7}\right)\right)}{\sqrt{1+\sqrt{7}}}",1,"((-I)*EllipticF[I*ArcSinh[Sqrt[2/(1 + Sqrt[7])]*x], -4/3 - Sqrt[7]/3])/Sqrt[-1 + Sqrt[7]]","C",0
37,1,52,43,0.0442021,"\int \frac{1}{\sqrt{3-3 x^2-2 x^4}} \, dx","Integrate[1/Sqrt[3 - 3*x^2 - 2*x^4],x]","-i \sqrt{\frac{2}{\sqrt{33}-3}} F\left(i \sinh ^{-1}\left(\frac{2 x}{\sqrt{3+\sqrt{33}}}\right)|-\frac{7}{4}-\frac{\sqrt{33}}{4}\right)","\sqrt{\frac{2}{3+\sqrt{33}}} F\left(\sin ^{-1}\left(\frac{2 x}{\sqrt{-3+\sqrt{33}}}\right)|\frac{1}{4} \left(-7+\sqrt{33}\right)\right)",1,"(-I)*Sqrt[2/(-3 + Sqrt[33])]*EllipticF[I*ArcSinh[(2*x)/Sqrt[3 + Sqrt[33]]], -7/4 - Sqrt[33]/4]","C",0
38,1,51,44,0.0479498,"\int \frac{1}{\sqrt{3-4 x^2-2 x^4}} \, dx","Integrate[1/Sqrt[3 - 4*x^2 - 2*x^4],x]","-\frac{i F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{2+\sqrt{10}}} x\right)|-\frac{7}{3}-\frac{2 \sqrt{10}}{3}\right)}{\sqrt{\sqrt{10}-2}}","\frac{F\left(\sin ^{-1}\left(\sqrt{\frac{2}{-2+\sqrt{10}}} x\right)|\frac{1}{3} \left(-7+2 \sqrt{10}\right)\right)}{\sqrt{2+\sqrt{10}}}",1,"((-I)*EllipticF[I*ArcSinh[Sqrt[2/(2 + Sqrt[10])]*x], -7/3 - (2*Sqrt[10])/3])/Sqrt[-2 + Sqrt[10]]","C",0
39,1,54,18,0.0255986,"\int \frac{1}{\sqrt{3-5 x^2-2 x^4}} \, dx","Integrate[1/Sqrt[3 - 5*x^2 - 2*x^4],x]","\frac{\sqrt{1-2 x^2} \sqrt{x^2+3} F\left(\sin ^{-1}\left(\sqrt{2} x\right)|-\frac{1}{6}\right)}{\sqrt{6} \sqrt{-2 x^4-5 x^2+3}}","\frac{F\left(\sin ^{-1}\left(\sqrt{2} x\right)|-\frac{1}{6}\right)}{\sqrt{6}}",1,"(Sqrt[1 - 2*x^2]*Sqrt[3 + x^2]*EllipticF[ArcSin[Sqrt[2]*x], -1/6])/(Sqrt[6]*Sqrt[3 - 5*x^2 - 2*x^4])","B",1
40,1,45,42,0.0596694,"\int \frac{1}{\sqrt{3-6 x^2-2 x^4}} \, dx","Integrate[1/Sqrt[3 - 6*x^2 - 2*x^4],x]","-\frac{i F\left(i \sinh ^{-1}\left(\sqrt{-1+\sqrt{\frac{5}{3}}} x\right)|-4-\sqrt{15}\right)}{\sqrt{\sqrt{15}-3}}","\sqrt{\frac{1}{6} \left(\sqrt{15}-3\right)} F\left(\sin ^{-1}\left(\sqrt{\frac{1}{3} \left(3+\sqrt{15}\right)} x\right)|-4+\sqrt{15}\right)",1,"((-I)*EllipticF[I*ArcSinh[Sqrt[-1 + Sqrt[5/3]]*x], -4 - Sqrt[15]])/Sqrt[-3 + Sqrt[15]]","C",0
41,1,52,45,0.0460737,"\int \frac{1}{\sqrt{3-7 x^2-2 x^4}} \, dx","Integrate[1/Sqrt[3 - 7*x^2 - 2*x^4],x]","-i \sqrt{\frac{2}{\sqrt{73}-7}} F\left(i \sinh ^{-1}\left(\frac{2 x}{\sqrt{7+\sqrt{73}}}\right)|\frac{1}{12} \left(-61-7 \sqrt{73}\right)\right)","\sqrt{\frac{2}{7+\sqrt{73}}} F\left(\sin ^{-1}\left(\frac{2 x}{\sqrt{-7+\sqrt{73}}}\right)|\frac{1}{12} \left(-61+7 \sqrt{73}\right)\right)",1,"(-I)*Sqrt[2/(-7 + Sqrt[73])]*EllipticF[I*ArcSinh[(2*x)/Sqrt[7 + Sqrt[73]]], (-61 - 7*Sqrt[73])/12]","C",0
42,1,54,67,0.0222007,"\int \frac{1}{\sqrt{-2+5 x^2+3 x^4}} \, dx","Integrate[1/Sqrt[-2 + 5*x^2 + 3*x^4],x]","\frac{\sqrt{1-3 x^2} \sqrt{x^2+2} F\left(\sin ^{-1}\left(\sqrt{3} x\right)|-\frac{1}{6}\right)}{\sqrt{6} \sqrt{3 x^4+5 x^2-2}}","\frac{\sqrt{x^2+2} \sqrt{3 x^2-1} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{7}{2}} x}{\sqrt{3 x^2-1}}\right)|\frac{6}{7}\right)}{\sqrt{7} \sqrt{3 x^4+5 x^2-2}}",1,"(Sqrt[1 - 3*x^2]*Sqrt[2 + x^2]*EllipticF[ArcSin[Sqrt[3]*x], -1/6])/(Sqrt[6]*Sqrt[-2 + 5*x^2 + 3*x^4])","A",1
43,1,81,141,0.0534381,"\int \frac{1}{\sqrt{-2+4 x^2+3 x^4}} \, dx","Integrate[1/Sqrt[-2 + 4*x^2 + 3*x^4],x]","-\frac{i \sqrt{-3 x^4-4 x^2+2} F\left(i \sinh ^{-1}\left(\sqrt{-1+\sqrt{\frac{5}{2}}} x\right)|\frac{1}{3} \left(-7-2 \sqrt{10}\right)\right)}{\sqrt{\sqrt{10}-2} \sqrt{3 x^4+4 x^2-2}}","\frac{\sqrt{\frac{2-\left(2-\sqrt{10}\right) x^2}{2-\left(2+\sqrt{10}\right) x^2}} \sqrt{\left(2+\sqrt{10}\right) x^2-2} F\left(\sin ^{-1}\left(\frac{2^{3/4} \sqrt[4]{5} x}{\sqrt{\left(2+\sqrt{10}\right) x^2-2}}\right)|\frac{1}{10} \left(5+\sqrt{10}\right)\right)}{2 \sqrt[4]{10} \sqrt{\frac{1}{2-\left(2+\sqrt{10}\right) x^2}} \sqrt{3 x^4+4 x^2-2}}",1,"((-I)*Sqrt[2 - 4*x^2 - 3*x^4]*EllipticF[I*ArcSinh[Sqrt[-1 + Sqrt[5/2]]*x], (-7 - 2*Sqrt[10])/3])/(Sqrt[-2 + Sqrt[10]]*Sqrt[-2 + 4*x^2 + 3*x^4])","C",0
44,1,83,146,0.062671,"\int \frac{1}{\sqrt{-2+3 x^2+3 x^4}} \, dx","Integrate[1/Sqrt[-2 + 3*x^2 + 3*x^4],x]","-\frac{i \sqrt{-6 x^4-6 x^2+4} F\left(i \sinh ^{-1}\left(\sqrt{\frac{6}{3+\sqrt{33}}} x\right)|-\frac{7}{4}-\frac{\sqrt{33}}{4}\right)}{\sqrt{\sqrt{33}-3} \sqrt{3 x^4+3 x^2-2}}","\frac{\sqrt{\frac{4-\left(3-\sqrt{33}\right) x^2}{4-\left(3+\sqrt{33}\right) x^2}} \sqrt{\left(3+\sqrt{33}\right) x^2-4} F\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{33} x}{\sqrt{\left(3+\sqrt{33}\right) x^2-4}}\right)|\frac{1}{22} \left(11+\sqrt{33}\right)\right)}{2 \sqrt{2} \sqrt[4]{33} \sqrt{\frac{1}{4-\left(3+\sqrt{33}\right) x^2}} \sqrt{3 x^4+3 x^2-2}}",1,"((-I)*Sqrt[4 - 6*x^2 - 6*x^4]*EllipticF[I*ArcSinh[Sqrt[6/(3 + Sqrt[33])]*x], -7/4 - Sqrt[33]/4])/(Sqrt[-3 + Sqrt[33]]*Sqrt[-2 + 3*x^2 + 3*x^4])","C",0
45,1,83,141,0.0502089,"\int \frac{1}{\sqrt{-2+2 x^2+3 x^4}} \, dx","Integrate[1/Sqrt[-2 + 2*x^2 + 3*x^4],x]","-\frac{i \sqrt{-3 x^4-2 x^2+2} F\left(i \sinh ^{-1}\left(\sqrt{\frac{3}{1+\sqrt{7}}} x\right)|-\frac{4}{3}-\frac{\sqrt{7}}{3}\right)}{\sqrt{\sqrt{7}-1} \sqrt{3 x^4+2 x^2-2}}","\frac{\sqrt{\frac{2-\left(1-\sqrt{7}\right) x^2}{2-\left(1+\sqrt{7}\right) x^2}} \sqrt{\left(1+\sqrt{7}\right) x^2-2} F\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{7} x}{\sqrt{\left(1+\sqrt{7}\right) x^2-2}}\right)|\frac{1}{14} \left(7+\sqrt{7}\right)\right)}{2 \sqrt[4]{7} \sqrt{\frac{1}{2-\left(1+\sqrt{7}\right) x^2}} \sqrt{3 x^4+2 x^2-2}}",1,"((-I)*Sqrt[2 - 2*x^2 - 3*x^4]*EllipticF[I*ArcSinh[Sqrt[3/(1 + Sqrt[7])]*x], -4/3 - Sqrt[7]/3])/(Sqrt[-1 + Sqrt[7]]*Sqrt[-2 + 2*x^2 + 3*x^4])","C",0
46,1,48,63,0.028411,"\int \frac{1}{\sqrt{-2+x^2+3 x^4}} \, dx","Integrate[1/Sqrt[-2 + x^2 + 3*x^4],x]","\frac{\sqrt{\left(\frac{2}{3}-x^2\right) \left(x^2+1\right)} F\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|-\frac{2}{3}\right)}{\sqrt{3 x^4+x^2-2}}","\frac{\sqrt{x^2+1} \sqrt{3 x^2-2} F\left(\sin ^{-1}\left(\frac{\sqrt{5} x}{\sqrt{3 x^2-2}}\right)|\frac{3}{5}\right)}{\sqrt{5} \sqrt{3 x^4+x^2-2}}",1,"(Sqrt[(2/3 - x^2)*(1 + x^2)]*EllipticF[ArcSin[Sqrt[3/2]*x], -2/3])/Sqrt[-2 + x^2 + 3*x^4]","A",0
47,1,40,115,0.0224892,"\int \frac{1}{\sqrt{-2+3 x^4}} \, dx","Integrate[1/Sqrt[-2 + 3*x^4],x]","\frac{\sqrt{2-3 x^4} F\left(\left.\sin ^{-1}\left(\sqrt[4]{\frac{3}{2}} x\right)\right|-1\right)}{\sqrt[4]{6} \sqrt{3 x^4-2}}","\frac{\sqrt{\sqrt{6} x^2-2} \sqrt{\frac{\sqrt{6} x^2+2}{2-\sqrt{6} x^2}} F\left(\sin ^{-1}\left(\frac{2^{3/4} \sqrt[4]{3} x}{\sqrt{\sqrt{6} x^2-2}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{6} \sqrt{\frac{1}{2-\sqrt{6} x^2}} \sqrt{3 x^4-2}}",1,"(Sqrt[2 - 3*x^4]*EllipticF[ArcSin[(3/2)^(1/4)*x], -1])/(6^(1/4)*Sqrt[-2 + 3*x^4])","A",1
48,1,60,65,0.0229035,"\int \frac{1}{\sqrt{-2-x^2+3 x^4}} \, dx","Integrate[1/Sqrt[-2 - x^2 + 3*x^4],x]","-\frac{i \sqrt{1-x^2} \sqrt{3 x^2+2} F\left(i \sinh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|-\frac{2}{3}\right)}{\sqrt{9 x^4-3 x^2-6}}","\frac{\sqrt{x^2-1} \sqrt{3 x^2+2} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{5}{2}} x}{\sqrt{x^2-1}}\right)|\frac{2}{5}\right)}{\sqrt{5} \sqrt{3 x^4-x^2-2}}",1,"((-I)*Sqrt[1 - x^2]*Sqrt[2 + 3*x^2]*EllipticF[I*ArcSinh[Sqrt[3/2]*x], -2/3])/Sqrt[-6 - 3*x^2 + 9*x^4]","C",1
49,1,81,148,0.038857,"\int \frac{1}{\sqrt{-2-2 x^2+3 x^4}} \, dx","Integrate[1/Sqrt[-2 - 2*x^2 + 3*x^4],x]","-\frac{i \sqrt{-3 x^4+2 x^2+2} F\left(i \sinh ^{-1}\left(\sqrt{\frac{3}{-1+\sqrt{7}}} x\right)|\frac{1}{3} \left(-4+\sqrt{7}\right)\right)}{\sqrt{1+\sqrt{7}} \sqrt{3 x^4-2 x^2-2}}","\frac{\sqrt{-\left(\left(1-\sqrt{7}\right) x^2\right)-2} \sqrt{\frac{\left(1+\sqrt{7}\right) x^2+2}{\left(1-\sqrt{7}\right) x^2+2}} F\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{7} x}{\sqrt{-\left(\left(1-\sqrt{7}\right) x^2\right)-2}}\right)|\frac{1}{14} \left(7-\sqrt{7}\right)\right)}{2 \sqrt[4]{7} \sqrt{\frac{1}{\left(1-\sqrt{7}\right) x^2+2}} \sqrt{3 x^4-2 x^2-2}}",1,"((-I)*Sqrt[2 + 2*x^2 - 3*x^4]*EllipticF[I*ArcSinh[Sqrt[3/(-1 + Sqrt[7])]*x], (-4 + Sqrt[7])/3])/(Sqrt[1 + Sqrt[7]]*Sqrt[-2 - 2*x^2 + 3*x^4])","C",0
50,1,81,153,0.0653256,"\int \frac{1}{\sqrt{-2-3 x^2+3 x^4}} \, dx","Integrate[1/Sqrt[-2 - 3*x^2 + 3*x^4],x]","-\frac{i \sqrt{-6 x^4+6 x^2+4} F\left(i \sinh ^{-1}\left(\sqrt{\frac{6}{-3+\sqrt{33}}} x\right)|\frac{1}{4} \left(-7+\sqrt{33}\right)\right)}{\sqrt{3+\sqrt{33}} \sqrt{3 x^4-3 x^2-2}}","\frac{\sqrt{-\left(\left(3-\sqrt{33}\right) x^2\right)-4} \sqrt{\frac{\left(3+\sqrt{33}\right) x^2+4}{\left(3-\sqrt{33}\right) x^2+4}} F\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{33} x}{\sqrt{-\left(\left(3-\sqrt{33}\right) x^2\right)-4}}\right)|\frac{1}{22} \left(11-\sqrt{33}\right)\right)}{2 \sqrt{2} \sqrt[4]{33} \sqrt{\frac{1}{\left(3-\sqrt{33}\right) x^2+4}} \sqrt{3 x^4-3 x^2-2}}",1,"((-I)*Sqrt[4 + 6*x^2 - 6*x^4]*EllipticF[I*ArcSinh[Sqrt[6/(-3 + Sqrt[33])]*x], (-7 + Sqrt[33])/4])/(Sqrt[3 + Sqrt[33]]*Sqrt[-2 - 3*x^2 + 3*x^4])","C",0
51,1,81,148,0.0649649,"\int \frac{1}{\sqrt{-2-4 x^2+3 x^4}} \, dx","Integrate[1/Sqrt[-2 - 4*x^2 + 3*x^4],x]","-\frac{i \sqrt{-3 x^4+4 x^2+2} F\left(i \sinh ^{-1}\left(\sqrt{1+\sqrt{\frac{5}{2}}} x\right)|\frac{1}{3} \left(-7+2 \sqrt{10}\right)\right)}{\sqrt{2+\sqrt{10}} \sqrt{3 x^4-4 x^2-2}}","\frac{\sqrt{-\left(\left(2-\sqrt{10}\right) x^2\right)-2} \sqrt{\frac{\left(2+\sqrt{10}\right) x^2+2}{\left(2-\sqrt{10}\right) x^2+2}} F\left(\sin ^{-1}\left(\frac{2^{3/4} \sqrt[4]{5} x}{\sqrt{-\left(\left(2-\sqrt{10}\right) x^2\right)-2}}\right)|\frac{1}{10} \left(5-\sqrt{10}\right)\right)}{2 \sqrt[4]{10} \sqrt{\frac{1}{\left(2-\sqrt{10}\right) x^2+2}} \sqrt{3 x^4-4 x^2-2}}",1,"((-I)*Sqrt[2 + 4*x^2 - 3*x^4]*EllipticF[I*ArcSinh[Sqrt[1 + Sqrt[5/2]]*x], (-7 + 2*Sqrt[10])/3])/(Sqrt[2 + Sqrt[10]]*Sqrt[-2 - 4*x^2 + 3*x^4])","C",0
52,1,65,63,0.0250052,"\int \frac{1}{\sqrt{-2-5 x^2+3 x^4}} \, dx","Integrate[1/Sqrt[-2 - 5*x^2 + 3*x^4],x]","-\frac{i \sqrt{1-\frac{x^2}{2}} \sqrt{3 x^2+1} F\left(i \sinh ^{-1}\left(\sqrt{3} x\right)|-\frac{1}{6}\right)}{\sqrt{3} \sqrt{3 x^4-5 x^2-2}}","\frac{\sqrt{x^2-2} \sqrt{3 x^2+1} F\left(\sin ^{-1}\left(\frac{\sqrt{7} x}{\sqrt{x^2-2}}\right)|\frac{1}{7}\right)}{\sqrt{7} \sqrt{3 x^4-5 x^2-2}}",1,"((-I)*Sqrt[1 - x^2/2]*Sqrt[1 + 3*x^2]*EllipticF[I*ArcSinh[Sqrt[3]*x], -1/6])/(Sqrt[3]*Sqrt[-2 - 5*x^2 + 3*x^4])","C",1
53,1,80,148,0.0524194,"\int \frac{1}{\sqrt{-3+7 x^2+2 x^4}} \, dx","Integrate[1/Sqrt[-3 + 7*x^2 + 2*x^4],x]","-\frac{i \sqrt{-4 x^4-14 x^2+6} F\left(i \sinh ^{-1}\left(\frac{2 x}{\sqrt{7+\sqrt{73}}}\right)|\frac{1}{12} \left(-61-7 \sqrt{73}\right)\right)}{\sqrt{\sqrt{73}-7} \sqrt{2 x^4+7 x^2-3}}","\frac{\sqrt{\frac{6-\left(7-\sqrt{73}\right) x^2}{6-\left(7+\sqrt{73}\right) x^2}} \sqrt{\left(7+\sqrt{73}\right) x^2-6} F\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{73} x}{\sqrt{\left(7+\sqrt{73}\right) x^2-6}}\right)|\frac{1}{146} \left(73+7 \sqrt{73}\right)\right)}{2 \sqrt{3} \sqrt[4]{73} \sqrt{\frac{1}{6-\left(7+\sqrt{73}\right) x^2}} \sqrt{2 x^4+7 x^2-3}}",1,"((-I)*Sqrt[6 - 14*x^2 - 4*x^4]*EllipticF[I*ArcSinh[(2*x)/Sqrt[7 + Sqrt[73]]], (-61 - 7*Sqrt[73])/12])/(Sqrt[-7 + Sqrt[73]]*Sqrt[-3 + 7*x^2 + 2*x^4])","C",0
54,1,77,148,0.0611708,"\int \frac{1}{\sqrt{-3+6 x^2+2 x^4}} \, dx","Integrate[1/Sqrt[-3 + 6*x^2 + 2*x^4],x]","-\frac{i \sqrt{-2 x^4-6 x^2+3} F\left(i \sinh ^{-1}\left(\sqrt{-1+\sqrt{\frac{5}{3}}} x\right)|-4-\sqrt{15}\right)}{\sqrt{\sqrt{15}-3} \sqrt{2 x^4+6 x^2-3}}","\frac{\sqrt{\frac{3-\left(3-\sqrt{15}\right) x^2}{3-\left(3+\sqrt{15}\right) x^2}} \sqrt{\left(3+\sqrt{15}\right) x^2-3} F\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{15} x}{\sqrt{\left(3+\sqrt{15}\right) x^2-3}}\right)|\frac{1}{10} \left(5+\sqrt{15}\right)\right)}{\sqrt{2} 3^{3/4} \sqrt[4]{5} \sqrt{\frac{1}{3-\left(3+\sqrt{15}\right) x^2}} \sqrt{2 x^4+6 x^2-3}}",1,"((-I)*Sqrt[3 - 6*x^2 - 2*x^4]*EllipticF[I*ArcSinh[Sqrt[-1 + Sqrt[5/3]]*x], -4 - Sqrt[15]])/(Sqrt[-3 + Sqrt[15]]*Sqrt[-3 + 6*x^2 + 2*x^4])","C",0
55,1,54,67,0.0234527,"\int \frac{1}{\sqrt{-3+5 x^2+2 x^4}} \, dx","Integrate[1/Sqrt[-3 + 5*x^2 + 2*x^4],x]","\frac{\sqrt{1-2 x^2} \sqrt{x^2+3} F\left(\sin ^{-1}\left(\sqrt{2} x\right)|-\frac{1}{6}\right)}{\sqrt{6} \sqrt{2 x^4+5 x^2-3}}","\frac{\sqrt{x^2+3} \sqrt{2 x^2-1} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{7}{3}} x}{\sqrt{2 x^2-1}}\right)|\frac{6}{7}\right)}{\sqrt{7} \sqrt{2 x^4+5 x^2-3}}",1,"(Sqrt[1 - 2*x^2]*Sqrt[3 + x^2]*EllipticF[ArcSin[Sqrt[2]*x], -1/6])/(Sqrt[6]*Sqrt[-3 + 5*x^2 + 2*x^4])","A",1
56,1,83,148,0.0616367,"\int \frac{1}{\sqrt{-3+4 x^2+2 x^4}} \, dx","Integrate[1/Sqrt[-3 + 4*x^2 + 2*x^4],x]","-\frac{i \sqrt{-2 x^4-4 x^2+3} F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{2+\sqrt{10}}} x\right)|-\frac{7}{3}-\frac{2 \sqrt{10}}{3}\right)}{\sqrt{\sqrt{10}-2} \sqrt{2 x^4+4 x^2-3}}","\frac{\sqrt{\frac{3-\left(2-\sqrt{10}\right) x^2}{3-\left(2+\sqrt{10}\right) x^2}} \sqrt{\left(2+\sqrt{10}\right) x^2-3} F\left(\sin ^{-1}\left(\frac{2^{3/4} \sqrt[4]{5} x}{\sqrt{\left(2+\sqrt{10}\right) x^2-3}}\right)|\frac{1}{10} \left(5+\sqrt{10}\right)\right)}{2^{3/4} \sqrt{3} \sqrt[4]{5} \sqrt{\frac{1}{3-\left(2+\sqrt{10}\right) x^2}} \sqrt{2 x^4+4 x^2-3}}",1,"((-I)*Sqrt[3 - 4*x^2 - 2*x^4]*EllipticF[I*ArcSinh[Sqrt[2/(2 + Sqrt[10])]*x], -7/3 - (2*Sqrt[10])/3])/(Sqrt[-2 + Sqrt[10]]*Sqrt[-3 + 4*x^2 + 2*x^4])","C",0
57,1,80,146,0.0682706,"\int \frac{1}{\sqrt{-3+3 x^2+2 x^4}} \, dx","Integrate[1/Sqrt[-3 + 3*x^2 + 2*x^4],x]","-\frac{i \sqrt{-4 x^4-6 x^2+6} F\left(i \sinh ^{-1}\left(\frac{2 x}{\sqrt{3+\sqrt{33}}}\right)|-\frac{7}{4}-\frac{\sqrt{33}}{4}\right)}{\sqrt{\sqrt{33}-3} \sqrt{2 x^4+3 x^2-3}}","\frac{\sqrt{\frac{6-\left(3-\sqrt{33}\right) x^2}{6-\left(3+\sqrt{33}\right) x^2}} \sqrt{\left(3+\sqrt{33}\right) x^2-6} F\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{33} x}{\sqrt{\left(3+\sqrt{33}\right) x^2-6}}\right)|\frac{1}{22} \left(11+\sqrt{33}\right)\right)}{2\ 3^{3/4} \sqrt[4]{11} \sqrt{\frac{1}{6-\left(3+\sqrt{33}\right) x^2}} \sqrt{2 x^4+3 x^2-3}}",1,"((-I)*Sqrt[6 - 6*x^2 - 4*x^4]*EllipticF[I*ArcSinh[(2*x)/Sqrt[3 + Sqrt[33]]], -7/4 - Sqrt[33]/4])/(Sqrt[-3 + Sqrt[33]]*Sqrt[-3 + 3*x^2 + 2*x^4])","C",0
58,1,83,143,0.0550045,"\int \frac{1}{\sqrt{-3+2 x^2+2 x^4}} \, dx","Integrate[1/Sqrt[-3 + 2*x^2 + 2*x^4],x]","-\frac{i \sqrt{-2 x^4-2 x^2+3} F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{7}}} x\right)|-\frac{4}{3}-\frac{\sqrt{7}}{3}\right)}{\sqrt{\sqrt{7}-1} \sqrt{2 x^4+2 x^2-3}}","\frac{\sqrt{\frac{3-\left(1-\sqrt{7}\right) x^2}{3-\left(1+\sqrt{7}\right) x^2}} \sqrt{\left(1+\sqrt{7}\right) x^2-3} F\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{7} x}{\sqrt{\left(1+\sqrt{7}\right) x^2-3}}\right)|\frac{1}{14} \left(7+\sqrt{7}\right)\right)}{\sqrt{6} \sqrt[4]{7} \sqrt{\frac{1}{3-\left(1+\sqrt{7}\right) x^2}} \sqrt{2 x^4+2 x^2-3}}",1,"((-I)*Sqrt[3 - 2*x^2 - 2*x^4]*EllipticF[I*ArcSinh[Sqrt[2/(1 + Sqrt[7])]*x], -4/3 - Sqrt[7]/3])/(Sqrt[-1 + Sqrt[7]]*Sqrt[-3 + 2*x^2 + 2*x^4])","C",0
59,1,63,63,0.0246045,"\int \frac{1}{\sqrt{-3+x^2+2 x^4}} \, dx","Integrate[1/Sqrt[-3 + x^2 + 2*x^4],x]","-\frac{i \sqrt{1-x^2} \sqrt{2 x^2+3} F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{3}} x\right)|-\frac{3}{2}\right)}{\sqrt{2} \sqrt{2 x^4+x^2-3}}","\frac{\sqrt{x^2-1} \sqrt{2 x^2+3} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{5}{3}} x}{\sqrt{x^2-1}}\right)|\frac{3}{5}\right)}{\sqrt{5} \sqrt{2 x^4+x^2-3}}",1,"((-I)*Sqrt[1 - x^2]*Sqrt[3 + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/3]*x], -3/2])/(Sqrt[2]*Sqrt[-3 + x^2 + 2*x^4])","C",1
60,1,40,112,0.0239958,"\int \frac{1}{\sqrt{-3+2 x^4}} \, dx","Integrate[1/Sqrt[-3 + 2*x^4],x]","\frac{\sqrt{3-2 x^4} F\left(\left.\sin ^{-1}\left(\sqrt[4]{\frac{2}{3}} x\right)\right|-1\right)}{\sqrt[4]{6} \sqrt{2 x^4-3}}","\frac{\sqrt{\sqrt{6} x^2-3} \sqrt{\frac{\sqrt{6} x^2+3}{3-\sqrt{6} x^2}} F\left(\sin ^{-1}\left(\frac{2^{3/4} \sqrt[4]{3} x}{\sqrt{\sqrt{6} x^2-3}}\right)|\frac{1}{2}\right)}{6^{3/4} \sqrt{\frac{1}{3-\sqrt{6} x^2}} \sqrt{2 x^4-3}}",1,"(Sqrt[3 - 2*x^4]*EllipticF[ArcSin[(2/3)^(1/4)*x], -1])/(6^(1/4)*Sqrt[-3 + 2*x^4])","A",1
61,1,51,65,0.0235174,"\int \frac{1}{\sqrt{-3-x^2+2 x^4}} \, dx","Integrate[1/Sqrt[-3 - x^2 + 2*x^4],x]","\frac{\sqrt{3-2 x^2} \sqrt{x^2+1} F\left(\sin ^{-1}\left(\sqrt{\frac{2}{3}} x\right)|-\frac{3}{2}\right)}{\sqrt{4 x^4-2 x^2-6}}","\frac{\sqrt{x^2+1} \sqrt{2 x^2-3} F\left(\sin ^{-1}\left(\frac{\sqrt{5} x}{\sqrt{2 x^2-3}}\right)|\frac{2}{5}\right)}{\sqrt{5} \sqrt{2 x^4-x^2-3}}",1,"(Sqrt[3 - 2*x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[Sqrt[2/3]*x], -3/2])/Sqrt[-6 - 2*x^2 + 4*x^4]","A",1
62,1,81,150,0.0469947,"\int \frac{1}{\sqrt{-3-2 x^2+2 x^4}} \, dx","Integrate[1/Sqrt[-3 - 2*x^2 + 2*x^4],x]","-\frac{i \sqrt{-2 x^4+2 x^2+3} F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{-1+\sqrt{7}}} x\right)|\frac{1}{3} \left(-4+\sqrt{7}\right)\right)}{\sqrt{1+\sqrt{7}} \sqrt{2 x^4-2 x^2-3}}","\frac{\sqrt{-\left(\left(1-\sqrt{7}\right) x^2\right)-3} \sqrt{\frac{\left(1+\sqrt{7}\right) x^2+3}{\left(1-\sqrt{7}\right) x^2+3}} F\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{7} x}{\sqrt{-\left(\left(1-\sqrt{7}\right) x^2\right)-3}}\right)|\frac{1}{14} \left(7-\sqrt{7}\right)\right)}{\sqrt{6} \sqrt[4]{7} \sqrt{\frac{1}{\left(1-\sqrt{7}\right) x^2+3}} \sqrt{2 x^4-2 x^2-3}}",1,"((-I)*Sqrt[3 + 2*x^2 - 2*x^4]*EllipticF[I*ArcSinh[Sqrt[2/(-1 + Sqrt[7])]*x], (-4 + Sqrt[7])/3])/(Sqrt[1 + Sqrt[7]]*Sqrt[-3 - 2*x^2 + 2*x^4])","C",0
63,1,78,153,0.0572876,"\int \frac{1}{\sqrt{-3-3 x^2+2 x^4}} \, dx","Integrate[1/Sqrt[-3 - 3*x^2 + 2*x^4],x]","-\frac{i \sqrt{-4 x^4+6 x^2+6} F\left(i \sinh ^{-1}\left(\frac{2 x}{\sqrt{-3+\sqrt{33}}}\right)|\frac{1}{4} \left(-7+\sqrt{33}\right)\right)}{\sqrt{3+\sqrt{33}} \sqrt{2 x^4-3 x^2-3}}","\frac{\sqrt{-\left(\left(3-\sqrt{33}\right) x^2\right)-6} \sqrt{\frac{\left(3+\sqrt{33}\right) x^2+6}{\left(3-\sqrt{33}\right) x^2+6}} F\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{33} x}{\sqrt{-\left(\left(3-\sqrt{33}\right) x^2\right)-6}}\right)|\frac{1}{22} \left(11-\sqrt{33}\right)\right)}{2\ 3^{3/4} \sqrt[4]{11} \sqrt{\frac{1}{\left(3-\sqrt{33}\right) x^2+6}} \sqrt{2 x^4-3 x^2-3}}",1,"((-I)*Sqrt[6 + 6*x^2 - 4*x^4]*EllipticF[I*ArcSinh[(2*x)/Sqrt[-3 + Sqrt[33]]], (-7 + Sqrt[33])/4])/(Sqrt[3 + Sqrt[33]]*Sqrt[-3 - 3*x^2 + 2*x^4])","C",0
64,1,83,155,0.063734,"\int \frac{1}{\sqrt{-3-4 x^2+2 x^4}} \, dx","Integrate[1/Sqrt[-3 - 4*x^2 + 2*x^4],x]","-\frac{i \sqrt{-2 x^4+4 x^2+3} F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{-2+\sqrt{10}}} x\right)|-\frac{7}{3}+\frac{2 \sqrt{10}}{3}\right)}{\sqrt{2+\sqrt{10}} \sqrt{2 x^4-4 x^2-3}}","\frac{\sqrt{-\left(\left(2-\sqrt{10}\right) x^2\right)-3} \sqrt{\frac{\left(2+\sqrt{10}\right) x^2+3}{\left(2-\sqrt{10}\right) x^2+3}} F\left(\sin ^{-1}\left(\frac{2^{3/4} \sqrt[4]{5} x}{\sqrt{-\left(\left(2-\sqrt{10}\right) x^2\right)-3}}\right)|\frac{1}{10} \left(5-\sqrt{10}\right)\right)}{2^{3/4} \sqrt{3} \sqrt[4]{5} \sqrt{\frac{1}{\left(2-\sqrt{10}\right) x^2+3}} \sqrt{2 x^4-4 x^2-3}}",1,"((-I)*Sqrt[3 + 4*x^2 - 2*x^4]*EllipticF[I*ArcSinh[Sqrt[2/(-2 + Sqrt[10])]*x], -7/3 + (2*Sqrt[10])/3])/(Sqrt[2 + Sqrt[10]]*Sqrt[-3 - 4*x^2 + 2*x^4])","C",0
65,1,65,63,0.0248913,"\int \frac{1}{\sqrt{-3-5 x^2+2 x^4}} \, dx","Integrate[1/Sqrt[-3 - 5*x^2 + 2*x^4],x]","-\frac{i \sqrt{1-\frac{x^2}{3}} \sqrt{2 x^2+1} F\left(i \sinh ^{-1}\left(\sqrt{2} x\right)|-\frac{1}{6}\right)}{\sqrt{2} \sqrt{2 x^4-5 x^2-3}}","\frac{\sqrt{x^2-3} \sqrt{2 x^2+1} F\left(\sin ^{-1}\left(\frac{\sqrt{7} x}{\sqrt{x^2-3}}\right)|\frac{1}{7}\right)}{\sqrt{7} \sqrt{2 x^4-5 x^2-3}}",1,"((-I)*Sqrt[1 - x^2/3]*Sqrt[1 + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2]*x], -1/6])/(Sqrt[2]*Sqrt[-3 - 5*x^2 + 2*x^4])","C",1
66,1,58,52,0.0245187,"\int \frac{1}{\sqrt{2+5 x^2+3 x^4}} \, dx","Integrate[1/Sqrt[2 + 5*x^2 + 3*x^4],x]","-\frac{i \sqrt{x^2+1} \sqrt{3 x^2+2} F\left(i \sinh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|\frac{2}{3}\right)}{\sqrt{9 x^4+15 x^2+6}}","\frac{\left(x^2+1\right) \sqrt{\frac{3 x^2+2}{x^2+1}} F\left(\tan ^{-1}(x)|-\frac{1}{2}\right)}{\sqrt{2} \sqrt{3 x^4+5 x^2+2}}",1,"((-I)*Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]*EllipticF[I*ArcSinh[Sqrt[3/2]*x], 2/3])/Sqrt[6 + 15*x^2 + 9*x^4]","C",1
67,1,144,90,0.083638,"\int \frac{1}{\sqrt{2+4 x^2+3 x^4}} \, dx","Integrate[1/Sqrt[2 + 4*x^2 + 3*x^4],x]","-\frac{i \sqrt{1-\frac{3 x^2}{-2-i \sqrt{2}}} \sqrt{1-\frac{3 x^2}{-2+i \sqrt{2}}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{3}{-2-i \sqrt{2}}} x\right)|\frac{-2-i \sqrt{2}}{-2+i \sqrt{2}}\right)}{\sqrt{3} \sqrt{-\frac{1}{-2-i \sqrt{2}}} \sqrt{3 x^4+4 x^2+2}}","\frac{\left(\sqrt{6} x^2+2\right) \sqrt{\frac{3 x^4+4 x^2+2}{\left(\sqrt{6} x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{3}{2}} x\right)|\frac{1}{2}-\frac{1}{\sqrt{6}}\right)}{2 \sqrt[4]{6} \sqrt{3 x^4+4 x^2+2}}",1,"((-I)*Sqrt[1 - (3*x^2)/(-2 - I*Sqrt[2])]*Sqrt[1 - (3*x^2)/(-2 + I*Sqrt[2])]*EllipticF[I*ArcSinh[Sqrt[-3/(-2 - I*Sqrt[2])]*x], (-2 - I*Sqrt[2])/(-2 + I*Sqrt[2])])/(Sqrt[3]*Sqrt[-(-2 - I*Sqrt[2])^(-1)]*Sqrt[2 + 4*x^2 + 3*x^4])","C",1
68,1,144,92,0.1128549,"\int \frac{1}{\sqrt{2+3 x^2+3 x^4}} \, dx","Integrate[1/Sqrt[2 + 3*x^2 + 3*x^4],x]","-\frac{i \sqrt{1-\frac{6 x^2}{-3-i \sqrt{15}}} \sqrt{1-\frac{6 x^2}{-3+i \sqrt{15}}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{6}{-3-i \sqrt{15}}} x\right)|\frac{-3-i \sqrt{15}}{-3+i \sqrt{15}}\right)}{\sqrt{6} \sqrt{-\frac{1}{-3-i \sqrt{15}}} \sqrt{3 x^4+3 x^2+2}}","\frac{\left(\sqrt{6} x^2+2\right) \sqrt{\frac{3 x^4+3 x^2+2}{\left(\sqrt{6} x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{3}{2}} x\right)|\frac{1}{8} \left(4-\sqrt{6}\right)\right)}{2 \sqrt[4]{6} \sqrt{3 x^4+3 x^2+2}}",1,"((-I)*Sqrt[1 - (6*x^2)/(-3 - I*Sqrt[15])]*Sqrt[1 - (6*x^2)/(-3 + I*Sqrt[15])]*EllipticF[I*ArcSinh[Sqrt[-6/(-3 - I*Sqrt[15])]*x], (-3 - I*Sqrt[15])/(-3 + I*Sqrt[15])])/(Sqrt[6]*Sqrt[-(-3 - I*Sqrt[15])^(-1)]*Sqrt[2 + 3*x^2 + 3*x^4])","C",1
69,1,144,92,0.0829284,"\int \frac{1}{\sqrt{2+2 x^2+3 x^4}} \, dx","Integrate[1/Sqrt[2 + 2*x^2 + 3*x^4],x]","-\frac{i \sqrt{1-\frac{3 x^2}{-1-i \sqrt{5}}} \sqrt{1-\frac{3 x^2}{-1+i \sqrt{5}}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{3}{-1-i \sqrt{5}}} x\right)|\frac{-1-i \sqrt{5}}{-1+i \sqrt{5}}\right)}{\sqrt{3} \sqrt{-\frac{1}{-1-i \sqrt{5}}} \sqrt{3 x^4+2 x^2+2}}","\frac{\left(\sqrt{6} x^2+2\right) \sqrt{\frac{3 x^4+2 x^2+2}{\left(\sqrt{6} x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{3}{2}} x\right)|\frac{1}{12} \left(6-\sqrt{6}\right)\right)}{2 \sqrt[4]{6} \sqrt{3 x^4+2 x^2+2}}",1,"((-I)*Sqrt[1 - (3*x^2)/(-1 - I*Sqrt[5])]*Sqrt[1 - (3*x^2)/(-1 + I*Sqrt[5])]*EllipticF[I*ArcSinh[Sqrt[-3/(-1 - I*Sqrt[5])]*x], (-1 - I*Sqrt[5])/(-1 + I*Sqrt[5])])/(Sqrt[3]*Sqrt[-(-1 - I*Sqrt[5])^(-1)]*Sqrt[2 + 2*x^2 + 3*x^4])","C",1
70,1,142,88,0.0799075,"\int \frac{1}{\sqrt{2+x^2+3 x^4}} \, dx","Integrate[1/Sqrt[2 + x^2 + 3*x^4],x]","-\frac{i \sqrt{1-\frac{6 x^2}{-1-i \sqrt{23}}} \sqrt{1-\frac{6 x^2}{-1+i \sqrt{23}}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{6}{-1-i \sqrt{23}}} x\right)|\frac{-1-i \sqrt{23}}{-1+i \sqrt{23}}\right)}{\sqrt{6} \sqrt{-\frac{1}{-1-i \sqrt{23}}} \sqrt{3 x^4+x^2+2}}","\frac{\left(\sqrt{6} x^2+2\right) \sqrt{\frac{3 x^4+x^2+2}{\left(\sqrt{6} x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{3}{2}} x\right)|\frac{1}{24} \left(12-\sqrt{6}\right)\right)}{2 \sqrt[4]{6} \sqrt{3 x^4+x^2+2}}",1,"((-I)*Sqrt[1 - (6*x^2)/(-1 - I*Sqrt[23])]*Sqrt[1 - (6*x^2)/(-1 + I*Sqrt[23])]*EllipticF[I*ArcSinh[Sqrt[-6/(-1 - I*Sqrt[23])]*x], (-1 - I*Sqrt[23])/(-1 + I*Sqrt[23])])/(Sqrt[6]*Sqrt[-(-1 - I*Sqrt[23])^(-1)]*Sqrt[2 + x^2 + 3*x^4])","C",1
71,1,25,72,0.0260989,"\int \frac{1}{\sqrt{2+3 x^4}} \, dx","Integrate[1/Sqrt[2 + 3*x^4],x]","-\sqrt[4]{-\frac{1}{6}} F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-\frac{3}{2}} x\right)\right|-1\right)","\frac{\left(\sqrt{6} x^2+2\right) \sqrt{\frac{3 x^4+2}{\left(\sqrt{6} x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{3}{2}} x\right)|\frac{1}{2}\right)}{2 \sqrt[4]{6} \sqrt{3 x^4+2}}",1,"-((-1/6)^(1/4)*EllipticF[I*ArcSinh[(-3/2)^(1/4)*x], -1])","C",1
72,1,144,90,0.0775463,"\int \frac{1}{\sqrt{2-x^2+3 x^4}} \, dx","Integrate[1/Sqrt[2 - x^2 + 3*x^4],x]","-\frac{i \sqrt{1-\frac{6 x^2}{1-i \sqrt{23}}} \sqrt{1-\frac{6 x^2}{1+i \sqrt{23}}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{6}{1-i \sqrt{23}}} x\right)|\frac{1-i \sqrt{23}}{1+i \sqrt{23}}\right)}{\sqrt{6} \sqrt{-\frac{1}{1-i \sqrt{23}}} \sqrt{3 x^4-x^2+2}}","\frac{\left(\sqrt{6} x^2+2\right) \sqrt{\frac{3 x^4-x^2+2}{\left(\sqrt{6} x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{3}{2}} x\right)|\frac{1}{24} \left(12+\sqrt{6}\right)\right)}{2 \sqrt[4]{6} \sqrt{3 x^4-x^2+2}}",1,"((-I)*Sqrt[1 - (6*x^2)/(1 - I*Sqrt[23])]*Sqrt[1 - (6*x^2)/(1 + I*Sqrt[23])]*EllipticF[I*ArcSinh[Sqrt[-6/(1 - I*Sqrt[23])]*x], (1 - I*Sqrt[23])/(1 + I*Sqrt[23])])/(Sqrt[6]*Sqrt[-(1 - I*Sqrt[23])^(-1)]*Sqrt[2 - x^2 + 3*x^4])","C",1
73,1,144,90,0.0752265,"\int \frac{1}{\sqrt{2-2 x^2+3 x^4}} \, dx","Integrate[1/Sqrt[2 - 2*x^2 + 3*x^4],x]","-\frac{i \sqrt{1-\frac{3 x^2}{1-i \sqrt{5}}} \sqrt{1-\frac{3 x^2}{1+i \sqrt{5}}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{3}{1-i \sqrt{5}}} x\right)|\frac{1-i \sqrt{5}}{1+i \sqrt{5}}\right)}{\sqrt{3} \sqrt{-\frac{1}{1-i \sqrt{5}}} \sqrt{3 x^4-2 x^2+2}}","\frac{\left(\sqrt{6} x^2+2\right) \sqrt{\frac{3 x^4-2 x^2+2}{\left(\sqrt{6} x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{3}{2}} x\right)|\frac{1}{12} \left(6+\sqrt{6}\right)\right)}{2 \sqrt[4]{6} \sqrt{3 x^4-2 x^2+2}}",1,"((-I)*Sqrt[1 - (3*x^2)/(1 - I*Sqrt[5])]*Sqrt[1 - (3*x^2)/(1 + I*Sqrt[5])]*EllipticF[I*ArcSinh[Sqrt[-3/(1 - I*Sqrt[5])]*x], (1 - I*Sqrt[5])/(1 + I*Sqrt[5])])/(Sqrt[3]*Sqrt[-(1 - I*Sqrt[5])^(-1)]*Sqrt[2 - 2*x^2 + 3*x^4])","C",1
74,1,144,90,0.1089934,"\int \frac{1}{\sqrt{2-3 x^2+3 x^4}} \, dx","Integrate[1/Sqrt[2 - 3*x^2 + 3*x^4],x]","-\frac{i \sqrt{1-\frac{6 x^2}{3-i \sqrt{15}}} \sqrt{1-\frac{6 x^2}{3+i \sqrt{15}}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{6}{3-i \sqrt{15}}} x\right)|\frac{3-i \sqrt{15}}{3+i \sqrt{15}}\right)}{\sqrt{6} \sqrt{-\frac{1}{3-i \sqrt{15}}} \sqrt{3 x^4-3 x^2+2}}","\frac{\left(\sqrt{6} x^2+2\right) \sqrt{\frac{3 x^4-3 x^2+2}{\left(\sqrt{6} x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{3}{2}} x\right)|\frac{1}{8} \left(4+\sqrt{6}\right)\right)}{2 \sqrt[4]{6} \sqrt{3 x^4-3 x^2+2}}",1,"((-I)*Sqrt[1 - (6*x^2)/(3 - I*Sqrt[15])]*Sqrt[1 - (6*x^2)/(3 + I*Sqrt[15])]*EllipticF[I*ArcSinh[Sqrt[-6/(3 - I*Sqrt[15])]*x], (3 - I*Sqrt[15])/(3 + I*Sqrt[15])])/(Sqrt[6]*Sqrt[-(3 - I*Sqrt[15])^(-1)]*Sqrt[2 - 3*x^2 + 3*x^4])","C",1
75,1,144,88,0.0810964,"\int \frac{1}{\sqrt{2-4 x^2+3 x^4}} \, dx","Integrate[1/Sqrt[2 - 4*x^2 + 3*x^4],x]","-\frac{i \sqrt{1-\frac{3 x^2}{2-i \sqrt{2}}} \sqrt{1-\frac{3 x^2}{2+i \sqrt{2}}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{3}{2-i \sqrt{2}}} x\right)|\frac{2-i \sqrt{2}}{2+i \sqrt{2}}\right)}{\sqrt{3} \sqrt{-\frac{1}{2-i \sqrt{2}}} \sqrt{3 x^4-4 x^2+2}}","\frac{\left(\sqrt{6} x^2+2\right) \sqrt{\frac{3 x^4-4 x^2+2}{\left(\sqrt{6} x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{3}{2}} x\right)|\frac{1}{2}+\frac{1}{\sqrt{6}}\right)}{2 \sqrt[4]{6} \sqrt{3 x^4-4 x^2+2}}",1,"((-I)*Sqrt[1 - (3*x^2)/(2 - I*Sqrt[2])]*Sqrt[1 - (3*x^2)/(2 + I*Sqrt[2])]*EllipticF[I*ArcSinh[Sqrt[-3/(2 - I*Sqrt[2])]*x], (2 - I*Sqrt[2])/(2 + I*Sqrt[2])])/(Sqrt[3]*Sqrt[-(2 - I*Sqrt[2])^(-1)]*Sqrt[2 - 4*x^2 + 3*x^4])","C",1
76,1,53,92,0.0239337,"\int \frac{1}{\sqrt{2-5 x^2+3 x^4}} \, dx","Integrate[1/Sqrt[2 - 5*x^2 + 3*x^4],x]","\frac{\sqrt{2-3 x^2} \sqrt{1-x^2} F\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|\frac{2}{3}\right)}{\sqrt{9 x^4-15 x^2+6}}","\frac{\left(\sqrt{6} x^2+2\right) \sqrt{\frac{3 x^4-5 x^2+2}{\left(\sqrt{6} x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{3}{2}} x\right)|\frac{1}{24} \left(12+5 \sqrt{6}\right)\right)}{2 \sqrt[4]{6} \sqrt{3 x^4-5 x^2+2}}",1,"(Sqrt[2 - 3*x^2]*Sqrt[1 - x^2]*EllipticF[ArcSin[Sqrt[3/2]*x], 2/3])/Sqrt[6 - 15*x^2 + 9*x^4]","A",1
77,1,85,90,0.0838123,"\int \frac{1}{\sqrt{2-6 x^2+3 x^4}} \, dx","Integrate[1/Sqrt[2 - 6*x^2 + 3*x^4],x]","\frac{\sqrt{-3 x^2-\sqrt{3}+3} \sqrt{\left(\sqrt{3}-3\right) x^2+2} F\left(\sin ^{-1}\left(\sqrt{\frac{1}{2} \left(3+\sqrt{3}\right)} x\right)|2-\sqrt{3}\right)}{\sqrt{6} \sqrt{3 x^4-6 x^2+2}}","\frac{\left(\sqrt{6} x^2+2\right) \sqrt{\frac{3 x^4-6 x^2+2}{\left(\sqrt{6} x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{3}{2}} x\right)|\frac{1}{4} \left(2+\sqrt{6}\right)\right)}{2 \sqrt[4]{6} \sqrt{3 x^4-6 x^2+2}}",1,"(Sqrt[3 - Sqrt[3] - 3*x^2]*Sqrt[2 + (-3 + Sqrt[3])*x^2]*EllipticF[ArcSin[Sqrt[(3 + Sqrt[3])/2]*x], 2 - Sqrt[3]])/(Sqrt[6]*Sqrt[2 - 6*x^2 + 3*x^4])","A",1
78,1,97,110,0.0713548,"\int \frac{1}{\sqrt{3+9 x^2+2 x^4}} \, dx","Integrate[1/Sqrt[3 + 9*x^2 + 2*x^4],x]","-\frac{i \sqrt{\frac{-4 x^2+\sqrt{57}-9}{\sqrt{57}-9}} \sqrt{4 x^2+\sqrt{57}+9} F\left(i \sinh ^{-1}\left(\frac{2 x}{\sqrt{9+\sqrt{57}}}\right)|\frac{23}{4}+\frac{3 \sqrt{57}}{4}\right)}{2 \sqrt{2 x^4+9 x^2+3}}","\frac{\sqrt{\frac{\left(9-\sqrt{57}\right) x^2+6}{\left(9+\sqrt{57}\right) x^2+6}} \left(\left(9+\sqrt{57}\right) x^2+6\right) F\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(9+\sqrt{57}\right)} x\right)|\frac{1}{4} \left(-19+3 \sqrt{57}\right)\right)}{\sqrt{6 \left(9+\sqrt{57}\right)} \sqrt{2 x^4+9 x^2+3}}",1,"((-1/2*I)*Sqrt[(-9 + Sqrt[57] - 4*x^2)/(-9 + Sqrt[57])]*Sqrt[9 + Sqrt[57] + 4*x^2]*EllipticF[I*ArcSinh[(2*x)/Sqrt[9 + Sqrt[57]]], 23/4 + (3*Sqrt[57])/4])/Sqrt[3 + 9*x^2 + 2*x^4]","C",0
79,1,98,110,0.079792,"\int \frac{1}{\sqrt{3+8 x^2+2 x^4}} \, dx","Integrate[1/Sqrt[3 + 8*x^2 + 2*x^4],x]","-\frac{i \sqrt{\frac{-2 x^2+\sqrt{10}-4}{\sqrt{10}-4}} \sqrt{2 x^2+\sqrt{10}+4} F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{4+\sqrt{10}}} x\right)|\frac{13}{3}+\frac{4 \sqrt{10}}{3}\right)}{\sqrt{4 x^4+16 x^2+6}}","\frac{\sqrt{\frac{\left(4-\sqrt{10}\right) x^2+3}{\left(4+\sqrt{10}\right) x^2+3}} \left(\left(4+\sqrt{10}\right) x^2+3\right) F\left(\tan ^{-1}\left(\sqrt{\frac{1}{3} \left(4+\sqrt{10}\right)} x\right)|-\frac{2}{3} \left(5-2 \sqrt{10}\right)\right)}{\sqrt{3 \left(4+\sqrt{10}\right)} \sqrt{2 x^4+8 x^2+3}}",1,"((-I)*Sqrt[(-4 + Sqrt[10] - 2*x^2)/(-4 + Sqrt[10])]*Sqrt[4 + Sqrt[10] + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/(4 + Sqrt[10])]*x], 13/3 + (4*Sqrt[10])/3])/Sqrt[6 + 16*x^2 + 4*x^4]","C",0
80,1,61,60,0.0257303,"\int \frac{1}{\sqrt{3+7 x^2+2 x^4}} \, dx","Integrate[1/Sqrt[3 + 7*x^2 + 2*x^4],x]","-\frac{i \sqrt{x^2+3} \sqrt{2 x^2+1} F\left(i \sinh ^{-1}\left(\sqrt{2} x\right)|\frac{1}{6}\right)}{\sqrt{6} \sqrt{2 x^4+7 x^2+3}}","\frac{\sqrt{\frac{x^2+3}{2 x^2+1}} \left(2 x^2+1\right) F\left(\tan ^{-1}\left(\sqrt{2} x\right)|\frac{5}{6}\right)}{\sqrt{6} \sqrt{2 x^4+7 x^2+3}}",1,"((-I)*Sqrt[3 + x^2]*Sqrt[1 + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2]*x], 1/6])/(Sqrt[6]*Sqrt[3 + 7*x^2 + 2*x^4])","C",1
81,1,90,104,0.058448,"\int \frac{1}{\sqrt{3+6 x^2+2 x^4}} \, dx","Integrate[1/Sqrt[3 + 6*x^2 + 2*x^4],x]","-\frac{i \sqrt{\frac{-2 x^2+\sqrt{3}-3}{\sqrt{3}-3}} \sqrt{2 x^2+\sqrt{3}+3} F\left(i \sinh ^{-1}\left(\sqrt{1-\frac{1}{\sqrt{3}}} x\right)|2+\sqrt{3}\right)}{\sqrt{4 x^4+12 x^2+6}}","\frac{\sqrt{\frac{\left(3-\sqrt{3}\right) x^2+3}{\left(3+\sqrt{3}\right) x^2+3}} \left(\left(3+\sqrt{3}\right) x^2+3\right) F\left(\tan ^{-1}\left(\sqrt{\frac{1}{3} \left(3+\sqrt{3}\right)} x\right)|-1+\sqrt{3}\right)}{\sqrt{3 \left(3+\sqrt{3}\right)} \sqrt{2 x^4+6 x^2+3}}",1,"((-I)*Sqrt[(-3 + Sqrt[3] - 2*x^2)/(-3 + Sqrt[3])]*Sqrt[3 + Sqrt[3] + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[1 - 1/Sqrt[3]]*x], 2 + Sqrt[3]])/Sqrt[6 + 12*x^2 + 4*x^4]","C",0
82,1,58,52,0.0246701,"\int \frac{1}{\sqrt{3+5 x^2+2 x^4}} \, dx","Integrate[1/Sqrt[3 + 5*x^2 + 2*x^4],x]","-\frac{i \sqrt{x^2+1} \sqrt{2 x^2+3} F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{3}} x\right)|\frac{3}{2}\right)}{\sqrt{4 x^4+10 x^2+6}}","\frac{\left(x^2+1\right) \sqrt{\frac{2 x^2+3}{x^2+1}} F\left(\tan ^{-1}(x)|\frac{1}{3}\right)}{\sqrt{3} \sqrt{2 x^4+5 x^2+3}}",1,"((-I)*Sqrt[1 + x^2]*Sqrt[3 + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/3]*x], 3/2])/Sqrt[6 + 10*x^2 + 4*x^4]","C",1
83,1,144,90,0.0838899,"\int \frac{1}{\sqrt{3+4 x^2+2 x^4}} \, dx","Integrate[1/Sqrt[3 + 4*x^2 + 2*x^4],x]","-\frac{i \sqrt{1-\frac{2 x^2}{-2-i \sqrt{2}}} \sqrt{1-\frac{2 x^2}{-2+i \sqrt{2}}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{2}{-2-i \sqrt{2}}} x\right)|\frac{-2-i \sqrt{2}}{-2+i \sqrt{2}}\right)}{\sqrt{2} \sqrt{-\frac{1}{-2-i \sqrt{2}}} \sqrt{2 x^4+4 x^2+3}}","\frac{\left(\sqrt{6} x^2+3\right) \sqrt{\frac{2 x^4+4 x^2+3}{\left(\sqrt{6} x^2+3\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{2}{3}} x\right)|\frac{1}{2}-\frac{1}{\sqrt{6}}\right)}{2 \sqrt[4]{6} \sqrt{2 x^4+4 x^2+3}}",1,"((-I)*Sqrt[1 - (2*x^2)/(-2 - I*Sqrt[2])]*Sqrt[1 - (2*x^2)/(-2 + I*Sqrt[2])]*EllipticF[I*ArcSinh[Sqrt[-2/(-2 - I*Sqrt[2])]*x], (-2 - I*Sqrt[2])/(-2 + I*Sqrt[2])])/(Sqrt[2]*Sqrt[-(-2 - I*Sqrt[2])^(-1)]*Sqrt[3 + 4*x^2 + 2*x^4])","C",1
84,1,142,92,0.1005458,"\int \frac{1}{\sqrt{3+3 x^2+2 x^4}} \, dx","Integrate[1/Sqrt[3 + 3*x^2 + 2*x^4],x]","-\frac{i \sqrt{1-\frac{4 x^2}{-3-i \sqrt{15}}} \sqrt{1-\frac{4 x^2}{-3+i \sqrt{15}}} F\left(i \sinh ^{-1}\left(2 \sqrt{-\frac{1}{-3-i \sqrt{15}}} x\right)|\frac{-3-i \sqrt{15}}{-3+i \sqrt{15}}\right)}{2 \sqrt{-\frac{1}{-3-i \sqrt{15}}} \sqrt{2 x^4+3 x^2+3}}","\frac{\left(\sqrt{6} x^2+3\right) \sqrt{\frac{2 x^4+3 x^2+3}{\left(\sqrt{6} x^2+3\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{2}{3}} x\right)|\frac{1}{8} \left(4-\sqrt{6}\right)\right)}{2 \sqrt[4]{6} \sqrt{2 x^4+3 x^2+3}}",1,"((-1/2*I)*Sqrt[1 - (4*x^2)/(-3 - I*Sqrt[15])]*Sqrt[1 - (4*x^2)/(-3 + I*Sqrt[15])]*EllipticF[I*ArcSinh[2*Sqrt[-(-3 - I*Sqrt[15])^(-1)]*x], (-3 - I*Sqrt[15])/(-3 + I*Sqrt[15])])/(Sqrt[-(-3 - I*Sqrt[15])^(-1)]*Sqrt[3 + 3*x^2 + 2*x^4])","C",1
85,1,144,92,0.0810845,"\int \frac{1}{\sqrt{3+2 x^2+2 x^4}} \, dx","Integrate[1/Sqrt[3 + 2*x^2 + 2*x^4],x]","-\frac{i \sqrt{1-\frac{2 x^2}{-1-i \sqrt{5}}} \sqrt{1-\frac{2 x^2}{-1+i \sqrt{5}}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{2}{-1-i \sqrt{5}}} x\right)|\frac{-1-i \sqrt{5}}{-1+i \sqrt{5}}\right)}{\sqrt{2} \sqrt{-\frac{1}{-1-i \sqrt{5}}} \sqrt{2 x^4+2 x^2+3}}","\frac{\left(\sqrt{6} x^2+3\right) \sqrt{\frac{2 x^4+2 x^2+3}{\left(\sqrt{6} x^2+3\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{2}{3}} x\right)|\frac{1}{12} \left(6-\sqrt{6}\right)\right)}{2 \sqrt[4]{6} \sqrt{2 x^4+2 x^2+3}}",1,"((-I)*Sqrt[1 - (2*x^2)/(-1 - I*Sqrt[5])]*Sqrt[1 - (2*x^2)/(-1 + I*Sqrt[5])]*EllipticF[I*ArcSinh[Sqrt[-2/(-1 - I*Sqrt[5])]*x], (-1 - I*Sqrt[5])/(-1 + I*Sqrt[5])])/(Sqrt[2]*Sqrt[-(-1 - I*Sqrt[5])^(-1)]*Sqrt[3 + 2*x^2 + 2*x^4])","C",1
86,1,140,88,0.070763,"\int \frac{1}{\sqrt{3+x^2+2 x^4}} \, dx","Integrate[1/Sqrt[3 + x^2 + 2*x^4],x]","-\frac{i \sqrt{1-\frac{4 x^2}{-1-i \sqrt{23}}} \sqrt{1-\frac{4 x^2}{-1+i \sqrt{23}}} F\left(i \sinh ^{-1}\left(2 \sqrt{-\frac{1}{-1-i \sqrt{23}}} x\right)|\frac{-1-i \sqrt{23}}{-1+i \sqrt{23}}\right)}{2 \sqrt{-\frac{1}{-1-i \sqrt{23}}} \sqrt{2 x^4+x^2+3}}","\frac{\left(\sqrt{6} x^2+3\right) \sqrt{\frac{2 x^4+x^2+3}{\left(\sqrt{6} x^2+3\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{2}{3}} x\right)|\frac{1}{24} \left(12-\sqrt{6}\right)\right)}{2 \sqrt[4]{6} \sqrt{2 x^4+x^2+3}}",1,"((-1/2*I)*Sqrt[1 - (4*x^2)/(-1 - I*Sqrt[23])]*Sqrt[1 - (4*x^2)/(-1 + I*Sqrt[23])]*EllipticF[I*ArcSinh[2*Sqrt[-(-1 - I*Sqrt[23])^(-1)]*x], (-1 - I*Sqrt[23])/(-1 + I*Sqrt[23])])/(Sqrt[-(-1 - I*Sqrt[23])^(-1)]*Sqrt[3 + x^2 + 2*x^4])","C",1
87,1,25,72,0.0268844,"\int \frac{1}{\sqrt{3+2 x^4}} \, dx","Integrate[1/Sqrt[3 + 2*x^4],x]","-\sqrt[4]{-\frac{1}{6}} F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-\frac{2}{3}} x\right)\right|-1\right)","\frac{\left(\sqrt{6} x^2+3\right) \sqrt{\frac{2 x^4+3}{\left(\sqrt{6} x^2+3\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{2}{3}} x\right)|\frac{1}{2}\right)}{2 \sqrt[4]{6} \sqrt{2 x^4+3}}",1,"-((-1/6)^(1/4)*EllipticF[I*ArcSinh[(-2/3)^(1/4)*x], -1])","C",1
88,1,142,90,0.0651432,"\int \frac{1}{\sqrt{3-x^2+2 x^4}} \, dx","Integrate[1/Sqrt[3 - x^2 + 2*x^4],x]","-\frac{i \sqrt{1-\frac{4 x^2}{1-i \sqrt{23}}} \sqrt{1-\frac{4 x^2}{1+i \sqrt{23}}} F\left(i \sinh ^{-1}\left(2 \sqrt{-\frac{1}{1-i \sqrt{23}}} x\right)|\frac{1-i \sqrt{23}}{1+i \sqrt{23}}\right)}{2 \sqrt{-\frac{1}{1-i \sqrt{23}}} \sqrt{2 x^4-x^2+3}}","\frac{\left(\sqrt{6} x^2+3\right) \sqrt{\frac{2 x^4-x^2+3}{\left(\sqrt{6} x^2+3\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{2}{3}} x\right)|\frac{1}{24} \left(12+\sqrt{6}\right)\right)}{2 \sqrt[4]{6} \sqrt{2 x^4-x^2+3}}",1,"((-1/2*I)*Sqrt[1 - (4*x^2)/(1 - I*Sqrt[23])]*Sqrt[1 - (4*x^2)/(1 + I*Sqrt[23])]*EllipticF[I*ArcSinh[2*Sqrt[-(1 - I*Sqrt[23])^(-1)]*x], (1 - I*Sqrt[23])/(1 + I*Sqrt[23])])/(Sqrt[-(1 - I*Sqrt[23])^(-1)]*Sqrt[3 - x^2 + 2*x^4])","C",1
89,1,144,90,0.074174,"\int \frac{1}{\sqrt{3-2 x^2+2 x^4}} \, dx","Integrate[1/Sqrt[3 - 2*x^2 + 2*x^4],x]","-\frac{i \sqrt{1-\frac{2 x^2}{1-i \sqrt{5}}} \sqrt{1-\frac{2 x^2}{1+i \sqrt{5}}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{2}{1-i \sqrt{5}}} x\right)|\frac{1-i \sqrt{5}}{1+i \sqrt{5}}\right)}{\sqrt{2} \sqrt{-\frac{1}{1-i \sqrt{5}}} \sqrt{2 x^4-2 x^2+3}}","\frac{\left(\sqrt{6} x^2+3\right) \sqrt{\frac{2 x^4-2 x^2+3}{\left(\sqrt{6} x^2+3\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{2}{3}} x\right)|\frac{1}{12} \left(6+\sqrt{6}\right)\right)}{2 \sqrt[4]{6} \sqrt{2 x^4-2 x^2+3}}",1,"((-I)*Sqrt[1 - (2*x^2)/(1 - I*Sqrt[5])]*Sqrt[1 - (2*x^2)/(1 + I*Sqrt[5])]*EllipticF[I*ArcSinh[Sqrt[-2/(1 - I*Sqrt[5])]*x], (1 - I*Sqrt[5])/(1 + I*Sqrt[5])])/(Sqrt[2]*Sqrt[-(1 - I*Sqrt[5])^(-1)]*Sqrt[3 - 2*x^2 + 2*x^4])","C",1
90,1,142,90,0.0965139,"\int \frac{1}{\sqrt{3-3 x^2+2 x^4}} \, dx","Integrate[1/Sqrt[3 - 3*x^2 + 2*x^4],x]","-\frac{i \sqrt{1-\frac{4 x^2}{3-i \sqrt{15}}} \sqrt{1-\frac{4 x^2}{3+i \sqrt{15}}} F\left(i \sinh ^{-1}\left(2 \sqrt{-\frac{1}{3-i \sqrt{15}}} x\right)|\frac{3-i \sqrt{15}}{3+i \sqrt{15}}\right)}{2 \sqrt{-\frac{1}{3-i \sqrt{15}}} \sqrt{2 x^4-3 x^2+3}}","\frac{\left(\sqrt{6} x^2+3\right) \sqrt{\frac{2 x^4-3 x^2+3}{\left(\sqrt{6} x^2+3\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{2}{3}} x\right)|\frac{1}{8} \left(4+\sqrt{6}\right)\right)}{2 \sqrt[4]{6} \sqrt{2 x^4-3 x^2+3}}",1,"((-1/2*I)*Sqrt[1 - (4*x^2)/(3 - I*Sqrt[15])]*Sqrt[1 - (4*x^2)/(3 + I*Sqrt[15])]*EllipticF[I*ArcSinh[2*Sqrt[-(3 - I*Sqrt[15])^(-1)]*x], (3 - I*Sqrt[15])/(3 + I*Sqrt[15])])/(Sqrt[-(3 - I*Sqrt[15])^(-1)]*Sqrt[3 - 3*x^2 + 2*x^4])","C",1
91,1,144,88,0.0813916,"\int \frac{1}{\sqrt{3-4 x^2+2 x^4}} \, dx","Integrate[1/Sqrt[3 - 4*x^2 + 2*x^4],x]","-\frac{i \sqrt{1-\frac{2 x^2}{2-i \sqrt{2}}} \sqrt{1-\frac{2 x^2}{2+i \sqrt{2}}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{2}{2-i \sqrt{2}}} x\right)|\frac{2-i \sqrt{2}}{2+i \sqrt{2}}\right)}{\sqrt{2} \sqrt{-\frac{1}{2-i \sqrt{2}}} \sqrt{2 x^4-4 x^2+3}}","\frac{\left(\sqrt{6} x^2+3\right) \sqrt{\frac{2 x^4-4 x^2+3}{\left(\sqrt{6} x^2+3\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{2}{3}} x\right)|\frac{1}{2}+\frac{1}{\sqrt{6}}\right)}{2 \sqrt[4]{6} \sqrt{2 x^4-4 x^2+3}}",1,"((-I)*Sqrt[1 - (2*x^2)/(2 - I*Sqrt[2])]*Sqrt[1 - (2*x^2)/(2 + I*Sqrt[2])]*EllipticF[I*ArcSinh[Sqrt[-2/(2 - I*Sqrt[2])]*x], (2 - I*Sqrt[2])/(2 + I*Sqrt[2])])/(Sqrt[2]*Sqrt[-(2 - I*Sqrt[2])^(-1)]*Sqrt[3 - 4*x^2 + 2*x^4])","C",1
92,1,53,92,0.0249755,"\int \frac{1}{\sqrt{3-5 x^2+2 x^4}} \, dx","Integrate[1/Sqrt[3 - 5*x^2 + 2*x^4],x]","\frac{\sqrt{3-2 x^2} \sqrt{1-x^2} F\left(\sin ^{-1}\left(\sqrt{\frac{2}{3}} x\right)|\frac{3}{2}\right)}{\sqrt{4 x^4-10 x^2+6}}","\frac{\left(\sqrt{6} x^2+3\right) \sqrt{\frac{2 x^4-5 x^2+3}{\left(\sqrt{6} x^2+3\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{2}{3}} x\right)|\frac{1}{24} \left(12+5 \sqrt{6}\right)\right)}{2 \sqrt[4]{6} \sqrt{2 x^4-5 x^2+3}}",1,"(Sqrt[3 - 2*x^2]*Sqrt[1 - x^2]*EllipticF[ArcSin[Sqrt[2/3]*x], 3/2])/Sqrt[6 - 10*x^2 + 4*x^4]","A",1
93,1,81,90,0.0757113,"\int \frac{1}{\sqrt{3-6 x^2+2 x^4}} \, dx","Integrate[1/Sqrt[3 - 6*x^2 + 2*x^4],x]","\frac{\sqrt{-2 x^2-\sqrt{3}+3} \sqrt{\left(\sqrt{3}-3\right) x^2+3} F\left(\sin ^{-1}\left(\sqrt{1+\frac{1}{\sqrt{3}}} x\right)|2-\sqrt{3}\right)}{\sqrt{6} \sqrt{2 x^4-6 x^2+3}}","\frac{\left(\sqrt{6} x^2+3\right) \sqrt{\frac{2 x^4-6 x^2+3}{\left(\sqrt{6} x^2+3\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{2}{3}} x\right)|\frac{1}{4} \left(2+\sqrt{6}\right)\right)}{2 \sqrt[4]{6} \sqrt{2 x^4-6 x^2+3}}",1,"(Sqrt[3 - Sqrt[3] - 2*x^2]*Sqrt[3 + (-3 + Sqrt[3])*x^2]*EllipticF[ArcSin[Sqrt[1 + 1/Sqrt[3]]*x], 2 - Sqrt[3]])/(Sqrt[6]*Sqrt[3 - 6*x^2 + 2*x^4])","A",1
94,1,58,92,0.0223468,"\int \frac{1}{\sqrt{3-7 x^2+2 x^4}} \, dx","Integrate[1/Sqrt[3 - 7*x^2 + 2*x^4],x]","\frac{\sqrt{1-2 x^2} \sqrt{1-\frac{x^2}{3}} F\left(\sin ^{-1}\left(\sqrt{2} x\right)|\frac{1}{6}\right)}{\sqrt{2} \sqrt{2 x^4-7 x^2+3}}","\frac{\left(\sqrt{6} x^2+3\right) \sqrt{\frac{2 x^4-7 x^2+3}{\left(\sqrt{6} x^2+3\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{2}{3}} x\right)|\frac{1}{24} \left(12+7 \sqrt{6}\right)\right)}{2 \sqrt[4]{6} \sqrt{2 x^4-7 x^2+3}}",1,"(Sqrt[1 - 2*x^2]*Sqrt[1 - x^2/3]*EllipticF[ArcSin[Sqrt[2]*x], 1/6])/(Sqrt[2]*Sqrt[3 - 7*x^2 + 2*x^4])","A",1
95,1,58,19,0.0221446,"\int \frac{1}{\sqrt{-3+7 x^2-2 x^4}} \, dx","Integrate[1/Sqrt[-3 + 7*x^2 - 2*x^4],x]","\frac{\sqrt{1-2 x^2} \sqrt{1-\frac{x^2}{3}} F\left(\sin ^{-1}\left(\sqrt{2} x\right)|\frac{1}{6}\right)}{\sqrt{2} \sqrt{-2 x^4+7 x^2-3}}","-\frac{F\left(\cos ^{-1}\left(\frac{x}{\sqrt{3}}\right)|\frac{6}{5}\right)}{\sqrt{5}}",1,"(Sqrt[1 - 2*x^2]*Sqrt[1 - x^2/3]*EllipticF[ArcSin[Sqrt[2]*x], 1/6])/(Sqrt[2]*Sqrt[-3 + 7*x^2 - 2*x^4])","B",1
96,1,81,44,0.032492,"\int \frac{1}{\sqrt{-3+6 x^2-2 x^4}} \, dx","Integrate[1/Sqrt[-3 + 6*x^2 - 2*x^4],x]","\frac{\sqrt{-2 x^2-\sqrt{3}+3} \sqrt{\left(\sqrt{3}-3\right) x^2+3} F\left(\sin ^{-1}\left(\sqrt{1+\frac{1}{\sqrt{3}}} x\right)|2-\sqrt{3}\right)}{\sqrt{6} \sqrt{-2 x^4+6 x^2-3}}","-\frac{F\left(\cos ^{-1}\left(\sqrt{\frac{1}{3} \left(3-\sqrt{3}\right)} x\right)|\frac{1}{2} \left(1+\sqrt{3}\right)\right)}{\sqrt{2} \sqrt[4]{3}}",1,"(Sqrt[3 - Sqrt[3] - 2*x^2]*Sqrt[3 + (-3 + Sqrt[3])*x^2]*EllipticF[ArcSin[Sqrt[1 + 1/Sqrt[3]]*x], 2 - Sqrt[3]])/(Sqrt[6]*Sqrt[-3 + 6*x^2 - 2*x^4])","A",1
97,1,53,14,0.0247821,"\int \frac{1}{\sqrt{-3+5 x^2-2 x^4}} \, dx","Integrate[1/Sqrt[-3 + 5*x^2 - 2*x^4],x]","\frac{\sqrt{3-2 x^2} \sqrt{1-x^2} F\left(\sin ^{-1}\left(\sqrt{\frac{2}{3}} x\right)|\frac{3}{2}\right)}{\sqrt{-4 x^4+10 x^2-6}}","-F\left(\left.\cos ^{-1}\left(\sqrt{\frac{2}{3}} x\right)\right|3\right)",1,"(Sqrt[3 - 2*x^2]*Sqrt[1 - x^2]*EllipticF[ArcSin[Sqrt[2/3]*x], 3/2])/Sqrt[-6 + 10*x^2 - 4*x^4]","B",1
98,1,144,88,0.0540895,"\int \frac{1}{\sqrt{-3+4 x^2-2 x^4}} \, dx","Integrate[1/Sqrt[-3 + 4*x^2 - 2*x^4],x]","-\frac{i \sqrt{1-\frac{2 x^2}{2-i \sqrt{2}}} \sqrt{1-\frac{2 x^2}{2+i \sqrt{2}}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{2}{2-i \sqrt{2}}} x\right)|\frac{2-i \sqrt{2}}{2+i \sqrt{2}}\right)}{\sqrt{2} \sqrt{-\frac{1}{2-i \sqrt{2}}} \sqrt{-2 x^4+4 x^2-3}}","\frac{\left(\sqrt{6} x^2+3\right) \sqrt{\frac{2 x^4-4 x^2+3}{\left(\sqrt{6} x^2+3\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{2}{3}} x\right)|\frac{1}{2}+\frac{1}{\sqrt{6}}\right)}{2 \sqrt[4]{6} \sqrt{-2 x^4+4 x^2-3}}",1,"((-I)*Sqrt[1 - (2*x^2)/(2 - I*Sqrt[2])]*Sqrt[1 - (2*x^2)/(2 + I*Sqrt[2])]*EllipticF[I*ArcSinh[Sqrt[-2/(2 - I*Sqrt[2])]*x], (2 - I*Sqrt[2])/(2 + I*Sqrt[2])])/(Sqrt[2]*Sqrt[-(2 - I*Sqrt[2])^(-1)]*Sqrt[-3 + 4*x^2 - 2*x^4])","C",1
99,1,142,90,0.0928377,"\int \frac{1}{\sqrt{-3+3 x^2-2 x^4}} \, dx","Integrate[1/Sqrt[-3 + 3*x^2 - 2*x^4],x]","-\frac{i \sqrt{1-\frac{4 x^2}{3-i \sqrt{15}}} \sqrt{1-\frac{4 x^2}{3+i \sqrt{15}}} F\left(i \sinh ^{-1}\left(2 \sqrt{-\frac{1}{3-i \sqrt{15}}} x\right)|\frac{3-i \sqrt{15}}{3+i \sqrt{15}}\right)}{2 \sqrt{-\frac{1}{3-i \sqrt{15}}} \sqrt{-2 x^4+3 x^2-3}}","\frac{\left(\sqrt{6} x^2+3\right) \sqrt{\frac{2 x^4-3 x^2+3}{\left(\sqrt{6} x^2+3\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{2}{3}} x\right)|\frac{1}{8} \left(4+\sqrt{6}\right)\right)}{2 \sqrt[4]{6} \sqrt{-2 x^4+3 x^2-3}}",1,"((-1/2*I)*Sqrt[1 - (4*x^2)/(3 - I*Sqrt[15])]*Sqrt[1 - (4*x^2)/(3 + I*Sqrt[15])]*EllipticF[I*ArcSinh[2*Sqrt[-(3 - I*Sqrt[15])^(-1)]*x], (3 - I*Sqrt[15])/(3 + I*Sqrt[15])])/(Sqrt[-(3 - I*Sqrt[15])^(-1)]*Sqrt[-3 + 3*x^2 - 2*x^4])","C",1
100,1,144,90,0.0711929,"\int \frac{1}{\sqrt{-3+2 x^2-2 x^4}} \, dx","Integrate[1/Sqrt[-3 + 2*x^2 - 2*x^4],x]","-\frac{i \sqrt{1-\frac{2 x^2}{1-i \sqrt{5}}} \sqrt{1-\frac{2 x^2}{1+i \sqrt{5}}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{2}{1-i \sqrt{5}}} x\right)|\frac{1-i \sqrt{5}}{1+i \sqrt{5}}\right)}{\sqrt{2} \sqrt{-\frac{1}{1-i \sqrt{5}}} \sqrt{-2 x^4+2 x^2-3}}","\frac{\left(\sqrt{6} x^2+3\right) \sqrt{\frac{2 x^4-2 x^2+3}{\left(\sqrt{6} x^2+3\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{2}{3}} x\right)|\frac{1}{12} \left(6+\sqrt{6}\right)\right)}{2 \sqrt[4]{6} \sqrt{-2 x^4+2 x^2-3}}",1,"((-I)*Sqrt[1 - (2*x^2)/(1 - I*Sqrt[5])]*Sqrt[1 - (2*x^2)/(1 + I*Sqrt[5])]*EllipticF[I*ArcSinh[Sqrt[-2/(1 - I*Sqrt[5])]*x], (1 - I*Sqrt[5])/(1 + I*Sqrt[5])])/(Sqrt[2]*Sqrt[-(1 - I*Sqrt[5])^(-1)]*Sqrt[-3 + 2*x^2 - 2*x^4])","C",1
101,1,140,88,0.0636997,"\int \frac{1}{\sqrt{-3+x^2-2 x^4}} \, dx","Integrate[1/Sqrt[-3 + x^2 - 2*x^4],x]","-\frac{i \sqrt{1-\frac{4 x^2}{1-i \sqrt{23}}} \sqrt{1-\frac{4 x^2}{1+i \sqrt{23}}} F\left(i \sinh ^{-1}\left(2 \sqrt{-\frac{1}{1-i \sqrt{23}}} x\right)|\frac{1-i \sqrt{23}}{1+i \sqrt{23}}\right)}{2 \sqrt{-\frac{1}{1-i \sqrt{23}}} \sqrt{-2 x^4+x^2-3}}","\frac{\left(\sqrt{6} x^2+3\right) \sqrt{\frac{2 x^4-x^2+3}{\left(\sqrt{6} x^2+3\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{2}{3}} x\right)|\frac{1}{24} \left(12+\sqrt{6}\right)\right)}{2 \sqrt[4]{6} \sqrt{-2 x^4+x^2-3}}",1,"((-1/2*I)*Sqrt[1 - (4*x^2)/(1 - I*Sqrt[23])]*Sqrt[1 - (4*x^2)/(1 + I*Sqrt[23])]*EllipticF[I*ArcSinh[2*Sqrt[-(1 - I*Sqrt[23])^(-1)]*x], (1 - I*Sqrt[23])/(1 + I*Sqrt[23])])/(Sqrt[-(1 - I*Sqrt[23])^(-1)]*Sqrt[-3 + x^2 - 2*x^4])","C",1
102,1,47,72,0.0260483,"\int \frac{1}{\sqrt{-3-2 x^4}} \, dx","Integrate[1/Sqrt[-3 - 2*x^4],x]","-\frac{\sqrt[4]{-\frac{1}{6}} \sqrt{2 x^4+3} F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-\frac{2}{3}} x\right)\right|-1\right)}{\sqrt{-2 x^4-3}}","\frac{\left(\sqrt{6} x^2+3\right) \sqrt{\frac{2 x^4+3}{\left(\sqrt{6} x^2+3\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{2}{3}} x\right)|\frac{1}{2}\right)}{2 \sqrt[4]{6} \sqrt{-2 x^4-3}}",1,"-(((-1/6)^(1/4)*Sqrt[3 + 2*x^4]*EllipticF[I*ArcSinh[(-2/3)^(1/4)*x], -1])/Sqrt[-3 - 2*x^4])","C",1
103,1,142,90,0.0621264,"\int \frac{1}{\sqrt{-3-x^2-2 x^4}} \, dx","Integrate[1/Sqrt[-3 - x^2 - 2*x^4],x]","-\frac{i \sqrt{1-\frac{4 x^2}{-1-i \sqrt{23}}} \sqrt{1-\frac{4 x^2}{-1+i \sqrt{23}}} F\left(i \sinh ^{-1}\left(2 \sqrt{-\frac{1}{-1-i \sqrt{23}}} x\right)|\frac{-1-i \sqrt{23}}{-1+i \sqrt{23}}\right)}{2 \sqrt{-\frac{1}{-1-i \sqrt{23}}} \sqrt{-2 x^4-x^2-3}}","\frac{\left(\sqrt{6} x^2+3\right) \sqrt{\frac{2 x^4+x^2+3}{\left(\sqrt{6} x^2+3\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{2}{3}} x\right)|\frac{1}{24} \left(12-\sqrt{6}\right)\right)}{2 \sqrt[4]{6} \sqrt{-2 x^4-x^2-3}}",1,"((-1/2*I)*Sqrt[1 - (4*x^2)/(-1 - I*Sqrt[23])]*Sqrt[1 - (4*x^2)/(-1 + I*Sqrt[23])]*EllipticF[I*ArcSinh[2*Sqrt[-(-1 - I*Sqrt[23])^(-1)]*x], (-1 - I*Sqrt[23])/(-1 + I*Sqrt[23])])/(Sqrt[-(-1 - I*Sqrt[23])^(-1)]*Sqrt[-3 - x^2 - 2*x^4])","C",1
104,1,144,92,0.0773005,"\int \frac{1}{\sqrt{-3-2 x^2-2 x^4}} \, dx","Integrate[1/Sqrt[-3 - 2*x^2 - 2*x^4],x]","-\frac{i \sqrt{1-\frac{2 x^2}{-1-i \sqrt{5}}} \sqrt{1-\frac{2 x^2}{-1+i \sqrt{5}}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{2}{-1-i \sqrt{5}}} x\right)|\frac{-1-i \sqrt{5}}{-1+i \sqrt{5}}\right)}{\sqrt{2} \sqrt{-\frac{1}{-1-i \sqrt{5}}} \sqrt{-2 x^4-2 x^2-3}}","\frac{\left(\sqrt{6} x^2+3\right) \sqrt{\frac{2 x^4+2 x^2+3}{\left(\sqrt{6} x^2+3\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{2}{3}} x\right)|\frac{1}{12} \left(6-\sqrt{6}\right)\right)}{2 \sqrt[4]{6} \sqrt{-2 x^4-2 x^2-3}}",1,"((-I)*Sqrt[1 - (2*x^2)/(-1 - I*Sqrt[5])]*Sqrt[1 - (2*x^2)/(-1 + I*Sqrt[5])]*EllipticF[I*ArcSinh[Sqrt[-2/(-1 - I*Sqrt[5])]*x], (-1 - I*Sqrt[5])/(-1 + I*Sqrt[5])])/(Sqrt[2]*Sqrt[-(-1 - I*Sqrt[5])^(-1)]*Sqrt[-3 - 2*x^2 - 2*x^4])","C",1
105,1,142,92,0.0972179,"\int \frac{1}{\sqrt{-3-3 x^2-2 x^4}} \, dx","Integrate[1/Sqrt[-3 - 3*x^2 - 2*x^4],x]","-\frac{i \sqrt{1-\frac{4 x^2}{-3-i \sqrt{15}}} \sqrt{1-\frac{4 x^2}{-3+i \sqrt{15}}} F\left(i \sinh ^{-1}\left(2 \sqrt{-\frac{1}{-3-i \sqrt{15}}} x\right)|\frac{-3-i \sqrt{15}}{-3+i \sqrt{15}}\right)}{2 \sqrt{-\frac{1}{-3-i \sqrt{15}}} \sqrt{-2 x^4-3 x^2-3}}","\frac{\left(\sqrt{6} x^2+3\right) \sqrt{\frac{2 x^4+3 x^2+3}{\left(\sqrt{6} x^2+3\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{2}{3}} x\right)|\frac{1}{8} \left(4-\sqrt{6}\right)\right)}{2 \sqrt[4]{6} \sqrt{-2 x^4-3 x^2-3}}",1,"((-1/2*I)*Sqrt[1 - (4*x^2)/(-3 - I*Sqrt[15])]*Sqrt[1 - (4*x^2)/(-3 + I*Sqrt[15])]*EllipticF[I*ArcSinh[2*Sqrt[-(-3 - I*Sqrt[15])^(-1)]*x], (-3 - I*Sqrt[15])/(-3 + I*Sqrt[15])])/(Sqrt[-(-3 - I*Sqrt[15])^(-1)]*Sqrt[-3 - 3*x^2 - 2*x^4])","C",1
106,1,144,90,0.0853025,"\int \frac{1}{\sqrt{-3-4 x^2-2 x^4}} \, dx","Integrate[1/Sqrt[-3 - 4*x^2 - 2*x^4],x]","-\frac{i \sqrt{1-\frac{2 x^2}{-2-i \sqrt{2}}} \sqrt{1-\frac{2 x^2}{-2+i \sqrt{2}}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{2}{-2-i \sqrt{2}}} x\right)|\frac{-2-i \sqrt{2}}{-2+i \sqrt{2}}\right)}{\sqrt{2} \sqrt{-\frac{1}{-2-i \sqrt{2}}} \sqrt{-2 x^4-4 x^2-3}}","\frac{\left(\sqrt{6} x^2+3\right) \sqrt{\frac{2 x^4+4 x^2+3}{\left(\sqrt{6} x^2+3\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{2}{3}} x\right)|\frac{1}{2}-\frac{1}{\sqrt{6}}\right)}{2 \sqrt[4]{6} \sqrt{-2 x^4-4 x^2-3}}",1,"((-I)*Sqrt[1 - (2*x^2)/(-2 - I*Sqrt[2])]*Sqrt[1 - (2*x^2)/(-2 + I*Sqrt[2])]*EllipticF[I*ArcSinh[Sqrt[-2/(-2 - I*Sqrt[2])]*x], (-2 - I*Sqrt[2])/(-2 + I*Sqrt[2])])/(Sqrt[2]*Sqrt[-(-2 - I*Sqrt[2])^(-1)]*Sqrt[-3 - 4*x^2 - 2*x^4])","C",1
107,1,63,53,0.0243842,"\int \frac{1}{\sqrt{-3-5 x^2-2 x^4}} \, dx","Integrate[1/Sqrt[-3 - 5*x^2 - 2*x^4],x]","-\frac{i \sqrt{x^2+1} \sqrt{2 x^2+3} F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{3}} x\right)|\frac{3}{2}\right)}{\sqrt{2} \sqrt{-2 x^4-5 x^2-3}}","\frac{\sqrt{2 x^2+3} F\left(\tan ^{-1}(x)|\frac{1}{3}\right)}{\sqrt{3} \sqrt{-x^2-1} \sqrt{\frac{2 x^2+3}{x^2+1}}}",1,"((-I)*Sqrt[1 + x^2]*Sqrt[3 + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/3]*x], 3/2])/(Sqrt[2]*Sqrt[-3 - 5*x^2 - 2*x^4])","C",1
108,1,85,42,0.076571,"\int \frac{1}{\sqrt{-2+6 x^2-3 x^4}} \, dx","Integrate[1/Sqrt[-2 + 6*x^2 - 3*x^4],x]","\frac{\sqrt{-3 x^2-\sqrt{3}+3} \sqrt{\left(\sqrt{3}-3\right) x^2+2} F\left(\sin ^{-1}\left(\sqrt{\frac{1}{2} \left(3+\sqrt{3}\right)} x\right)|2-\sqrt{3}\right)}{\sqrt{6} \sqrt{-3 x^4+6 x^2-2}}","-\frac{F\left(\cos ^{-1}\left(\sqrt{\frac{3}{3+\sqrt{3}}} x\right)|\frac{1}{2} \left(1+\sqrt{3}\right)\right)}{\sqrt{2} \sqrt[4]{3}}",1,"(Sqrt[3 - Sqrt[3] - 3*x^2]*Sqrt[2 + (-3 + Sqrt[3])*x^2]*EllipticF[ArcSin[Sqrt[(3 + Sqrt[3])/2]*x], 2 - Sqrt[3]])/(Sqrt[6]*Sqrt[-2 + 6*x^2 - 3*x^4])","B",1
109,1,53,6,0.0253856,"\int \frac{1}{\sqrt{-2+5 x^2-3 x^4}} \, dx","Integrate[1/Sqrt[-2 + 5*x^2 - 3*x^4],x]","\frac{\sqrt{2-3 x^2} \sqrt{1-x^2} F\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|\frac{2}{3}\right)}{\sqrt{-9 x^4+15 x^2-6}}","-F\left(\left.\cos ^{-1}(x)\right|3\right)",1,"(Sqrt[2 - 3*x^2]*Sqrt[1 - x^2]*EllipticF[ArcSin[Sqrt[3/2]*x], 2/3])/Sqrt[-6 + 15*x^2 - 9*x^4]","B",1
110,1,144,88,0.0784096,"\int \frac{1}{\sqrt{-2+4 x^2-3 x^4}} \, dx","Integrate[1/Sqrt[-2 + 4*x^2 - 3*x^4],x]","-\frac{i \sqrt{1-\frac{3 x^2}{2-i \sqrt{2}}} \sqrt{1-\frac{3 x^2}{2+i \sqrt{2}}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{3}{2-i \sqrt{2}}} x\right)|\frac{2-i \sqrt{2}}{2+i \sqrt{2}}\right)}{\sqrt{3} \sqrt{-\frac{1}{2-i \sqrt{2}}} \sqrt{-3 x^4+4 x^2-2}}","\frac{\left(\sqrt{6} x^2+2\right) \sqrt{\frac{3 x^4-4 x^2+2}{\left(\sqrt{6} x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{3}{2}} x\right)|\frac{1}{2}+\frac{1}{\sqrt{6}}\right)}{2 \sqrt[4]{6} \sqrt{-3 x^4+4 x^2-2}}",1,"((-I)*Sqrt[1 - (3*x^2)/(2 - I*Sqrt[2])]*Sqrt[1 - (3*x^2)/(2 + I*Sqrt[2])]*EllipticF[I*ArcSinh[Sqrt[-3/(2 - I*Sqrt[2])]*x], (2 - I*Sqrt[2])/(2 + I*Sqrt[2])])/(Sqrt[3]*Sqrt[-(2 - I*Sqrt[2])^(-1)]*Sqrt[-2 + 4*x^2 - 3*x^4])","C",1
111,1,144,90,0.1081472,"\int \frac{1}{\sqrt{-2+3 x^2-3 x^4}} \, dx","Integrate[1/Sqrt[-2 + 3*x^2 - 3*x^4],x]","-\frac{i \sqrt{1-\frac{6 x^2}{3-i \sqrt{15}}} \sqrt{1-\frac{6 x^2}{3+i \sqrt{15}}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{6}{3-i \sqrt{15}}} x\right)|\frac{3-i \sqrt{15}}{3+i \sqrt{15}}\right)}{\sqrt{6} \sqrt{-\frac{1}{3-i \sqrt{15}}} \sqrt{-3 x^4+3 x^2-2}}","\frac{\left(\sqrt{6} x^2+2\right) \sqrt{\frac{3 x^4-3 x^2+2}{\left(\sqrt{6} x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{3}{2}} x\right)|\frac{1}{8} \left(4+\sqrt{6}\right)\right)}{2 \sqrt[4]{6} \sqrt{-3 x^4+3 x^2-2}}",1,"((-I)*Sqrt[1 - (6*x^2)/(3 - I*Sqrt[15])]*Sqrt[1 - (6*x^2)/(3 + I*Sqrt[15])]*EllipticF[I*ArcSinh[Sqrt[-6/(3 - I*Sqrt[15])]*x], (3 - I*Sqrt[15])/(3 + I*Sqrt[15])])/(Sqrt[6]*Sqrt[-(3 - I*Sqrt[15])^(-1)]*Sqrt[-2 + 3*x^2 - 3*x^4])","C",1
112,1,144,90,0.0752426,"\int \frac{1}{\sqrt{-2+2 x^2-3 x^4}} \, dx","Integrate[1/Sqrt[-2 + 2*x^2 - 3*x^4],x]","-\frac{i \sqrt{1-\frac{3 x^2}{1-i \sqrt{5}}} \sqrt{1-\frac{3 x^2}{1+i \sqrt{5}}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{3}{1-i \sqrt{5}}} x\right)|\frac{1-i \sqrt{5}}{1+i \sqrt{5}}\right)}{\sqrt{3} \sqrt{-\frac{1}{1-i \sqrt{5}}} \sqrt{-3 x^4+2 x^2-2}}","\frac{\left(\sqrt{6} x^2+2\right) \sqrt{\frac{3 x^4-2 x^2+2}{\left(\sqrt{6} x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{3}{2}} x\right)|\frac{1}{12} \left(6+\sqrt{6}\right)\right)}{2 \sqrt[4]{6} \sqrt{-3 x^4+2 x^2-2}}",1,"((-I)*Sqrt[1 - (3*x^2)/(1 - I*Sqrt[5])]*Sqrt[1 - (3*x^2)/(1 + I*Sqrt[5])]*EllipticF[I*ArcSinh[Sqrt[-3/(1 - I*Sqrt[5])]*x], (1 - I*Sqrt[5])/(1 + I*Sqrt[5])])/(Sqrt[3]*Sqrt[-(1 - I*Sqrt[5])^(-1)]*Sqrt[-2 + 2*x^2 - 3*x^4])","C",1
113,1,142,88,0.0765089,"\int \frac{1}{\sqrt{-2+x^2-3 x^4}} \, dx","Integrate[1/Sqrt[-2 + x^2 - 3*x^4],x]","-\frac{i \sqrt{1-\frac{6 x^2}{1-i \sqrt{23}}} \sqrt{1-\frac{6 x^2}{1+i \sqrt{23}}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{6}{1-i \sqrt{23}}} x\right)|\frac{1-i \sqrt{23}}{1+i \sqrt{23}}\right)}{\sqrt{6} \sqrt{-\frac{1}{1-i \sqrt{23}}} \sqrt{-3 x^4+x^2-2}}","\frac{\left(\sqrt{6} x^2+2\right) \sqrt{\frac{3 x^4-x^2+2}{\left(\sqrt{6} x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{3}{2}} x\right)|\frac{1}{24} \left(12+\sqrt{6}\right)\right)}{2 \sqrt[4]{6} \sqrt{-3 x^4+x^2-2}}",1,"((-I)*Sqrt[1 - (6*x^2)/(1 - I*Sqrt[23])]*Sqrt[1 - (6*x^2)/(1 + I*Sqrt[23])]*EllipticF[I*ArcSinh[Sqrt[-6/(1 - I*Sqrt[23])]*x], (1 - I*Sqrt[23])/(1 + I*Sqrt[23])])/(Sqrt[6]*Sqrt[-(1 - I*Sqrt[23])^(-1)]*Sqrt[-2 + x^2 - 3*x^4])","C",1
114,1,47,72,0.0277631,"\int \frac{1}{\sqrt{-2-3 x^4}} \, dx","Integrate[1/Sqrt[-2 - 3*x^4],x]","-\frac{\sqrt[4]{-\frac{1}{6}} \sqrt{3 x^4+2} F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-\frac{3}{2}} x\right)\right|-1\right)}{\sqrt{-3 x^4-2}}","\frac{\left(\sqrt{6} x^2+2\right) \sqrt{\frac{3 x^4+2}{\left(\sqrt{6} x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{3}{2}} x\right)|\frac{1}{2}\right)}{2 \sqrt[4]{6} \sqrt{-3 x^4-2}}",1,"-(((-1/6)^(1/4)*Sqrt[2 + 3*x^4]*EllipticF[I*ArcSinh[(-3/2)^(1/4)*x], -1])/Sqrt[-2 - 3*x^4])","C",1
115,1,144,90,0.0774579,"\int \frac{1}{\sqrt{-2-x^2-3 x^4}} \, dx","Integrate[1/Sqrt[-2 - x^2 - 3*x^4],x]","-\frac{i \sqrt{1-\frac{6 x^2}{-1-i \sqrt{23}}} \sqrt{1-\frac{6 x^2}{-1+i \sqrt{23}}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{6}{-1-i \sqrt{23}}} x\right)|\frac{-1-i \sqrt{23}}{-1+i \sqrt{23}}\right)}{\sqrt{6} \sqrt{-\frac{1}{-1-i \sqrt{23}}} \sqrt{-3 x^4-x^2-2}}","\frac{\left(\sqrt{6} x^2+2\right) \sqrt{\frac{3 x^4+x^2+2}{\left(\sqrt{6} x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{3}{2}} x\right)|\frac{1}{24} \left(12-\sqrt{6}\right)\right)}{2 \sqrt[4]{6} \sqrt{-3 x^4-x^2-2}}",1,"((-I)*Sqrt[1 - (6*x^2)/(-1 - I*Sqrt[23])]*Sqrt[1 - (6*x^2)/(-1 + I*Sqrt[23])]*EllipticF[I*ArcSinh[Sqrt[-6/(-1 - I*Sqrt[23])]*x], (-1 - I*Sqrt[23])/(-1 + I*Sqrt[23])])/(Sqrt[6]*Sqrt[-(-1 - I*Sqrt[23])^(-1)]*Sqrt[-2 - x^2 - 3*x^4])","C",1
116,1,144,92,0.0785806,"\int \frac{1}{\sqrt{-2-2 x^2-3 x^4}} \, dx","Integrate[1/Sqrt[-2 - 2*x^2 - 3*x^4],x]","-\frac{i \sqrt{1-\frac{3 x^2}{-1-i \sqrt{5}}} \sqrt{1-\frac{3 x^2}{-1+i \sqrt{5}}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{3}{-1-i \sqrt{5}}} x\right)|\frac{-1-i \sqrt{5}}{-1+i \sqrt{5}}\right)}{\sqrt{3} \sqrt{-\frac{1}{-1-i \sqrt{5}}} \sqrt{-3 x^4-2 x^2-2}}","\frac{\left(\sqrt{6} x^2+2\right) \sqrt{\frac{3 x^4+2 x^2+2}{\left(\sqrt{6} x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{3}{2}} x\right)|\frac{1}{12} \left(6-\sqrt{6}\right)\right)}{2 \sqrt[4]{6} \sqrt{-3 x^4-2 x^2-2}}",1,"((-I)*Sqrt[1 - (3*x^2)/(-1 - I*Sqrt[5])]*Sqrt[1 - (3*x^2)/(-1 + I*Sqrt[5])]*EllipticF[I*ArcSinh[Sqrt[-3/(-1 - I*Sqrt[5])]*x], (-1 - I*Sqrt[5])/(-1 + I*Sqrt[5])])/(Sqrt[3]*Sqrt[-(-1 - I*Sqrt[5])^(-1)]*Sqrt[-2 - 2*x^2 - 3*x^4])","C",1
117,1,144,92,0.1117898,"\int \frac{1}{\sqrt{-2-3 x^2-3 x^4}} \, dx","Integrate[1/Sqrt[-2 - 3*x^2 - 3*x^4],x]","-\frac{i \sqrt{1-\frac{6 x^2}{-3-i \sqrt{15}}} \sqrt{1-\frac{6 x^2}{-3+i \sqrt{15}}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{6}{-3-i \sqrt{15}}} x\right)|\frac{-3-i \sqrt{15}}{-3+i \sqrt{15}}\right)}{\sqrt{6} \sqrt{-\frac{1}{-3-i \sqrt{15}}} \sqrt{-3 x^4-3 x^2-2}}","\frac{\left(\sqrt{6} x^2+2\right) \sqrt{\frac{3 x^4+3 x^2+2}{\left(\sqrt{6} x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{3}{2}} x\right)|\frac{1}{8} \left(4-\sqrt{6}\right)\right)}{2 \sqrt[4]{6} \sqrt{-3 x^4-3 x^2-2}}",1,"((-I)*Sqrt[1 - (6*x^2)/(-3 - I*Sqrt[15])]*Sqrt[1 - (6*x^2)/(-3 + I*Sqrt[15])]*EllipticF[I*ArcSinh[Sqrt[-6/(-3 - I*Sqrt[15])]*x], (-3 - I*Sqrt[15])/(-3 + I*Sqrt[15])])/(Sqrt[6]*Sqrt[-(-3 - I*Sqrt[15])^(-1)]*Sqrt[-2 - 3*x^2 - 3*x^4])","C",1
118,1,144,90,0.0807401,"\int \frac{1}{\sqrt{-2-4 x^2-3 x^4}} \, dx","Integrate[1/Sqrt[-2 - 4*x^2 - 3*x^4],x]","-\frac{i \sqrt{1-\frac{3 x^2}{-2-i \sqrt{2}}} \sqrt{1-\frac{3 x^2}{-2+i \sqrt{2}}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{3}{-2-i \sqrt{2}}} x\right)|\frac{-2-i \sqrt{2}}{-2+i \sqrt{2}}\right)}{\sqrt{3} \sqrt{-\frac{1}{-2-i \sqrt{2}}} \sqrt{-3 x^4-4 x^2-2}}","\frac{\left(\sqrt{6} x^2+2\right) \sqrt{\frac{3 x^4+4 x^2+2}{\left(\sqrt{6} x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{3}{2}} x\right)|\frac{1}{2}-\frac{1}{\sqrt{6}}\right)}{2 \sqrt[4]{6} \sqrt{-3 x^4-4 x^2-2}}",1,"((-I)*Sqrt[1 - (3*x^2)/(-2 - I*Sqrt[2])]*Sqrt[1 - (3*x^2)/(-2 + I*Sqrt[2])]*EllipticF[I*ArcSinh[Sqrt[-3/(-2 - I*Sqrt[2])]*x], (-2 - I*Sqrt[2])/(-2 + I*Sqrt[2])])/(Sqrt[3]*Sqrt[-(-2 - I*Sqrt[2])^(-1)]*Sqrt[-2 - 4*x^2 - 3*x^4])","C",1
119,1,63,52,0.0235693,"\int \frac{1}{\sqrt{-2-5 x^2-3 x^4}} \, dx","Integrate[1/Sqrt[-2 - 5*x^2 - 3*x^4],x]","-\frac{i \sqrt{x^2+1} \sqrt{3 x^2+2} F\left(i \sinh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|\frac{2}{3}\right)}{\sqrt{3} \sqrt{-3 x^4-5 x^2-2}}","-\frac{\sqrt{-3 x^2-2} F\left(\tan ^{-1}(x)|-\frac{1}{2}\right)}{\sqrt{2} \sqrt{x^2+1} \sqrt{\frac{3 x^2+2}{x^2+1}}}",1,"((-I)*Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]*EllipticF[I*ArcSinh[Sqrt[3/2]*x], 2/3])/(Sqrt[3]*Sqrt[-2 - 5*x^2 - 3*x^4])","C",1
120,1,144,92,0.1122985,"\int \frac{1}{\sqrt{2+5 x^2+5 x^4}} \, dx","Integrate[1/Sqrt[2 + 5*x^2 + 5*x^4],x]","-\frac{i \sqrt{1-\frac{10 x^2}{-5-i \sqrt{15}}} \sqrt{1-\frac{10 x^2}{-5+i \sqrt{15}}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{10}{-5-i \sqrt{15}}} x\right)|\frac{-5-i \sqrt{15}}{-5+i \sqrt{15}}\right)}{\sqrt{10} \sqrt{-\frac{1}{-5-i \sqrt{15}}} \sqrt{5 x^4+5 x^2+2}}","\frac{\left(\sqrt{10} x^2+2\right) \sqrt{\frac{5 x^4+5 x^2+2}{\left(\sqrt{10} x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{\frac{5}{2}} x\right)|\frac{1}{8} \left(4-\sqrt{10}\right)\right)}{2 \sqrt[4]{10} \sqrt{5 x^4+5 x^2+2}}",1,"((-I)*Sqrt[1 - (10*x^2)/(-5 - I*Sqrt[15])]*Sqrt[1 - (10*x^2)/(-5 + I*Sqrt[15])]*EllipticF[I*ArcSinh[Sqrt[-10/(-5 - I*Sqrt[15])]*x], (-5 - I*Sqrt[15])/(-5 + I*Sqrt[15])])/(Sqrt[10]*Sqrt[-(-5 - I*Sqrt[15])^(-1)]*Sqrt[2 + 5*x^2 + 5*x^4])","C",1
121,1,147,90,0.0848809,"\int \frac{1}{\sqrt{2+5 x^2+4 x^4}} \, dx","Integrate[1/Sqrt[2 + 5*x^2 + 4*x^4],x]","-\frac{i \sqrt{1-\frac{8 x^2}{-5-i \sqrt{7}}} \sqrt{1-\frac{8 x^2}{-5+i \sqrt{7}}} F\left(i \sinh ^{-1}\left(2 \sqrt{-\frac{2}{-5-i \sqrt{7}}} x\right)|\frac{-5-i \sqrt{7}}{-5+i \sqrt{7}}\right)}{2 \sqrt{2} \sqrt{-\frac{1}{-5-i \sqrt{7}}} \sqrt{4 x^4+5 x^2+2}}","\frac{\left(\sqrt{2} x^2+1\right) \sqrt{\frac{4 x^4+5 x^2+2}{\left(\sqrt{2} x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{16} \left(8-5 \sqrt{2}\right)\right)}{2\ 2^{3/4} \sqrt{4 x^4+5 x^2+2}}",1,"((-1/2*I)*Sqrt[1 - (8*x^2)/(-5 - I*Sqrt[7])]*Sqrt[1 - (8*x^2)/(-5 + I*Sqrt[7])]*EllipticF[I*ArcSinh[2*Sqrt[-2/(-5 - I*Sqrt[7])]*x], (-5 - I*Sqrt[7])/(-5 + I*Sqrt[7])])/(Sqrt[2]*Sqrt[-(-5 - I*Sqrt[7])^(-1)]*Sqrt[2 + 5*x^2 + 4*x^4])","C",1
122,1,58,52,0.0204184,"\int \frac{1}{\sqrt{2+5 x^2+3 x^4}} \, dx","Integrate[1/Sqrt[2 + 5*x^2 + 3*x^4],x]","-\frac{i \sqrt{x^2+1} \sqrt{3 x^2+2} F\left(i \sinh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)|\frac{2}{3}\right)}{\sqrt{9 x^4+15 x^2+6}}","\frac{\left(x^2+1\right) \sqrt{\frac{3 x^2+2}{x^2+1}} F\left(\tan ^{-1}(x)|-\frac{1}{2}\right)}{\sqrt{2} \sqrt{3 x^4+5 x^2+2}}",1,"((-I)*Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]*EllipticF[I*ArcSinh[Sqrt[3/2]*x], 2/3])/Sqrt[6 + 15*x^2 + 9*x^4]","C",1
123,1,58,58,0.0219327,"\int \frac{1}{\sqrt{2+5 x^2+2 x^4}} \, dx","Integrate[1/Sqrt[2 + 5*x^2 + 2*x^4],x]","-\frac{i \sqrt{x^2+2} \sqrt{2 x^2+1} F\left(i \sinh ^{-1}\left(\sqrt{2} x\right)|\frac{1}{4}\right)}{2 \sqrt{2 x^4+5 x^2+2}}","\frac{\sqrt{\frac{x^2+2}{2 x^2+1}} \left(2 x^2+1\right) F\left(\tan ^{-1}\left(\sqrt{2} x\right)|\frac{3}{4}\right)}{2 \sqrt{2 x^4+5 x^2+2}}",1,"((-1/2*I)*Sqrt[2 + x^2]*Sqrt[1 + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2]*x], 1/4])/Sqrt[2 + 5*x^2 + 2*x^4]","C",1
124,1,103,108,0.0820248,"\int \frac{1}{\sqrt{2+5 x^2+x^4}} \, dx","Integrate[1/Sqrt[2 + 5*x^2 + x^4],x]","-\frac{i \sqrt{2 x^2-\sqrt{17}+5} \sqrt{2 x^2+\sqrt{17}+5} F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{17}}} x\right)|\frac{21}{4}+\frac{5 \sqrt{17}}{4}\right)}{\sqrt{2 \left(5-\sqrt{17}\right)} \sqrt{x^4+5 x^2+2}}","\frac{\sqrt{\frac{\left(5-\sqrt{17}\right) x^2+4}{\left(5+\sqrt{17}\right) x^2+4}} \left(\left(5+\sqrt{17}\right) x^2+4\right) F\left(\tan ^{-1}\left(\frac{1}{2} \sqrt{5+\sqrt{17}} x\right)|\frac{1}{4} \left(-17+5 \sqrt{17}\right)\right)}{2 \sqrt{5+\sqrt{17}} \sqrt{x^4+5 x^2+2}}",1,"((-I)*Sqrt[5 - Sqrt[17] + 2*x^2]*Sqrt[5 + Sqrt[17] + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/(5 + Sqrt[17])]*x], 21/4 + (5*Sqrt[17])/4])/(Sqrt[2*(5 - Sqrt[17])]*Sqrt[2 + 5*x^2 + x^4])","C",0
125,1,55,48,0.0631975,"\int \frac{1}{\sqrt{2+5 x^2-x^4}} \, dx","Integrate[1/Sqrt[2 + 5*x^2 - x^4],x]","-i \sqrt{\frac{2}{5+\sqrt{33}}} F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{-5+\sqrt{33}}} x\right)|-\frac{29}{4}+\frac{5 \sqrt{33}}{4}\right)","\sqrt{\frac{2}{\sqrt{33}-5}} F\left(\sin ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{33}}} x\right)|\frac{1}{4} \left(-29-5 \sqrt{33}\right)\right)",1,"(-I)*Sqrt[2/(5 + Sqrt[33])]*EllipticF[I*ArcSinh[Sqrt[2/(-5 + Sqrt[33])]*x], -29/4 + (5*Sqrt[33])/4]","C",0
126,1,52,45,0.0464899,"\int \frac{1}{\sqrt{2+5 x^2-2 x^4}} \, dx","Integrate[1/Sqrt[2 + 5*x^2 - 2*x^4],x]","-i \sqrt{\frac{2}{5+\sqrt{41}}} F\left(i \sinh ^{-1}\left(\frac{2 x}{\sqrt{-5+\sqrt{41}}}\right)|-\frac{33}{8}+\frac{5 \sqrt{41}}{8}\right)","\sqrt{\frac{2}{\sqrt{41}-5}} F\left(\sin ^{-1}\left(\frac{2 x}{\sqrt{5+\sqrt{41}}}\right)|\frac{1}{8} \left(-33-5 \sqrt{41}\right)\right)",1,"(-I)*Sqrt[2/(5 + Sqrt[41])]*EllipticF[I*ArcSinh[(2*x)/Sqrt[-5 + Sqrt[41]]], -33/8 + (5*Sqrt[41])/8]","C",0
127,1,65,10,0.0232428,"\int \frac{1}{\sqrt{2+5 x^2-3 x^4}} \, dx","Integrate[1/Sqrt[2 + 5*x^2 - 3*x^4],x]","-\frac{i \sqrt{1-\frac{x^2}{2}} \sqrt{3 x^2+1} F\left(i \sinh ^{-1}\left(\sqrt{3} x\right)|-\frac{1}{6}\right)}{\sqrt{3} \sqrt{-3 x^4+5 x^2+2}}","F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-6\right)",1,"((-I)*Sqrt[1 - x^2/2]*Sqrt[1 + 3*x^2]*EllipticF[I*ArcSinh[Sqrt[3]*x], -1/6])/(Sqrt[3]*Sqrt[2 + 5*x^2 - 3*x^4])","C",1
128,1,56,49,0.0605514,"\int \frac{1}{\sqrt{2+5 x^2-4 x^4}} \, dx","Integrate[1/Sqrt[2 + 5*x^2 - 4*x^4],x]","-i \sqrt{\frac{2}{5+\sqrt{57}}} F\left(i \sinh ^{-1}\left(2 \sqrt{\frac{2}{-5+\sqrt{57}}} x\right)|\frac{1}{16} \left(-41+5 \sqrt{57}\right)\right)","\sqrt{\frac{2}{\sqrt{57}-5}} F\left(\sin ^{-1}\left(2 \sqrt{\frac{2}{5+\sqrt{57}}} x\right)|\frac{1}{16} \left(-41-5 \sqrt{57}\right)\right)",1,"(-I)*Sqrt[2/(5 + Sqrt[57])]*EllipticF[I*ArcSinh[2*Sqrt[2/(-5 + Sqrt[57])]*x], (-41 + 5*Sqrt[57])/16]","C",0
129,1,52,48,0.0587717,"\int \frac{1}{\sqrt{2+5 x^2-5 x^4}} \, dx","Integrate[1/Sqrt[2 + 5*x^2 - 5*x^4],x]","-i \sqrt{\frac{2}{5+\sqrt{65}}} F\left(i \sinh ^{-1}\left(\frac{1}{2} \sqrt{5+\sqrt{65}} x\right)|\frac{1}{4} \left(-9+\sqrt{65}\right)\right)","\sqrt{\frac{2}{\sqrt{65}-5}} F\left(\sin ^{-1}\left(\sqrt{\frac{10}{5+\sqrt{65}}} x\right)|\frac{1}{4} \left(-9-\sqrt{65}\right)\right)",1,"(-I)*Sqrt[2/(5 + Sqrt[65])]*EllipticF[I*ArcSinh[(Sqrt[5 + Sqrt[65]]*x)/2], (-9 + Sqrt[65])/4]","C",0
130,1,56,49,0.0490603,"\int \frac{1}{\sqrt{2+5 x^2-6 x^4}} \, dx","Integrate[1/Sqrt[2 + 5*x^2 - 6*x^4],x]","-i \sqrt{\frac{2}{5+\sqrt{73}}} F\left(i \sinh ^{-1}\left(2 \sqrt{\frac{3}{-5+\sqrt{73}}} x\right)|\frac{1}{24} \left(-49+5 \sqrt{73}\right)\right)","\sqrt{\frac{2}{\sqrt{73}-5}} F\left(\sin ^{-1}\left(2 \sqrt{\frac{3}{5+\sqrt{73}}} x\right)|\frac{1}{24} \left(-49-5 \sqrt{73}\right)\right)",1,"(-I)*Sqrt[2/(5 + Sqrt[73])]*EllipticF[I*ArcSinh[2*Sqrt[3/(-5 + Sqrt[73])]*x], (-49 + 5*Sqrt[73])/24]","C",0
131,1,65,12,0.0261242,"\int \frac{1}{\sqrt{2+5 x^2-7 x^4}} \, dx","Integrate[1/Sqrt[2 + 5*x^2 - 7*x^4],x]","-\frac{i \sqrt{1-x^2} \sqrt{7 x^2+2} F\left(i \sinh ^{-1}\left(\sqrt{\frac{7}{2}} x\right)|-\frac{2}{7}\right)}{\sqrt{7} \sqrt{-7 x^4+5 x^2+2}}","\frac{F\left(\sin ^{-1}(x)|-\frac{7}{2}\right)}{\sqrt{2}}",1,"((-I)*Sqrt[1 - x^2]*Sqrt[2 + 7*x^2]*EllipticF[I*ArcSinh[Sqrt[7/2]*x], -2/7])/(Sqrt[7]*Sqrt[2 + 5*x^2 - 7*x^4])","C",1
132,1,52,45,0.0435438,"\int \frac{1}{\sqrt{2+5 x^2-8 x^4}} \, dx","Integrate[1/Sqrt[2 + 5*x^2 - 8*x^4],x]","-i \sqrt{\frac{2}{5+\sqrt{89}}} F\left(i \sinh ^{-1}\left(\frac{4 x}{\sqrt{-5+\sqrt{89}}}\right)|\frac{1}{32} \left(-57+5 \sqrt{89}\right)\right)","\sqrt{\frac{2}{\sqrt{89}-5}} F\left(\sin ^{-1}\left(\frac{4 x}{\sqrt{5+\sqrt{89}}}\right)|\frac{1}{32} \left(-57-5 \sqrt{89}\right)\right)",1,"(-I)*Sqrt[2/(5 + Sqrt[89])]*EllipticF[I*ArcSinh[(4*x)/Sqrt[-5 + Sqrt[89]]], (-57 + 5*Sqrt[89])/32]","C",0
133,1,56,49,0.0491399,"\int \frac{1}{\sqrt{2+5 x^2-9 x^4}} \, dx","Integrate[1/Sqrt[2 + 5*x^2 - 9*x^4],x]","-i \sqrt{\frac{2}{5+\sqrt{97}}} F\left(i \sinh ^{-1}\left(3 \sqrt{\frac{2}{-5+\sqrt{97}}} x\right)|\frac{1}{36} \left(-61+5 \sqrt{97}\right)\right)","\sqrt{\frac{2}{\sqrt{97}-5}} F\left(\sin ^{-1}\left(3 \sqrt{\frac{2}{5+\sqrt{97}}} x\right)|\frac{1}{36} \left(-61-5 \sqrt{97}\right)\right)",1,"(-I)*Sqrt[2/(5 + Sqrt[97])]*EllipticF[I*ArcSinh[3*Sqrt[2/(-5 + Sqrt[97])]*x], (-61 + 5*Sqrt[97])/36]","C",0
134,1,17,17,0.0015859,"\int x^2 \left(b x^2+c x^4\right) \, dx","Integrate[x^2*(b*x^2 + c*x^4),x]","\frac{b x^5}{5}+\frac{c x^7}{7}","\frac{b x^5}{5}+\frac{c x^7}{7}",1,"(b*x^5)/5 + (c*x^7)/7","A",1
135,1,17,17,0.0015195,"\int x \left(b x^2+c x^4\right) \, dx","Integrate[x*(b*x^2 + c*x^4),x]","\frac{b x^4}{4}+\frac{c x^6}{6}","\frac{b x^4}{4}+\frac{c x^6}{6}",1,"(b*x^4)/4 + (c*x^6)/6","A",1
136,1,17,17,0.0000711,"\int \left(b x^2+c x^4\right) \, dx","Integrate[b*x^2 + c*x^4,x]","\frac{b x^3}{3}+\frac{c x^5}{5}","\frac{b x^3}{3}+\frac{c x^5}{5}",1,"(b*x^3)/3 + (c*x^5)/5","A",1
137,1,17,17,0.0012363,"\int \frac{b x^2+c x^4}{x} \, dx","Integrate[(b*x^2 + c*x^4)/x,x]","\frac{b x^2}{2}+\frac{c x^4}{4}","\frac{b x^2}{2}+\frac{c x^4}{4}",1,"(b*x^2)/2 + (c*x^4)/4","A",1
138,1,12,12,0.0003797,"\int \frac{b x^2+c x^4}{x^2} \, dx","Integrate[(b*x^2 + c*x^4)/x^2,x]","b x+\frac{c x^3}{3}","b x+\frac{c x^3}{3}",1,"b*x + (c*x^3)/3","A",1
139,1,13,13,0.0010422,"\int \frac{b x^2+c x^4}{x^3} \, dx","Integrate[(b*x^2 + c*x^4)/x^3,x]","b \log (x)+\frac{c x^2}{2}","b \log (x)+\frac{c x^2}{2}",1,"(c*x^2)/2 + b*Log[x]","A",1
140,1,10,10,0.0010544,"\int \frac{b x^2+c x^4}{x^4} \, dx","Integrate[(b*x^2 + c*x^4)/x^4,x]","c x-\frac{b}{x}","c x-\frac{b}{x}",1,"-(b/x) + c*x","A",1
141,1,13,13,0.0023228,"\int \frac{b x^2+c x^4}{x^5} \, dx","Integrate[(b*x^2 + c*x^4)/x^5,x]","c \log (x)-\frac{b}{2 x^2}","c \log (x)-\frac{b}{2 x^2}",1,"-1/2*b/x^2 + c*Log[x]","A",1
142,1,15,15,0.0024983,"\int \frac{b x^2+c x^4}{x^6} \, dx","Integrate[(b*x^2 + c*x^4)/x^6,x]","-\frac{b}{3 x^3}-\frac{c}{x}","-\frac{b}{3 x^3}-\frac{c}{x}",1,"-1/3*b/x^3 - c/x","A",1
143,1,17,17,0.0025498,"\int \frac{b x^2+c x^4}{x^7} \, dx","Integrate[(b*x^2 + c*x^4)/x^7,x]","-\frac{b}{4 x^4}-\frac{c}{2 x^2}","-\frac{b}{4 x^4}-\frac{c}{2 x^2}",1,"-1/4*b/x^4 - c/(2*x^2)","A",1
144,1,17,17,0.0020836,"\int \frac{b x^2+c x^4}{x^8} \, dx","Integrate[(b*x^2 + c*x^4)/x^8,x]","-\frac{b}{5 x^5}-\frac{c}{3 x^3}","-\frac{b}{5 x^5}-\frac{c}{3 x^3}",1,"-1/5*b/x^5 - c/(3*x^3)","A",1
145,1,30,30,0.0016529,"\int \left(b x^2+c x^4\right)^2 \, dx","Integrate[(b*x^2 + c*x^4)^2,x]","\frac{b^2 x^5}{5}+\frac{2}{7} b c x^7+\frac{c^2 x^9}{9}","\frac{b^2 x^5}{5}+\frac{2}{7} b c x^7+\frac{c^2 x^9}{9}",1,"(b^2*x^5)/5 + (2*b*c*x^7)/7 + (c^2*x^9)/9","A",1
146,1,30,30,0.0008245,"\int \frac{\left(b x^2+c x^4\right)^2}{x} \, dx","Integrate[(b*x^2 + c*x^4)^2/x,x]","\frac{b^2 x^4}{4}+\frac{1}{3} b c x^6+\frac{c^2 x^8}{8}","\frac{b^2 x^4}{4}+\frac{1}{3} b c x^6+\frac{c^2 x^8}{8}",1,"(b^2*x^4)/4 + (b*c*x^6)/3 + (c^2*x^8)/8","A",1
147,1,30,30,0.0008457,"\int \frac{\left(b x^2+c x^4\right)^2}{x^2} \, dx","Integrate[(b*x^2 + c*x^4)^2/x^2,x]","\frac{b^2 x^3}{3}+\frac{2}{5} b c x^5+\frac{c^2 x^7}{7}","\frac{b^2 x^3}{3}+\frac{2}{5} b c x^5+\frac{c^2 x^7}{7}",1,"(b^2*x^3)/3 + (2*b*c*x^5)/5 + (c^2*x^7)/7","A",1
148,1,16,16,0.0021104,"\int \frac{\left(b x^2+c x^4\right)^2}{x^3} \, dx","Integrate[(b*x^2 + c*x^4)^2/x^3,x]","\frac{\left(b+c x^2\right)^3}{6 c}","\frac{\left(b+c x^2\right)^3}{6 c}",1,"(b + c*x^2)^3/(6*c)","A",1
149,1,25,25,0.0010947,"\int \frac{\left(b x^2+c x^4\right)^2}{x^4} \, dx","Integrate[(b*x^2 + c*x^4)^2/x^4,x]","b^2 x+\frac{2}{3} b c x^3+\frac{c^2 x^5}{5}","b^2 x+\frac{2}{3} b c x^3+\frac{c^2 x^5}{5}",1,"b^2*x + (2*b*c*x^3)/3 + (c^2*x^5)/5","A",1
150,1,23,23,0.0008846,"\int \frac{\left(b x^2+c x^4\right)^2}{x^5} \, dx","Integrate[(b*x^2 + c*x^4)^2/x^5,x]","b^2 \log (x)+b c x^2+\frac{c^2 x^4}{4}","b^2 \log (x)+b c x^2+\frac{c^2 x^4}{4}",1,"b*c*x^2 + (c^2*x^4)/4 + b^2*Log[x]","A",1
151,1,24,24,0.0007257,"\int \frac{\left(b x^2+c x^4\right)^2}{x^6} \, dx","Integrate[(b*x^2 + c*x^4)^2/x^6,x]","-\frac{b^2}{x}+2 b c x+\frac{c^2 x^3}{3}","-\frac{b^2}{x}+2 b c x+\frac{c^2 x^3}{3}",1,"-(b^2/x) + 2*b*c*x + (c^2*x^3)/3","A",1
152,1,27,27,0.0009252,"\int \frac{\left(b x^2+c x^4\right)^2}{x^7} \, dx","Integrate[(b*x^2 + c*x^4)^2/x^7,x]","-\frac{b^2}{2 x^2}+2 b c \log (x)+\frac{c^2 x^2}{2}","-\frac{b^2}{2 x^2}+2 b c \log (x)+\frac{c^2 x^2}{2}",1,"-1/2*b^2/x^2 + (c^2*x^2)/2 + 2*b*c*Log[x]","A",1
153,1,23,23,0.0008373,"\int \frac{\left(b x^2+c x^4\right)^2}{x^8} \, dx","Integrate[(b*x^2 + c*x^4)^2/x^8,x]","-\frac{b^2}{3 x^3}-\frac{2 b c}{x}+c^2 x","-\frac{b^2}{3 x^3}-\frac{2 b c}{x}+c^2 x",1,"-1/3*b^2/x^3 - (2*b*c)/x + c^2*x","A",1
154,1,24,24,0.0008421,"\int \frac{\left(b x^2+c x^4\right)^2}{x^9} \, dx","Integrate[(b*x^2 + c*x^4)^2/x^9,x]","-\frac{b^2}{4 x^4}-\frac{b c}{x^2}+c^2 \log (x)","-\frac{b^2}{4 x^4}-\frac{b c}{x^2}+c^2 \log (x)",1,"-1/4*b^2/x^4 - (b*c)/x^2 + c^2*Log[x]","A",1
155,1,28,28,0.0008871,"\int \frac{\left(b x^2+c x^4\right)^2}{x^{10}} \, dx","Integrate[(b*x^2 + c*x^4)^2/x^10,x]","-\frac{b^2}{5 x^5}-\frac{2 b c}{3 x^3}-\frac{c^2}{x}","-\frac{b^2}{5 x^5}-\frac{2 b c}{3 x^3}-\frac{c^2}{x}",1,"-1/5*b^2/x^5 - (2*b*c)/(3*x^3) - c^2/x","A",1
156,1,30,19,0.0009085,"\int \frac{\left(b x^2+c x^4\right)^2}{x^{11}} \, dx","Integrate[(b*x^2 + c*x^4)^2/x^11,x]","-\frac{b^2}{6 x^6}-\frac{b c}{2 x^4}-\frac{c^2}{2 x^2}","-\frac{\left(b+c x^2\right)^3}{6 b x^6}",1,"-1/6*b^2/x^6 - (b*c)/(2*x^4) - c^2/(2*x^2)","A",1
157,1,30,30,0.000921,"\int \frac{\left(b x^2+c x^4\right)^2}{x^{12}} \, dx","Integrate[(b*x^2 + c*x^4)^2/x^12,x]","-\frac{b^2}{7 x^7}-\frac{2 b c}{5 x^5}-\frac{c^2}{3 x^3}","-\frac{b^2}{7 x^7}-\frac{2 b c}{5 x^5}-\frac{c^2}{3 x^3}",1,"-1/7*b^2/x^7 - (2*b*c)/(5*x^5) - c^2/(3*x^3)","A",1
158,1,43,43,0.0019637,"\int \frac{\left(b x^2+c x^4\right)^3}{x^2} \, dx","Integrate[(b*x^2 + c*x^4)^3/x^2,x]","\frac{b^3 x^5}{5}+\frac{3}{7} b^2 c x^7+\frac{1}{3} b c^2 x^9+\frac{c^3 x^{11}}{11}","\frac{b^3 x^5}{5}+\frac{3}{7} b^2 c x^7+\frac{1}{3} b c^2 x^9+\frac{c^3 x^{11}}{11}",1,"(b^3*x^5)/5 + (3*b^2*c*x^7)/7 + (b*c^2*x^9)/3 + (c^3*x^11)/11","A",1
159,1,43,34,0.0018393,"\int \frac{\left(b x^2+c x^4\right)^3}{x^3} \, dx","Integrate[(b*x^2 + c*x^4)^3/x^3,x]","\frac{b^3 x^4}{4}+\frac{1}{2} b^2 c x^6+\frac{3}{8} b c^2 x^8+\frac{c^3 x^{10}}{10}","\frac{\left(b+c x^2\right)^5}{10 c^2}-\frac{b \left(b+c x^2\right)^4}{8 c^2}",1,"(b^3*x^4)/4 + (b^2*c*x^6)/2 + (3*b*c^2*x^8)/8 + (c^3*x^10)/10","A",1
160,1,43,43,0.0022543,"\int \frac{\left(b x^2+c x^4\right)^3}{x^4} \, dx","Integrate[(b*x^2 + c*x^4)^3/x^4,x]","\frac{b^3 x^3}{3}+\frac{3}{5} b^2 c x^5+\frac{3}{7} b c^2 x^7+\frac{c^3 x^9}{9}","\frac{b^3 x^3}{3}+\frac{3}{5} b^2 c x^5+\frac{3}{7} b c^2 x^7+\frac{c^3 x^9}{9}",1,"(b^3*x^3)/3 + (3*b^2*c*x^5)/5 + (3*b*c^2*x^7)/7 + (c^3*x^9)/9","A",1
161,1,16,16,0.0022682,"\int \frac{\left(b x^2+c x^4\right)^3}{x^5} \, dx","Integrate[(b*x^2 + c*x^4)^3/x^5,x]","\frac{\left(b+c x^2\right)^4}{8 c}","\frac{\left(b+c x^2\right)^4}{8 c}",1,"(b + c*x^2)^4/(8*c)","A",1
162,1,35,35,0.0013458,"\int \frac{\left(b x^2+c x^4\right)^3}{x^6} \, dx","Integrate[(b*x^2 + c*x^4)^3/x^6,x]","b^3 x+b^2 c x^3+\frac{3}{5} b c^2 x^5+\frac{c^3 x^7}{7}","b^3 x+b^2 c x^3+\frac{3}{5} b c^2 x^5+\frac{c^3 x^7}{7}",1,"b^3*x + b^2*c*x^3 + (3*b*c^2*x^5)/5 + (c^3*x^7)/7","A",1
163,1,39,39,0.0040123,"\int \frac{\left(b x^2+c x^4\right)^3}{x^7} \, dx","Integrate[(b*x^2 + c*x^4)^3/x^7,x]","b^3 \log (x)+\frac{3}{2} b^2 c x^2+\frac{3}{4} b c^2 x^4+\frac{c^3 x^6}{6}","b^3 \log (x)+\frac{3}{2} b^2 c x^2+\frac{3}{4} b c^2 x^4+\frac{c^3 x^6}{6}",1,"(3*b^2*c*x^2)/2 + (3*b*c^2*x^4)/4 + (c^3*x^6)/6 + b^3*Log[x]","A",1
164,1,34,34,0.0037877,"\int \frac{\left(b x^2+c x^4\right)^3}{x^8} \, dx","Integrate[(b*x^2 + c*x^4)^3/x^8,x]","-\frac{b^3}{x}+3 b^2 c x+b c^2 x^3+\frac{c^3 x^5}{5}","-\frac{b^3}{x}+3 b^2 c x+b c^2 x^3+\frac{c^3 x^5}{5}",1,"-(b^3/x) + 3*b^2*c*x + b*c^2*x^3 + (c^3*x^5)/5","A",1
165,1,40,40,0.0066855,"\int \frac{\left(b x^2+c x^4\right)^3}{x^9} \, dx","Integrate[(b*x^2 + c*x^4)^3/x^9,x]","-\frac{b^3}{2 x^2}+3 b^2 c \log (x)+\frac{3}{2} b c^2 x^2+\frac{c^3 x^4}{4}","-\frac{b^3}{2 x^2}+3 b^2 c \log (x)+\frac{3}{2} b c^2 x^2+\frac{c^3 x^4}{4}",1,"-1/2*b^3/x^2 + (3*b*c^2*x^2)/2 + (c^3*x^4)/4 + 3*b^2*c*Log[x]","A",1
166,1,37,37,0.0038267,"\int \frac{\left(b x^2+c x^4\right)^3}{x^{10}} \, dx","Integrate[(b*x^2 + c*x^4)^3/x^10,x]","-\frac{b^3}{3 x^3}-\frac{3 b^2 c}{x}+3 b c^2 x+\frac{c^3 x^3}{3}","-\frac{b^3}{3 x^3}-\frac{3 b^2 c}{x}+3 b c^2 x+\frac{c^3 x^3}{3}",1,"-1/3*b^3/x^3 - (3*b^2*c)/x + 3*b*c^2*x + (c^3*x^3)/3","A",1
167,1,40,40,0.004445,"\int \frac{\left(b x^2+c x^4\right)^3}{x^{11}} \, dx","Integrate[(b*x^2 + c*x^4)^3/x^11,x]","-\frac{b^3}{4 x^4}-\frac{3 b^2 c}{2 x^2}+3 b c^2 \log (x)+\frac{c^3 x^2}{2}","-\frac{b^3}{4 x^4}-\frac{3 b^2 c}{2 x^2}+3 b c^2 \log (x)+\frac{c^3 x^2}{2}",1,"-1/4*b^3/x^4 - (3*b^2*c)/(2*x^2) + (c^3*x^2)/2 + 3*b*c^2*Log[x]","A",1
168,1,34,34,0.0051865,"\int \frac{\left(b x^2+c x^4\right)^3}{x^{12}} \, dx","Integrate[(b*x^2 + c*x^4)^3/x^12,x]","-\frac{b^3}{5 x^5}-\frac{b^2 c}{x^3}-\frac{3 b c^2}{x}+c^3 x","-\frac{b^3}{5 x^5}-\frac{b^2 c}{x^3}-\frac{3 b c^2}{x}+c^3 x",1,"-1/5*b^3/x^5 - (b^2*c)/x^3 - (3*b*c^2)/x + c^3*x","A",1
169,1,39,39,0.0042996,"\int \frac{\left(b x^2+c x^4\right)^3}{x^{13}} \, dx","Integrate[(b*x^2 + c*x^4)^3/x^13,x]","-\frac{b^3}{6 x^6}-\frac{3 b^2 c}{4 x^4}-\frac{3 b c^2}{2 x^2}+c^3 \log (x)","-\frac{b^3}{6 x^6}-\frac{3 b^2 c}{4 x^4}-\frac{3 b c^2}{2 x^2}+c^3 \log (x)",1,"-1/6*b^3/x^6 - (3*b^2*c)/(4*x^4) - (3*b*c^2)/(2*x^2) + c^3*Log[x]","A",1
170,1,39,39,0.0037952,"\int \frac{\left(b x^2+c x^4\right)^3}{x^{14}} \, dx","Integrate[(b*x^2 + c*x^4)^3/x^14,x]","-\frac{b^3}{7 x^7}-\frac{3 b^2 c}{5 x^5}-\frac{b c^2}{x^3}-\frac{c^3}{x}","-\frac{b^3}{7 x^7}-\frac{3 b^2 c}{5 x^5}-\frac{b c^2}{x^3}-\frac{c^3}{x}",1,"-1/7*b^3/x^7 - (3*b^2*c)/(5*x^5) - (b*c^2)/x^3 - c^3/x","A",1
171,1,43,19,0.0062963,"\int \frac{\left(b x^2+c x^4\right)^3}{x^{15}} \, dx","Integrate[(b*x^2 + c*x^4)^3/x^15,x]","-\frac{b^3}{8 x^8}-\frac{b^2 c}{2 x^6}-\frac{3 b c^2}{4 x^4}-\frac{c^3}{2 x^2}","-\frac{\left(b+c x^2\right)^4}{8 b x^8}",1,"-1/8*b^3/x^8 - (b^2*c)/(2*x^6) - (3*b*c^2)/(4*x^4) - c^3/(2*x^2)","B",1
172,1,43,43,0.0038022,"\int \frac{\left(b x^2+c x^4\right)^3}{x^{16}} \, dx","Integrate[(b*x^2 + c*x^4)^3/x^16,x]","-\frac{b^3}{9 x^9}-\frac{3 b^2 c}{7 x^7}-\frac{3 b c^2}{5 x^5}-\frac{c^3}{3 x^3}","-\frac{b^3}{9 x^9}-\frac{3 b^2 c}{7 x^7}-\frac{3 b c^2}{5 x^5}-\frac{c^3}{3 x^3}",1,"-1/9*b^3/x^9 - (3*b^2*c)/(7*x^7) - (3*b*c^2)/(5*x^5) - c^3/(3*x^3)","A",1
173,1,43,40,0.0038503,"\int \frac{\left(b x^2+c x^4\right)^3}{x^{17}} \, dx","Integrate[(b*x^2 + c*x^4)^3/x^17,x]","-\frac{b^3}{10 x^{10}}-\frac{3 b^2 c}{8 x^8}-\frac{b c^2}{2 x^6}-\frac{c^3}{4 x^4}","\frac{c \left(b+c x^2\right)^4}{40 b^2 x^8}-\frac{\left(b+c x^2\right)^4}{10 b x^{10}}",1,"-1/10*b^3/x^10 - (3*b^2*c)/(8*x^8) - (b*c^2)/(2*x^6) - c^3/(4*x^4)","A",1
174,1,68,68,0.025654,"\int \frac{x^{10}}{b x^2+c x^4} \, dx","Integrate[x^10/(b*x^2 + c*x^4),x]","\frac{b^{7/2} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{c^{9/2}}-\frac{b^3 x}{c^4}+\frac{b^2 x^3}{3 c^3}-\frac{b x^5}{5 c^2}+\frac{x^7}{7 c}","\frac{b^{7/2} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{c^{9/2}}-\frac{b^3 x}{c^4}+\frac{b^2 x^3}{3 c^3}-\frac{b x^5}{5 c^2}+\frac{x^7}{7 c}",1,"-((b^3*x)/c^4) + (b^2*x^3)/(3*c^3) - (b*x^5)/(5*c^2) + x^7/(7*c) + (b^(7/2)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/c^(9/2)","A",1
175,1,53,53,0.0059391,"\int \frac{x^9}{b x^2+c x^4} \, dx","Integrate[x^9/(b*x^2 + c*x^4),x]","-\frac{b^3 \log \left(b+c x^2\right)}{2 c^4}+\frac{b^2 x^2}{2 c^3}-\frac{b x^4}{4 c^2}+\frac{x^6}{6 c}","-\frac{b^3 \log \left(b+c x^2\right)}{2 c^4}+\frac{b^2 x^2}{2 c^3}-\frac{b x^4}{4 c^2}+\frac{x^6}{6 c}",1,"(b^2*x^2)/(2*c^3) - (b*x^4)/(4*c^2) + x^6/(6*c) - (b^3*Log[b + c*x^2])/(2*c^4)","A",1
176,1,55,55,0.0267838,"\int \frac{x^8}{b x^2+c x^4} \, dx","Integrate[x^8/(b*x^2 + c*x^4),x]","-\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{c^{7/2}}+\frac{b^2 x}{c^3}-\frac{b x^3}{3 c^2}+\frac{x^5}{5 c}","-\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{c^{7/2}}+\frac{b^2 x}{c^3}-\frac{b x^3}{3 c^2}+\frac{x^5}{5 c}",1,"(b^2*x)/c^3 - (b*x^3)/(3*c^2) + x^5/(5*c) - (b^(5/2)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/c^(7/2)","A",1
177,1,40,40,0.005542,"\int \frac{x^7}{b x^2+c x^4} \, dx","Integrate[x^7/(b*x^2 + c*x^4),x]","\frac{b^2 \log \left(b+c x^2\right)}{2 c^3}-\frac{b x^2}{2 c^2}+\frac{x^4}{4 c}","\frac{b^2 \log \left(b+c x^2\right)}{2 c^3}-\frac{b x^2}{2 c^2}+\frac{x^4}{4 c}",1,"-1/2*(b*x^2)/c^2 + x^4/(4*c) + (b^2*Log[b + c*x^2])/(2*c^3)","A",1
178,1,42,42,0.0187458,"\int \frac{x^6}{b x^2+c x^4} \, dx","Integrate[x^6/(b*x^2 + c*x^4),x]","\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{c^{5/2}}-\frac{b x}{c^2}+\frac{x^3}{3 c}","\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{c^{5/2}}-\frac{b x}{c^2}+\frac{x^3}{3 c}",1,"-((b*x)/c^2) + x^3/(3*c) + (b^(3/2)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/c^(5/2)","A",1
179,1,27,27,0.0046368,"\int \frac{x^5}{b x^2+c x^4} \, dx","Integrate[x^5/(b*x^2 + c*x^4),x]","\frac{x^2}{2 c}-\frac{b \log \left(b+c x^2\right)}{2 c^2}","\frac{x^2}{2 c}-\frac{b \log \left(b+c x^2\right)}{2 c^2}",1,"x^2/(2*c) - (b*Log[b + c*x^2])/(2*c^2)","A",1
180,1,31,31,0.0084035,"\int \frac{x^4}{b x^2+c x^4} \, dx","Integrate[x^4/(b*x^2 + c*x^4),x]","\frac{x}{c}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{c^{3/2}}","\frac{x}{c}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{c^{3/2}}",1,"x/c - (Sqrt[b]*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/c^(3/2)","A",1
181,1,15,15,0.0022345,"\int \frac{x^3}{b x^2+c x^4} \, dx","Integrate[x^3/(b*x^2 + c*x^4),x]","\frac{\log \left(b+c x^2\right)}{2 c}","\frac{\log \left(b+c x^2\right)}{2 c}",1,"Log[b + c*x^2]/(2*c)","A",1
182,1,24,24,0.00399,"\int \frac{x^2}{b x^2+c x^4} \, dx","Integrate[x^2/(b*x^2 + c*x^4),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{\sqrt{b} \sqrt{c}}","\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{\sqrt{b} \sqrt{c}}",1,"ArcTan[(Sqrt[c]*x)/Sqrt[b]]/(Sqrt[b]*Sqrt[c])","A",1
183,1,22,22,0.0048614,"\int \frac{x}{b x^2+c x^4} \, dx","Integrate[x/(b*x^2 + c*x^4),x]","\frac{\log (x)}{b}-\frac{\log \left(b+c x^2\right)}{2 b}","\frac{\log (x)}{b}-\frac{\log \left(b+c x^2\right)}{2 b}",1,"Log[x]/b - Log[b + c*x^2]/(2*b)","A",1
184,1,34,34,0.0120649,"\int \frac{1}{b x^2+c x^4} \, dx","Integrate[(b*x^2 + c*x^4)^(-1),x]","-\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{b^{3/2}}-\frac{1}{b x}","-\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{b^{3/2}}-\frac{1}{b x}",1,"-(1/(b*x)) - (Sqrt[c]*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/b^(3/2)","A",1
185,1,35,35,0.006835,"\int \frac{1}{x \left(b x^2+c x^4\right)} \, dx","Integrate[1/(x*(b*x^2 + c*x^4)),x]","\frac{c \log \left(b+c x^2\right)}{2 b^2}-\frac{c \log (x)}{b^2}-\frac{1}{2 b x^2}","\frac{c \log \left(b+c x^2\right)}{2 b^2}-\frac{c \log (x)}{b^2}-\frac{1}{2 b x^2}",1,"-1/2*1/(b*x^2) - (c*Log[x])/b^2 + (c*Log[b + c*x^2])/(2*b^2)","A",1
186,1,43,43,0.0204678,"\int \frac{1}{x^2 \left(b x^2+c x^4\right)} \, dx","Integrate[1/(x^2*(b*x^2 + c*x^4)),x]","\frac{c^{3/2} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{b^{5/2}}+\frac{c}{b^2 x}-\frac{1}{3 b x^3}","\frac{c^{3/2} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{b^{5/2}}+\frac{c}{b^2 x}-\frac{1}{3 b x^3}",1,"-1/3*1/(b*x^3) + c/(b^2*x) + (c^(3/2)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/b^(5/2)","A",1
187,1,49,49,0.0072683,"\int \frac{1}{x^3 \left(b x^2+c x^4\right)} \, dx","Integrate[1/(x^3*(b*x^2 + c*x^4)),x]","-\frac{c^2 \log \left(b+c x^2\right)}{2 b^3}+\frac{c^2 \log (x)}{b^3}+\frac{c}{2 b^2 x^2}-\frac{1}{4 b x^4}","-\frac{c^2 \log \left(b+c x^2\right)}{2 b^3}+\frac{c^2 \log (x)}{b^3}+\frac{c}{2 b^2 x^2}-\frac{1}{4 b x^4}",1,"-1/4*1/(b*x^4) + c/(2*b^2*x^2) + (c^2*Log[x])/b^3 - (c^2*Log[b + c*x^2])/(2*b^3)","A",1
188,1,58,58,0.0258993,"\int \frac{1}{x^4 \left(b x^2+c x^4\right)} \, dx","Integrate[1/(x^4*(b*x^2 + c*x^4)),x]","-\frac{c^{5/2} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{b^{7/2}}-\frac{c^2}{b^3 x}+\frac{c}{3 b^2 x^3}-\frac{1}{5 b x^5}","-\frac{c^{5/2} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{b^{7/2}}-\frac{c^2}{b^3 x}+\frac{c}{3 b^2 x^3}-\frac{1}{5 b x^5}",1,"-1/5*1/(b*x^5) + c/(3*b^2*x^3) - c^2/(b^3*x) - (c^(5/2)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/b^(7/2)","A",1
189,1,63,63,0.0073651,"\int \frac{1}{x^5 \left(b x^2+c x^4\right)} \, dx","Integrate[1/(x^5*(b*x^2 + c*x^4)),x]","\frac{c^3 \log \left(b+c x^2\right)}{2 b^4}-\frac{c^3 \log (x)}{b^4}-\frac{c^2}{2 b^3 x^2}+\frac{c}{4 b^2 x^4}-\frac{1}{6 b x^6}","\frac{c^3 \log \left(b+c x^2\right)}{2 b^4}-\frac{c^3 \log (x)}{b^4}-\frac{c^2}{2 b^3 x^2}+\frac{c}{4 b^2 x^4}-\frac{1}{6 b x^6}",1,"-1/6*1/(b*x^6) + c/(4*b^2*x^4) - c^2/(2*b^3*x^2) - (c^3*Log[x])/b^4 + (c^3*Log[b + c*x^2])/(2*b^4)","A",1
190,1,71,79,0.0495176,"\int \frac{x^{12}}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[x^12/(b*x^2 + c*x^4)^2,x]","\frac{x \left(\frac{15 b^3}{b+c x^2}+90 b^2-20 b c x^2+6 c^2 x^4\right)}{30 c^4}-\frac{7 b^{5/2} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{2 c^{9/2}}","-\frac{7 b^{5/2} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{2 c^{9/2}}+\frac{7 b^2 x}{2 c^4}-\frac{7 b x^3}{6 c^3}-\frac{x^7}{2 c \left(b+c x^2\right)}+\frac{7 x^5}{10 c^2}",1,"(x*(90*b^2 - 20*b*c*x^2 + 6*c^2*x^4 + (15*b^3)/(b + c*x^2)))/(30*c^4) - (7*b^(5/2)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(2*c^(9/2))","A",1
191,1,49,57,0.0171792,"\int \frac{x^{11}}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[x^11/(b*x^2 + c*x^4)^2,x]","\frac{\frac{2 b^3}{b+c x^2}+6 b^2 \log \left(b+c x^2\right)-4 b c x^2+c^2 x^4}{4 c^4}","\frac{b^3}{2 c^4 \left(b+c x^2\right)}+\frac{3 b^2 \log \left(b+c x^2\right)}{2 c^4}-\frac{b x^2}{c^3}+\frac{x^4}{4 c^2}",1,"(-4*b*c*x^2 + c^2*x^4 + (2*b^3)/(b + c*x^2) + 6*b^2*Log[b + c*x^2])/(4*c^4)","A",1
192,1,60,66,0.0412705,"\int \frac{x^{10}}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[x^10/(b*x^2 + c*x^4)^2,x]","\frac{5 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{2 c^{7/2}}+\frac{x \left(-\frac{3 b^2}{b+c x^2}-12 b+2 c x^2\right)}{6 c^3}","\frac{5 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{2 c^{7/2}}-\frac{5 b x}{2 c^3}-\frac{x^5}{2 c \left(b+c x^2\right)}+\frac{5 x^3}{6 c^2}",1,"(x*(-12*b + 2*c*x^2 - (3*b^2)/(b + c*x^2)))/(6*c^3) + (5*b^(3/2)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(2*c^(7/2))","A",1
193,1,38,44,0.0148032,"\int \frac{x^9}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[x^9/(b*x^2 + c*x^4)^2,x]","\frac{-\frac{b^2}{b+c x^2}-2 b \log \left(b+c x^2\right)+c x^2}{2 c^3}","-\frac{b^2}{2 c^3 \left(b+c x^2\right)}-\frac{b \log \left(b+c x^2\right)}{c^3}+\frac{x^2}{2 c^2}",1,"(c*x^2 - b^2/(b + c*x^2) - 2*b*Log[b + c*x^2])/(2*c^3)","A",1
194,1,51,55,0.0311493,"\int \frac{x^8}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[x^8/(b*x^2 + c*x^4)^2,x]","-\frac{3 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{2 c^{5/2}}+\frac{b x}{2 c^2 \left(b+c x^2\right)}+\frac{x}{c^2}","-\frac{3 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{2 c^{5/2}}-\frac{x^3}{2 c \left(b+c x^2\right)}+\frac{3 x}{2 c^2}",1,"x/c^2 + (b*x)/(2*c^2*(b + c*x^2)) - (3*Sqrt[b]*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(2*c^(5/2))","A",1
195,1,27,33,0.0078338,"\int \frac{x^7}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[x^7/(b*x^2 + c*x^4)^2,x]","\frac{\frac{b}{b+c x^2}+\log \left(b+c x^2\right)}{2 c^2}","\frac{b}{2 c^2 \left(b+c x^2\right)}+\frac{\log \left(b+c x^2\right)}{2 c^2}",1,"(b/(b + c*x^2) + Log[b + c*x^2])/(2*c^2)","A",1
196,1,45,45,0.0202,"\int \frac{x^6}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[x^6/(b*x^2 + c*x^4)^2,x]","\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{2 \sqrt{b} c^{3/2}}-\frac{x}{2 c \left(b+c x^2\right)}","\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{2 \sqrt{b} c^{3/2}}-\frac{x}{2 c \left(b+c x^2\right)}",1,"-1/2*x/(c*(b + c*x^2)) + ArcTan[(Sqrt[c]*x)/Sqrt[b]]/(2*Sqrt[b]*c^(3/2))","A",1
197,1,16,16,0.0020852,"\int \frac{x^5}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[x^5/(b*x^2 + c*x^4)^2,x]","-\frac{1}{2 c \left(b+c x^2\right)}","-\frac{1}{2 c \left(b+c x^2\right)}",1,"-1/2*1/(c*(b + c*x^2))","A",1
198,1,45,45,0.0244346,"\int \frac{x^4}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[x^4/(b*x^2 + c*x^4)^2,x]","\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{2 b^{3/2} \sqrt{c}}+\frac{x}{2 b \left(b+c x^2\right)}","\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{2 b^{3/2} \sqrt{c}}+\frac{x}{2 b \left(b+c x^2\right)}",1,"x/(2*b*(b + c*x^2)) + ArcTan[(Sqrt[c]*x)/Sqrt[b]]/(2*b^(3/2)*Sqrt[c])","A",1
199,1,33,38,0.0155134,"\int \frac{x^3}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[x^3/(b*x^2 + c*x^4)^2,x]","\frac{\frac{b}{b+c x^2}-\log \left(b+c x^2\right)+2 \log (x)}{2 b^2}","-\frac{\log \left(b+c x^2\right)}{2 b^2}+\frac{\log (x)}{b^2}+\frac{1}{2 b \left(b+c x^2\right)}",1,"(b/(b + c*x^2) + 2*Log[x] - Log[b + c*x^2])/(2*b^2)","A",1
200,1,54,57,0.0350911,"\int \frac{x^2}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[x^2/(b*x^2 + c*x^4)^2,x]","-\frac{3 \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{2 b^{5/2}}-\frac{c x}{2 b^2 \left(b+c x^2\right)}-\frac{1}{b^2 x}","-\frac{3 \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{2 b^{5/2}}-\frac{3}{2 b^2 x}+\frac{1}{2 b x \left(b+c x^2\right)}",1,"-(1/(b^2*x)) - (c*x)/(2*b^2*(b + c*x^2)) - (3*Sqrt[c]*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(2*b^(5/2))","A",1
201,1,41,49,0.0365017,"\int \frac{x}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[x/(b*x^2 + c*x^4)^2,x]","-\frac{b \left(\frac{c}{b+c x^2}+\frac{1}{x^2}\right)-2 c \log \left(b+c x^2\right)+4 c \log (x)}{2 b^3}","\frac{c \log \left(b+c x^2\right)}{b^3}-\frac{2 c \log (x)}{b^3}-\frac{c}{2 b^2 \left(b+c x^2\right)}-\frac{1}{2 b^2 x^2}",1,"-1/2*(b*(x^(-2) + c/(b + c*x^2)) + 4*c*Log[x] - 2*c*Log[b + c*x^2])/b^3","A",1
202,1,67,68,0.037673,"\int \frac{1}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[(b*x^2 + c*x^4)^(-2),x]","\frac{5 c^{3/2} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{2 b^{7/2}}+\frac{c^2 x}{2 b^3 \left(b+c x^2\right)}+\frac{2 c}{b^3 x}-\frac{1}{3 b^2 x^3}","\frac{5 c^{3/2} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{2 b^{7/2}}+\frac{5 c}{2 b^3 x}-\frac{5}{6 b^2 x^3}+\frac{1}{2 b x^3 \left(b+c x^2\right)}",1,"-1/3*1/(b^2*x^3) + (2*c)/(b^3*x) + (c^2*x)/(2*b^3*(b + c*x^2)) + (5*c^(3/2)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(2*b^(7/2))","A",1
203,1,57,66,0.0522322,"\int \frac{1}{x \left(b x^2+c x^4\right)^2} \, dx","Integrate[1/(x*(b*x^2 + c*x^4)^2),x]","\frac{-6 c^2 \log \left(b+c x^2\right)+b \left(\frac{2 c^2}{b+c x^2}-\frac{b}{x^4}+\frac{4 c}{x^2}\right)+12 c^2 \log (x)}{4 b^4}","-\frac{3 c^2 \log \left(b+c x^2\right)}{2 b^4}+\frac{3 c^2 \log (x)}{b^4}+\frac{c^2}{2 b^3 \left(b+c x^2\right)}+\frac{c}{b^3 x^2}-\frac{1}{4 b^2 x^4}",1,"(b*(-(b/x^4) + (4*c)/x^2 + (2*c^2)/(b + c*x^2)) + 12*c^2*Log[x] - 6*c^2*Log[b + c*x^2])/(4*b^4)","A",1
204,1,80,81,0.0448249,"\int \frac{1}{x^2 \left(b x^2+c x^4\right)^2} \, dx","Integrate[1/(x^2*(b*x^2 + c*x^4)^2),x]","-\frac{7 c^{5/2} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{2 b^{9/2}}-\frac{c^3 x}{2 b^4 \left(b+c x^2\right)}-\frac{3 c^2}{b^4 x}+\frac{2 c}{3 b^3 x^3}-\frac{1}{5 b^2 x^5}","-\frac{7 c^{5/2} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{2 b^{9/2}}-\frac{7 c^2}{2 b^4 x}+\frac{7 c}{6 b^3 x^3}-\frac{7}{10 b^2 x^5}+\frac{1}{2 b x^5 \left(b+c x^2\right)}",1,"-1/5*1/(b^2*x^5) + (2*c)/(3*b^3*x^3) - (3*c^2)/(b^4*x) - (c^3*x)/(2*b^4*(b + c*x^2)) - (7*c^(5/2)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(2*b^(9/2))","A",1
205,1,77,85,0.0467143,"\int \frac{x^{14}}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[x^14/(b*x^2 + c*x^4)^3,x]","\frac{35 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{8 c^{9/2}}-\frac{105 b^3 x+175 b^2 c x^3+56 b c^2 x^5-8 c^3 x^7}{24 c^4 \left(b+c x^2\right)^2}","\frac{35 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{8 c^{9/2}}-\frac{35 b x}{8 c^4}-\frac{7 x^5}{8 c^2 \left(b+c x^2\right)}-\frac{x^7}{4 c \left(b+c x^2\right)^2}+\frac{35 x^3}{24 c^3}",1,"-1/24*(105*b^3*x + 175*b^2*c*x^3 + 56*b*c^2*x^5 - 8*c^3*x^7)/(c^4*(b + c*x^2)^2) + (35*b^(3/2)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(8*c^(9/2))","A",1
206,1,48,65,0.0557229,"\int \frac{x^{13}}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[x^13/(b*x^2 + c*x^4)^3,x]","-\frac{\frac{b^2 \left(5 b+6 c x^2\right)}{\left(b+c x^2\right)^2}+6 b \log \left(b+c x^2\right)-2 c x^2}{4 c^4}","\frac{b^3}{4 c^4 \left(b+c x^2\right)^2}-\frac{3 b^2}{2 c^4 \left(b+c x^2\right)}-\frac{3 b \log \left(b+c x^2\right)}{2 c^4}+\frac{x^2}{2 c^3}",1,"-1/4*(-2*c*x^2 + (b^2*(5*b + 6*c*x^2))/(b + c*x^2)^2 + 6*b*Log[b + c*x^2])/c^4","A",1
207,1,66,74,0.045153,"\int \frac{x^{12}}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[x^12/(b*x^2 + c*x^4)^3,x]","\frac{15 b^2 x+25 b c x^3+8 c^2 x^5}{8 c^3 \left(b+c x^2\right)^2}-\frac{15 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{8 c^{7/2}}","-\frac{15 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{8 c^{7/2}}-\frac{5 x^3}{8 c^2 \left(b+c x^2\right)}-\frac{x^5}{4 c \left(b+c x^2\right)^2}+\frac{15 x}{8 c^3}",1,"(15*b^2*x + 25*b*c*x^3 + 8*c^2*x^5)/(8*c^3*(b + c*x^2)^2) - (15*Sqrt[b]*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(8*c^(7/2))","A",1
208,1,39,49,0.014432,"\int \frac{x^{11}}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[x^11/(b*x^2 + c*x^4)^3,x]","\frac{\frac{b \left(3 b+4 c x^2\right)}{\left(b+c x^2\right)^2}+2 \log \left(b+c x^2\right)}{4 c^3}","-\frac{b^2}{4 c^3 \left(b+c x^2\right)^2}+\frac{b}{c^3 \left(b+c x^2\right)}+\frac{\log \left(b+c x^2\right)}{2 c^3}",1,"((b*(3*b + 4*c*x^2))/(b + c*x^2)^2 + 2*Log[b + c*x^2])/(4*c^3)","A",1
209,1,55,64,0.0422908,"\int \frac{x^{10}}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[x^10/(b*x^2 + c*x^4)^3,x]","\frac{3 \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{8 \sqrt{b} c^{5/2}}-\frac{3 b x+5 c x^3}{8 c^2 \left(b+c x^2\right)^2}","\frac{3 \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{8 \sqrt{b} c^{5/2}}-\frac{3 x}{8 c^2 \left(b+c x^2\right)}-\frac{x^3}{4 c \left(b+c x^2\right)^2}",1,"-1/8*(3*b*x + 5*c*x^3)/(c^2*(b + c*x^2)^2) + (3*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(8*Sqrt[b]*c^(5/2))","A",1
210,1,24,19,0.0077173,"\int \frac{x^9}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[x^9/(b*x^2 + c*x^4)^3,x]","-\frac{b+2 c x^2}{4 c^2 \left(b+c x^2\right)^2}","\frac{x^4}{4 b \left(b+c x^2\right)^2}",1,"-1/4*(b + 2*c*x^2)/(c^2*(b + c*x^2)^2)","A",1
211,1,58,65,0.0280134,"\int \frac{x^8}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[x^8/(b*x^2 + c*x^4)^3,x]","\frac{\frac{\sqrt{b} \sqrt{c} x \left(c x^2-b\right)}{\left(b+c x^2\right)^2}+\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{8 b^{3/2} c^{3/2}}","\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{8 b^{3/2} c^{3/2}}+\frac{x}{8 b c \left(b+c x^2\right)}-\frac{x}{4 c \left(b+c x^2\right)^2}",1,"((Sqrt[b]*Sqrt[c]*x*(-b + c*x^2))/(b + c*x^2)^2 + ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(8*b^(3/2)*c^(3/2))","A",1
212,1,16,16,0.00237,"\int \frac{x^7}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[x^7/(b*x^2 + c*x^4)^3,x]","-\frac{1}{4 c \left(b+c x^2\right)^2}","-\frac{1}{4 c \left(b+c x^2\right)^2}",1,"-1/4*1/(c*(b + c*x^2)^2)","A",1
213,1,55,62,0.0327988,"\int \frac{x^6}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[x^6/(b*x^2 + c*x^4)^3,x]","\frac{3 \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{8 b^{5/2} \sqrt{c}}+\frac{5 b x+3 c x^3}{8 b^2 \left(b+c x^2\right)^2}","\frac{3 \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{8 b^{5/2} \sqrt{c}}+\frac{3 x}{8 b^2 \left(b+c x^2\right)}+\frac{x}{4 b \left(b+c x^2\right)^2}",1,"(5*b*x + 3*c*x^3)/(8*b^2*(b + c*x^2)^2) + (3*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(8*b^(5/2)*Sqrt[c])","A",1
214,1,43,54,0.0311208,"\int \frac{x^5}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[x^5/(b*x^2 + c*x^4)^3,x]","\frac{\frac{b \left(3 b+2 c x^2\right)}{\left(b+c x^2\right)^2}-2 \log \left(b+c x^2\right)+4 \log (x)}{4 b^3}","-\frac{\log \left(b+c x^2\right)}{2 b^3}+\frac{\log (x)}{b^3}+\frac{1}{2 b^2 \left(b+c x^2\right)}+\frac{1}{4 b \left(b+c x^2\right)^2}",1,"((b*(3*b + 2*c*x^2))/(b + c*x^2)^2 + 4*Log[x] - 2*Log[b + c*x^2])/(4*b^3)","A",1
215,1,68,76,0.0403481,"\int \frac{x^4}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[x^4/(b*x^2 + c*x^4)^3,x]","-\frac{15 \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{8 b^{7/2}}-\frac{8 b^2+25 b c x^2+15 c^2 x^4}{8 b^3 x \left(b+c x^2\right)^2}","-\frac{15 \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{8 b^{7/2}}-\frac{15}{8 b^3 x}+\frac{5}{8 b^2 x \left(b+c x^2\right)}+\frac{1}{4 b x \left(b+c x^2\right)^2}",1,"-1/8*(8*b^2 + 25*b*c*x^2 + 15*c^2*x^4)/(b^3*x*(b + c*x^2)^2) - (15*Sqrt[c]*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(8*b^(7/2))","A",1
216,1,59,67,0.0559671,"\int \frac{x^3}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[x^3/(b*x^2 + c*x^4)^3,x]","-\frac{\frac{b \left(2 b^2+9 b c x^2+6 c^2 x^4\right)}{x^2 \left(b+c x^2\right)^2}-6 c \log \left(b+c x^2\right)+12 c \log (x)}{4 b^4}","\frac{3 c \log \left(b+c x^2\right)}{2 b^4}-\frac{3 c \log (x)}{b^4}-\frac{c}{b^3 \left(b+c x^2\right)}-\frac{1}{2 b^3 x^2}-\frac{c}{4 b^2 \left(b+c x^2\right)^2}",1,"-1/4*((b*(2*b^2 + 9*b*c*x^2 + 6*c^2*x^4))/(x^2*(b + c*x^2)^2) + 12*c*Log[x] - 6*c*Log[b + c*x^2])/b^4","A",1
217,1,79,87,0.0422637,"\int \frac{x^2}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[x^2/(b*x^2 + c*x^4)^3,x]","\frac{35 c^{3/2} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{8 b^{9/2}}+\frac{-8 b^3+56 b^2 c x^2+175 b c^2 x^4+105 c^3 x^6}{24 b^4 x^3 \left(b+c x^2\right)^2}","\frac{35 c^{3/2} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{8 b^{9/2}}+\frac{35 c}{8 b^4 x}-\frac{35}{24 b^3 x^3}+\frac{7}{8 b^2 x^3 \left(b+c x^2\right)}+\frac{1}{4 b x^3 \left(b+c x^2\right)^2}",1,"(-8*b^3 + 56*b^2*c*x^2 + 175*b*c^2*x^4 + 105*c^3*x^6)/(24*b^4*x^3*(b + c*x^2)^2) + (35*c^(3/2)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(8*b^(9/2))","A",1
218,1,74,86,0.0473362,"\int \frac{x}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[x/(b*x^2 + c*x^4)^3,x]","\frac{\frac{b \left(-b^3+4 b^2 c x^2+18 b c^2 x^4+12 c^3 x^6\right)}{x^4 \left(b+c x^2\right)^2}-12 c^2 \log \left(b+c x^2\right)+24 c^2 \log (x)}{4 b^5}","-\frac{3 c^2 \log \left(b+c x^2\right)}{b^5}+\frac{6 c^2 \log (x)}{b^5}+\frac{3 c^2}{2 b^4 \left(b+c x^2\right)}+\frac{3 c}{2 b^4 x^2}+\frac{c^2}{4 b^3 \left(b+c x^2\right)^2}-\frac{1}{4 b^3 x^4}",1,"((b*(-b^3 + 4*b^2*c*x^2 + 18*b*c^2*x^4 + 12*c^3*x^6))/(x^4*(b + c*x^2)^2) + 24*c^2*Log[x] - 12*c^2*Log[b + c*x^2])/(4*b^5)","A",1
219,1,90,100,0.0533628,"\int \frac{1}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[(b*x^2 + c*x^4)^(-3),x]","-\frac{63 c^{5/2} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{8 b^{11/2}}-\frac{8 b^4-24 b^3 c x^2+168 b^2 c^2 x^4+525 b c^3 x^6+315 c^4 x^8}{40 b^5 x^5 \left(b+c x^2\right)^2}","-\frac{63 c^{5/2} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{8 b^{11/2}}-\frac{63 c^2}{8 b^5 x}+\frac{21 c}{8 b^4 x^3}-\frac{63}{40 b^3 x^5}+\frac{9}{8 b^2 x^5 \left(b+c x^2\right)}+\frac{1}{4 b x^5 \left(b+c x^2\right)^2}",1,"-1/40*(8*b^4 - 24*b^3*c*x^2 + 168*b^2*c^2*x^4 + 525*b*c^3*x^6 + 315*c^4*x^8)/(b^5*x^5*(b + c*x^2)^2) - (63*c^(5/2)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(8*b^(11/2))","A",1
