1,1,149,149,0.015823,"\int x^3 \left(d+e x^2\right) \left(a+c x^4\right)^5 \, dx","Integrate[x^3*(d + e*x^2)*(a + c*x^4)^5,x]","\frac{1}{4} a^5 d x^4+\frac{1}{6} a^5 e x^6+\frac{5}{8} a^4 c d x^8+\frac{1}{2} a^4 c e x^{10}+\frac{5}{6} a^3 c^2 d x^{12}+\frac{5}{7} a^3 c^2 e x^{14}+\frac{5}{8} a^2 c^3 d x^{16}+\frac{5}{9} a^2 c^3 e x^{18}+\frac{1}{4} a c^4 d x^{20}+\frac{5}{22} a c^4 e x^{22}+\frac{1}{24} c^5 d x^{24}+\frac{1}{26} c^5 e x^{26}","\frac{1}{4} a^5 d x^4+\frac{1}{6} a^5 e x^6+\frac{5}{8} a^4 c d x^8+\frac{1}{2} a^4 c e x^{10}+\frac{5}{6} a^3 c^2 d x^{12}+\frac{5}{7} a^3 c^2 e x^{14}+\frac{5}{8} a^2 c^3 d x^{16}+\frac{5}{9} a^2 c^3 e x^{18}+\frac{1}{4} a c^4 d x^{20}+\frac{5}{22} a c^4 e x^{22}+\frac{1}{24} c^5 d x^{24}+\frac{1}{26} c^5 e x^{26}",1,"(a^5*d*x^4)/4 + (a^5*e*x^6)/6 + (5*a^4*c*d*x^8)/8 + (a^4*c*e*x^10)/2 + (5*a^3*c^2*d*x^12)/6 + (5*a^3*c^2*e*x^14)/7 + (5*a^2*c^3*d*x^16)/8 + (5*a^2*c^3*e*x^18)/9 + (a*c^4*d*x^20)/4 + (5*a*c^4*e*x^22)/22 + (c^5*d*x^24)/24 + (c^5*e*x^26)/26","A",1
2,1,149,149,0.0036125,"\int x^2 \left(d+e x^2\right) \left(a+c x^4\right)^5 \, dx","Integrate[x^2*(d + e*x^2)*(a + c*x^4)^5,x]","\frac{1}{3} a^5 d x^3+\frac{1}{5} a^5 e x^5+\frac{5}{7} a^4 c d x^7+\frac{5}{9} a^4 c e x^9+\frac{10}{11} a^3 c^2 d x^{11}+\frac{10}{13} a^3 c^2 e x^{13}+\frac{2}{3} a^2 c^3 d x^{15}+\frac{10}{17} a^2 c^3 e x^{17}+\frac{5}{19} a c^4 d x^{19}+\frac{5}{21} a c^4 e x^{21}+\frac{1}{23} c^5 d x^{23}+\frac{1}{25} c^5 e x^{25}","\frac{1}{3} a^5 d x^3+\frac{1}{5} a^5 e x^5+\frac{5}{7} a^4 c d x^7+\frac{5}{9} a^4 c e x^9+\frac{10}{11} a^3 c^2 d x^{11}+\frac{10}{13} a^3 c^2 e x^{13}+\frac{2}{3} a^2 c^3 d x^{15}+\frac{10}{17} a^2 c^3 e x^{17}+\frac{5}{19} a c^4 d x^{19}+\frac{5}{21} a c^4 e x^{21}+\frac{1}{23} c^5 d x^{23}+\frac{1}{25} c^5 e x^{25}",1,"(a^5*d*x^3)/3 + (a^5*e*x^5)/5 + (5*a^4*c*d*x^7)/7 + (5*a^4*c*e*x^9)/9 + (10*a^3*c^2*d*x^11)/11 + (10*a^3*c^2*e*x^13)/13 + (2*a^2*c^3*d*x^15)/3 + (10*a^2*c^3*e*x^17)/17 + (5*a*c^4*d*x^19)/19 + (5*a*c^4*e*x^21)/21 + (c^5*d*x^23)/23 + (c^5*e*x^25)/25","A",1
3,1,146,89,0.0034688,"\int x \left(d+e x^2\right) \left(a+c x^4\right)^5 \, dx","Integrate[x*(d + e*x^2)*(a + c*x^4)^5,x]","\frac{1}{2} a^5 d x^2+\frac{1}{4} a^5 e x^4+\frac{5}{6} a^4 c d x^6+\frac{5}{8} a^4 c e x^8+a^3 c^2 d x^{10}+\frac{5}{6} a^3 c^2 e x^{12}+\frac{5}{7} a^2 c^3 d x^{14}+\frac{5}{8} a^2 c^3 e x^{16}+\frac{5}{18} a c^4 d x^{18}+\frac{1}{4} a c^4 e x^{20}+\frac{1}{22} c^5 d x^{22}+\frac{1}{24} c^5 e x^{24}","\frac{1}{2} a^5 d x^2+\frac{5}{6} a^4 c d x^6+a^3 c^2 d x^{10}+\frac{5}{7} a^2 c^3 d x^{14}+\frac{5}{18} a c^4 d x^{18}+\frac{e \left(a+c x^4\right)^6}{24 c}+\frac{1}{22} c^5 d x^{22}",1,"(a^5*d*x^2)/2 + (a^5*e*x^4)/4 + (5*a^4*c*d*x^6)/6 + (5*a^4*c*e*x^8)/8 + a^3*c^2*d*x^10 + (5*a^3*c^2*e*x^12)/6 + (5*a^2*c^3*d*x^14)/7 + (5*a^2*c^3*e*x^16)/8 + (5*a*c^4*d*x^18)/18 + (a*c^4*e*x^20)/4 + (c^5*d*x^22)/22 + (c^5*e*x^24)/24","A",1
4,1,141,141,0.0032143,"\int \left(d+e x^2\right) \left(a+c x^4\right)^5 \, dx","Integrate[(d + e*x^2)*(a + c*x^4)^5,x]","a^5 d x+\frac{1}{3} a^5 e x^3+a^4 c d x^5+\frac{5}{7} a^4 c e x^7+\frac{10}{9} a^3 c^2 d x^9+\frac{10}{11} a^3 c^2 e x^{11}+\frac{10}{13} a^2 c^3 d x^{13}+\frac{2}{3} a^2 c^3 e x^{15}+\frac{5}{17} a c^4 d x^{17}+\frac{5}{19} a c^4 e x^{19}+\frac{1}{21} c^5 d x^{21}+\frac{1}{23} c^5 e x^{23}","a^5 d x+\frac{1}{3} a^5 e x^3+a^4 c d x^5+\frac{5}{7} a^4 c e x^7+\frac{10}{9} a^3 c^2 d x^9+\frac{10}{11} a^3 c^2 e x^{11}+\frac{10}{13} a^2 c^3 d x^{13}+\frac{2}{3} a^2 c^3 e x^{15}+\frac{5}{17} a c^4 d x^{17}+\frac{5}{19} a c^4 e x^{19}+\frac{1}{21} c^5 d x^{21}+\frac{1}{23} c^5 e x^{23}",1,"a^5*d*x + (a^5*e*x^3)/3 + a^4*c*d*x^5 + (5*a^4*c*e*x^7)/7 + (10*a^3*c^2*d*x^9)/9 + (10*a^3*c^2*e*x^11)/11 + (10*a^2*c^3*d*x^13)/13 + (2*a^2*c^3*e*x^15)/3 + (5*a*c^4*d*x^17)/17 + (5*a*c^4*e*x^19)/19 + (c^5*d*x^21)/21 + (c^5*e*x^23)/23","A",1
5,1,142,142,0.011703,"\int \frac{\left(d+e x^2\right) \left(a+c x^4\right)^5}{x} \, dx","Integrate[((d + e*x^2)*(a + c*x^4)^5)/x,x]","a^5 d \log (x)+\frac{1}{2} a^5 e x^2+\frac{5}{4} a^4 c d x^4+\frac{5}{6} a^4 c e x^6+\frac{5}{4} a^3 c^2 d x^8+a^3 c^2 e x^{10}+\frac{5}{6} a^2 c^3 d x^{12}+\frac{5}{7} a^2 c^3 e x^{14}+\frac{5}{16} a c^4 d x^{16}+\frac{5}{18} a c^4 e x^{18}+\frac{1}{20} c^5 d x^{20}+\frac{1}{22} c^5 e x^{22}","a^5 d \log (x)+\frac{1}{2} a^5 e x^2+\frac{5}{4} a^4 c d x^4+\frac{5}{6} a^4 c e x^6+\frac{5}{4} a^3 c^2 d x^8+a^3 c^2 e x^{10}+\frac{5}{6} a^2 c^3 d x^{12}+\frac{5}{7} a^2 c^3 e x^{14}+\frac{5}{16} a c^4 d x^{16}+\frac{5}{18} a c^4 e x^{18}+\frac{1}{20} c^5 d x^{20}+\frac{1}{22} c^5 e x^{22}",1,"(a^5*e*x^2)/2 + (5*a^4*c*d*x^4)/4 + (5*a^4*c*e*x^6)/6 + (5*a^3*c^2*d*x^8)/4 + a^3*c^2*e*x^10 + (5*a^2*c^3*d*x^12)/6 + (5*a^2*c^3*e*x^14)/7 + (5*a*c^4*d*x^16)/16 + (5*a*c^4*e*x^18)/18 + (c^5*d*x^20)/20 + (c^5*e*x^22)/22 + a^5*d*Log[x]","A",1
6,1,139,139,0.0071497,"\int \frac{\left(d+e x^2\right) \left(a+c x^4\right)^5}{x^2} \, dx","Integrate[((d + e*x^2)*(a + c*x^4)^5)/x^2,x]","-\frac{a^5 d}{x}+a^5 e x+\frac{5}{3} a^4 c d x^3+a^4 c e x^5+\frac{10}{7} a^3 c^2 d x^7+\frac{10}{9} a^3 c^2 e x^9+\frac{10}{11} a^2 c^3 d x^{11}+\frac{10}{13} a^2 c^3 e x^{13}+\frac{1}{3} a c^4 d x^{15}+\frac{5}{17} a c^4 e x^{17}+\frac{1}{19} c^5 d x^{19}+\frac{1}{21} c^5 e x^{21}","-\frac{a^5 d}{x}+a^5 e x+\frac{5}{3} a^4 c d x^3+a^4 c e x^5+\frac{10}{7} a^3 c^2 d x^7+\frac{10}{9} a^3 c^2 e x^9+\frac{10}{11} a^2 c^3 d x^{11}+\frac{10}{13} a^2 c^3 e x^{13}+\frac{1}{3} a c^4 d x^{15}+\frac{5}{17} a c^4 e x^{17}+\frac{1}{19} c^5 d x^{19}+\frac{1}{21} c^5 e x^{21}",1,"-((a^5*d)/x) + a^5*e*x + (5*a^4*c*d*x^3)/3 + a^4*c*e*x^5 + (10*a^3*c^2*d*x^7)/7 + (10*a^3*c^2*e*x^9)/9 + (10*a^2*c^3*d*x^11)/11 + (10*a^2*c^3*e*x^13)/13 + (a*c^4*d*x^15)/3 + (5*a*c^4*e*x^17)/17 + (c^5*d*x^19)/19 + (c^5*e*x^21)/21","A",1
7,1,142,142,0.007545,"\int \frac{\left(d+e x^2\right) \left(a+c x^4\right)^5}{x^3} \, dx","Integrate[((d + e*x^2)*(a + c*x^4)^5)/x^3,x]","-\frac{a^5 d}{2 x^2}+a^5 e \log (x)+\frac{5}{2} a^4 c d x^2+\frac{5}{4} a^4 c e x^4+\frac{5}{3} a^3 c^2 d x^6+\frac{5}{4} a^3 c^2 e x^8+a^2 c^3 d x^{10}+\frac{5}{6} a^2 c^3 e x^{12}+\frac{5}{14} a c^4 d x^{14}+\frac{5}{16} a c^4 e x^{16}+\frac{1}{18} c^5 d x^{18}+\frac{1}{20} c^5 e x^{20}","-\frac{a^5 d}{2 x^2}+a^5 e \log (x)+\frac{5}{2} a^4 c d x^2+\frac{5}{4} a^4 c e x^4+\frac{5}{3} a^3 c^2 d x^6+\frac{5}{4} a^3 c^2 e x^8+a^2 c^3 d x^{10}+\frac{5}{6} a^2 c^3 e x^{12}+\frac{5}{14} a c^4 d x^{14}+\frac{5}{16} a c^4 e x^{16}+\frac{1}{18} c^5 d x^{18}+\frac{1}{20} c^5 e x^{20}",1,"-1/2*(a^5*d)/x^2 + (5*a^4*c*d*x^2)/2 + (5*a^4*c*e*x^4)/4 + (5*a^3*c^2*d*x^6)/3 + (5*a^3*c^2*e*x^8)/4 + a^2*c^3*d*x^10 + (5*a^2*c^3*e*x^12)/6 + (5*a*c^4*d*x^14)/14 + (5*a*c^4*e*x^16)/16 + (c^5*d*x^18)/18 + (c^5*e*x^20)/20 + a^5*e*Log[x]","A",1
8,1,50,67,0.0527603,"\int x^5 \left(2+3 x^2\right) \sqrt{5+x^4} \, dx","Integrate[x^5*(2 + 3*x^2)*Sqrt[5 + x^4],x]","\frac{1}{40} \sqrt{x^4+5} \left(12 x^8+10 x^6+20 x^4+25 x^2-200\right)-\frac{25}{8} \sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)","\frac{3}{10} \left(x^4+5\right)^{3/2} x^4-\frac{25}{8} \sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)-\frac{5}{8} \sqrt{x^4+5} x^2-\frac{1}{4} \left(4-x^2\right) \left(x^4+5\right)^{3/2}",1,"(Sqrt[5 + x^4]*(-200 + 25*x^2 + 20*x^4 + 10*x^6 + 12*x^8))/40 - (25*ArcSinh[x^2/Sqrt[5]])/8","A",1
9,1,44,51,0.0259879,"\int x^3 \left(2+3 x^2\right) \sqrt{5+x^4} \, dx","Integrate[x^3*(2 + 3*x^2)*Sqrt[5 + x^4],x]","\frac{1}{48} \left(\sqrt{x^4+5} \left(18 x^6+16 x^4+45 x^2+80\right)-225 \sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)\right)","-\frac{75}{16} \sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)-\frac{15}{16} \sqrt{x^4+5} x^2+\frac{1}{24} \left(9 x^2+8\right) \left(x^4+5\right)^{3/2}",1,"(Sqrt[5 + x^4]*(80 + 45*x^2 + 16*x^4 + 18*x^6) - 225*ArcSinh[x^2/Sqrt[5]])/48","A",1
10,1,36,44,0.0300947,"\int x \left(2+3 x^2\right) \sqrt{5+x^4} \, dx","Integrate[x*(2 + 3*x^2)*Sqrt[5 + x^4],x]","\frac{5}{2} \sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)+\frac{1}{2} \sqrt{x^4+5} \left(x^4+x^2+5\right)","\frac{1}{2} \left(x^4+5\right)^{3/2}+\frac{5}{2} \sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)+\frac{1}{2} \sqrt{x^4+5} x^2",1,"(Sqrt[5 + x^4]*(5 + x^2 + x^4))/2 + (5*ArcSinh[x^2/Sqrt[5]])/2","A",1
11,1,57,58,0.0594666,"\int \frac{\left(2+3 x^2\right) \sqrt{5+x^4}}{x} \, dx","Integrate[((2 + 3*x^2)*Sqrt[5 + x^4])/x,x]","\frac{1}{4} \left(-4 \sqrt{5} \tanh ^{-1}\left(\frac{\sqrt{x^4+5}}{\sqrt{5}}\right)+15 \sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)+\sqrt{x^4+5} \left(3 x^2+4\right)\right)","-\sqrt{5} \tanh ^{-1}\left(\frac{\sqrt{x^4+5}}{\sqrt{5}}\right)+\frac{15}{4} \sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)+\frac{1}{4} \sqrt{x^4+5} \left(3 x^2+4\right)",1,"((4 + 3*x^2)*Sqrt[5 + x^4] + 15*ArcSinh[x^2/Sqrt[5]] - 4*Sqrt[5]*ArcTanh[Sqrt[5 + x^4]/Sqrt[5]])/4","A",1
12,1,59,59,0.0439211,"\int \frac{\left(2+3 x^2\right) \sqrt{5+x^4}}{x^3} \, dx","Integrate[((2 + 3*x^2)*Sqrt[5 + x^4])/x^3,x]","\sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)+\frac{1}{2} \left(\frac{\left(3 x^2-2\right) \sqrt{x^4+5}}{x^2}-3 \sqrt{5} \tanh ^{-1}\left(\frac{\sqrt{x^4+5}}{\sqrt{5}}\right)\right)","-\frac{3}{2} \sqrt{5} \tanh ^{-1}\left(\frac{\sqrt{x^4+5}}{\sqrt{5}}\right)+\sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)-\frac{\sqrt{x^4+5} \left(2-3 x^2\right)}{2 x^2}",1,"ArcSinh[x^2/Sqrt[5]] + (((-2 + 3*x^2)*Sqrt[5 + x^4])/x^2 - 3*Sqrt[5]*ArcTanh[Sqrt[5 + x^4]/Sqrt[5]])/2","A",1
13,1,59,63,0.0670745,"\int \frac{\left(2+3 x^2\right) \sqrt{5+x^4}}{x^5} \, dx","Integrate[((2 + 3*x^2)*Sqrt[5 + x^4])/x^5,x]","\frac{1}{10} \left(-\sqrt{5} \tanh ^{-1}\left(\sqrt{\frac{x^4}{5}+1}\right)+15 \sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)-\frac{5 \sqrt{x^4+5} \left(3 x^2+1\right)}{x^4}\right)","-\frac{\tanh ^{-1}\left(\frac{\sqrt{x^4+5}}{\sqrt{5}}\right)}{2 \sqrt{5}}+\frac{3}{2} \sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)-\frac{\sqrt{x^4+5} \left(3 x^2+1\right)}{2 x^4}",1,"((-5*(1 + 3*x^2)*Sqrt[5 + x^4])/x^4 + 15*ArcSinh[x^2/Sqrt[5]] - Sqrt[5]*ArcTanh[Sqrt[1 + x^4/5]])/10","A",1
14,1,72,58,0.0341735,"\int \frac{\left(2+3 x^2\right) \sqrt{5+x^4}}{x^7} \, dx","Integrate[((2 + 3*x^2)*Sqrt[5 + x^4])/x^7,x]","-\frac{3 \left(5 x^4+\sqrt{5} \sqrt{x^4+5} x^4 \tanh ^{-1}\left(\sqrt{\frac{x^4}{5}+1}\right)+25\right)}{20 x^4 \sqrt{x^4+5}}-\frac{\left(x^4+5\right)^{3/2}}{15 x^6}","-\frac{3 \sqrt{x^4+5}}{4 x^4}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{x^4+5}}{\sqrt{5}}\right)}{4 \sqrt{5}}-\frac{\left(x^4+5\right)^{3/2}}{15 x^6}",1,"-1/15*(5 + x^4)^(3/2)/x^6 - (3*(25 + 5*x^4 + Sqrt[5]*x^4*Sqrt[5 + x^4]*ArcTanh[Sqrt[1 + x^4/5]]))/(20*x^4*Sqrt[5 + x^4])","A",1
15,1,82,208,0.0338688,"\int x^4 \left(2+3 x^2\right) \sqrt{5+x^4} \, dx","Integrate[x^4*(2 + 3*x^2)*Sqrt[5 + x^4],x]","\frac{1}{21} x \left(-30 \sqrt{5} \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{x^4}{5}\right)-35 \sqrt{5} x^2 \, _2F_1\left(-\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{x^4}{5}\right)+6 \left(x^4+5\right)^{3/2}+7 \left(x^4+5\right)^{3/2} x^2\right)","\frac{20}{21} \sqrt{x^4+5} x+\frac{2}{3} \sqrt{x^4+5} x^3-\frac{10 \sqrt{x^4+5} x}{x^2+\sqrt{5}}-\frac{5 \sqrt[4]{5} \left(21+2 \sqrt{5}\right) \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{21 \sqrt{x^4+5}}+\frac{10 \sqrt[4]{5} \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{\sqrt{x^4+5}}+\frac{1}{21} \left(7 x^2+6\right) \sqrt{x^4+5} x^5",1,"(x*(6*(5 + x^4)^(3/2) + 7*x^2*(5 + x^4)^(3/2) - 30*Sqrt[5]*Hypergeometric2F1[-1/2, 1/4, 5/4, -1/5*x^4] - 35*Sqrt[5]*x^2*Hypergeometric2F1[-1/2, 3/4, 7/4, -1/5*x^4]))/21","C",1
16,1,68,192,0.0242644,"\int x^2 \left(2+3 x^2\right) \sqrt{5+x^4} \, dx","Integrate[x^2*(2 + 3*x^2)*Sqrt[5 + x^4],x]","\frac{1}{21} x \left(-45 \sqrt{5} \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{x^4}{5}\right)+14 \sqrt{5} x^2 \, _2F_1\left(-\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{x^4}{5}\right)+9 \left(x^4+5\right)^{3/2}\right)","\frac{10}{7} \sqrt{x^4+5} x+\frac{4 \sqrt{x^4+5} x}{x^2+\sqrt{5}}+\frac{\sqrt[4]{5} \left(14-5 \sqrt{5}\right) \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{7 \sqrt{x^4+5}}-\frac{4 \sqrt[4]{5} \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{\sqrt{x^4+5}}+\frac{1}{35} \left(15 x^2+14\right) \sqrt{x^4+5} x^3",1,"(x*(9*(5 + x^4)^(3/2) - 45*Sqrt[5]*Hypergeometric2F1[-1/2, 1/4, 5/4, -1/5*x^4] + 14*Sqrt[5]*x^2*Hypergeometric2F1[-1/2, 3/4, 7/4, -1/5*x^4]))/21","C",1
17,1,48,176,0.0096715,"\int \left(2+3 x^2\right) \sqrt{5+x^4} \, dx","Integrate[(2 + 3*x^2)*Sqrt[5 + x^4],x]","\sqrt{5} x \left(2 \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{x^4}{5}\right)+x^2 \, _2F_1\left(-\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{x^4}{5}\right)\right)","\frac{1}{15} \left(9 x^2+10\right) \sqrt{x^4+5} x+\frac{6 \sqrt{x^4+5} x}{x^2+\sqrt{5}}+\frac{\sqrt[4]{5} \left(9+2 \sqrt{5}\right) \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{3 \sqrt{x^4+5}}-\frac{6 \sqrt[4]{5} \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{\sqrt{x^4+5}}",1,"Sqrt[5]*x*(2*Hypergeometric2F1[-1/2, 1/4, 5/4, -1/5*x^4] + x^2*Hypergeometric2F1[-1/2, 3/4, 7/4, -1/5*x^4])","C",1
18,1,53,171,0.0191323,"\int \frac{\left(2+3 x^2\right) \sqrt{5+x^4}}{x^2} \, dx","Integrate[((2 + 3*x^2)*Sqrt[5 + x^4])/x^2,x]","3 \sqrt{5} x \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{x^4}{5}\right)-\frac{2 \sqrt{5} \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};-\frac{x^4}{5}\right)}{x}","\frac{4 \sqrt{x^4+5} x}{x^2+\sqrt{5}}-\frac{\left(2-x^2\right) \sqrt{x^4+5}}{x}+\frac{\sqrt[4]{5} \left(2+\sqrt{5}\right) \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{\sqrt{x^4+5}}-\frac{4 \sqrt[4]{5} \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{\sqrt{x^4+5}}",1,"(-2*Sqrt[5]*Hypergeometric2F1[-1/2, -1/4, 3/4, -1/5*x^4])/x + 3*Sqrt[5]*x*Hypergeometric2F1[-1/2, 1/4, 5/4, -1/5*x^4]","C",1
19,1,54,192,0.0199293,"\int \frac{\left(2+3 x^2\right) \sqrt{5+x^4}}{x^4} \, dx","Integrate[((2 + 3*x^2)*Sqrt[5 + x^4])/x^4,x]","-\frac{\sqrt{5} \left(2 \, _2F_1\left(-\frac{3}{4},-\frac{1}{2};\frac{1}{4};-\frac{x^4}{5}\right)+9 x^2 \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};-\frac{x^4}{5}\right)\right)}{3 x^3}","-\frac{6 \sqrt{x^4+5}}{x}+\frac{6 \sqrt{x^4+5} x}{x^2+\sqrt{5}}+\frac{\left(2+9 \sqrt{5}\right) \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{3 \sqrt[4]{5} \sqrt{x^4+5}}-\frac{6 \sqrt[4]{5} \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{\sqrt{x^4+5}}-\frac{\left(2-9 x^2\right) \sqrt{x^4+5}}{3 x^3}",1,"-1/3*(Sqrt[5]*(2*Hypergeometric2F1[-3/4, -1/2, 1/4, -1/5*x^4] + 9*x^2*Hypergeometric2F1[-1/2, -1/4, 3/4, -1/5*x^4]))/x^3","C",1
20,1,72,83,0.0487329,"\int x^5 \left(2+3 x^2\right) \left(5+x^4\right)^{3/2} \, dx","Integrate[x^5*(2 + 3*x^2)*(5 + x^4)^(3/2),x]","\frac{3}{14} \left(x^4-2\right) \left(x^4+5\right)^{5/2}+\frac{1}{6} x^2 \left(x^4+5\right)^{5/2}-\frac{5}{48} \left(75 \sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)+\sqrt{x^4+5} \left(2 x^4+25\right) x^2\right)","\frac{3}{14} \left(x^4+5\right)^{5/2} x^4-\frac{125}{16} \sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)-\frac{5}{24} \left(x^4+5\right)^{3/2} x^2-\frac{25}{16} \sqrt{x^4+5} x^2-\frac{1}{42} \left(18-7 x^2\right) \left(x^4+5\right)^{5/2}",1,"(x^2*(5 + x^4)^(5/2))/6 + (3*(-2 + x^4)*(5 + x^4)^(5/2))/14 - (5*(x^2*Sqrt[5 + x^4]*(25 + 2*x^4) + 75*ArcSinh[x^2/Sqrt[5]]))/48","A",1
21,1,54,67,0.0363866,"\int x^3 \left(2+3 x^2\right) \left(5+x^4\right)^{3/2} \, dx","Integrate[x^3*(2 + 3*x^2)*(5 + x^4)^(3/2),x]","\frac{1}{160} \left(\sqrt{x^4+5} \left(40 x^{10}+32 x^8+350 x^6+320 x^4+375 x^2+800\right)-1875 \sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)\right)","-\frac{375}{32} \sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)+\frac{1}{20} \left(5 x^2+4\right) \left(x^4+5\right)^{5/2}-\frac{5}{16} x^2 \left(x^4+5\right)^{3/2}-\frac{75}{32} x^2 \sqrt{x^4+5}",1,"(Sqrt[5 + x^4]*(800 + 375*x^2 + 320*x^4 + 350*x^6 + 32*x^8 + 40*x^10) - 1875*ArcSinh[x^2/Sqrt[5]])/160","A",1
22,1,56,60,0.0268392,"\int x \left(2+3 x^2\right) \left(5+x^4\right)^{3/2} \, dx","Integrate[x*(2 + 3*x^2)*(5 + x^4)^(3/2),x]","\frac{75}{8} \sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)+\frac{1}{2} \sqrt{x^4+5} \left(\frac{3 x^8}{5}+\frac{x^6}{2}+6 x^4+\frac{25 x^2}{4}+15\right)","\frac{3}{10} \left(x^4+5\right)^{5/2}+\frac{75}{8} \sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)+\frac{1}{4} x^2 \left(x^4+5\right)^{3/2}+\frac{15}{8} x^2 \sqrt{x^4+5}",1,"(Sqrt[5 + x^4]*(15 + (25*x^2)/4 + 6*x^4 + x^6/2 + (3*x^8)/5))/2 + (75*ArcSinh[x^2/Sqrt[5]])/8","A",1
23,1,67,78,0.0343826,"\int \frac{\left(2+3 x^2\right) \left(5+x^4\right)^{3/2}}{x} \, dx","Integrate[((2 + 3*x^2)*(5 + x^4)^(3/2))/x,x]","\frac{1}{48} \left(-240 \sqrt{5} \tanh ^{-1}\left(\frac{\sqrt{x^4+5}}{\sqrt{5}}\right)+675 \sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)+\sqrt{x^4+5} \left(18 x^6+16 x^4+225 x^2+320\right)\right)","-5 \sqrt{5} \tanh ^{-1}\left(\frac{\sqrt{x^4+5}}{\sqrt{5}}\right)+\frac{225}{16} \sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)+\frac{1}{24} \left(9 x^2+8\right) \left(x^4+5\right)^{3/2}+\frac{5}{16} \left(9 x^2+16\right) \sqrt{x^4+5}",1,"(Sqrt[5 + x^4]*(320 + 225*x^2 + 16*x^4 + 18*x^6) + 675*ArcSinh[x^2/Sqrt[5]] - 240*Sqrt[5]*ArcTanh[Sqrt[5 + x^4]/Sqrt[5]])/48","A",1
24,1,71,81,0.0525812,"\int \frac{\left(2+3 x^2\right) \left(5+x^4\right)^{3/2}}{x^3} \, dx","Integrate[((2 + 3*x^2)*(5 + x^4)^(3/2))/x^3,x]","\frac{1}{2} \left(\sqrt{x^4+5} \left(x^4+20\right)-15 \sqrt{5} \tanh ^{-1}\left(\frac{\sqrt{x^4+5}}{\sqrt{5}}\right)\right)-\frac{5 \sqrt{5} \, _2F_1\left(-\frac{3}{2},-\frac{1}{2};\frac{1}{2};-\frac{x^4}{5}\right)}{x^2}","-\frac{15}{2} \sqrt{5} \tanh ^{-1}\left(\frac{\sqrt{x^4+5}}{\sqrt{5}}\right)+\frac{15}{2} \sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)-\frac{\left(2-x^2\right) \left(x^4+5\right)^{3/2}}{2 x^2}+\frac{3}{2} \left(x^2+5\right) \sqrt{x^4+5}",1,"(Sqrt[5 + x^4]*(20 + x^4) - 15*Sqrt[5]*ArcTanh[Sqrt[5 + x^4]/Sqrt[5]])/2 - (5*Sqrt[5]*Hypergeometric2F1[-3/2, -1/2, 1/2, -1/5*x^4])/x^2","C",1
25,1,60,86,0.026966,"\int \frac{\left(2+3 x^2\right) \left(5+x^4\right)^{3/2}}{x^5} \, dx","Integrate[((2 + 3*x^2)*(5 + x^4)^(3/2))/x^5,x]","\frac{1}{125} \left(x^4+5\right)^{5/2} \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};\frac{x^4}{5}+1\right)-\frac{15 \sqrt{5} \, _2F_1\left(-\frac{3}{2},-\frac{1}{2};\frac{1}{2};-\frac{x^4}{5}\right)}{2 x^2}","-\frac{3}{2} \sqrt{5} \tanh ^{-1}\left(\frac{\sqrt{x^4+5}}{\sqrt{5}}\right)+\frac{45}{4} \sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)-\frac{\left(2-3 x^2\right) \left(x^4+5\right)^{3/2}}{4 x^4}-\frac{3 \left(15-2 x^2\right) \sqrt{x^4+5}}{4 x^2}",1,"(-15*Sqrt[5]*Hypergeometric2F1[-3/2, -1/2, 1/2, -1/5*x^4])/(2*x^2) + ((5 + x^4)^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, 1 + x^4/5])/125","C",1
26,1,60,82,0.0246907,"\int \frac{\left(2+3 x^2\right) \left(5+x^4\right)^{3/2}}{x^7} \, dx","Integrate[((2 + 3*x^2)*(5 + x^4)^(3/2))/x^7,x]","\frac{3}{250} \left(x^4+5\right)^{5/2} \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};\frac{x^4}{5}+1\right)-\frac{5 \sqrt{5} \, _2F_1\left(-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};-\frac{x^4}{5}\right)}{3 x^6}","-\frac{9}{4} \sqrt{5} \tanh ^{-1}\left(\frac{\sqrt{x^4+5}}{\sqrt{5}}\right)+\sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)-\frac{\left(4-9 x^2\right) \sqrt{x^4+5}}{4 x^2}-\frac{\left(9 x^2+4\right) \left(x^4+5\right)^{3/2}}{12 x^6}",1,"(-5*Sqrt[5]*Hypergeometric2F1[-3/2, -3/2, -1/2, -1/5*x^4])/(3*x^6) + (3*(5 + x^4)^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, 1 + x^4/5])/250","C",1
27,1,74,235,0.0415275,"\int x^4 \left(2+3 x^2\right) \left(5+x^4\right)^{3/2} \, dx","Integrate[x^4*(2 + 3*x^2)*(5 + x^4)^(3/2),x]","\frac{1}{143} x \left(-650 \sqrt{5} \, _2F_1\left(-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{x^4}{5}\right)-825 \sqrt{5} x^2 \, _2F_1\left(-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{x^4}{5}\right)+\left(33 x^2+26\right) \left(x^4+5\right)^{5/2}\right)","\frac{200}{77} \sqrt{x^4+5} x+\frac{20}{13} \sqrt{x^4+5} x^3-\frac{300 \sqrt{x^4+5} x}{13 \left(x^2+\sqrt{5}\right)}-\frac{50 \sqrt[4]{5} \left(231+26 \sqrt{5}\right) \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{1001 \sqrt{x^4+5}}+\frac{300 \sqrt[4]{5} \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{13 \sqrt{x^4+5}}+\frac{1}{143} \left(33 x^2+26\right) \left(x^4+5\right)^{3/2} x^5+\frac{10 \left(77 x^2+78\right) \sqrt{x^4+5} x^5}{1001}",1,"(x*((26 + 33*x^2)*(5 + x^4)^(5/2) - 650*Sqrt[5]*Hypergeometric2F1[-3/2, 1/4, 5/4, -1/5*x^4] - 825*Sqrt[5]*x^2*Hypergeometric2F1[-3/2, 3/4, 7/4, -1/5*x^4]))/143","C",1
28,1,68,219,0.0274038,"\int x^2 \left(2+3 x^2\right) \left(5+x^4\right)^{3/2} \, dx","Integrate[x^2*(2 + 3*x^2)*(5 + x^4)^(3/2),x]","\frac{1}{33} x \left(-225 \sqrt{5} \, _2F_1\left(-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{x^4}{5}\right)+110 \sqrt{5} x^2 \, _2F_1\left(-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{x^4}{5}\right)+9 \left(x^4+5\right)^{5/2}\right)","\frac{300}{77} \sqrt{x^4+5} x+\frac{40 \sqrt{x^4+5} x}{3 \left(x^2+\sqrt{5}\right)}+\frac{10 \sqrt[4]{5} \left(154-45 \sqrt{5}\right) \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{231 \sqrt{x^4+5}}-\frac{40 \sqrt[4]{5} \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{3 \sqrt{x^4+5}}+\frac{1}{99} \left(27 x^2+22\right) \left(x^4+5\right)^{3/2} x^3+\frac{2}{231} \left(135 x^2+154\right) \sqrt{x^4+5} x^3",1,"(x*(9*(5 + x^4)^(5/2) - 225*Sqrt[5]*Hypergeometric2F1[-3/2, 1/4, 5/4, -1/5*x^4] + 110*Sqrt[5]*x^2*Hypergeometric2F1[-3/2, 3/4, 7/4, -1/5*x^4]))/33","C",1
29,1,49,197,0.0102184,"\int \left(2+3 x^2\right) \left(5+x^4\right)^{3/2} \, dx","Integrate[(2 + 3*x^2)*(5 + x^4)^(3/2),x]","5 \sqrt{5} x \left(2 \, _2F_1\left(-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{x^4}{5}\right)+x^2 \, _2F_1\left(-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{x^4}{5}\right)\right)","\frac{1}{21} x \left(7 x^2+6\right) \left(x^4+5\right)^{3/2}+\frac{2}{7} x \left(7 x^2+10\right) \sqrt{x^4+5}+\frac{20 x \sqrt{x^4+5}}{x^2+\sqrt{5}}+\frac{10 \sqrt[4]{5} \left(7+2 \sqrt{5}\right) \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{7 \sqrt{x^4+5}}-\frac{20 \sqrt[4]{5} \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{\sqrt{x^4+5}}",1,"5*Sqrt[5]*x*(2*Hypergeometric2F1[-3/2, 1/4, 5/4, -1/5*x^4] + x^2*Hypergeometric2F1[-3/2, 3/4, 7/4, -1/5*x^4])","C",1
30,1,53,199,0.0212391,"\int \frac{\left(2+3 x^2\right) \left(5+x^4\right)^{3/2}}{x^2} \, dx","Integrate[((2 + 3*x^2)*(5 + x^4)^(3/2))/x^2,x]","15 \sqrt{5} x \, _2F_1\left(-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{x^4}{5}\right)-\frac{10 \sqrt{5} \, _2F_1\left(-\frac{3}{2},-\frac{1}{4};\frac{3}{4};-\frac{x^4}{5}\right)}{x}","-\frac{\left(14-3 x^2\right) \left(x^4+5\right)^{3/2}}{7 x}+\frac{6}{35} x \left(14 x^2+25\right) \sqrt{x^4+5}+\frac{24 x \sqrt{x^4+5}}{x^2+\sqrt{5}}+\frac{6 \sqrt[4]{5} \left(14+5 \sqrt{5}\right) \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{7 \sqrt{x^4+5}}-\frac{24 \sqrt[4]{5} \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{\sqrt{x^4+5}}",1,"(-10*Sqrt[5]*Hypergeometric2F1[-3/2, -1/4, 3/4, -1/5*x^4])/x + 15*Sqrt[5]*x*Hypergeometric2F1[-3/2, 1/4, 5/4, -1/5*x^4]","C",1
31,1,54,201,0.0205936,"\int \frac{\left(2+3 x^2\right) \left(5+x^4\right)^{3/2}}{x^4} \, dx","Integrate[((2 + 3*x^2)*(5 + x^4)^(3/2))/x^4,x]","-\frac{5 \sqrt{5} \left(2 \, _2F_1\left(-\frac{3}{2},-\frac{3}{4};\frac{1}{4};-\frac{x^4}{5}\right)+9 x^2 \, _2F_1\left(-\frac{3}{2},-\frac{1}{4};\frac{3}{4};-\frac{x^4}{5}\right)\right)}{3 x^3}","-\frac{2 \left(27-2 x^2\right) \sqrt{x^4+5}}{3 x}+\frac{36 x \sqrt{x^4+5}}{x^2+\sqrt{5}}+\frac{2 \sqrt[4]{5} \left(27+2 \sqrt{5}\right) \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{3 \sqrt{x^4+5}}-\frac{36 \sqrt[4]{5} \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{\sqrt{x^4+5}}-\frac{\left(10-9 x^2\right) \left(x^4+5\right)^{3/2}}{15 x^3}",1,"(-5*Sqrt[5]*(2*Hypergeometric2F1[-3/2, -3/4, 1/4, -1/5*x^4] + 9*x^2*Hypergeometric2F1[-3/2, -1/4, 3/4, -1/5*x^4]))/(3*x^3)","C",1
32,1,44,67,0.02797,"\int \frac{x^7 \left(2+3 x^2\right)}{\sqrt{5+x^4}} \, dx","Integrate[(x^7*(2 + 3*x^2))/Sqrt[5 + x^4],x]","\frac{1}{48} \left(675 \sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)+\sqrt{x^4+5} \left(18 x^6+16 x^4-135 x^2-160\right)\right)","\frac{1}{3} \sqrt{x^4+5} x^4+\frac{225}{16} \sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)+\frac{3}{8} \sqrt{x^4+5} x^6-\frac{5}{48} \left(27 x^2+32\right) \sqrt{x^4+5}",1,"(Sqrt[5 + x^4]*(-160 - 135*x^2 + 16*x^4 + 18*x^6) + 675*ArcSinh[x^2/Sqrt[5]])/48","A",1
33,1,35,51,0.021756,"\int \frac{x^5 \left(2+3 x^2\right)}{\sqrt{5+x^4}} \, dx","Integrate[(x^5*(2 + 3*x^2))/Sqrt[5 + x^4],x]","\frac{1}{2} \left(\sqrt{x^4+5} \left(x^4+x^2-10\right)-5 \sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)\right)","\frac{1}{2} \sqrt{x^4+5} x^4-\frac{5}{2} \sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)-\frac{1}{2} \left(10-x^2\right) \sqrt{x^4+5}",1,"(Sqrt[5 + x^4]*(-10 + x^2 + x^4) - 5*ArcSinh[x^2/Sqrt[5]])/2","A",1
34,1,34,35,0.0168194,"\int \frac{x^3 \left(2+3 x^2\right)}{\sqrt{5+x^4}} \, dx","Integrate[(x^3*(2 + 3*x^2))/Sqrt[5 + x^4],x]","\frac{1}{4} \left(\left(3 x^2+4\right) \sqrt{x^4+5}-15 \sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)\right)","\frac{1}{4} \left(3 x^2+4\right) \sqrt{x^4+5}-\frac{15}{4} \sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)",1,"((4 + 3*x^2)*Sqrt[5 + x^4] - 15*ArcSinh[x^2/Sqrt[5]])/4","A",1
35,1,24,24,0.0193697,"\int \frac{x \left(2+3 x^2\right)}{\sqrt{5+x^4}} \, dx","Integrate[(x*(2 + 3*x^2))/Sqrt[5 + x^4],x]","\frac{3 \sqrt{x^4+5}}{2}+\sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)","\frac{3 \sqrt{x^4+5}}{2}+\sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)",1,"(3*Sqrt[5 + x^4])/2 + ArcSinh[x^2/Sqrt[5]]","A",1
36,1,38,38,0.0162644,"\int \frac{2+3 x^2}{x \sqrt{5+x^4}} \, dx","Integrate[(2 + 3*x^2)/(x*Sqrt[5 + x^4]),x]","\frac{3}{2} \sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt{x^4+5}}{\sqrt{5}}\right)}{\sqrt{5}}","\frac{3}{2} \sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt{x^4+5}}{\sqrt{5}}\right)}{\sqrt{5}}",1,"(3*ArcSinh[x^2/Sqrt[5]])/2 - ArcTanh[Sqrt[5 + x^4]/Sqrt[5]]/Sqrt[5]","A",1
37,1,42,42,0.0206011,"\int \frac{2+3 x^2}{x^3 \sqrt{5+x^4}} \, dx","Integrate[(2 + 3*x^2)/(x^3*Sqrt[5 + x^4]),x]","-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{x^4+5}}{\sqrt{5}}\right)}{2 \sqrt{5}}-\frac{\sqrt{x^4+5}}{5 x^2}","-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{x^4+5}}{\sqrt{5}}\right)}{2 \sqrt{5}}-\frac{\sqrt{x^4+5}}{5 x^2}",1,"-1/5*Sqrt[5 + x^4]/x^2 - (3*ArcTanh[Sqrt[5 + x^4]/Sqrt[5]])/(2*Sqrt[5])","A",1
38,1,49,58,0.0232469,"\int \frac{2+3 x^2}{x^5 \sqrt{5+x^4}} \, dx","Integrate[(2 + 3*x^2)/(x^5*Sqrt[5 + x^4]),x]","\frac{\sqrt{5} x^4 \tanh ^{-1}\left(\sqrt{\frac{x^4}{5}+1}\right)-5 \left(3 x^2+1\right) \sqrt{x^4+5}}{50 x^4}","-\frac{\sqrt{x^4+5}}{10 x^4}+\frac{\tanh ^{-1}\left(\frac{\sqrt{x^4+5}}{\sqrt{5}}\right)}{10 \sqrt{5}}-\frac{3 \sqrt{x^4+5}}{10 x^2}",1,"(-5*(1 + 3*x^2)*Sqrt[5 + x^4] + Sqrt[5]*x^4*ArcTanh[Sqrt[1 + x^4/5]])/(50*x^4)","A",1
39,1,74,185,0.0287453,"\int \frac{x^4 \left(2+3 x^2\right)}{\sqrt{5+x^4}} \, dx","Integrate[(x^4*(2 + 3*x^2))/Sqrt[5 + x^4],x]","\frac{1}{15} x \left(-10 \sqrt{5} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{x^4}{5}\right)-9 \sqrt{5} x^2 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{x^4}{5}\right)+\left(9 x^2+10\right) \sqrt{x^4+5}\right)","\frac{2}{3} \sqrt{x^4+5} x+\frac{3}{5} \sqrt{x^4+5} x^3-\frac{9 \sqrt{x^4+5} x}{x^2+\sqrt{5}}-\frac{\sqrt[4]{5} \left(27+2 \sqrt{5}\right) \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{6 \sqrt{x^4+5}}+\frac{9 \sqrt[4]{5} \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{\sqrt{x^4+5}}",1,"(x*((10 + 9*x^2)*Sqrt[5 + x^4] - 10*Sqrt[5]*Hypergeometric2F1[1/4, 1/2, 5/4, -1/5*x^4] - 9*Sqrt[5]*x^2*Hypergeometric2F1[1/2, 3/4, 7/4, -1/5*x^4]))/15","C",1
40,1,66,166,0.0212012,"\int \frac{x^2 \left(2+3 x^2\right)}{\sqrt{5+x^4}} \, dx","Integrate[(x^2*(2 + 3*x^2))/Sqrt[5 + x^4],x]","-\sqrt{5} x \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{x^4}{5}\right)+\frac{2 x^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{x^4}{5}\right)}{3 \sqrt{5}}+\sqrt{x^4+5} x","\sqrt{x^4+5} x+\frac{2 \sqrt{x^4+5} x}{x^2+\sqrt{5}}+\frac{\sqrt[4]{5} \left(2-\sqrt{5}\right) \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{2 \sqrt{x^4+5}}-\frac{2 \sqrt[4]{5} \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{\sqrt{x^4+5}}",1,"x*Sqrt[5 + x^4] - Sqrt[5]*x*Hypergeometric2F1[1/4, 1/2, 5/4, -1/5*x^4] + (2*x^3*Hypergeometric2F1[1/2, 3/4, 7/4, -1/5*x^4])/(3*Sqrt[5])","C",1
41,1,48,155,0.0112225,"\int \frac{2+3 x^2}{\sqrt{5+x^4}} \, dx","Integrate[(2 + 3*x^2)/Sqrt[5 + x^4],x]","\frac{x \left(2 \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{x^4}{5}\right)+x^2 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{x^4}{5}\right)\right)}{\sqrt{5}}","\frac{3 \sqrt{x^4+5} x}{x^2+\sqrt{5}}+\frac{\left(2+3 \sqrt{5}\right) \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{5} \sqrt{x^4+5}}-\frac{3 \sqrt[4]{5} \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{\sqrt{x^4+5}}",1,"(x*(2*Hypergeometric2F1[1/4, 1/2, 5/4, -1/5*x^4] + x^2*Hypergeometric2F1[1/2, 3/4, 7/4, -1/5*x^4]))/Sqrt[5]","C",1
42,1,53,173,0.0227391,"\int \frac{2+3 x^2}{x^2 \sqrt{5+x^4}} \, dx","Integrate[(2 + 3*x^2)/(x^2*Sqrt[5 + x^4]),x]","\frac{3 x \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{x^4}{5}\right)}{\sqrt{5}}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{x^4}{5}\right)}{\sqrt{5} x}","-\frac{2 \sqrt{x^4+5}}{5 x}+\frac{2 \sqrt{x^4+5} x}{5 \left(x^2+\sqrt{5}\right)}+\frac{\left(2+3 \sqrt{5}\right) \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{2\ 5^{3/4} \sqrt{x^4+5}}-\frac{2 \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{5^{3/4} \sqrt{x^4+5}}",1,"(-2*Hypergeometric2F1[-1/4, 1/2, 3/4, -1/5*x^4])/(Sqrt[5]*x) + (3*x*Hypergeometric2F1[1/4, 1/2, 5/4, -1/5*x^4])/Sqrt[5]","C",1
43,1,54,189,0.0207774,"\int \frac{2+3 x^2}{x^4 \sqrt{5+x^4}} \, dx","Integrate[(2 + 3*x^2)/(x^4*Sqrt[5 + x^4]),x]","-\frac{2 \, _2F_1\left(-\frac{3}{4},\frac{1}{2};\frac{1}{4};-\frac{x^4}{5}\right)+9 x^2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{x^4}{5}\right)}{3 \sqrt{5} x^3}","-\frac{3 \sqrt{x^4+5}}{5 x}-\frac{2 \sqrt{x^4+5}}{15 x^3}+\frac{3 \sqrt{x^4+5} x}{5 \left(x^2+\sqrt{5}\right)}-\frac{\left(2-9 \sqrt{5}\right) \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{30 \sqrt[4]{5} \sqrt{x^4+5}}-\frac{3 \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{5^{3/4} \sqrt{x^4+5}}",1,"-1/3*(2*Hypergeometric2F1[-3/4, 1/2, 1/4, -1/5*x^4] + 9*x^2*Hypergeometric2F1[-1/4, 1/2, 3/4, -1/5*x^4])/(Sqrt[5]*x^3)","C",1
44,1,51,58,0.0238212,"\int \frac{x^7 \left(2+3 x^2\right)}{\left(5+x^4\right)^{3/2}} \, dx","Integrate[(x^7*(2 + 3*x^2))/(5 + x^4)^(3/2),x]","\frac{3 x^6+4 x^4+45 x^2-45 \sqrt{x^4+5} \sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)+40}{4 \sqrt{x^4+5}}","-\frac{45}{4} \sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)-\frac{\left(3 x^2+2\right) x^4}{2 \sqrt{x^4+5}}+\frac{1}{4} \left(9 x^2+8\right) \sqrt{x^4+5}",1,"(40 + 45*x^2 + 4*x^4 + 3*x^6 - 45*Sqrt[5 + x^4]*ArcSinh[x^2/Sqrt[5]])/(4*Sqrt[5 + x^4])","A",1
45,1,46,45,0.0205288,"\int \frac{x^5 \left(2+3 x^2\right)}{\left(5+x^4\right)^{3/2}} \, dx","Integrate[(x^5*(2 + 3*x^2))/(5 + x^4)^(3/2),x]","\frac{3 x^4-2 x^2+2 \sqrt{x^4+5} \sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)+30}{2 \sqrt{x^4+5}}","3 \sqrt{x^4+5}+\sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)-\frac{\left(3 x^2+2\right) x^2}{2 \sqrt{x^4+5}}",1,"(30 - 2*x^2 + 3*x^4 + 2*Sqrt[5 + x^4]*ArcSinh[x^2/Sqrt[5]])/(2*Sqrt[5 + x^4])","A",1
46,1,41,35,0.017126,"\int \frac{x^3 \left(2+3 x^2\right)}{\left(5+x^4\right)^{3/2}} \, dx","Integrate[(x^3*(2 + 3*x^2))/(5 + x^4)^(3/2),x]","\frac{-3 x^2+3 \sqrt{x^4+5} \sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)-2}{2 \sqrt{x^4+5}}","\frac{3}{2} \sinh ^{-1}\left(\frac{x^2}{\sqrt{5}}\right)+\frac{-3 x^2-2}{2 \sqrt{x^4+5}}",1,"(-2 - 3*x^2 + 3*Sqrt[5 + x^4]*ArcSinh[x^2/Sqrt[5]])/(2*Sqrt[5 + x^4])","A",1
47,1,20,20,0.0093016,"\int \frac{x \left(2+3 x^2\right)}{\left(5+x^4\right)^{3/2}} \, dx","Integrate[(x*(2 + 3*x^2))/(5 + x^4)^(3/2),x]","\frac{2 x^2-15}{10 \sqrt{x^4+5}}","\frac{2 x^2-15}{10 \sqrt{x^4+5}}",1,"(-15 + 2*x^2)/(10*Sqrt[5 + x^4])","A",1
48,1,46,46,0.0320027,"\int \frac{2+3 x^2}{x \left(5+x^4\right)^{3/2}} \, dx","Integrate[(2 + 3*x^2)/(x*(5 + x^4)^(3/2)),x]","\frac{1}{50} \left(\frac{5 \left(3 x^2+2\right)}{\sqrt{x^4+5}}-2 \sqrt{5} \tanh ^{-1}\left(\frac{\sqrt{x^4+5}}{\sqrt{5}}\right)\right)","\frac{3 x^2+2}{10 \sqrt{x^4+5}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{x^4+5}}{\sqrt{5}}\right)}{5 \sqrt{5}}",1,"((5*(2 + 3*x^2))/Sqrt[5 + x^4] - 2*Sqrt[5]*ArcTanh[Sqrt[5 + x^4]/Sqrt[5]])/50","A",1
49,1,45,65,0.0249681,"\int \frac{2+3 x^2}{x^3 \left(5+x^4\right)^{3/2}} \, dx","Integrate[(2 + 3*x^2)/(x^3*(5 + x^4)^(3/2)),x]","\frac{15 x^2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{x^4}{5}+1\right)-4 x^4-10}{50 x^2 \sqrt{x^4+5}}","-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{x^4+5}}{\sqrt{5}}\right)}{10 \sqrt{5}}+\frac{3 x^2+2}{10 x^2 \sqrt{x^4+5}}-\frac{2 \sqrt{x^4+5}}{25 x^2}",1,"(-10 - 4*x^4 + 15*x^2*Hypergeometric2F1[-1/2, 1, 1/2, 1 + x^4/5])/(50*x^2*Sqrt[5 + x^4])","C",1
50,1,70,196,0.0389632,"\int \frac{x^4 \left(2+3 x^2\right)}{\left(5+x^4\right)^{3/2}} \, dx","Integrate[(x^4*(2 + 3*x^2))/(5 + x^4)^(3/2),x]","\frac{x \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{x^4}{5}\right)}{\sqrt{5}}-\frac{3 x^3 \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{x^4}{5}\right)}{\sqrt{5}}+\frac{\left(3 x^2-1\right) x}{\sqrt{x^4+5}}","-\frac{1}{5} \sqrt{x^4+5} x+\frac{9 \sqrt{x^4+5} x}{2 \left(x^2+\sqrt{5}\right)}+\frac{\left(2+9 \sqrt{5}\right) \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{4 \sqrt[4]{5} \sqrt{x^4+5}}-\frac{9 \sqrt[4]{5} \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{2 \sqrt{x^4+5}}-\frac{\left(15-2 x^2\right) x^3}{10 \sqrt{x^4+5}}",1,"(x*(-1 + 3*x^2))/Sqrt[5 + x^4] + (x*Hypergeometric2F1[1/4, 1/2, 5/4, -1/5*x^4])/Sqrt[5] - (3*x^3*Hypergeometric2F1[3/4, 3/2, 7/4, -1/5*x^4])/Sqrt[5]","C",1
51,1,68,177,0.0285106,"\int \frac{x^2 \left(2+3 x^2\right)}{\left(5+x^4\right)^{3/2}} \, dx","Integrate[(x^2*(2 + 3*x^2))/(5 + x^4)^(3/2),x]","\frac{1}{150} x \left(45 \sqrt{5} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{x^4}{5}\right)+4 \sqrt{5} x^2 \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{x^4}{5}\right)-\frac{225}{\sqrt{x^4+5}}\right)","-\frac{\sqrt{x^4+5} x}{5 \left(x^2+\sqrt{5}\right)}-\frac{\left(15-2 x^2\right) x}{10 \sqrt{x^4+5}}-\frac{\left(2-3 \sqrt{5}\right) \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{4\ 5^{3/4} \sqrt{x^4+5}}+\frac{\left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{5^{3/4} \sqrt{x^4+5}}",1,"(x*(-225/Sqrt[5 + x^4] + 45*Sqrt[5]*Hypergeometric2F1[1/4, 1/2, 5/4, -1/5*x^4] + 4*Sqrt[5]*x^2*Hypergeometric2F1[3/4, 3/2, 7/4, -1/5*x^4]))/150","C",1
52,1,66,180,0.0222211,"\int \frac{2+3 x^2}{\left(5+x^4\right)^{3/2}} \, dx","Integrate[(2 + 3*x^2)/(5 + x^4)^(3/2),x]","\frac{1}{25} x \left(\sqrt{5} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{x^4}{5}\right)+\sqrt{5} x^2 \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{x^4}{5}\right)+\frac{5}{\sqrt{x^4+5}}\right)","-\frac{3 \sqrt{x^4+5} x}{10 \left(x^2+\sqrt{5}\right)}+\frac{\left(3 x^2+2\right) x}{10 \sqrt{x^4+5}}+\frac{\left(2-3 \sqrt{5}\right) \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{20 \sqrt[4]{5} \sqrt{x^4+5}}+\frac{3 \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{2\ 5^{3/4} \sqrt{x^4+5}}",1,"(x*(5/Sqrt[5 + x^4] + Sqrt[5]*Hypergeometric2F1[1/4, 1/2, 5/4, -1/5*x^4] + Sqrt[5]*x^2*Hypergeometric2F1[3/4, 3/2, 7/4, -1/5*x^4]))/25","C",1
53,1,71,196,0.0509235,"\int \frac{2+3 x^2}{x^2 \left(5+x^4\right)^{3/2}} \, dx","Integrate[(2 + 3*x^2)/(x^2*(5 + x^4)^(3/2)),x]","\frac{3 x \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{x^4}{5}\right)}{10 \sqrt{5}}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{3}{2};\frac{3}{4};-\frac{x^4}{5}\right)}{5 \sqrt{5} x}+\frac{3 x}{10 \sqrt{x^4+5}}","-\frac{3 \sqrt{x^4+5}}{25 x}+\frac{3 \sqrt{x^4+5} x}{25 \left(x^2+\sqrt{5}\right)}+\frac{3 x^2+2}{10 \sqrt{x^4+5} x}+\frac{3 \left(2+\sqrt{5}\right) \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{20\ 5^{3/4} \sqrt{x^4+5}}-\frac{3 \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{5\ 5^{3/4} \sqrt{x^4+5}}",1,"(3*x)/(10*Sqrt[5 + x^4]) - (2*Hypergeometric2F1[-1/4, 3/2, 3/4, -1/5*x^4])/(5*Sqrt[5]*x) + (3*x*Hypergeometric2F1[1/4, 1/2, 5/4, -1/5*x^4])/(10*Sqrt[5])","C",1
54,1,54,214,0.023556,"\int \frac{2+3 x^2}{x^4 \left(5+x^4\right)^{3/2}} \, dx","Integrate[(2 + 3*x^2)/(x^4*(5 + x^4)^(3/2)),x]","-\frac{2 \, _2F_1\left(-\frac{3}{4},\frac{3}{2};\frac{1}{4};-\frac{x^4}{5}\right)+9 x^2 \, _2F_1\left(-\frac{1}{4},\frac{3}{2};\frac{3}{4};-\frac{x^4}{5}\right)}{15 \sqrt{5} x^3}","-\frac{9 \sqrt{x^4+5}}{50 x}-\frac{\sqrt{x^4+5}}{15 x^3}+\frac{9 \sqrt{x^4+5} x}{50 \left(x^2+\sqrt{5}\right)}+\frac{\left(27-2 \sqrt{5}\right) \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{60\ 5^{3/4} \sqrt{x^4+5}}-\frac{9 \left(x^2+\sqrt{5}\right) \sqrt{\frac{x^4+5}{\left(x^2+\sqrt{5}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{5}}\right)|\frac{1}{2}\right)}{10\ 5^{3/4} \sqrt{x^4+5}}+\frac{3 x^2+2}{10 \sqrt{x^4+5} x^3}",1,"-1/15*(2*Hypergeometric2F1[-3/4, 3/2, 1/4, -1/5*x^4] + 9*x^2*Hypergeometric2F1[-1/4, 3/2, 3/4, -1/5*x^4])/(Sqrt[5]*x^3)","C",1
55,1,189,269,0.5569335,"\int (f x)^m \left(d+e x^2\right) \left(1+2 x^2+x^4\right)^5 \, dx","Integrate[(f*x)^m*(d + e*x^2)*(1 + 2*x^2 + x^4)^5,x]","x (f x)^m \left(\frac{x^{20} (d+10 e)}{m+21}+\frac{5 x^{18} (2 d+9 e)}{m+19}+\frac{15 x^{16} (3 d+8 e)}{m+17}+\frac{30 x^{14} (4 d+7 e)}{m+15}+\frac{42 x^{12} (5 d+6 e)}{m+13}+\frac{42 x^{10} (6 d+5 e)}{m+11}+\frac{30 x^8 (7 d+4 e)}{m+9}+\frac{15 x^6 (8 d+3 e)}{m+7}+\frac{5 x^4 (9 d+2 e)}{m+5}+\frac{x^2 (10 d+e)}{m+3}+\frac{d}{m+1}+\frac{e x^{22}}{m+23}\right)","\frac{(d+10 e) (f x)^{m+21}}{f^{21} (m+21)}+\frac{5 (2 d+9 e) (f x)^{m+19}}{f^{19} (m+19)}+\frac{15 (3 d+8 e) (f x)^{m+17}}{f^{17} (m+17)}+\frac{30 (4 d+7 e) (f x)^{m+15}}{f^{15} (m+15)}+\frac{42 (5 d+6 e) (f x)^{m+13}}{f^{13} (m+13)}+\frac{42 (6 d+5 e) (f x)^{m+11}}{f^{11} (m+11)}+\frac{30 (7 d+4 e) (f x)^{m+9}}{f^9 (m+9)}+\frac{15 (8 d+3 e) (f x)^{m+7}}{f^7 (m+7)}+\frac{5 (9 d+2 e) (f x)^{m+5}}{f^5 (m+5)}+\frac{(10 d+e) (f x)^{m+3}}{f^3 (m+3)}+\frac{d (f x)^{m+1}}{f (m+1)}+\frac{e (f x)^{m+23}}{f^{23} (m+23)}",1,"x*(f*x)^m*(d/(1 + m) + ((10*d + e)*x^2)/(3 + m) + (5*(9*d + 2*e)*x^4)/(5 + m) + (15*(8*d + 3*e)*x^6)/(7 + m) + (30*(7*d + 4*e)*x^8)/(9 + m) + (42*(6*d + 5*e)*x^10)/(11 + m) + (42*(5*d + 6*e)*x^12)/(13 + m) + (30*(4*d + 7*e)*x^14)/(15 + m) + (15*(3*d + 8*e)*x^16)/(17 + m) + (5*(2*d + 9*e)*x^18)/(19 + m) + ((d + 10*e)*x^20)/(21 + m) + (e*x^22)/(23 + m))","A",1
56,1,153,63,0.0219277,"\int x^5 \left(d+e x^2\right) \left(1+2 x^2+x^4\right)^5 \, dx","Integrate[x^5*(d + e*x^2)*(1 + 2*x^2 + x^4)^5,x]","\frac{1}{26} x^{26} (d+10 e)+\frac{5}{24} x^{24} (2 d+9 e)+\frac{15}{22} x^{22} (3 d+8 e)+\frac{3}{2} x^{20} (4 d+7 e)+\frac{7}{3} x^{18} (5 d+6 e)+\frac{21}{8} x^{16} (6 d+5 e)+\frac{15}{7} x^{14} (7 d+4 e)+\frac{5}{4} x^{12} (8 d+3 e)+\frac{1}{2} x^{10} (9 d+2 e)+\frac{1}{8} x^8 (10 d+e)+\frac{d x^6}{6}+\frac{e x^{28}}{28}","\frac{1}{26} \left(x^2+1\right)^{13} (d-3 e)-\frac{1}{24} \left(x^2+1\right)^{12} (2 d-3 e)+\frac{1}{22} \left(x^2+1\right)^{11} (d-e)+\frac{1}{28} e \left(x^2+1\right)^{14}",1,"(d*x^6)/6 + ((10*d + e)*x^8)/8 + ((9*d + 2*e)*x^10)/2 + (5*(8*d + 3*e)*x^12)/4 + (15*(7*d + 4*e)*x^14)/7 + (21*(6*d + 5*e)*x^16)/8 + (7*(5*d + 6*e)*x^18)/3 + (3*(4*d + 7*e)*x^20)/2 + (15*(3*d + 8*e)*x^22)/22 + (5*(2*d + 9*e)*x^24)/24 + ((d + 10*e)*x^26)/26 + (e*x^28)/28","B",1
57,1,153,153,0.0225628,"\int x^4 \left(d+e x^2\right) \left(1+2 x^2+x^4\right)^5 \, dx","Integrate[x^4*(d + e*x^2)*(1 + 2*x^2 + x^4)^5,x]","\frac{1}{25} x^{25} (d+10 e)+\frac{5}{23} x^{23} (2 d+9 e)+\frac{5}{7} x^{21} (3 d+8 e)+\frac{30}{19} x^{19} (4 d+7 e)+\frac{42}{17} x^{17} (5 d+6 e)+\frac{14}{5} x^{15} (6 d+5 e)+\frac{30}{13} x^{13} (7 d+4 e)+\frac{15}{11} x^{11} (8 d+3 e)+\frac{5}{9} x^9 (9 d+2 e)+\frac{1}{7} x^7 (10 d+e)+\frac{d x^5}{5}+\frac{e x^{27}}{27}","\frac{1}{25} x^{25} (d+10 e)+\frac{5}{23} x^{23} (2 d+9 e)+\frac{5}{7} x^{21} (3 d+8 e)+\frac{30}{19} x^{19} (4 d+7 e)+\frac{42}{17} x^{17} (5 d+6 e)+\frac{14}{5} x^{15} (6 d+5 e)+\frac{30}{13} x^{13} (7 d+4 e)+\frac{15}{11} x^{11} (8 d+3 e)+\frac{5}{9} x^9 (9 d+2 e)+\frac{1}{7} x^7 (10 d+e)+\frac{d x^5}{5}+\frac{e x^{27}}{27}",1,"(d*x^5)/5 + ((10*d + e)*x^7)/7 + (5*(9*d + 2*e)*x^9)/9 + (15*(8*d + 3*e)*x^11)/11 + (30*(7*d + 4*e)*x^13)/13 + (14*(6*d + 5*e)*x^15)/5 + (42*(5*d + 6*e)*x^17)/17 + (30*(4*d + 7*e)*x^19)/19 + (5*(3*d + 8*e)*x^21)/7 + (5*(2*d + 9*e)*x^23)/23 + ((d + 10*e)*x^25)/25 + (e*x^27)/27","A",1
58,1,151,45,0.0172702,"\int x^3 \left(d+e x^2\right) \left(1+2 x^2+x^4\right)^5 \, dx","Integrate[x^3*(d + e*x^2)*(1 + 2*x^2 + x^4)^5,x]","\frac{1}{24} x^{24} (d+10 e)+\frac{5}{22} x^{22} (2 d+9 e)+\frac{3}{4} x^{20} (3 d+8 e)+\frac{5}{3} x^{18} (4 d+7 e)+\frac{21}{8} x^{16} (5 d+6 e)+3 x^{14} (6 d+5 e)+\frac{5}{2} x^{12} (7 d+4 e)+\frac{3}{2} x^{10} (8 d+3 e)+\frac{5}{8} x^8 (9 d+2 e)+\frac{1}{6} x^6 (10 d+e)+\frac{d x^4}{4}+\frac{e x^{26}}{26}","\frac{1}{24} \left(x^2+1\right)^{12} (d-2 e)-\frac{1}{22} \left(x^2+1\right)^{11} (d-e)+\frac{1}{26} e \left(x^2+1\right)^{13}",1,"(d*x^4)/4 + ((10*d + e)*x^6)/6 + (5*(9*d + 2*e)*x^8)/8 + (3*(8*d + 3*e)*x^10)/2 + (5*(7*d + 4*e)*x^12)/2 + 3*(6*d + 5*e)*x^14 + (21*(5*d + 6*e)*x^16)/8 + (5*(4*d + 7*e)*x^18)/3 + (3*(3*d + 8*e)*x^20)/4 + (5*(2*d + 9*e)*x^22)/22 + ((d + 10*e)*x^24)/24 + (e*x^26)/26","B",1
59,1,153,153,0.0166654,"\int x^2 \left(d+e x^2\right) \left(1+2 x^2+x^4\right)^5 \, dx","Integrate[x^2*(d + e*x^2)*(1 + 2*x^2 + x^4)^5,x]","\frac{1}{23} x^{23} (d+10 e)+\frac{5}{21} x^{21} (2 d+9 e)+\frac{15}{19} x^{19} (3 d+8 e)+\frac{30}{17} x^{17} (4 d+7 e)+\frac{14}{5} x^{15} (5 d+6 e)+\frac{42}{13} x^{13} (6 d+5 e)+\frac{30}{11} x^{11} (7 d+4 e)+\frac{5}{3} x^9 (8 d+3 e)+\frac{5}{7} x^7 (9 d+2 e)+\frac{1}{5} x^5 (10 d+e)+\frac{d x^3}{3}+\frac{e x^{25}}{25}","\frac{1}{23} x^{23} (d+10 e)+\frac{5}{21} x^{21} (2 d+9 e)+\frac{15}{19} x^{19} (3 d+8 e)+\frac{30}{17} x^{17} (4 d+7 e)+\frac{14}{5} x^{15} (5 d+6 e)+\frac{42}{13} x^{13} (6 d+5 e)+\frac{30}{11} x^{11} (7 d+4 e)+\frac{5}{3} x^9 (8 d+3 e)+\frac{5}{7} x^7 (9 d+2 e)+\frac{1}{5} x^5 (10 d+e)+\frac{d x^3}{3}+\frac{e x^{25}}{25}",1,"(d*x^3)/3 + ((10*d + e)*x^5)/5 + (5*(9*d + 2*e)*x^7)/7 + (5*(8*d + 3*e)*x^9)/3 + (30*(7*d + 4*e)*x^11)/11 + (42*(6*d + 5*e)*x^13)/13 + (14*(5*d + 6*e)*x^15)/5 + (30*(4*d + 7*e)*x^17)/17 + (15*(3*d + 8*e)*x^19)/19 + (5*(2*d + 9*e)*x^21)/21 + ((d + 10*e)*x^23)/23 + (e*x^25)/25","A",1
60,1,149,29,0.0124287,"\int x \left(d+e x^2\right) \left(1+2 x^2+x^4\right)^5 \, dx","Integrate[x*(d + e*x^2)*(1 + 2*x^2 + x^4)^5,x]","\frac{1}{22} x^{22} (d+10 e)+\frac{1}{4} x^{20} (2 d+9 e)+\frac{5}{6} x^{18} (3 d+8 e)+\frac{15}{8} x^{16} (4 d+7 e)+3 x^{14} (5 d+6 e)+\frac{7}{2} x^{12} (6 d+5 e)+3 x^{10} (7 d+4 e)+\frac{15}{8} x^8 (8 d+3 e)+\frac{5}{6} x^6 (9 d+2 e)+\frac{1}{4} x^4 (10 d+e)+\frac{d x^2}{2}+\frac{e x^{24}}{24}","\frac{1}{22} \left(x^2+1\right)^{11} (d-e)+\frac{1}{24} e \left(x^2+1\right)^{12}",1,"(d*x^2)/2 + ((10*d + e)*x^4)/4 + (5*(9*d + 2*e)*x^6)/6 + (15*(8*d + 3*e)*x^8)/8 + 3*(7*d + 4*e)*x^10 + (7*(6*d + 5*e)*x^12)/2 + 3*(5*d + 6*e)*x^14 + (15*(4*d + 7*e)*x^16)/8 + (5*(3*d + 8*e)*x^18)/6 + ((2*d + 9*e)*x^20)/4 + ((d + 10*e)*x^22)/22 + (e*x^24)/24","B",1
61,1,143,143,0.0157272,"\int \left(d+e x^2\right) \left(1+2 x^2+x^4\right)^5 \, dx","Integrate[(d + e*x^2)*(1 + 2*x^2 + x^4)^5,x]","\frac{1}{21} x^{21} (d+10 e)+\frac{5}{19} x^{19} (2 d+9 e)+\frac{15}{17} x^{17} (3 d+8 e)+2 x^{15} (4 d+7 e)+\frac{42}{13} x^{13} (5 d+6 e)+\frac{42}{11} x^{11} (6 d+5 e)+\frac{10}{3} x^9 (7 d+4 e)+\frac{15}{7} x^7 (8 d+3 e)+x^5 (9 d+2 e)+\frac{1}{3} x^3 (10 d+e)+d x+\frac{e x^{23}}{23}","\frac{1}{21} x^{21} (d+10 e)+\frac{5}{19} x^{19} (2 d+9 e)+\frac{15}{17} x^{17} (3 d+8 e)+2 x^{15} (4 d+7 e)+\frac{42}{13} x^{13} (5 d+6 e)+\frac{42}{11} x^{11} (6 d+5 e)+\frac{10}{3} x^9 (7 d+4 e)+\frac{15}{7} x^7 (8 d+3 e)+x^5 (9 d+2 e)+\frac{1}{3} x^3 (10 d+e)+d x+\frac{e x^{23}}{23}",1,"d*x + ((10*d + e)*x^3)/3 + (9*d + 2*e)*x^5 + (15*(8*d + 3*e)*x^7)/7 + (10*(7*d + 4*e)*x^9)/3 + (42*(6*d + 5*e)*x^11)/11 + (42*(5*d + 6*e)*x^13)/13 + 2*(4*d + 7*e)*x^15 + (15*(3*d + 8*e)*x^17)/17 + (5*(2*d + 9*e)*x^19)/19 + ((d + 10*e)*x^21)/21 + (e*x^23)/23","A",1
62,1,149,93,0.0251782,"\int \frac{\left(d+e x^2\right) \left(1+2 x^2+x^4\right)^5}{x} \, dx","Integrate[((d + e*x^2)*(1 + 2*x^2 + x^4)^5)/x,x]","\frac{1}{20} x^{20} (d+10 e)+\frac{5}{18} x^{18} (2 d+9 e)+\frac{15}{16} x^{16} (3 d+8 e)+\frac{15}{7} x^{14} (4 d+7 e)+\frac{7}{2} x^{12} (5 d+6 e)+\frac{21}{5} x^{10} (6 d+5 e)+\frac{15}{4} x^8 (7 d+4 e)+\frac{5}{2} x^6 (8 d+3 e)+\frac{5}{4} x^4 (9 d+2 e)+\frac{1}{2} x^2 (10 d+e)+d \log (x)+\frac{e x^{22}}{22}","\frac{d x^{20}}{20}+\frac{5 d x^{18}}{9}+\frac{45 d x^{16}}{16}+\frac{60 d x^{14}}{7}+\frac{35 d x^{12}}{2}+\frac{126 d x^{10}}{5}+\frac{105 d x^8}{4}+20 d x^6+\frac{45 d x^4}{4}+5 d x^2+d \log (x)+\frac{1}{22} e \left(x^2+1\right)^{11}",1,"((10*d + e)*x^2)/2 + (5*(9*d + 2*e)*x^4)/4 + (5*(8*d + 3*e)*x^6)/2 + (15*(7*d + 4*e)*x^8)/4 + (21*(6*d + 5*e)*x^10)/5 + (7*(5*d + 6*e)*x^12)/2 + (15*(4*d + 7*e)*x^14)/7 + (15*(3*d + 8*e)*x^16)/16 + (5*(2*d + 9*e)*x^18)/18 + ((d + 10*e)*x^20)/20 + (e*x^22)/22 + d*Log[x]","A",1
63,1,141,141,0.0262894,"\int \frac{\left(d+e x^2\right) \left(1+2 x^2+x^4\right)^5}{x^2} \, dx","Integrate[((d + e*x^2)*(1 + 2*x^2 + x^4)^5)/x^2,x]","\frac{1}{19} x^{19} (d+10 e)+\frac{5}{17} x^{17} (2 d+9 e)+x^{15} (3 d+8 e)+\frac{30}{13} x^{13} (4 d+7 e)+\frac{42}{11} x^{11} (5 d+6 e)+\frac{14}{3} x^9 (6 d+5 e)+\frac{30}{7} x^7 (7 d+4 e)+3 x^5 (8 d+3 e)+\frac{5}{3} x^3 (9 d+2 e)+x (10 d+e)-\frac{d}{x}+\frac{e x^{21}}{21}","\frac{1}{19} x^{19} (d+10 e)+\frac{5}{17} x^{17} (2 d+9 e)+x^{15} (3 d+8 e)+\frac{30}{13} x^{13} (4 d+7 e)+\frac{42}{11} x^{11} (5 d+6 e)+\frac{14}{3} x^9 (6 d+5 e)+\frac{30}{7} x^7 (7 d+4 e)+3 x^5 (8 d+3 e)+\frac{5}{3} x^3 (9 d+2 e)+x (10 d+e)-\frac{d}{x}+\frac{e x^{21}}{21}",1,"-(d/x) + (10*d + e)*x + (5*(9*d + 2*e)*x^3)/3 + 3*(8*d + 3*e)*x^5 + (30*(7*d + 4*e)*x^7)/7 + (14*(6*d + 5*e)*x^9)/3 + (42*(5*d + 6*e)*x^11)/11 + (30*(4*d + 7*e)*x^13)/13 + (3*d + 8*e)*x^15 + (5*(2*d + 9*e)*x^17)/17 + ((d + 10*e)*x^19)/19 + (e*x^21)/21","A",1
64,1,147,147,0.0344233,"\int \frac{\left(d+e x^2\right) \left(1+2 x^2+x^4\right)^5}{x^3} \, dx","Integrate[((d + e*x^2)*(1 + 2*x^2 + x^4)^5)/x^3,x]","\frac{1}{18} x^{18} (d+10 e)+\frac{5}{16} x^{16} (2 d+9 e)+\frac{15}{14} x^{14} (3 d+8 e)+\frac{5}{2} x^{12} (4 d+7 e)+\frac{21}{5} x^{10} (5 d+6 e)+\frac{21}{4} x^8 (6 d+5 e)+5 x^6 (7 d+4 e)+\frac{15}{4} x^4 (8 d+3 e)+\frac{5}{2} x^2 (9 d+2 e)+(10 d+e) \log (x)-\frac{d}{2 x^2}+\frac{e x^{20}}{20}","\frac{1}{18} x^{18} (d+10 e)+\frac{5}{16} x^{16} (2 d+9 e)+\frac{15}{14} x^{14} (3 d+8 e)+\frac{5}{2} x^{12} (4 d+7 e)+\frac{21}{5} x^{10} (5 d+6 e)+\frac{21}{4} x^8 (6 d+5 e)+5 x^6 (7 d+4 e)+\frac{15}{4} x^4 (8 d+3 e)+\frac{5}{2} x^2 (9 d+2 e)+(10 d+e) \log (x)-\frac{d}{2 x^2}+\frac{e x^{20}}{20}",1,"-1/2*d/x^2 + (5*(9*d + 2*e)*x^2)/2 + (15*(8*d + 3*e)*x^4)/4 + 5*(7*d + 4*e)*x^6 + (21*(6*d + 5*e)*x^8)/4 + (21*(5*d + 6*e)*x^10)/5 + (5*(4*d + 7*e)*x^12)/2 + (15*(3*d + 8*e)*x^14)/14 + (5*(2*d + 9*e)*x^16)/16 + ((d + 10*e)*x^18)/18 + (e*x^20)/20 + (10*d + e)*Log[x]","A",1
65,1,122,203,0.0443166,"\int (f x)^m \left(1+x^2\right) \left(1+2 x^2+x^4\right)^5 \, dx","Integrate[(f*x)^m*(1 + x^2)*(1 + 2*x^2 + x^4)^5,x]","x \left(\frac{x^{22}}{m+23}+\frac{11 x^{20}}{m+21}+\frac{55 x^{18}}{m+19}+\frac{165 x^{16}}{m+17}+\frac{330 x^{14}}{m+15}+\frac{462 x^{12}}{m+13}+\frac{462 x^{10}}{m+11}+\frac{330 x^8}{m+9}+\frac{165 x^6}{m+7}+\frac{55 x^4}{m+5}+\frac{11 x^2}{m+3}+\frac{1}{m+1}\right) (f x)^m","\frac{(f x)^{m+23}}{f^{23} (m+23)}+\frac{11 (f x)^{m+21}}{f^{21} (m+21)}+\frac{55 (f x)^{m+19}}{f^{19} (m+19)}+\frac{165 (f x)^{m+17}}{f^{17} (m+17)}+\frac{330 (f x)^{m+15}}{f^{15} (m+15)}+\frac{462 (f x)^{m+13}}{f^{13} (m+13)}+\frac{462 (f x)^{m+11}}{f^{11} (m+11)}+\frac{330 (f x)^{m+9}}{f^9 (m+9)}+\frac{165 (f x)^{m+7}}{f^7 (m+7)}+\frac{55 (f x)^{m+5}}{f^5 (m+5)}+\frac{11 (f x)^{m+3}}{f^3 (m+3)}+\frac{(f x)^{m+1}}{f (m+1)}",1,"x*(f*x)^m*((1 + m)^(-1) + (11*x^2)/(3 + m) + (55*x^4)/(5 + m) + (165*x^6)/(7 + m) + (330*x^8)/(9 + m) + (462*x^10)/(11 + m) + (462*x^12)/(13 + m) + (330*x^14)/(15 + m) + (165*x^16)/(17 + m) + (55*x^18)/(19 + m) + (11*x^20)/(21 + m) + x^22/(23 + m))","A",1
66,1,85,34,0.0018357,"\int x^5 \left(1+x^2\right) \left(1+2 x^2+x^4\right)^5 \, dx","Integrate[x^5*(1 + x^2)*(1 + 2*x^2 + x^4)^5,x]","\frac{x^{28}}{28}+\frac{11 x^{26}}{26}+\frac{55 x^{24}}{24}+\frac{15 x^{22}}{2}+\frac{33 x^{20}}{2}+\frac{77 x^{18}}{3}+\frac{231 x^{16}}{8}+\frac{165 x^{14}}{7}+\frac{55 x^{12}}{4}+\frac{11 x^{10}}{2}+\frac{11 x^8}{8}+\frac{x^6}{6}","\frac{1}{28} \left(x^2+1\right)^{14}-\frac{1}{13} \left(x^2+1\right)^{13}+\frac{1}{24} \left(x^2+1\right)^{12}",1,"x^6/6 + (11*x^8)/8 + (11*x^10)/2 + (55*x^12)/4 + (165*x^14)/7 + (231*x^16)/8 + (77*x^18)/3 + (33*x^20)/2 + (15*x^22)/2 + (55*x^24)/24 + (11*x^26)/26 + x^28/28","B",1
67,1,83,83,0.0015205,"\int x^4 \left(1+x^2\right) \left(1+2 x^2+x^4\right)^5 \, dx","Integrate[x^4*(1 + x^2)*(1 + 2*x^2 + x^4)^5,x]","\frac{x^{27}}{27}+\frac{11 x^{25}}{25}+\frac{55 x^{23}}{23}+\frac{55 x^{21}}{7}+\frac{330 x^{19}}{19}+\frac{462 x^{17}}{17}+\frac{154 x^{15}}{5}+\frac{330 x^{13}}{13}+15 x^{11}+\frac{55 x^9}{9}+\frac{11 x^7}{7}+\frac{x^5}{5}","\frac{x^{27}}{27}+\frac{11 x^{25}}{25}+\frac{55 x^{23}}{23}+\frac{55 x^{21}}{7}+\frac{330 x^{19}}{19}+\frac{462 x^{17}}{17}+\frac{154 x^{15}}{5}+\frac{330 x^{13}}{13}+15 x^{11}+\frac{55 x^9}{9}+\frac{11 x^7}{7}+\frac{x^5}{5}",1,"x^5/5 + (11*x^7)/7 + (55*x^9)/9 + 15*x^11 + (330*x^13)/13 + (154*x^15)/5 + (462*x^17)/17 + (330*x^19)/19 + (55*x^21)/7 + (55*x^23)/23 + (11*x^25)/25 + x^27/27","A",1
68,1,83,23,0.0017997,"\int x^3 \left(1+x^2\right) \left(1+2 x^2+x^4\right)^5 \, dx","Integrate[x^3*(1 + x^2)*(1 + 2*x^2 + x^4)^5,x]","\frac{x^{26}}{26}+\frac{11 x^{24}}{24}+\frac{5 x^{22}}{2}+\frac{33 x^{20}}{4}+\frac{55 x^{18}}{3}+\frac{231 x^{16}}{8}+33 x^{14}+\frac{55 x^{12}}{2}+\frac{33 x^{10}}{2}+\frac{55 x^8}{8}+\frac{11 x^6}{6}+\frac{x^4}{4}","\frac{1}{26} \left(x^2+1\right)^{13}-\frac{1}{24} \left(x^2+1\right)^{12}",1,"x^4/4 + (11*x^6)/6 + (55*x^8)/8 + (33*x^10)/2 + (55*x^12)/2 + 33*x^14 + (231*x^16)/8 + (55*x^18)/3 + (33*x^20)/4 + (5*x^22)/2 + (11*x^24)/24 + x^26/26","B",1
69,1,83,83,0.0014961,"\int x^2 \left(1+x^2\right) \left(1+2 x^2+x^4\right)^5 \, dx","Integrate[x^2*(1 + x^2)*(1 + 2*x^2 + x^4)^5,x]","\frac{x^{25}}{25}+\frac{11 x^{23}}{23}+\frac{55 x^{21}}{21}+\frac{165 x^{19}}{19}+\frac{330 x^{17}}{17}+\frac{154 x^{15}}{5}+\frac{462 x^{13}}{13}+30 x^{11}+\frac{55 x^9}{3}+\frac{55 x^7}{7}+\frac{11 x^5}{5}+\frac{x^3}{3}","\frac{x^{25}}{25}+\frac{11 x^{23}}{23}+\frac{55 x^{21}}{21}+\frac{165 x^{19}}{19}+\frac{330 x^{17}}{17}+\frac{154 x^{15}}{5}+\frac{462 x^{13}}{13}+30 x^{11}+\frac{55 x^9}{3}+\frac{55 x^7}{7}+\frac{11 x^5}{5}+\frac{x^3}{3}",1,"x^3/3 + (11*x^5)/5 + (55*x^7)/7 + (55*x^9)/3 + 30*x^11 + (462*x^13)/13 + (154*x^15)/5 + (330*x^17)/17 + (165*x^19)/19 + (55*x^21)/21 + (11*x^23)/23 + x^25/25","A",1
70,1,11,11,0.0017904,"\int x \left(1+x^2\right) \left(1+2 x^2+x^4\right)^5 \, dx","Integrate[x*(1 + x^2)*(1 + 2*x^2 + x^4)^5,x]","\frac{1}{24} \left(x^2+1\right)^{12}","\frac{1}{24} \left(x^2+1\right)^{12}",1,"(1 + x^2)^12/24","A",1
71,1,73,73,0.0009004,"\int \left(1+x^2\right) \left(1+2 x^2+x^4\right)^5 \, dx","Integrate[(1 + x^2)*(1 + 2*x^2 + x^4)^5,x]","\frac{x^{23}}{23}+\frac{11 x^{21}}{21}+\frac{55 x^{19}}{19}+\frac{165 x^{17}}{17}+22 x^{15}+\frac{462 x^{13}}{13}+42 x^{11}+\frac{110 x^9}{3}+\frac{165 x^7}{7}+11 x^5+\frac{11 x^3}{3}+x","\frac{x^{23}}{23}+\frac{11 x^{21}}{21}+\frac{55 x^{19}}{19}+\frac{165 x^{17}}{17}+22 x^{15}+\frac{462 x^{13}}{13}+42 x^{11}+\frac{110 x^9}{3}+\frac{165 x^7}{7}+11 x^5+\frac{11 x^3}{3}+x",1,"x + (11*x^3)/3 + 11*x^5 + (165*x^7)/7 + (110*x^9)/3 + 42*x^11 + (462*x^13)/13 + 22*x^15 + (165*x^17)/17 + (55*x^19)/19 + (11*x^21)/21 + x^23/23","A",1
72,1,80,80,0.0032858,"\int \frac{\left(1+x^2\right) \left(1+2 x^2+x^4\right)^5}{x} \, dx","Integrate[((1 + x^2)*(1 + 2*x^2 + x^4)^5)/x,x]","\frac{x^{22}}{22}+\frac{11 x^{20}}{20}+\frac{55 x^{18}}{18}+\frac{165 x^{16}}{16}+\frac{165 x^{14}}{7}+\frac{77 x^{12}}{2}+\frac{231 x^{10}}{5}+\frac{165 x^8}{4}+\frac{55 x^6}{2}+\frac{55 x^4}{4}+\frac{11 x^2}{2}+\log (x)","\frac{x^{22}}{22}+\frac{11 x^{20}}{20}+\frac{55 x^{18}}{18}+\frac{165 x^{16}}{16}+\frac{165 x^{14}}{7}+\frac{77 x^{12}}{2}+\frac{231 x^{10}}{5}+\frac{165 x^8}{4}+\frac{55 x^6}{2}+\frac{55 x^4}{4}+\frac{11 x^2}{2}+\log (x)",1,"(11*x^2)/2 + (55*x^4)/4 + (55*x^6)/2 + (165*x^8)/4 + (231*x^10)/5 + (77*x^12)/2 + (165*x^14)/7 + (165*x^16)/16 + (55*x^18)/18 + (11*x^20)/20 + x^22/22 + Log[x]","A",1
73,1,73,73,0.0034589,"\int \frac{\left(1+x^2\right) \left(1+2 x^2+x^4\right)^5}{x^2} \, dx","Integrate[((1 + x^2)*(1 + 2*x^2 + x^4)^5)/x^2,x]","\frac{x^{21}}{21}+\frac{11 x^{19}}{19}+\frac{55 x^{17}}{17}+11 x^{15}+\frac{330 x^{13}}{13}+42 x^{11}+\frac{154 x^9}{3}+\frac{330 x^7}{7}+33 x^5+\frac{55 x^3}{3}+11 x-\frac{1}{x}","\frac{x^{21}}{21}+\frac{11 x^{19}}{19}+\frac{55 x^{17}}{17}+11 x^{15}+\frac{330 x^{13}}{13}+42 x^{11}+\frac{154 x^9}{3}+\frac{330 x^7}{7}+33 x^5+\frac{55 x^3}{3}+11 x-\frac{1}{x}",1,"-x^(-1) + 11*x + (55*x^3)/3 + 33*x^5 + (330*x^7)/7 + (154*x^9)/3 + 42*x^11 + (330*x^13)/13 + 11*x^15 + (55*x^17)/17 + (11*x^19)/19 + x^21/21","A",1
74,1,80,80,0.0027526,"\int \frac{\left(1+x^2\right) \left(1+2 x^2+x^4\right)^5}{x^3} \, dx","Integrate[((1 + x^2)*(1 + 2*x^2 + x^4)^5)/x^3,x]","\frac{x^{20}}{20}+\frac{11 x^{18}}{18}+\frac{55 x^{16}}{16}+\frac{165 x^{14}}{14}+\frac{55 x^{12}}{2}+\frac{231 x^{10}}{5}+\frac{231 x^8}{4}+55 x^6+\frac{165 x^4}{4}+\frac{55 x^2}{2}-\frac{1}{2 x^2}+11 \log (x)","\frac{x^{20}}{20}+\frac{11 x^{18}}{18}+\frac{55 x^{16}}{16}+\frac{165 x^{14}}{14}+\frac{55 x^{12}}{2}+\frac{231 x^{10}}{5}+\frac{231 x^8}{4}+55 x^6+\frac{165 x^4}{4}+\frac{55 x^2}{2}-\frac{1}{2 x^2}+11 \log (x)",1,"-1/2*1/x^2 + (55*x^2)/2 + (165*x^4)/4 + 55*x^6 + (231*x^8)/4 + (231*x^10)/5 + (55*x^12)/2 + (165*x^14)/14 + (55*x^16)/16 + (11*x^18)/18 + x^20/20 + 11*Log[x]","A",1
75,1,80,145,0.0829222,"\int \frac{x^2 \left(d+e x^2\right)}{\sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx","Integrate[(x^2*(d + e*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 \left(-3 a e+3 b d+b e x^2\right)+3 \sqrt{a} (a e-b d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right)}{3 b^{5/2} \sqrt{\left(a+b x^2\right)^2}}","\frac{x \left(a+b x^2\right) (b d-a e)}{b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{e x^3 \left(a+b x^2\right)}{3 b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\sqrt{a} \left(a+b x^2\right) (b d-a e) \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*b*d - 3*a*e + b*e*x^2) + 3*Sqrt[a]*(-(b*d) + a*e)*ArcTan[(Sqrt[b]*x)/Sqrt[a]]))/(3*b^(5/2)*Sqrt[(a + b*x^2)^2])","A",1
76,1,51,83,0.0208072,"\int \frac{x \left(d+e x^2\right)}{\sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx","Integrate[(x*(d + e*x^2))/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]","\frac{\left(a+b x^2\right) \left((b d-a e) \log \left(a+b x^2\right)+b e x^2\right)}{2 b^2 \sqrt{\left(a+b x^2\right)^2}}","\frac{\left(a+b x^2\right) (b d-a e) \log \left(a+b x^2\right)}{2 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{e \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 b^2}",1,"((a + b*x^2)*(b*e*x^2 + (b*d - a*e)*Log[a + b*x^2]))/(2*b^2*Sqrt[(a + b*x^2)^2])","A",1
77,1,69,97,0.027234,"\int \frac{d+e x^2}{\sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx","Integrate[(d + e*x^2)/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]","-\frac{\left(a+b x^2\right) \left((a e-b d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)-\sqrt{a} \sqrt{b} e x\right)}{\sqrt{a} b^{3/2} \sqrt{\left(a+b x^2\right)^2}}","\frac{e x \left(a+b x^2\right)}{b \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left(a+b x^2\right) (b d-a e) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} b^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"-(((a + b*x^2)*(-(Sqrt[a]*Sqrt[b]*e*x) + (-(b*d) + a*e)*ArcTan[(Sqrt[b]*x)/Sqrt[a]]))/(Sqrt[a]*b^(3/2)*Sqrt[(a + b*x^2)^2]))","A",1
78,1,54,92,0.0207196,"\int \frac{d+e x^2}{x \sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx","Integrate[(d + e*x^2)/(x*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]),x]","\frac{\left(a+b x^2\right) \left((a e-b d) \log \left(a+b x^2\right)+2 b d \log (x)\right)}{2 a b \sqrt{\left(a+b x^2\right)^2}}","\frac{d \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) (b d-a e) \log \left(a+b x^2\right)}{2 a b \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"((a + b*x^2)*(2*b*d*Log[x] + (-(b*d) + a*e)*Log[a + b*x^2]))/(2*a*b*Sqrt[(a + b*x^2)^2])","A",1
79,1,72,101,0.0297076,"\int \frac{d+e x^2}{x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx","Integrate[(d + e*x^2)/(x^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]),x]","\frac{\left(a+b x^2\right) \left(\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) (a e x-b d x)-\sqrt{a} \sqrt{b} d\right)}{a^{3/2} \sqrt{b} x \sqrt{\left(a+b x^2\right)^2}}","-\frac{d \left(a+b x^2\right)}{a x \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left(a+b x^2\right) (b d-a e) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2} \sqrt{b} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"((a + b*x^2)*(-(Sqrt[a]*Sqrt[b]*d) + (-(b*d*x) + a*e*x)*ArcTan[(Sqrt[b]*x)/Sqrt[a]]))/(a^(3/2)*Sqrt[b]*x*Sqrt[(a + b*x^2)^2])","A",1
80,1,70,137,0.033172,"\int \frac{d+e x^2}{x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx","Integrate[(d + e*x^2)/(x^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]),x]","\frac{\left(a+b x^2\right) \left(2 x^2 \log (x) (a e-b d)+x^2 (b d-a e) \log \left(a+b x^2\right)-a d\right)}{2 a^2 x^2 \sqrt{\left(a+b x^2\right)^2}}","-\frac{\log (x) \left(a+b x^2\right) (b d-a e)}{a^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left(a+b x^2\right) (b d-a e) \log \left(a+b x^2\right)}{2 a^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d \left(a+b x^2\right)}{2 a x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"((a + b*x^2)*(-(a*d) + 2*(-(b*d) + a*e)*x^2*Log[x] + (b*d - a*e)*x^2*Log[a + b*x^2]))/(2*a^2*x^2*Sqrt[(a + b*x^2)^2])","A",1
81,1,108,153,0.0601266,"\int \frac{x^2 \left(d+e x^2\right)}{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}} \, dx","Integrate[(x^2*(d + e*x^2))/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","\frac{\left(a+b x^2\right)^2 (3 a e+b d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)-\sqrt{a} \sqrt{b} x \left(3 a^2 e+a b \left(d+5 e x^2\right)-b^2 d x^2\right)}{8 a^{3/2} b^{5/2} \left(a+b x^2\right) \sqrt{\left(a+b x^2\right)^2}}","\frac{x (b d-5 a e)}{8 a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{x (b d-a e)}{4 b^2 \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) (3 a e+b d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{3/2} b^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(-(Sqrt[a]*Sqrt[b]*x*(3*a^2*e - b^2*d*x^2 + a*b*(d + 5*e*x^2))) + (b*d + 3*a*e)*(a + b*x^2)^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(3/2)*b^(5/2)*(a + b*x^2)*Sqrt[(a + b*x^2)^2])","A",1
82,1,45,77,0.0186056,"\int \frac{x \left(d+e x^2\right)}{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}} \, dx","Integrate[(x*(d + e*x^2))/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","\frac{-a e-b \left(d+2 e x^2\right)}{4 b^2 \left(a+b x^2\right) \sqrt{\left(a+b x^2\right)^2}}","-\frac{b d-a e}{4 b^2 \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{e}{2 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(-(a*e) - b*(d + 2*e*x^2))/(4*b^2*(a + b*x^2)*Sqrt[(a + b*x^2)^2])","A",1
83,1,108,156,0.0510069,"\int \frac{d+e x^2}{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}} \, dx","Integrate[(d + e*x^2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","\frac{\sqrt{a} \sqrt{b} x \left(a^2 (-e)+a b \left(5 d+e x^2\right)+3 b^2 d x^2\right)+\left(a+b x^2\right)^2 (a e+3 b d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} b^{3/2} \left(a+b x^2\right) \sqrt{\left(a+b x^2\right)^2}}","\frac{x (a e+3 b d)}{8 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{x (b d-a e)}{4 a 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) (a e+3 b d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} b^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(Sqrt[a]*Sqrt[b]*x*(-(a^2*e) + 3*b^2*d*x^2 + a*b*(5*d + e*x^2)) + (3*b*d + a*e)*(a + b*x^2)^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*b^(3/2)*(a + b*x^2)*Sqrt[(a + b*x^2)^2])","A",1
84,1,92,161,0.0406562,"\int \frac{d+e x^2}{x \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}} \, dx","Integrate[(d + e*x^2)/(x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)),x]","\frac{a \left(a^2 (-e)+3 a b d+2 b^2 d x^2\right)+4 b d \log (x) \left(a+b x^2\right)^2-2 b d \left(a+b x^2\right)^2 \log \left(a+b x^2\right)}{4 a^3 b \left(a+b x^2\right) \sqrt{\left(a+b x^2\right)^2}}","\frac{b d-a e}{4 a b \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{d}{2 a^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{d \log (x) \left(a+b x^2\right)}{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d \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*b*d - a^2*e + 2*b^2*d*x^2) + 4*b*d*(a + b*x^2)^2*Log[x] - 2*b*d*(a + b*x^2)^2*Log[a + b*x^2])/(4*a^3*b*(a + b*x^2)*Sqrt[(a + b*x^2)^2])","A",1
