1,1,183,247,0.0783305,"\int \frac{c+d x^2}{a+b x^4} \, dx","Integrate[(c + d*x^2)/(a + b*x^4),x]","\frac{-\left(\sqrt{b} c-\sqrt{a} d\right) \left(\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)-\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)\right)-2 \left(\sqrt{a} d+\sqrt{b} c\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)+2 \left(\sqrt{a} d+\sqrt{b} c\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} a^{3/4} b^{3/4}}","-\frac{\left(\sqrt{b} c-\sqrt{a} d\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{4 \sqrt{2} a^{3/4} b^{3/4}}+\frac{\left(\sqrt{b} c-\sqrt{a} d\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{4 \sqrt{2} a^{3/4} b^{3/4}}-\frac{\left(\sqrt{a} d+\sqrt{b} c\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} a^{3/4} b^{3/4}}+\frac{\left(\sqrt{a} d+\sqrt{b} c\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} a^{3/4} b^{3/4}}",1,"(-2*(Sqrt[b]*c + Sqrt[a]*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*(Sqrt[b]*c + Sqrt[a]*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] - (Sqrt[b]*c - Sqrt[a]*d)*(Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] - Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2]))/(4*Sqrt[2]*a^(3/4)*b^(3/4))","A",1
2,1,184,247,0.0453261,"\int \frac{c-d x^2}{a+b x^4} \, dx","Integrate[(c - d*x^2)/(a + b*x^4),x]","\frac{-\left(\sqrt{a} d+\sqrt{b} c\right) \left(\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)-\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)\right)+\left(2 \sqrt{a} d-2 \sqrt{b} c\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)+2 \left(\sqrt{b} c-\sqrt{a} d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} a^{3/4} b^{3/4}}","-\frac{\left(\sqrt{a} d+\sqrt{b} c\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{4 \sqrt{2} a^{3/4} b^{3/4}}+\frac{\left(\sqrt{a} d+\sqrt{b} c\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{4 \sqrt{2} a^{3/4} b^{3/4}}-\frac{\left(\sqrt{b} c-\sqrt{a} d\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} a^{3/4} b^{3/4}}+\frac{\left(\sqrt{b} c-\sqrt{a} d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} a^{3/4} b^{3/4}}",1,"((-2*Sqrt[b]*c + 2*Sqrt[a]*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*(Sqrt[b]*c - Sqrt[a]*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] - (Sqrt[b]*c + Sqrt[a]*d)*(Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] - Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2]))/(4*Sqrt[2]*a^(3/4)*b^(3/4))","A",1
3,1,95,86,0.0298397,"\int \frac{c+d x^2}{a-b x^4} \, dx","Integrate[(c + d*x^2)/(a - b*x^4),x]","\frac{2 \left(\sqrt{b} c-\sqrt{a} d\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)-\left(\sqrt{a} d+\sqrt{b} c\right) \left(\log \left(\sqrt[4]{a}-\sqrt[4]{b} x\right)-\log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right)\right)}{4 a^{3/4} b^{3/4}}","\frac{\left(\sqrt{b} c-\sqrt{a} d\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^{3/4}}+\frac{\left(\sqrt{a} d+\sqrt{b} c\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^{3/4}}",1,"(2*(Sqrt[b]*c - Sqrt[a]*d)*ArcTan[(b^(1/4)*x)/a^(1/4)] - (Sqrt[b]*c + Sqrt[a]*d)*(Log[a^(1/4) - b^(1/4)*x] - Log[a^(1/4) + b^(1/4)*x]))/(4*a^(3/4)*b^(3/4))","A",1
4,1,95,86,0.0227694,"\int \frac{c-d x^2}{a-b x^4} \, dx","Integrate[(c - d*x^2)/(a - b*x^4),x]","\frac{2 \left(\sqrt{a} d+\sqrt{b} c\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)-\left(\sqrt{b} c-\sqrt{a} d\right) \left(\log \left(\sqrt[4]{a}-\sqrt[4]{b} x\right)-\log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right)\right)}{4 a^{3/4} b^{3/4}}","\frac{\left(\sqrt{a} d+\sqrt{b} c\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^{3/4}}+\frac{\left(\sqrt{b} c-\sqrt{a} d\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^{3/4}}",1,"(2*(Sqrt[b]*c + Sqrt[a]*d)*ArcTan[(b^(1/4)*x)/a^(1/4)] - (Sqrt[b]*c - Sqrt[a]*d)*(Log[a^(1/4) - b^(1/4)*x] - Log[a^(1/4) + b^(1/4)*x]))/(4*a^(3/4)*b^(3/4))","A",1
5,1,33,40,0.0136519,"\int \frac{2+3 x^2}{4+9 x^4} \, dx","Integrate[(2 + 3*x^2)/(4 + 9*x^4),x]","\frac{\tan ^{-1}\left(\sqrt{3} x+1\right)-\tan ^{-1}\left(1-\sqrt{3} x\right)}{2 \sqrt{3}}","\frac{\tan ^{-1}\left(\sqrt{3} x+1\right)}{2 \sqrt{3}}-\frac{\tan ^{-1}\left(1-\sqrt{3} x\right)}{2 \sqrt{3}}",1,"(-ArcTan[1 - Sqrt[3]*x] + ArcTan[1 + Sqrt[3]*x])/(2*Sqrt[3])","A",1
6,1,44,51,0.0142354,"\int \frac{2-3 x^2}{4+9 x^4} \, dx","Integrate[(2 - 3*x^2)/(4 + 9*x^4),x]","\frac{\log \left(3 x^2+2 \sqrt{3} x+2\right)-\log \left(-3 x^2+2 \sqrt{3} x-2\right)}{4 \sqrt{3}}","\frac{\log \left(3 x^2+2 \sqrt{3} x+2\right)}{4 \sqrt{3}}-\frac{\log \left(3 x^2-2 \sqrt{3} x+2\right)}{4 \sqrt{3}}",1,"(-Log[-2 + 2*Sqrt[3]*x - 3*x^2] + Log[2 + 2*Sqrt[3]*x + 3*x^2])/(4*Sqrt[3])","A",1
7,1,32,16,0.0146103,"\int \frac{2+3 x^2}{4-9 x^4} \, dx","Integrate[(2 + 3*x^2)/(4 - 9*x^4),x]","\frac{\log \left(3 x+\sqrt{6}\right)-\log \left(\sqrt{6}-3 x\right)}{2 \sqrt{6}}","\frac{\tanh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)}{\sqrt{6}}",1,"(-Log[Sqrt[6] - 3*x] + Log[Sqrt[6] + 3*x])/(2*Sqrt[6])","A",1
8,1,16,16,0.0052891,"\int \frac{2-3 x^2}{4-9 x^4} \, dx","Integrate[(2 - 3*x^2)/(4 - 9*x^4),x]","\frac{\tan ^{-1}\left(\sqrt{\frac{3}{2}} x\right)}{\sqrt{6}}","\frac{\tan ^{-1}\left(\sqrt{\frac{3}{2}} x\right)}{\sqrt{6}}",1,"ArcTan[Sqrt[3/2]*x]/Sqrt[6]","A",1
9,1,60,75,0.0212339,"\int \frac{\sqrt{a} \sqrt{b}+b x^2}{a+b x^4} \, dx","Integrate[(Sqrt[a]*Sqrt[b] + b*x^2)/(a + b*x^4),x]","\frac{\sqrt[4]{b} \left(\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)-\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right)}{\sqrt{2} \sqrt[4]{a}}","\frac{\sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{\sqrt{2} \sqrt[4]{a}}-\frac{\sqrt[4]{b} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{\sqrt{2} \sqrt[4]{a}}",1,"(b^(1/4)*(-ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)]))/(Sqrt[2]*a^(1/4))","A",1
10,1,91,106,0.0217515,"\int \frac{\sqrt{a} \sqrt{b}-b x^2}{a+b x^4} \, dx","Integrate[(Sqrt[a]*Sqrt[b] - b*x^2)/(a + b*x^4),x]","\frac{\sqrt[4]{b} \left(\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)-\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x-\sqrt{a}-\sqrt{b} x^2\right)\right)}{2 \sqrt{2} \sqrt[4]{a}}","\frac{\sqrt[4]{b} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{2 \sqrt{2} \sqrt[4]{a}}-\frac{\sqrt[4]{b} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{2 \sqrt{2} \sqrt[4]{a}}",1,"(b^(1/4)*(-Log[-Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x - Sqrt[b]*x^2] + Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2]))/(2*Sqrt[2]*a^(1/4))","A",1
11,1,60,75,0.0349049,"\int \frac{d+e x^2}{d^2+e^2 x^4} \, dx","Integrate[(d + e*x^2)/(d^2 + e^2*x^4),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d}}+1\right)-\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{2} \sqrt{d} \sqrt{e}}","\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d}}+1\right)}{\sqrt{2} \sqrt{d} \sqrt{e}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{2} \sqrt{d} \sqrt{e}}",1,"(-ArcTan[1 - (Sqrt[2]*Sqrt[e]*x)/Sqrt[d]] + ArcTan[1 + (Sqrt[2]*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[2]*Sqrt[d]*Sqrt[e])","A",1
12,1,75,90,0.022734,"\int \frac{d-e x^2}{d^2+e^2 x^4} \, dx","Integrate[(d - e*x^2)/(d^2 + e^2*x^4),x]","\frac{\log \left(\sqrt{2} \sqrt{d} \sqrt{e} x+d+e x^2\right)-\log \left(\sqrt{2} \sqrt{d} \sqrt{e} x-d-e x^2\right)}{2 \sqrt{2} \sqrt{d} \sqrt{e}}","\frac{\log \left(\sqrt{2} \sqrt{d} \sqrt{e} x+d+e x^2\right)}{2 \sqrt{2} \sqrt{d} \sqrt{e}}-\frac{\log \left(-\sqrt{2} \sqrt{d} \sqrt{e} x+d+e x^2\right)}{2 \sqrt{2} \sqrt{d} \sqrt{e}}",1,"(-Log[-d + Sqrt[2]*Sqrt[d]*Sqrt[e]*x - e*x^2] + Log[d + Sqrt[2]*Sqrt[d]*Sqrt[e]*x + e*x^2])/(2*Sqrt[2]*Sqrt[d]*Sqrt[e])","A",1
13,1,25,13,0.0062166,"\int \frac{5+2 x^2}{-1+x^4} \, dx","Integrate[(5 + 2*x^2)/(-1 + x^4),x]","\frac{7}{4} \log (1-x)-\frac{7}{4} \log (x+1)-\frac{3}{2} \tan ^{-1}(x)","-\frac{3}{2} \tan ^{-1}(x)-\frac{7}{2} \tanh ^{-1}(x)",1,"(-3*ArcTan[x])/2 + (7*Log[1 - x])/4 - (7*Log[1 + x])/4","A",1
14,1,45,16,0.0124344,"\int \frac{1+b x^2}{\sqrt{1-b^2 x^4}} \, dx","Integrate[(1 + b*x^2)/Sqrt[1 - b^2*x^4],x]","x \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};b^2 x^4\right)+\frac{1}{3} b x^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};b^2 x^4\right)","\frac{E\left(\left.\sin ^{-1}\left(\sqrt{b} x\right)\right|-1\right)}{\sqrt{b}}",1,"x*Hypergeometric2F1[1/4, 1/2, 5/4, b^2*x^4] + (b*x^3*Hypergeometric2F1[1/2, 3/4, 7/4, b^2*x^4])/3","C",1
15,1,45,35,0.0117958,"\int \frac{1-b x^2}{\sqrt{1-b^2 x^4}} \, dx","Integrate[(1 - b*x^2)/Sqrt[1 - b^2*x^4],x]","x \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};b^2 x^4\right)-\frac{1}{3} b x^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};b^2 x^4\right)","\frac{2 F\left(\left.\sin ^{-1}\left(\sqrt{b} x\right)\right|-1\right)}{\sqrt{b}}-\frac{E\left(\left.\sin ^{-1}\left(\sqrt{b} x\right)\right|-1\right)}{\sqrt{b}}",1,"x*Hypergeometric2F1[1/4, 1/2, 5/4, b^2*x^4] - (b*x^3*Hypergeometric2F1[1/2, 3/4, 7/4, b^2*x^4])/3","C",1
16,1,74,43,0.0232229,"\int \frac{1+b x^2}{\sqrt{-1+b^2 x^4}} \, dx","Integrate[(1 + b*x^2)/Sqrt[-1 + b^2*x^4],x]","\frac{\sqrt{1-b^2 x^4} \left(3 x \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};b^2 x^4\right)+b x^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};b^2 x^4\right)\right)}{3 \sqrt{b^2 x^4-1}}","\frac{\sqrt{1-b^2 x^4} E\left(\left.\sin ^{-1}\left(\sqrt{b} x\right)\right|-1\right)}{\sqrt{b} \sqrt{b^2 x^4-1}}",1,"(Sqrt[1 - b^2*x^4]*(3*x*Hypergeometric2F1[1/4, 1/2, 5/4, b^2*x^4] + b*x^3*Hypergeometric2F1[1/2, 3/4, 7/4, b^2*x^4]))/(3*Sqrt[-1 + b^2*x^4])","C",1
17,1,74,89,0.0199227,"\int \frac{1-b x^2}{\sqrt{-1+b^2 x^4}} \, dx","Integrate[(1 - b*x^2)/Sqrt[-1 + b^2*x^4],x]","-\frac{\sqrt{1-b^2 x^4} \left(b x^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};b^2 x^4\right)-3 x \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};b^2 x^4\right)\right)}{3 \sqrt{b^2 x^4-1}}","\frac{2 \sqrt{1-b^2 x^4} F\left(\left.\sin ^{-1}\left(\sqrt{b} x\right)\right|-1\right)}{\sqrt{b} \sqrt{b^2 x^4-1}}-\frac{\sqrt{1-b^2 x^4} E\left(\left.\sin ^{-1}\left(\sqrt{b} x\right)\right|-1\right)}{\sqrt{b} \sqrt{b^2 x^4-1}}",1,"-1/3*(Sqrt[1 - b^2*x^4]*(-3*x*Hypergeometric2F1[1/4, 1/2, 5/4, b^2*x^4] + b*x^3*Hypergeometric2F1[1/2, 3/4, 7/4, b^2*x^4]))/Sqrt[-1 + b^2*x^4]","C",1
18,1,47,89,0.0116504,"\int \frac{1-b x^2}{\sqrt{1+b^2 x^4}} \, dx","Integrate[(1 - b*x^2)/Sqrt[1 + b^2*x^4],x]","x \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-b^2 x^4\right)-\frac{1}{3} b x^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-b^2 x^4\right)","\frac{\left(b x^2+1\right) \sqrt{\frac{b^2 x^4+1}{\left(b x^2+1\right)^2}} E\left(2 \tan ^{-1}\left(\sqrt{b} x\right)|\frac{1}{2}\right)}{\sqrt{b} \sqrt{b^2 x^4+1}}-\frac{x \sqrt{b^2 x^4+1}}{b x^2+1}",1,"x*Hypergeometric2F1[1/4, 1/2, 5/4, -(b^2*x^4)] - (b*x^3*Hypergeometric2F1[1/2, 3/4, 7/4, -(b^2*x^4)])/3","C",1
19,1,47,152,0.0104564,"\int \frac{1+b x^2}{\sqrt{1+b^2 x^4}} \, dx","Integrate[(1 + b*x^2)/Sqrt[1 + b^2*x^4],x]","x \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-b^2 x^4\right)+\frac{1}{3} b x^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-b^2 x^4\right)","\frac{x \sqrt{b^2 x^4+1}}{b x^2+1}+\frac{\left(b x^2+1\right) \sqrt{\frac{b^2 x^4+1}{\left(b x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt{b} x\right)|\frac{1}{2}\right)}{\sqrt{b} \sqrt{b^2 x^4+1}}-\frac{\left(b x^2+1\right) \sqrt{\frac{b^2 x^4+1}{\left(b x^2+1\right)^2}} E\left(2 \tan ^{-1}\left(\sqrt{b} x\right)|\frac{1}{2}\right)}{\sqrt{b} \sqrt{b^2 x^4+1}}",1,"x*Hypergeometric2F1[1/4, 1/2, 5/4, -(b^2*x^4)] + (b*x^3*Hypergeometric2F1[1/2, 3/4, 7/4, -(b^2*x^4)])/3","C",1
20,1,76,90,0.0266502,"\int \frac{1-b x^2}{\sqrt{-1-b^2 x^4}} \, dx","Integrate[(1 - b*x^2)/Sqrt[-1 - b^2*x^4],x]","-\frac{\sqrt{b^2 x^4+1} \left(b x^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-b^2 x^4\right)-3 x \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-b^2 x^4\right)\right)}{3 \sqrt{-b^2 x^4-1}}","\frac{x \sqrt{-b^2 x^4-1}}{b x^2+1}+\frac{\left(b x^2+1\right) \sqrt{\frac{b^2 x^4+1}{\left(b x^2+1\right)^2}} E\left(2 \tan ^{-1}\left(\sqrt{b} x\right)|\frac{1}{2}\right)}{\sqrt{b} \sqrt{-b^2 x^4-1}}",1,"-1/3*(Sqrt[1 + b^2*x^4]*(-3*x*Hypergeometric2F1[1/4, 1/2, 5/4, -(b^2*x^4)] + b*x^3*Hypergeometric2F1[1/2, 3/4, 7/4, -(b^2*x^4)]))/Sqrt[-1 - b^2*x^4]","C",1
21,1,76,156,0.0184352,"\int \frac{1+b x^2}{\sqrt{-1-b^2 x^4}} \, dx","Integrate[(1 + b*x^2)/Sqrt[-1 - b^2*x^4],x]","\frac{\sqrt{b^2 x^4+1} \left(3 x \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-b^2 x^4\right)+b x^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-b^2 x^4\right)\right)}{3 \sqrt{-b^2 x^4-1}}","-\frac{x \sqrt{-b^2 x^4-1}}{b x^2+1}+\frac{\left(b x^2+1\right) \sqrt{\frac{b^2 x^4+1}{\left(b x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt{b} x\right)|\frac{1}{2}\right)}{\sqrt{b} \sqrt{-b^2 x^4-1}}-\frac{\left(b x^2+1\right) \sqrt{\frac{b^2 x^4+1}{\left(b x^2+1\right)^2}} E\left(2 \tan ^{-1}\left(\sqrt{b} x\right)|\frac{1}{2}\right)}{\sqrt{b} \sqrt{-b^2 x^4-1}}",1,"(Sqrt[1 + b^2*x^4]*(3*x*Hypergeometric2F1[1/4, 1/2, 5/4, -(b^2*x^4)] + b*x^3*Hypergeometric2F1[1/2, 3/4, 7/4, -(b^2*x^4)]))/(3*Sqrt[-1 - b^2*x^4])","C",1
22,1,10,10,0.0064599,"\int \frac{\sqrt{1+c^2 x^2}}{\sqrt{1-c^2 x^2}} \, dx","Integrate[Sqrt[1 + c^2*x^2]/Sqrt[1 - c^2*x^2],x]","\frac{E\left(\left.\sin ^{-1}(c x)\right|-1\right)}{c}","\frac{E\left(\left.\sin ^{-1}(c x)\right|-1\right)}{c}",1,"EllipticE[ArcSin[c*x], -1]/c","A",1
23,1,47,10,0.012364,"\int \frac{1+c^2 x^2}{\sqrt{1-c^4 x^4}} \, dx","Integrate[(1 + c^2*x^2)/Sqrt[1 - c^4*x^4],x]","x \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};c^4 x^4\right)+\frac{1}{3} c^2 x^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};c^4 x^4\right)","\frac{E\left(\left.\sin ^{-1}(c x)\right|-1\right)}{c}",1,"x*Hypergeometric2F1[1/4, 1/2, 5/4, c^4*x^4] + (c^2*x^3*Hypergeometric2F1[1/2, 3/4, 7/4, c^4*x^4])/3","C",1
24,1,24,23,0.0079414,"\int \frac{\sqrt{1-c^2 x^2}}{\sqrt{1+c^2 x^2}} \, dx","Integrate[Sqrt[1 - c^2*x^2]/Sqrt[1 + c^2*x^2],x]","\frac{E\left(\left.\sin ^{-1}\left(\sqrt{-c^2} x\right)\right|-1\right)}{\sqrt{-c^2}}","\frac{2 F\left(\left.\sin ^{-1}(c x)\right|-1\right)}{c}-\frac{E\left(\left.\sin ^{-1}(c x)\right|-1\right)}{c}",1,"EllipticE[ArcSin[Sqrt[-c^2]*x], -1]/Sqrt[-c^2]","A",1
25,1,47,23,0.0119999,"\int \frac{1-c^2 x^2}{\sqrt{1-c^4 x^4}} \, dx","Integrate[(1 - c^2*x^2)/Sqrt[1 - c^4*x^4],x]","x \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};c^4 x^4\right)-\frac{1}{3} c^2 x^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};c^4 x^4\right)","\frac{2 F\left(\left.\sin ^{-1}(c x)\right|-1\right)}{c}-\frac{E\left(\left.\sin ^{-1}(c x)\right|-1\right)}{c}",1,"x*Hypergeometric2F1[1/4, 1/2, 5/4, c^4*x^4] - (c^2*x^3*Hypergeometric2F1[1/2, 3/4, 7/4, c^4*x^4])/3","C",1
26,1,181,82,0.1101,"\int \frac{d+e x^2}{d^2+b x^2+e^2 x^4} \, dx","Integrate[(d + e*x^2)/(d^2 + b*x^2 + e^2*x^4),x]","\frac{\frac{\left(\sqrt{b^2-4 d^2 e^2}-b+2 d e\right) \tan ^{-1}\left(\frac{\sqrt{2} e x}{\sqrt{b-\sqrt{b^2-4 d^2 e^2}}}\right)}{\sqrt{b-\sqrt{b^2-4 d^2 e^2}}}+\frac{\left(\sqrt{b^2-4 d^2 e^2}+b-2 d e\right) \tan ^{-1}\left(\frac{\sqrt{2} e x}{\sqrt{\sqrt{b^2-4 d^2 e^2}+b}}\right)}{\sqrt{\sqrt{b^2-4 d^2 e^2}+b}}}{\sqrt{2} \sqrt{b^2-4 d^2 e^2}}","\frac{\tan ^{-1}\left(\frac{\sqrt{2 d e-b}+2 e x}{\sqrt{b+2 d e}}\right)}{\sqrt{b+2 d e}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2 d e-b}-2 e x}{\sqrt{b+2 d e}}\right)}{\sqrt{b+2 d e}}",1,"(((-b + 2*d*e + Sqrt[b^2 - 4*d^2*e^2])*ArcTan[(Sqrt[2]*e*x)/Sqrt[b - Sqrt[b^2 - 4*d^2*e^2]]])/Sqrt[b - Sqrt[b^2 - 4*d^2*e^2]] + ((b - 2*d*e + Sqrt[b^2 - 4*d^2*e^2])*ArcTan[(Sqrt[2]*e*x)/Sqrt[b + Sqrt[b^2 - 4*d^2*e^2]]])/Sqrt[b + Sqrt[b^2 - 4*d^2*e^2]])/(Sqrt[2]*Sqrt[b^2 - 4*d^2*e^2])","B",1
27,1,181,82,0.1113742,"\int \frac{d+e x^2}{d^2+f x^2+e^2 x^4} \, dx","Integrate[(d + e*x^2)/(d^2 + f*x^2 + e^2*x^4),x]","\frac{\frac{\left(\sqrt{f^2-4 d^2 e^2}+2 d e-f\right) \tan ^{-1}\left(\frac{\sqrt{2} e x}{\sqrt{f-\sqrt{f^2-4 d^2 e^2}}}\right)}{\sqrt{f-\sqrt{f^2-4 d^2 e^2}}}+\frac{\left(\sqrt{f^2-4 d^2 e^2}-2 d e+f\right) \tan ^{-1}\left(\frac{\sqrt{2} e x}{\sqrt{\sqrt{f^2-4 d^2 e^2}+f}}\right)}{\sqrt{\sqrt{f^2-4 d^2 e^2}+f}}}{\sqrt{2} \sqrt{f^2-4 d^2 e^2}}","\frac{\tan ^{-1}\left(\frac{\sqrt{2 d e-f}+2 e x}{\sqrt{2 d e+f}}\right)}{\sqrt{2 d e+f}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2 d e-f}-2 e x}{\sqrt{2 d e+f}}\right)}{\sqrt{2 d e+f}}",1,"(((2*d*e - f + Sqrt[-4*d^2*e^2 + f^2])*ArcTan[(Sqrt[2]*e*x)/Sqrt[f - Sqrt[-4*d^2*e^2 + f^2]]])/Sqrt[f - Sqrt[-4*d^2*e^2 + f^2]] + ((-2*d*e + f + Sqrt[-4*d^2*e^2 + f^2])*ArcTan[(Sqrt[2]*e*x)/Sqrt[f + Sqrt[-4*d^2*e^2 + f^2]]])/Sqrt[f + Sqrt[-4*d^2*e^2 + f^2]])/(Sqrt[2]*Sqrt[-4*d^2*e^2 + f^2])","B",1
28,1,189,78,0.1053049,"\int \frac{d+e x^2}{d^2-b x^2+e^2 x^4} \, dx","Integrate[(d + e*x^2)/(d^2 - b*x^2 + e^2*x^4),x]","\frac{\frac{\left(\sqrt{b^2-4 d^2 e^2}+b+2 d e\right) \tan ^{-1}\left(\frac{\sqrt{2} e x}{\sqrt{-\sqrt{b^2-4 d^2 e^2}-b}}\right)}{\sqrt{-\sqrt{b^2-4 d^2 e^2}-b}}+\frac{\left(\sqrt{b^2-4 d^2 e^2}-b-2 d e\right) \tan ^{-1}\left(\frac{\sqrt{2} e x}{\sqrt{\sqrt{b^2-4 d^2 e^2}-b}}\right)}{\sqrt{\sqrt{b^2-4 d^2 e^2}-b}}}{\sqrt{2} \sqrt{b^2-4 d^2 e^2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b+2 d e}-2 e x}{\sqrt{b-2 d e}}\right)}{\sqrt{b-2 d e}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b+2 d e}+2 e x}{\sqrt{b-2 d e}}\right)}{\sqrt{b-2 d e}}",1,"(((b + 2*d*e + Sqrt[b^2 - 4*d^2*e^2])*ArcTan[(Sqrt[2]*e*x)/Sqrt[-b - Sqrt[b^2 - 4*d^2*e^2]]])/Sqrt[-b - Sqrt[b^2 - 4*d^2*e^2]] + ((-b - 2*d*e + Sqrt[b^2 - 4*d^2*e^2])*ArcTan[(Sqrt[2]*e*x)/Sqrt[-b + Sqrt[b^2 - 4*d^2*e^2]]])/Sqrt[-b + Sqrt[b^2 - 4*d^2*e^2]])/(Sqrt[2]*Sqrt[b^2 - 4*d^2*e^2])","B",1
29,1,189,86,0.1069399,"\int \frac{d+e x^2}{d^2-f x^2+e^2 x^4} \, dx","Integrate[(d + e*x^2)/(d^2 - f*x^2 + e^2*x^4),x]","\frac{\frac{\left(\sqrt{f^2-4 d^2 e^2}+2 d e+f\right) \tan ^{-1}\left(\frac{\sqrt{2} e x}{\sqrt{-\sqrt{f^2-4 d^2 e^2}-f}}\right)}{\sqrt{-\sqrt{f^2-4 d^2 e^2}-f}}+\frac{\left(\sqrt{f^2-4 d^2 e^2}-2 d e-f\right) \tan ^{-1}\left(\frac{\sqrt{2} e x}{\sqrt{\sqrt{f^2-4 d^2 e^2}-f}}\right)}{\sqrt{\sqrt{f^2-4 d^2 e^2}-f}}}{\sqrt{2} \sqrt{f^2-4 d^2 e^2}}","\frac{\tan ^{-1}\left(\frac{\sqrt{2 d e+f}+2 e x}{\sqrt{2 d e-f}}\right)}{\sqrt{2 d e-f}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2 d e+f}-2 e x}{\sqrt{2 d e-f}}\right)}{\sqrt{2 d e-f}}",1,"(((2*d*e + f + Sqrt[-4*d^2*e^2 + f^2])*ArcTan[(Sqrt[2]*e*x)/Sqrt[-f - Sqrt[-4*d^2*e^2 + f^2]]])/Sqrt[-f - Sqrt[-4*d^2*e^2 + f^2]] + ((-2*d*e - f + Sqrt[-4*d^2*e^2 + f^2])*ArcTan[(Sqrt[2]*e*x)/Sqrt[-f + Sqrt[-4*d^2*e^2 + f^2]]])/Sqrt[-f + Sqrt[-4*d^2*e^2 + f^2]])/(Sqrt[2]*Sqrt[-4*d^2*e^2 + f^2])","B",1
30,1,182,78,0.1205939,"\int \frac{d-e x^2}{d^2+b x^2+e^2 x^4} \, dx","Integrate[(d - e*x^2)/(d^2 + b*x^2 + e^2*x^4),x]","\frac{\frac{\left(-\sqrt{b^2-4 d^2 e^2}+b+2 d e\right) \tan ^{-1}\left(\frac{\sqrt{2} e x}{\sqrt{b-\sqrt{b^2-4 d^2 e^2}}}\right)}{\sqrt{b-\sqrt{b^2-4 d^2 e^2}}}-\frac{\left(\sqrt{b^2-4 d^2 e^2}+b+2 d e\right) \tan ^{-1}\left(\frac{\sqrt{2} e x}{\sqrt{\sqrt{b^2-4 d^2 e^2}+b}}\right)}{\sqrt{\sqrt{b^2-4 d^2 e^2}+b}}}{\sqrt{2} \sqrt{b^2-4 d^2 e^2}}","\frac{\log \left(x \sqrt{2 d e-b}+d+e x^2\right)}{2 \sqrt{2 d e-b}}-\frac{\log \left(-x \sqrt{2 d e-b}+d+e x^2\right)}{2 \sqrt{2 d e-b}}",1,"(((b + 2*d*e - Sqrt[b^2 - 4*d^2*e^2])*ArcTan[(Sqrt[2]*e*x)/Sqrt[b - Sqrt[b^2 - 4*d^2*e^2]]])/Sqrt[b - Sqrt[b^2 - 4*d^2*e^2]] - ((b + 2*d*e + Sqrt[b^2 - 4*d^2*e^2])*ArcTan[(Sqrt[2]*e*x)/Sqrt[b + Sqrt[b^2 - 4*d^2*e^2]]])/Sqrt[b + Sqrt[b^2 - 4*d^2*e^2]])/(Sqrt[2]*Sqrt[b^2 - 4*d^2*e^2])","B",1
31,1,182,78,0.1247936,"\int \frac{d-e x^2}{d^2+f x^2+e^2 x^4} \, dx","Integrate[(d - e*x^2)/(d^2 + f*x^2 + e^2*x^4),x]","\frac{\frac{\left(-\sqrt{f^2-4 d^2 e^2}+2 d e+f\right) \tan ^{-1}\left(\frac{\sqrt{2} e x}{\sqrt{f-\sqrt{f^2-4 d^2 e^2}}}\right)}{\sqrt{f-\sqrt{f^2-4 d^2 e^2}}}-\frac{\left(\sqrt{f^2-4 d^2 e^2}+2 d e+f\right) \tan ^{-1}\left(\frac{\sqrt{2} e x}{\sqrt{\sqrt{f^2-4 d^2 e^2}+f}}\right)}{\sqrt{\sqrt{f^2-4 d^2 e^2}+f}}}{\sqrt{2} \sqrt{f^2-4 d^2 e^2}}","\frac{\log \left(x \sqrt{2 d e-f}+d+e x^2\right)}{2 \sqrt{2 d e-f}}-\frac{\log \left(-x \sqrt{2 d e-f}+d+e x^2\right)}{2 \sqrt{2 d e-f}}",1,"(((2*d*e + f - Sqrt[-4*d^2*e^2 + f^2])*ArcTan[(Sqrt[2]*e*x)/Sqrt[f - Sqrt[-4*d^2*e^2 + f^2]]])/Sqrt[f - Sqrt[-4*d^2*e^2 + f^2]] - ((2*d*e + f + Sqrt[-4*d^2*e^2 + f^2])*ArcTan[(Sqrt[2]*e*x)/Sqrt[f + Sqrt[-4*d^2*e^2 + f^2]]])/Sqrt[f + Sqrt[-4*d^2*e^2 + f^2]])/(Sqrt[2]*Sqrt[-4*d^2*e^2 + f^2])","B",1
32,1,190,70,0.1299197,"\int \frac{d-e x^2}{d^2-b x^2+e^2 x^4} \, dx","Integrate[(d - e*x^2)/(d^2 - b*x^2 + e^2*x^4),x]","\frac{\frac{\left(-\sqrt{b^2-4 d^2 e^2}+b-2 d e\right) \tan ^{-1}\left(\frac{\sqrt{2} e x}{\sqrt{\sqrt{b^2-4 d^2 e^2}-b}}\right)}{\sqrt{\sqrt{b^2-4 d^2 e^2}-b}}-\frac{\left(\sqrt{b^2-4 d^2 e^2}+b-2 d e\right) \tan ^{-1}\left(\frac{\sqrt{2} e x}{\sqrt{-\sqrt{b^2-4 d^2 e^2}-b}}\right)}{\sqrt{-\sqrt{b^2-4 d^2 e^2}-b}}}{\sqrt{2} \sqrt{b^2-4 d^2 e^2}}","\frac{\log \left(x \sqrt{b+2 d e}+d+e x^2\right)}{2 \sqrt{b+2 d e}}-\frac{\log \left(-x \sqrt{b+2 d e}+d+e x^2\right)}{2 \sqrt{b+2 d e}}",1,"(-(((b - 2*d*e + Sqrt[b^2 - 4*d^2*e^2])*ArcTan[(Sqrt[2]*e*x)/Sqrt[-b - Sqrt[b^2 - 4*d^2*e^2]]])/Sqrt[-b - Sqrt[b^2 - 4*d^2*e^2]]) + ((b - 2*d*e - Sqrt[b^2 - 4*d^2*e^2])*ArcTan[(Sqrt[2]*e*x)/Sqrt[-b + Sqrt[b^2 - 4*d^2*e^2]]])/Sqrt[-b + Sqrt[b^2 - 4*d^2*e^2]])/(Sqrt[2]*Sqrt[b^2 - 4*d^2*e^2])","B",1
33,1,190,70,0.1278497,"\int \frac{d-e x^2}{d^2-f x^2+e^2 x^4} \, dx","Integrate[(d - e*x^2)/(d^2 - f*x^2 + e^2*x^4),x]","\frac{\frac{\left(-\sqrt{f^2-4 d^2 e^2}-2 d e+f\right) \tan ^{-1}\left(\frac{\sqrt{2} e x}{\sqrt{\sqrt{f^2-4 d^2 e^2}-f}}\right)}{\sqrt{\sqrt{f^2-4 d^2 e^2}-f}}-\frac{\left(\sqrt{f^2-4 d^2 e^2}-2 d e+f\right) \tan ^{-1}\left(\frac{\sqrt{2} e x}{\sqrt{-\sqrt{f^2-4 d^2 e^2}-f}}\right)}{\sqrt{-\sqrt{f^2-4 d^2 e^2}-f}}}{\sqrt{2} \sqrt{f^2-4 d^2 e^2}}","\frac{\log \left(x \sqrt{2 d e+f}+d+e x^2\right)}{2 \sqrt{2 d e+f}}-\frac{\log \left(-x \sqrt{2 d e+f}+d+e x^2\right)}{2 \sqrt{2 d e+f}}",1,"(-(((-2*d*e + f + Sqrt[-4*d^2*e^2 + f^2])*ArcTan[(Sqrt[2]*e*x)/Sqrt[-f - Sqrt[-4*d^2*e^2 + f^2]]])/Sqrt[-f - Sqrt[-4*d^2*e^2 + f^2]]) + ((-2*d*e + f - Sqrt[-4*d^2*e^2 + f^2])*ArcTan[(Sqrt[2]*e*x)/Sqrt[-f + Sqrt[-4*d^2*e^2 + f^2]]])/Sqrt[-f + Sqrt[-4*d^2*e^2 + f^2]])/(Sqrt[2]*Sqrt[-4*d^2*e^2 + f^2])","B",1
34,1,250,134,0.161269,"\int \frac{d-e x^2}{\frac{c d^2}{e^2}+b x^2+c x^4} \, dx","Integrate[(d - e*x^2)/((c*d^2)/e^2 + b*x^2 + c*x^4),x]","\frac{e^{3/2} \left(-\frac{\left(\sqrt{b^2 e^2-4 c^2 d^2}-b e-2 c d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{e} x}{\sqrt{b e-\sqrt{b^2 e^2-4 c^2 d^2}}}\right)}{\sqrt{b e-\sqrt{b^2 e^2-4 c^2 d^2}}}-\frac{\left(\sqrt{b^2 e^2-4 c^2 d^2}+b e+2 c d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{e} x}{\sqrt{\sqrt{b^2 e^2-4 c^2 d^2}+b e}}\right)}{\sqrt{\sqrt{b^2 e^2-4 c^2 d^2}+b e}}\right)}{\sqrt{2} \sqrt{c} \sqrt{b^2 e^2-4 c^2 d^2}}","\frac{e^{3/2} \log \left(\sqrt{e} x \sqrt{2 c d-b e}+\sqrt{c} d+\sqrt{c} e x^2\right)}{2 \sqrt{c} \sqrt{2 c d-b e}}-\frac{e^{3/2} \log \left(-\sqrt{e} x \sqrt{2 c d-b e}+\sqrt{c} d+\sqrt{c} e x^2\right)}{2 \sqrt{c} \sqrt{2 c d-b e}}",1,"(e^(3/2)*(-(((-2*c*d - b*e + Sqrt[-4*c^2*d^2 + b^2*e^2])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[e]*x)/Sqrt[b*e - Sqrt[-4*c^2*d^2 + b^2*e^2]]])/Sqrt[b*e - Sqrt[-4*c^2*d^2 + b^2*e^2]]) - ((2*c*d + b*e + Sqrt[-4*c^2*d^2 + b^2*e^2])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[e]*x)/Sqrt[b*e + Sqrt[-4*c^2*d^2 + b^2*e^2]]])/Sqrt[b*e + Sqrt[-4*c^2*d^2 + b^2*e^2]]))/(Sqrt[2]*Sqrt[c]*Sqrt[-4*c^2*d^2 + b^2*e^2])","A",1
35,1,248,130,0.1198966,"\int \frac{d+e x^2}{\frac{c d^2}{e^2}+b x^2+c x^4} \, dx","Integrate[(d + e*x^2)/((c*d^2)/e^2 + b*x^2 + c*x^4),x]","\frac{e^{3/2} \left(\frac{\left(\sqrt{b^2 e^2-4 c^2 d^2}-b e+2 c d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{e} x}{\sqrt{b e-\sqrt{b^2 e^2-4 c^2 d^2}}}\right)}{\sqrt{b e-\sqrt{b^2 e^2-4 c^2 d^2}}}+\frac{\left(\sqrt{b^2 e^2-4 c^2 d^2}+b e-2 c d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{e} x}{\sqrt{\sqrt{b^2 e^2-4 c^2 d^2}+b e}}\right)}{\sqrt{\sqrt{b^2 e^2-4 c^2 d^2}+b e}}\right)}{\sqrt{2} \sqrt{c} \sqrt{b^2 e^2-4 c^2 d^2}}","\frac{e^{3/2} \tan ^{-1}\left(\frac{\sqrt{2 c d-b e}+2 \sqrt{c} \sqrt{e} x}{\sqrt{b e+2 c d}}\right)}{\sqrt{c} \sqrt{b e+2 c d}}-\frac{e^{3/2} \tan ^{-1}\left(\frac{\sqrt{2 c d-b e}-2 \sqrt{c} \sqrt{e} x}{\sqrt{b e+2 c d}}\right)}{\sqrt{c} \sqrt{b e+2 c d}}",1,"(e^(3/2)*(((2*c*d - b*e + Sqrt[-4*c^2*d^2 + b^2*e^2])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[e]*x)/Sqrt[b*e - Sqrt[-4*c^2*d^2 + b^2*e^2]]])/Sqrt[b*e - Sqrt[-4*c^2*d^2 + b^2*e^2]] + ((-2*c*d + b*e + Sqrt[-4*c^2*d^2 + b^2*e^2])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[e]*x)/Sqrt[b*e + Sqrt[-4*c^2*d^2 + b^2*e^2]]])/Sqrt[b*e + Sqrt[-4*c^2*d^2 + b^2*e^2]]))/(Sqrt[2]*Sqrt[c]*Sqrt[-4*c^2*d^2 + b^2*e^2])","A",1
36,1,248,130,0.0436987,"\int \frac{d+e x^2}{b x^2+c \left(\frac{d^2}{e^2}+x^4\right)} \, dx","Integrate[(d + e*x^2)/(b*x^2 + c*(d^2/e^2 + x^4)),x]","\frac{e^{3/2} \left(\frac{\left(\sqrt{b^2 e^2-4 c^2 d^2}-b e+2 c d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{e} x}{\sqrt{b e-\sqrt{b^2 e^2-4 c^2 d^2}}}\right)}{\sqrt{b e-\sqrt{b^2 e^2-4 c^2 d^2}}}+\frac{\left(\sqrt{b^2 e^2-4 c^2 d^2}+b e-2 c d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{e} x}{\sqrt{\sqrt{b^2 e^2-4 c^2 d^2}+b e}}\right)}{\sqrt{\sqrt{b^2 e^2-4 c^2 d^2}+b e}}\right)}{\sqrt{2} \sqrt{c} \sqrt{b^2 e^2-4 c^2 d^2}}","\frac{e^{3/2} \tan ^{-1}\left(\frac{\sqrt{2 c d-b e}+2 \sqrt{c} \sqrt{e} x}{\sqrt{b e+2 c d}}\right)}{\sqrt{c} \sqrt{b e+2 c d}}-\frac{e^{3/2} \tan ^{-1}\left(\frac{\sqrt{2 c d-b e}-2 \sqrt{c} \sqrt{e} x}{\sqrt{b e+2 c d}}\right)}{\sqrt{c} \sqrt{b e+2 c d}}",1,"(e^(3/2)*(((2*c*d - b*e + Sqrt[-4*c^2*d^2 + b^2*e^2])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[e]*x)/Sqrt[b*e - Sqrt[-4*c^2*d^2 + b^2*e^2]]])/Sqrt[b*e - Sqrt[-4*c^2*d^2 + b^2*e^2]] + ((-2*c*d + b*e + Sqrt[-4*c^2*d^2 + b^2*e^2])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[e]*x)/Sqrt[b*e + Sqrt[-4*c^2*d^2 + b^2*e^2]]])/Sqrt[b*e + Sqrt[-4*c^2*d^2 + b^2*e^2]]))/(Sqrt[2]*Sqrt[c]*Sqrt[-4*c^2*d^2 + b^2*e^2])","A",1
37,1,29,29,0.0191497,"\int \frac{a-b x^2}{a^2+(-1+2 a b) x^2+b^2 x^4} \, dx","Integrate[(a - b*x^2)/(a^2 + (-1 + 2*a*b)*x^2 + b^2*x^4),x]","\frac{1}{2} \log \left(a+b x^2+x\right)-\frac{1}{2} \log \left(a+b x^2-x\right)","\frac{1}{2} \log \left(a+b x^2+x\right)-\frac{1}{2} \log \left(a+b x^2-x\right)",1,"-1/2*Log[a - x + b*x^2] + Log[a + x + b*x^2]/2","A",1
38,1,138,60,0.2015208,"\int \frac{a+b x^2}{a^2+(-1+2 a b) x^2+b^2 x^4} \, dx","Integrate[(a + b*x^2)/(a^2 + (-1 + 2*a*b)*x^2 + b^2*x^4),x]","\frac{\frac{\left(\sqrt{1-4 a b}+1\right) \tan ^{-1}\left(\frac{b x}{\sqrt{a b-\frac{1}{2} \sqrt{1-4 a b}-\frac{1}{2}}}\right)}{\sqrt{2 a b-\sqrt{1-4 a b}-1}}+\frac{\left(\sqrt{1-4 a b}-1\right) \tan ^{-1}\left(\frac{\sqrt{2} b x}{\sqrt{2 a b+\sqrt{1-4 a b}-1}}\right)}{\sqrt{2 a b+\sqrt{1-4 a b}-1}}}{\sqrt{2-8 a b}}","\frac{\tanh ^{-1}\left(\frac{1-2 b x}{\sqrt{1-4 a b}}\right)}{\sqrt{1-4 a b}}-\frac{\tanh ^{-1}\left(\frac{2 b x+1}{\sqrt{1-4 a b}}\right)}{\sqrt{1-4 a b}}",1,"(((1 + Sqrt[1 - 4*a*b])*ArcTan[(b*x)/Sqrt[-1/2 + a*b - Sqrt[1 - 4*a*b]/2]])/Sqrt[-1 + 2*a*b - Sqrt[1 - 4*a*b]] + ((-1 + Sqrt[1 - 4*a*b])*ArcTan[(Sqrt[2]*b*x)/Sqrt[-1 + 2*a*b + Sqrt[1 - 4*a*b]]])/Sqrt[-1 + 2*a*b + Sqrt[1 - 4*a*b]])/Sqrt[2 - 8*a*b]","B",1
39,1,126,62,0.0598002,"\int \frac{1+2 x^2}{1+b x^2+4 x^4} \, dx","Integrate[(1 + 2*x^2)/(1 + b*x^2 + 4*x^4),x]","\frac{\frac{\left(\sqrt{b^2-16}-b+4\right) \tan ^{-1}\left(\frac{2 \sqrt{2} x}{\sqrt{b-\sqrt{b^2-16}}}\right)}{\sqrt{b-\sqrt{b^2-16}}}+\frac{\left(\sqrt{b^2-16}+b-4\right) \tan ^{-1}\left(\frac{2 \sqrt{2} x}{\sqrt{\sqrt{b^2-16}+b}}\right)}{\sqrt{\sqrt{b^2-16}+b}}}{\sqrt{2} \sqrt{b^2-16}}","\frac{\tan ^{-1}\left(\frac{\sqrt{4-b}+4 x}{\sqrt{b+4}}\right)}{\sqrt{b+4}}-\frac{\tan ^{-1}\left(\frac{\sqrt{4-b}-4 x}{\sqrt{b+4}}\right)}{\sqrt{b+4}}",1,"(((4 - b + Sqrt[-16 + b^2])*ArcTan[(2*Sqrt[2]*x)/Sqrt[b - Sqrt[-16 + b^2]]])/Sqrt[b - Sqrt[-16 + b^2]] + ((-4 + b + Sqrt[-16 + b^2])*ArcTan[(2*Sqrt[2]*x)/Sqrt[b + Sqrt[-16 + b^2]]])/Sqrt[b + Sqrt[-16 + b^2]])/(Sqrt[2]*Sqrt[-16 + b^2])","B",1
40,1,134,66,0.0585158,"\int \frac{1+2 x^2}{1-b x^2+4 x^4} \, dx","Integrate[(1 + 2*x^2)/(1 - b*x^2 + 4*x^4),x]","\frac{\frac{\left(\sqrt{b^2-16}+b+4\right) \tan ^{-1}\left(\frac{2 \sqrt{2} x}{\sqrt{-\sqrt{b^2-16}-b}}\right)}{\sqrt{-\sqrt{b^2-16}-b}}+\frac{\left(\sqrt{b^2-16}-b-4\right) \tan ^{-1}\left(\frac{2 \sqrt{2} x}{\sqrt{\sqrt{b^2-16}-b}}\right)}{\sqrt{\sqrt{b^2-16}-b}}}{\sqrt{2} \sqrt{b^2-16}}","\frac{\tan ^{-1}\left(\frac{\sqrt{b+4}+4 x}{\sqrt{4-b}}\right)}{\sqrt{4-b}}-\frac{\tan ^{-1}\left(\frac{\sqrt{b+4}-4 x}{\sqrt{4-b}}\right)}{\sqrt{4-b}}",1,"(((4 + b + Sqrt[-16 + b^2])*ArcTan[(2*Sqrt[2]*x)/Sqrt[-b - Sqrt[-16 + b^2]]])/Sqrt[-b - Sqrt[-16 + b^2]] + ((-4 - b + Sqrt[-16 + b^2])*ArcTan[(2*Sqrt[2]*x)/Sqrt[-b + Sqrt[-16 + b^2]]])/Sqrt[-b + Sqrt[-16 + b^2]])/(Sqrt[2]*Sqrt[-16 + b^2])","B",1
41,1,83,45,0.0766989,"\int \frac{1+2 x^2}{1+6 x^2+4 x^4} \, dx","Integrate[(1 + 2*x^2)/(1 + 6*x^2 + 4*x^4),x]","\frac{\left(\sqrt{5}-1\right) \tan ^{-1}\left(\frac{2 x}{\sqrt{3-\sqrt{5}}}\right)}{2 \sqrt{5 \left(3-\sqrt{5}\right)}}+\frac{\left(1+\sqrt{5}\right) \tan ^{-1}\left(\frac{2 x}{\sqrt{3+\sqrt{5}}}\right)}{2 \sqrt{5 \left(3+\sqrt{5}\right)}}","\frac{\tan ^{-1}\left(\frac{2 x}{\sqrt{3-\sqrt{5}}}\right)}{\sqrt{10}}+\frac{\tan ^{-1}\left(\frac{2 x}{\sqrt{3+\sqrt{5}}}\right)}{\sqrt{10}}",1,"((-1 + Sqrt[5])*ArcTan[(2*x)/Sqrt[3 - Sqrt[5]]])/(2*Sqrt[5*(3 - Sqrt[5])]) + ((1 + Sqrt[5])*ArcTan[(2*x)/Sqrt[3 + Sqrt[5]]])/(2*Sqrt[5*(3 + Sqrt[5])])","A",1
42,1,17,15,0.0074615,"\int \frac{1+2 x^2}{1+5 x^2+4 x^4} \, dx","Integrate[(1 + 2*x^2)/(1 + 5*x^2 + 4*x^4),x]","-\frac{1}{3} \tan ^{-1}\left(\frac{3 x}{2 x^2-1}\right)","\frac{1}{3} \tan ^{-1}(x)+\frac{1}{3} \tan ^{-1}(2 x)",1,"-1/3*ArcTan[(3*x)/(-1 + 2*x^2)]","A",1
43,1,14,14,0.0048432,"\int \frac{1+2 x^2}{1+4 x^2+4 x^4} \, dx","Integrate[(1 + 2*x^2)/(1 + 4*x^2 + 4*x^4),x]","\frac{\tan ^{-1}\left(\sqrt{2} x\right)}{\sqrt{2}}","\frac{\tan ^{-1}\left(\sqrt{2} x\right)}{\sqrt{2}}",1,"ArcTan[Sqrt[2]*x]/Sqrt[2]","A",1
44,1,97,38,0.1841448,"\int \frac{1+2 x^2}{1+3 x^2+4 x^4} \, dx","Integrate[(1 + 2*x^2)/(1 + 3*x^2 + 4*x^4),x]","\frac{\left(\sqrt{7}-i\right) \tan ^{-1}\left(\frac{2 x}{\sqrt{\frac{1}{2} \left(3-i \sqrt{7}\right)}}\right)}{\sqrt{42-14 i \sqrt{7}}}+\frac{\left(\sqrt{7}+i\right) \tan ^{-1}\left(\frac{2 x}{\sqrt{\frac{1}{2} \left(3+i \sqrt{7}\right)}}\right)}{\sqrt{42+14 i \sqrt{7}}}","\frac{\tan ^{-1}\left(\frac{4 x+1}{\sqrt{7}}\right)}{\sqrt{7}}-\frac{\tan ^{-1}\left(\frac{1-4 x}{\sqrt{7}}\right)}{\sqrt{7}}",1,"((-I + Sqrt[7])*ArcTan[(2*x)/Sqrt[(3 - I*Sqrt[7])/2]])/Sqrt[42 - (14*I)*Sqrt[7]] + ((I + Sqrt[7])*ArcTan[(2*x)/Sqrt[(3 + I*Sqrt[7])/2]])/Sqrt[42 + (14*I)*Sqrt[7]]","C",1
45,1,99,48,0.1036542,"\int \frac{1+2 x^2}{1+2 x^2+4 x^4} \, dx","Integrate[(1 + 2*x^2)/(1 + 2*x^2 + 4*x^4),x]","\frac{\left(\sqrt{3}-i\right) \tan ^{-1}\left(\frac{2 x}{\sqrt{1-i \sqrt{3}}}\right)}{2 \sqrt{3 \left(1-i \sqrt{3}\right)}}+\frac{\left(\sqrt{3}+i\right) \tan ^{-1}\left(\frac{2 x}{\sqrt{1+i \sqrt{3}}}\right)}{2 \sqrt{3 \left(1+i \sqrt{3}\right)}}","\frac{\tan ^{-1}\left(\frac{2 \sqrt{2} x+1}{\sqrt{3}}\right)}{\sqrt{6}}-\frac{\tan ^{-1}\left(\frac{1-2 \sqrt{2} x}{\sqrt{3}}\right)}{\sqrt{6}}",1,"((-I + Sqrt[3])*ArcTan[(2*x)/Sqrt[1 - I*Sqrt[3]]])/(2*Sqrt[3*(1 - I*Sqrt[3])]) + ((I + Sqrt[3])*ArcTan[(2*x)/Sqrt[1 + I*Sqrt[3]]])/(2*Sqrt[3*(1 + I*Sqrt[3])])","C",1
46,1,97,46,0.2244173,"\int \frac{1+2 x^2}{1+x^2+4 x^4} \, dx","Integrate[(1 + 2*x^2)/(1 + x^2 + 4*x^4),x]","\frac{\left(\sqrt{15}-3 i\right) \tan ^{-1}\left(\frac{2 x}{\sqrt{\frac{1}{2} \left(1-i \sqrt{15}\right)}}\right)}{\sqrt{30-30 i \sqrt{15}}}+\frac{\left(\sqrt{15}+3 i\right) \tan ^{-1}\left(\frac{2 x}{\sqrt{\frac{1}{2} \left(1+i \sqrt{15}\right)}}\right)}{\sqrt{30+30 i \sqrt{15}}}","\frac{\tan ^{-1}\left(\frac{4 x+\sqrt{3}}{\sqrt{5}}\right)}{\sqrt{5}}-\frac{\tan ^{-1}\left(\frac{\sqrt{3}-4 x}{\sqrt{5}}\right)}{\sqrt{5}}",1,"((-3*I + Sqrt[15])*ArcTan[(2*x)/Sqrt[(1 - I*Sqrt[15])/2]])/Sqrt[30 - (30*I)*Sqrt[15]] + ((3*I + Sqrt[15])*ArcTan[(2*x)/Sqrt[(1 + I*Sqrt[15])/2]])/Sqrt[30 + (30*I)*Sqrt[15]]","C",1
47,1,17,21,0.0060433,"\int \frac{1+2 x^2}{1+4 x^4} \, dx","Integrate[(1 + 2*x^2)/(1 + 4*x^4),x]","-\frac{1}{2} \tan ^{-1}\left(\frac{2 x}{2 x^2-1}\right)","\frac{1}{2} \tan ^{-1}(2 x+1)-\frac{1}{2} \tan ^{-1}(1-2 x)",1,"-1/2*ArcTan[(2*x)/(-1 + 2*x^2)]","A",1
48,1,101,46,0.2746641,"\int \frac{1+2 x^2}{1-x^2+4 x^4} \, dx","Integrate[(1 + 2*x^2)/(1 - x^2 + 4*x^4),x]","\frac{\left(\sqrt{15}-5 i\right) \tan ^{-1}\left(\frac{2 x}{\sqrt{\frac{1}{2} \left(-1-i \sqrt{15}\right)}}\right)}{\sqrt{30 \left(-1-i \sqrt{15}\right)}}+\frac{\left(\sqrt{15}+5 i\right) \tan ^{-1}\left(\frac{2 x}{\sqrt{\frac{1}{2} \left(-1+i \sqrt{15}\right)}}\right)}{\sqrt{30 \left(-1+i \sqrt{15}\right)}}","\frac{\tan ^{-1}\left(\frac{4 x+\sqrt{5}}{\sqrt{3}}\right)}{\sqrt{3}}-\frac{\tan ^{-1}\left(\frac{\sqrt{5}-4 x}{\sqrt{3}}\right)}{\sqrt{3}}",1,"((-5*I + Sqrt[15])*ArcTan[(2*x)/Sqrt[(-1 - I*Sqrt[15])/2]])/Sqrt[30*(-1 - I*Sqrt[15])] + ((5*I + Sqrt[15])*ArcTan[(2*x)/Sqrt[(-1 + I*Sqrt[15])/2]])/Sqrt[30*(-1 + I*Sqrt[15])]","C",1
49,1,99,44,0.1011968,"\int \frac{1+2 x^2}{1-2 x^2+4 x^4} \, dx","Integrate[(1 + 2*x^2)/(1 - 2*x^2 + 4*x^4),x]","\frac{\left(\sqrt{3}-3 i\right) \tan ^{-1}\left(\frac{2 x}{\sqrt{-1-i \sqrt{3}}}\right)}{2 \sqrt{3 \left(-1-i \sqrt{3}\right)}}+\frac{\left(\sqrt{3}+3 i\right) \tan ^{-1}\left(\frac{2 x}{\sqrt{-1+i \sqrt{3}}}\right)}{2 \sqrt{3 \left(-1+i \sqrt{3}\right)}}","\frac{\tan ^{-1}\left(2 \sqrt{2} x+\sqrt{3}\right)}{\sqrt{2}}-\frac{\tan ^{-1}\left(\sqrt{3}-2 \sqrt{2} x\right)}{\sqrt{2}}",1,"((-3*I + Sqrt[3])*ArcTan[(2*x)/Sqrt[-1 - I*Sqrt[3]]])/(2*Sqrt[3*(-1 - I*Sqrt[3])]) + ((3*I + Sqrt[3])*ArcTan[(2*x)/Sqrt[-1 + I*Sqrt[3]]])/(2*Sqrt[3*(-1 + I*Sqrt[3])])","C",1
50,1,14,23,0.0072003,"\int \frac{1+2 x^2}{1-3 x^2+4 x^4} \, dx","Integrate[(1 + 2*x^2)/(1 - 3*x^2 + 4*x^4),x]","-\tan ^{-1}\left(\frac{x}{2 x^2-1}\right)","\tan ^{-1}\left(4 x+\sqrt{7}\right)-\tan ^{-1}\left(\sqrt{7}-4 x\right)",1,"-ArcTan[x/(-1 + 2*x^2)]","A",1
51,1,12,11,0.0054561,"\int \frac{1+2 x^2}{1-4 x^2+4 x^4} \, dx","Integrate[(1 + 2*x^2)/(1 - 4*x^2 + 4*x^4),x]","-\frac{x}{2 x^2-1}","\frac{x}{1-2 x^2}",1,"-(x/(-1 + 2*x^2))","A",1
52,1,29,39,0.0061641,"\int \frac{1+2 x^2}{1-5 x^2+4 x^4} \, dx","Integrate[(1 + 2*x^2)/(1 - 5*x^2 + 4*x^4),x]","\frac{1}{2} \log \left(-2 x^2+x+1\right)-\frac{1}{2} \log \left(-2 x^2-x+1\right)","-\frac{1}{2} \log (1-2 x)+\frac{1}{2} \log (1-x)-\frac{1}{2} \log (x+1)+\frac{1}{2} \log (2 x+1)",1,"-1/2*Log[1 - x - 2*x^2] + Log[1 + x - 2*x^2]/2","A",1
53,1,42,44,0.0151739,"\int \frac{1+2 x^2}{1-6 x^2+4 x^4} \, dx","Integrate[(1 + 2*x^2)/(1 - 6*x^2 + 4*x^4),x]","\frac{\log \left(-2 x^2+\sqrt{2} x+1\right)-\log \left(2 x^2+\sqrt{2} x-1\right)}{2 \sqrt{2}}","\frac{\tanh ^{-1}\left(\sqrt{5}-2 \sqrt{2} x\right)}{\sqrt{2}}-\frac{\tanh ^{-1}\left(2 \sqrt{2} x+\sqrt{5}\right)}{\sqrt{2}}",1,"(Log[1 + Sqrt[2]*x - 2*x^2] - Log[-1 + Sqrt[2]*x + 2*x^2])/(2*Sqrt[2])","A",1
54,1,127,66,0.0730669,"\int \frac{1-2 x^2}{1+b x^2+4 x^4} \, dx","Integrate[(1 - 2*x^2)/(1 + b*x^2 + 4*x^4),x]","\frac{\frac{\left(-\sqrt{b^2-16}+b+4\right) \tan ^{-1}\left(\frac{2 \sqrt{2} x}{\sqrt{b-\sqrt{b^2-16}}}\right)}{\sqrt{b-\sqrt{b^2-16}}}-\frac{\left(\sqrt{b^2-16}+b+4\right) \tan ^{-1}\left(\frac{2 \sqrt{2} x}{\sqrt{\sqrt{b^2-16}+b}}\right)}{\sqrt{\sqrt{b^2-16}+b}}}{\sqrt{2} \sqrt{b^2-16}}","\frac{\log \left(\sqrt{4-b} x+2 x^2+1\right)}{2 \sqrt{4-b}}-\frac{\log \left(-\sqrt{4-b} x+2 x^2+1\right)}{2 \sqrt{4-b}}",1,"(((4 + b - Sqrt[-16 + b^2])*ArcTan[(2*Sqrt[2]*x)/Sqrt[b - Sqrt[-16 + b^2]]])/Sqrt[b - Sqrt[-16 + b^2]] - ((4 + b + Sqrt[-16 + b^2])*ArcTan[(2*Sqrt[2]*x)/Sqrt[b + Sqrt[-16 + b^2]]])/Sqrt[b + Sqrt[-16 + b^2]])/(Sqrt[2]*Sqrt[-16 + b^2])","A",1
55,1,84,46,0.0725804,"\int \frac{1-2 x^2}{1+6 x^2+4 x^4} \, dx","Integrate[(1 - 2*x^2)/(1 + 6*x^2 + 4*x^4),x]","\frac{-\left(\left(\sqrt{5}-5\right) \sqrt{3+\sqrt{5}} \tan ^{-1}\left(\frac{2 x}{\sqrt{3-\sqrt{5}}}\right)\right)-\sqrt{3-\sqrt{5}} \left(5+\sqrt{5}\right) \tan ^{-1}\left(\frac{2 x}{\sqrt{3+\sqrt{5}}}\right)}{4 \sqrt{5}}","\frac{\tan ^{-1}\left(\frac{2 x}{\sqrt{3-\sqrt{5}}}\right)}{\sqrt{2}}-\frac{\tan ^{-1}\left(\frac{2 x}{\sqrt{3+\sqrt{5}}}\right)}{\sqrt{2}}",1,"(-((-5 + Sqrt[5])*Sqrt[3 + Sqrt[5]]*ArcTan[(2*x)/Sqrt[3 - Sqrt[5]]]) - Sqrt[3 - Sqrt[5]]*(5 + Sqrt[5])*ArcTan[(2*x)/Sqrt[3 + Sqrt[5]]])/(4*Sqrt[5])","A",1
56,1,12,9,0.0069767,"\int \frac{1-2 x^2}{1+5 x^2+4 x^4} \, dx","Integrate[(1 - 2*x^2)/(1 + 5*x^2 + 4*x^4),x]","\tan ^{-1}\left(\frac{x}{2 x^2+1}\right)","\tan ^{-1}(2 x)-\tan ^{-1}(x)",1,"ArcTan[x/(1 + 2*x^2)]","A",1
57,1,11,11,0.0044661,"\int \frac{1-2 x^2}{1+4 x^2+4 x^4} \, dx","Integrate[(1 - 2*x^2)/(1 + 4*x^2 + 4*x^4),x]","\frac{x}{2 x^2+1}","\frac{x}{2 x^2+1}",1,"x/(1 + 2*x^2)","A",1
58,1,29,29,0.0056336,"\int \frac{1-2 x^2}{1+3 x^2+4 x^4} \, dx","Integrate[(1 - 2*x^2)/(1 + 3*x^2 + 4*x^4),x]","\frac{1}{2} \log \left(2 x^2+x+1\right)-\frac{1}{2} \log \left(2 x^2-x+1\right)","\frac{1}{2} \log \left(2 x^2+x+1\right)-\frac{1}{2} \log \left(2 x^2-x+1\right)",1,"-1/2*Log[1 - x + 2*x^2] + Log[1 + x + 2*x^2]/2","A",1
59,1,42,50,0.0138862,"\int \frac{1-2 x^2}{1+2 x^2+4 x^4} \, dx","Integrate[(1 - 2*x^2)/(1 + 2*x^2 + 4*x^4),x]","\frac{\log \left(2 x^2+\sqrt{2} x+1\right)-\log \left(-2 x^2+\sqrt{2} x-1\right)}{2 \sqrt{2}}","\frac{\log \left(2 x^2+\sqrt{2} x+1\right)}{2 \sqrt{2}}-\frac{\log \left(2 x^2-\sqrt{2} x+1\right)}{2 \sqrt{2}}",1,"(-Log[-1 + Sqrt[2]*x - 2*x^2] + Log[1 + Sqrt[2]*x + 2*x^2])/(2*Sqrt[2])","A",1
60,1,42,50,0.0148067,"\int \frac{1-2 x^2}{1+x^2+4 x^4} \, dx","Integrate[(1 - 2*x^2)/(1 + x^2 + 4*x^4),x]","\frac{\log \left(2 x^2+\sqrt{3} x+1\right)-\log \left(-2 x^2+\sqrt{3} x-1\right)}{2 \sqrt{3}}","\frac{\log \left(2 x^2+\sqrt{3} x+1\right)}{2 \sqrt{3}}-\frac{\log \left(2 x^2-\sqrt{3} x+1\right)}{2 \sqrt{3}}",1,"(-Log[-1 + Sqrt[3]*x - 2*x^2] + Log[1 + Sqrt[3]*x + 2*x^2])/(2*Sqrt[3])","A",1
61,1,31,31,0.0048013,"\int \frac{1-2 x^2}{1+4 x^4} \, dx","Integrate[(1 - 2*x^2)/(1 + 4*x^4),x]","\frac{1}{4} \log \left(2 x^2+2 x+1\right)-\frac{1}{4} \log \left(2 x^2-2 x+1\right)","\frac{1}{4} \log \left(2 x^2+2 x+1\right)-\frac{1}{4} \log \left(2 x^2-2 x+1\right)",1,"-1/4*Log[1 - 2*x + 2*x^2] + Log[1 + 2*x + 2*x^2]/4","A",1
62,1,42,50,0.0140751,"\int \frac{1-2 x^2}{1-x^2+4 x^4} \, dx","Integrate[(1 - 2*x^2)/(1 - x^2 + 4*x^4),x]","\frac{\log \left(2 x^2+\sqrt{5} x+1\right)-\log \left(-2 x^2+\sqrt{5} x-1\right)}{2 \sqrt{5}}","\frac{\log \left(2 x^2+\sqrt{5} x+1\right)}{2 \sqrt{5}}-\frac{\log \left(2 x^2-\sqrt{5} x+1\right)}{2 \sqrt{5}}",1,"(-Log[-1 + Sqrt[5]*x - 2*x^2] + Log[1 + Sqrt[5]*x + 2*x^2])/(2*Sqrt[5])","A",1
63,1,42,50,0.0196827,"\int \frac{1-2 x^2}{1-2 x^2+4 x^4} \, dx","Integrate[(1 - 2*x^2)/(1 - 2*x^2 + 4*x^4),x]","\frac{\log \left(2 x^2+\sqrt{6} x+1\right)-\log \left(-2 x^2+\sqrt{6} x-1\right)}{2 \sqrt{6}}","\frac{\log \left(2 x^2+\sqrt{6} x+1\right)}{2 \sqrt{6}}-\frac{\log \left(2 x^2-\sqrt{6} x+1\right)}{2 \sqrt{6}}",1,"(-Log[-1 + Sqrt[6]*x - 2*x^2] + Log[1 + Sqrt[6]*x + 2*x^2])/(2*Sqrt[6])","A",1
64,1,42,50,0.0160316,"\int \frac{1-2 x^2}{1-3 x^2+4 x^4} \, dx","Integrate[(1 - 2*x^2)/(1 - 3*x^2 + 4*x^4),x]","\frac{\log \left(2 x^2+\sqrt{7} x+1\right)-\log \left(-2 x^2+\sqrt{7} x-1\right)}{2 \sqrt{7}}","\frac{\log \left(2 x^2+\sqrt{7} x+1\right)}{2 \sqrt{7}}-\frac{\log \left(2 x^2-\sqrt{7} x+1\right)}{2 \sqrt{7}}",1,"(-Log[-1 + Sqrt[7]*x - 2*x^2] + Log[1 + Sqrt[7]*x + 2*x^2])/(2*Sqrt[7])","A",1
65,1,32,14,0.0085048,"\int \frac{1-2 x^2}{1-4 x^2+4 x^4} \, dx","Integrate[(1 - 2*x^2)/(1 - 4*x^2 + 4*x^4),x]","\frac{\log \left(2 x+\sqrt{2}\right)-\log \left(\sqrt{2}-2 x\right)}{2 \sqrt{2}}","\frac{\tanh ^{-1}\left(\sqrt{2} x\right)}{\sqrt{2}}",1,"(-Log[Sqrt[2] - 2*x] + Log[Sqrt[2] + 2*x])/(2*Sqrt[2])","B",1
66,1,31,39,0.0059012,"\int \frac{1-2 x^2}{1-5 x^2+4 x^4} \, dx","Integrate[(1 - 2*x^2)/(1 - 5*x^2 + 4*x^4),x]","\frac{1}{6} \log \left(2 x^2+3 x+1\right)-\frac{1}{6} \log \left(2 x^2-3 x+1\right)","-\frac{1}{6} \log (1-2 x)-\frac{1}{6} \log (1-x)+\frac{1}{6} \log (x+1)+\frac{1}{6} \log (2 x+1)",1,"-1/6*Log[1 - 3*x + 2*x^2] + Log[1 + 3*x + 2*x^2]/6","A",1
67,1,42,48,0.019485,"\int \frac{1-2 x^2}{1-6 x^2+4 x^4} \, dx","Integrate[(1 - 2*x^2)/(1 - 6*x^2 + 4*x^4),x]","\frac{\log \left(2 x^2+\sqrt{10} x+1\right)-\log \left(-2 x^2+\sqrt{10} x-1\right)}{2 \sqrt{10}}","\frac{\tanh ^{-1}\left(\frac{2 \sqrt{2} x+1}{\sqrt{5}}\right)}{\sqrt{10}}-\frac{\tanh ^{-1}\left(\frac{1-2 \sqrt{2} x}{\sqrt{5}}\right)}{\sqrt{10}}",1,"(-Log[-1 + Sqrt[10]*x - 2*x^2] + Log[1 + Sqrt[10]*x + 2*x^2])/(2*Sqrt[10])","A",1
68,1,124,62,0.0583623,"\int \frac{1+x^2}{1+b x^2+x^4} \, dx","Integrate[(1 + x^2)/(1 + b*x^2 + x^4),x]","\frac{\frac{\left(\sqrt{b^2-4}-b+2\right) \tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{b-\sqrt{b^2-4}}}\right)}{\sqrt{b-\sqrt{b^2-4}}}+\frac{\left(\sqrt{b^2-4}+b-2\right) \tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{\sqrt{b^2-4}+b}}\right)}{\sqrt{\sqrt{b^2-4}+b}}}{\sqrt{2} \sqrt{b^2-4}}","\frac{\tan ^{-1}\left(\frac{\sqrt{2-b}+2 x}{\sqrt{b+2}}\right)}{\sqrt{b+2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2-b}-2 x}{\sqrt{b+2}}\right)}{\sqrt{b+2}}",1,"(((2 - b + Sqrt[-4 + b^2])*ArcTan[(Sqrt[2]*x)/Sqrt[b - Sqrt[-4 + b^2]]])/Sqrt[b - Sqrt[-4 + b^2]] + ((-2 + b + Sqrt[-4 + b^2])*ArcTan[(Sqrt[2]*x)/Sqrt[b + Sqrt[-4 + b^2]]])/Sqrt[b + Sqrt[-4 + b^2]])/(Sqrt[2]*Sqrt[-4 + b^2])","A",1
69,1,83,49,0.139425,"\int \frac{1+x^2}{1+5 x^2+x^4} \, dx","Integrate[(1 + x^2)/(1 + 5*x^2 + x^4),x]","\frac{\left(\sqrt{21}-3\right) \tan ^{-1}\left(\sqrt{\frac{2}{5-\sqrt{21}}} x\right)}{\sqrt{42 \left(5-\sqrt{21}\right)}}+\frac{\left(3+\sqrt{21}\right) \tan ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{21}}} x\right)}{\sqrt{42 \left(5+\sqrt{21}\right)}}","\frac{\tan ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{21}}} x\right)}{\sqrt{7}}+\frac{\tan ^{-1}\left(\sqrt{\frac{1}{2} \left(5+\sqrt{21}\right)} x\right)}{\sqrt{7}}",1,"((-3 + Sqrt[21])*ArcTan[Sqrt[2/(5 - Sqrt[21])]*x])/Sqrt[42*(5 - Sqrt[21])] + ((3 + Sqrt[21])*ArcTan[Sqrt[2/(5 + Sqrt[21])]*x])/Sqrt[42*(5 + Sqrt[21])]","A",1
70,1,81,43,0.0701585,"\int \frac{1+x^2}{1+4 x^2+x^4} \, dx","Integrate[(1 + x^2)/(1 + 4*x^2 + x^4),x]","\frac{\left(\sqrt{3}-1\right) \tan ^{-1}\left(\frac{x}{\sqrt{2-\sqrt{3}}}\right)}{2 \sqrt{3 \left(2-\sqrt{3}\right)}}+\frac{\left(1+\sqrt{3}\right) \tan ^{-1}\left(\frac{x}{\sqrt{2+\sqrt{3}}}\right)}{2 \sqrt{3 \left(2+\sqrt{3}\right)}}","\frac{\tan ^{-1}\left(\frac{x}{\sqrt{2-\sqrt{3}}}\right)}{\sqrt{6}}+\frac{\tan ^{-1}\left(\frac{x}{\sqrt{2+\sqrt{3}}}\right)}{\sqrt{6}}",1,"((-1 + Sqrt[3])*ArcTan[x/Sqrt[2 - Sqrt[3]]])/(2*Sqrt[3*(2 - Sqrt[3])]) + ((1 + Sqrt[3])*ArcTan[x/Sqrt[2 + Sqrt[3]]])/(2*Sqrt[3*(2 + Sqrt[3])])","A",1
71,1,83,49,0.1037706,"\int \frac{1+x^2}{1+3 x^2+x^4} \, dx","Integrate[(1 + x^2)/(1 + 3*x^2 + x^4),x]","\frac{\left(\sqrt{5}-1\right) \tan ^{-1}\left(\sqrt{\frac{2}{3-\sqrt{5}}} x\right)}{\sqrt{10 \left(3-\sqrt{5}\right)}}+\frac{\left(1+\sqrt{5}\right) \tan ^{-1}\left(\sqrt{\frac{2}{3+\sqrt{5}}} x\right)}{\sqrt{10 \left(3+\sqrt{5}\right)}}","\frac{\tan ^{-1}\left(\sqrt{\frac{2}{3+\sqrt{5}}} x\right)}{\sqrt{5}}+\frac{\tan ^{-1}\left(\sqrt{\frac{1}{2} \left(3+\sqrt{5}\right)} x\right)}{\sqrt{5}}",1,"((-1 + Sqrt[5])*ArcTan[Sqrt[2/(3 - Sqrt[5])]*x])/Sqrt[10*(3 - Sqrt[5])] + ((1 + Sqrt[5])*ArcTan[Sqrt[2/(3 + Sqrt[5])]*x])/Sqrt[10*(3 + Sqrt[5])]","A",1
72,1,2,2,0.0027156,"\int \frac{1+x^2}{1+2 x^2+x^4} \, dx","Integrate[(1 + x^2)/(1 + 2*x^2 + x^4),x]","\tan ^{-1}(x)","\tan ^{-1}(x)",1,"ArcTan[x]","A",1
73,1,99,38,0.194894,"\int \frac{1+x^2}{1+x^2+x^4} \, dx","Integrate[(1 + x^2)/(1 + x^2 + x^4),x]","\frac{\left(\sqrt{3}-i\right) \tan ^{-1}\left(\frac{x}{\sqrt{\frac{1}{2} \left(1-i \sqrt{3}\right)}}\right)}{\sqrt{6 \left(1-i \sqrt{3}\right)}}+\frac{\left(\sqrt{3}+i\right) \tan ^{-1}\left(\frac{x}{\sqrt{\frac{1}{2} \left(1+i \sqrt{3}\right)}}\right)}{\sqrt{6 \left(1+i \sqrt{3}\right)}}","\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{\sqrt{3}}-\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{\sqrt{3}}",1,"((-I + Sqrt[3])*ArcTan[x/Sqrt[(1 - I*Sqrt[3])/2]])/Sqrt[6*(1 - I*Sqrt[3])] + ((I + Sqrt[3])*ArcTan[x/Sqrt[(1 + I*Sqrt[3])/2]])/Sqrt[6*(1 + I*Sqrt[3])]","C",1
74,1,30,35,0.0150067,"\int \frac{1+x^2}{1+x^4} \, dx","Integrate[(1 + x^2)/(1 + x^4),x]","\frac{\tan ^{-1}\left(\sqrt{2} x+1\right)-\tan ^{-1}\left(1-\sqrt{2} x\right)}{\sqrt{2}}","\frac{\tan ^{-1}\left(\sqrt{2} x+1\right)}{\sqrt{2}}-\frac{\tan ^{-1}\left(1-\sqrt{2} x\right)}{\sqrt{2}}",1,"(-ArcTan[1 - Sqrt[2]*x] + ArcTan[1 + Sqrt[2]*x])/Sqrt[2]","A",1
75,1,12,23,0.0068482,"\int \frac{1+x^2}{1-x^2+x^4} \, dx","Integrate[(1 + x^2)/(1 - x^2 + x^4),x]","-\tan ^{-1}\left(\frac{x}{x^2-1}\right)","\tan ^{-1}\left(2 x+\sqrt{3}\right)-\tan ^{-1}\left(\sqrt{3}-2 x\right)",1,"-ArcTan[x/(-1 + x^2)]","A",1
76,1,10,11,0.0041539,"\int \frac{1+x^2}{1-2 x^2+x^4} \, dx","Integrate[(1 + x^2)/(1 - 2*x^2 + x^4),x]","-\frac{x}{x^2-1}","\frac{x}{1-x^2}",1,"-(x/(-1 + x^2))","A",1
77,1,29,65,0.0056251,"\int \frac{1+x^2}{1-3 x^2+x^4} \, dx","Integrate[(1 + x^2)/(1 - 3*x^2 + x^4),x]","\frac{1}{2} \log \left(-x^2+x+1\right)-\frac{1}{2} \log \left(-x^2-x+1\right)","\frac{1}{2} \log \left(-2 x-\sqrt{5}+1\right)+\frac{1}{2} \log \left(-2 x+\sqrt{5}+1\right)-\frac{1}{2} \log \left(2 x-\sqrt{5}+1\right)-\frac{1}{2} \log \left(2 x+\sqrt{5}+1\right)",1,"-1/2*Log[1 - x - x^2] + Log[1 + x - x^2]/2","A",1
78,1,40,43,0.0136716,"\int \frac{1+x^2}{1-4 x^2+x^4} \, dx","Integrate[(1 + x^2)/(1 - 4*x^2 + x^4),x]","\frac{\log \left(-x^2+\sqrt{2} x+1\right)-\log \left(x^2+\sqrt{2} x-1\right)}{2 \sqrt{2}}","\frac{\tanh ^{-1}\left(\sqrt{3}-\sqrt{2} x\right)}{\sqrt{2}}-\frac{\tanh ^{-1}\left(\sqrt{2} x+\sqrt{3}\right)}{\sqrt{2}}",1,"(Log[1 + Sqrt[2]*x - x^2] - Log[-1 + Sqrt[2]*x + x^2])/(2*Sqrt[2])","A",1
79,1,40,46,0.0136465,"\int \frac{1+x^2}{1-5 x^2+x^4} \, dx","Integrate[(1 + x^2)/(1 - 5*x^2 + x^4),x]","\frac{\log \left(-x^2+\sqrt{3} x+1\right)-\log \left(x^2+\sqrt{3} x-1\right)}{2 \sqrt{3}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{7}-2 x}{\sqrt{3}}\right)}{\sqrt{3}}-\frac{\tanh ^{-1}\left(\frac{2 x+\sqrt{7}}{\sqrt{3}}\right)}{\sqrt{3}}",1,"(Log[1 + Sqrt[3]*x - x^2] - Log[-1 + Sqrt[3]*x + x^2])/(2*Sqrt[3])","A",1
80,1,125,62,0.0727002,"\int \frac{1-x^2}{1+b x^2+x^4} \, dx","Integrate[(1 - x^2)/(1 + b*x^2 + x^4),x]","\frac{\frac{\left(-\sqrt{b^2-4}+b+2\right) \tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{b-\sqrt{b^2-4}}}\right)}{\sqrt{b-\sqrt{b^2-4}}}-\frac{\left(\sqrt{b^2-4}+b+2\right) \tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{\sqrt{b^2-4}+b}}\right)}{\sqrt{\sqrt{b^2-4}+b}}}{\sqrt{2} \sqrt{b^2-4}}","\frac{\log \left(\sqrt{2-b} x+x^2+1\right)}{2 \sqrt{2-b}}-\frac{\log \left(-\sqrt{2-b} x+x^2+1\right)}{2 \sqrt{2-b}}",1,"(((2 + b - Sqrt[-4 + b^2])*ArcTan[(Sqrt[2]*x)/Sqrt[b - Sqrt[-4 + b^2]]])/Sqrt[b - Sqrt[-4 + b^2]] - ((2 + b + Sqrt[-4 + b^2])*ArcTan[(Sqrt[2]*x)/Sqrt[b + Sqrt[-4 + b^2]]])/Sqrt[b + Sqrt[-4 + b^2]])/(Sqrt[2]*Sqrt[-4 + b^2])","B",1
81,1,87,50,0.1351168,"\int \frac{1-x^2}{1+5 x^2+x^4} \, dx","Integrate[(1 - x^2)/(1 + 5*x^2 + x^4),x]","\frac{\left(7-\sqrt{21}\right) \tan ^{-1}\left(\sqrt{\frac{2}{5-\sqrt{21}}} x\right)}{\sqrt{42 \left(5-\sqrt{21}\right)}}+\frac{\left(-7-\sqrt{21}\right) \tan ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{21}}} x\right)}{\sqrt{42 \left(5+\sqrt{21}\right)}}","\frac{\tan ^{-1}\left(\sqrt{\frac{1}{2} \left(5+\sqrt{21}\right)} x\right)}{\sqrt{3}}-\frac{\tan ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{21}}} x\right)}{\sqrt{3}}",1,"((7 - Sqrt[21])*ArcTan[Sqrt[2/(5 - Sqrt[21])]*x])/Sqrt[42*(5 - Sqrt[21])] + ((-7 - Sqrt[21])*ArcTan[Sqrt[2/(5 + Sqrt[21])]*x])/Sqrt[42*(5 + Sqrt[21])]","A",1
82,1,82,44,0.0707289,"\int \frac{1-x^2}{1+4 x^2+x^4} \, dx","Integrate[(1 - x^2)/(1 + 4*x^2 + x^4),x]","\frac{-\left(\left(\sqrt{3}-3\right) \sqrt{2+\sqrt{3}} \tan ^{-1}\left(\frac{x}{\sqrt{2-\sqrt{3}}}\right)\right)-\sqrt{2-\sqrt{3}} \left(3+\sqrt{3}\right) \tan ^{-1}\left(\frac{x}{\sqrt{2+\sqrt{3}}}\right)}{2 \sqrt{3}}","\frac{\tan ^{-1}\left(\frac{x}{\sqrt{2-\sqrt{3}}}\right)}{\sqrt{2}}-\frac{\tan ^{-1}\left(\frac{x}{\sqrt{2+\sqrt{3}}}\right)}{\sqrt{2}}",1,"(-((-3 + Sqrt[3])*Sqrt[2 + Sqrt[3]]*ArcTan[x/Sqrt[2 - Sqrt[3]]]) - Sqrt[2 - Sqrt[3]]*(3 + Sqrt[3])*ArcTan[x/Sqrt[2 + Sqrt[3]]])/(2*Sqrt[3])","A",1
83,1,10,39,0.0069818,"\int \frac{1-x^2}{1+3 x^2+x^4} \, dx","Integrate[(1 - x^2)/(1 + 3*x^2 + x^4),x]","\tan ^{-1}\left(\frac{x}{x^2+1}\right)","\tan ^{-1}\left(\sqrt{\frac{1}{2} \left(3+\sqrt{5}\right)} x\right)-\tan ^{-1}\left(\sqrt{\frac{2}{3+\sqrt{5}}} x\right)",1,"ArcTan[x/(1 + x^2)]","A",1
84,1,9,9,0.0039829,"\int \frac{1-x^2}{1+2 x^2+x^4} \, dx","Integrate[(1 - x^2)/(1 + 2*x^2 + x^4),x]","\frac{x}{x^2+1}","\frac{x}{x^2+1}",1,"x/(1 + x^2)","A",1
85,1,25,25,0.0055381,"\int \frac{1-x^2}{1+x^2+x^4} \, dx","Integrate[(1 - x^2)/(1 + x^2 + x^4),x]","\frac{1}{2} \log \left(x^2+x+1\right)-\frac{1}{2} \log \left(x^2-x+1\right)","\frac{1}{2} \log \left(x^2+x+1\right)-\frac{1}{2} \log \left(x^2-x+1\right)",1,"-1/2*Log[1 - x + x^2] + Log[1 + x + x^2]/2","A",1
86,1,40,46,0.0129709,"\int \frac{1-x^2}{1+x^4} \, dx","Integrate[(1 - x^2)/(1 + x^4),x]","\frac{\log \left(x^2+\sqrt{2} x+1\right)-\log \left(-x^2+\sqrt{2} x-1\right)}{2 \sqrt{2}}","\frac{\log \left(x^2+\sqrt{2} x+1\right)}{2 \sqrt{2}}-\frac{\log \left(x^2-\sqrt{2} x+1\right)}{2 \sqrt{2}}",1,"(-Log[-1 + Sqrt[2]*x - x^2] + Log[1 + Sqrt[2]*x + x^2])/(2*Sqrt[2])","A",1
87,1,40,46,0.0127535,"\int \frac{1-x^2}{1-x^2+x^4} \, dx","Integrate[(1 - x^2)/(1 - x^2 + x^4),x]","\frac{\log \left(x^2+\sqrt{3} x+1\right)-\log \left(-x^2+\sqrt{3} x-1\right)}{2 \sqrt{3}}","\frac{\log \left(x^2+\sqrt{3} x+1\right)}{2 \sqrt{3}}-\frac{\log \left(x^2-\sqrt{3} x+1\right)}{2 \sqrt{3}}",1,"(-Log[-1 + Sqrt[3]*x - x^2] + Log[1 + Sqrt[3]*x + x^2])/(2*Sqrt[3])","A",1
88,1,19,2,0.0023291,"\int \frac{1-x^2}{1-2 x^2+x^4} \, dx","Integrate[(1 - x^2)/(1 - 2*x^2 + x^4),x]","\frac{1}{2} \log (x+1)-\frac{1}{2} \log (1-x)","\tanh ^{-1}(x)",1,"-1/2*Log[1 - x] + Log[1 + x]/2","B",1
89,1,40,38,0.0139809,"\int \frac{1-x^2}{1-3 x^2+x^4} \, dx","Integrate[(1 - x^2)/(1 - 3*x^2 + x^4),x]","\frac{\log \left(x^2+\sqrt{5} x+1\right)-\log \left(-x^2+\sqrt{5} x-1\right)}{2 \sqrt{5}}","\frac{\tanh ^{-1}\left(\frac{2 x+1}{\sqrt{5}}\right)}{\sqrt{5}}-\frac{\tanh ^{-1}\left(\frac{1-2 x}{\sqrt{5}}\right)}{\sqrt{5}}",1,"(-Log[-1 + Sqrt[5]*x - x^2] + Log[1 + Sqrt[5]*x + x^2])/(2*Sqrt[5])","A",1
90,1,40,47,0.017565,"\int \frac{1-x^2}{1-4 x^2+x^4} \, dx","Integrate[(1 - x^2)/(1 - 4*x^2 + x^4),x]","\frac{\log \left(x^2+\sqrt{6} x+1\right)-\log \left(-x^2+\sqrt{6} x-1\right)}{2 \sqrt{6}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x+1}{\sqrt{3}}\right)}{\sqrt{6}}-\frac{\tanh ^{-1}\left(\frac{1-\sqrt{2} x}{\sqrt{3}}\right)}{\sqrt{6}}",1,"(-Log[-1 + Sqrt[6]*x - x^2] + Log[1 + Sqrt[6]*x + x^2])/(2*Sqrt[6])","A",1
91,1,40,46,0.0158172,"\int \frac{1-x^2}{1-5 x^2+x^4} \, dx","Integrate[(1 - x^2)/(1 - 5*x^2 + x^4),x]","\frac{\log \left(x^2+\sqrt{7} x+1\right)-\log \left(-x^2+\sqrt{7} x-1\right)}{2 \sqrt{7}}","\frac{\tanh ^{-1}\left(\frac{2 x+\sqrt{3}}{\sqrt{7}}\right)}{\sqrt{7}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{3}-2 x}{\sqrt{7}}\right)}{\sqrt{7}}",1,"(-Log[-1 + Sqrt[7]*x - x^2] + Log[1 + Sqrt[7]*x + x^2])/(2*Sqrt[7])","A",1
92,1,99,43,0.1017528,"\int \frac{-1-3 x^2}{1+2 x^2+9 x^4} \, dx","Integrate[(-1 - 3*x^2)/(1 + 2*x^2 + 9*x^4),x]","-\frac{\left(\sqrt{2}-i\right) \tan ^{-1}\left(\frac{3 x}{\sqrt{1-2 i \sqrt{2}}}\right)}{2 \sqrt{2 \left(1-2 i \sqrt{2}\right)}}-\frac{\left(\sqrt{2}+i\right) \tan ^{-1}\left(\frac{3 x}{\sqrt{1+2 i \sqrt{2}}}\right)}{2 \sqrt{2 \left(1+2 i \sqrt{2}\right)}}","\frac{\tan ^{-1}\left(\frac{1-3 x}{\sqrt{2}}\right)}{2 \sqrt{2}}-\frac{\tan ^{-1}\left(\frac{3 x+1}{\sqrt{2}}\right)}{2 \sqrt{2}}",1,"-1/2*((-I + Sqrt[2])*ArcTan[(3*x)/Sqrt[1 - (2*I)*Sqrt[2]]])/Sqrt[2*(1 - (2*I)*Sqrt[2])] - ((I + Sqrt[2])*ArcTan[(3*x)/Sqrt[1 + (2*I)*Sqrt[2]]])/(2*Sqrt[2*(1 + (2*I)*Sqrt[2])])","C",1
