1,1,878,0,0.5094728,"\int \frac{\left(A+B x^2\right) \left(d+e x^2\right)^3}{\sqrt{a+c x^4}} \, dx","Int[((A + B*x^2)*(d + e*x^2)^3)/Sqrt[a + c*x^4],x]","\frac{B e^3 \sqrt{c x^4+a} x^5}{7 c}+\frac{e^2 (3 B d+A e) \sqrt{c x^4+a} x^3}{5 c}-\frac{5 a B e^3 \sqrt{c x^4+a} x}{21 c^2}+\frac{d e (B d+A e) \sqrt{c x^4+a} x}{c}-\frac{3 a e^2 (3 B d+A e) \sqrt{c x^4+a} x}{5 c^{3/2} \left(\sqrt{c} x^2+\sqrt{a}\right)}+\frac{d^2 (B d+3 A e) \sqrt{c x^4+a} x}{\sqrt{c} \left(\sqrt{c} x^2+\sqrt{a}\right)}+\frac{3 a^{5/4} e^2 (3 B d+A e) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{5 c^{7/4} \sqrt{c x^4+a}}-\frac{\sqrt[4]{a} d^2 (B d+3 A e) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\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{c x^4+a}}+\frac{A d^3 \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} \sqrt[4]{c} \sqrt{c x^4+a}}+\frac{5 a^{7/4} B e^3 \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{42 c^{9/4} \sqrt{c x^4+a}}-\frac{a^{3/4} d e (B d+A e) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 c^{5/4} \sqrt{c x^4+a}}-\frac{3 a^{5/4} e^2 (3 B d+A e) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{10 c^{7/4} \sqrt{c x^4+a}}+\frac{\sqrt[4]{a} d^2 (B d+3 A e) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 c^{3/4} \sqrt{c x^4+a}}","\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(-63 a^{3/2} \sqrt{c} e^2 (A e+3 B d)+25 a^2 B e^3+105 \sqrt{a} c^{3/2} d^2 (3 A e+B d)-105 a c d e (A e+B d)+105 A c^2 d^3\right)}{210 \sqrt[4]{a} c^{9/4} \sqrt{a+c x^4}}+\frac{e x \sqrt{a+c x^4} \left(-5 a B e^2+21 A c d e+21 B c d^2\right)}{21 c^2}+\frac{x \sqrt{a+c x^4} \left(-3 a A e^3-9 a B d e^2+15 A c d^2 e+5 B c d^3\right)}{5 c^{3/2} \left(\sqrt{a}+\sqrt{c} x^2\right)}-\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+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) \left(-3 a A e^3-9 a B d e^2+15 A c d^2 e+5 B c d^3\right)}{5 c^{7/4} \sqrt{a+c x^4}}+\frac{e^2 x^3 \sqrt{a+c x^4} (A e+3 B d)}{5 c}+\frac{B e^3 x^5 \sqrt{a+c x^4}}{7 c}",1,"(-5*a*B*e^3*x*Sqrt[a + c*x^4])/(21*c^2) + (d*e*(B*d + A*e)*x*Sqrt[a + c*x^4])/c + (e^2*(3*B*d + A*e)*x^3*Sqrt[a + c*x^4])/(5*c) + (B*e^3*x^5*Sqrt[a + c*x^4])/(7*c) - (3*a*e^2*(3*B*d + A*e)*x*Sqrt[a + c*x^4])/(5*c^(3/2)*(Sqrt[a] + Sqrt[c]*x^2)) + (d^2*(B*d + 3*A*e)*x*Sqrt[a + c*x^4])/(Sqrt[c]*(Sqrt[a] + Sqrt[c]*x^2)) + (3*a^(5/4)*e^2*(3*B*d + A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(5*c^(7/4)*Sqrt[a + c*x^4]) - (a^(1/4)*d^2*(B*d + 3*A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(c^(3/4)*Sqrt[a + c*x^4]) + (A*d^3*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*c^(1/4)*Sqrt[a + c*x^4]) + (5*a^(7/4)*B*e^3*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(42*c^(9/4)*Sqrt[a + c*x^4]) - (a^(3/4)*d*e*(B*d + A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*c^(5/4)*Sqrt[a + c*x^4]) - (3*a^(5/4)*e^2*(3*B*d + A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(10*c^(7/4)*Sqrt[a + c*x^4]) + (a^(1/4)*d^2*(B*d + 3*A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*c^(3/4)*Sqrt[a + c*x^4])","A",15,5,28,0.1786,1,"{1721, 220, 305, 1196, 321}"
2,1,706,0,0.3958827,"\int \frac{\left(A+B x^2\right) \left(d+e x^2\right)^2}{\sqrt{a+c x^4}} \, dx","Int[((A + B*x^2)*(d + e*x^2)^2)/Sqrt[a + c*x^4],x]","-\frac{a^{3/4} e \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} (A e+2 B d) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{6 c^{5/4} \sqrt{a+c x^4}}-\frac{3 a^{5/4} B e^2 \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)}{10 c^{7/4} \sqrt{a+c x^4}}+\frac{3 a^{5/4} B e^2 \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)}{5 c^{7/4} \sqrt{a+c x^4}}+\frac{\sqrt[4]{a} d \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} (2 A e+B d) 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} d \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} (2 A e+B d) 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} (A e+2 B d)}{3 c}+\frac{d x \sqrt{a+c x^4} (2 A e+B d)}{\sqrt{c} \left(\sqrt{a}+\sqrt{c} x^2\right)}+\frac{A d^2 \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} \sqrt[4]{c} \sqrt{a+c x^4}}-\frac{3 a B e^2 x \sqrt{a+c x^4}}{5 c^{3/2} \left(\sqrt{a}+\sqrt{c} x^2\right)}+\frac{B e^2 x^3 \sqrt{a+c x^4}}{5 c}","\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(-9 a^{3/2} B e^2+15 \sqrt{a} c d (2 A e+B d)-5 a \sqrt{c} e (A e+2 B d)+15 A c^{3/2} d^2\right)}{30 \sqrt[4]{a} c^{7/4} \sqrt{a+c x^4}}+\frac{x \sqrt{a+c x^4} \left(-3 a B e^2+10 A c d e+5 B c d^2\right)}{5 c^{3/2} \left(\sqrt{a}+\sqrt{c} x^2\right)}-\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+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) \left(-3 a B e^2+10 A c d e+5 B c d^2\right)}{5 c^{7/4} \sqrt{a+c x^4}}+\frac{e x \sqrt{a+c x^4} (A e+2 B d)}{3 c}+\frac{B e^2 x^3 \sqrt{a+c x^4}}{5 c}",1,"(e*(2*B*d + A*e)*x*Sqrt[a + c*x^4])/(3*c) + (B*e^2*x^3*Sqrt[a + c*x^4])/(5*c) - (3*a*B*e^2*x*Sqrt[a + c*x^4])/(5*c^(3/2)*(Sqrt[a] + Sqrt[c]*x^2)) + (d*(B*d + 2*A*e)*x*Sqrt[a + c*x^4])/(Sqrt[c]*(Sqrt[a] + Sqrt[c]*x^2)) + (3*a^(5/4)*B*e^2*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(5*c^(7/4)*Sqrt[a + c*x^4]) - (a^(1/4)*d*(B*d + 2*A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(c^(3/4)*Sqrt[a + c*x^4]) + (A*d^2*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*c^(1/4)*Sqrt[a + c*x^4]) - (3*a^(5/4)*B*e^2*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(10*c^(7/4)*Sqrt[a + c*x^4]) - (a^(3/4)*e*(2*B*d + A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(6*c^(5/4)*Sqrt[a + c*x^4]) + (a^(1/4)*d*(B*d + 2*A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*c^(3/4)*Sqrt[a + c*x^4])","A",12,5,28,0.1786,1,"{1721, 220, 305, 1196, 321}"
3,1,432,0,0.2245466,"\int \frac{\left(A+B x^2\right) \left(d+e x^2\right)}{\sqrt{a+c x^4}} \, dx","Int[((A + B*x^2)*(d + e*x^2))/Sqrt[a + c*x^4],x]","-\frac{a^{3/4} B e \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{6 c^{5/4} \sqrt{a+c x^4}}+\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}} (A e+B d) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 c^{3/4} \sqrt{a+c x^4}}-\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} (A e+B d) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{c^{3/4} \sqrt{a+c x^4}}+\frac{x \sqrt{a+c x^4} (A e+B d)}{\sqrt{c} \left(\sqrt{a}+\sqrt{c} x^2\right)}+\frac{A d \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} \sqrt[4]{c} \sqrt{a+c x^4}}+\frac{B e x \sqrt{a+c x^4}}{3 c}","\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) \left(\frac{3 A c d-a B e}{\sqrt{a}}+3 \sqrt{c} (A e+B d)\right)}{6 c^{5/4} \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}} (A e+B d) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{c^{3/4} \sqrt{a+c x^4}}+\frac{x \sqrt{a+c x^4} (A e+B d)}{\sqrt{c} \left(\sqrt{a}+\sqrt{c} x^2\right)}+\frac{B e x \sqrt{a+c x^4}}{3 c}",1,"(B*e*x*Sqrt[a + c*x^4])/(3*c) + ((B*d + A*e)*x*Sqrt[a + c*x^4])/(Sqrt[c]*(Sqrt[a] + Sqrt[c]*x^2)) - (a^(1/4)*(B*d + A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(c^(3/4)*Sqrt[a + c*x^4]) + (A*d*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*c^(1/4)*Sqrt[a + c*x^4]) - (a^(3/4)*B*e*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(6*c^(5/4)*Sqrt[a + c*x^4]) + (a^(1/4)*(B*d + A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*c^(3/4)*Sqrt[a + c*x^4])","A",8,5,26,0.1923,1,"{1721, 220, 305, 1196, 321}"
4,1,226,0,0.0656975,"\int \frac{A+B x^2}{\sqrt{a+c x^4}} \, dx","Int[(A + B*x^2)/Sqrt[a + c*x^4],x]","\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{A \sqrt{c}}{\sqrt{a}}+B\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} B \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{B x \sqrt{a+c x^4}}{\sqrt{c} \left(\sqrt{a}+\sqrt{c} x^2\right)}","\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{A \sqrt{c}}{\sqrt{a}}+B\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} B \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{B x \sqrt{a+c x^4}}{\sqrt{c} \left(\sqrt{a}+\sqrt{c} x^2\right)}",1,"(B*x*Sqrt[a + c*x^4])/(Sqrt[c]*(Sqrt[a] + Sqrt[c]*x^2)) - (a^(1/4)*B*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(c^(3/4)*Sqrt[a + c*x^4]) + (a^(1/4)*(B + (A*Sqrt[c])/Sqrt[a])*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*c^(3/4)*Sqrt[a + c*x^4])","A",3,3,19,0.1579,1,"{1198, 220, 1196}"
5,1,369,0,0.3601375,"\int \frac{A+B x^2}{\left(d+e x^2\right) \sqrt{a+c x^4}} \, dx","Int[(A + B*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]),x]","\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 (B d-A e) \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 e \sqrt{a+c x^4} \left(c d^2-a e^2\right)}-\frac{(B d-A 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{e} \sqrt{a e^2+c d^2}}-\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} B-A \sqrt{c}\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} \sqrt[4]{c} \sqrt{a+c x^4} \left(\sqrt{c} d-\sqrt{a} e\right)}","\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 (B d-A e) \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 e \sqrt{a+c x^4} \left(c d^2-a e^2\right)}-\frac{(B d-A 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{e} \sqrt{a e^2+c d^2}}-\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} B-A \sqrt{c}\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} \sqrt[4]{c} \sqrt{a+c x^4} \left(\sqrt{c} d-\sqrt{a} e\right)}",1,"-((B*d - A*e)*ArcTan[(Sqrt[c*d^2 + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + c*x^4])])/(2*Sqrt[d]*Sqrt[e]*Sqrt[c*d^2 + a*e^2]) - ((Sqrt[a]*B - A*Sqrt[c])*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*c^(1/4)*(Sqrt[c]*d - Sqrt[a]*e)*Sqrt[a + c*x^4]) + (a^(3/4)*((Sqrt[c]*d)/Sqrt[a] + e)^2*(B*d - A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[-(Sqrt[c]*d - Sqrt[a]*e)^2/(4*Sqrt[a]*Sqrt[c]*d*e), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(4*c^(1/4)*d*e*(c*d^2 - a*e^2)*Sqrt[a + c*x^4])","A",3,3,28,0.1071,1,"{1709, 220, 1707}"
6,1,641,0,1.1893732,"\int \frac{A+B x^2}{\left(d+e x^2\right)^2 \sqrt{a+c x^4}} \, dx","Int[(A + B*x^2)/((d + e*x^2)^2*Sqrt[a + c*x^4]),x]","\frac{\sqrt{c} x \sqrt{a+c x^4} (B d-A e)}{2 d \left(\sqrt{a}+\sqrt{c} x^2\right) \left(a e^2+c d^2\right)}-\frac{e x \sqrt{a+c x^4} (B d-A e)}{2 d \left(d+e x^2\right) \left(a e^2+c d^2\right)}-\frac{\left(-a A e^3-a B d e^2-3 A c d^2 e+B c d^3\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} \sqrt{e} \left(a e^2+c d^2\right)^{3/2}}-\frac{\sqrt[4]{a} \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}} (B d-A e) 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 A e^3-a B d e^2-3 A c d^2 e+B c d^3\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 e \sqrt{a+c x^4} \left(\sqrt{c} d-\sqrt{a} e\right) \left(a e^2+c d^2\right)}+\frac{A \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)}","\frac{\sqrt{c} x \sqrt{a+c x^4} (B d-A e)}{2 d \left(\sqrt{a}+\sqrt{c} x^2\right) \left(a e^2+c d^2\right)}-\frac{e x \sqrt{a+c x^4} (B d-A e)}{2 d \left(d+e x^2\right) \left(a e^2+c d^2\right)}-\frac{\left(-a A e^3-a B d e^2-3 A c d^2 e+B c d^3\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} \sqrt{e} \left(a e^2+c d^2\right)^{3/2}}-\frac{\sqrt[4]{a} \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}} (B d-A e) 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 A e^3-a B d e^2-3 A c d^2 e+B c d^3\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 e \sqrt{a+c x^4} \left(\sqrt{c} d-\sqrt{a} e\right) \left(a e^2+c d^2\right)}+\frac{A \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,"(Sqrt[c]*(B*d - A*e)*x*Sqrt[a + c*x^4])/(2*d*(c*d^2 + a*e^2)*(Sqrt[a] + Sqrt[c]*x^2)) - (e*(B*d - A*e)*x*Sqrt[a + c*x^4])/(2*d*(c*d^2 + a*e^2)*(d + e*x^2)) - ((B*c*d^3 - 3*A*c*d^2*e - a*B*d*e^2 - a*A*e^3)*ArcTan[(Sqrt[c*d^2 + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + c*x^4])])/(4*d^(3/2)*Sqrt[e]*(c*d^2 + a*e^2)^(3/2)) - (a^(1/4)*c^(1/4)*(B*d - A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*d*(c*d^2 + a*e^2)*Sqrt[a + c*x^4]) + (A*c^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*d*(Sqrt[c]*d - Sqrt[a]*e)*Sqrt[a + c*x^4]) + ((Sqrt[c]*d + Sqrt[a]*e)*(B*c*d^3 - 3*A*c*d^2*e - a*B*d*e^2 - a*A*e^3)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[-(Sqrt[c]*d - Sqrt[a]*e)^2/(4*Sqrt[a]*Sqrt[c]*d*e), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(8*a^(1/4)*c^(1/4)*d^2*e*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + a*e^2)*Sqrt[a + c*x^4])","A",6,6,28,0.2143,1,"{1697, 1715, 1196, 1709, 220, 1707}"