85,1,124,190,0.0687745,"\int \frac{d+e x^2}{x^2 \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}} \, dx","Integrate[(d + e*x^2)/(x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)),x]","\frac{\sqrt{a} \sqrt{b} \left(a^2 \left(5 e x^2-8 d\right)+a b \left(3 e x^4-25 d x^2\right)-15 b^2 d x^4\right)+3 x \left(a+b x^2\right)^2 (a e-5 b d) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{7/2} \sqrt{b} x \left(a+b x^2\right) \sqrt{\left(a+b x^2\right)^2}}","-\frac{x (b d-a e)}{4 a^2 \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) (5 b d-a e) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{7/2} \sqrt{b} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{x (7 b d-3 a e)}{8 a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d \left(a+b x^2\right)}{a^3 x \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(Sqrt[a]*Sqrt[b]*(-15*b^2*d*x^4 + a^2*(-8*d + 5*e*x^2) + a*b*(-25*d*x^2 + 3*e*x^4)) + 3*(-5*b*d + a*e)*x*(a + b*x^2)^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(7/2)*Sqrt[b]*x*(a + b*x^2)*Sqrt[(a + b*x^2)^2])","A",1
86,1,130,223,0.0702449,"\int \frac{d+e x^2}{x^3 \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}} \, dx","Integrate[(d + e*x^2)/(x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)),x]","\frac{a \left(a^2 \left(3 e x^2-2 d\right)+a b \left(2 e x^4-9 d x^2\right)-6 b^2 d x^4\right)+4 x^2 \log (x) \left(a+b x^2\right)^2 (a e-3 b d)+2 x^2 \left(a+b x^2\right)^2 (3 b d-a e) \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 d-a e}{4 a^2 \left(a+b x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\log (x) \left(a+b x^2\right) (3 b d-a e)}{a^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left(a+b x^2\right) (3 b d-a e) \log \left(a+b x^2\right)}{2 a^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{2 b d-a e}{2 a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d \left(a+b x^2\right)}{2 a^3 x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(a*(-6*b^2*d*x^4 + a^2*(-2*d + 3*e*x^2) + a*b*(-9*d*x^2 + 2*e*x^4)) + 4*(-3*b*d + a*e)*x^2*(a + b*x^2)^2*Log[x] + 2*(3*b*d - a*e)*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
87,1,160,400,0.2204647,"\int (f x)^m \left(d+e x^2\right) \left(a^2+2 a b x^2+b^2 x^4\right)^{5/2} \, dx","Integrate[(f*x)^m*(d + e*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]","\frac{x \sqrt{\left(a+b x^2\right)^2} (f x)^m \left(\frac{a^5 d}{m+1}+\frac{a^4 x^2 (a e+5 b d)}{m+3}+\frac{5 a^3 b x^4 (a e+2 b d)}{m+5}+\frac{10 a^2 b^2 x^6 (a e+b d)}{m+7}+\frac{b^4 x^{10} (5 a e+b d)}{m+11}+\frac{5 a b^3 x^8 (2 a e+b d)}{m+9}+\frac{b^5 e x^{12}}{m+13}\right)}{a+b x^2}","\frac{10 a^2 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+7} (a e+b d)}{f^7 (m+7) \left(a+b x^2\right)}+\frac{b^5 e \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+13}}{f^{13} (m+13) \left(a+b x^2\right)}+\frac{b^4 \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+11} (5 a e+b d)}{f^{11} (m+11) \left(a+b x^2\right)}+\frac{5 a b^3 \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+9} (2 a e+b d)}{f^9 (m+9) \left(a+b x^2\right)}+\frac{a^5 d \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+1}}{f (m+1) \left(a+b x^2\right)}+\frac{a^4 \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+3} (a e+5 b d)}{f^3 (m+3) \left(a+b x^2\right)}+\frac{5 a^3 b \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+5} (a e+2 b d)}{f^5 (m+5) \left(a+b x^2\right)}",1,"(x*(f*x)^m*Sqrt[(a + b*x^2)^2]*((a^5*d)/(1 + m) + (a^4*(5*b*d + a*e)*x^2)/(3 + m) + (5*a^3*b*(2*b*d + a*e)*x^4)/(5 + m) + (10*a^2*b^2*(b*d + a*e)*x^6)/(7 + m) + (5*a*b^3*(b*d + 2*a*e)*x^8)/(9 + m) + (b^4*(b*d + 5*a*e)*x^10)/(11 + m) + (b^5*e*x^12)/(13 + m)))/(a + b*x^2)","A",1
88,1,112,276,0.1103672,"\int (f x)^m \left(d+e x^2\right) \left(a^2+2 a b x^2+b^2 x^4\right)^{3/2} \, dx","Integrate[(f*x)^m*(d + e*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","\frac{x \left(\left(a+b x^2\right)^2\right)^{3/2} (f x)^m \left(\frac{a^3 d}{m+1}+\frac{a^2 x^2 (a e+3 b d)}{m+3}+\frac{b^2 x^6 (3 a e+b d)}{m+7}+\frac{3 a b x^4 (a e+b d)}{m+5}+\frac{b^3 e x^8}{m+9}\right)}{\left(a+b x^2\right)^3}","\frac{b^2 \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+7} (3 a e+b d)}{f^7 (m+7) \left(a+b x^2\right)}+\frac{3 a b \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+5} (a e+b d)}{f^5 (m+5) \left(a+b x^2\right)}+\frac{a^2 \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+3} (a e+3 b d)}{f^3 (m+3) \left(a+b x^2\right)}+\frac{b^3 e \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+9}}{f^9 (m+9) \left(a+b x^2\right)}+\frac{a^3 d \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+1}}{f (m+1) \left(a+b x^2\right)}",1,"(x*(f*x)^m*((a + b*x^2)^2)^(3/2)*((a^3*d)/(1 + m) + (a^2*(3*b*d + a*e)*x^2)/(3 + m) + (3*a*b*(b*d + a*e)*x^4)/(5 + m) + (b^2*(b*d + 3*a*e)*x^6)/(7 + m) + (b^3*e*x^8)/(9 + m)))/(a + b*x^2)^3","A",1
89,1,86,153,0.0550514,"\int (f x)^m \left(d+e x^2\right) \sqrt{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[(f*x)^m*(d + e*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]","\frac{x \sqrt{\left(a+b x^2\right)^2} (f x)^m \left(a (m+5) \left(d (m+3)+e (m+1) x^2\right)+b (m+1) x^2 \left(d (m+5)+e (m+3) x^2\right)\right)}{(m+1) (m+3) (m+5) \left(a+b x^2\right)}","\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+3} (a e+b d)}{f^3 (m+3) \left(a+b x^2\right)}+\frac{a d \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+1}}{f (m+1) \left(a+b x^2\right)}+\frac{b e \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+5}}{f^5 (m+5) \left(a+b x^2\right)}",1,"(x*(f*x)^m*Sqrt[(a + b*x^2)^2]*(a*(5 + m)*(d*(3 + m) + e*(1 + m)*x^2) + b*(1 + m)*x^2*(d*(5 + m) + e*(3 + m)*x^2)))/((1 + m)*(3 + m)*(5 + m)*(a + b*x^2))","A",1
90,1,78,134,0.0665107,"\int \frac{(f x)^m \left(d+e x^2\right)}{\sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx","Integrate[((f*x)^m*(d + e*x^2))/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]","-\frac{x \left(a+b x^2\right) (f x)^m \left((a e-b d) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)-a e\right)}{a b (m+1) \sqrt{\left(a+b x^2\right)^2}}","\frac{\left(a+b x^2\right) (f x)^{m+1} (b d-a e) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a b f (m+1) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{e \left(a+b x^2\right) (f x)^{m+1}}{b f (m+1) \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"-((x*(f*x)^m*(a + b*x^2)*(-(a*e) + (-(b*d) + a*e)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)]))/(a*b*(1 + m)*Sqrt[(a + b*x^2)^2]))","A",1
91,1,101,154,0.0722475,"\int \frac{(f x)^m \left(d+e x^2\right)}{\left(a^2+2 a b x^2+b^2 x^4\right)^{3/2}} \, dx","Integrate[((f*x)^m*(d + e*x^2))/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]","\frac{x \left(a+b x^2\right) (f x)^m \left((b d-a e) \, _2F_1\left(3,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)+a e \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)\right)}{a^3 b (m+1) \sqrt{\left(a+b x^2\right)^2}}","\frac{(f x)^{m+1} (b d-a e)}{4 a b f \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) (f x)^{m+1} (a e (m+1)+b d (3-m)) \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{4 a^3 b f (m+1) \sqrt{a^2+2 a b x^2+b^2 x^4}}",1,"(x*(f*x)^m*(a + b*x^2)*(a*e*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)] + (b*d - a*e)*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)]))/(a^3*b*(1 + m)*Sqrt[(a + b*x^2)^2])","A",1
92,1,25,34,0.0080744,"\int x \left(a+b x^2\right) \left(a^2+2 a b x^2+b^2 x^4\right)^p \, dx","Integrate[x*(a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^p,x]","\frac{\left(\left(a+b x^2\right)^2\right)^{p+1}}{4 b (p+1)}","\frac{\left(a^2+2 a b x^2+b^2 x^4\right)^{p+1}}{4 b (p+1)}",1,"((a + b*x^2)^2)^(1 + p)/(4*b*(1 + p))","A",1
93,1,45,86,0.024021,"\int x^3 \left(a+b x^2\right) \left(a^2+2 a b x^2+b^2 x^4\right)^p \, dx","Integrate[x^3*(a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^p,x]","\frac{\left(\left(a+b x^2\right)^2\right)^{p+1} \left(2 b (p+1) x^2-a\right)}{4 b^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^2 (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}{4 b^2 (p+1)}",1,"(((a + b*x^2)^2)^(1 + p)*(-a + 2*b*(1 + p)*x^2))/(4*b^2*(1 + p)*(3 + 2*p))","A",1
94,1,68,128,0.0352621,"\int x^5 \left(a+b x^2\right) \left(a^2+2 a b x^2+b^2 x^4\right)^p \, dx","Integrate[x^5*(a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^p,x]","\frac{\left(\left(a+b x^2\right)^2\right)^{p+1} \left(a^2-2 a b (p+1) x^2+b^2 \left(2 p^2+5 p+3\right) x^4\right)}{4 b^3 (p+1) (p+2) (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^3 (p+2)}-\frac{a \left(a+b x^2\right)^3 \left(a^2+2 a b x^2+b^2 x^4\right)^p}{b^3 (2 p+3)}+\frac{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^3 (p+1)}",1,"(((a + b*x^2)^2)^(1 + p)*(a^2 - 2*a*b*(1 + p)*x^2 + b^2*(3 + 5*p + 2*p^2)*x^4))/(4*b^3*(1 + p)*(2 + p)*(3 + 2*p))","A",1
95,1,166,166,0.0496877,"\int x^3 \left(A+B x^2\right) \left(a+b x^2+c x^4\right)^3 \, dx","Integrate[x^3*(A + B*x^2)*(a + b*x^2 + c*x^4)^3,x]","\frac{1}{4} a^3 A x^4+\frac{1}{6} a^2 x^6 (a B+3 A b)+\frac{3}{14} c x^{14} \left(a B c+A b c+b^2 B\right)+\frac{3}{8} a x^8 \left(A \left(a c+b^2\right)+a b B\right)+\frac{1}{12} x^{12} \left(3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right)+\frac{1}{10} x^{10} \left(A \left(6 a b c+b^3\right)+3 a B \left(a c+b^2\right)\right)+\frac{1}{16} c^2 x^{16} (A c+3 b B)+\frac{1}{18} B c^3 x^{18}","\frac{1}{4} a^3 A x^4+\frac{1}{6} a^2 x^6 (a B+3 A b)+\frac{3}{14} c x^{14} \left(a B c+A b c+b^2 B\right)+\frac{3}{8} a x^8 \left(A \left(a c+b^2\right)+a b B\right)+\frac{1}{12} x^{12} \left(3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right)+\frac{1}{10} x^{10} \left(A \left(6 a b c+b^3\right)+3 a B \left(a c+b^2\right)\right)+\frac{1}{16} c^2 x^{16} (A c+3 b B)+\frac{1}{18} B c^3 x^{18}",1,"(a^3*A*x^4)/4 + (a^2*(3*A*b + a*B)*x^6)/6 + (3*a*(a*b*B + A*(b^2 + a*c))*x^8)/8 + ((3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^10)/10 + ((b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^12)/12 + (3*c*(b^2*B + A*b*c + a*B*c)*x^14)/14 + (c^2*(3*b*B + A*c)*x^16)/16 + (B*c^3*x^18)/18","A",1
96,1,166,166,0.0535258,"\int x^2 \left(A+B x^2\right) \left(a+b x^2+c x^4\right)^3 \, dx","Integrate[x^2*(A + B*x^2)*(a + b*x^2 + c*x^4)^3,x]","\frac{1}{3} a^3 A x^3+\frac{1}{5} a^2 x^5 (a B+3 A b)+\frac{3}{13} c x^{13} \left(a B c+A b c+b^2 B\right)+\frac{3}{7} a x^7 \left(A \left(a c+b^2\right)+a b B\right)+\frac{1}{11} x^{11} \left(3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right)+\frac{1}{9} x^9 \left(A \left(6 a b c+b^3\right)+3 a B \left(a c+b^2\right)\right)+\frac{1}{15} c^2 x^{15} (A c+3 b B)+\frac{1}{17} B c^3 x^{17}","\frac{1}{3} a^3 A x^3+\frac{1}{5} a^2 x^5 (a B+3 A b)+\frac{3}{13} c x^{13} \left(a B c+A b c+b^2 B\right)+\frac{3}{7} a x^7 \left(A \left(a c+b^2\right)+a b B\right)+\frac{1}{11} x^{11} \left(3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right)+\frac{1}{9} x^9 \left(A \left(6 a b c+b^3\right)+3 a B \left(a c+b^2\right)\right)+\frac{1}{15} c^2 x^{15} (A c+3 b B)+\frac{1}{17} B c^3 x^{17}",1,"(a^3*A*x^3)/3 + (a^2*(3*A*b + a*B)*x^5)/5 + (3*a*(a*b*B + A*(b^2 + a*c))*x^7)/7 + ((3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^9)/9 + ((b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^11)/11 + (3*c*(b^2*B + A*b*c + a*B*c)*x^13)/13 + (c^2*(3*b*B + A*c)*x^15)/15 + (B*c^3*x^17)/17","A",1
97,1,154,166,0.0559594,"\int x \left(A+B x^2\right) \left(a+b x^2+c x^4\right)^3 \, dx","Integrate[x*(A + B*x^2)*(a + b*x^2 + c*x^4)^3,x]","\frac{1}{560} x^2 \left(280 a^3 A+140 a^2 x^2 (a B+3 A b)+140 c x^{10} \left(a B c+A b c+b^2 B\right)+280 a x^4 \left(A \left(a c+b^2\right)+a b B\right)+56 x^8 \left(3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right)+70 x^6 \left(A \left(6 a b c+b^3\right)+3 a B \left(a c+b^2\right)\right)+40 c^2 x^{12} (A c+3 b B)+35 B c^3 x^{14}\right)","\frac{1}{2} a^3 A x^2+\frac{1}{4} a^2 x^4 (a B+3 A b)+\frac{1}{4} c x^{12} \left(a B c+A b c+b^2 B\right)+\frac{1}{2} a x^6 \left(A \left(a c+b^2\right)+a b B\right)+\frac{1}{10} x^{10} \left(3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right)+\frac{1}{8} x^8 \left(A \left(6 a b c+b^3\right)+3 a B \left(a c+b^2\right)\right)+\frac{1}{14} c^2 x^{14} (A c+3 b B)+\frac{1}{16} B c^3 x^{16}",1,"(x^2*(280*a^3*A + 140*a^2*(3*A*b + a*B)*x^2 + 280*a*(a*b*B + A*(b^2 + a*c))*x^4 + 70*(3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^6 + 56*(b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^8 + 140*c*(b^2*B + A*b*c + a*B*c)*x^10 + 40*c^2*(3*b*B + A*c)*x^12 + 35*B*c^3*x^14))/560","A",1
98,1,161,161,0.0458773,"\int \left(A+B x^2\right) \left(a+b x^2+c x^4\right)^3 \, dx","Integrate[(A + B*x^2)*(a + b*x^2 + c*x^4)^3,x]","a^3 A x+\frac{1}{3} a^2 x^3 (a B+3 A b)+\frac{3}{11} c x^{11} \left(a B c+A b c+b^2 B\right)+\frac{3}{5} a x^5 \left(A \left(a c+b^2\right)+a b B\right)+\frac{1}{9} x^9 \left(3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right)+\frac{1}{7} x^7 \left(A \left(6 a b c+b^3\right)+3 a B \left(a c+b^2\right)\right)+\frac{1}{13} c^2 x^{13} (A c+3 b B)+\frac{1}{15} B c^3 x^{15}","a^3 A x+\frac{1}{3} a^2 x^3 (a B+3 A b)+\frac{3}{11} c x^{11} \left(a B c+A b c+b^2 B\right)+\frac{3}{5} a x^5 \left(A \left(a c+b^2\right)+a b B\right)+\frac{1}{9} x^9 \left(3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right)+\frac{1}{7} x^7 \left(A \left(6 a b c+b^3\right)+3 a B \left(a c+b^2\right)\right)+\frac{1}{13} c^2 x^{13} (A c+3 b B)+\frac{1}{15} B c^3 x^{15}",1,"a^3*A*x + (a^2*(3*A*b + a*B)*x^3)/3 + (3*a*(a*b*B + A*(b^2 + a*c))*x^5)/5 + ((3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^7)/7 + ((b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^9)/9 + (3*c*(b^2*B + A*b*c + a*B*c)*x^11)/11 + (c^2*(3*b*B + A*c)*x^13)/13 + (B*c^3*x^15)/15","A",1
99,1,162,162,0.0564914,"\int \frac{\left(A+B x^2\right) \left(a+b x^2+c x^4\right)^3}{x} \, dx","Integrate[((A + B*x^2)*(a + b*x^2 + c*x^4)^3)/x,x]","a^3 A \log (x)+\frac{1}{2} a^2 x^2 (a B+3 A b)+\frac{3}{10} c x^{10} \left(a B c+A b c+b^2 B\right)+\frac{3}{4} a x^4 \left(A \left(a c+b^2\right)+a b B\right)+\frac{1}{8} x^8 \left(3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right)+\frac{1}{6} x^6 \left(A \left(6 a b c+b^3\right)+3 a B \left(a c+b^2\right)\right)+\frac{1}{12} c^2 x^{12} (A c+3 b B)+\frac{1}{14} B c^3 x^{14}","a^3 A \log (x)+\frac{1}{2} a^2 x^2 (a B+3 A b)+\frac{3}{10} c x^{10} \left(a B c+A b c+b^2 B\right)+\frac{3}{4} a x^4 \left(A \left(a c+b^2\right)+a b B\right)+\frac{1}{8} x^8 \left(3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right)+\frac{1}{6} x^6 \left(A \left(6 a b c+b^3\right)+3 a B \left(a c+b^2\right)\right)+\frac{1}{12} c^2 x^{12} (A c+3 b B)+\frac{1}{14} B c^3 x^{14}",1,"(a^2*(3*A*b + a*B)*x^2)/2 + (3*a*(a*b*B + A*(b^2 + a*c))*x^4)/4 + ((3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^6)/6 + ((b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^8)/8 + (3*c*(b^2*B + A*b*c + a*B*c)*x^10)/10 + (c^2*(3*b*B + A*c)*x^12)/12 + (B*c^3*x^14)/14 + a^3*A*Log[x]","A",1
100,1,156,156,0.0818184,"\int \frac{\left(A+B x^2\right) \left(a+b x^2+c x^4\right)^3}{x^2} \, dx","Integrate[((A + B*x^2)*(a + b*x^2 + c*x^4)^3)/x^2,x]","-\frac{a^3 A}{x}+a^2 x (a B+3 A b)+\frac{1}{3} c x^9 \left(a B c+A b c+b^2 B\right)+a x^3 \left(A \left(a c+b^2\right)+a b B\right)+\frac{1}{7} x^7 \left(3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right)+\frac{1}{5} x^5 \left(A \left(6 a b c+b^3\right)+3 a B \left(a c+b^2\right)\right)+\frac{1}{11} c^2 x^{11} (A c+3 b B)+\frac{1}{13} B c^3 x^{13}","-\frac{a^3 A}{x}+a^2 x (a B+3 A b)+\frac{1}{3} c x^9 \left(a B c+A b c+b^2 B\right)+a x^3 \left(A \left(a c+b^2\right)+a b B\right)+\frac{1}{7} x^7 \left(3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right)+\frac{1}{5} x^5 \left(A \left(6 a b c+b^3\right)+3 a B \left(a c+b^2\right)\right)+\frac{1}{11} c^2 x^{11} (A c+3 b B)+\frac{1}{13} B c^3 x^{13}",1,"-((a^3*A)/x) + a^2*(3*A*b + a*B)*x + a*(a*b*B + A*(b^2 + a*c))*x^3 + ((3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^5)/5 + ((b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^7)/7 + (c*(b^2*B + A*b*c + a*B*c)*x^9)/3 + (c^2*(3*b*B + A*c)*x^11)/11 + (B*c^3*x^13)/13","A",1
101,1,162,162,0.0709292,"\int \frac{\left(A+B x^2\right) \left(a+b x^2+c x^4\right)^3}{x^3} \, dx","Integrate[((A + B*x^2)*(a + b*x^2 + c*x^4)^3)/x^3,x]","-\frac{a^3 A}{2 x^2}+a^2 \log (x) (a B+3 A b)+\frac{3}{8} c x^8 \left(a B c+A b c+b^2 B\right)+\frac{3}{2} a x^2 \left(A \left(a c+b^2\right)+a b B\right)+\frac{1}{6} x^6 \left(3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right)+\frac{1}{4} x^4 \left(A \left(6 a b c+b^3\right)+3 a B \left(a c+b^2\right)\right)+\frac{1}{10} c^2 x^{10} (A c+3 b B)+\frac{1}{12} B c^3 x^{12}","-\frac{a^3 A}{2 x^2}+a^2 \log (x) (a B+3 A b)+\frac{3}{8} c x^8 \left(a B c+A b c+b^2 B\right)+\frac{3}{2} a x^2 \left(A \left(a c+b^2\right)+a b B\right)+\frac{1}{6} x^6 \left(3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right)+\frac{1}{4} x^4 \left(A \left(6 a b c+b^3\right)+3 a B \left(a c+b^2\right)\right)+\frac{1}{10} c^2 x^{10} (A c+3 b B)+\frac{1}{12} B c^3 x^{12}",1,"-1/2*(a^3*A)/x^2 + (3*a*(a*b*B + A*(b^2 + a*c))*x^2)/2 + ((3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^4)/4 + ((b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^6)/6 + (3*c*(b^2*B + A*b*c + a*B*c)*x^8)/8 + (c^2*(3*b*B + A*c)*x^10)/10 + (B*c^3*x^12)/12 + a^2*(3*A*b + a*B)*Log[x]","A",1
102,1,126,133,0.0635441,"\int \frac{x^5 \left(A+B x^2\right)}{a+b x^2+c x^4} \, dx","Integrate[(x^5*(A + B*x^2))/(a + b*x^2 + c*x^4),x]","\frac{\left(-a B c-A b c+b^2 B\right) \log \left(a+b x^2+c x^4\right)+\frac{2 \left(-2 a A c^2+3 a b B c+A b^2 c+b^3 (-B)\right) \tan ^{-1}\left(\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+2 c x^2 (A c-b B)+B c^2 x^4}{4 c^3}","\frac{\left(-a B c-A b c+b^2 B\right) \log \left(a+b x^2+c x^4\right)}{4 c^3}+\frac{\left(2 a A c^2-3 a b B c-A b^2 c+b^3 B\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{x^2 (b B-A c)}{2 c^2}+\frac{B x^4}{4 c}",1,"(2*c*(-(b*B) + A*c)*x^2 + B*c^2*x^4 + (2*(-(b^3*B) + A*b^2*c + 3*a*b*B*c - 2*a*A*c^2)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + (b^2*B - A*b*c - a*B*c)*Log[a + b*x^2 + c*x^4])/(4*c^3)","A",1
103,1,93,97,0.0681578,"\int \frac{x^3 \left(A+B x^2\right)}{a+b x^2+c x^4} \, dx","Integrate[(x^3*(A + B*x^2))/(a + b*x^2 + c*x^4),x]","\frac{\frac{2 \left(-2 a B c-A b c+b^2 B\right) \tan ^{-1}\left(\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+(A c-b B) \log \left(a+b x^2+c x^4\right)+2 B c x^2}{4 c^2}","-\frac{\left(-2 a B c-A b c+b^2 B\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 B-A c) \log \left(a+b x^2+c x^4\right)}{4 c^2}+\frac{B x^2}{2 c}",1,"(2*B*c*x^2 + (2*(b^2*B - A*b*c - 2*a*B*c)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + (-(b*B) + A*c)*Log[a + b*x^2 + c*x^4])/(4*c^2)","A",1
104,1,71,71,0.0460646,"\int \frac{x \left(A+B x^2\right)}{a+b x^2+c x^4} \, dx","Integrate[(x*(A + B*x^2))/(a + b*x^2 + c*x^4),x]","\frac{B \log \left(a+b x^2+c x^4\right)-\frac{2 (b B-2 A c) \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 B-2 A c) \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{B \log \left(a+b x^2+c x^4\right)}{4 c}",1,"((-2*(b*B - 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)","A",1
105,1,128,78,0.1028146,"\int \frac{A+B x^2}{x \left(a+b x^2+c x^4\right)} \, dx","Integrate[(A + B*x^2)/(x*(a + b*x^2 + c*x^4)),x]","\frac{-\left(A \left(\sqrt{b^2-4 a c}+b\right)-2 a B\right) \log \left(-\sqrt{b^2-4 a c}+b+2 c x^2\right)+\left(A \left(b-\sqrt{b^2-4 a c}\right)-2 a B\right) \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)+4 A \log (x) \sqrt{b^2-4 a c}}{4 a \sqrt{b^2-4 a c}}","\frac{(A b-2 a 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{A \log \left(a+b x^2+c x^4\right)}{4 a}+\frac{A \log (x)}{a}",1,"(4*A*Sqrt[b^2 - 4*a*c]*Log[x] - (-2*a*B + A*(b + Sqrt[b^2 - 4*a*c]))*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2] + (-2*a*B + A*(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
106,1,186,112,0.2180552,"\int \frac{A+B x^2}{x^3 \left(a+b x^2+c x^4\right)} \, dx","Integrate[(A + B*x^2)/(x^3*(a + b*x^2 + c*x^4)),x]","\frac{\frac{\left(A \left(b \sqrt{b^2-4 a c}-2 a c+b^2\right)-a B \left(\sqrt{b^2-4 a c}+b\right)\right) \log \left(-\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\sqrt{b^2-4 a c}}+\frac{\left(A \left(b \sqrt{b^2-4 a c}+2 a c-b^2\right)+a B \left(b-\sqrt{b^2-4 a c}\right)\right) \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\sqrt{b^2-4 a c}}+4 \log (x) (a B-A b)-\frac{2 a A}{x^2}}{4 a^2}","-\frac{\left(-2 a A c-a b B+A b^2\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{(A b-a B) \log \left(a+b x^2+c x^4\right)}{4 a^2}-\frac{\log (x) (A b-a B)}{a^2}-\frac{A}{2 a x^2}",1,"((-2*a*A)/x^2 + 4*(-(A*b) + a*B)*Log[x] + ((-(a*B*(b + Sqrt[b^2 - 4*a*c])) + A*(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] + ((a*B*(b - Sqrt[b^2 - 4*a*c]) + A*(-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
107,1,327,261,0.4047617,"\int \frac{x^4 \left(A+B x^2\right)}{a+b x^2+c x^4} \, dx","Integrate[(x^4*(A + B*x^2))/(a + b*x^2 + c*x^4),x]","\frac{\left(-A b c \sqrt{b^2-4 a c}-2 a A c^2+b^2 B \sqrt{b^2-4 a c}-a B c \sqrt{b^2-4 a c}+3 a b B c+A b^2 c+b^3 (-B)\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(-A b c \sqrt{b^2-4 a c}+2 a A c^2+b^2 B \sqrt{b^2-4 a c}-a B c \sqrt{b^2-4 a c}-3 a b B c-A b^2 c+b^3 B\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{x (A c-b B)}{c^2}+\frac{B x^3}{3 c}","\frac{\left(-\frac{2 a A c^2-3 a b B c-A b^2 c+b^3 B}{\sqrt{b^2-4 a c}}-a B c-A b c+b^2 B\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{2 a A c^2-3 a b B c-A b^2 c+b^3 B}{\sqrt{b^2-4 a c}}-a B c-A b c+b^2 B\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{x (b B-A c)}{c^2}+\frac{B x^3}{3 c}",1,"((-(b*B) + A*c)*x)/c^2 + (B*x^3)/(3*c) + ((-(b^3*B) + A*b^2*c + 3*a*b*B*c - 2*a*A*c^2 + b^2*B*Sqrt[b^2 - 4*a*c] - A*b*c*Sqrt[b^2 - 4*a*c] - a*B*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*B - A*b^2*c - 3*a*b*B*c + 2*a*A*c^2 + b^2*B*Sqrt[b^2 - 4*a*c] - A*b*c*Sqrt[b^2 - 4*a*c] - a*B*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
108,1,251,208,0.1634238,"\int \frac{x^2 \left(A+B x^2\right)}{a+b x^2+c x^4} \, dx","Integrate[(x^2*(A + B*x^2))/(a + b*x^2 + c*x^4),x]","-\frac{\left(-A c \sqrt{b^2-4 a c}+b B \sqrt{b^2-4 a c}+2 a B c+A b c+b^2 (-B)\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(-A c \sqrt{b^2-4 a c}+b B \sqrt{b^2-4 a c}-2 a B c-A b c+b^2 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{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{B x}{c}","-\frac{\left(-\frac{-2 a B c-A b c+b^2 B}{\sqrt{b^2-4 a c}}-A c+b B\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{-2 a B c-A b c+b^2 B}{\sqrt{b^2-4 a c}}-A c+b 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{B x}{c}",1,"(B*x)/c - ((-(b^2*B) + A*b*c + 2*a*B*c + b*B*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^(3/2)*Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - ((b^2*B - A*b*c - 2*a*B*c + b*B*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^(3/2)*Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]])","A",1
109,1,173,172,0.0944532,"\int \frac{A+B x^2}{a+b x^2+c x^4} \, dx","Integrate[(A + B*x^2)/(a + b*x^2 + c*x^4),x]","\frac{\frac{\left(B \sqrt{b^2-4 a c}+2 A c-b B\right) \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{\left(B \sqrt{b^2-4 a c}-2 A c+b B\right) \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}}}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}","\frac{\left(B-\frac{b B-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} \sqrt{c} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left(\frac{b B-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} \sqrt{c} \sqrt{\sqrt{b^2-4 a c}+b}}",1,"(((-(b*B) + 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 - Sqrt[b^2 - 4*a*c]] + ((b*B - 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 + Sqrt[b^2 - 4*a*c]])/(Sqrt[2]*Sqrt[c]*Sqrt[b^2 - 4*a*c])","A",1
110,1,206,189,0.2913779,"\int \frac{A+B x^2}{x^2 \left(a+b x^2+c x^4\right)} \, dx","Integrate[(A + B*x^2)/(x^2*(a + b*x^2 + c*x^4)),x]","-\frac{\frac{\sqrt{2} \sqrt{c} \left(A \left(\sqrt{b^2-4 a c}+b\right)-2 a 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(A \left(\sqrt{b^2-4 a c}-b\right)+2 a 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 A}{x}}{2 a}","-\frac{\sqrt{c} \left(\frac{A b-2 a B}{\sqrt{b^2-4 a c}}+A\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(A-\frac{A b-2 a 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{A}{a x}",1,"-1/2*((2*A)/x + (Sqrt[2]*Sqrt[c]*(-2*a*B + A*(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]*(2*a*B + A*(-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
111,1,267,271,0.3162875,"\int \frac{A+B x^2}{x^4 \left(a+b x^2+c x^4\right)} \, dx","Integrate[(A + B*x^2)/(x^4*(a + b*x^2 + c*x^4)),x]","\frac{-\frac{3 \sqrt{2} \sqrt{c} \left(a B \left(\sqrt{b^2-4 a c}+b\right)-A \left(b \sqrt{b^2-4 a c}-2 a c+b^2\right)\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(A \left(b \sqrt{b^2-4 a c}+2 a c-b^2\right)+a B \left(b-\sqrt{b^2-4 a c}\right)\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{6 A b-6 a B}{x}-\frac{2 a A}{x^3}}{6 a^2}","-\frac{\sqrt{c} \left(a B \left(\sqrt{b^2-4 a c}+b\right)-A \left(b \sqrt{b^2-4 a c}-2 a c+b^2\right)\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^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left(a B \left(b-\sqrt{b^2-4 a c}\right)-A \left(-b \sqrt{b^2-4 a c}-2 a c+b^2\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} a^2 \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{A b-a B}{a^2 x}-\frac{A}{3 a x^3}",1,"((-2*a*A)/x^3 + (6*A*b - 6*a*B)/x - (3*Sqrt[2]*Sqrt[c]*(a*B*(b + Sqrt[b^2 - 4*a*c]) - A*(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]*(a*B*(b - Sqrt[b^2 - 4*a*c]) + A*(-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
112,1,208,212,0.2815416,"\int \frac{x^7 \left(A+B x^2\right)}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(x^7*(A + B*x^2))/(a + b*x^2 + c*x^4)^2,x]","\frac{-\frac{2 \left(a^2 c \left(2 c \left(A+B x^2\right)-3 b B\right)+a b \left(-b c \left(A+4 B x^2\right)+3 A c^2 x^2+b^2 B\right)+b^3 x^2 (b B-A c)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-\frac{2 \left(12 a^2 B c^2+6 a A b c^2-12 a b^2 B c-A b^3 c+2 b^4 B\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}}+(A c-2 b B) \log \left(a+b x^2+c x^4\right)+2 B c x^2}{4 c^3}","-\frac{\left(12 a^2 B c^2+6 a A b c^2-12 a b^2 B c-A b^3 c+2 b^4 B\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)^{3/2}}+\frac{x^2 \left(-6 a B c-A b c+2 b^2 B\right)}{2 c^2 \left(b^2-4 a c\right)}-\frac{x^4 \left(x^2 \left(-2 a B c-A b c+b^2 B\right)+a (b B-2 A c)\right)}{2 c \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-\frac{(2 b B-A c) \log \left(a+b x^2+c x^4\right)}{4 c^3}",1,"(2*B*c*x^2 - (2*(b^3*(b*B - A*c)*x^2 + a^2*c*(-3*b*B + 2*c*(A + B*x^2)) + a*b*(b^2*B + 3*A*c^2*x^2 - b*c*(A + 4*B*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - (2*(2*b^4*B - A*b^3*c - 12*a*b^2*B*c + 6*a*A*b*c^2 + 12*a^2*B*c^2)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + (-2*b*B + A*c)*Log[a + b*x^2 + c*x^4])/(4*c^3)","A",1
113,1,160,147,0.185895,"\int \frac{x^5 \left(A+B x^2\right)}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(x^5*(A + B*x^2))/(a + b*x^2 + c*x^4)^2,x]","\frac{-\frac{2 \left(2 a^2 B c+a \left(b c \left(A+3 B x^2\right)-2 A c^2 x^2+b^2 (-B)\right)+b^2 x^2 (A c-b B)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{2 \left(4 a A c^2-6 a b B c+b^3 B\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}}+B \log \left(a+b x^2+c x^4\right)}{4 c^2}","-\frac{x^2 \left(x^2 \left(-2 a B c-A b c+b^2 B\right)+a (b B-2 A c)\right)}{2 c \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\left(4 a A c^2-6 a b B c+b^3 B\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 \log \left(a+b x^2+c x^4\right)}{4 c^2}",1,"((-2*(2*a^2*B*c + b^2*(-(b*B) + A*c)*x^2 + a*(-(b^2*B) - 2*A*c^2*x^2 + b*c*(A + 3*B*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (2*(b^3*B - 6*a*b*B*c + 4*a*A*c^2)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + B*Log[a + b*x^2 + c*x^4])/(4*c^2)","A",1
114,1,111,107,0.0802663,"\int \frac{x^3 \left(A+B x^2\right)}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(x^3*(A + B*x^2))/(a + b*x^2 + c*x^4)^2,x]","\frac{-2 a c \left(A+B x^2\right)+a b B+b x^2 (b B-A c)}{2 c \left(4 a c-b^2\right) \left(a+b x^2+c x^4\right)}-\frac{(A b-2 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)^{3/2}}","-\frac{(A b-2 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)^{3/2}}-\frac{x^2 \left(-2 a B c-A b c+b^2 B\right)+a (b B-2 A c)}{2 c \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"(a*b*B + b*(b*B - A*c)*x^2 - 2*a*c*(A + B*x^2))/(2*c*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)) - ((A*b - 2*a*B)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2)","A",1
115,1,101,94,0.0740893,"\int \frac{x \left(A+B x^2\right)}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(x*(A + B*x^2))/(a + b*x^2 + c*x^4)^2,x]","\frac{\frac{2 (b B-2 A 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 \left(2 a+b x^2\right)-A \left(b+2 c x^2\right)}{a+b x^2+c x^4}}{2 \left(b^2-4 a c\right)}","-\frac{(b B-2 A 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{-2 a B-\left(x^2 (b B-2 A c)\right)+A b}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"((B*(2*a + b*x^2) - A*(b + 2*c*x^2))/(a + b*x^2 + c*x^4) + (2*(b*B - 2*A*c)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(2*(b^2 - 4*a*c))","A",1
116,1,243,150,0.3299037,"\int \frac{A+B x^2}{x \left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(A + B*x^2)/(x*(a + b*x^2 + c*x^4)^2),x]","\frac{-\frac{\left(4 a^2 B c+A \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)\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(4 a^2 B c+A \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)\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{2 a \left(a B \left(b+2 c x^2\right)-A \left(-2 a c+b^2+b c x^2\right)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+4 A \log (x)}{4 a^2}","\frac{\left(4 a^2 B c+A \left(b^3-6 a b c\right)\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{A \log \left(a+b x^2+c x^4\right)}{4 a^2}+\frac{A \log (x)}{a^2}-\frac{-A \left(b^2-2 a c\right)-\left(c x^2 (A b-2 a B)\right)+a b B}{2 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"((-2*a*(a*B*(b + 2*c*x^2) - A*(b^2 - 2*a*c + b*c*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + 4*A*Log[x] - ((4*a^2*B*c + A*(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*B*c + A*(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
117,1,379,223,0.5637121,"\int \frac{A+B x^2}{x^3 \left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(A + B*x^2)/(x^3*(a + b*x^2 + c*x^4)^2),x]","\frac{\frac{\left(2 A \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)+a B \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)\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(2 A \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)+a B \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)\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{2 a \left(A \left(-3 a b c-2 a c^2 x^2+b^3+b^2 c x^2\right)+a B \left(2 a c-b^2-b c x^2\right)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+4 \log (x) (a B-2 A b)-\frac{2 a A}{x^2}}{4 a^3}","\frac{(2 A b-a B) \log \left(a+b x^2+c x^4\right)}{4 a^3}-\frac{\log (x) (2 A b-a B)}{a^3}-\frac{-6 a A c-a b B+2 A b^2}{2 a^2 x^2 \left(b^2-4 a c\right)}+\frac{\left(a b B \left(b^2-6 a c\right)-2 A \left(6 a^2 c^2-6 a b^2 c+b^4\right)\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)^{3/2}}-\frac{-A \left(b^2-2 a c\right)-\left(c x^2 (A b-2 a B)\right)+a b B}{2 a x^2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"((-2*a*A)/x^2 - (2*a*(a*B*(-b^2 + 2*a*c - b*c*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*(-2*A*b + a*B)*Log[x] + ((a*B*(-b^3 + 6*a*b*c - b^2*Sqrt[b^2 - 4*a*c] + 4*a*c*Sqrt[b^2 - 4*a*c]) + 2*A*(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) + ((a*B*(b^3 - 6*a*b*c - b^2*Sqrt[b^2 - 4*a*c] + 4*a*c*Sqrt[b^2 - 4*a*c]) + 2*A*(-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))/(4*a^3)","A",1
118,1,455,425,1.2018641,"\int \frac{x^6 \left(A+B x^2\right)}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(x^6*(A + B*x^2))/(a + b*x^2 + c*x^4)^2,x]","\frac{\frac{2 \sqrt{c} x \left(-2 a^2 B c+a \left(-b c \left(A+3 B x^2\right)+2 A c^2 x^2+b^2 B\right)+b^2 x^2 (b B-A c)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-\frac{\sqrt{2} \left(2 a c^2 \left(3 A \sqrt{b^2-4 a c}-10 a B\right)+b^2 c \left(19 a B-A \sqrt{b^2-4 a c}\right)-a b c \left(13 B \sqrt{b^2-4 a c}+8 A c\right)+b^3 \left(3 B \sqrt{b^2-4 a c}+A c\right)-3 b^4 B\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(2 a c^2 \left(3 A \sqrt{b^2-4 a c}+10 a B\right)-b^2 c \left(A \sqrt{b^2-4 a c}+19 a B\right)+a b c \left(8 A c-13 B \sqrt{b^2-4 a c}\right)+b^3 \left(3 B \sqrt{b^2-4 a c}-A c\right)+3 b^4 B\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 B \sqrt{c} x}{4 c^{5/2}}","-\frac{\left(-\frac{20 a^2 B c^2+8 a A b c^2-19 a b^2 B c-A b^3 c+3 b^4 B}{\sqrt{b^2-4 a c}}+6 a A c^2-13 a b B c-A b^2 c+3 b^3 B\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 B c^2+8 a A b c^2-19 a b^2 B c-A b^3 c+3 b^4 B}{\sqrt{b^2-4 a c}}+6 a A c^2-13 a b B c-A b^2 c+3 b^3 B\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(-10 a B c-A b c+3 b^2 B\right)}{2 c^2 \left(b^2-4 a c\right)}-\frac{x^3 (b B-2 A c)}{2 c \left(b^2-4 a c\right)}-\frac{x^5 \left(-2 a B-\left(x^2 (b B-2 A c)\right)+A b\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"(4*B*Sqrt[c]*x + (2*Sqrt[c]*x*(-2*a^2*B*c + b^2*(b*B - A*c)*x^2 + a*(b^2*B + 2*A*c^2*x^2 - b*c*(A + 3*B*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - (Sqrt[2]*(-3*b^4*B + b^2*c*(19*a*B - A*Sqrt[b^2 - 4*a*c]) + 2*a*c^2*(-10*a*B + 3*A*Sqrt[b^2 - 4*a*c]) + b^3*(A*c + 3*B*Sqrt[b^2 - 4*a*c]) - a*b*c*(8*A*c + 13*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]*(3*b^4*B - b^2*c*(19*a*B + A*Sqrt[b^2 - 4*a*c]) + 2*a*c^2*(10*a*B + 3*A*Sqrt[b^2 - 4*a*c]) + a*b*c*(8*A*c - 13*B*Sqrt[b^2 - 4*a*c]) + b^3*(-(A*c) + 3*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*c^(5/2))","A",1
119,1,362,336,0.8526184,"\int \frac{x^4 \left(A+B x^2\right)}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(x^4*(A + B*x^2))/(a + b*x^2 + c*x^4)^2,x]","\frac{\frac{2 \sqrt{c} \left(2 a c x \left(A+B x^2\right)-a b B x+b x^3 (A c-b B)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \left(b^2 \left(B \sqrt{b^2-4 a c}-A c\right)+b c \left(A \sqrt{b^2-4 a c}+8 a B\right)-2 a c \left(3 B \sqrt{b^2-4 a c}+2 A c\right)+b^3 (-B)\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 \left(B \sqrt{b^2-4 a c}+A c\right)+b \left(A c \sqrt{b^2-4 a c}-8 a B c\right)+2 a c \left(2 A c-3 B \sqrt{b^2-4 a c}\right)+b^3 B\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{x^3 \left(-2 a B-\left(x^2 (b B-2 A c)\right)+A b\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-\frac{x (b B-2 A c)}{2 c \left(b^2-4 a c\right)}+\frac{\left(-\frac{4 a A c^2-8 a b B c+A b^2 c+b^3 B}{\sqrt{b^2-4 a c}}-6 a B c+A b c+b^2 B\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{4 a A c^2-8 a b B c+A b^2 c+b^3 B}{\sqrt{b^2-4 a c}}-6 a B c+A b c+b^2 B\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}}",1,"((2*Sqrt[c]*(-(a*b*B*x) + b*(-(b*B) + A*c)*x^3 + 2*a*c*x*(A + B*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*(-(b^3*B) + b*c*(8*a*B + A*Sqrt[b^2 - 4*a*c]) + b^2*(-(A*c) + B*Sqrt[b^2 - 4*a*c]) - 2*a*c*(2*A*c + 3*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]*(b^3*B + 2*a*c*(2*A*c - 3*B*Sqrt[b^2 - 4*a*c]) + b^2*(A*c + B*Sqrt[b^2 - 4*a*c]) + b*(-8*a*B*c + 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
120,1,298,276,0.6637515,"\int \frac{x^2 \left(A+B x^2\right)}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(x^2*(A + B*x^2))/(a + b*x^2 + c*x^4)^2,x]","\frac{1}{4} \left(\frac{2 x \left(B \left(2 a+b x^2\right)-A \left(b+2 c x^2\right)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \left(-2 A c \sqrt{b^2-4 a c}+b B \sqrt{b^2-4 a c}-4 a B c+4 A b c+b^2 (-B)\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(-2 A c \sqrt{b^2-4 a c}+b B \sqrt{b^2-4 a c}+4 a B c-4 A b c+b^2 B\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-\left(x^2 (b B-2 A c)\right)+A b\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\left(-\frac{4 a B c-4 A b c+b^2 B}{\sqrt{b^2-4 a c}}-2 A c+b B\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(\frac{4 a B c-4 A b c+b^2 B}{\sqrt{b^2-4 a c}}-2 A c+b B\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) \sqrt{\sqrt{b^2-4 a c}+b}}",1,"((2*x*(B*(2*a + b*x^2) - A*(b + 2*c*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*(-(b^2*B) + 4*A*b*c - 4*a*B*c + b*B*Sqrt[b^2 - 4*a*c] - 2*A*c*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*B - 4*A*b*c + 4*a*B*c + b*B*Sqrt[b^2 - 4*a*c] - 2*A*c*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
121,1,304,293,0.7857346,"\int \frac{A+B x^2}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(A + B*x^2)/(a + b*x^2 + c*x^4)^2,x]","\frac{\frac{2 x \left(A \left(-2 a c+b^2+b c x^2\right)-a B \left(b+2 c x^2\right)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \sqrt{c} \left(A \left(b \sqrt{b^2-4 a c}-12 a c+b^2\right)-2 a B \left(\sqrt{b^2-4 a c}-2 b\right)\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(A \left(b \sqrt{b^2-4 a c}+12 a c-b^2\right)-2 a B \left(\sqrt{b^2-4 a c}+2 b\right)\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(c x^2 (A b-2 a B)-2 a A c-a b B+A b^2\right)}{2 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{c} \left(\frac{A \left(b^2-12 a c\right)+4 a b B}{\sqrt{b^2-4 a c}}-2 a B+A b\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) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left(-\frac{-12 a A c+4 a b B+A b^2}{\sqrt{b^2-4 a c}}-2 a B+A b\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) \sqrt{\sqrt{b^2-4 a c}+b}}",1,"((2*x*(-(a*B*(b + 2*c*x^2)) + A*(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]*(-2*a*B*(-2*b + Sqrt[b^2 - 4*a*c]) + A*(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]*(-2*a*B*(2*b + Sqrt[b^2 - 4*a*c]) + A*(-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
122,1,382,389,1.0389025,"\int \frac{A+B x^2}{x^2 \left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(A + B*x^2)/(x^2*(a + b*x^2 + c*x^4)^2),x]","\frac{\frac{2 x \left(a B \left(-2 a c+b^2+b c x^2\right)-A \left(-3 a b c-2 a c^2 x^2+b^3+b^2 c x^2\right)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \sqrt{c} \left(A \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)+a B \left(b \sqrt{b^2-4 a c}-12 a c+b^2\right)\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(A \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)+a B \left(b \sqrt{b^2-4 a c}+12 a c-b^2\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{4 A}{x}}{4 a^2}","-\frac{-10 a A c-a b B+3 A b^2}{2 a^2 x \left(b^2-4 a c\right)}+\frac{\sqrt{c} \left(a B \left(b \sqrt{b^2-4 a c}-12 a c+b^2\right)-A \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)\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(\frac{a B \left(b^2-12 a c\right)-A \left(3 b^3-16 a b c\right)}{\sqrt{b^2-4 a c}}-10 a A c-a b B+3 A 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^2 \left(b^2-4 a c\right) \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{-A \left(b^2-2 a c\right)-\left(c x^2 (A b-2 a B)\right)+a b B}{2 a x \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"((-4*A)/x + (2*x*(a*B*(b^2 - 2*a*c + b*c*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)) + (Sqrt[2]*Sqrt[c]*(a*B*(b^2 - 12*a*c + b*Sqrt[b^2 - 4*a*c]) + A*(-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]*(a*B*(-b^2 + 12*a*c + b*Sqrt[b^2 - 4*a*c]) + A*(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
123,1,487,522,1.19917,"\int \frac{A+B x^2}{x^4 \left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(A + B*x^2)/(x^4*(a + b*x^2 + c*x^4)^2),x]","\frac{\frac{6 x \left(A \left(2 a^2 c^2-4 a b^2 c-3 a b c^2 x^2+b^4+b^3 c x^2\right)+a B \left(3 a b c+2 a c^2 x^2-b^3-b^2 c x^2\right)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{3 \sqrt{2} \sqrt{c} \left(A \left(28 a^2 c^2-29 a b^2 c-19 a b c \sqrt{b^2-4 a c}+5 b^3 \sqrt{b^2-4 a c}+5 b^4\right)+a B \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)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{3 \sqrt{2} \sqrt{c} \left(A \left(28 a^2 c^2-29 a b^2 c+19 a b c \sqrt{b^2-4 a c}-5 b^3 \sqrt{b^2-4 a c}+5 b^4\right)+a B \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)\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{24 A b-12 a B}{x}-\frac{4 a A}{x^3}}{12 a^3}","-\frac{a B \left(3 b^2-10 a c\right)-A \left(5 b^3-19 a b c\right)}{2 a^3 x \left(b^2-4 a c\right)}-\frac{-14 a A c-3 a b B+5 A b^2}{6 a^2 x^3 \left(b^2-4 a c\right)}-\frac{\sqrt{c} \left(a B \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)-A \left(28 a^2 c^2-29 a b^2 c-19 a b c \sqrt{b^2-4 a c}+5 b^3 \sqrt{b^2-4 a c}+5 b^4\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} a^3 \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left(a B \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)-A \left(28 a^2 c^2-29 a b^2 c+19 a b c \sqrt{b^2-4 a c}-5 b^3 \sqrt{b^2-4 a c}+5 b^4\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} a^3 \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{-A \left(b^2-2 a c\right)-\left(c x^2 (A b-2 a B)\right)+a b B}{2 a x^3 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"((-4*a*A)/x^3 + (24*A*b - 12*a*B)/x + (6*x*(a*B*(-b^3 + 3*a*b*c - b^2*c*x^2 + 2*a*c^2*x^2) + A*(b^4 - 4*a*b^2*c + 2*a^2*c^2 + b^3*c*x^2 - 3*a*b*c^2*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (3*Sqrt[2]*Sqrt[c]*(a*B*(-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]) + A*(5*b^4 - 29*a*b^2*c + 28*a^2*c^2 + 5*b^3*Sqrt[b^2 - 4*a*c] - 19*a*b*c*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (3*Sqrt[2]*Sqrt[c]*(a*B*(-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]) + A*(5*b^4 - 29*a*b^2*c + 28*a^2*c^2 - 5*b^3*Sqrt[b^2 - 4*a*c] + 19*a*b*c*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(12*a^3)","A",1
124,1,435,365,0.6634068,"\int \frac{x^{11} \left(A+B x^2\right)}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[(x^11*(A + B*x^2))/(a + b*x^2 + c*x^4)^3,x]","\frac{\frac{a^3 c^2 \left(2 c \left(A+B x^2\right)-5 b B\right)+a^2 b c \left(-b c \left(4 A+9 B x^2\right)+5 A c^2 x^2+5 b^2 B\right)+a b^3 \left(b c \left(A+6 B x^2\right)-5 A c^2 x^2+b^2 (-B)\right)+b^5 x^2 (A c-b B)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}-\frac{2 c \left(60 a^3 B c^3+30 a^2 A b c^3-90 a^2 b^2 B c^2-10 a A b^3 c^2+30 a b^4 B c+A b^5 c-3 b^6 B\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{4 a^3 c^4 \left(8 A+9 B x^2\right)-3 a^2 b^2 c^3 \left(13 A+34 B x^2\right)+2 a^2 b c^3 \left(25 A c x^2-39 a B\right)+2 b^5 c \left(2 A c x^2-7 a B\right)+a b^4 c^2 \left(11 A+48 B x^2\right)+a b^3 c^2 \left(61 a B-30 A c x^2\right)-b^6 c \left(A+6 B x^2\right)+b^7 B}{\left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+c (A c-3 b B) \log \left(a+b x^2+c x^4\right)+2 B c^2 x^2}{4 c^5}","-\frac{x^4 \left(x^2 \left(20 a^2 B c^2+10 a A b c^2-20 a b^2 B c-A b^3 c+3 b^4 B\right)+a \left(16 a A c^2-18 a b B c-A b^2 c+3 b^3 B\right)\right)}{4 c^2 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{x^2 \left(30 a^2 B c^2+7 a A b c^2-21 a b^2 B c-A b^3 c+3 b^4 B\right)}{2 c^3 \left(b^2-4 a c\right)^2}-\frac{\left(-60 a^3 B c^3-30 a^2 A b c^3+90 a^2 b^2 B c^2+10 a A b^3 c^2-30 a b^4 B c-A b^5 c+3 b^6 B\right) \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{2 c^4 \left(b^2-4 a c\right)^{5/2}}-\frac{x^8 \left(x^2 \left(-2 a B c-A b c+b^2 B\right)+a (b B-2 A c)\right)}{4 c \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}-\frac{(3 b B-A c) \log \left(a+b x^2+c x^4\right)}{4 c^4}",1,"(2*B*c^2*x^2 + (b^7*B - b^6*c*(A + 6*B*x^2) + 4*a^3*c^4*(8*A + 9*B*x^2) - 3*a^2*b^2*c^3*(13*A + 34*B*x^2) + a*b^4*c^2*(11*A + 48*B*x^2) + a*b^3*c^2*(61*a*B - 30*A*c*x^2) + 2*b^5*c*(-7*a*B + 2*A*c*x^2) + 2*a^2*b*c^3*(-39*a*B + 25*A*c*x^2))/((b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (b^5*(-(b*B) + A*c)*x^2 + a^3*c^2*(-5*b*B + 2*c*(A + B*x^2)) + a*b^3*(-(b^2*B) - 5*A*c^2*x^2 + b*c*(A + 6*B*x^2)) + a^2*b*c*(5*b^2*B + 5*A*c^2*x^2 - b*c*(4*A + 9*B*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (2*c*(-3*b^6*B + A*b^5*c + 30*a*b^4*B*c - 10*a*A*b^3*c^2 - 90*a^2*b^2*B*c^2 + 30*a^2*A*b*c^3 + 60*a^3*B*c^3)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2) + c*(-3*b*B + A*c)*Log[a + b*x^2 + c*x^4])/(4*c^5)","A",1
125,1,354,254,0.4789911,"\int \frac{x^9 \left(A+B x^2\right)}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[(x^9*(A + B*x^2))/(a + b*x^2 + c*x^4)^3,x]","\frac{-\frac{2 c \left(-12 a^2 A c^3+30 a^2 b B c^2-10 a b^3 B c+b^5 B\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^2 b c^3 \left(11 A+25 B x^2\right)+4 a^2 c^3 \left(8 a B-5 A c x^2\right)+b^4 c \left(11 a B-2 A c x^2\right)-2 a b^3 c^2 \left(4 A+15 B x^2\right)+a b^2 c^2 \left(16 A c x^2-39 a B\right)+b^5 c \left(A+4 B x^2\right)+b^6 (-B)}{\left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{2 a^3 B c^2+a^2 c \left(b c \left(3 A+5 B x^2\right)-2 A c^2 x^2-4 b^2 B\right)+a b^2 \left(-b c \left(A+5 B x^2\right)+4 A c^2 x^2+b^2 B\right)+b^4 x^2 (b B-A c)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}+B c \log \left(a+b x^2+c x^4\right)}{4 c^4}","\frac{\left(-12 a^2 A c^3+30 a^2 b B c^2-10 a b^3 B c+b^5 B\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{x^2 \left(x^2 \left(16 a^2 B c^2+6 a A b c^2-15 a b^2 B c+2 b^4 B\right)+2 a \left(6 a A c^2-7 a b B c+b^3 B\right)\right)}{4 c^2 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}-\frac{x^6 \left(x^2 \left(-2 a B c-A b c+b^2 B\right)+a (b B-2 A c)\right)}{4 c \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}+\frac{B \log \left(a+b x^2+c x^4\right)}{4 c^3}",1,"((-(b^6*B) + b^5*c*(A + 4*B*x^2) - 2*a*b^3*c^2*(4*A + 15*B*x^2) + 2*a^2*b*c^3*(11*A + 25*B*x^2) + 4*a^2*c^3*(8*a*B - 5*A*c*x^2) + b^4*c*(11*a*B - 2*A*c*x^2) + a*b^2*c^2*(-39*a*B + 16*A*c*x^2))/((b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (2*a^3*B*c^2 + b^4*(b*B - A*c)*x^2 + a*b^2*(b^2*B + 4*A*c^2*x^2 - b*c*(A + 5*B*x^2)) + a^2*c*(-4*b^2*B - 2*A*c^2*x^2 + b*c*(3*A + 5*B*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (2*c*(b^5*B - 10*a*b^3*B*c + 30*a^2*b*B*c^2 - 12*a^2*A*c^3)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2) + B*c*Log[a + b*x^2 + c*x^4])/(4*c^4)","A",1
126,1,261,146,0.2682553,"\int \frac{x^7 \left(A+B x^2\right)}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[(x^7*(A + B*x^2))/(a + b*x^2 + c*x^4)^3,x]","\frac{1}{4} \left(\frac{a^2 c \left(2 c \left(A+B x^2\right)-3 b B\right)+a b \left(-b c \left(A+4 B x^2\right)+3 A c^2 x^2+b^2 B\right)+b^3 x^2 (b B-A c)}{c^3 \left(4 a c-b^2\right) \left(a+b x^2+c x^4\right)^2}+\frac{-4 a^2 c^3 \left(4 A+5 B x^2\right)+a b^2 c^2 \left(5 A+16 B x^2\right)+2 a b c^2 \left(11 a B-3 A c x^2\right)-8 a b^3 B c-b^4 c \left(A+2 B x^2\right)+b^5 B}{c^3 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}-\frac{12 a (A b-2 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}}\right)","\frac{3 a (A b-2 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 x^2 \left(2 a+b x^2\right) (A b-2 a B)}{4 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}-\frac{x^6 \left(-2 a B-\left(x^2 (b B-2 A c)\right)+A b\right)}{4 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}",1,"((b^5*B - 8*a*b^3*B*c - b^4*c*(A + 2*B*x^2) - 4*a^2*c^3*(4*A + 5*B*x^2) + a*b^2*c^2*(5*A + 16*B*x^2) + 2*a*b*c^2*(11*a*B - 3*A*c*x^2))/(c^3*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (b^3*(b*B - A*c)*x^2 + a^2*c*(-3*b*B + 2*c*(A + B*x^2)) + a*b*(b^2*B + 3*A*c^2*x^2 - b*c*(A + 4*B*x^2)))/(c^3*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)^2) - (12*a*(A*b - 2*a*B)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2))/4","A",1
127,1,233,185,0.2360914,"\int \frac{x^5 \left(A+B x^2\right)}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[(x^5*(A + B*x^2))/(a + b*x^2 + c*x^4)^3,x]","\frac{1}{4} \left(\frac{2 a^2 B c+a \left(b c \left(A+3 B x^2\right)-2 A c^2 x^2+b^2 (-B)\right)+b^2 x^2 (A c-b B)}{c^2 \left(4 a c-b^2\right) \left(a+b x^2+c x^4\right)^2}+\frac{4 \left(A \left(2 a c+b^2\right)-3 a b B\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{b^2 c \left(5 a B+2 A c x^2\right)+2 a b c^2 \left(A-3 B x^2\right)+4 a c^2 \left(A c x^2-4 a B\right)+A b^3 c+b^4 (-B)}{c^2 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}\right)","\frac{\left(3 a b B-A \left(2 a c+b^2\right)\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^4 \left(-2 a B-\left(x^2 (b B-2 A c)\right)+A b\right)}{4 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}-\frac{a \left(8 a B c-6 A b c+b^2 B\right)+x^2 \left(4 a A c^2+2 a b B c-4 A b^2 c+b^3 B\right)}{4 c \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}",1,"((-(b^4*B) + A*b^3*c + 2*a*b*c^2*(A - 3*B*x^2) + 4*a*c^2*(-4*a*B + A*c*x^2) + b^2*c*(5*a*B + 2*A*c*x^2))/(c^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (2*a^2*B*c + b^2*(-(b*B) + A*c)*x^2 + a*(-(b^2*B) - 2*A*c^2*x^2 + b*c*(A + 3*B*x^2)))/(c^2*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)^2) + (4*(-3*a*b*B + A*(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
128,1,172,170,0.2042609,"\int \frac{x^3 \left(A+B x^2\right)}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[(x^3*(A + B*x^2))/(a + b*x^2 + c*x^4)^3,x]","\frac{1}{4} \left(\frac{4 \left(2 a B c-3 A b c+b^2 B\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(b+2 c x^2\right) \left(2 a B c-3 A b c+b^2 B\right)}{c \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{-2 a c \left(A+B x^2\right)+a b B+b x^2 (b B-A c)}{c \left(4 a c-b^2\right) \left(a+b x^2+c x^4\right)^2}\right)","-\frac{\left(2 a B c-3 A b c+b^2 B\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{\left(b+2 c x^2\right) \left(2 a B c-3 A b c+b^2 B\right)}{4 c \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}-\frac{x^2 \left(-2 a B c-A b c+b^2 B\right)+a (b B-2 A c)}{4 c \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}",1,"(((b^2*B - 3*A*b*c + 2*a*B*c)*(b + 2*c*x^2))/(c*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (a*b*B + b*(b*B - A*c)*x^2 - 2*a*c*(A + B*x^2))/(c*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)^2) + (4*(b^2*B - 3*A*b*c + 2*a*B*c)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2))/4","A",1
129,1,142,139,0.1349452,"\int \frac{x \left(A+B x^2\right)}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[(x*(A + B*x^2))/(a + b*x^2 + c*x^4)^3,x]","\frac{-\frac{12 c (b B-2 A 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(B \left(2 a+b x^2\right)-A \left(b+2 c x^2\right)\right)}{\left(a+b x^2+c x^4\right)^2}-\frac{3 \left(b+2 c x^2\right) (b B-2 A c)}{a+b x^2+c x^4}}{4 \left(b^2-4 a c\right)^2}","\frac{3 c (b B-2 A 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{3 \left(b+2 c x^2\right) (b B-2 A c)}{4 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}-\frac{-2 a B-\left(x^2 (b B-2 A c)\right)+A b}{4 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}",1,"((-3*(b*B - 2*A*c)*(b + 2*c*x^2))/(a + b*x^2 + c*x^4) + ((b^2 - 4*a*c)*(B*(2*a + b*x^2) - A*(b + 2*c*x^2)))/(a + b*x^2 + c*x^4)^2 - (12*c*(b*B - 2*A*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
130,1,396,252,0.6912198,"\int \frac{A+B x^2}{x \left(a+b x^2+c x^4\right)^3} \, dx","Integrate[(A + B*x^2)/(x*(a + b*x^2 + c*x^4)^3),x]","\frac{\frac{a^2 \left(A \left(-2 a c+b^2+b c x^2\right)-a B \left(b+2 c x^2\right)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}+\frac{a \left(2 a^2 c \left(8 A c+3 b B+6 B c x^2\right)-a A b c \left(15 b+14 c x^2\right)+2 A b^3 \left(b+c x^2\right)\right)}{\left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}-\frac{\left(A \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)-12 a^3 B c^2\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(12 a^3 B c^2+A \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)\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 A \log (x)}{4 a^3}","-\frac{A \log \left(a+b x^2+c x^4\right)}{4 a^3}+\frac{A \log (x)}{a^3}+\frac{2 c x^2 \left(6 a^2 B c+A \left(b^3-7 a b c\right)\right)+A \left(16 a^2 c^2-15 a b^2 c+2 b^4\right)+6 a^2 b B c}{4 a^2 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}-\frac{\left(12 a^3 B c^2-A \left(30 a^2 b c^2-10 a b^3 c+b^5\right)\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{-A \left(b^2-2 a c\right)-\left(c x^2 (A b-2 a B)\right)+a b B}{4 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}",1,"((a^2*(-(a*B*(b + 2*c*x^2)) + A*(b^2 - 2*a*c + b*c*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (a*(2*A*b^3*(b + c*x^2) - a*A*b*c*(15*b + 14*c*x^2) + 2*a^2*c*(3*b*B + 8*A*c + 6*B*c*x^2)))/((b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + 4*A*Log[x] - ((-12*a^3*B*c^2 + A*(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) - ((12*a^3*B*c^2 + A*(-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
131,1,642,363,1.500173,"\int \frac{A+B x^2}{x^3 \left(a+b x^2+c x^4\right)^3} \, dx","Integrate[(A + B*x^2)/(x^3*(a + b*x^2 + c*x^4)^3),x]","\frac{-\frac{a^2 \left(A \left(-3 a b c-2 a c^2 x^2+b^3+b^2 c x^2\right)+a B \left(2 a c-b^2-b c x^2\right)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}+\frac{a \left(a B \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)-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)\right)}{\left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{\left(3 A \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)-a B \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)\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(a B \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)+3 A \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)\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) (a B-3 A b)-\frac{2 a A}{x^2}}{4 a^4}","\frac{(3 A b-a B) \log \left(a+b x^2+c x^4\right)}{4 a^4}-\frac{\log (x) (3 A b-a B)}{a^4}-\frac{-A \left(20 a^2 c^2-20 a b^2 c+3 b^4\right)+c x^2 \left(a B \left(b^2-16 a c\right)-3 A \left(b^3-6 a b c\right)\right)+a b B \left(b^2-10 a c\right)}{4 a^2 x^2 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{a b B \left(b^2-7 a c\right)-3 A \left(10 a^2 c^2-7 a b^2 c+b^4\right)}{2 a^3 x^2 \left(b^2-4 a c\right)^2}+\frac{\left(a b B \left(30 a^2 c^2-10 a b^2 c+b^4\right)-3 A \left(-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right)\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{-A \left(b^2-2 a c\right)-\left(c x^2 (A b-2 a B)\right)+a b B}{4 a x^2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}",1,"((-2*a*A)/x^2 - (a^2*(a*B*(-b^2 + 2*a*c - b*c*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)^2) + (a*(a*B*(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) - 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)) + 4*(-3*A*b + a*B)*Log[x] + ((-(a*B*(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])) + 3*A*(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) + ((a*B*(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]) + 3*A*(-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
132,1,644,554,2.3983609,"\int \frac{x^8 \left(A+B x^2\right)}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[(x^8*(A + B*x^2))/(a + b*x^2 + c*x^4)^3,x]","\frac{-\frac{4 x \left(a^2 c \left(2 c \left(A+B x^2\right)-3 b B\right)+a b \left(-b c \left(A+4 B x^2\right)+3 A c^2 x^2+b^2 B\right)+b^3 x^2 (b B-A c)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}+\frac{\sqrt{2} \sqrt{c} \left(4 a^2 c^2 \left(21 B \sqrt{b^2-4 a c}+10 A c\right)-4 a b c^2 \left(4 A \sqrt{b^2-4 a c}+33 a B\right)+9 a b^2 c \left(2 A c-3 B \sqrt{b^2-4 a c}\right)+b^4 \left(3 B \sqrt{b^2-4 a c}-A c\right)+b^3 c \left(A \sqrt{b^2-4 a c}+33 a B\right)-3 b^5 B\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(4 a^2 c^2 \left(21 B \sqrt{b^2-4 a c}-10 A c\right)+4 a b c^2 \left(33 a B-4 A \sqrt{b^2-4 a c}\right)-9 a b^2 c \left(3 B \sqrt{b^2-4 a c}+2 A c\right)+b^4 \left(3 B \sqrt{b^2-4 a c}+A c\right)+b^3 \left(A c \sqrt{b^2-4 a c}-33 a B c\right)+3 b^5 B\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(-4 a^2 c^3 \left(9 A+11 B x^2\right)+b^3 c \left(A c x^2-17 a B\right)+a b^2 c^2 \left(11 A+37 B x^2\right)+16 a b c^2 \left(3 a B-A c x^2\right)-b^4 c \left(2 A+5 B x^2\right)+2 b^5 B\right)}{\left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}}{16 c^3}","\frac{\left(-\frac{-40 a^2 A c^3+132 a^2 b B c^2-18 a A b^2 c^2-33 a b^3 B c+A b^4 c+3 b^5 B}{\sqrt{b^2-4 a c}}+84 a^2 B c^2-16 a A b c^2-27 a b^2 B c+A b^3 c+3 b^4 B\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{\left(\frac{-40 a^2 A c^3+132 a^2 b B c^2-18 a A b^2 c^2-33 a b^3 B c+A b^4 c+3 b^5 B}{\sqrt{b^2-4 a c}}+84 a^2 B c^2-16 a A b c^2-27 a b^2 B c+A b^3 c+3 b^4 B\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{x^3 \left(-28 a B c+12 A b c+b^2 B\right)}{8 c \left(b^2-4 a c\right)^2}-\frac{x^7 \left(-2 a B-\left(x^2 (b B-2 A c)\right)+A b\right)}{4 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}-\frac{x^5 \left(x^2 \left(-28 a B c+12 A b c+b^2 B\right)-4 a A c-12 a b B+7 A b^2\right)}{8 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}-\frac{x \left(20 a A c^2-24 a b B c+A b^2 c+3 b^3 B\right)}{8 c^2 \left(b^2-4 a c\right)^2}",1,"((2*x*(2*b^5*B - b^4*c*(2*A + 5*B*x^2) - 4*a^2*c^3*(9*A + 11*B*x^2) + a*b^2*c^2*(11*A + 37*B*x^2) + 16*a*b*c^2*(3*a*B - A*c*x^2) + b^3*c*(-17*a*B + A*c*x^2)))/((b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - (4*x*(b^3*(b*B - A*c)*x^2 + a^2*c*(-3*b*B + 2*c*(A + B*x^2)) + a*b*(b^2*B + 3*A*c^2*x^2 - b*c*(A + 4*B*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (Sqrt[2]*Sqrt[c]*(-3*b^5*B + b^3*c*(33*a*B + A*Sqrt[b^2 - 4*a*c]) - 4*a*b*c^2*(33*a*B + 4*A*Sqrt[b^2 - 4*a*c]) + 9*a*b^2*c*(2*A*c - 3*B*Sqrt[b^2 - 4*a*c]) + b^4*(-(A*c) + 3*B*Sqrt[b^2 - 4*a*c]) + 4*a^2*c^2*(10*A*c + 21*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]]) + (Sqrt[2]*Sqrt[c]*(3*b^5*B + 4*a*b*c^2*(33*a*B - 4*A*Sqrt[b^2 - 4*a*c]) + b^4*(A*c + 3*B*Sqrt[b^2 - 4*a*c]) - 9*a*b^2*c*(2*A*c + 3*B*Sqrt[b^2 - 4*a*c]) + 4*a^2*c^2*(-10*A*c + 21*B*Sqrt[b^2 - 4*a*c]) + b^3*(-33*a*B*c + 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^3)","A",1
133,1,543,461,2.0214443,"\int \frac{x^6 \left(A+B x^2\right)}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[(x^6*(A + B*x^2))/(a + b*x^2 + c*x^4)^3,x]","\frac{-\frac{4 x \left(2 a^2 B c+a \left(b c \left(A+3 B x^2\right)-2 A c^2 x^2+b^2 (-B)\right)+b^2 x^2 (A c-b B)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}+\frac{2 x \left(b^2 c \left(11 a B+3 A c x^2\right)+4 a b c^2 \left(A-4 B x^2\right)+12 a c^2 \left(A c x^2-3 a B\right)+b^3 c \left(2 A+B x^2\right)-2 b^4 B\right)}{\left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \sqrt{c} \left(4 a c^2 \left(3 A \sqrt{b^2-4 a c}+10 a B\right)+3 b^2 c \left(A \sqrt{b^2-4 a c}+6 a B\right)-4 a b c \left(4 B \sqrt{b^2-4 a c}+9 A c\right)+b^3 \left(B \sqrt{b^2-4 a c}-3 A c\right)+b^4 (-B)\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(4 a c^2 \left(3 A \sqrt{b^2-4 a c}-10 a B\right)+3 b^2 c \left(A \sqrt{b^2-4 a c}-6 a B\right)+4 a b c \left(9 A c-4 B \sqrt{b^2-4 a c}\right)+b^3 \left(B \sqrt{b^2-4 a c}+3 A c\right)+b^4 B\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 B c^2+36 a A b c^2-18 a b^2 B c+3 A b^3 c+b^4 B}{\sqrt{b^2-4 a c}}+12 a A c^2-16 a b B c+3 A b^2 c+b^3 B\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 B c^2+36 a A b c^2-18 a b^2 B c+3 A b^3 c+b^4 B}{\sqrt{b^2-4 a c}}+12 a A c^2-16 a b B c+3 A b^2 c+b^3 B\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-\left(x^2 (b B-2 A c)\right)+A b\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 B c-12 A b c+b^2 B\right)+4 a A c-12 a b B+5 A b^2\right)}{8 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}-\frac{x \left(20 a B c-12 A b c+b^2 B\right)}{8 c \left(b^2-4 a c\right)^2}",1,"((2*x*(-2*b^4*B + 4*a*b*c^2*(A - 4*B*x^2) + b^3*c*(2*A + B*x^2) + 12*a*c^2*(-3*a*B + A*c*x^2) + b^2*c*(11*a*B + 3*A*c*x^2)))/((b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - (4*x*(2*a^2*B*c + b^2*(-(b*B) + A*c)*x^2 + a*(-(b^2*B) - 2*A*c^2*x^2 + b*c*(A + 3*B*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (Sqrt[2]*Sqrt[c]*(-(b^4*B) + 3*b^2*c*(6*a*B + A*Sqrt[b^2 - 4*a*c]) + 4*a*c^2*(10*a*B + 3*A*Sqrt[b^2 - 4*a*c]) + b^3*(-3*A*c + B*Sqrt[b^2 - 4*a*c]) - 4*a*b*c*(9*A*c + 4*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]]) + (Sqrt[2]*Sqrt[c]*(b^4*B + 3*b^2*c*(-6*a*B + A*Sqrt[b^2 - 4*a*c]) + 4*a*c^2*(-10*a*B + 3*A*Sqrt[b^2 - 4*a*c]) + 4*a*b*c*(9*A*c - 4*B*Sqrt[b^2 - 4*a*c]) + b^3*(3*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)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(16*c^2)","A",1
134,1,447,380,1.6964271,"\int \frac{x^4 \left(A+B x^2\right)}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[(x^4*(A + B*x^2))/(a + b*x^2 + c*x^4)^3,x]","\frac{\frac{8 a c x \left(A+B x^2\right)-4 a b B x+4 b x^3 (A c-b B)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}+\frac{2 x \left(4 b c \left(a B-3 A c x^2\right)+4 a c^2 \left(A+3 B x^2\right)+b^2 \left(3 B c x^2-7 A c\right)+2 b^3 B\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(b^2 \left(B \sqrt{b^2-4 a c}+6 A c\right)-4 b c \left(A \sqrt{b^2-4 a c}+3 a B\right)+4 a c \left(B \sqrt{b^2-4 a c}+2 A c\right)+b^3 (-B)\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 \left(B \sqrt{b^2-4 a c}-6 A c\right)+4 b c \left(3 a B-A \sqrt{b^2-4 a c}\right)+4 a c \left(B \sqrt{b^2-4 a c}-2 A c\right)+b^3 B\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 B c-4 A b c+b^2 B\right)-A \left(4 a c+b^2\right)+4 a b 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-\left(x^2 (b B-2 A c)\right)+A b\right)}{4 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}+\frac{3 \left(-\frac{-8 a A c^2+12 a b B c-6 A b^2 c+b^3 B}{\sqrt{b^2-4 a c}}+4 a B c-4 A b c+b^2 B\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{-8 a A c^2+12 a b B c-6 A b^2 c+b^3 B}{\sqrt{b^2-4 a c}}+4 a B c-4 A b c+b^2 B\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*a*b*B*x + 4*b*(-(b*B) + A*c)*x^3 + 8*a*c*x*(A + B*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (2*x*(2*b^3*B + 4*a*c^2*(A + 3*B*x^2) + 4*b*c*(a*B - 3*A*c*x^2) + b^2*(-7*A*c + 3*B*c*x^2)))/((b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (3*Sqrt[2]*Sqrt[c]*(-(b^3*B) - 4*b*c*(3*a*B + A*Sqrt[b^2 - 4*a*c]) + 4*a*c*(2*A*c + B*Sqrt[b^2 - 4*a*c]) + b^2*(6*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)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[2]*Sqrt[c]*(b^3*B + 4*b*c*(3*a*B - A*Sqrt[b^2 - 4*a*c]) + b^2*(-6*A*c + B*Sqrt[b^2 - 4*a*c]) + 4*a*c*(-2*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)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(16*c)","A",1