93,1,99,43,0.0327368,"\int \frac{1+3 x^2}{-1-2 x^2-9 x^4} \, dx","Integrate[(1 + 3*x^2)/(-1 - 2*x^2 - 9*x^4),x]","-\frac{\left(\sqrt{2}-i\right) \tan ^{-1}\left(\frac{3 x}{\sqrt{1-2 i \sqrt{2}}}\right)}{2 \sqrt{2 \left(1-2 i \sqrt{2}\right)}}-\frac{\left(\sqrt{2}+i\right) \tan ^{-1}\left(\frac{3 x}{\sqrt{1+2 i \sqrt{2}}}\right)}{2 \sqrt{2 \left(1+2 i \sqrt{2}\right)}}","\frac{\tan ^{-1}\left(\frac{1-3 x}{\sqrt{2}}\right)}{2 \sqrt{2}}-\frac{\tan ^{-1}\left(\frac{3 x+1}{\sqrt{2}}\right)}{2 \sqrt{2}}",1,"-1/2*((-I + Sqrt[2])*ArcTan[(3*x)/Sqrt[1 - (2*I)*Sqrt[2]]])/Sqrt[2*(1 - (2*I)*Sqrt[2])] - ((I + Sqrt[2])*ArcTan[(3*x)/Sqrt[1 + (2*I)*Sqrt[2]]])/(2*Sqrt[2*(1 + (2*I)*Sqrt[2])])","C",1
94,1,27,21,0.0104319,"\int \frac{3+2 x^2}{1-2 x^2+x^4} \, dx","Integrate[(3 + 2*x^2)/(1 - 2*x^2 + x^4),x]","\frac{1}{4} \left(-\frac{10 x}{x^2-1}-\log (1-x)+\log (x+1)\right)","\frac{5 x}{2 \left(1-x^2\right)}+\frac{1}{2} \tanh ^{-1}(x)",1,"((-10*x)/(-1 + x^2) - Log[1 - x] + Log[1 + x])/4","A",1
95,1,53,28,0.0194936,"\int \frac{2+3 x^2}{5-8 x^2+3 x^4} \, dx","Integrate[(2 + 3*x^2)/(5 - 8*x^2 + 3*x^4),x]","\frac{1}{20} \left(7 \sqrt{15} \log \left(\sqrt{15}-3 x\right)-25 \log (1-x)+25 \log (x+1)-7 \sqrt{15} \log \left(3 x+\sqrt{15}\right)\right)","\frac{5}{2} \tanh ^{-1}(x)-\frac{7}{2} \sqrt{\frac{3}{5}} \tanh ^{-1}\left(\sqrt{\frac{3}{5}} x\right)",1,"(7*Sqrt[15]*Log[Sqrt[15] - 3*x] - 25*Log[1 - x] + 25*Log[1 + x] - 7*Sqrt[15]*Log[Sqrt[15] + 3*x])/20","A",1
96,1,72,36,0.0410635,"\int \frac{d+e x^2}{5-8 x^2+3 x^4} \, dx","Integrate[(d + e*x^2)/(5 - 8*x^2 + 3*x^4),x]","\frac{1}{60} \left(\sqrt{15} (3 d+5 e) \log \left(\sqrt{15}-3 x\right)-15 (d+e) \log (1-x)+15 (d+e) \log (x+1)-\sqrt{15} (3 d+5 e) \log \left(3 x+\sqrt{15}\right)\right)","\frac{1}{2} (d+e) \tanh ^{-1}(x)-\frac{(3 d+5 e) \tanh ^{-1}\left(\sqrt{\frac{3}{5}} x\right)}{2 \sqrt{15}}",1,"(Sqrt[15]*(3*d + 5*e)*Log[Sqrt[15] - 3*x] - 15*(d + e)*Log[1 - x] + 15*(d + e)*Log[1 + x] - Sqrt[15]*(3*d + 5*e)*Log[Sqrt[15] + 3*x])/60","A",1
97,1,73,74,0.0979341,"\int \frac{3+x^2}{1+3 x^2+x^4} \, dx","Integrate[(3 + x^2)/(1 + 3*x^2 + x^4),x]","\frac{\left(3+\sqrt{5}\right)^{3/2} \tan ^{-1}\left(\sqrt{\frac{1}{2} \left(3+\sqrt{5}\right)} x\right)-\left(3-\sqrt{5}\right)^{3/2} \tan ^{-1}\left(\sqrt{\frac{2}{3+\sqrt{5}}} x\right)}{2 \sqrt{10}}","\frac{\left(3+\sqrt{5}\right)^{3/2} \tan ^{-1}\left(\sqrt{\frac{1}{2} \left(3+\sqrt{5}\right)} x\right)}{2 \sqrt{10}}-\frac{1}{10} \sqrt{180-80 \sqrt{5}} \tan ^{-1}\left(\sqrt{\frac{2}{3+\sqrt{5}}} x\right)",1,"(-((3 - Sqrt[5])^(3/2)*ArcTan[Sqrt[2/(3 + Sqrt[5])]*x]) + (3 + Sqrt[5])^(3/2)*ArcTan[Sqrt[(3 + Sqrt[5])/2]*x])/(2*Sqrt[10])","A",1
98,1,97,83,0.1270541,"\int \frac{a+b x^2}{1+x^2+x^4} \, dx","Integrate[(a + b*x^2)/(1 + x^2 + x^4),x]","\frac{\left(2 i a+\left(\sqrt{3}-i\right) b\right) \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}-i\right) x\right)}{\sqrt{6+6 i \sqrt{3}}}+\frac{\left(\left(\sqrt{3}+i\right) b-2 i a\right) \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}+i\right) x\right)}{\sqrt{6-6 i \sqrt{3}}}","-\frac{1}{4} (a-b) \log \left(x^2-x+1\right)+\frac{1}{4} (a-b) \log \left(x^2+x+1\right)-\frac{(a+b) \tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{2 \sqrt{3}}+\frac{(a+b) \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{2 \sqrt{3}}",1,"(((2*I)*a + (-I + Sqrt[3])*b)*ArcTan[((-I + Sqrt[3])*x)/2])/Sqrt[6 + (6*I)*Sqrt[3]] + (((-2*I)*a + (I + Sqrt[3])*b)*ArcTan[((I + Sqrt[3])*x)/2])/Sqrt[6 - (6*I)*Sqrt[3]]","C",0
99,1,147,119,0.24945,"\int \frac{a+b x^2}{\left(1+x^2+x^4\right)^2} \, dx","Integrate[(a + b*x^2)/(1 + x^2 + x^4)^2,x]","\frac{x \left(-a x^2+a+2 b x^2+b\right)}{6 \left(x^4+x^2+1\right)}-\frac{\left(\left(\sqrt{3}-11 i\right) a-2 \left(\sqrt{3}-2 i\right) b\right) \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}-i\right) x\right)}{6 \sqrt{6+6 i \sqrt{3}}}-\frac{\left(\left(\sqrt{3}+11 i\right) a-2 \left(\sqrt{3}+2 i\right) b\right) \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}+i\right) x\right)}{6 \sqrt{6-6 i \sqrt{3}}}","-\frac{1}{8} (2 a-b) \log \left(x^2-x+1\right)+\frac{1}{8} (2 a-b) \log \left(x^2+x+1\right)+\frac{x \left(-\left(x^2 (a-2 b)\right)+a+b\right)}{6 \left(x^4+x^2+1\right)}-\frac{(4 a+b) \tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{12 \sqrt{3}}+\frac{(4 a+b) \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{12 \sqrt{3}}",1,"(x*(a + b - a*x^2 + 2*b*x^2))/(6*(1 + x^2 + x^4)) - (((-11*I + Sqrt[3])*a - 2*(-2*I + Sqrt[3])*b)*ArcTan[((-I + Sqrt[3])*x)/2])/(6*Sqrt[6 + (6*I)*Sqrt[3]]) - (((11*I + Sqrt[3])*a - 2*(2*I + Sqrt[3])*b)*ArcTan[((I + Sqrt[3])*x)/2])/(6*Sqrt[6 - (6*I)*Sqrt[3]])","C",0
100,1,111,234,0.1163037,"\int \frac{a+b x^2}{2+x^2+x^4} \, dx","Integrate[(a + b*x^2)/(2 + x^2 + x^4),x]","\frac{\left(\left(\sqrt{7}+i\right) b-2 i a\right) \tan ^{-1}\left(\frac{x}{\sqrt{\frac{1}{2} \left(1-i \sqrt{7}\right)}}\right)}{\sqrt{14-14 i \sqrt{7}}}+\frac{\left(2 i a+\left(\sqrt{7}-i\right) b\right) \tan ^{-1}\left(\frac{x}{\sqrt{\frac{1}{2} \left(1+i \sqrt{7}\right)}}\right)}{\sqrt{14+14 i \sqrt{7}}}","-\frac{\left(a-\sqrt{2} b\right) \log \left(x^2-\sqrt{2 \sqrt{2}-1} x+\sqrt{2}\right)}{4 \sqrt{2 \left(2 \sqrt{2}-1\right)}}+\frac{\left(a-\sqrt{2} b\right) \log \left(x^2+\sqrt{2 \sqrt{2}-1} x+\sqrt{2}\right)}{4 \sqrt{2 \left(2 \sqrt{2}-1\right)}}-\frac{1}{2} \sqrt{\frac{1}{14} \left(2 \sqrt{2}-1\right)} \left(a+\sqrt{2} b\right) \tan ^{-1}\left(\frac{\sqrt{2 \sqrt{2}-1}-2 x}{\sqrt{1+2 \sqrt{2}}}\right)+\frac{1}{2} \sqrt{\frac{1}{14} \left(2 \sqrt{2}-1\right)} \left(a+\sqrt{2} b\right) \tan ^{-1}\left(\frac{2 x+\sqrt{2 \sqrt{2}-1}}{\sqrt{1+2 \sqrt{2}}}\right)",1,"(((-2*I)*a + (I + Sqrt[7])*b)*ArcTan[x/Sqrt[(1 - I*Sqrt[7])/2]])/Sqrt[14 - (14*I)*Sqrt[7]] + (((2*I)*a + (-I + Sqrt[7])*b)*ArcTan[x/Sqrt[(1 + I*Sqrt[7])/2]])/Sqrt[14 + (14*I)*Sqrt[7]]","C",1
101,1,165,316,0.2163077,"\int \frac{a+b x^2}{\left(2+x^2+x^4\right)^2} \, dx","Integrate[(a + b*x^2)/(2 + x^2 + x^4)^2,x]","\frac{2 b \left(2 x^3+x\right)-a x \left(x^2-3\right)}{28 \left(x^4+x^2+2\right)}-\frac{\left(\left(\sqrt{7}+23 i\right) a-4 \left(\sqrt{7}+2 i\right) b\right) \tan ^{-1}\left(\frac{x}{\sqrt{\frac{1}{2} \left(1-i \sqrt{7}\right)}}\right)}{28 \sqrt{14-14 i \sqrt{7}}}-\frac{\left(\left(\sqrt{7}-23 i\right) a-4 \left(\sqrt{7}-2 i\right) b\right) \tan ^{-1}\left(\frac{x}{\sqrt{\frac{1}{2} \left(1+i \sqrt{7}\right)}}\right)}{28 \sqrt{14+14 i \sqrt{7}}}","-\frac{\left(\sqrt{2} (a-4 b)+11 a-2 b\right) \log \left(x^2-\sqrt{2 \sqrt{2}-1} x+\sqrt{2}\right)}{112 \sqrt{2 \left(2 \sqrt{2}-1\right)}}+\frac{\left(\left(11+\sqrt{2}\right) a-2 \left(2 \sqrt{2} b+b\right)\right) \log \left(x^2+\sqrt{2 \sqrt{2}-1} x+\sqrt{2}\right)}{112 \sqrt{2 \left(2 \sqrt{2}-1\right)}}+\frac{x \left(-\left(x^2 (a-4 b)\right)+3 a+2 b\right)}{28 \left(x^4+x^2+2\right)}-\frac{1}{56} \sqrt{\frac{1}{14} \left(2 \sqrt{2}-1\right)} \left(\left(11-\sqrt{2}\right) a-\left(2-4 \sqrt{2}\right) b\right) \tan ^{-1}\left(\frac{\sqrt{2 \sqrt{2}-1}-2 x}{\sqrt{1+2 \sqrt{2}}}\right)+\frac{1}{56} \sqrt{\frac{1}{14} \left(2 \sqrt{2}-1\right)} \left(\left(11-\sqrt{2}\right) a-\left(2-4 \sqrt{2}\right) b\right) \tan ^{-1}\left(\frac{2 x+\sqrt{2 \sqrt{2}-1}}{\sqrt{1+2 \sqrt{2}}}\right)",1,"(-(a*x*(-3 + x^2)) + 2*b*(x + 2*x^3))/(28*(2 + x^2 + x^4)) - (((23*I + Sqrt[7])*a - 4*(2*I + Sqrt[7])*b)*ArcTan[x/Sqrt[(1 - I*Sqrt[7])/2]])/(28*Sqrt[14 - (14*I)*Sqrt[7]]) - (((-23*I + Sqrt[7])*a - 4*(-2*I + Sqrt[7])*b)*ArcTan[x/Sqrt[(1 + I*Sqrt[7])/2]])/(28*Sqrt[14 + (14*I)*Sqrt[7]])","C",1
102,1,53,160,0.0449706,"\int \frac{\sqrt{2}-x^2}{1-\sqrt{2} x^2+x^4} \, dx","Integrate[(Sqrt[2] - x^2)/(1 - Sqrt[2]*x^2 + x^4),x]","\frac{\sqrt{-1-i} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{-1-i}}\right)+\sqrt{-1+i} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{-1+i}}\right)}{2^{3/4}}","-\frac{1}{4} \sqrt{1+\frac{1}{\sqrt{2}}} \log \left(x^2-\sqrt{2+\sqrt{2}} x+1\right)+\frac{1}{4} \sqrt{1+\frac{1}{\sqrt{2}}} \log \left(x^2+\sqrt{2+\sqrt{2}} x+1\right)-\frac{\tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}}-2 x}{\sqrt{2-\sqrt{2}}}\right)}{2 \sqrt{2+\sqrt{2}}}+\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right)}{2 \sqrt{2+\sqrt{2}}}",1,"(Sqrt[-1 - I]*ArcTan[(2^(1/4)*x)/Sqrt[-1 - I]] + Sqrt[-1 + I]*ArcTan[(2^(1/4)*x)/Sqrt[-1 + I]])/2^(3/4)","C",1
103,1,53,172,0.0352084,"\int \frac{\sqrt{2}+x^2}{1+\sqrt{2} x^2+x^4} \, dx","Integrate[(Sqrt[2] + x^2)/(1 + Sqrt[2]*x^2 + x^4),x]","\frac{\sqrt{1-i} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{1-i}}\right)+\sqrt{1+i} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{1+i}}\right)}{2^{3/4}}","-\frac{1}{4} \sqrt{1-\frac{1}{\sqrt{2}}} \log \left(x^2-\sqrt{2-\sqrt{2}} x+1\right)+\frac{1}{4} \sqrt{1-\frac{1}{\sqrt{2}}} \log \left(x^2+\sqrt{2-\sqrt{2}} x+1\right)-\frac{\tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}}-2 x}{\sqrt{2+\sqrt{2}}}\right)}{2 \sqrt{2-\sqrt{2}}}+\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right)}{2 \sqrt{2-\sqrt{2}}}",1,"(Sqrt[1 - I]*ArcTan[(2^(1/4)*x)/Sqrt[1 - I]] + Sqrt[1 + I]*ArcTan[(2^(1/4)*x)/Sqrt[1 + I]])/2^(3/4)","C",1
104,1,137,160,0.0895614,"\int \frac{\sqrt{2}-x^2}{1+b x^2+x^4} \, dx","Integrate[(Sqrt[2] - x^2)/(1 + b*x^2 + x^4),x]","\frac{\frac{\left(-\sqrt{b^2-4}+b+2 \sqrt{2}\right) \tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{b-\sqrt{b^2-4}}}\right)}{\sqrt{b-\sqrt{b^2-4}}}-\frac{\left(\sqrt{b^2-4}+b+2 \sqrt{2}\right) \tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{\sqrt{b^2-4}+b}}\right)}{\sqrt{\sqrt{b^2-4}+b}}}{\sqrt{2} \sqrt{b^2-4}}","-\frac{\left(1+\sqrt{2}\right) \log \left(-\sqrt{2-b} x+x^2+1\right)}{4 \sqrt{2-b}}+\frac{\left(1+\sqrt{2}\right) \log \left(\sqrt{2-b} x+x^2+1\right)}{4 \sqrt{2-b}}+\frac{\left(1-\sqrt{2}\right) \tan ^{-1}\left(\frac{\sqrt{2-b}-2 x}{\sqrt{b+2}}\right)}{2 \sqrt{b+2}}-\frac{\left(1-\sqrt{2}\right) \tan ^{-1}\left(\frac{\sqrt{2-b}+2 x}{\sqrt{b+2}}\right)}{2 \sqrt{b+2}}",1,"(((2*Sqrt[2] + b - Sqrt[-4 + b^2])*ArcTan[(Sqrt[2]*x)/Sqrt[b - Sqrt[-4 + b^2]]])/Sqrt[b - Sqrt[-4 + b^2]] - ((2*Sqrt[2] + b + Sqrt[-4 + b^2])*ArcTan[(Sqrt[2]*x)/Sqrt[b + Sqrt[-4 + b^2]]])/Sqrt[b + Sqrt[-4 + b^2]])/(Sqrt[2]*Sqrt[-4 + b^2])","A",1
105,1,136,160,0.0572386,"\int \frac{\sqrt{2}+x^2}{1+b x^2+x^4} \, dx","Integrate[(Sqrt[2] + x^2)/(1 + b*x^2 + x^4),x]","\frac{\frac{\left(\sqrt{b^2-4}-b+2 \sqrt{2}\right) \tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{b-\sqrt{b^2-4}}}\right)}{\sqrt{b-\sqrt{b^2-4}}}+\frac{\left(\sqrt{b^2-4}+b-2 \sqrt{2}\right) \tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{\sqrt{b^2-4}+b}}\right)}{\sqrt{\sqrt{b^2-4}+b}}}{\sqrt{2} \sqrt{b^2-4}}","\frac{\left(1-\sqrt{2}\right) \log \left(-\sqrt{2-b} x+x^2+1\right)}{4 \sqrt{2-b}}-\frac{\left(1-\sqrt{2}\right) \log \left(\sqrt{2-b} x+x^2+1\right)}{4 \sqrt{2-b}}-\frac{\left(1+\sqrt{2}\right) \tan ^{-1}\left(\frac{\sqrt{2-b}-2 x}{\sqrt{b+2}}\right)}{2 \sqrt{b+2}}+\frac{\left(1+\sqrt{2}\right) \tan ^{-1}\left(\frac{\sqrt{2-b}+2 x}{\sqrt{b+2}}\right)}{2 \sqrt{b+2}}",1,"(((2*Sqrt[2] - b + Sqrt[-4 + b^2])*ArcTan[(Sqrt[2]*x)/Sqrt[b - Sqrt[-4 + b^2]]])/Sqrt[b - Sqrt[-4 + b^2]] + ((-2*Sqrt[2] + b + Sqrt[-4 + b^2])*ArcTan[(Sqrt[2]*x)/Sqrt[b + Sqrt[-4 + b^2]]])/Sqrt[b + Sqrt[-4 + b^2]])/(Sqrt[2]*Sqrt[-4 + b^2])","A",1
106,1,115,114,0.1717952,"\int \frac{2 a-x^2}{a^2-a x^2+x^4} \, dx","Integrate[(2*a - x^2)/(a^2 - a*x^2 + x^4),x]","\frac{\sqrt[4]{-1} \left(\sqrt{\sqrt{3}-i} \left(\sqrt{3}-3 i\right) \tanh ^{-1}\left(\frac{(1+i) x}{\sqrt{\sqrt{3}+i} \sqrt{a}}\right)-\sqrt{\sqrt{3}+i} \left(\sqrt{3}+3 i\right) \tan ^{-1}\left(\frac{(1+i) x}{\sqrt{\sqrt{3}-i} \sqrt{a}}\right)\right)}{2 \sqrt{6} \sqrt{a}}","-\frac{\sqrt{3} \log \left(-\sqrt{3} \sqrt{a} x+a+x^2\right)}{4 \sqrt{a}}+\frac{\sqrt{3} \log \left(\sqrt{3} \sqrt{a} x+a+x^2\right)}{4 \sqrt{a}}-\frac{\tan ^{-1}\left(\sqrt{3}-\frac{2 x}{\sqrt{a}}\right)}{2 \sqrt{a}}+\frac{\tan ^{-1}\left(\frac{2 x}{\sqrt{a}}+\sqrt{3}\right)}{2 \sqrt{a}}",1,"((-1)^(1/4)*(-(Sqrt[I + Sqrt[3]]*(3*I + Sqrt[3])*ArcTan[((1 + I)*x)/(Sqrt[-I + Sqrt[3]]*Sqrt[a])]) + Sqrt[-I + Sqrt[3]]*(-3*I + Sqrt[3])*ArcTanh[((1 + I)*x)/(Sqrt[I + Sqrt[3]]*Sqrt[a])]))/(2*Sqrt[6]*Sqrt[a])","C",1
107,1,115,122,0.1533206,"\int \frac{2 \sqrt{a}-x^2}{a-\sqrt{a} x^2+x^4} \, dx","Integrate[(2*Sqrt[a] - x^2)/(a - Sqrt[a]*x^2 + x^4),x]","\frac{\sqrt[4]{-1} \left(\sqrt{\sqrt{3}-i} \left(\sqrt{3}-3 i\right) \tanh ^{-1}\left(\frac{(1+i) x}{\sqrt{\sqrt{3}+i} \sqrt[4]{a}}\right)-\sqrt{\sqrt{3}+i} \left(\sqrt{3}+3 i\right) \tan ^{-1}\left(\frac{(1+i) x}{\sqrt{\sqrt{3}-i} \sqrt[4]{a}}\right)\right)}{2 \sqrt{6} \sqrt[4]{a}}","-\frac{\sqrt{3} \log \left(-\sqrt{3} \sqrt[4]{a} x+\sqrt{a}+x^2\right)}{4 \sqrt[4]{a}}+\frac{\sqrt{3} \log \left(\sqrt{3} \sqrt[4]{a} x+\sqrt{a}+x^2\right)}{4 \sqrt[4]{a}}-\frac{\tan ^{-1}\left(\sqrt{3}-\frac{2 x}{\sqrt[4]{a}}\right)}{2 \sqrt[4]{a}}+\frac{\tan ^{-1}\left(\frac{2 x}{\sqrt[4]{a}}+\sqrt{3}\right)}{2 \sqrt[4]{a}}",1,"((-1)^(1/4)*(-(Sqrt[I + Sqrt[3]]*(3*I + Sqrt[3])*ArcTan[((1 + I)*x)/(Sqrt[-I + Sqrt[3]]*a^(1/4))]) + Sqrt[-I + Sqrt[3]]*(-3*I + Sqrt[3])*ArcTanh[((1 + I)*x)/(Sqrt[I + Sqrt[3]]*a^(1/4))]))/(2*Sqrt[6]*a^(1/4))","C",1
108,1,115,124,0.130751,"\int \frac{2 b^{2/3}+x^2}{b^{4/3}+b^{2/3} x^2+x^4} \, dx","Integrate[(2*b^(2/3) + x^2)/(b^(4/3) + b^(2/3)*x^2 + x^4),x]","\frac{\sqrt[4]{-1} \left(\sqrt{\sqrt{3}-i} \left(\sqrt{3}-3 i\right) \tan ^{-1}\left(\frac{(1+i) x}{\sqrt{\sqrt{3}+i} \sqrt[3]{b}}\right)-\sqrt{\sqrt{3}+i} \left(\sqrt{3}+3 i\right) \tanh ^{-1}\left(\frac{(1+i) x}{\sqrt{\sqrt{3}-i} \sqrt[3]{b}}\right)\right)}{2 \sqrt{6} \sqrt[3]{b}}","-\frac{\log \left(b^{2/3}-\sqrt[3]{b} x+x^2\right)}{4 \sqrt[3]{b}}+\frac{\log \left(b^{2/3}+\sqrt[3]{b} x+x^2\right)}{4 \sqrt[3]{b}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{b}-2 x}{\sqrt{3} \sqrt[3]{b}}\right)}{2 \sqrt[3]{b}}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{b}+2 x}{\sqrt{3} \sqrt[3]{b}}\right)}{2 \sqrt[3]{b}}",1,"((-1)^(1/4)*(Sqrt[-I + Sqrt[3]]*(-3*I + Sqrt[3])*ArcTan[((1 + I)*x)/(Sqrt[I + Sqrt[3]]*b^(1/3))] - Sqrt[I + Sqrt[3]]*(3*I + Sqrt[3])*ArcTanh[((1 + I)*x)/(Sqrt[-I + Sqrt[3]]*b^(1/3))]))/(2*Sqrt[6]*b^(1/3))","C",1
109,1,130,136,0.1503003,"\int \frac{A+B x^2}{a^2-a x^2+x^4} \, dx","Integrate[(A + B*x^2)/(a^2 - a*x^2 + x^4),x]","\frac{\sqrt[4]{-1} \left(\frac{\left(\left(\sqrt{3}-i\right) a B-2 i A\right) \tan ^{-1}\left(\frac{(1+i) x}{\sqrt{\sqrt{3}-i} \sqrt{a}}\right)}{\sqrt{\sqrt{3}-i}}-\frac{\left(\left(\sqrt{3}+i\right) a B+2 i A\right) \tanh ^{-1}\left(\frac{(1+i) x}{\sqrt{\sqrt{3}+i} \sqrt{a}}\right)}{\sqrt{\sqrt{3}+i}}\right)}{\sqrt{6} a^{3/2}}","-\frac{(A-a B) \log \left(-\sqrt{3} \sqrt{a} x+a+x^2\right)}{4 \sqrt{3} a^{3/2}}+\frac{(A-a B) \log \left(\sqrt{3} \sqrt{a} x+a+x^2\right)}{4 \sqrt{3} a^{3/2}}-\frac{(a B+A) \tan ^{-1}\left(\sqrt{3}-\frac{2 x}{\sqrt{a}}\right)}{2 a^{3/2}}+\frac{(a B+A) \tan ^{-1}\left(\frac{2 x}{\sqrt{a}}+\sqrt{3}\right)}{2 a^{3/2}}",1,"((-1)^(1/4)*((((-2*I)*A + (-I + Sqrt[3])*a*B)*ArcTan[((1 + I)*x)/(Sqrt[-I + Sqrt[3]]*Sqrt[a])])/Sqrt[-I + Sqrt[3]] - (((2*I)*A + (I + Sqrt[3])*a*B)*ArcTanh[((1 + I)*x)/(Sqrt[I + Sqrt[3]]*Sqrt[a])])/Sqrt[I + Sqrt[3]]))/(Sqrt[6]*a^(3/2))","C",1
110,1,138,160,0.1349175,"\int \frac{A+B x^2}{a-\sqrt{a} x^2+x^4} \, dx","Integrate[(A + B*x^2)/(a - Sqrt[a]*x^2 + x^4),x]","\frac{\sqrt[4]{-1} \left(\frac{\left(\left(\sqrt{3}-i\right) \sqrt{a} B-2 i A\right) \tan ^{-1}\left(\frac{(1+i) x}{\sqrt{\sqrt{3}-i} \sqrt[4]{a}}\right)}{\sqrt{\sqrt{3}-i}}-\frac{\left(\left(\sqrt{3}+i\right) \sqrt{a} B+2 i A\right) \tanh ^{-1}\left(\frac{(1+i) x}{\sqrt{\sqrt{3}+i} \sqrt[4]{a}}\right)}{\sqrt{\sqrt{3}+i}}\right)}{\sqrt{6} a^{3/4}}","-\frac{\left(A-\sqrt{a} B\right) \log \left(-\sqrt{3} \sqrt[4]{a} x+\sqrt{a}+x^2\right)}{4 \sqrt{3} a^{3/4}}+\frac{\left(A-\sqrt{a} B\right) \log \left(\sqrt{3} \sqrt[4]{a} x+\sqrt{a}+x^2\right)}{4 \sqrt{3} a^{3/4}}-\frac{\left(\sqrt{a} B+A\right) \tan ^{-1}\left(\sqrt{3}-\frac{2 x}{\sqrt[4]{a}}\right)}{2 a^{3/4}}+\frac{\left(\sqrt{a} B+A\right) \tan ^{-1}\left(\frac{2 x}{\sqrt[4]{a}}+\sqrt{3}\right)}{2 a^{3/4}}",1,"((-1)^(1/4)*((((-2*I)*A + (-I + Sqrt[3])*Sqrt[a]*B)*ArcTan[((1 + I)*x)/(Sqrt[-I + Sqrt[3]]*a^(1/4))])/Sqrt[-I + Sqrt[3]] - (((2*I)*A + (I + Sqrt[3])*Sqrt[a]*B)*ArcTanh[((1 + I)*x)/(Sqrt[I + Sqrt[3]]*a^(1/4))])/Sqrt[I + Sqrt[3]]))/(Sqrt[6]*a^(3/4))","C",1
111,1,247,414,0.2017175,"\int \frac{A+B x^2}{a-\sqrt{a c} x^2+c x^4} \, dx","Integrate[(A + B*x^2)/(a - Sqrt[a*c]*x^2 + c*x^4),x]","\frac{\frac{\left(\sqrt{3} \sqrt{a} B \sqrt{c}-i \left(B \sqrt{a c}+2 A c\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{-\sqrt{a c}-i \sqrt{3} \sqrt{a} \sqrt{c}}}\right)}{\sqrt{-\sqrt{a c}-i \sqrt{3} \sqrt{a} \sqrt{c}}}+\frac{\left(\sqrt{3} \sqrt{a} B \sqrt{c}+i \left(B \sqrt{a c}+2 A c\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{-\sqrt{a c}+i \sqrt{3} \sqrt{a} \sqrt{c}}}\right)}{\sqrt{-\sqrt{a c}+i \sqrt{3} \sqrt{a} \sqrt{c}}}}{\sqrt{6} \sqrt{a} c}","-\frac{\left(A-\frac{\sqrt{a} B}{\sqrt{c}}\right) \log \left(-x \sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{a} \sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}}+\frac{\left(A-\frac{\sqrt{a} B}{\sqrt{c}}\right) \log \left(x \sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{a} \sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}}-\frac{\left(\sqrt{a} B+A \sqrt{c}\right) \tan ^{-1}\left(\frac{\sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}-2 \sqrt{c} x}{\sqrt{2 \sqrt{a} \sqrt{c}-\sqrt{a c}}}\right)}{2 \sqrt{a} \sqrt{c} \sqrt{2 \sqrt{a} \sqrt{c}-\sqrt{a c}}}+\frac{\left(\sqrt{a} B+A \sqrt{c}\right) \tan ^{-1}\left(\frac{\sqrt{2 \sqrt{a} \sqrt{c}+\sqrt{a c}}+2 \sqrt{c} x}{\sqrt{2 \sqrt{a} \sqrt{c}-\sqrt{a c}}}\right)}{2 \sqrt{a} \sqrt{c} \sqrt{2 \sqrt{a} \sqrt{c}-\sqrt{a c}}}",1,"(((Sqrt[3]*Sqrt[a]*B*Sqrt[c] - I*(2*A*c + B*Sqrt[a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[(-I)*Sqrt[3]*Sqrt[a]*Sqrt[c] - Sqrt[a*c]]])/Sqrt[(-I)*Sqrt[3]*Sqrt[a]*Sqrt[c] - Sqrt[a*c]] + ((Sqrt[3]*Sqrt[a]*B*Sqrt[c] + I*(2*A*c + B*Sqrt[a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[I*Sqrt[3]*Sqrt[a]*Sqrt[c] - Sqrt[a*c]]])/Sqrt[I*Sqrt[3]*Sqrt[a]*Sqrt[c] - Sqrt[a*c]])/(Sqrt[6]*Sqrt[a]*c)","C",1
112,1,163,234,0.1863061,"\int \frac{A+B x^2}{a-\sqrt{a} \sqrt{c} x^2+c x^4} \, dx","Integrate[(A + B*x^2)/(a - Sqrt[a]*Sqrt[c]*x^2 + c*x^4),x]","\frac{\sqrt[4]{-1} \left(\frac{\left(\left(\sqrt{3}-i\right) \sqrt{a} B-2 i A \sqrt{c}\right) \tan ^{-1}\left(\frac{(1+i) \sqrt[4]{c} x}{\sqrt{\sqrt{3}-i} \sqrt[4]{a}}\right)}{\sqrt{\sqrt{3}-i}}-\frac{\left(\left(\sqrt{3}+i\right) \sqrt{a} B+2 i A \sqrt{c}\right) \tanh ^{-1}\left(\frac{(1+i) \sqrt[4]{c} x}{\sqrt{\sqrt{3}+i} \sqrt[4]{a}}\right)}{\sqrt{\sqrt{3}+i}}\right)}{\sqrt{6} a^{3/4} c^{3/4}}","-\frac{\left(\sqrt{a} B+A \sqrt{c}\right) \tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{2 a^{3/4} c^{3/4}}+\frac{\left(\sqrt{a} B+A \sqrt{c}\right) \tan ^{-1}\left(\frac{2 \sqrt[4]{c} x}{\sqrt[4]{a}}+\sqrt{3}\right)}{2 a^{3/4} c^{3/4}}-\frac{\left(A-\frac{\sqrt{a} B}{\sqrt{c}}\right) \log \left(-\sqrt{3} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{3} a^{3/4} \sqrt[4]{c}}+\frac{\left(A-\frac{\sqrt{a} B}{\sqrt{c}}\right) \log \left(\sqrt{3} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{3} a^{3/4} \sqrt[4]{c}}",1,"((-1)^(1/4)*((((-I + Sqrt[3])*Sqrt[a]*B - (2*I)*A*Sqrt[c])*ArcTan[((1 + I)*c^(1/4)*x)/(Sqrt[-I + Sqrt[3]]*a^(1/4))])/Sqrt[-I + Sqrt[3]] - (((I + Sqrt[3])*Sqrt[a]*B + (2*I)*A*Sqrt[c])*ArcTanh[((1 + I)*c^(1/4)*x)/(Sqrt[I + Sqrt[3]]*a^(1/4))])/Sqrt[I + Sqrt[3]]))/(Sqrt[6]*a^(3/4)*c^(3/4))","C",1
113,1,103,96,0.135161,"\int \frac{3-x^2}{\sqrt{3+x^2-x^4}} \, dx","Integrate[(3 - x^2)/Sqrt[3 + x^2 - x^4],x]","-\frac{i \left(\left(1+\sqrt{13}\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{-1+\sqrt{13}}} x\right)|\frac{1}{6} \left(-7+\sqrt{13}\right)\right)-\left(\sqrt{13}-5\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{-1+\sqrt{13}}} x\right)|\frac{1}{6} \left(-7+\sqrt{13}\right)\right)\right)}{\sqrt{2 \left(1+\sqrt{13}\right)}}","\sqrt{7+2 \sqrt{13}} F\left(\sin ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{13}}} x\right)|\frac{1}{6} \left(-7-\sqrt{13}\right)\right)-\sqrt{\frac{1}{2} \left(\sqrt{13}-1\right)} E\left(\sin ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{13}}} x\right)|\frac{1}{6} \left(-7-\sqrt{13}\right)\right)",1,"((-I)*((1 + Sqrt[13])*EllipticE[I*ArcSinh[Sqrt[2/(-1 + Sqrt[13])]*x], (-7 + Sqrt[13])/6] - (-5 + Sqrt[13])*EllipticF[I*ArcSinh[Sqrt[2/(-1 + Sqrt[13])]*x], (-7 + Sqrt[13])/6]))/Sqrt[2*(1 + Sqrt[13])]","C",0
114,1,19,25,0.0553175,"\int \frac{3-x^2}{\sqrt{3+2 x^2-x^4}} \, dx","Integrate[(3 - x^2)/Sqrt[3 + 2*x^2 - x^4],x]","-i \sqrt{3} E\left(i \sinh ^{-1}(x)|-\frac{1}{3}\right)","4 F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{3}}\right)\right|-3\right)-E\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{3}}\right)\right|-3\right)",1,"(-I)*Sqrt[3]*EllipticE[I*ArcSinh[x], -1/3]","C",0
115,1,103,96,0.1679191,"\int \frac{3-x^2}{\sqrt{3+3 x^2-x^4}} \, dx","Integrate[(3 - x^2)/Sqrt[3 + 3*x^2 - x^4],x]","-\frac{i \left(\left(3+\sqrt{21}\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{-3+\sqrt{21}}} x\right)|\frac{1}{2} \left(-5+\sqrt{21}\right)\right)-\left(\sqrt{21}-3\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{-3+\sqrt{21}}} x\right)|\frac{1}{2} \left(-5+\sqrt{21}\right)\right)\right)}{\sqrt{2 \left(3+\sqrt{21}\right)}}","\sqrt{9+2 \sqrt{21}} F\left(\sin ^{-1}\left(\sqrt{\frac{2}{3+\sqrt{21}}} x\right)|\frac{1}{2} \left(-5-\sqrt{21}\right)\right)-\sqrt{\frac{1}{2} \left(\sqrt{21}-3\right)} E\left(\sin ^{-1}\left(\sqrt{\frac{2}{3+\sqrt{21}}} x\right)|\frac{1}{2} \left(-5-\sqrt{21}\right)\right)",1,"((-I)*((3 + Sqrt[21])*EllipticE[I*ArcSinh[Sqrt[2/(-3 + Sqrt[21])]*x], (-5 + Sqrt[21])/2] - (-3 + Sqrt[21])*EllipticF[I*ArcSinh[Sqrt[2/(-3 + Sqrt[21])]*x], (-5 + Sqrt[21])/2]))/Sqrt[2*(3 + Sqrt[21])]","C",0
116,1,107,92,0.1340433,"\int \frac{3-x^2}{\sqrt{3-x^2-x^4}} \, dx","Integrate[(3 - x^2)/Sqrt[3 - x^2 - x^4],x]","-\frac{i \left(\left(\sqrt{13}-1\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{13}}} x\right)|-\frac{7}{6}-\frac{\sqrt{13}}{6}\right)-\left(\sqrt{13}-7\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{13}}} x\right)|-\frac{7}{6}-\frac{\sqrt{13}}{6}\right)\right)}{\sqrt{2 \left(\sqrt{13}-1\right)}}","\sqrt{5+2 \sqrt{13}} F\left(\sin ^{-1}\left(\sqrt{\frac{2}{-1+\sqrt{13}}} x\right)|\frac{1}{6} \left(-7+\sqrt{13}\right)\right)-\sqrt{\frac{1}{2} \left(1+\sqrt{13}\right)} E\left(\sin ^{-1}\left(\sqrt{\frac{2}{-1+\sqrt{13}}} x\right)|\frac{1}{6} \left(-7+\sqrt{13}\right)\right)",1,"((-I)*((-1 + Sqrt[13])*EllipticE[I*ArcSinh[Sqrt[2/(1 + Sqrt[13])]*x], -7/6 - Sqrt[13]/6] - (-7 + Sqrt[13])*EllipticF[I*ArcSinh[Sqrt[2/(1 + Sqrt[13])]*x], -7/6 - Sqrt[13]/6]))/Sqrt[2*(-1 + Sqrt[13])]","C",0
117,1,35,27,0.062759,"\int \frac{3-x^2}{\sqrt{3-2 x^2-x^4}} \, dx","Integrate[(3 - x^2)/Sqrt[3 - 2*x^2 - x^4],x]","-i \left(2 F\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{3}}\right)\right|-3\right)+E\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{3}}\right)\right|-3\right)\right)","2 \sqrt{3} F\left(\sin ^{-1}(x)|-\frac{1}{3}\right)-\sqrt{3} E\left(\sin ^{-1}(x)|-\frac{1}{3}\right)",1,"(-I)*(EllipticE[I*ArcSinh[x/Sqrt[3]], -3] + 2*EllipticF[I*ArcSinh[x/Sqrt[3]], -3])","C",1
118,1,107,92,0.1667533,"\int \frac{3-x^2}{\sqrt{3-3 x^2-x^4}} \, dx","Integrate[(3 - x^2)/Sqrt[3 - 3*x^2 - x^4],x]","-\frac{i \left(\left(\sqrt{21}-3\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{3+\sqrt{21}}} x\right)|-\frac{5}{2}-\frac{\sqrt{21}}{2}\right)-\left(\sqrt{21}-9\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{2}{3+\sqrt{21}}} x\right)|-\frac{5}{2}-\frac{\sqrt{21}}{2}\right)\right)}{\sqrt{2 \left(\sqrt{21}-3\right)}}","\sqrt{3+2 \sqrt{21}} F\left(\sin ^{-1}\left(\sqrt{\frac{2}{-3+\sqrt{21}}} x\right)|\frac{1}{2} \left(-5+\sqrt{21}\right)\right)-\sqrt{\frac{1}{2} \left(3+\sqrt{21}\right)} E\left(\sin ^{-1}\left(\sqrt{\frac{2}{-3+\sqrt{21}}} x\right)|\frac{1}{2} \left(-5+\sqrt{21}\right)\right)",1,"((-I)*((-3 + Sqrt[21])*EllipticE[I*ArcSinh[Sqrt[2/(3 + Sqrt[21])]*x], -5/2 - Sqrt[21]/2] - (-9 + Sqrt[21])*EllipticF[I*ArcSinh[Sqrt[2/(3 + Sqrt[21])]*x], -5/2 - Sqrt[21]/2]))/Sqrt[2*(-3 + Sqrt[21])]","C",0
119,1,187,296,0.3010467,"\int \frac{b-\sqrt{b^2-4 a c}+2 c x^2}{\sqrt{a+b x^2+c x^4}} \, dx","Integrate[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/Sqrt[a + b*x^2 + c*x^4],x]","-\frac{2 i \sqrt{2} a \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \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} 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)}{\sqrt{a+b x^2+c x^4}}","\frac{\left(-\sqrt{b^2-4 a c}+2 \sqrt{a} \sqrt{c}+b\right) \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{2 \sqrt[4]{a} \sqrt[4]{c} \sqrt{a+b x^2+c x^4}}+\frac{2 \sqrt{c} x \sqrt{a+b x^2+c x^4}}{\sqrt{a}+\sqrt{c} x^2}-\frac{2 \sqrt[4]{a} \sqrt[4]{c} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{\sqrt{a+b x^2+c x^4}}",1,"((-2*I)*Sqrt[2]*a*Sqrt[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[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/Sqrt[a + b*x^2 + c*x^4]","C",1
120,1,106,106,0.0199402,"\int \left(d+e x^2\right)^4 \left(a+c x^4\right) \, dx","Integrate[(d + e*x^2)^4*(a + c*x^4),x]","\frac{1}{9} e^2 x^9 \left(a e^2+6 c d^2\right)+\frac{4}{7} d e x^7 \left(a e^2+c d^2\right)+\frac{1}{5} d^2 x^5 \left(6 a e^2+c d^2\right)+a d^4 x+\frac{4}{3} a d^3 e x^3+\frac{4}{11} c d e^3 x^{11}+\frac{1}{13} c e^4 x^{13}","\frac{1}{9} e^2 x^9 \left(a e^2+6 c d^2\right)+\frac{4}{7} d e x^7 \left(a e^2+c d^2\right)+\frac{1}{5} d^2 x^5 \left(6 a e^2+c d^2\right)+a d^4 x+\frac{4}{3} a d^3 e x^3+\frac{4}{11} c d e^3 x^{11}+\frac{1}{13} c e^4 x^{13}",1,"a*d^4*x + (4*a*d^3*e*x^3)/3 + (d^2*(c*d^2 + 6*a*e^2)*x^5)/5 + (4*d*e*(c*d^2 + a*e^2)*x^7)/7 + (e^2*(6*c*d^2 + a*e^2)*x^9)/9 + (4*c*d*e^3*x^11)/11 + (c*e^4*x^13)/13","A",1
121,1,79,79,0.0164288,"\int \left(d+e x^2\right)^3 \left(a+c x^4\right) \, dx","Integrate[(d + e*x^2)^3*(a + c*x^4),x]","\frac{1}{7} e x^7 \left(a e^2+3 c d^2\right)+\frac{1}{5} d x^5 \left(3 a e^2+c d^2\right)+a d^3 x+a d^2 e x^3+\frac{1}{3} c d e^2 x^9+\frac{1}{11} c e^3 x^{11}","\frac{1}{7} e x^7 \left(a e^2+3 c d^2\right)+\frac{1}{5} d x^5 \left(3 a e^2+c d^2\right)+a d^3 x+a d^2 e x^3+\frac{1}{3} c d e^2 x^9+\frac{1}{11} c e^3 x^{11}",1,"a*d^3*x + a*d^2*e*x^3 + (d*(c*d^2 + 3*a*e^2)*x^5)/5 + (e*(3*c*d^2 + a*e^2)*x^7)/7 + (c*d*e^2*x^9)/3 + (c*e^3*x^11)/11","A",1
122,1,56,56,0.0108747,"\int \left(d+e x^2\right)^2 \left(a+c x^4\right) \, dx","Integrate[(d + e*x^2)^2*(a + c*x^4),x]","\frac{1}{5} x^5 \left(a e^2+c d^2\right)+a d^2 x+\frac{2}{3} a d e x^3+\frac{2}{7} c d e x^7+\frac{1}{9} c e^2 x^9","\frac{1}{5} x^5 \left(a e^2+c d^2\right)+a d^2 x+\frac{2}{3} a d e x^3+\frac{2}{7} c d e x^7+\frac{1}{9} c e^2 x^9",1,"a*d^2*x + (2*a*d*e*x^3)/3 + ((c*d^2 + a*e^2)*x^5)/5 + (2*c*d*e*x^7)/7 + (c*e^2*x^9)/9","A",1
123,1,32,32,0.0017152,"\int \left(d+e x^2\right) \left(a+c x^4\right) \, dx","Integrate[(d + e*x^2)*(a + c*x^4),x]","a d x+\frac{1}{3} a e x^3+\frac{1}{5} c d x^5+\frac{1}{7} c e x^7","a d x+\frac{1}{3} a e x^3+\frac{1}{5} c d x^5+\frac{1}{7} c e x^7",1,"a*d*x + (a*e*x^3)/3 + (c*d*x^5)/5 + (c*e*x^7)/7","A",1
124,1,55,55,0.0357702,"\int \frac{a+c x^4}{d+e x^2} \, dx","Integrate[(a + c*x^4)/(d + e*x^2),x]","\frac{\left(a e^2+c d^2\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{d} e^{5/2}}-\frac{c d x}{e^2}+\frac{c x^3}{3 e}","\frac{\left(a e^2+c d^2\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{d} e^{5/2}}-\frac{c d x}{e^2}+\frac{c x^3}{3 e}",1,"-((c*d*x)/e^2) + (c*x^3)/(3*e) + ((c*d^2 + a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*e^(5/2))","A",1
125,1,78,74,0.0506467,"\int \frac{a+c x^4}{\left(d+e x^2\right)^2} \, dx","Integrate[(a + c*x^4)/(d + e*x^2)^2,x]","\frac{x \left(a e^2+c d^2\right)}{2 d e^2 \left(d+e x^2\right)}-\frac{\left(3 c d^2-a e^2\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{2 d^{3/2} e^{5/2}}+\frac{c x}{e^2}","\frac{x \left(a+\frac{c d^2}{e^2}\right)}{2 d \left(d+e x^2\right)}-\frac{\left(3 c d^2-a e^2\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{2 d^{3/2} e^{5/2}}+\frac{c x}{e^2}",1,"(c*x)/e^2 + ((c*d^2 + a*e^2)*x)/(2*d*e^2*(d + e*x^2)) - ((3*c*d^2 - a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(3/2)*e^(5/2))","A",1
126,1,92,93,0.0636527,"\int \frac{a+c x^4}{\left(d+e x^2\right)^3} \, dx","Integrate[(a + c*x^4)/(d + e*x^2)^3,x]","\frac{a e^2 x \left(5 d+3 e x^2\right)-c d^2 x \left(3 d+5 e x^2\right)}{8 d^2 e^2 \left(d+e x^2\right)^2}+\frac{3 \left(a e^2+c d^2\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{8 d^{5/2} e^{5/2}}","\frac{x \left(\frac{3 a}{d^2}-\frac{5 c}{e^2}\right)}{8 \left(d+e x^2\right)}+\frac{x \left(a+\frac{c d^2}{e^2}\right)}{4 d \left(d+e x^2\right)^2}+\frac{3 \left(a e^2+c d^2\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{8 d^{5/2} e^{5/2}}",1,"(a*e^2*x*(5*d + 3*e*x^2) - c*d^2*x*(3*d + 5*e*x^2))/(8*d^2*e^2*(d + e*x^2)^2) + (3*(c*d^2 + a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(5/2)*e^(5/2))","A",1
127,1,113,123,0.0807321,"\int \frac{a+c x^4}{\left(d+e x^2\right)^4} \, dx","Integrate[(a + c*x^4)/(d + e*x^2)^4,x]","\frac{\left(5 a e^2+c d^2\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{16 d^{7/2} e^{5/2}}+\frac{x \left(a e^2 \left(33 d^2+40 d e x^2+15 e^2 x^4\right)+c d^2 \left(-3 d^2-8 d e x^2+3 e^2 x^4\right)\right)}{48 d^3 e^2 \left(d+e x^2\right)^3}","\frac{x \left(\frac{5 a}{d^2}+\frac{c}{e^2}\right)}{16 d \left(d+e x^2\right)}+\frac{x \left(\frac{5 a}{d^2}-\frac{7 c}{e^2}\right)}{24 \left(d+e x^2\right)^2}+\frac{x \left(a+\frac{c d^2}{e^2}\right)}{6 d \left(d+e x^2\right)^3}+\frac{\left(5 a e^2+c d^2\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{16 d^{7/2} e^{5/2}}",1,"(x*(c*d^2*(-3*d^2 - 8*d*e*x^2 + 3*e^2*x^4) + a*e^2*(33*d^2 + 40*d*e*x^2 + 15*e^2*x^4)))/(48*d^3*e^2*(d + e*x^2)^3) + ((c*d^2 + 5*a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(16*d^(7/2)*e^(5/2))","A",1
128,1,133,133,0.0218224,"\int \left(d+e x^2\right)^3 \left(a+c x^4\right)^2 \, dx","Integrate[(d + e*x^2)^3*(a + c*x^4)^2,x]","a^2 d^3 x+a^2 d^2 e x^3+\frac{1}{11} c e x^{11} \left(2 a e^2+3 c d^2\right)+\frac{1}{9} c d x^9 \left(6 a e^2+c d^2\right)+\frac{1}{7} a e x^7 \left(a e^2+6 c d^2\right)+\frac{1}{5} a d x^5 \left(3 a e^2+2 c d^2\right)+\frac{3}{13} c^2 d e^2 x^{13}+\frac{1}{15} c^2 e^3 x^{15}","a^2 d^3 x+a^2 d^2 e x^3+\frac{1}{11} c e x^{11} \left(2 a e^2+3 c d^2\right)+\frac{1}{9} c d x^9 \left(6 a e^2+c d^2\right)+\frac{1}{7} a e x^7 \left(a e^2+6 c d^2\right)+\frac{1}{5} a d x^5 \left(3 a e^2+2 c d^2\right)+\frac{3}{13} c^2 d e^2 x^{13}+\frac{1}{15} c^2 e^3 x^{15}",1,"a^2*d^3*x + a^2*d^2*e*x^3 + (a*d*(2*c*d^2 + 3*a*e^2)*x^5)/5 + (a*e*(6*c*d^2 + a*e^2)*x^7)/7 + (c*d*(c*d^2 + 6*a*e^2)*x^9)/9 + (c*e*(3*c*d^2 + 2*a*e^2)*x^11)/11 + (3*c^2*d*e^2*x^13)/13 + (c^2*e^3*x^15)/15","A",1
129,1,97,97,0.0175153,"\int \left(d+e x^2\right)^2 \left(a+c x^4\right)^2 \, dx","Integrate[(d + e*x^2)^2*(a + c*x^4)^2,x]","a^2 d^2 x+\frac{2}{3} a^2 d e x^3+\frac{1}{9} c x^9 \left(2 a e^2+c d^2\right)+\frac{1}{5} a x^5 \left(a e^2+2 c d^2\right)+\frac{4}{7} a c d e x^7+\frac{2}{11} c^2 d e x^{11}+\frac{1}{13} c^2 e^2 x^{13}","a^2 d^2 x+\frac{2}{3} a^2 d e x^3+\frac{1}{9} c x^9 \left(2 a e^2+c d^2\right)+\frac{1}{5} a x^5 \left(a e^2+2 c d^2\right)+\frac{4}{7} a c d e x^7+\frac{2}{11} c^2 d e x^{11}+\frac{1}{13} c^2 e^2 x^{13}",1,"a^2*d^2*x + (2*a^2*d*e*x^3)/3 + (a*(2*c*d^2 + a*e^2)*x^5)/5 + (4*a*c*d*e*x^7)/7 + (c*(c*d^2 + 2*a*e^2)*x^9)/9 + (2*c^2*d*e*x^11)/11 + (c^2*e^2*x^13)/13","A",1
130,1,60,60,0.0025216,"\int \left(d+e x^2\right) \left(a+c x^4\right)^2 \, dx","Integrate[(d + e*x^2)*(a + c*x^4)^2,x]","a^2 d x+\frac{1}{3} a^2 e x^3+\frac{2}{5} a c d x^5+\frac{2}{7} a c e x^7+\frac{1}{9} c^2 d x^9+\frac{1}{11} c^2 e x^{11}","a^2 d x+\frac{1}{3} a^2 e x^3+\frac{2}{5} a c d x^5+\frac{2}{7} a c e x^7+\frac{1}{9} c^2 d x^9+\frac{1}{11} c^2 e x^{11}",1,"a^2*d*x + (a^2*e*x^3)/3 + (2*a*c*d*x^5)/5 + (2*a*c*e*x^7)/7 + (c^2*d*x^9)/9 + (c^2*e*x^11)/11","A",1
131,1,25,25,0.0009925,"\int \left(a+c x^4\right)^2 \, dx","Integrate[(a + c*x^4)^2,x]","a^2 x+\frac{2}{5} a c x^5+\frac{c^2 x^9}{9}","a^2 x+\frac{2}{5} a c x^5+\frac{c^2 x^9}{9}",1,"a^2*x + (2*a*c*x^5)/5 + (c^2*x^9)/9","A",1
132,1,97,108,0.0786791,"\int \frac{\left(a+c x^4\right)^2}{d+e x^2} \, dx","Integrate[(a + c*x^4)^2/(d + e*x^2),x]","\frac{\left(a e^2+c d^2\right)^2 \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{d} e^{9/2}}+\frac{c x \left(70 a e^2 \left(e x^2-3 d\right)+c \left(-105 d^3+35 d^2 e x^2-21 d e^2 x^4+15 e^3 x^6\right)\right)}{105 e^4}","\frac{\left(a e^2+c d^2\right)^2 \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{d} e^{9/2}}-\frac{c d x \left(2 a e^2+c d^2\right)}{e^4}+\frac{c x^3 \left(2 a e^2+c d^2\right)}{3 e^3}-\frac{c^2 d x^5}{5 e^2}+\frac{c^2 x^7}{7 e}",1,"(c*x*(70*a*e^2*(-3*d + e*x^2) + c*(-105*d^3 + 35*d^2*e*x^2 - 21*d*e^2*x^4 + 15*e^3*x^6)))/(105*e^4) + ((c*d^2 + a*e^2)^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*e^(9/2))","A",1
133,1,134,131,0.1078078,"\int \frac{\left(a+c x^4\right)^2}{\left(d+e x^2\right)^2} \, dx","Integrate[(a + c*x^4)^2/(d + e*x^2)^2,x]","-\frac{\left(-a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{2 d^{3/2} e^{9/2}}+\frac{x \left(a e^2+c d^2\right)^2}{2 d e^4 \left(d+e x^2\right)}+\frac{c x \left(2 a e^2+3 c d^2\right)}{e^4}-\frac{2 c^2 d x^3}{3 e^3}+\frac{c^2 x^5}{5 e^2}","\frac{x \left(a e^2+c d^2\right)^2}{2 d e^4 \left(d+e x^2\right)}+\frac{c x \left(2 a e^2+3 c d^2\right)}{e^4}-\frac{\left(7 c d^2-a e^2\right) \left(a e^2+c d^2\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{2 d^{3/2} e^{9/2}}-\frac{2 c^2 d x^3}{3 e^3}+\frac{c^2 x^5}{5 e^2}",1,"(c*(3*c*d^2 + 2*a*e^2)*x)/e^4 - (2*c^2*d*x^3)/(3*e^3) + (c^2*x^5)/(5*e^2) + ((c*d^2 + a*e^2)^2*x)/(2*d*e^4*(d + e*x^2)) - ((7*c^2*d^4 + 6*a*c*d^2*e^2 - a^2*e^4)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(3/2)*e^(9/2))","A",1
134,1,154,155,0.1102824,"\int \frac{\left(a+c x^4\right)^2}{\left(d+e x^2\right)^3} \, dx","Integrate[(a + c*x^4)^2/(d + e*x^2)^3,x]","\frac{x \left(3 a^2 e^4 \left(5 d+3 e x^2\right)-6 a c d^2 e^2 \left(3 d+5 e x^2\right)-c^2 d^2 \left(105 d^3+175 d^2 e x^2+56 d e^2 x^4-8 e^3 x^6\right)\right)}{24 d^2 e^4 \left(d+e x^2\right)^2}+\frac{\left(3 a^2 e^4+6 a c d^2 e^2+35 c^2 d^4\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{8 d^{5/2} e^{9/2}}","\frac{x \left(3 a^2-\frac{10 a c d^2}{e^2}-\frac{13 c^2 d^4}{e^4}\right)}{8 d^2 \left(d+e x^2\right)}+\frac{\left(3 a^2 e^4+6 a c d^2 e^2+35 c^2 d^4\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{8 d^{5/2} e^{9/2}}+\frac{x \left(a e^2+c d^2\right)^2}{4 d e^4 \left(d+e x^2\right)^2}-\frac{3 c^2 d x}{e^4}+\frac{c^2 x^3}{3 e^3}",1,"(x*(3*a^2*e^4*(5*d + 3*e*x^2) - 6*a*c*d^2*e^2*(3*d + 5*e*x^2) - c^2*d^2*(105*d^3 + 175*d^2*e*x^2 + 56*d*e^2*x^4 - 8*e^3*x^6)))/(24*d^2*e^4*(d + e*x^2)^2) + ((35*c^2*d^4 + 6*a*c*d^2*e^2 + 3*a^2*e^4)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(5/2)*e^(9/2))","A",1
135,1,174,184,0.1382474,"\int \frac{\left(a+c x^4\right)^2}{\left(d+e x^2\right)^4} \, dx","Integrate[(a + c*x^4)^2/(d + e*x^2)^4,x]","\frac{x \left(a^2 e^4 \left(33 d^2+40 d e x^2+15 e^2 x^4\right)-2 a c d^2 e^2 \left(3 d^2+8 d e x^2-3 e^2 x^4\right)+c^2 d^3 \left(105 d^3+280 d^2 e x^2+231 d e^2 x^4+48 e^3 x^6\right)\right)}{48 d^3 e^4 \left(d+e x^2\right)^3}-\frac{\left(-5 a^2 e^4-2 a c d^2 e^2+35 c^2 d^4\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{16 d^{7/2} e^{9/2}}","\frac{x \left(5 a^2-\frac{14 a c d^2}{e^2}-\frac{19 c^2 d^4}{e^4}\right)}{24 d^2 \left(d+e x^2\right)^2}-\frac{\left(-5 a^2 e^4-2 a c d^2 e^2+35 c^2 d^4\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{16 d^{7/2} e^{9/2}}+\frac{x \left(5 a^2+\frac{2 a c d^2}{e^2}+\frac{29 c^2 d^4}{e^4}\right)}{16 d^3 \left(d+e x^2\right)}+\frac{x \left(a e^2+c d^2\right)^2}{6 d e^4 \left(d+e x^2\right)^3}+\frac{c^2 x}{e^4}",1,"(x*(-2*a*c*d^2*e^2*(3*d^2 + 8*d*e*x^2 - 3*e^2*x^4) + a^2*e^4*(33*d^2 + 40*d*e*x^2 + 15*e^2*x^4) + c^2*d^3*(105*d^3 + 280*d^2*e*x^2 + 231*d*e^2*x^4 + 48*e^3*x^6)))/(48*d^3*e^4*(d + e*x^2)^3) - ((35*c^2*d^4 - 2*a*c*d^2*e^2 - 5*a^2*e^4)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(16*d^(7/2)*e^(9/2))","A",1
136,1,200,223,0.1888214,"\int \frac{\left(a+c x^4\right)^2}{\left(d+e x^2\right)^5} \, dx","Integrate[(a + c*x^4)^2/(d + e*x^2)^5,x]","\frac{3 \left(35 a^2 e^4+6 a c d^2 e^2+35 c^2 d^4\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)+\frac{\sqrt{d} \sqrt{e} x \left(a^2 e^4 \left(279 d^3+511 d^2 e x^2+385 d e^2 x^4+105 e^3 x^6\right)-6 a c d^2 e^2 \left(3 d^3+11 d^2 e x^2-11 d e^2 x^4-3 e^3 x^6\right)-c^2 d^4 \left(105 d^3+385 d^2 e x^2+511 d e^2 x^4+279 e^3 x^6\right)\right)}{\left(d+e x^2\right)^4}}{384 d^{9/2} e^{9/2}}","-\frac{x \left(-35 a^2 e^4-6 a c d^2 e^2+93 c^2 d^4\right)}{128 d^4 e^4 \left(d+e x^2\right)}+\frac{x \left(7 a^2-\frac{18 a c d^2}{e^2}-\frac{25 c^2 d^4}{e^4}\right)}{48 d^2 \left(d+e x^2\right)^3}+\frac{\left(35 a^2 e^4+6 a c d^2 e^2+35 c^2 d^4\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{128 d^{9/2} e^{9/2}}+\frac{x \left(35 a^2+\frac{6 a c d^2}{e^2}+\frac{163 c^2 d^4}{e^4}\right)}{192 d^3 \left(d+e x^2\right)^2}+\frac{x \left(a e^2+c d^2\right)^2}{8 d e^4 \left(d+e x^2\right)^4}",1,"((Sqrt[d]*Sqrt[e]*x*(-6*a*c*d^2*e^2*(3*d^3 + 11*d^2*e*x^2 - 11*d*e^2*x^4 - 3*e^3*x^6) + a^2*e^4*(279*d^3 + 511*d^2*e*x^2 + 385*d*e^2*x^4 + 105*e^3*x^6) - c^2*d^4*(105*d^3 + 385*d^2*e*x^2 + 511*d*e^2*x^4 + 279*e^3*x^6)))/(d + e*x^2)^4 + 3*(35*c^2*d^4 + 6*a*c*d^2*e^2 + 35*a^2*e^4)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(384*d^(9/2)*e^(9/2))","A",1
137,1,444,437,0.3385263,"\int \frac{\left(d+e x^2\right)^4}{a+c x^4} \, dx","Integrate[(d + e*x^2)^4/(a + c*x^4),x]","\frac{160 a^{3/4} c^{5/4} d e^3 x^3+24 a^{3/4} c^{5/4} e^4 x^5-120 a^{3/4} \sqrt[4]{c} e^2 x \left(a e^2-6 c d^2\right)-15 \sqrt{2} \left(4 a^{3/2} \sqrt{c} d e^3+a^2 e^4-4 \sqrt{a} c^{3/2} d^3 e-6 a c d^2 e^2+c^2 d^4\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)+15 \sqrt{2} \left(4 a^{3/2} \sqrt{c} d e^3+a^2 e^4-4 \sqrt{a} c^{3/2} d^3 e-6 a c d^2 e^2+c^2 d^4\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)-30 \sqrt{2} \left(-4 a^{3/2} \sqrt{c} d e^3+a^2 e^4+4 \sqrt{a} c^{3/2} d^3 e-6 a c d^2 e^2+c^2 d^4\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)+30 \sqrt{2} \left(-4 a^{3/2} \sqrt{c} d e^3+a^2 e^4+4 \sqrt{a} c^{3/2} d^3 e-6 a c d^2 e^2+c^2 d^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{120 a^{3/4} c^{9/4}}","-\frac{\left(a^2 e^4-6 a c d^2 e^2-4 \sqrt{a} \sqrt{c} d e \left(c d^2-a e^2\right)+c^2 d^4\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} c^{9/4}}+\frac{\left(a^2 e^4-6 a c d^2 e^2-4 \sqrt{a} \sqrt{c} d e \left(c d^2-a e^2\right)+c^2 d^4\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} c^{9/4}}-\frac{\left(a^2 e^4-6 a c d^2 e^2+4 \sqrt{a} \sqrt{c} d e \left(c d^2-a e^2\right)+c^2 d^4\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} a^{3/4} c^{9/4}}+\frac{\left(a^2 e^4-6 a c d^2 e^2+4 \sqrt{a} \sqrt{c} d e \left(c d^2-a e^2\right)+c^2 d^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} a^{3/4} c^{9/4}}+\frac{e^2 x \left(6 c d^2-a e^2\right)}{c^2}+\frac{4 d e^3 x^3}{3 c}+\frac{e^4 x^5}{5 c}",1,"(-120*a^(3/4)*c^(1/4)*e^2*(-6*c*d^2 + a*e^2)*x + 160*a^(3/4)*c^(5/4)*d*e^3*x^3 + 24*a^(3/4)*c^(5/4)*e^4*x^5 - 30*Sqrt[2]*(c^2*d^4 + 4*Sqrt[a]*c^(3/2)*d^3*e - 6*a*c*d^2*e^2 - 4*a^(3/2)*Sqrt[c]*d*e^3 + a^2*e^4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 30*Sqrt[2]*(c^2*d^4 + 4*Sqrt[a]*c^(3/2)*d^3*e - 6*a*c*d^2*e^2 - 4*a^(3/2)*Sqrt[c]*d*e^3 + a^2*e^4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)] - 15*Sqrt[2]*(c^2*d^4 - 4*Sqrt[a]*c^(3/2)*d^3*e - 6*a*c*d^2*e^2 + 4*a^(3/2)*Sqrt[c]*d*e^3 + a^2*e^4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] + 15*Sqrt[2]*(c^2*d^4 - 4*Sqrt[a]*c^(3/2)*d^3*e - 6*a*c*d^2*e^2 + 4*a^(3/2)*Sqrt[c]*d*e^3 + a^2*e^4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(120*a^(3/4)*c^(9/4))","A",1
138,1,360,370,0.2768613,"\int \frac{\left(d+e x^2\right)^3}{a+c x^4} \, dx","Integrate[(d + e*x^2)^3/(a + c*x^4),x]","\frac{-3 \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)+3 \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)+6 \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)+6 \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)+72 a^{3/4} c^{3/4} d e^2 x+8 a^{3/4} c^{3/4} e^3 x^3}{24 a^{3/4} c^{7/4}}","-\frac{\left(\sqrt{c} d \left(c d^2-3 a e^2\right)-\sqrt{a} e \left(3 c d^2-a e^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^{3/4} c^{7/4}}+\frac{\left(\sqrt{c} d \left(c d^2-3 a e^2\right)-\sqrt{a} e \left(3 c d^2-a e^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^{3/4} c^{7/4}}-\frac{\left(\sqrt{c} d \left(c d^2-3 a e^2\right)+\sqrt{a} e \left(3 c d^2-a e^2\right)\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} a^{3/4} c^{7/4}}+\frac{\left(\sqrt{c} d \left(c d^2-3 a e^2\right)+\sqrt{a} e \left(3 c d^2-a e^2\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} a^{3/4} c^{7/4}}+\frac{3 d e^2 x}{c}+\frac{e^3 x^3}{3 c}",1,"(72*a^(3/4)*c^(3/4)*d*e^2*x + 8*a^(3/4)*c^(3/4)*e^3*x^3 + 6*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)] + 6*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)] - 3*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] + 3*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])/(24*a^(3/4)*c^(7/4))","A",1
139,1,269,297,0.257162,"\int \frac{\left(d+e x^2\right)^2}{a+c x^4} \, dx","Integrate[(d + e*x^2)^2/(a + c*x^4),x]","\frac{8 a^{3/4} \sqrt[4]{c} e^2 x+\sqrt{2} \left(2 \sqrt{a} \sqrt{c} d e+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)+\sqrt{2} \left(-2 \sqrt{a} \sqrt{c} d e-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)-2 \sqrt{2} \left(2 \sqrt{a} \sqrt{c} d e-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} \left(2 \sqrt{a} \sqrt{c} d e-a e^2+c d^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{8 a^{3/4} c^{5/4}}","-\frac{\left(-2 \sqrt{a} \sqrt{c} d e-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^{3/4} c^{5/4}}+\frac{\left(-2 \sqrt{a} \sqrt{c} d e-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^{3/4} c^{5/4}}-\frac{\left(2 \sqrt{a} \sqrt{c} d e-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^{3/4} c^{5/4}}+\frac{\left(2 \sqrt{a} \sqrt{c} d e-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^{3/4} c^{5/4}}+\frac{e^2 x}{c}",1,"(8*a^(3/4)*c^(1/4)*e^2*x - 2*Sqrt[2]*(c*d^2 + 2*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 2*Sqrt[2]*(c*d^2 + 2*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + Sqrt[2]*(-(c*d^2) + 2*Sqrt[a]*Sqrt[c]*d*e + a*e^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] + Sqrt[2]*(c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(8*a^(3/4)*c^(5/4))","A",1
140,1,183,247,0.0540684,"\int \frac{d+e x^2}{a+c x^4} \, dx","Integrate[(d + e*x^2)/(a + c*x^4),x]","\frac{-\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)}{4 \sqrt{2} a^{3/4} c^{3/4}}","-\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} a^{3/4} c^{3/4}}+\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} a^{3/4} c^{3/4}}-\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} a^{3/4} c^{3/4}}+\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} a^{3/4} c^{3/4}}",1,"(-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]))/(4*Sqrt[2]*a^(3/4)*c^(3/4))","A",1
141,1,134,185,0.0184911,"\int \frac{1}{a+c x^4} \, dx","Integrate[(a + c*x^4)^(-1),x]","\frac{-\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)-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} a^{3/4} \sqrt[4]{c}}","-\frac{\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} a^{3/4} \sqrt[4]{c}}+\frac{\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} a^{3/4} \sqrt[4]{c}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} a^{3/4} \sqrt[4]{c}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} a^{3/4} \sqrt[4]{c}}",1,"(-2*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)] - 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])/(4*Sqrt[2]*a^(3/4)*c^(1/4))","A",1
142,1,234,336,0.1540978,"\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
143,1,362,453,0.4684595,"\int \frac{1}{\left(d+e x^2\right)^2 \left(a+c x^4\right)} \, dx","Integrate[1/((d + e*x^2)^2*(a + c*x^4)),x]","\frac{\frac{\sqrt{2} c^{3/4} \left(-2 \sqrt{a} \sqrt{c} d e+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)}{a^{3/4}}+\frac{\sqrt{2} c^{3/4} \left(2 \sqrt{a} \sqrt{c} d e-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)}{a^{3/4}}+\frac{2 \sqrt{2} c^{3/4} \left(2 \sqrt{a} \sqrt{c} d e+a e^2-c d^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{a^{3/4}}-\frac{2 \sqrt{2} c^{3/4} \left(2 \sqrt{a} \sqrt{c} d e+a e^2-c d^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{a^{3/4}}+\frac{4 e^2 x \left(a e^2+c d^2\right)}{d \left(d+e x^2\right)}+\frac{4 e^{3/2} \left(a e^2+5 c d^2\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{d^{3/2}}}{8 \left(a e^2+c d^2\right)^2}","-\frac{c^{3/4} \left(2 \sqrt{a} \sqrt{c} d e-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^{3/4} \left(a e^2+c d^2\right)^2}+\frac{c^{3/4} \left(2 \sqrt{a} \sqrt{c} d e-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^{3/4} \left(a e^2+c d^2\right)^2}-\frac{c^{3/4} \left(-2 \sqrt{a} \sqrt{c} d e-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^{3/4} \left(a e^2+c d^2\right)^2}+\frac{c^{3/4} \left(-2 \sqrt{a} \sqrt{c} d e-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^{3/4} \left(a e^2+c d^2\right)^2}+\frac{e^2 x}{2 d \left(d+e x^2\right) \left(a e^2+c d^2\right)}+\frac{2 c \sqrt{d} e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\left(a e^2+c d^2\right)^2}+\frac{e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{2 d^{3/2} \left(a e^2+c d^2\right)}",1,"((4*e^2*(c*d^2 + a*e^2)*x)/(d*(d + e*x^2)) + (4*e^(3/2)*(5*c*d^2 + a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/d^(3/2) + (2*Sqrt[2]*c^(3/4)*(-(c*d^2) + 2*Sqrt[a]*Sqrt[c]*d*e + a*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(3/4) - (2*Sqrt[2]*c^(3/4)*(-(c*d^2) + 2*Sqrt[a]*Sqrt[c]*d*e + a*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(3/4) + (Sqrt[2]*c^(3/4)*(-(c*d^2) - 2*Sqrt[a]*Sqrt[c]*d*e + a*e^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/a^(3/4) + (Sqrt[2]*c^(3/4)*(c*d^2 + 2*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/a^(3/4))/(8*(c*d^2 + a*e^2)^2)","A",1
144,1,371,363,0.2604675,"\int \frac{\left(d+e x^2\right)^3}{\left(a+c x^4\right)^2} \, dx","Integrate[(d + e*x^2)^3/(a + c*x^4)^2,x]","\frac{-\frac{8 a^{3/4} c^{3/4} \left(a e^2 x \left(3 d+e x^2\right)-c d^2 x \left(d+3 e x^2\right)\right)}{a+c x^4}+3 \sqrt{2} \left(a^{3/2} e^3+\sqrt{a} c d^2 e-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)+3 \sqrt{2} \left(-a^{3/2} e^3-\sqrt{a} c d^2 e+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)-6 \sqrt{2} \left(a^{3/2} e^3+\sqrt{a} c d^2 e+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)+6 \sqrt{2} \left(a^{3/2} e^3+\sqrt{a} c d^2 e+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)}{32 a^{7/4} c^{7/4}}","-\frac{3 \left(\sqrt{c} d-\sqrt{a} e\right) \left(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)}{16 \sqrt{2} a^{7/4} c^{7/4}}+\frac{3 \left(\sqrt{c} d-\sqrt{a} e\right) \left(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)}{16 \sqrt{2} a^{7/4} c^{7/4}}-\frac{3 \left(\sqrt{a} e+\sqrt{c} d\right) \left(a e^2+c d^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{8 \sqrt{2} a^{7/4} c^{7/4}}+\frac{3 \left(\sqrt{a} e+\sqrt{c} d\right) \left(a e^2+c d^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{8 \sqrt{2} a^{7/4} c^{7/4}}+\frac{x \left(3 e x^2 \left(a e^2+c d^2\right)+d \left(c d^2-3 a e^2\right)\right)}{4 a c \left(a+c x^4\right)}-\frac{e^3 x^3}{c \left(a+c x^4\right)}",1,"((-8*a^(3/4)*c^(3/4)*(a*e^2*x*(3*d + e*x^2) - c*d^2*x*(d + 3*e*x^2)))/(a + c*x^4) - 6*Sqrt[2]*(c^(3/2)*d^3 + 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)] + 6*Sqrt[2]*(c^(3/2)*d^3 + 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)] + 3*Sqrt[2]*(-(c^(3/2)*d^3) + 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] + 3*Sqrt[2]*(c^(3/2)*d^3 - 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])/(32*a^(7/4)*c^(7/4))","A",1
145,1,295,349,0.1669517,"\int \frac{\left(d+e x^2\right)^2}{\left(a+c x^4\right)^2} \, dx","Integrate[(d + e*x^2)^2/(a + c*x^4)^2,x]","\frac{-\frac{8 a^{3/4} \sqrt[4]{c} \left(a e^2 x-c d x \left(d+2 e x^2\right)\right)}{a+c x^4}-\sqrt{2} \left(-2 \sqrt{a} \sqrt{c} d e+a e^2+3 c d^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)+\sqrt{2} \left(-2 \sqrt{a} \sqrt{c} d e+a e^2+3 c d^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)-2 \sqrt{2} \left(2 \sqrt{a} \sqrt{c} d e+a e^2+3 c d^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)+2 \sqrt{2} \left(2 \sqrt{a} \sqrt{c} d e+a e^2+3 c d^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{32 a^{7/4} c^{5/4}}","-\frac{\left(-2 \sqrt{a} \sqrt{c} d e+a e^2+3 c d^2\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} c^{5/4}}+\frac{\left(-2 \sqrt{a} \sqrt{c} d e+a e^2+3 c d^2\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} c^{5/4}}-\frac{\left(2 \sqrt{a} \sqrt{c} d e+a e^2+3 c d^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{8 \sqrt{2} a^{7/4} c^{5/4}}+\frac{\left(2 \sqrt{a} \sqrt{c} d e+a e^2+3 c d^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{8 \sqrt{2} a^{7/4} c^{5/4}}+\frac{x \left(a e^2+3 c d^2+6 c d e x^2\right)}{12 a c \left(a+c x^4\right)}-\frac{e^2 x}{3 c \left(a+c x^4\right)}",1,"((-8*a^(3/4)*c^(1/4)*(a*e^2*x - c*d*x*(d + 2*e*x^2)))/(a + c*x^4) - 2*Sqrt[2]*(3*c*d^2 + 2*Sqrt[a]*Sqrt[c]*d*e + a*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 2*Sqrt[2]*(3*c*d^2 + 2*Sqrt[a]*Sqrt[c]*d*e + a*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)] - Sqrt[2]*(3*c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e + a*e^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] + Sqrt[2]*(3*c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e + a*e^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(32*a^(7/4)*c^(5/4))","A",1
146,1,267,275,0.2734774,"\int \frac{d+e x^2}{\left(a+c x^4\right)^2} \, dx","Integrate[(d + e*x^2)/(a + c*x^4)^2,x]","\frac{\frac{\sqrt{2} \left(a^{3/4} e-3 \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)}{c^{3/4}}+\frac{\sqrt{2} \left(3 \sqrt[4]{a} \sqrt{c} d-a^{3/4} e\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{c^{3/4}}-\frac{2 \sqrt{2} \sqrt[4]{a} \left(\sqrt{a} e+3 \sqrt{c} d\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{c^{3/4}}+\frac{2 \sqrt{2} \sqrt[4]{a} \left(\sqrt{a} e+3 \sqrt{c} d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{c^{3/4}}+\frac{8 a x \left(d+e x^2\right)}{a+c x^4}}{32 a^2}","-\frac{\left(3 \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)}{16 \sqrt{2} a^{7/4} c^{3/4}}+\frac{\left(3 \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)}{16 \sqrt{2} a^{7/4} c^{3/4}}-\frac{\left(\sqrt{a} e+3 \sqrt{c} d\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{8 \sqrt{2} a^{7/4} c^{3/4}}+\frac{\left(\sqrt{a} e+3 \sqrt{c} d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{8 \sqrt{2} a^{7/4} c^{3/4}}+\frac{x \left(d+e x^2\right)}{4 a \left(a+c x^4\right)}",1,"((8*a*x*(d + e*x^2))/(a + c*x^4) - (2*Sqrt[2]*a^(1/4)*(3*Sqrt[c]*d + Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/c^(3/4) + (2*Sqrt[2]*a^(1/4)*(3*Sqrt[c]*d + Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/c^(3/4) + (Sqrt[2]*(-3*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])/c^(3/4) + (Sqrt[2]*(3*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])/c^(3/4))/(32*a^2)","A",1