7,1,875,0,1.7841507,"\int \frac{A+B x^2}{\left(d+e x^2\right)^3 \sqrt{a+c x^4}} \, dx","Int[(A + B*x^2)/((d + e*x^2)^3*Sqrt[a + c*x^4]),x]","\frac{\sqrt{c} \left(5 B c d^3-9 A c e d^2-a B e^2 d-3 a A e^3\right) \sqrt{c x^4+a} x}{8 d^2 \left(c d^2+a e^2\right)^2 \left(\sqrt{c} x^2+\sqrt{a}\right)}-\frac{e \left(5 B c d^3-9 A c e d^2-a B e^2 d-3 a A e^3\right) \sqrt{c x^4+a} x}{8 d^2 \left(c d^2+a e^2\right)^2 \left(e x^2+d\right)}-\frac{e (B d-A e) \sqrt{c x^4+a} x}{4 d \left(c d^2+a e^2\right) \left(e x^2+d\right)^2}+\frac{\left(3 A e \left(5 c^2 d^4+2 a c e^2 d^2+a^2 e^4\right)-B \left(3 c^2 d^5-10 a c e^2 d^3-a^2 e^4 d\right)\right) \tan ^{-1}\left(\frac{\sqrt{c d^2+a e^2} x}{\sqrt{d} \sqrt{e} \sqrt{c x^4+a}}\right)}{16 d^{5/2} \sqrt{e} \left(c d^2+a e^2\right)^{5/2}}-\frac{\sqrt[4]{a} \sqrt[4]{c} \left(5 B c d^3-9 A c e d^2-a B e^2 d-3 a A e^3\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{8 d^2 \left(c d^2+a e^2\right)^2 \sqrt{c x^4+a}}+\frac{\sqrt[4]{c} \left(4 A c d^2+\sqrt{a} \sqrt{c} (B d-A e) d+a e (B d+3 A e)\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} 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 \left(\sqrt{c} d-\sqrt{a} e\right) \left(c d^2+a e^2\right) \sqrt{c x^4+a}}-\frac{\left(\sqrt{c} d+\sqrt{a} e\right) \left(3 A e \left(5 c^2 d^4+2 a c e^2 d^2+a^2 e^4\right)-B \left(3 c^2 d^5-10 a c e^2 d^3-a^2 e^4 d\right)\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\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)}{32 \sqrt[4]{a} \sqrt[4]{c} d^3 e \left(\sqrt{c} d-\sqrt{a} e\right) \left(c d^2+a e^2\right)^2 \sqrt{c x^4+a}}","\frac{\sqrt{c} \left(5 B c d^3-9 A c e d^2-a B e^2 d-3 a A e^3\right) \sqrt{c x^4+a} x}{8 d^2 \left(c d^2+a e^2\right)^2 \left(\sqrt{c} x^2+\sqrt{a}\right)}-\frac{e \left(5 B c d^3-9 A c e d^2-a B e^2 d-3 a A e^3\right) \sqrt{c x^4+a} x}{8 d^2 \left(c d^2+a e^2\right)^2 \left(e x^2+d\right)}-\frac{e (B d-A e) \sqrt{c x^4+a} x}{4 d \left(c d^2+a e^2\right) \left(e x^2+d\right)^2}+\frac{\left(3 A e \left(5 c^2 d^4+2 a c e^2 d^2+a^2 e^4\right)-B \left(3 c^2 d^5-10 a c e^2 d^3-a^2 e^4 d\right)\right) \tan ^{-1}\left(\frac{\sqrt{c d^2+a e^2} x}{\sqrt{d} \sqrt{e} \sqrt{c x^4+a}}\right)}{16 d^{5/2} \sqrt{e} \left(c d^2+a e^2\right)^{5/2}}-\frac{\sqrt[4]{a} \sqrt[4]{c} \left(5 B c d^3-9 A c e d^2-a B e^2 d-3 a A e^3\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{8 d^2 \left(c d^2+a e^2\right)^2 \sqrt{c x^4+a}}+\frac{\sqrt[4]{c} \left(4 A c d^2+\sqrt{a} \sqrt{c} (B d-A e) d+a e (B d+3 A e)\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} 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 \left(\sqrt{c} d-\sqrt{a} e\right) \left(c d^2+a e^2\right) \sqrt{c x^4+a}}-\frac{\left(\sqrt{c} d+\sqrt{a} e\right) \left(3 A e \left(5 c^2 d^4+2 a c e^2 d^2+a^2 e^4\right)-B \left(3 c^2 d^5-10 a c e^2 d^3-a^2 e^4 d\right)\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\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)}{32 \sqrt[4]{a} \sqrt[4]{c} d^3 e \left(\sqrt{c} d-\sqrt{a} e\right) \left(c d^2+a e^2\right)^2 \sqrt{c x^4+a}}",1,"(Sqrt[c]*(5*B*c*d^3 - 9*A*c*d^2*e - a*B*d*e^2 - 3*a*A*e^3)*x*Sqrt[a + c*x^4])/(8*d^2*(c*d^2 + a*e^2)^2*(Sqrt[a] + Sqrt[c]*x^2)) - (e*(B*d - A*e)*x*Sqrt[a + c*x^4])/(4*d*(c*d^2 + a*e^2)*(d + e*x^2)^2) - (e*(5*B*c*d^3 - 9*A*c*d^2*e - a*B*d*e^2 - 3*a*A*e^3)*x*Sqrt[a + c*x^4])/(8*d^2*(c*d^2 + a*e^2)^2*(d + e*x^2)) + ((3*A*e*(5*c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4) - B*(3*c^2*d^5 - 10*a*c*d^3*e^2 - a^2*d*e^4))*ArcTan[(Sqrt[c*d^2 + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + c*x^4])])/(16*d^(5/2)*Sqrt[e]*(c*d^2 + a*e^2)^(5/2)) - (a^(1/4)*c^(1/4)*(5*B*c*d^3 - 9*A*c*d^2*e - a*B*d*e^2 - 3*a*A*e^3)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(8*d^2*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4]) + (c^(1/4)*(4*A*c*d^2 + Sqrt[a]*Sqrt[c]*d*(B*d - A*e) + a*e*(B*d + 3*A*e))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(8*a^(1/4)*d^2*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + a*e^2)*Sqrt[a + c*x^4]) - ((Sqrt[c]*d + Sqrt[a]*e)*(3*A*e*(5*c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4) - B*(3*c^2*d^5 - 10*a*c*d^3*e^2 - a^2*d*e^4))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[-(Sqrt[c]*d - Sqrt[a]*e)^2/(4*Sqrt[a]*Sqrt[c]*d*e), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(32*a^(1/4)*c^(1/4)*d^3*e*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4])","A",7,6,28,0.2143,1,"{1697, 1715, 1196, 1709, 220, 1707}"
8,1,912,0,0.7255596,"\int \frac{\left(A+B x^2\right) \left(d+e x^2\right)^3}{\left(a+c x^4\right)^{3/2}} \, dx","Int[((A + B*x^2)*(d + e*x^2)^3)/(a + c*x^4)^(3/2),x]","-\frac{a^{3/4} B \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) e^3}{6 c^{9/4} \sqrt{c x^4+a}}+\frac{B x \sqrt{c x^4+a} e^3}{3 c^2}-\frac{\sqrt[4]{a} (3 B d+A e) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) e^2}{c^{7/4} \sqrt{c x^4+a}}+\frac{\sqrt[4]{a} (3 B d+A e) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) e^2}{2 c^{7/4} \sqrt{c x^4+a}}+\frac{(3 B d+A e) x \sqrt{c x^4+a} e^2}{c^{3/2} \left(\sqrt{c} x^2+\sqrt{a}\right)}+\frac{\left(3 B c d^2+3 A c e d-a B e^2\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) e}{2 \sqrt[4]{a} c^{9/4} \sqrt{c x^4+a}}+\frac{\left(B c d^3+3 A c e d^2-3 a B e^2 d-a A e^3\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 a^{3/4} c^{7/4} \sqrt{c x^4+a}}+\frac{\left(A c^2 d^3-\sqrt{a} c^{3/2} (B d+3 A e) d^2-3 a c e (B d+A e) d+a^2 B e^3+a^{3/2} \sqrt{c} e^2 (3 B d+A e)\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 a^{5/4} c^{9/4} \sqrt{c x^4+a}}-\frac{\left(B c d^3+3 A c e d^2-3 a B e^2 d-a A e^3\right) x \sqrt{c x^4+a}}{2 a c^{3/2} \left(\sqrt{c} x^2+\sqrt{a}\right)}+\frac{x \left(c \left(B c d^3+3 A c e d^2-3 a B e^2 d-a A e^3\right) x^2+A c d \left(c d^2-3 a e^2\right)-a B e \left(3 c d^2-a e^2\right)\right)}{2 a c^2 \sqrt{c x^4+a}}","-\frac{a^{3/4} B \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) e^3}{6 c^{9/4} \sqrt{c x^4+a}}+\frac{B x \sqrt{c x^4+a} e^3}{3 c^2}-\frac{\sqrt[4]{a} (3 B d+A e) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) e^2}{c^{7/4} \sqrt{c x^4+a}}+\frac{\sqrt[4]{a} (3 B d+A e) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) e^2}{2 c^{7/4} \sqrt{c x^4+a}}+\frac{(3 B d+A e) x \sqrt{c x^4+a} e^2}{c^{3/2} \left(\sqrt{c} x^2+\sqrt{a}\right)}+\frac{\left(3 B c d^2+3 A c e d-a B e^2\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) e}{2 \sqrt[4]{a} c^{9/4} \sqrt{c x^4+a}}+\frac{\left(B c d^3+3 A c e d^2-3 a B e^2 d-a A e^3\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 a^{3/4} c^{7/4} \sqrt{c x^4+a}}+\frac{\left(A c^2 d^3-\sqrt{a} c^{3/2} (B d+3 A e) d^2-3 a c e (B d+A e) d+a^2 B e^3+a^{3/2} \sqrt{c} e^2 (3 B d+A e)\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 a^{5/4} c^{9/4} \sqrt{c x^4+a}}-\frac{\left(B c d^3+3 A c e d^2-3 a B e^2 d-a A e^3\right) x \sqrt{c x^4+a}}{2 a c^{3/2} \left(\sqrt{c} x^2+\sqrt{a}\right)}+\frac{x \left(c \left(B c d^3+3 A c e d^2-3 a B e^2 d-a A e^3\right) x^2+A c d \left(c d^2-3 a e^2\right)-a B e \left(3 c d^2-a e^2\right)\right)}{2 a c^2 \sqrt{c x^4+a}}",1,"(x*(A*c*d*(c*d^2 - 3*a*e^2) - a*B*e*(3*c*d^2 - a*e^2) + c*(B*c*d^3 + 3*A*c*d^2*e - 3*a*B*d*e^2 - a*A*e^3)*x^2))/(2*a*c^2*Sqrt[a + c*x^4]) + (B*e^3*x*Sqrt[a + c*x^4])/(3*c^2) + (e^2*(3*B*d + A*e)*x*Sqrt[a + c*x^4])/(c^(3/2)*(Sqrt[a] + Sqrt[c]*x^2)) - ((B*c*d^3 + 3*A*c*d^2*e - 3*a*B*d*e^2 - a*A*e^3)*x*Sqrt[a + c*x^4])/(2*a*c^(3/2)*(Sqrt[a] + Sqrt[c]*x^2)) - (a^(1/4)*e^2*(3*B*d + A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(c^(7/4)*Sqrt[a + c*x^4]) + ((B*c*d^3 + 3*A*c*d^2*e - 3*a*B*d*e^2 - a*A*e^3)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(3/4)*c^(7/4)*Sqrt[a + c*x^4]) - (a^(3/4)*B*e^3*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(6*c^(9/4)*Sqrt[a + c*x^4]) + (a^(1/4)*e^2*(3*B*d + A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*c^(7/4)*Sqrt[a + c*x^4]) + (e*(3*B*c*d^2 + 3*A*c*d*e - a*B*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*c^(9/4)*Sqrt[a + c*x^4]) + ((A*c^2*d^3 + a^2*B*e^3 - 3*a*c*d*e*(B*d + A*e) + a^(3/2)*Sqrt[c]*e^2*(3*B*d + A*e) - Sqrt[a]*c^(3/2)*d^2*(B*d + 3*A*e))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(5/4)*c^(9/4)*Sqrt[a + c*x^4])","A",12,7,28,0.2500,1,"{1721, 1179, 1198, 220, 1196, 305, 321}"
9,1,694,0,0.5315175,"\int \frac{\left(A+B x^2\right) \left(d+e x^2\right)^2}{\left(a+c x^4\right)^{3/2}} \, dx","Int[((A + B*x^2)*(d + e*x^2)^2)/(a + c*x^4)^(3/2),x]","-\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(-\frac{\sqrt{c} \left(-a A e^2-2 a B d e+A c d^2\right)}{\sqrt{a}}-a B e^2+2 A c d e+B c d^2\right)}{4 a^{3/4} c^{7/4} \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}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) \left(-a B e^2+2 A c d e+B c d^2\right)}{2 a^{3/4} c^{7/4} \sqrt{a+c x^4}}-\frac{x \sqrt{a+c x^4} \left(-a B e^2+2 A c d e+B c d^2\right)}{2 a c^{3/2} \left(\sqrt{a}+\sqrt{c} x^2\right)}+\frac{e \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} (A e+2 B d) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} c^{5/4} \sqrt{a+c x^4}}+\frac{x \left(x^2 \left(-a B e^2+2 A c d e+B c d^2\right)-a A e^2-2 a B d e+A c d^2\right)}{2 a c \sqrt{a+c x^4}}+\frac{B e^2 x \sqrt{a+c x^4}}{c^{3/2} \left(\sqrt{a}+\sqrt{c} x^2\right)}+\frac{\sqrt[4]{a} B e^2 \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 c^{7/4} \sqrt{a+c x^4}}-\frac{\sqrt[4]{a} B e^2 \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^{7/4} \sqrt{a+c x^4}}","-\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) \left(-\frac{\sqrt{c} \left(-a A e^2-2 a B d e+A c d^2\right)}{\sqrt{a}}-a B e^2+2 A c d e+B c d^2\right)}{4 a^{3/4} c^{7/4} \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}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) \left(-a B e^2+2 A c d e+B c d^2\right)}{2 a^{3/4} c^{7/4} \sqrt{a+c x^4}}-\frac{x \sqrt{a+c x^4} \left(-a B e^2+2 A c d e+B c d^2\right)}{2 a c^{3/2} \left(\sqrt{a}+\sqrt{c} x^2\right)}+\frac{e \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} (A e+2 B d) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} c^{5/4} \sqrt{a+c x^4}}+\frac{x \left(x^2 \left(-a B e^2+2 A c d e+B c d^2\right)-a A e^2-2 a B d e+A c d^2\right)}{2 a c \sqrt{a+c x^4}}+\frac{B e^2 x \sqrt{a+c x^4}}{c^{3/2} \left(\sqrt{a}+\sqrt{c} x^2\right)}+\frac{\sqrt[4]{a} B e^2 \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 c^{7/4} \sqrt{a+c x^4}}-\frac{\sqrt[4]{a} B e^2 \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^{7/4} \sqrt{a+c x^4}}",1,"(x*(A*c*d^2 - 2*a*B*d*e - a*A*e^2 + (B*c*d^2 + 2*A*c*d*e - a*B*e^2)*x^2))/(2*a*c*Sqrt[a + c*x^4]) + (B*e^2*x*Sqrt[a + c*x^4])/(c^(3/2)*(Sqrt[a] + Sqrt[c]*x^2)) - ((B*c*d^2 + 2*A*c*d*e - a*B*e^2)*x*Sqrt[a + c*x^4])/(2*a*c^(3/2)*(Sqrt[a] + Sqrt[c]*x^2)) - (a^(1/4)*B*e^2*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(c^(7/4)*Sqrt[a + c*x^4]) + ((B*c*d^2 + 2*A*c*d*e - a*B*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(3/4)*c^(7/4)*Sqrt[a + c*x^4]) + (a^(1/4)*B*e^2*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*c^(7/4)*Sqrt[a + c*x^4]) + (e*(2*B*d + A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*c^(5/4)*Sqrt[a + c*x^4]) - ((B*c*d^2 + 2*A*c*d*e - a*B*e^2 - (Sqrt[c]*(A*c*d^2 - 2*a*B*d*e - a*A*e^2))/Sqrt[a])*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(3/4)*c^(7/4)*Sqrt[a + c*x^4])","A",10,6,28,0.2143,1,"{1721, 1179, 1198, 220, 1196, 305}"
10,1,395,0,0.3271191,"\int \frac{\left(A+B x^2\right) \left(d+e x^2\right)}{\left(a+c x^4\right)^{3/2}} \, dx","Int[((A + B*x^2)*(d + e*x^2))/(a + c*x^4)^(3/2),x]","\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} \sqrt{c} (A e+B d)-a B e+A c d\right)}{4 a^{5/4} c^{5/4} \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}} (A e+B d) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 a^{3/4} c^{3/4} \sqrt{a+c x^4}}-\frac{x \sqrt{a+c x^4} (A e+B d)}{2 a \sqrt{c} \left(\sqrt{a}+\sqrt{c} x^2\right)}+\frac{x \left(-a B e+c x^2 (A e+B d)+A c d\right)}{2 a c \sqrt{a+c x^4}}+\frac{B e \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} c^{5/4} \sqrt{a+c x^4}}","\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) \left(-\sqrt{a} \sqrt{c} (A e+B d)-a B e+A c d\right)}{4 a^{5/4} c^{5/4} \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}} (A e+B d) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 a^{3/4} c^{3/4} \sqrt{a+c x^4}}-\frac{x \sqrt{a+c x^4} (A e+B d)}{2 a \sqrt{c} \left(\sqrt{a}+\sqrt{c} x^2\right)}+\frac{x \left(-a B e+c x^2 (A e+B d)+A c d\right)}{2 a c \sqrt{a+c x^4}}+\frac{B e \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} c^{5/4} \sqrt{a+c x^4}}",1,"(x*(A*c*d - a*B*e + c*(B*d + A*e)*x^2))/(2*a*c*Sqrt[a + c*x^4]) - ((B*d + A*e)*x*Sqrt[a + c*x^4])/(2*a*Sqrt[c]*(Sqrt[a] + Sqrt[c]*x^2)) + ((B*d + A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(3/4)*c^(3/4)*Sqrt[a + c*x^4]) + (B*e*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*c^(5/4)*Sqrt[a + c*x^4]) + ((A*c*d - a*B*e - Sqrt[a]*Sqrt[c]*(B*d + A*e))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(5/4)*c^(5/4)*Sqrt[a + c*x^4])","A",7,5,26,0.1923,1,"{1721, 1179, 1198, 220, 1196}"
11,1,262,0,0.0964995,"\int \frac{A+B x^2}{\left(a+c x^4\right)^{3/2}} \, dx","Int[(A + B*x^2)/(a + c*x^4)^(3/2),x]","-\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} B-A \sqrt{c}\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 a^{5/4} c^{3/4} \sqrt{a+c x^4}}+\frac{B \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 a^{3/4} c^{3/4} \sqrt{a+c x^4}}+\frac{x \left(A+B x^2\right)}{2 a \sqrt{a+c x^4}}-\frac{B x \sqrt{a+c x^4}}{2 a \sqrt{c} \left(\sqrt{a}+\sqrt{c} x^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} B-A \sqrt{c}\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 a^{5/4} c^{3/4} \sqrt{a+c x^4}}+\frac{B \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 a^{3/4} c^{3/4} \sqrt{a+c x^4}}+\frac{x \left(A+B x^2\right)}{2 a \sqrt{a+c x^4}}-\frac{B x \sqrt{a+c x^4}}{2 a \sqrt{c} \left(\sqrt{a}+\sqrt{c} x^2\right)}",1,"(x*(A + B*x^2))/(2*a*Sqrt[a + c*x^4]) - (B*x*Sqrt[a + c*x^4])/(2*a*Sqrt[c]*(Sqrt[a] + Sqrt[c]*x^2)) + (B*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(3/4)*c^(3/4)*Sqrt[a + c*x^4]) - ((Sqrt[a]*B - A*Sqrt[c])*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(5/4)*c^(3/4)*Sqrt[a + c*x^4])","A",4,4,19,0.2105,1,"{1179, 1198, 220, 1196}"
12,1,732,0,0.7810605,"\int \frac{A+B x^2}{\left(d+e x^2\right) \left(a+c x^4\right)^{3/2}} \, dx","Int[(A + B*x^2)/((d + e*x^2)*(a + c*x^4)^(3/2)),x]","\frac{a^{3/4} 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(\frac{\sqrt{c} d}{\sqrt{a}}+e\right)^2 (B d-A e) \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^2 d^4-a^2 e^4\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}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) \left(-\sqrt{a} \sqrt{c} (B d-A e)+a B e+A c d\right)}{4 a^{5/4} \sqrt[4]{c} \sqrt{a+c x^4} \left(a e^2+c d^2\right)}+\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}} (B d-A e) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 a^{3/4} \sqrt{a+c x^4} \left(a e^2+c d^2\right)}-\frac{\sqrt{c} x \sqrt{a+c x^4} (B d-A e)}{2 a \left(\sqrt{a}+\sqrt{c} x^2\right) \left(a e^2+c d^2\right)}+\frac{x \left(a B e+c x^2 (B d-A e)+A c d\right)}{2 a \sqrt{a+c x^4} \left(a e^2+c d^2\right)}-\frac{e^{3/2} (B d-A 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} \left(a e^2+c d^2\right)^{3/2}}-\frac{\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}} (B d-A e) 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) \left(a e^2+c d^2\right)}","\frac{a^{3/4} 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(\frac{\sqrt{c} d}{\sqrt{a}}+e\right)^2 (B d-A e) \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^2 d^4-a^2 e^4\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}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) \left(-\sqrt{a} \sqrt{c} (B d-A e)+a B e+A c d\right)}{4 a^{5/4} \sqrt[4]{c} \sqrt{a+c x^4} \left(a e^2+c d^2\right)}+\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}} (B d-A e) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 a^{3/4} \sqrt{a+c x^4} \left(a e^2+c d^2\right)}-\frac{\sqrt{c} x \sqrt{a+c x^4} (B d-A e)}{2 a \left(\sqrt{a}+\sqrt{c} x^2\right) \left(a e^2+c d^2\right)}+\frac{x \left(a B e+c x^2 (B d-A e)+A c d\right)}{2 a \sqrt{a+c x^4} \left(a e^2+c d^2\right)}-\frac{e^{3/2} (B d-A 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} \left(a e^2+c d^2\right)^{3/2}}-\frac{\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}} (B d-A e) 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) \left(a e^2+c d^2\right)}",1,"(x*(A*c*d + a*B*e + c*(B*d - A*e)*x^2))/(2*a*(c*d^2 + a*e^2)*Sqrt[a + c*x^4]) - (Sqrt[c]*(B*d - A*e)*x*Sqrt[a + c*x^4])/(2*a*(c*d^2 + a*e^2)*(Sqrt[a] + Sqrt[c]*x^2)) - (e^(3/2)*(B*d - A*e)*ArcTan[(Sqrt[c*d^2 + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + c*x^4])])/(2*Sqrt[d]*(c*d^2 + a*e^2)^(3/2)) + (c^(1/4)*(B*d - A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(3/4)*(c*d^2 + a*e^2)*Sqrt[a + c*x^4]) - (c^(1/4)*e*(B*d - A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + a*e^2)*Sqrt[a + c*x^4]) + ((A*c*d + a*B*e - Sqrt[a]*Sqrt[c]*(B*d - A*e))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(5/4)*c^(1/4)*(c*d^2 + a*e^2)*Sqrt[a + c*x^4]) + (a^(3/4)*e*((Sqrt[c]*d)/Sqrt[a] + e)^2*(B*d - A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[-(Sqrt[c]*d - Sqrt[a]*e)^2/(4*Sqrt[a]*Sqrt[c]*d*e), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(4*c^(1/4)*d*(c^2*d^4 - a^2*e^4)*Sqrt[a + c*x^4])","A",9,7,28,0.2500,1,"{1721, 1179, 1198, 220, 1196, 1217, 1707}"