135,1,436,438,1.6458522,"\int \frac{x^2 \left(A+B x^2\right)}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[(x^2*(A + B*x^2))/(a + b*x^2 + c*x^4)^3,x]","\frac{1}{16} \left(\frac{4 x \left(B \left(2 a+b x^2\right)-A \left(b+2 c x^2\right)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}+\frac{2 x \left(A \left(8 a b c+20 a c^2 x^2+b^3+b^2 c x^2\right)+a B \left(4 a c-7 b^2-12 b c x^2\right)\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(A \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)+6 a B \left(-2 b \sqrt{b^2-4 a c}+4 a c+3 b^2\right)\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(A \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)-6 a B \left(2 b \sqrt{b^2-4 a c}+4 a c+3 b^2\right)\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(-2 a B-\left(x^2 (b B-2 A c)\right)+A b\right)}{4 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}-\frac{x \left(-A \left(8 a b c+b^3\right)+c x^2 \left(12 a b B-A \left(20 a c+b^2\right)\right)+a B \left(7 b^2-4 a c\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(A \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)+6 a B \left(-2 b \sqrt{b^2-4 a c}+4 a c+3 b^2\right)\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)^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left(A \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)+6 a B \left(2 b \sqrt{b^2-4 a c}+4 a c+3 b^2\right)\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)^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}",1,"((4*x*(B*(2*a + b*x^2) - A*(b + 2*c*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (2*x*(a*B*(-7*b^2 + 4*a*c - 12*b*c*x^2) + A*(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]*(6*a*B*(3*b^2 + 4*a*c - 2*b*Sqrt[b^2 - 4*a*c]) + A*(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]*(-6*a*B*(3*b^2 + 4*a*c + 2*b*Sqrt[b^2 - 4*a*c]) + A*(-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
136,1,516,460,2.1894806,"\int \frac{A+B x^2}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[(A + B*x^2)/(a + b*x^2 + c*x^4)^3,x]","\frac{\frac{2 x \left(A \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)+a B \left(8 a b c+20 a c^2 x^2+b^3+b^2 c x^2\right)\right)}{\left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \sqrt{c} \left(3 A \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)+a B \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)\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(3 A \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)+a B \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)\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(a B \left(b+2 c x^2\right)-A \left(-2 a c+b^2+b c x^2\right)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}}{16 a^2}","\frac{x \left(A \left(28 a^2 c^2-25 a b^2 c+3 b^4\right)+c x^2 \left(3 A \left(b^3-8 a b c\right)+a B \left(20 a c+b^2\right)\right)+a b B \left(8 a c+b^2\right)\right)}{8 a^2 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{\sqrt{c} \left(\frac{3 A \left(56 a^2 c^2-10 a b^2 c+b^4\right)+a b B \left(b^2-52 a c\right)}{\sqrt{b^2-4 a c}}+3 A \left(b^3-8 a b c\right)+a B \left(20 a c+b^2\right)\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)^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left(-\frac{3 A \left(56 a^2 c^2-10 a b^2 c+b^4\right)+a b B \left(b^2-52 a c\right)}{\sqrt{b^2-4 a c}}+3 A \left(b^3-8 a b c\right)+a B \left(20 a c+b^2\right)\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(c x^2 (A b-2 a B)-2 a A c-a b B+A b^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*(a*B*(b + 2*c*x^2) - A*(b^2 - 2*a*c + b*c*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (2*x*(a*B*(b^3 + 8*a*b*c + b^2*c*x^2 + 20*a*c^2*x^2) + A*(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)) + (Sqrt[2]*Sqrt[c]*(a*B*(b^3 - 52*a*b*c + b^2*Sqrt[b^2 - 4*a*c] + 20*a*c*Sqrt[b^2 - 4*a*c]) + 3*A*(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]]) + (Sqrt[2]*Sqrt[c]*(a*B*(-b^3 + 52*a*b*c + b^2*Sqrt[b^2 - 4*a*c] + 20*a*c*Sqrt[b^2 - 4*a*c]) + 3*A*(-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
137,1,25,25,0.0093301,"\int \frac{x \left(-7+4 x^2\right)}{4-5 x^2+x^4} \, dx","Integrate[(x*(-7 + 4*x^2))/(4 - 5*x^2 + x^4),x]","\frac{1}{2} \log \left(1-x^2\right)+\frac{3}{2} \log \left(4-x^2\right)","\frac{1}{2} \log \left(1-x^2\right)+\frac{3}{2} \log \left(4-x^2\right)",1,"Log[1 - x^2]/2 + (3*Log[4 - x^2])/2","A",1
138,1,25,25,0.005514,"\int \frac{-7 x+4 x^3}{4-5 x^2+x^4} \, dx","Integrate[(-7*x + 4*x^3)/(4 - 5*x^2 + x^4),x]","\frac{1}{2} \log \left(1-x^2\right)+\frac{3}{2} \log \left(4-x^2\right)","\frac{1}{2} \log \left(1-x^2\right)+\frac{3}{2} \log \left(4-x^2\right)",1,"Log[1 - x^2]/2 + (3*Log[4 - x^2])/2","A",1
139,1,37,37,0.0134395,"\int \frac{x \left(2+x^2\right)}{1+x^2+x^4} \, dx","Integrate[(x*(2 + x^2))/(1 + x^2 + x^4),x]","\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{2 x^2+1}{\sqrt{3}}\right)+\frac{1}{4} \log \left(x^4+x^2+1\right)","\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{2 x^2+1}{\sqrt{3}}\right)+\frac{1}{4} \log \left(x^4+x^2+1\right)",1,"(Sqrt[3]*ArcTan[(1 + 2*x^2)/Sqrt[3]])/2 + Log[1 + x^2 + x^4]/4","A",1
140,1,37,37,0.0058235,"\int \frac{2 x+x^3}{1+x^2+x^4} \, dx","Integrate[(2*x + x^3)/(1 + x^2 + x^4),x]","\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{2 x^2+1}{\sqrt{3}}\right)+\frac{1}{4} \log \left(x^4+x^2+1\right)","\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{2 x^2+1}{\sqrt{3}}\right)+\frac{1}{4} \log \left(x^4+x^2+1\right)",1,"(Sqrt[3]*ArcTan[(1 + 2*x^2)/Sqrt[3]])/2 + Log[1 + x^2 + x^4]/4","A",1
141,1,45,45,0.0262778,"\int \frac{11 x+2 x^3}{\left(3+2 x^2+x^4\right)^2} \, dx","Integrate[(11*x + 2*x^3)/(3 + 2*x^2 + x^4)^2,x]","\frac{9 \tan ^{-1}\left(\frac{x^2+1}{\sqrt{2}}\right)}{8 \sqrt{2}}+\frac{9 x^2+5}{8 \left(x^4+2 x^2+3\right)}","\frac{9 \tan ^{-1}\left(\frac{x^2+1}{\sqrt{2}}\right)}{8 \sqrt{2}}+\frac{9 x^2+5}{8 \left(x^4+2 x^2+3\right)}",1,"(5 + 9*x^2)/(8*(3 + 2*x^2 + x^4)) + (9*ArcTan[(1 + x^2)/Sqrt[2]])/(8*Sqrt[2])","A",1
142,1,71,102,0.0291134,"\int x^5 \left(2+3 x^2\right) \sqrt{3+5 x^2+x^4} \, dx","Integrate[x^5*(2 + 3*x^2)*Sqrt[3 + 5*x^2 + x^4],x]","\frac{318435 \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)+2 \sqrt{x^4+5 x^2+3} \left(1152 x^8+1680 x^6-2248 x^4+12250 x^2-78387\right)}{7680}","\frac{3}{10} \left(x^4+5 x^2+3\right)^{3/2} x^4+\frac{1}{480} \left(1837-510 x^2\right) \left(x^4+5 x^2+3\right)^{3/2}-\frac{1633}{256} \left(2 x^2+5\right) \sqrt{x^4+5 x^2+3}+\frac{21229}{512} \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)",1,"(2*Sqrt[3 + 5*x^2 + x^4]*(-78387 + 12250*x^2 - 2248*x^4 + 1680*x^6 + 1152*x^8) + 318435*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/7680","A",1
143,1,66,81,0.0206685,"\int x^3 \left(2+3 x^2\right) \sqrt{3+5 x^2+x^4} \, dx","Integrate[x^3*(2 + 3*x^2)*Sqrt[3 + 5*x^2 + x^4],x]","\frac{1}{768} \left(2 \sqrt{x^4+5 x^2+3} \left(144 x^6+248 x^4-374 x^2+2469\right)-10101 \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)\right)","-\frac{1}{48} \left(59-18 x^2\right) \left(x^4+5 x^2+3\right)^{3/2}+\frac{259}{128} \left(2 x^2+5\right) \sqrt{x^4+5 x^2+3}-\frac{3367}{256} \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)",1,"(2*Sqrt[3 + 5*x^2 + x^4]*(2469 - 374*x^2 + 248*x^4 + 144*x^6) - 10101*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/768","A",1
144,1,61,74,0.0166832,"\int x \left(2+3 x^2\right) \sqrt{3+5 x^2+x^4} \, dx","Integrate[x*(2 + 3*x^2)*Sqrt[3 + 5*x^2 + x^4],x]","\frac{1}{32} \left(2 \sqrt{x^4+5 x^2+3} \left(8 x^4+18 x^2-31\right)+143 \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)\right)","\frac{1}{2} \left(x^4+5 x^2+3\right)^{3/2}-\frac{11}{16} \left(2 x^2+5\right) \sqrt{x^4+5 x^2+3}+\frac{143}{32} \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)",1,"(2*Sqrt[3 + 5*x^2 + x^4]*(-31 + 18*x^2 + 8*x^4) + 143*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/32","A",1
145,1,92,94,0.0377516,"\int \frac{\left(2+3 x^2\right) \sqrt{3+5 x^2+x^4}}{x} \, dx","Integrate[((2 + 3*x^2)*Sqrt[3 + 5*x^2 + x^4])/x,x]","\frac{1}{16} \left(2 \sqrt{x^4+5 x^2+3} \left(6 x^2+23\right)+\tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)-16 \sqrt{3} \tanh ^{-1}\left(\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right)\right)","\frac{1}{8} \sqrt{x^4+5 x^2+3} \left(6 x^2+23\right)+\frac{1}{16} \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)-\sqrt{3} \tanh ^{-1}\left(\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right)",1,"(2*(23 + 6*x^2)*Sqrt[3 + 5*x^2 + x^4] + ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])] - 16*Sqrt[3]*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])])/16","A",1
146,1,97,97,0.040877,"\int \frac{\left(2+3 x^2\right) \sqrt{3+5 x^2+x^4}}{x^3} \, dx","Integrate[((2 + 3*x^2)*Sqrt[3 + 5*x^2 + x^4])/x^3,x]","\frac{\sqrt{x^4+5 x^2+3} \left(3 x^2-2\right)}{2 x^2}+\frac{19}{4} \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)-\frac{7 \tanh ^{-1}\left(\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right)}{\sqrt{3}}","-\frac{\sqrt{x^4+5 x^2+3} \left(2-3 x^2\right)}{2 x^2}+\frac{19}{4} \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)-\frac{7 \tanh ^{-1}\left(\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right)}{\sqrt{3}}",1,"((-2 + 3*x^2)*Sqrt[3 + 5*x^2 + x^4])/(2*x^2) + (19*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/4 - (7*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])])/Sqrt[3]","A",1
147,1,97,99,0.0393975,"\int \frac{\left(2+3 x^2\right) \sqrt{3+5 x^2+x^4}}{x^5} \, dx","Integrate[((2 + 3*x^2)*Sqrt[3 + 5*x^2 + x^4])/x^5,x]","\frac{1}{72} \left(-\frac{6 \sqrt{x^4+5 x^2+3} \left(23 x^2+6\right)}{x^4}+108 \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)-77 \sqrt{3} \tanh ^{-1}\left(\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right)\right)","-\frac{\sqrt{x^4+5 x^2+3} \left(23 x^2+6\right)}{12 x^4}+\frac{3}{2} \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)-\frac{77 \tanh ^{-1}\left(\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right)}{24 \sqrt{3}}",1,"((-6*(6 + 23*x^2)*Sqrt[3 + 5*x^2 + x^4])/x^4 + 108*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])] - 77*Sqrt[3]*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])])/72","A",1
148,1,74,90,0.0255547,"\int \frac{\left(2+3 x^2\right) \sqrt{3+5 x^2+x^4}}{x^7} \, dx","Integrate[((2 + 3*x^2)*Sqrt[3 + 5*x^2 + x^4])/x^7,x]","\frac{1}{108} \left(13 \sqrt{3} \tanh ^{-1}\left(\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right)-\frac{6 \sqrt{x^4+5 x^2+3} \left(7 x^4+16 x^2+6\right)}{x^6}\right)","-\frac{\left(5 x^2+6\right) \sqrt{x^4+5 x^2+3}}{18 x^4}+\frac{13 \tanh ^{-1}\left(\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right)}{36 \sqrt{3}}-\frac{\left(x^4+5 x^2+3\right)^{3/2}}{9 x^6}",1,"((-6*Sqrt[3 + 5*x^2 + x^4]*(6 + 16*x^2 + 7*x^4))/x^6 + 13*Sqrt[3]*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])])/108","A",1
149,1,82,111,0.0270145,"\int \frac{\left(2+3 x^2\right) \sqrt{3+5 x^2+x^4}}{x^9} \, dx","Integrate[((2 + 3*x^2)*Sqrt[3 + 5*x^2 + x^4])/x^9,x]","\frac{6 \sqrt{x^4+5 x^2+3} \left(247 x^6-182 x^4-984 x^2-432\right)-871 \sqrt{3} x^8 \tanh ^{-1}\left(\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right)}{10368 x^8}","\frac{67 \left(5 x^2+6\right) \sqrt{x^4+5 x^2+3}}{1728 x^4}-\frac{871 \tanh ^{-1}\left(\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right)}{3456 \sqrt{3}}-\frac{\left(x^4+5 x^2+3\right)^{3/2}}{12 x^8}-\frac{11 \left(x^4+5 x^2+3\right)^{3/2}}{216 x^6}",1,"(6*Sqrt[3 + 5*x^2 + x^4]*(-432 - 984*x^2 - 182*x^4 + 247*x^6) - 871*Sqrt[3]*x^8*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])])/(10368*x^8)","A",1
150,1,84,132,0.0374223,"\int \frac{\left(2+3 x^2\right) \sqrt{3+5 x^2+x^4}}{x^{11}} \, dx","Integrate[((2 + 3*x^2)*Sqrt[3 + 5*x^2 + x^4])/x^11,x]","\frac{10465 \sqrt{3} \tanh ^{-1}\left(\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right)-\frac{6 \sqrt{x^4+5 x^2+3} \left(2641 x^8-1370 x^6+1176 x^4+10800 x^2+5184\right)}{x^{10}}}{155520}","-\frac{161 \left(5 x^2+6\right) \sqrt{x^4+5 x^2+3}}{5184 x^4}+\frac{2093 \tanh ^{-1}\left(\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right)}{10368 \sqrt{3}}-\frac{\left(x^4+5 x^2+3\right)^{3/2}}{15 x^{10}}-\frac{\left(x^4+5 x^2+3\right)^{3/2}}{36 x^8}+\frac{173 \left(x^4+5 x^2+3\right)^{3/2}}{3240 x^6}",1,"((-6*Sqrt[3 + 5*x^2 + x^4]*(5184 + 10800*x^2 + 1176*x^4 - 1370*x^6 + 2641*x^8))/x^10 + 10465*Sqrt[3]*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])])/155520","A",1
151,1,237,322,0.3775193,"\int x^4 \left(2+3 x^2\right) \sqrt{3+5 x^2+x^4} \, dx","Integrate[x^4*(2 + 3*x^2)*Sqrt[3 + 5*x^2 + x^4],x]","\frac{70 x^{11}+460 x^9+604 x^7+460 x^5+4082 x^3+13 i \sqrt{2} \left(148 \sqrt{13}-635\right) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)-1924 i \sqrt{2} \left(\sqrt{13}-5\right) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} E\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)+2730 x}{210 \sqrt{x^4+5 x^2+3}}","\frac{13}{3} \sqrt{x^4+5 x^2+3} x-\frac{1924 \left(2 x^2+\sqrt{13}+5\right) x}{105 \sqrt{x^4+5 x^2+3}}-\frac{13 \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) F\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{\sqrt{6 \left(5+\sqrt{13}\right)} \sqrt{x^4+5 x^2+3}}+\frac{962 \sqrt{\frac{2}{3} \left(5+\sqrt{13}\right)} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) E\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{105 \sqrt{x^4+5 x^2+3}}+\frac{1}{21} \left(7 x^2+11\right) \sqrt{x^4+5 x^2+3} x^5-\frac{26}{35} \sqrt{x^4+5 x^2+3} x^3",1,"(2730*x + 4082*x^3 + 460*x^5 + 604*x^7 + 460*x^9 + 70*x^11 - (1924*I)*Sqrt[2]*(-5 + Sqrt[13])*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticE[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6] + (13*I)*Sqrt[2]*(-635 + 148*Sqrt[13])*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6])/(210*Sqrt[3 + 5*x^2 + x^4])","C",0
152,1,234,305,0.2561461,"\int x^2 \left(2+3 x^2\right) \sqrt{3+5 x^2+x^4} \, dx","Integrate[x^2*(2 + 3*x^2)*Sqrt[3 + 5*x^2 + x^4],x]","\frac{-i \sqrt{2} \left(1247 \sqrt{13}-5395\right) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)+1247 i \sqrt{2} \left(\sqrt{13}-5\right) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} E\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)+4 x \left(45 x^8+312 x^6+430 x^4-439 x^2-420\right)}{420 \sqrt{x^4+5 x^2+3}}","-\frac{4}{3} \sqrt{x^4+5 x^2+3} x+\frac{1247 \left(2 x^2+\sqrt{13}+5\right) x}{210 \sqrt{x^4+5 x^2+3}}+\frac{2 \sqrt{\frac{2}{3 \left(5+\sqrt{13}\right)}} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) F\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{\sqrt{x^4+5 x^2+3}}-\frac{1247 \sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) E\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{210 \sqrt{x^4+5 x^2+3}}+\frac{1}{35} \left(15 x^2+29\right) \sqrt{x^4+5 x^2+3} x^3",1,"(4*x*(-420 - 439*x^2 + 430*x^4 + 312*x^6 + 45*x^8) + (1247*I)*Sqrt[2]*(-5 + Sqrt[13])*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticE[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6] - I*Sqrt[2]*(-5395 + 1247*Sqrt[13])*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6])/(420*Sqrt[3 + 5*x^2 + x^4])","C",0
153,1,229,279,0.2431077,"\int \left(2+3 x^2\right) \sqrt{3+5 x^2+x^4} \, dx","Integrate[(2 + 3*x^2)*Sqrt[3 + 5*x^2 + x^4],x]","\frac{i \sqrt{2} \left(23 \sqrt{13}-130\right) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)-23 i \sqrt{2} \left(\sqrt{13}-5\right) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} E\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)+2 x \left(9 x^6+70 x^4+152 x^2+75\right)}{30 \sqrt{x^4+5 x^2+3}}","-\frac{23 x \left(2 x^2+\sqrt{13}+5\right)}{15 \sqrt{x^4+5 x^2+3}}+\frac{1}{15} x \left(9 x^2+25\right) \sqrt{x^4+5 x^2+3}+\frac{\sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) F\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{\sqrt{6 \left(5+\sqrt{13}\right)} \sqrt{x^4+5 x^2+3}}+\frac{23 \sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) E\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{15 \sqrt{x^4+5 x^2+3}}",1,"(2*x*(75 + 152*x^2 + 70*x^4 + 9*x^6) - (23*I)*Sqrt[2]*(-5 + Sqrt[13])*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticE[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6] + I*Sqrt[2]*(-130 + 23*Sqrt[13])*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6])/(30*Sqrt[3 + 5*x^2 + x^4])","C",0
154,1,231,284,0.2619488,"\int \frac{\left(2+3 x^2\right) \sqrt{3+5 x^2+x^4}}{x^2} \, dx","Integrate[((2 + 3*x^2)*Sqrt[3 + 5*x^2 + x^4])/x^2,x]","\frac{-i \sqrt{2} \left(9 \sqrt{13}-13\right) x \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)+9 i \sqrt{2} \left(\sqrt{13}-5\right) x \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} E\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)+4 \left(x^6+3 x^4-7 x^2-6\right)}{4 x \sqrt{x^4+5 x^2+3}}","-\frac{\sqrt{x^4+5 x^2+3} \left(2-x^2\right)}{x}+\frac{9 x \left(2 x^2+\sqrt{13}+5\right)}{2 \sqrt{x^4+5 x^2+3}}+\frac{8 \sqrt{\frac{2}{3 \left(5+\sqrt{13}\right)}} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) F\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{\sqrt{x^4+5 x^2+3}}-\frac{3 \sqrt{\frac{3}{2} \left(5+\sqrt{13}\right)} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) E\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{2 \sqrt{x^4+5 x^2+3}}",1,"(4*(-6 - 7*x^2 + 3*x^4 + x^6) + (9*I)*Sqrt[2]*(-5 + Sqrt[13])*x*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticE[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6] - I*Sqrt[2]*(-13 + 9*Sqrt[13])*x*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6])/(4*x*Sqrt[3 + 5*x^2 + x^4])","C",0
155,1,237,305,0.2600743,"\int \frac{\left(2+3 x^2\right) \sqrt{3+5 x^2+x^4}}{x^4} \, dx","Integrate[((2 + 3*x^2)*Sqrt[3 + 5*x^2 + x^4])/x^4,x]","\frac{-i \sqrt{2} \left(32 \sqrt{13}-13\right) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} x^3 F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)+32 i \sqrt{2} \left(\sqrt{13}-5\right) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} x^3 E\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)-2 \left(37 x^6+191 x^4+141 x^2+18\right)}{18 x^3 \sqrt{x^4+5 x^2+3}}","-\frac{64 \sqrt{x^4+5 x^2+3}}{9 x}+\frac{32 x \left(2 x^2+\sqrt{13}+5\right)}{9 \sqrt{x^4+5 x^2+3}}+\frac{49 \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) F\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{3 \sqrt{6 \left(5+\sqrt{13}\right)} \sqrt{x^4+5 x^2+3}}-\frac{16 \sqrt{\frac{2}{3} \left(5+\sqrt{13}\right)} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) E\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{9 \sqrt{x^4+5 x^2+3}}-\frac{\sqrt{x^4+5 x^2+3} \left(2-9 x^2\right)}{3 x^3}",1,"(-2*(18 + 141*x^2 + 191*x^4 + 37*x^6) + (32*I)*Sqrt[2]*(-5 + Sqrt[13])*x^3*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticE[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6] - I*Sqrt[2]*(-13 + 32*Sqrt[13])*x^3*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6])/(18*x^3*Sqrt[3 + 5*x^2 + x^4])","C",0
156,1,81,127,0.0354231,"\int x^5 \left(2+3 x^2\right) \left(3+5 x^2+x^4\right)^{3/2} \, dx","Integrate[x^5*(2 + 3*x^2)*(3 + 5*x^2 + x^4)^(3/2),x]","\frac{2 \sqrt{x^4+5 x^2+3} \left(46080 x^{12}+323840 x^{10}+482944 x^8+154800 x^6+283304 x^4-1499570 x^2+9546951\right)-38737335 \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)}{430080}","\frac{3}{14} \left(x^4+5 x^2+3\right)^{5/2} x^4+\frac{\left(3313-1070 x^2\right) \left(x^4+5 x^2+3\right)^{5/2}}{1680}-\frac{2183}{768} \left(2 x^2+5\right) \left(x^4+5 x^2+3\right)^{3/2}+\frac{28379 \left(2 x^2+5\right) \sqrt{x^4+5 x^2+3}}{2048}-\frac{368927 \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)}{4096}",1,"(2*Sqrt[3 + 5*x^2 + x^4]*(9546951 - 1499570*x^2 + 283304*x^4 + 154800*x^6 + 482944*x^8 + 323840*x^10 + 46080*x^12) - 38737335*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/430080","A",1
157,1,76,106,0.0307019,"\int x^3 \left(2+3 x^2\right) \left(3+5 x^2+x^4\right)^{3/2} \, dx","Integrate[x^3*(2 + 3*x^2)*(3 + 5*x^2 + x^4)^(3/2),x]","\frac{311805 \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)+2 \sqrt{x^4+5 x^2+3} \left(1280 x^{10}+9344 x^8+14960 x^6+5064 x^4+12390 x^2-77229\right)}{10240}","-\frac{1}{40} \left(27-10 x^2\right) \left(x^4+5 x^2+3\right)^{5/2}+\frac{123}{128} \left(2 x^2+5\right) \left(x^4+5 x^2+3\right)^{3/2}-\frac{4797 \left(2 x^2+5\right) \sqrt{x^4+5 x^2+3}}{1024}+\frac{62361 \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)}{2048}",1,"(2*Sqrt[3 + 5*x^2 + x^4]*(-77229 + 12390*x^2 + 5064*x^4 + 14960*x^6 + 9344*x^8 + 1280*x^10) + 311805*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/10240","A",1
158,1,71,99,0.0245345,"\int x \left(2+3 x^2\right) \left(3+5 x^2+x^4\right)^{3/2} \, dx","Integrate[x*(2 + 3*x^2)*(3 + 5*x^2 + x^4)^(3/2),x]","\frac{2 \sqrt{x^4+5 x^2+3} \left(384 x^8+2960 x^6+5304 x^4+2170 x^2+7581\right)-27885 \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)}{2560}","\frac{3}{10} \left(x^4+5 x^2+3\right)^{5/2}-\frac{11}{32} \left(2 x^2+5\right) \left(x^4+5 x^2+3\right)^{3/2}+\frac{429}{256} \left(2 x^2+5\right) \sqrt{x^4+5 x^2+3}-\frac{5577}{512} \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)",1,"(2*Sqrt[3 + 5*x^2 + x^4]*(7581 + 2170*x^2 + 5304*x^4 + 2960*x^6 + 384*x^8) - 27885*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/2560","A",1
159,1,104,119,0.0578903,"\int \frac{\left(2+3 x^2\right) \left(3+5 x^2+x^4\right)^{3/2}}{x} \, dx","Integrate[((2 + 3*x^2)*(3 + 5*x^2 + x^4)^(3/2))/x,x]","\frac{2401}{256} \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)-3 \sqrt{3} \tanh ^{-1}\left(\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right)+\frac{1}{384} \sqrt{x^4+5 x^2+3} \left(144 x^6+1208 x^4+2650 x^2+2061\right)","\frac{1}{48} \left(18 x^2+61\right) \left(x^4+5 x^2+3\right)^{3/2}+\frac{1}{128} \left(199-74 x^2\right) \sqrt{x^4+5 x^2+3}+\frac{2401}{256} \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)-3 \sqrt{3} \tanh ^{-1}\left(\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right)",1,"(Sqrt[3 + 5*x^2 + x^4]*(2061 + 2650*x^2 + 1208*x^4 + 144*x^6))/384 + (2401*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/256 - 3*Sqrt[3]*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])]","A",1
160,1,107,122,0.0536123,"\int \frac{\left(2+3 x^2\right) \left(3+5 x^2+x^4\right)^{3/2}}{x^3} \, dx","Integrate[((2 + 3*x^2)*(3 + 5*x^2 + x^4)^(3/2))/x^3,x]","\frac{609}{32} \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)-12 \sqrt{3} \tanh ^{-1}\left(\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right)+\frac{\sqrt{x^4+5 x^2+3} \left(8 x^6+78 x^4+271 x^2-48\right)}{16 x^2}","-\frac{\left(2-x^2\right) \left(x^4+5 x^2+3\right)^{3/2}}{2 x^2}+\frac{3}{16} \left(18 x^2+109\right) \sqrt{x^4+5 x^2+3}+\frac{609}{32} \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)-12 \sqrt{3} \tanh ^{-1}\left(\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right)",1,"(Sqrt[3 + 5*x^2 + x^4]*(-48 + 271*x^2 + 78*x^4 + 8*x^6))/(16*x^2) + (609*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/32 - 12*Sqrt[3]*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])]","A",1
161,1,107,127,0.0621805,"\int \frac{\left(2+3 x^2\right) \left(3+5 x^2+x^4\right)^{3/2}}{x^5} \, dx","Integrate[((2 + 3*x^2)*(3 + 5*x^2 + x^4)^(3/2))/x^5,x]","\frac{1}{16} \left(453 \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)-254 \sqrt{3} \tanh ^{-1}\left(\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right)+\frac{2 \sqrt{x^4+5 x^2+3} \left(6 x^6+83 x^4-86 x^2-12\right)}{x^4}\right)","-\frac{\left(2-3 x^2\right) \left(x^4+5 x^2+3\right)^{3/2}}{4 x^4}-\frac{3 \left(28-19 x^2\right) \sqrt{x^4+5 x^2+3}}{8 x^2}+\frac{453}{16} \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)-\frac{127}{8} \sqrt{3} \tanh ^{-1}\left(\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right)",1,"((2*Sqrt[3 + 5*x^2 + x^4]*(-12 - 86*x^2 + 83*x^4 + 6*x^6))/x^4 + 453*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])] - 254*Sqrt[3]*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])])/16","A",1
162,1,107,127,0.0512476,"\int \frac{\left(2+3 x^2\right) \left(3+5 x^2+x^4\right)^{3/2}}{x^7} \, dx","Integrate[((2 + 3*x^2)*(3 + 5*x^2 + x^4)^(3/2))/x^7,x]","\frac{1}{72} \left(882 \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)-527 \sqrt{3} \tanh ^{-1}\left(\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right)+\frac{6 \sqrt{x^4+5 x^2+3} \left(18 x^6-141 x^4-62 x^2-12\right)}{x^6}\right)","-\frac{\left(67-32 x^2\right) \sqrt{x^4+5 x^2+3}}{12 x^2}+\frac{49}{4} \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)-\frac{527 \tanh ^{-1}\left(\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right)}{24 \sqrt{3}}-\frac{\left(7 x^2+2\right) \left(x^4+5 x^2+3\right)^{3/2}}{6 x^6}",1,"((6*Sqrt[3 + 5*x^2 + x^4]*(-12 - 62*x^2 - 141*x^4 + 18*x^6))/x^6 + 882*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])] - 527*Sqrt[3]*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])])/72","A",1
163,1,249,356,0.2702871,"\int x^4 \left(2+3 x^2\right) \left(3+5 x^2+x^4\right)^{3/2} \, dx","Integrate[x^4*(2 + 3*x^2)*(3 + 5*x^2 + x^4)^(3/2),x]","\frac{-i \sqrt{2} \left(176723 \sqrt{13}-757315\right) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)+176723 i \sqrt{2} \left(\sqrt{13}-5\right) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} E\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)+4 x \left(495 x^{14}+6015 x^{12}+24635 x^{10}+39650 x^8+29003 x^6+3055 x^4-93991 x^2-63150\right)}{8580 \sqrt{x^4+5 x^2+3}}","-\frac{4210}{429} \sqrt{x^4+5 x^2+3} x+\frac{176723 \left(2 x^2+\sqrt{13}+5\right) x}{4290 \sqrt{x^4+5 x^2+3}}+\frac{2105 \sqrt{\frac{2}{3 \left(5+\sqrt{13}\right)}} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) F\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{143 \sqrt{x^4+5 x^2+3}}-\frac{176723 \sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) E\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{4290 \sqrt{x^4+5 x^2+3}}+\frac{1}{143} \left(33 x^2+71\right) \left(x^4+5 x^2+3\right)^{3/2} x^5-\frac{1}{429} \left(272 x^2+283\right) \sqrt{x^4+5 x^2+3} x^5+\frac{1251}{715} \sqrt{x^4+5 x^2+3} x^3",1,"(4*x*(-63150 - 93991*x^2 + 3055*x^4 + 29003*x^6 + 39650*x^8 + 24635*x^10 + 6015*x^12 + 495*x^14) + (176723*I)*Sqrt[2]*(-5 + Sqrt[13])*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticE[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6] - I*Sqrt[2]*(-757315 + 176723*Sqrt[13])*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6])/(8580*Sqrt[3 + 5*x^2 + x^4])","C",0
164,1,244,331,0.2535887,"\int x^2 \left(2+3 x^2\right) \left(3+5 x^2+x^4\right)^{3/2} \, dx","Integrate[x^2*(2 + 3*x^2)*(3 + 5*x^2 + x^4)^(3/2),x]","\frac{i \sqrt{2} \left(49949 \sqrt{13}-212680\right) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)-49949 i \sqrt{2} \left(\sqrt{13}-5\right) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} E\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)+2 x \left(945 x^{12}+11795 x^{10}+50075 x^8+84962 x^6+69535 x^4+74681 x^2+37065\right)}{6930 \sqrt{x^4+5 x^2+3}}","\frac{353}{99} \sqrt{x^4+5 x^2+3} x-\frac{49949 \left(2 x^2+\sqrt{13}+5\right) x}{3465 \sqrt{x^4+5 x^2+3}}-\frac{353 \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) F\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{33 \sqrt{6 \left(5+\sqrt{13}\right)} \sqrt{x^4+5 x^2+3}}+\frac{49949 \sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) E\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{3465 \sqrt{x^4+5 x^2+3}}+\frac{1}{99} \left(27 x^2+67\right) \left(x^4+5 x^2+3\right)^{3/2} x^3-\frac{\left(890 x^2+911\right) \sqrt{x^4+5 x^2+3} x^3}{1155}",1,"(2*x*(37065 + 74681*x^2 + 69535*x^4 + 84962*x^6 + 50075*x^8 + 11795*x^10 + 945*x^12) - (49949*I)*Sqrt[2]*(-5 + Sqrt[13])*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticE[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6] + I*Sqrt[2]*(-212680 + 49949*Sqrt[13])*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6])/(6930*Sqrt[3 + 5*x^2 + x^4])","C",0
165,-1,0,308,0,"\int \left(2+3 x^2\right) \left(3+5 x^2+x^4\right)^{3/2} \, dx","Integrate[(2 + 3*x^2)*(3 + 5*x^2 + x^4)^(3/2),x]","\text{\$Aborted}","\frac{1}{3} x \left(x^2+3\right) \left(x^4+5 x^2+3\right)^{3/2}-\frac{1}{15} x \left(12 x^2+5\right) \sqrt{x^4+5 x^2+3}+\frac{203 x \left(2 x^2+\sqrt{13}+5\right)}{30 \sqrt{x^4+5 x^2+3}}+\frac{5 \sqrt{\frac{2}{3 \left(5+\sqrt{13}\right)}} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) F\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{\sqrt{x^4+5 x^2+3}}-\frac{203 \sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) E\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{30 \sqrt{x^4+5 x^2+3}}",1,"$Aborted","F",-1
166,1,235,312,0.2611834,"\int \frac{\left(2+3 x^2\right) \left(3+5 x^2+x^4\right)^{3/2}}{x^2} \, dx","Integrate[((2 + 3*x^2)*(3 + 5*x^2 + x^4)^(3/2))/x^2,x]","\frac{30 x^{10}+418 x^8+2130 x^6+3884 x^4-i \sqrt{2} \left(412 \sqrt{13}-65\right) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} x F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)+412 i \sqrt{2} \left(\sqrt{13}-5\right) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} x E\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)-1260}{70 x \sqrt{x^4+5 x^2+3}}","-\frac{\left(14-3 x^2\right) \left(x^4+5 x^2+3\right)^{3/2}}{7 x}+\frac{1}{35} x \left(129 x^2+655\right) \sqrt{x^4+5 x^2+3}+\frac{412 x \left(2 x^2+\sqrt{13}+5\right)}{35 \sqrt{x^4+5 x^2+3}}+\frac{19 \sqrt{\frac{3}{2 \left(5+\sqrt{13}\right)}} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) F\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{\sqrt{x^4+5 x^2+3}}-\frac{206 \sqrt{\frac{2}{3} \left(5+\sqrt{13}\right)} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) E\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{35 \sqrt{x^4+5 x^2+3}}",1,"(-1260 + 3884*x^4 + 2130*x^6 + 418*x^8 + 30*x^10 + (412*I)*Sqrt[2]*(-5 + Sqrt[13])*x*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticE[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6] - I*Sqrt[2]*(-65 + 412*Sqrt[13])*x*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6])/(70*x*Sqrt[3 + 5*x^2 + x^4])","C",0
167,1,247,314,0.2940312,"\int \frac{\left(2+3 x^2\right) \left(3+5 x^2+x^4\right)^{3/2}}{x^4} \, dx","Integrate[((2 + 3*x^2)*(3 + 5*x^2 + x^4)^(3/2))/x^4,x]","\frac{-13 i \sqrt{2} \left(73 \sqrt{13}-65\right) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} x^3 F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)+949 i \sqrt{2} \left(\sqrt{13}-5\right) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} x^3 E\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)+4 \left(9 x^{10}+145 x^8+192 x^6-1405 x^4-1155 x^2-90\right)}{60 x^3 \sqrt{x^4+5 x^2+3}}","-\frac{13 \left(24-5 x^2\right) \sqrt{x^4+5 x^2+3}}{15 x}+\frac{949 x \left(2 x^2+\sqrt{13}+5\right)}{30 \sqrt{x^4+5 x^2+3}}+\frac{65 \sqrt{\frac{2}{3 \left(5+\sqrt{13}\right)}} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) F\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{\sqrt{x^4+5 x^2+3}}-\frac{949 \sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) E\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{30 \sqrt{x^4+5 x^2+3}}-\frac{\left(10-9 x^2\right) \left(x^4+5 x^2+3\right)^{3/2}}{15 x^3}",1,"(4*(-90 - 1155*x^2 - 1405*x^4 + 192*x^6 + 145*x^8 + 9*x^10) + (949*I)*Sqrt[2]*(-5 + Sqrt[13])*x^3*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticE[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6] - (13*I)*Sqrt[2]*(-65 + 73*Sqrt[13])*x^3*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6])/(60*x^3*Sqrt[3 + 5*x^2 + x^4])","C",0
168,1,244,331,0.2809383,"\int \frac{\left(2+3 x^2\right) \left(3+5 x^2+x^4\right)^{3/2}}{x^6} \, dx","Integrate[((2 + 3*x^2)*(3 + 5*x^2 + x^4)^(3/2))/x^6,x]","\frac{30 x^{10}-634 x^8-4040 x^6-3438 x^4-810 x^2-i \sqrt{2} \left(361 \sqrt{13}-260\right) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} x^5 F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)+361 i \sqrt{2} \left(\sqrt{13}-5\right) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} x^5 E\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)-108}{30 x^5 \sqrt{x^4+5 x^2+3}}","-\frac{722 \sqrt{x^4+5 x^2+3}}{15 x}+\frac{361 x \left(2 x^2+\sqrt{13}+5\right)}{15 \sqrt{x^4+5 x^2+3}}+\frac{103 \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) F\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{\sqrt{6 \left(5+\sqrt{13}\right)} \sqrt{x^4+5 x^2+3}}-\frac{361 \sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) E\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{15 \sqrt{x^4+5 x^2+3}}-\frac{\left(2-5 x^2\right) \left(x^4+5 x^2+3\right)^{3/2}}{5 x^5}-\frac{\left(40-87 x^2\right) \sqrt{x^4+5 x^2+3}}{5 x^3}",1,"(-108 - 810*x^2 - 3438*x^4 - 4040*x^6 - 634*x^8 + 30*x^10 + (361*I)*Sqrt[2]*(-5 + Sqrt[13])*x^5*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticE[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6] - I*Sqrt[2]*(-260 + 361*Sqrt[13])*x^5*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6])/(30*x^5*Sqrt[3 + 5*x^2 + x^4])","C",0
169,1,139,153,0.1109093,"\int \frac{x^5 \left(A+B x^2\right)}{\sqrt{a+b x^2+c x^4}} \, dx","Integrate[(x^5*(A + B*x^2))/Sqrt[a + b*x^2 + c*x^4],x]","\frac{2 \sqrt{c} \sqrt{a+b x^2+c x^4} \left(4 c \left(-4 a B+3 A c x^2+2 B c x^4\right)-2 b c \left(9 A+5 B x^2\right)+15 b^2 B\right)-3 \left(8 a A c^2-12 a b B c-6 A b^2 c+5 b^3 B\right) \tanh ^{-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{\sqrt{a+b x^2+c x^4} \left(-16 a B c-2 c x^2 (5 b B-6 A c)-18 A b c+15 b^2 B\right)}{48 c^3}-\frac{\left(8 a A c^2-12 a b B c-6 A b^2 c+5 b^3 B\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{B 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*B - 2*b*c*(9*A + 5*B*x^2) + 4*c*(-4*a*B + 3*A*c*x^2 + 2*B*c*x^4)) - 3*(5*b^3*B - 6*A*b^2*c - 12*a*b*B*c + 8*a*A*c^2)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(96*c^(7/2))","A",1
170,1,101,100,0.046781,"\int \frac{x^3 \left(A+B x^2\right)}{\sqrt{a+b x^2+c x^4}} \, dx","Integrate[(x^3*(A + B*x^2))/Sqrt[a + b*x^2 + c*x^4],x]","\frac{\left(-4 a B c-4 A b c+3 b^2 B\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(4 A c-3 b B+2 B c x^2\right)}{16 c^{5/2}}","\frac{\left(-4 a B c-4 A b c+3 b^2 B\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{\sqrt{a+b x^2+c x^4} \left(-4 A c+3 b B-2 B c x^2\right)}{8 c^2}",1,"(2*Sqrt[c]*(-3*b*B + 4*A*c + 2*B*c*x^2)*Sqrt[a + b*x^2 + c*x^4] + (3*b^2*B - 4*A*b*c - 4*a*B*c)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(16*c^(5/2))","A",1
171,1,78,76,0.0253689,"\int \frac{x \left(A+B x^2\right)}{\sqrt{a+b x^2+c x^4}} \, dx","Integrate[(x*(A + B*x^2))/Sqrt[a + b*x^2 + c*x^4],x]","\frac{1}{2} \left(\frac{(2 A c-b B) \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}}+\frac{B \sqrt{a+b x^2+c x^4}}{c}\right)","\frac{B \sqrt{a+b x^2+c x^4}}{2 c}-\frac{(b B-2 A c) \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,"((B*Sqrt[a + b*x^2 + c*x^4])/c + ((-(b*B) + 2*A*c)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(2*c^(3/2)))/2","A",1
172,1,89,90,0.0277127,"\int \frac{A+B x^2}{x \sqrt{a+b x^2+c x^4}} \, dx","Integrate[(A + B*x^2)/(x*Sqrt[a + b*x^2 + c*x^4]),x]","\frac{1}{2} \left(\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}}-\frac{A \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{\sqrt{a}}\right)","\frac{B \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{A \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,"(-((A*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/Sqrt[a]) + (B*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/Sqrt[c])/2","A",1
173,1,82,80,0.0351965,"\int \frac{A+B x^2}{x^3 \sqrt{a+b x^2+c x^4}} \, dx","Integrate[(A + B*x^2)/(x^3*Sqrt[a + b*x^2 + c*x^4]),x]","\frac{1}{2} \left(\frac{(A b-2 a B) \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{A \sqrt{a+b x^2+c x^4}}{a x^2}\right)","\frac{(A b-2 a 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{A \sqrt{a+b x^2+c x^4}}{2 a x^2}",1,"(-((A*Sqrt[a + b*x^2 + c*x^4])/(a*x^2)) + ((A*b - 2*a*B)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/(2*a^(3/2)))/2","A",1
174,1,107,124,0.0754414,"\int \frac{A+B x^2}{x^5 \sqrt{a+b x^2+c x^4}} \, dx","Integrate[(A + B*x^2)/(x^5*Sqrt[a + b*x^2 + c*x^4]),x]","\frac{\left(4 a A c+4 a b B-3 A 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{\sqrt{a+b x^2+c x^4} \left(3 A b x^2-2 a \left(A+2 B x^2\right)\right)}{8 a^2 x^4}","-\frac{\left(-4 a A c-4 a b B+3 A 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{(3 A b-4 a B) \sqrt{a+b x^2+c x^4}}{8 a^2 x^2}-\frac{A \sqrt{a+b x^2+c x^4}}{4 a x^4}",1,"(Sqrt[a + b*x^2 + c*x^4]*(3*A*b*x^2 - 2*a*(A + 2*B*x^2)))/(8*a^2*x^4) + ((-3*A*b^2 + 4*a*b*B + 4*a*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
175,1,148,177,0.1083971,"\int \frac{A+B x^2}{x^7 \sqrt{a+b x^2+c x^4}} \, dx","Integrate[(A + B*x^2)/(x^7*Sqrt[a + b*x^2 + c*x^4]),x]","\frac{\left(8 a^2 B c-12 a A b c-6 a b^2 B+5 A b^3\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(-4 a^2 \left(2 A+3 B x^2\right)+2 a \left(5 A b x^2+8 A c x^4+9 b B x^4\right)-15 A b^2 x^4\right)}{48 a^3 x^6}","-\frac{\sqrt{a+b x^2+c x^4} \left(-16 a A c-18 a b B+15 A b^2\right)}{48 a^3 x^2}+\frac{(5 A b-6 a B) \sqrt{a+b x^2+c x^4}}{24 a^2 x^4}+\frac{\left(8 a^2 B c-12 a A b c-6 a b^2 B+5 A b^3\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{A \sqrt{a+b x^2+c x^4}}{6 a x^6}",1,"(Sqrt[a + b*x^2 + c*x^4]*(-15*A*b^2*x^4 - 4*a^2*(2*A + 3*B*x^2) + 2*a*(5*A*b*x^2 + 9*b*B*x^4 + 8*A*c*x^4)))/(48*a^3*x^6) + ((5*A*b^3 - 6*a*b^2*B - 12*a*A*b*c + 8*a^2*B*c)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/(32*a^(7/2))","A",1
176,1,532,403,2.2110678,"\int \frac{x^4 \left(A+B x^2\right)}{\sqrt{a+b x^2+c x^4}} \, dx","Integrate[(x^4*(A + B*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 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} \left(-9 a B c-10 A b c+8 b^2 B\right) E\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)+4 c x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left(a+b x^2+c x^4\right) \left(5 A c-4 b B+3 B c x^2\right)-i \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}}} \left(2 b^2 \left(4 B \sqrt{b^2-4 a c}+5 A c\right)+b c \left(17 a B-10 A \sqrt{b^2-4 a c}\right)-a c \left(9 B \sqrt{b^2-4 a c}+10 A c\right)-8 b^3 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}}{b-\sqrt{b^2-4 a c}}\right)}{60 c^3 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","\frac{x \sqrt{a+b x^2+c x^4} \left(-9 a B c-10 A b c+8 b^2 B\right)}{15 c^{5/2} \left(\sqrt{a}+\sqrt{c} x^2\right)}+\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}} \left(\sqrt{a} \sqrt{c} (4 b B-5 A c)-9 a B c-10 A b c+8 b^2 B\right) 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^{11/4} \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}} \left(-9 a B c-10 A b c+8 b^2 B\right) 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^{11/4} \sqrt{a+b x^2+c x^4}}-\frac{x \sqrt{a+b x^2+c x^4} (4 b B-5 A c)}{15 c^2}+\frac{B x^3 \sqrt{a+b x^2+c x^4}}{5 c}",1,"(4*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*(-4*b*B + 5*A*c + 3*B*c*x^2)*(a + b*x^2 + c*x^4) + I*(8*b^2*B - 10*A*b*c - 9*a*B*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^3*B + b*c*(17*a*B - 10*A*Sqrt[b^2 - 4*a*c]) + 2*b^2*(5*A*c + 4*B*Sqrt[b^2 - 4*a*c]) - a*c*(10*A*c + 9*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])])/(60*c^3*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[a + b*x^2 + c*x^4])","C",1
177,1,479,336,1.3734496,"\int \frac{x^2 \left(A+B x^2\right)}{\sqrt{a+b x^2+c x^4}} \, dx","Integrate[(x^2*(A + B*x^2))/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 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} \left(-3 A c \sqrt{b^2-4 a c}+2 b B \sqrt{b^2-4 a c}+2 a B c+3 A b c-2 b^2 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}}{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}}} (2 b B-3 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 B c x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left(a+b x^2+c x^4\right)}{12 c^2 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","-\frac{x \sqrt{a+b x^2+c x^4} (2 b B-3 A c)}{3 c^{3/2} \left(\sqrt{a}+\sqrt{c} x^2\right)}-\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}} \left(\sqrt{a} B \sqrt{c}-3 A c+2 b B\right) 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{\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}} (2 b B-3 A c) 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{B x \sqrt{a+b x^2+c x^4}}{3 c}",1,"(4*B*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*(a + b*x^2 + c*x^4) - I*(2*b*B - 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*(-2*b^2*B + 3*A*b*c + 2*a*B*c + 2*b*B*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])])/(12*c^2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[a + b*x^2 + c*x^4])","C",1
178,1,302,283,0.2543563,"\int \frac{A+B x^2}{\sqrt{a+b x^2+c x^4}} \, dx","Integrate[(A + B*x^2)/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} \left(\left(-B \sqrt{b^2-4 a c}-2 A c+b 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}}{b-\sqrt{b^2-4 a c}}\right)+B \left(\sqrt{b^2-4 a c}-b\right) E\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)\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) \left(\frac{A \sqrt{c}}{\sqrt{a}}+B\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} 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} \sqrt{a+b x^2+c x^4}}+\frac{B x \sqrt{a+b x^2+c x^4}}{\sqrt{c} \left(\sqrt{a}+\sqrt{c} x^2\right)}",1,"((I/2)*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])]*(B*(-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])] + (b*B - 2*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
179,1,448,312,1.0696902,"\int \frac{A+B x^2}{x^2 \sqrt{a+b x^2+c x^4}} \, dx","Integrate[(A + B*x^2)/(x^2*Sqrt[a + b*x^2 + c*x^4]),x]","\frac{-i x \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}}} \left(A \left(\sqrt{b^2-4 a c}-b\right)+2 a 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}}{b-\sqrt{b^2-4 a c}}\right)+i A 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)-4 A \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left(a+b x^2+c x^4\right)}{4 a x \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) \left(\sqrt{a} B+A \sqrt{c}\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[4]{c} \sqrt{a+b x^2+c x^4}}-\frac{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)}{a^{3/4} \sqrt{a+b x^2+c x^4}}-\frac{A \sqrt{a+b x^2+c x^4}}{a x}+\frac{A \sqrt{c} x \sqrt{a+b x^2+c x^4}}{a \left(\sqrt{a}+\sqrt{c} x^2\right)}",1,"(-4*A*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*(a + b*x^2 + c*x^4) + I*A*(-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*(2*a*B + A*(-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])]*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*a*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*Sqrt[a + b*x^2 + c*x^4])","C",1
180,1,373,376,0.6954743,"\int \frac{A+B x^2}{x^4 \sqrt{a+b x^2+c x^4}} \, dx","Integrate[(A + B*x^2)/(x^4*Sqrt[a + b*x^2 + c*x^4]),x]","\frac{-\frac{4 \left(a+b x^2+c x^4\right) \left(a \left(A+3 B x^2\right)-2 A b x^2\right)}{x^3}+\frac{i \sqrt{2} \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(\left(2 A \left(b \sqrt{b^2-4 a c}+a c-b^2\right)+3 a B \left(b-\sqrt{b^2-4 a c}\right)\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)-\left(\sqrt{b^2-4 a c}-b\right) (2 A b-3 a 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)\right)}{\sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}}}}{12 a^2 \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}} \left(\sqrt{a} A \sqrt{c}-3 a B+2 A b\right) 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{\sqrt[4]{c} \left(\sqrt{a}+\sqrt{c} x^2\right) (2 A b-3 a B) \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 A b-3 a B) \sqrt{a+b x^2+c x^4}}{3 a^2 x}-\frac{\sqrt{c} x (2 A b-3 a B) \sqrt{a+b x^2+c x^4}}{3 a^2 \left(\sqrt{a}+\sqrt{c} x^2\right)}-\frac{A \sqrt{a+b x^2+c x^4}}{3 a x^3}",1,"((-4*(a + b*x^2 + c*x^4)*(-2*A*b*x^2 + a*(A + 3*B*x^2)))/x^3 + (I*Sqrt[2]*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])]*(-((2*A*b - 3*a*B)*(-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])]) + (3*a*B*(b - Sqrt[b^2 - 4*a*c]) + 2*A*(-b^2 + 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])])/(12*a^2*Sqrt[a + b*x^2 + c*x^4])","C",1
181,1,66,98,0.0280852,"\int \frac{x^7 \left(2+3 x^2\right)}{\sqrt{3+5 x^2+x^4}} \, dx","Integrate[(x^7*(2 + 3*x^2))/Sqrt[3 + 5*x^2 + x^4],x]","\frac{1}{768} \left(98403 \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)+2 \sqrt{x^4+5 x^2+3} \left(144 x^6-712 x^4+3802 x^2-24243\right)\right)","-\frac{89}{48} \sqrt{x^4+5 x^2+3} x^4-\frac{1}{384} \left(24243-3802 x^2\right) \sqrt{x^4+5 x^2+3}+\frac{32801}{256} \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)+\frac{3}{8} \sqrt{x^4+5 x^2+3} x^6",1,"(2*Sqrt[3 + 5*x^2 + x^4]*(-24243 + 3802*x^2 - 712*x^4 + 144*x^6) + 98403*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/768","A",1
182,1,61,77,0.020033,"\int \frac{x^5 \left(2+3 x^2\right)}{\sqrt{3+5 x^2+x^4}} \, dx","Integrate[(x^5*(2 + 3*x^2))/Sqrt[3 + 5*x^2 + x^4],x]","\frac{1}{32} \left(2 \sqrt{x^4+5 x^2+3} \left(8 x^4-42 x^2+267\right)-1083 \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)\right)","\frac{1}{2} \sqrt{x^4+5 x^2+3} x^4+\frac{3}{16} \left(89-14 x^2\right) \sqrt{x^4+5 x^2+3}-\frac{1083}{32} \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)",1,"(2*Sqrt[3 + 5*x^2 + x^4]*(267 - 42*x^2 + 8*x^4) - 1083*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/32","A",1
183,1,56,56,0.0158798,"\int \frac{x^3 \left(2+3 x^2\right)}{\sqrt{3+5 x^2+x^4}} \, dx","Integrate[(x^3*(2 + 3*x^2))/Sqrt[3 + 5*x^2 + x^4],x]","\frac{1}{16} \left(2 \sqrt{x^4+5 x^2+3} \left(6 x^2-37\right)+149 \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)\right)","\frac{149}{16} \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)-\frac{1}{8} \left(37-6 x^2\right) \sqrt{x^4+5 x^2+3}",1,"(2*(-37 + 6*x^2)*Sqrt[3 + 5*x^2 + x^4] + 149*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/16","A",1
184,1,49,49,0.0094318,"\int \frac{x \left(2+3 x^2\right)}{\sqrt{3+5 x^2+x^4}} \, dx","Integrate[(x*(2 + 3*x^2))/Sqrt[3 + 5*x^2 + x^4],x]","\frac{3}{2} \sqrt{x^4+5 x^2+3}-\frac{11}{4} \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)","\frac{3}{2} \sqrt{x^4+5 x^2+3}-\frac{11}{4} \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)",1,"(3*Sqrt[3 + 5*x^2 + x^4])/2 - (11*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/4","A",1
185,1,69,69,0.0134627,"\int \frac{2+3 x^2}{x \sqrt{3+5 x^2+x^4}} \, dx","Integrate[(2 + 3*x^2)/(x*Sqrt[3 + 5*x^2 + x^4]),x]","\frac{3}{2} \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)-\frac{\tanh ^{-1}\left(\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right)}{\sqrt{3}}","\frac{3}{2} \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)-\frac{\tanh ^{-1}\left(\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right)}{\sqrt{3}}",1,"(3*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/2 - ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])]/Sqrt[3]","A",1
186,1,62,62,0.0146008,"\int \frac{2+3 x^2}{x^3 \sqrt{3+5 x^2+x^4}} \, dx","Integrate[(2 + 3*x^2)/(x^3*Sqrt[3 + 5*x^2 + x^4]),x]","-\frac{\sqrt{x^4+5 x^2+3}}{3 x^2}-\frac{2 \tanh ^{-1}\left(\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right)}{3 \sqrt{3}}","-\frac{\sqrt{x^4+5 x^2+3}}{3 x^2}-\frac{2 \tanh ^{-1}\left(\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right)}{3 \sqrt{3}}",1,"-1/3*Sqrt[3 + 5*x^2 + x^4]/x^2 - (2*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])])/(3*Sqrt[3])","A",1
187,1,67,83,0.0223793,"\int \frac{2+3 x^2}{x^5 \sqrt{3+5 x^2+x^4}} \, dx","Integrate[(2 + 3*x^2)/(x^5*Sqrt[3 + 5*x^2 + x^4]),x]","\frac{1}{8} \sqrt{3} \tanh ^{-1}\left(\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right)-\frac{\left(x^2+2\right) \sqrt{x^4+5 x^2+3}}{12 x^4}","-\frac{\sqrt{x^4+5 x^2+3}}{12 x^2}-\frac{\sqrt{x^4+5 x^2+3}}{6 x^4}+\frac{1}{8} \sqrt{3} \tanh ^{-1}\left(\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right)",1,"-1/12*((2 + x^2)*Sqrt[3 + 5*x^2 + x^4])/x^4 + (Sqrt[3]*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])])/8","A",1
188,1,77,104,0.0235016,"\int \frac{2+3 x^2}{x^7 \sqrt{3+5 x^2+x^4}} \, dx","Integrate[(2 + 3*x^2)/(x^7*Sqrt[3 + 5*x^2 + x^4]),x]","\frac{6 \sqrt{x^4+5 x^2+3} \left(13 x^4-2 x^2-12\right)-61 \sqrt{3} x^6 \tanh ^{-1}\left(\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right)}{648 x^6}","\frac{13 \sqrt{x^4+5 x^2+3}}{108 x^2}-\frac{\sqrt{x^4+5 x^2+3}}{54 x^4}-\frac{61 \tanh ^{-1}\left(\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right)}{216 \sqrt{3}}-\frac{\sqrt{x^4+5 x^2+3}}{9 x^6}",1,"(6*Sqrt[3 + 5*x^2 + x^4]*(-12 - 2*x^2 + 13*x^4) - 61*Sqrt[3]*x^6*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])])/(648*x^6)","A",1
189,1,229,298,0.2713988,"\int \frac{x^4 \left(2+3 x^2\right)}{\sqrt{3+5 x^2+x^4}} \, dx","Integrate[(x^4*(2 + 3*x^2))/Sqrt[3 + 5*x^2 + x^4],x]","\frac{-i \sqrt{2} \left(419 \sqrt{13}-1795\right) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)+419 i \sqrt{2} \left(\sqrt{13}-5\right) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} E\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)+4 x \left(9 x^6-5 x^4-223 x^2-150\right)}{60 \sqrt{x^4+5 x^2+3}}","-\frac{10}{3} \sqrt{x^4+5 x^2+3} x+\frac{419 \left(2 x^2+\sqrt{13}+5\right) x}{30 \sqrt{x^4+5 x^2+3}}+\frac{5 \sqrt{\frac{2}{3 \left(5+\sqrt{13}\right)}} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) F\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{\sqrt{x^4+5 x^2+3}}-\frac{419 \sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) E\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{30 \sqrt{x^4+5 x^2+3}}+\frac{3}{5} \sqrt{x^4+5 x^2+3} x^3",1,"(4*x*(-150 - 223*x^2 - 5*x^4 + 9*x^6) + (419*I)*Sqrt[2]*(-5 + Sqrt[13])*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticE[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6] - I*Sqrt[2]*(-1795 + 419*Sqrt[13])*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6])/(60*Sqrt[3 + 5*x^2 + x^4])","C",0
190,1,222,270,0.2437143,"\int \frac{x^2 \left(2+3 x^2\right)}{\sqrt{3+5 x^2+x^4}} \, dx","Integrate[(x^2*(2 + 3*x^2))/Sqrt[3 + 5*x^2 + x^4],x]","\frac{i \sqrt{2} \left(4 \sqrt{13}-17\right) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)-4 i \sqrt{2} \left(\sqrt{13}-5\right) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} E\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)+2 x \left(x^4+5 x^2+3\right)}{2 \sqrt{x^4+5 x^2+3}}","\sqrt{x^4+5 x^2+3} x-\frac{4 \left(2 x^2+\sqrt{13}+5\right) x}{\sqrt{x^4+5 x^2+3}}-\frac{\sqrt{\frac{3}{2 \left(5+\sqrt{13}\right)}} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) F\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{\sqrt{x^4+5 x^2+3}}+\frac{2 \sqrt{\frac{2}{3} \left(5+\sqrt{13}\right)} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) E\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{\sqrt{x^4+5 x^2+3}}",1,"(2*x*(3 + 5*x^2 + x^4) - (4*I)*Sqrt[2]*(-5 + Sqrt[13])*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticE[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6] + I*Sqrt[2]*(-17 + 4*Sqrt[13])*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6])/(2*Sqrt[3 + 5*x^2 + x^4])","C",0
191,1,159,257,0.118537,"\int \frac{2+3 x^2}{\sqrt{3+5 x^2+x^4}} \, dx","Integrate[(2 + 3*x^2)/Sqrt[3 + 5*x^2 + x^4],x]","\frac{i \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} \left(\left(11-3 \sqrt{13}\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)+3 \left(\sqrt{13}-5\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)\right)}{2 \sqrt{2} \sqrt{x^4+5 x^2+3}}","\frac{3 x \left(2 x^2+\sqrt{13}+5\right)}{2 \sqrt{x^4+5 x^2+3}}+\frac{\sqrt{\frac{2}{3 \left(5+\sqrt{13}\right)}} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) F\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{\sqrt{x^4+5 x^2+3}}-\frac{\sqrt{\frac{3}{2} \left(5+\sqrt{13}\right)} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) E\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{2 \sqrt{x^4+5 x^2+3}}",1,"((I/2)*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*(3*(-5 + Sqrt[13])*EllipticE[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6] + (11 - 3*Sqrt[13])*EllipticF[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6]))/(Sqrt[2]*Sqrt[3 + 5*x^2 + x^4])","C",0
192,1,224,278,0.2471277,"\int \frac{2+3 x^2}{x^2 \sqrt{3+5 x^2+x^4}} \, dx","Integrate[(2 + 3*x^2)/(x^2*Sqrt[3 + 5*x^2 + x^4]),x]","\frac{-i \sqrt{2} \left(4+\sqrt{13}\right) x \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)+i \sqrt{2} \left(\sqrt{13}-5\right) x \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} E\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)-4 \left(x^4+5 x^2+3\right)}{6 x \sqrt{x^4+5 x^2+3}}","\frac{x \left(2 x^2+\sqrt{13}+5\right)}{3 \sqrt{x^4+5 x^2+3}}-\frac{2 \sqrt{x^4+5 x^2+3}}{3 x}+\frac{\sqrt{\frac{3}{2 \left(5+\sqrt{13}\right)}} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) F\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{\sqrt{x^4+5 x^2+3}}-\frac{\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) E\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{3 \sqrt{x^4+5 x^2+3}}",1,"(-4*(3 + 5*x^2 + x^4) + I*Sqrt[2]*(-5 + Sqrt[13])*x*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticE[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6] - I*Sqrt[2]*(4 + Sqrt[13])*x*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6])/(6*x*Sqrt[3 + 5*x^2 + x^4])","C",0
193,1,237,302,0.262789,"\int \frac{2+3 x^2}{x^4 \sqrt{3+5 x^2+x^4}} \, dx","Integrate[(2 + 3*x^2)/(x^4*Sqrt[3 + 5*x^2 + x^4]),x]","\frac{-i \sqrt{2} \left(7 \sqrt{13}-47\right) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} x^3 F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)+7 i \sqrt{2} \left(\sqrt{13}-5\right) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} x^3 E\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)-4 \left(7 x^6+41 x^4+51 x^2+18\right)}{108 x^3 \sqrt{x^4+5 x^2+3}}","\frac{7 x \left(2 x^2+\sqrt{13}+5\right)}{54 \sqrt{x^4+5 x^2+3}}-\frac{7 \sqrt{x^4+5 x^2+3}}{27 x}-\frac{\sqrt{\frac{2}{3 \left(5+\sqrt{13}\right)}} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) F\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{9 \sqrt{x^4+5 x^2+3}}-\frac{7 \sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) E\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{54 \sqrt{x^4+5 x^2+3}}-\frac{2 \sqrt{x^4+5 x^2+3}}{9 x^3}",1,"(-4*(18 + 51*x^2 + 41*x^4 + 7*x^6) + (7*I)*Sqrt[2]*(-5 + Sqrt[13])*x^3*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticE[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6] - I*Sqrt[2]*(-47 + 7*Sqrt[13])*x^3*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6])/(108*x^3*Sqrt[3 + 5*x^2 + x^4])","C",0
194,1,72,77,0.0182068,"\int \frac{x^5 \left(2+3 x^2\right)}{\left(3+5 x^2+x^4\right)^{3/2}} \, dx","Integrate[(x^5*(2 + 3*x^2))/(3 + 5*x^2 + x^4)^(3/2),x]","\frac{78 x^4+1198 x^2-533 \sqrt{x^4+5 x^2+3} \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)+798}{52 \sqrt{x^4+5 x^2+3}}","-\frac{\left(47 x^2+33\right) x^2}{13 \sqrt{x^4+5 x^2+3}}+\frac{133}{26} \sqrt{x^4+5 x^2+3}-\frac{41}{4} \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)",1,"(798 + 1198*x^2 + 78*x^4 - 533*Sqrt[3 + 5*x^2 + x^4]*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/(52*Sqrt[3 + 5*x^2 + x^4])","A",1
195,1,54,56,0.1210901,"\int \frac{x^3 \left(2+3 x^2\right)}{\left(3+5 x^2+x^4\right)^{3/2}} \, dx","Integrate[(x^3*(2 + 3*x^2))/(3 + 5*x^2 + x^4)^(3/2),x]","\frac{3}{2} \log \left(2 x^2+2 \sqrt{x^4+5 x^2+3}+5\right)-\frac{47 x^2+33}{13 \sqrt{x^4+5 x^2+3}}","\frac{-47 x^2-33}{13 \sqrt{x^4+5 x^2+3}}+\frac{3}{2} \tanh ^{-1}\left(\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right)",1,"-1/13*(33 + 47*x^2)/Sqrt[3 + 5*x^2 + x^4] + (3*Log[5 + 2*x^2 + 2*Sqrt[3 + 5*x^2 + x^4]])/2","A",1
196,1,25,25,0.0917156,"\int \frac{x \left(2+3 x^2\right)}{\left(3+5 x^2+x^4\right)^{3/2}} \, dx","Integrate[(x*(2 + 3*x^2))/(3 + 5*x^2 + x^4)^(3/2),x]","\frac{11 x^2+8}{13 \sqrt{x^4+5 x^2+3}}","\frac{11 x^2+8}{13 \sqrt{x^4+5 x^2+3}}",1,"(8 + 11*x^2)/(13*Sqrt[3 + 5*x^2 + x^4])","A",1
197,1,66,66,0.0242085,"\int \frac{2+3 x^2}{x \left(3+5 x^2+x^4\right)^{3/2}} \, dx","Integrate[(2 + 3*x^2)/(x*(3 + 5*x^2 + x^4)^(3/2)),x]","-\frac{8 x^2+7}{39 \sqrt{x^4+5 x^2+3}}-\frac{\tanh ^{-1}\left(\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right)}{3 \sqrt{3}}","\frac{-8 x^2-7}{39 \sqrt{x^4+5 x^2+3}}-\frac{\tanh ^{-1}\left(\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right)}{3 \sqrt{3}}",1,"-1/39*(7 + 8*x^2)/Sqrt[3 + 5*x^2 + x^4] - ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])]/(3*Sqrt[3])","A",1
198,1,88,90,0.0177631,"\int \frac{2+3 x^2}{x^3 \left(3+5 x^2+x^4\right)^{3/2}} \, dx","Integrate[(2 + 3*x^2)/(x^3*(3 + 5*x^2 + x^4)^(3/2)),x]","\frac{-6 x^4-54 x^2+13 \sqrt{3} \sqrt{x^4+5 x^2+3} x^2 \tanh ^{-1}\left(\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right)-39}{117 x^2 \sqrt{x^4+5 x^2+3}}","\frac{-8 x^2-7}{39 x^2 \sqrt{x^4+5 x^2+3}}-\frac{2 \sqrt{x^4+5 x^2+3}}{39 x^2}+\frac{\tanh ^{-1}\left(\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right)}{3 \sqrt{3}}",1,"(-39 - 54*x^2 - 6*x^4 + 13*Sqrt[3]*x^2*Sqrt[3 + 5*x^2 + x^4]*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])])/(117*x^2*Sqrt[3 + 5*x^2 + x^4])","A",1
199,1,219,307,0.2576018,"\int \frac{x^4 \left(2+3 x^2\right)}{\left(3+5 x^2+x^4\right)^{3/2}} \, dx","Integrate[(x^4*(2 + 3*x^2))/(3 + 5*x^2 + x^4)^(3/2),x]","\frac{-2 x \left(47 x^2+33\right)-i \sqrt{2} \left(43 \sqrt{13}-182\right) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)+43 i \sqrt{2} \left(\sqrt{13}-5\right) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} E\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)}{26 \sqrt{x^4+5 x^2+3}}","-\frac{11}{13} \sqrt{x^4+5 x^2+3} x+\frac{43 \left(2 x^2+\sqrt{13}+5\right) x}{13 \sqrt{x^4+5 x^2+3}}+\frac{11 \sqrt{\frac{3}{2 \left(5+\sqrt{13}\right)}} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) F\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{13 \sqrt{x^4+5 x^2+3}}-\frac{43 \sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) E\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{13 \sqrt{x^4+5 x^2+3}}+\frac{\left(11 x^2+8\right) x^3}{13 \sqrt{x^4+5 x^2+3}}",1,"(-2*x*(33 + 47*x^2) + (43*I)*Sqrt[2]*(-5 + Sqrt[13])*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticE[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6] - I*Sqrt[2]*(-182 + 43*Sqrt[13])*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6])/(26*Sqrt[3 + 5*x^2 + x^4])","C",0
200,1,219,286,0.2521354,"\int \frac{x^2 \left(2+3 x^2\right)}{\left(3+5 x^2+x^4\right)^{3/2}} \, dx","Integrate[(x^2*(2 + 3*x^2))/(3 + 5*x^2 + x^4)^(3/2),x]","\frac{4 x \left(11 x^2+8\right)+i \sqrt{2} \left(11 \sqrt{13}-39\right) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)-11 i \sqrt{2} \left(\sqrt{13}-5\right) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} E\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)}{52 \sqrt{x^4+5 x^2+3}}","-\frac{11 x \left(2 x^2+\sqrt{13}+5\right)}{26 \sqrt{x^4+5 x^2+3}}+\frac{x \left(11 x^2+8\right)}{13 \sqrt{x^4+5 x^2+3}}-\frac{4 \sqrt{\frac{2}{3 \left(5+\sqrt{13}\right)}} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) F\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{13 \sqrt{x^4+5 x^2+3}}+\frac{11 \sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) E\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{26 \sqrt{x^4+5 x^2+3}}",1,"(4*x*(8 + 11*x^2) - (11*I)*Sqrt[2]*(-5 + Sqrt[13])*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticE[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6] + I*Sqrt[2]*(-39 + 11*Sqrt[13])*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6])/(52*Sqrt[3 + 5*x^2 + x^4])","C",0
201,-1,0,282,0,"\int \frac{2+3 x^2}{\left(3+5 x^2+x^4\right)^{3/2}} \, dx","Integrate[(2 + 3*x^2)/(3 + 5*x^2 + x^4)^(3/2),x]","\text{\$Aborted}","\frac{4 x \left(2 x^2+\sqrt{13}+5\right)}{39 \sqrt{x^4+5 x^2+3}}-\frac{x \left(8 x^2+7\right)}{39 \sqrt{x^4+5 x^2+3}}+\frac{11 \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) F\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{13 \sqrt{6 \left(5+\sqrt{13}\right)} \sqrt{x^4+5 x^2+3}}-\frac{2 \sqrt{\frac{2}{3} \left(5+\sqrt{13}\right)} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) E\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{39 \sqrt{x^4+5 x^2+3}}",1,"$Aborted","F",-1
202,1,228,309,0.2698256,"\int \frac{2+3 x^2}{x^2 \left(3+5 x^2+x^4\right)^{3/2}} \, dx","Integrate[(2 + 3*x^2)/(x^2*(3 + 5*x^2 + x^4)^(3/2)),x]","\frac{-i \sqrt{2} \left(19 \sqrt{13}-143\right) x \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)+19 i \sqrt{2} \left(\sqrt{13}-5\right) x \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} E\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)-4 \left(19 x^4+119 x^2+78\right)}{468 x \sqrt{x^4+5 x^2+3}}","\frac{19 x \left(2 x^2+\sqrt{13}+5\right)}{234 \sqrt{x^4+5 x^2+3}}-\frac{19 \sqrt{x^4+5 x^2+3}}{117 x}-\frac{8 x^2+7}{39 x \sqrt{x^4+5 x^2+3}}-\frac{4 \sqrt{\frac{2}{3 \left(5+\sqrt{13}\right)}} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) F\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{39 \sqrt{x^4+5 x^2+3}}-\frac{19 \sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) E\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{234 \sqrt{x^4+5 x^2+3}}",1,"(-4*(78 + 119*x^2 + 19*x^4) + (19*I)*Sqrt[2]*(-5 + Sqrt[13])*x*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticE[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6] - I*Sqrt[2]*(-143 + 19*Sqrt[13])*x*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6])/(468*x*Sqrt[3 + 5*x^2 + x^4])","C",0
