1,1,200,405,0.2750634,"\int \frac{1}{(d+e x) \sqrt{a+c x^4}} \, dx","Integrate[1/((d + e*x)*Sqrt[a + c*x^4]),x]","\frac{\sqrt{\frac{c x^4}{a}+1} \left(\sqrt[4]{c} d \log \left(\frac{e^2 x^2-d^2}{a e^2 \left(\sqrt{\frac{c x^4}{a}+1} \sqrt{\frac{c d^4}{a e^4}+1}+1\right)+c d^2 x^2}\right)-2 \sqrt[4]{-1} \sqrt[4]{a} e \sqrt{\frac{c d^4}{a e^4}+1} \Pi \left(\frac{i \sqrt{a} e^2}{\sqrt{c} d^2};\left.\sin ^{-1}\left(\frac{(-1)^{3/4} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)\right|-1\right)\right)}{2 \sqrt[4]{c} d e \sqrt{a+c x^4} \sqrt{\frac{c d^4}{a e^4}+1}}","\frac{e \tan ^{-1}\left(\frac{x \sqrt{-a e^4-c d^4}}{d e \sqrt{a+c x^4}}\right)}{2 \sqrt{-a e^4-c d^4}}+\frac{\sqrt[4]{c} 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{a+c x^4} \left(\sqrt{a} e^2+\sqrt{c} d^2\right)}-\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(\sqrt{c} d^2-\sqrt{a} e^2\right) \Pi \left(\frac{\left(\sqrt{c} d^2+\sqrt{a} e^2\right)^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 \sqrt[4]{a} \sqrt[4]{c} d \sqrt{a+c x^4} \left(\sqrt{a} e^2+\sqrt{c} d^2\right)}-\frac{e \tanh ^{-1}\left(\frac{a e^2+c d^2 x^2}{\sqrt{a+c x^4} \sqrt{a e^4+c d^4}}\right)}{2 \sqrt{a e^4+c d^4}}",1,"(Sqrt[1 + (c*x^4)/a]*(-2*(-1)^(1/4)*a^(1/4)*Sqrt[1 + (c*d^4)/(a*e^4)]*e*EllipticPi[(I*Sqrt[a]*e^2)/(Sqrt[c]*d^2), ArcSin[((-1)^(3/4)*c^(1/4)*x)/a^(1/4)], -1] + c^(1/4)*d*Log[(-d^2 + e^2*x^2)/(c*d^2*x^2 + a*e^2*(1 + Sqrt[1 + (c*d^4)/(a*e^4)]*Sqrt[1 + (c*x^4)/a]))]))/(2*c^(1/4)*d*Sqrt[1 + (c*d^4)/(a*e^4)]*e*Sqrt[a + c*x^4])","C",1
2,1,425,610,1.1512498,"\int \frac{1}{(d+e x)^2 \sqrt{a+c x^4}} \, dx","Integrate[1/((d + e*x)^2*Sqrt[a + c*x^4]),x]","\frac{-\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \left(2 \sqrt[4]{-1} \sqrt[4]{a} c^{3/4} d^2 \sqrt{\frac{c x^4}{a}+1} (d+e x) \sqrt{a e^4+c d^4} \Pi \left(\frac{i \sqrt{a} e^2}{\sqrt{c} d^2};\left.\sin ^{-1}\left(\frac{(-1)^{3/4} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)\right|-1\right)+e^3 \left(a+c x^4\right) \sqrt{a e^4+c d^4}+c d^3 e \sqrt{a+c x^4} (d+e x) \tanh ^{-1}\left(\frac{a e^2+c d^2 x^2}{\sqrt{a+c x^4} \sqrt{a e^4+c d^4}}\right)\right)+\sqrt{a} \sqrt{c} e^2 \sqrt{\frac{c x^4}{a}+1} (d+e x) \sqrt{a e^4+c d^4} E\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)+i \sqrt{c} \sqrt{\frac{c x^4}{a}+1} (d+e x) \left(\sqrt{c} d^2+i \sqrt{a} e^2\right) \sqrt{a e^4+c d^4} F\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \sqrt{a+c x^4} (d+e x) \left(a e^4+c d^4\right)^{3/2}}","-\frac{c^{3/4} 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}} \left(\sqrt{c} d^2-\sqrt{a} e^2\right) \Pi \left(\frac{\left(\sqrt{c} d^2+\sqrt{a} e^2\right)^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} \sqrt{a+c x^4} \left(\sqrt{a} e^2+\sqrt{c} d^2\right) \left(a e^4+c d^4\right)}-\frac{e^3 \sqrt{a+c x^4}}{(d+e x) \left(a e^4+c d^4\right)}+\frac{\sqrt{c} e^2 x \sqrt{a+c x^4}}{\left(\sqrt{a}+\sqrt{c} x^2\right) \left(a e^4+c d^4\right)}-\frac{\sqrt[4]{a} \sqrt[4]{c} 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)}{\sqrt{a+c x^4} \left(a e^4+c d^4\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}} 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{a} e^2+\sqrt{c} d^2\right)}-\frac{c d^3 e \tan ^{-1}\left(\frac{x \sqrt{-a e^4-c d^4}}{d e \sqrt{a+c x^4}}\right)}{\left(-a e^4-c