13,1,1494,0,2.0065043,"\int \frac{A+B x^2}{\left(d+e x^2\right)^2 \left(a+c x^4\right)^{3/2}} \, dx","Int[(A + B*x^2)/((d + e*x^2)^2*(a + c*x^4)^(3/2)),x]","-\frac{(B d-A e) x \sqrt{c x^4+a} e^3}{2 d \left(c d^2+a e^2\right)^2 \left(e x^2+d\right)}-\frac{\sqrt[4]{a} \sqrt[4]{c} (B d-A e) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) e^2}{2 d \left(c d^2+a e^2\right)^2 \sqrt{c x^4+a}}+\frac{\sqrt{c} (B d-A e) x \sqrt{c x^4+a} e^2}{2 d \left(c d^2+a e^2\right)^2 \left(\sqrt{c} x^2+\sqrt{a}\right)}-\frac{(B d-A e) \left(3 c d^2+a e^2\right) \tan ^{-1}\left(\frac{\sqrt{c d^2+a e^2} x}{\sqrt{d} \sqrt{e} \sqrt{c x^4+a}}\right) e^{3/2}}{4 d^{3/2} \left(c d^2+a e^2\right)^{5/2}}-\frac{\left(B c d^2-2 A c e d-a B e^2\right) \tan ^{-1}\left(\frac{\sqrt{c d^2+a e^2} x}{\sqrt{d} \sqrt{e} \sqrt{c x^4+a}}\right) e^{3/2}}{2 \sqrt{d} \left(c d^2+a e^2\right)^{5/2}}-\frac{\sqrt[4]{c} \left(B c d^2-2 A c e d-a B e^2\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) e}{2 \sqrt[4]{a} \left(\sqrt{c} d-\sqrt{a} e\right) \left(c d^2+a e^2\right)^2 \sqrt{c x^4+a}}-\frac{\sqrt[4]{c} (B d-A e) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) e}{2 \sqrt[4]{a} d \left(\sqrt{c} d-\sqrt{a} e\right) \left(c d^2+a e^2\right) \sqrt{c x^4+a}}+\frac{\left(\sqrt{c} d+\sqrt{a} e\right) (B d-A e) \left(3 c d^2+a e^2\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\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) e}{8 \sqrt[4]{a} \sqrt[4]{c} d^2 \left(\sqrt{c} d-\sqrt{a} e\right) \left(c d^2+a e^2\right)^2 \sqrt{c x^4+a}}+\frac{a^{3/4} \left(\frac{\sqrt{c} d}{\sqrt{a}}+e\right)^2 \left(B c d^2-2 A c e d-a B e^2\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\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) e}{4 \sqrt[4]{c} d \left(c d^2-a e^2\right) \left(c d^2+a e^2\right)^2 \sqrt{c x^4+a}}+\frac{\sqrt[4]{c} \left(B c d^2-2 A c e d-a B e^2\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 a^{3/4} \left(c d^2+a e^2\right)^2 \sqrt{c x^4+a}}-\frac{\sqrt[4]{c} \left(B c d^2-2 A c e d-a B e^2-\frac{\sqrt{c} \left(A c d^2+2 a B e d-a A e^2\right)}{\sqrt{a}}\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 a^{3/4} \left(c d^2+a e^2\right)^2 \sqrt{c x^4+a}}-\frac{\sqrt{c} \left(B c d^2-2 A c e d-a B e^2\right) x \sqrt{c x^4+a}}{2 a \left(c d^2+a e^2\right)^2 \left(\sqrt{c} x^2+\sqrt{a}\right)}+\frac{c x \left(A c d^2+2 a B e d-a A e^2+\left(B c d^2-2 A c e d-a B e^2\right) x^2\right)}{2 a \left(c d^2+a e^2\right)^2 \sqrt{c x^4+a}}","-\frac{(B d-A e) x \sqrt{c x^4+a} e^3}{2 d \left(c d^2+a e^2\right)^2 \left(e x^2+d\right)}-\frac{\sqrt[4]{a} \sqrt[4]{c} (B d-A e) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) e^2}{2 d \left(c d^2+a e^2\right)^2 \sqrt{c x^4+a}}+\frac{\sqrt{c} (B d-A e) x \sqrt{c x^4+a} e^2}{2 d \left(c d^2+a e^2\right)^2 \left(\sqrt{c} x^2+\sqrt{a}\right)}-\frac{(B d-A e) \left(3 c d^2+a e^2\right) \tan ^{-1}\left(\frac{\sqrt{c d^2+a e^2} x}{\sqrt{d} \sqrt{e} \sqrt{c x^4+a}}\right) e^{3/2}}{4 d^{3/2} \left(c d^2+a e^2\right)^{5/2}}-\frac{\left(B c d^2-2 A c e d-a B e^2\right) \tan ^{-1}\left(\frac{\sqrt{c d^2+a e^2} x}{\sqrt{d} \sqrt{e} \sqrt{c x^4+a}}\right) e^{3/2}}{2 \sqrt{d} \left(c d^2+a e^2\right)^{5/2}}-\frac{\sqrt[4]{c} \left(B c d^2-2 A c e d-a B e^2\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) e}{2 \sqrt[4]{a} \left(\sqrt{c} d-\sqrt{a} e\right) \left(c d^2+a e^2\right)^2 \sqrt{c x^4+a}}-\frac{\sqrt[4]{c} (B d-A e) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) e}{2 \sqrt[4]{a} d \left(\sqrt{c} d-\sqrt{a} e\right) \left(c d^2+a e^2\right) \sqrt{c x^4+a}}+\frac{\left(\sqrt{c} d+\sqrt{a} e\right) (B d-A e) \left(3 c d^2+a e^2\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\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) e}{8 \sqrt[4]{a} \sqrt[4]{c} d^2 \left(\sqrt{c} d-\sqrt{a} e\right) \left(c d^2+a e^2\right)^2 \sqrt{c x^4+a}}+\frac{a^{3/4} \left(\frac{\sqrt{c} d}{\sqrt{a}}+e\right)^2 \left(B c d^2-2 A c e d-a B e^2\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\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) e}{4 \sqrt[4]{c} d \left(c d^2-a e^2\right) \left(c d^2+a e^2\right)^2 \sqrt{c x^4+a}}+\frac{\sqrt[4]{c} \left(B c d^2-2 A c e d-a B e^2\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 a^{3/4} \left(c d^2+a e^2\right)^2 \sqrt{c x^4+a}}-\frac{\sqrt[4]{c} \left(B c d^2-2 A c e d-a B e^2-\frac{\sqrt{c} \left(A c d^2+2 a B e d-a A e^2\right)}{\sqrt{a}}\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 a^{3/4} \left(c d^2+a e^2\right)^2 \sqrt{c x^4+a}}-\frac{\sqrt{c} \left(B c d^2-2 A c e d-a B e^2\right) x \sqrt{c x^4+a}}{2 a \left(c d^2+a e^2\right)^2 \left(\sqrt{c} x^2+\sqrt{a}\right)}+\frac{c x \left(A c d^2+2 a B e d-a A e^2+\left(B c d^2-2 A c e d-a B e^2\right) x^2\right)}{2 a \left(c d^2+a e^2\right)^2 \sqrt{c x^4+a}}",1,"(c*x*(A*c*d^2 + 2*a*B*d*e - a*A*e^2 + (B*c*d^2 - 2*A*c*d*e - a*B*e^2)*x^2))/(2*a*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4]) + (Sqrt[c]*e^2*(B*d - A*e)*x*Sqrt[a + c*x^4])/(2*d*(c*d^2 + a*e^2)^2*(Sqrt[a] + Sqrt[c]*x^2)) - (Sqrt[c]*(B*c*d^2 - 2*A*c*d*e - a*B*e^2)*x*Sqrt[a + c*x^4])/(2*a*(c*d^2 + a*e^2)^2*(Sqrt[a] + Sqrt[c]*x^2)) - (e^3*(B*d - A*e)*x*Sqrt[a + c*x^4])/(2*d*(c*d^2 + a*e^2)^2*(d + e*x^2)) - (e^(3/2)*(B*d - A*e)*(3*c*d^2 + a*e^2)*ArcTan[(Sqrt[c*d^2 + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + c*x^4])])/(4*d^(3/2)*(c*d^2 + a*e^2)^(5/2)) - (e^(3/2)*(B*c*d^2 - 2*A*c*d*e - a*B*e^2)*ArcTan[(Sqrt[c*d^2 + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + c*x^4])])/(2*Sqrt[d]*(c*d^2 + a*e^2)^(5/2)) - (a^(1/4)*c^(1/4)*e^2*(B*d - A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*d*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4]) + (c^(1/4)*(B*c*d^2 - 2*A*c*d*e - a*B*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(3/4)*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4]) - (c^(1/4)*e*(B*d - A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*d*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + a*e^2)*Sqrt[a + c*x^4]) - (c^(1/4)*e*(B*c*d^2 - 2*A*c*d*e - a*B*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4]) - (c^(1/4)*(B*c*d^2 - 2*A*c*d*e - a*B*e^2 - (Sqrt[c]*(A*c*d^2 + 2*a*B*d*e - a*A*e^2))/Sqrt[a])*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(3/4)*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4]) + (e*(Sqrt[c]*d + Sqrt[a]*e)*(B*d - A*e)*(3*c*d^2 + a*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[-(Sqrt[c]*d - Sqrt[a]*e)^2/(4*Sqrt[a]*Sqrt[c]*d*e), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(8*a^(1/4)*c^(1/4)*d^2*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4]) + (a^(3/4)*e*((Sqrt[c]*d)/Sqrt[a] + e)^2*(B*c*d^2 - 2*A*c*d*e - a*B*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[-(Sqrt[c]*d - Sqrt[a]*e)^2/(4*Sqrt[a]*Sqrt[c]*d*e), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(4*c^(1/4)*d*(c*d^2 - a*e^2)*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4])","A",15,10,28,0.3571,1,"{1721, 1179, 1198, 220, 1196, 1224, 1715, 1709, 1707, 1217}"
14,1,2452,0,3.9712631,"\int \frac{A+B x^2}{\left(d+e x^2\right)^3 \left(a+c x^4\right)^{3/2}} \, dx","Int[(A + B*x^2)/((d + e*x^2)^3*(a + c*x^4)^(3/2)),x]","-\frac{3 (B d-A e) \left(3 c d^2+a e^2\right) x \sqrt{c x^4+a} e^3}{8 d^2 \left(c d^2+a e^2\right)^3 \left(e x^2+d\right)}-\frac{\left(B c d^2-2 A c e d-a B e^2\right) x \sqrt{c x^4+a} e^3}{2 d \left(c d^2+a e^2\right)^3 \left(e x^2+d\right)}-\frac{(B d-A e) x \sqrt{c x^4+a} e^3}{4 d \left(c d^2+a e^2\right)^2 \left(e x^2+d\right)^2}-\frac{3 \sqrt[4]{a} \sqrt[4]{c} (B d-A e) \left(3 c d^2+a e^2\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) e^2}{8 d^2 \left(c d^2+a e^2\right)^3 \sqrt{c x^4+a}}-\frac{\sqrt[4]{a} \sqrt[4]{c} \left(B c d^2-2 A c e d-a B e^2\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) e^2}{2 d \left(c d^2+a e^2\right)^3 \sqrt{c x^4+a}}+\frac{3 \sqrt{c} (B d-A e) \left(3 c d^2+a e^2\right) x \sqrt{c x^4+a} e^2}{8 d^2 \left(c d^2+a e^2\right)^3 \left(\sqrt{c} x^2+\sqrt{a}\right)}+\frac{\sqrt{c} \left(B c d^2-2 A c e d-a B e^2\right) x \sqrt{c x^4+a} e^2}{2 d \left(c d^2+a e^2\right)^3 \left(\sqrt{c} x^2+\sqrt{a}\right)}-\frac{\left(3 c d^2+a e^2\right) \left(B c d^2-2 A c e d-a B e^2\right) \tan ^{-1}\left(\frac{\sqrt{c d^2+a e^2} x}{\sqrt{d} \sqrt{e} \sqrt{c x^4+a}}\right) e^{3/2}}{4 d^{3/2} \left(c d^2+a e^2\right)^{7/2}}-\frac{c \left(B c d^3-3 A c e d^2-3 a B e^2 d+a A e^3\right) \tan ^{-1}\left(\frac{\sqrt{c d^2+a e^2} x}{\sqrt{d} \sqrt{e} \sqrt{c x^4+a}}\right) e^{3/2}}{2 \sqrt{d} \left(c d^2+a e^2\right)^{7/2}}-\frac{3 (B d-A e) \left(5 c^2 d^4+2 a c e^2 d^2+a^2 e^4\right) \tan ^{-1}\left(\frac{\sqrt{c d^2+a e^2} x}{\sqrt{d} \sqrt{e} \sqrt{c x^4+a}}\right) e^{3/2}}{16 d^{5/2} \left(c d^2+a e^2\right)^{7/2}}-\frac{\sqrt[4]{c} (B d-A e) \left(4 c d^2-\sqrt{a} \sqrt{c} e d+3 a e^2\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) e}{8 \sqrt[4]{a} d^2 \left(\sqrt{c} d-\sqrt{a} e\right) \left(c d^2+a e^2\right)^2 \sqrt{c x^4+a}}-\frac{\sqrt[4]{c} \left(B c d^2-2 A c e d-a B e^2\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) e}{2 \sqrt[4]{a} d \left(\sqrt{c} d-\sqrt{a} e\right) \left(c d^2+a e^2\right)^2 \sqrt{c x^4+a}}-\frac{c^{5/4} \left(B c d^3-3 A c e d^2-3 a B e^2 d+a A e^3\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) e}{2 \sqrt[4]{a} \left(\sqrt{c} d-\sqrt{a} e\right) \left(c d^2+a e^2\right)^3 \sqrt{c x^4+a}}+\frac{\left(\sqrt{c} d+\sqrt{a} e\right) \left(3 c d^2+a e^2\right) \left(B c d^2-2 A c e d-a B e^2\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\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) e}{8 \sqrt[4]{a} \sqrt[4]{c} d^2 \left(\sqrt{c} d-\sqrt{a} e\right) \left(c d^2+a e^2\right)^3 \sqrt{c x^4+a}}+\frac{a^{3/4} c^{3/4} \left(\frac{\sqrt{c} d}{\sqrt{a}}+e\right)^2 \left(B c d^3-3 A c e d^2-3 a B e^2 d+a A e^3\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\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) e}{4 d \left(c d^2-a e^2\right) \left(c d^2+a e^2\right)^3 \sqrt{c x^4+a}}+\frac{3 \left(\sqrt{c} d+\sqrt{a} e\right) (B d-A e) \left(5 c^2 d^4+2 a c e^2 d^2+a^2 e^4\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\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) e}{32 \sqrt[4]{a} \sqrt[4]{c} d^3 \left(\sqrt{c} d-\sqrt{a} e\right) \left(c d^2+a e^2\right)^3 \sqrt{c x^4+a}}+\frac{c^{5/4} \left(B c d^3-3 A c e d^2-3 a B e^2 d+a A e^3\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 a^{3/4} \left(c d^2+a e^2\right)^3 \sqrt{c x^4+a}}+\frac{c^{3/4} \left(A c^2 d^3-\sqrt{a} c^{3/2} (B d-3 A e) d^2+3 a c e (B d-A e) d-a^2 B e^3+a^{3/2} \sqrt{c} e^2 (3 B d-A e)\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 a^{5/4} \left(c d^2+a e^2\right)^3 \sqrt{c x^4+a}}-\frac{c^{3/2} \left(B c d^3-3 A c e d^2-3 a B e^2 d+a A e^3\right) x \sqrt{c x^4+a}}{2 a \left(c d^2+a e^2\right)^3 \left(\sqrt{c} x^2+\sqrt{a}\right)}+\frac{c x \left(c \left(B c d^3-3 A c e d^2-3 a B e^2 d+a A e^3\right) x^2+A c d \left(c d^2-3 a e^2\right)+a B e \left(3 c d^2-a e^2\right)\right)}{2 a \left(c d^2+a e^2\right)^3 \sqrt{c x^4+a}}","-\frac{3 (B d-A e) \left(3 c d^2+a e^2\right) x \sqrt{c x^4+a} e^3}{8 d^2 \left(c d^2+a e^2\right)^3 \left(e x^2+d\right)}-\frac{\left(B c d^2-2 A c e d-a B e^2\right) x \sqrt{c x^4+a} e^3}{2 d \left(c d^2+a e^2\right)^3 \left(e x^2+d\right)}-\frac{(B d-A e) x \sqrt{c x^4+a} e^3}{4 d \left(c d^2+a e^2\right)^2 \left(e x^2+d\right)^2}-\frac{3 \sqrt[4]{a} \sqrt[4]{c} (B d-A e) \left(3 c d^2+a e^2\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) e^2}{8 d^2 \left(c d^2+a e^2\right)^3 \sqrt{c x^4+a}}-\frac{\sqrt[4]{a} \sqrt[4]{c} \left(B c d^2-2 A c e d-a B e^2\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) e^2}{2 d \left(c d^2+a e^2\right)^3 \sqrt{c x^4+a}}+\frac{3 \sqrt{c} (B d-A e) \left(3 c d^2+a e^2\right) x \sqrt{c x^4+a} e^2}{8 d^2 \left(c d^2+a e^2\right)^3 \left(\sqrt{c} x^2+\sqrt{a}\right)}+\frac{\sqrt{c} \left(B c d^2-2 A c e d-a B e^2\right) x \sqrt{c x^4+a} e^2}{2 d \left(c d^2+a e^2\right)^3 \left(\sqrt{c} x^2+\sqrt{a}\right)}-\frac{\left(3 c d^2+a e^2\right) \left(B c d^2-2 A c e d-a B e^2\right) \tan ^{-1}\left(\frac{\sqrt{c d^2+a e^2} x}{\sqrt{d} \sqrt{e} \sqrt{c x^4+a}}\right) e^{3/2}}{4 d^{3/2} \left(c d^2+a e^2\right)^{7/2}}-\frac{c \left(B c d^3-3 A c e d^2-3 a B e^2 d+a A e^3\right) \tan ^{-1}\left(\frac{\sqrt{c d^2+a e^2} x}{\sqrt{d} \sqrt{e} \sqrt{c x^4+a}}\right) e^{3/2}}{2 \sqrt{d} \left(c d^2+a e^2\right)^{7/2}}-\frac{3 (B d-A e) \left(5 c^2 d^4+2 a c e^2 d^2+a^2 e^4\right) \tan ^{-1}\left(\frac{\sqrt{c d^2+a e^2} x}{\sqrt{d} \sqrt{e} \sqrt{c x^4+a}}\right) e^{3/2}}{16 d^{5/2} \left(c d^2+a e^2\right)^{7/2}}-\frac{\sqrt[4]{c} (B d-A e) \left(4 c d^2-\sqrt{a} \sqrt{c} e d+3 a e^2\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) e}{8 \sqrt[4]{a} d^2 \left(\sqrt{c} d-\sqrt{a} e\right) \left(c d^2+a e^2\right)^2 \sqrt{c x^4+a}}-\frac{\sqrt[4]{c} \left(B c d^2-2 A c e d-a B e^2\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) e}{2 \sqrt[4]{a} d \left(\sqrt{c} d-\sqrt{a} e\right) \left(c d^2+a e^2\right)^2 \sqrt{c x^4+a}}-\frac{c^{5/4} \left(B c d^3-3 A c e d^2-3 a B e^2 d+a A e^3\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) e}{2 \sqrt[4]{a} \left(\sqrt{c} d-\sqrt{a} e\right) \left(c d^2+a e^2\right)^3 \sqrt{c x^4+a}}+\frac{\left(\sqrt{c} d+\sqrt{a} e\right) \left(3 c d^2+a e^2\right) \left(B c d^2-2 A c e d-a B e^2\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\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) e}{8 \sqrt[4]{a} \sqrt[4]{c} d^2 \left(\sqrt{c} d-\sqrt{a} e\right) \left(c d^2+a e^2\right)^3 \sqrt{c