220,1,85,95,0.0708169,"\int \frac{1}{x \left(b x^2+c x^4\right)^3} \, dx","Integrate[1/(x*(b*x^2 + c*x^4)^3),x]","-\frac{\frac{b \left(2 b^4-5 b^3 c x^2+20 b^2 c^2 x^4+90 b c^3 x^6+60 c^4 x^8\right)}{x^6 \left(b+c x^2\right)^2}-60 c^3 \log \left(b+c x^2\right)+120 c^3 \log (x)}{12 b^6}","\frac{5 c^3 \log \left(b+c x^2\right)}{b^6}-\frac{10 c^3 \log (x)}{b^6}-\frac{2 c^3}{b^5 \left(b+c x^2\right)}-\frac{3 c^2}{b^5 x^2}-\frac{c^3}{4 b^4 \left(b+c x^2\right)^2}+\frac{3 c}{4 b^4 x^4}-\frac{1}{6 b^3 x^6}",1,"-1/12*((b*(2*b^4 - 5*b^3*c*x^2 + 20*b^2*c^2*x^4 + 90*b*c^3*x^6 + 60*c^4*x^8))/(x^6*(b + c*x^2)^2) + 120*c^3*Log[x] - 60*c^3*Log[b + c*x^2])/b^6","A",1
221,1,114,119,0.0760992,"\int x^5 \sqrt{b x^2+c x^4} \, dx","Integrate[x^5*Sqrt[b*x^2 + c*x^4],x]","\frac{x \sqrt{b+c x^2} \left(\sqrt{c} x \sqrt{b+c x^2} \left(15 b^3-10 b^2 c x^2+8 b c^2 x^4+48 c^3 x^6\right)-15 b^4 \log \left(\sqrt{c} \sqrt{b+c x^2}+c x\right)\right)}{384 c^{7/2} \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{5 b^4 \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{128 c^{7/2}}+\frac{5 b^2 \left(b+2 c x^2\right) \sqrt{b x^2+c x^4}}{128 c^3}-\frac{5 b \left(b x^2+c x^4\right)^{3/2}}{48 c^2}+\frac{x^2 \left(b x^2+c x^4\right)^{3/2}}{8 c}",1,"(x*Sqrt[b + c*x^2]*(Sqrt[c]*x*Sqrt[b + c*x^2]*(15*b^3 - 10*b^2*c*x^2 + 8*b*c^2*x^4 + 48*c^3*x^6) - 15*b^4*Log[c*x + Sqrt[c]*Sqrt[b + c*x^2]]))/(384*c^(7/2)*Sqrt[x^2*(b + c*x^2)])","A",1
222,1,103,91,0.0612241,"\int x^3 \sqrt{b x^2+c x^4} \, dx","Integrate[x^3*Sqrt[b*x^2 + c*x^4],x]","\frac{x \sqrt{b+c x^2} \left(3 b^3 \log \left(\sqrt{c} \sqrt{b+c x^2}+c x\right)+\sqrt{c} x \sqrt{b+c x^2} \left(-3 b^2+2 b c x^2+8 c^2 x^4\right)\right)}{48 c^{5/2} \sqrt{x^2 \left(b+c x^2\right)}}","\frac{b^3 \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{16 c^{5/2}}-\frac{b \left(b+2 c x^2\right) \sqrt{b x^2+c x^4}}{16 c^2}+\frac{\left(b x^2+c x^4\right)^{3/2}}{6 c}",1,"(x*Sqrt[b + c*x^2]*(Sqrt[c]*x*Sqrt[b + c*x^2]*(-3*b^2 + 2*b*c*x^2 + 8*c^2*x^4) + 3*b^3*Log[c*x + Sqrt[c]*Sqrt[b + c*x^2]]))/(48*c^(5/2)*Sqrt[x^2*(b + c*x^2)])","A",1
223,1,90,68,0.0482895,"\int x \sqrt{b x^2+c x^4} \, dx","Integrate[x*Sqrt[b*x^2 + c*x^4],x]","\frac{x \sqrt{b+c x^2} \left(\sqrt{c} x \sqrt{b+c x^2} \left(b+2 c x^2\right)-b^2 \log \left(\sqrt{c} \sqrt{b+c x^2}+c x\right)\right)}{8 c^{3/2} \sqrt{x^2 \left(b+c x^2\right)}}","\frac{\left(b+2 c x^2\right) \sqrt{b x^2+c x^4}}{8 c}-\frac{b^2 \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{8 c^{3/2}}",1,"(x*Sqrt[b + c*x^2]*(Sqrt[c]*x*Sqrt[b + c*x^2]*(b + 2*c*x^2) - b^2*Log[c*x + Sqrt[c]*Sqrt[b + c*x^2]]))/(8*c^(3/2)*Sqrt[x^2*(b + c*x^2)])","A",1
224,1,64,55,0.0251216,"\int \frac{\sqrt{b x^2+c x^4}}{x} \, dx","Integrate[Sqrt[b*x^2 + c*x^4]/x,x]","\frac{1}{2} \sqrt{x^2 \left(b+c x^2\right)} \left(\frac{b \log \left(\sqrt{c} \sqrt{b+c x^2}+c x\right)}{\sqrt{c} x \sqrt{b+c x^2}}+1\right)","\frac{1}{2} \sqrt{b x^2+c x^4}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{2 \sqrt{c}}",1,"(Sqrt[x^2*(b + c*x^2)]*(1 + (b*Log[c*x + Sqrt[c]*Sqrt[b + c*x^2]])/(Sqrt[c]*x*Sqrt[b + c*x^2])))/2","A",1
225,1,60,52,0.0933488,"\int \frac{\sqrt{b x^2+c x^4}}{x^3} \, dx","Integrate[Sqrt[b*x^2 + c*x^4]/x^3,x]","\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(\frac{\sqrt{c} x \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{\sqrt{b} \sqrt{\frac{c x^2}{b}+1}}-1\right)}{x^2}","\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)-\frac{\sqrt{b x^2+c x^4}}{x^2}",1,"(Sqrt[x^2*(b + c*x^2)]*(-1 + (Sqrt[c]*x*ArcSinh[(Sqrt[c]*x)/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (c*x^2)/b])))/x^2","A",1
226,1,25,25,0.0103569,"\int \frac{\sqrt{b x^2+c x^4}}{x^5} \, dx","Integrate[Sqrt[b*x^2 + c*x^4]/x^5,x]","-\frac{\left(x^2 \left(b+c x^2\right)\right)^{3/2}}{3 b x^6}","-\frac{\left(b x^2+c x^4\right)^{3/2}}{3 b x^6}",1,"-1/3*(x^2*(b + c*x^2))^(3/2)/(b*x^6)","A",1
227,1,35,52,0.0114279,"\int \frac{\sqrt{b x^2+c x^4}}{x^7} \, dx","Integrate[Sqrt[b*x^2 + c*x^4]/x^7,x]","\frac{\left(x^2 \left(b+c x^2\right)\right)^{3/2} \left(2 c x^2-3 b\right)}{15 b^2 x^8}","\frac{2 c \left(b x^2+c x^4\right)^{3/2}}{15 b^2 x^6}-\frac{\left(b x^2+c x^4\right)^{3/2}}{5 b x^8}",1,"((x^2*(b + c*x^2))^(3/2)*(-3*b + 2*c*x^2))/(15*b^2*x^8)","A",1
228,1,46,80,0.0126522,"\int \frac{\sqrt{b x^2+c x^4}}{x^9} \, dx","Integrate[Sqrt[b*x^2 + c*x^4]/x^9,x]","-\frac{\left(x^2 \left(b+c x^2\right)\right)^{3/2} \left(15 b^2-12 b c x^2+8 c^2 x^4\right)}{105 b^3 x^{10}}","-\frac{8 c^2 \left(b x^2+c x^4\right)^{3/2}}{105 b^3 x^6}+\frac{4 c \left(b x^2+c x^4\right)^{3/2}}{35 b^2 x^8}-\frac{\left(b x^2+c x^4\right)^{3/2}}{7 b x^{10}}",1,"-1/105*((x^2*(b + c*x^2))^(3/2)*(15*b^2 - 12*b*c*x^2 + 8*c^2*x^4))/(b^3*x^10)","A",1
229,1,57,108,0.0137199,"\int \frac{\sqrt{b x^2+c x^4}}{x^{11}} \, dx","Integrate[Sqrt[b*x^2 + c*x^4]/x^11,x]","\frac{\left(x^2 \left(b+c x^2\right)\right)^{3/2} \left(-35 b^3+30 b^2 c x^2-24 b c^2 x^4+16 c^3 x^6\right)}{315 b^4 x^{12}}","\frac{16 c^3 \left(b x^2+c x^4\right)^{3/2}}{315 b^4 x^6}-\frac{8 c^2 \left(b x^2+c x^4\right)^{3/2}}{105 b^3 x^8}+\frac{2 c \left(b x^2+c x^4\right)^{3/2}}{21 b^2 x^{10}}-\frac{\left(b x^2+c x^4\right)^{3/2}}{9 b x^{12}}",1,"((x^2*(b + c*x^2))^(3/2)*(-35*b^3 + 30*b^2*c*x^2 - 24*b*c^2*x^4 + 16*c^3*x^6))/(315*b^4*x^12)","A",1
230,1,68,136,0.0152307,"\int \frac{\sqrt{b x^2+c x^4}}{x^{13}} \, dx","Integrate[Sqrt[b*x^2 + c*x^4]/x^13,x]","-\frac{\left(x^2 \left(b+c x^2\right)\right)^{3/2} \left(315 b^4-280 b^3 c x^2+240 b^2 c^2 x^4-192 b c^3 x^6+128 c^4 x^8\right)}{3465 b^5 x^{14}}","-\frac{128 c^4 \left(b x^2+c x^4\right)^{3/2}}{3465 b^5 x^6}+\frac{64 c^3 \left(b x^2+c x^4\right)^{3/2}}{1155 b^4 x^8}-\frac{16 c^2 \left(b x^2+c x^4\right)^{3/2}}{231 b^3 x^{10}}+\frac{8 c \left(b x^2+c x^4\right)^{3/2}}{99 b^2 x^{12}}-\frac{\left(b x^2+c x^4\right)^{3/2}}{11 b x^{14}}",1,"-1/3465*((x^2*(b + c*x^2))^(3/2)*(315*b^4 - 280*b^3*c*x^2 + 240*b^2*c^2*x^4 - 192*b*c^3*x^6 + 128*c^4*x^8))/(b^5*x^14)","A",1
231,1,46,78,0.0225204,"\int x^4 \sqrt{b x^2+c x^4} \, dx","Integrate[x^4*Sqrt[b*x^2 + c*x^4],x]","\frac{\left(x^2 \left(b+c x^2\right)\right)^{3/2} \left(8 b^2-12 b c x^2+15 c^2 x^4\right)}{105 c^3 x^3}","\frac{8 b^2 \left(b x^2+c x^4\right)^{3/2}}{105 c^3 x^3}-\frac{4 b \left(b x^2+c x^4\right)^{3/2}}{35 c^2 x}+\frac{x \left(b x^2+c x^4\right)^{3/2}}{7 c}",1,"((x^2*(b + c*x^2))^(3/2)*(8*b^2 - 12*b*c*x^2 + 15*c^2*x^4))/(105*c^3*x^3)","A",1
232,1,35,52,0.017701,"\int x^2 \sqrt{b x^2+c x^4} \, dx","Integrate[x^2*Sqrt[b*x^2 + c*x^4],x]","\frac{\left(x^2 \left(b+c x^2\right)\right)^{3/2} \left(3 c x^2-2 b\right)}{15 c^2 x^3}","\frac{\left(b x^2+c x^4\right)^{3/2}}{5 c x}-\frac{2 b \left(b x^2+c x^4\right)^{3/2}}{15 c^2 x^3}",1,"((x^2*(b + c*x^2))^(3/2)*(-2*b + 3*c*x^2))/(15*c^2*x^3)","A",1
233,1,25,25,0.0051915,"\int \sqrt{b x^2+c x^4} \, dx","Integrate[Sqrt[b*x^2 + c*x^4],x]","\frac{\left(x^2 \left(b+c x^2\right)\right)^{3/2}}{3 c x^3}","\frac{\left(b x^2+c x^4\right)^{3/2}}{3 c x^3}",1,"(x^2*(b + c*x^2))^(3/2)/(3*c*x^3)","A",1
234,1,60,50,0.0299955,"\int \frac{\sqrt{b x^2+c x^4}}{x^2} \, dx","Integrate[Sqrt[b*x^2 + c*x^4]/x^2,x]","\frac{x \left(-\sqrt{b} \sqrt{b+c x^2} \tanh ^{-1}\left(\frac{\sqrt{b+c x^2}}{\sqrt{b}}\right)+b+c x^2\right)}{\sqrt{x^2 \left(b+c x^2\right)}}","\frac{\sqrt{b x^2+c x^4}}{x}-\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)",1,"(x*(b + c*x^2 - Sqrt[b]*Sqrt[b + c*x^2]*ArcTanh[Sqrt[b + c*x^2]/Sqrt[b]]))/Sqrt[x^2*(b + c*x^2)]","A",1
235,1,63,56,0.0419195,"\int \frac{\sqrt{b x^2+c x^4}}{x^4} \, dx","Integrate[Sqrt[b*x^2 + c*x^4]/x^4,x]","-\frac{c x^2 \sqrt{\frac{c x^2}{b}+1} \tanh ^{-1}\left(\sqrt{\frac{c x^2}{b}+1}\right)+b+c x^2}{2 x \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{c \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)}{2 \sqrt{b}}-\frac{\sqrt{b x^2+c x^4}}{2 x^3}",1,"-1/2*(b + c*x^2 + c*x^2*Sqrt[1 + (c*x^2)/b]*ArcTanh[Sqrt[1 + (c*x^2)/b]])/(x*Sqrt[x^2*(b + c*x^2)])","A",1
236,1,46,84,0.0140964,"\int \frac{\sqrt{b x^2+c x^4}}{x^6} \, dx","Integrate[Sqrt[b*x^2 + c*x^4]/x^6,x]","-\frac{c^2 \left(x^2 \left(b+c x^2\right)\right)^{3/2} \, _2F_1\left(\frac{3}{2},3;\frac{5}{2};\frac{c x^2}{b}+1\right)}{3 b^3 x^3}","\frac{c^2 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)}{8 b^{3/2}}-\frac{\sqrt{b x^2+c x^4}}{4 x^5}-\frac{c \sqrt{b x^2+c x^4}}{8 b x^3}",1,"-1/3*(c^2*(x^2*(b + c*x^2))^(3/2)*Hypergeometric2F1[3/2, 3, 5/2, 1 + (c*x^2)/b])/(b^3*x^3)","C",1
237,1,46,112,0.0140432,"\int \frac{\sqrt{b x^2+c x^4}}{x^8} \, dx","Integrate[Sqrt[b*x^2 + c*x^4]/x^8,x]","\frac{c^3 \left(x^2 \left(b+c x^2\right)\right)^{3/2} \, _2F_1\left(\frac{3}{2},4;\frac{5}{2};\frac{c x^2}{b}+1\right)}{3 b^4 x^3}","-\frac{c^3 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)}{16 b^{5/2}}+\frac{c^2 \sqrt{b x^2+c x^4}}{16 b^2 x^3}-\frac{\sqrt{b x^2+c x^4}}{6 x^7}-\frac{c \sqrt{b x^2+c x^4}}{24 b x^5}",1,"(c^3*(x^2*(b + c*x^2))^(3/2)*Hypergeometric2F1[3/2, 4, 5/2, 1 + (c*x^2)/b])/(3*b^4*x^3)","C",1
238,1,126,124,0.0993413,"\int x^3 \left(b x^2+c x^4\right)^{3/2} \, dx","Integrate[x^3*(b*x^2 + c*x^4)^(3/2),x]","\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(\sqrt{c} x \sqrt{\frac{c x^2}{b}+1} \left(15 b^4-10 b^3 c x^2+8 b^2 c^2 x^4+176 b c^3 x^6+128 c^4 x^8\right)-15 b^{9/2} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)\right)}{1280 c^{7/2} x \sqrt{\frac{c x^2}{b}+1}}","-\frac{3 b^5 \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{256 c^{7/2}}+\frac{3 b^3 \left(b+2 c x^2\right) \sqrt{b x^2+c x^4}}{256 c^3}-\frac{b \left(b+2 c x^2\right) \left(b x^2+c x^4\right)^{3/2}}{32 c^2}+\frac{\left(b x^2+c x^4\right)^{5/2}}{10 c}",1,"(Sqrt[x^2*(b + c*x^2)]*(Sqrt[c]*x*Sqrt[1 + (c*x^2)/b]*(15*b^4 - 10*b^3*c*x^2 + 8*b^2*c^2*x^4 + 176*b*c^3*x^6 + 128*c^4*x^8) - 15*b^(9/2)*ArcSinh[(Sqrt[c]*x)/Sqrt[b]]))/(1280*c^(7/2)*x*Sqrt[1 + (c*x^2)/b])","A",1
239,1,115,101,0.0963083,"\int x \left(b x^2+c x^4\right)^{3/2} \, dx","Integrate[x*(b*x^2 + c*x^4)^(3/2),x]","\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(3 b^{7/2} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)+\sqrt{c} x \sqrt{\frac{c x^2}{b}+1} \left(-3 b^3+2 b^2 c x^2+24 b c^2 x^4+16 c^3 x^6\right)\right)}{128 c^{5/2} x \sqrt{\frac{c x^2}{b}+1}}","\frac{3 b^4 \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{128 c^{5/2}}-\frac{3 b^2 \left(b+2 c x^2\right) \sqrt{b x^2+c x^4}}{128 c^2}+\frac{\left(b+2 c x^2\right) \left(b x^2+c x^4\right)^{3/2}}{16 c}",1,"(Sqrt[x^2*(b + c*x^2)]*(Sqrt[c]*x*Sqrt[1 + (c*x^2)/b]*(-3*b^3 + 2*b^2*c*x^2 + 24*b*c^2*x^4 + 16*c^3*x^6) + 3*b^(7/2)*ArcSinh[(Sqrt[c]*x)/Sqrt[b]]))/(128*c^(5/2)*x*Sqrt[1 + (c*x^2)/b])","A",1
240,1,104,88,0.0747458,"\int \frac{\left(b x^2+c x^4\right)^{3/2}}{x} \, dx","Integrate[(b*x^2 + c*x^4)^(3/2)/x,x]","\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(\sqrt{c} x \sqrt{\frac{c x^2}{b}+1} \left(3 b^2+14 b c x^2+8 c^2 x^4\right)-3 b^{5/2} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)\right)}{48 c^{3/2} x \sqrt{\frac{c x^2}{b}+1}}","-\frac{b^3 \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{16 c^{3/2}}+\frac{b \left(b+2 c x^2\right) \sqrt{b x^2+c x^4}}{16 c}+\frac{1}{6} \left(b x^2+c x^4\right)^{3/2}",1,"(Sqrt[x^2*(b + c*x^2)]*(Sqrt[c]*x*Sqrt[1 + (c*x^2)/b]*(3*b^2 + 14*b*c*x^2 + 8*c^2*x^4) - 3*b^(5/2)*ArcSinh[(Sqrt[c]*x)/Sqrt[b]]))/(48*c^(3/2)*x*Sqrt[1 + (c*x^2)/b])","A",1
241,1,71,80,0.1027451,"\int \frac{\left(b x^2+c x^4\right)^{3/2}}{x^3} \, dx","Integrate[(b*x^2 + c*x^4)^(3/2)/x^3,x]","\frac{1}{8} \sqrt{x^2 \left(b+c x^2\right)} \left(\frac{3 b^{3/2} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)}{\sqrt{c} x \sqrt{\frac{c x^2}{b}+1}}+5 b+2 c x^2\right)","\frac{3 b^2 \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{8 \sqrt{c}}+\frac{3}{8} b \sqrt{b x^2+c x^4}+\frac{\left(b x^2+c x^4\right)^{3/2}}{4 x^2}",1,"(Sqrt[x^2*(b + c*x^2)]*(5*b + 2*c*x^2 + (3*b^(3/2)*ArcSinh[(Sqrt[c]*x)/Sqrt[b]])/(Sqrt[c]*x*Sqrt[1 + (c*x^2)/b])))/8","A",1
242,1,54,76,0.0135225,"\int \frac{\left(b x^2+c x^4\right)^{3/2}}{x^5} \, dx","Integrate[(b*x^2 + c*x^4)^(3/2)/x^5,x]","-\frac{b \sqrt{x^2 \left(b+c x^2\right)} \, _2F_1\left(-\frac{3}{2},-\frac{1}{2};\frac{1}{2};-\frac{c x^2}{b}\right)}{x^2 \sqrt{\frac{c x^2}{b}+1}}","-\frac{\left(b x^2+c x^4\right)^{3/2}}{x^4}+\frac{3}{2} c \sqrt{b x^2+c x^4}+\frac{3}{2} b \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)",1,"-((b*Sqrt[x^2*(b + c*x^2)]*Hypergeometric2F1[-3/2, -1/2, 1/2, -((c*x^2)/b)])/(x^2*Sqrt[1 + (c*x^2)/b]))","C",1
243,1,56,75,0.0160787,"\int \frac{\left(b x^2+c x^4\right)^{3/2}}{x^7} \, dx","Integrate[(b*x^2 + c*x^4)^(3/2)/x^7,x]","-\frac{b \sqrt{x^2 \left(b+c x^2\right)} \, _2F_1\left(-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};-\frac{c x^2}{b}\right)}{3 x^4 \sqrt{\frac{c x^2}{b}+1}}","c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)-\frac{c \sqrt{b x^2+c x^4}}{x^2}-\frac{\left(b x^2+c x^4\right)^{3/2}}{3 x^6}",1,"-1/3*(b*Sqrt[x^2*(b + c*x^2)]*Hypergeometric2F1[-3/2, -3/2, -1/2, -((c*x^2)/b)])/(x^4*Sqrt[1 + (c*x^2)/b])","C",1
244,1,25,25,0.0142261,"\int \frac{\left(b x^2+c x^4\right)^{3/2}}{x^9} \, dx","Integrate[(b*x^2 + c*x^4)^(3/2)/x^9,x]","-\frac{\left(x^2 \left(b+c x^2\right)\right)^{5/2}}{5 b x^{10}}","-\frac{\left(b x^2+c x^4\right)^{5/2}}{5 b x^{10}}",1,"-1/5*(x^2*(b + c*x^2))^(5/2)/(b*x^10)","A",1
245,1,35,52,0.0135471,"\int \frac{\left(b x^2+c x^4\right)^{3/2}}{x^{11}} \, dx","Integrate[(b*x^2 + c*x^4)^(3/2)/x^11,x]","\frac{\left(x^2 \left(b+c x^2\right)\right)^{5/2} \left(2 c x^2-5 b\right)}{35 b^2 x^{12}}","\frac{2 c \left(b x^2+c x^4\right)^{5/2}}{35 b^2 x^{10}}-\frac{\left(b x^2+c x^4\right)^{5/2}}{7 b x^{12}}",1,"((x^2*(b + c*x^2))^(5/2)*(-5*b + 2*c*x^2))/(35*b^2*x^12)","A",1
246,1,46,80,0.0148966,"\int \frac{\left(b x^2+c x^4\right)^{3/2}}{x^{13}} \, dx","Integrate[(b*x^2 + c*x^4)^(3/2)/x^13,x]","-\frac{\left(x^2 \left(b+c x^2\right)\right)^{5/2} \left(35 b^2-20 b c x^2+8 c^2 x^4\right)}{315 b^3 x^{14}}","-\frac{8 c^2 \left(b x^2+c x^4\right)^{5/2}}{315 b^3 x^{10}}+\frac{4 c \left(b x^2+c x^4\right)^{5/2}}{63 b^2 x^{12}}-\frac{\left(b x^2+c x^4\right)^{5/2}}{9 b x^{14}}",1,"-1/315*((x^2*(b + c*x^2))^(5/2)*(35*b^2 - 20*b*c*x^2 + 8*c^2*x^4))/(b^3*x^14)","A",1
247,1,57,108,0.0169835,"\int \frac{\left(b x^2+c x^4\right)^{3/2}}{x^{15}} \, dx","Integrate[(b*x^2 + c*x^4)^(3/2)/x^15,x]","\frac{\left(x^2 \left(b+c x^2\right)\right)^{5/2} \left(-105 b^3+70 b^2 c x^2-40 b c^2 x^4+16 c^3 x^6\right)}{1155 b^4 x^{16}}","\frac{16 c^3 \left(b x^2+c x^4\right)^{5/2}}{1155 b^4 x^{10}}-\frac{8 c^2 \left(b x^2+c x^4\right)^{5/2}}{231 b^3 x^{12}}+\frac{2 c \left(b x^2+c x^4\right)^{5/2}}{33 b^2 x^{14}}-\frac{\left(b x^2+c x^4\right)^{5/2}}{11 b x^{16}}",1,"((x^2*(b + c*x^2))^(5/2)*(-105*b^3 + 70*b^2*c*x^2 - 40*b*c^2*x^4 + 16*c^3*x^6))/(1155*b^4*x^16)","A",1
248,1,68,136,0.0176199,"\int \frac{\left(b x^2+c x^4\right)^{3/2}}{x^{17}} \, dx","Integrate[(b*x^2 + c*x^4)^(3/2)/x^17,x]","-\frac{\left(x^2 \left(b+c x^2\right)\right)^{5/2} \left(1155 b^4-840 b^3 c x^2+560 b^2 c^2 x^4-320 b c^3 x^6+128 c^4 x^8\right)}{15015 b^5 x^{18}}","-\frac{128 c^4 \left(b x^2+c x^4\right)^{5/2}}{15015 b^5 x^{10}}+\frac{64 c^3 \left(b x^2+c x^4\right)^{5/2}}{3003 b^4 x^{12}}-\frac{16 c^2 \left(b x^2+c x^4\right)^{5/2}}{429 b^3 x^{14}}+\frac{8 c \left(b x^2+c x^4\right)^{5/2}}{143 b^2 x^{16}}-\frac{\left(b x^2+c x^4\right)^{5/2}}{13 b x^{18}}",1,"-1/15015*((x^2*(b + c*x^2))^(5/2)*(1155*b^4 - 840*b^3*c*x^2 + 560*b^2*c^2*x^4 - 320*b*c^3*x^6 + 128*c^4*x^8))/(b^5*x^18)","A",1
249,1,75,134,0.0361387,"\int x^6 \left(b x^2+c x^4\right)^{3/2} \, dx","Integrate[x^6*(b*x^2 + c*x^4)^(3/2),x]","\frac{x \left(b+c x^2\right)^3 \left(128 b^4-320 b^3 c x^2+560 b^2 c^2 x^4-840 b c^3 x^6+1155 c^4 x^8\right)}{15015 c^5 \sqrt{x^2 \left(b+c x^2\right)}}","\frac{128 b^4 \left(b x^2+c x^4\right)^{5/2}}{15015 c^5 x^5}-\frac{64 b^3 \left(b x^2+c x^4\right)^{5/2}}{3003 c^4 x^3}+\frac{16 b^2 \left(b x^2+c x^4\right)^{5/2}}{429 c^3 x}-\frac{8 b x \left(b x^2+c x^4\right)^{5/2}}{143 c^2}+\frac{x^3 \left(b x^2+c x^4\right)^{5/2}}{13 c}",1,"(x*(b + c*x^2)^3*(128*b^4 - 320*b^3*c*x^2 + 560*b^2*c^2*x^4 - 840*b*c^3*x^6 + 1155*c^4*x^8))/(15015*c^5*Sqrt[x^2*(b + c*x^2)])","A",1
250,1,64,106,0.0306376,"\int x^4 \left(b x^2+c x^4\right)^{3/2} \, dx","Integrate[x^4*(b*x^2 + c*x^4)^(3/2),x]","\frac{x \left(b+c x^2\right)^3 \left(-16 b^3+40 b^2 c x^2-70 b c^2 x^4+105 c^3 x^6\right)}{1155 c^4 \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{16 b^3 \left(b x^2+c x^4\right)^{5/2}}{1155 c^4 x^5}+\frac{8 b^2 \left(b x^2+c x^4\right)^{5/2}}{231 c^3 x^3}-\frac{2 b \left(b x^2+c x^4\right)^{5/2}}{33 c^2 x}+\frac{x \left(b x^2+c x^4\right)^{5/2}}{11 c}",1,"(x*(b + c*x^2)^3*(-16*b^3 + 40*b^2*c*x^2 - 70*b*c^2*x^4 + 105*c^3*x^6))/(1155*c^4*Sqrt[x^2*(b + c*x^2)])","A",1
251,1,53,80,0.0242336,"\int x^2 \left(b x^2+c x^4\right)^{3/2} \, dx","Integrate[x^2*(b*x^2 + c*x^4)^(3/2),x]","\frac{x \left(b+c x^2\right)^3 \left(8 b^2-20 b c x^2+35 c^2 x^4\right)}{315 c^3 \sqrt{x^2 \left(b+c x^2\right)}}","\frac{8 b^2 \left(b x^2+c x^4\right)^{5/2}}{315 c^3 x^5}-\frac{4 b \left(b x^2+c x^4\right)^{5/2}}{63 c^2 x^3}+\frac{\left(b x^2+c x^4\right)^{5/2}}{9 c x}",1,"(x*(b + c*x^2)^3*(8*b^2 - 20*b*c*x^2 + 35*c^2*x^4))/(315*c^3*Sqrt[x^2*(b + c*x^2)])","A",1
252,1,42,52,0.0197008,"\int \left(b x^2+c x^4\right)^{3/2} \, dx","Integrate[(b*x^2 + c*x^4)^(3/2),x]","\frac{x \left(b+c x^2\right)^3 \left(5 c x^2-2 b\right)}{35 c^2 \sqrt{x^2 \left(b+c x^2\right)}}","\frac{\left(b x^2+c x^4\right)^{5/2}}{7 c x^3}-\frac{2 b \left(b x^2+c x^4\right)^{5/2}}{35 c^2 x^5}",1,"(x*(b + c*x^2)^3*(-2*b + 5*c*x^2))/(35*c^2*Sqrt[x^2*(b + c*x^2)])","A",1
253,1,25,25,0.0082083,"\int \frac{\left(b x^2+c x^4\right)^{3/2}}{x^2} \, dx","Integrate[(b*x^2 + c*x^4)^(3/2)/x^2,x]","\frac{\left(x^2 \left(b+c x^2\right)\right)^{5/2}}{5 c x^5}","\frac{\left(b x^2+c x^4\right)^{5/2}}{5 c x^5}",1,"(x^2*(b + c*x^2))^(5/2)/(5*c*x^5)","A",1
254,1,76,73,0.0520086,"\int \frac{\left(b x^2+c x^4\right)^{3/2}}{x^4} \, dx","Integrate[(b*x^2 + c*x^4)^(3/2)/x^4,x]","\frac{x \left(-3 b^{3/2} \sqrt{b+c x^2} \tanh ^{-1}\left(\frac{\sqrt{b+c x^2}}{\sqrt{b}}\right)+4 b^2+5 b c x^2+c^2 x^4\right)}{3 \sqrt{x^2 \left(b+c x^2\right)}}","b^{3/2} \left(-\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)\right)+\frac{b \sqrt{b x^2+c x^4}}{x}+\frac{\left(b x^2+c x^4\right)^{3/2}}{3 x^3}",1,"(x*(4*b^2 + 5*b*c*x^2 + c^2*x^4 - 3*b^(3/2)*Sqrt[b + c*x^2]*ArcTanh[Sqrt[b + c*x^2]/Sqrt[b]]))/(3*Sqrt[x^2*(b + c*x^2)])","A",1
255,1,44,79,0.0161627,"\int \frac{\left(b x^2+c x^4\right)^{3/2}}{x^6} \, dx","Integrate[(b*x^2 + c*x^4)^(3/2)/x^6,x]","\frac{c \left(x^2 \left(b+c x^2\right)\right)^{5/2} \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};\frac{c x^2}{b}+1\right)}{5 b^2 x^5}","\frac{3 c \sqrt{b x^2+c x^4}}{2 x}-\frac{3}{2} \sqrt{b} c \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)-\frac{\left(b x^2+c x^4\right)^{3/2}}{2 x^5}",1,"(c*(x^2*(b + c*x^2))^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, 1 + (c*x^2)/b])/(5*b^2*x^5)","C",1
256,1,80,81,0.050722,"\int \frac{\left(b x^2+c x^4\right)^{3/2}}{x^8} \, dx","Integrate[(b*x^2 + c*x^4)^(3/2)/x^8,x]","-\frac{2 b^2+3 c^2 x^4 \sqrt{\frac{c x^2}{b}+1} \tanh ^{-1}\left(\sqrt{\frac{c x^2}{b}+1}\right)+7 b c x^2+5 c^2 x^4}{8 x^3 \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{3 c^2 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)}{8 \sqrt{b}}-\frac{\left(b x^2+c x^4\right)^{3/2}}{4 x^7}-\frac{3 c \sqrt{b x^2+c x^4}}{8 x^3}",1,"-1/8*(2*b^2 + 7*b*c*x^2 + 5*c^2*x^4 + 3*c^2*x^4*Sqrt[1 + (c*x^2)/b]*ArcTanh[Sqrt[1 + (c*x^2)/b]])/(x^3*Sqrt[x^2*(b + c*x^2)])","A",1
257,1,46,109,0.0178126,"\int \frac{\left(b x^2+c x^4\right)^{3/2}}{x^{10}} \, dx","Integrate[(b*x^2 + c*x^4)^(3/2)/x^10,x]","\frac{c^3 \left(x^2 \left(b+c x^2\right)\right)^{5/2} \, _2F_1\left(\frac{5}{2},4;\frac{7}{2};\frac{c x^2}{b}+1\right)}{5 b^4 x^5}","\frac{c^3 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)}{16 b^{3/2}}-\frac{c^2 \sqrt{b x^2+c x^4}}{16 b x^3}-\frac{\left(b x^2+c x^4\right)^{3/2}}{6 x^9}-\frac{c \sqrt{b x^2+c x^4}}{8 x^5}",1,"(c^3*(x^2*(b + c*x^2))^(5/2)*Hypergeometric2F1[5/2, 4, 7/2, 1 + (c*x^2)/b])/(5*b^4*x^5)","C",1
258,1,46,137,0.0168372,"\int \frac{\left(b x^2+c x^4\right)^{3/2}}{x^{12}} \, dx","Integrate[(b*x^2 + c*x^4)^(3/2)/x^12,x]","-\frac{c^4 \left(x^2 \left(b+c x^2\right)\right)^{5/2} \, _2F_1\left(\frac{5}{2},5;\frac{7}{2};\frac{c x^2}{b}+1\right)}{5 b^5 x^5}","-\frac{3 c^4 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)}{128 b^{5/2}}+\frac{3 c^3 \sqrt{b x^2+c x^4}}{128 b^2 x^3}-\frac{c^2 \sqrt{b x^2+c x^4}}{64 b x^5}-\frac{\left(b x^2+c x^4\right)^{3/2}}{8 x^{11}}-\frac{c \sqrt{b x^2+c x^4}}{16 x^7}",1,"-1/5*(c^4*(x^2*(b + c*x^2))^(5/2)*Hypergeometric2F1[5/2, 5, 7/2, 1 + (c*x^2)/b])/(b^5*x^5)","C",1
259,1,46,165,0.0165708,"\int \frac{\left(b x^2+c x^4\right)^{3/2}}{x^{14}} \, dx","Integrate[(b*x^2 + c*x^4)^(3/2)/x^14,x]","\frac{c^5 \left(x^2 \left(b+c x^2\right)\right)^{5/2} \, _2F_1\left(\frac{5}{2},6;\frac{7}{2};\frac{c x^2}{b}+1\right)}{5 b^6 x^5}","\frac{3 c^5 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)}{256 b^{7/2}}-\frac{3 c^4 \sqrt{b x^2+c x^4}}{256 b^3 x^3}+\frac{c^3 \sqrt{b x^2+c x^4}}{128 b^2 x^5}-\frac{c^2 \sqrt{b x^2+c x^4}}{160 b x^7}-\frac{\left(b x^2+c x^4\right)^{3/2}}{10 x^{13}}-\frac{3 c \sqrt{b x^2+c x^4}}{80 x^9}",1,"(c^5*(x^2*(b + c*x^2))^(5/2)*Hypergeometric2F1[5/2, 6, 7/2, 1 + (c*x^2)/b])/(5*b^6*x^5)","C",1
260,1,100,114,0.0467916,"\int \frac{x^7}{\sqrt{b x^2+c x^4}} \, dx","Integrate[x^7/Sqrt[b*x^2 + c*x^4],x]","\frac{x \left(\sqrt{c} x \left(15 b^3+5 b^2 c x^2-2 b c^2 x^4+8 c^3 x^6\right)-15 b^3 \sqrt{b+c x^2} \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b+c x^2}}\right)\right)}{48 c^{7/2} \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{5 b^3 \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{16 c^{7/2}}+\frac{5 b^2 \sqrt{b x^2+c x^4}}{16 c^3}-\frac{5 b x^2 \sqrt{b x^2+c x^4}}{24 c^2}+\frac{x^4 \sqrt{b x^2+c x^4}}{6 c}",1,"(x*(Sqrt[c]*x*(15*b^3 + 5*b^2*c*x^2 - 2*b*c^2*x^4 + 8*c^3*x^6) - 15*b^3*Sqrt[b + c*x^2]*ArcTanh[(Sqrt[c]*x)/Sqrt[b + c*x^2]]))/(48*c^(7/2)*Sqrt[x^2*(b + c*x^2)])","A",1
261,1,89,86,0.0358256,"\int \frac{x^5}{\sqrt{b x^2+c x^4}} \, dx","Integrate[x^5/Sqrt[b*x^2 + c*x^4],x]","\frac{x \left(\sqrt{c} x \left(-3 b^2-b c x^2+2 c^2 x^4\right)+3 b^2 \sqrt{b+c x^2} \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b+c x^2}}\right)\right)}{8 c^{5/2} \sqrt{x^2 \left(b+c x^2\right)}}","\frac{3 b^2 \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{8 c^{5/2}}-\frac{3 b \sqrt{b x^2+c x^4}}{8 c^2}+\frac{x^2 \sqrt{b x^2+c x^4}}{4 c}",1,"(x*(Sqrt[c]*x*(-3*b^2 - b*c*x^2 + 2*c^2*x^4) + 3*b^2*Sqrt[b + c*x^2]*ArcTanh[(Sqrt[c]*x)/Sqrt[b + c*x^2]]))/(8*c^(5/2)*Sqrt[x^2*(b + c*x^2)])","A",1
262,1,73,58,0.0267561,"\int \frac{x^3}{\sqrt{b x^2+c x^4}} \, dx","Integrate[x^3/Sqrt[b*x^2 + c*x^4],x]","\frac{x \left(\sqrt{c} x \left(b+c x^2\right)-b \sqrt{b+c x^2} \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b+c x^2}}\right)\right)}{2 c^{3/2} \sqrt{x^2 \left(b+c x^2\right)}}","\frac{\sqrt{b x^2+c x^4}}{2 c}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{2 c^{3/2}}",1,"(x*(Sqrt[c]*x*(b + c*x^2) - b*Sqrt[b + c*x^2]*ArcTanh[(Sqrt[c]*x)/Sqrt[b + c*x^2]]))/(2*c^(3/2)*Sqrt[x^2*(b + c*x^2)])","A",1
263,1,52,31,0.0130645,"\int \frac{x}{\sqrt{b x^2+c x^4}} \, dx","Integrate[x/Sqrt[b*x^2 + c*x^4],x]","\frac{x \sqrt{b+c x^2} \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b+c x^2}}\right)}{\sqrt{c} \sqrt{x^2 \left(b+c x^2\right)}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{\sqrt{c}}",1,"(x*Sqrt[b + c*x^2]*ArcTanh[(Sqrt[c]*x)/Sqrt[b + c*x^2]])/(Sqrt[c]*Sqrt[x^2*(b + c*x^2)])","A",1
264,1,23,23,0.0080126,"\int \frac{1}{x \sqrt{b x^2+c x^4}} \, dx","Integrate[1/(x*Sqrt[b*x^2 + c*x^4]),x]","-\frac{\sqrt{x^2 \left(b+c x^2\right)}}{b x^2}","-\frac{\sqrt{b x^2+c x^4}}{b x^2}",1,"-(Sqrt[x^2*(b + c*x^2)]/(b*x^2))","A",1
265,1,35,52,0.0150285,"\int \frac{1}{x^3 \sqrt{b x^2+c x^4}} \, dx","Integrate[1/(x^3*Sqrt[b*x^2 + c*x^4]),x]","\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(2 c x^2-b\right)}{3 b^2 x^4}","\frac{2 c \sqrt{b x^2+c x^4}}{3 b^2 x^2}-\frac{\sqrt{b x^2+c x^4}}{3 b x^4}",1,"(Sqrt[x^2*(b + c*x^2)]*(-b + 2*c*x^2))/(3*b^2*x^4)","A",1
266,1,46,80,0.0144517,"\int \frac{1}{x^5 \sqrt{b x^2+c x^4}} \, dx","Integrate[1/(x^5*Sqrt[b*x^2 + c*x^4]),x]","-\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(3 b^2-4 b c x^2+8 c^2 x^4\right)}{15 b^3 x^6}","-\frac{8 c^2 \sqrt{b x^2+c x^4}}{15 b^3 x^2}+\frac{4 c \sqrt{b x^2+c x^4}}{15 b^2 x^4}-\frac{\sqrt{b x^2+c x^4}}{5 b x^6}",1,"-1/15*(Sqrt[x^2*(b + c*x^2)]*(3*b^2 - 4*b*c*x^2 + 8*c^2*x^4))/(b^3*x^6)","A",1
267,1,57,108,0.0148157,"\int \frac{1}{x^7 \sqrt{b x^2+c x^4}} \, dx","Integrate[1/(x^7*Sqrt[b*x^2 + c*x^4]),x]","\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(-5 b^3+6 b^2 c x^2-8 b c^2 x^4+16 c^3 x^6\right)}{35 b^4 x^8}","\frac{16 c^3 \sqrt{b x^2+c x^4}}{35 b^4 x^2}-\frac{8 c^2 \sqrt{b x^2+c x^4}}{35 b^3 x^4}+\frac{6 c \sqrt{b x^2+c x^4}}{35 b^2 x^6}-\frac{\sqrt{b x^2+c x^4}}{7 b x^8}",1,"(Sqrt[x^2*(b + c*x^2)]*(-5*b^3 + 6*b^2*c*x^2 - 8*b*c^2*x^4 + 16*c^3*x^6))/(35*b^4*x^8)","A",1
268,1,34,50,0.0163425,"\int \frac{x^4}{\sqrt{b x^2+c x^4}} \, dx","Integrate[x^4/Sqrt[b*x^2 + c*x^4],x]","\frac{\left(c x^2-2 b\right) \sqrt{x^2 \left(b+c x^2\right)}}{3 c^2 x}","\frac{x \sqrt{b x^2+c x^4}}{3 c}-\frac{2 b \sqrt{b x^2+c x^4}}{3 c^2 x}",1,"((-2*b + c*x^2)*Sqrt[x^2*(b + c*x^2)])/(3*c^2*x)","A",1
269,1,22,22,0.0051993,"\int \frac{x^2}{\sqrt{b x^2+c x^4}} \, dx","Integrate[x^2/Sqrt[b*x^2 + c*x^4],x]","\frac{\sqrt{x^2 \left(b+c x^2\right)}}{c x}","\frac{\sqrt{b x^2+c x^4}}{c x}",1,"Sqrt[x^2*(b + c*x^2)]/(c*x)","A",1
270,1,52,30,0.0091904,"\int \frac{1}{\sqrt{b x^2+c x^4}} \, dx","Integrate[1/Sqrt[b*x^2 + c*x^4],x]","-\frac{x \sqrt{b+c x^2} \tanh ^{-1}\left(\frac{\sqrt{b+c x^2}}{\sqrt{b}}\right)}{\sqrt{b} \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)}{\sqrt{b}}",1,"-((x*Sqrt[b + c*x^2]*ArcTanh[Sqrt[b + c*x^2]/Sqrt[b]])/(Sqrt[b]*Sqrt[x^2*(b + c*x^2)]))","A",1
271,1,68,59,0.0603901,"\int \frac{1}{x^2 \sqrt{b x^2+c x^4}} \, dx","Integrate[1/(x^2*Sqrt[b*x^2 + c*x^4]),x]","\frac{c \sqrt{x^2 \left(b+c x^2\right)} \left(\frac{\tanh ^{-1}\left(\sqrt{\frac{c x^2}{b}+1}\right)}{2 \sqrt{\frac{c x^2}{b}+1}}-\frac{b}{2 c x^2}\right)}{b^2 x}","\frac{c \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)}{2 b^{3/2}}-\frac{\sqrt{b x^2+c x^4}}{2 b x^3}",1,"(c*Sqrt[x^2*(b + c*x^2)]*(-1/2*b/(c*x^2) + ArcTanh[Sqrt[1 + (c*x^2)/b]]/(2*Sqrt[1 + (c*x^2)/b])))/(b^2*x)","A",1
272,1,44,87,0.012378,"\int \frac{1}{x^4 \sqrt{b x^2+c x^4}} \, dx","Integrate[1/(x^4*Sqrt[b*x^2 + c*x^4]),x]","-\frac{c^2 \sqrt{x^2 \left(b+c x^2\right)} \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};\frac{c x^2}{b}+1\right)}{b^3 x}","-\frac{3 c^2 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)}{8 b^{5/2}}+\frac{3 c \sqrt{b x^2+c x^4}}{8 b^2 x^3}-\frac{\sqrt{b x^2+c x^4}}{4 b x^5}",1,"-((c^2*Sqrt[x^2*(b + c*x^2)]*Hypergeometric2F1[1/2, 3, 3/2, 1 + (c*x^2)/b])/(b^3*x))","C",1
273,1,88,109,0.0496585,"\int \frac{x^9}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[x^9/(b*x^2 + c*x^4)^(3/2),x]","\frac{x \left(15 b^{5/2} \sqrt{\frac{c x^2}{b}+1} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)+\sqrt{c} x \left(-15 b^2-5 b c x^2+2 c^2 x^4\right)\right)}{8 c^{7/2} \sqrt{x^2 \left(b+c x^2\right)}}","\frac{15 b^2 \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{8 c^{7/2}}-\frac{15 b \sqrt{b x^2+c x^4}}{8 c^3}+\frac{5 x^2 \sqrt{b x^2+c x^4}}{4 c^2}-\frac{x^6}{c \sqrt{b x^2+c x^4}}",1,"(x*(Sqrt[c]*x*(-15*b^2 - 5*b*c*x^2 + 2*c^2*x^4) + 15*b^(5/2)*Sqrt[1 + (c*x^2)/b]*ArcSinh[(Sqrt[c]*x)/Sqrt[b]]))/(8*c^(7/2)*Sqrt[x^2*(b + c*x^2)])","A",1
274,1,76,81,0.0406221,"\int \frac{x^7}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[x^7/(b*x^2 + c*x^4)^(3/2),x]","\frac{x \left(\sqrt{c} x \left(3 b+c x^2\right)-3 b^{3/2} \sqrt{\frac{c x^2}{b}+1} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)\right)}{2 c^{5/2} \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{3 b \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{2 c^{5/2}}+\frac{3 \sqrt{b x^2+c x^4}}{2 c^2}-\frac{x^4}{c \sqrt{b x^2+c x^4}}",1,"(x*(Sqrt[c]*x*(3*b + c*x^2) - 3*b^(3/2)*Sqrt[1 + (c*x^2)/b]*ArcSinh[(Sqrt[c]*x)/Sqrt[b]]))/(2*c^(5/2)*Sqrt[x^2*(b + c*x^2)])","A",1
275,1,66,55,0.0747357,"\int \frac{x^5}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[x^5/(b*x^2 + c*x^4)^(3/2),x]","\frac{\sqrt{b} x \sqrt{\frac{c x^2}{b}+1} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b}}\right)-\sqrt{c} x^2}{c^{3/2} \sqrt{x^2 \left(b+c x^2\right)}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{c^{3/2}}-\frac{x^2}{c \sqrt{b x^2+c x^4}}",1,"(-(Sqrt[c]*x^2) + Sqrt[b]*x*Sqrt[1 + (c*x^2)/b]*ArcSinh[(Sqrt[c]*x)/Sqrt[b]])/(c^(3/2)*Sqrt[x^2*(b + c*x^2)])","A",1
276,1,22,22,0.0076074,"\int \frac{x^3}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[x^3/(b*x^2 + c*x^4)^(3/2),x]","\frac{x^2}{b \sqrt{x^2 \left(b+c x^2\right)}}","\frac{x^2}{b \sqrt{b x^2+c x^4}}",1,"x^2/(b*Sqrt[x^2*(b + c*x^2)])","A",1
277,1,29,28,0.0095572,"\int \frac{x}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[x/(b*x^2 + c*x^4)^(3/2),x]","\frac{-b-2 c x^2}{b^2 \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{b+2 c x^2}{b^2 \sqrt{b x^2+c x^4}}",1,"(-b - 2*c*x^2)/(b^2*Sqrt[x^2*(b + c*x^2)])","A",1
278,1,48,74,0.0101341,"\int \frac{1}{x \left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[1/(x*(b*x^2 + c*x^4)^(3/2)),x]","-\frac{\left(b+c x^2\right) \left(b^2-4 b c x^2-8 c^2 x^4\right)}{3 b^3 \left(x^2 \left(b+c x^2\right)\right)^{3/2}}","\frac{8 c \sqrt{b x^2+c x^4}}{3 b^3 x^2}-\frac{4 \sqrt{b x^2+c x^4}}{3 b^2 x^4}+\frac{1}{b x^2 \sqrt{b x^2+c x^4}}",1,"-1/3*((b + c*x^2)*(b^2 - 4*b*c*x^2 - 8*c^2*x^4))/(b^3*(x^2*(b + c*x^2))^(3/2))","A",1
279,1,57,102,0.0114727,"\int \frac{1}{x^3 \left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[1/(x^3*(b*x^2 + c*x^4)^(3/2)),x]","\frac{-b^3+2 b^2 c x^2-8 b c^2 x^4-16 c^3 x^6}{5 b^4 x^4 \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{16 c^2 \sqrt{b x^2+c x^4}}{5 b^4 x^2}+\frac{8 c \sqrt{b x^2+c x^4}}{5 b^3 x^4}-\frac{6 \sqrt{b x^2+c x^4}}{5 b^2 x^6}+\frac{1}{b x^4 \sqrt{b x^2+c x^4}}",1,"(-b^3 + 2*b^2*c*x^2 - 8*b*c^2*x^4 - 16*c^3*x^6)/(5*b^4*x^4*Sqrt[x^2*(b + c*x^2)])","A",1
280,1,68,130,0.0131035,"\int \frac{1}{x^5 \left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[1/(x^5*(b*x^2 + c*x^4)^(3/2)),x]","\frac{-5 b^4+8 b^3 c x^2-16 b^2 c^2 x^4+64 b c^3 x^6+128 c^4 x^8}{35 b^5 x^6 \sqrt{x^2 \left(b+c x^2\right)}}","\frac{128 c^3 \sqrt{b x^2+c x^4}}{35 b^5 x^2}-\frac{64 c^2 \sqrt{b x^2+c x^4}}{35 b^4 x^4}+\frac{48 c \sqrt{b x^2+c x^4}}{35 b^3 x^6}-\frac{8 \sqrt{b x^2+c x^4}}{7 b^2 x^8}+\frac{1}{b x^6 \sqrt{b x^2+c x^4}}",1,"(-5*b^4 + 8*b^3*c*x^2 - 16*b^2*c^2*x^4 + 64*b*c^3*x^6 + 128*c^4*x^8)/(35*b^5*x^6*Sqrt[x^2*(b + c*x^2)])","A",1
281,1,29,47,0.0142999,"\int \frac{x^6}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[x^6/(b*x^2 + c*x^4)^(3/2),x]","\frac{x \left(2 b+c x^2\right)}{c^2 \sqrt{x^2 \left(b+c x^2\right)}}","\frac{2 \sqrt{b x^2+c x^4}}{c^2 x}-\frac{x^3}{c \sqrt{b x^2+c x^4}}",1,"(x*(2*b + c*x^2))/(c^2*Sqrt[x^2*(b + c*x^2)])","A",1
282,1,21,21,0.0046588,"\int \frac{x^4}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[x^4/(b*x^2 + c*x^4)^(3/2),x]","-\frac{x}{c \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{x}{c \sqrt{b x^2+c x^4}}",1,"-(x/(c*Sqrt[x^2*(b + c*x^2)]))","A",1
283,1,38,51,0.0092858,"\int \frac{x^2}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[x^2/(b*x^2 + c*x^4)^(3/2),x]","\frac{x \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{c x^2}{b}+1\right)}{b \sqrt{x^2 \left(b+c x^2\right)}}","\frac{x}{b \sqrt{b x^2+c x^4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)}{b^{3/2}}",1,"(x*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (c*x^2)/b])/(b*Sqrt[x^2*(b + c*x^2)])","C",1
284,1,40,81,0.0080631,"\int \frac{1}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(b*x^2 + c*x^4)^(-3/2),x]","-\frac{c x \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};\frac{c x^2}{b}+1\right)}{b^2 \sqrt{x^2 \left(b+c x^2\right)}}","\frac{3 c \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)}{2 b^{5/2}}-\frac{3 \sqrt{b x^2+c x^4}}{2 b^2 x^3}+\frac{1}{b x \sqrt{b x^2+c x^4}}",1,"-((c*x*Hypergeometric2F1[-1/2, 2, 1/2, 1 + (c*x^2)/b])/(b^2*Sqrt[x^2*(b + c*x^2)]))","C",1
285,1,41,109,0.0108211,"\int \frac{1}{x^2 \left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[1/(x^2*(b*x^2 + c*x^4)^(3/2)),x]","\frac{c^2 x \, _2F_1\left(-\frac{1}{2},3;\frac{1}{2};\frac{c x^2}{b}+1\right)}{b^3 \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{15 c^2 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)}{8 b^{7/2}}+\frac{15 c \sqrt{b x^2+c x^4}}{8 b^3 x^3}-\frac{5 \sqrt{b x^2+c x^4}}{4 b^2 x^5}+\frac{1}{b x^3 \sqrt{b x^2+c x^4}}",1,"(c^2*x*Hypergeometric2F1[-1/2, 3, 1/2, 1 + (c*x^2)/b])/(b^3*Sqrt[x^2*(b + c*x^2)])","C",1
286,1,57,34,0.0177594,"\int \frac{x^3}{\sqrt{3 x^2-4 x^4}} \, dx","Integrate[x^3/Sqrt[3*x^2 - 4*x^4],x]","\frac{x \left(8 x^3+3 \sqrt{4 x^2-3} \tanh ^{-1}\left(\frac{2 x}{\sqrt{4 x^2-3}}\right)-6 x\right)}{16 \sqrt{3 x^2-4 x^4}}","-\frac{3}{32} \sin ^{-1}\left(1-\frac{8 x^2}{3}\right)-\frac{1}{8} \sqrt{3 x^2-4 x^4}",1,"(x*(-6*x + 8*x^3 + 3*Sqrt[-3 + 4*x^2]*ArcTanh[(2*x)/Sqrt[-3 + 4*x^2]]))/(16*Sqrt[3*x^2 - 4*x^4])","A",1
287,1,52,34,0.0175837,"\int \frac{x^3}{\sqrt{-3 x^2-4 x^4}} \, dx","Integrate[x^3/Sqrt[-3*x^2 - 4*x^4],x]","\frac{x \left(8 x^3-3 \sqrt{4 x^2+3} \sinh ^{-1}\left(\frac{2 x}{\sqrt{3}}\right)+6 x\right)}{16 \sqrt{-x^2 \left(4 x^2+3\right)}}","-\frac{3}{32} \sin ^{-1}\left(\frac{8 x^2}{3}+1\right)-\frac{1}{8} \sqrt{-4 x^4-3 x^2}",1,"(x*(6*x + 8*x^3 - 3*Sqrt[3 + 4*x^2]*ArcSinh[(2*x)/Sqrt[3]]))/(16*Sqrt[-(x^2*(3 + 4*x^2))])","A",1
288,1,51,45,0.0094855,"\int \frac{x^3}{\sqrt{3 x^2+4 x^4}} \, dx","Integrate[x^3/Sqrt[3*x^2 + 4*x^4],x]","\frac{x \left(8 x^3-3 \sqrt{4 x^2+3} \sinh ^{-1}\left(\frac{2 x}{\sqrt{3}}\right)+6 x\right)}{16 \sqrt{x^2 \left(4 x^2+3\right)}}","\frac{1}{8} \sqrt{4 x^4+3 x^2}-\frac{3}{16} \tanh ^{-1}\left(\frac{2 x^2}{\sqrt{4 x^4+3 x^2}}\right)",1,"(x*(6*x + 8*x^3 - 3*Sqrt[3 + 4*x^2]*ArcSinh[(2*x)/Sqrt[3]]))/(16*Sqrt[x^2*(3 + 4*x^2)])","A",1
289,1,57,45,0.0123175,"\int \frac{x^3}{\sqrt{-3 x^2+4 x^4}} \, dx","Integrate[x^3/Sqrt[-3*x^2 + 4*x^4],x]","\frac{x \left(8 x^3+3 \sqrt{4 x^2-3} \tanh ^{-1}\left(\frac{2 x}{\sqrt{4 x^2-3}}\right)-6 x\right)}{16 \sqrt{x^2 \left(4 x^2-3\right)}}","\frac{1}{8} \sqrt{4 x^4-3 x^2}+\frac{3}{16} \tanh ^{-1}\left(\frac{2 x^2}{\sqrt{4 x^4-3 x^2}}\right)",1,"(x*(-6*x + 8*x^3 + 3*Sqrt[-3 + 4*x^2]*ArcTanh[(2*x)/Sqrt[-3 + 4*x^2]]))/(16*Sqrt[x^2*(-3 + 4*x^2)])","A",1
290,1,73,58,0.0331176,"\int \frac{x^3}{\sqrt{a x^2+b x^4}} \, dx","Integrate[x^3/Sqrt[a*x^2 + b*x^4],x]","\frac{x \left(\sqrt{b} x \left(a+b x^2\right)-a \sqrt{a+b x^2} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)\right)}{2 b^{3/2} \sqrt{x^2 \left(a+b x^2\right)}}","\frac{\sqrt{a x^2+b x^4}}{2 b}-\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a x^2+b x^4}}\right)}{2 b^{3/2}}",1,"(x*(Sqrt[b]*x*(a + b*x^2) - a*Sqrt[a + b*x^2]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]]))/(2*b^(3/2)*Sqrt[x^2*(a + b*x^2)])","A",1
291,1,77,60,0.0424971,"\int \frac{x^3}{\sqrt{a x^2-b x^4}} \, dx","Integrate[x^3/Sqrt[a*x^2 - b*x^4],x]","\frac{x \left(\sqrt{b} x \left(b x^2-a\right)+a \sqrt{a-b x^2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a-b x^2}}\right)\right)}{2 b^{3/2} \sqrt{x^2 \left(a-b x^2\right)}}","\frac{a \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a x^2-b x^4}}\right)}{2 b^{3/2}}-\frac{\sqrt{a x^2-b x^4}}{2 b}",1,"(x*(Sqrt[b]*x*(-a + b*x^2) + a*Sqrt[a - b*x^2]*ArcTan[(Sqrt[b]*x)/Sqrt[a - b*x^2]]))/(2*b^(3/2)*Sqrt[x^2*(a - b*x^2)])","A",1
292,1,21,21,0.0055422,"\int x^{7/2} \left(b x^2+c x^4\right) \, dx","Integrate[x^(7/2)*(b*x^2 + c*x^4),x]","\frac{2}{13} b x^{13/2}+\frac{2}{17} c x^{17/2}","\frac{2}{13} b x^{13/2}+\frac{2}{17} c x^{17/2}",1,"(2*b*x^(13/2))/13 + (2*c*x^(17/2))/17","A",1
293,1,21,21,0.0049479,"\int x^{5/2} \left(b x^2+c x^4\right) \, dx","Integrate[x^(5/2)*(b*x^2 + c*x^4),x]","\frac{2}{11} b x^{11/2}+\frac{2}{15} c x^{15/2}","\frac{2}{11} b x^{11/2}+\frac{2}{15} c x^{15/2}",1,"(2*b*x^(11/2))/11 + (2*c*x^(15/2))/15","A",1
294,1,21,21,0.0048184,"\int x^{3/2} \left(b x^2+c x^4\right) \, dx","Integrate[x^(3/2)*(b*x^2 + c*x^4),x]","\frac{2}{9} b x^{9/2}+\frac{2}{13} c x^{13/2}","\frac{2}{9} b x^{9/2}+\frac{2}{13} c x^{13/2}",1,"(2*b*x^(9/2))/9 + (2*c*x^(13/2))/13","A",1
295,1,21,21,0.00475,"\int \sqrt{x} \left(b x^2+c x^4\right) \, dx","Integrate[Sqrt[x]*(b*x^2 + c*x^4),x]","\frac{2}{7} b x^{7/2}+\frac{2}{11} c x^{11/2}","\frac{2}{7} b x^{7/2}+\frac{2}{11} c x^{11/2}",1,"(2*b*x^(7/2))/7 + (2*c*x^(11/2))/11","A",1
296,1,21,21,0.0045946,"\int \frac{b x^2+c x^4}{\sqrt{x}} \, dx","Integrate[(b*x^2 + c*x^4)/Sqrt[x],x]","\frac{2}{5} b x^{5/2}+\frac{2}{9} c x^{9/2}","\frac{2}{5} b x^{5/2}+\frac{2}{9} c x^{9/2}",1,"(2*b*x^(5/2))/5 + (2*c*x^(9/2))/9","A",1
297,1,21,21,0.005241,"\int \frac{b x^2+c x^4}{x^{3/2}} \, dx","Integrate[(b*x^2 + c*x^4)/x^(3/2),x]","\frac{2}{3} b x^{3/2}+\frac{2}{7} c x^{7/2}","\frac{2}{3} b x^{3/2}+\frac{2}{7} c x^{7/2}",1,"(2*b*x^(3/2))/3 + (2*c*x^(7/2))/7","A",1
298,1,19,19,0.0044536,"\int \frac{b x^2+c x^4}{x^{5/2}} \, dx","Integrate[(b*x^2 + c*x^4)/x^(5/2),x]","2 b \sqrt{x}+\frac{2}{5} c x^{5/2}","2 b \sqrt{x}+\frac{2}{5} c x^{5/2}",1,"2*b*Sqrt[x] + (2*c*x^(5/2))/5","A",1
299,1,19,19,0.0053139,"\int \frac{b x^2+c x^4}{x^{7/2}} \, dx","Integrate[(b*x^2 + c*x^4)/x^(7/2),x]","\frac{2}{3} c x^{3/2}-\frac{2 b}{\sqrt{x}}","\frac{2}{3} c x^{3/2}-\frac{2 b}{\sqrt{x}}",1,"(-2*b)/Sqrt[x] + (2*c*x^(3/2))/3","A",1
300,1,30,36,0.0087312,"\int x^{7/2} \left(b x^2+c x^4\right)^2 \, dx","Integrate[x^(7/2)*(b*x^2 + c*x^4)^2,x]","\frac{2 x^{17/2} \left(525 b^2+850 b c x^2+357 c^2 x^4\right)}{8925}","\frac{2}{17} b^2 x^{17/2}+\frac{4}{21} b c x^{21/2}+\frac{2}{25} c^2 x^{25/2}",1,"(2*x^(17/2)*(525*b^2 + 850*b*c*x^2 + 357*c^2*x^4))/8925","A",1
301,1,30,36,0.008016,"\int x^{5/2} \left(b x^2+c x^4\right)^2 \, dx","Integrate[x^(5/2)*(b*x^2 + c*x^4)^2,x]","\frac{2 x^{15/2} \left(437 b^2+690 b c x^2+285 c^2 x^4\right)}{6555}","\frac{2}{15} b^2 x^{15/2}+\frac{4}{19} b c x^{19/2}+\frac{2}{23} c^2 x^{23/2}",1,"(2*x^(15/2)*(437*b^2 + 690*b*c*x^2 + 285*c^2*x^4))/6555","A",1
302,1,30,36,0.0081079,"\int x^{3/2} \left(b x^2+c x^4\right)^2 \, dx","Integrate[x^(3/2)*(b*x^2 + c*x^4)^2,x]","\frac{2 x^{13/2} \left(357 b^2+546 b c x^2+221 c^2 x^4\right)}{4641}","\frac{2}{13} b^2 x^{13/2}+\frac{4}{17} b c x^{17/2}+\frac{2}{21} c^2 x^{21/2}",1,"(2*x^(13/2)*(357*b^2 + 546*b*c*x^2 + 221*c^2*x^4))/4641","A",1
303,1,30,36,0.008631,"\int \sqrt{x} \left(b x^2+c x^4\right)^2 \, dx","Integrate[Sqrt[x]*(b*x^2 + c*x^4)^2,x]","\frac{2 x^{11/2} \left(285 b^2+418 b c x^2+165 c^2 x^4\right)}{3135}","\frac{2}{11} b^2 x^{11/2}+\frac{4}{15} b c x^{15/2}+\frac{2}{19} c^2 x^{19/2}",1,"(2*x^(11/2)*(285*b^2 + 418*b*c*x^2 + 165*c^2*x^4))/3135","A",1
304,1,30,36,0.0077521,"\int \frac{\left(b x^2+c x^4\right)^2}{\sqrt{x}} \, dx","Integrate[(b*x^2 + c*x^4)^2/Sqrt[x],x]","\frac{2 x^{9/2} \left(221 b^2+306 b c x^2+117 c^2 x^4\right)}{1989}","\frac{2}{9} b^2 x^{9/2}+\frac{4}{13} b c x^{13/2}+\frac{2}{17} c^2 x^{17/2}",1,"(2*x^(9/2)*(221*b^2 + 306*b*c*x^2 + 117*c^2*x^4))/1989","A",1
305,1,30,36,0.0082837,"\int \frac{\left(b x^2+c x^4\right)^2}{x^{3/2}} \, dx","Integrate[(b*x^2 + c*x^4)^2/x^(3/2),x]","\frac{2 x^{7/2} \left(165 b^2+210 b c x^2+77 c^2 x^4\right)}{1155}","\frac{2}{7} b^2 x^{7/2}+\frac{4}{11} b c x^{11/2}+\frac{2}{15} c^2 x^{15/2}",1,"(2*x^(7/2)*(165*b^2 + 210*b*c*x^2 + 77*c^2*x^4))/1155","A",1
306,1,30,36,0.0075704,"\int \frac{\left(b x^2+c x^4\right)^2}{x^{5/2}} \, dx","Integrate[(b*x^2 + c*x^4)^2/x^(5/2),x]","\frac{2}{585} x^{5/2} \left(117 b^2+130 b c x^2+45 c^2 x^4\right)","\frac{2}{5} b^2 x^{5/2}+\frac{4}{9} b c x^{9/2}+\frac{2}{13} c^2 x^{13/2}",1,"(2*x^(5/2)*(117*b^2 + 130*b*c*x^2 + 45*c^2*x^4))/585","A",1
307,1,30,36,0.0081918,"\int \frac{\left(b x^2+c x^4\right)^2}{x^{7/2}} \, dx","Integrate[(b*x^2 + c*x^4)^2/x^(7/2),x]","\frac{2}{231} x^{3/2} \left(77 b^2+66 b c x^2+21 c^2 x^4\right)","\frac{2}{3} b^2 x^{3/2}+\frac{4}{7} b c x^{7/2}+\frac{2}{11} c^2 x^{11/2}",1,"(2*x^(3/2)*(77*b^2 + 66*b*c*x^2 + 21*c^2*x^4))/231","A",1
308,1,51,51,0.0130268,"\int x^{7/2} \left(b x^2+c x^4\right)^3 \, dx","Integrate[x^(7/2)*(b*x^2 + c*x^4)^3,x]","\frac{2}{21} b^3 x^{21/2}+\frac{6}{25} b^2 c x^{25/2}+\frac{6}{29} b c^2 x^{29/2}+\frac{2}{33} c^3 x^{33/2}","\frac{2}{21} b^3 x^{21/2}+\frac{6}{25} b^2 c x^{25/2}+\frac{6}{29} b c^2 x^{29/2}+\frac{2}{33} c^3 x^{33/2}",1,"(2*b^3*x^(21/2))/21 + (6*b^2*c*x^(25/2))/25 + (6*b*c^2*x^(29/2))/29 + (2*c^3*x^(33/2))/33","A",1
309,1,51,51,0.0120196,"\int x^{5/2} \left(b x^2+c x^4\right)^3 \, dx","Integrate[x^(5/2)*(b*x^2 + c*x^4)^3,x]","\frac{2}{19} b^3 x^{19/2}+\frac{6}{23} b^2 c x^{23/2}+\frac{2}{9} b c^2 x^{27/2}+\frac{2}{31} c^3 x^{31/2}","\frac{2}{19} b^3 x^{19/2}+\frac{6}{23} b^2 c x^{23/2}+\frac{2}{9} b c^2 x^{27/2}+\frac{2}{31} c^3 x^{31/2}",1,"(2*b^3*x^(19/2))/19 + (6*b^2*c*x^(23/2))/23 + (2*b*c^2*x^(27/2))/9 + (2*c^3*x^(31/2))/31","A",1
310,1,51,51,0.010705,"\int x^{3/2} \left(b x^2+c x^4\right)^3 \, dx","Integrate[x^(3/2)*(b*x^2 + c*x^4)^3,x]","\frac{2}{17} b^3 x^{17/2}+\frac{2}{7} b^2 c x^{21/2}+\frac{6}{25} b c^2 x^{25/2}+\frac{2}{29} c^3 x^{29/2}","\frac{2}{17} b^3 x^{17/2}+\frac{2}{7} b^2 c x^{21/2}+\frac{6}{25} b c^2 x^{25/2}+\frac{2}{29} c^3 x^{29/2}",1,"(2*b^3*x^(17/2))/17 + (2*b^2*c*x^(21/2))/7 + (6*b*c^2*x^(25/2))/25 + (2*c^3*x^(29/2))/29","A",1
311,1,41,51,0.0099923,"\int \sqrt{x} \left(b x^2+c x^4\right)^3 \, dx","Integrate[Sqrt[x]*(b*x^2 + c*x^4)^3,x]","\frac{2 x^{15/2} \left(3933 b^3+9315 b^2 c x^2+7695 b c^2 x^4+2185 c^3 x^6\right)}{58995}","\frac{2}{15} b^3 x^{15/2}+\frac{6}{19} b^2 c x^{19/2}+\frac{6}{23} b c^2 x^{23/2}+\frac{2}{27} c^3 x^{27/2}",1,"(2*x^(15/2)*(3933*b^3 + 9315*b^2*c*x^2 + 7695*b*c^2*x^4 + 2185*c^3*x^6))/58995","A",1
312,1,41,51,0.0106921,"\int \frac{\left(b x^2+c x^4\right)^3}{\sqrt{x}} \, dx","Integrate[(b*x^2 + c*x^4)^3/Sqrt[x],x]","\frac{2 x^{13/2} \left(2975 b^3+6825 b^2 c x^2+5525 b c^2 x^4+1547 c^3 x^6\right)}{38675}","\frac{2}{13} b^3 x^{13/2}+\frac{6}{17} b^2 c x^{17/2}+\frac{2}{7} b c^2 x^{21/2}+\frac{2}{25} c^3 x^{25/2}",1,"(2*x^(13/2)*(2975*b^3 + 6825*b^2*c*x^2 + 5525*b*c^2*x^4 + 1547*c^3*x^6))/38675","A",1
313,1,41,51,0.0110431,"\int \frac{\left(b x^2+c x^4\right)^3}{x^{3/2}} \, dx","Integrate[(b*x^2 + c*x^4)^3/x^(3/2),x]","\frac{2 x^{11/2} \left(2185 b^3+4807 b^2 c x^2+3795 b c^2 x^4+1045 c^3 x^6\right)}{24035}","\frac{2}{11} b^3 x^{11/2}+\frac{2}{5} b^2 c x^{15/2}+\frac{6}{19} b c^2 x^{19/2}+\frac{2}{23} c^3 x^{23/2}",1,"(2*x^(11/2)*(2185*b^3 + 4807*b^2*c*x^2 + 3795*b*c^2*x^4 + 1045*c^3*x^6))/24035","A",1
314,1,41,51,0.0103819,"\int \frac{\left(b x^2+c x^4\right)^3}{x^{5/2}} \, dx","Integrate[(b*x^2 + c*x^4)^3/x^(5/2),x]","\frac{2 x^{9/2} \left(1547 b^3+3213 b^2 c x^2+2457 b c^2 x^4+663 c^3 x^6\right)}{13923}","\frac{2}{9} b^3 x^{9/2}+\frac{6}{13} b^2 c x^{13/2}+\frac{6}{17} b c^2 x^{17/2}+\frac{2}{21} c^3 x^{21/2}",1,"(2*x^(9/2)*(1547*b^3 + 3213*b^2*c*x^2 + 2457*b*c^2*x^4 + 663*c^3*x^6))/13923","A",1
315,1,41,51,0.009872,"\int \frac{\left(b x^2+c x^4\right)^3}{x^{7/2}} \, dx","Integrate[(b*x^2 + c*x^4)^3/x^(7/2),x]","\frac{2 x^{7/2} \left(1045 b^3+1995 b^2 c x^2+1463 b c^2 x^4+385 c^3 x^6\right)}{7315}","\frac{2}{7} b^3 x^{7/2}+\frac{6}{11} b^2 c x^{11/2}+\frac{2}{5} b c^2 x^{15/2}+\frac{2}{19} c^3 x^{19/2}",1,"(2*x^(7/2)*(1045*b^3 + 1995*b^2*c*x^2 + 1463*b*c^2*x^4 + 385*c^3*x^6))/7315","A",1
316,1,89,217,0.0437707,"\int \frac{x^{13/2}}{b x^2+c x^4} \, dx","Integrate[x^(13/2)/(b*x^2 + c*x^4),x]","\frac{2 c^{3/4} x^{3/2} \left(3 c x^2-7 b\right)+21 (-b)^{7/4} \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-b}}\right)+21 b (-b)^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-b}}\right)}{21 c^{11/4}}","\frac{b^{7/4} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} c^{11/4}}-\frac{b^{7/4} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} c^{11/4}}-\frac{b^{7/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{\sqrt{2} c^{11/4}}+\frac{b^{7/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{\sqrt{2} c^{11/4}}-\frac{2 b x^{3/2}}{3 c^2}+\frac{2 x^{7/2}}{7 c}",1,"(2*c^(3/4)*x^(3/2)*(-7*b + 3*c*x^2) + 21*(-b)^(7/4)*ArcTan[(c^(1/4)*Sqrt[x])/(-b)^(1/4)] + 21*(-b)^(3/4)*b*ArcTanh[(c^(1/4)*Sqrt[x])/(-b)^(1/4)])/(21*c^(11/4))","A",1
317,1,203,215,0.064157,"\int \frac{x^{11/2}}{b x^2+c x^4} \, dx","Integrate[x^(11/2)/(b*x^2 + c*x^4),x]","\frac{-5 \sqrt{2} b^{5/4} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)+5 \sqrt{2} b^{5/4} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)-10 \sqrt{2} b^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)+10 \sqrt{2} b^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)-40 b \sqrt[4]{c} \sqrt{x}+8 c^{5/4} x^{5/2}}{20 c^{9/4}}","-\frac{b^{5/4} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} c^{9/4}}+\frac{b^{5/4} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} c^{9/4}}-\frac{b^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{\sqrt{2} c^{9/4}}+\frac{b^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{\sqrt{2} c^{9/4}}-\frac{2 b \sqrt{x}}{c^2}+\frac{2 x^{5/2}}{5 c}",1,"(-40*b*c^(1/4)*Sqrt[x] + 8*c^(5/4)*x^(5/2) - 10*Sqrt[2]*b^(5/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 10*Sqrt[2]*b^(5/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] - 5*Sqrt[2]*b^(5/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + 5*Sqrt[2]*b^(5/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(20*c^(9/4))","A",1
318,1,78,204,0.01848,"\int \frac{x^{9/2}}{b x^2+c x^4} \, dx","Integrate[x^(9/2)/(b*x^2 + c*x^4),x]","\frac{(-b)^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-b}}\right)}{c^{7/4}}-\frac{(-b)^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-b}}\right)}{c^{7/4}}+\frac{2 x^{3/2}}{3 c}","-\frac{b^{3/4} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} c^{7/4}}+\frac{b^{3/4} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} c^{7/4}}+\frac{b^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{\sqrt{2} c^{7/4}}-\frac{b^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{\sqrt{2} c^{7/4}}+\frac{2 x^{3/2}}{3 c}",1,"(2*x^(3/2))/(3*c) + ((-b)^(3/4)*ArcTan[(c^(1/4)*Sqrt[x])/(-b)^(1/4)])/c^(7/4) - ((-b)^(3/4)*ArcTanh[(c^(1/4)*Sqrt[x])/(-b)^(1/4)])/c^(7/4)","A",1
319,1,189,202,0.0357803,"\int \frac{x^{7/2}}{b x^2+c x^4} \, dx","Integrate[x^(7/2)/(b*x^2 + c*x^4),x]","\frac{\sqrt{2} \sqrt[4]{b} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)-\sqrt{2} \sqrt[4]{b} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)+2 \sqrt{2} \sqrt[4]{b} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)-2 \sqrt{2} \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)+8 \sqrt[4]{c} \sqrt{x}}{4 c^{5/4}}","\frac{\sqrt[4]{b} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} c^{5/4}}-\frac{\sqrt[4]{b} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} c^{5/4}}+\frac{\sqrt[4]{b} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{\sqrt{2} c^{5/4}}-\frac{\sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{\sqrt{2} c^{5/4}}+\frac{2 \sqrt{x}}{c}",1,"(8*c^(1/4)*Sqrt[x] + 2*Sqrt[2]*b^(1/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] - 2*Sqrt[2]*b^(1/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + Sqrt[2]*b^(1/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - Sqrt[2]*b^(1/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(4*c^(5/4))","A",1
320,1,54,192,0.0244729,"\int \frac{x^{5/2}}{b x^2+c x^4} \, dx","Integrate[x^(5/2)/(b*x^2 + c*x^4),x]","\frac{b \left(\tan ^{-1}\left(\frac{b \sqrt[4]{c} \sqrt{x}}{(-b)^{5/4}}\right)+\tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-b}}\right)\right)}{(-b)^{5/4} c^{3/4}}","\frac{\log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} \sqrt[4]{b} c^{3/4}}-\frac{\log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} \sqrt[4]{b} c^{3/4}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{\sqrt{2} \sqrt[4]{b} c^{3/4}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{\sqrt{2} \sqrt[4]{b} c^{3/4}}",1,"(b*(ArcTan[(b*c^(1/4)*Sqrt[x])/(-b)^(5/4)] + ArcTanh[(c^(1/4)*Sqrt[x])/(-b)^(1/4)]))/((-b)^(5/4)*c^(3/4))","A",1
321,1,146,192,0.0349953,"\int \frac{x^{3/2}}{b x^2+c x^4} \, dx","Integrate[x^(3/2)/(b*x^2 + c*x^4),x]","\frac{-\log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)+\log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{2 \sqrt{2} b^{3/4} \sqrt[4]{c}}","-\frac{\log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} b^{3/4} \sqrt[4]{c}}+\frac{\log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} b^{3/4} \sqrt[4]{c}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{\sqrt{2} b^{3/4} \sqrt[4]{c}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{\sqrt{2} b^{3/4} \sqrt[4]{c}}",1,"(-2*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] - Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(2*Sqrt[2]*b^(3/4)*c^(1/4))","A",1
322,1,27,202,0.0053989,"\int \frac{\sqrt{x}}{b x^2+c x^4} \, dx","Integrate[Sqrt[x]/(b*x^2 + c*x^4),x]","-\frac{2 \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\frac{c x^2}{b}\right)}{b \sqrt{x}}","-\frac{\sqrt[4]{c} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} b^{5/4}}+\frac{\sqrt[4]{c} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} b^{5/4}}+\frac{\sqrt[4]{c} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{\sqrt{2} b^{5/4}}-\frac{\sqrt[4]{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{\sqrt{2} b^{5/4}}-\frac{2}{b \sqrt{x}}",1,"(-2*Hypergeometric2F1[-1/4, 1, 3/4, -((c*x^2)/b)])/(b*Sqrt[x])","C",1
323,1,29,204,0.006493,"\int \frac{1}{\sqrt{x} \left(b x^2+c x^4\right)} \, dx","Integrate[1/(Sqrt[x]*(b*x^2 + c*x^4)),x]","-\frac{2 \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\frac{c x^2}{b}\right)}{3 b x^{3/2}}","\frac{c^{3/4} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} b^{7/4}}-\frac{c^{3/4} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} b^{7/4}}+\frac{c^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{\sqrt{2} b^{7/4}}-\frac{c^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{\sqrt{2} b^{7/4}}-\frac{2}{3 b x^{3/2}}",1,"(-2*Hypergeometric2F1[-3/4, 1, 1/4, -((c*x^2)/b)])/(3*b*x^(3/2))","C",1
324,1,29,215,0.0063667,"\int \frac{1}{x^{3/2} \left(b x^2+c x^4\right)} \, dx","Integrate[1/(x^(3/2)*(b*x^2 + c*x^4)),x]","-\frac{2 \, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};-\frac{c x^2}{b}\right)}{5 b x^{5/2}}","\frac{c^{5/4} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} b^{9/4}}-\frac{c^{5/4} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} b^{9/4}}-\frac{c^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{\sqrt{2} b^{9/4}}+\frac{c^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{\sqrt{2} b^{9/4}}+\frac{2 c}{b^2 \sqrt{x}}-\frac{2}{5 b x^{5/2}}",1,"(-2*Hypergeometric2F1[-5/4, 1, -1/4, -((c*x^2)/b)])/(5*b*x^(5/2))","C",1
325,1,29,217,0.0058653,"\int \frac{1}{x^{5/2} \left(b x^2+c x^4\right)} \, dx","Integrate[1/(x^(5/2)*(b*x^2 + c*x^4)),x]","-\frac{2 \, _2F_1\left(-\frac{7}{4},1;-\frac{3}{4};-\frac{c x^2}{b}\right)}{7 b x^{7/2}}","-\frac{c^{7/4} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} b^{11/4}}+\frac{c^{7/4} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} b^{11/4}}-\frac{c^{7/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{\sqrt{2} b^{11/4}}+\frac{c^{7/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{\sqrt{2} b^{11/4}}+\frac{2 c}{3 b^2 x^{3/2}}-\frac{2}{7 b x^{7/2}}",1,"(-2*Hypergeometric2F1[-7/4, 1, -3/4, -((c*x^2)/b)])/(7*b*x^(7/2))","C",1
326,1,29,230,0.0065771,"\int \frac{1}{x^{7/2} \left(b x^2+c x^4\right)} \, dx","Integrate[1/(x^(7/2)*(b*x^2 + c*x^4)),x]","-\frac{2 \, _2F_1\left(-\frac{9}{4},1;-\frac{5}{4};-\frac{c x^2}{b}\right)}{9 b x^{9/2}}","-\frac{c^{9/4} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} b^{13/4}}+\frac{c^{9/4} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{2 \sqrt{2} b^{13/4}}+\frac{c^{9/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{\sqrt{2} b^{13/4}}-\frac{c^{9/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{\sqrt{2} b^{13/4}}-\frac{2 c^2}{b^3 \sqrt{x}}+\frac{2 c}{5 b^2 x^{5/2}}-\frac{2}{9 b x^{9/2}}",1,"(-2*Hypergeometric2F1[-9/4, 1, -5/4, -((c*x^2)/b)])/(9*b*x^(9/2))","C",1
327,1,220,243,0.2058489,"\int \frac{x^{19/2}}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[x^(19/2)/(b*x^2 + c*x^4)^2,x]","\frac{-45 \sqrt{2} b^{5/4} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)+45 \sqrt{2} b^{5/4} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)-90 \sqrt{2} b^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)+90 \sqrt{2} b^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)+\frac{8 \sqrt[4]{c} \sqrt{x} \left(-45 b^2-36 b c x^2+4 c^2 x^4\right)}{b+c x^2}}{80 c^{13/4}}","-\frac{9 b^{5/4} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} c^{13/4}}+\frac{9 b^{5/4} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} c^{13/4}}-\frac{9 b^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{4 \sqrt{2} c^{13/4}}+\frac{9 b^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{4 \sqrt{2} c^{13/4}}-\frac{9 b \sqrt{x}}{2 c^3}-\frac{x^{9/2}}{2 c \left(b+c x^2\right)}+\frac{9 x^{5/2}}{10 c^2}",1,"((8*c^(1/4)*Sqrt[x]*(-45*b^2 - 36*b*c*x^2 + 4*c^2*x^4))/(b + c*x^2) - 90*Sqrt[2]*b^(5/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 90*Sqrt[2]*b^(5/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] - 45*Sqrt[2]*b^(5/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + 45*Sqrt[2]*b^(5/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(80*c^(13/4))","A",1
328,1,57,230,0.0152759,"\int \frac{x^{17/2}}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[x^(17/2)/(b*x^2 + c*x^4)^2,x]","-\frac{2 x^{3/2} \left(7 \left(b+c x^2\right) \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-\frac{c x^2}{b}\right)-7 b-c x^2\right)}{3 c^2 \left(b+c x^2\right)}","-\frac{7 b^{3/4} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} c^{11/4}}+\frac{7 b^{3/4} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} c^{11/4}}+\frac{7 b^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{4 \sqrt{2} c^{11/4}}-\frac{7 b^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{4 \sqrt{2} c^{11/4}}-\frac{x^{7/2}}{2 c \left(b+c x^2\right)}+\frac{7 x^{3/2}}{6 c^2}",1,"(-2*x^(3/2)*(-7*b - c*x^2 + 7*(b + c*x^2)*Hypergeometric2F1[3/4, 2, 7/4, -((c*x^2)/b)]))/(3*c^2*(b + c*x^2))","C",1
329,1,221,230,0.1043383,"\int \frac{x^{15/2}}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[x^(15/2)/(b*x^2 + c*x^4)^2,x]","\frac{\frac{32 c^{5/4} x^{5/2}}{b+c x^2}+\frac{40 b \sqrt[4]{c} \sqrt{x}}{b+c x^2}+5 \sqrt{2} \sqrt[4]{b} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)-5 \sqrt{2} \sqrt[4]{b} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)+10 \sqrt{2} \sqrt[4]{b} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)-10 \sqrt{2} \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{16 c^{9/4}}","\frac{5 \sqrt[4]{b} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} c^{9/4}}-\frac{5 \sqrt[4]{b} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} c^{9/4}}+\frac{5 \sqrt[4]{b} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{4 \sqrt{2} c^{9/4}}-\frac{5 \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{4 \sqrt{2} c^{9/4}}-\frac{x^{5/2}}{2 c \left(b+c x^2\right)}+\frac{5 \sqrt{x}}{2 c^2}",1,"((40*b*c^(1/4)*Sqrt[x])/(b + c*x^2) + (32*c^(5/4)*x^(5/2))/(b + c*x^2) + 10*Sqrt[2]*b^(1/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] - 10*Sqrt[2]*b^(1/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 5*Sqrt[2]*b^(1/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - 5*Sqrt[2]*b^(1/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(16*c^(9/4))","A",1
330,1,43,218,0.0138358,"\int \frac{x^{13/2}}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[x^(13/2)/(b*x^2 + c*x^4)^2,x]","\frac{2 x^{3/2} \left(\frac{\, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-\frac{c x^2}{b}\right)}{b}-\frac{1}{b+c x^2}\right)}{c}","\frac{3 \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} \sqrt[4]{b} c^{7/4}}-\frac{3 \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} \sqrt[4]{b} c^{7/4}}-\frac{3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{4 \sqrt{2} \sqrt[4]{b} c^{7/4}}+\frac{3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{4 \sqrt{2} \sqrt[4]{b} c^{7/4}}-\frac{x^{3/2}}{2 c \left(b+c x^2\right)}",1,"(2*x^(3/2)*(-(b + c*x^2)^(-1) + Hypergeometric2F1[3/4, 2, 7/4, -((c*x^2)/b)]/b))/c","C",1
331,1,198,218,0.0969243,"\int \frac{x^{11/2}}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[x^(11/2)/(b*x^2 + c*x^4)^2,x]","\frac{-\frac{\sqrt{2} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{b^{3/4}}+\frac{\sqrt{2} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{b^{3/4}}-\frac{2 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{b^{3/4}}+\frac{2 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{b^{3/4}}-\frac{8 \sqrt[4]{c} \sqrt{x}}{b+c x^2}}{16 c^{5/4}}","-\frac{\log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} b^{3/4} c^{5/4}}+\frac{\log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} b^{3/4} c^{5/4}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{4 \sqrt{2} b^{3/4} c^{5/4}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{4 \sqrt{2} b^{3/4} c^{5/4}}-\frac{\sqrt{x}}{2 c \left(b+c x^2\right)}",1,"((-8*c^(1/4)*Sqrt[x])/(b + c*x^2) - (2*Sqrt[2]*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/b^(3/4) + (2*Sqrt[2]*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/b^(3/4) - (Sqrt[2]*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/b^(3/4) + (Sqrt[2]*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/b^(3/4))/(16*c^(5/4))","A",1
332,1,29,218,0.0053742,"\int \frac{x^{9/2}}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[x^(9/2)/(b*x^2 + c*x^4)^2,x]","\frac{2 x^{3/2} \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-\frac{c x^2}{b}\right)}{3 b^2}","\frac{\log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} b^{5/4} c^{3/4}}-\frac{\log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} b^{5/4} c^{3/4}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{4 \sqrt{2} b^{5/4} c^{3/4}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{4 \sqrt{2} b^{5/4} c^{3/4}}+\frac{x^{3/2}}{2 b \left(b+c x^2\right)}",1,"(2*x^(3/2)*Hypergeometric2F1[3/4, 2, 7/4, -((c*x^2)/b)])/(3*b^2)","C",1
333,1,199,218,0.0934241,"\int \frac{x^{7/2}}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[x^(7/2)/(b*x^2 + c*x^4)^2,x]","\frac{\frac{8 b^{3/4} \sqrt{x}}{b+c x^2}-\frac{3 \sqrt{2} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{\sqrt[4]{c}}+\frac{3 \sqrt{2} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{\sqrt[4]{c}}-\frac{6 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{\sqrt[4]{c}}+\frac{6 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{\sqrt[4]{c}}}{16 b^{7/4}}","-\frac{3 \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} b^{7/4} \sqrt[4]{c}}+\frac{3 \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} b^{7/4} \sqrt[4]{c}}-\frac{3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{4 \sqrt{2} b^{7/4} \sqrt[4]{c}}+\frac{3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{4 \sqrt{2} b^{7/4} \sqrt[4]{c}}+\frac{\sqrt{x}}{2 b \left(b+c x^2\right)}",1,"((8*b^(3/4)*Sqrt[x])/(b + c*x^2) - (6*Sqrt[2]*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/c^(1/4) + (6*Sqrt[2]*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/c^(1/4) - (3*Sqrt[2]*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/c^(1/4) + (3*Sqrt[2]*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/c^(1/4))/(16*b^(7/4))","A",1
334,1,27,230,0.006332,"\int \frac{x^{5/2}}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[x^(5/2)/(b*x^2 + c*x^4)^2,x]","-\frac{2 \, _2F_1\left(-\frac{1}{4},2;\frac{3}{4};-\frac{c x^2}{b}\right)}{b^2 \sqrt{x}}","-\frac{5 \sqrt[4]{c} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} b^{9/4}}+\frac{5 \sqrt[4]{c} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} b^{9/4}}+\frac{5 \sqrt[4]{c} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{4 \sqrt{2} b^{9/4}}-\frac{5 \sqrt[4]{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{4 \sqrt{2} b^{9/4}}-\frac{5}{2 b^2 \sqrt{x}}+\frac{1}{2 b \sqrt{x} \left(b+c x^2\right)}",1,"(-2*Hypergeometric2F1[-1/4, 2, 3/4, -((c*x^2)/b)])/(b^2*Sqrt[x])","C",1
335,1,29,230,0.0066574,"\int \frac{x^{3/2}}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[x^(3/2)/(b*x^2 + c*x^4)^2,x]","-\frac{2 \, _2F_1\left(-\frac{3}{4},2;\frac{1}{4};-\frac{c x^2}{b}\right)}{3 b^2 x^{3/2}}","\frac{7 c^{3/4} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} b^{11/4}}-\frac{7 c^{3/4} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} b^{11/4}}+\frac{7 c^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{4 \sqrt{2} b^{11/4}}-\frac{7 c^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{4 \sqrt{2} b^{11/4}}-\frac{7}{6 b^2 x^{3/2}}+\frac{1}{2 b x^{3/2} \left(b+c x^2\right)}",1,"(-2*Hypergeometric2F1[-3/4, 2, 1/4, -((c*x^2)/b)])/(3*b^2*x^(3/2))","C",1
336,1,29,243,0.0064679,"\int \frac{\sqrt{x}}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[Sqrt[x]/(b*x^2 + c*x^4)^2,x]","-\frac{2 \, _2F_1\left(-\frac{5}{4},2;-\frac{1}{4};-\frac{c x^2}{b}\right)}{5 b^2 x^{5/2}}","\frac{9 c^{5/4} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} b^{13/4}}-\frac{9 c^{5/4} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} b^{13/4}}-\frac{9 c^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{4 \sqrt{2} b^{13/4}}+\frac{9 c^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{4 \sqrt{2} b^{13/4}}+\frac{9 c}{2 b^3 \sqrt{x}}-\frac{9}{10 b^2 x^{5/2}}+\frac{1}{2 b x^{5/2} \left(b+c x^2\right)}",1,"(-2*Hypergeometric2F1[-5/4, 2, -1/4, -((c*x^2)/b)])/(5*b^2*x^(5/2))","C",1
337,1,29,243,0.0062887,"\int \frac{1}{\sqrt{x} \left(b x^2+c x^4\right)^2} \, dx","Integrate[1/(Sqrt[x]*(b*x^2 + c*x^4)^2),x]","-\frac{2 \, _2F_1\left(-\frac{7}{4},2;-\frac{3}{4};-\frac{c x^2}{b}\right)}{7 b^2 x^{7/2}}","-\frac{11 c^{7/4} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} b^{15/4}}+\frac{11 c^{7/4} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} b^{15/4}}-\frac{11 c^{7/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{4 \sqrt{2} b^{15/4}}+\frac{11 c^{7/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{4 \sqrt{2} b^{15/4}}+\frac{11 c}{6 b^3 x^{3/2}}-\frac{11}{14 b^2 x^{7/2}}+\frac{1}{2 b x^{7/2} \left(b+c x^2\right)}",1,"(-2*Hypergeometric2F1[-7/4, 2, -3/4, -((c*x^2)/b)])/(7*b^2*x^(7/2))","C",1
338,1,29,258,0.0068777,"\int \frac{1}{x^{3/2} \left(b x^2+c x^4\right)^2} \, dx","Integrate[1/(x^(3/2)*(b*x^2 + c*x^4)^2),x]","-\frac{2 \, _2F_1\left(-\frac{9}{4},2;-\frac{5}{4};-\frac{c x^2}{b}\right)}{9 b^2 x^{9/2}}","-\frac{13 c^{9/4} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} b^{17/4}}+\frac{13 c^{9/4} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{8 \sqrt{2} b^{17/4}}+\frac{13 c^{9/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{4 \sqrt{2} b^{17/4}}-\frac{13 c^{9/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{4 \sqrt{2} b^{17/4}}-\frac{13 c^2}{2 b^4 \sqrt{x}}+\frac{13 c}{10 b^3 x^{5/2}}-\frac{13}{18 b^2 x^{9/2}}+\frac{1}{2 b x^{9/2} \left(b+c x^2\right)}",1,"(-2*Hypergeometric2F1[-9/4, 2, -5/4, -((c*x^2)/b)])/(9*b^2*x^(9/2))","C",1
339,1,220,251,0.2499038,"\int \frac{x^{23/2}}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[x^(23/2)/(b*x^2 + c*x^4)^3,x]","\frac{\frac{8 \sqrt[4]{c} \sqrt{x} \left(45 b^2+81 b c x^2+32 c^2 x^4\right)}{\left(b+c x^2\right)^2}+45 \sqrt{2} \sqrt[4]{b} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)-45 \sqrt{2} \sqrt[4]{b} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)+90 \sqrt{2} \sqrt[4]{b} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)-90 \sqrt{2} \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{128 c^{13/4}}","\frac{45 \sqrt[4]{b} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} c^{13/4}}-\frac{45 \sqrt[4]{b} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} c^{13/4}}+\frac{45 \sqrt[4]{b} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{32 \sqrt{2} c^{13/4}}-\frac{45 \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{32 \sqrt{2} c^{13/4}}-\frac{9 x^{5/2}}{16 c^2 \left(b+c x^2\right)}-\frac{x^{9/2}}{4 c \left(b+c x^2\right)^2}+\frac{45 \sqrt{x}}{16 c^3}",1,"((8*c^(1/4)*Sqrt[x]*(45*b^2 + 81*b*c*x^2 + 32*c^2*x^4))/(b + c*x^2)^2 + 90*Sqrt[2]*b^(1/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] - 90*Sqrt[2]*b^(1/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 45*Sqrt[2]*b^(1/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - 45*Sqrt[2]*b^(1/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(128*c^(13/4))","A",1
340,1,66,239,0.0194882,"\int \frac{x^{21/2}}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[x^(21/2)/(b*x^2 + c*x^4)^3,x]","\frac{2 x^{3/2} \left(7 \left(b+c x^2\right)^2 \, _2F_1\left(\frac{3}{4},3;\frac{7}{4};-\frac{c x^2}{b}\right)-b \left(7 b+5 c x^2\right)\right)}{5 b c^2 \left(b+c x^2\right)^2}","\frac{21 \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} \sqrt[4]{b} c^{11/4}}-\frac{21 \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} \sqrt[4]{b} c^{11/4}}-\frac{21 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{32 \sqrt{2} \sqrt[4]{b} c^{11/4}}+\frac{21 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{32 \sqrt{2} \sqrt[4]{b} c^{11/4}}-\frac{7 x^{3/2}}{16 c^2 \left(b+c x^2\right)}-\frac{x^{7/2}}{4 c \left(b+c x^2\right)^2}",1,"(2*x^(3/2)*(-(b*(7*b + 5*c*x^2)) + 7*(b + c*x^2)^2*Hypergeometric2F1[3/4, 3, 7/4, -((c*x^2)/b)]))/(5*b*c^2*(b + c*x^2)^2)","C",1
341,1,242,239,0.1037578,"\int \frac{x^{19/2}}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[x^(19/2)/(b*x^2 + c*x^4)^3,x]","\frac{-\frac{15 \sqrt{2} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{b^{3/4}}+\frac{15 \sqrt{2} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{b^{3/4}}-\frac{30 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{b^{3/4}}+\frac{30 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{b^{3/4}}-\frac{256 c^{5/4} x^{5/2}}{\left(b+c x^2\right)^2}+\frac{40 \sqrt[4]{c} \sqrt{x}}{b+c x^2}-\frac{160 b \sqrt[4]{c} \sqrt{x}}{\left(b+c x^2\right)^2}}{384 c^{9/4}}","-\frac{5 \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{3/4} c^{9/4}}+\frac{5 \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{3/4} c^{9/4}}-\frac{5 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{32 \sqrt{2} b^{3/4} c^{9/4}}+\frac{5 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{32 \sqrt{2} b^{3/4} c^{9/4}}-\frac{5 \sqrt{x}}{16 c^2 \left(b+c x^2\right)}-\frac{x^{5/2}}{4 c \left(b+c x^2\right)^2}",1,"((-160*b*c^(1/4)*Sqrt[x])/(b + c*x^2)^2 - (256*c^(5/4)*x^(5/2))/(b + c*x^2)^2 + (40*c^(1/4)*Sqrt[x])/(b + c*x^2) - (30*Sqrt[2]*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/b^(3/4) + (30*Sqrt[2]*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/b^(3/4) - (15*Sqrt[2]*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/b^(3/4) + (15*Sqrt[2]*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/b^(3/4))/(384*c^(9/4))","A",1
342,1,45,242,0.0160235,"\int \frac{x^{17/2}}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[x^(17/2)/(b*x^2 + c*x^4)^3,x]","\frac{2 x^{3/2} \left(\frac{\, _2F_1\left(\frac{3}{4},3;\frac{7}{4};-\frac{c x^2}{b}\right)}{b^2}-\frac{1}{\left(b+c x^2\right)^2}\right)}{5 c}","\frac{3 \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{5/4} c^{7/4}}-\frac{3 \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{5/4} c^{7/4}}-\frac{3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{32 \sqrt{2} b^{5/4} c^{7/4}}+\frac{3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{32 \sqrt{2} b^{5/4} c^{7/4}}+\frac{3 x^{3/2}}{16 b c \left(b+c x^2\right)}-\frac{x^{3/2}}{4 c \left(b+c x^2\right)^2}",1,"(2*x^(3/2)*(-(b + c*x^2)^(-2) + Hypergeometric2F1[3/4, 3, 7/4, -((c*x^2)/b)]/b^2))/(5*c)","C",1
343,1,223,242,0.1100164,"\int \frac{x^{15/2}}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[x^(15/2)/(b*x^2 + c*x^4)^3,x]","\frac{-\frac{3 \sqrt{2} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{b^{7/4}}+\frac{3 \sqrt{2} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{b^{7/4}}-\frac{6 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{b^{7/4}}+\frac{6 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{b^{7/4}}+\frac{8 \sqrt[4]{c} \sqrt{x}}{b^2+b c x^2}-\frac{32 \sqrt[4]{c} \sqrt{x}}{\left(b+c x^2\right)^2}}{128 c^{5/4}}","-\frac{3 \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{7/4} c^{5/4}}+\frac{3 \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{7/4} c^{5/4}}-\frac{3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{32 \sqrt{2} b^{7/4} c^{5/4}}+\frac{3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{32 \sqrt{2} b^{7/4} c^{5/4}}+\frac{\sqrt{x}}{16 b c \left(b+c x^2\right)}-\frac{\sqrt{x}}{4 c \left(b+c x^2\right)^2}",1,"((-32*c^(1/4)*Sqrt[x])/(b + c*x^2)^2 + (8*c^(1/4)*Sqrt[x])/(b^2 + b*c*x^2) - (6*Sqrt[2]*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/b^(7/4) + (6*Sqrt[2]*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/b^(7/4) - (3*Sqrt[2]*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/b^(7/4) + (3*Sqrt[2]*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/b^(7/4))/(128*c^(5/4))","A",1
344,1,29,239,0.0051011,"\int \frac{x^{13/2}}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[x^(13/2)/(b*x^2 + c*x^4)^3,x]","\frac{2 x^{3/2} \, _2F_1\left(\frac{3}{4},3;\frac{7}{4};-\frac{c x^2}{b}\right)}{3 b^3}","\frac{5 \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{9/4} c^{3/4}}-\frac{5 \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{9/4} c^{3/4}}-\frac{5 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{32 \sqrt{2} b^{9/4} c^{3/4}}+\frac{5 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{32 \sqrt{2} b^{9/4} c^{3/4}}+\frac{5 x^{3/2}}{16 b^2 \left(b+c x^2\right)}+\frac{x^{3/2}}{4 b \left(b+c x^2\right)^2}",1,"(2*x^(3/2)*Hypergeometric2F1[3/4, 3, 7/4, -((c*x^2)/b)])/(3*b^3)","C",1
345,1,220,239,0.0815079,"\int \frac{x^{11/2}}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[x^(11/2)/(b*x^2 + c*x^4)^3,x]","\frac{\frac{32 b^{7/4} \sqrt{x}}{\left(b+c x^2\right)^2}+\frac{56 b^{3/4} \sqrt{x}}{b+c x^2}-\frac{21 \sqrt{2} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{\sqrt[4]{c}}+\frac{21 \sqrt{2} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{\sqrt[4]{c}}-\frac{42 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{\sqrt[4]{c}}+\frac{42 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{\sqrt[4]{c}}}{128 b^{11/4}}","-\frac{21 \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{11/4} \sqrt[4]{c}}+\frac{21 \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{11/4} \sqrt[4]{c}}-\frac{21 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{32 \sqrt{2} b^{11/4} \sqrt[4]{c}}+\frac{21 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{32 \sqrt{2} b^{11/4} \sqrt[4]{c}}+\frac{7 \sqrt{x}}{16 b^2 \left(b+c x^2\right)}+\frac{\sqrt{x}}{4 b \left(b+c x^2\right)^2}",1,"((32*b^(7/4)*Sqrt[x])/(b + c*x^2)^2 + (56*b^(3/4)*Sqrt[x])/(b + c*x^2) - (42*Sqrt[2]*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/c^(1/4) + (42*Sqrt[2]*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/c^(1/4) - (21*Sqrt[2]*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/c^(1/4) + (21*Sqrt[2]*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/c^(1/4))/(128*b^(11/4))","A",1
346,1,27,251,0.005826,"\int \frac{x^{9/2}}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[x^(9/2)/(b*x^2 + c*x^4)^3,x]","-\frac{2 \, _2F_1\left(-\frac{1}{4},3;\frac{3}{4};-\frac{c x^2}{b}\right)}{b^3 \sqrt{x}}","-\frac{45 \sqrt[4]{c} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{13/4}}+\frac{45 \sqrt[4]{c} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{13/4}}+\frac{45 \sqrt[4]{c} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{32 \sqrt{2} b^{13/4}}-\frac{45 \sqrt[4]{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{32 \sqrt{2} b^{13/4}}-\frac{45}{16 b^3 \sqrt{x}}+\frac{9}{16 b^2 \sqrt{x} \left(b+c x^2\right)}+\frac{1}{4 b \sqrt{x} \left(b+c x^2\right)^2}",1,"(-2*Hypergeometric2F1[-1/4, 3, 3/4, -((c*x^2)/b)])/(b^3*Sqrt[x])","C",1
347,1,29,251,0.0066584,"\int \frac{x^{7/2}}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[x^(7/2)/(b*x^2 + c*x^4)^3,x]","-\frac{2 \, _2F_1\left(-\frac{3}{4},3;\frac{1}{4};-\frac{c x^2}{b}\right)}{3 b^3 x^{3/2}}","\frac{77 c^{3/4} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{15/4}}-\frac{77 c^{3/4} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{15/4}}+\frac{77 c^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{32 \sqrt{2} b^{15/4}}-\frac{77 c^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{32 \sqrt{2} b^{15/4}}-\frac{77}{48 b^3 x^{3/2}}+\frac{11}{16 b^2 x^{3/2} \left(b+c x^2\right)}+\frac{1}{4 b x^{3/2} \left(b+c x^2\right)^2}",1,"(-2*Hypergeometric2F1[-3/4, 3, 1/4, -((c*x^2)/b)])/(3*b^3*x^(3/2))","C",1
348,1,29,264,0.0068196,"\int \frac{x^{5/2}}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[x^(5/2)/(b*x^2 + c*x^4)^3,x]","-\frac{2 \, _2F_1\left(-\frac{5}{4},3;-\frac{1}{4};-\frac{c x^2}{b}\right)}{5 b^3 x^{5/2}}","\frac{117 c^{5/4} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{17/4}}-\frac{117 c^{5/4} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{17/4}}-\frac{117 c^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{32 \sqrt{2} b^{17/4}}+\frac{117 c^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{32 \sqrt{2} b^{17/4}}+\frac{117 c}{16 b^4 \sqrt{x}}-\frac{117}{80 b^3 x^{5/2}}+\frac{13}{16 b^2 x^{5/2} \left(b+c x^2\right)}+\frac{1}{4 b x^{5/2} \left(b+c x^2\right)^2}",1,"(-2*Hypergeometric2F1[-5/4, 3, -1/4, -((c*x^2)/b)])/(5*b^3*x^(5/2))","C",1
349,1,29,264,0.0061516,"\int \frac{x^{3/2}}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[x^(3/2)/(b*x^2 + c*x^4)^3,x]","-\frac{2 \, _2F_1\left(-\frac{7}{4},3;-\frac{3}{4};-\frac{c x^2}{b}\right)}{7 b^3 x^{7/2}}","-\frac{165 c^{7/4} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{19/4}}+\frac{165 c^{7/4} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{19/4}}-\frac{165 c^{7/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{32 \sqrt{2} b^{19/4}}+\frac{165 c^{7/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{32 \sqrt{2} b^{19/4}}+\frac{55 c}{16 b^4 x^{3/2}}-\frac{165}{112 b^3 x^{7/2}}+\frac{15}{16 b^2 x^{7/2} \left(b+c x^2\right)}+\frac{1}{4 b x^{7/2} \left(b+c x^2\right)^2}",1,"(-2*Hypergeometric2F1[-7/4, 3, -3/4, -((c*x^2)/b)])/(7*b^3*x^(7/2))","C",1
350,1,29,279,0.0067954,"\int \frac{\sqrt{x}}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[Sqrt[x]/(b*x^2 + c*x^4)^3,x]","-\frac{2 \, _2F_1\left(-\frac{9}{4},3;-\frac{5}{4};-\frac{c x^2}{b}\right)}{9 b^3 x^{9/2}}","-\frac{221 c^{9/4} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{21/4}}+\frac{221 c^{9/4} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{21/4}}+\frac{221 c^{9/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{32 \sqrt{2} b^{21/4}}-\frac{221 c^{9/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{32 \sqrt{2} b^{21/4}}-\frac{221 c^2}{16 b^5 \sqrt{x}}+\frac{221 c}{80 b^4 x^{5/2}}-\frac{221}{144 b^3 x^{9/2}}+\frac{17}{16 b^2 x^{9/2} \left(b+c x^2\right)}+\frac{1}{4 b x^{9/2} \left(b+c x^2\right)^2}",1,"(-2*Hypergeometric2F1[-9/4, 3, -5/4, -((c*x^2)/b)])/(9*b^3*x^(9/2))","C",1
351,1,29,279,0.0072981,"\int \frac{1}{\sqrt{x} \left(b x^2+c x^4\right)^3} \, dx","Integrate[1/(Sqrt[x]*(b*x^2 + c*x^4)^3),x]","-\frac{2 \, _2F_1\left(-\frac{11}{4},3;-\frac{7}{4};-\frac{c x^2}{b}\right)}{11 b^3 x^{11/2}}","\frac{285 c^{11/4} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{23/4}}-\frac{285 c^{11/4} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right)}{64 \sqrt{2} b^{23/4}}+\frac{285 c^{11/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)}{32 \sqrt{2} b^{23/4}}-\frac{285 c^{11/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}+1\right)}{32 \sqrt{2} b^{23/4}}-\frac{95 c^2}{16 b^5 x^{3/2}}+\frac{285 c}{112 b^4 x^{7/2}}-\frac{285}{176 b^3 x^{11/2}}+\frac{19}{16 b^2 x^{11/2} \left(b+c x^2\right)}+\frac{1}{4 b x^{11/2} \left(b+c x^2\right)^2}",1,"(-2*Hypergeometric2F1[-11/4, 3, -7/4, -((c*x^2)/b)])/(11*b^3*x^(11/2))","C",1
352,1,102,323,0.0690903,"\int x^{7/2} \sqrt{b x^2+c x^4} \, dx","Integrate[x^(7/2)*Sqrt[b*x^2 + c*x^4],x]","\frac{2 \sqrt{x} \sqrt{x^2 \left(b+c x^2\right)} \left(\sqrt{\frac{c x^2}{b}+1} \left(-7 b^2+2 b c x^2+9 c^2 x^4\right)+7 b^2 \, _2F_1\left(-\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{b}\right)\right)}{117 c^2 \sqrt{\frac{c x^2}{b}+1}}","\frac{14 b^{13/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{195 c^{11/4} \sqrt{b x^2+c x^4}}-\frac{28 b^{13/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{195 c^{11/4} \sqrt{b x^2+c x^4}}+\frac{28 b^3 x^{3/2} \left(b+c x^2\right)}{195 c^{5/2} \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}-\frac{28 b^2 \sqrt{x} \sqrt{b x^2+c x^4}}{585 c^2}+\frac{2}{13} x^{9/2} \sqrt{b x^2+c x^4}+\frac{4 b x^{5/2} \sqrt{b x^2+c x^4}}{117 c}",1,"(2*Sqrt[x]*Sqrt[x^2*(b + c*x^2)]*(Sqrt[1 + (c*x^2)/b]*(-7*b^2 + 2*b*c*x^2 + 9*c^2*x^4) + 7*b^2*Hypergeometric2F1[-1/2, 3/4, 7/4, -((c*x^2)/b)]))/(117*c^2*Sqrt[1 + (c*x^2)/b])","C",1
353,1,102,176,0.0554575,"\int x^{5/2} \sqrt{b x^2+c x^4} \, dx","Integrate[x^(5/2)*Sqrt[b*x^2 + c*x^4],x]","\frac{2 \sqrt{x^2 \left(b+c x^2\right)} \left(\sqrt{\frac{c x^2}{b}+1} \left(-5 b^2+2 b c x^2+7 c^2 x^4\right)+5 b^2 \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^2}{b}\right)\right)}{77 c^2 \sqrt{x} \sqrt{\frac{c x^2}{b}+1}}","\frac{10 b^{11/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{231 c^{9/4} \sqrt{b x^2+c x^4}}-\frac{20 b^2 \sqrt{b x^2+c x^4}}{231 c^2 \sqrt{x}}+\frac{2}{11} x^{7/2} \sqrt{b x^2+c x^4}+\frac{4 b x^{3/2} \sqrt{b x^2+c x^4}}{77 c}",1,"(2*Sqrt[x^2*(b + c*x^2)]*(Sqrt[1 + (c*x^2)/b]*(-5*b^2 + 2*b*c*x^2 + 7*c^2*x^4) + 5*b^2*Hypergeometric2F1[-1/2, 1/4, 5/4, -((c*x^2)/b)]))/(77*c^2*Sqrt[x]*Sqrt[1 + (c*x^2)/b])","C",1
354,1,86,293,0.0361356,"\int x^{3/2} \sqrt{b x^2+c x^4} \, dx","Integrate[x^(3/2)*Sqrt[b*x^2 + c*x^4],x]","\frac{2 \sqrt{x} \sqrt{x^2 \left(b+c x^2\right)} \left(\left(b+c x^2\right) \sqrt{\frac{c x^2}{b}+1}-b \, _2F_1\left(-\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{b}\right)\right)}{9 c \sqrt{\frac{c x^2}{b}+1}}","-\frac{2 b^{9/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{15 c^{7/4} \sqrt{b x^2+c x^4}}+\frac{4 b^{9/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{15 c^{7/4} \sqrt{b x^2+c x^4}}-\frac{4 b^2 x^{3/2} \left(b+c x^2\right)}{15 c^{3/2} \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}+\frac{4 b \sqrt{x} \sqrt{b x^2+c x^4}}{45 c}+\frac{2}{9} x^{5/2} \sqrt{b x^2+c x^4}",1,"(2*Sqrt[x]*Sqrt[x^2*(b + c*x^2)]*((b + c*x^2)*Sqrt[1 + (c*x^2)/b] - b*Hypergeometric2F1[-1/2, 3/4, 7/4, -((c*x^2)/b)]))/(9*c*Sqrt[1 + (c*x^2)/b])","C",1
355,1,86,146,0.0341763,"\int \sqrt{x} \sqrt{b x^2+c x^4} \, dx","Integrate[Sqrt[x]*Sqrt[b*x^2 + c*x^4],x]","\frac{2 \sqrt{x^2 \left(b+c x^2\right)} \left(\left(b+c x^2\right) \sqrt{\frac{c x^2}{b}+1}-b \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^2}{b}\right)\right)}{7 c \sqrt{x} \sqrt{\frac{c x^2}{b}+1}}","-\frac{2 b^{7/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{21 c^{5/4} \sqrt{b x^2+c x^4}}+\frac{4 b \sqrt{b x^2+c x^4}}{21 c \sqrt{x}}+\frac{2}{7} x^{3/2} \sqrt{b x^2+c x^4}",1,"(2*Sqrt[x^2*(b + c*x^2)]*((b + c*x^2)*Sqrt[1 + (c*x^2)/b] - b*Hypergeometric2F1[-1/2, 1/4, 5/4, -((c*x^2)/b)]))/(7*c*Sqrt[x]*Sqrt[1 + (c*x^2)/b])","C",1
356,1,57,263,0.0148608,"\int \frac{\sqrt{b x^2+c x^4}}{\sqrt{x}} \, dx","Integrate[Sqrt[b*x^2 + c*x^4]/Sqrt[x],x]","\frac{2 \sqrt{x} \sqrt{x^2 \left(b+c x^2\right)} \, _2F_1\left(-\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{b}\right)}{3 \sqrt{\frac{c x^2}{b}+1}}","\frac{2 b^{5/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{5 c^{3/4} \sqrt{b x^2+c x^4}}-\frac{4 b^{5/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{5 c^{3/4} \sqrt{b x^2+c x^4}}+\frac{2}{5} \sqrt{x} \sqrt{b x^2+c x^4}+\frac{4 b x^{3/2} \left(b+c x^2\right)}{5 \sqrt{c} \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}",1,"(2*Sqrt[x]*Sqrt[x^2*(b + c*x^2)]*Hypergeometric2F1[-1/2, 3/4, 7/4, -((c*x^2)/b)])/(3*Sqrt[1 + (c*x^2)/b])","C",1
357,1,55,118,0.0130281,"\int \frac{\sqrt{b x^2+c x^4}}{x^{3/2}} \, dx","Integrate[Sqrt[b*x^2 + c*x^4]/x^(3/2),x]","\frac{2 \sqrt{x^2 \left(b+c x^2\right)} \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^2}{b}\right)}{\sqrt{x} \sqrt{\frac{c x^2}{b}+1}}","\frac{2 b^{3/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{3 \sqrt[4]{c} \sqrt{b x^2+c x^4}}+\frac{2 \sqrt{b x^2+c x^4}}{3 \sqrt{x}}",1,"(2*Sqrt[x^2*(b + c*x^2)]*Hypergeometric2F1[-1/2, 1/4, 5/4, -((c*x^2)/b)])/(Sqrt[x]*Sqrt[1 + (c*x^2)/b])","C",1
358,1,55,254,0.0142402,"\int \frac{\sqrt{b x^2+c x^4}}{x^{5/2}} \, dx","Integrate[Sqrt[b*x^2 + c*x^4]/x^(5/2),x]","-\frac{2 \sqrt{x^2 \left(b+c x^2\right)} \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};-\frac{c x^2}{b}\right)}{x^{3/2} \sqrt{\frac{c x^2}{b}+1}}","\frac{2 \sqrt[4]{b} \sqrt[4]{c} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{\sqrt{b x^2+c x^4}}-\frac{4 \sqrt[4]{b} \sqrt[4]{c} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{\sqrt{b x^2+c x^4}}+\frac{4 \sqrt{c} x^{3/2} \left(b+c x^2\right)}{\left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}-\frac{2 \sqrt{b x^2+c x^4}}{x^{3/2}}",1,"(-2*Sqrt[x^2*(b + c*x^2)]*Hypergeometric2F1[-1/2, -1/4, 3/4, -((c*x^2)/b)])/(x^(3/2)*Sqrt[1 + (c*x^2)/b])","C",1
359,1,57,118,0.0134221,"\int \frac{\sqrt{b x^2+c x^4}}{x^{7/2}} \, dx","Integrate[Sqrt[b*x^2 + c*x^4]/x^(7/2),x]","-\frac{2 \sqrt{x^2 \left(b+c x^2\right)} \, _2F_1\left(-\frac{3}{4},-\frac{1}{2};\frac{1}{4};-\frac{c x^2}{b}\right)}{3 x^{5/2} \sqrt{\frac{c x^2}{b}+1}}","\frac{2 c^{3/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{3 \sqrt[4]{b} \sqrt{b x^2+c x^4}}-\frac{2 \sqrt{b x^2+c x^4}}{3 x^{5/2}}",1,"(-2*Sqrt[x^2*(b + c*x^2)]*Hypergeometric2F1[-3/4, -1/2, 1/4, -((c*x^2)/b)])/(3*x^(5/2)*Sqrt[1 + (c*x^2)/b])","C",1
360,1,57,293,0.0142068,"\int \frac{\sqrt{b x^2+c x^4}}{x^{9/2}} \, dx","Integrate[Sqrt[b*x^2 + c*x^4]/x^(9/2),x]","-\frac{2 \sqrt{x^2 \left(b+c x^2\right)} \, _2F_1\left(-\frac{5}{4},-\frac{1}{2};-\frac{1}{4};-\frac{c x^2}{b}\right)}{5 x^{7/2} \sqrt{\frac{c x^2}{b}+1}}","\frac{2 c^{5/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{5 b^{3/4} \sqrt{b x^2+c x^4}}-\frac{4 c^{5/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{5 b^{3/4} \sqrt{b x^2+c x^4}}+\frac{4 c^{3/2} x^{3/2} \left(b+c x^2\right)}{5 b \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}-\frac{4 c \sqrt{b x^2+c x^4}}{5 b x^{3/2}}-\frac{2 \sqrt{b x^2+c x^4}}{5 x^{7/2}}",1,"(-2*Sqrt[x^2*(b + c*x^2)]*Hypergeometric2F1[-5/4, -1/2, -1/4, -((c*x^2)/b)])/(5*x^(7/2)*Sqrt[1 + (c*x^2)/b])","C",1
361,1,57,146,0.0133396,"\int \frac{\sqrt{b x^2+c x^4}}{x^{11/2}} \, dx","Integrate[Sqrt[b*x^2 + c*x^4]/x^(11/2),x]","-\frac{2 \sqrt{x^2 \left(b+c x^2\right)} \, _2F_1\left(-\frac{7}{4},-\frac{1}{2};-\frac{3}{4};-\frac{c x^2}{b}\right)}{7 x^{9/2} \sqrt{\frac{c x^2}{b}+1}}","-\frac{2 c^{7/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{21 b^{5/4} \sqrt{b x^2+c x^4}}-\frac{4 c \sqrt{b x^2+c x^4}}{21 b x^{5/2}}-\frac{2 \sqrt{b x^2+c x^4}}{7 x^{9/2}}",1,"(-2*Sqrt[x^2*(b + c*x^2)]*Hypergeometric2F1[-7/4, -1/2, -3/4, -((c*x^2)/b)])/(7*x^(9/2)*Sqrt[1 + (c*x^2)/b])","C",1
362,1,57,323,0.0149055,"\int \frac{\sqrt{b x^2+c x^4}}{x^{13/2}} \, dx","Integrate[Sqrt[b*x^2 + c*x^4]/x^(13/2),x]","-\frac{2 \sqrt{x^2 \left(b+c x^2\right)} \, _2F_1\left(-\frac{9}{4},-\frac{1}{2};-\frac{5}{4};-\frac{c x^2}{b}\right)}{9 x^{11/2} \sqrt{\frac{c x^2}{b}+1}}","-\frac{2 c^{9/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{15 b^{7/4} \sqrt{b x^2+c x^4}}+\frac{4 c^{9/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{15 b^{7/4} \sqrt{b x^2+c x^4}}-\frac{4 c^{5/2} x^{3/2} \left(b+c x^2\right)}{15 b^2 \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}+\frac{4 c^2 \sqrt{b x^2+c x^4}}{15 b^2 x^{3/2}}-\frac{4 c \sqrt{b x^2+c x^4}}{45 b x^{7/2}}-\frac{2 \sqrt{b x^2+c x^4}}{9 x^{11/2}}",1,"(-2*Sqrt[x^2*(b + c*x^2)]*Hypergeometric2F1[-9/4, -1/2, -5/4, -((c*x^2)/b)])/(9*x^(11/2)*Sqrt[1 + (c*x^2)/b])","C",1
363,1,57,176,0.0143959,"\int \frac{\sqrt{b x^2+c x^4}}{x^{15/2}} \, dx","Integrate[Sqrt[b*x^2 + c*x^4]/x^(15/2),x]","-\frac{2 \sqrt{x^2 \left(b+c x^2\right)} \, _2F_1\left(-\frac{11}{4},-\frac{1}{2};-\frac{7}{4};-\frac{c x^2}{b}\right)}{11 x^{13/2} \sqrt{\frac{c x^2}{b}+1}}","\frac{10 c^{11/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{231 b^{9/4} \sqrt{b x^2+c x^4}}+\frac{20 c^2 \sqrt{b x^2+c x^4}}{231 b^2 x^{5/2}}-\frac{4 c \sqrt{b x^2+c x^4}}{77 b x^{9/2}}-\frac{2 \sqrt{b x^2+c x^4}}{11 x^{13/2}}",1,"(-2*Sqrt[x^2*(b + c*x^2)]*Hypergeometric2F1[-11/4, -1/2, -7/4, -((c*x^2)/b)])/(11*x^(13/2)*Sqrt[1 + (c*x^2)/b])","C",1
364,1,101,350,0.0612503,"\int x^{3/2} \left(b x^2+c x^4\right)^{3/2} \, dx","Integrate[x^(3/2)*(b*x^2 + c*x^4)^(3/2),x]","\frac{2 \sqrt{x} \sqrt{x^2 \left(b+c x^2\right)} \left(7 b^3 \, _2F_1\left(-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{b}\right)-\left(7 b-13 c x^2\right) \left(b+c x^2\right)^2 \sqrt{\frac{c x^2}{b}+1}\right)}{221 c^2 \sqrt{\frac{c x^2}{b}+1}}","\frac{28 b^{17/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{1105 c^{11/4} \sqrt{b x^2+c x^4}}-\frac{56 b^{17/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{1105 c^{11/4} \sqrt{b x^2+c x^4}}+\frac{56 b^4 x^{3/2} \left(b+c x^2\right)}{1105 c^{5/2} \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}-\frac{56 b^3 \sqrt{x} \sqrt{b x^2+c x^4}}{3315 c^2}+\frac{8 b^2 x^{5/2} \sqrt{b x^2+c x^4}}{663 c}+\frac{12}{221} b x^{9/2} \sqrt{b x^2+c x^4}+\frac{2}{17} x^{5/2} \left(b x^2+c x^4\right)^{3/2}",1,"(2*Sqrt[x]*Sqrt[x^2*(b + c*x^2)]*(-((7*b - 13*c*x^2)*(b + c*x^2)^2*Sqrt[1 + (c*x^2)/b]) + 7*b^3*Hypergeometric2F1[-3/2, 3/4, 7/4, -((c*x^2)/b)]))/(221*c^2*Sqrt[1 + (c*x^2)/b])","C",1
365,1,101,203,0.0616629,"\int \sqrt{x} \left(b x^2+c x^4\right)^{3/2} \, dx","Integrate[Sqrt[x]*(b*x^2 + c*x^4)^(3/2),x]","\frac{2 \sqrt{x^2 \left(b+c x^2\right)} \left(5 b^3 \, _2F_1\left(-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^2}{b}\right)-\left(5 b-11 c x^2\right) \left(b+c x^2\right)^2 \sqrt{\frac{c x^2}{b}+1}\right)}{165 c^2 \sqrt{x} \sqrt{\frac{c x^2}{b}+1}}","\frac{4 b^{15/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{231 c^{9/4} \sqrt{b x^2+c x^4}}-\frac{8 b^3 \sqrt{b x^2+c x^4}}{231 c^2 \sqrt{x}}+\frac{8 b^2 x^{3/2} \sqrt{b x^2+c x^4}}{385 c}+\frac{4}{55} b x^{7/2} \sqrt{b x^2+c x^4}+\frac{2}{15} x^{3/2} \left(b x^2+c x^4\right)^{3/2}",1,"(2*Sqrt[x^2*(b + c*x^2)]*(-((5*b - 11*c*x^2)*(b + c*x^2)^2*Sqrt[1 + (c*x^2)/b]) + 5*b^3*Hypergeometric2F1[-3/2, 1/4, 5/4, -((c*x^2)/b)]))/(165*c^2*Sqrt[x]*Sqrt[1 + (c*x^2)/b])","C",1
366,1,90,320,0.0447788,"\int \frac{\left(b x^2+c x^4\right)^{3/2}}{\sqrt{x}} \, dx","Integrate[(b*x^2 + c*x^4)^(3/2)/Sqrt[x],x]","\frac{2 \sqrt{x} \sqrt{x^2 \left(b+c x^2\right)} \left(\left(b+c x^2\right)^2 \sqrt{\frac{c x^2}{b}+1}-b^2 \, _2F_1\left(-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{b}\right)\right)}{13 c \sqrt{\frac{c x^2}{b}+1}}","-\frac{4 b^{13/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{65 c^{7/4} \sqrt{b x^2+c x^4}}+\frac{8 b^{13/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{65 c^{7/4} \sqrt{b x^2+c x^4}}-\frac{8 b^3 x^{3/2} \left(b+c x^2\right)}{65 c^{3/2} \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}+\frac{8 b^2 \sqrt{x} \sqrt{b x^2+c x^4}}{195 c}+\frac{2}{13} \sqrt{x} \left(b x^2+c x^4\right)^{3/2}+\frac{4}{39} b x^{5/2} \sqrt{b x^2+c x^4}",1,"(2*Sqrt[x]*Sqrt[x^2*(b + c*x^2)]*((b + c*x^2)^2*Sqrt[1 + (c*x^2)/b] - b^2*Hypergeometric2F1[-3/2, 3/4, 7/4, -((c*x^2)/b)]))/(13*c*Sqrt[1 + (c*x^2)/b])","C",1
367,1,90,173,0.0369438,"\int \frac{\left(b x^2+c x^4\right)^{3/2}}{x^{3/2}} \, dx","Integrate[(b*x^2 + c*x^4)^(3/2)/x^(3/2),x]","\frac{2 \sqrt{x^2 \left(b+c x^2\right)} \left(\left(b+c x^2\right)^2 \sqrt{\frac{c x^2}{b}+1}-b^2 \, _2F_1\left(-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^2}{b}\right)\right)}{11 c \sqrt{x} \sqrt{\frac{c x^2}{b}+1}}","-\frac{4 b^{11/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{77 c^{5/4} \sqrt{b x^2+c x^4}}+\frac{8 b^2 \sqrt{b x^2+c x^4}}{77 c \sqrt{x}}+\frac{2 \left(b x^2+c x^4\right)^{3/2}}{11 \sqrt{x}}+\frac{12}{77} b x^{3/2} \sqrt{b x^2+c x^4}",1,"(2*Sqrt[x^2*(b + c*x^2)]*((b + c*x^2)^2*Sqrt[1 + (c*x^2)/b] - b^2*Hypergeometric2F1[-3/2, 1/4, 5/4, -((c*x^2)/b)]))/(11*c*Sqrt[x]*Sqrt[1 + (c*x^2)/b])","C",1
368,1,58,290,0.0170128,"\int \frac{\left(b x^2+c x^4\right)^{3/2}}{x^{5/2}} \, dx","Integrate[(b*x^2 + c*x^4)^(3/2)/x^(5/2),x]","\frac{2 b \sqrt{x} \sqrt{x^2 \left(b+c x^2\right)} \, _2F_1\left(-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{b}\right)}{3 \sqrt{\frac{c x^2}{b}+1}}","\frac{4 b^{9/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{15 c^{3/4} \sqrt{b x^2+c x^4}}-\frac{8 b^{9/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{15 c^{3/4} \sqrt{b x^2+c x^4}}+\frac{8 b^2 x^{3/2} \left(b+c x^2\right)}{15 \sqrt{c} \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}+\frac{4}{15} b \sqrt{x} \sqrt{b x^2+c x^4}+\frac{2 \left(b x^2+c x^4\right)^{3/2}}{9 x^{3/2}}",1,"(2*b*Sqrt[x]*Sqrt[x^2*(b + c*x^2)]*Hypergeometric2F1[-3/2, 3/4, 7/4, -((c*x^2)/b)])/(3*Sqrt[1 + (c*x^2)/b])","C",1
369,1,56,143,0.0165929,"\int \frac{\left(b x^2+c x^4\right)^{3/2}}{x^{7/2}} \, dx","Integrate[(b*x^2 + c*x^4)^(3/2)/x^(7/2),x]","\frac{2 b \sqrt{x^2 \left(b+c x^2\right)} \, _2F_1\left(-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^2}{b}\right)}{\sqrt{x} \sqrt{\frac{c x^2}{b}+1}}","\frac{4 b^{7/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{7 \sqrt[4]{c} \sqrt{b x^2+c x^4}}+\frac{4 b \sqrt{b x^2+c x^4}}{7 \sqrt{x}}+\frac{2 \left(b x^2+c x^4\right)^{3/2}}{7 x^{5/2}}",1,"(2*b*Sqrt[x^2*(b + c*x^2)]*Hypergeometric2F1[-3/2, 1/4, 5/4, -((c*x^2)/b)])/(Sqrt[x]*Sqrt[1 + (c*x^2)/b])","C",1
370,1,56,286,0.0157396,"\int \frac{\left(b x^2+c x^4\right)^{3/2}}{x^{9/2}} \, dx","Integrate[(b*x^2 + c*x^4)^(3/2)/x^(9/2),x]","-\frac{2 b \sqrt{x^2 \left(b+c x^2\right)} \, _2F_1\left(-\frac{3}{2},-\frac{1}{4};\frac{3}{4};-\frac{c x^2}{b}\right)}{x^{3/2} \sqrt{\frac{c x^2}{b}+1}}","\frac{12 b^{5/4} \sqrt[4]{c} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{5 \sqrt{b x^2+c x^4}}-\frac{24 b^{5/4} \sqrt[4]{c} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{5 \sqrt{b x^2+c x^4}}+\frac{12}{5} c \sqrt{x} \sqrt{b x^2+c x^4}+\frac{24 b \sqrt{c} x^{3/2} \left(b+c x^2\right)}{5 \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}-\frac{2 \left(b x^2+c x^4\right)^{3/2}}{x^{7/2}}",1,"(-2*b*Sqrt[x^2*(b + c*x^2)]*Hypergeometric2F1[-3/2, -1/4, 3/4, -((c*x^2)/b)])/(x^(3/2)*Sqrt[1 + (c*x^2)/b])","C",1
371,1,58,143,0.0174924,"\int \frac{\left(b x^2+c x^4\right)^{3/2}}{x^{11/2}} \, dx","Integrate[(b*x^2 + c*x^4)^(3/2)/x^(11/2),x]","-\frac{2 b \sqrt{x^2 \left(b+c x^2\right)} \, _2F_1\left(-\frac{3}{2},-\frac{3}{4};\frac{1}{4};-\frac{c x^2}{b}\right)}{3 x^{5/2} \sqrt{\frac{c x^2}{b}+1}}","\frac{4 b^{3/4} c^{3/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{3 \sqrt{b x^2+c x^4}}+\frac{4 c \sqrt{b x^2+c x^4}}{3 \sqrt{x}}-\frac{2 \left(b x^2+c x^4\right)^{3/2}}{3 x^{9/2}}",1,"(-2*b*Sqrt[x^2*(b + c*x^2)]*Hypergeometric2F1[-3/2, -3/4, 1/4, -((c*x^2)/b)])/(3*x^(5/2)*Sqrt[1 + (c*x^2)/b])","C",1
372,1,58,287,0.0160332,"\int \frac{\left(b x^2+c x^4\right)^{3/2}}{x^{13/2}} \, dx","Integrate[(b*x^2 + c*x^4)^(3/2)/x^(13/2),x]","-\frac{2 b \sqrt{x^2 \left(b+c x^2\right)} \, _2F_1\left(-\frac{3}{2},-\frac{5}{4};-\frac{1}{4};-\frac{c x^2}{b}\right)}{5 x^{7/2} \sqrt{\frac{c x^2}{b}+1}}","\frac{12 \sqrt[4]{b} c^{5/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{5 \sqrt{b x^2+c x^4}}-\frac{24 \sqrt[4]{b} c^{5/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{5 \sqrt{b x^2+c x^4}}+\frac{24 c^{3/2} x^{3/2} \left(b+c x^2\right)}{5 \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}-\frac{12 c \sqrt{b x^2+c x^4}}{5 x^{3/2}}-\frac{2 \left(b x^2+c x^4\right)^{3/2}}{5 x^{11/2}}",1,"(-2*b*Sqrt[x^2*(b + c*x^2)]*Hypergeometric2F1[-3/2, -5/4, -1/4, -((c*x^2)/b)])/(5*x^(7/2)*Sqrt[1 + (c*x^2)/b])","C",1
373,1,58,143,0.0167823,"\int \frac{\left(b x^2+c x^4\right)^{3/2}}{x^{15/2}} \, dx","Integrate[(b*x^2 + c*x^4)^(3/2)/x^(15/2),x]","-\frac{2 b \sqrt{x^2 \left(b+c x^2\right)} \, _2F_1\left(-\frac{7}{4},-\frac{3}{2};-\frac{3}{4};-\frac{c x^2}{b}\right)}{7 x^{9/2} \sqrt{\frac{c x^2}{b}+1}}","\frac{4 c^{7/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{7 \sqrt[4]{b} \sqrt{b x^2+c x^4}}-\frac{4 c \sqrt{b x^2+c x^4}}{7 x^{5/2}}-\frac{2 \left(b x^2+c x^4\right)^{3/2}}{7 x^{13/2}}",1,"(-2*b*Sqrt[x^2*(b + c*x^2)]*Hypergeometric2F1[-7/4, -3/2, -3/4, -((c*x^2)/b)])/(7*x^(9/2)*Sqrt[1 + (c*x^2)/b])","C",1
374,1,58,320,0.0178154,"\int \frac{\left(b x^2+c x^4\right)^{3/2}}{x^{17/2}} \, dx","Integrate[(b*x^2 + c*x^4)^(3/2)/x^(17/2),x]","-\frac{2 b \sqrt{x^2 \left(b+c x^2\right)} \, _2F_1\left(-\frac{9}{4},-\frac{3}{2};-\frac{5}{4};-\frac{c x^2}{b}\right)}{9 x^{11/2} \sqrt{\frac{c x^2}{b}+1}}","\frac{4 c^{9/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{15 b^{3/4} \sqrt{b x^2+c x^4}}-\frac{8 c^{9/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{15 b^{3/4} \sqrt{b x^2+c x^4}}+\frac{8 c^{5/2} x^{3/2} \left(b+c x^2\right)}{15 b \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}-\frac{8 c^2 \sqrt{b x^2+c x^4}}{15 b x^{3/2}}-\frac{4 c \sqrt{b x^2+c x^4}}{15 x^{7/2}}-\frac{2 \left(b x^2+c x^4\right)^{3/2}}{9 x^{15/2}}",1,"(-2*b*Sqrt[x^2*(b + c*x^2)]*Hypergeometric2F1[-9/4, -3/2, -5/4, -((c*x^2)/b)])/(9*x^(11/2)*Sqrt[1 + (c*x^2)/b])","C",1
375,1,58,173,0.0176732,"\int \frac{\left(b x^2+c x^4\right)^{3/2}}{x^{19/2}} \, dx","Integrate[(b*x^2 + c*x^4)^(3/2)/x^(19/2),x]","-\frac{2 b \sqrt{x^2 \left(b+c x^2\right)} \, _2F_1\left(-\frac{11}{4},-\frac{3}{2};-\frac{7}{4};-\frac{c x^2}{b}\right)}{11 x^{13/2} \sqrt{\frac{c x^2}{b}+1}}","-\frac{4 c^{11/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{77 b^{5/4} \sqrt{b x^2+c x^4}}-\frac{8 c^2 \sqrt{b x^2+c x^4}}{77 b x^{5/2}}-\frac{12 c \sqrt{b x^2+c x^4}}{77 x^{9/2}}-\frac{2 \left(b x^2+c x^4\right)^{3/2}}{11 x^{17/2}}",1,"(-2*b*Sqrt[x^2*(b + c*x^2)]*Hypergeometric2F1[-11/4, -3/2, -7/4, -((c*x^2)/b)])/(11*x^(13/2)*Sqrt[1 + (c*x^2)/b])","C",1
376,1,58,350,0.0185384,"\int \frac{\left(b x^2+c x^4\right)^{3/2}}{x^{21/2}} \, dx","Integrate[(b*x^2 + c*x^4)^(3/2)/x^(21/2),x]","-\frac{2 b \sqrt{x^2 \left(b+c x^2\right)} \, _2F_1\left(-\frac{13}{4},-\frac{3}{2};-\frac{9}{4};-\frac{c x^2}{b}\right)}{13 x^{15/2} \sqrt{\frac{c x^2}{b}+1}}","-\frac{4 c^{13/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{65 b^{7/4} \sqrt{b x^2+c x^4}}+\frac{8 c^{13/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{65 b^{7/4} \sqrt{b x^2+c x^4}}-\frac{8 c^{7/2} x^{3/2} \left(b+c x^2\right)}{65 b^2 \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}+\frac{8 c^3 \sqrt{b x^2+c x^4}}{65 b^2 x^{3/2}}-\frac{8 c^2 \sqrt{b x^2+c x^4}}{195 b x^{7/2}}-\frac{4 c \sqrt{b x^2+c x^4}}{39 x^{11/2}}-\frac{2 \left(b x^2+c x^4\right)^{3/2}}{13 x^{19/2}}",1,"(-2*b*Sqrt[x^2*(b + c*x^2)]*Hypergeometric2F1[-13/4, -3/2, -9/4, -((c*x^2)/b)])/(13*x^(15/2)*Sqrt[1 + (c*x^2)/b])","C",1
377,1,58,203,0.0175492,"\int \frac{\left(b x^2+c x^4\right)^{3/2}}{x^{23/2}} \, dx","Integrate[(b*x^2 + c*x^4)^(3/2)/x^(23/2),x]","-\frac{2 b \sqrt{x^2 \left(b+c x^2\right)} \, _2F_1\left(-\frac{15}{4},-\frac{3}{2};-\frac{11}{4};-\frac{c x^2}{b}\right)}{15 x^{17/2} \sqrt{\frac{c x^2}{b}+1}}","\frac{4 c^{15/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{231 b^{9/4} \sqrt{b x^2+c x^4}}+\frac{8 c^3 \sqrt{b x^2+c x^4}}{231 b^2 x^{5/2}}-\frac{8 c^2 \sqrt{b x^2+c x^4}}{385 b x^{9/2}}-\frac{4 c \sqrt{b x^2+c x^4}}{55 x^{13/2}}-\frac{2 \left(b x^2+c x^4\right)^{3/2}}{15 x^{21/2}}",1,"(-2*b*Sqrt[x^2*(b + c*x^2)]*Hypergeometric2F1[-15/4, -3/2, -11/4, -((c*x^2)/b)])/(15*x^(17/2)*Sqrt[1 + (c*x^2)/b])","C",1
378,1,97,179,0.036215,"\int \frac{x^{13/2}}{\sqrt{b x^2+c x^4}} \, dx","Integrate[x^(13/2)/Sqrt[b*x^2 + c*x^4],x]","\frac{2 x^{3/2} \left(-15 b^3 \sqrt{\frac{c x^2}{b}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^2}{b}\right)+15 b^3+6 b^2 c x^2-2 b c^2 x^4+7 c^3 x^6\right)}{77 c^3 \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{15 b^{11/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{77 c^{13/4} \sqrt{b x^2+c x^4}}+\frac{30 b^2 \sqrt{b x^2+c x^4}}{77 c^3 \sqrt{x}}-\frac{18 b x^{3/2} \sqrt{b x^2+c x^4}}{77 c^2}+\frac{2 x^{7/2} \sqrt{b x^2+c x^4}}{11 c}",1,"(2*x^(3/2)*(15*b^3 + 6*b^2*c*x^2 - 2*b*c^2*x^4 + 7*c^3*x^6 - 15*b^3*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^2)/b)]))/(77*c^3*Sqrt[x^2*(b + c*x^2)])","C",1
379,1,86,296,0.0362734,"\int \frac{x^{11/2}}{\sqrt{b x^2+c x^4}} \, dx","Integrate[x^(11/2)/Sqrt[b*x^2 + c*x^4],x]","\frac{2 x^{5/2} \left(7 b^2 \sqrt{\frac{c x^2}{b}+1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{b}\right)-7 b^2-2 b c x^2+5 c^2 x^4\right)}{45 c^2 \sqrt{x^2 \left(b+c x^2\right)}}","\frac{7 b^{9/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{15 c^{11/4} \sqrt{b x^2+c x^4}}-\frac{14 b^{9/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{15 c^{11/4} \sqrt{b x^2+c x^4}}+\frac{14 b^2 x^{3/2} \left(b+c x^2\right)}{15 c^{5/2} \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}-\frac{14 b \sqrt{x} \sqrt{b x^2+c x^4}}{45 c^2}+\frac{2 x^{5/2} \sqrt{b x^2+c x^4}}{9 c}",1,"(2*x^(5/2)*(-7*b^2 - 2*b*c*x^2 + 5*c^2*x^4 + 7*b^2*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[1/2, 3/4, 7/4, -((c*x^2)/b)]))/(45*c^2*Sqrt[x^2*(b + c*x^2)])","C",1