203,1,234,326,0.2745229,"\int \frac{2+3 x^2}{x^4 \left(3+5 x^2+x^4\right)^{3/2}} \, dx","Integrate[(2 + 3*x^2)/(x^4*(3 + 5*x^2 + x^4)^(3/2)),x]","\frac{532 x^6+2630 x^4+1014 x^2+i \sqrt{2} \left(133 \sqrt{13}-650\right) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} x^3 F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)-133 i \sqrt{2} \left(\sqrt{13}-5\right) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} x^3 E\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{13}}} x\right)|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right)-468}{2106 x^3 \sqrt{x^4+5 x^2+3}}","-\frac{133 x \left(2 x^2+\sqrt{13}+5\right)}{1053 \sqrt{x^4+5 x^2+3}}+\frac{266 \sqrt{x^4+5 x^2+3}}{1053 x}-\frac{5 \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) F\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{351 \sqrt{6 \left(5+\sqrt{13}\right)} \sqrt{x^4+5 x^2+3}}+\frac{133 \sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} \sqrt{\frac{\left(5-\sqrt{13}\right) x^2+6}{\left(5+\sqrt{13}\right) x^2+6}} \left(\left(5+\sqrt{13}\right) x^2+6\right) E\left(\tan ^{-1}\left(\sqrt{\frac{1}{6} \left(5+\sqrt{13}\right)} x\right)|\frac{1}{6} \left(-13+5 \sqrt{13}\right)\right)}{1053 \sqrt{x^4+5 x^2+3}}-\frac{5 \sqrt{x^4+5 x^2+3}}{351 x^3}-\frac{8 x^2+7}{39 x^3 \sqrt{x^4+5 x^2+3}}",1,"(-468 + 1014*x^2 + 2630*x^4 + 532*x^6 - (133*I)*Sqrt[2]*(-5 + Sqrt[13])*x^3*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticE[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6] + I*Sqrt[2]*(-650 + 133*Sqrt[13])*x^3*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6])/(2106*x^3*Sqrt[3 + 5*x^2 + x^4])","C",0
204,1,430,297,0.9826238,"\int (f x)^{3/2} \left(d+e x^2\right) \sqrt{a+b x^2+c x^4} \, dx","Integrate[(f*x)^(3/2)*(d + e*x^2)*Sqrt[a + b*x^2 + c*x^4],x]","\frac{2 f \sqrt{f x} \left(10 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) \left(-18 a c e+7 b^2 e-13 b c d\right)+2 x^2 \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{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) \left(-79 a b c e+130 a c^2 d+21 b^3 e-39 b^2 c d\right)+5 \left(a+b x^2+c x^4\right) \left(c \left(36 a e+65 c d x^2+45 c e x^4\right)-14 b^2 e+2 b c \left(13 d+5 e x^2\right)\right)\right)}{2925 c^2 \sqrt{a+b x^2+c x^4}}","\frac{2 d (f 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 f \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}}+\frac{2 e (f x)^{9/2} \sqrt{a+b x^2+c x^4} F_1\left(\frac{9}{4};-\frac{1}{2},-\frac{1}{2};\frac{13}{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)}{9 f^3 \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*f*Sqrt[f*x]*(5*(a + b*x^2 + c*x^4)*(-14*b^2*e + 2*b*c*(13*d + 5*e*x^2) + c*(36*a*e + 65*c*d*x^2 + 45*c*e*x^4)) + 10*a*(-13*b*c*d + 7*b^2*e - 18*a*c*e)*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*(-39*b^2*c*d + 130*a*c^2*d + 21*b^3*e - 79*a*b*c*e)*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])]))/(2925*c^2*Sqrt[a + b*x^2 + c*x^4])","A",0
205,1,386,297,5.7142451,"\int \sqrt{f x} \left(d+e x^2\right) \sqrt{a+b x^2+c x^4} \, dx","Integrate[Sqrt[f*x]*(d + e*x^2)*Sqrt[a + b*x^2 + c*x^4],x]","\frac{2 x \sqrt{f x} \left(6 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) \left(14 a c e-5 b^2 e+11 b c d\right)+14 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}} (22 c d-3 b e) 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) \left(2 b e+11 c d+7 c e x^2\right)\right)}{1617 c \sqrt{a+b x^2+c x^4}}","\frac{2 d (f 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 f \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}}+\frac{2 e (f x)^{7/2} \sqrt{a+b x^2+c x^4} F_1\left(\frac{7}{4};-\frac{1}{2},-\frac{1}{2};\frac{11}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right)}{7 f^3 \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[f*x]*(21*(11*c*d + 2*b*e + 7*c*e*x^2)*(a + b*x^2 + c*x^4) + 14*a*(22*c*d - 3*b*e)*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*(11*b*c*d - 5*b^2*e + 14*a*c*e)*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])]))/(1617*c*Sqrt[a + b*x^2 + c*x^4])","A",0
206,1,386,295,0.7089346,"\int \frac{\left(d+e x^2\right) \sqrt{a+b x^2+c x^4}}{\sqrt{f x}} \, dx","Integrate[((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4])/Sqrt[f*x],x]","\frac{2 x \left(2 x^2 \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{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) \left(10 a c e-3 b^2 e+9 b c d\right)+10 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}} (18 c d-b e) 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) \left(2 b e+9 c d+5 c e x^2\right)\right)}{225 c \sqrt{f x} \sqrt{a+b x^2+c x^4}}","\frac{2 d \sqrt{f 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)}{f \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}}+\frac{2 e (f 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 f^3 \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*(9*c*d + 2*b*e + 5*c*e*x^2)*(a + b*x^2 + c*x^4) + 10*a*(18*c*d - b*e)*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*(9*b*c*d - 3*b^2*e + 10*a*c*e)*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[f*x]*Sqrt[a + b*x^2 + c*x^4])","A",0
207,1,370,295,0.8586689,"\int \frac{\left(d+e x^2\right) \sqrt{a+b x^2+c x^4}}{(f x)^{3/2}} \, dx","Integrate[((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4])/(f*x)^(3/2),x]","\frac{x \left(28 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}} (2 a e+7 b d) 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)+12 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}} (b e+14 c d) 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(7 d-e x^2\right) \left(a+b x^2+c x^4\right)\right)}{147 (f x)^{3/2} \sqrt{a+b x^2+c x^4}}","\frac{2 e (f 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 f^3 \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}}-\frac{2 d \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)}{f \sqrt{f 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*(7*d - e*x^2)*(a + b*x^2 + c*x^4) + 28*(7*b*d + 2*a*e)*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])] + 12*(14*c*d + b*e)*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])]))/(147*(f*x)^(3/2)*Sqrt[a + b*x^2 + c*x^4])","A",0
208,-1,0,299,0,"\int (f x)^{3/2} \left(d+e x^2\right) \left(a+b x^2+c x^4\right)^{3/2} \, dx","Integrate[(f*x)^(3/2)*(d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2),x]","\text{\$Aborted}","\frac{2 a d (f 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 f \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}}+\frac{2 a e (f x)^{9/2} \sqrt{a+b x^2+c x^4} F_1\left(\frac{9}{4};-\frac{3}{2},-\frac{3}{2};\frac{13}{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)}{9 f^3 \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,"$Aborted","F",-1
209,1,490,299,6.1442781,"\int \sqrt{f x} \left(d+e x^2\right) \left(a+b x^2+c x^4\right)^{3/2} \, dx","Integrate[Sqrt[f*x]*(d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2),x]","\frac{2 x \sqrt{f x} \left(12 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) \left(420 a^2 c^2 e-309 a b^2 c e+684 a b c^2 d+45 b^4 e-95 b^3 c d\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) \left(-156 a b c e+836 a c^2 d+27 b^3 e-57 b^2 c d\right)+7 \left(a+b x^2+c x^4\right) \left(b c \left(624 a e+7 c x^2 \left(323 d+231 e x^2\right)\right)+c^2 \left(a \left(3971 d+2415 e x^2\right)+77 c x^4 \left(19 d+15 e x^2\right)\right)-108 b^3 e+12 b^2 c \left(19 d+7 e x^2\right)\right)\right)}{153615 c^2 \sqrt{a+b x^2+c x^4}}","\frac{2 a d (f 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 f \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}}+\frac{2 a e (f x)^{7/2} \sqrt{a+b x^2+c x^4} F_1\left(\frac{7}{4};-\frac{3}{2},-\frac{3}{2};\frac{11}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right)}{7 f^3 \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[f*x]*(7*(a + b*x^2 + c*x^4)*(-108*b^3*e + 12*b^2*c*(19*d + 7*e*x^2) + b*c*(624*a*e + 7*c*x^2*(323*d + 231*e*x^2)) + c^2*(77*c*x^4*(19*d + 15*e*x^2) + a*(3971*d + 2415*e*x^2))) + 28*a*(-57*b^2*c*d + 836*a*c^2*d + 27*b^3*e - 156*a*b*c*e)*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*(-95*b^3*c*d + 684*a*b*c^2*d + 45*b^4*e - 309*a*b^2*c*e + 420*a^2*c^2*e)*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])]))/(153615*c^2*Sqrt[a + b*x^2 + c*x^4])","A",0
210,-1,0,297,0,"\int \frac{\left(d+e x^2\right) \left(a+b x^2+c x^4\right)^{3/2}}{\sqrt{f x}} \, dx","Integrate[((d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2))/Sqrt[f*x],x]","\text{\$Aborted}","\frac{2 a d \sqrt{f 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)}{f \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}}+\frac{2 a e (f 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 f^3 \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,"$Aborted","F",-1
211,1,447,297,0.992821,"\int \frac{\left(d+e x^2\right) \left(a+b x^2+c x^4\right)^{3/2}}{(f x)^{3/2}} \, dx","Integrate[((d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(f*x)^(3/2),x]","\frac{x \left(-56 a 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) \left(-44 a c e+3 b^2 e-240 b c d\right)+24 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) \left(36 a b c e+420 a c^2 d-5 b^3 e+15 b^2 c d\right)+14 \left(a+b x^2+c x^4\right) \left(a c \left(209 e x^2-1155 d\right)+x^2 \left(12 b^2 e+b c \left(195 d+119 e x^2\right)+7 c^2 x^2 \left(15 d+11 e x^2\right)\right)\right)\right)}{8085 c (f x)^{3/2} \sqrt{a+b x^2+c x^4}}","\frac{2 a e (f 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 f^3 \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}}-\frac{2 a d \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)}{f \sqrt{f 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*(a + b*x^2 + c*x^4)*(a*c*(-1155*d + 209*e*x^2) + x^2*(12*b^2*e + 7*c^2*x^2*(15*d + 11*e*x^2) + b*c*(195*d + 119*e*x^2))) - 56*a*(-240*b*c*d + 3*b^2*e - 44*a*c*e)*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*(15*b^2*c*d + 420*a*c^2*d - 5*b^3*e + 36*a*b*c*e)*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])]))/(8085*c*(f*x)^(3/2)*Sqrt[a + b*x^2 + c*x^4])","A",0
212,1,354,297,0.5892152,"\int \frac{(f x)^{3/2} \left(d+e x^2\right)}{\sqrt{a+b x^2+c x^4}} \, dx","Integrate[((f*x)^(3/2)*(d + e*x^2))/Sqrt[a + b*x^2 + c*x^4],x]","\frac{2 f \sqrt{f x} \left(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}} (5 c d-3 b e) 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 a e \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 e \left(a+b x^2+c x^4\right)\right)}{25 c \sqrt{a+b x^2+c x^4}}","\frac{2 d (f 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 f \sqrt{a+b x^2+c x^4}}+\frac{2 e (f x)^{9/2} \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{9}{4};\frac{1}{2},\frac{1}{2};\frac{13}{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)}{9 f^3 \sqrt{a+b x^2+c x^4}}",1,"(2*f*Sqrt[f*x]*(5*e*(a + b*x^2 + c*x^4) - 5*a*e*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])] + (5*c*d - 3*b*e)*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*c*Sqrt[a + b*x^2 + c*x^4])","A",0
213,1,242,297,5.1407305,"\int \frac{\sqrt{f x} \left(d+e x^2\right)}{\sqrt{a+b x^2+c x^4}} \, dx","Integrate[(Sqrt[f*x]*(d + e*x^2))/Sqrt[a + b*x^2 + c*x^4],x]","\frac{2 \sqrt{f 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}} \left(7 d x F_1\left(\frac{3}{4};\frac{1}{2},\frac{1}{2};\frac{7}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right)+3 e x^3 F_1\left(\frac{7}{4};\frac{1}{2},\frac{1}{2};\frac{11}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right)\right)}{21 \sqrt{a+b x^2+c x^4}}","\frac{2 d (f 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 f \sqrt{a+b x^2+c x^4}}+\frac{2 e (f x)^{7/2} \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{7}{4};\frac{1}{2},\frac{1}{2};\frac{11}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right)}{7 f^3 \sqrt{a+b x^2+c x^4}}",1,"(2*Sqrt[f*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])]*(7*d*x*AppellF1[3/4, 1/2, 1/2, 7/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])] + 3*e*x^3*AppellF1[7/4, 1/2, 1/2, 11/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])]))/(21*Sqrt[a + b*x^2 + c*x^4])","A",0
214,1,241,295,0.1950605,"\int \frac{d+e x^2}{\sqrt{f x} \sqrt{a+b x^2+c x^4}} \, dx","Integrate[(d + e*x^2)/(Sqrt[f*x]*Sqrt[a + b*x^2 + c*x^4]),x]","\frac{2 \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \left(5 d x 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)+e x^3 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)\right)}{5 \sqrt{f x} \sqrt{a+b x^2+c x^4}}","\frac{2 d \sqrt{f 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)}{f \sqrt{a+b x^2+c x^4}}+\frac{2 e (f 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 f^3 \sqrt{a+b x^2+c x^4}}",1,"(2*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*(5*d*x*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])] + e*x^3*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[f*x]*Sqrt[a + b*x^2 + c*x^4])","A",0
215,1,356,295,0.6182406,"\int \frac{d+e x^2}{(f x)^{3/2} \sqrt{a+b x^2+c x^4}} \, dx","Integrate[(d + e*x^2)/((f*x)^(3/2)*Sqrt[a + b*x^2 + c*x^4]),x]","\frac{2 x \left(7 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}} (a e+b d) 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)+9 c d 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)-21 d \left(a+b x^2+c x^4\right)\right)}{21 a (f x)^{3/2} \sqrt{a+b x^2+c x^4}}","\frac{2 e (f 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 f^3 \sqrt{a+b x^2+c x^4}}-\frac{2 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} 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)}{f \sqrt{f x} \sqrt{a+b x^2+c x^4}}",1,"(2*x*(-21*d*(a + b*x^2 + c*x^4) + 7*(b*d + a*e)*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])] + 9*c*d*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*(f*x)^(3/2)*Sqrt[a + b*x^2 + c*x^4])","A",0
216,-1,0,303,0,"\int \frac{(f x)^{3/2} \left(d+e x^2\right)}{\left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[((f*x)^(3/2)*(d + e*x^2))/(a + b*x^2 + c*x^4)^(3/2),x]","\text{\$Aborted}","\frac{2 d (f 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 f \sqrt{a+b x^2+c x^4}}+\frac{2 e (f x)^{9/2} \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{9}{4};\frac{3}{2},\frac{3}{2};\frac{13}{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)}{9 a f^3 \sqrt{a+b x^2+c x^4}}",1,"$Aborted","F",-1
217,1,397,303,5.7001127,"\int \frac{\sqrt{f x} \left(d+e x^2\right)}{\left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(Sqrt[f*x]*(d + e*x^2))/(a + b*x^2 + c*x^4)^(3/2),x]","\frac{x \sqrt{f x} \left(7 \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) \left(-3 a b e+2 a c d+b^2 d\right)+9 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}} (b d-2 a e) 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)+21 b \left(a e-c d x^2\right)+42 a c \left(d+e x^2\right)-21 b^2 d\right)}{21 a \left(4 a c-b^2\right) \sqrt{a+b x^2+c x^4}}","\frac{2 d (f 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 f \sqrt{a+b x^2+c x^4}}+\frac{2 e (f x)^{7/2} \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{7}{4};\frac{3}{2},\frac{3}{2};\frac{11}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right)}{7 a f^3 \sqrt{a+b x^2+c x^4}}",1,"(x*Sqrt[f*x]*(-21*b^2*d + 21*b*(a*e - c*d*x^2) + 42*a*c*(d + e*x^2) + 7*(b^2*d + 2*a*c*d - 3*a*b*e)*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*c*(b*d - 2*a*e)*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])","A",0
218,-1,0,301,0,"\int \frac{d+e x^2}{\sqrt{f x} \left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(d + e*x^2)/(Sqrt[f*x]*(a + b*x^2 + c*x^4)^(3/2)),x]","\text{\$Aborted}","\frac{2 d \sqrt{f 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 f \sqrt{a+b x^2+c x^4}}+\frac{2 e (f 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 f^3 \sqrt{a+b x^2+c x^4}}",1,"$Aborted","F",-1
219,1,460,301,0.9727123,"\int \frac{d+e x^2}{(f x)^{3/2} \left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(d + e*x^2)/((f*x)^(3/2)*(a + b*x^2 + c*x^4)^(3/2)),x]","-\frac{x \left(7 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) \left(2 a^2 c e+a b^2 e+9 a b c d-3 b^3 d\right)-21 \left(a^2 c \left(8 d-2 e x^2\right)+a \left(b^2 \left(e x^2-2 d\right)+b c x^2 \left(11 d+e x^2\right)+10 c^2 d x^4\right)-3 b^2 d x^2 \left(b+c x^2\right)\right)-9 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) \left(-a b e-10 a c d+3 b^2 d\right)\right)}{21 a^2 (f x)^{3/2} \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}","\frac{2 e (f 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 f^3 \sqrt{a+b x^2+c x^4}}-\frac{2 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} 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 f \sqrt{f x} \sqrt{a+b x^2+c x^4}}",1,"-1/21*(x*(-21*(-3*b^2*d*x^2*(b + c*x^2) + a^2*c*(8*d - 2*e*x^2) + a*(10*c^2*d*x^4 + b^2*(-2*d + e*x^2) + b*c*x^2*(11*d + e*x^2))) + 7*(-3*b^3*d + 9*a*b*c*d + a*b^2*e + 2*a^2*c*e)*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])] - 9*c*(3*b^2*d - 10*a*c*d - a*b*e)*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)*(f*x)^(3/2)*Sqrt[a + b*x^2 + c*x^4])","A",0
220,1,191,243,0.2977342,"\int (f x)^m \left(d+e x^2\right) \left(a+b x^2+c x^4\right)^3 \, dx","Integrate[(f*x)^m*(d + e*x^2)*(a + b*x^2 + c*x^4)^3,x]","x (f x)^m \left(\frac{a^3 d}{m+1}+\frac{x^6 \left(3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right)}{m+7}+\frac{a^2 x^2 (a e+3 b d)}{m+3}+\frac{3 c x^{10} \left(a c e+b^2 e+b c d\right)}{m+11}+\frac{3 a x^4 \left(a b e+a c d+b^2 d\right)}{m+5}+\frac{x^8 \left(6 a b c e+3 a c^2 d+b^3 e+3 b^2 c d\right)}{m+9}+\frac{c^2 x^{12} (3 b e+c d)}{m+13}+\frac{c^3 e x^{14}}{m+15}\right)","\frac{a^3 d (f x)^{m+1}}{f (m+1)}+\frac{(f x)^{m+7} \left(3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right)}{f^7 (m+7)}+\frac{a^2 (f x)^{m+3} (a e+3 b d)}{f^3 (m+3)}+\frac{3 c (f x)^{m+11} \left(a c e+b^2 e+b c d\right)}{f^{11} (m+11)}+\frac{3 a (f x)^{m+5} \left(a b e+a c d+b^2 d\right)}{f^5 (m+5)}+\frac{(f x)^{m+9} \left(6 a b c e+3 a c^2 d+b^3 e+3 b^2 c d\right)}{f^9 (m+9)}+\frac{c^2 (f x)^{m+13} (3 b e+c d)}{f^{13} (m+13)}+\frac{c^3 e (f x)^{m+15}}{f^{15} (m+15)}",1,"x*(f*x)^m*((a^3*d)/(1 + m) + (a^2*(3*b*d + a*e)*x^2)/(3 + m) + (3*a*(b^2*d + a*c*d + a*b*e)*x^4)/(5 + m) + ((b^3*d + 6*a*b*c*d + 3*a*b^2*e + 3*a^2*c*e)*x^6)/(7 + m) + ((3*b^2*c*d + 3*a*c^2*d + b^3*e + 6*a*b*c*e)*x^8)/(9 + m) + (3*c*(b*c*d + b^2*e + a*c*e)*x^10)/(11 + m) + (c^2*(c*d + 3*b*e)*x^12)/(13 + m) + (c^3*e*x^14)/(15 + m))","A",1
221,1,117,155,0.1231191,"\int (f x)^m \left(d+e x^2\right) \left(a+b x^2+c x^4\right)^2 \, dx","Integrate[(f*x)^m*(d + e*x^2)*(a + b*x^2 + c*x^4)^2,x]","x (f x)^m \left(\frac{a^2 d}{m+1}+\frac{x^6 \left(2 a c e+b^2 e+2 b c d\right)}{m+7}+\frac{x^4 \left(2 a b e+2 a c d+b^2 d\right)}{m+5}+\frac{a x^2 (a e+2 b d)}{m+3}+\frac{c x^8 (2 b e+c d)}{m+9}+\frac{c^2 e x^{10}}{m+11}\right)","\frac{a^2 d (f x)^{m+1}}{f (m+1)}+\frac{(f x)^{m+7} \left(2 a c e+b^2 e+2 b c d\right)}{f^7 (m+7)}+\frac{(f x)^{m+5} \left(2 a b e+2 a c d+b^2 d\right)}{f^5 (m+5)}+\frac{a (f x)^{m+3} (a e+2 b d)}{f^3 (m+3)}+\frac{c (f x)^{m+9} (2 b e+c d)}{f^9 (m+9)}+\frac{c^2 e (f x)^{m+11}}{f^{11} (m+11)}",1,"x*(f*x)^m*((a^2*d)/(1 + m) + (a*(2*b*d + a*e)*x^2)/(3 + m) + ((b^2*d + 2*a*c*d + 2*a*b*e)*x^4)/(5 + m) + ((2*b*c*d + b^2*e + 2*a*c*e)*x^6)/(7 + m) + (c*(c*d + 2*b*e)*x^8)/(9 + m) + (c^2*e*x^10)/(11 + m))","A",1
222,1,59,83,0.0486683,"\int (f x)^m \left(d+e x^2\right) \left(a+b x^2+c x^4\right) \, dx","Integrate[(f*x)^m*(d + e*x^2)*(a + b*x^2 + c*x^4),x]","x (f x)^m \left(\frac{x^2 (a e+b d)}{m+3}+\frac{a d}{m+1}+\frac{x^4 (b e+c d)}{m+5}+\frac{c e x^6}{m+7}\right)","\frac{(f x)^{m+3} (a e+b d)}{f^3 (m+3)}+\frac{a d (f x)^{m+1}}{f (m+1)}+\frac{(f x)^{m+5} (b e+c d)}{f^5 (m+5)}+\frac{c e (f x)^{m+7}}{f^7 (m+7)}",1,"x*(f*x)^m*((a*d)/(1 + m) + ((b*d + a*e)*x^2)/(3 + m) + ((c*d + b*e)*x^4)/(5 + m) + (c*e*x^6)/(7 + m))","A",1
223,1,156,194,0.2440806,"\int \frac{(f x)^m \left(d+e x^2\right)}{a+b x^2+c x^4} \, dx","Integrate[((f*x)^m*(d + e*x^2))/(a + b*x^2 + c*x^4),x]","\frac{x (f x)^m \left(\left(d \sqrt{b^2-4 a c}-2 a e+b d\right) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right)+\left(d \sqrt{b^2-4 a c}+2 a e-b d\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)\right)}{2 a (m+1) \sqrt{b^2-4 a c}}","\frac{(f x)^{m+1} \left(\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\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)}{f (m+1) \left(b-\sqrt{b^2-4 a c}\right)}+\frac{(f x)^{m+1} \left(e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\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)}{f (m+1) \left(\sqrt{b^2-4 a c}+b\right)}",1,"(x*(f*x)^m*((b*d + Sqrt[b^2 - 4*a*c]*d - 2*a*e)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])] + (-(b*d) + Sqrt[b^2 - 4*a*c]*d + 2*a*e)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]))/(2*a*Sqrt[b^2 - 4*a*c]*(1 + m))","A",1
224,1,160,392,0.2262141,"\int \frac{(f x)^m \left(d+e x^2\right)}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[((f*x)^m*(d + e*x^2))/(a + b*x^2 + c*x^4)^2,x]","\frac{x (f x)^m \left(d (m+3) 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)+e (m+1) x^2 F_1\left(\frac{m+3}{2};2,2;\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)}{a^2 (m+1) (m+3)}","\frac{c (f x)^{m+1} \left(b \left(d (1-m) \sqrt{b^2-4 a c}+4 a e\right)-2 a \left(e (1-m) \sqrt{b^2-4 a c}+2 c d (3-m)\right)+b^2 (d-d 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 f (m+1) \left(b^2-4 a c\right)^{3/2} \left(b-\sqrt{b^2-4 a c}\right)}-\frac{c (f x)^{m+1} \left(b \left(4 a e-d (1-m) \sqrt{b^2-4 a c}\right)+2 a \left(e (1-m) \sqrt{b^2-4 a c}-2 c d (3-m)\right)+b^2 d (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 f (m+1) \left(b^2-4 a c\right)^{3/2} \left(\sqrt{b^2-4 a c}+b\right)}+\frac{(f x)^{m+1} \left(c x^2 (b d-2 a e)-a b e-2 a c d+b^2 d\right)}{2 a f \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"(x*(f*x)^m*(d*(3 + 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])] + e*(1 + m)*x^2*AppellF1[(3 + m)/2, 2, 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])]))/(a^2*(1 + m)*(3 + m))","C",0
225,1,466,319,0.53144,"\int (f x)^m \left(d+e x^2\right) \left(a+b x^2+c x^4\right)^{3/2} \, dx","Integrate[(f*x)^m*(d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2),x]","\frac{x (f x)^m \sqrt{a+b x^2+c x^4} \left((m+1) x^2 \left(\left(m^2+12 m+35\right) (a e+b d) 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)+(m+3) x^2 \left((m+7) (b e+c d) 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)+c e (m+5) x^2 F_1\left(\frac{m+7}{2};-\frac{1}{2},-\frac{1}{2};\frac{m+9}{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)+a d \left(m^3+15 m^2+71 m+105\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)\right)}{(m+1) (m+3) (m+5) (m+7) \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 (f 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)}{f (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}}+\frac{a e (f x)^{m+3} \sqrt{a+b x^2+c x^4} F_1\left(\frac{m+3}{2};-\frac{3}{2},-\frac{3}{2};\frac{m+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)}{f^3 (m+3) \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*(f*x)^m*Sqrt[a + b*x^2 + c*x^4]*(a*d*(105 + 71*m + 15*m^2 + m^3)*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*d + a*e)*(35 + 12*m + m^2)*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])] + (3 + m)*x^2*((c*d + b*e)*(7 + m)*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])] + c*e*(5 + m)*x^2*AppellF1[(7 + m)/2, -1/2, -1/2, (9 + 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)*(7 + 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
226,1,267,317,0.1982983,"\int (f x)^m \left(d+e x^2\right) \sqrt{a+b x^2+c x^4} \, dx","Integrate[(f*x)^m*(d + e*x^2)*Sqrt[a + b*x^2 + c*x^4],x]","\frac{x (f x)^m \sqrt{a+b x^2+c x^4} \left(d (m+3) 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)+e (m+1) x^2 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)}{(m+1) (m+3) \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 (f 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)}{f (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}}+\frac{e (f x)^{m+3} \sqrt{a+b x^2+c x^4} 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}{b+\sqrt{b^2-4 a c}}\right)}{f^3 (m+3) \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*(f*x)^m*Sqrt[a + b*x^2 + c*x^4]*(d*(3 + m)*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])] + e*(1 + m)*x^2*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])]))/((1 + m)*(3 + 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
227,1,267,317,0.264246,"\int \frac{(f x)^m \left(d+e x^2\right)}{\sqrt{a+b x^2+c x^4}} \, dx","Integrate[((f*x)^m*(d + e*x^2))/Sqrt[a + b*x^2 + c*x^4],x]","\frac{x (f 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}} \left(d (m+3) 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)+e (m+1) x^2 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)}{(m+1) (m+3) \sqrt{a+b x^2+c x^4}}","\frac{d (f 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)}{f (m+1) \sqrt{a+b x^2+c x^4}}+\frac{e (f x)^{m+3} \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+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}{b+\sqrt{b^2-4 a c}}\right)}{f^3 (m+3) \sqrt{a+b x^2+c x^4}}",1,"(x*(f*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])]*(d*(3 + m)*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])] + e*(1 + m)*x^2*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])]))/((1 + m)*(3 + m)*Sqrt[a + b*x^2 + c*x^4])","A",0
228,1,307,323,0.405066,"\int \frac{(f x)^m \left(d+e x^2\right)}{\left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[((f*x)^m*(d + e*x^2))/(a + b*x^2 + c*x^4)^(3/2),x]","\frac{x (f 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} \left(d (m+3) 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)+e (m+1) x^2 F_1\left(\frac{m+3}{2};\frac{3}{2},\frac{3}{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)}{(m+1) (m+3) \left(\sqrt{b^2-4 a c}-b\right) \left(a+b x^2+c x^4\right)^{3/2}}","\frac{d (f 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 f (m+1) \sqrt{a+b x^2+c x^4}}+\frac{e (f x)^{m+3} \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+3}{2};\frac{3}{2},\frac{3}{2};\frac{m+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)}{a f^3 (m+3) \sqrt{a+b x^2+c x^4}}",1,"(x*(f*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)*(d*(3 + m)*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])] + e*(1 + m)*x^2*AppellF1[(3 + m)/2, 3/2, 3/2, (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])]))/((-b + Sqrt[b^2 - 4*a*c])*(1 + m)*(3 + m)*(a + b*x^2 + c*x^4)^(3/2))","A",0
229,1,134,134,0.0625355,"\int \frac{x^9}{\left(d+e x^2\right) \left(a+c x^4\right)} \, dx","Integrate[x^9/((d + e*x^2)*(a + c*x^4)),x]","\frac{a^{3/2} d \tan ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a}}\right)}{2 c^{3/2} \left(a e^2+c d^2\right)}-\frac{a^2 e \log \left(a+c x^4\right)}{4 c^2 \left(a e^2+c d^2\right)}+\frac{d^4 \log \left(d+e x^2\right)}{2 e^3 \left(a e^2+c d^2\right)}-\frac{d x^2}{2 c e^2}+\frac{x^4}{4 c e}","\frac{a^{3/2} d \tan ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a}}\right)}{2 c^{3/2} \left(a e^2+c d^2\right)}-\frac{a^2 e \log \left(a+c x^4\right)}{4 c^2 \left(a e^2+c d^2\right)}+\frac{d^4 \log \left(d+e x^2\right)}{2 e^3 \left(a e^2+c d^2\right)}-\frac{d x^2}{2 c e^2}+\frac{x^4}{4 c e}",1,"-1/2*(d*x^2)/(c*e^2) + x^4/(4*c*e) + (a^(3/2)*d*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*c^(3/2)*(c*d^2 + a*e^2)) + (d^4*Log[d + e*x^2])/(2*e^3*(c*d^2 + a*e^2)) - (a^2*e*Log[a + c*x^4])/(4*c^2*(c*d^2 + a*e^2))","A",1
230,1,99,118,0.0938768,"\int \frac{x^7}{\left(d+e x^2\right) \left(a+c x^4\right)} \, dx","Integrate[x^7/((d + e*x^2)*(a + c*x^4)),x]","\frac{-\frac{2 a^{3/2} e^3 \tan ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a}}\right)}{c^{3/2}}+\frac{e \left(2 x^2 \left(a e^2+c d^2\right)-a d e \log \left(a+c x^4\right)\right)}{c}-2 d^3 \log \left(d+e x^2\right)}{4 e^2 \left(a e^2+c d^2\right)}","-\frac{a^{3/2} e \tan ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a}}\right)}{2 c^{3/2} \left(a e^2+c d^2\right)}-\frac{a d \log \left(a+c x^4\right)}{4 c \left(a e^2+c d^2\right)}-\frac{d^3 \log \left(d+e x^2\right)}{2 e^2 \left(a e^2+c d^2\right)}+\frac{x^2}{2 c e}",1,"((-2*a^(3/2)*e^3*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/c^(3/2) - 2*d^3*Log[d + e*x^2] + (e*(2*(c*d^2 + a*e^2)*x^2 - a*d*e*Log[a + c*x^4]))/c)/(4*e^2*(c*d^2 + a*e^2))","A",1
231,1,77,105,0.0335685,"\int \frac{x^5}{\left(d+e x^2\right) \left(a+c x^4\right)} \, dx","Integrate[x^5/((d + e*x^2)*(a + c*x^4)),x]","\frac{-2 \sqrt{a} \sqrt{c} d e \tan ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a}}\right)+a e^2 \log \left(a+c x^4\right)+2 c d^2 \log \left(d+e x^2\right)}{4 a c e^3+4 c^2 d^2 e}","\frac{a e \log \left(a+c x^4\right)}{4 c \left(a e^2+c d^2\right)}+\frac{d^2 \log \left(d+e x^2\right)}{2 e \left(a e^2+c d^2\right)}-\frac{\sqrt{a} d \tan ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a}}\right)}{2 \sqrt{c} \left(a e^2+c d^2\right)}",1,"(-2*Sqrt[a]*Sqrt[c]*d*e*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]] + 2*c*d^2*Log[d + e*x^2] + a*e^2*Log[a + c*x^4])/(4*c^2*d^2*e + 4*a*c*e^3)","A",1
232,1,66,96,0.0396581,"\int \frac{x^3}{\left(d+e x^2\right) \left(a+c x^4\right)} \, dx","Integrate[x^3/((d + e*x^2)*(a + c*x^4)),x]","\frac{d \log \left(a+c x^4\right)+\frac{2 \sqrt{a} e \tan ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a}}\right)}{\sqrt{c}}-2 d \log \left(d+e x^2\right)}{4 a e^2+4 c d^2}","\frac{d \log \left(a+c x^4\right)}{4 \left(a e^2+c d^2\right)}-\frac{d \log \left(d+e x^2\right)}{2 \left(a e^2+c d^2\right)}+\frac{\sqrt{a} e \tan ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a}}\right)}{2 \sqrt{c} \left(a e^2+c d^2\right)}",1,"((2*Sqrt[a]*e*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/Sqrt[c] - 2*d*Log[d + e*x^2] + d*Log[a + c*x^4])/(4*c*d^2 + 4*a*e^2)","A",1
233,1,67,96,0.0367908,"\int \frac{x}{\left(d+e x^2\right) \left(a+c x^4\right)} \, dx","Integrate[x/((d + e*x^2)*(a + c*x^4)),x]","\frac{\frac{2 \sqrt{c} d \tan ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a}}\right)}{\sqrt{a}}-e \log \left(a+c x^4\right)+2 e \log \left(d+e x^2\right)}{4 a e^2+4 c d^2}","-\frac{e \log \left(a+c x^4\right)}{4 \left(a e^2+c d^2\right)}+\frac{e \log \left(d+e x^2\right)}{2 \left(a e^2+c d^2\right)}+\frac{\sqrt{c} d \tan ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a}}\right)}{2 \sqrt{a} \left(a e^2+c d^2\right)}",1,"((2*Sqrt[c]*d*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/Sqrt[a] + 2*e*Log[d + e*x^2] - e*Log[a + c*x^4])/(4*c*d^2 + 4*a*e^2)","A",1
234,1,134,114,0.0707708,"\int \frac{1}{x \left(d+e x^2\right) \left(a+c x^4\right)} \, dx","Integrate[1/(x*(d + e*x^2)*(a + c*x^4)),x]","\frac{-c d^2 \log \left(a+c x^4\right)+2 \sqrt{a} \sqrt{c} d e \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)+2 \sqrt{a} \sqrt{c} d e \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)-2 a e^2 \log \left(d+e x^2\right)+4 a e^2 \log (x)+4 c d^2 \log (x)}{4 a^2 d e^2+4 a c d^3}","-\frac{c d \log \left(a+c x^4\right)}{4 a \left(a e^2+c d^2\right)}-\frac{e^2 \log \left(d+e x^2\right)}{2 d \left(a e^2+c d^2\right)}-\frac{\sqrt{c} e \tan ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a}}\right)}{2 \sqrt{a} \left(a e^2+c d^2\right)}+\frac{\log (x)}{a d}",1,"(2*Sqrt[a]*Sqrt[c]*d*e*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 2*Sqrt[a]*Sqrt[c]*d*e*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 4*c*d^2*Log[x] + 4*a*e^2*Log[x] - 2*a*e^2*Log[d + e*x^2] - c*d^2*Log[a + c*x^4])/(4*a*c*d^3 + 4*a^2*d*e^2)","A",1
235,1,169,129,0.1010573,"\int \frac{1}{x^3 \left(d+e x^2\right) \left(a+c x^4\right)} \, dx","Integrate[1/(x^3*(d + e*x^2)*(a + c*x^4)),x]","\frac{2 c^{3/2} d^3 x^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)+2 c^{3/2} d^3 x^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)+\sqrt{a} \left(-4 e x^2 \log (x) \left(a e^2+c d^2\right)+c d^2 e x^2 \log \left(a+c x^4\right)+2 a e^3 x^2 \log \left(d+e x^2\right)-2 a d e^2-2 c d^3\right)}{4 a^{3/2} d^2 x^2 \left(a e^2+c d^2\right)}","-\frac{c^{3/2} d \tan ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a}}\right)}{2 a^{3/2} \left(a e^2+c d^2\right)}+\frac{c e \log \left(a+c x^4\right)}{4 a \left(a e^2+c d^2\right)}+\frac{e^3 \log \left(d+e x^2\right)}{2 d^2 \left(a e^2+c d^2\right)}-\frac{e \log (x)}{a d^2}-\frac{1}{2 a d x^2}",1,"(2*c^(3/2)*d^3*x^2*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 2*c^(3/2)*d^3*x^2*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + Sqrt[a]*(-2*c*d^3 - 2*a*d*e^2 - 4*e*(c*d^2 + a*e^2)*x^2*Log[x] + 2*a*e^3*x^2*Log[d + e*x^2] + c*d^2*e*x^2*Log[a + c*x^4]))/(4*a^(3/2)*d^2*(c*d^2 + a*e^2)*x^2)","A",1
236,1,209,156,0.093313,"\int \frac{1}{x^5 \left(d+e x^2\right) \left(a+c x^4\right)} \, dx","Integrate[1/(x^5*(d + e*x^2)*(a + c*x^4)),x]","-\frac{a^2 d^2 e^2+2 a^2 e^4 x^4 \log \left(d+e x^2\right)-2 a^2 d e^3 x^2-4 a^2 e^4 x^4 \log (x)+2 \sqrt{a} c^{3/2} d^3 e x^4 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)+2 \sqrt{a} c^{3/2} d^3 e x^4 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)-c^2 d^4 x^4 \log \left(a+c x^4\right)+a c d^4-2 a c d^3 e x^2+4 c^2 d^4 x^4 \log (x)}{4 a^2 d^3 x^4 \left(a e^2+c d^2\right)}","\frac{c^{3/2} e \tan ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a}}\right)}{2 a^{3/2} \left(a e^2+c d^2\right)}+\frac{c^2 d \log \left(a+c x^4\right)}{4 a^2 \left(a e^2+c d^2\right)}-\frac{\log (x) \left(c d^2-a e^2\right)}{a^2 d^3}-\frac{e^4 \log \left(d+e x^2\right)}{2 d^3 \left(a e^2+c d^2\right)}+\frac{e}{2 a d^2 x^2}-\frac{1}{4 a d x^4}",1,"-1/4*(a*c*d^4 + a^2*d^2*e^2 - 2*a*c*d^3*e*x^2 - 2*a^2*d*e^3*x^2 + 2*Sqrt[a]*c^(3/2)*d^3*e*x^4*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 2*Sqrt[a]*c^(3/2)*d^3*e*x^4*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 4*c^2*d^4*x^4*Log[x] - 4*a^2*e^4*x^4*Log[x] + 2*a^2*e^4*x^4*Log[d + e*x^2] - c^2*d^4*x^4*Log[a + c*x^4])/(a^2*d^3*(c*d^2 + a*e^2)*x^4)","A",1
237,1,344,359,0.3598445,"\int \frac{x^8}{\left(d+e x^2\right) \left(a+c x^4\right)} \, dx","Integrate[x^8/((d + e*x^2)*(a + c*x^4)),x]","\frac{-3 \sqrt{2} a e^{5/2} \left(a^{3/4} e+\sqrt[4]{a} \sqrt{c} d\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)+3 \sqrt{2} a e^{5/2} \left(a^{3/4} e+\sqrt[4]{a} \sqrt{c} d\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)+6 \sqrt{2} a^{5/4} e^{5/2} \left(\sqrt{a} e-\sqrt{c} d\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)-6 \sqrt{2} a^{5/4} e^{5/2} \left(\sqrt{a} e-\sqrt{c} d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)-24 c^{3/4} d \sqrt{e} x \left(a e^2+c d^2\right)+8 c^{3/4} e^{3/2} x^3 \left(a e^2+c d^2\right)+24 c^{7/4} d^{7/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{24 c^{7/4} e^{5/2} \left(a e^2+c d^2\right)}","-\frac{a^{5/4} \left(\sqrt{a} e+\sqrt{c} d\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} c^{7/4} \left(a e^2+c d^2\right)}+\frac{a^{5/4} \left(\sqrt{a} e+\sqrt{c} d\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} c^{7/4} \left(a e^2+c d^2\right)}-\frac{a^{5/4} \left(\sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} c^{7/4} \left(a e^2+c d^2\right)}+\frac{a^{5/4} \left(\sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} c^{7/4} \left(a e^2+c d^2\right)}+\frac{d^{7/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{e^{5/2} \left(a e^2+c d^2\right)}-\frac{d x}{c e^2}+\frac{x^3}{3 c e}",1,"(-24*c^(3/4)*d*Sqrt[e]*(c*d^2 + a*e^2)*x + 8*c^(3/4)*e^(3/2)*(c*d^2 + a*e^2)*x^3 + 24*c^(7/4)*d^(7/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + 6*Sqrt[2]*a^(5/4)*e^(5/2)*(-(Sqrt[c]*d) + Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] - 6*Sqrt[2]*a^(5/4)*e^(5/2)*(-(Sqrt[c]*d) + Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)] - 3*Sqrt[2]*a*e^(5/2)*(a^(1/4)*Sqrt[c]*d + a^(3/4)*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] + 3*Sqrt[2]*a*e^(5/2)*(a^(1/4)*Sqrt[c]*d + a^(3/4)*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(24*c^(7/4)*e^(5/2)*(c*d^2 + a*e^2))","A",1
238,1,373,345,0.2381099,"\int \frac{x^6}{\left(d+e x^2\right) \left(a+c x^4\right)} \, dx","Integrate[x^6/((d + e*x^2)*(a + c*x^4)),x]","-\frac{\left(a^{3/4} c d-a^{5/4} \sqrt{c} e\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} c^{7/4} \left(a e^2+c d^2\right)}+\frac{\left(a^{3/4} c d-a^{5/4} \sqrt{c} e\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} c^{7/4} \left(a e^2+c d^2\right)}-\frac{\left(a^{3/4} c d+a^{5/4} \sqrt{c} e\right) \tan ^{-1}\left(\frac{2 \sqrt[4]{c} x-\sqrt{2} \sqrt[4]{a}}{\sqrt{2} \sqrt[4]{a}}\right)}{2 \sqrt{2} c^{7/4} \left(a e^2+c d^2\right)}-\frac{\left(a^{3/4} c d+a^{5/4} \sqrt{c} e\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a}+2 \sqrt[4]{c} x}{\sqrt{2} \sqrt[4]{a}}\right)}{2 \sqrt{2} c^{7/4} \left(a e^2+c d^2\right)}-\frac{d^{5/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{e^{3/2} \left(a e^2+c d^2\right)}+\frac{x}{c e}","-\frac{a^{3/4} \left(\sqrt{c} d-\sqrt{a} e\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} c^{5/4} \left(a e^2+c d^2\right)}+\frac{a^{3/4} \left(\sqrt{c} d-\sqrt{a} e\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} c^{5/4} \left(a e^2+c d^2\right)}+\frac{a^{3/4} \left(\sqrt{a} e+\sqrt{c} d\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} c^{5/4} \left(a e^2+c d^2\right)}-\frac{a^{3/4} \left(\sqrt{a} e+\sqrt{c} d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} c^{5/4} \left(a e^2+c d^2\right)}-\frac{d^{5/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{e^{3/2} \left(a e^2+c d^2\right)}+\frac{x}{c e}",1,"x/(c*e) - (d^(5/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(e^(3/2)*(c*d^2 + a*e^2)) - ((a^(3/4)*c*d + a^(5/4)*Sqrt[c]*e)*ArcTan[(-(Sqrt[2]*a^(1/4)) + 2*c^(1/4)*x)/(Sqrt[2]*a^(1/4))])/(2*Sqrt[2]*c^(7/4)*(c*d^2 + a*e^2)) - ((a^(3/4)*c*d + a^(5/4)*Sqrt[c]*e)*ArcTan[(Sqrt[2]*a^(1/4) + 2*c^(1/4)*x)/(Sqrt[2]*a^(1/4))])/(2*Sqrt[2]*c^(7/4)*(c*d^2 + a*e^2)) - ((a^(3/4)*c*d - a^(5/4)*Sqrt[c]*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*c^(7/4)*(c*d^2 + a*e^2)) + ((a^(3/4)*c*d - a^(5/4)*Sqrt[c]*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*c^(7/4)*(c*d^2 + a*e^2))","A",1
239,1,233,336,0.1509155,"\int \frac{x^4}{\left(d+e x^2\right) \left(a+c x^4\right)} \, dx","Integrate[x^4/((d + e*x^2)*(a + c*x^4)),x]","\frac{\sqrt{2} \sqrt[4]{a} \sqrt{e} \left(\left(\sqrt{a} e+\sqrt{c} d\right) \left(\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)-\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)\right)+2 \left(\sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)+\left(2 \sqrt{a} e-2 \sqrt{c} d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)\right)+8 c^{3/4} d^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{8 c^{3/4} \sqrt{e} \left(a e^2+c d^2\right)}","\frac{\sqrt[4]{a} \left(\sqrt{a} e+\sqrt{c} d\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} c^{3/4} \left(a e^2+c d^2\right)}-\frac{\sqrt[4]{a} \left(\sqrt{a} e+\sqrt{c} d\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} c^{3/4} \left(a e^2+c d^2\right)}+\frac{\sqrt[4]{a} \left(\sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} c^{3/4} \left(a e^2+c d^2\right)}-\frac{\sqrt[4]{a} \left(\sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} c^{3/4} \left(a e^2+c d^2\right)}+\frac{d^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{e} \left(a e^2+c d^2\right)}",1,"(8*c^(3/4)*d^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + Sqrt[2]*a^(1/4)*Sqrt[e]*(2*(Sqrt[c]*d - Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + (-2*Sqrt[c]*d + 2*Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + (Sqrt[c]*d + Sqrt[a]*e)*(Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] - Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])))/(8*c^(3/4)*Sqrt[e]*(c*d^2 + a*e^2))","A",1
240,1,232,337,0.1261467,"\int \frac{x^2}{\left(d+e x^2\right) \left(a+c x^4\right)} \, dx","Integrate[x^2/((d + e*x^2)*(a + c*x^4)),x]","\frac{\sqrt{2} \left(\left(\sqrt{c} d-\sqrt{a} e\right) \left(\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)-\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)\right)-2 \left(\sqrt{a} e+\sqrt{c} d\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)+2 \left(\sqrt{a} e+\sqrt{c} d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)\right)-8 \sqrt[4]{a} \sqrt[4]{c} \sqrt{d} \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{8 \sqrt[4]{a} \sqrt[4]{c} \left(a e^2+c d^2\right)}","\frac{\left(\sqrt{c} d-\sqrt{a} e\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} \sqrt[4]{a} \sqrt[4]{c} \left(a e^2+c d^2\right)}-\frac{\left(\sqrt{c} d-\sqrt{a} e\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} \sqrt[4]{a} \sqrt[4]{c} \left(a e^2+c d^2\right)}-\frac{\sqrt{d} \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{a e^2+c d^2}-\frac{\left(\sqrt{a} e+\sqrt{c} d\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} \sqrt[4]{a} \sqrt[4]{c} \left(a e^2+c d^2\right)}+\frac{\left(\sqrt{a} e+\sqrt{c} d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} \sqrt[4]{a} \sqrt[4]{c} \left(a e^2+c d^2\right)}",1,"(-8*a^(1/4)*c^(1/4)*Sqrt[d]*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + Sqrt[2]*(-2*(Sqrt[c]*d + Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 2*(Sqrt[c]*d + Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + (Sqrt[c]*d - Sqrt[a]*e)*(Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] - Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])))/(8*a^(1/4)*c^(1/4)*(c*d^2 + a*e^2))","A",1
241,1,234,336,0.1450418,"\int \frac{1}{\left(d+e x^2\right) \left(a+c x^4\right)} \, dx","Integrate[1/((d + e*x^2)*(a + c*x^4)),x]","\frac{8 a^{3/4} e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)+\sqrt{2} \sqrt[4]{c} \sqrt{d} \left(-\left(\sqrt{a} e+\sqrt{c} d\right) \left(\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)-\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)\right)+\left(2 \sqrt{a} e-2 \sqrt{c} d\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)+2 \left(\sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)\right)}{8 a^{3/4} \sqrt{d} \left(a e^2+c d^2\right)}","-\frac{\sqrt[4]{c} \left(\sqrt{a} e+\sqrt{c} d\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} a^{3/4} \left(a e^2+c d^2\right)}+\frac{\sqrt[4]{c} \left(\sqrt{a} e+\sqrt{c} d\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} a^{3/4} \left(a e^2+c d^2\right)}-\frac{\sqrt[4]{c} \left(\sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} a^{3/4} \left(a e^2+c d^2\right)}+\frac{\sqrt[4]{c} \left(\sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} a^{3/4} \left(a e^2+c d^2\right)}+\frac{e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{d} \left(a e^2+c d^2\right)}",1,"(8*a^(3/4)*e^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + Sqrt[2]*c^(1/4)*Sqrt[d]*((-2*Sqrt[c]*d + 2*Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 2*(Sqrt[c]*d - Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)] - (Sqrt[c]*d + Sqrt[a]*e)*(Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] - Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])))/(8*a^(3/4)*Sqrt[d]*(c*d^2 + a*e^2))","A",1
242,1,389,348,0.2531648,"\int \frac{1}{x^2 \left(d+e x^2\right) \left(a+c x^4\right)} \, dx","Integrate[1/(x^2*(d + e*x^2)*(a + c*x^4)),x]","\frac{-\sqrt{d} \left(8 a^{5/4} e^2+\sqrt{2} c^{5/4} d^2 x \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)-\sqrt{2} c^{5/4} d^2 x \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)-\sqrt{2} \sqrt{a} c^{3/4} d e x \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)+\sqrt{2} \sqrt{a} c^{3/4} d e x \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)-2 \sqrt{2} c^{3/4} d x \left(\sqrt{a} e+\sqrt{c} d\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)+2 \sqrt{2} c^{3/4} d x \left(\sqrt{a} e+\sqrt{c} d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)+8 \sqrt[4]{a} c d^2\right)-8 a^{5/4} e^{5/2} x \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{8 a^{5/4} d^{3/2} x \left(a e^2+c d^2\right)}","-\frac{c^{3/4} \left(\sqrt{c} d-\sqrt{a} e\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} a^{5/4} \left(a e^2+c d^2\right)}+\frac{c^{3/4} \left(\sqrt{c} d-\sqrt{a} e\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} a^{5/4} \left(a e^2+c d^2\right)}+\frac{c^{3/4} \left(\sqrt{a} e+\sqrt{c} d\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} a^{5/4} \left(a e^2+c d^2\right)}-\frac{c^{3/4} \left(\sqrt{a} e+\sqrt{c} d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} a^{5/4} \left(a e^2+c d^2\right)}-\frac{e^{5/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{d^{3/2} \left(a e^2+c d^2\right)}-\frac{1}{a d x}",1,"(-8*a^(5/4)*e^(5/2)*x*ArcTan[(Sqrt[e]*x)/Sqrt[d]] - Sqrt[d]*(8*a^(1/4)*c*d^2 + 8*a^(5/4)*e^2 - 2*Sqrt[2]*c^(3/4)*d*(Sqrt[c]*d + Sqrt[a]*e)*x*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 2*Sqrt[2]*c^(3/4)*d*(Sqrt[c]*d + Sqrt[a]*e)*x*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + Sqrt[2]*c^(5/4)*d^2*x*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] - Sqrt[2]*Sqrt[a]*c^(3/4)*d*e*x*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] - Sqrt[2]*c^(5/4)*d^2*x*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] + Sqrt[2]*Sqrt[a]*c^(3/4)*d*e*x*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2]))/(8*a^(5/4)*d^(3/2)*(c*d^2 + a*e^2)*x)","A",1
243,1,367,360,0.4005298,"\int \frac{1}{x^4 \left(d+e x^2\right) \left(a+c x^4\right)} \, dx","Integrate[1/(x^4*(d + e*x^2)*(a + c*x^4)),x]","\frac{3 \sqrt{2} c^{5/4} d^{5/2} x^3 \left(a^{3/4} e+\sqrt[4]{a} \sqrt{c} d\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)-3 \sqrt{2} c^{5/4} d^{5/2} x^3 \left(a^{3/4} e+\sqrt[4]{a} \sqrt{c} d\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)+24 a^2 e^{7/2} x^3 \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)+6 \sqrt{2} \sqrt[4]{a} c^{5/4} d^{5/2} x^3 \left(\sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)+6 \sqrt{2} \sqrt[4]{a} c^{5/4} d^{5/2} x^3 \left(\sqrt{a} e-\sqrt{c} d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)+24 a \sqrt{d} e x^2 \left(a e^2+c d^2\right)-8 a d^{3/2} \left(a e^2+c d^2\right)}{24 a^2 d^{5/2} x^3 \left(a e^2+c d^2\right)}","\frac{c^{5/4} \left(\sqrt{a} e+\sqrt{c} d\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} a^{7/4} \left(a e^2+c d^2\right)}-\frac{c^{5/4} \left(\sqrt{a} e+\sqrt{c} d\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} a^{7/4} \left(a e^2+c d^2\right)}+\frac{c^{5/4} \left(\sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} a^{7/4} \left(a e^2+c d^2\right)}-\frac{c^{5/4} \left(\sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} a^{7/4} \left(a e^2+c d^2\right)}+\frac{e^{7/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{d^{5/2} \left(a e^2+c d^2\right)}+\frac{e}{a d^2 x}-\frac{1}{3 a d x^3}",1,"(-8*a*d^(3/2)*(c*d^2 + a*e^2) + 24*a*Sqrt[d]*e*(c*d^2 + a*e^2)*x^2 + 24*a^2*e^(7/2)*x^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + 6*Sqrt[2]*a^(1/4)*c^(5/4)*d^(5/2)*(Sqrt[c]*d - Sqrt[a]*e)*x^3*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 6*Sqrt[2]*a^(1/4)*c^(5/4)*d^(5/2)*(-(Sqrt[c]*d) + Sqrt[a]*e)*x^3*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 3*Sqrt[2]*c^(5/4)*d^(5/2)*(a^(1/4)*Sqrt[c]*d + a^(3/4)*e)*x^3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] - 3*Sqrt[2]*c^(5/4)*d^(5/2)*(a^(1/4)*Sqrt[c]*d + a^(3/4)*e)*x^3*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(24*a^2*d^(5/2)*(c*d^2 + a*e^2)*x^3)","A",1
244,1,135,169,0.209868,"\int \frac{x^9}{\left(d+e x^2\right) \left(a+c x^4\right)^2} \, dx","Integrate[x^9/((d + e*x^2)*(a + c*x^4)^2),x]","\frac{-\frac{\sqrt{a} d \left(a e^2+3 c d^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a}}\right)}{c^{3/2}}+\frac{a e \left(a e^2+2 c d^2\right) \log \left(a+c x^4\right)}{c^2}+\frac{a \left(a e^2+c d^2\right) \left(a e+c d x^2\right)}{c^2 \left(a+c x^4\right)}+\frac{2 d^4 \log \left(d+e x^2\right)}{e}}{4 \left(a e^2+c d^2\right)^2}","-\frac{\sqrt{a} d \left(a e^2+3 c d^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a}}\right)}{4 c^{3/2} \left(a e^2+c d^2\right)^2}+\frac{a e \left(a e^2+2 c d^2\right) \log \left(a+c x^4\right)}{4 c^2 \left(a e^2+c d^2\right)^2}+\frac{a \left(a e+c d x^2\right)}{4 c^2 \left(a+c x^4\right) \left(a e^2+c d^2\right)}+\frac{d^4 \log \left(d+e x^2\right)}{2 e \left(a e^2+c d^2\right)^2}",1,"((a*(c*d^2 + a*e^2)*(a*e + c*d*x^2))/(c^2*(a + c*x^4)) - (Sqrt[a]*d*(3*c*d^2 + a*e^2)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/c^(3/2) + (2*d^4*Log[d + e*x^2])/e + (a*e*(2*c*d^2 + a*e^2)*Log[a + c*x^4])/c^2)/(4*(c*d^2 + a*e^2)^2)","A",1
245,1,142,150,0.1121719,"\int \frac{x^7}{\left(d+e x^2\right) \left(a+c x^4\right)^2} \, dx","Integrate[x^7/((d + e*x^2)*(a + c*x^4)^2),x]","\frac{\sqrt{a} e \left(a+c x^4\right) \left(a e^2+3 c d^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a}}\right)+\sqrt{c} \left(-2 c d^3 \left(a+c x^4\right) \log \left(d+e x^2\right)+c d^3 \left(a+c x^4\right) \log \left(a+c x^4\right)+a \left(d-e x^2\right) \left(a e^2+c d^2\right)\right)}{4 c^{3/2} \left(a+c x^4\right) \left(a e^2+c d^2\right)^2}","\frac{\sqrt{a} e \left(a e^2+3 c d^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a}}\right)}{4 c^{3/2} \left(a e^2+c d^2\right)^2}+\frac{a \left(d-e x^2\right)}{4 c \left(a+c x^4\right) \left(a e^2+c d^2\right)}+\frac{d^3 \log \left(a+c x^4\right)}{4 \left(a e^2+c d^2\right)^2}-\frac{d^3 \log \left(d+e x^2\right)}{2 \left(a e^2+c d^2\right)^2}",1,"(Sqrt[a]*e*(3*c*d^2 + a*e^2)*(a + c*x^4)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]] + Sqrt[c]*(a*(c*d^2 + a*e^2)*(d - e*x^2) - 2*c*d^3*(a + c*x^4)*Log[d + e*x^2] + c*d^3*(a + c*x^4)*Log[a + c*x^4]))/(4*c^(3/2)*(c*d^2 + a*e^2)^2*(a + c*x^4))","A",1
246,1,120,155,0.1534652,"\int \frac{x^5}{\left(d+e x^2\right) \left(a+c x^4\right)^2} \, dx","Integrate[x^5/((d + e*x^2)*(a + c*x^4)^2),x]","\frac{\frac{d \left(c d^2-a e^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{c}}-\frac{\left(a e^2+c d^2\right) \left(a e+c d x^2\right)}{c \left(a+c x^4\right)}-d^2 e \log \left(a+c x^4\right)+2 d^2 e \log \left(d+e x^2\right)}{4 \left(a e^2+c d^2\right)^2}","-\frac{d^2 e \log \left(a+c x^4\right)}{4 \left(a e^2+c d^2\right)^2}+\frac{d^2 e \log \left(d+e x^2\right)}{2 \left(a e^2+c d^2\right)^2}+\frac{d \left(c d^2-a e^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a}}\right)}{4 \sqrt{a} \sqrt{c} \left(a e^2+c d^2\right)^2}+\frac{-a e-c d x^2}{4 c \left(a+c x^4\right) \left(a e^2+c d^2\right)}",1,"(-(((c*d^2 + a*e^2)*(a*e + c*d*x^2))/(c*(a + c*x^4))) + (d*(c*d^2 - a*e^2)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(Sqrt[a]*Sqrt[c]) + 2*d^2*e*Log[d + e*x^2] - d^2*e*Log[a + c*x^4])/(4*(c*d^2 + a*e^2)^2)","A",1
247,1,114,149,0.1413183,"\int \frac{x^3}{\left(d+e x^2\right) \left(a+c x^4\right)^2} \, dx","Integrate[x^3/((d + e*x^2)*(a + c*x^4)^2),x]","\frac{\frac{e \left(a e^2-c d^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{c}}+\frac{\left(e x^2-d\right) \left(a e^2+c d^2\right)}{a+c x^4}+d e^2 \log \left(a+c x^4\right)-2 d e^2 \log \left(d+e x^2\right)}{4 \left(a e^2+c d^2\right)^2}","\frac{d e^2 \log \left(a+c x^4\right)}{4 \left(a e^2+c d^2\right)^2}-\frac{d e^2 \log \left(d+e x^2\right)}{2 \left(a e^2+c d^2\right)^2}-\frac{e \left(c d^2-a e^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a}}\right)}{4 \sqrt{a} \sqrt{c} \left(a e^2+c d^2\right)^2}+\frac{e x^2-d}{4 \left(a+c x^4\right) \left(a e^2+c d^2\right)}",1,"(((c*d^2 + a*e^2)*(-d + e*x^2))/(a + c*x^4) + (e*(-(c*d^2) + a*e^2)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(Sqrt[a]*Sqrt[c]) - 2*d*e^2*Log[d + e*x^2] + d*e^2*Log[a + c*x^4])/(4*(c*d^2 + a*e^2)^2)","A",1
248,1,117,151,0.1333855,"\int \frac{x}{\left(d+e x^2\right) \left(a+c x^4\right)^2} \, dx","Integrate[x/((d + e*x^2)*(a + c*x^4)^2),x]","\frac{\frac{\sqrt{c} d \left(3 a e^2+c d^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a}}\right)}{a^{3/2}}+\frac{\left(a e^2+c d^2\right) \left(a e+c d x^2\right)}{a \left(a+c x^4\right)}-e^3 \log \left(a+c x^4\right)+2 e^3 \log \left(d+e x^2\right)}{4 \left(a e^2+c d^2\right)^2}","\frac{\sqrt{c} d \left(3 a e^2+c d^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a}}\right)}{4 a^{3/2} \left(a e^2+c d^2\right)^2}+\frac{a e+c d x^2}{4 a \left(a+c x^4\right) \left(a e^2+c d^2\right)}-\frac{e^3 \log \left(a+c x^4\right)}{4 \left(a e^2+c d^2\right)^2}+\frac{e^3 \log \left(d+e x^2\right)}{2 \left(a e^2+c d^2\right)^2}",1,"(((c*d^2 + a*e^2)*(a*e + c*d*x^2))/(a*(a + c*x^4)) + (Sqrt[c]*d*(c*d^2 + 3*a*e^2)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/a^(3/2) + 2*e^3*Log[d + e*x^2] - e^3*Log[a + c*x^4])/(4*(c*d^2 + a*e^2)^2)","A",1
249,1,241,209,0.1826153,"\int \frac{1}{x \left(d+e x^2\right) \left(a+c x^4\right)^2} \, dx","Integrate[1/(x*(d + e*x^2)*(a + c*x^4)^2),x]","\frac{-2 a^2 e^4 \left(a+c x^4\right) \log \left(d+e x^2\right)+4 \log (x) \left(a+c x^4\right) \left(a e^2+c d^2\right)^2-c d^2 \left(a+c x^4\right) \left(2 a e^2+c d^2\right) \log \left(a+c x^4\right)+\sqrt{a} \sqrt{c} d e \left(a+c x^4\right) \left(3 a e^2+c d^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)+\sqrt{a} \sqrt{c} d e \left(a+c x^4\right) \left(3 a e^2+c d^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)+a c d \left(d-e x^2\right) \left(a e^2+c d^2\right)}{4 a^2 d \left(a+c x^4\right) \left(a e^2+c d^2\right)^2}","-\frac{\sqrt{c} e \tan ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a}}\right)}{4 a^{3/2} \left(a e^2+c d^2\right)}-\frac{c d \left(2 a e^2+c d^2\right) \log \left(a+c x^4\right)}{4 a^2 \left(a e^2+c d^2\right)^2}+\frac{\log (x)}{a^2 d}+\frac{c \left(d-e x^2\right)}{4 a \left(a+c x^4\right) \left(a e^2+c d^2\right)}-\frac{e^4 \log \left(d+e x^2\right)}{2 d \left(a e^2+c d^2\right)^2}-\frac{\sqrt{c} e^3 \tan ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a}}\right)}{2 \sqrt{a} \left(a e^2+c d^2\right)^2}",1,"(a*c*d*(c*d^2 + a*e^2)*(d - e*x^2) + Sqrt[a]*Sqrt[c]*d*e*(c*d^2 + 3*a*e^2)*(a + c*x^4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + Sqrt[a]*Sqrt[c]*d*e*(c*d^2 + 3*a*e^2)*(a + c*x^4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 4*(c*d^2 + a*e^2)^2*(a + c*x^4)*Log[x] - 2*a^2*e^4*(a + c*x^4)*Log[d + e*x^2] - c*d^2*(c*d^2 + 2*a*e^2)*(a + c*x^4)*Log[a + c*x^4])/(4*a^2*d*(c*d^2 + a*e^2)^2*(a + c*x^4))","A",1
250,1,248,236,0.4271711,"\int \frac{1}{x^3 \left(d+e x^2\right) \left(a+c x^4\right)^2} \, dx","Integrate[1/(x^3*(d + e*x^2)*(a + c*x^4)^2),x]","\frac{1}{4} \left(\frac{c^{3/2} d \left(5 a e^2+3 c d^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{a^{5/2} \left(a e^2+c d^2\right)^2}+\frac{c^{3/2} d \left(5 a e^2+3 c d^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{a^{5/2} \left(a e^2+c d^2\right)^2}-\frac{c \left(a e+c d x^2\right)}{a^2 \left(a+c x^4\right) \left(a e^2+c d^2\right)}+\frac{c \left(2 a e^3+c d^2 e\right) \log \left(a+c x^4\right)}{a^2 \left(a e^2+c d^2\right)^2}-\frac{4 e \log (x)}{a^2 d^2}-\frac{2}{a^2 d x^2}+\frac{2 e^5 \log \left(d+e x^2\right)}{\left(a d e^2+c d^3\right)^2}\right)","-\frac{c^{3/2} d \left(2 a e^2+c d^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a}}\right)}{2 a^{5/2} \left(a e^2+c d^2\right)^2}-\frac{c^{3/2} d \tan ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a}}\right)}{4 a^{5/2} \left(a e^2+c d^2\right)}+\frac{c e \left(2 a e^2+c d^2\right) \log \left(a+c x^4\right)}{4 a^2 \left(a e^2+c d^2\right)^2}-\frac{c \left(a e+c d x^2\right)}{4 a^2 \left(a+c x^4\right) \left(a e^2+c d^2\right)}-\frac{e \log (x)}{a^2 d^2}-\frac{1}{2 a^2 d x^2}+\frac{e^5 \log \left(d+e x^2\right)}{2 d^2 \left(a e^2+c d^2\right)^2}",1,"(-2/(a^2*d*x^2) - (c*(a*e + c*d*x^2))/(a^2*(c*d^2 + a*e^2)*(a + c*x^4)) + (c^(3/2)*d*(3*c*d^2 + 5*a*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(a^(5/2)*(c*d^2 + a*e^2)^2) + (c^(3/2)*d*(3*c*d^2 + 5*a*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(a^(5/2)*(c*d^2 + a*e^2)^2) - (4*e*Log[x])/(a^2*d^2) + (2*e^5*Log[d + e*x^2])/(c*d^3 + a*d*e^2)^2 + (c*(c*d^2*e + 2*a*e^3)*Log[a + c*x^4])/(a^2*(c*d^2 + a*e^2)^2))/4","A",1
251,1,278,265,0.3913979,"\int \frac{1}{x^5 \left(d+e x^2\right) \left(a+c x^4\right)^2} \, dx","Integrate[1/(x^5*(d + e*x^2)*(a + c*x^4)^2),x]","\frac{1}{4} \left(-\frac{c^{3/2} e \left(5 a e^2+3 c d^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{a^{5/2} \left(a e^2+c d^2\right)^2}-\frac{c^{3/2} e \left(5 a e^2+3 c d^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{a^{5/2} \left(a e^2+c d^2\right)^2}+\frac{c^2 \left(3 a d e^2+2 c d^3\right) \log \left(a+c x^4\right)}{a^3 \left(a e^2+c d^2\right)^2}+\frac{4 \log (x) \left(a e^2-2 c d^2\right)}{a^3 d^3}+\frac{c^2 \left(e x^2-d\right)}{a^2 \left(a+c x^4\right) \left(a e^2+c d^2\right)}+\frac{2 e}{a^2 d^2 x^2}-\frac{1}{a^2 d x^4}-\frac{2 e^6 \log \left(d+e x^2\right)}{d^3 \left(a e^2+c d^2\right)^2}\right)","\frac{c^{3/2} e \left(2 a e^2+c d^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a}}\right)}{2 a^{5/2} \left(a e^2+c d^2\right)^2}+\frac{c^{3/2} e \tan ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a}}\right)}{4 a^{5/2} \left(a e^2+c d^2\right)}+\frac{c^2 d \left(3 a e^2+2 c d^2\right) \log \left(a+c x^4\right)}{4 a^3 \left(a e^2+c d^2\right)^2}-\frac{\log (x) \left(2 c d^2-a e^2\right)}{a^3 d^3}-\frac{c^2 \left(d-e x^2\right)}{4 a^2 \left(a+c x^4\right) \left(a e^2+c d^2\right)}+\frac{e}{2 a^2 d^2 x^2}-\frac{1}{4 a^2 d x^4}-\frac{e^6 \log \left(d+e x^2\right)}{2 d^3 \left(a e^2+c d^2\right)^2}",1,"(-(1/(a^2*d*x^4)) + (2*e)/(a^2*d^2*x^2) + (c^2*(-d + e*x^2))/(a^2*(c*d^2 + a*e^2)*(a + c*x^4)) - (c^(3/2)*e*(3*c*d^2 + 5*a*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(a^(5/2)*(c*d^2 + a*e^2)^2) - (c^(3/2)*e*(3*c*d^2 + 5*a*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(a^(5/2)*(c*d^2 + a*e^2)^2) + (4*(-2*c*d^2 + a*e^2)*Log[x])/(a^3*d^3) - (2*e^6*Log[d + e*x^2])/(d^3*(c*d^2 + a*e^2)^2) + (c^2*(2*c*d^3 + 3*a*d*e^2)*Log[a + c*x^4])/(a^3*(c*d^2 + a*e^2)^2))/4","A",1
252,1,431,712,0.3039592,"\int \frac{x^8}{\left(d+e x^2\right) \left(a+c x^4\right)^2} \, dx","Integrate[x^8/((d + e*x^2)*(a + c*x^4)^2),x]","\frac{\frac{\sqrt{2} \sqrt[4]{a} \left(3 a^{3/2} e^3+7 \sqrt{a} c d^2 e+a \sqrt{c} d e^2+5 c^{3/2} d^3\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{c^{7/4}}-\frac{\sqrt{2} \sqrt[4]{a} \left(3 a^{3/2} e^3+7 \sqrt{a} c d^2 e+a \sqrt{c} d e^2+5 c^{3/2} d^3\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{c^{7/4}}-\frac{2 \sqrt{2} \sqrt[4]{a} \left(3 a^{3/2} e^3+7 \sqrt{a} c d^2 e-a \sqrt{c} d e^2-5 c^{3/2} d^3\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{c^{7/4}}+\frac{2 \sqrt{2} \sqrt[4]{a} \left(3 a^{3/2} e^3+7 \sqrt{a} c d^2 e-a \sqrt{c} d e^2-5 c^{3/2} d^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{c^{7/4}}+\frac{8 a x \left(d-e x^2\right) \left(a e^2+c d^2\right)}{c \left(a+c x^4\right)}+\frac{32 d^{7/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{e}}}{32 \left(a e^2+c d^2\right)^2}","\frac{\sqrt[4]{a} d^2 \left(\sqrt{a} e+\sqrt{c} d\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} c^{3/4} \left(a e^2+c d^2\right)^2}-\frac{\sqrt[4]{a} d^2 \left(\sqrt{a} e+\sqrt{c} d\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} c^{3/4} \left(a e^2+c d^2\right)^2}+\frac{\sqrt[4]{a} \left(3 \sqrt{a} e+\sqrt{c} d\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{16 \sqrt{2} c^{7/4} \left(a e^2+c d^2\right)}-\frac{\sqrt[4]{a} \left(3 \sqrt{a} e+\sqrt{c} d\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{16 \sqrt{2} c^{7/4} \left(a e^2+c d^2\right)}+\frac{\sqrt[4]{a} d^2 \left(\sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} c^{3/4} \left(a e^2+c d^2\right)^2}-\frac{\sqrt[4]{a} d^2 \left(\sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} c^{3/4} \left(a e^2+c d^2\right)^2}+\frac{\sqrt[4]{a} \left(\sqrt{c} d-3 \sqrt{a} e\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{8 \sqrt{2} c^{7/4} \left(a e^2+c d^2\right)}-\frac{\sqrt[4]{a} \left(\sqrt{c} d-3 \sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{8 \sqrt{2} c^{7/4} \left(a e^2+c d^2\right)}-\frac{x^3 \left(a e+c d x^2\right)}{4 c \left(a+c x^4\right) \left(a e^2+c d^2\right)}+\frac{d x}{4 c \left(a e^2+c d^2\right)}+\frac{d^{7/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{e} \left(a e^2+c d^2\right)^2}",1,"((8*a*(c*d^2 + a*e^2)*x*(d - e*x^2))/(c*(a + c*x^4)) + (32*d^(7/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] - (2*Sqrt[2]*a^(1/4)*(-5*c^(3/2)*d^3 + 7*Sqrt[a]*c*d^2*e - a*Sqrt[c]*d*e^2 + 3*a^(3/2)*e^3)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/c^(7/4) + (2*Sqrt[2]*a^(1/4)*(-5*c^(3/2)*d^3 + 7*Sqrt[a]*c*d^2*e - a*Sqrt[c]*d*e^2 + 3*a^(3/2)*e^3)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/c^(7/4) + (Sqrt[2]*a^(1/4)*(5*c^(3/2)*d^3 + 7*Sqrt[a]*c*d^2*e + a*Sqrt[c]*d*e^2 + 3*a^(3/2)*e^3)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/c^(7/4) - (Sqrt[2]*a^(1/4)*(5*c^(3/2)*d^3 + 7*Sqrt[a]*c*d^2*e + a*Sqrt[c]*d*e^2 + 3*a^(3/2)*e^3)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/c^(7/4))/(32*(c*d^2 + a*e^2)^2)","A",1
253,1,428,687,0.3685574,"\int \frac{x^6}{\left(d+e x^2\right) \left(a+c x^4\right)^2} \, dx","Integrate[x^6/((d + e*x^2)*(a + c*x^4)^2),x]","-\frac{\frac{\sqrt{2} \left(a^{3/2} e^3+5 \sqrt{a} c d^2 e+a \sqrt{c} d e^2-3 c^{3/2} d^3\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{\sqrt[4]{a} c^{5/4}}-\frac{\sqrt{2} \left(a^{3/2} e^3+5 \sqrt{a} c d^2 e+a \sqrt{c} d e^2-3 c^{3/2} d^3\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{\sqrt[4]{a} c^{5/4}}+\frac{2 \sqrt{2} \left(a^{3/2} e^3+5 \sqrt{a} c d^2 e-a \sqrt{c} d e^2+3 c^{3/2} d^3\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{\sqrt[4]{a} c^{5/4}}-\frac{2 \sqrt{2} \left(a^{3/2} e^3+5 \sqrt{a} c d^2 e-a \sqrt{c} d e^2+3 c^{3/2} d^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{\sqrt[4]{a} c^{5/4}}+\frac{8 \left(a e^2+c d^2\right) \left(a e x+c d x^3\right)}{c \left(a+c x^4\right)}+32 d^{5/2} \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{32 \left(a e^2+c d^2\right)^2}","-\frac{\left(\sqrt{a} e+\sqrt{c} d\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{16 \sqrt{2} \sqrt[4]{a} c^{5/4} \left(a e^2+c d^2\right)}+\frac{\left(\sqrt{a} e+\sqrt{c} d\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{16 \sqrt{2} \sqrt[4]{a} c^{5/4} \left(a e^2+c d^2\right)}+\frac{\left(\sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{8 \sqrt{2} \sqrt[4]{a} c^{5/4} \left(a e^2+c d^2\right)}-\frac{\left(\sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{8 \sqrt{2} \sqrt[4]{a} c^{5/4} \left(a e^2+c d^2\right)}+\frac{d^2 \left(\sqrt{c} d-\sqrt{a} e\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} \sqrt[4]{a} \sqrt[4]{c} \left(a e^2+c d^2\right)^2}-\frac{d^2 \left(\sqrt{c} d-\sqrt{a} e\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} \sqrt[4]{a} \sqrt[4]{c} \left(a e^2+c d^2\right)^2}-\frac{x \left(a e+c d x^2\right)}{4 c \left(a+c x^4\right) \left(a e^2+c d^2\right)}-\frac{d^2 \left(\sqrt{a} e+\sqrt{c} d\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} \sqrt[4]{a} \sqrt[4]{c} \left(a e^2+c d^2\right)^2}+\frac{d^2 \left(\sqrt{a} e+\sqrt{c} d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} \sqrt[4]{a} \sqrt[4]{c} \left(a e^2+c d^2\right)^2}-\frac{d^{5/2} \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\left(a e^2+c d^2\right)^2}",1,"-1/32*((8*(c*d^2 + a*e^2)*(a*e*x + c*d*x^3))/(c*(a + c*x^4)) + 32*d^(5/2)*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + (2*Sqrt[2]*(3*c^(3/2)*d^3 + 5*Sqrt[a]*c*d^2*e - a*Sqrt[c]*d*e^2 + a^(3/2)*e^3)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(a^(1/4)*c^(5/4)) - (2*Sqrt[2]*(3*c^(3/2)*d^3 + 5*Sqrt[a]*c*d^2*e - a*Sqrt[c]*d*e^2 + a^(3/2)*e^3)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(a^(1/4)*c^(5/4)) + (Sqrt[2]*(-3*c^(3/2)*d^3 + 5*Sqrt[a]*c*d^2*e + a*Sqrt[c]*d*e^2 + a^(3/2)*e^3)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(a^(1/4)*c^(5/4)) - (Sqrt[2]*(-3*c^(3/2)*d^3 + 5*Sqrt[a]*c*d^2*e + a*Sqrt[c]*d*e^2 + a^(3/2)*e^3)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(a^(1/4)*c^(5/4)))/(c*d^2 + a*e^2)^2","A",1