x^4+a}}+\frac{a^{3/4} c^{3/4} \left(\frac{\sqrt{c} d}{\sqrt{a}}+e\right)^2 \left(B c d^3-3 A c e d^2-3 a B e^2 d+a A e^3\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\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) e}{4 d \left(c d^2-a e^2\right) \left(c d^2+a e^2\right)^3 \sqrt{c x^4+a}}+\frac{3 \left(\sqrt{c} d+\sqrt{a} e\right) (B d-A e) \left(5 c^2 d^4+2 a c e^2 d^2+a^2 e^4\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\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) e}{32 \sqrt[4]{a} \sqrt[4]{c} d^3 \left(\sqrt{c} d-\sqrt{a} e\right) \left(c d^2+a e^2\right)^3 \sqrt{c x^4+a}}+\frac{c^{5/4} \left(B c d^3-3 A c e d^2-3 a B e^2 d+a A e^3\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 a^{3/4} \left(c d^2+a e^2\right)^3 \sqrt{c x^4+a}}+\frac{c^{3/4} \left(A c^2 d^3-\sqrt{a} c^{3/2} (B d-3 A e) d^2+3 a c e (B d-A e) d-a^2 B e^3+a^{3/2} \sqrt{c} e^2 (3 B d-A e)\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 a^{5/4} \left(c d^2+a e^2\right)^3 \sqrt{c x^4+a}}-\frac{c^{3/2} \left(B c d^3-3 A c e d^2-3 a B e^2 d+a A e^3\right) x \sqrt{c x^4+a}}{2 a \left(c d^2+a e^2\right)^3 \left(\sqrt{c} x^2+\sqrt{a}\right)}+\frac{c x \left(c \left(B c d^3-3 A c e d^2-3 a B e^2 d+a A e^3\right) x^2+A c d \left(c d^2-3 a e^2\right)+a B e \left(3 c d^2-a e^2\right)\right)}{2 a \left(c d^2+a e^2\right)^3 \sqrt{c x^4+a}}",1,"(c*x*(A*c*d*(c*d^2 - 3*a*e^2) + a*B*e*(3*c*d^2 - a*e^2) + c*(B*c*d^3 - 3*A*c*d^2*e - 3*a*B*d*e^2 + a*A*e^3)*x^2))/(2*a*(c*d^2 + a*e^2)^3*Sqrt[a + c*x^4]) + (3*Sqrt[c]*e^2*(B*d - A*e)*(3*c*d^2 + a*e^2)*x*Sqrt[a + c*x^4])/(8*d^2*(c*d^2 + a*e^2)^3*(Sqrt[a] + Sqrt[c]*x^2)) + (Sqrt[c]*e^2*(B*c*d^2 - 2*A*c*d*e - a*B*e^2)*x*Sqrt[a + c*x^4])/(2*d*(c*d^2 + a*e^2)^3*(Sqrt[a] + Sqrt[c]*x^2)) - (c^(3/2)*(B*c*d^3 - 3*A*c*d^2*e - 3*a*B*d*e^2 + a*A*e^3)*x*Sqrt[a + c*x^4])/(2*a*(c*d^2 + a*e^2)^3*(Sqrt[a] + Sqrt[c]*x^2)) - (e^3*(B*d - A*e)*x*Sqrt[a + c*x^4])/(4*d*(c*d^2 + a*e^2)^2*(d + e*x^2)^2) - (3*e^3*(B*d - A*e)*(3*c*d^2 + a*e^2)*x*Sqrt[a + c*x^4])/(8*d^2*(c*d^2 + a*e^2)^3*(d + e*x^2)) - (e^3*(B*c*d^2 - 2*A*c*d*e - a*B*e^2)*x*Sqrt[a + c*x^4])/(2*d*(c*d^2 + a*e^2)^3*(d + e*x^2)) - (e^(3/2)*(3*c*d^2 + a*e^2)*(B*c*d^2 - 2*A*c*d*e - a*B*e^2)*ArcTan[(Sqrt[c*d^2 + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + c*x^4])])/(4*d^(3/2)*(c*d^2 + a*e^2)^(7/2)) - (c*e^(3/2)*(B*c*d^3 - 3*A*c*d^2*e - 3*a*B*d*e^2 + a*A*e^3)*ArcTan[(Sqrt[c*d^2 + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + c*x^4])])/(2*Sqrt[d]*(c*d^2 + a*e^2)^(7/2)) - (3*e^(3/2)*(B*d - A*e)*(5*c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*ArcTan[(Sqrt[c*d^2 + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + c*x^4])])/(16*d^(5/2)*(c*d^2 + a*e^2)^(7/2)) - (3*a^(1/4)*c^(1/4)*e^2*(B*d - A*e)*(3*c*d^2 + a*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(8*d^2*(c*d^2 + a*e^2)^3*Sqrt[a + c*x^4]) - (a^(1/4)*c^(1/4)*e^2*(B*c*d^2 - 2*A*c*d*e - a*B*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*d*(c*d^2 + a*e^2)^3*Sqrt[a + c*x^4]) + (c^(5/4)*(B*c*d^3 - 3*A*c*d^2*e - 3*a*B*d*e^2 + a*A*e^3)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(3/4)*(c*d^2 + a*e^2)^3*Sqrt[a + c*x^4]) - (c^(1/4)*e*(B*d - A*e)*(4*c*d^2 - Sqrt[a]*Sqrt[c]*d*e + 3*a*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(8*a^(1/4)*d^2*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4]) - (c^(1/4)*e*(B*c*d^2 - 2*A*c*d*e - a*B*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*d*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4]) - (c^(5/4)*e*(B*c*d^3 - 3*A*c*d^2*e - 3*a*B*d*e^2 + a*A*e^3)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + a*e^2)^3*Sqrt[a + c*x^4]) + (c^(3/4)*(A*c^2*d^3 - a^2*B*e^3 - Sqrt[a]*c^(3/2)*d^2*(B*d - 3*A*e) + 3*a*c*d*e*(B*d - A*e) + a^(3/2)*Sqrt[c]*e^2*(3*B*d - A*e))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(5/4)*(c*d^2 + a*e^2)^3*Sqrt[a + c*x^4]) + (e*(Sqrt[c]*d + Sqrt[a]*e)*(3*c*d^2 + a*e^2)*(B*c*d^2 - 2*A*c*d*e - a*B*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[-(Sqrt[c]*d - Sqrt[a]*e)^2/(4*Sqrt[a]*Sqrt[c]*d*e), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(8*a^(1/4)*c^(1/4)*d^2*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + a*e^2)^3*Sqrt[a + c*x^4]) + (a^(3/4)*c^(3/4)*e*((Sqrt[c]*d)/Sqrt[a] + e)^2*(B*c*d^3 - 3*A*c*d^2*e - 3*a*B*d*e^2 + a*A*e^3)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[-(Sqrt[c]*d - Sqrt[a]*e)^2/(4*Sqrt[a]*Sqrt[c]*d*e), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(4*d*(c*d^2 - a*e^2)*(c*d^2 + a*e^2)^3*Sqrt[a + c*x^4]) + (3*e*(Sqrt[c]*d + Sqrt[a]*e)*(B*d - A*e)*(5*c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[-(Sqrt[c]*d - Sqrt[a]*e)^2/(4*Sqrt[a]*Sqrt[c]*d*e), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(32*a^(1/4)*c^(1/4)*d^3*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + a*e^2)^3*Sqrt[a + c*x^4])","A",22,11,28,0.3929,1,"{1721, 1179, 1198, 220, 1196, 1224, 1697, 1715, 1709, 1707, 1217}"
15,1,169,0,0.2189093,"\int \frac{\left(A+B x^2\right) \left(d+e x^2\right)^q}{a+c x^4} \, dx","Int[((A + B*x^2)*(d + e*x^2)^q)/(a + c*x^4),x]","\frac{x \left(A-\frac{\sqrt{-a} B}{\sqrt{c}}\right) \left(d+e x^2\right)^q \left(\frac{e x^2}{d}+1\right)^{-q} F_1\left(\frac{1}{2};1,-q;\frac{3}{2};-\frac{\sqrt{c} x^2}{\sqrt{-a}},-\frac{e x^2}{d}\right)}{2 a}+\frac{x \left(\frac{\sqrt{-a} B}{\sqrt{c}}+A\right) \left(d+e x^2\right)^q \left(\frac{e x^2}{d}+1\right)^{-q} F_1\left(\frac{1}{2};1,-q;\frac{3}{2};\frac{\sqrt{c} x^2}{\sqrt{-a}},-\frac{e x^2}{d}\right)}{2 a}","\frac{x \left(A-\frac{\sqrt{-a} B}{\sqrt{c}}\right) \left(d+e x^2\right)^q \left(\frac{e x^2}{d}+1\right)^{-q} F_1\left(\frac{1}{2};1,-q;\frac{3}{2};-\frac{\sqrt{c} x^2}{\sqrt{-a}},-\frac{e x^2}{d}\right)}{2 a}+\frac{x \left(\frac{\sqrt{-a} B}{\sqrt{c}}+A\right) \left(d+e x^2\right)^q \left(\frac{e x^2}{d}+1\right)^{-q} F_1\left(\frac{1}{2};1,-q;\frac{3}{2};\frac{\sqrt{c} x^2}{\sqrt{-a}},-\frac{e x^2}{d}\right)}{2 a}",1,"((A - (Sqrt[-a]*B)/Sqrt[c])*x*(d + e*x^2)^q*AppellF1[1/2, 1, -q, 3/2, -((Sqrt[c]*x^2)/Sqrt[-a]), -((e*x^2)/d)])/(2*a*(1 + (e*x^2)/d)^q) + ((A + (Sqrt[-a]*B)/Sqrt[c])*x*(d + e*x^2)^q*AppellF1[1/2, 1, -q, 3/2, (Sqrt[c]*x^2)/Sqrt[-a], -((e*x^2)/d)])/(2*a*(1 + (e*x^2)/d)^q)","A",6,3,26,0.1154,1,"{1693, 430, 429}"
16,1,48,0,0.0290589,"\int \frac{2+x^2}{\left(1+x^2\right) \sqrt{2+3 x^2+x^4}} \, dx","Int[(2 + x^2)/((1 + x^2)*Sqrt[2 + 3*x^2 + x^4]),x]","\frac{\sqrt{2} \left(x^2+2\right) E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}}","\frac{\sqrt{2} \left(x^2+2\right) E\left(\tan ^{-1}(x)|\frac{1}{2}\right)}{\sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}}",1,"(Sqrt[2]*(2 + x^2)*EllipticE[ArcTan[x], 1/2])/(Sqrt[(2 + x^2)/(1 + x^2)]*Sqrt[2 + 3*x^2 + x^4])","A",2,2,27,0.07407,1,"{1456, 411}"
17,1,755,0,1.3568848,"\int \frac{\left(A+B x^2\right) \left(d+e x^2\right)^3}{\sqrt{a+b x^2+c x^4}} \, dx","Int[((A + B*x^2)*(d + e*x^2)^3)/Sqrt[a + b*x^2 + c*x^4],x]","\frac{e x \sqrt{a+b x^2+c x^4} \left(B \left(-c e (25 a e+84 b d)+24 b^2 e^2+105 c^2 d^2\right)+7 A c e (15 c d-4 b e)\right)}{105 c^3}+\frac{x \sqrt{a+b x^2+c x^4} \left(7 A c e \left(-3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right)+B \left(-21 c^2 d e (9 a e+10 b d)+8 b c e^2 (13 a e+21 b d)-48 b^3 e^3+105 c^3 d^3\right)\right)}{105 c^{7/2} \left(\sqrt{a}+\sqrt{c} x^2\right)}+\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} 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(\frac{\sqrt{c} \left(7 A c \left(4 a b e^3-15 a c d e^2+15 c^2 d^3\right)-a B e \left(-c e (25 a e+84 b d)+24 b^2 e^2+105 c^2 d^2\right)\right)}{\sqrt{a}}+7 A c e \left(-3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right)+B \left(-21 c^2 d e (9 a e+10 b d)+8 b c e^2 (13 a e+21 b d)-48 b^3 e^3+105 c^3 d^3\right)\right)}{210 c^{15/4} \sqrt{a+b x^2+c x^4}}-\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) \left(7 A c e \left(-3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right)+B \left(-21 c^2 d e (9 a e+10 b d)+8 b c e^2 (13 a e+21 b d)-48 b^3 e^3+105 c^3 d^3\right)\right)}{105 c^{15/4} \sqrt{a+b x^2+c x^4}}+\frac{e^2 x^3 \sqrt{a+b x^2+c x^4} (7 A c e-6 b B e+21 B c d)}{35 c^2}+\frac{B e^3 x^5 \sqrt{a+b x^2+c x^4}}{7 c}","\frac{e x \sqrt{a+b x^2+c x^4} \left(B \left(-c e (25 a e+84 b d)+24 b^2 e^2+105 c^2 d^2\right)+7 A c e (15 c d-4 b e)\right)}{105 c^3}+\frac{x \sqrt{a+b x^2+c x^4} \left(7 A c e \left(-3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right)+B \left(-21 c^2 d e (9 a e+10 b d)+8 b c e^2 (13 a e+21 b d)-48 b^3 e^3+105 c^3 d^3\right)\right)}{105 c^{7/2} \left(\sqrt{a}+\sqrt{c} x^2\right)}+\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} 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(\frac{\sqrt{c} \left(7 A c \left(4 a b e^3-15 a c d e^2+15 c^2 d^3\right)-a B e \left(-c e (25 a e+84 b d)+24 b^2 e^2+105 c^2 d^2\right)\right)}{\sqrt{a}}+7 A c e \left(-3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right)+B \left(-21 c^2 d e (9 a e+10 b d)+8 b c e^2 (13 a e+21 b d)-48 b^3 e^3+105 c^3 d^3\right)\right)}{210 c^{15/4} \sqrt{a+b x^2+c x^4}}-\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) \left(7 A c e \left(-3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right)+B \left(-21 c^2 d e (9 a e+10 b d)+8 b c e^2 (13 a e+21 b d)-48 b^3 e^3+105 c^3 d^3\right)\right)}{105 c^{15/4} \sqrt{a+b x^2+c x^4}}+\frac{e^2 x^3 \sqrt{a+b x^2+c x^4} (7 A c e-6 b B e+21 B c d)}{35 c^2}+\frac{B e^3 x^5 \sqrt{a+b x^2+c x^4}}{7 c}",1,"(e*(7*A*c*e*(15*c*d - 4*b*e) + B*(105*c^2*d^2 + 24*b^2*e^2 - c*e*(84*b*d + 25*a*e)))*x*Sqrt[a + b*x^2 + c*x^4])/(105*c^3) + (e^2*(21*B*c*d - 6*b*B*e + 7*A*c*e)*x^3*Sqrt[a + b*x^2 + c*x^4])/(35*c^2) + (B*e^3*x^5*Sqrt[a + b*x^2 + c*x^4])/(7*c) + ((7*A*c*e*(45*c^2*d^2 + 8*b^2*e^2 - 3*c*e*(10*b*d + 3*a*e)) + B*(105*c^3*d^3 - 48*b^3*e^3 - 21*c^2*d*e*(10*b*d + 9*a*e) + 8*b*c*e^2*(21*b*d + 13*a*e)))*x*Sqrt[a + b*x^2 + c*x^4])/(105*c^(7/2)*(Sqrt[a] + Sqrt[c]*x^2)) - (a^(1/4)*(7*A*c*e*(45*c^2*d^2 + 8*b^2*e^2 - 3*c*e*(10*b*d + 3*a*e)) + B*(105*c^3*d^3 - 48*b^3*e^3 - 21*c^2*d*e*(10*b*d + 9*a*e) + 8*b*c*e^2*(21*b*d + 13*a*e)))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(105*c^(15/4)*Sqrt[a + b*x^2 + c*x^4]) + (a^(1/4)*(7*A*c*e*(45*c^2*d^2 + 8*b^2*e^2 - 3*c*e*(10*b*d + 3*a*e)) + B*(105*c^3*d^3 - 48*b^3*e^3 - 21*c^2*d*e*(10*b*d + 9*a*e) + 8*b*c*e^2*(21*b*d + 13*a*e)) + (Sqrt[c]*(7*A*c*(15*c^2*d^3 - 15*a*c*d*e^2 + 4*a*b*e^3) - a*B*e*(105*c^2*d^2 + 24*b^2*e^2 - c*e*(84*b*d + 25*a*e))))/Sqrt[a])*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(210*c^(15/4)*Sqrt[a + b*x^2 + c*x^4])","A",6,4,33,0.1212,1,"{1679, 1197, 1103, 1195}"
18,1,528,0,0.7263666,"\int \frac{\left(A+B x^2\right) \left(d+e x^2\right)^2}{\sqrt{a+b x^2+c x^4}} \, dx","Int[((A + B*x^2)*(d + e*x^2)^2)/Sqrt[a + b*x^2 + c*x^4],x]","\frac{x \sqrt{a+b x^2+c x^4} \left(B \left(-c e (9 a e+20 b d)+8 b^2 e^2+15 c^2 d^2\right)+10 A c e (3 c d-b e)\right)}{15 c^{5/2} \left(\sqrt{a}+\sqrt{c} x^2\right)}+\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} 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(-\frac{\sqrt{c} \left(2 a B e (5 c d-2 b e)-5 A c \left(3 c d^2-a e^2\right)\right)}{\sqrt{a}}+B \left(-c e (9 a e+20 b d)+8 b^2 e^2+15 c^2 d^2\right)+10 A c e (3 c d-b e)\right)}{30 c^{11/4} \sqrt{a+b x^2+c x^4}}-\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) \left(B \left(-c e (9 a e+20 b d)+8 b^2 e^2+15 c^2 d^2\right)+10 A c e (3 c d-b e)\right)}{15 c^{11/4} \sqrt{a+b x^2+c x^4}}+\frac{e x \sqrt{a+b x^2+c x^4} (5 A c e-4 b B e+10 B c d)}{15 c^2}+\frac{B e^2 x^3 \sqrt{a+b x^2+c x^4}}{5 c}","\frac{x \sqrt{a+b x^2+c x^4} \left(B \left(-c e (9 a e+20 b d)+8 b^2 e^2+15 c^2 d^2\right)+10 A c e (3 c d-b e)\right)}{15 c^{5/2} \left(\sqrt{a}+\sqrt{c} x^2\right)}+\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} 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(-\frac{\sqrt{c} \left(2 a B e (5 c d-2 b e)-5 A c \left(3 c d^2-a e^2\right)\right)}{\sqrt{a}}+B \left(-c e (9 a e+20 b d)+8 b^2 e^2+15 c^2 d^2\right)+10 A c e (3 c d-b e)\right)}{30 c^{11/4} \sqrt{a+b x^2+c x^4}}-\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) \left(B \left(-c e (9 a e+20 b d)+8 b^2 e^2+15 c^2 d^2\right)+10 A c e (3 c d-b e)\right)}{15 c^{11/4} \sqrt{a+b x^2+c x^4}}+\frac{e x \sqrt{a+b x^2+c x^4} (5 A c e-4 b B e+10 B c d)}{15 c^2}+\frac{B e^2 x^3 \sqrt{a+b x^2+c x^4}}{5 c}",1,"(e*(10*B*c*d - 4*b*B*e + 5*A*c*e)*x*Sqrt[a + b*x^2 + c*x^4])/(15*c^2) + (B*e^2*x^3*Sqrt[a + b*x^2 + c*x^4])/(5*c) + ((10*A*c*e*(3*c*d - b*e) + B*(15*c^2*d^2 + 8*b^2*e^2 - c*e*(20*b*d + 9*a*e)))*x*Sqrt[a + b*x^2 + c*x^4])/(15*c^(5/2)*(Sqrt[a] + Sqrt[c]*x^2)) - (a^(1/4)*(10*A*c*e*(3*c*d - b*e) + B*(15*c^2*d^2 + 8*b^2*e^2 - c*e*(20*b*d + 9*a*e)))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(15*c^(11/4)*Sqrt[a + b*x^2 + c*x^4]) + (a^(1/4)*(10*A*c*e*(3*c*d - b*e) + B*(15*c^2*d^2 + 8*b^2*e^2 - c*e*(20*b*d + 9*a*e)) - (Sqrt[c]*(2*a*B*e*(5*c*d - 2*b*e) - 5*A*c*(3*c*d^2 - a*e^2)))/Sqrt[a])*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(30*c^(11/4)*Sqrt[a + b*x^2 + c*x^4])","A",5,4,33,0.1212,1,"{1679, 1197, 1103, 1195}"
19,1,368,0,0.25398,"\int \frac{\left(A+B x^2\right) \left(d+e x^2\right)}{\sqrt{a+b x^2+c x^4}} \, dx","Int[((A + B*x^2)*(d + e*x^2))/Sqrt[a + b*x^2 + c*x^4],x]","\frac{x \sqrt{a+b x^2+c x^4} (3 A c e-2 b B e+3 B c d)}{3 c^{3/2} \left(\sqrt{a}+\sqrt{c} x^2\right)}+\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} 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(\frac{\sqrt{c} (3 A c d-a B e)}{\sqrt{a}}+3 A c e-2 b B e+3 B c d\right)}{6 c^{7/4} \sqrt{a+b x^2+c x^4}}-\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) (3 A c e-2 b B e+3 B c d)}{3 c^{7/4} \sqrt{a+b x^2+c x^4}}+\frac{B e x \sqrt{a+b x^2+c x^4}}{3 c}","\frac{x \sqrt{a+b x^2+c x^4} (3 A c e-2 b B e+3 B c d)}{3 c^{3/2} \left(\sqrt{a}+\sqrt{c} x^2\right)}+\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} 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(\frac{\sqrt{c} (3 A c d-a B e)}{\sqrt{a}}+3 A c e-2 b B e+3 B c d\right)}{6 c^{7/4} \sqrt{a+b x^2+c x^4}}-\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) (3 A c e-2 b B e+3 B c d)}{3 c^{7/4} \sqrt{a+b x^2+c x^4}}+\frac{B e x \sqrt{a+b x^2+c x^4}}{3 c}",1,"(B*e*x*Sqrt[a + b*x^2 + c*x^4])/(3*c) + ((3*B*c*d - 2*b*B*e + 3*A*c*e)*x*Sqrt[a + b*x^2 + c*x^4])/(3*c^(3/2)*(Sqrt[a] + Sqrt[c]*x^2)) - (a^(1/4)*(3*B*c*d - 2*b*B*e + 3*A*c*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(3*c^(7/4)*Sqrt[a + b*x^2 + c*x^4]) + (a^(1/4)*(3*B*c*d - 2*b*B*e + 3*A*c*e + (Sqrt[c]*(3*A*c*d - a*B*e))/Sqrt[a])*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(6*c^(7/4)*Sqrt[a + b*x^2 + c*x^4])","A",4,4,31,0.1290,1,"{1679, 1197, 1103, 1195}"