147,1,183,202,0.1150284,"\int \frac{1}{\left(a+c x^4\right)^2} \, dx","Integrate[(a + c*x^4)^(-2),x]","\frac{\frac{8 a^{3/4} x}{a+c x^4}-\frac{3 \sqrt{2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{\sqrt[4]{c}}+\frac{3 \sqrt{2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{\sqrt[4]{c}}-\frac{6 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{\sqrt[4]{c}}+\frac{6 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{\sqrt[4]{c}}}{32 a^{7/4}}","-\frac{3 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{16 \sqrt{2} a^{7/4} \sqrt[4]{c}}+\frac{3 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{16 \sqrt{2} a^{7/4} \sqrt[4]{c}}-\frac{3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{8 \sqrt{2} a^{7/4} \sqrt[4]{c}}+\frac{3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{8 \sqrt{2} a^{7/4} \sqrt[4]{c}}+\frac{x}{4 a \left(a+c x^4\right)}",1,"((8*a^(3/4)*x)/(a + c*x^4) - (6*Sqrt[2]*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/c^(1/4) + (6*Sqrt[2]*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/c^(1/4) - (3*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/c^(1/4) + (3*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/c^(1/4))/(32*a^(7/4))","A",1
148,1,429,689,0.2950231,"\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
149,1,540,864,0.5842599,"\int \frac{1}{\left(d+e x^2\right)^2 \left(a+c x^4\right)^2} \, dx","Integrate[1/((d + e*x^2)^2*(a + c*x^4)^2),x]","\frac{-\frac{\sqrt{2} c^{3/4} \left(18 a^{3/2} \sqrt{c} d e^3-7 a^2 e^4+2 \sqrt{a} c^{3/2} d^3 e+12 a c d^2 e^2+3 c^2 d^4\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} c^{3/4} \left(18 a^{3/2} \sqrt{c} d e^3-7 a^2 e^4+2 \sqrt{a} c^{3/2} d^3 e+12 a c d^2 e^2+3 c^2 d^4\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} c^{3/4} \left(18 a^{3/2} \sqrt{c} d e^3+7 a^2 e^4+2 \sqrt{a} c^{3/2} d^3 e-12 a c d^2 e^2-3 c^2 d^4\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{a^{7/4}}-\frac{2 \sqrt{2} c^{3/4} \left(18 a^{3/2} \sqrt{c} d e^3+7 a^2 e^4+2 \sqrt{a} c^{3/2} d^3 e-12 a c d^2 e^2-3 c^2 d^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{a^{7/4}}+\frac{8 c x \left(a e^2+c d^2\right) \left(c d \left(d-2 e x^2\right)-a e^2\right)}{a \left(a+c x^4\right)}+\frac{16 e^4 x \left(a e^2+c d^2\right)}{d \left(d+e x^2\right)}+\frac{16 e^{7/2} \left(a e^2+9 c d^2\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{d^{3/2}}}{32 \left(a e^2+c d^2\right)^3}","\frac{x e^4}{2 d \left(c d^2+a e^2\right)^2 \left(e x^2+d\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) e^{7/2}}{2 d^{3/2} \left(c d^2+a e^2\right)^2}+\frac{4 c \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) e^{7/2}}{\left(c d^2+a e^2\right)^3}-\frac{c^{3/4} \left(3 c d^2-4 \sqrt{a} \sqrt{c} e d-a e^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right) e^2}{2 \sqrt{2} a^{3/4} \left(c d^2+a e^2\right)^3}+\frac{c^{3/4} \left(3 c d^2-4 \sqrt{a} \sqrt{c} e d-a e^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right) e^2}{2 \sqrt{2} a^{3/4} \left(c d^2+a e^2\right)^3}-\frac{c^{3/4} \left(3 c d^2+4 \sqrt{a} \sqrt{c} e d-a e^2\right) \log \left(\sqrt{c} x^2-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}\right) e^2}{4 \sqrt{2} a^{3/4} \left(c d^2+a e^2\right)^3}+\frac{c^{3/4} \left(3 c d^2+4 \sqrt{a} \sqrt{c} e d-a e^2\right) \log \left(\sqrt{c} x^2+\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}\right) e^2}{4 \sqrt{2} a^{3/4} \left(c d^2+a e^2\right)^3}-\frac{c^{3/4} \left(3 c d^2-2 \sqrt{a} \sqrt{c} e d-3 a e^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{8 \sqrt{2} a^{7/4} \left(c d^2+a e^2\right)^2}+\frac{c^{3/4} \left(3 c d^2-2 \sqrt{a} \sqrt{c} e d-3 a e^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{8 \sqrt{2} a^{7/4} \left(c d^2+a e^2\right)^2}-\frac{c^{3/4} \left(3 c d^2+2 \sqrt{a} \sqrt{c} e d-3 a e^2\right) \log \left(\sqrt{c} x^2-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}\right)}{16 \sqrt{2} a^{7/4} \left(c d^2+a e^2\right)^2}+\frac{c^{3/4} \left(3 c d^2+2 \sqrt{a} \sqrt{c} e d-3 a e^2\right) \log \left(\sqrt{c} x^2+\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}\right)}{16 \sqrt{2} a^{7/4} \left(c d^2+a e^2\right)^2}+\frac{c x \left(c d^2-2 c e x^2 d-a e^2\right)}{4 a \left(c d^2+a e^2\right)^2 \left(c x^4+a\right)}",1,"((16*e^4*(c*d^2 + a*e^2)*x)/(d*(d + e*x^2)) + (8*c*(c*d^2 + a*e^2)*x*(-(a*e^2) + c*d*(d - 2*e*x^2)))/(a*(a + c*x^4)) + (16*e^(7/2)*(9*c*d^2 + a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/d^(3/2) + (2*Sqrt[2]*c^(3/4)*(-3*c^2*d^4 + 2*Sqrt[a]*c^(3/2)*d^3*e - 12*a*c*d^2*e^2 + 18*a^(3/2)*Sqrt[c]*d*e^3 + 7*a^2*e^4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(7/4) - (2*Sqrt[2]*c^(3/4)*(-3*c^2*d^4 + 2*Sqrt[a]*c^(3/2)*d^3*e - 12*a*c*d^2*e^2 + 18*a^(3/2)*Sqrt[c]*d*e^3 + 7*a^2*e^4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(7/4) - (Sqrt[2]*c^(3/4)*(3*c^2*d^4 + 2*Sqrt[a]*c^(3/2)*d^3*e + 12*a*c*d^2*e^2 + 18*a^(3/2)*Sqrt[c]*d*e^3 - 7*a^2*e^4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/a^(7/4) + (Sqrt[2]*c^(3/4)*(3*c^2*d^4 + 2*Sqrt[a]*c^(3/2)*d^3*e + 12*a*c*d^2*e^2 + 18*a^(3/2)*Sqrt[c]*d*e^3 - 7*a^2*e^4)*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)^3)","A",1
150,1,203,388,0.211696,"\int \frac{\left(d+e x^2\right)^4}{\sqrt{a+c x^4}} \, dx","Integrate[(d + e*x^2)^4/Sqrt[a + c*x^4],x]","\frac{e x \left(-25 a^2 e^3+28 c d x^2 \sqrt{\frac{c x^4}{a}+1} \left(5 c d^2-3 a e^2\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^4}{a}\right)+2 a c e \left(105 d^2+42 d e x^2-5 e^2 x^4\right)+3 c^2 e x^4 \left(70 d^2+28 d e x^2+5 e^2 x^4\right)\right)+5 x \sqrt{\frac{c x^4}{a}+1} \left(5 a^2 e^4-42 a c d^2 e^2+21 c^2 d^4\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^4}{a}\right)}{105 c^2 \sqrt{a+c x^4}}","\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(-252 a^{3/2} \sqrt{c} d e^3+25 a^2 e^4+420 \sqrt{a} c^{3/2} d^3 e-210 a c d^2 e^2+105 c^2 d^4\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{210 \sqrt[4]{a} c^{9/4} \sqrt{a+c x^4}}+\frac{4 d e x \sqrt{a+c x^4} \left(5 c d^2-3 a e^2\right)}{5 c^{3/2} \left(\sqrt{a}+\sqrt{c} x^2\right)}-\frac{4 \sqrt[4]{a} d e \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(5 c d^2-3 a e^2\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{5 c^{7/4} \sqrt{a+c x^4}}+\frac{e^2 x \sqrt{a+c x^4} \left(42 c d^2-5 a e^2\right)}{21 c^2}+\frac{4 d e^3 x^3 \sqrt{a+c x^4}}{5 c}+\frac{e^4 x^5 \sqrt{a+c x^4}}{7 c}",1,"(5*(21*c^2*d^4 - 42*a*c*d^2*e^2 + 5*a^2*e^4)*x*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^4)/a)] + e*x*(-25*a^2*e^3 + 2*a*c*e*(105*d^2 + 42*d*e*x^2 - 5*e^2*x^4) + 3*c^2*e*x^4*(70*d^2 + 28*d*e*x^2 + 5*e^2*x^4) + 28*c*d*(5*c*d^2 - 3*a*e^2)*x^2*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, -((c*x^4)/a)]))/(105*c^2*Sqrt[a + c*x^4])","C",1
151,1,140,326,0.1384487,"\int \frac{\left(d+e x^2\right)^3}{\sqrt{a+c x^4}} \, dx","Integrate[(d + e*x^2)^3/Sqrt[a + c*x^4],x]","\frac{5 d x \sqrt{\frac{c x^4}{a}+1} \left(c d^2-a e^2\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^4}{a}\right)+e x \left(x^2 \sqrt{\frac{c x^4}{a}+1} \left(5 c d^2-a e^2\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^4}{a}\right)+e \left(a+c x^4\right) \left(5 d+e x^2\right)\right)}{5 c \sqrt{a+c x^4}}","\frac{3 e x \sqrt{a+c x^4} \left(5 c d^2-a e^2\right)}{5 c^{3/2} \left(\sqrt{a}+\sqrt{c} x^2\right)}-\frac{3 \sqrt[4]{a} e \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(5 c d^2-a e^2\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{5 c^{7/4} \sqrt{a+c x^4}}+\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(\frac{5 \sqrt{c} d \left(c d^2-a e^2\right)}{\sqrt{a}}-3 a e^3+15 c d^2 e\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{10 c^{7/4} \sqrt{a+c x^4}}+\frac{d e^2 x \sqrt{a+c x^4}}{c}+\frac{e^3 x^3 \sqrt{a+c x^4}}{5 c}",1,"(5*d*(c*d^2 - a*e^2)*x*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^4)/a)] + e*x*(e*(5*d + e*x^2)*(a + c*x^4) + (5*c*d^2 - a*e^2)*x^2*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, -((c*x^4)/a)]))/(5*c*Sqrt[a + c*x^4])","C",1
152,1,120,264,0.0889814,"\int \frac{\left(d+e x^2\right)^2}{\sqrt{a+c x^4}} \, dx","Integrate[(d + e*x^2)^2/Sqrt[a + c*x^4],x]","\frac{x \sqrt{\frac{c x^4}{a}+1} \left(3 c d^2-a e^2\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^4}{a}\right)+e x \left(2 c d x^2 \sqrt{\frac{c x^4}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^4}{a}\right)+e \left(a+c x^4\right)\right)}{3 c \sqrt{a+c x^4}}","\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(6 \sqrt{a} \sqrt{c} d e-a e^2+3 c d^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{6 \sqrt[4]{a} c^{5/4} \sqrt{a+c x^4}}-\frac{2 \sqrt[4]{a} d e \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{c^{3/4} \sqrt{a+c x^4}}+\frac{2 d e x \sqrt{a+c x^4}}{\sqrt{c} \left(\sqrt{a}+\sqrt{c} x^2\right)}+\frac{e^2 x \sqrt{a+c x^4}}{3 c}",1,"((3*c*d^2 - a*e^2)*x*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^4)/a)] + e*x*(e*(a + c*x^4) + 2*c*d*x^2*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, -((c*x^4)/a)]))/(3*c*Sqrt[a + c*x^4])","C",1
153,1,77,226,0.0318247,"\int \frac{d+e x^2}{\sqrt{a+c x^4}} \, dx","Integrate[(d + e*x^2)/Sqrt[a + c*x^4],x]","\frac{\sqrt{\frac{c x^4}{a}+1} \left(3 d x \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^4}{a}\right)+e x^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^4}{a}\right)\right)}{3 \sqrt{a+c x^4}}","\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(\frac{\sqrt{c} d}{\sqrt{a}}+e\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 c^{3/4} \sqrt{a+c x^4}}-\frac{\sqrt[4]{a} e \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{c^{3/4} \sqrt{a+c x^4}}+\frac{e x \sqrt{a+c x^4}}{\sqrt{c} \left(\sqrt{a}+\sqrt{c} x^2\right)}",1,"(Sqrt[1 + (c*x^4)/a]*(3*d*x*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^4)/a)] + e*x^3*Hypergeometric2F1[1/2, 3/4, 7/4, -((c*x^4)/a)]))/(3*Sqrt[a + c*x^4])","C",1
154,1,95,334,0.1499486,"\int \frac{1}{\left(d+e x^2\right) \sqrt{a+c x^4}} \, dx","Integrate[1/((d + e*x^2)*Sqrt[a + c*x^4]),x]","-\frac{i \sqrt{\frac{c x^4}{a}+1} \Pi \left(-\frac{i \sqrt{a} e}{\sqrt{c} d};\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)}{d \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \sqrt{a+c x^4}}","-\frac{a^{3/4} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(\frac{\sqrt{c} d}{\sqrt{a}}+e\right)^2 \Pi \left(-\frac{\left(\sqrt{c} d-\sqrt{a} e\right)^2}{4 \sqrt{a} \sqrt{c} d e};2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 \sqrt[4]{c} d \sqrt{a+c x^4} \left(c d^2-a e^2\right)}+\frac{\sqrt{e} \tan ^{-1}\left(\frac{x \sqrt{a e^2+c d^2}}{\sqrt{d} \sqrt{e} \sqrt{a+c x^4}}\right)}{2 \sqrt{d} \sqrt{a e^2+c d^2}}+\frac{\sqrt[4]{c} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} \sqrt{a+c x^4} \left(\sqrt{c} d-\sqrt{a} e\right)}",1,"((-I)*Sqrt[1 + (c*x^4)/a]*EllipticPi[((-I)*Sqrt[a]*e)/(Sqrt[c]*d), I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])/(Sqrt[(I*Sqrt[c])/Sqrt[a]]*d*Sqrt[a + c*x^4])","C",1
155,1,522,581,0.7639438,"\int \frac{1}{\left(d+e x^2\right)^2 \sqrt{a+c x^4}} \, dx","Integrate[1/((d + e*x^2)^2*Sqrt[a + c*x^4]),x]","\frac{-3 i c d^3 \sqrt{\frac{c x^4}{a}+1} \Pi \left(-\frac{i \sqrt{a} e}{\sqrt{c} d};\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)-3 i c d^2 e x^2 \sqrt{\frac{c x^4}{a}+1} \Pi \left(-\frac{i \sqrt{a} e}{\sqrt{c} d};\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)-i a e^3 x^2 \sqrt{\frac{c x^4}{a}+1} \Pi \left(-\frac{i \sqrt{a} e}{\sqrt{c} d};\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)+c d e^2 x^5 \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}-i a d e^2 \sqrt{\frac{c x^4}{a}+1} \Pi \left(-\frac{i \sqrt{a} e}{\sqrt{c} d};\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)+a d e^2 x \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}+\sqrt{c} d \sqrt{\frac{c x^4}{a}+1} \left(d+e x^2\right) \left(\sqrt{a} e+i \sqrt{c} d\right) F\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)-\sqrt{a} \sqrt{c} d e \sqrt{\frac{c x^4}{a}+1} \left(d+e x^2\right) E\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)}{2 d^2 \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \sqrt{a+c x^4} \left(d+e x^2\right) \left(a e^2+c d^2\right)}","\frac{e^2 x \sqrt{a+c x^4}}{2 d \left(d+e x^2\right) \left(a e^2+c d^2\right)}-\frac{\sqrt{c} e x \sqrt{a+c x^4}}{2 d \left(\sqrt{a}+\sqrt{c} x^2\right) \left(a e^2+c d^2\right)}+\frac{\sqrt[4]{a} \sqrt[4]{c} e \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 d \sqrt{a+c x^4} \left(a e^2+c d^2\right)}-\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(\sqrt{a} e+\sqrt{c} d\right) \left(a e^2+3 c d^2\right) \Pi \left(-\frac{\left(\sqrt{c} d-\sqrt{a} e\right)^2}{4 \sqrt{a} \sqrt{c} d e};2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{8 \sqrt[4]{a} \sqrt[4]{c} d^2 \sqrt{a+c x^4} \left(\sqrt{c} d-\sqrt{a} e\right) \left(a e^2+c d^2\right)}+\frac{\sqrt{e} \left(a e^2+3 c d^2\right) \tan ^{-1}\left(\frac{x \sqrt{a e^2+c d^2}}{\sqrt{d} \sqrt{e} \sqrt{a+c x^4}}\right)}{4 d^{3/2} \left(a e^2+c d^2\right)^{3/2}}+\frac{\sqrt[4]{c} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} d \sqrt{a+c x^4} \left(\sqrt{c} d-\sqrt{a} e\right)}",1,"(a*Sqrt[(I*Sqrt[c])/Sqrt[a]]*d*e^2*x + Sqrt[(I*Sqrt[c])/Sqrt[a]]*c*d*e^2*x^5 - Sqrt[a]*Sqrt[c]*d*e*(d + e*x^2)*Sqrt[1 + (c*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + Sqrt[c]*d*(I*Sqrt[c]*d + Sqrt[a]*e)*(d + e*x^2)*Sqrt[1 + (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] - (3*I)*c*d^3*Sqrt[1 + (c*x^4)/a]*EllipticPi[((-I)*Sqrt[a]*e)/(Sqrt[c]*d), I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] - I*a*d*e^2*Sqrt[1 + (c*x^4)/a]*EllipticPi[((-I)*Sqrt[a]*e)/(Sqrt[c]*d), I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] - (3*I)*c*d^2*e*x^2*Sqrt[1 + (c*x^4)/a]*EllipticPi[((-I)*Sqrt[a]*e)/(Sqrt[c]*d), I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] - I*a*e^3*x^2*Sqrt[1 + (c*x^4)/a]*EllipticPi[((-I)*Sqrt[a]*e)/(Sqrt[c]*d), I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])/(2*Sqrt[(I*Sqrt[c])/Sqrt[a]]*d^2*(c*d^2 + a*e^2)*(d + e*x^2)*Sqrt[a + c*x^4])","C",1
156,1,332,729,1.096635,"\int \frac{1}{\left(d+e x^2\right)^3 \sqrt{a+c x^4}} \, dx","Integrate[1/((d + e*x^2)^3*Sqrt[a + c*x^4]),x]","\frac{\frac{d e^2 x \left(a+c x^4\right) \left(a e^2 \left(5 d+3 e x^2\right)+c d^2 \left(11 d+9 e x^2\right)\right)}{\left(d+e x^2\right)^2}+\frac{\sqrt{\frac{c x^4}{a}+1} \left(i \left(\sqrt{c} d \left(-3 i a^{3/2} e^3-9 i \sqrt{a} c d^2 e+a \sqrt{c} d e^2+7 c^{3/2} d^3\right) F\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)-3 \left(a^2 e^4+2 a c d^2 e^2+5 c^2 d^4\right) \Pi \left(-\frac{i \sqrt{a} e}{\sqrt{c} d};\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)\right)-3 \sqrt{a} \sqrt{c} d e \left(a e^2+3 c d^2\right) E\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)\right)}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}}}{8 d^3 \sqrt{a+c x^4} \left(a e^2+c d^2\right)^2}","\frac{3 \sqrt{e} \left(a^2 e^4+2 a c d^2 e^2+5 c^2 d^4\right) \tan ^{-1}\left(\frac{x \sqrt{a e^2+c d^2}}{\sqrt{d} \sqrt{e} \sqrt{a+c x^4}}\right)}{16 d^{5/2} \left(a e^2+c d^2\right)^{5/2}}-\frac{3 \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(\sqrt{a} e+\sqrt{c} d\right) \left(a^2 e^4+2 a c d^2 e^2+5 c^2 d^4\right) \Pi \left(-\frac{\left(\sqrt{c} d-\sqrt{a} e\right)^2}{4 \sqrt{a} \sqrt{c} d e};2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{32 \sqrt[4]{a} \sqrt[4]{c} d^3 \sqrt{a+c x^4} \left(\sqrt{c} d-\sqrt{a} e\right) \left(a e^2+c d^2\right)^2}+\frac{3 e^2 x \sqrt{a+c x^4} \left(a e^2+3 c d^2\right)}{8 d^2 \left(d+e x^2\right) \left(a e^2+c d^2\right)^2}+\frac{e^2 x \sqrt{a+c x^4}}{4 d \left(d+e x^2\right)^2 \left(a e^2+c d^2\right)}-\frac{3 \sqrt{c} e x \sqrt{a+c x^4} \left(a e^2+3 c d^2\right)}{8 d^2 \left(\sqrt{a}+\sqrt{c} x^2\right) \left(a e^2+c d^2\right)^2}+\frac{\sqrt[4]{c} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(-\sqrt{a} \sqrt{c} d e+3 a e^2+4 c d^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{8 \sqrt[4]{a} d^2 \sqrt{a+c x^4} \left(\sqrt{c} d-\sqrt{a} e\right) \left(a e^2+c d^2\right)}+\frac{3 \sqrt[4]{a} \sqrt[4]{c} e \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(a e^2+3 c d^2\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{8 d^2 \sqrt{a+c x^4} \left(a e^2+c d^2\right)^2}",1,"((d*e^2*x*(a + c*x^4)*(a*e^2*(5*d + 3*e*x^2) + c*d^2*(11*d + 9*e*x^2)))/(d + e*x^2)^2 + (Sqrt[1 + (c*x^4)/a]*(-3*Sqrt[a]*Sqrt[c]*d*e*(3*c*d^2 + a*e^2)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + I*(Sqrt[c]*d*(7*c^(3/2)*d^3 - (9*I)*Sqrt[a]*c*d^2*e + a*Sqrt[c]*d*e^2 - (3*I)*a^(3/2)*e^3)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] - 3*(5*c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*EllipticPi[((-I)*Sqrt[a]*e)/(Sqrt[c]*d), I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])))/Sqrt[(I*Sqrt[c])/Sqrt[a]])/(8*d^3*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4])","C",1
157,1,141,213,0.1600097,"\int \frac{\left(d+e x^2\right)^3}{\sqrt{a-c x^4}} \, dx","Integrate[(d + e*x^2)^3/Sqrt[a - c*x^4],x]","\frac{5 d x \sqrt{1-\frac{c x^4}{a}} \left(a e^2+c d^2\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{c x^4}{a}\right)+e x \left(x^2 \sqrt{1-\frac{c x^4}{a}} \left(a e^2+5 c d^2\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{c x^4}{a}\right)+e \left(c x^4-a\right) \left(5 d+e x^2\right)\right)}{5 c \sqrt{a-c x^4}}","\frac{a^{3/4} \sqrt{1-\frac{c x^4}{a}} \left(\frac{5 \sqrt{c} d \left(a e^2+c d^2\right)}{\sqrt{a}}-3 e \left(a e^2+5 c d^2\right)\right) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)\right|-1\right)}{5 c^{7/4} \sqrt{a-c x^4}}+\frac{3 a^{3/4} e \sqrt{1-\frac{c x^4}{a}} \left(a e^2+5 c d^2\right) E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)\right|-1\right)}{5 c^{7/4} \sqrt{a-c x^4}}-\frac{d e^2 x \sqrt{a-c x^4}}{c}-\frac{e^3 x^3 \sqrt{a-c x^4}}{5 c}",1,"(5*d*(c*d^2 + a*e^2)*x*Sqrt[1 - (c*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, (c*x^4)/a] + e*x*(e*(5*d + e*x^2)*(-a + c*x^4) + (5*c*d^2 + a*e^2)*x^2*Sqrt[1 - (c*x^4)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, (c*x^4)/a]))/(5*c*Sqrt[a - c*x^4])","C",1
158,1,121,162,0.102149,"\int \frac{\left(d+e x^2\right)^2}{\sqrt{a-c x^4}} \, dx","Integrate[(d + e*x^2)^2/Sqrt[a - c*x^4],x]","\frac{x \sqrt{1-\frac{c x^4}{a}} \left(a e^2+3 c d^2\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{c x^4}{a}\right)+e x \left(2 c d x^2 \sqrt{1-\frac{c x^4}{a}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{c x^4}{a}\right)-a e+c e x^4\right)}{3 c \sqrt{a-c x^4}}","\frac{2 a^{3/4} d e \sqrt{1-\frac{c x^4}{a}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)\right|-1\right)}{c^{3/4} \sqrt{a-c x^4}}+\frac{\sqrt[4]{a} \sqrt{1-\frac{c x^4}{a}} \left(-6 \sqrt{a} \sqrt{c} d e+a e^2+3 c d^2\right) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)\right|-1\right)}{3 c^{5/4} \sqrt{a-c x^4}}-\frac{e^2 x \sqrt{a-c x^4}}{3 c}",1,"((3*c*d^2 + a*e^2)*x*Sqrt[1 - (c*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, (c*x^4)/a] + e*x*(-(a*e) + c*e*x^4 + 2*c*d*x^2*Sqrt[1 - (c*x^4)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, (c*x^4)/a]))/(3*c*Sqrt[a - c*x^4])","C",1
159,1,77,124,0.0310813,"\int \frac{d+e x^2}{\sqrt{a-c x^4}} \, dx","Integrate[(d + e*x^2)/Sqrt[a - c*x^4],x]","\frac{\sqrt{1-\frac{c x^4}{a}} \left(3 d x \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{c x^4}{a}\right)+e x^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{c x^4}{a}\right)\right)}{3 \sqrt{a-c x^4}}","\frac{a^{3/4} \sqrt{1-\frac{c x^4}{a}} \left(\frac{\sqrt{c} d}{\sqrt{a}}-e\right) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)\right|-1\right)}{c^{3/4} \sqrt{a-c x^4}}+\frac{a^{3/4} e \sqrt{1-\frac{c x^4}{a}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)\right|-1\right)}{c^{3/4} \sqrt{a-c x^4}}",1,"(Sqrt[1 - (c*x^4)/a]*(3*d*x*Hypergeometric2F1[1/4, 1/2, 5/4, (c*x^4)/a] + e*x^3*Hypergeometric2F1[1/2, 3/4, 7/4, (c*x^4)/a]))/(3*Sqrt[a - c*x^4])","C",1
160,1,91,72,0.1490263,"\int \frac{1}{\left(d+e x^2\right) \sqrt{a-c x^4}} \, dx","Integrate[1/((d + e*x^2)*Sqrt[a - c*x^4]),x]","-\frac{i \sqrt{1-\frac{c x^4}{a}} \Pi \left(-\frac{\sqrt{a} e}{\sqrt{c} d};\left.i \sinh ^{-1}\left(\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)}{d \sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} \sqrt{a-c x^4}}","\frac{\sqrt[4]{a} \sqrt{1-\frac{c x^4}{a}} \Pi \left(-\frac{\sqrt{a} e}{\sqrt{c} d};\left.\sin ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)\right|-1\right)}{\sqrt[4]{c} d \sqrt{a-c x^4}}",1,"((-I)*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1])/(Sqrt[-(Sqrt[c]/Sqrt[a])]*d*Sqrt[a - c*x^4])","C",1
161,1,508,299,0.9579867,"\int \frac{1}{\left(d+e x^2\right)^2 \sqrt{a-c x^4}} \, dx","Integrate[1/((d + e*x^2)^2*Sqrt[a - c*x^4]),x]","\frac{-3 i c d^3 \sqrt{1-\frac{c x^4}{a}} \Pi \left(-\frac{\sqrt{a} e}{\sqrt{c} d};\left.i \sinh ^{-1}\left(\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)-3 i c d^2 e x^2 \sqrt{1-\frac{c x^4}{a}} \Pi \left(-\frac{\sqrt{a} e}{\sqrt{c} d};\left.i \sinh ^{-1}\left(\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)+i a e^3 x^2 \sqrt{1-\frac{c x^4}{a}} \Pi \left(-\frac{\sqrt{a} e}{\sqrt{c} d};\left.i \sinh ^{-1}\left(\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)+c d e^2 x^5 \sqrt{-\frac{\sqrt{c}}{\sqrt{a}}}+i a d e^2 \sqrt{1-\frac{c x^4}{a}} \Pi \left(-\frac{\sqrt{a} e}{\sqrt{c} d};\left.i \sinh ^{-1}\left(\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)-a d e^2 x \sqrt{-\frac{\sqrt{c}}{\sqrt{a}}}-i \sqrt{c} d \sqrt{1-\frac{c x^4}{a}} \left(d+e x^2\right) \left(\sqrt{a} e-\sqrt{c} d\right) F\left(\left.i \sinh ^{-1}\left(\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)+i \sqrt{a} \sqrt{c} d e \sqrt{1-\frac{c x^4}{a}} \left(d+e x^2\right) E\left(\left.i \sinh ^{-1}\left(\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)}{2 d^2 \sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} \sqrt{a-c x^4} \left(d+e x^2\right) \left(c d^2-a e^2\right)}","-\frac{a^{3/4} \sqrt[4]{c} e \sqrt{1-\frac{c x^4}{a}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)\right|-1\right)}{2 d \sqrt{a-c x^4} \left(c d^2-a e^2\right)}+\frac{\sqrt[4]{a} \sqrt{1-\frac{c x^4}{a}} \left(3 c d^2-a e^2\right) \Pi \left(-\frac{\sqrt{a} e}{\sqrt{c} d};\left.\sin ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)\right|-1\right)}{2 \sqrt[4]{c} d^2 \sqrt{a-c x^4} \left(c d^2-a e^2\right)}-\frac{e^2 x \sqrt{a-c x^4}}{2 d \left(d+e x^2\right) \left(c d^2-a e^2\right)}-\frac{\sqrt[4]{a} \sqrt[4]{c} \sqrt{1-\frac{c x^4}{a}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)\right|-1\right)}{2 d \sqrt{a-c x^4} \left(\sqrt{a} e+\sqrt{c} d\right)}",1,"(-(a*Sqrt[-(Sqrt[c]/Sqrt[a])]*d*e^2*x) + Sqrt[-(Sqrt[c]/Sqrt[a])]*c*d*e^2*x^5 + I*Sqrt[a]*Sqrt[c]*d*e*(d + e*x^2)*Sqrt[1 - (c*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - I*Sqrt[c]*d*(-(Sqrt[c]*d) + Sqrt[a]*e)*(d + e*x^2)*Sqrt[1 - (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - (3*I)*c*d^3*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + I*a*d*e^2*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - (3*I)*c*d^2*e*x^2*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + I*a*e^3*x^2*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1])/(2*Sqrt[-(Sqrt[c]/Sqrt[a])]*d^2*(c*d^2 - a*e^2)*(d + e*x^2)*Sqrt[a - c*x^4])","C",1
162,1,321,425,1.2361543,"\int \frac{1}{\left(d+e x^2\right)^3 \sqrt{a-c x^4}} \, dx","Integrate[1/((d + e*x^2)^3*Sqrt[a - c*x^4]),x]","\frac{\frac{d e^2 x \left(a-c x^4\right) \left(a e^2 \left(5 d+3 e x^2\right)-c d^2 \left(11 d+9 e x^2\right)\right)}{\left(d+e x^2\right)^2}-\frac{i \sqrt{1-\frac{c x^4}{a}} \left(\left(-3 a^{3/2} \sqrt{c} d e^3+9 \sqrt{a} c^{3/2} d^3 e+a c d^2 e^2-7 c^2 d^4\right) F\left(\left.i \sinh ^{-1}\left(\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)+3 \left(a^2 e^4-2 a c d^2 e^2+5 c^2 d^4\right) \Pi \left(-\frac{\sqrt{a} e}{\sqrt{c} d};\left.i \sinh ^{-1}\left(\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)+3 \sqrt{a} \sqrt{c} d e \left(a e^2-3 c d^2\right) E\left(\left.i \sinh ^{-1}\left(\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)\right)}{\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}}}}{8 d^3 \sqrt{a-c x^4} \left(c d^2-a e^2\right)^2}","-\frac{3 a^{3/4} \sqrt[4]{c} e \sqrt{1-\frac{c x^4}{a}} \left(3 c d^2-a e^2\right) E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)\right|-1\right)}{8 d^2 \sqrt{a-c x^4} \left(c d^2-a e^2\right)^2}+\frac{3 \sqrt[4]{a} \sqrt{1-\frac{c x^4}{a}} \left(a^2 e^4-2 a c d^2 e^2+5 c^2 d^4\right) \Pi \left(-\frac{\sqrt{a} e}{\sqrt{c} d};\left.\sin ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)\right|-1\right)}{8 \sqrt[4]{c} d^3 \sqrt{a-c x^4} \left(c d^2-a e^2\right)^2}-\frac{\sqrt[4]{a} \sqrt[4]{c} \sqrt{1-\frac{c x^4}{a}} \left(-2 \sqrt{a} \sqrt{c} d e-3 a e^2+7 c d^2\right) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)\right|-1\right)}{8 d^2 \sqrt{a-c x^4} \left(\sqrt{a} e+\sqrt{c} d\right) \left(c d^2-a e^2\right)}-\frac{3 e^2 x \sqrt{a-c x^4} \left(3 c d^2-a e^2\right)}{8 d^2 \left(d+e x^2\right) \left(c d^2-a e^2\right)^2}-\frac{e^2 x \sqrt{a-c x^4}}{4 d \left(d+e x^2\right)^2 \left(c d^2-a e^2\right)}",1,"((d*e^2*x*(a - c*x^4)*(a*e^2*(5*d + 3*e*x^2) - c*d^2*(11*d + 9*e*x^2)))/(d + e*x^2)^2 - (I*Sqrt[1 - (c*x^4)/a]*(3*Sqrt[a]*Sqrt[c]*d*e*(-3*c*d^2 + a*e^2)*EllipticE[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + (-7*c^2*d^4 + 9*Sqrt[a]*c^(3/2)*d^3*e + a*c*d^2*e^2 - 3*a^(3/2)*Sqrt[c]*d*e^3)*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + 3*(5*c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1]))/Sqrt[-(Sqrt[c]/Sqrt[a])])/(8*d^3*(c*d^2 - a*e^2)^2*Sqrt[a - c*x^4])","C",1
163,1,458,563,1.9273895,"\int \frac{1}{\left(d+e x^2\right)^4 \sqrt{a-c x^4}} \, dx","Integrate[1/((d + e*x^2)^4*Sqrt[a - c*x^4]),x]","\frac{-\frac{d e^2 x \left(a-c x^4\right) \left(3 \left(d+e x^2\right)^2 \left(5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right)+8 \left(c d^3-a d e^2\right)^2+10 d \left(d+e x^2\right) \left(c d^2-a e^2\right) \left(3 c d^2-a e^2\right)\right)}{\left(d+e x^2\right)^3 \left(c d^2-a e^2\right)^3}-\frac{i \sqrt{1-\frac{c x^4}{a}} \left(3 \sqrt{a} \sqrt{c} d e \left(5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right) E\left(\left.i \sinh ^{-1}\left(\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)+\sqrt{c} d \left(42 a^{3/2} c d^2 e^3-15 a^{5/2} e^5+5 a^2 \sqrt{c} d e^4-2 a c^{3/2} d^3 e^2-87 \sqrt{a} c^2 d^4 e+57 c^{5/2} d^5\right) F\left(\left.i \sinh ^{-1}\left(\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)+3 \left(5 a^3 e^6-17 a^2 c d^2 e^4+7 a c^2 d^4 e^2-35 c^3 d^6\right) \Pi \left(-\frac{\sqrt{a} e}{\sqrt{c} d};\left.i \sinh ^{-1}\left(\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)\right)}{\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} \left(a e^2-c d^2\right)^3}}{48 d^4 \sqrt{a-c x^4}}","-\frac{e^2 x \sqrt{a-c x^4} \left(5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right)}{16 d^3 \left(d+e x^2\right) \left(c d^2-a e^2\right)^3}-\frac{a^{3/4} \sqrt[4]{c} e \sqrt{1-\frac{c x^4}{a}} \left(5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right) E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)\right|-1\right)}{16 d^3 \sqrt{a-c x^4} \left(c d^2-a e^2\right)^3}-\frac{\sqrt[4]{a} \sqrt[4]{c} \sqrt{1-\frac{c x^4}{a}} \left(10 a^{3/2} \sqrt{c} d e^3+15 a^2 e^4-30 \sqrt{a} c^{3/2} d^3 e-32 a c d^2 e^2+57 c^2 d^4\right) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)\right|-1\right)}{48 d^3 \sqrt{a-c x^4} \left(\sqrt{c} d-\sqrt{a} e\right)^2 \left(\sqrt{a} e+\sqrt{c} d\right)^3}+\frac{\sqrt[4]{a} \sqrt{1-\frac{c x^4}{a}} \left(-5 a^3 e^6+17 a^2 c d^2 e^4-7 a c^2 d^4 e^2+35 c^3 d^6\right) \Pi \left(-\frac{\sqrt{a} e}{\sqrt{c} d};\left.\sin ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)\right|-1\right)}{16 \sqrt[4]{c} d^4 \sqrt{a-c x^4} \left(c d^2-a e^2\right)^3}-\frac{5 e^2 x \sqrt{a-c x^4} \left(3 c d^2-a e^2\right)}{24 d^2 \left(d+e x^2\right)^2 \left(c d^2-a e^2\right)^2}-\frac{e^2 x \sqrt{a-c x^4}}{6 d \left(d+e x^2\right)^3 \left(c d^2-a e^2\right)}",1,"(-((d*e^2*x*(a - c*x^4)*(8*(c*d^3 - a*d*e^2)^2 + 10*d*(c*d^2 - a*e^2)*(3*c*d^2 - a*e^2)*(d + e*x^2) + 3*(29*c^2*d^4 - 14*a*c*d^2*e^2 + 5*a^2*e^4)*(d + e*x^2)^2))/((c*d^2 - a*e^2)^3*(d + e*x^2)^3)) - (I*Sqrt[1 - (c*x^4)/a]*(3*Sqrt[a]*Sqrt[c]*d*e*(29*c^2*d^4 - 14*a*c*d^2*e^2 + 5*a^2*e^4)*EllipticE[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + Sqrt[c]*d*(57*c^(5/2)*d^5 - 87*Sqrt[a]*c^2*d^4*e - 2*a*c^(3/2)*d^3*e^2 + 42*a^(3/2)*c*d^2*e^3 + 5*a^2*Sqrt[c]*d*e^4 - 15*a^(5/2)*e^5)*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + 3*(-35*c^3*d^6 + 7*a*c^2*d^4*e^2 - 17*a^2*c*d^2*e^4 + 5*a^3*e^6)*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1]))/(Sqrt[-(Sqrt[c]/Sqrt[a])]*(-(c*d^2) + a*e^2)^3))/(48*d^4*Sqrt[a - c*x^4])","C",1
164,1,78,126,0.0347028,"\int \frac{d+e x^2}{\sqrt{-a+c x^4}} \, dx","Integrate[(d + e*x^2)/Sqrt[-a + c*x^4],x]","\frac{\sqrt{1-\frac{c x^4}{a}} \left(3 d x \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{c x^4}{a}\right)+e x^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{c x^4}{a}\right)\right)}{3 \sqrt{c x^4-a}}","\frac{a^{3/4} \sqrt{1-\frac{c x^4}{a}} \left(\frac{\sqrt{c} d}{\sqrt{a}}-e\right) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)\right|-1\right)}{c^{3/4} \sqrt{c x^4-a}}+\frac{a^{3/4} e \sqrt{1-\frac{c x^4}{a}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)\right|-1\right)}{c^{3/4} \sqrt{c x^4-a}}",1,"(Sqrt[1 - (c*x^4)/a]*(3*d*x*Hypergeometric2F1[1/4, 1/2, 5/4, (c*x^4)/a] + e*x^3*Hypergeometric2F1[1/2, 3/4, 7/4, (c*x^4)/a]))/(3*Sqrt[-a + c*x^4])","C",1
165,1,92,73,0.1463536,"\int \frac{1}{\left(d+e x^2\right) \sqrt{-a+c x^4}} \, dx","Integrate[1/((d + e*x^2)*Sqrt[-a + c*x^4]),x]","-\frac{i \sqrt{1-\frac{c x^4}{a}} \Pi \left(-\frac{\sqrt{a} e}{\sqrt{c} d};\left.i \sinh ^{-1}\left(\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)}{d \sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} \sqrt{c x^4-a}}","\frac{\sqrt[4]{a} \sqrt{1-\frac{c x^4}{a}} \Pi \left(-\frac{\sqrt{a} e}{\sqrt{c} d};\left.\sin ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)\right|-1\right)}{\sqrt[4]{c} d \sqrt{c x^4-a}}",1,"((-I)*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1])/(Sqrt[-(Sqrt[c]/Sqrt[a])]*d*Sqrt[-a + c*x^4])","C",1
166,1,86,54,0.0374872,"\int \frac{\sqrt{a}+\sqrt{c} x^2}{\sqrt{-a+c x^4}} \, dx","Integrate[(Sqrt[a] + Sqrt[c]*x^2)/Sqrt[-a + c*x^4],x]","\frac{\sqrt{1-\frac{c x^4}{a}} \left(3 \sqrt{a} x \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{c x^4}{a}\right)+\sqrt{c} x^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{c x^4}{a}\right)\right)}{3 \sqrt{c x^4-a}}","\frac{a^{3/4} \sqrt{1-\frac{c x^4}{a}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)\right|-1\right)}{\sqrt[4]{c} \sqrt{c x^4-a}}",1,"(Sqrt[1 - (c*x^4)/a]*(3*Sqrt[a]*x*Hypergeometric2F1[1/4, 1/2, 5/4, (c*x^4)/a] + Sqrt[c]*x^3*Hypergeometric2F1[1/2, 3/4, 7/4, (c*x^4)/a]))/(3*Sqrt[-a + c*x^4])","C",1
167,1,85,52,0.0302623,"\int \frac{1+\sqrt{\frac{c}{a}} x^2}{\sqrt{-a+c x^4}} \, dx","Integrate[(1 + Sqrt[c/a]*x^2)/Sqrt[-a + c*x^4],x]","\frac{\sqrt{1-\frac{c x^4}{a}} \left(3 x \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{c x^4}{a}\right)+x^3 \sqrt{\frac{c}{a}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{c x^4}{a}\right)\right)}{3 \sqrt{c x^4-a}}","\frac{\sqrt{1-\frac{c x^4}{a}} E\left(\left.\sin ^{-1}\left(\sqrt[4]{\frac{c}{a}} x\right)\right|-1\right)}{\sqrt[4]{\frac{c}{a}} \sqrt{c x^4-a}}",1,"(Sqrt[1 - (c*x^4)/a]*(3*x*Hypergeometric2F1[1/4, 1/2, 5/4, (c*x^4)/a] + Sqrt[c/a]*x^3*Hypergeometric2F1[1/2, 3/4, 7/4, (c*x^4)/a]))/(3*Sqrt[-a + c*x^4])","C",1
168,1,80,236,0.0355679,"\int \frac{d+e x^2}{\sqrt{-a-c x^4}} \, dx","Integrate[(d + e*x^2)/Sqrt[-a - c*x^4],x]","\frac{\sqrt{\frac{c x^4}{a}+1} \left(3 d x \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^4}{a}\right)+e x^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^4}{a}\right)\right)}{3 \sqrt{-a-c x^4}}","\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(\frac{\sqrt{c} d}{\sqrt{a}}+e\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 c^{3/4} \sqrt{-a-c x^4}}-\frac{\sqrt[4]{a} e \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{c^{3/4} \sqrt{-a-c x^4}}-\frac{e x \sqrt{-a-c x^4}}{\sqrt{c} \left(\sqrt{a}+\sqrt{c} x^2\right)}",1,"(Sqrt[1 + (c*x^4)/a]*(3*d*x*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^4)/a)] + e*x^3*Hypergeometric2F1[1/2, 3/4, 7/4, -((c*x^4)/a)]))/(3*Sqrt[-a - c*x^4])","C",1
169,1,98,347,0.1493051,"\int \frac{1}{\left(d+e x^2\right) \sqrt{-a-c x^4}} \, dx","Integrate[1/((d + e*x^2)*Sqrt[-a - c*x^4]),x]","-\frac{i \sqrt{\frac{c x^4}{a}+1} \Pi \left(-\frac{i \sqrt{a} e}{\sqrt{c} d};\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)}{d \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \sqrt{-a-c x^4}}","-\frac{a^{3/4} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(\frac{\sqrt{c} d}{\sqrt{a}}+e\right)^2 \Pi \left(-\frac{\left(\sqrt{c} d-\sqrt{a} e\right)^2}{4 \sqrt{a} \sqrt{c} d e};2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 \sqrt[4]{c} d \sqrt{-a-c x^4} \left(c d^2-a e^2\right)}+\frac{\sqrt{e} \tan ^{-1}\left(\frac{x \sqrt{-a e^2-c d^2}}{\sqrt{d} \sqrt{e} \sqrt{-a-c x^4}}\right)}{2 \sqrt{d} \sqrt{-a e^2-c d^2}}+\frac{\sqrt[4]{c} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} \sqrt{-a-c x^4} \left(\sqrt{c} d-\sqrt{a} e\right)}",1,"((-I)*Sqrt[1 + (c*x^4)/a]*EllipticPi[((-I)*Sqrt[a]*e)/(Sqrt[c]*d), I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])/(Sqrt[(I*Sqrt[c])/Sqrt[a]]*d*Sqrt[-a - c*x^4])","C",1
170,1,40,40,0.1261008,"\int \frac{1}{\left(a+b x^2\right) \sqrt{4-5 x^4}} \, dx","Integrate[1/((a + b*x^2)*Sqrt[4 - 5*x^4]),x]","\frac{\Pi \left(-\frac{2 b}{\sqrt{5} a};\left.\sin ^{-1}\left(\frac{\sqrt[4]{5} x}{\sqrt{2}}\right)\right|-1\right)}{\sqrt{2} \sqrt[4]{5} a}","\frac{\Pi \left(-\frac{2 b}{\sqrt{5} a};\left.\sin ^{-1}\left(\frac{\sqrt[4]{5} x}{\sqrt{2}}\right)\right|-1\right)}{\sqrt{2} \sqrt[4]{5} a}",1,"EllipticPi[(-2*b)/(Sqrt[5]*a), ArcSin[(5^(1/4)*x)/Sqrt[2]], -1]/(Sqrt[2]*5^(1/4)*a)","A",1
171,1,50,310,0.1023885,"\int \frac{1}{\left(a+b x^2\right) \sqrt{4+5 x^4}} \, dx","Integrate[1/((a + b*x^2)*Sqrt[4 + 5*x^4]),x]","-\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \Pi \left(-\frac{2 i b}{\sqrt{5} a};\left.i \sinh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt[4]{5} x\right)\right|-1\right)}{\sqrt[4]{5} a}","\frac{\sqrt{b} \tan ^{-1}\left(\frac{x \sqrt{5 a^2+4 b^2}}{\sqrt{a} \sqrt{b} \sqrt{5 x^4+4}}\right)}{2 \sqrt{a} \sqrt{5 a^2+4 b^2}}+\frac{\sqrt[4]{5} \left(\sqrt{5} x^2+2\right) \sqrt{\frac{5 x^4+4}{\left(\sqrt{5} x^2+2\right)^2}} \left(\sqrt{5} a+2 b\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{5} x}{\sqrt{2}}\right)|\frac{1}{2}\right)}{2 \sqrt{2} \sqrt{5 x^4+4} \left(5 a^2-4 b^2\right)}-\frac{\left(\sqrt{5} x^2+2\right) \sqrt{\frac{5 x^4+4}{\left(\sqrt{5} x^2+2\right)^2}} \left(\sqrt{5} a+2 b\right)^2 \Pi \left(-\frac{\left(\sqrt{5} a-2 b\right)^2}{8 \sqrt{5} a b};2 \tan ^{-1}\left(\frac{\sqrt[4]{5} x}{\sqrt{2}}\right)|\frac{1}{2}\right)}{4 \sqrt{2} \sqrt[4]{5} a \sqrt{5 x^4+4} \left(5 a^2-4 b^2\right)}",1,"((-1/2 - I/2)*EllipticPi[((-2*I)*b)/(Sqrt[5]*a), I*ArcSinh[(1/2 + I/2)*5^(1/4)*x], -1])/(5^(1/4)*a)","C",1
172,1,59,40,0.1237813,"\int \frac{1}{\left(a+b x^2\right) \sqrt{4-d x^4}} \, dx","Integrate[1/((a + b*x^2)*Sqrt[4 - d*x^4]),x]","-\frac{i \Pi \left(-\frac{2 b}{a \sqrt{d}};\left.i \sinh ^{-1}\left(\frac{\sqrt{-\sqrt{d}} x}{\sqrt{2}}\right)\right|-1\right)}{\sqrt{2} a \sqrt{-\sqrt{d}}}","\frac{\Pi \left(-\frac{2 b}{a \sqrt{d}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt{2}}\right)\right|-1\right)}{\sqrt{2} a \sqrt[4]{d}}",1,"((-I)*EllipticPi[(-2*b)/(a*Sqrt[d]), I*ArcSinh[(Sqrt[-Sqrt[d]]*x)/Sqrt[2]], -1])/(Sqrt[2]*a*Sqrt[-Sqrt[d]])","C",1
173,1,65,300,0.1109669,"\int \frac{1}{\left(a+b x^2\right) \sqrt{4+d x^4}} \, dx","Integrate[1/((a + b*x^2)*Sqrt[4 + d*x^4]),x]","-\frac{i \Pi \left(-\frac{2 i b}{a \sqrt{d}};\left.i \sinh ^{-1}\left(\frac{\sqrt{i \sqrt{d}} x}{\sqrt{2}}\right)\right|-1\right)}{\sqrt{2} a \sqrt{i \sqrt{d}}}","\frac{\sqrt{b} \tan ^{-1}\left(\frac{x \sqrt{a^2 d+4 b^2}}{\sqrt{a} \sqrt{b} \sqrt{d x^4+4}}\right)}{2 \sqrt{a} \sqrt{a^2 d+4 b^2}}-\frac{\sqrt[4]{d} \left(\sqrt{d} x^2+2\right) \sqrt{\frac{d x^4+4}{\left(\sqrt{d} x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt{2}}\right)|\frac{1}{2}\right)}{2 \sqrt{2} \sqrt{d x^4+4} \left(2 b-a \sqrt{d}\right)}+\frac{\left(\sqrt{d} x^2+2\right) \sqrt{\frac{d x^4+4}{\left(\sqrt{d} x^2+2\right)^2}} \left(a \sqrt{d}+2 b\right) \Pi \left(-\frac{\left(2 b-a \sqrt{d}\right)^2}{8 a b \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt{2}}\right)|\frac{1}{2}\right)}{4 \sqrt{2} a \sqrt[4]{d} \sqrt{d x^4+4} \left(2 b-a \sqrt{d}\right)}",1,"((-I)*EllipticPi[((-2*I)*b)/(a*Sqrt[d]), I*ArcSinh[(Sqrt[I*Sqrt[d]]*x)/Sqrt[2]], -1])/(Sqrt[2]*a*Sqrt[I*Sqrt[d]])","C",1
174,0,0,112,0.0597719,"\int \frac{\sqrt{a+b x^2}}{\sqrt{1-x^4}} \, dx","Integrate[Sqrt[a + b*x^2]/Sqrt[1 - x^4],x]","\int \frac{\sqrt{a+b x^2}}{\sqrt{1-x^4}} \, dx","\frac{a \sqrt{1-x^2} \sqrt{\frac{a \left(x^2+1\right)}{a+b x^2}} \Pi \left(\frac{b}{a+b};\sin ^{-1}\left(\frac{\sqrt{a+b} x}{\sqrt{b x^2+a}}\right)|-\frac{a-b}{a+b}\right)}{\sqrt{x^2+1} \sqrt{a+b} \sqrt{\frac{a \left(1-x^2\right)}{a+b x^2}}}",1,"Integrate[Sqrt[a + b*x^2]/Sqrt[1 - x^4], x]","F",-1
175,0,0,22,0.0760427,"\int \left(c+e x^2\right)^q \left(a+b x^4\right)^p \, dx","Integrate[(c + e*x^2)^q*(a + b*x^4)^p,x]","\int \left(c+e x^2\right)^q \left(a+b x^4\right)^p \, dx","\text{Int}\left(\left(a+b x^4\right)^p \left(c+e x^2\right)^q,x\right)",0,"Integrate[(c + e*x^2)^q*(a + b*x^4)^p, x]","A",-1
176,1,136,204,0.0740828,"\int \left(c+e x^2\right)^3 \left(a+b x^4\right)^p \, dx","Integrate[(c + e*x^2)^3*(a + b*x^4)^p,x]","\frac{1}{35} x \left(a+b x^4\right)^p \left(\frac{b x^4}{a}+1\right)^{-p} \left(35 c^3 \, _2F_1\left(\frac{1}{4},-p;\frac{5}{4};-\frac{b x^4}{a}\right)+e x^2 \left(35 c^2 \, _2F_1\left(\frac{3}{4},-p;\frac{7}{4};-\frac{b x^4}{a}\right)+e x^2 \left(21 c \, _2F_1\left(\frac{5}{4},-p;\frac{9}{4};-\frac{b x^4}{a}\right)+5 e x^2 \, _2F_1\left(\frac{7}{4},-p;\frac{11}{4};-\frac{b x^4}{a}\right)\right)\right)\right)","c^3 x \left(a+b x^4\right)^p \left(\frac{b x^4}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{4},-p;\frac{5}{4};-\frac{b x^4}{a}\right)-\frac{e x^3 \left(a+b x^4\right)^p \left(\frac{b x^4}{a}+1\right)^{-p} \left(a e^2-b c^2 (4 p+7)\right) \, _2F_1\left(\frac{3}{4},-p;\frac{7}{4};-\frac{b x^4}{a}\right)}{b (4 p+7)}+\frac{3}{5} c e^2 x^5 \left(a+b x^4\right)^p \left(\frac{b x^4}{a}+1\right)^{-p} \, _2F_1\left(\frac{5}{4},-p;\frac{9}{4};-\frac{b x^4}{a}\right)+\frac{e^3 x^3 \left(a+b x^4\right)^{p+1}}{b (4 p+7)}",1,"(x*(a + b*x^4)^p*(35*c^3*Hypergeometric2F1[1/4, -p, 5/4, -((b*x^4)/a)] + e*x^2*(35*c^2*Hypergeometric2F1[3/4, -p, 7/4, -((b*x^4)/a)] + e*x^2*(21*c*Hypergeometric2F1[5/4, -p, 9/4, -((b*x^4)/a)] + 5*e*x^2*Hypergeometric2F1[7/4, -p, 11/4, -((b*x^4)/a)]))))/(35*(1 + (b*x^4)/a)^p)","A",1
177,1,106,150,0.0427369,"\int \left(c+e x^2\right)^2 \left(a+b x^4\right)^p \, dx","Integrate[(c + e*x^2)^2*(a + b*x^4)^p,x]","\frac{1}{15} x \left(a+b x^4\right)^p \left(\frac{b x^4}{a}+1\right)^{-p} \left(15 c^2 \, _2F_1\left(\frac{1}{4},-p;\frac{5}{4};-\frac{b x^4}{a}\right)+e x^2 \left(10 c \, _2F_1\left(\frac{3}{4},-p;\frac{7}{4};-\frac{b x^4}{a}\right)+3 e x^2 \, _2F_1\left(\frac{5}{4},-p;\frac{9}{4};-\frac{b x^4}{a}\right)\right)\right)","-\frac{x \left(a+b x^4\right)^p \left(\frac{b x^4}{a}+1\right)^{-p} \left(a e^2-b c^2 (4 p+5)\right) \, _2F_1\left(\frac{1}{4},-p;\frac{5}{4};-\frac{b x^4}{a}\right)}{b (4 p+5)}+\frac{2}{3} c e x^3 \left(a+b x^4\right)^p \left(\frac{b x^4}{a}+1\right)^{-p} \, _2F_1\left(\frac{3}{4},-p;\frac{7}{4};-\frac{b x^4}{a}\right)+\frac{e^2 x \left(a+b x^4\right)^{p+1}}{b (4 p+5)}",1,"(x*(a + b*x^4)^p*(15*c^2*Hypergeometric2F1[1/4, -p, 5/4, -((b*x^4)/a)] + e*x^2*(10*c*Hypergeometric2F1[3/4, -p, 7/4, -((b*x^4)/a)] + 3*e*x^2*Hypergeometric2F1[5/4, -p, 9/4, -((b*x^4)/a)])))/(15*(1 + (b*x^4)/a)^p)","A",1
178,1,75,96,0.022909,"\int \left(c+e x^2\right) \left(a+b x^4\right)^p \, dx","Integrate[(c + e*x^2)*(a + b*x^4)^p,x]","\frac{1}{3} x \left(a+b x^4\right)^p \left(\frac{b x^4}{a}+1\right)^{-p} \left(3 c \, _2F_1\left(\frac{1}{4},-p;\frac{5}{4};-\frac{b x^4}{a}\right)+e x^2 \, _2F_1\left(\frac{3}{4},-p;\frac{7}{4};-\frac{b x^4}{a}\right)\right)","c x \left(a+b x^4\right)^p \left(\frac{b x^4}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{4},-p;\frac{5}{4};-\frac{b x^4}{a}\right)+\frac{1}{3} e x^3 \left(a+b x^4\right)^p \left(\frac{b x^4}{a}+1\right)^{-p} \, _2F_1\left(\frac{3}{4},-p;\frac{7}{4};-\frac{b x^4}{a}\right)",1,"(x*(a + b*x^4)^p*(3*c*Hypergeometric2F1[1/4, -p, 5/4, -((b*x^4)/a)] + e*x^2*Hypergeometric2F1[3/4, -p, 7/4, -((b*x^4)/a)]))/(3*(1 + (b*x^4)/a)^p)","A",1
179,1,44,44,0.0034888,"\int \left(a+b x^4\right)^p \, dx","Integrate[(a + b*x^4)^p,x]","x \left(a+b x^4\right)^p \left(\frac{b x^4}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{4},-p;\frac{5}{4};-\frac{b x^4}{a}\right)","x \left(a+b x^4\right)^p \left(\frac{b x^4}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{4},-p;\frac{5}{4};-\frac{b x^4}{a}\right)",1,"(x*(a + b*x^4)^p*Hypergeometric2F1[1/4, -p, 5/4, -((b*x^4)/a)])/(1 + (b*x^4)/a)^p","A",1
180,0,0,123,0.1266622,"\int \frac{\left(a+b x^4\right)^p}{c+e x^2} \, dx","Integrate[(a + b*x^4)^p/(c + e*x^2),x]","\int \frac{\left(a+b x^4\right)^p}{c+e x^2} \, dx","\frac{x \left(a+b x^4\right)^p \left(\frac{b x^4}{a}+1\right)^{-p} F_1\left(\frac{1}{4};-p,1;\frac{5}{4};-\frac{b x^4}{a},\frac{e^2 x^4}{c^2}\right)}{c}-\frac{e x^3 \left(a+b x^4\right)^p \left(\frac{b x^4}{a}+1\right)^{-p} F_1\left(\frac{3}{4};-p,1;\frac{7}{4};-\frac{b x^4}{a},\frac{e^2 x^4}{c^2}\right)}{3 c^2}",1,"Integrate[(a + b*x^4)^p/(c + e*x^2), x]","F",-1
181,0,0,189,0.243161,"\int \frac{\left(a+b x^4\right)^p}{\left(c+e x^2\right)^2} \, dx","Integrate[(a + b*x^4)^p/(c + e*x^2)^2,x]","\int \frac{\left(a+b x^4\right)^p}{\left(c+e x^2\right)^2} \, dx","\frac{x \left(a+b x^4\right)^p \left(\frac{b x^4}{a}+1\right)^{-p} F_1\left(\frac{1}{4};-p,2;\frac{5}{4};-\frac{b x^4}{a},\frac{e^2 x^4}{c^2}\right)}{c^2}+\frac{e^2 x^5 \left(a+b x^4\right)^p \left(\frac{b x^4}{a}+1\right)^{-p} F_1\left(\frac{5}{4};-p,2;\frac{9}{4};-\frac{b x^4}{a},\frac{e^2 x^4}{c^2}\right)}{5 c^4}-\frac{2 e x^3 \left(a+b x^4\right)^p \left(\frac{b x^4}{a}+1\right)^{-p} F_1\left(\frac{3}{4};-p,2;\frac{7}{4};-\frac{b x^4}{a},\frac{e^2 x^4}{c^2}\right)}{3 c^3}",1,"Integrate[(a + b*x^4)^p/(c + e*x^2)^2, x]","F",-1
182,1,86,108,0.0171906,"\int \left(1-x^2\right)^3 \left(1+b x^4\right)^p \, dx","Integrate[(1 - x^2)^3*(1 + b*x^4)^p,x]","x \, _2F_1\left(\frac{1}{4},-p;\frac{5}{4};-b x^4\right)-\frac{1}{7} x^7 \, _2F_1\left(\frac{7}{4},-p;\frac{11}{4};-b x^4\right)+\frac{3}{5} x^5 \, _2F_1\left(\frac{5}{4},-p;\frac{9}{4};-b x^4\right)-x^3 \, _2F_1\left(\frac{3}{4},-p;\frac{7}{4};-b x^4\right)","x \, _2F_1\left(\frac{1}{4},-p;\frac{5}{4};-b x^4\right)+\frac{3}{5} x^5 \, _2F_1\left(\frac{5}{4},-p;\frac{9}{4};-b x^4\right)+\frac{x^3 (1-b (4 p+7)) \, _2F_1\left(\frac{3}{4},-p;\frac{7}{4};-b x^4\right)}{b (4 p+7)}-\frac{x^3 \left(b x^4+1\right)^{p+1}}{b (4 p+7)}",1,"x*Hypergeometric2F1[1/4, -p, 5/4, -(b*x^4)] - x^3*Hypergeometric2F1[3/4, -p, 7/4, -(b*x^4)] + (3*x^5*Hypergeometric2F1[5/4, -p, 9/4, -(b*x^4)])/5 - (x^7*Hypergeometric2F1[7/4, -p, 11/4, -(b*x^4)])/7","A",1
183,1,65,86,0.0109536,"\int \left(1-x^2\right)^2 \left(1+b x^4\right)^p \, dx","Integrate[(1 - x^2)^2*(1 + b*x^4)^p,x]","x \, _2F_1\left(\frac{1}{4},-p;\frac{5}{4};-b x^4\right)+\frac{1}{5} x^5 \, _2F_1\left(\frac{5}{4},-p;\frac{9}{4};-b x^4\right)-\frac{2}{3} x^3 \, _2F_1\left(\frac{3}{4},-p;\frac{7}{4};-b x^4\right)","-\frac{x (1-b (4 p+5)) \, _2F_1\left(\frac{1}{4},-p;\frac{5}{4};-b x^4\right)}{b (4 p+5)}-\frac{2}{3} x^3 \, _2F_1\left(\frac{3}{4},-p;\frac{7}{4};-b x^4\right)+\frac{x \left(b x^4+1\right)^{p+1}}{b (4 p+5)}",1,"x*Hypergeometric2F1[1/4, -p, 5/4, -(b*x^4)] - (2*x^3*Hypergeometric2F1[3/4, -p, 7/4, -(b*x^4)])/3 + (x^5*Hypergeometric2F1[5/4, -p, 9/4, -(b*x^4)])/5","A",1
184,1,42,42,0.0068766,"\int \left(1-x^2\right) \left(1+b x^4\right)^p \, dx","Integrate[(1 - x^2)*(1 + b*x^4)^p,x]","x \, _2F_1\left(\frac{1}{4},-p;\frac{5}{4};-b x^4\right)-\frac{1}{3} x^3 \, _2F_1\left(\frac{3}{4},-p;\frac{7}{4};-b x^4\right)","x \, _2F_1\left(\frac{1}{4},-p;\frac{5}{4};-b x^4\right)-\frac{1}{3} x^3 \, _2F_1\left(\frac{3}{4},-p;\frac{7}{4};-b x^4\right)",1,"x*Hypergeometric2F1[1/4, -p, 5/4, -(b*x^4)] - (x^3*Hypergeometric2F1[3/4, -p, 7/4, -(b*x^4)])/3","A",1
185,1,18,18,0.0017071,"\int \left(1+b x^4\right)^p \, dx","Integrate[(1 + b*x^4)^p,x]","x \, _2F_1\left(\frac{1}{4},-p;\frac{5}{4};-b x^4\right)","x \, _2F_1\left(\frac{1}{4},-p;\frac{5}{4};-b x^4\right)",1,"x*Hypergeometric2F1[1/4, -p, 5/4, -(b*x^4)]","A",1
186,0,0,50,0.081032,"\int \frac{\left(1+b x^4\right)^p}{1-x^2} \, dx","Integrate[(1 + b*x^4)^p/(1 - x^2),x]","\int \frac{\left(1+b x^4\right)^p}{1-x^2} \, dx","x F_1\left(\frac{1}{4};1,-p;\frac{5}{4};x^4,-b x^4\right)+\frac{1}{3} x^3 F_1\left(\frac{3}{4};1,-p;\frac{7}{4};x^4,-b x^4\right)",1,"Integrate[(1 + b*x^4)^p/(1 - x^2), x]","F",-1
187,0,0,77,0.1430101,"\int \frac{\left(1+b x^4\right)^p}{\left(1-x^2\right)^2} \, dx","Integrate[(1 + b*x^4)^p/(1 - x^2)^2,x]","\int \frac{\left(1+b x^4\right)^p}{\left(1-x^2\right)^2} \, dx","x F_1\left(\frac{1}{4};2,-p;\frac{5}{4};x^4,-b x^4\right)+\frac{1}{5} x^5 F_1\left(\frac{5}{4};2,-p;\frac{9}{4};x^4,-b x^4\right)+\frac{2}{3} x^3 F_1\left(\frac{3}{4};2,-p;\frac{7}{4};x^4,-b x^4\right)",1,"Integrate[(1 + b*x^4)^p/(1 - x^2)^2, x]","F",-1
188,0,0,101,0.2566529,"\int \frac{\left(1+b x^4\right)^p}{\left(1-x^2\right)^3} \, dx","Integrate[(1 + b*x^4)^p/(1 - x^2)^3,x]","\int \frac{\left(1+b x^4\right)^p}{\left(1-x^2\right)^3} \, dx","x F_1\left(\frac{1}{4};3,-p;\frac{5}{4};x^4,-b x^4\right)+\frac{1}{7} x^7 F_1\left(\frac{7}{4};3,-p;\frac{11}{4};x^4,-b x^4\right)+\frac{3}{5} x^5 F_1\left(\frac{5}{4};3,-p;\frac{9}{4};x^4,-b x^4\right)+x^3 F_1\left(\frac{3}{4};3,-p;\frac{7}{4};x^4,-b x^4\right)",1,"Integrate[(1 + b*x^4)^p/(1 - x^2)^3, x]","F",-1
189,1,51,51,0.0233746,"\int \frac{\left(d+e x^2\right)^4}{d^2-e^2 x^4} \, dx","Integrate[(d + e*x^2)^4/(d^2 - e^2*x^4),x]","\frac{8 d^{5/2} \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{e}}-7 d^2 x-\frac{4}{3} d e x^3-\frac{1}{5} e^2 x^5","\frac{8 d^{5/2} \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{e}}-7 d^2 x-\frac{4}{3} d e x^3-\frac{1}{5} e^2 x^5",1,"-7*d^2*x - (4*d*e*x^3)/3 - (e^2*x^5)/5 + (8*d^(5/2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e]","A",1
190,1,38,38,0.0176525,"\int \frac{\left(d+e x^2\right)^3}{d^2-e^2 x^4} \, dx","Integrate[(d + e*x^2)^3/(d^2 - e^2*x^4),x]","\frac{4 d^{3/2} \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{e}}-3 d x-\frac{e x^3}{3}","\frac{4 d^{3/2} \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{e}}-3 d x-\frac{e x^3}{3}",1,"-3*d*x - (e*x^3)/3 + (4*d^(3/2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e]","A",1
191,1,29,29,0.0086655,"\int \frac{\left(d+e x^2\right)^2}{d^2-e^2 x^4} \, dx","Integrate[(d + e*x^2)^2/(d^2 - e^2*x^4),x]","\frac{2 \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{e}}-x","\frac{2 \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{e}}-x",1,"-x + (2*Sqrt[d]*ArcTanh[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e]","A",1
192,1,24,24,0.0045596,"\int \frac{d+e x^2}{d^2-e^2 x^4} \, dx","Integrate[(d + e*x^2)/(d^2 - e^2*x^4),x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{d} \sqrt{e}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{d} \sqrt{e}}",1,"ArcTanh[(Sqrt[e]*x)/Sqrt[d]]/(Sqrt[d]*Sqrt[e])","A",1
193,1,65,72,0.037038,"\int \frac{1}{\left(d+e x^2\right) \left(d^2-e^2 x^4\right)} \, dx","Integrate[1/((d + e*x^2)*(d^2 - e^2*x^4)),x]","\frac{\frac{\sqrt{d} x}{d+e x^2}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{e}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{e}}}{4 d^{5/2}}","\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{2 d^{5/2} \sqrt{e}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{4 d^{5/2} \sqrt{e}}+\frac{x}{4 d^2 \left(d+e x^2\right)}",1,"((Sqrt[d]*x)/(d + e*x^2) + (2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] + ArcTanh[(Sqrt[e]*x)/Sqrt[d]]/Sqrt[e])/(4*d^(5/2))","A",1
194,1,76,89,0.0592033,"\int \frac{1}{\left(d+e x^2\right)^2 \left(d^2-e^2 x^4\right)} \, dx","Integrate[1/((d + e*x^2)^2*(d^2 - e^2*x^4)),x]","\frac{\frac{\sqrt{d} x \left(7 d+5 e x^2\right)}{\left(d+e x^2\right)^2}+\frac{7 \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{e}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{e}}}{16 d^{7/2}}","\frac{7 \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{16 d^{7/2} \sqrt{e}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{8 d^{7/2} \sqrt{e}}+\frac{5 x}{16 d^3 \left(d+e x^2\right)}+\frac{x}{8 d^2 \left(d+e x^2\right)^2}",1,"((Sqrt[d]*x*(7*d + 5*e*x^2))/(d + e*x^2)^2 + (7*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] + (2*ArcTanh[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e])/(16*d^(7/2))","A",1
195,1,61,62,0.0287497,"\int \frac{\left(d+e x^2\right)^{3/2}}{d^2-e^2 x^4} \, dx","Integrate[(d + e*x^2)^(3/2)/(d^2 - e^2*x^4),x]","\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d+e x^2}}\right)-\log \left(\sqrt{e} \sqrt{d+e x^2}+e x\right)}{\sqrt{e}}","\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d+e x^2}}\right)}{\sqrt{e}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{\sqrt{e}}",1,"(Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[e]*x)/Sqrt[d + e*x^2]] - Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/Sqrt[e]","A",1
196,1,38,38,0.154804,"\int \frac{\sqrt{d+e x^2}}{d^2-e^2 x^4} \, dx","Integrate[Sqrt[d + e*x^2]/(d^2 - e^2*x^4),x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d+e x^2}}\right)}{\sqrt{2} d \sqrt{e}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d+e x^2}}\right)}{\sqrt{2} d \sqrt{e}}",1,"ArcTanh[(Sqrt[2]*Sqrt[e]*x)/Sqrt[d + e*x^2]]/(Sqrt[2]*d*Sqrt[e])","A",1
197,1,108,61,0.1260708,"\int \frac{1}{\sqrt{d+e x^2} \left(d^2-e^2 x^4\right)} \, dx","Integrate[1/(Sqrt[d + e*x^2]*(d^2 - e^2*x^4)),x]","\frac{\frac{4 x}{\sqrt{d+e x^2}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{d}-\sqrt{e} x}{\sqrt{2} \sqrt{d+e x^2}}\right)}{\sqrt{e}}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{d}+\sqrt{e} x}{\sqrt{2} \sqrt{d+e x^2}}\right)}{\sqrt{e}}}{8 d^2}","\frac{x}{2 d^2 \sqrt{d+e x^2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d+e x^2}}\right)}{2 \sqrt{2} d^2 \sqrt{e}}",1,"((4*x)/Sqrt[d + e*x^2] - (Sqrt[2]*ArcTanh[(Sqrt[d] - Sqrt[e]*x)/(Sqrt[2]*Sqrt[d + e*x^2])])/Sqrt[e] + (Sqrt[2]*ArcTanh[(Sqrt[d] + Sqrt[e]*x)/(Sqrt[2]*Sqrt[d + e*x^2])])/Sqrt[e])/(8*d^2)","A",1
198,1,345,80,3.3428669,"\int \frac{1}{\left(d+e x^2\right)^{3/2} \left(d^2-e^2 x^4\right)} \, dx","Integrate[1/((d + e*x^2)^(3/2)*(d^2 - e^2*x^4)),x]","\frac{\frac{384 e^4 x^8 \left(d+e x^2\right)^2 \, _3F_2\left(2,2,2;1,\frac{9}{2};-\frac{2 e x^2}{d-e x^2}\right)}{e x^2-d}+\frac{384 e^4 x^8 \left(4 d^2+7 d e x^2+3 e^2 x^4\right) \, _2F_1\left(2,2;\frac{9}{2};-\frac{2 e x^2}{d-e x^2}\right)}{e x^2-d}+\frac{35 \sqrt{2} \sqrt{\frac{e x^2}{e x^2-d}} \left(-15 d^3-5 d^2 e x^2+12 d e^2 x^4+8 e^3 x^6\right) \left(\sqrt{2} \sqrt{\frac{e x^2}{e x^2-d}} \sqrt{\frac{d+e x^2}{d-e x^2}} \left(-3 d^2-2 d e x^2+5 e^2 x^4\right)+3 \left(d+e x^2\right)^2 \sin ^{-1}\left(\sqrt{2} \sqrt{\frac{e x^2}{e x^2-d}}\right)\right)}{\sqrt{\frac{d+e x^2}{d-e x^2}}}}{2520 d^5 e^3 x^5 \sqrt{d+e x^2} \left(1-\frac{e^2 x^4}{d^2}\right)}","\frac{7 x}{12 d^3 \sqrt{d+e x^2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d+e x^2}}\right)}{4 \sqrt{2} d^3 \sqrt{e}}+\frac{x}{6 d^2 \left(d+e x^2\right)^{3/2}}",1,"((35*Sqrt[2]*Sqrt[(e*x^2)/(-d + e*x^2)]*(-15*d^3 - 5*d^2*e*x^2 + 12*d*e^2*x^4 + 8*e^3*x^6)*(Sqrt[2]*Sqrt[(e*x^2)/(-d + e*x^2)]*Sqrt[(d + e*x^2)/(d - e*x^2)]*(-3*d^2 - 2*d*e*x^2 + 5*e^2*x^4) + 3*(d + e*x^2)^2*ArcSin[Sqrt[2]*Sqrt[(e*x^2)/(-d + e*x^2)]]))/Sqrt[(d + e*x^2)/(d - e*x^2)] + (384*e^4*x^8*(4*d^2 + 7*d*e*x^2 + 3*e^2*x^4)*Hypergeometric2F1[2, 2, 9/2, (-2*e*x^2)/(d - e*x^2)])/(-d + e*x^2) + (384*e^4*x^8*(d + e*x^2)^2*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, (-2*e*x^2)/(d - e*x^2)])/(-d + e*x^2))/(2520*d^5*e^3*x^5*Sqrt[d + e*x^2]*(1 - (e^2*x^4)/d^2))","C",0