254,1,423,685,0.2717409,"\int \frac{x^4}{\left(d+e x^2\right) \left(a+c x^4\right)^2} \, dx","Integrate[x^4/((d + e*x^2)*(a + c*x^4)^2),x]","\frac{\frac{\sqrt{2} \left(a^{3/2} e^3-3 \sqrt{a} c d^2 e+3 a \sqrt{c} d e^2-c^{3/2} d^3\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{a^{3/4} c^{3/4}}+\frac{\sqrt{2} \left(-a^{3/2} e^3+3 \sqrt{a} c d^2 e-3 a \sqrt{c} d e^2+c^{3/2} d^3\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{a^{3/4} c^{3/4}}-\frac{2 \sqrt{2} \left(a^{3/2} e^3-3 \sqrt{a} c d^2 e-3 a \sqrt{c} d e^2+c^{3/2} d^3\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{a^{3/4} c^{3/4}}+\frac{2 \sqrt{2} \left(a^{3/2} e^3-3 \sqrt{a} c d^2 e-3 a \sqrt{c} d e^2+c^{3/2} d^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{a^{3/4} c^{3/4}}+\frac{8 \left(e x^3-d x\right) \left(a e^2+c d^2\right)}{a+c x^4}+32 d^{3/2} e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{32 \left(a e^2+c d^2\right)^2}","\frac{\left(\sqrt{a} e+3 \sqrt{c} d\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{16 \sqrt{2} a^{3/4} c^{3/4} \left(a e^2+c d^2\right)}-\frac{\left(\sqrt{a} e+3 \sqrt{c} d\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{16 \sqrt{2} a^{3/4} c^{3/4} \left(a e^2+c d^2\right)}+\frac{\left(3 \sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{8 \sqrt{2} a^{3/4} c^{3/4} \left(a e^2+c d^2\right)}-\frac{\left(3 \sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{8 \sqrt{2} a^{3/4} c^{3/4} \left(a e^2+c d^2\right)}-\frac{\sqrt[4]{c} d^2 \left(\sqrt{a} e+\sqrt{c} d\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} a^{3/4} \left(a e^2+c d^2\right)^2}+\frac{\sqrt[4]{c} d^2 \left(\sqrt{a} e+\sqrt{c} d\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} a^{3/4} \left(a e^2+c d^2\right)^2}-\frac{\sqrt[4]{c} d^2 \left(\sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} a^{3/4} \left(a e^2+c d^2\right)^2}+\frac{\sqrt[4]{c} d^2 \left(\sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} a^{3/4} \left(a e^2+c d^2\right)^2}-\frac{x \left(d-e x^2\right)}{4 \left(a+c x^4\right) \left(a e^2+c d^2\right)}+\frac{d^{3/2} e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\left(a e^2+c d^2\right)^2}",1,"((8*(c*d^2 + a*e^2)*(-(d*x) + e*x^3))/(a + c*x^4) + 32*d^(3/2)*e^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]] - (2*Sqrt[2]*(c^(3/2)*d^3 - 3*Sqrt[a]*c*d^2*e - 3*a*Sqrt[c]*d*e^2 + a^(3/2)*e^3)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(a^(3/4)*c^(3/4)) + (2*Sqrt[2]*(c^(3/2)*d^3 - 3*Sqrt[a]*c*d^2*e - 3*a*Sqrt[c]*d*e^2 + a^(3/2)*e^3)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(a^(3/4)*c^(3/4)) + (Sqrt[2]*(-(c^(3/2)*d^3) - 3*Sqrt[a]*c*d^2*e + 3*a*Sqrt[c]*d*e^2 + a^(3/2)*e^3)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(a^(3/4)*c^(3/4)) + (Sqrt[2]*(c^(3/2)*d^3 + 3*Sqrt[a]*c*d^2*e - 3*a*Sqrt[c]*d*e^2 - a^(3/2)*e^3)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(a^(3/4)*c^(3/4)))/(32*(c*d^2 + a*e^2)^2)","A",1
255,1,428,685,0.2830048,"\int \frac{x^2}{\left(d+e x^2\right) \left(a+c x^4\right)^2} \, dx","Integrate[x^2/((d + e*x^2)*(a + c*x^4)^2),x]","\frac{\frac{\sqrt{2} \left(-3 a^{3/2} e^3+\sqrt{a} c d^2 e+5 a \sqrt{c} d e^2+c^{3/2} d^3\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{a^{5/4} \sqrt[4]{c}}-\frac{\sqrt{2} \left(-3 a^{3/2} e^3+\sqrt{a} c d^2 e+5 a \sqrt{c} d e^2+c^{3/2} d^3\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{a^{5/4} \sqrt[4]{c}}-\frac{2 \sqrt{2} \left(3 a^{3/2} e^3-\sqrt{a} c d^2 e+5 a \sqrt{c} d e^2+c^{3/2} d^3\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{a^{5/4} \sqrt[4]{c}}+\frac{2 \sqrt{2} \left(3 a^{3/2} e^3-\sqrt{a} c d^2 e+5 a \sqrt{c} d e^2+c^{3/2} d^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{a^{5/4} \sqrt[4]{c}}+\frac{8 \left(a e^2+c d^2\right) \left(a e x+c d x^3\right)}{a \left(a+c x^4\right)}-32 \sqrt{d} e^{5/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{32 \left(a e^2+c d^2\right)^2}","\frac{\sqrt[4]{c} d e \left(\sqrt{a} e+\sqrt{c} d\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} a^{3/4} \left(a e^2+c d^2\right)^2}-\frac{\sqrt[4]{c} d e \left(\sqrt{a} e+\sqrt{c} d\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} a^{3/4} \left(a e^2+c d^2\right)^2}+\frac{\left(\sqrt{c} d-3 \sqrt{a} e\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{16 \sqrt{2} a^{5/4} \sqrt[4]{c} \left(a e^2+c d^2\right)}-\frac{\left(\sqrt{c} d-3 \sqrt{a} e\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{16 \sqrt{2} a^{5/4} \sqrt[4]{c} \left(a e^2+c d^2\right)}+\frac{\sqrt[4]{c} d e \left(\sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} a^{3/4} \left(a e^2+c d^2\right)^2}-\frac{\sqrt[4]{c} d e \left(\sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} a^{3/4} \left(a e^2+c d^2\right)^2}-\frac{\left(3 \sqrt{a} e+\sqrt{c} d\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{8 \sqrt{2} a^{5/4} \sqrt[4]{c} \left(a e^2+c d^2\right)}+\frac{\left(3 \sqrt{a} e+\sqrt{c} d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{8 \sqrt{2} a^{5/4} \sqrt[4]{c} \left(a e^2+c d^2\right)}+\frac{x \left(a e+c d x^2\right)}{4 a \left(a+c x^4\right) \left(a e^2+c d^2\right)}-\frac{\sqrt{d} e^{5/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\left(a e^2+c d^2\right)^2}",1,"((8*(c*d^2 + a*e^2)*(a*e*x + c*d*x^3))/(a*(a + c*x^4)) - 32*Sqrt[d]*e^(5/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]] - (2*Sqrt[2]*(c^(3/2)*d^3 - Sqrt[a]*c*d^2*e + 5*a*Sqrt[c]*d*e^2 + 3*a^(3/2)*e^3)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(a^(5/4)*c^(1/4)) + (2*Sqrt[2]*(c^(3/2)*d^3 - Sqrt[a]*c*d^2*e + 5*a*Sqrt[c]*d*e^2 + 3*a^(3/2)*e^3)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(a^(5/4)*c^(1/4)) + (Sqrt[2]*(c^(3/2)*d^3 + Sqrt[a]*c*d^2*e + 5*a*Sqrt[c]*d*e^2 - 3*a^(3/2)*e^3)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(a^(5/4)*c^(1/4)) - (Sqrt[2]*(c^(3/2)*d^3 + Sqrt[a]*c*d^2*e + 5*a*Sqrt[c]*d*e^2 - 3*a^(3/2)*e^3)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(a^(5/4)*c^(1/4)))/(32*(c*d^2 + a*e^2)^2)","A",1
256,1,429,689,0.2908353,"\int \frac{1}{\left(d+e x^2\right) \left(a+c x^4\right)^2} \, dx","Integrate[1/((d + e*x^2)*(a + c*x^4)^2),x]","\frac{-\frac{\sqrt{2} \sqrt[4]{c} \left(5 a^{3/2} e^3+\sqrt{a} c d^2 e+7 a \sqrt{c} d e^2+3 c^{3/2} d^3\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{a^{7/4}}+\frac{\sqrt{2} \sqrt[4]{c} \left(5 a^{3/2} e^3+\sqrt{a} c d^2 e+7 a \sqrt{c} d e^2+3 c^{3/2} d^3\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{a^{7/4}}+\frac{2 \sqrt{2} \sqrt[4]{c} \left(5 a^{3/2} e^3+\sqrt{a} c d^2 e-7 a \sqrt{c} d e^2-3 c^{3/2} d^3\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{a^{7/4}}-\frac{2 \sqrt{2} \sqrt[4]{c} \left(5 a^{3/2} e^3+\sqrt{a} c d^2 e-7 a \sqrt{c} d e^2-3 c^{3/2} d^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{a^{7/4}}+\frac{8 c x \left(d-e x^2\right) \left(a e^2+c d^2\right)}{a \left(a+c x^4\right)}+\frac{32 e^{7/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{d}}}{32 \left(a e^2+c d^2\right)^2}","-\frac{\sqrt[4]{c} e^2 \left(\sqrt{a} e+\sqrt{c} d\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} a^{3/4} \left(a e^2+c d^2\right)^2}+\frac{\sqrt[4]{c} e^2 \left(\sqrt{a} e+\sqrt{c} d\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} a^{3/4} \left(a e^2+c d^2\right)^2}-\frac{\sqrt[4]{c} \left(\sqrt{a} e+3 \sqrt{c} d\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{16 \sqrt{2} a^{7/4} \left(a e^2+c d^2\right)}+\frac{\sqrt[4]{c} \left(\sqrt{a} e+3 \sqrt{c} d\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{16 \sqrt{2} a^{7/4} \left(a e^2+c d^2\right)}-\frac{\sqrt[4]{c} e^2 \left(\sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} a^{3/4} \left(a e^2+c d^2\right)^2}+\frac{\sqrt[4]{c} e^2 \left(\sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} a^{3/4} \left(a e^2+c d^2\right)^2}-\frac{\sqrt[4]{c} \left(3 \sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{8 \sqrt{2} a^{7/4} \left(a e^2+c d^2\right)}+\frac{\sqrt[4]{c} \left(3 \sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{8 \sqrt{2} a^{7/4} \left(a e^2+c d^2\right)}+\frac{c x \left(d-e x^2\right)}{4 a \left(a+c x^4\right) \left(a e^2+c d^2\right)}+\frac{e^{7/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{d} \left(a e^2+c d^2\right)^2}",1,"((8*c*(c*d^2 + a*e^2)*x*(d - e*x^2))/(a*(a + c*x^4)) + (32*e^(7/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d] + (2*Sqrt[2]*c^(1/4)*(-3*c^(3/2)*d^3 + Sqrt[a]*c*d^2*e - 7*a*Sqrt[c]*d*e^2 + 5*a^(3/2)*e^3)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(7/4) - (2*Sqrt[2]*c^(1/4)*(-3*c^(3/2)*d^3 + Sqrt[a]*c*d^2*e - 7*a*Sqrt[c]*d*e^2 + 5*a^(3/2)*e^3)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(7/4) - (Sqrt[2]*c^(1/4)*(3*c^(3/2)*d^3 + Sqrt[a]*c*d^2*e + 7*a*Sqrt[c]*d*e^2 + 5*a^(3/2)*e^3)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/a^(7/4) + (Sqrt[2]*c^(1/4)*(3*c^(3/2)*d^3 + Sqrt[a]*c*d^2*e + 7*a*Sqrt[c]*d*e^2 + 5*a^(3/2)*e^3)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/a^(7/4))/(32*(c*d^2 + a*e^2)^2)","A",1
257,1,499,745,0.3701039,"\int \frac{1}{x^2 \left(d+e x^2\right) \left(a+c x^4\right)^2} \, dx","Integrate[1/(x^2*(d + e*x^2)*(a + c*x^4)^2),x]","\frac{1}{32} \left(\frac{\sqrt{2} c^{3/4} \left(7 a^{3/2} e^3+3 \sqrt{a} c d^2 e-9 a \sqrt{c} d e^2-5 c^{3/2} d^3\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{a^{9/4} \left(a e^2+c d^2\right)^2}+\frac{\sqrt{2} c^{3/4} \left(-7 a^{3/2} e^3-3 \sqrt{a} c d^2 e+9 a \sqrt{c} d e^2+5 c^{3/2} d^3\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{a^{9/4} \left(a e^2+c d^2\right)^2}+\frac{2 \sqrt{2} c^{3/4} \left(7 a^{3/2} e^3+3 \sqrt{a} c d^2 e+9 a \sqrt{c} d e^2+5 c^{3/2} d^3\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{a^{9/4} \left(a e^2+c d^2\right)^2}-\frac{2 \sqrt{2} c^{3/4} \left(7 a^{3/2} e^3+3 \sqrt{a} c d^2 e+9 a \sqrt{c} d e^2+5 c^{3/2} d^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{a^{9/4} \left(a e^2+c d^2\right)^2}-\frac{8 c x \left(a e+c d x^2\right)}{a^2 \left(a+c x^4\right) \left(a e^2+c d^2\right)}-\frac{32}{a^2 d x}-\frac{32 e^{9/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{d^{3/2} \left(a e^2+c d^2\right)^2}\right)","-\frac{c^{3/4} \left(\sqrt{c} d-3 \sqrt{a} e\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{16 \sqrt{2} a^{9/4} \left(a e^2+c d^2\right)}+\frac{c^{3/4} \left(\sqrt{c} d-3 \sqrt{a} e\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{16 \sqrt{2} a^{9/4} \left(a e^2+c d^2\right)}+\frac{c^{3/4} \left(3 \sqrt{a} e+\sqrt{c} d\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{8 \sqrt{2} a^{9/4} \left(a e^2+c d^2\right)}-\frac{c^{3/4} \left(3 \sqrt{a} e+\sqrt{c} d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{8 \sqrt{2} a^{9/4} \left(a e^2+c d^2\right)}+\frac{c^{3/4} \left(a^{3/2} e^3-\sqrt{c} d \left(2 a e^2+c d^2\right)\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} a^{9/4} \left(a e^2+c d^2\right)^2}-\frac{c^{3/4} \left(a^{3/2} e^3-\sqrt{c} d \left(2 a e^2+c d^2\right)\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} a^{9/4} \left(a e^2+c d^2\right)^2}+\frac{c^{3/4} \left(a^{3/2} e^3+\sqrt{c} d \left(2 a e^2+c d^2\right)\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} a^{9/4} \left(a e^2+c d^2\right)^2}-\frac{c^{3/4} \left(a^{3/2} e^3+\sqrt{c} d \left(2 a e^2+c d^2\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} a^{9/4} \left(a e^2+c d^2\right)^2}-\frac{c x \left(a e+c d x^2\right)}{4 a^2 \left(a+c x^4\right) \left(a e^2+c d^2\right)}-\frac{1}{a^2 d x}-\frac{e^{9/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{d^{3/2} \left(a e^2+c d^2\right)^2}",1,"(-32/(a^2*d*x) - (8*c*x*(a*e + c*d*x^2))/(a^2*(c*d^2 + a*e^2)*(a + c*x^4)) - (32*e^(9/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(3/2)*(c*d^2 + a*e^2)^2) + (2*Sqrt[2]*c^(3/4)*(5*c^(3/2)*d^3 + 3*Sqrt[a]*c*d^2*e + 9*a*Sqrt[c]*d*e^2 + 7*a^(3/2)*e^3)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(a^(9/4)*(c*d^2 + a*e^2)^2) - (2*Sqrt[2]*c^(3/4)*(5*c^(3/2)*d^3 + 3*Sqrt[a]*c*d^2*e + 9*a*Sqrt[c]*d*e^2 + 7*a^(3/2)*e^3)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(a^(9/4)*(c*d^2 + a*e^2)^2) + (Sqrt[2]*c^(3/4)*(-5*c^(3/2)*d^3 + 3*Sqrt[a]*c*d^2*e - 9*a*Sqrt[c]*d*e^2 + 7*a^(3/2)*e^3)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(a^(9/4)*(c*d^2 + a*e^2)^2) + (Sqrt[2]*c^(3/4)*(5*c^(3/2)*d^3 - 3*Sqrt[a]*c*d^2*e + 9*a*Sqrt[c]*d*e^2 - 7*a^(3/2)*e^3)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(a^(9/4)*(c*d^2 + a*e^2)^2))/32","A",1
258,1,513,751,0.3856115,"\int \frac{1}{x^4 \left(d+e x^2\right) \left(a+c x^4\right)^2} \, dx","Integrate[1/(x^4*(d + e*x^2)*(a + c*x^4)^2),x]","\frac{1}{96} \left(\frac{3 \sqrt{2} c^{5/4} \left(9 a^{3/2} e^3+5 \sqrt{a} c d^2 e+11 a \sqrt{c} d e^2+7 c^{3/2} d^3\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{a^{11/4} \left(a e^2+c d^2\right)^2}-\frac{3 \sqrt{2} c^{5/4} \left(9 a^{3/2} e^3+5 \sqrt{a} c d^2 e+11 a \sqrt{c} d e^2+7 c^{3/2} d^3\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{a^{11/4} \left(a e^2+c d^2\right)^2}+\frac{6 \sqrt{2} c^{5/4} \left(-9 a^{3/2} e^3-5 \sqrt{a} c d^2 e+11 a \sqrt{c} d e^2+7 c^{3/2} d^3\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{a^{11/4} \left(a e^2+c d^2\right)^2}+\frac{6 \sqrt{2} c^{5/4} \left(9 a^{3/2} e^3+5 \sqrt{a} c d^2 e-11 a \sqrt{c} d e^2-7 c^{3/2} d^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{a^{11/4} \left(a e^2+c d^2\right)^2}-\frac{24 c^2 x \left(d-e x^2\right)}{a^2 \left(a+c x^4\right) \left(a e^2+c d^2\right)}+\frac{96 e}{a^2 d^2 x}-\frac{32}{a^2 d x^3}+\frac{96 e^{11/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{d^{5/2} \left(a e^2+c d^2\right)^2}\right)","\frac{c^{5/4} \left(\sqrt{a} e+\sqrt{c} d\right) \left(2 a e^2+c d^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} a^{11/4} \left(a e^2+c d^2\right)^2}+\frac{c^{5/4} \left(\sqrt{a} e+3 \sqrt{c} d\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{16 \sqrt{2} a^{11/4} \left(a e^2+c d^2\right)}-\frac{c^{5/4} \left(\sqrt{a} e+\sqrt{c} d\right) \left(2 a e^2+c d^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} a^{11/4} \left(a e^2+c d^2\right)^2}-\frac{c^{5/4} \left(\sqrt{a} e+3 \sqrt{c} d\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{16 \sqrt{2} a^{11/4} \left(a e^2+c d^2\right)}+\frac{c^{5/4} \left(\sqrt{c} d-\sqrt{a} e\right) \left(2 a e^2+c d^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} a^{11/4} \left(a e^2+c d^2\right)^2}+\frac{c^{5/4} \left(3 \sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{8 \sqrt{2} a^{11/4} \left(a e^2+c d^2\right)}-\frac{c^{5/4} \left(\sqrt{c} d-\sqrt{a} e\right) \left(2 a e^2+c d^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} a^{11/4} \left(a e^2+c d^2\right)^2}-\frac{c^{5/4} \left(3 \sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{8 \sqrt{2} a^{11/4} \left(a e^2+c d^2\right)}-\frac{c^2 x \left(d-e x^2\right)}{4 a^2 \left(a+c x^4\right) \left(a e^2+c d^2\right)}+\frac{e}{a^2 d^2 x}-\frac{1}{3 a^2 d x^3}+\frac{e^{11/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{d^{5/2} \left(a e^2+c d^2\right)^2}",1,"(-32/(a^2*d*x^3) + (96*e)/(a^2*d^2*x) - (24*c^2*x*(d - e*x^2))/(a^2*(c*d^2 + a*e^2)*(a + c*x^4)) + (96*e^(11/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(5/2)*(c*d^2 + a*e^2)^2) + (6*Sqrt[2]*c^(5/4)*(7*c^(3/2)*d^3 - 5*Sqrt[a]*c*d^2*e + 11*a*Sqrt[c]*d*e^2 - 9*a^(3/2)*e^3)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(a^(11/4)*(c*d^2 + a*e^2)^2) + (6*Sqrt[2]*c^(5/4)*(-7*c^(3/2)*d^3 + 5*Sqrt[a]*c*d^2*e - 11*a*Sqrt[c]*d*e^2 + 9*a^(3/2)*e^3)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(a^(11/4)*(c*d^2 + a*e^2)^2) + (3*Sqrt[2]*c^(5/4)*(7*c^(3/2)*d^3 + 5*Sqrt[a]*c*d^2*e + 11*a*Sqrt[c]*d*e^2 + 9*a^(3/2)*e^3)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(a^(11/4)*(c*d^2 + a*e^2)^2) - (3*Sqrt[2]*c^(5/4)*(7*c^(3/2)*d^3 + 5*Sqrt[a]*c*d^2*e + 11*a*Sqrt[c]*d*e^2 + 9*a^(3/2)*e^3)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(a^(11/4)*(c*d^2 + a*e^2)^2))/96","A",1
259,1,40,70,0.0956073,"\int \frac{x^2}{\left(1+x^2\right) \sqrt{1+x^4}} \, dx","Integrate[x^2/((1 + x^2)*Sqrt[1 + x^4]),x]","\sqrt[4]{-1} \left(\Pi \left(-i;\left.i \sinh ^{-1}\left(\sqrt[4]{-1} x\right)\right|-1\right)-F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-1} x\right)\right|-1\right)\right)","\frac{\left(x^2+1\right) \sqrt{\frac{x^4+1}{\left(x^2+1\right)^2}} F\left(2 \tan ^{-1}(x)|\frac{1}{2}\right)}{4 \sqrt{x^4+1}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{x^4+1}}\right)}{2 \sqrt{2}}",1,"(-1)^(1/4)*(-EllipticF[I*ArcSinh[(-1)^(1/4)*x], -1] + EllipticPi[-I, I*ArcSinh[(-1)^(1/4)*x], -1])","C",1
260,1,36,70,0.0920804,"\int \frac{x^2}{\left(1-x^2\right) \sqrt{1+x^4}} \, dx","Integrate[x^2/((1 - x^2)*Sqrt[1 + x^4]),x]","\sqrt[4]{-1} \left(F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-1} x\right)\right|-1\right)-\Pi \left(i;\left.\sin ^{-1}\left((-1)^{3/4} x\right)\right|-1\right)\right)","\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{x^4+1}}\right)}{2 \sqrt{2}}-\frac{\left(x^2+1\right) \sqrt{\frac{x^4+1}{\left(x^2+1\right)^2}} F\left(2 \tan ^{-1}(x)|\frac{1}{2}\right)}{4 \sqrt{x^4+1}}",1,"(-1)^(1/4)*(EllipticF[I*ArcSinh[(-1)^(1/4)*x], -1] - EllipticPi[I, ArcSin[(-1)^(3/4)*x], -1])","C",1
261,1,46,99,0.1055745,"\int \frac{x^2}{\left(1+x^2\right) \sqrt{1-x^4}} \, dx","Integrate[x^2/((1 + x^2)*Sqrt[1 - x^4]),x]","\frac{1}{2} \left(-\frac{x}{\sqrt{1-x^4}}+\frac{x^3}{\sqrt{1-x^4}}+2 F\left(\left.\sin ^{-1}(x)\right|-1\right)-E\left(\left.\sin ^{-1}(x)\right|-1\right)\right)","-\frac{x \left(1-x^2\right)}{2 \sqrt{1-x^4}}+\frac{\sqrt{x^2+1} \sqrt{1-x^2} F\left(\left.\sin ^{-1}(x)\right|-1\right)}{\sqrt{1-x^4}}-\frac{\sqrt{x^2+1} \sqrt{1-x^2} E\left(\left.\sin ^{-1}(x)\right|-1\right)}{2 \sqrt{1-x^4}}",1,"(-(x/Sqrt[1 - x^4]) + x^3/Sqrt[1 - x^4] - EllipticE[ArcSin[x], -1] + 2*EllipticF[ArcSin[x], -1])/2","A",1
262,1,37,61,0.0787196,"\int \frac{x^2}{\left(1-x^2\right) \sqrt{1-x^4}} \, dx","Integrate[x^2/((1 - x^2)*Sqrt[1 - x^4]),x]","\frac{-\sqrt{1-x^4} E\left(\left.\sin ^{-1}(x)\right|-1\right)+x^3+x}{2 \sqrt{1-x^4}}","\frac{x \left(x^2+1\right)}{2 \sqrt{1-x^4}}-\frac{\sqrt{1-x^2} \sqrt{x^2+1} E\left(\left.\sin ^{-1}(x)\right|-1\right)}{2 \sqrt{1-x^4}}",1,"(x + x^3 - Sqrt[1 - x^4]*EllipticE[ArcSin[x], -1])/(2*Sqrt[1 - x^4])","A",1
263,1,54,113,0.0751793,"\int \frac{x^2}{\left(1+x^2\right) \sqrt{-1+x^4}} \, dx","Integrate[x^2/((1 + x^2)*Sqrt[-1 + x^4]),x]","\frac{2 \sqrt{1-x^4} F\left(\left.\sin ^{-1}(x)\right|-1\right)-\sqrt{1-x^4} E\left(\left.\sin ^{-1}(x)\right|-1\right)+x^3-x}{2 \sqrt{x^4-1}}","-\frac{x \left(1-x^2\right)}{2 \sqrt{x^4-1}}+\frac{\sqrt{x^2-1} \sqrt{x^2+1} F\left(\sin ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right)|\frac{1}{2}\right)}{\sqrt{2} \sqrt{x^4-1}}-\frac{\sqrt{x^2+1} \sqrt{1-x^2} E\left(\left.\sin ^{-1}(x)\right|-1\right)}{2 \sqrt{x^4-1}}",1,"(-x + x^3 - Sqrt[1 - x^4]*EllipticE[ArcSin[x], -1] + 2*Sqrt[1 - x^4]*EllipticF[ArcSin[x], -1])/(2*Sqrt[-1 + x^4])","A",1
264,1,35,57,0.0720686,"\int \frac{x^2}{\left(1-x^2\right) \sqrt{-1+x^4}} \, dx","Integrate[x^2/((1 - x^2)*Sqrt[-1 + x^4]),x]","\frac{-\sqrt{1-x^4} E\left(\left.\sin ^{-1}(x)\right|-1\right)+x^3+x}{2 \sqrt{x^4-1}}","\frac{x \left(x^2+1\right)}{2 \sqrt{x^4-1}}-\frac{\sqrt{1-x^2} \sqrt{x^2+1} E\left(\left.\sin ^{-1}(x)\right|-1\right)}{2 \sqrt{x^4-1}}",1,"(x + x^3 - Sqrt[1 - x^4]*EllipticE[ArcSin[x], -1])/(2*Sqrt[-1 + x^4])","A",1
265,1,60,74,0.0907919,"\int \frac{x^2}{\left(1+x^2\right) \sqrt{-1-x^4}} \, dx","Integrate[x^2/((1 + x^2)*Sqrt[-1 - x^4]),x]","\frac{\sqrt[4]{-1} \sqrt{x^4+1} \left(\Pi \left(-i;\left.i \sinh ^{-1}\left(\sqrt[4]{-1} x\right)\right|-1\right)-F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-1} x\right)\right|-1\right)\right)}{\sqrt{-x^4-1}}","\frac{\left(x^2+1\right) \sqrt{\frac{x^4+1}{\left(x^2+1\right)^2}} F\left(2 \tan ^{-1}(x)|\frac{1}{2}\right)}{4 \sqrt{-x^4-1}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{-x^4-1}}\right)}{2 \sqrt{2}}",1,"((-1)^(1/4)*Sqrt[1 + x^4]*(-EllipticF[I*ArcSinh[(-1)^(1/4)*x], -1] + EllipticPi[-I, I*ArcSinh[(-1)^(1/4)*x], -1]))/Sqrt[-1 - x^4]","C",1
266,1,56,74,0.0875388,"\int \frac{x^2}{\left(1-x^2\right) \sqrt{-1-x^4}} \, dx","Integrate[x^2/((1 - x^2)*Sqrt[-1 - x^4]),x]","\frac{\sqrt[4]{-1} \sqrt{x^4+1} \left(F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-1} x\right)\right|-1\right)-\Pi \left(i;\left.\sin ^{-1}\left((-1)^{3/4} x\right)\right|-1\right)\right)}{\sqrt{-x^4-1}}","\frac{\tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{-x^4-1}}\right)}{2 \sqrt{2}}-\frac{\left(x^2+1\right) \sqrt{\frac{x^4+1}{\left(x^2+1\right)^2}} F\left(2 \tan ^{-1}(x)|\frac{1}{2}\right)}{4 \sqrt{-x^4-1}}",1,"((-1)^(1/4)*Sqrt[1 + x^4]*(EllipticF[I*ArcSinh[(-1)^(1/4)*x], -1] - EllipticPi[I, ArcSin[(-1)^(3/4)*x], -1]))/Sqrt[-1 - x^4]","C",1
267,1,142,243,0.1840972,"\int x^2 \sqrt{c+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[x^2*Sqrt[c + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]","\frac{\sqrt{\left(a+b x^2\right)^2} \sqrt{c+d x^2} \left(3 c^{3/2} (b c-2 a d) \sinh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)+\sqrt{d} x \sqrt{\frac{d x^2}{c}+1} \left(6 a d \left(c+2 d x^2\right)+b \left(-3 c^2+2 c d x^2+8 d^2 x^4\right)\right)\right)}{48 d^{5/2} \left(a+b x^2\right) \sqrt{\frac{d x^2}{c}+1}}","\frac{c^2 \sqrt{a^2+2 a b x^2+b^2 x^4} (b c-2 a d) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{16 d^{5/2} \left(a+b x^2\right)}-\frac{c x \sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2} (b c-2 a d)}{16 d^2 \left(a+b x^2\right)}+\frac{b x^3 \sqrt{a^2+2 a b x^2+b^2 x^4} \left(c+d x^2\right)^{3/2}}{6 d \left(a+b x^2\right)}-\frac{x^3 \sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2} (b c-2 a d)}{8 d \left(a+b x^2\right)}",1,"(Sqrt[(a + b*x^2)^2]*Sqrt[c + d*x^2]*(Sqrt[d]*x*Sqrt[1 + (d*x^2)/c]*(6*a*d*(c + 2*d*x^2) + b*(-3*c^2 + 2*c*d*x^2 + 8*d^2*x^4)) + 3*c^(3/2)*(b*c - 2*a*d)*ArcSinh[(Sqrt[d]*x)/Sqrt[c]]))/(48*d^(5/2)*(a + b*x^2)*Sqrt[1 + (d*x^2)/c])","A",1
268,1,56,108,0.0298663,"\int x \sqrt{c+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[x*Sqrt[c + d*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(c+d x^2\right)^{3/2} \left(5 a d-2 b c+3 b d x^2\right)}{15 d^2 \left(a+b x^2\right)}","\frac{b \sqrt{a^2+2 a b x^2+b^2 x^4} \left(c+d x^2\right)^{5/2}}{5 d^2 \left(a+b x^2\right)}-\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left(c+d x^2\right)^{3/2} (b c-a d)}{3 d^2 \left(a+b x^2\right)}",1,"(Sqrt[(a + b*x^2)^2]*(c + d*x^2)^(3/2)*(-2*b*c + 5*a*d + 3*b*d*x^2))/(15*d^2*(a + b*x^2))","A",1
269,1,121,178,0.1128065,"\int \sqrt{c+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[Sqrt[c + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]","\frac{\sqrt{\left(a+b x^2\right)^2} \sqrt{c+d x^2} \left(\sqrt{d} x \sqrt{\frac{d x^2}{c}+1} \left(4 a d+b \left(c+2 d x^2\right)\right)-\sqrt{c} (b c-4 a d) \sinh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)\right)}{8 d^{3/2} \left(a+b x^2\right) \sqrt{\frac{d x^2}{c}+1}}","-\frac{c \sqrt{a^2+2 a b x^2+b^2 x^4} (b c-4 a d) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{8 d^{3/2} \left(a+b x^2\right)}+\frac{b x \sqrt{a^2+2 a b x^2+b^2 x^4} \left(c+d x^2\right)^{3/2}}{4 d \left(a+b x^2\right)}-\frac{x \sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2} (b c-4 a d)}{8 d \left(a+b x^2\right)}",1,"(Sqrt[(a + b*x^2)^2]*Sqrt[c + d*x^2]*(Sqrt[d]*x*Sqrt[1 + (d*x^2)/c]*(4*a*d + b*(c + 2*d*x^2)) - Sqrt[c]*(b*c - 4*a*d)*ArcSinh[(Sqrt[d]*x)/Sqrt[c]]))/(8*d^(3/2)*(a + b*x^2)*Sqrt[1 + (d*x^2)/c])","A",1
270,1,83,152,0.0462166,"\int \frac{\sqrt{c+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{x} \, dx","Integrate[(Sqrt[c + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/x,x]","\frac{\sqrt{\left(a+b x^2\right)^2} \left(\sqrt{c+d x^2} \left(3 a d+b \left(c+d x^2\right)\right)-3 a \sqrt{c} d \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)\right)}{3 d \left(a+b x^2\right)}","\frac{b \sqrt{a^2+2 a b x^2+b^2 x^4} \left(c+d x^2\right)^{3/2}}{3 d \left(a+b x^2\right)}+\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2}}{a+b x^2}-\frac{a \sqrt{c} \sqrt{a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{a+b x^2}",1,"(Sqrt[(a + b*x^2)^2]*(Sqrt[c + d*x^2]*(3*a*d + b*(c + d*x^2)) - 3*a*Sqrt[c]*d*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]]))/(3*d*(a + b*x^2))","A",1
271,1,122,177,0.1087765,"\int \frac{\sqrt{c+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{x^2} \, dx","Integrate[(Sqrt[c + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/x^2,x]","\frac{\sqrt{\left(a+b x^2\right)^2} \sqrt{c+d x^2} \left(\sqrt{c} \sqrt{d} \left(b x^2-2 a\right) \sqrt{\frac{d x^2}{c}+1}+x (2 a d+b c) \sinh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)\right)}{2 \sqrt{c} \sqrt{d} x \left(a+b x^2\right) \sqrt{\frac{d x^2}{c}+1}}","-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4} \left(c+d x^2\right)^{3/2}}{c x \left(a+b x^2\right)}+\frac{x \sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2} (2 a d+b c)}{2 c \left(a+b x^2\right)}+\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} (2 a d+b c) \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{2 \sqrt{d} \left(a+b x^2\right)}",1,"(Sqrt[(a + b*x^2)^2]*Sqrt[c + d*x^2]*(Sqrt[c]*Sqrt[d]*(-2*a + b*x^2)*Sqrt[1 + (d*x^2)/c] + (b*c + 2*a*d)*x*ArcSinh[(Sqrt[d]*x)/Sqrt[c]]))/(2*Sqrt[c]*Sqrt[d]*x*(a + b*x^2)*Sqrt[1 + (d*x^2)/c])","A",1
272,1,90,177,0.0452055,"\int \frac{\sqrt{c+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{x^3} \, dx","Integrate[(Sqrt[c + d*x^2]*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(\sqrt{c} \left(a-2 b x^2\right) \sqrt{c+d x^2}+x^2 (a d+2 b c) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)\right)}{2 \sqrt{c} x^2 \left(a+b x^2\right)}","-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4} \left(c+d x^2\right)^{3/2}}{2 c x^2 \left(a+b x^2\right)}+\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2} (a d+2 b c)}{2 c \left(a+b x^2\right)}-\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} (a d+2 b c) \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{2 \sqrt{c} \left(a+b x^2\right)}",1,"-1/2*(Sqrt[(a + b*x^2)^2]*(Sqrt[c]*(a - 2*b*x^2)*Sqrt[c + d*x^2] + (2*b*c + a*d)*x^2*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]]))/(Sqrt[c]*x^2*(a + b*x^2))","A",1
273,1,72,78,0.0259074,"\int x^3 \left(d+e x^2\right)^2 \left(a+b x^2+c x^4\right) \, dx","Integrate[x^3*(d + e*x^2)^2*(a + b*x^2 + c*x^4),x]","\frac{1}{120} x^4 \left(15 x^4 \left(e (a e+2 b d)+c d^2\right)+20 d x^2 (2 a e+b d)+30 a d^2+12 e x^6 (b e+2 c d)+10 c e^2 x^8\right)","\frac{1}{8} x^8 \left(e (a e+2 b d)+c d^2\right)+\frac{1}{6} d x^6 (2 a e+b d)+\frac{1}{4} a d^2 x^4+\frac{1}{10} e x^{10} (b e+2 c d)+\frac{1}{12} c e^2 x^{12}",1,"(x^4*(30*a*d^2 + 20*d*(b*d + 2*a*e)*x^2 + 15*(c*d^2 + e*(2*b*d + a*e))*x^4 + 12*e*(2*c*d + b*e)*x^6 + 10*c*e^2*x^8))/120","A",1
274,1,78,78,0.0159623,"\int x^2 \left(d+e x^2\right)^2 \left(a+b x^2+c x^4\right) \, dx","Integrate[x^2*(d + e*x^2)^2*(a + b*x^2 + c*x^4),x]","\frac{1}{7} x^7 \left(a e^2+2 b d e+c d^2\right)+\frac{1}{5} d x^5 (2 a e+b d)+\frac{1}{3} a d^2 x^3+\frac{1}{9} e x^9 (b e+2 c d)+\frac{1}{11} c e^2 x^{11}","\frac{1}{7} x^7 \left(e (a e+2 b d)+c d^2\right)+\frac{1}{5} d x^5 (2 a e+b d)+\frac{1}{3} a d^2 x^3+\frac{1}{9} e x^9 (b e+2 c d)+\frac{1}{11} c e^2 x^{11}",1,"(a*d^2*x^3)/3 + (d*(b*d + 2*a*e)*x^5)/5 + ((c*d^2 + 2*b*d*e + a*e^2)*x^7)/7 + (e*(2*c*d + b*e)*x^9)/9 + (c*e^2*x^11)/11","A",1
275,1,72,75,0.0218202,"\int x \left(d+e x^2\right)^2 \left(a+b x^2+c x^4\right) \, dx","Integrate[x*(d + e*x^2)^2*(a + b*x^2 + c*x^4),x]","\frac{1}{120} x^2 \left(20 x^4 \left(e (a e+2 b d)+c d^2\right)+30 d x^2 (2 a e+b d)+60 a d^2+15 e x^6 (b e+2 c d)+12 c e^2 x^8\right)","\frac{\left(d+e x^2\right)^3 \left(a e^2-b d e+c d^2\right)}{6 e^3}-\frac{\left(d+e x^2\right)^4 (2 c d-b e)}{8 e^3}+\frac{c \left(d+e x^2\right)^5}{10 e^3}",1,"(x^2*(60*a*d^2 + 30*d*(b*d + 2*a*e)*x^2 + 20*(c*d^2 + e*(2*b*d + a*e))*x^4 + 15*e*(2*c*d + b*e)*x^6 + 12*c*e^2*x^8))/120","A",1
276,1,73,73,0.0153924,"\int \left(d+e x^2\right)^2 \left(a+b x^2+c x^4\right) \, dx","Integrate[(d + e*x^2)^2*(a + b*x^2 + c*x^4),x]","\frac{1}{5} x^5 \left(a e^2+2 b d e+c d^2\right)+\frac{1}{3} d x^3 (2 a e+b d)+a d^2 x+\frac{1}{7} e x^7 (b e+2 c d)+\frac{1}{9} c e^2 x^9","\frac{1}{5} x^5 \left(e (a e+2 b d)+c d^2\right)+\frac{1}{3} d x^3 (2 a e+b d)+a d^2 x+\frac{1}{7} e x^7 (b e+2 c d)+\frac{1}{9} c e^2 x^9",1,"a*d^2*x + (d*(b*d + 2*a*e)*x^3)/3 + ((c*d^2 + 2*b*d*e + a*e^2)*x^5)/5 + (e*(2*c*d + b*e)*x^7)/7 + (c*e^2*x^9)/9","A",1
277,1,74,74,0.0200494,"\int \frac{\left(d+e x^2\right)^2 \left(a+b x^2+c x^4\right)}{x} \, dx","Integrate[((d + e*x^2)^2*(a + b*x^2 + c*x^4))/x,x]","\frac{1}{4} x^4 \left(a e^2+2 b d e+c d^2\right)+\frac{1}{2} d x^2 (2 a e+b d)+a d^2 \log (x)+\frac{1}{6} e x^6 (b e+2 c d)+\frac{1}{8} c e^2 x^8","\frac{1}{4} x^4 \left(e (a e+2 b d)+c d^2\right)+\frac{1}{2} d x^2 (2 a e+b d)+a d^2 \log (x)+\frac{1}{6} e x^6 (b e+2 c d)+\frac{1}{8} c e^2 x^8",1,"(d*(b*d + 2*a*e)*x^2)/2 + ((c*d^2 + 2*b*d*e + a*e^2)*x^4)/4 + (e*(2*c*d + b*e)*x^6)/6 + (c*e^2*x^8)/8 + a*d^2*Log[x]","A",1
278,1,71,71,0.0320953,"\int \frac{\left(d+e x^2\right)^2 \left(a+b x^2+c x^4\right)}{x^2} \, dx","Integrate[((d + e*x^2)^2*(a + b*x^2 + c*x^4))/x^2,x]","\frac{1}{3} x^3 \left(a e^2+2 b d e+c d^2\right)+d x (2 a e+b d)-\frac{a d^2}{x}+\frac{1}{5} e x^5 (b e+2 c d)+\frac{1}{7} c e^2 x^7","\frac{1}{3} x^3 \left(e (a e+2 b d)+c d^2\right)+d x (2 a e+b d)-\frac{a d^2}{x}+\frac{1}{5} e x^5 (b e+2 c d)+\frac{1}{7} c e^2 x^7",1,"-((a*d^2)/x) + d*(b*d + 2*a*e)*x + ((c*d^2 + 2*b*d*e + a*e^2)*x^3)/3 + (e*(2*c*d + b*e)*x^5)/5 + (c*e^2*x^7)/7","A",1
279,1,71,74,0.0418995,"\int \frac{\left(d+e x^2\right)^2 \left(a+b x^2+c x^4\right)}{x^3} \, dx","Integrate[((d + e*x^2)^2*(a + b*x^2 + c*x^4))/x^3,x]","\frac{1}{12} \left(6 x^2 \left(e (a e+2 b d)+c d^2\right)+12 d \log (x) (2 a e+b d)-\frac{6 a d^2}{x^2}+3 e x^4 (b e+2 c d)+2 c e^2 x^6\right)","\frac{1}{2} x^2 \left(e (a e+2 b d)+c d^2\right)+d \log (x) (2 a e+b d)-\frac{a d^2}{2 x^2}+\frac{1}{4} e x^4 (b e+2 c d)+\frac{1}{6} c e^2 x^6",1,"((-6*a*d^2)/x^2 + 6*(c*d^2 + e*(2*b*d + a*e))*x^2 + 3*e*(2*c*d + b*e)*x^4 + 2*c*e^2*x^6 + 12*d*(b*d + 2*a*e)*Log[x])/12","A",1
280,1,165,168,0.1377845,"\int \frac{x^6 \left(a+b x^2+c x^4\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x^6*(a + b*x^2 + c*x^4))/(d + e*x^2)^2,x]","-\frac{d x \left(2 a e^2-3 b d e+4 c d^2\right)}{e^5}+\frac{x^3 \left(a e^2-2 b d e+3 c d^2\right)}{3 e^4}+\frac{d^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(5 a e^2-7 b d e+9 c d^2\right)}{2 e^{11/2}}-\frac{x \left(a d^2 e^2-b d^3 e+c d^4\right)}{2 e^5 \left(d+e x^2\right)}+\frac{x^5 (b e-2 c d)}{5 e^3}+\frac{c x^7}{7 e^2}","-\frac{d x \left(4 c d^2-e (3 b d-2 a e)\right)}{e^5}+\frac{x^3 \left(3 c d^2-e (2 b d-a e)\right)}{3 e^4}-\frac{d^2 x \left(a e^2-b d e+c d^2\right)}{2 e^5 \left(d+e x^2\right)}+\frac{d^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(9 c d^2-e (7 b d-5 a e)\right)}{2 e^{11/2}}-\frac{x^5 (2 c d-b e)}{5 e^3}+\frac{c x^7}{7 e^2}",1,"-((d*(4*c*d^2 - 3*b*d*e + 2*a*e^2)*x)/e^5) + ((3*c*d^2 - 2*b*d*e + a*e^2)*x^3)/(3*e^4) + ((-2*c*d + b*e)*x^5)/(5*e^3) + (c*x^7)/(7*e^2) - ((c*d^4 - b*d^3*e + a*d^2*e^2)*x)/(2*e^5*(d + e*x^2)) + (d^(3/2)*(9*c*d^2 - 7*b*d*e + 5*a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*e^(11/2))","A",1
281,1,133,135,0.0777868,"\int \frac{x^4 \left(a+b x^2+c x^4\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x^4*(a + b*x^2 + c*x^4))/(d + e*x^2)^2,x]","-\frac{\sqrt{d} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(3 a e^2-5 b d e+7 c d^2\right)}{2 e^{9/2}}+\frac{x \left(a e^2-2 b d e+3 c d^2\right)}{e^4}+\frac{x \left(a d e^2-b d^2 e+c d^3\right)}{2 e^4 \left(d+e x^2\right)}+\frac{x^3 (b e-2 c d)}{3 e^3}+\frac{c x^5}{5 e^2}","-\frac{\sqrt{d} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(7 c d^2-e (5 b d-3 a e)\right)}{2 e^{9/2}}+\frac{x \left(3 c d^2-e (2 b d-a e)\right)}{e^4}+\frac{d x \left(a e^2-b d e+c d^2\right)}{2 e^4 \left(d+e x^2\right)}-\frac{x^3 (2 c d-b e)}{3 e^3}+\frac{c x^5}{5 e^2}",1,"((3*c*d^2 - 2*b*d*e + a*e^2)*x)/e^4 + ((-2*c*d + b*e)*x^3)/(3*e^3) + (c*x^5)/(5*e^2) + ((c*d^3 - b*d^2*e + a*d*e^2)*x)/(2*e^4*(d + e*x^2)) - (Sqrt[d]*(7*c*d^2 - 5*b*d*e + 3*a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*e^(9/2))","A",1
282,1,102,106,0.0645392,"\int \frac{x^2 \left(a+b x^2+c x^4\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x^2*(a + b*x^2 + c*x^4))/(d + e*x^2)^2,x]","\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(a e^2-3 b d e+5 c d^2\right)}{2 \sqrt{d} e^{7/2}}-\frac{x \left(a e^2-b d e+c d^2\right)}{2 e^3 \left(d+e x^2\right)}+\frac{x (b e-2 c d)}{e^3}+\frac{c x^3}{3 e^2}","\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(5 c d^2-e (3 b d-a e)\right)}{2 \sqrt{d} e^{7/2}}-\frac{x \left(a e^2-b d e+c d^2\right)}{2 e^3 \left(d+e x^2\right)}-\frac{x (2 c d-b e)}{e^3}+\frac{c x^3}{3 e^2}",1,"((-2*c*d + b*e)*x)/e^3 + (c*x^3)/(3*e^2) - ((c*d^2 - b*d*e + a*e^2)*x)/(2*e^3*(d + e*x^2)) + ((5*c*d^2 - 3*b*d*e + a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*Sqrt[d]*e^(7/2))","A",1
283,1,88,83,0.0536687,"\int \frac{a+b x^2+c x^4}{\left(d+e x^2\right)^2} \, dx","Integrate[(a + b*x^2 + c*x^4)/(d + e*x^2)^2,x]","\frac{x \left(a e^2-b d e+c d^2\right)}{2 d e^2 \left(d+e x^2\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(-a e^2-b d e+3 c d^2\right)}{2 d^{3/2} e^{5/2}}+\frac{c x}{e^2}","-\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(3 c d^2-e (a e+b d)\right)}{2 d^{3/2} e^{5/2}}+\frac{x \left(a+\frac{d (c d-b e)}{e^2}\right)}{2 d \left(d+e x^2\right)}+\frac{c x}{e^2}",1,"(c*x)/e^2 + ((c*d^2 - b*d*e + a*e^2)*x)/(2*d*e^2*(d + e*x^2)) - ((3*c*d^2 - b*d*e - a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(3/2)*e^(5/2))","A",1
284,1,89,89,0.0607305,"\int \frac{a+b x^2+c x^4}{x^2 \left(d+e x^2\right)^2} \, dx","Integrate[(a + b*x^2 + c*x^4)/(x^2*(d + e*x^2)^2),x]","-\frac{x \left(a e^2-b d e+c d^2\right)}{2 d^2 e \left(d+e x^2\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(-3 a e^2+b d e+c d^2\right)}{2 d^{5/2} e^{3/2}}-\frac{a}{d^2 x}","-\frac{x \left(a e^2-b d e+c d^2\right)}{2 d^2 e \left(d+e x^2\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(e (b d-3 a e)+c d^2\right)}{2 d^{5/2} e^{3/2}}-\frac{a}{d^2 x}",1,"-(a/(d^2*x)) - ((c*d^2 - b*d*e + a*e^2)*x)/(2*d^2*e*(d + e*x^2)) + ((c*d^2 + b*d*e - 3*a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(5/2)*e^(3/2))","A",1
285,1,105,106,0.0629121,"\int \frac{a+b x^2+c x^4}{x^4 \left(d+e x^2\right)^2} \, dx","Integrate[(a + b*x^2 + c*x^4)/(x^4*(d + e*x^2)^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(5 a e^2-3 b d e+c d^2\right)}{2 d^{7/2} \sqrt{e}}+\frac{x \left(a e^2-b d e+c d^2\right)}{2 d^3 \left(d+e x^2\right)}+\frac{2 a e-b d}{d^3 x}-\frac{a}{3 d^2 x^3}","\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(c d^2-e (3 b d-5 a e)\right)}{2 d^{7/2} \sqrt{e}}+\frac{x \left(a e^2-b d e+c d^2\right)}{2 d^3 \left(d+e x^2\right)}-\frac{b d-2 a e}{d^3 x}-\frac{a}{3 d^2 x^3}",1,"-1/3*a/(d^2*x^3) + (-(b*d) + 2*a*e)/(d^3*x) + ((c*d^2 - b*d*e + a*e^2)*x)/(2*d^3*(d + e*x^2)) + ((c*d^2 - 3*b*d*e + 5*a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(7/2)*Sqrt[e])","A",1
286,1,135,136,0.087585,"\int \frac{a+b x^2+c x^4}{x^6 \left(d+e x^2\right)^2} \, dx","Integrate[(a + b*x^2 + c*x^4)/(x^6*(d + e*x^2)^2),x]","-\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(7 a e^2-5 b d e+3 c d^2\right)}{2 d^{9/2}}-\frac{e x \left(a e^2-b d e+c d^2\right)}{2 d^4 \left(d+e x^2\right)}+\frac{-3 a e^2+2 b d e-c d^2}{d^4 x}+\frac{2 a e-b d}{3 d^3 x^3}-\frac{a}{5 d^2 x^5}","-\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(3 c d^2-e (5 b d-7 a e)\right)}{2 d^{9/2}}-\frac{e x \left(a e^2-b d e+c d^2\right)}{2 d^4 \left(d+e x^2\right)}-\frac{c d^2-e (2 b d-3 a e)}{d^4 x}-\frac{b d-2 a e}{3 d^3 x^3}-\frac{a}{5 d^2 x^5}",1,"-1/5*a/(d^2*x^5) + (-(b*d) + 2*a*e)/(3*d^3*x^3) + (-(c*d^2) + 2*b*d*e - 3*a*e^2)/(d^4*x) - (e*(c*d^2 - b*d*e + a*e^2)*x)/(2*d^4*(d + e*x^2)) - (Sqrt[e]*(3*c*d^2 - 5*b*d*e + 7*a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(9/2))","A",1
287,1,166,167,0.0950762,"\int \frac{a+b x^2+c x^4}{x^8 \left(d+e x^2\right)^2} \, dx","Integrate[(a + b*x^2 + c*x^4)/(x^8*(d + e*x^2)^2),x]","\frac{e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(9 a e^2-7 b d e+5 c d^2\right)}{2 d^{11/2}}+\frac{e^2 x \left(a e^2-b d e+c d^2\right)}{2 d^5 \left(d+e x^2\right)}+\frac{e \left(4 a e^2-3 b d e+2 c d^2\right)}{d^5 x}+\frac{-3 a e^2+2 b d e-c d^2}{3 d^4 x^3}+\frac{2 a e-b d}{5 d^3 x^5}-\frac{a}{7 d^2 x^7}","\frac{e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(5 c d^2-e (7 b d-9 a e)\right)}{2 d^{11/2}}+\frac{e^2 x \left(a e^2-b d e+c d^2\right)}{2 d^5 \left(d+e x^2\right)}+\frac{e \left(2 c d^2-e (3 b d-4 a e)\right)}{d^5 x}-\frac{c d^2-e (2 b d-3 a e)}{3 d^4 x^3}-\frac{b d-2 a e}{5 d^3 x^5}-\frac{a}{7 d^2 x^7}",1,"-1/7*a/(d^2*x^7) + (-(b*d) + 2*a*e)/(5*d^3*x^5) + (-(c*d^2) + 2*b*d*e - 3*a*e^2)/(3*d^4*x^3) + (e*(2*c*d^2 - 3*b*d*e + 4*a*e^2))/(d^5*x) + (e^2*(c*d^2 - b*d*e + a*e^2)*x)/(2*d^5*(d + e*x^2)) + (e^(3/2)*(5*c*d^2 - 7*b*d*e + 9*a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(11/2))","A",1
288,1,170,173,0.1108342,"\int \frac{x^6 \left(a+b x^2+c x^4\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x^6*(a + b*x^2 + c*x^4))/(d + e*x^2)^3,x]","\frac{x \left(d e (9 a e-13 b d)+17 c d^3\right)}{8 e^5 \left(d+e x^2\right)}-\frac{\sqrt{d} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(5 e (3 a e-7 b d)+63 c d^2\right)}{8 e^{11/2}}+\frac{x \left(e (a e-3 b d)+6 c d^2\right)}{e^5}-\frac{x \left(d^2 e (a e-b d)+c d^4\right)}{4 e^5 \left(d+e x^2\right)^2}+\frac{x^3 (b e-3 c d)}{3 e^4}+\frac{c x^5}{5 e^3}","\frac{d x \left(17 c d^2-e (13 b d-9 a e)\right)}{8 e^5 \left(d+e x^2\right)}+\frac{x \left(6 c d^2-e (3 b d-a e)\right)}{e^5}-\frac{\sqrt{d} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(15 a e^2-35 b d e+63 c d^2\right)}{8 e^{11/2}}-\frac{d^2 x \left(a e^2-b d e+c d^2\right)}{4 e^5 \left(d+e x^2\right)^2}-\frac{x^3 (3 c d-b e)}{3 e^4}+\frac{c x^5}{5 e^3}",1,"((6*c*d^2 + e*(-3*b*d + a*e))*x)/e^5 + ((-3*c*d + b*e)*x^3)/(3*e^4) + (c*x^5)/(5*e^3) - ((c*d^4 + d^2*e*(-(b*d) + a*e))*x)/(4*e^5*(d + e*x^2)^2) + ((17*c*d^3 + d*e*(-13*b*d + 9*a*e))*x)/(8*e^5*(d + e*x^2)) - (Sqrt[d]*(63*c*d^2 + 5*e*(-7*b*d + 3*a*e))*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*e^(11/2))","A",1
289,1,141,143,0.0874653,"\int \frac{x^4 \left(a+b x^2+c x^4\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x^4*(a + b*x^2 + c*x^4))/(d + e*x^2)^3,x]","\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(3 a e^2-15 b d e+35 c d^2\right)}{8 \sqrt{d} e^{9/2}}-\frac{x \left(5 a e^2-9 b d e+13 c d^2\right)}{8 e^4 \left(d+e x^2\right)}+\frac{x \left(a d e^2-b d^2 e+c d^3\right)}{4 e^4 \left(d+e x^2\right)^2}+\frac{x (b e-3 c d)}{e^4}+\frac{c x^3}{3 e^3}","\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(35 c d^2-3 e (5 b d-a e)\right)}{8 \sqrt{d} e^{9/2}}-\frac{x \left(13 c d^2-e (9 b d-5 a e)\right)}{8 e^4 \left(d+e x^2\right)}+\frac{d x \left(a e^2-b d e+c d^2\right)}{4 e^4 \left(d+e x^2\right)^2}-\frac{x (3 c d-b e)}{e^4}+\frac{c x^3}{3 e^3}",1,"((-3*c*d + b*e)*x)/e^4 + (c*x^3)/(3*e^3) + ((c*d^3 - b*d^2*e + a*d*e^2)*x)/(4*e^4*(d + e*x^2)^2) - ((13*c*d^2 - 9*b*d*e + 5*a*e^2)*x)/(8*e^4*(d + e*x^2)) + ((35*c*d^2 - 15*b*d*e + 3*a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*Sqrt[d]*e^(9/2))","A",1
290,1,122,124,0.103617,"\int \frac{x^2 \left(a+b x^2+c x^4\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x^2*(a + b*x^2 + c*x^4))/(d + e*x^2)^3,x]","\frac{x \left(a e^2-5 b d e+9 c d^2\right)}{8 d e^3 \left(d+e x^2\right)}-\frac{x \left(a e^2-b d e+c d^2\right)}{4 e^3 \left(d+e x^2\right)^2}-\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(-a e^2-3 b d e+15 c d^2\right)}{8 d^{3/2} e^{7/2}}+\frac{c x}{e^3}","\frac{x \left(9 c d^2-e (5 b d-a e)\right)}{8 d e^3 \left(d+e x^2\right)}-\frac{x \left(a e^2-b d e+c d^2\right)}{4 e^3 \left(d+e x^2\right)^2}-\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(15 c d^2-e (a e+3 b d)\right)}{8 d^{3/2} e^{7/2}}+\frac{c x}{e^3}",1,"(c*x)/e^3 - ((c*d^2 - b*d*e + a*e^2)*x)/(4*e^3*(d + e*x^2)^2) + ((9*c*d^2 - 5*b*d*e + a*e^2)*x)/(8*d*e^3*(d + e*x^2)) - ((15*c*d^2 - 3*b*d*e - a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(3/2)*e^(7/2))","A",1
291,1,110,115,0.0958629,"\int \frac{a+b x^2+c x^4}{\left(d+e x^2\right)^3} \, dx","Integrate[(a + b*x^2 + c*x^4)/(d + e*x^2)^3,x]","\frac{x \left(e \left(a e \left(5 d+3 e x^2\right)+b d \left(e x^2-d\right)\right)-c d^2 \left(3 d+5 e x^2\right)\right)}{8 d^2 e^2 \left(d+e x^2\right)^2}+\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(e (3 a e+b d)+3 c d^2\right)}{8 d^{5/2} e^{5/2}}","-\frac{x \left(5 c d^2-e (3 a e+b d)\right)}{8 d^2 e^2 \left(d+e x^2\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(e (3 a e+b d)+3 c d^2\right)}{8 d^{5/2} e^{5/2}}+\frac{x \left(a+\frac{d (c d-b e)}{e^2}\right)}{4 d \left(d+e x^2\right)^2}",1,"(x*(-(c*d^2*(3*d + 5*e*x^2)) + e*(b*d*(-d + e*x^2) + a*e*(5*d + 3*e*x^2))))/(8*d^2*e^2*(d + e*x^2)^2) + ((3*c*d^2 + e*(b*d + 3*a*e))*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(5/2)*e^(5/2))","A",1
292,1,124,127,0.1349616,"\int \frac{a+b x^2+c x^4}{x^2 \left(d+e x^2\right)^3} \, dx","Integrate[(a + b*x^2 + c*x^4)/(x^2*(d + e*x^2)^3),x]","\frac{\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(3 e (b d-5 a e)+c d^2\right)}{e^{3/2}}+\frac{\sqrt{d} \left(d x^2 \left(b e \left(5 d+3 e x^2\right)+c d \left(e x^2-d\right)\right)-a e \left(8 d^2+25 d e x^2+15 e^2 x^4\right)\right)}{e x \left(d+e x^2\right)^2}}{8 d^{7/2}}","-\frac{x \left(a e^2-b d e+c d^2\right)}{4 d^2 e \left(d+e x^2\right)^2}+\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(3 e (b d-5 a e)+c d^2\right)}{8 d^{7/2} e^{3/2}}+\frac{x \left(e (3 b d-7 a e)+c d^2\right)}{8 d^3 e \left(d+e x^2\right)}-\frac{a}{d^3 x}",1,"((Sqrt[d]*(-(a*e*(8*d^2 + 25*d*e*x^2 + 15*e^2*x^4)) + d*x^2*(c*d*(-d + e*x^2) + b*e*(5*d + 3*e*x^2))))/(e*x*(d + e*x^2)^2) + ((c*d^2 + 3*e*(b*d - 5*a*e))*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/e^(3/2))/(8*d^(7/2))","A",1
293,1,141,142,0.0854191,"\int \frac{a+b x^2+c x^4}{x^4 \left(d+e x^2\right)^3} \, dx","Integrate[(a + b*x^2 + c*x^4)/(x^4*(d + e*x^2)^3),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(35 a e^2-15 b d e+3 c d^2\right)}{8 d^{9/2} \sqrt{e}}+\frac{x \left(11 a e^2-7 b d e+3 c d^2\right)}{8 d^4 \left(d+e x^2\right)}+\frac{x \left(a e^2-b d e+c d^2\right)}{4 d^3 \left(d+e x^2\right)^2}+\frac{3 a e-b d}{d^4 x}-\frac{a}{3 d^3 x^3}","\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(35 a e^2-15 b d e+3 c d^2\right)}{8 d^{9/2} \sqrt{e}}+\frac{x \left(3 c d^2-e (7 b d-11 a e)\right)}{8 d^4 \left(d+e x^2\right)}+\frac{x \left(a e^2-b d e+c d^2\right)}{4 d^3 \left(d+e x^2\right)^2}-\frac{b d-3 a e}{d^4 x}-\frac{a}{3 d^3 x^3}",1,"-1/3*a/(d^3*x^3) + (-(b*d) + 3*a*e)/(d^4*x) + ((c*d^2 - b*d*e + a*e^2)*x)/(4*d^3*(d + e*x^2)^2) + ((3*c*d^2 - 7*b*d*e + 11*a*e^2)*x)/(8*d^4*(d + e*x^2)) + ((3*c*d^2 - 15*b*d*e + 35*a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(9/2)*Sqrt[e])","A",1
294,1,173,171,0.1138523,"\int \frac{a+b x^2+c x^4}{x^6 \left(d+e x^2\right)^3} \, dx","Integrate[(a + b*x^2 + c*x^4)/(x^6*(d + e*x^2)^3),x]","-\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(63 a e^2-35 b d e+15 c d^2\right)}{8 d^{11/2}}+\frac{-6 a e^2+3 b d e-c d^2}{d^5 x}-\frac{x \left(15 a e^3-11 b d e^2+7 c d^2 e\right)}{8 d^5 \left(d+e x^2\right)}-\frac{e x \left(a e^2-b d e+c d^2\right)}{4 d^4 \left(d+e x^2\right)^2}+\frac{3 a e-b d}{3 d^4 x^3}-\frac{a}{5 d^3 x^5}","-\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(63 a e^2-35 b d e+15 c d^2\right)}{8 d^{11/2}}-\frac{6 a e^2-3 b d e+c d^2}{d^5 x}-\frac{e x \left(7 c d^2-e (11 b d-15 a e)\right)}{8 d^5 \left(d+e x^2\right)}-\frac{e x \left(a e^2-b d e+c d^2\right)}{4 d^4 \left(d+e x^2\right)^2}-\frac{b d-3 a e}{3 d^4 x^3}-\frac{a}{5 d^3 x^5}",1,"-1/5*a/(d^3*x^5) + (-(b*d) + 3*a*e)/(3*d^4*x^3) + (-(c*d^2) + 3*b*d*e - 6*a*e^2)/(d^5*x) - (e*(c*d^2 - b*d*e + a*e^2)*x)/(4*d^4*(d + e*x^2)^2) - ((7*c*d^2*e - 11*b*d*e^2 + 15*a*e^3)*x)/(8*d^5*(d + e*x^2)) - (Sqrt[e]*(15*c*d^2 - 35*b*d*e + 63*a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(11/2))","A",1
295,1,228,230,0.2432388,"\int \frac{x^9}{\left(d+e x^2\right) \left(a+b x^2+c x^4\right)} \, dx","Integrate[x^9/((d + e*x^2)*(a + b*x^2 + c*x^4)),x]","\frac{1}{4} \left(\frac{\left(-a^2 c e+a b^2 e+2 a b c d+b^3 (-d)\right) \log \left(a+b x^2+c x^4\right)}{c^3 \left(e (a e-b d)+c d^2\right)}-\frac{2 \left(3 a^2 b c e+2 a^2 c^2 d-a b^3 e-4 a b^2 c d+b^4 d\right) \tan ^{-1}\left(\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right)}{c^3 \sqrt{4 a c-b^2} \left(e (b d-a e)-c d^2\right)}+\frac{2 d^4 \log \left(d+e x^2\right)}{e^3 \left(e (a e-b d)+c d^2\right)}-\frac{2 x^2 (b e+c d)}{c^2 e^2}+\frac{x^4}{c e}\right)","-\frac{\left(a^2 c e-a b^2 e-2 a b c d+b^3 d\right) \log \left(a+b x^2+c x^4\right)}{4 c^3 \left(a e^2-b d e+c d^2\right)}-\frac{\left(3 a^2 b c e+2 a^2 c^2 d-a b^3 e-4 a b^2 c d+b^4 d\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} \left(a e^2-b d e+c d^2\right)}+\frac{d^4 \log \left(d+e x^2\right)}{2 e^3 \left(a e^2-b d e+c d^2\right)}-\frac{x^2 (b e+c d)}{2 c^2 e^2}+\frac{x^4}{4 c e}",1,"((-2*(c*d + b*e)*x^2)/(c^2*e^2) + x^4/(c*e) - (2*(b^4*d - 4*a*b^2*c*d + 2*a^2*c^2*d - a*b^3*e + 3*a^2*b*c*e)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(c^3*Sqrt[-b^2 + 4*a*c]*(-(c*d^2) + e*(b*d - a*e))) + (2*d^4*Log[d + e*x^2])/(e^3*(c*d^2 + e*(-(b*d) + a*e))) + ((-(b^3*d) + 2*a*b*c*d + a*b^2*e - a^2*c*e)*Log[a + b*x^2 + c*x^4])/(c^3*(c*d^2 + e*(-(b*d) + a*e))))/4","A",1
296,1,186,189,0.1874534,"\int \frac{x^7}{\left(d+e x^2\right) \left(a+b x^2+c x^4\right)} \, dx","Integrate[x^7/((d + e*x^2)*(a + b*x^2 + c*x^4)),x]","\frac{2 e^2 \left(2 a^2 c e-a b^2 e-3 a b c d+b^3 d\right) \tan ^{-1}\left(\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right)+\sqrt{4 a c-b^2} \left(e \left(e \left(a b e+a c d+b^2 (-d)\right) \log \left(a+b x^2+c x^4\right)-2 c x^2 \left(a e^2-b d e+c d^2\right)\right)+2 c^2 d^3 \log \left(d+e x^2\right)\right)}{4 c^2 e^2 \sqrt{4 a c-b^2} \left(e (b d-a e)-c d^2\right)}","\frac{\left(2 a^2 c e-a b^2 e-3 a b c d+b^3 d\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} \left(a e^2-b d e+c d^2\right)}+\frac{\left(-a b e-a c d+b^2 d\right) \log \left(a+b x^2+c x^4\right)}{4 c^2 \left(a e^2-b d e+c d^2\right)}-\frac{d^3 \log \left(d+e x^2\right)}{2 e^2 \left(a e^2-b d e+c d^2\right)}+\frac{x^2}{2 c e}",1,"(2*e^2*(b^3*d - 3*a*b*c*d - a*b^2*e + 2*a^2*c*e)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]] + Sqrt[-b^2 + 4*a*c]*(2*c^2*d^3*Log[d + e*x^2] + e*(-2*c*(c*d^2 - b*d*e + a*e^2)*x^2 + e*(-(b^2*d) + a*c*d + a*b*e)*Log[a + b*x^2 + c*x^4])))/(4*c^2*Sqrt[-b^2 + 4*a*c]*e^2*(-(c*d^2) + e*(b*d - a*e)))","A",1
297,1,139,158,0.1063276,"\int \frac{x^5}{\left(d+e x^2\right) \left(a+b x^2+c x^4\right)} \, dx","Integrate[x^5/((d + e*x^2)*(a + b*x^2 + c*x^4)),x]","-\frac{\sqrt{4 a c-b^2} \left(e (b d-a e) \log \left(a+b x^2+c x^4\right)-2 c d^2 \log \left(d+e x^2\right)\right)+2 e \left(a b e+2 a c d+b^2 (-d)\right) \tan ^{-1}\left(\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right)}{4 c e \sqrt{4 a c-b^2} \left(e (a e-b d)+c d^2\right)}","-\frac{\left(-a b e-2 a c d+b^2 d\right) \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{2 c \sqrt{b^2-4 a c} \left(a e^2-b d e+c d^2\right)}+\frac{d^2 \log \left(d+e x^2\right)}{2 e \left(a e^2-b d e+c d^2\right)}-\frac{(b d-a e) \log \left(a+b x^2+c x^4\right)}{4 c \left(a e^2-b d e+c d^2\right)}",1,"-1/4*(2*e*(-(b^2*d) + 2*a*c*d + a*b*e)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]] + Sqrt[-b^2 + 4*a*c]*(-2*c*d^2*Log[d + e*x^2] + e*(b*d - a*e)*Log[a + b*x^2 + c*x^4]))/(c*Sqrt[-b^2 + 4*a*c]*e*(c*d^2 + e*(-(b*d) + a*e)))","A",1
298,1,114,132,0.0695139,"\int \frac{x^3}{\left(d+e x^2\right) \left(a+b x^2+c x^4\right)} \, dx","Integrate[x^3/((d + e*x^2)*(a + b*x^2 + c*x^4)),x]","\frac{2 (b d-2 a e) \tan ^{-1}\left(\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right)+d \sqrt{4 a c-b^2} \left(2 \log \left(d+e x^2\right)-\log \left(a+b x^2+c x^4\right)\right)}{4 \sqrt{4 a c-b^2} \left(e (b d-a e)-c d^2\right)}","\frac{(b d-2 a e) \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{2 \sqrt{b^2-4 a c} \left(a e^2-b d e+c d^2\right)}-\frac{d \log \left(d+e x^2\right)}{2 \left(a e^2-b d e+c d^2\right)}+\frac{d \log \left(a+b x^2+c x^4\right)}{4 \left(a e^2-b d e+c d^2\right)}",1,"(2*(b*d - 2*a*e)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]] + Sqrt[-b^2 + 4*a*c]*d*(2*Log[d + e*x^2] - Log[a + b*x^2 + c*x^4]))/(4*Sqrt[-b^2 + 4*a*c]*(-(c*d^2) + e*(b*d - a*e)))","A",1
299,1,112,133,0.067666,"\int \frac{x}{\left(d+e x^2\right) \left(a+b x^2+c x^4\right)} \, dx","Integrate[x/((d + e*x^2)*(a + b*x^2 + c*x^4)),x]","\frac{(2 b e-4 c d) \tan ^{-1}\left(\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right)+e \sqrt{4 a c-b^2} \left(\log \left(a+b x^2+c x^4\right)-2 \log \left(d+e x^2\right)\right)}{4 \sqrt{4 a c-b^2} \left(e (b d-a e)-c d^2\right)}","-\frac{(2 c d-b e) \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{2 \sqrt{b^2-4 a c} \left(a e^2-b d e+c d^2\right)}+\frac{e \log \left(d+e x^2\right)}{2 \left(a e^2-b d e+c d^2\right)}-\frac{e \log \left(a+b x^2+c x^4\right)}{4 \left(a e^2-b d e+c d^2\right)}",1,"((-4*c*d + 2*b*e)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]] + Sqrt[-b^2 + 4*a*c]*e*(-2*Log[d + e*x^2] + Log[a + b*x^2 + c*x^4]))/(4*Sqrt[-b^2 + 4*a*c]*(-(c*d^2) + e*(b*d - a*e)))","A",1
300,1,242,167,0.3206224,"\int \frac{1}{x \left(d+e x^2\right) \left(a+b x^2+c x^4\right)} \, dx","Integrate[1/(x*(d + e*x^2)*(a + b*x^2 + c*x^4)),x]","\frac{4 \log (x) \sqrt{b^2-4 a c} \left(e (a e-b d)+c d^2\right)-2 a e^2 \sqrt{b^2-4 a c} \log \left(d+e x^2\right)-d \left(c d \sqrt{b^2-4 a c}-b e \sqrt{b^2-4 a c}+2 a c e+b^2 (-e)+b c d\right) \log \left(-\sqrt{b^2-4 a c}+b+2 c x^2\right)+d \left(-c d \sqrt{b^2-4 a c}+b e \sqrt{b^2-4 a c}+2 a c e+b^2 (-e)+b c d\right) \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{4 a d \sqrt{b^2-4 a c} \left(e (a e-b d)+c d^2\right)}","\frac{\left(2 a c e+b^2 (-e)+b c d\right) \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{2 a \sqrt{b^2-4 a c} \left(a e^2-b d e+c d^2\right)}-\frac{e^2 \log \left(d+e x^2\right)}{2 d \left(a e^2-b d e+c d^2\right)}-\frac{(c d-b e) \log \left(a+b x^2+c x^4\right)}{4 a \left(a e^2-b d e+c d^2\right)}+\frac{\log (x)}{a d}",1,"(4*Sqrt[b^2 - 4*a*c]*(c*d^2 + e*(-(b*d) + a*e))*Log[x] - d*(b*c*d + c*Sqrt[b^2 - 4*a*c]*d - b^2*e + 2*a*c*e - b*Sqrt[b^2 - 4*a*c]*e)*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2] + d*(b*c*d - c*Sqrt[b^2 - 4*a*c]*d - b^2*e + 2*a*c*e + b*Sqrt[b^2 - 4*a*c]*e)*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2] - 2*a*Sqrt[b^2 - 4*a*c]*e^2*Log[d + e*x^2])/(4*a*Sqrt[b^2 - 4*a*c]*d*(c*d^2 + e*(-(b*d) + a*e)))","A",1
301,1,331,205,0.3429825,"\int \frac{1}{x^3 \left(d+e x^2\right) \left(a+b x^2+c x^4\right)} \, dx","Integrate[1/(x^3*(d + e*x^2)*(a + b*x^2 + c*x^4)),x]","\frac{1}{4} \left(\frac{\left(b^2 \left(e \sqrt{b^2-4 a c}-c d\right)-b c \left(d \sqrt{b^2-4 a c}+3 a e\right)+a c \left(2 c d-e \sqrt{b^2-4 a c}\right)+b^3 e\right) \log \left(-\sqrt{b^2-4 a c}+b+2 c x^2\right)}{a^2 \sqrt{b^2-4 a c} \left(e (b d-a e)-c d^2\right)}+\frac{\left(b^2 \left(e \sqrt{b^2-4 a c}+c d\right)+b c \left(3 a e-d \sqrt{b^2-4 a c}\right)-a c \left(e \sqrt{b^2-4 a c}+2 c d\right)+b^3 (-e)\right) \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{a^2 \sqrt{b^2-4 a c} \left(e (b d-a e)-c d^2\right)}-\frac{4 \log (x) (a e+b d)}{a^2 d^2}+\frac{2 e^3 \log \left(d+e x^2\right)}{d^2 e (a e-b d)+c d^4}-\frac{2}{a d x^2}\right)","\frac{\left(a c e+b^2 (-e)+b c d\right) \log \left(a+b x^2+c x^4\right)}{4 a^2 \left(a e^2-b d e+c d^2\right)}-\frac{\left(3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d\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} \left(a e^2-b d e+c d^2\right)}-\frac{\log (x) (a e+b d)}{a^2 d^2}+\frac{e^3 \log \left(d+e x^2\right)}{2 d^2 \left(a e^2-b d e+c d^2\right)}-\frac{1}{2 a d x^2}",1,"(-2/(a*d*x^2) - (4*(b*d + a*e)*Log[x])/(a^2*d^2) + ((b^3*e - b*c*(Sqrt[b^2 - 4*a*c]*d + 3*a*e) + a*c*(2*c*d - Sqrt[b^2 - 4*a*c]*e) + b^2*(-(c*d) + Sqrt[b^2 - 4*a*c]*e))*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(a^2*Sqrt[b^2 - 4*a*c]*(-(c*d^2) + e*(b*d - a*e))) + ((-(b^3*e) + b*c*(-(Sqrt[b^2 - 4*a*c]*d) + 3*a*e) + b^2*(c*d + Sqrt[b^2 - 4*a*c]*e) - a*c*(2*c*d + Sqrt[b^2 - 4*a*c]*e))*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(a^2*Sqrt[b^2 - 4*a*c]*(-(c*d^2) + e*(b*d - a*e))) + (2*e^3*Log[d + e*x^2])/(c*d^4 + d^2*e*(-(b*d) + a*e)))/4","A",1
302,1,426,268,0.433702,"\int \frac{1}{x^5 \left(d+e x^2\right) \left(a+b x^2+c x^4\right)} \, dx","Integrate[1/(x^5*(d + e*x^2)*(a + b*x^2 + c*x^4)),x]","\frac{1}{4} \left(\frac{4 \log (x) \left(a b d e+a \left(a e^2-c d^2\right)+b^2 d^2\right)}{a^3 d^3}-\frac{\left(a c^2 \left(d \sqrt{b^2-4 a c}+2 a e\right)-b^2 c \left(d \sqrt{b^2-4 a c}+4 a e\right)+a b c \left(3 c d-2 e \sqrt{b^2-4 a c}\right)+b^3 \left(e \sqrt{b^2-4 a c}-c d\right)+b^4 e\right) \log \left(-\sqrt{b^2-4 a c}+b+2 c x^2\right)}{a^3 \sqrt{b^2-4 a c} \left(e (b d-a e)-c d^2\right)}-\frac{\left(a c^2 \left(d \sqrt{b^2-4 a c}-2 a e\right)+b^2 c \left(4 a e-d \sqrt{b^2-4 a c}\right)-a b c \left(2 e \sqrt{b^2-4 a c}+3 c d\right)+b^3 \left(e \sqrt{b^2-4 a c}+c d\right)+b^4 (-e)\right) \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{a^3 \sqrt{b^2-4 a c} \left(e (b d-a e)-c d^2\right)}+\frac{2 (a e+b d)}{a^2 d^2 x^2}-\frac{2 e^4 \log \left(d+e x^2\right)}{d^3 e (a e-b d)+c d^5}-\frac{1}{a d x^4}\right)","\frac{\log (x) \left(a b d e-a \left(c d^2-a e^2\right)+b^2 d^2\right)}{a^3 d^3}-\frac{\left(2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right) \log \left(a+b x^2+c x^4\right)}{4 a^3 \left(a e^2-b d e+c d^2\right)}+\frac{a e+b d}{2 a^2 d^2 x^2}+\frac{\left(-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^4 (-e)+b^3 c d\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} \left(a e^2-b d e+c d^2\right)}-\frac{e^4 \log \left(d+e x^2\right)}{2 d^3 \left(a e^2-b d e+c d^2\right)}-\frac{1}{4 a d x^4}",1,"(-(1/(a*d*x^4)) + (2*(b*d + a*e))/(a^2*d^2*x^2) + (4*(b^2*d^2 + a*b*d*e + a*(-(c*d^2) + a*e^2))*Log[x])/(a^3*d^3) - ((b^4*e + a*c^2*(Sqrt[b^2 - 4*a*c]*d + 2*a*e) - b^2*c*(Sqrt[b^2 - 4*a*c]*d + 4*a*e) + a*b*c*(3*c*d - 2*Sqrt[b^2 - 4*a*c]*e) + b^3*(-(c*d) + Sqrt[b^2 - 4*a*c]*e))*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(a^3*Sqrt[b^2 - 4*a*c]*(-(c*d^2) + e*(b*d - a*e))) - ((-(b^4*e) + a*c^2*(Sqrt[b^2 - 4*a*c]*d - 2*a*e) + b^2*c*(-(Sqrt[b^2 - 4*a*c]*d) + 4*a*e) + b^3*(c*d + Sqrt[b^2 - 4*a*c]*e) - a*b*c*(3*c*d + 2*Sqrt[b^2 - 4*a*c]*e))*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(a^3*Sqrt[b^2 - 4*a*c]*(-(c*d^2) + e*(b*d - a*e))) - (2*e^4*Log[d + e*x^2])/(c*d^5 + d^3*e*(-(b*d) + a*e)))/4","A",1
303,1,463,387,0.6142519,"\int \frac{x^8}{\left(d+e x^2\right) \left(a+b x^2+c x^4\right)} \, dx","Integrate[x^8/((d + e*x^2)*(a + b*x^2 + c*x^4)),x]","\frac{\left(a^2 c \left(e \sqrt{b^2-4 a c}-2 c d\right)+a b^2 \left(4 c d-e \sqrt{b^2-4 a c}\right)-a b c \left(2 d \sqrt{b^2-4 a c}+3 a e\right)+b^3 \left(d \sqrt{b^2-4 a c}+a e\right)+b^4 (-d)\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}} \left(e (b d-a e)-c d^2\right)}+\frac{\left(a^2 c \left(e \sqrt{b^2-4 a c}+2 c d\right)-a b^2 \left(e \sqrt{b^2-4 a c}+4 c d\right)+a b c \left(3 a e-2 d \sqrt{b^2-4 a c}\right)+b^3 \left(d \sqrt{b^2-4 a c}-a e\right)+b^4 d\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} \left(e (b d-a e)-c d^2\right)}+\frac{d^{7/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{e^{5/2} \left(a e^2-b d e+c d^2\right)}-\frac{x (b e+c d)}{c^2 e^2}+\frac{x^3}{3 c e}","-\frac{\left(-\frac{3 a^2 b c e+2 a^2 c^2 d-a b^3 e-4 a b^2 c d+b^4 d}{\sqrt{b^2-4 a c}}+a^2 c e-a b^2 e-2 a b c d+b^3 d\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}} \left(a e^2-b d e+c d^2\right)}-\frac{\left(\frac{3 a^2 b c e+2 a^2 c^2 d-a b^3 e-4 a b^2 c d+b^4 d}{\sqrt{b^2-4 a c}}+a^2 c e-a b^2 e-2 a b c d+b^3 d\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} \left(a e^2-b d e+c d^2\right)}+\frac{d^{7/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{e^{5/2} \left(a e^2-b d e+c d^2\right)}-\frac{x (b e+c d)}{c^2 e^2}+\frac{x^3}{3 c e}",1,"-(((c*d + b*e)*x)/(c^2*e^2)) + x^3/(3*c*e) + ((-(b^4*d) + b^3*(Sqrt[b^2 - 4*a*c]*d + a*e) - a*b*c*(2*Sqrt[b^2 - 4*a*c]*d + 3*a*e) + a*b^2*(4*c*d - Sqrt[b^2 - 4*a*c]*e) + a^2*c*(-2*c*d + Sqrt[b^2 - 4*a*c]*e))*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]]*(-(c*d^2) + e*(b*d - a*e))) + ((b^4*d + b^3*(Sqrt[b^2 - 4*a*c]*d - a*e) + a*b*c*(-2*Sqrt[b^2 - 4*a*c]*d + 3*a*e) + a^2*c*(2*c*d + Sqrt[b^2 - 4*a*c]*e) - a*b^2*(4*c*d + Sqrt[b^2 - 4*a*c]*e))*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]]*(-(c*d^2) + e*(b*d - a*e))) + (d^(7/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(e^(5/2)*(c*d^2 - b*d*e + a*e^2))","A",1
304,1,385,323,0.5199481,"\int \frac{x^6}{\left(d+e x^2\right) \left(a+b x^2+c x^4\right)} \, dx","Integrate[x^6/((d + e*x^2)*(a + b*x^2 + c*x^4)),x]","\frac{\left(-b^2 \left(d \sqrt{b^2-4 a c}+a e\right)+a b \left(e \sqrt{b^2-4 a c}-3 c d\right)+a c \left(d \sqrt{b^2-4 a c}+2 a e\right)+b^3 d\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}} \left(e (b d-a e)-c d^2\right)}+\frac{\left(b^2 \left(d \sqrt{b^2-4 a c}-a e\right)-a b \left(e \sqrt{b^2-4 a c}+3 c d\right)+a c \left(2 a e-d \sqrt{b^2-4 a c}\right)+b^3 d\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} \left(e (a e-b d)+c d^2\right)}-\frac{d^{5/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{e^{3/2} \left(a e^2-b d e+c d^2\right)}+\frac{x}{c e}","\frac{\left(-\frac{2 a^2 c e-a b^2 e-3 a b c d+b^3 d}{\sqrt{b^2-4 a c}}-a b e-a c d+b^2 d\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}} \left(a e^2-b d e+c d^2\right)}+\frac{\left(\frac{2 a^2 c e-a b^2 e-3 a b c d+b^3 d}{\sqrt{b^2-4 a c}}-a b e-a c d+b^2 d\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} \left(a e^2-b d e+c d^2\right)}-\frac{d^{5/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{e^{3/2} \left(a e^2-b d e+c d^2\right)}+\frac{x}{c e}",1,"x/(c*e) + ((b^3*d - b^2*(Sqrt[b^2 - 4*a*c]*d + a*e) + a*c*(Sqrt[b^2 - 4*a*c]*d + 2*a*e) + a*b*(-3*c*d + Sqrt[b^2 - 4*a*c]*e))*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]]*(-(c*d^2) + e*(b*d - a*e))) + ((b^3*d + b^2*(Sqrt[b^2 - 4*a*c]*d - a*e) + a*c*(-(Sqrt[b^2 - 4*a*c]*d) + 2*a*e) - a*b*(3*c*d + Sqrt[b^2 - 4*a*c]*e))*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]]*(c*d^2 + e*(-(b*d) + a*e))) - (d^(5/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(e^(3/2)*(c*d^2 - b*d*e + a*e^2))","A",1
305,1,323,280,0.3339621,"\int \frac{x^4}{\left(d+e x^2\right) \left(a+b x^2+c x^4\right)} \, dx","Integrate[x^4/((d + e*x^2)*(a + b*x^2 + c*x^4)),x]","\frac{\left(b d \sqrt{b^2-4 a c}-a e \sqrt{b^2-4 a c}+a b e+2 a c d+b^2 (-d)\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}} \left(-a e^2+b d e-c d^2\right)}+\frac{\left(b d \sqrt{b^2-4 a c}-a e \sqrt{b^2-4 a c}-a b e-2 a c d+b^2 d\right) \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} \sqrt{\sqrt{b^2-4 a c}+b} \left(-a e^2+b d e-c d^2\right)}+\frac{d^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{e} \left(a e^2-b d e+c d^2\right)}","-\frac{\left(-\frac{-a b e-2 a c d+b^2 d}{\sqrt{b^2-4 a c}}-a e+b d\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-\sqrt{b^2-4 a c}} \left(a e^2-b d e+c d^2\right)}-\frac{\left(\frac{-a b e-2 a c d+b^2 d}{\sqrt{b^2-4 a c}}-a e+b d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} \sqrt{c} \sqrt{\sqrt{b^2-4 a c}+b} \left(a e^2-b d e+c d^2\right)}+\frac{d^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{e} \left(a e^2-b d e+c d^2\right)}",1,"((-(b^2*d) + 2*a*c*d + b*Sqrt[b^2 - 4*a*c]*d + a*b*e - a*Sqrt[b^2 - 4*a*c]*e)*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]]*(-(c*d^2) + b*d*e - a*e^2)) + ((b^2*d - 2*a*c*d + b*Sqrt[b^2 - 4*a*c]*d - a*b*e - a*Sqrt[b^2 - 4*a*c]*e)*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]]*(-(c*d^2) + b*d*e - a*e^2)) + (d^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[e]*(c*d^2 - b*d*e + a*e^2))","A",1
306,1,277,251,0.498217,"\int \frac{x^2}{\left(d+e x^2\right) \left(a+b x^2+c x^4\right)} \, dx","Integrate[x^2/((d + e*x^2)*(a + b*x^2 + c*x^4)),x]","-\frac{\sqrt{c} \left(d \sqrt{b^2-4 a c}+2 a e-b d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}} \left(-a e^2+b d e-c d^2\right)}-\frac{\sqrt{c} \left(d \sqrt{b^2-4 a c}-2 a e+b d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b} \left(-a e^2+b d e-c d^2\right)}-\frac{\sqrt{d} \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{a e^2-b d e+c d^2}","\frac{\sqrt{c} \left(d-\frac{b d-2 a e}{\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} \sqrt{b-\sqrt{b^2-4 a c}} \left(a e^2-b d e+c d^2\right)}+\frac{\sqrt{c} \left(\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} \sqrt{\sqrt{b^2-4 a c}+b} \left(a e^2-b d e+c d^2\right)}-\frac{\sqrt{d} \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{a e^2-b d e+c d^2}",1,"-((Sqrt[c]*(-(b*d) + Sqrt[b^2 - 4*a*c]*d + 2*a*e)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]*(-(c*d^2) + b*d*e - a*e^2))) - (Sqrt[c]*(b*d + Sqrt[b^2 - 4*a*c]*d - 2*a*e)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]*(-(c*d^2) + b*d*e - a*e^2)) - (Sqrt[d]*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(c*d^2 - b*d*e + a*e^2)","A",1