20,1,283,0,0.0808877,"\int \frac{A+B x^2}{\sqrt{a+b x^2+c x^4}} \, dx","Int[(A + B*x^2)/Sqrt[a + b*x^2 + c*x^4],x]","\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{c} x^2\right) \left(\frac{A \sqrt{c}}{\sqrt{a}}+B\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{2 c^{3/4} \sqrt{a+b x^2+c x^4}}-\frac{\sqrt[4]{a} B \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{c^{3/4} \sqrt{a+b x^2+c x^4}}+\frac{B x \sqrt{a+b x^2+c x^4}}{\sqrt{c} \left(\sqrt{a}+\sqrt{c} x^2\right)}","\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{c} x^2\right) \left(\frac{A \sqrt{c}}{\sqrt{a}}+B\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{2 c^{3/4} \sqrt{a+b x^2+c x^4}}-\frac{\sqrt[4]{a} B \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{c^{3/4} \sqrt{a+b x^2+c x^4}}+\frac{B x \sqrt{a+b x^2+c x^4}}{\sqrt{c} \left(\sqrt{a}+\sqrt{c} x^2\right)}",1,"(B*x*Sqrt[a + b*x^2 + c*x^4])/(Sqrt[c]*(Sqrt[a] + Sqrt[c]*x^2)) - (a^(1/4)*B*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(c^(3/4)*Sqrt[a + b*x^2 + c*x^4]) + (a^(1/4)*(B + (A*Sqrt[c])/Sqrt[a])*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*c^(3/4)*Sqrt[a + b*x^2 + c*x^4])","A",3,3,24,0.1250,1,"{1197, 1103, 1195}"
21,1,436,0,0.4085818,"\int \frac{A+B x^2}{\left(d+e x^2\right) \sqrt{a+b x^2+c x^4}} \, dx","Int[(A + B*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]),x]","\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 (B d-A e) \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 e \sqrt{a+b x^2+c x^4} \left(c d^2-a e^2\right)}-\frac{(B d-A 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{e} \sqrt{a e^2-b d e+c d^2}}-\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \left(\sqrt{a} B-A \sqrt{c}\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{2 \sqrt[4]{a} \sqrt[4]{c} \sqrt{a+b x^2+c x^4} \left(\sqrt{c} d-\sqrt{a} e\right)}","\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 (B d-A e) \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 e \sqrt{a+b x^2+c x^4} \left(c d^2-a e^2\right)}-\frac{(B d-A 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{e} \sqrt{a e^2-b d e+c d^2}}-\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \left(\sqrt{a} B-A \sqrt{c}\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{2 \sqrt[4]{a} \sqrt[4]{c} \sqrt{a+b x^2+c x^4} \left(\sqrt{c} d-\sqrt{a} e\right)}",1,"-((B*d - A*e)*ArcTan[(Sqrt[c*d^2 - b*d*e + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + b*x^2 + c*x^4])])/(2*Sqrt[d]*Sqrt[e]*Sqrt[c*d^2 - b*d*e + a*e^2]) - ((Sqrt[a]*B - A*Sqrt[c])*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*a^(1/4)*c^(1/4)*(Sqrt[c]*d - Sqrt[a]*e)*Sqrt[a + b*x^2 + c*x^4]) + (a^(3/4)*((Sqrt[c]*d)/Sqrt[a] + e)^2*(B*d - A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[-(Sqrt[c]*d - Sqrt[a]*e)^2/(4*Sqrt[a]*Sqrt[c]*d*e), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(4*c^(1/4)*d*e*(c*d^2 - a*e^2)*Sqrt[a + b*x^2 + c*x^4])","A",3,3,33,0.09091,1,"{1708, 1103, 1706}"
22,1,782,0,1.466902,"\int \frac{A+B x^2}{\left(d+e x^2\right)^2 \sqrt{a+b x^2+c x^4}} \, dx","Int[(A + B*x^2)/((d + e*x^2)^2*Sqrt[a + b*x^2 + c*x^4]),x]","\frac{\sqrt{c} x \sqrt{a+b x^2+c x^4} (B d-A e)}{2 d \left(\sqrt{a}+\sqrt{c} x^2\right) \left(a e^2-b d e+c d^2\right)}-\frac{e x \sqrt{a+b x^2+c x^4} (B d-A e)}{2 d \left(d+e x^2\right) \left(a e^2-b d e+c d^2\right)}-\frac{\left(B \left(c d^3-a d e^2\right)-A e \left(3 c d^2-e (2 b d-a e)\right)\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} \sqrt{e} \left(a e^2-b d e+c d^2\right)^{3/2}}-\frac{\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}} (B d-A 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)}{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(B \left(c d^3-a d e^2\right)-A e \left(3 c d^2-e (2 b d-a e)\right)\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 e \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{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}} 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)}","\frac{\sqrt{c} x \sqrt{a+b x^2+c x^4} (B d-A e)}{2 d \left(\sqrt{a}+\sqrt{c} x^2\right) \left(a e^2-b d e+c d^2\right)}-\frac{e x \sqrt{a+b x^2+c x^4} (B d-A e)}{2 d \left(d+e x^2\right) \left(a e^2-b d e+c d^2\right)}-\frac{\left(B \left(c d^3-a d e^2\right)-A e \left(3 c d^2-e (2 b d-a e)\right)\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} \sqrt{e} \left(a e^2-b d e+c d^2\right)^{3/2}}-\frac{\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}} (B d-A 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)}{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(B \left(c d^3-a d e^2\right)-A e \left(3 c d^2-e (2 b d-a e)\right)\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 e \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{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}} 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,"(Sqrt[c]*(B*d - A*e)*x*Sqrt[a + b*x^2 + c*x^4])/(2*d*(c*d^2 - b*d*e + a*e^2)*(Sqrt[a] + Sqrt[c]*x^2)) - (e*(B*d - A*e)*x*Sqrt[a + b*x^2 + c*x^4])/(2*d*(c*d^2 - b*d*e + a*e^2)*(d + e*x^2)) - ((B*(c*d^3 - a*d*e^2) - A*e*(3*c*d^2 - e*(2*b*d - a*e)))*ArcTan[(Sqrt[c*d^2 - b*d*e + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + b*x^2 + c*x^4])])/(4*d^(3/2)*Sqrt[e]*(c*d^2 - b*d*e + a*e^2)^(3/2)) - (a^(1/4)*c^(1/4)*(B*d - A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*d*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x^2 + c*x^4]) + (A*c^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*a^(1/4)*d*(Sqrt[c]*d - Sqrt[a]*e)*Sqrt[a + b*x^2 + c*x^4]) + ((Sqrt[c]*d + Sqrt[a]*e)*(B*(c*d^3 - a*d*e^2) - A*e*(3*c*d^2 - e*(2*b*d - a*e)))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[-(Sqrt[c]*d - Sqrt[a]*e)^2/(4*Sqrt[a]*Sqrt[c]*d*e), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(8*a^(1/4)*c^(1/4)*d^2*e*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x^2 + c*x^4])","A",6,6,33,0.1818,1,"{1696, 1714, 1195, 1708, 1103, 1706}"
23,1,1125,0,3.5943101,"\int \frac{A+B x^2}{\left(d+e x^2\right)^3 \sqrt{a+b x^2+c x^4}} \, dx","Int[(A + B*x^2)/((d + e*x^2)^3*Sqrt[a + b*x^2 + c*x^4]),x]","-\frac{\sqrt{c} \left(3 A e \left(3 c d^2-e (2 b d-a e)\right)-B \left(5 c d^3-d e (2 b d+a e)\right)\right) \sqrt{c x^4+b x^2+a} x}{8 d^2 \left(c d^2-b e d+a e^2\right)^2 \left(\sqrt{c} x^2+\sqrt{a}\right)}+\frac{e \left(3 A e \left(3 c d^2-e (2 b d-a e)\right)-B \left(5 c d^3-d e (2 b d+a e)\right)\right) \sqrt{c x^4+b x^2+a} x}{8 d^2 \left(c d^2-b e d+a e^2\right)^2 \left(e x^2+d\right)}-\frac{e (B d-A e) \sqrt{c x^4+b x^2+a} x}{4 d \left(c d^2-b e d+a e^2\right) \left(e x^2+d\right)^2}-\frac{\left(B \left(3 c^2 d^5-10 a c e^2 d^3+a e^3 (4 b d-a e) d\right)-A e \left(15 c^2 d^4-2 c e (10 b d-3 a e) d^2+e^2 \left(8 b^2 d^2-8 a b e d+3 a^2 e^2\right)\right)\right) \tan ^{-1}\left(\frac{\sqrt{c d^2-b e d+a e^2} x}{\sqrt{d} \sqrt{e} \sqrt{c x^4+b x^2+a}}\right)}{16 d^{5/2} \sqrt{e} \left(c d^2-b e d+a e^2\right)^{5/2}}+\frac{\sqrt[4]{a} \sqrt[4]{c} \left(3 A e \left(3 c d^2-e (2 b d-a e)\right)-B \left(5 c d^3-d e (2 b d+a e)\right)\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+b x^2+a}{\left(\sqrt{c} x^2+\sqrt{a}\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)}{8 d^2 \left(c d^2-b e d+a e^2\right)^2 \sqrt{c x^4+b x^2+a}}+\frac{\sqrt[4]{c} \left(\sqrt{a} \sqrt{c} d (B d-A e)+a e (B d+3 A e)+4 A d (c d-b e)\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+b x^2+a}{\left(\sqrt{c} x^2+\sqrt{a}\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)}{8 \sqrt[4]{a} d^2 \left(\sqrt{c} d-\sqrt{a} e\right) \left(c d^2-b e d+a e^2\right) \sqrt{c x^4+b x^2+a}}+\frac{\left(\sqrt{c} d+\sqrt{a} e\right) \left(B \left(3 c^2 d^5-10 a c e^2 d^3+a e^3 (4 b d-a e) d\right)-A e \left(15 c^2 d^4-2 c e (10 b d-3 a e) d^2+e^2 \left(8 b^2 d^2-8 a b e d+3 a^2 e^2\right)\right)\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+b x^2+a}{\left(\sqrt{c} x^2+\sqrt{a}\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)}{32 \sqrt[4]{a} \sqrt[4]{c} d^3 e \left(\sqrt{c} d-\sqrt{a} e\right) \left(c d^2-b e d+a e^2\right)^2 \sqrt{c x^4+b x^2+a}}","-\frac{\sqrt{c} \left(3 A e \left(3 c d^2-e (2 b d-a e)\right)-B d \left(5 c d^2-e (2 b d+a e)\right)\right) \sqrt{c x^4+b x^2+a} x}{8 d^2 \left(c d^2-b e d+a e^2\right)^2 \left(\sqrt{c} x^2+\sqrt{a}\right)}+\frac{e \left(3 A e \left(3 c d^2-e (2 b d-a e)\right)-B d \left(5 c d^2-e (2 b d+a e)\right)\right) \sqrt{c x^4+b x^2+a} x}{8 d^2 \left(c d^2-b e d+a e^2\right)^2 \left(e x^2+d\right)}-\frac{e (B d-A e) \sqrt{c x^4+b x^2+a} x}{4 d \left(c d^2-b e d+a e^2\right) \left(e x^2+d\right)^2}-\frac{\left(B d \left(3 c^2 d^4-10 a c e^2 d^2+a e^3 (4 b d-a e)\right)-A e \left(15 c^2 d^4-2 c e (10 b d-3 a e) d^2+e^2 \left(8 b^2 d^2-8 a b e d+3 a^2 e^2\right)\right)\right) \tan ^{-1}\left(\frac{\sqrt{c d^2-b e d+a e^2} x}{\sqrt{d} \sqrt{e} \sqrt{c x^4+b x^2+a}}\right)}{16 d^{5/2} \sqrt{e} \left(c d^2-b e d+a e^2\right)^{5/2}}+\frac{\sqrt[4]{a} \sqrt[4]{c} \left(3 A e \left(3 c d^2-e (2 b d-a e)\right)-B d \left(5 c d^2-e (2 b d+a e)\right)\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+b x^2+a}{\left(\sqrt{c} x^2+\sqrt{a}\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)}{8 d^2 \left(c d^2-b e d+a e^2\right)^2 \sqrt{c x^4+b x^2+a}}+\frac{\sqrt[4]{c} \left(\sqrt{a} \sqrt{c} d (B d-A e)+a e (B d+3 A e)+4 A d (c d-b e)\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+b x^2+a}{\left(\sqrt{c} x^2+\sqrt{a}\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)}{8 \sqrt[4]{a} d^2 \left(\sqrt{c} d-\sqrt{a} e\right) \left(c d^2-b e d+a e^2\right) \sqrt{c x^4+b x^2+a}}+\frac{\left(\sqrt{c} d+\sqrt{a} e\right) \left(B d \left(3 c^2 d^4-10 a c e^2 d^2+a e^3 (4 b d-a e)\right)-A e \left(15 c^2 d^4-2 c e (10 b d-3 a e) d^2+e^2 \left(8 b^2 d^2-8 a b e d+3 a^2 e^2\right)\right)\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+b x^2+a}{\left(\sqrt{c} x^2+\sqrt{a}\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)}{32 \sqrt[4]{a} \sqrt[4]{c} d^3 e \left(\sqrt{c} d-\sqrt{a} e\right) \left(c d^2-b e d+a e^2\right)^2 \sqrt{c x^4+b x^2+a}}",1,"-(Sqrt[c]*(3*A*e*(3*c*d^2 - e*(2*b*d - a*e)) - B*(5*c*d^3 - d*e*(2*b*d + a*e)))*x*Sqrt[a + b*x^2 + c*x^4])/(8*d^2*(c*d^2 - b*d*e + a*e^2)^2*(Sqrt[a] + Sqrt[c]*x^2)) - (e*(B*d - A*e)*x*Sqrt[a + b*x^2 + c*x^4])/(4*d*(c*d^2 - b*d*e + a*e^2)*(d + e*x^2)^2) + (e*(3*A*e*(3*c*d^2 - e*(2*b*d - a*e)) - B*(5*c*d^3 - d*e*(2*b*d + a*e)))*x*Sqrt[a + b*x^2 + c*x^4])/(8*d^2*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x^2)) - ((B*(3*c^2*d^5 - 10*a*c*d^3*e^2 + a*d*e^3*(4*b*d - a*e)) - A*e*(15*c^2*d^4 - 2*c*d^2*e*(10*b*d - 3*a*e) + e^2*(8*b^2*d^2 - 8*a*b*d*e + 3*a^2*e^2)))*ArcTan[(Sqrt[c*d^2 - b*d*e + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + b*x^2 + c*x^4])])/(16*d^(5/2)*Sqrt[e]*(c*d^2 - b*d*e + a*e^2)^(5/2)) + (a^(1/4)*c^(1/4)*(3*A*e*(3*c*d^2 - e*(2*b*d - a*e)) - B*(5*c*d^3 - d*e*(2*b*d + a*e)))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(8*d^2*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[a + b*x^2 + c*x^4]) + (c^(1/4)*(Sqrt[a]*Sqrt[c]*d*(B*d - A*e) + a*e*(B*d + 3*A*e) + 4*A*d*(c*d - b*e))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(8*a^(1/4)*d^2*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x^2 + c*x^4]) + ((Sqrt[c]*d + Sqrt[a]*e)*(B*(3*c^2*d^5 - 10*a*c*d^3*e^2 + a*d*e^3*(4*b*d - a*e)) - A*e*(15*c^2*d^4 - 2*c*d^2*e*(10*b*d - 3*a*e) + e^2*(8*b^2*d^2 - 8*a*b*d*e + 3*a^2*e^2)))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[-(Sqrt[c]*d - Sqrt[a]*e)^2/(4*Sqrt[a]*Sqrt[c]*d*e), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(32*a^(1/4)*c^(1/4)*d^3*e*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[a + b*x^2 + c*x^4])","A",7,6,33,0.1818,1,"{1696, 1714, 1195, 1708, 1103, 1706}"
24,1,859,0,1.3796346,"\int \frac{\left(A+B x^2\right) \left(d+e x^2\right)^3}{\left(a+b x^2+c x^4\right)^{3/2}} \, dx","Int[((A + B*x^2)*(d + e*x^2)^3)/(a + b*x^2 + c*x^4)^(3/2),x]","\frac{B x \sqrt{c x^4+b x^2+a} e^3}{3 c^2}-\frac{\left(a B \left(6 c^3 d^3-9 c^2 e (b d+6 a e) d-8 b^3 e^3+b c e^2 (18 b d+29 a e)\right)+3 A c \left(2 a b^2 e^3+6 a c \left(c d^2-a e^2\right) e-b c d \left(c d^2+3 a e^2\right)\right)\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+b x^2+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{3 a^{3/4} c^{11/4} \left(b^2-4 a c\right) \sqrt{c x^4+b x^2+a}}-\frac{\left(3 A c^3 d^3-3 \sqrt{a} c^{5/2} (B d+3 A e) d^2-5 a^2 B c e^3+a e (3 c d-2 b e) (3 B c d-4 b B e+3 A c e)+3 a^{3/2} \sqrt{c} e^2 (9 B c d-4 b B e+3 A c e)\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+b x^2+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{6 a^{3/4} \left(b-2 \sqrt{a} \sqrt{c}\right) c^{11/4} \sqrt{c x^4+b x^2+a}}+\frac{\left(a B \left(6 c^3 d^3-9 c^2 e (b d+6 a e) d-8 b^3 e^3+b c e^2 (18 b d+29 a e)\right)+3 A c \left(2 a b^2 e^3+6 a c \left(c d^2-a e^2\right) e-b c d \left(c d^2+3 a e^2\right)\right)\right) x \sqrt{c x^4+b x^2+a}}{3 a c^{5/2} \left(b^2-4 a c\right) \left(\sqrt{c} x^2+\sqrt{a}\right)}+\frac{x \left(-\left(a B (2 c d-b e) \left(c^2 d^2+b^2 e^2-c e (b d+3 a e)\right)+A c \left(a b^2 e^3+2 a c \left(3 c d^2-a e^2\right) e-b c d \left(c d^2+3 a e^2\right)\right)\right) x^2+A c \left(b^2 c d^3-2 a c \left(c d^2-3 a e^2\right) d-a b e \left(3 c d^2+a e^2\right)\right)+a B \left(a b^2 e^3+2 a c \left(3 c d^2-a e^2\right) e-b c d \left(c d^2+3 a e^2\right)\right)\right)}{a c^2 \left(b^2-4 a c\right) \sqrt{c x^4+b x^2+a}}","\frac{B x \sqrt{c x^4+b x^2+a} e^3}{3 c^2}-\frac{\left(a B \left(6 c^3 d^3-9 c^2 e (b d+6 a e) d-8 b^3 e^3+b c e^2 (18 b d+29 a e)\right)+3 A c \left(2 a b^2 e^3+6 a c \left(c d^2-a e^2\right) e-b c d \left(c d^2+3 a e^2\right)\right)\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+b x^2+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{3 a^{3/4} c^{11/4} \left(b^2-4 a c\right) \sqrt{c x^4+b x^2+a}}-\frac{\left(3 A c^3 d^3-3 \sqrt{a} c^{5/2} (B d+3 A e) d^2-5 a^2 B c e^3+a e (3 c d-2 b e) (3 B c d-4 b B e+3 A c e)+3 a^{3/2} \sqrt{c} e^2 (9 B c d-4 b B e+3 A c e)\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+b x^2+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{6 a^{3/4} \left(b-2 \sqrt{a} \sqrt{c}\right) c^{11/4} \sqrt{c x^4+b x^2+a}}+\frac{\left(a B \left(6 c^3 d^3-9 c^2 e (b d+6 a e) d-8 b^3 e^3+b c e^2 (18 b d+29 a e)\right)+3 A c \left(2 a b^2 e^3+6 a c \left(c d^2-a e^2\right) e-b c d \left(c d^2+3 a e^2\right)\right)\right) x \sqrt{c x^4+b x^2+a}}{3 a c^{5/2} \left(b^2-4 a c\right) \left(\sqrt{c} x^2+\sqrt{a}\right)}+\frac{x \left(-\left(a B (2 c d-b e) \left(c^2 d^2+b^2 e^2-c e (b d+3 a e)\right)+A c \left(a b^2 e^3+2 a c \left(3 c d^2-a e^2\right) e-b c d \left(c d^2+3 a e^2\right)\right)\right) x^2+A c \left(b^2 c d^3-2 a c \left(c d^2-3 a e^2\right) d-a b e \left(3 c d^2+a e^2\right)\right)+a B \left(a b^2 e^3+2 a c \left(3 c d^2-a e^2\right) e-b c d \left(c d^2+3 a e^2\right)\right)\right)}{a c^2 \left(b^2-4 a c\right) \sqrt{c x^4+b x^2+a}}",1,"(x*(A*c*(b^2*c*d^3 - 2*a*c*d*(c*d^2 - 3*a*e^2) - a*b*e*(3*c*d^2 + a*e^2)) + a*B*(a*b^2*e^3 + 2*a*c*e*(3*c*d^2 - a*e^2) - b*c*d*(c*d^2 + 3*a*e^2)) - (a*B*(2*c*d - b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e)) + A*c*(a*b^2*e^3 + 2*a*c*e*(3*c*d^2 - a*e^2) - b*c*d*(c*d^2 + 3*a*e^2)))*x^2))/(a*c^2*(b^2 - 4*a*c)*Sqrt[a + b*x^2 + c*x^4]) + (B*e^3*x*Sqrt[a + b*x^2 + c*x^4])/(3*c^2) + ((a*B*(6*c^3*d^3 - 8*b^3*e^3 - 9*c^2*d*e*(b*d + 6*a*e) + b*c*e^2*(18*b*d + 29*a*e)) + 3*A*c*(2*a*b^2*e^3 + 6*a*c*e*(c*d^2 - a*e^2) - b*c*d*(c*d^2 + 3*a*e^2)))*x*Sqrt[a + b*x^2 + c*x^4])/(3*a*c^(5/2)*(b^2 - 4*a*c)*(Sqrt[a] + Sqrt[c]*x^2)) - ((a*B*(6*c^3*d^3 - 8*b^3*e^3 - 9*c^2*d*e*(b*d + 6*a*e) + b*c*e^2*(18*b*d + 29*a*e)) + 3*A*c*(2*a*b^2*e^3 + 6*a*c*e*(c*d^2 - a*e^2) - b*c*d*(c*d^2 + 3*a*e^2)))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(3*a^(3/4)*c^(11/4)*(b^2 - 4*a*c)*Sqrt[a + b*x^2 + c*x^4]) - ((3*A*c^3*d^3 - 5*a^2*B*c*e^3 - 3*Sqrt[a]*c^(5/2)*d^2*(B*d + 3*A*e) + a*e*(3*c*d - 2*b*e)*(3*B*c*d - 4*b*B*e + 3*A*c*e) + 3*a^(3/2)*Sqrt[c]*e^2*(9*B*c*d - 4*b*B*e + 3*A*c*e))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(6*a^(3/4)*(b - 2*Sqrt[a]*Sqrt[c])*c^(11/4)*Sqrt[a + b*x^2 + c*x^4])","A",5,5,33,0.1515,1,"{1678, 1679, 1197, 1103, 1195}"