380,1,86,149,0.0315359,"\int \frac{x^{9/2}}{\sqrt{b x^2+c x^4}} \, dx","Integrate[x^(9/2)/Sqrt[b*x^2 + c*x^4],x]","\frac{2 x^{3/2} \left(5 b^2 \sqrt{\frac{c x^2}{b}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^2}{b}\right)-5 b^2-2 b c x^2+3 c^2 x^4\right)}{21 c^2 \sqrt{x^2 \left(b+c x^2\right)}}","\frac{5 b^{7/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{21 c^{9/4} \sqrt{b x^2+c x^4}}-\frac{10 b \sqrt{b x^2+c x^4}}{21 c^2 \sqrt{x}}+\frac{2 x^{3/2} \sqrt{b x^2+c x^4}}{7 c}",1,"(2*x^(3/2)*(-5*b^2 - 2*b*c*x^2 + 3*c^2*x^4 + 5*b^2*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^2)/b)]))/(21*c^2*Sqrt[x^2*(b + c*x^2)])","C",1
381,1,70,266,0.0258011,"\int \frac{x^{7/2}}{\sqrt{b x^2+c x^4}} \, dx","Integrate[x^(7/2)/Sqrt[b*x^2 + c*x^4],x]","\frac{2 x^{5/2} \left(-b \sqrt{\frac{c x^2}{b}+1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{b}\right)+b+c x^2\right)}{5 c \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{3 b^{5/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{5 c^{7/4} \sqrt{b x^2+c x^4}}+\frac{6 b^{5/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{5 c^{7/4} \sqrt{b x^2+c x^4}}-\frac{6 b x^{3/2} \left(b+c x^2\right)}{5 c^{3/2} \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}+\frac{2 \sqrt{x} \sqrt{b x^2+c x^4}}{5 c}",1,"(2*x^(5/2)*(b + c*x^2 - b*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[1/2, 3/4, 7/4, -((c*x^2)/b)]))/(5*c*Sqrt[x^2*(b + c*x^2)])","C",1
382,1,70,121,0.0256653,"\int \frac{x^{5/2}}{\sqrt{b x^2+c x^4}} \, dx","Integrate[x^(5/2)/Sqrt[b*x^2 + c*x^4],x]","\frac{2 x^{3/2} \left(-b \sqrt{\frac{c x^2}{b}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^2}{b}\right)+b+c x^2\right)}{3 c \sqrt{x^2 \left(b+c x^2\right)}}","\frac{2 \sqrt{b x^2+c x^4}}{3 c \sqrt{x}}-\frac{b^{3/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{3 c^{5/4} \sqrt{b x^2+c x^4}}",1,"(2*x^(3/2)*(b + c*x^2 - b*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^2)/b)]))/(3*c*Sqrt[x^2*(b + c*x^2)])","C",1
383,1,57,231,0.0121486,"\int \frac{x^{3/2}}{\sqrt{b x^2+c x^4}} \, dx","Integrate[x^(3/2)/Sqrt[b*x^2 + c*x^4],x]","\frac{2 x^{5/2} \sqrt{\frac{c x^2}{b}+1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{b}\right)}{3 \sqrt{x^2 \left(b+c x^2\right)}}","\frac{\sqrt[4]{b} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{c^{3/4} \sqrt{b x^2+c x^4}}-\frac{2 \sqrt[4]{b} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{c^{3/4} \sqrt{b x^2+c x^4}}+\frac{2 x^{3/2} \left(b+c x^2\right)}{\sqrt{c} \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}",1,"(2*x^(5/2)*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[1/2, 3/4, 7/4, -((c*x^2)/b)])/(3*Sqrt[x^2*(b + c*x^2)])","C",1
384,1,55,90,0.0127229,"\int \frac{\sqrt{x}}{\sqrt{b x^2+c x^4}} \, dx","Integrate[Sqrt[x]/Sqrt[b*x^2 + c*x^4],x]","\frac{2 x^{3/2} \sqrt{\frac{c x^2}{b}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^2}{b}\right)}{\sqrt{x^2 \left(b+c x^2\right)}}","\frac{x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{\sqrt[4]{b} \sqrt[4]{c} \sqrt{b x^2+c x^4}}",1,"(2*x^(3/2)*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^2)/b)])/Sqrt[x^2*(b + c*x^2)]","C",1
385,1,55,259,0.0150966,"\int \frac{1}{\sqrt{x} \sqrt{b x^2+c x^4}} \, dx","Integrate[1/(Sqrt[x]*Sqrt[b*x^2 + c*x^4]),x]","-\frac{2 \sqrt{x} \sqrt{\frac{c x^2}{b}+1} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{c x^2}{b}\right)}{\sqrt{x^2 \left(b+c x^2\right)}}","\frac{\sqrt[4]{c} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{b^{3/4} \sqrt{b x^2+c x^4}}-\frac{2 \sqrt[4]{c} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{b^{3/4} \sqrt{b x^2+c x^4}}+\frac{2 \sqrt{c} x^{3/2} \left(b+c x^2\right)}{b \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}-\frac{2 \sqrt{b x^2+c x^4}}{b x^{3/2}}",1,"(-2*Sqrt[x]*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[-1/4, 1/2, 3/4, -((c*x^2)/b)])/Sqrt[x^2*(b + c*x^2)]","C",1
386,1,57,121,0.0158452,"\int \frac{1}{x^{3/2} \sqrt{b x^2+c x^4}} \, dx","Integrate[1/(x^(3/2)*Sqrt[b*x^2 + c*x^4]),x]","-\frac{2 \sqrt{\frac{c x^2}{b}+1} \, _2F_1\left(-\frac{3}{4},\frac{1}{2};\frac{1}{4};-\frac{c x^2}{b}\right)}{3 \sqrt{x} \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{c^{3/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{3 b^{5/4} \sqrt{b x^2+c x^4}}-\frac{2 \sqrt{b x^2+c x^4}}{3 b x^{5/2}}",1,"(-2*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[-3/4, 1/2, 1/4, -((c*x^2)/b)])/(3*Sqrt[x]*Sqrt[x^2*(b + c*x^2)])","C",1
387,1,57,296,0.0139937,"\int \frac{1}{x^{5/2} \sqrt{b x^2+c x^4}} \, dx","Integrate[1/(x^(5/2)*Sqrt[b*x^2 + c*x^4]),x]","-\frac{2 \sqrt{\frac{c x^2}{b}+1} \, _2F_1\left(-\frac{5}{4},\frac{1}{2};-\frac{1}{4};-\frac{c x^2}{b}\right)}{5 x^{3/2} \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{3 c^{5/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{5 b^{7/4} \sqrt{b x^2+c x^4}}+\frac{6 c^{5/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{5 b^{7/4} \sqrt{b x^2+c x^4}}-\frac{6 c^{3/2} x^{3/2} \left(b+c x^2\right)}{5 b^2 \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}+\frac{6 c \sqrt{b x^2+c x^4}}{5 b^2 x^{3/2}}-\frac{2 \sqrt{b x^2+c x^4}}{5 b x^{7/2}}",1,"(-2*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[-5/4, 1/2, -1/4, -((c*x^2)/b)])/(5*x^(3/2)*Sqrt[x^2*(b + c*x^2)])","C",1
388,1,57,149,0.0139603,"\int \frac{1}{x^{7/2} \sqrt{b x^2+c x^4}} \, dx","Integrate[1/(x^(7/2)*Sqrt[b*x^2 + c*x^4]),x]","-\frac{2 \sqrt{\frac{c x^2}{b}+1} \, _2F_1\left(-\frac{7}{4},\frac{1}{2};-\frac{3}{4};-\frac{c x^2}{b}\right)}{7 x^{5/2} \sqrt{x^2 \left(b+c x^2\right)}}","\frac{5 c^{7/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{21 b^{9/4} \sqrt{b x^2+c x^4}}+\frac{10 c \sqrt{b x^2+c x^4}}{21 b^2 x^{5/2}}-\frac{2 \sqrt{b x^2+c x^4}}{7 b x^{9/2}}",1,"(-2*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[-7/4, 1/2, -3/4, -((c*x^2)/b)])/(7*x^(5/2)*Sqrt[x^2*(b + c*x^2)])","C",1
389,1,57,326,0.0146383,"\int \frac{1}{x^{9/2} \sqrt{b x^2+c x^4}} \, dx","Integrate[1/(x^(9/2)*Sqrt[b*x^2 + c*x^4]),x]","-\frac{2 \sqrt{\frac{c x^2}{b}+1} \, _2F_1\left(-\frac{9}{4},\frac{1}{2};-\frac{5}{4};-\frac{c x^2}{b}\right)}{9 x^{7/2} \sqrt{x^2 \left(b+c x^2\right)}}","\frac{7 c^{9/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{15 b^{11/4} \sqrt{b x^2+c x^4}}-\frac{14 c^{9/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{15 b^{11/4} \sqrt{b x^2+c x^4}}+\frac{14 c^{5/2} x^{3/2} \left(b+c x^2\right)}{15 b^3 \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}-\frac{14 c^2 \sqrt{b x^2+c x^4}}{15 b^3 x^{3/2}}+\frac{14 c \sqrt{b x^2+c x^4}}{45 b^2 x^{7/2}}-\frac{2 \sqrt{b x^2+c x^4}}{9 b x^{11/2}}",1,"(-2*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[-9/4, 1/2, -5/4, -((c*x^2)/b)])/(9*x^(7/2)*Sqrt[x^2*(b + c*x^2)])","C",1
390,1,57,179,0.0148062,"\int \frac{1}{x^{11/2} \sqrt{b x^2+c x^4}} \, dx","Integrate[1/(x^(11/2)*Sqrt[b*x^2 + c*x^4]),x]","-\frac{2 \sqrt{\frac{c x^2}{b}+1} \, _2F_1\left(-\frac{11}{4},\frac{1}{2};-\frac{7}{4};-\frac{c x^2}{b}\right)}{11 x^{9/2} \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{15 c^{11/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{77 b^{13/4} \sqrt{b x^2+c x^4}}-\frac{30 c^2 \sqrt{b x^2+c x^4}}{77 b^3 x^{5/2}}+\frac{18 c \sqrt{b x^2+c x^4}}{77 b^2 x^{9/2}}-\frac{2 \sqrt{b x^2+c x^4}}{11 b x^{13/2}}",1,"(-2*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[-11/4, 1/2, -7/4, -((c*x^2)/b)])/(11*x^(9/2)*Sqrt[x^2*(b + c*x^2)])","C",1
391,1,86,174,0.0320612,"\int \frac{x^{17/2}}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[x^(17/2)/(b*x^2 + c*x^4)^(3/2),x]","\frac{x^{3/2} \left(15 b^2 \sqrt{\frac{c x^2}{b}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^2}{b}\right)-15 b^2-6 b c x^2+2 c^2 x^4\right)}{7 c^3 \sqrt{x^2 \left(b+c x^2\right)}}","\frac{15 b^{7/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{14 c^{13/4} \sqrt{b x^2+c x^4}}-\frac{15 b \sqrt{b x^2+c x^4}}{7 c^3 \sqrt{x}}+\frac{9 x^{3/2} \sqrt{b x^2+c x^4}}{7 c^2}-\frac{x^{11/2}}{c \sqrt{b x^2+c x^4}}",1,"(x^(3/2)*(-15*b^2 - 6*b*c*x^2 + 2*c^2*x^4 + 15*b^2*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^2)/b)]))/(7*c^3*Sqrt[x^2*(b + c*x^2)])","C",1
392,1,72,291,0.0271659,"\int \frac{x^{15/2}}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[x^(15/2)/(b*x^2 + c*x^4)^(3/2),x]","\frac{2 x^{5/2} \left(7 b \sqrt{\frac{c x^2}{b}+1} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{c x^2}{b}\right)-7 b+c x^2\right)}{5 c^2 \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{21 b^{5/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{10 c^{11/4} \sqrt{b x^2+c x^4}}+\frac{21 b^{5/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{5 c^{11/4} \sqrt{b x^2+c x^4}}-\frac{21 b x^{3/2} \left(b+c x^2\right)}{5 c^{5/2} \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}+\frac{7 \sqrt{x} \sqrt{b x^2+c x^4}}{5 c^2}-\frac{x^{9/2}}{c \sqrt{b x^2+c x^4}}",1,"(2*x^(5/2)*(-7*b + c*x^2 + 7*b*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[3/4, 3/2, 7/4, -((c*x^2)/b)]))/(5*c^2*Sqrt[x^2*(b + c*x^2)])","C",1
393,1,73,146,0.0276911,"\int \frac{x^{13/2}}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[x^(13/2)/(b*x^2 + c*x^4)^(3/2),x]","\frac{x^{3/2} \left(-5 b \sqrt{\frac{c x^2}{b}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^2}{b}\right)+5 b+2 c x^2\right)}{3 c^2 \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{5 b^{3/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{6 c^{9/4} \sqrt{b x^2+c x^4}}+\frac{5 \sqrt{b x^2+c x^4}}{3 c^2 \sqrt{x}}-\frac{x^{7/2}}{c \sqrt{b x^2+c x^4}}",1,"(x^(3/2)*(5*b + 2*c*x^2 - 5*b*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^2)/b)]))/(3*c^2*Sqrt[x^2*(b + c*x^2)])","C",1
394,1,61,259,0.0235194,"\int \frac{x^{11/2}}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[x^(11/2)/(b*x^2 + c*x^4)^(3/2),x]","-\frac{2 x^{5/2} \left(\sqrt{\frac{c x^2}{b}+1} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{c x^2}{b}\right)-1\right)}{c \sqrt{x^2 \left(b+c x^2\right)}}","\frac{3 \sqrt[4]{b} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{2 c^{7/4} \sqrt{b x^2+c x^4}}-\frac{3 \sqrt[4]{b} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{c^{7/4} \sqrt{b x^2+c x^4}}+\frac{3 x^{3/2} \left(b+c x^2\right)}{c^{3/2} \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}-\frac{x^{5/2}}{c \sqrt{b x^2+c x^4}}",1,"(-2*x^(5/2)*(-1 + Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[3/4, 3/2, 7/4, -((c*x^2)/b)]))/(c*Sqrt[x^2*(b + c*x^2)])","C",1
395,1,60,119,0.0221644,"\int \frac{x^{9/2}}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[x^(9/2)/(b*x^2 + c*x^4)^(3/2),x]","\frac{x^{3/2} \left(\sqrt{\frac{c x^2}{b}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^2}{b}\right)-1\right)}{c \sqrt{x^2 \left(b+c x^2\right)}}","\frac{x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{b} c^{5/4} \sqrt{b x^2+c x^4}}-\frac{x^{3/2}}{c \sqrt{b x^2+c x^4}}",1,"(x^(3/2)*(-1 + Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^2)/b)]))/(c*Sqrt[x^2*(b + c*x^2)])","C",1
396,1,60,260,0.0150957,"\int \frac{x^{7/2}}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[x^(7/2)/(b*x^2 + c*x^4)^(3/2),x]","\frac{2 x^{5/2} \sqrt{\frac{c x^2}{b}+1} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{c x^2}{b}\right)}{3 b \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{2 b^{3/4} c^{3/4} \sqrt{b x^2+c x^4}}+\frac{x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{b^{3/4} c^{3/4} \sqrt{b x^2+c x^4}}+\frac{x^{5/2}}{b \sqrt{b x^2+c x^4}}-\frac{x^{3/2} \left(b+c x^2\right)}{b \sqrt{c} \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}",1,"(2*x^(5/2)*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[3/4, 3/2, 7/4, -((c*x^2)/b)])/(3*b*Sqrt[x^2*(b + c*x^2)])","C",1
397,1,60,118,0.0194546,"\int \frac{x^{5/2}}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[x^(5/2)/(b*x^2 + c*x^4)^(3/2),x]","\frac{x^{3/2} \left(\sqrt{\frac{c x^2}{b}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^2}{b}\right)+1\right)}{b \sqrt{x^2 \left(b+c x^2\right)}}","\frac{x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{2 b^{5/4} \sqrt[4]{c} \sqrt{b x^2+c x^4}}+\frac{x^{3/2}}{b \sqrt{b x^2+c x^4}}",1,"(x^(3/2)*(1 + Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^2)/b)]))/(b*Sqrt[x^2*(b + c*x^2)])","C",1
398,1,58,286,0.0167286,"\int \frac{x^{3/2}}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[x^(3/2)/(b*x^2 + c*x^4)^(3/2),x]","-\frac{2 \sqrt{x} \sqrt{\frac{c x^2}{b}+1} \, _2F_1\left(-\frac{1}{4},\frac{3}{2};\frac{3}{4};-\frac{c x^2}{b}\right)}{b \sqrt{x^2 \left(b+c x^2\right)}}","\frac{3 \sqrt[4]{c} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{2 b^{7/4} \sqrt{b x^2+c x^4}}-\frac{3 \sqrt[4]{c} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{b^{7/4} \sqrt{b x^2+c x^4}}+\frac{3 \sqrt{c} x^{3/2} \left(b+c x^2\right)}{b^2 \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}-\frac{3 \sqrt{b x^2+c x^4}}{b^2 x^{3/2}}+\frac{\sqrt{x}}{b \sqrt{b x^2+c x^4}}",1,"(-2*Sqrt[x]*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[-1/4, 3/2, 3/4, -((c*x^2)/b)])/(b*Sqrt[x^2*(b + c*x^2)])","C",1
399,1,60,145,0.017952,"\int \frac{\sqrt{x}}{\left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[Sqrt[x]/(b*x^2 + c*x^4)^(3/2),x]","-\frac{2 \sqrt{\frac{c x^2}{b}+1} \, _2F_1\left(-\frac{3}{4},\frac{3}{2};\frac{1}{4};-\frac{c x^2}{b}\right)}{3 b \sqrt{x} \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{5 c^{3/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{6 b^{9/4} \sqrt{b x^2+c x^4}}-\frac{5 \sqrt{b x^2+c x^4}}{3 b^2 x^{5/2}}+\frac{1}{b \sqrt{x} \sqrt{b x^2+c x^4}}",1,"(-2*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[-3/4, 3/2, 1/4, -((c*x^2)/b)])/(3*b*Sqrt[x]*Sqrt[x^2*(b + c*x^2)])","C",1
400,1,60,320,0.0162321,"\int \frac{1}{\sqrt{x} \left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[1/(Sqrt[x]*(b*x^2 + c*x^4)^(3/2)),x]","-\frac{2 \sqrt{\frac{c x^2}{b}+1} \, _2F_1\left(-\frac{5}{4},\frac{3}{2};-\frac{1}{4};-\frac{c x^2}{b}\right)}{5 b x^{3/2} \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{21 c^{5/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{10 b^{11/4} \sqrt{b x^2+c x^4}}+\frac{21 c^{5/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{5 b^{11/4} \sqrt{b x^2+c x^4}}-\frac{21 c^{3/2} x^{3/2} \left(b+c x^2\right)}{5 b^3 \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}+\frac{21 c \sqrt{b x^2+c x^4}}{5 b^3 x^{3/2}}-\frac{7 \sqrt{b x^2+c x^4}}{5 b^2 x^{7/2}}+\frac{1}{b x^{3/2} \sqrt{b x^2+c x^4}}",1,"(-2*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[-5/4, 3/2, -1/4, -((c*x^2)/b)])/(5*b*x^(3/2)*Sqrt[x^2*(b + c*x^2)])","C",1
401,1,60,173,0.0182214,"\int \frac{1}{x^{3/2} \left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[1/(x^(3/2)*(b*x^2 + c*x^4)^(3/2)),x]","-\frac{2 \sqrt{\frac{c x^2}{b}+1} \, _2F_1\left(-\frac{7}{4},\frac{3}{2};-\frac{3}{4};-\frac{c x^2}{b}\right)}{7 b x^{5/2} \sqrt{x^2 \left(b+c x^2\right)}}","\frac{15 c^{7/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{14 b^{13/4} \sqrt{b x^2+c x^4}}+\frac{15 c \sqrt{b x^2+c x^4}}{7 b^3 x^{5/2}}-\frac{9 \sqrt{b x^2+c x^4}}{7 b^2 x^{9/2}}+\frac{1}{b x^{5/2} \sqrt{b x^2+c x^4}}",1,"(-2*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[-7/4, 3/2, -3/4, -((c*x^2)/b)])/(7*b*x^(5/2)*Sqrt[x^2*(b + c*x^2)])","C",1
402,1,60,350,0.0165649,"\int \frac{1}{x^{5/2} \left(b x^2+c x^4\right)^{3/2}} \, dx","Integrate[1/(x^(5/2)*(b*x^2 + c*x^4)^(3/2)),x]","-\frac{2 \sqrt{\frac{c x^2}{b}+1} \, _2F_1\left(-\frac{9}{4},\frac{3}{2};-\frac{5}{4};-\frac{c x^2}{b}\right)}{9 b x^{7/2} \sqrt{x^2 \left(b+c x^2\right)}}","\frac{77 c^{9/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{30 b^{15/4} \sqrt{b x^2+c x^4}}-\frac{77 c^{9/4} x \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{\frac{b+c x^2}{\left(\sqrt{b}+\sqrt{c} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{b}}\right)|\frac{1}{2}\right)}{15 b^{15/4} \sqrt{b x^2+c x^4}}+\frac{77 c^{5/2} x^{3/2} \left(b+c x^2\right)}{15 b^4 \left(\sqrt{b}+\sqrt{c} x\right) \sqrt{b x^2+c x^4}}-\frac{77 c^2 \sqrt{b x^2+c x^4}}{15 b^4 x^{3/2}}+\frac{77 c \sqrt{b x^2+c x^4}}{45 b^3 x^{7/2}}-\frac{11 \sqrt{b x^2+c x^4}}{9 b^2 x^{11/2}}+\frac{1}{b x^{7/2} \sqrt{b x^2+c x^4}}",1,"(-2*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[-9/4, 3/2, -5/4, -((c*x^2)/b)])/(9*b*x^(7/2)*Sqrt[x^2*(b + c*x^2)])","C",1
403,1,59,73,0.039396,"\int (c x)^m \left(b x^2+c x^4\right)^3 \, dx","Integrate[(c*x)^m*(b*x^2 + c*x^4)^3,x]","x^7 (c x)^m \left(\frac{b^3}{m+7}+\frac{3 b^2 c x^2}{m+9}+\frac{3 b c^2 x^4}{m+11}+\frac{c^3 x^6}{m+13}\right)","\frac{b^3 x^7 (c x)^m}{m+7}+\frac{3 b^2 c x^9 (c x)^m}{m+9}+\frac{3 b c^2 x^{11} (c x)^m}{m+11}+\frac{c^3 x^{13} (c x)^m}{m+13}",1,"x^7*(c*x)^m*(b^3/(7 + m) + (3*b^2*c*x^2)/(9 + m) + (3*b*c^2*x^4)/(11 + m) + (c^3*x^6)/(13 + m))","A",1
404,1,43,52,0.0394686,"\int (c x)^m \left(b x^2+c x^4\right)^2 \, dx","Integrate[(c*x)^m*(b*x^2 + c*x^4)^2,x]","x^5 (c x)^m \left(\frac{b^2}{m+5}+\frac{2 b c x^2}{m+7}+\frac{c^2 x^4}{m+9}\right)","\frac{b^2 x^5 (c x)^m}{m+5}+\frac{2 b c x^7 (c x)^m}{m+7}+\frac{c^2 x^9 (c x)^m}{m+9}",1,"x^5*(c*x)^m*(b^2/(5 + m) + (2*b*c*x^2)/(7 + m) + (c^2*x^4)/(9 + m))","A",1
405,1,27,34,0.0156767,"\int (c x)^m \left(b x^2+c x^4\right) \, dx","Integrate[(c*x)^m*(b*x^2 + c*x^4),x]","x^3 (c x)^m \left(\frac{b}{m+3}+\frac{c x^2}{m+5}\right)","\frac{b (c x)^{m+3}}{c^3 (m+3)}+\frac{(c x)^{m+5}}{c^4 (m+5)}",1,"x^3*(c*x)^m*(b/(3 + m) + (c*x^2)/(5 + m))","A",1
406,1,42,45,0.0112191,"\int \frac{(c x)^m}{b x^2+c x^4} \, dx","Integrate[(c*x)^m/(b*x^2 + c*x^4),x]","\frac{(c x)^m \, _2F_1\left(1,\frac{m-1}{2};\frac{m+1}{2};-\frac{c x^2}{b}\right)}{b (m-1) x}","-\frac{(c x)^m \, _2F_1\left(1,\frac{m-1}{2};\frac{m+1}{2};-\frac{c x^2}{b}\right)}{b (1-m) x}",1,"((c*x)^m*Hypergeometric2F1[1, (-1 + m)/2, (1 + m)/2, -((c*x^2)/b)])/(b*(-1 + m)*x)","A",1
407,1,44,45,0.0114854,"\int \frac{(c x)^m}{\left(b x^2+c x^4\right)^2} \, dx","Integrate[(c*x)^m/(b*x^2 + c*x^4)^2,x]","\frac{(c x)^m \, _2F_1\left(2,\frac{m-3}{2};\frac{m-3}{2}+1;-\frac{c x^2}{b}\right)}{b^2 (m-3) x^3}","-\frac{(c x)^m \, _2F_1\left(2,\frac{m-3}{2};\frac{m-1}{2};-\frac{c x^2}{b}\right)}{b^2 (3-m) x^3}",1,"((c*x)^m*Hypergeometric2F1[2, (-3 + m)/2, 1 + (-3 + m)/2, -((c*x^2)/b)])/(b^2*(-3 + m)*x^3)","A",1
408,1,44,45,0.0123297,"\int \frac{(c x)^m}{\left(b x^2+c x^4\right)^3} \, dx","Integrate[(c*x)^m/(b*x^2 + c*x^4)^3,x]","\frac{(c x)^m \, _2F_1\left(3,\frac{m-5}{2};\frac{m-5}{2}+1;-\frac{c x^2}{b}\right)}{b^3 (m-5) x^5}","-\frac{(c x)^m \, _2F_1\left(3,\frac{m-5}{2};\frac{m-3}{2};-\frac{c x^2}{b}\right)}{b^3 (5-m) x^5}",1,"((c*x)^m*Hypergeometric2F1[3, (-5 + m)/2, 1 + (-5 + m)/2, -((c*x^2)/b)])/(b^3*(-5 + m)*x^5)","A",1
409,1,30,30,0.0015246,"\int x^3 \left(a^2+2 a b x^2+b^2 x^4\right) \, dx","Integrate[x^3*(a^2 + 2*a*b*x^2 + b^2*x^4),x]","\frac{a^2 x^4}{4}+\frac{1}{3} a b x^6+\frac{b^2 x^8}{8}","\frac{a^2 x^4}{4}+\frac{1}{3} a b x^6+\frac{b^2 x^8}{8}",1,"(a^2*x^4)/4 + (a*b*x^6)/3 + (b^2*x^8)/8","A",1
410,1,30,30,0.0009957,"\int x^2 \left(a^2+2 a b x^2+b^2 x^4\right) \, dx","Integrate[x^2*(a^2 + 2*a*b*x^2 + b^2*x^4),x]","\frac{a^2 x^3}{3}+\frac{2}{5} a b x^5+\frac{b^2 x^7}{7}","\frac{a^2 x^3}{3}+\frac{2}{5} a b x^5+\frac{b^2 x^7}{7}",1,"(a^2*x^3)/3 + (2*a*b*x^5)/5 + (b^2*x^7)/7","A",1
411,1,16,30,0.0020742,"\int x \left(a^2+2 a b x^2+b^2 x^4\right) \, dx","Integrate[x*(a^2 + 2*a*b*x^2 + b^2*x^4),x]","\frac{\left(a+b x^2\right)^3}{6 b}","\frac{a^2 x^2}{2}+\frac{1}{2} a b x^4+\frac{b^2 x^6}{6}",1,"(a + b*x^2)^3/(6*b)","A",1
412,1,25,25,0.0001055,"\int \left(a^2+2 a b x^2+b^2 x^4\right) \, dx","Integrate[a^2 + 2*a*b*x^2 + b^2*x^4,x]","a^2 x+\frac{2}{3} a b x^3+\frac{b^2 x^5}{5}","a^2 x+\frac{2}{3} a b x^3+\frac{b^2 x^5}{5}",1,"a^2*x + (2*a*b*x^3)/3 + (b^2*x^5)/5","A",1
413,1,23,23,0.0012803,"\int \frac{a^2+2 a b x^2+b^2 x^4}{x} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)/x,x]","a^2 \log (x)+a b x^2+\frac{b^2 x^4}{4}","a^2 \log (x)+a b x^2+\frac{b^2 x^4}{4}",1,"a*b*x^2 + (b^2*x^4)/4 + a^2*Log[x]","A",1
414,1,24,24,0.0010986,"\int \frac{a^2+2 a b x^2+b^2 x^4}{x^2} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)/x^2,x]","-\frac{a^2}{x}+2 a b x+\frac{b^2 x^3}{3}","-\frac{a^2}{x}+2 a b x+\frac{b^2 x^3}{3}",1,"-(a^2/x) + 2*a*b*x + (b^2*x^3)/3","A",1
415,1,27,27,0.0014683,"\int \frac{a^2+2 a b x^2+b^2 x^4}{x^3} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)/x^3,x]","-\frac{a^2}{2 x^2}+2 a b \log (x)+\frac{b^2 x^2}{2}","-\frac{a^2}{2 x^2}+2 a b \log (x)+\frac{b^2 x^2}{2}",1,"-1/2*a^2/x^2 + (b^2*x^2)/2 + 2*a*b*Log[x]","A",1
416,1,23,23,0.001286,"\int \frac{a^2+2 a b x^2+b^2 x^4}{x^4} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)/x^4,x]","-\frac{a^2}{3 x^3}-\frac{2 a b}{x}+b^2 x","-\frac{a^2}{3 x^3}-\frac{2 a b}{x}+b^2 x",1,"-1/3*a^2/x^3 - (2*a*b)/x + b^2*x","A",1
417,1,24,24,0.00092,"\int \frac{a^2+2 a b x^2+b^2 x^4}{x^5} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)/x^5,x]","-\frac{a^2}{4 x^4}-\frac{a b}{x^2}+b^2 \log (x)","-\frac{a^2}{4 x^4}-\frac{a b}{x^2}+b^2 \log (x)",1,"-1/4*a^2/x^4 - (a*b)/x^2 + b^2*Log[x]","A",1
418,1,28,28,0.0008861,"\int \frac{a^2+2 a b x^2+b^2 x^4}{x^6} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)/x^6,x]","-\frac{a^2}{5 x^5}-\frac{2 a b}{3 x^3}-\frac{b^2}{x}","-\frac{a^2}{5 x^5}-\frac{2 a b}{3 x^3}-\frac{b^2}{x}",1,"-1/5*a^2/x^5 - (2*a*b)/(3*x^3) - b^2/x","A",1
419,1,30,30,0.001015,"\int \frac{a^2+2 a b x^2+b^2 x^4}{x^7} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)/x^7,x]","-\frac{a^2}{6 x^6}-\frac{a b}{2 x^4}-\frac{b^2}{2 x^2}","-\frac{a^2}{6 x^6}-\frac{a b}{2 x^4}-\frac{b^2}{2 x^2}",1,"-1/6*a^2/x^6 - (a*b)/(2*x^4) - b^2/(2*x^2)","A",1
420,1,30,30,0.0009496,"\int \frac{a^2+2 a b x^2+b^2 x^4}{x^8} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)/x^8,x]","-\frac{a^2}{7 x^7}-\frac{2 a b}{5 x^5}-\frac{b^2}{3 x^3}","-\frac{a^2}{7 x^7}-\frac{2 a b}{5 x^5}-\frac{b^2}{3 x^3}",1,"-1/7*a^2/x^7 - (2*a*b)/(5*x^5) - b^2/(3*x^3)","A",1
421,1,56,56,0.0025372,"\int x^6 \left(a^2+2 a b x^2+b^2 x^4\right)^2 \, dx","Integrate[x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","\frac{a^4 x^7}{7}+\frac{4}{9} a^3 b x^9+\frac{6}{11} a^2 b^2 x^{11}+\frac{4}{13} a b^3 x^{13}+\frac{b^4 x^{15}}{15}","\frac{a^4 x^7}{7}+\frac{4}{9} a^3 b x^9+\frac{6}{11} a^2 b^2 x^{11}+\frac{4}{13} a b^3 x^{13}+\frac{b^4 x^{15}}{15}",1,"(a^4*x^7)/7 + (4*a^3*b*x^9)/9 + (6*a^2*b^2*x^11)/11 + (4*a*b^3*x^13)/13 + (b^4*x^15)/15","A",1
422,1,56,53,0.0022607,"\int x^5 \left(a^2+2 a b x^2+b^2 x^4\right)^2 \, dx","Integrate[x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","\frac{a^4 x^6}{6}+\frac{1}{2} a^3 b x^8+\frac{3}{5} a^2 b^2 x^{10}+\frac{1}{3} a b^3 x^{12}+\frac{b^4 x^{14}}{14}","\frac{a^2 \left(a+b x^2\right)^5}{10 b^3}+\frac{\left(a+b x^2\right)^7}{14 b^3}-\frac{a \left(a+b x^2\right)^6}{6 b^3}",1,"(a^4*x^6)/6 + (a^3*b*x^8)/2 + (3*a^2*b^2*x^10)/5 + (a*b^3*x^12)/3 + (b^4*x^14)/14","A",1
423,1,56,56,0.0021823,"\int x^4 \left(a^2+2 a b x^2+b^2 x^4\right)^2 \, dx","Integrate[x^4*(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","\frac{a^4 x^5}{5}+\frac{4}{7} a^3 b x^7+\frac{2}{3} a^2 b^2 x^9+\frac{4}{11} a b^3 x^{11}+\frac{b^4 x^{13}}{13}","\frac{a^4 x^5}{5}+\frac{4}{7} a^3 b x^7+\frac{2}{3} a^2 b^2 x^9+\frac{4}{11} a b^3 x^{11}+\frac{b^4 x^{13}}{13}",1,"(a^4*x^5)/5 + (4*a^3*b*x^7)/7 + (2*a^2*b^2*x^9)/3 + (4*a*b^3*x^11)/11 + (b^4*x^13)/13","A",1
424,1,56,34,0.0021042,"\int x^3 \left(a^2+2 a b x^2+b^2 x^4\right)^2 \, dx","Integrate[x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","\frac{a^4 x^4}{4}+\frac{2}{3} a^3 b x^6+\frac{3}{4} a^2 b^2 x^8+\frac{2}{5} a b^3 x^{10}+\frac{b^4 x^{12}}{12}","\frac{\left(a+b x^2\right)^6}{12 b^2}-\frac{a \left(a+b x^2\right)^5}{10 b^2}",1,"(a^4*x^4)/4 + (2*a^3*b*x^6)/3 + (3*a^2*b^2*x^8)/4 + (2*a*b^3*x^10)/5 + (b^4*x^12)/12","A",1
425,1,56,56,0.0023015,"\int x^2 \left(a^2+2 a b x^2+b^2 x^4\right)^2 \, dx","Integrate[x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","\frac{a^4 x^3}{3}+\frac{4}{5} a^3 b x^5+\frac{6}{7} a^2 b^2 x^7+\frac{4}{9} a b^3 x^9+\frac{b^4 x^{11}}{11}","\frac{a^4 x^3}{3}+\frac{4}{5} a^3 b x^5+\frac{6}{7} a^2 b^2 x^7+\frac{4}{9} a b^3 x^9+\frac{b^4 x^{11}}{11}",1,"(a^4*x^3)/3 + (4*a^3*b*x^5)/5 + (6*a^2*b^2*x^7)/7 + (4*a*b^3*x^9)/9 + (b^4*x^11)/11","A",1
426,1,16,16,0.0024138,"\int x \left(a^2+2 a b x^2+b^2 x^4\right)^2 \, dx","Integrate[x*(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","\frac{\left(a+b x^2\right)^5}{10 b}","\frac{\left(a+b x^2\right)^5}{10 b}",1,"(a + b*x^2)^5/(10*b)","A",1
427,1,51,51,0.0012231,"\int \left(a^2+2 a b x^2+b^2 x^4\right)^2 \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","a^4 x+\frac{4}{3} a^3 b x^3+\frac{6}{5} a^2 b^2 x^5+\frac{4}{7} a b^3 x^7+\frac{b^4 x^9}{9}","a^4 x+\frac{4}{3} a^3 b x^3+\frac{6}{5} a^2 b^2 x^5+\frac{4}{7} a b^3 x^7+\frac{b^4 x^9}{9}",1,"a^4*x + (4*a^3*b*x^3)/3 + (6*a^2*b^2*x^5)/5 + (4*a*b^3*x^7)/7 + (b^4*x^9)/9","A",1
428,1,50,50,0.0039216,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^2}{x} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x,x]","a^4 \log (x)+2 a^3 b x^2+\frac{3}{2} a^2 b^2 x^4+\frac{2}{3} a b^3 x^6+\frac{b^4 x^8}{8}","a^4 \log (x)+2 a^3 b x^2+\frac{3}{2} a^2 b^2 x^4+\frac{2}{3} a b^3 x^6+\frac{b^4 x^8}{8}",1,"2*a^3*b*x^2 + (3*a^2*b^2*x^4)/2 + (2*a*b^3*x^6)/3 + (b^4*x^8)/8 + a^4*Log[x]","A",1
429,1,48,48,0.0073355,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^2}{x^2} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^2,x]","-\frac{a^4}{x}+4 a^3 b x+2 a^2 b^2 x^3+\frac{4}{5} a b^3 x^5+\frac{b^4 x^7}{7}","-\frac{a^4}{x}+4 a^3 b x+2 a^2 b^2 x^3+\frac{4}{5} a b^3 x^5+\frac{b^4 x^7}{7}",1,"-(a^4/x) + 4*a^3*b*x + 2*a^2*b^2*x^3 + (4*a*b^3*x^5)/5 + (b^4*x^7)/7","A",1
430,1,48,48,0.0045131,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^2}{x^3} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^3,x]","-\frac{a^4}{2 x^2}+4 a^3 b \log (x)+3 a^2 b^2 x^2+a b^3 x^4+\frac{b^4 x^6}{6}","-\frac{a^4}{2 x^2}+4 a^3 b \log (x)+3 a^2 b^2 x^2+a b^3 x^4+\frac{b^4 x^6}{6}",1,"-1/2*a^4/x^2 + 3*a^2*b^2*x^2 + a*b^3*x^4 + (b^4*x^6)/6 + 4*a^3*b*Log[x]","A",1
431,1,50,50,0.0062718,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^2}{x^4} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^4,x]","-\frac{a^4}{3 x^3}-\frac{4 a^3 b}{x}+6 a^2 b^2 x+\frac{4}{3} a b^3 x^3+\frac{b^4 x^5}{5}","-\frac{a^4}{3 x^3}-\frac{4 a^3 b}{x}+6 a^2 b^2 x+\frac{4}{3} a b^3 x^3+\frac{b^4 x^5}{5}",1,"-1/3*a^4/x^3 - (4*a^3*b)/x + 6*a^2*b^2*x + (4*a*b^3*x^3)/3 + (b^4*x^5)/5","A",1
432,1,49,49,0.0046092,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^2}{x^5} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^5,x]","-\frac{a^4}{4 x^4}-\frac{2 a^3 b}{x^2}+6 a^2 b^2 \log (x)+2 a b^3 x^2+\frac{b^4 x^4}{4}","-\frac{a^4}{4 x^4}-\frac{2 a^3 b}{x^2}+6 a^2 b^2 \log (x)+2 a b^3 x^2+\frac{b^4 x^4}{4}",1,"-1/4*a^4/x^4 - (2*a^3*b)/x^2 + 2*a*b^3*x^2 + (b^4*x^4)/4 + 6*a^2*b^2*Log[x]","A",1
433,1,50,50,0.00945,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^2}{x^6} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^6,x]","-\frac{a^4}{5 x^5}-\frac{4 a^3 b}{3 x^3}-\frac{6 a^2 b^2}{x}+4 a b^3 x+\frac{b^4 x^3}{3}","-\frac{a^4}{5 x^5}-\frac{4 a^3 b}{3 x^3}-\frac{6 a^2 b^2}{x}+4 a b^3 x+\frac{b^4 x^3}{3}",1,"-1/5*a^4/x^5 - (4*a^3*b)/(3*x^3) - (6*a^2*b^2)/x + 4*a*b^3*x + (b^4*x^3)/3","A",1
434,1,49,49,0.0050749,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^2}{x^7} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^7,x]","-\frac{a^4}{6 x^6}-\frac{a^3 b}{x^4}-\frac{3 a^2 b^2}{x^2}+4 a b^3 \log (x)+\frac{b^4 x^2}{2}","-\frac{a^4}{6 x^6}-\frac{a^3 b}{x^4}-\frac{3 a^2 b^2}{x^2}+4 a b^3 \log (x)+\frac{b^4 x^2}{2}",1,"-1/6*a^4/x^6 - (a^3*b)/x^4 - (3*a^2*b^2)/x^2 + (b^4*x^2)/2 + 4*a*b^3*Log[x]","A",1
435,1,47,47,0.0056971,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^2}{x^8} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^8,x]","-\frac{a^4}{7 x^7}-\frac{4 a^3 b}{5 x^5}-\frac{2 a^2 b^2}{x^3}-\frac{4 a b^3}{x}+b^4 x","-\frac{a^4}{7 x^7}-\frac{4 a^3 b}{5 x^5}-\frac{2 a^2 b^2}{x^3}-\frac{4 a b^3}{x}+b^4 x",1,"-1/7*a^4/x^7 - (4*a^3*b)/(5*x^5) - (2*a^2*b^2)/x^3 - (4*a*b^3)/x + b^4*x","A",1
436,1,50,50,0.0046053,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^2}{x^9} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^9,x]","-\frac{a^4}{8 x^8}-\frac{2 a^3 b}{3 x^6}-\frac{3 a^2 b^2}{2 x^4}-\frac{2 a b^3}{x^2}+b^4 \log (x)","-\frac{a^4}{8 x^8}-\frac{2 a^3 b}{3 x^6}-\frac{3 a^2 b^2}{2 x^4}-\frac{2 a b^3}{x^2}+b^4 \log (x)",1,"-1/8*a^4/x^8 - (2*a^3*b)/(3*x^6) - (3*a^2*b^2)/(2*x^4) - (2*a*b^3)/x^2 + b^4*Log[x]","A",1
437,1,54,54,0.0082427,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^2}{x^{10}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^10,x]","-\frac{a^4}{9 x^9}-\frac{4 a^3 b}{7 x^7}-\frac{6 a^2 b^2}{5 x^5}-\frac{4 a b^3}{3 x^3}-\frac{b^4}{x}","-\frac{a^4}{9 x^9}-\frac{4 a^3 b}{7 x^7}-\frac{6 a^2 b^2}{5 x^5}-\frac{4 a b^3}{3 x^3}-\frac{b^4}{x}",1,"-1/9*a^4/x^9 - (4*a^3*b)/(7*x^7) - (6*a^2*b^2)/(5*x^5) - (4*a*b^3)/(3*x^3) - b^4/x","A",1
438,1,52,19,0.0045472,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^2}{x^{11}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^11,x]","-\frac{a^4}{10 x^{10}}-\frac{a^3 b}{2 x^8}-\frac{a^2 b^2}{x^6}-\frac{a b^3}{x^4}-\frac{b^4}{2 x^2}","-\frac{\left(a+b x^2\right)^5}{10 a x^{10}}",1,"-1/10*a^4/x^10 - (a^3*b)/(2*x^8) - (a^2*b^2)/x^6 - (a*b^3)/x^4 - b^4/(2*x^2)","B",1
439,1,56,56,0.006954,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^2}{x^{12}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^12,x]","-\frac{a^4}{11 x^{11}}-\frac{4 a^3 b}{9 x^9}-\frac{6 a^2 b^2}{7 x^7}-\frac{4 a b^3}{5 x^5}-\frac{b^4}{3 x^3}","-\frac{a^4}{11 x^{11}}-\frac{4 a^3 b}{9 x^9}-\frac{6 a^2 b^2}{7 x^7}-\frac{4 a b^3}{5 x^5}-\frac{b^4}{3 x^3}",1,"-1/11*a^4/x^11 - (4*a^3*b)/(9*x^9) - (6*a^2*b^2)/(7*x^7) - (4*a*b^3)/(5*x^5) - b^4/(3*x^3)","A",1
440,1,56,40,0.0043156,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^2}{x^{13}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^13,x]","-\frac{a^4}{12 x^{12}}-\frac{2 a^3 b}{5 x^{10}}-\frac{3 a^2 b^2}{4 x^8}-\frac{2 a b^3}{3 x^6}-\frac{b^4}{4 x^4}","\frac{b \left(a+b x^2\right)^5}{60 a^2 x^{10}}-\frac{\left(a+b x^2\right)^5}{12 a x^{12}}",1,"-1/12*a^4/x^12 - (2*a^3*b)/(5*x^10) - (3*a^2*b^2)/(4*x^8) - (2*a*b^3)/(3*x^6) - b^4/(4*x^4)","A",1
441,1,56,56,0.008543,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^2}{x^{14}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^14,x]","-\frac{a^4}{13 x^{13}}-\frac{4 a^3 b}{11 x^{11}}-\frac{2 a^2 b^2}{3 x^9}-\frac{4 a b^3}{7 x^7}-\frac{b^4}{5 x^5}","-\frac{a^4}{13 x^{13}}-\frac{4 a^3 b}{11 x^{11}}-\frac{2 a^2 b^2}{3 x^9}-\frac{4 a b^3}{7 x^7}-\frac{b^4}{5 x^5}",1,"-1/13*a^4/x^13 - (4*a^3*b)/(11*x^11) - (2*a^2*b^2)/(3*x^9) - (4*a*b^3)/(7*x^7) - b^4/(5*x^5)","A",1
442,1,56,56,0.0043512,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^2}{x^{15}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^15,x]","-\frac{a^4}{14 x^{14}}-\frac{a^3 b}{3 x^{12}}-\frac{3 a^2 b^2}{5 x^{10}}-\frac{a b^3}{2 x^8}-\frac{b^4}{6 x^6}","-\frac{a^4}{14 x^{14}}-\frac{a^3 b}{3 x^{12}}-\frac{3 a^2 b^2}{5 x^{10}}-\frac{a b^3}{2 x^8}-\frac{b^4}{6 x^6}",1,"-1/14*a^4/x^14 - (a^3*b)/(3*x^12) - (3*a^2*b^2)/(5*x^10) - (a*b^3)/(2*x^8) - b^4/(6*x^6)","A",1
443,1,56,56,0.0070786,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^2}{x^{16}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^16,x]","-\frac{a^4}{15 x^{15}}-\frac{4 a^3 b}{13 x^{13}}-\frac{6 a^2 b^2}{11 x^{11}}-\frac{4 a b^3}{9 x^9}-\frac{b^4}{7 x^7}","-\frac{a^4}{15 x^{15}}-\frac{4 a^3 b}{13 x^{13}}-\frac{6 a^2 b^2}{11 x^{11}}-\frac{4 a b^3}{9 x^9}-\frac{b^4}{7 x^7}",1,"-1/15*a^4/x^15 - (4*a^3*b)/(13*x^13) - (6*a^2*b^2)/(11*x^11) - (4*a*b^3)/(9*x^9) - b^4/(7*x^7)","A",1
444,1,82,82,0.0031523,"\int x^8 \left(a^2+2 a b x^2+b^2 x^4\right)^3 \, dx","Integrate[x^8*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{a^6 x^9}{9}+\frac{6}{11} a^5 b x^{11}+\frac{15}{13} a^4 b^2 x^{13}+\frac{4}{3} a^3 b^3 x^{15}+\frac{15}{17} a^2 b^4 x^{17}+\frac{6}{19} a b^5 x^{19}+\frac{b^6 x^{21}}{21}","\frac{a^6 x^9}{9}+\frac{6}{11} a^5 b x^{11}+\frac{15}{13} a^4 b^2 x^{13}+\frac{4}{3} a^3 b^3 x^{15}+\frac{15}{17} a^2 b^4 x^{17}+\frac{6}{19} a b^5 x^{19}+\frac{b^6 x^{21}}{21}",1,"(a^6*x^9)/9 + (6*a^5*b*x^11)/11 + (15*a^4*b^2*x^13)/13 + (4*a^3*b^3*x^15)/3 + (15*a^2*b^4*x^17)/17 + (6*a*b^5*x^19)/19 + (b^6*x^21)/21","A",1
445,1,82,72,0.0026666,"\int x^7 \left(a^2+2 a b x^2+b^2 x^4\right)^3 \, dx","Integrate[x^7*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{a^6 x^8}{8}+\frac{3}{5} a^5 b x^{10}+\frac{5}{4} a^4 b^2 x^{12}+\frac{10}{7} a^3 b^3 x^{14}+\frac{15}{16} a^2 b^4 x^{16}+\frac{1}{3} a b^5 x^{18}+\frac{b^6 x^{20}}{20}","-\frac{a^3 \left(a+b x^2\right)^7}{14 b^4}+\frac{3 a^2 \left(a+b x^2\right)^8}{16 b^4}+\frac{\left(a+b x^2\right)^{10}}{20 b^4}-\frac{a \left(a+b x^2\right)^9}{6 b^4}",1,"(a^6*x^8)/8 + (3*a^5*b*x^10)/5 + (5*a^4*b^2*x^12)/4 + (10*a^3*b^3*x^14)/7 + (15*a^2*b^4*x^16)/16 + (a*b^5*x^18)/3 + (b^6*x^20)/20","A",1
446,1,79,79,0.0027887,"\int x^6 \left(a^2+2 a b x^2+b^2 x^4\right)^3 \, dx","Integrate[x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{a^6 x^7}{7}+\frac{2}{3} a^5 b x^9+\frac{15}{11} a^4 b^2 x^{11}+\frac{20}{13} a^3 b^3 x^{13}+a^2 b^4 x^{15}+\frac{6}{17} a b^5 x^{17}+\frac{b^6 x^{19}}{19}","\frac{a^6 x^7}{7}+\frac{2}{3} a^5 b x^9+\frac{15}{11} a^4 b^2 x^{11}+\frac{20}{13} a^3 b^3 x^{13}+a^2 b^4 x^{15}+\frac{6}{17} a b^5 x^{17}+\frac{b^6 x^{19}}{19}",1,"(a^6*x^7)/7 + (2*a^5*b*x^9)/3 + (15*a^4*b^2*x^11)/11 + (20*a^3*b^3*x^13)/13 + a^2*b^4*x^15 + (6*a*b^5*x^17)/17 + (b^6*x^19)/19","A",1
447,1,82,53,0.0025589,"\int x^5 \left(a^2+2 a b x^2+b^2 x^4\right)^3 \, dx","Integrate[x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{a^6 x^6}{6}+\frac{3}{4} a^5 b x^8+\frac{3}{2} a^4 b^2 x^{10}+\frac{5}{3} a^3 b^3 x^{12}+\frac{15}{14} a^2 b^4 x^{14}+\frac{3}{8} a b^5 x^{16}+\frac{b^6 x^{18}}{18}","\frac{a^2 \left(a+b x^2\right)^7}{14 b^3}+\frac{\left(a+b x^2\right)^9}{18 b^3}-\frac{a \left(a+b x^2\right)^8}{8 b^3}",1,"(a^6*x^6)/6 + (3*a^5*b*x^8)/4 + (3*a^4*b^2*x^10)/2 + (5*a^3*b^3*x^12)/3 + (15*a^2*b^4*x^14)/14 + (3*a*b^5*x^16)/8 + (b^6*x^18)/18","A",1
448,1,82,82,0.0025562,"\int x^4 \left(a^2+2 a b x^2+b^2 x^4\right)^3 \, dx","Integrate[x^4*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{a^6 x^5}{5}+\frac{6}{7} a^5 b x^7+\frac{5}{3} a^4 b^2 x^9+\frac{20}{11} a^3 b^3 x^{11}+\frac{15}{13} a^2 b^4 x^{13}+\frac{2}{5} a b^5 x^{15}+\frac{b^6 x^{17}}{17}","\frac{a^6 x^5}{5}+\frac{6}{7} a^5 b x^7+\frac{5}{3} a^4 b^2 x^9+\frac{20}{11} a^3 b^3 x^{11}+\frac{15}{13} a^2 b^4 x^{13}+\frac{2}{5} a b^5 x^{15}+\frac{b^6 x^{17}}{17}",1,"(a^6*x^5)/5 + (6*a^5*b*x^7)/7 + (5*a^4*b^2*x^9)/3 + (20*a^3*b^3*x^11)/11 + (15*a^2*b^4*x^13)/13 + (2*a*b^5*x^15)/5 + (b^6*x^17)/17","A",1
449,1,77,34,0.0028272,"\int x^3 \left(a^2+2 a b x^2+b^2 x^4\right)^3 \, dx","Integrate[x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{a^6 x^4}{4}+a^5 b x^6+\frac{15}{8} a^4 b^2 x^8+2 a^3 b^3 x^{10}+\frac{5}{4} a^2 b^4 x^{12}+\frac{3}{7} a b^5 x^{14}+\frac{b^6 x^{16}}{16}","\frac{\left(a+b x^2\right)^8}{16 b^2}-\frac{a \left(a+b x^2\right)^7}{14 b^2}",1,"(a^6*x^4)/4 + a^5*b*x^6 + (15*a^4*b^2*x^8)/8 + 2*a^3*b^3*x^10 + (5*a^2*b^4*x^12)/4 + (3*a*b^5*x^14)/7 + (b^6*x^16)/16","B",1
450,1,82,82,0.0028156,"\int x^2 \left(a^2+2 a b x^2+b^2 x^4\right)^3 \, dx","Integrate[x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{a^6 x^3}{3}+\frac{6}{5} a^5 b x^5+\frac{15}{7} a^4 b^2 x^7+\frac{20}{9} a^3 b^3 x^9+\frac{15}{11} a^2 b^4 x^{11}+\frac{6}{13} a b^5 x^{13}+\frac{b^6 x^{15}}{15}","\frac{a^6 x^3}{3}+\frac{6}{5} a^5 b x^5+\frac{15}{7} a^4 b^2 x^7+\frac{20}{9} a^3 b^3 x^9+\frac{15}{11} a^2 b^4 x^{11}+\frac{6}{13} a b^5 x^{13}+\frac{b^6 x^{15}}{15}",1,"(a^6*x^3)/3 + (6*a^5*b*x^5)/5 + (15*a^4*b^2*x^7)/7 + (20*a^3*b^3*x^9)/9 + (15*a^2*b^4*x^11)/11 + (6*a*b^5*x^13)/13 + (b^6*x^15)/15","A",1
451,1,16,16,0.0022073,"\int x \left(a^2+2 a b x^2+b^2 x^4\right)^3 \, dx","Integrate[x*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{\left(a+b x^2\right)^7}{14 b}","\frac{\left(a+b x^2\right)^7}{14 b}",1,"(a + b*x^2)^7/(14*b)","A",1
452,1,73,73,0.0013502,"\int \left(a^2+2 a b x^2+b^2 x^4\right)^3 \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","a^6 x+2 a^5 b x^3+3 a^4 b^2 x^5+\frac{20}{7} a^3 b^3 x^7+\frac{5}{3} a^2 b^4 x^9+\frac{6}{11} a b^5 x^{11}+\frac{b^6 x^{13}}{13}","a^6 x+2 a^5 b x^3+3 a^4 b^2 x^5+\frac{20}{7} a^3 b^3 x^7+\frac{5}{3} a^2 b^4 x^9+\frac{6}{11} a b^5 x^{11}+\frac{b^6 x^{13}}{13}",1,"a^6*x + 2*a^5*b*x^3 + 3*a^4*b^2*x^5 + (20*a^3*b^3*x^7)/7 + (5*a^2*b^4*x^9)/3 + (6*a*b^5*x^11)/11 + (b^6*x^13)/13","A",1
453,1,76,76,0.0043927,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^3}{x} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x,x]","a^6 \log (x)+3 a^5 b x^2+\frac{15}{4} a^4 b^2 x^4+\frac{10}{3} a^3 b^3 x^6+\frac{15}{8} a^2 b^4 x^8+\frac{3}{5} a b^5 x^{10}+\frac{b^6 x^{12}}{12}","a^6 \log (x)+3 a^5 b x^2+\frac{15}{4} a^4 b^2 x^4+\frac{10}{3} a^3 b^3 x^6+\frac{15}{8} a^2 b^4 x^8+\frac{3}{5} a b^5 x^{10}+\frac{b^6 x^{12}}{12}",1,"3*a^5*b*x^2 + (15*a^4*b^2*x^4)/4 + (10*a^3*b^3*x^6)/3 + (15*a^2*b^4*x^8)/8 + (3*a*b^5*x^10)/5 + (b^6*x^12)/12 + a^6*Log[x]","A",1
454,1,72,72,0.0084205,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^3}{x^2} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^2,x]","-\frac{a^6}{x}+6 a^5 b x+5 a^4 b^2 x^3+4 a^3 b^3 x^5+\frac{15}{7} a^2 b^4 x^7+\frac{2}{3} a b^5 x^9+\frac{b^6 x^{11}}{11}","-\frac{a^6}{x}+6 a^5 b x+5 a^4 b^2 x^3+4 a^3 b^3 x^5+\frac{15}{7} a^2 b^4 x^7+\frac{2}{3} a b^5 x^9+\frac{b^6 x^{11}}{11}",1,"-(a^6/x) + 6*a^5*b*x + 5*a^4*b^2*x^3 + 4*a^3*b^3*x^5 + (15*a^2*b^4*x^7)/7 + (2*a*b^5*x^9)/3 + (b^6*x^11)/11","A",1
455,1,77,77,0.0077004,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^3}{x^3} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^3,x]","-\frac{a^6}{2 x^2}+6 a^5 b \log (x)+\frac{15}{2} a^4 b^2 x^2+5 a^3 b^3 x^4+\frac{5}{2} a^2 b^4 x^6+\frac{3}{4} a b^5 x^8+\frac{b^6 x^{10}}{10}","-\frac{a^6}{2 x^2}+6 a^5 b \log (x)+\frac{15}{2} a^4 b^2 x^2+5 a^3 b^3 x^4+\frac{5}{2} a^2 b^4 x^6+\frac{3}{4} a b^5 x^8+\frac{b^6 x^{10}}{10}",1,"-1/2*a^6/x^2 + (15*a^4*b^2*x^2)/2 + 5*a^3*b^3*x^4 + (5*a^2*b^4*x^6)/2 + (3*a*b^5*x^8)/4 + (b^6*x^10)/10 + 6*a^5*b*Log[x]","A",1
456,1,74,74,0.0087801,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^3}{x^4} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^4,x]","-\frac{a^6}{3 x^3}-\frac{6 a^5 b}{x}+15 a^4 b^2 x+\frac{20}{3} a^3 b^3 x^3+3 a^2 b^4 x^5+\frac{6}{7} a b^5 x^7+\frac{b^6 x^9}{9}","-\frac{a^6}{3 x^3}-\frac{6 a^5 b}{x}+15 a^4 b^2 x+\frac{20}{3} a^3 b^3 x^3+3 a^2 b^4 x^5+\frac{6}{7} a b^5 x^7+\frac{b^6 x^9}{9}",1,"-1/3*a^6/x^3 - (6*a^5*b)/x + 15*a^4*b^2*x + (20*a^3*b^3*x^3)/3 + 3*a^2*b^4*x^5 + (6*a*b^5*x^7)/7 + (b^6*x^9)/9","A",1
457,1,72,72,0.0049316,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^3}{x^5} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^5,x]","-\frac{a^6}{4 x^4}-\frac{3 a^5 b}{x^2}+15 a^4 b^2 \log (x)+10 a^3 b^3 x^2+\frac{15}{4} a^2 b^4 x^4+a b^5 x^6+\frac{b^6 x^8}{8}","-\frac{a^6}{4 x^4}-\frac{3 a^5 b}{x^2}+15 a^4 b^2 \log (x)+10 a^3 b^3 x^2+\frac{15}{4} a^2 b^4 x^4+a b^5 x^6+\frac{b^6 x^8}{8}",1,"-1/4*a^6/x^4 - (3*a^5*b)/x^2 + 10*a^3*b^3*x^2 + (15*a^2*b^4*x^4)/4 + a*b^5*x^6 + (b^6*x^8)/8 + 15*a^4*b^2*Log[x]","A",1
458,1,72,72,0.0065318,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^3}{x^6} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^6,x]","-\frac{a^6}{5 x^5}-\frac{2 a^5 b}{x^3}-\frac{15 a^4 b^2}{x}+20 a^3 b^3 x+5 a^2 b^4 x^3+\frac{6}{5} a b^5 x^5+\frac{b^6 x^7}{7}","-\frac{a^6}{5 x^5}-\frac{2 a^5 b}{x^3}-\frac{15 a^4 b^2}{x}+20 a^3 b^3 x+5 a^2 b^4 x^3+\frac{6}{5} a b^5 x^5+\frac{b^6 x^7}{7}",1,"-1/5*a^6/x^5 - (2*a^5*b)/x^3 - (15*a^4*b^2)/x + 20*a^3*b^3*x + 5*a^2*b^4*x^3 + (6*a*b^5*x^5)/5 + (b^6*x^7)/7","A",1
459,1,79,79,0.0052406,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^3}{x^7} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^7,x]","-\frac{a^6}{6 x^6}-\frac{3 a^5 b}{2 x^4}-\frac{15 a^4 b^2}{2 x^2}+20 a^3 b^3 \log (x)+\frac{15}{2} a^2 b^4 x^2+\frac{3}{2} a b^5 x^4+\frac{b^6 x^6}{6}","-\frac{a^6}{6 x^6}-\frac{3 a^5 b}{2 x^4}-\frac{15 a^4 b^2}{2 x^2}+20 a^3 b^3 \log (x)+\frac{15}{2} a^2 b^4 x^2+\frac{3}{2} a b^5 x^4+\frac{b^6 x^6}{6}",1,"-1/6*a^6/x^6 - (3*a^5*b)/(2*x^4) - (15*a^4*b^2)/(2*x^2) + (15*a^2*b^4*x^2)/2 + (3*a*b^5*x^4)/2 + (b^6*x^6)/6 + 20*a^3*b^3*Log[x]","A",1
460,1,72,72,0.0088946,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^3}{x^8} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^8,x]","-\frac{a^6}{7 x^7}-\frac{6 a^5 b}{5 x^5}-\frac{5 a^4 b^2}{x^3}-\frac{20 a^3 b^3}{x}+15 a^2 b^4 x+2 a b^5 x^3+\frac{b^6 x^5}{5}","-\frac{a^6}{7 x^7}-\frac{6 a^5 b}{5 x^5}-\frac{5 a^4 b^2}{x^3}-\frac{20 a^3 b^3}{x}+15 a^2 b^4 x+2 a b^5 x^3+\frac{b^6 x^5}{5}",1,"-1/7*a^6/x^7 - (6*a^5*b)/(5*x^5) - (5*a^4*b^2)/x^3 - (20*a^3*b^3)/x + 15*a^2*b^4*x + 2*a*b^5*x^3 + (b^6*x^5)/5","A",1
461,1,73,73,0.0081257,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^3}{x^9} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^9,x]","-\frac{a^6}{8 x^8}-\frac{a^5 b}{x^6}-\frac{15 a^4 b^2}{4 x^4}-\frac{10 a^3 b^3}{x^2}+15 a^2 b^4 \log (x)+3 a b^5 x^2+\frac{b^6 x^4}{4}","-\frac{a^6}{8 x^8}-\frac{a^5 b}{x^6}-\frac{15 a^4 b^2}{4 x^4}-\frac{10 a^3 b^3}{x^2}+15 a^2 b^4 \log (x)+3 a b^5 x^2+\frac{b^6 x^4}{4}",1,"-1/8*a^6/x^8 - (a^5*b)/x^6 - (15*a^4*b^2)/(4*x^4) - (10*a^3*b^3)/x^2 + 3*a*b^5*x^2 + (b^6*x^4)/4 + 15*a^2*b^4*Log[x]","A",1
462,1,74,74,0.0088946,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^3}{x^{10}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^10,x]","-\frac{a^6}{9 x^9}-\frac{6 a^5 b}{7 x^7}-\frac{3 a^4 b^2}{x^5}-\frac{20 a^3 b^3}{3 x^3}-\frac{15 a^2 b^4}{x}+6 a b^5 x+\frac{b^6 x^3}{3}","-\frac{a^6}{9 x^9}-\frac{6 a^5 b}{7 x^7}-\frac{3 a^4 b^2}{x^5}-\frac{20 a^3 b^3}{3 x^3}-\frac{15 a^2 b^4}{x}+6 a b^5 x+\frac{b^6 x^3}{3}",1,"-1/9*a^6/x^9 - (6*a^5*b)/(7*x^7) - (3*a^4*b^2)/x^5 - (20*a^3*b^3)/(3*x^3) - (15*a^2*b^4)/x + 6*a*b^5*x + (b^6*x^3)/3","A",1
463,1,77,77,0.0054138,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^3}{x^{11}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^11,x]","-\frac{a^6}{10 x^{10}}-\frac{3 a^5 b}{4 x^8}-\frac{5 a^4 b^2}{2 x^6}-\frac{5 a^3 b^3}{x^4}-\frac{15 a^2 b^4}{2 x^2}+6 a b^5 \log (x)+\frac{b^6 x^2}{2}","-\frac{a^6}{10 x^{10}}-\frac{3 a^5 b}{4 x^8}-\frac{5 a^4 b^2}{2 x^6}-\frac{5 a^3 b^3}{x^4}-\frac{15 a^2 b^4}{2 x^2}+6 a b^5 \log (x)+\frac{b^6 x^2}{2}",1,"-1/10*a^6/x^10 - (3*a^5*b)/(4*x^8) - (5*a^4*b^2)/(2*x^6) - (5*a^3*b^3)/x^4 - (15*a^2*b^4)/(2*x^2) + (b^6*x^2)/2 + 6*a*b^5*Log[x]","A",1
464,1,71,71,0.0055296,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^3}{x^{12}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^12,x]","-\frac{a^6}{11 x^{11}}-\frac{2 a^5 b}{3 x^9}-\frac{15 a^4 b^2}{7 x^7}-\frac{4 a^3 b^3}{x^5}-\frac{5 a^2 b^4}{x^3}-\frac{6 a b^5}{x}+b^6 x","-\frac{a^6}{11 x^{11}}-\frac{2 a^5 b}{3 x^9}-\frac{15 a^4 b^2}{7 x^7}-\frac{4 a^3 b^3}{x^5}-\frac{5 a^2 b^4}{x^3}-\frac{6 a b^5}{x}+b^6 x",1,"-1/11*a^6/x^11 - (2*a^5*b)/(3*x^9) - (15*a^4*b^2)/(7*x^7) - (4*a^3*b^3)/x^5 - (5*a^2*b^4)/x^3 - (6*a*b^5)/x + b^6*x","A",1
465,1,76,76,0.0049793,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^3}{x^{13}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^13,x]","-\frac{a^6}{12 x^{12}}-\frac{3 a^5 b}{5 x^{10}}-\frac{15 a^4 b^2}{8 x^8}-\frac{10 a^3 b^3}{3 x^6}-\frac{15 a^2 b^4}{4 x^4}-\frac{3 a b^5}{x^2}+b^6 \log (x)","-\frac{a^6}{12 x^{12}}-\frac{3 a^5 b}{5 x^{10}}-\frac{15 a^4 b^2}{8 x^8}-\frac{10 a^3 b^3}{3 x^6}-\frac{15 a^2 b^4}{4 x^4}-\frac{3 a b^5}{x^2}+b^6 \log (x)",1,"-1/12*a^6/x^12 - (3*a^5*b)/(5*x^10) - (15*a^4*b^2)/(8*x^8) - (10*a^3*b^3)/(3*x^6) - (15*a^2*b^4)/(4*x^4) - (3*a*b^5)/x^2 + b^6*Log[x]","A",1
466,1,76,76,0.0093532,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^3}{x^{14}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^14,x]","-\frac{a^6}{13 x^{13}}-\frac{6 a^5 b}{11 x^{11}}-\frac{5 a^4 b^2}{3 x^9}-\frac{20 a^3 b^3}{7 x^7}-\frac{3 a^2 b^4}{x^5}-\frac{2 a b^5}{x^3}-\frac{b^6}{x}","-\frac{a^6}{13 x^{13}}-\frac{6 a^5 b}{11 x^{11}}-\frac{5 a^4 b^2}{3 x^9}-\frac{20 a^3 b^3}{7 x^7}-\frac{3 a^2 b^4}{x^5}-\frac{2 a b^5}{x^3}-\frac{b^6}{x}",1,"-1/13*a^6/x^13 - (6*a^5*b)/(11*x^11) - (5*a^4*b^2)/(3*x^9) - (20*a^3*b^3)/(7*x^7) - (3*a^2*b^4)/x^5 - (2*a*b^5)/x^3 - b^6/x","A",1
467,1,82,19,0.0075327,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^3}{x^{15}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^15,x]","-\frac{a^6}{14 x^{14}}-\frac{a^5 b}{2 x^{12}}-\frac{3 a^4 b^2}{2 x^{10}}-\frac{5 a^3 b^3}{2 x^8}-\frac{5 a^2 b^4}{2 x^6}-\frac{3 a b^5}{2 x^4}-\frac{b^6}{2 x^2}","-\frac{\left(a+b x^2\right)^7}{14 a x^{14}}",1,"-1/14*a^6/x^14 - (a^5*b)/(2*x^12) - (3*a^4*b^2)/(2*x^10) - (5*a^3*b^3)/(2*x^8) - (5*a^2*b^4)/(2*x^6) - (3*a*b^5)/(2*x^4) - b^6/(2*x^2)","B",1
468,1,82,82,0.0094948,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^3}{x^{16}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^16,x]","-\frac{a^6}{15 x^{15}}-\frac{6 a^5 b}{13 x^{13}}-\frac{15 a^4 b^2}{11 x^{11}}-\frac{20 a^3 b^3}{9 x^9}-\frac{15 a^2 b^4}{7 x^7}-\frac{6 a b^5}{5 x^5}-\frac{b^6}{3 x^3}","-\frac{a^6}{15 x^{15}}-\frac{6 a^5 b}{13 x^{13}}-\frac{15 a^4 b^2}{11 x^{11}}-\frac{20 a^3 b^3}{9 x^9}-\frac{15 a^2 b^4}{7 x^7}-\frac{6 a b^5}{5 x^5}-\frac{b^6}{3 x^3}",1,"-1/15*a^6/x^15 - (6*a^5*b)/(13*x^13) - (15*a^4*b^2)/(11*x^11) - (20*a^3*b^3)/(9*x^9) - (15*a^2*b^4)/(7*x^7) - (6*a*b^5)/(5*x^5) - b^6/(3*x^3)","A",1
469,1,78,40,0.0045978,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^3}{x^{17}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^17,x]","-\frac{a^6}{16 x^{16}}-\frac{3 a^5 b}{7 x^{14}}-\frac{5 a^4 b^2}{4 x^{12}}-\frac{2 a^3 b^3}{x^{10}}-\frac{15 a^2 b^4}{8 x^8}-\frac{a b^5}{x^6}-\frac{b^6}{4 x^4}","\frac{b \left(a+b x^2\right)^7}{112 a^2 x^{14}}-\frac{\left(a+b x^2\right)^7}{16 a x^{16}}",1,"-1/16*a^6/x^16 - (3*a^5*b)/(7*x^14) - (5*a^4*b^2)/(4*x^12) - (2*a^3*b^3)/x^10 - (15*a^2*b^4)/(8*x^8) - (a*b^5)/x^6 - b^6/(4*x^4)","A",1
470,1,82,82,0.0067371,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^3}{x^{18}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^18,x]","-\frac{a^6}{17 x^{17}}-\frac{2 a^5 b}{5 x^{15}}-\frac{15 a^4 b^2}{13 x^{13}}-\frac{20 a^3 b^3}{11 x^{11}}-\frac{5 a^2 b^4}{3 x^9}-\frac{6 a b^5}{7 x^7}-\frac{b^6}{5 x^5}","-\frac{a^6}{17 x^{17}}-\frac{2 a^5 b}{5 x^{15}}-\frac{15 a^4 b^2}{13 x^{13}}-\frac{20 a^3 b^3}{11 x^{11}}-\frac{5 a^2 b^4}{3 x^9}-\frac{6 a b^5}{7 x^7}-\frac{b^6}{5 x^5}",1,"-1/17*a^6/x^17 - (2*a^5*b)/(5*x^15) - (15*a^4*b^2)/(13*x^13) - (20*a^3*b^3)/(11*x^11) - (5*a^2*b^4)/(3*x^9) - (6*a*b^5)/(7*x^7) - b^6/(5*x^5)","A",1
471,1,82,62,0.004629,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^3}{x^{19}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^19,x]","-\frac{a^6}{18 x^{18}}-\frac{3 a^5 b}{8 x^{16}}-\frac{15 a^4 b^2}{14 x^{14}}-\frac{5 a^3 b^3}{3 x^{12}}-\frac{3 a^2 b^4}{2 x^{10}}-\frac{3 a b^5}{4 x^8}-\frac{b^6}{6 x^6}","-\frac{b^2 \left(a+b x^2\right)^7}{504 a^3 x^{14}}+\frac{b \left(a+b x^2\right)^7}{72 a^2 x^{16}}-\frac{\left(a+b x^2\right)^7}{18 a x^{18}}",1,"-1/18*a^6/x^18 - (3*a^5*b)/(8*x^16) - (15*a^4*b^2)/(14*x^14) - (5*a^3*b^3)/(3*x^12) - (3*a^2*b^4)/(2*x^10) - (3*a*b^5)/(4*x^8) - b^6/(6*x^6)","A",1
472,1,80,80,0.0094996,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^3}{x^{20}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^20,x]","-\frac{a^6}{19 x^{19}}-\frac{6 a^5 b}{17 x^{17}}-\frac{a^4 b^2}{x^{15}}-\frac{20 a^3 b^3}{13 x^{13}}-\frac{15 a^2 b^4}{11 x^{11}}-\frac{2 a b^5}{3 x^9}-\frac{b^6}{7 x^7}","-\frac{a^6}{19 x^{19}}-\frac{6 a^5 b}{17 x^{17}}-\frac{a^4 b^2}{x^{15}}-\frac{20 a^3 b^3}{13 x^{13}}-\frac{15 a^2 b^4}{11 x^{11}}-\frac{2 a b^5}{3 x^9}-\frac{b^6}{7 x^7}",1,"-1/19*a^6/x^19 - (6*a^5*b)/(17*x^17) - (a^4*b^2)/x^15 - (20*a^3*b^3)/(13*x^13) - (15*a^2*b^4)/(11*x^11) - (2*a*b^5)/(3*x^9) - b^6/(7*x^7)","A",1
473,1,82,84,0.007638,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^3}{x^{21}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^21,x]","-\frac{a^6}{20 x^{20}}-\frac{a^5 b}{3 x^{18}}-\frac{15 a^4 b^2}{16 x^{16}}-\frac{10 a^3 b^3}{7 x^{14}}-\frac{5 a^2 b^4}{4 x^{12}}-\frac{3 a b^5}{5 x^{10}}-\frac{b^6}{8 x^8}","\frac{b^3 \left(a+b x^2\right)^7}{1680 a^4 x^{14}}-\frac{b^2 \left(a+b x^2\right)^7}{240 a^3 x^{16}}+\frac{b \left(a+b x^2\right)^7}{60 a^2 x^{18}}-\frac{\left(a+b x^2\right)^7}{20 a x^{20}}",1,"-1/20*a^6/x^20 - (a^5*b)/(3*x^18) - (15*a^4*b^2)/(16*x^16) - (10*a^3*b^3)/(7*x^14) - (5*a^2*b^4)/(4*x^12) - (3*a*b^5)/(5*x^10) - b^6/(8*x^8)","A",1
474,1,82,82,0.0096368,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^3}{x^{22}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^22,x]","-\frac{a^6}{21 x^{21}}-\frac{6 a^5 b}{19 x^{19}}-\frac{15 a^4 b^2}{17 x^{17}}-\frac{4 a^3 b^3}{3 x^{15}}-\frac{15 a^2 b^4}{13 x^{13}}-\frac{6 a b^5}{11 x^{11}}-\frac{b^6}{9 x^9}","-\frac{a^6}{21 x^{21}}-\frac{6 a^5 b}{19 x^{19}}-\frac{15 a^4 b^2}{17 x^{17}}-\frac{4 a^3 b^3}{3 x^{15}}-\frac{15 a^2 b^4}{13 x^{13}}-\frac{6 a b^5}{11 x^{11}}-\frac{b^6}{9 x^9}",1,"-1/21*a^6/x^21 - (6*a^5*b)/(19*x^19) - (15*a^4*b^2)/(17*x^17) - (4*a^3*b^3)/(3*x^15) - (15*a^2*b^4)/(13*x^13) - (6*a*b^5)/(11*x^11) - b^6/(9*x^9)","A",1
475,1,72,83,0.0213495,"\int \frac{x^{11}}{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[x^11/(a^2 + 2*a*b*x^2 + b^2*x^4),x]","\frac{\frac{12 a^5}{a+b x^2}+60 a^4 \log \left(a+b x^2\right)-48 a^3 b x^2+18 a^2 b^2 x^4-8 a b^3 x^6+3 b^4 x^8}{24 b^6}","\frac{a^5}{2 b^6 \left(a+b x^2\right)}+\frac{5 a^4 \log \left(a+b x^2\right)}{2 b^6}-\frac{2 a^3 x^2}{b^5}+\frac{3 a^2 x^4}{4 b^4}-\frac{a x^6}{3 b^3}+\frac{x^8}{8 b^2}",1,"(-48*a^3*b*x^2 + 18*a^2*b^2*x^4 - 8*a*b^3*x^6 + 3*b^4*x^8 + (12*a^5)/(a + b*x^2) + 60*a^4*Log[a + b*x^2])/(24*b^6)","A",1
476,1,60,70,0.0216167,"\int \frac{x^9}{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[x^9/(a^2 + 2*a*b*x^2 + b^2*x^4),x]","\frac{-\frac{3 a^4}{a+b x^2}-12 a^3 \log \left(a+b x^2\right)+9 a^2 b x^2-3 a b^2 x^4+b^3 x^6}{6 b^5}","-\frac{a^4}{2 b^5 \left(a+b x^2\right)}-\frac{2 a^3 \log \left(a+b x^2\right)}{b^5}+\frac{3 a^2 x^2}{2 b^4}-\frac{a x^4}{2 b^3}+\frac{x^6}{6 b^2}",1,"(9*a^2*b*x^2 - 3*a*b^2*x^4 + b^3*x^6 - (3*a^4)/(a + b*x^2) - 12*a^3*Log[a + b*x^2])/(6*b^5)","A",1
477,1,49,57,0.0153088,"\int \frac{x^7}{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[x^7/(a^2 + 2*a*b*x^2 + b^2*x^4),x]","\frac{\frac{2 a^3}{a+b x^2}+6 a^2 \log \left(a+b x^2\right)-4 a b x^2+b^2 x^4}{4 b^4}","\frac{a^3}{2 b^4 \left(a+b x^2\right)}+\frac{3 a^2 \log \left(a+b x^2\right)}{2 b^4}-\frac{a x^2}{b^3}+\frac{x^4}{4 b^2}",1,"(-4*a*b*x^2 + b^2*x^4 + (2*a^3)/(a + b*x^2) + 6*a^2*Log[a + b*x^2])/(4*b^4)","A",1
478,1,38,44,0.0150479,"\int \frac{x^5}{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[x^5/(a^2 + 2*a*b*x^2 + b^2*x^4),x]","\frac{-\frac{a^2}{a+b x^2}-2 a \log \left(a+b x^2\right)+b x^2}{2 b^3}","-\frac{a^2}{2 b^3 \left(a+b x^2\right)}-\frac{a \log \left(a+b x^2\right)}{b^3}+\frac{x^2}{2 b^2}",1,"(b*x^2 - a^2/(a + b*x^2) - 2*a*Log[a + b*x^2])/(2*b^3)","A",1
479,1,27,33,0.0084658,"\int \frac{x^3}{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[x^3/(a^2 + 2*a*b*x^2 + b^2*x^4),x]","\frac{\frac{a}{a+b x^2}+\log \left(a+b x^2\right)}{2 b^2}","\frac{a}{2 b^2 \left(a+b x^2\right)}+\frac{\log \left(a+b x^2\right)}{2 b^2}",1,"(a/(a + b*x^2) + Log[a + b*x^2])/(2*b^2)","A",1
480,1,16,16,0.0026526,"\int \frac{x}{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[x/(a^2 + 2*a*b*x^2 + b^2*x^4),x]","-\frac{1}{2 b \left(a+b x^2\right)}","-\frac{1}{2 b \left(a+b x^2\right)}",1,"-1/2*1/(b*(a + b*x^2))","A",1
481,1,33,38,0.0154004,"\int \frac{1}{x \left(a^2+2 a b x^2+b^2 x^4\right)} \, dx","Integrate[1/(x*(a^2 + 2*a*b*x^2 + b^2*x^4)),x]","\frac{\frac{a}{a+b x^2}-\log \left(a+b x^2\right)+2 \log (x)}{2 a^2}","-\frac{\log \left(a+b x^2\right)}{2 a^2}+\frac{\log (x)}{a^2}+\frac{1}{2 a \left(a+b x^2\right)}",1,"(a/(a + b*x^2) + 2*Log[x] - Log[a + b*x^2])/(2*a^2)","A",1
482,1,41,49,0.0358088,"\int \frac{1}{x^3 \left(a^2+2 a b x^2+b^2 x^4\right)} \, dx","Integrate[1/(x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)),x]","-\frac{a \left(\frac{b}{a+b x^2}+\frac{1}{x^2}\right)-2 b \log \left(a+b x^2\right)+4 b \log (x)}{2 a^3}","\frac{b \log \left(a+b x^2\right)}{a^3}-\frac{2 b \log (x)}{a^3}-\frac{b}{2 a^2 \left(a+b x^2\right)}-\frac{1}{2 a^2 x^2}",1,"-1/2*(a*(x^(-2) + b/(a + b*x^2)) + 4*b*Log[x] - 2*b*Log[a + b*x^2])/a^3","A",1
483,1,57,66,0.051282,"\int \frac{1}{x^5 \left(a^2+2 a b x^2+b^2 x^4\right)} \, dx","Integrate[1/(x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)),x]","\frac{-6 b^2 \log \left(a+b x^2\right)+a \left(\frac{2 b^2}{a+b x^2}-\frac{a}{x^4}+\frac{4 b}{x^2}\right)+12 b^2 \log (x)}{4 a^4}","-\frac{3 b^2 \log \left(a+b x^2\right)}{2 a^4}+\frac{3 b^2 \log (x)}{a^4}+\frac{b^2}{2 a^3 \left(a+b x^2\right)}+\frac{b}{a^3 x^2}-\frac{1}{4 a^2 x^4}",1,"(a*(-(a/x^4) + (4*b)/x^2 + (2*b^2)/(a + b*x^2)) + 12*b^2*Log[x] - 6*b^2*Log[a + b*x^2])/(4*a^4)","A",1
484,1,82,92,0.0530659,"\int \frac{x^{10}}{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[x^10/(a^2 + 2*a*b*x^2 + b^2*x^4),x]","\frac{9 a^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{11/2}}+\frac{x \left(-\frac{35 a^4}{a+b x^2}-280 a^3+70 a^2 b x^2-28 a b^2 x^4+10 b^3 x^6\right)}{70 b^5}","\frac{9 a^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{11/2}}-\frac{9 a^3 x}{2 b^5}+\frac{3 a^2 x^3}{2 b^4}-\frac{9 a x^5}{10 b^3}-\frac{x^9}{2 b \left(a+b x^2\right)}+\frac{9 x^7}{14 b^2}",1,"(x*(-280*a^3 + 70*a^2*b*x^2 - 28*a*b^2*x^4 + 10*b^3*x^6 - (35*a^4)/(a + b*x^2)))/(70*b^5) + (9*a^(7/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(11/2))","A",1
485,1,71,79,0.0463717,"\int \frac{x^8}{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[x^8/(a^2 + 2*a*b*x^2 + b^2*x^4),x]","\frac{x \left(\frac{15 a^3}{a+b x^2}+90 a^2-20 a b x^2+6 b^2 x^4\right)}{30 b^4}-\frac{7 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{9/2}}","-\frac{7 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{9/2}}+\frac{7 a^2 x}{2 b^4}-\frac{7 a x^3}{6 b^3}-\frac{x^7}{2 b \left(a+b x^2\right)}+\frac{7 x^5}{10 b^2}",1,"(x*(90*a^2 - 20*a*b*x^2 + 6*b^2*x^4 + (15*a^3)/(a + b*x^2)))/(30*b^4) - (7*a^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(9/2))","A",1
486,1,60,66,0.0411358,"\int \frac{x^6}{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[x^6/(a^2 + 2*a*b*x^2 + b^2*x^4),x]","\frac{5 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{7/2}}+\frac{x \left(-\frac{3 a^2}{a+b x^2}-12 a+2 b x^2\right)}{6 b^3}","\frac{5 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{7/2}}-\frac{5 a x}{2 b^3}-\frac{x^5}{2 b \left(a+b x^2\right)}+\frac{5 x^3}{6 b^2}",1,"(x*(-12*a + 2*b*x^2 - (3*a^2)/(a + b*x^2)))/(6*b^3) + (5*a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(7/2))","A",1
487,1,51,55,0.0315015,"\int \frac{x^4}{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[x^4/(a^2 + 2*a*b*x^2 + b^2*x^4),x]","-\frac{3 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{5/2}}+\frac{a x}{2 b^2 \left(a+b x^2\right)}+\frac{x}{b^2}","-\frac{3 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{5/2}}-\frac{x^3}{2 b \left(a+b x^2\right)}+\frac{3 x}{2 b^2}",1,"x/b^2 + (a*x)/(2*b^2*(a + b*x^2)) - (3*Sqrt[a]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(5/2))","A",1
488,1,45,45,0.0206064,"\int \frac{x^2}{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[x^2/(a^2 + 2*a*b*x^2 + b^2*x^4),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 \sqrt{a} b^{3/2}}-\frac{x}{2 b \left(a+b x^2\right)}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 \sqrt{a} b^{3/2}}-\frac{x}{2 b \left(a+b x^2\right)}",1,"-1/2*x/(b*(a + b*x^2)) + ArcTan[(Sqrt[b]*x)/Sqrt[a]]/(2*Sqrt[a]*b^(3/2))","A",1
489,1,45,45,0.0235484,"\int \frac{1}{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(-1),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} \sqrt{b}}+\frac{x}{2 a \left(a+b x^2\right)}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} \sqrt{b}}+\frac{x}{2 a \left(a+b x^2\right)}",1,"x/(2*a*(a + b*x^2)) + ArcTan[(Sqrt[b]*x)/Sqrt[a]]/(2*a^(3/2)*Sqrt[b])","A",1
490,1,54,57,0.0362961,"\int \frac{1}{x^2 \left(a^2+2 a b x^2+b^2 x^4\right)} \, dx","Integrate[1/(x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)),x]","-\frac{3 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{5/2}}-\frac{b x}{2 a^2 \left(a+b x^2\right)}-\frac{1}{a^2 x}","-\frac{3 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{5/2}}-\frac{3}{2 a^2 x}+\frac{1}{2 a x \left(a+b x^2\right)}",1,"-(1/(a^2*x)) - (b*x)/(2*a^2*(a + b*x^2)) - (3*Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(5/2))","A",1
491,1,67,68,0.0374823,"\int \frac{1}{x^4 \left(a^2+2 a b x^2+b^2 x^4\right)} \, dx","Integrate[1/(x^4*(a^2 + 2*a*b*x^2 + b^2*x^4)),x]","\frac{5 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{7/2}}+\frac{b^2 x}{2 a^3 \left(a+b x^2\right)}+\frac{2 b}{a^3 x}-\frac{1}{3 a^2 x^3}","\frac{5 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{7/2}}+\frac{5 b}{2 a^3 x}-\frac{5}{6 a^2 x^3}+\frac{1}{2 a x^3 \left(a+b x^2\right)}",1,"-1/3*1/(a^2*x^3) + (2*b)/(a^3*x) + (b^2*x)/(2*a^3*(a + b*x^2)) + (5*b^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(7/2))","A",1
492,1,80,81,0.0443247,"\int \frac{1}{x^6 \left(a^2+2 a b x^2+b^2 x^4\right)} \, dx","Integrate[1/(x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)),x]","-\frac{7 b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{9/2}}-\frac{b^3 x}{2 a^4 \left(a+b x^2\right)}-\frac{3 b^2}{a^4 x}+\frac{2 b}{3 a^3 x^3}-\frac{1}{5 a^2 x^5}","-\frac{7 b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{9/2}}-\frac{7 b^2}{2 a^4 x}+\frac{7 b}{6 a^3 x^3}-\frac{7}{10 a^2 x^5}+\frac{1}{2 a x^5 \left(a+b x^2\right)}",1,"-1/5*1/(a^2*x^5) + (2*b)/(3*a^3*x^3) - (3*b^2)/(a^4*x) - (b^3*x)/(2*a^4*(a + b*x^2)) - (7*b^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(9/2))","A",1
493,1,78,91,0.0305239,"\int \frac{x^{11}}{\left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[x^11/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","\frac{\frac{2 a^5}{\left(a+b x^2\right)^3}-\frac{15 a^4}{\left(a+b x^2\right)^2}+\frac{60 a^3}{a+b x^2}+60 a^2 \log \left(a+b x^2\right)-24 a b x^2+3 b^2 x^4}{12 b^6}","\frac{a^5}{6 b^6 \left(a+b x^2\right)^3}-\frac{5 a^4}{4 b^6 \left(a+b x^2\right)^2}+\frac{5 a^3}{b^6 \left(a+b x^2\right)}+\frac{5 a^2 \log \left(a+b x^2\right)}{b^6}-\frac{2 a x^2}{b^5}+\frac{x^4}{4 b^4}",1,"(-24*a*b*x^2 + 3*b^2*x^4 + (2*a^5)/(a + b*x^2)^3 - (15*a^4)/(a + b*x^2)^2 + (60*a^3)/(a + b*x^2) + 60*a^2*Log[a + b*x^2])/(12*b^6)","A",1
494,1,59,77,0.0496572,"\int \frac{x^9}{\left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[x^9/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","-\frac{\frac{a^2 \left(13 a^2+30 a b x^2+18 b^2 x^4\right)}{\left(a+b x^2\right)^3}+12 a \log \left(a+b x^2\right)-3 b x^2}{6 b^5}","-\frac{a^4}{6 b^5 \left(a+b x^2\right)^3}+\frac{a^3}{b^5 \left(a+b x^2\right)^2}-\frac{3 a^2}{b^5 \left(a+b x^2\right)}-\frac{2 a \log \left(a+b x^2\right)}{b^5}+\frac{x^2}{2 b^4}",1,"-1/6*(-3*b*x^2 + (a^2*(13*a^2 + 30*a*b*x^2 + 18*b^2*x^4))/(a + b*x^2)^3 + 12*a*Log[a + b*x^2])/b^5","A",1
495,1,50,71,0.0184961,"\int \frac{x^7}{\left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[x^7/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","\frac{\frac{a \left(11 a^2+27 a b x^2+18 b^2 x^4\right)}{\left(a+b x^2\right)^3}+6 \log \left(a+b x^2\right)}{12 b^4}","\frac{a^3}{6 b^4 \left(a+b x^2\right)^3}-\frac{3 a^2}{4 b^4 \left(a+b x^2\right)^2}+\frac{3 a}{2 b^4 \left(a+b x^2\right)}+\frac{\log \left(a+b x^2\right)}{2 b^4}",1,"((a*(11*a^2 + 27*a*b*x^2 + 18*b^2*x^4))/(a + b*x^2)^3 + 6*Log[a + b*x^2])/(12*b^4)","A",1
496,1,35,19,0.0126594,"\int \frac{x^5}{\left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[x^5/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","-\frac{a^2+3 a b x^2+3 b^2 x^4}{6 b^3 \left(a+b x^2\right)^3}","\frac{x^6}{6 a \left(a+b x^2\right)^3}",1,"-1/6*(a^2 + 3*a*b*x^2 + 3*b^2*x^4)/(b^3*(a + b*x^2)^3)","A",1
497,1,24,34,0.0087021,"\int \frac{x^3}{\left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[x^3/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","-\frac{a+3 b x^2}{12 b^2 \left(a+b x^2\right)^3}","\frac{a}{6 b^2 \left(a+b x^2\right)^3}-\frac{1}{4 b^2 \left(a+b x^2\right)^2}",1,"-1/12*(a + 3*b*x^2)/(b^2*(a + b*x^2)^3)","A",1
498,1,16,16,0.0026724,"\int \frac{x}{\left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[x/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","-\frac{1}{6 b \left(a+b x^2\right)^3}","-\frac{1}{6 b \left(a+b x^2\right)^3}",1,"-1/6*1/(b*(a + b*x^2)^3)","A",1
499,1,54,70,0.0413103,"\int \frac{1}{x \left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[1/(x*(a^2 + 2*a*b*x^2 + b^2*x^4)^2),x]","\frac{\frac{a \left(11 a^2+15 a b x^2+6 b^2 x^4\right)}{\left(a+b x^2\right)^3}-6 \log \left(a+b x^2\right)+12 \log (x)}{12 a^4}","-\frac{\log \left(a+b x^2\right)}{2 a^4}+\frac{\log (x)}{a^4}+\frac{1}{2 a^3 \left(a+b x^2\right)}+\frac{1}{4 a^2 \left(a+b x^2\right)^2}+\frac{1}{6 a \left(a+b x^2\right)^3}",1,"((a*(11*a^2 + 15*a*b*x^2 + 6*b^2*x^4))/(a + b*x^2)^3 + 12*Log[x] - 6*Log[a + b*x^2])/(12*a^4)","A",1
500,1,70,84,0.0670295,"\int \frac{1}{x^3 \left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[1/(x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^2),x]","-\frac{\frac{a \left(3 a^3+22 a^2 b x^2+30 a b^2 x^4+12 b^3 x^6\right)}{x^2 \left(a+b x^2\right)^3}-12 b \log \left(a+b x^2\right)+24 b \log (x)}{6 a^5}","\frac{2 b \log \left(a+b x^2\right)}{a^5}-\frac{4 b \log (x)}{a^5}-\frac{3 b}{2 a^4 \left(a+b x^2\right)}-\frac{1}{2 a^4 x^2}-\frac{b}{2 a^3 \left(a+b x^2\right)^2}-\frac{b}{6 a^2 \left(a+b x^2\right)^3}",1,"-1/6*((a*(3*a^3 + 22*a^2*b*x^2 + 30*a*b^2*x^4 + 12*b^3*x^6))/(x^2*(a + b*x^2)^3) + 24*b*Log[x] - 12*b*Log[a + b*x^2])/a^5","A",1
501,1,85,101,0.0582459,"\int \frac{1}{x^5 \left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[1/(x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)^2),x]","\frac{\frac{a \left(-3 a^4+15 a^3 b x^2+110 a^2 b^2 x^4+150 a b^3 x^6+60 b^4 x^8\right)}{x^4 \left(a+b x^2\right)^3}-60 b^2 \log \left(a+b x^2\right)+120 b^2 \log (x)}{12 a^6}","-\frac{5 b^2 \log \left(a+b x^2\right)}{a^6}+\frac{10 b^2 \log (x)}{a^6}+\frac{3 b^2}{a^5 \left(a+b x^2\right)}+\frac{2 b}{a^5 x^2}+\frac{3 b^2}{4 a^4 \left(a+b x^2\right)^2}-\frac{1}{4 a^4 x^4}+\frac{b^2}{6 a^3 \left(a+b x^2\right)^3}",1,"((a*(-3*a^4 + 15*a^3*b*x^2 + 110*a^2*b^2*x^4 + 150*a*b^3*x^6 + 60*b^4*x^8))/(x^4*(a + b*x^2)^3) + 120*b^2*Log[x] - 60*b^2*Log[a + b*x^2])/(12*a^6)","A",1
502,1,99,117,0.0558491,"\int \frac{x^{12}}{\left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[x^12/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","\frac{3465 a^5 x+9240 a^4 b x^3+7623 a^3 b^2 x^5+1584 a^2 b^3 x^7-176 a b^4 x^9+48 b^5 x^{11}}{240 b^6 \left(a+b x^2\right)^3}-\frac{231 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{16 b^{13/2}}","-\frac{231 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{16 b^{13/2}}+\frac{231 a^2 x}{16 b^6}-\frac{77 a x^3}{16 b^5}-\frac{33 x^7}{16 b^3 \left(a+b x^2\right)}-\frac{11 x^9}{24 b^2 \left(a+b x^2\right)^2}-\frac{x^{11}}{6 b \left(a+b x^2\right)^3}+\frac{231 x^5}{80 b^4}",1,"(3465*a^5*x + 9240*a^4*b*x^3 + 7623*a^3*b^2*x^5 + 1584*a^2*b^3*x^7 - 176*a*b^4*x^9 + 48*b^5*x^11)/(240*b^6*(a + b*x^2)^3) - (231*a^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(16*b^(13/2))","A",1
503,1,89,104,0.0464212,"\int \frac{x^{10}}{\left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[x^10/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","\frac{315 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)+\frac{\sqrt{b} x \left(-315 a^4-840 a^3 b x^2-693 a^2 b^2 x^4-144 a b^3 x^6+16 b^4 x^8\right)}{\left(a+b x^2\right)^3}}{48 b^{11/2}}","\frac{105 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{16 b^{11/2}}-\frac{105 a x}{16 b^5}-\frac{21 x^5}{16 b^3 \left(a+b x^2\right)}-\frac{3 x^7}{8 b^2 \left(a+b x^2\right)^2}-\frac{x^9}{6 b \left(a+b x^2\right)^3}+\frac{35 x^3}{16 b^4}",1,"((Sqrt[b]*x*(-315*a^4 - 840*a^3*b*x^2 - 693*a^2*b^2*x^4 - 144*a*b^3*x^6 + 16*b^4*x^8))/(a + b*x^2)^3 + 315*a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(48*b^(11/2))","A",1
504,1,77,93,0.0449766,"\int \frac{x^8}{\left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[x^8/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","\frac{105 a^3 x+280 a^2 b x^3+231 a b^2 x^5+48 b^3 x^7}{48 b^4 \left(a+b x^2\right)^3}-\frac{35 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{16 b^{9/2}}","-\frac{35 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{16 b^{9/2}}-\frac{35 x^3}{48 b^3 \left(a+b x^2\right)}-\frac{7 x^5}{24 b^2 \left(a+b x^2\right)^2}-\frac{x^7}{6 b \left(a+b x^2\right)^3}+\frac{35 x}{16 b^4}",1,"(105*a^3*x + 280*a^2*b*x^3 + 231*a*b^2*x^5 + 48*b^3*x^7)/(48*b^4*(a + b*x^2)^3) - (35*Sqrt[a]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(16*b^(9/2))","A",1
505,1,66,83,0.0377948,"\int \frac{x^6}{\left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[x^6/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","\frac{5 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{16 \sqrt{a} b^{7/2}}-\frac{x \left(15 a^2+40 a b x^2+33 b^2 x^4\right)}{48 b^3 \left(a+b x^2\right)^3}","\frac{5 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{16 \sqrt{a} b^{7/2}}-\frac{5 x}{16 b^3 \left(a+b x^2\right)}-\frac{5 x^3}{24 b^2 \left(a+b x^2\right)^2}-\frac{x^5}{6 b \left(a+b x^2\right)^3}",1,"-1/48*(x*(15*a^2 + 40*a*b*x^2 + 33*b^2*x^4))/(b^3*(a + b*x^2)^3) + (5*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(16*Sqrt[a]*b^(7/2))","A",1
506,1,69,84,0.0439916,"\int \frac{x^4}{\left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[x^4/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{16 a^{3/2} b^{5/2}}+\frac{-3 a^2 x-8 a b x^3+3 b^2 x^5}{48 a b^2 \left(a+b x^2\right)^3}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{16 a^{3/2} b^{5/2}}+\frac{x}{16 a b^2 \left(a+b x^2\right)}-\frac{x}{8 b^2 \left(a+b x^2\right)^2}-\frac{x^3}{6 b \left(a+b x^2\right)^3}",1,"(-3*a^2*x - 8*a*b*x^3 + 3*b^2*x^5)/(48*a*b^2*(a + b*x^2)^3) + ArcTan[(Sqrt[b]*x)/Sqrt[a]]/(16*a^(3/2)*b^(5/2))","A",1
507,1,69,85,0.037894,"\int \frac{x^2}{\left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[x^2/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{16 a^{5/2} b^{3/2}}+\frac{-3 a^2 x+8 a b x^3+3 b^2 x^5}{48 a^2 b \left(a+b x^2\right)^3}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{16 a^{5/2} b^{3/2}}+\frac{x}{16 a^2 b \left(a+b x^2\right)}+\frac{x}{24 a b \left(a+b x^2\right)^2}-\frac{x}{6 b \left(a+b x^2\right)^3}",1,"(-3*a^2*x + 8*a*b*x^3 + 3*b^2*x^5)/(48*a^2*b*(a + b*x^2)^3) + ArcTan[(Sqrt[b]*x)/Sqrt[a]]/(16*a^(5/2)*b^(3/2))","A",1
508,1,66,79,0.0345623,"\int \frac{1}{\left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(-2),x]","\frac{5 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{16 a^{7/2} \sqrt{b}}+\frac{33 a^2 x+40 a b x^3+15 b^2 x^5}{48 a^3 \left(a+b x^2\right)^3}","\frac{5 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{16 a^{7/2} \sqrt{b}}+\frac{5 x}{16 a^3 \left(a+b x^2\right)}+\frac{5 x}{24 a^2 \left(a+b x^2\right)^2}+\frac{x}{6 a \left(a+b x^2\right)^3}",1,"(33*a^2*x + 40*a*b*x^3 + 15*b^2*x^5)/(48*a^3*(a + b*x^2)^3) + (5*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(16*a^(7/2)*Sqrt[b])","A",1
509,1,79,95,0.0431653,"\int \frac{1}{x^2 \left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[1/(x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^2),x]","-\frac{35 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{16 a^{9/2}}-\frac{48 a^3+231 a^2 b x^2+280 a b^2 x^4+105 b^3 x^6}{48 a^4 x \left(a+b x^2\right)^3}","-\frac{35 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{16 a^{9/2}}-\frac{35}{16 a^4 x}+\frac{35}{48 a^3 x \left(a+b x^2\right)}+\frac{7}{24 a^2 x \left(a+b x^2\right)^2}+\frac{1}{6 a x \left(a+b x^2\right)^3}",1,"-1/48*(48*a^3 + 231*a^2*b*x^2 + 280*a*b^2*x^4 + 105*b^3*x^6)/(a^4*x*(a + b*x^2)^3) - (35*Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(16*a^(9/2))","A",1
510,1,91,106,0.0489021,"\int \frac{1}{x^4 \left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[1/(x^4*(a^2 + 2*a*b*x^2 + b^2*x^4)^2),x]","\frac{\frac{\sqrt{a} \left(-16 a^4+144 a^3 b x^2+693 a^2 b^2 x^4+840 a b^3 x^6+315 b^4 x^8\right)}{x^3 \left(a+b x^2\right)^3}+315 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{48 a^{11/2}}","\frac{105 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{16 a^{11/2}}+\frac{105 b}{16 a^5 x}-\frac{35}{16 a^4 x^3}+\frac{21}{16 a^3 x^3 \left(a+b x^2\right)}+\frac{3}{8 a^2 x^3 \left(a+b x^2\right)^2}+\frac{1}{6 a x^3 \left(a+b x^2\right)^3}",1,"((Sqrt[a]*(-16*a^4 + 144*a^3*b*x^2 + 693*a^2*b^2*x^4 + 840*a*b^3*x^6 + 315*b^4*x^8))/(x^3*(a + b*x^2)^3) + 315*b^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(48*a^(11/2))","A",1
511,1,101,119,0.0520776,"\int \frac{1}{x^6 \left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[1/(x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)^2),x]","-\frac{231 b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{16 a^{13/2}}-\frac{48 a^5-176 a^4 b x^2+1584 a^3 b^2 x^4+7623 a^2 b^3 x^6+9240 a b^4 x^8+3465 b^5 x^{10}}{240 a^6 x^5 \left(a+b x^2\right)^3}","-\frac{231 b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{16 a^{13/2}}-\frac{231 b^2}{16 a^6 x}+\frac{77 b}{16 a^5 x^3}-\frac{231}{80 a^4 x^5}+\frac{33}{16 a^3 x^5 \left(a+b x^2\right)}+\frac{11}{24 a^2 x^5 \left(a+b x^2\right)^2}+\frac{1}{6 a x^5 \left(a+b x^2\right)^3}",1,"-1/240*(48*a^5 - 176*a^4*b*x^2 + 1584*a^3*b^2*x^4 + 7623*a^2*b^3*x^6 + 9240*a*b^4*x^8 + 3465*b^5*x^10)/(a^6*x^5*(a + b*x^2)^3) - (231*b^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(16*a^(13/2))","A",1
512,1,114,133,0.0250193,"\int \frac{x^{15}}{\left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[x^15/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{459 a^7+1875 a^6 b x^2+2700 a^5 b^2 x^4+1300 a^4 b^3 x^6-400 a^3 b^4 x^8-500 a^2 b^5 x^{10}+420 a^2 \left(a+b x^2\right)^5 \log \left(a+b x^2\right)-70 a b^6 x^{12}+10 b^7 x^{14}}{40 b^8 \left(a+b x^2\right)^5}","\frac{a^7}{10 b^8 \left(a+b x^2\right)^5}-\frac{7 a^6}{8 b^8 \left(a+b x^2\right)^4}+\frac{7 a^5}{2 b^8 \left(a+b x^2\right)^3}-\frac{35 a^4}{4 b^8 \left(a+b x^2\right)^2}+\frac{35 a^3}{2 b^8 \left(a+b x^2\right)}+\frac{21 a^2 \log \left(a+b x^2\right)}{2 b^8}-\frac{3 a x^2}{b^7}+\frac{x^4}{4 b^6}",1,"(459*a^7 + 1875*a^6*b*x^2 + 2700*a^5*b^2*x^4 + 1300*a^4*b^3*x^6 - 400*a^3*b^4*x^8 - 500*a^2*b^5*x^10 - 70*a*b^6*x^12 + 10*b^7*x^14 + 420*a^2*(a + b*x^2)^5*Log[a + b*x^2])/(40*b^8*(a + b*x^2)^5)","A",1
513,1,101,118,0.0280764,"\int \frac{x^{13}}{\left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[x^13/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","-\frac{87 a^6+375 a^5 b x^2+600 a^4 b^2 x^4+400 a^3 b^3 x^6+50 a^2 b^4 x^8-50 a b^5 x^{10}+60 a \left(a+b x^2\right)^5 \log \left(a+b x^2\right)-10 b^6 x^{12}}{20 b^7 \left(a+b x^2\right)^5}","-\frac{a^6}{10 b^7 \left(a+b x^2\right)^5}+\frac{3 a^5}{4 b^7 \left(a+b x^2\right)^4}-\frac{5 a^4}{2 b^7 \left(a+b x^2\right)^3}+\frac{5 a^3}{b^7 \left(a+b x^2\right)^2}-\frac{15 a^2}{2 b^7 \left(a+b x^2\right)}-\frac{3 a \log \left(a+b x^2\right)}{b^7}+\frac{x^2}{2 b^6}",1,"-1/20*(87*a^6 + 375*a^5*b*x^2 + 600*a^4*b^2*x^4 + 400*a^3*b^3*x^6 + 50*a^2*b^4*x^8 - 50*a*b^5*x^10 - 10*b^6*x^12 + 60*a*(a + b*x^2)^5*Log[a + b*x^2])/(b^7*(a + b*x^2)^5)","A",1
514,1,72,109,0.0243207,"\int \frac{x^{11}}{\left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[x^11/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{\frac{a \left(137 a^4+625 a^3 b x^2+1100 a^2 b^2 x^4+900 a b^3 x^6+300 b^4 x^8\right)}{\left(a+b x^2\right)^5}+60 \log \left(a+b x^2\right)}{120 b^6}","\frac{a^5}{10 b^6 \left(a+b x^2\right)^5}-\frac{5 a^4}{8 b^6 \left(a+b x^2\right)^4}+\frac{5 a^3}{3 b^6 \left(a+b x^2\right)^3}-\frac{5 a^2}{2 b^6 \left(a+b x^2\right)^2}+\frac{5 a}{2 b^6 \left(a+b x^2\right)}+\frac{\log \left(a+b x^2\right)}{2 b^6}",1,"((a*(137*a^4 + 625*a^3*b*x^2 + 1100*a^2*b^2*x^4 + 900*a*b^3*x^6 + 300*b^4*x^8))/(a + b*x^2)^5 + 60*Log[a + b*x^2])/(120*b^6)","A",1
515,1,57,19,0.0157746,"\int \frac{x^9}{\left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[x^9/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","-\frac{a^4+5 a^3 b x^2+10 a^2 b^2 x^4+10 a b^3 x^6+5 b^4 x^8}{10 b^5 \left(a+b x^2\right)^5}","\frac{x^{10}}{10 a \left(a+b x^2\right)^5}",1,"-1/10*(a^4 + 5*a^3*b*x^2 + 10*a^2*b^2*x^4 + 10*a*b^3*x^6 + 5*b^4*x^8)/(b^5*(a + b*x^2)^5)","B",1
516,1,46,39,0.0139007,"\int \frac{x^7}{\left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[x^7/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","-\frac{a^3+5 a^2 b x^2+10 a b^2 x^4+10 b^3 x^6}{40 b^4 \left(a+b x^2\right)^5}","\frac{x^8}{40 a^2 \left(a+b x^2\right)^4}+\frac{x^8}{10 a \left(a+b x^2\right)^5}",1,"-1/40*(a^3 + 5*a^2*b*x^2 + 10*a*b^2*x^4 + 10*b^3*x^6)/(b^4*(a + b*x^2)^5)","A",1
517,1,35,53,0.0131153,"\int \frac{x^5}{\left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[x^5/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","-\frac{a^2+5 a b x^2+10 b^2 x^4}{60 b^3 \left(a+b x^2\right)^5}","-\frac{a^2}{10 b^3 \left(a+b x^2\right)^5}+\frac{a}{4 b^3 \left(a+b x^2\right)^4}-\frac{1}{6 b^3 \left(a+b x^2\right)^3}",1,"-1/60*(a^2 + 5*a*b*x^2 + 10*b^2*x^4)/(b^3*(a + b*x^2)^5)","A",1
518,1,24,34,0.0079362,"\int \frac{x^3}{\left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[x^3/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","-\frac{a+5 b x^2}{40 b^2 \left(a+b x^2\right)^5}","\frac{a}{10 b^2 \left(a+b x^2\right)^5}-\frac{1}{8 b^2 \left(a+b x^2\right)^4}",1,"-1/40*(a + 5*b*x^2)/(b^2*(a + b*x^2)^5)","A",1
519,1,16,16,0.0030093,"\int \frac{x}{\left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[x/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","-\frac{1}{10 b \left(a+b x^2\right)^5}","-\frac{1}{10 b \left(a+b x^2\right)^5}",1,"-1/10*1/(b*(a + b*x^2)^5)","A",1
520,1,76,102,0.0591944,"\int \frac{1}{x \left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[1/(x*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]","\frac{\frac{a \left(137 a^4+385 a^3 b x^2+470 a^2 b^2 x^4+270 a b^3 x^6+60 b^4 x^8\right)}{\left(a+b x^2\right)^5}-60 \log \left(a+b x^2\right)+120 \log (x)}{120 a^6}","-\frac{\log \left(a+b x^2\right)}{2 a^6}+\frac{\log (x)}{a^6}+\frac{1}{2 a^5 \left(a+b x^2\right)}+\frac{1}{4 a^4 \left(a+b x^2\right)^2}+\frac{1}{6 a^3 \left(a+b x^2\right)^3}+\frac{1}{8 a^2 \left(a+b x^2\right)^4}+\frac{1}{10 a \left(a+b x^2\right)^5}",1,"((a*(137*a^4 + 385*a^3*b*x^2 + 470*a^2*b^2*x^4 + 270*a*b^3*x^6 + 60*b^4*x^8))/(a + b*x^2)^5 + 120*Log[x] - 60*Log[a + b*x^2])/(120*a^6)","A",1
521,1,92,116,0.0844684,"\int \frac{1}{x^3 \left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[1/(x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]","-\frac{\frac{a \left(10 a^5+137 a^4 b x^2+385 a^3 b^2 x^4+470 a^2 b^3 x^6+270 a b^4 x^8+60 b^5 x^{10}\right)}{x^2 \left(a+b x^2\right)^5}-60 b \log \left(a+b x^2\right)+120 b \log (x)}{20 a^7}","\frac{3 b \log \left(a+b x^2\right)}{a^7}-\frac{6 b \log (x)}{a^7}-\frac{5 b}{2 a^6 \left(a+b x^2\right)}-\frac{1}{2 a^6 x^2}-\frac{b}{a^5 \left(a+b x^2\right)^2}-\frac{b}{2 a^4 \left(a+b x^2\right)^3}-\frac{b}{4 a^3 \left(a+b x^2\right)^4}-\frac{b}{10 a^2 \left(a+b x^2\right)^5}",1,"-1/20*((a*(10*a^5 + 137*a^4*b*x^2 + 385*a^3*b^2*x^4 + 470*a^2*b^3*x^6 + 270*a*b^4*x^8 + 60*b^5*x^10))/(x^2*(a + b*x^2)^5) + 120*b*Log[x] - 60*b*Log[a + b*x^2])/a^7","A",1
522,1,107,140,0.061247,"\int \frac{1}{x^5 \left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[1/(x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]","\frac{\frac{a \left(-10 a^6+70 a^5 b x^2+959 a^4 b^2 x^4+2695 a^3 b^3 x^6+3290 a^2 b^4 x^8+1890 a b^5 x^{10}+420 b^6 x^{12}\right)}{x^4 \left(a+b x^2\right)^5}-420 b^2 \log \left(a+b x^2\right)+840 b^2 \log (x)}{40 a^8}","-\frac{21 b^2 \log \left(a+b x^2\right)}{2 a^8}+\frac{21 b^2 \log (x)}{a^8}+\frac{15 b^2}{2 a^7 \left(a+b x^2\right)}+\frac{3 b}{a^7 x^2}+\frac{5 b^2}{2 a^6 \left(a+b x^2\right)^2}-\frac{1}{4 a^6 x^4}+\frac{b^2}{a^5 \left(a+b x^2\right)^3}+\frac{3 b^2}{8 a^4 \left(a+b x^2\right)^4}+\frac{b^2}{10 a^3 \left(a+b x^2\right)^5}",1,"((a*(-10*a^6 + 70*a^5*b*x^2 + 959*a^4*b^2*x^4 + 2695*a^3*b^3*x^6 + 3290*a^2*b^4*x^8 + 1890*a*b^5*x^10 + 420*b^6*x^12))/(x^4*(a + b*x^2)^5) + 840*b^2*Log[x] - 420*b^2*Log[a + b*x^2])/(40*a^8)","A",1
523,1,122,155,0.0641874,"\int \frac{x^{16}}{\left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[x^16/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{\frac{\sqrt{b} x \left(45045 a^7+210210 a^6 b x^2+384384 a^5 b^2 x^4+338910 a^4 b^3 x^6+137995 a^3 b^4 x^8+16640 a^2 b^5 x^{10}-1280 a b^6 x^{12}+256 b^7 x^{14}\right)}{\left(a+b x^2\right)^5}-45045 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{1280 b^{17/2}}","-\frac{9009 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{256 b^{17/2}}+\frac{9009 a^2 x}{256 b^8}-\frac{3003 a x^3}{256 b^7}-\frac{1287 x^7}{256 b^5 \left(a+b x^2\right)}-\frac{143 x^9}{128 b^4 \left(a+b x^2\right)^2}-\frac{13 x^{11}}{32 b^3 \left(a+b x^2\right)^3}-\frac{3 x^{13}}{16 b^2 \left(a+b x^2\right)^4}-\frac{x^{15}}{10 b \left(a+b x^2\right)^5}+\frac{9009 x^5}{1280 b^6}",1,"((Sqrt[b]*x*(45045*a^7 + 210210*a^6*b*x^2 + 384384*a^5*b^2*x^4 + 338910*a^4*b^3*x^6 + 137995*a^3*b^4*x^8 + 16640*a^2*b^5*x^10 - 1280*a*b^6*x^12 + 256*b^7*x^14))/(a + b*x^2)^5 - 45045*a^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(1280*b^(17/2))","A",1
524,1,111,142,0.0579645,"\int \frac{x^{14}}{\left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[x^14/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{45045 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)+\frac{\sqrt{b} x \left(-45045 a^6-210210 a^5 b x^2-384384 a^4 b^2 x^4-338910 a^3 b^3 x^6-137995 a^2 b^4 x^8-16640 a b^5 x^{10}+1280 b^6 x^{12}\right)}{\left(a+b x^2\right)^5}}{3840 b^{15/2}}","\frac{3003 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{256 b^{15/2}}-\frac{3003 a x}{256 b^7}-\frac{3003 x^5}{1280 b^5 \left(a+b x^2\right)}-\frac{429 x^7}{640 b^4 \left(a+b x^2\right)^2}-\frac{143 x^9}{480 b^3 \left(a+b x^2\right)^3}-\frac{13 x^{11}}{80 b^2 \left(a+b x^2\right)^4}-\frac{x^{13}}{10 b \left(a+b x^2\right)^5}+\frac{1001 x^3}{256 b^6}",1,"((Sqrt[b]*x*(-45045*a^6 - 210210*a^5*b*x^2 - 384384*a^4*b^2*x^4 - 338910*a^3*b^3*x^6 - 137995*a^2*b^4*x^8 - 16640*a*b^5*x^10 + 1280*b^6*x^12))/(a + b*x^2)^5 + 45045*a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(3840*b^(15/2))","A",1
525,1,100,131,0.0516618,"\int \frac{x^{12}}{\left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[x^12/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{\frac{\sqrt{b} x \left(3465 a^5+16170 a^4 b x^2+29568 a^3 b^2 x^4+26070 a^2 b^3 x^6+10615 a b^4 x^8+1280 b^5 x^{10}\right)}{\left(a+b x^2\right)^5}-3465 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{1280 b^{13/2}}","-\frac{693 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{256 b^{13/2}}-\frac{231 x^3}{256 b^5 \left(a+b x^2\right)}-\frac{231 x^5}{640 b^4 \left(a+b x^2\right)^2}-\frac{33 x^7}{160 b^3 \left(a+b x^2\right)^3}-\frac{11 x^9}{80 b^2 \left(a+b x^2\right)^4}-\frac{x^{11}}{10 b \left(a+b x^2\right)^5}+\frac{693 x}{256 b^6}",1,"((Sqrt[b]*x*(3465*a^5 + 16170*a^4*b*x^2 + 29568*a^3*b^2*x^4 + 26070*a^2*b^3*x^6 + 10615*a*b^4*x^8 + 1280*b^5*x^10))/(a + b*x^2)^5 - 3465*Sqrt[a]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(1280*b^(13/2))","A",1
526,1,88,121,0.049739,"\int \frac{x^{10}}{\left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[x^10/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{63 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{256 \sqrt{a} b^{11/2}}-\frac{x \left(315 a^4+1470 a^3 b x^2+2688 a^2 b^2 x^4+2370 a b^3 x^6+965 b^4 x^8\right)}{1280 b^5 \left(a+b x^2\right)^5}","\frac{63 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{256 \sqrt{a} b^{11/2}}-\frac{63 x}{256 b^5 \left(a+b x^2\right)}-\frac{21 x^3}{128 b^4 \left(a+b x^2\right)^2}-\frac{21 x^5}{160 b^3 \left(a+b x^2\right)^3}-\frac{9 x^7}{80 b^2 \left(a+b x^2\right)^4}-\frac{x^9}{10 b \left(a+b x^2\right)^5}",1,"-1/1280*(x*(315*a^4 + 1470*a^3*b*x^2 + 2688*a^2*b^2*x^4 + 2370*a*b^3*x^6 + 965*b^4*x^8))/(b^5*(a + b*x^2)^5) + (63*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(256*Sqrt[a]*b^(11/2))","A",1
527,1,91,122,0.0577961,"\int \frac{x^8}{\left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[x^8/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{7 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{256 a^{3/2} b^{9/2}}-\frac{x \left(105 a^4+490 a^3 b x^2+896 a^2 b^2 x^4+790 a b^3 x^6-105 b^4 x^8\right)}{3840 a b^4 \left(a+b x^2\right)^5}","\frac{7 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{256 a^{3/2} b^{9/2}}+\frac{7 x}{256 a b^4 \left(a+b x^2\right)}-\frac{7 x}{128 b^4 \left(a+b x^2\right)^2}-\frac{7 x^3}{96 b^3 \left(a+b x^2\right)^3}-\frac{7 x^5}{80 b^2 \left(a+b x^2\right)^4}-\frac{x^7}{10 b \left(a+b x^2\right)^5}",1,"-1/3840*(x*(105*a^4 + 490*a^3*b*x^2 + 896*a^2*b^2*x^4 + 790*a*b^3*x^6 - 105*b^4*x^8))/(a*b^4*(a + b*x^2)^5) + (7*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(256*a^(3/2)*b^(9/2))","A",1
528,1,91,123,0.0597282,"\int \frac{x^6}{\left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[x^6/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{3 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{256 a^{5/2} b^{7/2}}+\frac{-15 a^4 x-70 a^3 b x^3-128 a^2 b^2 x^5+70 a b^3 x^7+15 b^4 x^9}{1280 a^2 b^3 \left(a+b x^2\right)^5}","\frac{3 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{256 a^{5/2} b^{7/2}}+\frac{3 x}{256 a^2 b^3 \left(a+b x^2\right)}+\frac{x}{128 a b^3 \left(a+b x^2\right)^2}-\frac{x}{32 b^3 \left(a+b x^2\right)^3}-\frac{x^3}{16 b^2 \left(a+b x^2\right)^4}-\frac{x^5}{10 b \left(a+b x^2\right)^5}",1,"(-15*a^4*x - 70*a^3*b*x^3 - 128*a^2*b^2*x^5 + 70*a*b^3*x^7 + 15*b^4*x^9)/(1280*a^2*b^3*(a + b*x^2)^5) + (3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(256*a^(5/2)*b^(7/2))","A",1
529,1,91,124,0.0496239,"\int \frac{x^4}{\left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[x^4/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{3 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{256 a^{7/2} b^{5/2}}+\frac{-15 a^4 x-70 a^3 b x^3+128 a^2 b^2 x^5+70 a b^3 x^7+15 b^4 x^9}{1280 a^3 b^2 \left(a+b x^2\right)^5}","\frac{3 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{256 a^{7/2} b^{5/2}}+\frac{3 x}{256 a^3 b^2 \left(a+b x^2\right)}+\frac{x}{128 a^2 b^2 \left(a+b x^2\right)^2}+\frac{x}{160 a b^2 \left(a+b x^2\right)^3}-\frac{3 x}{80 b^2 \left(a+b x^2\right)^4}-\frac{x^3}{10 b \left(a+b x^2\right)^5}",1,"(-15*a^4*x - 70*a^3*b*x^3 + 128*a^2*b^2*x^5 + 70*a*b^3*x^7 + 15*b^4*x^9)/(1280*a^3*b^2*(a + b*x^2)^5) + (3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(256*a^(7/2)*b^(5/2))","A",1
530,1,91,125,0.0513143,"\int \frac{x^2}{\left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[x^2/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{7 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{256 a^{9/2} b^{3/2}}+\frac{-105 a^4 x+790 a^3 b x^3+896 a^2 b^2 x^5+490 a b^3 x^7+105 b^4 x^9}{3840 a^4 b \left(a+b x^2\right)^5}","\frac{7 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{256 a^{9/2} b^{3/2}}+\frac{7 x}{256 a^4 b \left(a+b x^2\right)}+\frac{7 x}{384 a^3 b \left(a+b x^2\right)^2}+\frac{7 x}{480 a^2 b \left(a+b x^2\right)^3}+\frac{x}{80 a b \left(a+b x^2\right)^4}-\frac{x}{10 b \left(a+b x^2\right)^5}",1,"(-105*a^4*x + 790*a^3*b*x^3 + 896*a^2*b^2*x^5 + 490*a*b^3*x^7 + 105*b^4*x^9)/(3840*a^4*b*(a + b*x^2)^5) + (7*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(256*a^(9/2)*b^(3/2))","A",1
531,1,89,113,0.0450213,"\int \frac{1}{\left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(-3),x]","\frac{\frac{\sqrt{a} x \left(965 a^4+2370 a^3 b x^2+2688 a^2 b^2 x^4+1470 a b^3 x^6+315 b^4 x^8\right)}{\left(a+b x^2\right)^5}+\frac{315 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{b}}}{1280 a^{11/2}}","\frac{63 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{256 a^{11/2} \sqrt{b}}+\frac{63 x}{256 a^5 \left(a+b x^2\right)}+\frac{21 x}{128 a^4 \left(a+b x^2\right)^2}+\frac{21 x}{160 a^3 \left(a+b x^2\right)^3}+\frac{9 x}{80 a^2 \left(a+b x^2\right)^4}+\frac{x}{10 a \left(a+b x^2\right)^5}",1,"((Sqrt[a]*x*(965*a^4 + 2370*a^3*b*x^2 + 2688*a^2*b^2*x^4 + 1470*a*b^3*x^6 + 315*b^4*x^8))/(a + b*x^2)^5 + (315*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[b])/(1280*a^(11/2))","A",1
532,1,101,133,0.0553185,"\int \frac{1}{x^2 \left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[1/(x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]","-\frac{693 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{256 a^{13/2}}-\frac{1280 a^5+10615 a^4 b x^2+26070 a^3 b^2 x^4+29568 a^2 b^3 x^6+16170 a b^4 x^8+3465 b^5 x^{10}}{1280 a^6 x \left(a+b x^2\right)^5}","-\frac{693 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{256 a^{13/2}}-\frac{693}{256 a^6 x}+\frac{231}{256 a^5 x \left(a+b x^2\right)}+\frac{231}{640 a^4 x \left(a+b x^2\right)^2}+\frac{33}{160 a^3 x \left(a+b x^2\right)^3}+\frac{11}{80 a^2 x \left(a+b x^2\right)^4}+\frac{1}{10 a x \left(a+b x^2\right)^5}",1,"-1/1280*(1280*a^5 + 10615*a^4*b*x^2 + 26070*a^3*b^2*x^4 + 29568*a^2*b^3*x^6 + 16170*a*b^4*x^8 + 3465*b^5*x^10)/(a^6*x*(a + b*x^2)^5) - (693*Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(256*a^(13/2))","A",1
533,1,113,144,0.060429,"\int \frac{1}{x^4 \left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[1/(x^4*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]","\frac{\frac{\sqrt{a} \left(-1280 a^6+16640 a^5 b x^2+137995 a^4 b^2 x^4+338910 a^3 b^3 x^6+384384 a^2 b^4 x^8+210210 a b^5 x^{10}+45045 b^6 x^{12}\right)}{x^3 \left(a+b x^2\right)^5}+45045 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{3840 a^{15/2}}","\frac{3003 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{256 a^{15/2}}+\frac{3003 b}{256 a^7 x}-\frac{1001}{256 a^6 x^3}+\frac{3003}{1280 a^5 x^3 \left(a+b x^2\right)}+\frac{429}{640 a^4 x^3 \left(a+b x^2\right)^2}+\frac{143}{480 a^3 x^3 \left(a+b x^2\right)^3}+\frac{13}{80 a^2 x^3 \left(a+b x^2\right)^4}+\frac{1}{10 a x^3 \left(a+b x^2\right)^5}",1,"((Sqrt[a]*(-1280*a^6 + 16640*a^5*b*x^2 + 137995*a^4*b^2*x^4 + 338910*a^3*b^3*x^6 + 384384*a^2*b^4*x^8 + 210210*a*b^5*x^10 + 45045*b^6*x^12))/(x^3*(a + b*x^2)^5) + 45045*b^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(3840*a^(15/2))","A",1
534,1,123,157,0.0631589,"\int \frac{1}{x^6 \left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[1/(x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]","-\frac{9009 b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{256 a^{17/2}}-\frac{256 a^7-1280 a^6 b x^2+16640 a^5 b^2 x^4+137995 a^4 b^3 x^6+338910 a^3 b^4 x^8+384384 a^2 b^5 x^{10}+210210 a b^6 x^{12}+45045 b^7 x^{14}}{1280 a^8 x^5 \left(a+b x^2\right)^5}","-\frac{9009 b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{256 a^{17/2}}-\frac{9009 b^2}{256 a^8 x}+\frac{3003 b}{256 a^7 x^3}-\frac{9009}{1280 a^6 x^5}+\frac{1287}{256 a^5 x^5 \left(a+b x^2\right)}+\frac{143}{128 a^4 x^5 \left(a+b x^2\right)^2}+\frac{13}{32 a^3 x^5 \left(a+b x^2\right)^3}+\frac{3}{16 a^2 x^5 \left(a+b x^2\right)^4}+\frac{1}{10 a x^5 \left(a+b x^2\right)^5}",1,"-1/1280*(256*a^7 - 1280*a^6*b*x^2 + 16640*a^5*b^2*x^4 + 137995*a^4*b^3*x^6 + 338910*a^3*b^4*x^8 + 384384*a^2*b^5*x^10 + 210210*a*b^6*x^12 + 45045*b^7*x^14)/(a^8*x^5*(a + b*x^2)^5) - (9009*b^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(256*a^(17/2))","A",1
535,1,16,19,0.0060081,"\int \frac{1}{1+2 x^2+x^4} \, dx","Integrate[(1 + 2*x^2 + x^4)^(-1),x]","\frac{1}{2} \left(\frac{x}{x^2+1}+\tan ^{-1}(x)\right)","\frac{x}{2 \left(x^2+1\right)}+\frac{1}{2} \tan ^{-1}(x)",1,"(x/(1 + x^2) + ArcTan[x])/2","A",1
536,1,11,11,0.0011984,"\int \frac{x}{1+2 x^2+x^4} \, dx","Integrate[x/(1 + 2*x^2 + x^4),x]","-\frac{1}{2 \left(x^2+1\right)}","-\frac{1}{2 \left(x^2+1\right)}",1,"-1/2*1/(1 + x^2)","A",1
537,1,19,19,0.0078622,"\int \frac{x^2}{1+2 x^2+x^4} \, dx","Integrate[x^2/(1 + 2*x^2 + x^4),x]","\frac{1}{2} \tan ^{-1}(x)-\frac{x}{2 \left(x^2+1\right)}","\frac{1}{2} \tan ^{-1}(x)-\frac{x}{2 \left(x^2+1\right)}",1,"-1/2*x/(1 + x^2) + ArcTan[x]/2","A",1
538,1,18,22,0.0046558,"\int \frac{x^3}{1+2 x^2+x^4} \, dx","Integrate[x^3/(1 + 2*x^2 + x^4),x]","\frac{1}{2} \left(\frac{1}{x^2+1}+\log \left(x^2+1\right)\right)","\frac{1}{2 \left(x^2+1\right)}+\frac{1}{2} \log \left(x^2+1\right)",1,"((1 + x^2)^(-1) + Log[1 + x^2])/2","A",1
539,1,11,13,0.0019186,"\int \frac{x}{81-18 x^2+x^4} \, dx","Integrate[x/(81 - 18*x^2 + x^4),x]","-\frac{1}{2 \left(x^2-9\right)}","\frac{1}{2 \left(9-x^2\right)}",1,"-1/2*1/(-9 + x^2)","A",1
540,1,20,24,0.0056055,"\int \frac{x^3}{16-8 x^2+x^4} \, dx","Integrate[x^3/(16 - 8*x^2 + x^4),x]","\frac{1}{2} \log \left(x^2-4\right)-\frac{2}{x^2-4}","\frac{2}{4-x^2}+\frac{1}{2} \log \left(4-x^2\right)",1,"-2/(-4 + x^2) + Log[-4 + x^2]/2","A",1
541,1,39,79,0.014849,"\int x^5 \sqrt{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[x^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]","\frac{\sqrt{\left(a+b x^2\right)^2} \left(4 a x^6+3 b x^8\right)}{24 \left(a+b x^2\right)}","\frac{b x^8 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 \left(a+b x^2\right)}+\frac{a x^6 \sqrt{a^2+2 a b x^2+b^2 x^4}}{6 \left(a+b x^2\right)}",1,"(Sqrt[(a + b*x^2)^2]*(4*a*x^6 + 3*b*x^8))/(24*(a + b*x^2))","A",1
542,1,39,67,0.0074979,"\int x^3 \sqrt{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[x^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]","\frac{\sqrt{\left(a+b x^2\right)^2} \left(3 a x^4+2 b x^6\right)}{12 \left(a+b x^2\right)}","\frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}{6 b^2}-\frac{a \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 b^2}",1,"(Sqrt[(a + b*x^2)^2]*(3*a*x^4 + 2*b*x^6))/(12*(a + b*x^2))","A",1
543,1,38,36,0.0072984,"\int x \sqrt{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[x*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]","\frac{\sqrt{\left(a+b x^2\right)^2} \left(2 a x^2+b x^4\right)}{4 \left(a+b x^2\right)}","\frac{\left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 b}",1,"(Sqrt[(a + b*x^2)^2]*(2*a*x^2 + b*x^4))/(4*(a + b*x^2))","A",1
544,1,37,75,0.0115136,"\int \frac{\sqrt{a^2+2 a b x^2+b^2 x^4}}{x} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/x,x]","\frac{\sqrt{\left(a+b x^2\right)^2} \left(2 a \log (x)+b x^2\right)}{2 \left(a+b x^2\right)}","\frac{b x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 \left(a+b x^2\right)}+\frac{a \log (x) \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}",1,"(Sqrt[(a + b*x^2)^2]*(b*x^2 + 2*a*Log[x]))/(2*(a + b*x^2))","A",1
545,1,39,75,0.0103291,"\int \frac{\sqrt{a^2+2 a b x^2+b^2 x^4}}{x^3} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/x^3,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(a-2 b x^2 \log (x)\right)}{2 x^2 \left(a+b x^2\right)}","\frac{b \log (x) \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 x^2 \left(a+b x^2\right)}",1,"-1/2*(Sqrt[(a + b*x^2)^2]*(a - 2*b*x^2*Log[x]))/(x^2*(a + b*x^2))","A",1
546,1,37,39,0.0077422,"\int \frac{\sqrt{a^2+2 a b x^2+b^2 x^4}}{x^5} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/x^5,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(a+2 b x^2\right)}{4 x^4 \left(a+b x^2\right)}","-\frac{\left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 a x^4}",1,"-1/4*(Sqrt[(a + b*x^2)^2]*(a + 2*b*x^2))/(x^4*(a + b*x^2))","A",1
547,1,39,72,0.0080075,"\int \frac{\sqrt{a^2+2 a b x^2+b^2 x^4}}{x^7} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/x^7,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(2 a+3 b x^2\right)}{12 x^6 \left(a+b x^2\right)}","\frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}{12 a^2 x^6}-\frac{\left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 a x^6}",1,"-1/12*(Sqrt[(a + b*x^2)^2]*(2*a + 3*b*x^2))/(x^6*(a + b*x^2))","A",1
548,1,39,79,0.007908,"\int \frac{\sqrt{a^2+2 a b x^2+b^2 x^4}}{x^9} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/x^9,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(3 a+4 b x^2\right)}{24 x^8 \left(a+b x^2\right)}","-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 x^8 \left(a+b x^2\right)}-\frac{b \sqrt{a^2+2 a b x^2+b^2 x^4}}{6 x^6 \left(a+b x^2\right)}",1,"-1/24*(Sqrt[(a + b*x^2)^2]*(3*a + 4*b*x^2))/(x^8*(a + b*x^2))","A",1
549,1,39,79,0.008221,"\int \frac{\sqrt{a^2+2 a b x^2+b^2 x^4}}{x^{11}} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/x^11,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(4 a+5 b x^2\right)}{40 x^{10} \left(a+b x^2\right)}","-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4}}{10 x^{10} \left(a+b x^2\right)}-\frac{b \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 x^8 \left(a+b x^2\right)}",1,"-1/40*(Sqrt[(a + b*x^2)^2]*(4*a + 5*b*x^2))/(x^10*(a + b*x^2))","A",1
550,1,39,79,0.006704,"\int x^4 \sqrt{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[x^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]","\frac{\sqrt{\left(a+b x^2\right)^2} \left(7 a x^5+5 b x^7\right)}{35 \left(a+b x^2\right)}","\frac{b x^7 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 \left(a+b x^2\right)}+\frac{a x^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 \left(a+b x^2\right)}",1,"(Sqrt[(a + b*x^2)^2]*(7*a*x^5 + 5*b*x^7))/(35*(a + b*x^2))","A",1
551,1,39,79,0.0070555,"\int x^2 \sqrt{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[x^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]","\frac{\sqrt{\left(a+b x^2\right)^2} \left(5 a x^3+3 b x^5\right)}{15 \left(a+b x^2\right)}","\frac{b x^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 \left(a+b x^2\right)}+\frac{a x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left(a+b x^2\right)}",1,"(Sqrt[(a + b*x^2)^2]*(5*a*x^3 + 3*b*x^5))/(15*(a + b*x^2))","A",1
552,1,36,74,0.0068063,"\int \sqrt{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]","\frac{\sqrt{\left(a+b x^2\right)^2} \left(3 a x+b x^3\right)}{3 \left(a+b x^2\right)}","\frac{a x \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac{b x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left(a+b x^2\right)}",1,"(Sqrt[(a + b*x^2)^2]*(3*a*x + b*x^3))/(3*(a + b*x^2))","A",1
553,1,35,72,0.0083681,"\int \frac{\sqrt{a^2+2 a b x^2+b^2 x^4}}{x^2} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/x^2,x]","\frac{\left(b x^2-a\right) \sqrt{\left(a+b x^2\right)^2}}{x \left(a+b x^2\right)}","\frac{b x \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left(a+b x^2\right)}",1,"((-a + b*x^2)*Sqrt[(a + b*x^2)^2])/(x*(a + b*x^2))","A",1
554,1,37,77,0.0069362,"\int \frac{\sqrt{a^2+2 a b x^2+b^2 x^4}}{x^4} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/x^4,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(a+3 b x^2\right)}{3 x^3 \left(a+b x^2\right)}","-\frac{b \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left(a+b x^2\right)}-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left(a+b x^2\right)}",1,"-1/3*(Sqrt[(a + b*x^2)^2]*(a + 3*b*x^2))/(x^3*(a + b*x^2))","A",1
555,1,39,79,0.0084404,"\int \frac{\sqrt{a^2+2 a b x^2+b^2 x^4}}{x^6} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/x^6,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(3 a+5 b x^2\right)}{15 x^5 \left(a+b x^2\right)}","-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left(a+b x^2\right)}-\frac{b \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left(a+b x^2\right)}",1,"-1/15*(Sqrt[(a + b*x^2)^2]*(3*a + 5*b*x^2))/(x^5*(a + b*x^2))","A",1
556,1,39,79,0.0083739,"\int \frac{\sqrt{a^2+2 a b x^2+b^2 x^4}}{x^8} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/x^8,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(5 a+7 b x^2\right)}{35 x^7 \left(a+b x^2\right)}","-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left(a+b x^2\right)}-\frac{b \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left(a+b x^2\right)}",1,"-1/35*(Sqrt[(a + b*x^2)^2]*(5*a + 7*b*x^2))/(x^7*(a + b*x^2))","A",1
557,1,39,79,0.0083395,"\int \frac{\sqrt{a^2+2 a b x^2+b^2 x^4}}{x^{10}} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/x^10,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(7 a+9 b x^2\right)}{63 x^9 \left(a+b x^2\right)}","-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 x^9 \left(a+b x^2\right)}-\frac{b \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left(a+b x^2\right)}",1,"-1/63*(Sqrt[(a + b*x^2)^2]*(7*a + 9*b*x^2))/(x^9*(a + b*x^2))","A",1
558,1,61,167,0.0191219,"\int x^9 \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2} \, dx","Integrate[x^9*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","\frac{x^{10} \sqrt{\left(a+b x^2\right)^2} \left(56 a^3+140 a^2 b x^2+120 a b^2 x^4+35 b^3 x^6\right)}{560 \left(a+b x^2\right)}","\frac{3 a b^2 x^{14} \sqrt{a^2+2 a b x^2+b^2 x^4}}{14 \left(a+b x^2\right)}+\frac{a^2 b x^{12} \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 \left(a+b x^2\right)}+\frac{b^3 x^{16} \sqrt{a^2+2 a b x^2+b^2 x^4}}{16 \left(a+b x^2\right)}+\frac{a^3 x^{10} \sqrt{a^2+2 a b x^2+b^2 x^4}}{10 \left(a+b x^2\right)}",1,"(x^10*Sqrt[(a + b*x^2)^2]*(56*a^3 + 140*a^2*b*x^2 + 120*a*b^2*x^4 + 35*b^3*x^6))/(560*(a + b*x^2))","A",1
559,1,61,167,0.0156552,"\int x^7 \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2} \, dx","Integrate[x^7*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","\frac{x^8 \sqrt{\left(a+b x^2\right)^2} \left(35 a^3+84 a^2 b x^2+70 a b^2 x^4+20 b^3 x^6\right)}{280 \left(a+b x^2\right)}","\frac{a b^2 x^{12} \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 \left(a+b x^2\right)}+\frac{3 a^2 b x^{10} \sqrt{a^2+2 a b x^2+b^2 x^4}}{10 \left(a+b x^2\right)}+\frac{b^3 x^{14} \sqrt{a^2+2 a b x^2+b^2 x^4}}{14 \left(a+b x^2\right)}+\frac{a^3 x^8 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 \left(a+b x^2\right)}",1,"(x^8*Sqrt[(a + b*x^2)^2]*(35*a^3 + 84*a^2*b*x^2 + 70*a*b^2*x^4 + 20*b^3*x^6))/(280*(a + b*x^2))","A",1