199,1,98,153,0.1654972,"\int \frac{\left(a+b x^2\right)^{5/2}}{\sqrt{a^2-b^2 x^4}} \, dx","Integrate[(a + b*x^2)^(5/2)/Sqrt[a^2 - b^2*x^4],x]","-\frac{\left(11 a x+2 b x^3\right) \sqrt{a^2-b^2 x^4}}{8 \sqrt{a+b x^2}}+\frac{19 i a^2 \log \left(\frac{2 \sqrt{a^2-b^2 x^4}}{\sqrt{a+b x^2}}-2 i \sqrt{b} x\right)}{8 \sqrt{b}}","-\frac{9 a x \left(a-b x^2\right) \sqrt{a+b x^2}}{8 \sqrt{a^2-b^2 x^4}}-\frac{x \left(a-b x^2\right) \left(a+b x^2\right)^{3/2}}{4 \sqrt{a^2-b^2 x^4}}+\frac{19 a^2 \sqrt{a-b x^2} \sqrt{a+b x^2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a-b x^2}}\right)}{8 \sqrt{b} \sqrt{a^2-b^2 x^4}}",1,"-1/8*((11*a*x + 2*b*x^3)*Sqrt[a^2 - b^2*x^4])/Sqrt[a + b*x^2] + (((19*I)/8)*a^2*Log[(-2*I)*Sqrt[b]*x + (2*Sqrt[a^2 - b^2*x^4])/Sqrt[a + b*x^2]])/Sqrt[b]","C",1
200,1,86,110,0.072388,"\int \frac{\left(a+b x^2\right)^{3/2}}{\sqrt{a^2-b^2 x^4}} \, dx","Integrate[(a + b*x^2)^(3/2)/Sqrt[a^2 - b^2*x^4],x]","-\frac{x \sqrt{a^2-b^2 x^4}}{2 \sqrt{a+b x^2}}+\frac{3 i a \log \left(\frac{2 \sqrt{a^2-b^2 x^4}}{\sqrt{a+b x^2}}-2 i \sqrt{b} x\right)}{2 \sqrt{b}}","\frac{3 a \sqrt{a-b x^2} \sqrt{a+b x^2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a-b x^2}}\right)}{2 \sqrt{b} \sqrt{a^2-b^2 x^4}}-\frac{x \left(a-b x^2\right) \sqrt{a+b x^2}}{2 \sqrt{a^2-b^2 x^4}}",1,"-1/2*(x*Sqrt[a^2 - b^2*x^4])/Sqrt[a + b*x^2] + (((3*I)/2)*a*Log[(-2*I)*Sqrt[b]*x + (2*Sqrt[a^2 - b^2*x^4])/Sqrt[a + b*x^2]])/Sqrt[b]","C",1
201,1,50,65,0.043177,"\int \frac{\sqrt{a+b x^2}}{\sqrt{a^2-b^2 x^4}} \, dx","Integrate[Sqrt[a + b*x^2]/Sqrt[a^2 - b^2*x^4],x]","\frac{i \log \left(\frac{2 \sqrt{a^2-b^2 x^4}}{\sqrt{a+b x^2}}-2 i \sqrt{b} x\right)}{\sqrt{b}}","\frac{\sqrt{a-b x^2} \sqrt{a+b x^2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a-b x^2}}\right)}{\sqrt{b} \sqrt{a^2-b^2 x^4}}",1,"(I*Log[(-2*I)*Sqrt[b]*x + (2*Sqrt[a^2 - b^2*x^4])/Sqrt[a + b*x^2]])/Sqrt[b]","C",1
202,1,78,78,0.0521562,"\int \frac{1}{\sqrt{a+b x^2} \sqrt{a^2-b^2 x^4}} \, dx","Integrate[1/(Sqrt[a + b*x^2]*Sqrt[a^2 - b^2*x^4]),x]","\frac{\sqrt{a^2-b^2 x^4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a-b x^2}}\right)}{\sqrt{2} a \sqrt{b} \sqrt{a-b x^2} \sqrt{a+b x^2}}","\frac{\sqrt{a-b x^2} \sqrt{a+b x^2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a-b x^2}}\right)}{\sqrt{2} a \sqrt{b} \sqrt{a^2-b^2 x^4}}",1,"(Sqrt[a^2 - b^2*x^4]*ArcTan[(Sqrt[2]*Sqrt[b]*x)/Sqrt[a - b*x^2]])/(Sqrt[2]*a*Sqrt[b]*Sqrt[a - b*x^2]*Sqrt[a + b*x^2])","A",1
203,1,111,125,0.0847324,"\int \frac{1}{\left(a+b x^2\right)^{3/2} \sqrt{a^2-b^2 x^4}} \, dx","Integrate[1/((a + b*x^2)^(3/2)*Sqrt[a^2 - b^2*x^4]),x]","\frac{\sqrt{a^2-b^2 x^4} \left(2 \sqrt{b} x \sqrt{a-b x^2}+3 \sqrt{2} \left(a+b x^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a-b x^2}}\right)\right)}{8 a^2 \sqrt{b} \sqrt{a-b x^2} \left(a+b x^2\right)^{3/2}}","\frac{x \left(a-b x^2\right)}{4 a^2 \sqrt{a+b x^2} \sqrt{a^2-b^2 x^4}}+\frac{3 \sqrt{a+b x^2} \sqrt{a-b x^2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a-b x^2}}\right)}{4 \sqrt{2} a^2 \sqrt{b} \sqrt{a^2-b^2 x^4}}",1,"(Sqrt[a^2 - b^2*x^4]*(2*Sqrt[b]*x*Sqrt[a - b*x^2] + 3*Sqrt[2]*(a + b*x^2)*ArcTan[(Sqrt[2]*Sqrt[b]*x)/Sqrt[a - b*x^2]]))/(8*a^2*Sqrt[b]*Sqrt[a - b*x^2]*(a + b*x^2)^(3/2))","A",1
204,1,123,168,0.1068104,"\int \frac{1}{\left(a+b x^2\right)^{5/2} \sqrt{a^2-b^2 x^4}} \, dx","Integrate[1/((a + b*x^2)^(5/2)*Sqrt[a^2 - b^2*x^4]),x]","\frac{\sqrt{a^2-b^2 x^4} \left(2 \sqrt{b} x \sqrt{a-b x^2} \left(13 a+9 b x^2\right)+19 \sqrt{2} \left(a+b x^2\right)^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a-b x^2}}\right)\right)}{64 a^3 \sqrt{b} \sqrt{a-b x^2} \left(a+b x^2\right)^{5/2}}","\frac{x \left(a-b x^2\right)}{8 a^2 \left(a+b x^2\right)^{3/2} \sqrt{a^2-b^2 x^4}}+\frac{9 x \left(a-b x^2\right)}{32 a^3 \sqrt{a+b x^2} \sqrt{a^2-b^2 x^4}}+\frac{19 \sqrt{a+b x^2} \sqrt{a-b x^2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a-b x^2}}\right)}{32 \sqrt{2} a^3 \sqrt{b} \sqrt{a^2-b^2 x^4}}",1,"(Sqrt[a^2 - b^2*x^4]*(2*Sqrt[b]*x*Sqrt[a - b*x^2]*(13*a + 9*b*x^2) + 19*Sqrt[2]*(a + b*x^2)^2*ArcTan[(Sqrt[2]*Sqrt[b]*x)/Sqrt[a - b*x^2]]))/(64*a^3*Sqrt[b]*Sqrt[a - b*x^2]*(a + b*x^2)^(5/2))","A",1
205,1,123,152,0.1997267,"\int \frac{\left(a-b x^2\right)^{5/2}}{\sqrt{a^2-b^2 x^4}} \, dx","Integrate[(a - b*x^2)^(5/2)/Sqrt[a^2 - b^2*x^4],x]","\frac{1}{8} \left(\frac{x \left(2 b x^2-11 a\right) \sqrt{a^2-b^2 x^4}}{\sqrt{a-b x^2}}+\frac{19 a^2 \log \left(\sqrt{b} \sqrt{a-b x^2} \sqrt{a^2-b^2 x^4}+a b x-b^2 x^3\right)}{\sqrt{b}}-\frac{19 a^2 \log \left(b x^2-a\right)}{\sqrt{b}}\right)","-\frac{9 a x \sqrt{a-b x^2} \left(a+b x^2\right)}{8 \sqrt{a^2-b^2 x^4}}-\frac{x \left(a-b x^2\right)^{3/2} \left(a+b x^2\right)}{4 \sqrt{a^2-b^2 x^4}}+\frac{19 a^2 \sqrt{a-b x^2} \sqrt{a+b x^2} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 \sqrt{b} \sqrt{a^2-b^2 x^4}}",1,"((x*(-11*a + 2*b*x^2)*Sqrt[a^2 - b^2*x^4])/Sqrt[a - b*x^2] - (19*a^2*Log[-a + b*x^2])/Sqrt[b] + (19*a^2*Log[a*b*x - b^2*x^3 + Sqrt[b]*Sqrt[a - b*x^2]*Sqrt[a^2 - b^2*x^4]])/Sqrt[b])/8","A",1
206,1,110,109,0.1098929,"\int \frac{\left(a-b x^2\right)^{3/2}}{\sqrt{a^2-b^2 x^4}} \, dx","Integrate[(a - b*x^2)^(3/2)/Sqrt[a^2 - b^2*x^4],x]","\frac{1}{2} \left(-\frac{x \sqrt{a^2-b^2 x^4}}{\sqrt{a-b x^2}}+\frac{3 a \log \left(\sqrt{b} \sqrt{a-b x^2} \sqrt{a^2-b^2 x^4}+a b x-b^2 x^3\right)}{\sqrt{b}}-\frac{3 a \log \left(b x^2-a\right)}{\sqrt{b}}\right)","\frac{3 a \sqrt{a-b x^2} \sqrt{a+b x^2} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 \sqrt{b} \sqrt{a^2-b^2 x^4}}-\frac{x \sqrt{a-b x^2} \left(a+b x^2\right)}{2 \sqrt{a^2-b^2 x^4}}",1,"(-((x*Sqrt[a^2 - b^2*x^4])/Sqrt[a - b*x^2]) - (3*a*Log[-a + b*x^2])/Sqrt[b] + (3*a*Log[a*b*x - b^2*x^3 + Sqrt[b]*Sqrt[a - b*x^2]*Sqrt[a^2 - b^2*x^4]])/Sqrt[b])/2","A",1
207,1,67,64,0.0395447,"\int \frac{\sqrt{a-b x^2}}{\sqrt{a^2-b^2 x^4}} \, dx","Integrate[Sqrt[a - b*x^2]/Sqrt[a^2 - b^2*x^4],x]","\frac{\log \left(\sqrt{b} \sqrt{a-b x^2} \sqrt{a^2-b^2 x^4}+a b x-b^2 x^3\right)-\log \left(b x^2-a\right)}{\sqrt{b}}","\frac{\sqrt{a-b x^2} \sqrt{a+b x^2} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{\sqrt{b} \sqrt{a^2-b^2 x^4}}",1,"(-Log[-a + b*x^2] + Log[a*b*x - b^2*x^3 + Sqrt[b]*Sqrt[a - b*x^2]*Sqrt[a^2 - b^2*x^4]])/Sqrt[b]","A",1
208,1,77,77,0.0504071,"\int \frac{1}{\sqrt{a-b x^2} \sqrt{a^2-b^2 x^4}} \, dx","Integrate[1/(Sqrt[a - b*x^2]*Sqrt[a^2 - b^2*x^4]),x]","\frac{\sqrt{a^2-b^2 x^4} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a+b x^2}}\right)}{\sqrt{2} a \sqrt{b} \sqrt{a-b x^2} \sqrt{a+b x^2}}","\frac{\sqrt{a-b x^2} \sqrt{a+b x^2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a+b x^2}}\right)}{\sqrt{2} a \sqrt{b} \sqrt{a^2-b^2 x^4}}",1,"(Sqrt[a^2 - b^2*x^4]*ArcTanh[(Sqrt[2]*Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[2]*a*Sqrt[b]*Sqrt[a - b*x^2]*Sqrt[a + b*x^2])","A",1
209,1,110,124,0.0817984,"\int \frac{1}{\left(a-b x^2\right)^{3/2} \sqrt{a^2-b^2 x^4}} \, dx","Integrate[1/((a - b*x^2)^(3/2)*Sqrt[a^2 - b^2*x^4]),x]","\frac{\sqrt{a^2-b^2 x^4} \left(2 \sqrt{b} x \sqrt{a+b x^2}+3 \sqrt{2} \left(a-b x^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a+b x^2}}\right)\right)}{8 a^2 \sqrt{b} \left(a-b x^2\right)^{3/2} \sqrt{a+b x^2}}","\frac{x \left(a+b x^2\right)}{4 a^2 \sqrt{a-b x^2} \sqrt{a^2-b^2 x^4}}+\frac{3 \sqrt{a-b x^2} \sqrt{a+b x^2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a+b x^2}}\right)}{4 \sqrt{2} a^2 \sqrt{b} \sqrt{a^2-b^2 x^4}}",1,"(Sqrt[a^2 - b^2*x^4]*(2*Sqrt[b]*x*Sqrt[a + b*x^2] + 3*Sqrt[2]*(a - b*x^2)*ArcTanh[(Sqrt[2]*Sqrt[b]*x)/Sqrt[a + b*x^2]]))/(8*a^2*Sqrt[b]*(a - b*x^2)^(3/2)*Sqrt[a + b*x^2])","A",1
210,1,122,167,0.1060794,"\int \frac{1}{\left(a-b x^2\right)^{5/2} \sqrt{a^2-b^2 x^4}} \, dx","Integrate[1/((a - b*x^2)^(5/2)*Sqrt[a^2 - b^2*x^4]),x]","\frac{\sqrt{a^2-b^2 x^4} \left(2 \sqrt{b} x \left(13 a-9 b x^2\right) \sqrt{a+b x^2}+19 \sqrt{2} \left(a-b x^2\right)^2 \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a+b x^2}}\right)\right)}{64 a^3 \sqrt{b} \left(a-b x^2\right)^{5/2} \sqrt{a+b x^2}}","\frac{x \left(a+b x^2\right)}{8 a^2 \left(a-b x^2\right)^{3/2} \sqrt{a^2-b^2 x^4}}+\frac{9 x \left(a+b x^2\right)}{32 a^3 \sqrt{a-b x^2} \sqrt{a^2-b^2 x^4}}+\frac{19 \sqrt{a-b x^2} \sqrt{a+b x^2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a+b x^2}}\right)}{32 \sqrt{2} a^3 \sqrt{b} \sqrt{a^2-b^2 x^4}}",1,"(Sqrt[a^2 - b^2*x^4]*(2*Sqrt[b]*x*(13*a - 9*b*x^2)*Sqrt[a + b*x^2] + 19*Sqrt[2]*(a - b*x^2)^2*ArcTanh[(Sqrt[2]*Sqrt[b]*x)/Sqrt[a + b*x^2]]))/(64*a^3*Sqrt[b]*(a - b*x^2)^(5/2)*Sqrt[a + b*x^2])","A",1
211,1,38,30,0.0232494,"\int \frac{\sqrt{-1+x^2}}{\sqrt{-1+x^4}} \, dx","Integrate[Sqrt[-1 + x^2]/Sqrt[-1 + x^4],x]","\log \left(x^3+\sqrt{x^2-1} \sqrt{x^4-1}-x\right)-\log \left(1-x^2\right)","\frac{\sqrt{x^2-1} \sqrt{x^2+1} \sinh ^{-1}(x)}{\sqrt{x^4-1}}",1,"-Log[1 - x^2] + Log[-x + x^3 + Sqrt[-1 + x^2]*Sqrt[-1 + x^4]]","A",1
212,1,34,24,0.0198799,"\int \frac{\sqrt{1+x^2}}{\sqrt{-1+x^4}} \, dx","Integrate[Sqrt[1 + x^2]/Sqrt[-1 + x^4],x]","\log \left(x^3+\sqrt{x^2+1} \sqrt{x^4-1}+x\right)-\log \left(x^2+1\right)","-\frac{\sqrt{x^4-1} \sin ^{-1}(x)}{\sqrt{1-x^4}}",1,"-Log[1 + x^2] + Log[x + x^3 + Sqrt[1 + x^2]*Sqrt[-1 + x^4]]","A",1
213,1,71,73,0.0559922,"\int \frac{-\sqrt{-1+x^2}+\sqrt{1+x^2}}{\sqrt{-1+x^4}} \, dx","Integrate[(-Sqrt[-1 + x^2] + Sqrt[1 + x^2])/Sqrt[-1 + x^4],x]","\log \left(1-x^2\right)-\log \left(x^2+1\right)-\log \left(x^3+\sqrt{x^2-1} \sqrt{x^4-1}-x\right)+\log \left(x^3+\sqrt{x^2+1} \sqrt{x^4-1}+x\right)","\frac{\sqrt{x^2-1} \sqrt{x^4-1} \sinh ^{-1}(x)}{\left(1-x^2\right) \sqrt{x^2+1}}-\frac{\sqrt{x^4-1} \sin ^{-1}(x)}{\sqrt{1-x^2} \sqrt{x^2+1}}",1,"Log[1 - x^2] - Log[1 + x^2] - Log[-x + x^3 + Sqrt[-1 + x^2]*Sqrt[-1 + x^4]] + Log[x + x^3 + Sqrt[1 + x^2]*Sqrt[-1 + x^4]]","A",1
214,1,121,121,0.0749213,"\int \frac{\left(d+e x^2\right)^4}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx","Integrate[(d + e*x^2)^4/(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4),x]","-\frac{x \left(-b^2 e^2+5 b c d e-7 c^2 d^2\right)}{c^3}-\frac{(b e-2 c d)^3 \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{e} x}{\sqrt{b e-c d}}\right)}{c^{7/2} \sqrt{e} \sqrt{b e-c d}}-\frac{e x^3 (b e-4 c d)}{3 c^2}+\frac{e^2 x^5}{5 c}","\frac{x \left(b^2 e^2-5 b c d e+7 c^2 d^2\right)}{c^3}-\frac{(2 c d-b e)^3 \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right)}{c^{7/2} \sqrt{e} \sqrt{c d-b e}}+\frac{e x^3 (4 c d-b e)}{3 c^2}+\frac{e^2 x^5}{5 c}",1,"-(((-7*c^2*d^2 + 5*b*c*d*e - b^2*e^2)*x)/c^3) - (e*(-4*c*d + b*e)*x^3)/(3*c^2) + (e^2*x^5)/(5*c) - ((-2*c*d + b*e)^3*ArcTan[(Sqrt[c]*Sqrt[e]*x)/Sqrt[-(c*d) + b*e]])/(c^(7/2)*Sqrt[e]*Sqrt[-(c*d) + b*e])","A",1
215,1,84,86,0.0456714,"\int \frac{\left(d+e x^2\right)^3}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx","Integrate[(d + e*x^2)^3/(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4),x]","\frac{(b e-2 c d)^2 \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{e} x}{\sqrt{b e-c d}}\right)}{c^{5/2} \sqrt{e} \sqrt{b e-c d}}-\frac{x (b e-3 c d)}{c^2}+\frac{e x^3}{3 c}","-\frac{(2 c d-b e)^2 \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right)}{c^{5/2} \sqrt{e} \sqrt{c d-b e}}+\frac{x (3 c d-b e)}{c^2}+\frac{e x^3}{3 c}",1,"-(((-3*c*d + b*e)*x)/c^2) + (e*x^3)/(3*c) + ((-2*c*d + b*e)^2*ArcTan[(Sqrt[c]*Sqrt[e]*x)/Sqrt[-(c*d) + b*e]])/(c^(5/2)*Sqrt[e]*Sqrt[-(c*d) + b*e])","A",1
216,1,63,64,0.0550686,"\int \frac{\left(d+e x^2\right)^2}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx","Integrate[(d + e*x^2)^2/(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4),x]","\frac{x}{c}-\frac{(b e-2 c d) \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{e} x}{\sqrt{b e-c d}}\right)}{c^{3/2} \sqrt{e} \sqrt{b e-c d}}","\frac{x}{c}-\frac{(2 c d-b e) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right)}{c^{3/2} \sqrt{e} \sqrt{c d-b e}}",1,"x/c - ((-2*c*d + b*e)*ArcTan[(Sqrt[c]*Sqrt[e]*x)/Sqrt[-(c*d) + b*e]])/(c^(3/2)*Sqrt[e]*Sqrt[-(c*d) + b*e])","A",1
217,1,48,49,0.0121592,"\int \frac{d+e x^2}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx","Integrate[(d + e*x^2)/(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{c} \sqrt{e} x}{\sqrt{b e-c d}}\right)}{\sqrt{c} \sqrt{e} \sqrt{b e-c d}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right)}{\sqrt{c} \sqrt{e} \sqrt{c d-b e}}",1,"ArcTan[(Sqrt[c]*Sqrt[e]*x)/Sqrt[-(c*d) + b*e]]/(Sqrt[c]*Sqrt[e]*Sqrt[-(c*d) + b*e])","A",1
218,1,133,136,0.2014988,"\int \frac{1}{\left(d+e x^2\right) \left(-c d^2+b d e+b e^2 x^2+c e^2 x^4\right)} \, dx","Integrate[1/((d + e*x^2)*(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4)),x]","\frac{c^{3/2} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{e} x}{\sqrt{b e-c d}}\right)}{\sqrt{e} (b e-2 c d)^2 \sqrt{b e-c d}}+\frac{(b e-4 c d) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{2 d^{3/2} \sqrt{e} (2 c d-b e)^2}-\frac{x}{2 d \left(d+e x^2\right) (2 c d-b e)}","-\frac{c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right)}{\sqrt{e} \sqrt{c d-b e} (2 c d-b e)^2}-\frac{(4 c d-b e) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{2 d^{3/2} \sqrt{e} (2 c d-b e)^2}-\frac{x}{2 d \left(d+e x^2\right) (2 c d-b e)}",1,"-1/2*x/(d*(2*c*d - b*e)*(d + e*x^2)) + ((-4*c*d + b*e)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(3/2)*Sqrt[e]*(2*c*d - b*e)^2) + (c^(3/2)*ArcTan[(Sqrt[c]*Sqrt[e]*x)/Sqrt[-(c*d) + b*e]])/(Sqrt[e]*(-2*c*d + b*e)^2*Sqrt[-(c*d) + b*e])","A",1
219,1,177,187,0.4110636,"\int \frac{1}{\left(d+e x^2\right)^2 \left(-c d^2+b d e+b e^2 x^2+c e^2 x^4\right)} \, dx","Integrate[1/((d + e*x^2)^2*(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4)),x]","\frac{1}{8} \left(-\frac{\left(3 b^2 e^2-16 b c d e+28 c^2 d^2\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{d^{5/2} \sqrt{e} (2 c d-b e)^3}-\frac{\frac{8 c^{5/2} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{e} x}{\sqrt{b e-c d}}\right)}{\sqrt{e} \sqrt{b e-c d}}+\frac{x (b e-2 c d) \left(2 c d \left(7 d+5 e x^2\right)-b e \left(5 d+3 e x^2\right)\right)}{d^2 \left(d+e x^2\right)^2}}{(b e-2 c d)^3}\right)","-\frac{\left(3 b^2 e^2-16 b c d e+28 c^2 d^2\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{8 d^{5/2} \sqrt{e} (2 c d-b e)^3}-\frac{c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right)}{\sqrt{e} \sqrt{c d-b e} (2 c d-b e)^3}-\frac{x (10 c d-3 b e)}{8 d^2 \left(d+e x^2\right) (2 c d-b e)^2}-\frac{x}{4 d \left(d+e x^2\right)^2 (2 c d-b e)}",1,"(-(((28*c^2*d^2 - 16*b*c*d*e + 3*b^2*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(5/2)*Sqrt[e]*(2*c*d - b*e)^3)) - (((-2*c*d + b*e)*x*(-(b*e*(5*d + 3*e*x^2)) + 2*c*d*(7*d + 5*e*x^2)))/(d^2*(d + e*x^2)^2) + (8*c^(5/2)*ArcTan[(Sqrt[c]*Sqrt[e]*x)/Sqrt[-(c*d) + b*e]])/(Sqrt[e]*Sqrt[-(c*d) + b*e]))/(-2*c*d + b*e)^3)/8","A",1
220,1,134,139,0.2574455,"\int \frac{\left(d+e x^2\right)^{5/2}}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx","Integrate[(d + e*x^2)^(5/2)/(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4),x]","-\frac{\frac{(2 b e-5 c d) \log \left(\sqrt{e} \sqrt{d+e x^2}+e x\right)}{\sqrt{e}}-\frac{2 (b e-2 c d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{e} x \sqrt{b e-2 c d}}{\sqrt{d+e x^2} \sqrt{b e-c d}}\right)}{\sqrt{e} \sqrt{b e-c d}}-c x \sqrt{d+e x^2}}{2 c^2}","-\frac{(2 c d-b e)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{e} x \sqrt{2 c d-b e}}{\sqrt{d+e x^2} \sqrt{c d-b e}}\right)}{c^2 \sqrt{e} \sqrt{c d-b e}}+\frac{(5 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,"-1/2*(-(c*x*Sqrt[d + e*x^2]) - (2*(-2*c*d + b*e)^(3/2)*ArcTanh[(Sqrt[e]*Sqrt[-2*c*d + b*e]*x)/(Sqrt[-(c*d) + b*e]*Sqrt[d + e*x^2])])/(Sqrt[e]*Sqrt[-(c*d) + b*e]) + ((-5*c*d + 2*b*e)*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/Sqrt[e])/c^2","A",1
221,1,103,108,0.0862345,"\int \frac{\left(d+e x^2\right)^{3/2}}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx","Integrate[(d + e*x^2)^(3/2)/(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4),x]","-\frac{\frac{\sqrt{b e-2 c d} \tanh ^{-1}\left(\frac{\sqrt{e} x \sqrt{b e-2 c d}}{\sqrt{d+e x^2} \sqrt{b e-c d}}\right)}{\sqrt{b e-c d}}-\log \left(\sqrt{e} \sqrt{d+e x^2}+e x\right)}{c \sqrt{e}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{c \sqrt{e}}-\frac{\sqrt{2 c d-b e} \tanh ^{-1}\left(\frac{\sqrt{e} x \sqrt{2 c d-b e}}{\sqrt{d+e x^2} \sqrt{c d-b e}}\right)}{c \sqrt{e} \sqrt{c d-b e}}",1,"-(((Sqrt[-2*c*d + b*e]*ArcTanh[(Sqrt[e]*Sqrt[-2*c*d + b*e]*x)/(Sqrt[-(c*d) + b*e]*Sqrt[d + e*x^2])])/Sqrt[-(c*d) + b*e] - Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/(c*Sqrt[e]))","A",1
222,1,76,76,0.0653614,"\int \frac{\sqrt{d+e x^2}}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx","Integrate[Sqrt[d + e*x^2]/(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4),x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{e} x \sqrt{2 c d-b e}}{\sqrt{d+e x^2} \sqrt{c d-b e}}\right)}{\sqrt{e} \sqrt{c d-b e} \sqrt{2 c d-b e}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{e} x \sqrt{2 c d-b e}}{\sqrt{d+e x^2} \sqrt{c d-b e}}\right)}{\sqrt{e} \sqrt{c d-b e} \sqrt{2 c d-b e}}",1,"-(ArcTanh[(Sqrt[e]*Sqrt[2*c*d - b*e]*x)/(Sqrt[c*d - b*e]*Sqrt[d + e*x^2])]/(Sqrt[e]*Sqrt[c*d - b*e]*Sqrt[2*c*d - b*e]))","A",1
223,1,418,106,1.04939,"\int \frac{1}{\sqrt{d+e x^2} \left(-c d^2+b d e+b e^2 x^2+c e^2 x^4\right)} \, dx","Integrate[1/(Sqrt[d + e*x^2]*(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4)),x]","-\frac{x \left(-\frac{2 c e x^2 \left(\frac{e x^2 (b e-2 c d)}{\left(d+e x^2\right) (b e-c d)}\right)^{5/2} \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};\frac{e (b e-2 c d) x^2}{(b e-c d) \left(e x^2+d\right)}\right)}{c d-b e}+2 \left(\frac{e x^2 (b e-2 c d)}{\left(d+e x^2\right) (b e-c d)}\right)^{5/2} \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};\frac{e (b e-2 c d) x^2}{(b e-c d) \left(e x^2+d\right)}\right)+\frac{10 c e x^2 \sqrt{\frac{e x^2 (b e-2 c d)}{\left(d+e x^2\right) (b e-c d)}}}{c d-b e}-15 \sqrt{\frac{e x^2 (b e-2 c d)}{\left(d+e x^2\right) (b e-c d)}}-\frac{10 c e x^2 \tanh ^{-1}\left(\sqrt{\frac{e x^2 (b e-2 c d)}{\left(d+e x^2\right) (b e-c d)}}\right)}{c d-b e}+15 \tanh ^{-1}\left(\sqrt{\frac{e x^2 (b e-2 c d)}{\left(d+e x^2\right) (b e-c d)}}\right)\right)}{5 \left(d+e x^2\right)^{3/2} (c d-b e) \left(\frac{e x^2 (b e-2 c d)}{\left(d+e x^2\right) (b e-c d)}\right)^{3/2}}","-\frac{x}{d \sqrt{d+e x^2} (2 c d-b e)}-\frac{c \tanh ^{-1}\left(\frac{\sqrt{e} x \sqrt{2 c d-b e}}{\sqrt{d+e x^2} \sqrt{c d-b e}}\right)}{\sqrt{e} \sqrt{c d-b e} (2 c d-b e)^{3/2}}",1,"-1/5*(x*(-15*Sqrt[(e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2))] + (10*c*e*x^2*Sqrt[(e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2))])/(c*d - b*e) + 15*ArcTanh[Sqrt[(e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2))]] - (10*c*e*x^2*ArcTanh[Sqrt[(e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2))]])/(c*d - b*e) + 2*((e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2)))^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, (e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2))] - (2*c*e*x^2*((e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2)))^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, (e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2))])/(c*d - b*e)))/((c*d - b*e)*((e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2)))^(3/2)*(d + e*x^2)^(3/2))","C",0
224,1,1058,149,4.1360295,"\int \frac{1}{\left(d+e x^2\right)^{3/2} \left(-c d^2+b d e+b e^2 x^2+c e^2 x^4\right)} \, dx","Integrate[1/((d + e*x^2)^(3/2)*(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4)),x]","-\frac{x \left(-\frac{56 c^2 e^2 \left(\frac{e (b e-2 c d) x^2}{(b e-c d) \left(e x^2+d\right)}\right)^{3/2} x^4}{(c d-b e)^2}+\frac{168 c^2 e^2 \tanh ^{-1}\left(\sqrt{\frac{e (b e-2 c d) x^2}{(b e-c d) \left(e x^2+d\right)}}\right) x^4}{(c d-b e)^2}+\frac{36 c^2 e^2 \left(\frac{e (b e-2 c d) x^2}{(b e-c d) \left(e x^2+d\right)}\right)^{7/2} \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};\frac{e (b e-2 c d) x^2}{(b e-c d) \left(e x^2+d\right)}\right) x^4}{(c d-b e)^2}+\frac{12 c^2 e^2 \left(\frac{e (b e-2 c d) x^2}{(b e-c d) \left(e x^2+d\right)}\right)^{7/2} \, _3F_2\left(2,2,\frac{7}{2};1,\frac{9}{2};\frac{e (b e-2 c d) x^2}{(b e-c d) \left(e x^2+d\right)}\right) x^4}{(c d-b e)^2}-\frac{168 c^2 e^2 \sqrt{\frac{e (b e-2 c d) x^2}{(b e-c d) \left(e x^2+d\right)}} x^4}{(c d-b e)^2}+\frac{140 c e \left(\frac{e (b e-2 c d) x^2}{(b e-c d) \left(e x^2+d\right)}\right)^{3/2} x^2}{c d-b e}-\frac{420 c e \tanh ^{-1}\left(\sqrt{\frac{e (b e-2 c d) x^2}{(b e-c d) \left(e x^2+d\right)}}\right) x^2}{c d-b e}-\frac{84 c e \left(\frac{e (b e-2 c d) x^2}{(b e-c d) \left(e x^2+d\right)}\right)^{7/2} \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};\frac{e (b e-2 c d) x^2}{(b e-c d) \left(e x^2+d\right)}\right) x^2}{c d-b e}-\frac{24 c e \left(\frac{e (b e-2 c d) x^2}{(b e-c d) \left(e x^2+d\right)}\right)^{7/2} \, _3F_2\left(2,2,\frac{7}{2};1,\frac{9}{2};\frac{e (b e-2 c d) x^2}{(b e-c d) \left(e x^2+d\right)}\right) x^2}{c d-b e}+\frac{420 c e \sqrt{\frac{e (b e-2 c d) x^2}{(b e-c d) \left(e x^2+d\right)}} x^2}{c d-b e}-105 \left(\frac{e (b e-2 c d) x^2}{(b e-c d) \left(e x^2+d\right)}\right)^{3/2}+315 \tanh ^{-1}\left(\sqrt{\frac{e (b e-2 c d) x^2}{(b e-c d) \left(e x^2+d\right)}}\right)+48 \left(\frac{e (b e-2 c d) x^2}{(b e-c d) \left(e x^2+d\right)}\right)^{7/2} \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};\frac{e (b e-2 c d) x^2}{(b e-c d) \left(e x^2+d\right)}\right)+12 \left(\frac{e (b e-2 c d) x^2}{(b e-c d) \left(e x^2+d\right)}\right)^{7/2} \, _3F_2\left(2,2,\frac{7}{2};1,\frac{9}{2};\frac{e (b e-2 c d) x^2}{(b e-c d) \left(e x^2+d\right)}\right)-315 \sqrt{\frac{e (b e-2 c d) x^2}{(b e-c d) \left(e x^2+d\right)}}\right)}{63 (c d-b e) \left(\frac{e (b e-2 c d) x^2}{(b e-c d) \left(e x^2+d\right)}\right)^{5/2} \left(e x^2+d\right)^{5/2}}","-\frac{c^2 \tanh ^{-1}\left(\frac{\sqrt{e} x \sqrt{2 c d-b e}}{\sqrt{d+e x^2} \sqrt{c d-b e}}\right)}{\sqrt{e} \sqrt{c d-b e} (2 c d-b e)^{5/2}}-\frac{x (7 c d-2 b e)}{3 d^2 \sqrt{d+e x^2} (2 c d-b e)^2}-\frac{x}{3 d \left(d+e x^2\right)^{3/2} (2 c d-b e)}",1,"-1/63*(x*(-315*Sqrt[(e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2))] + (420*c*e*x^2*Sqrt[(e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2))])/(c*d - b*e) - (168*c^2*e^2*x^4*Sqrt[(e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2))])/(c*d - b*e)^2 - 105*((e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2)))^(3/2) + (140*c*e*x^2*((e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2)))^(3/2))/(c*d - b*e) - (56*c^2*e^2*x^4*((e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2)))^(3/2))/(c*d - b*e)^2 + 315*ArcTanh[Sqrt[(e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2))]] - (420*c*e*x^2*ArcTanh[Sqrt[(e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2))]])/(c*d - b*e) + (168*c^2*e^2*x^4*ArcTanh[Sqrt[(e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2))]])/(c*d - b*e)^2 + 48*((e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2)))^(7/2)*Hypergeometric2F1[2, 7/2, 9/2, (e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2))] - (84*c*e*x^2*((e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2)))^(7/2)*Hypergeometric2F1[2, 7/2, 9/2, (e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2))])/(c*d - b*e) + (36*c^2*e^2*x^4*((e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2)))^(7/2)*Hypergeometric2F1[2, 7/2, 9/2, (e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2))])/(c*d - b*e)^2 + 12*((e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2)))^(7/2)*HypergeometricPFQ[{2, 2, 7/2}, {1, 9/2}, (e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2))] - (24*c*e*x^2*((e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2)))^(7/2)*HypergeometricPFQ[{2, 2, 7/2}, {1, 9/2}, (e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2))])/(c*d - b*e) + (12*c^2*e^2*x^4*((e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2)))^(7/2)*HypergeometricPFQ[{2, 2, 7/2}, {1, 9/2}, (e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2))])/(c*d - b*e)^2))/((c*d - b*e)*((e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2)))^(5/2)*(d + e*x^2)^(5/2))","C",0
225,1,169,183,0.3157448,"\int \left(1+x^2\right)^3 \sqrt{1+x^2+x^4} \, dx","Integrate[(1 + x^2)^3*Sqrt[1 + x^2 + x^4],x]","\frac{2 (-1)^{5/6} \left(4 \sqrt{3}+9 i\right) \sqrt{\sqrt[3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} F\left(i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)+26 \sqrt[3]{-1} \sqrt{\sqrt[3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} E\left(i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)+x \left(5 x^{10}+25 x^8+57 x^6+81 x^4+61 x^2+29\right)}{45 \sqrt{x^4+x^2+1}}","\frac{1}{3} \left(x^4+x^2+1\right)^{3/2} x+\frac{2}{45} \left(6 x^2+7\right) \sqrt{x^4+x^2+1} x+\frac{26 \sqrt{x^4+x^2+1} x}{45 \left(x^2+1\right)}+\frac{7 \left(x^2+1\right) \sqrt{\frac{x^4+x^2+1}{\left(x^2+1\right)^2}} F\left(2 \tan ^{-1}(x)|\frac{1}{4}\right)}{15 \sqrt{x^4+x^2+1}}-\frac{26 \left(x^2+1\right) \sqrt{\frac{x^4+x^2+1}{\left(x^2+1\right)^2}} E\left(2 \tan ^{-1}(x)|\frac{1}{4}\right)}{45 \sqrt{x^4+x^2+1}}+\frac{1}{9} \left(x^4+x^2+1\right)^{3/2} x^3",1,"(x*(29 + 61*x^2 + 81*x^4 + 57*x^6 + 25*x^8 + 5*x^10) + 26*(-1)^(1/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticE[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] + 2*(-1)^(5/6)*(9*I + 4*Sqrt[3])*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticF[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)])/(45*Sqrt[1 + x^2 + x^4])","C",1
226,1,162,164,0.1553246,"\int \left(1+x^2\right)^2 \sqrt{1+x^2+x^4} \, dx","Integrate[(1 + x^2)^2*Sqrt[1 + x^2 + x^4],x]","\frac{2 \sqrt[3]{-1} \left(5 \sqrt[3]{-1}-7\right) \sqrt{\sqrt[3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} F\left(i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)+14 \sqrt[3]{-1} \sqrt{\sqrt[3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} E\left(i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)+x \left(3 x^8+12 x^6+23 x^4+20 x^2+11\right)}{21 \sqrt{x^4+x^2+1}}","\frac{1}{7} x \left(x^4+x^2+1\right)^{3/2}+\frac{2}{21} x \left(3 x^2+4\right) \sqrt{x^4+x^2+1}+\frac{2 x \sqrt{x^4+x^2+1}}{3 \left(x^2+1\right)}+\frac{4 \left(x^2+1\right) \sqrt{\frac{x^4+x^2+1}{\left(x^2+1\right)^2}} F\left(2 \tan ^{-1}(x)|\frac{1}{4}\right)}{7 \sqrt{x^4+x^2+1}}-\frac{2 \left(x^2+1\right) \sqrt{\frac{x^4+x^2+1}{\left(x^2+1\right)^2}} E\left(2 \tan ^{-1}(x)|\frac{1}{4}\right)}{3 \sqrt{x^4+x^2+1}}",1,"(x*(11 + 20*x^2 + 23*x^4 + 12*x^6 + 3*x^8) + 14*(-1)^(1/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticE[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] + 2*(-1)^(1/3)*(-7 + 5*(-1)^(1/3))*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticF[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)])/(21*Sqrt[1 + x^2 + x^4])","C",1
227,1,168,145,0.1816317,"\int \left(1+x^2\right) \sqrt{1+x^2+x^4} \, dx","Integrate[(1 + x^2)*Sqrt[1 + x^2 + x^4],x]","\frac{x^7+3 x^5+3 x^3+\frac{3}{2} \sqrt{2+\left(1-i \sqrt{3}\right) x^2} \sqrt{2+\left(1+i \sqrt{3}\right) x^2} F\left(\sin ^{-1}\left(\frac{1}{2} \left(i \sqrt{3} x+x\right)\right)|\frac{1}{2} i \left(i+\sqrt{3}\right)\right)+3 \sqrt[3]{-1} \sqrt{\sqrt[3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} E\left(i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)+2 x}{5 \sqrt{x^4+x^2+1}}","\frac{1}{5} \left(x^2+2\right) \sqrt{x^4+x^2+1} x+\frac{3 \sqrt{x^4+x^2+1} x}{5 \left(x^2+1\right)}+\frac{3 \left(x^2+1\right) \sqrt{\frac{x^4+x^2+1}{\left(x^2+1\right)^2}} F\left(2 \tan ^{-1}(x)|\frac{1}{4}\right)}{5 \sqrt{x^4+x^2+1}}-\frac{3 \left(x^2+1\right) \sqrt{\frac{x^4+x^2+1}{\left(x^2+1\right)^2}} E\left(2 \tan ^{-1}(x)|\frac{1}{4}\right)}{5 \sqrt{x^4+x^2+1}}",1,"(2*x + 3*x^3 + 3*x^5 + x^7 + 3*(-1)^(1/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticE[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] + (3*Sqrt[2 + (1 - I*Sqrt[3])*x^2]*Sqrt[2 + (1 + I*Sqrt[3])*x^2]*EllipticF[ArcSin[(x + I*Sqrt[3]*x)/2], (I/2)*(I + Sqrt[3])])/2)/(5*Sqrt[1 + x^2 + x^4])","C",1
228,1,117,137,0.0948721,"\int \frac{\sqrt{1+x^2+x^4}}{1+x^2} \, dx","Integrate[Sqrt[1 + x^2 + x^4]/(1 + x^2),x]","\frac{\sqrt[3]{-1} \sqrt{\sqrt[3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} \left(-F\left(i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)+E\left(i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)+\sqrt[3]{-1} \Pi \left(\sqrt[3]{-1};i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)\right)}{\sqrt{x^4+x^2+1}}","\frac{\sqrt{x^4+x^2+1} x}{x^2+1}+\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt{x^4+x^2+1}}\right)+\frac{3 \left(x^2+1\right) \sqrt{\frac{x^4+x^2+1}{\left(x^2+1\right)^2}} F\left(2 \tan ^{-1}(x)|\frac{1}{4}\right)}{4 \sqrt{x^4+x^2+1}}-\frac{\left(x^2+1\right) \sqrt{\frac{x^4+x^2+1}{\left(x^2+1\right)^2}} E\left(2 \tan ^{-1}(x)|\frac{1}{4}\right)}{\sqrt{x^4+x^2+1}}",1,"((-1)^(1/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*(EllipticE[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] - EllipticF[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] + (-1)^(1/3)*EllipticPi[(-1)^(1/3), I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)]))/Sqrt[1 + x^2 + x^4]","C",1
229,1,164,49,0.3466859,"\int \frac{\sqrt{1+x^2+x^4}}{\left(1+x^2\right)^2} \, dx","Integrate[Sqrt[1 + x^2 + x^4]/(1 + x^2)^2,x]","\frac{(-1)^{2/3} \sqrt{\sqrt[3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} F\left(i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)+\sqrt[3]{-1} \sqrt{\sqrt[3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} \left(F\left(i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)-E\left(i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)\right)+\frac{x^5+x^3+x}{x^2+1}}{2 \sqrt{x^4+x^2+1}}","\frac{\left(x^2+1\right) \sqrt{\frac{x^4+x^2+1}{\left(x^2+1\right)^2}} E\left(2 \tan ^{-1}(x)|\frac{1}{4}\right)}{2 \sqrt{x^4+x^2+1}}",1,"((x + x^3 + x^5)/(1 + x^2) + (-1)^(2/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticF[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] + (-1)^(1/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*(-EllipticE[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] + EllipticF[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)]))/(2*Sqrt[1 + x^2 + x^4])","C",1
230,1,176,93,0.3038135,"\int \frac{\sqrt{1+x^2+x^4}}{\left(1+x^2\right)^3} \, dx","Integrate[Sqrt[1 + x^2 + x^4]/(1 + x^2)^3,x]","\frac{2 (-1)^{2/3} \sqrt{\sqrt[3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} \Pi \left(\sqrt[3]{-1};i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)+\sqrt[3]{-1} \sqrt{\sqrt[3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} \left(F\left(i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)-E\left(i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)\right)+\frac{x \left(x^2+2\right) \left(x^4+x^2+1\right)}{\left(x^2+1\right)^2}}{4 \sqrt{x^4+x^2+1}}","\frac{\sqrt{x^4+x^2+1} x}{4 \left(x^2+1\right)^2}+\frac{1}{4} \tan ^{-1}\left(\frac{x}{\sqrt{x^4+x^2+1}}\right)+\frac{\left(x^2+1\right) \sqrt{\frac{x^4+x^2+1}{\left(x^2+1\right)^2}} E\left(2 \tan ^{-1}(x)|\frac{1}{4}\right)}{4 \sqrt{x^4+x^2+1}}",1,"((x*(2 + x^2)*(1 + x^2 + x^4))/(1 + x^2)^2 + (-1)^(1/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*(-EllipticE[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] + EllipticF[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)]) + 2*(-1)^(2/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticPi[(-1)^(1/3), I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)])/(4*Sqrt[1 + x^2 + x^4])","C",1
231,1,240,166,0.4075513,"\int \frac{\sqrt{1+x^2+x^4}}{\left(1+x^2\right)^4} \, dx","Integrate[Sqrt[1 + x^2 + x^4]/(1 + x^2)^4,x]","\frac{-(-1)^{2/3} \sqrt{\sqrt[3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} F\left(i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)+3 (-1)^{2/3} \sqrt{\sqrt[3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} \Pi \left(\sqrt[3]{-1};i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)-2 \sqrt[3]{-1} \sqrt{\sqrt[3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} \left(E\left(i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)-F\left(i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)\right)+\frac{x \left(x^4+x^2+1\right) \left(2 x^4+5 x^2+4\right)}{\left(x^2+1\right)^3}}{6 \sqrt{x^4+x^2+1}}","\frac{\sqrt{x^4+x^2+1} x}{6 \left(x^2+1\right)^2}+\frac{\sqrt{x^4+x^2+1} x}{6 \left(x^2+1\right)^3}+\frac{1}{4} \tan ^{-1}\left(\frac{x}{\sqrt{x^4+x^2+1}}\right)-\frac{\left(x^2+1\right) \sqrt{\frac{x^4+x^2+1}{\left(x^2+1\right)^2}} F\left(2 \tan ^{-1}(x)|\frac{1}{4}\right)}{8 \sqrt{x^4+x^2+1}}+\frac{\left(x^2+1\right) \sqrt{\frac{x^4+x^2+1}{\left(x^2+1\right)^2}} E\left(2 \tan ^{-1}(x)|\frac{1}{4}\right)}{3 \sqrt{x^4+x^2+1}}",1,"((x*(1 + x^2 + x^4)*(4 + 5*x^2 + 2*x^4))/(1 + x^2)^3 - 2*(-1)^(1/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*(EllipticE[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] - EllipticF[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)]) - (-1)^(2/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticF[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] + 3*(-1)^(2/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticPi[(-1)^(1/3), I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)])/(6*Sqrt[1 + x^2 + x^4])","C",1
232,1,157,159,0.1704116,"\int \frac{\left(1+x^2\right)^3}{\sqrt{1+x^2+x^4}} \, dx","Integrate[(1 + x^2)^3/Sqrt[1 + x^2 + x^4],x]","\frac{2 \sqrt[3]{-1} \left(2 \sqrt[3]{-1}-7\right) \sqrt{\sqrt[3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} F\left(i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)+14 \sqrt[3]{-1} \sqrt{\sqrt[3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} E\left(i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)+x \left(3 x^6+14 x^4+14 x^2+11\right)}{15 \sqrt{x^4+x^2+1}}","\frac{14 \sqrt{x^4+x^2+1} x}{15 \left(x^2+1\right)}+\frac{11}{15} \sqrt{x^4+x^2+1} x+\frac{3 \left(x^2+1\right) \sqrt{\frac{x^4+x^2+1}{\left(x^2+1\right)^2}} F\left(2 \tan ^{-1}(x)|\frac{1}{4}\right)}{5 \sqrt{x^4+x^2+1}}-\frac{14 \left(x^2+1\right) \sqrt{\frac{x^4+x^2+1}{\left(x^2+1\right)^2}} E\left(2 \tan ^{-1}(x)|\frac{1}{4}\right)}{15 \sqrt{x^4+x^2+1}}+\frac{1}{5} \sqrt{x^4+x^2+1} x^3",1,"(x*(11 + 14*x^2 + 14*x^4 + 3*x^6) + 14*(-1)^(1/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticE[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] + 2*(-1)^(1/3)*(-7 + 2*(-1)^(1/3))*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticF[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)])/(15*Sqrt[1 + x^2 + x^4])","C",1
233,1,143,137,0.1399651,"\int \frac{\left(1+x^2\right)^2}{\sqrt{1+x^2+x^4}} \, dx","Integrate[(1 + x^2)^2/Sqrt[1 + x^2 + x^4],x]","\frac{x^5+x^3+2 \sqrt[3]{-1} \left(\sqrt[3]{-1}-2\right) \sqrt{\sqrt[3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} F\left(i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)+4 \sqrt[3]{-1} \sqrt{\sqrt[3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} E\left(i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)+x}{3 \sqrt{x^4+x^2+1}}","\frac{4 \sqrt{x^4+x^2+1} x}{3 \left(x^2+1\right)}+\frac{1}{3} \sqrt{x^4+x^2+1} x+\frac{\left(x^2+1\right) \sqrt{\frac{x^4+x^2+1}{\left(x^2+1\right)^2}} F\left(2 \tan ^{-1}(x)|\frac{1}{4}\right)}{\sqrt{x^4+x^2+1}}-\frac{4 \left(x^2+1\right) \sqrt{\frac{x^4+x^2+1}{\left(x^2+1\right)^2}} E\left(2 \tan ^{-1}(x)|\frac{1}{4}\right)}{3 \sqrt{x^4+x^2+1}}",1,"(x + x^3 + x^5 + 4*(-1)^(1/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticE[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] + 2*(-1)^(1/3)*(-2 + (-1)^(1/3))*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticF[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)])/(3*Sqrt[1 + x^2 + x^4])","C",1
234,1,94,115,0.0705928,"\int \frac{1+x^2}{\sqrt{1+x^2+x^4}} \, dx","Integrate[(1 + x^2)/Sqrt[1 + x^2 + x^4],x]","\frac{\sqrt[3]{-1} \sqrt{\sqrt[3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} \left(\left(\sqrt[3]{-1}-1\right) F\left(i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)+E\left(i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)\right)}{\sqrt{x^4+x^2+1}}","\frac{\sqrt{x^4+x^2+1} x}{x^2+1}+\frac{\left(x^2+1\right) \sqrt{\frac{x^4+x^2+1}{\left(x^2+1\right)^2}} F\left(2 \tan ^{-1}(x)|\frac{1}{4}\right)}{\sqrt{x^4+x^2+1}}-\frac{\left(x^2+1\right) \sqrt{\frac{x^4+x^2+1}{\left(x^2+1\right)^2}} E\left(2 \tan ^{-1}(x)|\frac{1}{4}\right)}{\sqrt{x^4+x^2+1}}",1,"((-1)^(1/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*(EllipticE[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] + (-1 + (-1)^(1/3))*EllipticF[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)]))/Sqrt[1 + x^2 + x^4]","C",1
235,1,72,69,0.0650985,"\int \frac{1}{\left(1+x^2\right) \sqrt{1+x^2+x^4}} \, dx","Integrate[1/((1 + x^2)*Sqrt[1 + x^2 + x^4]),x]","\frac{(-1)^{2/3} \sqrt{\sqrt[3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} \Pi \left(\sqrt[3]{-1};i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)}{\sqrt{x^4+x^2+1}}","\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt{x^4+x^2+1}}\right)+\frac{\left(x^2+1\right) \sqrt{\frac{x^4+x^2+1}{\left(x^2+1\right)^2}} F\left(2 \tan ^{-1}(x)|\frac{1}{4}\right)}{4 \sqrt{x^4+x^2+1}}",1,"((-1)^(2/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticPi[(-1)^(1/3), I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)])/Sqrt[1 + x^2 + x^4]","C",1
236,1,226,118,0.4049418,"\int \frac{1}{\left(1+x^2\right)^2 \sqrt{1+x^2+x^4}} \, dx","Integrate[1/((1 + x^2)^2*Sqrt[1 + x^2 + x^4]),x]","\frac{-(-1)^{2/3} \sqrt{\sqrt[3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} F\left(i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)+2 (-1)^{2/3} \sqrt{\sqrt[3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} \Pi \left(\sqrt[3]{-1};i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)+\sqrt[3]{-1} \sqrt{\sqrt[3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} \left(F\left(i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)-E\left(i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)\right)+\frac{x^5+x^3+x}{x^2+1}}{2 \sqrt{x^4+x^2+1}}","\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt{x^4+x^2+1}}\right)-\frac{\left(x^2+1\right) \sqrt{\frac{x^4+x^2+1}{\left(x^2+1\right)^2}} F\left(2 \tan ^{-1}(x)|\frac{1}{4}\right)}{4 \sqrt{x^4+x^2+1}}+\frac{\left(x^2+1\right) \sqrt{\frac{x^4+x^2+1}{\left(x^2+1\right)^2}} E\left(2 \tan ^{-1}(x)|\frac{1}{4}\right)}{2 \sqrt{x^4+x^2+1}}",1,"((x + x^3 + x^5)/(1 + x^2) - (-1)^(2/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticF[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] + (-1)^(1/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*(-EllipticE[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] + EllipticF[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)]) + 2*(-1)^(2/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticPi[(-1)^(1/3), I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)])/(2*Sqrt[1 + x^2 + x^4])","C",1
237,1,235,142,0.342553,"\int \frac{1}{\left(1+x^2\right)^3 \sqrt{1+x^2+x^4}} \, dx","Integrate[1/((1 + x^2)^3*Sqrt[1 + x^2 + x^4]),x]","\frac{-2 (-1)^{2/3} \sqrt{\sqrt[3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} F\left(i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)+2 (-1)^{2/3} \sqrt{\sqrt[3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} \Pi \left(\sqrt[3]{-1};i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)-3 \sqrt[3]{-1} \sqrt{\sqrt[3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} \left(E\left(i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)-F\left(i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)\right)+\frac{x \left(3 x^2+4\right) \left(x^4+x^2+1\right)}{\left(x^2+1\right)^2}}{4 \sqrt{x^4+x^2+1}}","\frac{\sqrt{x^4+x^2+1} x}{4 \left(x^2+1\right)^2}+\frac{1}{4} \tan ^{-1}\left(\frac{x}{\sqrt{x^4+x^2+1}}\right)-\frac{\left(x^2+1\right) \sqrt{\frac{x^4+x^2+1}{\left(x^2+1\right)^2}} F\left(2 \tan ^{-1}(x)|\frac{1}{4}\right)}{2 \sqrt{x^4+x^2+1}}+\frac{3 \left(x^2+1\right) \sqrt{\frac{x^4+x^2+1}{\left(x^2+1\right)^2}} E\left(2 \tan ^{-1}(x)|\frac{1}{4}\right)}{4 \sqrt{x^4+x^2+1}}",1,"((x*(4 + 3*x^2)*(1 + x^2 + x^4))/(1 + x^2)^2 - 3*(-1)^(1/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*(EllipticE[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] - EllipticF[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)]) - 2*(-1)^(2/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticF[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] + 2*(-1)^(2/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticPi[(-1)^(1/3), I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)])/(4*Sqrt[1 + x^2 + x^4])","C",1
238,-1,0,144,0,"\int \frac{\left(1+x^2\right)^3}{\left(1+x^2+x^4\right)^{3/2}} \, dx","Integrate[(1 + x^2)^3/(1 + x^2 + x^4)^(3/2),x]","\text{\$Aborted}","\frac{2 \sqrt{x^4+x^2+1} x}{3 \left(x^2+1\right)}-\frac{\left(1-x^2\right) x}{3 \sqrt{x^4+x^2+1}}+\frac{\left(x^2+1\right) \sqrt{\frac{x^4+x^2+1}{\left(x^2+1\right)^2}} F\left(2 \tan ^{-1}(x)|\frac{1}{4}\right)}{\sqrt{x^4+x^2+1}}-\frac{2 \left(x^2+1\right) \sqrt{\frac{x^4+x^2+1}{\left(x^2+1\right)^2}} E\left(2 \tan ^{-1}(x)|\frac{1}{4}\right)}{3 \sqrt{x^4+x^2+1}}",1,"$Aborted","F",-1
239,-1,0,98,0,"\int \frac{\left(1+x^2\right)^2}{\left(1+x^2+x^4\right)^{3/2}} \, dx","Integrate[(1 + x^2)^2/(1 + x^2 + x^4)^(3/2),x]","\text{\$Aborted}","-\frac{2 \sqrt{x^4+x^2+1} x}{3 \left(x^2+1\right)}+\frac{\left(2 x^2+1\right) x}{3 \sqrt{x^4+x^2+1}}+\frac{2 \left(x^2+1\right) \sqrt{\frac{x^4+x^2+1}{\left(x^2+1\right)^2}} E\left(2 \tan ^{-1}(x)|\frac{1}{4}\right)}{3 \sqrt{x^4+x^2+1}}",1,"$Aborted","F",-1
240,-1,0,96,0,"\int \frac{1+x^2}{\left(1+x^2+x^4\right)^{3/2}} \, dx","Integrate[(1 + x^2)/(1 + x^2 + x^4)^(3/2),x]","\text{\$Aborted}","-\frac{\sqrt{x^4+x^2+1} x}{3 \left(x^2+1\right)}+\frac{\left(x^2+2\right) x}{3 \sqrt{x^4+x^2+1}}+\frac{\left(x^2+1\right) \sqrt{\frac{x^4+x^2+1}{\left(x^2+1\right)^2}} E\left(2 \tan ^{-1}(x)|\frac{1}{4}\right)}{3 \sqrt{x^4+x^2+1}}",1,"$Aborted","F",-1
241,1,204,166,0.205134,"\int \frac{1}{\left(1+x^2\right) \left(1+x^2+x^4\right)^{3/2}} \, dx","Integrate[1/((1 + x^2)*(1 + x^2 + x^4)^(3/2)),x]","\frac{-2 x^3+\sqrt[3]{-1} \left(\sqrt[3]{-1}-2\right) \sqrt{\sqrt[3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} F\left(i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)+2 \sqrt[3]{-1} \sqrt{\sqrt[3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} E\left(i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)+3 (-1)^{2/3} \sqrt{\sqrt[3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} \Pi \left(\sqrt[3]{-1};i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)-x}{3 \sqrt{x^4+x^2+1}}","\frac{2 \sqrt{x^4+x^2+1} x}{3 \left(x^2+1\right)}-\frac{\left(2 x^2+1\right) x}{3 \sqrt{x^4+x^2+1}}+\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt{x^4+x^2+1}}\right)+\frac{3 \left(x^2+1\right) \sqrt{\frac{x^4+x^2+1}{\left(x^2+1\right)^2}} F\left(2 \tan ^{-1}(x)|\frac{1}{4}\right)}{4 \sqrt{x^4+x^2+1}}-\frac{2 \left(x^2+1\right) \sqrt{\frac{x^4+x^2+1}{\left(x^2+1\right)^2}} E\left(2 \tan ^{-1}(x)|\frac{1}{4}\right)}{3 \sqrt{x^4+x^2+1}}",1,"(-x - 2*x^3 + 2*(-1)^(1/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticE[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] + (-1)^(1/3)*(-2 + (-1)^(1/3))*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticF[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] + 3*(-1)^(2/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticPi[(-1)^(1/3), I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)])/(3*Sqrt[1 + x^2 + x^4])","C",1
242,1,168,111,0.3846426,"\int \frac{1}{\left(1+x^2\right)^2 \left(1+x^2+x^4\right)^{3/2}} \, dx","Integrate[1/((1 + x^2)^2*(1 + x^2 + x^4)^(3/2)),x]","\frac{-2 x \left(x^2+1\right) \left(x^2+2\right)-\sqrt[3]{-1} \left(x^2+1\right) \sqrt{\sqrt[3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} \left(\left(5 \sqrt[3]{-1}-1\right) F\left(i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)+E\left(i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)-12 \sqrt[3]{-1} \Pi \left(\sqrt[3]{-1};i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)\right)+3 x \left(x^4+x^2+1\right)}{6 \left(x^2+1\right) \sqrt{x^4+x^2+1}}","\frac{\sqrt{x^4+x^2+1} x}{3 \left(x^2+1\right)}-\frac{\left(x^2+2\right) x}{3 \sqrt{x^4+x^2+1}}+\tan ^{-1}\left(\frac{x}{\sqrt{x^4+x^2+1}}\right)+\frac{\left(x^2+1\right) \sqrt{\frac{x^4+x^2+1}{\left(x^2+1\right)^2}} E\left(2 \tan ^{-1}(x)|\frac{1}{4}\right)}{6 \sqrt{x^4+x^2+1}}",1,"(-2*x*(1 + x^2)*(2 + x^2) + 3*x*(1 + x^2 + x^4) - (-1)^(1/3)*(1 + x^2)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*(EllipticE[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] + (-1 + 5*(-1)^(1/3))*EllipticF[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] - 12*(-1)^(1/3)*EllipticPi[(-1)^(1/3), I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)]))/(6*(1 + x^2)*Sqrt[1 + x^2 + x^4])","C",1
243,1,192,190,0.3372245,"\int \frac{1}{\left(1+x^2\right)^3 \left(1+x^2+x^4\right)^{3/2}} \, dx","Integrate[1/((1 + x^2)^3*(1 + x^2 + x^4)^(3/2)),x]","\frac{4 x \left(x^2-1\right) \left(x^2+1\right)^2-\sqrt[3]{-1} \sqrt{\sqrt[3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} \left(x^2+1\right)^2 \left(\left(-9+10 i \sqrt{3}\right) F\left(i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)+19 E\left(i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)-18 \sqrt[3]{-1} \Pi \left(\sqrt[3]{-1};i \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)\right)+15 x \left(x^4+x^2+1\right) \left(x^2+1\right)+3 x \left(x^4+x^2+1\right)}{12 \left(x^2+1\right)^2 \sqrt{x^4+x^2+1}}","-\frac{\sqrt{x^4+x^2+1} x}{3 \left(x^2+1\right)}+\frac{\sqrt{x^4+x^2+1} x}{4 \left(x^2+1\right)^2}-\frac{\left(1-x^2\right) x}{3 \sqrt{x^4+x^2+1}}+\frac{3}{4} \tan ^{-1}\left(\frac{x}{\sqrt{x^4+x^2+1}}\right)-\frac{5 \left(x^2+1\right) \sqrt{\frac{x^4+x^2+1}{\left(x^2+1\right)^2}} F\left(2 \tan ^{-1}(x)|\frac{1}{4}\right)}{4 \sqrt{x^4+x^2+1}}+\frac{19 \left(x^2+1\right) \sqrt{\frac{x^4+x^2+1}{\left(x^2+1\right)^2}} E\left(2 \tan ^{-1}(x)|\frac{1}{4}\right)}{12 \sqrt{x^4+x^2+1}}",1,"(4*x*(-1 + x^2)*(1 + x^2)^2 + 3*x*(1 + x^2 + x^4) + 15*x*(1 + x^2)*(1 + x^2 + x^4) - (-1)^(1/3)*(1 + x^2)^2*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*(19*EllipticE[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] + (-9 + (10*I)*Sqrt[3])*EllipticF[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] - 18*(-1)^(1/3)*EllipticPi[(-1)^(1/3), I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)]))/(12*(1 + x^2)^2*Sqrt[1 + x^2 + x^4])","C",1
244,1,135,135,0.0369742,"\int \left(d+e x^2\right)^4 \left(a+b x^2+c x^4\right) \, dx","Integrate[(d + e*x^2)^4*(a + b*x^2 + c*x^4),x]","\frac{1}{9} e^2 x^9 \left(a e^2+4 b d e+6 c d^2\right)+\frac{2}{7} d e x^7 \left(2 a e^2+3 b d e+2 c d^2\right)+\frac{1}{5} d^2 x^5 \left(6 a e^2+4 b d e+c d^2\right)+\frac{1}{3} d^3 x^3 (4 a e+b d)+a d^4 x+\frac{1}{11} e^3 x^{11} (b e+4 c d)+\frac{1}{13} c e^4 x^{13}","\frac{1}{9} e^2 x^9 \left(e (a e+4 b d)+6 c d^2\right)+\frac{1}{5} d^2 x^5 \left(6 a e^2+4 b d e+c d^2\right)+\frac{2}{7} d e x^7 \left(e (2 a e+3 b d)+2 c d^2\right)+\frac{1}{3} d^3 x^3 (4 a e+b d)+a d^4 x+\frac{1}{11} e^3 x^{11} (b e+4 c d)+\frac{1}{13} c e^4 x^{13}",1,"a*d^4*x + (d^3*(b*d + 4*a*e)*x^3)/3 + (d^2*(c*d^2 + 4*b*d*e + 6*a*e^2)*x^5)/5 + (2*d*e*(2*c*d^2 + 3*b*d*e + 2*a*e^2)*x^7)/7 + (e^2*(6*c*d^2 + 4*b*d*e + a*e^2)*x^9)/9 + (e^3*(4*c*d + b*e)*x^11)/11 + (c*e^4*x^13)/13","A",1
245,1,104,103,0.0276205,"\int \left(d+e x^2\right)^3 \left(a+b x^2+c x^4\right) \, dx","Integrate[(d + e*x^2)^3*(a + b*x^2 + c*x^4),x]","\frac{1}{7} e x^7 \left(a e^2+3 b d e+3 c d^2\right)+\frac{1}{5} d x^5 \left(3 a e^2+3 b d e+c d^2\right)+\frac{1}{3} d^2 x^3 (3 a e+b d)+a d^3 x+\frac{1}{9} e^2 x^9 (b e+3 c d)+\frac{1}{11} c e^3 x^{11}","\frac{1}{7} e x^7 \left(e (a e+3 b d)+3 c d^2\right)+\frac{1}{5} d x^5 \left(3 e (a e+b d)+c d^2\right)+\frac{1}{3} d^2 x^3 (3 a e+b d)+a d^3 x+\frac{1}{9} e^2 x^9 (b e+3 c d)+\frac{1}{11} c e^3 x^{11}",1,"a*d^3*x + (d^2*(b*d + 3*a*e)*x^3)/3 + (d*(c*d^2 + 3*b*d*e + 3*a*e^2)*x^5)/5 + (e*(3*c*d^2 + 3*b*d*e + a*e^2)*x^7)/7 + (e^2*(3*c*d + b*e)*x^9)/9 + (c*e^3*x^11)/11","A",1
246,1,73,73,0.020076,"\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
247,1,42,42,0.008818,"\int \left(d+e x^2\right) \left(a+b x^2+c x^4\right) \, dx","Integrate[(d + e*x^2)*(a + b*x^2 + c*x^4),x]","\frac{1}{3} x^3 (a e+b d)+a d x+\frac{1}{5} x^5 (b e+c d)+\frac{1}{7} c e x^7","\frac{1}{3} x^3 (a e+b d)+a d x+\frac{1}{5} x^5 (b e+c d)+\frac{1}{7} c e x^7",1,"a*d*x + ((b*d + a*e)*x^3)/3 + ((c*d + b*e)*x^5)/5 + (c*e*x^7)/7","A",1
248,1,65,66,0.0516487,"\int \frac{a+b x^2+c x^4}{d+e x^2} \, dx","Integrate[(a + b*x^2 + c*x^4)/(d + e*x^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(a e^2-b d e+c d^2\right)}{\sqrt{d} e^{5/2}}+\frac{x (b e-c d)}{e^2}+\frac{c x^3}{3 e}","\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(a e^2-b d e+c d^2\right)}{\sqrt{d} e^{5/2}}-\frac{x (c d-b e)}{e^2}+\frac{c x^3}{3 e}",1,"((-(c*d) + b*e)*x)/e^2 + (c*x^3)/(3*e) + ((c*d^2 - b*d*e + a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*e^(5/2))","A",1
249,1,88,83,0.0559843,"\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
250,1,110,115,0.0970293,"\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
251,1,142,150,0.1330023,"\int \frac{a+b x^2+c x^4}{\left(d+e x^2\right)^4} \, dx","Integrate[(a + b*x^2 + c*x^4)/(d + e*x^2)^4,x]","\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(e (5 a e+b d)+c d^2\right)}{16 d^{7/2} e^{5/2}}+\frac{x \left(e \left(a e \left(33 d^2+40 d e x^2+15 e^2 x^4\right)+b d \left(-3 d^2+8 d e x^2+3 e^2 x^4\right)\right)+c d^2 \left(-3 d^2-8 d e x^2+3 e^2 x^4\right)\right)}{48 d^3 e^2 \left(d+e x^2\right)^3}","-\frac{x \left(7 c d^2-e (5 a e+b d)\right)}{24 d^2 e^2 \left(d+e x^2\right)^2}+\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(e (5 a e+b d)+c d^2\right)}{16 d^{7/2} e^{5/2}}+\frac{x \left(e (5 a e+b d)+c d^2\right)}{16 d^3 e^2 \left(d+e x^2\right)}+\frac{x \left(a+\frac{d (c d-b e)}{e^2}\right)}{6 d \left(d+e x^2\right)^3}",1,"(x*(c*d^2*(-3*d^2 - 8*d*e*x^2 + 3*e^2*x^4) + e*(b*d*(-3*d^2 + 8*d*e*x^2 + 3*e^2*x^4) + a*e*(33*d^2 + 40*d*e*x^2 + 15*e^2*x^4))))/(48*d^3*e^2*(d + e*x^2)^3) + ((c*d^2 + e*(b*d + 5*a*e))*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(16*d^(7/2)*e^(5/2))","A",1
252,1,223,223,0.0866447,"\int \left(d+e x^2\right)^3 \left(a+b x^2+c x^4\right)^2 \, dx","Integrate[(d + e*x^2)^3*(a + b*x^2 + c*x^4)^2,x]","\frac{1}{7} x^7 \left(a^2 e^3+6 a b d e^2+6 a c d^2 e+3 b^2 d^2 e+2 b c d^3\right)+a^2 d^3 x+\frac{1}{11} e x^{11} \left(2 c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right)+\frac{1}{5} d x^5 \left(6 a b d e+a \left(3 a e^2+2 c d^2\right)+b^2 d^2\right)+\frac{1}{9} x^9 \left(6 c d e (a e+b d)+b e^2 (2 a e+3 b d)+c^2 d^3\right)+\frac{1}{3} a d^2 x^3 (3 a e+2 b d)+\frac{1}{13} c e^2 x^{13} (2 b e+3 c d)+\frac{1}{15} c^2 e^3 x^{15}","\frac{1}{7} x^7 \left(a^2 e^3+6 a b d e^2+6 a c d^2 e+3 b^2 d^2 e+2 b c d^3\right)+a^2 d^3 x+\frac{1}{11} e x^{11} \left(2 c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right)+\frac{1}{5} d x^5 \left(6 a b d e+a \left(3 a e^2+2 c d^2\right)+b^2 d^2\right)+\frac{1}{9} x^9 \left(6 c d e (a e+b d)+b e^2 (2 a e+3 b d)+c^2 d^3\right)+\frac{1}{3} a d^2 x^3 (3 a e+2 b d)+\frac{1}{13} c e^2 x^{13} (2 b e+3 c d)+\frac{1}{15} c^2 e^3 x^{15}",1,"a^2*d^3*x + (a*d^2*(2*b*d + 3*a*e)*x^3)/3 + (d*(b^2*d^2 + 6*a*b*d*e + a*(2*c*d^2 + 3*a*e^2))*x^5)/5 + ((2*b*c*d^3 + 3*b^2*d^2*e + 6*a*c*d^2*e + 6*a*b*d*e^2 + a^2*e^3)*x^7)/7 + ((c^2*d^3 + 6*c*d*e*(b*d + a*e) + b*e^2*(3*b*d + 2*a*e))*x^9)/9 + (e*(3*c^2*d^2 + b^2*e^2 + 2*c*e*(3*b*d + a*e))*x^11)/11 + (c*e^2*(3*c*d + 2*b*e)*x^13)/13 + (c^2*e^3*x^15)/15","A",1