25,1,633,0,0.5844271,"\int \frac{\left(A+B x^2\right) \left(d+e x^2\right)^2}{\left(a+b x^2+c x^4\right)^{3/2}} \, dx","Int[((A + B*x^2)*(d + e*x^2)^2)/(a + b*x^2 + c*x^4)^(3/2),x]","\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}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) \left(A c \left(a b e^2-4 a c d e+b c d^2\right)-2 a B \left(-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right)\right)}{a^{3/4} c^{7/4} \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}-\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) \left(3 a^{3/2} B \sqrt{c} e^2+a e (A c e-2 b B e+2 B c d)-\sqrt{a} c^{3/2} d (2 A e+B d)+A c^2 d^2\right)}{2 a^{3/4} c^{7/4} \left(b-2 \sqrt{a} \sqrt{c}\right) \sqrt{a+b x^2+c x^4}}-\frac{x \sqrt{a+b x^2+c x^4} \left(A c \left(a b e^2-4 a c d e+b c d^2\right)-2 a B \left(-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right)\right)}{a c^{3/2} \left(b^2-4 a c\right) \left(\sqrt{a}+\sqrt{c} x^2\right)}-\frac{x \left(c \left(\frac{a B \left(a b e^2-4 a c d e+b c d^2\right)}{c}-A \left(-2 a b d e-2 a \left(c d^2-a e^2\right)+b^2 d^2\right)\right)-x^2 \left(A c \left(a b e^2-4 a c d e+b c d^2\right)-a B \left(-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right)\right)\right)}{a c \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}","\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) \left(A c \left(a b e^2-4 a c d e+b c d^2\right)-2 a B \left(-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right)\right)}{a^{3/4} c^{7/4} \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}-\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) \left(3 a^{3/2} B \sqrt{c} e^2+a e (A c e-2 b B e+2 B c d)-\sqrt{a} c^{3/2} d (2 A e+B d)+A c^2 d^2\right)}{2 a^{3/4} c^{7/4} \left(b-2 \sqrt{a} \sqrt{c}\right) \sqrt{a+b x^2+c x^4}}-\frac{x \sqrt{a+b x^2+c x^4} \left(A c \left(a b e^2-4 a c d e+b c d^2\right)-2 a B \left(-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right)\right)}{a c^{3/2} \left(b^2-4 a c\right) \left(\sqrt{a}+\sqrt{c} x^2\right)}-\frac{x \left(x^2 \left(-\left(A c \left(a b e^2-4 a c d e+b c d^2\right)-a B \left(-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right)\right)\right)-A c \left(-2 a b d e-2 a \left(c d^2-a e^2\right)+b^2 d^2\right)+a B \left(a b e^2-4 a c d e+b c d^2\right)\right)}{a c \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}",1,"-((x*(c*((a*B*(b*c*d^2 - 4*a*c*d*e + a*b*e^2))/c - A*(b^2*d^2 - 2*a*b*d*e - 2*a*(c*d^2 - a*e^2))) - (A*c*(b*c*d^2 - 4*a*c*d*e + a*b*e^2) - a*B*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e)))*x^2))/(a*c*(b^2 - 4*a*c)*Sqrt[a + b*x^2 + c*x^4])) - ((A*c*(b*c*d^2 - 4*a*c*d*e + a*b*e^2) - 2*a*B*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e)))*x*Sqrt[a + b*x^2 + c*x^4])/(a*c^(3/2)*(b^2 - 4*a*c)*(Sqrt[a] + Sqrt[c]*x^2)) + ((A*c*(b*c*d^2 - 4*a*c*d*e + a*b*e^2) - 2*a*B*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e)))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(a^(3/4)*c^(7/4)*(b^2 - 4*a*c)*Sqrt[a + b*x^2 + c*x^4]) - ((A*c^2*d^2 + 3*a^(3/2)*B*Sqrt[c]*e^2 - Sqrt[a]*c^(3/2)*d*(B*d + 2*A*e) + a*e*(2*B*c*d - 2*b*B*e + A*c*e))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*a^(3/4)*(b - 2*Sqrt[a]*Sqrt[c])*c^(7/4)*Sqrt[a + b*x^2 + c*x^4])","A",4,4,33,0.1212,1,"{1678, 1197, 1103, 1195}"
26,1,481,0,0.3872615,"\int \frac{\left(A+B x^2\right) \left(d+e x^2\right)}{\left(a+b x^2+c x^4\right)^{3/2}} \, dx","Int[((A + B*x^2)*(d + e*x^2))/(a + b*x^2 + c*x^4)^(3/2),x]","\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}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) (A c (b d-2 a e)-a B (2 c d-b e))}{a^{3/4} c^{3/4} \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}+\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \left(\sqrt{a} B-A \sqrt{c}\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(\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 a^{3/4} c^{3/4} \left(b-2 \sqrt{a} \sqrt{c}\right) \sqrt{a+b x^2+c x^4}}-\frac{x \sqrt{a+b x^2+c x^4} (A c (b d-2 a e)-a B (2 c d-b e))}{a \sqrt{c} \left(b^2-4 a c\right) \left(\sqrt{a}+\sqrt{c} x^2\right)}-\frac{x \left(-A \left(-a b e-2 a c d+b^2 d\right)+x^2 (-(A c (b d-2 a e)-a B (2 c d-b e)))+a B (b d-2 a e)\right)}{a \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}","\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right) (A c (b d-2 a e)-a B (2 c d-b e))}{a^{3/4} c^{3/4} \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}+\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \left(\sqrt{a} B-A \sqrt{c}\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(\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 a^{3/4} c^{3/4} \left(b-2 \sqrt{a} \sqrt{c}\right) \sqrt{a+b x^2+c x^4}}-\frac{x \sqrt{a+b x^2+c x^4} (A c (b d-2 a e)-a B (2 c d-b e))}{a \sqrt{c} \left(b^2-4 a c\right) \left(\sqrt{a}+\sqrt{c} x^2\right)}-\frac{x \left(-A \left(-a b e-2 a c d+b^2 d\right)+x^2 (-(A c (b d-2 a e)-a B (2 c d-b e)))+a B (b d-2 a e)\right)}{a \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}",1,"-((x*(a*B*(b*d - 2*a*e) - A*(b^2*d - 2*a*c*d - a*b*e) - (A*c*(b*d - 2*a*e) - a*B*(2*c*d - b*e))*x^2))/(a*(b^2 - 4*a*c)*Sqrt[a + b*x^2 + c*x^4])) - ((A*c*(b*d - 2*a*e) - a*B*(2*c*d - b*e))*x*Sqrt[a + b*x^2 + c*x^4])/(a*Sqrt[c]*(b^2 - 4*a*c)*(Sqrt[a] + Sqrt[c]*x^2)) + ((A*c*(b*d - 2*a*e) - a*B*(2*c*d - b*e))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(a^(3/4)*c^(3/4)*(b^2 - 4*a*c)*Sqrt[a + b*x^2 + c*x^4]) + ((Sqrt[a]*B - A*Sqrt[c])*(Sqrt[c]*d - Sqrt[a]*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*a^(3/4)*(b - 2*Sqrt[a]*Sqrt[c])*c^(3/4)*Sqrt[a + b*x^2 + c*x^4])","A",4,4,31,0.1290,1,"{1678, 1197, 1103, 1195}"
27,1,398,0,0.1950448,"\int \frac{A+B x^2}{\left(a+b x^2+c x^4\right)^{3/2}} \, dx","Int[(A + B*x^2)/(a + b*x^2 + c*x^4)^(3/2),x]","\frac{\sqrt[4]{c} \left(\sqrt{a}+\sqrt{c} x^2\right) (A b-2 a B) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{a^{3/4} \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}+\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \left(\sqrt{a} B-A \sqrt{c}\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{2 a^{3/4} \sqrt[4]{c} \left(b-2 \sqrt{a} \sqrt{c}\right) \sqrt{a+b x^2+c x^4}}-\frac{\sqrt{c} x (A b-2 a B) \sqrt{a+b x^2+c x^4}}{a \left(b^2-4 a c\right) \left(\sqrt{a}+\sqrt{c} x^2\right)}+\frac{x \left(c x^2 (A b-2 a B)-2 a A c-a b B+A b^2\right)}{a \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}","\frac{\sqrt[4]{c} \left(\sqrt{a}+\sqrt{c} x^2\right) (A b-2 a B) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{a^{3/4} \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}+\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \left(\sqrt{a} B-A \sqrt{c}\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{2 a^{3/4} \sqrt[4]{c} \left(b-2 \sqrt{a} \sqrt{c}\right) \sqrt{a+b x^2+c x^4}}-\frac{\sqrt{c} x (A b-2 a B) \sqrt{a+b x^2+c x^4}}{a \left(b^2-4 a c\right) \left(\sqrt{a}+\sqrt{c} x^2\right)}+\frac{x \left(c x^2 (A b-2 a B)-2 a A c-a b B+A b^2\right)}{a \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}",1,"(x*(A*b^2 - a*b*B - 2*a*A*c + (A*b - 2*a*B)*c*x^2))/(a*(b^2 - 4*a*c)*Sqrt[a + b*x^2 + c*x^4]) - ((A*b - 2*a*B)*Sqrt[c]*x*Sqrt[a + b*x^2 + c*x^4])/(a*(b^2 - 4*a*c)*(Sqrt[a] + Sqrt[c]*x^2)) + ((A*b - 2*a*B)*c^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(a^(3/4)*(b^2 - 4*a*c)*Sqrt[a + b*x^2 + c*x^4]) + ((Sqrt[a]*B - A*Sqrt[c])*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*a^(3/4)*(b - 2*Sqrt[a]*Sqrt[c])*c^(1/4)*Sqrt[a + b*x^2 + c*x^4])","A",4,4,24,0.1667,1,"{1178, 1197, 1103, 1195}"
28,1,1045,0,1.4235582,"\int \frac{A+B x^2}{\left(d+e x^2\right) \left(a+b x^2+c x^4\right)^{3/2}} \, dx","Int[(A + B*x^2)/((d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2)),x]","\frac{a^{3/4} e (B d-A e) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+b x^2+a}{\left(\sqrt{c} x^2+\sqrt{a}\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) \left(\frac{\sqrt{c} d}{\sqrt{a}}+e\right)^2}{4 \sqrt[4]{c} d \left(c d^2-a e^2\right) \left(c d^2-b e d+a e^2\right) \sqrt{c x^4+b x^2+a}}-\frac{e^{3/2} (B d-A e) \tan ^{-1}\left(\frac{\sqrt{c d^2-b e d+a e^2} x}{\sqrt{d} \sqrt{e} \sqrt{c x^4+b x^2+a}}\right)}{2 \sqrt{d} \left(c d^2-b e d+a e^2\right)^{3/2}}-\frac{\sqrt[4]{c} \left(a B (2 c d-b e)-A \left(-e b^2+c d b+2 a c e\right)\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+b x^2+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{a^{3/4} \left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right) \sqrt{c x^4+b x^2+a}}-\frac{\sqrt[4]{c} \left(a B e-\sqrt{a} \sqrt{c} (B d-A e)+A (c d-b e)\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+b x^2+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{2 a^{3/4} \left(b-2 \sqrt{a} \sqrt{c}\right) \left(c d^2-b e d+a e^2\right) \sqrt{c x^4+b x^2+a}}-\frac{\sqrt[4]{c} e (B d-A e) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+b x^2+a}{\left(\sqrt{c} x^2+\sqrt{a}\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} \left(\sqrt{c} d-\sqrt{a} e\right) \left(c d^2-b e d+a e^2\right) \sqrt{c x^4+b x^2+a}}+\frac{\sqrt{c} \left(a B (2 c d-b e)-A \left(-e b^2+c d b+2 a c e\right)\right) x \sqrt{c x^4+b x^2+a}}{a \left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right) \left(\sqrt{c} x^2+\sqrt{a}\right)}-\frac{x \left(c \left(a B (2 c d-b e)-A \left(-e b^2+c d b+2 a c e\right)\right) x^2+a b c (B d-A e)-\left(b^2-2 a c\right) (A c d-A b e+a B e)\right)}{a \left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right) \sqrt{c x^4+b x^2+a}}","\frac{a^{3/4} e (B d-A e) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+b x^2+a}{\left(\sqrt{c} x^2+\sqrt{a}\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) \left(\frac{\sqrt{c} d}{\sqrt{a}}+e\right)^2}{4 \sqrt[4]{c} d \left(c d^2-a e^2\right) \left(c d^2-b e d+a e^2\right) \sqrt{c x^4+b x^2+a}}-\frac{e^{3/2} (B d-A e) \tan ^{-1}\left(\frac{\sqrt{c d^2-b e d+a e^2} x}{\sqrt{d} \sqrt{e} \sqrt{c x^4+b x^2+a}}\right)}{2 \sqrt{d} \left(c d^2-b e d+a e^2\right)^{3/2}}-\frac{\sqrt[4]{c} \left(a B (2 c d-b e)-A \left(-e b^2+c d b+2 a c e\right)\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+b x^2+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{a^{3/4} \left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right) \sqrt{c x^4+b x^2+a}}+\frac{\left(\sqrt{a} B-A \sqrt{c}\right) \sqrt[4]{c} \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+b x^2+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}-\frac{b}{4 \sqrt{a} \sqrt{c}}\right)}{2 a^{3/4} \left(b-2 \sqrt{a} \sqrt{c}\right) \left(\sqrt{c} d-\sqrt{a} e\right) \sqrt{c x^4+b x^2+a}}+\frac{\sqrt{c} \left(a B (2 c d-b e)-A \left(-e b^2+c d b+2 a c e\right)\right) x \sqrt{c x^4+b x^2+a}}{a \left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right) \left(\sqrt{c} x^2+\sqrt{a}\right)}-\frac{x \left(c \left(a B (2 c d-b e)-A \left(-e b^2+c d b+2 a c e\right)\right) x^2+a b c (B d-A e)-\left(b^2-2 a c\right) (A c d-A b e+a B e)\right)}{a \left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right) \sqrt{c x^4+b x^2+a}}",1,"-((x*(a*b*c*(B*d - A*e) - (b^2 - 2*a*c)*(A*c*d - A*b*e + a*B*e) + c*(a*B*(2*c*d - b*e) - A*(b*c*d - b^2*e + 2*a*c*e))*x^2))/(a*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x^2 + c*x^4])) + (Sqrt[c]*(a*B*(2*c*d - b*e) - A*(b*c*d - b^2*e + 2*a*c*e))*x*Sqrt[a + b*x^2 + c*x^4])/(a*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(Sqrt[a] + Sqrt[c]*x^2)) - (e^(3/2)*(B*d - A*e)*ArcTan[(Sqrt[c*d^2 - b*d*e + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + b*x^2 + c*x^4])])/(2*Sqrt[d]*(c*d^2 - b*d*e + a*e^2)^(3/2)) - (c^(1/4)*(a*B*(2*c*d - b*e) - A*(b*c*d - b^2*e + 2*a*c*e))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(a^(3/4)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x^2 + c*x^4]) - (c^(1/4)*e*(B*d - A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*a^(1/4)*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x^2 + c*x^4]) - (c^(1/4)*(a*B*e - Sqrt[a]*Sqrt[c]*(B*d - A*e) + A*(c*d - b*e))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*a^(3/4)*(b - 2*Sqrt[a]*Sqrt[c])*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x^2 + c*x^4]) + (a^(3/4)*e*((Sqrt[c]*d)/Sqrt[a] + e)^2*(B*d - A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[-(Sqrt[c]*d - Sqrt[a]*e)^2/(4*Sqrt[a]*Sqrt[c]*d*e), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(4*c^(1/4)*d*(c*d^2 - a*e^2)*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x^2 + c*x^4])","A",9,7,33,0.2121,1,"{1720, 1178, 1197, 1103, 1195, 1216, 1706}"