560,1,61,106,0.0157844,"\int x^5 \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2} \, dx","Integrate[x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","\frac{x^6 \sqrt{\left(a+b x^2\right)^2} \left(20 a^3+45 a^2 b x^2+36 a b^2 x^4+10 b^3 x^6\right)}{120 \left(a+b x^2\right)}","\frac{\left(a+b x^2\right) \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{12 b^3}-\frac{a \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{5 b^3}+\frac{a^2 \left(a+b x^2\right) \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}{8 b^3}",1,"(x^6*Sqrt[(a + b*x^2)^2]*(20*a^3 + 45*a^2*b*x^2 + 36*a*b^2*x^4 + 10*b^3*x^6))/(120*(a + b*x^2))","A",1
561,1,61,67,0.0159923,"\int x^3 \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2} \, dx","Integrate[x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","\frac{x^4 \sqrt{\left(a+b x^2\right)^2} \left(10 a^3+20 a^2 b x^2+15 a b^2 x^4+4 b^3 x^6\right)}{40 \left(a+b x^2\right)}","\frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{10 b^2}-\frac{a \left(a+b x^2\right) \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}{8 b^2}",1,"(x^4*Sqrt[(a + b*x^2)^2]*(10*a^3 + 20*a^2*b*x^2 + 15*a*b^2*x^4 + 4*b^3*x^6))/(40*(a + b*x^2))","A",1
562,1,27,36,0.0121654,"\int x \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2} \, dx","Integrate[x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","\frac{\left(a+b x^2\right) \left(\left(a+b x^2\right)^2\right)^{3/2}}{8 b}","\frac{\left(a+b x^2\right) \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}{8 b}",1,"((a + b*x^2)*((a + b*x^2)^2)^(3/2))/(8*b)","A",1
563,1,60,163,0.0211578,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}{x} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x,x]","\frac{\sqrt{\left(a+b x^2\right)^2} \left(12 a^3 \log (x)+b x^2 \left(18 a^2+9 a b x^2+2 b^2 x^4\right)\right)}{12 \left(a+b x^2\right)}","\frac{3 a b^2 x^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 \left(a+b x^2\right)}+\frac{3 a^2 b x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 \left(a+b x^2\right)}+\frac{b^3 x^6 \sqrt{a^2+2 a b x^2+b^2 x^4}}{6 \left(a+b x^2\right)}+\frac{a^3 \log (x) \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}",1,"(Sqrt[(a + b*x^2)^2]*(b*x^2*(18*a^2 + 9*a*b*x^2 + 2*b^2*x^4) + 12*a^3*Log[x]))/(12*(a + b*x^2))","A",1
564,1,62,164,0.0207536,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}{x^3} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^3,x]","\frac{\sqrt{\left(a+b x^2\right)^2} \left(-2 a^3+12 a^2 b x^2 \log (x)+6 a b^2 x^4+b^3 x^6\right)}{4 x^2 \left(a+b x^2\right)}","\frac{3 a b^2 x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 \left(a+b x^2\right)}+\frac{3 a^2 b \log (x) \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac{b^3 x^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 \left(a+b x^2\right)}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 x^2 \left(a+b x^2\right)}",1,"(Sqrt[(a + b*x^2)^2]*(-2*a^3 + 6*a*b^2*x^4 + b^3*x^6 + 12*a^2*b*x^2*Log[x]))/(4*x^2*(a + b*x^2))","A",1
565,1,61,164,0.0156846,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}{x^5} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^5,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(a^3+6 a^2 b x^2-12 a b^2 x^4 \log (x)-2 b^3 x^6\right)}{4 x^4 \left(a+b x^2\right)}","-\frac{3 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 x^2 \left(a+b x^2\right)}+\frac{3 a b^2 \log (x) \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac{b^3 x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 \left(a+b x^2\right)}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 x^4 \left(a+b x^2\right)}",1,"-1/4*(Sqrt[(a + b*x^2)^2]*(a^3 + 6*a^2*b*x^2 - 2*b^3*x^6 - 12*a*b^2*x^4*Log[x]))/(x^4*(a + b*x^2))","A",1
566,1,63,163,0.0211783,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}{x^7} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^7,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(a \left(2 a^2+9 a b x^2+18 b^2 x^4\right)-12 b^3 x^6 \log (x)\right)}{12 x^6 \left(a+b x^2\right)}","-\frac{3 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 x^4 \left(a+b x^2\right)}-\frac{3 a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 x^2 \left(a+b x^2\right)}+\frac{b^3 \log (x) \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{6 x^6 \left(a+b x^2\right)}",1,"-1/12*(Sqrt[(a + b*x^2)^2]*(a*(2*a^2 + 9*a*b*x^2 + 18*b^2*x^4) - 12*b^3*x^6*Log[x]))/(x^6*(a + b*x^2))","A",1
567,1,59,41,0.0157532,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}{x^9} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^9,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(a^3+4 a^2 b x^2+6 a b^2 x^4+4 b^3 x^6\right)}{8 x^8 \left(a+b x^2\right)}","-\frac{\left(a+b x^2\right)^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 a x^8}",1,"-1/8*(Sqrt[(a + b*x^2)^2]*(a^3 + 4*a^2*b*x^2 + 6*a*b^2*x^4 + 4*b^3*x^6))/(x^8*(a + b*x^2))","A",1
568,1,61,72,0.0143783,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}{x^{11}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^11,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(4 a^3+15 a^2 b x^2+20 a b^2 x^4+10 b^3 x^6\right)}{40 x^{10} \left(a+b x^2\right)}","\frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{40 a^2 x^{10}}-\frac{\left(a+b x^2\right) \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}{8 a x^{10}}",1,"-1/40*(Sqrt[(a + b*x^2)^2]*(4*a^3 + 15*a^2*b*x^2 + 20*a*b^2*x^4 + 10*b^3*x^6))/(x^10*(a + b*x^2))","A",1
569,1,61,167,0.0140615,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}{x^{13}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^13,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(10 a^3+36 a^2 b x^2+45 a b^2 x^4+20 b^3 x^6\right)}{120 x^{12} \left(a+b x^2\right)}","-\frac{3 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{10 x^{10} \left(a+b x^2\right)}-\frac{3 a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 x^8 \left(a+b x^2\right)}-\frac{b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{6 x^6 \left(a+b x^2\right)}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{12 x^{12} \left(a+b x^2\right)}",1,"-1/120*(Sqrt[(a + b*x^2)^2]*(10*a^3 + 36*a^2*b*x^2 + 45*a*b^2*x^4 + 20*b^3*x^6))/(x^12*(a + b*x^2))","A",1
570,1,61,167,0.0166876,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}{x^{15}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^15,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(20 a^3+70 a^2 b x^2+84 a b^2 x^4+35 b^3 x^6\right)}{280 x^{14} \left(a+b x^2\right)}","-\frac{a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 x^{12} \left(a+b x^2\right)}-\frac{3 a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{10 x^{10} \left(a+b x^2\right)}-\frac{b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 x^8 \left(a+b x^2\right)}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{14 x^{14} \left(a+b x^2\right)}",1,"-1/280*(Sqrt[(a + b*x^2)^2]*(20*a^3 + 70*a^2*b*x^2 + 84*a*b^2*x^4 + 35*b^3*x^6))/(x^14*(a + b*x^2))","A",1
571,1,61,167,0.0143548,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}{x^{17}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^17,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(35 a^3+120 a^2 b x^2+140 a b^2 x^4+56 b^3 x^6\right)}{560 x^{16} \left(a+b x^2\right)}","-\frac{3 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{14 x^{14} \left(a+b x^2\right)}-\frac{a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 x^{12} \left(a+b x^2\right)}-\frac{b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{10 x^{10} \left(a+b x^2\right)}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{16 x^{16} \left(a+b x^2\right)}",1,"-1/560*(Sqrt[(a + b*x^2)^2]*(35*a^3 + 120*a^2*b*x^2 + 140*a*b^2*x^4 + 56*b^3*x^6))/(x^16*(a + b*x^2))","A",1
572,1,61,167,0.0152283,"\int x^8 \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2} \, dx","Integrate[x^8*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","\frac{x^9 \sqrt{\left(a+b x^2\right)^2} \left(715 a^3+1755 a^2 b x^2+1485 a b^2 x^4+429 b^3 x^6\right)}{6435 \left(a+b x^2\right)}","\frac{3 a b^2 x^{13} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 \left(a+b x^2\right)}+\frac{3 a^2 b x^{11} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 \left(a+b x^2\right)}+\frac{b^3 x^{15} \sqrt{a^2+2 a b x^2+b^2 x^4}}{15 \left(a+b x^2\right)}+\frac{a^3 x^9 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 \left(a+b x^2\right)}",1,"(x^9*Sqrt[(a + b*x^2)^2]*(715*a^3 + 1755*a^2*b*x^2 + 1485*a*b^2*x^4 + 429*b^3*x^6))/(6435*(a + b*x^2))","A",1
573,1,61,167,0.0153146,"\int x^6 \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2} \, dx","Integrate[x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","\frac{x^7 \sqrt{\left(a+b x^2\right)^2} \left(429 a^3+1001 a^2 b x^2+819 a b^2 x^4+231 b^3 x^6\right)}{3003 \left(a+b x^2\right)}","\frac{3 a b^2 x^{11} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 \left(a+b x^2\right)}+\frac{a^2 b x^9 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left(a+b x^2\right)}+\frac{b^3 x^{13} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 \left(a+b x^2\right)}+\frac{a^3 x^7 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 \left(a+b x^2\right)}",1,"(x^7*Sqrt[(a + b*x^2)^2]*(429*a^3 + 1001*a^2*b*x^2 + 819*a*b^2*x^4 + 231*b^3*x^6))/(3003*(a + b*x^2))","A",1
574,1,61,167,0.0151338,"\int x^4 \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2} \, dx","Integrate[x^4*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","\frac{x^5 \sqrt{\left(a+b x^2\right)^2} \left(231 a^3+495 a^2 b x^2+385 a b^2 x^4+105 b^3 x^6\right)}{1155 \left(a+b x^2\right)}","\frac{a b^2 x^9 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left(a+b x^2\right)}+\frac{3 a^2 b x^7 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 \left(a+b x^2\right)}+\frac{b^3 x^{11} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 \left(a+b x^2\right)}+\frac{a^3 x^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 \left(a+b x^2\right)}",1,"(x^5*Sqrt[(a + b*x^2)^2]*(231*a^3 + 495*a^2*b*x^2 + 385*a*b^2*x^4 + 105*b^3*x^6))/(1155*(a + b*x^2))","A",1
575,1,61,167,0.0118954,"\int x^2 \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2} \, dx","Integrate[x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","\frac{\sqrt{\left(a+b x^2\right)^2} \left(105 a^3 x^3+189 a^2 b x^5+135 a b^2 x^7+35 b^3 x^9\right)}{315 \left(a+b x^2\right)}","\frac{3 a b^2 x^7 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 \left(a+b x^2\right)}+\frac{3 a^2 b x^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 \left(a+b x^2\right)}+\frac{b^3 x^9 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 \left(a+b x^2\right)}+\frac{a^3 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left(a+b x^2\right)}",1,"(Sqrt[(a + b*x^2)^2]*(105*a^3*x^3 + 189*a^2*b*x^5 + 135*a*b^2*x^7 + 35*b^3*x^9))/(315*(a + b*x^2))","A",1
576,1,59,159,0.0122315,"\int \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","\frac{\sqrt{\left(a+b x^2\right)^2} \left(35 a^3 x+35 a^2 b x^3+21 a b^2 x^5+5 b^3 x^7\right)}{35 \left(a+b x^2\right)}","\frac{3 a b^2 x^5 \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}{5 \left(a+b x^2\right)^3}+\frac{a^2 b x^3 \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}{\left(a+b x^2\right)^3}+\frac{b^3 x^7 \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}{7 \left(a+b x^2\right)^3}+\frac{a^3 x \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}{\left(a+b x^2\right)^3}",1,"(Sqrt[(a + b*x^2)^2]*(35*a^3*x + 35*a^2*b*x^3 + 21*a*b^2*x^5 + 5*b^3*x^7))/(35*(a + b*x^2))","A",1
577,1,60,158,0.0152761,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}{x^2} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^2,x]","\frac{\sqrt{\left(a+b x^2\right)^2} \left(-5 a^3+15 a^2 b x^2+5 a b^2 x^4+b^3 x^6\right)}{5 x \left(a+b x^2\right)}","\frac{3 a^2 b x \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac{a b^2 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac{b^3 x^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 \left(a+b x^2\right)}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left(a+b x^2\right)}",1,"(Sqrt[(a + b*x^2)^2]*(-5*a^3 + 15*a^2*b*x^2 + 5*a*b^2*x^4 + b^3*x^6))/(5*x*(a + b*x^2))","A",1
578,1,59,161,0.0139241,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}{x^4} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^4,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(a^3+9 a^2 b x^2-9 a b^2 x^4-b^3 x^6\right)}{3 x^3 \left(a+b x^2\right)}","-\frac{3 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left(a+b x^2\right)}+\frac{3 a b^2 x \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac{b^3 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left(a+b x^2\right)}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left(a+b x^2\right)}",1,"-1/3*(Sqrt[(a + b*x^2)^2]*(a^3 + 9*a^2*b*x^2 - 9*a*b^2*x^4 - b^3*x^6))/(x^3*(a + b*x^2))","A",1
579,1,59,158,0.0142543,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}{x^6} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^6,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(a^3+5 a^2 b x^2+15 a b^2 x^4-5 b^3 x^6\right)}{5 x^5 \left(a+b x^2\right)}","-\frac{3 a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left(a+b x^2\right)}-\frac{a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{x^3 \left(a+b x^2\right)}+\frac{b^3 x \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left(a+b x^2\right)}",1,"-1/5*(Sqrt[(a + b*x^2)^2]*(a^3 + 5*a^2*b*x^2 + 15*a*b^2*x^4 - 5*b^3*x^6))/(x^5*(a + b*x^2))","A",1
580,1,61,163,0.0135121,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}{x^8} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^8,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(5 a^3+21 a^2 b x^2+35 a b^2 x^4+35 b^3 x^6\right)}{35 x^7 \left(a+b x^2\right)}","-\frac{3 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left(a+b x^2\right)}-\frac{a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x^3 \left(a+b x^2\right)}-\frac{b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left(a+b x^2\right)}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left(a+b x^2\right)}",1,"-1/35*(Sqrt[(a + b*x^2)^2]*(5*a^3 + 21*a^2*b*x^2 + 35*a*b^2*x^4 + 35*b^3*x^6))/(x^7*(a + b*x^2))","A",1
581,1,61,167,0.0135082,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}{x^{10}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^10,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(35 a^3+135 a^2 b x^2+189 a b^2 x^4+105 b^3 x^6\right)}{315 x^9 \left(a+b x^2\right)}","-\frac{3 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left(a+b x^2\right)}-\frac{3 a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left(a+b x^2\right)}-\frac{b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left(a+b x^2\right)}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 x^9 \left(a+b x^2\right)}",1,"-1/315*(Sqrt[(a + b*x^2)^2]*(35*a^3 + 135*a^2*b*x^2 + 189*a*b^2*x^4 + 105*b^3*x^6))/(x^9*(a + b*x^2))","A",1
582,1,61,167,0.0156215,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}{x^{12}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^12,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(105 a^3+385 a^2 b x^2+495 a b^2 x^4+231 b^3 x^6\right)}{1155 x^{11} \left(a+b x^2\right)}","-\frac{a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^9 \left(a+b x^2\right)}-\frac{3 a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left(a+b x^2\right)}-\frac{b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left(a+b x^2\right)}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 x^{11} \left(a+b x^2\right)}",1,"-1/1155*(Sqrt[(a + b*x^2)^2]*(105*a^3 + 385*a^2*b*x^2 + 495*a*b^2*x^4 + 231*b^3*x^6))/(x^11*(a + b*x^2))","A",1
583,1,61,167,0.0136223,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}{x^{14}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^14,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(231 a^3+819 a^2 b x^2+1001 a b^2 x^4+429 b^3 x^6\right)}{3003 x^{13} \left(a+b x^2\right)}","-\frac{3 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 x^{11} \left(a+b x^2\right)}-\frac{a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^9 \left(a+b x^2\right)}-\frac{b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left(a+b x^2\right)}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 x^{13} \left(a+b x^2\right)}",1,"-1/3003*(Sqrt[(a + b*x^2)^2]*(231*a^3 + 819*a^2*b*x^2 + 1001*a*b^2*x^4 + 429*b^3*x^6))/(x^13*(a + b*x^2))","A",1
584,1,61,167,0.0137569,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}{x^{16}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^16,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(429 a^3+1485 a^2 b x^2+1755 a b^2 x^4+715 b^3 x^6\right)}{6435 x^{15} \left(a+b x^2\right)}","-\frac{3 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 x^{13} \left(a+b x^2\right)}-\frac{3 a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 x^{11} \left(a+b x^2\right)}-\frac{b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 x^9 \left(a+b x^2\right)}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{15 x^{15} \left(a+b x^2\right)}",1,"-1/6435*(Sqrt[(a + b*x^2)^2]*(429*a^3 + 1485*a^2*b*x^2 + 1755*a*b^2*x^4 + 715*b^3*x^6))/(x^15*(a + b*x^2))","A",1
585,1,83,255,0.0266275,"\int x^{13} \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2} \, dx","Integrate[x^13*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{x^{14} \sqrt{\left(a+b x^2\right)^2} \left(792 a^5+3465 a^4 b x^2+6160 a^3 b^2 x^4+5544 a^2 b^3 x^6+2520 a b^4 x^8+462 b^5 x^{10}\right)}{11088 \left(a+b x^2\right)}","\frac{b^5 x^{24} \sqrt{a^2+2 a b x^2+b^2 x^4}}{24 \left(a+b x^2\right)}+\frac{5 a b^4 x^{22} \sqrt{a^2+2 a b x^2+b^2 x^4}}{22 \left(a+b x^2\right)}+\frac{a^2 b^3 x^{20} \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 \left(a+b x^2\right)}+\frac{a^5 x^{14} \sqrt{a^2+2 a b x^2+b^2 x^4}}{14 \left(a+b x^2\right)}+\frac{5 a^4 b x^{16} \sqrt{a^2+2 a b x^2+b^2 x^4}}{16 \left(a+b x^2\right)}+\frac{5 a^3 b^2 x^{18} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 \left(a+b x^2\right)}",1,"(x^14*Sqrt[(a + b*x^2)^2]*(792*a^5 + 3465*a^4*b*x^2 + 6160*a^3*b^2*x^4 + 5544*a^2*b^3*x^6 + 2520*a*b^4*x^8 + 462*b^5*x^10))/(11088*(a + b*x^2))","A",1
586,1,83,255,0.0227447,"\int x^{11} \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2} \, dx","Integrate[x^11*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{x^{12} \sqrt{\left(a+b x^2\right)^2} \left(462 a^5+1980 a^4 b x^2+3465 a^3 b^2 x^4+3080 a^2 b^3 x^6+1386 a b^4 x^8+252 b^5 x^{10}\right)}{5544 \left(a+b x^2\right)}","\frac{b^5 x^{22} \sqrt{a^2+2 a b x^2+b^2 x^4}}{22 \left(a+b x^2\right)}+\frac{a b^4 x^{20} \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 \left(a+b x^2\right)}+\frac{5 a^2 b^3 x^{18} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 \left(a+b x^2\right)}+\frac{a^5 x^{12} \sqrt{a^2+2 a b x^2+b^2 x^4}}{12 \left(a+b x^2\right)}+\frac{5 a^4 b x^{14} \sqrt{a^2+2 a b x^2+b^2 x^4}}{14 \left(a+b x^2\right)}+\frac{5 a^3 b^2 x^{16} \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 \left(a+b x^2\right)}",1,"(x^12*Sqrt[(a + b*x^2)^2]*(462*a^5 + 1980*a^4*b*x^2 + 3465*a^3*b^2*x^4 + 3080*a^2*b^3*x^6 + 1386*a*b^4*x^8 + 252*b^5*x^10))/(5544*(a + b*x^2))","A",1
587,1,83,201,0.0204226,"\int x^9 \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2} \, dx","Integrate[x^9*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{x^{10} \sqrt{\left(a+b x^2\right)^2} \left(252 a^5+1050 a^4 b x^2+1800 a^3 b^2 x^4+1575 a^2 b^3 x^6+700 a b^4 x^8+126 b^5 x^{10}\right)}{2520 \left(a+b x^2\right)}","\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left(a+b x^2\right)^9}{20 b^5}-\frac{2 a \sqrt{a^2+2 a b x^2+b^2 x^4} \left(a+b x^2\right)^8}{9 b^5}+\frac{3 a^2 \sqrt{a^2+2 a b x^2+b^2 x^4} \left(a+b x^2\right)^7}{8 b^5}+\frac{a^4 \sqrt{a^2+2 a b x^2+b^2 x^4} \left(a+b x^2\right)^5}{12 b^5}-\frac{2 a^3 \sqrt{a^2+2 a b x^2+b^2 x^4} \left(a+b x^2\right)^6}{7 b^5}",1,"(x^10*Sqrt[(a + b*x^2)^2]*(252*a^5 + 1050*a^4*b*x^2 + 1800*a^3*b^2*x^4 + 1575*a^2*b^3*x^6 + 700*a*b^4*x^8 + 126*b^5*x^10))/(2520*(a + b*x^2))","A",1
588,1,83,160,0.0222081,"\int x^7 \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2} \, dx","Integrate[x^7*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{x^8 \sqrt{\left(a+b x^2\right)^2} \left(126 a^5+504 a^4 b x^2+840 a^3 b^2 x^4+720 a^2 b^3 x^6+315 a b^4 x^8+56 b^5 x^{10}\right)}{1008 \left(a+b x^2\right)}","\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left(a+b x^2\right)^8}{18 b^4}-\frac{3 a \sqrt{a^2+2 a b x^2+b^2 x^4} \left(a+b x^2\right)^7}{16 b^4}+\frac{3 a^2 \sqrt{a^2+2 a b x^2+b^2 x^4} \left(a+b x^2\right)^6}{14 b^4}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4} \left(a+b x^2\right)^5}{12 b^4}",1,"(x^8*Sqrt[(a + b*x^2)^2]*(126*a^5 + 504*a^4*b*x^2 + 840*a^3*b^2*x^4 + 720*a^2*b^3*x^6 + 315*a*b^4*x^8 + 56*b^5*x^10))/(1008*(a + b*x^2))","A",1
589,1,83,119,0.0216085,"\int x^5 \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2} \, dx","Integrate[x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{x^6 \sqrt{\left(a+b x^2\right)^2} \left(56 a^5+210 a^4 b x^2+336 a^3 b^2 x^4+280 a^2 b^3 x^6+120 a b^4 x^8+21 b^5 x^{10}\right)}{336 \left(a+b x^2\right)}","\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left(a+b x^2\right)^7}{16 b^3}-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4} \left(a+b x^2\right)^6}{7 b^3}+\frac{a^2 \sqrt{a^2+2 a b x^2+b^2 x^4} \left(a+b x^2\right)^5}{12 b^3}",1,"(x^6*Sqrt[(a + b*x^2)^2]*(56*a^5 + 210*a^4*b*x^2 + 336*a^3*b^2*x^4 + 280*a^2*b^3*x^6 + 120*a*b^4*x^8 + 21*b^5*x^10))/(336*(a + b*x^2))","A",1
590,1,83,67,0.0206447,"\int x^3 \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2} \, dx","Integrate[x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{x^4 \sqrt{\left(a+b x^2\right)^2} \left(21 a^5+70 a^4 b x^2+105 a^3 b^2 x^4+84 a^2 b^3 x^6+35 a b^4 x^8+6 b^5 x^{10}\right)}{84 \left(a+b x^2\right)}","\frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{7/2}}{14 b^2}-\frac{a \left(a+b x^2\right) \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{12 b^2}",1,"(x^4*Sqrt[(a + b*x^2)^2]*(21*a^5 + 70*a^4*b*x^2 + 105*a^3*b^2*x^4 + 84*a^2*b^3*x^6 + 35*a*b^4*x^8 + 6*b^5*x^10))/(84*(a + b*x^2))","A",1
591,1,27,36,0.0138242,"\int x \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2} \, dx","Integrate[x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{\left(a+b x^2\right) \left(\left(a+b x^2\right)^2\right)^{5/2}}{12 b}","\frac{\left(a+b x^2\right) \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{12 b}",1,"((a + b*x^2)*((a + b*x^2)^2)^(5/2))/(12*b)","A",1
592,1,82,251,0.0247298,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{x} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x,x]","\frac{\sqrt{\left(a+b x^2\right)^2} \left(120 a^5 \log (x)+b x^2 \left(300 a^4+300 a^3 b x^2+200 a^2 b^2 x^4+75 a b^3 x^6+12 b^4 x^8\right)\right)}{120 \left(a+b x^2\right)}","\frac{b^5 x^{10} \sqrt{a^2+2 a b x^2+b^2 x^4}}{10 \left(a+b x^2\right)}+\frac{5 a b^4 x^8 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 \left(a+b x^2\right)}+\frac{5 a^2 b^3 x^6 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left(a+b x^2\right)}+\frac{a^5 \log (x) \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac{5 a^4 b x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 \left(a+b x^2\right)}+\frac{5 a^3 b^2 x^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 \left(a+b x^2\right)}",1,"(Sqrt[(a + b*x^2)^2]*(b*x^2*(300*a^4 + 300*a^3*b*x^2 + 200*a^2*b^2*x^4 + 75*a*b^3*x^6 + 12*b^4*x^8) + 120*a^5*Log[x]))/(120*(a + b*x^2))","A",1
593,1,85,250,0.0260637,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{x^3} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^3,x]","\frac{\sqrt{\left(a+b x^2\right)^2} \left(-12 a^5+120 a^4 b x^2 \log (x)+120 a^3 b^2 x^4+60 a^2 b^3 x^6+20 a b^4 x^8+3 b^5 x^{10}\right)}{24 x^2 \left(a+b x^2\right)}","\frac{b^5 x^8 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 \left(a+b x^2\right)}+\frac{5 a b^4 x^6 \sqrt{a^2+2 a b x^2+b^2 x^4}}{6 \left(a+b x^2\right)}+\frac{5 a^2 b^3 x^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 \left(a+b x^2\right)}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 x^2 \left(a+b x^2\right)}+\frac{5 a^4 b \log (x) \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac{5 a^3 b^2 x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}",1,"(Sqrt[(a + b*x^2)^2]*(-12*a^5 + 120*a^3*b^2*x^4 + 60*a^2*b^3*x^6 + 20*a*b^4*x^8 + 3*b^5*x^10 + 120*a^4*b*x^2*Log[x]))/(24*x^2*(a + b*x^2))","A",1
594,1,85,250,0.0259507,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{x^5} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^5,x]","\frac{\sqrt{\left(a+b x^2\right)^2} \left(-3 a^5-30 a^4 b x^2+120 a^3 b^2 x^4 \log (x)+60 a^2 b^3 x^6+15 a b^4 x^8+2 b^5 x^{10}\right)}{12 x^4 \left(a+b x^2\right)}","\frac{b^5 x^6 \sqrt{a^2+2 a b x^2+b^2 x^4}}{6 \left(a+b x^2\right)}+\frac{5 a b^4 x^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 \left(a+b x^2\right)}+\frac{5 a^2 b^3 x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 x^4 \left(a+b x^2\right)}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 x^2 \left(a+b x^2\right)}+\frac{10 a^3 b^2 \log (x) \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}",1,"(Sqrt[(a + b*x^2)^2]*(-3*a^5 - 30*a^4*b*x^2 + 60*a^2*b^3*x^6 + 15*a*b^4*x^8 + 2*b^5*x^10 + 120*a^3*b^2*x^4*Log[x]))/(12*x^4*(a + b*x^2))","A",1
595,1,85,250,0.0239121,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{x^7} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^7,x]","\frac{\sqrt{\left(a+b x^2\right)^2} \left(-2 a^5-15 a^4 b x^2-60 a^3 b^2 x^4+120 a^2 b^3 x^6 \log (x)+30 a b^4 x^8+3 b^5 x^{10}\right)}{12 x^6 \left(a+b x^2\right)}","\frac{b^5 x^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 \left(a+b x^2\right)}+\frac{5 a b^4 x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 \left(a+b x^2\right)}+\frac{10 a^2 b^3 \log (x) \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{6 x^6 \left(a+b x^2\right)}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 x^4 \left(a+b x^2\right)}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x^2 \left(a+b x^2\right)}",1,"(Sqrt[(a + b*x^2)^2]*(-2*a^5 - 15*a^4*b*x^2 - 60*a^3*b^2*x^4 + 30*a*b^4*x^8 + 3*b^5*x^10 + 120*a^2*b^3*x^6*Log[x]))/(12*x^6*(a + b*x^2))","A",1
596,1,85,250,0.0200328,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{x^9} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^9,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(3 a^5+20 a^4 b x^2+60 a^3 b^2 x^4+120 a^2 b^3 x^6-120 a b^4 x^8 \log (x)-12 b^5 x^{10}\right)}{24 x^8 \left(a+b x^2\right)}","\frac{b^5 x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 \left(a+b x^2\right)}+\frac{5 a b^4 \log (x) \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x^2 \left(a+b x^2\right)}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 x^8 \left(a+b x^2\right)}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{6 x^6 \left(a+b x^2\right)}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 x^4 \left(a+b x^2\right)}",1,"-1/24*(Sqrt[(a + b*x^2)^2]*(3*a^5 + 20*a^4*b*x^2 + 60*a^3*b^2*x^4 + 120*a^2*b^3*x^6 - 12*b^5*x^10 - 120*a*b^4*x^8*Log[x]))/(x^8*(a + b*x^2))","A",1
597,1,85,251,0.0274102,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{x^{11}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^11,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(a \left(12 a^4+75 a^3 b x^2+200 a^2 b^2 x^4+300 a b^3 x^6+300 b^4 x^8\right)-120 b^5 x^{10} \log (x)\right)}{120 x^{10} \left(a+b x^2\right)}","\frac{b^5 \log (x) \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}-\frac{5 a b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 x^2 \left(a+b x^2\right)}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 x^4 \left(a+b x^2\right)}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{10 x^{10} \left(a+b x^2\right)}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 x^8 \left(a+b x^2\right)}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^6 \left(a+b x^2\right)}",1,"-1/120*(Sqrt[(a + b*x^2)^2]*(a*(12*a^4 + 75*a^3*b*x^2 + 200*a^2*b^2*x^4 + 300*a*b^3*x^6 + 300*b^4*x^8) - 120*b^5*x^10*Log[x]))/(x^10*(a + b*x^2))","A",1
598,1,81,41,0.0178852,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{x^{13}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^13,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(a^5+6 a^4 b x^2+15 a^3 b^2 x^4+20 a^2 b^3 x^6+15 a b^4 x^8+6 b^5 x^{10}\right)}{12 x^{12} \left(a+b x^2\right)}","-\frac{\left(a+b x^2\right)^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{12 a x^{12}}",1,"-1/12*(Sqrt[(a + b*x^2)^2]*(a^5 + 6*a^4*b*x^2 + 15*a^3*b^2*x^4 + 20*a^2*b^3*x^6 + 15*a*b^4*x^8 + 6*b^5*x^10))/(x^12*(a + b*x^2))","A",1
599,1,83,72,0.020522,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{x^{15}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^15,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(6 a^5+35 a^4 b x^2+84 a^3 b^2 x^4+105 a^2 b^3 x^6+70 a b^4 x^8+21 b^5 x^{10}\right)}{84 x^{14} \left(a+b x^2\right)}","\frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{7/2}}{84 a^2 x^{14}}-\frac{\left(a+b x^2\right) \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{12 a x^{14}}",1,"-1/84*(Sqrt[(a + b*x^2)^2]*(6*a^5 + 35*a^4*b*x^2 + 84*a^3*b^2*x^4 + 105*a^2*b^3*x^6 + 70*a*b^4*x^8 + 21*b^5*x^10))/(x^14*(a + b*x^2))","A",1
600,1,83,128,0.0178267,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{x^{17}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^17,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(21 a^5+120 a^4 b x^2+280 a^3 b^2 x^4+336 a^2 b^3 x^6+210 a b^4 x^8+56 b^5 x^{10}\right)}{336 x^{16} \left(a+b x^2\right)}","-\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left(a+b x^2\right)^5}{16 a x^{16}}+\frac{b \sqrt{a^2+2 a b x^2+b^2 x^4} \left(a+b x^2\right)^5}{56 a^2 x^{14}}-\frac{b^2 \sqrt{a^2+2 a b x^2+b^2 x^4} \left(a+b x^2\right)^5}{336 a^3 x^{12}}",1,"-1/336*(Sqrt[(a + b*x^2)^2]*(21*a^5 + 120*a^4*b*x^2 + 280*a^3*b^2*x^4 + 336*a^2*b^3*x^6 + 210*a*b^4*x^8 + 56*b^5*x^10))/(x^16*(a + b*x^2))","A",1
601,1,83,255,0.0172625,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{x^{19}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^19,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(56 a^5+315 a^4 b x^2+720 a^3 b^2 x^4+840 a^2 b^3 x^6+504 a b^4 x^8+126 b^5 x^{10}\right)}{1008 x^{18} \left(a+b x^2\right)}","-\frac{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 x^8 \left(a+b x^2\right)}-\frac{a b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 x^{10} \left(a+b x^2\right)}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{6 x^{12} \left(a+b x^2\right)}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{18 x^{18} \left(a+b x^2\right)}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{16 x^{16} \left(a+b x^2\right)}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^{14} \left(a+b x^2\right)}",1,"-1/1008*(Sqrt[(a + b*x^2)^2]*(56*a^5 + 315*a^4*b*x^2 + 720*a^3*b^2*x^4 + 840*a^2*b^3*x^6 + 504*a*b^4*x^8 + 126*b^5*x^10))/(x^18*(a + b*x^2))","A",1
602,1,83,255,0.0178276,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{x^{21}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^21,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(126 a^5+700 a^4 b x^2+1575 a^3 b^2 x^4+1800 a^2 b^3 x^6+1050 a b^4 x^8+252 b^5 x^{10}\right)}{2520 x^{20} \left(a+b x^2\right)}","-\frac{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{10 x^{10} \left(a+b x^2\right)}-\frac{5 a b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{12 x^{12} \left(a+b x^2\right)}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^{14} \left(a+b x^2\right)}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{20 x^{20} \left(a+b x^2\right)}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{18 x^{18} \left(a+b x^2\right)}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 x^{16} \left(a+b x^2\right)}",1,"-1/2520*(Sqrt[(a + b*x^2)^2]*(126*a^5 + 700*a^4*b*x^2 + 1575*a^3*b^2*x^4 + 1800*a^2*b^3*x^6 + 1050*a*b^4*x^8 + 252*b^5*x^10))/(x^20*(a + b*x^2))","A",1
603,1,83,255,0.0178548,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{x^{23}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^23,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(252 a^5+1386 a^4 b x^2+3080 a^3 b^2 x^4+3465 a^2 b^3 x^6+1980 a b^4 x^8+462 b^5 x^{10}\right)}{5544 x^{22} \left(a+b x^2\right)}","-\frac{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{12 x^{12} \left(a+b x^2\right)}-\frac{5 a b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{14 x^{14} \left(a+b x^2\right)}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 x^{16} \left(a+b x^2\right)}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{22 x^{22} \left(a+b x^2\right)}-\frac{a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 x^{20} \left(a+b x^2\right)}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 x^{18} \left(a+b x^2\right)}",1,"-1/5544*(Sqrt[(a + b*x^2)^2]*(252*a^5 + 1386*a^4*b*x^2 + 3080*a^3*b^2*x^4 + 3465*a^2*b^3*x^6 + 1980*a*b^4*x^8 + 462*b^5*x^10))/(x^22*(a + b*x^2))","A",1
604,1,83,255,0.022342,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{x^{25}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^25,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(462 a^5+2520 a^4 b x^2+5544 a^3 b^2 x^4+6160 a^2 b^3 x^6+3465 a b^4 x^8+792 b^5 x^{10}\right)}{11088 x^{24} \left(a+b x^2\right)}","-\frac{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{14 x^{14} \left(a+b x^2\right)}-\frac{5 a b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{16 x^{16} \left(a+b x^2\right)}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 x^{18} \left(a+b x^2\right)}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{24 x^{24} \left(a+b x^2\right)}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{22 x^{22} \left(a+b x^2\right)}-\frac{a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 x^{20} \left(a+b x^2\right)}",1,"-1/11088*(Sqrt[(a + b*x^2)^2]*(462*a^5 + 2520*a^4*b*x^2 + 5544*a^3*b^2*x^4 + 6160*a^2*b^3*x^6 + 3465*a*b^4*x^8 + 792*b^5*x^10))/(x^24*(a + b*x^2))","A",1
605,1,83,255,0.0219552,"\int x^{12} \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2} \, dx","Integrate[x^12*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{x^{13} \sqrt{\left(a+b x^2\right)^2} \left(156009 a^5+676039 a^4 b x^2+1193010 a^3 b^2 x^4+1067430 a^2 b^3 x^6+482885 a b^4 x^8+88179 b^5 x^{10}\right)}{2028117 \left(a+b x^2\right)}","\frac{b^5 x^{23} \sqrt{a^2+2 a b x^2+b^2 x^4}}{23 \left(a+b x^2\right)}+\frac{5 a b^4 x^{21} \sqrt{a^2+2 a b x^2+b^2 x^4}}{21 \left(a+b x^2\right)}+\frac{10 a^2 b^3 x^{19} \sqrt{a^2+2 a b x^2+b^2 x^4}}{19 \left(a+b x^2\right)}+\frac{a^5 x^{13} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 \left(a+b x^2\right)}+\frac{a^4 b x^{15} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left(a+b x^2\right)}+\frac{10 a^3 b^2 x^{17} \sqrt{a^2+2 a b x^2+b^2 x^4}}{17 \left(a+b x^2\right)}",1,"(x^13*Sqrt[(a + b*x^2)^2]*(156009*a^5 + 676039*a^4*b*x^2 + 1193010*a^3*b^2*x^4 + 1067430*a^2*b^3*x^6 + 482885*a*b^4*x^8 + 88179*b^5*x^10))/(2028117*(a + b*x^2))","A",1
606,1,83,255,0.0199232,"\int x^{10} \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2} \, dx","Integrate[x^10*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{x^{11} \sqrt{\left(a+b x^2\right)^2} \left(88179 a^5+373065 a^4 b x^2+646646 a^3 b^2 x^4+570570 a^2 b^3 x^6+255255 a b^4 x^8+46189 b^5 x^{10}\right)}{969969 \left(a+b x^2\right)}","\frac{b^5 x^{21} \sqrt{a^2+2 a b x^2+b^2 x^4}}{21 \left(a+b x^2\right)}+\frac{5 a b^4 x^{19} \sqrt{a^2+2 a b x^2+b^2 x^4}}{19 \left(a+b x^2\right)}+\frac{10 a^2 b^3 x^{17} \sqrt{a^2+2 a b x^2+b^2 x^4}}{17 \left(a+b x^2\right)}+\frac{a^5 x^{11} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 \left(a+b x^2\right)}+\frac{5 a^4 b x^{13} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 \left(a+b x^2\right)}+\frac{2 a^3 b^2 x^{15} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left(a+b x^2\right)}",1,"(x^11*Sqrt[(a + b*x^2)^2]*(88179*a^5 + 373065*a^4*b*x^2 + 646646*a^3*b^2*x^4 + 570570*a^2*b^3*x^6 + 255255*a*b^4*x^8 + 46189*b^5*x^10))/(969969*(a + b*x^2))","A",1
607,1,83,255,0.0199316,"\int x^8 \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2} \, dx","Integrate[x^8*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{x^9 \sqrt{\left(a+b x^2\right)^2} \left(46189 a^5+188955 a^4 b x^2+319770 a^3 b^2 x^4+277134 a^2 b^3 x^6+122265 a b^4 x^8+21879 b^5 x^{10}\right)}{415701 \left(a+b x^2\right)}","\frac{b^5 x^{19} \sqrt{a^2+2 a b x^2+b^2 x^4}}{19 \left(a+b x^2\right)}+\frac{5 a b^4 x^{17} \sqrt{a^2+2 a b x^2+b^2 x^4}}{17 \left(a+b x^2\right)}+\frac{2 a^2 b^3 x^{15} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left(a+b x^2\right)}+\frac{a^5 x^9 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 \left(a+b x^2\right)}+\frac{5 a^4 b x^{11} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 \left(a+b x^2\right)}+\frac{10 a^3 b^2 x^{13} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 \left(a+b x^2\right)}",1,"(x^9*Sqrt[(a + b*x^2)^2]*(46189*a^5 + 188955*a^4*b*x^2 + 319770*a^3*b^2*x^4 + 277134*a^2*b^3*x^6 + 122265*a*b^4*x^8 + 21879*b^5*x^10))/(415701*(a + b*x^2))","A",1
608,1,83,255,0.0199526,"\int x^6 \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2} \, dx","Integrate[x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{x^7 \sqrt{\left(a+b x^2\right)^2} \left(21879 a^5+85085 a^4 b x^2+139230 a^3 b^2 x^4+117810 a^2 b^3 x^6+51051 a b^4 x^8+9009 b^5 x^{10}\right)}{153153 \left(a+b x^2\right)}","\frac{b^5 x^{17} \sqrt{a^2+2 a b x^2+b^2 x^4}}{17 \left(a+b x^2\right)}+\frac{a b^4 x^{15} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left(a+b x^2\right)}+\frac{10 a^2 b^3 x^{13} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 \left(a+b x^2\right)}+\frac{a^5 x^7 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 \left(a+b x^2\right)}+\frac{5 a^4 b x^9 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 \left(a+b x^2\right)}+\frac{10 a^3 b^2 x^{11} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 \left(a+b x^2\right)}",1,"(x^7*Sqrt[(a + b*x^2)^2]*(21879*a^5 + 85085*a^4*b*x^2 + 139230*a^3*b^2*x^4 + 117810*a^2*b^3*x^6 + 51051*a*b^4*x^8 + 9009*b^5*x^10))/(153153*(a + b*x^2))","A",1
609,1,83,255,0.020472,"\int x^4 \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2} \, dx","Integrate[x^4*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{x^5 \sqrt{\left(a+b x^2\right)^2} \left(9009 a^5+32175 a^4 b x^2+50050 a^3 b^2 x^4+40950 a^2 b^3 x^6+17325 a b^4 x^8+3003 b^5 x^{10}\right)}{45045 \left(a+b x^2\right)}","\frac{b^5 x^{15} \sqrt{a^2+2 a b x^2+b^2 x^4}}{15 \left(a+b x^2\right)}+\frac{5 a b^4 x^{13} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 \left(a+b x^2\right)}+\frac{10 a^2 b^3 x^{11} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 \left(a+b x^2\right)}+\frac{a^5 x^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 \left(a+b x^2\right)}+\frac{5 a^4 b x^7 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 \left(a+b x^2\right)}+\frac{10 a^3 b^2 x^9 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 \left(a+b x^2\right)}",1,"(x^5*Sqrt[(a + b*x^2)^2]*(9009*a^5 + 32175*a^4*b*x^2 + 50050*a^3*b^2*x^4 + 40950*a^2*b^3*x^6 + 17325*a*b^4*x^8 + 3003*b^5*x^10))/(45045*(a + b*x^2))","A",1
610,1,83,252,0.0204101,"\int x^2 \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2} \, dx","Integrate[x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{x^3 \sqrt{\left(a+b x^2\right)^2} \left(3003 a^5+9009 a^4 b x^2+12870 a^3 b^2 x^4+10010 a^2 b^3 x^6+4095 a b^4 x^8+693 b^5 x^{10}\right)}{9009 \left(a+b x^2\right)}","\frac{b^5 x^{13} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 \left(a+b x^2\right)}+\frac{5 a b^4 x^{11} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 \left(a+b x^2\right)}+\frac{10 a^2 b^3 x^9 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 \left(a+b x^2\right)}+\frac{a^5 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left(a+b x^2\right)}+\frac{a^4 b x^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac{10 a^3 b^2 x^7 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 \left(a+b x^2\right)}",1,"(x^3*Sqrt[(a + b*x^2)^2]*(3003*a^5 + 9009*a^4*b*x^2 + 12870*a^3*b^2*x^4 + 10010*a^2*b^3*x^6 + 4095*a*b^4*x^8 + 693*b^5*x^10))/(9009*(a + b*x^2))","A",1
611,1,81,248,0.0164732,"\int \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{\sqrt{\left(a+b x^2\right)^2} \left(693 a^5 x+1155 a^4 b x^3+1386 a^3 b^2 x^5+990 a^2 b^3 x^7+385 a b^4 x^9+63 b^5 x^{11}\right)}{693 \left(a+b x^2\right)}","\frac{b^5 x^{11} \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{11 \left(a+b x^2\right)^5}+\frac{5 a b^4 x^9 \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{9 \left(a+b x^2\right)^5}+\frac{10 a^2 b^3 x^7 \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{7 \left(a+b x^2\right)^5}+\frac{a^5 x \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{\left(a+b x^2\right)^5}+\frac{5 a^4 b x^3 \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{3 \left(a+b x^2\right)^5}+\frac{2 a^3 b^2 x^5 \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{\left(a+b x^2\right)^5}",1,"(Sqrt[(a + b*x^2)^2]*(693*a^5*x + 1155*a^4*b*x^3 + 1386*a^3*b^2*x^5 + 990*a^2*b^3*x^7 + 385*a*b^4*x^9 + 63*b^5*x^11))/(693*(a + b*x^2))","A",1
612,1,83,247,0.0202352,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{x^2} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^2,x]","\frac{\sqrt{\left(a+b x^2\right)^2} \left(-63 a^5+315 a^4 b x^2+210 a^3 b^2 x^4+126 a^2 b^3 x^6+45 a b^4 x^8+7 b^5 x^{10}\right)}{63 x \left(a+b x^2\right)}","\frac{b^5 x^9 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 \left(a+b x^2\right)}+\frac{5 a b^4 x^7 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 \left(a+b x^2\right)}+\frac{2 a^2 b^3 x^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left(a+b x^2\right)}+\frac{5 a^4 b x \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac{10 a^3 b^2 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left(a+b x^2\right)}",1,"(Sqrt[(a + b*x^2)^2]*(-63*a^5 + 315*a^4*b*x^2 + 210*a^3*b^2*x^4 + 126*a^2*b^3*x^6 + 45*a*b^4*x^8 + 7*b^5*x^10))/(63*x*(a + b*x^2))","A",1
613,1,83,246,0.0232536,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{x^4} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^4,x]","\frac{\sqrt{\left(a+b x^2\right)^2} \left(-7 a^5-105 a^4 b x^2+210 a^3 b^2 x^4+70 a^2 b^3 x^6+21 a b^4 x^8+3 b^5 x^{10}\right)}{21 x^3 \left(a+b x^2\right)}","\frac{b^5 x^7 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 \left(a+b x^2\right)}+\frac{a b^4 x^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac{10 a^2 b^3 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left(a+b x^2\right)}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left(a+b x^2\right)}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left(a+b x^2\right)}+\frac{10 a^3 b^2 x \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}",1,"(Sqrt[(a + b*x^2)^2]*(-7*a^5 - 105*a^4*b*x^2 + 210*a^3*b^2*x^4 + 70*a^2*b^3*x^6 + 21*a*b^4*x^8 + 3*b^5*x^10))/(21*x^3*(a + b*x^2))","A",1
614,1,83,249,0.0200852,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{x^6} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^6,x]","\frac{\sqrt{\left(a+b x^2\right)^2} \left(-3 a^5-25 a^4 b x^2-150 a^3 b^2 x^4+150 a^2 b^3 x^6+25 a b^4 x^8+3 b^5 x^{10}\right)}{15 x^5 \left(a+b x^2\right)}","\frac{b^5 x^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 \left(a+b x^2\right)}+\frac{5 a b^4 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left(a+b x^2\right)}+\frac{10 a^2 b^3 x \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left(a+b x^2\right)}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left(a+b x^2\right)}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left(a+b x^2\right)}",1,"(Sqrt[(a + b*x^2)^2]*(-3*a^5 - 25*a^4*b*x^2 - 150*a^3*b^2*x^4 + 150*a^2*b^3*x^6 + 25*a*b^4*x^8 + 3*b^5*x^10))/(15*x^5*(a + b*x^2))","A",1
615,1,83,247,0.0175974,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{x^8} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^8,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(3 a^5+21 a^4 b x^2+70 a^3 b^2 x^4+210 a^2 b^3 x^6-105 a b^4 x^8-7 b^5 x^{10}\right)}{21 x^7 \left(a+b x^2\right)}","\frac{b^5 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left(a+b x^2\right)}+\frac{5 a b^4 x \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left(a+b x^2\right)}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left(a+b x^2\right)}-\frac{a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{x^5 \left(a+b x^2\right)}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left(a+b x^2\right)}",1,"-1/21*(Sqrt[(a + b*x^2)^2]*(3*a^5 + 21*a^4*b*x^2 + 70*a^3*b^2*x^4 + 210*a^2*b^3*x^6 - 105*a*b^4*x^8 - 7*b^5*x^10))/(x^7*(a + b*x^2))","A",1
616,1,83,246,0.0178145,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{x^{10}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^10,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(7 a^5+45 a^4 b x^2+126 a^3 b^2 x^4+210 a^2 b^3 x^6+315 a b^4 x^8-63 b^5 x^{10}\right)}{63 x^9 \left(a+b x^2\right)}","\frac{b^5 x \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}-\frac{5 a b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left(a+b x^2\right)}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left(a+b x^2\right)}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 x^9 \left(a+b x^2\right)}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left(a+b x^2\right)}-\frac{2 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x^5 \left(a+b x^2\right)}",1,"-1/63*(Sqrt[(a + b*x^2)^2]*(7*a^5 + 45*a^4*b*x^2 + 126*a^3*b^2*x^4 + 210*a^2*b^3*x^6 + 315*a*b^4*x^8 - 63*b^5*x^10))/(x^9*(a + b*x^2))","A",1
617,1,83,251,0.016833,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{x^{12}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^12,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(63 a^5+385 a^4 b x^2+990 a^3 b^2 x^4+1386 a^2 b^3 x^6+1155 a b^4 x^8+693 b^5 x^{10}\right)}{693 x^{11} \left(a+b x^2\right)}","-\frac{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left(a+b x^2\right)}-\frac{5 a b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left(a+b x^2\right)}-\frac{2 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x^5 \left(a+b x^2\right)}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 x^{11} \left(a+b x^2\right)}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 x^9 \left(a+b x^2\right)}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left(a+b x^2\right)}",1,"-1/693*(Sqrt[(a + b*x^2)^2]*(63*a^5 + 385*a^4*b*x^2 + 990*a^3*b^2*x^4 + 1386*a^2*b^3*x^6 + 1155*a*b^4*x^8 + 693*b^5*x^10))/(x^11*(a + b*x^2))","A",1
618,1,83,253,0.0172417,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{x^{14}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^14,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(693 a^5+4095 a^4 b x^2+10010 a^3 b^2 x^4+12870 a^2 b^3 x^6+9009 a b^4 x^8+3003 b^5 x^{10}\right)}{9009 x^{13} \left(a+b x^2\right)}","-\frac{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left(a+b x^2\right)}-\frac{a b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x^5 \left(a+b x^2\right)}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left(a+b x^2\right)}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 x^{13} \left(a+b x^2\right)}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 x^{11} \left(a+b x^2\right)}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 x^9 \left(a+b x^2\right)}",1,"-1/9009*(Sqrt[(a + b*x^2)^2]*(693*a^5 + 4095*a^4*b*x^2 + 10010*a^3*b^2*x^4 + 12870*a^2*b^3*x^6 + 9009*a*b^4*x^8 + 3003*b^5*x^10))/(x^13*(a + b*x^2))","A",1
619,1,83,255,0.0173155,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{x^{16}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^16,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(3003 a^5+17325 a^4 b x^2+40950 a^3 b^2 x^4+50050 a^2 b^3 x^6+32175 a b^4 x^8+9009 b^5 x^{10}\right)}{45045 x^{15} \left(a+b x^2\right)}","-\frac{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left(a+b x^2\right)}-\frac{5 a b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left(a+b x^2\right)}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 x^9 \left(a+b x^2\right)}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{15 x^{15} \left(a+b x^2\right)}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 x^{13} \left(a+b x^2\right)}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 x^{11} \left(a+b x^2\right)}",1,"-1/45045*(Sqrt[(a + b*x^2)^2]*(3003*a^5 + 17325*a^4*b*x^2 + 40950*a^3*b^2*x^4 + 50050*a^2*b^3*x^6 + 32175*a*b^4*x^8 + 9009*b^5*x^10))/(x^15*(a + b*x^2))","A",1
620,1,83,255,0.0172038,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{x^{18}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^18,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(9009 a^5+51051 a^4 b x^2+117810 a^3 b^2 x^4+139230 a^2 b^3 x^6+85085 a b^4 x^8+21879 b^5 x^{10}\right)}{153153 x^{17} \left(a+b x^2\right)}","-\frac{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left(a+b x^2\right)}-\frac{5 a b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 x^9 \left(a+b x^2\right)}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 x^{11} \left(a+b x^2\right)}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{17 x^{17} \left(a+b x^2\right)}-\frac{a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^{15} \left(a+b x^2\right)}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 x^{13} \left(a+b x^2\right)}",1,"-1/153153*(Sqrt[(a + b*x^2)^2]*(9009*a^5 + 51051*a^4*b*x^2 + 117810*a^3*b^2*x^4 + 139230*a^2*b^3*x^6 + 85085*a*b^4*x^8 + 21879*b^5*x^10))/(x^17*(a + b*x^2))","A",1
621,1,83,255,0.0194548,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{x^{20}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^20,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(21879 a^5+122265 a^4 b x^2+277134 a^3 b^2 x^4+319770 a^2 b^3 x^6+188955 a b^4 x^8+46189 b^5 x^{10}\right)}{415701 x^{19} \left(a+b x^2\right)}","-\frac{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 x^9 \left(a+b x^2\right)}-\frac{5 a b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 x^{11} \left(a+b x^2\right)}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 x^{13} \left(a+b x^2\right)}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{19 x^{19} \left(a+b x^2\right)}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{17 x^{17} \left(a+b x^2\right)}-\frac{2 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^{15} \left(a+b x^2\right)}",1,"-1/415701*(Sqrt[(a + b*x^2)^2]*(21879*a^5 + 122265*a^4*b*x^2 + 277134*a^3*b^2*x^4 + 319770*a^2*b^3*x^6 + 188955*a*b^4*x^8 + 46189*b^5*x^10))/(x^19*(a + b*x^2))","A",1
622,1,83,255,0.0175961,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{x^{22}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^22,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(46189 a^5+255255 a^4 b x^2+570570 a^3 b^2 x^4+646646 a^2 b^3 x^6+373065 a b^4 x^8+88179 b^5 x^{10}\right)}{969969 x^{21} \left(a+b x^2\right)}","-\frac{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 x^{11} \left(a+b x^2\right)}-\frac{5 a b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 x^{13} \left(a+b x^2\right)}-\frac{2 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^{15} \left(a+b x^2\right)}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{21 x^{21} \left(a+b x^2\right)}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{19 x^{19} \left(a+b x^2\right)}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{17 x^{17} \left(a+b x^2\right)}",1,"-1/969969*(Sqrt[(a + b*x^2)^2]*(46189*a^5 + 255255*a^4*b*x^2 + 570570*a^3*b^2*x^4 + 646646*a^2*b^3*x^6 + 373065*a*b^4*x^8 + 88179*b^5*x^10))/(x^21*(a + b*x^2))","A",1
623,1,83,255,0.0175724,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{x^{24}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^24,x]","-\frac{\sqrt{\left(a+b x^2\right)^2} \left(88179 a^5+482885 a^4 b x^2+1067430 a^3 b^2 x^4+1193010 a^2 b^3 x^6+676039 a b^4 x^8+156009 b^5 x^{10}\right)}{2028117 x^{23} \left(a+b x^2\right)}","-\frac{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 x^{13} \left(a+b x^2\right)}-\frac{a b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^{15} \left(a+b x^2\right)}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{17 x^{17} \left(a+b x^2\right)}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{23 x^{23} \left(a+b x^2\right)}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{21 x^{21} \left(a+b x^2\right)}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{19 x^{19} \left(a+b x^2\right)}",1,"-1/2028117*(Sqrt[(a + b*x^2)^2]*(88179*a^5 + 482885*a^4*b*x^2 + 1067430*a^3*b^2*x^4 + 1193010*a^2*b^3*x^6 + 676039*a*b^4*x^8 + 156009*b^5*x^10))/(x^23*(a + b*x^2))","A",1
624,1,55,127,0.0239357,"\int \frac{x^5}{\sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx","Integrate[x^5/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]","\frac{\left(a+b x^2\right) \left(2 a^2 \log \left(a+b x^2\right)+b x^2 \left(b x^2-2 a\right)\right)}{4 b^3 \sqrt{\left(a+b x^2\right)^2}}","\frac{x^4 \left(a+b x^2\right)}{4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a x^2 \left(a+b x^2\right)}{2 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{a^2 \left(a+b x^2\right) \log \left(a+b x^2\right)}{2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"((a + b*x^2)*(b*x^2*(-2*a + b*x^2) + 2*a^2*Log[a + b*x^2]))/(4*b^3*Sqrt[(a + b*x^2)^2])","A",1
625,1,44,75,0.0123994,"\int \frac{x^3}{\sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx","Integrate[x^3/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]","\frac{\left(a+b x^2\right) \left(b x^2-a \log \left(a+b x^2\right)\right)}{2 b^2 \sqrt{\left(a+b x^2\right)^2}}","\frac{\sqrt{a^2+2 a b x^2+b^2 x^4}}{2 b^2}-\frac{a \left(a+b x^2\right) \log \left(a+b x^2\right)}{2 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"((a + b*x^2)*(b*x^2 - a*Log[a + b*x^2]))/(2*b^2*Sqrt[(a + b*x^2)^2])","A",1
626,1,35,44,0.0083039,"\int \frac{x}{\sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx","Integrate[x/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]","\frac{\left(a+b x^2\right) \log \left(a+b x^2\right)}{2 b \sqrt{\left(a+b x^2\right)^2}}","\frac{\left(a+b x^2\right) \log \left(a+b x^2\right)}{2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"((a + b*x^2)*Log[a + b*x^2])/(2*b*Sqrt[(a + b*x^2)^2])","A",1
627,1,42,80,0.0112178,"\int \frac{1}{x \sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx","Integrate[1/(x*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]),x]","\frac{\left(a+b x^2\right) \left(2 \log (x)-\log \left(a+b x^2\right)\right)}{2 a \sqrt{\left(a+b x^2\right)^2}}","\frac{\log (x) \left(a+b x^2\right)}{a \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left(a+b x^2\right) \log \left(a+b x^2\right)}{2 a \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"((a + b*x^2)*(2*Log[x] - Log[a + b*x^2]))/(2*a*Sqrt[(a + b*x^2)^2])","A",1
628,1,54,125,0.0154998,"\int \frac{1}{x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx","Integrate[1/(x^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]),x]","-\frac{\left(a+b x^2\right) \left(-b x^2 \log \left(a+b x^2\right)+a+2 b x^2 \log (x)\right)}{2 a^2 x^2 \sqrt{\left(a+b x^2\right)^2}}","\frac{-a-b x^2}{2 a x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b \log (x) \left(a+b x^2\right)}{a^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{b \left(a+b x^2\right) \log \left(a+b x^2\right)}{2 a^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"-1/2*((a + b*x^2)*(a + 2*b*x^2*Log[x] - b*x^2*Log[a + b*x^2]))/(a^2*x^2*Sqrt[(a + b*x^2)^2])","A",1
629,1,66,129,0.0259129,"\int \frac{x^4}{\sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx","Integrate[x^4/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]","\frac{\left(a+b x^2\right) \left(3 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)+\sqrt{b} x \left(b x^2-3 a\right)\right)}{3 b^{5/2} \sqrt{\left(a+b x^2\right)^2}}","-\frac{a x \left(a+b x^2\right)}{b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{x^3 \left(a+b x^2\right)}{3 b \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{a^{3/2} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"((a + b*x^2)*(Sqrt[b]*x*(-3*a + b*x^2) + 3*a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]]))/(3*b^(5/2)*Sqrt[(a + b*x^2)^2])","A",1
630,1,54,89,0.0138556,"\int \frac{x^2}{\sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx","Integrate[x^2/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]","\frac{\left(a+b x^2\right) \left(\sqrt{b} x-\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right)}{b^{3/2} \sqrt{\left(a+b x^2\right)^2}}","\frac{x \left(a+b x^2\right)}{b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\sqrt{a} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"((a + b*x^2)*(Sqrt[b]*x - Sqrt[a]*ArcTan[(Sqrt[b]*x)/Sqrt[a]]))/(b^(3/2)*Sqrt[(a + b*x^2)^2])","A",1
631,1,44,53,0.0120561,"\int \frac{1}{\sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx","Integrate[1/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]","\frac{\left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b} \sqrt{\left(a+b x^2\right)^2}}","\frac{\left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"((a + b*x^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*Sqrt[(a + b*x^2)^2])","A",1
632,1,56,92,0.0144153,"\int \frac{1}{x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx","Integrate[1/(x^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]),x]","-\frac{\left(a+b x^2\right) \left(\sqrt{b} x \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)+\sqrt{a}\right)}{a^{3/2} x \sqrt{\left(a+b x^2\right)^2}}","-\frac{a+b x^2}{a x \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\sqrt{b} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"-(((a + b*x^2)*(Sqrt[a] + Sqrt[b]*x*ArcTan[(Sqrt[b]*x)/Sqrt[a]]))/(a^(3/2)*x*Sqrt[(a + b*x^2)^2]))","A",1
633,1,70,133,0.0249235,"\int \frac{1}{x^4 \sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx","Integrate[1/(x^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]),x]","-\frac{\left(a+b x^2\right) \left(\sqrt{a} \left(a-3 b x^2\right)-3 b^{3/2} x^3 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right)}{3 a^{5/2} x^3 \sqrt{\left(a+b x^2\right)^2}}","\frac{b \left(a+b x^2\right)}{a^2 x \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{-a-b x^2}{3 a x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{b^{3/2} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"-1/3*((a + b*x^2)*(Sqrt[a]*(a - 3*b*x^2) - 3*b^(3/2)*x^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]]))/(a^(5/2)*x^3*Sqrt[(a + b*x^2)^2])","A",1
634,1,81,158,0.0309397,"\int \frac{x^7}{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}} \, dx","Integrate[x^7/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","\frac{-5 a^3-4 a^2 b x^2+4 a b^2 x^4-6 a \left(a+b x^2\right)^2 \log \left(a+b x^2\right)+2 b^3 x^6}{4 b^4 \left(a+b x^2\right) \sqrt{\left(a+b x^2\right)^2}}","-\frac{3 a^2}{2 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 a \left(a+b x^2\right) \log \left(a+b x^2\right)}{2 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{x^2 \left(a+b x^2\right)}{2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{a^3}{4 b^4 \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(-5*a^3 - 4*a^2*b*x^2 + 4*a*b^2*x^4 + 2*b^3*x^6 - 6*a*(a + b*x^2)^2*Log[a + b*x^2])/(4*b^4*(a + b*x^2)*Sqrt[(a + b*x^2)^2])","A",1
635,1,61,113,0.0216351,"\int \frac{x^5}{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}} \, dx","Integrate[x^5/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","\frac{a \left(3 a+4 b x^2\right)+2 \left(a+b x^2\right)^2 \log \left(a+b x^2\right)}{4 b^3 \left(a+b x^2\right) \sqrt{\left(a+b x^2\right)^2}}","-\frac{a^2}{4 b^3 \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{a}{b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left(a+b x^2\right) \log \left(a+b x^2\right)}{2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(a*(3*a + 4*b*x^2) + 2*(a + b*x^2)^2*Log[a + b*x^2])/(4*b^3*(a + b*x^2)*Sqrt[(a + b*x^2)^2])","A",1
636,1,39,41,0.0122862,"\int \frac{x^3}{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}} \, dx","Integrate[x^3/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","\frac{-a-2 b x^2}{4 b^2 \left(a+b x^2\right) \sqrt{\left(a+b x^2\right)^2}}","\frac{x^4}{4 a \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(-a - 2*b*x^2)/(4*b^2*(a + b*x^2)*Sqrt[(a + b*x^2)^2])","A",1
637,1,27,38,0.0081757,"\int \frac{x}{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}} \, dx","Integrate[x/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","-\frac{a+b x^2}{4 b \left(\left(a+b x^2\right)^2\right)^{3/2}}","-\frac{1}{4 b \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"-1/4*(a + b*x^2)/(b*((a + b*x^2)^2)^(3/2))","A",1
638,1,74,147,0.0274317,"\int \frac{1}{x \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}} \, dx","Integrate[1/(x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)),x]","\frac{a \left(3 a+2 b x^2\right)+4 \log (x) \left(a+b x^2\right)^2-2 \left(a+b x^2\right)^2 \log \left(a+b x^2\right)}{4 a^3 \left(a+b x^2\right) \sqrt{\left(a+b x^2\right)^2}}","\frac{1}{4 a \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{2 a^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\log (x) \left(a+b x^2\right)}{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left(a+b x^2\right) \log \left(a+b x^2\right)}{2 a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(a*(3*a + 2*b*x^2) + 4*(a + b*x^2)^2*Log[x] - 2*(a + b*x^2)^2*Log[a + b*x^2])/(4*a^3*(a + b*x^2)*Sqrt[(a + b*x^2)^2])","A",1
639,1,97,189,0.0374058,"\int \frac{1}{x^3 \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}} \, dx","Integrate[1/(x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)),x]","\frac{-a \left(2 a^2+9 a b x^2+6 b^2 x^4\right)-12 b x^2 \log (x) \left(a+b x^2\right)^2+6 b x^2 \left(a+b x^2\right)^2 \log \left(a+b x^2\right)}{4 a^4 x^2 \left(a+b x^2\right) \sqrt{\left(a+b x^2\right)^2}}","-\frac{b}{4 a^2 \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 b \log (x) \left(a+b x^2\right)}{a^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3 b \left(a+b x^2\right) \log \left(a+b x^2\right)}{2 a^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b}{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a+b x^2}{2 a^3 x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(-(a*(2*a^2 + 9*a*b*x^2 + 6*b^2*x^4)) - 12*b*x^2*(a + b*x^2)^2*Log[x] + 6*b*x^2*(a + b*x^2)^2*Log[a + b*x^2])/(4*a^4*x^2*(a + b*x^2)*Sqrt[(a + b*x^2)^2])","A",1
640,1,84,128,0.0316395,"\int \frac{x^4}{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}} \, dx","Integrate[x^4/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","\frac{3 \left(a+b x^2\right)^2 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)-\sqrt{a} \sqrt{b} x \left(3 a+5 b x^2\right)}{8 \sqrt{a} b^{5/2} \left(a+b x^2\right) \sqrt{\left(a+b x^2\right)^2}}","-\frac{3 x}{8 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{x^3}{4 b \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3 \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 \sqrt{a} b^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(-(Sqrt[a]*Sqrt[b]*x*(3*a + 5*b*x^2)) + 3*(a + b*x^2)^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*Sqrt[a]*b^(5/2)*(a + b*x^2)*Sqrt[(a + b*x^2)^2])","A",1
641,1,81,129,0.0283515,"\int \frac{x^2}{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}} \, dx","Integrate[x^2/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","\frac{\sqrt{a} \sqrt{b} x \left(b x^2-a\right)+\left(a+b x^2\right)^2 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{3/2} b^{3/2} \left(a+b x^2\right) \sqrt{\left(a+b x^2\right)^2}}","\frac{x}{8 a b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{x}{4 b \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{3/2} b^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(Sqrt[a]*Sqrt[b]*x*(-a + b*x^2) + (a + b*x^2)^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(3/2)*b^(3/2)*(a + b*x^2)*Sqrt[(a + b*x^2)^2])","A",1
642,1,83,135,0.0242364,"\int \frac{1}{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(-3/2),x]","\frac{\sqrt{a} \sqrt{b} x \left(5 a+3 b x^2\right)+3 \left(a+b x^2\right)^2 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} \sqrt{b} \left(a+b x^2\right) \sqrt{\left(a+b x^2\right)^2}}","\frac{3 x \left(a+b x^2\right)^2}{8 a^2 \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}+\frac{x \left(a+b x^2\right)}{4 a \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}+\frac{3 \left(a+b x^2\right)^3 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} \sqrt{b} \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}",1,"(Sqrt[a]*Sqrt[b]*x*(5*a + 3*b*x^2) + 3*(a + b*x^2)^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*Sqrt[b]*(a + b*x^2)*Sqrt[(a + b*x^2)^2])","A",1
643,1,93,169,0.0321497,"\int \frac{1}{x^2 \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}} \, dx","Integrate[1/(x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)),x]","\frac{-\sqrt{a} \left(8 a^2+25 a b x^2+15 b^2 x^4\right)-15 \sqrt{b} x \left(a+b x^2\right)^2 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{7/2} x \left(a+b x^2\right) \sqrt{\left(a+b x^2\right)^2}}","\frac{5}{8 a^2 x \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a x \sqrt{a^2+2 a b x^2+b^2 x^4} \left(a+b x^2\right)}-\frac{15 \sqrt{b} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{15 \left(a+b x^2\right)}{8 a^3 x \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(-(Sqrt[a]*(8*a^2 + 25*a*b*x^2 + 15*b^2*x^4)) - 15*Sqrt[b]*x*(a + b*x^2)^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(7/2)*x*(a + b*x^2)*Sqrt[(a + b*x^2)^2])","A",1
644,1,105,209,0.0380888,"\int \frac{1}{x^4 \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}} \, dx","Integrate[1/(x^4*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)),x]","\frac{\sqrt{a} \left(-8 a^3+56 a^2 b x^2+175 a b^2 x^4+105 b^3 x^6\right)+105 b^{3/2} x^3 \left(a+b x^2\right)^2 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{24 a^{9/2} x^3 \left(a+b x^2\right) \sqrt{\left(a+b x^2\right)^2}}","\frac{1}{4 a x^3 \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{7}{8 a^2 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{35 b^{3/2} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{35 b \left(a+b x^2\right)}{8 a^4 x \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{35 \left(a+b x^2\right)}{24 a^3 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(Sqrt[a]*(-8*a^3 + 56*a^2*b*x^2 + 175*a*b^2*x^4 + 105*b^3*x^6) + 105*b^(3/2)*x^3*(a + b*x^2)^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(24*a^(9/2)*x^3*(a + b*x^2)*Sqrt[(a + b*x^2)^2])","A",1
645,1,103,238,0.0385252,"\int \frac{x^{11}}{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}} \, dx","Integrate[x^11/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{-77 a^5-248 a^4 b x^2-252 a^3 b^2 x^4-48 a^2 b^3 x^6+48 a b^4 x^8-60 a \left(a+b x^2\right)^4 \log \left(a+b x^2\right)+12 b^5 x^{10}}{24 b^6 \left(a+b x^2\right)^3 \sqrt{\left(a+b x^2\right)^2}}","-\frac{5 a^2}{b^6 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 a \left(a+b x^2\right) \log \left(a+b x^2\right)}{2 b^6 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{x^2 \left(a+b x^2\right)}{2 b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{a^5}{8 b^6 \left(a+b x^2\right)^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 a^4}{6 b^6 \left(a+b x^2\right)^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 a^3}{2 b^6 \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(-77*a^5 - 248*a^4*b*x^2 - 252*a^3*b^2*x^4 - 48*a^2*b^3*x^6 + 48*a*b^4*x^8 + 12*b^5*x^10 - 60*a*(a + b*x^2)^4*Log[a + b*x^2])/(24*b^6*(a + b*x^2)^3*Sqrt[(a + b*x^2)^2])","A",1
646,1,83,196,0.028886,"\int \frac{x^9}{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}} \, dx","Integrate[x^9/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{a \left(25 a^3+88 a^2 b x^2+108 a b^2 x^4+48 b^3 x^6\right)+12 \left(a+b x^2\right)^4 \log \left(a+b x^2\right)}{24 b^5 \left(a+b x^2\right)^3 \sqrt{\left(a+b x^2\right)^2}}","-\frac{3 a^2}{2 b^5 \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{2 a}{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left(a+b x^2\right) \log \left(a+b x^2\right)}{2 b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a^4}{8 b^5 \left(a+b x^2\right)^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{2 a^3}{3 b^5 \left(a+b x^2\right)^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(a*(25*a^3 + 88*a^2*b*x^2 + 108*a*b^2*x^4 + 48*b^3*x^6) + 12*(a + b*x^2)^4*Log[a + b*x^2])/(24*b^5*(a + b*x^2)^3*Sqrt[(a + b*x^2)^2])","A",1
647,1,61,41,0.0168024,"\int \frac{x^7}{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}} \, dx","Integrate[x^7/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{-a^3-4 a^2 b x^2-6 a b^2 x^4-4 b^3 x^6}{8 b^4 \left(a+b x^2\right)^3 \sqrt{\left(a+b x^2\right)^2}}","\frac{x^8}{8 a \left(a+b x^2\right)^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(-a^3 - 4*a^2*b*x^2 - 6*a*b^2*x^4 - 4*b^3*x^6)/(8*b^4*(a + b*x^2)^3*Sqrt[(a + b*x^2)^2])","A",1
648,1,50,74,0.0163656,"\int \frac{x^5}{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}} \, dx","Integrate[x^5/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{-a^2-4 a b x^2-6 b^2 x^4}{24 b^3 \left(a+b x^2\right)^3 \sqrt{\left(a+b x^2\right)^2}}","\frac{x^6}{8 a \left(a+b x^2\right) \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}+\frac{x^6}{24 a^2 \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}",1,"(-a^2 - 4*a*b*x^2 - 6*b^2*x^4)/(24*b^3*(a + b*x^2)^3*Sqrt[(a + b*x^2)^2])","A",1
649,1,39,69,0.0134581,"\int \frac{x^3}{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}} \, dx","Integrate[x^3/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{-a-4 b x^2}{24 b^2 \left(a+b x^2\right)^3 \sqrt{\left(a+b x^2\right)^2}}","\frac{a}{8 b^2 \left(a+b x^2\right) \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}-\frac{1}{6 b^2 \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}",1,"(-a - 4*b*x^2)/(24*b^2*(a + b*x^2)^3*Sqrt[(a + b*x^2)^2])","A",1
650,1,27,38,0.0103182,"\int \frac{x}{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}} \, dx","Integrate[x/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","-\frac{a+b x^2}{8 b \left(\left(a+b x^2\right)^2\right)^{5/2}}","-\frac{1}{8 b \left(a+b x^2\right) \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}",1,"-1/8*(a + b*x^2)/(b*((a + b*x^2)^2)^(5/2))","A",1
651,1,96,223,0.0429079,"\int \frac{1}{x \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}} \, dx","Integrate[1/(x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]","\frac{a \left(25 a^3+52 a^2 b x^2+42 a b^2 x^4+12 b^3 x^6\right)+24 \log (x) \left(a+b x^2\right)^4-12 \left(a+b x^2\right)^4 \log \left(a+b x^2\right)}{24 a^5 \left(a+b x^2\right)^3 \sqrt{\left(a+b x^2\right)^2}}","\frac{1}{6 a^2 \left(a+b x^2\right)^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{8 a \left(a+b x^2\right)^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\log (x) \left(a+b x^2\right)}{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left(a+b x^2\right) \log \left(a+b x^2\right)}{2 a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{2 a^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a^3 \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(a*(25*a^3 + 52*a^2*b*x^2 + 42*a*b^2*x^4 + 12*b^3*x^6) + 24*(a + b*x^2)^4*Log[x] - 12*(a + b*x^2)^4*Log[a + b*x^2])/(24*a^5*(a + b*x^2)^3*Sqrt[(a + b*x^2)^2])","A",1
652,1,119,267,0.0471575,"\int \frac{1}{x^3 \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}} \, dx","Integrate[1/(x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]","\frac{-a \left(12 a^4+125 a^3 b x^2+260 a^2 b^2 x^4+210 a b^3 x^6+60 b^4 x^8\right)-120 b x^2 \log (x) \left(a+b x^2\right)^4+60 b x^2 \left(a+b x^2\right)^4 \log \left(a+b x^2\right)}{24 a^6 x^2 \left(a+b x^2\right)^3 \sqrt{\left(a+b x^2\right)^2}}","-\frac{b}{8 a^2 \left(a+b x^2\right)^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 b \log (x) \left(a+b x^2\right)}{a^6 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 b \left(a+b x^2\right) \log \left(a+b x^2\right)}{2 a^6 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{2 b}{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a+b x^2}{2 a^5 x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 b}{4 a^4 \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b}{3 a^3 \left(a+b x^2\right)^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(-(a*(12*a^4 + 125*a^3*b*x^2 + 260*a^2*b^2*x^4 + 210*a*b^3*x^6 + 60*b^4*x^8)) - 120*b*x^2*(a + b*x^2)^4*Log[x] + 60*b*x^2*(a + b*x^2)^4*Log[a + b*x^2])/(24*a^6*x^2*(a + b*x^2)^3*Sqrt[(a + b*x^2)^2])","A",1
653,1,105,211,0.0429659,"\int \frac{x^6}{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}} \, dx","Integrate[x^6/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{\sqrt{a} \sqrt{b} x \left(-15 a^3-55 a^2 b x^2-73 a b^2 x^4+15 b^3 x^6\right)+15 \left(a+b x^2\right)^4 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{384 a^{3/2} b^{7/2} \left(a+b x^2\right)^3 \sqrt{\left(a+b x^2\right)^2}}","-\frac{x^5}{8 b \left(a+b x^2\right)^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 x^3}{48 b^2 \left(a+b x^2\right)^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 x}{64 b^3 \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 x}{128 a b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{128 a^{3/2} b^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(Sqrt[a]*Sqrt[b]*x*(-15*a^3 - 55*a^2*b*x^2 - 73*a*b^2*x^4 + 15*b^3*x^6) + 15*(a + b*x^2)^4*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(384*a^(3/2)*b^(7/2)*(a + b*x^2)^3*Sqrt[(a + b*x^2)^2])","A",1
654,1,105,212,0.0403023,"\int \frac{x^4}{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}} \, dx","Integrate[x^4/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{\sqrt{a} \sqrt{b} x \left(-3 a^3-11 a^2 b x^2+11 a b^2 x^4+3 b^3 x^6\right)+3 \left(a+b x^2\right)^4 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{128 a^{5/2} b^{5/2} \left(a+b x^2\right)^3 \sqrt{\left(a+b x^2\right)^2}}","\frac{x}{64 a b^2 \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3 x}{128 a^2 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{x}{16 b^2 \left(a+b x^2\right)^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{x^3}{8 b \left(a+b x^2\right)^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3 \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{128 a^{5/2} b^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(Sqrt[a]*Sqrt[b]*x*(-3*a^3 - 11*a^2*b*x^2 + 11*a*b^2*x^4 + 3*b^3*x^6) + 3*(a + b*x^2)^4*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(128*a^(5/2)*b^(5/2)*(a + b*x^2)^3*Sqrt[(a + b*x^2)^2])","A",1
655,1,105,213,0.0339708,"\int \frac{x^2}{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}} \, dx","Integrate[x^2/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{\sqrt{a} \sqrt{b} x \left(-15 a^3+73 a^2 b x^2+55 a b^2 x^4+15 b^3 x^6\right)+15 \left(a+b x^2\right)^4 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{384 a^{7/2} b^{3/2} \left(a+b x^2\right)^3 \sqrt{\left(a+b x^2\right)^2}}","\frac{5 x}{192 a^2 b \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{x}{48 a b \left(a+b x^2\right)^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{x}{8 b \left(a+b x^2\right)^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{128 a^{7/2} b^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 x}{128 a^3 b \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(Sqrt[a]*Sqrt[b]*x*(-15*a^3 + 73*a^2*b*x^2 + 55*a*b^2*x^4 + 15*b^3*x^6) + 15*(a + b*x^2)^4*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(384*a^(7/2)*b^(3/2)*(a + b*x^2)^3*Sqrt[(a + b*x^2)^2])","A",1
656,1,105,213,0.0385242,"\int \frac{1}{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(-5/2),x]","\frac{\sqrt{a} \sqrt{b} x \left(279 a^3+511 a^2 b x^2+385 a b^2 x^4+105 b^3 x^6\right)+105 \left(a+b x^2\right)^4 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{384 a^{9/2} \sqrt{b} \left(a+b x^2\right)^3 \sqrt{\left(a+b x^2\right)^2}}","\frac{7 x \left(a+b x^2\right)^2}{48 a^2 \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}+\frac{x \left(a+b x^2\right)}{8 a \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}+\frac{35 \left(a+b x^2\right)^5 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{128 a^{9/2} \sqrt{b} \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}+\frac{35 x \left(a+b x^2\right)^4}{128 a^4 \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}+\frac{35 x \left(a+b x^2\right)^3}{192 a^3 \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}",1,"(Sqrt[a]*Sqrt[b]*x*(279*a^3 + 511*a^2*b*x^2 + 385*a*b^2*x^4 + 105*b^3*x^6) + 105*(a + b*x^2)^4*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(384*a^(9/2)*Sqrt[b]*(a + b*x^2)^3*Sqrt[(a + b*x^2)^2])","A",1
657,1,115,251,0.043591,"\int \frac{1}{x^2 \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}} \, dx","Integrate[1/(x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]","\frac{-\sqrt{a} \left(128 a^4+837 a^3 b x^2+1533 a^2 b^2 x^4+1155 a b^3 x^6+315 b^4 x^8\right)-315 \sqrt{b} x \left(a+b x^2\right)^4 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{128 a^{11/2} x \left(a+b x^2\right)^3 \sqrt{\left(a+b x^2\right)^2}}","\frac{3}{16 a^2 x \sqrt{a^2+2 a b x^2+b^2 x^4} \left(a+b x^2\right)^2}+\frac{1}{8 a x \sqrt{a^2+2 a b x^2+b^2 x^4} \left(a+b x^2\right)^3}-\frac{315 \sqrt{b} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{128 a^{11/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{315 \left(a+b x^2\right)}{128 a^5 x \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{105}{128 a^4 x \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{21}{64 a^3 x \sqrt{a^2+2 a b x^2+b^2 x^4} \left(a+b x^2\right)}",1,"(-(Sqrt[a]*(128*a^4 + 837*a^3*b*x^2 + 1533*a^2*b^2*x^4 + 1155*a*b^3*x^6 + 315*b^4*x^8)) - 315*Sqrt[b]*x*(a + b*x^2)^4*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(128*a^(11/2)*x*(a + b*x^2)^3*Sqrt[(a + b*x^2)^2])","A",1
658,1,127,291,0.0480954,"\int \frac{1}{x^4 \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}} \, dx","Integrate[1/(x^4*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]","\frac{\sqrt{a} \left(-128 a^5+1408 a^4 b x^2+9207 a^3 b^2 x^4+16863 a^2 b^3 x^6+12705 a b^4 x^8+3465 b^5 x^{10}\right)+3465 b^{3/2} x^3 \left(a+b x^2\right)^4 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{384 a^{13/2} x^3 \left(a+b x^2\right)^3 \sqrt{\left(a+b x^2\right)^2}}","\frac{11}{48 a^2 x^3 \left(a+b x^2\right)^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{8 a x^3 \left(a+b x^2\right)^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1155 b^{3/2} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{128 a^{13/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1155 b \left(a+b x^2\right)}{128 a^6 x \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{385 \left(a+b x^2\right)}{128 a^5 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{231}{128 a^4 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{33}{64 a^3 x^3 \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(Sqrt[a]*(-128*a^5 + 1408*a^4*b*x^2 + 9207*a^3*b^2*x^4 + 16863*a^2*b^3*x^6 + 12705*a*b^4*x^8 + 3465*b^5*x^10) + 3465*b^(3/2)*x^3*(a + b*x^2)^4*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(384*a^(13/2)*x^3*(a + b*x^2)^3*Sqrt[(a + b*x^2)^2])","A",1
659,1,64,298,0.0327773,"\int \frac{x^2}{\sqrt[3]{a^2+2 a b x^2+b^2 x^4}} \, dx","Integrate[x^2/(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/3),x]","\frac{3 x \left(-a \left(\frac{b x^2}{a}+1\right)^{2/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{3}{2};-\frac{b x^2}{a}\right)+a+b x^2\right)}{5 b \sqrt[3]{\left(a+b x^2\right)^2}}","\frac{3 x \left(a+b x^2\right)}{5 b \sqrt[3]{a^2+2 a b x^2+b^2 x^4}}+\frac{3\ 3^{3/4} \sqrt{2-\sqrt{3}} a^2 \left(\frac{b x^2}{a}+1\right)^{2/3} \left(1-\sqrt[3]{\frac{b x^2}{a}+1}\right) \sqrt{\frac{\left(\frac{b x^2}{a}+1\right)^{2/3}+\sqrt[3]{\frac{b x^2}{a}+1}+1}{\left(-\sqrt[3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{b x^2}{a}+1}+\sqrt{3}+1}{-\sqrt[3]{\frac{b x^2}{a}+1}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{5 b^2 x \sqrt[3]{a^2+2 a b x^2+b^2 x^4} \sqrt{-\frac{1-\sqrt[3]{\frac{b x^2}{a}+1}}{\left(-\sqrt[3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right)^2}}}",1,"(3*x*(a + b*x^2 - a*(1 + (b*x^2)/a)^(2/3)*Hypergeometric2F1[1/2, 2/3, 3/2, -((b*x^2)/a)]))/(5*b*((a + b*x^2)^2)^(1/3))","C",1
660,1,48,256,0.0114762,"\int \frac{1}{\sqrt[3]{a^2+2 a b x^2+b^2 x^4}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(-1/3),x]","\frac{x \left(\frac{b x^2}{a}+1\right)^{2/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{3}{2};-\frac{b x^2}{a}\right)}{\sqrt[3]{\left(a+b x^2\right)^2}}","-\frac{3^{3/4} \sqrt{2-\sqrt{3}} a \left(\frac{b x^2}{a}+1\right)^{2/3} \left(1-\sqrt[3]{\frac{b x^2}{a}+1}\right) \sqrt{\frac{\left(\frac{b x^2}{a}+1\right)^{2/3}+\sqrt[3]{\frac{b x^2}{a}+1}+1}{\left(-\sqrt[3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{b x^2}{a}+1}+\sqrt{3}+1}{-\sqrt[3]{\frac{b x^2}{a}+1}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{b x \sqrt[3]{a^2+2 a b x^2+b^2 x^4} \sqrt{-\frac{1-\sqrt[3]{\frac{b x^2}{a}+1}}{\left(-\sqrt[3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right)^2}}}",1,"(x*(1 + (b*x^2)/a)^(2/3)*Hypergeometric2F1[1/2, 2/3, 3/2, -((b*x^2)/a)])/((a + b*x^2)^2)^(1/3)","C",1
661,1,51,289,0.0100654,"\int \frac{1}{x^2 \sqrt[3]{a^2+2 a b x^2+b^2 x^4}} \, dx","Integrate[1/(x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/3)),x]","-\frac{\left(\frac{b x^2}{a}+1\right)^{2/3} \, _2F_1\left(-\frac{1}{2},\frac{2}{3};\frac{1}{2};-\frac{b x^2}{a}\right)}{x \sqrt[3]{\left(a+b x^2\right)^2}}","\frac{\sqrt{2-\sqrt{3}} \left(\frac{b x^2}{a}+1\right)^{2/3} \left(1-\sqrt[3]{\frac{b x^2}{a}+1}\right) \sqrt{\frac{\left(\frac{b x^2}{a}+1\right)^{2/3}+\sqrt[3]{\frac{b x^2}{a}+1}+1}{\left(-\sqrt[3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{b x^2}{a}+1}+\sqrt{3}+1}{-\sqrt[3]{\frac{b x^2}{a}+1}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt[4]{3} x \sqrt[3]{a^2+2 a b x^2+b^2 x^4} \sqrt{-\frac{1-\sqrt[3]{\frac{b x^2}{a}+1}}{\left(-\sqrt[3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right)^2}}}-\frac{a+b x^2}{a x \sqrt[3]{a^2+2 a b x^2+b^2 x^4}}",1,"-(((1 + (b*x^2)/a)^(2/3)*Hypergeometric2F1[-1/2, 2/3, 1/2, -((b*x^2)/a)])/(x*((a + b*x^2)^2)^(1/3)))","C",1
662,1,64,618,0.0333197,"\int \frac{x^2}{\left(a^2+2 a b x^2+b^2 x^4\right)^{2/3}} \, dx","Integrate[x^2/(a^2 + 2*a*b*x^2 + b^2*x^4)^(2/3),x]","\frac{3 x \left(a+b x^2\right) \left(\sqrt[3]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)-1\right)}{2 b \left(\left(a+b x^2\right)^2\right)^{2/3}}","-\frac{9 a x \left(\frac{b x^2}{a}+1\right)^{4/3}}{2 b \left(a^2+2 a b x^2+b^2 x^4\right)^{2/3} \left(-\sqrt[3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right)}-\frac{3 x \left(a+b x^2\right)}{2 b \left(a^2+2 a b x^2+b^2 x^4\right)^{2/3}}-\frac{3\ 3^{3/4} a^2 \left(\frac{b x^2}{a}+1\right)^{4/3} \left(1-\sqrt[3]{\frac{b x^2}{a}+1}\right) \sqrt{\frac{\left(\frac{b x^2}{a}+1\right)^{2/3}+\sqrt[3]{\frac{b x^2}{a}+1}+1}{\left(-\sqrt[3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{b x^2}{a}+1}+\sqrt{3}+1}{-\sqrt[3]{\frac{b x^2}{a}+1}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt{2} b^2 x \left(a^2+2 a b x^2+b^2 x^4\right)^{2/3} \sqrt{-\frac{1-\sqrt[3]{\frac{b x^2}{a}+1}}{\left(-\sqrt[3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right)^2}}}+\frac{9 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^2 \left(\frac{b x^2}{a}+1\right)^{4/3} \left(1-\sqrt[3]{\frac{b x^2}{a}+1}\right) \sqrt{\frac{\left(\frac{b x^2}{a}+1\right)^{2/3}+\sqrt[3]{\frac{b x^2}{a}+1}+1}{\left(-\sqrt[3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{b x^2}{a}+1}+\sqrt{3}+1}{-\sqrt[3]{\frac{b x^2}{a}+1}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{4 b^2 x \left(a^2+2 a b x^2+b^2 x^4\right)^{2/3} \sqrt{-\frac{1-\sqrt[3]{\frac{b x^2}{a}+1}}{\left(-\sqrt[3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right)^2}}}",1,"(3*x*(a + b*x^2)*(-1 + (1 + (b*x^2)/a)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, -((b*x^2)/a)]))/(2*b*((a + b*x^2)^2)^(2/3))","C",1
663,1,64,609,0.0198878,"\int \frac{1}{\left(a^2+2 a b x^2+b^2 x^4\right)^{2/3}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(-2/3),x]","-\frac{x \left(a+b x^2\right) \left(\sqrt[3]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)-3\right)}{2 a \left(\left(a+b x^2\right)^2\right)^{2/3}}","\frac{3 x \left(\frac{b x^2}{a}+1\right)^{4/3}}{2 \left(a^2+2 a b x^2+b^2 x^4\right)^{2/3} \left(-\sqrt[3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right)}+\frac{3 x \left(a+b x^2\right)}{2 a \left(a^2+2 a b x^2+b^2 x^4\right)^{2/3}}+\frac{3^{3/4} a \left(1-\sqrt[3]{\frac{b x^2}{a}+1}\right) \sqrt{\frac{\left(\frac{b x^2}{a}+1\right)^{2/3}+\sqrt[3]{\frac{b x^2}{a}+1}+1}{\left(-\sqrt[3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right)^2}} \left(\frac{b x^2}{a}+1\right)^{4/3} F\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{b x^2}{a}+1}+\sqrt{3}+1}{-\sqrt[3]{\frac{b x^2}{a}+1}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt{2} b x \left(a^2+2 a b x^2+b^2 x^4\right)^{2/3} \sqrt{-\frac{1-\sqrt[3]{\frac{b x^2}{a}+1}}{\left(-\sqrt[3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right)^2}}}-\frac{3 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a \left(1-\sqrt[3]{\frac{b x^2}{a}+1}\right) \sqrt{\frac{\left(\frac{b x^2}{a}+1\right)^{2/3}+\sqrt[3]{\frac{b x^2}{a}+1}+1}{\left(-\sqrt[3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right)^2}} \left(\frac{b x^2}{a}+1\right)^{4/3} E\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{b x^2}{a}+1}+\sqrt{3}+1}{-\sqrt[3]{\frac{b x^2}{a}+1}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{4 b x \left(a^2+2 a b x^2+b^2 x^4\right)^{2/3} \sqrt{-\frac{1-\sqrt[3]{\frac{b x^2}{a}+1}}{\left(-\sqrt[3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right)^2}}}",1,"-1/2*(x*(a + b*x^2)*(-3 + (1 + (b*x^2)/a)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, -((b*x^2)/a)]))/(a*((a + b*x^2)^2)^(2/3))","C",1
664,1,61,649,0.0154358,"\int \frac{1}{x^2 \left(a^2+2 a b x^2+b^2 x^4\right)^{2/3}} \, dx","Integrate[1/(x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(2/3)),x]","-\frac{\left(a+b x^2\right) \sqrt[3]{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{1}{2},\frac{4}{3};\frac{1}{2};-\frac{b x^2}{a}\right)}{a x \left(\left(a+b x^2\right)^2\right)^{2/3}}","-\frac{5 \left(a+b x^2\right)^2}{2 a^2 x \left(a^2+2 a b x^2+b^2 x^4\right)^{2/3}}+\frac{3 \left(a+b x^2\right)}{2 a x \left(a^2+2 a b x^2+b^2 x^4\right)^{2/3}}-\frac{5 b x \left(\frac{b x^2}{a}+1\right)^{4/3}}{2 a \left(a^2+2 a b x^2+b^2 x^4\right)^{2/3} \left(-\sqrt[3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right)}-\frac{5 \left(\frac{b x^2}{a}+1\right)^{4/3} \left(1-\sqrt[3]{\frac{b x^2}{a}+1}\right) \sqrt{\frac{\left(\frac{b x^2}{a}+1\right)^{2/3}+\sqrt[3]{\frac{b x^2}{a}+1}+1}{\left(-\sqrt[3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{b x^2}{a}+1}+\sqrt{3}+1}{-\sqrt[3]{\frac{b x^2}{a}+1}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt{2} \sqrt[4]{3} x \left(a^2+2 a b x^2+b^2 x^4\right)^{2/3} \sqrt{-\frac{1-\sqrt[3]{\frac{b x^2}{a}+1}}{\left(-\sqrt[3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right)^2}}}+\frac{5 \sqrt[4]{3} \sqrt{2+\sqrt{3}} \left(\frac{b x^2}{a}+1\right)^{4/3} \left(1-\sqrt[3]{\frac{b x^2}{a}+1}\right) \sqrt{\frac{\left(\frac{b x^2}{a}+1\right)^{2/3}+\sqrt[3]{\frac{b x^2}{a}+1}+1}{\left(-\sqrt[3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{b x^2}{a}+1}+\sqrt{3}+1}{-\sqrt[3]{\frac{b x^2}{a}+1}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{4 x \left(a^2+2 a b x^2+b^2 x^4\right)^{2/3} \sqrt{-\frac{1-\sqrt[3]{\frac{b x^2}{a}+1}}{\left(-\sqrt[3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right)^2}}}",1,"-(((a + b*x^2)*(1 + (b*x^2)/a)^(1/3)*Hypergeometric2F1[-1/2, 4/3, 1/2, -((b*x^2)/a)])/(a*x*((a + b*x^2)^2)^(2/3)))","C",1
665,1,33,51,0.0151018,"\int (d x)^{5/2} \left(a^2+2 a b x^2+b^2 x^4\right) \, dx","Integrate[(d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4),x]","\frac{2 x (d x)^{5/2} \left(165 a^2+210 a b x^2+77 b^2 x^4\right)}{1155}","\frac{2 a^2 (d x)^{7/2}}{7 d}+\frac{4 a b (d x)^{11/2}}{11 d^3}+\frac{2 b^2 (d x)^{15/2}}{15 d^5}",1,"(2*x*(d*x)^(5/2)*(165*a^2 + 210*a*b*x^2 + 77*b^2*x^4))/1155","A",1
666,1,33,51,0.0115912,"\int (d x)^{3/2} \left(a^2+2 a b x^2+b^2 x^4\right) \, dx","Integrate[(d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4),x]","\frac{2}{585} x (d x)^{3/2} \left(117 a^2+130 a b x^2+45 b^2 x^4\right)","\frac{2 a^2 (d x)^{5/2}}{5 d}+\frac{4 a b (d x)^{9/2}}{9 d^3}+\frac{2 b^2 (d x)^{13/2}}{13 d^5}",1,"(2*x*(d*x)^(3/2)*(117*a^2 + 130*a*b*x^2 + 45*b^2*x^4))/585","A",1
667,1,33,51,0.0087807,"\int \sqrt{d x} \left(a^2+2 a b x^2+b^2 x^4\right) \, dx","Integrate[Sqrt[d*x]*(a^2 + 2*a*b*x^2 + b^2*x^4),x]","\frac{2}{231} x \sqrt{d x} \left(77 a^2+66 a b x^2+21 b^2 x^4\right)","\frac{2 a^2 (d x)^{3/2}}{3 d}+\frac{4 a b (d x)^{7/2}}{7 d^3}+\frac{2 b^2 (d x)^{11/2}}{11 d^5}",1,"(2*x*Sqrt[d*x]*(77*a^2 + 66*a*b*x^2 + 21*b^2*x^4))/231","A",1
668,1,33,49,0.0101299,"\int \frac{a^2+2 a b x^2+b^2 x^4}{\sqrt{d x}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)/Sqrt[d*x],x]","\frac{2 \left(45 a^2 x+18 a b x^3+5 b^2 x^5\right)}{45 \sqrt{d x}}","\frac{2 a^2 \sqrt{d x}}{d}+\frac{4 a b (d x)^{5/2}}{5 d^3}+\frac{2 b^2 (d x)^{9/2}}{9 d^5}",1,"(2*(45*a^2*x + 18*a*b*x^3 + 5*b^2*x^5))/(45*Sqrt[d*x])","A",1
669,1,33,49,0.0116711,"\int \frac{a^2+2 a b x^2+b^2 x^4}{(d x)^{3/2}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)/(d*x)^(3/2),x]","\frac{2 x \left(-21 a^2+14 a b x^2+3 b^2 x^4\right)}{21 (d x)^{3/2}}","-\frac{2 a^2}{d \sqrt{d x}}+\frac{4 a b (d x)^{3/2}}{3 d^3}+\frac{2 b^2 (d x)^{7/2}}{7 d^5}",1,"(2*x*(-21*a^2 + 14*a*b*x^2 + 3*b^2*x^4))/(21*(d*x)^(3/2))","A",1
670,1,33,49,0.0115834,"\int \frac{a^2+2 a b x^2+b^2 x^4}{(d x)^{5/2}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)/(d*x)^(5/2),x]","\frac{x \left(-10 a^2+60 a b x^2+6 b^2 x^4\right)}{15 (d x)^{5/2}}","-\frac{2 a^2}{3 d (d x)^{3/2}}+\frac{4 a b \sqrt{d x}}{d^3}+\frac{2 b^2 (d x)^{5/2}}{5 d^5}",1,"(x*(-10*a^2 + 60*a*b*x^2 + 6*b^2*x^4))/(15*(d*x)^(5/2))","A",1
671,1,38,49,0.0130641,"\int \frac{a^2+2 a b x^2+b^2 x^4}{(d x)^{7/2}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)/(d*x)^(7/2),x]","\frac{2 \sqrt{d x} \left(-3 a^2-30 a b x^2+5 b^2 x^4\right)}{15 d^4 x^3}","-\frac{2 a^2}{5 d (d x)^{5/2}}-\frac{4 a b}{d^3 \sqrt{d x}}+\frac{2 b^2 (d x)^{3/2}}{3 d^5}",1,"(2*Sqrt[d*x]*(-3*a^2 - 30*a*b*x^2 + 5*b^2*x^4))/(15*d^4*x^3)","A",1
672,1,55,91,0.022097,"\int (d x)^{5/2} \left(a^2+2 a b x^2+b^2 x^4\right)^2 \, dx","Integrate[(d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","\frac{2 x (d x)^{5/2} \left(24035 a^4+61180 a^3 b x^2+67298 a^2 b^2 x^4+35420 a b^3 x^6+7315 b^4 x^8\right)}{168245}","\frac{2 a^4 (d x)^{7/2}}{7 d}+\frac{8 a^3 b (d x)^{11/2}}{11 d^3}+\frac{4 a^2 b^2 (d x)^{15/2}}{5 d^5}+\frac{8 a b^3 (d x)^{19/2}}{19 d^7}+\frac{2 b^4 (d x)^{23/2}}{23 d^9}",1,"(2*x*(d*x)^(5/2)*(24035*a^4 + 61180*a^3*b*x^2 + 67298*a^2*b^2*x^4 + 35420*a*b^3*x^6 + 7315*b^4*x^8))/168245","A",1
673,1,55,91,0.0185123,"\int (d x)^{3/2} \left(a^2+2 a b x^2+b^2 x^4\right)^2 \, dx","Integrate[(d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","\frac{2 x (d x)^{3/2} \left(13923 a^4+30940 a^3 b x^2+32130 a^2 b^2 x^4+16380 a b^3 x^6+3315 b^4 x^8\right)}{69615}","\frac{2 a^4 (d x)^{5/2}}{5 d}+\frac{8 a^3 b (d x)^{9/2}}{9 d^3}+\frac{12 a^2 b^2 (d x)^{13/2}}{13 d^5}+\frac{8 a b^3 (d x)^{17/2}}{17 d^7}+\frac{2 b^4 (d x)^{21/2}}{21 d^9}",1,"(2*x*(d*x)^(3/2)*(13923*a^4 + 30940*a^3*b*x^2 + 32130*a^2*b^2*x^4 + 16380*a*b^3*x^6 + 3315*b^4*x^8))/69615","A",1
674,1,55,91,0.0144186,"\int \sqrt{d x} \left(a^2+2 a b x^2+b^2 x^4\right)^2 \, dx","Integrate[Sqrt[d*x]*(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","\frac{2 x \sqrt{d x} \left(7315 a^4+12540 a^3 b x^2+11970 a^2 b^2 x^4+5852 a b^3 x^6+1155 b^4 x^8\right)}{21945}","\frac{2 a^4 (d x)^{3/2}}{3 d}+\frac{8 a^3 b (d x)^{7/2}}{7 d^3}+\frac{12 a^2 b^2 (d x)^{11/2}}{11 d^5}+\frac{8 a b^3 (d x)^{15/2}}{15 d^7}+\frac{2 b^4 (d x)^{19/2}}{19 d^9}",1,"(2*x*Sqrt[d*x]*(7315*a^4 + 12540*a^3*b*x^2 + 11970*a^2*b^2*x^4 + 5852*a*b^3*x^6 + 1155*b^4*x^8))/21945","A",1
675,1,55,89,0.0160503,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^2}{\sqrt{d x}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/Sqrt[d*x],x]","\frac{2 \left(3315 a^4 x+2652 a^3 b x^3+2210 a^2 b^2 x^5+1020 a b^3 x^7+195 b^4 x^9\right)}{3315 \sqrt{d x}}","\frac{2 a^4 \sqrt{d x}}{d}+\frac{8 a^3 b (d x)^{5/2}}{5 d^3}+\frac{4 a^2 b^2 (d x)^{9/2}}{3 d^5}+\frac{8 a b^3 (d x)^{13/2}}{13 d^7}+\frac{2 b^4 (d x)^{17/2}}{17 d^9}",1,"(2*(3315*a^4*x + 2652*a^3*b*x^3 + 2210*a^2*b^2*x^5 + 1020*a*b^3*x^7 + 195*b^4*x^9))/(3315*Sqrt[d*x])","A",1
676,1,55,89,0.0166086,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^2}{(d x)^{3/2}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/(d*x)^(3/2),x]","\frac{2 x \left(-1155 a^4+1540 a^3 b x^2+990 a^2 b^2 x^4+420 a b^3 x^6+77 b^4 x^8\right)}{1155 (d x)^{3/2}}","-\frac{2 a^4}{d \sqrt{d x}}+\frac{8 a^3 b (d x)^{3/2}}{3 d^3}+\frac{12 a^2 b^2 (d x)^{7/2}}{7 d^5}+\frac{8 a b^3 (d x)^{11/2}}{11 d^7}+\frac{2 b^4 (d x)^{15/2}}{15 d^9}",1,"(2*x*(-1155*a^4 + 1540*a^3*b*x^2 + 990*a^2*b^2*x^4 + 420*a*b^3*x^6 + 77*b^4*x^8))/(1155*(d*x)^(3/2))","A",1
677,1,55,89,0.0165529,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^2}{(d x)^{5/2}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/(d*x)^(5/2),x]","\frac{x \left(-390 a^4+4680 a^3 b x^2+1404 a^2 b^2 x^4+520 a b^3 x^6+90 b^4 x^8\right)}{585 (d x)^{5/2}}","-\frac{2 a^4}{3 d (d x)^{3/2}}+\frac{8 a^3 b \sqrt{d x}}{d^3}+\frac{12 a^2 b^2 (d x)^{5/2}}{5 d^5}+\frac{8 a b^3 (d x)^{9/2}}{9 d^7}+\frac{2 b^4 (d x)^{13/2}}{13 d^9}",1,"(x*(-390*a^4 + 4680*a^3*b*x^2 + 1404*a^2*b^2*x^4 + 520*a*b^3*x^6 + 90*b^4*x^8))/(585*(d*x)^(5/2))","A",1
678,1,60,87,0.0196471,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^2}{(d x)^{7/2}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/(d*x)^(7/2),x]","\frac{2 \sqrt{d x} \left(-77 a^4-1540 a^3 b x^2+770 a^2 b^2 x^4+220 a b^3 x^6+35 b^4 x^8\right)}{385 d^4 x^3}","-\frac{2 a^4}{5 d (d x)^{5/2}}-\frac{8 a^3 b}{d^3 \sqrt{d x}}+\frac{4 a^2 b^2 (d x)^{3/2}}{d^5}+\frac{8 a b^3 (d x)^{7/2}}{7 d^7}+\frac{2 b^4 (d x)^{11/2}}{11 d^9}",1,"(2*Sqrt[d*x]*(-77*a^4 - 1540*a^3*b*x^2 + 770*a^2*b^2*x^4 + 220*a*b^3*x^6 + 35*b^4*x^8))/(385*d^4*x^3)","A",1
679,1,77,129,0.0297403,"\int (d x)^{5/2} \left(a^2+2 a b x^2+b^2 x^4\right)^3 \, dx","Integrate[(d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{2 x (d x)^{5/2} \left(1341153 a^6+5120766 a^5 b x^2+9388071 a^4 b^2 x^4+9882180 a^3 b^3 x^6+6122655 a^2 b^4 x^8+2086238 a b^5 x^{10}+302841 b^6 x^{12}\right)}{9388071}","\frac{2 a^6 (d x)^{7/2}}{7 d}+\frac{12 a^5 b (d x)^{11/2}}{11 d^3}+\frac{2 a^4 b^2 (d x)^{15/2}}{d^5}+\frac{40 a^3 b^3 (d x)^{19/2}}{19 d^7}+\frac{30 a^2 b^4 (d x)^{23/2}}{23 d^9}+\frac{4 a b^5 (d x)^{27/2}}{9 d^{11}}+\frac{2 b^6 (d x)^{31/2}}{31 d^{13}}",1,"(2*x*(d*x)^(5/2)*(1341153*a^6 + 5120766*a^5*b*x^2 + 9388071*a^4*b^2*x^4 + 9882180*a^3*b^3*x^6 + 6122655*a^2*b^4*x^8 + 2086238*a*b^5*x^10 + 302841*b^6*x^12))/9388071","A",1
680,1,77,131,0.0253483,"\int (d x)^{3/2} \left(a^2+2 a b x^2+b^2 x^4\right)^3 \, dx","Integrate[(d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{2 x (d x)^{3/2} \left(672945 a^6+2243150 a^5 b x^2+3882375 a^4 b^2 x^4+3958500 a^3 b^3 x^6+2403375 a^2 b^4 x^8+807534 a b^5 x^{10}+116025 b^6 x^{12}\right)}{3364725}","\frac{2 a^6 (d x)^{5/2}}{5 d}+\frac{4 a^5 b (d x)^{9/2}}{3 d^3}+\frac{30 a^4 b^2 (d x)^{13/2}}{13 d^5}+\frac{40 a^3 b^3 (d x)^{17/2}}{17 d^7}+\frac{10 a^2 b^4 (d x)^{21/2}}{7 d^9}+\frac{12 a b^5 (d x)^{25/2}}{25 d^{11}}+\frac{2 b^6 (d x)^{29/2}}{29 d^{13}}",1,"(2*x*(d*x)^(3/2)*(672945*a^6 + 2243150*a^5*b*x^2 + 3882375*a^4*b^2*x^4 + 3958500*a^3*b^3*x^6 + 2403375*a^2*b^4*x^8 + 807534*a*b^5*x^10 + 116025*b^6*x^12))/3364725","A",1
681,1,77,131,0.0211453,"\int \sqrt{d x} \left(a^2+2 a b x^2+b^2 x^4\right)^3 \, dx","Integrate[Sqrt[d*x]*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{2 x \sqrt{d x} \left(302841 a^6+778734 a^5 b x^2+1238895 a^4 b^2 x^4+1211364 a^3 b^3 x^6+717255 a^2 b^4 x^8+237006 a b^5 x^{10}+33649 b^6 x^{12}\right)}{908523}","\frac{2 a^6 (d x)^{3/2}}{3 d}+\frac{12 a^5 b (d x)^{7/2}}{7 d^3}+\frac{30 a^4 b^2 (d x)^{11/2}}{11 d^5}+\frac{8 a^3 b^3 (d x)^{15/2}}{3 d^7}+\frac{30 a^2 b^4 (d x)^{19/2}}{19 d^9}+\frac{12 a b^5 (d x)^{23/2}}{23 d^{11}}+\frac{2 b^6 (d x)^{27/2}}{27 d^{13}}",1,"(2*x*Sqrt[d*x]*(302841*a^6 + 778734*a^5*b*x^2 + 1238895*a^4*b^2*x^4 + 1211364*a^3*b^3*x^6 + 717255*a^2*b^4*x^8 + 237006*a*b^5*x^10 + 33649*b^6*x^12))/908523","A",1
682,1,77,129,0.0229983,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^3}{\sqrt{d x}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/Sqrt[d*x],x]","\frac{2 \left(116025 a^6 x+139230 a^5 b x^3+193375 a^4 b^2 x^5+178500 a^3 b^3 x^7+102375 a^2 b^4 x^9+33150 a b^5 x^{11}+4641 b^6 x^{13}\right)}{116025 \sqrt{d x}}","\frac{2 a^6 \sqrt{d x}}{d}+\frac{12 a^5 b (d x)^{5/2}}{5 d^3}+\frac{10 a^4 b^2 (d x)^{9/2}}{3 d^5}+\frac{40 a^3 b^3 (d x)^{13/2}}{13 d^7}+\frac{30 a^2 b^4 (d x)^{17/2}}{17 d^9}+\frac{4 a b^5 (d x)^{21/2}}{7 d^{11}}+\frac{2 b^6 (d x)^{25/2}}{25 d^{13}}",1,"(2*(116025*a^6*x + 139230*a^5*b*x^3 + 193375*a^4*b^2*x^5 + 178500*a^3*b^3*x^7 + 102375*a^2*b^4*x^9 + 33150*a*b^5*x^11 + 4641*b^6*x^13))/(116025*Sqrt[d*x])","A",1
683,1,77,125,0.0222127,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^3}{(d x)^{3/2}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/(d*x)^(3/2),x]","\frac{2 x \left(-33649 a^6+67298 a^5 b x^2+72105 a^4 b^2 x^4+61180 a^3 b^3 x^6+33649 a^2 b^4 x^8+10626 a b^5 x^{10}+1463 b^6 x^{12}\right)}{33649 (d x)^{3/2}}","-\frac{2 a^6}{d \sqrt{d x}}+\frac{4 a^5 b (d x)^{3/2}}{d^3}+\frac{30 a^4 b^2 (d x)^{7/2}}{7 d^5}+\frac{40 a^3 b^3 (d x)^{11/2}}{11 d^7}+\frac{2 a^2 b^4 (d x)^{15/2}}{d^9}+\frac{12 a b^5 (d x)^{19/2}}{19 d^{11}}+\frac{2 b^6 (d x)^{23/2}}{23 d^{13}}",1,"(2*x*(-33649*a^6 + 67298*a^5*b*x^2 + 72105*a^4*b^2*x^4 + 61180*a^3*b^3*x^6 + 33649*a^2*b^4*x^8 + 10626*a*b^5*x^10 + 1463*b^6*x^12))/(33649*(d*x)^(3/2))","A",1