d^4\right)^{3/2}}-\frac{c d^3 e \tanh ^{-1}\left(\frac{a e^2+c d^2 x^2}{\sqrt{a+c x^4} \sqrt{a e^4+c d^4}}\right)}{\left(a e^4+c d^4\right)^{3/2}}",1,"(Sqrt[a]*Sqrt[c]*e^2*Sqrt[c*d^4 + a*e^4]*(d + e*x)*Sqrt[1 + (c*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + I*Sqrt[c]*(Sqrt[c]*d^2 + I*Sqrt[a]*e^2)*Sqrt[c*d^4 + a*e^4]*(d + e*x)*Sqrt[1 + (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] - Sqrt[(I*Sqrt[c])/Sqrt[a]]*(e^3*Sqrt[c*d^4 + a*e^4]*(a + c*x^4) + c*d^3*e*(d + e*x)*Sqrt[a + c*x^4]*ArcTanh[(a*e^2 + c*d^2*x^2)/(Sqrt[c*d^4 + a*e^4]*Sqrt[a + c*x^4])] + 2*(-1)^(1/4)*a^(1/4)*c^(3/4)*d^2*Sqrt[c*d^4 + a*e^4]*(d + e*x)*Sqrt[1 + (c*x^4)/a]*EllipticPi[(I*Sqrt[a]*e^2)/(Sqrt[c]*d^2), ArcSin[((-1)^(3/4)*c^(1/4)*x)/a^(1/4)], -1]))/(Sqrt[(I*Sqrt[c])/Sqrt[a]]*(c*d^4 + a*e^4)^(3/2)*(d + e*x)*Sqrt[a + c*x^4])","C",1
3,1,1725,518,6.9003643,"\int \frac{1}{(d+e x) \sqrt{a+b x^2+c x^4}} \, dx","Integrate[1/((d + e*x)*Sqrt[a + b*x^2 + c*x^4]),x]","\frac{2 \left(\frac{\sqrt{-\frac{b}{c}-\frac{\sqrt{b^2-4 a c}}{c}}}{\sqrt{2}}+\frac{\sqrt{\frac{\sqrt{b^2-4 a c}}{c}-\frac{b}{c}}}{\sqrt{2}}\right) \left(x-\frac{\sqrt{-\frac{b}{c}-\frac{\sqrt{b^2-4 a c}}{c}}}{\sqrt{2}}\right)^2 \sqrt{\frac{\sqrt{\frac{-b-\sqrt{b^2-4 a c}}{c}} \left(x-\frac{\sqrt{\frac{\sqrt{b^2-4 a c}}{c}-\frac{b}{c}}}{\sqrt{2}}\right)}{\left(\frac{\sqrt{-\frac{b}{c}-\frac{\sqrt{b^2-4 a c}}{c}}}{\sqrt{2}}+\frac{\sqrt{\frac{\sqrt{b^2-4 a c}}{c}-\frac{b}{c}}}{\sqrt{2}}\right) \left(x-\frac{\sqrt{-\frac{b}{c}-\frac{\sqrt{b^2-4 a c}}{c}}}{\sqrt{2}}\right)}} \sqrt{\frac{\sqrt{\frac{-b-\sqrt{b^2-4 a c}}{c}} \left(x+\frac{\sqrt{\frac{\sqrt{b^2-4 a c}}{c}-\frac{b}{c}}}{\sqrt{2}}\right)}{\left(\frac{\sqrt{-\frac{b}{c}-\frac{\sqrt{b^2-4 a c}}{c}}}{\sqrt{2}}-\frac{\sqrt{\frac{\sqrt{b^2-4 a c}}{c}-\frac{b}{c}}}{\sqrt{2}}\right) \left(x-\frac{\sqrt{-\frac{b}{c}-\frac{\sqrt{b^2-4 a c}}{c}}}{\sqrt{2}}\right)}} \sqrt{\frac{\left(\sqrt{\frac{-b-\sqrt{b^2-4 a c}}{c}}-\sqrt{\frac{\sqrt{b^2-4 a c}-b}{c}}\right) \left(2 x+\sqrt{2} \sqrt{\frac{-b-\sqrt{b^2-4 a c}}{c}}\right)}{\left(\sqrt{\frac{-b-\sqrt{b^2-4 a c}}{c}}+\sqrt{\frac{\sqrt{b^2-4 a c}-b}{c}}\right) \left(\sqrt{2} \sqrt{\frac{-b-\sqrt{b^2-4 a c}}{c}}-2 x\right)}} \left(\left(\frac{\sqrt{-\frac{b}{c}-\frac{\sqrt{b^2-4 a c}}{c}} e}{\sqrt{2}}-d\right) F\left(\sin ^{-1}\left(\sqrt{\frac{\left(\sqrt{\frac{-b-\sqrt{b^2-4 a c}}{c}}-\sqrt{\frac{\sqrt{b^2-4 a c}-b}{c}}\right) \left(2 x+\sqrt{2} \sqrt{\frac{-b-\sqrt{b^2-4 a c}}{c}}\right)}{\left(\sqrt{\frac{-b-\sqrt{b^2-4 a c}}{c}}+\sqrt{\frac{\sqrt{b^2-4 a c}-b}{c}}\right) \left(\sqrt{2} \sqrt{\frac{-b-\sqrt{b^2-4 a c}}{c}}-2 x\right)}}\right)|\frac{\left(\sqrt{\frac{-b-\sqrt{b^2-4 a c}}{c}}+\sqrt{\frac{\sqrt{b^2-4 a c}-b}{c}}\right)^2}{\left(\sqrt{\frac{-b-\sqrt{b^2-4 a c}}{c}}-\sqrt{\frac{\sqrt{b^2-4 a c}-b}{c}}\right)^2}\right)-\sqrt{2} \sqrt{\frac{-b-\sqrt{b^2-4 a c}}{c}} e \Pi \left(\frac{\left(\frac{\sqrt{-\frac{b}{c}-\frac{\sqrt{b^2-4 a c}}{c}}}{\sqrt{2}}+\frac{\sqrt{\frac{\sqrt{b^2-4 a c}}{c}-\frac{b}{c}}}{\sqrt{2}}\right) \left(d+\frac{\sqrt{-\frac{b}{c}-\frac{\sqrt{b^2-4 a c}}{c}} e}{\sqrt{2}}\right)}{\left(\frac{\sqrt{\frac{\sqrt{b^2-4 a c}}{c}-\frac{b}{c}}}{\sqrt{2}}-\frac{\sqrt{-\frac{b}{c}-\frac{\sqrt{b^2-4 a c}}{c}}}{\sqrt{2}}\right) \left(d-\frac{\sqrt{-\frac{b}{c}-\frac{\sqrt{b^2-4 