1,1,217,453,0.2664083,"\int \frac{\left(A+B x^2\right) \left(d+e x^2\right)^3}{\sqrt{a+c x^4}} \, dx","Integrate[((A + B*x^2)*(d + e*x^2)^3)/Sqrt[a + c*x^4],x]","\frac{5 x \sqrt{\frac{c x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^4}{a}\right) \left(21 A c d \left(c d^2-a e^2\right)+a B e \left(5 a e^2-21 c d^2\right)\right)+e x \left(a+c x^4\right) \left(-25 a B e^2+21 A c e \left(5 d+e x^2\right)+3 B c \left(35 d^2+21 d e x^2+5 e^2 x^4\right)\right)+7 c x^3 \sqrt{\frac{c x^4}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^4}{a}\right) \left(-3 a A e^3-9 a B d e^2+15 A c d^2 e+5 B c d^3\right)}{105 c^2 \sqrt{a+c x^4}}","\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} 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{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 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{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,"(e*x*(a + c*x^4)*(-25*a*B*e^2 + 21*A*c*e*(5*d + e*x^2) + 3*B*c*(35*d^2 + 21*d*e*x^2 + 5*e^2*x^4)) + 5*(21*A*c*d*(c*d^2 - a*e^2) + a*B*e*(-21*c*d^2 + 5*a*e^2))*x*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^4)/a)] + 7*c*(5*B*c*d^3 + 15*A*c*d^2*e - 9*a*B*d*e^2 - 3*a*A*e^3)*x^3*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, -((c*x^4)/a)])/(105*c^2*Sqrt[a + c*x^4])","C",1
2,1,159,367,0.1680105,"\int \frac{\left(A+B x^2\right) \left(d+e x^2\right)^2}{\sqrt{a+c x^4}} \, dx","Integrate[((A + B*x^2)*(d + e*x^2)^2)/Sqrt[a + c*x^4],x]","\frac{-5 x \sqrt{\frac{c x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^4}{a}\right) \left(a A e^2+2 a B d e-3 A c d^2\right)+x^3 \sqrt{\frac{c x^4}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^4}{a}\right) \left(-3 a B e^2+10 A c d e+5 B c d^2\right)+e x \left(a+c x^4\right) \left(5 A e+10 B d+3 B e x^2\right)}{15 c \sqrt{a+c x^4}}","\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} 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*x*(10*B*d + 5*A*e + 3*B*e*x^2)*(a + c*x^4) - 5*(-3*A*c*d^2 + 2*a*B*d*e + a*A*e^2)*x*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^4)/a)] + (5*B*c*d^2 + 10*A*c*d*e - 3*a*B*e^2)*x^3*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, -((c*x^4)/a)])/(15*c*Sqrt[a + c*x^4])","C",1
3,1,121,277,0.0969567,"\int \frac{\left(A+B x^2\right) \left(d+e x^2\right)}{\sqrt{a+c x^4}} \, dx","Integrate[((A + B*x^2)*(d + e*x^2))/Sqrt[a + c*x^4],x]","\frac{x \sqrt{\frac{c x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^4}{a}\right) (3 A c d-a B e)+c x^3 \sqrt{\frac{c x^4}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^4}{a}\right) (A e+B d)+B e x \left(a+c x^4\right)}{3 c \sqrt{a+c x^4}}","\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) \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*(a + c*x^4) + (3*A*c*d - a*B*e)*x*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^4)/a)] + c*(B*d + A*e)*x^3*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, -((c*x^4)/a)])/(3*c*Sqrt[a + c*x^4])","C",1
4,1,77,226,0.0295155,"\int \frac{A+B x^2}{\sqrt{a+c x^4}} \, dx","Integrate[(A + B*x^2)/Sqrt[a + c*x^4],x]","\frac{\sqrt{\frac{c x^4}{a}+1} \left(3 A x \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^4}{a}\right)+B x^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^4}{a}\right)\right)}{3 \sqrt{a+c x^4}}","\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(\frac{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,"(Sqrt[1 + (c*x^4)/a]*(3*A*x*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^4)/a)] + B*x^3*Hypergeometric2F1[1/2, 3/4, 7/4, -((c*x^4)/a)]))/(3*Sqrt[a + c*x^4])","C",1
5,1,138,369,0.2995382,"\int \frac{A+B x^2}{\left(d+e x^2\right) \sqrt{a+c x^4}} \, dx","Integrate[(A + B*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]),x]","-\frac{i \sqrt{\frac{c x^4}{a}+1} \left((A e-B d) \Pi \left(-\frac{i \sqrt{a} e}{\sqrt{c} d};\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)+B d F\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)\right)}{d e \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \sqrt{a+c x^4}}","\frac{a^{3/4} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(\frac{\sqrt{c} d}{\sqrt{a}}+e\right)^2 (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,"((-I)*Sqrt[1 + (c*x^4)/a]*(B*d*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + (-(B*d) + A*e)*EllipticPi[((-I)*Sqrt[a]*e)/(Sqrt[c]*d), I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1]))/(Sqrt[(I*Sqrt[c])/Sqrt[a]]*d*e*Sqrt[a + c*x^4])","C",1
6,1,297,641,1.1506224,"\int \frac{A+B x^2}{\left(d+e x^2\right)^2 \sqrt{a+c x^4}} \, dx","Integrate[(A + B*x^2)/((d + e*x^2)^2*Sqrt[a + c*x^4]),x]","\frac{\frac{d e x \left(a+c x^4\right) (A e-B d)}{\left(d+e x^2\right) \left(a e^2+c d^2\right)}-\frac{i \sqrt{\frac{c x^4}{a}+1} \left(\left(a A e^3+a B d e^2+3 A c d^2 e-B c d^3\right) \Pi \left(-\frac{i \sqrt{a} e}{\sqrt{c} d};\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)+\sqrt{c} d \left(\sqrt{c} d-i \sqrt{a} e\right) (B d-A e) F\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)+i \sqrt{a} \sqrt{c} d e (B d-A e) E\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)\right)}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \left(a e^3+c d^2 e\right)}}{2 d^2 \sqrt{a+c x^4}}","\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{\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{\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{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,"((d*e*(-(B*d) + A*e)*x*(a + c*x^4))/((c*d^2 + a*e^2)*(d + e*x^2)) - (I*Sqrt[1 + (c*x^4)/a]*(I*Sqrt[a]*Sqrt[c]*d*e*(B*d - A*e)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + Sqrt[c]*d*(Sqrt[c]*d - I*Sqrt[a]*e)*(B*d - A*e)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + (-(B*c*d^3) + 3*A*c*d^2*e + a*B*d*e^2 + a*A*e^3)*EllipticPi[((-I)*Sqrt[a]*e)/(Sqrt[c]*d), I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1]))/(Sqrt[(I*Sqrt[c])/Sqrt[a]]*(c*d^2*e + a*e^3)))/(2*d^2*Sqrt[a + c*x^4])","C",1
7,1,453,875,2.1326584,"\int \frac{A+B x^2}{\left(d+e x^2\right)^3 \sqrt{a+c x^4}} \, dx","Integrate[(A + B*x^2)/((d + e*x^2)^3*Sqrt[a + c*x^4]),x]","\frac{-\frac{d e^2 x \left(a+c x^4\right) \left(2 d \left(a e^2+c d^2\right) (B d-A e)+\left(d+e x^2\right) \left(-3 a A e^3-a B d e^2-9 A c d^2 e+5 B c d^3\right)\right)}{\left(d+e x^2\right)^2}-\frac{i \sqrt{\frac{c x^4}{a}+1} \left(\left(3 A e \left(a^2 e^4+2 a c d^2 e^2+5 c^2 d^4\right)+B \left(a^2 d e^4+10 a c d^3 e^2-3 c^2 d^5\right)\right) \Pi \left(-\frac{i \sqrt{a} e}{\sqrt{c} d};\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)+\sqrt{c} d \left(\sqrt{c} d-i \sqrt{a} e\right) F\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right) \left(A e \left(2 i \sqrt{a} \sqrt{c} d e-3 a e^2-7 c d^2\right)+B d \left(-2 i \sqrt{a} \sqrt{c} d e-a e^2+3 c d^2\right)\right)-i \sqrt{a} \sqrt{c} d e E\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right) \left(3 a A e^3+a B d e^2+9 A c d^2 e-5 B c d^3\right)\right)}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}}}{8 d^3 e \sqrt{a+c x^4} \left(a e^2+c d^2\right)^2}","\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,"(-((d*e^2*x*(a + c*x^4)*(2*d*(B*d - A*e)*(c*d^2 + a*e^2) + (5*B*c*d^3 - 9*A*c*d^2*e - a*B*d*e^2 - 3*a*A*e^3)*(d + e*x^2)))/(d + e*x^2)^2) - (I*Sqrt[1 + (c*x^4)/a]*((-I)*Sqrt[a]*Sqrt[c]*d*e*(-5*B*c*d^3 + 9*A*c*d^2*e + a*B*d*e^2 + 3*a*A*e^3)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + Sqrt[c]*d*(Sqrt[c]*d - I*Sqrt[a]*e)*(A*e*(-7*c*d^2 + (2*I)*Sqrt[a]*Sqrt[c]*d*e - 3*a*e^2) + B*d*(3*c*d^2 - (2*I)*Sqrt[a]*Sqrt[c]*d*e - a*e^2))*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + (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))*EllipticPi[((-I)*Sqrt[a]*e)/(Sqrt[c]*d), I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1]))/Sqrt[(I*Sqrt[c])/Sqrt[a]])/(8*d^3*e*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4])","C",1