684,1,77,127,0.0223666,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^3}{(d x)^{5/2}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/(d*x)^(5/2),x]","\frac{2 x \left(-4641 a^6+83538 a^5 b x^2+41769 a^4 b^2 x^4+30940 a^3 b^3 x^6+16065 a^2 b^4 x^8+4914 a b^5 x^{10}+663 b^6 x^{12}\right)}{13923 (d x)^{5/2}}","-\frac{2 a^6}{3 d (d x)^{3/2}}+\frac{12 a^5 b \sqrt{d x}}{d^3}+\frac{6 a^4 b^2 (d x)^{5/2}}{d^5}+\frac{40 a^3 b^3 (d x)^{9/2}}{9 d^7}+\frac{30 a^2 b^4 (d x)^{13/2}}{13 d^9}+\frac{12 a b^5 (d x)^{17/2}}{17 d^{11}}+\frac{2 b^6 (d x)^{21/2}}{21 d^{13}}",1,"(2*x*(-4641*a^6 + 83538*a^5*b*x^2 + 41769*a^4*b^2*x^4 + 30940*a^3*b^3*x^6 + 16065*a^2*b^4*x^8 + 4914*a*b^5*x^10 + 663*b^6*x^12))/(13923*(d*x)^(5/2))","A",1
685,1,82,127,0.0266183,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^3}{(d x)^{7/2}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/(d*x)^(7/2),x]","\frac{2 \sqrt{d x} \left(-1463 a^6-43890 a^5 b x^2+36575 a^4 b^2 x^4+20900 a^3 b^3 x^6+9975 a^2 b^4 x^8+2926 a b^5 x^{10}+385 b^6 x^{12}\right)}{7315 d^4 x^3}","-\frac{2 a^6}{5 d (d x)^{5/2}}-\frac{12 a^5 b}{d^3 \sqrt{d x}}+\frac{10 a^4 b^2 (d x)^{3/2}}{d^5}+\frac{40 a^3 b^3 (d x)^{7/2}}{7 d^7}+\frac{30 a^2 b^4 (d x)^{11/2}}{11 d^9}+\frac{4 a b^5 (d x)^{15/2}}{5 d^{11}}+\frac{2 b^6 (d x)^{19/2}}{19 d^{13}}",1,"(2*Sqrt[d*x]*(-1463*a^6 - 43890*a^5*b*x^2 + 36575*a^4*b^2*x^4 + 20900*a^3*b^3*x^6 + 9975*a^2*b^4*x^8 + 2926*a*b^5*x^10 + 385*b^6*x^12))/(7315*d^4*x^3)","A",1
686,1,235,316,0.3400989,"\int \frac{(d x)^{11/2}}{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[(d*x)^(11/2)/(a^2 + 2*a*b*x^2 + b^2*x^4),x]","\frac{d^5 \sqrt{d x} \left(-45 \sqrt{2} a^{5/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+45 \sqrt{2} a^{5/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-90 \sqrt{2} a^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)+90 \sqrt{2} a^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)+\frac{8 \sqrt[4]{b} \sqrt{x} \left(-45 a^2-36 a b x^2+4 b^2 x^4\right)}{a+b x^2}\right)}{80 b^{13/4} \sqrt{x}}","-\frac{9 a^{5/4} d^{11/2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{8 \sqrt{2} b^{13/4}}+\frac{9 a^{5/4} d^{11/2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{8 \sqrt{2} b^{13/4}}-\frac{9 a^{5/4} d^{11/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{4 \sqrt{2} b^{13/4}}+\frac{9 a^{5/4} d^{11/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{4 \sqrt{2} b^{13/4}}-\frac{9 a d^5 \sqrt{d x}}{2 b^3}-\frac{d (d x)^{9/2}}{2 b \left(a+b x^2\right)}+\frac{9 d^3 (d x)^{5/2}}{10 b^2}",1,"(d^5*Sqrt[d*x]*((8*b^(1/4)*Sqrt[x]*(-45*a^2 - 36*a*b*x^2 + 4*b^2*x^4))/(a + b*x^2) - 90*Sqrt[2]*a^(5/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 90*Sqrt[2]*a^(5/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 45*Sqrt[2]*a^(5/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 45*Sqrt[2]*a^(5/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]))/(80*b^(13/4)*Sqrt[x])","A",1
687,1,63,298,0.0188158,"\int \frac{(d x)^{9/2}}{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[(d*x)^(9/2)/(a^2 + 2*a*b*x^2 + b^2*x^4),x]","-\frac{2 d^4 x \sqrt{d x} \left(7 \left(a+b x^2\right) \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-\frac{b x^2}{a}\right)-7 a-b x^2\right)}{3 b^2 \left(a+b x^2\right)}","-\frac{7 a^{3/4} d^{9/2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{8 \sqrt{2} b^{11/4}}+\frac{7 a^{3/4} d^{9/2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{8 \sqrt{2} b^{11/4}}+\frac{7 a^{3/4} d^{9/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{4 \sqrt{2} b^{11/4}}-\frac{7 a^{3/4} d^{9/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{4 \sqrt{2} b^{11/4}}-\frac{d (d x)^{7/2}}{2 b \left(a+b x^2\right)}+\frac{7 d^3 (d x)^{3/2}}{6 b^2}",1,"(-2*d^4*x*Sqrt[d*x]*(-7*a - b*x^2 + 7*(a + b*x^2)*Hypergeometric2F1[3/4, 2, 7/4, -((b*x^2)/a)]))/(3*b^2*(a + b*x^2))","C",1
688,1,244,298,0.1626571,"\int \frac{(d x)^{7/2}}{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[(d*x)^(7/2)/(a^2 + 2*a*b*x^2 + b^2*x^4),x]","\frac{d^3 \sqrt{d x} \left(\frac{32 b^{5/4} x^2}{a+b x^2}+\frac{40 a \sqrt[4]{b}}{a+b x^2}+\frac{5 \sqrt{2} \sqrt[4]{a} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{\sqrt{x}}-\frac{5 \sqrt{2} \sqrt[4]{a} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{\sqrt{x}}+\frac{10 \sqrt{2} \sqrt[4]{a} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{x}}-\frac{10 \sqrt{2} \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{x}}\right)}{16 b^{9/4}}","\frac{5 \sqrt[4]{a} d^{7/2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{8 \sqrt{2} b^{9/4}}-\frac{5 \sqrt[4]{a} d^{7/2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{8 \sqrt{2} b^{9/4}}+\frac{5 \sqrt[4]{a} d^{7/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{4 \sqrt{2} b^{9/4}}-\frac{5 \sqrt[4]{a} d^{7/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{4 \sqrt{2} b^{9/4}}-\frac{d (d x)^{5/2}}{2 b \left(a+b x^2\right)}+\frac{5 d^3 \sqrt{d x}}{2 b^2}",1,"(d^3*Sqrt[d*x]*((40*a*b^(1/4))/(a + b*x^2) + (32*b^(5/4)*x^2)/(a + b*x^2) + (10*Sqrt[2]*a^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/Sqrt[x] - (10*Sqrt[2]*a^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/Sqrt[x] + (5*Sqrt[2]*a^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/Sqrt[x] - (5*Sqrt[2]*a^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/Sqrt[x]))/(16*b^(9/4))","A",1
689,1,54,281,0.0155883,"\int \frac{(d x)^{5/2}}{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[(d*x)^(5/2)/(a^2 + 2*a*b*x^2 + b^2*x^4),x]","\frac{2 d (d x)^{3/2} \left(\left(a+b x^2\right) \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-\frac{b x^2}{a}\right)-a\right)}{a b \left(a+b x^2\right)}","\frac{3 d^{5/2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{8 \sqrt{2} \sqrt[4]{a} b^{7/4}}-\frac{3 d^{5/2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{8 \sqrt{2} \sqrt[4]{a} b^{7/4}}-\frac{3 d^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{4 \sqrt{2} \sqrt[4]{a} b^{7/4}}+\frac{3 d^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{4 \sqrt{2} \sqrt[4]{a} b^{7/4}}-\frac{d (d x)^{3/2}}{2 b \left(a+b x^2\right)}",1,"(2*d*(d*x)^(3/2)*(-a + (a + b*x^2)*Hypergeometric2F1[3/4, 2, 7/4, -((b*x^2)/a)]))/(a*b*(a + b*x^2))","C",1
690,1,210,281,0.1698634,"\int \frac{(d x)^{3/2}}{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[(d*x)^(3/2)/(a^2 + 2*a*b*x^2 + b^2*x^4),x]","\frac{(d x)^{3/2} \left(-\frac{\sqrt{2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{3/4}}+\frac{\sqrt{2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{3/4}}-\frac{2 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{a^{3/4}}+\frac{2 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{a^{3/4}}-\frac{8 \sqrt[4]{b} \sqrt{x}}{a+b x^2}\right)}{16 b^{5/4} x^{3/2}}","-\frac{d^{3/2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{8 \sqrt{2} a^{3/4} b^{5/4}}+\frac{d^{3/2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{8 \sqrt{2} a^{3/4} b^{5/4}}-\frac{d^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{4 \sqrt{2} a^{3/4} b^{5/4}}+\frac{d^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{4 \sqrt{2} a^{3/4} b^{5/4}}-\frac{d \sqrt{d x}}{2 b \left(a+b x^2\right)}",1,"((d*x)^(3/2)*((-8*b^(1/4)*Sqrt[x])/(a + b*x^2) - (2*Sqrt[2]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(3/4) + (2*Sqrt[2]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(3/4) - (Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(3/4) + (Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(3/4)))/(16*b^(5/4)*x^(3/2))","A",1
691,1,32,283,0.006225,"\int \frac{\sqrt{d x}}{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[Sqrt[d*x]/(a^2 + 2*a*b*x^2 + b^2*x^4),x]","\frac{2 x \sqrt{d x} \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-\frac{b x^2}{a}\right)}{3 a^2}","\frac{\sqrt{d} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{8 \sqrt{2} a^{5/4} b^{3/4}}-\frac{\sqrt{d} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{8 \sqrt{2} a^{5/4} b^{3/4}}-\frac{\sqrt{d} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{4 \sqrt{2} a^{5/4} b^{3/4}}+\frac{\sqrt{d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{4 \sqrt{2} a^{5/4} b^{3/4}}+\frac{(d x)^{3/2}}{2 a d \left(a+b x^2\right)}",1,"(2*x*Sqrt[d*x]*Hypergeometric2F1[3/4, 2, 7/4, -((b*x^2)/a)])/(3*a^2)","C",1
692,1,211,283,0.1708166,"\int \frac{1}{\sqrt{d x} \left(a^2+2 a b x^2+b^2 x^4\right)} \, dx","Integrate[1/(Sqrt[d*x]*(a^2 + 2*a*b*x^2 + b^2*x^4)),x]","\frac{\sqrt{x} \left(\frac{8 a^{3/4} \sqrt{x}}{a+b x^2}-\frac{3 \sqrt{2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{\sqrt[4]{b}}+\frac{3 \sqrt{2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{\sqrt[4]{b}}-\frac{6 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt[4]{b}}+\frac{6 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt[4]{b}}\right)}{16 a^{7/4} \sqrt{d x}}","-\frac{3 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{8 \sqrt{2} a^{7/4} \sqrt[4]{b} \sqrt{d}}+\frac{3 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{8 \sqrt{2} a^{7/4} \sqrt[4]{b} \sqrt{d}}-\frac{3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{4 \sqrt{2} a^{7/4} \sqrt[4]{b} \sqrt{d}}+\frac{3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{4 \sqrt{2} a^{7/4} \sqrt[4]{b} \sqrt{d}}+\frac{\sqrt{d x}}{2 a d \left(a+b x^2\right)}",1,"(Sqrt[x]*((8*a^(3/4)*Sqrt[x])/(a + b*x^2) - (6*Sqrt[2]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(1/4) + (6*Sqrt[2]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(1/4) - (3*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(1/4) + (3*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(1/4)))/(16*a^(7/4)*Sqrt[d*x])","A",1
693,1,30,300,0.0100915,"\int \frac{1}{(d x)^{3/2} \left(a^2+2 a b x^2+b^2 x^4\right)} \, dx","Integrate[1/((d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)),x]","-\frac{2 x \, _2F_1\left(-\frac{1}{4},2;\frac{3}{4};-\frac{b x^2}{a}\right)}{a^2 (d x)^{3/2}}","-\frac{5 \sqrt[4]{b} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{8 \sqrt{2} a^{9/4} d^{3/2}}+\frac{5 \sqrt[4]{b} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{8 \sqrt{2} a^{9/4} d^{3/2}}+\frac{5 \sqrt[4]{b} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{4 \sqrt{2} a^{9/4} d^{3/2}}-\frac{5 \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{4 \sqrt{2} a^{9/4} d^{3/2}}-\frac{5}{2 a^2 d \sqrt{d x}}+\frac{1}{2 a d \sqrt{d x} \left(a+b x^2\right)}",1,"(-2*x*Hypergeometric2F1[-1/4, 2, 3/4, -((b*x^2)/a)])/(a^2*(d*x)^(3/2))","C",1
694,1,32,300,0.0113087,"\int \frac{1}{(d x)^{5/2} \left(a^2+2 a b x^2+b^2 x^4\right)} \, dx","Integrate[1/((d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)),x]","-\frac{2 x \, _2F_1\left(-\frac{3}{4},2;\frac{1}{4};-\frac{b x^2}{a}\right)}{3 a^2 (d x)^{5/2}}","\frac{7 b^{3/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{8 \sqrt{2} a^{11/4} d^{5/2}}-\frac{7 b^{3/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{8 \sqrt{2} a^{11/4} d^{5/2}}+\frac{7 b^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{4 \sqrt{2} a^{11/4} d^{5/2}}-\frac{7 b^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{4 \sqrt{2} a^{11/4} d^{5/2}}-\frac{7}{6 a^2 d (d x)^{3/2}}+\frac{1}{2 a d (d x)^{3/2} \left(a+b x^2\right)}",1,"(-2*x*Hypergeometric2F1[-3/4, 2, 1/4, -((b*x^2)/a)])/(3*a^2*(d*x)^(5/2))","C",1
695,1,37,318,0.0102528,"\int \frac{1}{(d x)^{7/2} \left(a^2+2 a b x^2+b^2 x^4\right)} \, dx","Integrate[1/((d*x)^(7/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)),x]","-\frac{2 \sqrt{d x} \, _2F_1\left(-\frac{5}{4},2;-\frac{1}{4};-\frac{b x^2}{a}\right)}{5 a^2 d^4 x^3}","\frac{9 b^{5/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{8 \sqrt{2} a^{13/4} d^{7/2}}-\frac{9 b^{5/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{8 \sqrt{2} a^{13/4} d^{7/2}}-\frac{9 b^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{4 \sqrt{2} a^{13/4} d^{7/2}}+\frac{9 b^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{4 \sqrt{2} a^{13/4} d^{7/2}}+\frac{9 b}{2 a^3 d^3 \sqrt{d x}}-\frac{9}{10 a^2 d (d x)^{5/2}}+\frac{1}{2 a d (d x)^{5/2} \left(a+b x^2\right)}",1,"(-2*Sqrt[d*x]*Hypergeometric2F1[-5/4, 2, -1/4, -((b*x^2)/a)])/(5*a^2*d^4*x^3)","C",1
696,1,347,368,0.2272599,"\int \frac{(d x)^{19/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[(d*x)^(19/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","\frac{d^9 \sqrt{d x} \left(\frac{-69615 \sqrt{2} a^{5/4} \left(a+b x^2\right)^3 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+69615 \sqrt{2} a^{5/4} \left(a+b x^2\right)^3 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+139230 \sqrt{2} a^{5/4} \left(a+b x^2\right)^3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)-848640 a^4 \sqrt[4]{b} \sqrt{x}-2036736 a^3 b^{5/4} x^{5/2}+106080 a^3 \sqrt[4]{b} \sqrt{x} \left(a+b x^2\right)-1584128 a^2 b^{9/4} x^{9/2}+185640 a^2 \sqrt[4]{b} \sqrt{x} \left(a+b x^2\right)^2-365568 a b^{13/4} x^{13/2}+21504 b^{17/4} x^{17/2}}{\left(a+b x^2\right)^3}-139230 \sqrt{2} a^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)\right)}{53760 b^{21/4} \sqrt{x}}","-\frac{663 a^{5/4} d^{19/2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{256 \sqrt{2} b^{21/4}}+\frac{663 a^{5/4} d^{19/2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{256 \sqrt{2} b^{21/4}}-\frac{663 a^{5/4} d^{19/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{128 \sqrt{2} b^{21/4}}+\frac{663 a^{5/4} d^{19/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{128 \sqrt{2} b^{21/4}}-\frac{663 a d^9 \sqrt{d x}}{64 b^5}-\frac{221 d^5 (d x)^{9/2}}{192 b^3 \left(a+b x^2\right)}-\frac{17 d^3 (d x)^{13/2}}{48 b^2 \left(a+b x^2\right)^2}-\frac{d (d x)^{17/2}}{6 b \left(a+b x^2\right)^3}+\frac{663 d^7 (d x)^{5/2}}{320 b^4}",1,"(d^9*Sqrt[d*x]*(-139230*Sqrt[2]*a^(5/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + (-848640*a^4*b^(1/4)*Sqrt[x] - 2036736*a^3*b^(5/4)*x^(5/2) - 1584128*a^2*b^(9/4)*x^(9/2) - 365568*a*b^(13/4)*x^(13/2) + 21504*b^(17/4)*x^(17/2) + 106080*a^3*b^(1/4)*Sqrt[x]*(a + b*x^2) + 185640*a^2*b^(1/4)*Sqrt[x]*(a + b*x^2)^2 + 139230*Sqrt[2]*a^(5/4)*(a + b*x^2)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 69615*Sqrt[2]*a^(5/4)*(a + b*x^2)^3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 69615*Sqrt[2]*a^(5/4)*(a + b*x^2)^3*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a + b*x^2)^3))/(53760*b^(21/4)*Sqrt[x])","A",1
697,1,87,350,0.0279118,"\int \frac{(d x)^{17/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[(d*x)^(17/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","-\frac{2 d^8 x \sqrt{d x} \left(-77 a^3-99 a^2 b x^2-45 a b^2 x^4+77 \left(a+b x^2\right)^3 \, _2F_1\left(\frac{3}{4},4;\frac{7}{4};-\frac{b x^2}{a}\right)-3 b^3 x^6\right)}{9 b^4 \left(a+b x^2\right)^3}","-\frac{385 a^{3/4} d^{17/2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{256 \sqrt{2} b^{19/4}}+\frac{385 a^{3/4} d^{17/2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{256 \sqrt{2} b^{19/4}}+\frac{385 a^{3/4} d^{17/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{128 \sqrt{2} b^{19/4}}-\frac{385 a^{3/4} d^{17/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{128 \sqrt{2} b^{19/4}}-\frac{55 d^5 (d x)^{7/2}}{64 b^3 \left(a+b x^2\right)}-\frac{5 d^3 (d x)^{11/2}}{16 b^2 \left(a+b x^2\right)^2}-\frac{d (d x)^{15/2}}{6 b \left(a+b x^2\right)^3}+\frac{385 d^7 (d x)^{3/2}}{192 b^4}",1,"(-2*d^8*x*Sqrt[d*x]*(-77*a^3 - 99*a^2*b*x^2 - 45*a*b^2*x^4 - 3*b^3*x^6 + 77*(a + b*x^2)^3*Hypergeometric2F1[3/4, 4, 7/4, -((b*x^2)/a)]))/(9*b^4*(a + b*x^2)^3)","C",1
698,1,324,350,0.1399169,"\int \frac{(d x)^{15/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[(d*x)^(15/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","\frac{d^7 \sqrt{d x} \left(\frac{49920 a^3 \sqrt[4]{b}}{\left(a+b x^2\right)^3}+\frac{119808 a^2 b^{5/4} x^2}{\left(a+b x^2\right)^3}-\frac{6240 a^2 \sqrt[4]{b}}{\left(a+b x^2\right)^2}+\frac{21504 b^{13/4} x^6}{\left(a+b x^2\right)^3}+\frac{93184 a b^{9/4} x^4}{\left(a+b x^2\right)^3}-\frac{10920 a \sqrt[4]{b}}{a+b x^2}+\frac{4095 \sqrt{2} \sqrt[4]{a} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{\sqrt{x}}-\frac{4095 \sqrt{2} \sqrt[4]{a} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{\sqrt{x}}+\frac{8190 \sqrt{2} \sqrt[4]{a} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{x}}-\frac{8190 \sqrt{2} \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{x}}\right)}{10752 b^{17/4}}","\frac{195 \sqrt[4]{a} d^{15/2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{256 \sqrt{2} b^{17/4}}-\frac{195 \sqrt[4]{a} d^{15/2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{256 \sqrt{2} b^{17/4}}+\frac{195 \sqrt[4]{a} d^{15/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{128 \sqrt{2} b^{17/4}}-\frac{195 \sqrt[4]{a} d^{15/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{128 \sqrt{2} b^{17/4}}-\frac{39 d^5 (d x)^{5/2}}{64 b^3 \left(a+b x^2\right)}-\frac{13 d^3 (d x)^{9/2}}{48 b^2 \left(a+b x^2\right)^2}-\frac{d (d x)^{13/2}}{6 b \left(a+b x^2\right)^3}+\frac{195 d^7 \sqrt{d x}}{64 b^4}",1,"(d^7*Sqrt[d*x]*((49920*a^3*b^(1/4))/(a + b*x^2)^3 + (119808*a^2*b^(5/4)*x^2)/(a + b*x^2)^3 + (93184*a*b^(9/4)*x^4)/(a + b*x^2)^3 + (21504*b^(13/4)*x^6)/(a + b*x^2)^3 - (6240*a^2*b^(1/4))/(a + b*x^2)^2 - (10920*a*b^(1/4))/(a + b*x^2) + (8190*Sqrt[2]*a^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/Sqrt[x] - (8190*Sqrt[2]*a^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/Sqrt[x] + (4095*Sqrt[2]*a^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/Sqrt[x] - (4095*Sqrt[2]*a^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/Sqrt[x]))/(10752*b^(17/4))","A",1
699,1,83,333,0.0266953,"\int \frac{(d x)^{13/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[(d*x)^(13/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","\frac{2 d^6 x \sqrt{d x} \left(77 \left(a+b x^2\right)^3 \, _2F_1\left(\frac{3}{4},4;\frac{7}{4};-\frac{b x^2}{a}\right)-a \left(77 a^2+99 a b x^2+45 b^2 x^4\right)\right)}{45 a b^3 \left(a+b x^2\right)^3}","\frac{77 d^{13/2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{256 \sqrt{2} \sqrt[4]{a} b^{15/4}}-\frac{77 d^{13/2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{256 \sqrt{2} \sqrt[4]{a} b^{15/4}}-\frac{77 d^{13/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{128 \sqrt{2} \sqrt[4]{a} b^{15/4}}+\frac{77 d^{13/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{128 \sqrt{2} \sqrt[4]{a} b^{15/4}}-\frac{77 d^5 (d x)^{3/2}}{192 b^3 \left(a+b x^2\right)}-\frac{11 d^3 (d x)^{7/2}}{48 b^2 \left(a+b x^2\right)^2}-\frac{d (d x)^{11/2}}{6 b \left(a+b x^2\right)^3}",1,"(2*d^6*x*Sqrt[d*x]*(-(a*(77*a^2 + 99*a*b*x^2 + 45*b^2*x^4)) + 77*(a + b*x^2)^3*Hypergeometric2F1[3/4, 4, 7/4, -((b*x^2)/a)]))/(45*a*b^3*(a + b*x^2)^3)","C",1
700,1,299,333,0.1303398,"\int \frac{(d x)^{11/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[(d*x)^(11/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","\frac{d^5 \sqrt{d x} \left(-\frac{315 \sqrt{2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{3/4} \sqrt{x}}+\frac{315 \sqrt{2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{3/4} \sqrt{x}}-\frac{630 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{a^{3/4} \sqrt{x}}+\frac{630 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{a^{3/4} \sqrt{x}}-\frac{3840 a^2 \sqrt[4]{b}}{\left(a+b x^2\right)^3}-\frac{9216 a b^{5/4} x^2}{\left(a+b x^2\right)^3}-\frac{7168 b^{9/4} x^4}{\left(a+b x^2\right)^3}+\frac{840 \sqrt[4]{b}}{a+b x^2}+\frac{480 a \sqrt[4]{b}}{\left(a+b x^2\right)^2}\right)}{10752 b^{13/4}}","-\frac{15 d^{11/2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{256 \sqrt{2} a^{3/4} b^{13/4}}+\frac{15 d^{11/2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{256 \sqrt{2} a^{3/4} b^{13/4}}-\frac{15 d^{11/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{128 \sqrt{2} a^{3/4} b^{13/4}}+\frac{15 d^{11/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{128 \sqrt{2} a^{3/4} b^{13/4}}-\frac{15 d^5 \sqrt{d x}}{64 b^3 \left(a+b x^2\right)}-\frac{3 d^3 (d x)^{5/2}}{16 b^2 \left(a+b x^2\right)^2}-\frac{d (d x)^{9/2}}{6 b \left(a+b x^2\right)^3}",1,"(d^5*Sqrt[d*x]*((-3840*a^2*b^(1/4))/(a + b*x^2)^3 - (9216*a*b^(5/4)*x^2)/(a + b*x^2)^3 - (7168*b^(9/4)*x^4)/(a + b*x^2)^3 + (480*a*b^(1/4))/(a + b*x^2)^2 + (840*b^(1/4))/(a + b*x^2) - (630*Sqrt[2]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(3/4)*Sqrt[x]) + (630*Sqrt[2]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(3/4)*Sqrt[x]) - (315*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(3/4)*Sqrt[x]) + (315*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(3/4)*Sqrt[x])))/(10752*b^(13/4))","A",1
701,1,74,336,0.0223433,"\int \frac{(d x)^{9/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[(d*x)^(9/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","\frac{2 d^4 x \sqrt{d x} \left(7 \left(a+b x^2\right)^3 \, _2F_1\left(\frac{3}{4},4;\frac{7}{4};-\frac{b x^2}{a}\right)-a^2 \left(7 a+9 b x^2\right)\right)}{45 a^2 b^2 \left(a+b x^2\right)^3}","\frac{7 d^{9/2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{256 \sqrt{2} a^{5/4} b^{11/4}}-\frac{7 d^{9/2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{256 \sqrt{2} a^{5/4} b^{11/4}}-\frac{7 d^{9/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{128 \sqrt{2} a^{5/4} b^{11/4}}+\frac{7 d^{9/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{128 \sqrt{2} a^{5/4} b^{11/4}}+\frac{7 d^3 (d x)^{3/2}}{64 a b^2 \left(a+b x^2\right)}-\frac{7 d^3 (d x)^{3/2}}{48 b^2 \left(a+b x^2\right)^2}-\frac{d (d x)^{7/2}}{6 b \left(a+b x^2\right)^3}",1,"(2*d^4*x*Sqrt[d*x]*(-(a^2*(7*a + 9*b*x^2)) + 7*(a + b*x^2)^3*Hypergeometric2F1[3/4, 4, 7/4, -((b*x^2)/a)]))/(45*a^2*b^2*(a + b*x^2)^3)","C",1
702,1,279,336,0.1656727,"\int \frac{(d x)^{7/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[(d*x)^(7/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","\frac{d^3 \sqrt{d x} \left(-\frac{105 \sqrt{2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{7/4} \sqrt{x}}+\frac{105 \sqrt{2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{7/4} \sqrt{x}}-\frac{210 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{a^{7/4} \sqrt{x}}+\frac{210 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{a^{7/4} \sqrt{x}}+\frac{280 \sqrt[4]{b}}{a^2+a b x^2}-\frac{3072 b^{5/4} x^2}{\left(a+b x^2\right)^3}+\frac{160 \sqrt[4]{b}}{\left(a+b x^2\right)^2}-\frac{1280 a \sqrt[4]{b}}{\left(a+b x^2\right)^3}\right)}{10752 b^{9/4}}","-\frac{5 d^{7/2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{256 \sqrt{2} a^{7/4} b^{9/4}}+\frac{5 d^{7/2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{256 \sqrt{2} a^{7/4} b^{9/4}}-\frac{5 d^{7/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{128 \sqrt{2} a^{7/4} b^{9/4}}+\frac{5 d^{7/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{128 \sqrt{2} a^{7/4} b^{9/4}}+\frac{5 d^3 \sqrt{d x}}{192 a b^2 \left(a+b x^2\right)}-\frac{5 d^3 \sqrt{d x}}{48 b^2 \left(a+b x^2\right)^2}-\frac{d (d x)^{5/2}}{6 b \left(a+b x^2\right)^3}",1,"(d^3*Sqrt[d*x]*((-1280*a*b^(1/4))/(a + b*x^2)^3 - (3072*b^(5/4)*x^2)/(a + b*x^2)^3 + (160*b^(1/4))/(a + b*x^2)^2 + (280*b^(1/4))/(a^2 + a*b*x^2) - (210*Sqrt[2]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(7/4)*Sqrt[x]) + (210*Sqrt[2]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(7/4)*Sqrt[x]) - (105*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(7/4)*Sqrt[x]) + (105*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(7/4)*Sqrt[x])))/(10752*b^(9/4))","A",1
703,1,60,335,0.0197634,"\int \frac{(d x)^{5/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[(d*x)^(5/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","\frac{2 d (d x)^{3/2} \left(\left(a+b x^2\right)^3 \, _2F_1\left(\frac{3}{4},4;\frac{7}{4};-\frac{b x^2}{a}\right)-a^3\right)}{9 a^3 b \left(a+b x^2\right)^3}","\frac{5 d^{5/2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{256 \sqrt{2} a^{9/4} b^{7/4}}-\frac{5 d^{5/2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{256 \sqrt{2} a^{9/4} b^{7/4}}-\frac{5 d^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{128 \sqrt{2} a^{9/4} b^{7/4}}+\frac{5 d^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{128 \sqrt{2} a^{9/4} b^{7/4}}+\frac{5 d (d x)^{3/2}}{64 a^2 b \left(a+b x^2\right)}+\frac{d (d x)^{3/2}}{16 a b \left(a+b x^2\right)^2}-\frac{d (d x)^{3/2}}{6 b \left(a+b x^2\right)^3}",1,"(2*d*(d*x)^(3/2)*(-a^3 + (a + b*x^2)^3*Hypergeometric2F1[3/4, 4, 7/4, -((b*x^2)/a)]))/(9*a^3*b*(a + b*x^2)^3)","C",1
704,1,260,335,0.1306738,"\int \frac{(d x)^{3/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[(d*x)^(3/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","\frac{d \sqrt{d x} \left(-\frac{21 \sqrt{2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{11/4} \sqrt{x}}+\frac{21 \sqrt{2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{11/4} \sqrt{x}}-\frac{42 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{a^{11/4} \sqrt{x}}+\frac{42 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{a^{11/4} \sqrt{x}}+\frac{56 \sqrt[4]{b}}{a^2 \left(a+b x^2\right)}+\frac{32 \sqrt[4]{b}}{a \left(a+b x^2\right)^2}-\frac{256 \sqrt[4]{b}}{\left(a+b x^2\right)^3}\right)}{1536 b^{5/4}}","-\frac{7 d^{3/2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{256 \sqrt{2} a^{11/4} b^{5/4}}+\frac{7 d^{3/2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{256 \sqrt{2} a^{11/4} b^{5/4}}-\frac{7 d^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{128 \sqrt{2} a^{11/4} b^{5/4}}+\frac{7 d^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{128 \sqrt{2} a^{11/4} b^{5/4}}+\frac{7 d \sqrt{d x}}{192 a^2 b \left(a+b x^2\right)}+\frac{d \sqrt{d x}}{48 a b \left(a+b x^2\right)^2}-\frac{d \sqrt{d x}}{6 b \left(a+b x^2\right)^3}",1,"(d*Sqrt[d*x]*((-256*b^(1/4))/(a + b*x^2)^3 + (32*b^(1/4))/(a*(a + b*x^2)^2) + (56*b^(1/4))/(a^2*(a + b*x^2)) - (42*Sqrt[2]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(11/4)*Sqrt[x]) + (42*Sqrt[2]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(11/4)*Sqrt[x]) - (21*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(11/4)*Sqrt[x]) + (21*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(11/4)*Sqrt[x])))/(1536*b^(5/4))","A",1
705,1,32,335,0.0062813,"\int \frac{\sqrt{d x}}{\left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[Sqrt[d*x]/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","\frac{2 x \sqrt{d x} \, _2F_1\left(\frac{3}{4},4;\frac{7}{4};-\frac{b x^2}{a}\right)}{3 a^4}","\frac{15 \sqrt{d} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{256 \sqrt{2} a^{13/4} b^{3/4}}-\frac{15 \sqrt{d} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{256 \sqrt{2} a^{13/4} b^{3/4}}-\frac{15 \sqrt{d} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{128 \sqrt{2} a^{13/4} b^{3/4}}+\frac{15 \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{128 \sqrt{2} a^{13/4} b^{3/4}}+\frac{15 (d x)^{3/2}}{64 a^3 d \left(a+b x^2\right)}+\frac{3 (d x)^{3/2}}{16 a^2 d \left(a+b x^2\right)^2}+\frac{(d x)^{3/2}}{6 a d \left(a+b x^2\right)^3}",1,"(2*x*Sqrt[d*x]*Hypergeometric2F1[3/4, 4, 7/4, -((b*x^2)/a)])/(3*a^4)","C",1
706,1,253,335,0.1121772,"\int \frac{1}{\sqrt{d x} \left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[1/(Sqrt[d*x]*(a^2 + 2*a*b*x^2 + b^2*x^4)^2),x]","\frac{\sqrt{x} \left(\frac{256 a^{11/4} \sqrt{x}}{\left(a+b x^2\right)^3}+\frac{352 a^{7/4} \sqrt{x}}{\left(a+b x^2\right)^2}+\frac{616 a^{3/4} \sqrt{x}}{a+b x^2}-\frac{231 \sqrt{2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{\sqrt[4]{b}}+\frac{231 \sqrt{2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{\sqrt[4]{b}}-\frac{462 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt[4]{b}}+\frac{462 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt[4]{b}}\right)}{1536 a^{15/4} \sqrt{d x}}","-\frac{77 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{256 \sqrt{2} a^{15/4} \sqrt[4]{b} \sqrt{d}}+\frac{77 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{256 \sqrt{2} a^{15/4} \sqrt[4]{b} \sqrt{d}}-\frac{77 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{128 \sqrt{2} a^{15/4} \sqrt[4]{b} \sqrt{d}}+\frac{77 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{128 \sqrt{2} a^{15/4} \sqrt[4]{b} \sqrt{d}}+\frac{77 \sqrt{d x}}{192 a^3 d \left(a+b x^2\right)}+\frac{11 \sqrt{d x}}{48 a^2 d \left(a+b x^2\right)^2}+\frac{\sqrt{d x}}{6 a d \left(a+b x^2\right)^3}",1,"(Sqrt[x]*((256*a^(11/4)*Sqrt[x])/(a + b*x^2)^3 + (352*a^(7/4)*Sqrt[x])/(a + b*x^2)^2 + (616*a^(3/4)*Sqrt[x])/(a + b*x^2) - (462*Sqrt[2]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(1/4) + (462*Sqrt[2]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(1/4) - (231*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(1/4) + (231*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(1/4)))/(1536*a^(15/4)*Sqrt[d*x])","A",1
707,1,30,352,0.0108865,"\int \frac{1}{(d x)^{3/2} \left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[1/((d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^2),x]","-\frac{2 x \, _2F_1\left(-\frac{1}{4},4;\frac{3}{4};-\frac{b x^2}{a}\right)}{a^4 (d x)^{3/2}}","-\frac{195 \sqrt[4]{b} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{256 \sqrt{2} a^{17/4} d^{3/2}}+\frac{195 \sqrt[4]{b} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{256 \sqrt{2} a^{17/4} d^{3/2}}+\frac{195 \sqrt[4]{b} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{128 \sqrt{2} a^{17/4} d^{3/2}}-\frac{195 \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{128 \sqrt{2} a^{17/4} d^{3/2}}-\frac{195}{64 a^4 d \sqrt{d x}}+\frac{39}{64 a^3 d \sqrt{d x} \left(a+b x^2\right)}+\frac{13}{48 a^2 d \sqrt{d x} \left(a+b x^2\right)^2}+\frac{1}{6 a d \sqrt{d x} \left(a+b x^2\right)^3}",1,"(-2*x*Hypergeometric2F1[-1/4, 4, 3/4, -((b*x^2)/a)])/(a^4*(d*x)^(3/2))","C",1
708,1,32,352,0.0129155,"\int \frac{1}{(d x)^{5/2} \left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[1/((d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^2),x]","-\frac{2 x \, _2F_1\left(-\frac{3}{4},4;\frac{1}{4};-\frac{b x^2}{a}\right)}{3 a^4 (d x)^{5/2}}","\frac{385 b^{3/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{256 \sqrt{2} a^{19/4} d^{5/2}}-\frac{385 b^{3/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{256 \sqrt{2} a^{19/4} d^{5/2}}+\frac{385 b^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{128 \sqrt{2} a^{19/4} d^{5/2}}-\frac{385 b^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{128 \sqrt{2} a^{19/4} d^{5/2}}-\frac{385}{192 a^4 d (d x)^{3/2}}+\frac{55}{64 a^3 d (d x)^{3/2} \left(a+b x^2\right)}+\frac{5}{16 a^2 d (d x)^{3/2} \left(a+b x^2\right)^2}+\frac{1}{6 a d (d x)^{3/2} \left(a+b x^2\right)^3}",1,"(-2*x*Hypergeometric2F1[-3/4, 4, 1/4, -((b*x^2)/a)])/(3*a^4*(d*x)^(5/2))","C",1
709,1,37,370,0.0110922,"\int \frac{1}{(d x)^{7/2} \left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[1/((d*x)^(7/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^2),x]","-\frac{2 \sqrt{d x} \, _2F_1\left(-\frac{5}{4},4;-\frac{1}{4};-\frac{b x^2}{a}\right)}{5 a^4 d^4 x^3}","\frac{663 b^{5/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{256 \sqrt{2} a^{21/4} d^{7/2}}-\frac{663 b^{5/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{256 \sqrt{2} a^{21/4} d^{7/2}}-\frac{663 b^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{128 \sqrt{2} a^{21/4} d^{7/2}}+\frac{663 b^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{128 \sqrt{2} a^{21/4} d^{7/2}}+\frac{663 b}{64 a^5 d^3 \sqrt{d x}}-\frac{663}{320 a^4 d (d x)^{5/2}}+\frac{221}{192 a^3 d (d x)^{5/2} \left(a+b x^2\right)}+\frac{17}{48 a^2 d (d x)^{5/2} \left(a+b x^2\right)^2}+\frac{1}{6 a d (d x)^{5/2} \left(a+b x^2\right)^3}",1,"(-2*Sqrt[d*x]*Hypergeometric2F1[-5/4, 4, -1/4, -((b*x^2)/a)])/(5*a^4*d^4*x^3)","C",1
710,1,432,420,0.1801876,"\int \frac{(d x)^{27/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[(d*x)^(27/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{d^{13} \sqrt{d x} \left(-3828825 \sqrt{2} a^{5/4} \left(a+b x^2\right)^5 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+3828825 \sqrt{2} a^{5/4} \left(a+b x^2\right)^5 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-7657650 \sqrt{2} a^{5/4} \left(a+b x^2\right)^5 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)+7657650 \sqrt{2} a^{5/4} \left(a+b x^2\right)^5 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)-54312960 a^6 \sqrt[4]{b} \sqrt{x}-217251840 a^5 b^{5/4} x^{5/2}+3394560 a^5 \sqrt[4]{b} \sqrt{x} \left(a+b x^2\right)-362086400 a^4 b^{9/4} x^{9/2}+4243200 a^4 \sqrt[4]{b} \sqrt{x} \left(a+b x^2\right)^2-306380800 a^3 b^{13/4} x^{13/2}+5834400 a^3 \sqrt[4]{b} \sqrt{x} \left(a+b x^2\right)^3-126156800 a^2 b^{17/4} x^{17/2}+10210200 a^2 \sqrt[4]{b} \sqrt{x} \left(a+b x^2\right)^4-18022400 a b^{21/4} x^{21/2}+720896 b^{25/4} x^{25/2}\right)}{1802240 b^{29/4} \sqrt{x} \left(a+b x^2\right)^5}","-\frac{69615 a^{5/4} d^{27/2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} b^{29/4}}+\frac{69615 a^{5/4} d^{27/2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} b^{29/4}}-\frac{69615 a^{5/4} d^{27/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{8192 \sqrt{2} b^{29/4}}+\frac{69615 a^{5/4} d^{27/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{8192 \sqrt{2} b^{29/4}}-\frac{69615 a d^{13} \sqrt{d x}}{4096 b^7}-\frac{7735 d^9 (d x)^{9/2}}{4096 b^5 \left(a+b x^2\right)}-\frac{595 d^7 (d x)^{13/2}}{1024 b^4 \left(a+b x^2\right)^2}-\frac{35 d^5 (d x)^{17/2}}{128 b^3 \left(a+b x^2\right)^3}-\frac{5 d^3 (d x)^{21/2}}{32 b^2 \left(a+b x^2\right)^4}-\frac{d (d x)^{25/2}}{10 b \left(a+b x^2\right)^5}+\frac{13923 d^{11} (d x)^{5/2}}{4096 b^6}",1,"(d^13*Sqrt[d*x]*(-54312960*a^6*b^(1/4)*Sqrt[x] - 217251840*a^5*b^(5/4)*x^(5/2) - 362086400*a^4*b^(9/4)*x^(9/2) - 306380800*a^3*b^(13/4)*x^(13/2) - 126156800*a^2*b^(17/4)*x^(17/2) - 18022400*a*b^(21/4)*x^(21/2) + 720896*b^(25/4)*x^(25/2) + 3394560*a^5*b^(1/4)*Sqrt[x]*(a + b*x^2) + 4243200*a^4*b^(1/4)*Sqrt[x]*(a + b*x^2)^2 + 5834400*a^3*b^(1/4)*Sqrt[x]*(a + b*x^2)^3 + 10210200*a^2*b^(1/4)*Sqrt[x]*(a + b*x^2)^4 - 7657650*Sqrt[2]*a^(5/4)*(a + b*x^2)^5*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 7657650*Sqrt[2]*a^(5/4)*(a + b*x^2)^5*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 3828825*Sqrt[2]*a^(5/4)*(a + b*x^2)^5*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 3828825*Sqrt[2]*a^(5/4)*(a + b*x^2)^5*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]))/(1802240*b^(29/4)*Sqrt[x]*(a + b*x^2)^5)","A",1
711,1,109,402,0.0342295,"\int \frac{(d x)^{25/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[(d*x)^(25/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","-\frac{2 d^{12} x \sqrt{d x} \left(-168245 a^5-408595 a^4 b x^2-482885 a^3 b^2 x^4-289731 a^2 b^3 x^6-76245 a b^4 x^8+168245 \left(a+b x^2\right)^5 \, _2F_1\left(\frac{3}{4},6;\frac{7}{4};-\frac{b x^2}{a}\right)-3315 b^5 x^{10}\right)}{9945 b^6 \left(a+b x^2\right)^5}","-\frac{33649 a^{3/4} d^{25/2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} b^{27/4}}+\frac{33649 a^{3/4} d^{25/2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} b^{27/4}}+\frac{33649 a^{3/4} d^{25/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{8192 \sqrt{2} b^{27/4}}-\frac{33649 a^{3/4} d^{25/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{8192 \sqrt{2} b^{27/4}}-\frac{4807 d^9 (d x)^{7/2}}{4096 b^5 \left(a+b x^2\right)}-\frac{437 d^7 (d x)^{11/2}}{1024 b^4 \left(a+b x^2\right)^2}-\frac{437 d^5 (d x)^{15/2}}{1920 b^3 \left(a+b x^2\right)^3}-\frac{23 d^3 (d x)^{19/2}}{160 b^2 \left(a+b x^2\right)^4}-\frac{d (d x)^{23/2}}{10 b \left(a+b x^2\right)^5}+\frac{33649 d^{11} (d x)^{3/2}}{12288 b^6}",1,"(-2*d^12*x*Sqrt[d*x]*(-168245*a^5 - 408595*a^4*b*x^2 - 482885*a^3*b^2*x^4 - 289731*a^2*b^3*x^6 - 76245*a*b^4*x^8 - 3315*b^5*x^10 + 168245*(a + b*x^2)^5*Hypergeometric2F1[3/4, 6, 7/4, -((b*x^2)/a)]))/(9945*b^6*(a + b*x^2)^5)","C",1
712,1,408,402,0.2974466,"\int \frac{(d x)^{23/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[(d*x)^(23/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{d^{11} \sqrt{d x} \left(\frac{10862592 a^5 \sqrt[4]{b} \sqrt{x}+43450368 a^4 b^{5/4} x^{5/2}-678912 a^4 \sqrt[4]{b} \sqrt{x} \left(a+b x^2\right)+72417280 a^3 b^{9/4} x^{9/2}-848640 a^3 \sqrt[4]{b} \sqrt{x} \left(a+b x^2\right)^2+61276160 a^2 b^{13/4} x^{13/2}-1166880 a^2 \sqrt[4]{b} \sqrt{x} \left(a+b x^2\right)^3+25231360 a b^{17/4} x^{17/2}-2042040 a \sqrt[4]{b} \sqrt{x} \left(a+b x^2\right)^4+765765 \sqrt{2} \sqrt[4]{a} \left(a+b x^2\right)^5 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-765765 \sqrt{2} \sqrt[4]{a} \left(a+b x^2\right)^5 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-1531530 \sqrt{2} \sqrt[4]{a} \left(a+b x^2\right)^5 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)+3604480 b^{21/4} x^{21/2}}{\left(a+b x^2\right)^5}+1531530 \sqrt{2} \sqrt[4]{a} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)\right)}{1802240 b^{25/4} \sqrt{x}}","\frac{13923 \sqrt[4]{a} d^{23/2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} b^{25/4}}-\frac{13923 \sqrt[4]{a} d^{23/2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} b^{25/4}}+\frac{13923 \sqrt[4]{a} d^{23/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{8192 \sqrt{2} b^{25/4}}-\frac{13923 \sqrt[4]{a} d^{23/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{8192 \sqrt{2} b^{25/4}}-\frac{13923 d^9 (d x)^{5/2}}{20480 b^5 \left(a+b x^2\right)}-\frac{1547 d^7 (d x)^{9/2}}{5120 b^4 \left(a+b x^2\right)^2}-\frac{119 d^5 (d x)^{13/2}}{640 b^3 \left(a+b x^2\right)^3}-\frac{21 d^3 (d x)^{17/2}}{160 b^2 \left(a+b x^2\right)^4}-\frac{d (d x)^{21/2}}{10 b \left(a+b x^2\right)^5}+\frac{13923 d^{11} \sqrt{d x}}{4096 b^6}",1,"(d^11*Sqrt[d*x]*(1531530*Sqrt[2]*a^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + (10862592*a^5*b^(1/4)*Sqrt[x] + 43450368*a^4*b^(5/4)*x^(5/2) + 72417280*a^3*b^(9/4)*x^(9/2) + 61276160*a^2*b^(13/4)*x^(13/2) + 25231360*a*b^(17/4)*x^(17/2) + 3604480*b^(21/4)*x^(21/2) - 678912*a^4*b^(1/4)*Sqrt[x]*(a + b*x^2) - 848640*a^3*b^(1/4)*Sqrt[x]*(a + b*x^2)^2 - 1166880*a^2*b^(1/4)*Sqrt[x]*(a + b*x^2)^3 - 2042040*a*b^(1/4)*Sqrt[x]*(a + b*x^2)^4 - 1531530*Sqrt[2]*a^(1/4)*(a + b*x^2)^5*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 765765*Sqrt[2]*a^(1/4)*(a + b*x^2)^5*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - 765765*Sqrt[2]*a^(1/4)*(a + b*x^2)^5*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a + b*x^2)^5))/(1802240*b^(25/4)*Sqrt[x])","A",1
713,1,104,385,0.0358594,"\int \frac{(d x)^{21/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[(d*x)^(21/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{2 d^9 (d x)^{3/2} \left(7315 \left(a+b x^2\right)^5 \, _2F_1\left(\frac{3}{4},6;\frac{7}{4};-\frac{b x^2}{a}\right)-a \left(7315 a^4+17765 a^3 b x^2+20995 a^2 b^2 x^4+12597 a b^3 x^6+3315 b^4 x^8\right)\right)}{3315 a b^5 \left(a+b x^2\right)^5}","\frac{4389 d^{21/2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} \sqrt[4]{a} b^{23/4}}-\frac{4389 d^{21/2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} \sqrt[4]{a} b^{23/4}}-\frac{4389 d^{21/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{8192 \sqrt{2} \sqrt[4]{a} b^{23/4}}+\frac{4389 d^{21/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{8192 \sqrt{2} \sqrt[4]{a} b^{23/4}}-\frac{1463 d^9 (d x)^{3/2}}{4096 b^5 \left(a+b x^2\right)}-\frac{209 d^7 (d x)^{7/2}}{1024 b^4 \left(a+b x^2\right)^2}-\frac{19 d^5 (d x)^{11/2}}{128 b^3 \left(a+b x^2\right)^3}-\frac{19 d^3 (d x)^{15/2}}{160 b^2 \left(a+b x^2\right)^4}-\frac{d (d x)^{19/2}}{10 b \left(a+b x^2\right)^5}",1,"(2*d^9*(d*x)^(3/2)*(-(a*(7315*a^4 + 17765*a^3*b*x^2 + 20995*a^2*b^2*x^4 + 12597*a*b^3*x^6 + 3315*b^4*x^8)) + 7315*(a + b*x^2)^5*Hypergeometric2F1[3/4, 6, 7/4, -((b*x^2)/a)]))/(3315*a*b^5*(a + b*x^2)^5)","C",1
714,1,381,385,0.1954162,"\int \frac{(d x)^{19/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[(d*x)^(19/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{d^9 \sqrt{d x} \left(-\frac{765765 \sqrt{2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{3/4} \sqrt{x}}+\frac{765765 \sqrt{2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{3/4} \sqrt{x}}-\frac{1531530 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{a^{3/4} \sqrt{x}}+\frac{1531530 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{a^{3/4} \sqrt{x}}-\frac{10862592 a^4 \sqrt[4]{b}}{\left(a+b x^2\right)^5}-\frac{43450368 a^3 b^{5/4} x^2}{\left(a+b x^2\right)^5}+\frac{678912 a^3 \sqrt[4]{b}}{\left(a+b x^2\right)^4}-\frac{72417280 a^2 b^{9/4} x^4}{\left(a+b x^2\right)^5}+\frac{848640 a^2 \sqrt[4]{b}}{\left(a+b x^2\right)^3}-\frac{25231360 b^{17/4} x^8}{\left(a+b x^2\right)^5}-\frac{61276160 a b^{13/4} x^6}{\left(a+b x^2\right)^5}+\frac{2042040 \sqrt[4]{b}}{a+b x^2}+\frac{1166880 a \sqrt[4]{b}}{\left(a+b x^2\right)^2}\right)}{37847040 b^{21/4}}","-\frac{663 d^{19/2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} a^{3/4} b^{21/4}}+\frac{663 d^{19/2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} a^{3/4} b^{21/4}}-\frac{663 d^{19/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{8192 \sqrt{2} a^{3/4} b^{21/4}}+\frac{663 d^{19/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{8192 \sqrt{2} a^{3/4} b^{21/4}}-\frac{663 d^9 \sqrt{d x}}{4096 b^5 \left(a+b x^2\right)}-\frac{663 d^7 (d x)^{5/2}}{5120 b^4 \left(a+b x^2\right)^2}-\frac{221 d^5 (d x)^{9/2}}{1920 b^3 \left(a+b x^2\right)^3}-\frac{17 d^3 (d x)^{13/2}}{160 b^2 \left(a+b x^2\right)^4}-\frac{d (d x)^{17/2}}{10 b \left(a+b x^2\right)^5}",1,"(d^9*Sqrt[d*x]*((-10862592*a^4*b^(1/4))/(a + b*x^2)^5 - (43450368*a^3*b^(5/4)*x^2)/(a + b*x^2)^5 - (72417280*a^2*b^(9/4)*x^4)/(a + b*x^2)^5 - (61276160*a*b^(13/4)*x^6)/(a + b*x^2)^5 - (25231360*b^(17/4)*x^8)/(a + b*x^2)^5 + (678912*a^3*b^(1/4))/(a + b*x^2)^4 + (848640*a^2*b^(1/4))/(a + b*x^2)^3 + (1166880*a*b^(1/4))/(a + b*x^2)^2 + (2042040*b^(1/4))/(a + b*x^2) - (1531530*Sqrt[2]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(3/4)*Sqrt[x]) + (1531530*Sqrt[2]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(3/4)*Sqrt[x]) - (765765*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(3/4)*Sqrt[x]) + (765765*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(3/4)*Sqrt[x])))/(37847040*b^(21/4))","A",1
715,1,96,388,0.0308875,"\int \frac{(d x)^{17/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[(d*x)^(17/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{2 d^8 x \sqrt{d x} \left(385 \left(a+b x^2\right)^5 \, _2F_1\left(\frac{3}{4},6;\frac{7}{4};-\frac{b x^2}{a}\right)-a^2 \left(385 a^3+935 a^2 b x^2+1105 a b^2 x^4+663 b^3 x^6\right)\right)}{3315 a^2 b^4 \left(a+b x^2\right)^5}","\frac{231 d^{17/2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} a^{5/4} b^{19/4}}-\frac{231 d^{17/2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} a^{5/4} b^{19/4}}-\frac{231 d^{17/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{8192 \sqrt{2} a^{5/4} b^{19/4}}+\frac{231 d^{17/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{8192 \sqrt{2} a^{5/4} b^{19/4}}+\frac{231 d^7 (d x)^{3/2}}{4096 a b^4 \left(a+b x^2\right)}-\frac{77 d^7 (d x)^{3/2}}{1024 b^4 \left(a+b x^2\right)^2}-\frac{11 d^5 (d x)^{7/2}}{128 b^3 \left(a+b x^2\right)^3}-\frac{3 d^3 (d x)^{11/2}}{32 b^2 \left(a+b x^2\right)^4}-\frac{d (d x)^{15/2}}{10 b \left(a+b x^2\right)^5}",1,"(2*d^8*x*Sqrt[d*x]*(-(a^2*(385*a^3 + 935*a^2*b*x^2 + 1105*a*b^2*x^4 + 663*b^3*x^6)) + 385*(a + b*x^2)^5*Hypergeometric2F1[3/4, 6, 7/4, -((b*x^2)/a)]))/(3315*a^2*b^4*(a + b*x^2)^5)","C",1
716,1,359,388,0.2594013,"\int \frac{(d x)^{15/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[(d*x)^(15/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{d^7 \sqrt{d x} \left(-\frac{45045 \sqrt{2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{7/4} \sqrt{x}}+\frac{45045 \sqrt{2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{7/4} \sqrt{x}}-\frac{90090 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{a^{7/4} \sqrt{x}}+\frac{90090 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{a^{7/4} \sqrt{x}}-\frac{638976 a^3 \sqrt[4]{b}}{\left(a+b x^2\right)^5}-\frac{2555904 a^2 b^{5/4} x^2}{\left(a+b x^2\right)^5}+\frac{120120 \sqrt[4]{b}}{a^2+a b x^2}+\frac{39936 a^2 \sqrt[4]{b}}{\left(a+b x^2\right)^4}-\frac{3604480 b^{13/4} x^6}{\left(a+b x^2\right)^5}-\frac{4259840 a b^{9/4} x^4}{\left(a+b x^2\right)^5}+\frac{68640 \sqrt[4]{b}}{\left(a+b x^2\right)^2}+\frac{49920 a \sqrt[4]{b}}{\left(a+b x^2\right)^3}\right)}{12615680 b^{17/4}}","-\frac{117 d^{15/2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} a^{7/4} b^{17/4}}+\frac{117 d^{15/2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} a^{7/4} b^{17/4}}-\frac{117 d^{15/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{8192 \sqrt{2} a^{7/4} b^{17/4}}+\frac{117 d^{15/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{8192 \sqrt{2} a^{7/4} b^{17/4}}+\frac{39 d^7 \sqrt{d x}}{4096 a b^4 \left(a+b x^2\right)}-\frac{39 d^7 \sqrt{d x}}{1024 b^4 \left(a+b x^2\right)^2}-\frac{39 d^5 (d x)^{5/2}}{640 b^3 \left(a+b x^2\right)^3}-\frac{13 d^3 (d x)^{9/2}}{160 b^2 \left(a+b x^2\right)^4}-\frac{d (d x)^{13/2}}{10 b \left(a+b x^2\right)^5}",1,"(d^7*Sqrt[d*x]*((-638976*a^3*b^(1/4))/(a + b*x^2)^5 - (2555904*a^2*b^(5/4)*x^2)/(a + b*x^2)^5 - (4259840*a*b^(9/4)*x^4)/(a + b*x^2)^5 - (3604480*b^(13/4)*x^6)/(a + b*x^2)^5 + (39936*a^2*b^(1/4))/(a + b*x^2)^4 + (49920*a*b^(1/4))/(a + b*x^2)^3 + (68640*b^(1/4))/(a + b*x^2)^2 + (120120*b^(1/4))/(a^2 + a*b*x^2) - (90090*Sqrt[2]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(7/4)*Sqrt[x]) + (90090*Sqrt[2]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(7/4)*Sqrt[x]) - (45045*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(7/4)*Sqrt[x]) + (45045*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(7/4)*Sqrt[x])))/(12615680*b^(17/4))","A",1
717,1,85,391,0.0310376,"\int \frac{(d x)^{13/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[(d*x)^(13/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{2 d^6 x \sqrt{d x} \left(77 \left(a+b x^2\right)^5 \, _2F_1\left(\frac{3}{4},6;\frac{7}{4};-\frac{b x^2}{a}\right)-a^3 \left(77 a^2+187 a b x^2+221 b^2 x^4\right)\right)}{1989 a^3 b^3 \left(a+b x^2\right)^5}","\frac{77 d^{13/2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} a^{9/4} b^{15/4}}-\frac{77 d^{13/2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} a^{9/4} b^{15/4}}-\frac{77 d^{13/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{8192 \sqrt{2} a^{9/4} b^{15/4}}+\frac{77 d^{13/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{8192 \sqrt{2} a^{9/4} b^{15/4}}+\frac{77 d^5 (d x)^{3/2}}{4096 a^2 b^3 \left(a+b x^2\right)}+\frac{77 d^5 (d x)^{3/2}}{5120 a b^3 \left(a+b x^2\right)^2}-\frac{77 d^5 (d x)^{3/2}}{1920 b^3 \left(a+b x^2\right)^3}-\frac{11 d^3 (d x)^{7/2}}{160 b^2 \left(a+b x^2\right)^4}-\frac{d (d x)^{11/2}}{10 b \left(a+b x^2\right)^5}",1,"(2*d^6*x*Sqrt[d*x]*(-(a^3*(77*a^2 + 187*a*b*x^2 + 221*b^2*x^4)) + 77*(a + b*x^2)^5*Hypergeometric2F1[3/4, 6, 7/4, -((b*x^2)/a)]))/(1989*a^3*b^3*(a + b*x^2)^5)","C",1
718,1,337,391,0.1820542,"\int \frac{(d x)^{11/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[(d*x)^(11/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{d^5 \sqrt{d x} \left(-\frac{3465 \sqrt{2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{11/4} \sqrt{x}}+\frac{3465 \sqrt{2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{11/4} \sqrt{x}}-\frac{6930 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{a^{11/4} \sqrt{x}}+\frac{6930 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{a^{11/4} \sqrt{x}}+\frac{9240 \sqrt[4]{b}}{a^2 \left(a+b x^2\right)}-\frac{49152 a^2 \sqrt[4]{b}}{\left(a+b x^2\right)^5}-\frac{196608 a b^{5/4} x^2}{\left(a+b x^2\right)^5}-\frac{327680 b^{9/4} x^4}{\left(a+b x^2\right)^5}+\frac{5280 \sqrt[4]{b}}{a \left(a+b x^2\right)^2}+\frac{3840 \sqrt[4]{b}}{\left(a+b x^2\right)^3}+\frac{3072 a \sqrt[4]{b}}{\left(a+b x^2\right)^4}\right)}{1802240 b^{13/4}}","-\frac{63 d^{11/2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} a^{11/4} b^{13/4}}+\frac{63 d^{11/2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} a^{11/4} b^{13/4}}-\frac{63 d^{11/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{8192 \sqrt{2} a^{11/4} b^{13/4}}+\frac{63 d^{11/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{8192 \sqrt{2} a^{11/4} b^{13/4}}+\frac{21 d^5 \sqrt{d x}}{4096 a^2 b^3 \left(a+b x^2\right)}+\frac{3 d^5 \sqrt{d x}}{1024 a b^3 \left(a+b x^2\right)^2}-\frac{3 d^5 \sqrt{d x}}{128 b^3 \left(a+b x^2\right)^3}-\frac{9 d^3 (d x)^{5/2}}{160 b^2 \left(a+b x^2\right)^4}-\frac{d (d x)^{9/2}}{10 b \left(a+b x^2\right)^5}",1,"(d^5*Sqrt[d*x]*((-49152*a^2*b^(1/4))/(a + b*x^2)^5 - (196608*a*b^(5/4)*x^2)/(a + b*x^2)^5 - (327680*b^(9/4)*x^4)/(a + b*x^2)^5 + (3072*a*b^(1/4))/(a + b*x^2)^4 + (3840*b^(1/4))/(a + b*x^2)^3 + (5280*b^(1/4))/(a*(a + b*x^2)^2) + (9240*b^(1/4))/(a^2*(a + b*x^2)) - (6930*Sqrt[2]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(11/4)*Sqrt[x]) + (6930*Sqrt[2]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(11/4)*Sqrt[x]) - (3465*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(11/4)*Sqrt[x]) + (3465*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(11/4)*Sqrt[x])))/(1802240*b^(13/4))","A",1
719,1,61,394,0.0258014,"\int \frac{(d x)^{9/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[(d*x)^(9/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{2 d^4 x \sqrt{d x} \left(\frac{7 \, _2F_1\left(\frac{3}{4},6;\frac{7}{4};-\frac{b x^2}{a}\right)}{a^4}+\frac{-7 a-17 b x^2}{\left(a+b x^2\right)^5}\right)}{221 b^2}","\frac{63 d^{9/2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} a^{13/4} b^{11/4}}-\frac{63 d^{9/2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} a^{13/4} b^{11/4}}-\frac{63 d^{9/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{8192 \sqrt{2} a^{13/4} b^{11/4}}+\frac{63 d^{9/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{8192 \sqrt{2} a^{13/4} b^{11/4}}+\frac{63 d^3 (d x)^{3/2}}{4096 a^3 b^2 \left(a+b x^2\right)}+\frac{63 d^3 (d x)^{3/2}}{5120 a^2 b^2 \left(a+b x^2\right)^2}+\frac{7 d^3 (d x)^{3/2}}{640 a b^2 \left(a+b x^2\right)^3}-\frac{7 d^3 (d x)^{3/2}}{160 b^2 \left(a+b x^2\right)^4}-\frac{d (d x)^{7/2}}{10 b \left(a+b x^2\right)^5}",1,"(2*d^4*x*Sqrt[d*x]*((-7*a - 17*b*x^2)/(a + b*x^2)^5 + (7*Hypergeometric2F1[3/4, 6, 7/4, -((b*x^2)/a)])/a^4))/(221*b^2)","C",1
720,1,317,394,0.1703025,"\int \frac{(d x)^{7/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[(d*x)^(7/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{d^3 \sqrt{d x} \left(-\frac{1155 \sqrt{2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{15/4} \sqrt{x}}+\frac{1155 \sqrt{2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{15/4} \sqrt{x}}-\frac{2310 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{a^{15/4} \sqrt{x}}+\frac{2310 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{a^{15/4} \sqrt{x}}+\frac{3080 \sqrt[4]{b}}{a^3 \left(a+b x^2\right)}+\frac{1760 \sqrt[4]{b}}{a^2 \left(a+b x^2\right)^2}-\frac{65536 b^{5/4} x^2}{\left(a+b x^2\right)^5}+\frac{1280 \sqrt[4]{b}}{a \left(a+b x^2\right)^3}+\frac{1024 \sqrt[4]{b}}{\left(a+b x^2\right)^4}-\frac{16384 a \sqrt[4]{b}}{\left(a+b x^2\right)^5}\right)}{491520 b^{9/4}}","-\frac{77 d^{7/2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} a^{15/4} b^{9/4}}+\frac{77 d^{7/2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} a^{15/4} b^{9/4}}-\frac{77 d^{7/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{8192 \sqrt{2} a^{15/4} b^{9/4}}+\frac{77 d^{7/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{8192 \sqrt{2} a^{15/4} b^{9/4}}+\frac{77 d^3 \sqrt{d x}}{12288 a^3 b^2 \left(a+b x^2\right)}+\frac{11 d^3 \sqrt{d x}}{3072 a^2 b^2 \left(a+b x^2\right)^2}+\frac{d^3 \sqrt{d x}}{384 a b^2 \left(a+b x^2\right)^3}-\frac{d^3 \sqrt{d x}}{32 b^2 \left(a+b x^2\right)^4}-\frac{d (d x)^{5/2}}{10 b \left(a+b x^2\right)^5}",1,"(d^3*Sqrt[d*x]*((-16384*a*b^(1/4))/(a + b*x^2)^5 - (65536*b^(5/4)*x^2)/(a + b*x^2)^5 + (1024*b^(1/4))/(a + b*x^2)^4 + (1280*b^(1/4))/(a*(a + b*x^2)^3) + (1760*b^(1/4))/(a^2*(a + b*x^2)^2) + (3080*b^(1/4))/(a^3*(a + b*x^2)) - (2310*Sqrt[2]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(15/4)*Sqrt[x]) + (2310*Sqrt[2]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(15/4)*Sqrt[x]) - (1155*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(15/4)*Sqrt[x]) + (1155*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(15/4)*Sqrt[x])))/(491520*b^(9/4))","A",1
721,1,48,389,0.0201378,"\int \frac{(d x)^{5/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[(d*x)^(5/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{2 d (d x)^{3/2} \left(\frac{\, _2F_1\left(\frac{3}{4},6;\frac{7}{4};-\frac{b x^2}{a}\right)}{a^5}-\frac{1}{\left(a+b x^2\right)^5}\right)}{17 b}","\frac{117 d^{5/2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} a^{17/4} b^{7/4}}-\frac{117 d^{5/2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} a^{17/4} b^{7/4}}-\frac{117 d^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{8192 \sqrt{2} a^{17/4} b^{7/4}}+\frac{117 d^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{8192 \sqrt{2} a^{17/4} b^{7/4}}+\frac{117 d (d x)^{3/2}}{4096 a^4 b \left(a+b x^2\right)}+\frac{117 d (d x)^{3/2}}{5120 a^3 b \left(a+b x^2\right)^2}+\frac{13 d (d x)^{3/2}}{640 a^2 b \left(a+b x^2\right)^3}+\frac{3 d (d x)^{3/2}}{160 a b \left(a+b x^2\right)^4}-\frac{d (d x)^{3/2}}{10 b \left(a+b x^2\right)^5}",1,"(2*d*(d*x)^(3/2)*(-(a + b*x^2)^(-5) + Hypergeometric2F1[3/4, 6, 7/4, -((b*x^2)/a)]/a^5))/(17*b)","C",1
722,1,298,389,0.1725954,"\int \frac{(d x)^{3/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[(d*x)^(3/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{d \sqrt{d x} \left(-\frac{1155 \sqrt{2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{19/4} \sqrt{x}}+\frac{1155 \sqrt{2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{19/4} \sqrt{x}}-\frac{2310 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{a^{19/4} \sqrt{x}}+\frac{2310 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{a^{19/4} \sqrt{x}}+\frac{3080 \sqrt[4]{b}}{a^4 \left(a+b x^2\right)}+\frac{1760 \sqrt[4]{b}}{a^3 \left(a+b x^2\right)^2}+\frac{1280 \sqrt[4]{b}}{a^2 \left(a+b x^2\right)^3}+\frac{1024 \sqrt[4]{b}}{a \left(a+b x^2\right)^4}-\frac{16384 \sqrt[4]{b}}{\left(a+b x^2\right)^5}\right)}{163840 b^{5/4}}","-\frac{231 d^{3/2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} a^{19/4} b^{5/4}}+\frac{231 d^{3/2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} a^{19/4} b^{5/4}}-\frac{231 d^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{8192 \sqrt{2} a^{19/4} b^{5/4}}+\frac{231 d^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{8192 \sqrt{2} a^{19/4} b^{5/4}}+\frac{77 d \sqrt{d x}}{4096 a^4 b \left(a+b x^2\right)}+\frac{11 d \sqrt{d x}}{1024 a^3 b \left(a+b x^2\right)^2}+\frac{d \sqrt{d x}}{128 a^2 b \left(a+b x^2\right)^3}+\frac{d \sqrt{d x}}{160 a b \left(a+b x^2\right)^4}-\frac{d \sqrt{d x}}{10 b \left(a+b x^2\right)^5}",1,"(d*Sqrt[d*x]*((-16384*b^(1/4))/(a + b*x^2)^5 + (1024*b^(1/4))/(a*(a + b*x^2)^4) + (1280*b^(1/4))/(a^2*(a + b*x^2)^3) + (1760*b^(1/4))/(a^3*(a + b*x^2)^2) + (3080*b^(1/4))/(a^4*(a + b*x^2)) - (2310*Sqrt[2]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(19/4)*Sqrt[x]) + (2310*Sqrt[2]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(19/4)*Sqrt[x]) - (1155*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(19/4)*Sqrt[x]) + (1155*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(19/4)*Sqrt[x])))/(163840*b^(5/4))","A",1
723,1,32,387,0.0064023,"\int \frac{\sqrt{d x}}{\left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[Sqrt[d*x]/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{2 x \sqrt{d x} \, _2F_1\left(\frac{3}{4},6;\frac{7}{4};-\frac{b x^2}{a}\right)}{3 a^6}","\frac{663 \sqrt{d} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} a^{21/4} b^{3/4}}-\frac{663 \sqrt{d} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} a^{21/4} b^{3/4}}-\frac{663 \sqrt{d} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{8192 \sqrt{2} a^{21/4} b^{3/4}}+\frac{663 \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{8192 \sqrt{2} a^{21/4} b^{3/4}}+\frac{663 (d x)^{3/2}}{4096 a^5 d \left(a+b x^2\right)}+\frac{663 (d x)^{3/2}}{5120 a^4 d \left(a+b x^2\right)^2}+\frac{221 (d x)^{3/2}}{1920 a^3 d \left(a+b x^2\right)^3}+\frac{17 (d x)^{3/2}}{160 a^2 d \left(a+b x^2\right)^4}+\frac{(d x)^{3/2}}{10 a d \left(a+b x^2\right)^5}",1,"(2*x*Sqrt[d*x]*Hypergeometric2F1[3/4, 6, 7/4, -((b*x^2)/a)])/(3*a^6)","C",1
724,1,295,387,0.1598731,"\int \frac{1}{\sqrt{d x} \left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[1/(Sqrt[d*x]*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]","\frac{\sqrt{x} \left(\frac{16384 a^{19/4} \sqrt{x}}{\left(a+b x^2\right)^5}+\frac{19456 a^{15/4} \sqrt{x}}{\left(a+b x^2\right)^4}+\frac{24320 a^{11/4} \sqrt{x}}{\left(a+b x^2\right)^3}+\frac{33440 a^{7/4} \sqrt{x}}{\left(a+b x^2\right)^2}+\frac{58520 a^{3/4} \sqrt{x}}{a+b x^2}-\frac{21945 \sqrt{2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{\sqrt[4]{b}}+\frac{21945 \sqrt{2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{\sqrt[4]{b}}-\frac{43890 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt[4]{b}}+\frac{43890 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt[4]{b}}\right)}{163840 a^{23/4} \sqrt{d x}}","-\frac{4389 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} a^{23/4} \sqrt[4]{b} \sqrt{d}}+\frac{4389 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} a^{23/4} \sqrt[4]{b} \sqrt{d}}-\frac{4389 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{8192 \sqrt{2} a^{23/4} \sqrt[4]{b} \sqrt{d}}+\frac{4389 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{8192 \sqrt{2} a^{23/4} \sqrt[4]{b} \sqrt{d}}+\frac{1463 \sqrt{d x}}{4096 a^5 d \left(a+b x^2\right)}+\frac{209 \sqrt{d x}}{1024 a^4 d \left(a+b x^2\right)^2}+\frac{19 \sqrt{d x}}{128 a^3 d \left(a+b x^2\right)^3}+\frac{19 \sqrt{d x}}{160 a^2 d \left(a+b x^2\right)^4}+\frac{\sqrt{d x}}{10 a d \left(a+b x^2\right)^5}",1,"(Sqrt[x]*((16384*a^(19/4)*Sqrt[x])/(a + b*x^2)^5 + (19456*a^(15/4)*Sqrt[x])/(a + b*x^2)^4 + (24320*a^(11/4)*Sqrt[x])/(a + b*x^2)^3 + (33440*a^(7/4)*Sqrt[x])/(a + b*x^2)^2 + (58520*a^(3/4)*Sqrt[x])/(a + b*x^2) - (43890*Sqrt[2]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(1/4) + (43890*Sqrt[2]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(1/4) - (21945*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(1/4) + (21945*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(1/4)))/(163840*a^(23/4)*Sqrt[d*x])","A",1
725,1,30,404,0.0123,"\int \frac{1}{(d x)^{3/2} \left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[1/((d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]","-\frac{2 x \, _2F_1\left(-\frac{1}{4},6;\frac{3}{4};-\frac{b x^2}{a}\right)}{a^6 (d x)^{3/2}}","-\frac{13923 \sqrt[4]{b} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} a^{25/4} d^{3/2}}+\frac{13923 \sqrt[4]{b} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} a^{25/4} d^{3/2}}+\frac{13923 \sqrt[4]{b} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{8192 \sqrt{2} a^{25/4} d^{3/2}}-\frac{13923 \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{8192 \sqrt{2} a^{25/4} d^{3/2}}-\frac{13923}{4096 a^6 d \sqrt{d x}}+\frac{13923}{20480 a^5 d \sqrt{d x} \left(a+b x^2\right)}+\frac{1547}{5120 a^4 d \sqrt{d x} \left(a+b x^2\right)^2}+\frac{119}{640 a^3 d \sqrt{d x} \left(a+b x^2\right)^3}+\frac{21}{160 a^2 d \sqrt{d x} \left(a+b x^2\right)^4}+\frac{1}{10 a d \sqrt{d x} \left(a+b x^2\right)^5}",1,"(-2*x*Hypergeometric2F1[-1/4, 6, 3/4, -((b*x^2)/a)])/(a^6*(d*x)^(3/2))","C",1
726,1,32,404,0.0144154,"\int \frac{1}{(d x)^{5/2} \left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[1/((d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]","-\frac{2 x \, _2F_1\left(-\frac{3}{4},6;\frac{1}{4};-\frac{b x^2}{a}\right)}{3 a^6 (d x)^{5/2}}","\frac{33649 b^{3/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} a^{27/4} d^{5/2}}-\frac{33649 b^{3/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} a^{27/4} d^{5/2}}+\frac{33649 b^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{8192 \sqrt{2} a^{27/4} d^{5/2}}-\frac{33649 b^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{8192 \sqrt{2} a^{27/4} d^{5/2}}-\frac{33649}{12288 a^6 d (d x)^{3/2}}+\frac{4807}{4096 a^5 d (d x)^{3/2} \left(a+b x^2\right)}+\frac{437}{1024 a^4 d (d x)^{3/2} \left(a+b x^2\right)^2}+\frac{437}{1920 a^3 d (d x)^{3/2} \left(a+b x^2\right)^3}+\frac{23}{160 a^2 d (d x)^{3/2} \left(a+b x^2\right)^4}+\frac{1}{10 a d (d x)^{3/2} \left(a+b x^2\right)^5}",1,"(-2*x*Hypergeometric2F1[-3/4, 6, 1/4, -((b*x^2)/a)])/(3*a^6*(d*x)^(5/2))","C",1
727,1,37,422,0.0131579,"\int \frac{1}{(d x)^{7/2} \left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[1/((d*x)^(7/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]","-\frac{2 \sqrt{d x} \, _2F_1\left(-\frac{5}{4},6;-\frac{1}{4};-\frac{b x^2}{a}\right)}{5 a^6 d^4 x^3}","\frac{69615 b^{5/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} a^{29/4} d^{7/2}}-\frac{69615 b^{5/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{16384 \sqrt{2} a^{29/4} d^{7/2}}-\frac{69615 b^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{8192 \sqrt{2} a^{29/4} d^{7/2}}+\frac{69615 b^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{8192 \sqrt{2} a^{29/4} d^{7/2}}+\frac{69615 b}{4096 a^7 d^3 \sqrt{d x}}-\frac{13923}{4096 a^6 d (d x)^{5/2}}+\frac{7735}{4096 a^5 d (d x)^{5/2} \left(a+b x^2\right)}+\frac{595}{1024 a^4 d (d x)^{5/2} \left(a+b x^2\right)^2}+\frac{35}{128 a^3 d (d x)^{5/2} \left(a+b x^2\right)^3}+\frac{5}{32 a^2 d (d x)^{5/2} \left(a+b x^2\right)^4}+\frac{1}{10 a d (d x)^{5/2} \left(a+b x^2\right)^5}",1,"(-2*Sqrt[d*x]*Hypergeometric2F1[-5/4, 6, -1/4, -((b*x^2)/a)])/(5*a^6*d^4*x^3)","C",1
728,1,44,93,0.0201541,"\int (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[(d*x)^(5/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]","\frac{2 x (d x)^{5/2} \sqrt{\left(a+b x^2\right)^2} \left(11 a+7 b x^2\right)}{77 \left(a+b x^2\right)}","\frac{2 b (d x)^{11/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 d^3 \left(a+b x^2\right)}+\frac{2 a (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d \left(a+b x^2\right)}",1,"(2*x*(d*x)^(5/2)*Sqrt[(a + b*x^2)^2]*(11*a + 7*b*x^2))/(77*(a + b*x^2))","A",1
729,1,44,93,0.0139677,"\int (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[(d*x)^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]","\frac{2 x (d x)^{3/2} \sqrt{\left(a+b x^2\right)^2} \left(9 a+5 b x^2\right)}{45 \left(a+b x^2\right)}","\frac{2 b (d x)^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 d^3 \left(a+b x^2\right)}+\frac{2 a (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d \left(a+b x^2\right)}",1,"(2*x*(d*x)^(3/2)*Sqrt[(a + b*x^2)^2]*(9*a + 5*b*x^2))/(45*(a + b*x^2))","A",1
730,1,44,93,0.0132425,"\int \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[Sqrt[d*x]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]","\frac{2 \sqrt{d x} \sqrt{\left(a+b x^2\right)^2} \left(7 a x+3 b x^3\right)}{21 \left(a+b x^2\right)}","\frac{2 b (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d^3 \left(a+b x^2\right)}+\frac{2 a (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d \left(a+b x^2\right)}",1,"(2*Sqrt[d*x]*Sqrt[(a + b*x^2)^2]*(7*a*x + 3*b*x^3))/(21*(a + b*x^2))","A",1
731,1,43,91,0.0124493,"\int \frac{\sqrt{a^2+2 a b x^2+b^2 x^4}}{\sqrt{d x}} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/Sqrt[d*x],x]","\frac{2 \sqrt{\left(a+b x^2\right)^2} \left(5 a x+b x^3\right)}{5 \sqrt{d x} \left(a+b x^2\right)}","\frac{2 b (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d^3 \left(a+b x^2\right)}+\frac{2 a \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d \left(a+b x^2\right)}",1,"(2*Sqrt[(a + b*x^2)^2]*(5*a*x + b*x^3))/(5*Sqrt[d*x]*(a + b*x^2))","A",1
732,1,43,91,0.0142281,"\int \frac{\sqrt{a^2+2 a b x^2+b^2 x^4}}{(d x)^{3/2}} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/(d*x)^(3/2),x]","\frac{2 x \left(b x^2-3 a\right) \sqrt{\left(a+b x^2\right)^2}}{3 (d x)^{3/2} \left(a+b x^2\right)}","\frac{2 b (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d^3 \left(a+b x^2\right)}-\frac{2 a \sqrt{a^2+2 a b x^2+b^2 x^4}}{d \sqrt{d x} \left(a+b x^2\right)}",1,"(2*x*(-3*a + b*x^2)*Sqrt[(a + b*x^2)^2])/(3*(d*x)^(3/2)*(a + b*x^2))","A",1
733,1,42,91,0.0164999,"\int \frac{\sqrt{a^2+2 a b x^2+b^2 x^4}}{(d x)^{5/2}} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/(d*x)^(5/2),x]","-\frac{2 x \left(a-3 b x^2\right) \sqrt{\left(a+b x^2\right)^2}}{3 (d x)^{5/2} \left(a+b x^2\right)}","\frac{2 b \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^3 \left(a+b x^2\right)}-\frac{2 a \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d (d x)^{3/2} \left(a+b x^2\right)}",1,"(-2*x*(a - 3*b*x^2)*Sqrt[(a + b*x^2)^2])/(3*(d*x)^(5/2)*(a + b*x^2))","A",1
734,1,42,91,0.0160333,"\int \frac{\sqrt{a^2+2 a b x^2+b^2 x^4}}{(d x)^{7/2}} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/(d*x)^(7/2),x]","-\frac{2 x \sqrt{\left(a+b x^2\right)^2} \left(a+5 b x^2\right)}{5 (d x)^{7/2} \left(a+b x^2\right)}","-\frac{2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^3 \sqrt{d x} \left(a+b x^2\right)}-\frac{2 a \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d (d x)^{5/2} \left(a+b x^2\right)}",1,"(-2*x*Sqrt[(a + b*x^2)^2]*(a + 5*b*x^2))/(5*(d*x)^(7/2)*(a + b*x^2))","A",1
735,1,66,195,0.0300848,"\int (d x)^{5/2} \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2} \, dx","Integrate[(d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","\frac{2 x (d x)^{5/2} \sqrt{\left(a+b x^2\right)^2} \left(1045 a^3+1995 a^2 b x^2+1463 a b^2 x^4+385 b^3 x^6\right)}{7315 \left(a+b x^2\right)}","\frac{2 a b^2 (d x)^{15/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d^5 \left(a+b x^2\right)}+\frac{6 a^2 b (d x)^{11/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 d^3 \left(a+b x^2\right)}+\frac{2 b^3 (d x)^{19/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{19 d^7 \left(a+b x^2\right)}+\frac{2 a^3 (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d \left(a+b x^2\right)}",1,"(2*x*(d*x)^(5/2)*Sqrt[(a + b*x^2)^2]*(1045*a^3 + 1995*a^2*b*x^2 + 1463*a*b^2*x^4 + 385*b^3*x^6))/(7315*(a + b*x^2))","A",1
736,1,66,195,0.0236021,"\int (d x)^{3/2} \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2} \, dx","Integrate[(d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","\frac{2 x (d x)^{3/2} \sqrt{\left(a+b x^2\right)^2} \left(663 a^3+1105 a^2 b x^2+765 a b^2 x^4+195 b^3 x^6\right)}{3315 \left(a+b x^2\right)}","\frac{6 a b^2 (d x)^{13/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 d^5 \left(a+b x^2\right)}+\frac{2 a^2 b (d x)^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d^3 \left(a+b x^2\right)}+\frac{2 b^3 (d x)^{17/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{17 d^7 \left(a+b x^2\right)}+\frac{2 a^3 (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d \left(a+b x^2\right)}",1,"(2*x*(d*x)^(3/2)*Sqrt[(a + b*x^2)^2]*(663*a^3 + 1105*a^2*b*x^2 + 765*a*b^2*x^4 + 195*b^3*x^6))/(3315*(a + b*x^2))","A",1
737,1,66,195,0.0214835,"\int \sqrt{d x} \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2} \, dx","Integrate[Sqrt[d*x]*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","\frac{2 \sqrt{d x} \sqrt{\left(a+b x^2\right)^2} \left(385 a^3 x+495 a^2 b x^3+315 a b^2 x^5+77 b^3 x^7\right)}{1155 \left(a+b x^2\right)}","\frac{6 a b^2 (d x)^{11/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 d^5 \left(a+b x^2\right)}+\frac{6 a^2 b (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d^3 \left(a+b x^2\right)}+\frac{2 b^3 (d x)^{15/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{15 d^7 \left(a+b x^2\right)}+\frac{2 a^3 (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d \left(a+b x^2\right)}",1,"(2*Sqrt[d*x]*Sqrt[(a + b*x^2)^2]*(385*a^3*x + 495*a^2*b*x^3 + 315*a*b^2*x^5 + 77*b^3*x^7))/(1155*(a + b*x^2))","A",1
738,1,66,193,0.0208279,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}{\sqrt{d x}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/Sqrt[d*x],x]","\frac{2 \sqrt{\left(a+b x^2\right)^2} \left(195 a^3 x+117 a^2 b x^3+65 a b^2 x^5+15 b^3 x^7\right)}{195 \sqrt{d x} \left(a+b x^2\right)}","\frac{2 a b^2 (d x)^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d^5 \left(a+b x^2\right)}+\frac{6 a^2 b (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d^3 \left(a+b x^2\right)}+\frac{2 b^3 (d x)^{13/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 d^7 \left(a+b x^2\right)}+\frac{2 a^3 \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d \left(a+b x^2\right)}",1,"(2*Sqrt[(a + b*x^2)^2]*(195*a^3*x + 117*a^2*b*x^3 + 65*a*b^2*x^5 + 15*b^3*x^7))/(195*Sqrt[d*x]*(a + b*x^2))","A",1
739,1,66,191,0.0242304,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}{(d x)^{3/2}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/(d*x)^(3/2),x]","\frac{2 x \sqrt{\left(a+b x^2\right)^2} \left(-77 a^3+77 a^2 b x^2+33 a b^2 x^4+7 b^3 x^6\right)}{77 (d x)^{3/2} \left(a+b x^2\right)}","\frac{6 a b^2 (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d^5 \left(a+b x^2\right)}+\frac{2 a^2 b (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^3 \left(a+b x^2\right)}+\frac{2 b^3 (d x)^{11/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 d^7 \left(a+b x^2\right)}-\frac{2 a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{d \sqrt{d x} \left(a+b x^2\right)}",1,"(2*x*Sqrt[(a + b*x^2)^2]*(-77*a^3 + 77*a^2*b*x^2 + 33*a*b^2*x^4 + 7*b^3*x^6))/(77*(d*x)^(3/2)*(a + b*x^2))","A",1
740,1,66,193,0.0263286,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}{(d x)^{5/2}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/(d*x)^(5/2),x]","\frac{2 x \sqrt{\left(a+b x^2\right)^2} \left(-15 a^3+135 a^2 b x^2+27 a b^2 x^4+5 b^3 x^6\right)}{45 (d x)^{5/2} \left(a+b x^2\right)}","\frac{6 a b^2 (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d^5 \left(a+b x^2\right)}+\frac{6 a^2 b \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^3 \left(a+b x^2\right)}+\frac{2 b^3 (d x)^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 d^7 \left(a+b x^2\right)}-\frac{2 a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d (d x)^{3/2} \left(a+b x^2\right)}",1,"(2*x*Sqrt[(a + b*x^2)^2]*(-15*a^3 + 135*a^2*b*x^2 + 27*a*b^2*x^4 + 5*b^3*x^6))/(45*(d*x)^(5/2)*(a + b*x^2))","A",1
741,1,66,191,0.0265461,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}}{(d x)^{7/2}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/(d*x)^(7/2),x]","\frac{2 x \sqrt{\left(a+b x^2\right)^2} \left(-7 a^3-105 a^2 b x^2+35 a b^2 x^4+5 b^3 x^6\right)}{35 (d x)^{7/2} \left(a+b x^2\right)}","\frac{2 a b^2 (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^5 \left(a+b x^2\right)}-\frac{6 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^3 \sqrt{d x} \left(a+b x^2\right)}+\frac{2 b^3 (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d^7 \left(a+b x^2\right)}-\frac{2 a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d (d x)^{5/2} \left(a+b x^2\right)}",1,"(2*x*Sqrt[(a + b*x^2)^2]*(-7*a^3 - 105*a^2*b*x^2 + 35*a*b^2*x^4 + 5*b^3*x^6))/(35*(d*x)^(7/2)*(a + b*x^2))","A",1
742,1,88,297,0.042915,"\int (d x)^{5/2} \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2} \, dx","Integrate[(d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{2 x (d x)^{5/2} \sqrt{\left(a+b x^2\right)^2} \left(129789 a^5+412965 a^4 b x^2+605682 a^3 b^2 x^4+478170 a^2 b^3 x^6+197505 a b^4 x^8+33649 b^5 x^{10}\right)}{908523 \left(a+b x^2\right)}","\frac{2 b^5 (d x)^{27/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{27 d^{11} \left(a+b x^2\right)}+\frac{10 a b^4 (d x)^{23/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{23 d^9 \left(a+b x^2\right)}+\frac{20 a^2 b^3 (d x)^{19/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{19 d^7 \left(a+b x^2\right)}+\frac{2 a^5 (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d \left(a+b x^2\right)}+\frac{10 a^4 b (d x)^{11/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 d^3 \left(a+b x^2\right)}+\frac{4 a^3 b^2 (d x)^{15/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d^5 \left(a+b x^2\right)}",1,"(2*x*(d*x)^(5/2)*Sqrt[(a + b*x^2)^2]*(129789*a^5 + 412965*a^4*b*x^2 + 605682*a^3*b^2*x^4 + 478170*a^2*b^3*x^6 + 197505*a*b^4*x^8 + 33649*b^5*x^10))/(908523*(a + b*x^2))","A",1
743,1,88,297,0.0349964,"\int (d x)^{3/2} \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2} \, dx","Integrate[(d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{2 x (d x)^{3/2} \sqrt{\left(a+b x^2\right)^2} \left(69615 a^5+193375 a^4 b x^2+267750 a^3 b^2 x^4+204750 a^2 b^3 x^6+82875 a b^4 x^8+13923 b^5 x^{10}\right)}{348075 \left(a+b x^2\right)}","\frac{2 b^5 (d x)^{25/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{25 d^{11} \left(a+b x^2\right)}+\frac{10 a b^4 (d x)^{21/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{21 d^9 \left(a+b x^2\right)}+\frac{20 a^2 b^3 (d x)^{17/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{17 d^7 \left(a+b x^2\right)}+\frac{2 a^5 (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d \left(a+b x^2\right)}+\frac{10 a^4 b (d x)^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 d^3 \left(a+b x^2\right)}+\frac{20 a^3 b^2 (d x)^{13/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 d^5 \left(a+b x^2\right)}",1,"(2*x*(d*x)^(3/2)*Sqrt[(a + b*x^2)^2]*(69615*a^5 + 193375*a^4*b*x^2 + 267750*a^3*b^2*x^4 + 204750*a^2*b^3*x^6 + 82875*a*b^4*x^8 + 13923*b^5*x^10))/(348075*(a + b*x^2))","A",1
744,1,88,297,0.0301257,"\int \sqrt{d x} \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2} \, dx","Integrate[Sqrt[d*x]*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{2 \sqrt{d x} \sqrt{\left(a+b x^2\right)^2} \left(33649 a^5 x+72105 a^4 b x^3+91770 a^3 b^2 x^5+67298 a^2 b^3 x^7+26565 a b^4 x^9+4389 b^5 x^{11}\right)}{100947 \left(a+b x^2\right)}","\frac{2 b^5 (d x)^{23/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{23 d^{11} \left(a+b x^2\right)}+\frac{10 a b^4 (d x)^{19/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{19 d^9 \left(a+b x^2\right)}+\frac{4 a^2 b^3 (d x)^{15/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d^7 \left(a+b x^2\right)}+\frac{2 a^5 (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d \left(a+b x^2\right)}+\frac{10 a^4 b (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d^3 \left(a+b x^2\right)}+\frac{20 a^3 b^2 (d x)^{11/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 d^5 \left(a+b x^2\right)}",1,"(2*Sqrt[d*x]*Sqrt[(a + b*x^2)^2]*(33649*a^5*x + 72105*a^4*b*x^3 + 91770*a^3*b^2*x^5 + 67298*a^2*b^3*x^7 + 26565*a*b^4*x^9 + 4389*b^5*x^11))/(100947*(a + b*x^2))","A",1
745,1,88,293,0.0306989,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{\sqrt{d x}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/Sqrt[d*x],x]","\frac{2 \sqrt{\left(a+b x^2\right)^2} \left(13923 a^5 x+13923 a^4 b x^3+15470 a^3 b^2 x^5+10710 a^2 b^3 x^7+4095 a b^4 x^9+663 b^5 x^{11}\right)}{13923 \sqrt{d x} \left(a+b x^2\right)}","\frac{2 b^5 (d x)^{21/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{21 d^{11} \left(a+b x^2\right)}+\frac{10 a b^4 (d x)^{17/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{17 d^9 \left(a+b x^2\right)}+\frac{20 a^2 b^3 (d x)^{13/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 d^7 \left(a+b x^2\right)}+\frac{2 a^5 \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d \left(a+b x^2\right)}+\frac{2 a^4 b (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^3 \left(a+b x^2\right)}+\frac{20 a^3 b^2 (d x)^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 d^5 \left(a+b x^2\right)}",1,"(2*Sqrt[(a + b*x^2)^2]*(13923*a^5*x + 13923*a^4*b*x^3 + 15470*a^3*b^2*x^5 + 10710*a^2*b^3*x^7 + 4095*a*b^4*x^9 + 663*b^5*x^11))/(13923*Sqrt[d*x]*(a + b*x^2))","A",1
746,1,88,295,0.0339051,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{(d x)^{3/2}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/(d*x)^(3/2),x]","\frac{2 x \sqrt{\left(a+b x^2\right)^2} \left(-4389 a^5+7315 a^4 b x^2+6270 a^3 b^2 x^4+3990 a^2 b^3 x^6+1463 a b^4 x^8+231 b^5 x^{10}\right)}{4389 (d x)^{3/2} \left(a+b x^2\right)}","\frac{2 b^5 (d x)^{19/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{19 d^{11} \left(a+b x^2\right)}+\frac{2 a b^4 (d x)^{15/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d^9 \left(a+b x^2\right)}+\frac{20 a^2 b^3 (d x)^{11/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 d^7 \left(a+b x^2\right)}-\frac{2 a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{d \sqrt{d x} \left(a+b x^2\right)}+\frac{10 a^4 b (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d^3 \left(a+b x^2\right)}+\frac{20 a^3 b^2 (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d^5 \left(a+b x^2\right)}",1,"(2*x*Sqrt[(a + b*x^2)^2]*(-4389*a^5 + 7315*a^4*b*x^2 + 6270*a^3*b^2*x^4 + 3990*a^2*b^3*x^6 + 1463*a*b^4*x^8 + 231*b^5*x^10))/(4389*(d*x)^(3/2)*(a + b*x^2))","A",1
747,1,88,293,0.0350561,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{(d x)^{5/2}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/(d*x)^(5/2),x]","\frac{2 x \sqrt{\left(a+b x^2\right)^2} \left(-663 a^5+9945 a^4 b x^2+3978 a^3 b^2 x^4+2210 a^2 b^3 x^6+765 a b^4 x^8+117 b^5 x^{10}\right)}{1989 (d x)^{5/2} \left(a+b x^2\right)}","\frac{2 b^5 (d x)^{17/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{17 d^{11} \left(a+b x^2\right)}+\frac{10 a b^4 (d x)^{13/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 d^9 \left(a+b x^2\right)}+\frac{20 a^2 b^3 (d x)^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 d^7 \left(a+b x^2\right)}-\frac{2 a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d (d x)^{3/2} \left(a+b x^2\right)}+\frac{10 a^4 b \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^3 \left(a+b x^2\right)}+\frac{4 a^3 b^2 (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^5 \left(a+b x^2\right)}",1,"(2*x*Sqrt[(a + b*x^2)^2]*(-663*a^5 + 9945*a^4*b*x^2 + 3978*a^3*b^2*x^4 + 2210*a^2*b^3*x^6 + 765*a*b^4*x^8 + 117*b^5*x^10))/(1989*(d*x)^(5/2)*(a + b*x^2))","A",1
748,1,88,295,0.0365768,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}}{(d x)^{7/2}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/(d*x)^(7/2),x]","\frac{2 x \sqrt{\left(a+b x^2\right)^2} \left(-231 a^5-5775 a^4 b x^2+3850 a^3 b^2 x^4+1650 a^2 b^3 x^6+525 a b^4 x^8+77 b^5 x^{10}\right)}{1155 (d x)^{7/2} \left(a+b x^2\right)}","\frac{2 b^5 (d x)^{15/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{15 d^{11} \left(a+b x^2\right)}+\frac{10 a b^4 (d x)^{11/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 d^9 \left(a+b x^2\right)}+\frac{20 a^2 b^3 (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d^7 \left(a+b x^2\right)}-\frac{2 a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d (d x)^{5/2} \left(a+b x^2\right)}-\frac{10 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^3 \sqrt{d x} \left(a+b x^2\right)}+\frac{20 a^3 b^2 (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d^5 \left(a+b x^2\right)}",1,"(2*x*Sqrt[(a + b*x^2)^2]*(-231*a^5 - 5775*a^4*b*x^2 + 3850*a^3*b^2*x^4 + 1650*a^2*b^3*x^6 + 525*a*b^4*x^8 + 77*b^5*x^10))/(1155*(d*x)^(7/2)*(a + b*x^2))","A",1
749,1,238,457,0.0908782,"\int \frac{(d x)^{7/2}}{\sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx","Integrate[(d*x)^(7/2)/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]","\frac{d^3 \sqrt{d x} \left(a+b x^2\right) \left(-5 \sqrt{2} a^{5/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+5 \sqrt{2} a^{5/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-10 \sqrt{2} a^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)+10 \sqrt{2} a^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)-40 a \sqrt[4]{b} \sqrt{x}+8 b^{5/4} x^{5/2}\right)}{20 b^{9/4} \sqrt{x} \sqrt{\left(a+b x^2\right)^2}}","-\frac{2 a d^3 \sqrt{d x} \left(a+b x^2\right)}{b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{2 d (d x)^{5/2} \left(a+b x^2\right)}{5 b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a^{5/4} d^{7/2} \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{2 \sqrt{2} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{a^{5/4} d^{7/2} \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{2 \sqrt{2} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a^{5/4} d^{7/2} \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{\sqrt{2} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{a^{5/4} d^{7/2} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{\sqrt{2} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(d^3*Sqrt[d*x]*(a + b*x^2)*(-40*a*b^(1/4)*Sqrt[x] + 8*b^(5/4)*x^(5/2) - 10*Sqrt[2]*a^(5/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 10*Sqrt[2]*a^(5/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 5*Sqrt[2]*a^(5/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 5*Sqrt[2]*a^(5/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]))/(20*b^(9/4)*Sqrt[x]*Sqrt[(a + b*x^2)^2])","A",1
750,1,110,412,0.0540032,"\int \frac{(d x)^{5/2}}{\sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx","Integrate[(d*x)^(5/2)/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]","\frac{(d x)^{5/2} \left(a+b x^2\right) \left(3 (-a)^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{-a}}\right)-3 (-a)^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{-a}}\right)+2 b^{3/4} x^{3/2}\right)}{3 b^{7/4} x^{5/2} \sqrt{\left(a+b x^2\right)^2}}","\frac{2 d (d x)^{3/2} \left(a+b x^2\right)}{3 b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a^{3/4} d^{5/2} \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{2 \sqrt{2} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{a^{3/4} d^{5/2} \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{2 \sqrt{2} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{a^{3/4} d^{5/2} \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{\sqrt{2} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a^{3/4} d^{5/2} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{\sqrt{2} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"((d*x)^(5/2)*(a + b*x^2)*(2*b^(3/4)*x^(3/2) + 3*(-a)^(3/4)*ArcTan[(b^(1/4)*Sqrt[x])/(-a)^(1/4)] - 3*(-a)^(3/4)*ArcTanh[(b^(1/4)*Sqrt[x])/(-a)^(1/4)]))/(3*b^(7/4)*x^(5/2)*Sqrt[(a + b*x^2)^2])","A",1
751,1,221,410,0.0755493,"\int \frac{(d x)^{3/2}}{\sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx","Integrate[(d*x)^(3/2)/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]","\frac{(d x)^{3/2} \left(a+b x^2\right) \left(\sqrt{2} \sqrt[4]{a} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-\sqrt{2} \sqrt[4]{a} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+2 \sqrt{2} \sqrt[4]{a} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)-2 \sqrt{2} \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)+8 \sqrt[4]{b} \sqrt{x}\right)}{4 b^{5/4} x^{3/2} \sqrt{\left(a+b x^2\right)^2}}","\frac{2 d \sqrt{d x} \left(a+b x^2\right)}{b \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\sqrt[4]{a} d^{3/2} \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{2 \sqrt{2} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\sqrt[4]{a} d^{3/2} \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{2 \sqrt{2} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\sqrt[4]{a} d^{3/2} \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{\sqrt{2} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\sqrt[4]{a} d^{3/2} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{\sqrt{2} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"((d*x)^(3/2)*(a + b*x^2)*(8*b^(1/4)*Sqrt[x] + 2*Sqrt[2]*a^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 2*Sqrt[2]*a^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + Sqrt[2]*a^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - Sqrt[2]*a^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]))/(4*b^(5/4)*x^(3/2)*Sqrt[(a + b*x^2)^2])","A",1
752,1,85,368,0.0435887,"\int \frac{\sqrt{d x}}{\sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx","Integrate[Sqrt[d*x]/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]","\frac{\sqrt{d x} \left(a+b x^2\right) \left(\tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{-a}}\right)+\tanh ^{-1}\left(\frac{a \sqrt[4]{b} \sqrt{x}}{(-a)^{5/4}}\right)\right)}{\sqrt[4]{-a} b^{3/4} \sqrt{x} \sqrt{\left(a+b x^2\right)^2}}","\frac{\sqrt{d} \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{2 \sqrt{2} \sqrt[4]{a} b^{3/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\sqrt{d} \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{2 \sqrt{2} \sqrt[4]{a} b^{3/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\sqrt{d} \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{\sqrt{2} \sqrt[4]{a} b^{3/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\sqrt{d} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{\sqrt{2} \sqrt[4]{a} b^{3/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(Sqrt[d*x]*(a + b*x^2)*(ArcTan[(b^(1/4)*Sqrt[x])/(-a)^(1/4)] + ArcTanh[(a*b^(1/4)*Sqrt[x])/(-a)^(5/4)]))/((-a)^(1/4)*b^(3/4)*Sqrt[x]*Sqrt[(a + b*x^2)^2])","A",1
753,1,178,368,0.0486773,"\int \frac{1}{\sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx","Integrate[1/(Sqrt[d*x]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]),x]","-\frac{\sqrt{x} \left(a+b x^2\right) \left(\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)\right)}{2 \sqrt{2} a^{3/4} \sqrt[4]{b} \sqrt{d x} \sqrt{\left(a+b x^2\right)^2}}","-\frac{\left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{2 \sqrt{2} a^{3/4} \sqrt[4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{2 \sqrt{2} a^{3/4} \sqrt[4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{\sqrt{2} a^{3/4} \sqrt[4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{\sqrt{2} a^{3/4} \sqrt[4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"-1/2*(Sqrt[x]*(a + b*x^2)*(2*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]))/(Sqrt[2]*a^(3/4)*b^(1/4)*Sqrt[d*x]*Sqrt[(a + b*x^2)^2])","A",1
754,1,50,412,0.0121831,"\int \frac{1}{(d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx","Integrate[1/((d*x)^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]),x]","-\frac{2 x \left(a+b x^2\right) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\frac{b x^2}{a}\right)}{a (d x)^{3/2} \sqrt{\left(a+b x^2\right)^2}}","-\frac{2 \left(a+b x^2\right)}{a d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\sqrt[4]{b} \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{2 \sqrt{2} a^{5/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\sqrt[4]{b} \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{2 \sqrt{2} a^{5/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\sqrt[4]{b} \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{\sqrt{2} a^{5/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\sqrt[4]{b} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{\sqrt{2} a^{5/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(-2*x*(a + b*x^2)*Hypergeometric2F1[-1/4, 1, 3/4, -((b*x^2)/a)])/(a*(d*x)^(3/2)*Sqrt[(a + b*x^2)^2])","C",1
755,1,52,414,0.0130516,"\int \frac{1}{(d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx","Integrate[1/((d*x)^(5/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]),x]","-\frac{2 x \left(a+b x^2\right) \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\frac{b x^2}{a}\right)}{3 a (d x)^{5/2} \sqrt{\left(a+b x^2\right)^2}}","-\frac{2 \left(a+b x^2\right)}{3 a d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{b^{3/4} \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{2 \sqrt{2} a^{7/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b^{3/4} \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{2 \sqrt{2} a^{7/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{b^{3/4} \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{\sqrt{2} a^{7/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b^{3/4} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{\sqrt{2} a^{7/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(-2*x*(a + b*x^2)*Hypergeometric2F1[-3/4, 1, 1/4, -((b*x^2)/a)])/(3*a*(d*x)^(5/2)*Sqrt[(a + b*x^2)^2])","C",1
756,1,52,459,0.0131905,"\int \frac{1}{(d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx","Integrate[1/((d*x)^(7/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]),x]","-\frac{2 x \left(a+b x^2\right) \, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};-\frac{b x^2}{a}\right)}{5 a (d x)^{7/2} \sqrt{\left(a+b x^2\right)^2}}","\frac{2 b \left(a+b x^2\right)}{a^2 d^3 \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{2 \left(a+b x^2\right)}{5 a d (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{b^{5/4} \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{2 \sqrt{2} a^{9/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b^{5/4} \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{2 \sqrt{2} a^{9/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b^{5/4} \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{\sqrt{2} a^{9/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{b^{5/4} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{\sqrt{2} a^{9/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(-2*x*(a + b*x^2)*Hypergeometric2F1[-5/4, 1, -1/4, -((b*x^2)/a)])/(5*a*(d*x)^(7/2)*Sqrt[(a + b*x^2)^2])","C",1
757,1,498,551,0.1647407,"\int \frac{(d x)^{15/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}} \, dx","Integrate[(d*x)^(15/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","\frac{d^7 \sqrt{d x} \left(-585 \sqrt{2} a^{5/4} b^2 x^4 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+585 \sqrt{2} a^{5/4} b^2 x^4 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-1170 \sqrt{2} a^{9/4} b x^2 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+1170 \sqrt{2} a^{9/4} b x^2 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-1170 \sqrt{2} a^{5/4} \left(a+b x^2\right)^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)+1170 \sqrt{2} a^{5/4} \left(a+b x^2\right)^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)-585 \sqrt{2} a^{13/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+585 \sqrt{2} a^{13/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-4680 a^3 \sqrt[4]{b} \sqrt{x}-8424 a^2 b^{5/4} x^{5/2}-3328 a b^{9/4} x^{9/2}+256 b^{13/4} x^{13/2}\right)}{640 b^{17/4} \sqrt{x} \left(a+b x^2\right) \sqrt{\left(a+b x^2\right)^2}}","-\frac{13 d^3 (d x)^{9/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{13/2}}{4 b \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{117 a d^7 \sqrt{d x} \left(a+b x^2\right)}{16 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{117 d^5 (d x)^{5/2} \left(a+b x^2\right)}{80 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{117 a^{5/4} d^{15/2} \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{64 \sqrt{2} b^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{117 a^{5/4} d^{15/2} \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{64 \sqrt{2} b^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{117 a^{5/4} d^{15/2} \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{32 \sqrt{2} b^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{117 a^{5/4} d^{15/2} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{32 \sqrt{2} b^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(d^7*Sqrt[d*x]*(-4680*a^3*b^(1/4)*Sqrt[x] - 8424*a^2*b^(5/4)*x^(5/2) - 3328*a*b^(9/4)*x^(9/2) + 256*b^(13/4)*x^(13/2) - 1170*Sqrt[2]*a^(5/4)*(a + b*x^2)^2*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 1170*Sqrt[2]*a^(5/4)*(a + b*x^2)^2*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 585*Sqrt[2]*a^(13/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - 1170*Sqrt[2]*a^(9/4)*b*x^2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - 585*Sqrt[2]*a^(5/4)*b^2*x^4*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 585*Sqrt[2]*a^(13/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 1170*Sqrt[2]*a^(9/4)*b*x^2*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 585*Sqrt[2]*a^(5/4)*b^2*x^4*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]))/(640*b^(17/4)*Sqrt[x]*(a + b*x^2)*Sqrt[(a + b*x^2)^2])","A",1
758,1,88,504,0.0337738,"\int \frac{(d x)^{13/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}} \, dx","Integrate[(d*x)^(13/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","-\frac{2 d^5 (d x)^{3/2} \left(-77 a^2+77 \left(a+b x^2\right)^2 \, _2F_1\left(\frac{3}{4},3;\frac{7}{4};-\frac{b x^2}{a}\right)-55 a b x^2-5 b^2 x^4\right)}{15 b^3 \left(a+b x^2\right) \sqrt{\left(a+b x^2\right)^2}}","-\frac{11 d^3 (d x)^{7/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{11/2}}{4 b \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{77 d^5 (d x)^{3/2} \left(a+b x^2\right)}{48 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{77 a^{3/4} d^{13/2} \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{64 \sqrt{2} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{77 a^{3/4} d^{13/2} \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{64 \sqrt{2} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{77 a^{3/4} d^{13/2} \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{32 \sqrt{2} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{77 a^{3/4} d^{13/2} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{32 \sqrt{2} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(-2*d^5*(d*x)^(3/2)*(-77*a^2 - 55*a*b*x^2 - 5*b^2*x^4 + 77*(a + b*x^2)^2*Hypergeometric2F1[3/4, 3, 7/4, -((b*x^2)/a)]))/(15*b^3*(a + b*x^2)*Sqrt[(a + b*x^2)^2])","C",1
759,1,484,504,0.1656649,"\int \frac{(d x)^{11/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}} \, dx","Integrate[(d*x)^(11/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","\frac{15 a^2 (d x)^{11/2} \left(a+b x^2\right)}{4 b^3 x^5 \left(\left(a+b x^2\right)^2\right)^{3/2}}+\frac{45 \sqrt[4]{a} (d x)^{11/2} \left(a+b x^2\right)^3 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} b^{13/4} x^{11/2} \left(\left(a+b x^2\right)^2\right)^{3/2}}-\frac{45 \sqrt[4]{a} (d x)^{11/2} \left(a+b x^2\right)^3 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} b^{13/4} x^{11/2} \left(\left(a+b x^2\right)^2\right)^{3/2}}+\frac{45 \sqrt[4]{a} (d x)^{11/2} \left(a+b x^2\right)^3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{32 \sqrt{2} b^{13/4} x^{11/2} \left(\left(a+b x^2\right)^2\right)^{3/2}}-\frac{45 \sqrt[4]{a} (d x)^{11/2} \left(a+b x^2\right)^3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{32 \sqrt{2} b^{13/4} x^{11/2} \left(\left(a+b x^2\right)^2\right)^{3/2}}-\frac{15 a (d x)^{11/2} \left(a+b x^2\right)^2}{16 b^3 x^5 \left(\left(a+b x^2\right)^2\right)^{3/2}}+\frac{6 a (d x)^{11/2} \left(a+b x^2\right)}{b^2 x^3 \left(\left(a+b x^2\right)^2\right)^{3/2}}+\frac{2 (d x)^{11/2} \left(a+b x^2\right)}{b x \left(\left(a+b x^2\right)^2\right)^{3/2}}","-\frac{9 d^3 (d x)^{5/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{9/2}}{4 b \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 \sqrt[4]{a} d^{11/2} \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{64 \sqrt{2} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \sqrt[4]{a} d^{11/2} \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{64 \sqrt{2} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 \sqrt[4]{a} d^{11/2} \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{32 \sqrt{2} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \sqrt[4]{a} d^{11/2} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{32 \sqrt{2} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 d^5 \sqrt{d x} \left(a+b x^2\right)}{16 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(15*a^2*(d*x)^(11/2)*(a + b*x^2))/(4*b^3*x^5*((a + b*x^2)^2)^(3/2)) + (6*a*(d*x)^(11/2)*(a + b*x^2))/(b^2*x^3*((a + b*x^2)^2)^(3/2)) + (2*(d*x)^(11/2)*(a + b*x^2))/(b*x*((a + b*x^2)^2)^(3/2)) - (15*a*(d*x)^(11/2)*(a + b*x^2)^2)/(16*b^3*x^5*((a + b*x^2)^2)^(3/2)) + (45*a^(1/4)*(d*x)^(11/2)*(a + b*x^2)^3*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*b^(13/4)*x^(11/2)*((a + b*x^2)^2)^(3/2)) - (45*a^(1/4)*(d*x)^(11/2)*(a + b*x^2)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*b^(13/4)*x^(11/2)*((a + b*x^2)^2)^(3/2)) + (45*a^(1/4)*(d*x)^(11/2)*(a + b*x^2)^3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*b^(13/4)*x^(11/2)*((a + b*x^2)^2)^(3/2)) - (45*a^(1/4)*(d*x)^(11/2)*(a + b*x^2)^3*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*b^(13/4)*x^(11/2)*((a + b*x^2)^2)^(3/2))","A",1
760,1,84,458,0.0316903,"\int \frac{(d x)^{9/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}} \, dx","Integrate[(d*x)^(9/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","\frac{2 d^3 (d x)^{3/2} \left(7 \left(a+b x^2\right)^2 \, _2F_1\left(\frac{3}{4},3;\frac{7}{4};-\frac{b x^2}{a}\right)-a \left(7 a+5 b x^2\right)\right)}{5 a b^2 \left(a+b x^2\right) \sqrt{\left(a+b x^2\right)^2}}","-\frac{7 d^3 (d x)^{3/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{7/2}}{4 b \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{21 d^{9/2} \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{64 \sqrt{2} \sqrt[4]{a} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{21 d^{9/2} \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{64 \sqrt{2} \sqrt[4]{a} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{21 d^{9/2} \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{32 \sqrt{2} \sqrt[4]{a} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{21 d^{9/2} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{32 \sqrt{2} \sqrt[4]{a} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(2*d^3*(d*x)^(3/2)*(-(a*(7*a + 5*b*x^2)) + 7*(a + b*x^2)^2*Hypergeometric2F1[3/4, 3, 7/4, -((b*x^2)/a)]))/(5*a*b^2*(a + b*x^2)*Sqrt[(a + b*x^2)^2])","C",1