29,1,2112,0,3.3862112,"\int \frac{A+B x^2}{\left(d+e x^2\right)^2 \left(a+b x^2+c x^4\right)^{3/2}} \, dx","Int[(A + B*x^2)/((d + e*x^2)^2*(a + b*x^2 + c*x^4)^(3/2)),x]","-\frac{(B d-A e) x \sqrt{c x^4+b x^2+a} e^3}{2 d \left(c d^2-b e d+a e^2\right)^2 \left(e x^2+d\right)}-\frac{\sqrt[4]{a} \sqrt[4]{c} (B d-A e) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+b x^2+a}{\left(\sqrt{c} x^2+\sqrt{a}\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) e^2}{2 d \left(c d^2-b e d+a e^2\right)^2 \sqrt{c x^4+b x^2+a}}+\frac{\sqrt{c} (B d-A e) x \sqrt{c x^4+b x^2+a} e^2}{2 d \left(c d^2-b e d+a e^2\right)^2 \left(\sqrt{c} x^2+\sqrt{a}\right)}-\frac{(B d-A e) \left(3 c d^2-e (2 b d-a e)\right) \tan ^{-1}\left(\frac{\sqrt{c d^2-b e d+a e^2} x}{\sqrt{d} \sqrt{e} \sqrt{c x^4+b x^2+a}}\right) e^{3/2}}{4 d^{3/2} \left(c d^2-b e d+a e^2\right)^{5/2}}+\frac{\left(A e (2 c d-b e)-B \left(c d^2-a e^2\right)\right) \tan ^{-1}\left(\frac{\sqrt{c d^2-b e d+a e^2} x}{\sqrt{d} \sqrt{e} \sqrt{c x^4+b x^2+a}}\right) e^{3/2}}{2 \sqrt{d} \left(c d^2-b e d+a e^2\right)^{5/2}}+\frac{\sqrt[4]{c} \left(A e (2 c d-b e)-B \left(c d^2-a e^2\right)\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+b x^2+a}{\left(\sqrt{c} x^2+\sqrt{a}\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) e}{2 \sqrt[4]{a} \left(\sqrt{c} d-\sqrt{a} e\right) \left(c d^2-b e d+a e^2\right)^2 \sqrt{c x^4+b x^2+a}}-\frac{\sqrt[4]{c} (B d-A e) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+b x^2+a}{\left(\sqrt{c} x^2+\sqrt{a}\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) e}{2 \sqrt[4]{a} d \left(\sqrt{c} d-\sqrt{a} e\right) \left(c d^2-b e d+a e^2\right) \sqrt{c x^4+b x^2+a}}+\frac{\left(\sqrt{c} d+\sqrt{a} e\right) (B d-A e) \left(3 c d^2-e (2 b d-a e)\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+b x^2+a}{\left(\sqrt{c} x^2+\sqrt{a}\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) e}{8 \sqrt[4]{a} \sqrt[4]{c} d^2 \left(\sqrt{c} d-\sqrt{a} e\right) \left(c d^2-b e d+a e^2\right)^2 \sqrt{c x^4+b x^2+a}}-\frac{a^{3/4} \left(\frac{\sqrt{c} d}{\sqrt{a}}+e\right)^2 \left(A e (2 c d-b e)-B \left(c d^2-a e^2\right)\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+b x^2+a}{\left(\sqrt{c} x^2+\sqrt{a}\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) e}{4 \sqrt[4]{c} d \left(c d^2-a e^2\right) \left(c d^2-b e d+a e^2\right)^2 \sqrt{c x^4+b x^2+a}}-\frac{\sqrt[4]{c} \left(a B \left(2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right)+A \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)\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+b x^2+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{a^{3/4} \left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right)^2 \sqrt{c x^4+b x^2+a}}-\frac{\sqrt[4]{c} \left(a^{3/2} B \sqrt{c} e^2+a (2 B c d-b B e-A c e) e+A (c d-b e)^2-\sqrt{a} \sqrt{c} \left(B c d^2-A e (2 c d-b e)\right)\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+b x^2+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{2 a^{3/4} \left(b-2 \sqrt{a} \sqrt{c}\right) \left(c d^2-b e d+a e^2\right)^2 \sqrt{c x^4+b x^2+a}}+\frac{\sqrt{c} \left(a B \left(2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right)+A \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)\right) x \sqrt{c x^4+b x^2+a}}{a \left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right)^2 \left(\sqrt{c} x^2+\sqrt{a}\right)}+\frac{x \left(-c \left(a B \left(2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right)+A \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)\right) x^2+a b c \left(A e (2 c d-b e)-B \left(c d^2-a e^2\right)\right)+\left(b^2-2 a c\right) \left(a B e (2 c d-b e)+A \left(c^2 d^2+b^2 e^2-c e (2 b d+a e)\right)\right)\right)}{a \left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right)^2 \sqrt{c x^4+b x^2+a}}","-\frac{(B d-A e) x \sqrt{c x^4+b x^2+a} e^3}{2 d \left(c d^2-b e d+a e^2\right)^2 \left(e x^2+d\right)}+\frac{\left(A e \left(7 c d^2-e (4 b d-a e)\right)-B d \left(5 c d^2-e (2 b d+a e)\right)\right) \tan ^{-1}\left(\frac{\sqrt{c d^2-b e d+a e^2} x}{\sqrt{d} \sqrt{e} \sqrt{c x^4+b x^2+a}}\right) e^{3/2}}{4 d^{3/2} \left(c d^2-b e d+a e^2\right)^{5/2}}-\frac{\left(\sqrt{c} d+\sqrt{a} e\right) \left(A e \left(7 c d^2-e (4 b d-a e)\right)-B d \left(5 c d^2-e (2 b d+a e)\right)\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+b x^2+a}{\left(\sqrt{c} x^2+\sqrt{a}\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) e}{8 \sqrt[4]{a} \sqrt[4]{c} d^2 \left(\sqrt{c} d-\sqrt{a} e\right) \left(c d^2-b e d+a e^2\right)^2 \sqrt{c x^4+b x^2+a}}-\frac{\sqrt[4]{c} \left(a B d \left(4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right)-A \left(2 d e^2 b^3+\left(a e^3-4 c d^2 e\right) b^2+2 c d \left(c d^2-3 a e^2\right) b-4 a c e \left(a e^2-2 c d^2\right)\right)\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+b x^2+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}-\frac{b}{4 \sqrt{a} \sqrt{c}}\right)}{2 a^{3/4} \left(b^2-4 a c\right) d \left(c d^2+e (a e-b d)\right)^2 \sqrt{c x^4+b x^2+a}}+\frac{\sqrt[4]{c} \left(a \sqrt{c} e (B d-2 A e)+\sqrt{a} (B d-A e) (c d-b e)+A \sqrt{c} d (b e-c d)\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+b x^2+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}-\frac{b}{4 \sqrt{a} \sqrt{c}}\right)}{2 a^{3/4} \left(b-2 \sqrt{a} \sqrt{c}\right) d \left(\sqrt{a} e-\sqrt{c} d\right) \left(e (b d-a e)-c d^2\right) \sqrt{c x^4+b x^2+a}}+\frac{\sqrt{c} \left(a B d \left(-4 c^2 d^2-3 b^2 e^2+4 c e (b d+2 a e)\right)+A \left(2 d e^2 b^3+\left(a e^3-4 c d^2 e\right) b^2+2 c d \left(c d^2-3 a e^2\right) b-4 a c e \left(a e^2-2 c d^2\right)\right)\right) x \sqrt{c x^4+b x^2+a}}{2 a \left(4 a c-b^2\right) d \left(c d^2+e (a e-b d)\right)^2 \left(\sqrt{c} x^2+\sqrt{a}\right)}+\frac{x \left(-c \left(a B \left(2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right)+A \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)\right) x^2+a b c \left(A e (2 c d-b e)-B \left(c d^2-a e^2\right)\right)+\left(b^2-2 a c\right) \left(a B e (2 c d-b e)+A \left(c^2 d^2+b^2 e^2-c e (2 b d+a e)\right)\right)\right)}{a \left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right)^2 \sqrt{c x^4+b x^2+a}}",1,"(x*(a*b*c*(A*e*(2*c*d - b*e) - B*(c*d^2 - a*e^2)) + (b^2 - 2*a*c)*(a*B*e*(2*c*d - b*e) + A*(c^2*d^2 + b^2*e^2 - c*e*(2*b*d + a*e))) - c*(a*B*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e)) + A*(2*b^2*c*d*e - 4*a*c^2*d*e - b^3*e^2 - b*c*(c*d^2 - 3*a*e^2)))*x^2))/(a*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[a + b*x^2 + c*x^4]) + (Sqrt[c]*e^2*(B*d - A*e)*x*Sqrt[a + b*x^2 + c*x^4])/(2*d*(c*d^2 - b*d*e + a*e^2)^2*(Sqrt[a] + Sqrt[c]*x^2)) + (Sqrt[c]*(a*B*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e)) + A*(2*b^2*c*d*e - 4*a*c^2*d*e - b^3*e^2 - b*c*(c*d^2 - 3*a*e^2)))*x*Sqrt[a + b*x^2 + c*x^4])/(a*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*(Sqrt[a] + Sqrt[c]*x^2)) - (e^3*(B*d - A*e)*x*Sqrt[a + b*x^2 + c*x^4])/(2*d*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x^2)) - (e^(3/2)*(B*d - A*e)*(3*c*d^2 - e*(2*b*d - a*e))*ArcTan[(Sqrt[c*d^2 - b*d*e + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + b*x^2 + c*x^4])])/(4*d^(3/2)*(c*d^2 - b*d*e + a*e^2)^(5/2)) + (e^(3/2)*(A*e*(2*c*d - b*e) - B*(c*d^2 - a*e^2))*ArcTan[(Sqrt[c*d^2 - b*d*e + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + b*x^2 + c*x^4])])/(2*Sqrt[d]*(c*d^2 - b*d*e + a*e^2)^(5/2)) - (a^(1/4)*c^(1/4)*e^2*(B*d - A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*d*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[a + b*x^2 + c*x^4]) - (c^(1/4)*(a*B*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e)) + A*(2*b^2*c*d*e - 4*a*c^2*d*e - b^3*e^2 - b*c*(c*d^2 - 3*a*e^2)))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(a^(3/4)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[a + b*x^2 + c*x^4]) - (c^(1/4)*e*(B*d - A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*a^(1/4)*d*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x^2 + c*x^4]) + (c^(1/4)*e*(A*e*(2*c*d - b*e) - B*(c*d^2 - a*e^2))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*a^(1/4)*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[a + b*x^2 + c*x^4]) - (c^(1/4)*(a^(3/2)*B*Sqrt[c]*e^2 + A*(c*d - b*e)^2 + a*e*(2*B*c*d - b*B*e - A*c*e) - Sqrt[a]*Sqrt[c]*(B*c*d^2 - A*e*(2*c*d - b*e)))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*a^(3/4)*(b - 2*Sqrt[a]*Sqrt[c])*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[a + b*x^2 + c*x^4]) + (e*(Sqrt[c]*d + Sqrt[a]*e)*(B*d - A*e)*(3*c*d^2 - e*(2*b*d - a*e))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[-(Sqrt[c]*d - Sqrt[a]*e)^2/(4*Sqrt[a]*Sqrt[c]*d*e), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(8*a^(1/4)*c^(1/4)*d^2*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[a + b*x^2 + c*x^4]) - (a^(3/4)*e*((Sqrt[c]*d)/Sqrt[a] + e)^2*(A*e*(2*c*d - b*e) - B*(c*d^2 - a*e^2))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[-(Sqrt[c]*d - Sqrt[a]*e)^2/(4*Sqrt[a]*Sqrt[c]*d*e), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(4*c^(1/4)*d*(c*d^2 - a*e^2)*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[a + b*x^2 + c*x^4])","A",15,10,33,0.3030,1,"{1720, 1178, 1197, 1103, 1195, 1223, 1714, 1708, 1706, 1216}"
30,1,273,0,0.1356835,"\int \frac{\sqrt{a}+\sqrt{c} x^2}{\left(d+e x^2\right) \sqrt{a+b x^2+c x^4}} \, dx","Int[(Sqrt[a] + Sqrt[c]*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]),x]","\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) \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]{a} \sqrt[4]{c} d e \sqrt{a+b x^2+c x^4}}-\frac{\left(\sqrt{c} d-\sqrt{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)}{2 \sqrt{d} \sqrt{e} \sqrt{a e^2-b d e+c d^2}}","\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) \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]{a} \sqrt[4]{c} d e \sqrt{a+b x^2+c x^4}}-\frac{\left(\sqrt{c} d-\sqrt{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)}{2 \sqrt{d} \sqrt{e} \sqrt{a e^2-b d e+c d^2}}",1,"-((Sqrt[c]*d - Sqrt[a]*e)*ArcTan[(Sqrt[c*d^2 - b*d*e + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + b*x^2 + c*x^4])])/(2*Sqrt[d]*Sqrt[e]*Sqrt[c*d^2 - b*d*e + a*e^2]) + ((Sqrt[c]*d + Sqrt[a]*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[-(Sqrt[c]*d - Sqrt[a]*e)^2/(4*Sqrt[a]*Sqrt[c]*d*e), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(4*a^(1/4)*c^(1/4)*d*e*Sqrt[a + b*x^2 + c*x^4])","A",1,1,41,0.02439,1,"{1706}"
31,1,271,0,0.1869937,"\int \frac{1+\sqrt{\frac{c}{a}} x^2}{\left(d+e x^2\right) \sqrt{a+b x^2+c x^4}} \, dx","Int[(1 + Sqrt[c/a]*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]),x]","\frac{\left(x^2 \sqrt{\frac{c}{a}}+1\right) \sqrt{\frac{a+b x^2+c x^4}{a \left(x^2 \sqrt{\frac{c}{a}}+1\right)^2}} \left(d \sqrt{\frac{c}{a}}+e\right) \Pi \left(-\frac{\left(\sqrt{\frac{c}{a}} d-e\right)^2}{4 \sqrt{\frac{c}{a}} d e};2 \tan ^{-1}\left(\sqrt[4]{\frac{c}{a}} x\right)|\frac{1}{4} \left(2-\frac{b \sqrt{\frac{c}{a}}}{c}\right)\right)}{4 d e \sqrt[4]{\frac{c}{a}} \sqrt{a+b x^2+c x^4}}-\frac{\left(d \sqrt{\frac{c}{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)}{2 \sqrt{d} \sqrt{e} \sqrt{a e^2-b d e+c d^2}}","\frac{\left(x^2 \sqrt{\frac{c}{a}}+1\right) \sqrt{\frac{a+b x^2+c x^4}{a \left(x^2 \sqrt{\frac{c}{a}}+1\right)^2}} \left(d \sqrt{\frac{c}{a}}+e\right) \Pi \left(-\frac{\left(\sqrt{\frac{c}{a}} d-e\right)^2}{4 \sqrt{\frac{c}{a}} d e};2 \tan ^{-1}\left(\sqrt[4]{\frac{c}{a}} x\right)|\frac{1}{4} \left(2-\frac{b \sqrt{\frac{c}{a}}}{c}\right)\right)}{4 d e \sqrt[4]{\frac{c}{a}} \sqrt{a+b x^2+c x^4}}-\frac{\left(d \sqrt{\frac{c}{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)}{2 \sqrt{d} \sqrt{e} \sqrt{a e^2-b d e+c d^2}}",1,"-((Sqrt[c/a]*d - e)*ArcTan[(Sqrt[c*d^2 - b*d*e + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + b*x^2 + c*x^4])])/(2*Sqrt[d]*Sqrt[e]*Sqrt[c*d^2 - b*d*e + a*e^2]) + ((Sqrt[c/a]*d + e)*(1 + Sqrt[c/a]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + Sqrt[c/a]*x^2)^2)]*EllipticPi[-(Sqrt[c/a]*d - e)^2/(4*Sqrt[c/a]*d*e), 2*ArcTan[(c/a)^(1/4)*x], (2 - (b*Sqrt[c/a])/c)/4])/(4*(c/a)^(1/4)*d*e*Sqrt[a + b*x^2 + c*x^4])","A",1,1,41,0.02439,1,"{1706}"
32,1,106,0,0.1053433,"\int \frac{946+315 x^2}{\left(7+5 x^2\right) \sqrt{2+3 x^2+x^4}} \, dx","Int[(946 + 315*x^2)/((7 + 5*x^2)*Sqrt[2 + 3*x^2 + x^4]),x]","\frac{631 \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{2525 \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}}","\frac{631 \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{2525 \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,"(631*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/2])/(2*Sqrt[2]*Sqrt[2 + 3*x^2 + x^4]) - (2525*(2 + x^2)*EllipticPi[2/7, ArcTan[x], 1/2])/(14*Sqrt[2]*Sqrt[(2 + x^2)/(1 + x^2)]*Sqrt[2 + 3*x^2 + x^4])","A",4,4,31,0.1290,1,"{1704, 1099, 1456, 539}"
33,1,218,0,0.4634491,"\int \frac{\left(A+B x^2\right) \left(d+e x^2\right)^q}{a+b x^2+c x^4} \, dx","Int[((A + B*x^2)*(d + e*x^2)^q)/(a + b*x^2 + c*x^4),x]","\frac{x \left(d+e x^2\right)^q \left(\frac{e x^2}{d}+1\right)^{-q} \left(B-\frac{b B-2 A c}{\sqrt{b^2-4 a c}}\right) F_1\left(\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right)}{b-\sqrt{b^2-4 a c}}+\frac{x \left(d+e x^2\right)^q \left(\frac{e x^2}{d}+1\right)^{-q} \left(\frac{b B-2 A c}{\sqrt{b^2-4 a c}}+B\right) F_1\left(\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right)}{\sqrt{b^2-4 a c}+b}","\frac{x \left(d+e x^2\right)^q \left(\frac{e x^2}{d}+1\right)^{-q} \left(B-\frac{b B-2 A c}{\sqrt{b^2-4 a c}}\right) F_1\left(\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right)}{b-\sqrt{b^2-4 a c}}+\frac{x \left(d+e x^2\right)^q \left(\frac{e x^2}{d}+1\right)^{-q} \left(\frac{b B-2 A c}{\sqrt{b^2-4 a c}}+B\right) F_1\left(\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right)}{\sqrt{b^2-4 a c}+b}",1,"((B - (b*B - 2*A*c)/Sqrt[b^2 - 4*a*c])*x*(d + e*x^2)^q*AppellF1[1/2, 1, -q, 3/2, (-2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]), -((e*x^2)/d)])/((b - Sqrt[b^2 - 4*a*c])*(1 + (e*x^2)/d)^q) + ((B + (b*B - 2*A*c)/Sqrt[b^2 - 4*a*c])*x*(d + e*x^2)^q*AppellF1[1/2, 1, -q, 3/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), -((e*x^2)/d)])/((b + Sqrt[b^2 - 4*a*c])*(1 + (e*x^2)/d)^q)","A",6,3,31,0.09677,1,"{1692, 430, 429}"
34,1,106,0,0.1285215,"\int \frac{x \left(1+2 x^2\right)}{\sqrt{1+x^2} \left(1+x^2+x^4\right)} \, dx","Int[(x*(1 + 2*x^2))/(Sqrt[1 + x^2]*(1 + x^2 + x^4)),x]","\frac{1}{4} \sqrt{3} \log \left(x^2-\sqrt{3} \sqrt{x^2+1}+2\right)-\frac{1}{4} \sqrt{3} \log \left(x^2+\sqrt{3} \sqrt{x^2+1}+2\right)-\frac{1}{2} \tan ^{-1}\left(\sqrt{3}-2 \sqrt{x^2+1}\right)+\frac{1}{2} \tan ^{-1}\left(2 \sqrt{x^2+1}+\sqrt{3}\right)","\frac{1}{4} \sqrt{3} \log \left(x^2-\sqrt{3} \sqrt{x^2+1}+2\right)-\frac{1}{4} \sqrt{3} \log \left(x^2+\sqrt{3} \sqrt{x^2+1}+2\right)-\frac{1}{2} \tan ^{-1}\left(\sqrt{3}-2 \sqrt{x^2+1}\right)+\frac{1}{2} \tan ^{-1}\left(2 \sqrt{x^2+1}+\sqrt{3}\right)",1,"-ArcTan[Sqrt[3] - 2*Sqrt[1 + x^2]]/2 + ArcTan[Sqrt[3] + 2*Sqrt[1 + x^2]]/2 + (Sqrt[3]*Log[2 + x^2 - Sqrt[3]*Sqrt[1 + x^2]])/4 - (Sqrt[3]*Log[2 + x^2 + Sqrt[3]*Sqrt[1 + x^2]])/4","A",11,7,28,0.2500,1,"{1685, 826, 1169, 634, 618, 204, 628}"