8,1,222,912,0.2812606,"\int \frac{\left(A+B x^2\right) \left(d+e x^2\right)^3}{\left(a+c x^4\right)^{3/2}} \, dx","Integrate[((A + B*x^2)*(d + e*x^2)^3)/(a + c*x^4)^(3/2),x]","\frac{x \sqrt{\frac{c x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^4}{a}\right) \left(3 A c d \left(3 a e^2+c d^2\right)+a B e \left(9 c d^2-5 a e^2\right)\right)+2 c x^3 \sqrt{\frac{c x^4}{a}+1} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{c x^4}{a}\right) \left(-3 a A e^3-9 a B d e^2+3 A c d^2 e+B c d^3\right)+3 A c x \left(a e^2 \left(2 e x^2-3 d\right)+c d^3\right)+a B e x \left(5 a e^2+c \left(-9 d^2+18 d e x^2+2 e^2 x^4\right)\right)}{6 a c^2 \sqrt{a+c x^4}}","-\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,"(3*A*c*x*(c*d^3 + a*e^2*(-3*d + 2*e*x^2)) + a*B*e*x*(5*a*e^2 + c*(-9*d^2 + 18*d*e*x^2 + 2*e^2*x^4)) + (a*B*e*(9*c*d^2 - 5*a*e^2) + 3*A*c*d*(c*d^2 + 3*a*e^2))*x*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^4)/a)] + 2*c*(B*c*d^3 + 3*A*c*d^2*e - 9*a*B*d*e^2 - 3*a*A*e^3)*x^3*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[3/4, 3/2, 7/4, -((c*x^4)/a)])/(6*a*c^2*Sqrt[a + c*x^4])","C",1
9,1,166,694,0.1722997,"\int \frac{\left(A+B x^2\right) \left(d+e x^2\right)^2}{\left(a+c x^4\right)^{3/2}} \, dx","Integrate[((A + B*x^2)*(d + e*x^2)^2)/(a + c*x^4)^(3/2),x]","\frac{3 x \sqrt{\frac{c x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^4}{a}\right) \left(a A e^2+2 a B d e+A c d^2\right)+2 x^3 \sqrt{\frac{c x^4}{a}+1} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{c x^4}{a}\right) \left(-3 a B e^2+2 A c d e+B c d^2\right)+3 A x \left(c d^2-a e^2\right)+6 a B e x \left(e x^2-d\right)}{6 a c \sqrt{a+c x^4}}","-\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} 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,"(3*A*(c*d^2 - a*e^2)*x + 6*a*B*e*x*(-d + e*x^2) + 3*(A*c*d^2 + 2*a*B*d*e + a*A*e^2)*x*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^4)/a)] + 2*(B*c*d^2 + 2*A*c*d*e - 3*a*B*e^2)*x^3*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[3/4, 3/2, 7/4, -((c*x^4)/a)])/(6*a*c*Sqrt[a + c*x^4])","C",1
10,1,126,395,0.1062425,"\int \frac{\left(A+B x^2\right) \left(d+e x^2\right)}{\left(a+c x^4\right)^{3/2}} \, dx","Integrate[((A + B*x^2)*(d + e*x^2))/(a + c*x^4)^(3/2),x]","\frac{3 x \sqrt{\frac{c x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^4}{a}\right) (a B e+A c d)+2 c x^3 \sqrt{\frac{c x^4}{a}+1} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{c x^4}{a}\right) (A e+B d)+3 x (A c d-a B e)}{6 a c \sqrt{a+c x^4}}","\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} 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,"(3*(A*c*d - a*B*e)*x + 3*(A*c*d + a*B*e)*x*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^4)/a)] + 2*c*(B*d + A*e)*x^3*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[3/4, 3/2, 7/4, -((c*x^4)/a)])/(6*a*c*Sqrt[a + c*x^4])","C",1
11,1,99,262,0.0440273,"\int \frac{A+B x^2}{\left(a+c x^4\right)^{3/2}} \, dx","Integrate[(A + B*x^2)/(a + c*x^4)^(3/2),x]","\frac{3 A x \sqrt{\frac{c x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^4}{a}\right)+2 B x^3 \sqrt{\frac{c x^4}{a}+1} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{c x^4}{a}\right)+3 A x}{6 a \sqrt{a+c x^4}}","-\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \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,"(3*A*x + 3*A*x*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^4)/a)] + 2*B*x^3*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[3/4, 3/2, 7/4, -((c*x^4)/a)])/(6*a*Sqrt[a + c*x^4])","C",1
12,1,432,732,0.7911548,"\int \frac{A+B x^2}{\left(d+e x^2\right) \left(a+c x^4\right)^{3/2}} \, dx","Integrate[(A + B*x^2)/((d + e*x^2)*(a + c*x^4)^(3/2)),x]","\frac{d \sqrt{\frac{c x^4}{a}+1} \left(\sqrt{a} B-i A \sqrt{c}\right) \left(\sqrt{c} d-i \sqrt{a} e\right) F\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)-\sqrt{a} \sqrt{c} d \sqrt{\frac{c x^4}{a}+1} (B d-A e) E\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)+A c d^2 x \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}-2 i a A e^2 \sqrt{\frac{c x^4}{a}+1} \Pi \left(-\frac{i \sqrt{a} e}{\sqrt{c} d};\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)-A c d e x^3 \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}+B c d^2 x^3 \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}+2 i a B d e \sqrt{\frac{c x^4}{a}+1} \Pi \left(-\frac{i \sqrt{a} e}{\sqrt{c} d};\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)+a B d e x \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}}{2 a d \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \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}} 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{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{\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{\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{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}}",1,"(A*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c*d^2*x + a*B*Sqrt[(I*Sqrt[c])/Sqrt[a]]*d*e*x + B*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c*d^2*x^3 - A*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c*d*e*x^3 - Sqrt[a]*Sqrt[c]*d*(B*d - A*e)*Sqrt[1 + (c*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + (Sqrt[a]*B - I*A*Sqrt[c])*d*(Sqrt[c]*d - I*Sqrt[a]*e)*Sqrt[1 + (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + (2*I)*a*B*d*e*Sqrt[1 + (c*x^4)/a]*EllipticPi[((-I)*Sqrt[a]*e)/(Sqrt[c]*d), I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] - (2*I)*a*A*e^2*Sqrt[1 + (c*x^4)/a]*EllipticPi[((-I)*Sqrt[a]*e)/(Sqrt[c]*d), I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])/(2*a*Sqrt[(I*Sqrt[c])/Sqrt[a]]*d*(c*d^2 + a*e^2)*Sqrt[a + c*x^4])","C",1