761,1,447,458,0.1444711,"\int \frac{(d x)^{7/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}} \, dx","Integrate[(d*x)^(7/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","-\frac{5 (d x)^{7/2} \left(a+b x^2\right)^3 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{3/4} b^{9/4} x^{7/2} \left(\left(a+b x^2\right)^2\right)^{3/2}}+\frac{5 (d x)^{7/2} \left(a+b x^2\right)^3 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{3/4} b^{9/4} x^{7/2} \left(\left(a+b x^2\right)^2\right)^{3/2}}-\frac{5 (d x)^{7/2} \left(a+b x^2\right)^3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{32 \sqrt{2} a^{3/4} b^{9/4} x^{7/2} \left(\left(a+b x^2\right)^2\right)^{3/2}}+\frac{5 (d x)^{7/2} \left(a+b x^2\right)^3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{32 \sqrt{2} a^{3/4} b^{9/4} x^{7/2} \left(\left(a+b x^2\right)^2\right)^{3/2}}+\frac{5 (d x)^{7/2} \left(a+b x^2\right)^2}{48 b^2 x^3 \left(\left(a+b x^2\right)^2\right)^{3/2}}-\frac{5 a (d x)^{7/2} \left(a+b x^2\right)}{12 b^2 x^3 \left(\left(a+b x^2\right)^2\right)^{3/2}}-\frac{2 (d x)^{7/2} \left(a+b x^2\right)}{3 b x \left(\left(a+b x^2\right)^2\right)^{3/2}}","-\frac{5 d^3 \sqrt{d x}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{5/2}}{4 b \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 d^{7/2} \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{64 \sqrt{2} a^{3/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 d^{7/2} \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{64 \sqrt{2} a^{3/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 d^{7/2} \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{32 \sqrt{2} a^{3/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 d^{7/2} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{32 \sqrt{2} a^{3/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(-5*a*(d*x)^(7/2)*(a + b*x^2))/(12*b^2*x^3*((a + b*x^2)^2)^(3/2)) - (2*(d*x)^(7/2)*(a + b*x^2))/(3*b*x*((a + b*x^2)^2)^(3/2)) + (5*(d*x)^(7/2)*(a + b*x^2)^2)/(48*b^2*x^3*((a + b*x^2)^2)^(3/2)) - (5*(d*x)^(7/2)*(a + b*x^2)^3*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(3/4)*b^(9/4)*x^(7/2)*((a + b*x^2)^2)^(3/2)) + (5*(d*x)^(7/2)*(a + b*x^2)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(3/4)*b^(9/4)*x^(7/2)*((a + b*x^2)^2)^(3/2)) - (5*(d*x)^(7/2)*(a + b*x^2)^3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(3/4)*b^(9/4)*x^(7/2)*((a + b*x^2)^2)^(3/2)) + (5*(d*x)^(7/2)*(a + b*x^2)^3*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(3/4)*b^(9/4)*x^(7/2)*((a + b*x^2)^2)^(3/2))","A",1
762,1,73,459,0.0256774,"\int \frac{(d x)^{5/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}} \, dx","Integrate[(d*x)^(5/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","\frac{2 d (d x)^{3/2} \left(\left(a+b x^2\right)^2 \, _2F_1\left(\frac{3}{4},3;\frac{7}{4};-\frac{b x^2}{a}\right)-a^2\right)}{5 a^2 b \left(a+b x^2\right) \sqrt{\left(a+b x^2\right)^2}}","\frac{3 d (d x)^{3/2}}{16 a b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{3/2}}{4 b \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3 d^{5/2} \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{64 \sqrt{2} a^{5/4} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 d^{5/2} \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{64 \sqrt{2} a^{5/4} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 d^{5/2} \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{32 \sqrt{2} a^{5/4} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3 d^{5/2} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{32 \sqrt{2} a^{5/4} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(2*d*(d*x)^(3/2)*(-a^2 + (a + b*x^2)^2*Hypergeometric2F1[3/4, 3, 7/4, -((b*x^2)/a)]))/(5*a^2*b*(a + b*x^2)*Sqrt[(a + b*x^2)^2])","C",1
763,1,272,459,0.2050086,"\int \frac{(d x)^{3/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}} \, dx","Integrate[(d*x)^(3/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","\frac{(d x)^{3/2} \left(a+b x^2\right) \left(8 a^{3/4} \sqrt[4]{b} \sqrt{x} \left(a+b x^2\right)-32 a^{7/4} \sqrt[4]{b} \sqrt{x}-3 \sqrt{2} \left(a+b x^2\right)^2 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+3 \sqrt{2} \left(a+b x^2\right)^2 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-6 \sqrt{2} \left(a+b x^2\right)^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)+6 \sqrt{2} \left(a+b x^2\right)^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)\right)}{128 a^{7/4} b^{5/4} x^{3/2} \left(\left(a+b x^2\right)^2\right)^{3/2}}","\frac{d \sqrt{d x}}{16 a b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d \sqrt{d x}}{4 b \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 d^{3/2} \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{64 \sqrt{2} a^{7/4} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3 d^{3/2} \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{64 \sqrt{2} a^{7/4} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 d^{3/2} \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{32 \sqrt{2} a^{7/4} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3 d^{3/2} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{32 \sqrt{2} a^{7/4} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"((d*x)^(3/2)*(a + b*x^2)*(-32*a^(7/4)*b^(1/4)*Sqrt[x] + 8*a^(3/4)*b^(1/4)*Sqrt[x]*(a + b*x^2) - 6*Sqrt[2]*(a + b*x^2)^2*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 6*Sqrt[2]*(a + b*x^2)^2*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 3*Sqrt[2]*(a + b*x^2)^2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 3*Sqrt[2]*(a + b*x^2)^2*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]))/(128*a^(7/4)*b^(5/4)*x^(3/2)*((a + b*x^2)^2)^(3/2))","A",1
764,1,54,460,0.0144135,"\int \frac{\sqrt{d x}}{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}} \, dx","Integrate[Sqrt[d*x]/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","\frac{2 x \sqrt{d x} \left(a+b x^2\right)^3 \, _2F_1\left(\frac{3}{4},3;\frac{7}{4};-\frac{b x^2}{a}\right)}{3 a^3 \left(\left(a+b x^2\right)^2\right)^{3/2}}","\frac{5 (d x)^{3/2}}{16 a^2 d \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{(d x)^{3/2}}{4 a d \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 \sqrt{d} \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{64 \sqrt{2} a^{9/4} b^{3/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 \sqrt{d} \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{64 \sqrt{2} a^{9/4} b^{3/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 \sqrt{d} \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{32 \sqrt{2} a^{9/4} b^{3/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 \sqrt{d} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{32 \sqrt{2} a^{9/4} b^{3/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(2*x*Sqrt[d*x]*(a + b*x^2)^3*Hypergeometric2F1[3/4, 3, 7/4, -((b*x^2)/a)])/(3*a^3*((a + b*x^2)^2)^(3/2))","C",1
765,1,272,460,0.187637,"\int \frac{1}{\sqrt{d x} \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}} \, dx","Integrate[1/(Sqrt[d*x]*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)),x]","\frac{\sqrt{x} \left(a+b x^2\right) \left(56 a^{3/4} \sqrt[4]{b} \sqrt{x} \left(a+b x^2\right)+32 a^{7/4} \sqrt[4]{b} \sqrt{x}-21 \sqrt{2} \left(a+b x^2\right)^2 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+21 \sqrt{2} \left(a+b x^2\right)^2 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-42 \sqrt{2} \left(a+b x^2\right)^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)+42 \sqrt{2} \left(a+b x^2\right)^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)\right)}{128 a^{11/4} \sqrt[4]{b} \sqrt{d x} \left(\left(a+b x^2\right)^2\right)^{3/2}}","\frac{7 \sqrt{d x}}{16 a^2 d \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\sqrt{d x}}{4 a d \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{21 \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{64 \sqrt{2} a^{11/4} \sqrt[4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{21 \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{64 \sqrt{2} a^{11/4} \sqrt[4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{21 \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{32 \sqrt{2} a^{11/4} \sqrt[4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{21 \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{32 \sqrt{2} a^{11/4} \sqrt[4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(Sqrt[x]*(a + b*x^2)*(32*a^(7/4)*b^(1/4)*Sqrt[x] + 56*a^(3/4)*b^(1/4)*Sqrt[x]*(a + b*x^2) - 42*Sqrt[2]*(a + b*x^2)^2*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 42*Sqrt[2]*(a + b*x^2)^2*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 21*Sqrt[2]*(a + b*x^2)^2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 21*Sqrt[2]*(a + b*x^2)^2*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]))/(128*a^(11/4)*b^(1/4)*Sqrt[d*x]*((a + b*x^2)^2)^(3/2))","A",1
766,1,52,506,0.0143646,"\int \frac{1}{(d x)^{3/2} \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}} \, dx","Integrate[1/((d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)),x]","-\frac{2 x \left(a+b x^2\right)^3 \, _2F_1\left(-\frac{1}{4},3;\frac{3}{4};-\frac{b x^2}{a}\right)}{a^3 (d x)^{3/2} \left(\left(a+b x^2\right)^2\right)^{3/2}}","\frac{9}{16 a^2 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4} \left(a+b x^2\right)}-\frac{45 \sqrt[4]{b} \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{64 \sqrt{2} a^{13/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 \sqrt[4]{b} \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{64 \sqrt{2} a^{13/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 \sqrt[4]{b} \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{32 \sqrt{2} a^{13/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \sqrt[4]{b} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{32 \sqrt{2} a^{13/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \left(a+b x^2\right)}{16 a^3 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(-2*x*(a + b*x^2)^3*Hypergeometric2F1[-1/4, 3, 3/4, -((b*x^2)/a)])/(a^3*(d*x)^(3/2)*((a + b*x^2)^2)^(3/2))","C",1
767,1,54,506,0.0138454,"\int \frac{1}{(d x)^{5/2} \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}} \, dx","Integrate[1/((d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)),x]","-\frac{2 x \left(a+b x^2\right)^3 \, _2F_1\left(-\frac{3}{4},3;\frac{1}{4};-\frac{b x^2}{a}\right)}{3 a^3 (d x)^{5/2} \left(\left(a+b x^2\right)^2\right)^{3/2}}","\frac{11}{16 a^2 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4} \left(a+b x^2\right)}+\frac{77 b^{3/4} \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{64 \sqrt{2} a^{15/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{77 b^{3/4} \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{64 \sqrt{2} a^{15/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{77 b^{3/4} \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{32 \sqrt{2} a^{15/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{77 b^{3/4} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{32 \sqrt{2} a^{15/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{77 \left(a+b x^2\right)}{48 a^3 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(-2*x*(a + b*x^2)^3*Hypergeometric2F1[-3/4, 3, 1/4, -((b*x^2)/a)])/(3*a^3*(d*x)^(5/2)*((a + b*x^2)^2)^(3/2))","C",1
768,1,54,553,0.0150913,"\int \frac{1}{(d x)^{7/2} \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}} \, dx","Integrate[1/((d*x)^(7/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)),x]","-\frac{2 x \left(a+b x^2\right)^3 \, _2F_1\left(-\frac{5}{4},3;-\frac{1}{4};-\frac{b x^2}{a}\right)}{5 a^3 (d x)^{7/2} \left(\left(a+b x^2\right)^2\right)^{3/2}}","\frac{1}{4 a d (d x)^{5/2} \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{13}{16 a^2 d (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{117 b^{5/4} \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{64 \sqrt{2} a^{17/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{117 b^{5/4} \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{64 \sqrt{2} a^{17/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{117 b^{5/4} \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{32 \sqrt{2} a^{17/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{117 b^{5/4} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{32 \sqrt{2} a^{17/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{117 b \left(a+b x^2\right)}{16 a^4 d^3 \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{117 \left(a+b x^2\right)}{80 a^3 d (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(-2*x*(a + b*x^2)^3*Hypergeometric2F1[-5/4, 3, -1/4, -((b*x^2)/a)])/(5*a^3*(d*x)^(7/2)*((a + b*x^2)^2)^(3/2))","C",1
769,1,401,647,0.3042135,"\int \frac{(d x)^{23/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}} \, dx","Integrate[(d*x)^(23/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{(d x)^{23/2} \left(a+b x^2\right) \left(-765765 \sqrt{2} a^{5/4} \left(a+b x^2\right)^4 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+765765 \sqrt{2} a^{5/4} \left(a+b x^2\right)^4 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-1531530 \sqrt{2} a^{5/4} \left(a+b x^2\right)^4 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)+1531530 \sqrt{2} a^{5/4} \left(a+b x^2\right)^4 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)-10183680 a^5 \sqrt[4]{b} \sqrt{x}-32587776 a^4 b^{5/4} x^{5/2}+848640 a^4 \sqrt[4]{b} \sqrt{x} \left(a+b x^2\right)-39829504 a^3 b^{9/4} x^{9/2}+1166880 a^3 \sqrt[4]{b} \sqrt{x} \left(a+b x^2\right)^2-21446656 a^2 b^{13/4} x^{13/2}+2042040 a^2 \sqrt[4]{b} \sqrt{x} \left(a+b x^2\right)^3-3784704 a b^{17/4} x^{17/2}+180224 b^{21/4} x^{21/2}\right)}{450560 b^{25/4} x^{23/2} \left(\left(a+b x^2\right)^2\right)^{5/2}}","-\frac{7 d^3 (d x)^{17/2}}{32 b^2 \left(a+b x^2\right)^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{21/2}}{8 b \left(a+b x^2\right)^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{13923 a d^{11} \sqrt{d x} \left(a+b x^2\right)}{1024 b^6 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{13923 d^9 (d x)^{5/2} \left(a+b x^2\right)}{5120 b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{1547 d^7 (d x)^{9/2}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{119 d^5 (d x)^{13/2}}{256 b^3 \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{13923 a^{5/4} d^{23/2} \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{4096 \sqrt{2} b^{25/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{13923 a^{5/4} d^{23/2} \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{4096 \sqrt{2} b^{25/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{13923 a^{5/4} d^{23/2} \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{2048 \sqrt{2} b^{25/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{13923 a^{5/4} d^{23/2} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{2048 \sqrt{2} b^{25/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"((d*x)^(23/2)*(a + b*x^2)*(-10183680*a^5*b^(1/4)*Sqrt[x] - 32587776*a^4*b^(5/4)*x^(5/2) - 39829504*a^3*b^(9/4)*x^(9/2) - 21446656*a^2*b^(13/4)*x^(13/2) - 3784704*a*b^(17/4)*x^(17/2) + 180224*b^(21/4)*x^(21/2) + 848640*a^4*b^(1/4)*Sqrt[x]*(a + b*x^2) + 1166880*a^3*b^(1/4)*Sqrt[x]*(a + b*x^2)^2 + 2042040*a^2*b^(1/4)*Sqrt[x]*(a + b*x^2)^3 - 1531530*Sqrt[2]*a^(5/4)*(a + b*x^2)^4*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 1531530*Sqrt[2]*a^(5/4)*(a + b*x^2)^4*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 765765*Sqrt[2]*a^(5/4)*(a + b*x^2)^4*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 765765*Sqrt[2]*a^(5/4)*(a + b*x^2)^4*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]))/(450560*b^(25/4)*x^(23/2)*((a + b*x^2)^2)^(5/2))","A",1
770,1,110,600,0.0477597,"\int \frac{(d x)^{21/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}} \, dx","Integrate[(d*x)^(21/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","-\frac{2 d^9 (d x)^{3/2} \left(-1463 a^4-2717 a^3 b x^2-2223 a^2 b^2 x^4-741 a b^3 x^6+1463 \left(a+b x^2\right)^4 \, _2F_1\left(\frac{3}{4},5;\frac{7}{4};-\frac{b x^2}{a}\right)-39 b^4 x^8\right)}{117 b^5 \left(a+b x^2\right)^3 \sqrt{\left(a+b x^2\right)^2}}","-\frac{19 d^3 (d x)^{15/2}}{96 b^2 \left(a+b x^2\right)^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{19/2}}{8 b \left(a+b x^2\right)^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{7315 d^9 (d x)^{3/2} \left(a+b x^2\right)}{3072 b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{1045 d^7 (d x)^{7/2}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{95 d^5 (d x)^{11/2}}{256 b^3 \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 a^{3/4} d^{21/2} \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{4096 \sqrt{2} b^{23/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{7315 a^{3/4} d^{21/2} \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{4096 \sqrt{2} b^{23/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{7315 a^{3/4} d^{21/2} \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{2048 \sqrt{2} b^{23/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 a^{3/4} d^{21/2} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{2048 \sqrt{2} b^{23/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(-2*d^9*(d*x)^(3/2)*(-1463*a^4 - 2717*a^3*b*x^2 - 2223*a^2*b^2*x^4 - 741*a*b^3*x^6 - 39*b^4*x^8 + 1463*(a + b*x^2)^4*Hypergeometric2F1[3/4, 5, 7/4, -((b*x^2)/a)]))/(117*b^5*(a + b*x^2)^3*Sqrt[(a + b*x^2)^2])","C",1
771,1,384,600,0.2828305,"\int \frac{(d x)^{19/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}} \, dx","Integrate[(d*x)^(19/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{(d x)^{19/2} \left(a+b x^2\right) \left(10183680 a^4 \sqrt[4]{b} \sqrt{x}+32587776 a^3 b^{5/4} x^{5/2}-848640 a^3 \sqrt[4]{b} \sqrt{x} \left(a+b x^2\right)+39829504 a^2 b^{9/4} x^{9/2}-1166880 a^2 \sqrt[4]{b} \sqrt{x} \left(a+b x^2\right)^2+21446656 a b^{13/4} x^{13/2}-2042040 a \sqrt[4]{b} \sqrt{x} \left(a+b x^2\right)^3+765765 \sqrt{2} \sqrt[4]{a} \left(a+b x^2\right)^4 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-765765 \sqrt{2} \sqrt[4]{a} \left(a+b x^2\right)^4 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+1531530 \sqrt{2} \sqrt[4]{a} \left(a+b x^2\right)^4 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)-1531530 \sqrt{2} \sqrt[4]{a} \left(a+b x^2\right)^4 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)+3784704 b^{17/4} x^{17/2}\right)}{1892352 b^{21/4} x^{19/2} \left(\left(a+b x^2\right)^2\right)^{5/2}}","-\frac{17 d^3 (d x)^{13/2}}{96 b^2 \left(a+b x^2\right)^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{17/2}}{8 b \left(a+b x^2\right)^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3315 \sqrt[4]{a} d^{19/2} \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{4096 \sqrt{2} b^{21/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3315 \sqrt[4]{a} d^{19/2} \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{4096 \sqrt{2} b^{21/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3315 \sqrt[4]{a} d^{19/2} \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{2048 \sqrt{2} b^{21/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3315 \sqrt[4]{a} d^{19/2} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{2048 \sqrt{2} b^{21/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3315 d^9 \sqrt{d x} \left(a+b x^2\right)}{1024 b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{663 d^7 (d x)^{5/2}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{221 d^5 (d x)^{9/2}}{768 b^3 \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"((d*x)^(19/2)*(a + b*x^2)*(10183680*a^4*b^(1/4)*Sqrt[x] + 32587776*a^3*b^(5/4)*x^(5/2) + 39829504*a^2*b^(9/4)*x^(9/2) + 21446656*a*b^(13/4)*x^(13/2) + 3784704*b^(17/4)*x^(17/2) - 848640*a^3*b^(1/4)*Sqrt[x]*(a + b*x^2) - 1166880*a^2*b^(1/4)*Sqrt[x]*(a + b*x^2)^2 - 2042040*a*b^(1/4)*Sqrt[x]*(a + b*x^2)^3 + 1531530*Sqrt[2]*a^(1/4)*(a + b*x^2)^4*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 1531530*Sqrt[2]*a^(1/4)*(a + b*x^2)^4*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 765765*Sqrt[2]*a^(1/4)*(a + b*x^2)^4*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - 765765*Sqrt[2]*a^(1/4)*(a + b*x^2)^4*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]))/(1892352*b^(21/4)*x^(19/2)*((a + b*x^2)^2)^(5/2))","A",1
772,1,106,554,0.045074,"\int \frac{(d x)^{17/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}} \, dx","Integrate[(d*x)^(17/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{2 d^7 (d x)^{3/2} \left(77 \left(a+b x^2\right)^4 \, _2F_1\left(\frac{3}{4},5;\frac{7}{4};-\frac{b x^2}{a}\right)-a \left(77 a^3+143 a^2 b x^2+117 a b^2 x^4+39 b^3 x^6\right)\right)}{39 a b^4 \left(a+b x^2\right)^3 \sqrt{\left(a+b x^2\right)^2}}","-\frac{5 d^3 (d x)^{11/2}}{32 b^2 \left(a+b x^2\right)^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{15/2}}{8 b \left(a+b x^2\right)^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1155 d^{17/2} \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{4096 \sqrt{2} \sqrt[4]{a} b^{19/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{1155 d^{17/2} \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{4096 \sqrt{2} \sqrt[4]{a} b^{19/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{1155 d^{17/2} \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{2048 \sqrt{2} \sqrt[4]{a} b^{19/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1155 d^{17/2} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{2048 \sqrt{2} \sqrt[4]{a} b^{19/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{385 d^7 (d x)^{3/2}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{55 d^5 (d x)^{7/2}}{256 b^3 \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(2*d^7*(d*x)^(3/2)*(-(a*(77*a^3 + 143*a^2*b*x^2 + 117*a*b^2*x^4 + 39*b^3*x^6)) + 77*(a + b*x^2)^4*Hypergeometric2F1[3/4, 5, 7/4, -((b*x^2)/a)]))/(39*a*b^4*(a + b*x^2)^3*Sqrt[(a + b*x^2)^2])","C",1
773,1,366,554,0.2638399,"\int \frac{(d x)^{15/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}} \, dx","Integrate[(d*x)^(15/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{(d x)^{15/2} \left(a+b x^2\right) \left(-\frac{45045 \sqrt{2} \left(a+b x^2\right)^4 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{3/4}}+\frac{45045 \sqrt{2} \left(a+b x^2\right)^4 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{3/4}}-\frac{90090 \sqrt{2} \left(a+b x^2\right)^4 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{a^{3/4}}+\frac{90090 \sqrt{2} \left(a+b x^2\right)^4 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{a^{3/4}}-599040 a^3 \sqrt[4]{b} \sqrt{x}-1916928 a^2 b^{5/4} x^{5/2}+49920 a^2 \sqrt[4]{b} \sqrt{x} \left(a+b x^2\right)-2342912 a b^{9/4} x^{9/2}+120120 \sqrt[4]{b} \sqrt{x} \left(a+b x^2\right)^3+68640 a \sqrt[4]{b} \sqrt{x} \left(a+b x^2\right)^2-1261568 b^{13/4} x^{13/2}\right)}{1892352 b^{17/4} x^{15/2} \left(\left(a+b x^2\right)^2\right)^{5/2}}","-\frac{13 d^3 (d x)^{9/2}}{96 b^2 \left(a+b x^2\right)^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{13/2}}{8 b \left(a+b x^2\right)^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{195 d^7 \sqrt{d x}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{39 d^5 (d x)^{5/2}}{256 b^3 \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{195 d^{15/2} \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{4096 \sqrt{2} a^{3/4} b^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{195 d^{15/2} \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{4096 \sqrt{2} a^{3/4} b^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{195 d^{15/2} \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{2048 \sqrt{2} a^{3/4} b^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{195 d^{15/2} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{2048 \sqrt{2} a^{3/4} b^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"((d*x)^(15/2)*(a + b*x^2)*(-599040*a^3*b^(1/4)*Sqrt[x] - 1916928*a^2*b^(5/4)*x^(5/2) - 2342912*a*b^(9/4)*x^(9/2) - 1261568*b^(13/4)*x^(13/2) + 49920*a^2*b^(1/4)*Sqrt[x]*(a + b*x^2) + 68640*a*b^(1/4)*Sqrt[x]*(a + b*x^2)^2 + 120120*b^(1/4)*Sqrt[x]*(a + b*x^2)^3 - (90090*Sqrt[2]*(a + b*x^2)^4*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(3/4) + (90090*Sqrt[2]*(a + b*x^2)^4*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(3/4) - (45045*Sqrt[2]*(a + b*x^2)^4*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(3/4) + (45045*Sqrt[2]*(a + b*x^2)^4*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(3/4)))/(1892352*b^(17/4)*x^(15/2)*((a + b*x^2)^2)^(5/2))","A",1
774,1,97,557,0.0375412,"\int \frac{(d x)^{13/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}} \, dx","Integrate[(d*x)^(13/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{2 d^5 (d x)^{3/2} \left(77 \left(a+b x^2\right)^4 \, _2F_1\left(\frac{3}{4},5;\frac{7}{4};-\frac{b x^2}{a}\right)-a^2 \left(77 a^2+143 a b x^2+117 b^2 x^4\right)\right)}{585 a^2 b^3 \left(a+b x^2\right)^3 \sqrt{\left(a+b x^2\right)^2}}","-\frac{11 d^3 (d x)^{7/2}}{96 b^2 \left(a+b x^2\right)^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{11/2}}{8 b \left(a+b x^2\right)^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{77 d^5 (d x)^{3/2}}{1024 a b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{77 d^5 (d x)^{3/2}}{768 b^3 \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{77 d^{13/2} \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{4096 \sqrt{2} a^{5/4} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{77 d^{13/2} \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{4096 \sqrt{2} a^{5/4} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{77 d^{13/2} \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{2048 \sqrt{2} a^{5/4} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{77 d^{13/2} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{2048 \sqrt{2} a^{5/4} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(2*d^5*(d*x)^(3/2)*(-(a^2*(77*a^2 + 143*a*b*x^2 + 117*b^2*x^4)) + 77*(a + b*x^2)^4*Hypergeometric2F1[3/4, 5, 7/4, -((b*x^2)/a)]))/(585*a^2*b^3*(a + b*x^2)^3*Sqrt[(a + b*x^2)^2])","C",1
775,1,352,557,0.2876105,"\int \frac{(d x)^{11/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}} \, dx","Integrate[(d*x)^(11/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{d (d x)^{9/2} \left(a+b x^2\right) \left(-\frac{3465 \sqrt{2} \left(a+b x^2\right)^4 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{7/4}}+\frac{3465 \sqrt{2} \left(a+b x^2\right)^4 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{7/4}}-\frac{6930 \sqrt{2} \left(a+b x^2\right)^4 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{a^{7/4}}+\frac{6930 \sqrt{2} \left(a+b x^2\right)^4 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{a^{7/4}}-46080 a^2 \sqrt[4]{b} \sqrt{x}-147456 a b^{5/4} x^{5/2}+\frac{9240 \sqrt[4]{b} \sqrt{x} \left(a+b x^2\right)^3}{a}+5280 \sqrt[4]{b} \sqrt{x} \left(a+b x^2\right)^2+3840 a \sqrt[4]{b} \sqrt{x} \left(a+b x^2\right)-180224 b^{9/4} x^{9/2}\right)}{630784 b^{13/4} x^{9/2} \left(\left(a+b x^2\right)^2\right)^{5/2}}","-\frac{3 d^3 (d x)^{5/2}}{32 b^2 \left(a+b x^2\right)^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{9/2}}{8 b \left(a+b x^2\right)^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{15 d^5 \sqrt{d x}}{1024 a b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{15 d^5 \sqrt{d x}}{256 b^3 \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 d^{11/2} \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{4096 \sqrt{2} a^{7/4} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 d^{11/2} \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{4096 \sqrt{2} a^{7/4} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 d^{11/2} \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{2048 \sqrt{2} a^{7/4} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 d^{11/2} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{2048 \sqrt{2} a^{7/4} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(d*(d*x)^(9/2)*(a + b*x^2)*(-46080*a^2*b^(1/4)*Sqrt[x] - 147456*a*b^(5/4)*x^(5/2) - 180224*b^(9/4)*x^(9/2) + 3840*a*b^(1/4)*Sqrt[x]*(a + b*x^2) + 5280*b^(1/4)*Sqrt[x]*(a + b*x^2)^2 + (9240*b^(1/4)*Sqrt[x]*(a + b*x^2)^3)/a - (6930*Sqrt[2]*(a + b*x^2)^4*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(7/4) + (6930*Sqrt[2]*(a + b*x^2)^4*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(7/4) - (3465*Sqrt[2]*(a + b*x^2)^4*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(7/4) + (3465*Sqrt[2]*(a + b*x^2)^4*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(7/4)))/(630784*b^(13/4)*x^(9/2)*((a + b*x^2)^2)^(5/2))","A",1
776,1,86,560,0.0380018,"\int \frac{(d x)^{9/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}} \, dx","Integrate[(d*x)^(9/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{2 d^3 (d x)^{3/2} \left(7 \left(a+b x^2\right)^4 \, _2F_1\left(\frac{3}{4},5;\frac{7}{4};-\frac{b x^2}{a}\right)-a^3 \left(7 a+13 b x^2\right)\right)}{117 a^3 b^2 \left(a+b x^2\right)^3 \sqrt{\left(a+b x^2\right)^2}}","\frac{35 d^3 (d x)^{3/2}}{1024 a^2 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{7 d^3 (d x)^{3/2}}{256 a b^2 \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7 d^3 (d x)^{3/2}}{96 b^2 \left(a+b x^2\right)^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{7/2}}{8 b \left(a+b x^2\right)^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{35 d^{9/2} \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{4096 \sqrt{2} a^{9/4} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{35 d^{9/2} \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{4096 \sqrt{2} a^{9/4} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{35 d^{9/2} \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{2048 \sqrt{2} a^{9/4} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{35 d^{9/2} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{2048 \sqrt{2} a^{9/4} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(2*d^3*(d*x)^(3/2)*(-(a^3*(7*a + 13*b*x^2)) + 7*(a + b*x^2)^4*Hypergeometric2F1[3/4, 5, 7/4, -((b*x^2)/a)]))/(117*a^3*b^2*(a + b*x^2)^3*Sqrt[(a + b*x^2)^2])","C",1
777,1,341,560,0.3052996,"\int \frac{(d x)^{7/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}} \, dx","Integrate[(d*x)^(7/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{(d x)^{7/2} \left(a+b x^2\right) \left(-49152 a^{11/4} b^{5/4} x^{5/2}+3080 a^{3/4} \sqrt[4]{b} \sqrt{x} \left(a+b x^2\right)^3+1760 a^{7/4} \sqrt[4]{b} \sqrt{x} \left(a+b x^2\right)^2+1280 a^{11/4} \sqrt[4]{b} \sqrt{x} \left(a+b x^2\right)-15360 a^{15/4} \sqrt[4]{b} \sqrt{x}-1155 \sqrt{2} \left(a+b x^2\right)^4 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+1155 \sqrt{2} \left(a+b x^2\right)^4 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-2310 \sqrt{2} \left(a+b x^2\right)^4 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)+2310 \sqrt{2} \left(a+b x^2\right)^4 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)\right)}{270336 a^{11/4} b^{9/4} x^{7/2} \left(\left(a+b x^2\right)^2\right)^{5/2}}","\frac{35 d^3 \sqrt{d x}}{3072 a^2 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 d^3 \sqrt{d x}}{768 a b^2 \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 d^3 \sqrt{d x}}{96 b^2 \left(a+b x^2\right)^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{5/2}}{8 b \left(a+b x^2\right)^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{35 d^{7/2} \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{4096 \sqrt{2} a^{11/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{35 d^{7/2} \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{4096 \sqrt{2} a^{11/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{35 d^{7/2} \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{2048 \sqrt{2} a^{11/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{35 d^{7/2} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{2048 \sqrt{2} a^{11/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"((d*x)^(7/2)*(a + b*x^2)*(-15360*a^(15/4)*b^(1/4)*Sqrt[x] - 49152*a^(11/4)*b^(5/4)*x^(5/2) + 1280*a^(11/4)*b^(1/4)*Sqrt[x]*(a + b*x^2) + 1760*a^(7/4)*b^(1/4)*Sqrt[x]*(a + b*x^2)^2 + 3080*a^(3/4)*b^(1/4)*Sqrt[x]*(a + b*x^2)^3 - 2310*Sqrt[2]*(a + b*x^2)^4*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 2310*Sqrt[2]*(a + b*x^2)^4*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 1155*Sqrt[2]*(a + b*x^2)^4*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 1155*Sqrt[2]*(a + b*x^2)^4*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]))/(270336*a^(11/4)*b^(9/4)*x^(7/2)*((a + b*x^2)^2)^(5/2))","A",1
778,1,73,557,0.0317384,"\int \frac{(d x)^{5/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}} \, dx","Integrate[(d*x)^(5/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{2 d (d x)^{3/2} \left(\left(a+b x^2\right)^4 \, _2F_1\left(\frac{3}{4},5;\frac{7}{4};-\frac{b x^2}{a}\right)-a^4\right)}{13 a^4 b \left(a+b x^2\right)^3 \sqrt{\left(a+b x^2\right)^2}}","\frac{9 d (d x)^{3/2}}{256 a^2 b \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{d (d x)^{3/2}}{32 a b \left(a+b x^2\right)^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{3/2}}{8 b \left(a+b x^2\right)^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 d^{5/2} \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{4096 \sqrt{2} a^{13/4} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 d^{5/2} \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{4096 \sqrt{2} a^{13/4} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 d^{5/2} \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{2048 \sqrt{2} a^{13/4} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 d^{5/2} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{2048 \sqrt{2} a^{13/4} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 d (d x)^{3/2}}{1024 a^3 b \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(2*d*(d*x)^(3/2)*(-a^4 + (a + b*x^2)^4*Hypergeometric2F1[3/4, 5, 7/4, -((b*x^2)/a)]))/(13*a^4*b*(a + b*x^2)^3*Sqrt[(a + b*x^2)^2])","C",1
779,1,324,557,0.2956678,"\int \frac{(d x)^{3/2}}{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}} \, dx","Integrate[(d*x)^(3/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{(d x)^{3/2} \left(a+b x^2\right) \left(616 a^{3/4} \sqrt[4]{b} \sqrt{x} \left(a+b x^2\right)^3+352 a^{7/4} \sqrt[4]{b} \sqrt{x} \left(a+b x^2\right)^2+256 a^{11/4} \sqrt[4]{b} \sqrt{x} \left(a+b x^2\right)-3072 a^{15/4} \sqrt[4]{b} \sqrt{x}-231 \sqrt{2} \left(a+b x^2\right)^4 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+231 \sqrt{2} \left(a+b x^2\right)^4 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-462 \sqrt{2} \left(a+b x^2\right)^4 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)+462 \sqrt{2} \left(a+b x^2\right)^4 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)\right)}{24576 a^{15/4} b^{5/4} x^{3/2} \left(\left(a+b x^2\right)^2\right)^{5/2}}","\frac{11 d \sqrt{d x}}{768 a^2 b \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{d \sqrt{d x}}{96 a b \left(a+b x^2\right)^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d \sqrt{d x}}{8 b \left(a+b x^2\right)^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{77 d^{3/2} \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{4096 \sqrt{2} a^{15/4} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{77 d^{3/2} \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{4096 \sqrt{2} a^{15/4} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{77 d^{3/2} \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{2048 \sqrt{2} a^{15/4} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{77 d^{3/2} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{2048 \sqrt{2} a^{15/4} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{77 d \sqrt{d x}}{3072 a^3 b \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"((d*x)^(3/2)*(a + b*x^2)*(-3072*a^(15/4)*b^(1/4)*Sqrt[x] + 256*a^(11/4)*b^(1/4)*Sqrt[x]*(a + b*x^2) + 352*a^(7/4)*b^(1/4)*Sqrt[x]*(a + b*x^2)^2 + 616*a^(3/4)*b^(1/4)*Sqrt[x]*(a + b*x^2)^3 - 462*Sqrt[2]*(a + b*x^2)^4*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 462*Sqrt[2]*(a + b*x^2)^4*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 231*Sqrt[2]*(a + b*x^2)^4*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 231*Sqrt[2]*(a + b*x^2)^4*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]))/(24576*a^(15/4)*b^(5/4)*x^(3/2)*((a + b*x^2)^2)^(5/2))","A",1
780,1,54,556,0.0128675,"\int \frac{\sqrt{d x}}{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}} \, dx","Integrate[Sqrt[d*x]/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{2 x \sqrt{d x} \left(a+b x^2\right)^5 \, _2F_1\left(\frac{3}{4},5;\frac{7}{4};-\frac{b x^2}{a}\right)}{3 a^5 \left(\left(a+b x^2\right)^2\right)^{5/2}}","\frac{13 (d x)^{3/2}}{96 a^2 d \left(a+b x^2\right)^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{(d x)^{3/2}}{8 a d \left(a+b x^2\right)^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{195 \sqrt{d} \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{4096 \sqrt{2} a^{17/4} b^{3/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{195 \sqrt{d} \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{4096 \sqrt{2} a^{17/4} b^{3/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{195 \sqrt{d} \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{2048 \sqrt{2} a^{17/4} b^{3/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{195 \sqrt{d} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{2048 \sqrt{2} a^{17/4} b^{3/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{195 (d x)^{3/2}}{1024 a^4 d \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{39 (d x)^{3/2}}{256 a^3 d \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(2*x*Sqrt[d*x]*(a + b*x^2)^5*Hypergeometric2F1[3/4, 5, 7/4, -((b*x^2)/a)])/(3*a^5*((a + b*x^2)^2)^(5/2))","C",1
781,1,319,556,0.1344863,"\int \frac{1}{\sqrt{d x} \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}} \, dx","Integrate[1/(Sqrt[d*x]*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]","\frac{\sqrt{x} \left(a+b x^2\right) \left(3080 a^{3/4} \sqrt{x} \left(a+b x^2\right)^3+1760 a^{7/4} \sqrt{x} \left(a+b x^2\right)^2+1280 a^{11/4} \sqrt{x} \left(a+b x^2\right)+1024 a^{15/4} \sqrt{x}-\frac{1155 \sqrt{2} \left(a+b x^2\right)^4 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{\sqrt[4]{b}}+\frac{1155 \sqrt{2} \left(a+b x^2\right)^4 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{\sqrt[4]{b}}-\frac{2310 \sqrt{2} \left(a+b x^2\right)^4 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt[4]{b}}+\frac{2310 \sqrt{2} \left(a+b x^2\right)^4 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt[4]{b}}\right)}{8192 a^{19/4} \sqrt{d x} \left(\left(a+b x^2\right)^2\right)^{5/2}}","\frac{5 \sqrt{d x}}{32 a^2 d \left(a+b x^2\right)^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\sqrt{d x}}{8 a d \left(a+b x^2\right)^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{1155 \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{4096 \sqrt{2} a^{19/4} \sqrt[4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1155 \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{4096 \sqrt{2} a^{19/4} \sqrt[4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{1155 \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{2048 \sqrt{2} a^{19/4} \sqrt[4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1155 \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{2048 \sqrt{2} a^{19/4} \sqrt[4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{385 \sqrt{d x}}{1024 a^4 d \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{55 \sqrt{d x}}{256 a^3 d \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(Sqrt[x]*(a + b*x^2)*(1024*a^(15/4)*Sqrt[x] + 1280*a^(11/4)*Sqrt[x]*(a + b*x^2) + 1760*a^(7/4)*Sqrt[x]*(a + b*x^2)^2 + 3080*a^(3/4)*Sqrt[x]*(a + b*x^2)^3 - (2310*Sqrt[2]*(a + b*x^2)^4*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(1/4) + (2310*Sqrt[2]*(a + b*x^2)^4*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(1/4) - (1155*Sqrt[2]*(a + b*x^2)^4*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(1/4) + (1155*Sqrt[2]*(a + b*x^2)^4*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(1/4)))/(8192*a^(19/4)*Sqrt[d*x]*((a + b*x^2)^2)^(5/2))","A",1
782,1,52,602,0.0143866,"\int \frac{1}{(d x)^{3/2} \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}} \, dx","Integrate[1/((d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]","-\frac{2 x \left(a+b x^2\right)^5 \, _2F_1\left(-\frac{1}{4},5;\frac{3}{4};-\frac{b x^2}{a}\right)}{a^5 (d x)^{3/2} \left(\left(a+b x^2\right)^2\right)^{5/2}}","\frac{17}{96 a^2 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4} \left(a+b x^2\right)^2}+\frac{1}{8 a d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4} \left(a+b x^2\right)^3}-\frac{3315 \sqrt[4]{b} \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{4096 \sqrt{2} a^{21/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3315 \sqrt[4]{b} \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{4096 \sqrt{2} a^{21/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3315 \sqrt[4]{b} \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{2048 \sqrt{2} a^{21/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3315 \sqrt[4]{b} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{2048 \sqrt{2} a^{21/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3315 \left(a+b x^2\right)}{1024 a^5 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{663}{1024 a^4 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{221}{768 a^3 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4} \left(a+b x^2\right)}",1,"(-2*x*(a + b*x^2)^5*Hypergeometric2F1[-1/4, 5, 3/4, -((b*x^2)/a)])/(a^5*(d*x)^(3/2)*((a + b*x^2)^2)^(5/2))","C",1
783,1,54,602,0.0179868,"\int \frac{1}{(d x)^{5/2} \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}} \, dx","Integrate[1/((d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]","-\frac{2 x \left(a+b x^2\right)^5 \, _2F_1\left(-\frac{3}{4},5;\frac{1}{4};-\frac{b x^2}{a}\right)}{3 a^5 (d x)^{5/2} \left(\left(a+b x^2\right)^2\right)^{5/2}}","\frac{19}{96 a^2 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4} \left(a+b x^2\right)^2}+\frac{1}{8 a d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4} \left(a+b x^2\right)^3}+\frac{7315 b^{3/4} \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{4096 \sqrt{2} a^{23/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 b^{3/4} \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{4096 \sqrt{2} a^{23/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{7315 b^{3/4} \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{2048 \sqrt{2} a^{23/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 b^{3/4} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{2048 \sqrt{2} a^{23/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 \left(a+b x^2\right)}{3072 a^5 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1045}{1024 a^4 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{95}{256 a^3 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4} \left(a+b x^2\right)}",1,"(-2*x*(a + b*x^2)^5*Hypergeometric2F1[-3/4, 5, 1/4, -((b*x^2)/a)])/(3*a^5*(d*x)^(5/2)*((a + b*x^2)^2)^(5/2))","C",1
784,1,54,649,0.0168312,"\int \frac{1}{(d x)^{7/2} \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}} \, dx","Integrate[1/((d*x)^(7/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]","-\frac{2 x \left(a+b x^2\right)^5 \, _2F_1\left(-\frac{5}{4},5;-\frac{1}{4};-\frac{b x^2}{a}\right)}{5 a^5 (d x)^{7/2} \left(\left(a+b x^2\right)^2\right)^{5/2}}","\frac{7}{32 a^2 d (d x)^{5/2} \left(a+b x^2\right)^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{8 a d (d x)^{5/2} \left(a+b x^2\right)^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{13923 b^{5/4} \left(a+b x^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{4096 \sqrt{2} a^{25/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{13923 b^{5/4} \left(a+b x^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right)}{4096 \sqrt{2} a^{25/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{13923 b^{5/4} \left(a+b x^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}\right)}{2048 \sqrt{2} a^{25/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{13923 b^{5/4} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{d x}}{\sqrt[4]{a} \sqrt{d}}+1\right)}{2048 \sqrt{2} a^{25/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{13923 b \left(a+b x^2\right)}{1024 a^6 d^3 \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{13923 \left(a+b x^2\right)}{5120 a^5 d (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1547}{1024 a^4 d (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{119}{256 a^3 d (d x)^{5/2} \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(-2*x*(a + b*x^2)^5*Hypergeometric2F1[-5/4, 5, -1/4, -((b*x^2)/a)])/(5*a^5*(d*x)^(7/2)*((a + b*x^2)^2)^(5/2))","C",1
785,1,105,150,0.068831,"\int (d x)^m \left(a^2+2 a b x^2+b^2 x^4\right)^3 \, dx","Integrate[(d*x)^m*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","x (d x)^m \left(\frac{a^6}{m+1}+\frac{6 a^5 b x^2}{m+3}+\frac{15 a^4 b^2 x^4}{m+5}+\frac{20 a^3 b^3 x^6}{m+7}+\frac{15 a^2 b^4 x^8}{m+9}+\frac{6 a b^5 x^{10}}{m+11}+\frac{b^6 x^{12}}{m+13}\right)","\frac{a^6 (d x)^{m+1}}{d (m+1)}+\frac{6 a^5 b (d x)^{m+3}}{d^3 (m+3)}+\frac{15 a^4 b^2 (d x)^{m+5}}{d^5 (m+5)}+\frac{20 a^3 b^3 (d x)^{m+7}}{d^7 (m+7)}+\frac{15 a^2 b^4 (d x)^{m+9}}{d^9 (m+9)}+\frac{6 a b^5 (d x)^{m+11}}{d^{11} (m+11)}+\frac{b^6 (d x)^{m+13}}{d^{13} (m+13)}",1,"x*(d*x)^m*(a^6/(1 + m) + (6*a^5*b*x^2)/(3 + m) + (15*a^4*b^2*x^4)/(5 + m) + (20*a^3*b^3*x^6)/(7 + m) + (15*a^2*b^4*x^8)/(9 + m) + (6*a*b^5*x^10)/(11 + m) + (b^6*x^12)/(13 + m))","A",1
786,1,73,104,0.0353212,"\int (d x)^m \left(a^2+2 a b x^2+b^2 x^4\right)^2 \, dx","Integrate[(d*x)^m*(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","x (d x)^m \left(\frac{a^4}{m+1}+\frac{4 a^3 b x^2}{m+3}+\frac{6 a^2 b^2 x^4}{m+5}+\frac{4 a b^3 x^6}{m+7}+\frac{b^4 x^8}{m+9}\right)","\frac{a^4 (d x)^{m+1}}{d (m+1)}+\frac{4 a^3 b (d x)^{m+3}}{d^3 (m+3)}+\frac{6 a^2 b^2 (d x)^{m+5}}{d^5 (m+5)}+\frac{4 a b^3 (d x)^{m+7}}{d^7 (m+7)}+\frac{b^4 (d x)^{m+9}}{d^9 (m+9)}",1,"x*(d*x)^m*(a^4/(1 + m) + (4*a^3*b*x^2)/(3 + m) + (6*a^2*b^2*x^4)/(5 + m) + (4*a*b^3*x^6)/(7 + m) + (b^4*x^8)/(9 + m))","A",1
787,1,41,58,0.0325069,"\int (d x)^m \left(a^2+2 a b x^2+b^2 x^4\right) \, dx","Integrate[(d*x)^m*(a^2 + 2*a*b*x^2 + b^2*x^4),x]","x (d x)^m \left(\frac{a^2}{m+1}+\frac{2 a b x^2}{m+3}+\frac{b^2 x^4}{m+5}\right)","\frac{a^2 (d x)^{m+1}}{d (m+1)}+\frac{2 a b (d x)^{m+3}}{d^3 (m+3)}+\frac{b^2 (d x)^{m+5}}{d^5 (m+5)}",1,"x*(d*x)^m*(a^2/(1 + m) + (2*a*b*x^2)/(3 + m) + (b^2*x^4)/(5 + m))","A",1
788,1,42,44,0.0087888,"\int \frac{(d x)^m}{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[(d*x)^m/(a^2 + 2*a*b*x^2 + b^2*x^4),x]","\frac{x (d x)^m \, _2F_1\left(2,\frac{m+1}{2};\frac{m+1}{2}+1;-\frac{b x^2}{a}\right)}{a^2 (m+1)}","\frac{(d x)^{m+1} \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a^2 d (m+1)}",1,"(x*(d*x)^m*Hypergeometric2F1[2, (1 + m)/2, 1 + (1 + m)/2, -((b*x^2)/a)])/(a^2*(1 + m))","A",1
789,1,42,44,0.0087071,"\int \frac{(d x)^m}{\left(a^2+2 a b x^2+b^2 x^4\right)^2} \, dx","Integrate[(d*x)^m/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]","\frac{x (d x)^m \, _2F_1\left(4,\frac{m+1}{2};\frac{m+1}{2}+1;-\frac{b x^2}{a}\right)}{a^4 (m+1)}","\frac{(d x)^{m+1} \, _2F_1\left(4,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a^4 d (m+1)}",1,"(x*(d*x)^m*Hypergeometric2F1[4, (1 + m)/2, 1 + (1 + m)/2, -((b*x^2)/a)])/(a^4*(1 + m))","A",1
790,1,42,44,0.0093825,"\int \frac{(d x)^m}{\left(a^2+2 a b x^2+b^2 x^4\right)^3} \, dx","Integrate[(d*x)^m/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]","\frac{x (d x)^m \, _2F_1\left(6,\frac{m+1}{2};\frac{m+1}{2}+1;-\frac{b x^2}{a}\right)}{a^6 (m+1)}","\frac{(d x)^{m+1} \, _2F_1\left(6,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a^6 d (m+1)}",1,"(x*(d*x)^m*Hypergeometric2F1[6, (1 + m)/2, 1 + (1 + m)/2, -((b*x^2)/a)])/(a^6*(1 + m))","A",1
791,1,111,313,0.0916257,"\int (d x)^m \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2} \, dx","Integrate[(d*x)^m*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{x \left(\left(a+b x^2\right)^2\right)^{5/2} (d x)^m \left(\frac{a^5}{m+1}+\frac{5 a^4 b x^2}{m+3}+\frac{10 a^3 b^2 x^4}{m+5}+\frac{10 a^2 b^3 x^6}{m+7}+\frac{5 a b^4 x^8}{m+9}+\frac{b^5 x^{10}}{m+11}\right)}{\left(a+b x^2\right)^5}","\frac{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+11}}{d^{11} (m+11) \left(a+b x^2\right)}+\frac{5 a b^4 \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+9}}{d^9 (m+9) \left(a+b x^2\right)}+\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+7}}{d^7 (m+7) \left(a+b x^2\right)}+\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+1}}{d (m+1) \left(a+b x^2\right)}+\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+3}}{d^3 (m+3) \left(a+b x^2\right)}+\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+5}}{d^5 (m+5) \left(a+b x^2\right)}",1,"(x*(d*x)^m*((a + b*x^2)^2)^(5/2)*(a^5/(1 + m) + (5*a^4*b*x^2)/(3 + m) + (10*a^3*b^2*x^4)/(5 + m) + (10*a^2*b^3*x^6)/(7 + m) + (5*a*b^4*x^8)/(9 + m) + (b^5*x^10)/(11 + m)))/(a + b*x^2)^5","A",1
792,1,131,205,0.0700892,"\int (d x)^m \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2} \, dx","Integrate[(d*x)^m*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","\frac{x \sqrt{\left(a+b x^2\right)^2} (d x)^m \left(a^3 \left(m^3+15 m^2+71 m+105\right)+3 a^2 b \left(m^3+13 m^2+47 m+35\right) x^2+3 a b^2 \left(m^3+11 m^2+31 m+21\right) x^4+b^3 \left(m^3+9 m^2+23 m+15\right) x^6\right)}{(m+1) (m+3) (m+5) (m+7) \left(a+b x^2\right)}","\frac{3 a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+5}}{d^5 (m+5) \left(a+b x^2\right)}+\frac{3 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+3}}{d^3 (m+3) \left(a+b x^2\right)}+\frac{b^3 \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+7}}{d^7 (m+7) \left(a+b x^2\right)}+\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+1}}{d (m+1) \left(a+b x^2\right)}",1,"(x*(d*x)^m*Sqrt[(a + b*x^2)^2]*(a^3*(105 + 71*m + 15*m^2 + m^3) + 3*a^2*b*(35 + 47*m + 13*m^2 + m^3)*x^2 + 3*a*b^2*(21 + 31*m + 11*m^2 + m^3)*x^4 + b^3*(15 + 23*m + 9*m^2 + m^3)*x^6))/((1 + m)*(3 + m)*(5 + m)*(7 + m)*(a + b*x^2))","A",1
793,1,53,97,0.0237259,"\int (d x)^m \sqrt{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[(d*x)^m*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]","\frac{x \sqrt{\left(a+b x^2\right)^2} (d x)^m \left(a (m+3)+b (m+1) x^2\right)}{(m+1) (m+3) \left(a+b x^2\right)}","\frac{b \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+3}}{d^3 (m+3) \left(a+b x^2\right)}+\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+1}}{d (m+1) \left(a+b x^2\right)}",1,"(x*(d*x)^m*Sqrt[(a + b*x^2)^2]*(a*(3 + m) + b*(1 + m)*x^2))/((1 + m)*(3 + m)*(a + b*x^2))","A",1
794,1,62,73,0.0198861,"\int \frac{(d x)^m}{\sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx","Integrate[(d*x)^m/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]","\frac{x \left(a+b x^2\right) (d x)^m \, _2F_1\left(1,\frac{m+1}{2};\frac{m+1}{2}+1;-\frac{b x^2}{a}\right)}{a (m+1) \sqrt{\left(a+b x^2\right)^2}}","\frac{\left(a+b x^2\right) (d x)^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a d (m+1) \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(x*(d*x)^m*(a + b*x^2)*Hypergeometric2F1[1, (1 + m)/2, 1 + (1 + m)/2, -((b*x^2)/a)])/(a*(1 + m)*Sqrt[(a + b*x^2)^2])","A",1
795,1,60,73,0.0217837,"\int \frac{(d x)^m}{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}} \, dx","Integrate[(d*x)^m/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","\frac{x \left(a+b x^2\right) (d x)^m \, _2F_1\left(3,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a^3 (m+1) \sqrt{\left(a+b x^2\right)^2}}","\frac{\left(a+b x^2\right) (d x)^{m+1} \, _2F_1\left(3,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a^3 d (m+1) \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(x*(d*x)^m*(a + b*x^2)*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(a^3*(1 + m)*Sqrt[(a + b*x^2)^2])","A",1
796,1,60,73,0.0194927,"\int \frac{(d x)^m}{\left(a^2+2 a b x^2+b^2 x^4\right)^{5/2}} \, dx","Integrate[(d*x)^m/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{x \left(a+b x^2\right) (d x)^m \, _2F_1\left(5,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a^5 (m+1) \sqrt{\left(a+b x^2\right)^2}}","\frac{\left(a+b x^2\right) (d x)^{m+1} \, _2F_1\left(5,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a^5 d (m+1) \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(x*(d*x)^m*(a + b*x^2)*Hypergeometric2F1[5, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(a^5*(1 + m)*Sqrt[(a + b*x^2)^2])","A",1
797,1,66,74,0.0187073,"\int (d x)^m \left(a^2+2 a b x^2+b^2 x^4\right)^p \, dx","Integrate[(d*x)^m*(a^2 + 2*a*b*x^2 + b^2*x^4)^p,x]","\frac{x (d x)^m \left(\left(a+b x^2\right)^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-2 p} \, _2F_1\left(\frac{m+1}{2},-2 p;\frac{m+1}{2}+1;-\frac{b x^2}{a}\right)}{m+1}","\frac{\left(a+b x^2\right) (d x)^{m+1} \left(a^2+2 a b x^2+b^2 x^4\right)^p \, _2F_1\left(1,\frac{1}{2} (m+4 p+3);\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a d (m+1)}",1,"(x*(d*x)^m*((a + b*x^2)^2)^p*Hypergeometric2F1[(1 + m)/2, -2*p, 1 + (1 + m)/2, -((b*x^2)/a)])/((1 + m)*(1 + (b*x^2)/a)^(2*p))","A",1
798,1,110,174,0.0588677,"\int x^7 \left(a^2+2 a b x^2+b^2 x^4\right)^p \, dx","Integrate[x^7*(a^2 + 2*a*b*x^2 + b^2*x^4)^p,x]","\frac{\left(a+b x^2\right) \left(\left(a+b x^2\right)^2\right)^p \left(-3 a^3+3 a^2 b (2 p+1) x^2-3 a b^2 \left(2 p^2+3 p+1\right) x^4+b^3 \left(4 p^3+12 p^2+11 p+3\right) x^6\right)}{4 b^4 (p+1) (p+2) (2 p+1) (2 p+3)}","\frac{\left(a+b x^2\right)^4 \left(a^2+2 a b x^2+b^2 x^4\right)^p}{4 b^4 (p+2)}-\frac{3 a \left(a+b x^2\right)^3 \left(a^2+2 a b x^2+b^2 x^4\right)^p}{2 b^4 (2 p+3)}+\frac{3 a^2 \left(a+b x^2\right)^2 \left(a^2+2 a b x^2+b^2 x^4\right)^p}{4 b^4 (p+1)}-\frac{a^3 \left(a+b x^2\right) \left(a^2+2 a b x^2+b^2 x^4\right)^p}{2 b^4 (2 p+1)}",1,"((a + b*x^2)*((a + b*x^2)^2)^p*(-3*a^3 + 3*a^2*b*(1 + 2*p)*x^2 - 3*a*b^2*(1 + 3*p + 2*p^2)*x^4 + b^3*(3 + 11*p + 12*p^2 + 4*p^3)*x^6))/(4*b^4*(1 + p)*(2 + p)*(1 + 2*p)*(3 + 2*p))","A",1
799,1,77,130,0.0343056,"\int x^5 \left(a^2+2 a b x^2+b^2 x^4\right)^p \, dx","Integrate[x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)^p,x]","\frac{\left(a+b x^2\right) \left(\left(a+b x^2\right)^2\right)^p \left(a^2-a b (2 p+1) x^2+b^2 \left(2 p^2+3 p+1\right) x^4\right)}{2 b^3 (p+1) (2 p+1) (2 p+3)}","\frac{\left(a+b x^2\right)^3 \left(a^2+2 a b x^2+b^2 x^4\right)^p}{2 b^3 (2 p+3)}-\frac{a \left(a+b x^2\right)^2 \left(a^2+2 a b x^2+b^2 x^4\right)^p}{2 b^3 (p+1)}+\frac{a^2 \left(a+b x^2\right) \left(a^2+2 a b x^2+b^2 x^4\right)^p}{2 b^3 (2 p+1)}",1,"((a + b*x^2)*((a + b*x^2)^2)^p*(a^2 - a*b*(1 + 2*p)*x^2 + b^2*(1 + 3*p + 2*p^2)*x^4))/(2*b^3*(1 + p)*(1 + 2*p)*(3 + 2*p))","A",1
800,1,51,84,0.0212921,"\int x^3 \left(a^2+2 a b x^2+b^2 x^4\right)^p \, dx","Integrate[x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^p,x]","\frac{\left(a+b x^2\right) \left(\left(a+b x^2\right)^2\right)^p \left(b (2 p+1) x^2-a\right)}{4 b^2 (p+1) (2 p+1)}","\frac{\left(a+b x^2\right)^2 \left(a^2+2 a b x^2+b^2 x^4\right)^p}{4 b^2 (p+1)}-\frac{a \left(a+b x^2\right) \left(a^2+2 a b x^2+b^2 x^4\right)^p}{2 b^2 (2 p+1)}",1,"((a + b*x^2)*((a + b*x^2)^2)^p*(-a + b*(1 + 2*p)*x^2))/(4*b^2*(1 + p)*(1 + 2*p))","A",1
801,1,29,41,0.0044565,"\int x \left(a^2+2 a b x^2+b^2 x^4\right)^p \, dx","Integrate[x*(a^2 + 2*a*b*x^2 + b^2*x^4)^p,x]","\frac{\left(a+b x^2\right) \left(\left(a+b x^2\right)^2\right)^p}{4 b p+2 b}","\frac{\left(a+b x^2\right) \left(a^2+2 a b x^2+b^2 x^4\right)^p}{2 b (2 p+1)}",1,"((a + b*x^2)*((a + b*x^2)^2)^p)/(2*b + 4*b*p)","A",1
802,1,54,63,0.0105652,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^p}{x} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^p/x,x]","-\frac{\left(a+b x^2\right) \left(\left(a+b x^2\right)^2\right)^p \, _2F_1\left(1,2 p+1;2 p+2;\frac{b x^2}{a}+1\right)}{2 a (2 p+1)}","-\frac{\left(a+b x^2\right) \left(a^2+2 a b x^2+b^2 x^4\right)^p \, _2F_1\left(1,2 p+1;2 (p+1);\frac{b x^2}{a}+1\right)}{2 a (2 p+1)}",1,"-1/2*((a + b*x^2)*((a + b*x^2)^2)^p*Hypergeometric2F1[1, 1 + 2*p, 2 + 2*p, 1 + (b*x^2)/a])/(a*(1 + 2*p))","A",1
803,1,55,64,0.0119543,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^p}{x^3} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^p/x^3,x]","\frac{b \left(a+b x^2\right) \left(\left(a+b x^2\right)^2\right)^p \, _2F_1\left(2,2 p+1;2 p+2;\frac{b x^2}{a}+1\right)}{2 a^2 (2 p+1)}","\frac{b \left(a+b x^2\right) \left(a^2+2 a b x^2+b^2 x^4\right)^p \, _2F_1\left(2,2 p+1;2 (p+1);\frac{b x^2}{a}+1\right)}{2 a^2 (2 p+1)}",1,"(b*(a + b*x^2)*((a + b*x^2)^2)^p*Hypergeometric2F1[2, 1 + 2*p, 2 + 2*p, 1 + (b*x^2)/a])/(2*a^2*(1 + 2*p))","A",1
804,1,51,60,0.0067832,"\int x^4 \left(a^2+2 a b x^2+b^2 x^4\right)^p \, dx","Integrate[x^4*(a^2 + 2*a*b*x^2 + b^2*x^4)^p,x]","\frac{1}{5} x^5 \left(\left(a+b x^2\right)^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-2 p} \, _2F_1\left(\frac{5}{2},-2 p;\frac{7}{2};-\frac{b x^2}{a}\right)","\frac{1}{5} x^5 \left(\frac{b x^2}{a}+1\right)^{-2 p} \left(a^2+2 a b x^2+b^2 x^4\right)^p \, _2F_1\left(\frac{5}{2},-2 p;\frac{7}{2};-\frac{b x^2}{a}\right)",1,"(x^5*((a + b*x^2)^2)^p*Hypergeometric2F1[5/2, -2*p, 7/2, -((b*x^2)/a)])/(5*(1 + (b*x^2)/a)^(2*p))","A",1
805,1,51,60,0.0062739,"\int x^2 \left(a^2+2 a b x^2+b^2 x^4\right)^p \, dx","Integrate[x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^p,x]","\frac{1}{3} x^3 \left(\left(a+b x^2\right)^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-2 p} \, _2F_1\left(\frac{3}{2},-2 p;\frac{5}{2};-\frac{b x^2}{a}\right)","\frac{1}{3} x^3 \left(\frac{b x^2}{a}+1\right)^{-2 p} \left(a^2+2 a b x^2+b^2 x^4\right)^p \, _2F_1\left(\frac{3}{2},-2 p;\frac{5}{2};-\frac{b x^2}{a}\right)",1,"(x^3*((a + b*x^2)^2)^p*Hypergeometric2F1[3/2, -2*p, 5/2, -((b*x^2)/a)])/(3*(1 + (b*x^2)/a)^(2*p))","A",1
806,1,46,55,0.005901,"\int \left(a^2+2 a b x^2+b^2 x^4\right)^p \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^p,x]","x \left(\left(a+b x^2\right)^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-2 p} \, _2F_1\left(\frac{1}{2},-2 p;\frac{3}{2};-\frac{b x^2}{a}\right)","x \left(\frac{b x^2}{a}+1\right)^{-2 p} \left(a^2+2 a b x^2+b^2 x^4\right)^p \, _2F_1\left(\frac{1}{2},-2 p;\frac{3}{2};-\frac{b x^2}{a}\right)",1,"(x*((a + b*x^2)^2)^p*Hypergeometric2F1[1/2, -2*p, 3/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^(2*p)","A",1
807,1,49,58,0.0063696,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^p}{x^2} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^p/x^2,x]","-\frac{\left(\left(a+b x^2\right)^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-2 p} \, _2F_1\left(-\frac{1}{2},-2 p;\frac{1}{2};-\frac{b x^2}{a}\right)}{x}","-\frac{\left(\frac{b x^2}{a}+1\right)^{-2 p} \left(a^2+2 a b x^2+b^2 x^4\right)^p \, _2F_1\left(-\frac{1}{2},-2 p;\frac{1}{2};-\frac{b x^2}{a}\right)}{x}",1,"-((((a + b*x^2)^2)^p*Hypergeometric2F1[-1/2, -2*p, 1/2, -((b*x^2)/a)])/(x*(1 + (b*x^2)/a)^(2*p)))","A",1
808,1,51,60,0.0064783,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^p}{x^4} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^p/x^4,x]","-\frac{\left(\left(a+b x^2\right)^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-2 p} \, _2F_1\left(-\frac{3}{2},-2 p;-\frac{1}{2};-\frac{b x^2}{a}\right)}{3 x^3}","-\frac{\left(\frac{b x^2}{a}+1\right)^{-2 p} \left(a^2+2 a b x^2+b^2 x^4\right)^p \, _2F_1\left(-\frac{3}{2},-2 p;-\frac{1}{2};-\frac{b x^2}{a}\right)}{3 x^3}",1,"-1/3*(((a + b*x^2)^2)^p*Hypergeometric2F1[-3/2, -2*p, -1/2, -((b*x^2)/a)])/(x^3*(1 + (b*x^2)/a)^(2*p))","A",1
809,1,56,67,0.0085206,"\int (d x)^{3/2} \left(a^2+2 a b x^2+b^2 x^4\right)^p \, dx","Integrate[(d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^p,x]","\frac{2}{5} x (d x)^{3/2} \left(\left(a+b x^2\right)^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-2 p} \, _2F_1\left(\frac{5}{4},-2 p;\frac{9}{4};-\frac{b x^2}{a}\right)","\frac{2 (d x)^{5/2} \left(\frac{b x^2}{a}+1\right)^{-2 p} \left(a^2+2 a b x^2+b^2 x^4\right)^p \, _2F_1\left(\frac{5}{4},-2 p;\frac{9}{4};-\frac{b x^2}{a}\right)}{5 d}",1,"(2*x*(d*x)^(3/2)*((a + b*x^2)^2)^p*Hypergeometric2F1[5/4, -2*p, 9/4, -((b*x^2)/a)])/(5*(1 + (b*x^2)/a)^(2*p))","A",1
810,1,56,67,0.0080615,"\int \sqrt{d x} \left(a^2+2 a b x^2+b^2 x^4\right)^p \, dx","Integrate[Sqrt[d*x]*(a^2 + 2*a*b*x^2 + b^2*x^4)^p,x]","\frac{2}{3} x \sqrt{d x} \left(\left(a+b x^2\right)^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-2 p} \, _2F_1\left(\frac{3}{4},-2 p;\frac{7}{4};-\frac{b x^2}{a}\right)","\frac{2 (d x)^{3/2} \left(\frac{b x^2}{a}+1\right)^{-2 p} \left(a^2+2 a b x^2+b^2 x^4\right)^p \, _2F_1\left(\frac{3}{4},-2 p;\frac{7}{4};-\frac{b x^2}{a}\right)}{3 d}",1,"(2*x*Sqrt[d*x]*((a + b*x^2)^2)^p*Hypergeometric2F1[3/4, -2*p, 7/4, -((b*x^2)/a)])/(3*(1 + (b*x^2)/a)^(2*p))","A",1
811,1,54,65,0.0084605,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^p}{\sqrt{d x}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^p/Sqrt[d*x],x]","\frac{2 x \left(\left(a+b x^2\right)^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-2 p} \, _2F_1\left(\frac{1}{4},-2 p;\frac{5}{4};-\frac{b x^2}{a}\right)}{\sqrt{d x}}","\frac{2 \sqrt{d x} \left(\frac{b x^2}{a}+1\right)^{-2 p} \left(a^2+2 a b x^2+b^2 x^4\right)^p \, _2F_1\left(\frac{1}{4},-2 p;\frac{5}{4};-\frac{b x^2}{a}\right)}{d}",1,"(2*x*((a + b*x^2)^2)^p*Hypergeometric2F1[1/4, -2*p, 5/4, -((b*x^2)/a)])/(Sqrt[d*x]*(1 + (b*x^2)/a)^(2*p))","A",1
812,1,54,65,0.0089586,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^p}{(d x)^{3/2}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^p/(d*x)^(3/2),x]","-\frac{2 x \left(\left(a+b x^2\right)^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-2 p} \, _2F_1\left(-\frac{1}{4},-2 p;\frac{3}{4};-\frac{b x^2}{a}\right)}{(d x)^{3/2}}","-\frac{2 \left(\frac{b x^2}{a}+1\right)^{-2 p} \left(a^2+2 a b x^2+b^2 x^4\right)^p \, _2F_1\left(-\frac{1}{4},-2 p;\frac{3}{4};-\frac{b x^2}{a}\right)}{d \sqrt{d x}}",1,"(-2*x*((a + b*x^2)^2)^p*Hypergeometric2F1[-1/4, -2*p, 3/4, -((b*x^2)/a)])/((d*x)^(3/2)*(1 + (b*x^2)/a)^(2*p))","A",1
813,1,56,67,0.0088862,"\int \frac{\left(a^2+2 a b x^2+b^2 x^4\right)^p}{(d x)^{5/2}} \, dx","Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^p/(d*x)^(5/2),x]","-\frac{2 x \left(\left(a+b x^2\right)^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-2 p} \, _2F_1\left(-\frac{3}{4},-2 p;\frac{1}{4};-\frac{b x^2}{a}\right)}{3 (d x)^{5/2}}","-\frac{2 \left(\frac{b x^2}{a}+1\right)^{-2 p} \left(a^2+2 a b x^2+b^2 x^4\right)^p \, _2F_1\left(-\frac{3}{4},-2 p;\frac{1}{4};-\frac{b x^2}{a}\right)}{3 d (d x)^{3/2}}",1,"(-2*x*((a + b*x^2)^2)^p*Hypergeometric2F1[-3/4, -2*p, 1/4, -((b*x^2)/a)])/(3*(d*x)^(5/2)*(1 + (b*x^2)/a)^(2*p))","A",1
814,1,25,25,0.0021889,"\int x^2 \left(a+b x^2+c x^4\right) \, dx","Integrate[x^2*(a + b*x^2 + c*x^4),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
815,1,25,25,0.0012928,"\int x \left(a+b x^2+c x^4\right) \, dx","Integrate[x*(a + b*x^2 + c*x^4),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
816,1,20,20,0.0000815,"\int \left(a+b x^2+c x^4\right) \, dx","Integrate[a + b*x^2 + c*x^4,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
817,1,21,21,0.0015905,"\int \frac{a+b x^2+c x^4}{x} \, dx","Integrate[(a + b*x^2 + c*x^4)/x,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
818,1,18,18,0.0018337,"\int \frac{a+b x^2+c x^4}{x^2} \, dx","Integrate[(a + b*x^2 + c*x^4)/x^2,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
819,1,21,21,0.0019064,"\int \frac{a+b x^2+c x^4}{x^3} \, dx","Integrate[(a + b*x^2 + c*x^4)/x^3,x]","-\frac{a}{2 x^2}+b \log (x)+\frac{c x^2}{2}","-\frac{a}{2 x^2}+b \log (x)+\frac{c x^2}{2}",1,"-1/2*a/x^2 + (c*x^2)/2 + b*Log[x]","A",1
820,1,18,18,0.0040503,"\int \frac{a+b x^2+c x^4}{x^4} \, dx","Integrate[(a + b*x^2 + c*x^4)/x^4,x]","-\frac{a}{3 x^3}-\frac{b}{x}+c x","-\frac{a}{3 x^3}-\frac{b}{x}+c x",1,"-1/3*a/x^3 - b/x + c*x","A",1
821,1,21,21,0.0026979,"\int \frac{a+b x^2+c x^4}{x^5} \, dx","Integrate[(a + b*x^2 + c*x^4)/x^5,x]","-\frac{a}{4 x^4}-\frac{b}{2 x^2}+c \log (x)","-\frac{a}{4 x^4}-\frac{b}{2 x^2}+c \log (x)",1,"-1/4*a/x^4 - b/(2*x^2) + c*Log[x]","A",1
822,1,23,23,0.0024738,"\int \frac{a+b x^2+c x^4}{x^6} \, dx","Integrate[(a + b*x^2 + c*x^4)/x^6,x]","-\frac{a}{5 x^5}-\frac{b}{3 x^3}-\frac{c}{x}","-\frac{a}{5 x^5}-\frac{b}{3 x^3}-\frac{c}{x}",1,"-1/5*a/x^5 - b/(3*x^3) - c/x","A",1
823,1,25,25,0.0024167,"\int \frac{a+b x^2+c x^4}{x^7} \, dx","Integrate[(a + b*x^2 + c*x^4)/x^7,x]","-\frac{a}{6 x^6}-\frac{b}{4 x^4}-\frac{c}{2 x^2}","-\frac{a}{6 x^6}-\frac{b}{4 x^4}-\frac{c}{2 x^2}",1,"-1/6*a/x^6 - b/(4*x^4) - c/(2*x^2)","A",1
824,1,25,25,0.0024263,"\int \frac{a+b x^2+c x^4}{x^8} \, dx","Integrate[(a + b*x^2 + c*x^4)/x^8,x]","-\frac{a}{7 x^7}-\frac{b}{5 x^5}-\frac{c}{3 x^3}","-\frac{a}{7 x^7}-\frac{b}{5 x^5}-\frac{c}{3 x^3}",1,"-1/7*a/x^7 - b/(5*x^5) - c/(3*x^3)","A",1
825,1,54,54,0.0070815,"\int x^2 \left(a+b x^2+c x^4\right)^2 \, dx","Integrate[x^2*(a + b*x^2 + c*x^4)^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
826,1,48,54,0.0076529,"\int x \left(a+b x^2+c x^4\right)^2 \, dx","Integrate[x*(a + b*x^2 + c*x^4)^2,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
827,1,49,49,0.004742,"\int \left(a+b x^2+c x^4\right)^2 \, dx","Integrate[(a + b*x^2 + c*x^4)^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
828,1,47,47,0.0121937,"\int \frac{\left(a+b x^2+c x^4\right)^2}{x} \, dx","Integrate[(a + b*x^2 + c*x^4)^2/x,x]","a^2 \log (x)+\frac{1}{4} x^4 \left(2 a c+b^2\right)+a b x^2+\frac{1}{3} b c x^6+\frac{c^2 x^8}{8}","a^2 \log (x)+\frac{1}{4} x^4 \left(2 a c+b^2\right)+a b x^2+\frac{1}{3} b c x^6+\frac{c^2 x^8}{8}",1,"a*b*x^2 + ((b^2 + 2*a*c)*x^4)/4 + (b*c*x^6)/3 + (c^2*x^8)/8 + a^2*Log[x]","A",1
829,1,48,48,0.0177285,"\int \frac{\left(a+b x^2+c x^4\right)^2}{x^2} \, dx","Integrate[(a + b*x^2 + c*x^4)^2/x^2,x]","-\frac{a^2}{x}+\frac{1}{3} x^3 \left(2 a c+b^2\right)+2 a b x+\frac{2}{5} b c x^5+\frac{c^2 x^7}{7}","-\frac{a^2}{x}+\frac{1}{3} x^3 \left(2 a c+b^2\right)+2 a b x+\frac{2}{5} b c x^5+\frac{c^2 x^7}{7}",1,"-(a^2/x) + 2*a*b*x + ((b^2 + 2*a*c)*x^3)/3 + (2*b*c*x^5)/5 + (c^2*x^7)/7","A",1
830,1,46,51,0.0155941,"\int \frac{\left(a+b x^2+c x^4\right)^2}{x^3} \, dx","Integrate[(a + b*x^2 + c*x^4)^2/x^3,x]","\frac{1}{6} \left(-\frac{3 a^2}{x^2}+3 x^2 \left(2 a c+b^2\right)+12 a b \log (x)+3 b c x^4+c^2 x^6\right)","-\frac{a^2}{2 x^2}+\frac{1}{2} x^2 \left(2 a c+b^2\right)+2 a b \log (x)+\frac{1}{2} b c x^4+\frac{c^2 x^6}{6}",1,"((-3*a^2)/x^2 + 3*(b^2 + 2*a*c)*x^2 + 3*b*c*x^4 + c^2*x^6 + 12*a*b*Log[x])/6","A",1
831,1,47,47,0.0192538,"\int \frac{\left(a+b x^2+c x^4\right)^2}{x^4} \, dx","Integrate[(a + b*x^2 + c*x^4)^2/x^4,x]","-\frac{a^2}{3 x^3}+x \left(2 a c+b^2\right)-\frac{2 a b}{x}+\frac{2}{3} b c x^3+\frac{c^2 x^5}{5}","-\frac{a^2}{3 x^3}+x \left(2 a c+b^2\right)-\frac{2 a b}{x}+\frac{2}{3} b c x^3+\frac{c^2 x^5}{5}",1,"-1/3*a^2/x^3 - (2*a*b)/x + (b^2 + 2*a*c)*x + (2*b*c*x^3)/3 + (c^2*x^5)/5","A",1
832,1,41,45,0.0188588,"\int \frac{\left(a+b x^2+c x^4\right)^2}{x^5} \, dx","Integrate[(a + b*x^2 + c*x^4)^2/x^5,x]","\log (x) \left(2 a c+b^2\right)+\frac{\left(c x^4-a\right) \left(a+4 b x^2+c x^4\right)}{4 x^4}","-\frac{a^2}{4 x^4}+\log (x) \left(2 a c+b^2\right)-\frac{a b}{x^2}+b c x^2+\frac{c^2 x^4}{4}",1,"((-a + c*x^4)*(a + 4*b*x^2 + c*x^4))/(4*x^4) + (b^2 + 2*a*c)*Log[x]","A",1
833,1,49,48,0.0210435,"\int \frac{\left(a+b x^2+c x^4\right)^2}{x^6} \, dx","Integrate[(a + b*x^2 + c*x^4)^2/x^6,x]","-\frac{a^2}{5 x^5}+\frac{-2 a c-b^2}{x}-\frac{2 a b}{3 x^3}+2 b c x+\frac{c^2 x^3}{3}","-\frac{a^2}{5 x^5}-\frac{2 a c+b^2}{x}-\frac{2 a b}{3 x^3}+2 b c x+\frac{c^2 x^3}{3}",1,"-1/5*a^2/x^5 - (2*a*b)/(3*x^3) + (-b^2 - 2*a*c)/x + 2*b*c*x + (c^2*x^3)/3","A",1
834,1,50,51,0.017649,"\int \frac{\left(a+b x^2+c x^4\right)^2}{x^7} \, dx","Integrate[(a + b*x^2 + c*x^4)^2/x^7,x]","-\frac{a^2+3 a b x^2+6 a c x^4+3 b^2 x^4-12 b c x^6 \log (x)-3 c^2 x^8}{6 x^6}","-\frac{a^2}{6 x^6}-\frac{2 a c+b^2}{2 x^2}-\frac{a b}{2 x^4}+2 b c \log (x)+\frac{c^2 x^2}{2}",1,"-1/6*(a^2 + 3*a*b*x^2 + 3*b^2*x^4 + 6*a*c*x^4 - 3*c^2*x^8 - 12*b*c*x^6*Log[x])/x^6","A",1
835,1,49,47,0.0229035,"\int \frac{\left(a+b x^2+c x^4\right)^2}{x^8} \, dx","Integrate[(a + b*x^2 + c*x^4)^2/x^8,x]","-\frac{a^2}{7 x^7}+\frac{-2 a c-b^2}{3 x^3}-\frac{2 a b}{5 x^5}-\frac{2 b c}{x}+c^2 x","-\frac{a^2}{7 x^7}-\frac{2 a c+b^2}{3 x^3}-\frac{2 a b}{5 x^5}-\frac{2 b c}{x}+c^2 x",1,"-1/7*a^2/x^7 - (2*a*b)/(5*x^5) + (-b^2 - 2*a*c)/(3*x^3) - (2*b*c)/x + c^2*x","A",1
836,1,50,48,0.0255856,"\int \frac{\left(a+b x^2+c x^4\right)^2}{x^9} \, dx","Integrate[(a + b*x^2 + c*x^4)^2/x^9,x]","-\frac{a^2}{8 x^8}+\frac{-2 a c-b^2}{4 x^4}-\frac{a b}{3 x^6}-\frac{b c}{x^2}+c^2 \log (x)","-\frac{a^2}{8 x^8}-\frac{2 a c+b^2}{4 x^4}-\frac{a b}{3 x^6}-\frac{b c}{x^2}+c^2 \log (x)",1,"-1/8*a^2/x^8 - (a*b)/(3*x^6) + (-b^2 - 2*a*c)/(4*x^4) - (b*c)/x^2 + c^2*Log[x]","A",1
837,1,50,52,0.0196941,"\int \frac{\left(a+b x^2+c x^4\right)^2}{x^{10}} \, dx","Integrate[(a + b*x^2 + c*x^4)^2/x^10,x]","-\frac{35 a^2+90 a b x^2+126 a c x^4+63 b^2 x^4+210 b c x^6+315 c^2 x^8}{315 x^9}","-\frac{a^2}{9 x^9}-\frac{2 a c+b^2}{5 x^5}-\frac{2 a b}{7 x^7}-\frac{2 b c}{3 x^3}-\frac{c^2}{x}",1,"-1/315*(35*a^2 + 90*a*b*x^2 + 63*b^2*x^4 + 126*a*c*x^4 + 210*b*c*x^6 + 315*c^2*x^8)/x^9","A",1
838,1,53,54,0.015151,"\int \frac{\left(a+b x^2+c x^4\right)^2}{x^{11}} \, dx","Integrate[(a + b*x^2 + c*x^4)^2/x^11,x]","-\frac{6 a^2+5 a \left(3 b x^2+4 c x^4\right)+10 x^4 \left(b^2+3 b c x^2+3 c^2 x^4\right)}{60 x^{10}}","-\frac{a^2}{10 x^{10}}-\frac{2 a c+b^2}{6 x^6}-\frac{a b}{4 x^8}-\frac{b c}{2 x^4}-\frac{c^2}{2 x^2}",1,"-1/60*(6*a^2 + 5*a*(3*b*x^2 + 4*c*x^4) + 10*x^4*(b^2 + 3*b*c*x^2 + 3*c^2*x^4))/x^10","A",1
839,1,56,54,0.0245553,"\int \frac{\left(a+b x^2+c x^4\right)^2}{x^{12}} \, dx","Integrate[(a + b*x^2 + c*x^4)^2/x^12,x]","-\frac{a^2}{11 x^{11}}+\frac{-2 a c-b^2}{7 x^7}-\frac{2 a b}{9 x^9}-\frac{2 b c}{5 x^5}-\frac{c^2}{3 x^3}","-\frac{a^2}{11 x^{11}}-\frac{2 a c+b^2}{7 x^7}-\frac{2 a b}{9 x^9}-\frac{2 b c}{5 x^5}-\frac{c^2}{3 x^3}",1,"-1/11*a^2/x^11 - (2*a*b)/(9*x^9) + (-b^2 - 2*a*c)/(7*x^7) - (2*b*c)/(5*x^5) - c^2/(3*x^3)","A",1
840,1,50,54,0.0164766,"\int \frac{\left(a+b x^2+c x^4\right)^2}{x^{13}} \, dx","Integrate[(a + b*x^2 + c*x^4)^2/x^13,x]","-\frac{10 a^2+24 a b x^2+30 a c x^4+15 b^2 x^4+40 b c x^6+30 c^2 x^8}{120 x^{12}}","-\frac{a^2}{12 x^{12}}-\frac{2 a c+b^2}{8 x^8}-\frac{a b}{5 x^{10}}-\frac{b c}{3 x^6}-\frac{c^2}{4 x^4}",1,"-1/120*(10*a^2 + 24*a*b*x^2 + 15*b^2*x^4 + 30*a*c*x^4 + 40*b*c*x^6 + 30*c^2*x^8)/x^12","A",1
841,1,89,89,0.0134108,"\int x^2 \left(a+b x^2+c x^4\right)^3 \, dx","Integrate[x^2*(a + b*x^2 + c*x^4)^3,x]","\frac{a^3 x^3}{3}+\frac{3}{5} a^2 b x^5+\frac{3}{11} c x^{11} \left(a c+b^2\right)+\frac{1}{9} b x^9 \left(6 a c+b^2\right)+\frac{3}{7} a x^7 \left(a c+b^2\right)+\frac{3}{13} b c^2 x^{13}+\frac{c^3 x^{15}}{15}","\frac{a^3 x^3}{3}+\frac{3}{5} a^2 b x^5+\frac{3}{11} c x^{11} \left(a c+b^2\right)+\frac{1}{9} b x^9 \left(6 a c+b^2\right)+\frac{3}{7} a x^7 \left(a c+b^2\right)+\frac{3}{13} b c^2 x^{13}+\frac{c^3 x^{15}}{15}",1,"(a^3*x^3)/3 + (3*a^2*b*x^5)/5 + (3*a*(b^2 + a*c)*x^7)/7 + (b*(b^2 + 6*a*c)*x^9)/9 + (3*c*(b^2 + a*c)*x^11)/11 + (3*b*c^2*x^13)/13 + (c^3*x^15)/15","A",1
842,1,79,89,0.0164432,"\int x \left(a+b x^2+c x^4\right)^3 \, dx","Integrate[x*(a + b*x^2 + c*x^4)^3,x]","\frac{1}{280} x^2 \left(140 a^3+210 a^2 b x^2+84 c x^8 \left(a c+b^2\right)+35 b x^6 \left(6 a c+b^2\right)+140 a x^4 \left(a c+b^2\right)+70 b c^2 x^{10}+20 c^3 x^{12}\right)","\frac{a^3 x^2}{2}+\frac{3}{4} a^2 b x^4+\frac{3}{10} c x^{10} \left(a c+b^2\right)+\frac{1}{8} b x^8 \left(6 a c+b^2\right)+\frac{1}{2} a x^6 \left(a c+b^2\right)+\frac{1}{4} b c^2 x^{12}+\frac{c^3 x^{14}}{14}",1,"(x^2*(140*a^3 + 210*a^2*b*x^2 + 140*a*(b^2 + a*c)*x^4 + 35*b*(b^2 + 6*a*c)*x^6 + 84*c*(b^2 + a*c)*x^8 + 70*b*c^2*x^10 + 20*c^3*x^12))/280","A",1
843,1,81,81,0.0101297,"\int \left(a+b x^2+c x^4\right)^3 \, dx","Integrate[(a + b*x^2 + c*x^4)^3,x]","a^3 x+a^2 b x^3+\frac{1}{3} c x^9 \left(a c+b^2\right)+\frac{1}{7} b x^7 \left(6 a c+b^2\right)+\frac{3}{5} a x^5 \left(a c+b^2\right)+\frac{3}{11} b c^2 x^{11}+\frac{c^3 x^{13}}{13}","a^3 x+a^2 b x^3+\frac{1}{3} c x^9 \left(a c+b^2\right)+\frac{1}{7} b x^7 \left(6 a c+b^2\right)+\frac{3}{5} a x^5 \left(a c+b^2\right)+\frac{3}{11} b c^2 x^{11}+\frac{c^3 x^{13}}{13}",1,"a^3*x + a^2*b*x^3 + (3*a*(b^2 + a*c)*x^5)/5 + (b*(b^2 + 6*a*c)*x^7)/7 + (c*(b^2 + a*c)*x^9)/3 + (3*b*c^2*x^11)/11 + (c^3*x^13)/13","A",1
844,1,85,85,0.0213457,"\int \frac{\left(a+b x^2+c x^4\right)^3}{x} \, dx","Integrate[(a + b*x^2 + c*x^4)^3/x,x]","a^3 \log (x)+\frac{3}{2} a^2 b x^2+\frac{3}{8} c x^8 \left(a c+b^2\right)+\frac{1}{6} b x^6 \left(6 a c+b^2\right)+\frac{3}{4} a x^4 \left(a c+b^2\right)+\frac{3}{10} b c^2 x^{10}+\frac{c^3 x^{12}}{12}","a^3 \log (x)+\frac{3}{2} a^2 b x^2+\frac{3}{8} c x^8 \left(a c+b^2\right)+\frac{1}{6} b x^6 \left(6 a c+b^2\right)+\frac{3}{4} a x^4 \left(a c+b^2\right)+\frac{3}{10} b c^2 x^{10}+\frac{c^3 x^{12}}{12}",1,"(3*a^2*b*x^2)/2 + (3*a*(b^2 + a*c)*x^4)/4 + (b*(b^2 + 6*a*c)*x^6)/6 + (3*c*(b^2 + a*c)*x^8)/8 + (3*b*c^2*x^10)/10 + (c^3*x^12)/12 + a^3*Log[x]","A",1
845,1,80,80,0.024097,"\int \frac{\left(a+b x^2+c x^4\right)^3}{x^2} \, dx","Integrate[(a + b*x^2 + c*x^4)^3/x^2,x]","-\frac{a^3}{x}+3 a^2 b x+\frac{3}{7} c x^7 \left(a c+b^2\right)+\frac{1}{5} b x^5 \left(6 a c+b^2\right)+a x^3 \left(a c+b^2\right)+\frac{1}{3} b c^2 x^9+\frac{c^3 x^{11}}{11}","-\frac{a^3}{x}+3 a^2 b x+\frac{3}{7} c x^7 \left(a c+b^2\right)+\frac{1}{5} b x^5 \left(6 a c+b^2\right)+a x^3 \left(a c+b^2\right)+\frac{1}{3} b c^2 x^9+\frac{c^3 x^{11}}{11}",1,"-(a^3/x) + 3*a^2*b*x + a*(b^2 + a*c)*x^3 + (b*(b^2 + 6*a*c)*x^5)/5 + (3*c*(b^2 + a*c)*x^7)/7 + (b*c^2*x^9)/3 + (c^3*x^11)/11","A",1
846,1,78,86,0.0342189,"\int \frac{\left(a+b x^2+c x^4\right)^3}{x^3} \, dx","Integrate[(a + b*x^2 + c*x^4)^3/x^3,x]","\frac{1}{40} \left(-\frac{20 a^3}{x^2}+120 a^2 b \log (x)+20 c x^6 \left(a c+b^2\right)+10 b x^4 \left(6 a c+b^2\right)+60 a x^2 \left(a c+b^2\right)+15 b c^2 x^8+4 c^3 x^{10}\right)","-\frac{a^3}{2 x^2}+3 a^2 b \log (x)+\frac{1}{2} c x^6 \left(a c+b^2\right)+\frac{1}{4} b x^4 \left(6 a c+b^2\right)+\frac{3}{2} a x^2 \left(a c+b^2\right)+\frac{3}{8} b c^2 x^8+\frac{c^3 x^{10}}{10}",1,"((-20*a^3)/x^2 + 60*a*(b^2 + a*c)*x^2 + 10*b*(b^2 + 6*a*c)*x^4 + 20*c*(b^2 + a*c)*x^6 + 15*b*c^2*x^8 + 4*c^3*x^10 + 120*a^2*b*Log[x])/40","A",1
847,1,83,83,0.0250754,"\int \frac{\left(a+b x^2+c x^4\right)^3}{x^4} \, dx","Integrate[(a + b*x^2 + c*x^4)^3/x^4,x]","-\frac{a^3}{3 x^3}-\frac{3 a^2 b}{x}+\frac{3}{5} c x^5 \left(a c+b^2\right)+\frac{1}{3} b x^3 \left(6 a c+b^2\right)+3 a x \left(a c+b^2\right)+\frac{3}{7} b c^2 x^7+\frac{c^3 x^9}{9}","-\frac{a^3}{3 x^3}-\frac{3 a^2 b}{x}+\frac{3}{5} c x^5 \left(a c+b^2\right)+\frac{1}{3} b x^3 \left(6 a c+b^2\right)+3 a x \left(a c+b^2\right)+\frac{3}{7} b c^2 x^7+\frac{c^3 x^9}{9}",1,"-1/3*a^3/x^3 - (3*a^2*b)/x + 3*a*(b^2 + a*c)*x + (b*(b^2 + 6*a*c)*x^3)/3 + (3*c*(b^2 + a*c)*x^5)/5 + (3*b*c^2*x^7)/7 + (c^3*x^9)/9","A",1
848,1,93,100,0.088958,"\int \frac{x^7}{a+b x^2+c x^4} \, dx","Integrate[x^7/(a + b*x^2 + c*x^4),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
849,1,78,81,0.0441877,"\int \frac{x^5}{a+b x^2+c x^4} \, dx","Integrate[x^5/(a + b*x^2 + c*x^4),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
850,1,62,63,0.0237972,"\int \frac{x^3}{a+b x^2+c x^4} \, dx","Integrate[x^3/(a + b*x^2 + c*x^4),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
851,1,39,36,0.0085672,"\int \frac{x}{a+b x^2+c x^4} \, dx","Integrate[x/(a + b*x^2 + c*x^4),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
852,1,113,69,0.0690863,"\int \frac{1}{x \left(a+b x^2+c x^4\right)} \, dx","Integrate[1/(x*(a + b*x^2 + c*x^4)),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
853,1,135,89,0.1254278,"\int \frac{1}{x^3 \left(a+b x^2+c x^4\right)} \, dx","Integrate[1/(x^3*(a + b*x^2 + c*x^4)),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
854,1,188,114,0.2345525,"\int \frac{1}{x^5 \left(a+b x^2+c x^4\right)} \, dx","Integrate[1/(x^5*(a + b*x^2 + c*x^4)),x]","\frac{-\frac{a^2}{x^4}+4 \log (x) \left(b^2-a c\right)-\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) \log \left(-\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\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) \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\sqrt{b^2-4 a c}}+\frac{2 a b}{x^2}}{4 a^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 a^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 a^3}+\frac{\log (x) \left(b^2-a c\right)}{a^3}+\frac{b}{2 a^2 x^2}-\frac{1}{4 a x^4}",1,"(-(a^2/x^4) + (2*a*b)/x^2 + 4*(b^2 - a*c)*Log[x] - ((b^3 - 3*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - a*c*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^3 - 3*a*b*c - b^2*Sqrt[b^2 - 4*a*c] + a*c*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^3)","A",1
855,1,250,203,0.1502191,"\int \frac{x^6}{a+b x^2+c x^4} \, dx","Integrate[x^6/(a + b*x^2 + c*x^4),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
856,1,202,179,0.1069188,"\int \frac{x^4}{a+b x^2+c x^4} \, dx","Integrate[x^4/(a + b*x^2 + c*x^4),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
857,1,165,150,0.0818486,"\int \frac{x^2}{a+b x^2+c x^4} \, dx","Integrate[x^2/(a + b*x^2 + c*x^4),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
858,1,129,150,0.0753248,"\int \frac{1}{a+b x^2+c x^4} \, dx","Integrate[(a + b*x^2 + c*x^4)^(-1),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
859,1,191,174,0.3826612,"\int \frac{1}{x^2 \left(a+b x^2+c x^4\right)} \, dx","Integrate[1/(x^2*(a + b*x^2 + c*x^4)),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
860,1,216,196,0.137741,"\int \frac{1}{x^4 \left(a+b x^2+c x^4\right)} \, dx","Integrate[1/(x^4*(a + b*x^2 + c*x^4)),x]","\frac{\frac{3 \sqrt{2} \sqrt{c} \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{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{2} \sqrt{c} \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{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{2 a}{x^3}+\frac{6 b}{x}}{6 a^2}","\frac{\sqrt{c} \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{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} a^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \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{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} a^2 \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{b}{a^2 x}-\frac{1}{3 a x^3}",1,"((-2*a)/x^3 + (6*b)/x + (3*Sqrt[2]*Sqrt[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[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[2]*Sqrt[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[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(6*a^2)","A",1
861,1,121,132,0.1684709,"\int \frac{x^7}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[x^7/(a + b*x^2 + c*x^4)^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
862,1,93,78,0.0851607,"\int \frac{x^5}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[x^5/(a + b*x^2 + c*x^4)^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
863,1,79,75,0.0627145,"\int \frac{x^3}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[x^3/(a + b*x^2 + c*x^4)^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
864,1,79,74,0.0780546,"\int \frac{x}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[x/(a + b*x^2 + c*x^4)^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
865,1,207,122,0.3259319,"\int \frac{1}{x \left(a+b x^2+c x^4\right)^2} \, dx","Integrate[1/(x*(a + b*x^2 + c*x^4)^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
866,1,248,162,0.268107,"\int \frac{1}{x^3 \left(a+b x^2+c x^4\right)^2} \, dx","Integrate[1/(x^3*(a + b*x^2 + c*x^4)^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
867,1,327,331,0.6571119,"\int \frac{x^8}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[x^8/(a + b*x^2 + c*x^4)^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
868,1,282,271,0.503957,"\int \frac{x^6}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[x^6/(a + b*x^2 + c*x^4)^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
869,1,235,237,0.4016987,"\int \frac{x^4}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[x^4/(a + b*x^2 + c*x^4)^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
870,1,222,221,0.4320737,"\int \frac{x^2}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[x^2/(a + b*x^2 + c*x^4)^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
871,1,243,252,0.4186294,"\int \frac{1}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(a + b*x^2 + c*x^4)^(-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
872,1,302,308,0.6014185,"\int \frac{1}{x^2 \left(a+b x^2+c x^4\right)^2} \, dx","Integrate[1/(x^2*(a + b*x^2 + c*x^4)^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
873,1,244,209,0.3247414,"\int \frac{x^{11}}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[x^11/(a + b*x^2 + c*x^4)^3,x]","\frac{-\frac{2 b c \left(30 a^2 c^2-10 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)^{5/2}}+\frac{2 a^3 c^2+a^2 b c \left(5 c x^2-4 b\right)+a b^3 \left(b-5 c x^2\right)+b^5 x^2}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}+\frac{32 a^3 c^3-39 a^2 b^2 c^2+50 a^2 b c^3 x^2+11 a b^4 c-30 a b^3 c^2 x^2-b^6+4 b^5 c x^2}{\left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+c \log \left(a+b x^2+c x^4\right)}{4 c^4}","\frac{b \left(30 a^2 c^2-10 a b^2 c+b^4\right) \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{2 c^3 \left(b^2-4 a c\right)^{5/2}}-\frac{b x^2 \left(b^2-7 a c\right)}{2 c^2 \left(b^2-4 a c\right)^2}+\frac{x^4 \left(b x^2 \left(b^2-10 a c\right)+a \left(b^2-16 a c\right)\right)}{4 c \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{x^8 \left(2 a+b x^2\right)}{4 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}+\frac{\log \left(a+b x^2+c x^4\right)}{4 c^3}",1,"((-b^6 + 11*a*b^4*c - 39*a^2*b^2*c^2 + 32*a^3*c^3 + 4*b^5*c*x^2 - 30*a*b^3*c^2*x^2 + 50*a^2*b*c^3*x^2)/((b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (2*a^3*c^2 + b^5*x^2 + a*b^3*(b - 5*c*x^2) + a^2*b*c*(-4*b + 5*c*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (2*b*c*(b^4 - 10*a*b^2*c + 30*a^2*c^2)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2) + c*Log[a + b*x^2 + c*x^4])/(4*c^4)","A",1
874,1,194,121,0.1723167,"\int \frac{x^9}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[x^9/(a + b*x^2 + c*x^4)^3,x]","\frac{1}{4} \left(\frac{24 a^2 \tan ^{-1}\left(\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{5/2}}+\frac{a^2 c \left(2 c x^2-3 b\right)+a b^2 \left(b-4 c x^2\right)+b^4 x^2}{c^3 \left(4 a c-b^2\right) \left(a+b x^2+c x^4\right)^2}+\frac{22 a^2 b c^2-20 a^2 c^3 x^2-8 a b^3 c+16 a b^2 c^2 x^2+b^5-2 b^4 c x^2}{c^3 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}\right)","-\frac{6 a^2 \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{5/2}}-\frac{3 a x^2 \left(2 a+b x^2\right)}{2 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{x^6 \left(2 a+b x^2\right)}{4 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}",1,"((b^5 - 8*a*b^3*c + 22*a^2*b*c^2 - 2*b^4*c*x^2 + 16*a*b^2*c^2*x^2 - 20*a^2*c^3*x^2)/(c^3*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (b^4*x^2 + a*b^2*(b - 4*c*x^2) + a^2*c*(-3*b + 2*c*x^2))/(c^3*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)^2) + (24*a^2*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2))/4","A",1
875,1,137,119,0.1930111,"\int \frac{x^7}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[x^7/(a + b*x^2 + c*x^4)^3,x]","-\frac{8 a^3 c+a^2 \left(b^2+10 b c x^2+16 c^2 x^4\right)+a b x^2 \left(2 b^2+b c x^2+6 c^2 x^4\right)+b^4 x^4}{4 c \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)^2}-\frac{3 a b \tan ^{-1}\left(\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{5/2}}","\frac{3 a b \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{5/2}}+\frac{3 b x^2 \left(2 a+b x^2\right)}{4 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}-\frac{x^6 \left(b+2 c x^2\right)}{4 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}",1,"-1/4*(8*a^3*c + b^4*x^4 + a*b*x^2*(2*b^2 + b*c*x^2 + 6*c^2*x^4) + a^2*(b^2 + 10*b*c*x^2 + 16*c^2*x^4))/(c*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)^2) - (3*a*b*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2)","A",1
876,1,145,130,0.134004,"\int \frac{x^5}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[x^5/(a + b*x^2 + c*x^4)^3,x]","\frac{1}{4} \left(\frac{4 \left(2 a c+b^2\right) \tan ^{-1}\left(\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{5/2}}+\frac{\left(2 a c+b^2\right) \left(b+2 c x^2\right)}{c \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{a \left(b-2 c x^2\right)+b^2 x^2}{c \left(4 a c-b^2\right) \left(a+b x^2+c x^4\right)^2}\right)","-\frac{\left(2 a c+b^2\right) \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{5/2}}+\frac{x^2 \left(2 a+b x^2\right)}{4 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}+\frac{x^2 \left(2 a c+b^2\right)+3 a b}{2 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}",1,"(((b^2 + 2*a*c)*(b + 2*c*x^2))/(c*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (b^2*x^2 + a*(b - 2*c*x^2))/(c*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)^2) + (4*(b^2 + 2*a*c)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2))/4","A",1
877,1,114,113,0.0993518,"\int \frac{x^3}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[x^3/(a + b*x^2 + c*x^4)^3,x]","\frac{-\frac{12 b c \tan ^{-1}\left(\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+\frac{\left(b^2-4 a c\right) \left(2 a+b x^2\right)}{\left(a+b x^2+c x^4\right)^2}-\frac{3 b \left(b+2 c x^2\right)}{a+b x^2+c x^4}}{4 \left(b^2-4 a c\right)^2}","\frac{3 b c \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{5/2}}+\frac{2 a+b x^2}{4 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}-\frac{3 b \left(b+2 c x^2\right)}{4 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}",1,"(((b^2 - 4*a*c)*(2*a + b*x^2))/(a + b*x^2 + c*x^4)^2 - (3*b*(b + 2*c*x^2))/(a + b*x^2 + c*x^4) - (12*b*c*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(4*(b^2 - 4*a*c)^2)","A",1
878,1,106,113,0.0988115,"\int \frac{x}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[x/(a + b*x^2 + c*x^4)^3,x]","\frac{\frac{24 c^2 \tan ^{-1}\left(\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}-\frac{\left(b+2 c x^2\right) \left(-2 c \left(5 a+3 c x^4\right)+b^2-6 b c x^2\right)}{\left(a+b x^2+c x^4\right)^2}}{4 \left(b^2-4 a c\right)^2}","-\frac{6 c^2 \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{5/2}}+\frac{3 c \left(b+2 c x^2\right)}{2 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}-\frac{b+2 c x^2}{4 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}",1,"(-(((b + 2*c*x^2)*(b^2 - 6*b*c*x^2 - 2*c*(5*a + 3*c*x^4)))/(a + b*x^2 + c*x^4)^2) + (24*c^2*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(4*(b^2 - 4*a*c)^2)","A",1
879,1,342,200,0.5224375,"\int \frac{1}{x \left(a+b x^2+c x^4\right)^3} \, dx","Integrate[1/(x*(a + b*x^2 + c*x^4)^3),x]","\frac{\frac{a^2 \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)^2}+\frac{a \left(16 a^2 c^2-15 a b^2 c-14 a b c^2 x^2+2 b^4+2 b^3 c x^2\right)}{\left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}-\frac{\left(16 a^2 c^2 \sqrt{b^2-4 a c}+30 a^2 b c^2-10 a b^3 c-8 a b^2 c \sqrt{b^2-4 a c}+b^4 \sqrt{b^2-4 a c}+b^5\right) \log \left(-\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\left(b^2-4 a c\right)^{5/2}}+\frac{\left(-16 a^2 c^2 \sqrt{b^2-4 a c}+30 a^2 b c^2-10 a b^3 c+8 a b^2 c \sqrt{b^2-4 a c}-b^4 \sqrt{b^2-4 a c}+b^5\right) \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\left(b^2-4 a c\right)^{5/2}}+4 \log (x)}{4 a^3}","-\frac{\log \left(a+b x^2+c x^4\right)}{4 a^3}+\frac{\log (x)}{a^3}+\frac{16 a^2 c^2+2 b c x^2 \left(b^2-7 a c\right)-15 a b^2 c+2 b^4}{4 a^2 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{b \left(30 a^2 c^2-10 a b^2 c+b^4\right) \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{2 a^3 \left(b^2-4 a c\right)^{5/2}}+\frac{-2 a c+b^2+b c x^2}{4 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}",1,"((a^2*(b^2 - 2*a*c + b*c*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (a*(2*b^4 - 15*a*b^2*c + 16*a^2*c^2 + 2*b^3*c*x^2 - 14*a*b*c^2*x^2))/((b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + 4*Log[x] - ((b^5 - 10*a*b^3*c + 30*a^2*b*c^2 + b^4*Sqrt[b^2 - 4*a*c] - 8*a*b^2*c*Sqrt[b^2 - 4*a*c] + 16*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)^(5/2) + ((b^5 - 10*a*b^3*c + 30*a^2*b*c^2 - b^4*Sqrt[b^2 - 4*a*c] + 8*a*b^2*c*Sqrt[b^2 - 4*a*c] - 16*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)^(5/2))/(4*a^3)","A",1
880,1,402,255,0.6205495,"\int \frac{1}{x^3 \left(a+b x^2+c x^4\right)^3} \, dx","Integrate[1/(x^3*(a + b*x^2 + c*x^4)^3),x]","\frac{\frac{a^2 \left(-3 a b c-2 a c^2 x^2+b^3+b^2 c x^2\right)}{\left(4 a c-b^2\right) \left(a+b x^2+c x^4\right)^2}-\frac{a \left(46 a^2 b c^2+28 a^2 c^3 x^2-29 a b^3 c-26 a b^2 c^2 x^2+4 b^5+4 b^4 c x^2\right)}{\left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{3 \left(-20 a^3 c^3+30 a^2 b^2 c^2+16 a^2 b c^2 \sqrt{b^2-4 a c}-10 a b^4 c+b^5 \sqrt{b^2-4 a c}-8 a b^3 c \sqrt{b^2-4 a c}+b^6\right) \log \left(-\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\left(b^2-4 a c\right)^{5/2}}+\frac{3 \left(20 a^3 c^3-30 a^2 b^2 c^2+16 a^2 b c^2 \sqrt{b^2-4 a c}+10 a b^4 c+b^5 \sqrt{b^2-4 a c}-8 a b^3 c \sqrt{b^2-4 a c}-b^6\right) \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\left(b^2-4 a c\right)^{5/2}}-\frac{2 a}{x^2}-12 b \log (x)}{4 a^4}","\frac{3 b \log \left(a+b x^2+c x^4\right)}{4 a^4}-\frac{3 b \log (x)}{a^4}-\frac{3 \left(b^2-5 a c\right) \left(b^2-2 a c\right)}{2 a^3 x^2 \left(b^2-4 a c\right)^2}+\frac{20 a^2 c^2+3 b c x^2 \left(b^2-6 a c\right)-20 a b^2 c+3 b^4}{4 a^2 x^2 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}-\frac{3 \left(-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\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)^{5/2}}+\frac{-2 a c+b^2+b c x^2}{4 a x^2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}",1,"((-2*a)/x^2 + (a^2*(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)^2) - (a*(4*b^5 - 29*a*b^3*c + 46*a^2*b*c^2 + 4*b^4*c*x^2 - 26*a*b^2*c^2*x^2 + 28*a^2*c^3*x^2))/((b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - 12*b*Log[x] + (3*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3 + b^5*Sqrt[b^2 - 4*a*c] - 8*a*b^3*c*Sqrt[b^2 - 4*a*c] + 16*a^2*b*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)^(5/2) + (3*(-b^6 + 10*a*b^4*c - 30*a^2*b^2*c^2 + 20*a^3*c^3 + b^5*Sqrt[b^2 - 4*a*c] - 8*a*b^3*c*Sqrt[b^2 - 4*a*c] + 16*a^2*b*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)^(5/2))/(4*a^4)","A",1
881,1,455,400,1.1663778,"\int \frac{x^{10}}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[x^10/(a + b*x^2 + c*x^4)^3,x]","\frac{-\frac{4 \left(a^2 c x \left(2 c x^2-3 b\right)+a b^2 x \left(b-4 c x^2\right)+b^4 x^3\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}+\frac{3 \sqrt{2} \sqrt{c} \left(28 a^2 c^2 \sqrt{b^2-4 a c}-44 a^2 b c^2+11 a b^3 c-9 a b^2 c \sqrt{b^2-4 a c}+b^4 \sqrt{b^2-4 a c}-b^5\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)^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{2} \sqrt{c} \left(28 a^2 c^2 \sqrt{b^2-4 a c}+44 a^2 b c^2-11 a b^3 c-9 a b^2 c \sqrt{b^2-4 a c}+b^4 \sqrt{b^2-4 a c}+b^5\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)^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 x \left(48 a^2 b c^2-44 a^2 c^3 x^2-17 a b^3 c+37 a b^2 c^2 x^2+2 b^5-5 b^4 c x^2\right)}{\left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}}{16 c^3}","\frac{3 \left(-\frac{44 a^2 b c^2-11 a b^3 c+b^5}{\sqrt{b^2-4 a c}}+28 a^2 c^2-9 a b^2 c+b^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{8 \sqrt{2} c^{5/2} \left(b^2-4 a c\right)^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \left(\frac{44 a^2 b c^2-11 a b^3 c+b^5}{\sqrt{b^2-4 a c}}+28 a^2 c^2-9 a b^2 c+b^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{8 \sqrt{2} c^{5/2} \left(b^2-4 a c\right)^2 \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{3 b x \left(b^2-8 a c\right)}{8 c^2 \left(b^2-4 a c\right)^2}+\frac{x^3 \left(b^2-28 a c\right)}{8 c \left(b^2-4 a c\right)^2}+\frac{x^7 \left(2 a+b x^2\right)}{4 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}+\frac{x^5 \left(12 a b-x^2 \left(b^2-28 a c\right)\right)}{8 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}",1,"((2*x*(2*b^5 - 17*a*b^3*c + 48*a^2*b*c^2 - 5*b^4*c*x^2 + 37*a*b^2*c^2*x^2 - 44*a^2*c^3*x^2))/((b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - (4*(b^4*x^3 + a*b^2*x*(b - 4*c*x^2) + a^2*c*x*(-3*b + 2*c*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (3*Sqrt[2]*Sqrt[c]*(-b^5 + 11*a*b^3*c - 44*a^2*b*c^2 + b^4*Sqrt[b^2 - 4*a*c] - 9*a*b^2*c*Sqrt[b^2 - 4*a*c] + 28*a^2*c^2*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)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[2]*Sqrt[c]*(b^5 - 11*a*b^3*c + 44*a^2*b*c^2 + b^4*Sqrt[b^2 - 4*a*c] - 9*a*b^2*c*Sqrt[b^2 - 4*a*c] + 28*a^2*c^2*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)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(16*c^3)","A",1
882,1,381,348,0.9607013,"\int \frac{x^8}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[x^8/(a + b*x^2 + c*x^4)^3,x]","\frac{\frac{4 \left(-2 a^2 c x+a b x \left(b-3 c x^2\right)+b^3 x^3\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}+\frac{2 x \left(-36 a^2 c^2+11 a b^2 c-16 a b c^2 x^2-2 b^4+b^3 c x^2\right)}{\left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \sqrt{c} \left(40 a^2 c^2+18 a b^2 c-16 a b c \sqrt{b^2-4 a c}+b^3 \sqrt{b^2-4 a c}-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)^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left(-40 a^2 c^2-18 a b^2 c-16 a b c \sqrt{b^2-4 a c}+b^3 \sqrt{b^2-4 a c}+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)^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}}{16 c^2}","\frac{\left(-\frac{-40 a^2 c^2-18 a b^2 c+b^4}{\sqrt{b^2-4 a c}}-16 a b c+b^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{8 \sqrt{2} c^{3/2} \left(b^2-4 a c\right)^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left(\frac{-40 a^2 c^2-18 a b^2 c+b^4}{\sqrt{b^2-4 a c}}-16 a b c+b^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{8 \sqrt{2} c^{3/2} \left(b^2-4 a c\right)^2 \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x^5 \left(2 a+b x^2\right)}{4 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}+\frac{x^3 \left(x^2 \left(20 a c+b^2\right)+12 a b\right)}{8 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}-\frac{x \left(20 a c+b^2\right)}{8 c \left(b^2-4 a c\right)^2}",1,"((2*x*(-2*b^4 + 11*a*b^2*c - 36*a^2*c^2 + b^3*c*x^2 - 16*a*b*c^2*x^2))/((b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (4*(-2*a^2*c*x + b^3*x^3 + a*b*x*(b - 3*c*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (Sqrt[2]*Sqrt[c]*(-b^4 + 18*a*b^2*c + 40*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 16*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)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(b^4 - 18*a*b^2*c - 40*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 16*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)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(16*c^2)","A",1
883,1,343,298,0.8452482,"\int \frac{x^6}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[x^6/(a + b*x^2 + c*x^4)^3,x]","\frac{-\frac{4 \left(a x \left(b-2 c x^2\right)+b^2 x^3\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}+\frac{8 a b c x+24 a c^2 x^3+4 b^3 x+6 b^2 c x^3}{\left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{3 \sqrt{2} \sqrt{c} \left(b^2 \sqrt{b^2-4 a c}+4 a c \sqrt{b^2-4 a c}-12 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)^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{2} \sqrt{c} \left(b^2 \sqrt{b^2-4 a c}+4 a c \sqrt{b^2-4 a c}+12 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)^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}}{16 c}","\frac{3 x \left(x^2 \left(4 a c+b^2\right)+4 a b\right)}{8 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{x^3 \left(2 a+b x^2\right)}{4 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}+\frac{3 \left(-\frac{b \left(12 a c+b^2\right)}{\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)}{8 \sqrt{2} \sqrt{c} \left(b^2-4 a c\right)^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \left(\frac{b \left(12 a c+b^2\right)}{\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)}{8 \sqrt{2} \sqrt{c} \left(b^2-4 a c\right)^2 \sqrt{\sqrt{b^2-4 a c}+b}}",1,"((4*b^3*x + 8*a*b*c*x + 6*b^2*c*x^3 + 24*a*c^2*x^3)/((b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - (4*(b^2*x^3 + a*x*(b - 2*c*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (3*Sqrt[2]*Sqrt[c]*(-b^3 - 12*a*b*c + b^2*Sqrt[b^2 - 4*a*c] + 4*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)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[2]*Sqrt[c]*(b^3 + 12*a*b*c + b^2*Sqrt[b^2 - 4*a*c] + 4*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)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(16*c)","A",1
884,1,285,289,0.7104267,"\int \frac{x^4}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[x^4/(a + b*x^2 + c*x^4)^3,x]","\frac{1}{8} \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)^2}+\frac{4 a c x-7 b^2 x-12 b c x^3}{\left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{3 \sqrt{2} \sqrt{c} \left(-2 b \sqrt{b^2-4 a c}+4 a c+3 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)^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{3 \sqrt{2} \sqrt{c} \left(2 b \sqrt{b^2-4 a c}+4 a c+3 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)^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}\right)","\frac{x \left(2 a+b x^2\right)}{4 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}-\frac{x \left(-4 a c+7 b^2+12 b c x^2\right)}{8 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{3 \sqrt{c} \left(-2 b \sqrt{b^2-4 a c}+4 a c+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{4 \sqrt{2} \left(b^2-4 a c\right)^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{3 \sqrt{c} \left(2 b \sqrt{b^2-4 a c}+4 a c+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{4 \sqrt{2} \left(b^2-4 a c\right)^{5/2} \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)^2) + (-7*b^2*x + 4*a*c*x - 12*b*c*x^3)/((b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (3*Sqrt[2]*Sqrt[c]*(3*b^2 + 4*a*c - 2*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)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (3*Sqrt[2]*Sqrt[c]*(3*b^2 + 4*a*c + 2*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)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/8","A",1