307,1,274,254,0.2476828,"\int \frac{1}{\left(d+e x^2\right) \left(a+b x^2+c x^4\right)} \, dx","Integrate[1/((d + e*x^2)*(a + b*x^2 + c*x^4)),x]","\frac{\sqrt{c} \left(e \sqrt{b^2-4 a c}+b e-2 c d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}} \left(-a e^2+b d e-c d^2\right)}+\frac{\sqrt{c} \left(e \sqrt{b^2-4 a c}-b e+2 c d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b} \left(-a e^2+b d e-c d^2\right)}+\frac{e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{d} \left(a e^2-b d e+c d^2\right)}","-\frac{\sqrt{c} \left(e-\frac{2 c d-b e}{\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} \sqrt{b-\sqrt{b^2-4 a c}} \left(a e^2-b d e+c d^2\right)}-\frac{\sqrt{c} \left(\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} \sqrt{\sqrt{b^2-4 a c}+b} \left(a e^2-b d e+c d^2\right)}+\frac{e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{d} \left(a e^2-b d e+c d^2\right)}",1,"(Sqrt[c]*(-2*c*d + b*e + Sqrt[b^2 - 4*a*c]*e)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]*(-(c*d^2) + b*d*e - a*e^2)) + (Sqrt[c]*(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]*(-(c*d^2) + b*d*e - a*e^2)) + (e^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*(c*d^2 - b*d*e + a*e^2))","A",1
308,1,340,298,0.39879,"\int \frac{1}{x^2 \left(d+e x^2\right) \left(a+b x^2+c x^4\right)} \, dx","Integrate[1/(x^2*(d + e*x^2)*(a + b*x^2 + c*x^4)),x]","-\frac{\sqrt{c} \left(c d \sqrt{b^2-4 a c}-b e \sqrt{b^2-4 a c}+2 a c e+b^2 (-e)+b c d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} a \sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}} \left(e (a e-b d)+c d^2\right)}+\frac{\sqrt{c} \left(-c d \sqrt{b^2-4 a c}+b e \sqrt{b^2-4 a c}+2 a c e+b^2 (-e)+b c d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} a \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b} \left(e (a e-b d)+c d^2\right)}-\frac{e^{5/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{d^{3/2} \left(a e^2-b d e+c d^2\right)}-\frac{1}{a d x}","-\frac{\sqrt{c} \left(\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\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}} \left(a e^2-b d e+c d^2\right)}-\frac{\sqrt{c} \left(-\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\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} \left(a e^2-b d e+c d^2\right)}-\frac{e^{5/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{d^{3/2} \left(a e^2-b d e+c d^2\right)}-\frac{1}{a d x}",1,"-(1/(a*d*x)) - (Sqrt[c]*(b*c*d + c*Sqrt[b^2 - 4*a*c]*d - b^2*e + 2*a*c*e - b*Sqrt[b^2 - 4*a*c]*e)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*a*Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]*(c*d^2 + e*(-(b*d) + a*e))) + (Sqrt[c]*(b*c*d - c*Sqrt[b^2 - 4*a*c]*d - b^2*e + 2*a*c*e + b*Sqrt[b^2 - 4*a*c]*e)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*a*Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]*(c*d^2 + e*(-(b*d) + a*e))) - (e^(5/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(3/2)*(c*d^2 - b*d*e + a*e^2))","A",1
309,1,410,348,0.5247782,"\int \frac{1}{x^4 \left(d+e x^2\right) \left(a+b x^2+c x^4\right)} \, dx","Integrate[1/(x^4*(d + e*x^2)*(a + b*x^2 + c*x^4)),x]","\frac{\sqrt{c} \left(b^2 \left(c d-e \sqrt{b^2-4 a c}\right)+b c \left(d \sqrt{b^2-4 a c}+3 a e\right)+a c \left(e \sqrt{b^2-4 a c}-2 c d\right)+b^3 (-e)\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^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}} \left(e (a e-b d)+c d^2\right)}+\frac{\sqrt{c} \left(-b^2 \left(e \sqrt{b^2-4 a c}+c d\right)+b c \left(d \sqrt{b^2-4 a c}-3 a e\right)+a c \left(e \sqrt{b^2-4 a c}+2 c d\right)+b^3 e\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} a^2 \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b} \left(e (a e-b d)+c d^2\right)}+\frac{a e+b d}{a^2 d^2 x}+\frac{e^{7/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{d^{5/2} \left(a e^2-b d e+c d^2\right)}-\frac{1}{3 a d x^3}","\frac{\sqrt{c} \left(\frac{3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\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}} \left(a e^2-b d e+c d^2\right)}+\frac{\sqrt{c} \left(-\frac{3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\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} \left(a e^2-b d e+c d^2\right)}+\frac{a e+b d}{a^2 d^2 x}+\frac{e^{7/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{d^{5/2} \left(a e^2-b d e+c d^2\right)}-\frac{1}{3 a d x^3}",1,"-1/3*1/(a*d*x^3) + (b*d + a*e)/(a^2*d^2*x) + (Sqrt[c]*(-(b^3*e) + b*c*(Sqrt[b^2 - 4*a*c]*d + 3*a*e) + b^2*(c*d - Sqrt[b^2 - 4*a*c]*e) + a*c*(-2*c*d + Sqrt[b^2 - 4*a*c]*e))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*a^2*Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]*(c*d^2 + e*(-(b*d) + a*e))) + (Sqrt[c]*(b^3*e + b*c*(Sqrt[b^2 - 4*a*c]*d - 3*a*e) - b^2*(c*d + Sqrt[b^2 - 4*a*c]*e) + a*c*(2*c*d + Sqrt[b^2 - 4*a*c]*e))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*a^2*Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]*(c*d^2 + e*(-(b*d) + a*e))) + (e^(7/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(5/2)*(c*d^2 - b*d*e + a*e^2))","A",1
310,1,267,866,0.3766174,"\int \frac{1}{\sqrt{f x} \left(d+e x^2\right) \left(a+b x^2+c x^4\right)} \, dx","Integrate[1/(Sqrt[f*x]*(d + e*x^2)*(a + b*x^2 + c*x^4)),x]","\frac{\sqrt{x} \left(\sqrt{2} e^{7/4} \left(-\log \left(-\sqrt{2} \sqrt[4]{d} \sqrt[4]{e} \sqrt{x}+\sqrt{d}+\sqrt{e} x\right)+\log \left(\sqrt{2} \sqrt[4]{d} \sqrt[4]{e} \sqrt{x}+\sqrt{d}+\sqrt{e} x\right)-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{e} \sqrt{x}}{\sqrt[4]{d}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{e} \sqrt{x}}{\sqrt[4]{d}}+1\right)\right)-2 d^{3/4} \text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^4 c e \log \left(\sqrt{x}-\text{$\#$1}\right)+b e \log \left(\sqrt{x}-\text{$\#$1}\right)-c d \log \left(\sqrt{x}-\text{$\#$1}\right)}{2 \text{$\#$1}^7 c+\text{$\#$1}^3 b}\&\right]\right)}{4 d^{3/4} \sqrt{f x} \left(e (a e-b d)+c d^2\right)}","-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{e} \sqrt{f x}}{\sqrt[4]{d} \sqrt{f}}\right) e^{7/4}}{\sqrt{2} d^{3/4} \left(c d^2-b e d+a e^2\right) \sqrt{f}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{e} \sqrt{f x}}{\sqrt[4]{d} \sqrt{f}}+1\right) e^{7/4}}{\sqrt{2} d^{3/4} \left(c d^2-b e d+a e^2\right) \sqrt{f}}-\frac{\log \left(\sqrt{e} \sqrt{f} x+\sqrt{d} \sqrt{f}-\sqrt{2} \sqrt[4]{d} \sqrt[4]{e} \sqrt{f x}\right) e^{7/4}}{2 \sqrt{2} d^{3/4} \left(c d^2-b e d+a e^2\right) \sqrt{f}}+\frac{\log \left(\sqrt{e} \sqrt{f} x+\sqrt{d} \sqrt{f}+\sqrt{2} \sqrt[4]{d} \sqrt[4]{e} \sqrt{f x}\right) e^{7/4}}{2 \sqrt{2} d^{3/4} \left(c d^2-b e d+a e^2\right) \sqrt{f}}+\frac{c^{3/4} \left(2 c d-\left(b-\sqrt{b^2-4 a c}\right) e\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{f x}}{\sqrt[4]{-b-\sqrt{b^2-4 a c}} \sqrt{f}}\right)}{\sqrt[4]{2} \sqrt{b^2-4 a c} \left(-b-\sqrt{b^2-4 a c}\right)^{3/4} \left(c d^2-b e d+a e^2\right) \sqrt{f}}-\frac{c^{3/4} \left(2 c d-\left(b+\sqrt{b^2-4 a c}\right) e\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{f x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b} \sqrt{f}}\right)}{\sqrt[4]{2} \sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}-b\right)^{3/4} \left(c d^2-b e d+a e^2\right) \sqrt{f}}+\frac{c^{3/4} \left(2 c d-\left(b-\sqrt{b^2-4 a c}\right) e\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{f x}}{\sqrt[4]{-b-\sqrt{b^2-4 a c}} \sqrt{f}}\right)}{\sqrt[4]{2} \sqrt{b^2-4 a c} \left(-b-\sqrt{b^2-4 a c}\right)^{3/4} \left(c d^2-b e d+a e^2\right) \sqrt{f}}-\frac{c^{3/4} \left(2 c d-\left(b+\sqrt{b^2-4 a c}\right) e\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} \sqrt{f x}}{\sqrt[4]{\sqrt{b^2-4 a c}-b} \sqrt{f}}\right)}{\sqrt[4]{2} \sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}-b\right)^{3/4} \left(c d^2-b e d+a e^2\right) \sqrt{f}}",1,"(Sqrt[x]*(Sqrt[2]*e^(7/4)*(-2*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4)] - Log[Sqrt[d] - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqrt[e]*x] + Log[Sqrt[d] + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqrt[e]*x]) - 2*d^(3/4)*RootSum[a + b*#1^4 + c*#1^8 & , (-(c*d*Log[Sqrt[x] - #1]) + b*e*Log[Sqrt[x] - #1] + c*e*Log[Sqrt[x] - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ]))/(4*d^(3/4)*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[f*x])","C",1
311,1,267,272,0.4348889,"\int \frac{x^5 \sqrt{a+b x^2+c x^4}}{d+e x^2} \, dx","Integrate[(x^5*Sqrt[a + b*x^2 + c*x^4])/(d + e*x^2),x]","\frac{2 \sqrt{c} \left(e \sqrt{a+b x^2+c x^4} \left(2 c e \left(4 a e-3 b d+b e x^2\right)-3 b^2 e^2+4 c^2 \left(6 d^2-3 d e x^2+2 e^2 x^4\right)\right)+24 c^2 d^2 \sqrt{e (a e-b d)+c d^2} \tanh ^{-1}\left(\frac{-2 a e+b \left(d-e x^2\right)+2 c d x^2}{2 \sqrt{a+b x^2+c x^4} \sqrt{e (a e-b d)+c d^2}}\right)\right)-3 \left(8 c^2 d e (a e-b d)-2 b c e^2 (b d-2 a e)-b^3 e^3+16 c^3 d^3\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} e^4}","-\frac{\left(-8 c^2 d e (b d-a e)-2 b c e^2 (b d-2 a e)-b^3 e^3+16 c^3 d^3\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} e^4}+\frac{\sqrt{a+b x^2+c x^4} \left((2 c d-b e) (b e+4 c d)-2 c e x^2 (b e+2 c d)\right)}{16 c^2 e^3}+\frac{d^2 \sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left(\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right)}{2 e^4}+\frac{\left(a+b x^2+c x^4\right)^{3/2}}{6 c e}",1,"(-3*(16*c^3*d^3 - b^3*e^3 - 2*b*c*e^2*(b*d - 2*a*e) + 8*c^2*d*e*(-(b*d) + a*e))*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])] + 2*Sqrt[c]*(e*Sqrt[a + b*x^2 + c*x^4]*(-3*b^2*e^2 + 2*c*e*(-3*b*d + 4*a*e + b*e*x^2) + 4*c^2*(6*d^2 - 3*d*e*x^2 + 2*e^2*x^4)) + 24*c^2*d^2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*ArcTanh[(-2*a*e + 2*c*d*x^2 + b*(d - e*x^2))/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + b*x^2 + c*x^4])]))/(96*c^(5/2)*e^4)","A",1
312,1,205,208,0.2617113,"\int \frac{x^3 \sqrt{a+b x^2+c x^4}}{d+e x^2} \, dx","Integrate[(x^3*Sqrt[a + b*x^2 + c*x^4])/(d + e*x^2),x]","\frac{\left(4 c e (a e-b d)-b^2 e^2+8 c^2 d^2\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(4 c d \sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left(\frac{2 a e-b d+b e x^2-2 c d x^2}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right)+e \sqrt{a+b x^2+c x^4} \left(b e-4 c d+2 c e x^2\right)\right)}{16 c^{3/2} e^3}","\frac{\left(-4 c e (b d-a e)-b^2 e^2+8 c^2 d^2\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} e^3}-\frac{d \sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left(\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right)}{2 e^3}-\frac{\sqrt{a+b x^2+c x^4} \left(-b e+4 c d-2 c e x^2\right)}{8 c e^2}",1,"((8*c^2*d^2 - b^2*e^2 + 4*c*e*(-(b*d) + a*e))*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])] + 2*Sqrt[c]*(e*(-4*c*d + b*e + 2*c*e*x^2)*Sqrt[a + b*x^2 + c*x^4] + 4*c*d*Sqrt[c*d^2 - b*d*e + a*e^2]*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x^2 + b*e*x^2)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x^2 + c*x^4])]))/(16*c^(3/2)*e^3)","A",1
313,1,167,168,0.1276154,"\int \frac{x \sqrt{a+b x^2+c x^4}}{d+e x^2} \, dx","Integrate[(x*Sqrt[a + b*x^2 + c*x^4])/(d + e*x^2),x]","\frac{2 \sqrt{c} \left(e \sqrt{a+b x^2+c x^4}-\sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left(\frac{2 a e-b d+b e x^2-2 c d x^2}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right)\right)+(b e-2 c d) \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{4 \sqrt{c} e^2}","\frac{\sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left(\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right)}{2 e^2}-\frac{(2 c d-b e) \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{4 \sqrt{c} e^2}+\frac{\sqrt{a+b x^2+c x^4}}{2 e}",1,"((-2*c*d + b*e)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])] + 2*Sqrt[c]*(e*Sqrt[a + b*x^2 + c*x^4] - Sqrt[c*d^2 - b*d*e + a*e^2]*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x^2 + b*e*x^2)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x^2 + c*x^4])]))/(4*Sqrt[c]*e^2)","A",1
314,1,179,186,0.1493822,"\int \frac{\sqrt{a+b x^2+c x^4}}{x \left(d+e x^2\right)} \, dx","Integrate[Sqrt[a + b*x^2 + c*x^4]/(x*(d + e*x^2)),x]","-\frac{\sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left(\frac{-2 a e+b d-b e x^2+2 c d x^2}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right)-\sqrt{c} d \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)+\sqrt{a} e \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{2 d e}","-\frac{\sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left(\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right)}{2 d e}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{2 d}+\frac{\sqrt{c} \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{2 e}",1,"-1/2*(Sqrt[a]*e*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])] - Sqrt[c]*d*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])] + Sqrt[c*d^2 - b*d*e + a*e^2]*ArcTanh[(b*d - 2*a*e + 2*c*d*x^2 - b*e*x^2)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x^2 + c*x^4])])/(d*e)","A",1
315,1,165,361,0.3152978,"\int \frac{\sqrt{a+b x^2+c x^4}}{x^3 \left(d+e x^2\right)} \, dx","Integrate[Sqrt[a + b*x^2 + c*x^4]/(x^3*(d + e*x^2)),x]","\frac{2 \sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left(\frac{-2 a e+b d-b e x^2+2 c d x^2}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right)+\frac{(2 a e-b d) \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 d \sqrt{a+b x^2+c x^4}}{x^2}}{4 d^2}","\frac{\sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left(\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right)}{2 d^2}+\frac{\sqrt{a} e \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{2 d^2}-\frac{b e \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{4 \sqrt{c} d^2}-\frac{(2 c d-b e) \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{4 \sqrt{c} d^2}-\frac{\sqrt{a+b x^2+c x^4}}{2 d 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} d}+\frac{\sqrt{c} \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{2 d}",1,"((-2*d*Sqrt[a + b*x^2 + c*x^4])/x^2 + ((-(b*d) + 2*a*e)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/Sqrt[a] + 2*Sqrt[c*d^2 - b*d*e + a*e^2]*ArcTanh[(b*d - 2*a*e + 2*c*d*x^2 - b*e*x^2)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x^2 + c*x^4])])/(4*d^2)","A",1
316,1,209,424,0.2593246,"\int \frac{x^4 \sqrt{1+2 x^2+2 x^4}}{3+2 x^2} \, dx","Integrate[(x^4*Sqrt[1 + 2*x^2 + 2*x^4])/(3 + 2*x^2),x]","\frac{48 x^7-56 x^5-80 x^3-(199-417 i) \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} F\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)-218 i \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} E\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)+225 (1-i)^{3/2} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} \Pi \left(\frac{1}{3}+\frac{i}{3};\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)-52 x}{240 \sqrt{2 x^4+2 x^2+1}}","-\frac{1}{60} \left(13-6 x^2\right) \sqrt{2 x^4+2 x^2+1} x+\frac{109 \sqrt{2 x^4+2 x^2+1} x}{60 \sqrt{2} \left(\sqrt{2} x^2+1\right)}+\frac{3}{16} \sqrt{15} \tan ^{-1}\left(\frac{\sqrt{\frac{5}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right)+\frac{\left(263 \sqrt{2}-70\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{60\ 2^{3/4} \left(3 \sqrt{2}-2\right) \sqrt{2 x^4+2 x^2+1}}-\frac{109 \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} E\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{60\ 2^{3/4} \sqrt{2 x^4+2 x^2+1}}+\frac{15 \left(3+\sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} \Pi \left(\frac{1}{24} \left(12-11 \sqrt{2}\right);2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{16\ 2^{3/4} \left(2-3 \sqrt{2}\right) \sqrt{2 x^4+2 x^2+1}}",1,"(-52*x - 80*x^3 - 56*x^5 + 48*x^7 - (218*I)*Sqrt[1 - I]*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticE[I*ArcSinh[Sqrt[1 - I]*x], I] - (199 - 417*I)*Sqrt[1 - I]*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticF[I*ArcSinh[Sqrt[1 - I]*x], I] + 225*(1 - I)^(3/2)*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticPi[1/3 + I/3, I*ArcSinh[Sqrt[1 - I]*x], I])/(240*Sqrt[1 + 2*x^2 + 2*x^4])","C",1
317,1,204,417,0.1954566,"\int \frac{x^2 \sqrt{1+2 x^2+2 x^4}}{3+2 x^2} \, dx","Integrate[(x^2*Sqrt[1 + 2*x^2 + 2*x^4])/(3 + 2*x^2),x]","\frac{8 x^5+8 x^3+(13-27 i) \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} F\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)+14 i \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} E\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)-15 (1-i)^{3/2} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} \Pi \left(\frac{1}{3}+\frac{i}{3};\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)+4 x}{24 \sqrt{2 x^4+2 x^2+1}}","-\frac{7 \sqrt{2 x^4+2 x^2+1} x}{6 \sqrt{2} \left(\sqrt{2} x^2+1\right)}+\frac{1}{6} \sqrt{2 x^4+2 x^2+1} x-\frac{1}{8} \sqrt{15} \tan ^{-1}\left(\frac{\sqrt{\frac{5}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right)-\frac{\left(17 \sqrt{2}-4\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{6\ 2^{3/4} \left(3 \sqrt{2}-2\right) \sqrt{2 x^4+2 x^2+1}}+\frac{7 \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} E\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{6\ 2^{3/4} \sqrt{2 x^4+2 x^2+1}}-\frac{5 \left(3+\sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} \Pi \left(\frac{1}{24} \left(12-11 \sqrt{2}\right);2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{8\ 2^{3/4} \left(2-3 \sqrt{2}\right) \sqrt{2 x^4+2 x^2+1}}",1,"(4*x + 8*x^3 + 8*x^5 + (14*I)*Sqrt[1 - I]*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticE[I*ArcSinh[Sqrt[1 - I]*x], I] + (13 - 27*I)*Sqrt[1 - I]*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticF[I*ArcSinh[Sqrt[1 - I]*x], I] - 15*(1 - I)^(3/2)*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticPi[1/3 + I/3, I*ArcSinh[Sqrt[1 - I]*x], I])/(24*Sqrt[1 + 2*x^2 + 2*x^4])","C",1
318,1,127,381,0.1085344,"\int \frac{\sqrt{1+2 x^2+2 x^4}}{3+2 x^2} \, dx","Integrate[Sqrt[1 + 2*x^2 + 2*x^4]/(3 + 2*x^2),x]","-\frac{\sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} \left(-(3+6 i) F\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)+(3+3 i) E\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)+5 i \Pi \left(\frac{1}{3}+\frac{i}{3};\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)\right)}{6 \sqrt{1-i} \sqrt{2 x^4+2 x^2+1}}","\frac{\sqrt{2 x^4+2 x^2+1} x}{\sqrt{2} \left(\sqrt{2} x^2+1\right)}+\frac{1}{4} \sqrt{\frac{5}{3}} \tan ^{-1}\left(\frac{\sqrt{\frac{5}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right)+\frac{2^{3/4} \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{\left(3 \sqrt{2}-2\right) \sqrt{2 x^4+2 x^2+1}}-\frac{\left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} E\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{2^{3/4} \sqrt{2 x^4+2 x^2+1}}+\frac{5 \left(3+\sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} \Pi \left(\frac{1}{24} \left(12-11 \sqrt{2}\right);2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{12\ 2^{3/4} \left(2-3 \sqrt{2}\right) \sqrt{2 x^4+2 x^2+1}}",1,"-1/6*(Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*((3 + 3*I)*EllipticE[I*ArcSinh[Sqrt[1 - I]*x], I] - (3 + 6*I)*EllipticF[I*ArcSinh[Sqrt[1 - I]*x], I] + (5*I)*EllipticPi[1/3 + I/3, I*ArcSinh[Sqrt[1 - I]*x], I]))/(Sqrt[1 - I]*Sqrt[1 + 2*x^2 + 2*x^4])","C",1
319,1,208,399,0.1903659,"\int \frac{\sqrt{1+2 x^2+2 x^4}}{x^2 \left(3+2 x^2\right)} \, dx","Integrate[Sqrt[1 + 2*x^2 + 2*x^4]/(x^2*(3 + 2*x^2)),x]","\frac{-12 x^4-12 x^2+(9-3 i) \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} x F\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)-6 i \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} x E\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)-5 (1-i)^{3/2} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} x \Pi \left(\frac{1}{3}+\frac{i}{3};\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)-6}{18 x \sqrt{2 x^4+2 x^2+1}}","\frac{\sqrt{2} \sqrt{2 x^4+2 x^2+1} x}{3 \left(\sqrt{2} x^2+1\right)}-\frac{\sqrt{2 x^4+2 x^2+1}}{3 x}-\frac{1}{6} \sqrt{\frac{5}{3}} \tan ^{-1}\left(\frac{\sqrt{\frac{5}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right)+\frac{\left(3+\sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{21 \sqrt[4]{2} \sqrt{2 x^4+2 x^2+1}}-\frac{\sqrt[4]{2} \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} E\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{3 \sqrt{2 x^4+2 x^2+1}}+\frac{5 \left(3+\sqrt{2}\right)^2 \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} \Pi \left(\frac{1}{24} \left(12-11 \sqrt{2}\right);2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{252 \sqrt[4]{2} \sqrt{2 x^4+2 x^2+1}}",1,"(-6 - 12*x^2 - 12*x^4 - (6*I)*Sqrt[1 - I]*x*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticE[I*ArcSinh[Sqrt[1 - I]*x], I] + (9 - 3*I)*Sqrt[1 - I]*x*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticF[I*ArcSinh[Sqrt[1 - I]*x], I] - 5*(1 - I)^(3/2)*x*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticPi[1/3 + I/3, I*ArcSinh[Sqrt[1 - I]*x], I])/(18*x*Sqrt[1 + 2*x^2 + 2*x^4])","C",1
320,1,154,360,0.1725009,"\int \frac{\sqrt{1+2 x^2+2 x^4}}{x^4 \left(3+2 x^2\right)} \, dx","Integrate[Sqrt[1 + 2*x^2 + 2*x^4]/(x^4*(3 + 2*x^2)),x]","-\frac{6 x^4+6 x^2+3 (1-i)^{3/2} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} x^3 F\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)-5 (1-i)^{3/2} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} x^3 \Pi \left(\frac{1}{3}+\frac{i}{3};\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)+3}{27 x^3 \sqrt{2 x^4+2 x^2+1}}","\frac{1}{9} \sqrt{\frac{5}{3}} \tan ^{-1}\left(\frac{\sqrt{\frac{5}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right)+\frac{5 \left(3+\sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{63 \sqrt[4]{2} \sqrt{2 x^4+2 x^2+1}}-\frac{\left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{9 \sqrt[4]{2} \sqrt{2 x^4+2 x^2+1}}-\frac{5 \left(3+\sqrt{2}\right)^2 \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} \Pi \left(\frac{1}{24} \left(12-11 \sqrt{2}\right);2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{378 \sqrt[4]{2} \sqrt{2 x^4+2 x^2+1}}-\frac{\sqrt{2 x^4+2 x^2+1}}{9 x^3}",1,"-1/27*(3 + 6*x^2 + 6*x^4 + 3*(1 - I)^(3/2)*x^3*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticF[I*ArcSinh[Sqrt[1 - I]*x], I] - 5*(1 - I)^(3/2)*x^3*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticPi[1/3 + I/3, I*ArcSinh[Sqrt[1 - I]*x], I])/(x^3*Sqrt[1 + 2*x^2 + 2*x^4])","C",1
321,1,224,546,0.2376041,"\int \frac{\sqrt{1+2 x^2+2 x^4}}{x^6 \left(3+2 x^2\right)} \, dx","Integrate[Sqrt[1 + 2*x^2 + 2*x^4]/(x^6*(3 + 2*x^2)),x]","-\frac{72 x^8+48 x^6+66 x^4+42 x^2-(12+24 i) \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} x^5 F\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)+36 i \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} x^5 E\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)+50 (1-i)^{3/2} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} x^5 \Pi \left(\frac{1}{3}+\frac{i}{3};\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)+27}{405 x^5 \sqrt{2 x^4+2 x^2+1}}","\frac{4 \sqrt{2} \sqrt{2 x^4+2 x^2+1} x}{45 \left(\sqrt{2} x^2+1\right)}-\frac{4 \sqrt{2 x^4+2 x^2+1}}{45 x}-\frac{2}{27} \sqrt{\frac{5}{3}} \tan ^{-1}\left(\frac{\sqrt{\frac{5}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right)-\frac{\sqrt[4]{2} \left(19-2 \sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{135 \sqrt{2 x^4+2 x^2+1}}+\frac{5 \sqrt[4]{2} \left(5-3 \sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{189 \sqrt{2 x^4+2 x^2+1}}-\frac{4 \sqrt[4]{2} \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} E\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{45 \sqrt{2 x^4+2 x^2+1}}+\frac{5 \left(3+\sqrt{2}\right)^2 \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} \Pi \left(\frac{1}{24} \left(12-11 \sqrt{2}\right);2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{567 \sqrt[4]{2} \sqrt{2 x^4+2 x^2+1}}-\frac{\sqrt{2 x^4+2 x^2+1}}{15 x^5}+\frac{4 \sqrt{2 x^4+2 x^2+1}}{135 x^3}",1,"-1/405*(27 + 42*x^2 + 66*x^4 + 48*x^6 + 72*x^8 + (36*I)*Sqrt[1 - I]*x^5*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticE[I*ArcSinh[Sqrt[1 - I]*x], I] - (12 + 24*I)*Sqrt[1 - I]*x^5*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticF[I*ArcSinh[Sqrt[1 - I]*x], I] + 50*(1 - I)^(3/2)*x^5*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticPi[1/3 + I/3, I*ArcSinh[Sqrt[1 - I]*x], I])/(x^5*Sqrt[1 + 2*x^2 + 2*x^4])","C",1
322,1,545,482,0.9756722,"\int \frac{x^5 \left(a+b x^2+c x^4\right)^{3/2}}{d+e x^2} \, dx","Integrate[(x^5*(a + b*x^2 + c*x^4)^(3/2))/(d + e*x^2),x]","\frac{-\frac{90 d e \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)}{c^{5/2}}-\frac{240 d^2 \left((2 c d-b e) \left(4 c e (3 a e-2 b d)-b^2 e^2+8 c^2 d^2\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(e \sqrt{a+b x^2+c x^4} \left(-2 c e \left(4 a e-5 b d+b e x^2\right)-b^2 e^2+4 c^2 d \left(e x^2-2 d\right)\right)+8 c \left(e (a e-b d)+c d^2\right)^{3/2} \tanh ^{-1}\left(\frac{2 a e-b d+b e x^2-2 c d x^2}{2 \sqrt{a+b x^2+c x^4} \sqrt{e (a e-b d)+c d^2}}\right)\right)\right)}{c^{3/2} e^3}+\frac{15 b e^2 \left(3 \left(b^2-4 a c\right) \left(\frac{\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)}{c^{3/2}}+\frac{2 \left(b+2 c x^2\right) \sqrt{a+b x^2+c x^4}}{c}\right)-16 \left(b+2 c x^2\right) \left(a+b x^2+c x^4\right)^{3/2}\right)}{c^2}+1280 d^2 \left(a+b x^2+c x^4\right)^{3/2}-\frac{480 d e \left(b+2 c x^2\right) \left(a+b x^2+c x^4\right)^{3/2}}{c}+\frac{768 e^2 \left(a+b x^2+c x^4\right)^{5/2}}{c}}{7680 e^3}","-\frac{\left(16 b c^2 e^3 \left(3 a^2 e^2-3 a b d e+b^2 d^2\right)+6 b^3 c e^4 (b d-4 a e)-384 c^4 d^3 e (b d-a e)+96 c^3 d e^2 (b d-a e)^2+3 b^5 e^5+256 c^5 d^5\right) \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} e^6}+\frac{\left(a+b x^2+c x^4\right)^{3/2} \left(-3 b^2 e^2-6 c e x^2 (b e+2 c d)-6 b c d e+16 c^2 d^2\right)}{96 c^2 e^3}+\frac{\sqrt{a+b x^2+c x^4} \left(-2 c e x^2 \left(-8 c^2 d e (2 b d-3 a e)-6 b c e^2 (b d-2 a e)-3 b^3 e^3+32 c^3 d^3\right)+6 b^2 c e^3 (b d-2 a e)-32 c^3 d^2 e (5 b d-4 a e)+8 b c^2 d e^2 (2 b d-3 a e)+3 b^4 e^4+128 c^4 d^4\right)}{256 c^3 e^5}+\frac{d^2 \left(a e^2-b d e+c d^2\right)^{3/2} \tanh ^{-1}\left(\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right)}{2 e^6}+\frac{\left(a+b x^2+c x^4\right)^{5/2}}{10 c e}",1,"(1280*d^2*(a + b*x^2 + c*x^4)^(3/2) - (480*d*e*(b + 2*c*x^2)*(a + b*x^2 + c*x^4)^(3/2))/c + (768*e^2*(a + b*x^2 + c*x^4)^(5/2))/c - (90*(b^2 - 4*a*c)*d*e*(-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])]))/c^(5/2) + (15*b*e^2*(-16*(b + 2*c*x^2)*(a + b*x^2 + c*x^4)^(3/2) + 3*(b^2 - 4*a*c)*((2*(b + 2*c*x^2)*Sqrt[a + b*x^2 + c*x^4])/c + ((-b^2 + 4*a*c)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/c^(3/2))))/c^2 - (240*d^2*((2*c*d - b*e)*(8*c^2*d^2 - b^2*e^2 + 4*c*e*(-2*b*d + 3*a*e))*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])] + 2*Sqrt[c]*(e*Sqrt[a + b*x^2 + c*x^4]*(-(b^2*e^2) + 4*c^2*d*(-2*d + e*x^2) - 2*c*e*(-5*b*d + 4*a*e + b*e*x^2)) + 8*c*(c*d^2 + e*(-(b*d) + a*e))^(3/2)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x^2 + b*e*x^2)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + b*x^2 + c*x^4])])))/(c^(3/2)*e^3))/(7680*e^3)","A",1
323,1,344,360,0.5995807,"\int \frac{x^3 \left(a+b x^2+c x^4\right)^{3/2}}{d+e x^2} \, dx","Integrate[(x^3*(a + b*x^2 + c*x^4)^(3/2))/(d + e*x^2),x]","\frac{2 \sqrt{c} \left(e \sqrt{a+b x^2+c x^4} \left(8 c^2 e \left(a e \left(15 e x^2-32 d\right)+b \left(30 d^2-14 d e x^2+9 e^2 x^4\right)\right)+6 b c e^2 \left(10 a e-4 b d+b e x^2\right)-9 b^3 e^3-16 c^3 \left(12 d^3-6 d^2 e x^2+4 d e^2 x^4-3 e^3 x^6\right)\right)+192 c^2 d \left(e (a e-b d)+c d^2\right)^{3/2} \tanh ^{-1}\left(\frac{2 a e-b d+b e x^2-2 c d x^2}{2 \sqrt{a+b x^2+c x^4} \sqrt{e (a e-b d)+c d^2}}\right)\right)+3 \left(8 b^2 c e^3 (b d-3 a e)-192 c^3 d^2 e (b d-a e)+48 c^2 e^2 (b d-a e)^2+3 b^4 e^4+128 c^4 d^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^{5/2} e^5}","\frac{\left(8 b^2 c e^3 (b d-3 a e)-192 c^3 d^2 e (b d-a e)+48 c^2 e^2 (b d-a e)^2+3 b^4 e^4+128 c^4 d^4\right) \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} e^5}-\frac{\sqrt{a+b x^2+c x^4} \left(-2 c e x^2 \left(-4 c e (2 b d-3 a e)-3 b^2 e^2+16 c^2 d^2\right)-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (2 b d-3 a e)+3 b^3 e^3+64 c^3 d^3\right)}{128 c^2 e^4}-\frac{d \left(a e^2-b d e+c d^2\right)^{3/2} \tanh ^{-1}\left(\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right)}{2 e^5}-\frac{\left(a+b x^2+c x^4\right)^{3/2} \left(-3 b e+8 c d-6 c e x^2\right)}{48 c e^2}",1,"(3*(128*c^4*d^4 + 3*b^4*e^4 + 8*b^2*c*e^3*(b*d - 3*a*e) - 192*c^3*d^2*e*(b*d - a*e) + 48*c^2*e^2*(b*d - a*e)^2)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])] + 2*Sqrt[c]*(e*Sqrt[a + b*x^2 + c*x^4]*(-9*b^3*e^3 + 6*b*c*e^2*(-4*b*d + 10*a*e + b*e*x^2) - 16*c^3*(12*d^3 - 6*d^2*e*x^2 + 4*d*e^2*x^4 - 3*e^3*x^6) + 8*c^2*e*(a*e*(-32*d + 15*e*x^2) + b*(30*d^2 - 14*d*e*x^2 + 9*e^2*x^4))) + 192*c^2*d*(c*d^2 + e*(-(b*d) + a*e))^(3/2)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x^2 + b*e*x^2)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + b*x^2 + c*x^4])]))/(768*c^(5/2)*e^5)","A",1
324,1,255,269,0.3511154,"\int \frac{x \left(a+b x^2+c x^4\right)^{3/2}}{d+e x^2} \, dx","Integrate[(x*(a + b*x^2 + c*x^4)^(3/2))/(d + e*x^2),x]","\frac{2 \sqrt{c} \left(e \sqrt{a+b x^2+c x^4} \left(2 c e \left(16 a e-15 b d+7 b e x^2\right)+3 b^2 e^2+4 c^2 \left(6 d^2-3 d e x^2+2 e^2 x^4\right)\right)-24 c \left(e (a e-b d)+c d^2\right)^{3/2} \tanh ^{-1}\left(\frac{2 a e-b d+b e x^2-2 c d x^2}{2 \sqrt{a+b x^2+c x^4} \sqrt{e (a e-b d)+c d^2}}\right)\right)-3 (2 c d-b e) \left(4 c e (3 a e-2 b d)-b^2 e^2+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{96 c^{3/2} e^4}","\frac{\sqrt{a+b x^2+c x^4} \left(-2 c e (5 b d-4 a e)+b^2 e^2-2 c e x^2 (2 c d-b e)+8 c^2 d^2\right)}{16 c e^3}-\frac{(2 c d-b e) \left(-4 c e (2 b d-3 a e)-b^2 e^2+8 c^2 d^2\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} e^4}+\frac{\left(a e^2-b d e+c d^2\right)^{3/2} \tanh ^{-1}\left(\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right)}{2 e^4}+\frac{\left(a+b x^2+c x^4\right)^{3/2}}{6 e}",1,"(-3*(2*c*d - b*e)*(8*c^2*d^2 - b^2*e^2 + 4*c*e*(-2*b*d + 3*a*e))*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])] + 2*Sqrt[c]*(e*Sqrt[a + b*x^2 + c*x^4]*(3*b^2*e^2 + 2*c*e*(-15*b*d + 16*a*e + 7*b*e*x^2) + 4*c^2*(6*d^2 - 3*d*e*x^2 + 2*e^2*x^4)) - 24*c*(c*d^2 + e*(-(b*d) + a*e))^(3/2)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x^2 + b*e*x^2)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + b*x^2 + c*x^4])]))/(96*c^(3/2)*e^4)","A",1
325,1,251,350,0.5335017,"\int \frac{\left(a+b x^2+c x^4\right)^{3/2}}{x \left(d+e x^2\right)} \, dx","Integrate[(a + b*x^2 + c*x^4)^(3/2)/(x*(d + e*x^2)),x]","\frac{1}{16} \left(-\frac{8 a^{3/2} \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{d}+\frac{\left(12 c e (a e-b d)+3 b^2 e^2+8 c^2 d^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} e^3}+\frac{2 \left(4 \left(e (a e-b d)+c d^2\right)^{3/2} \tanh ^{-1}\left(\frac{2 a e-b d+b e x^2-2 c d x^2}{2 \sqrt{a+b x^2+c x^4} \sqrt{e (a e-b d)+c d^2}}\right)+d e \sqrt{a+b x^2+c x^4} \left(5 b e-4 c d+2 c e x^2\right)\right)}{d e^3}\right)","-\frac{a^{3/2} \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{2 d}+\frac{\left(-12 c d e (b d-a e)+b e^2 (3 b d-4 a e)+8 c^2 d^3\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} d e^3}-\frac{\sqrt{a+b x^2+c x^4} \left(-e (5 b d-4 a e)+4 c d^2-2 c d e x^2\right)}{8 d e^2}-\frac{\left(a e^2-b d e+c d^2\right)^{3/2} \tanh ^{-1}\left(\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right)}{2 d e^3}+\frac{a \sqrt{a+b x^2+c x^4}}{2 d}+\frac{a 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} d}",1,"((-8*a^(3/2)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/d + ((8*c^2*d^2 + 3*b^2*e^2 + 12*c*e*(-(b*d) + a*e))*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(Sqrt[c]*e^3) + (2*(d*e*(-4*c*d + 5*b*e + 2*c*e*x^2)*Sqrt[a + b*x^2 + c*x^4] + 4*(c*d^2 + e*(-(b*d) + a*e))^(3/2)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x^2 + b*e*x^2)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + b*x^2 + c*x^4])]))/(d*e^3))/16","A",1
326,1,240,562,0.5032579,"\int \frac{\left(a+b x^2+c x^4\right)^{3/2}}{x^3 \left(d+e x^2\right)} \, dx","Integrate[(a + b*x^2 + c*x^4)^(3/2)/(x^3*(d + e*x^2)),x]","\frac{1}{4} \left(-\frac{2 \left(x^2 \left(e (a e-b d)+c d^2\right)^{3/2} \tanh ^{-1}\left(\frac{2 a e-b d+b e x^2-2 c d x^2}{2 \sqrt{a+b x^2+c x^4} \sqrt{e (a e-b d)+c d^2}}\right)+d e \sqrt{a+b x^2+c x^4} \left(a e-c d x^2\right)\right)}{d^2 e^2 x^2}+\frac{\sqrt{a} (2 a e-3 b d) \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{d^2}-\frac{\sqrt{c} (2 c d-3 b e) \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{e^2}\right)","\frac{a^{3/2} e \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{2 d^2}+\frac{b e \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} d^2}+\frac{\sqrt{a+b x^2+c x^4} \left(-2 c e (5 b d-4 a e)+b^2 e^2-2 c e x^2 (2 c d-b e)+8 c^2 d^2\right)}{16 c d^2 e}-\frac{(2 c d-b e) \left(-4 c e (2 b d-3 a e)-b^2 e^2+8 c^2 d^2\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} d^2 e^2}-\frac{e \left(8 a c+b^2+2 b c x^2\right) \sqrt{a+b x^2+c x^4}}{16 c d^2}+\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} d}+\frac{\left(a e^2-b d e+c d^2\right)^{3/2} \tanh ^{-1}\left(\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right)}{2 d^2 e^2}-\frac{\left(a+b x^2+c x^4\right)^{3/2}}{2 d x^2}+\frac{3 \left(3 b+2 c x^2\right) \sqrt{a+b x^2+c x^4}}{8 d}-\frac{3 \sqrt{a} b \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{4 d}",1,"((Sqrt[a]*(-3*b*d + 2*a*e)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/d^2 - (Sqrt[c]*(2*c*d - 3*b*e)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/e^2 - (2*(d*e*(a*e - c*d*x^2)*Sqrt[a + b*x^2 + c*x^4] + (c*d^2 + e*(-(b*d) + a*e))^(3/2)*x^2*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x^2 + b*e*x^2)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + b*x^2 + c*x^4])]))/(d^2*e^2*x^2))/4","A",1
327,1,214,463,0.2723241,"\int \frac{x^2 \left(1+2 x^2+2 x^4\right)^{3/2}}{3-2 x^2} \, dx","Integrate[(x^2*(1 + 2*x^2 + 2*x^4)^(3/2))/(3 - 2*x^2),x]","\frac{-160 x^9-752 x^7-2456 x^5-2080 x^3-(9669-5247 i) \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} F\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)+4422 i \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} E\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)+10115 (1-i)^{3/2} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} \Pi \left(-\frac{1}{3}-\frac{i}{3};\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)-892 x}{560 \sqrt{2 x^4+2 x^2+1}}","-\frac{1}{14} \left(2 x^4+2 x^2+1\right)^{3/2} x-\frac{2211 \sqrt{2 x^4+2 x^2+1} x}{140 \sqrt{2} \left(\sqrt{2} x^2+1\right)}-\frac{213}{140} \sqrt{2 x^4+2 x^2+1} x+\frac{17}{16} \sqrt{51} \tanh ^{-1}\left(\frac{\sqrt{\frac{17}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right)-\frac{3 \left(514+2717 \sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{140\ 2^{3/4} \left(2+3 \sqrt{2}\right) \sqrt{2 x^4+2 x^2+1}}+\frac{2211 \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} E\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{140\ 2^{3/4} \sqrt{2 x^4+2 x^2+1}}-\frac{289 \left(3-\sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} \Pi \left(\frac{1}{24} \left(12+11 \sqrt{2}\right);2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{16\ 2^{3/4} \left(2+3 \sqrt{2}\right) \sqrt{2 x^4+2 x^2+1}}-\frac{27}{70} \sqrt{2 x^4+2 x^2+1} x^3",1,"(-892*x - 2080*x^3 - 2456*x^5 - 752*x^7 - 160*x^9 + (4422*I)*Sqrt[1 - I]*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticE[I*ArcSinh[Sqrt[1 - I]*x], I] - (9669 - 5247*I)*Sqrt[1 - I]*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticF[I*ArcSinh[Sqrt[1 - I]*x], I] + 10115*(1 - I)^(3/2)*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticPi[-1/3 - I/3, I*ArcSinh[Sqrt[1 - I]*x], I])/(560*Sqrt[1 + 2*x^2 + 2*x^4])","C",1
328,1,209,428,0.1494561,"\int \frac{\left(1+2 x^2+2 x^4\right)^{3/2}}{3-2 x^2} \, dx","Integrate[(1 + 2*x^2 + 2*x^4)^(3/2)/(3 - 2*x^2),x]","\frac{-48 x^7-264 x^5-240 x^3-(1371-753 i) \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} F\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)+618 i \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} E\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)+1445 (1-i)^{3/2} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} \Pi \left(-\frac{1}{3}-\frac{i}{3};\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)-108 x}{120 \sqrt{2 x^4+2 x^2+1}}","-\frac{1}{10} \left(2 x^2+9\right) \sqrt{2 x^4+2 x^2+1} x-\frac{103 \sqrt{2 x^4+2 x^2+1} x}{10 \sqrt{2} \left(\sqrt{2} x^2+1\right)}+\frac{17}{8} \sqrt{\frac{17}{3}} \tanh ^{-1}\left(\frac{\sqrt{\frac{17}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right)-\frac{\left(66+383 \sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{10\ 2^{3/4} \left(2+3 \sqrt{2}\right) \sqrt{2 x^4+2 x^2+1}}+\frac{103 \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} E\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{10\ 2^{3/4} \sqrt{2 x^4+2 x^2+1}}-\frac{289 \left(3-\sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} \Pi \left(\frac{1}{24} \left(12+11 \sqrt{2}\right);2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{24\ 2^{3/4} \left(2+3 \sqrt{2}\right) \sqrt{2 x^4+2 x^2+1}}",1,"(-108*x - 240*x^3 - 264*x^5 - 48*x^7 + (618*I)*Sqrt[1 - I]*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticE[I*ArcSinh[Sqrt[1 - I]*x], I] - (1371 - 753*I)*Sqrt[1 - I]*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticF[I*ArcSinh[Sqrt[1 - I]*x], I] + 1445*(1 - I)^(3/2)*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticPi[-1/3 - I/3, I*ArcSinh[Sqrt[1 - I]*x], I])/(120*Sqrt[1 + 2*x^2 + 2*x^4])","C",1
329,1,213,722,0.2057416,"\int \frac{\left(1+2 x^2+2 x^4\right)^{3/2}}{x^2 \left(3-2 x^2\right)} \, dx","Integrate[(1 + 2*x^2 + 2*x^4)^(3/2)/(x^2*(3 - 2*x^2)),x]","\frac{-24 x^6-48 x^4-36 x^2-(255-165 i) \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} x F\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)+90 i \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} x E\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)+289 (1-i)^{3/2} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} x \Pi \left(-\frac{1}{3}-\frac{i}{3};\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)-12}{36 x \sqrt{2 x^4+2 x^2+1}}","\frac{\sqrt{2} \sqrt{2 x^4+2 x^2+1} x}{3 \left(\sqrt{2} x^2+1\right)}-\frac{17 \sqrt{2 x^4+2 x^2+1} x}{3 \sqrt{2} \left(\sqrt{2} x^2+1\right)}-\frac{\left(x^2+1\right) \sqrt{2 x^4+2 x^2+1}}{3 x}+\frac{17}{12} \sqrt{\frac{17}{3}} \tanh ^{-1}\left(\frac{\sqrt{\frac{17}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right)-\frac{17 \left(5+\sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{12 \sqrt[4]{2} \sqrt{2 x^4+2 x^2+1}}+\frac{289 \left(3-\sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{84 \sqrt[4]{2} \sqrt{2 x^4+2 x^2+1}}+\frac{\left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{3\ 2^{3/4} \sqrt{2 x^4+2 x^2+1}}-\frac{\sqrt[4]{2} \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} E\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{3 \sqrt{2 x^4+2 x^2+1}}+\frac{17 \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} E\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{3\ 2^{3/4} \sqrt{2 x^4+2 x^2+1}}-\frac{289 \left(11-6 \sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} \Pi \left(\frac{1}{24} \left(12+11 \sqrt{2}\right);2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{504 \sqrt[4]{2} \sqrt{2 x^4+2 x^2+1}}",1,"(-12 - 36*x^2 - 48*x^4 - 24*x^6 + (90*I)*Sqrt[1 - I]*x*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticE[I*ArcSinh[Sqrt[1 - I]*x], I] - (255 - 165*I)*Sqrt[1 - I]*x*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticF[I*ArcSinh[Sqrt[1 - I]*x], I] + 289*(1 - I)^(3/2)*x*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticPi[-1/3 - I/3, I*ArcSinh[Sqrt[1 - I]*x], I])/(36*x*Sqrt[1 + 2*x^2 + 2*x^4])","C",1
330,1,219,625,0.2179016,"\int \frac{\left(1+2 x^2+2 x^4\right)^{3/2}}{x^4 \left(3-2 x^2\right)} \, dx","Integrate[(1 + 2*x^2 + 2*x^4)^(3/2)/(x^4*(3 - 2*x^2)),x]","\frac{-120 x^6-132 x^4-72 x^2-(195-201 i) \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} x^3 F\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)-6 i \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} x^3 E\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)+289 (1-i)^{3/2} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} x^3 \Pi \left(-\frac{1}{3}-\frac{i}{3};\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)-6}{54 x^3 \sqrt{2 x^4+2 x^2+1}}","\frac{\sqrt{2} \sqrt{2 x^4+2 x^2+1} x}{9 \left(\sqrt{2} x^2+1\right)}-\frac{2 \sqrt{2 x^4+2 x^2+1}}{x}+\frac{17}{18} \sqrt{\frac{17}{3}} \tanh ^{-1}\left(\frac{\sqrt{\frac{17}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right)+\frac{\sqrt[4]{2} \left(9+5 \sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{9 \sqrt{2 x^4+2 x^2+1}}-\frac{17 \left(5+\sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{18 \sqrt[4]{2} \sqrt{2 x^4+2 x^2+1}}+\frac{289 \left(3-\sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{126 \sqrt[4]{2} \sqrt{2 x^4+2 x^2+1}}-\frac{\sqrt[4]{2} \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} E\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{9 \sqrt{2 x^4+2 x^2+1}}-\frac{289 \left(11-6 \sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} \Pi \left(\frac{1}{24} \left(12+11 \sqrt{2}\right);2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{756 \sqrt[4]{2} \sqrt{2 x^4+2 x^2+1}}-\frac{\left(1-8 x^2\right) \sqrt{2 x^4+2 x^2+1}}{9 x^3}",1,"(-6 - 72*x^2 - 132*x^4 - 120*x^6 - (6*I)*Sqrt[1 - I]*x^3*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticE[I*ArcSinh[Sqrt[1 - I]*x], I] - (195 - 201*I)*Sqrt[1 - I]*x^3*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticF[I*ArcSinh[Sqrt[1 - I]*x], I] + 289*(1 - I)^(3/2)*x^3*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticPi[-1/3 - I/3, I*ArcSinh[Sqrt[1 - I]*x], I])/(54*x^3*Sqrt[1 + 2*x^2 + 2*x^4])","C",1
331,1,224,553,0.2445696,"\int \frac{\left(1+2 x^2+2 x^4\right)^{3/2}}{x^6 \left(3-2 x^2\right)} \, dx","Integrate[(1 + 2*x^2 + 2*x^4)^(3/2)/(x^6*(3 - 2*x^2)),x]","-\frac{1572 x^8+1848 x^6+1116 x^4+192 x^2+(543-1329 i) \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} x^5 F\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)+786 i \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} x^5 E\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)-1445 (1-i)^{3/2} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} x^5 \Pi \left(-\frac{1}{3}-\frac{i}{3};\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)+27}{405 x^5 \sqrt{2 x^4+2 x^2+1}}","\frac{262 \sqrt{2} \sqrt{2 x^4+2 x^2+1} x}{135 \left(\sqrt{2} x^2+1\right)}-\frac{262 \sqrt{2 x^4+2 x^2+1}}{135 x}+\frac{17}{27} \sqrt{\frac{17}{3}} \tanh ^{-1}\left(\frac{\sqrt{\frac{17}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right)+\frac{2^{3/4} \left(37+23 \sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{135 \sqrt{2 x^4+2 x^2+1}}+\frac{85\ 2^{3/4} \left(3-\sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{189 \sqrt{2 x^4+2 x^2+1}}-\frac{262 \sqrt[4]{2} \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} E\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{135 \sqrt{2 x^4+2 x^2+1}}-\frac{289 \left(11-6 \sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} \Pi \left(\frac{1}{24} \left(12+11 \sqrt{2}\right);2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{1134 \sqrt[4]{2} \sqrt{2 x^4+2 x^2+1}}-\frac{\left(40 x^2+3\right) \sqrt{2 x^4+2 x^2+1}}{45 x^5}+\frac{74 \sqrt{2 x^4+2 x^2+1}}{135 x^3}",1,"-1/405*(27 + 192*x^2 + 1116*x^4 + 1848*x^6 + 1572*x^8 + (786*I)*Sqrt[1 - I]*x^5*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticE[I*ArcSinh[Sqrt[1 - I]*x], I] + (543 - 1329*I)*Sqrt[1 - I]*x^5*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticF[I*ArcSinh[Sqrt[1 - I]*x], I] - 1445*(1 - I)^(3/2)*x^5*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticPi[-1/3 - I/3, I*ArcSinh[Sqrt[1 - I]*x], I])/(x^5*Sqrt[1 + 2*x^2 + 2*x^4])","C",1
332,1,171,173,0.3669623,"\int \frac{x^5}{\left(d+e x^2\right) \sqrt{a+b x^2+c x^4}} \, dx","Integrate[x^5/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]),x]","\frac{2 \sqrt{c} \left(\frac{c d^2 \tanh ^{-1}\left(\frac{-2 a e+b d-b e x^2+2 c d x^2}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right)}{\sqrt{a e^2-b d e+c d^2}}+e \sqrt{a+b x^2+c x^4}\right)-(b e+2 c d) \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} e^2}","-\frac{(b e+2 c d) \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} e^2}+\frac{d^2 \tanh ^{-1}\left(\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right)}{2 e^2 \sqrt{a e^2-b d e+c d^2}}+\frac{\sqrt{a+b x^2+c x^4}}{2 c e}",1,"(-((2*c*d + b*e)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])]) + 2*Sqrt[c]*(e*Sqrt[a + b*x^2 + c*x^4] + (c*d^2*ArcTanh[(b*d - 2*a*e + 2*c*d*x^2 - b*e*x^2)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x^2 + c*x^4])])/Sqrt[c*d^2 - b*d*e + a*e^2]))/(4*c^(3/2)*e^2)","A",1
333,1,133,137,0.1182407,"\int \frac{x^3}{\left(d+e x^2\right) \sqrt{a+b x^2+c x^4}} \, dx","Integrate[x^3/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]),x]","\frac{\frac{d \tanh ^{-1}\left(\frac{2 a e-b d+b e x^2-2 c d x^2}{2 \sqrt{a+b x^2+c x^4} \sqrt{e (a e-b d)+c d^2}}\right)}{\sqrt{e (a e-b d)+c d^2}}+\frac{\tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{\sqrt{c}}}{2 e}","\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} e}-\frac{d \tanh ^{-1}\left(\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right)}{2 e \sqrt{a e^2-b d e+c d^2}}",1,"(ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])]/Sqrt[c] + (d*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x^2 + b*e*x^2)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + b*x^2 + c*x^4])])/Sqrt[c*d^2 + e*(-(b*d) + a*e)])/(2*e)","A",1
334,1,87,86,0.0184882,"\int \frac{x}{\left(d+e x^2\right) \sqrt{a+b x^2+c x^4}} \, dx","Integrate[x/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]),x]","-\frac{\tanh ^{-1}\left(\frac{2 a e-b d+b e x^2-2 c d x^2}{2 \sqrt{a+b x^2+c x^4} \sqrt{e (a e-b d)+c d^2}}\right)}{2 \sqrt{e (a e-b d)+c d^2}}","\frac{\tanh ^{-1}\left(\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right)}{2 \sqrt{a e^2-b d e+c d^2}}",1,"-1/2*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x^2 + b*e*x^2)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + b*x^2 + c*x^4])]/Sqrt[c*d^2 + e*(-(b*d) + a*e)]","A",1
335,1,134,138,0.1398004,"\int \frac{1}{x \left(d+e x^2\right) \sqrt{a+b x^2+c x^4}} \, dx","Integrate[1/(x*(d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]),x]","-\frac{\frac{e \tanh ^{-1}\left(\frac{-2 a e+b \left(d-e x^2\right)+2 c d x^2}{2 \sqrt{a+b x^2+c x^4} \sqrt{e (a e-b d)+c d^2}}\right)}{\sqrt{e (a e-b d)+c d^2}}+\frac{\tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{\sqrt{a}}}{2 d}","-\frac{e \tanh ^{-1}\left(\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right)}{2 d \sqrt{a e^2-b d e+c d^2}}-\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} d}",1,"-1/2*(ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])]/Sqrt[a] + (e*ArcTanh[(-2*a*e + 2*c*d*x^2 + b*(d - e*x^2))/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + b*x^2 + c*x^4])])/Sqrt[c*d^2 + e*(-(b*d) + a*e)])/d","A",1
336,1,175,218,0.3671819,"\int \frac{1}{x^3 \left(d+e x^2\right) \sqrt{a+b x^2+c x^4}} \, dx","Integrate[1/(x^3*(d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]),x]","\frac{2 \sqrt{a} \left(\frac{a e^2 \tanh ^{-1}\left(\frac{-2 a e+b d-b e x^2+2 c d x^2}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right)}{\sqrt{a e^2-b d e+c d^2}}-\frac{d \sqrt{a+b x^2+c x^4}}{x^2}\right)+(2 a e+b d) \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} d^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} d}+\frac{e^2 \tanh ^{-1}\left(\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right)}{2 d^2 \sqrt{a e^2-b d e+c d^2}}+\frac{e \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{2 \sqrt{a} d^2}-\frac{\sqrt{a+b x^2+c x^4}}{2 a d x^2}",1,"((b*d + 2*a*e)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])] + 2*Sqrt[a]*(-((d*Sqrt[a + b*x^2 + c*x^4])/x^2) + (a*e^2*ArcTanh[(b*d - 2*a*e + 2*c*d*x^2 - b*e*x^2)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x^2 + c*x^4])])/Sqrt[c*d^2 - b*d*e + a*e^2]))/(4*a^(3/2)*d^2)","A",1
337,1,127,418,0.2069399,"\int \frac{x^4}{\left(3+2 x^2\right) \sqrt{1+2 x^2+2 x^4}} \, dx","Integrate[x^4/((3 + 2*x^2)*Sqrt[1 + 2*x^2 + 2*x^4]),x]","-\frac{\sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} \left(-(1+4 i) F\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)+(1+i) E\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)+3 i \Pi \left(\frac{1}{3}+\frac{i}{3};\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)\right)}{4 \sqrt{1-i} \sqrt{2 x^4+2 x^2+1}}","\frac{\sqrt{2 x^4+2 x^2+1} x}{2 \sqrt{2} \left(\sqrt{2} x^2+1\right)}-\frac{3 \sqrt{\frac{3}{10}} \left(3-\sqrt{2}\right) \tan ^{-1}\left(\frac{\sqrt{\frac{5}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right)}{4 \left(2-3 \sqrt{2}\right)}+\frac{\left(1-3 \sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{2\ 2^{3/4} \left(2-3 \sqrt{2}\right) \sqrt{2 x^4+2 x^2+1}}-\frac{\left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} E\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{2\ 2^{3/4} \sqrt{2 x^4+2 x^2+1}}+\frac{3 \left(3+\sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} \Pi \left(\frac{1}{24} \left(12-11 \sqrt{2}\right);2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{8\ 2^{3/4} \left(2-3 \sqrt{2}\right) \sqrt{2 x^4+2 x^2+1}}",1,"-1/4*(Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*((1 + I)*EllipticE[I*ArcSinh[Sqrt[1 - I]*x], I] - (1 + 4*I)*EllipticF[I*ArcSinh[Sqrt[1 - I]*x], I] + (3*I)*EllipticPi[1/3 + I/3, I*ArcSinh[Sqrt[1 - I]*x], I]))/(Sqrt[1 - I]*Sqrt[1 + 2*x^2 + 2*x^4])","C",1
338,1,99,247,0.1239392,"\int \frac{x^2}{\left(3+2 x^2\right) \sqrt{1+2 x^2+2 x^4}} \, dx","Integrate[x^2/((3 + 2*x^2)*Sqrt[1 + 2*x^2 + 2*x^4]),x]","\frac{(1-i)^{3/2} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} \left(F\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)-\Pi \left(\frac{1}{3}+\frac{i}{3};\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)\right)}{4 \sqrt{2 x^4+2 x^2+1}}","-\frac{1}{4} \sqrt{\frac{3}{5}} \tan ^{-1}\left(\frac{\sqrt{\frac{5}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right)-\frac{\left(3+\sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{14\ 2^{3/4} \sqrt{2 x^4+2 x^2+1}}+\frac{\left(3+\sqrt{2}\right)^2 \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} \Pi \left(\frac{1}{24} \left(12-11 \sqrt{2}\right);2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{56 \sqrt[4]{2} \sqrt{2 x^4+2 x^2+1}}",1,"((1 - I)^(3/2)*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*(EllipticF[I*ArcSinh[Sqrt[1 - I]*x], I] - EllipticPi[1/3 + I/3, I*ArcSinh[Sqrt[1 - I]*x], I]))/(4*Sqrt[1 + 2*x^2 + 2*x^4])","C",1
339,1,80,245,0.0567511,"\int \frac{1}{\left(3+2 x^2\right) \sqrt{1+2 x^2+2 x^4}} \, dx","Integrate[1/((3 + 2*x^2)*Sqrt[1 + 2*x^2 + 2*x^4]),x]","-\frac{i \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} \Pi \left(\frac{1}{3}+\frac{i}{3};\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)}{3 \sqrt{1-i} \sqrt{2 x^4+2 x^2+1}}","\frac{\tan ^{-1}\left(\frac{\sqrt{\frac{5}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right)}{2 \sqrt{15}}+\frac{\left(3+\sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{14 \sqrt[4]{2} \sqrt{2 x^4+2 x^2+1}}-\frac{\left(3+\sqrt{2}\right)^2 \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} \Pi \left(\frac{1}{24} \left(12-11 \sqrt{2}\right);2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{84 \sqrt[4]{2} \sqrt{2 x^4+2 x^2+1}}",1,"((-1/3*I)*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticPi[1/3 + I/3, I*ArcSinh[Sqrt[1 - I]*x], I])/(Sqrt[1 - I]*Sqrt[1 + 2*x^2 + 2*x^4])","C",1
340,1,147,399,0.2183368,"\int \frac{1}{x^2 \left(3+2 x^2\right) \sqrt{1+2 x^2+2 x^4}} \, dx","Integrate[1/(x^2*(3 + 2*x^2)*Sqrt[1 + 2*x^2 + 2*x^4]),x]","-\frac{i \left(\sqrt{1-i} x \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} \left(-3 F\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)+3 E\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)-(1+i) \Pi \left(\frac{1}{3}+\frac{i}{3};\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)\right)-3 i \left(2 x^4+2 x^2+1\right)\right)}{9 x \sqrt{2 x^4+2 x^2+1}}","\frac{\sqrt{2} \sqrt{2 x^4+2 x^2+1} x}{3 \left(\sqrt{2} x^2+1\right)}-\frac{\sqrt{2 x^4+2 x^2+1}}{3 x}-\frac{\tan ^{-1}\left(\frac{\sqrt{\frac{5}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right)}{3 \sqrt{15}}+\frac{\left(5-3 \sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{21\ 2^{3/4} \sqrt{2 x^4+2 x^2+1}}-\frac{\sqrt[4]{2} \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} E\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{3 \sqrt{2 x^4+2 x^2+1}}+\frac{\left(3+\sqrt{2}\right)^2 \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} \Pi \left(\frac{1}{24} \left(12-11 \sqrt{2}\right);2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{126 \sqrt[4]{2} \sqrt{2 x^4+2 x^2+1}}",1,"((-1/9*I)*((-3*I)*(1 + 2*x^2 + 2*x^4) + Sqrt[1 - I]*x*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*(3*EllipticE[I*ArcSinh[Sqrt[1 - I]*x], I] - 3*EllipticF[I*ArcSinh[Sqrt[1 - I]*x], I] - (1 + I)*EllipticPi[1/3 + I/3, I*ArcSinh[Sqrt[1 - I]*x], I])))/(x*Sqrt[1 + 2*x^2 + 2*x^4])","C",1
341,1,219,422,0.1841016,"\int \frac{1}{x^4 \left(3+2 x^2\right) \sqrt{1+2 x^2+2 x^4}} \, dx","Integrate[1/(x^4*(3 + 2*x^2)*Sqrt[1 + 2*x^2 + 2*x^4]),x]","\frac{36 x^6+30 x^4+12 x^2-(3+15 i) \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} x^3 F\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)+18 i \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} x^3 E\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)+2 (1-i)^{3/2} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} x^3 \Pi \left(\frac{1}{3}+\frac{i}{3};\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)-3}{27 x^3 \sqrt{2 x^4+2 x^2+1}}","-\frac{2 \sqrt{2} \sqrt{2 x^4+2 x^2+1} x}{3 \left(\sqrt{2} x^2+1\right)}+\frac{2 \sqrt{2 x^4+2 x^2+1}}{3 x}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{\frac{5}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right)}{9 \sqrt{15}}-\frac{\left(1+19 \sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{63 \sqrt[4]{2} \sqrt{2 x^4+2 x^2+1}}+\frac{2 \sqrt[4]{2} \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} E\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{3 \sqrt{2 x^4+2 x^2+1}}-\frac{\left(3+\sqrt{2}\right)^2 \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} \Pi \left(\frac{1}{24} \left(12-11 \sqrt{2}\right);2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{189 \sqrt[4]{2} \sqrt{2 x^4+2 x^2+1}}-\frac{\sqrt{2 x^4+2 x^2+1}}{9 x^3}",1,"(-3 + 12*x^2 + 30*x^4 + 36*x^6 + (18*I)*Sqrt[1 - I]*x^3*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticE[I*ArcSinh[Sqrt[1 - I]*x], I] - (3 + 15*I)*Sqrt[1 - I]*x^3*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticF[I*ArcSinh[Sqrt[1 - I]*x], I] + 2*(1 - I)^(3/2)*x^3*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticPi[1/3 + I/3, I*ArcSinh[Sqrt[1 - I]*x], I])/(27*x^3*Sqrt[1 + 2*x^2 + 2*x^4])","C",1
342,1,271,236,0.821362,"\int \frac{x^7}{\left(d+e x^2\right) \left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[x^7/((d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2)),x]","\frac{1}{2} \left(\frac{2 \left(a^2 \left(b e+2 c \left(d-e x^2\right)\right)+a b \left(-b d+b e x^2+3 c d x^2\right)+b^3 (-d) x^2\right)}{c \left(4 a c-b^2\right) \sqrt{a+b x^2+c x^4} \left(e (a e-b d)+c d^2\right)}+\frac{\log \left(2 \sqrt{c} \sqrt{a+b x^2+c x^4}+b+2 c x^2\right)}{c^{3/2} e}+\frac{d^3 \log \left(2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}+2 a e-b d+b e x^2-2 c d x^2\right)}{e \left(e (a e-b d)+c d^2\right)^{3/2}}-\frac{d^3 \log \left(d+e x^2\right)}{e \left(e (a e-b d)+c d^2\right)^{3/2}}\right)","\frac{x^2 \left(2 a^2 c e-a b^2 e-3 a b c d+b^3 d\right)+a \left(-a b e-2 a c d+b^2 d\right)}{c \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4} \left(a e^2-b d e+c d^2\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} e}-\frac{d^3 \tanh ^{-1}\left(\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right)}{2 e \left(a e^2-b d e+c d^2\right)^{3/2}}",1,"((2*(-(b^3*d*x^2) + a*b*(-(b*d) + 3*c*d*x^2 + b*e*x^2) + a^2*(b*e + 2*c*(d - e*x^2))))/(c*(-b^2 + 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[a + b*x^2 + c*x^4]) - (d^3*Log[d + e*x^2])/(e*(c*d^2 + e*(-(b*d) + a*e))^(3/2)) + Log[b + 2*c*x^2 + 2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]]/(c^(3/2)*e) + (d^3*Log[-(b*d) + 2*a*e - 2*c*d*x^2 + b*e*x^2 + 2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x^2 + c*x^4]])/(e*(c*d^2 + e*(-(b*d) + a*e))^(3/2)))/2","A",1
343,1,204,167,0.6163346,"\int \frac{x^5}{\left(d+e x^2\right) \left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[x^5/((d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2)),x]","\frac{1}{2} \left(\frac{2 \left(-2 a^2 e+a b \left(d-e x^2\right)-2 a c d x^2+b^2 d x^2\right)}{\left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4} \left(e (b d-a e)-c d^2\right)}-\frac{d^2 \log \left(2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}+2 a e-b d+b e x^2-2 c d x^2\right)}{\left(e (a e-b d)+c d^2\right)^{3/2}}+\frac{d^2 \log \left(d+e x^2\right)}{\left(e (a e-b d)+c d^2\right)^{3/2}}\right)","\frac{d^2 \tanh ^{-1}\left(\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right)}{2 \left(a e^2-b d e+c d^2\right)^{3/2}}-\frac{x^2 \left(-a b e-2 a c d+b^2 d\right)+a (b d-2 a e)}{\left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4} \left(a e^2-b d e+c d^2\right)}",1,"((2*(-2*a^2*e + b^2*d*x^2 - 2*a*c*d*x^2 + a*b*(d - e*x^2)))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*Sqrt[a + b*x^2 + c*x^4]) + (d^2*Log[d + e*x^2])/(c*d^2 + e*(-(b*d) + a*e))^(3/2) - (d^2*Log[-(b*d) + 2*a*e - 2*c*d*x^2 + b*e*x^2 + 2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x^2 + c*x^4]])/(c*d^2 + e*(-(b*d) + a*e))^(3/2))/2","A",1
344,1,162,159,0.171952,"\int \frac{x^3}{\left(d+e x^2\right) \left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[x^3/((d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2)),x]","\frac{a \left(b e-2 c d+2 c e x^2\right)-b c d x^2}{\left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4} \left(e (b d-a e)-c d^2\right)}+\frac{d e \tanh ^{-1}\left(\frac{2 a e-b d+b e x^2-2 c d x^2}{2 \sqrt{a+b x^2+c x^4} \sqrt{e (a e-b d)+c d^2}}\right)}{2 \left(e (a e-b d)+c d^2\right)^{3/2}}","\frac{c x^2 (b d-2 a e)+a (2 c d-b e)}{\left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4} \left(a e^2-b d e+c d^2\right)}-\frac{d e \tanh ^{-1}\left(\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right)}{2 \left(a e^2-b d e+c d^2\right)^{3/2}}",1,"(-(b*c*d*x^2) + a*(-2*c*d + b*e + 2*c*e*x^2))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*Sqrt[a + b*x^2 + c*x^4]) + (d*e*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x^2 + b*e*x^2)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + b*x^2 + c*x^4])])/(2*(c*d^2 + e*(-(b*d) + a*e))^(3/2))","A",1
345,1,167,166,0.1492528,"\int \frac{x}{\left(d+e x^2\right) \left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[x/((d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2)),x]","-\frac{2 a c e+b^2 (-e)+c x^2 (2 c d-b e)+b c d}{\left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4} \left(e (a e-b d)+c d^2\right)}-\frac{e^2 \tanh ^{-1}\left(\frac{2 a e-b d+b e x^2-2 c d x^2}{2 \sqrt{a+b x^2+c x^4} \sqrt{e (a e-b d)+c d^2}}\right)}{2 \left(e (a e-b d)+c d^2\right)^{3/2}}","\frac{e^2 \tanh ^{-1}\left(\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right)}{2 \left(a e^2-b d e+c d^2\right)^{3/2}}-\frac{2 a c e+b^2 (-e)+c x^2 (2 c d-b e)+b c d}{\left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4} \left(a e^2-b d e+c d^2\right)}",1,"-((b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x^2)/((b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[a + b*x^2 + c*x^4])) - (e^2*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x^2 + b*e*x^2)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + b*x^2 + c*x^4])])/(2*(c*d^2 + e*(-(b*d) + a*e))^(3/2))","A",1
346,1,236,266,0.7313193,"\int \frac{1}{x \left(d+e x^2\right) \left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[1/(x*(d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2)),x]","-\frac{\frac{\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 d \left(b c \left(3 a e+c d x^2\right)-2 a c^2 \left(d-e x^2\right)+b^3 (-e)+b^2 c \left(d-e x^2\right)\right)}{a \left(4 a c-b^2\right) \sqrt{a+b x^2+c x^4} \left(e (a e-b d)+c d^2\right)}+\frac{e^3 \tanh ^{-1}\left(\frac{-2 a e+b \left(d-e x^2\right)+2 c d x^2}{2 \sqrt{a+b x^2+c x^4} \sqrt{e (a e-b d)+c d^2}}\right)}{\left(e (a e-b d)+c d^2\right)^{3/2}}}{2 d}","-\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} d}+\frac{e \left(2 a c e+b^2 (-e)+c x^2 (2 c d-b e)+b c d\right)}{d \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4} \left(a e^2-b d e+c d^2\right)}+\frac{-2 a c+b^2+b c x^2}{a d \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}-\frac{e^3 \tanh ^{-1}\left(\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right)}{2 d \left(a e^2-b d e+c d^2\right)^{3/2}}",1,"-1/2*((2*d*(-(b^3*e) + b*c*(3*a*e + c*d*x^2) + b^2*c*(d - e*x^2) - 2*a*c^2*(d - e*x^2)))/(a*(-b^2 + 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))*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])]/a^(3/2) + (e^3*ArcTanh[(-2*a*e + 2*c*d*x^2 + b*(d - e*x^2))/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + b*x^2 + c*x^4])])/(c*d^2 + e*(-(b*d) + a*e))^(3/2))/d","A",1