13,1,427,1494,1.572104,"\int \frac{A+B x^2}{\left(d+e x^2\right)^2 \left(a+c x^4\right)^{3/2}} \, dx","Integrate[(A + B*x^2)/((d + e*x^2)^2*(a + c*x^4)^(3/2)),x]","\frac{d \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \left(c d x \left(d+e x^2\right) \left(-a A e^2+a B e \left(2 d-e x^2\right)+A c d \left(d-2 e x^2\right)+B c d^2 x^2\right)+a e^3 x \left(a+c x^4\right) (A e-B d)\right)-\sqrt{\frac{c x^4}{a}+1} \left(d+e x^2\right) \left(i \left(\sqrt{c} d \left(\sqrt{c} d-i \sqrt{a} e\right) F\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right) \left(i \sqrt{a} \sqrt{c} d (B d-A e)+a e (2 B d-A e)+A c d^2\right)+a e \left(a A e^3+a B d e^2+7 A c d^2 e-5 B c d^3\right) \Pi \left(-\frac{i \sqrt{a} e}{\sqrt{c} d};\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)\right)-\sqrt{a} \sqrt{c} d E\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right) \left(-a A e^3+2 a B d e^2+2 A c d^2 e-B c d^3\right)\right)}{2 a \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \sqrt{a+c x^4} \left(d+e x^2\right) \left(a d e^2+c d^3\right)^2}","-\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,"(Sqrt[(I*Sqrt[c])/Sqrt[a]]*d*(a*e^3*(-(B*d) + A*e)*x*(a + c*x^4) + c*d*x*(d + e*x^2)*(-(a*A*e^2) + B*c*d^2*x^2 + A*c*d*(d - 2*e*x^2) + a*B*e*(2*d - e*x^2))) - (d + e*x^2)*Sqrt[1 + (c*x^4)/a]*(-(Sqrt[a]*Sqrt[c]*d*(-(B*c*d^3) + 2*A*c*d^2*e + 2*a*B*d*e^2 - a*A*e^3)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1]) + I*(Sqrt[c]*d*(Sqrt[c]*d - I*Sqrt[a]*e)*(A*c*d^2 + I*Sqrt[a]*Sqrt[c]*d*(B*d - A*e) + a*e*(2*B*d - A*e))*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + a*e*(-5*B*c*d^3 + 7*A*c*d^2*e + a*B*d*e^2 + a*A*e^3)*EllipticPi[((-I)*Sqrt[a]*e)/(Sqrt[c]*d), I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])))/(2*a*Sqrt[(I*Sqrt[c])/Sqrt[a]]*(c*d^3 + a*d*e^2)^2*(d + e*x^2)*Sqrt[a + c*x^4])","C",1
14,1,630,2452,2.744913,"\int \frac{A+B x^2}{\left(d+e x^2\right)^3 \left(a+c x^4\right)^{3/2}} \, dx","Integrate[(A + B*x^2)/((d + e*x^2)^3*(a + c*x^4)^(3/2)),x]","\frac{d x \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \left(4 c d^2 \left(d+e x^2\right)^2 \left(B \left(-a^2 e^3+3 a c d e \left(d-e x^2\right)+c^2 d^3 x^2\right)+A c \left(a e^2 \left(e x^2-3 d\right)+c d^2 \left(d-3 e x^2\right)\right)\right)-2 a d e^3 \left(a+c x^4\right) \left(a e^2+c d^2\right) (B d-A e)+a e^3 \left(a+c x^4\right) \left(d+e x^2\right) \left(3 a A e^3+a B d e^2+17 A c d^2 e-13 B c d^3\right)\right)-\sqrt{\frac{c x^4}{a}+1} \left(d+e x^2\right)^2 \left(\sqrt{a} \sqrt{c} d E\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right) \left(3 A e \left(a^2 e^4+7 a c d^2 e^2-4 c^2 d^4\right)+B \left(a^2 d e^4-25 a c d^3 e^2+4 c^2 d^5\right)\right)+i \left(a e \left(3 A e \left(a^2 e^4+2 a c d^2 e^2+21 c^2 d^4\right)+B \left(a^2 d e^4+26 a c d^3 e^2-35 c^2 d^5\right)\right) \Pi \left(-\frac{i \sqrt{a} e}{\sqrt{c} d};\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)+\sqrt{c} d \left(\sqrt{c} d-i \sqrt{a} e\right) F\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right) \left(-2 i a^{3/2} \sqrt{c} d e^2 (3 B d-A e)-a^2 e^3 (3 A e+B d)+4 i \sqrt{a} c^{3/2} d^3 (B d-2 A e)+19 a c d^2 e (B d-A e)+4 A c^2 d^4\right)\right)\right)}{8 a \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \sqrt{a+c x^4} \left(d+e x^2\right)^2 \left(a d e^2+c d^3\right)^3}","-\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,"(Sqrt[(I*Sqrt[c])/Sqrt[a]]*d*x*(-2*a*d*e^3*(B*d - A*e)*(c*d^2 + a*e^2)*(a + c*x^4) + a*e^3*(-13*B*c*d^3 + 17*A*c*d^2*e + a*B*d*e^2 + 3*a*A*e^3)*(d + e*x^2)*(a + c*x^4) + 4*c*d^2*(d + e*x^2)^2*(B*(-(a^2*e^3) + c^2*d^3*x^2 + 3*a*c*d*e*(d - e*x^2)) + A*c*(c*d^2*(d - 3*e*x^2) + a*e^2*(-3*d + e*x^2)))) - (d + e*x^2)^2*Sqrt[1 + (c*x^4)/a]*(Sqrt[a]*Sqrt[c]*d*(3*A*e*(-4*c^2*d^4 + 7*a*c*d^2*e^2 + a^2*e^4) + B*(4*c^2*d^5 - 25*a*c*d^3*e^2 + a^2*d*e^4))*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + I*(Sqrt[c]*d*(Sqrt[c]*d - I*Sqrt[a]*e)*(4*A*c^2*d^4 + (4*I)*Sqrt[a]*c^(3/2)*d^3*(B*d - 2*A*e) + 19*a*c*d^2*e*(B*d - A*e) - (2*I)*a^(3/2)*Sqrt[c]*d*e^2*(3*B*d - A*e) - a^2*e^3*(B*d + 3*A*e))*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + a*e*(3*A*e*(21*c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4) + B*(-35*c^2*d^5 + 26*a*c*d^3*e^2 + a^2*d*e^4))*EllipticPi[((-I)*Sqrt[a]*e)/(Sqrt[c]*d), I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])))/(8*a*Sqrt[(I*Sqrt[c])/Sqrt[a]]*(c*d^3 + a*d*e^2)^3*(d + e*x^2)^2*Sqrt[a + c*x^4])","C",1
15,0,0,169,0.3762539,"\int \frac{\left(A+B x^2\right) \left(d+e x^2\right)^q}{a+c x^4} \, dx","Integrate[((A + B*x^2)*(d + e*x^2)^q)/(a + c*x^4),x]","\int \frac{\left(A+B x^2\right) \left(d+e x^2\right)^q}{a+c x^4} \, dx","\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,"Integrate[((A + B*x^2)*(d + e*x^2)^q)/(a + c*x^4), x]","F",-1
16,1,94,48,0.1267593,"\int \frac{2+x^2}{\left(1+x^2\right) \sqrt{2+3 x^2+x^4}} \, dx","Integrate[(2 + x^2)/((1 + x^2)*Sqrt[2 + 3*x^2 + x^4]),x]","\frac{x^3-i \sqrt{x^2+1} \sqrt{x^2+2} F\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+i \sqrt{x^2+1} \sqrt{x^2+2} E\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+2 x}{\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,"(2*x + x^3 + I*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - I*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt[2]], 2])/Sqrt[2 + 3*x^2 + x^4]","C",1
17,1,4473,755,6.4769172,"\int \frac{\left(A+B x^2\right) \left(d+e x^2\right)^3}{\sqrt{a+b x^2+c x^4}} \, dx","Integrate[((A + B*x^2)*(d + e*x^2)^3)/Sqrt[a + b*x^2 + c*x^4],x]","\text{Result too large to show}","\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,"Sqrt[a + b*x^2 + c*x^4]*(-1/105*(e*(-105*B*c^2*d^2 + 84*b*B*c*d*e - 105*A*c^2*d*e - 24*b^2*B*e^2 + 28*A*b*c*e^2 + 25*a*B*c*e^2)*x)/c^3 + (e^2*(21*B*c*d - 6*b*B*e + 7*A*c*e)*x^3)/(35*c^2) + (B*e^3*x^5)/(7*c)) + ((((105*I)/2)*B*c^2*(-b + Sqrt[b^2 - 4*a*c])*d^3*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]) - ((105*I)*b*B*c*(-b + Sqrt[b^2 - 4*a*c])*d^2*e*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]) + (((315*I)/2)*A*c^2*(-b + Sqrt[b^2 - 4*a*c])*d^2*e*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]) + ((42*I)*Sqrt[2]*b^2*B*(-b + Sqrt[b^2 - 4*a*c])*d*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[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*Sqrt[a + b*x^2 + c*x^4]) - ((105*I)*A*b*c*(-b + Sqrt[b^2 - 4*a*c])*d*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]) - (((189*I)/2)*a*B*c*(-b + Sqrt[b^2 - 4*a*c])*d*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]) + ((14*I)*Sqrt[2]*A*b^2*(-b + Sqrt[b^2 - 4*a*c])*e^3*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[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*Sqrt[a + b*x^2 + c*x^4]) + ((26*I)*Sqrt[2]*a*b*B*(-b + Sqrt[b^2 - 4*a*c])*e^3*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[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*Sqrt[a + b*x^2 + c*x^4]) - ((12*I)*Sqrt[2]*b^3*B*(-b + Sqrt[b^2 - 4*a*c])*e^3*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])]))/(c*Sqrt[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*Sqrt[a + b*x^2 + c*x^4]) - (((63*I)/2)*a*A*c*(-b + Sqrt[b^2 - 4*a*c])*e^3*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]) - ((105*I)*A*c^3*d^3*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]) + ((105*I)*a*B*c^2*d^2*e*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]) - ((42*I)*Sqrt[2]*a*b*B*c*d*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])]*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*x], (-b - Sqrt[b^2 - 4*a*c])/(-b + Sqrt[b^2 - 4*a*c])])/(Sqrt[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*Sqrt[a + b*x^2 + c*x^4]) + ((105*I)*a*A*c^2*d*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])]*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]) + ((12*I)*Sqrt[2]*a*b^2*B*e^3*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[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*Sqrt[a + b*x^2 + c*x^4]) - ((14*I)*Sqrt[2]*a*A*b*c*e^3*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[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*Sqrt[a + b*x^2 + c*x^4]) - ((25*I)*a^2*B*c*e^3*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]))/(105*c^3)","C",1