253,1,156,155,0.0540374,"\int \left(d+e x^2\right)^2 \left(a+b x^2+c x^4\right)^2 \, dx","Integrate[(d + e*x^2)^2*(a + b*x^2 + c*x^4)^2,x]","\frac{1}{5} x^5 \left(a^2 e^2+4 a b d e+2 a c d^2+b^2 d^2\right)+a^2 d^2 x+\frac{1}{9} x^9 \left(2 a c e^2+b^2 e^2+4 b c d e+c^2 d^2\right)+\frac{2}{7} x^7 \left(a b e^2+2 a c d e+b^2 d e+b c d^2\right)+\frac{2}{3} a d x^3 (a e+b d)+\frac{2}{11} c e x^{11} (b e+c d)+\frac{1}{13} c^2 e^2 x^{13}","a^2 d^2 x+\frac{1}{9} x^9 \left(2 c e (a e+2 b d)+b^2 e^2+c^2 d^2\right)+\frac{2}{7} x^7 \left(a b e^2+2 a c d e+b^2 d e+b c d^2\right)+\frac{1}{5} x^5 \left(4 a b d e+a \left(a e^2+2 c d^2\right)+b^2 d^2\right)+\frac{2}{3} a d x^3 (a e+b d)+\frac{2}{11} c e x^{11} (b e+c d)+\frac{1}{13} c^2 e^2 x^{13}",1,"a^2*d^2*x + (2*a*d*(b*d + a*e)*x^3)/3 + ((b^2*d^2 + 2*a*c*d^2 + 4*a*b*d*e + a^2*e^2)*x^5)/5 + (2*(b*c*d^2 + b^2*d*e + 2*a*c*d*e + a*b*e^2)*x^7)/7 + ((c^2*d^2 + 4*b*c*d*e + b^2*e^2 + 2*a*c*e^2)*x^9)/9 + (2*c*e*(c*d + b*e)*x^11)/11 + (c^2*e^2*x^13)/13","A",1
254,1,96,96,0.0241777,"\int \left(d+e x^2\right) \left(a+b x^2+c x^4\right)^2 \, dx","Integrate[(d + e*x^2)*(a + b*x^2 + c*x^4)^2,x]","a^2 d x+\frac{1}{7} x^7 \left(2 a c e+b^2 e+2 b c d\right)+\frac{1}{5} x^5 \left(2 a b e+2 a c d+b^2 d\right)+\frac{1}{3} a x^3 (a e+2 b d)+\frac{1}{9} c x^9 (2 b e+c d)+\frac{1}{11} c^2 e x^{11}","a^2 d x+\frac{1}{7} x^7 \left(2 a c e+b^2 e+2 b c d\right)+\frac{1}{5} x^5 \left(2 a b e+2 a c d+b^2 d\right)+\frac{1}{3} a x^3 (a e+2 b d)+\frac{1}{9} c x^9 (2 b e+c d)+\frac{1}{11} c^2 e x^{11}",1,"a^2*d*x + (a*(2*b*d + a*e)*x^3)/3 + ((b^2*d + 2*a*c*d + 2*a*b*e)*x^5)/5 + ((2*b*c*d + b^2*e + 2*a*c*e)*x^7)/7 + (c*(c*d + 2*b*e)*x^9)/9 + (c^2*e*x^11)/11","A",1
255,1,49,49,0.0056251,"\int \left(a+b x^2+c x^4\right)^2 \, dx","Integrate[(a + b*x^2 + c*x^4)^2,x]","a^2 x+\frac{1}{5} x^5 \left(2 a c+b^2\right)+\frac{2}{3} a b x^3+\frac{2}{7} b c x^7+\frac{c^2 x^9}{9}","a^2 x+\frac{1}{5} x^5 \left(2 a c+b^2\right)+\frac{2}{3} a b x^3+\frac{2}{7} b c x^7+\frac{c^2 x^9}{9}",1,"a^2*x + (2*a*b*x^3)/3 + ((b^2 + 2*a*c)*x^5)/5 + (2*b*c*x^7)/7 + (c^2*x^9)/9","A",1
256,1,144,143,0.0660244,"\int \frac{\left(a+b x^2+c x^4\right)^2}{d+e x^2} \, dx","Integrate[(a + b*x^2 + c*x^4)^2/(d + e*x^2),x]","\frac{x^3 \left(2 a c e^2+b^2 e^2-2 b c d e+c^2 d^2\right)}{3 e^3}+\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(a e^2-b d e+c d^2\right)^2}{\sqrt{d} e^{9/2}}+\frac{x (b e-c d) \left(2 a e^2-b d e+c d^2\right)}{e^4}+\frac{c x^5 (2 b e-c d)}{5 e^2}+\frac{c^2 x^7}{7 e}","\frac{x^3 \left(-2 c e (b d-a e)+b^2 e^2+c^2 d^2\right)}{3 e^3}-\frac{x (c d-b e) \left(c d^2-e (b d-2 a e)\right)}{e^4}+\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(a e^2-b d e+c d^2\right)^2}{\sqrt{d} e^{9/2}}-\frac{c x^5 (c d-2 b e)}{5 e^2}+\frac{c^2 x^7}{7 e}",1,"((-(c*d) + b*e)*(c*d^2 - b*d*e + 2*a*e^2)*x)/e^4 + ((c^2*d^2 - 2*b*c*d*e + b^2*e^2 + 2*a*c*e^2)*x^3)/(3*e^3) + (c*(-(c*d) + 2*b*e)*x^5)/(5*e^2) + (c^2*x^7)/(7*e) + ((c*d^2 - b*d*e + a*e^2)^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*e^(9/2))","A",1
257,1,183,166,0.1006178,"\int \frac{\left(a+b x^2+c x^4\right)^2}{\left(d+e x^2\right)^2} \, dx","Integrate[(a + b*x^2 + c*x^4)^2/(d + e*x^2)^2,x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(-e^2 \left(a^2 e^2+2 a b d e-3 b^2 d^2\right)+2 c d^2 e (3 a e-5 b d)+7 c^2 d^4\right)}{2 d^{3/2} e^{9/2}}+\frac{x \left(2 c e (a e-2 b d)+b^2 e^2+3 c^2 d^2\right)}{e^4}+\frac{x \left(e (a e-b d)+c d^2\right)^2}{2 d e^4 \left(d+e x^2\right)}+\frac{2 c x^3 (b e-c d)}{3 e^3}+\frac{c^2 x^5}{5 e^2}","\frac{x \left(-2 c e (2 b d-a e)+b^2 e^2+3 c^2 d^2\right)}{e^4}+\frac{x \left(a e^2-b d e+c d^2\right)^2}{2 d e^4 \left(d+e x^2\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(a e^2-b d e+c d^2\right) \left(7 c d^2-e (a e+3 b d)\right)}{2 d^{3/2} e^{9/2}}-\frac{2 c x^3 (c d-b e)}{3 e^3}+\frac{c^2 x^5}{5 e^2}",1,"((3*c^2*d^2 + b^2*e^2 + 2*c*e*(-2*b*d + a*e))*x)/e^4 + (2*c*(-(c*d) + b*e)*x^3)/(3*e^3) + (c^2*x^5)/(5*e^2) + ((c*d^2 + e*(-(b*d) + a*e))^2*x)/(2*d*e^4*(d + e*x^2)) - ((7*c^2*d^4 + 2*c*d^2*e*(-5*b*d + 3*a*e) - e^2*(-3*b^2*d^2 + 2*a*b*d*e + a^2*e^2))*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(3/2)*e^(9/2))","A",1
258,1,217,201,0.1126166,"\int \frac{\left(a+b x^2+c x^4\right)^2}{\left(d+e x^2\right)^3} \, dx","Integrate[(a + b*x^2 + c*x^4)^2/(d + e*x^2)^3,x]","-\frac{x \left(e^2 \left(-3 a^2 e^2-2 a b d e+5 b^2 d^2\right)-2 c d^2 e (9 b d-5 a e)+13 c^2 d^4\right)}{8 d^2 e^4 \left(d+e x^2\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(e^2 \left(3 a^2 e^2+2 a b d e+3 b^2 d^2\right)+6 c d^2 e (a e-5 b d)+35 c^2 d^4\right)}{8 d^{5/2} e^{9/2}}+\frac{x \left(e (a e-b d)+c d^2\right)^2}{4 d e^4 \left(d+e x^2\right)^2}+\frac{c x (2 b e-3 c d)}{e^4}+\frac{c^2 x^3}{3 e^3}","\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(e^2 \left(3 a^2 e^2+2 a b d e+3 b^2 d^2\right)-6 c d^2 e (5 b d-a e)+35 c^2 d^4\right)}{8 d^{5/2} e^{9/2}}-\frac{x \left(-3 a e^2-5 b d e+13 c d^2\right) \left(a e^2-b d e+c d^2\right)}{8 d^2 e^4 \left(d+e x^2\right)}+\frac{x \left(a e^2-b d e+c d^2\right)^2}{4 d e^4 \left(d+e x^2\right)^2}-\frac{c x (3 c d-2 b e)}{e^4}+\frac{c^2 x^3}{3 e^3}",1,"(c*(-3*c*d + 2*b*e)*x)/e^4 + (c^2*x^3)/(3*e^3) + ((c*d^2 + e*(-(b*d) + a*e))^2*x)/(4*d*e^4*(d + e*x^2)^2) - ((13*c^2*d^4 - 2*c*d^2*e*(9*b*d - 5*a*e) + e^2*(5*b^2*d^2 - 2*a*b*d*e - 3*a^2*e^2))*x)/(8*d^2*e^4*(d + e*x^2)) + ((35*c^2*d^4 + 6*c*d^2*e*(-5*b*d + a*e) + e^2*(3*b^2*d^2 + 2*a*b*d*e + 3*a^2*e^2))*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(5/2)*e^(9/2))","A",1
259,1,267,250,0.1482896,"\int \frac{\left(a+b x^2+c x^4\right)^2}{\left(d+e x^2\right)^4} \, dx","Integrate[(a + b*x^2 + c*x^4)^2/(d + e*x^2)^4,x]","-\frac{x \left(e^2 \left(-5 a^2 e^2-2 a b d e+7 b^2 d^2\right)+2 c d^2 e (7 a e-13 b d)+19 c^2 d^4\right)}{24 d^2 e^4 \left(d+e x^2\right)^2}-\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(-e^2 \left(5 a^2 e^2+2 a b d e+b^2 d^2\right)-2 c d^2 e (a e+5 b d)+35 c^2 d^4\right)}{16 d^{7/2} e^{9/2}}+\frac{x \left(e^2 \left(5 a^2 e^2+2 a b d e+b^2 d^2\right)+2 c d^2 e (a e-11 b d)+29 c^2 d^4\right)}{16 d^3 e^4 \left(d+e x^2\right)}+\frac{x \left(e (a e-b d)+c d^2\right)^2}{6 d e^4 \left(d+e x^2\right)^3}+\frac{c^2 x}{e^4}","-\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(-e^2 \left(5 a^2 e^2+2 a b d e+b^2 d^2\right)-2 c d^2 e (a e+5 b d)+35 c^2 d^4\right)}{16 d^{7/2} e^{9/2}}+\frac{x \left(e^2 \left(5 a^2 e^2+2 a b d e+b^2 d^2\right)-2 c d^2 e (11 b d-a e)+29 c^2 d^4\right)}{16 d^3 e^4 \left(d+e x^2\right)}-\frac{x \left(-5 a e^2-7 b d e+19 c d^2\right) \left(a e^2-b d e+c d^2\right)}{24 d^2 e^4 \left(d+e x^2\right)^2}+\frac{x \left(a e^2-b d e+c d^2\right)^2}{6 d e^4 \left(d+e x^2\right)^3}+\frac{c^2 x}{e^4}",1,"(c^2*x)/e^4 + ((c*d^2 + e*(-(b*d) + a*e))^2*x)/(6*d*e^4*(d + e*x^2)^3) - ((19*c^2*d^4 + 2*c*d^2*e*(-13*b*d + 7*a*e) + e^2*(7*b^2*d^2 - 2*a*b*d*e - 5*a^2*e^2))*x)/(24*d^2*e^4*(d + e*x^2)^2) + ((29*c^2*d^4 + 2*c*d^2*e*(-11*b*d + a*e) + e^2*(b^2*d^2 + 2*a*b*d*e + 5*a^2*e^2))*x)/(16*d^3*e^4*(d + e*x^2)) - ((35*c^2*d^4 - 2*c*d^2*e*(5*b*d + a*e) - e^2*(b^2*d^2 + 2*a*b*d*e + 5*a^2*e^2))*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(16*d^(7/2)*e^(9/2))","A",1
260,1,345,317,0.2253234,"\int \frac{\left(a+b x^2+c x^4\right)^2}{\left(d+e x^2\right)^5} \, dx","Integrate[(a + b*x^2 + c*x^4)^2/(d + e*x^2)^5,x]","\frac{-\frac{3 \sqrt{d} \sqrt{e} x \left(-e^2 \left(35 a^2 e^2+10 a b d e+3 b^2 d^2\right)-2 c d^2 e (3 a e+5 b d)+93 c^2 d^4\right)}{d+e x^2}+3 \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(e^2 \left(35 a^2 e^2+10 a b d e+3 b^2 d^2\right)+2 c d^2 e (3 a e+5 b d)+35 c^2 d^4\right)-\frac{8 d^{5/2} \sqrt{e} x \left(e^2 \left(-7 a^2 e^2-2 a b d e+9 b^2 d^2\right)+2 c d^2 e (9 a e-17 b d)+25 c^2 d^4\right)}{\left(d+e x^2\right)^3}+\frac{2 d^{3/2} \sqrt{e} x \left(e^2 \left(35 a^2 e^2+10 a b d e+3 b^2 d^2\right)+2 c d^2 e (3 a e-59 b d)+163 c^2 d^4\right)}{\left(d+e x^2\right)^2}+\frac{48 d^{7/2} \sqrt{e} x \left(e (a e-b d)+c d^2\right)^2}{\left(d+e x^2\right)^4}}{384 d^{9/2} e^{9/2}}","-\frac{x \left(-e^2 \left(35 a^2 e^2+10 a b d e+3 b^2 d^2\right)-2 c d^2 e (3 a e+5 b d)+93 c^2 d^4\right)}{128 d^4 e^4 \left(d+e x^2\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(e^2 \left(35 a^2 e^2+10 a b d e+3 b^2 d^2\right)+2 c d^2 e (3 a e+5 b d)+35 c^2 d^4\right)}{128 d^{9/2} e^{9/2}}+\frac{x \left(e^2 \left(35 a^2 e^2+10 a b d e+3 b^2 d^2\right)-2 c d^2 e (59 b d-3 a e)+163 c^2 d^4\right)}{192 d^3 e^4 \left(d+e x^2\right)^2}+\frac{x \left(a e^2-b d e+c d^2\right)^2}{8 d e^4 \left(d+e x^2\right)^4}-\frac{x \left(-7 a e^2-9 b d e+25 c d^2\right) \left(a e^2-b d e+c d^2\right)}{48 d^2 e^4 \left(d+e x^2\right)^3}",1,"((48*d^(7/2)*Sqrt[e]*(c*d^2 + e*(-(b*d) + a*e))^2*x)/(d + e*x^2)^4 - (8*d^(5/2)*Sqrt[e]*(25*c^2*d^4 + 2*c*d^2*e*(-17*b*d + 9*a*e) + e^2*(9*b^2*d^2 - 2*a*b*d*e - 7*a^2*e^2))*x)/(d + e*x^2)^3 + (2*d^(3/2)*Sqrt[e]*(163*c^2*d^4 + 2*c*d^2*e*(-59*b*d + 3*a*e) + e^2*(3*b^2*d^2 + 10*a*b*d*e + 35*a^2*e^2))*x)/(d + e*x^2)^2 - (3*Sqrt[d]*Sqrt[e]*(93*c^2*d^4 - 2*c*d^2*e*(5*b*d + 3*a*e) - e^2*(3*b^2*d^2 + 10*a*b*d*e + 35*a^2*e^2))*x)/(d + e*x^2) + 3*(35*c^2*d^4 + 2*c*d^2*e*(5*b*d + 3*a*e) + e^2*(3*b^2*d^2 + 10*a*b*d*e + 35*a^2*e^2))*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(384*d^(9/2)*e^(9/2))","A",1
261,1,88,83,0.0168993,"\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
262,1,88,83,0.0169846,"\int \frac{a+x^2 \left(b+c x^2\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(a + x^2*(b + c*x^2))/(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
263,1,570,459,0.687195,"\int \frac{\left(d+e x^2\right)^4}{a+b x^2+c x^4} \, dx","Integrate[(d + e*x^2)^4/(a + b*x^2 + c*x^4),x]","\frac{e^2 x \left(-c e (a e+4 b d)+b^2 e^2+6 c^2 d^2\right)}{c^3}+\frac{\left(4 c^3 d^2 e \left(d \sqrt{b^2-4 a c}-3 a e-b d\right)+2 c^2 e^2 \left(-3 b d \left(d \sqrt{b^2-4 a c}-2 a e\right)+a e \left(a e-2 d \sqrt{b^2-4 a c}\right)+3 b^2 d^2\right)+2 b c e^3 \left(2 b d \sqrt{b^2-4 a c}+a e \sqrt{b^2-4 a c}-2 a b e-2 b^2 d\right)+b^3 e^4 \left(b-\sqrt{b^2-4 a c}\right)+2 c^4 d^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} c^{7/2} \sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left(-4 c^3 d^2 e \left(d \sqrt{b^2-4 a c}+3 a e+b d\right)+2 c^2 e^2 \left(3 b d \left(d \sqrt{b^2-4 a c}+2 a e\right)+a e \left(2 d \sqrt{b^2-4 a c}+a e\right)+3 b^2 d^2\right)-2 b c e^3 \left(2 b \left(d \sqrt{b^2-4 a c}+a e\right)+a e \sqrt{b^2-4 a c}+2 b^2 d\right)+b^3 e^4 \left(\sqrt{b^2-4 a c}+b\right)+2 c^4 d^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} c^{7/2} \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{e^3 x^3 (4 c d-b e)}{3 c^2}+\frac{e^4 x^5}{5 c}","\frac{\left(\frac{2 c^2 e^2 \left(a^2 e^2+6 a b d e+3 b^2 d^2\right)-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4}{\sqrt{b^2-4 a c}}+e (2 c d-b e) \left(-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} c^{7/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left(e (2 c d-b e) \left(-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right)-\frac{2 c^2 e^2 \left(a^2 e^2+6 a b d e+3 b^2 d^2\right)-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4}{\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} c^{7/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{e^2 x \left(-c e (a e+4 b d)+b^2 e^2+6 c^2 d^2\right)}{c^3}+\frac{e^3 x^3 (4 c d-b e)}{3 c^2}+\frac{e^4 x^5}{5 c}",1,"(e^2*(6*c^2*d^2 + b^2*e^2 - c*e*(4*b*d + a*e))*x)/c^3 + (e^3*(4*c*d - b*e)*x^3)/(3*c^2) + (e^4*x^5)/(5*c) + ((2*c^4*d^4 + b^3*(b - Sqrt[b^2 - 4*a*c])*e^4 + 4*c^3*d^2*e*(-(b*d) + Sqrt[b^2 - 4*a*c]*d - 3*a*e) + 2*b*c*e^3*(-2*b^2*d + 2*b*Sqrt[b^2 - 4*a*c]*d - 2*a*b*e + a*Sqrt[b^2 - 4*a*c]*e) + 2*c^2*e^2*(3*b^2*d^2 - 3*b*d*(Sqrt[b^2 - 4*a*c]*d - 2*a*e) + a*e*(-2*Sqrt[b^2 - 4*a*c]*d + a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(7/2)*Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - ((2*c^4*d^4 + b^3*(b + Sqrt[b^2 - 4*a*c])*e^4 - 4*c^3*d^2*e*(b*d + Sqrt[b^2 - 4*a*c]*d + 3*a*e) - 2*b*c*e^3*(2*b^2*d + a*Sqrt[b^2 - 4*a*c]*e + 2*b*(Sqrt[b^2 - 4*a*c]*d + a*e)) + 2*c^2*e^2*(3*b^2*d^2 + a*e*(2*Sqrt[b^2 - 4*a*c]*d + a*e) + 3*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]*c^(7/2)*Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]])","A",1
264,1,402,316,0.550139,"\int \frac{\left(d+e x^2\right)^3}{a+b x^2+c x^4} \, dx","Integrate[(d + e*x^2)^3/(a + b*x^2 + c*x^4),x]","\frac{\frac{3 \sqrt{2} \left(3 c^2 d e \left(d \sqrt{b^2-4 a c}-2 a e-b d\right)+c e^2 \left(-3 b d \sqrt{b^2-4 a c}-a e \sqrt{b^2-4 a c}+3 a b e+3 b^2 d\right)+b^2 e^3 \left(\sqrt{b^2-4 a c}-b\right)+2 c^3 d^3\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} \left(3 c^2 d e \left(d \sqrt{b^2-4 a c}+2 a e+b d\right)-c e^2 \left(3 b \left(d \sqrt{b^2-4 a c}+a e\right)+a e \sqrt{b^2-4 a c}+3 b^2 d\right)+b^2 e^3 \left(\sqrt{b^2-4 a c}+b\right)-2 c^3 d^3\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}}+6 \sqrt{c} e^2 x (3 c d-b e)+2 c^{3/2} e^3 x^3}{6 c^{5/2}}","\frac{\left(e \left(-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right)+\frac{(2 c d-b e) \left(-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right)}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} c^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left(e \left(-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right)-\frac{(2 c d-b e) \left(-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right)}{\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} c^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{e^2 x (3 c d-b e)}{c^2}+\frac{e^3 x^3}{3 c}",1,"(6*Sqrt[c]*e^2*(3*c*d - b*e)*x + 2*c^(3/2)*e^3*x^3 + (3*Sqrt[2]*(2*c^3*d^3 + b^2*(-b + Sqrt[b^2 - 4*a*c])*e^3 + 3*c^2*d*e*(-(b*d) + Sqrt[b^2 - 4*a*c]*d - 2*a*e) + c*e^2*(3*b^2*d - 3*b*Sqrt[b^2 - 4*a*c]*d + 3*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[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[2]*(-2*c^3*d^3 + b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 + 3*c^2*d*e*(b*d + Sqrt[b^2 - 4*a*c]*d + 2*a*e) - c*e^2*(3*b^2*d + a*Sqrt[b^2 - 4*a*c]*e + 3*b*(Sqrt[b^2 - 4*a*c]*d + a*e)))*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*c^(5/2))","A",1
265,1,269,238,0.3216263,"\int \frac{\left(d+e x^2\right)^2}{a+b x^2+c x^4} \, dx","Integrate[(d + e*x^2)^2/(a + b*x^2 + c*x^4),x]","\frac{\frac{\sqrt{2} \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{\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} \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{\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}}+2 \sqrt{c} e^2 x}{2 c^{3/2}}","\frac{\left(\frac{-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2}{\sqrt{b^2-4 a c}}+e (2 c d-b e)\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(e (2 c d-b e)-\frac{-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2}{\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} c^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{e^2 x}{c}",1,"(2*Sqrt[c]*e^2*x + (Sqrt[2]*(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))*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]*(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))*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]]))/(2*c^(3/2))","A",1
266,1,172,174,0.1390431,"\int \frac{d+e x^2}{a+b x^2+c x^4} \, dx","Integrate[(d + e*x^2)/(a + b*x^2 + c*x^4),x]","\frac{\frac{\left(e \left(\sqrt{b^2-4 a c}-b\right)+2 c d\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(e \left(\sqrt{b^2-4 a c}+b\right)-2 c d\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(\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\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(e-\frac{2 c d-b e}{\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} \sqrt{c} \sqrt{\sqrt{b^2-4 a c}+b}}",1,"(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/Sqrt[b - Sqrt[b^2 - 4*a*c]] + ((-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*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
267,1,129,150,0.085134,"\int \frac{1}{a+b x^2+c x^4} \, dx","Integrate[(a + b*x^2 + c*x^4)^(-1),x]","\frac{\sqrt{2} \sqrt{c} \left(\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{b^2-4 a c}}","\frac{\sqrt{2} \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}",1,"(Sqrt[2]*Sqrt[c]*(ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]]/Sqrt[b - Sqrt[b^2 - 4*a*c]] - ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]]/Sqrt[b + Sqrt[b^2 - 4*a*c]]))/Sqrt[b^2 - 4*a*c]","A",1
268,1,274,254,0.271593,"\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
269,1,354,429,0.7530101,"\int \frac{1}{\left(d+e x^2\right)^2 \left(a+b x^2+c x^4\right)} \, dx","Integrate[1/((d + e*x^2)^2*(a + b*x^2 + c*x^4)),x]","\frac{\frac{\sqrt{2} \sqrt{c} \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{\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(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{\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{e^2 x \left(e (a e-b d)+c d^2\right)}{d \left(d+e x^2\right)}+\frac{e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(e (a e-3 b d)+5 c d^2\right)}{d^{3/2}}}{2 \left(e (a e-b d)+c d^2\right)^2}","\frac{\sqrt{c} \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{\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)^2}-\frac{\sqrt{c} \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{\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)^2}+\frac{e^2 x}{2 d \left(d+e x^2\right) \left(a e^2-b d e+c d^2\right)}+\frac{e^{3/2} (2 c d-b e) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{d} \left(a e^2-b d e+c d^2\right)^2}+\frac{e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{2 d^{3/2} \left(a e^2-b d e+c d^2\right)}",1,"((e^2*(c*d^2 + e*(-(b*d) + a*e))*x)/(d*(d + e*x^2)) + (Sqrt[2]*Sqrt[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))*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*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))*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]]) + (e^(3/2)*(5*c*d^2 + e*(-3*b*d + a*e))*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/d^(3/2))/(2*(c*d^2 + e*(-(b*d) + a*e))^2)","A",1
270,1,540,563,1.6300243,"\int \frac{\left(d+e x^2\right)^3}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(d + e*x^2)^3/(a + b*x^2 + c*x^4)^2,x]","\frac{\frac{2 \sqrt{c} x \left(b \left(-a^2 e^3-3 a c d e \left(d-e x^2\right)+c^2 d^3 x^2\right)+b^2 \left(c d^3-a e^3 x^2\right)+2 a c \left(a e^2 \left(3 d+e x^2\right)-c d^2 \left(d+3 e x^2\right)\right)\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \left(-a b^3 e^3+b^2 \left(a e^3 \sqrt{b^2-4 a c}-3 a c d e^2+c^2 d^3\right)-6 a c \left(a e^2+c d^2\right) \left(e \sqrt{b^2-4 a c}+2 c d\right)+b c \left(c d^2 \left(d \sqrt{b^2-4 a c}+12 a e\right)+a e^2 \left(3 d \sqrt{b^2-4 a c}+8 a e\right)\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} \left(a b^3 e^3+b^2 \left(a e^3 \sqrt{b^2-4 a c}+3 a c d e^2-c^2 d^3\right)+6 a c \left(a e^2+c d^2\right) \left(2 c d-e \sqrt{b^2-4 a c}\right)+b c \left(c d^2 \left(d \sqrt{b^2-4 a c}-12 a e\right)+a e^2 \left(3 d \sqrt{b^2-4 a c}-8 a e\right)\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 c^{3/2}}","\frac{x \left(c \left(-\frac{a b e \left(a e^2+3 c d^2\right)}{c}-2 a d \left(c d^2-3 a e^2\right)+b^2 d^3\right)-x^2 \left(a b^2 e^3-b c d \left(3 a e^2+c d^2\right)+2 a c e \left(3 c d^2-a e^2\right)\right)\right)}{2 a c \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}-\frac{\left(a b^3 e^3-b^2 \left(a e^3 \sqrt{b^2-4 a c}-3 a c d e^2+c^2 d^3\right)+6 a c \left(a e^2+c d^2\right) \left(e \sqrt{b^2-4 a c}+2 c d\right)-b c \left(c d^2 \left(d \sqrt{b^2-4 a c}+12 a e\right)+a e^2 \left(3 d \sqrt{b^2-4 a c}+8 a e\right)\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} a c^{3/2} \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left(a b^3 e^3-b^2 \left(-a e^3 \sqrt{b^2-4 a c}-3 a c d e^2+c^2 d^3\right)+6 a c \left(a e^2+c d^2\right) \left(2 c d-e \sqrt{b^2-4 a c}\right)+b c \left(c d^2 \left(d \sqrt{b^2-4 a c}-12 a e\right)+a e^2 \left(3 d \sqrt{b^2-4 a c}-8 a e\right)\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} a c^{3/2} \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}",1,"((2*Sqrt[c]*x*(b^2*(c*d^3 - a*e^3*x^2) + b*(-(a^2*e^3) + c^2*d^3*x^2 - 3*a*c*d*e*(d - e*x^2)) + 2*a*c*(a*e^2*(3*d + e*x^2) - c*d^2*(d + 3*e*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*(-(a*b^3*e^3) - 6*a*c*(2*c*d + Sqrt[b^2 - 4*a*c]*e)*(c*d^2 + a*e^2) + b^2*(c^2*d^3 - 3*a*c*d*e^2 + a*Sqrt[b^2 - 4*a*c]*e^3) + b*c*(a*e^2*(3*Sqrt[b^2 - 4*a*c]*d + 8*a*e) + c*d^2*(Sqrt[b^2 - 4*a*c]*d + 12*a*e)))*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]*(a*b^3*e^3 + 6*a*c*(2*c*d - Sqrt[b^2 - 4*a*c]*e)*(c*d^2 + a*e^2) + b^2*(-(c^2*d^3) + 3*a*c*d*e^2 + a*Sqrt[b^2 - 4*a*c]*e^3) + b*c*(c*d^2*(Sqrt[b^2 - 4*a*c]*d - 12*a*e) + a*e^2*(3*Sqrt[b^2 - 4*a*c]*d - 8*a*e)))*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*c^(3/2))","A",1
271,1,415,386,1.1113342,"\int \frac{\left(d+e x^2\right)^2}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(d + e*x^2)^2/(a + b*x^2 + c*x^4)^2,x]","\frac{\frac{2 x \left(2 a^2 e^2+a b e \left(e x^2-2 d\right)-2 a c d \left(d+2 e x^2\right)+b^2 d^2+b c d^2 x^2\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \left(b^2 \left(c d^2-a e^2\right)-4 a c \left(e \left(d \sqrt{b^2-4 a c}+a e\right)+3 c d^2\right)+b \left(c d \left(d \sqrt{b^2-4 a c}+8 a e\right)+a e^2 \sqrt{b^2-4 a c}\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{c} \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \left(b^2 \left(a e^2-c d^2\right)+4 a c \left(e \left(a e-d \sqrt{b^2-4 a c}\right)+3 c d^2\right)+b \left(c d \left(d \sqrt{b^2-4 a c}-8 a e\right)+a e^2 \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{c} \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}}{4 a}","\frac{x \left(x^2 \left(a b e^2-4 a c d e+b c d^2\right)-2 a b d e-2 a \left(c d^2-a e^2\right)+b^2 d^2\right)}{2 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\left(\frac{b^2 \left(c d^2-a e^2\right)+8 a b c d e-4 a c \left(a e^2+3 c d^2\right)}{\sqrt{b^2-4 a c}}+a b e^2-4 a c d e+b c d^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} a \sqrt{c} \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left(-\frac{b^2 \left(c d^2-a e^2\right)+8 a b c d e-4 a c \left(a e^2+3 c d^2\right)}{\sqrt{b^2-4 a c}}+a b e^2-4 a c d e+b c d^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} a \sqrt{c} \left(b^2-4 a c\right) \sqrt{\sqrt{b^2-4 a c}+b}}",1,"((2*x*(b^2*d^2 + 2*a^2*e^2 + b*c*d^2*x^2 + a*b*e*(-2*d + e*x^2) - 2*a*c*d*(d + 2*e*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*(b^2*(c*d^2 - a*e^2) - 4*a*c*(3*c*d^2 + e*(Sqrt[b^2 - 4*a*c]*d + a*e)) + b*(a*Sqrt[b^2 - 4*a*c]*e^2 + c*d*(Sqrt[b^2 - 4*a*c]*d + 8*a*e)))*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*(-(c*d^2) + a*e^2) + b*(a*Sqrt[b^2 - 4*a*c]*e^2 + c*d*(Sqrt[b^2 - 4*a*c]*d - 8*a*e)) + 4*a*c*(3*c*d^2 + e*(-(Sqrt[b^2 - 4*a*c]*d) + a*e)))*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)","A",1
272,1,310,293,0.7489426,"\int \frac{d+e x^2}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(d + e*x^2)/(a + b*x^2 + c*x^4)^2,x]","\frac{\frac{2 x \left(b \left(c d x^2-a e\right)-2 a c \left(d+e x^2\right)+b^2 d\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \sqrt{c} \left(b \left(d \sqrt{b^2-4 a c}+4 a e\right)-2 a \left(e \sqrt{b^2-4 a c}+6 c d\right)+b^2 d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left(b d \sqrt{b^2-4 a c}-2 a e \sqrt{b^2-4 a c}-4 a b e+12 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)}{\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 (b d-2 a e)-a b e-2 a c d+b^2 d\right)}{2 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{c} \left(\frac{4 a b e-12 a c d+b^2 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)}{2 \sqrt{2} a \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left(-\frac{4 a b e-12 a c d+b^2 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)}{2 \sqrt{2} a \left(b^2-4 a c\right) \sqrt{\sqrt{b^2-4 a c}+b}}",1,"((2*x*(b^2*d + b*(-(a*e) + c*d*x^2) - 2*a*c*(d + e*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*Sqrt[c]*(b^2*d + b*(Sqrt[b^2 - 4*a*c]*d + 4*a*e) - 2*a*(6*c*d + Sqrt[b^2 - 4*a*c]*e))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(-(b^2*d) + 12*a*c*d + b*Sqrt[b^2 - 4*a*c]*d - 4*a*b*e - 2*a*Sqrt[b^2 - 4*a*c]*e)*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
273,1,243,252,0.4247374,"\int \frac{1}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(a + b*x^2 + c*x^4)^(-2),x]","\frac{\frac{2 x \left(-2 a c+b^2+b c x^2\right)}{\left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \sqrt{c} \left(b \sqrt{b^2-4 a c}-12 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left(b \sqrt{b^2-4 a c}+12 a c-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}}{4 a}","\frac{x \left(-2 a c+b^2+b c x^2\right)}{2 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{c} \left(b \sqrt{b^2-4 a c}-12 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} a \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left(-b \sqrt{b^2-4 a c}-12 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} a \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}",1,"((2*x*(b^2 - 2*a*c + b*c*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*Sqrt[c]*(b^2 - 12*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(-b^2 + 12*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(4*a)","A",1
274,1,708,660,2.7889438,"\int \frac{1}{\left(d+e x^2\right) \left(a+b x^2+c x^4\right)^2} \, dx","Integrate[1/((d + e*x^2)*(a + b*x^2 + c*x^4)^2),x]","\frac{\frac{2 x \left(e (a e-b d)+c d^2\right) \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(e x^2-d\right)\right)}{a \left(4 a c-b^2\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \sqrt{c} \left(b^2 \left(-c d e \left(2 d \sqrt{b^2-4 a c}+3 a e\right)-3 a e^3 \sqrt{b^2-4 a c}+c^2 d^3\right)+2 a c \left(c d e \left(d \sqrt{b^2-4 a c}-14 a e\right)+5 a e^3 \sqrt{b^2-4 a c}-6 c^2 d^3\right)+b c \left(c d^2 \left(d \sqrt{b^2-4 a c}+20 a e\right)+a e^2 \left(16 a e-d \sqrt{b^2-4 a c}\right)\right)+b^3 e \left(e \left(d \sqrt{b^2-4 a c}-3 a e\right)-2 c d^2\right)+b^4 d e^2\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)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left(-b^2 \left(c d e \left(2 d \sqrt{b^2-4 a c}-3 a e\right)+3 a e^3 \sqrt{b^2-4 a c}+c^2 d^3\right)+2 a c \left(c d e \left(d \sqrt{b^2-4 a c}+14 a e\right)+5 a e^3 \sqrt{b^2-4 a c}+6 c^2 d^3\right)+b c \left(c d^2 \left(d \sqrt{b^2-4 a c}-20 a e\right)-a e^2 \left(d \sqrt{b^2-4 a c}+16 a e\right)\right)+b^3 e \left(e \left(d \sqrt{b^2-4 a c}+3 a e\right)+2 c d^2\right)+b^4 (-d) e^2\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)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{4 e^{7/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{d}}}{4 \left(e (a e-b d)+c d^2\right)^2}","-\frac{\sqrt{c} e^2 \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)^2}-\frac{\sqrt{c} e^2 \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)^2}+\frac{x \left(c x^2 \left(2 a c e+b^2 (-e)+b c d\right)+3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d\right)}{2 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right) \left(a e^2-b d e+c d^2\right)}+\frac{\sqrt{c} \left(\frac{8 a b c e-12 a c^2 d+b^3 (-e)+b^2 c d}{\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)}{2 \sqrt{2} a \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}} \left(a e^2-b d e+c d^2\right)}+\frac{\sqrt{c} \left(-\frac{8 a b c e-12 a c^2 d+b^3 (-e)+b^2 c d}{\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)}{2 \sqrt{2} a \left(b^2-4 a c\right) \sqrt{\sqrt{b^2-4 a c}+b} \left(a e^2-b d e+c d^2\right)}+\frac{e^{7/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{d} \left(a e^2-b d e+c d^2\right)^2}",1,"((2*(c*d^2 + e*(-(b*d) + a*e))*x*(b^3*e - b*c*(3*a*e + c*d*x^2) + 2*a*c^2*(d - e*x^2) + b^2*c*(-d + e*x^2)))/(a*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*Sqrt[c]*(b^4*d*e^2 + 2*a*c*(-6*c^2*d^3 + 5*a*Sqrt[b^2 - 4*a*c]*e^3 + c*d*e*(Sqrt[b^2 - 4*a*c]*d - 14*a*e)) + b^3*e*(-2*c*d^2 + e*(Sqrt[b^2 - 4*a*c]*d - 3*a*e)) + b^2*(c^2*d^3 - 3*a*Sqrt[b^2 - 4*a*c]*e^3 - c*d*e*(2*Sqrt[b^2 - 4*a*c]*d + 3*a*e)) + b*c*(a*e^2*(-(Sqrt[b^2 - 4*a*c]*d) + 16*a*e) + c*d^2*(Sqrt[b^2 - 4*a*c]*d + 20*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(-(b^4*d*e^2) - b^2*(c^2*d^3 + 3*a*Sqrt[b^2 - 4*a*c]*e^3 + c*d*e*(2*Sqrt[b^2 - 4*a*c]*d - 3*a*e)) + b^3*e*(2*c*d^2 + e*(Sqrt[b^2 - 4*a*c]*d + 3*a*e)) + 2*a*c*(6*c^2*d^3 + 5*a*Sqrt[b^2 - 4*a*c]*e^3 + c*d*e*(Sqrt[b^2 - 4*a*c]*d + 14*a*e)) + b*c*(c*d^2*(Sqrt[b^2 - 4*a*c]*d - 20*a*e) - a*e^2*(Sqrt[b^2 - 4*a*c]*d + 16*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (4*e^(7/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d])/(4*(c*d^2 + e*(-(b*d) + a*e))^2)","A",1
275,1,1020,1077,5.8429302,"\int \frac{1}{\left(d+e x^2\right)^2 \left(a+b x^2+c x^4\right)^2} \, dx","Integrate[1/((d + e*x^2)^2*(a + b*x^2 + c*x^4)^2),x]","\frac{1}{4} \left(\frac{2 x e^4}{d \left(c d^2+e (a e-b d)\right)^2 \left(e x^2+d\right)}+\frac{2 \left(9 c d^2+e (a e-5 b d)\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) e^{7/2}}{d^{3/2} \left(c d^2+e (a e-b d)\right)^3}-\frac{2 x \left(e^2 b^4+c e \left(e x^2-2 d\right) b^3+c \left(c d \left(d-2 e x^2\right)-4 a e^2\right) b^2+c^2 \left(c d^2 x^2-3 a e \left(e x^2-2 d\right)\right) b+2 a c^2 \left(a e^2-c d \left(d-2 e x^2\right)\right)\right)}{a \left(4 a c-b^2\right) \left(c d^2+e (a e-b d)\right)^2 \left(c x^4+b x^2+a\right)}+\frac{\sqrt{2} \sqrt{c} \left(d e^3 b^5+e^2 \left(e \left(\sqrt{b^2-4 a c} d-5 a e\right)-3 c d^2\right) b^4+e \left(c d-\sqrt{b^2-4 a c} e\right) \left(3 c d^2+5 a e^2\right) b^3+c \left(-c^2 d^4+3 c e \left(\sqrt{b^2-4 a c} d+4 a e\right) d^2+a e^3 \left(7 \sqrt{b^2-4 a c} d+29 a e\right)\right) b^2-c \left(-19 a^2 \sqrt{b^2-4 a c} e^4+2 a c d \left(26 a e-3 \sqrt{b^2-4 a c} d\right) e^2+c^2 d^3 \left(\sqrt{b^2-4 a c} d+28 a e\right)\right) b-4 a c^2 \left(-3 c^2 d^4+c e \left(\sqrt{b^2-4 a c} d-12 a e\right) d^2+a e^3 \left(9 \sqrt{b^2-4 a c} d+7 a e\right)\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)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}} \left(e (b d-a e)-c d^2\right)^3}-\frac{\sqrt{2} \sqrt{c} \left(d e^3 b^5-e^2 \left(3 c d^2+e \left(\sqrt{b^2-4 a c} d+5 a e\right)\right) b^4+e \left(c d+\sqrt{b^2-4 a c} e\right) \left(3 c d^2+5 a e^2\right) b^3-c \left(c^2 d^4+3 c e \left(\sqrt{b^2-4 a c} d-4 a e\right) d^2+a e^3 \left(7 \sqrt{b^2-4 a c} d-29 a e\right)\right) b^2+c \left(-19 a^2 \sqrt{b^2-4 a c} e^4-2 a c d \left(3 \sqrt{b^2-4 a c} d+26 a e\right) e^2+c^2 d^3 \left(\sqrt{b^2-4 a c} d-28 a e\right)\right) b+4 a c^2 \left(3 c^2 d^4+c e \left(\sqrt{b^2-4 a c} d+12 a e\right) d^2+a e^3 \left(9 \sqrt{b^2-4 a c} d-7 a e\right)\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)^{3/2} \sqrt{b+\sqrt{b^2-4 a c}} \left(e (b d-a e)-c d^2\right)^3}\right)","\frac{x e^4}{2 d \left(c d^2-b e d+a e^2\right)^2 \left(e x^2+d\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) e^{7/2}}{2 d^{3/2} \left(c d^2-b e d+a e^2\right)^2}+\frac{2 (2 c d-b e) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) e^{7/2}}{\sqrt{d} \left(c d^2-b e d+a e^2\right)^3}+\frac{\sqrt{2} \sqrt{c} \left(3 c^2 d^2+b \left(b+\sqrt{b^2-4 a c}\right) e^2-c e \left(3 b d+2 \sqrt{b^2-4 a c} d+a e\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) e^2}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}} \left(c d^2-b e d+a e^2\right)^3}-\frac{\sqrt{2} \sqrt{c} \left(3 c^2 d^2+b \left(b-\sqrt{b^2-4 a c}\right) e^2-c e \left(3 b d-2 \sqrt{b^2-4 a c} d+a e\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right) e^2}{\sqrt{b^2-4 a c} \sqrt{b+\sqrt{b^2-4 a c}} \left(c d^2-b e d+a e^2\right)^3}+\frac{\sqrt{c} \left(e^2 b^4-e \left(2 c d-\sqrt{b^2-4 a c} e\right) b^3+c \left(c d^2-e \left(2 \sqrt{b^2-4 a c} d+9 a e\right)\right) b^2-c \left(3 a \sqrt{b^2-4 a c} e^2-c d \left(\sqrt{b^2-4 a c} d+16 a e\right)\right) b-4 a c^2 \left(3 c d^2-e \left(\sqrt{b^2-4 a c} d+3 a e\right)\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} a \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}} \left(c d^2-b e d+a e^2\right)^2}-\frac{\sqrt{c} \left(e^2 b^4-e \left(2 c d+\sqrt{b^2-4 a c} e\right) b^3+c \left(c d^2+e \left(2 \sqrt{b^2-4 a c} d-9 a e\right)\right) b^2+c \left(3 a \sqrt{b^2-4 a c} e^2-c d \left(\sqrt{b^2-4 a c} d-16 a e\right)\right) b-4 a c^2 \left(3 c d^2+e \left(\sqrt{b^2-4 a c} d-3 a e\right)\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} a \left(b^2-4 a c\right)^{3/2} \sqrt{b+\sqrt{b^2-4 a c}} \left(c d^2-b e d+a e^2\right)^2}+\frac{x \left(-c \left(-e^2 b^3+2 c d e b^2-c \left(c d^2-3 a e^2\right) b-4 a c^2 d e\right) x^2+a b c e (2 c d-b e)+\left(b^2-2 a c\right) \left(c^2 d^2+b^2 e^2-c e (2 b d+a e)\right)\right)}{2 a \left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right)^2 \left(c x^4+b x^2+a\right)}",1,"((2*e^4*x)/(d*(c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x^2)) - (2*x*(b^4*e^2 + b^3*c*e*(-2*d + e*x^2) + 2*a*c^2*(a*e^2 - c*d*(d - 2*e*x^2)) + b^2*c*(-4*a*e^2 + c*d*(d - 2*e*x^2)) + b*c^2*(c*d^2*x^2 - 3*a*e*(-2*d + e*x^2))))/(a*(-b^2 + 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))^2*(a + b*x^2 + c*x^4)) + (Sqrt[2]*Sqrt[c]*(b^5*d*e^3 + b^3*e*(c*d - Sqrt[b^2 - 4*a*c]*e)*(3*c*d^2 + 5*a*e^2) + b^4*e^2*(-3*c*d^2 + e*(Sqrt[b^2 - 4*a*c]*d - 5*a*e)) - 4*a*c^2*(-3*c^2*d^4 + c*d^2*e*(Sqrt[b^2 - 4*a*c]*d - 12*a*e) + a*e^3*(9*Sqrt[b^2 - 4*a*c]*d + 7*a*e)) - b*c*(-19*a^2*Sqrt[b^2 - 4*a*c]*e^4 + 2*a*c*d*e^2*(-3*Sqrt[b^2 - 4*a*c]*d + 26*a*e) + c^2*d^3*(Sqrt[b^2 - 4*a*c]*d + 28*a*e)) + b^2*c*(-(c^2*d^4) + 3*c*d^2*e*(Sqrt[b^2 - 4*a*c]*d + 4*a*e) + a*e^3*(7*Sqrt[b^2 - 4*a*c]*d + 29*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]*(-(c*d^2) + e*(b*d - a*e))^3) - (Sqrt[2]*Sqrt[c]*(b^5*d*e^3 + b^3*e*(c*d + Sqrt[b^2 - 4*a*c]*e)*(3*c*d^2 + 5*a*e^2) - b^2*c*(c^2*d^4 + a*e^3*(7*Sqrt[b^2 - 4*a*c]*d - 29*a*e) + 3*c*d^2*e*(Sqrt[b^2 - 4*a*c]*d - 4*a*e)) - b^4*e^2*(3*c*d^2 + e*(Sqrt[b^2 - 4*a*c]*d + 5*a*e)) + 4*a*c^2*(3*c^2*d^4 + a*e^3*(9*Sqrt[b^2 - 4*a*c]*d - 7*a*e) + c*d^2*e*(Sqrt[b^2 - 4*a*c]*d + 12*a*e)) + b*c*(-19*a^2*Sqrt[b^2 - 4*a*c]*e^4 + c^2*d^3*(Sqrt[b^2 - 4*a*c]*d - 28*a*e) - 2*a*c*d*e^2*(3*Sqrt[b^2 - 4*a*c]*d + 26*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]*(-(c*d^2) + e*(b*d - a*e))^3) + (2*e^(7/2)*(9*c*d^2 + e*(-5*b*d + a*e))*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(3/2)*(c*d^2 + e*(-(b*d) + a*e))^3))/4","A",1
276,1,190,215,0.3882484,"\int \left(d+e x^2\right)^{5/2} \left(a+b x^2+c x^4\right) \, dx","Integrate[(d + e*x^2)^(5/2)*(a + b*x^2 + c*x^4),x]","\frac{\sqrt{d+e x^2} \left(\frac{15 d^{5/2} \sinh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(10 e (8 a e-b d)+3 c d^2\right)}{\sqrt{\frac{e x^2}{d}+1}}+\sqrt{e} x \left(10 e \left(8 a e \left(33 d^2+26 d e x^2+8 e^2 x^4\right)+b \left(15 d^3+118 d^2 e x^2+136 d e^2 x^4+48 e^3 x^6\right)\right)+c \left(-45 d^4+30 d^3 e x^2+744 d^2 e^2 x^4+1008 d e^3 x^6+384 e^4 x^8\right)\right)\right)}{3840 e^{5/2}}","\frac{x \left(d+e x^2\right)^{5/2} \left(80 a e^2-10 b d e+3 c d^2\right)}{480 e^2}+\frac{d x \left(d+e x^2\right)^{3/2} \left(80 a e^2-10 b d e+3 c d^2\right)}{384 e^2}+\frac{d^2 x \sqrt{d+e x^2} \left(80 a e^2-10 b d e+3 c d^2\right)}{256 e^2}+\frac{d^3 \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right) \left(80 a e^2-10 b d e+3 c d^2\right)}{256 e^{5/2}}-\frac{x \left(d+e x^2\right)^{7/2} (3 c d-10 b e)}{80 e^2}+\frac{c x^3 \left(d+e x^2\right)^{7/2}}{10 e}",1,"(Sqrt[d + e*x^2]*(Sqrt[e]*x*(c*(-45*d^4 + 30*d^3*e*x^2 + 744*d^2*e^2*x^4 + 1008*d*e^3*x^6 + 384*e^4*x^8) + 10*e*(8*a*e*(33*d^2 + 26*d*e*x^2 + 8*e^2*x^4) + b*(15*d^3 + 118*d^2*e*x^2 + 136*d*e^2*x^4 + 48*e^3*x^6))) + (15*d^(5/2)*(3*c*d^2 + 10*e*(-(b*d) + 8*a*e))*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[1 + (e*x^2)/d]))/(3840*e^(5/2))","A",1
277,1,157,175,0.3199335,"\int \left(d+e x^2\right)^{3/2} \left(a+b x^2+c x^4\right) \, dx","Integrate[(d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4),x]","\frac{\sqrt{d+e x^2} \left(\frac{3 d^{3/2} \sinh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(8 e (6 a e-b d)+3 c d^2\right)}{\sqrt{\frac{e x^2}{d}+1}}+\sqrt{e} x \left(8 e \left(6 a e \left(5 d+2 e x^2\right)+b \left(3 d^2+14 d e x^2+8 e^2 x^4\right)\right)+c \left(-9 d^3+6 d^2 e x^2+72 d e^2 x^4+48 e^3 x^6\right)\right)\right)}{384 e^{5/2}}","\frac{x \left(d+e x^2\right)^{3/2} \left(48 a e^2-8 b d e+3 c d^2\right)}{192 e^2}+\frac{d x \sqrt{d+e x^2} \left(48 a e^2-8 b d e+3 c d^2\right)}{128 e^2}+\frac{d^2 \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right) \left(48 a e^2-8 b d e+3 c d^2\right)}{128 e^{5/2}}-\frac{x \left(d+e x^2\right)^{5/2} (3 c d-8 b e)}{48 e^2}+\frac{c x^3 \left(d+e x^2\right)^{5/2}}{8 e}",1,"(Sqrt[d + e*x^2]*(Sqrt[e]*x*(c*(-9*d^3 + 6*d^2*e*x^2 + 72*d*e^2*x^4 + 48*e^3*x^6) + 8*e*(6*a*e*(5*d + 2*e*x^2) + b*(3*d^2 + 14*d*e*x^2 + 8*e^2*x^4))) + (3*d^(3/2)*(3*c*d^2 + 8*e*(-(b*d) + 6*a*e))*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[1 + (e*x^2)/d]))/(384*e^(5/2))","A",1
278,1,121,132,0.2337604,"\int \sqrt{d+e x^2} \left(a+b x^2+c x^4\right) \, dx","Integrate[Sqrt[d + e*x^2]*(a + b*x^2 + c*x^4),x]","\frac{\sqrt{d+e x^2} \left(\frac{3 \sqrt{d} \sinh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(8 a e^2-2 b d e+c d^2\right)}{\sqrt{\frac{e x^2}{d}+1}}+\sqrt{e} x \left(6 e \left(4 a e+b \left(d+2 e x^2\right)\right)+c \left(-3 d^2+2 d e x^2+8 e^2 x^4\right)\right)\right)}{48 e^{5/2}}","\frac{x \sqrt{d+e x^2} \left(8 a e^2-2 b d e+c d^2\right)}{16 e^2}+\frac{d \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right) \left(8 a e^2-2 b d e+c d^2\right)}{16 e^{5/2}}-\frac{x \left(d+e x^2\right)^{3/2} (c d-2 b e)}{8 e^2}+\frac{c x^3 \left(d+e x^2\right)^{3/2}}{6 e}",1,"(Sqrt[d + e*x^2]*(Sqrt[e]*x*(c*(-3*d^2 + 2*d*e*x^2 + 8*e^2*x^4) + 6*e*(4*a*e + b*(d + 2*e*x^2))) + (3*Sqrt[d]*(c*d^2 - 2*b*d*e + 8*a*e^2)*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[1 + (e*x^2)/d]))/(48*e^(5/2))","A",1
279,1,82,97,0.0638318,"\int \frac{a+b x^2+c x^4}{\sqrt{d+e x^2}} \, dx","Integrate[(a + b*x^2 + c*x^4)/Sqrt[d + e*x^2],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right) \left(8 a e^2-4 b d e+3 c d^2\right)+\sqrt{e} x \sqrt{d+e x^2} \left(4 b e-3 c d+2 c e x^2\right)}{8 e^{5/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right) \left(8 a e^2-4 b d e+3 c d^2\right)}{8 e^{5/2}}-\frac{x \sqrt{d+e x^2} (3 c d-4 b e)}{8 e^2}+\frac{c x^3 \sqrt{d+e x^2}}{4 e}",1,"(Sqrt[e]*x*Sqrt[d + e*x^2]*(-3*c*d + 4*b*e + 2*c*e*x^2) + (3*c*d^2 - 4*b*d*e + 8*a*e^2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(8*e^(5/2))","A",1
280,1,98,89,0.1051349,"\int \frac{a+b x^2+c x^4}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(a + b*x^2 + c*x^4)/(d + e*x^2)^(3/2),x]","\frac{\sqrt{e} x \left(2 e (a e-b d)+c d \left(3 d+e x^2\right)\right)-d^{3/2} \sqrt{\frac{e x^2}{d}+1} (3 c d-2 b e) \sinh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{2 d e^{5/2} \sqrt{d+e x^2}}","\frac{x \left(a+\frac{d (c d-b e)}{e^2}\right)}{d \sqrt{d+e x^2}}-\frac{(3 c d-2 b e) \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{2 e^{5/2}}+\frac{c x \sqrt{d+e x^2}}{2 e^2}",1,"(Sqrt[e]*x*(2*e*(-(b*d) + a*e) + c*d*(3*d + e*x^2)) - d^(3/2)*(3*c*d - 2*b*e)*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])/(2*d*e^(5/2)*Sqrt[d + e*x^2])","A",1
281,1,112,101,0.1900485,"\int \frac{a+b x^2+c x^4}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(a + b*x^2 + c*x^4)/(d + e*x^2)^(5/2),x]","\frac{\sqrt{e} x \left(e^2 \left(3 a d+2 a e x^2+b d x^2\right)-c d^2 \left(3 d+4 e x^2\right)\right)+3 c d^{5/2} \left(d+e x^2\right) \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{3 d^2 e^{5/2} \left(d+e x^2\right)^{3/2}}","-\frac{x \left(4 c d^2-e (2 a e+b d)\right)}{3 d^2 e^2 \sqrt{d+e x^2}}+\frac{x \left(a+\frac{d (c d-b e)}{e^2}\right)}{3 d \left(d+e x^2\right)^{3/2}}+\frac{c \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{e^{5/2}}",1,"(Sqrt[e]*x*(-(c*d^2*(3*d + 4*e*x^2)) + e^2*(3*a*d + b*d*x^2 + 2*a*e*x^2)) + 3*c*d^(5/2)*(d + e*x^2)*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])/(3*d^2*e^(5/2)*(d + e*x^2)^(3/2))","A",1
282,1,67,86,0.05171,"\int \frac{a+b x^2+c x^4}{\left(d+e x^2\right)^{7/2}} \, dx","Integrate[(a + b*x^2 + c*x^4)/(d + e*x^2)^(7/2),x]","\frac{a \left(15 d^2 x+20 d e x^3+8 e^2 x^5\right)+d x^3 \left(5 b d+2 b e x^2+3 c d x^2\right)}{15 d^3 \left(d+e x^2\right)^{5/2}}","\frac{x^5 \left(2 e (4 a e+b d)+3 c d^2\right)}{15 d^3 \left(d+e x^2\right)^{5/2}}+\frac{x^3 (4 a e+b d)}{3 d^2 \left(d+e x^2\right)^{5/2}}+\frac{a x}{d \left(d+e x^2\right)^{5/2}}",1,"(d*x^3*(5*b*d + 3*c*d*x^2 + 2*b*e*x^2) + a*(15*d^2*x + 20*d*e*x^3 + 8*e^2*x^5))/(15*d^3*(d + e*x^2)^(5/2))","A",1
283,1,101,126,0.0929931,"\int \frac{a+b x^2+c x^4}{\left(d+e x^2\right)^{9/2}} \, dx","Integrate[(a + b*x^2 + c*x^4)/(d + e*x^2)^(9/2),x]","\frac{3 a \left(35 d^3 x+70 d^2 e x^3+56 d e^2 x^5+16 e^3 x^7\right)+d x^3 \left(b \left(35 d^2+28 d e x^2+8 e^2 x^4\right)+3 c d x^2 \left(7 d+2 e x^2\right)\right)}{105 d^4 \left(d+e x^2\right)^{7/2}}","\frac{2 e x^7 \left(4 e (6 a e+b d)+3 c d^2\right)}{105 d^4 \left(d+e x^2\right)^{7/2}}+\frac{x^5 \left(4 e (6 a e+b d)+3 c d^2\right)}{15 d^3 \left(d+e x^2\right)^{7/2}}+\frac{x^3 (6 a e+b d)}{3 d^2 \left(d+e x^2\right)^{7/2}}+\frac{a x}{d \left(d+e x^2\right)^{7/2}}",1,"(3*a*(35*d^3*x + 70*d^2*e*x^3 + 56*d*e^2*x^5 + 16*e^3*x^7) + d*x^3*(3*c*d*x^2*(7*d + 2*e*x^2) + b*(35*d^2 + 28*d*e*x^2 + 8*e^2*x^4)))/(105*d^4*(d + e*x^2)^(7/2))","A",1
284,1,132,165,0.1170973,"\int \frac{a+b x^2+c x^4}{\left(d+e x^2\right)^{11/2}} \, dx","Integrate[(a + b*x^2 + c*x^4)/(d + e*x^2)^(11/2),x]","\frac{a \left(315 d^4 x+840 d^3 e x^3+1008 d^2 e^2 x^5+576 d e^3 x^7+128 e^4 x^9\right)+d x^3 \left(b \left(105 d^3+126 d^2 e x^2+72 d e^2 x^4+16 e^3 x^6\right)+c d x^2 \left(63 d^2+36 d e x^2+8 e^2 x^4\right)\right)}{315 d^5 \left(d+e x^2\right)^{9/2}}","\frac{8 e^2 x^9 \left(2 e (8 a e+b d)+c d^2\right)}{315 d^5 \left(d+e x^2\right)^{9/2}}+\frac{4 e x^7 \left(2 e (8 a e+b d)+c d^2\right)}{35 d^4 \left(d+e x^2\right)^{9/2}}+\frac{x^5 \left(2 e (8 a e+b d)+c d^2\right)}{5 d^3 \left(d+e x^2\right)^{9/2}}+\frac{x^3 (8 a e+b d)}{3 d^2 \left(d+e x^2\right)^{9/2}}+\frac{a x}{d \left(d+e x^2\right)^{9/2}}",1,"(a*(315*d^4*x + 840*d^3*e*x^3 + 1008*d^2*e^2*x^5 + 576*d*e^3*x^7 + 128*e^4*x^9) + d*x^3*(c*d*x^2*(63*d^2 + 36*d*e*x^2 + 8*e^2*x^4) + b*(105*d^3 + 126*d^2*e*x^2 + 72*d*e^2*x^4 + 16*e^3*x^6)))/(315*d^5*(d + e*x^2)^(9/2))","A",1
285,1,167,210,0.1418234,"\int \frac{a+b x^2+c x^4}{\left(d+e x^2\right)^{13/2}} \, dx","Integrate[(a + b*x^2 + c*x^4)/(d + e*x^2)^(13/2),x]","\frac{5 a \left(693 d^5 x+2310 d^4 e x^3+3696 d^3 e^2 x^5+3168 d^2 e^3 x^7+1408 d e^4 x^9+256 e^5 x^{11}\right)+d x^3 \left(b \left(1155 d^4+1848 d^3 e x^2+1584 d^2 e^2 x^4+704 d e^3 x^6+128 e^4 x^8\right)+3 c d x^2 \left(231 d^3+198 d^2 e x^2+88 d e^2 x^4+16 e^3 x^6\right)\right)}{3465 d^6 \left(d+e x^2\right)^{11/2}}","\frac{16 e^3 x^{11} \left(8 e (10 a e+b d)+3 c d^2\right)}{3465 d^6 \left(d+e x^2\right)^{11/2}}+\frac{8 e^2 x^9 \left(8 e (10 a e+b d)+3 c d^2\right)}{315 d^5 \left(d+e x^2\right)^{11/2}}+\frac{2 e x^7 \left(8 e (10 a e+b d)+3 c d^2\right)}{35 d^4 \left(d+e x^2\right)^{11/2}}+\frac{x^5 \left(8 e (10 a e+b d)+3 c d^2\right)}{15 d^3 \left(d+e x^2\right)^{11/2}}+\frac{x^3 (10 a e+b d)}{3 d^2 \left(d+e x^2\right)^{11/2}}+\frac{a x}{d \left(d+e x^2\right)^{11/2}}",1,"(5*a*(693*d^5*x + 2310*d^4*e*x^3 + 3696*d^3*e^2*x^5 + 3168*d^2*e^3*x^7 + 1408*d*e^4*x^9 + 256*e^5*x^11) + d*x^3*(3*c*d*x^2*(231*d^3 + 198*d^2*e*x^2 + 88*d*e^2*x^4 + 16*e^3*x^6) + b*(1155*d^4 + 1848*d^3*e*x^2 + 1584*d^2*e^2*x^4 + 704*d*e^3*x^6 + 128*e^4*x^8)))/(3465*d^6*(d + e*x^2)^(11/2))","A",1
286,1,119,193,0.1024526,"\int \left(7+5 x^2\right)^3 \sqrt{2+3 x^2+x^4} \, dx","Integrate[(7 + 5*x^2)^3*Sqrt[2 + 3*x^2 + x^4],x]","\frac{875 x^{11}+7725 x^9+28496 x^7+57312 x^5+61214 x^3-5553 i \sqrt{x^2+1} \sqrt{x^2+2} F\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)-12117 i \sqrt{x^2+1} \sqrt{x^2+2} E\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+25548 x}{63 \sqrt{x^4+3 x^2+2}}","\frac{275}{7} \left(x^4+3 x^2+2\right)^{3/2} x+\frac{1}{21} \left(757 x^2+2608\right) \sqrt{x^4+3 x^2+2} x+\frac{577 \left(x^2+2\right) x}{3 \sqrt{x^4+3 x^2+2}}+\frac{2945 \sqrt{2} \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{21 \sqrt{x^4+3 x^2+2}}-\frac{577 \sqrt{2} \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{3 \sqrt{x^4+3 x^2+2}}+\frac{125}{9} \left(x^4+3 x^2+2\right)^{3/2} x^3",1,"(25548*x + 61214*x^3 + 57312*x^5 + 28496*x^7 + 7725*x^9 + 875*x^11 - (12117*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - (5553*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt[2]], 2])/(63*Sqrt[2 + 3*x^2 + x^4])","C",1
287,1,114,168,0.079187,"\int \left(7+5 x^2\right)^2 \sqrt{2+3 x^2+x^4} \, dx","Integrate[(7 + 5*x^2)^2*Sqrt[2 + 3*x^2 + x^4],x]","\frac{75 x^9+564 x^7+1724 x^5+2349 x^3-293 i \sqrt{x^2+1} \sqrt{x^2+2} F\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)-651 i \sqrt{x^2+1} \sqrt{x^2+2} E\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+1114 x}{21 \sqrt{x^4+3 x^2+2}}","\frac{25}{7} x \left(x^4+3 x^2+2\right)^{3/2}+\frac{1}{21} x \left(114 x^2+407\right) \sqrt{x^4+3 x^2+2}+\frac{31 x \left(x^2+2\right)}{\sqrt{x^4+3 x^2+2}}+\frac{472 \sqrt{2} \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{21 \sqrt{x^4+3 x^2+2}}-\frac{31 \sqrt{2} \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{x^4+3 x^2+2}}",1,"(1114*x + 2349*x^3 + 1724*x^5 + 564*x^7 + 75*x^9 - (651*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - (293*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt[2]], 2])/(21*Sqrt[2 + 3*x^2 + x^4])","C",1
288,1,109,149,0.0625025,"\int \left(7+5 x^2\right) \sqrt{2+3 x^2+x^4} \, dx","Integrate[(7 + 5*x^2)*Sqrt[2 + 3*x^2 + x^4],x]","\frac{3 x^7+19 x^5+36 x^3-7 i \sqrt{x^2+1} \sqrt{x^2+2} F\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)-15 i \sqrt{x^2+1} \sqrt{x^2+2} E\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+20 x}{3 \sqrt{x^4+3 x^2+2}}","\frac{5 x \left(x^2+2\right)}{\sqrt{x^4+3 x^2+2}}+\frac{1}{3} x \left(3 x^2+10\right) \sqrt{x^4+3 x^2+2}+\frac{11 \sqrt{2} \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{3 \sqrt{x^4+3 x^2+2}}-\frac{5 \sqrt{2} \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{x^4+3 x^2+2}}",1,"(20*x + 36*x^3 + 19*x^5 + 3*x^7 - (15*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - (7*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt[2]], 2])/(3*Sqrt[2 + 3*x^2 + x^4])","C",1
289,1,102,141,0.0355905,"\int \sqrt{2+3 x^2+x^4} \, dx","Integrate[Sqrt[2 + 3*x^2 + x^4],x]","\frac{x^5+3 x^3-i \sqrt{x^2+1} \sqrt{x^2+2} F\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)-3 i \sqrt{x^2+1} \sqrt{x^2+2} E\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+2 x}{3 \sqrt{x^4+3 x^2+2}}","\frac{1}{3} \sqrt{x^4+3 x^2+2} x+\frac{\left(x^2+2\right) x}{\sqrt{x^4+3 x^2+2}}+\frac{2 \sqrt{2} \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{3 \sqrt{x^4+3 x^2+2}}-\frac{\sqrt{2} \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{x^4+3 x^2+2}}",1,"(2*x + 3*x^3 + x^5 - (3*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - I*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt[2]], 2])/(3*Sqrt[2 + 3*x^2 + x^4])","C",1
290,1,90,178,0.1453446,"\int \frac{\sqrt{2+3 x^2+x^4}}{7+5 x^2} \, dx","Integrate[Sqrt[2 + 3*x^2 + x^4]/(7 + 5*x^2),x]","-\frac{i \sqrt{x^2+1} \sqrt{x^2+2} \left(21 F\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+35 E\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)-6 \Pi \left(\frac{10}{7};\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)\right)}{175 \sqrt{x^4+3 x^2+2}}","\frac{x \left(x^2+2\right)}{5 \sqrt{x^4+3 x^2+2}}+\frac{\left(x^2+1\right) \sqrt{\frac{x^2+2}{2 x^2+2}} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{5 \sqrt{x^4+3 x^2+2}}-\frac{\sqrt{2} \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{5 \sqrt{x^4+3 x^2+2}}+\frac{3 \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} \Pi \left(\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right)}{35 \sqrt{2} \sqrt{x^4+3 x^2+2}}",1,"((-1/175*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*(35*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] + 21*EllipticF[I*ArcSinh[x/Sqrt[2]], 2] - 6*EllipticPi[10/7, I*ArcSinh[x/Sqrt[2]], 2]))/Sqrt[2 + 3*x^2 + x^4]","C",1
291,1,208,209,0.2754946,"\int \frac{\sqrt{2+3 x^2+x^4}}{\left(7+5 x^2\right)^2} \, dx","Integrate[Sqrt[2 + 3*x^2 + x^4]/(7 + 5*x^2)^2,x]","\frac{175 x^5+525 x^3-84 i \sqrt{x^2+1} \sqrt{x^2+2} \left(5 x^2+7\right) F\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+35 i \sqrt{x^2+1} \sqrt{x^2+2} \left(5 x^2+7\right) E\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)-5 i \sqrt{x^2+1} \sqrt{x^2+2} x^2 \Pi \left(\frac{10}{7};\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)-7 i \sqrt{x^2+1} \sqrt{x^2+2} \Pi \left(\frac{10}{7};\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+350 x}{2450 \left(5 x^2+7\right) \sqrt{x^4+3 x^2+2}}","\frac{\sqrt{x^4+3 x^2+2} x}{14 \left(5 x^2+7\right)}-\frac{\left(x^2+2\right) x}{70 \sqrt{x^4+3 x^2+2}}+\frac{3 \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{140 \sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{\left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{35 \sqrt{2} \sqrt{x^4+3 x^2+2}}-\frac{\left(x^2+2\right) \Pi \left(\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right)}{980 \sqrt{2} \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}}",1,"(350*x + 525*x^3 + 175*x^5 + (35*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*(7 + 5*x^2)*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - (84*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*(7 + 5*x^2)*EllipticF[I*ArcSinh[x/Sqrt[2]], 2] - (7*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticPi[10/7, I*ArcSinh[x/Sqrt[2]], 2] - (5*I)*x^2*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticPi[10/7, I*ArcSinh[x/Sqrt[2]], 2])/(2450*(7 + 5*x^2)*Sqrt[2 + 3*x^2 + x^4])","C",1
292,1,174,237,0.3547163,"\int \frac{\sqrt{2+3 x^2+x^4}}{\left(7+5 x^2\right)^3} \, dx","Integrate[Sqrt[2 + 3*x^2 + x^4]/(7 + 5*x^2)^3,x]","\frac{-434 i \sqrt{x^2+1} \sqrt{x^2+2} F\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+385 i \sqrt{x^2+1} \sqrt{x^2+2} E\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)-1201 i \sqrt{x^2+1} \sqrt{x^2+2} \Pi \left(\frac{10}{7};\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+\frac{1925 x \left(x^4+3 x^2+2\right)}{5 x^2+7}+\frac{14700 x \left(x^4+3 x^2+2\right)}{\left(5 x^2+7\right)^2}}{411600 \sqrt{x^4+3 x^2+2}}","\frac{11 \sqrt{x^4+3 x^2+2} x}{2352 \left(5 x^2+7\right)}+\frac{\sqrt{x^4+3 x^2+2} x}{28 \left(5 x^2+7\right)^2}-\frac{11 \left(x^2+2\right) x}{11760 \sqrt{x^4+3 x^2+2}}+\frac{81 \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{7840 \sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{11 \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{5880 \sqrt{2} \sqrt{x^4+3 x^2+2}}-\frac{1201 \left(x^2+2\right) \Pi \left(\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right)}{164640 \sqrt{2} \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}}",1,"((14700*x*(2 + 3*x^2 + x^4))/(7 + 5*x^2)^2 + (1925*x*(2 + 3*x^2 + x^4))/(7 + 5*x^2) + (385*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - (434*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt[2]], 2] - (1201*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticPi[10/7, I*ArcSinh[x/Sqrt[2]], 2])/(411600*Sqrt[2 + 3*x^2 + x^4])","C",1
293,-1,0,219,0,"\int \left(7+5 x^2\right)^3 \left(2+3 x^2+x^4\right)^{3/2} \, dx","Integrate[(7 + 5*x^2)^3*(2 + 3*x^2 + x^4)^(3/2),x]","\text{\$Aborted}","\frac{3825}{143} \left(x^4+3 x^2+2\right)^{5/2} x+\frac{\left(65345 x^2+208212\right) \left(x^4+3 x^2+2\right)^{3/2} x}{3003}+\frac{\left(297911 x^2+1032541\right) \sqrt{x^4+3 x^2+2} x}{5005}+\frac{20884 \left(x^2+2\right) x}{65 \sqrt{x^4+3 x^2+2}}+\frac{1171349 \sqrt{2} \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{5005 \sqrt{x^4+3 x^2+2}}-\frac{20884 \sqrt{2} \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{65 \sqrt{x^4+3 x^2+2}}+\frac{125}{13} \left(x^4+3 x^2+2\right)^{5/2} x^3",1,"$Aborted","F",-1