a c}}{c}} e}{\sqrt{2}}\right)};\sin ^{-1}\left(\sqrt{\frac{\left(\sqrt{\frac{-b-\sqrt{b^2-4 a c}}{c}}-\sqrt{\frac{\sqrt{b^2-4 a c}-b}{c}}\right) \left(2 x+\sqrt{2} \sqrt{\frac{-b-\sqrt{b^2-4 a c}}{c}}\right)}{\left(\sqrt{\frac{-b-\sqrt{b^2-4 a c}}{c}}+\sqrt{\frac{\sqrt{b^2-4 a c}-b}{c}}\right) \left(\sqrt{2} \sqrt{\frac{-b-\sqrt{b^2-4 a c}}{c}}-2 x\right)}}\right)|\frac{\left(\sqrt{\frac{-b-\sqrt{b^2-4 a c}}{c}}+\sqrt{\frac{\sqrt{b^2-4 a c}-b}{c}}\right)^2}{\left(\sqrt{\frac{-b-\sqrt{b^2-4 a c}}{c}}-\sqrt{\frac{\sqrt{b^2-4 a c}-b}{c}}\right)^2}\right)\right)}{\sqrt{\frac{-b-\sqrt{b^2-4 a c}}{c}} \left(\frac{\sqrt{-\frac{b}{c}-\frac{\sqrt{b^2-4 a c}}{c}}}{\sqrt{2}}-\frac{\sqrt{\frac{\sqrt{b^2-4 a c}}{c}-\frac{b}{c}}}{\sqrt{2}}\right) \left(-d-\frac{\sqrt{-\frac{b}{c}-\frac{\sqrt{b^2-4 a c}}{c}} e}{\sqrt{2}}\right) \left(d-\frac{\sqrt{-\frac{b}{c}-\frac{\sqrt{b^2-4 a c}}{c}} e}{\sqrt{2}}\right) \sqrt{c x^4+b x^2+a}}","\frac{\sqrt[4]{c} d \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{2 \sqrt[4]{a} \sqrt{a+b x^2+c x^4} \left(\sqrt{a} e^2+\sqrt{c} d^2\right)}-\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(\sqrt{c} d^2-\sqrt{a} e^2\right) \Pi \left(\frac{\left(\sqrt{c} d^2+\sqrt{a} e^2\right)^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{4 \sqrt[4]{a} \sqrt[4]{c} d \sqrt{a+b x^2+c x^4} \left(\sqrt{a} e^2+\sqrt{c} d^2\right)}+\frac{e \tan ^{-1}\left(\frac{x \sqrt{-a e^4-b d^2 e^2-c d^4}}{d e \sqrt{a+b x^2+c x^4}}\right)}{2 \sqrt{-a e^4-b d^2 e^2-c d^4}}-\frac{e \tanh ^{-1}\left(\frac{2 a e^2+x^2 \left(b e^2+2 c d^2\right)+b d^2}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^4+b d^2 e^2+c d^4}}\right)}{2 \sqrt{a e^4+b d^2 e^2+c d^4}}",1,"(2*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + x)^2*Sqrt[(Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c]*(-(Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + x))/((Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + x))]*Sqrt[(Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c]*(Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + x))/((Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] - Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + x))]*Sqrt[((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + 2*x))/((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - 2*x))]*((-d + (Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e)/Sqrt[2])*EllipticF[ArcSin[Sqrt[((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + 2*x))/((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - 2*x))]], (Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])^2/(Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])^2] - Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c]*e*EllipticPi[((Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(d + (Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e)/Sqrt[2]))/((-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(d - (Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e)/Sqrt[2])), ArcSin[Sqrt[((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + 2*x))/((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - 2*x))]], (Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])^2/(Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])^2]))/(Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c]*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] - Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(-d - (Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e)/Sqrt[2])*(d - (Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e)/Sqrt[2])*Sqrt[a + b*x^2 + c*x^4])","B",1