18,1,674,528,4.4137764,"\int \frac{\left(A+B x^2\right) \left(d+e x^2\right)^2}{\sqrt{a+b x^2+c x^4}} \, dx","Integrate[((A + B*x^2)*(d + e*x^2)^2)/Sqrt[a + b*x^2 + c*x^4],x]","\frac{i \left(\sqrt{b^2-4 a c}-b\right) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} E\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right) \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)+4 c e x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left(a+b x^2+c x^4\right) \left(5 A c e+B \left(-4 b e+10 c d+3 c e x^2\right)\right)-i \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right) \left(c \left(10 A c \left(e \left(3 d \sqrt{b^2-4 a c}-a e\right)+3 c d^2\right)+B \left(5 c d \left(3 d \sqrt{b^2-4 a c}-4 a e\right)-9 a e^2 \sqrt{b^2-4 a c}\right)\right)-b c \left(10 A e \left(e \sqrt{b^2-4 a c}+3 c d\right)+B e \left(20 d \sqrt{b^2-4 a c}-17 a e\right)+15 B c d^2\right)+2 b^2 e \left(4 B e \sqrt{b^2-4 a c}+5 A c e+10 B c d\right)-8 b^3 B e^2\right)}{60 c^3 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","\frac{x \sqrt{a+b x^2+c x^4} \left(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,"(4*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*e*x*(a + b*x^2 + c*x^4)*(5*A*c*e + B*(10*c*d - 4*b*e + 3*c*e*x^2)) + I*(-b + Sqrt[b^2 - 4*a*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)))*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - I*(-8*b^3*B*e^2 + 2*b^2*e*(10*B*c*d + 5*A*c*e + 4*B*Sqrt[b^2 - 4*a*c]*e) - b*c*(15*B*c*d^2 + B*e*(20*Sqrt[b^2 - 4*a*c]*d - 17*a*e) + 10*A*e*(3*c*d + Sqrt[b^2 - 4*a*c]*e)) + c*(B*(-9*a*Sqrt[b^2 - 4*a*c]*e^2 + 5*c*d*(3*Sqrt[b^2 - 4*a*c]*d - 4*a*e)) + 10*A*c*(3*c*d^2 + e*(3*Sqrt[b^2 - 4*a*c]*d - a*e))))*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/(60*c^3*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[a + b*x^2 + c*x^4])","C",1
19,1,521,368,2.1603501,"\int \frac{\left(A+B x^2\right) \left(d+e x^2\right)}{\sqrt{a+b x^2+c x^4}} \, dx","Integrate[((A + B*x^2)*(d + e*x^2))/Sqrt[a + b*x^2 + c*x^4],x]","\frac{i \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right) \left(b \left(2 B e \sqrt{b^2-4 a c}+3 A c e+3 B c d\right)-c \left(3 A e \sqrt{b^2-4 a c}+3 B d \sqrt{b^2-4 a c}-2 a B e+6 A c d\right)-2 b^2 B e\right)-i \left(\sqrt{b^2-4 a c}-b\right) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} E\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right) (-3 A c e+2 b B e-3 B c d)+4 B c e x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left(a+b x^2+c x^4\right)}{12 c^2 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","\frac{x \sqrt{a+b x^2+c x^4} (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,"(4*B*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*e*x*(a + b*x^2 + c*x^4) - I*(-b + Sqrt[b^2 - 4*a*c])*(-3*B*c*d + 2*b*B*e - 3*A*c*e)*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + I*(-2*b^2*B*e - c*(6*A*c*d + 3*B*Sqrt[b^2 - 4*a*c]*d - 2*a*B*e + 3*A*Sqrt[b^2 - 4*a*c]*e) + b*(3*B*c*d + 3*A*c*e + 2*B*Sqrt[b^2 - 4*a*c]*e))*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/(12*c^2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[a + b*x^2 + c*x^4])","C",1
20,1,302,283,0.2449977,"\int \frac{A+B x^2}{\sqrt{a+b x^2+c x^4}} \, dx","Integrate[(A + B*x^2)/Sqrt[a + b*x^2 + c*x^4],x]","\frac{i \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left(\left(-B \sqrt{b^2-4 a c}-2 A c+b B\right) F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)+B \left(\sqrt{b^2-4 a c}-b\right) E\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)\right)}{2 \sqrt{2} c \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{c} x^2\right) \left(\frac{A \sqrt{c}}{\sqrt{a}}+B\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{2 c^{3/4} \sqrt{a+b x^2+c x^4}}-\frac{\sqrt[4]{a} B \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{c^{3/4} \sqrt{a+b x^2+c x^4}}+\frac{B x \sqrt{a+b x^2+c x^4}}{\sqrt{c} \left(\sqrt{a}+\sqrt{c} x^2\right)}",1,"((I/2)*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*(B*(-b + Sqrt[b^2 - 4*a*c])*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + (b*B - 2*A*c - B*Sqrt[b^2 - 4*a*c])*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]))/(Sqrt[2]*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[a + b*x^2 + c*x^4])","C",1
21,1,298,436,0.4850211,"\int \frac{A+B x^2}{\left(d+e x^2\right) \sqrt{a+b x^2+c x^4}} \, dx","Integrate[(A + B*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]),x]","-\frac{i \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left((A e-B d) \Pi \left(\frac{\left(b+\sqrt{b^2-4 a c}\right) e}{2 c d};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)+B d F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)\right)}{\sqrt{2} d e \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","\frac{a^{3/4} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(\frac{\sqrt{c} d}{\sqrt{a}}+e\right)^2 (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,"((-I)*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*(B*d*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])] + (-(B*d) + A*e)*EllipticPi[((b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]))/(Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*e*Sqrt[a + b*x^2 + c*x^4])","C",1