294,-1,0,198,0,"\int \left(7+5 x^2\right)^2 \left(2+3 x^2+x^4\right)^{3/2} \, dx","Integrate[(7 + 5*x^2)^2*(2 + 3*x^2 + x^4)^(3/2),x]","\text{\$Aborted}","\frac{25}{11} x \left(x^4+3 x^2+2\right)^{5/2}+\frac{1}{693} x \left(2240 x^2+7281\right) \left(x^4+3 x^2+2\right)^{3/2}+\frac{x \left(10643 x^2+36783\right) \sqrt{x^4+3 x^2+2}}{1155}+\frac{742 x \left(x^2+2\right)}{15 \sqrt{x^4+3 x^2+2}}+\frac{13879 \sqrt{2} \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{385 \sqrt{x^4+3 x^2+2}}-\frac{742 \sqrt{2} \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{15 \sqrt{x^4+3 x^2+2}}",1,"$Aborted","F",-1
295,-1,0,179,0,"\int \left(7+5 x^2\right) \left(2+3 x^2+x^4\right)^{3/2} \, dx","Integrate[(7 + 5*x^2)*(2 + 3*x^2 + x^4)^(3/2),x]","\text{\$Aborted}","\frac{1}{63} x \left(35 x^2+108\right) \left(x^4+3 x^2+2\right)^{3/2}+\frac{1}{105} x \left(149 x^2+519\right) \sqrt{x^4+3 x^2+2}+\frac{116 x \left(x^2+2\right)}{15 \sqrt{x^4+3 x^2+2}}+\frac{197 \sqrt{2} \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{35 \sqrt{x^4+3 x^2+2}}-\frac{116 \sqrt{2} \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{15 \sqrt{x^4+3 x^2+2}}",1,"$Aborted","F",-1
296,1,114,172,0.0430816,"\int \left(2+3 x^2+x^4\right)^{3/2} \, dx","Integrate[(2 + 3*x^2 + x^4)^(3/2),x]","\frac{5 x^9+39 x^7+121 x^5+165 x^3-20 i \sqrt{x^2+1} \sqrt{x^2+2} F\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)-42 i \sqrt{x^2+1} \sqrt{x^2+2} E\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+78 x}{35 \sqrt{x^4+3 x^2+2}}","\frac{1}{7} x \left(x^4+3 x^2+2\right)^{3/2}+\frac{1}{35} x \left(9 x^2+29\right) \sqrt{x^4+3 x^2+2}+\frac{6 x \left(x^2+2\right)}{5 \sqrt{x^4+3 x^2+2}}+\frac{31 \sqrt{2} \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{35 \sqrt{x^4+3 x^2+2}}-\frac{6 \sqrt{2} \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{5 \sqrt{x^4+3 x^2+2}}",1,"(78*x + 165*x^3 + 121*x^5 + 39*x^7 + 5*x^9 - (42*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - (20*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt[2]], 2])/(35*Sqrt[2 + 3*x^2 + x^4])","C",1
297,1,148,207,0.1770847,"\int \frac{\left(2+3 x^2+x^4\right)^{3/2}}{7+5 x^2} \, dx","Integrate[(2 + 3*x^2 + x^4)^(3/2)/(7 + 5*x^2),x]","\frac{525 x^7+3500 x^5+6825 x^3-1022 i \sqrt{x^2+1} \sqrt{x^2+2} F\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)-2520 i \sqrt{x^2+1} \sqrt{x^2+2} E\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)-108 i \sqrt{x^2+1} \sqrt{x^2+2} \Pi \left(\frac{10}{7};\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+3850 x}{13125 \sqrt{x^4+3 x^2+2}}","\frac{24 x \left(x^2+2\right)}{125 \sqrt{x^4+3 x^2+2}}+\frac{1}{75} x \left(3 x^2+11\right) \sqrt{x^4+3 x^2+2}+\frac{56 \sqrt{2} \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{375 \sqrt{x^4+3 x^2+2}}-\frac{24 \sqrt{2} \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{125 \sqrt{x^4+3 x^2+2}}-\frac{9 \sqrt{2} \left(x^2+2\right) \Pi \left(\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right)}{875 \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}}",1,"(3850*x + 6825*x^3 + 3500*x^5 + 525*x^7 - (2520*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - (1022*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt[2]], 2] - (108*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticPi[10/7, I*ArcSinh[x/Sqrt[2]], 2])/(13125*Sqrt[2 + 3*x^2 + x^4])","C",1
298,1,213,222,0.2931394,"\int \frac{\left(2+3 x^2+x^4\right)^{3/2}}{\left(7+5 x^2\right)^2} \, dx","Integrate[(2 + 3*x^2 + x^4)^(3/2)/(7 + 5*x^2)^2,x]","\frac{1225 x^7+5075 x^5+6650 x^3-182 i \sqrt{x^2+1} \sqrt{x^2+2} \left(5 x^2+7\right) F\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)-945 i \sqrt{x^2+1} \sqrt{x^2+2} \left(5 x^2+7\right) E\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+135 i \sqrt{x^2+1} \sqrt{x^2+2} x^2 \Pi \left(\frac{10}{7};\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+189 i \sqrt{x^2+1} \sqrt{x^2+2} \Pi \left(\frac{10}{7};\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+2800 x}{18375 \left(5 x^2+7\right) \sqrt{x^4+3 x^2+2}}","-\frac{3 \sqrt{x^4+3 x^2+2} x}{175 \left(5 x^2+7\right)}+\frac{1}{75} \sqrt{x^4+3 x^2+2} x+\frac{9 \left(x^2+2\right) x}{175 \sqrt{x^4+3 x^2+2}}+\frac{59 \left(x^2+1\right) \sqrt{\frac{x^2+2}{2 x^2+2}} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{1050 \sqrt{x^4+3 x^2+2}}-\frac{9 \sqrt{2} \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{175 \sqrt{x^4+3 x^2+2}}+\frac{9 \left(x^2+1\right) \sqrt{\frac{x^2+2}{2 x^2+2}} \Pi \left(\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right)}{2450 \sqrt{x^4+3 x^2+2}}",1,"(2800*x + 6650*x^3 + 5075*x^5 + 1225*x^7 - (945*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*(7 + 5*x^2)*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - (182*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*(7 + 5*x^2)*EllipticF[I*ArcSinh[x/Sqrt[2]], 2] + (189*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticPi[10/7, I*ArcSinh[x/Sqrt[2]], 2] + (135*I)*x^2*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticPi[10/7, I*ArcSinh[x/Sqrt[2]], 2])/(18375*(7 + 5*x^2)*Sqrt[2 + 3*x^2 + x^4])","C",1
299,1,174,231,0.3790021,"\int \frac{\left(2+3 x^2+x^4\right)^{3/2}}{\left(7+5 x^2\right)^3} \, dx","Integrate[(2 + 3*x^2 + x^4)^(3/2)/(7 + 5*x^2)^3,x]","\frac{-406 i \sqrt{x^2+1} \sqrt{x^2+2} F\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)-525 i \sqrt{x^2+1} \sqrt{x^2+2} E\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+141 i \sqrt{x^2+1} \sqrt{x^2+2} \Pi \left(\frac{10}{7};\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+\frac{119 x \left(x^4+3 x^2+2\right)}{5 x^2+7}-\frac{588 x \left(x^4+3 x^2+2\right)}{\left(5 x^2+7\right)^2}}{68600 \sqrt{x^4+3 x^2+2}}","\frac{17 \sqrt{x^4+3 x^2+2} x}{9800 \left(5 x^2+7\right)}-\frac{3 \sqrt{x^4+3 x^2+2} x}{350 \left(5 x^2+7\right)^2}+\frac{3 \left(x^2+2\right) x}{392 \sqrt{x^4+3 x^2+2}}+\frac{5 \left(x^2+1\right) \sqrt{\frac{x^2+2}{2 x^2+2}} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{784 \sqrt{x^4+3 x^2+2}}-\frac{3 \left(x^2+1\right) \sqrt{\frac{x^2+2}{2 x^2+2}} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{196 \sqrt{x^4+3 x^2+2}}+\frac{141 \left(x^2+2\right) \Pi \left(\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right)}{27440 \sqrt{2} \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}}",1,"((-588*x*(2 + 3*x^2 + x^4))/(7 + 5*x^2)^2 + (119*x*(2 + 3*x^2 + x^4))/(7 + 5*x^2) - (525*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - (406*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt[2]], 2] + (141*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticPi[10/7, I*ArcSinh[x/Sqrt[2]], 2])/(68600*Sqrt[2 + 3*x^2 + x^4])","C",1
300,1,106,157,0.1030829,"\int \frac{\left(7+5 x^2\right)^3}{\sqrt{2+3 x^2+x^4}} \, dx","Integrate[(7 + 5*x^2)^3/Sqrt[2 + 3*x^2 + x^4],x]","\frac{-58 i \sqrt{x^2+1} \sqrt{x^2+2} F\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)-135 i \sqrt{x^2+1} \sqrt{x^2+2} E\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+25 x \left(x^6+6 x^4+11 x^2+6\right)}{\sqrt{x^4+3 x^2+2}}","75 \sqrt{x^4+3 x^2+2} x+\frac{135 \left(x^2+2\right) x}{\sqrt{x^4+3 x^2+2}}+\frac{193 \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{2} \sqrt{x^4+3 x^2+2}}-\frac{135 \sqrt{2} \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{x^4+3 x^2+2}}+25 \sqrt{x^4+3 x^2+2} x^3",1,"(25*x*(6 + 11*x^2 + 6*x^4 + x^6) - (135*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - (58*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt[2]], 2])/Sqrt[2 + 3*x^2 + x^4]","C",1
301,1,104,142,0.0875117,"\int \frac{\left(7+5 x^2\right)^2}{\sqrt{2+3 x^2+x^4}} \, dx","Integrate[(7 + 5*x^2)^2/Sqrt[2 + 3*x^2 + x^4],x]","\frac{-37 i \sqrt{x^2+1} \sqrt{x^2+2} F\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)-60 i \sqrt{x^2+1} \sqrt{x^2+2} E\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+25 x \left(x^4+3 x^2+2\right)}{3 \sqrt{x^4+3 x^2+2}}","\frac{25}{3} \sqrt{x^4+3 x^2+2} x+\frac{20 \left(x^2+2\right) x}{\sqrt{x^4+3 x^2+2}}+\frac{97 \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{3 \sqrt{2} \sqrt{x^4+3 x^2+2}}-\frac{20 \sqrt{2} \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{x^4+3 x^2+2}}",1,"(25*x*(2 + 3*x^2 + x^4) - (60*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - (37*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt[2]], 2])/(3*Sqrt[2 + 3*x^2 + x^4])","C",1
302,1,69,121,0.0603953,"\int \frac{7+5 x^2}{\sqrt{2+3 x^2+x^4}} \, dx","Integrate[(7 + 5*x^2)/Sqrt[2 + 3*x^2 + x^4],x]","-\frac{i \sqrt{x^2+1} \sqrt{x^2+2} \left(2 F\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+5 E\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)\right)}{\sqrt{x^4+3 x^2+2}}","\frac{5 x \left(x^2+2\right)}{\sqrt{x^4+3 x^2+2}}+\frac{7 \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{2} \sqrt{x^4+3 x^2+2}}-\frac{5 \sqrt{2} \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{x^4+3 x^2+2}}",1,"((-I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*(5*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] + 2*EllipticF[I*ArcSinh[x/Sqrt[2]], 2]))/Sqrt[2 + 3*x^2 + x^4]","C",1
303,1,50,48,0.0149899,"\int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx","Integrate[1/Sqrt[2 + 3*x^2 + x^4],x]","-\frac{i \sqrt{x^2+1} \sqrt{x^2+2} F\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)}{\sqrt{x^4+3 x^2+2}}","\frac{\left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{2} \sqrt{x^4+3 x^2+2}}",1,"((-I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt[2]], 2])/Sqrt[2 + 3*x^2 + x^4]","C",1
304,1,55,106,0.0957037,"\int \frac{1}{\left(7+5 x^2\right) \sqrt{2+3 x^2+x^4}} \, dx","Integrate[1/((7 + 5*x^2)*Sqrt[2 + 3*x^2 + x^4]),x]","-\frac{i \sqrt{x^2+1} \sqrt{x^2+2} \Pi \left(\frac{10}{7};\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)}{7 \sqrt{x^4+3 x^2+2}}","\frac{\left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{2 \sqrt{2} \sqrt{x^4+3 x^2+2}}-\frac{5 \left(x^2+2\right) \Pi \left(\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right)}{14 \sqrt{2} \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}}",1,"((-1/7*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticPi[10/7, I*ArcSinh[x/Sqrt[2]], 2])/Sqrt[2 + 3*x^2 + x^4]","C",1
305,1,208,209,0.2572755,"\int \frac{1}{\left(7+5 x^2\right)^2 \sqrt{2+3 x^2+x^4}} \, dx","Integrate[1/((7 + 5*x^2)^2*Sqrt[2 + 3*x^2 + x^4]),x]","\frac{-175 x^5-525 x^3-14 i \sqrt{x^2+1} \sqrt{x^2+2} \left(5 x^2+7\right) F\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)-35 i \sqrt{x^2+1} \sqrt{x^2+2} \left(5 x^2+7\right) E\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)-65 i \sqrt{x^2+1} \sqrt{x^2+2} x^2 \Pi \left(\frac{10}{7};\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)-91 i \sqrt{x^2+1} \sqrt{x^2+2} \Pi \left(\frac{10}{7};\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)-350 x}{588 \left(5 x^2+7\right) \sqrt{x^4+3 x^2+2}}","-\frac{25 \sqrt{x^4+3 x^2+2} x}{84 \left(5 x^2+7\right)}+\frac{5 \left(x^2+2\right) x}{84 \sqrt{x^4+3 x^2+2}}+\frac{9 \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{56 \sqrt{2} \sqrt{x^4+3 x^2+2}}-\frac{5 \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{42 \sqrt{2} \sqrt{x^4+3 x^2+2}}-\frac{65 \left(x^2+2\right) \Pi \left(\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right)}{1176 \sqrt{2} \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}}",1,"(-350*x - 525*x^3 - 175*x^5 - (35*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*(7 + 5*x^2)*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - (14*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*(7 + 5*x^2)*EllipticF[I*ArcSinh[x/Sqrt[2]], 2] - (91*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticPi[10/7, I*ArcSinh[x/Sqrt[2]], 2] - (65*I)*x^2*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticPi[10/7, I*ArcSinh[x/Sqrt[2]], 2])/(588*(7 + 5*x^2)*Sqrt[2 + 3*x^2 + x^4])","C",1
306,1,186,237,0.3433333,"\int \frac{1}{\left(7+5 x^2\right)^3 \sqrt{2+3 x^2+x^4}} \, dx","Integrate[1/((7 + 5*x^2)^3*Sqrt[2 + 3*x^2 + x^4]),x]","\frac{14 i \sqrt{x^2+1} \sqrt{x^2+2} \left(5 x^2+7\right)^2 F\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)-455 i \sqrt{x^2+1} \sqrt{x^2+2} \left(5 x^2+7\right)^2 E\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)-505 i \sqrt{x^2+1} \sqrt{x^2+2} \left(5 x^2+7\right)^2 \Pi \left(\frac{10}{7};\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)-175 x \left(65 x^6+314 x^4+487 x^2+238\right)}{32928 \left(5 x^2+7\right)^2 \sqrt{x^4+3 x^2+2}}","-\frac{325 \sqrt{x^4+3 x^2+2} x}{4704 \left(5 x^2+7\right)}-\frac{25 \sqrt{x^4+3 x^2+2} x}{168 \left(5 x^2+7\right)^2}+\frac{65 \left(x^2+2\right) x}{4704 \sqrt{x^4+3 x^2+2}}+\frac{631 \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{9408 \sqrt{2} \sqrt{x^4+3 x^2+2}}-\frac{65 \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{2352 \sqrt{2} \sqrt{x^4+3 x^2+2}}-\frac{2525 \left(x^2+2\right) \Pi \left(\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right)}{65856 \sqrt{2} \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}}",1,"(-175*x*(238 + 487*x^2 + 314*x^4 + 65*x^6) - (455*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*(7 + 5*x^2)^2*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] + (14*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*(7 + 5*x^2)^2*EllipticF[I*ArcSinh[x/Sqrt[2]], 2] - (505*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*(7 + 5*x^2)^2*EllipticPi[10/7, I*ArcSinh[x/Sqrt[2]], 2])/(32928*(7 + 5*x^2)^2*Sqrt[2 + 3*x^2 + x^4])","C",1
307,-1,0,189,0,"\int \frac{\left(7+5 x^2\right)^5}{\left(2+3 x^2+x^4\right)^{3/2}} \, dx","Integrate[(7 + 5*x^2)^5/(2 + 3*x^2 + x^4)^(3/2),x]","\text{\$Aborted}","\frac{5000}{3} \sqrt{x^4+3 x^2+2} x+\frac{7679 \left(x^2+2\right) x}{2 \sqrt{x^4+3 x^2+2}}-\frac{\left(179 x^2+115\right) x}{2 \sqrt{x^4+3 x^2+2}}+\frac{15383 \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{3 \sqrt{2} \sqrt{x^4+3 x^2+2}}-\frac{7679 \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{2} \sqrt{x^4+3 x^2+2}}+625 \sqrt{x^4+3 x^2+2} x^3",1,"$Aborted","F",-1
308,-1,0,170,0,"\int \frac{\left(7+5 x^2\right)^4}{\left(2+3 x^2+x^4\right)^{3/2}} \, dx","Integrate[(7 + 5*x^2)^4/(2 + 3*x^2 + x^4)^(3/2),x]","\text{\$Aborted}","\frac{625}{3} \sqrt{x^4+3 x^2+2} x+\frac{637 \left(x^2+2\right) x}{2 \sqrt{x^4+3 x^2+2}}+\frac{\left(113 x^2+145\right) x}{2 \sqrt{x^4+3 x^2+2}}+\frac{1067 \sqrt{2} \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{3 \sqrt{x^4+3 x^2+2}}-\frac{637 \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{2} \sqrt{x^4+3 x^2+2}}",1,"$Aborted","F",-1
309,-1,0,149,0,"\int \frac{\left(7+5 x^2\right)^3}{\left(2+3 x^2+x^4\right)^{3/2}} \, dx","Integrate[(7 + 5*x^2)^3/(2 + 3*x^2 + x^4)^(3/2),x]","\text{\$Aborted}","\frac{x \left(5-11 x^2\right)}{2 \sqrt{x^4+3 x^2+2}}+\frac{261 x \left(x^2+2\right)}{2 \sqrt{x^4+3 x^2+2}}+\frac{169 \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{2} \sqrt{x^4+3 x^2+2}}-\frac{261 \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{2} \sqrt{x^4+3 x^2+2}}",1,"$Aborted","F",-1
310,-1,0,149,0,"\int \frac{\left(7+5 x^2\right)^2}{\left(2+3 x^2+x^4\right)^{3/2}} \, dx","Integrate[(7 + 5*x^2)^2/(2 + 3*x^2 + x^4)^(3/2),x]","\text{\$Aborted}","-\frac{17 x \left(x^2+2\right)}{2 \sqrt{x^4+3 x^2+2}}+\frac{x \left(17 x^2+25\right)}{2 \sqrt{x^4+3 x^2+2}}+\frac{6 \sqrt{2} \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{x^4+3 x^2+2}}+\frac{17 \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{2} \sqrt{x^4+3 x^2+2}}",1,"$Aborted","F",-1
311,-1,0,145,0,"\int \frac{7+5 x^2}{\left(2+3 x^2+x^4\right)^{3/2}} \, dx","Integrate[(7 + 5*x^2)/(2 + 3*x^2 + x^4)^(3/2),x]","\text{\$Aborted}","-\frac{x \left(x^2+2\right)}{2 \sqrt{x^4+3 x^2+2}}+\frac{x \left(x^2+5\right)}{2 \sqrt{x^4+3 x^2+2}}+\frac{\left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{\left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{2} \sqrt{x^4+3 x^2+2}}",1,"$Aborted","F",-1
312,1,99,149,0.0410131,"\int \frac{1}{\left(2+3 x^2+x^4\right)^{3/2}} \, dx","Integrate[(2 + 3*x^2 + x^4)^(-3/2),x]","\frac{3 x^3+i \sqrt{x^2+1} \sqrt{x^2+2} F\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+3 i \sqrt{x^2+1} \sqrt{x^2+2} E\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+5 x}{2 \sqrt{x^4+3 x^2+2}}","-\frac{3 x \left(x^2+2\right)}{2 \sqrt{x^4+3 x^2+2}}+\frac{x \left(3 x^2+5\right)}{2 \sqrt{x^4+3 x^2+2}}-\frac{\sqrt{2} \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{x^4+3 x^2+2}}+\frac{3 \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{2} \sqrt{x^4+3 x^2+2}}",1,"(5*x + 3*x^3 + (3*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] + I*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt[2]], 2])/(2*Sqrt[2 + 3*x^2 + x^4])","C",1
313,1,138,173,0.1691265,"\int \frac{1}{\left(7+5 x^2\right) \left(2+3 x^2+x^4\right)^{3/2}} \, dx","Integrate[1/((7 + 5*x^2)*(2 + 3*x^2 + x^4)^(3/2)),x]","\frac{14 x^3-7 i \sqrt{x^2+1} \sqrt{x^2+2} F\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+14 i \sqrt{x^2+1} \sqrt{x^2+2} E\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+25 i \sqrt{x^2+1} \sqrt{x^2+2} \Pi \left(\frac{10}{7};\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+35 x}{42 \sqrt{x^4+3 x^2+2}}","\frac{x}{6 \sqrt{x^4+3 x^2+2}}-\frac{9 \left(x^2+1\right) \sqrt{\frac{x^2+2}{2 x^2+2}} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{4 \sqrt{x^4+3 x^2+2}}+\frac{\sqrt{2} \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{3 \sqrt{x^4+3 x^2+2}}+\frac{125 \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} \Pi \left(\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right)}{84 \sqrt{2} \sqrt{x^4+3 x^2+2}}",1,"(35*x + 14*x^3 + (14*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - (7*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt[2]], 2] + (25*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticPi[10/7, I*ArcSinh[x/Sqrt[2]], 2])/(42*Sqrt[2 + 3*x^2 + x^4])","C",1
314,1,208,235,0.2811129,"\int \frac{1}{\left(7+5 x^2\right)^2 \left(2+3 x^2+x^4\right)^{3/2}} \, dx","Integrate[1/((7 + 5*x^2)^2*(2 + 3*x^2 + x^4)^(3/2)),x]","\frac{3255 x^5+10157 x^3+182 i \sqrt{x^2+1} \sqrt{x^2+2} \left(5 x^2+7\right) F\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+651 i \sqrt{x^2+1} \sqrt{x^2+2} \left(5 x^2+7\right) E\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+1125 i \sqrt{x^2+1} \sqrt{x^2+2} x^2 \Pi \left(\frac{10}{7};\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+1575 i \sqrt{x^2+1} \sqrt{x^2+2} \Pi \left(\frac{10}{7};\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+7490 x}{1176 \left(5 x^2+7\right) \sqrt{x^4+3 x^2+2}}","\frac{625 \sqrt{x^4+3 x^2+2} x}{504 \left(5 x^2+7\right)}-\frac{31 \left(x^2+2\right) x}{56 \sqrt{x^4+3 x^2+2}}+\frac{\left(11 x^2+20\right) x}{36 \sqrt{x^4+3 x^2+2}}-\frac{463 \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{336 \sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{31 \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{28 \sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{375 \left(x^2+2\right) \Pi \left(\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right)}{784 \sqrt{2} \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}}",1,"(7490*x + 10157*x^3 + 3255*x^5 + (651*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*(7 + 5*x^2)*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] + (182*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*(7 + 5*x^2)*EllipticF[I*ArcSinh[x/Sqrt[2]], 2] + (1575*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticPi[10/7, I*ArcSinh[x/Sqrt[2]], 2] + (1125*I)*x^2*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticPi[10/7, I*ArcSinh[x/Sqrt[2]], 2])/(1176*(7 + 5*x^2)*Sqrt[2 + 3*x^2 + x^4])","C",1
315,1,159,263,0.5008424,"\int \frac{1}{\left(7+5 x^2\right)^3 \left(2+3 x^2+x^4\right)^{3/2}} \, dx","Integrate[1/((7 + 5*x^2)^3*(2 + 3*x^2 + x^4)^(3/2)),x]","\frac{-742 i \sqrt{x^2+1} \sqrt{x^2+2} F\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+40579 i \sqrt{x^2+1} \sqrt{x^2+2} E\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+38525 i \sqrt{x^2+1} \sqrt{x^2+2} \Pi \left(\frac{10}{7};\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+\frac{7 x \left(144925 x^6+698290 x^4+1089803 x^2+550550\right)}{\left(5 x^2+7\right)^2}}{197568 \sqrt{x^4+3 x^2+2}}","\frac{41875 \sqrt{x^4+3 x^2+2} x}{84672 \left(5 x^2+7\right)}+\frac{625 \sqrt{x^4+3 x^2+2} x}{1008 \left(5 x^2+7\right)^2}-\frac{5797 \left(x^2+2\right) x}{28224 \sqrt{x^4+3 x^2+2}}+\frac{\left(23 x^2+50\right) x}{216 \sqrt{x^4+3 x^2+2}}-\frac{49907 \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{56448 \sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{5797 \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{14112 \sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{192625 \left(x^2+2\right) \Pi \left(\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right)}{395136 \sqrt{2} \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}}",1,"((7*x*(550550 + 1089803*x^2 + 698290*x^4 + 144925*x^6))/(7 + 5*x^2)^2 + (40579*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - (742*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt[2]], 2] + (38525*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticPi[10/7, I*ArcSinh[x/Sqrt[2]], 2])/(197568*Sqrt[2 + 3*x^2 + x^4])","C",1
316,1,112,116,0.1324561,"\int \left(7+5 x^2\right)^4 \sqrt{2+x^2-x^4} \, dx","Integrate[(7 + 5*x^2)^4*Sqrt[2 + x^2 - x^4],x]","\frac{-13125 x^{13}-75250 x^{11}-105925 x^9+231228 x^7+1125819 x^5-186503 x^3-4838091 i \sqrt{-2 x^4+2 x^2+4} F\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)+3764813 i \sqrt{-2 x^4+2 x^2+4} E\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)-1037294 x}{231 \sqrt{-x^4+x^2+2}}","-\frac{116100}{77} \left(-x^4+x^2+2\right)^{3/2} x+\frac{1}{231} \left(717372 x^2+177953\right) \sqrt{-x^4+x^2+2} x-\frac{625}{11} \left(-x^4+x^2+2\right)^{3/2} x^5-\frac{14500}{33} \left(-x^4+x^2+2\right)^{3/2} x^3-\frac{539419}{77} F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)+\frac{3764813}{231} E\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)",1,"(-1037294*x - 186503*x^3 + 1125819*x^5 + 231228*x^7 - 105925*x^9 - 75250*x^11 - 13125*x^13 + (3764813*I)*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticE[I*ArcSinh[x], -1/2] - (4838091*I)*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticF[I*ArcSinh[x], -1/2])/(231*Sqrt[2 + x^2 - x^4])","C",1
317,1,107,95,0.1035587,"\int \left(7+5 x^2\right)^3 \sqrt{2+x^2-x^4} \, dx","Integrate[(7 + 5*x^2)^3*Sqrt[2 + x^2 - x^4],x]","\frac{-875 x^{11}-3725 x^9-1116 x^7+21660 x^5+9938 x^3-106014 i \sqrt{-2 x^4+2 x^2+4} F\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)+79411 i \sqrt{-2 x^4+2 x^2+4} E\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)-9988 x}{63 \sqrt{-x^4+x^2+2}}","-\frac{1825}{21} \left(-x^4+x^2+2\right)^{3/2} x+\frac{1}{63} \left(14691 x^2+5956\right) \sqrt{-x^4+x^2+2} x-\frac{125}{9} \left(-x^4+x^2+2\right)^{3/2} x^3-\frac{8735}{21} F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)+\frac{79411}{63} E\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)",1,"(-9988*x + 9938*x^3 + 21660*x^5 - 1116*x^7 - 3725*x^9 - 875*x^11 + (79411*I)*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticE[I*ArcSinh[x], -1/2] - (106014*I)*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticF[I*ArcSinh[x], -1/2])/(63*Sqrt[2 + x^2 - x^4])","C",1
318,1,102,74,0.0887172,"\int \left(7+5 x^2\right)^2 \sqrt{2+x^2-x^4} \, dx","Integrate[(7 + 5*x^2)^2*Sqrt[2 + x^2 - x^4],x]","\frac{-75 x^9-204 x^7+304 x^5+683 x^3-2949 i \sqrt{-2 x^4+2 x^2+4} F\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)+2045 i \sqrt{-2 x^4+2 x^2+4} E\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)+250 x}{21 \sqrt{-x^4+x^2+2}}","-\frac{25}{7} x \left(-x^4+x^2+2\right)^{3/2}+\frac{1}{21} x \left(354 x^2+275\right) \sqrt{-x^4+x^2+2}-\frac{79}{7} F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)+\frac{2045}{21} E\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)",1,"(250*x + 683*x^3 + 304*x^5 - 204*x^7 - 75*x^9 + (2045*I)*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticE[I*ArcSinh[x], -1/2] - (2949*I)*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticF[I*ArcSinh[x], -1/2])/(21*Sqrt[2 + x^2 - x^4])","C",1
319,1,94,46,0.0806587,"\int \left(7+5 x^2\right) \sqrt{2+x^2-x^4} \, dx","Integrate[(7 + 5*x^2)*Sqrt[2 + x^2 - x^4],x]","\frac{-x^7-x^5+4 x^3-12 i \sqrt{-2 x^4+2 x^2+4} F\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)+7 i \sqrt{-2 x^4+2 x^2+4} E\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)+4 x}{\sqrt{-x^4+x^2+2}}","x \sqrt{-x^4+x^2+2} \left(x^2+2\right)+3 F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)+7 E\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)",1,"(4*x + 4*x^3 - x^5 - x^7 + (7*I)*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticE[I*ArcSinh[x], -1/2] - (12*I)*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticF[I*ArcSinh[x], -1/2])/Sqrt[2 + x^2 - x^4]","C",1
320,1,90,44,0.045242,"\int \sqrt{2+x^2-x^4} \, dx","Integrate[Sqrt[2 + x^2 - x^4],x]","\frac{-x^5+x^3-3 i \sqrt{-2 x^4+2 x^2+4} F\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)+i \sqrt{-2 x^4+2 x^2+4} E\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)+2 x}{3 \sqrt{-x^4+x^2+2}}","\frac{1}{3} \sqrt{-x^4+x^2+2} x+F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)+\frac{1}{3} E\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)",1,"(2*x + x^3 - x^5 + I*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticE[I*ArcSinh[x], -1/2] - (3*I)*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticF[I*ArcSinh[x], -1/2])/(3*Sqrt[2 + x^2 - x^4])","C",1
321,1,51,46,0.1311382,"\int \frac{\sqrt{2+x^2-x^4}}{7+5 x^2} \, dx","Integrate[Sqrt[2 + x^2 - x^4]/(7 + 5*x^2),x]","-\frac{1}{175} i \sqrt{2} \left(7 F\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)+35 E\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)-17 \Pi \left(\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right)\right)","\frac{17}{25} F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)-\frac{1}{5} E\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)-\frac{34}{175} \Pi \left(-\frac{10}{7};\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)",1,"(-1/175*I)*Sqrt[2]*(35*EllipticE[I*ArcSinh[x], -1/2] + 7*EllipticF[I*ArcSinh[x], -1/2] - 17*EllipticPi[5/7, I*ArcSinh[x], -1/2])","C",1
322,1,196,74,0.2825058,"\int \frac{\sqrt{2+x^2-x^4}}{\left(7+5 x^2\right)^2} \, dx","Integrate[Sqrt[2 + x^2 - x^4]/(7 + 5*x^2)^2,x]","\frac{-350 x^5+350 x^3-21 i \sqrt{2} \left(5 x^2+7\right) \sqrt{-x^4+x^2+2} F\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)+70 i \sqrt{2} \left(5 x^2+7\right) \sqrt{-x^4+x^2+2} E\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)-495 i \sqrt{2} \sqrt{-x^4+x^2+2} x^2 \Pi \left(\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right)-693 i \sqrt{2} \sqrt{-x^4+x^2+2} \Pi \left(\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right)+700 x}{4900 \left(5 x^2+7\right) \sqrt{-x^4+x^2+2}}","\frac{\sqrt{-x^4+x^2+2} x}{14 \left(5 x^2+7\right)}-\frac{6}{175} F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)+\frac{1}{70} E\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)+\frac{99 \Pi \left(-\frac{10}{7};\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)}{2450}",1,"(700*x + 350*x^3 - 350*x^5 + (70*I)*Sqrt[2]*(7 + 5*x^2)*Sqrt[2 + x^2 - x^4]*EllipticE[I*ArcSinh[x], -1/2] - (21*I)*Sqrt[2]*(7 + 5*x^2)*Sqrt[2 + x^2 - x^4]*EllipticF[I*ArcSinh[x], -1/2] - (693*I)*Sqrt[2]*Sqrt[2 + x^2 - x^4]*EllipticPi[5/7, I*ArcSinh[x], -1/2] - (495*I)*Sqrt[2]*x^2*Sqrt[2 + x^2 - x^4]*EllipticPi[5/7, I*ArcSinh[x], -1/2])/(4900*(7 + 5*x^2)*Sqrt[2 + x^2 - x^4])","C",1
323,1,244,102,0.3571456,"\int \frac{\sqrt{2+x^2-x^4}}{\left(7+5 x^2\right)^3} \, dx","Integrate[Sqrt[2 + x^2 - x^4]/(7 + 5*x^2)^3,x]","\frac{54250 x^7-144900 x^5-17850 x^3+7021 i \sqrt{2} \left(5 x^2+7\right)^2 \sqrt{-x^4+x^2+2} F\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)-2170 i \sqrt{2} \left(5 x^2+7\right)^2 \sqrt{-x^4+x^2+2} E\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)-415025 i \sqrt{2} \sqrt{-x^4+x^2+2} x^4 \Pi \left(\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right)-1162070 i \sqrt{2} \sqrt{-x^4+x^2+2} x^2 \Pi \left(\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right)-813449 i \sqrt{2} \sqrt{-x^4+x^2+2} \Pi \left(\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right)+181300 x}{4664800 \left(5 x^2+7\right)^2 \sqrt{-x^4+x^2+2}}","-\frac{31 \sqrt{-x^4+x^2+2} x}{13328 \left(5 x^2+7\right)}+\frac{\sqrt{-x^4+x^2+2} x}{28 \left(5 x^2+7\right)^2}-\frac{269 F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)}{166600}-\frac{31 E\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)}{66640}+\frac{16601 \Pi \left(-\frac{10}{7};\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)}{2332400}",1,"(181300*x - 17850*x^3 - 144900*x^5 + 54250*x^7 - (2170*I)*Sqrt[2]*(7 + 5*x^2)^2*Sqrt[2 + x^2 - x^4]*EllipticE[I*ArcSinh[x], -1/2] + (7021*I)*Sqrt[2]*(7 + 5*x^2)^2*Sqrt[2 + x^2 - x^4]*EllipticF[I*ArcSinh[x], -1/2] - (813449*I)*Sqrt[2]*Sqrt[2 + x^2 - x^4]*EllipticPi[5/7, I*ArcSinh[x], -1/2] - (1162070*I)*Sqrt[2]*x^2*Sqrt[2 + x^2 - x^4]*EllipticPi[5/7, I*ArcSinh[x], -1/2] - (415025*I)*Sqrt[2]*x^4*Sqrt[2 + x^2 - x^4]*EllipticPi[5/7, I*ArcSinh[x], -1/2])/(4664800*(7 + 5*x^2)^2*Sqrt[2 + x^2 - x^4])","C",1
324,-1,0,142,0,"\int \left(7+5 x^2\right)^4 \left(2+x^2-x^4\right)^{3/2} \, dx","Integrate[(7 + 5*x^2)^4*(2 + x^2 - x^4)^(3/2),x]","\text{\$Aborted}","-\frac{132300}{143} \left(-x^4+x^2+2\right)^{5/2} x-\frac{\left(69817-1581440 x^2\right) \left(-x^4+x^2+2\right)^{3/2} x}{1001}+\frac{3 \left(7837383 x^2+2193559\right) \sqrt{-x^4+x^2+2} x}{5005}-\frac{125}{3} \left(-x^4+x^2+2\right)^{5/2} x^5-\frac{11750}{39} \left(-x^4+x^2+2\right)^{5/2} x^3-\frac{50794416 F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)}{5005}+\frac{124141422 E\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)}{5005}",1,"$Aborted","F",-1
325,-1,0,121,0,"\int \left(7+5 x^2\right)^3 \left(2+x^2-x^4\right)^{3/2} \, dx","Integrate[(7 + 5*x^2)^3*(2 + x^2 - x^4)^(3/2),x]","\text{\$Aborted}","-\frac{7825}{143} \left(-x^4+x^2+2\right)^{5/2} x+\frac{\left(374045 x^2+33792\right) \left(-x^4+x^2+2\right)^{3/2} x}{3003}+\frac{\left(5712051 x^2+2512273\right) \sqrt{-x^4+x^2+2} x}{15015}-\frac{125}{13} \left(-x^4+x^2+2\right)^{5/2} x^3-\frac{3199778 F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)}{5005}+\frac{31072528 E\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)}{15015}",1,"$Aborted","F",-1
326,-1,0,100,0,"\int \left(7+5 x^2\right)^2 \left(2+x^2-x^4\right)^{3/2} \, dx","Integrate[(7 + 5*x^2)^2*(2 + x^2 - x^4)^(3/2),x]","\text{\$Aborted}","-\frac{25}{11} x \left(-x^4+x^2+2\right)^{5/2}+\frac{1}{99} x \left(920 x^2+363\right) \left(-x^4+x^2+2\right)^{3/2}+\frac{1}{495} x \left(14889 x^2+11497\right) \sqrt{-x^4+x^2+2}-\frac{3392}{165} F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)+\frac{85942}{495} E\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)",1,"$Aborted","F",-1
327,-1,0,81,0,"\int \left(7+5 x^2\right) \left(2+x^2-x^4\right)^{3/2} \, dx","Integrate[(7 + 5*x^2)*(2 + x^2 - x^4)^(3/2),x]","\text{\$Aborted}","\frac{1}{63} x \left(35 x^2+48\right) \left(-x^4+x^2+2\right)^{3/2}+\frac{1}{315} x \left(669 x^2+1087\right) \sqrt{-x^4+x^2+2}+\frac{418}{105} F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)+\frac{4432}{315} E\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)",1,"$Aborted","F",-1
328,1,102,74,0.0546074,"\int \left(2+x^2-x^4\right)^{3/2} \, dx","Integrate[(2 + x^2 - x^4)^(3/2),x]","\frac{5 x^9-13 x^7-31 x^5+45 x^3-75 i \sqrt{-2 x^4+2 x^2+4} F\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)+34 i \sqrt{-2 x^4+2 x^2+4} E\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)+58 x}{35 \sqrt{-x^4+x^2+2}}","\frac{1}{7} x \left(-x^4+x^2+2\right)^{3/2}+\frac{1}{35} x \left(3 x^2+19\right) \sqrt{-x^4+x^2+2}+\frac{48}{35} F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)+\frac{34}{35} E\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)",1,"(58*x + 45*x^3 - 31*x^5 - 13*x^7 + 5*x^9 + (34*I)*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticE[I*ArcSinh[x], -1/2] - (75*I)*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticF[I*ArcSinh[x], -1/2])/(35*Sqrt[2 + x^2 - x^4])","C",1
329,1,130,72,0.1955193,"\int \frac{\left(2+x^2-x^4\right)^{3/2}}{7+5 x^2} \, dx","Integrate[(2 + x^2 - x^4)^(3/2)/(7 + 5*x^2),x]","\frac{525 x^7-2800 x^5+1225 x^3-2961 i \sqrt{-2 x^4+2 x^2+4} F\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)+3220 i \sqrt{-2 x^4+2 x^2+4} E\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)-1734 i \sqrt{-2 x^4+2 x^2+4} \Pi \left(\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right)+4550 x}{13125 \sqrt{-x^4+x^2+2}}","\frac{1}{75} x \sqrt{-x^4+x^2+2} \left(13-3 x^2\right)-\frac{178}{625} F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)+\frac{92}{375} E\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)+\frac{1156 \Pi \left(-\frac{10}{7};\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)}{4375}",1,"(4550*x + 1225*x^3 - 2800*x^5 + 525*x^7 + (3220*I)*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticE[I*ArcSinh[x], -1/2] - (2961*I)*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticF[I*ArcSinh[x], -1/2] - (1734*I)*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticPi[5/7, I*ArcSinh[x], -1/2])/(13125*Sqrt[2 + x^2 - x^4])","C",1
330,1,201,93,0.3136718,"\int \frac{\left(2+x^2-x^4\right)^{3/2}}{\left(7+5 x^2\right)^2} \, dx","Integrate[(2 + x^2 - x^4)^(3/2)/(7 + 5*x^2)^2,x]","\frac{2450 x^7+4550 x^5-11900 x^3+567 i \sqrt{2} \left(5 x^2+7\right) \sqrt{-x^4+x^2+2} F\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)-6790 i \sqrt{2} \left(5 x^2+7\right) \sqrt{-x^4+x^2+2} E\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)+18615 i \sqrt{2} \sqrt{-x^4+x^2+2} x^2 \Pi \left(\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right)+26061 i \sqrt{2} \sqrt{-x^4+x^2+2} \Pi \left(\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right)-14000 x}{36750 \left(5 x^2+7\right) \sqrt{-x^4+x^2+2}}","-\frac{17 \sqrt{-x^4+x^2+2} x}{175 \left(5 x^2+7\right)}-\frac{1}{75} \sqrt{-x^4+x^2+2} x+\frac{458}{875} F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)-\frac{97}{525} E\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)-\frac{1241 \Pi \left(-\frac{10}{7};\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)}{6125}",1,"(-14000*x - 11900*x^3 + 4550*x^5 + 2450*x^7 - (6790*I)*Sqrt[2]*(7 + 5*x^2)*Sqrt[2 + x^2 - x^4]*EllipticE[I*ArcSinh[x], -1/2] + (567*I)*Sqrt[2]*(7 + 5*x^2)*Sqrt[2 + x^2 - x^4]*EllipticF[I*ArcSinh[x], -1/2] + (26061*I)*Sqrt[2]*Sqrt[2 + x^2 - x^4]*EllipticPi[5/7, I*ArcSinh[x], -1/2] + (18615*I)*Sqrt[2]*x^2*Sqrt[2 + x^2 - x^4]*EllipticPi[5/7, I*ArcSinh[x], -1/2])/(36750*(7 + 5*x^2)*Sqrt[2 + x^2 - x^4])","C",1
331,1,244,102,0.4071519,"\int \frac{\left(2+x^2-x^4\right)^{3/2}}{\left(7+5 x^2\right)^3} \, dx","Integrate[(2 + x^2 - x^4)^(3/2)/(7 + 5*x^2)^3,x]","\frac{-197050 x^7-45500 x^5+636650 x^3-2541 i \sqrt{2} \left(5 x^2+7\right)^2 \sqrt{-x^4+x^2+2} F\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)+13370 i \sqrt{2} \left(5 x^2+7\right)^2 \sqrt{-x^4+x^2+2} E\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)-246975 i \sqrt{2} \sqrt{-x^4+x^2+2} x^4 \Pi \left(\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right)-691530 i \sqrt{2} \sqrt{-x^4+x^2+2} x^2 \Pi \left(\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right)-484071 i \sqrt{2} \sqrt{-x^4+x^2+2} \Pi \left(\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right)+485100 x}{686000 \left(5 x^2+7\right)^2 \sqrt{-x^4+x^2+2}}","\frac{563 \sqrt{-x^4+x^2+2} x}{9800 \left(5 x^2+7\right)}-\frac{17 \sqrt{-x^4+x^2+2} x}{350 \left(5 x^2+7\right)^2}-\frac{1251 F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)}{24500}+\frac{191 E\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)}{9800}+\frac{9879 \Pi \left(-\frac{10}{7};\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)}{343000}",1,"(485100*x + 636650*x^3 - 45500*x^5 - 197050*x^7 + (13370*I)*Sqrt[2]*(7 + 5*x^2)^2*Sqrt[2 + x^2 - x^4]*EllipticE[I*ArcSinh[x], -1/2] - (2541*I)*Sqrt[2]*(7 + 5*x^2)^2*Sqrt[2 + x^2 - x^4]*EllipticF[I*ArcSinh[x], -1/2] - (484071*I)*Sqrt[2]*Sqrt[2 + x^2 - x^4]*EllipticPi[5/7, I*ArcSinh[x], -1/2] - (691530*I)*Sqrt[2]*x^2*Sqrt[2 + x^2 - x^4]*EllipticPi[5/7, I*ArcSinh[x], -1/2] - (246975*I)*Sqrt[2]*x^4*Sqrt[2 + x^2 - x^4]*EllipticPi[5/7, I*ArcSinh[x], -1/2])/(686000*(7 + 5*x^2)^2*Sqrt[2 + x^2 - x^4])","C",1
332,1,97,65,0.1143953,"\int \frac{\left(7+5 x^2\right)^3}{\sqrt{2+x^2-x^4}} \, dx","Integrate[(7 + 5*x^2)^3/Sqrt[2 + x^2 - x^4],x]","\frac{150 x^7+1100 x^5-1550 x^3-10089 i \sqrt{-2 x^4+2 x^2+4} F\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)+7810 i \sqrt{-2 x^4+2 x^2+4} E\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)-2500 x}{6 \sqrt{-x^4+x^2+2}}","-\frac{625}{3} \sqrt{-x^4+x^2+2} x-25 \sqrt{-x^4+x^2+2} x^3-542 F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)+\frac{3905}{3} E\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)",1,"(-2500*x - 1550*x^3 + 1100*x^5 + 150*x^7 + (7810*I)*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticE[I*ArcSinh[x], -1/2] - (10089*I)*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticF[I*ArcSinh[x], -1/2])/(6*Sqrt[2 + x^2 - x^4])","C",1
333,1,92,46,0.0967168,"\int \frac{\left(7+5 x^2\right)^2}{\sqrt{2+x^2-x^4}} \, dx","Integrate[(7 + 5*x^2)^2/Sqrt[2 + x^2 - x^4],x]","\frac{50 x^5-50 x^3-717 i \sqrt{-2 x^4+2 x^2+4} F\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)+520 i \sqrt{-2 x^4+2 x^2+4} E\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)-100 x}{6 \sqrt{-x^4+x^2+2}}","-\frac{25}{3} \sqrt{-x^4+x^2+2} x-21 F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)+\frac{260}{3} E\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)",1,"(-100*x - 50*x^3 + 50*x^5 + (520*I)*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticE[I*ArcSinh[x], -1/2] - (717*I)*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticF[I*ArcSinh[x], -1/2])/(6*Sqrt[2 + x^2 - x^4])","C",1
334,1,34,25,0.0592508,"\int \frac{7+5 x^2}{\sqrt{2+x^2-x^4}} \, dx","Integrate[(7 + 5*x^2)/Sqrt[2 + x^2 - x^4],x]","\frac{i \left(10 E\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)-17 F\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)\right)}{\sqrt{2}}","2 F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)+5 E\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)",1,"(I*(10*EllipticE[I*ArcSinh[x], -1/2] - 17*EllipticF[I*ArcSinh[x], -1/2]))/Sqrt[2]","C",1
335,1,19,10,0.0156281,"\int \frac{1}{\sqrt{2+x^2-x^4}} \, dx","Integrate[1/Sqrt[2 + x^2 - x^4],x]","-\frac{i F\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)}{\sqrt{2}}","F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)",1,"((-I)*EllipticF[I*ArcSinh[x], -1/2])/Sqrt[2]","C",1
336,1,24,17,0.0981033,"\int \frac{1}{\left(7+5 x^2\right) \sqrt{2+x^2-x^4}} \, dx","Integrate[1/((7 + 5*x^2)*Sqrt[2 + x^2 - x^4]),x]","-\frac{i \Pi \left(\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right)}{7 \sqrt{2}}","\frac{1}{7} \Pi \left(-\frac{10}{7};\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)",1,"((-1/7*I)*EllipticPi[5/7, I*ArcSinh[x], -1/2])/Sqrt[2]","C",1
337,1,196,74,0.285125,"\int \frac{1}{\left(7+5 x^2\right)^2 \sqrt{2+x^2-x^4}} \, dx","Integrate[1/((7 + 5*x^2)^2*Sqrt[2 + x^2 - x^4]),x]","\frac{350 x^5-350 x^3+119 i \sqrt{2} \left(5 x^2+7\right) \sqrt{-x^4+x^2+2} F\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)-70 i \sqrt{2} \left(5 x^2+7\right) \sqrt{-x^4+x^2+2} E\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)-835 i \sqrt{2} \sqrt{-x^4+x^2+2} x^2 \Pi \left(\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right)-1169 i \sqrt{2} \sqrt{-x^4+x^2+2} \Pi \left(\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right)-700 x}{6664 \left(5 x^2+7\right) \sqrt{-x^4+x^2+2}}","-\frac{25 \sqrt{-x^4+x^2+2} x}{476 \left(5 x^2+7\right)}-\frac{1}{238} F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)-\frac{5}{476} E\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)+\frac{167 \Pi \left(-\frac{10}{7};\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)}{3332}",1,"(-700*x - 350*x^3 + 350*x^5 - (70*I)*Sqrt[2]*(7 + 5*x^2)*Sqrt[2 + x^2 - x^4]*EllipticE[I*ArcSinh[x], -1/2] + (119*I)*Sqrt[2]*(7 + 5*x^2)*Sqrt[2 + x^2 - x^4]*EllipticF[I*ArcSinh[x], -1/2] - (1169*I)*Sqrt[2]*Sqrt[2 + x^2 - x^4]*EllipticPi[5/7, I*ArcSinh[x], -1/2] - (835*I)*Sqrt[2]*x^2*Sqrt[2 + x^2 - x^4]*EllipticPi[5/7, I*ArcSinh[x], -1/2])/(6664*(7 + 5*x^2)*Sqrt[2 + x^2 - x^4])","C",1
338,1,108,102,0.4155654,"\int \frac{1}{\left(7+5 x^2\right)^3 \sqrt{2+x^2-x^4}} \, dx","Integrate[1/((7 + 5*x^2)^3*Sqrt[2 + x^2 - x^4]),x]","\frac{\frac{350 x \left(2505 x^6+1478 x^4-8993 x^2-7966\right)}{\left(5 x^2+7\right)^2 \sqrt{-x^4+x^2+2}}+56287 i \sqrt{2} F\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)-35070 i \sqrt{2} E\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)-58915 i \sqrt{2} \Pi \left(\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right)}{6344128}","-\frac{12525 \sqrt{-x^4+x^2+2} x}{453152 \left(5 x^2+7\right)}-\frac{25 \sqrt{-x^4+x^2+2} x}{952 \left(5 x^2+7\right)^2}-\frac{263 F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)}{226576}-\frac{2505 E\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)}{453152}+\frac{58915 \Pi \left(-\frac{10}{7};\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)}{3172064}",1,"((350*x*(-7966 - 8993*x^2 + 1478*x^4 + 2505*x^6))/((7 + 5*x^2)^2*Sqrt[2 + x^2 - x^4]) - (35070*I)*Sqrt[2]*EllipticE[I*ArcSinh[x], -1/2] + (56287*I)*Sqrt[2]*EllipticF[I*ArcSinh[x], -1/2] - (58915*I)*Sqrt[2]*EllipticPi[5/7, I*ArcSinh[x], -1/2])/6344128","C",1
339,-1,0,93,0,"\int \frac{\left(7+5 x^2\right)^5}{\left(2+x^2-x^4\right)^{3/2}} \, dx","Integrate[(7 + 5*x^2)^5/(2 + x^2 - x^4)^(3/2),x]","\text{\$Aborted}","\frac{27500}{3} \sqrt{-x^4+x^2+2} x+\frac{\left(1419793 x^2+1419985\right) x}{18 \sqrt{-x^4+x^2+2}}+625 \sqrt{-x^4+x^2+2} x^3+\frac{627857}{6} F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)-\frac{3482293}{18} E\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)",1,"$Aborted","F",-1
340,-1,0,74,0,"\int \frac{\left(7+5 x^2\right)^4}{\left(2+x^2-x^4\right)^{3/2}} \, dx","Integrate[(7 + 5*x^2)^4/(2 + x^2 - x^4)^(3/2),x]","\text{\$Aborted}","\frac{625}{3} \sqrt{-x^4+x^2+2} x+\frac{\left(83489 x^2+83585\right) x}{18 \sqrt{-x^4+x^2+2}}+\frac{31921}{6} F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)-\frac{165239}{18} E\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)",1,"$Aborted","F",-1
341,-1,0,55,0,"\int \frac{\left(7+5 x^2\right)^3}{\left(2+x^2-x^4\right)^{3/2}} \, dx","Integrate[(7 + 5*x^2)^3/(2 + x^2 - x^4)^(3/2),x]","\text{\$Aborted}","\frac{x \left(4897 x^2+4945\right)}{18 \sqrt{-x^4+x^2+2}}+\frac{1763}{6} F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)-\frac{7147}{18} E\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)",1,"$Aborted","F",-1
342,-1,0,55,0,"\int \frac{\left(7+5 x^2\right)^2}{\left(2+x^2-x^4\right)^{3/2}} \, dx","Integrate[(7 + 5*x^2)^2/(2 + x^2 - x^4)^(3/2),x]","\text{\$Aborted}","\frac{x \left(281 x^2+305\right)}{18 \sqrt{-x^4+x^2+2}}+\frac{139}{6} F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)-\frac{281}{18} E\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)",1,"$Aborted","F",-1
343,-1,0,55,0,"\int \frac{7+5 x^2}{\left(2+x^2-x^4\right)^{3/2}} \, dx","Integrate[(7 + 5*x^2)/(2 + x^2 - x^4)^(3/2),x]","\text{\$Aborted}","\frac{x \left(13 x^2+25\right)}{18 \sqrt{-x^4+x^2+2}}+\frac{17}{6} F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)-\frac{13}{18} E\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)",1,"$Aborted","F",-1
344,-1,0,55,0,"\int \frac{1}{\left(2+x^2-x^4\right)^{3/2}} \, dx","Integrate[(2 + x^2 - x^4)^(-3/2),x]","\text{\$Aborted}","\frac{x \left(5-x^2\right)}{18 \sqrt{-x^4+x^2+2}}+\frac{1}{6} F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)+\frac{1}{18} E\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)",1,"$Aborted","F",-1
345,1,101,72,0.2180035,"\int \frac{1}{\left(7+5 x^2\right) \left(2+x^2-x^4\right)^{3/2}} \, dx","Integrate[1/((7 + 5*x^2)*(2 + x^2 - x^4)^(3/2)),x]","\frac{\frac{490 x}{\sqrt{-x^4+x^2+2}}-\frac{224 x^3}{\sqrt{-x^4+x^2+2}}-357 i \sqrt{2} F\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)+224 i \sqrt{2} E\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)+225 i \sqrt{2} \Pi \left(\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right)}{4284}","\frac{x \left(35-16 x^2\right)}{306 \sqrt{-x^4+x^2+2}}+\frac{1}{102} F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)+\frac{8}{153} E\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)-\frac{25}{238} \Pi \left(-\frac{10}{7};\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)",1,"((490*x)/Sqrt[2 + x^2 - x^4] - (224*x^3)/Sqrt[2 + x^2 - x^4] + (224*I)*Sqrt[2]*EllipticE[I*ArcSinh[x], -1/2] - (357*I)*Sqrt[2]*EllipticF[I*ArcSinh[x], -1/2] + (225*I)*Sqrt[2]*EllipticPi[5/7, I*ArcSinh[x], -1/2])/4284","C",1
346,1,196,100,0.3293229,"\int \frac{1}{\left(7+5 x^2\right)^2 \left(2+x^2-x^4\right)^{3/2}} \, dx","Integrate[1/((7 + 5*x^2)^2*(2 + x^2 - x^4)^(3/2)),x]","\frac{-360010 x^5+253386 x^3-111741 i \sqrt{2} \left(5 x^2+7\right) \sqrt{-x^4+x^2+2} F\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)+72002 i \sqrt{2} \left(5 x^2+7\right) \sqrt{-x^4+x^2+2} E\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)+487125 i \sqrt{2} \sqrt{-x^4+x^2+2} x^2 \Pi \left(\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right)+681975 i \sqrt{2} \sqrt{-x^4+x^2+2} \Pi \left(\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right)+953260 x}{2039184 \left(5 x^2+7\right) \sqrt{-x^4+x^2+2}}","\frac{625 \sqrt{-x^4+x^2+2} x}{16184 \left(5 x^2+7\right)}+\frac{\left(580-287 x^2\right) x}{10404 \sqrt{-x^4+x^2+2}}+\frac{89 F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)}{24276}+\frac{5143 E\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)}{145656}-\frac{10825 \Pi \left(-\frac{10}{7};\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)}{113288}",1,"(953260*x + 253386*x^3 - 360010*x^5 + (72002*I)*Sqrt[2]*(7 + 5*x^2)*Sqrt[2 + x^2 - x^4]*EllipticE[I*ArcSinh[x], -1/2] - (111741*I)*Sqrt[2]*(7 + 5*x^2)*Sqrt[2 + x^2 - x^4]*EllipticF[I*ArcSinh[x], -1/2] + (681975*I)*Sqrt[2]*Sqrt[2 + x^2 - x^4]*EllipticPi[5/7, I*ArcSinh[x], -1/2] + (487125*I)*Sqrt[2]*x^2*Sqrt[2 + x^2 - x^4]*EllipticPi[5/7, I*ArcSinh[x], -1/2])/(2039184*(7 + 5*x^2)*Sqrt[2 + x^2 - x^4])","C",1
347,1,244,128,0.3973345,"\int \frac{1}{\left(7+5 x^2\right)^3 \left(2+x^2-x^4\right)^{3/2}} \, dx","Integrate[1/((7 + 5*x^2)^3*(2 + x^2 - x^4)^(3/2)),x]","\frac{-1080258550 x^7-737347940 x^5+3876617542 x^3-67352691 i \sqrt{2} \left(5 x^2+7\right)^2 \sqrt{-x^4+x^2+2} F\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)+43210342 i \sqrt{2} \left(5 x^2+7\right)^2 \sqrt{-x^4+x^2+2} E\left(i \sinh ^{-1}(x)|-\frac{1}{2}\right)+1552179375 i \sqrt{2} \sqrt{-x^4+x^2+2} x^4 \Pi \left(\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right)+4346102250 i \sqrt{2} \sqrt{-x^4+x^2+2} x^2 \Pi \left(\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right)+3042271575 i \sqrt{2} \sqrt{-x^4+x^2+2} \Pi \left(\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right)+3857257460 x}{1941303168 \left(5 x^2+7\right)^2 \sqrt{-x^4+x^2+2}}","\frac{645625 \sqrt{-x^4+x^2+2} x}{15407168 \left(5 x^2+7\right)}+\frac{625 \sqrt{-x^4+x^2+2} x}{32368 \left(5 x^2+7\right)^2}+\frac{\left(9830-4909 x^2\right) x}{353736 \sqrt{-x^4+x^2+2}}+\frac{60409 F\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)}{23110752}+\frac{3086453 E\left(\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)}{138664512}-\frac{6898575 \Pi \left(-\frac{10}{7};\left.\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|-2\right)}{107850176}",1,"(3857257460*x + 3876617542*x^3 - 737347940*x^5 - 1080258550*x^7 + (43210342*I)*Sqrt[2]*(7 + 5*x^2)^2*Sqrt[2 + x^2 - x^4]*EllipticE[I*ArcSinh[x], -1/2] - (67352691*I)*Sqrt[2]*(7 + 5*x^2)^2*Sqrt[2 + x^2 - x^4]*EllipticF[I*ArcSinh[x], -1/2] + (3042271575*I)*Sqrt[2]*Sqrt[2 + x^2 - x^4]*EllipticPi[5/7, I*ArcSinh[x], -1/2] + (4346102250*I)*Sqrt[2]*x^2*Sqrt[2 + x^2 - x^4]*EllipticPi[5/7, I*ArcSinh[x], -1/2] + (1552179375*I)*Sqrt[2]*x^4*Sqrt[2 + x^2 - x^4]*EllipticPi[5/7, I*ArcSinh[x], -1/2])/(1941303168*(7 + 5*x^2)^2*Sqrt[2 + x^2 - x^4])","C",1
348,1,354,242,0.5854144,"\int \left(7+5 x^2\right)^4 \sqrt{4+3 x^2+x^4} \, dx","Integrate[(7 + 5*x^2)^4*Sqrt[4 + 3*x^2 + x^4],x]","\frac{3 \sqrt{2} \left(51665 \sqrt{7}-36253 i\right) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)-154995 \sqrt{2} \left(\sqrt{7}+3 i\right) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} E\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)+4 \sqrt{-\frac{i}{\sqrt{7}-3 i}} x \left(5625 x^{12}+57250 x^{10}+264075 x^8+712748 x^6+1217475 x^4+1257535 x^2+663924\right)}{396 \sqrt{-\frac{i}{\sqrt{7}-3 i}} \sqrt{x^4+3 x^2+4}}","\frac{3050}{11} \left(x^4+3 x^2+4\right)^{3/2} x+\frac{1}{33} \left(4516 x^2+18727\right) \sqrt{x^4+3 x^2+4} x+\frac{51665 \sqrt{x^4+3 x^2+4} x}{33 \left(x^2+2\right)}+\frac{33159 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{11 \sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{51665 \sqrt{2} \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{33 \sqrt{x^4+3 x^2+4}}+\frac{625}{11} \left(x^4+3 x^2+4\right)^{3/2} x^5+\frac{23500}{99} \left(x^4+3 x^2+4\right)^{3/2} x^3",1,"(4*Sqrt[(-I)/(-3*I + Sqrt[7])]*x*(663924 + 1257535*x^2 + 1217475*x^4 + 712748*x^6 + 264075*x^8 + 57250*x^10 + 5625*x^12) - 154995*Sqrt[2]*(3*I + Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + 3*Sqrt[2]*(-36253*I + 51665*Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])])/(396*Sqrt[(-I)/(-3*I + Sqrt[7])]*Sqrt[4 + 3*x^2 + x^4])","C",1
349,1,349,221,0.5048586,"\int \left(7+5 x^2\right)^3 \sqrt{4+3 x^2+x^4} \, dx","Integrate[(7 + 5*x^2)^3*Sqrt[4 + 3*x^2 + x^4],x]","\frac{3 \sqrt{2} \left(4717 \sqrt{7}-3409 i\right) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)-14151 \sqrt{2} \left(\sqrt{7}+3 i\right) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} E\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)+4 \sqrt{-\frac{i}{\sqrt{7}-3 i}} x \left(875 x^{10}+7725 x^8+30946 x^6+71862 x^4+93656 x^2+60096\right)}{252 \sqrt{-\frac{i}{\sqrt{7}-3 i}} \sqrt{x^4+3 x^2+4}}","\frac{275}{7} \left(x^4+3 x^2+4\right)^{3/2} x+\frac{1}{21} \left(407 x^2+1708\right) \sqrt{x^4+3 x^2+4} x+\frac{4717 \sqrt{x^4+3 x^2+4} x}{21 \left(x^2+2\right)}+\frac{1301 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{3 \sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{4717 \sqrt{2} \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{21 \sqrt{x^4+3 x^2+4}}+\frac{125}{9} \left(x^4+3 x^2+4\right)^{3/2} x^3",1,"(4*Sqrt[(-I)/(-3*I + Sqrt[7])]*x*(60096 + 93656*x^2 + 71862*x^4 + 30946*x^6 + 7725*x^8 + 875*x^10) - 14151*Sqrt[2]*(3*I + Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + 3*Sqrt[2]*(-3409*I + 4717*Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])])/(252*Sqrt[(-I)/(-3*I + Sqrt[7])]*Sqrt[4 + 3*x^2 + x^4])","C",1
350,1,343,198,0.4703577,"\int \left(7+5 x^2\right)^2 \sqrt{4+3 x^2+x^4} \, dx","Integrate[(7 + 5*x^2)^2*Sqrt[4 + 3*x^2 + x^4],x]","\frac{\sqrt{2} \left(319 \sqrt{7}-35 i\right) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)-319 \sqrt{2} \left(\sqrt{7}+3 i\right) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} E\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)+4 \sqrt{-\frac{i}{\sqrt{7}-3 i}} x \left(25 x^8+188 x^6+658 x^4+1109 x^2+876\right)}{28 \sqrt{-\frac{i}{\sqrt{7}-3 i}} \sqrt{x^4+3 x^2+4}}","\frac{25}{7} x \left(x^4+3 x^2+4\right)^{3/2}+\frac{1}{7} x \left(38 x^2+119\right) \sqrt{x^4+3 x^2+4}+\frac{319 x \sqrt{x^4+3 x^2+4}}{7 \left(x^2+2\right)}+\frac{81 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{\sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{319 \sqrt{2} \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{7 \sqrt{x^4+3 x^2+4}}",1,"(4*Sqrt[(-I)/(-3*I + Sqrt[7])]*x*(876 + 1109*x^2 + 658*x^4 + 188*x^6 + 25*x^8) - 319*Sqrt[2]*(3*I + Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + Sqrt[2]*(-35*I + 319*Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])])/(28*Sqrt[(-I)/(-3*I + Sqrt[7])]*Sqrt[4 + 3*x^2 + x^4])","C",1