4,1,4019,822,7.7716878,"\int \frac{1}{(d+e x)^2 \sqrt{a+b x^2+c x^4}} \, dx","Integrate[1/((d + e*x)^2*Sqrt[a + b*x^2 + c*x^4]),x]","\text{Result too large to show}","-\frac{\sqrt{c x^4+b x^2+a} e^3}{\left(c d^4+b e^2 d^2+a e^4\right) (d+e x)}-\frac{\sqrt[4]{a} \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}} 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}{\left(c d^4+b e^2 d^2+a e^4\right) \sqrt{c x^4+b x^2+a}}+\frac{\sqrt{c} x \sqrt{c x^4+b x^2+a} e^2}{\left(c d^4+b e^2 d^2+a e^4\right) \left(\sqrt{c} x^2+\sqrt{a}\right)}-\frac{d \left(2 c d^2+b e^2\right) \tan ^{-1}\left(\frac{\sqrt{-c d^4-b e^2 d^2-a e^4} x}{d e \sqrt{c x^4+b x^2+a}}\right) e}{2 \left(-c d^4-b e^2 d^2-a e^4\right)^{3/2}}-\frac{d \left(2 c d^2+b e^2\right) \tanh ^{-1}\left(\frac{b d^2+2 a e^2+\left(2 c d^2+b e^2\right) x^2}{2 \sqrt{c d^4+b e^2 d^2+a e^4} \sqrt{c x^4+b x^2+a}}\right) e}{2 \left(c d^4+b e^2 d^2+a e^4\right)^{3/2}}+\frac{\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}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{2 \sqrt[4]{a} \left(\sqrt{c} d^2+\sqrt{a} e^2\right) \sqrt{c x^4+b x^2+a}}-\frac{\left(\sqrt{c} d^2-\sqrt{a} e^2\right) \left(2 c d^2+b e^2\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^2+\sqrt{a} e^2\right)^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{4 \sqrt[4]{a} \sqrt[4]{c} \left(\sqrt{c} d^2+\sqrt{a} e^2\right) \left(c d^4+b e^2 d^2+a e^4\right) \sqrt{c x^4+b x^2+a}}",1,"-((e^3*Sqrt[a + b*x^2 + c*x^4])/((c*d^4 + b*d^2*e^2 + a*e^4)*(d + e*x))) + (((I/2)*(-b + Sqrt[b^2 - 4*a*c])*e^2*Sqrt[1 - (2*c*x^2)/(-b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 - (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])]*(EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*x], (-b - Sqrt[b^2 - 4*a*c])/(-b + Sqrt[b^2 - 4*a*c])] - EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*x], (-b - Sqrt[b^2 - 4*a*c])/(-b + Sqrt[b^2 - 4*a*c])]))/(Sqrt[2]*Sqrt[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*Sqrt[a + b*x^2 + c*x^4]) + (I*c*d^2*Sqrt[1 - (2*c*x^2)/(-b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 - (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])]*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*x], (-b - Sqrt[b^2 - 4*a*c])/(-b + Sqrt[b^2 - 4*a*c])])/(Sqrt[2]*Sqrt[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*Sqrt[a + b*x^2 + c*x^4]) + (4*c*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*d^3*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + x)^2*Sqrt[(Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c]*(-(Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + x))/((Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + x))]*Sqrt[(Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c]*(Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + x))/((Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] - Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + x))]*Sqrt[((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + 2*x))/((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - 2*x))]*((-d + (Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e)/Sqrt[2])*EllipticF[ArcSin[Sqrt[((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + 2*x))/((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - 2*x))]], (Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])^2/(Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])^2] - Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c]*e*EllipticPi[((Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(d + (Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e)/Sqrt[2]))/((-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(d - (Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e)/Sqrt[2])), ArcSin[Sqrt[((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + 2*x))/((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - 2*x))]], (Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])^2/(Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])^2]))/(Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c]*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] - Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(-d - (Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e)/Sqrt[2])*(d - (Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e)/Sqrt[2])*Sqrt[a + b*x^2 + c*x^4]) + (2*b*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*d*e^2*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + x)^2*Sqrt[(Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c]*(-(Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + x))/((Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + x))]*Sqrt[(Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c]*(Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + x))/((Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] - Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + x))]*Sqrt[((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + 2*x))/((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - 2*x))]*((-d + (Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e)/Sqrt[2])*EllipticF[ArcSin[Sqrt[((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + 2*x))/((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - 2*x))]], (Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])^2/(Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])^2] - Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c]*e*EllipticPi[((Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(d + (Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e)/Sqrt[2]))/((-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(d - (Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e)/Sqrt[2])), ArcSin[Sqrt[((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + 2*x))/((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - 2*x))]], (Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])^2/(Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])^2]))/(Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c]*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] - Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(-d - (Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e)/Sqrt[2])*(d - (Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e)/Sqrt[2])*Sqrt[a + b*x^2 + c*x^4]))/(c*d^4 + b*d^2*e^2 + a*e^4)","C",0