22,1,1853,782,3.4698959,"\int \frac{A+B x^2}{\left(d+e x^2\right)^2 \sqrt{a+b x^2+c x^4}} \, dx","Integrate[(A + B*x^2)/((d + e*x^2)^2*Sqrt[a + b*x^2 + c*x^4]),x]","-\frac{i \left(2 \sqrt{2} B c \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left(e x^2+d\right) F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right) d^3-2 \sqrt{2} B c \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left(e x^2+d\right) \Pi \left(\frac{\left(b+\sqrt{b^2-4 a c}\right) e}{2 c d};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right) d^3+\sqrt{2} B \left(b-\sqrt{b^2-4 a c}\right) e \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left(e x^2+d\right) \left(E\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)-F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)\right) d^2-2 \sqrt{2} A c e \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left(e x^2+d\right) F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right) d^2+6 \sqrt{2} A c e \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left(e x^2+d\right) \Pi \left(\frac{\left(b+\sqrt{b^2-4 a c}\right) e}{2 c d};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right) d^2-4 i \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} e^2 (B d-A e) x \left(c x^4+b x^2+a\right) d+\sqrt{2} A \left(\sqrt{b^2-4 a c}-b\right) e^2 \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left(e x^2+d\right) \left(E\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)-F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)\right) d-4 \sqrt{2} A b e^2 \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left(e x^2+d\right) \Pi \left(\frac{\left(b+\sqrt{b^2-4 a c}\right) e}{2 c d};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right) d+2 \sqrt{2} a B e^2 \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left(e x^2+d\right) \Pi \left(\frac{\left(b+\sqrt{b^2-4 a c}\right) e}{2 c d};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right) d+2 \sqrt{2} a A e^3 \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left(e x^2+d\right) \Pi \left(\frac{\left(b+\sqrt{b^2-4 a c}\right) e}{2 c d};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)\right)}{8 \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} d e \left(c d^3+e (a e-b d) d\right) \left(e x^2+d\right) \sqrt{c x^4+b x^2+a}}","\frac{\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{\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{\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{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,"((-1/8*I)*((-4*I)*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*e^2*(B*d - A*e)*x*(a + b*x^2 + c*x^4) + Sqrt[2]*B*(b - Sqrt[b^2 - 4*a*c])*d^2*e*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*(d + e*x^2)*(EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]) + Sqrt[2]*A*(-b + Sqrt[b^2 - 4*a*c])*d*e^2*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*(d + e*x^2)*(EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]) + 2*Sqrt[2]*B*c*d^3*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*(d + e*x^2)*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - 2*Sqrt[2]*A*c*d^2*e*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*(d + e*x^2)*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - 2*Sqrt[2]*B*c*d^3*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*(d + e*x^2)*EllipticPi[((b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + 6*Sqrt[2]*A*c*d^2*e*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*(d + e*x^2)*EllipticPi[((b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - 4*Sqrt[2]*A*b*d*e^2*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*(d + e*x^2)*EllipticPi[((b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + 2*Sqrt[2]*a*B*d*e^2*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*(d + e*x^2)*EllipticPi[((b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + 2*Sqrt[2]*a*A*e^3*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*(d + e*x^2)*EllipticPi[((b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]))/(Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*e*(c*d^3 + d*e*(-(b*d) + a*e))*(d + e*x^2)*Sqrt[a + b*x^2 + c*x^4])","C",1
23,1,781,1125,5.7316557,"\int \frac{A+B x^2}{\left(d+e x^2\right)^3 \sqrt{a+b x^2+c x^4}} \, dx","Integrate[(A + B*x^2)/((d + e*x^2)^3*Sqrt[a + b*x^2 + c*x^4]),x]","\frac{-\frac{4 d e^2 x \left(a+b x^2+c x^4\right) \left(2 d (B d-A e) \left(e (a e-b d)+c d^2\right)+\left(d+e x^2\right) \left(B \left(5 c d^3-d e (a e+2 b d)\right)-3 A e \left(e (a e-2 b d)+3 c d^2\right)\right)\right)}{\left(d+e x^2\right)^2}-\frac{i \sqrt{2} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left(2 \left(A e \left(e^2 \left(3 a^2 e^2-8 a b d e+8 b^2 d^2\right)+2 c d^2 e (3 a e-10 b d)+15 c^2 d^4\right)+B \left(a d e^3 (a e-4 b d)+10 a c d^3 e^2-3 c^2 d^5\right)\right) \Pi \left(\frac{\left(b+\sqrt{b^2-4 a c}\right) e}{2 c d};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)+d F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right) \left(B d \left(-e^2 \left(\sqrt{b^2-4 a c}-b\right) (a e+2 b d)+c d e \left(5 d \sqrt{b^2-4 a c}-6 a e-5 b d\right)+6 c^2 d^3\right)-A e \left(-3 e^2 \left(\sqrt{b^2-4 a c}-b\right) (2 b d-a e)+c d e \left(9 d \sqrt{b^2-4 a c}+2 a e-17 b d\right)+14 c^2 d^3\right)\right)+d e \left(\sqrt{b^2-4 a c}-b\right) E\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right) \left(3 A e \left(e (a e-2 b d)+3 c d^2\right)+B \left(d e (a e+2 b d)-5 c d^3\right)\right)\right)}{\sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}}}}{32 d^3 e \sqrt{a+b x^2+c x^4} \left(e (a e-b d)+c d^2\right)^2}","-\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,"((-4*d*e^2*x*(a + b*x^2 + c*x^4)*(2*d*(B*d - A*e)*(c*d^2 + e*(-(b*d) + a*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)))*(d + e*x^2)))/(d + e*x^2)^2 - (I*Sqrt[2]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*((-b + Sqrt[b^2 - 4*a*c])*d*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)))*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + d*(B*d*(6*c^2*d^3 + c*d*e*(-5*b*d + 5*Sqrt[b^2 - 4*a*c]*d - 6*a*e) - (-b + Sqrt[b^2 - 4*a*c])*e^2*(2*b*d + a*e)) - A*e*(14*c^2*d^3 - 3*(-b + Sqrt[b^2 - 4*a*c])*e^2*(2*b*d - a*e) + c*d*e*(-17*b*d + 9*Sqrt[b^2 - 4*a*c]*d + 2*a*e)))*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + 2*(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)))*EllipticPi[((b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]))/Sqrt[c/(b + Sqrt[b^2 - 4*a*c])])/(32*d^3*e*(c*d^2 + e*(-(b*d) + a*e))^2*Sqrt[a + b*x^2 + c*x^4])","C",1
24,1,5432,859,6.6528949,"\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","Integrate[((A + B*x^2)*(d + e*x^2)^3)/(a + b*x^2 + c*x^4)^(3/2),x]","\text{Result too large to show}","\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(\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\right)+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,"Result too large to show","C",1