885,1,334,311,0.8506284,"\int \frac{x^2}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[x^2/(a + b*x^2 + c*x^4)^3,x]","\frac{1}{16} \left(-\frac{4 x \left(b+2 c x^2\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}+\frac{2 x \left(8 a b c+20 a c^2 x^2+b^3+b^2 c x^2\right)}{a \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \sqrt{c} \left(b^2 \sqrt{b^2-4 a c}+20 a c \sqrt{b^2-4 a c}-52 a b c+b^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{a \left(b^2-4 a c\right)^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left(b^2 \sqrt{b^2-4 a c}+20 a c \sqrt{b^2-4 a c}+52 a b c-b^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{a \left(b^2-4 a c\right)^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}\right)","-\frac{x \left(b+2 c x^2\right)}{4 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}+\frac{x \left(c x^2 \left(20 a c+b^2\right)+b \left(8 a c+b^2\right)\right)}{8 a \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{\sqrt{c} \left(\frac{b \left(b^2-52 a c\right)}{\sqrt{b^2-4 a c}}+20 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{8 \sqrt{2} a \left(b^2-4 a c\right)^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left(-\frac{b \left(b^2-52 a c\right)}{\sqrt{b^2-4 a c}}+20 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{8 \sqrt{2} a \left(b^2-4 a c\right)^2 \sqrt{\sqrt{b^2-4 a c}+b}}",1,"((-4*x*(b + 2*c*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (2*x*(b^3 + 8*a*b*c + b^2*c*x^2 + 20*a*c^2*x^2))/(a*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (Sqrt[2]*Sqrt[c]*(b^3 - 52*a*b*c + b^2*Sqrt[b^2 - 4*a*c] + 20*a*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(-b^3 + 52*a*b*c + b^2*Sqrt[b^2 - 4*a*c] + 20*a*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/16","A",1
886,1,372,355,1.0233396,"\int \frac{1}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[(a + b*x^2 + c*x^4)^(-3),x]","\frac{\frac{2 x \left(28 a^2 c^2-25 a b^2 c-24 a b c^2 x^2+3 b^4+3 b^3 c x^2\right)}{\left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{3 \sqrt{2} \sqrt{c} \left(56 a^2 c^2-10 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) \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)^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{3 \sqrt{2} \sqrt{c} \left(56 a^2 c^2-10 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) \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)^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{4 a 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)^2}}{16 a^2}","\frac{x \left(3 b c x^2 \left(b^2-8 a c\right)+\left(b^2-7 a c\right) \left(3 b^2-4 a c\right)\right)}{8 a^2 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{3 \sqrt{c} \left(56 a^2 c^2-10 a b^2 c+b \left(b^2-8 a c\right) \sqrt{b^2-4 a c}+b^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{8 \sqrt{2} a^2 \left(b^2-4 a c\right)^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{c} \left(-\frac{56 a^2 c^2-10 a b^2 c+b^4}{\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)}{8 \sqrt{2} a^2 \left(b^2-4 a c\right)^2 \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x \left(-2 a c+b^2+b c x^2\right)}{4 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}",1,"((4*a*x*(b^2 - 2*a*c + b*c*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (2*x*(3*b^4 - 25*a*b^2*c + 28*a^2*c^2 + 3*b^3*c*x^2 - 24*a*b*c^2*x^2))/((b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (3*Sqrt[2]*Sqrt[c]*(b^4 - 10*a*b^2*c + 56*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 8*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)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (3*Sqrt[2]*Sqrt[c]*(b^4 - 10*a*b^2*c + 56*a^2*c^2 - b^3*Sqrt[b^2 - 4*a*c] + 8*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)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(16*a^2)","A",1
887,1,454,425,1.7581755,"\int \frac{1}{x^2 \left(a+b x^2+c x^4\right)^3} \, dx","Integrate[1/(x^2*(a + b*x^2 + c*x^4)^3),x]","-\frac{\frac{3 \sqrt{2} \sqrt{c} \left(60 a^2 c^2 \sqrt{b^2-4 a c}+124 a^2 b c^2-47 a b^3 c-37 a b^2 c \sqrt{b^2-4 a c}+5 b^4 \sqrt{b^2-4 a c}+5 b^5\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)^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{2} \sqrt{c} \left(60 a^2 c^2 \sqrt{b^2-4 a c}-124 a^2 b c^2+47 a b^3 c-37 a b^2 c \sqrt{b^2-4 a c}+5 b^4 \sqrt{b^2-4 a c}-5 b^5\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)^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 x \left(84 a^2 b c^2+52 a^2 c^3 x^2-52 a b^3 c-47 a b^2 c^2 x^2+7 b^5+7 b^4 c x^2\right)}{\left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{4 a 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)^2}+\frac{16}{x}}{16 a^3}","-\frac{3 \left(5 b^2-12 a c\right) \left(b^2-5 a c\right)}{8 a^3 x \left(b^2-4 a c\right)^2}+\frac{36 a^2 c^2+b c x^2 \left(5 b^2-32 a c\right)-35 a b^2 c+5 b^4}{8 a^2 x \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}-\frac{3 \sqrt{c} \left(\frac{b \left(124 a^2 c^2-47 a b^2 c+5 b^4\right)}{\sqrt{b^2-4 a c}}+\left(5 b^2-12 a c\right) \left(b^2-5 a c\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{8 \sqrt{2} a^3 \left(b^2-4 a c\right)^2 \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{3 \sqrt{c} \left(\left(5 b^2-12 a c\right) \left(b^2-5 a c\right)-\frac{124 a^2 b c^2-47 a b^3 c+5 b^5}{\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)}{8 \sqrt{2} a^3 \left(b^2-4 a c\right)^2 \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{-2 a c+b^2+b c x^2}{4 a x \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}",1,"-1/16*(16/x + (4*a*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)^2) + (2*x*(7*b^5 - 52*a*b^3*c + 84*a^2*b*c^2 + 7*b^4*c*x^2 - 47*a*b^2*c^2*x^2 + 52*a^2*c^3*x^2))/((b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (3*Sqrt[2]*Sqrt[c]*(5*b^5 - 47*a*b^3*c + 124*a^2*b*c^2 + 5*b^4*Sqrt[b^2 - 4*a*c] - 37*a*b^2*c*Sqrt[b^2 - 4*a*c] + 60*a^2*c^2*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)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[2]*Sqrt[c]*(-5*b^5 + 47*a*b^3*c - 124*a^2*b*c^2 + 5*b^4*Sqrt[b^2 - 4*a*c] - 37*a*b^2*c*Sqrt[b^2 - 4*a*c] + 60*a^2*c^2*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)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/a^3","A",1
888,1,80,82,0.0521735,"\int \frac{x^5}{a-b x^2+c x^4} \, dx","Integrate[x^5/(a - b*x^2 + c*x^4),x]","\frac{\frac{2 \left(b^2-2 a c\right) \tan ^{-1}\left(\frac{2 c x^2-b}{\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
889,1,65,64,0.0228705,"\int \frac{x^3}{a-b x^2+c x^4} \, dx","Integrate[x^3/(a - b*x^2 + c*x^4),x]","\frac{\frac{2 b \tan ^{-1}\left(\frac{2 c x^2-b}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+\log \left(a-b x^2+c x^4\right)}{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
890,1,41,35,0.0083871,"\int \frac{x}{a-b x^2+c x^4} \, dx","Integrate[x/(a - b*x^2 + c*x^4),x]","\frac{\tan ^{-1}\left(\frac{2 c x^2-b}{\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
891,1,117,70,0.0737007,"\int \frac{1}{x \left(a-b x^2+c x^4\right)} \, dx","Integrate[1/(x*(a - b*x^2 + c*x^4)),x]","\frac{\left(b-\sqrt{b^2-4 a c}\right) \log \left(-\sqrt{b^2-4 a c}-b+2 c x^2\right)-\left(\sqrt{b^2-4 a c}+b\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
892,1,139,89,0.1409584,"\int \frac{1}{x^3 \left(a-b x^2+c x^4\right)} \, dx","Integrate[1/(x^3*(a - b*x^2 + c*x^4)),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
893,1,208,179,0.1233741,"\int \frac{x^4}{a-b x^2+c x^4} \, dx","Integrate[x^4/(a - b*x^2 + c*x^4),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{-\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{\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) \tanh ^{-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) \tanh ^{-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
894,1,137,150,0.1065512,"\int \frac{x^2}{a-b x^2+c x^4} \, dx","Integrate[x^2/(a - b*x^2 + c*x^4),x]","\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{-\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}} \tanh ^{-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}}-\frac{\sqrt{\sqrt{b^2-4 a c}+b} \tanh ^{-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}}",1,"(-(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]]*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
895,1,137,150,0.0792444,"\int \frac{1}{a-b x^2+c x^4} \, dx","Integrate[(a - b*x^2 + c*x^4)^(-1),x]","\frac{\sqrt{2} \sqrt{c} \left(\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}}-\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} \tanh ^{-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} \tanh ^{-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
896,1,199,172,0.3994824,"\int \frac{1}{x^2 \left(a-b x^2+c x^4\right)} \, dx","Integrate[1/(x^2*(a - b*x^2 + c*x^4)),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{-\sqrt{b^2-4 a c}-b}}\right)}{\sqrt{b^2-4 a c} \sqrt{-\sqrt{b^2-4 a c}-b}}+\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) \tanh ^{-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) \tanh ^{-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
897,1,62,69,0.037889,"\int \frac{x^5}{a-b+2 a x^2+a x^4} \, dx","Integrate[x^5/(a - b + 2*a*x^2 + a*x^4),x]","\frac{x^2-\log \left(a \left(x^2+1\right)^2-b\right)}{2 a}-\frac{(a+b) \tanh ^{-1}\left(\frac{\sqrt{a} \left(x^2+1\right)}{\sqrt{b}}\right)}{2 a^{3/2} \sqrt{b}}","-\frac{(a+b) \tanh ^{-1}\left(\frac{\sqrt{a} \left(x^2+1\right)}{\sqrt{b}}\right)}{2 a^{3/2} \sqrt{b}}-\frac{\log \left(a x^4+2 a x^2+a-b\right)}{2 a}+\frac{x^2}{2 a}",1,"-1/2*((a + b)*ArcTanh[(Sqrt[a]*(1 + x^2))/Sqrt[b]])/(a^(3/2)*Sqrt[b]) + (x^2 - Log[-b + a*(1 + x^2)^2])/(2*a)","A",1
898,1,51,56,0.0186677,"\int \frac{x^3}{a-b+2 a x^2+a x^4} \, dx","Integrate[x^3/(a - b + 2*a*x^2 + a*x^4),x]","\frac{\log \left(a \left(x^2+1\right)^2-b\right)+\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \left(x^2+1\right)}{\sqrt{b}}\right)}{\sqrt{b}}}{4 a}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \left(x^2+1\right)}{\sqrt{b}}\right)}{2 \sqrt{a} \sqrt{b}}+\frac{\log \left(a x^4+2 a x^2+a-b\right)}{4 a}",1,"((2*Sqrt[a]*ArcTanh[(Sqrt[a]*(1 + x^2))/Sqrt[b]])/Sqrt[b] + Log[-b + a*(1 + x^2)^2])/(4*a)","A",1
899,1,31,31,0.0083877,"\int \frac{x}{a-b+2 a x^2+a x^4} \, dx","Integrate[x/(a - b + 2*a*x^2 + a*x^4),x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \left(x^2+1\right)}{\sqrt{b}}\right)}{2 \sqrt{a} \sqrt{b}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \left(x^2+1\right)}{\sqrt{b}}\right)}{2 \sqrt{a} \sqrt{b}}",1,"-1/2*ArcTanh[(Sqrt[a]*(1 + x^2))/Sqrt[b]]/(Sqrt[a]*Sqrt[b])","A",1
900,1,90,77,0.0489528,"\int \frac{1}{x \left(a-b+2 a x^2+a x^4\right)} \, dx","Integrate[1/(x*(a - b + 2*a*x^2 + a*x^4)),x]","\frac{\left(\sqrt{a}+\sqrt{b}\right) \log \left(\sqrt{a} \left(x^2+1\right)-\sqrt{b}\right)+\left(\sqrt{b}-\sqrt{a}\right) \log \left(\sqrt{a} \left(x^2+1\right)+\sqrt{b}\right)-4 \sqrt{b} \log (x)}{4 \sqrt{b} (b-a)}","\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \left(x^2+1\right)}{\sqrt{b}}\right)}{2 \sqrt{b} (a-b)}-\frac{\log \left(a x^4+2 a x^2+a-b\right)}{4 (a-b)}+\frac{\log (x)}{a-b}",1,"(-4*Sqrt[b]*Log[x] + (Sqrt[a] + Sqrt[b])*Log[-Sqrt[b] + Sqrt[a]*(1 + x^2)] + (-Sqrt[a] + Sqrt[b])*Log[Sqrt[b] + Sqrt[a]*(1 + x^2)])/(4*Sqrt[b]*(-a + b))","A",1
901,1,146,97,0.0935508,"\int \frac{1}{x^3 \left(a-b+2 a x^2+a x^4\right)} \, dx","Integrate[1/(x^3*(a - b + 2*a*x^2 + a*x^4)),x]","\frac{-8 a \sqrt{b} x^2 \log (x)+\sqrt{a} x^2 \left(\sqrt{a}+\sqrt{b}\right)^2 \log \left(\sqrt{a} \left(x^2+1\right)-\sqrt{b}\right)-\left(\sqrt{a}-\sqrt{b}\right) \left(\left(a x^2-\sqrt{a} \sqrt{b} x^2\right) \log \left(\sqrt{a} \left(x^2+1\right)+\sqrt{b}\right)+2 \left(\sqrt{a} \sqrt{b}+b\right)\right)}{4 \sqrt{b} x^2 (a-b)^2}","-\frac{1}{2 x^2 (a-b)}-\frac{\sqrt{a} (a+b) \tanh ^{-1}\left(\frac{\sqrt{a} \left(x^2+1\right)}{\sqrt{b}}\right)}{2 \sqrt{b} (a-b)^2}+\frac{a \log \left(a x^4+2 a x^2+a-b\right)}{2 (a-b)^2}-\frac{2 a \log (x)}{(a-b)^2}",1,"(-8*a*Sqrt[b]*x^2*Log[x] + Sqrt[a]*(Sqrt[a] + Sqrt[b])^2*x^2*Log[-Sqrt[b] + Sqrt[a]*(1 + x^2)] - (Sqrt[a] - Sqrt[b])*(2*(Sqrt[a]*Sqrt[b] + b) + (a*x^2 - Sqrt[a]*Sqrt[b]*x^2)*Log[Sqrt[b] + Sqrt[a]*(1 + x^2)]))/(4*(a - b)^2*Sqrt[b]*x^2)","A",1
902,1,144,114,0.0855322,"\int \frac{x^4}{a-b+2 a x^2+a x^4} \, dx","Integrate[x^4/(a - b + 2*a*x^2 + a*x^4),x]","\frac{\left(\sqrt{a}-\sqrt{b}\right)^2 \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{a-\sqrt{a} \sqrt{b}}}\right)}{2 a \sqrt{b} \sqrt{a-\sqrt{a} \sqrt{b}}}-\frac{\left(\sqrt{a}+\sqrt{b}\right)^2 \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{2 a \sqrt{b} \sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{x}{a}","\frac{\left(\sqrt{a}-\sqrt{b}\right)^{3/2} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 a^{5/4} \sqrt{b}}-\frac{\left(\sqrt{a}+\sqrt{b}\right)^{3/2} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 a^{5/4} \sqrt{b}}+\frac{x}{a}",1,"x/a + ((Sqrt[a] - Sqrt[b])^2*ArcTan[(Sqrt[a]*x)/Sqrt[a - Sqrt[a]*Sqrt[b]]])/(2*a*Sqrt[a - Sqrt[a]*Sqrt[b]]*Sqrt[b]) - ((Sqrt[a] + Sqrt[b])^2*ArcTan[(Sqrt[a]*x)/Sqrt[a + Sqrt[a]*Sqrt[b]]])/(2*a*Sqrt[a + Sqrt[a]*Sqrt[b]]*Sqrt[b])","A",1
903,1,128,109,0.1040661,"\int \frac{x^2}{a-b+2 a x^2+a x^4} \, dx","Integrate[x^2/(a - b + 2*a*x^2 + a*x^4),x]","\frac{\frac{\left(\sqrt{a}+\sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}+a}}-\frac{\left(\sqrt{a}-\sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{a-\sqrt{a} \sqrt{b}}}\right)}{\sqrt{a-\sqrt{a} \sqrt{b}}}}{2 \sqrt{a} \sqrt{b}}","\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 a^{3/4} \sqrt{b}}-\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 a^{3/4} \sqrt{b}}",1,"(-(((Sqrt[a] - Sqrt[b])*ArcTan[(Sqrt[a]*x)/Sqrt[a - Sqrt[a]*Sqrt[b]]])/Sqrt[a - Sqrt[a]*Sqrt[b]]) + ((Sqrt[a] + Sqrt[b])*ArcTan[(Sqrt[a]*x)/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]])/(2*Sqrt[a]*Sqrt[b])","A",1
904,1,105,109,0.0628617,"\int \frac{1}{a-b+2 a x^2+a x^4} \, dx","Integrate[(a - b + 2*a*x^2 + a*x^4)^(-1),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{a-\sqrt{a} \sqrt{b}}}\right)}{2 \sqrt{b} \sqrt{a-\sqrt{a} \sqrt{b}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{2 \sqrt{b} \sqrt{\sqrt{a} \sqrt{b}+a}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 \sqrt[4]{a} \sqrt{b} \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 \sqrt[4]{a} \sqrt{b} \sqrt{\sqrt{a}+\sqrt{b}}}",1,"ArcTan[(Sqrt[a]*x)/Sqrt[a - Sqrt[a]*Sqrt[b]]]/(2*Sqrt[a - Sqrt[a]*Sqrt[b]]*Sqrt[b]) - ArcTan[(Sqrt[a]*x)/Sqrt[a + Sqrt[a]*Sqrt[b]]]/(2*Sqrt[a + Sqrt[a]*Sqrt[b]]*Sqrt[b])","A",1
905,1,143,121,0.1455317,"\int \frac{1}{x^2 \left(a-b+2 a x^2+a x^4\right)} \, dx","Integrate[1/(x^2*(a - b + 2*a*x^2 + a*x^4)),x]","\frac{\frac{\left(\sqrt{a} \sqrt{b}+a\right) \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{a-\sqrt{a} \sqrt{b}}}\right)}{\sqrt{b} \sqrt{a-\sqrt{a} \sqrt{b}}}-\frac{\left(a-\sqrt{a} \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{b} \sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{2}{x}}{2 (b-a)}","-\frac{1}{x (a-b)}-\frac{\sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 \sqrt{b} \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}+\frac{\sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 \sqrt{b} \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}",1,"(2/x + ((a + Sqrt[a]*Sqrt[b])*ArcTan[(Sqrt[a]*x)/Sqrt[a - Sqrt[a]*Sqrt[b]]])/(Sqrt[a - Sqrt[a]*Sqrt[b]]*Sqrt[b]) - ((a - Sqrt[a]*Sqrt[b])*ArcTan[(Sqrt[a]*x)/Sqrt[a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a + Sqrt[a]*Sqrt[b]]*Sqrt[b]))/(2*(-a + b))","A",1
906,1,62,69,0.0362919,"\int \frac{x^5}{a+b+2 a x^2+a x^4} \, dx","Integrate[x^5/(a + b + 2*a*x^2 + a*x^4),x]","\frac{\sqrt{a} \left(x^2-\log \left(a \left(x^2+1\right)^2+b\right)\right)+\frac{(a-b) \tan ^{-1}\left(\frac{\sqrt{a} \left(x^2+1\right)}{\sqrt{b}}\right)}{\sqrt{b}}}{2 a^{3/2}}","\frac{(a-b) \tan ^{-1}\left(\frac{\sqrt{a} \left(x^2+1\right)}{\sqrt{b}}\right)}{2 a^{3/2} \sqrt{b}}-\frac{\log \left(a x^4+2 a x^2+a+b\right)}{2 a}+\frac{x^2}{2 a}",1,"(((a - b)*ArcTan[(Sqrt[a]*(1 + x^2))/Sqrt[b]])/Sqrt[b] + Sqrt[a]*(x^2 - Log[b + a*(1 + x^2)^2]))/(2*a^(3/2))","A",1
907,1,49,54,0.0181301,"\int \frac{x^3}{a+b+2 a x^2+a x^4} \, dx","Integrate[x^3/(a + b + 2*a*x^2 + a*x^4),x]","\frac{\log \left(a \left(x^2+1\right)^2+b\right)-\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \left(x^2+1\right)}{\sqrt{b}}\right)}{\sqrt{b}}}{4 a}","\frac{\log \left(a x^4+2 a x^2+a+b\right)}{4 a}-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \left(x^2+1\right)}{\sqrt{b}}\right)}{2 \sqrt{a} \sqrt{b}}",1,"((-2*Sqrt[a]*ArcTan[(Sqrt[a]*(1 + x^2))/Sqrt[b]])/Sqrt[b] + Log[b + a*(1 + x^2)^2])/(4*a)","A",1
908,1,31,31,0.0074419,"\int \frac{x}{a+b+2 a x^2+a x^4} \, dx","Integrate[x/(a + b + 2*a*x^2 + a*x^4),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \left(x^2+1\right)}{\sqrt{b}}\right)}{2 \sqrt{a} \sqrt{b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \left(x^2+1\right)}{\sqrt{b}}\right)}{2 \sqrt{a} \sqrt{b}}",1,"ArcTan[(Sqrt[a]*(1 + x^2))/Sqrt[b]]/(2*Sqrt[a]*Sqrt[b])","A",1
909,1,105,69,0.0565909,"\int \frac{1}{x \left(a+b+2 a x^2+a x^4\right)} \, dx","Integrate[1/(x*(a + b + 2*a*x^2 + a*x^4)),x]","\frac{i \left(\sqrt{a}+i \sqrt{b}\right) \log \left(\sqrt{a} \left(x^2+1\right)-i \sqrt{b}\right)+\left(-\sqrt{b}-i \sqrt{a}\right) \log \left(\sqrt{a} \left(x^2+1\right)+i \sqrt{b}\right)+4 \sqrt{b} \log (x)}{4 \sqrt{b} (a+b)}","-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \left(x^2+1\right)}{\sqrt{b}}\right)}{2 \sqrt{b} (a+b)}-\frac{\log \left(a x^4+2 a x^2+a+b\right)}{4 (a+b)}+\frac{\log (x)}{a+b}",1,"(4*Sqrt[b]*Log[x] + I*(Sqrt[a] + I*Sqrt[b])*Log[(-I)*Sqrt[b] + Sqrt[a]*(1 + x^2)] + ((-I)*Sqrt[a] - Sqrt[b])*Log[I*Sqrt[b] + Sqrt[a]*(1 + x^2)])/(4*Sqrt[b]*(a + b))","C",1
910,1,163,89,0.0981714,"\int \frac{1}{x^3 \left(a+b+2 a x^2+a x^4\right)} \, dx","Integrate[1/(x^3*(a + b + 2*a*x^2 + a*x^4)),x]","\frac{\left(2 a^{3/2} \sqrt{b}-i a^2+i a b\right) \log \left(\sqrt{a} x^2+\sqrt{a}-i \sqrt{b}\right)}{4 \sqrt{a} \sqrt{b} (a+b)^2}+\frac{\left(2 a^{3/2} \sqrt{b}+i a^2-i a b\right) \log \left(\sqrt{a} x^2+\sqrt{a}+i \sqrt{b}\right)}{4 \sqrt{a} \sqrt{b} (a+b)^2}-\frac{1}{2 x^2 (a+b)}-\frac{2 a \log (x)}{(a+b)^2}","-\frac{1}{2 x^2 (a+b)}+\frac{\sqrt{a} (a-b) \tan ^{-1}\left(\frac{\sqrt{a} \left(x^2+1\right)}{\sqrt{b}}\right)}{2 \sqrt{b} (a+b)^2}+\frac{a \log \left(a x^4+2 a x^2+a+b\right)}{2 (a+b)^2}-\frac{2 a \log (x)}{(a+b)^2}",1,"-1/2*1/((a + b)*x^2) - (2*a*Log[x])/(a + b)^2 + (((-I)*a^2 + 2*a^(3/2)*Sqrt[b] + I*a*b)*Log[Sqrt[a] - I*Sqrt[b] + Sqrt[a]*x^2])/(4*Sqrt[a]*Sqrt[b]*(a + b)^2) + ((I*a^2 + 2*a^(3/2)*Sqrt[b] - I*a*b)*Log[Sqrt[a] + I*Sqrt[b] + Sqrt[a]*x^2])/(4*Sqrt[a]*Sqrt[b]*(a + b)^2)","C",1
911,1,164,432,0.0972775,"\int \frac{x^4}{a+b+2 a x^2+a x^4} \, dx","Integrate[x^4/(a + b + 2*a*x^2 + a*x^4),x]","-\frac{i \left(\sqrt{a}-i \sqrt{b}\right)^2 \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{a-i \sqrt{a} \sqrt{b}}}\right)}{2 a \sqrt{b} \sqrt{a-i \sqrt{a} \sqrt{b}}}+\frac{i \left(\sqrt{a}+i \sqrt{b}\right)^2 \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{a+i \sqrt{a} \sqrt{b}}}\right)}{2 a \sqrt{b} \sqrt{a+i \sqrt{a} \sqrt{b}}}+\frac{x}{a}","\frac{\left(-2 \sqrt{a} \sqrt{a+b}+a+b\right) \log \left(-\sqrt{2} \sqrt[4]{a} x \sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{a+b}+\sqrt{a} x^2\right)}{4 \sqrt{2} a^{5/4} \sqrt{a+b} \sqrt{\sqrt{a+b}-\sqrt{a}}}-\frac{\left(-2 \sqrt{a} \sqrt{a+b}+a+b\right) \log \left(\sqrt{2} \sqrt[4]{a} x \sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{a+b}+\sqrt{a} x^2\right)}{4 \sqrt{2} a^{5/4} \sqrt{a+b} \sqrt{\sqrt{a+b}-\sqrt{a}}}+\frac{\left(2 \sqrt{a} \sqrt{a+b}+a+b\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a+b}-\sqrt{a}}-\sqrt{2} \sqrt[4]{a} x}{\sqrt{\sqrt{a+b}+\sqrt{a}}}\right)}{2 \sqrt{2} a^{5/4} \sqrt{a+b} \sqrt{\sqrt{a+b}+\sqrt{a}}}-\frac{\left(2 \sqrt{a} \sqrt{a+b}+a+b\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{2} \sqrt[4]{a} x}{\sqrt{\sqrt{a+b}+\sqrt{a}}}\right)}{2 \sqrt{2} a^{5/4} \sqrt{a+b} \sqrt{\sqrt{a+b}+\sqrt{a}}}+\frac{x}{a}",1,"x/a - ((I/2)*(Sqrt[a] - I*Sqrt[b])^2*ArcTan[(Sqrt[a]*x)/Sqrt[a - I*Sqrt[a]*Sqrt[b]]])/(a*Sqrt[a - I*Sqrt[a]*Sqrt[b]]*Sqrt[b]) + ((I/2)*(Sqrt[a] + I*Sqrt[b])^2*ArcTan[(Sqrt[a]*x)/Sqrt[a + I*Sqrt[a]*Sqrt[b]]])/(a*Sqrt[a + I*Sqrt[a]*Sqrt[b]]*Sqrt[b])","C",1
912,1,143,331,0.1157114,"\int \frac{x^2}{a+b+2 a x^2+a x^4} \, dx","Integrate[x^2/(a + b + 2*a*x^2 + a*x^4),x]","\frac{\frac{\left(\sqrt{b}+i \sqrt{a}\right) \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{a-i \sqrt{a} \sqrt{b}}}\right)}{\sqrt{a-i \sqrt{a} \sqrt{b}}}+\frac{\left(\sqrt{b}-i \sqrt{a}\right) \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{a+i \sqrt{a} \sqrt{b}}}\right)}{\sqrt{a+i \sqrt{a} \sqrt{b}}}}{2 \sqrt{a} \sqrt{b}}","\frac{\log \left(-\sqrt{2} \sqrt[4]{a} x \sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{a+b}+\sqrt{a} x^2\right)}{4 \sqrt{2} a^{3/4} \sqrt{\sqrt{a+b}-\sqrt{a}}}-\frac{\log \left(\sqrt{2} \sqrt[4]{a} x \sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{a+b}+\sqrt{a} x^2\right)}{4 \sqrt{2} a^{3/4} \sqrt{\sqrt{a+b}-\sqrt{a}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{a+b}-\sqrt{a}}-\sqrt{2} \sqrt[4]{a} x}{\sqrt{\sqrt{a+b}+\sqrt{a}}}\right)}{2 \sqrt{2} a^{3/4} \sqrt{\sqrt{a+b}+\sqrt{a}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{2} \sqrt[4]{a} x}{\sqrt{\sqrt{a+b}+\sqrt{a}}}\right)}{2 \sqrt{2} a^{3/4} \sqrt{\sqrt{a+b}+\sqrt{a}}}",1,"(((I*Sqrt[a] + Sqrt[b])*ArcTan[(Sqrt[a]*x)/Sqrt[a - I*Sqrt[a]*Sqrt[b]]])/Sqrt[a - I*Sqrt[a]*Sqrt[b]] + (((-I)*Sqrt[a] + Sqrt[b])*ArcTan[(Sqrt[a]*x)/Sqrt[a + I*Sqrt[a]*Sqrt[b]]])/Sqrt[a + I*Sqrt[a]*Sqrt[b]])/(2*Sqrt[a]*Sqrt[b])","C",1
913,1,119,359,0.0700068,"\int \frac{1}{a+b+2 a x^2+a x^4} \, dx","Integrate[(a + b + 2*a*x^2 + a*x^4)^(-1),x]","\frac{i \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{a+i \sqrt{a} \sqrt{b}}}\right)}{2 \sqrt{b} \sqrt{a+i \sqrt{a} \sqrt{b}}}-\frac{i \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{a-i \sqrt{a} \sqrt{b}}}\right)}{2 \sqrt{b} \sqrt{a-i \sqrt{a} \sqrt{b}}}","-\frac{\log \left(-\sqrt{2} \sqrt[4]{a} x \sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{a+b}+\sqrt{a} x^2\right)}{4 \sqrt{2} \sqrt[4]{a} \sqrt{a+b} \sqrt{\sqrt{a+b}-\sqrt{a}}}+\frac{\log \left(\sqrt{2} \sqrt[4]{a} x \sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{a+b}+\sqrt{a} x^2\right)}{4 \sqrt{2} \sqrt[4]{a} \sqrt{a+b} \sqrt{\sqrt{a+b}-\sqrt{a}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{a+b}-\sqrt{a}}-\sqrt{2} \sqrt[4]{a} x}{\sqrt{\sqrt{a+b}+\sqrt{a}}}\right)}{2 \sqrt{2} \sqrt[4]{a} \sqrt{a+b} \sqrt{\sqrt{a+b}+\sqrt{a}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{2} \sqrt[4]{a} x}{\sqrt{\sqrt{a+b}+\sqrt{a}}}\right)}{2 \sqrt{2} \sqrt[4]{a} \sqrt{a+b} \sqrt{\sqrt{a+b}+\sqrt{a}}}",1,"((-1/2*I)*ArcTan[(Sqrt[a]*x)/Sqrt[a - I*Sqrt[a]*Sqrt[b]]])/(Sqrt[a - I*Sqrt[a]*Sqrt[b]]*Sqrt[b]) + ((I/2)*ArcTan[(Sqrt[a]*x)/Sqrt[a + I*Sqrt[a]*Sqrt[b]]])/(Sqrt[a + I*Sqrt[a]*Sqrt[b]]*Sqrt[b])","C",1
914,1,174,433,0.1496809,"\int \frac{1}{x^2 \left(a+b+2 a x^2+a x^4\right)} \, dx","Integrate[1/(x^2*(a + b + 2*a*x^2 + a*x^4)),x]","\frac{1}{x (-a-b)}+\frac{\left(-\sqrt{a} \sqrt{b}+i a\right) \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{a-i \sqrt{a} \sqrt{b}}}\right)}{2 \sqrt{b} \sqrt{a-i \sqrt{a} \sqrt{b}} (a+b)}+\frac{\left(-\sqrt{a} \sqrt{b}-i a\right) \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{a+i \sqrt{a} \sqrt{b}}}\right)}{2 \sqrt{b} \sqrt{a+i \sqrt{a} \sqrt{b}} (a+b)}","\frac{\sqrt[4]{a} \left(2 \sqrt{a}-\sqrt{a+b}\right) \log \left(-\sqrt{2} \sqrt[4]{a} x \sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{a+b}+\sqrt{a} x^2\right)}{4 \sqrt{2} (a+b)^{3/2} \sqrt{\sqrt{a+b}-\sqrt{a}}}-\frac{\sqrt[4]{a} \left(2 \sqrt{a}-\sqrt{a+b}\right) \log \left(\sqrt{2} \sqrt[4]{a} x \sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{a+b}+\sqrt{a} x^2\right)}{4 \sqrt{2} (a+b)^{3/2} \sqrt{\sqrt{a+b}-\sqrt{a}}}-\frac{1}{x (a+b)}+\frac{\sqrt[4]{a} \left(\sqrt{a+b}+2 \sqrt{a}\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a+b}-\sqrt{a}}-\sqrt{2} \sqrt[4]{a} x}{\sqrt{\sqrt{a+b}+\sqrt{a}}}\right)}{2 \sqrt{2} (a+b)^{3/2} \sqrt{\sqrt{a+b}+\sqrt{a}}}-\frac{\sqrt[4]{a} \left(\sqrt{a+b}+2 \sqrt{a}\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{2} \sqrt[4]{a} x}{\sqrt{\sqrt{a+b}+\sqrt{a}}}\right)}{2 \sqrt{2} (a+b)^{3/2} \sqrt{\sqrt{a+b}+\sqrt{a}}}",1,"1/((-a - b)*x) + ((I*a - Sqrt[a]*Sqrt[b])*ArcTan[(Sqrt[a]*x)/Sqrt[a - I*Sqrt[a]*Sqrt[b]]])/(2*Sqrt[a - I*Sqrt[a]*Sqrt[b]]*Sqrt[b]*(a + b)) + (((-I)*a - Sqrt[a]*Sqrt[b])*ArcTan[(Sqrt[a]*x)/Sqrt[a + I*Sqrt[a]*Sqrt[b]]])/(2*Sqrt[a + I*Sqrt[a]*Sqrt[b]]*Sqrt[b]*(a + b))","C",1
915,1,20,20,0.0063439,"\int \frac{x}{1+x^2+x^4} \, dx","Integrate[x/(1 + x^2 + x^4),x]","\frac{\tan ^{-1}\left(\frac{2 x^2+1}{\sqrt{3}}\right)}{\sqrt{3}}","\frac{\tan ^{-1}\left(\frac{2 x^2+1}{\sqrt{3}}\right)}{\sqrt{3}}",1,"ArcTan[(1 + 2*x^2)/Sqrt[3]]/Sqrt[3]","A",1
916,1,14,14,0.0053323,"\int \frac{x}{10+2 x^2+x^4} \, dx","Integrate[x/(10 + 2*x^2 + x^4),x]","\frac{1}{6} \tan ^{-1}\left(\frac{1}{3} \left(x^2+1\right)\right)","\frac{1}{6} \tan ^{-1}\left(\frac{1}{3} \left(x^2+1\right)\right)",1,"ArcTan[(1 + x^2)/3]/6","A",1
917,1,23,23,0.0143816,"\int \frac{x^2}{20+9 x^2+x^4} \, dx","Integrate[x^2/(20 + 9*x^2 + x^4),x]","\sqrt{5} \tan ^{-1}\left(\frac{x}{\sqrt{5}}\right)-2 \tan ^{-1}\left(\frac{x}{2}\right)","\sqrt{5} \tan ^{-1}\left(\frac{x}{\sqrt{5}}\right)-2 \tan ^{-1}\left(\frac{x}{2}\right)",1,"-2*ArcTan[x/2] + Sqrt[5]*ArcTan[x/Sqrt[5]]","A",1
918,1,94,74,0.1438689,"\int \frac{x^2}{1-x^2+x^4} \, dx","Integrate[x^2/(1 - x^2 + x^4),x]","\frac{\sqrt{-1-i \sqrt{3}} \left(\sqrt{3}+i\right) \tan ^{-1}\left(\frac{1}{2} \left(1-i \sqrt{3}\right) x\right)+\sqrt{-1+i \sqrt{3}} \left(\sqrt{3}-i\right) \tan ^{-1}\left(\frac{1}{2} \left(1+i \sqrt{3}\right) x\right)}{2 \sqrt{6}}","\frac{\log \left(x^2-\sqrt{3} x+1\right)}{4 \sqrt{3}}-\frac{\log \left(x^2+\sqrt{3} x+1\right)}{4 \sqrt{3}}-\frac{1}{2} \tan ^{-1}\left(\sqrt{3}-2 x\right)+\frac{1}{2} \tan ^{-1}\left(2 x+\sqrt{3}\right)",1,"(Sqrt[-1 - I*Sqrt[3]]*(I + Sqrt[3])*ArcTan[((1 - I*Sqrt[3])*x)/2] + Sqrt[-1 + I*Sqrt[3]]*(-I + Sqrt[3])*ArcTan[((1 + I*Sqrt[3])*x)/2])/(2*Sqrt[6])","C",0
919,1,39,188,0.0314819,"\int \frac{x^2}{2-2 x^2+x^4} \, dx","Integrate[x^2/(2 - 2*x^2 + x^4),x]","-\frac{\tan ^{-1}\left(\frac{x}{\sqrt{-1-i}}\right)}{(-1-i)^{3/2}}-\frac{\tan ^{-1}\left(\frac{x}{\sqrt{-1+i}}\right)}{(-1+i)^{3/2}}","\frac{\log \left(x^2-\sqrt{2 \left(1+\sqrt{2}\right)} x+\sqrt{2}\right)}{4 \sqrt{2 \left(1+\sqrt{2}\right)}}-\frac{\log \left(x^2+\sqrt{2 \left(1+\sqrt{2}\right)} x+\sqrt{2}\right)}{4 \sqrt{2 \left(1+\sqrt{2}\right)}}-\frac{1}{2} \sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)}-2 x}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)+\frac{1}{2} \sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{2 x+\sqrt{2 \left(1+\sqrt{2}\right)}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)",1,"-(ArcTan[x/Sqrt[-1 - I]]/(-1 - I)^(3/2)) - ArcTan[x/Sqrt[-1 + I]]/(-1 + I)^(3/2)","C",1
920,1,164,171,0.1468402,"\int x^7 \sqrt{a+b x^2+c x^4} \, dx","Integrate[x^7*Sqrt[a + b*x^2 + c*x^4],x]","\frac{-\frac{\left(32 a c-35 b^2+42 b c x^2\right) \left(a+b x^2+c x^4\right)^{3/2}}{48 c^2}+\frac{5 \left(12 a b c-7 b^3\right) \left(2 \sqrt{c} \left(b+2 c x^2\right) \sqrt{a+b x^2+c x^4}-\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)\right)}{256 c^{7/2}}+x^4 \left(a+b x^2+c x^4\right)^{3/2}}{10 c}","\frac{b \left(7 b^2-12 a c\right) \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)}{512 c^{9/2}}-\frac{b \left(7 b^2-12 a c\right) \left(b+2 c x^2\right) \sqrt{a+b x^2+c x^4}}{256 c^4}+\frac{\left(-32 a c+35 b^2-42 b c x^2\right) \left(a+b x^2+c x^4\right)^{3/2}}{480 c^3}+\frac{x^4 \left(a+b x^2+c x^4\right)^{3/2}}{10 c}",1,"(x^4*(a + b*x^2 + c*x^4)^(3/2) - ((-35*b^2 + 32*a*c + 42*b*c*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(48*c^2) + (5*(-7*b^3 + 12*a*b*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)))/(10*c)","A",1
921,1,136,153,0.0671973,"\int x^5 \sqrt{a+b x^2+c x^4} \, dx","Integrate[x^5*Sqrt[a + b*x^2 + c*x^4],x]","\frac{2 \sqrt{c} \sqrt{a+b x^2+c x^4} \left(b \left(8 c^2 x^4-52 a c\right)+24 c^2 x^2 \left(a+2 c x^4\right)+15 b^3-10 b^2 c x^2\right)-3 \left(16 a^2 c^2-24 a b^2 c+5 b^4\right) \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{768 c^{7/2}}","-\frac{\left(b^2-4 a c\right) \left(5 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)}{256 c^{7/2}}+\frac{\left(5 b^2-4 a c\right) \left(b+2 c x^2\right) \sqrt{a+b x^2+c x^4}}{128 c^3}-\frac{5 b \left(a+b x^2+c x^4\right)^{3/2}}{48 c^2}+\frac{x^2 \left(a+b x^2+c x^4\right)^{3/2}}{8 c}",1,"(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]*(15*b^3 - 10*b^2*c*x^2 + 24*c^2*x^2*(a + 2*c*x^4) + b*(-52*a*c + 8*c^2*x^4)) - 3*(5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(768*c^(7/2))","A",1
922,1,101,108,0.0487551,"\int x^3 \sqrt{a+b x^2+c x^4} \, dx","Integrate[x^3*Sqrt[a + b*x^2 + c*x^4],x]","\frac{2 \sqrt{c} \sqrt{a+b x^2+c x^4} \left(8 c \left(a+c x^4\right)-3 b^2+2 b c x^2\right)+3 b \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)}{96 c^{5/2}}","\frac{b \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)}{32 c^{5/2}}-\frac{b \left(b+2 c x^2\right) \sqrt{a+b x^2+c x^4}}{16 c^2}+\frac{\left(a+b x^2+c x^4\right)^{3/2}}{6 c}",1,"(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]*(-3*b^2 + 2*b*c*x^2 + 8*c*(a + c*x^4)) + 3*b*(b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(96*c^(5/2))","A",1
923,1,83,83,0.0221381,"\int x \sqrt{a+b x^2+c x^4} \, dx","Integrate[x*Sqrt[a + b*x^2 + c*x^4],x]","\frac{\left(b+2 c x^2\right) \sqrt{a+b x^2+c x^4}}{8 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)}{16 c^{3/2}}","\frac{\left(b+2 c x^2\right) \sqrt{a+b x^2+c x^4}}{8 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)}{16 c^{3/2}}",1,"((b + 2*c*x^2)*Sqrt[a + b*x^2 + c*x^4])/(8*c) - ((b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(16*c^(3/2))","A",1
924,1,106,109,0.0426244,"\int \frac{\sqrt{a+b x^2+c x^4}}{x} \, dx","Integrate[Sqrt[a + b*x^2 + c*x^4]/x,x]","\frac{1}{4} \left(2 \sqrt{a+b x^2+c x^4}-2 \sqrt{a} \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)+\frac{b \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{\sqrt{c}}\right)","\frac{1}{2} \sqrt{a+b x^2+c x^4}-\frac{1}{2} \sqrt{a} \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)+\frac{b \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{4 \sqrt{c}}",1,"(2*Sqrt[a + b*x^2 + c*x^4] - 2*Sqrt[a]*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])])/Sqrt[c])/4","A",1
925,1,112,112,0.0488222,"\int \frac{\sqrt{a+b x^2+c x^4}}{x^3} \, dx","Integrate[Sqrt[a + b*x^2 + c*x^4]/x^3,x]","-\frac{\sqrt{a+b x^2+c x^4}}{2 x^2}-\frac{b \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{4 \sqrt{a}}+\frac{1}{2} \sqrt{c} \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)","-\frac{\sqrt{a+b x^2+c x^4}}{2 x^2}-\frac{b \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{4 \sqrt{a}}+\frac{1}{2} \sqrt{c} \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)",1,"-1/2*Sqrt[a + b*x^2 + c*x^4]/x^2 - (b*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/(4*Sqrt[a]) + (Sqrt[c]*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/2","A",1
926,1,88,88,0.0396609,"\int \frac{\sqrt{a+b x^2+c x^4}}{x^5} \, dx","Integrate[Sqrt[a + b*x^2 + c*x^4]/x^5,x]","\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{16 a^{3/2}}-\frac{\left(2 a+b x^2\right) \sqrt{a+b x^2+c x^4}}{8 a x^4}","\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{16 a^{3/2}}-\frac{\left(2 a+b x^2\right) \sqrt{a+b x^2+c x^4}}{8 a x^4}",1,"-1/8*((2*a + b*x^2)*Sqrt[a + b*x^2 + c*x^4])/(a*x^4) + ((b^2 - 4*a*c)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/(16*a^(3/2))","A",1
927,1,108,116,0.0749749,"\int \frac{\sqrt{a+b x^2+c x^4}}{x^7} \, dx","Integrate[Sqrt[a + b*x^2 + c*x^4]/x^7,x]","-\frac{b \left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{32 a^{5/2}}-\frac{\sqrt{a+b x^2+c x^4} \left(8 a^2+2 a x^2 \left(b+4 c x^2\right)-3 b^2 x^4\right)}{48 a^2 x^6}","-\frac{b \left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{32 a^{5/2}}+\frac{b \left(2 a+b x^2\right) \sqrt{a+b x^2+c x^4}}{16 a^2 x^4}-\frac{\left(a+b x^2+c x^4\right)^{3/2}}{6 a x^6}",1,"-1/48*(Sqrt[a + b*x^2 + c*x^4]*(8*a^2 - 3*b^2*x^4 + 2*a*x^2*(b + 4*c*x^2)))/(a^2*x^6) - (b*(b^2 - 4*a*c)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/(32*a^(5/2))","A",1
928,1,141,161,0.0951317,"\int \frac{\sqrt{a+b x^2+c x^4}}{x^9} \, dx","Integrate[Sqrt[a + b*x^2 + c*x^4]/x^9,x]","\frac{3 \left(16 a^2 c^2-24 a b^2 c+5 b^4\right) \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)-\frac{2 \sqrt{a} \sqrt{a+b x^2+c x^4} \left(48 a^3+8 a^2 x^2 \left(b+3 c x^2\right)-2 a b x^4 \left(5 b+26 c x^2\right)+15 b^3 x^6\right)}{x^8}}{768 a^{7/2}}","\frac{\left(b^2-4 a c\right) \left(5 b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{256 a^{7/2}}-\frac{\left(5 b^2-4 a c\right) \left(2 a+b x^2\right) \sqrt{a+b x^2+c x^4}}{128 a^3 x^4}+\frac{5 b \left(a+b x^2+c x^4\right)^{3/2}}{48 a^2 x^6}-\frac{\left(a+b x^2+c x^4\right)^{3/2}}{8 a x^8}",1,"((-2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4]*(48*a^3 + 15*b^3*x^6 + 8*a^2*x^2*(b + 3*c*x^2) - 2*a*b*x^4*(5*b + 26*c*x^2)))/x^8 + 3*(5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/(768*a^(7/2))","A",1
929,1,173,199,0.1196542,"\int \frac{\sqrt{a+b x^2+c x^4}}{x^{11}} \, dx","Integrate[Sqrt[a + b*x^2 + c*x^4]/x^11,x]","-\frac{b \left(48 a^2 c^2-40 a b^2 c+7 b^4\right) \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{512 a^{9/2}}-\frac{\sqrt{a+b x^2+c x^4} \left(384 a^4+16 a^3 \left(3 b x^2+8 c x^4\right)-8 a^2 \left(7 b^2 x^4+29 b c x^6+32 c^2 x^8\right)+10 a b^2 x^6 \left(7 b+46 c x^2\right)-105 b^4 x^8\right)}{3840 a^4 x^{10}}","-\frac{b \left(7 b^2-12 a c\right) \left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{512 a^{9/2}}+\frac{b \left(7 b^2-12 a c\right) \left(2 a+b x^2\right) \sqrt{a+b x^2+c x^4}}{256 a^4 x^4}-\frac{\left(35 b^2-32 a c\right) \left(a+b x^2+c x^4\right)^{3/2}}{480 a^3 x^6}+\frac{7 b \left(a+b x^2+c x^4\right)^{3/2}}{80 a^2 x^8}-\frac{\left(a+b x^2+c x^4\right)^{3/2}}{10 a x^{10}}",1,"-1/3840*(Sqrt[a + b*x^2 + c*x^4]*(384*a^4 - 105*b^4*x^8 + 10*a*b^2*x^6*(7*b + 46*c*x^2) + 16*a^3*(3*b*x^2 + 8*c*x^4) - 8*a^2*(7*b^2*x^4 + 29*b*c*x^6 + 32*c^2*x^8)))/(a^4*x^10) - (b*(7*b^4 - 40*a*b^2*c + 48*a^2*c^2)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/(512*a^(9/2))","A",1
930,1,538,395,1.5490114,"\int x^4 \sqrt{a+b x^2+c x^4} \, dx","Integrate[x^4*Sqrt[a + b*x^2 + c*x^4],x]","\frac{4 c x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left(10 a^2 c+a \left(-4 b^2+13 b c x^2+25 c^2 x^4\right)-4 b^3 x^2-b^2 c x^4+18 b c^2 x^6+15 c^3 x^8\right)-i \left(-20 a^2 c^2+37 a b^2 c-29 a b c \sqrt{b^2-4 a c}+8 b^3 \sqrt{b^2-4 a c}-8 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(8 b^2-29 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)}{420 c^3 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","\frac{b x \left(8 b^2-29 a c\right) \sqrt{a+b x^2+c x^4}}{105 c^{5/2} \left(\sqrt{a}+\sqrt{c} x^2\right)}-\frac{\sqrt[4]{a} b \left(8 b^2-29 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)}{105 c^{11/4} \sqrt{a+b x^2+c x^4}}-\frac{2 x \left(2 b^2-5 a c\right) \sqrt{a+b x^2+c x^4}}{105 c^2}+\frac{\sqrt[4]{a} \left(2 \sqrt{a} \sqrt{c} \left(2 b^2-5 a c\right)-29 a b c+8 b^3\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)}{210 c^{11/4} \sqrt{a+b x^2+c x^4}}+\frac{x^3 \left(b+5 c x^2\right) \sqrt{a+b x^2+c x^4}}{35 c}",1,"(4*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*(10*a^2*c - 4*b^3*x^2 - b^2*c*x^4 + 18*b*c^2*x^6 + 15*c^3*x^8 + a*(-4*b^2 + 13*b*c*x^2 + 25*c^2*x^4)) + I*b*(8*b^2 - 29*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*(-8*b^4 + 37*a*b^2*c - 20*a^2*c^2 + 8*b^3*Sqrt[b^2 - 4*a*c] - 29*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])])/(420*c^3*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[a + b*x^2 + c*x^4])","C",1
931,1,479,342,1.2504618,"\int x^2 \sqrt{a+b x^2+c x^4} \, dx","Integrate[x^2*Sqrt[a + b*x^2 + c*x^4],x]","\frac{-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)}{30 c^2 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","-\frac{2 x \left(b^2-3 a c\right) \sqrt{a+b x^2+c x^4}}{15 c^{3/2} \left(\sqrt{a}+\sqrt{c} x^2\right)}-\frac{\sqrt[4]{a} \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+b x^2+c x^4}}+\frac{2 \sqrt[4]{a} \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+b x^2+c x^4}}+\frac{x \left(b+3 c x^2\right) \sqrt{a+b x^2+c x^4}}{15 c}",1,"(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[a + b*x^2 + c*x^4])","C",1
932,1,445,309,0.8222488,"\int \sqrt{a+b x^2+c x^4} \, dx","Integrate[Sqrt[a + b*x^2 + c*x^4],x]","\frac{-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)}{12 c \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","\frac{\sqrt[4]{a} \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+b x^2+c x^4}}-\frac{\sqrt[4]{a} b \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+b x^2+c x^4}}+\frac{b x \sqrt{a+b x^2+c x^4}}{3 \sqrt{c} \left(\sqrt{a}+\sqrt{c} x^2\right)}+\frac{1}{3} x \sqrt{a+b x^2+c x^4}",1,"(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[a + b*x^2 + c*x^4])","C",1
933,1,435,303,0.8105649,"\int \frac{\sqrt{a+b x^2+c x^4}}{x^2} \, dx","Integrate[Sqrt[a + b*x^2 + c*x^4]/x^2,x]","\frac{-i \sqrt{2} x \sqrt{b^2-4 a c} \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\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(\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 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left(a+b x^2+c x^4\right)}{2 x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","\frac{2 \sqrt{c} x \sqrt{a+b x^2+c x^4}}{\sqrt{a}+\sqrt{c} x^2}-\frac{\sqrt{a+b x^2+c x^4}}{x}+\frac{\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)}{2 \sqrt[4]{a} \sqrt[4]{c} \sqrt{a+b x^2+c x^4}}-\frac{2 \sqrt[4]{a} \sqrt[4]{c} \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)}{\sqrt{a+b x^2+c x^4}}",1,"(-2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*(a + b*x^2 + c*x^4) + I*(-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*Sqrt[2]*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[(b + Sqrt[b^2 - 4*a*c] + 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])])/(2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*Sqrt[a + b*x^2 + c*x^4])","C",1
934,1,459,341,0.8959239,"\int \frac{\sqrt{a+b x^2+c x^4}}{x^4} \, dx","Integrate[Sqrt[a + b*x^2 + c*x^4]/x^4,x]","\frac{-4 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left(a+b x^2\right) \left(a+b x^2+c x^4\right)-i x^3 \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 x^3 \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)}{12 a x^3 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","\frac{\sqrt[4]{c} \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 a^{3/4} \sqrt{a+b x^2+c x^4}}-\frac{b \sqrt[4]{c} \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 a^{3/4} \sqrt{a+b x^2+c x^4}}-\frac{b \sqrt{a+b x^2+c x^4}}{3 a x}+\frac{b \sqrt{c} x \sqrt{a+b x^2+c x^4}}{3 a \left(\sqrt{a}+\sqrt{c} x^2\right)}-\frac{\sqrt{a+b x^2+c x^4}}{3 x^3}",1,"(-4*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*(a + b*x^2)*(a + b*x^2 + c*x^4) + I*b*(-b + Sqrt[b^2 - 4*a*c])*x^3*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])*x^3*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*a*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x^3*Sqrt[a + b*x^2 + c*x^4])","C",1
935,1,530,397,1.340744,"\int \frac{\sqrt{a+b x^2+c x^4}}{x^6} \, dx","Integrate[Sqrt[a + b*x^2 + c*x^4]/x^6,x]","\frac{-2 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left(3 a^3+a^2 \left(4 b x^2+9 c x^4\right)+a \left(-b^2 x^4+7 b c x^6+6 c^2 x^8\right)-2 b^2 x^6 \left(b+c x^2\right)\right)-i x^5 \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^5 \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)}{30 a^2 x^5 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","-\frac{\sqrt[4]{c} \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 a^{7/4} \sqrt{a+b x^2+c x^4}}+\frac{2 \sqrt[4]{c} \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 a^{7/4} \sqrt{a+b x^2+c x^4}}+\frac{2 \left(b^2-3 a c\right) \sqrt{a+b x^2+c x^4}}{15 a^2 x}-\frac{2 \sqrt{c} x \left(b^2-3 a c\right) \sqrt{a+b x^2+c x^4}}{15 a^2 \left(\sqrt{a}+\sqrt{c} x^2\right)}-\frac{\sqrt{a+b x^2+c x^4}}{5 x^5}-\frac{b \sqrt{a+b x^2+c x^4}}{15 a x^3}",1,"(-2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*(3*a^3 - 2*b^2*x^6*(b + c*x^2) + a^2*(4*b*x^2 + 9*c*x^4) + a*(-(b^2*x^4) + 7*b*c*x^6 + 6*c^2*x^8)) - I*(b^2 - 3*a*c)*(-b + Sqrt[b^2 - 4*a*c])*x^5*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^5*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*a^2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x^5*Sqrt[a + b*x^2 + c*x^4])","C",1
936,1,192,223,0.2528512,"\int x^7 \left(a+b x^2+c x^4\right)^{3/2} \, dx","Integrate[x^7*(a + b*x^2 + c*x^4)^(3/2),x]","\frac{-\frac{\left(16 a c-21 b^2+30 b c x^2\right) \left(a+b x^2+c x^4\right)^{5/2}}{40 c^2}+\frac{7 \left(4 a b c-3 b^3\right) \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)}{2048 c^{9/2}}+x^4 \left(a+b x^2+c x^4\right)^{5/2}}{14 c}","-\frac{3 b \left(b^2-4 a c\right)^2 \left(3 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)}{4096 c^{11/2}}+\frac{3 b \left(b^2-4 a c\right) \left(3 b^2-4 a c\right) \left(b+2 c x^2\right) \sqrt{a+b x^2+c x^4}}{2048 c^5}-\frac{b \left(3 b^2-4 a c\right) \left(b+2 c x^2\right) \left(a+b x^2+c x^4\right)^{3/2}}{256 c^4}+\frac{\left(-16 a c+21 b^2-30 b c x^2\right) \left(a+b x^2+c x^4\right)^{5/2}}{560 c^3}+\frac{x^4 \left(a+b x^2+c x^4\right)^{5/2}}{14 c}",1,"(x^4*(a + b*x^2 + c*x^4)^(5/2) - ((-21*b^2 + 16*a*c + 30*b*c*x^2)*(a + b*x^2 + c*x^4)^(5/2))/(40*c^2) + (7*(-3*b^3 + 4*a*b*c)*(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])]))/(2048*c^(9/2)))/(14*c)","A",1
937,1,175,204,0.1647397,"\int x^5 \left(a+b x^2+c x^4\right)^{3/2} \, dx","Integrate[x^5*(a + b*x^2 + c*x^4)^(3/2),x]","\frac{\frac{\left(7 b^2-4 a c\right) \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)}{512 c^{7/2}}+x^2 \left(a+b x^2+c x^4\right)^{5/2}-\frac{7 b \left(a+b x^2+c x^4\right)^{5/2}}{10 c}}{12 c}","\frac{\left(b^2-4 a c\right)^2 \left(7 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)}{2048 c^{9/2}}-\frac{\left(b^2-4 a c\right) \left(7 b^2-4 a c\right) \left(b+2 c x^2\right) \sqrt{a+b x^2+c x^4}}{1024 c^4}+\frac{\left(7 b^2-4 a c\right) \left(b+2 c x^2\right) \left(a+b x^2+c x^4\right)^{3/2}}{384 c^3}-\frac{7 b \left(a+b x^2+c x^4\right)^{5/2}}{120 c^2}+\frac{x^2 \left(a+b x^2+c x^4\right)^{5/2}}{12 c}",1,"((-7*b*(a + b*x^2 + c*x^4)^(5/2))/(10*c) + x^2*(a + b*x^2 + c*x^4)^(5/2) + ((7*b^2 - 4*a*c)*(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])]))/(512*c^(7/2)))/(12*c)","A",1
938,1,149,150,0.1422314,"\int x^3 \left(a+b x^2+c x^4\right)^{3/2} \, dx","Integrate[x^3*(a + b*x^2 + c*x^4)^(3/2),x]","-\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)}{512 c^{7/2}}-\frac{b \left(b+2 c x^2\right) \left(a+b x^2+c x^4\right)^{3/2}}{32 c^2}+\frac{\left(a+b x^2+c x^4\right)^{5/2}}{10 c}","-\frac{3 b \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)}{512 c^{7/2}}+\frac{3 b \left(b^2-4 a c\right) \left(b+2 c x^2\right) \sqrt{a+b x^2+c x^4}}{256 c^3}-\frac{b \left(b+2 c x^2\right) \left(a+b x^2+c x^4\right)^{3/2}}{32 c^2}+\frac{\left(a+b x^2+c x^4\right)^{5/2}}{10 c}",1,"-1/32*(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)/(10*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])]))/(512*c^(7/2))","A",1
939,1,126,124,0.0855414,"\int x \left(a+b x^2+c x^4\right)^{3/2} \, dx","Integrate[x*(a + b*x^2 + c*x^4)^(3/2),x]","\frac{\frac{3 \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)}{8 c^{3/2}}+2 \left(b+2 c x^2\right) \left(a+b x^2+c x^4\right)^{3/2}}{32 c}","\frac{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)}{256 c^{5/2}}-\frac{3 \left(b^2-4 a c\right) \left(b+2 c x^2\right) \sqrt{a+b x^2+c x^4}}{128 c^2}+\frac{\left(b+2 c x^2\right) \left(a+b x^2+c x^4\right)^{3/2}}{16 c}",1,"(2*(b + 2*c*x^2)*(a + b*x^2 + c*x^4)^(3/2) + (3*(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])]))/(8*c^(3/2)))/(32*c)","A",1
940,1,143,155,0.1454024,"\int \frac{\left(a+b x^2+c x^4\right)^{3/2}}{x} \, dx","Integrate[(a + b*x^2 + c*x^4)^(3/2)/x,x]","\frac{1}{96} \left(-48 a^{3/2} \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)-\frac{3 b \left(b^2-12 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{c^{3/2}}+\frac{2 \sqrt{a+b x^2+c x^4} \left(8 c \left(4 a+c x^4\right)+3 b^2+14 b c x^2\right)}{c}\right)","-\frac{1}{2} a^{3/2} \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)-\frac{b \left(b^2-12 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{32 c^{3/2}}+\frac{\left(8 a c+b^2+2 b c x^2\right) \sqrt{a+b x^2+c x^4}}{16 c}+\frac{1}{6} \left(a+b x^2+c x^4\right)^{3/2}",1,"((2*Sqrt[a + b*x^2 + c*x^4]*(3*b^2 + 14*b*c*x^2 + 8*c*(4*a + c*x^4)))/c - 48*a^(3/2)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])] - (3*b*(b^2 - 12*a*c)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/c^(3/2))/96","A",1
941,1,134,150,0.1182677,"\int \frac{\left(a+b x^2+c x^4\right)^{3/2}}{x^3} \, dx","Integrate[(a + b*x^2 + c*x^4)^(3/2)/x^3,x]","\frac{1}{16} \left(\frac{3 \left(4 a c+b^2\right) \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{\sqrt{c}}+\frac{2 \sqrt{a+b x^2+c x^4} \left(-4 a+5 b x^2+2 c x^4\right)}{x^2}-12 \sqrt{a} b \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)\right)","\frac{3 \left(4 a c+b^2\right) \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{16 \sqrt{c}}-\frac{\left(a+b x^2+c x^4\right)^{3/2}}{2 x^2}+\frac{3}{8} \left(3 b+2 c x^2\right) \sqrt{a+b x^2+c x^4}-\frac{3}{4} \sqrt{a} b \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)",1,"((2*Sqrt[a + b*x^2 + c*x^4]*(-4*a + 5*b*x^2 + 2*c*x^4))/x^2 - 12*Sqrt[a]*b*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])] + (3*(b^2 + 4*a*c)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/Sqrt[c])/16","A",1
942,1,134,151,0.1695013,"\int \frac{\left(a+b x^2+c x^4\right)^{3/2}}{x^5} \, dx","Integrate[(a + b*x^2 + c*x^4)^(3/2)/x^5,x]","\frac{1}{16} \left(-\frac{3 \left(4 a c+b^2\right) \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{\sqrt{a}}-\frac{2 \sqrt{a+b x^2+c x^4} \left(2 a+5 b x^2-4 c x^4\right)}{x^4}+12 b \sqrt{c} \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)\right)","-\frac{3 \left(4 a c+b^2\right) \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{16 \sqrt{a}}-\frac{\left(a+b x^2+c x^4\right)^{3/2}}{4 x^4}-\frac{3 \left(b-2 c x^2\right) \sqrt{a+b x^2+c x^4}}{8 x^2}+\frac{3}{4} b \sqrt{c} \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)",1,"((-2*(2*a + 5*b*x^2 - 4*c*x^4)*Sqrt[a + b*x^2 + c*x^4])/x^4 - (3*(b^2 + 4*a*c)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/Sqrt[a] + 12*b*Sqrt[c]*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/16","A",1
943,1,149,163,0.2158711,"\int \frac{\left(a+b x^2+c x^4\right)^{3/2}}{x^7} \, dx","Integrate[(a + b*x^2 + c*x^4)^(3/2)/x^7,x]","\frac{1}{96} \left(\frac{3 b \left(b^2-12 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{a^{3/2}}-\frac{2 \sqrt{a+b x^2+c x^4} \left(8 a^2+14 a b x^2+32 a c x^4+3 b^2 x^4\right)}{a x^6}+48 c^{3/2} \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)\right)","\frac{b \left(b^2-12 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{32 a^{3/2}}-\frac{\left(x^2 \left(8 a c+b^2\right)+2 a b\right) \sqrt{a+b x^2+c x^4}}{16 a x^4}+\frac{1}{2} c^{3/2} \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)-\frac{\left(a+b x^2+c x^4\right)^{3/2}}{6 x^6}",1,"((-2*Sqrt[a + b*x^2 + c*x^4]*(8*a^2 + 14*a*b*x^2 + 3*b^2*x^4 + 32*a*c*x^4))/(a*x^6) + (3*b*(b^2 - 12*a*c)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/a^(3/2) + 48*c^(3/2)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/96","A",1
944,1,138,133,0.1667439,"\int \frac{\left(a+b x^2+c x^4\right)^{3/2}}{x^9} \, dx","Integrate[(a + b*x^2 + c*x^4)^(3/2)/x^9,x]","-\frac{\frac{3 \left(b^2-4 a c\right) \left(x^4 \left(b^2-4 a c\right) \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+b x^2\right) \sqrt{a+b x^2+c x^4}\right)}{8 a^{3/2} x^4}+\frac{2 \left(2 a+b x^2\right) \left(a+b x^2+c x^4\right)^{3/2}}{x^8}}{32 a}","-\frac{3 \left(b^2-4 a c\right)^2 \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{256 a^{5/2}}+\frac{3 \left(b^2-4 a c\right) \left(2 a+b x^2\right) \sqrt{a+b x^2+c x^4}}{128 a^2 x^4}-\frac{\left(2 a+b x^2\right) \left(a+b x^2+c x^4\right)^{3/2}}{16 a x^8}",1,"-1/32*((2*(2*a + b*x^2)*(a + b*x^2 + c*x^4)^(3/2))/x^8 + (3*(b^2 - 4*a*c)*(-2*Sqrt[a]*(2*a + b*x^2)*Sqrt[a + b*x^2 + c*x^4] + (b^2 - 4*a*c)*x^4*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])]))/(8*a^(3/2)*x^4))/a","A",1
945,1,167,162,0.1402812,"\int \frac{\left(a+b x^2+c x^4\right)^{3/2}}{x^{11}} \, dx","Integrate[(a + b*x^2 + c*x^4)^(3/2)/x^11,x]","\frac{b \left(16 a^{3/2} \left(2 a+b x^2\right) \left(a+b x^2+c x^4\right)^{3/2}-3 x^4 \left(b^2-4 a c\right) \left(2 \sqrt{a} \left(2 a+b x^2\right) \sqrt{a+b x^2+c x^4}-x^4 \left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)\right)\right)}{512 a^{7/2} x^8}-\frac{\left(a+b x^2+c x^4\right)^{5/2}}{10 a x^{10}}","\frac{3 b \left(b^2-4 a c\right)^2 \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{512 a^{7/2}}-\frac{3 b \left(b^2-4 a c\right) \left(2 a+b x^2\right) \sqrt{a+b x^2+c x^4}}{256 a^3 x^4}+\frac{b \left(2 a+b x^2\right) \left(a+b x^2+c x^4\right)^{3/2}}{32 a^2 x^8}-\frac{\left(a+b x^2+c x^4\right)^{5/2}}{10 a x^{10}}",1,"-1/10*(a + b*x^2 + c*x^4)^(5/2)/(a*x^10) + (b*(16*a^(3/2)*(2*a + b*x^2)*(a + b*x^2 + c*x^4)^(3/2) - 3*(b^2 - 4*a*c)*x^4*(2*Sqrt[a]*(2*a + b*x^2)*Sqrt[a + b*x^2 + c*x^4] - (b^2 - 4*a*c)*x^4*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])))/(512*a^(7/2)*x^8)","A",1
946,1,206,216,0.2123593,"\int \frac{\left(a+b x^2+c x^4\right)^{3/2}}{x^{13}} \, dx","Integrate[(a + b*x^2 + c*x^4)^(3/2)/x^13,x]","-\frac{\frac{\left(\frac{7 b^2}{2}-2 a c\right) \left(16 a^{3/2} \left(2 a+b x^2\right) \left(a+b x^2+c x^4\right)^{3/2}-3 x^4 \left(b^2-4 a c\right) \left(2 \sqrt{a} \left(2 a+b x^2\right) \sqrt{a+b x^2+c x^4}-x^4 \left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)\right)\right)}{256 a^{7/2} x^8}+\frac{\left(a+b x^2+c x^4\right)^{5/2}}{x^{12}}-\frac{7 b \left(a+b x^2+c x^4\right)^{5/2}}{10 a x^{10}}}{12 a}","-\frac{\left(b^2-4 a c\right)^2 \left(7 b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{2048 a^{9/2}}+\frac{\left(b^2-4 a c\right) \left(7 b^2-4 a c\right) \left(2 a+b x^2\right) \sqrt{a+b x^2+c x^4}}{1024 a^4 x^4}-\frac{\left(7 b^2-4 a c\right) \left(2 a+b x^2\right) \left(a+b x^2+c x^4\right)^{3/2}}{384 a^3 x^8}+\frac{7 b \left(a+b x^2+c x^4\right)^{5/2}}{120 a^2 x^{10}}-\frac{\left(a+b x^2+c x^4\right)^{5/2}}{12 a x^{12}}",1,"-1/12*((a + b*x^2 + c*x^4)^(5/2)/x^12 - (7*b*(a + b*x^2 + c*x^4)^(5/2))/(10*a*x^10) + (((7*b^2)/2 - 2*a*c)*(16*a^(3/2)*(2*a + b*x^2)*(a + b*x^2 + c*x^4)^(3/2) - 3*(b^2 - 4*a*c)*x^4*(2*Sqrt[a]*(2*a + b*x^2)*Sqrt[a + b*x^2 + c*x^4] - (b^2 - 4*a*c)*x^4*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])))/(256*a^(7/2)*x^8))/a","A",1
947,1,657,495,2.2076863,"\int x^4 \left(a+b x^2+c x^4\right)^{3/2} \, dx","Integrate[x^4*(a + b*x^2 + c*x^4)^(3/2),x]","\frac{-4 i b \left(27 a^2 c^2-15 a b^2 c+2 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)+2 c x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left(60 a^3 c^2+a^2 c \left(-51 b^2+92 b c x^2+255 c^2 x^4\right)+a \left(8 b^4-57 b^3 c x^2-14 b^2 c^2 x^4+367 b c^3 x^6+300 c^4 x^8\right)+x^2 \left(8 b^5+2 b^4 c x^2-b^3 c^2 x^4+145 b^2 c^3 x^6+245 b c^4 x^8+105 c^5 x^{10}\right)\right)+i \left(60 a^3 c^3-159 a^2 b^2 c^2+108 a^2 b c^2 \sqrt{b^2-4 a c}+68 a b^4 c+8 b^5 \sqrt{b^2-4 a c}-60 a b^3 c \sqrt{b^2-4 a c}-8 b^6\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)}{2310 c^4 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","-\frac{\sqrt[4]{a} \left(\sqrt{a} \sqrt{c} \left(60 a^2 c^2-51 a b^2 c+8 b^4\right)+8 b \left(2 b^2-9 a c\right) \left(b^2-3 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)}{2310 c^{15/4} \sqrt{a+b x^2+c x^4}}+\frac{x \left(60 a^2 c^2-51 a b^2 c+8 b^4\right) \sqrt{a+b x^2+c x^4}}{1155 c^3}-\frac{8 b x \left(2 b^2-9 a c\right) \left(b^2-3 a c\right) \sqrt{a+b x^2+c x^4}}{1155 c^{7/2} \left(\sqrt{a}+\sqrt{c} x^2\right)}+\frac{8 \sqrt[4]{a} b \left(2 b^2-9 a c\right) \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)}{1155 c^{15/4} \sqrt{a+b x^2+c x^4}}-\frac{x^3 \left(10 c x^2 \left(b^2-3 a c\right)+b \left(a c+2 b^2\right)\right) \sqrt{a+b x^2+c x^4}}{385 c^2}+\frac{x^3 \left(b+3 c x^2\right) \left(a+b x^2+c x^4\right)^{3/2}}{33 c}",1,"(2*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*(60*a^3*c^2 + a^2*c*(-51*b^2 + 92*b*c*x^2 + 255*c^2*x^4) + a*(8*b^4 - 57*b^3*c*x^2 - 14*b^2*c^2*x^4 + 367*b*c^3*x^6 + 300*c^4*x^8) + x^2*(8*b^5 + 2*b^4*c*x^2 - b^3*c^2*x^4 + 145*b^2*c^3*x^6 + 245*b*c^4*x^8 + 105*c^5*x^10)) - (4*I)*b*(2*b^4 - 15*a*b^2*c + 27*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^6 + 68*a*b^4*c - 159*a^2*b^2*c^2 + 60*a^3*c^3 + 8*b^5*Sqrt[b^2 - 4*a*c] - 60*a*b^3*c*Sqrt[b^2 - 4*a*c] + 108*a^2*b*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])])/(2310*c^4*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[a + b*x^2 + c*x^4])","C",1
948,1,602,443,1.8894937,"\int x^2 \left(a+b x^2+c x^4\right)^{3/2} \, dx","Integrate[x^2*(a + b*x^2 + c*x^4)^(3/2),x]","\frac{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)}{1260 c^3 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","\frac{x \left(84 a^2 c^2-57 a b^2 c+8 b^4\right) \sqrt{a+b x^2+c x^4}}{315 c^{5/2} \left(\sqrt{a}+\sqrt{c} x^2\right)}+\frac{\sqrt[4]{a} \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+b x^2+c x^4}}-\frac{\sqrt[4]{a} \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+b x^2+c x^4}}-\frac{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+b x^2+c x^4}}{315 c^2}+\frac{x \left(3 b+7 c x^2\right) \left(a+b x^2+c x^4\right)^{3/2}}{63 c}",1,"(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[a + b*x^2 + c*x^4])","C",1
949,1,533,381,1.5169618,"\int \left(a+b x^2+c x^4\right)^{3/2} \, dx","Integrate[(a + b*x^2 + c*x^4)^(3/2),x]","\frac{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)}{70 c^2 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","-\frac{2 b x \left(b^2-8 a c\right) \sqrt{a+b x^2+c x^4}}{35 c^{3/2} \left(\sqrt{a}+\sqrt{c} x^2\right)}-\frac{\sqrt[4]{a} \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+b x^2+c x^4}}+\frac{2 \sqrt[4]{a} b \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+b x^2+c x^4}}+\frac{x \left(10 a c+b^2+3 b c x^2\right) \sqrt{a+b x^2+c x^4}}{35 c}+\frac{1}{7} x \left(a+b x^2+c x^4\right)^{3/2}",1,"(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[a + b*x^2 + c*x^4])","C",1
950,1,505,361,1.2735846,"\int \frac{\left(a+b x^2+c x^4\right)^{3/2}}{x^2} \, dx","Integrate[(a + b*x^2 + c*x^4)^(3/2)/x^2,x]","\frac{4 c \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left(-5 a^2-3 a b x^2-4 a c x^4+2 b^2 x^4+3 b c x^6+c^2 x^8\right)+i x \left(12 a c+b^2\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}+12 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)}{20 c x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","\frac{\sqrt[4]{a} \left(8 \sqrt{a} b \sqrt{c}+12 a c+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)}{10 c^{3/4} \sqrt{a+b x^2+c x^4}}-\frac{\sqrt[4]{a} \left(12 a c+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}} 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)}{5 c^{3/4} \sqrt{a+b x^2+c x^4}}+\frac{x \left(12 a c+b^2\right) \sqrt{a+b x^2+c x^4}}{5 \sqrt{c} \left(\sqrt{a}+\sqrt{c} x^2\right)}-\frac{\left(a+b x^2+c x^4\right)^{3/2}}{x}+\frac{1}{5} x \left(7 b+6 c x^2\right) \sqrt{a+b x^2+c x^4}",1,"(4*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*(-5*a^2 - 3*a*b*x^2 + 2*b^2*x^4 - 4*a*c*x^4 + 3*b*c*x^6 + c^2*x^8) + I*(b^2 + 12*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] + 12*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])])/(20*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*Sqrt[a + b*x^2 + c*x^4])","C",1
951,1,473,353,0.9503225,"\int \frac{\left(a+b x^2+c x^4\right)^{3/2}}{x^4} \, dx","Integrate[(a + b*x^2 + c*x^4)^(3/2)/x^4,x]","\frac{2 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left(-a^2-5 a b x^2-4 b^2 x^4-3 b c x^6+c^2 x^8\right)-i x^3 \left(4 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)+4 i b x^3 \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)}{6 x^3 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","\frac{\left(8 \sqrt{a} b \sqrt{c}+4 a c+3 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)}{6 \sqrt[4]{a} \sqrt[4]{c} \sqrt{a+b x^2+c x^4}}-\frac{\left(3 b-2 c x^2\right) \sqrt{a+b x^2+c x^4}}{3 x}+\frac{8 b \sqrt{c} x \sqrt{a+b x^2+c x^4}}{3 \left(\sqrt{a}+\sqrt{c} x^2\right)}-\frac{8 \sqrt[4]{a} b \sqrt[4]{c} \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 \sqrt{a+b x^2+c x^4}}-\frac{\left(a+b x^2+c x^4\right)^{3/2}}{3 x^3}",1,"(2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*(-a^2 - 5*a*b*x^2 - 4*b^2*x^4 - 3*b*c*x^6 + c^2*x^8) + (4*I)*b*(-b + Sqrt[b^2 - 4*a*c])*x^3*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 + 4*b*Sqrt[b^2 - 4*a*c])*x^3*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])])/(6*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x^3*Sqrt[a + b*x^2 + c*x^4])","C",1
952,1,527,400,1.3645954,"\int \frac{\left(a+b x^2+c x^4\right)^{3/2}}{x^6} \, dx","Integrate[(a + b*x^2 + c*x^4)^(3/2)/x^6,x]","\frac{-4 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left(a^3+a^2 \left(3 b x^2+8 c x^4\right)+a \left(3 b^2 x^4+9 b c x^6+7 c^2 x^8\right)+b^2 x^6 \left(b+c x^2\right)\right)+i x^5 \left(12 a c+b^2\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^5 \left(b^2 \sqrt{b^2-4 a c}+12 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)}{20 a x^5 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","\frac{\sqrt[4]{c} \left(8 \sqrt{a} b \sqrt{c}+12 a c+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)}{10 a^{3/4} \sqrt{a+b x^2+c x^4}}-\frac{\sqrt[4]{c} \left(12 a c+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}} 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)}{5 a^{3/4} \sqrt{a+b x^2+c x^4}}-\frac{\left(12 a c+b^2\right) \sqrt{a+b x^2+c x^4}}{5 a x}+\frac{\sqrt{c} x \left(12 a c+b^2\right) \sqrt{a+b x^2+c x^4}}{5 a \left(\sqrt{a}+\sqrt{c} x^2\right)}-\frac{\left(a+b x^2+c x^4\right)^{3/2}}{5 x^5}-\frac{\left(b-6 c x^2\right) \sqrt{a+b x^2+c x^4}}{5 x^3}",1,"(-4*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*(a^3 + b^2*x^6*(b + c*x^2) + a^2*(3*b*x^2 + 8*c*x^4) + a*(3*b^2*x^4 + 9*b*c*x^6 + 7*c^2*x^8)) + I*(b^2 + 12*a*c)*(-b + Sqrt[b^2 - 4*a*c])*x^5*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] + 12*a*c*Sqrt[b^2 - 4*a*c])*x^5*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])])/(20*a*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x^5*Sqrt[a + b*x^2 + c*x^4])","C",1
953,1,572,447,1.5616505,"\int \frac{\left(a+b x^2+c x^4\right)^{3/2}}{x^8} \, dx","Integrate[(a + b*x^2 + c*x^4)^(3/2)/x^8,x]","\frac{i x^7 \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)-2 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left(5 a^4+a^3 \left(13 b x^2+20 c x^4\right)+3 a^2 \left(3 b^2 x^4+13 b c x^6+5 c^2 x^8\right)+a b x^6 \left(-b^2+17 b c x^2+16 c^2 x^4\right)-2 b^3 x^8 \left(b+c x^2\right)\right)-i b x^7 \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)}{70 a^2 x^7 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","-\frac{\sqrt[4]{c} \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 a^{7/4} \sqrt{a+b x^2+c x^4}}+\frac{2 b \sqrt[4]{c} \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 a^{7/4} \sqrt{a+b x^2+c x^4}}+\frac{2 b \left(b^2-8 a c\right) \sqrt{a+b x^2+c x^4}}{35 a^2 x}-\frac{2 b \sqrt{c} x \left(b^2-8 a c\right) \sqrt{a+b x^2+c x^4}}{35 a^2 \left(\sqrt{a}+\sqrt{c} x^2\right)}-\frac{\left(b^2-20 a c\right) \sqrt{a+b x^2+c x^4}}{35 a x^3}-\frac{\left(a+b x^2+c x^4\right)^{3/2}}{7 x^7}-\frac{3 \left(b+10 c x^2\right) \sqrt{a+b x^2+c x^4}}{35 x^5}",1,"(-2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*(5*a^4 - 2*b^3*x^8*(b + c*x^2) + a^3*(13*b*x^2 + 20*c*x^4) + a*b*x^6*(-b^2 + 17*b*c*x^2 + 16*c^2*x^4) + 3*a^2*(3*b^2*x^4 + 13*b*c*x^6 + 5*c^2*x^8)) - I*b*(b^2 - 8*a*c)*(-b + Sqrt[b^2 - 4*a*c])*x^7*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])*x^7*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*a^2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x^7*Sqrt[a + b*x^2 + c*x^4])","C",1
954,1,59,48,0.0639555,"\int \sqrt{3-2 x^2-x^4} \, dx","Integrate[Sqrt[3 - 2*x^2 - x^4],x]","\frac{1}{3} \left(\sqrt{-x^4-2 x^2+3} x-4 i F\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{3}}\right)\right|-3\right)-2 i E\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{3}}\right)\right|-3\right)\right)","\frac{1}{3} \sqrt{-x^4-2 x^2+3} x+\frac{4 F\left(\sin ^{-1}(x)|-\frac{1}{3}\right)}{\sqrt{3}}-\frac{2 E\left(\sin ^{-1}(x)|-\frac{1}{3}\right)}{\sqrt{3}}",1,"(x*Sqrt[3 - 2*x^2 - x^4] - (2*I)*EllipticE[I*ArcSinh[x/Sqrt[3]], -3] - (4*I)*EllipticF[I*ArcSinh[x/Sqrt[3]], -3])/3","C",1
955,1,104,121,0.063106,"\int \frac{x^7}{\sqrt{a+b x^2+c x^4}} \, dx","Integrate[x^7/Sqrt[a + b*x^2 + c*x^4],x]","\frac{\left(36 a b c-15 b^3\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} \sqrt{a+b x^2+c x^4} \left(8 c \left(c x^4-2 a\right)+15 b^2-10 b c x^2\right)}{96 c^{7/2}}","-\frac{b \left(5 b^2-12 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{32 c^{7/2}}+\frac{\left(-16 a c+15 b^2-10 b c x^2\right) \sqrt{a+b x^2+c x^4}}{48 c^3}+\frac{x^4 \sqrt{a+b x^2+c x^4}}{6 c}",1,"(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]*(15*b^2 - 10*b*c*x^2 + 8*c*(-2*a + c*x^4)) + (-15*b^3 + 36*a*b*c)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(96*c^(7/2))","A",1
956,1,88,104,0.0342996,"\int \frac{x^5}{\sqrt{a+b x^2+c x^4}} \, dx","Integrate[x^5/Sqrt[a + b*x^2 + c*x^4],x]","\frac{\left(3 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(2 c x^2-3 b\right) \sqrt{a+b x^2+c x^4}}{16 c^{5/2}}","\frac{\left(3 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)}{16 c^{5/2}}-\frac{3 b \sqrt{a+b x^2+c x^4}}{8 c^2}+\frac{x^2 \sqrt{a+b x^2+c x^4}}{4 c}",1,"(2*Sqrt[c]*(-3*b + 2*c*x^2)*Sqrt[a + b*x^2 + c*x^4] + (3*b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(16*c^(5/2))","A",1
957,1,68,68,0.0140038,"\int \frac{x^3}{\sqrt{a+b x^2+c x^4}} \, dx","Integrate[x^3/Sqrt[a + b*x^2 + c*x^4],x]","\frac{\sqrt{a+b x^2+c x^4}}{2 c}-\frac{b \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{4 c^{3/2}}","\frac{\sqrt{a+b x^2+c x^4}}{2 c}-\frac{b \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{4 c^{3/2}}",1,"Sqrt[a + b*x^2 + c*x^4]/(2*c) - (b*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(4*c^(3/2))","A",1
958,1,43,43,0.0057857,"\int \frac{x}{\sqrt{a+b x^2+c x^4}} \, dx","Integrate[x/Sqrt[a + b*x^2 + c*x^4],x]","\frac{\tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{2 \sqrt{c}}","\frac{\tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{2 \sqrt{c}}",1,"ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])]/(2*Sqrt[c])","A",1
959,1,44,44,0.010554,"\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
960,1,72,72,0.0210714,"\int \frac{1}{x^3 \sqrt{a+b x^2+c x^4}} \, dx","Integrate[1/(x^3*Sqrt[a + b*x^2 + c*x^4]),x]","\frac{b \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{4 a^{3/2}}-\frac{\sqrt{a+b x^2+c x^4}}{2 a x^2}","\frac{b \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{4 a^{3/2}}-\frac{\sqrt{a+b x^2+c x^4}}{2 a x^2}",1,"-1/2*Sqrt[a + b*x^2 + c*x^4]/(a*x^2) + (b*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/(4*a^(3/2))","A",1
961,1,91,108,0.0538662,"\int \frac{1}{x^5 \sqrt{a+b x^2+c x^4}} \, dx","Integrate[1/(x^5*Sqrt[a + b*x^2 + c*x^4]),x]","\frac{\left(4 a c-3 b^2\right) \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{16 a^{5/2}}+\frac{\left(3 b x^2-2 a\right) \sqrt{a+b x^2+c x^4}}{8 a^2 x^4}","-\frac{\left(3 b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{16 a^{5/2}}+\frac{3 b \sqrt{a+b x^2+c x^4}}{8 a^2 x^2}-\frac{\sqrt{a+b x^2+c x^4}}{4 a x^4}",1,"((-2*a + 3*b*x^2)*Sqrt[a + b*x^2 + c*x^4])/(8*a^2*x^4) + ((-3*b^2 + 4*a*c)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/(16*a^(5/2))","A",1
962,1,112,145,0.0788309,"\int \frac{1}{x^7 \sqrt{a+b x^2+c x^4}} \, dx","Integrate[1/(x^7*Sqrt[a + b*x^2 + c*x^4]),x]","\frac{b \left(5 b^2-12 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{32 a^{7/2}}+\frac{\sqrt{a+b x^2+c x^4} \left(-8 a^2+2 a \left(5 b x^2+8 c x^4\right)-15 b^2 x^4\right)}{48 a^3 x^6}","\frac{b \left(5 b^2-12 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{32 a^{7/2}}-\frac{\left(15 b^2-16 a c\right) \sqrt{a+b x^2+c x^4}}{48 a^3 x^2}+\frac{5 b \sqrt{a+b x^2+c x^4}}{24 a^2 x^4}-\frac{\sqrt{a+b x^2+c x^4}}{6 a x^6}",1,"(Sqrt[a + b*x^2 + c*x^4]*(-8*a^2 - 15*b^2*x^4 + 2*a*(5*b*x^2 + 8*c*x^4)))/(48*a^3*x^6) + (b*(5*b^2 - 12*a*c)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/(32*a^(7/2))","A",1
963,1,444,313,0.8603219,"\int \frac{x^4}{\sqrt{a+b x^2+c x^4}} \, dx","Integrate[x^4/Sqrt[a + b*x^2 + c*x^4],x]","\frac{i \left(b \sqrt{b^2-4 a c}+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)+2 c x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left(a+b x^2+c x^4\right)}{6 c^2 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","-\frac{2 b x \sqrt{a+b x^2+c x^4}}{3 c^{3/2} \left(\sqrt{a}+\sqrt{c} x^2\right)}-\frac{\sqrt[4]{a} \left(\sqrt{a} \sqrt{c}+2 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^{7/4} \sqrt{a+b x^2+c x^4}}+\frac{2 \sqrt[4]{a} b \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^{7/4} \sqrt{a+b x^2+c x^4}}+\frac{x \sqrt{a+b x^2+c x^4}}{3 c}",1,"(2*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 + 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])])/(6*c^2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[a + b*x^2 + c*x^4])","C",1
964,1,278,267,0.1326991,"\int \frac{x^2}{\sqrt{a+b x^2+c x^4}} \, dx","Integrate[x^2/Sqrt[a + b*x^2 + c*x^4],x]","\frac{i \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)}{2 \sqrt{2} c \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","\frac{\sqrt[4]{a} \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 c^{3/4} \sqrt{a+b x^2+c x^4}}-\frac{\sqrt[4]{a} \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)}{c^{3/4} \sqrt{a+b x^2+c x^4}}+\frac{x \sqrt{a+b x^2+c x^4}}{\sqrt{c} \left(\sqrt{a}+\sqrt{c} x^2\right)}",1,"((I/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[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[2]*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[a + b*x^2 + c*x^4])","C",1
965,1,186,114,0.0990431,"\int \frac{1}{\sqrt{a+b x^2+c x^4}} \, dx","Integrate[1/Sqrt[a + b*x^2 + c*x^4],x]","-\frac{i \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{a+b x^2+c x^4}}","\frac{\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+b x^2+c x^4}}",1,"((-I)*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[a + b*x^2 + c*x^4])","C",1
966,1,298,294,0.475589,"\int \frac{1}{x^2 \sqrt{a+b x^2+c x^4}} \, dx","Integrate[1/(x^2*Sqrt[a + b*x^2 + c*x^4]),x]","\frac{-\frac{4 \left(a+b x^2+c x^4\right)}{x}+\frac{i \sqrt{2} \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{a+b x^2+c x^4}}","\frac{\sqrt[4]{c} \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+b x^2+c x^4}}-\frac{\sqrt[4]{c} \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+b x^2+c x^4}}+\frac{\sqrt{c} x \sqrt{a+b x^2+c x^4}}{a \left(\sqrt{a}+\sqrt{c} x^2\right)}-\frac{\sqrt{a+b x^2+c x^4}}{a x}",1,"((-4*(a + b*x^2 + c*x^4))/x + (I*Sqrt[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[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[a + b*x^2 + c*x^4])","C",1
967,1,459,345,0.89069,"\int \frac{1}{x^4 \sqrt{a+b x^2+c x^4}} \, dx","Integrate[1/(x^4*Sqrt[a + b*x^2 + c*x^4]),x]","\frac{-2 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left(a-2 b x^2\right) \left(a+b x^2+c x^4\right)+i x^3 \left(b \sqrt{b^2-4 a c}+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 x^3 \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)}{6 a^2 x^3 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","-\frac{\sqrt[4]{c} \left(\sqrt{a} \sqrt{c}+2 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 a^{7/4} \sqrt{a+b x^2+c x^4}}+\frac{2 b \sqrt[4]{c} \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 a^{7/4} \sqrt{a+b x^2+c x^4}}+\frac{2 b \sqrt{a+b x^2+c x^4}}{3 a^2 x}-\frac{2 b \sqrt{c} x \sqrt{a+b x^2+c x^4}}{3 a^2 \left(\sqrt{a}+\sqrt{c} x^2\right)}-\frac{\sqrt{a+b x^2+c x^4}}{3 a x^3}",1,"(-2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*(a - 2*b*x^2)*(a + b*x^2 + c*x^4) - I*b*(-b + Sqrt[b^2 - 4*a*c])*x^3*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 + a*c + b*Sqrt[b^2 - 4*a*c])*x^3*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])])/(6*a^2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x^3*Sqrt[a + b*x^2 + c*x^4])","C",1
968,1,107,124,0.0770732,"\int \frac{x^7}{\sqrt{a+b x^2-c x^4}} \, dx","Integrate[x^7/Sqrt[a + b*x^2 - c*x^4],x]","\frac{-2 \sqrt{c} \sqrt{a+b x^2-c x^4} \left(8 c \left(2 a+c x^4\right)+15 b^2+10 b c x^2\right)-3 b \left(12 a c+5 b^2\right) \tan ^{-1}\left(\frac{b-2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2-c x^4}}\right)}{96 c^{7/2}}","-\frac{b \left(12 a c+5 b^2\right) \tan ^{-1}\left(\frac{b-2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2-c x^4}}\right)}{32 c^{7/2}}-\frac{\left(16 a c+15 b^2+10 b c x^2\right) \sqrt{a+b x^2-c x^4}}{48 c^3}-\frac{x^4 \sqrt{a+b x^2-c x^4}}{6 c}",1,"(-2*Sqrt[c]*Sqrt[a + b*x^2 - c*x^4]*(15*b^2 + 10*b*c*x^2 + 8*c*(2*a + c*x^4)) - 3*b*(5*b^2 + 12*a*c)*ArcTan[(b - 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 - c*x^4])])/(96*c^(7/2))","A",1
969,1,89,107,0.0441656,"\int \frac{x^5}{\sqrt{a+b x^2-c x^4}} \, dx","Integrate[x^5/Sqrt[a + b*x^2 - c*x^4],x]","-\frac{\left(4 a c+3 b^2\right) \tan ^{-1}\left(\frac{b-2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2-c x^4}}\right)}{16 c^{5/2}}-\frac{\left(3 b+2 c x^2\right) \sqrt{a+b x^2-c x^4}}{8 c^2}","-\frac{\left(4 a c+3 b^2\right) \tan ^{-1}\left(\frac{b-2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2-c x^4}}\right)}{16 c^{5/2}}-\frac{3 b \sqrt{a+b x^2-c x^4}}{8 c^2}-\frac{x^2 \sqrt{a+b x^2-c x^4}}{4 c}",1,"-1/8*((3*b + 2*c*x^2)*Sqrt[a + b*x^2 - c*x^4])/c^2 - ((3*b^2 + 4*a*c)*ArcTan[(b - 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 - c*x^4])])/(16*c^(5/2))","A",1
970,1,70,70,0.0163131,"\int \frac{x^3}{\sqrt{a+b x^2-c x^4}} \, dx","Integrate[x^3/Sqrt[a + b*x^2 - c*x^4],x]","-\frac{b \tan ^{-1}\left(\frac{b-2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2-c x^4}}\right)}{4 c^{3/2}}-\frac{\sqrt{a+b x^2-c x^4}}{2 c}","-\frac{b \tan ^{-1}\left(\frac{b-2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2-c x^4}}\right)}{4 c^{3/2}}-\frac{\sqrt{a+b x^2-c x^4}}{2 c}",1,"-1/2*Sqrt[a + b*x^2 - c*x^4]/c - (b*ArcTan[(b - 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 - c*x^4])])/(4*c^(3/2))","A",1
971,1,44,44,0.005587,"\int \frac{x}{\sqrt{a+b x^2-c x^4}} \, dx","Integrate[x/Sqrt[a + b*x^2 - c*x^4],x]","-\frac{\tan ^{-1}\left(\frac{b-2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2-c x^4}}\right)}{2 \sqrt{c}}","-\frac{\tan ^{-1}\left(\frac{b-2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2-c x^4}}\right)}{2 \sqrt{c}}",1,"-1/2*ArcTan[(b - 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 - c*x^4])]/Sqrt[c]","A",1
972,1,46,47,0.014174,"\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{\tan ^{-1}\left(\frac{b x^2-2 a}{2 \sqrt{a} \sqrt{-a+b x^2+c x^4}}\right)}{2 \sqrt{a}}","-\frac{\tan ^{-1}\left(\frac{2 a-b x^2}{2 \sqrt{a} \sqrt{-a+b x^2+c x^4}}\right)}{2 \sqrt{a}}",1,"ArcTan[(-2*a + b*x^2)/(2*Sqrt[a]*Sqrt[-a + b*x^2 + c*x^4])]/(2*Sqrt[a])","A",1
973,1,76,77,0.0221765,"\int \frac{1}{x^3 \sqrt{-a+b x^2+c x^4}} \, dx","Integrate[1/(x^3*Sqrt[-a + b*x^2 + c*x^4]),x]","\frac{b \tan ^{-1}\left(\frac{b x^2-2 a}{2 \sqrt{a} \sqrt{-a+b x^2+c x^4}}\right)}{4 a^{3/2}}+\frac{\sqrt{-a+b x^2+c x^4}}{2 a x^2}","\frac{\sqrt{-a+b x^2+c x^4}}{2 a x^2}-\frac{b \tan ^{-1}\left(\frac{2 a-b x^2}{2 \sqrt{a} \sqrt{-a+b x^2+c x^4}}\right)}{4 a^{3/2}}",1,"Sqrt[-a + b*x^2 + c*x^4]/(2*a*x^2) + (b*ArcTan[(-2*a + b*x^2)/(2*Sqrt[a]*Sqrt[-a + b*x^2 + c*x^4])])/(4*a^(3/2))","A",1
974,1,95,115,0.0455632,"\int \frac{1}{x^5 \sqrt{-a+b x^2+c x^4}} \, dx","Integrate[1/(x^5*Sqrt[-a + b*x^2 + c*x^4]),x]","\frac{\left(4 a c+3 b^2\right) \tan ^{-1}\left(\frac{b x^2-2 a}{2 \sqrt{a} \sqrt{-a+b x^2+c x^4}}\right)}{16 a^{5/2}}+\frac{\left(2 a+3 b x^2\right) \sqrt{-a+b x^2+c x^4}}{8 a^2 x^4}","-\frac{\left(4 a c+3 b^2\right) \tan ^{-1}\left(\frac{2 a-b x^2}{2 \sqrt{a} \sqrt{-a+b x^2+c x^4}}\right)}{16 a^{5/2}}+\frac{3 b \sqrt{-a+b x^2+c x^4}}{8 a^2 x^2}+\frac{\sqrt{-a+b x^2+c x^4}}{4 a x^4}",1,"((2*a + 3*b*x^2)*Sqrt[-a + b*x^2 + c*x^4])/(8*a^2*x^4) + ((3*b^2 + 4*a*c)*ArcTan[(-2*a + b*x^2)/(2*Sqrt[a]*Sqrt[-a + b*x^2 + c*x^4])])/(16*a^(5/2))","A",1
975,1,116,154,0.0758796,"\int \frac{1}{x^7 \sqrt{-a+b x^2+c x^4}} \, dx","Integrate[1/(x^7*Sqrt[-a + b*x^2 + c*x^4]),x]","\frac{b \left(12 a c+5 b^2\right) \tan ^{-1}\left(\frac{b x^2-2 a}{2 \sqrt{a} \sqrt{-a+b x^2+c x^4}}\right)}{32 a^{7/2}}+\frac{\sqrt{-a+b x^2+c x^4} \left(8 a^2+2 a \left(5 b x^2+8 c x^4\right)+15 b^2 x^4\right)}{48 a^3 x^6}","-\frac{b \left(12 a c+5 b^2\right) \tan ^{-1}\left(\frac{2 a-b x^2}{2 \sqrt{a} \sqrt{-a+b x^2+c x^4}}\right)}{32 a^{7/2}}+\frac{\left(16 a c+15 b^2\right) \sqrt{-a+b x^2+c x^4}}{48 a^3 x^2}+\frac{5 b \sqrt{-a+b x^2+c x^4}}{24 a^2 x^4}+\frac{\sqrt{-a+b x^2+c x^4}}{6 a x^6}",1,"(Sqrt[-a + b*x^2 + c*x^4]*(8*a^2 + 15*b^2*x^4 + 2*a*(5*b*x^2 + 8*c*x^4)))/(48*a^3*x^6) + (b*(5*b^2 + 12*a*c)*ArcTan[(-2*a + b*x^2)/(2*Sqrt[a]*Sqrt[-a + b*x^2 + c*x^4])])/(32*a^(7/2))","A",1
976,1,459,409,0.7690469,"\int \frac{x^4}{\sqrt{a+b x^2-c x^4}} \, dx","Integrate[x^4/Sqrt[a + b*x^2 - c*x^4],x]","\frac{i \sqrt{2} \left(b \sqrt{4 a c+b^2}-a c-b^2\right) \sqrt{\frac{\sqrt{4 a c+b^2}+b-2 c x^2}{\sqrt{4 a c+b^2}+b}} \sqrt{\frac{\sqrt{4 a c+b^2}-b+2 c x^2}{\sqrt{4 a c+b^2}-b}} 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 \sqrt{2} b \left(\sqrt{4 a c+b^2}-b\right) \sqrt{\frac{\sqrt{4 a c+b^2}+b-2 c x^2}{\sqrt{4 a c+b^2}+b}} \sqrt{\frac{\sqrt{4 a c+b^2}-b+2 c x^2}{\sqrt{4 a c+b^2}-b}} 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{4 a c+b^2}+b}} \left(-a-b x^2+c x^4\right)}{6 c^2 \sqrt{-\frac{c}{\sqrt{4 a c+b^2}+b}} \sqrt{a+b x^2-c x^4}}","\frac{\sqrt{\sqrt{4 a c+b^2}+b} \left(-b \sqrt{4 a c+b^2}+a c+b^2\right) \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} F\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right)|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right)}{3 \sqrt{2} c^{5/2} \sqrt{a+b x^2-c x^4}}-\frac{b \left(b-\sqrt{4 a c+b^2}\right) \sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} E\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right)|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right)}{3 \sqrt{2} c^{5/2} \sqrt{a+b x^2-c x^4}}-\frac{x \sqrt{a+b x^2-c x^4}}{3 c}",1,"(2*c*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*x*(-a - b*x^2 + c*x^4) - I*Sqrt[2]*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[(-b + Sqrt[b^2 + 4*a*c] + 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])] + I*Sqrt[2]*(-b^2 - 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[(-b + Sqrt[b^2 + 4*a*c] + 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])])/(6*c^2*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*Sqrt[a + b*x^2 - c*x^4])","C",1
977,1,271,377,0.1182367,"\int \frac{x^2}{\sqrt{a+b x^2-c x^4}} \, dx","Integrate[x^2/Sqrt[a + b*x^2 - c*x^4],x]","-\frac{i \left(\sqrt{4 a c+b^2}-b\right) \sqrt{\frac{2 c x^2}{\sqrt{4 a c+b^2}-b}+1} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} \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}}{\sqrt{b^2+4 a c}-b}\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}}{\sqrt{b^2+4 a c}-b}\right)\right)}{2 \sqrt{2} c \sqrt{-\frac{c}{\sqrt{4 a c+b^2}+b}} \sqrt{a+b x^2-c x^4}}","\frac{\left(b-\sqrt{4 a c+b^2}\right) \sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} F\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right)|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right)}{2 \sqrt{2} c^{3/2} \sqrt{a+b x^2-c x^4}}-\frac{\left(b-\sqrt{4 a c+b^2}\right) \sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} E\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right)|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right)}{2 \sqrt{2} c^{3/2} \sqrt{a+b x^2-c x^4}}",1,"((-1/2*I)*(-b + Sqrt[b^2 + 4*a*c])*Sqrt[1 + (2*c*x^2)/(-b + Sqrt[b^2 + 4*a*c])]*Sqrt[1 - (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*(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[2]*c*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*Sqrt[a + b*x^2 - c*x^4])","C",1
978,1,177,169,0.0844059,"\int \frac{1}{\sqrt{a+b x^2-c x^4}} \, dx","Integrate[1/Sqrt[a + b*x^2 - c*x^4],x]","-\frac{i \sqrt{\frac{2 c x^2}{\sqrt{4 a c+b^2}-b}+1} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} 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}}{\sqrt{b^2+4 a c}-b}\right)}{\sqrt{2} \sqrt{-\frac{c}{\sqrt{4 a c+b^2}+b}} \sqrt{a+b x^2-c x^4}}","\frac{\sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} F\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right)|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right)}{\sqrt{2} \sqrt{c} \sqrt{a+b x^2-c x^4}}",1,"((-I)*Sqrt[1 + (2*c*x^2)/(-b + Sqrt[b^2 + 4*a*c])]*Sqrt[1 - (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*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[a + b*x^2 - c*x^4])","C",1
979,1,283,408,0.4279793,"\int \frac{1}{x^2 \sqrt{a+b x^2-c x^4}} \, dx","Integrate[1/(x^2*Sqrt[a + b*x^2 - c*x^4]),x]","\frac{\frac{i \left(\sqrt{4 a c+b^2}-b\right) \sqrt{\frac{4 c x^2}{\sqrt{4 a c+b^2}-b}+2} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} \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{4 a c+b^2}+b}}}-\frac{4 a}{x}-4 b x+4 c x^3}{4 a \sqrt{a+b x^2-c x^4}}","-\frac{\left(b-\sqrt{4 a c+b^2}\right) \sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} F\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right)|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right)}{2 \sqrt{2} a \sqrt{c} \sqrt{a+b x^2-c x^4}}+\frac{\left(b-\sqrt{4 a c+b^2}\right) \sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} E\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right)|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right)}{2 \sqrt{2} a \sqrt{c} \sqrt{a+b x^2-c x^4}}-\frac{\sqrt{a+b x^2-c x^4}}{a x}",1,"((-4*a)/x - 4*b*x + 4*c*x^3 + (I*(-b + Sqrt[b^2 + 4*a*c])*Sqrt[2 + (4*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[a + b*x^2 - c*x^4])","C",1
980,1,472,445,0.7628812,"\int \frac{1}{x^4 \sqrt{a+b x^2-c x^4}} \, dx","Integrate[1/(x^4*Sqrt[a + b*x^2 - c*x^4]),x]","\frac{-2 \sqrt{-\frac{c}{\sqrt{4 a c+b^2}+b}} \left(a-2 b x^2\right) \left(a+b x^2-c x^4\right)+i \sqrt{2} x^3 \left(b \sqrt{4 a c+b^2}-a c-b^2\right) \sqrt{\frac{\sqrt{4 a c+b^2}+b-2 c x^2}{\sqrt{4 a c+b^2}+b}} \sqrt{\frac{\sqrt{4 a c+b^2}-b+2 c x^2}{\sqrt{4 a c+b^2}-b}} 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 \sqrt{2} b x^3 \left(\sqrt{4 a c+b^2}-b\right) \sqrt{\frac{\sqrt{4 a c+b^2}+b-2 c x^2}{\sqrt{4 a c+b^2}+b}} \sqrt{\frac{\sqrt{4 a c+b^2}-b+2 c x^2}{\sqrt{4 a c+b^2}-b}} 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)}{6 a^2 x^3 \sqrt{-\frac{c}{\sqrt{4 a c+b^2}+b}} \sqrt{a+b x^2-c x^4}}","\frac{\sqrt{\sqrt{4 a c+b^2}+b} \left(-b \sqrt{4 a c+b^2}+a c+b^2\right) \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} F\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right)|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right)}{3 \sqrt{2} a^2 \sqrt{c} \sqrt{a+b x^2-c x^4}}-\frac{b \left(b-\sqrt{4 a c+b^2}\right) \sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} E\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right)|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right)}{3 \sqrt{2} a^2 \sqrt{c} \sqrt{a+b x^2-c x^4}}+\frac{2 b \sqrt{a+b x^2-c x^4}}{3 a^2 x}-\frac{\sqrt{a+b x^2-c x^4}}{3 a x^3}",1,"(-2*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*(a - 2*b*x^2)*(a + b*x^2 - c*x^4) - I*Sqrt[2]*b*(-b + Sqrt[b^2 + 4*a*c])*x^3*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])]*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*Sqrt[2]*(-b^2 - a*c + b*Sqrt[b^2 + 4*a*c])*x^3*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])]*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])])/(6*a^2*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*x^3*Sqrt[a + b*x^2 - c*x^4])","C",1
981,1,181,190,0.1937514,"\int \frac{x^9}{\left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[x^9/(a + b*x^2 + c*x^4)^(3/2),x]","\frac{\frac{2 \sqrt{c} \left(4 a^2 c \left(6 c x^2-13 b\right)+a \left(15 b^3-62 b^2 c x^2-20 b c^2 x^4+8 c^3 x^6\right)+b^2 x^2 \left(15 b^2+5 b c x^2-2 c^2 x^4\right)\right)}{\sqrt{a+b x^2+c x^4}}-3 \left(16 a^2 c^2-24 a b^2 c+5 b^4\right) \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{16 c^{7/2} \left(4 a c-b^2\right)}","\frac{3 \left(5 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)}{16 c^{7/2}}-\frac{\left(b \left(15 b^2-52 a c\right)-2 c x^2 \left(5 b^2-12 a c\right)\right) \sqrt{a+b x^2+c x^4}}{8 c^3 \left(b^2-4 a c\right)}-\frac{b x^4 \sqrt{a+b x^2+c x^4}}{c \left(b^2-4 a c\right)}+\frac{x^6 \left(2 a+b x^2\right)}{\left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}",1,"((2*Sqrt[c]*(4*a^2*c*(-13*b + 6*c*x^2) + b^2*x^2*(15*b^2 + 5*b*c*x^2 - 2*c^2*x^4) + a*(15*b^3 - 62*b^2*c*x^2 - 20*b*c^2*x^4 + 8*c^3*x^6)))/Sqrt[a + b*x^2 + c*x^4] - 3*(5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(16*c^(7/2)*(-b^2 + 4*a*c))","A",1
982,1,137,134,0.1172597,"\int \frac{x^7}{\left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[x^7/(a + b*x^2 + c*x^4)^(3/2),x]","\frac{\frac{2 \sqrt{c} \left(8 a^2 c+a \left(-3 b^2+10 b c x^2+4 c^2 x^4\right)-b^2 x^2 \left(3 b+c x^2\right)\right)}{\sqrt{a+b x^2+c x^4}}+3 b \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)}{4 c^{5/2} \left(4 a c-b^2\right)}","\frac{\left(-8 a c+3 b^2-2 b c x^2\right) \sqrt{a+b x^2+c x^4}}{2 c^2 \left(b^2-4 a c\right)}+\frac{x^4 \left(2 a+b x^2\right)}{\left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}-\frac{3 b \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{4 c^{5/2}}",1,"((2*Sqrt[c]*(8*a^2*c - b^2*x^2*(3*b + c*x^2) + a*(-3*b^2 + 10*b*c*x^2 + 4*c^2*x^4)))/Sqrt[a + b*x^2 + c*x^4] + 3*b*(b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(4*c^(5/2)*(-b^2 + 4*a*c))","A",1
983,1,107,115,0.0977731,"\int \frac{x^5}{\left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[x^5/(a + b*x^2 + c*x^4)^(3/2),x]","\frac{\frac{2 \sqrt{c} \left(a \left(b-2 c x^2\right)+b^2 x^2\right)}{\sqrt{a+b x^2+c x^4}}-\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 c^{3/2} \left(4 a c-b^2\right)}","\frac{x^2 \left(2 a+b x^2\right)}{\left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}-\frac{b \sqrt{a+b x^2+c x^4}}{c \left(b^2-4 a c\right)}+\frac{\tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{2 c^{3/2}}",1,"((2*Sqrt[c]*(b^2*x^2 + a*(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])])/(2*c^(3/2)*(-b^2 + 4*a*c))","A",1
984,1,36,36,0.0961062,"\int \frac{x^3}{\left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[x^3/(a + b*x^2 + c*x^4)^(3/2),x]","\frac{2 a+b x^2}{\left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}","\frac{2 a+b x^2}{\left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}",1,"(2*a + b*x^2)/((b^2 - 4*a*c)*Sqrt[a + b*x^2 + c*x^4])","A",1
985,1,37,36,0.0204959,"\int \frac{x}{\left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[x/(a + b*x^2 + c*x^4)^(3/2),x]","\frac{b+2 c x^2}{\left(4 a c-b^2\right) \sqrt{a+b x^2+c x^4}}","-\frac{b+2 c x^2}{\left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}",1,"(b + 2*c*x^2)/((-b^2 + 4*a*c)*Sqrt[a + b*x^2 + c*x^4])","A",1
986,1,89,89,0.1029349,"\int \frac{1}{x \left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[1/(x*(a + b*x^2 + c*x^4)^(3/2)),x]","\frac{-2 a c+b^2+b c x^2}{a \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}-\frac{\tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{2 a^{3/2}}","\frac{-2 a c+b^2+b c x^2}{a \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}-\frac{\tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{2 a^{3/2}}",1,"(b^2 - 2*a*c + b*c*x^2)/(a*(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))","A",1
987,1,137,139,0.084185,"\int \frac{1}{x^3 \left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[1/(x^3*(a + b*x^2 + c*x^4)^(3/2)),x]","\frac{\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)}{x^2 \sqrt{a+b x^2+c x^4}}-3 b \left(b^2-4 a c\right) \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} \left(4 a c-b^2\right)}","\frac{3 b \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}}-\frac{\left(3 b^2-8 a c\right) \sqrt{a+b x^2+c x^4}}{2 a^2 x^2 \left(b^2-4 a c\right)}+\frac{-2 a c+b^2+b c x^2}{a x^2 \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}",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)))/(x^2*Sqrt[a + b*x^2 + c*x^4]) - 3*b*(b^2 - 4*a*c)*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))","A",1
988,1,179,195,0.1259977,"\int \frac{1}{x^5 \left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[1/(x^5*(a + b*x^2 + c*x^4)^(3/2)),x]","\frac{3 \left(16 a^2 c^2-24 a b^2 c+5 b^4\right) \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)+\frac{2 \sqrt{a} \left(-8 a^3 c+2 a^2 \left(b^2+10 b c x^2-12 c^2 x^4\right)+a b x^2 \left(-5 b^2+62 b c x^2+52 c^2 x^4\right)-15 b^3 x^4 \left(b+c x^2\right)\right)}{x^4 \sqrt{a+b x^2+c x^4}}}{16 a^{7/2} \left(4 a c-b^2\right)}","-\frac{3 \left(5 b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{16 a^{7/2}}+\frac{b \left(15 b^2-52 a c\right) \sqrt{a+b x^2+c x^4}}{8 a^3 x^2 \left(b^2-4 a c\right)}-\frac{\left(5 b^2-12 a c\right) \sqrt{a+b x^2+c x^4}}{4 a^2 x^4 \left(b^2-4 a c\right)}+\frac{-2 a c+b^2+b c x^2}{a x^4 \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}",1,"((2*Sqrt[a]*(-8*a^3*c - 15*b^3*x^4*(b + c*x^2) + 2*a^2*(b^2 + 10*b*c*x^2 - 12*c^2*x^4) + a*b*x^2*(-5*b^2 + 62*b*c*x^2 + 52*c^2*x^4)))/(x^4*Sqrt[a + b*x^2 + c*x^4]) + 3*(5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/(16*a^(7/2)*(-b^2 + 4*a*c))","A",1
989,1,489,408,1.2251515,"\int \frac{x^6}{\left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[x^6/(a + b*x^2 + c*x^4)^(3/2),x]","\frac{2 c x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left(a \left(b-2 c x^2\right)+b^2 x^2\right)-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)+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)}{2 c^2 \left(4 a c-b^2\right) \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","\frac{2 x \left(b^2-3 a c\right) \sqrt{a+b x^2+c x^4}}{c^{3/2} \left(b^2-4 a c\right) \left(\sqrt{a}+\sqrt{c} x^2\right)}+\frac{\sqrt[4]{a} \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 c^{7/4} \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}-\frac{2 \sqrt[4]{a} \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)}{c^{7/4} \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}-\frac{b x \sqrt{a+b x^2+c x^4}}{c \left(b^2-4 a c\right)}+\frac{x^3 \left(2 a+b x^2\right)}{\left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}",1,"(2*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*(b^2*x^2 + a*(b - 2*c*x^2)) - 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])])/(2*c^2*(-b^2 + 4*a*c)*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[a + b*x^2 + c*x^4])","C",1
990,1,452,342,0.8389618,"\int \frac{x^4}{\left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[x^4/(a + b*x^2 + c*x^4)^(3/2),x]","\frac{4 c x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left(2 a+b 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)}{4 c \left(b^2-4 a c\right) \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","\frac{\sqrt[4]{a} b \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)}{c^{3/4} \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}-\frac{b x \sqrt{a+b x^2+c x^4}}{\sqrt{c} \left(b^2-4 a c\right) \left(\sqrt{a}+\sqrt{c} x^2\right)}+\frac{x \left(2 a+b x^2\right)}{\left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}-\frac{\sqrt[4]{a} \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 c^{3/4} \left(b-2 \sqrt{a} \sqrt{c}\right) \sqrt{a+b x^2+c x^4}}",1,"(4*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*(2*a + b*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])])/(4*c*(b^2 - 4*a*c)*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[a + b*x^2 + c*x^4])","C",1
991,1,437,341,0.782386,"\int \frac{x^2}{\left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[x^2/(a + b*x^2 + c*x^4)^(3/2),x]","\frac{-2 x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left(b+2 c x^2\right)-i \sqrt{2} \sqrt{b^2-4 a c} \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\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 \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 \left(b^2-4 a c\right) \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","\frac{2 \sqrt{c} x \sqrt{a+b x^2+c x^4}}{\left(b^2-4 a c\right) \left(\sqrt{a}+\sqrt{c} x^2\right)}-\frac{x \left(b+2 c x^2\right)}{\left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}-\frac{2 \sqrt[4]{a} \sqrt[4]{c} \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)}{\left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}+\frac{\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} \left(b-2 \sqrt{a} \sqrt{c}\right) \sqrt{a+b x^2+c x^4}}",1,"(-2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*(b + 2*c*x^2) + I*(-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*Sqrt[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[(b + Sqrt[b^2 - 4*a*c] + 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])])/(2*(b^2 - 4*a*c)*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[a + b*x^2 + c*x^4])","C",1
992,1,456,353,0.8441514,"\int \frac{1}{\left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(a + b*x^2 + c*x^4)^(-3/2),x]","-\frac{-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)}{4 a \left(b^2-4 a c\right) \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","\frac{b \sqrt[4]{c} \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+b x^2+c x^4}}-\frac{\sqrt[4]{c} \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+b x^2+c x^4}}-\frac{b \sqrt{c} x \sqrt{a+b x^2+c x^4}}{a \left(b^2-4 a c\right) \left(\sqrt{a}+\sqrt{c} x^2\right)}+\frac{x \left(-2 a c+b^2+b c x^2\right)}{a \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}",1,"-1/4*(-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[a + b*x^2 + c*x^4])","C",1
993,1,515,428,1.3286772,"\int \frac{1}{x^2 \left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[1/(x^2*(a + b*x^2 + c*x^4)^(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 x \left(b^2-4 a c\right) \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","\frac{\sqrt[4]{c} \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+b x^2+c x^4}}-\frac{2 \sqrt[4]{c} \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+b x^2+c x^4}}-\frac{2 \left(b^2-3 a c\right) \sqrt{a+b x^2+c x^4}}{a^2 x \left(b^2-4 a c\right)}+\frac{2 \sqrt{c} x \left(b^2-3 a c\right) \sqrt{a+b x^2+c x^4}}{a^2 \left(b^2-4 a c\right) \left(\sqrt{a}+\sqrt{c} x^2\right)}+\frac{-2 a c+b^2+b c x^2}{a x \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}",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])]*x*Sqrt[a + b*x^2 + c*x^4])","C",1
994,1,34,50,0.0206386,"\int \frac{x^4}{\sqrt{2+2 a-2 (1+a)+b x^2+c x^4}} \, dx","Integrate[x^4/Sqrt[2 + 2*a - 2*(1 + a) + b*x^2 + c*x^4],x]","\frac{\left(c x^2-2 b\right) \sqrt{x^2 \left(b+c x^2\right)}}{3 c^2 x}","\frac{x \sqrt{b x^2+c x^4}}{3 c}-\frac{2 b \sqrt{b x^2+c x^4}}{3 c^2 x}",1,"((-2*b + c*x^2)*Sqrt[x^2*(b + c*x^2)])/(3*c^2*x)","A",1
995,1,73,58,0.0364752,"\int \frac{x^3}{\sqrt{2+2 a-2 (1+a)+b x^2+c x^4}} \, dx","Integrate[x^3/Sqrt[2 + 2*a - 2*(1 + a) + b*x^2 + c*x^4],x]","\frac{x \left(\sqrt{c} x \left(b+c x^2\right)-b \sqrt{b+c x^2} \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b+c x^2}}\right)\right)}{2 c^{3/2} \sqrt{x^2 \left(b+c x^2\right)}}","\frac{\sqrt{b x^2+c x^4}}{2 c}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{2 c^{3/2}}",1,"(x*(Sqrt[c]*x*(b + c*x^2) - b*Sqrt[b + c*x^2]*ArcTanh[(Sqrt[c]*x)/Sqrt[b + c*x^2]]))/(2*c^(3/2)*Sqrt[x^2*(b + c*x^2)])","A",1
996,1,22,22,0.0054195,"\int \frac{x^2}{\sqrt{2+2 a-2 (1+a)+b x^2+c x^4}} \, dx","Integrate[x^2/Sqrt[2 + 2*a - 2*(1 + a) + b*x^2 + c*x^4],x]","\frac{\sqrt{x^2 \left(b+c x^2\right)}}{c x}","\frac{\sqrt{b x^2+c x^4}}{c x}",1,"Sqrt[x^2*(b + c*x^2)]/(c*x)","A",1
997,1,52,31,0.0119666,"\int \frac{x}{\sqrt{2+2 a-2 (1+a)+b x^2+c x^4}} \, dx","Integrate[x/Sqrt[2 + 2*a - 2*(1 + a) + b*x^2 + c*x^4],x]","\frac{x \sqrt{b+c x^2} \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b+c x^2}}\right)}{\sqrt{c} \sqrt{x^2 \left(b+c x^2\right)}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right)}{\sqrt{c}}",1,"(x*Sqrt[b + c*x^2]*ArcTanh[(Sqrt[c]*x)/Sqrt[b + c*x^2]])/(Sqrt[c]*Sqrt[x^2*(b + c*x^2)])","A",1
998,1,52,30,0.010796,"\int \frac{1}{\sqrt{2+2 a-2 (1+a)+b x^2+c x^4}} \, dx","Integrate[1/Sqrt[2 + 2*a - 2*(1 + a) + b*x^2 + c*x^4],x]","-\frac{x \sqrt{b+c x^2} \tanh ^{-1}\left(\frac{\sqrt{b+c x^2}}{\sqrt{b}}\right)}{\sqrt{b} \sqrt{x^2 \left(b+c x^2\right)}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)}{\sqrt{b}}",1,"-((x*Sqrt[b + c*x^2]*ArcTanh[Sqrt[b + c*x^2]/Sqrt[b]])/(Sqrt[b]*Sqrt[x^2*(b + c*x^2)]))","A",1
999,1,23,23,0.0074047,"\int \frac{1}{x \sqrt{2+2 a-2 (1+a)+b x^2+c x^4}} \, dx","Integrate[1/(x*Sqrt[2 + 2*a - 2*(1 + a) + b*x^2 + c*x^4]),x]","-\frac{\sqrt{x^2 \left(b+c x^2\right)}}{b x^2}","-\frac{\sqrt{b x^2+c x^4}}{b x^2}",1,"-(Sqrt[x^2*(b + c*x^2)]/(b*x^2))","A",1
1000,1,68,59,0.0644791,"\int \frac{1}{x^2 \sqrt{2+2 a-2 (1+a)+b x^2+c x^4}} \, dx","Integrate[1/(x^2*Sqrt[2 + 2*a - 2*(1 + a) + b*x^2 + c*x^4]),x]","\frac{c \sqrt{x^2 \left(b+c x^2\right)} \left(\frac{\tanh ^{-1}\left(\sqrt{\frac{c x^2}{b}+1}\right)}{2 \sqrt{\frac{c x^2}{b}+1}}-\frac{b}{2 c x^2}\right)}{b^2 x}","\frac{c \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)}{2 b^{3/2}}-\frac{\sqrt{b x^2+c x^4}}{2 b x^3}",1,"(c*Sqrt[x^2*(b + c*x^2)]*(-1/2*b/(c*x^2) + ArcTanh[Sqrt[1 + (c*x^2)/b]]/(2*Sqrt[1 + (c*x^2)/b])))/(b^2*x)","A",1
1001,1,35,52,0.0147252,"\int \frac{1}{x^3 \sqrt{2+2 a-2 (1+a)+b x^2+c x^4}} \, dx","Integrate[1/(x^3*Sqrt[2 + 2*a - 2*(1 + a) + b*x^2 + c*x^4]),x]","\frac{\sqrt{x^2 \left(b+c x^2\right)} \left(2 c x^2-b\right)}{3 b^2 x^4}","\frac{2 c \sqrt{b x^2+c x^4}}{3 b^2 x^2}-\frac{\sqrt{b x^2+c x^4}}{3 b x^4}",1,"(Sqrt[x^2*(b + c*x^2)]*(-b + 2*c*x^2))/(3*b^2*x^4)","A",1
1002,1,44,87,0.0118876,"\int \frac{1}{x^4 \sqrt{2+2 a-2 (1+a)+b x^2+c x^4}} \, dx","Integrate[1/(x^4*Sqrt[2 + 2*a - 2*(1 + a) + b*x^2 + c*x^4]),x]","-\frac{c^2 \sqrt{x^2 \left(b+c x^2\right)} \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};\frac{c x^2}{b}+1\right)}{b^3 x}","-\frac{3 c^2 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right)}{8 b^{5/2}}+\frac{3 c \sqrt{b x^2+c x^4}}{8 b^2 x^3}-\frac{\sqrt{b x^2+c x^4}}{4 b x^5}",1,"-((c^2*Sqrt[x^2*(b + c*x^2)]*Hypergeometric2F1[1/2, 3, 3/2, 1 + (c*x^2)/b])/(b^3*x))","C",1
1003,1,62,108,0.0233263,"\int \frac{x^4}{\sqrt{a+(2+2 b-2 (1+b)) x^2+c x^4}} \, dx","Integrate[x^4/Sqrt[a + (2 + 2*b - 2*(1 + b))*x^2 + c*x^4],x]","\frac{x \left(-a \sqrt{\frac{c x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^4}{a}\right)+a+c x^4\right)}{3 c \sqrt{a+c x^4}}","\frac{x \sqrt{a+c x^4}}{3 c}-\frac{a^{3/4} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+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}{2}\right)}{6 c^{5/4} \sqrt{a+c x^4}}",1,"(x*(a + c*x^4 - a*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^4)/a)]))/(3*c*Sqrt[a + c*x^4])","C",1
1004,1,18,18,0.0036879,"\int \frac{x^3}{\sqrt{a+(2+2 b-2 (1+b)) x^2+c x^4}} \, dx","Integrate[x^3/Sqrt[a + (2 + 2*b - 2*(1 + b))*x^2 + c*x^4],x]","\frac{\sqrt{a+c x^4}}{2 c}","\frac{\sqrt{a+c x^4}}{2 c}",1,"Sqrt[a + c*x^4]/(2*c)","A",1
1005,1,51,210,0.0091048,"\int \frac{x^2}{\sqrt{a+(2+2 b-2 (1+b)) x^2+c x^4}} \, dx","Integrate[x^2/Sqrt[a + (2 + 2*b - 2*(1 + b))*x^2 + c*x^4],x]","\frac{x^3 \sqrt{\frac{c x^4}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^4}{a}\right)}{3 \sqrt{a+c x^4}}","\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+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}{2}\right)}{2 c^{3/4} \sqrt{a+c x^4}}-\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+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}{2}\right)}{c^{3/4} \sqrt{a+c x^4}}+\frac{x \sqrt{a+c x^4}}{\sqrt{c} \left(\sqrt{a}+\sqrt{c} x^2\right)}",1,"(x^3*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, -((c*x^4)/a)])/(3*Sqrt[a + c*x^4])","C",1
1006,1,30,30,0.005648,"\int \frac{x}{\sqrt{a+(2+2 b-2 (1+b)) x^2+c x^4}} \, dx","Integrate[x/Sqrt[a + (2 + 2*b - 2*(1 + b))*x^2 + c*x^4],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right)}{2 \sqrt{c}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right)}{2 \sqrt{c}}",1,"ArcTanh[(Sqrt[c]*x^2)/Sqrt[a + c*x^4]]/(2*Sqrt[c])","A",1
1007,1,74,88,0.0345614,"\int \frac{1}{\sqrt{a+(2+2 b-2 (1+b)) x^2+c x^4}} \, dx","Integrate[1/Sqrt[a + (2 + 2*b - 2*(1 + b))*x^2 + c*x^4],x]","-\frac{i \sqrt{\frac{c x^4}{a}+1} F\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \sqrt{a+c x^4}}","\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+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}{2}\right)}{2 \sqrt[4]{a} \sqrt[4]{c} \sqrt{a+c x^4}}",1,"((-I)*Sqrt[1 + (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])/(Sqrt[(I*Sqrt[c])/Sqrt[a]]*Sqrt[a + c*x^4])","C",1
1008,1,27,27,0.0055921,"\int \frac{1}{x \sqrt{a+(2+2 b-2 (1+b)) x^2+c x^4}} \, dx","Integrate[1/(x*Sqrt[a + (2 + 2*b - 2*(1 + b))*x^2 + c*x^4]),x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right)}{2 \sqrt{a}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right)}{2 \sqrt{a}}",1,"-1/2*ArcTanh[Sqrt[a + c*x^4]/Sqrt[a]]/Sqrt[a]","A",1