347,1,350,419,1.4766159,"\int \frac{1}{x^3 \left(d+e x^2\right) \left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[1/(x^3*(d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2)),x]","\frac{\frac{(2 a e+3 b d) \tanh ^{-1}\left(\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right)}{a^{5/2}}+\frac{2 d \left(-4 a^3 c e^2+a^2 \left(b^2 e^2+4 b c e \left(d-e x^2\right)-4 c^2 \left(d^2+d e x^2+e^2 x^4\right)\right)+a \left(b^3 e \left(e x^2-d\right)+b^2 c \left(d^2+12 d e x^2+e^2 x^4\right)-10 b c^2 d x^2 \left(d-e x^2\right)-8 c^3 d^2 x^4\right)+3 b^2 d x^2 \left(b+c x^2\right) (c d-b e)\right)}{a^2 x^2 \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4} \left(e (b d-a e)-c d^2\right)}+\frac{2 e^4 \tanh ^{-1}\left(\frac{-2 a e+b \left(d-e x^2\right)+2 c d x^2}{2 \sqrt{a+b x^2+c x^4} \sqrt{e (a e-b d)+c d^2}}\right)}{\left(e (a e-b d)+c d^2\right)^{3/2}}}{4 d^2}","\frac{e \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} d^2}+\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} d}-\frac{\left(3 b^2-8 a c\right) \sqrt{a+b x^2+c x^4}}{2 a^2 d x^2 \left(b^2-4 a c\right)}-\frac{e^2 \left(2 a c e+b^2 (-e)+c x^2 (2 c d-b e)+b c d\right)}{d^2 \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4} \left(a e^2-b d e+c d^2\right)}-\frac{e \left(-2 a c+b^2+b c x^2\right)}{a d^2 \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}+\frac{-2 a c+b^2+b c x^2}{a d x^2 \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}+\frac{e^4 \tanh ^{-1}\left(\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right)}{2 d^2 \left(a e^2-b d e+c d^2\right)^{3/2}}",1,"((2*d*(-4*a^3*c*e^2 + 3*b^2*d*(c*d - b*e)*x^2*(b + c*x^2) + a^2*(b^2*e^2 + 4*b*c*e*(d - e*x^2) - 4*c^2*(d^2 + d*e*x^2 + e^2*x^4)) + a*(-8*c^3*d^2*x^4 - 10*b*c^2*d*x^2*(d - e*x^2) + b^3*e*(-d + e*x^2) + b^2*c*(d^2 + 12*d*e*x^2 + e^2*x^4))))/(a^2*(b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*x^2*Sqrt[a + b*x^2 + c*x^4]) + ((3*b*d + 2*a*e)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/a^(5/2) + (2*e^4*ArcTanh[(-2*a*e + 2*c*d*x^2 + b*(d - e*x^2))/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + b*x^2 + c*x^4])])/(c*d^2 + e*(-(b*d) + a*e))^(3/2))/(4*d^2)","A",1
348,1,199,449,0.2771035,"\int \frac{x^8}{\left(3+2 x^2\right) \left(1+2 x^2+2 x^4\right)^{3/2}} \, dx","Integrate[x^8/((3 + 2*x^2)*(1 + 2*x^2 + 2*x^4)^(3/2)),x]","\frac{12 x^3-(29-33 i) \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} F\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)-4 i \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} E\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)+27 (1-i)^{3/2} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} \Pi \left(\frac{1}{3}+\frac{i}{3};\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)+4 x}{80 \sqrt{2 x^4+2 x^2+1}}","\frac{\sqrt{2 x^4+2 x^2+1} x}{10 \sqrt{2} \left(\sqrt{2} x^2+1\right)}+\frac{1}{20} \sqrt{2 x^4+2 x^2+1} x+\frac{27}{80} \sqrt{\frac{3}{5}} \tan ^{-1}\left(\frac{\sqrt{\frac{5}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right)+\frac{\left(7 \sqrt{2}-2\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{8\ 2^{3/4} \left(3 \sqrt{2}-2\right) \sqrt{2 x^4+2 x^2+1}}-\frac{\left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} E\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{10\ 2^{3/4} \sqrt{2 x^4+2 x^2+1}}+\frac{27 \left(3+\sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} \Pi \left(\frac{1}{24} \left(12-11 \sqrt{2}\right);2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{80\ 2^{3/4} \left(2-3 \sqrt{2}\right) \sqrt{2 x^4+2 x^2+1}}+\frac{\left(1-2 x^2\right) x^3}{20 \sqrt{2 x^4+2 x^2+1}}",1,"(4*x + 12*x^3 - (4*I)*Sqrt[1 - I]*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticE[I*ArcSinh[Sqrt[1 - I]*x], I] - (29 - 33*I)*Sqrt[1 - I]*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticF[I*ArcSinh[Sqrt[1 - I]*x], I] + 27*(1 - I)^(3/2)*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticPi[1/3 + I/3, I*ArcSinh[Sqrt[1 - I]*x], I])/(80*Sqrt[1 + 2*x^2 + 2*x^4])","C",1
349,1,199,423,0.205341,"\int \frac{x^6}{\left(3+2 x^2\right) \left(1+2 x^2+2 x^4\right)^{3/2}} \, dx","Integrate[x^6/((3 + 2*x^2)*(1 + 2*x^2 + 2*x^4)^(3/2)),x]","\frac{-4 x^3+(8-6 i) \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} F\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)-2 i \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} E\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)-9 (1-i)^{3/2} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} \Pi \left(\frac{1}{3}+\frac{i}{3};\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)+2 x}{40 \sqrt{2 x^4+2 x^2+1}}","\frac{\sqrt{2 x^4+2 x^2+1} x}{10 \sqrt{2} \left(\sqrt{2} x^2+1\right)}+\frac{\left(1-2 x^2\right) x}{20 \sqrt{2 x^4+2 x^2+1}}-\frac{9}{40} \sqrt{\frac{3}{5}} \tan ^{-1}\left(\frac{\sqrt{\frac{5}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right)-\frac{\left(\sqrt[4]{2}+2^{3/4}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{8 \left(3 \sqrt{2}-2\right) \sqrt{2 x^4+2 x^2+1}}-\frac{\left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} E\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{10\ 2^{3/4} \sqrt{2 x^4+2 x^2+1}}-\frac{9 \left(3+\sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} \Pi \left(\frac{1}{24} \left(12-11 \sqrt{2}\right);2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{40\ 2^{3/4} \left(2-3 \sqrt{2}\right) \sqrt{2 x^4+2 x^2+1}}",1,"(2*x - 4*x^3 - (2*I)*Sqrt[1 - I]*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticE[I*ArcSinh[Sqrt[1 - I]*x], I] + (8 - 6*I)*Sqrt[1 - I]*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticF[I*ArcSinh[Sqrt[1 - I]*x], I] - 9*(1 - I)^(3/2)*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticPi[1/3 + I/3, I*ArcSinh[Sqrt[1 - I]*x], I])/(40*Sqrt[1 + 2*x^2 + 2*x^4])","C",1
350,1,199,422,0.2057906,"\int \frac{x^4}{\left(3+2 x^2\right) \left(1+2 x^2+2 x^4\right)^{3/2}} \, dx","Integrate[x^4/((3 + 2*x^2)*(1 + 2*x^2 + 2*x^4)^(3/2)),x]","-\frac{2 x^3+(1-2 i) \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} F\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)+i \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} E\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)-3 (1-i)^{3/2} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} \Pi \left(\frac{1}{3}+\frac{i}{3};\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)+4 x}{20 \sqrt{2 x^4+2 x^2+1}}","\frac{\sqrt{2 x^4+2 x^2+1} x}{10 \sqrt{2} \left(\sqrt{2} x^2+1\right)}-\frac{\left(x^2+2\right) x}{10 \sqrt{2 x^4+2 x^2+1}}+\frac{3}{20} \sqrt{\frac{3}{5}} \tan ^{-1}\left(\frac{\sqrt{\frac{5}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right)+\frac{\left(2+\sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{4\ 2^{3/4} \left(3 \sqrt{2}-2\right) \sqrt{2 x^4+2 x^2+1}}-\frac{\left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} E\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{10\ 2^{3/4} \sqrt{2 x^4+2 x^2+1}}+\frac{3 \left(3+\sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} \Pi \left(\frac{1}{24} \left(12-11 \sqrt{2}\right);2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{20\ 2^{3/4} \left(2-3 \sqrt{2}\right) \sqrt{2 x^4+2 x^2+1}}",1,"-1/20*(4*x + 2*x^3 + I*Sqrt[1 - I]*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticE[I*ArcSinh[Sqrt[1 - I]*x], I] + (1 - 2*I)*Sqrt[1 - I]*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticF[I*ArcSinh[Sqrt[1 - I]*x], I] - 3*(1 - I)^(3/2)*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticPi[1/3 + I/3, I*ArcSinh[Sqrt[1 - I]*x], I])/Sqrt[1 + 2*x^2 + 2*x^4]","C",1
351,1,199,423,0.1804537,"\int \frac{x^2}{\left(3+2 x^2\right) \left(1+2 x^2+2 x^4\right)^{3/2}} \, dx","Integrate[x^2/((3 + 2*x^2)*(1 + 2*x^2 + 2*x^4)^(3/2)),x]","\frac{8 x^3-(1+3 i) \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} F\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)+4 i \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} E\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)-2 (1-i)^{3/2} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} \Pi \left(\frac{1}{3}+\frac{i}{3};\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)+6 x}{20 \sqrt{2 x^4+2 x^2+1}}","-\frac{\sqrt{2} \sqrt{2 x^4+2 x^2+1} x}{5 \left(\sqrt{2} x^2+1\right)}+\frac{\left(4 x^2+3\right) x}{10 \sqrt{2 x^4+2 x^2+1}}-\frac{1}{10} \sqrt{\frac{3}{5}} \tan ^{-1}\left(\frac{\sqrt{\frac{5}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right)-\frac{\left(\sqrt[4]{2}+2^{3/4}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{4 \left(3 \sqrt{2}-2\right) \sqrt{2 x^4+2 x^2+1}}+\frac{\sqrt[4]{2} \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} E\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{5 \sqrt{2 x^4+2 x^2+1}}-\frac{\left(3+\sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} \Pi \left(\frac{1}{24} \left(12-11 \sqrt{2}\right);2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{10\ 2^{3/4} \left(2-3 \sqrt{2}\right) \sqrt{2 x^4+2 x^2+1}}",1,"(6*x + 8*x^3 + (4*I)*Sqrt[1 - I]*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticE[I*ArcSinh[Sqrt[1 - I]*x], I] - (1 + 3*I)*Sqrt[1 - I]*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticF[I*ArcSinh[Sqrt[1 - I]*x], I] - 2*(1 - I)^(3/2)*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticPi[1/3 + I/3, I*ArcSinh[Sqrt[1 - I]*x], I])/(20*Sqrt[1 + 2*x^2 + 2*x^4])","C",1
352,1,199,422,0.1372583,"\int \frac{1}{\left(3+2 x^2\right) \left(1+2 x^2+2 x^4\right)^{3/2}} \, dx","Integrate[1/((3 + 2*x^2)*(1 + 2*x^2 + 2*x^4)^(3/2)),x]","\frac{-18 x^3+(6+3 i) \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} F\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)-9 i \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} E\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)+2 (1-i)^{3/2} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} \Pi \left(\frac{1}{3}+\frac{i}{3};\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)-6 x}{30 \sqrt{2 x^4+2 x^2+1}}","\frac{3 \sqrt{2 x^4+2 x^2+1} x}{5 \sqrt{2} \left(\sqrt{2} x^2+1\right)}-\frac{\left(3 x^2+1\right) x}{5 \sqrt{2 x^4+2 x^2+1}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\frac{5}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right)}{5 \sqrt{15}}+\frac{\left(2+\sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{2\ 2^{3/4} \left(3 \sqrt{2}-2\right) \sqrt{2 x^4+2 x^2+1}}-\frac{3 \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} E\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{5\ 2^{3/4} \sqrt{2 x^4+2 x^2+1}}+\frac{\left(3+\sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} \Pi \left(\frac{1}{24} \left(12-11 \sqrt{2}\right);2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{15\ 2^{3/4} \left(2-3 \sqrt{2}\right) \sqrt{2 x^4+2 x^2+1}}",1,"(-6*x - 18*x^3 - (9*I)*Sqrt[1 - I]*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticE[I*ArcSinh[Sqrt[1 - I]*x], I] + (6 + 3*I)*Sqrt[1 - I]*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticF[I*ArcSinh[Sqrt[1 - I]*x], I] + 2*(1 - I)^(3/2)*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticPi[1/3 + I/3, I*ArcSinh[Sqrt[1 - I]*x], I])/(30*Sqrt[1 + 2*x^2 + 2*x^4])","C",1
353,1,211,468,0.2252006,"\int \frac{1}{x^2 \left(3+2 x^2\right) \left(1+2 x^2+2 x^4\right)^{3/2}} \, dx","Integrate[1/(x^2*(3 + 2*x^2)*(1 + 2*x^2 + 2*x^4)^(3/2)),x]","\frac{-(27-39 i) \sqrt{1-i} x \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} F\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)-12 i \sqrt{1-i} x \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} E\left(\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)-2 \left(12 x^4+39 x^2+2 (1-i)^{3/2} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} x \Pi \left(\frac{1}{3}+\frac{i}{3};\left.i \sinh ^{-1}\left(\sqrt{1-i} x\right)\right|i\right)+15\right)}{90 x \sqrt{2 x^4+2 x^2+1}}","\frac{2 \sqrt{2} \sqrt{2 x^4+2 x^2+1} x}{15 \left(\sqrt{2} x^2+1\right)}+\frac{2 \left(3 x^2+1\right) x}{15 \sqrt{2 x^4+2 x^2+1}}-\frac{x}{3 \sqrt{2 x^4+2 x^2+1}}-\frac{\sqrt{2 x^4+2 x^2+1}}{3 x}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{\frac{5}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right)}{15 \sqrt{15}}+\frac{\left(3 \sqrt{2}-7\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{3\ 2^{3/4} \left(3 \sqrt{2}-2\right) \sqrt{2 x^4+2 x^2+1}}-\frac{2 \sqrt[4]{2} \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} E\left(2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{15 \sqrt{2 x^4+2 x^2+1}}-\frac{\sqrt[4]{2} \left(3+\sqrt{2}\right) \left(\sqrt{2} x^2+1\right) \sqrt{\frac{2 x^4+2 x^2+1}{\left(\sqrt{2} x^2+1\right)^2}} \Pi \left(\frac{1}{24} \left(12-11 \sqrt{2}\right);2 \tan ^{-1}\left(\sqrt[4]{2} x\right)|\frac{1}{4} \left(2-\sqrt{2}\right)\right)}{45 \left(2-3 \sqrt{2}\right) \sqrt{2 x^4+2 x^2+1}}",1,"((-12*I)*Sqrt[1 - I]*x*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticE[I*ArcSinh[Sqrt[1 - I]*x], I] - (27 - 39*I)*Sqrt[1 - I]*x*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticF[I*ArcSinh[Sqrt[1 - I]*x], I] - 2*(15 + 39*x^2 + 12*x^4 + 2*(1 - I)^(3/2)*x*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticPi[1/3 + I/3, I*ArcSinh[Sqrt[1 - I]*x], I]))/(90*x*Sqrt[1 + 2*x^2 + 2*x^4])","C",1
354,1,943,406,10.8416643,"\int \frac{x^7 \sqrt{d+e x^2}}{a+b x^2+c x^4} \, dx","Integrate[(x^7*Sqrt[d + e*x^2])/(a + b*x^2 + c*x^4),x]","\frac{\frac{c \left(105 \left(-b^3+\sqrt{b^2-4 a c} b^2+3 a c b-a c \sqrt{b^2-4 a c}\right) \tanh ^{-1}\left(\sqrt{2} \sqrt{\frac{c \left(e x^2+d\right)}{2 c d-b e+\sqrt{b^2-4 a c} e}}\right) e^3+\sqrt{2} \sqrt{\frac{c \left(e x^2+d\right)}{2 c d-b e+\sqrt{b^2-4 a c} e}} \left(105 b^3 e^3-35 b^2 \left(3 \sqrt{b^2-4 a c} e+c \left(e x^2+d\right)\right) e^2+7 b c \left(\left(e x^2+d\right) \left(-5 c d+5 \sqrt{b^2-4 a c} e+3 c \left(e x^2+d\right)\right)-45 a e^2\right) e+c \left(35 a \left(3 \sqrt{b^2-4 a c} e+2 c \left(e x^2+d\right)\right) e^2+c \left(e x^2+d\right) \left(7 \sqrt{b^2-4 a c} e \left(5 d-3 \left(e x^2+d\right)\right)+c \left(-70 d^2+84 \left(e x^2+d\right) d-30 \left(e x^2+d\right)^2\right)\right)\right)\right)\right) \left(e x^2+d\right)^{9/2}}{210 \sqrt{2} \sqrt{b^2-4 a c} e^4 \left(2 c d+\left(\sqrt{b^2-4 a c}-b\right) e\right)^3 \left(-\frac{2 c d-b e}{e^2}-\frac{\sqrt{b^2-4 a c}}{e}\right) \left(\frac{c \left(e x^2+d\right)}{2 c d-b e+\sqrt{b^2-4 a c} e}\right)^{9/2}}+\frac{2 c d^3 \left(\frac{3 \left(b^3+\sqrt{b^2-4 a c} b^2-3 a c b-a c \sqrt{b^2-4 a c}\right) e^3 \tanh ^{-1}\left(\sqrt{2} \sqrt{\frac{c \left(e x^2+d\right)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}}\right) \left(e x^2+d\right)^3}{4 \sqrt{2} d^3 \left(2 c d-\left(b+\sqrt{b^2-4 a c}\right) e\right)^3 \left(\frac{c \left(e x^2+d\right)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)^{9/2}}+\frac{\left(2 c d-\left(b+\sqrt{b^2-4 a c}\right) e\right) \left(-105 b^3 e^3+35 b^2 \left(c \left(e x^2+d\right)-3 \sqrt{b^2-4 a c} e\right) e^2-7 b c \left(\left(e x^2+d\right) \left(-5 c d-5 \sqrt{b^2-4 a c} e+3 c \left(e x^2+d\right)\right)-45 a e^2\right) e+c \left(35 a \left(3 \sqrt{b^2-4 a c} e-2 c \left(e x^2+d\right)\right) e^2+c \left(e x^2+d\right) \left(7 \sqrt{b^2-4 a c} e \left(5 d-3 \left(e x^2+d\right)\right)+c \left(70 d^2-84 \left(e x^2+d\right) d+30 \left(e x^2+d\right)^2\right)\right)\right)\right)}{140 c^4 d^3 \left(e x^2+d\right)}\right) \left(e x^2+d\right)^{3/2}}{3 \sqrt{b^2-4 a c} e^4 \left(\frac{\sqrt{b^2-4 a c}}{e}-\frac{2 c d-b e}{e^2}\right)}}{e}","-\frac{\left(-\frac{-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^4 (-e)+b^3 c d}{\sqrt{b^2-4 a c}}+2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{\sqrt{2} c^{7/2} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}-\frac{\left(\frac{-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^4 (-e)+b^3 c d}{\sqrt{b^2-4 a c}}+2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2} c^{7/2} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{\left(b^2-a c\right) \sqrt{d+e x^2}}{c^3}-\frac{\left(d+e x^2\right)^{3/2} (b e+c d)}{3 c^2 e^2}+\frac{\left(d+e x^2\right)^{5/2}}{5 c e^2}",1,"((c*(d + e*x^2)^(9/2)*(Sqrt[2]*Sqrt[(c*(d + e*x^2))/(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)]*(105*b^3*e^3 - 35*b^2*e^2*(3*Sqrt[b^2 - 4*a*c]*e + c*(d + e*x^2)) + 7*b*c*e*(-45*a*e^2 + (d + e*x^2)*(-5*c*d + 5*Sqrt[b^2 - 4*a*c]*e + 3*c*(d + e*x^2))) + c*(35*a*e^2*(3*Sqrt[b^2 - 4*a*c]*e + 2*c*(d + e*x^2)) + c*(d + e*x^2)*(7*Sqrt[b^2 - 4*a*c]*e*(5*d - 3*(d + e*x^2)) + c*(-70*d^2 + 84*d*(d + e*x^2) - 30*(d + e*x^2)^2)))) + 105*(-b^3 + 3*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - a*c*Sqrt[b^2 - 4*a*c])*e^3*ArcTanh[Sqrt[2]*Sqrt[(c*(d + e*x^2))/(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)]]))/(210*Sqrt[2]*Sqrt[b^2 - 4*a*c]*e^4*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)^3*(-(Sqrt[b^2 - 4*a*c]/e) - (2*c*d - b*e)/e^2)*((c*(d + e*x^2))/(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e))^(9/2)) + (2*c*d^3*(d + e*x^2)^(3/2)*(((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*(-105*b^3*e^3 + 35*b^2*e^2*(-3*Sqrt[b^2 - 4*a*c]*e + c*(d + e*x^2)) - 7*b*c*e*(-45*a*e^2 + (d + e*x^2)*(-5*c*d - 5*Sqrt[b^2 - 4*a*c]*e + 3*c*(d + e*x^2))) + c*(35*a*e^2*(3*Sqrt[b^2 - 4*a*c]*e - 2*c*(d + e*x^2)) + c*(d + e*x^2)*(7*Sqrt[b^2 - 4*a*c]*e*(5*d - 3*(d + e*x^2)) + c*(70*d^2 - 84*d*(d + e*x^2) + 30*(d + e*x^2)^2)))))/(140*c^4*d^3*(d + e*x^2)) + (3*(b^3 - 3*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - a*c*Sqrt[b^2 - 4*a*c])*e^3*(d + e*x^2)^3*ArcTanh[Sqrt[2]*Sqrt[(c*(d + e*x^2))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]])/(4*Sqrt[2]*d^3*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)^3*((c*(d + e*x^2))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e))^(9/2))))/(3*Sqrt[b^2 - 4*a*c]*e^4*(Sqrt[b^2 - 4*a*c]/e - (2*c*d - b*e)/e^2)))/e","B",0
355,1,591,324,7.2508245,"\int \frac{x^5 \sqrt{d+e x^2}}{a+b x^2+c x^4} \, dx","Integrate[(x^5*Sqrt[d + e*x^2])/(a + b*x^2 + c*x^4),x]","\frac{c \left(d+e x^2\right)^{7/2} \left(\frac{e^2 \left(\sqrt{2} \sqrt{\frac{c \left(d+e x^2\right)}{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}} \left(5 b e \left(3 e \sqrt{b^2-4 a c}+c \left(d+e x^2\right)\right)+c \left(d+e x^2\right) \left(-5 e \sqrt{b^2-4 a c}+4 c d-6 c e x^2\right)+30 a c e^2-15 b^2 e^2\right)-15 e^2 \left(b \sqrt{b^2-4 a c}+2 a c-b^2\right) \tanh ^{-1}\left(\sqrt{2} \sqrt{\frac{c \left(d+e x^2\right)}{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)\right)}{\left(e \left(b-\sqrt{b^2-4 a c}\right)-2 c d\right) \left(e \left(\sqrt{b^2-4 a c}-b\right)+2 c d\right)^2 \left(\frac{c \left(d+e x^2\right)}{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}\right)^{7/2}}-\frac{e^2 \left(\sqrt{2} \sqrt{\frac{c \left(d+e x^2\right)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} \left(-5 b e \left(3 e \sqrt{b^2-4 a c}-c \left(d+e x^2\right)\right)+c \left(d+e x^2\right) \left(5 e \sqrt{b^2-4 a c}+4 c d-6 c e x^2\right)+30 a c e^2-15 b^2 e^2\right)+15 e^2 \left(b \sqrt{b^2-4 a c}-2 a c+b^2\right) \tanh ^{-1}\left(\sqrt{2} \sqrt{\frac{c \left(d+e x^2\right)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)\right)}{\left(e \left(\sqrt{b^2-4 a c}+b\right)-2 c d\right)^3 \left(\frac{c \left(d+e x^2\right)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}\right)^{7/2}}\right)}{30 \sqrt{2} e^4 \sqrt{b^2-4 a c}}","\frac{\left(-\frac{3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{\sqrt{2} c^{5/2} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}+\frac{\left(\frac{3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2} c^{5/2} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{b \sqrt{d+e x^2}}{c^2}+\frac{\left(d+e x^2\right)^{3/2}}{3 c e}",1,"(c*(d + e*x^2)^(7/2)*((e^2*(Sqrt[2]*Sqrt[(c*(d + e*x^2))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)]*(-15*b^2*e^2 + 30*a*c*e^2 + c*(d + e*x^2)*(4*c*d - 5*Sqrt[b^2 - 4*a*c]*e - 6*c*e*x^2) + 5*b*e*(3*Sqrt[b^2 - 4*a*c]*e + c*(d + e*x^2))) - 15*(-b^2 + 2*a*c + b*Sqrt[b^2 - 4*a*c])*e^2*ArcTanh[Sqrt[2]*Sqrt[(c*(d + e*x^2))/(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)]]))/((-2*c*d + (b - Sqrt[b^2 - 4*a*c])*e)*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)^2*((c*(d + e*x^2))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e))^(7/2)) - (e^2*(Sqrt[2]*Sqrt[(c*(d + e*x^2))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*(-15*b^2*e^2 + 30*a*c*e^2 + c*(d + e*x^2)*(4*c*d + 5*Sqrt[b^2 - 4*a*c]*e - 6*c*e*x^2) - 5*b*e*(3*Sqrt[b^2 - 4*a*c]*e - c*(d + e*x^2))) + 15*(b^2 - 2*a*c + b*Sqrt[b^2 - 4*a*c])*e^2*ArcTanh[Sqrt[2]*Sqrt[(c*(d + e*x^2))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]]))/((-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)^3*((c*(d + e*x^2))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e))^(7/2))))/(30*Sqrt[2]*Sqrt[b^2 - 4*a*c]*e^4)","A",1
356,1,308,292,0.5470655,"\int \frac{x^3 \sqrt{d+e x^2}}{a+b x^2+c x^4} \, dx","Integrate[(x^3*Sqrt[d + e*x^2])/(a + b*x^2 + c*x^4),x]","\frac{\frac{\left(-c d \sqrt{b^2-4 a c}+b e \sqrt{b^2-4 a c}+2 a c e+b^2 (-e)+b c d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c} \sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}+\frac{\left(-c d \sqrt{b^2-4 a c}+b e \sqrt{b^2-4 a c}-2 a c e+b^2 e-b c d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\sqrt{d+e x^2}}{c}","\frac{\left(-\sqrt{b^2-4 a c} (c d-b e)+2 a c e+b^2 (-e)+b c d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{\sqrt{2} c^{3/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}-\frac{\left(\sqrt{b^2-4 a c} (c d-b e)+2 a c e+b^2 (-e)+b c d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2} c^{3/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{\sqrt{d+e x^2}}{c}",1,"(Sqrt[d + e*x^2] + ((b*c*d - c*Sqrt[b^2 - 4*a*c]*d - b^2*e + 2*a*c*e + b*Sqrt[b^2 - 4*a*c]*e)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[2]*Sqrt[c]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) + ((-(b*c*d) - c*Sqrt[b^2 - 4*a*c]*d + b^2*e - 2*a*c*e + b*Sqrt[b^2 - 4*a*c]*e)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*Sqrt[c]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/c","A",1
357,1,179,202,0.3275898,"\int \frac{x \sqrt{d+e x^2}}{a+b x^2+c x^4} \, dx","Integrate[(x*Sqrt[d + e*x^2])/(a + b*x^2 + c*x^4),x]","\frac{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)-\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}","\frac{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}-\frac{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}",1,"(-(Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]]) + Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*Sqrt[c]*Sqrt[b^2 - 4*a*c])","A",1
358,1,241,281,0.8138727,"\int \frac{\sqrt{d+e x^2}}{x \left(a+b x^2+c x^4\right)} \, dx","Integrate[Sqrt[d + e*x^2]/(x*(a + b*x^2 + c*x^4)),x]","\frac{\frac{\sqrt{2} \left(\left(\sqrt{b^2-4 a c}+b\right) \sqrt{e \sqrt{b^2-4 a c}-b e+2 c d} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)+\left(\sqrt{b^2-4 a c}-b\right) \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)\right)}{\sqrt{c} \sqrt{b^2-4 a c}}-4 \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{4 a}","\frac{\sqrt{c} \left(d \sqrt{b^2-4 a c}-2 a e+b d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{\sqrt{2} a \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}-\frac{\sqrt{c} \left(-d \sqrt{b^2-4 a c}-2 a e+b d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2} a \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{\sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{a}",1,"(-4*Sqrt[d]*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]] + (Sqrt[2]*((b + Sqrt[b^2 - 4*a*c])*Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]] + (-b + Sqrt[b^2 - 4*a*c])*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]]))/(Sqrt[c]*Sqrt[b^2 - 4*a*c]))/(4*a)","A",1
359,1,349,382,1.3885295,"\int \frac{\sqrt{d+e x^2}}{x^3 \left(a+b x^2+c x^4\right)} \, dx","Integrate[Sqrt[d + e*x^2]/(x^3*(a + b*x^2 + c*x^4)),x]","\frac{\frac{\sqrt{2} \sqrt{c} \left(\frac{\left(-b d \sqrt{b^2-4 a c}+a e \sqrt{b^2-4 a c}+a b e+2 a c d+b^2 (-d)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{\sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}-\frac{\left(b d \sqrt{b^2-4 a c}-a e \sqrt{b^2-4 a c}+a b e+2 a c d+b^2 (-d)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{b^2-4 a c}}+\frac{(2 b d-a e) \log \left(\sqrt{d} \sqrt{d+e x^2}+d\right)}{\sqrt{d}}+\frac{\log (x) (a e-2 b d)}{\sqrt{d}}-\frac{a \sqrt{d+e x^2}}{x^2}}{2 a^2}","-\frac{\sqrt{c} \left(\sqrt{b^2-4 a c} (b d-a e)-a b e-2 a c d+b^2 d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{\sqrt{2} a^2 \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}+\frac{\sqrt{c} \left(-b \left(d \sqrt{b^2-4 a c}+a e\right)-a \left(2 c d-e \sqrt{b^2-4 a c}\right)+b^2 d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2} a^2 \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{(b d-a e) \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{a^2 \sqrt{d}}-\frac{\sqrt{d+e x^2}}{2 a x^2}+\frac{e \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{2 a \sqrt{d}}",1,"(-((a*Sqrt[d + e*x^2])/x^2) + (Sqrt[2]*Sqrt[c]*(((-(b^2*d) + 2*a*c*d - b*Sqrt[b^2 - 4*a*c]*d + a*b*e + a*Sqrt[b^2 - 4*a*c]*e)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e] - ((-(b^2*d) + 2*a*c*d + b*Sqrt[b^2 - 4*a*c]*d + a*b*e - a*Sqrt[b^2 - 4*a*c]*e)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/Sqrt[b^2 - 4*a*c] + ((-2*b*d + a*e)*Log[x])/Sqrt[d] + ((2*b*d - a*e)*Log[d + Sqrt[d]*Sqrt[d + e*x^2]])/Sqrt[d])/(2*a^2)","A",1
360,1,466,552,1.9665651,"\int \frac{\sqrt{d+e x^2}}{x^5 \left(a+b x^2+c x^4\right)} \, dx","Integrate[Sqrt[d + e*x^2]/(x^5*(a + b*x^2 + c*x^4)),x]","\frac{\frac{\log \left(\sqrt{d} \sqrt{d+e x^2}+d\right) \left(4 a b d e+a \left(a e^2+8 c d^2\right)-8 b^2 d^2\right)}{d^{3/2}}-\frac{\log (x) \left(4 a b d e+a \left(a e^2+8 c d^2\right)-8 b^2 d^2\right)}{d^{3/2}}-\frac{4 \sqrt{2} \sqrt{c} \left(\frac{\left(b^2 \left(a e-d \sqrt{b^2-4 a c}\right)+a b \left(e \sqrt{b^2-4 a c}+3 c d\right)+a c \left(d \sqrt{b^2-4 a c}-2 a e\right)+b^3 (-d)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{\sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}+\frac{\left(-b^2 \left(d \sqrt{b^2-4 a c}+a e\right)+a b \left(e \sqrt{b^2-4 a c}-3 c d\right)+a c \left(d \sqrt{b^2-4 a c}+2 a e\right)+b^3 d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{b^2-4 a c}}+\frac{a \sqrt{d+e x^2} \left(4 b d x^2-a \left(2 d+e x^2\right)\right)}{d x^4}}{8 a^3}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right) \left(-a b e-a c d+b^2 d\right)}{a^3 \sqrt{d}}+\frac{\sqrt{c} \left(b^2 \left(d \sqrt{b^2-4 a c}-a e\right)-a b \left(e \sqrt{b^2-4 a c}+3 c d\right)-a c \left(d \sqrt{b^2-4 a c}-2 a e\right)+b^3 d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{\sqrt{2} a^3 \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}-\frac{\sqrt{c} \left(-b^2 \left(d \sqrt{b^2-4 a c}+a e\right)-a b \left(3 c d-e \sqrt{b^2-4 a c}\right)+a c \left(d \sqrt{b^2-4 a c}+2 a e\right)+b^3 d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2} a^3 \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{e (b d-a e) \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{2 a^2 d^{3/2}}+\frac{\sqrt{d+e x^2} (b d-a e)}{2 a^2 d x^2}-\frac{3 e^2 \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{8 a d^{3/2}}+\frac{3 e \sqrt{d+e x^2}}{8 a d x^2}-\frac{\sqrt{d+e x^2}}{4 a x^4}",1,"((a*Sqrt[d + e*x^2]*(4*b*d*x^2 - a*(2*d + e*x^2)))/(d*x^4) - (4*Sqrt[2]*Sqrt[c]*(((-(b^3*d) + a*c*(Sqrt[b^2 - 4*a*c]*d - 2*a*e) + b^2*(-(Sqrt[b^2 - 4*a*c]*d) + a*e) + a*b*(3*c*d + Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e] + ((b^3*d - b^2*(Sqrt[b^2 - 4*a*c]*d + a*e) + a*c*(Sqrt[b^2 - 4*a*c]*d + 2*a*e) + a*b*(-3*c*d + Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/Sqrt[b^2 - 4*a*c] - ((-8*b^2*d^2 + 4*a*b*d*e + a*(8*c*d^2 + a*e^2))*Log[x])/d^(3/2) + ((-8*b^2*d^2 + 4*a*b*d*e + a*(8*c*d^2 + a*e^2))*Log[d + Sqrt[d]*Sqrt[d + e*x^2]])/d^(3/2))/(8*a^3)","A",1
361,1,10915,390,6.3965072,"\int \frac{x^4 \sqrt{d+e x^2}}{a+b x^2+c x^4} \, dx","Integrate[(x^4*Sqrt[d + e*x^2])/(a + b*x^2 + c*x^4),x]","\text{Result too large to show}","-\frac{\left(-\frac{3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right)}{c^2 \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}-\frac{\left(\frac{3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right)}{c^2 \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{(c d-2 b e) \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{2 c^2 \sqrt{e}}+\frac{x \sqrt{d+e x^2}}{2 c}",1,"Result too large to show","B",0
362,1,7768,324,6.1712675,"\int \frac{x^2 \sqrt{d+e x^2}}{a+b x^2+c x^4} \, dx","Integrate[(x^2*Sqrt[d + e*x^2])/(a + b*x^2 + c*x^4),x]","\text{Result too large to show}","\frac{\left(-\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right)}{c \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}+\frac{\left(\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right)}{c \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{\sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{c}",1,"Result too large to show","B",0
363,1,2585,240,5.1246905,"\int \frac{\sqrt{d+e x^2}}{a+b x^2+c x^4} \, dx","Integrate[Sqrt[d + e*x^2]/(a + b*x^2 + c*x^4),x]","\text{Result too large to show}","\frac{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)} \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right)}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)} \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right)}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}",1,"(Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*Sqrt[2*d - ((b + Sqrt[b^2 - 4*a*c])*e)/c]*Log[-(Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]/Sqrt[2]) + x] - Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*Sqrt[2*d - ((b + Sqrt[b^2 - 4*a*c])*e)/c]*Log[Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]/Sqrt[2] + x] - 2*c*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*d*Sqrt[(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/c]*Log[-(Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]/Sqrt[2]) + x] + b*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*e*Sqrt[(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/c]*Log[-(Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]/Sqrt[2]) + x] + Sqrt[b^2 - 4*a*c]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*e*Sqrt[(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/c]*Log[-(Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]/Sqrt[2]) + x] + 2*c*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*d*Sqrt[(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/c]*Log[Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]/Sqrt[2] + x] - b*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*e*Sqrt[(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/c]*Log[Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]/Sqrt[2] + x] - Sqrt[b^2 - 4*a*c]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*e*Sqrt[(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/c]*Log[Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]/Sqrt[2] + x] + 2*c*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*d*Sqrt[2*d - ((b + Sqrt[b^2 - 4*a*c])*e)/c]*Log[2*d - Sqrt[2]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*e*x + Sqrt[(4*c*d - 2*b*e + 2*Sqrt[b^2 - 4*a*c]*e)/c]*Sqrt[d + e*x^2]] - b*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*e*Sqrt[2*d - ((b + Sqrt[b^2 - 4*a*c])*e)/c]*Log[2*d - Sqrt[2]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*e*x + Sqrt[(4*c*d - 2*b*e + 2*Sqrt[b^2 - 4*a*c]*e)/c]*Sqrt[d + e*x^2]] + Sqrt[b^2 - 4*a*c]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*e*Sqrt[2*d - ((b + Sqrt[b^2 - 4*a*c])*e)/c]*Log[2*d - Sqrt[2]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*e*x + Sqrt[(4*c*d - 2*b*e + 2*Sqrt[b^2 - 4*a*c]*e)/c]*Sqrt[d + e*x^2]] - 2*c*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*d*Sqrt[2*d - ((b + Sqrt[b^2 - 4*a*c])*e)/c]*Log[2*d + Sqrt[2]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*e*x + Sqrt[(4*c*d - 2*b*e + 2*Sqrt[b^2 - 4*a*c]*e)/c]*Sqrt[d + e*x^2]] + b*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*e*Sqrt[2*d - ((b + Sqrt[b^2 - 4*a*c])*e)/c]*Log[2*d + Sqrt[2]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*e*x + Sqrt[(4*c*d - 2*b*e + 2*Sqrt[b^2 - 4*a*c]*e)/c]*Sqrt[d + e*x^2]] - Sqrt[b^2 - 4*a*c]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*e*Sqrt[2*d - ((b + Sqrt[b^2 - 4*a*c])*e)/c]*Log[2*d + Sqrt[2]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*e*x + Sqrt[(4*c*d - 2*b*e + 2*Sqrt[b^2 - 4*a*c]*e)/c]*Sqrt[d + e*x^2]] - 2*c*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*d*Sqrt[(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/c]*Log[2*d - Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*e*x + Sqrt[4*d - (2*(b + Sqrt[b^2 - 4*a*c])*e)/c]*Sqrt[d + e*x^2]] + b*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*e*Sqrt[(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/c]*Log[2*d - Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*e*x + Sqrt[4*d - (2*(b + Sqrt[b^2 - 4*a*c])*e)/c]*Sqrt[d + e*x^2]] + Sqrt[b^2 - 4*a*c]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*e*Sqrt[(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/c]*Log[2*d - Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*e*x + Sqrt[4*d - (2*(b + Sqrt[b^2 - 4*a*c])*e)/c]*Sqrt[d + e*x^2]] + 2*c*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*d*Sqrt[(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/c]*Log[2*d + Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*e*x + Sqrt[4*d - (2*(b + Sqrt[b^2 - 4*a*c])*e)/c]*Sqrt[d + e*x^2]] - b*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*e*Sqrt[(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/c]*Log[2*d + Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*e*x + Sqrt[4*d - (2*(b + Sqrt[b^2 - 4*a*c])*e)/c]*Sqrt[d + e*x^2]] - Sqrt[b^2 - 4*a*c]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*e*Sqrt[(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/c]*Log[2*d + Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*e*x + Sqrt[4*d - (2*(b + Sqrt[b^2 - 4*a*c])*e)/c]*Sqrt[d + e*x^2]])/(2*c*Sqrt[b^2 - 4*a*c]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*Sqrt[(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/c]*Sqrt[2*d - ((b + Sqrt[b^2 - 4*a*c])*e)/c])","B",1
364,1,4644,291,6.322369,"\int \frac{\sqrt{d+e x^2}}{x^2 \left(a+b x^2+c x^4\right)} \, dx","Integrate[Sqrt[d + e*x^2]/(x^2*(a + b*x^2 + c*x^4)),x]","\text{Result too large to show}","-\frac{c \left(\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right)}{a \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}-\frac{c \left(d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right)}{a \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{\sqrt{d+e x^2}}{a x}",1,"-(Sqrt[d + e*x^2]/(a*x)) - (-1/2*(b*d*(Log[Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + x]/Sqrt[d + ((-(b/c) - Sqrt[b^2 - 4*a*c]/c)*e)/2] - Log[2*d - Sqrt[2]*Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e*x + 2*Sqrt[d + ((-(b/c) - Sqrt[b^2 - 4*a*c]/c)*e)/2]*Sqrt[d + e*x^2]]/Sqrt[d + ((-(b/c) - Sqrt[b^2 - 4*a*c]/c)*e)/2]))/(Sqrt[2]*c*Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) - Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])) + (Sqrt[b^2 - 4*a*c]*d*(Log[Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + x]/Sqrt[d + ((-(b/c) - Sqrt[b^2 - 4*a*c]/c)*e)/2] - Log[2*d - Sqrt[2]*Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e*x + 2*Sqrt[d + ((-(b/c) - Sqrt[b^2 - 4*a*c]/c)*e)/2]*Sqrt[d + e*x^2]]/Sqrt[d + ((-(b/c) - Sqrt[b^2 - 4*a*c]/c)*e)/2]))/(2*Sqrt[2]*c*Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) - Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])) + (a*e*(Log[Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + x]/Sqrt[d + ((-(b/c) - Sqrt[b^2 - 4*a*c]/c)*e)/2] - Log[2*d - Sqrt[2]*Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e*x + 2*Sqrt[d + ((-(b/c) - Sqrt[b^2 - 4*a*c]/c)*e)/2]*Sqrt[d + e*x^2]]/Sqrt[d + ((-(b/c) - Sqrt[b^2 - 4*a*c]/c)*e)/2]))/(Sqrt[2]*c*Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) - Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])) + (b*d*(Log[-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + x]/Sqrt[d + ((-(b/c) - Sqrt[b^2 - 4*a*c]/c)*e)/2] - Log[2*d + Sqrt[2]*Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e*x + 2*Sqrt[d + ((-(b/c) - Sqrt[b^2 - 4*a*c]/c)*e)/2]*Sqrt[d + e*x^2]]/Sqrt[d + ((-(b/c) - Sqrt[b^2 - 4*a*c]/c)*e)/2]))/(2*Sqrt[2]*c*Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] - Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])) - (Sqrt[b^2 - 4*a*c]*d*(Log[-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + x]/Sqrt[d + ((-(b/c) - Sqrt[b^2 - 4*a*c]/c)*e)/2] - Log[2*d + Sqrt[2]*Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e*x + 2*Sqrt[d + ((-(b/c) - Sqrt[b^2 - 4*a*c]/c)*e)/2]*Sqrt[d + e*x^2]]/Sqrt[d + ((-(b/c) - Sqrt[b^2 - 4*a*c]/c)*e)/2]))/(2*Sqrt[2]*c*Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] - Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])) - (a*e*(Log[-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + x]/Sqrt[d + ((-(b/c) - Sqrt[b^2 - 4*a*c]/c)*e)/2] - Log[2*d + Sqrt[2]*Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e*x + 2*Sqrt[d + ((-(b/c) - Sqrt[b^2 - 4*a*c]/c)*e)/2]*Sqrt[d + e*x^2]]/Sqrt[d + ((-(b/c) - Sqrt[b^2 - 4*a*c]/c)*e)/2]))/(Sqrt[2]*c*Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] - Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])) - (b*d*(Log[Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + x]/Sqrt[d + ((-(b/c) + Sqrt[b^2 - 4*a*c]/c)*e)/2] - Log[2*d - Sqrt[2]*Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]*e*x + 2*Sqrt[d + ((-(b/c) + Sqrt[b^2 - 4*a*c]/c)*e)/2]*Sqrt[d + e*x^2]]/Sqrt[d + ((-(b/c) + Sqrt[b^2 - 4*a*c]/c)*e)/2]))/(2*Sqrt[2]*c*Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) - Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] - Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])) - (Sqrt[b^2 - 4*a*c]*d*(Log[Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + x]/Sqrt[d + ((-(b/c) + Sqrt[b^2 - 4*a*c]/c)*e)/2] - Log[2*d - Sqrt[2]*Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]*e*x + 2*Sqrt[d + ((-(b/c) + Sqrt[b^2 - 4*a*c]/c)*e)/2]*Sqrt[d + e*x^2]]/Sqrt[d + ((-(b/c) + Sqrt[b^2 - 4*a*c]/c)*e)/2]))/(2*Sqrt[2]*c*Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) - Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] - Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])) + (a*e*(Log[Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + x]/Sqrt[d + ((-(b/c) + Sqrt[b^2 - 4*a*c]/c)*e)/2] - Log[2*d - Sqrt[2]*Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]*e*x + 2*Sqrt[d + ((-(b/c) + Sqrt[b^2 - 4*a*c]/c)*e)/2]*Sqrt[d + e*x^2]]/Sqrt[d + ((-(b/c) + Sqrt[b^2 - 4*a*c]/c)*e)/2]))/(Sqrt[2]*c*Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) - Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] - Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])) + (b*d*(Log[-(Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + x]/Sqrt[d + ((-(b/c) + Sqrt[b^2 - 4*a*c]/c)*e)/2] - Log[2*d + Sqrt[2]*Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]*e*x + 2*Sqrt[d + ((-(b/c) + Sqrt[b^2 - 4*a*c]/c)*e)/2]*Sqrt[d + e*x^2]]/Sqrt[d + ((-(b/c) + Sqrt[b^2 - 4*a*c]/c)*e)/2]))/(2*Sqrt[2]*c*Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])) + (Sqrt[b^2 - 4*a*c]*d*(Log[-(Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + x]/Sqrt[d + ((-(b/c) + Sqrt[b^2 - 4*a*c]/c)*e)/2] - Log[2*d + Sqrt[2]*Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]*e*x + 2*Sqrt[d + ((-(b/c) + Sqrt[b^2 - 4*a*c]/c)*e)/2]*Sqrt[d + e*x^2]]/Sqrt[d + ((-(b/c) + Sqrt[b^2 - 4*a*c]/c)*e)/2]))/(2*Sqrt[2]*c*Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])) - (a*e*(Log[-(Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + x]/Sqrt[d + ((-(b/c) + Sqrt[b^2 - 4*a*c]/c)*e)/2] - Log[2*d + Sqrt[2]*Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]*e*x + 2*Sqrt[d + ((-(b/c) + Sqrt[b^2 - 4*a*c]/c)*e)/2]*Sqrt[d + e*x^2]]/Sqrt[d + ((-(b/c) + Sqrt[b^2 - 4*a*c]/c)*e)/2]))/(Sqrt[2]*c*Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])))/a","B",0
365,1,7777,373,6.3905778,"\int \frac{\sqrt{d+e x^2}}{x^4 \left(a+b x^2+c x^4\right)} \, dx","Integrate[Sqrt[d + e*x^2]/(x^4*(a + b*x^2 + c*x^4)),x]","\text{Result too large to show}","\frac{c \left(\frac{-a b e-2 a c d+b^2 d}{\sqrt{b^2-4 a c}}-a e+b d\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right)}{a^2 \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}+\frac{c \left(-\frac{-a b e-2 a c d+b^2 d}{\sqrt{b^2-4 a c}}-a e+b d\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right)}{a^2 \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{\sqrt{d+e x^2} (b d-a e)}{a^2 d x}+\frac{2 e \sqrt{d+e x^2}}{3 a d x}-\frac{\sqrt{d+e x^2}}{3 a x^3}",1,"Result too large to show","B",0
366,1,10933,512,6.586122,"\int \frac{\sqrt{d+e x^2}}{x^6 \left(a+b x^2+c x^4\right)} \, dx","Integrate[Sqrt[d + e*x^2]/(x^6*(a + b*x^2 + c*x^4)),x]","\text{Result too large to show}","-\frac{\sqrt{d+e x^2} \left(-a b e-a c d+b^2 d\right)}{a^3 d x}-\frac{2 e \sqrt{d+e x^2} (b d-a e)}{3 a^2 d^2 x}+\frac{\sqrt{d+e x^2} (b d-a e)}{3 a^2 d x^3}-\frac{c \left(\frac{2 a^2 c e-a b^2 e-3 a b c d+b^3 d}{\sqrt{b^2-4 a c}}-a b e-a c d+b^2 d\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right)}{a^3 \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}-\frac{c \left(-\frac{2 a^2 c e-a b^2 e-3 a b c d+b^3 d}{\sqrt{b^2-4 a c}}-a b e-a c d+b^2 d\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right)}{a^3 \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{8 e^2 \sqrt{d+e x^2}}{15 a d^2 x}-\frac{\sqrt{d+e x^2}}{5 a x^5}+\frac{4 e \sqrt{d+e x^2}}{15 a d x^3}",1,"Result too large to show","B",0
367,1,457,460,0.9663684,"\int \frac{x^3 \left(d+e x^2\right)^{3/2}}{a+b x^2+c x^4} \, dx","Integrate[(x^3*(d + e*x^2)^(3/2))/(a + b*x^2 + c*x^4),x]","-\frac{\left(-b c \left(e \left(2 d \sqrt{b^2-4 a c}-3 a e\right)+c d^2\right)+c \left(c d \left(d \sqrt{b^2-4 a c}-4 a e\right)-a e^2 \sqrt{b^2-4 a c}\right)+b^2 e \left(e \sqrt{b^2-4 a c}+2 c d\right)+b^3 \left(-e^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{\sqrt{2} c^{5/2} \sqrt{b^2-4 a c} \sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}-\frac{\left(b c \left(c d^2-e \left(2 d \sqrt{b^2-4 a c}+3 a e\right)\right)+c \left(c d \left(d \sqrt{b^2-4 a c}+4 a e\right)-a e^2 \sqrt{b^2-4 a c}\right)+b^2 e \left(e \sqrt{b^2-4 a c}-2 c d\right)+b^3 e^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2} c^{5/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{\sqrt{d+e x^2} (c d-b e)}{c^2}+\frac{\left(d+e x^2\right)^{3/2}}{3 c}","\frac{\left(b c \left(e \left(2 d \sqrt{b^2-4 a c}-3 a e\right)+c d^2\right)+c \left(a e^2 \sqrt{b^2-4 a c}-c d \left(d \sqrt{b^2-4 a c}-4 a e\right)\right)-b^2 e \left(e \sqrt{b^2-4 a c}+2 c d\right)+b^3 e^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{\sqrt{2} c^{5/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}-\frac{\left(b c \left(c d^2-e \left(2 d \sqrt{b^2-4 a c}+3 a e\right)\right)-c \left(a e^2 \sqrt{b^2-4 a c}-c d \left(d \sqrt{b^2-4 a c}+4 a e\right)\right)-b^2 e \left(2 c d-e \sqrt{b^2-4 a c}\right)+b^3 e^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2} c^{5/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{\sqrt{d+e x^2} (c d-b e)}{c^2}+\frac{\left(d+e x^2\right)^{3/2}}{3 c}",1,"((c*d - b*e)*Sqrt[d + e*x^2])/c^2 + (d + e*x^2)^(3/2)/(3*c) - ((-(b^3*e^2) + b^2*e*(2*c*d + Sqrt[b^2 - 4*a*c]*e) + c*(-(a*Sqrt[b^2 - 4*a*c]*e^2) + c*d*(Sqrt[b^2 - 4*a*c]*d - 4*a*e)) - b*c*(c*d^2 + e*(2*Sqrt[b^2 - 4*a*c]*d - 3*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[2]*c^(5/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) - ((b^3*e^2 + b^2*e*(-2*c*d + Sqrt[b^2 - 4*a*c]*e) + b*c*(c*d^2 - e*(2*Sqrt[b^2 - 4*a*c]*d + 3*a*e)) + c*(-(a*Sqrt[b^2 - 4*a*c]*e^2) + c*d*(Sqrt[b^2 - 4*a*c]*d + 4*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*c^(5/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])","A",1
368,1,324,327,0.5683171,"\int \frac{x \left(d+e x^2\right)^{3/2}}{a+b x^2+c x^4} \, dx","Integrate[(x*(d + e*x^2)^(3/2))/(a + b*x^2 + c*x^4),x]","\frac{\left(2 c e \left(-d \sqrt{b^2-4 a c}+a e+b d\right)+b e^2 \left(\sqrt{b^2-4 a c}-b\right)-2 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{\sqrt{2} c^{3/2} \sqrt{b^2-4 a c} \sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}+\frac{\left(-2 c e \left(d \sqrt{b^2-4 a c}+a e+b d\right)+b e^2 \left(\sqrt{b^2-4 a c}+b\right)+2 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2} c^{3/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{e \sqrt{d+e x^2}}{c}","-\frac{\left(-2 c e \left(-d \sqrt{b^2-4 a c}+a e+b d\right)+b e^2 \left(b-\sqrt{b^2-4 a c}\right)+2 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{\sqrt{2} c^{3/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}+\frac{\left(-2 c e \left(d \sqrt{b^2-4 a c}+a e+b d\right)+b e^2 \left(\sqrt{b^2-4 a c}+b\right)+2 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2} c^{3/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{e \sqrt{d+e x^2}}{c}",1,"(e*Sqrt[d + e*x^2])/c + ((-2*c^2*d^2 + b*(-b + Sqrt[b^2 - 4*a*c])*e^2 + 2*c*e*(b*d - Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[2]*c^(3/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) + ((2*c^2*d^2 + b*(b + Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(b*d + Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*c^(3/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])","A",1
369,1,333,346,1.3794459,"\int \frac{\left(d+e x^2\right)^{3/2}}{x \left(a+b x^2+c x^4\right)} \, dx","Integrate[(d + e*x^2)^(3/2)/(x*(a + b*x^2 + c*x^4)),x]","-\frac{\frac{\left(-c d \left(d \sqrt{b^2-4 a c}+4 a e\right)+a e^2 \sqrt{b^2-4 a c}+b \left(a e^2+c d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{\left(c d \left(d \sqrt{b^2-4 a c}-4 a e\right)-a e^2 \sqrt{b^2-4 a c}+b \left(a e^2+c d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{\sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}}{\sqrt{2} a \sqrt{c} \sqrt{b^2-4 a c}}-\frac{d^{3/2} \log \left(\sqrt{d} \sqrt{d+e x^2}+d\right)}{a}+\frac{d^{3/2} \log (x)}{a}","-\frac{\left(-c d \left(d \sqrt{b^2-4 a c}-4 a e\right)+a e^2 \sqrt{b^2-4 a c}-b \left(a e^2+c d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{\sqrt{2} a \sqrt{c} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}-\frac{\left(-c d \left(d \sqrt{b^2-4 a c}+4 a e\right)+a e^2 \sqrt{b^2-4 a c}+b \left(a e^2+c d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2} a \sqrt{c} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{d^{3/2} \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{a}",1,"-((-(((-(a*Sqrt[b^2 - 4*a*c]*e^2) + c*d*(Sqrt[b^2 - 4*a*c]*d - 4*a*e) + b*(c*d^2 + a*e^2))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) + ((a*Sqrt[b^2 - 4*a*c]*e^2 - c*d*(Sqrt[b^2 - 4*a*c]*d + 4*a*e) + b*(c*d^2 + a*e^2))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])/(Sqrt[2]*a*Sqrt[c]*Sqrt[b^2 - 4*a*c])) + (d^(3/2)*Log[x])/a - (d^(3/2)*Log[d + Sqrt[d]*Sqrt[d + e*x^2]])/a","A",1
370,1,380,417,1.5995533,"\int \frac{\left(d+e x^2\right)^{3/2}}{x^3 \left(a+b x^2+c x^4\right)} \, dx","Integrate[(d + e*x^2)^(3/2)/(x^3*(a + b*x^2 + c*x^4)),x]","\frac{\frac{\sqrt{2} \sqrt{c} \left(\frac{\left(2 a \left(e \left(d \sqrt{b^2-4 a c}-a e\right)+c d^2\right)+b d \left(2 a e-d \sqrt{b^2-4 a c}\right)-b^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{\sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}-\frac{\left(b d \left(d \sqrt{b^2-4 a c}+2 a e\right)-2 a e \left(d \sqrt{b^2-4 a c}+a e\right)+2 a c d^2-b^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{b^2-4 a c}}+\sqrt{d} (2 b d-3 a e) \log \left(\sqrt{d} \sqrt{d+e x^2}+d\right)-\sqrt{d} \log (x) (2 b d-3 a e)-\frac{a d \sqrt{d+e x^2}}{x^2}}{2 a^2}","-\frac{\sqrt{c} \left(-2 a \left(e \left(d \sqrt{b^2-4 a c}-a e\right)+c d^2\right)+b d \left(d \sqrt{b^2-4 a c}-2 a e\right)+b^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{\sqrt{2} a^2 \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}+\frac{\sqrt{c} \left(-2 a \left(c d^2-e \left(d \sqrt{b^2-4 a c}+a e\right)\right)-b d \left(d \sqrt{b^2-4 a c}+2 a e\right)+b^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2} a^2 \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{\sqrt{d} (b d-2 a e) \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{a^2}-\frac{d \sqrt{d+e x^2}}{2 a x^2}+\frac{\sqrt{d} e \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{2 a}",1,"(-((a*d*Sqrt[d + e*x^2])/x^2) + (Sqrt[2]*Sqrt[c]*(((-(b^2*d^2) + b*d*(-(Sqrt[b^2 - 4*a*c]*d) + 2*a*e) + 2*a*(c*d^2 + e*(Sqrt[b^2 - 4*a*c]*d - a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e] - ((-(b^2*d^2) + 2*a*c*d^2 - 2*a*e*(Sqrt[b^2 - 4*a*c]*d + a*e) + b*d*(Sqrt[b^2 - 4*a*c]*d + 2*a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/Sqrt[b^2 - 4*a*c] - Sqrt[d]*(2*b*d - 3*a*e)*Log[x] + Sqrt[d]*(2*b*d - 3*a*e)*Log[d + Sqrt[d]*Sqrt[d + e*x^2]])/(2*a^2)","A",1
371,1,18689,595,6.4887264,"\int \frac{x^4 \left(d+e x^2\right)^{3/2}}{a+b x^2+c x^4} \, dx","Integrate[(x^4*(d + e*x^2)^(3/2))/(a + b*x^2 + c*x^4),x]","\text{Result too large to show}","-\frac{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)} \left(-\frac{3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right)}{2 c^3 \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)} \left(\frac{3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right)}{2 c^3 \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{\sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right) \left(-\frac{3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right)}{2 c^3}-\frac{\sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right) \left(\frac{3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right)}{2 c^3}+\frac{x \sqrt{d+e x^2} (3 c d-4 b e)}{8 c^2}+\frac{d (3 c d-4 b e) \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{8 c^2 \sqrt{e}}+\frac{x \left(d+e x^2\right)^{3/2}}{4 c}",1,"Result too large to show","B",0
372,1,14032,491,6.2605392,"\int \frac{x^2 \left(d+e x^2\right)^{3/2}}{a+b x^2+c x^4} \, dx","Integrate[(x^2*(d + e*x^2)^(3/2))/(a + b*x^2 + c*x^4),x]","\text{Result too large to show}","\frac{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)} \left(-\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right)}{2 c^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)} \left(\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right)}{2 c^2 \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{\sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right) \left(-\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right)}{2 c^2}+\frac{\sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right) \left(\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right)}{2 c^2}+\frac{e x \sqrt{d+e x^2}}{2 c}+\frac{d \sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{2 c}",1,"Result too large to show","B",0
373,1,9290,487,6.160197,"\int \frac{\left(d+e x^2\right)^{3/2}}{a+b x^2+c x^4} \, dx","Integrate[(d + e*x^2)^(3/2)/(a + b*x^2 + c*x^4),x]","\text{Result too large to show}","\frac{\left(-2 c e \left(-d \sqrt{b^2-4 a c}+a e+b d\right)+b e^2 \left(b-\sqrt{b^2-4 a c}\right)+2 c^2 d^2\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right)}{c \sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}-\frac{\left(-2 c e \left(d \sqrt{b^2-4 a c}+a e+b d\right)+b e^2 \left(\sqrt{b^2-4 a c}+b\right)+2 c^2 d^2\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right)}{c \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{\sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right) \left(3 c d-e \left(b-\sqrt{b^2-4 a c}\right)\right)}{2 c \sqrt{b^2-4 a c}}-\frac{\sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right) \left(3 c d-e \left(\sqrt{b^2-4 a c}+b\right)\right)}{2 c \sqrt{b^2-4 a c}}",1,"Result too large to show","B",0
374,1,7789,260,6.3024138,"\int \frac{\left(d+e x^2\right)^{3/2}}{x^2 \left(a+b x^2+c x^4\right)} \, dx","Integrate[(d + e*x^2)^(3/2)/(x^2*(a + b*x^2 + c*x^4)),x]","\text{Result too large to show}","-\frac{\left(2 c d-e \left(b-\sqrt{b^2-4 a c}\right)\right)^{3/2} \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right)}{\sqrt{b^2-4 a c} \left(b-\sqrt{b^2-4 a c}\right)^{3/2}}+\frac{\left(2 c d-e \left(\sqrt{b^2-4 a c}+b\right)\right)^{3/2} \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right)}{\sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}+b\right)^{3/2}}-\frac{d \sqrt{d+e x^2}}{a x}",1,"Result too large to show","B",0
375,1,9321,523,6.4076683,"\int \frac{\left(d+e x^2\right)^{3/2}}{x^4 \left(a+b x^2+c x^4\right)} \, dx","Integrate[(d + e*x^2)^(3/2)/(x^4*(a + b*x^2 + c*x^4)),x]","\text{Result too large to show}","\frac{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)} \left(\frac{-a b e-2 a c d+b^2 d}{\sqrt{b^2-4 a c}}-a e+b d\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right)}{2 a^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)} \left(-\frac{-a b e-2 a c d+b^2 d}{\sqrt{b^2-4 a c}}-a e+b d\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right)}{2 a^2 \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{\sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right) \left(-\frac{-a b e-2 a c d+b^2 d}{\sqrt{b^2-4 a c}}-a e+b d\right)}{2 a^2}+\frac{\sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right) \left(\frac{-a b e-2 a c d+b^2 d}{\sqrt{b^2-4 a c}}-a e+b d\right)}{2 a^2}+\frac{\sqrt{d+e x^2} (b d-a e)}{a^2 x}-\frac{\sqrt{e} (b d-a e) \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{a^2}-\frac{\left(d+e x^2\right)^{3/2}}{3 a x^3}",1,"Result too large to show","B",0
376,1,354,281,0.5417206,"\int \frac{x^5 \sqrt{1-x^2}}{a+b x^2+c x^4} \, dx","Integrate[(x^5*Sqrt[1 - x^2])/(a + b*x^2 + c*x^4),x]","\frac{-\frac{3 \sqrt{2} \left(b^2 \left(\sqrt{b^2-4 a c}+c\right)+b c \left(\sqrt{b^2-4 a c}-3 a\right)-a c \left(\sqrt{b^2-4 a c}+2 c\right)+b^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{1-x^2}}{\sqrt{-\sqrt{b^2-4 a c}-b-2 c}}\right)}{\sqrt{b^2-4 a c} \sqrt{-\sqrt{b^2-4 a c}-b-2 c}}-\frac{3 \sqrt{2} \left(b^2 \left(\sqrt{b^2-4 a c}-c\right)+b c \left(\sqrt{b^2-4 a c}+3 a\right)+a c \left(2 c-\sqrt{b^2-4 a c}\right)-b^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{1-x^2}}{\sqrt{\sqrt{b^2-4 a c}-b-2 c}}\right)}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}-b-2 c}}-6 b \sqrt{c} \sqrt{1-x^2}-2 c^{3/2} \left(1-x^2\right)^{3/2}}{6 c^{5/2}}","\frac{\left(-\frac{-3 a b c-2 a c^2+b^3+b^2 c}{\sqrt{b^2-4 a c}}-a c+b^2+b c\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{1-x^2}}{\sqrt{-\sqrt{b^2-4 a c}+b+2 c}}\right)}{\sqrt{2} c^{5/2} \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}+\frac{\left(\frac{-3 a b c-2 a c^2+b^3+b^2 c}{\sqrt{b^2-4 a c}}-a c+b^2+b c\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{1-x^2}}{\sqrt{\sqrt{b^2-4 a c}+b+2 c}}\right)}{\sqrt{2} c^{5/2} \sqrt{\sqrt{b^2-4 a c}+b+2 c}}-\frac{b \sqrt{1-x^2}}{c^2}-\frac{\left(1-x^2\right)^{3/2}}{3 c}",1,"(-6*b*Sqrt[c]*Sqrt[1 - x^2] - 2*c^(3/2)*(1 - x^2)^(3/2) - (3*Sqrt[2]*(b^3 + b*c*(-3*a + Sqrt[b^2 - 4*a*c]) + b^2*(c + Sqrt[b^2 - 4*a*c]) - a*c*(2*c + Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[1 - x^2])/Sqrt[-b - 2*c - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[-b - 2*c - Sqrt[b^2 - 4*a*c]]) - (3*Sqrt[2]*(-b^3 + a*c*(2*c - Sqrt[b^2 - 4*a*c]) + b*c*(3*a + Sqrt[b^2 - 4*a*c]) + b^2*(-c + Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[1 - x^2])/Sqrt[-b - 2*c + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[-b - 2*c + Sqrt[b^2 - 4*a*c]]))/(6*c^(5/2))","A",1
377,1,276,229,0.3867928,"\int \frac{x^3 \sqrt{1-x^2}}{a+b x^2+c x^4} \, dx","Integrate[(x^3*Sqrt[1 - x^2])/(a + b*x^2 + c*x^4),x]","\frac{\frac{\left(b \left(c-\sqrt{b^2-4 a c}\right)-c \left(\sqrt{b^2-4 a c}+2 a\right)+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{1-x^2}}{\sqrt{-\sqrt{b^2-4 a c}+b+2 c}}\right)}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c} \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}-\frac{\left(b \left(\sqrt{b^2-4 a c}+c\right)+c \left(\sqrt{b^2-4 a c}-2 a\right)+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{1-x^2}}{\sqrt{\sqrt{b^2-4 a c}+b+2 c}}\right)}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b+2 c}}+\sqrt{1-x^2}}{c}","-\frac{\left(-\frac{-2 a c+b^2+b c}{\sqrt{b^2-4 a c}}+b+c\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{1-x^2}}{\sqrt{-\sqrt{b^2-4 a c}+b+2 c}}\right)}{\sqrt{2} c^{3/2} \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}-\frac{\left(\frac{-2 a c+b^2+b c}{\sqrt{b^2-4 a c}}+b+c\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{1-x^2}}{\sqrt{\sqrt{b^2-4 a c}+b+2 c}}\right)}{\sqrt{2} c^{3/2} \sqrt{\sqrt{b^2-4 a c}+b+2 c}}+\frac{\sqrt{1-x^2}}{c}",1,"(Sqrt[1 - x^2] + ((b^2 + b*(c - Sqrt[b^2 - 4*a*c]) - c*(2*a + Sqrt[b^2 - 4*a*c]))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[1 - x^2])/Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[c]*Sqrt[b^2 - 4*a*c]*Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]) - ((b^2 + c*(-2*a + Sqrt[b^2 - 4*a*c]) + b*(c + Sqrt[b^2 - 4*a*c]))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[1 - x^2])/Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[c]*Sqrt[b^2 - 4*a*c]*Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]))/c","A",1
378,1,169,182,0.24298,"\int \frac{x \sqrt{1-x^2}}{a+b x^2+c x^4} \, dx","Integrate[(x*Sqrt[1 - x^2])/(a + b*x^2 + c*x^4),x]","\frac{\sqrt{-\sqrt{b^2-4 a c}-b-2 c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{1-x^2}}{\sqrt{-\sqrt{b^2-4 a c}-b-2 c}}\right)-\sqrt{\sqrt{b^2-4 a c}-b-2 c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{1-x^2}}{\sqrt{\sqrt{b^2-4 a c}-b-2 c}}\right)}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}","\frac{\sqrt{\sqrt{b^2-4 a c}+b+2 c} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{1-x^2}}{\sqrt{\sqrt{b^2-4 a c}+b+2 c}}\right)}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}-\frac{\sqrt{-\sqrt{b^2-4 a c}+b+2 c} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{1-x^2}}{\sqrt{-\sqrt{b^2-4 a c}+b+2 c}}\right)}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}",1,"(Sqrt[-b - 2*c - Sqrt[b^2 - 4*a*c]]*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[1 - x^2])/Sqrt[-b - 2*c - Sqrt[b^2 - 4*a*c]]] - Sqrt[-b - 2*c + Sqrt[b^2 - 4*a*c]]*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[1 - x^2])/Sqrt[-b - 2*c + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[c]*Sqrt[b^2 - 4*a*c])","A",1
379,1,212,241,0.4320724,"\int \frac{\sqrt{1-x^2}}{x \left(a+b x^2+c x^4\right)} \, dx","Integrate[Sqrt[1 - x^2]/(x*(a + b*x^2 + c*x^4)),x]","\frac{\frac{\sqrt{2} \left(\sqrt{-\sqrt{b^2-4 a c}+b+2 c} \left(\sqrt{b^2-4 a c}+b\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{1-x^2}}{\sqrt{-\sqrt{b^2-4 a c}+b+2 c}}\right)+\left(\sqrt{b^2-4 a c}-b\right) \sqrt{\sqrt{b^2-4 a c}+b+2 c} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{1-x^2}}{\sqrt{\sqrt{b^2-4 a c}+b+2 c}}\right)\right)}{\sqrt{c} \sqrt{b^2-4 a c}}-4 \tanh ^{-1}\left(\sqrt{1-x^2}\right)}{4 a}","\frac{\sqrt{c} \left(\sqrt{b^2-4 a c}+2 a+b\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{1-x^2}}{\sqrt{-\sqrt{b^2-4 a c}+b+2 c}}\right)}{\sqrt{2} a \sqrt{b^2-4 a c} \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}-\frac{\sqrt{c} \left(-\sqrt{b^2-4 a c}+2 a+b\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{1-x^2}}{\sqrt{\sqrt{b^2-4 a c}+b+2 c}}\right)}{\sqrt{2} a \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b+2 c}}-\frac{\tanh ^{-1}\left(\sqrt{1-x^2}\right)}{a}",1,"(-4*ArcTanh[Sqrt[1 - x^2]] + (Sqrt[2]*(Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]*(b + Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[1 - x^2])/Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]] + (-b + Sqrt[b^2 - 4*a*c])*Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[1 - x^2])/Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]]))/(Sqrt[c]*Sqrt[b^2 - 4*a*c]))/(4*a)","A",1
380,1,292,290,0.7616576,"\int \frac{\sqrt{1-x^2}}{x^3 \left(a+b x^2+c x^4\right)} \, dx","Integrate[Sqrt[1 - x^2]/(x^3*(a + b*x^2 + c*x^4)),x]","\frac{\frac{\sqrt{2} \sqrt{c} \left(-\frac{\left(b \left(\sqrt{b^2-4 a c}+b\right)+a \left(\sqrt{b^2-4 a c}+b-2 c\right)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{1-x^2}}{\sqrt{-\sqrt{b^2-4 a c}+b+2 c}}\right)}{\sqrt{-\sqrt{b^2-4 a c}+b+2 c}}-\frac{\left(b \left(\sqrt{b^2-4 a c}-b\right)+a \left(\sqrt{b^2-4 a c}-b+2 c\right)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{1-x^2}}{\sqrt{\sqrt{b^2-4 a c}+b+2 c}}\right)}{\sqrt{\sqrt{b^2-4 a c}+b+2 c}}\right)}{\sqrt{b^2-4 a c}}+(a+2 b) \log \left(\sqrt{1-x^2}+1\right)-(a+2 b) \log (x)-\frac{a \sqrt{1-x^2}}{x^2}}{2 a^2}","-\frac{\sqrt{c} \left(\frac{a (b-2 c)+b^2}{\sqrt{b^2-4 a c}}+a+b\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{1-x^2}}{\sqrt{-\sqrt{b^2-4 a c}+b+2 c}}\right)}{\sqrt{2} a^2 \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}-\frac{\sqrt{c} \left(-\frac{a (b-2 c)+b^2}{\sqrt{b^2-4 a c}}+a+b\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{1-x^2}}{\sqrt{\sqrt{b^2-4 a c}+b+2 c}}\right)}{\sqrt{2} a^2 \sqrt{\sqrt{b^2-4 a c}+b+2 c}}+\frac{(a+2 b) \tanh ^{-1}\left(\sqrt{1-x^2}\right)}{2 a^2}-\frac{1}{4 a \left(1-\sqrt{1-x^2}\right)}+\frac{1}{4 a \left(\sqrt{1-x^2}+1\right)}",1,"(-((a*Sqrt[1 - x^2])/x^2) + (Sqrt[2]*Sqrt[c]*(-(((b*(b + Sqrt[b^2 - 4*a*c]) + a*(b - 2*c + Sqrt[b^2 - 4*a*c]))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[1 - x^2])/Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]])/Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]) - ((b*(-b + Sqrt[b^2 - 4*a*c]) + a*(-b + 2*c + Sqrt[b^2 - 4*a*c]))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[1 - x^2])/Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]])/Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]))/Sqrt[b^2 - 4*a*c] - (a + 2*b)*Log[x] + (a + 2*b)*Log[1 + Sqrt[1 - x^2]])/(2*a^2)","A",1
381,1,10606,325,6.3070868,"\int \frac{x^4 \sqrt{1-x^2}}{a+b x^2+c x^4} \, dx","Integrate[(x^4*Sqrt[1 - x^2])/(a + b*x^2 + c*x^4),x]","\text{Result too large to show}","-\frac{\left(-\frac{-3 a b c-2 a c^2+b^3+b^2 c}{\sqrt{b^2-4 a c}}-a c+b^2+b c\right) \tan ^{-1}\left(\frac{x \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}{\sqrt{1-x^2} \sqrt{b-\sqrt{b^2-4 a c}}}\right)}{c^2 \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}-\frac{\left(\frac{-3 a b c-2 a c^2+b^3+b^2 c}{\sqrt{b^2-4 a c}}-a c+b^2+b c\right) \tan ^{-1}\left(\frac{x \sqrt{\sqrt{b^2-4 a c}+b+2 c}}{\sqrt{1-x^2} \sqrt{\sqrt{b^2-4 a c}+b}}\right)}{c^2 \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{\sqrt{b^2-4 a c}+b+2 c}}+\frac{(2 b+c) \sin ^{-1}(x)}{2 c^2}+\frac{\sqrt{1-x^2} x}{2 c}",1,"Result too large to show","B",0