25,1,766,628,4.7195543,"\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","Integrate[((A + B*x^2)*(d + e*x^2)^2)/(a + b*x^2 + c*x^4)^(3/2),x]","\frac{-4 c x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left(A c \left(2 a^2 e^2+a b e \left(e x^2-2 d\right)-2 a c d \left(d+2 e x^2\right)+b^2 d^2+b c d^2 x^2\right)-a B \left(a b e^2-2 a c e \left(2 d+e x^2\right)+b^2 e^2 x^2+b c d \left(d-2 e x^2\right)+2 c^2 d^2 x^2\right)\right)+i \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right) \left(A c \left(b^2 \left(a e^2+c d^2\right)-b \sqrt{b^2-4 a c} \left(a e^2+c d^2\right)-4 a c \left(-d e \sqrt{b^2-4 a c}+a e^2+c d^2\right)\right)+2 a B \left(c^2 d \left(d \sqrt{b^2-4 a c}-4 a e\right)+c e \left(-b d \sqrt{b^2-4 a c}-3 a e \sqrt{b^2-4 a c}+4 a b e+b^2 d\right)+b^2 e^2 \left(\sqrt{b^2-4 a c}-b\right)\right)\right)-i \left(\sqrt{b^2-4 a c}-b\right) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} E\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right) \left(2 a B \left(-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right)-A c \left(a b e^2-4 a c d e+b c d^2\right)\right)}{4 a c^2 \left(4 a c-b^2\right) \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","\frac{\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 \left(-\left(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(-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}}-\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)}",1,"(-4*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*(-(a*B*(a*b*e^2 + 2*c^2*d^2*x^2 + b^2*e^2*x^2 + b*c*d*(d - 2*e*x^2) - 2*a*c*e*(2*d + e*x^2))) + A*c*(b^2*d^2 + 2*a^2*e^2 + b*c*d^2*x^2 + a*b*e*(-2*d + e*x^2) - 2*a*c*d*(d + 2*e*x^2))) - I*(-b + Sqrt[b^2 - 4*a*c])*(-(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[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + I*(2*a*B*(b^2*(-b + Sqrt[b^2 - 4*a*c])*e^2 + c^2*d*(Sqrt[b^2 - 4*a*c]*d - 4*a*e) + c*e*(b^2*d - b*Sqrt[b^2 - 4*a*c]*d + 4*a*b*e - 3*a*Sqrt[b^2 - 4*a*c]*e)) + A*c*(b^2*(c*d^2 + a*e^2) - b*Sqrt[b^2 - 4*a*c]*(c*d^2 + a*e^2) - 4*a*c*(c*d^2 - Sqrt[b^2 - 4*a*c]*d*e + a*e^2)))*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/(4*a*c^2*(-b^2 + 4*a*c)*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[a + b*x^2 + c*x^4])","C",1
26,-1,0,481,0,"\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","Integrate[((A + B*x^2)*(d + e*x^2))/(a + b*x^2 + c*x^4)^(3/2),x]","\text{\$Aborted}","\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)-\left(x^2 (A c (b d-2 a e)-a B (2 c d-b e))\right)+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,"$Aborted","F",-1
27,-1,0,398,0,"\int \frac{A+B x^2}{\left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(A + B*x^2)/(a + b*x^2 + c*x^4)^(3/2),x]","\text{\$Aborted}","\frac{\sqrt[4]{c} \left(\sqrt{a}+\sqrt{c} x^2\right) (A b-2 a B) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{a^{3/4} \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}+\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \left(\sqrt{a} B-A \sqrt{c}\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{2 a^{3/4} \sqrt[4]{c} \left(b-2 \sqrt{a} \sqrt{c}\right) \sqrt{a+b x^2+c x^4}}-\frac{\sqrt{c} x (A b-2 a B) \sqrt{a+b x^2+c x^4}}{a \left(b^2-4 a c\right) \left(\sqrt{a}+\sqrt{c} x^2\right)}+\frac{x \left(c x^2 (A b-2 a B)-2 a A c-a b B+A b^2\right)}{a \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}",1,"$Aborted","F",-1
28,1,1736,867,5.0571338,"\int \frac{A+B x^2}{\left(d+e x^2\right) \left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(A + B*x^2)/((d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2)),x]","\frac{4 A \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} d e x b^3+4 A c \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} d e x^3 b^2-4 A c \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} d^2 x b^2-4 a B \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} d e x b^2+2 i a A e^2 \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{4 c x^2+2 b-2 \sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}} \Pi \left(\frac{\left(b+\sqrt{b^2-4 a c}\right) e}{2 c d};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right) b^2-2 i a B d e \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{4 c x^2+2 b-2 \sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}} \Pi \left(\frac{\left(b+\sqrt{b^2-4 a c}\right) e}{2 c d};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right) b^2-4 A c^2 \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} d^2 x^3 b-4 a B c \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} d e x^3 b+4 a B c \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} d^2 x b-12 a A c \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} d e x b+8 a B c^2 \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} d^2 x^3-8 a A c^2 \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} d e x^3+8 a A c^2 \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} d^2 x+8 a^2 B c \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} d e x-i \left(\sqrt{b^2-4 a c}-b\right) d \left(a B (2 c d-b e)+A \left(e b^2-c d b-2 a c e\right)\right) \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{4 c x^2+2 b-2 \sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}} E\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)+i d \left(a B \left(b \left(b-\sqrt{b^2-4 a c}\right) e+2 c \left(\sqrt{b^2-4 a c} d-2 a e\right)\right)+A \left(-e b^3+\left(c d+\sqrt{b^2-4 a c} e\right) b^2+c \left(4 a e-\sqrt{b^2-4 a c} d\right) b-2 a c \left(2 c d+\sqrt{b^2-4 a c} e\right)\right)\right) \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{4 c x^2+2 b-2 \sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}} F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)-8 i a^2 A c e^2 \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{4 c x^2+2 b-2 \sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}} \Pi \left(\frac{\left(b+\sqrt{b^2-4 a c}\right) e}{2 c d};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)+8 i a^2 B c d e \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{4 c x^2+2 b-2 \sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}} \Pi \left(\frac{\left(b+\sqrt{b^2-4 a c}\right) e}{2 c d};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)}{4 a \left(4 a c-b^2\right) \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} d \left(c d^2+e (a e-b d)\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,"(-4*A*b^2*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d^2*x + 4*a*b*B*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d^2*x + 8*a*A*c^2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d^2*x + 4*A*b^3*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*e*x - 4*a*b^2*B*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*e*x - 12*a*A*b*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*e*x + 8*a^2*B*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*e*x - 4*A*b*c^2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d^2*x^3 + 8*a*B*c^2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d^2*x^3 + 4*A*b^2*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*e*x^3 - 4*a*b*B*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*e*x^3 - 8*a*A*c^2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*e*x^3 - I*(-b + Sqrt[b^2 - 4*a*c])*d*(a*B*(2*c*d - b*e) + A*(-(b*c*d) + b^2*e - 2*a*c*e))*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + I*d*(a*B*(b*(b - Sqrt[b^2 - 4*a*c])*e + 2*c*(Sqrt[b^2 - 4*a*c]*d - 2*a*e)) + A*(-(b^3*e) + b*c*(-(Sqrt[b^2 - 4*a*c]*d) + 4*a*e) + b^2*(c*d + Sqrt[b^2 - 4*a*c]*e) - 2*a*c*(2*c*d + Sqrt[b^2 - 4*a*c]*e)))*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - (2*I)*a*b^2*B*d*e*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticPi[((b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + (8*I)*a^2*B*c*d*e*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticPi[((b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + (2*I)*a*A*b^2*e^2*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticPi[((b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - (8*I)*a^2*A*c*e^2*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticPi[((b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/(4*a*(-b^2 + 4*a*c)*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[a + b*x^2 + c*x^4])","C",1