35,1,145,0,0.2074191,"\int \frac{\sqrt{a+b x^2+c x^4}}{a d-c d x^4} \, dx","Int[Sqrt[a + b*x^2 + c*x^4]/(a*d - c*d*x^4),x]","\frac{\sqrt{2 \sqrt{a} \sqrt{c}+b} \tanh ^{-1}\left(\frac{x \sqrt{2 \sqrt{a} \sqrt{c}+b}}{\sqrt{a+b x^2+c x^4}}\right)}{4 \sqrt{a} \sqrt{c} d}-\frac{\sqrt{b-2 \sqrt{a} \sqrt{c}} \tanh ^{-1}\left(\frac{x \sqrt{b-2 \sqrt{a} \sqrt{c}}}{\sqrt{a+b x^2+c x^4}}\right)}{4 \sqrt{a} \sqrt{c} d}","\frac{\sqrt{2 \sqrt{a} \sqrt{c}+b} \tanh ^{-1}\left(\frac{x \sqrt{2 \sqrt{a} \sqrt{c}+b}}{\sqrt{a+b x^2+c x^4}}\right)}{4 \sqrt{a} \sqrt{c} d}-\frac{\sqrt{b-2 \sqrt{a} \sqrt{c}} \tanh ^{-1}\left(\frac{x \sqrt{b-2 \sqrt{a} \sqrt{c}}}{\sqrt{a+b x^2+c x^4}}\right)}{4 \sqrt{a} \sqrt{c} d}",1,"-(Sqrt[b - 2*Sqrt[a]*Sqrt[c]]*ArcTanh[(Sqrt[b - 2*Sqrt[a]*Sqrt[c]]*x)/Sqrt[a + b*x^2 + c*x^4]])/(4*Sqrt[a]*Sqrt[c]*d) + (Sqrt[b + 2*Sqrt[a]*Sqrt[c]]*ArcTanh[(Sqrt[b + 2*Sqrt[a]*Sqrt[c]]*x)/Sqrt[a + b*x^2 + c*x^4]])/(4*Sqrt[a]*Sqrt[c]*d)","A",4,3,30,0.1000,1,"{2071, 1093, 208}"
36,1,239,0,0.1843687,"\int \frac{\sqrt{a+b x^2-c x^4}}{a d+c d x^4} \, dx","Int[Sqrt[a + b*x^2 - c*x^4]/(a*d + c*d*x^4),x]","\frac{\sqrt{\sqrt{4 a c+b^2}-b} \tanh ^{-1}\left(\frac{x \sqrt{\sqrt{4 a c+b^2}-b} \left(\sqrt{4 a c+b^2}+b-2 c x^2\right)}{2 \sqrt{2} \sqrt{a} \sqrt{c} \sqrt{a+b x^2-c x^4}}\right)}{2 \sqrt{2} \sqrt{a} \sqrt{c} d}-\frac{\sqrt{\sqrt{4 a c+b^2}+b} \tan ^{-1}\left(\frac{x \sqrt{\sqrt{4 a c+b^2}+b} \left(-\sqrt{4 a c+b^2}+b-2 c x^2\right)}{2 \sqrt{2} \sqrt{a} \sqrt{c} \sqrt{a+b x^2-c x^4}}\right)}{2 \sqrt{2} \sqrt{a} \sqrt{c} d}","\frac{\sqrt{\sqrt{4 a c+b^2}-b} \tanh ^{-1}\left(\frac{x \sqrt{\sqrt{4 a c+b^2}-b} \left(\sqrt{4 a c+b^2}+b-2 c x^2\right)}{2 \sqrt{2} \sqrt{a} \sqrt{c} \sqrt{a+b x^2-c x^4}}\right)}{2 \sqrt{2} \sqrt{a} \sqrt{c} d}-\frac{\sqrt{\sqrt{4 a c+b^2}+b} \tan ^{-1}\left(\frac{x \sqrt{\sqrt{4 a c+b^2}+b} \left(-\sqrt{4 a c+b^2}+b-2 c x^2\right)}{2 \sqrt{2} \sqrt{a} \sqrt{c} \sqrt{a+b x^2-c x^4}}\right)}{2 \sqrt{2} \sqrt{a} \sqrt{c} d}",1,"-(Sqrt[b + Sqrt[b^2 + 4*a*c]]*ArcTan[(Sqrt[b + Sqrt[b^2 + 4*a*c]]*x*(b - Sqrt[b^2 + 4*a*c] - 2*c*x^2))/(2*Sqrt[2]*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x^2 - c*x^4])])/(2*Sqrt[2]*Sqrt[a]*Sqrt[c]*d) + (Sqrt[-b + Sqrt[b^2 + 4*a*c]]*ArcTanh[(Sqrt[-b + Sqrt[b^2 + 4*a*c]]*x*(b + Sqrt[b^2 + 4*a*c] - 2*c*x^2))/(2*Sqrt[2]*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x^2 - c*x^4])])/(2*Sqrt[2]*Sqrt[a]*Sqrt[c]*d)","A",1,1,30,0.03333,1,"{2072}"
37,1,309,0,0.6252238,"\int x \sqrt{c+e x+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4} \, dx","Int[x*Sqrt[c + e*x + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]","-\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left(c+d x^2+e x\right)^{3/2} \left(-80 a d^2+32 b c d+42 b d e x-35 b e^2\right)}{240 d^3 \left(a+b x^2\right)}+\frac{e \sqrt{a^2+2 a b x^2+b^2 x^4} (2 d x+e) \sqrt{c+d x^2+e x} \left(-16 a d^2+12 b c d-7 b e^2\right)}{128 d^4 \left(a+b x^2\right)}+\frac{e \sqrt{a^2+2 a b x^2+b^2 x^4} \left(4 c d-e^2\right) \left(-16 a d^2+12 b c d-7 b e^2\right) \tanh ^{-1}\left(\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right)}{256 d^{9/2} \left(a+b x^2\right)}+\frac{b x^2 \sqrt{a^2+2 a b x^2+b^2 x^4} \left(c+d x^2+e x\right)^{3/2}}{5 d \left(a+b x^2\right)}","-\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left(c+d x^2+e x\right)^{3/2} \left(-80 a d^2+32 b c d+42 b d e x-35 b e^2\right)}{240 d^3 \left(a+b x^2\right)}+\frac{e \sqrt{a^2+2 a b x^2+b^2 x^4} (2 d x+e) \sqrt{c+d x^2+e x} \left(-16 a d^2+12 b c d-7 b e^2\right)}{128 d^4 \left(a+b x^2\right)}+\frac{e \sqrt{a^2+2 a b x^2+b^2 x^4} \left(4 c d-e^2\right) \left(-16 a d^2+12 b c d-7 b e^2\right) \tanh ^{-1}\left(\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right)}{256 d^{9/2} \left(a+b x^2\right)}+\frac{b x^2 \sqrt{a^2+2 a b x^2+b^2 x^4} \left(c+d x^2+e x\right)^{3/2}}{5 d \left(a+b x^2\right)}",1,"(e*(12*b*c*d - 16*a*d^2 - 7*b*e^2)*(e + 2*d*x)*Sqrt[c + e*x + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(128*d^4*(a + b*x^2)) + (b*x^2*(c + e*x + d*x^2)^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(5*d*(a + b*x^2)) - ((32*b*c*d - 80*a*d^2 - 35*b*e^2 + 42*b*d*e*x)*(c + e*x + d*x^2)^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(240*d^3*(a + b*x^2)) + (e*(4*c*d - e^2)*(12*b*c*d - 16*a*d^2 - 7*b*e^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]*Sqrt[c + e*x + d*x^2])])/(256*d^(9/2)*(a + b*x^2))","A",6,6,38,0.1579,1,"{6744, 1653, 779, 612, 621, 206}"
38,1,283,0,0.3405348,"\int \sqrt{c+e x+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4} \, dx","Int[Sqrt[c + e*x + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]","-\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} (2 d x+e) \sqrt{c+d x^2+e x} \left(-16 a d^2+4 b c d-5 b e^2\right)}{64 d^3 \left(a+b x^2\right)}-\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left(4 c d-e^2\right) \left(-16 a d^2+4 b c d-5 b e^2\right) \tanh ^{-1}\left(\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right)}{128 d^{7/2} \left(a+b x^2\right)}-\frac{5 b e \sqrt{a^2+2 a b x^2+b^2 x^4} \left(c+d x^2+e x\right)^{3/2}}{24 d^2 \left(a+b x^2\right)}+\frac{b x \sqrt{a^2+2 a b x^2+b^2 x^4} \left(c+d x^2+e x\right)^{3/2}}{4 d \left(a+b x^2\right)}","-\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} (2 d x+e) \sqrt{c+d x^2+e x} \left(-16 a d^2+4 b c d-5 b e^2\right)}{64 d^3 \left(a+b x^2\right)}-\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left(4 c d-e^2\right) \left(-16 a d^2+4 b c d-5 b e^2\right) \tanh ^{-1}\left(\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right)}{128 d^{7/2} \left(a+b x^2\right)}-\frac{5 b e \sqrt{a^2+2 a b x^2+b^2 x^4} \left(c+d x^2+e x\right)^{3/2}}{24 d^2 \left(a+b x^2\right)}+\frac{b x \sqrt{a^2+2 a b x^2+b^2 x^4} \left(c+d x^2+e x\right)^{3/2}}{4 d \left(a+b x^2\right)}",1,"-((4*b*c*d - 16*a*d^2 - 5*b*e^2)*(e + 2*d*x)*Sqrt[c + e*x + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(64*d^3*(a + b*x^2)) - (5*b*e*(c + e*x + d*x^2)^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(24*d^2*(a + b*x^2)) + (b*x*(c + e*x + d*x^2)^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(4*d*(a + b*x^2)) - ((4*c*d - e^2)*(4*b*c*d - 16*a*d^2 - 5*b*e^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]*Sqrt[c + e*x + d*x^2])])/(128*d^(7/2)*(a + b*x^2))","A",6,6,37,0.1622,1,"{6744, 1661, 640, 612, 621, 206}"
39,1,286,0,0.75443,"\int \frac{\sqrt{c+e x+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{x} \, dx","Int[(Sqrt[c + e*x + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/x,x]","\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2+e x} \left(8 a d^2-2 b d e x-b e^2\right)}{8 d^2 \left(a+b x^2\right)}+\frac{e \sqrt{a^2+2 a b x^2+b^2 x^4} \left(8 a d^2-b \left(4 c d-e^2\right)\right) \tanh ^{-1}\left(\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right)}{16 d^{5/2} \left(a+b x^2\right)}+\frac{b \sqrt{a^2+2 a b x^2+b^2 x^4} \left(c+d x^2+e x\right)^{3/2}}{3 d \left(a+b x^2\right)}-\frac{a \sqrt{c} \sqrt{a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left(\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+d x^2+e x}}\right)}{a+b x^2}","\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2+e x} \left(8 a d^2-2 b d e x-b e^2\right)}{8 d^2 \left(a+b x^2\right)}+\frac{e \sqrt{a^2+2 a b x^2+b^2 x^4} \left(8 a d^2-b \left(4 c d-e^2\right)\right) \tanh ^{-1}\left(\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right)}{16 d^{5/2} \left(a+b x^2\right)}+\frac{b \sqrt{a^2+2 a b x^2+b^2 x^4} \left(c+d x^2+e x\right)^{3/2}}{3 d \left(a+b x^2\right)}-\frac{a \sqrt{c} \sqrt{a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left(\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+d x^2+e x}}\right)}{a+b x^2}",1,"((8*a*d^2 - b*e^2 - 2*b*d*e*x)*Sqrt[c + e*x + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(8*d^2*(a + b*x^2)) + (b*(c + e*x + d*x^2)^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*d*(a + b*x^2)) + (e*(8*a*d^2 - b*(4*c*d - e^2))*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]*Sqrt[c + e*x + d*x^2])])/(16*d^(5/2)*(a + b*x^2)) - (a*Sqrt[c]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*ArcTanh[(2*c + e*x)/(2*Sqrt[c]*Sqrt[c + e*x + d*x^2])])/(a + b*x^2)","A",8,7,40,0.1750,1,"{6744, 1653, 814, 843, 621, 206, 724}"
40,1,294,0,0.7671181,"\int \frac{\sqrt{c+e x+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{x^2} \, dx","Int[(Sqrt[c + e*x + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/x^2,x]","\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left(8 a d^2+4 b c d-b e^2\right) \tanh ^{-1}\left(\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right)}{8 d^{3/2} \left(a+b x^2\right)}-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4} \left(c+d x^2+e x\right)^{3/2}}{c x \left(a+b x^2\right)}+\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2+e x} (e (4 a d+b c)+2 d x (2 a d+b c))}{4 c d \left(a+b x^2\right)}-\frac{a e \sqrt{a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left(\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+d x^2+e x}}\right)}{2 \sqrt{c} \left(a+b x^2\right)}","\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left(8 a d^2+4 b c d-b e^2\right) \tanh ^{-1}\left(\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right)}{8 d^{3/2} \left(a+b x^2\right)}-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4} \left(c+d x^2+e x\right)^{3/2}}{c x \left(a+b x^2\right)}+\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2+e x} (e (4 a d+b c)+2 d x (2 a d+b c))}{4 c d \left(a+b x^2\right)}-\frac{a e \sqrt{a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left(\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+d x^2+e x}}\right)}{2 \sqrt{c} \left(a+b x^2\right)}",1,"(((b*c + 4*a*d)*e + 2*d*(b*c + 2*a*d)*x)*Sqrt[c + e*x + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(4*c*d*(a + b*x^2)) - (a*(c + e*x + d*x^2)^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(c*x*(a + b*x^2)) + ((4*b*c*d + 8*a*d^2 - b*e^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]*Sqrt[c + e*x + d*x^2])])/(8*d^(3/2)*(a + b*x^2)) - (a*e*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*ArcTanh[(2*c + e*x)/(2*Sqrt[c]*Sqrt[c + e*x + d*x^2])])/(2*Sqrt[c]*(a + b*x^2))","A",8,7,40,0.1750,1,"{6744, 1650, 814, 843, 621, 206, 724}"
41,1,288,0,0.7841784,"\int \frac{\sqrt{c+e x+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{x^3} \, dx","Int[(Sqrt[c + e*x + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/x^3,x]","-\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left(4 a c d-a e^2+8 b c^2\right) \tanh ^{-1}\left(\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+d x^2+e x}}\right)}{8 c^{3/2} \left(a+b x^2\right)}-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4} \left(c+d x^2+e x\right)^{3/2}}{2 c x^2 \left(a+b x^2\right)}+\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2+e x} (2 x (a d+2 b c)+a e)}{4 c x \left(a+b x^2\right)}+\frac{b e \sqrt{a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left(\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right)}{2 \sqrt{d} \left(a+b x^2\right)}","-\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left(4 a c d-a e^2+8 b c^2\right) \tanh ^{-1}\left(\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+d x^2+e x}}\right)}{8 c^{3/2} \left(a+b x^2\right)}-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4} \left(c+d x^2+e x\right)^{3/2}}{2 c x^2 \left(a+b x^2\right)}+\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2+e x} (2 x (a d+2 b c)+a e)}{4 c x \left(a+b x^2\right)}+\frac{b e \sqrt{a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left(\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right)}{2 \sqrt{d} \left(a+b x^2\right)}",1,"((a*e + 2*(2*b*c + a*d)*x)*Sqrt[c + e*x + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(4*c*x*(a + b*x^2)) - (a*(c + e*x + d*x^2)^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(2*c*x^2*(a + b*x^2)) + (b*e*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]*Sqrt[c + e*x + d*x^2])])/(2*Sqrt[d]*(a + b*x^2)) - ((8*b*c^2 + 4*a*c*d - a*e^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*ArcTanh[(2*c + e*x)/(2*Sqrt[c]*Sqrt[c + e*x + d*x^2])])/(8*c^(3/2)*(a + b*x^2))","A",8,7,40,0.1750,1,"{6744, 1650, 812, 843, 621, 206, 724}"
42,1,294,0,0.7868892,"\int \frac{\sqrt{c+e x+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{x^4} \, dx","Int[(Sqrt[c + e*x + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/x^4,x]","\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2+e x} \left(2 a c e-x \left(8 b c^2-a e^2\right)\right)}{8 c^2 x^2 \left(a+b x^2\right)}-\frac{e \sqrt{a^2+2 a b x^2+b^2 x^4} \left(8 b c^2-a \left(4 c d-e^2\right)\right) \tanh ^{-1}\left(\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+d x^2+e x}}\right)}{16 c^{5/2} \left(a+b x^2\right)}-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4} \left(c+d x^2+e x\right)^{3/2}}{3 c x^3 \left(a+b x^2\right)}+\frac{b \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left(\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right)}{a+b x^2}","\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2+e x} \left(2 a c e-x \left(8 b c^2-a e^2\right)\right)}{8 c^2 x^2 \left(a+b x^2\right)}-\frac{e \sqrt{a^2+2 a b x^2+b^2 x^4} \left(8 b c^2-a \left(4 c d-e^2\right)\right) \tanh ^{-1}\left(\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+d x^2+e x}}\right)}{16 c^{5/2} \left(a+b x^2\right)}-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4} \left(c+d x^2+e x\right)^{3/2}}{3 c x^3 \left(a+b x^2\right)}+\frac{b \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left(\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right)}{a+b x^2}",1,"((2*a*c*e - (8*b*c^2 - a*e^2)*x)*Sqrt[c + e*x + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(8*c^2*x^2*(a + b*x^2)) - (a*(c + e*x + d*x^2)^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*c*x^3*(a + b*x^2)) + (b*Sqrt[d]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]*Sqrt[c + e*x + d*x^2])])/(a + b*x^2) - (e*(8*b*c^2 - a*(4*c*d - e^2))*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*ArcTanh[(2*c + e*x)/(2*Sqrt[c]*Sqrt[c + e*x + d*x^2])])/(16*c^(5/2)*(a + b*x^2))","A",8,7,40,0.1750,1,"{6744, 1650, 810, 843, 621, 206, 724}"