1009,1,49,232,0.0089422,"\int \frac{1}{x^2 \sqrt{a+(2+2 b-2 (1+b)) x^2+c x^4}} \, dx","Integrate[1/(x^2*Sqrt[a + (2 + 2*b - 2*(1 + b))*x^2 + c*x^4]),x]","-\frac{\sqrt{\frac{c x^4}{a}+1} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{c x^4}{a}\right)}{x \sqrt{a+c x^4}}","\frac{\sqrt[4]{c} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+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}{2}\right)}{2 a^{3/4} \sqrt{a+c x^4}}-\frac{\sqrt[4]{c} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+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}{2}\right)}{a^{3/4} \sqrt{a+c x^4}}-\frac{\sqrt{a+c x^4}}{a x}+\frac{\sqrt{c} x \sqrt{a+c x^4}}{a \left(\sqrt{a}+\sqrt{c} x^2\right)}",1,"-((Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[-1/4, 1/2, 3/4, -((c*x^4)/a)])/(x*Sqrt[a + c*x^4]))","C",1
1010,1,21,21,0.0046729,"\int \frac{1}{x^3 \sqrt{a+(2+2 b-2 (1+b)) x^2+c x^4}} \, dx","Integrate[1/(x^3*Sqrt[a + (2 + 2*b - 2*(1 + b))*x^2 + c*x^4]),x]","-\frac{\sqrt{a+c x^4}}{2 a x^2}","-\frac{\sqrt{a+c x^4}}{2 a x^2}",1,"-1/2*Sqrt[a + c*x^4]/(a*x^2)","A",1
1011,1,51,110,0.0090105,"\int \frac{1}{x^4 \sqrt{a+(2+2 b-2 (1+b)) x^2+c x^4}} \, dx","Integrate[1/(x^4*Sqrt[a + (2 + 2*b - 2*(1 + b))*x^2 + c*x^4]),x]","-\frac{\sqrt{\frac{c x^4}{a}+1} \, _2F_1\left(-\frac{3}{4},\frac{1}{2};\frac{1}{4};-\frac{c x^4}{a}\right)}{3 x^3 \sqrt{a+c x^4}}","-\frac{c^{3/4} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+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}{2}\right)}{6 a^{5/4} \sqrt{a+c x^4}}-\frac{\sqrt{a+c x^4}}{3 a x^3}",1,"-1/3*(Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[-3/4, 1/2, 1/4, -((c*x^4)/a)])/(x^3*Sqrt[a + c*x^4])","C",1
1012,1,62,73,0.0280721,"\int \frac{x^4}{\sqrt{a+b x^2+(2+2 c-2 (1+c)) x^4}} \, dx","Integrate[x^4/Sqrt[a + b*x^2 + (2 + 2*c - 2*(1 + c))*x^4],x]","\frac{3 a^2 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)+\sqrt{b} x \sqrt{a+b x^2} \left(2 b x^2-3 a\right)}{8 b^{5/2}}","\frac{3 a^2 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 b^{5/2}}-\frac{3 a x \sqrt{a+b x^2}}{8 b^2}+\frac{x^3 \sqrt{a+b x^2}}{4 b}",1,"(Sqrt[b]*x*Sqrt[a + b*x^2]*(-3*a + 2*b*x^2) + 3*a^2*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*b^(5/2))","A",1
1013,1,27,36,0.0139016,"\int \frac{x^3}{\sqrt{a+b x^2+(2+2 c-2 (1+c)) x^4}} \, dx","Integrate[x^3/Sqrt[a + b*x^2 + (2 + 2*c - 2*(1 + c))*x^4],x]","\frac{\left(b x^2-2 a\right) \sqrt{a+b x^2}}{3 b^2}","\frac{\left(a+b x^2\right)^{3/2}}{3 b^2}-\frac{a \sqrt{a+b x^2}}{b^2}",1,"((-2*a + b*x^2)*Sqrt[a + b*x^2])/(3*b^2)","A",1
1014,1,49,49,0.018808,"\int \frac{x^2}{\sqrt{a+b x^2+(2+2 c-2 (1+c)) x^4}} \, dx","Integrate[x^2/Sqrt[a + b*x^2 + (2 + 2*c - 2*(1 + c))*x^4],x]","\frac{x \sqrt{a+b x^2}}{2 b}-\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 b^{3/2}}","\frac{x \sqrt{a+b x^2}}{2 b}-\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 b^{3/2}}",1,"(x*Sqrt[a + b*x^2])/(2*b) - (a*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(3/2))","A",1
1015,1,15,15,0.0021748,"\int \frac{x}{\sqrt{a+b x^2+(2+2 c-2 (1+c)) x^4}} \, dx","Integrate[x/Sqrt[a + b*x^2 + (2 + 2*c - 2*(1 + c))*x^4],x]","\frac{\sqrt{a+b x^2}}{b}","\frac{\sqrt{a+b x^2}}{b}",1,"Sqrt[a + b*x^2]/b","A",1
1016,1,25,25,0.004894,"\int \frac{1}{\sqrt{a+b x^2+(2+2 c-2 (1+c)) x^4}} \, dx","Integrate[1/Sqrt[a + b*x^2 + (2 + 2*c - 2*(1 + c))*x^4],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{\sqrt{b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{\sqrt{b}}",1,"ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]]/Sqrt[b]","A",1
1017,1,25,25,0.0056435,"\int \frac{1}{x \sqrt{a+b x^2+(2+2 c-2 (1+c)) x^4}} \, dx","Integrate[1/(x*Sqrt[a + b*x^2 + (2 + 2*c - 2*(1 + c))*x^4]),x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{\sqrt{a}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{\sqrt{a}}",1,"-(ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]/Sqrt[a])","A",1
1018,1,19,19,0.0037359,"\int \frac{1}{x^2 \sqrt{a+b x^2+(2+2 c-2 (1+c)) x^4}} \, dx","Integrate[1/(x^2*Sqrt[a + b*x^2 + (2 + 2*c - 2*(1 + c))*x^4]),x]","-\frac{\sqrt{a+b x^2}}{a x}","-\frac{\sqrt{a+b x^2}}{a x}",1,"-(Sqrt[a + b*x^2]/(a*x))","A",1
1019,1,61,50,0.0540627,"\int \frac{1}{x^3 \sqrt{a+b x^2+(2+2 c-2 (1+c)) x^4}} \, dx","Integrate[1/(x^3*Sqrt[a + b*x^2 + (2 + 2*c - 2*(1 + c))*x^4]),x]","\frac{b \sqrt{a+b x^2} \left(\frac{\tanh ^{-1}\left(\sqrt{\frac{b x^2}{a}+1}\right)}{2 \sqrt{\frac{b x^2}{a}+1}}-\frac{a}{2 b x^2}\right)}{a^2}","\frac{b \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{2 a^{3/2}}-\frac{\sqrt{a+b x^2}}{2 a x^2}",1,"(b*Sqrt[a + b*x^2]*(-1/2*a/(b*x^2) + ArcTanh[Sqrt[1 + (b*x^2)/a]]/(2*Sqrt[1 + (b*x^2)/a])))/a^2","A",1
1020,1,29,44,0.0063149,"\int \frac{1}{x^4 \sqrt{a+b x^2+(2+2 c-2 (1+c)) x^4}} \, dx","Integrate[1/(x^4*Sqrt[a + b*x^2 + (2 + 2*c - 2*(1 + c))*x^4]),x]","-\frac{\left(a-2 b x^2\right) \sqrt{a+b x^2}}{3 a^2 x^3}","\frac{2 b \sqrt{a+b x^2}}{3 a^2 x}-\frac{\sqrt{a+b x^2}}{3 a x^3}",1,"-1/3*((a - 2*b*x^2)*Sqrt[a + b*x^2])/(a^2*x^3)","A",1
1021,1,16,16,0.0023635,"\int \frac{x^4}{\sqrt{2+2 a-2 (1+a)+c x^4}} \, dx","Integrate[x^4/Sqrt[2 + 2*a - 2*(1 + a) + c*x^4],x]","\frac{x^5}{3 \sqrt{c x^4}}","\frac{x^5}{3 \sqrt{c x^4}}",1,"x^5/(3*Sqrt[c*x^4])","A",1
1022,1,16,16,0.0018489,"\int \frac{x^3}{\sqrt{2+2 a-2 (1+a)+c x^4}} \, dx","Integrate[x^3/Sqrt[2 + 2*a - 2*(1 + a) + c*x^4],x]","\frac{x^4}{2 \sqrt{c x^4}}","\frac{x^4}{2 \sqrt{c x^4}}",1,"x^4/(2*Sqrt[c*x^4])","A",1
1023,1,13,13,0.001329,"\int \frac{x^2}{\sqrt{2+2 a-2 (1+a)+c x^4}} \, dx","Integrate[x^2/Sqrt[2 + 2*a - 2*(1 + a) + c*x^4],x]","\frac{x^3}{\sqrt{c x^4}}","\frac{x^3}{\sqrt{c x^4}}",1,"x^3/Sqrt[c*x^4]","A",1
1024,1,15,15,0.0016321,"\int \frac{x}{\sqrt{2+2 a-2 (1+a)+c x^4}} \, dx","Integrate[x/Sqrt[2 + 2*a - 2*(1 + a) + c*x^4],x]","\frac{x^2 \log (x)}{\sqrt{c x^4}}","\frac{x^2 \log (x)}{\sqrt{c x^4}}",1,"(x^2*Log[x])/Sqrt[c*x^4]","A",1
1025,1,12,12,0.0012829,"\int \frac{1}{\sqrt{2+2 a-2 (1+a)+c x^4}} \, dx","Integrate[1/Sqrt[2 + 2*a - 2*(1 + a) + c*x^4],x]","-\frac{x}{\sqrt{c x^4}}","-\frac{x}{\sqrt{c x^4}}",1,"-(x/Sqrt[c*x^4])","A",1
1026,1,13,13,0.0013684,"\int \frac{1}{x \sqrt{2+2 a-2 (1+a)+c x^4}} \, dx","Integrate[1/(x*Sqrt[2 + 2*a - 2*(1 + a) + c*x^4]),x]","-\frac{1}{2 \sqrt{c x^4}}","-\frac{1}{2 \sqrt{c x^4}}",1,"-1/2*1/Sqrt[c*x^4]","A",1
1027,1,16,16,0.0018604,"\int \frac{1}{x^2 \sqrt{2+2 a-2 (1+a)+c x^4}} \, dx","Integrate[1/(x^2*Sqrt[2 + 2*a - 2*(1 + a) + c*x^4]),x]","-\frac{1}{3 x \sqrt{c x^4}}","-\frac{1}{3 x \sqrt{c x^4}}",1,"-1/3*1/(x*Sqrt[c*x^4])","A",1
1028,1,17,16,0.0036028,"\int \frac{1}{x^3 \sqrt{2+2 a-2 (1+a)+c x^4}} \, dx","Integrate[1/(x^3*Sqrt[2 + 2*a - 2*(1 + a) + c*x^4]),x]","-\frac{c x^2}{4 \left(c x^4\right)^{3/2}}","-\frac{1}{4 x^2 \sqrt{c x^4}}",1,"-1/4*(c*x^2)/(c*x^4)^(3/2)","A",1
1029,1,15,16,0.0023519,"\int \frac{1}{x^4 \sqrt{2+2 a-2 (1+a)+c x^4}} \, dx","Integrate[1/(x^4*Sqrt[2 + 2*a - 2*(1 + a) + c*x^4]),x]","-\frac{c x}{5 \left(c x^4\right)^{3/2}}","-\frac{1}{5 x^3 \sqrt{c x^4}}",1,"-1/5*(c*x)/(c*x^4)^(3/2)","A",1
1030,1,12,12,0.0007876,"\int \frac{x^4}{\sqrt{a+(2+2 c-2 (1+c)) x^4}} \, dx","Integrate[x^4/Sqrt[a + (2 + 2*c - 2*(1 + c))*x^4],x]","\frac{x^5}{5 \sqrt{a}}","\frac{x^5}{5 \sqrt{a}}",1,"x^5/(5*Sqrt[a])","A",1
1031,1,12,12,0.0006321,"\int \frac{x^3}{\sqrt{a+(2+2 c-2 (1+c)) x^4}} \, dx","Integrate[x^3/Sqrt[a + (2 + 2*c - 2*(1 + c))*x^4],x]","\frac{x^4}{4 \sqrt{a}}","\frac{x^4}{4 \sqrt{a}}",1,"x^4/(4*Sqrt[a])","A",1
1032,1,12,12,0.0006143,"\int \frac{x^2}{\sqrt{a+(2+2 c-2 (1+c)) x^4}} \, dx","Integrate[x^2/Sqrt[a + (2 + 2*c - 2*(1 + c))*x^4],x]","\frac{x^3}{3 \sqrt{a}}","\frac{x^3}{3 \sqrt{a}}",1,"x^3/(3*Sqrt[a])","A",1
1033,1,12,12,0.0006151,"\int \frac{x}{\sqrt{a+(2+2 c-2 (1+c)) x^4}} \, dx","Integrate[x/Sqrt[a + (2 + 2*c - 2*(1 + c))*x^4],x]","\frac{x^2}{2 \sqrt{a}}","\frac{x^2}{2 \sqrt{a}}",1,"x^2/(2*Sqrt[a])","A",1
1034,1,7,7,0.0004746,"\int \frac{1}{\sqrt{a+(2+2 c-2 (1+c)) x^4}} \, dx","Integrate[1/Sqrt[a + (2 + 2*c - 2*(1 + c))*x^4],x]","\frac{x}{\sqrt{a}}","\frac{x}{\sqrt{a}}",1,"x/Sqrt[a]","A",1
1035,1,8,8,0.0004769,"\int \frac{1}{x \sqrt{a+(2+2 c-2 (1+c)) x^4}} \, dx","Integrate[1/(x*Sqrt[a + (2 + 2*c - 2*(1 + c))*x^4]),x]","\frac{\log (x)}{\sqrt{a}}","\frac{\log (x)}{\sqrt{a}}",1,"Log[x]/Sqrt[a]","A",1
1036,1,10,10,0.0005403,"\int \frac{1}{x^2 \sqrt{a+(2+2 c-2 (1+c)) x^4}} \, dx","Integrate[1/(x^2*Sqrt[a + (2 + 2*c - 2*(1 + c))*x^4]),x]","-\frac{1}{\sqrt{a} x}","-\frac{1}{\sqrt{a} x}",1,"-(1/(Sqrt[a]*x))","A",1
1037,1,12,12,0.0005353,"\int \frac{1}{x^3 \sqrt{a+(2+2 c-2 (1+c)) x^4}} \, dx","Integrate[1/(x^3*Sqrt[a + (2 + 2*c - 2*(1 + c))*x^4]),x]","-\frac{1}{2 \sqrt{a} x^2}","-\frac{1}{2 \sqrt{a} x^2}",1,"-1/2*1/(Sqrt[a]*x^2)","A",1
1038,1,12,12,0.0005702,"\int \frac{1}{x^4 \sqrt{a+(2+2 c-2 (1+c)) x^4}} \, dx","Integrate[1/(x^4*Sqrt[a + (2 + 2*c - 2*(1 + c))*x^4]),x]","-\frac{1}{3 \sqrt{a} x^3}","-\frac{1}{3 \sqrt{a} x^3}",1,"-1/3*1/(Sqrt[a]*x^3)","A",1
1039,1,18,12,0.0196322,"\int \frac{1}{\sqrt{3-2 x^2-x^4}} \, dx","Integrate[1/Sqrt[3 - 2*x^2 - x^4],x]","-i F\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{3}}\right)\right|-3\right)","\frac{F\left(\sin ^{-1}(x)|-\frac{1}{3}\right)}{\sqrt{3}}",1,"(-I)*EllipticF[I*ArcSinh[x/Sqrt[3]], -3]","C",1
1040,1,87,39,0.1133247,"\int \frac{1}{\sqrt{-1+5 x^2-x^4}} \, dx","Integrate[1/Sqrt[-1 + 5*x^2 - x^4],x]","\frac{\sqrt{-2 x^2-\sqrt{21}+5} \sqrt{\left(\sqrt{21}-5\right) x^2+2} F\left(\sin ^{-1}\left(\sqrt{\frac{1}{2} \left(5+\sqrt{21}\right)} x\right)|\frac{23}{2}-\frac{5 \sqrt{21}}{2}\right)}{2 \sqrt{-x^4+5 x^2-1}}","-\frac{F\left(\cos ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{21}}} x\right)|\frac{1}{42} \left(21+5 \sqrt{21}\right)\right)}{\sqrt[4]{21}}",1,"(Sqrt[5 - Sqrt[21] - 2*x^2]*Sqrt[2 + (-5 + Sqrt[21])*x^2]*EllipticF[ArcSin[Sqrt[(5 + Sqrt[21])/2]*x], 23/2 - (5*Sqrt[21])/2])/(2*Sqrt[-1 + 5*x^2 - x^4])","B",1
1041,1,25,31,0.0069616,"\int x^{5/2} \left(a+b x^2+c x^4\right) \, dx","Integrate[x^(5/2)*(a + b*x^2 + c*x^4),x]","\frac{2 x^{7/2} \left(165 a+105 b x^2+77 c x^4\right)}{1155}","\frac{2}{7} a x^{7/2}+\frac{2}{11} b x^{11/2}+\frac{2}{15} c x^{15/2}",1,"(2*x^(7/2)*(165*a + 105*b*x^2 + 77*c*x^4))/1155","A",1
1042,1,25,31,0.0087265,"\int x^{3/2} \left(a+b x^2+c x^4\right) \, dx","Integrate[x^(3/2)*(a + b*x^2 + c*x^4),x]","\frac{2}{585} x^{5/2} \left(117 a+65 b x^2+45 c x^4\right)","\frac{2}{5} a x^{5/2}+\frac{2}{9} b x^{9/2}+\frac{2}{13} c x^{13/2}",1,"(2*x^(5/2)*(117*a + 65*b*x^2 + 45*c*x^4))/585","A",1
1043,1,25,31,0.0072816,"\int \sqrt{x} \left(a+b x^2+c x^4\right) \, dx","Integrate[Sqrt[x]*(a + b*x^2 + c*x^4),x]","\frac{2}{231} x^{3/2} \left(77 a+33 b x^2+21 c x^4\right)","\frac{2}{3} a x^{3/2}+\frac{2}{7} b x^{7/2}+\frac{2}{11} c x^{11/2}",1,"(2*x^(3/2)*(77*a + 33*b*x^2 + 21*c*x^4))/231","A",1
1044,1,25,29,0.0079649,"\int \frac{a+b x^2+c x^4}{\sqrt{x}} \, dx","Integrate[(a + b*x^2 + c*x^4)/Sqrt[x],x]","\frac{2}{45} \sqrt{x} \left(45 a+9 b x^2+5 c x^4\right)","2 a \sqrt{x}+\frac{2}{5} b x^{5/2}+\frac{2}{9} c x^{9/2}",1,"(2*Sqrt[x]*(45*a + 9*b*x^2 + 5*c*x^4))/45","A",1
1045,1,25,29,0.0091189,"\int \frac{a+b x^2+c x^4}{x^{3/2}} \, dx","Integrate[(a + b*x^2 + c*x^4)/x^(3/2),x]","\frac{2 \left(-21 a+7 b x^2+3 c x^4\right)}{21 \sqrt{x}}","-\frac{2 a}{\sqrt{x}}+\frac{2}{3} b x^{3/2}+\frac{2}{7} c x^{7/2}",1,"(2*(-21*a + 7*b*x^2 + 3*c*x^4))/(21*Sqrt[x])","A",1
1046,1,25,29,0.0085025,"\int \frac{a+b x^2+c x^4}{x^{5/2}} \, dx","Integrate[(a + b*x^2 + c*x^4)/x^(5/2),x]","\frac{2 \left(-5 a+15 b x^2+3 c x^4\right)}{15 x^{3/2}}","-\frac{2 a}{3 x^{3/2}}+2 b \sqrt{x}+\frac{2}{5} c x^{5/2}",1,"(2*(-5*a + 15*b*x^2 + 3*c*x^4))/(15*x^(3/2))","A",1
1047,1,25,29,0.0090084,"\int \frac{a+b x^2+c x^4}{x^{7/2}} \, dx","Integrate[(a + b*x^2 + c*x^4)/x^(7/2),x]","\frac{2 \left(-3 a-15 b x^2+5 c x^4\right)}{15 x^{5/2}}","-\frac{2 a}{5 x^{5/2}}-\frac{2 b}{\sqrt{x}}+\frac{2}{3} c x^{3/2}",1,"(2*(-3*a - 15*b*x^2 + 5*c*x^4))/(15*x^(5/2))","A",1
1048,1,64,64,3.6961632,"\int x^{5/2} \left(a+b x^2+c x^4\right)^2 \, dx","Integrate[x^(5/2)*(a + b*x^2 + c*x^4)^2,x]","\frac{2}{7} a^2 x^{7/2}+\frac{2}{15} x^{15/2} \left(2 a c+b^2\right)+\frac{4}{11} a b x^{11/2}+\frac{4}{19} b c x^{19/2}+\frac{2}{23} c^2 x^{23/2}","\frac{2}{7} a^2 x^{7/2}+\frac{2}{15} x^{15/2} \left(2 a c+b^2\right)+\frac{4}{11} a b x^{11/2}+\frac{4}{19} b c x^{19/2}+\frac{2}{23} c^2 x^{23/2}",1,"(2*a^2*x^(7/2))/7 + (4*a*b*x^(11/2))/11 + (2*(b^2 + 2*a*c)*x^(15/2))/15 + (4*b*c*x^(19/2))/19 + (2*c^2*x^(23/2))/23","A",1
1049,1,66,64,0.0566576,"\int x^{3/2} \left(a+b x^2+c x^4\right)^2 \, dx","Integrate[x^(3/2)*(a + b*x^2 + c*x^4)^2,x]","2 \left(\frac{1}{5} a^2 x^{5/2}+\frac{1}{13} x^{13/2} \left(2 a c+b^2\right)+\frac{2}{9} a b x^{9/2}+\frac{2}{17} b c x^{17/2}+\frac{1}{21} c^2 x^{21/2}\right)","\frac{2}{5} a^2 x^{5/2}+\frac{2}{13} x^{13/2} \left(2 a c+b^2\right)+\frac{4}{9} a b x^{9/2}+\frac{4}{17} b c x^{17/2}+\frac{2}{21} c^2 x^{21/2}",1,"2*((a^2*x^(5/2))/5 + (2*a*b*x^(9/2))/9 + ((b^2 + 2*a*c)*x^(13/2))/13 + (2*b*c*x^(17/2))/17 + (c^2*x^(21/2))/21)","A",1
1050,1,50,64,3.3838952,"\int \sqrt{x} \left(a+b x^2+c x^4\right)^2 \, dx","Integrate[Sqrt[x]*(a + b*x^2 + c*x^4)^2,x]","\frac{2 x^{3/2} \left(7315 a^2+1995 x^4 \left(2 a c+b^2\right)+6270 a b x^2+2926 b c x^6+1155 c^2 x^8\right)}{21945}","\frac{2}{3} a^2 x^{3/2}+\frac{2}{11} x^{11/2} \left(2 a c+b^2\right)+\frac{4}{7} a b x^{7/2}+\frac{4}{15} b c x^{15/2}+\frac{2}{19} c^2 x^{19/2}",1,"(2*x^(3/2)*(7315*a^2 + 6270*a*b*x^2 + 1995*(b^2 + 2*a*c)*x^4 + 2926*b*c*x^6 + 1155*c^2*x^8))/21945","A",1
1051,1,63,62,0.0424461,"\int \frac{\left(a+b x^2+c x^4\right)^2}{\sqrt{x}} \, dx","Integrate[(a + b*x^2 + c*x^4)^2/Sqrt[x],x]","2 \left(a^2 \sqrt{x}+\frac{1}{9} x^{9/2} \left(2 a c+b^2\right)+\frac{2}{5} a b x^{5/2}+\frac{2}{13} b c x^{13/2}+\frac{1}{17} c^2 x^{17/2}\right)","2 a^2 \sqrt{x}+\frac{2}{9} x^{9/2} \left(2 a c+b^2\right)+\frac{4}{5} a b x^{5/2}+\frac{4}{13} b c x^{13/2}+\frac{2}{17} c^2 x^{17/2}",1,"2*(a^2*Sqrt[x] + (2*a*b*x^(5/2))/5 + ((b^2 + 2*a*c)*x^(9/2))/9 + (2*b*c*x^(13/2))/13 + (c^2*x^(17/2))/17)","A",1
1052,1,54,62,0.0443769,"\int \frac{\left(a+b x^2+c x^4\right)^2}{x^{3/2}} \, dx","Integrate[(a + b*x^2 + c*x^4)^2/x^(3/2),x]","\frac{2 \left(-1155 a^2+110 a \left(7 b x^2+3 c x^4\right)+165 b^2 x^4+210 b c x^6+77 c^2 x^8\right)}{1155 \sqrt{x}}","-\frac{2 a^2}{\sqrt{x}}+\frac{2}{7} x^{7/2} \left(2 a c+b^2\right)+\frac{4}{3} a b x^{3/2}+\frac{4}{11} b c x^{11/2}+\frac{2}{15} c^2 x^{15/2}",1,"(2*(-1155*a^2 + 165*b^2*x^4 + 210*b*c*x^6 + 77*c^2*x^8 + 110*a*(7*b*x^2 + 3*c*x^4)))/(1155*Sqrt[x])","A",1
1053,1,53,62,0.0653165,"\int \frac{\left(a+b x^2+c x^4\right)^2}{x^{5/2}} \, dx","Integrate[(a + b*x^2 + c*x^4)^2/x^(5/2),x]","\frac{-390 a^2+468 a \left(5 b x^2+c x^4\right)+234 b^2 x^4+260 b c x^6+90 c^2 x^8}{585 x^{3/2}}","-\frac{2 a^2}{3 x^{3/2}}+\frac{2}{5} x^{5/2} \left(2 a c+b^2\right)+4 a b \sqrt{x}+\frac{4}{9} b c x^{9/2}+\frac{2}{13} c^2 x^{13/2}",1,"(-390*a^2 + 234*b^2*x^4 + 260*b*c*x^6 + 90*c^2*x^8 + 468*a*(5*b*x^2 + c*x^4))/(585*x^(3/2))","A",1
1054,1,50,62,0.0487915,"\int \frac{\left(a+b x^2+c x^4\right)^2}{x^{7/2}} \, dx","Integrate[(a + b*x^2 + c*x^4)^2/x^(7/2),x]","\frac{2 \left(-231 a^2+385 x^4 \left(2 a c+b^2\right)-2310 a b x^2+330 b c x^6+105 c^2 x^8\right)}{1155 x^{5/2}}","-\frac{2 a^2}{5 x^{5/2}}+\frac{2}{3} x^{3/2} \left(2 a c+b^2\right)-\frac{4 a b}{\sqrt{x}}+\frac{4}{7} b c x^{7/2}+\frac{2}{11} c^2 x^{11/2}",1,"(2*(-231*a^2 - 2310*a*b*x^2 + 385*(b^2 + 2*a*c)*x^4 + 330*b*c*x^6 + 105*c^2*x^8))/(1155*x^(5/2))","A",1
1055,1,103,103,3.7243745,"\int x^{5/2} \left(a+b x^2+c x^4\right)^3 \, dx","Integrate[x^(5/2)*(a + b*x^2 + c*x^4)^3,x]","\frac{2}{7} a^3 x^{7/2}+\frac{6}{11} a^2 b x^{11/2}+\frac{6}{23} c x^{23/2} \left(a c+b^2\right)+\frac{2}{19} b x^{19/2} \left(6 a c+b^2\right)+\frac{2}{5} a x^{15/2} \left(a c+b^2\right)+\frac{2}{9} b c^2 x^{27/2}+\frac{2}{31} c^3 x^{31/2}","\frac{2}{7} a^3 x^{7/2}+\frac{6}{11} a^2 b x^{11/2}+\frac{6}{23} c x^{23/2} \left(a c+b^2\right)+\frac{2}{19} b x^{19/2} \left(6 a c+b^2\right)+\frac{2}{5} a x^{15/2} \left(a c+b^2\right)+\frac{2}{9} b c^2 x^{27/2}+\frac{2}{31} c^3 x^{31/2}",1,"(2*a^3*x^(7/2))/7 + (6*a^2*b*x^(11/2))/11 + (2*a*(b^2 + a*c)*x^(15/2))/5 + (2*b*(b^2 + 6*a*c)*x^(19/2))/19 + (6*c*(b^2 + a*c)*x^(23/2))/23 + (2*b*c^2*x^(27/2))/9 + (2*c^3*x^(31/2))/31","A",1
1056,1,105,103,0.097646,"\int x^{3/2} \left(a+b x^2+c x^4\right)^3 \, dx","Integrate[x^(3/2)*(a + b*x^2 + c*x^4)^3,x]","2 \left(\frac{1}{5} a^3 x^{5/2}+\frac{1}{3} a^2 b x^{9/2}+\frac{1}{7} c x^{21/2} \left(a c+b^2\right)+\frac{1}{17} b x^{17/2} \left(6 a c+b^2\right)+\frac{3}{13} a x^{13/2} \left(a c+b^2\right)+\frac{3}{25} b c^2 x^{25/2}+\frac{1}{29} c^3 x^{29/2}\right)","\frac{2}{5} a^3 x^{5/2}+\frac{2}{3} a^2 b x^{9/2}+\frac{2}{7} c x^{21/2} \left(a c+b^2\right)+\frac{2}{17} b x^{17/2} \left(6 a c+b^2\right)+\frac{6}{13} a x^{13/2} \left(a c+b^2\right)+\frac{6}{25} b c^2 x^{25/2}+\frac{2}{29} c^3 x^{29/2}",1,"2*((a^3*x^(5/2))/5 + (a^2*b*x^(9/2))/3 + (3*a*(b^2 + a*c)*x^(13/2))/13 + (b*(b^2 + 6*a*c)*x^(17/2))/17 + (c*(b^2 + a*c)*x^(21/2))/7 + (3*b*c^2*x^(25/2))/25 + (c^3*x^(29/2))/29)","A",1
1057,1,103,103,3.3793779,"\int \sqrt{x} \left(a+b x^2+c x^4\right)^3 \, dx","Integrate[Sqrt[x]*(a + b*x^2 + c*x^4)^3,x]","\frac{2}{3} a^3 x^{3/2}+\frac{6}{7} a^2 b x^{7/2}+\frac{6}{19} c x^{19/2} \left(a c+b^2\right)+\frac{2}{15} b x^{15/2} \left(6 a c+b^2\right)+\frac{6}{11} a x^{11/2} \left(a c+b^2\right)+\frac{6}{23} b c^2 x^{23/2}+\frac{2}{27} c^3 x^{27/2}","\frac{2}{3} a^3 x^{3/2}+\frac{6}{7} a^2 b x^{7/2}+\frac{6}{19} c x^{19/2} \left(a c+b^2\right)+\frac{2}{15} b x^{15/2} \left(6 a c+b^2\right)+\frac{6}{11} a x^{11/2} \left(a c+b^2\right)+\frac{6}{23} b c^2 x^{23/2}+\frac{2}{27} c^3 x^{27/2}",1,"(2*a^3*x^(3/2))/3 + (6*a^2*b*x^(7/2))/7 + (6*a*(b^2 + a*c)*x^(11/2))/11 + (2*b*(b^2 + 6*a*c)*x^(15/2))/15 + (6*c*(b^2 + a*c)*x^(19/2))/19 + (6*b*c^2*x^(23/2))/23 + (2*c^3*x^(27/2))/27","A",1
1058,1,102,101,0.0735538,"\int \frac{\left(a+b x^2+c x^4\right)^3}{\sqrt{x}} \, dx","Integrate[(a + b*x^2 + c*x^4)^3/Sqrt[x],x]","2 \left(a^3 \sqrt{x}+\frac{3}{5} a^2 b x^{5/2}+\frac{3}{17} c x^{17/2} \left(a c+b^2\right)+\frac{1}{13} b x^{13/2} \left(6 a c+b^2\right)+\frac{1}{3} a x^{9/2} \left(a c+b^2\right)+\frac{1}{7} b c^2 x^{21/2}+\frac{1}{25} c^3 x^{25/2}\right)","2 a^3 \sqrt{x}+\frac{6}{5} a^2 b x^{5/2}+\frac{6}{17} c x^{17/2} \left(a c+b^2\right)+\frac{2}{13} b x^{13/2} \left(6 a c+b^2\right)+\frac{2}{3} a x^{9/2} \left(a c+b^2\right)+\frac{2}{7} b c^2 x^{21/2}+\frac{2}{25} c^3 x^{25/2}",1,"2*(a^3*Sqrt[x] + (3*a^2*b*x^(5/2))/5 + (a*(b^2 + a*c)*x^(9/2))/3 + (b*(b^2 + 6*a*c)*x^(13/2))/13 + (3*c*(b^2 + a*c)*x^(17/2))/17 + (b*c^2*x^(21/2))/7 + (c^3*x^(25/2))/25)","A",1
1059,1,100,99,0.0843307,"\int \frac{\left(a+b x^2+c x^4\right)^3}{x^{3/2}} \, dx","Integrate[(a + b*x^2 + c*x^4)^3/x^(3/2),x]","2 \left(-\frac{a^3}{\sqrt{x}}+a^2 b x^{3/2}+\frac{1}{5} c x^{15/2} \left(a c+b^2\right)+\frac{1}{11} b x^{11/2} \left(6 a c+b^2\right)+\frac{3}{7} a x^{7/2} \left(a c+b^2\right)+\frac{3}{19} b c^2 x^{19/2}+\frac{1}{23} c^3 x^{23/2}\right)","-\frac{2 a^3}{\sqrt{x}}+2 a^2 b x^{3/2}+\frac{2}{5} c x^{15/2} \left(a c+b^2\right)+\frac{2}{11} b x^{11/2} \left(6 a c+b^2\right)+\frac{6}{7} a x^{7/2} \left(a c+b^2\right)+\frac{6}{19} b c^2 x^{19/2}+\frac{2}{23} c^3 x^{23/2}",1,"2*(-(a^3/Sqrt[x]) + a^2*b*x^(3/2) + (3*a*(b^2 + a*c)*x^(7/2))/7 + (b*(b^2 + 6*a*c)*x^(11/2))/11 + (c*(b^2 + a*c)*x^(15/2))/5 + (3*b*c^2*x^(19/2))/19 + (c^3*x^(23/2))/23)","A",1
1060,1,103,101,0.0747407,"\int \frac{\left(a+b x^2+c x^4\right)^3}{x^{5/2}} \, dx","Integrate[(a + b*x^2 + c*x^4)^3/x^(5/2),x]","2 \left(-\frac{a^3}{3 x^{3/2}}+3 a^2 b \sqrt{x}+\frac{3}{13} c x^{13/2} \left(a c+b^2\right)+\frac{1}{9} b x^{9/2} \left(6 a c+b^2\right)+\frac{3}{5} a x^{5/2} \left(a c+b^2\right)+\frac{3}{17} b c^2 x^{17/2}+\frac{1}{21} c^3 x^{21/2}\right)","-\frac{2 a^3}{3 x^{3/2}}+6 a^2 b \sqrt{x}+\frac{6}{13} c x^{13/2} \left(a c+b^2\right)+\frac{2}{9} b x^{9/2} \left(6 a c+b^2\right)+\frac{6}{5} a x^{5/2} \left(a c+b^2\right)+\frac{6}{17} b c^2 x^{17/2}+\frac{2}{21} c^3 x^{21/2}",1,"2*(-1/3*a^3/x^(3/2) + 3*a^2*b*Sqrt[x] + (3*a*(b^2 + a*c)*x^(5/2))/5 + (b*(b^2 + 6*a*c)*x^(9/2))/9 + (3*c*(b^2 + a*c)*x^(13/2))/13 + (3*b*c^2*x^(17/2))/17 + (c^3*x^(21/2))/21)","A",1
1061,1,100,99,0.0771561,"\int \frac{\left(a+b x^2+c x^4\right)^3}{x^{7/2}} \, dx","Integrate[(a + b*x^2 + c*x^4)^3/x^(7/2),x]","2 \left(-\frac{a^3}{5 x^{5/2}}-\frac{3 a^2 b}{\sqrt{x}}+\frac{3}{11} c x^{11/2} \left(a c+b^2\right)+\frac{1}{7} b x^{7/2} \left(6 a c+b^2\right)+a x^{3/2} \left(a c+b^2\right)+\frac{1}{5} b c^2 x^{15/2}+\frac{1}{19} c^3 x^{19/2}\right)","-\frac{2 a^3}{5 x^{5/2}}-\frac{6 a^2 b}{\sqrt{x}}+\frac{6}{11} c x^{11/2} \left(a c+b^2\right)+\frac{2}{7} b x^{7/2} \left(6 a c+b^2\right)+2 a x^{3/2} \left(a c+b^2\right)+\frac{2}{5} b c^2 x^{15/2}+\frac{2}{19} c^3 x^{19/2}",1,"2*(-1/5*a^3/x^(5/2) - (3*a^2*b)/Sqrt[x] + a*(b^2 + a*c)*x^(3/2) + (b*(b^2 + 6*a*c)*x^(7/2))/7 + (3*c*(b^2 + a*c)*x^(11/2))/11 + (b*c^2*x^(15/2))/5 + (c^3*x^(19/2))/19)","A",1
1062,1,80,389,0.0526437,"\int \frac{x^{9/2}}{a+b x^2+c x^4} \, dx","Integrate[x^(9/2)/(a + b*x^2 + c*x^4),x]","\frac{4 x^{3/2}-3 \text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^4 b \log \left(\sqrt{x}-\text{$\#$1}\right)+a \log \left(\sqrt{x}-\text{$\#$1}\right)}{2 \text{$\#$1}^5 c+\text{$\#$1} b}\&\right]}{6 c}","-\frac{\left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2^{3/4} c^{7/4} \sqrt[4]{-\sqrt{b^2-4 a c}-b}}-\frac{\left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2^{3/4} c^{7/4} \sqrt[4]{\sqrt{b^2-4 a c}-b}}+\frac{\left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2^{3/4} c^{7/4} \sqrt[4]{-\sqrt{b^2-4 a c}-b}}+\frac{\left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2^{3/4} c^{7/4} \sqrt[4]{\sqrt{b^2-4 a c}-b}}+\frac{2 x^{3/2}}{3 c}",1,"(4*x^(3/2) - 3*RootSum[a + b*#1^4 + c*#1^8 & , (a*Log[Sqrt[x] - #1] + b*Log[Sqrt[x] - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ])/(6*c)","C",1
1063,1,80,385,0.0446962,"\int \frac{x^{7/2}}{a+b x^2+c x^4} \, dx","Integrate[x^(7/2)/(a + b*x^2 + c*x^4),x]","-\frac{\text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^4 b \log \left(\sqrt{x}-\text{$\#$1}\right)+a \log \left(\sqrt{x}-\text{$\#$1}\right)}{2 \text{$\#$1}^7 c+\text{$\#$1}^3 b}\&\right]-4 \sqrt{x}}{2 c}","\frac{\left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{\sqrt[4]{2} c^{5/4} \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{\left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{\sqrt[4]{2} c^{5/4} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{\left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{\sqrt[4]{2} c^{5/4} \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{\left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{\sqrt[4]{2} c^{5/4} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{2 \sqrt{x}}{c}",1,"-1/2*(-4*Sqrt[x] + RootSum[a + b*#1^4 + c*#1^8 & , (a*Log[Sqrt[x] - #1] + b*Log[Sqrt[x] - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ])/c","C",1
1064,1,48,331,0.0293777,"\int \frac{x^{5/2}}{a+b x^2+c x^4} \, dx","Integrate[x^(5/2)/(a + b*x^2 + c*x^4),x]","\frac{1}{2} \text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^3 \log \left(\sqrt{x}-\text{$\#$1}\right)}{2 \text{$\#$1}^4 c+b}\&\right]","-\frac{\left(-\sqrt{b^2-4 a c}-b\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2^{3/4} c^{3/4} \sqrt{b^2-4 a c}}+\frac{\left(\sqrt{b^2-4 a c}-b\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2^{3/4} c^{3/4} \sqrt{b^2-4 a c}}+\frac{\left(-\sqrt{b^2-4 a c}-b\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2^{3/4} c^{3/4} \sqrt{b^2-4 a c}}-\frac{\left(\sqrt{b^2-4 a c}-b\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2^{3/4} c^{3/4} \sqrt{b^2-4 a c}}",1,"RootSum[a + b*#1^4 + c*#1^8 & , (Log[Sqrt[x] - #1]*#1^3)/(b + 2*c*#1^4) & ]/2","C",1
1065,1,46,331,0.0277086,"\int \frac{x^{3/2}}{a+b x^2+c x^4} \, dx","Integrate[x^(3/2)/(a + b*x^2 + c*x^4),x]","\frac{1}{2} \text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1} \log \left(\sqrt{x}-\text{$\#$1}\right)}{2 \text{$\#$1}^4 c+b}\&\right]","\frac{\sqrt[4]{-\sqrt{b^2-4 a c}-b} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{\sqrt[4]{2} \sqrt[4]{c} \sqrt{b^2-4 a c}}-\frac{\sqrt[4]{\sqrt{b^2-4 a c}-b} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{\sqrt[4]{2} \sqrt[4]{c} \sqrt{b^2-4 a c}}+\frac{\sqrt[4]{-\sqrt{b^2-4 a c}-b} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{\sqrt[4]{2} \sqrt[4]{c} \sqrt{b^2-4 a c}}-\frac{\sqrt[4]{\sqrt{b^2-4 a c}-b} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{\sqrt[4]{2} \sqrt[4]{c} \sqrt{b^2-4 a c}}",1,"RootSum[a + b*#1^4 + c*#1^8 & , (Log[Sqrt[x] - #1]*#1)/(b + 2*c*#1^4) & ]/2","C",1
1066,1,47,331,0.0290896,"\int \frac{\sqrt{x}}{a+b x^2+c x^4} \, dx","Integrate[Sqrt[x]/(a + b*x^2 + c*x^4),x]","\frac{1}{2} \text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\log \left(\sqrt{x}-\text{$\#$1}\right)}{2 \text{$\#$1}^5 c+\text{$\#$1} b}\&\right]","-\frac{\sqrt[4]{2} \sqrt[4]{c} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{\sqrt{b^2-4 a c} \sqrt[4]{-\sqrt{b^2-4 a c}-b}}+\frac{\sqrt[4]{2} \sqrt[4]{c} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{\sqrt{b^2-4 a c} \sqrt[4]{\sqrt{b^2-4 a c}-b}}+\frac{\sqrt[4]{2} \sqrt[4]{c} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{\sqrt{b^2-4 a c} \sqrt[4]{-\sqrt{b^2-4 a c}-b}}-\frac{\sqrt[4]{2} \sqrt[4]{c} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{\sqrt{b^2-4 a c} \sqrt[4]{\sqrt{b^2-4 a c}-b}}",1,"RootSum[a + b*#1^4 + c*#1^8 & , Log[Sqrt[x] - #1]/(b*#1 + 2*c*#1^5) & ]/2","C",1
1067,1,49,331,0.0315565,"\int \frac{1}{\sqrt{x} \left(a+b x^2+c x^4\right)} \, dx","Integrate[1/(Sqrt[x]*(a + b*x^2 + c*x^4)),x]","\frac{1}{2} \text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\log \left(\sqrt{x}-\text{$\#$1}\right)}{2 \text{$\#$1}^7 c+\text{$\#$1}^3 b}\&\right]","\frac{2^{3/4} c^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{\sqrt{b^2-4 a c} \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{2^{3/4} c^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{\sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{2^{3/4} c^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{\sqrt{b^2-4 a c} \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{2^{3/4} c^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{\sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}",1,"RootSum[a + b*#1^4 + c*#1^8 & , Log[Sqrt[x] - #1]/(b*#1^3 + 2*c*#1^7) & ]/2","C",1
1068,1,78,371,0.0492359,"\int \frac{1}{x^{3/2} \left(a+b x^2+c x^4\right)} \, dx","Integrate[1/(x^(3/2)*(a + b*x^2 + c*x^4)),x]","-\frac{\text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^4 c \log \left(\sqrt{x}-\text{$\#$1}\right)+b \log \left(\sqrt{x}-\text{$\#$1}\right)}{2 \text{$\#$1}^5 c+\text{$\#$1} b}\&\right]+\frac{4}{\sqrt{x}}}{2 a}","-\frac{\sqrt[4]{c} \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2^{3/4} a \sqrt[4]{-\sqrt{b^2-4 a c}-b}}-\frac{\sqrt[4]{c} \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2^{3/4} a \sqrt[4]{\sqrt{b^2-4 a c}-b}}+\frac{\sqrt[4]{c} \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2^{3/4} a \sqrt[4]{-\sqrt{b^2-4 a c}-b}}+\frac{\sqrt[4]{c} \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2^{3/4} a \sqrt[4]{\sqrt{b^2-4 a c}-b}}-\frac{2}{a \sqrt{x}}",1,"-1/2*(4/Sqrt[x] + RootSum[a + b*#1^4 + c*#1^8 & , (b*Log[Sqrt[x] - #1] + c*Log[Sqrt[x] - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ])/a","C",1
1069,1,82,371,0.05366,"\int \frac{1}{x^{5/2} \left(a+b x^2+c x^4\right)} \, dx","Integrate[1/(x^(5/2)*(a + b*x^2 + c*x^4)),x]","-\frac{3 \text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^4 c \log \left(\sqrt{x}-\text{$\#$1}\right)+b \log \left(\sqrt{x}-\text{$\#$1}\right)}{2 \text{$\#$1}^7 c+\text{$\#$1}^3 b}\&\right]+\frac{4}{x^{3/2}}}{6 a}","\frac{c^{3/4} \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{\sqrt[4]{2} a \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{c^{3/4} \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{\sqrt[4]{2} a \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{c^{3/4} \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{\sqrt[4]{2} a \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{c^{3/4} \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{\sqrt[4]{2} a \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{2}{3 a x^{3/2}}",1,"-1/6*(4/x^(3/2) + 3*RootSum[a + b*#1^4 + c*#1^8 & , (b*Log[Sqrt[x] - #1] + c*Log[Sqrt[x] - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ])/a","C",1
1070,1,107,412,0.073353,"\int \frac{1}{x^{7/2} \left(a+b x^2+c x^4\right)} \, dx","Integrate[1/(x^(7/2)*(a + b*x^2 + c*x^4)),x]","-\frac{-5 \text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^4 b c \log \left(\sqrt{x}-\text{$\#$1}\right)-a c \log \left(\sqrt{x}-\text{$\#$1}\right)+b^2 \log \left(\sqrt{x}-\text{$\#$1}\right)}{2 \text{$\#$1}^5 c+\text{$\#$1} b}\&\right]+\frac{4 a}{x^{5/2}}-\frac{20 b}{\sqrt{x}}}{10 a^2}","\frac{\sqrt[4]{c} \left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2^{3/4} a^2 \sqrt[4]{-\sqrt{b^2-4 a c}-b}}+\frac{\sqrt[4]{c} \left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2^{3/4} a^2 \sqrt[4]{\sqrt{b^2-4 a c}-b}}-\frac{\sqrt[4]{c} \left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2^{3/4} a^2 \sqrt[4]{-\sqrt{b^2-4 a c}-b}}-\frac{\sqrt[4]{c} \left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2^{3/4} a^2 \sqrt[4]{\sqrt{b^2-4 a c}-b}}+\frac{2 b}{a^2 \sqrt{x}}-\frac{2}{5 a x^{5/2}}",1,"-1/10*((4*a)/x^(5/2) - (20*b)/Sqrt[x] - 5*RootSum[a + b*#1^4 + c*#1^8 & , (b^2*Log[Sqrt[x] - #1] - a*c*Log[Sqrt[x] - #1] + b*c*Log[Sqrt[x] - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ])/a^2","C",1
1071,1,144,544,0.2813171,"\int \frac{x^{13/2}}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[x^(13/2)/(a + b*x^2 + c*x^4)^2,x]","\frac{\text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{-14 \text{$\#$1}^4 a c \log \left(\sqrt{x}-\text{$\#$1}\right)+3 \text{$\#$1}^4 b^2 \log \left(\sqrt{x}-\text{$\#$1}\right)+3 a b \log \left(\sqrt{x}-\text{$\#$1}\right)}{2 \text{$\#$1}^5 c+\text{$\#$1} b}\&\right]-\frac{4 x^{3/2} \left(a \left(b-2 c x^2\right)+b^2 x^2\right)}{a+b x^2+c x^4}}{8 c \left(b^2-4 a c\right)}","-\frac{b x^{3/2}}{2 c \left(b^2-4 a c\right)}+\frac{x^{7/2} \left(2 a+b x^2\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\left(\left(3 b^2-14 a c\right) \sqrt{b^2-4 a c}-20 a b c+3 b^3\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{4\ 2^{3/4} c^{7/4} \left(b^2-4 a c\right)^{3/2} \sqrt[4]{-\sqrt{b^2-4 a c}-b}}-\frac{\left(-\left(3 b^2-14 a c\right) \sqrt{b^2-4 a c}-20 a b c+3 b^3\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{4\ 2^{3/4} c^{7/4} \left(b^2-4 a c\right)^{3/2} \sqrt[4]{\sqrt{b^2-4 a c}-b}}-\frac{\left(\left(3 b^2-14 a c\right) \sqrt{b^2-4 a c}-20 a b c+3 b^3\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{4\ 2^{3/4} c^{7/4} \left(b^2-4 a c\right)^{3/2} \sqrt[4]{-\sqrt{b^2-4 a c}-b}}+\frac{\left(-\left(3 b^2-14 a c\right) \sqrt{b^2-4 a c}-20 a b c+3 b^3\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{4\ 2^{3/4} c^{7/4} \left(b^2-4 a c\right)^{3/2} \sqrt[4]{\sqrt{b^2-4 a c}-b}}",1,"((-4*x^(3/2)*(b^2*x^2 + a*(b - 2*c*x^2)))/(a + b*x^2 + c*x^4) + RootSum[a + b*#1^4 + c*#1^8 & , (3*a*b*Log[Sqrt[x] - #1] + 3*b^2*Log[Sqrt[x] - #1]*#1^4 - 14*a*c*Log[Sqrt[x] - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ])/(8*c*(b^2 - 4*a*c))","C",1
1072,1,144,520,0.2658587,"\int \frac{x^{11/2}}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[x^(11/2)/(a + b*x^2 + c*x^4)^2,x]","\frac{\text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{-10 \text{$\#$1}^4 a c \log \left(\sqrt{x}-\text{$\#$1}\right)+\text{$\#$1}^4 b^2 \log \left(\sqrt{x}-\text{$\#$1}\right)+a b \log \left(\sqrt{x}-\text{$\#$1}\right)}{2 \text{$\#$1}^7 c+\text{$\#$1}^3 b}\&\right]-\frac{4 \sqrt{x} \left(a \left(b-2 c x^2\right)+b^2 x^2\right)}{a+b x^2+c x^4}}{8 c \left(b^2-4 a c\right)}","-\frac{\left(\frac{b \left(b^2-12 a c\right)}{\sqrt{b^2-4 a c}}-10 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{4 \sqrt[4]{2} c^{5/4} \left(b^2-4 a c\right) \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{\left(-\frac{b \left(b^2-12 a c\right)}{\sqrt{b^2-4 a c}}-10 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{4 \sqrt[4]{2} c^{5/4} \left(b^2-4 a c\right) \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{\left(\frac{b \left(b^2-12 a c\right)}{\sqrt{b^2-4 a c}}-10 a c+b^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{4 \sqrt[4]{2} c^{5/4} \left(b^2-4 a c\right) \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{\left(-\frac{b \left(b^2-12 a c\right)}{\sqrt{b^2-4 a c}}-10 a c+b^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{4 \sqrt[4]{2} c^{5/4} \left(b^2-4 a c\right) \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{x^{5/2} \left(2 a+b x^2\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-\frac{b \sqrt{x}}{2 c \left(b^2-4 a c\right)}",1,"((-4*Sqrt[x]*(b^2*x^2 + a*(b - 2*c*x^2)))/(a + b*x^2 + c*x^4) + RootSum[a + b*#1^4 + c*#1^8 & , (a*b*Log[Sqrt[x] - #1] + b^2*Log[Sqrt[x] - #1]*#1^4 - 10*a*c*Log[Sqrt[x] - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ])/(8*c*(b^2 - 4*a*c))","C",1
1073,1,124,471,0.2131785,"\int \frac{x^{9/2}}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[x^(9/2)/(a + b*x^2 + c*x^4)^2,x]","\frac{\text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^4 b \log \left(\sqrt{x}-\text{$\#$1}\right)-6 a \log \left(\sqrt{x}-\text{$\#$1}\right)}{2 \text{$\#$1}^5 c+\text{$\#$1} b}\&\right]}{8 \left(b^2-4 a c\right)}-\frac{-2 a x^{3/2}-b x^{7/2}}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}","\frac{\left(b \sqrt{b^2-4 a c}+12 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{4\ 2^{3/4} c^{3/4} \left(b^2-4 a c\right)^{3/2} \sqrt[4]{-\sqrt{b^2-4 a c}-b}}+\frac{\left(b-\frac{12 a c+b^2}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{4\ 2^{3/4} c^{3/4} \left(b^2-4 a c\right) \sqrt[4]{\sqrt{b^2-4 a c}-b}}-\frac{\left(b \sqrt{b^2-4 a c}+12 a c+b^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{4\ 2^{3/4} c^{3/4} \left(b^2-4 a c\right)^{3/2} \sqrt[4]{-\sqrt{b^2-4 a c}-b}}-\frac{\left(b-\frac{12 a c+b^2}{\sqrt{b^2-4 a c}}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{4\ 2^{3/4} c^{3/4} \left(b^2-4 a c\right) \sqrt[4]{\sqrt{b^2-4 a c}-b}}+\frac{x^{3/2} \left(2 a+b x^2\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"-1/2*(-2*a*x^(3/2) - b*x^(7/2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + RootSum[a + b*#1^4 + c*#1^8 & , (-6*a*Log[Sqrt[x] - #1] + b*Log[Sqrt[x] - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ]/(8*(b^2 - 4*a*c))","C",1
1074,1,127,483,0.2129109,"\int \frac{x^{7/2}}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[x^(7/2)/(a + b*x^2 + c*x^4)^2,x]","\frac{\text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{3 \text{$\#$1}^4 b \log \left(\sqrt{x}-\text{$\#$1}\right)-2 a \log \left(\sqrt{x}-\text{$\#$1}\right)}{2 \text{$\#$1}^7 c+\text{$\#$1}^3 b}\&\right]}{8 \left(b^2-4 a c\right)}-\frac{-2 a \sqrt{x}-b x^{5/2}}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}","\frac{\sqrt{x} \left(2 a+b x^2\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-\frac{\left(3 b \sqrt{b^2-4 a c}+4 a c+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{4 \sqrt[4]{2} \sqrt[4]{c} \left(b^2-4 a c\right)^{3/2} \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{\left(-3 b \sqrt{b^2-4 a c}+4 a c+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{4 \sqrt[4]{2} \sqrt[4]{c} \left(b^2-4 a c\right)^{3/2} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{\left(3 b \sqrt{b^2-4 a c}+4 a c+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{4 \sqrt[4]{2} \sqrt[4]{c} \left(b^2-4 a c\right)^{3/2} \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{\left(-3 b \sqrt{b^2-4 a c}+4 a c+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{4 \sqrt[4]{2} \sqrt[4]{c} \left(b^2-4 a c\right)^{3/2} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}",1,"-1/2*(-2*a*Sqrt[x] - b*x^(5/2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + RootSum[a + b*#1^4 + c*#1^8 & , (-2*a*Log[Sqrt[x] - #1] + 3*b*Log[Sqrt[x] - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ]/(8*(b^2 - 4*a*c))","C",1
1075,1,109,450,0.2136545,"\int \frac{x^{5/2}}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[x^(5/2)/(a + b*x^2 + c*x^4)^2,x]","-\frac{\text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{2 \text{$\#$1}^4 c \log \left(\sqrt{x}-\text{$\#$1}\right)-3 b \log \left(\sqrt{x}-\text{$\#$1}\right)}{2 \text{$\#$1}^5 c+\text{$\#$1} b}\&\right]+\frac{4 x^{3/2} \left(b+2 c x^2\right)}{a+b x^2+c x^4}}{8 \left(b^2-4 a c\right)}","-\frac{x^{3/2} \left(b+2 c x^2\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-\frac{\sqrt[4]{c} \left(\sqrt{b^2-4 a c}+4 b\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2\ 2^{3/4} \left(b^2-4 a c\right)^{3/2} \sqrt[4]{-\sqrt{b^2-4 a c}-b}}+\frac{\sqrt[4]{c} \left(4 b-\sqrt{b^2-4 a c}\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2\ 2^{3/4} \left(b^2-4 a c\right)^{3/2} \sqrt[4]{\sqrt{b^2-4 a c}-b}}+\frac{\sqrt[4]{c} \left(\sqrt{b^2-4 a c}+4 b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2\ 2^{3/4} \left(b^2-4 a c\right)^{3/2} \sqrt[4]{-\sqrt{b^2-4 a c}-b}}-\frac{\sqrt[4]{c} \left(4 b-\sqrt{b^2-4 a c}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2\ 2^{3/4} \left(b^2-4 a c\right)^{3/2} \sqrt[4]{\sqrt{b^2-4 a c}-b}}",1,"-1/8*((4*x^(3/2)*(b + 2*c*x^2))/(a + b*x^2 + c*x^4) + RootSum[a + b*#1^4 + c*#1^8 & , (-3*b*Log[Sqrt[x] - #1] + 2*c*Log[Sqrt[x] - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ])/(b^2 - 4*a*c)","C",1
1076,1,111,442,0.2304412,"\int \frac{x^{3/2}}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[x^(3/2)/(a + b*x^2 + c*x^4)^2,x]","-\frac{\text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{6 \text{$\#$1}^4 c \log \left(\sqrt{x}-\text{$\#$1}\right)-b \log \left(\sqrt{x}-\text{$\#$1}\right)}{2 \text{$\#$1}^7 c+\text{$\#$1}^3 b}\&\right]+\frac{4 \sqrt{x} \left(b+2 c x^2\right)}{a+b x^2+c x^4}}{8 \left(b^2-4 a c\right)}","\frac{c^{3/4} \left(\frac{4 b}{\sqrt{b^2-4 a c}}+3\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} \left(b^2-4 a c\right) \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{c^{3/4} \left(3-\frac{4 b}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} \left(b^2-4 a c\right) \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{c^{3/4} \left(\frac{4 b}{\sqrt{b^2-4 a c}}+3\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} \left(b^2-4 a c\right) \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{c^{3/4} \left(3-\frac{4 b}{\sqrt{b^2-4 a c}}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} \left(b^2-4 a c\right) \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{\sqrt{x} \left(b+2 c x^2\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"-1/8*((4*Sqrt[x]*(b + 2*c*x^2))/(a + b*x^2 + c*x^4) + RootSum[a + b*#1^4 + c*#1^8 & , (-(b*Log[Sqrt[x] - #1]) + 6*c*Log[Sqrt[x] - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ])/(b^2 - 4*a*c)","C",1
1077,1,149,489,0.2432498,"\int \frac{\sqrt{x}}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[Sqrt[x]/(a + b*x^2 + c*x^4)^2,x]","-\frac{\left(a+b x^2+c x^4\right) \text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^4 b c \log \left(\sqrt{x}-\text{$\#$1}\right)-10 a c \log \left(\sqrt{x}-\text{$\#$1}\right)+b^2 \log \left(\sqrt{x}-\text{$\#$1}\right)}{2 \text{$\#$1}^5 c+\text{$\#$1} b}\&\right]+4 x^{3/2} \left(-2 a c+b^2+b c x^2\right)}{8 a \left(4 a c-b^2\right) \left(a+b x^2+c x^4\right)}","\frac{x^{3/2} \left(-2 a c+b^2+b c x^2\right)}{2 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt[4]{c} \left(b-\frac{b^2-20 a c}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{4\ 2^{3/4} a \left(b^2-4 a c\right) \sqrt[4]{-\sqrt{b^2-4 a c}-b}}+\frac{\sqrt[4]{c} \left(\frac{b^2-20 a c}{\sqrt{b^2-4 a c}}+b\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{4\ 2^{3/4} a \left(b^2-4 a c\right) \sqrt[4]{\sqrt{b^2-4 a c}-b}}-\frac{\sqrt[4]{c} \left(b-\frac{b^2-20 a c}{\sqrt{b^2-4 a c}}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{4\ 2^{3/4} a \left(b^2-4 a c\right) \sqrt[4]{-\sqrt{b^2-4 a c}-b}}-\frac{\sqrt[4]{c} \left(\frac{b^2-20 a c}{\sqrt{b^2-4 a c}}+b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{4\ 2^{3/4} a \left(b^2-4 a c\right) \sqrt[4]{\sqrt{b^2-4 a c}-b}}",1,"-1/8*(4*x^(3/2)*(b^2 - 2*a*c + b*c*x^2) + (a + b*x^2 + c*x^4)*RootSum[a + b*#1^4 + c*#1^8 & , (b^2*Log[Sqrt[x] - #1] - 10*a*c*Log[Sqrt[x] - #1] + b*c*Log[Sqrt[x] - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ])/(a*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4))","C",1
1078,1,153,503,0.2487637,"\int \frac{1}{\sqrt{x} \left(a+b x^2+c x^4\right)^2} \, dx","Integrate[1/(Sqrt[x]*(a + b*x^2 + c*x^4)^2),x]","-\frac{\left(a+b x^2+c x^4\right) \text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{3 \text{$\#$1}^4 b c \log \left(\sqrt{x}-\text{$\#$1}\right)-14 a c \log \left(\sqrt{x}-\text{$\#$1}\right)+3 b^2 \log \left(\sqrt{x}-\text{$\#$1}\right)}{2 \text{$\#$1}^7 c+\text{$\#$1}^3 b}\&\right]+4 \sqrt{x} \left(-2 a c+b^2+b c x^2\right)}{8 a \left(4 a c-b^2\right) \left(a+b x^2+c x^4\right)}","\frac{c^{3/4} \left(-3 b \sqrt{b^2-4 a c}-28 a c+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{4 \sqrt[4]{2} a \left(b^2-4 a c\right)^{3/2} \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{c^{3/4} \left(3 b \sqrt{b^2-4 a c}-28 a c+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{4 \sqrt[4]{2} a \left(b^2-4 a c\right)^{3/2} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{c^{3/4} \left(-3 b \sqrt{b^2-4 a c}-28 a c+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{4 \sqrt[4]{2} a \left(b^2-4 a c\right)^{3/2} \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{c^{3/4} \left(3 b \sqrt{b^2-4 a c}-28 a c+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{4 \sqrt[4]{2} a \left(b^2-4 a c\right)^{3/2} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{\sqrt{x} \left(-2 a c+b^2+b c x^2\right)}{2 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"-1/8*(4*Sqrt[x]*(b^2 - 2*a*c + b*c*x^2) + (a + b*x^2 + c*x^4)*RootSum[a + b*#1^4 + c*#1^8 & , (3*b^2*Log[Sqrt[x] - #1] - 14*a*c*Log[Sqrt[x] - #1] + 3*b*c*Log[Sqrt[x] - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ])/(a*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4))","C",1
1079,1,190,573,0.3390081,"\int \frac{1}{x^{3/2} \left(a+b x^2+c x^4\right)^2} \, dx","Integrate[1/(x^(3/2)*(a + b*x^2 + c*x^4)^2),x]","-\frac{\frac{\text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{-18 \text{$\#$1}^4 a c^2 \log \left(\sqrt{x}-\text{$\#$1}\right)+5 \text{$\#$1}^4 b^2 c \log \left(\sqrt{x}-\text{$\#$1}\right)-23 a b c \log \left(\sqrt{x}-\text{$\#$1}\right)+5 b^3 \log \left(\sqrt{x}-\text{$\#$1}\right)}{2 \text{$\#$1}^5 c+\text{$\#$1} b}\&\right]}{b^2-4 a c}+\frac{4 x^{3/2} \left(-3 a b c-2 a c^2 x^2+b^3+b^2 c x^2\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{16}{\sqrt{x}}}{8 a^2}","-\frac{5 b^2-18 a c}{2 a^2 \sqrt{x} \left(b^2-4 a c\right)}+\frac{\sqrt[4]{c} \left(-\left(5 b^2-18 a c\right) \sqrt{b^2-4 a c}-28 a b c+5 b^3\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{4\ 2^{3/4} a^2 \left(b^2-4 a c\right)^{3/2} \sqrt[4]{-\sqrt{b^2-4 a c}-b}}-\frac{\sqrt[4]{c} \left(\left(5 b^2-18 a c\right) \sqrt{b^2-4 a c}-28 a b c+5 b^3\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{4\ 2^{3/4} a^2 \left(b^2-4 a c\right)^{3/2} \sqrt[4]{\sqrt{b^2-4 a c}-b}}-\frac{\sqrt[4]{c} \left(-\left(5 b^2-18 a c\right) \sqrt{b^2-4 a c}-28 a b c+5 b^3\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{4\ 2^{3/4} a^2 \left(b^2-4 a c\right)^{3/2} \sqrt[4]{-\sqrt{b^2-4 a c}-b}}+\frac{\sqrt[4]{c} \left(\left(5 b^2-18 a c\right) \sqrt{b^2-4 a c}-28 a b c+5 b^3\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{4\ 2^{3/4} a^2 \left(b^2-4 a c\right)^{3/2} \sqrt[4]{\sqrt{b^2-4 a c}-b}}+\frac{-2 a c+b^2+b c x^2}{2 a \sqrt{x} \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"-1/8*(16/Sqrt[x] + (4*x^(3/2)*(b^3 - 3*a*b*c + b^2*c*x^2 - 2*a*c^2*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + RootSum[a + b*#1^4 + c*#1^8 & , (5*b^3*Log[Sqrt[x] - #1] - 23*a*b*c*Log[Sqrt[x] - #1] + 5*b^2*c*Log[Sqrt[x] - #1]*#1^4 - 18*a*c^2*Log[Sqrt[x] - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ]/(b^2 - 4*a*c))/a^2","C",1
1080,1,254,621,0.4415043,"\int \frac{x^{15/2}}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[x^(15/2)/(a + b*x^2 + c*x^4)^3,x]","\frac{3 c \left(a+b x^2+c x^4\right)^2 \text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{-28 \text{$\#$1}^4 a b c \log \left(\sqrt{x}-\text{$\#$1}\right)+\text{$\#$1}^4 b^3 \log \left(\sqrt{x}-\text{$\#$1}\right)+12 a^2 c \log \left(\sqrt{x}-\text{$\#$1}\right)+a b^2 \log \left(\sqrt{x}-\text{$\#$1}\right)}{2 \text{$\#$1}^7 c+\text{$\#$1}^3 b}\&\right]+16 \sqrt{x} \left(b^2-4 a c\right) \left(-2 a^2 c+a b \left(b-3 c x^2\right)+b^3 x^2\right)+4 \sqrt{x} \left(-68 a^2 c^2+21 a b^2 c-28 a b c^2 x^2-4 b^4+b^3 c x^2\right) \left(a+b x^2+c x^4\right)}{64 c^2 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)^2}","-\frac{3 \left(\frac{-24 a^2 c^2-30 a b^2 c+b^4}{\sqrt{b^2-4 a c}}-28 a b c+b^3\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{32 \sqrt[4]{2} c^{5/4} \left(b^2-4 a c\right)^2 \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{3 \left(-\frac{-24 a^2 c^2-30 a b^2 c+b^4}{\sqrt{b^2-4 a c}}-28 a b c+b^3\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{32 \sqrt[4]{2} c^{5/4} \left(b^2-4 a c\right)^2 \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{3 \left(\frac{-24 a^2 c^2-30 a b^2 c+b^4}{\sqrt{b^2-4 a c}}-28 a b c+b^3\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{32 \sqrt[4]{2} c^{5/4} \left(b^2-4 a c\right)^2 \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{3 \left(-\frac{-24 a^2 c^2-30 a b^2 c+b^4}{\sqrt{b^2-4 a c}}-28 a b c+b^3\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{32 \sqrt[4]{2} c^{5/4} \left(b^2-4 a c\right)^2 \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{x^{9/2} \left(2 a+b x^2\right)}{4 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}+\frac{3 x^{5/2} \left(x^2 \left(12 a c+b^2\right)+8 a b\right)}{16 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}-\frac{3 \sqrt{x} \left(12 a c+b^2\right)}{16 c \left(b^2-4 a c\right)^2}",1,"(4*Sqrt[x]*(-4*b^4 + 21*a*b^2*c - 68*a^2*c^2 + b^3*c*x^2 - 28*a*b*c^2*x^2)*(a + b*x^2 + c*x^4) + 16*(b^2 - 4*a*c)*Sqrt[x]*(-2*a^2*c + b^3*x^2 + a*b*(b - 3*c*x^2)) + 3*c*(a + b*x^2 + c*x^4)^2*RootSum[a + b*#1^4 + c*#1^8 & , (a*b^2*Log[Sqrt[x] - #1] + 12*a^2*c*Log[Sqrt[x] - #1] + b^3*Log[Sqrt[x] - #1]*#1^4 - 28*a*b*c*Log[Sqrt[x] - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ])/(64*c^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)^2)","C",1
1081,1,216,569,0.4080049,"\int \frac{x^{13/2}}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[x^(13/2)/(a + b*x^2 + c*x^4)^3,x]","\frac{c \left(a+b x^2+c x^4\right)^2 \text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{28 \text{$\#$1}^4 a c \log \left(\sqrt{x}-\text{$\#$1}\right)+5 \text{$\#$1}^4 b^2 \log \left(\sqrt{x}-\text{$\#$1}\right)-72 a b \log \left(\sqrt{x}-\text{$\#$1}\right)}{2 \text{$\#$1}^5 c+\text{$\#$1} b}\&\right]-16 x^{3/2} \left(b^2-4 a c\right) \left(a \left(b-2 c x^2\right)+b^2 x^2\right)+4 x^{3/2} \left(8 a b c+28 a c^2 x^2+4 b^3+5 b^2 c x^2\right) \left(a+b x^2+c x^4\right)}{64 c \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)^2}","\frac{x^{7/2} \left(2 a+b x^2\right)}{4 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}+\frac{x^{3/2} \left(x^2 \left(28 a c+5 b^2\right)+24 a b\right)}{16 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{\left(\sqrt{b^2-4 a c} \left(28 a c+5 b^2\right)+172 a b c+5 b^3\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{32\ 2^{3/4} c^{3/4} \left(b^2-4 a c\right)^{5/2} \sqrt[4]{-\sqrt{b^2-4 a c}-b}}+\frac{\left(-\frac{172 a b c+5 b^3}{\sqrt{b^2-4 a c}}+28 a c+5 b^2\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{32\ 2^{3/4} c^{3/4} \left(b^2-4 a c\right)^2 \sqrt[4]{\sqrt{b^2-4 a c}-b}}-\frac{\left(\sqrt{b^2-4 a c} \left(28 a c+5 b^2\right)+172 a b c+5 b^3\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{32\ 2^{3/4} c^{3/4} \left(b^2-4 a c\right)^{5/2} \sqrt[4]{-\sqrt{b^2-4 a c}-b}}-\frac{\left(-\frac{172 a b c+5 b^3}{\sqrt{b^2-4 a c}}+28 a c+5 b^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{32\ 2^{3/4} c^{3/4} \left(b^2-4 a c\right)^2 \sqrt[4]{\sqrt{b^2-4 a c}-b}}",1,"(4*x^(3/2)*(4*b^3 + 8*a*b*c + 5*b^2*c*x^2 + 28*a*c^2*x^2)*(a + b*x^2 + c*x^4) - 16*(b^2 - 4*a*c)*x^(3/2)*(b^2*x^2 + a*(b - 2*c*x^2)) + c*(a + b*x^2 + c*x^4)^2*RootSum[a + b*#1^4 + c*#1^8 & , (-72*a*b*Log[Sqrt[x] - #1] + 5*b^2*Log[Sqrt[x] - #1]*#1^4 + 28*a*c*Log[Sqrt[x] - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ])/(64*c*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)^2)","C",1
1082,1,219,569,0.3874375,"\int \frac{x^{11/2}}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[x^(11/2)/(a + b*x^2 + c*x^4)^3,x]","\frac{3 c \left(a+b x^2+c x^4\right)^2 \text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{20 \text{$\#$1}^4 a c \log \left(\sqrt{x}-\text{$\#$1}\right)+7 \text{$\#$1}^4 b^2 \log \left(\sqrt{x}-\text{$\#$1}\right)-8 a b \log \left(\sqrt{x}-\text{$\#$1}\right)}{2 \text{$\#$1}^7 c+\text{$\#$1}^3 b}\&\right]-16 \sqrt{x} \left(b^2-4 a c\right) \left(a \left(b-2 c x^2\right)+b^2 x^2\right)+4 \sqrt{x} \left(8 a b c+20 a c^2 x^2+4 b^3+7 b^2 c x^2\right) \left(a+b x^2+c x^4\right)}{64 c \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)^2}","\frac{\sqrt{x} \left(x^2 \left(20 a c+7 b^2\right)+24 a b\right)}{16 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{x^{5/2} \left(2 a+b x^2\right)}{4 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}-\frac{3 \left(\sqrt{b^2-4 a c} \left(20 a c+7 b^2\right)+36 a b c+7 b^3\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{32 \sqrt[4]{2} \sqrt[4]{c} \left(b^2-4 a c\right)^{5/2} \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{3 \left(-\frac{36 a b c+7 b^3}{\sqrt{b^2-4 a c}}+20 a c+7 b^2\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{32 \sqrt[4]{2} \sqrt[4]{c} \left(b^2-4 a c\right)^2 \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{3 \left(\sqrt{b^2-4 a c} \left(20 a c+7 b^2\right)+36 a b c+7 b^3\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{32 \sqrt[4]{2} \sqrt[4]{c} \left(b^2-4 a c\right)^{5/2} \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{3 \left(-\frac{36 a b c+7 b^3}{\sqrt{b^2-4 a c}}+20 a c+7 b^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{32 \sqrt[4]{2} \sqrt[4]{c} \left(b^2-4 a c\right)^2 \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}",1,"(4*Sqrt[x]*(4*b^3 + 8*a*b*c + 7*b^2*c*x^2 + 20*a*c^2*x^2)*(a + b*x^2 + c*x^4) - 16*(b^2 - 4*a*c)*Sqrt[x]*(b^2*x^2 + a*(b - 2*c*x^2)) + 3*c*(a + b*x^2 + c*x^4)^2*RootSum[a + b*#1^4 + c*#1^8 & , (-8*a*b*Log[Sqrt[x] - #1] + 7*b^2*Log[Sqrt[x] - #1]*#1^4 + 20*a*c*Log[Sqrt[x] - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ])/(64*c*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)^2)","C",1
1083,1,176,533,0.3797677,"\int \frac{x^{9/2}}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[x^(9/2)/(a + b*x^2 + c*x^4)^3,x]","\frac{-3 \text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{8 \text{$\#$1}^4 b c \log \left(\sqrt{x}-\text{$\#$1}\right)-20 a c \log \left(\sqrt{x}-\text{$\#$1}\right)-7 b^2 \log \left(\sqrt{x}-\text{$\#$1}\right)}{2 \text{$\#$1}^5 c+\text{$\#$1} b}\&\right]-\frac{12 x^{3/2} \left(-4 a c+5 b^2+8 b c x^2\right)}{a+b x^2+c x^4}+\frac{16 x^{3/2} \left(b^2-4 a c\right) \left(2 a+b x^2\right)}{\left(a+b x^2+c x^4\right)^2}}{64 \left(b^2-4 a c\right)^2}","-\frac{3 x^{3/2} \left(-4 a c+5 b^2+8 b c x^2\right)}{16 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{x^{3/2} \left(2 a+b x^2\right)}{4 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}-\frac{3 \sqrt[4]{c} \left(4 b \sqrt{b^2-4 a c}+20 a c+11 b^2\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{16\ 2^{3/4} \left(b^2-4 a c\right)^{5/2} \sqrt[4]{-\sqrt{b^2-4 a c}-b}}+\frac{3 \sqrt[4]{c} \left(-4 b \sqrt{b^2-4 a c}+20 a c+11 b^2\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{16\ 2^{3/4} \left(b^2-4 a c\right)^{5/2} \sqrt[4]{\sqrt{b^2-4 a c}-b}}+\frac{3 \sqrt[4]{c} \left(4 b \sqrt{b^2-4 a c}+20 a c+11 b^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{16\ 2^{3/4} \left(b^2-4 a c\right)^{5/2} \sqrt[4]{-\sqrt{b^2-4 a c}-b}}-\frac{3 \sqrt[4]{c} \left(-4 b \sqrt{b^2-4 a c}+20 a c+11 b^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{16\ 2^{3/4} \left(b^2-4 a c\right)^{5/2} \sqrt[4]{\sqrt{b^2-4 a c}-b}}",1,"((16*(b^2 - 4*a*c)*x^(3/2)*(2*a + b*x^2))/(a + b*x^2 + c*x^4)^2 - (12*x^(3/2)*(5*b^2 - 4*a*c + 8*b*c*x^2))/(a + b*x^2 + c*x^4) - 3*RootSum[a + b*#1^4 + c*#1^8 & , (-7*b^2*Log[Sqrt[x] - #1] - 20*a*c*Log[Sqrt[x] - #1] + 8*b*c*Log[Sqrt[x] - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ])/(64*(b^2 - 4*a*c)^2)","C",1
1084,1,177,533,0.440862,"\int \frac{x^{7/2}}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[x^(7/2)/(a + b*x^2 + c*x^4)^3,x]","-\frac{\text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{72 \text{$\#$1}^4 b c \log \left(\sqrt{x}-\text{$\#$1}\right)-28 a c \log \left(\sqrt{x}-\text{$\#$1}\right)-5 b^2 \log \left(\sqrt{x}-\text{$\#$1}\right)}{2 \text{$\#$1}^7 c+\text{$\#$1}^3 b}\&\right]+\frac{4 \sqrt{x} \left(28 a^2 c+a \left(5 b^2+36 b c x^2-4 c^2 x^4\right)+b x^2 \left(9 b^2+37 b c x^2+24 c^2 x^4\right)\right)}{\left(a+b x^2+c x^4\right)^2}}{64 \left(b^2-4 a c\right)^2}","\frac{c^{3/4} \left(36 b \sqrt{b^2-4 a c}+28 a c+41 b^2\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{16 \sqrt[4]{2} \left(b^2-4 a c\right)^{5/2} \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{c^{3/4} \left(-36 b \sqrt{b^2-4 a c}+28 a c+41 b^2\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{16 \sqrt[4]{2} \left(b^2-4 a c\right)^{5/2} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{c^{3/4} \left(36 b \sqrt{b^2-4 a c}+28 a c+41 b^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{16 \sqrt[4]{2} \left(b^2-4 a c\right)^{5/2} \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{c^{3/4} \left(-36 b \sqrt{b^2-4 a c}+28 a c+41 b^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{16 \sqrt[4]{2} \left(b^2-4 a c\right)^{5/2} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{\sqrt{x} \left(2 a+b x^2\right)}{4 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}-\frac{\sqrt{x} \left(-4 a c+13 b^2+24 b c x^2\right)}{16 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}",1,"-1/64*((4*Sqrt[x]*(28*a^2*c + a*(5*b^2 + 36*b*c*x^2 - 4*c^2*x^4) + b*x^2*(9*b^2 + 37*b*c*x^2 + 24*c^2*x^4)))/(a + b*x^2 + c*x^4)^2 + RootSum[a + b*#1^4 + c*#1^8 & , (-5*b^2*Log[Sqrt[x] - #1] - 28*a*c*Log[Sqrt[x] - #1] + 72*b*c*Log[Sqrt[x] - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ])/(b^2 - 4*a*c)^2","C",1
1085,1,222,594,0.4056751,"\int \frac{x^{5/2}}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[x^(5/2)/(a + b*x^2 + c*x^4)^3,x]","\frac{3 \left(a+b x^2+c x^4\right)^2 \text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{12 \text{$\#$1}^4 a c^2 \log \left(\sqrt{x}-\text{$\#$1}\right)+\text{$\#$1}^4 b^2 c \log \left(\sqrt{x}-\text{$\#$1}\right)-28 a b c \log \left(\sqrt{x}-\text{$\#$1}\right)+b^3 \log \left(\sqrt{x}-\text{$\#$1}\right)}{2 \text{$\#$1}^5 c+\text{$\#$1} b}\&\right]-16 a x^{3/2} \left(b^2-4 a c\right) \left(b+2 c x^2\right)+12 x^{3/2} \left(4 a b c+12 a c^2 x^2+b^3+b^2 c x^2\right) \left(a+b x^2+c x^4\right)}{64 a \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)^2}","\frac{3 x^{3/2} \left(c x^2 \left(12 a c+b^2\right)+b \left(4 a c+b^2\right)\right)}{16 a \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}-\frac{x^{3/2} \left(b+2 c x^2\right)}{4 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}+\frac{3 \sqrt[4]{c} \left(\frac{68 a b c}{\sqrt{b^2-4 a c}}-\frac{b^3}{\sqrt{b^2-4 a c}}+12 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{32\ 2^{3/4} a \left(b^2-4 a c\right)^2 \sqrt[4]{-\sqrt{b^2-4 a c}-b}}+\frac{3 \sqrt[4]{c} \left(\sqrt{b^2-4 a c} \left(12 a c+b^2\right)-68 a b c+b^3\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{32\ 2^{3/4} a \left(b^2-4 a c\right)^{5/2} \sqrt[4]{\sqrt{b^2-4 a c}-b}}-\frac{3 \sqrt[4]{c} \left(\frac{68 a b c}{\sqrt{b^2-4 a c}}-\frac{b^3}{\sqrt{b^2-4 a c}}+12 a c+b^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{32\ 2^{3/4} a \left(b^2-4 a c\right)^2 \sqrt[4]{-\sqrt{b^2-4 a c}-b}}-\frac{3 \sqrt[4]{c} \left(\sqrt{b^2-4 a c} \left(12 a c+b^2\right)-68 a b c+b^3\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{32\ 2^{3/4} a \left(b^2-4 a c\right)^{5/2} \sqrt[4]{\sqrt{b^2-4 a c}-b}}",1,"(-16*a*(b^2 - 4*a*c)*x^(3/2)*(b + 2*c*x^2) + 12*x^(3/2)*(b^3 + 4*a*b*c + b^2*c*x^2 + 12*a*c^2*x^2)*(a + b*x^2 + c*x^4) + 3*(a + b*x^2 + c*x^4)^2*RootSum[a + b*#1^4 + c*#1^8 & , (b^3*Log[Sqrt[x] - #1] - 28*a*b*c*Log[Sqrt[x] - #1] + b^2*c*Log[Sqrt[x] - #1]*#1^4 + 12*a*c^2*Log[Sqrt[x] - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ])/(64*a*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)^2)","C",1
1086,1,224,594,0.4119099,"\int \frac{x^{3/2}}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[x^(3/2)/(a + b*x^2 + c*x^4)^3,x]","\frac{3 \left(a+b x^2+c x^4\right)^2 \text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{44 \text{$\#$1}^4 a c^2 \log \left(\sqrt{x}-\text{$\#$1}\right)+\text{$\#$1}^4 b^2 c \log \left(\sqrt{x}-\text{$\#$1}\right)-12 a b c \log \left(\sqrt{x}-\text{$\#$1}\right)+b^3 \log \left(\sqrt{x}-\text{$\#$1}\right)}{2 \text{$\#$1}^7 c+\text{$\#$1}^3 b}\&\right]-16 a \sqrt{x} \left(b^2-4 a c\right) \left(b+2 c x^2\right)+4 \sqrt{x} \left(20 a b c+44 a c^2 x^2+b^3+b^2 c x^2\right) \left(a+b x^2+c x^4\right)}{64 a \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)^2}","-\frac{\sqrt{x} \left(b+2 c x^2\right)}{4 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}+\frac{\sqrt{x} \left(c x^2 \left(44 a c+b^2\right)+b \left(20 a c+b^2\right)\right)}{16 a \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}-\frac{3 c^{3/4} \left(\frac{68 a b c}{\sqrt{b^2-4 a c}}-\frac{b^3}{\sqrt{b^2-4 a c}}+44 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{32 \sqrt[4]{2} a \left(b^2-4 a c\right)^2 \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{3 c^{3/4} \left(\sqrt{b^2-4 a c} \left(44 a c+b^2\right)-68 a b c+b^3\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{32 \sqrt[4]{2} a \left(b^2-4 a c\right)^{5/2} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{3 c^{3/4} \left(\frac{68 a b c}{\sqrt{b^2-4 a c}}-\frac{b^3}{\sqrt{b^2-4 a c}}+44 a c+b^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{32 \sqrt[4]{2} a \left(b^2-4 a c\right)^2 \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{3 c^{3/4} \left(\sqrt{b^2-4 a c} \left(44 a c+b^2\right)-68 a b c+b^3\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{32 \sqrt[4]{2} a \left(b^2-4 a c\right)^{5/2} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}",1,"(-16*a*(b^2 - 4*a*c)*Sqrt[x]*(b + 2*c*x^2) + 4*Sqrt[x]*(b^3 + 20*a*b*c + b^2*c*x^2 + 44*a*c^2*x^2)*(a + b*x^2 + c*x^4) + 3*(a + b*x^2 + c*x^4)^2*RootSum[a + b*#1^4 + c*#1^8 & , (b^3*Log[Sqrt[x] - #1] - 12*a*b*c*Log[Sqrt[x] - #1] + b^2*c*Log[Sqrt[x] - #1]*#1^4 + 44*a*c^2*Log[Sqrt[x] - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ])/(64*a*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)^2)","C",1
1087,1,254,658,0.494699,"\int \frac{\sqrt{x}}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[Sqrt[x]/(a + b*x^2 + c*x^4)^3,x]","\frac{\text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{-44 \text{$\#$1}^4 a b c^2 \log \left(\sqrt{x}-\text{$\#$1}\right)+5 \text{$\#$1}^4 b^3 c \log \left(\sqrt{x}-\text{$\#$1}\right)+260 a^2 c^2 \log \left(\sqrt{x}-\text{$\#$1}\right)-49 a b^2 c \log \left(\sqrt{x}-\text{$\#$1}\right)+5 b^4 \log \left(\sqrt{x}-\text{$\#$1}\right)}{2 \text{$\#$1}^5 c+\text{$\#$1} b}\&\right]+\frac{4 x^{3/2} \left(52 a^2 c^2-45 a b^2 c-44 a b c^2 x^2+5 b^4+5 b^3 c x^2\right)}{a+b x^2+c x^4}-\frac{16 a x^{3/2} \left(4 a c-b^2\right) \left(-2 a c+b^2+b c x^2\right)}{\left(a+b x^2+c x^4\right)^2}}{64 a^2 \left(b^2-4 a c\right)^2}","\frac{x^{3/2} \left(52 a^2 c^2+b c x^2 \left(5 b^2-44 a c\right)-45 a b^2 c+5 b^4\right)}{16 a^2 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}-\frac{\sqrt[4]{c} \left(520 a^2 c^2-54 a b^2 c-b \left(5 b^2-44 a c\right) \sqrt{b^2-4 a c}+5 b^4\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{32\ 2^{3/4} a^2 \left(b^2-4 a c\right)^{5/2} \sqrt[4]{-\sqrt{b^2-4 a c}-b}}+\frac{\sqrt[4]{c} \left(520 a^2 c^2-54 a b^2 c+b \left(5 b^2-44 a c\right) \sqrt{b^2-4 a c}+5 b^4\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{32\ 2^{3/4} a^2 \left(b^2-4 a c\right)^{5/2} \sqrt[4]{\sqrt{b^2-4 a c}-b}}+\frac{\sqrt[4]{c} \left(520 a^2 c^2-54 a b^2 c-b \left(5 b^2-44 a c\right) \sqrt{b^2-4 a c}+5 b^4\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{32\ 2^{3/4} a^2 \left(b^2-4 a c\right)^{5/2} \sqrt[4]{-\sqrt{b^2-4 a c}-b}}-\frac{\sqrt[4]{c} \left(520 a^2 c^2-54 a b^2 c+b \left(5 b^2-44 a c\right) \sqrt{b^2-4 a c}+5 b^4\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{32\ 2^{3/4} a^2 \left(b^2-4 a c\right)^{5/2} \sqrt[4]{\sqrt{b^2-4 a c}-b}}+\frac{x^{3/2} \left(-2 a c+b^2+b c x^2\right)}{4 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}",1,"((-16*a*(-b^2 + 4*a*c)*x^(3/2)*(b^2 - 2*a*c + b*c*x^2))/(a + b*x^2 + c*x^4)^2 + (4*x^(3/2)*(5*b^4 - 45*a*b^2*c + 52*a^2*c^2 + 5*b^3*c*x^2 - 44*a*b*c^2*x^2))/(a + b*x^2 + c*x^4) + RootSum[a + b*#1^4 + c*#1^8 & , (5*b^4*Log[Sqrt[x] - #1] - 49*a*b^2*c*Log[Sqrt[x] - #1] + 260*a^2*c^2*Log[Sqrt[x] - #1] + 5*b^3*c*Log[Sqrt[x] - #1]*#1^4 - 44*a*b*c^2*Log[Sqrt[x] - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ])/(64*a^2*(b^2 - 4*a*c)^2)","C",1
1088,1,258,658,0.4640829,"\int \frac{1}{\sqrt{x} \left(a+b x^2+c x^4\right)^3} \, dx","Integrate[1/(Sqrt[x]*(a + b*x^2 + c*x^4)^3),x]","\frac{3 \text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{-52 \text{$\#$1}^4 a b c^2 \log \left(\sqrt{x}-\text{$\#$1}\right)+7 \text{$\#$1}^4 b^3 c \log \left(\sqrt{x}-\text{$\#$1}\right)+140 a^2 c^2 \log \left(\sqrt{x}-\text{$\#$1}\right)-59 a b^2 c \log \left(\sqrt{x}-\text{$\#$1}\right)+7 b^4 \log \left(\sqrt{x}-\text{$\#$1}\right)}{2 \text{$\#$1}^7 c+\text{$\#$1}^3 b}\&\right]+\frac{4 \sqrt{x} \left(60 a^2 c^2-55 a b^2 c-52 a b c^2 x^2+7 b^4+7 b^3 c x^2\right)}{a+b x^2+c x^4}-\frac{16 a \sqrt{x} \left(4 a c-b^2\right) \left(-2 a c+b^2+b c x^2\right)}{\left(a+b x^2+c x^4\right)^2}}{64 a^2 \left(b^2-4 a c\right)^2}","\frac{\sqrt{x} \left(60 a^2 c^2+b c x^2 \left(7 b^2-52 a c\right)-55 a b^2 c+7 b^4\right)}{16 a^2 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{3 c^{3/4} \left(280 a^2 c^2-66 a b^2 c-b \left(7 b^2-52 a c\right) \sqrt{b^2-4 a c}+7 b^4\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{32 \sqrt[4]{2} a^2 \left(b^2-4 a c\right)^{5/2} \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{3 c^{3/4} \left(280 a^2 c^2-66 a b^2 c+b \left(7 b^2-52 a c\right) \sqrt{b^2-4 a c}+7 b^4\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{32 \sqrt[4]{2} a^2 \left(b^2-4 a c\right)^{5/2} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{3 c^{3/4} \left(280 a^2 c^2-66 a b^2 c-b \left(7 b^2-52 a c\right) \sqrt{b^2-4 a c}+7 b^4\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{32 \sqrt[4]{2} a^2 \left(b^2-4 a c\right)^{5/2} \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{3 c^{3/4} \left(280 a^2 c^2-66 a b^2 c+b \left(7 b^2-52 a c\right) \sqrt{b^2-4 a c}+7 b^4\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{32 \sqrt[4]{2} a^2 \left(b^2-4 a c\right)^{5/2} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{\sqrt{x} \left(-2 a c+b^2+b c x^2\right)}{4 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}",1,"((-16*a*(-b^2 + 4*a*c)*Sqrt[x]*(b^2 - 2*a*c + b*c*x^2))/(a + b*x^2 + c*x^4)^2 + (4*Sqrt[x]*(7*b^4 - 55*a*b^2*c + 60*a^2*c^2 + 7*b^3*c*x^2 - 52*a*b*c^2*x^2))/(a + b*x^2 + c*x^4) + 3*RootSum[a + b*#1^4 + c*#1^8 & , (7*b^4*Log[Sqrt[x] - #1] - 59*a*b^2*c*Log[Sqrt[x] - #1] + 140*a^2*c^2*Log[Sqrt[x] - #1] + 7*b^3*c*Log[Sqrt[x] - #1]*#1^4 - 52*a*b*c^2*Log[Sqrt[x] - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ])/(64*a^2*(b^2 - 4*a*c)^2)","C",1
1089,1,365,147,0.5791524,"\int (d x)^{3/2} \sqrt{a+b x^2+c x^4} \, dx","Integrate[(d*x)^(3/2)*Sqrt[a + b*x^2 + c*x^4],x]","\frac{2 d \sqrt{d x} \left(2 x^2 \left(10 a c-3 b^2\right) \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{5}{4};\frac{1}{2},\frac{1}{2};\frac{9}{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)-10 a b \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{1}{4};\frac{1}{2},\frac{1}{2};\frac{5}{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)+5 \left(2 b+5 c x^2\right) \left(a+b x^2+c x^4\right)\right)}{225 c \sqrt{a+b x^2+c x^4}}","\frac{2 (d x)^{5/2} \sqrt{a+b x^2+c x^4} F_1\left(\frac{5}{4};-\frac{1}{2},-\frac{1}{2};\frac{9}{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)}{5 d \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}}",1,"(2*d*Sqrt[d*x]*(5*(2*b + 5*c*x^2)*(a + b*x^2 + c*x^4) - 10*a*b*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[1/4, 1/2, 1/2, 5/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])] + 2*(-3*b^2 + 10*a*c)*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[5/4, 1/2, 1/2, 9/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])]))/(225*c*Sqrt[a + b*x^2 + c*x^4])","B",0
1090,1,342,147,0.4318685,"\int \sqrt{d x} \sqrt{a+b x^2+c x^4} \, dx","Integrate[Sqrt[d*x]*Sqrt[a + b*x^2 + c*x^4],x]","\frac{2 x \sqrt{d x} \left(6 b 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{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)+28 a \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)+21 \left(a+b x^2+c x^4\right)\right)}{147 \sqrt{a+b x^2+c x^4}}","\frac{2 (d x)^{3/2} \sqrt{a+b x^2+c x^4} 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 d \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}}",1,"(2*x*Sqrt[d*x]*(21*(a + b*x^2 + c*x^4) + 28*a*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])] + 6*b*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[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])]))/(147*Sqrt[a + b*x^2 + c*x^4])","B",0
1091,1,342,145,0.3802445,"\int \frac{\sqrt{a+b x^2+c x^4}}{\sqrt{d x}} \, dx","Integrate[Sqrt[a + b*x^2 + c*x^4]/Sqrt[d*x],x]","\frac{2 x \left(2 b 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{5}{4};\frac{1}{2},\frac{1}{2};\frac{9}{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)+20 a \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{1}{4};\frac{1}{2},\frac{1}{2};\frac{5}{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)+5 \left(a+b x^2+c x^4\right)\right)}{25 \sqrt{d x} \sqrt{a+b x^2+c x^4}}","\frac{2 \sqrt{d x} \sqrt{a+b x^2+c x^4} F_1\left(\frac{1}{4};-\frac{1}{2},-\frac{1}{2};\frac{5}{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)}{d \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}}",1,"(2*x*(5*(a + b*x^2 + c*x^4) + 20*a*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[1/4, 1/2, 1/2, 5/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])] + 2*b*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[5/4, 1/2, 1/2, 9/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])]))/(25*Sqrt[d*x]*Sqrt[a + b*x^2 + c*x^4])","B",0
1092,1,345,145,0.3556724,"\int \frac{\sqrt{a+b x^2+c x^4}}{(d x)^{3/2}} \, dx","Integrate[Sqrt[a + b*x^2 + c*x^4]/(d*x)^(3/2),x]","\frac{x \left(28 b 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)+24 c x^4 \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{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)-42 \left(a+b x^2+c x^4\right)\right)}{21 (d x)^{3/2} \sqrt{a+b x^2+c x^4}}","-\frac{2 \sqrt{a+b x^2+c x^4} F_1\left(-\frac{1}{4};-\frac{1}{2},-\frac{1}{2};\frac{3}{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)}{d \sqrt{d x} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}}",1,"(x*(-42*(a + b*x^2 + c*x^4) + 28*b*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])] + 24*c*x^4*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[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*(d*x)^(3/2)*Sqrt[a + b*x^2 + c*x^4])","B",0
1093,1,459,148,0.8787914,"\int (d x)^{3/2} \left(a+b x^2+c x^4\right)^{3/2} \, dx","Integrate[(d*x)^(3/2)*(a + b*x^2 + c*x^4)^(3/2),x]","\frac{2 d \sqrt{d x} \left(4 x^2 \left(260 a^2 c^2-157 a b^2 c+21 b^4\right) \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{5}{4};\frac{1}{2},\frac{1}{2};\frac{9}{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)+5 \left(a^2 c \left(176 b+455 c x^2\right)+a \left(-28 b^3+196 b^2 c x^2+916 b c^2 x^4+650 c^3 x^6\right)-28 b^4 x^2-8 b^3 c x^4+305 b^2 c^2 x^6+480 b c^3 x^8+195 c^4 x^{10}\right)+20 a b \left(7 b^2-44 a c\right) \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{1}{4};\frac{1}{2},\frac{1}{2};\frac{5}{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)}{16575 c^2 \sqrt{a+b x^2+c x^4}}","\frac{2 a (d x)^{5/2} \sqrt{a+b x^2+c x^4} F_1\left(\frac{5}{4};-\frac{3}{2},-\frac{3}{2};\frac{9}{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)}{5 d \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}}",1,"(2*d*Sqrt[d*x]*(5*(-28*b^4*x^2 - 8*b^3*c*x^4 + 305*b^2*c^2*x^6 + 480*b*c^3*x^8 + 195*c^4*x^10 + a^2*c*(176*b + 455*c*x^2) + a*(-28*b^3 + 196*b^2*c*x^2 + 916*b*c^2*x^4 + 650*c^3*x^6)) + 20*a*b*(7*b^2 - 44*a*c)*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[1/4, 1/2, 1/2, 5/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])] + 4*(21*b^4 - 157*a*b^2*c + 260*a^2*c^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[5/4, 1/2, 1/2, 9/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])]))/(16575*c^2*Sqrt[a + b*x^2 + c*x^4])","B",0
1094,1,417,148,0.7097764,"\int \sqrt{d x} \left(a+b x^2+c x^4\right)^{3/2} \, dx","Integrate[Sqrt[d*x]*(a + b*x^2 + c*x^4)^(3/2),x]","\frac{2 x \sqrt{d x} \left(7 \left(209 a^2 c+12 a b^2+328 a b c x^2+286 a c^2 x^4+12 b^3 x^2+131 b^2 c x^4+196 b c^2 x^6+77 c^3 x^8\right)+12 b x^2 \left(36 a c-5 b^2\right) \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{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)-28 a \left(3 b^2-44 a c\right) \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)\right)}{8085 c \sqrt{a+b x^2+c x^4}}","\frac{2 a (d x)^{3/2} \sqrt{a+b x^2+c x^4} F_1\left(\frac{3}{4};-\frac{3}{2},-\frac{3}{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 d \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}}",1,"(2*x*Sqrt[d*x]*(7*(12*a*b^2 + 209*a^2*c + 12*b^3*x^2 + 328*a*b*c*x^2 + 131*b^2*c*x^4 + 286*a*c^2*x^4 + 196*b*c^2*x^6 + 77*c^3*x^8) - 28*a*(3*b^2 - 44*a*c)*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])] + 12*b*(-5*b^2 + 36*a*c)*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[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])]))/(8085*c*Sqrt[a + b*x^2 + c*x^4])","B",0
1095,1,415,146,0.7057975,"\int \frac{\left(a+b x^2+c x^4\right)^{3/2}}{\sqrt{d x}} \, dx","Integrate[(a + b*x^2 + c*x^4)^(3/2)/Sqrt[d*x],x]","\frac{2 x \left(5 \left(51 a^2 c+4 a b^2+76 a b c x^2+66 a c^2 x^4+4 b^3 x^2+29 b^2 c x^4+40 b c^2 x^6+15 c^3 x^8\right)-4 b x^2 \left(3 b^2-28 a c\right) \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{5}{4};\frac{1}{2},\frac{1}{2};\frac{9}{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)-20 a \left(b^2-36 a c\right) \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{1}{4};\frac{1}{2},\frac{1}{2};\frac{5}{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)}{975 c \sqrt{d x} \sqrt{a+b x^2+c x^4}}","\frac{2 a \sqrt{d x} \sqrt{a+b x^2+c x^4} F_1\left(\frac{1}{4};-\frac{3}{2},-\frac{3}{2};\frac{5}{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)}{d \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}}",1,"(2*x*(5*(4*a*b^2 + 51*a^2*c + 4*b^3*x^2 + 76*a*b*c*x^2 + 29*b^2*c*x^4 + 66*a*c^2*x^4 + 40*b*c^2*x^6 + 15*c^3*x^8) - 20*a*(b^2 - 36*a*c)*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[1/4, 1/2, 1/2, 5/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])] - 4*b*(3*b^2 - 28*a*c)*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[5/4, 1/2, 1/2, 9/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])]))/(975*c*Sqrt[d*x]*Sqrt[a + b*x^2 + c*x^4])","B",0
1096,1,384,146,0.7313882,"\int \frac{\left(a+b x^2+c x^4\right)^{3/2}}{(d x)^{3/2}} \, dx","Integrate[(a + b*x^2 + c*x^4)^(3/2)/(d*x)^(3/2),x]","\frac{x \left(14 \left(-77 a^2-64 a b x^2-70 a c x^4+13 b^2 x^4+20 b c x^6+7 c^2 x^8\right)+896 a b 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)+24 x^4 \left(28 a c+b^2\right) \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{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)}{539 (d x)^{3/2} \sqrt{a+b x^2+c x^4}}","-\frac{2 a \sqrt{a+b x^2+c x^4} F_1\left(-\frac{1}{4};-\frac{3}{2},-\frac{3}{2};\frac{3}{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)}{d \sqrt{d x} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}}",1,"(x*(14*(-77*a^2 - 64*a*b*x^2 + 13*b^2*x^4 - 70*a*c*x^4 + 20*b*c*x^6 + 7*c^2*x^8) + 896*a*b*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])] + 24*(b^2 + 28*a*c)*x^4*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[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])]))/(539*(d*x)^(3/2)*Sqrt[a + b*x^2 + c*x^4])","B",0
1097,1,173,147,0.0915265,"\int \frac{(d x)^{3/2}}{\sqrt{a+b x^2+c x^4}} \, dx","Integrate[(d*x)^(3/2)/Sqrt[a + b*x^2 + c*x^4],x]","\frac{2 x (d x)^{3/2} \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{5}{4};\frac{1}{2},\frac{1}{2};\frac{9}{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)}{5 \sqrt{a+b x^2+c x^4}}","\frac{2 (d x)^{5/2} \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{5}{4};\frac{1}{2},\frac{1}{2};\frac{9}{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)}{5 d \sqrt{a+b x^2+c x^4}}",1,"(2*x*(d*x)^(3/2)*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[5/4, 1/2, 1/2, 9/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/(5*Sqrt[a + b*x^2 + c*x^4])","A",0
1098,1,173,147,0.0985187,"\int \frac{\sqrt{d x}}{\sqrt{a+b x^2+c x^4}} \, dx","Integrate[Sqrt[d*x]/Sqrt[a + b*x^2 + c*x^4],x]","\frac{2 x \sqrt{d x} \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{a+b x^2+c x^4}}","\frac{2 (d x)^{3/2} \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 d \sqrt{a+b x^2+c x^4}}",1,"(2*x*Sqrt[d*x]*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[a + b*x^2 + c*x^4])","A",0
1099,1,171,145,0.0801632,"\int \frac{1}{\sqrt{d x} \sqrt{a+b x^2+c x^4}} \, dx","Integrate[1/(Sqrt[d*x]*Sqrt[a + b*x^2 + c*x^4]),x]","\frac{2 x \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{1}{4};\frac{1}{2},\frac{1}{2};\frac{5}{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)}{\sqrt{d x} \sqrt{a+b x^2+c x^4}}","\frac{2 \sqrt{d x} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1} F_1\left(\frac{1}{4};\frac{1}{2},\frac{1}{2};\frac{5}{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)}{d \sqrt{a+b x^2+c x^4}}",1,"(2*x*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[1/4, 1/2, 1/2, 5/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/(Sqrt[d*x]*Sqrt[a + b*x^2 + c*x^4])","A",0
1100,1,348,145,0.478356,"\int \frac{1}{(d x)^{3/2} \sqrt{a+b x^2+c x^4}} \, dx","Integrate[1/((d*x)^(3/2)*Sqrt[a + b*x^2 + c*x^4]),x]","\frac{x \left(14 b 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)+18 c x^4 \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{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)-42 \left(a+b x^2+c x^4\right)\right)}{21 a (d x)^{3/2} \sqrt{a+b x^2+c x^4}}","-\frac{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{1}{4};\frac{1}{2},\frac{1}{2};\frac{3}{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)}{d \sqrt{d x} \sqrt{a+b x^2+c x^4}}",1,"(x*(-42*(a + b*x^2 + c*x^4) + 14*b*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])] + 18*c*x^4*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[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*a*(d*x)^(3/2)*Sqrt[a + b*x^2 + c*x^4])","B",0
1101,1,348,150,0.3436936,"\int \frac{(d x)^{3/2}}{\left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(d*x)^(3/2)/(a + b*x^2 + c*x^4)^(3/2),x]","\frac{d \sqrt{d x} \left(2 c 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{5}{4};\frac{1}{2},\frac{1}{2};\frac{9}{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)+5 b \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{1}{4};\frac{1}{2},\frac{1}{2};\frac{5}{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)-5 \left(b+2 c x^2\right)\right)}{5 \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}","\frac{2 (d x)^{5/2} \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{5}{4};\frac{3}{2},\frac{3}{2};\frac{9}{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)}{5 a d \sqrt{a+b x^2+c x^4}}",1,"(d*Sqrt[d*x]*(-5*(b + 2*c*x^2) + 5*b*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[1/4, 1/2, 1/2, 5/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])] + 2*c*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[5/4, 1/2, 1/2, 9/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])]))/(5*(b^2 - 4*a*c)*Sqrt[a + b*x^2 + c*x^4])","B",0
1102,1,367,150,0.5101967,"\int \frac{\sqrt{d x}}{\left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[Sqrt[d*x]/(a + b*x^2 + c*x^4)^(3/2),x]","\frac{x \sqrt{d x} \left(9 b c 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{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)+7 \left(2 a c+b^2\right) \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)-21 \left(-2 a c+b^2+b c x^2\right)\right)}{21 a \left(4 a c-b^2\right) \sqrt{a+b x^2+c x^4}}","\frac{2 (d x)^{3/2} \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{3}{2},\frac{3}{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 a d \sqrt{a+b x^2+c x^4}}",1,"(x*Sqrt[d*x]*(-21*(b^2 - 2*a*c + b*c*x^2) + 7*(b^2 + 2*a*c)*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])] + 9*b*c*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[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*a*(-b^2 + 4*a*c)*Sqrt[a + b*x^2 + c*x^4])","B",0
1103,1,366,148,0.4721076,"\int \frac{1}{\sqrt{d x} \left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[1/(Sqrt[d*x]*(a + b*x^2 + c*x^4)^(3/2)),x]","\frac{x \left(b c 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{5}{4};\frac{1}{2},\frac{1}{2};\frac{9}{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)-5 \left(b^2-6 a c\right) \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{1}{4};\frac{1}{2},\frac{1}{2};\frac{5}{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)-5 \left(-2 a c+b^2+b c x^2\right)\right)}{5 a \sqrt{d x} \left(4 a c-b^2\right) \sqrt{a+b x^2+c x^4}}","\frac{2 \sqrt{d x} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1} F_1\left(\frac{1}{4};\frac{3}{2},\frac{3}{2};\frac{5}{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)}{a d \sqrt{a+b x^2+c x^4}}",1,"(x*(-5*(b^2 - 2*a*c + b*c*x^2) - 5*(b^2 - 6*a*c)*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[1/4, 1/2, 1/2, 5/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])] + b*c*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[5/4, 1/2, 1/2, 9/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])]))/(5*a*(-b^2 + 4*a*c)*Sqrt[d*x]*Sqrt[a + b*x^2 + c*x^4])","B",0
1104,1,409,148,0.6932327,"\int \frac{1}{(d x)^{3/2} \left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[1/((d*x)^(3/2)*(a + b*x^2 + c*x^4)^(3/2)),x]","-\frac{x \left(-7 \left(8 a^2 c+a \left(-2 b^2+11 b c x^2+10 c^2 x^4\right)-3 b^2 x^2 \left(b+c x^2\right)\right)-7 b x^2 \left(b^2-3 a c\right) \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 c x^4 \left(10 a c-3 b^2\right) \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{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)}{7 a^2 (d x)^{3/2} \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}","-\frac{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{1}{4};\frac{3}{2},\frac{3}{2};\frac{3}{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)}{a d \sqrt{d x} \sqrt{a+b x^2+c x^4}}",1,"-1/7*(x*(-7*(8*a^2*c - 3*b^2*x^2*(b + c*x^2) + a*(-2*b^2 + 11*b*c*x^2 + 10*c^2*x^4)) - 7*b*(b^2 - 3*a*c)*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*c*(-3*b^2 + 10*a*c)*x^4*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[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])]))/(a^2*(b^2 - 4*a*c)*(d*x)^(3/2)*Sqrt[a + b*x^2 + c*x^4])","B",0
1105,1,111,156,0.129294,"\int (d x)^m \left(a+b x^2+c x^4\right)^3 \, dx","Integrate[(d*x)^m*(a + b*x^2 + c*x^4)^3,x]","x (d x)^m \left(\frac{a^3}{m+1}+\frac{3 a^2 b x^2}{m+3}+\frac{3 c x^8 \left(a c+b^2\right)}{m+9}+\frac{b x^6 \left(6 a c+b^2\right)}{m+7}+\frac{3 a x^4 \left(a c+b^2\right)}{m+5}+\frac{3 b c^2 x^{10}}{m+11}+\frac{c^3 x^{12}}{m+13}\right)","\frac{a^3 (d x)^{m+1}}{d (m+1)}+\frac{3 a^2 b (d x)^{m+3}}{d^3 (m+3)}+\frac{3 c \left(a c+b^2\right) (d x)^{m+9}}{d^9 (m+9)}+\frac{b \left(6 a c+b^2\right) (d x)^{m+7}}{d^7 (m+7)}+\frac{3 a \left(a c+b^2\right) (d x)^{m+5}}{d^5 (m+5)}+\frac{3 b c^2 (d x)^{m+11}}{d^{11} (m+11)}+\frac{c^3 (d x)^{m+13}}{d^{13} (m+13)}",1,"x*(d*x)^m*(a^3/(1 + m) + (3*a^2*b*x^2)/(3 + m) + (3*a*(b^2 + a*c)*x^4)/(5 + m) + (b*(b^2 + 6*a*c)*x^6)/(7 + m) + (3*c*(b^2 + a*c)*x^8)/(9 + m) + (3*b*c^2*x^10)/(11 + m) + (c^3*x^12)/(13 + m))","A",1
1106,1,70,101,0.0539032,"\int (d x)^m \left(a+b x^2+c x^4\right)^2 \, dx","Integrate[(d*x)^m*(a + b*x^2 + c*x^4)^2,x]","x (d x)^m \left(\frac{a^2}{m+1}+\frac{x^4 \left(2 a c+b^2\right)}{m+5}+\frac{2 a b x^2}{m+3}+\frac{2 b c x^6}{m+7}+\frac{c^2 x^8}{m+9}\right)","\frac{a^2 (d x)^{m+1}}{d (m+1)}+\frac{\left(2 a c+b^2\right) (d x)^{m+5}}{d^5 (m+5)}+\frac{2 a b (d x)^{m+3}}{d^3 (m+3)}+\frac{2 b c (d x)^{m+7}}{d^7 (m+7)}+\frac{c^2 (d x)^{m+9}}{d^9 (m+9)}",1,"x*(d*x)^m*(a^2/(1 + m) + (2*a*b*x^2)/(3 + m) + ((b^2 + 2*a*c)*x^4)/(5 + m) + (2*b*c*x^6)/(7 + m) + (c^2*x^8)/(9 + m))","A",1
1107,1,35,52,0.0291373,"\int (d x)^m \left(a+b x^2+c x^4\right) \, dx","Integrate[(d*x)^m*(a + b*x^2 + c*x^4),x]","x (d x)^m \left(\frac{a}{m+1}+\frac{b x^2}{m+3}+\frac{c x^4}{m+5}\right)","\frac{a (d x)^{m+1}}{d (m+1)}+\frac{b (d x)^{m+3}}{d^3 (m+3)}+\frac{c (d x)^{m+5}}{d^5 (m+5)}",1,"x*(d*x)^m*(a/(1 + m) + (b*x^2)/(3 + m) + (c*x^4)/(5 + m))","A",1
1108,1,82,173,0.0572284,"\int \frac{(d x)^m}{a+b x^2+c x^4} \, dx","Integrate[(d*x)^m/(a + b*x^2 + c*x^4),x]","\frac{(d x)^m \text{RootSum}\left[\text{$\#$1}^4 c+\text{$\#$1}^2 b+a\&,\frac{\left(\frac{x}{x-\text{$\#$1}}\right)^{-m} \, _2F_1\left(-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right)}{2 \text{$\#$1}^3 c+\text{$\#$1} b}\&\right]}{2 m}","\frac{2 c (d x)^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right)}{d (m+1) \sqrt{b^2-4 a c} \left(b-\sqrt{b^2-4 a c}\right)}-\frac{2 c (d x)^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right)}{d (m+1) \sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}+b\right)}",1,"((d*x)^m*RootSum[a + b*#1^2 + c*#1^4 & , Hypergeometric2F1[-m, -m, 1 - m, -(#1/(x - #1))]/((x/(x - #1))^m*(b*#1 + 2*c*#1^3)) & ])/(2*m)","C",0
1109,1,78,315,0.0803871,"\int \frac{(d x)^m}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(d*x)^m/(a + b*x^2 + c*x^4)^2,x]","\frac{x (d x)^m F_1\left(\frac{m+1}{2};2,2;\frac{m+3}{2};-\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)}{a^2 (m+1)}","\frac{c (d x)^{m+1} \left(b (1-m) \sqrt{b^2-4 a c}-4 a c (3-m)+b^2 (1-m)\right) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right)}{2 a d (m+1) \left(b^2-4 a c\right)^{3/2} \left(b-\sqrt{b^2-4 a c}\right)}-\frac{c (d x)^{m+1} \left(-b (1-m) \sqrt{b^2-4 a c}-4 a c (3-m)+b^2 (1-m)\right) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right)}{2 a d (m+1) \left(b^2-4 a c\right)^{3/2} \left(\sqrt{b^2-4 a c}+b\right)}+\frac{(d x)^{m+1} \left(-2 a c+b^2+b c x^2\right)}{2 a d \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"(x*(d*x)^m*AppellF1[(1 + m)/2, 2, 2, (3 + m)/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/(a^2*(1 + m))","C",0
1110,1,357,158,0.2807817,"\int (d x)^m \left(a+b x^2+c x^4\right)^{3/2} \, dx","Integrate[(d*x)^m*(a + b*x^2 + c*x^4)^(3/2),x]","\frac{x (d x)^m \sqrt{a+b x^2+c x^4} \left(a \left(m^2+8 m+15\right) F_1\left(\frac{m+1}{2};-\frac{1}{2},-\frac{1}{2};\frac{m+3}{2};-\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)+(m+1) x^2 \left(c (m+3) x^2 F_1\left(\frac{m+5}{2};-\frac{1}{2},-\frac{1}{2};\frac{m+7}{2};-\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)+b (m+5) F_1\left(\frac{m+3}{2};-\frac{1}{2},-\frac{1}{2};\frac{m+5}{2};-\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)\right)}{(m+1) (m+3) (m+5) \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}}}","\frac{a (d x)^{m+1} \sqrt{a+b x^2+c x^4} F_1\left(\frac{m+1}{2};-\frac{3}{2},-\frac{3}{2};\frac{m+3}{2};-\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)}{d (m+1) \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}}",1,"(x*(d*x)^m*Sqrt[a + b*x^2 + c*x^4]*(a*(15 + 8*m + m^2)*AppellF1[(1 + m)/2, -1/2, -1/2, (3 + m)/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])] + (1 + m)*x^2*(b*(5 + m)*AppellF1[(3 + m)/2, -1/2, -1/2, (5 + m)/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])] + c*(3 + m)*x^2*AppellF1[(5 + m)/2, -1/2, -1/2, (7 + m)/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])))/((1 + m)*(3 + m)*(5 + m)*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])])","B",0
1111,1,181,157,0.0989531,"\int (d x)^m \sqrt{a+b x^2+c x^4} \, dx","Integrate[(d*x)^m*Sqrt[a + b*x^2 + c*x^4],x]","\frac{x (d x)^m \sqrt{a+b x^2+c x^4} F_1\left(\frac{m+1}{2};-\frac{1}{2},-\frac{1}{2};\frac{m+3}{2};-\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)}{(m+1) \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}}}","\frac{(d x)^{m+1} \sqrt{a+b x^2+c x^4} F_1\left(\frac{m+1}{2};-\frac{1}{2},-\frac{1}{2};\frac{m+3}{2};-\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)}{d (m+1) \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}}",1,"(x*(d*x)^m*Sqrt[a + b*x^2 + c*x^4]*AppellF1[(1 + m)/2, -1/2, -1/2, (3 + m)/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/((1 + m)*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])])","A",0
1112,1,181,157,0.1119068,"\int \frac{(d x)^m}{\sqrt{a+b x^2+c x^4}} \, dx","Integrate[(d*x)^m/Sqrt[a + b*x^2 + c*x^4],x]","\frac{x (d x)^m \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{m+1}{2};\frac{1}{2},\frac{1}{2};\frac{m+3}{2};-\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)}{(m+1) \sqrt{a+b x^2+c x^4}}","\frac{(d x)^{m+1} \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{m+1}{2};\frac{1}{2},\frac{1}{2};\frac{m+3}{2};-\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)}{d (m+1) \sqrt{a+b x^2+c x^4}}",1,"(x*(d*x)^m*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[(1 + m)/2, 1/2, 1/2, (3 + m)/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/((1 + m)*Sqrt[a + b*x^2 + c*x^4])","A",0
1113,1,221,160,0.1716749,"\int \frac{(d x)^m}{\left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(d*x)^m/(a + b*x^2 + c*x^4)^(3/2),x]","\frac{x (d x)^m \left(\sqrt{b^2-4 a c}-b-2 c x^2\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{b-\sqrt{b^2-4 a c}}} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}\right)^{3/2} F_1\left(\frac{m+1}{2};\frac{3}{2},\frac{3}{2};\frac{m+3}{2};-\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)}{(m+1) \left(\sqrt{b^2-4 a c}-b\right) \left(a+b x^2+c x^4\right)^{3/2}}","\frac{(d x)^{m+1} \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{m+1}{2};\frac{3}{2},\frac{3}{2};\frac{m+3}{2};-\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)}{a d (m+1) \sqrt{a+b x^2+c x^4}}",1,"(x*(d*x)^m*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]))^(3/2)*AppellF1[(1 + m)/2, 3/2, 3/2, (3 + m)/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/((-b + Sqrt[b^2 - 4*a*c])*(1 + m)*(a + b*x^2 + c*x^4)^(3/2))","A",0
1114,1,179,155,0.240921,"\int (d x)^m \left(a+b x^2+c x^4\right)^p \, dx","Integrate[(d*x)^m*(a + b*x^2 + c*x^4)^p,x]","\frac{x (d x)^m \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}\right)^{-p} \left(a+b x^2+c x^4\right)^p F_1\left(\frac{m+1}{2};-p,-p;\frac{m+3}{2};-\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)}{m+1}","\frac{(d x)^{m+1} \left(\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^2+c x^4\right)^p F_1\left(\frac{m+1}{2};-p,-p;\frac{m+3}{2};-\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)}{d (m+1)}",1,"(x*(d*x)^m*(a + b*x^2 + c*x^4)^p*AppellF1[(1 + m)/2, -p, -p, (3 + m)/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/((1 + m)*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]))^p)","A",0
1115,1,162,257,0.2340816,"\int x^7 \left(a+b x^2+c x^4\right)^p \, dx","Integrate[x^7*(a + b*x^2 + c*x^4)^p,x]","\frac{1}{8} x^8 \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}\right)^{-p} \left(a+b x^2+c x^4\right)^p F_1\left(4;-p,-p;5;-\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)","-\frac{b 2^{p-2} \left(6 a c-b^2 (p+3)\right) \left(a+b x^2+c x^4\right)^{p+1} \left(-\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}}\right)^{-p-1} \, _2F_1\left(-p,p+1;p+2;\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right)}{c^3 (p+1) (2 p+3) \sqrt{b^2-4 a c}}+\frac{\left(-2 a c (2 p+3)+b^2 (p+2) (p+3)-2 b c (p+1) (p+3) x^2\right) \left(a+b x^2+c x^4\right)^{p+1}}{8 c^3 (p+1) (p+2) (2 p+3)}+\frac{x^4 \left(a+b x^2+c x^4\right)^{p+1}}{4 c (p+2)}",1,"(x^8*(a + b*x^2 + c*x^4)^p*AppellF1[4, -p, -p, 5, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/(8*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]))^p)","C",0
1116,1,162,223,0.2243245,"\int x^5 \left(a+b x^2+c x^4\right)^p \, dx","Integrate[x^5*(a + b*x^2 + c*x^4)^p,x]","\frac{1}{6} x^6 \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}\right)^{-p} \left(a+b x^2+c x^4\right)^p F_1\left(3;-p,-p;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)","\frac{2^{p-1} \left(2 a c-b^2 (p+2)\right) \left(a+b x^2+c x^4\right)^{p+1} \left(-\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}}\right)^{-p-1} \, _2F_1\left(-p,p+1;p+2;\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right)}{c^2 (p+1) (2 p+3) \sqrt{b^2-4 a c}}-\frac{b (p+2) \left(a+b x^2+c x^4\right)^{p+1}}{4 c^2 (p+1) (2 p+3)}+\frac{x^2 \left(a+b x^2+c x^4\right)^{p+1}}{2 c (2 p+3)}",1,"(x^6*(a + b*x^2 + c*x^4)^p*AppellF1[3, -p, -p, 4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/(6*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]))^p)","C",0
1117,1,162,160,0.1984835,"\int x^3 \left(a+b x^2+c x^4\right)^p \, dx","Integrate[x^3*(a + b*x^2 + c*x^4)^p,x]","\frac{1}{4} x^4 \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}\right)^{-p} \left(a+b x^2+c x^4\right)^p F_1\left(2;-p,-p;3;-\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)","\frac{b 2^{p-1} \left(a+b x^2+c x^4\right)^{p+1} \left(-\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}}\right)^{-p-1} \, _2F_1\left(-p,p+1;p+2;\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right)}{c (p+1) \sqrt{b^2-4 a c}}+\frac{\left(a+b x^2+c x^4\right)^{p+1}}{4 c (p+1)}",1,"(x^4*(a + b*x^2 + c*x^4)^p*AppellF1[2, -p, -p, 3, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/(4*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]))^p)","C",0
1118,1,135,126,0.0979307,"\int x \left(a+b x^2+c x^4\right)^p \, dx","Integrate[x*(a + b*x^2 + c*x^4)^p,x]","\frac{2^{p-2} \left(-\sqrt{b^2-4 a c}+b+2 c x^2\right) \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}}\right)^{-p} \left(a+b x^2+c x^4\right)^p \, _2F_1\left(-p,p+1;p+2;\frac{-2 c x^2-b+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right)}{c (p+1)}","-\frac{2^p \left(-\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}}\right)^{-p-1} \left(a+b x^2+c x^4\right)^{p+1} \, _2F_1\left(-p,p+1;p+2;\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right)}{(p+1) \sqrt{b^2-4 a c}}",1,"(2^(-2 + p)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)*(a + b*x^2 + c*x^4)^p*Hypergeometric2F1[-p, 1 + p, 2 + p, (-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)/(2*Sqrt[b^2 - 4*a*c])])/(c*(1 + p)*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/Sqrt[b^2 - 4*a*c])^p)","A",1
1119,1,152,152,0.204434,"\int \frac{\left(a+b x^2+c x^4\right)^p}{x} \, dx","Integrate[(a + b*x^2 + c*x^4)^p/x,x]","\frac{4^{p-1} \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{c x^2}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{c x^2}\right)^{-p} \left(a+b x^2+c x^4\right)^p F_1\left(-2 p;-p,-p;1-2 p;-\frac{b+\sqrt{b^2-4 a c}}{2 c x^2},\frac{\sqrt{b^2-4 a c}-b}{2 c x^2}\right)}{p}","\frac{4^{p-1} \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{c x^2}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{c x^2}\right)^{-p} \left(a+b x^2+c x^4\right)^p F_1\left(-2 p;-p,-p;1-2 p;-\frac{b-\sqrt{b^2-4 a c}}{2 c x^2},-\frac{b+\sqrt{b^2-4 a c}}{2 c x^2}\right)}{p}",1,"(4^(-1 + p)*(a + b*x^2 + c*x^4)^p*AppellF1[-2*p, -p, -p, 1 - 2*p, -1/2*(b + Sqrt[b^2 - 4*a*c])/(c*x^2), (-b + Sqrt[b^2 - 4*a*c])/(2*c*x^2)])/(p*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(c*x^2))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(c*x^2))^p)","A",0
1120,1,163,166,0.203137,"\int \frac{\left(a+b x^2+c x^4\right)^p}{x^3} \, dx","Integrate[(a + b*x^2 + c*x^4)^p/x^3,x]","\frac{2^{2 p-1} \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{c x^2}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{c x^2}\right)^{-p} \left(a+b x^2+c x^4\right)^p F_1\left(1-2 p;-p,-p;2-2 p;-\frac{b+\sqrt{b^2-4 a c}}{2 c x^2},\frac{\sqrt{b^2-4 a c}-b}{2 c x^2}\right)}{(2 p-1) x^2}","-\frac{2^{2 p-1} \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{c x^2}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{c x^2}\right)^{-p} \left(a+b x^2+c x^4\right)^p F_1\left(1-2 p;-p,-p;2 (1-p);-\frac{b-\sqrt{b^2-4 a c}}{2 c x^2},-\frac{b+\sqrt{b^2-4 a c}}{2 c x^2}\right)}{(1-2 p) x^2}",1,"(2^(-1 + 2*p)*(a + b*x^2 + c*x^4)^p*AppellF1[1 - 2*p, -p, -p, 2 - 2*p, -1/2*(b + Sqrt[b^2 - 4*a*c])/(c*x^2), (-b + Sqrt[b^2 - 4*a*c])/(2*c*x^2)])/((-1 + 2*p)*x^2*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(c*x^2))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(c*x^2))^p)","A",0
1121,1,159,164,0.2211301,"\int \frac{\left(a+b x^2+c x^4\right)^p}{x^5} \, dx","Integrate[(a + b*x^2 + c*x^4)^p/x^5,x]","\frac{4^{p-1} \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{c x^2}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{c x^2}\right)^{-p} \left(a+b x^2+c x^4\right)^p F_1\left(2-2 p;-p,-p;3-2 p;-\frac{b+\sqrt{b^2-4 a c}}{2 c x^2},\frac{\sqrt{b^2-4 a c}-b}{2 c x^2}\right)}{(p-1) x^4}","-\frac{4^{p-1} \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{c x^2}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{c x^2}\right)^{-p} \left(a+b x^2+c x^4\right)^p F_1\left(2 (1-p);-p,-p;3-2 p;-\frac{b-\sqrt{b^2-4 a c}}{2 c x^2},-\frac{b+\sqrt{b^2-4 a c}}{2 c x^2}\right)}{(1-p) x^4}",1,"(4^(-1 + p)*(a + b*x^2 + c*x^4)^p*AppellF1[2 - 2*p, -p, -p, 3 - 2*p, -1/2*(b + Sqrt[b^2 - 4*a*c])/(c*x^2), (-b + Sqrt[b^2 - 4*a*c])/(2*c*x^2)])/((-1 + p)*x^4*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(c*x^2))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(c*x^2))^p)","A",0
1122,1,166,138,0.1696816,"\int x^4 \left(a+b x^2+c x^4\right)^p \, dx","Integrate[x^4*(a + b*x^2 + c*x^4)^p,x]","\frac{1}{5} x^5 \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}\right)^{-p} \left(a+b x^2+c x^4\right)^p F_1\left(\frac{5}{2};-p,-p;\frac{7}{2};-\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)","\frac{1}{5} x^5 \left(\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^2+c x^4\right)^p F_1\left(\frac{5}{2};-p,-p;\frac{7}{2};-\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)",1,"(x^5*(a + b*x^2 + c*x^4)^p*AppellF1[5/2, -p, -p, 7/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/(5*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]))^p)","A",0
1123,1,166,138,0.1540268,"\int x^2 \left(a+b x^2+c x^4\right)^p \, dx","Integrate[x^2*(a + b*x^2 + c*x^4)^p,x]","\frac{1}{3} x^3 \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}\right)^{-p} \left(a+b x^2+c x^4\right)^p F_1\left(\frac{3}{2};-p,-p;\frac{5}{2};-\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)","\frac{1}{3} x^3 \left(\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^2+c x^4\right)^p F_1\left(\frac{3}{2};-p,-p;\frac{5}{2};-\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)",1,"(x^3*(a + b*x^2 + c*x^4)^p*AppellF1[3/2, -p, -p, 5/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/(3*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]))^p)","A",0
1124,1,161,133,0.156315,"\int \left(a+b x^2+c x^4\right)^p \, dx","Integrate[(a + b*x^2 + c*x^4)^p,x]","x \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}\right)^{-p} \left(a+b x^2+c x^4\right)^p F_1\left(\frac{1}{2};-p,-p;\frac{3}{2};-\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)","x \left(\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^2+c x^4\right)^p F_1\left(\frac{1}{2};-p,-p;\frac{3}{2};-\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)",1,"(x*(a + b*x^2 + c*x^4)^p*AppellF1[1/2, -p, -p, 3/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/(((b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]))^p)","A",0
1125,1,164,136,0.1586771,"\int \frac{\left(a+b x^2+c x^4\right)^p}{x^2} \, dx","Integrate[(a + b*x^2 + c*x^4)^p/x^2,x]","-\frac{\left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}\right)^{-p} \left(a+b x^2+c x^4\right)^p F_1\left(-\frac{1}{2};-p,-p;\frac{1}{2};-\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)}{x}","-\frac{\left(\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^2+c x^4\right)^p F_1\left(-\frac{1}{2};-p,-p;\frac{1}{2};-\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)}{x}",1,"-(((a + b*x^2 + c*x^4)^p*AppellF1[-1/2, -p, -p, 1/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/(x*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]))^p))","A",0
1126,1,166,138,0.1688066,"\int \frac{\left(a+b x^2+c x^4\right)^p}{x^4} \, dx","Integrate[(a + b*x^2 + c*x^4)^p/x^4,x]","-\frac{\left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}\right)^{-p} \left(a+b x^2+c x^4\right)^p F_1\left(-\frac{3}{2};-p,-p;-\frac{1}{2};-\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 x^3}","-\frac{\left(\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^2+c x^4\right)^p F_1\left(-\frac{3}{2};-p,-p;-\frac{1}{2};-\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 x^3}",1,"-1/3*((a + b*x^2 + c*x^4)^p*AppellF1[-3/2, -p, -p, -1/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/(x^3*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]))^p)","A",0