29,1,8031,1301,7.8005884,"\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","Integrate[(A + B*x^2)/((d + e*x^2)^2*(a + b*x^2 + c*x^4)^(3/2)),x]","\text{Result too large to show}","-\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,"Result too large to show","C",0
30,1,310,273,0.5044839,"\int \frac{\sqrt{a}+\sqrt{c} x^2}{\left(d+e x^2\right) \sqrt{a+b x^2+c x^4}} \, dx","Integrate[(Sqrt[a] + Sqrt[c]*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]),x]","-\frac{i \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left(\left(\sqrt{a} e-\sqrt{c} d\right) \Pi \left(\frac{\left(b+\sqrt{b^2-4 a c}\right) e}{2 c d};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)+\sqrt{c} d F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)\right)}{\sqrt{2} d e \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \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,"((-I)*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*(Sqrt[c]*d*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + (-(Sqrt[c]*d) + Sqrt[a]*e)*EllipticPi[((b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]))/(Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*e*Sqrt[a + b*x^2 + c*x^4])","C",1
31,1,312,271,0.4265104,"\int \frac{1+\sqrt{\frac{c}{a}} x^2}{\left(d+e x^2\right) \sqrt{a+b x^2+c x^4}} \, dx","Integrate[(1 + Sqrt[c/a]*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]),x]","-\frac{i \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left(\left(e-d \sqrt{\frac{c}{a}}\right) \Pi \left(\frac{\left(b+\sqrt{b^2-4 a c}\right) e}{2 c d};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)+d \sqrt{\frac{c}{a}} F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)\right)}{\sqrt{2} d e \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","\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,"((-I)*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*(Sqrt[c/a]*d*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + (-(Sqrt[c/a]*d) + e)*EllipticPi[((b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]))/(Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*e*Sqrt[a + b*x^2 + c*x^4])","C",1
32,1,74,106,0.1820918,"\int \frac{946+315 x^2}{\left(7+5 x^2\right) \sqrt{2+3 x^2+x^4}} \, dx","Integrate[(946 + 315*x^2)/((7 + 5*x^2)*Sqrt[2 + 3*x^2 + x^4]),x]","-\frac{i \sqrt{x^2+1} \sqrt{x^2+2} \left(441 F\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)+505 \Pi \left(\frac{10}{7};\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)\right|2\right)\right)}{7 \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,"((-1/7*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*(441*EllipticF[I*ArcSinh[x/Sqrt[2]], 2] + 505*EllipticPi[10/7, I*ArcSinh[x/Sqrt[2]], 2]))/Sqrt[2 + 3*x^2 + x^4]","C",1
33,0,0,218,0.2249263,"\int \frac{\left(A+B x^2\right) \left(d+e x^2\right)^q}{a+b x^2+c x^4} \, dx","Integrate[((A + B*x^2)*(d + e*x^2)^q)/(a + b*x^2 + c*x^4),x]","\int \frac{\left(A+B x^2\right) \left(d+e x^2\right)^q}{a+b x^2+c x^4} \, dx","\frac{x \left(d+e x^2\right)^q \left(\frac{e x^2}{d}+1\right)^{-q} \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,"Integrate[((A + B*x^2)*(d + e*x^2)^q)/(a + b*x^2 + c*x^4), x]","F",-1
34,1,125,106,0.2010295,"\int \frac{x \left(1+2 x^2\right)}{\sqrt{1+x^2} \left(1+x^2+x^4\right)} \, dx","Integrate[(x*(1 + 2*x^2))/(Sqrt[1 + x^2]*(1 + x^2 + x^4)),x]","\frac{\sqrt{2 \left(1-i \sqrt{3}\right)} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{x^2+1}}{\sqrt{1-i \sqrt{3}}}\right)}{-1+i \sqrt{3}}+\frac{\sqrt{2 \left(1+i \sqrt{3}\right)} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{x^2+1}}{\sqrt{1+i \sqrt{3}}}\right)}{-1-i \sqrt{3}}","\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,"(Sqrt[2*(1 - I*Sqrt[3])]*ArcTanh[(Sqrt[2]*Sqrt[1 + x^2])/Sqrt[1 - I*Sqrt[3]]])/(-1 + I*Sqrt[3]) + (Sqrt[2*(1 + I*Sqrt[3])]*ArcTanh[(Sqrt[2]*Sqrt[1 + x^2])/Sqrt[1 + I*Sqrt[3]]])/(-1 - I*Sqrt[3])","C",1
35,1,441,145,0.6080117,"\int \frac{\sqrt{a+b x^2+c x^4}}{a d-c d x^4} \, dx","Integrate[Sqrt[a + b*x^2 + c*x^4]/(a*d - c*d*x^4),x]","\frac{i \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left(2 \sqrt{a} \sqrt{c} F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)-\left(2 \sqrt{a} \sqrt{c}+b\right) \Pi \left(\frac{-b-\sqrt{b^2-4 a c}}{2 \sqrt{a} \sqrt{c}};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)+\left(b-2 \sqrt{a} \sqrt{c}\right) \Pi \left(\frac{b+\sqrt{b^2-4 a c}}{2 \sqrt{a} \sqrt{c}};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)\right)}{2 \sqrt{2} \sqrt{a} \sqrt{c} d \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","\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,"((I/2)*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*(2*Sqrt[a]*Sqrt[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])] - (b + 2*Sqrt[a]*Sqrt[c])*EllipticPi[(-b - Sqrt[b^2 - 4*a*c])/(2*Sqrt[a]*Sqrt[c]), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + (b - 2*Sqrt[a]*Sqrt[c])*EllipticPi[(b + Sqrt[b^2 - 4*a*c])/(2*Sqrt[a]*Sqrt[c]), 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[a]*Sqrt[c]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*Sqrt[a + b*x^2 + c*x^4])","C",1
36,1,432,239,0.6846967,"\int \frac{\sqrt{a+b x^2-c x^4}}{a d+c d x^4} \, dx","Integrate[Sqrt[a + b*x^2 - c*x^4]/(a*d + c*d*x^4),x]","\frac{\sqrt{\frac{4 c x^2}{\sqrt{4 a c+b^2}-b}+2} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} \left(2 i \sqrt{a} \sqrt{c} F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{-\frac{c}{b+\sqrt{b^2+4 a c}}} x\right)|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right)+\left(b-2 i \sqrt{a} \sqrt{c}\right) \Pi \left(-\frac{i \left(b+\sqrt{b^2+4 a c}\right)}{2 \sqrt{a} \sqrt{c}};i \sinh ^{-1}\left(\sqrt{2} \sqrt{-\frac{c}{b+\sqrt{b^2+4 a c}}} x\right)|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right)-\left(b+2 i \sqrt{a} \sqrt{c}\right) \Pi \left(\frac{i \left(b+\sqrt{b^2+4 a c}\right)}{2 \sqrt{a} \sqrt{c}};i \sinh ^{-1}\left(\sqrt{2} \sqrt{-\frac{c}{b+\sqrt{b^2+4 a c}}} x\right)|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right)\right)}{4 \sqrt{a} \sqrt{c} d \sqrt{-\frac{c}{\sqrt{4 a c+b^2}+b}} \sqrt{a+b x^2-c x^4}}","\frac{\sqrt{\sqrt{4 a c+b^2}-b} \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[2 + (4*c*x^2)/(-b + Sqrt[b^2 + 4*a*c])]*Sqrt[1 - (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*((2*I)*Sqrt[a]*Sqrt[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])] + (b - (2*I)*Sqrt[a]*Sqrt[c])*EllipticPi[((-1/2*I)*(b + Sqrt[b^2 + 4*a*c]))/(Sqrt[a]*Sqrt[c]), I*ArcSinh[Sqrt[2]*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*x], (b + Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])] - (b + (2*I)*Sqrt[a]*Sqrt[c])*EllipticPi[((I/2)*(b + Sqrt[b^2 + 4*a*c]))/(Sqrt[a]*Sqrt[c]), I*ArcSinh[Sqrt[2]*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*x], (b + Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])]))/(4*Sqrt[a]*Sqrt[c]*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*d*Sqrt[a + b*x^2 - c*x^4])","C",1