351,1,338,177,0.4307857,"\int \left(7+5 x^2\right) \sqrt{4+3 x^2+x^4} \, dx","Integrate[(7 + 5*x^2)*Sqrt[4 + 3*x^2 + x^4],x]","\frac{\sqrt{2} \left(27 \sqrt{7}-7 i\right) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)-27 \sqrt{2} \left(\sqrt{7}+3 i\right) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} E\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)+4 \sqrt{-\frac{i}{\sqrt{7}-3 i}} x \left(3 x^6+19 x^4+42 x^2+40\right)}{12 \sqrt{-\frac{i}{\sqrt{7}-3 i}} \sqrt{x^4+3 x^2+4}}","\frac{1}{3} \left(3 x^2+10\right) \sqrt{x^4+3 x^2+4} x+\frac{9 \sqrt{x^4+3 x^2+4} x}{x^2+2}+\frac{49 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{3 \sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{9 \sqrt{2} \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{\sqrt{x^4+3 x^2+4}}",1,"(4*Sqrt[(-I)/(-3*I + Sqrt[7])]*x*(40 + 42*x^2 + 19*x^4 + 3*x^6) - 27*Sqrt[2]*(3*I + Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + Sqrt[2]*(-7*I + 27*Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])])/(12*Sqrt[(-I)/(-3*I + Sqrt[7])]*Sqrt[4 + 3*x^2 + x^4])","C",1
352,1,331,169,0.3518406,"\int \sqrt{4+3 x^2+x^4} \, dx","Integrate[Sqrt[4 + 3*x^2 + x^4],x]","\frac{\sqrt{2} \left(3 \sqrt{7}-7 i\right) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)-3 \sqrt{2} \left(\sqrt{7}+3 i\right) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} E\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)+4 \sqrt{-\frac{i}{\sqrt{7}-3 i}} x \left(x^4+3 x^2+4\right)}{12 \sqrt{-\frac{i}{\sqrt{7}-3 i}} \sqrt{x^4+3 x^2+4}}","\frac{\sqrt{x^4+3 x^2+4} x}{x^2+2}+\frac{1}{3} \sqrt{x^4+3 x^2+4} x+\frac{7 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{3 \sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{\sqrt{2} \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{\sqrt{x^4+3 x^2+4}}",1,"(4*Sqrt[(-I)/(-3*I + Sqrt[7])]*x*(4 + 3*x^2 + x^4) - 3*Sqrt[2]*(3*I + Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + Sqrt[2]*(-7*I + 3*Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])])/(12*Sqrt[(-I)/(-3*I + Sqrt[7])]*Sqrt[4 + 3*x^2 + x^4])","C",1
353,1,283,322,0.2514946,"\int \frac{\sqrt{4+3 x^2+x^4}}{7+5 x^2} \, dx","Integrate[Sqrt[4 + 3*x^2 + x^4]/(7 + 5*x^2),x]","-\frac{\sqrt{1-\frac{2 i x^2}{\sqrt{7}-3 i}} \sqrt{1+\frac{2 i x^2}{\sqrt{7}+3 i}} \left(\left(-35 \sqrt{7}+7 i\right) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)+35 \left(\sqrt{7}+3 i\right) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)+88 i \Pi \left(\frac{5}{14} \left(3+i \sqrt{7}\right);i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)\right)}{350 \sqrt{2} \sqrt{-\frac{i}{\sqrt{7}-3 i}} \sqrt{x^4+3 x^2+4}}","\frac{\sqrt{x^4+3 x^2+4} x}{5 \left(x^2+2\right)}+\frac{1}{5} \sqrt{\frac{11}{35}} \tan ^{-1}\left(\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{x^4+3 x^2+4}}\right)-\frac{11 \sqrt{2} \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{75 \sqrt{x^4+3 x^2+4}}+\frac{9 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{25 \sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{\sqrt{2} \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{5 \sqrt{x^4+3 x^2+4}}+\frac{187 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} \Pi \left(-\frac{9}{280};2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{525 \sqrt{2} \sqrt{x^4+3 x^2+4}}",1,"-1/350*(Sqrt[1 - ((2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[1 + ((2*I)*x^2)/(3*I + Sqrt[7])]*(35*(3*I + Sqrt[7])*EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + (7*I - 35*Sqrt[7])*EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + (88*I)*EllipticPi[(5*(3 + I*Sqrt[7]))/14, I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])]))/(Sqrt[2]*Sqrt[(-I)/(-3*I + Sqrt[7])]*Sqrt[4 + 3*x^2 + x^4])","C",1
354,1,481,284,0.7683245,"\int \frac{\sqrt{4+3 x^2+x^4}}{\left(7+5 x^2\right)^2} \, dx","Integrate[Sqrt[4 + 3*x^2 + x^4]/(7 + 5*x^2)^2,x]","\frac{-98 i \left(5 x^2+7\right) \sqrt{2-\frac{4 i x^2}{\sqrt{7}-3 i}} \sqrt{1+\frac{2 i x^2}{\sqrt{7}+3 i}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)-102 i \left(5 x^2+7\right) \sqrt{2-\frac{4 i x^2}{\sqrt{7}-3 i}} \sqrt{1+\frac{2 i x^2}{\sqrt{7}+3 i}} \Pi \left(\frac{5}{14} \left(3+i \sqrt{7}\right);i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)+35 \left(\sqrt{7}+3 i\right) \left(5 x^2+7\right) \sqrt{2-\frac{4 i x^2}{\sqrt{7}-3 i}} \sqrt{1+\frac{2 i x^2}{\sqrt{7}+3 i}} \left(E\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)-F\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)\right)+700 \sqrt{-\frac{i}{\sqrt{7}-3 i}} x \left(x^4+3 x^2+4\right)}{9800 \sqrt{-\frac{i}{\sqrt{7}-3 i}} \left(5 x^2+7\right) \sqrt{x^4+3 x^2+4}}","-\frac{\sqrt{x^4+3 x^2+4} x}{70 \left(x^2+2\right)}+\frac{\sqrt{x^4+3 x^2+4} x}{14 \left(5 x^2+7\right)}+\frac{51 \tan ^{-1}\left(\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{x^4+3 x^2+4}}\right)}{280 \sqrt{385}}-\frac{\left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{35 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{\left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{35 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{289 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} \Pi \left(-\frac{9}{280};2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{9800 \sqrt{2} \sqrt{x^4+3 x^2+4}}",1,"(700*Sqrt[(-I)/(-3*I + Sqrt[7])]*x*(4 + 3*x^2 + x^4) + 35*(3*I + Sqrt[7])*(7 + 5*x^2)*Sqrt[2 - ((4*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[1 + ((2*I)*x^2)/(3*I + Sqrt[7])]*(EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] - EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])]) - (98*I)*(7 + 5*x^2)*Sqrt[2 - ((4*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[1 + ((2*I)*x^2)/(3*I + Sqrt[7])]*EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] - (102*I)*(7 + 5*x^2)*Sqrt[2 - ((4*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[1 + ((2*I)*x^2)/(3*I + Sqrt[7])]*EllipticPi[(5*(3 + I*Sqrt[7]))/14, I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])])/(9800*Sqrt[(-I)/(-3*I + Sqrt[7])]*(7 + 5*x^2)*Sqrt[4 + 3*x^2 + x^4])","C",1
355,1,308,312,0.679516,"\int \frac{\sqrt{4+3 x^2+x^4}}{\left(7+5 x^2\right)^3} \, dx","Integrate[Sqrt[4 + 3*x^2 + x^4]/(7 + 5*x^2)^3,x]","\frac{\frac{700 x \left(695 x^2+1589\right) \left(x^4+3 x^2+4\right)}{\left(5 x^2+7\right)^2}+i \sqrt{6+2 i \sqrt{7}} \sqrt{1-\frac{2 i x^2}{\sqrt{7}-3 i}} \sqrt{1+\frac{2 i x^2}{\sqrt{7}+3 i}} \left(\left(-9597+4865 i \sqrt{7}\right) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)+4865 \left(3-i \sqrt{7}\right) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)-29998 \Pi \left(\frac{5}{14} \left(3+i \sqrt{7}\right);i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)\right)}{12073600 \sqrt{x^4+3 x^2+4}}","-\frac{139 \sqrt{x^4+3 x^2+4} x}{86240 \left(x^2+2\right)}+\frac{139 \sqrt{x^4+3 x^2+4} x}{17248 \left(5 x^2+7\right)}+\frac{\sqrt{x^4+3 x^2+4} x}{28 \left(5 x^2+7\right)^2}+\frac{14999 \tan ^{-1}\left(\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{x^4+3 x^2+4}}\right)}{344960 \sqrt{385}}-\frac{23 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{2940 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{139 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{43120 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{254983 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} \Pi \left(-\frac{9}{280};2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{36220800 \sqrt{2} \sqrt{x^4+3 x^2+4}}",1,"((700*x*(1589 + 695*x^2)*(4 + 3*x^2 + x^4))/(7 + 5*x^2)^2 + I*Sqrt[6 + (2*I)*Sqrt[7]]*Sqrt[1 - ((2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[1 + ((2*I)*x^2)/(3*I + Sqrt[7])]*(4865*(3 - I*Sqrt[7])*EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + (-9597 + (4865*I)*Sqrt[7])*EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] - 29998*EllipticPi[(5*(3 + I*Sqrt[7]))/14, I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])]))/(12073600*Sqrt[4 + 3*x^2 + x^4])","C",1
356,-1,0,268,0,"\int \left(7+5 x^2\right)^4 \left(4+3 x^2+x^4\right)^{3/2} \, dx","Integrate[(7 + 5*x^2)^4*(4 + 3*x^2 + x^4)^(3/2),x]","\text{\$Aborted}","\frac{92150}{429} \left(x^4+3 x^2+4\right)^{5/2} x+\frac{\left(131080 x^2+452001\right) \left(x^4+3 x^2+4\right)^{3/2} x}{1287}+\frac{7 \left(174989 x^2+661429\right) \sqrt{x^4+3 x^2+4} x}{2145}+\frac{12665086 \sqrt{x^4+3 x^2+4} x}{2145 \left(x^2+2\right)}+\frac{2383556 \sqrt{2} \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{429 \sqrt{x^4+3 x^2+4}}-\frac{12665086 \sqrt{2} \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{2145 \sqrt{x^4+3 x^2+4}}+\frac{125}{3} \left(x^4+3 x^2+4\right)^{5/2} x^5+\frac{2250}{13} \left(x^4+3 x^2+4\right)^{5/2} x^3",1,"$Aborted","F",-1
357,-1,0,247,0,"\int \left(7+5 x^2\right)^3 \left(4+3 x^2+x^4\right)^{3/2} \, dx","Integrate[(7 + 5*x^2)^3*(4 + 3*x^2 + x^4)^(3/2),x]","\text{\$Aborted}","\frac{3825}{143} \left(x^4+3 x^2+4\right)^{5/2} x+\frac{\left(15365 x^2+53504\right) \left(x^4+3 x^2+4\right)^{3/2} x}{1001}+\frac{\left(435441 x^2+1653701\right) \sqrt{x^4+3 x^2+4} x}{5005}+\frac{4525662 \sqrt{x^4+3 x^2+4} x}{5005 \left(x^2+2\right)}+\frac{121826 \sqrt{2} \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{143 \sqrt{x^4+3 x^2+4}}-\frac{4525662 \sqrt{2} \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{5005 \sqrt{x^4+3 x^2+4}}+\frac{125}{13} \left(x^4+3 x^2+4\right)^{5/2} x^3",1,"$Aborted","F",-1
358,-1,0,226,0,"\int \left(7+5 x^2\right)^2 \left(4+3 x^2+x^4\right)^{3/2} \, dx","Integrate[(7 + 5*x^2)^2*(4 + 3*x^2 + x^4)^(3/2),x]","\text{\$Aborted}","\frac{25}{11} x \left(x^4+3 x^2+4\right)^{5/2}+\frac{1}{693} x \left(2240 x^2+6831\right) \left(x^4+3 x^2+4\right)^{3/2}+\frac{x \left(18253 x^2+64533\right) \sqrt{x^4+3 x^2+4}}{1155}+\frac{175346 x \sqrt{x^4+3 x^2+4}}{1155 \left(x^2+2\right)}+\frac{4628 \sqrt{2} \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{33 \sqrt{x^4+3 x^2+4}}-\frac{175346 \sqrt{2} \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{1155 \sqrt{x^4+3 x^2+4}}",1,"$Aborted","F",-1
359,-1,0,207,0,"\int \left(7+5 x^2\right) \left(4+3 x^2+x^4\right)^{3/2} \, dx","Integrate[(7 + 5*x^2)*(4 + 3*x^2 + x^4)^(3/2),x]","\text{\$Aborted}","\frac{1}{63} x \left(35 x^2+108\right) \left(x^4+3 x^2+4\right)^{3/2}+\frac{1}{105} x \left(289 x^2+1029\right) \sqrt{x^4+3 x^2+4}+\frac{2798 x \sqrt{x^4+3 x^2+4}}{105 \left(x^2+2\right)}+\frac{74 \sqrt{2} \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{3 \sqrt{x^4+3 x^2+4}}-\frac{2798 \sqrt{2} \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{105 \sqrt{x^4+3 x^2+4}}",1,"$Aborted","F",-1
360,1,343,198,0.4195888,"\int \left(4+3 x^2+x^4\right)^{3/2} \, dx","Integrate[(4 + 3*x^2 + x^4)^(3/2),x]","\frac{\sqrt{2} \left(69 \sqrt{7}-77 i\right) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)-69 \sqrt{2} \left(\sqrt{7}+3 i\right) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} E\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)+2 \sqrt{-\frac{i}{\sqrt{7}-3 i}} x \left(5 x^8+39 x^6+161 x^4+303 x^2+276\right)}{70 \sqrt{-\frac{i}{\sqrt{7}-3 i}} \sqrt{x^4+3 x^2+4}}","\frac{1}{7} x \left(x^4+3 x^2+4\right)^{3/2}+\frac{1}{35} x \left(9 x^2+49\right) \sqrt{x^4+3 x^2+4}+\frac{138 x \sqrt{x^4+3 x^2+4}}{35 \left(x^2+2\right)}+\frac{4 \sqrt{2} \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{\sqrt{x^4+3 x^2+4}}-\frac{138 \sqrt{2} \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{35 \sqrt{x^4+3 x^2+4}}",1,"(2*Sqrt[(-I)/(-3*I + Sqrt[7])]*x*(276 + 303*x^2 + 161*x^4 + 39*x^6 + 5*x^8) - 69*Sqrt[2]*(3*I + Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + Sqrt[2]*(-77*I + 69*Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])])/(70*Sqrt[(-I)/(-3*I + Sqrt[7])]*Sqrt[4 + 3*x^2 + x^4])","C",1
361,1,477,284,0.7240646,"\int \frac{\left(4+3 x^2+x^4\right)^{3/2}}{7+5 x^2} \, dx","Integrate[(4 + 3*x^2 + x^4)^(3/2)/(7 + 5*x^2),x]","\frac{7 \sqrt{2} \left(705 \sqrt{7}-241 i\right) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)-4935 \sqrt{2} \left(\sqrt{7}+3 i\right) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} E\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)-5808 i \sqrt{2} \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} \Pi \left(\frac{5}{14} \left(3+i \sqrt{7}\right);i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)+350 \sqrt{-\frac{i}{\sqrt{7}-3 i}} x \left(3 x^6+20 x^4+45 x^2+44\right)}{26250 \sqrt{-\frac{i}{\sqrt{7}-3 i}} \sqrt{x^4+3 x^2+4}}","\frac{1}{75} \left(3 x^2+11\right) \sqrt{x^4+3 x^2+4} x+\frac{94 \sqrt{x^4+3 x^2+4} x}{125 \left(x^2+2\right)}+\frac{44}{125} \sqrt{\frac{11}{35}} \tan ^{-1}\left(\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{x^4+3 x^2+4}}\right)+\frac{54 \sqrt{2} \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{125 \sqrt{x^4+3 x^2+4}}-\frac{94 \sqrt{2} \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{125 \sqrt{x^4+3 x^2+4}}+\frac{4114 \sqrt{2} \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} \Pi \left(-\frac{9}{280};2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{13125 \sqrt{x^4+3 x^2+4}}",1,"(350*Sqrt[(-I)/(-3*I + Sqrt[7])]*x*(44 + 45*x^2 + 20*x^4 + 3*x^6) - 4935*Sqrt[2]*(3*I + Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + 7*Sqrt[2]*(-241*I + 705*Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] - (5808*I)*Sqrt[2]*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticPi[(5*(3 + I*Sqrt[7]))/14, I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])])/(26250*Sqrt[(-I)/(-3*I + Sqrt[7])]*Sqrt[4 + 3*x^2 + x^4])","C",1
362,1,309,305,0.5690189,"\int \frac{\left(4+3 x^2+x^4\right)^{3/2}}{\left(7+5 x^2\right)^2} \, dx","Integrate[(4 + 3*x^2 + x^4)^(3/2)/(7 + 5*x^2)^2,x]","\frac{\frac{175 x \left(7 x^2+23\right) \left(x^4+3 x^2+4\right)}{5 x^2+7}-i \sqrt{6+2 i \sqrt{7}} \sqrt{1-\frac{2 i x^2}{\sqrt{7}-3 i}} \sqrt{1+\frac{2 i x^2}{\sqrt{7}+3 i}} \left(7 \left(158+15 i \sqrt{7}\right) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)+105 \left(3-i \sqrt{7}\right) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)+429 \Pi \left(\frac{5}{14} \left(3+i \sqrt{7}\right);i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)\right)}{18375 \sqrt{x^4+3 x^2+4}}","\frac{4 \sqrt{x^4+3 x^2+4} x}{175 \left(x^2+2\right)}+\frac{22 \sqrt{x^4+3 x^2+4} x}{175 \left(5 x^2+7\right)}+\frac{1}{75} \sqrt{x^4+3 x^2+4} x+\frac{13}{350} \sqrt{\frac{11}{35}} \tan ^{-1}\left(\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{x^4+3 x^2+4}}\right)+\frac{4 \sqrt{2} \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{175 \sqrt{x^4+3 x^2+4}}-\frac{4 \sqrt{2} \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{175 \sqrt{x^4+3 x^2+4}}+\frac{2431 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} \Pi \left(-\frac{9}{280};2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{36750 \sqrt{2} \sqrt{x^4+3 x^2+4}}",1,"((175*x*(23 + 7*x^2)*(4 + 3*x^2 + x^4))/(7 + 5*x^2) - I*Sqrt[6 + (2*I)*Sqrt[7]]*Sqrt[1 - ((2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[1 + ((2*I)*x^2)/(3*I + Sqrt[7])]*(105*(3 - I*Sqrt[7])*EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + 7*(158 + (15*I)*Sqrt[7])*EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + 429*EllipticPi[(5*(3 + I*Sqrt[7]))/14, I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])]))/(18375*Sqrt[4 + 3*x^2 + x^4])","C",1
363,1,309,440,0.687884,"\int \frac{\left(4+3 x^2+x^4\right)^{3/2}}{\left(7+5 x^2\right)^3} \, dx","Integrate[(4 + 3*x^2 + x^4)^(3/2)/(7 + 5*x^2)^3,x]","\frac{\frac{140 x \left(167 x^2+357\right) \left(x^4+3 x^2+4\right)}{\left(5 x^2+7\right)^2}-i \sqrt{6+2 i \sqrt{7}} \sqrt{1-\frac{2 i x^2}{\sqrt{7}-3 i}} \sqrt{1+\frac{2 i x^2}{\sqrt{7}+3 i}} \left(7 \left(103+45 i \sqrt{7}\right) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)+315 \left(3-i \sqrt{7}\right) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)+2694 \Pi \left(\frac{5}{14} \left(3+i \sqrt{7}\right);i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)\right)}{274400 \sqrt{x^4+3 x^2+4}}","\frac{9 \sqrt{x^4+3 x^2+4} x}{1960 \left(x^2+2\right)}+\frac{167 \sqrt{x^4+3 x^2+4} x}{9800 \left(5 x^2+7\right)}+\frac{11 \sqrt{x^4+3 x^2+4} x}{175 \left(5 x^2+7\right)^2}+\frac{1347 \tan ^{-1}\left(\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{x^4+3 x^2+4}}\right)}{7840 \sqrt{385}}-\frac{22 \sqrt{2} \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{13125 \sqrt{x^4+3 x^2+4}}-\frac{817 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{91875 \sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{6 \sqrt{2} \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{875 \sqrt{x^4+3 x^2+4}}+\frac{111 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{24500 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{7633 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} \Pi \left(-\frac{9}{280};2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{274400 \sqrt{2} \sqrt{x^4+3 x^2+4}}",1,"((140*x*(357 + 167*x^2)*(4 + 3*x^2 + x^4))/(7 + 5*x^2)^2 - I*Sqrt[6 + (2*I)*Sqrt[7]]*Sqrt[1 - ((2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[1 + ((2*I)*x^2)/(3*I + Sqrt[7])]*(315*(3 - I*Sqrt[7])*EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + 7*(103 + (45*I)*Sqrt[7])*EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + 2694*EllipticPi[(5*(3 + I*Sqrt[7]))/14, I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])]))/(274400*Sqrt[4 + 3*x^2 + x^4])","C",1
364,1,337,187,0.4792661,"\int \frac{\left(7+5 x^2\right)^3}{\sqrt{4+3 x^2+x^4}} \, dx","Integrate[(7 + 5*x^2)^3/Sqrt[4 + 3*x^2 + x^4],x]","\frac{-\sqrt{2} \left(15 \sqrt{7}+131 i\right) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)+15 \sqrt{2} \left(\sqrt{7}+3 i\right) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} E\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)+100 \sqrt{-\frac{i}{\sqrt{7}-3 i}} x \left(x^6+6 x^4+13 x^2+12\right)}{4 \sqrt{-\frac{i}{\sqrt{7}-3 i}} \sqrt{x^4+3 x^2+4}}","-\frac{15 \sqrt{x^4+3 x^2+4} x}{x^2+2}+75 \sqrt{x^4+3 x^2+4} x+\frac{13 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{2 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{15 \sqrt{2} \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{\sqrt{x^4+3 x^2+4}}+25 \sqrt{x^4+3 x^2+4} x^3",1,"(100*Sqrt[(-I)/(-3*I + Sqrt[7])]*x*(12 + 13*x^2 + 6*x^4 + x^6) + 15*Sqrt[2]*(3*I + Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] - Sqrt[2]*(131*I + 15*Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])])/(4*Sqrt[(-I)/(-3*I + Sqrt[7])]*Sqrt[4 + 3*x^2 + x^4])","C",1
365,1,331,170,0.4269625,"\int \frac{\left(7+5 x^2\right)^2}{\sqrt{4+3 x^2+x^4}} \, dx","Integrate[(7 + 5*x^2)^2/Sqrt[4 + 3*x^2 + x^4],x]","\frac{\sqrt{2} \left(30 \sqrt{7}+43 i\right) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)-30 \sqrt{2} \left(\sqrt{7}+3 i\right) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} E\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)+50 \sqrt{-\frac{i}{\sqrt{7}-3 i}} x \left(x^4+3 x^2+4\right)}{6 \sqrt{-\frac{i}{\sqrt{7}-3 i}} \sqrt{x^4+3 x^2+4}}","\frac{20 \sqrt{x^4+3 x^2+4} x}{x^2+2}+\frac{25}{3} \sqrt{x^4+3 x^2+4} x+\frac{167 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{6 \sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{20 \sqrt{2} \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{\sqrt{x^4+3 x^2+4}}",1,"(50*Sqrt[(-I)/(-3*I + Sqrt[7])]*x*(4 + 3*x^2 + x^4) - 30*Sqrt[2]*(3*I + Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + Sqrt[2]*(43*I + 30*Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])])/(6*Sqrt[(-I)/(-3*I + Sqrt[7])]*Sqrt[4 + 3*x^2 + x^4])","C",1
366,1,214,151,0.1763163,"\int \frac{7+5 x^2}{\sqrt{4+3 x^2+x^4}} \, dx","Integrate[(7 + 5*x^2)/Sqrt[4 + 3*x^2 + x^4],x]","\frac{\sqrt{1-\frac{2 i x^2}{\sqrt{7}-3 i}} \sqrt{1+\frac{2 i x^2}{\sqrt{7}+3 i}} \left(\left(5 \sqrt{7}+i\right) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)-5 \left(\sqrt{7}+3 i\right) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)\right)}{2 \sqrt{2} \sqrt{-\frac{i}{\sqrt{7}-3 i}} \sqrt{x^4+3 x^2+4}}","\frac{5 \sqrt{x^4+3 x^2+4} x}{x^2+2}+\frac{17 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{2 \sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{5 \sqrt{2} \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{\sqrt{x^4+3 x^2+4}}",1,"(Sqrt[1 - ((2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[1 + ((2*I)*x^2)/(3*I + Sqrt[7])]*(-5*(3*I + Sqrt[7])*EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + (I + 5*Sqrt[7])*EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])]))/(2*Sqrt[2]*Sqrt[(-I)/(-3*I + Sqrt[7])]*Sqrt[4 + 3*x^2 + x^4])","C",1
367,1,142,64,0.0546058,"\int \frac{1}{\sqrt{4+3 x^2+x^4}} \, dx","Integrate[1/Sqrt[4 + 3*x^2 + x^4],x]","-\frac{i \sqrt{1-\frac{2 x^2}{-3-i \sqrt{7}}} \sqrt{1-\frac{2 x^2}{-3+i \sqrt{7}}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{2}{-3-i \sqrt{7}}} x\right)|\frac{-3-i \sqrt{7}}{-3+i \sqrt{7}}\right)}{\sqrt{2} \sqrt{-\frac{1}{-3-i \sqrt{7}}} \sqrt{x^4+3 x^2+4}}","\frac{\left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{2 \sqrt{2} \sqrt{x^4+3 x^2+4}}",1,"((-I)*Sqrt[1 - (2*x^2)/(-3 - I*Sqrt[7])]*Sqrt[1 - (2*x^2)/(-3 + I*Sqrt[7])]*EllipticF[I*ArcSinh[Sqrt[-2/(-3 - I*Sqrt[7])]*x], (-3 - I*Sqrt[7])/(-3 + I*Sqrt[7])])/(Sqrt[2]*Sqrt[-(-3 - I*Sqrt[7])^(-1)]*Sqrt[4 + 3*x^2 + x^4])","C",1
368,1,159,168,0.1389071,"\int \frac{1}{\left(7+5 x^2\right) \sqrt{4+3 x^2+x^4}} \, dx","Integrate[1/((7 + 5*x^2)*Sqrt[4 + 3*x^2 + x^4]),x]","-\frac{i \sqrt{1-\frac{2 x^2}{-3-i \sqrt{7}}} \sqrt{1-\frac{2 x^2}{-3+i \sqrt{7}}} \Pi \left(-\frac{5}{14} \left(-3-i \sqrt{7}\right);i \sinh ^{-1}\left(\sqrt{-\frac{2}{-3-i \sqrt{7}}} x\right)|\frac{-3-i \sqrt{7}}{-3+i \sqrt{7}}\right)}{7 \sqrt{2} \sqrt{-\frac{1}{-3-i \sqrt{7}}} \sqrt{x^4+3 x^2+4}}","\frac{1}{4} \sqrt{\frac{5}{77}} \tan ^{-1}\left(\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{x^4+3 x^2+4}}\right)-\frac{\left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{6 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{17 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} \Pi \left(-\frac{9}{280};2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{84 \sqrt{2} \sqrt{x^4+3 x^2+4}}",1,"((-1/7*I)*Sqrt[1 - (2*x^2)/(-3 - I*Sqrt[7])]*Sqrt[1 - (2*x^2)/(-3 + I*Sqrt[7])]*EllipticPi[(-5*(-3 - I*Sqrt[7]))/14, I*ArcSinh[Sqrt[-2/(-3 - I*Sqrt[7])]*x], (-3 - I*Sqrt[7])/(-3 + I*Sqrt[7])])/(Sqrt[2]*Sqrt[-(-3 - I*Sqrt[7])^(-1)]*Sqrt[4 + 3*x^2 + x^4])","C",1
369,1,481,286,0.7794813,"\int \frac{1}{\left(7+5 x^2\right)^2 \sqrt{4+3 x^2+x^4}} \, dx","Integrate[1/((7 + 5*x^2)^2*Sqrt[4 + 3*x^2 + x^4]),x]","\frac{98 i \left(5 x^2+7\right) \sqrt{2-\frac{4 i x^2}{\sqrt{7}-3 i}} \sqrt{1+\frac{2 i x^2}{\sqrt{7}+3 i}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)-74 i \left(5 x^2+7\right) \sqrt{2-\frac{4 i x^2}{\sqrt{7}-3 i}} \sqrt{1+\frac{2 i x^2}{\sqrt{7}+3 i}} \Pi \left(\frac{5}{14} \left(3+i \sqrt{7}\right);i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)+35 \left(\sqrt{7}+3 i\right) \left(5 x^2+7\right) \sqrt{2-\frac{4 i x^2}{\sqrt{7}-3 i}} \sqrt{1+\frac{2 i x^2}{\sqrt{7}+3 i}} \left(E\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)-F\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)\right)+700 \sqrt{-\frac{i}{\sqrt{7}-3 i}} x \left(x^4+3 x^2+4\right)}{17248 \sqrt{-\frac{i}{\sqrt{7}-3 i}} \left(5 x^2+7\right) \sqrt{x^4+3 x^2+4}}","-\frac{5 \sqrt{x^4+3 x^2+4} x}{616 \left(x^2+2\right)}+\frac{25 \sqrt{x^4+3 x^2+4} x}{616 \left(5 x^2+7\right)}+\frac{37 \sqrt{\frac{5}{77}} \tan ^{-1}\left(\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{x^4+3 x^2+4}}\right)}{2464}-\frac{\left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{42 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{5 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{308 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{629 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} \Pi \left(-\frac{9}{280};2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{51744 \sqrt{2} \sqrt{x^4+3 x^2+4}}",1,"(700*Sqrt[(-I)/(-3*I + Sqrt[7])]*x*(4 + 3*x^2 + x^4) + 35*(3*I + Sqrt[7])*(7 + 5*x^2)*Sqrt[2 - ((4*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[1 + ((2*I)*x^2)/(3*I + Sqrt[7])]*(EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] - EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])]) + (98*I)*(7 + 5*x^2)*Sqrt[2 - ((4*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[1 + ((2*I)*x^2)/(3*I + Sqrt[7])]*EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] - (74*I)*(7 + 5*x^2)*Sqrt[2 - ((4*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[1 + ((2*I)*x^2)/(3*I + Sqrt[7])]*EllipticPi[(5*(3 + I*Sqrt[7]))/14, I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])])/(17248*Sqrt[(-I)/(-3*I + Sqrt[7])]*(7 + 5*x^2)*Sqrt[4 + 3*x^2 + x^4])","C",1
370,1,308,314,0.8908516,"\int \frac{1}{\left(7+5 x^2\right)^3 \sqrt{4+3 x^2+x^4}} \, dx","Integrate[1/((7 + 5*x^2)^3*Sqrt[4 + 3*x^2 + x^4]),x]","\frac{\frac{700 x \left(555 x^2+1393\right) \left(x^4+3 x^2+4\right)}{\left(5 x^2+7\right)^2}+i \sqrt{6+2 i \sqrt{7}} \sqrt{1-\frac{2 i x^2}{\sqrt{7}-3 i}} \sqrt{1+\frac{2 i x^2}{\sqrt{7}+3 i}} \left(\left(-9401+3885 i \sqrt{7}\right) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)+3885 \left(3-i \sqrt{7}\right) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)+6570 \Pi \left(\frac{5}{14} \left(3+i \sqrt{7}\right);i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)\right)}{21249536 \sqrt{x^4+3 x^2+4}}","-\frac{555 \sqrt{x^4+3 x^2+4} x}{758912 \left(x^2+2\right)}+\frac{2775 \sqrt{x^4+3 x^2+4} x}{758912 \left(5 x^2+7\right)}+\frac{25 \sqrt{x^4+3 x^2+4} x}{1232 \left(5 x^2+7\right)^2}-\frac{3285 \sqrt{\frac{5}{77}} \tan ^{-1}\left(\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{x^4+3 x^2+4}}\right)}{3035648}-\frac{\left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{8624 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{555 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{379456 \sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{18615 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} \Pi \left(-\frac{9}{280};2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{21249536 \sqrt{2} \sqrt{x^4+3 x^2+4}}",1,"((700*x*(1393 + 555*x^2)*(4 + 3*x^2 + x^4))/(7 + 5*x^2)^2 + I*Sqrt[6 + (2*I)*Sqrt[7]]*Sqrt[1 - ((2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[1 + ((2*I)*x^2)/(3*I + Sqrt[7])]*(3885*(3 - I*Sqrt[7])*EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + (-9401 + (3885*I)*Sqrt[7])*EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + 6570*EllipticPi[(5*(3 + I*Sqrt[7]))/14, I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])]))/(21249536*Sqrt[4 + 3*x^2 + x^4])","C",1
371,1,339,219,0.5181449,"\int \frac{\left(7+5 x^2\right)^5}{\left(4+3 x^2+x^4\right)^{3/2}} \, dx","Integrate[(7 + 5*x^2)^5/(4 + 3*x^2 + x^4)^(3/2),x]","\frac{-\sqrt{2} \left(662337 \sqrt{7}+975947 i\right) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)+662337 \sqrt{2} \left(\sqrt{7}+3 i\right) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} E\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)+4 \sqrt{-\frac{i}{\sqrt{7}-3 i}} x \left(52500 x^6+297500 x^4+767337 x^2+858479\right)}{336 \sqrt{-\frac{i}{\sqrt{7}-3 i}} \sqrt{x^4+3 x^2+4}}","-\frac{220779 \sqrt{x^4+3 x^2+4} x}{28 \left(x^2+2\right)}+\frac{5000}{3} \sqrt{x^4+3 x^2+4} x+\frac{\left(45779 x^2+99493\right) x}{28 \sqrt{x^4+3 x^2+4}}-\frac{130729 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{12 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{220779 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{14 \sqrt{2} \sqrt{x^4+3 x^2+4}}+625 \sqrt{x^4+3 x^2+4} x^3",1,"(4*Sqrt[(-I)/(-3*I + Sqrt[7])]*x*(858479 + 767337*x^2 + 297500*x^4 + 52500*x^6) + 662337*Sqrt[2]*(3*I + Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] - Sqrt[2]*(975947*I + 662337*Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])])/(336*Sqrt[(-I)/(-3*I + Sqrt[7])]*Sqrt[4 + 3*x^2 + x^4])","C",1
372,-1,0,200,0,"\int \frac{\left(7+5 x^2\right)^4}{\left(4+3 x^2+x^4\right)^{3/2}} \, dx","Integrate[(7 + 5*x^2)^4/(4 + 3*x^2 + x^4)^(3/2),x]","\text{\$Aborted}","\frac{14523 \sqrt{x^4+3 x^2+4} x}{28 \left(x^2+2\right)}+\frac{625}{3} \sqrt{x^4+3 x^2+4} x+\frac{\left(2719-4023 x^2\right) x}{28 \sqrt{x^4+3 x^2+4}}+\frac{4243 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{12 \sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{14523 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{14 \sqrt{2} \sqrt{x^4+3 x^2+4}}",1,"$Aborted","F",-1
373,-1,0,181,0,"\int \frac{\left(7+5 x^2\right)^3}{\left(4+3 x^2+x^4\right)^{3/2}} \, dx","Integrate[(7 + 5*x^2)^3/(4 + 3*x^2 + x^4)^(3/2),x]","\text{\$Aborted}","\frac{4449 \sqrt{x^4+3 x^2+4} x}{28 \left(x^2+2\right)}-\frac{\left(949 x^2+2323\right) x}{28 \sqrt{x^4+3 x^2+4}}+\frac{973 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{4 \sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{4449 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{14 \sqrt{2} \sqrt{x^4+3 x^2+4}}",1,"$Aborted","F",-1
374,-1,0,181,0,"\int \frac{\left(7+5 x^2\right)^2}{\left(4+3 x^2+x^4\right)^{3/2}} \, dx","Integrate[(7 + 5*x^2)^2/(4 + 3*x^2 + x^4)^(3/2),x]","\text{\$Aborted}","-\frac{113 \sqrt{x^4+3 x^2+4} x}{28 \left(x^2+2\right)}-\frac{\left(9-113 x^2\right) x}{28 \sqrt{x^4+3 x^2+4}}+\frac{9 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{4 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{113 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{14 \sqrt{2} \sqrt{x^4+3 x^2+4}}",1,"$Aborted","F",-1
375,-1,0,181,0,"\int \frac{7+5 x^2}{\left(4+3 x^2+x^4\right)^{3/2}} \, dx","Integrate[(7 + 5*x^2)/(4 + 3*x^2 + x^4)^(3/2),x]","\text{\$Aborted}","-\frac{19 \sqrt{x^4+3 x^2+4} x}{28 \left(x^2+2\right)}+\frac{\left(19 x^2+53\right) x}{28 \sqrt{x^4+3 x^2+4}}-\frac{3 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{4 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{19 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{14 \sqrt{2} \sqrt{x^4+3 x^2+4}}",1,"$Aborted","F",-1
376,1,328,181,0.3541098,"\int \frac{1}{\left(4+3 x^2+x^4\right)^{3/2}} \, dx","Integrate[(4 + 3*x^2 + x^4)^(-3/2),x]","\frac{-4 \sqrt{-\frac{i}{\sqrt{7}-3 i}} x \left(3 x^2+1\right)+\sqrt{2} \left(3 \sqrt{7}-7 i\right) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)-3 \sqrt{2} \left(\sqrt{7}+3 i\right) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} E\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)}{112 \sqrt{-\frac{i}{\sqrt{7}-3 i}} \sqrt{x^4+3 x^2+4}}","\frac{3 \sqrt{x^4+3 x^2+4} x}{28 \left(x^2+2\right)}-\frac{\left(3 x^2+1\right) x}{28 \sqrt{x^4+3 x^2+4}}+\frac{\left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{4 \sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{3 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{14 \sqrt{2} \sqrt{x^4+3 x^2+4}}",1,"(-4*Sqrt[(-I)/(-3*I + Sqrt[7])]*x*(1 + 3*x^2) - 3*Sqrt[2]*(3*I + Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + Sqrt[2]*(-7*I + 3*Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])])/(112*Sqrt[(-I)/(-3*I + Sqrt[7])]*Sqrt[4 + 3*x^2 + x^4])","C",1
377,1,483,284,0.5502654,"\int \frac{1}{\left(7+5 x^2\right) \left(4+3 x^2+x^4\right)^{3/2}} \, dx","Integrate[1/((7 + 5*x^2)*(4 + 3*x^2 + x^4)^(3/2)),x]","\frac{-8 \sqrt{-\frac{i}{\sqrt{7}-3 i}} x^3+\sqrt{2} \left(2 \sqrt{7}+7 i\right) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)-2 \sqrt{2} \left(\sqrt{7}+3 i\right) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} E\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)-25 i \sqrt{2} \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} \Pi \left(\frac{5}{14} \left(3+i \sqrt{7}\right);i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)-26 \sqrt{-\frac{i}{\sqrt{7}-3 i}} x}{616 \sqrt{-\frac{i}{\sqrt{7}-3 i}} \sqrt{x^4+3 x^2+4}}","\frac{\sqrt{x^4+3 x^2+4} x}{77 \left(x^2+2\right)}-\frac{\left(4 x^2+13\right) x}{308 \sqrt{x^4+3 x^2+4}}+\frac{25}{176} \sqrt{\frac{5}{77}} \tan ^{-1}\left(\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{x^4+3 x^2+4}}\right)-\frac{\left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{12 \sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{\sqrt{2} \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{77 \sqrt{x^4+3 x^2+4}}+\frac{425 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} \Pi \left(-\frac{9}{280};2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{3696 \sqrt{2} \sqrt{x^4+3 x^2+4}}",1,"(-26*Sqrt[(-I)/(-3*I + Sqrt[7])]*x - 8*Sqrt[(-I)/(-3*I + Sqrt[7])]*x^3 - 2*Sqrt[2]*(3*I + Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + Sqrt[2]*(7*I + 2*Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] - (25*I)*Sqrt[2]*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticPi[(5*(3 + I*Sqrt[7]))/14, I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])])/(616*Sqrt[(-I)/(-3*I + Sqrt[7])]*Sqrt[4 + 3*x^2 + x^4])","C",1
378,1,311,312,0.5998087,"\int \frac{1}{\left(7+5 x^2\right)^2 \left(4+3 x^2+x^4\right)^{3/2}} \, dx","Integrate[1/((7 + 5*x^2)^2*(4 + 3*x^2 + x^4)^(3/2)),x]","\frac{28 x \left(995 x^4+2633 x^2+2836\right)+i \sqrt{6+2 i \sqrt{7}} \left(5 x^2+7\right) \sqrt{1-\frac{2 i x^2}{\sqrt{7}-3 i}} \sqrt{1+\frac{2 i x^2}{\sqrt{7}+3 i}} \left(7 \left(101+199 i \sqrt{7}\right) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)+1393 \left(3-i \sqrt{7}\right) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)-1150 \Pi \left(\frac{5}{14} \left(3+i \sqrt{7}\right);i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)\right)}{758912 \left(5 x^2+7\right) \sqrt{x^4+3 x^2+4}}","-\frac{199 \sqrt{x^4+3 x^2+4} x}{27104 \left(x^2+2\right)}+\frac{625 \sqrt{x^4+3 x^2+4} x}{27104 \left(5 x^2+7\right)}+\frac{\left(37 x^2+24\right) x}{13552 \sqrt{x^4+3 x^2+4}}+\frac{575 \sqrt{\frac{5}{77}} \tan ^{-1}\left(\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{x^4+3 x^2+4}}\right)}{108416}-\frac{2 \sqrt{2} \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{231 \sqrt{x^4+3 x^2+4}}+\frac{199 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{13552 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{9775 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} \Pi \left(-\frac{9}{280};2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{2276736 \sqrt{2} \sqrt{x^4+3 x^2+4}}",1,"(28*x*(2836 + 2633*x^2 + 995*x^4) + I*Sqrt[6 + (2*I)*Sqrt[7]]*(7 + 5*x^2)*Sqrt[1 - ((2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[1 + ((2*I)*x^2)/(3*I + Sqrt[7])]*(1393*(3 - I*Sqrt[7])*EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + 7*(101 + (199*I)*Sqrt[7])*EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] - 1150*EllipticPi[(5*(3 + I*Sqrt[7]))/14, I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])]))/(758912*(7 + 5*x^2)*Sqrt[4 + 3*x^2 + x^4])","C",1
379,1,320,340,0.7496479,"\int \frac{1}{\left(7+5 x^2\right)^3 \left(4+3 x^2+x^4\right)^{3/2}} \, dx","Integrate[1/((7 + 5*x^2)^3*(4 + 3*x^2 + x^4)^(3/2)),x]","\frac{28 x \left(453975 x^6+2838330 x^4+5811451 x^2+4496212\right)+3 i \sqrt{6+2 i \sqrt{7}} \sqrt{1-\frac{2 i x^2}{\sqrt{7}-3 i}} \sqrt{1+\frac{2 i x^2}{\sqrt{7}+3 i}} \left(5 x^2+7\right)^2 \left(7 i \left(6053 \sqrt{7}+23633 i\right) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)+42371 \left(3-i \sqrt{7}\right) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)+352950 \Pi \left(\frac{5}{14} \left(3+i \sqrt{7}\right);i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right)|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right)\right)}{934979584 \left(5 x^2+7\right)^2 \sqrt{x^4+3 x^2+4}}","-\frac{18159 \sqrt{x^4+3 x^2+4} x}{33392128 \left(x^2+2\right)}+\frac{51875 \sqrt{x^4+3 x^2+4} x}{33392128 \left(5 x^2+7\right)}+\frac{625 \sqrt{x^4+3 x^2+4} x}{54208 \left(5 x^2+7\right)^2}+\frac{\left(139 x^2+548\right) x}{596288 \sqrt{x^4+3 x^2+4}}-\frac{529425 \sqrt{\frac{5}{77}} \tan ^{-1}\left(\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{x^4+3 x^2+4}}\right)}{133568512}+\frac{843 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{379456 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{18159 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{16696064 \sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{3000075 \left(x^2+2\right) \sqrt{\frac{x^4+3 x^2+4}{\left(x^2+2\right)^2}} \Pi \left(-\frac{9}{280};2 \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)|\frac{1}{8}\right)}{934979584 \sqrt{2} \sqrt{x^4+3 x^2+4}}",1,"(28*x*(4496212 + 5811451*x^2 + 2838330*x^4 + 453975*x^6) + (3*I)*Sqrt[6 + (2*I)*Sqrt[7]]*(7 + 5*x^2)^2*Sqrt[1 - ((2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[1 + ((2*I)*x^2)/(3*I + Sqrt[7])]*(42371*(3 - I*Sqrt[7])*EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + (7*I)*(23633*I + 6053*Sqrt[7])*EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + 352950*EllipticPi[(5*(3 + I*Sqrt[7]))/14, I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])]))/(934979584*(7 + 5*x^2)^2*Sqrt[4 + 3*x^2 + x^4])","C",1
380,1,584,467,2.8708304,"\int \frac{\left(d+e x^2\right)^3}{\sqrt{a+b x^2+c x^4}} \, dx","Integrate[(d + e*x^2)^3/Sqrt[a + b*x^2 + c*x^4],x]","\frac{i e \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(-3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\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)-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(15 c^2 d e \left(3 d \sqrt{b^2-4 a c}-2 a e-3 b d\right)+c e^2 \left(-30 b d \sqrt{b^2-4 a c}-9 a e \sqrt{b^2-4 a c}+17 a b e+30 b^2 d\right)+8 b^2 e^3 \left(\sqrt{b^2-4 a c}-b\right)+30 c^3 d^3\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)+4 c e^2 x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left(a+b x^2+c x^4\right) \left(3 c \left(5 d+e x^2\right)-4 b e\right)}{60 c^3 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","\frac{e x \sqrt{a+b x^2+c x^4} \left(-3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right)}{15 c^{5/2} \left(\sqrt{a}+\sqrt{c} x^2\right)}-\frac{\sqrt[4]{a} e \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(-3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\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{\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(e \left(-3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right)+\frac{\sqrt{c} \left(4 a b e^3-15 a c d e^2+15 c^2 d^3\right)}{\sqrt{a}}\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{e^2 x \sqrt{a+b x^2+c x^4} (15 c d-4 b e)}{15 c^2}+\frac{e^3 x^3 \sqrt{a+b x^2+c x^4}}{5 c}",1,"(4*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*e^2*x*(a + b*x^2 + c*x^4)*(-4*b*e + 3*c*(5*d + e*x^2)) + I*(-b + Sqrt[b^2 - 4*a*c])*e*(45*c^2*d^2 + 8*b^2*e^2 - 3*c*e*(10*b*d + 3*a*e))*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*(30*c^3*d^3 + 8*b^2*(-b + Sqrt[b^2 - 4*a*c])*e^3 + 15*c^2*d*e*(-3*b*d + 3*Sqrt[b^2 - 4*a*c]*d - 2*a*e) + c*e^2*(30*b^2*d - 30*b*Sqrt[b^2 - 4*a*c]*d + 17*a*b*e - 9*a*Sqrt[b^2 - 4*a*c]*e))*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
381,1,488,356,1.6150609,"\int \frac{\left(d+e x^2\right)^2}{\sqrt{a+b x^2+c x^4}} \, dx","Integrate[(d + e*x^2)^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(c e \left(-3 d \sqrt{b^2-4 a c}+a e+3 b d\right)+b e^2 \left(\sqrt{b^2-4 a c}-b\right)-3 c^2 d^2\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 e \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}}} (b e-3 c d) E\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)+2 c e^2 x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left(a+b x^2+c x^4\right)}{6 c^2 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","\frac{\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(\frac{\sqrt{c} \left(3 c d^2-a e^2\right)}{\sqrt{a}}+2 e (3 c d-b e)\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{2 e x \sqrt{a+b x^2+c x^4} (3 c d-b e)}{3 c^{3/2} \left(\sqrt{a}+\sqrt{c} x^2\right)}-\frac{2 \sqrt[4]{a} e \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}} (3 c d-b e) 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{e^2 x \sqrt{a+b x^2+c x^4}}{3 c}",1,"(2*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*e^2*x*(a + b*x^2 + c*x^4) - I*(-b + Sqrt[b^2 - 4*a*c])*e*(-3*c*d + b*e)*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*(-3*c^2*d^2 + b*(-b + Sqrt[b^2 - 4*a*c])*e^2 + c*e*(3*b*d - 3*Sqrt[b^2 - 4*a*c]*d + a*e))*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/(6*c^2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[a + b*x^2 + c*x^4])","C",1
382,1,302,283,0.2565622,"\int \frac{d+e x^2}{\sqrt{a+b x^2+c x^4}} \, dx","Integrate[(d + e*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(e \left(b-\sqrt{b^2-4 a c}\right)-2 c d\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)+e \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) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(\frac{\sqrt{c} d}{\sqrt{a}}+e\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)}{2 c^{3/4} \sqrt{a+b x^2+c x^4}}-\frac{\sqrt[4]{a} e \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{e 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 + Sqrt[b^2 - 4*a*c])*e*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + (-2*c*d + (b - Sqrt[b^2 - 4*a*c])*e)*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
383,1,214,401,0.2201033,"\int \frac{1}{\left(d+e x^2\right) \sqrt{a+b x^2+c x^4}} \, dx","Integrate[1/((d + e*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} \Pi \left(\frac{\left(b+\sqrt{b^2-4 a c}\right) e}{2 c d};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)}{\sqrt{2} d \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","-\frac{a^{3/4} \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(\frac{\sqrt{c} d}{\sqrt{a}}+e\right)^2 \Pi \left(-\frac{\left(\sqrt{c} d-\sqrt{a} e\right)^2}{4 \sqrt{a} \sqrt{c} d e};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)}{4 \sqrt[4]{c} d \sqrt{a+b x^2+c x^4} \left(c d^2-a e^2\right)}+\frac{\sqrt{e} \tan ^{-1}\left(\frac{x \sqrt{a e^2-b d e+c d^2}}{\sqrt{d} \sqrt{e} \sqrt{a+b x^2+c x^4}}\right)}{2 \sqrt{d} \sqrt{a e^2-b d e+c d^2}}+\frac{\sqrt[4]{c} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{2 \sqrt[4]{a} \sqrt{a+b x^2+c x^4} \left(\sqrt{c} d-\sqrt{a} e\right)}",1,"((-I)*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticPi[((b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/(Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*Sqrt[a + b*x^2 + c*x^4])","C",1
384,1,1069,718,1.8631591,"\int \frac{1}{\left(d+e x^2\right)^2 \sqrt{a+b x^2+c x^4}} \, dx","Integrate[1/((d + e*x^2)^2*Sqrt[a + b*x^2 + c*x^4]),x]","\frac{2 i \sqrt{2} c \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left(e x^2+d\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) d^2-6 i \sqrt{2} c \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left(e x^2+d\right) \Pi \left(\frac{\left(b+\sqrt{b^2-4 a c}\right) e}{2 c d};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) d^2+4 \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} e^2 x \left(c x^4+b x^2+a\right) d+i \sqrt{2} \left(b-\sqrt{b^2-4 a c}\right) e \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left(e x^2+d\right) \left(E\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)-F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)\right) d+4 i \sqrt{2} b e \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left(e x^2+d\right) \Pi \left(\frac{\left(b+\sqrt{b^2-4 a c}\right) e}{2 c d};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) d-2 i \sqrt{2} a e^2 \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left(e x^2+d\right) \Pi \left(\frac{\left(b+\sqrt{b^2-4 a c}\right) e}{2 c d};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)}{8 \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} d \left(c d^3+e (a e-b d) d\right) \left(e x^2+d\right) \sqrt{c x^4+b x^2+a}}","\frac{e^2 x \sqrt{a+b x^2+c x^4}}{2 d \left(d+e x^2\right) \left(a e^2-b d e+c d^2\right)}-\frac{\sqrt{c} e x \sqrt{a+b x^2+c x^4}}{2 d \left(\sqrt{a}+\sqrt{c} x^2\right) \left(a e^2-b d e+c d^2\right)}+\frac{\sqrt[4]{a} \sqrt[4]{c} e \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)}{2 d \sqrt{a+b x^2+c x^4} \left(a e^2-b d e+c d^2\right)}-\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(\sqrt{a} e+\sqrt{c} d\right) \left(3 c d^2-e (2 b d-a e)\right) \Pi \left(-\frac{\left(\sqrt{c} d-\sqrt{a} e\right)^2}{4 \sqrt{a} \sqrt{c} d e};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)}{8 \sqrt[4]{a} \sqrt[4]{c} d^2 \sqrt{a+b x^2+c x^4} \left(\sqrt{c} d-\sqrt{a} e\right) \left(a e^2-b d e+c d^2\right)}+\frac{\sqrt{e} \left(3 c d^2-e (2 b d-a e)\right) \tan ^{-1}\left(\frac{x \sqrt{a e^2-b d e+c d^2}}{\sqrt{d} \sqrt{e} \sqrt{a+b x^2+c x^4}}\right)}{4 d^{3/2} \left(a e^2-b d e+c d^2\right)^{3/2}}+\frac{\sqrt[4]{c} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{2 \sqrt[4]{a} d \sqrt{a+b x^2+c x^4} \left(\sqrt{c} d-\sqrt{a} e\right)}",1,"(4*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*e^2*x*(a + b*x^2 + c*x^4) + I*Sqrt[2]*(b - Sqrt[b^2 - 4*a*c])*d*e*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])]*(d + e*x^2)*(EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]) + (2*I)*Sqrt[2]*c*d^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])]*(d + e*x^2)*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - (6*I)*Sqrt[2]*c*d^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])]*(d + e*x^2)*EllipticPi[((b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d), 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*I)*Sqrt[2]*b*d*e*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])]*(d + e*x^2)*EllipticPi[((b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - (2*I)*Sqrt[2]*a*e^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])]*(d + e*x^2)*EllipticPi[((b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d), 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])])/(8*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*(c*d^3 + d*e*(-(b*d) + a*e))*(d + e*x^2)*Sqrt[a + b*x^2 + c*x^4])","C",1
385,1,596,553,2.4700138,"\int \frac{\left(d+e x^2\right)^3}{\sqrt{a+b x^2-c x^4}} \, dx","Integrate[(d + e*x^2)^3/Sqrt[a + b*x^2 - c*x^4],x]","\frac{-i \sqrt{2} e \left(\sqrt{4 a c+b^2}-b\right) \sqrt{\frac{\sqrt{4 a c+b^2}+b-2 c x^2}{\sqrt{4 a c+b^2}+b}} \sqrt{\frac{\sqrt{4 a c+b^2}-b+2 c x^2}{\sqrt{4 a c+b^2}-b}} \left(3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\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)+i \sqrt{2} \sqrt{\frac{\sqrt{4 a c+b^2}+b-2 c x^2}{\sqrt{4 a c+b^2}+b}} \sqrt{\frac{\sqrt{4 a c+b^2}-b+2 c x^2}{\sqrt{4 a c+b^2}-b}} \left(15 c^2 d e \left(3 d \sqrt{4 a c+b^2}-2 a e-3 b d\right)+c e^2 \left(30 b d \sqrt{4 a c+b^2}+9 a e \sqrt{4 a c+b^2}-17 a b e-30 b^2 d\right)+8 b^2 e^3 \left(\sqrt{4 a c+b^2}-b\right)-30 c^3 d^3\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)-4 c e^2 x \sqrt{-\frac{c}{\sqrt{4 a c+b^2}+b}} \left(a+b x^2-c x^4\right) \left(4 b e+3 c \left(5 d+e x^2\right)\right)}{60 c^3 \sqrt{-\frac{c}{\sqrt{4 a c+b^2}+b}} \sqrt{a+b x^2-c x^4}}","-\frac{e \left(b-\sqrt{4 a c+b^2}\right) \sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} \left(3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right)|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right)}{30 \sqrt{2} c^{7/2} \sqrt{a+b x^2-c x^4}}+\frac{\left(b-\sqrt{4 a c+b^2}\right) \sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} \left(\frac{2 c \left(4 a b e^3+15 a c d e^2+15 c^2 d^3\right)}{b-\sqrt{4 a c+b^2}}+e \left(3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right)|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right)}{30 \sqrt{2} c^{7/2} \sqrt{a+b x^2-c x^4}}-\frac{e^2 x \sqrt{a+b x^2-c x^4} (4 b e+15 c d)}{15 c^2}-\frac{e^3 x^3 \sqrt{a+b x^2-c x^4}}{5 c}",1,"(-4*c*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*e^2*x*(a + b*x^2 - c*x^4)*(4*b*e + 3*c*(5*d + e*x^2)) - I*Sqrt[2]*(-b + Sqrt[b^2 + 4*a*c])*e*(45*c^2*d^2 + 8*b^2*e^2 + 3*c*e*(10*b*d + 3*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])]*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*x], (b + Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])] + I*Sqrt[2]*(-30*c^3*d^3 + 8*b^2*(-b + Sqrt[b^2 + 4*a*c])*e^3 + 15*c^2*d*e*(-3*b*d + 3*Sqrt[b^2 + 4*a*c]*d - 2*a*e) + c*e^2*(-30*b^2*d + 30*b*Sqrt[b^2 + 4*a*c]*d - 17*a*b*e + 9*a*Sqrt[b^2 + 4*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])]*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
386,1,503,454,1.3896531,"\int \frac{\left(d+e x^2\right)^2}{\sqrt{a+b x^2-c x^4}} \, dx","Integrate[(d + e*x^2)^2/Sqrt[a + b*x^2 - c*x^4],x]","\frac{i \sqrt{2} \sqrt{\frac{\sqrt{4 a c+b^2}+b-2 c x^2}{\sqrt{4 a c+b^2}+b}} \sqrt{\frac{\sqrt{4 a c+b^2}-b+2 c x^2}{\sqrt{4 a c+b^2}-b}} \left(-c e \left(-3 d \sqrt{4 a c+b^2}+a e+3 b d\right)+b e^2 \left(\sqrt{4 a c+b^2}-b\right)-3 c^2 d^2\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 \sqrt{2} e \left(\sqrt{4 a c+b^2}-b\right) \sqrt{\frac{\sqrt{4 a c+b^2}+b-2 c x^2}{\sqrt{4 a c+b^2}+b}} \sqrt{\frac{\sqrt{4 a c+b^2}-b+2 c x^2}{\sqrt{4 a c+b^2}-b}} (b e+3 c d) E\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{-\frac{c}{b+\sqrt{b^2+4 a c}}} x\right)|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right)+2 c e^2 x \sqrt{-\frac{c}{\sqrt{4 a c+b^2}+b}} \left(-a-b x^2+c x^4\right)}{6 c^2 \sqrt{-\frac{c}{\sqrt{4 a c+b^2}+b}} \sqrt{a+b x^2-c x^4}}","-\frac{e \left(b-\sqrt{4 a c+b^2}\right) \sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} (b e+3 c d) E\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right)|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right)}{3 \sqrt{2} c^{5/2} \sqrt{a+b x^2-c x^4}}+\frac{\sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} \left(c e \left(-3 d \sqrt{4 a c+b^2}+a e+3 b d\right)+b e^2 \left(b-\sqrt{4 a c+b^2}\right)+3 c^2 d^2\right) F\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right)|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right)}{3 \sqrt{2} c^{5/2} \sqrt{a+b x^2-c x^4}}-\frac{e^2 x \sqrt{a+b x^2-c x^4}}{3 c}",1,"(2*c*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*e^2*x*(-a - b*x^2 + c*x^4) - I*Sqrt[2]*(-b + Sqrt[b^2 + 4*a*c])*e*(3*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])]*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*x], (b + Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])] + I*Sqrt[2]*(-3*c^2*d^2 + b*(-b + Sqrt[b^2 + 4*a*c])*e^2 - c*e*(3*b*d - 3*Sqrt[b^2 + 4*a*c]*d + 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])]*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*x], (b + Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])])/(6*c^2*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*Sqrt[a + b*x^2 - c*x^4])","C",1
387,1,293,385,0.2592133,"\int \frac{d+e x^2}{\sqrt{a+b x^2-c x^4}} \, dx","Integrate[(d + e*x^2)/Sqrt[a + b*x^2 - c*x^4],x]","-\frac{i \sqrt{\frac{2 c x^2}{\sqrt{4 a c+b^2}-b}+1} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} \left(\left(e \left(b-\sqrt{4 a c+b^2}\right)+2 c d\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)+e \left(\sqrt{4 a c+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)\right)}{2 \sqrt{2} c \sqrt{-\frac{c}{\sqrt{4 a c+b^2}+b}} \sqrt{a+b x^2-c x^4}}","\frac{\sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} \left(e \left(b-\sqrt{4 a c+b^2}\right)+2 c d\right) F\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right)|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right)}{2 \sqrt{2} c^{3/2} \sqrt{a+b x^2-c x^4}}-\frac{e \left(b-\sqrt{4 a c+b^2}\right) \sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} E\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right)|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right)}{2 \sqrt{2} c^{3/2} \sqrt{a+b x^2-c x^4}}",1,"((-1/2*I)*Sqrt[1 + (2*c*x^2)/(-b + Sqrt[b^2 + 4*a*c])]*Sqrt[1 - (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*((-b + Sqrt[b^2 + 4*a*c])*e*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*x], (b + Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])] + (2*c*d + (b - Sqrt[b^2 + 4*a*c])*e)*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