382,1,7543,263,6.1401966,"\int \frac{x^2 \sqrt{1-x^2}}{a+b x^2+c x^4} \, dx","Integrate[(x^2*Sqrt[1 - x^2])/(a + b*x^2 + c*x^4),x]","\text{Result too large to show}","\frac{\left(-\frac{-2 a c+b^2+b c}{\sqrt{b^2-4 a c}}+b+c\right) \tan ^{-1}\left(\frac{x \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}{\sqrt{1-x^2} \sqrt{b-\sqrt{b^2-4 a c}}}\right)}{c \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}+\frac{\left(\frac{-2 a c+b^2+b c}{\sqrt{b^2-4 a c}}+b+c\right) \tan ^{-1}\left(\frac{x \sqrt{\sqrt{b^2-4 a c}+b+2 c}}{\sqrt{1-x^2} \sqrt{\sqrt{b^2-4 a c}+b}}\right)}{c \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{\sqrt{b^2-4 a c}+b+2 c}}-\frac{\sin ^{-1}(x)}{c}",1,"Result too large to show","B",1
383,1,2266,220,5.4181505,"\int \frac{\sqrt{1-x^2}}{a+b x^2+c x^4} \, dx","Integrate[Sqrt[1 - x^2]/(a + b*x^2 + c*x^4),x]","\text{Result too large to show}","\frac{\sqrt{-\sqrt{b^2-4 a c}+b+2 c} \tan ^{-1}\left(\frac{x \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}{\sqrt{1-x^2} \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{\sqrt{b^2-4 a c}+b+2 c} \tan ^{-1}\left(\frac{x \sqrt{\sqrt{b^2-4 a c}+b+2 c}}{\sqrt{1-x^2} \sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}",1,"-1/2*(Sqrt[2]*(-b - 2*c + Sqrt[b^2 - 4*a*c])*Sqrt[-((b^2 + c*(-2*a + Sqrt[b^2 - 4*a*c]) + b*(c + Sqrt[b^2 - 4*a*c]))/c^2)]*Log[-(Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]/Sqrt[2]) + x] + Sqrt[2]*(-((-b - 2*c + Sqrt[b^2 - 4*a*c])*Sqrt[-((b^2 + c*(-2*a + Sqrt[b^2 - 4*a*c]) + b*(c + Sqrt[b^2 - 4*a*c]))/c^2)]*Log[Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]/Sqrt[2] + x]) + (b + 2*c + Sqrt[b^2 - 4*a*c])*Sqrt[(-b^2 + c*(2*a + Sqrt[b^2 - 4*a*c]) + b*(-c + Sqrt[b^2 - 4*a*c]))/c^2]*Log[-(Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]/Sqrt[2]) + x] - b*Sqrt[(-b^2 + c*(2*a + Sqrt[b^2 - 4*a*c]) + b*(-c + Sqrt[b^2 - 4*a*c]))/c^2]*Log[Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]/Sqrt[2] + x] - 2*c*Sqrt[(-b^2 + c*(2*a + Sqrt[b^2 - 4*a*c]) + b*(-c + Sqrt[b^2 - 4*a*c]))/c^2]*Log[Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]/Sqrt[2] + x] - Sqrt[b^2 - 4*a*c]*Sqrt[(-b^2 + c*(2*a + Sqrt[b^2 - 4*a*c]) + b*(-c + Sqrt[b^2 - 4*a*c]))/c^2]*Log[Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]/Sqrt[2] + x] + b*Sqrt[-((b^2 + c*(-2*a + Sqrt[b^2 - 4*a*c]) + b*(c + Sqrt[b^2 - 4*a*c]))/c^2)]*Log[2 - Sqrt[2]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*x + Sqrt[2]*Sqrt[(b + 2*c - Sqrt[b^2 - 4*a*c])/c]*Sqrt[1 - x^2]] + 2*c*Sqrt[-((b^2 + c*(-2*a + Sqrt[b^2 - 4*a*c]) + b*(c + Sqrt[b^2 - 4*a*c]))/c^2)]*Log[2 - Sqrt[2]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*x + Sqrt[2]*Sqrt[(b + 2*c - Sqrt[b^2 - 4*a*c])/c]*Sqrt[1 - x^2]] - Sqrt[b^2 - 4*a*c]*Sqrt[-((b^2 + c*(-2*a + Sqrt[b^2 - 4*a*c]) + b*(c + Sqrt[b^2 - 4*a*c]))/c^2)]*Log[2 - Sqrt[2]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*x + Sqrt[2]*Sqrt[(b + 2*c - Sqrt[b^2 - 4*a*c])/c]*Sqrt[1 - x^2]] - b*Sqrt[-((b^2 + c*(-2*a + Sqrt[b^2 - 4*a*c]) + b*(c + Sqrt[b^2 - 4*a*c]))/c^2)]*Log[2 + Sqrt[2]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*x + Sqrt[2]*Sqrt[(b + 2*c - Sqrt[b^2 - 4*a*c])/c]*Sqrt[1 - x^2]] - 2*c*Sqrt[-((b^2 + c*(-2*a + Sqrt[b^2 - 4*a*c]) + b*(c + Sqrt[b^2 - 4*a*c]))/c^2)]*Log[2 + Sqrt[2]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*x + Sqrt[2]*Sqrt[(b + 2*c - Sqrt[b^2 - 4*a*c])/c]*Sqrt[1 - x^2]] + Sqrt[b^2 - 4*a*c]*Sqrt[-((b^2 + c*(-2*a + Sqrt[b^2 - 4*a*c]) + b*(c + Sqrt[b^2 - 4*a*c]))/c^2)]*Log[2 + Sqrt[2]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*x + Sqrt[2]*Sqrt[(b + 2*c - Sqrt[b^2 - 4*a*c])/c]*Sqrt[1 - x^2]] - b*Sqrt[(-b^2 + c*(2*a + Sqrt[b^2 - 4*a*c]) + b*(-c + Sqrt[b^2 - 4*a*c]))/c^2]*Log[2 - Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*x + Sqrt[2]*Sqrt[(b + 2*c + Sqrt[b^2 - 4*a*c])/c]*Sqrt[1 - x^2]] - 2*c*Sqrt[(-b^2 + c*(2*a + Sqrt[b^2 - 4*a*c]) + b*(-c + Sqrt[b^2 - 4*a*c]))/c^2]*Log[2 - Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*x + Sqrt[2]*Sqrt[(b + 2*c + Sqrt[b^2 - 4*a*c])/c]*Sqrt[1 - x^2]] - Sqrt[b^2 - 4*a*c]*Sqrt[(-b^2 + c*(2*a + Sqrt[b^2 - 4*a*c]) + b*(-c + Sqrt[b^2 - 4*a*c]))/c^2]*Log[2 - Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*x + Sqrt[2]*Sqrt[(b + 2*c + Sqrt[b^2 - 4*a*c])/c]*Sqrt[1 - x^2]] + b*Sqrt[(-b^2 + c*(2*a + Sqrt[b^2 - 4*a*c]) + b*(-c + Sqrt[b^2 - 4*a*c]))/c^2]*Log[2 + Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*x + Sqrt[2]*Sqrt[(b + 2*c + Sqrt[b^2 - 4*a*c])/c]*Sqrt[1 - x^2]] + 2*c*Sqrt[(-b^2 + c*(2*a + Sqrt[b^2 - 4*a*c]) + b*(-c + Sqrt[b^2 - 4*a*c]))/c^2]*Log[2 + Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*x + Sqrt[2]*Sqrt[(b + 2*c + Sqrt[b^2 - 4*a*c])/c]*Sqrt[1 - x^2]] + Sqrt[b^2 - 4*a*c]*Sqrt[(-b^2 + c*(2*a + Sqrt[b^2 - 4*a*c]) + b*(-c + Sqrt[b^2 - 4*a*c]))/c^2]*Log[2 + Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*x + Sqrt[2]*Sqrt[(b + 2*c + Sqrt[b^2 - 4*a*c])/c]*Sqrt[1 - x^2]]))/(c*Sqrt[b^2 - 4*a*c]*Sqrt[((b + 2*c - Sqrt[b^2 - 4*a*c])*(-b + Sqrt[b^2 - 4*a*c]))/c^2]*Sqrt[-(((b + Sqrt[b^2 - 4*a*c])*(b + 2*c + Sqrt[b^2 - 4*a*c]))/c^2)])","B",0
384,1,2661,265,4.9462339,"\int \frac{\sqrt{1-x^2}}{x^2 \left(a+b x^2+c x^4\right)} \, dx","Integrate[Sqrt[1 - x^2]/(x^2*(a + b*x^2 + c*x^4)),x]","\text{Result too large to show}","-\frac{c \left(\frac{2 a+b}{\sqrt{b^2-4 a c}}+1\right) \tan ^{-1}\left(\frac{x \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}{\sqrt{1-x^2} \sqrt{b-\sqrt{b^2-4 a c}}}\right)}{a \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}-\frac{c \left(1-\frac{2 a+b}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{x \sqrt{\sqrt{b^2-4 a c}+b+2 c}}{\sqrt{1-x^2} \sqrt{\sqrt{b^2-4 a c}+b}}\right)}{a \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{\sqrt{b^2-4 a c}+b+2 c}}-\frac{\sqrt{1-x^2}}{a x}",1,"-1/2*(4*Sqrt[b^2 - 4*a*c]*Sqrt[(-b^2 + c*(2*a + Sqrt[b^2 - 4*a*c]) + b*(-c + Sqrt[b^2 - 4*a*c]))/c^2]*Sqrt[-((b^2 + c*(-2*a + Sqrt[b^2 - 4*a*c]) + b*(c + Sqrt[b^2 - 4*a*c]))/c^2)]*Sqrt[1 - x^2] + Sqrt[2]*(2*a + b + Sqrt[b^2 - 4*a*c])*Sqrt[-((b^2 + c*(-2*a + Sqrt[b^2 - 4*a*c]) + b*(c + Sqrt[b^2 - 4*a*c]))/c^2)]*x*Log[-(Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]/Sqrt[2]) + x] - Sqrt[2]*(2*a + b + Sqrt[b^2 - 4*a*c])*Sqrt[-((b^2 + c*(-2*a + Sqrt[b^2 - 4*a*c]) + b*(c + Sqrt[b^2 - 4*a*c]))/c^2)]*x*Log[Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]/Sqrt[2] + x] - 2*Sqrt[2]*a*Sqrt[(-b^2 + c*(2*a + Sqrt[b^2 - 4*a*c]) + b*(-c + Sqrt[b^2 - 4*a*c]))/c^2]*x*Log[-(Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]/Sqrt[2]) + x] - Sqrt[2]*b*Sqrt[(-b^2 + c*(2*a + Sqrt[b^2 - 4*a*c]) + b*(-c + Sqrt[b^2 - 4*a*c]))/c^2]*x*Log[-(Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]/Sqrt[2]) + x] + Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[(-b^2 + c*(2*a + Sqrt[b^2 - 4*a*c]) + b*(-c + Sqrt[b^2 - 4*a*c]))/c^2]*x*Log[-(Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]/Sqrt[2]) + x] + 2*Sqrt[2]*a*Sqrt[(-b^2 + c*(2*a + Sqrt[b^2 - 4*a*c]) + b*(-c + Sqrt[b^2 - 4*a*c]))/c^2]*x*Log[Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]/Sqrt[2] + x] + Sqrt[2]*b*Sqrt[(-b^2 + c*(2*a + Sqrt[b^2 - 4*a*c]) + b*(-c + Sqrt[b^2 - 4*a*c]))/c^2]*x*Log[Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]/Sqrt[2] + x] - Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[(-b^2 + c*(2*a + Sqrt[b^2 - 4*a*c]) + b*(-c + Sqrt[b^2 - 4*a*c]))/c^2]*x*Log[Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]/Sqrt[2] + x] - 2*Sqrt[2]*a*Sqrt[-((b^2 + c*(-2*a + Sqrt[b^2 - 4*a*c]) + b*(c + Sqrt[b^2 - 4*a*c]))/c^2)]*x*Log[2 - Sqrt[2]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*x + Sqrt[2]*Sqrt[(b + 2*c - Sqrt[b^2 - 4*a*c])/c]*Sqrt[1 - x^2]] - Sqrt[2]*b*Sqrt[-((b^2 + c*(-2*a + Sqrt[b^2 - 4*a*c]) + b*(c + Sqrt[b^2 - 4*a*c]))/c^2)]*x*Log[2 - Sqrt[2]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*x + Sqrt[2]*Sqrt[(b + 2*c - Sqrt[b^2 - 4*a*c])/c]*Sqrt[1 - x^2]] - Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[-((b^2 + c*(-2*a + Sqrt[b^2 - 4*a*c]) + b*(c + Sqrt[b^2 - 4*a*c]))/c^2)]*x*Log[2 - Sqrt[2]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*x + Sqrt[2]*Sqrt[(b + 2*c - Sqrt[b^2 - 4*a*c])/c]*Sqrt[1 - x^2]] + 2*Sqrt[2]*a*Sqrt[-((b^2 + c*(-2*a + Sqrt[b^2 - 4*a*c]) + b*(c + Sqrt[b^2 - 4*a*c]))/c^2)]*x*Log[2 + Sqrt[2]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*x + Sqrt[2]*Sqrt[(b + 2*c - Sqrt[b^2 - 4*a*c])/c]*Sqrt[1 - x^2]] + Sqrt[2]*b*Sqrt[-((b^2 + c*(-2*a + Sqrt[b^2 - 4*a*c]) + b*(c + Sqrt[b^2 - 4*a*c]))/c^2)]*x*Log[2 + Sqrt[2]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*x + Sqrt[2]*Sqrt[(b + 2*c - Sqrt[b^2 - 4*a*c])/c]*Sqrt[1 - x^2]] + Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[-((b^2 + c*(-2*a + Sqrt[b^2 - 4*a*c]) + b*(c + Sqrt[b^2 - 4*a*c]))/c^2)]*x*Log[2 + Sqrt[2]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*x + Sqrt[2]*Sqrt[(b + 2*c - Sqrt[b^2 - 4*a*c])/c]*Sqrt[1 - x^2]] + 2*Sqrt[2]*a*Sqrt[(-b^2 + c*(2*a + Sqrt[b^2 - 4*a*c]) + b*(-c + Sqrt[b^2 - 4*a*c]))/c^2]*x*Log[2 - Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*x + Sqrt[2]*Sqrt[(b + 2*c + Sqrt[b^2 - 4*a*c])/c]*Sqrt[1 - x^2]] + Sqrt[2]*b*Sqrt[(-b^2 + c*(2*a + Sqrt[b^2 - 4*a*c]) + b*(-c + Sqrt[b^2 - 4*a*c]))/c^2]*x*Log[2 - Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*x + Sqrt[2]*Sqrt[(b + 2*c + Sqrt[b^2 - 4*a*c])/c]*Sqrt[1 - x^2]] - Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[(-b^2 + c*(2*a + Sqrt[b^2 - 4*a*c]) + b*(-c + Sqrt[b^2 - 4*a*c]))/c^2]*x*Log[2 - Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*x + Sqrt[2]*Sqrt[(b + 2*c + Sqrt[b^2 - 4*a*c])/c]*Sqrt[1 - x^2]] - 2*Sqrt[2]*a*Sqrt[(-b^2 + c*(2*a + Sqrt[b^2 - 4*a*c]) + b*(-c + Sqrt[b^2 - 4*a*c]))/c^2]*x*Log[2 + Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*x + Sqrt[2]*Sqrt[(b + 2*c + Sqrt[b^2 - 4*a*c])/c]*Sqrt[1 - x^2]] - Sqrt[2]*b*Sqrt[(-b^2 + c*(2*a + Sqrt[b^2 - 4*a*c]) + b*(-c + Sqrt[b^2 - 4*a*c]))/c^2]*x*Log[2 + Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*x + Sqrt[2]*Sqrt[(b + 2*c + Sqrt[b^2 - 4*a*c])/c]*Sqrt[1 - x^2]] + Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[(-b^2 + c*(2*a + Sqrt[b^2 - 4*a*c]) + b*(-c + Sqrt[b^2 - 4*a*c]))/c^2]*x*Log[2 + Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*x + Sqrt[2]*Sqrt[(b + 2*c + Sqrt[b^2 - 4*a*c])/c]*Sqrt[1 - x^2]])/(a*Sqrt[b^2 - 4*a*c]*Sqrt[((b + 2*c - Sqrt[b^2 - 4*a*c])*(-b + Sqrt[b^2 - 4*a*c]))/c^2]*Sqrt[-(((b + Sqrt[b^2 - 4*a*c])*(b + 2*c + Sqrt[b^2 - 4*a*c]))/c^2)]*x)","B",0
385,1,743,96,0.4959173,"\int \frac{x^2 \sqrt{1-x^2}}{-1+x^2+x^4} \, dx","Integrate[(x^2*Sqrt[1 - x^2])/(-1 + x^2 + x^4),x]","\frac{i \sqrt{5 \left(\sqrt{5}-2\right)} \log \left(\sqrt{2 \left(3+\sqrt{5}\right)} \sqrt{1-x^2}-i \sqrt{2 \left(1+\sqrt{5}\right)} x+2\right)+2 i \sqrt{\sqrt{5}-2} \log \left(\sqrt{2 \left(3+\sqrt{5}\right)} \sqrt{1-x^2}-i \sqrt{2 \left(1+\sqrt{5}\right)} x+2\right)-i \sqrt{5 \left(\sqrt{5}-2\right)} \log \left(\sqrt{2 \left(3+\sqrt{5}\right)} \sqrt{1-x^2}+i \sqrt{2 \left(1+\sqrt{5}\right)} x+2\right)-2 i \sqrt{\sqrt{5}-2} \log \left(\sqrt{2 \left(3+\sqrt{5}\right)} \sqrt{1-x^2}+i \sqrt{2 \left(1+\sqrt{5}\right)} x+2\right)-\sqrt{5 \left(2+\sqrt{5}\right)} \log \left(\sqrt{2} \sqrt{\left(\sqrt{5}-3\right) \left(x^2-1\right)}-\sqrt{2 \left(\sqrt{5}-1\right)} x+2\right)+2 \sqrt{2+\sqrt{5}} \log \left(\sqrt{2} \sqrt{\left(\sqrt{5}-3\right) \left(x^2-1\right)}-\sqrt{2 \left(\sqrt{5}-1\right)} x+2\right)+\sqrt{5 \left(2+\sqrt{5}\right)} \log \left(\sqrt{2} \sqrt{\left(\sqrt{5}-3\right) \left(x^2-1\right)}+\sqrt{2 \left(\sqrt{5}-1\right)} x+2\right)-2 \sqrt{2+\sqrt{5}} \log \left(\sqrt{2} \sqrt{\left(\sqrt{5}-3\right) \left(x^2-1\right)}+\sqrt{2 \left(\sqrt{5}-1\right)} x+2\right)+\left(\sqrt{5}-2\right) \sqrt{2+\sqrt{5}} \log \left(x-\sqrt{\frac{1}{2} \left(\sqrt{5}-1\right)}\right)-\sqrt{5 \left(2+\sqrt{5}\right)} \log \left(x+\sqrt{\frac{1}{2} \left(\sqrt{5}-1\right)}\right)+2 \sqrt{2+\sqrt{5}} \log \left(x+\sqrt{\frac{1}{2} \left(\sqrt{5}-1\right)}\right)-i \sqrt{5 \left(\sqrt{5}-2\right)} \log \left(x-i \sqrt{\frac{1}{2} \left(1+\sqrt{5}\right)}\right)-2 i \sqrt{\sqrt{5}-2} \log \left(x-i \sqrt{\frac{1}{2} \left(1+\sqrt{5}\right)}\right)+i \sqrt{5 \left(\sqrt{5}-2\right)} \log \left(x+i \sqrt{\frac{1}{2} \left(1+\sqrt{5}\right)}\right)+2 i \sqrt{\sqrt{5}-2} \log \left(x+i \sqrt{\frac{1}{2} \left(1+\sqrt{5}\right)}\right)-2 \sqrt{5} \sin ^{-1}(x)}{2 \sqrt{5}}","\sqrt{\frac{1}{5} \left(2+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2} \left(1+\sqrt{5}\right)} x}{\sqrt{1-x^2}}\right)-\sqrt{\frac{1}{5} \left(\sqrt{5}-2\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2} \left(\sqrt{5}-1\right)} x}{\sqrt{1-x^2}}\right)-\sin ^{-1}(x)",1,"(-2*Sqrt[5]*ArcSin[x] + (-2 + Sqrt[5])*Sqrt[2 + Sqrt[5]]*Log[-Sqrt[(-1 + Sqrt[5])/2] + x] + 2*Sqrt[2 + Sqrt[5]]*Log[Sqrt[(-1 + Sqrt[5])/2] + x] - Sqrt[5*(2 + Sqrt[5])]*Log[Sqrt[(-1 + Sqrt[5])/2] + x] - (2*I)*Sqrt[-2 + Sqrt[5]]*Log[(-I)*Sqrt[(1 + Sqrt[5])/2] + x] - I*Sqrt[5*(-2 + Sqrt[5])]*Log[(-I)*Sqrt[(1 + Sqrt[5])/2] + x] + (2*I)*Sqrt[-2 + Sqrt[5]]*Log[I*Sqrt[(1 + Sqrt[5])/2] + x] + I*Sqrt[5*(-2 + Sqrt[5])]*Log[I*Sqrt[(1 + Sqrt[5])/2] + x] + (2*I)*Sqrt[-2 + Sqrt[5]]*Log[2 - I*Sqrt[2*(1 + Sqrt[5])]*x + Sqrt[2*(3 + Sqrt[5])]*Sqrt[1 - x^2]] + I*Sqrt[5*(-2 + Sqrt[5])]*Log[2 - I*Sqrt[2*(1 + Sqrt[5])]*x + Sqrt[2*(3 + Sqrt[5])]*Sqrt[1 - x^2]] - (2*I)*Sqrt[-2 + Sqrt[5]]*Log[2 + I*Sqrt[2*(1 + Sqrt[5])]*x + Sqrt[2*(3 + Sqrt[5])]*Sqrt[1 - x^2]] - I*Sqrt[5*(-2 + Sqrt[5])]*Log[2 + I*Sqrt[2*(1 + Sqrt[5])]*x + Sqrt[2*(3 + Sqrt[5])]*Sqrt[1 - x^2]] + 2*Sqrt[2 + Sqrt[5]]*Log[2 - Sqrt[2*(-1 + Sqrt[5])]*x + Sqrt[2]*Sqrt[(-3 + Sqrt[5])*(-1 + x^2)]] - Sqrt[5*(2 + Sqrt[5])]*Log[2 - Sqrt[2*(-1 + Sqrt[5])]*x + Sqrt[2]*Sqrt[(-3 + Sqrt[5])*(-1 + x^2)]] - 2*Sqrt[2 + Sqrt[5]]*Log[2 + Sqrt[2*(-1 + Sqrt[5])]*x + Sqrt[2]*Sqrt[(-3 + Sqrt[5])*(-1 + x^2)]] + Sqrt[5*(2 + Sqrt[5])]*Log[2 + Sqrt[2*(-1 + Sqrt[5])]*x + Sqrt[2]*Sqrt[(-3 + Sqrt[5])*(-1 + x^2)]])/(2*Sqrt[5])","C",0
386,1,461,479,1.8676024,"\int \frac{x^8}{\sqrt{d+e x^2} \left(a+b x^2+c x^4\right)} \, dx","Integrate[x^8/(Sqrt[d + e*x^2]*(a + b*x^2 + c*x^4)),x]","\frac{-\frac{8 \left(-\frac{2 a^2 c^2-4 a b^2 c+b^4}{\sqrt{b^2-4 a c}}-2 a b c+b^3\right) \tan ^{-1}\left(\frac{x \sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right)}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}-\frac{8 \left(\frac{2 a^2 c^2-4 a b^2 c+b^4}{\sqrt{b^2-4 a c}}-2 a b c+b^3\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right)}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{8 \left(b^2-a c\right) \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{\sqrt{e}}+\frac{4 b c d \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{e^{3/2}}-\frac{4 b c x \sqrt{d+e x^2}}{e}+\frac{3 c^2 d \left(d \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)-\sqrt{e} x \sqrt{d+e x^2}\right)}{e^{5/2}}+\frac{2 c^2 x^3 \sqrt{d+e x^2}}{e}}{8 c^3}","-\frac{\left(-\frac{2 a^2 c^2-4 a b^2 c+b^4}{\sqrt{b^2-4 a c}}-2 a b c+b^3\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right)}{c^3 \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}-\frac{\left(\frac{2 a^2 c^2-4 a b^2 c+b^4}{\sqrt{b^2-4 a c}}-2 a b c+b^3\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right)}{c^3 \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{\left(b^2-a c\right) \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{c^3 \sqrt{e}}+\frac{b d \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{2 c^2 e^{3/2}}-\frac{b x \sqrt{d+e x^2}}{2 c^2 e}+\frac{3 d^2 \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{8 c e^{5/2}}-\frac{3 d x \sqrt{d+e x^2}}{8 c e^2}+\frac{x^3 \sqrt{d+e x^2}}{4 c e}",1,"((-4*b*c*x*Sqrt[d + e*x^2])/e + (2*c^2*x^3*Sqrt[d + e*x^2])/e - (8*(b^3 - 2*a*b*c - (b^4 - 4*a*b^2*c + 2*a^2*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]*x)/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) - (8*(b^3 - 2*a*b*c + (b^4 - 4*a*b^2*c + 2*a^2*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]) + (4*b*c*d*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/e^(3/2) + (8*(b^2 - a*c)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/Sqrt[e] + (3*c^2*d*(-(Sqrt[e]*x*Sqrt[d + e*x^2]) + d*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]))/e^(5/2))/(8*c^3)","A",1
387,1,355,366,1.0304946,"\int \frac{x^6}{\sqrt{d+e x^2} \left(a+b x^2+c x^4\right)} \, dx","Integrate[x^6/(Sqrt[d + e*x^2]*(a + b*x^2 + c*x^4)),x]","\frac{\frac{2 \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{x \sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right)}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}+\frac{2 \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{x \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right)}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{2 b \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{\sqrt{e}}-\frac{c d \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{e^{3/2}}+\frac{c x \sqrt{d+e x^2}}{e}}{2 c^2}","\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{x \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right)}{c^2 \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}+\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{x \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right)}{c^2 \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{c^2 \sqrt{e}}-\frac{d \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{2 c e^{3/2}}+\frac{x \sqrt{d+e x^2}}{2 c e}",1,"((c*x*Sqrt[d + e*x^2])/e + (2*(b^2 - a*c - (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]*x)/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) + (2*(b^2 - a*c + (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]) - (c*d*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/e^(3/2) - (2*b*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/Sqrt[e])/(2*c^2)","A",1
388,1,292,298,0.5875188,"\int \frac{x^4}{\sqrt{d+e x^2} \left(a+b x^2+c x^4\right)} \, dx","Integrate[x^4/(Sqrt[d + e*x^2]*(a + b*x^2 + c*x^4)),x]","\frac{-\frac{\left(\frac{2 a c-b^2}{\sqrt{b^2-4 a c}}+b\right) \tan ^{-1}\left(\frac{x \sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right)}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}-\frac{\left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right)}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{\sqrt{e}}}{c}","-\frac{\left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right)}{c \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}-\frac{\left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right)}{c \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{c \sqrt{e}}",1,"(-(((b + (-b^2 + 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]*x)/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e])) - ((b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]) + ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]/Sqrt[e])/c","A",1
389,1,227,240,0.4781199,"\int \frac{x^2}{\sqrt{d+e x^2} \left(a+b x^2+c x^4\right)} \, dx","Integrate[x^2/(Sqrt[d + e*x^2]*(a + b*x^2 + c*x^4)),x]","\frac{\frac{\sqrt{\sqrt{b^2-4 a c}+b} \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right)}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{\sqrt{b-\sqrt{b^2-4 a c}} \tan ^{-1}\left(\frac{x \sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right)}{\sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}}{\sqrt{b^2-4 a c}}","\frac{\sqrt{\sqrt{b^2-4 a c}+b} \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right)}{\sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{\sqrt{b-\sqrt{b^2-4 a c}} \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right)}{\sqrt{b^2-4 a c} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}",1,"(-((Sqrt[b - Sqrt[b^2 - 4*a*c]]*ArcTan[(Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]*x)/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) + (Sqrt[b + Sqrt[b^2 - 4*a*c]]*ArcTan[(Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])/Sqrt[b^2 - 4*a*c]","A",1
390,1,229,243,0.4056109,"\int \frac{1}{\sqrt{d+e x^2} \left(a+b x^2+c x^4\right)} \, dx","Integrate[1/(Sqrt[d + e*x^2]*(a + b*x^2 + c*x^4)),x]","\frac{2 c \left(\frac{\tan ^{-1}\left(\frac{x \sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right)}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}-\frac{\tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right)}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{b^2-4 a c}}","\frac{2 c \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right)}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}-\frac{2 c \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right)}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}",1,"(2*c*(ArcTan[(Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]*x)/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])]/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) - ArcTan[(Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])]/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])))/Sqrt[b^2 - 4*a*c]","A",1
391,1,271,280,1.0130672,"\int \frac{1}{x^2 \sqrt{d+e x^2} \left(a+b x^2+c x^4\right)} \, dx","Integrate[1/(x^2*Sqrt[d + e*x^2]*(a + b*x^2 + c*x^4)),x]","-\frac{\frac{c \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \tan ^{-1}\left(\frac{x \sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right)}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}+\frac{c \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right)}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{\sqrt{d+e x^2}}{d x}}{a}","-\frac{c \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right)}{a \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}-\frac{c \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right)}{a \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{\sqrt{d+e x^2}}{a d x}",1,"-((Sqrt[d + e*x^2]/(d*x) + (c*(1 + b/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]*x)/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) + (c*(1 - b/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/a)","A",1
392,1,320,341,0.685669,"\int \frac{1}{x^4 \sqrt{d+e x^2} \left(a+b x^2+c x^4\right)} \, dx","Integrate[1/(x^4*Sqrt[d + e*x^2]*(a + b*x^2 + c*x^4)),x]","\frac{\frac{3 c \left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tan ^{-1}\left(\frac{x \sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right)}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}+\frac{3 c \left(\frac{2 a c-b^2}{\sqrt{b^2-4 a c}}+b\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right)}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{a \left(d-2 e x^2\right) \sqrt{d+e x^2}}{d^2 x^3}+\frac{3 b \sqrt{d+e x^2}}{d x}}{3 a^2}","\frac{c \left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right)}{a^2 \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}+\frac{c \left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right)}{a^2 \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{b \sqrt{d+e x^2}}{a^2 d x}+\frac{2 e \sqrt{d+e x^2}}{3 a d^2 x}-\frac{\sqrt{d+e x^2}}{3 a d x^3}",1,"((3*b*Sqrt[d + e*x^2])/(d*x) - (a*(d - 2*e*x^2)*Sqrt[d + e*x^2])/(d^2*x^3) + (3*c*(b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]*x)/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) + (3*c*(b + (-b^2 + 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/(3*a^2)","A",1
393,1,383,443,1.6362549,"\int \frac{1}{x^6 \sqrt{d+e x^2} \left(a+b x^2+c x^4\right)} \, dx","Integrate[1/(x^6*Sqrt[d + e*x^2]*(a + b*x^2 + c*x^4)),x]","-\frac{\frac{a^2 \sqrt{d+e x^2} \left(3 d^2-4 d e x^2+8 e^2 x^4\right)}{d^3 x^5}+\frac{15 \left(b^2-a c\right) \sqrt{d+e x^2}}{d x}+\frac{15 c \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{x \sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right)}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}+\frac{15 c \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{x \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right)}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{5 a b \left(d-2 e x^2\right) \sqrt{d+e x^2}}{d^2 x^3}}{15 a^3}","-\frac{\left(b^2-a c\right) \sqrt{d+e x^2}}{a^3 d x}-\frac{c \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{x \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right)}{a^3 \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}-\frac{c \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{x \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right)}{a^3 \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{2 b e \sqrt{d+e x^2}}{3 a^2 d^2 x}+\frac{b \sqrt{d+e x^2}}{3 a^2 d x^3}-\frac{8 e^2 \sqrt{d+e x^2}}{15 a d^3 x}+\frac{4 e \sqrt{d+e x^2}}{15 a d^2 x^3}-\frac{\sqrt{d+e x^2}}{5 a d x^5}",1,"-1/15*((15*(b^2 - a*c)*Sqrt[d + e*x^2])/(d*x) - (5*a*b*(d - 2*e*x^2)*Sqrt[d + e*x^2])/(d^2*x^3) + (a^2*Sqrt[d + e*x^2]*(3*d^2 - 4*d*e*x^2 + 8*e^2*x^4))/(d^3*x^5) + (15*c*(b^2 - a*c + (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]*x)/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) + (15*c*(b^2 - a*c - (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/a^3","A",1
394,1,10968,350,11.2619932,"\int \frac{x^6}{\left(d+e x^2\right)^{3/2} \left(a+b x^2+c x^4\right)} \, dx","Integrate[x^6/((d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)),x]","\text{Result too large to show}","\frac{2 \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{x \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right)}{c \sqrt{b-\sqrt{b^2-4 a c}} \left(2 c d-e \left(b-\sqrt{b^2-4 a c}\right)\right)^{3/2}}+\frac{2 \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{x \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right)}{c \sqrt{\sqrt{b^2-4 a c}+b} \left(2 c d-e \left(\sqrt{b^2-4 a c}+b\right)\right)^{3/2}}-\frac{d^2 x}{e \sqrt{d+e x^2} \left(a e^2-b d e+c d^2\right)}+\frac{\tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{c e^{3/2}}",1,"Result too large to show","B",0
395,1,2162,360,7.8974077,"\int \frac{x^4}{\left(d+e x^2\right)^{3/2} \left(a+b x^2+c x^4\right)} \, dx","Integrate[x^4/((d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)),x]","\text{Result too large to show}","-\frac{\left(-\frac{-a b e-2 a c d+b^2 d}{\sqrt{b^2-4 a c}}-a e+b d\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right)}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)} \left(a e^2-b d e+c d^2\right)}-\frac{\left(\frac{-a b e-2 a c d+b^2 d}{\sqrt{b^2-4 a c}}-a e+b d\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right)}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)} \left(a e^2-b d e+c d^2\right)}+\frac{d x}{\sqrt{d+e x^2} \left(a e^2-b d e+c d^2\right)}",1,"x/(c*d*Sqrt[d + e*x^2]) - ((b + (-b^2 + 2*a*c)/Sqrt[b^2 - 4*a*c])*x*(45*Sqrt[-(((-b + Sqrt[b^2 - 4*a*c])*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2*(d + e*x^2))/(d^2*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)^2))] + (30*e*x^2*Sqrt[-(((-b + Sqrt[b^2 - 4*a*c])*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2*(d + e*x^2))/(d^2*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)^2))])/d - 45*ArcSin[Sqrt[-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))]] - (30*e*x^2*ArcSin[Sqrt[-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))]])/d - (45*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2*ArcSin[Sqrt[-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))]])/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)) - (30*e*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^4*ArcSin[Sqrt[-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))]])/(d^2*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)) + 4*(-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))))^(5/2)*Sqrt[((-b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))]*Hypergeometric2F1[2, 2, 7/2, -(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))] + (4*e*x^2*(-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))))^(5/2)*Sqrt[((-b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))]*Hypergeometric2F1[2, 2, 7/2, -(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))])/d))/(15*c*(b - Sqrt[b^2 - 4*a*c])*d*(-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))))^(3/2)*(1 - (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c]))*Sqrt[d + e*x^2]*Sqrt[((-b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))]) - ((b - (-b^2 + 2*a*c)/Sqrt[b^2 - 4*a*c])*x*(45*Sqrt[-(((b + Sqrt[b^2 - 4*a*c])*(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*x^2*(d + e*x^2))/(d^2*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)^2))] + (30*e*x^2*Sqrt[-(((b + Sqrt[b^2 - 4*a*c])*(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*x^2*(d + e*x^2))/(d^2*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)^2))])/d - 45*ArcSin[Sqrt[((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]] - (30*e*x^2*ArcSin[Sqrt[((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]])/d + (45*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2*ArcSin[Sqrt[((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]])/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)) - (30*e*(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*x^4*ArcSin[Sqrt[((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]])/(d^2*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)) + 4*(((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)))^(5/2)*Sqrt[((b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]*Hypergeometric2F1[2, 2, 7/2, ((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))] + (4*e*x^2*(((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)))^(5/2)*Sqrt[((b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]*Hypergeometric2F1[2, 2, 7/2, ((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))])/d))/(15*c*(b + Sqrt[b^2 - 4*a*c])*d*(((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)))^(3/2)*(1 + (2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]))*Sqrt[d + e*x^2]*Sqrt[((b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))])","C",0
396,1,2119,333,6.719486,"\int \frac{x^2}{\left(d+e x^2\right)^{3/2} \left(a+b x^2+c x^4\right)} \, dx","Integrate[x^2/((d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)),x]","\text{Result too large to show}","\frac{c \left(d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right)}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)} \left(a e^2-b d e+c d^2\right)}+\frac{c \left(\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right)}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)} \left(a e^2-b d e+c d^2\right)}-\frac{e x}{\sqrt{d+e x^2} \left(a e^2-b d e+c d^2\right)}",1,"((1 - b/Sqrt[b^2 - 4*a*c])*x*(45*Sqrt[-(((-b + Sqrt[b^2 - 4*a*c])*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2*(d + e*x^2))/(d^2*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)^2))] + (30*e*x^2*Sqrt[-(((-b + Sqrt[b^2 - 4*a*c])*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2*(d + e*x^2))/(d^2*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)^2))])/d - 45*ArcSin[Sqrt[-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))]] - (30*e*x^2*ArcSin[Sqrt[-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))]])/d - (45*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2*ArcSin[Sqrt[-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))]])/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)) - (30*e*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^4*ArcSin[Sqrt[-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))]])/(d^2*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)) + 4*(-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))))^(5/2)*Sqrt[((-b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))]*Hypergeometric2F1[2, 2, 7/2, -(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))] + (4*e*x^2*(-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))))^(5/2)*Sqrt[((-b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))]*Hypergeometric2F1[2, 2, 7/2, -(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))])/d))/(15*(b - Sqrt[b^2 - 4*a*c])*d*(-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))))^(3/2)*(1 - (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c]))*Sqrt[d + e*x^2]*Sqrt[((-b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))]) + ((1 + b/Sqrt[b^2 - 4*a*c])*x*(45*Sqrt[-(((b + Sqrt[b^2 - 4*a*c])*(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*x^2*(d + e*x^2))/(d^2*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)^2))] + (30*e*x^2*Sqrt[-(((b + Sqrt[b^2 - 4*a*c])*(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*x^2*(d + e*x^2))/(d^2*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)^2))])/d - 45*ArcSin[Sqrt[((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]] - (30*e*x^2*ArcSin[Sqrt[((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]])/d + (45*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2*ArcSin[Sqrt[((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]])/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)) - (30*e*(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*x^4*ArcSin[Sqrt[((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]])/(d^2*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)) + 4*(((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)))^(5/2)*Sqrt[((b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]*Hypergeometric2F1[2, 2, 7/2, ((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))] + (4*e*x^2*(((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)))^(5/2)*Sqrt[((b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]*Hypergeometric2F1[2, 2, 7/2, ((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))])/d))/(15*(b + Sqrt[b^2 - 4*a*c])*d*(((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)))^(3/2)*(1 + (2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]))*Sqrt[d + e*x^2]*Sqrt[((b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))])","C",0
397,1,2112,341,7.151599,"\int \frac{1}{\left(d+e x^2\right)^{3/2} \left(a+b x^2+c x^4\right)} \, dx","Integrate[1/((d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)),x]","\text{Result too large to show}","-\frac{c \left(e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right)}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)} \left(a e^2-b d e+c d^2\right)}-\frac{c \left(\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right)}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)} \left(a e^2-b d e+c d^2\right)}+\frac{e^2 x}{d \sqrt{d+e x^2} \left(a e^2-b d e+c d^2\right)}",1,"(2*c*x*(45*Sqrt[-(((-b + Sqrt[b^2 - 4*a*c])*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2*(d + e*x^2))/(d^2*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)^2))] + (30*e*x^2*Sqrt[-(((-b + Sqrt[b^2 - 4*a*c])*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2*(d + e*x^2))/(d^2*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)^2))])/d - 45*ArcSin[Sqrt[-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))]] - (30*e*x^2*ArcSin[Sqrt[-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))]])/d - (45*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2*ArcSin[Sqrt[-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))]])/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)) - (30*e*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^4*ArcSin[Sqrt[-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))]])/(d^2*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)) + 4*(-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))))^(5/2)*Sqrt[((-b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))]*Hypergeometric2F1[2, 2, 7/2, -(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))] + (4*e*x^2*(-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))))^(5/2)*Sqrt[((-b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))]*Hypergeometric2F1[2, 2, 7/2, -(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))])/d))/(15*Sqrt[b^2 - 4*a*c]*(b - Sqrt[b^2 - 4*a*c])*d*(-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))))^(3/2)*(1 - (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c]))*Sqrt[d + e*x^2]*Sqrt[((-b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))]) - (2*c*x*(45*Sqrt[-(((b + Sqrt[b^2 - 4*a*c])*(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*x^2*(d + e*x^2))/(d^2*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)^2))] + (30*e*x^2*Sqrt[-(((b + Sqrt[b^2 - 4*a*c])*(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*x^2*(d + e*x^2))/(d^2*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)^2))])/d - 45*ArcSin[Sqrt[((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]] - (30*e*x^2*ArcSin[Sqrt[((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]])/d + (45*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2*ArcSin[Sqrt[((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]])/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)) - (30*e*(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*x^4*ArcSin[Sqrt[((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]])/(d^2*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)) + 4*(((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)))^(5/2)*Sqrt[((b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]*Hypergeometric2F1[2, 2, 7/2, ((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))] + (4*e*x^2*(((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)))^(5/2)*Sqrt[((b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]*Hypergeometric2F1[2, 2, 7/2, ((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))])/d))/(15*Sqrt[b^2 - 4*a*c]*(b + Sqrt[b^2 - 4*a*c])*d*(((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)))^(3/2)*(1 + (2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]))*Sqrt[d + e*x^2]*Sqrt[((b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))])","C",0
398,1,2158,339,6.7737036,"\int \frac{1}{x^2 \left(d+e x^2\right)^{3/2} \left(a+b x^2+c x^4\right)} \, dx","Integrate[1/(x^2*(d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)),x]","\text{Result too large to show}","-\frac{2 c^2 \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right)}{a \sqrt{b-\sqrt{b^2-4 a c}} \left(2 c d-e \left(b-\sqrt{b^2-4 a c}\right)\right)^{3/2}}-\frac{2 c^2 \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right)}{a \sqrt{\sqrt{b^2-4 a c}+b} \left(2 c d-e \left(\sqrt{b^2-4 a c}+b\right)\right)^{3/2}}+\frac{e x (c d-b e)}{a d \sqrt{d+e x^2} \left(e (a e-b d)+c d^2\right)}+\frac{-d-2 e x^2}{a d^2 x \sqrt{d+e x^2}}",1,"-((d + 2*e*x^2)/(a*d^2*x*Sqrt[d + e*x^2])) - ((c + (b*c)/Sqrt[b^2 - 4*a*c])*x*(45*Sqrt[-(((-b + Sqrt[b^2 - 4*a*c])*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2*(d + e*x^2))/(d^2*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)^2))] + (30*e*x^2*Sqrt[-(((-b + Sqrt[b^2 - 4*a*c])*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2*(d + e*x^2))/(d^2*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)^2))])/d - 45*ArcSin[Sqrt[-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))]] - (30*e*x^2*ArcSin[Sqrt[-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))]])/d - (45*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2*ArcSin[Sqrt[-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))]])/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)) - (30*e*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^4*ArcSin[Sqrt[-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))]])/(d^2*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)) + 4*(-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))))^(5/2)*Sqrt[((-b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))]*Hypergeometric2F1[2, 2, 7/2, -(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))] + (4*e*x^2*(-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))))^(5/2)*Sqrt[((-b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))]*Hypergeometric2F1[2, 2, 7/2, -(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))])/d))/(15*a*(b - Sqrt[b^2 - 4*a*c])*d*(-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))))^(3/2)*(1 - (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c]))*Sqrt[d + e*x^2]*Sqrt[((-b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))]) + ((-c + (b*c)/Sqrt[b^2 - 4*a*c])*x*(45*Sqrt[-(((b + Sqrt[b^2 - 4*a*c])*(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*x^2*(d + e*x^2))/(d^2*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)^2))] + (30*e*x^2*Sqrt[-(((b + Sqrt[b^2 - 4*a*c])*(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*x^2*(d + e*x^2))/(d^2*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)^2))])/d - 45*ArcSin[Sqrt[((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]] - (30*e*x^2*ArcSin[Sqrt[((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]])/d + (45*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2*ArcSin[Sqrt[((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]])/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)) - (30*e*(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*x^4*ArcSin[Sqrt[((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]])/(d^2*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)) + 4*(((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)))^(5/2)*Sqrt[((b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]*Hypergeometric2F1[2, 2, 7/2, ((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))] + (4*e*x^2*(((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)))^(5/2)*Sqrt[((b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]*Hypergeometric2F1[2, 2, 7/2, ((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))])/d))/(15*a*(b + Sqrt[b^2 - 4*a*c])*d*(((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)))^(3/2)*(1 + (2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]))*Sqrt[d + e*x^2]*Sqrt[((b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))])","C",0
399,1,2218,419,6.7956983,"\int \frac{1}{x^4 \left(d+e x^2\right)^{3/2} \left(a+b x^2+c x^4\right)} \, dx","Integrate[1/(x^4*(d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)),x]","\text{Result too large to show}","\frac{2 c^2 \left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right)}{a^2 \sqrt{b-\sqrt{b^2-4 a c}} \left(2 c d-e \left(b-\sqrt{b^2-4 a c}\right)\right)^{3/2}}+\frac{2 c^2 \left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{x \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right)}{a^2 \sqrt{\sqrt{b^2-4 a c}+b} \left(2 c d-e \left(\sqrt{b^2-4 a c}+b\right)\right)^{3/2}}-\frac{e x \left(a c e+b^2 (-e)+b c d\right)}{a^2 d \sqrt{d+e x^2} \left(e (a e-b d)+c d^2\right)}+\frac{2 e x (4 a e+3 b d)}{3 a^2 d^3 \sqrt{d+e x^2}}+\frac{4 a e+3 b d}{3 a^2 d^2 x \sqrt{d+e x^2}}-\frac{1}{3 a d x^3 \sqrt{d+e x^2}}",1,"(b*(d + 2*e*x^2))/(a^2*d^2*x*Sqrt[d + e*x^2]) - (d^2 - 4*d*e*x^2 - 8*e^2*x^4)/(3*a*d^3*x^3*Sqrt[d + e*x^2]) + ((b*c + (c*(b^2 - 2*a*c))/Sqrt[b^2 - 4*a*c])*x*(45*Sqrt[-(((-b + Sqrt[b^2 - 4*a*c])*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2*(d + e*x^2))/(d^2*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)^2))] + (30*e*x^2*Sqrt[-(((-b + Sqrt[b^2 - 4*a*c])*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2*(d + e*x^2))/(d^2*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)^2))])/d - 45*ArcSin[Sqrt[-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))]] - (30*e*x^2*ArcSin[Sqrt[-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))]])/d - (45*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2*ArcSin[Sqrt[-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))]])/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)) - (30*e*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^4*ArcSin[Sqrt[-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))]])/(d^2*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)) + 4*(-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))))^(5/2)*Sqrt[((-b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))]*Hypergeometric2F1[2, 2, 7/2, -(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))] + (4*e*x^2*(-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))))^(5/2)*Sqrt[((-b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))]*Hypergeometric2F1[2, 2, 7/2, -(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))])/d))/(15*a^2*(b - Sqrt[b^2 - 4*a*c])*d*(-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))))^(3/2)*(1 - (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c]))*Sqrt[d + e*x^2]*Sqrt[((-b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))]) + ((b*c - (c*(b^2 - 2*a*c))/Sqrt[b^2 - 4*a*c])*x*(45*Sqrt[-(((b + Sqrt[b^2 - 4*a*c])*(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*x^2*(d + e*x^2))/(d^2*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)^2))] + (30*e*x^2*Sqrt[-(((b + Sqrt[b^2 - 4*a*c])*(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*x^2*(d + e*x^2))/(d^2*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)^2))])/d - 45*ArcSin[Sqrt[((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]] - (30*e*x^2*ArcSin[Sqrt[((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]])/d + (45*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2*ArcSin[Sqrt[((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]])/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)) - (30*e*(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*x^4*ArcSin[Sqrt[((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]])/(d^2*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)) + 4*(((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)))^(5/2)*Sqrt[((b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]*Hypergeometric2F1[2, 2, 7/2, ((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))] + (4*e*x^2*(((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)))^(5/2)*Sqrt[((b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]*Hypergeometric2F1[2, 2, 7/2, ((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))])/d))/(15*a^2*(b + Sqrt[b^2 - 4*a*c])*d*(((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)))^(3/2)*(1 + (2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]))*Sqrt[d + e*x^2]*Sqrt[((b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))])","C",0
400,0,0,243,0.2062211,"\int \frac{(f x)^m \left(d+e x^2\right)^q}{a+b x^2+c x^4} \, dx","Integrate[((f*x)^m*(d + e*x^2)^q)/(a + b*x^2 + c*x^4),x]","\int \frac{(f x)^m \left(d+e x^2\right)^q}{a+b x^2+c x^4} \, dx","\frac{2 c (f x)^{m+1} \left(d+e x^2\right)^q \left(\frac{e x^2}{d}+1\right)^{-q} F_1\left(\frac{m+1}{2};1,-q;\frac{m+3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right)}{f (m+1) \sqrt{b^2-4 a c} \left(b-\sqrt{b^2-4 a c}\right)}-\frac{2 c (f x)^{m+1} \left(d+e x^2\right)^q \left(\frac{e x^2}{d}+1\right)^{-q} F_1\left(\frac{m+1}{2};1,-q;\frac{m+3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right)}{f (m+1) \sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}+b\right)}",1,"Integrate[((f*x)^m*(d + e*x^2)^q)/(a + b*x^2 + c*x^4), x]","F",-1
401,1,272,313,0.7906883,"\int \frac{x^7 \left(d+e x^2\right)^q}{a+b x^2+c x^4} \, dx","Integrate[(x^7*(d + e*x^2)^q)/(a + b*x^2 + c*x^4),x]","\frac{\left(d+e x^2\right)^{q+1} \left(\frac{c \left(\frac{b \left(b^2-3 a c\right)}{c \sqrt{b^2-4 a c}}+a-\frac{b^2}{c}\right) \, _2F_1\left(1,q+1;q+2;\frac{2 c \left(e x^2+d\right)}{2 c d+\left(\sqrt{b^2-4 a c}-b\right) e}\right)}{(q+1) \left(e \left(\sqrt{b^2-4 a c}-b\right)+2 c d\right)}+\frac{c \left(-\frac{b \left(b^2-3 a c\right)}{c \sqrt{b^2-4 a c}}+a-\frac{b^2}{c}\right) \, _2F_1\left(1,q+1;q+2;\frac{2 c \left(e x^2+d\right)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{(q+1) \left(2 c d-e \left(\sqrt{b^2-4 a c}+b\right)\right)}-\frac{b e+c d}{e^2 (q+1)}+\frac{c \left(d+e x^2\right)}{e^2 (q+2)}\right)}{2 c^2}","\frac{\left(\frac{b \left(b^2-3 a c\right)}{c \sqrt{b^2-4 a c}}+a-\frac{b^2}{c}\right) \left(d+e x^2\right)^{q+1} \, _2F_1\left(1,q+1;q+2;\frac{2 c \left(e x^2+d\right)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e}\right)}{2 c (q+1) \left(2 c d-e \left(b-\sqrt{b^2-4 a c}\right)\right)}+\frac{\left(-\frac{b \left(b^2-3 a c\right)}{c \sqrt{b^2-4 a c}}+a-\frac{b^2}{c}\right) \left(d+e x^2\right)^{q+1} \, _2F_1\left(1,q+1;q+2;\frac{2 c \left(e x^2+d\right)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{2 c (q+1) \left(2 c d-e \left(\sqrt{b^2-4 a c}+b\right)\right)}-\frac{(b e+c d) \left(d+e x^2\right)^{q+1}}{2 c^2 e^2 (q+1)}+\frac{\left(d+e x^2\right)^{q+2}}{2 c e^2 (q+2)}",1,"((d + e*x^2)^(1 + q)*(-((c*d + b*e)/(e^2*(1 + q))) + (c*(d + e*x^2))/(e^2*(2 + q)) + (c*(a - b^2/c + (b*(b^2 - 3*a*c))/(c*Sqrt[b^2 - 4*a*c]))*Hypergeometric2F1[1, 1 + q, 2 + q, (2*c*(d + e*x^2))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)])/((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*(1 + q)) + (c*(a - b^2/c - (b*(b^2 - 3*a*c))/(c*Sqrt[b^2 - 4*a*c]))*Hypergeometric2F1[1, 1 + q, 2 + q, (2*c*(d + e*x^2))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*(1 + q))))/(2*c^2)","A",1
402,1,211,256,0.3429782,"\int \frac{x^5 \left(d+e x^2\right)^q}{a+b x^2+c x^4} \, dx","Integrate[(x^5*(d + e*x^2)^q)/(a + b*x^2 + c*x^4),x]","\frac{\left(d+e x^2\right)^{q+1} \left(\frac{\left(\frac{2 a c-b^2}{\sqrt{b^2-4 a c}}+b\right) \, _2F_1\left(1,q+1;q+2;\frac{2 c \left(e x^2+d\right)}{2 c d+\left(\sqrt{b^2-4 a c}-b\right) e}\right)}{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}+\frac{\left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \, _2F_1\left(1,q+1;q+2;\frac{2 c \left(e x^2+d\right)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}+\frac{1}{e}\right)}{2 c (q+1)}","\frac{\left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \left(d+e x^2\right)^{q+1} \, _2F_1\left(1,q+1;q+2;\frac{2 c \left(e x^2+d\right)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e}\right)}{2 c (q+1) \left(2 c d-e \left(b-\sqrt{b^2-4 a c}\right)\right)}+\frac{\left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \left(d+e x^2\right)^{q+1} \, _2F_1\left(1,q+1;q+2;\frac{2 c \left(e x^2+d\right)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{2 c (q+1) \left(2 c d-e \left(\sqrt{b^2-4 a c}+b\right)\right)}+\frac{\left(d+e x^2\right)^{q+1}}{2 c e (q+1)}",1,"((d + e*x^2)^(1 + q)*(e^(-1) + ((b + (-b^2 + 2*a*c)/Sqrt[b^2 - 4*a*c])*Hypergeometric2F1[1, 1 + q, 2 + q, (2*c*(d + e*x^2))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)])/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e) + ((b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*Hypergeometric2F1[1, 1 + q, 2 + q, (2*c*(d + e*x^2))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)))/(2*c*(1 + q))","A",1
403,1,183,210,0.2037322,"\int \frac{x^3 \left(d+e x^2\right)^q}{a+b x^2+c x^4} \, dx","Integrate[(x^3*(d + e*x^2)^q)/(a + b*x^2 + c*x^4),x]","-\frac{\left(d+e x^2\right)^{q+1} \left(\left(d \sqrt{b^2-4 a c}+2 a e-b d\right) \, _2F_1\left(1,q+1;q+2;\frac{2 c \left(e x^2+d\right)}{2 c d+\left(\sqrt{b^2-4 a c}-b\right) e}\right)+\left(d \sqrt{b^2-4 a c}-2 a e+b d\right) \, _2F_1\left(1,q+1;q+2;\frac{2 c \left(e x^2+d\right)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)\right)}{4 (q+1) \sqrt{b^2-4 a c} \left(e (a e-b d)+c d^2\right)}","-\frac{\left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \left(d+e x^2\right)^{q+1} \, _2F_1\left(1,q+1;q+2;\frac{2 c \left(e x^2+d\right)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e}\right)}{2 (q+1) \left(2 c d-e \left(b-\sqrt{b^2-4 a c}\right)\right)}-\frac{\left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \left(d+e x^2\right)^{q+1} \, _2F_1\left(1,q+1;q+2;\frac{2 c \left(e x^2+d\right)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{2 (q+1) \left(2 c d-e \left(\sqrt{b^2-4 a c}+b\right)\right)}",1,"-1/4*((d + e*x^2)^(1 + q)*((-(b*d) + Sqrt[b^2 - 4*a*c]*d + 2*a*e)*Hypergeometric2F1[1, 1 + q, 2 + q, (2*c*(d + e*x^2))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)] + (b*d + Sqrt[b^2 - 4*a*c]*d - 2*a*e)*Hypergeometric2F1[1, 1 + q, 2 + q, (2*c*(d + e*x^2))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]))/(Sqrt[b^2 - 4*a*c]*(c*d^2 + e*(-(b*d) + a*e))*(1 + q))","A",1
404,1,168,198,0.2640454,"\int \frac{x \left(d+e x^2\right)^q}{a+b x^2+c x^4} \, dx","Integrate[(x*(d + e*x^2)^q)/(a + b*x^2 + c*x^4),x]","\frac{c \left(d+e x^2\right)^{q+1} \left(\frac{\, _2F_1\left(1,q+1;q+2;\frac{2 c \left(e x^2+d\right)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}-\frac{\, _2F_1\left(1,q+1;q+2;\frac{2 c \left(e x^2+d\right)}{2 c d+\left(\sqrt{b^2-4 a c}-b\right) e}\right)}{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}\right)}{(q+1) \sqrt{b^2-4 a c}}","\frac{c \left(d+e x^2\right)^{q+1} \, _2F_1\left(1,q+1;q+2;\frac{2 c \left(e x^2+d\right)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{(q+1) \sqrt{b^2-4 a c} \left(2 c d-e \left(\sqrt{b^2-4 a c}+b\right)\right)}-\frac{c \left(d+e x^2\right)^{q+1} \, _2F_1\left(1,q+1;q+2;\frac{2 c \left(e x^2+d\right)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e}\right)}{(q+1) \sqrt{b^2-4 a c} \left(2 c d-e \left(b-\sqrt{b^2-4 a c}\right)\right)}",1,"(c*(d + e*x^2)^(1 + q)*(-(Hypergeometric2F1[1, 1 + q, 2 + q, (2*c*(d + e*x^2))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)]/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)) + Hypergeometric2F1[1, 1 + q, 2 + q, (2*c*(d + e*x^2))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)))/(Sqrt[b^2 - 4*a*c]*(1 + q))","A",1
405,1,218,262,0.35064,"\int \frac{\left(d+e x^2\right)^q}{x \left(a+b x^2+c x^4\right)} \, dx","Integrate[(d + e*x^2)^q/(x*(a + b*x^2 + c*x^4)),x]","\frac{\left(d+e x^2\right)^{q+1} \left(\frac{c \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \, _2F_1\left(1,q+1;q+2;\frac{2 c \left(e x^2+d\right)}{2 c d+\left(\sqrt{b^2-4 a c}-b\right) e}\right)}{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}+\frac{c \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \, _2F_1\left(1,q+1;q+2;\frac{2 c \left(e x^2+d\right)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}-\frac{\, _2F_1\left(1,q+1;q+2;\frac{e x^2}{d}+1\right)}{d}\right)}{2 a (q+1)}","\frac{c \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \left(d+e x^2\right)^{q+1} \, _2F_1\left(1,q+1;q+2;\frac{2 c \left(e x^2+d\right)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e}\right)}{2 a (q+1) \left(2 c d-e \left(b-\sqrt{b^2-4 a c}\right)\right)}+\frac{c \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \left(d+e x^2\right)^{q+1} \, _2F_1\left(1,q+1;q+2;\frac{2 c \left(e x^2+d\right)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{2 a (q+1) \left(2 c d-e \left(\sqrt{b^2-4 a c}+b\right)\right)}-\frac{\left(d+e x^2\right)^{q+1} \, _2F_1\left(1,q+1;q+2;\frac{e x^2}{d}+1\right)}{2 a d (q+1)}",1,"((d + e*x^2)^(1 + q)*((c*(1 + b/Sqrt[b^2 - 4*a*c])*Hypergeometric2F1[1, 1 + q, 2 + q, (2*c*(d + e*x^2))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)])/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e) + (c*(1 - b/Sqrt[b^2 - 4*a*c])*Hypergeometric2F1[1, 1 + q, 2 + q, (2*c*(d + e*x^2))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e) - Hypergeometric2F1[1, 1 + q, 2 + q, 1 + (e*x^2)/d]/d))/(2*a*(1 + q))","A",1
406,1,259,322,0.3476712,"\int \frac{\left(d+e x^2\right)^q}{x^3 \left(a+b x^2+c x^4\right)} \, dx","Integrate[(d + e*x^2)^q/(x^3*(a + b*x^2 + c*x^4)),x]","\frac{\left(d+e x^2\right)^{q+1} \left(-\frac{c \left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \, _2F_1\left(1,q+1;q+2;\frac{2 c \left(e x^2+d\right)}{2 c d+\left(\sqrt{b^2-4 a c}-b\right) e}\right)}{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}-\frac{c \left(\frac{2 a c-b^2}{\sqrt{b^2-4 a c}}+b\right) \, _2F_1\left(1,q+1;q+2;\frac{2 c \left(e x^2+d\right)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}+\frac{a e \, _2F_1\left(2,q+1;q+2;\frac{e x^2}{d}+1\right)}{d^2}+\frac{b \, _2F_1\left(1,q+1;q+2;\frac{e x^2}{d}+1\right)}{d}\right)}{2 a^2 (q+1)}","-\frac{c \left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \left(d+e x^2\right)^{q+1} \, _2F_1\left(1,q+1;q+2;\frac{2 c \left(e x^2+d\right)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e}\right)}{2 a^2 (q+1) \left(2 c d-e \left(b-\sqrt{b^2-4 a c}\right)\right)}-\frac{c \left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \left(d+e x^2\right)^{q+1} \, _2F_1\left(1,q+1;q+2;\frac{2 c \left(e x^2+d\right)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{2 a^2 (q+1) \left(2 c d-e \left(\sqrt{b^2-4 a c}+b\right)\right)}+\frac{b \left(d+e x^2\right)^{q+1} \, _2F_1\left(1,q+1;q+2;\frac{e x^2}{d}+1\right)}{2 a^2 d (q+1)}+\frac{e \left(d+e x^2\right)^{q+1} \, _2F_1\left(2,q+1;q+2;\frac{e x^2}{d}+1\right)}{2 a d^2 (q+1)}",1,"((d + e*x^2)^(1 + q)*(-((c*(b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*Hypergeometric2F1[1, 1 + q, 2 + q, (2*c*(d + e*x^2))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)])/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)) - (c*(b + (-b^2 + 2*a*c)/Sqrt[b^2 - 4*a*c])*Hypergeometric2F1[1, 1 + q, 2 + q, (2*c*(d + e*x^2))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e) + (b*Hypergeometric2F1[1, 1 + q, 2 + q, 1 + (e*x^2)/d])/d + (a*e*Hypergeometric2F1[2, 1 + q, 2 + q, 1 + (e*x^2)/d])/d^2))/(2*a^2*(1 + q))","A",1
407,0,0,339,0.4936276,"\int \frac{x^6 \left(d+e x^2\right)^q}{a+b x^2+c x^4} \, dx","Integrate[(x^6*(d + e*x^2)^q)/(a + b*x^2 + c*x^4),x]","\int \frac{x^6 \left(d+e x^2\right)^q}{a+b x^2+c x^4} \, dx","\frac{x \left(-\frac{b \left(b^2-3 a c\right)}{\sqrt{b^2-4 a c}}-a c+b^2\right) \left(d+e x^2\right)^q \left(\frac{e x^2}{d}+1\right)^{-q} F_1\left(\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right)}{c^2 \left(b-\sqrt{b^2-4 a c}\right)}+\frac{x \left(\frac{b \left(b^2-3 a c\right)}{\sqrt{b^2-4 a c}}-a c+b^2\right) \left(d+e x^2\right)^q \left(\frac{e x^2}{d}+1\right)^{-q} F_1\left(\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right)}{c^2 \left(\sqrt{b^2-4 a c}+b\right)}-\frac{b x \left(d+e x^2\right)^q \left(\frac{e x^2}{d}+1\right)^{-q} \, _2F_1\left(\frac{1}{2},-q;\frac{3}{2};-\frac{e x^2}{d}\right)}{c^2}+\frac{x^3 \left(d+e x^2\right)^q \left(\frac{e x^2}{d}+1\right)^{-q} \, _2F_1\left(\frac{3}{2},-q;\frac{5}{2};-\frac{e x^2}{d}\right)}{3 c}",1,"Integrate[(x^6*(d + e*x^2)^q)/(a + b*x^2 + c*x^4), x]","F",-1
408,0,0,273,0.2345495,"\int \frac{x^4 \left(d+e x^2\right)^q}{a+b x^2+c x^4} \, dx","Integrate[(x^4*(d + e*x^2)^q)/(a + b*x^2 + c*x^4),x]","\int \frac{x^4 \left(d+e x^2\right)^q}{a+b x^2+c x^4} \, dx","-\frac{x \left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \left(d+e x^2\right)^q \left(\frac{e x^2}{d}+1\right)^{-q} F_1\left(\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right)}{c \left(b-\sqrt{b^2-4 a c}\right)}-\frac{x \left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \left(d+e x^2\right)^q \left(\frac{e x^2}{d}+1\right)^{-q} F_1\left(\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right)}{c \left(\sqrt{b^2-4 a c}+b\right)}+\frac{x \left(d+e x^2\right)^q \left(\frac{e x^2}{d}+1\right)^{-q} \, _2F_1\left(\frac{1}{2},-q;\frac{3}{2};-\frac{e x^2}{d}\right)}{c}",1,"Integrate[(x^4*(d + e*x^2)^q)/(a + b*x^2 + c*x^4), x]","F",-1
409,0,0,162,0.0994611,"\int \frac{x^2 \left(d+e x^2\right)^q}{a+b x^2+c x^4} \, dx","Integrate[(x^2*(d + e*x^2)^q)/(a + b*x^2 + c*x^4),x]","\int \frac{x^2 \left(d+e x^2\right)^q}{a+b x^2+c x^4} \, dx","\frac{x \left(d+e x^2\right)^q \left(\frac{e x^2}{d}+1\right)^{-q} F_1\left(\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right)}{\sqrt{b^2-4 a c}}-\frac{x \left(d+e x^2\right)^q \left(\frac{e x^2}{d}+1\right)^{-q} F_1\left(\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right)}{\sqrt{b^2-4 a c}}",1,"Integrate[(x^2*(d + e*x^2)^q)/(a + b*x^2 + c*x^4), x]","F",-1
410,0,0,190,0.0353921,"\int \frac{\left(d+e x^2\right)^q}{a+b x^2+c x^4} \, dx","Integrate[(d + e*x^2)^q/(a + b*x^2 + c*x^4),x]","\int \frac{\left(d+e x^2\right)^q}{a+b x^2+c x^4} \, dx","-\frac{2 c x \left(d+e x^2\right)^q \left(\frac{e x^2}{d}+1\right)^{-q} F_1\left(\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right)}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{2 c x \left(d+e x^2\right)^q \left(\frac{e x^2}{d}+1\right)^{-q} F_1\left(\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right)}{b \sqrt{b^2-4 a c}-4 a c+b^2}",1,"Integrate[(d + e*x^2)^q/(a + b*x^2 + c*x^4), x]","F",-1
411,0,0,264,0.2050437,"\int \frac{\left(d+e x^2\right)^q}{x^2 \left(a+b x^2+c x^4\right)} \, dx","Integrate[(d + e*x^2)^q/(x^2*(a + b*x^2 + c*x^4)),x]","\int \frac{\left(d+e x^2\right)^q}{x^2 \left(a+b x^2+c x^4\right)} \, dx","-\frac{c x \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \left(d+e x^2\right)^q \left(\frac{e x^2}{d}+1\right)^{-q} F_1\left(\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right)}{a \left(b-\sqrt{b^2-4 a c}\right)}-\frac{c x \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \left(d+e x^2\right)^q \left(\frac{e x^2}{d}+1\right)^{-q} F_1\left(\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right)}{a \left(\sqrt{b^2-4 a c}+b\right)}-\frac{\left(d+e x^2\right)^q \left(\frac{e x^2}{d}+1\right)^{-q} \, _2F_1\left(-\frac{1}{2},-q;\frac{1}{2};-\frac{e x^2}{d}\right)}{a x}",1,"Integrate[(d + e*x^2)^q/(x^2*(a + b*x^2 + c*x^4)), x]","F",-1
412,0,0,328,0.4996146,"\int \frac{\left(d+e x^2\right)^q}{x^4 \left(a+b x^2+c x^4\right)} \, dx","Integrate[(d + e*x^2)^q/(x^4*(a + b*x^2 + c*x^4)),x]","\int \frac{\left(d+e x^2\right)^q}{x^4 \left(a+b x^2+c x^4\right)} \, dx","\frac{c x \left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \left(d+e x^2\right)^q \left(\frac{e x^2}{d}+1\right)^{-q} F_1\left(\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right)}{a^2 \left(b-\sqrt{b^2-4 a c}\right)}+\frac{c x \left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \left(d+e x^2\right)^q \left(\frac{e x^2}{d}+1\right)^{-q} F_1\left(\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right)}{a^2 \left(\sqrt{b^2-4 a c}+b\right)}+\frac{b \left(d+e x^2\right)^q \left(\frac{e x^2}{d}+1\right)^{-q} \, _2F_1\left(-\frac{1}{2},-q;\frac{1}{2};-\frac{e x^2}{d}\right)}{a^2 x}-\frac{\left(d+e x^2\right)^q \left(\frac{e x^2}{d}+1\right)^{-q} \, _2F_1\left(-\frac{3}{2},-q;-\frac{1}{2};-\frac{e x^2}{d}\right)}{3 a x^3}",1,"Integrate[(d + e*x^2)^q/(x^4*(a + b*x^2 + c*x^4)), x]","F",-1
413,1,44,40,0.0522198,"\int \frac{\sqrt{1+\frac{1}{c^2 x^2}}}{\sqrt{1-c^4 x^4}} \, dx","Integrate[Sqrt[1 + 1/(c^2*x^2)]/Sqrt[1 - c^4*x^4],x]","-\frac{x \sqrt{\frac{1}{c^2 x^2}+1} \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{\sqrt{c^2 x^2+1}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{1-c^4 x^4}}{c x \sqrt{\frac{1}{c^2 x^2}+1}}\right)}{c}",1,"-((Sqrt[1 + 1/(c^2*x^2)]*x*ArcTanh[Sqrt[1 - c^2*x^2]])/Sqrt[1 + c^2*x^2])","A",1