37,1,214,309,0.3269588,"\int x \sqrt{c+e x+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[x*Sqrt[c + e*x + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]","\frac{\sqrt{\left(a+b x^2\right)^2} \left(2 \sqrt{d} \sqrt{c+x (d x+e)} \left(80 a d^2 \left(8 c d+8 d^2 x^2+2 d e x-3 e^2\right)+b \left(-256 c^2 d^2+4 c d \left(32 d^2 x^2-58 d e x+115 e^2\right)+384 d^4 x^4+48 d^3 e x^3-56 d^2 e^2 x^2+70 d e^3 x-105 e^4\right)\right)+15 e \left(e^2-4 c d\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+x (d x+e)}}\right)\right)}{3840 d^{9/2} \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{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{\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{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,"(Sqrt[(a + b*x^2)^2]*(2*Sqrt[d]*Sqrt[c + x*(e + d*x)]*(80*a*d^2*(8*c*d - 3*e^2 + 2*d*e*x + 8*d^2*x^2) + b*(-256*c^2*d^2 - 105*e^4 + 70*d*e^3*x - 56*d^2*e^2*x^2 + 48*d^3*e*x^3 + 384*d^4*x^4 + 4*c*d*(115*e^2 - 58*d*e*x + 32*d^2*x^2))) + 15*e*(-4*c*d + e^2)*(-12*b*c*d + 16*a*d^2 + 7*b*e^2)*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]*Sqrt[c + x*(e + d*x)])]))/(3840*d^(9/2)*(a + b*x^2))","A",1
38,1,168,283,0.222962,"\int \sqrt{c+e x+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4} \, dx","Integrate[Sqrt[c + e*x + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]","\frac{\sqrt{\left(a+b x^2\right)^2} \left(2 \sqrt{d} \sqrt{c+x (d x+e)} \left(48 a d^2 (2 d x+e)+b \left(4 c d (6 d x-13 e)+48 d^3 x^3+8 d^2 e x^2-10 d e^2 x+15 e^3\right)\right)-3 \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+x (d x+e)}}\right)\right)}{384 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{\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{\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{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,"(Sqrt[(a + b*x^2)^2]*(2*Sqrt[d]*Sqrt[c + x*(e + d*x)]*(48*a*d^2*(e + 2*d*x) + b*(15*e^3 - 10*d*e^2*x + 8*d^2*e*x^2 + 48*d^3*x^3 + 4*c*d*(-13*e + 6*d*x))) - 3*(4*c*d - e^2)*(4*b*c*d - 16*a*d^2 - 5*b*e^2)*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]*Sqrt[c + x*(e + d*x)])]))/(384*d^(7/2)*(a + b*x^2))","A",1
39,1,176,286,0.3100927,"\int \frac{\sqrt{c+e x+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{x} \, dx","Integrate[(Sqrt[c + e*x + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/x,x]","\frac{\sqrt{\left(a+b x^2\right)^2} \left(2 \sqrt{d} \left(\sqrt{c+x (d x+e)} \left(24 a d^2+b \left(8 c d+8 d^2 x^2+2 d e x-3 e^2\right)\right)-24 a \sqrt{c} d^2 \tanh ^{-1}\left(\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+x (d x+e)}}\right)\right)+3 e \left(8 a d^2+b \left(e^2-4 c d\right)\right) \tanh ^{-1}\left(\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+x (d x+e)}}\right)\right)}{48 d^{5/2} \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} \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,"(Sqrt[(a + b*x^2)^2]*(3*e*(8*a*d^2 + b*(-4*c*d + e^2))*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]*Sqrt[c + x*(e + d*x)])] + 2*Sqrt[d]*(Sqrt[c + x*(e + d*x)]*(24*a*d^2 + b*(8*c*d - 3*e^2 + 2*d*e*x + 8*d^2*x^2)) - 24*a*Sqrt[c]*d^2*ArcTanh[(2*c + e*x)/(2*Sqrt[c]*Sqrt[c + x*(e + d*x)])])))/(48*d^(5/2)*(a + b*x^2))","A",1
40,1,171,294,0.2514717,"\int \frac{\sqrt{c+e x+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{x^2} \, dx","Integrate[(Sqrt[c + e*x + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/x^2,x]","\frac{\sqrt{\left(a+b x^2\right)^2} \left(\sqrt{c} x \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+x (d x+e)}}\right)+2 \sqrt{d} \left(\sqrt{c} \sqrt{c+x (d x+e)} (b x (2 d x+e)-4 a d)-2 a d e x \tanh ^{-1}\left(\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+x (d x+e)}}\right)\right)\right)}{8 \sqrt{c} d^{3/2} x \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,"(Sqrt[(a + b*x^2)^2]*(Sqrt[c]*(4*b*c*d + 8*a*d^2 - b*e^2)*x*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]*Sqrt[c + x*(e + d*x)])] + 2*Sqrt[d]*(Sqrt[c]*Sqrt[c + x*(e + d*x)]*(-4*a*d + b*x*(e + 2*d*x)) - 2*a*d*e*x*ArcTanh[(2*c + e*x)/(2*Sqrt[c]*Sqrt[c + x*(e + d*x)])])))/(8*Sqrt[c]*d^(3/2)*x*(a + b*x^2))","A",1
41,1,177,288,0.3389151,"\int \frac{\sqrt{c+e x+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{x^3} \, dx","Integrate[(Sqrt[c + e*x + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/x^3,x]","\frac{\sqrt{\left(a+b x^2\right)^2} \left(4 b c^{3/2} e x^2 \tanh ^{-1}\left(\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+x (d x+e)}}\right)-\sqrt{d} \left(x^2 \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+x (d x+e)}}\right)+2 \sqrt{c} \sqrt{c+x (d x+e)} \left(2 a c+a e x-4 b c x^2\right)\right)\right)}{8 c^{3/2} \sqrt{d} x^2 \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,"(Sqrt[(a + b*x^2)^2]*(4*b*c^(3/2)*e*x^2*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]*Sqrt[c + x*(e + d*x)])] - Sqrt[d]*(2*Sqrt[c]*(2*a*c + a*e*x - 4*b*c*x^2)*Sqrt[c + x*(e + d*x)] + (8*b*c^2 + 4*a*c*d - a*e^2)*x^2*ArcTanh[(2*c + e*x)/(2*Sqrt[c]*Sqrt[c + x*(e + d*x)])])))/(8*c^(3/2)*Sqrt[d]*x^2*(a + b*x^2))","A",1
42,1,189,294,0.2872972,"\int \frac{\sqrt{c+e x+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{x^4} \, dx","Integrate[(Sqrt[c + e*x + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/x^4,x]","\frac{\sqrt{\left(a+b x^2\right)^2} \left(-3 e x^3 \left(a \left(e^2-4 c d\right)+8 b c^2\right) \tanh ^{-1}\left(\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+x (d x+e)}}\right)-2 \sqrt{c} \sqrt{c+x (d x+e)} \left(a \left(8 c^2+2 c x (4 d x+e)-3 e^2 x^2\right)+24 b c^2 x^2\right)+48 b c^{5/2} \sqrt{d} x^3 \tanh ^{-1}\left(\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+x (d x+e)}}\right)\right)}{48 c^{5/2} x^3 \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} \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{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{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)}",1,"(Sqrt[(a + b*x^2)^2]*(-2*Sqrt[c]*Sqrt[c + x*(e + d*x)]*(24*b*c^2*x^2 + a*(8*c^2 - 3*e^2*x^2 + 2*c*x*(e + 4*d*x))) + 48*b*c^(5/2)*Sqrt[d]*x^3*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]*Sqrt[c + x*(e + d*x)])] - 3*e*(8*b*c^2 + a*(-4*c*d + e^2))*x^3*ArcTanh[(2*c + e*x)/(2*Sqrt[c]*Sqrt[c + x*(e + d*x)])]))/(48*c^(5/2)*x^3*(a + b*x^2))","A",1