388,1,205,197,0.2272517,"\int \frac{1}{\left(d+e x^2\right) \sqrt{a+b x^2-c x^4}} \, dx","Integrate[1/((d + e*x^2)*Sqrt[a + b*x^2 - c*x^4]),x]","-\frac{i \sqrt{\frac{2 c x^2}{\sqrt{4 a c+b^2}-b}+1} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} \Pi \left(-\frac{\left(b+\sqrt{b^2+4 a c}\right) e}{2 c d};i \sinh ^{-1}\left(\sqrt{2} \sqrt{-\frac{c}{b+\sqrt{b^2+4 a c}}} x\right)|-\frac{b+\sqrt{b^2+4 a c}}{\sqrt{b^2+4 a c}-b}\right)}{\sqrt{2} d \sqrt{-\frac{c}{\sqrt{4 a c+b^2}+b}} \sqrt{a+b x^2-c x^4}}","\frac{\sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} \Pi \left(-\frac{\left(b+\sqrt{b^2+4 a c}\right) e}{2 c d};\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right)|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right)}{\sqrt{2} \sqrt{c} d \sqrt{a+b x^2-c x^4}}",1,"((-I)*Sqrt[1 + (2*c*x^2)/(-b + Sqrt[b^2 + 4*a*c])]*Sqrt[1 - (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*EllipticPi[-1/2*((b + Sqrt[b^2 + 4*a*c])*e)/(c*d), I*ArcSinh[Sqrt[2]*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*x], -((b + Sqrt[b^2 + 4*a*c])/(-b + Sqrt[b^2 + 4*a*c]))])/(Sqrt[2]*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*d*Sqrt[a + b*x^2 - c*x^4])","C",1
389,1,464,718,5.5300798,"\int \frac{1}{\left(d+e x^2\right)^2 \sqrt{a+b x^2-c x^4}} \, dx","Integrate[1/((d + e*x^2)^2*Sqrt[a + b*x^2 - c*x^4]),x]","-\frac{\sqrt{a+b x^2-c x^4} \left(4 d e^2 x+\frac{i \left(d+e x^2\right) \sqrt{\frac{4 c x^2}{\sqrt{4 a c+b^2}-b}+2} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} \left(2 \left(e (a e-2 b d)-3 c d^2\right) \Pi \left(-\frac{\left(b+\sqrt{b^2+4 a c}\right) e}{2 c d};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)+d \left(e \left(b-\sqrt{4 a c+b^2}\right)+2 c d\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)+d e \left(\sqrt{4 a c+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)\right)}{\sqrt{-\frac{c}{\sqrt{4 a c+b^2}+b}} \left(-a-b x^2+c x^4\right)}\right)}{8 d^2 \left(d+e x^2\right) \left(e (b d-a e)+c d^2\right)}","-\frac{\sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} \left(e \left(b-\sqrt{4 a c+b^2}\right)+2 c d\right) F\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right)|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right)}{4 \sqrt{2} \sqrt{c} d \sqrt{a+b x^2-c x^4} \left(e (b d-a e)+c d^2\right)}+\frac{e \left(b-\sqrt{4 a c+b^2}\right) \sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} E\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right)|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right)}{4 \sqrt{2} \sqrt{c} d \sqrt{a+b x^2-c x^4} \left(e (b d-a e)+c d^2\right)}+\frac{\sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} \left(e (2 b d-a e)+3 c d^2\right) \Pi \left(-\frac{\left(b+\sqrt{b^2+4 a c}\right) e}{2 c d};\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right)|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right)}{2 \sqrt{2} \sqrt{c} d^2 \sqrt{a+b x^2-c x^4} \left(e (b d-a e)+c d^2\right)}-\frac{e^2 x \sqrt{a+b x^2-c x^4}}{2 d \left(d+e x^2\right) \left(-a e^2+b d e+c d^2\right)}",1,"-1/8*(Sqrt[a + b*x^2 - c*x^4]*(4*d*e^2*x + (I*Sqrt[2 + (4*c*x^2)/(-b + Sqrt[b^2 + 4*a*c])]*Sqrt[1 - (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*(d + e*x^2)*((-b + Sqrt[b^2 + 4*a*c])*d*e*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])] + d*(2*c*d + (b - Sqrt[b^2 + 4*a*c])*e)*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*x], (b + Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])] + 2*(-3*c*d^2 + e*(-2*b*d + a*e))*EllipticPi[-1/2*((b + Sqrt[b^2 + 4*a*c])*e)/(c*d), 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]))]*(-a - b*x^2 + c*x^4))))/(d^2*(c*d^2 + e*(b*d - a*e))*(d + e*x^2))","C",1
390,1,304,479,0.3021564,"\int \frac{d+e x^2}{\sqrt{-a+b x^2+c x^4}} \, dx","Integrate[(d + e*x^2)/Sqrt[-a + b*x^2 + c*x^4],x]","\frac{i \sqrt{\frac{\sqrt{4 a c+b^2}+b+2 c x^2}{\sqrt{4 a c+b^2}+b}} \sqrt{\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1} \left(\left(e \left(b-\sqrt{4 a c+b^2}\right)-2 c d\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)+e \left(\sqrt{4 a c+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)\right)}{2 \sqrt{2} c \sqrt{\frac{c}{\sqrt{4 a c+b^2}+b}} \sqrt{-a+b x^2+c x^4}}","-\frac{e \left(b-\sqrt{4 a c+b^2}\right) \sqrt{\sqrt{4 a c+b^2}+b} \left(\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1\right) E\left(\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right)|-\frac{2 \sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right)}{2 \sqrt{2} c^{3/2} \sqrt{\frac{\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1}{\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}+1}} \sqrt{-a+b x^2+c x^4}}+\frac{d \sqrt{\sqrt{4 a c+b^2}+b} \left(\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1\right) F\left(\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right)|-\frac{2 \sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right)}{\sqrt{2} \sqrt{c} \sqrt{\frac{\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1}{\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}+1}} \sqrt{-a+b x^2+c x^4}}+\frac{e x \left(b-\sqrt{4 a c+b^2}\right) \left(\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1\right)}{2 c \sqrt{-a+b x^2+c x^4}}",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 + Sqrt[b^2 + 4*a*c])*e*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 + 4*a*c])]*x], (b + Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])] + (-2*c*d + (b - Sqrt[b^2 + 4*a*c])*e)*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
391,1,216,204,0.2233425,"\int \frac{1}{\left(d+e x^2\right) \sqrt{-a+b x^2+c x^4}} \, dx","Integrate[1/((d + e*x^2)*Sqrt[-a + b*x^2 + c*x^4]),x]","-\frac{i \sqrt{\frac{\sqrt{4 a c+b^2}+b+2 c x^2}{\sqrt{4 a c+b^2}+b}} \sqrt{\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1} \Pi \left(\frac{\left(b+\sqrt{b^2+4 a c}\right) e}{2 c d};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2+4 a c}}} x\right)|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right)}{\sqrt{2} d \sqrt{\frac{c}{\sqrt{4 a c+b^2}+b}} \sqrt{-a+b x^2+c x^4}}","\frac{\sqrt{\sqrt{4 a c+b^2}-b} \sqrt{\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1} \sqrt{\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}+1} \Pi \left(\frac{\left(b-\sqrt{b^2+4 a c}\right) e}{2 c d};\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2+4 a c}-b}}\right)|\frac{b-\sqrt{b^2+4 a c}}{b+\sqrt{b^2+4 a c}}\right)}{\sqrt{2} \sqrt{c} d \sqrt{-a+b x^2+c x^4}}",1,"((-I)*Sqrt[(b + Sqrt[b^2 + 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 + 4*a*c])]*EllipticPi[((b + Sqrt[b^2 + 4*a*c])*e)/(2*c*d), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 + 4*a*c])]*x], (b + Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])])/(Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 + 4*a*c])]*d*Sqrt[-a + b*x^2 + c*x^4])","C",1
392,1,295,293,0.3063188,"\int \frac{d+e x^2}{\sqrt{-a+b x^2-c x^4}} \, dx","Integrate[(d + e*x^2)/Sqrt[-a + b*x^2 - c*x^4],x]","-\frac{i \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}+1} \sqrt{1-\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}} \left(\left(e \left(b-\sqrt{b^2-4 a c}\right)+2 c d\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)+e \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) \sqrt{\frac{a-b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(\frac{\sqrt{c} d}{\sqrt{a}}+e\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(\frac{b}{\sqrt{a} \sqrt{c}}+2\right)\right)}{2 c^{3/4} \sqrt{-a+b x^2-c x^4}}-\frac{\sqrt[4]{a} e \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(\frac{b}{\sqrt{a} \sqrt{c}}+2\right)\right)}{c^{3/4} \sqrt{-a+b x^2-c x^4}}-\frac{e x \sqrt{-a+b x^2-c x^4}}{\sqrt{c} \left(\sqrt{a}+\sqrt{c} x^2\right)}",1,"((-1/2*I)*Sqrt[1 + (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 - (2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*((-b + Sqrt[b^2 - 4*a*c])*e*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[-(c/(b + Sqrt[b^2 - 4*a*c]))]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + (2*c*d + (b - Sqrt[b^2 - 4*a*c])*e)*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
393,1,207,412,0.2213085,"\int \frac{1}{\left(d+e x^2\right) \sqrt{-a+b x^2-c x^4}} \, dx","Integrate[1/((d + e*x^2)*Sqrt[-a + b*x^2 - c*x^4]),x]","-\frac{i \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}+1} \sqrt{1-\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}} \Pi \left(-\frac{\left(b+\sqrt{b^2-4 a c}\right) e}{2 c d};i \sinh ^{-1}\left(\sqrt{2} \sqrt{-\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|-\frac{b+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}-b}\right)}{\sqrt{2} d \sqrt{-\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{-a+b x^2-c x^4}}","-\frac{a^{3/4} \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(\frac{\sqrt{c} d}{\sqrt{a}}+e\right)^2 \Pi \left(-\frac{\left(\sqrt{c} d-\sqrt{a} e\right)^2}{4 \sqrt{a} \sqrt{c} d e};2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(\frac{b}{\sqrt{a} \sqrt{c}}+2\right)\right)}{4 \sqrt[4]{c} d \sqrt{-a+b x^2-c x^4} \left(c d^2-a e^2\right)}+\frac{\sqrt{e} \tan ^{-1}\left(\frac{x \sqrt{-e (a e+b d)-c d^2}}{\sqrt{d} \sqrt{e} \sqrt{-a+b x^2-c x^4}}\right)}{2 \sqrt{d} \sqrt{-e (a e+b d)-c d^2}}+\frac{\sqrt[4]{c} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a-b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(\frac{b}{\sqrt{a} \sqrt{c}}+2\right)\right)}{2 \sqrt[4]{a} \sqrt{-a+b x^2-c x^4} \left(\sqrt{c} d-\sqrt{a} e\right)}",1,"((-I)*Sqrt[1 + (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 - (2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*EllipticPi[-1/2*((b + Sqrt[b^2 - 4*a*c])*e)/(c*d), I*ArcSinh[Sqrt[2]*Sqrt[-(c/(b + Sqrt[b^2 - 4*a*c]))]*x], -((b + Sqrt[b^2 - 4*a*c])/(-b + Sqrt[b^2 - 4*a*c]))])/(Sqrt[2]*Sqrt[-(c/(b + Sqrt[b^2 - 4*a*c]))]*d*Sqrt[-a + b*x^2 - c*x^4])","C",1
394,1,154,229,0.2303272,"\int \frac{\left(d+e x^2\right)^3}{\sqrt{2+3 x^2+x^4}} \, dx","Integrate[(d + e*x^2)^3/Sqrt[2 + 3*x^2 + x^4],x]","\frac{-3 i e \sqrt{x^2+1} \sqrt{x^2+2} \left(5 d^2-10 d e+6 e^2\right) E\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)-5 i \sqrt{x^2+1} \sqrt{x^2+2} \left(d^3-3 d^2 e+4 d e^2-2 e^3\right) F\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+e^2 x \left(x^4+3 x^2+2\right) \left(5 d+e \left(x^2-4\right)\right)}{5 \sqrt{x^4+3 x^2+2}}","\frac{\left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} \left(5 d^3-10 d e^2+8 e^3\right) F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{5 \sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{3 e \left(x^2+2\right) x \left(5 d^2-10 d e+6 e^2\right)}{5 \sqrt{x^4+3 x^2+2}}-\frac{3 \sqrt{2} e \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} \left(5 d^2-10 d e+6 e^2\right) E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{5 \sqrt{x^4+3 x^2+2}}+\frac{1}{5} e^2 \sqrt{x^4+3 x^2+2} x (5 d-4 e)+\frac{1}{5} e^3 \sqrt{x^4+3 x^2+2} x^3",1,"(e^2*x*(2 + 3*x^2 + x^4)*(5*d + e*(-4 + x^2)) - (3*I)*e*(5*d^2 - 10*d*e + 6*e^2)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - (5*I)*(d^3 - 3*d^2*e + 4*d*e^2 - 2*e^3)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt[2]], 2])/(5*Sqrt[2 + 3*x^2 + x^4])","C",1
395,1,127,168,0.1569499,"\int \frac{\left(d+e x^2\right)^2}{\sqrt{2+3 x^2+x^4}} \, dx","Integrate[(d + e*x^2)^2/Sqrt[2 + 3*x^2 + x^4],x]","\frac{-i \sqrt{x^2+1} \sqrt{x^2+2} \left(3 d^2-6 d e+4 e^2\right) F\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)-6 i e \sqrt{x^2+1} \sqrt{x^2+2} (d-e) E\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+e^2 x \left(x^4+3 x^2+2\right)}{3 \sqrt{x^4+3 x^2+2}}","\frac{\left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} \left(3 d^2-2 e^2\right) F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{3 \sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{2 e x \left(x^2+2\right) (d-e)}{\sqrt{x^4+3 x^2+2}}-\frac{2 \sqrt{2} e \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} (d-e) E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{x^4+3 x^2+2}}+\frac{1}{3} e^2 x \sqrt{x^4+3 x^2+2}",1,"(e^2*x*(2 + 3*x^2 + x^4) - (6*I)*(d - e)*e*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - I*(3*d^2 - 6*d*e + 4*e^2)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt[2]], 2])/(3*Sqrt[2 + 3*x^2 + x^4])","C",1
396,1,73,122,0.0738095,"\int \frac{d+e x^2}{\sqrt{2+3 x^2+x^4}} \, dx","Integrate[(d + e*x^2)/Sqrt[2 + 3*x^2 + x^4],x]","-\frac{i \sqrt{x^2+1} \sqrt{x^2+2} \left((d-e) F\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+e E\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)\right)}{\sqrt{x^4+3 x^2+2}}","\frac{d \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{e x \left(x^2+2\right)}{\sqrt{x^4+3 x^2+2}}-\frac{\sqrt{2} e \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{x^4+3 x^2+2}}",1,"((-I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*(e*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] + (d - e)*EllipticF[I*ArcSinh[x/Sqrt[2]], 2]))/Sqrt[2 + 3*x^2 + x^4]","C",1
397,1,59,124,0.1100095,"\int \frac{1}{\left(d+e x^2\right) \sqrt{2+3 x^2+x^4}} \, dx","Integrate[1/((d + e*x^2)*Sqrt[2 + 3*x^2 + x^4]),x]","-\frac{i \sqrt{x^2+1} \sqrt{x^2+2} \Pi \left(\frac{2 e}{d};\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)}{d \sqrt{x^4+3 x^2+2}}","\frac{\left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{2} \sqrt{x^4+3 x^2+2} (d-e)}-\frac{e \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} \Pi \left(1-\frac{e}{d};\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{2} d \sqrt{x^4+3 x^2+2} (d-e)}",1,"((-I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticPi[(2*e)/d, I*ArcSinh[x/Sqrt[2]], 2])/(d*Sqrt[2 + 3*x^2 + x^4])","C",1
398,1,175,316,0.5987962,"\int \frac{1}{\left(d+e x^2\right)^2 \sqrt{2+3 x^2+x^4}} \, dx","Integrate[1/((d + e*x^2)^2*Sqrt[2 + 3*x^2 + x^4]),x]","\frac{\frac{e^2 x \left(x^4+3 x^2+2\right)}{\left(d^2-3 d e+2 e^2\right) \left(d+e x^2\right)}+\frac{i \sqrt{x^2+1} \sqrt{x^2+2} \left(\left(-3 d^2+6 d e-2 e^2\right) \Pi \left(\frac{2 e}{d};\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+d (d-e) F\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+d e E\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)\right)}{d (d-2 e) (d-e)}}{2 d \sqrt{x^4+3 x^2+2}}","\frac{e^2 x \sqrt{x^4+3 x^2+2}}{2 d \left(d^2-3 d e+2 e^2\right) \left(d+e x^2\right)}-\frac{e x \left(x^2+2\right)}{2 d \sqrt{x^4+3 x^2+2} \left(d^2-3 d e+2 e^2\right)}-\frac{e \left(x^2+2\right) \left(3 d^2-6 d e+2 e^2\right) \Pi \left(1-\frac{e}{d};\tan ^{-1}(x)|\frac{1}{2}\right)}{2 \sqrt{2} d^2 \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2} (d-2 e) (d-e)^2}+\frac{\left(x^2+1\right) \sqrt{\frac{x^2+2}{2 x^2+2}} (2 d-e) F\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{2 d \sqrt{x^4+3 x^2+2} (d-e)^2}+\frac{e \left(x^2+1\right) \sqrt{\frac{x^2+2}{x^2+1}} E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{2} d \sqrt{x^4+3 x^2+2} (d-2 e) (d-e)}",1,"((e^2*x*(2 + 3*x^2 + x^4))/((d^2 - 3*d*e + 2*e^2)*(d + e*x^2)) + (I*Sqrt[1 + x^2]*Sqrt[2 + x^2]*(d*e*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] + d*(d - e)*EllipticF[I*ArcSinh[x/Sqrt[2]], 2] + (-3*d^2 + 6*d*e - 2*e^2)*EllipticPi[(2*e)/d, I*ArcSinh[x/Sqrt[2]], 2]))/(d*(d - 2*e)*(d - e)))/(2*d*Sqrt[2 + 3*x^2 + x^4])","C",1
399,0,0,27,0.097706,"\int \left(c+e x^2\right)^q \left(a+c x^2+b x^4\right)^p \, dx","Integrate[(c + e*x^2)^q*(a + c*x^2 + b*x^4)^p,x]","\int \left(c+e x^2\right)^q \left(a+c x^2+b x^4\right)^p \, dx","\text{Int}\left(\left(c+e x^2\right)^q \left(a+b x^4+c x^2\right)^p,x\right)",0,"Integrate[(c + e*x^2)^q*(a + c*x^2 + b*x^4)^p, x]","A",-1
400,1,373,498,0.5090104,"\int \left(c+e x^2\right)^3 \left(a+c x^2+b x^4\right)^p \, dx","Integrate[(c + e*x^2)^3*(a + c*x^2 + b*x^4)^p,x]","\frac{1}{35} x \left(\frac{-\sqrt{c^2-4 a b}+2 b x^2+c}{c-\sqrt{c^2-4 a b}}\right)^{-p} \left(\frac{\sqrt{c^2-4 a b}+2 b x^2+c}{\sqrt{c^2-4 a b}+c}\right)^{-p} \left(a+b x^4+c x^2\right)^p \left(e x^2 \left(e x^2 \left(5 e x^2 F_1\left(\frac{7}{2};-p,-p;\frac{9}{2};-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}},\frac{2 b x^2}{\sqrt{c^2-4 a b}-c}\right)+21 c F_1\left(\frac{5}{2};-p,-p;\frac{7}{2};-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}},\frac{2 b x^2}{\sqrt{c^2-4 a b}-c}\right)\right)+35 c^2 F_1\left(\frac{3}{2};-p,-p;\frac{5}{2};-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}},\frac{2 b x^2}{\sqrt{c^2-4 a b}-c}\right)\right)+35 c^3 F_1\left(\frac{1}{2};-p,-p;\frac{3}{2};-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}},\frac{2 b x^2}{\sqrt{c^2-4 a b}-c}\right)\right)","\frac{e x^3 \left(-3 b e \left(a e (4 p+5)+c^2 \left(8 p^2+26 p+21\right)\right)+3 b^2 c^2 \left(16 p^2+48 p+35\right)+c^2 e^2 \left(4 p^2+16 p+15\right)\right) \left(\frac{2 b x^2}{c-\sqrt{c^2-4 a b}}+1\right)^{-p} \left(a+b x^4+c x^2\right)^p \left(\frac{2 b x^2}{\sqrt{c^2-4 a b}+c}+1\right)^{-p} F_1\left(\frac{3}{2};-p,-p;\frac{5}{2};-\frac{2 b x^2}{c-\sqrt{c^2-4 a b}},-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}}\right)}{3 b^2 (4 p+5) (4 p+7)}+\frac{c x \left(-3 a b e^2 (4 p+7)+a e^3 (2 p+5)+b^2 c^2 \left(16 p^2+48 p+35\right)\right) \left(\frac{2 b x^2}{c-\sqrt{c^2-4 a b}}+1\right)^{-p} \left(a+b x^4+c x^2\right)^p \left(\frac{2 b x^2}{\sqrt{c^2-4 a b}+c}+1\right)^{-p} F_1\left(\frac{1}{2};-p,-p;\frac{3}{2};-\frac{2 b x^2}{c-\sqrt{c^2-4 a b}},-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}}\right)}{b^2 (4 p+5) (4 p+7)}+\frac{c e^2 x (12 b p+21 b-2 e p-5 e) \left(a+b x^4+c x^2\right)^{p+1}}{b^2 (4 p+5) (4 p+7)}+\frac{e^3 x^3 \left(a+b x^4+c x^2\right)^{p+1}}{b (4 p+7)}",1,"(x*(a + c*x^2 + b*x^4)^p*(35*c^3*AppellF1[1/2, -p, -p, 3/2, (-2*b*x^2)/(c + Sqrt[-4*a*b + c^2]), (2*b*x^2)/(-c + Sqrt[-4*a*b + c^2])] + e*x^2*(35*c^2*AppellF1[3/2, -p, -p, 5/2, (-2*b*x^2)/(c + Sqrt[-4*a*b + c^2]), (2*b*x^2)/(-c + Sqrt[-4*a*b + c^2])] + e*x^2*(21*c*AppellF1[5/2, -p, -p, 7/2, (-2*b*x^2)/(c + Sqrt[-4*a*b + c^2]), (2*b*x^2)/(-c + Sqrt[-4*a*b + c^2])] + 5*e*x^2*AppellF1[7/2, -p, -p, 9/2, (-2*b*x^2)/(c + Sqrt[-4*a*b + c^2]), (2*b*x^2)/(-c + Sqrt[-4*a*b + c^2])]))))/(35*((c - Sqrt[-4*a*b + c^2] + 2*b*x^2)/(c - Sqrt[-4*a*b + c^2]))^p*((c + Sqrt[-4*a*b + c^2] + 2*b*x^2)/(c + Sqrt[-4*a*b + c^2]))^p)","A",0
401,1,303,358,0.3700087,"\int \left(c+e x^2\right)^2 \left(a+c x^2+b x^4\right)^p \, dx","Integrate[(c + e*x^2)^2*(a + c*x^2 + b*x^4)^p,x]","\frac{1}{15} x \left(\frac{-\sqrt{c^2-4 a b}+2 b x^2+c}{c-\sqrt{c^2-4 a b}}\right)^{-p} \left(\frac{\sqrt{c^2-4 a b}+2 b x^2+c}{\sqrt{c^2-4 a b}+c}\right)^{-p} \left(a+b x^4+c x^2\right)^p \left(e x^2 \left(3 e x^2 F_1\left(\frac{5}{2};-p,-p;\frac{7}{2};-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}},\frac{2 b x^2}{\sqrt{c^2-4 a b}-c}\right)+10 c F_1\left(\frac{3}{2};-p,-p;\frac{5}{2};-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}},\frac{2 b x^2}{\sqrt{c^2-4 a b}-c}\right)\right)+15 c^2 F_1\left(\frac{1}{2};-p,-p;\frac{3}{2};-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}},\frac{2 b x^2}{\sqrt{c^2-4 a b}-c}\right)\right)","-\frac{x \left(a e^2-b c^2 (4 p+5)\right) \left(\frac{2 b x^2}{c-\sqrt{c^2-4 a b}}+1\right)^{-p} \left(a+b x^4+c x^2\right)^p \left(\frac{2 b x^2}{\sqrt{c^2-4 a b}+c}+1\right)^{-p} F_1\left(\frac{1}{2};-p,-p;\frac{3}{2};-\frac{2 b x^2}{c-\sqrt{c^2-4 a b}},-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}}\right)}{b (4 p+5)}+\frac{c e x^3 (8 b p+10 b-2 e p-3 e) \left(\frac{2 b x^2}{c-\sqrt{c^2-4 a b}}+1\right)^{-p} \left(a+b x^4+c x^2\right)^p \left(\frac{2 b x^2}{\sqrt{c^2-4 a b}+c}+1\right)^{-p} F_1\left(\frac{3}{2};-p,-p;\frac{5}{2};-\frac{2 b x^2}{c-\sqrt{c^2-4 a b}},-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}}\right)}{3 b (4 p+5)}+\frac{e^2 x \left(a+b x^4+c x^2\right)^{p+1}}{b (4 p+5)}",1,"(x*(a + c*x^2 + b*x^4)^p*(15*c^2*AppellF1[1/2, -p, -p, 3/2, (-2*b*x^2)/(c + Sqrt[-4*a*b + c^2]), (2*b*x^2)/(-c + Sqrt[-4*a*b + c^2])] + e*x^2*(10*c*AppellF1[3/2, -p, -p, 5/2, (-2*b*x^2)/(c + Sqrt[-4*a*b + c^2]), (2*b*x^2)/(-c + Sqrt[-4*a*b + c^2])] + 3*e*x^2*AppellF1[5/2, -p, -p, 7/2, (-2*b*x^2)/(c + Sqrt[-4*a*b + c^2]), (2*b*x^2)/(-c + Sqrt[-4*a*b + c^2])])))/(15*((c - Sqrt[-4*a*b + c^2] + 2*b*x^2)/(c - Sqrt[-4*a*b + c^2]))^p*((c + Sqrt[-4*a*b + c^2] + 2*b*x^2)/(c + Sqrt[-4*a*b + c^2]))^p)","A",0
402,1,232,274,0.2545019,"\int \left(c+e x^2\right) \left(a+c x^2+b x^4\right)^p \, dx","Integrate[(c + e*x^2)*(a + c*x^2 + b*x^4)^p,x]","\frac{1}{3} x \left(\frac{-\sqrt{c^2-4 a b}+2 b x^2+c}{c-\sqrt{c^2-4 a b}}\right)^{-p} \left(\frac{\sqrt{c^2-4 a b}+2 b x^2+c}{\sqrt{c^2-4 a b}+c}\right)^{-p} \left(a+b x^4+c x^2\right)^p \left(e x^2 F_1\left(\frac{3}{2};-p,-p;\frac{5}{2};-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}},\frac{2 b x^2}{\sqrt{c^2-4 a b}-c}\right)+3 c F_1\left(\frac{1}{2};-p,-p;\frac{3}{2};-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}},\frac{2 b x^2}{\sqrt{c^2-4 a b}-c}\right)\right)","\frac{1}{3} e x^3 \left(\frac{2 b x^2}{c-\sqrt{c^2-4 a b}}+1\right)^{-p} \left(a+b x^4+c x^2\right)^p \left(\frac{2 b x^2}{\sqrt{c^2-4 a b}+c}+1\right)^{-p} F_1\left(\frac{3}{2};-p,-p;\frac{5}{2};-\frac{2 b x^2}{c-\sqrt{c^2-4 a b}},-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}}\right)+c x \left(\frac{2 b x^2}{c-\sqrt{c^2-4 a b}}+1\right)^{-p} \left(a+b x^4+c x^2\right)^p \left(\frac{2 b x^2}{\sqrt{c^2-4 a b}+c}+1\right)^{-p} F_1\left(\frac{1}{2};-p,-p;\frac{3}{2};-\frac{2 b x^2}{c-\sqrt{c^2-4 a b}},-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}}\right)",1,"(x*(a + c*x^2 + b*x^4)^p*(3*c*AppellF1[1/2, -p, -p, 3/2, (-2*b*x^2)/(c + Sqrt[-4*a*b + c^2]), (2*b*x^2)/(-c + Sqrt[-4*a*b + c^2])] + e*x^2*AppellF1[3/2, -p, -p, 5/2, (-2*b*x^2)/(c + Sqrt[-4*a*b + c^2]), (2*b*x^2)/(-c + Sqrt[-4*a*b + c^2])]))/(3*((c - Sqrt[-4*a*b + c^2] + 2*b*x^2)/(c - Sqrt[-4*a*b + c^2]))^p*((c + Sqrt[-4*a*b + c^2] + 2*b*x^2)/(c + Sqrt[-4*a*b + c^2]))^p)","A",0
403,1,161,133,0.1558581,"\int \left(a+c x^2+b x^4\right)^p \, dx","Integrate[(a + c*x^2 + b*x^4)^p,x]","x \left(\frac{-\sqrt{c^2-4 a b}+2 b x^2+c}{c-\sqrt{c^2-4 a b}}\right)^{-p} \left(\frac{\sqrt{c^2-4 a b}+2 b x^2+c}{\sqrt{c^2-4 a b}+c}\right)^{-p} \left(a+b x^4+c x^2\right)^p F_1\left(\frac{1}{2};-p,-p;\frac{3}{2};-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}},\frac{2 b x^2}{\sqrt{c^2-4 a b}-c}\right)","x \left(\frac{2 b x^2}{c-\sqrt{c^2-4 a b}}+1\right)^{-p} \left(\frac{2 b x^2}{\sqrt{c^2-4 a b}+c}+1\right)^{-p} \left(a+b x^4+c x^2\right)^p F_1\left(\frac{1}{2};-p,-p;\frac{3}{2};-\frac{2 b x^2}{c-\sqrt{c^2-4 a b}},-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}}\right)",1,"(x*(a + c*x^2 + b*x^4)^p*AppellF1[1/2, -p, -p, 3/2, (-2*b*x^2)/(c + Sqrt[-4*a*b + c^2]), (2*b*x^2)/(-c + Sqrt[-4*a*b + c^2])])/(((c - Sqrt[-4*a*b + c^2] + 2*b*x^2)/(c - Sqrt[-4*a*b + c^2]))^p*((c + Sqrt[-4*a*b + c^2] + 2*b*x^2)/(c + Sqrt[-4*a*b + c^2]))^p)","A",0
404,0,0,27,0.129085,"\int \frac{\left(a+c x^2+b x^4\right)^p}{c+e x^2} \, dx","Integrate[(a + c*x^2 + b*x^4)^p/(c + e*x^2),x]","\int \frac{\left(a+c x^2+b x^4\right)^p}{c+e x^2} \, dx","\text{Int}\left(\frac{\left(a+b x^4+c x^2\right)^p}{c+e x^2},x\right)",0,"Integrate[(a + c*x^2 + b*x^4)^p/(c + e*x^2), x]","A",-1
405,0,0,27,0.2602257,"\int \frac{\left(a+c x^2+b x^4\right)^p}{\left(c+e x^2\right)^2} \, dx","Integrate[(a + c*x^2 + b*x^4)^p/(c + e*x^2)^2,x]","\int \frac{\left(a+c x^2+b x^4\right)^p}{\left(c+e x^2\right)^2} \, dx","\text{Int}\left(\frac{\left(a+b x^4+c x^2\right)^p}{\left(c+e x^2\right)^2},x\right)",0,"Integrate[(a + c*x^2 + b*x^4)^p/(c + e*x^2)^2, x]","A",-1
406,1,258,446,0.6837362,"\int \frac{f+g x}{(d+e x) \sqrt{a+c x^4}} \, dx","Integrate[(f + g*x)/((d + e*x)*Sqrt[a + c*x^4]),x]","\frac{\frac{(d g-e f) \left(\sqrt[4]{c} d e \sqrt{a+c x^4} \tanh ^{-1}\left(\frac{a e^2+c d^2 x^2}{\sqrt{a+c x^4} \sqrt{a e^4+c d^4}}\right)+2 \sqrt[4]{-1} \sqrt[4]{a} \sqrt{\frac{c x^4}{a}+1} \sqrt{a e^4+c d^4} \Pi \left(\frac{i \sqrt{a} e^2}{\sqrt{c} d^2};\left.\sin ^{-1}\left(\frac{(-1)^{3/4} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)\right|-1\right)\right)}{\sqrt[4]{c} d \sqrt{a e^4+c d^4}}-\frac{2 i g \sqrt{\frac{c x^4}{a}+1} F\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}}}{2 e \sqrt{a+c x^4}}","\frac{(e f-d g) \tan ^{-1}\left(\frac{x \sqrt{-a e^4-c d^4}}{d e \sqrt{a+c x^4}}\right)}{2 \sqrt{-a e^4-c d^4}}+\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) \left(\sqrt{a} e g+\sqrt{c} d f\right)}{2 \sqrt[4]{a} \sqrt[4]{c} \sqrt{a+c x^4} \left(\sqrt{a} e^2+\sqrt{c} d^2\right)}-\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(\sqrt{c} d^2-\sqrt{a} e^2\right) (e f-d g) \Pi \left(\frac{\left(\sqrt{c} d^2+\sqrt{a} e^2\right)^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 \sqrt[4]{a} \sqrt[4]{c} d e \sqrt{a+c x^4} \left(\sqrt{a} e^2+\sqrt{c} d^2\right)}-\frac{(e f-d g) \tanh ^{-1}\left(\frac{a e^2+c d^2 x^2}{\sqrt{a+c x^4} \sqrt{a e^4+c d^4}}\right)}{2 \sqrt{a e^4+c d^4}}",1,"(((-2*I)*g*Sqrt[1 + (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])/Sqrt[(I*Sqrt[c])/Sqrt[a]] + ((-(e*f) + d*g)*(c^(1/4)*d*e*Sqrt[a + c*x^4]*ArcTanh[(a*e^2 + c*d^2*x^2)/(Sqrt[c*d^4 + a*e^4]*Sqrt[a + c*x^4])] + 2*(-1)^(1/4)*a^(1/4)*Sqrt[c*d^4 + a*e^4]*Sqrt[1 + (c*x^4)/a]*EllipticPi[(I*Sqrt[a]*e^2)/(Sqrt[c]*d^2), ArcSin[((-1)^(3/4)*c^(1/4)*x)/a^(1/4)], -1]))/(c^(1/4)*d*Sqrt[c*d^4 + a*e^4]))/(2*e*Sqrt[a + c*x^4])","C",1
407,1,719,218,1.2602958,"\int \frac{f+g x}{(d+e x) \sqrt{-a+c x^4}} \, dx","Integrate[(f + g*x)/((d + e*x)*Sqrt[-a + c*x^4]),x]","\frac{\frac{i f \sqrt{-\frac{(1-i) \left(\sqrt[4]{a}-\sqrt[4]{c} x\right)}{\sqrt[4]{c} x+i \sqrt[4]{a}}} \sqrt{\frac{(1+i) \left(\sqrt[4]{a}+i \sqrt[4]{c} x\right) \left(\sqrt[4]{a}+\sqrt[4]{c} x\right)}{\left(\sqrt[4]{a}-i \sqrt[4]{c} x\right)^2}} \left(\sqrt[4]{a}-i \sqrt[4]{c} x\right)^2 \left(\left(\sqrt[4]{a} e-\sqrt[4]{c} d\right) F\left(\left.\sin ^{-1}\left(\sqrt{\frac{(1+i) \left(\sqrt[4]{c} x+\sqrt[4]{a}\right)}{2 \sqrt[4]{c} x+2 i \sqrt[4]{a}}}\right)\right|2\right)-(1-i) \sqrt[4]{a} e \Pi \left(\frac{(1-i) \left(\sqrt[4]{c} d-i \sqrt[4]{a} e\right)}{\sqrt[4]{c} d-\sqrt[4]{a} e};\left.\sin ^{-1}\left(\sqrt{\frac{(1+i) \left(\sqrt[4]{c} x+\sqrt[4]{a}\right)}{2 \sqrt[4]{c} x+2 i \sqrt[4]{a}}}\right)\right|2\right)\right)}{\sqrt[4]{a} \left(\sqrt[4]{a} e-\sqrt[4]{c} d\right) \left(\sqrt[4]{a} e+i \sqrt[4]{c} d\right)}+\frac{d g \sqrt{-\frac{(1-i) \left(\sqrt[4]{a}-\sqrt[4]{c} x\right)}{\sqrt[4]{c} x+i \sqrt[4]{a}}} \sqrt{\frac{(1+i) \left(\sqrt[4]{a}+i \sqrt[4]{c} x\right) \left(\sqrt[4]{a}+\sqrt[4]{c} x\right)}{\left(\sqrt[4]{a}-i \sqrt[4]{c} x\right)^2}} \left(\sqrt[4]{a}-i \sqrt[4]{c} x\right)^2 \left(i \left(\sqrt[4]{c} d-\sqrt[4]{a} e\right) F\left(\left.\sin ^{-1}\left(\sqrt{\frac{(1+i) \left(\sqrt[4]{c} x+\sqrt[4]{a}\right)}{2 \sqrt[4]{c} x+2 i \sqrt[4]{a}}}\right)\right|2\right)+(1+i) \sqrt[4]{a} e \Pi \left(\frac{(1-i) \left(\sqrt[4]{c} d-i \sqrt[4]{a} e\right)}{\sqrt[4]{c} d-\sqrt[4]{a} e};\left.\sin ^{-1}\left(\sqrt{\frac{(1+i) \left(\sqrt[4]{c} x+\sqrt[4]{a}\right)}{2 \sqrt[4]{c} x+2 i \sqrt[4]{a}}}\right)\right|2\right)\right)}{\sqrt[4]{a} e \left(\sqrt[4]{a} e-\sqrt[4]{c} d\right) \left(\sqrt[4]{a} e+i \sqrt[4]{c} d\right)}-\frac{i g \sqrt{1-\frac{c x^4}{a}} F\left(\left.i \sinh ^{-1}\left(\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)}{e \sqrt{-\frac{\sqrt{c}}{\sqrt{a}}}}}{\sqrt{c x^4-a}}","\frac{\sqrt[4]{a} \sqrt{1-\frac{c x^4}{a}} (e f-d g) \Pi \left(\frac{\sqrt{a} e^2}{\sqrt{c} d^2};\left.\sin ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)\right|-1\right)}{\sqrt[4]{c} d e \sqrt{c x^4-a}}+\frac{(e f-d g) \tanh ^{-1}\left(\frac{a e^2-c d^2 x^2}{\sqrt{c x^4-a} \sqrt{c d^4-a e^4}}\right)}{2 \sqrt{c d^4-a e^4}}+\frac{\sqrt[4]{a} g \sqrt{1-\frac{c x^4}{a}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)\right|-1\right)}{\sqrt[4]{c} e \sqrt{c x^4-a}}",1,"(((-I)*g*Sqrt[1 - (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1])/(Sqrt[-(Sqrt[c]/Sqrt[a])]*e) + (I*f*(a^(1/4) - I*c^(1/4)*x)^2*Sqrt[((-1 + I)*(a^(1/4) - c^(1/4)*x))/(I*a^(1/4) + c^(1/4)*x)]*Sqrt[((1 + I)*(a^(1/4) + I*c^(1/4)*x)*(a^(1/4) + c^(1/4)*x))/(a^(1/4) - I*c^(1/4)*x)^2]*((-(c^(1/4)*d) + a^(1/4)*e)*EllipticF[ArcSin[Sqrt[((1 + I)*(a^(1/4) + c^(1/4)*x))/((2*I)*a^(1/4) + 2*c^(1/4)*x)]], 2] - (1 - I)*a^(1/4)*e*EllipticPi[((1 - I)*(c^(1/4)*d - I*a^(1/4)*e))/(c^(1/4)*d - a^(1/4)*e), ArcSin[Sqrt[((1 + I)*(a^(1/4) + c^(1/4)*x))/((2*I)*a^(1/4) + 2*c^(1/4)*x)]], 2]))/(a^(1/4)*(-(c^(1/4)*d) + a^(1/4)*e)*(I*c^(1/4)*d + a^(1/4)*e)) + (d*g*(a^(1/4) - I*c^(1/4)*x)^2*Sqrt[((-1 + I)*(a^(1/4) - c^(1/4)*x))/(I*a^(1/4) + c^(1/4)*x)]*Sqrt[((1 + I)*(a^(1/4) + I*c^(1/4)*x)*(a^(1/4) + c^(1/4)*x))/(a^(1/4) - I*c^(1/4)*x)^2]*(I*(c^(1/4)*d - a^(1/4)*e)*EllipticF[ArcSin[Sqrt[((1 + I)*(a^(1/4) + c^(1/4)*x))/((2*I)*a^(1/4) + 2*c^(1/4)*x)]], 2] + (1 + I)*a^(1/4)*e*EllipticPi[((1 - I)*(c^(1/4)*d - I*a^(1/4)*e))/(c^(1/4)*d - a^(1/4)*e), ArcSin[Sqrt[((1 + I)*(a^(1/4) + c^(1/4)*x))/((2*I)*a^(1/4) + 2*c^(1/4)*x)]], 2]))/(a^(1/4)*e*(-(c^(1/4)*d) + a^(1/4)*e)*(I*c^(1/4)*d + a^(1/4)*e)))/Sqrt[-a + c*x^4]","C",1
408,1,685,65,3.0681701,"\int \frac{1-\sqrt{3}+x}{\left(1+\sqrt{3}+x\right) \sqrt{-4+4 \sqrt{3} x^2+x^4}} \, dx","Integrate[(1 - Sqrt[3] + x)/((1 + Sqrt[3] + x)*Sqrt[-4 + 4*Sqrt[3]*x^2 + x^4]),x]","\frac{\left(x+\sqrt{3}-1\right)^2 \sqrt{-x^3+\left(\sqrt{3}-1\right) x^2-2 \left(2+\sqrt{3}\right) x+2 \left(1+\sqrt{3}\right)} \sqrt{\frac{-\frac{4}{x+\sqrt{3}-1}+\sqrt{3}+1}{3+\sqrt{3}+i \sqrt{2 \left(2+\sqrt{3}\right)}}} \left(2 \sqrt{6} \sqrt{\frac{x^2+2 \sqrt{3}+4}{\left(x+\sqrt{3}-1\right)^2}} \sqrt{\sqrt{2 \left(2+\sqrt{3}\right)}-i \left(\frac{8}{x+\sqrt{3}-1}-\sqrt{3}+1\right)} \Pi \left(\frac{2 \sqrt{2 \left(2+\sqrt{3}\right)}}{\sqrt{2 \left(2+\sqrt{3}\right)}+i \left(3+\sqrt{3}\right)};\sin ^{-1}\left(\frac{\sqrt{\sqrt{2 \left(2+\sqrt{3}\right)}-i \left(-\sqrt{3}+1+\frac{8}{x+\sqrt{3}-1}\right)}}{2^{3/4} \sqrt[4]{2+\sqrt{3}}}\right)|\frac{2 i \sqrt{2 \left(2+\sqrt{3}\right)}}{3+\sqrt{3}+i \sqrt{2 \left(2+\sqrt{3}\right)}}\right)+\left(\frac{2 \left(2 i \sqrt{3}-\sqrt{2 \left(2+\sqrt{3}\right)}+\sqrt{6 \left(2+\sqrt{3}\right)}\right)}{x+\sqrt{3}-1}+i \left(-1+\sqrt{3}+i \sqrt{2 \left(2+\sqrt{3}\right)}\right)\right) \sqrt{\sqrt{2 \left(2+\sqrt{3}\right)}+i \left(\frac{8}{x+\sqrt{3}-1}-\sqrt{3}+1\right)} F\left(\sin ^{-1}\left(\frac{\sqrt{\sqrt{2 \left(2+\sqrt{3}\right)}-i \left(-\sqrt{3}+1+\frac{8}{x+\sqrt{3}-1}\right)}}{2^{3/4} \sqrt[4]{2+\sqrt{3}}}\right)|\frac{2 i \sqrt{2 \left(2+\sqrt{3}\right)}}{3+\sqrt{3}+i \sqrt{2 \left(2+\sqrt{3}\right)}}\right)\right)}{\left(\sqrt{2 \left(2+\sqrt{3}\right)}+i \left(3+\sqrt{3}\right)\right) \sqrt{-\frac{x^3}{2}+\frac{1}{2} \left(\sqrt{3}-1\right) x^2-\left(2+\sqrt{3}\right) x+\sqrt{3}+1} \sqrt{x^4+4 \sqrt{3} x^2-4} \sqrt{\sqrt{2 \left(2+\sqrt{3}\right)}-i \left(\frac{8}{x+\sqrt{3}-1}-\sqrt{3}+1\right)}}","\frac{1}{3} \sqrt{2 \sqrt{3}-3} \tanh ^{-1}\left(\frac{\left(x-\sqrt{3}+1\right)^2}{\sqrt{3 \left(2 \sqrt{3}-3\right)} \sqrt{x^4+4 \sqrt{3} x^2-4}}\right)",1,"((-1 + Sqrt[3] + x)^2*Sqrt[2*(1 + Sqrt[3]) - 2*(2 + Sqrt[3])*x + (-1 + Sqrt[3])*x^2 - x^3]*Sqrt[(1 + Sqrt[3] - 4/(-1 + Sqrt[3] + x))/(3 + Sqrt[3] + I*Sqrt[2*(2 + Sqrt[3])])]*((I*(-1 + Sqrt[3] + I*Sqrt[2*(2 + Sqrt[3])]) + (2*((2*I)*Sqrt[3] - Sqrt[2*(2 + Sqrt[3])] + Sqrt[6*(2 + Sqrt[3])]))/(-1 + Sqrt[3] + x))*Sqrt[Sqrt[2*(2 + Sqrt[3])] + I*(1 - Sqrt[3] + 8/(-1 + Sqrt[3] + x))]*EllipticF[ArcSin[Sqrt[Sqrt[2*(2 + Sqrt[3])] - I*(1 - Sqrt[3] + 8/(-1 + Sqrt[3] + x))]/(2^(3/4)*(2 + Sqrt[3])^(1/4))], ((2*I)*Sqrt[2*(2 + Sqrt[3])])/(3 + Sqrt[3] + I*Sqrt[2*(2 + Sqrt[3])])] + 2*Sqrt[6]*Sqrt[(4 + 2*Sqrt[3] + x^2)/(-1 + Sqrt[3] + x)^2]*Sqrt[Sqrt[2*(2 + Sqrt[3])] - I*(1 - Sqrt[3] + 8/(-1 + Sqrt[3] + x))]*EllipticPi[(2*Sqrt[2*(2 + Sqrt[3])])/(Sqrt[2*(2 + Sqrt[3])] + I*(3 + Sqrt[3])), ArcSin[Sqrt[Sqrt[2*(2 + Sqrt[3])] - I*(1 - Sqrt[3] + 8/(-1 + Sqrt[3] + x))]/(2^(3/4)*(2 + Sqrt[3])^(1/4))], ((2*I)*Sqrt[2*(2 + Sqrt[3])])/(3 + Sqrt[3] + I*Sqrt[2*(2 + Sqrt[3])])]))/((Sqrt[2*(2 + Sqrt[3])] + I*(3 + Sqrt[3]))*Sqrt[1 + Sqrt[3] - (2 + Sqrt[3])*x + ((-1 + Sqrt[3])*x^2)/2 - x^3/2]*Sqrt[-4 + 4*Sqrt[3]*x^2 + x^4]*Sqrt[Sqrt[2*(2 + Sqrt[3])] - I*(1 - Sqrt[3] + 8/(-1 + Sqrt[3] + x))])","C",0
409,1,876,63,7.8271977,"\int \frac{1+\sqrt{3}+x}{\left(1-\sqrt{3}+x\right) \sqrt{-4-4 \sqrt{3} x^2+x^4}} \, dx","Integrate[(1 + Sqrt[3] + x)/((1 - Sqrt[3] + x)*Sqrt[-4 - 4*Sqrt[3]*x^2 + x^4]),x]","-\frac{\sqrt{2} \sqrt{\frac{\sqrt{3}-1-\frac{4}{-x+\sqrt{3}+1}}{-3+\sqrt{3}-i \sqrt{4-2 \sqrt{3}}}} \left(-x+\sqrt{3}+1\right)^2 \left(\left(\frac{2 \left(2 i \sqrt{3} \sqrt{i \left(\sqrt{3}+1-\frac{8}{-x+\sqrt{3}+1}\right)+\sqrt{4-2 \sqrt{3}}}+\sqrt{6} \sqrt{2 \sqrt{4-2 \sqrt{3}}-\sqrt{12-6 \sqrt{3}}+i \sqrt{3}-i+\frac{8 i \left(-2+\sqrt{3}\right)}{-x+\sqrt{3}+1}}+\sqrt{-\frac{2 i \left(\left(-1+\sqrt{3}\right) x-8 \sqrt{3}+14\right)}{-x+\sqrt{3}+1}+4 \sqrt{4-2 \sqrt{3}}-2 \sqrt{12-6 \sqrt{3}}}\right)}{x-\sqrt{3}-1}+i \sqrt{3} \sqrt{i \left(\sqrt{3}+1-\frac{8}{-x+\sqrt{3}+1}\right)+\sqrt{4-2 \sqrt{3}}}+i \sqrt{i \left(\sqrt{3}+1-\frac{8}{-x+\sqrt{3}+1}\right)+\sqrt{4-2 \sqrt{3}}}+\sqrt{-\frac{2 i \left(\left(-1+\sqrt{3}\right) x-8 \sqrt{3}+14\right)}{-x+\sqrt{3}+1}+4 \sqrt{4-2 \sqrt{3}}-2 \sqrt{12-6 \sqrt{3}}}\right) F\left(\sin ^{-1}\left(\frac{\sqrt{\sqrt{4-2 \sqrt{3}}-i \left(\sqrt{3}+1-\frac{8}{-x+\sqrt{3}+1}\right)}}{2^{3/4} \sqrt[4]{2-\sqrt{3}}}\right)|\frac{2 \sqrt{4-2 \sqrt{3}}}{\sqrt{4-2 \sqrt{3}}+i \left(-3+\sqrt{3}\right)}\right)+2 \sqrt{6} \sqrt{\sqrt{4-2 \sqrt{3}}-i \left(\sqrt{3}+1-\frac{8}{-x+\sqrt{3}+1}\right)} \sqrt{\frac{x^2-2 \sqrt{3}+4}{\left(-x+\sqrt{3}+1\right)^2}} \Pi \left(\frac{2 \sqrt{4-2 \sqrt{3}}}{\sqrt{4-2 \sqrt{3}}-i \left(-3+\sqrt{3}\right)};\sin ^{-1}\left(\frac{\sqrt{\sqrt{4-2 \sqrt{3}}-i \left(\sqrt{3}+1-\frac{8}{-x+\sqrt{3}+1}\right)}}{2^{3/4} \sqrt[4]{2-\sqrt{3}}}\right)|\frac{2 \sqrt{4-2 \sqrt{3}}}{\sqrt{4-2 \sqrt{3}}+i \left(-3+\sqrt{3}\right)}\right)\right)}{\left(\sqrt{4-2 \sqrt{3}}-i \left(-3+\sqrt{3}\right)\right) \sqrt{\sqrt{4-2 \sqrt{3}}-i \left(\sqrt{3}+1-\frac{8}{-x+\sqrt{3}+1}\right)} \sqrt{x^4-4 \sqrt{3} x^2-4}}","-\frac{1}{3} \sqrt{3+2 \sqrt{3}} \tan ^{-1}\left(\frac{\left(x+\sqrt{3}+1\right)^2}{\sqrt{3 \left(3+2 \sqrt{3}\right)} \sqrt{x^4-4 \sqrt{3} x^2-4}}\right)",1,"-((Sqrt[2]*Sqrt[(-1 + Sqrt[3] - 4/(1 + Sqrt[3] - x))/(-3 + Sqrt[3] - I*Sqrt[4 - 2*Sqrt[3]])]*(1 + Sqrt[3] - x)^2*((I*Sqrt[Sqrt[4 - 2*Sqrt[3]] + I*(1 + Sqrt[3] - 8/(1 + Sqrt[3] - x))] + I*Sqrt[3]*Sqrt[Sqrt[4 - 2*Sqrt[3]] + I*(1 + Sqrt[3] - 8/(1 + Sqrt[3] - x))] + Sqrt[-2*Sqrt[12 - 6*Sqrt[3]] + 4*Sqrt[4 - 2*Sqrt[3]] - ((2*I)*(14 - 8*Sqrt[3] + (-1 + Sqrt[3])*x))/(1 + Sqrt[3] - x)] + (2*((2*I)*Sqrt[3]*Sqrt[Sqrt[4 - 2*Sqrt[3]] + I*(1 + Sqrt[3] - 8/(1 + Sqrt[3] - x))] + Sqrt[6]*Sqrt[-I + I*Sqrt[3] - Sqrt[12 - 6*Sqrt[3]] + 2*Sqrt[4 - 2*Sqrt[3]] + ((8*I)*(-2 + Sqrt[3]))/(1 + Sqrt[3] - x)] + Sqrt[-2*Sqrt[12 - 6*Sqrt[3]] + 4*Sqrt[4 - 2*Sqrt[3]] - ((2*I)*(14 - 8*Sqrt[3] + (-1 + Sqrt[3])*x))/(1 + Sqrt[3] - x)]))/(-1 - Sqrt[3] + x))*EllipticF[ArcSin[Sqrt[Sqrt[4 - 2*Sqrt[3]] - I*(1 + Sqrt[3] - 8/(1 + Sqrt[3] - x))]/(2^(3/4)*(2 - Sqrt[3])^(1/4))], (2*Sqrt[4 - 2*Sqrt[3]])/(Sqrt[4 - 2*Sqrt[3]] + I*(-3 + Sqrt[3]))] + 2*Sqrt[6]*Sqrt[Sqrt[4 - 2*Sqrt[3]] - I*(1 + Sqrt[3] - 8/(1 + Sqrt[3] - x))]*Sqrt[(4 - 2*Sqrt[3] + x^2)/(1 + Sqrt[3] - x)^2]*EllipticPi[(2*Sqrt[4 - 2*Sqrt[3]])/(Sqrt[4 - 2*Sqrt[3]] - I*(-3 + Sqrt[3])), ArcSin[Sqrt[Sqrt[4 - 2*Sqrt[3]] - I*(1 + Sqrt[3] - 8/(1 + Sqrt[3] - x))]/(2^(3/4)*(2 - Sqrt[3])^(1/4))], (2*Sqrt[4 - 2*Sqrt[3]])/(Sqrt[4 - 2*Sqrt[3]] + I*(-3 + Sqrt[3]))]))/((Sqrt[4 - 2*Sqrt[3]] - I*(-3 + Sqrt[3]))*Sqrt[Sqrt[4 - 2*Sqrt[3]] - I*(1 + Sqrt[3] - 8/(1 + Sqrt[3] - x))]*Sqrt[-4 - 4*Sqrt[3]*x^2 + x^4]))","C",0
410,1,623,72,1.7321911,"\int \frac{1-\sqrt{3}+2 x}{\left(1+\sqrt{3}+2 x\right) \sqrt{-1+4 \sqrt{3} x^2+4 x^4}} \, dx","Integrate[(1 - Sqrt[3] + 2*x)/((1 + Sqrt[3] + 2*x)*Sqrt[-1 + 4*Sqrt[3]*x^2 + 4*x^4]),x]","\frac{\left(2 x+\sqrt{3}-1\right)^2 \sqrt{\frac{-\frac{4}{2 x+\sqrt{3}-1}+\sqrt{3}+1}{3+\sqrt{3}+i \sqrt{2 \left(2+\sqrt{3}\right)}}} \left(4 \sqrt{3} \sqrt{\frac{2 x^2+\sqrt{3}+2}{\left(2 x+\sqrt{3}-1\right)^2}} \sqrt{\sqrt{2 \left(2+\sqrt{3}\right)}-i \left(\frac{8}{2 x+\sqrt{3}-1}-\sqrt{3}+1\right)} \Pi \left(\frac{2 \sqrt{2 \left(2+\sqrt{3}\right)}}{\sqrt{2 \left(2+\sqrt{3}\right)}+i \left(3+\sqrt{3}\right)};\sin ^{-1}\left(\frac{\sqrt{\sqrt{2 \left(2+\sqrt{3}\right)}-i \left(-\sqrt{3}+1+\frac{8}{2 x+\sqrt{3}-1}\right)}}{2^{3/4} \sqrt[4]{2+\sqrt{3}}}\right)|\frac{2 i \sqrt{2 \left(2+\sqrt{3}\right)}}{3+\sqrt{3}+i \sqrt{2 \left(2+\sqrt{3}\right)}}\right)+\left(\frac{2 \left(2 i \sqrt{3}-\sqrt{2 \left(2+\sqrt{3}\right)}+\sqrt{6 \left(2+\sqrt{3}\right)}\right)}{2 x+\sqrt{3}-1}+i \left(-1+\sqrt{3}+i \sqrt{2 \left(2+\sqrt{3}\right)}\right)\right) \sqrt{\sqrt{2 \left(2+\sqrt{3}\right)}+i \left(\frac{8}{2 x+\sqrt{3}-1}-\sqrt{3}+1\right)} F\left(\sin ^{-1}\left(\frac{\sqrt{\sqrt{2 \left(2+\sqrt{3}\right)}-i \left(-\sqrt{3}+1+\frac{8}{2 x+\sqrt{3}-1}\right)}}{2^{3/4} \sqrt[4]{2+\sqrt{3}}}\right)|\frac{2 i \sqrt{2 \left(2+\sqrt{3}\right)}}{3+\sqrt{3}+i \sqrt{2 \left(2+\sqrt{3}\right)}}\right)\right)}{\left(\sqrt{2 \left(2+\sqrt{3}\right)}+i \left(3+\sqrt{3}\right)\right) \sqrt{8 x^4+8 \sqrt{3} x^2-2} \sqrt{\sqrt{2 \left(2+\sqrt{3}\right)}-i \left(\frac{8}{2 x+\sqrt{3}-1}-\sqrt{3}+1\right)}}","\frac{1}{3} \sqrt{2 \sqrt{3}-3} \tanh ^{-1}\left(\frac{\left(2 x-\sqrt{3}+1\right)^2}{2 \sqrt{3 \left(2 \sqrt{3}-3\right)} \sqrt{4 x^4+4 \sqrt{3} x^2-1}}\right)",1,"((-1 + Sqrt[3] + 2*x)^2*Sqrt[(1 + Sqrt[3] - 4/(-1 + Sqrt[3] + 2*x))/(3 + Sqrt[3] + I*Sqrt[2*(2 + Sqrt[3])])]*((I*(-1 + Sqrt[3] + I*Sqrt[2*(2 + Sqrt[3])]) + (2*((2*I)*Sqrt[3] - Sqrt[2*(2 + Sqrt[3])] + Sqrt[6*(2 + Sqrt[3])]))/(-1 + Sqrt[3] + 2*x))*Sqrt[Sqrt[2*(2 + Sqrt[3])] + I*(1 - Sqrt[3] + 8/(-1 + Sqrt[3] + 2*x))]*EllipticF[ArcSin[Sqrt[Sqrt[2*(2 + Sqrt[3])] - I*(1 - Sqrt[3] + 8/(-1 + Sqrt[3] + 2*x))]/(2^(3/4)*(2 + Sqrt[3])^(1/4))], ((2*I)*Sqrt[2*(2 + Sqrt[3])])/(3 + Sqrt[3] + I*Sqrt[2*(2 + Sqrt[3])])] + 4*Sqrt[3]*Sqrt[(2 + Sqrt[3] + 2*x^2)/(-1 + Sqrt[3] + 2*x)^2]*Sqrt[Sqrt[2*(2 + Sqrt[3])] - I*(1 - Sqrt[3] + 8/(-1 + Sqrt[3] + 2*x))]*EllipticPi[(2*Sqrt[2*(2 + Sqrt[3])])/(Sqrt[2*(2 + Sqrt[3])] + I*(3 + Sqrt[3])), ArcSin[Sqrt[Sqrt[2*(2 + Sqrt[3])] - I*(1 - Sqrt[3] + 8/(-1 + Sqrt[3] + 2*x))]/(2^(3/4)*(2 + Sqrt[3])^(1/4))], ((2*I)*Sqrt[2*(2 + Sqrt[3])])/(3 + Sqrt[3] + I*Sqrt[2*(2 + Sqrt[3])])]))/((Sqrt[2*(2 + Sqrt[3])] + I*(3 + Sqrt[3]))*Sqrt[-2 + 8*Sqrt[3]*x^2 + 8*x^4]*Sqrt[Sqrt[2*(2 + Sqrt[3])] - I*(1 - Sqrt[3] + 8/(-1 + Sqrt[3] + 2*x))])","C",0
411,1,881,70,6.3860121,"\int \frac{1+\sqrt{3}+2 x}{\left(1-\sqrt{3}+2 x\right) \sqrt{-1-4 \sqrt{3} x^2+4 x^4}} \, dx","Integrate[(1 + Sqrt[3] + 2*x)/((1 - Sqrt[3] + 2*x)*Sqrt[-1 - 4*Sqrt[3]*x^2 + 4*x^4]),x]","-\frac{\sqrt{\frac{\sqrt{3}-1-\frac{4}{-2 x+\sqrt{3}+1}}{-3+\sqrt{3}-i \sqrt{4-2 \sqrt{3}}}} \left(-2 x+\sqrt{3}+1\right)^2 \left(\left(-\frac{2 i \left(2 \sqrt{3} \sqrt{i \left(\sqrt{3}+1-\frac{8}{-2 x+\sqrt{3}+1}\right)+\sqrt{4-2 \sqrt{3}}}-i \sqrt{6} \sqrt{-\frac{2 i \left(\left(-1+\sqrt{3}\right) x-4 \sqrt{3}+7\right)}{-2 x+\sqrt{3}+1}+2 \sqrt{4-2 \sqrt{3}}-\sqrt{12-6 \sqrt{3}}}-i \sqrt{-\frac{4 i \left(\left(-1+\sqrt{3}\right) x-4 \sqrt{3}+7\right)}{-2 x+\sqrt{3}+1}+4 \sqrt{4-2 \sqrt{3}}-2 \sqrt{12-6 \sqrt{3}}}\right)}{-2 x+\sqrt{3}+1}+i \sqrt{3} \sqrt{i \left(\sqrt{3}+1-\frac{8}{-2 x+\sqrt{3}+1}\right)+\sqrt{4-2 \sqrt{3}}}+i \sqrt{i \left(\sqrt{3}+1-\frac{8}{-2 x+\sqrt{3}+1}\right)+\sqrt{4-2 \sqrt{3}}}+\sqrt{-\frac{4 i \left(\left(-1+\sqrt{3}\right) x-4 \sqrt{3}+7\right)}{-2 x+\sqrt{3}+1}+4 \sqrt{4-2 \sqrt{3}}-2 \sqrt{12-6 \sqrt{3}}}\right) F\left(\sin ^{-1}\left(\frac{\sqrt{\sqrt{4-2 \sqrt{3}}-i \left(\sqrt{3}+1-\frac{8}{-2 x+\sqrt{3}+1}\right)}}{2^{3/4} \sqrt[4]{2-\sqrt{3}}}\right)|\frac{2 \sqrt{4-2 \sqrt{3}}}{\sqrt{4-2 \sqrt{3}}+i \left(-3+\sqrt{3}\right)}\right)+4 \sqrt{\sqrt{4-2 \sqrt{3}}-i \left(\sqrt{3}+1-\frac{8}{-2 x+\sqrt{3}+1}\right)} \sqrt{\frac{6 x^2-3 \sqrt{3}+6}{\left(-2 x+\sqrt{3}+1\right)^2}} \Pi \left(\frac{2 \sqrt{4-2 \sqrt{3}}}{\sqrt{4-2 \sqrt{3}}-i \left(-3+\sqrt{3}\right)};\sin ^{-1}\left(\frac{\sqrt{\sqrt{4-2 \sqrt{3}}-i \left(\sqrt{3}+1-\frac{8}{-2 x+\sqrt{3}+1}\right)}}{2^{3/4} \sqrt[4]{2-\sqrt{3}}}\right)|\frac{2 \sqrt{4-2 \sqrt{3}}}{\sqrt{4-2 \sqrt{3}}+i \left(-3+\sqrt{3}\right)}\right)\right)}{\sqrt{2} \left(\sqrt{4-2 \sqrt{3}}-i \left(-3+\sqrt{3}\right)\right) \sqrt{\sqrt{4-2 \sqrt{3}}-i \left(\sqrt{3}+1-\frac{8}{-2 x+\sqrt{3}+1}\right)} \sqrt{4 x^4-4 \sqrt{3} x^2-1}}","-\frac{1}{3} \sqrt{3+2 \sqrt{3}} \tan ^{-1}\left(\frac{\left(2 x+\sqrt{3}+1\right)^2}{2 \sqrt{3 \left(3+2 \sqrt{3}\right)} \sqrt{4 x^4-4 \sqrt{3} x^2-1}}\right)",1,"-((Sqrt[(-1 + Sqrt[3] - 4/(1 + Sqrt[3] - 2*x))/(-3 + Sqrt[3] - I*Sqrt[4 - 2*Sqrt[3]])]*(1 + Sqrt[3] - 2*x)^2*((I*Sqrt[Sqrt[4 - 2*Sqrt[3]] + I*(1 + Sqrt[3] - 8/(1 + Sqrt[3] - 2*x))] + I*Sqrt[3]*Sqrt[Sqrt[4 - 2*Sqrt[3]] + I*(1 + Sqrt[3] - 8/(1 + Sqrt[3] - 2*x))] + Sqrt[-2*Sqrt[12 - 6*Sqrt[3]] + 4*Sqrt[4 - 2*Sqrt[3]] - ((4*I)*(7 - 4*Sqrt[3] + (-1 + Sqrt[3])*x))/(1 + Sqrt[3] - 2*x)] - ((2*I)*(2*Sqrt[3]*Sqrt[Sqrt[4 - 2*Sqrt[3]] + I*(1 + Sqrt[3] - 8/(1 + Sqrt[3] - 2*x))] - I*Sqrt[6]*Sqrt[-Sqrt[12 - 6*Sqrt[3]] + 2*Sqrt[4 - 2*Sqrt[3]] - ((2*I)*(7 - 4*Sqrt[3] + (-1 + Sqrt[3])*x))/(1 + Sqrt[3] - 2*x)] - I*Sqrt[-2*Sqrt[12 - 6*Sqrt[3]] + 4*Sqrt[4 - 2*Sqrt[3]] - ((4*I)*(7 - 4*Sqrt[3] + (-1 + Sqrt[3])*x))/(1 + Sqrt[3] - 2*x)]))/(1 + Sqrt[3] - 2*x))*EllipticF[ArcSin[Sqrt[Sqrt[4 - 2*Sqrt[3]] - I*(1 + Sqrt[3] - 8/(1 + Sqrt[3] - 2*x))]/(2^(3/4)*(2 - Sqrt[3])^(1/4))], (2*Sqrt[4 - 2*Sqrt[3]])/(Sqrt[4 - 2*Sqrt[3]] + I*(-3 + Sqrt[3]))] + 4*Sqrt[Sqrt[4 - 2*Sqrt[3]] - I*(1 + Sqrt[3] - 8/(1 + Sqrt[3] - 2*x))]*Sqrt[(6 - 3*Sqrt[3] + 6*x^2)/(1 + Sqrt[3] - 2*x)^2]*EllipticPi[(2*Sqrt[4 - 2*Sqrt[3]])/(Sqrt[4 - 2*Sqrt[3]] - I*(-3 + Sqrt[3])), ArcSin[Sqrt[Sqrt[4 - 2*Sqrt[3]] - I*(1 + Sqrt[3] - 8/(1 + Sqrt[3] - 2*x))]/(2^(3/4)*(2 - Sqrt[3])^(1/4))], (2*Sqrt[4 - 2*Sqrt[3]])/(Sqrt[4 - 2*Sqrt[3]] + I*(-3 + Sqrt[3]))]))/(Sqrt[2]*(Sqrt[4 - 2*Sqrt[3]] - I*(-3 + Sqrt[3]))*Sqrt[Sqrt[4 - 2*Sqrt[3]] - I*(1 + Sqrt[3] - 8/(1 + Sqrt[3] - 2*x))]*Sqrt[-1 - 4*Sqrt[3]*x^2 + 4*x^4]))","C",0
412,1,3652,560,7.869484,"\int \frac{f+g x}{(d+e x) \sqrt{a+b x^2+c x^4}} \, dx","Integrate[(f + g*x)/((d + e*x)*Sqrt[a + b*x^2 + c*x^4]),x]","\text{Result too large to show}","\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) \left(\sqrt{a} e g+\sqrt{c} d f\right)}{2 \sqrt[4]{a} \sqrt[4]{c} \sqrt{a+b x^2+c x^4} \left(\sqrt{a} e^2+\sqrt{c} d^2\right)}-\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(\sqrt{c} d^2-\sqrt{a} e^2\right) (e f-d g) \Pi \left(\frac{\left(\sqrt{c} d^2+\sqrt{a} e^2\right)^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};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)}{4 \sqrt[4]{a} \sqrt[4]{c} d e \sqrt{a+b x^2+c x^4} \left(\sqrt{a} e^2+\sqrt{c} d^2\right)}+\frac{(e f-d g) \tan ^{-1}\left(\frac{x \sqrt{-a e^4-b d^2 e^2-c d^4}}{d e \sqrt{a+b x^2+c x^4}}\right)}{2 \sqrt{-e^2 \left(a e^2+b d^2\right)-c d^4}}-\frac{(e f-d g) \tanh ^{-1}\left(\frac{2 a e^2+x^2 \left(b e^2+2 c d^2\right)+b d^2}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^4+b d^2 e^2+c d^4}}\right)}{2 \sqrt{a e^4+b d^2 e^2+c d^4}}",1,"((-I)*g*Sqrt[1 - (2*c*x^2)/(-b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 - (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])]*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*x], (-b - Sqrt[b^2 - 4*a*c])/(-b + Sqrt[b^2 - 4*a*c])])/(Sqrt[2]*Sqrt[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*e*Sqrt[a + b*x^2 + c*x^4]) + (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])*f*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + x)^2*Sqrt[(Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c]*(-(Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + x))/((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]) + x))]*Sqrt[(Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c]*(Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + x))/((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]) + x))]*Sqrt[((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + 2*x))/((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - 2*x))]*((-d + (Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e)/Sqrt[2])*EllipticF[ArcSin[Sqrt[((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + 2*x))/((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - 2*x))]], (Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])^2/(Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])^2] - Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c]*e*EllipticPi[((Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(d + (Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e)/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])*(d - (Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e)/Sqrt[2])), ArcSin[Sqrt[((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + 2*x))/((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - 2*x))]], (Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])^2/(Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])^2]))/(Sqrt[(-b - 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])*(-d - (Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e)/Sqrt[2])*(d - (Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e)/Sqrt[2])*Sqrt[a + b*x^2 + c*x^4]) - (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])*d*g*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + x)^2*Sqrt[(Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c]*(-(Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + x))/((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]) + x))]*Sqrt[(Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c]*(Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + x))/((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]) + x))]*Sqrt[((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + 2*x))/((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - 2*x))]*((-d + (Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e)/Sqrt[2])*EllipticF[ArcSin[Sqrt[((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + 2*x))/((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - 2*x))]], (Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])^2/(Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])^2] - Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c]*e*EllipticPi[((Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(d + (Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e)/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])*(d - (Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e)/Sqrt[2])), ArcSin[Sqrt[((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + 2*x))/((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - 2*x))]], (Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])^2/(Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])^2]))/(Sqrt[(-b - 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])*e*(-d - (Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e)/Sqrt[2])*(d - (Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e)/Sqrt[2])*Sqrt[a + b*x^2 + c*x^4])","C",1
413,1,3658,527,7.8658483,"\int \frac{f+g x}{(d+e x) \sqrt{-a+b x^2+c x^4}} \, dx","Integrate[(f + g*x)/((d + e*x)*Sqrt[-a + b*x^2 + c*x^4]),x]","\text{Result too large to show}","\frac{\sqrt{\sqrt{4 a c+b^2}-b} \sqrt{\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1} \sqrt{\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}+1} (e f-d g) \Pi \left(-\frac{\left(b-\sqrt{b^2+4 a c}\right) e^2}{2 c d^2};\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2+4 a c}-b}}\right)|\frac{b-\sqrt{b^2+4 a c}}{b+\sqrt{b^2+4 a c}}\right)}{\sqrt{2} \sqrt{c} d e \sqrt{-a+b x^2+c x^4}}+\frac{g \sqrt{\sqrt{4 a c+b^2}+b} \left(\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1\right) F\left(\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right)|-\frac{2 \sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right)}{\sqrt{2} \sqrt{c} e \sqrt{\frac{\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1}{\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}+1}} \sqrt{-a+b x^2+c x^4}}-\frac{(e f-d g) \tanh ^{-1}\left(\frac{-2 a e^2+x^2 \left(b e^2+2 c d^2\right)+b d^2}{2 \sqrt{-a+b x^2+c x^4} \sqrt{-a e^4+b d^2 e^2+c d^4}}\right)}{2 \sqrt{-a e^4+b d^2 e^2+c d^4}}",1,"((-I)*g*Sqrt[1 - (2*c*x^2)/(-b - Sqrt[b^2 + 4*a*c])]*Sqrt[1 - (2*c*x^2)/(-b + Sqrt[b^2 + 4*a*c])]*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[-(c/(-b - Sqrt[b^2 + 4*a*c]))]*x], (-b - Sqrt[b^2 + 4*a*c])/(-b + Sqrt[b^2 + 4*a*c])])/(Sqrt[2]*Sqrt[-(c/(-b - Sqrt[b^2 + 4*a*c]))]*e*Sqrt[-a + b*x^2 + c*x^4]) + (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])*f*(-(Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c]/Sqrt[2]) + x)^2*Sqrt[(Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c]*(-(Sqrt[-(b/c) + Sqrt[b^2 + 4*a*c]/c]/Sqrt[2]) + x))/((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]) + x))]*Sqrt[(Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c]*(Sqrt[-(b/c) + Sqrt[b^2 + 4*a*c]/c]/Sqrt[2] + x))/((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]) + x))]*Sqrt[((Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] + 2*x))/((Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] - 2*x))]*((-d + (Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c]*e)/Sqrt[2])*EllipticF[ArcSin[Sqrt[((Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] + 2*x))/((Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] - 2*x))]], (Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])^2/(Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])^2] - Sqrt[2]*Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c]*e*EllipticPi[((Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 + 4*a*c]/c]/Sqrt[2])*(d + (Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c]*e)/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])*(d - (Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c]*e)/Sqrt[2])), ArcSin[Sqrt[((Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] + 2*x))/((Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] - 2*x))]], (Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])^2/(Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])^2]))/(Sqrt[(-b - 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])*(-d - (Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c]*e)/Sqrt[2])*(d - (Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c]*e)/Sqrt[2])*Sqrt[-a + b*x^2 + c*x^4]) - (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])*d*g*(-(Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c]/Sqrt[2]) + x)^2*Sqrt[(Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c]*(-(Sqrt[-(b/c) + Sqrt[b^2 + 4*a*c]/c]/Sqrt[2]) + x))/((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]) + x))]*Sqrt[(Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c]*(Sqrt[-(b/c) + Sqrt[b^2 + 4*a*c]/c]/Sqrt[2] + x))/((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]) + x))]*Sqrt[((Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] + 2*x))/((Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] - 2*x))]*((-d + (Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c]*e)/Sqrt[2])*EllipticF[ArcSin[Sqrt[((Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] + 2*x))/((Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] - 2*x))]], (Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])^2/(Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])^2] - Sqrt[2]*Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c]*e*EllipticPi[((Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 + 4*a*c]/c]/Sqrt[2])*(d + (Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c]*e)/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])*(d - (Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c]*e)/Sqrt[2])), ArcSin[Sqrt[((Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] + 2*x))/((Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] - 2*x))]], (Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])^2/(Sqrt[(-b - Sqrt[b^2 + 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 + 4*a*c])/c])^2]))/(Sqrt[(-b - 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])*e*(-d - (Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c]*e)/Sqrt[2])*(d - (Sqrt[-(b/c) - Sqrt[b^2 + 4*a*c]/c]*e)/Sqrt[2])*Sqrt[-a + b*x^2 + c*x^4])","C",1